**Meet the editor**

Waldemar Alfredo Monterio, MSc (Solid State Physics), DSc (Nuclear Technology), is a physicist, at the University of São Paulo, São Paulo, Brazil. His expertise areas are Physical Metallurgy, Powder Metallurgy, and Materials Characterization (Optical and Electron Microscopy; Microanalysis Techniques). He is senior researcher in metallic materials at the Nuclear and Energetic Researches

Institute, São Paulo, Brazil. He is professor (graduate course) on Materials Characterization (Electron Microscopy Techniques and Microanalysis) and researcher and lecturer (undergraduate and graduate courses) at the Presbyterian Mackenzie University, São Paulo, Brazil. He has published almost 3 books, 5 book chapters, and 156 articles (scientific journal or congress proceedings). His academic advisories include 45 scientific initiation (undergraduate students), 27 MSc, and 15 DSc students.

### Contents

**Preface XI**


## Preface

This book brings suitable data concerning theory and experiments of neutron interactions with different materials. Since the neutron discovery by Chadwick in 1932 (Proc Roy Soc A136 692), researchers of the entire world begin to make studies about it. It is well-known that neutron have no charge, and their electric dipole moment is either zero or too small to measure, but theories show that neutron has spin (presence of magnetic moment), and po‐ larization neutron scattering is plausible.

It is known that neutron scattering is a "non-destructive microscope" for seeing inside mat‐ ter. The theories and experiments show us that solid matter is not very dense since the size of a scattering center (i.e., a nucleus) is typically 100,000 times smaller than the distance be‐ tween centers.

Furthermore, neutrons interact with atoms via nuclear rather than electrical forces, and nucle‐ ar forces are very short range—on the order of scarce femtometers. As a consequence, neutrons can move large distances through most materials without being scattered or absorbed. For these reasons, neutron can penetrate matter far better than charged particles even if it first has to pass through a container, such as would be required for a liquid or powder sample, for example, or if the sample had to be maintained at low temperature or high pressure.

In this book, it is acquired information concerning inelastic scattering cross-sections for neu‐ tron; polarized neutron reflectivity; scattering methods; neutron reflectometry tool to probe the chemical structures; neutron scattering for amino acid crystals; and small-angle neutron scattering nanoemulsion heat transfer fluids.

### **Prof. Dr. Waldemar Alfredo Monteiro**

Materials Science and Technology Center Nuclear and Energetic Researches Institute University of São Paulo

> Engineering School Presbyterian Mackenzie University São Paulo, SP Brazil

**Chapter 1**

## **Using X-ray as a Probe of the Terahertz Dynamics of Disordered Systems – Complementarity with Inelastic Neutron Scattering and Future Perspectives**

Alessandro Cunsolo

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62844

### **Abstract**

This Chapter is devoted to a general introduction of the high resolution (meV) inelastic xray scattering (IXS) technique. This starts from a theoretical derivation of the IXS cross section and the demostration of its link with the spectrum of density fluctuations. The complementarity of this technique with the other mesocopic spectrocopy method, inelas‐ tic neutron scattering (INS) is discussed in detail by emphasizing differences and similari‐ ties both from the practical and the teorethical points of view. Along with this general discussion, examples are given of both an existing IXS spectrometer and a new-concept one to be soon in operation. Finally, the result of a recent joint INS and IXS experiment on water, taking advantage of the complementarity of these two techniques, are described in the last paragraphs.

**Keywords:** Inelastic neutron and X-ray scattering, dynamics of disorder materials, THz spectrocopy

### **1. Introduction**

The study of the collective dynamics of disordered systems has been a vibrant field of research since the dawn of modern science, and yet, despite an intensive theoretical, experimental, and computational scrutiny, it still presents many unsettled aspects. The main reasons are the lack of a translational invariance in the microscopic structure of these systems and the often exceptionally complex movements of their microscopic constituents.

For instance, the interplay between collective modes and inter- and intramolecular degrees of freedom, as rotations, internal vibrations, or structural relaxation processes, is a fundamental property of the fluid, which still eludes a comprehensive understanding.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The spectrum of the density fluctuations, *S*(*Q,ω*), is a well-suited variable to test various theoretical models of the liquids' dynamics because it can be directly accessed by both conventional spectroscopic methods and molecular dynamics (MD) computer simulations.

Although the shape of *S*(*Q,ω*) is reasonably understood at macroscopic or quasi-macroscopic scales, over which the fluid appears as a continuum, its evolution beyond the continuous limit still represents a theoretical challenge. In particular, this applies to the so-called "mesoscopic" regime, corresponding to distances and timescales matching with first neighboring molecules' separations and cage oscillation periods, respectively.

From the experimental side, the study of *S*(*Q,ω*) in liquids at mesoscopic scales has been an exclusive duty of inelastic neutron scattering (INS) until almost the end of the last millennium. Nowadays, INS is a rather mature technique, which first saw the light in the mid-1950s [1]. Conversely, the other mesoscopic inelastic spectroscopy, inelastic X-ray scattering (IXS), is relatively young, having been developed only a couple of decades ago, thanks to the advent of synchrotron sources with unprecedented brilliance and parallel advances in crystal optic fabrication. This time lag mainly owes to the exceptionally small energy resolution, ∆*E*/*E*, required by IXS studies of the mesoscopic [nm, millielectron volt (meV)] dynamics of fluids*.* Specifically, being current IXS spectrometers operated at incident energies larger than ≈ 20 keV, their ability to resolve meV energies imposes a relative energy resolution, ∆*E*/*E*, at least as small as 10−7.

Conversely, neutrons at low-to-moderate temperatures have 1 to 10 meV energies; therefore, even moderate ∆*E*/*E* values (10−2 – 10−1) are sufficient to resolve meV collective excitations in the *S*(*Q,ω*) of liquids.

Furthermore, the IX Sintensity of materials with atomic number *Z* < 4 is dominated by photoelectric absorption, which makes this technique rather inefficient when dealing with systems with high *Z*.

Finally, the rapid decay of the IXS cross-section upon increasing the exchanged wave vector (*Q*) imposes severe intensity penalties even at intermediate *Q* values.

Nonetheless, the superior photon fluxes delivered by new-generation undulator sources, coupled with substantial advances in the design/fabrication of IXS spectrometers' optics, can nowadays overcompensate for the mentioned intensity limitations. Consequently, the statistical accuracy currently achieved in routine IXS measurements is unmatched by other complementary terahertz (THz) methods.

A general discussion of theoretical and practical aspects of IXS technique and in particular its applications to the study of the THz dynamics of liquids is the main purpose of this chapter, the rest of which is organized as follows. Section 2 describes brief derivation of the IXS crosssection. Section 3 discusses complementary aspects of IXS and INS methods. Section 4 proposes an example of current and future IXS spectrometers to illustrate how the latter hold the promise of a superior energy resolution. Finally, Section 5 concludes this chapter with an example of a joint INS and IXS experiment taking advantage of the complementarity of the two techniques.

### **2. The theory of IXS spectroscopy**

The spectrum of the density fluctuations, *S*(*Q,ω*), is a well-suited variable to test various theoretical models of the liquids' dynamics because it can be directly accessed by both conventional spectroscopic methods and molecular dynamics (MD) computer simulations.

Although the shape of *S*(*Q,ω*) is reasonably understood at macroscopic or quasi-macroscopic scales, over which the fluid appears as a continuum, its evolution beyond the continuous limit still represents a theoretical challenge. In particular, this applies to the so-called "mesoscopic" regime, corresponding to distances and timescales matching with first neighboring molecules'

From the experimental side, the study of *S*(*Q,ω*) in liquids at mesoscopic scales has been an exclusive duty of inelastic neutron scattering (INS) until almost the end of the last millennium. Nowadays, INS is a rather mature technique, which first saw the light in the mid-1950s [1]. Conversely, the other mesoscopic inelastic spectroscopy, inelastic X-ray scattering (IXS), is relatively young, having been developed only a couple of decades ago, thanks to the advent of synchrotron sources with unprecedented brilliance and parallel advances in crystal optic fabrication. This time lag mainly owes to the exceptionally small energy resolution, ∆*E*/*E*, required by IXS studies of the mesoscopic [nm, millielectron volt (meV)] dynamics of fluids*.* Specifically, being current IXS spectrometers operated at incident energies larger than ≈ 20 keV, their ability to resolve meV energies imposes a relative energy resolution, ∆*E*/*E*, at least

Conversely, neutrons at low-to-moderate temperatures have 1 to 10 meV energies; therefore, even moderate ∆*E*/*E* values (10−2 – 10−1) are sufficient to resolve meV collective excitations in

Furthermore, the IX Sintensity of materials with atomic number *Z* < 4 is dominated by photoelectric absorption, which makes this technique rather inefficient when dealing with

Finally, the rapid decay of the IXS cross-section upon increasing the exchanged wave vector

Nonetheless, the superior photon fluxes delivered by new-generation undulator sources, coupled with substantial advances in the design/fabrication of IXS spectrometers' optics, can nowadays overcompensate for the mentioned intensity limitations. Consequently, the statistical accuracy currently achieved in routine IXS measurements is unmatched by other

A general discussion of theoretical and practical aspects of IXS technique and in particular its applications to the study of the THz dynamics of liquids is the main purpose of this chapter, the rest of which is organized as follows. Section 2 describes brief derivation of the IXS crosssection. Section 3 discusses complementary aspects of IXS and INS methods. Section 4 proposes an example of current and future IXS spectrometers to illustrate how the latter hold the promise of a superior energy resolution. Finally, Section 5 concludes this chapter with an example of a joint INS and IXS experiment taking advantage of the complementarity of the two techniques.

(*Q*) imposes severe intensity penalties even at intermediate *Q* values.

separations and cage oscillation periods, respectively.

as small as 10−7.

2 Neutron Scattering

the *S*(*Q,ω*) of liquids.

systems with high *Z*.

complementary terahertz (THz) methods.

Spectroscopy experiments are among the most powerful tools to investigate the properties of matter and have a very straightforward working principle, schematically outlined below.

A beam of particles–waves (e.g. neutron or X-rays) having narrow energy spread and angular divergence and a well-defined polarization impinges on a sample and is scattered in all directions.

A detector is used to count the particles deviated by an angle 2*θ* within a small solid angle dΩ. Before the whole flight to the detector is accomplished, the particles may pass through some elements filtering their energy (monochromators and analyzers), angular divergence (colli‐ mators), and, in some cases, polarization (polarizers).

The whole instrument used to perform a scattering measurement is referred to as spectrometer, and its degree of complexity strongly depends, of course, on the physical problem the instrument is meant to investigate and the related requirement in collimation/monochroma‐ tization. Indeed, some optical elements cannot be included in the layout as irrelevant to the purpose of the measurement; for instance, when probing static, or structural, properties of the sample, as in diffraction experiments, there is no energy filter for the scattered beam (energy analyzer) because energy-integrated intensity is to be measured.

Regardless on the complexity of the specific experiment, the ultimate aim is to investigate the properties of the target sample through the counting of the probe particles having experienced an interaction with it through the scattering event. By virtue of this interaction event, the energy of the probe's particles may either be exchanged or remain constant, which, respectively, corresponds to the case of elastic or inelastic scattering measurements. Here the main focus is on the latter class of experiments and, in particular, on those involving X-rays as a probe and liquids as samples.

In general, it can be safely assumed that the interaction time between probe and target system is much shorter than any other timescale relevant to the experiment. In other terms, the scattering event can be schematized as an instantaneous collision between the X-ray beam and the scatterers, for example, electrons belonging to atoms or molecules of the sample.

If the scatterer is assumed fixed at the origin, infinitely massive, and at rest, the scattered electrical field at a distance *r* = |*r* <sup>→</sup> | from the origin can be derived as the solution of the inhomogeneous Helmholtz equation [2]:

$$
\sigma\left(\nabla^2 + k^2\right)\mu\left(\vec{r}\right) = -\delta\left(\vec{r}\right),
\tag{1}
$$

where " ∇<sup>2</sup> " is the Laplacian operator, whereas *k* is the wave number of the incident electro‐ magnetic wave. At large distances from the origin, the outgoing wave is the sum of a plane (transmitted) and a spherical wave. At finite scattering angles, the state of the scattered photons can be described by a spherical wave:

$$
\boldsymbol{\nu}\_{\rm sc} \propto \frac{\mathbf{e}^{\rm ltr}}{r}.\tag{2}
$$

In summary, one can see that by virtue of the scattering event, photons are partly removed from the plane wave and re-radiated in a spherical wave, which can be thought as a simple manifestation of the Huygens' principle [3]. The photon scattered at an angle 2*θ* within a solid angle dΩ passes though the energy filter (analyzer) and is ultimately counted by the detector after impinging on its sensitive area d*A*=*r* <sup>2</sup> dΩ, which is here intended to be small enough to safely approximate the scattered wave impinging on it as a plane wave (see Figure 1). The intensity scattered within a solid angle dΩ and an energy spread d*E* <sup>F</sup> has the following general form:

$$I = K I\_0 \frac{\partial^2 \sigma}{\partial \Omega \ \partial E\_{\text{F}}} \text{d}\Omega \text{ d}E\_{\text{F}'} \tag{3}$$

where *I* 0 is the intensity of the beam impinging on the sample and *K* is a coefficient taking into account detector efficiency, sample self-absorption, and all geometrical and/or spurious intensity effects. In the above formula, the double differential scattering cross-section was introduced:

$$\frac{\partial^2 \sigma}{\partial \Omega \,\partial \mathbf{E}'} = \frac{\text{Rate of photons scattered into d}\Omega \text{ with final energy between } \mathbf{E}' \text{ and } \mathbf{E}' + \mathbf{d} \mathbf{E}'}{I\_0 \partial \Omega \,\partial \mathbf{E}'}. \tag{4}$$

To derive an explicit expression for the double the variable above, it is useful to start from the Hamiltonian describing the interaction between the electrons of the target system and the time dependent electromagnetic field impinging on them. In the non-relativistic case, such an Hamiltonian reads as follows [4]:

$$H = \frac{1}{2m\_e} \sum\_i \left[ \vec{P}\_i - \frac{e}{c} \vec{A} \left( r\_i \right) \delta \left( \vec{r} - \vec{r}\_i \right) \right] \cdot \left[ \vec{P}\_i - \frac{e}{c} \vec{A} \left( r\_i \right) \delta \left( \vec{r} - \vec{r}\_i \right) \right] + \sum\_i V \left( r\_i \right) + V\_{\text{int}}^{e-e} \tag{5}$$

where *P* → *i* and *r* → *<sup>i</sup>* represent the momentum and the position of the *i*th electron, respectively; *c*, *e*, and *m* <sup>e</sup> are the speed of light in vacuum, the electron charge, and mass, respectively; *A* → (*r* <sup>→</sup> ) is the vector potential at the position *r* <sup>→</sup> and finally, *V*int e−e is the electron–electron interaction integrated over the electron clouds of target atoms.

Using X-ray as a Probe of the Terahertz Dynamics of Disordered Systems – Complementarity with Inelastic... http://dx.doi.org/10.5772/62844 5

**Figure 1.** Schematic representation of the scattering geometry.

*i*

(transmitted) and a spherical wave. At finite scattering angles, the state of the scattered photons

i

<sup>e</sup> . *kr r*

In summary, one can see that by virtue of the scattering event, photons are partly removed from the plane wave and re-radiated in a spherical wave, which can be thought as a simple manifestation of the Huygens' principle [3]. The photon scattered at an angle 2*θ* within a solid angle dΩ passes though the energy filter (analyzer) and is ultimately counted by the detector

safely approximate the scattered wave impinging on it as a plane wave (see Figure 1). The intensity scattered within a solid angle dΩ and an energy spread d*E* <sup>F</sup> has the following general

F

0 Rate of photons scattered into d with final energy between and d = .

= , <sup>2</sup> *i i ii i i i i i e e e H P Ar r r P Ar r r V r V*

*e*, and *m* <sup>e</sup> are the speed of light in vacuum, the electron charge, and mass, respectively; *A*

<sup>→</sup> and finally, *V*int


To derive an explicit expression for the double the variable above, it is useful to start from the Hamiltonian describing the interaction between the electrons of the target system and the time dependent electromagnetic field impinging on them. In the non-relativistic case, such an

()( ) ()( ) () e e

 d


*<sup>i</sup>* represent the momentum and the position of the *i*th electron, respectively; *c*,

e−e

 W + ¢ ¢¢ ¶W ¶ ¢ ¢ ¶W ¶ (4)

*E I E*

*m c c* d

Hamiltonian reads as follows [4]:

the vector potential at the position *r*

integrated over the electron clouds of target atoms.

1

where *I* 0 is the intensity of the beam impinging on the sample and *K* is a coefficient taking into account detector efficiency, sample self-absorption, and all geometrical and/or spurious intensity effects. In the above formula, the double differential scattering cross-section was

2 <sup>0</sup> *I KI* <sup>=</sup> d d , ¶ sW

¶W ¶

F

*E*

µ (2)

dΩ, which is here intended to be small enough to

*<sup>E</sup>* (3)

*E EE*

int

is the electron–electron interaction

→ (*r* <sup>→</sup> ) is

sc

y

can be described by a spherical wave:

after impinging on its sensitive area d*A*=*r* <sup>2</sup>

form:

4 Neutron Scattering

introduced:

2

¶ s

where *P* → *i* and *r* → In Eq. 5, all the spin-dependent contributions are omitted because, in general, those couple very weakly with the incident electromagnetic field. The Hamiltonian in Eq. 5 can be cast as follows:

$$\begin{aligned} \label{eq:10} H &= H\_{\text{el}} + H\_{\text{int}}^{(1)} + H\_{\text{int}}^{(2)}\\ \text{where} \\ H\_{\text{el}} &= \sum\_{i} \left[ \frac{P\_{i}^{2}}{2M} + V(r\_{i}) \right] \\ H\_{\text{int}}^{1} &= -\frac{e}{2mc} \sum\_{i} \left[ \vec{P}\_{i} \cdot \vec{A}(\vec{r}) \delta\left(\vec{r} - \vec{r}\_{i}\right) + \vec{A} \delta\left(\vec{r} - \vec{r}\_{i}\right) \cdot \vec{P}\_{i} \right] \\ H\_{\text{int}}^{2} &= \frac{e^{2}}{2mc^{2}} \sum\_{l} \delta\left(\vec{r} - \vec{r}\_{l}\right) \vec{A}(\vec{r}) \cdot \vec{A}(\vec{r}) \end{aligned} \tag{6}$$

On a general ground, the parts containing the square of the perturbing term, i.e. the vector potential, contribute to two-photon processes such as the scattering event. The two terms entering in *H*int 1 (Eq. 6b), both being linear in *A* → (*r* <sup>→</sup> ), describe to the leading order one-photon processes, such as absorption and emission, while they account for the scattering process to the second order only. Conversely, the so-called Thomson term, i.e. the term *H*int 2 in Eq. 6c, being quadratic in the vector potential, accounts to the first order for a two-photon process, such as the scattering event. It can be shown that, away from an energy resonance, the Thomson term dominates over the second-order expansion of Eq. 6b, thus providing the leading contribution to the scattering process. In the following, an explicit derivation of the double differential IXS cross section is carried out within the assumption that the Thomson term entirely describes the scattering process and further assuming that:


The following derivation can be found in various textbooks and is also clearly illustrated in a theoretical work by Sinha [4], whose main results are discussed here.

In principle, one could use the perturbation theory to derive the rate of scattering events associated to an incident plane wave having wave vector *k* → I and to all plane waves having wave vector *k* → F, pointing to a given direction 2*θ* within the solid angle dΩ intercepted by the detector. However, the normalization of the scattered wave would require its amplitude to decrease at least as *r* <sup>−</sup><sup>1</sup> at large distances *r* from the scattering event, and this hardly fits the case of a plane wave. To circumvent this problem, it is useful to define the scattering process within a box of size *L* with periodic boundary conditions and eventually consider the limit of large *L*. This expedient enables to easily count the states and properly normalize the wave functions.

Within such a box, the vector potential can be expressed as a linear combination of normalized plane waves of the form 1 / *L*3/2exp(*ik* <sup>→</sup> ⋅ *r* <sup>→</sup> ). Namely:

$$\vec{A}(\vec{r}) = \sum\_{k,\varepsilon} \sqrt{\frac{\hbar}{\phi\_k L^3}} \left[ \varepsilon \hat{\vec{a}} \left[ \vec{a}\_{k,\varepsilon}^{\ast} \exp(i \vec{k} \cdot \vec{r}) + \vec{a}\_{k,\varepsilon}^{\cdot} \exp(-i \vec{k} \cdot \vec{r}) \right] \right] \tag{7}$$

where the exponential with either the sign "+" or "−" describes downstream and upstream propagating plane waves, respectively. Here η *= h*/2*π*, where *h* is the Planck constant, while the indexes *k* and ε label, respectively, the wave vector and the polarization states of the wave, as identified by the unit vector *ε* ^ and the vector *<sup>k</sup>* <sup>→</sup> , respectively. The coefficients *a* → *k* ,*ε* + and *a* → *k* ,*ε* <sup>−</sup> in Eq. 7 represent, respectively, the annihilation and the creation operators for the initial photon state |*k*, *ε* , while *ωk* is the angular frequency.

The scattering process can be depicted as a simultaneous transition between the |*kI* ,*εI* and |*kF* ,*εF* photon states and between the |*λI* and |*λF* sample states. The rate of such a transition is predicted by Fermi's golden rule [5]:

Using X-ray as a Probe of the Terahertz Dynamics of Disordered Systems – Complementarity with Inelastic... http://dx.doi.org/10.5772/62844 7

$$\sum\_{k\_{\rm r}}^{\rm d\Omega} W\_{(l)\to\rm l\{F\}} = \frac{2\pi}{\hbar} \nu\_{k\_{\rm r}, c\_{\rm tr}} \left| \left\langle F \upharpoonright H\_{\rm int}^{(2)} \mid I \right\rangle \right|^2 \,,\tag{8}$$

where, to ease the notation, it is assumed | *I* = |*k*<sup>I</sup> ,*ε*I , *<sup>λ</sup>*<sup>I</sup> , | *<sup>F</sup>* <sup>=</sup> <sup>|</sup>*k*F,*ε*F, *<sup>λ</sup>*<sup>F</sup> . The parameters *νk*F,*ε*<sup>F</sup> represent the density of final photon states and can be derived by evaluating the density of *k* F points in the reciprocal space. For this purpose, one can consider all plane waves having energies included in between *E* F and *E* <sup>F</sup> *+* d*E* <sup>F</sup> and pointing toward a direction 2*θ* within the solid angle dΩ, the number of such plane waves being *νk*F,*ε*<sup>F</sup> d*E*F. In the reciprocal space, the energy spread d*E* F corresponds to the volume d*V* (*k*F) of the spherical shell (see Figure 2), which is identified by vectors *k* → <sup>F</sup> having amplitude included in between *k* F and *k*<sup>F</sup> + d*k*F and direction included within a solid angle dΩ. This elemental volume reads as follows:

contribution to the scattering process. In the following, an explicit derivation of the double differential IXS cross section is carried out within the assumption that the Thomson term

**•** The center of mass of the electronic cloud follows the nuclear motion as a slow drift, i.e. with no delay (adiabatic approximation). This justifies the expression of the initial and final states of scatterer as the product of two terms containing either only electronic or only nuclear coordinates. This approximation becomes particularly accurate at energies smaller than excitation energies of electrons in bound core states, i.e. for nearly all cases of practical interest for high resolution IXS measurements. In liquid metals, this approximation only

**•** The electronic part of the total wave function is unaffected by the scattering process, thus the difference between the initial (before scattering) state and the final (after scattering) state

The following derivation can be found in various textbooks and is also clearly illustrated in a

In principle, one could use the perturbation theory to derive the rate of scattering events

However, the normalization of the scattered wave would require its amplitude to decrease at

wave. To circumvent this problem, it is useful to define the scattering process within a box of size *L* with periodic boundary conditions and eventually consider the limit of large *L*. This expedient enables to easily count the states and properly normalize the wave functions.

Within such a box, the vector potential can be expressed as a linear combination of normalized

where the exponential with either the sign "+" or "−" describes downstream and upstream propagating plane waves, respectively. Here η *= h*/2*π*, where *h* is the Planck constant, while the indexes *k* and ε label, respectively, the wave vector and the polarization states of the wave,

Eq. 7 represent, respectively, the annihilation and the creation operators for the initial photon

The scattering process can be depicted as a simultaneous transition between the |*kI* ,*εI* and |*kF* ,*εF* photon states and between the |*λI* and |*λF* sample states. The rate of such a transition

<sup>3</sup> , , () ( )

 e

<sup>→</sup> , respectively. The coefficients *a*

+ - æ ö é ù ç ÷ × + -× ç ÷ ë û è ø <sup>å</sup> <sup>r</sup> r r r h r rr r (7)

<sup>→</sup> ). Namely:

( )= ˆ exp exp , *k k*

*A r c a ik r a ik r*

e

^ and the vector *<sup>k</sup>*

F, pointing to a given direction 2*θ* within the solid angle dΩ intercepted by the detector.

at large distances *r* from the scattering event, and this hardly fits the case of a plane

→ I

and to all plane waves having wave

→ *k* ,*ε* + and *a* → *k* ,*ε* <sup>−</sup> in

is due only to excitations associated with atomic density fluctuations.

<sup>→</sup> ⋅ *r*

e

theoretical work by Sinha [4], whose main results are discussed here.

entirely describes the scattering process and further assuming that:

excludes electron densities near the Fermi level.

associated to an incident plane wave having wave vector *k*

plane waves of the form 1 / *L*3/2exp(*ik*

as identified by the unit vector *ε*

,

state |*k*, *ε* , while *ωk* is the angular frequency.

is predicted by Fermi's golden rule [5]:

e

*k k*

*L*

w

vector *k* →

least as *r* <sup>−</sup><sup>1</sup>

6 Neutron Scattering

$$\mathbf{d}V\left(k\_{\mathrm{p}}\right) = \mathbf{d}\Omega \, k\_{\mathrm{p}}^2 \mathbf{d}k\_{\mathrm{p}}... $$

The *kF* values included within the box defined above are given by 2*π* / *L*(*nx*,*ny*,*nz*), with *ns*=*x*,*y*,*<sup>z</sup>* generic integers. This defines a lattice in the *k* space whose points can be associated with a unit cell of volume *V*min =(2*π* / *L*) 3 . Assuming a small cell (a large size box *L)*, the total number of lattice points inside the elemental volume *dV* (*k*F) is given by the ratio between such a volume and *V*min, and this must be equal to the number of plane waves defined above, that is:

$$\nu\_{k\_r, c\_r} \mathrm{d}E\_\mathrm{F} = \mathrm{d}\Omega \, k\_\mathrm{F}^2 \left(\frac{L}{2\pi}\right)^3 \mathrm{d}k\_\mathrm{F} \,. \tag{9}$$

$$\mathrm{d}\Omega \, \mathrm{K}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega \, \mathrm{m}\Omega$$

**Figure 2.** Schematic representation of the elemental volumes in the reciprocal space. The cube enclosing a lattice point represents the unit cell of volume *V*min =(2*π* / *L*) 3 .

$$
\lambda \nu\_{k\_F, c\_F} dE\_F = d\Omega \, k\_F^2 \left(\frac{L}{2\pi}\right)^3 dk\_F \, . \tag{10}
$$

Furthermore, being *E*<sup>F</sup> =ℏ*ck*F, or, equivalently, d*E*<sup>F</sup> =ℏ*c*d*k*F, one has:

$$
\hbar \mathbf{v}\_{k\_{\mathbf{r}}, \varepsilon\_{\mathbf{r}}} \mathbf{d} E\_{\mathbf{F}} = \frac{k\_{\mathbf{r}}^2 \mathbf{L}^3}{\left(2\pi\right)^3 \hbar c} \mathbf{d} \Omega \,\,\,. \tag{11}
$$

Also, the incident photon flux, i.e. the number of photons passing through a unit surface in the unit time, is given by:

$$
\varphi = \frac{c}{V} \,. \tag{12}
$$

Furthermore, the double differential cross section for the scattering process can be related to such a flux through:

$$
\left(\frac{\mathrm{d}\,\sigma}{\mathrm{d}\Omega \mathrm{d}E\_{\mathrm{F}}}\right)\_{(l)\to(l^{\prime})} = \frac{1}{\rho} \frac{1}{\mathrm{d}\Omega} \sum\_{k\_{\mathrm{r}}}^{\mathrm{d}\Omega} W\_{\lambda\_{\mathrm{l}}\to\lambda\_{\mathrm{r}}}.\tag{13}
$$

Using Eqs. 8–12, the matrix element defining Fermi's golden rule (Eq. 8) can be thus expressed as:

$$\left\langle F \middle| H\_{\text{int}}^{(2)} \middle| I \right\rangle = \frac{1}{L^3} \frac{e^2}{mc^2} \frac{\hbar c}{\left[ \alpha \left( k\_{\text{l}} \right) \alpha \left( k\_{\text{l}} \right) \right]^{1/2}} \left\langle \hat{c}\_{\text{l}} \cdot \hat{c}\_{\text{F}} \right\rangle \left\langle \hat{\lambda}\_{\text{F}} \left| \sum\_{j} \exp \left( -i (\bar{\mathbf{Q}} \cdot \vec{r}\_{j}) \right) \hat{\lambda}\_{\text{I}} \right\rangle \delta \left( \hbar \alpha - E\_{\text{l}} - E\_{\text{F}} \right) \right\rangle,\tag{14}$$

Here the conservation laws of momentum and energy in the scattering event have been superimposed by assuming ℏ*Q* <sup>→</sup> <sup>=</sup>ℏ(*<sup>k</sup>* → <sup>F</sup> −*k* → I ) and ℏ*ω* =*E*<sup>F</sup> −*E*<sup>I</sup> , with ℏ*Q* → and ℏ*<sup>ω</sup>* being the momen‐ tum and energy transferred from the photon to the target sample. In particular, the energy conservation is accounted by the *δ*(ℏ*ω* −*E*I−*E*F) term in Eq. 13. Considering that the frequency of the incident (scattered) wave is *ω*(*k*<sup>I</sup> ) =*ck*<sup>I</sup> (*ω*(*k*F) =*ck*F), the double differential cross-section reduces to:

$$\begin{split} \left(\frac{\mathbf{d}^2 \sigma}{\mathbf{d}\Omega \mathbf{d}E}\right)\_{(l)\to(F)} &= \frac{e^2}{mc^2} \left(\frac{k'}{k}\right) \left(\hat{\boldsymbol{\omega}}\_1 \cdot \hat{\boldsymbol{\varepsilon}}\_\mathbb{F}\right) \times \\ \times & \sum\_{m'} \left[ \left<\hat{\boldsymbol{\lambda}}\_1 \middle| \exp\left(-i\vec{Q}\cdot\vec{\boldsymbol{r}}\_m\right) \middle| \hat{\boldsymbol{\lambda}}\_\mathbb{F}\right> \left<\hat{\boldsymbol{\lambda}}\_\mathbb{F}\middle| \exp\left(i\vec{Q}\cdot\vec{\boldsymbol{r}}\_\gamma\right) \middle| \hat{\boldsymbol{\lambda}}\_\mathbb{r}\right>\right] \delta\left(\hbar o + \boldsymbol{E}\_1 - \boldsymbol{E}\_\mathbb{F}\right) . \end{split} \tag{15}$$

It has to be noticed that the sum involves states of the target system only. Here the indexes "*m*" and "*j*" label the coordinates of the scatterers, i.e. the electrons interacting with the electro‐ magnetic field.

In a scattering experiment, the states of the target system |*λ*I and |*λ*F are generally unknown, because only the final and initial states of the photons are measured. However, it is usually assumed that, before the scattering event, the sample is in its thermodynamic equilibrium and, therefore, a "thermal" average can be performed over its initial state. Furthermore, if the sample can be treated as a many-body classical system, the probability of the |*λ*<sup>I</sup> state is *p*I=exp( −*βE*<sup>I</sup> ) / ∑ I exp( −*βE*<sup>I</sup> ), where *β =* 1/*k* <sup>B</sup> *T* and *k* B and *T* are the Boltzmann constant and the temperature of the sample, respectively. At this stage, the cross section associated to the transition between the photon states |*k*<sup>I</sup> , *ε*I and |*k*F, *ε*F can be written as:

$$\begin{split} \left(\frac{\mathbf{d}^2 \sigma}{\mathbf{d}\Omega \mathbf{d} E\_{\mathbf{F}}}\right)\_{\parallel k\_z, \mathbf{c}\_1 \wedge \dots \wedge \mathbf{c}\_{\mathbf{F}}, \mathbf{c}\_{\mathbf{F}}} &= \frac{e^2}{mc^2} \Biggl( \frac{k\_{\mathbf{F}}}{k\_{\mathbf{1}}} \bigg) \left(\hat{\boldsymbol{\varepsilon}}\_{\mathbf{l}} \cdot \hat{\boldsymbol{\varepsilon}}\_{\mathbf{r}}\right) \times \\ \times \sum\_{m \atop \mathbf{m}} \sum\_{\lambda\_{\mathbf{r}} l\_{\mathbf{r}}} \bigg[ p\_z \left\langle \lambda\_{\mathbf{r}} \bigg| \exp\left(-i\vec{Q} \cdot \vec{r}\_m\right) \middle| \lambda\_{\mathbf{r}} \right\rangle \bigg\langle \lambda\_{\mathbf{r}} \bigg| \exp\left(i\vec{Q} \cdot \vec{r}\_{\vec{j}}\right) \bigg| \lambda\_{\mathbf{r}} \rangle \bigg] \delta \left(\hbar o + E\_{\mathbf{l}} - E\_{\mathbf{F}}\right) \Biggr. \end{split} \tag{16}$$

This formula can be further simplified by:


3

(10)

p

2 , = . <sup>2</sup> *F F kF F F <sup>L</sup> dE d k dk*

æ ö <sup>W</sup> ç ÷ è ø

2 3 F , F <sup>3</sup> d = d .

Also, the incident photon flux, i.e. the number of photons passing through a unit surface in

Furthermore, the double differential cross section for the scattering process can be related to

I F

è ø W W <sup>å</sup> (13)

× - × --

å h r <sup>v</sup> ) <sup>h</sup> (14)

, with ℏ*Q*

®

F

()() ( ) ( ) ( ) <sup>2</sup>

<sup>1</sup> <sup>=</sup> <sup>ˆ</sup> exp ( , *j I j*

*W*

l l

d

W

p

= . *<sup>c</sup> V* j

d 11 = . dd d *<sup>k</sup> I F*

j

int 3 2 1/2 IF F I F

 l

Here the conservation laws of momentum and energy in the scattering event have been

tum and energy transferred from the photon to the target sample. In particular, the energy conservation is accounted by the *δ*(ℏ*ω* −*E*I−*E*F) term in Eq. 13. Considering that the frequency

) and ℏ*ω* =*E*<sup>F</sup> −*E*<sup>I</sup>

( ) ( ) ( )

*iQ r iQ r E E*

 ldw

exp exp .

I F F I I F

r r r r <sup>h</sup>

*e c FH I iQ r E E*

) =*ck*<sup>I</sup>

( )

*m j*

é ù ´ -× × +- åê ú ë û

2 I F

e e

)

ee

Using Eqs. 8–12, the matrix element defining Fermi's golden rule (Eq. 8) can be thus expressed

*c*

W

<sup>h</sup> (11)

(12)

ld

 w

(*ω*(*k*F) =*ck*F), the double differential cross-section

→ and ℏ*<sup>ω</sup>* being the momen‐

(15)

e n

Furthermore, being *E*<sup>F</sup> =ℏ*ck*F, or, equivalently, d*E*<sup>F</sup> =ℏ*c*d*k*F, one has:

F F

the unit time, is given by:

8 Neutron Scattering

such a flux through:

(2)

reduces to:

as:

e n

F | |

ñ® ñ

*E*

s

æ ö ç ÷

I I

<sup>→</sup> <sup>=</sup>ℏ(*<sup>k</sup>* → <sup>F</sup> −*k* → I

*e k*

l l

é ù ë û

*L mc k k*

superimposed by assuming ℏ*Q*

of the incident (scattered) wave is *ω*(*k*<sup>I</sup>

*'*

l

s

*mj*

2 2

d ' <sup>ˆ</sup> d d

æ ö æ ö ç ÷ = ×´ ç ÷ è ø <sup>W</sup> è ø

*E mc k*


*I F*

ñ® ñ

w w (2 ) *<sup>k</sup> k L <sup>E</sup>*

$$\delta(\hbar\omega + E\_{\rm I} - E\_{\rm F}) = 1/2\pi\hbar \underbrace{\stackrel{\cdots}{\int}}\_{\to} \exp(i\omega t) \exp(E\_{\rm I} - E\_{\rm F})/\hbar\omega^2$$


With the above manipulations, the double differential cross section eventually reduces to:

$$\begin{split} \left(\frac{\mathbf{d}^2 \sigma}{\mathbf{d}\Omega \mathbf{d} \vec{E}}\right)\_{\left[\mathbf{k}\_\parallel,\varepsilon\_\parallel\right]\to\left[\mathbf{k}\_\parallel,\varepsilon\_\parallel\right]} &= \frac{e^2}{mc^2} \Big(\frac{\mathbf{k}^\prime}{k}\Big) (\hat{\boldsymbol{\varepsilon}}\_\parallel \cdot \hat{\boldsymbol{\varepsilon}}\_\parallel) \times \\ & \times \frac{1}{2\pi\hbar} \Big\| \sum\_{n\neq \mathbf{0}} \Big\langle \exp\Big\langle -i\vec{Q} \cdot \left[\vec{r}\_\parallel(t) - \vec{r}\_n(0)\right] \Big\rangle \Big\rangle \exp\left(i\alpha t\right) \mathbf{d}t,\end{split} \tag{17}$$

where the correlation function between two generic variables was introduced as:

*A* 0 *B*(*t*) =∑ *λ p<sup>λ</sup> λ* | *A* 0 *B*(*t*)|*λ* .was introduced.

### **2.1. Cross section and density correlation function**

The double differential cross section can be now simply connected to the Fourier transform of the density–density correlation function, as discussed in [6].

To show this explicitly, it is useful to start from an appropriate definition of the atomic density in the mesoscopic regime probed by IXS. There, the density of a single atom can be approxi‐ mated by Dirac's *δ* function of the space coordinate:

$$m\_{\dot{\gamma}}\left(\vec{r},t\right) = \mathcal{S}\left[\vec{r} - \vec{R}\_{\dot{\gamma}}(t)\right].\tag{18}$$

Clearly, this is a highly discontinuous function diverging at the center of mass position of the *j*th atom *R* ⇀ *j* (*t*) and vanishing elsewhere. The normalization condition *∫ V nj* (*r* ⇀ ,*t*)d*r* <sup>→</sup> =1 simply means that, if a sufficiently large volume is covered, the particle should be counted one time inside such a volume. A coarse estimate of the density variable is given by the mean value *nj* (*r* <sup>→</sup> ,*t*) =1 / *<sup>V</sup> <sup>∫</sup> V nj* (*r* <sup>→</sup> ,*t*)d*r* <sup>→</sup> =1 / *<sup>V</sup>* obtained as a statistical average over a certain volume. The density function of a system of *N* atoms, the density function can be expressed as follows:

$$m(\vec{r}, t) = \sum\_{i=1}^{N} \delta \left[ \vec{r} - \vec{R}\_i(t) \right].$$

One can perform the average density by integrating over the volume, eventually obtaining *n*(*r* ⇀ ,*t*) =1/ *V ∫ V n*(*r* ⇀ ,*t*)d*r* ⇀ =*n*, where *n=N*/*V* is the number density (number of atoms in the unit volume) of the system. However, in the present context is more useful to deal with the density fluctuation:

$$\mathcal{S}n(\vec{r},t) = \sum\_{i=1}^{N} \delta \left[\vec{r} - \vec{R}\_i(t)\right] - n. \tag{19}$$

At this stage, the atomic Van Hove correlation function can be introduced as:

$$G(\vec{r},t) = \frac{1}{N} \sum\_{k,j=1}^{N} \left\langle \delta\left[\vec{r} - \vec{R}\_k(0)\right] \delta\left[\vec{r} - \vec{R}\_j(t)\right] \right\rangle - n = \frac{1}{N} \sum\_{k,j=1}^{N} \left\langle \delta\left[\vec{r} - \vec{R}\_k(0) - \vec{R}\_j(t)\right] \right\rangle - n. \tag{20}$$

Since a scattering experiment probes the reciprocal rather than the real space, it is here convenient to introduce the Fourier transform of the density in Eq. 18:

$$\delta n\left(\vec{Q},t\right) = \sum\_{\vec{r}} \exp\left[\vec{r} - \vec{R}\_{\cdot}(t)\right] - \left(2\pi\right)^{3} n\delta\left(\vec{Q}\right). \tag{21}$$

and its time-correlation, the intermediate scattering function:

where the correlation function between two generic variables was introduced as:

The double differential cross section can be now simply connected to the Fourier transform of

To show this explicitly, it is useful to start from an appropriate definition of the atomic density in the mesoscopic regime probed by IXS. There, the density of a single atom can be approxi‐

é ù - ë û

Clearly, this is a highly discontinuous function diverging at the center of mass position of the

means that, if a sufficiently large volume is covered, the particle should be counted one time inside such a volume. A coarse estimate of the density variable is given by the mean value

<sup>→</sup> =1 / *<sup>V</sup>* obtained as a statistical average over a certain volume. The density

⇀ =*n*, where *n=N*/*V* is the number density (number of atoms in the unit

r r <sup>v</sup> (19)

*i*

<sup>r</sup> <sup>v</sup> <sup>v</sup> (18)

*V nj* (*r* ⇀ ,*t*)d*r*

<sup>→</sup> =1 simply

( ) , = (). *j j n rt r R t* d

(*t*) and vanishing elsewhere. The normalization condition *∫*

function of a system of *N* atoms, the density function can be expressed as follows:

=1 ( , )= ( ) . *N*

One can perform the average density by integrating over the volume, eventually obtaining

volume) of the system. However, in the present context is more useful to deal with the density

*i*

*k j k j*

é ù - - -= - - - éù é ù

d

=1 ( , )= ( ) .

 d*nrt r R t n* é ù - - å ë û

1 1 ( , )= (0) ( ) (0) ( ) .

 å å ë û ëû ë û r r r r r rr <sup>r</sup> (20)

*Grt r R r R t n r R R t n*

*i*

At this stage, the atomic Van Hove correlation function can be introduced as:

, =1 , =1

*k j k j*

*N N*

 d

d

*N N*

d

*N*

r r v

*i nrt r R t* dé ù - å ë û

*p<sup>λ</sup> λ* | *A* 0 *B*(*t*)|*λ* .was introduced.

the density–density correlation function, as discussed in [6].

**2.1. Cross section and density correlation function**

mated by Dirac's *δ* function of the space coordinate:

*A* 0 *B*(*t*) =∑

10 Neutron Scattering

*j*th atom *R*

*nj* (*r*

*n*(*r*

⇀ *j*

<sup>→</sup> ,*t*) =1 / *<sup>V</sup> <sup>∫</sup>*

⇀ ,*t*) =1/ *V ∫*

fluctuation:

*V n*(*r* ⇀ ,*t*)d*r*

*V nj* (*r* <sup>→</sup> ,*t*)d*r*

*λ*

$$\tilde{F}(\vec{Q},t) = \langle \delta n^\dagger(\vec{Q},0) \delta n(\vec{Q},t) \rangle = \frac{1}{N} \sum\_{n \neq} \exp\left\langle i\vec{Q} \cdot \left[\vec{R}\_{\cdot}(t) - \vec{R}\_{n}(0)\right] \right\rangle - \left(2\pi\right)^3 \delta\left(\vec{Q}\right). \tag{22}$$

Finally, the dynamic structure factor, which is the variable actually measured in a scattering experiment, reads as:

$$\begin{split} \tilde{S}(\vec{Q}, o) &= \frac{1}{2\pi\hbar} \int\_{-\alpha}^{+\alpha} e^{iat} \left< \delta\hat{u}^\* \left( \vec{Q}, 0 \right) \delta\hat{u} \left( \vec{Q}, t \right) \right> dt = \\ &\frac{1}{2\pi\hbar} \left[ \sum\_{m} \exp\{i\vec{Q} \cdot \left[ \vec{R}\_{\cdot}(t) - \vec{R}\_{m}(0) \right] \} - \left( 2\pi\right)^3 \delta\left( \vec{Q} \right) \right] \exp(iat) dt \ . \end{split} \tag{23}$$

One can easily recognize that the second term under integration (having the sign "-" as a prefactor) is proportional to *δ*(*ω*)*δ*(*Q* <sup>→</sup>), thus is not relevant in a real IXS or INS experiment, as it describes the merely elastic (ω=0) scattering in the forward direction (*Q* <sup>→</sup> =0). Therefore, one can make the following identification:

$$\tilde{S}(\bar{Q}, o) \to S(\bar{Q}, o) = \frac{1}{2\pi\hbar} \left[ \left[ \sum\_{m \neq} \exp\left\{ i\bar{Q} \cdot \left[ \bar{R}\_{\uparrow}(t) - \bar{R}\_{m}(0) \right] \right\} \right] \exp\left( i\alpha t \right) \text{d}t \right]$$

It is important to recognize that Eqs. 17–23 depend on the center of mass (nuclear) coordinates of the atoms. The integral in Eq. 16 is thus to be identified with the spectrum of the density fluctuation autocorrelation function, provided the electronic coordinates *r* → *j* (*t*) are replaced by the nuclear ones *R* ⇀ *j* (*t*). This identification relies on the validity of the Born–Oppenheimer approximation, which ultimately justifies the factorization between exponential terms containing either merely electronic or merely nuclear coordinates. With such a factorization, the double differential cross section reduces to:

$$
\left(\frac{\mathbf{d}^2 \sigma}{\mathbf{d}\Omega \mathbf{d}E'}\right) = \frac{e^2}{mc^2} \left(\frac{k'}{k}\right) \left(\hat{\boldsymbol{\varepsilon}}\_1 \cdot \hat{\boldsymbol{\varepsilon}}\_\mathbf{r}\right) \left|f\left(\mathbf{Q}\right)\right|^2 S\left(\mathbf{Q}\_\nu \boldsymbol{\alpha}\right),\tag{24}
$$

where

$$f\left(Q\right) = \sum\_{a} \exp(-i\vec{Q} \cdot \vec{\rho}\_a) \tag{25}$$

is the atomic structure factor, which depends on the coordinate *ρ* → *<sup>α</sup>* of the *α*th electron in the reference frame with the nucleus at the origin. It is worth stressing that the sum in Eq. 25 runs over all electrons in the atom. Furthermore, in Eq. 24 it was assumed the isotropic character of a liquid system, which implies that physical properties do not depend on the direction of *Q* ⇀ but only on its amplitude *Q* = |*Q* ⇀ |.

### **2.2. The incoherent contribution**

The derivation discussed above strictly applies to systems of *N* identical atoms within the validity of the Born–Oppenheimer approximation. For a system containing different atoms, the derivation of the scattering cross section is only slightly more complex. However, a factorization between a merely electronic and a nuclear term is still possible and leads to the conclusion that the double differential cross section splits into two components: a coherent and an incoherent one. The former is proportional to the average value of the form factor, whereas the latter arises from its mean square fluctuations.

To illustrate this point, it is useful to start from the assumption that atoms in the target system have reasonably symmetric electronic clouds. The double differential cross section can thus be expressed as follows:

$$\overline{\left(\frac{\partial \sigma}{\partial \mathbf{Q} \mathbf{d} E'}\right)} = r\_0^2 \left(\frac{k'}{k}\right) \{\hat{\boldsymbol{\varepsilon}}\_{\boldsymbol{\beta}} \cdot \hat{\boldsymbol{\varepsilon}}\_{\boldsymbol{\beta}}\}^2 \frac{1}{N} \sum\_{a \neq \boldsymbol{0}} \sum\_{n \neq} \left\langle \exp\left[i \boldsymbol{Q} \cdot \left(\vec{\boldsymbol{R}}\_{\boldsymbol{\beta}} \left(t\right) - \vec{\boldsymbol{R}}\_{\boldsymbol{n}} \left(0\right)\right)\right] \overline{\exp\left[i \boldsymbol{\vec{Q}} \cdot \left(\vec{\boldsymbol{\rho}}\_{a}^{n} - \vec{\boldsymbol{\rho}}\_{\boldsymbol{\beta}}^{n}\right)\right]}\right.\tag{26}$$

where the *ρ* → *α <sup>i</sup>* is the position the αth electron of the *i*th atom in the reference frame of atomic center of atomic mass. Here the bar on the top of the exponential term indicates an average over all atoms of the system. We can thus distinguish two cases in the following:

**1.** The term under average involves distinct atoms (*n ≠ m*), then:

$$\sum\_{a,\boldsymbol{\rho}} \overline{\exp\{i\boldsymbol{\mathbb{Q}} \cdot \vec{\boldsymbol{\rho}}\_a^{\boldsymbol{\eta}}\}} \exp\{i\boldsymbol{\bar{\mathbb{Q}} \cdot \vec{\boldsymbol{\rho}}\_{\boldsymbol{\rho}}^{\boldsymbol{\eta}}\}} = \sum\_{a} \overline{\exp\{i\boldsymbol{\bar{\mathbb{Q}} \cdot \vec{\boldsymbol{\rho}}\_a^{\boldsymbol{\eta}}\}} \sum\_{\boldsymbol{\rho}} \overline{\exp\{i\boldsymbol{\bar{\mathbb{Q}} \cdot \vec{\boldsymbol{\rho}}\_{\boldsymbol{\rho}}^{\boldsymbol{\eta}}\}}} = \overline{f\left(\boldsymbol{\mathbb{Q}}\right)^2}.$$

**2.** The term under average involves the same atom (*n =m*), then:

$$\sum\_{\alpha,\beta} \overline{\exp\left(i\vec{Q}\cdot\vec{\rho}\_{\alpha}^{\n}\right)\exp\left(i\vec{Q}\cdot\vec{\rho}\_{\beta}^{\n\prime}\right)} = \overline{f\left(Q\right)^{2}}.$$

Using X-ray as a Probe of the Terahertz Dynamics of Disordered Systems – Complementarity with Inelastic... http://dx.doi.org/10.5772/62844 13

Thus, in general:

where

12 Neutron Scattering

but only on its amplitude *Q* = |*Q*

**2.2. The incoherent contribution**

expressed as follows:

→ *α*

,

a b

s

where the *ρ*

0

*k*

e e *f Q iQ* ( ) exp(- )

= × å

reference frame with the nucleus at the origin. It is worth stressing that the sum in Eq. 25 runs over all electrons in the atom. Furthermore, in Eq. 24 it was assumed the isotropic character of a liquid system, which implies that physical properties do not depend on the direction of *Q*

The derivation discussed above strictly applies to systems of *N* identical atoms within the validity of the Born–Oppenheimer approximation. For a system containing different atoms, the derivation of the scattering cross section is only slightly more complex. However, a factorization between a merely electronic and a nuclear term is still possible and leads to the conclusion that the double differential cross section splits into two components: a coherent and an incoherent one. The former is proportional to the average value of the form factor,

To illustrate this point, it is useful to start from the assumption that atoms in the target system have reasonably symmetric electronic clouds. The double differential cross section can thus be

( () ( )) ( ) 2 2

center of atomic mass. Here the bar on the top of the exponential term indicates an average

( )( ) ( ) ( ) ( )<sup>2</sup>

( )( ) ( )<sup>2</sup>

exp exp . *n m iQ iQ f Q*

 a

 r

> b

 r b

exp exp exp exp . *nm n m iQ iQ*

a

a

å × ×= r

r r r r

å åå × ×= × ×=

rr r r rr r r

æ ö æö ¶ ¢ é ù é ù ç ÷ ç÷ <sup>×</sup> × - × - ¢ ë û ë û è ø èø ¶W åå r r <sup>r</sup> r r (26)

*<sup>i</sup>* is the position the αth electron of the *i*th atom in the reference frame of atomic

<sup>1</sup> = ( ) exp ˆ ˆ 0 xp , <sup>d</sup>

*f i j m*

over all atoms of the system. We can thus distinguish two cases in the following:

*<sup>r</sup> iQ R t R e iQ Ek N*

*mj*

ab

**1.** The term under average involves distinct atoms (*n ≠ m*), then:

ab

**2.** The term under average involves the same atom (*n =m*), then:

rr

,

a b a

is the atomic structure factor, which depends on the coordinate *ρ*

⇀ |.

whereas the latter arises from its mean square fluctuations.

a

→

(25)

⇀

*<sup>α</sup>* of the *α*th electron in the

*n m*

a b

r r

 b

 r

*iQ iQ f Q*

r<sup>r</sup> <sup>r</sup>

$$
\sum\_{a,\boldsymbol{\rho}} \overline{\exp\left(i\boldsymbol{\bar{Q}} \cdot \bar{\boldsymbol{\rho}}\_{a}^{n}\right)} \exp\left(i\boldsymbol{\bar{Q}} \cdot \bar{\boldsymbol{\rho}}\_{\boldsymbol{\rho}}^{n}\right) = \overline{f\left(\boldsymbol{Q}\right)^{2}} + \delta\_{n,n} \overline{\left(f\left(\boldsymbol{Q}\right)^{2} - \overline{f\left(\boldsymbol{Q}\right)}^{2}\right)} = \overline{f\left(\boldsymbol{Q}\right)^{2}} + \delta\_{n,n} \overline{\left(f\left(\boldsymbol{Q}\right) - \overline{f\left(\boldsymbol{Q}\right)}\right)^{2}}.
$$

$$
\overline{f\left(\boldsymbol{Q}\right)^{2}} + \delta\_{n,n} \overline{\left(f\left(\boldsymbol{Q}\right)^{2} - \overline{f\left(\boldsymbol{Q}\right)}^{2}\right)} = \overline{f\left(\boldsymbol{Q}\right)^{2}} + \delta\_{n,n} \overline{\left(f\left(\boldsymbol{Q}\right) - \overline{f\left(\boldsymbol{Q}\right)}\right)^{2}}
$$

which eventually leads to the following expression for the IXS cross section:

$$\frac{\hat{\sigma}^2 \sigma}{\hat{\sigma} \Omega \hat{\sigma} \mathbb{E}\_{\text{F}}} = r\_0^2 \left( \frac{k\_{\text{F}}}{k\_{\text{1}}} \right) \left( \hat{\varepsilon}\_{\text{1}} \cdot \hat{\varepsilon}\_{\text{F}} \right) \left[ \overline{f\left(\mathbb{Q}\right)}^2 \text{S}\_{\text{C}}(\mathbb{Q}, o) + \left[ \overline{\left(f\left(\mathbb{Q}\right) - \overline{f\left(\mathbb{Q}\right)} \right)}^2 \right] \text{S}\_{\text{1}}(\mathbb{Q}, o) \right]. \tag{27}$$

The above formula shows that the scattering of the sample is composed of a coherent and an incoherent part, respectively, labeled by the "*C*" and "*I*" suffixes. While the latter results from the independent scattering of isolated atoms, the former also contains *the* interference between photon waves scattered by different atoms. Insight on the collective dynamics of atoms can only be sought for in the coherent scattering, whereas both coherent and incoherent scattering convey information on the single-atom dynamics.

#### **2.3. The absorption coefficient**

When actually performing the IXS measurement, it is important to optimize the sample thickness crossed by the beam (*d*) so as to maximize the scattering intensity*.* In fact, a simple *d* increase, albeit increasing the number of scatterers, does not necessarily enhance the scattered intensity because it also increases the sample (self) absorption. Therefore, the optimal choice of *d* is the result of a trade-off between competing scattering and absorption requirements. To handle the problem on a more quantitative basis, attenuation effects should be included in the general expression of the scattering intensity as prescribed by the following formula [7]:

$$I = I\_0 \frac{\partial^2 \sigma}{\partial \Omega \,\, \partial E\_{\text{F}}} \text{d}\Omega \,\, \text{d}E\_{\text{F}} \text{ } n \,\, d \, \exp\left(-\mu d\right), \tag{28}$$

where *I*0 is the number of incident photons per second and *μ* is the total absorption coefficient.

The intensity *I* in Eq. 27 reaches a maximum when ∂ *I* / ∂*d* =0, i.e. for *d =* 1/*μ* and, consequently, *I* ∝1 / *μ*.

At the energies typical of IXS experiments, attenuation effects are dominated by the photo‐ electric absorption, for which *μ* ∝*Z* <sup>4</sup> . Consequently, for high-*Z* elements, *μ* is very large, and therefore, the optimal sample thickness is small. Of course, the optimal *d* value strongly depends on the incident energy and on *Z*. This makes IXS, in principle, ideal to work with extremely small high *Z* samples, and it turns out to be a decisive advantage while dealing with samples only available in modest quantities and/or when extreme thermodynamic conditions need to be explored.

### **3. A closer comparison between IXS and INS techniques**

In general, INS and IXS present several analogies:


Other similarities emerge from the comparison of the IXS double differential cross section in Eq. 26 and its INS counterparts as derived, for instance, by [8]:

$$\frac{\mathbf{d}^2 \sigma}{\mathbf{d} \mathbf{d} \mathbf{Q} \mathbf{d} E\_{\mathrm{F}}}\bigg|\_{\mathrm{NIS}} = \frac{k\_{\mathrm{F}}}{k\_{\mathrm{l}}} N \left[ \overline{\left\| \overline{b} \right\|}^2 \, \mathrm{S}\_{\mathrm{C}} \left( \mathrm{Q}\_{\prime} \alpha \right) + \overline{\left\| b - \left\| b \right\|}^2 \, \mathrm{S}\_{\mathrm{l}} \left( \mathrm{Q}\_{\prime} \alpha \right) \right]. \tag{29}$$

It can be readily noticed that:


Despite these formal similarities, the two techniques have several complementary aspects, as discussed below in some detail. This makes each of them better suited to some experiments and less to other. A practical example of the coordinate use of these complementary methods is discussed at the end of this chapter.

### **3.1. Part A: advantages of IXS**

### *3.1.1. Kinematic limitations*

therefore, the optimal sample thickness is small. Of course, the optimal *d* value strongly depends on the incident energy and on *Z*. This makes IXS, in principle, ideal to work with extremely small high *Z* samples, and it turns out to be a decisive advantage while dealing with samples only available in modest quantities and/or when extreme thermodynamic conditions

**1.** They can be used to investigate bulk properties of materials, as opposite to more strongly interacting probes, such as electrons, which essentially convey insight on materials'

**2.** They probe the dynamic response of density fluctuations through the Fourier transform

**3.** They are "mesoscopic" probes, i.e. they cover distances and timescales matching, respectively, first neighboring atoms' (molecules') separations and cage oscillation

Other similarities emerge from the comparison of the IXS double differential cross section in

w

**•** For both X-ray and neutron scattering, the cross section depends on the ratio *k* F/*k* I. However, as discussed in the following, for IXS, this factor does not depend on frequency (energy)

**•** The role of the form factor *f* (*Q*) in the IXS cross section mirrors the one of the scattering length *b* in the INS one. The main difference is their physical origin: *f* (*Q*) relates to the photon–electron electromagnetic interaction, while *b* is connected to the neutron–nuclei interaction. As a consequence, *f* (*Q*) depends on *Z* yet not on the atomic mass, i.e. it depends on the chemical rather than isotopic specie of the target sample. As discussed below, another fundamental difference is that *f* (*Q*) sharply decreases at high exchanged momenta while *b*

Despite these formal similarities, the two techniques have several complementary aspects, as discussed below in some detail. This makes each of them better suited to some experiments

C I

é ù = ê ú + -

 w (29)

( ) ( ) <sup>2</sup> \_\_\_\_\_\_\_\_\_\_\_ <sup>2</sup> \_\_\_\_ <sup>2</sup> <sup>F</sup>

<sup>d</sup> , , . d d *<sup>k</sup> N b S Q b b SQ E k*

<sup>W</sup> ë û

of their autocorrelation function, i.e. the dynamic structure factor, *S*(*Q*, *ω*).

**3. A closer comparison between IXS and INS techniques**

Eq. 26 and its INS counterparts as derived, for instance, by [8]:

F I INS

s

and can be safely approximated by 1.

remains essentially constant.

It can be readily noticed that:

In general, INS and IXS present several analogies:

need to be explored.

14 Neutron Scattering

surfaces.

periods.

The conservation laws of momentum and energy lead to very different results when dealing with either neutron or X-ray probes. Let us consider the neutron case first. As mentioned, the momentum conservation law reads as:

$$
\vec{Q} = \vec{k}\_{\text{F}} - \vec{k}\_{1\prime} \tag{30}
$$

which can be rearranged as follows:

$$\left(\frac{\mathcal{Q}}{k\_{\rm l}}\right)^2 = \left(\frac{k\_{\rm f}}{k\_{\rm l}}\right)^2 + 1 - 2\frac{k\_{\rm f}}{k\_{\rm l}}\cos 2\theta\_{\prime} \tag{31}$$

whereas the energy conservation law for the scattering event reads as:

$$E\_{\rm F} - E\_{\rm l} = \hbar \alpha \epsilon.\tag{32}$$

Considering that the kinetic energy of the freely flying neutron is:

$$E = \hbar^2 k^2 \Big/ 2m\_n\tag{33}$$

with *m* n being the neutron mass, Eq. 27 can be written as:

$$\left(\frac{k\_{\rm F}}{k\_{\rm I}}\right)^{2} = \frac{\hbar \alpha \prime}{E\_{\rm F}} + 1.\tag{34}$$

By combining Eqs. 29 and 26 and taking the square root of both members, one eventually obtains:

$$
\left(\frac{\mathcal{Q}}{k\_1}\right) = \sqrt{2 - \left(\frac{\hbar \alpha}{E\_1}\right) - 2 \cdot \cos\left(2\theta\right)} \sqrt{1 - \left(\frac{\hbar \alpha}{E\_1}\right)}.\tag{35}
$$

It is thus readily noticed that the dynamic domain (*Q*,*ω*) explored by an INS measurement is limited by the condition that both the arguments under the two square roots of Eq. 30 are positive. In particular, the argument of the "nested" square root, having pre-factor 2 cos(2*θ)*, is positive whenever *E* I < *ℏω*. This reflects the obvious requirement that the energy transferred from the neutron to the sample cannot exceed the energy *E* <sup>I</sup> initially carried by the neutron. Furthermore, the requirement for the argument of the larger square root in Eq. 26 to be positive introduces further restrictions to the explored dynamic range.

An example of these restrictions, customarily referred to as kinematic limitations, is provided by Figure 3, which refers to the specific case of INS experiments on water.

It is worth recalling that, since the pioneering MD work of [9] in the mid-1970s, it is well-known that the THz spectrum of water is dominated by two inelastic peaks, whose *Q* dispersions are reported in Figure 3 as computed by MD simulations [10]. However, the spectrum of water measured by two successive INS works (also reported in the plot) did not bear any evidence of a double peaked shape. The reason of the apparent discrepancy between computational and experimental results mainly owed to kinematic and resolution limitations affecting the latter. These limitations were substantially different in the two measurements, owing to the different incident energies: 36 meV and 80 meV respectively for [11] and [12]. In the former experiment [11] these limitations prevented the proper coverage of the high energy modes. Conversely, the higher incident energy (barely) permitted the second INS measurement [12] to cover the high energy mode, yet it entailed a coarsening of the energy resolution, which became too broad to properly resolve the low frequency mode. For reference, Figure 3 displays the portion of the dynamic plane covered by Bosi and coworkers' measurement (shadowed area) along with the boundary of the dynamic region explored by Teixeira and coworkers' one.

Briefly, Figure 3 exemplifies a situation in which the dynamic range covered by the experiment resembles a "blanket too short" preventing the simultaneous access to both high and low *ω* or *Q* values. A similar problem is commonly encountered in INS measurements,

The development of high energy resolution (meV) IXS in the mid-1990s offered the opportunity of performing THz spectroscopy measurements virtually free from kinematic limitations, apart from the obvious ones related to the finite energy and momentum resolutions.

Such a virtual absence of kinematic constraints follows from these two circumstances:

**1.** the energy–momentum relationship for photons is linear:

$$E\_{\parallel} = ck\_{\parallel} \tag{36}$$

instead of quadratic, as for INS (see Eq. 28) and

**2.** for IXS ℏ*ω* < <*E*<sup>I</sup> (typically *E*<sup>I</sup> ≈ 20 keV, while |ℏ*ω* | ≤ ≈80 meV), or, equivalently *E* <sup>F</sup> *= E* I, which, through Eq. 31, implies *k* <sup>F</sup> *= k* I.

Using X-ray as a Probe of the Terahertz Dynamics of Disordered Systems – Complementarity with Inelastic... http://dx.doi.org/10.5772/62844 17

It is thus readily noticed that the dynamic domain (*Q*,*ω*) explored by an INS measurement is limited by the condition that both the arguments under the two square roots of Eq. 30 are positive. In particular, the argument of the "nested" square root, having pre-factor 2 cos(2*θ)*, is positive whenever *E* I < *ℏω*. This reflects the obvious requirement that the energy transferred from the neutron to the sample cannot exceed the energy *E* <sup>I</sup> initially carried by the neutron. Furthermore, the requirement for the argument of the larger square root in Eq. 26 to be positive

An example of these restrictions, customarily referred to as kinematic limitations, is provided

It is worth recalling that, since the pioneering MD work of [9] in the mid-1970s, it is well-known that the THz spectrum of water is dominated by two inelastic peaks, whose *Q* dispersions are reported in Figure 3 as computed by MD simulations [10]. However, the spectrum of water measured by two successive INS works (also reported in the plot) did not bear any evidence of a double peaked shape. The reason of the apparent discrepancy between computational and experimental results mainly owed to kinematic and resolution limitations affecting the latter. These limitations were substantially different in the two measurements, owing to the different incident energies: 36 meV and 80 meV respectively for [11] and [12]. In the former experiment [11] these limitations prevented the proper coverage of the high energy modes. Conversely, the higher incident energy (barely) permitted the second INS measurement [12] to cover the high energy mode, yet it entailed a coarsening of the energy resolution, which became too broad to properly resolve the low frequency mode. For reference, Figure 3 displays the portion of the dynamic plane covered by Bosi and coworkers' measurement (shadowed area) along

with the boundary of the dynamic region explored by Teixeira and coworkers' one.

apart from the obvious ones related to the finite energy and momentum resolutions.

Such a virtual absence of kinematic constraints follows from these two circumstances:

*Q* values. A similar problem is commonly encountered in INS measurements,

**1.** the energy–momentum relationship for photons is linear:

instead of quadratic, as for INS (see Eq. 28) and

which, through Eq. 31, implies *k* <sup>F</sup> *= k* I.

(typically *E*<sup>I</sup>

**2.** for IXS ℏ*ω* < <*E*<sup>I</sup>

Briefly, Figure 3 exemplifies a situation in which the dynamic range covered by the experiment resembles a "blanket too short" preventing the simultaneous access to both high and low *ω* or

The development of high energy resolution (meV) IXS in the mid-1990s offered the opportunity of performing THz spectroscopy measurements virtually free from kinematic limitations,

=*j j E ck* (36)

≈ 20 keV, while |ℏ*ω* | ≤ ≈80 meV), or, equivalently *E* <sup>F</sup> *= E* I,

introduces further restrictions to the explored dynamic range.

16 Neutron Scattering

by Figure 3, which refers to the specific case of INS experiments on water.

**Figure 3.** INS results in heavy water by [11] and [12] are reported as red and black dots, respectively, along with the MD simulation results of the two modes in the spectrum of water (dashed lines of corresponding color). The boundary of the explored dynamic region for the indicated scattering angle is reported with reference to the INS measurement by Bosi and coworkers. The shadowed area represents the dynamic domain accessible by such a measurement. The boundary of the dynamic plane explored by the Teixeira and coworkers' experiment is also indicated as a dot-dashed black line.

In particular, from the condition (2), it follows that, after the scattering event, the momentum of the photon ℏ*k* <sup>→</sup> changes in direction, yet not in amplitude; therefore, one has:

$$\mathbf{Q}^2 = k\_\mathrm{l}^2 + k\_\mathrm{p}^2 - 2k\_\mathrm{l}k\_\mathrm{p} \cos \left(2\theta\right) = 2k\_\mathrm{l}^2 \left(1 - \cos 2\theta\right) = \left(\frac{4\pi}{\lambda}\right)^2 \sin^2 \left(\theta\right),\tag{37}$$

where *k* <sup>F</sup> *= k* <sup>I</sup> *=* 2*π*/*λ*, *λ* is the incident wavelength. The above formula shows that, for IXS, *Q* and *ω* are uncoupled, and, specifically, *Q* depends only on scattering angle and incident wavelength. This implies that the accessible portion of the dynamic plane is virtually unlimited at high (*Q,ω*) (virtual absence of kinematic limitations), while at small (*Q,ω*), it is still limited by the finite energy and momentum resolution of the spectrometer.

Taking advantage of the extension in the covered dynamic domain as well as of the statistical accuracy achievable with new synchrotron sources, recent IXS measurements on water droplets [13] evidenced a rather complex spectral shape. This contains both high and low frequency inelastic modes, reminiscent of the sharper peaks dominating the phonon spectrum of ice, also measured in the same work. This clearly emerges from the two panels of Figure 4: the upper one, comparing the IXS measurement on a water droplet [13] to the INS one on heavy water [11] and lower one, which displays the IXS spectrum of ice.

ItclearlyappearsthatthesuperiorspectralcontrastoftheINSmeasurementisthekeytoproperly resolve the low energy mode, although this can only be achieved in a restricted dynamic range excluding all high frequency modes.

**Figure 4.** IXS spectra measured in a droplet of supercooled water (upper panel) and in ice (lower panel). Data are re‐ drawn from [13] and are reported along with best fit line shape (thick black lines) and the various inelastic components (dotted lines). The INS spectrum of bulk heavy water at ambient temperature and same Q values is reported in the upper plot as redrawn from [11], after rescaling for an arbitrary intensity factor (red dots).

### *3.1.2. Constant Q scans*

One of the major drawbacks of INS is that scans of *E* performed at constant *θ* in general do not provide a fixed-*Q* measurement, yet they change the *Q* value through Eq. 30. This *E–Q*coupling is usually inconvenient because most theoretical models deal with constant *Q* (rather than constant *θ*) cuts of *S*(*Q,ω*). Two strategies are commonly used in INS experiments to measure constant *Q* spectra. The one firstly developed [1] is based on the use of triple axis spectrometers [14] in which *θ* and *E* movements can be coordinated to keep *Q* constant (through Eq. 30). This technique is rather time consuming since no more than one *Q* value can be measured in each scan. Conversely, time of flight (*ToF*) spectrometers [15] enable to map *S*(*Q*,*ω*) surfaces with points sparse in the (*Q,ω*) plane, which makes this technique much less time consuming. However, constant *Q* cuts of *S*(*Q*,*ω*) can only be interpolated using analytical models, yielding questionable results when *S*(*Q*,*ω*) surfaces are not sufficiently smooth.

### *3.1.3. Incoherent scattering*

frequency inelastic modes, reminiscent of the sharper peaks dominating the phonon spectrum of ice, also measured in the same work. This clearly emerges from the two panels of Figure 4: the upper one, comparing the IXS measurement on a water droplet [13] to the INS one on heavy

ItclearlyappearsthatthesuperiorspectralcontrastoftheINSmeasurementisthekeytoproperly resolve the low energy mode, although this can only be achieved in a restricted dynamic range

**Figure 4.** IXS spectra measured in a droplet of supercooled water (upper panel) and in ice (lower panel). Data are re‐ drawn from [13] and are reported along with best fit line shape (thick black lines) and the various inelastic components (dotted lines). The INS spectrum of bulk heavy water at ambient temperature and same Q values is reported in the

One of the major drawbacks of INS is that scans of *E* performed at constant *θ* in general do not provide a fixed-*Q* measurement, yet they change the *Q* value through Eq. 30. This *E–Q*coupling

upper plot as redrawn from [11], after rescaling for an arbitrary intensity factor (red dots).

*3.1.2. Constant Q scans*

water [11] and lower one, which displays the IXS spectrum of ice.

excluding all high frequency modes.

18 Neutron Scattering

As discussed above, the incoherent scattering contribution to IXS intensity essentially arises from random fluctuations of the electronic form factor, *f* (*Q*), while for INS it originates from fluctuations of the scattering length *b*. The latter depends on *Z*, *A* (the atomic number), and *J* <sup>→</sup> , the total (electron + nuclear) magnetic moment. The resulting scattering strength can thus be very different not only for different isotopes of the same material, but also for the same isotope species if these have total magnetic moment with different orientations. This can cause substantial mean square fluctuations of *b*, which provide a significant contribution to the incoherent part of the neutron cross section (indeed ∝ | *b* |<sup>2</sup> − |*b*|<sup>2</sup> ). Obviously, this is a serious drawback if the experiment aims at investigating the collective dynamics of the sample, as for the examples discussed in this chapter. However, it represents a valuable resource when the experiment focuses on single molecule's dynamics of, because the incoherent scattering provides direct insight on it. In this respect, highly hydrogenated systems are ideal sample for incoherent INS measurements, owing to almost fully incoherent character of the INS cross section of hydrogen.

### *3.1.4. Incident flux on the sample*

Although IXS can be considered a "photon hungry" technique, state of art IXS spectrometers have incident fluxes much higher than that their INS counterparts. Typically, IXS beamlines can produce a photon flux of 109 –1010 photons/s within a focal spot as narrow as 100 µm2 , while for ToF spectrometers, a flux of ≈10<sup>5</sup> photons/s is typically achieved within focal spot of a few square centimeters. Although the collection/spectral analysis of IXS spectrometers ultimately imposes severe intensity penalties, mostly due to the relatively low angular acceptance of IXS analyzers, the final count rate of meV-resolution IXS measurements is usually much higher than that of their INS counterparts.

### *3.1.5. Multiple scattering*

As mentioned, the leading contribution to the attenuation of IXS signal is the photoelectric absorption, which, on the bright side, also causes a strong suppression of multiple scattering events. As a consequence, in IXS measurements the multiple scattering intensity is often negligible compared to the single scattering signal, which is the one proportional to *S*(*Q,ω*). The situation is radically different for INS measurements, for which the attenuation is dominated by the scattering process itself, and, therefore, multiple scattering events substan‐ tially contribute to the inelastic signal.

### *3.1.6. Transverse beam size*

The incident beam of typical IXS spectrometers has an extremely small focal spot, usually few tens of square micrometer or even less. This permits IXS experiments to deal with small-sized samples, thus potentially disclosing the access to extreme thermodynamic conditions, such as high pressure and extremely low or high temperature. This is also a valuable asset when dealing with samples only available in small volumes. The use of smaller samples in IXS measurements is also fostered by the usually high absorption, especially for high *Z* materials. Conversely, the neutron beam is much larger in size and, also, more deeply penetrating inside the matter and this makes small samples relatively "transparent" to this technique.

### **3.2. Part B: advantages of INS**

### *3.2.1. Resolution shape*

Probably, the most important advantage of INS over IXS is the narrower and sharper energy resolution function. Most importantly, INS offers the opportunity of tailoring the energy resolution width .In fact, the latter can be narrowed according to the specific needs of experi‐ menters, although, unavoidably, to the cost of a shrinkage of the dynamic range covered. The use of a narrow and sharp resolution is crucial in many applications including, for instance, the study of the slow dynamic response of highly viscous systems as glass formers approaching the melting. In fact, in this case all relevant spectral features concentrate in the so-called quasielastic region of the spectrum. Resolution functions as narrow as a few meV can be reached using cold neutrons as a probe. Furthermore, INS resolution profiles are usually sharp (nearly Gaussian) as opposite to the broad (mostly Lorentzian) IXS resolution wings.

Although new concept IXS spectrometers discussed in the following promise a drastic improvement of both resolution width (sub-meV) and spectral contrast, the performance of quasi-elastic neutron scattering instruments [16] is unlikely to be ever matched by IXS spectrometers.

### *3.2.2. Q-decay of the cross section*

Due to the highly localized interaction of neutrons with the target nuclei, no appreciable *Q* decay of the scattering length can be observed up to *Q* values of the order of the inverse of the nuclear size. This value is nearly a factor 10<sup>3</sup> larger than the typical *Q* value at which the IXS cross halves its *Q =* 0 value. For this reason, the use of neutrons is particularly convenient for extremely high *Q* measurements probing the so-called single particle, or impulse approxima‐ tion, regime [17]. Indeed, the access to this regime has been an exclusive task of deep inelastic neutron scattering (DINS) for decades ([18]. Although some successful attempts to reach the impulse approximation with IXS have been reported in the literature [19], the maximum *Q* reached by these measurements is still smaller than the range covered by DINS for more than two orders of magnitude.

### *3.2.3. Absorption*

dominated by the scattering process itself, and, therefore, multiple scattering events substan‐

The incident beam of typical IXS spectrometers has an extremely small focal spot, usually few tens of square micrometer or even less. This permits IXS experiments to deal with small-sized samples, thus potentially disclosing the access to extreme thermodynamic conditions, such as high pressure and extremely low or high temperature. This is also a valuable asset when dealing with samples only available in small volumes. The use of smaller samples in IXS measurements is also fostered by the usually high absorption, especially for high *Z* materials. Conversely, the neutron beam is much larger in size and, also, more deeply penetrating inside

Probably, the most important advantage of INS over IXS is the narrower and sharper energy resolution function. Most importantly, INS offers the opportunity of tailoring the energy resolution width .In fact, the latter can be narrowed according to the specific needs of experi‐ menters, although, unavoidably, to the cost of a shrinkage of the dynamic range covered. The use of a narrow and sharp resolution is crucial in many applications including, for instance, the study of the slow dynamic response of highly viscous systems as glass formers approaching the melting. In fact, in this case all relevant spectral features concentrate in the so-called quasielastic region of the spectrum. Resolution functions as narrow as a few meV can be reached using cold neutrons as a probe. Furthermore, INS resolution profiles are usually sharp (nearly

Although new concept IXS spectrometers discussed in the following promise a drastic improvement of both resolution width (sub-meV) and spectral contrast, the performance of quasi-elastic neutron scattering instruments [16] is unlikely to be ever matched by IXS

Due to the highly localized interaction of neutrons with the target nuclei, no appreciable *Q* decay of the scattering length can be observed up to *Q* values of the order of the inverse of the nuclear size. This value is nearly a factor 10<sup>3</sup> larger than the typical *Q* value at which the IXS cross halves its *Q =* 0 value. For this reason, the use of neutrons is particularly convenient for extremely high *Q* measurements probing the so-called single particle, or impulse approxima‐ tion, regime [17]. Indeed, the access to this regime has been an exclusive task of deep inelastic neutron scattering (DINS) for decades ([18]. Although some successful attempts to reach the impulse approximation with IXS have been reported in the literature [19], the maximum *Q*

the matter and this makes small samples relatively "transparent" to this technique.

Gaussian) as opposite to the broad (mostly Lorentzian) IXS resolution wings.

tially contribute to the inelastic signal.

*3.1.6. Transverse beam size*

20 Neutron Scattering

**3.2. Part B: advantages of INS**

*3.2.1. Resolution shape*

spectrometers.

*3.2.2. Q-decay of the cross section*

As mentioned, at the typical incident energies of IXS, the scattered intensity is mostly attenu‐ ated by the photoelectric absorption, whose coefficient µ is proportional to *Z* <sup>4</sup> . In high *Z* samples the absorption dominates over the Thomson scattering term, thus making IXS of limited efficiency for these materials. Conversely, the neutron absorption cross section is, in most cases, relatively low, which makes neutrons an ideal, non-destructive, probe of bulk properties of materials. The low absorption is also a crucial pre-requisite to probe biological systems because a low incident flux reduces the risk of radiation-induced damage in these samples [20]. Consequently, complex or delicate biological materials can be studied less destructively than any other forms of high energy radiation. Even in vivo studies are now being carried out using neutron probes, an area that has great potential for further develop‐ ment and applications in medical research.

On the other hand, the larger penetration depth of neutron probes comes in handy in some large volume high-pressure applications, as it makes the design of high-pressure vessels more straightforward. In fact the cell body can be often penetrated by the neutron beam with no need of using neutron-transparent windows, thereby drastically reducing the risk of highpressure leaks.

### *3.2.4. Single particle dynamics*

The neutron cross section of hydrogen is substantially higher than the one of other atomic species and is almost completely incoherent. This represents a key advantage when studying the single particle response of hydrogen compounds as well as hydration patterns in proteins and other macromolecules [21].

### *3.2.5. Contrast variation*

When the sample is embedded or floating in a substrate, as for confined systems, solutions, and so on, it is often useful to improve the contrast of the measurement, i.e. the difference between the scattering intensity from the confined sample and the one from the substrate. In principle, the contrast can be enhanced by manipulating the scattering density of a specific component.

However, the IXS intensity of a given component can only be manipulated changing the atomic number, i.e. altering the chemical species. Conversely, the neutron scattering cross section is isotope-specific. Therefore, the scattering can be substantially manipulated through the change of the isotopic composition in the sample, i.e. without altering its chemical properties. This suggests to modulate the cross section by means of a mere isotopic substitution to optimize the contrast [22].

### *3.2.6. Isotopic substitution*

As discussed just above, the INS cross section is isotope-specific, i.e. it depends on the nuclear mass, but not on the atomic number, or, equivalently, the chemical specie of the target sample. One method taking full advantage of this property is based upon parallel INS measure‐ ments on chemically equivalent systems in which an "isotopic substitution" has been implemented. In fact, by using a smart isotope manipulation, one in principle can selective‐ ly enhance the partial scattering contribution of different atomic species in a molecular sample, or in a mixture [22].

### **4. Present and future of IXS: ID28 beamline at ESRF and 10ID beamline at NSLS II**

### **4.1. An example of state-of-art IXS spectrometer: ID28 beamline**

As discussed in the introductory section, the investigation of collective modes in the spectrum of density fluctuations of fluids imposes a challenging requirement on the relative energy resolution (∆*E*/*E* ≤ 10−7). The implementation of IXS monochromators with the required ∆*E*/*E* demands the use of extremely high quality crystals, i.e. crystals having a relative lattice parameter variation, ∆*d*/*d*, smaller than 10−7, at least within their active part (diffracting volume).

Furthermore, to preserve the desired energy resolution without significantly reducing the photon flux, the whole divergence of the incoming photon beam must fall within the intrinsic angular acceptance of the considered monochromator reflection, i.e. the Darwin width, *w* D. Within the dynamical theory of X-ray diffraction [23], the latter is given by:

$$w\_{\rm D} = (AE/E)h \tan \theta\_{\rm B'}$$

where *h* and *θ* B are the Bragg reflection order and angle, respectively. Since tan*θ* <sup>B</sup> diverges for *θ* <sup>B</sup> *≈* 90*°*, upon approaching a backscattering geometry *w* D can be enhanced enough to accept the X-ray beams delivered by standard undulators' sources.

Therefore, it was predicted almost 50 years ago that the use of extreme backscattering geome‐ tries coupled with the small ∆*E*/*E* typical of high order (high *h*) Bragg reflections could in principle enable high-resolution (meV) IXS measurements. Historically, the first demonstra‐ tion that the needed 10−7 relative resolution can be achieved with silicon crystals in backscat‐ tering geometry was given by [24] and [25]; these results were further developed by [26] for applications to IXS.

The ∆*E*/*E* ultimately achieved in these works was 5×10−7 with a photon flux of ~10<sup>6</sup> photons/s on the sample. However, the best performance achieved back at those times (*∆E*/*E* ≈ 3.5 × 10−7 within 2.l µrad angular acceptance) exploited a "four bounce" crystal array, originally developed for resonant nuclear scattering with synchrotron radiation [27].

The first complete meV-resolution IXS spectrometer, ID16 beamline, was built on 1995 at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Few years later another IXS spectrometer, ID28 beamline [31], was developed at *ESRF* with a similar design, yet enabling an extended Q-range coverage. This is the only meV-resolution spectrometer currently active at ESRF and its general description is concisely reported in the following. It is worth stressing, that other meV-resolution IXS spectrometers with slightly different layouts are currently available in Japan [32] and United States [33, 34]. nergyofincidentphotons.Therotationsaroundthesecondaxis,locatedatthesampleposition,changeins teadthemomentumtransfer *Q* 

*3.2.6. Isotopic substitution*

22 Neutron Scattering

sample, or in a mixture [22].

**NSLS II**

volume).

applications to IXS.

As discussed just above, the INS cross section is isotope-specific, i.e. it depends on the nuclear mass, but not on the atomic number, or, equivalently, the chemical specie of the target sample. One method taking full advantage of this property is based upon parallel INS measure‐ ments on chemically equivalent systems in which an "isotopic substitution" has been implemented. In fact, by using a smart isotope manipulation, one in principle can selective‐ ly enhance the partial scattering contribution of different atomic species in a molecular

**4. Present and future of IXS: ID28 beamline at ESRF and 10ID beamline at**

As discussed in the introductory section, the investigation of collective modes in the spectrum of density fluctuations of fluids imposes a challenging requirement on the relative energy resolution (∆*E*/*E* ≤ 10−7). The implementation of IXS monochromators with the required ∆*E*/*E* demands the use of extremely high quality crystals, i.e. crystals having a relative lattice parameter variation, ∆*d*/*d*, smaller than 10−7, at least within their active part (diffracting

Furthermore, to preserve the desired energy resolution without significantly reducing the photon flux, the whole divergence of the incoming photon beam must fall within the intrinsic angular acceptance of the considered monochromator reflection, i.e. the Darwin width, *w* D.

D B *w AE E h* = ( / ) tan ,

where *h* and *θ* B are the Bragg reflection order and angle, respectively. Since tan*θ* <sup>B</sup> diverges for *θ* <sup>B</sup> *≈* 90*°*, upon approaching a backscattering geometry *w* D can be enhanced enough to

Therefore, it was predicted almost 50 years ago that the use of extreme backscattering geome‐ tries coupled with the small ∆*E*/*E* typical of high order (high *h*) Bragg reflections could in principle enable high-resolution (meV) IXS measurements. Historically, the first demonstra‐ tion that the needed 10−7 relative resolution can be achieved with silicon crystals in backscat‐ tering geometry was given by [24] and [25]; these results were further developed by [26] for

on the sample. However, the best performance achieved back at those times (*∆E*/*E* ≈ 3.5 × 10−7 within 2.l µrad angular acceptance) exploited a "four bounce" crystal array, originally

The ∆*E*/*E* ultimately achieved in these works was 5×10−7 with a photon flux of ~10<sup>6</sup>

developed for resonant nuclear scattering with synchrotron radiation [27].

q

photons/s

Within the dynamical theory of X-ray diffraction [23], the latter is given by:

accept the X-ray beams delivered by standard undulators' sources.

**4.1. An example of state-of-art IXS spectrometer: ID28 beamline**

As evident from Figure 5, the design of ID28 (and all current IXS spectrometers) is based on the triple axis concept, first developed in the mid-1950s [1]: the first axis locates at the mono‐ chromator crystal, and rotations around it are used to select the energy of incident photons. Rotations around the second axis, located at the sample position, change instead the momen‐ tum transfer *Q* <sup>→</sup>. Finally, the third axis is the one of the analyzer crystals, and, in principle, rotations around it determine the energy scattered photons, ℏ*ω<sup>f</sup>* . A more detailed layout and a picture of ID28 beamline can be found in Figures 6 and 7, respectively. *f*

At variance of traditional triple axis schemes, implementing energy scans of scattered intensity through rotations of the analyzer crystal(s), at ID28 beamline, these are performed by changing the *d*-spacing of reflecting crystals through the scan of their temperature. The achievement of a 10<sup>−</sup><sup>7</sup> ÷10−<sup>8</sup> relative energy resolution demands a millikelvin accuracy in the temperature control of the monochromator/analyzer optics. This is obtained by using a carefully designed temperature bath controlled by an active feedback system with a precision of 0.2 mK at ambient conditions.

**Figure 5.** The general scheme of the ID28 IXS spectrometer at ESRF.

The X-ray source consists of three undulators having 32 mm magnetic period, placed in a straight high-beta *β* section of the electron storage ring. The utilized X-ray radiation energies correspond to the undulator emission of the third or fifth harmonic. The X-ray beam delivered at these harmonics has ∆*E*/*E* ≈ 10−2, a 40×15 µRad (horizontal × vertical) angular divergence and about 200 W integrated power. Such a beam is pre-monochromatized to Δ*E* / *E* ≈2×10−<sup>4</sup> Δ*E* / *E* ≈2⋅10−<sup>4</sup> using a silicon, Si(1,1,1), in vacuum and cryogenically cooled channel-cut crystal. Its main task is to absorb the relevant portion of the irradiated power that will be otherwise impinging on the main monochromator, thus drastically reducing its heat load. This is required to minimize thermal deformations induced on the monochromator crystal that can drastically degrade the energy resolution. The x-photon beam filtered by the pre-monochromator is then back reflected by the high-resolution monochromator, consisting of an asymmetrically cut silicon crystal oriented along the [111] direction and operated at a Bragg angle of 89.98°. This extreme backscattering geometry minimizes geometrical contributions to the total energy resolution broadening. Under these conditions, the angular acceptance of the monochromator is larger than the X-ray beam divergence and all photons are thus reflected within the desired energy bandwidth. Both the use of high-order Bragg reflections (*h* ,*h* ,*h* ) with (*h* =7,8,9,11,12,13) and the superior quality of the active optics are required to ultimately achieve the needed relative resolution Δ*E* / *E* ≈10−<sup>7</sup> ÷10−<sup>8</sup> . The monochromatic beam is focused on the horizontal and vertical planes by a platinum-coated toroidal mirror, located 25 m upstream of the sample, which provides a 250×80 *μm*<sup>2</sup> full width at half maximum (FWHM). More performing optical elements are also available at ID28 to focus the beam down to 30 × 40 µm2 .

**Figure 6.** The schematic layout of ID28 beamline at ESRF (courtesy of F. Bencivenga).

The energy of the radiation scattered by the sample is analyzed by five spherical analyzers (nine in the upgraded instrument) mounted on the tip of an horizontal arm, which can be rotated by an angle 2*θ* so as to reach the desired exchanged momentum. These five (nine) analyzer units have a mutual angular offset, which enables the simultaneous collection of IXS spectra at five (nine) different *Q*s. Although the problems connected to the energy resolution are conceptually the same for monochromator and analyzer, the required angular acceptance is very different in the two cases. For the analyzer, the optimal angular acceptance results from the best compromise between conflicting count rate and Q-resolution requirements. In fact, albeit a reduction of the angular acceptance improves the momentum resolution, it reduces the overall count rate.

correspond to the undulator emission of the third or fifth harmonic. The X-ray beam delivered at these harmonics has ∆*E*/*E* ≈ 10−2, a 40×15 µRad (horizontal × vertical) angular divergence and about 200 W integrated power. Such a beam is pre-monochromatized to Δ*E* / *E* ≈2×10−<sup>4</sup>

Its main task is to absorb the relevant portion of the irradiated power that will be otherwise impinging on the main monochromator, thus drastically reducing its heat load. This is required to minimize thermal deformations induced on the monochromator crystal that can drastically degrade the energy resolution. The x-photon beam filtered by the pre-monochromator is then back reflected by the high-resolution monochromator, consisting of an asymmetrically cut silicon crystal oriented along the [111] direction and operated at a Bragg angle of 89.98°. This extreme backscattering geometry minimizes geometrical contributions to the total energy resolution broadening. Under these conditions, the angular acceptance of the monochromator is larger than the X-ray beam divergence and all photons are thus reflected within the desired energy bandwidth. Both the use of high-order Bragg reflections (*h* ,*h* ,*h* ) with (*h* =7,8,9,11,12,13) and the superior quality of the active optics are required to ultimately achieve

horizontal and vertical planes by a platinum-coated toroidal mirror, located 25 m upstream of

performing optical elements are also available at ID28 to focus the beam down to 30 × 40 µm2

The energy of the radiation scattered by the sample is analyzed by five spherical analyzers (nine in the upgraded instrument) mounted on the tip of an horizontal arm, which can be rotated by an angle 2*θ* so as to reach the desired exchanged momentum. These five (nine)

using a silicon, Si(1,1,1), in vacuum and cryogenically cooled channel-cut crystal.

. The monochromatic beam is focused on the

.

full width at half maximum (FWHM). More

Δ*E* / *E* ≈2⋅10−<sup>4</sup>

24 Neutron Scattering

the needed relative resolution Δ*E* / *E* ≈10−<sup>7</sup> ÷10−<sup>8</sup>

**Figure 6.** The schematic layout of ID28 beamline at ESRF (courtesy of F. Bencivenga).

the sample, which provides a 250×80 *μm*<sup>2</sup>

**Figure 7.** The ID28 beamline's spectrometer arm of at ESRF (from the ID28 beamline's website, courtesy of A. Bosak)

Considering that for typical IXS measurements an acceptable *Q* spread is, at the lowest Q's, <sup>Δ</sup>*Q*<sup>M</sup> <sup>≤</sup>0.1÷0.5 nm−<sup>1</sup> 0.1÷0.5 *nm*−<sup>1</sup> , the angular acceptance of the analyzer is correspondingly set to few milliradians, a value much larger than the Darwin width of the Bragg reflections typically used. The only option to achieve such a large acceptance is the design of a focusing system, which, however, must minimize possible distortions of the *d-*spacing ultimately degrading the energy resolution. The solution actually adopted at ID28 beamline is a mosaic of ≈12,000 silicon perfect single square crystals glued on a spherical surface. These analyzer arrays are operated in a 1:1 pseudo-Rowland circle geometry with aberration low enough not to degrade appreciably the desired energy resolution [28].

Each analyzer is equipped with motorized entrance slits setting the desired Q-resolution, its back-reflection being intercepted by Peltier-cooled silicon diode with extremely low dark count (≈1 count over 30 minutes).

The spectrometer arm can be rotated around a vertical axis passing through the sample by an < 55°, which corresponds to Q < 100 nm−1, for Si(11,11,11) monochromator/analyzer Bragg reflection.

The measured resolution profile of the instrument has a FWHM of approximately 1.5 meV, as required to perform spectroscopy measurements of the high frequency dynamics of disordered systems.

### **4.2. Toward a new generation IXS spectrometer**

On a general ground, high resolution measurements as the one discussed above suffer from major drawbacks: the relatively large width of the resolution (≥1 meV) and, perhaps more importantly, the poor spectral contrast, i.e. the slowly decaying (essentially Lorentzian) wings of the instrumental resolution profile. The development of next generation IXS instruments using a new monochromatization/energy analysis holds the promise of effectively tackling these problems.

Currently, state-of-art IXS spectrometers have a working principle similar to the one of the ID28 beamline of ESRF illustrated above, i.e. they use extreme backscattering geometries in which the energy analysis is implemented through temperature scans [28].

In these schemes, the final achievement of a meV-resolution resolution is challenged by two major difficulties: (1) the energy analysis imposes tight constraints on the temperature stability of the optics, as only <10−4 K temperature gradients can be tolerated and (2) the higher order Bragg reflections are more deeply penetrating into reflecting crystals, thus being severely affected by extinction losses.

A few alternative schemes based on the dispersion character of Bragg back reflection from asymmetrically cut crystals [35] have been successfully tested. Namely:


Both optical schemes demonstrated to provide sub-meV broad and extremely sharp, essen‐ tially Gaussian, resolution profiles. Furthermore, these optical schemes have a wealth of advantages when compared to current backscattering IXS spectrometers:


**3.** They are extremely flexible, offering the opportunity of changing incident energy and tailoring the resolution to any specific needs.

Each analyzer is equipped with motorized entrance slits setting the desired Q-resolution, its back-reflection being intercepted by Peltier-cooled silicon diode with extremely low dark

The spectrometer arm can be rotated around a vertical axis passing through the sample by an < 55°, which corresponds to Q < 100 nm−1, for Si(11,11,11) monochromator/analyzer Bragg

The measured resolution profile of the instrument has a FWHM of approximately 1.5 meV, as required to perform spectroscopy measurements of the high frequency dynamics of disordered

On a general ground, high resolution measurements as the one discussed above suffer from major drawbacks: the relatively large width of the resolution (≥1 meV) and, perhaps more importantly, the poor spectral contrast, i.e. the slowly decaying (essentially Lorentzian) wings of the instrumental resolution profile. The development of next generation IXS instruments using a new monochromatization/energy analysis holds the promise of effectively tackling

Currently, state-of-art IXS spectrometers have a working principle similar to the one of the ID28 beamline of ESRF illustrated above, i.e. they use extreme backscattering geometries in

In these schemes, the final achievement of a meV-resolution resolution is challenged by two major difficulties: (1) the energy analysis imposes tight constraints on the temperature stability of the optics, as only <10−4 K temperature gradients can be tolerated and (2) the higher order Bragg reflections are more deeply penetrating into reflecting crystals, thus being severely

A few alternative schemes based on the dispersion character of Bragg back reflection from

**1.** an array of four silicon crystals, customarily referred to as 4-bounce monochromator [36]

**2.** an assembly of collimator (*C*), dispersive (*D*), and wavelength (*W*) selector crystals, usually referred to as *CDW*, or *CDDW*, if two dispersive crystals are used instead [37].

Both optical schemes demonstrated to provide sub-meV broad and extremely sharp, essen‐ tially Gaussian, resolution profiles. Furthermore, these optical schemes have a wealth of

**1.** They do not use high order Bragg reflections, which greatly improves their overall efficiency. In fact, they are operated at moderate energy values at which synchrotron

**2.** The spectral contrast, i.e. the sharpness of the resolution function, is greatly enhanced by

which the energy analysis is implemented through temperature scans [28].

asymmetrically cut crystals [35] have been successfully tested. Namely:

advantages when compared to current backscattering IXS spectrometers:

count (≈1 count over 30 minutes).

**4.2. Toward a new generation IXS spectrometer**

reflection.

26 Neutron Scattering

systems.

these problems.

and

affected by extinction losses.

undulators usually perform best.

the multiple reflection bounces.

The feasibility of these schemes for IXS application was recently demonstrated [38]. Most importantly, a novel instrument (10ID beamline) with a very narrow (sub-meV) and sharp (essentially Gaussian) resolution function [39, 40] will be soon operated at the new synchrotron source NSLS-II in Upton, New York. A schematic representation of the various components of such a beamline is provided in Figure 8.

**Figure 8.** A schematic layout of the new high resolution IXS spectrometer soon available at NSLS II, taken from [40].

### **5. An example of a joint IXS and IXS experiment: the study of a shear mode propagation in water**

Among various topics emerging from the vast literature on THz measurements of S(Q,ω) of liquids, one of the most controversial focuses on the presence of a second, low frequency, mode in the spectrum of water. More than other research subjects, this topic lends itself to the synergic use of the two complementary THz spectroscopic methods discussed above: IXS and INS.

As mentioned, the presence of the the low-frequency peak of water was also reported in the INS work of [11] and in the later INS measurement by [41]. Both these works, owing to the mentioned kinematic limitations, could not access to the high frequency back then referred, somehow misleadingly (see [42, 43]), to as the "fast sound mode", which was instead observed in the INS work of Teixeira and coworkers [12]. For a long time, the low frequency mode was interpreted as the finite-Q extension of the macroscopic sound mode, based on the overall consistency between the slope of its low Q dispersion curve and the known value of the adiabatic sound speed [44]. This interpretation was confuted years later by both MD [10] and IXS [45] studies of the water spectrum, which demonstrated that such low frequency mode has instead a transverse acoustic origin. This interpretation initially raised some controversy [46, 47], but found important validations in successive IXS works [13, 45]; a concise account of these studies can be found in [48].

In a way, the onset of a transverse mode in the spectrum of a disorder material may not be too surprising as one recognizes that quasi-polarized (transverse or longitudinal) modes are routinely observed in the inelastic spectra of poly-crystals in the first-Brillouin zone. Therefore, the hypothesis that a definite polarization, either longitudinal or transverse, are intertwined has in principle some ground for a liquid system, especially over distances matching the size of molecular disorder. However, it is broadly accepted that a longitudinal or transverse polarization can be still assigned to a mode in the mesoscopic regime, provided such a mode dominates the current spectra of corresponding – longitudinal or transverse – polarization. Longitudinal current spectra CL(Q,ω) are directly determined from the experimentally measured S(Q,ω) using CL(Q,ω) = (ω/Q)2 S(Q,ω). Conversely, transverse current spectra CL(Q,ω) can only be determined by computer simulations as the Fourier transform of the autocorrelation function between transverse components of atomic velocities.

It is worth recalling that the (transverse or longitudinal) polarization of the acoustic mode is defined in reference to the direction of the exchanged momentum ℏ*Q* <sup>→</sup>. In this respect, it is useful to recall that S(Q,ω) is the Fourier transform of the correlation function ∑ *jk* exp[*iQ* <sup>→</sup> <sup>⋅</sup>*<sup>R</sup>* → *<sup>k</sup>* (*t*)]exp[*iQ* <sup>→</sup> <sup>⋅</sup>*<sup>R</sup>* → *<sup>k</sup>* (*t*) . Clearly, the presence of the *Q* <sup>→</sup> <sup>⋅</sup>*<sup>R</sup>* → *i* (*t*) term in the exponents shows that only longitudinal movements, i.e. movements along the *Q* <sup>→</sup> direction, are relevant to S(Q,ω). Transverse modes are therefore visible in the S(Q,ω) shape only indirectly, via the so-called longitudinal–transverse (L–T) coupling. This mainly consists in a coupling between transverse and longitudinal waves, which occurs above some Q threshold. For instance, in water, this threshold was located at 4 or ≈6 nm−1 by [49] and [50], respectively.

The onset of an L–T coupling in liquids has been often associated to the intrinsically "open" (large free volume) tetrahedral structure of water, as opposite to the essentially close-packed arrangement of "normal" fluids. In fact, an L–T coupling has been observed in tetrahedral systems such as water itself as well as GeO2 [52] and GeSe2 [53].

However, nowadays, this interpretation needs some reconsideration because the L–T coupling was also observed in non-tetrahedral systems, such as glassy glycerol [54] and liquid metals [55-59]. Furthermore, a similar effect was reported in more complex systems such as binary mixtures [51] and biophysical samples [60-62].

On a general ground, the ability to support shear propagation can be considered as a mani‐ festation of the solid-like response of a fluid at short times and distances. This seems consistent to the early IXS work of [49], where substantial similarities between the THz dynamic response of water and ice were suggested. Also, based on previous lattice vibrations, i.e. calculations and INS measurements [64], this additional spectral feature in water was assigned to the transverse optical (TO) visible in ice. Consistently with this scenario, the IXS work of Ponte‐ corvo and coworkers [45] also demonstrated that an L–T coupling is only visible when density fluctuations propagate at frequencies larger than the inverse of the relaxation time 1/τ. Under these conditions, acoustic waves "perceive" the propagation medium as frozen as expected for a glass. These slows degrees of freedom cannot energetically couple with the acoustic wave, which, therefore, travel "elastically," that is keeping its energy constant. The virtual lack of acoustic dissipation of this elastic regime is reflected by the higher sound velocity and the lower sound damping. Furthermore, restoring forces become more effective in fostering a sound propagation in the shear plane. This seems consistent with the result of a successive IXS work [13], where a study of shear mode propagation was jointly performed in normal liquid, solid, and supercooled phase of water.

### **5.1. The joint use of IXS and INS**

consistency between the slope of its low Q dispersion curve and the known value of the adiabatic sound speed [44]. This interpretation was confuted years later by both MD [10] and IXS [45] studies of the water spectrum, which demonstrated that such low frequency mode has instead a transverse acoustic origin. This interpretation initially raised some controversy [46, 47], but found important validations in successive IXS works [13, 45]; a concise account of

In a way, the onset of a transverse mode in the spectrum of a disorder material may not be too surprising as one recognizes that quasi-polarized (transverse or longitudinal) modes are routinely observed in the inelastic spectra of poly-crystals in the first-Brillouin zone. Therefore, the hypothesis that a definite polarization, either longitudinal or transverse, are intertwined has in principle some ground for a liquid system, especially over distances matching the size of molecular disorder. However, it is broadly accepted that a longitudinal or transverse polarization can be still assigned to a mode in the mesoscopic regime, provided such a mode dominates the current spectra of corresponding – longitudinal or transverse – polarization. Longitudinal current spectra CL(Q,ω) are directly determined from the experimentally

CL(Q,ω) can only be determined by computer simulations as the Fourier transform of the

It is worth recalling that the (transverse or longitudinal) polarization of the acoustic mode is

useful to recall that S(Q,ω) is the Fourier transform of the correlation function

to S(Q,ω). Transverse modes are therefore visible in the S(Q,ω) shape only indirectly, via the so-called longitudinal–transverse (L–T) coupling. This mainly consists in a coupling between transverse and longitudinal waves, which occurs above some Q threshold. For instance, in

The onset of an L–T coupling in liquids has been often associated to the intrinsically "open" (large free volume) tetrahedral structure of water, as opposite to the essentially close-packed arrangement of "normal" fluids. In fact, an L–T coupling has been observed in tetrahedral

However, nowadays, this interpretation needs some reconsideration because the L–T coupling was also observed in non-tetrahedral systems, such as glassy glycerol [54] and liquid metals [55-59]. Furthermore, a similar effect was reported in more complex systems such as binary

On a general ground, the ability to support shear propagation can be considered as a mani‐ festation of the solid-like response of a fluid at short times and distances. This seems consistent to the early IXS work of [49], where substantial similarities between the THz dynamic response of water and ice were suggested. Also, based on previous lattice vibrations, i.e. calculations and INS measurements [64], this additional spectral feature in water was assigned to the transverse optical (TO) visible in ice. Consistently with this scenario, the IXS work of Ponte‐

*<sup>k</sup>* (*t*) . Clearly, the presence of the *Q*

autocorrelation function between transverse components of atomic velocities.

water, this threshold was located at 4 or ≈6 nm−1 by [49] and [50], respectively.

defined in reference to the direction of the exchanged momentum ℏ*Q*

shows that only longitudinal movements, i.e. movements along the *Q*

systems such as water itself as well as GeO2 [52] and GeSe2 [53].

S(Q,ω). Conversely, transverse current spectra

<sup>→</sup> <sup>⋅</sup>*<sup>R</sup>* → *i* <sup>→</sup>. In this respect, it is

(*t*) term in the exponents

<sup>→</sup> direction, are relevant

these studies can be found in [48].

measured S(Q,ω) using CL(Q,ω) = (ω/Q)2

<sup>→</sup> <sup>⋅</sup>*<sup>R</sup>* →

mixtures [51] and biophysical samples [60-62].

*<sup>k</sup>* (*t*)]exp[*iQ*

∑ *jk*

exp[*iQ* <sup>→</sup> <sup>⋅</sup>*<sup>R</sup>* →

28 Neutron Scattering

The various IXS investigations of transverse propagation in water mentioned above often pushed to limit the capability of this spectroscopic method; as a matter of fact, these studies could be successfully accomplished either reaching somehow extreme thermodynamic conditions (high density and/or low temperature) or accessing to large Q values.

The observation of a transverse mode at low Q's or at high T's would have required a sharp and narrow resolution function, definitely beyond the current capability of IXS. In principle, INS measurements can offer the required resolution performance, but only by confining the covered dynamic range within a restricted region excluding the longitudinal, high frequency, mode (see data reported in Figure 3). A possible method to circumvent these complementary limitations foresees simultaneous IXS and INS measurements on the same heavy water sample.

This is the strategy followed by joint INS–IXS measurement in heavy water [50] and in a successive one on deuterated water–glycerol mixtures [51]. The use of deuterated samples was, of course, imposed by the highly incoherent INS cross section of hydrogen, which would have prevented the INS measurement to reliably determine the coherent spectrum. In these two measurements, the INS spectrometer was operated using an extremely narrow (<0.1 meV) and essentially Gaussian resolution profile, as required for an optimal characterization of the low frequency portion of the spectrum. The parallel use of IXS allowed to dramatically extending the low energy range covered by the neutron experiment. Figure 9 provides an example of spectra measured in heavy water by these two complementary techniques.

Consistent to the other works in the literature [65], the spectrum was approximated by a viscoelastic model accounting for the longitudinal component of density fluctuations, while the additional peak in the spectrum generated by the L–T coupling was approximated by a simple Damped Harmonic Oscillator (DHO) term [66]:

$$\frac{S(\mathbf{Q},oo)}{S(\mathbf{Q})} = (1 - \tilde{T})\mathbf{S}\_{\text{VE}}(\mathbf{Q},oo) + \tilde{T}\frac{1}{\pi} \frac{2\Omega\_t^2 \Gamma\_t}{\left[oo^2 - \Omega\_t^2\right]^2 + 4\Gamma\_t^2 o^2} \tag{38}$$

where the suffix "VE" labels the viscoelastic model, while Ω*t* and Γ*t* represent, respectively, the characteristic frequency and damping of the low-frequency excitation. Finally, *T*˜ is a *Q*-

**Figure 9.** Representative IXS (black dots) and INS (black dots) measurements of water spectra are reported along with the corresponding current spectra (lines of corresponding color). All reported curves are normalized to the S(Q) for consistency. Data are redrawn with permission from Ref. [51], which is copyrighted by the American Physical Society.

dependent scaling factor, yielding the relative weight of the DHO component. Within the hypothesis that the DHO term arises from the L–T coupling, *T*˜ provides a measure of the strength of such coupling. An important outcome of this joint measurement is reported in Figure 10.

It clearly appears that at low *Q*, the transverse mode becomes heavily damped, being its frequency nearly equal to the damping, and therefore, the transverse perturbation assumes a essentially non-propagating character. Interestingly, when this low *Q* regime is reached, both acoustic frequency and damping are of the same order and the elastic (solid-like) character of the dynamics is lost. At even lower *Q* values, no clear evidence of a transverse mode in the spectrum can be inferred from measured spectral shape.

The trend discussed above indicates that below the *Q* threshold of the L–T mode, the inelastic peak merges into the quasi-elastic mode typical of all fluids exhibiting a viscoelastic behavior. This is the well-known Mountain [67] mode, which arises from the coupling of density fluctuations with active relaxation processes.

provides a measure of the strength of such coupling. An important outcome of this joint measurement is reported in Figure 10. Using X-ray as a Probe of the Terahertz Dynamics of Disordered Systems – Complementarity with Inelastic... http://dx.doi.org/10.5772/62844 31

relative weight of the DHO component. Within the hypothesis that the DHO term arises from the L–T coupling, T<sup>ɶ</sup>

Figure 10. The Q dependence of the frequency Ωt (red dots) and the damping Γt (blue dots) of the transverse acoustic mode in water spectrum are reported along with the inverse of the relaxation time (open circles + line). Data are redrawn with permission from Ref. [51], which is copyrighted by the American Physical Society. **Figure 10.** The Q dependence of the frequency Ω<sup>t</sup> (red dots) and the damping Γ<sup>t</sup> (blue dots) of the transverse acoustic mode in water spectrum are reported along with the inverse of the relaxation time (open circles + line). Data are re‐ drawn with permission from Ref. [51], which is copyrighted by the American Physical Society.

#### **Acknowledgements** It clearly appears that at low *Q*, the transverse mode becomes heavily damped because the frequency equals the damping, and therefore, the transverse perturbation assumes a diffusive character. Interestingly, when this low *Q*

The work performed at National Synchrotron Light Source II, Brookhaven National Labora‐ tory, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC0012704. of the dynamics is lost. At even lower *Q* values, no clear evidence of a transverse mode in the spectrum can be inferred from measured spectral shape. The trend discussed above likely indicates that below the *Q* threshold of the L–T mode, the inelastic peak merges into the quasi-elastic mode typical of all fluids exhibiting a viscoelastic behavior. This is the well-known Mountain

regime is reached, both acoustic frequency and damping are of the same order and the elastic (solid-like) character

or relaxation mode, which arises from the coupling of density fluctuations with active relaxation processes.

the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-SC0012704.

[5] Dirac, P.A. In *The quantum theory of the emission and absorption of radiation*, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1927; The Royal Society: pp 243-265.

#### **Author details** The work performed at National Synchrotron Light Source II, Brookhaven National Laboratory, was supported by

dependent scaling factor, yielding the relative weight of the DHO component. Within the hypothesis that the DHO term arises from the L–T coupling, *T*˜ provides a measure of the strength of such coupling. An important outcome of this joint measurement is reported in

**Figure 9.** Representative IXS (black dots) and INS (black dots) measurements of water spectra are reported along with the corresponding current spectra (lines of corresponding color). All reported curves are normalized to the S(Q) for consistency. Data are redrawn with permission from Ref. [51], which is copyrighted by the American Physical Society.

It clearly appears that at low *Q*, the transverse mode becomes heavily damped, being its frequency nearly equal to the damping, and therefore, the transverse perturbation assumes a essentially non-propagating character. Interestingly, when this low *Q* regime is reached, both acoustic frequency and damping are of the same order and the elastic (solid-like) character of the dynamics is lost. At even lower *Q* values, no clear evidence of a transverse mode in the

The trend discussed above indicates that below the *Q* threshold of the L–T mode, the inelastic peak merges into the quasi-elastic mode typical of all fluids exhibiting a viscoelastic behavior. This is the well-known Mountain [67] mode, which arises from the coupling of density

spectrum can be inferred from measured spectral shape.

fluctuations with active relaxation processes.

Figure 10.

30 Neutron Scattering

Alessandro Cunsolo References

Address all correspondence to: acunsolo@bnl.gov [1] Brockhouse, B.N.; Stewart, A.T. Scattering of neutrons by phonons in an aluminum single crystal. *Physical Review* 1955, *100*, 756-757.

Acknowledgment

National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, NY, USA [3] Jackson, J.D.; Jackson, J.D. *Classical electrodynamics*. Wiley New York etc.: 1962; Vol. 3. [4] Sinha, S.K. Theory of inelastic x-ray scattering from condensed matter. *J Phys-Condens Mat* 2001, *13*, 7511-7523.

[2] Goodman, J.W. *Introduction to fourier optics*. Roberts and Company Publishers: 2005.

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62184

### **Abstract**

Scattering methods are powerful tools used for the examination of condensed mat‐ ter. They offer the possibility to analyze particles without disturbing their natural environment. Small angle neutron scattering (SANS) can also be used to analyze solid and liquid systems, phase transformations, germination, growth flaws and de‐ fects, as well as generally any inhomogeneity occurring in a range of 10–1000 Å. The most common experimental SANS methods and patterns are reviewed and ex‐ plained.

**Keywords:** SANS, methods, patterns

### **1. Introduction**

### **1.1. Material science issues accessible through small angle scattering of X-rays (SAXS) and neutrons (SANS)**

The fundamental properties of the neutron make it a powerful tool for materials science investigations. It is useful to introduce neutrons by comparison with X-rays because both probes are used for diffraction investigations, and more workers are familiar with X-ray methods. X-rays can be obtained with a generator tube or, in the past few years, at a synchro‐ tron X-ray laboratory.

Neutrons for neutron beam research arise either from nuclear fission in reactors or from the spallation process. In the latter, accelerator-produced high-energy charged particles yield a large number of energetic neutrons from collisions with heavy-atom nuclei in the accelerator target.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

X-ray diffraction was utilized for the first time in establishing the atomic structure of the crystals. Later, this method was used in other applications, extended to the study of imper‐ fections in crystals, of the size of the crystallites, and even to the study of amorphous objects. From the classical discovery by Laue, in 1912, that X-rays are diffracted on their passage through crystals, the development of the applied domain of this technique was due to the subsequent progress of the X-ray diffraction theory and the improvement of experimental methods. The first observations on small angle X-ray scattering (SAXS) were performed in the beginning of the 1930s [1], and its domain was developed later.

In the following years, the small angle X-ray scattering technique has known to be an important development, proved by a great number of publications over the years. SAXS technique was developed from the necessity to observe long lattice spacing in crystals, in comparison with wavelengths of the X-rays used in structural analyses. Such distances are encountered in certain minerals and certain complex molecules such as polymers or proteins. For example, in the study of macromolecular crystals, the systems for X-ray diffraction should be extended to include the very small angles. The fundamental equation which describes the X-ray diffraction in the crystalline substance is *λ* = 2*d*sin *θ*, which shows that the diffraction angle *θ* inversely varies with the separation distance of the crystalline lattice planes (see Fig. 1) which generates the considered diffraction maximum.

**Figure 1.** Bragg diffraction of the X-rays.

In ordinary crystals, especially those of inorganic matter, most inter-lattice distances observed are on the same magnitude with the wavelength of utilized X-rays; therefore the used *θ* angles are quite large. For example, let us consider the radiation Cu *k*<sup>α</sup> and an inter-lattice distance of 100 Å, which would result in a diffraction *θ* angle of 0.45°; for a distance of 1000 Å, *θ* would be 0.045° or 2.6'. All these examples had shown the importance of SAXS techniques in biology, biochemistry, etc. Using X-rays with long wavelength to obtain larger diffraction angles for a specific inter-lattice distance is strictly limited by the variation of the material X-ray linear absorption coefficient with radiation wavelength. Because X-rays are strongly absorbed by the traversed substance, it also became necessary for the utilization of other particles with higher penetrability due to their weak interaction with the matter they are passing through. Elsasser first observed in 1936 that neutron movement could be determined by wave mechanism, and later two experiments were conducted by Halban and Preiswerk and Mitchell and Powers, respectively, which validated this theory [2]. These experiments were performed using source emitting neutrons of different energies, without considering collimation or monochromation of that beam. However, all these conditions were sufficient to emphasize the neutron diffrac‐ tion but without providing quantitative data. The development of nuclear reactors made possible the availability of neutrons in a very large number to allow collimation in beams, and separation by energy in a narrow band. This was the beginning of the neutron diffraction technique, very similar to X-ray diffraction.

X-ray diffraction was utilized for the first time in establishing the atomic structure of the crystals. Later, this method was used in other applications, extended to the study of imper‐ fections in crystals, of the size of the crystallites, and even to the study of amorphous objects. From the classical discovery by Laue, in 1912, that X-rays are diffracted on their passage through crystals, the development of the applied domain of this technique was due to the subsequent progress of the X-ray diffraction theory and the improvement of experimental methods. The first observations on small angle X-ray scattering (SAXS) were performed in the

In the following years, the small angle X-ray scattering technique has known to be an important development, proved by a great number of publications over the years. SAXS technique was developed from the necessity to observe long lattice spacing in crystals, in comparison with wavelengths of the X-rays used in structural analyses. Such distances are encountered in certain minerals and certain complex molecules such as polymers or proteins. For example, in the study of macromolecular crystals, the systems for X-ray diffraction should be extended to include the very small angles. The fundamental equation which describes the X-ray diffraction in the crystalline substance is *λ* = 2*d*sin *θ*, which shows that the diffraction angle *θ* inversely varies with the separation distance of the crystalline lattice planes (see Fig. 1) which generates

In ordinary crystals, especially those of inorganic matter, most inter-lattice distances observed are on the same magnitude with the wavelength of utilized X-rays; therefore the used *θ* angles are quite large. For example, let us consider the radiation Cu *k*<sup>α</sup> and an inter-lattice distance of 100 Å, which would result in a diffraction *θ* angle of 0.45°; for a distance of 1000 Å, *θ* would be 0.045° or 2.6'. All these examples had shown the importance of SAXS techniques in biology, biochemistry, etc. Using X-rays with long wavelength to obtain larger diffraction angles for a specific inter-lattice distance is strictly limited by the variation of the material X-ray linear

beginning of the 1930s [1], and its domain was developed later.

the considered diffraction maximum.

38 Neutron Scattering

**Figure 1.** Bragg diffraction of the X-rays.

The first device of this type, initially called "neutron spectrometer," was built for the first time in 1945 at Argonne National Laboratory in USA. Since then, many instruments with different shapes and conception were built around the world, and they are mostly known as "neutron diffractometer." The investigation of condensed matter with the help of neutrons is not a substitute for classical X-ray diffractions, but it is a completion of these techniques, because the special nature of neutron interaction with the substance allows the neutron diffraction to provide information that cannot be revealed by other means. The spectrum of neutrons emitted by a reactor is dependent on the moderator temperature, and in most reactors, the spectrum peak is around 1.5 Å. This spectrum is Maxwellian and has a maximum which depends on the temperature. Therefore, for *T* = 273 K, *λ*max is around 1.55 Å, and for *T* = 373 K *λ*max is 1.33 Å. Using "hot sources," it is possible to move the maximum of the neutron spectrum to short wavelengths up to 0.4 Å. In a similar manner, by cooling the moderator, the maximum of the neutron spectrum can be moved to long wavelengths. We can observe that neutron beams, as spectrum broadening, become comparable with X-ray beams, where the normal range of wavelengths is from 2.2 Å (Cr Kα) to 0.56 Å (Ag Kα), thereby having the advantage of being less absorbed by the matter than X-rays. In the last years, in the competition between X-rays and neutrons, the synchrotron sources have been developed without compromising on the advantages of neutron diffraction. Today, the progress of X-rays and thermal neutron radiation sources makes it possible to use diffraction experiment radiations with *λ* up to 10 Å, and the development of the technology to obtain crystals with "pre-established inter-lattice" distances assures the conditions to extract monochromatic beams from the radiation emitted by the source. In their competition with X-rays, the neutrons have the huge advantage of being less absorbed by the matter in comparison with X-rays, which allow them to be used in the investigation of not only volume effects but also surface effects. In the case of neutrons, there are two ways of interaction with condensed matter, with the nuclei or with the atomic magnetic moments of the investigated material. Over the years, several methods were developed to investigate the partial order of analyzed systems. Among these methods, the technique of small angle neutron scattering is the most valuable. Technically speaking, the term "small angle scattering" (SAS) does not refer always to small diffraction angles but to small values of the quantity *Q* which is the scattering vector, *Q* <sup>→</sup> <sup>=</sup>*<sup>k</sup>* <sup>→</sup> −*k* → 0, where *k* → 0 and *k* <sup>→</sup> are the wave vectors of the incident and scattered radiations. For elastic scattering *Q* = (4*π*/*λ*)sin *θ*, where *θ* is half of the scattering angle and *λ* is the radiation wavelength. In the angular range of SAS, the interested intensity is produced by inhomogeneities extended from 10 to 1000 Å. The quantity *D* of these inhomogeneities determines the most interesting angular range [3]:

$$\frac{1}{D} < Q < \frac{10}{D}$$

There are many processes that lead to scattering length variations, extended to zones up to 100 Å. Among these, there are the processes in solid phase namely aging, precipitating processes in solid phase, repeated thermal shock, neutron irradiation, the processes in mixed systems, for example the catalysts, where the two coexisting solid crystalline components are of enormous interest, as well as the ferofluids or the polymers in which neither one component is crystalline. The application of SAXS and SANS to material science has its origin from the experiments performed by Guinier in 1938, to study different types of aluminum alloys with various components such as Al–Cu and Al–Ag. He proved the existence of extremely small precipitate zones (so-called Guinier–Preston (GP) zones) as the first step in solidification of the precipitate of these alloys [3]. The GP zones represent submicroscopic regions of the matrix in which are produced a concentration of the initial solvate element. From one point of view, the GP zones could be considered as pre-precipitates because their formation represents a preliminary stage of precipitations set up from ultra-saturated matrix, but from another point of view it could be considered "clusters," in the sense that represent preferential associations of atoms.

In Al–Cu alloys, it has a diameter of about 80 Å and a thickness of 3–6 Å. With an average composition of 90% Cu, it comes out that the GP zones in these alloys are predominant constituted from Cu atoms. The physical nature and structural characteristics of the formed precipitates are dependent on temperature and heating duration because the precipitates are formed by germination and growing up processes, based on diffusion phenomena. The most important modification of alloy proprieties occur in the initial stages of the aging, with low temperatures and short heating durations when microstructural transformations are to a scale size under the resolution limit of optical microscopy (submicron precipitates). All these phenomena could be emphasized by SAXS and SANS methods. Regrouping of various atoms species during the precipitation process modifies the dispersion strength of X-rays or neutrons in different zones of aged alloy, due to the existence of an important density difference between the precipitated zones and the rest of alloy matrix. Among the structural processes produced through thermal treatment, which is applied to solid crystalline materials, the spinodal transition is something special [3]. The first phases of coherent precipitation took place through phase separation inside the crystalline lattice. Phase separation could start only through a nucleation process followed by nuclei growing. The germination of a new phase in a solid solution needs that in some regions the solid solution should considerably modify its compo‐ sition. To be transformed into stable germs, these regions should have a specific dimension (about 10 Å). In these conditions, the interface between the germ and the solid solution (matrix) shows a distinct structural discontinuity, and therefore it has a free positive energy which inhibits the germination. This germination difficulty is more evident in spinodal decomposi‐ tion. In this case, the matrix regions, where the composition is modified, became more extended than classical germs and are named clusters. The interface between the matrix and the regions with modified composition became diffusive and does not present anymore a distinct structural discontinuity characterized by a positive free energy. To produce regions with modified composition, extended on large volumes, long-range composition fluctuations are necessary that would tend to decompose the original solid solution. The probability of these long-range composition fluctuations is much more increased when the temperature of original solid solution is situated close to the alloy "spinodal temperature." Above the spinodal temperature *T*<sup>s</sup> (where the second derivative of the Gibbs free energy for the solid solution is positive), the supersaturated solid solution is metastable. Below the *T*s spinodal temperature, the second derivative of the Gibbs free energy is negative, resulting in an unstable solid solution, which separates immediately by spinodal decomposition. Concentration fluctuations of spinodal transition are detected by their geometrical parameter values, in SAS methods structural observation domain. By these methods, it can be defined as a wavelength *λ*c and obviously its reciprocal *Q* = 2*π*/*λ*<sup>c</sup> that depend on different factors as temperature difference (*T*s − *T*) or energy gradient. Cahn provided a quantitative description of the primary phase's kinetics of spinodal decomposition, in his linear approximation theory. In accordance with this theory, the growing rate has a maximum for the wavelength *λ*<sup>m</sup> = 21/2*λ*c. Thus both wave vectors *Q*c and *Q*m can be experimentally deduced by SAS because the intensity *I*(*Q*) is proportional with the amplitude square of the suitable concentration waves. SAS methods provide quan‐ titative information not only about particle dimensions but also about volume fraction of the second phase as well as metastable mixture limits. Therefore, the idea of using SANS as a nondestructive testing method for the industry came to reality. Because neutron absorption in the substance is very low, machine components of relatively large thickness can be easily investigated by SANS. The initial steps in this direction began in Italy by a research group from Fiat Corporation. They built a cold neutron source for different material studies. They began in 1974 when Pizzi and coworkers performed several tests on various materials used in machine components and spare parts [3]. For example, they tested the palettes of a turbine that works for a very long time in a power plant. The base material was a Ni alloy, with the particle dimensions between 200 and 300 Å, which confer some mechanical properties to that alloy. After 16,000 of hours working time, they noticed a radial growth of the constituent particles, in the middle and top of the palettes, reaching 900 Å when the breaking occurred after approximately 60,000 working hours. Similar effects were observed in the aviation turbine palettes, where the heavy working conditions affected the particles situated mostly in the middle of the palettes. Other tests were carried out on different machine parts built of heterogeneous materials subject to cyclic deformations or having areas affected by strong heating (welding zones). The conclusion was that any change in the dimension of constituent particles (up to 1000 Å) could be easily observed using the SANS technique. Another field of SAS applicability is the study of flows (voids) appearing in some materials irradiated with neutrons [3]. The components of nuclear reactors are subject to massive exposure to high rapid neutron fluxes, therefore resulting in the necessity of the determination and study of defects that might appear in those materials, having effects on the modification of initial proprieties.

scattering angle and *λ* is the radiation wavelength. In the angular range of SAS, the interested intensity is produced by inhomogeneities extended from 10 to 1000 Å. The quantity *D* of these

> 1 10 *<sup>Q</sup> D D* < <

There are many processes that lead to scattering length variations, extended to zones up to 100 Å. Among these, there are the processes in solid phase namely aging, precipitating processes in solid phase, repeated thermal shock, neutron irradiation, the processes in mixed systems, for example the catalysts, where the two coexisting solid crystalline components are of enormous interest, as well as the ferofluids or the polymers in which neither one component is crystalline. The application of SAXS and SANS to material science has its origin from the experiments performed by Guinier in 1938, to study different types of aluminum alloys with various components such as Al–Cu and Al–Ag. He proved the existence of extremely small precipitate zones (so-called Guinier–Preston (GP) zones) as the first step in solidification of the precipitate of these alloys [3]. The GP zones represent submicroscopic regions of the matrix in which are produced a concentration of the initial solvate element. From one point of view, the GP zones could be considered as pre-precipitates because their formation represents a preliminary stage of precipitations set up from ultra-saturated matrix, but from another point of view it could be considered "clusters," in the sense that represent preferential associations

In Al–Cu alloys, it has a diameter of about 80 Å and a thickness of 3–6 Å. With an average composition of 90% Cu, it comes out that the GP zones in these alloys are predominant constituted from Cu atoms. The physical nature and structural characteristics of the formed precipitates are dependent on temperature and heating duration because the precipitates are formed by germination and growing up processes, based on diffusion phenomena. The most important modification of alloy proprieties occur in the initial stages of the aging, with low temperatures and short heating durations when microstructural transformations are to a scale size under the resolution limit of optical microscopy (submicron precipitates). All these phenomena could be emphasized by SAXS and SANS methods. Regrouping of various atoms species during the precipitation process modifies the dispersion strength of X-rays or neutrons in different zones of aged alloy, due to the existence of an important density difference between the precipitated zones and the rest of alloy matrix. Among the structural processes produced through thermal treatment, which is applied to solid crystalline materials, the spinodal transition is something special [3]. The first phases of coherent precipitation took place through phase separation inside the crystalline lattice. Phase separation could start only through a nucleation process followed by nuclei growing. The germination of a new phase in a solid solution needs that in some regions the solid solution should considerably modify its compo‐ sition. To be transformed into stable germs, these regions should have a specific dimension (about 10 Å). In these conditions, the interface between the germ and the solid solution (matrix) shows a distinct structural discontinuity, and therefore it has a free positive energy which

inhomogeneities determines the most interesting angular range [3]:

of atoms.

40 Neutron Scattering

The stress yielding of the material is correlated with the occurrence of a high concentration of micropores having a diameter between 200 and 1000 Å. Pores having smaller dimensions (≤ 25 Å) are usually present in a very large number and influence the material density. The intermediary sized pores give the gas permeability and the stability to irradiation of the covering layers. All these become visible by SAS technique utilization. SANS is a very useful instrument in the study of microscopic distribution of the magnetic field and the static interaction of spins from different magnetic system as well. Magnetic scattering is caused by the interaction between the neutron magnetic moment and the local magnetization of the sample. If this occurs at a small *Q*, it will show the magnetization fluctuations adequate for large distances in the studied sample and also the magnetic periodicities extended on large distances.

Another important field of study is offered by complex disordered materials such as ceramics, clays, cement, or glasses. Among these, the importance of cement in modern society cannot be underestimated. Cement is found in concrete structures everywhere such as buildings, roads, bridges, dams, and even for conditioning radioactive wastes. There is no escape from the impact of cement in our everyday life. Due to its great importance in the following, we will point only on its applications. Cement, as it is commonly known, is a mixture of compounds made by burning limestone and clay together at very high temperatures ranging from 1400 to 1600°C. Water is the key ingredient, which when mixed with cement forms a paste that evolves as a hard product. The water causes the hardening of the cement by a process called hydration. Hydration is a chemical reaction in which the major compounds in cement form chemical bonds with water molecules to yield hydrates or hydration products.

The heterogeneous nature and chemical complexity of the cement make the characterization of this system difficult. In spite of the inherent difficulties of the system, considerable under‐ standing of the nature of cement and concrete has been obtained through the use of a wide variety of tools.

Studies of the cement pore liquid as a function of time, X-ray diffraction investigations, electron microscope, and NMR experiments have all contributed to a better understanding of the behavior of this material.

The greater penetrating power of neutrons (in comparison to X-rays), the opportunities for in situ measurement of hydrating pastes, the ability to use very long wavelength incident neutrons and the unique opportunities to change contrast offered by D2O–H2O exchange make SANS attractive as a means to probe the internal microstructure of cement pastes. The issue to be resolved, however, is the interpretation of the small angle scattering data. While the actual measurements are generally straightforward, the complexity of phases, contrasts, object morphologies, and size ranges forms a barrier to simple interpretations.

Small angle neutron scattering was first used to study the microstructure of cements in the early 1980s by Allen and coworkers [4]. They measured the small angle neutron scattering from a set of cement specimens hydrated at different water to cement ratios. They also examined the effect of soaking the specimen in heavy water and the change in SANS obtained when the specimen was dried. In their analysis of the SANS data, they assumed a simplified model of the cement hydration reactions. In this model, the hydrated compounds C6AFH12, C4AH13, C3ACSH12, and C3S2H2.5 and the hydroxides Mg(OH)2, Ca(OH)2, NaOH, and KOH in appropriate amounts were the products of cement hydration. Each of these compounds represented a different volume fraction of the cement paste. The strength of the SANS signal from the specimen in the small *Q* Guinier region was directly proportional to the square of the scattering length density difference (the contrast, see Equation 18) and to the total volume of the scattering particles. If the scattering in the two-phase model is to be interpreted as between the matrix C3S2H2.5 and pores in the C3S2H2.5 gel, it matters whether the pores are filled with light water, heavy water, or with air. The contrast between C3S2H2.5 and water will vanish where the scattering length density curves intersect, at approximately 60% D2O. If SANS data from cement paste are to be interpreted on the basis of a two-phase model, one issue is the identification of the appropriate phases. While the contrast is proportional to the SANS intensity, the detailed shape of the SANS curves will depend on the size distribution and shape of the scattering objects.

The SANS is presumed to be caused by a size distribution of objects of a known shape. To calculate the form factor for objects of known shape we can use Equation 17.

The parameters describing the object and its size distribution are then varied to minimize the difference between the calculated scattering and the data. Spheres, cylinders, and disks were considered by Allen and coworkers, and it was ultimately concluded that the SANS from their cement paste specimens was created by a distribution of water-filled spherical pores approx‐ imately 5 nm in diameter and a smaller component of pores with diameters of about 10 nm. SANS from the specimen that had been dried and not rewetted was interpreted as the one due to scattering from a much broader distribution of pore sizes with a peak of about 5 nm but extending to much larger diameters.

SANS from the specimens immersed in D2O showed that the heavy water rapidly exchanged with the water in the pores and with the water in the surrounding C–S–H gel. The total porosity was of the order 1% of the volume.

All these are only a few examples that illustrate the large field of SAS application, where the immediate observation of the phenomena produced in materials lead to solving many practical problems and to extend the theoretical knowledge as well.

### **2. Physical principle of SANS**

The stress yielding of the material is correlated with the occurrence of a high concentration of micropores having a diameter between 200 and 1000 Å. Pores having smaller dimensions (≤ 25 Å) are usually present in a very large number and influence the material density. The intermediary sized pores give the gas permeability and the stability to irradiation of the covering layers. All these become visible by SAS technique utilization. SANS is a very useful instrument in the study of microscopic distribution of the magnetic field and the static interaction of spins from different magnetic system as well. Magnetic scattering is caused by the interaction between the neutron magnetic moment and the local magnetization of the sample. If this occurs at a small *Q*, it will show the magnetization fluctuations adequate for large distances in the studied sample and also the magnetic periodicities extended on large

Another important field of study is offered by complex disordered materials such as ceramics, clays, cement, or glasses. Among these, the importance of cement in modern society cannot be underestimated. Cement is found in concrete structures everywhere such as buildings, roads, bridges, dams, and even for conditioning radioactive wastes. There is no escape from the impact of cement in our everyday life. Due to its great importance in the following, we will point only on its applications. Cement, as it is commonly known, is a mixture of compounds made by burning limestone and clay together at very high temperatures ranging from 1400 to 1600°C. Water is the key ingredient, which when mixed with cement forms a paste that evolves as a hard product. The water causes the hardening of the cement by a process called hydration. Hydration is a chemical reaction in which the major compounds in cement form chemical

The heterogeneous nature and chemical complexity of the cement make the characterization of this system difficult. In spite of the inherent difficulties of the system, considerable under‐ standing of the nature of cement and concrete has been obtained through the use of a wide

Studies of the cement pore liquid as a function of time, X-ray diffraction investigations, electron microscope, and NMR experiments have all contributed to a better understanding of the

The greater penetrating power of neutrons (in comparison to X-rays), the opportunities for in situ measurement of hydrating pastes, the ability to use very long wavelength incident neutrons and the unique opportunities to change contrast offered by D2O–H2O exchange make SANS attractive as a means to probe the internal microstructure of cement pastes. The issue to be resolved, however, is the interpretation of the small angle scattering data. While the actual measurements are generally straightforward, the complexity of phases, contrasts, object

Small angle neutron scattering was first used to study the microstructure of cements in the early 1980s by Allen and coworkers [4]. They measured the small angle neutron scattering from a set of cement specimens hydrated at different water to cement ratios. They also examined the effect of soaking the specimen in heavy water and the change in SANS obtained when the specimen was dried. In their analysis of the SANS data, they assumed a simplified

bonds with water molecules to yield hydrates or hydration products.

morphologies, and size ranges forms a barrier to simple interpretations.

distances.

42 Neutron Scattering

variety of tools.

behavior of this material.

SANS is a method used to obtain information regarding the shape, dimensions, and internal configuration of the zones with scattering density significantly different from the average value, having relatively small dimensions (up to 1000 Å), unevenly distributed in some environment, where the distance arrangement is present or not. In a considered sample, all scattering centers participate in the scattering process. The scattering centers could be atoms, molecules, or particles of the analyzed system. System response could be a spatial distribution of scattered radiation reducible to a succession of maximums, determined by the existence of distance arrangement in a material, or a succession of overlapping maximums over a contin‐ uous distribution when the distance arrangement is accompanied by disordered processes, such as amorphization, precipitation in solid phase, flaws, etc., or by a continuous distribution which present one or two isolated maximums like in the case of liquids (in this case the maximum is owed to the next close vicinity of each scattering center) or a continuous curve in case of SANS. Because X-ray scattering is similar to neutron scattering, in the following presentation I will start with the theoretical principles of SAXS accordingly adapted for SANS. It is well known that a diffraction image of a sample could be easily described in terms of reciprocal space or Fourier space [1]. If we consider *ρ*(*x* <sup>→</sup> ) as the electronic density of the scattering body in a point defined by the vector *x* <sup>→</sup> , then *A*(*Q* <sup>→</sup>) is defined as the transform of *ρ*(*<sup>x</sup>* → ) in the point defined by the vector *Q* <sup>→</sup> in the reciprocal space, and it is given by

$$A\left(\vec{Q}\right) = \int \rho\left(\vec{x}\right) \exp\left(-i\vec{Q}\cdot\vec{x}\right)d\vec{x} \tag{1}$$

The X-ray diffraction theory is based on the fact that *A*(*Q* <sup>→</sup>) represents the amplitude of scattered radiation when *Q* <sup>→</sup> is defined as

$$
\vec{Q} = \left(\frac{2\pi}{\lambda}\right) \left(\vec{s} - \vec{s}\_0\right) \tag{2}
$$

where *λ* is the wavelength of the radiation, *s* → <sup>0</sup> and *s* <sup>→</sup> are the unit vectors on the incident radiation direction and scattered radiation direction, respectively. The magnitude of *Q* <sup>→</sup> is equal to (4*π*/ *λ*)sin *θ*, where 2*θ* = *θ*s is the scattering angle (the angle between incident ray and scattered ray). Therefore, very small scattering angles correspond to small values of the *Q* quantity. Analyzing Equation (1), we find that the intensity of the scattered radiation observed for a specific value of *Q* <sup>→</sup> is equal to the square value of *A*(*<sup>Q</sup>* <sup>→</sup>), where *A*(*<sup>Q</sup>* <sup>→</sup>) is the suitable component of *<sup>Q</sup>* <sup>→</sup> in the Fourier serial development of *ρ*(*x* <sup>→</sup> ). For small values of *Q* <sup>→</sup> that are produced at very small angles, the terms in *ρ*(*x* <sup>→</sup> ) that define the quantity *A*(*Q* <sup>→</sup>) give a periodicity of *x* = 2*π*/*Q*, which is large enough in comparison with X-ray wavelength. These general considerations show that the very small diffraction angle (smaller than a few degrees) provides information on the structure of the matter to a relatively large scale in comparison with X-ray wavelength. It was experi‐ mentally observed that some samples generate a continuous and intense scattering at angles under 2°, without producing the normal diffraction effects from the usual X-ray experiments. This was first time noticed in specific types of fine carbon, black carbon powders, and different types of other substances, all having in common the fine particles of submicroscopic sizes. Later, it was concluded that continuous scattering around the direct beam is owed to the existence of the matter composed of small particles or, more general, to the existence of heterogeneities in the analyzed substance that can have dimensions from a few tens to a few hundred times the wavelength of the X-ray. The qualitative description of the central scatter‐ ing, due to the presence of very small particles, could be done by analogy with the well-known light diffraction phenomena, which lead to the appearance of a halo when the light ray traverses a powder with grains having the dimensions 10 times larger than the wavelength of the light radiation.

Let us consider a particle bathed in an X-ray beam; thus all the electrons are wave scattering sources. If the scattering direction is the same as the incident ray, we can say that the scattered rays are all in phase and, if the scattering angle increases, the phase difference between the different scattered waves will also increase. Therefore, the amplitude of the resulting scattered wave will have values bigger or smaller in accordance only with the existing phase differences. This happens for a scattering angle about 2*θ* = *λ*/*D*, *D* being the average size of the particle, proving in this way that the study of central scattering offers a method to obtain the particle dimensions. This method is applicable only to the particles with dimensions between some specific limits. If *D* is too big, the scattering is limited to angles so small that are experimentally inaccessible. If *D* is too small, of the order of a few wavelengths, the scattering is too broad and too weak to be observed. To show more precisely which factors depend on the small angle scattering, let us consider a small particle of electronic density *ρ*(*x* <sup>→</sup> ). Let us define a form factor *s*(*x* <sup>→</sup> ) of this particle, which has a value of 1 when *x* <sup>→</sup> vector is inside the particle, and a value of 0 when *x* <sup>→</sup> is outside the particle. Thus, according to Equation (1), the amplitude of the scattered radiation from this particle will be

$$A\_1(\vec{Q}) = \int \rho(\vec{x})\mathbf{s}(\vec{x})\exp\left(-i\vec{Q}\cdot\vec{x}\right)d\vec{x} \tag{3}$$

If *A*(*Q* <sup>→</sup>) and *S*(*<sup>Q</sup>* <sup>→</sup>) are the Fourier transforms of *ρ*(*<sup>x</sup>* <sup>→</sup> ) and *s*(*x* <sup>→</sup> ), respectively, then

$$A\_1\left(\vec{Q}\right) = A\left(\vec{Q}\right) \cdot S\left(\vec{Q}\right) \tag{4}$$

This can be demonstrated in the following manner. Let us consider the functions

$$a(\vec{\boldsymbol{x}}) = \rho\left(\vec{\boldsymbol{x}}\right) \* \operatorname{s}\left(\vec{\boldsymbol{x}}\right) \text{ and } a\left(\vec{\boldsymbol{x}}\right) = \int \rho\left(\vec{\boldsymbol{y}}\right) \cdot \operatorname{s}\left(\vec{\boldsymbol{x}} - \vec{\boldsymbol{y}}\right) d\boldsymbol{v}\_{\boldsymbol{y}}$$

Thus, the Fourier transform of *a*(*x* <sup>→</sup> ) is *A*1(*Q* <sup>→</sup>) ;hence

$$A\_1(\vec{Q}) = \int a(\vec{x}) \exp\{2\pi i \vec{Q} \cdot \vec{x}\} dv\_x = \iint \rho(\vec{y}) \mathbf{s}(\vec{x} - \vec{y}) \exp\{2\pi i \vec{Q} \cdot \vec{x}\} dv\_x dv\_y$$

Using the notations *Y = y, Z = x – y*

distance arrangement in a material, or a succession of overlapping maximums over a contin‐ uous distribution when the distance arrangement is accompanied by disordered processes, such as amorphization, precipitation in solid phase, flaws, etc., or by a continuous distribution which present one or two isolated maximums like in the case of liquids (in this case the maximum is owed to the next close vicinity of each scattering center) or a continuous curve in case of SANS. Because X-ray scattering is similar to neutron scattering, in the following presentation I will start with the theoretical principles of SAXS accordingly adapted for SANS. It is well known that a diffraction image of a sample could be easily described in terms of

> *A Q x iQ x dx* ( ) = -× ò r

> > <sup>2</sup> *Q ss* p

æ ö = - ç ÷ è ø

l

direction and scattered radiation direction, respectively. The magnitude of *Q*

( ) <sup>0</sup>

<sup>→</sup> 0 and *s*

*λ*)sin *θ*, where 2*θ* = *θ*s is the scattering angle (the angle between incident ray and scattered ray). Therefore, very small scattering angles correspond to small values of the *Q* quantity. Analyzing Equation (1), we find that the intensity of the scattered radiation observed for a specific value

<sup>→</sup>), where *A*(*<sup>Q</sup>*

enough in comparison with X-ray wavelength. These general considerations show that the very small diffraction angle (smaller than a few degrees) provides information on the structure of the matter to a relatively large scale in comparison with X-ray wavelength. It was experi‐ mentally observed that some samples generate a continuous and intense scattering at angles under 2°, without producing the normal diffraction effects from the usual X-ray experiments. This was first time noticed in specific types of fine carbon, black carbon powders, and different types of other substances, all having in common the fine particles of submicroscopic sizes. Later, it was concluded that continuous scattering around the direct beam is owed to the existence of the matter composed of small particles or, more general, to the existence of heterogeneities in the analyzed substance that can have dimensions from a few tens to a few hundred times the wavelength of the X-ray. The qualitative description of the central scatter‐ ing, due to the presence of very small particles, could be done by analogy with the well-known light diffraction phenomena, which lead to the appearance of a halo when the light ray traverses a powder with grains having the dimensions 10 times larger than the wavelength of

<sup>→</sup> ). For small values of *Q*

<sup>→</sup> , then *A*(*Q*

<sup>→</sup> in the reciprocal space, and it is given by

( )exp( ) r r r rr (1)

<sup>r</sup> r r (2)

<sup>→</sup> ) as the electronic density of the

→ )

<sup>→</sup>) is defined as the transform of *ρ*(*<sup>x</sup>*

<sup>→</sup>) represents the amplitude of scattered

<sup>→</sup> are the unit vectors on the incident radiation

<sup>→</sup>) is the suitable component of *<sup>Q</sup>*

<sup>→</sup>) give a periodicity of *x* = 2*π*/*Q*, which is large

<sup>→</sup> that are produced at very small angles,

<sup>→</sup> is equal to (4*π*/

<sup>→</sup> in the

reciprocal space or Fourier space [1]. If we consider *ρ*(*x*

The X-ray diffraction theory is based on the fact that *A*(*Q*

scattering body in a point defined by the vector *x*

<sup>→</sup> is defined as

where *λ* is the wavelength of the radiation, *s*

<sup>→</sup> is equal to the square value of *A*(*<sup>Q</sup>*

<sup>→</sup> ) that define the quantity *A*(*Q*

Fourier serial development of *ρ*(*x*

in the point defined by the vector *Q*

radiation when *Q*

44 Neutron Scattering

of *Q*

the terms in *ρ*(*x*

the light radiation.

we replace the elementary differentials *dvxdvy* with *dvYdvZ* and to obtain

$$\begin{split} A\_{1}\left(\vec{Q}\right) &= \iint \rho\left(\vec{y}\right) s\left(\vec{Z}\right) \exp\left[2\pi i \vec{Q} \cdot \left(\vec{Y} + \vec{Z}\right)\right] dv\_{1} dv\_{Z} = \\ &= \int \rho\left(\vec{Y}\right) \exp\left(2\pi i \vec{Q} \cdot \vec{Y}\right) dv\_{Y} \cdot \int s\left(\vec{Z}\right) \exp\left(2\pi i \vec{Q} \cdot \vec{Z}\right) dv\_{Z} = A\left(\vec{Q}\right) \cdot \mathcal{S}\left(\vec{Q}\right) \\ A\_{1}\left(\vec{Q}\right) &= A\left(\vec{Q}\right) \cdot \mathcal{S}\left(\vec{Q}\right) \end{split}$$

Giving the dimensions of the region where *s*(*x* <sup>→</sup> ) is different from zero, its transform *S*(*Q* <sup>→</sup>) is completely determinated and, if the particle has dimensions from some few tens to a few hundreds of atomic diameters, *S*(*Q* <sup>→</sup>) will be different from zero only for very small values of *Q* <sup>→</sup>. Now, let us consider the function *A*(*<sup>Q</sup>* <sup>→</sup>). If we suppose that the sample has a constant electronic density *ρ*(*x* <sup>→</sup> )=*k*, then the transform *A*(*Q* <sup>→</sup>) will behave like a Dirac delta function, being zero overall with the exception of the point *Q* <sup>→</sup> = 0, where it is infinite and *∫δ*(*<sup>x</sup>* <sup>→</sup> )*d x*<sup>→</sup> =1. We can conclude that small angle scattering is the method to study the dimensions of the system constituent particles, but the supposition of the existence of the identical particles separated by long distances between them is not quite accurate in the case of real samples. Therefore, the generalization of the theory is needed to take into consideration the diversity of particle sizes and shapes as well as the effect of particle close packing. Without considering the particles, a mathematical expression should be obtained for the scattered intensity in the vicinity of the center that should take into account the electronic density in all the points of the sample. Further, we will discuss only the process of single scattering, neglecting the multiple scattering phenomena and Compton scattering. We will consider an X-ray beam, sufficiently large to irradiate a large number of particles, but narrow enough in comparison with the sampledetector distance, to be like a ray in the experimental device geometry. The amplitude of scattered radiation from the *Pn* point (Fig. 2) will depend on the scattering factor *un* and the direction defined by the unit vector *s* →

$$A\_n = A\_\varepsilon \mu\_n \exp\left[-i\frac{2\pi}{\lambda}(\vec{s} - \vec{s}\_0)\cdot\left(\overrightarrow{OP}\_n\right)\right] \tag{5}$$

where *Ae* is the amplitude scattered by a single electron in the same conditions, *O* is an arbitrary origin that serves to describe the pathway difference between different rays, and *s* → <sup>0</sup> is the unit vector defining the incident radiation direction.

**Figure 2.** Diffraction by a scattering body.

The total amplitude of scattered radiation will then be

$$A\left(\vec{Q}\right) = \sum\_{\boldsymbol{\mu}} A\_{\boldsymbol{\mu}} = A\_{\boldsymbol{\epsilon}}\left(\vec{Q}\right) \sum\_{\boldsymbol{\mu}} \mu\_{\boldsymbol{\mu}} \exp\left(-i\vec{Q} \cdot \overrightarrow{OP}\_{\boldsymbol{\mu}}\right) \tag{6}$$

and the scattered intensity, the product between the amplitude *A,* and its complex conjugate *A*\* , will be

$$I\left(\vec{Q}\right) = \left|A\_{\epsilon}\left(\vec{Q}\right)\right|^{2} \sum\_{\boldsymbol{\mu}} \sum\_{\boldsymbol{\slash}} \mu\_{\boldsymbol{\mu}} \mu\_{\boldsymbol{\slash}} \exp\left[-i\vec{Q} \cdot \left(\overrightarrow{OP}\_{\boldsymbol{\mu}} - \overrightarrow{OP}\_{\boldsymbol{\slash}}\right)\right] \tag{7}$$

The intensity scattered by an electron is

Giving the dimensions of the region where *s*(*x*

<sup>→</sup> )=*k*, then the transform *A*(*Q*

→

exp *<sup>n</sup> A A u i s s OP n en* p

origin that serves to describe the pathway difference between different rays, and *s*

l

é ù = - -× ê ú

where *Ae* is the amplitude scattered by a single electron in the same conditions, *O* is an arbitrary

<sup>→</sup>. Now, let us consider the function *A*(*<sup>Q</sup>*

zero overall with the exception of the point *Q*

hundreds of atomic diameters, *S*(*Q*

direction defined by the unit vector *s*

**Figure 2.** Diffraction by a scattering body.

vector defining the incident radiation direction.

electronic density *ρ*(*x*

*Q*

46 Neutron Scattering

<sup>→</sup> ) is different from zero, its transform *S*(*Q*

<sup>→</sup>). If we suppose that the sample has a constant

<sup>→</sup>) will behave like a Dirac delta function, being

uuur r r (5)

→

<sup>0</sup> is the unit

<sup>→</sup>) will be different from zero only for very small values of

<sup>→</sup> = 0, where it is infinite and *∫δ*(*<sup>x</sup>*

completely determinated and, if the particle has dimensions from some few tens to a few

conclude that small angle scattering is the method to study the dimensions of the system constituent particles, but the supposition of the existence of the identical particles separated by long distances between them is not quite accurate in the case of real samples. Therefore, the generalization of the theory is needed to take into consideration the diversity of particle sizes and shapes as well as the effect of particle close packing. Without considering the particles, a mathematical expression should be obtained for the scattered intensity in the vicinity of the center that should take into account the electronic density in all the points of the sample. Further, we will discuss only the process of single scattering, neglecting the multiple scattering phenomena and Compton scattering. We will consider an X-ray beam, sufficiently large to irradiate a large number of particles, but narrow enough in comparison with the sampledetector distance, to be like a ray in the experimental device geometry. The amplitude of scattered radiation from the *Pn* point (Fig. 2) will depend on the scattering factor *un* and the

( ) <sup>0</sup> ( ) <sup>2</sup>

ë û

<sup>→</sup>) is

<sup>→</sup> )*d x*<sup>→</sup> =1. We can

$$I\_{\epsilon} \left( \mathbb{Q} \right) = A\_{\epsilon}^{2} \left( \mathbb{Q} \right) = A\_{0}^{2} \frac{e^{4}}{m^{2}c^{4}} \cdot \frac{1}{r^{2}} \sin 2\theta = I\_{0} \frac{e^{4}}{m^{2}c^{4}} \cdot \frac{1}{r^{2}} \sin 2\theta \tag{8}$$

which is a three-variable function, the incident intensity, the scattering angle, and the distance of observation; *I*0 is the intensity of incident beam; and *r* is the distance from the particle to detector. If the analyzed environment has a center of symmetry, and this center is placed in the origin *O*, then to each *OPn* <sup>→</sup> vector will correspond another vector −*OPn* →. Thus, we will obtain a simplified expression for the scattered amplitude:

$$A\left(\vec{Q}\right) = \sum\_{\boldsymbol{n}} A\_{\boldsymbol{n}} = A\_{\boldsymbol{\epsilon}}\left(\vec{Q}\right) \sum\_{\boldsymbol{n}} \mu\_{\boldsymbol{n}} \cos\left(\vec{Q} \cdot \overrightarrow{\boldsymbol{OP}}\_{\boldsymbol{n}}\right) \tag{9}$$

We define the structure factor of the environment as the report between the total scattered amplitude and the radiation amplitude scattered by a single electron in the same conditions:

$$F\left(\vec{Q}\right) = \frac{\sum\_{\boldsymbol{\pi}} A\_{\boldsymbol{\pi}}(\vec{Q})}{A\_{\boldsymbol{\pi}}(\vec{Q})} = \sum\_{\boldsymbol{\pi}} \mu\_{\boldsymbol{\pi}} \cos\left(\vec{Q} \cdot \overrightarrow{OP}\_{\boldsymbol{\pi}}\right) \tag{10}$$

The scattered intensity will then be

$$I\left(\vec{Q}\right) = I\_e\left(\vec{Q}\right)\left[\sum\_n u\_n \cos(\vec{Q} \cdot \overrightarrow{OP}\_n)\right]^2 = I\_e\left(\vec{Q}\right)F^2\left(\vec{Q}\right) \tag{11}$$

The term "*Pn* point" was used to define the structure of a particle. If we consider a large particle, the base element in its description is the atom; in this case, the point *Pn* refers to the center of the *n*th atom, and the scattering factor *un* is the scattering factor of this *n*th atom. In the interested angular domain for SAS, *un*(*Q* <sup>→</sup>) can be considered as a constant equal to *un*(0). If we consider a small region, then *Pn* point refers to a small volume element that surrounds it. The scattering factor *un* is then equal to *ρndvn,* where *ρn* is the electronic density of the particle in the vicinity of *Pn* point, and *dvn* is the volume of considered element. Generally, it is convenient to describe the particle structure taking into account enough small elements to consider the scattering factors of these elements to be constant, independent of scattering angle over the whole domain where the structure factor of the considered particle is different from zero. This theory of small angle X-ray scattering can be adapted very easily to neutrons.

Small angle neutron scattering is used where the X-rays cannot provide the desired informa‐ tion either of the lack of scattering contrast or of severe absorption of the studied material. The choice of thermal neutrons as a means of substance investigation was owed to their wave‐ lengths and their energies corresponding to the inter atomic distances and excitation energies of condensed matter. Neutron absorption from the matter is very low, thus resulting in the samples that can have thicknesses larger than in the X-rays case which are strongly absorbed with an increase in the volume of analyzed samples. We can say that SANS allows the investigation of materials in various conditions, such as in containers, furnaces, cryostats, etc. Due to its magnetic moment, the neutron provides unique possibilities to the study of magnetic structures, magnetic moment distribution, and magnetic excitations. Being a nuclear propriety, the nuclear scattering amplitude can be considerably different between various isotopes of a specific chemical species. For example, the big difference between the coherent scattering lengths of hydrogen and deuterium lead to the usage of phase contrast in the study of hydrogenate materials, allowing a good resolution in analyzing polymers and biological substances in general. Because thermal neutrons interact very low with the matter, this interaction can be theoretically treated on the base of first Born approximation [5, 6]. We will show further that the variation of the scattering density is very important in the study of condensed matter through SANS. In the following, we will consider the "static approxima‐ tion"; the scattering process is produced only at nuclear level on fixed targets letting those targets unchanged after the collision, and without considering the polarization effects.

Thus, we can write the coherent elastic scattering differential cross section on a single atom:

$$\frac{d\sigma}{d\Omega} = \frac{1}{N} \left| \sum\_{\vec{R}} b\_{\vec{R}} \exp\left(i\vec{Q} \cdot \vec{R}\right) \right|^2 \tag{12}$$

where *N* is the number of scattering nuclei exposed to the beam, and *bR* is the coherent scattering length of the chemical species that occupy a specific place determined by the position vector *R* <sup>→</sup> . If we replace *bR* through an average local scattering length density *ρb*(*<sup>r</sup>* <sup>→</sup> ), where *r* <sup>→</sup> is a continuously variable position vector, we can write

$$\frac{d\sigma}{d\Omega} = \frac{1}{N} \left| \int\_{V} \rho\_{\flat} \left( \vec{r} \right) \exp \left( i \vec{Q} \cdot \vec{r} \right) d^{3} \vec{r} \right|^{2} \tag{13}$$

where the integration is performed over the whole sample volume *V*.

The scattering length density *ρb*(*r* <sup>→</sup> ) can now vary on distances of the order of *d*min, where *d*min ≅ *π*/*Q*max, *Q*max being the maximal value accessible by experiment. Thus, we can write that

$$
\rho\_b \left( \vec{r} \right) = \Delta \rho\_b \left( \vec{r} \right) + \vec{\rho}\_b \tag{14}
$$

where *ρb* is averaged on volumes larger than the instrument resolution volume, determined by the minimum observed value of *Q*. Therefore, the cross section that contains the useful information about the studied system is dependent on Δ*ρb*(*r* <sup>→</sup> ). By substituting Equation (14) in Equation (13) we obtain

$$\frac{d\sigma}{d\Omega} = \frac{1}{N} \left| \int\_{V} \rho\_{\flat} \left( \vec{r} \right) - \overline{\rho\_{\flat}} \right| \exp \left( i \vec{Q} \cdot \vec{r} \right) d^{3} \vec{r} \right|^{2} \tag{15}$$

For a specific distribution of scattering length, the cross section can be analytically calculated with the help of Equations (13) and (15). Because the cross section values are not known everywhere in reciprocal space, *ρb*(*r* <sup>→</sup> ) determination by experimental measurements of *d*Σ/*d*Ω values is very hard to be carried out, or even impossible. Let us take a sample containing *Np* particles with scattering length densities *ρbp* = *bp*/*vap*, where *bp* is the scattering length averaged on the particle volume, and *vap* is the atomic volume from the particle. Let us consider that these particles are packed in a homogeneous matrix having scattering length densities *ρbm* = *bm*/*vam*.

Using Equation (13) we obtain

consider a small region, then *Pn* point refers to a small volume element that surrounds it. The scattering factor *un* is then equal to *ρndvn,* where *ρn* is the electronic density of the particle in the vicinity of *Pn* point, and *dvn* is the volume of considered element. Generally, it is convenient to describe the particle structure taking into account enough small elements to consider the scattering factors of these elements to be constant, independent of scattering angle over the whole domain where the structure factor of the considered particle is different from zero. This

Small angle neutron scattering is used where the X-rays cannot provide the desired informa‐ tion either of the lack of scattering contrast or of severe absorption of the studied material. The choice of thermal neutrons as a means of substance investigation was owed to their wave‐ lengths and their energies corresponding to the inter atomic distances and excitation energies of condensed matter. Neutron absorption from the matter is very low, thus resulting in the samples that can have thicknesses larger than in the X-rays case which are strongly absorbed with an increase in the volume of analyzed samples. We can say that SANS allows the investigation of materials in various conditions, such as in containers, furnaces, cryostats, etc. Due to its magnetic moment, the neutron provides unique possibilities to the study of magnetic structures, magnetic moment distribution, and magnetic excitations. Being a nuclear propriety, the nuclear scattering amplitude can be considerably different between various isotopes of a specific chemical species. For example, the big difference between the coherent scattering lengths of hydrogen and deuterium lead to the usage of phase contrast in the study of hydrogenate materials, allowing a good resolution in analyzing polymers and biological substances in general. Because thermal neutrons interact very low with the matter, this interaction can be theoretically treated on the base of first Born approximation [5, 6]. We will show further that the variation of the scattering density is very important in the study of condensed matter through SANS. In the following, we will consider the "static approxima‐ tion"; the scattering process is produced only at nuclear level on fixed targets letting those targets unchanged after the collision, and without considering the polarization effects.

Thus, we can write the coherent elastic scattering differential cross section on a single atom:

exp *<sup>R</sup>*

where *N* is the number of scattering nuclei exposed to the beam, and *bR* is the coherent scattering length of the chemical species that occupy a specific place determined by the position

> ( ) ( ) <sup>2</sup> <sup>1</sup> <sup>3</sup> exp *<sup>b</sup>*

<sup>→</sup> . If we replace *bR* through an average local scattering length density *ρb*(*<sup>r</sup>*

*<sup>d</sup> r iQ r d r d N*

1

*V*

r

s

continuously variable position vector, we can write

s=

vector *R*

48 Neutron Scattering

*R <sup>d</sup> b iQ R d N*

( ) <sup>2</sup>

<sup>=</sup> <sup>×</sup> <sup>W</sup> <sup>å</sup> <sup>r</sup> <sup>r</sup> (12)

<sup>×</sup> <sup>W</sup> <sup>ò</sup> <sup>r</sup> r rr (13)

<sup>→</sup> ), where *r*

<sup>→</sup> is a

theory of small angle X-ray scattering can be adapted very easily to neutrons.

$$\frac{d\Sigma}{d\Omega}(\vec{Q}) = \frac{1}{N} (\rho\_{bp} - \rho\_{bm})^2 \left| \int\_{V\_t} \exp(i\vec{Q} \cdot \vec{r}) d^3\vec{r} \right|^2 \tag{16}$$

where the integral is extended over the volume *Vt* occupied by all the particles. In the most general case, this integral will contain spatial and orientation correlation elements between particles, and effects owed to the size distribution of the particles. Taking into account the form factor expression for a single particle

$$F\_p\left(\vec{Q}\right) = \frac{1}{V\_p} \int \exp\left(i\vec{Q}\cdot\vec{r}\right) d^3\vec{r} \tag{17}$$

where *Vp* is the particle volume so that │*Fp*(0) │<sup>2</sup> = 1. We will now consider *Np* identical particles for which we can write the following expression

$$\frac{d\Sigma}{d\Omega}(\bar{Q}) = \frac{V\_p^2 N\_p}{N} \left(\rho\_{bp} - \rho\_{bm}\right)^2 \left| F\_p \left(\bar{Q}\right) \right|^2 \tag{18}$$

that is valid in the case of binary systems, without contributions from the interferences between particles.

The interference term from Equation (16) that was neglected in Equation (18) is the Fourier transform Ψ(*Q* <sup>→</sup>) of the correlation function:

$$\Psi^{\Psi}(\vec{Q}) = \frac{1}{N\_p} \sum\_{i \neq j} \exp\left[i\vec{\mathcal{Q}} \cdot (\vec{r}\_{0i} - \vec{r}\_{0j})\right] \tag{19}$$

where *r* → 0*i* and *r* → 0 *j* are the position vectors of the particle centers denoted by *i* and *j*.

The Ψ(*Q* <sup>→</sup>) function will be zero for all the values of *<sup>Q</sup>* <sup>→</sup> <sup>≠</sup>0 if the distance distribution between the particles is completely random, approximately like in the case of very diluted systems. Equation (18) is valid for particles anisotropically oriented, but all are identically oriented. In most cases of random distribution of the orientations or in discrete but multiple orientations of anisotropic particles, suitable averages of | *Fp*(*Q* <sup>→</sup>)| 2 are used. Guinier shows that at small values of *Q*, the scattering function *S*(*Q* <sup>→</sup>)= <sup>|</sup> *<sup>F</sup>* (*<sup>Q</sup>* <sup>→</sup>)| <sup>2</sup> for a specific physical system can be written as

$$S\left(\vec{Q}\right) = \exp\left(-Q^2 R\_D^2\right) \tag{20}$$

where *RD* means the gyration radius which is a material parameter having the length dimen‐ sion defined through

$$R\_D^2 = \frac{1}{V\_p} \int\_{\mathcal{V}\_p} r\_D^2 q\left(r\_D\right) dr\_D \tag{21}$$

where *q*(*rD*) is the geometrical cross section of the particle, through the length of a plane normal to a direction *D* and at a distance *rD* from the origin inside the particle. For a system of particles randomly oriented

$$S(\vec{Q}) = \exp\left(-Q^2 R\_{\odot}^2 / 3\right) \tag{22}$$

with

$$R\_C^2 = \frac{1}{V\_p} \int r^2 d^3 \vec{r}$$

For spheres of radius *RS*, *RG* = (3/5)1/2*RS*, and Equation (21) coincides with the expression

$$\mathcal{S}\_{\mathbb{S}}\left(\mathcal{Q}\mathcal{R}\_{\mathbb{S}}\right) \\ = \left[ \Im\left(\sin\left(\mathcal{Q}\mathcal{R}\_{\mathbb{S}}\right) \ -\mathcal{Q}\mathcal{R}\_{\mathbb{S}}\cos\left(\mathcal{Q}\mathcal{R}\_{\mathbb{S}}\right) / \mathcal{Q}^3 \mathcal{R}\_{\mathbb{S}}^{\cdot 3}\right) \right]^2$$

to the term proportional with *Q*<sup>4</sup> . The subsequent terms in the developments of the two series are in a good concordance, but the corresponding terms have the same sign. Guinier approx‐ imation is thus accepted over a large range of *QRG*(<1.2).

For rotational ellipsoids, the Guinier approximation coincides with the extension of scattering functions until *Q*<sup>6</sup> . For other shapes of the particles, it might be necessary for an investigation of the small values of *Q* to be able to determine *RG* with sufficient accuracy. In the case of homogeneous particles, with pronounced margins and area *Ap*, Porod has proved that

$$\overline{S(\bar{Q})} \equiv 2\pi A\_p \, / \, V\_p^2 \mathcal{Q}^4 \tag{24}$$

This shows the average decrease in the scattering function at large *Q*. *Q* should be larger than the reciprocal value of the smallest dimension of the particle.

### **3. Experimental methods used in the study of condensed matter by SANS**

The goal of SANS method is to measure *I*(*Q*) and to extract from that measurement the "structural" data regarding the studied system. The quantity *Q* is defined as *Q* = 4*π*sin*θ<sup>s</sup> <sup>λ</sup>* <sup>≈</sup> <sup>2</sup>*<sup>π</sup> <sup>λ</sup> <sup>θ</sup>*, if *<sup>θ</sup>* <sup>=</sup> *θs* <sup>2</sup> is small enough.

Therefore, to measure *I*(*Q*), it is sufficient to measure *I*(*θ*) at constant *λ* or *I*(*λ*) at constant *θ*. In the following, we will discuss both experimental alternatives.

### **3.1. The method** *I***(***θ***) at constant** *λ*

that is valid in the case of binary systems, without contributions from the interferences between

The interference term from Equation (16) that was neglected in Equation (18) is the Fourier

are the position vectors of the particle centers denoted by *i* and *j*.

the particles is completely random, approximately like in the case of very diluted systems. Equation (18) is valid for particles anisotropically oriented, but all are identically oriented. In most cases of random distribution of the orientations or in discrete but multiple orientations

<sup>→</sup>)| <sup>2</sup>

where *RD* means the gyration radius which is a material parameter having the length dimen‐

where *q*(*rD*) is the geometrical cross section of the particle, through the length of a plane normal to a direction *D* and at a distance *rD* from the origin inside the particle. For a system of particles

( ) 2 2 <sup>1</sup> *p D DD D p V*

<sup>2</sup> 1 2 3

*p V R rdr*

*G*

*p*

r r r r (19)

( ) ( ) 2 2 exp *<sup>D</sup> SQ QR* = - <sup>r</sup> (20)

*R r q r dr <sup>V</sup>* <sup>=</sup> ò (21)

( ) ( ) 2 2 exp / 3 *<sup>G</sup> SQ QR* = - <sup>r</sup> (22)

*<sup>V</sup>* <sup>=</sup> <sup>ò</sup> <sup>r</sup> (23)

<sup>→</sup> <sup>≠</sup>0 if the distance distribution between

<sup>→</sup>)| 2 are used. Guinier shows that at small

for a specific physical system can be written

( ) 0 0 <sup>1</sup> exp ( ) *i j*

<sup>→</sup>)= <sup>|</sup> *<sup>F</sup>* (*<sup>Q</sup>*

*p i j <sup>Q</sup> iQ r r <sup>N</sup>* <sup>¹</sup> Y = ×- é ù å ë û

<sup>→</sup>) of the correlation function:

<sup>→</sup>) function will be zero for all the values of *<sup>Q</sup>*

of anisotropic particles, suitable averages of | *Fp*(*Q*

values of *Q*, the scattering function *S*(*Q*

sion defined through

randomly oriented

with

particles.

50 Neutron Scattering

where *r* → 0*i* and *r* → 0 *j*

The Ψ(*Q*

as

transform Ψ(*Q*

This method is based on monochromatic (*λ* = constant) beam extraction from the white thermal neutron spectrum provided by a research reactor. The thermal neutrons released from the reactor are first collimated with a Soller collimator (with plane parallel slits) and after that they are monochromatized with either a mechanical monochromator or a crystal. Monochromati‐ zation is the method that allows the extraction of desired neutrons from the neutron beam provided by the source; in other words, to obtain thermal neutrons with energies in a very narrow interval centered around a value *E*<sup>0</sup> of the energy. Neutron monochromation is based either on their corpuscular proprieties (mechanical monochromators) or on their wave proprieties (crystal monochromator). The mechanical monochromator is a cylinder endowed with helicoidal slits on the whole length of its generatrix.

When the cylinder is rotating with the angular speed *ω*, the neutrons with their motion direction parallel with the rotor axis would traverse the monochromator slits without modi‐ fying their distance to the slits walls only if they would have a specific speed *v*0, directly correlated with *ω* and which is corresponding to the wavelength *λ*<sup>0</sup> of the neutrons transmitted by the monochromator. Another type of mechanical monochromator uses disks instead of a cylinder. Its functional principle is identical with the cylindrical monochromator, but having the difference that the place of helicoidal slits is taken by straight slits cut off on a rigid mounted disc on a horizontal axis, which rotates with an angular speed *ω*. The disks are positioned on the axis having the slits displaced to provide a selective opening in accordance with the axis rotation speed *ω*, only to the neutrons moving with a specific speed *v*0. Nevertheless, the most used monochromatization method is by using a crystal. Crystal utilization in spectrometer devices is based on the neutron diffraction phenomena.

Monochromatic neutron beam separation from the thermal neutron beam provided by the reactor is realized through its Bragg reflection on a monocrystal. If we note the distance between the crystalline planes with *dhkl* determined by the *h, k, l* indexes, and *θB* the neutron incident angle on these planes, then the reflected neutron wavelength is given by the Bragg law

$$n\vec{\omega} = \mathbf{2}d\_{hkl}\sin\theta\_{\oplus} = \frac{nh}{\sqrt{2mE}}$$

where *m* and *E* are the neutron mass and neutron energy, respectively, but *n* is the reflection order.

In other words, if a monocrystal is used to select neutrons with a specific wavelength *λ* that satisfies the Bragg relation, then the neutrons selected by the crystal would also have the wavelengths *λ*/2, *λ*/3, *λ*/4,..... etc., which means that reflections with a higher order take place, with *n* = 1, 2, 3,.... In the case of monochromators with crystal, the presence of high-order reflections could be a serious problem because sometimes, depending on the spectral distri‐ bution of the incident beam and the crystal structure factor, the reflection intensities become comparable and even greater than the first-order reflection intensity. In most neutron physics experiments, first-order reflection (*n* = 1) is used, but other reflections (high-order contamina‐ tions) should be eliminated by different means.

Analyzing the Bragg relation, we can see that by using of monochromator crystals, only monochromatic neutrons with the wavelength of *λ*max ≤2*d*max can be selected, where *d*max is the maximum distance between the crystal planes. The neutrons having the wavelength *λ* > 2*d*max are not presenting Bragg reflections.

This observation suggests a simple way to eliminate the second-order reflections, utilized mostly when the monochromatic neutrons desired to be extracted from the reactor emergent beam have wavelengths relatively large, for example, in the range *λ* > 6 Å. It is sufficient to introduce a polycrystalline filter of convenient size in the pathway of neutron beam falling on the monochromator crystal, with its crystalline structure characterized by *d*max< *λ*0/2, where *λ*<sup>0</sup> is the wavelength of the neutrons to be extracted. There is another experimental solution to eliminate second-order reflections, for example, using a monochromator of special monocrys‐ tals with an internal composition that cancels the structural factor for some specific groups of crystalline indexes. Because the problem of high-order reflections is overpassing the goal of this chapter, we will not insist on this subject.

When the cylinder is rotating with the angular speed *ω*, the neutrons with their motion direction parallel with the rotor axis would traverse the monochromator slits without modi‐ fying their distance to the slits walls only if they would have a specific speed *v*0, directly correlated with *ω* and which is corresponding to the wavelength *λ*<sup>0</sup> of the neutrons transmitted by the monochromator. Another type of mechanical monochromator uses disks instead of a cylinder. Its functional principle is identical with the cylindrical monochromator, but having the difference that the place of helicoidal slits is taken by straight slits cut off on a rigid mounted disc on a horizontal axis, which rotates with an angular speed *ω*. The disks are positioned on the axis having the slits displaced to provide a selective opening in accordance with the axis rotation speed *ω*, only to the neutrons moving with a specific speed *v*0. Nevertheless, the most used monochromatization method is by using a crystal. Crystal utilization in spectrometer

Monochromatic neutron beam separation from the thermal neutron beam provided by the reactor is realized through its Bragg reflection on a monocrystal. If we note the distance between the crystalline planes with *dhkl* determined by the *h, k, l* indexes, and *θB* the neutron incident angle on these planes, then the reflected neutron wavelength is given by the Bragg

2 sin

l

2 *hkl B nh n d*

 q= =

where *m* and *E* are the neutron mass and neutron energy, respectively, but *n* is the reflection

In other words, if a monocrystal is used to select neutrons with a specific wavelength *λ* that satisfies the Bragg relation, then the neutrons selected by the crystal would also have the wavelengths *λ*/2, *λ*/3, *λ*/4,..... etc., which means that reflections with a higher order take place, with *n* = 1, 2, 3,.... In the case of monochromators with crystal, the presence of high-order reflections could be a serious problem because sometimes, depending on the spectral distri‐ bution of the incident beam and the crystal structure factor, the reflection intensities become comparable and even greater than the first-order reflection intensity. In most neutron physics experiments, first-order reflection (*n* = 1) is used, but other reflections (high-order contamina‐

Analyzing the Bragg relation, we can see that by using of monochromator crystals, only monochromatic neutrons with the wavelength of *λ*max ≤2*d*max can be selected, where *d*max is the maximum distance between the crystal planes. The neutrons having the wavelength *λ* > 2*d*max

This observation suggests a simple way to eliminate the second-order reflections, utilized mostly when the monochromatic neutrons desired to be extracted from the reactor emergent beam have wavelengths relatively large, for example, in the range *λ* > 6 Å. It is sufficient to introduce a polycrystalline filter of convenient size in the pathway of neutron beam falling on the monochromator crystal, with its crystalline structure characterized by *d*max< *λ*0/2, where *λ*<sup>0</sup>

*mE*

devices is based on the neutron diffraction phenomena.

tions) should be eliminated by different means.

are not presenting Bragg reflections.

law

52 Neutron Scattering

order.

Frequently, the experimental pattern used to obtain monochromatic neutron beams in this way contains a Soller collimator which spatially delimitates the incident neutron beam coming out from the reactor, the monochromator crystal, and the second Soller collimator which delimi‐ tates the diffracted beam. Simultaneous rotations of the second collimator and the crystal with angular speeds staying in a ratio of 2:1 allows to extract monochromatic neutron beams of different wavelengths from the continuous spectrum of the incident thermal neutron beam. The monochromatic neutrons are recorded by a detector placed after the second collimator rotating in the same time with it.

The devices built on this principle, the so-called neutron spectrometer with crystal, allow the determination of total cross section by the measurement of sample transmission in accordance with neutron wavelength. To study the angular distribution of scattered neutrons by a certain target and implicitly to measure the differential cross sections, a pattern with a third Soller collimator between the sample and detector is used. To investigate the neutron scattering of different wavelengths, the whole system built from the second collimator, the studied sample, the third collimator, and the neutron detector should be rotated around the first fixed axis that is going through the middle of the monochromator crystal and together with it; the angular speed ratio of the crystal and the mentioned system should be 1/2. For a certain position of the monochromator crystal and the second collimator, the third collimator and the detector could be rotated around the second fixed axis (that is going through the middle of the sample) allowing the measurement of sample differential cross section for thermal neutrons having the wavelengths determined by the monochromator preceding the sample. This experimental pattern is named neutron spectrometer with two axes. Many experimental systems use optical patterns derived from the two axis spectrometer described earlier. Anticipating the discussion in the next section, we would mention that from the simple presentation of the optical pattern of considered spectrometer arises a specific limitation of its efficiency. Actually, to measure *I*(*Q*) with sufficient precision in the optical pattern described before, it is necessary to increase the distances (*R*) between the studied sample and the detector, as much as possible; increasing this distance obviously will lead to decreasing the lower limit of the explored *Q* range. Because the intensity of the scattered beam varies with 1/*R*<sup>2</sup> , the obtained resolution gain is drastically affected by losing the luminosity of the experimental system. If we add to this inconvenience the construction of experimental devices having excessive dimensions, the continuous search for more efficient SANS optical patterns becomes explicable. One of the identified optical patterns which is considered most acceptable is by using two crystals, with the sample staying between them.

Two crystal systems were used as monochromators in spectrometric assemblies long time ago, first in X-ray physics and then in neutron physics. To discuss the utilization manner of two monocrystals systems in SANS, we have to reveal the optical characteristics of these systems.

**Figure 3.** Parallel pattern.

**Figure 4.** Antiparallel pattern.

To simplify the explanation, we will limit our case to consider that the two monocrystals are ideal identical crystals, from the point of view of their internal crystalline structure and their cutting method (has the same active reflecting planes). A two-monocrystal system could work in two optical patterns: parallel (Fig. 3) and antiparallel (Fig. 4).

In the parallel pattern, any selected wavelength from the incident polychromatic beam through Bragg reflection on the first crystal leads to the propagation of the double reflected beam in parallel direction with the incident beam. Therefore, this pattern is not dispersive (does not Experimental Methods in the Study of Neutron Scattering at Small Angles http://dx.doi.org/10.5772/62184 55

**Figure 5.** Double crystal parallel pattern.

To simplify the explanation, we will limit our case to consider that the two monocrystals are ideal identical crystals, from the point of view of their internal crystalline structure and their cutting method (has the same active reflecting planes). A two-monocrystal system could work

In the parallel pattern, any selected wavelength from the incident polychromatic beam through Bragg reflection on the first crystal leads to the propagation of the double reflected beam in parallel direction with the incident beam. Therefore, this pattern is not dispersive (does not

in two optical patterns: parallel (Fig. 3) and antiparallel (Fig. 4).

**Figure 3.** Parallel pattern.

54 Neutron Scattering

**Figure 4.** Antiparallel pattern.

spatially separate the beams of different wavelengths). In the second optical pattern, the Bragg double reflected beam is propagated on a direction oriented toward the region containing the radiation source and is named antiparallel. To notice that the wavelength change of the selected radiation beam through double reflection is realized along two directions depending on the wavelengths, revealing the dispersive feature of the antiparallel pattern. In the case of the parallel pattern, the thermal neutron beam falling on the first monochromator crystal under the incident angle *θB*, it will be Bragg reflected on the second crystal, the double reflected ray making the same angle *θB* with the second crystal (Fig. 5).

If a sample is interposed between the two crystals, then the beam reflected by the first crystal will be scattered by the sample under the *θS* angle and will fall on the second crystal under the *θ* incident angle (Fig. 6).

To detect the scattered beam, we have to rotate the second monochromator crystal together with the detector by an angle *α*; thus the Bragg relation is satisfied by obtaining the same incident *θB* angle for the sample scattered ray. Among these angles, we can write the following relations:

$$\begin{aligned} \theta - \alpha &= \theta\_{\text{\\$}} \implies \theta - \alpha = \theta\_{\text{\\$}}\\ \theta\_{\text{\\$}} + \theta\_{\text{\\$}} &= \theta \implies \theta - \theta\_{\text{\\$}} = \theta\_{\text{\\$}} \implies \alpha = \theta\_{\text{\\$}} \end{aligned}$$

**Figure 6.** Three axes parallel pattern.

Because the sample is scattering, the neutrons under different angles *θS*(*θS*1, *θS*2,..., *θSn*) arise that the system composed from the second monochromator and the detector should be rotated in accordance with the suitable *α*(*α*1, *α*2,..., *αn*) angles to be able to detect all scattered neutrons. Let us now consider the case of the antiparallel pattern. The thermal neutron beam falls on the first monochromator crystal under the incident *θB* angle and is reflected on the second monochromator crystal, the double reflected ray leaving the system under the same *θB* angle (Fig. 7).

If we have a sample between the two crystals, then the reflected beam by the first crystal will be scattered by the sample under a *θS* angle and will fall on the second crystal under the *θ* angle (Fig. 8).

The neutrons scattered by the sample *S* could be detected only by rotating the second crystal with the *α* angle, thereby reestablishing the Bragg law by remaking the initial incident *θB* angle. Also, we found that

$$\begin{aligned} \theta + \alpha &= \theta\_{\text{B}} \implies \theta + \alpha = \theta\_{\text{B}}\\ \theta\_{\text{B}} - \theta\_{\text{S}} &= \theta \Longrightarrow \theta + \theta\_{\text{S}} = \theta\_{\text{B}} \Longrightarrow \alpha = \theta\_{\text{S}} \end{aligned}$$

Experimental Methods in the Study of Neutron Scattering at Small Angles http://dx.doi.org/10.5772/62184 57

**Figure 7.** Double crystal antiparallel pattern.

**Figure 8.** Three-axe antiparallel pattern.

Because the sample is scattering, the neutrons under different angles *θS*(*θS*1, *θS*2,..., *θSn*) arise that the system composed from the second monochromator and the detector should be rotated in accordance with the suitable *α*(*α*1, *α*2,..., *αn*) angles to be able to detect all scattered neutrons. Let us now consider the case of the antiparallel pattern. The thermal neutron beam falls on the first monochromator crystal under the incident *θB* angle and is reflected on the second monochromator crystal, the double reflected ray leaving the system under the same *θB* angle

If we have a sample between the two crystals, then the reflected beam by the first crystal will be scattered by the sample under a *θS* angle and will fall on the second crystal under the *θ*

The neutrons scattered by the sample *S* could be detected only by rotating the second crystal with the *α* angle, thereby reestablishing the Bragg law by remaking the initial incident *θB* angle.

> *B B B S SB S*


 q aq

 qaq

+ = Þ+ =

qaq

q q q qq

(Fig. 7).

56 Neutron Scattering

angle (Fig. 8).

Also, we found that

**Figure 6.** Three axes parallel pattern.

To notice that this pattern is the most frequently used due to its dispersive feature this allows a better measurement of the interest neutrons.

### **3.2. The method** *I***(***λ***) at constant** *θ*

Another method is to make the observation only on a certain scattering direction 2*θ* = constant. This time the directly measurable quantity is the wavelength (or the energy) of the coherent radiation scattered by the sample. This method is based on *λ* determination for every detected neutron, by its time of flight measurement along a known distance. According to the De Broglie equation

$$\mathcal{A} = \frac{h}{m\upsilon} = \frac{ht}{mL}$$

where *L* is the distance between the neutron source and detector, *h* is the Planck constant, and *m* is the neutron mass. The neutrons with a wavelength *λ* will have a speed *v* = *h*/*mλ* so that to cover an *L* distance, the time *t* = *L*/*v* will be necessary.

Neutrons of different wavelengths will need different time to cover the same distance *L*. If the moment of neutron pulse emission coincides with the time scale origin of the neutron detector impulse analyzer, the detector impulse selection in accordance with the impulse appearance time (practically by neutron arrival time) allows the spectral distribution measurement of the neutron beam. The electronic devices performing this selection are "the time of flight ana‐ lyzers." The analyzer has a number of channels in which the impulses are accumulated due to the neutrons arriving to the detector in the time intervals *t*0 + *τ*, *t*0 + 2*τ*, *t*0 + 3*τ,* and so on.

Here, *t*0 represents a "delay time" of the impulse temporally analyzer system start-up, which could be varied with the used experimental pattern particularities, and *τ* is the channel temporal width, a quantity that can be varied in stages for a specific instrument. The meas‐ urements by the time of flight method can be made using both steady-state reactors and pulsed reactors as neutron sources.

In the case of steady-state reactors, the neutron beam leaving the reactor channel meets an obturator in its way which opens the way only for very short time intervals. In this manner, the detector "notices" a pulsed neutron beam with a determined time length, for a specific obturator–detector distance, by the neutron energetic spectrum. The detector is connected with a time of flight analyzer, which can be unleashed by the obturator opening. To the pulsed reactors, the obturator function is undertaken by the neutron source itself. The time of flight analyzer start-up command is given by the neutron source when the pulse is accomplished. In every SANS experiment, the main goal is to determine with a good precision and in a reasonably short time the *Q* dependence of the analyzed sample coherent elastic scattering differential cross section in a large as possible *Q* variation range. From the point of view of the *Q* range explored value extension, the two SANS investigation methods indicate the following possible performances:

$$\frac{Q\_{\text{max}}}{Q\_{\text{min}}} = \left[\frac{\theta\_{\text{max}}}{\theta\_{\text{min}}}\right]\_{\lambda = \text{const.}} \quad \text{and respective } \frac{Q\_{\text{max}}}{Q\_{\text{min}}} = \left[\frac{\lambda\_{\text{min}}}{\lambda\_{\text{max}}}\right]\_{\theta = \text{const.}}$$

For any *Q*max – *Q*min, the interest interval for a certain sample, the first method seems to be more reasonable because the *Q* limitation apparently is dictated only by the measurement condi‐ tions, but in the case of the second method, the *λ* limitation is dictated by the spectral charac‐ teristics of the neutron source. If the neutron source is in steady state, then the first method utilization is certainly more advantageous because pulsing the neutron beam with a chopper leads to a usage of 10−4–10−3 from the produced neutrons even if we consider the advantage of the second method to replace the sequential measurement of *I*(*Q*) with the simultaneous one. The advantage of the first method toward the second in the case of utilization of a steady-state source is more evident if we estimate the consequences resulting in the obtained precisions and the experiment duration from the efficient utilization of the neutron beam emitted by the source. In the case of pulsed sources, the same criteria of neutron beam efficient utilization impose the application of the second working method. The approximate equivalence of the two methods, the first by the steady-state reactors and the second the pulsed sources, can be motivated that in present, at realized impulse fluxes and obtained pulses frequencies, the pulsed sources lead to getting middle fluxes comparable with the one obtained from the steady-state reactors of the same category. All these reasons justify the topics discussed in the next section, the presentation of SANS equipment used by the first method at the steady-state reactors or the second method used at the pulsed sources.

### **4. Experimental patterns**

*h ht mv mL*

where *L* is the distance between the neutron source and detector, *h* is the Planck constant, and *m* is the neutron mass. The neutrons with a wavelength *λ* will have a speed *v* = *h*/*mλ* so that

Neutrons of different wavelengths will need different time to cover the same distance *L*. If the moment of neutron pulse emission coincides with the time scale origin of the neutron detector impulse analyzer, the detector impulse selection in accordance with the impulse appearance time (practically by neutron arrival time) allows the spectral distribution measurement of the neutron beam. The electronic devices performing this selection are "the time of flight ana‐ lyzers." The analyzer has a number of channels in which the impulses are accumulated due to the neutrons arriving to the detector in the time intervals *t*0 + *τ*, *t*0 + 2*τ*, *t*0 + 3*τ,* and so on.

Here, *t*0 represents a "delay time" of the impulse temporally analyzer system start-up, which could be varied with the used experimental pattern particularities, and *τ* is the channel temporal width, a quantity that can be varied in stages for a specific instrument. The meas‐ urements by the time of flight method can be made using both steady-state reactors and pulsed

In the case of steady-state reactors, the neutron beam leaving the reactor channel meets an obturator in its way which opens the way only for very short time intervals. In this manner, the detector "notices" a pulsed neutron beam with a determined time length, for a specific obturator–detector distance, by the neutron energetic spectrum. The detector is connected with a time of flight analyzer, which can be unleashed by the obturator opening. To the pulsed reactors, the obturator function is undertaken by the neutron source itself. The time of flight analyzer start-up command is given by the neutron source when the pulse is accomplished. In every SANS experiment, the main goal is to determine with a good precision and in a reasonably short time the *Q* dependence of the analyzed sample coherent elastic scattering differential cross section in a large as possible *Q* variation range. From the point of view of the *Q* range explored value extension, the two SANS investigation methods indicate the following

max max max min

and respective *Q Q Q Q*

l

q

q

min min const. min max const.

é ù é ù = = ê ú ê ú ë û ë û

For any *Q*max – *Q*min, the interest interval for a certain sample, the first method seems to be more reasonable because the *Q* limitation apparently is dictated only by the measurement condi‐ tions, but in the case of the second method, the *λ* limitation is dictated by the spectral charac‐ teristics of the neutron source. If the neutron source is in steady state, then the first method utilization is certainly more advantageous because pulsing the neutron beam with a chopper leads to a usage of 10−4–10−3 from the produced neutrons even if we consider the advantage of

= =

q

l

 l

l= =

to cover an *L* distance, the time *t* = *L*/*v* will be necessary.

reactors as neutron sources.

58 Neutron Scattering

possible performances:

### **4.1. The method** *I***(***Q***) at constant** *λ*

As it was discussed in Section 3, from among the optical patterns, the one with two crystals and the sample situated between them is much more appreciated. This pattern was used in many reactors, and a typical description of it is given in Fig. 9.

**Figure 9.** Three-axe spectrometer.

The monochromator group built by *C*0 and *C*1 collimators, and the monocrystal *M* sets up a monochromatic beam falling on the studied sample *S*. The analyzing group formed by the monocrystal *A*, the collimator *C*2, and the detector *D* is initially tuned to detect the incident neutrons in the sample. To determine the neutron angular distribution produced by small angle scattering in the sample, we have to perform a combined scanning of the detector and the analyzer crystal around a vertical axis going through the center of the *A* crystal. Let us consider a small reactor providing a neutron flux of 1 × 1013 neutrons/cm2 s at 2 MW maximal power. From the white thermal neutron flux transmitted through the *C*0 collimator, the *M* monochromator selects a very narrow band centered around the *λ* = 1.15 Å, with a wavelength resolution of *Δλ*/*λ* = 4.5 × 10−3. This beam is transmitted further through the *C*1 collimator to the analyzed *S* sample and scattered at small angles (between 0 and 2 minutes). The *A* crystal collects the scattered beam and transmits it to the *D* detector through the *C*2 collimator.

We consider that both crystals are copper monocrystals identically cut, with the reflecting plane orientation in the (002) direction and a mosaic divergence structure of 14 minutes. The *C*0 and *C*<sup>1</sup> collimators have an angular horizontal divergence of 10 minutes, while that of the *C*2 collimator is only of 5 minutes. The 2*θ* monochromator angular range is 50° but the scattering range of monochromator and analyzer is maximum 120°. The detection system is an ordinary BF3 detector. Another type of neutron spectrometer is by using a cold source to allow neutron moderation at low temperatures, such as the VVR-SM reactor from Budapest Neutron Center (BNC). With an active core providing an average power of 10 MW, the neutron balance is improved by the mean of a beryllium reflector up to 1014 neutrons/cm2 s. The small angle scattering spectrometer described (Fig. 10) in the following is a conventional one. It uses a mechanical monochromator instead of a crystal monochromator. The mechanical selector with helical slits gives a wavelength resolution *Δλ*/*λ* between 5 and 20% for a *λ* lying between 3.5 and 8 Å.

The neutron beam is geometrically built up by a 3 m-long vacuum collimator. The maximal beam dimension in the sample is 2 cm diameter. The distance from the sample to detector can vary from 1 to 5 m. The spectrometer is provided with an automatic sample changer controlled by a computer. The scattering vector range lies between 5 × 10−3 Å−1 < *Q* < 2 Å−1. The scattered neutrons are detected by an XY positional detector with BF3 having the dimension 64 × 64 cm2 .

Measurement instrument control and experimental data collection are performed by a computer.

### **4.2. The method** *I***(***λ***) at constant** *θ*

The monochromator group built by *C*0 and *C*1 collimators, and the monocrystal *M* sets up a monochromatic beam falling on the studied sample *S*. The analyzing group formed by the monocrystal *A*, the collimator *C*2, and the detector *D* is initially tuned to detect the incident neutrons in the sample. To determine the neutron angular distribution produced by small angle scattering in the sample, we have to perform a combined scanning of the detector and the analyzer crystal around a vertical axis going through the center of the *A* crystal. Let us

power. From the white thermal neutron flux transmitted through the *C*0 collimator, the *M* monochromator selects a very narrow band centered around the *λ* = 1.15 Å, with a wavelength resolution of *Δλ*/*λ* = 4.5 × 10−3. This beam is transmitted further through the *C*1 collimator to the analyzed *S* sample and scattered at small angles (between 0 and 2 minutes). The *A* crystal collects the scattered beam and transmits it to the *D* detector through the *C*2 collimator.

We consider that both crystals are copper monocrystals identically cut, with the reflecting plane orientation in the (002) direction and a mosaic divergence structure of 14 minutes. The *C*0 and *C*<sup>1</sup> collimators have an angular horizontal divergence of 10 minutes, while that of the *C*2 collimator is only of 5 minutes. The 2*θ* monochromator angular range is 50° but the scattering range of monochromator and analyzer is maximum 120°. The detection system is an ordinary BF3 detector. Another type of neutron spectrometer is by using a cold source to allow neutron moderation at low temperatures, such as the VVR-SM reactor from Budapest Neutron Center (BNC). With an active core providing an average power of 10 MW, the neutron

angle scattering spectrometer described (Fig. 10) in the following is a conventional one. It uses a mechanical monochromator instead of a crystal monochromator. The mechanical selector with helical slits gives a wavelength resolution *Δλ*/*λ* between 5 and 20% for a *λ* lying between

The neutron beam is geometrically built up by a 3 m-long vacuum collimator. The maximal beam dimension in the sample is 2 cm diameter. The distance from the sample to detector can vary from 1 to 5 m. The spectrometer is provided with an automatic sample changer controlled by a computer. The scattering vector range lies between 5 × 10−3 Å−1 < *Q* < 2 Å−1. The scattered neutrons are detected by an XY positional detector with BF3 having the dimension 64 × 64 cm2

balance is improved by the mean of a beryllium reflector up to 1014 neutrons/cm2

3.5 and 8 Å.

60 Neutron Scattering

**Figure 10.** Spectrometer with mechanical monochromator.

s at 2 MW maximal

s. The small

.

consider a small reactor providing a neutron flux of 1 × 1013 neutrons/cm2

This method is used mostly for the neutron pulsed sources. This type of spectrometer built on the time of flight principle is installed to the IBR-2 pulsed nuclear reactor at Joint Institute for Nuclear Research (JINR) in Dubna, Russian Federation. The name of the spectrometer is YuMo [7], given from its constructor, the scientist Yu. M. Ostanevich (Fig. 11). The reactor fuel is plutonium oxide, and uses liquid sodium as a coolant. The reactor reflector is composed of a static part and a rotational one with constructive and movement characteristics that determine the issue of fast neutron pulsed flux. The produced neutrons are moderated in water and through the horizontal beam tubes that surround the reactor active core in the experimental hall.

In the current working conditions, the reactor provides a maximum flux of thermal neutrons of 1016 neutrons/cm2 s per pulse at the moderator surface. The frequency of pulse repetition is 5 Hz.

**Figure 11.** SANS spectrometer YuMo.1) two reflectors, 2) zone of reactor with moderator, 3) chopper, 4) first collimator (C1), 5) vacuum tube, 6) second collimator (C2), 7) thermostat, 8) samples table, 9) goniometer, 10)–11) vanadium standard, 12) ring-wire detector, 13) position-sensitive detector "Volga," 14) direct beam detector.

The chopper placed after the neutron moderator plays the main role to remove the neutrons from other reactor pulses and to assure that the neutrons from the interested pulse arrive at the detector. The neutron tube is made by steel pipes of different diameters and is kept under vacuum of about 10−2 Torr to avoid neutron absorption.

It is connected to the central vacuum system which is controlled from reactor control room and is provided with a mechanical beam stop that automatically closes the beam. A complete cycle of closing–opening the beam stop lasts about 6 minutes. The chopper is followed by a collimator with discrete variable aperture (C1) that assures a geometrical pre-shape of the neutron beam and by the monitor whose signals are used for the normalization of the data obtained from different reactor pulses. The next collimator (C2) assures the geometrical profile of the incident neutron beam in the sample and defines its dimensions. The collimator (C1) is composed of four cylinders of different diameters (100 mm, 80 mm, 60 mm, and 40 mm), but with the same length which can be remote-controlled to be aligned in the neutron beam by a rotational movement in accordance with the experimental requirements of the beam geomet‐ rical pre-shape. The collimator (C2) which limits the geometrical profile of the incident neutron beam in the sample has a remote-controlled variable opening (28 mm, 14 mm, and 7 mm). The holder with the collimation orifices of the (C2) collimator are made from boron polyethylene having an insignificant scattering cross section in comparison with the absorption cross section. All collimators and the most important shielding elements are covered with a mixture of polyester and boron carbonate or boron acid. The sample holder having six slots is connected to a special remote-controlled mechanism of sample changing that allows each sample to be introduced in the neutron beam according to the experimental program. The sample is placed at 18.74 m from the moderator surface. The sample holder is provided with a cooling and warming system that allows adjusting the sample temperature between 20 to 150°C.

To record and analyze the scattered neutrons in accordance with their time of flight, the experimental equipment is provided with a measurement system having two detectors. The detectors are functionally identical but different by dimensions that assure their efficiency in observing the *Q* values at small and large ranges. Therefore, both detectors are provided with eight active ring detectors; one detector having the central opening of 80 mm while the second of 200 mm having outside diameters of 730 mm and 560 mm, respectively. Each detector can be settled in one of the eight positions having the sample detector distance fixed in the interval of 4.5 –12.5 m. The active part of each detector is divided into eight independent ring detectors by means of a concentric copper partition (the cathodes); at the center of each ring exists a tungsten wire of 25 µm thickness (the anode). This type of assembly provides eight independ‐ ent ring detectors that offer more accuracy of the measurements by partial overlapping of the detection range of each ring with the range of the previous ring. The detector efficiency for neutrons with *λ* = 1.8Ǻ is about 70%.

The detector room is located after the samples and is made from a steel pipe of 1200 mm diameter and 12 m length having the inside surface covered with a cadmium layer of 0.5 mm thickness. Inside the room are mounted two rails for the movement of the electric-driven detectors trucks. The incident neutron beam is transmitted through the central hole of the detectors. At a distance of about 1.5 m from the reactor side, the detector is placed on a truckdriven mechanism in the vanadium standard (a vanadium metal sheet of about 0.3 mm thickness) that allows the introduction of the beam according to a program. This standard is used for the obtained experimental data calibration. The incident beam detector placed after the annular detectors measures the spectral distribution of the incident neutrons in the sample and forwards the normalization to this spectra of the scattered neutron spectra.

### **5. Conclusions**

The chopper placed after the neutron moderator plays the main role to remove the neutrons from other reactor pulses and to assure that the neutrons from the interested pulse arrive at the detector. The neutron tube is made by steel pipes of different diameters and is kept under

It is connected to the central vacuum system which is controlled from reactor control room and is provided with a mechanical beam stop that automatically closes the beam. A complete cycle of closing–opening the beam stop lasts about 6 minutes. The chopper is followed by a collimator with discrete variable aperture (C1) that assures a geometrical pre-shape of the neutron beam and by the monitor whose signals are used for the normalization of the data obtained from different reactor pulses. The next collimator (C2) assures the geometrical profile of the incident neutron beam in the sample and defines its dimensions. The collimator (C1) is composed of four cylinders of different diameters (100 mm, 80 mm, 60 mm, and 40 mm), but with the same length which can be remote-controlled to be aligned in the neutron beam by a rotational movement in accordance with the experimental requirements of the beam geomet‐ rical pre-shape. The collimator (C2) which limits the geometrical profile of the incident neutron beam in the sample has a remote-controlled variable opening (28 mm, 14 mm, and 7 mm). The holder with the collimation orifices of the (C2) collimator are made from boron polyethylene having an insignificant scattering cross section in comparison with the absorption cross section. All collimators and the most important shielding elements are covered with a mixture of polyester and boron carbonate or boron acid. The sample holder having six slots is connected to a special remote-controlled mechanism of sample changing that allows each sample to be introduced in the neutron beam according to the experimental program. The sample is placed at 18.74 m from the moderator surface. The sample holder is provided with a cooling and

warming system that allows adjusting the sample temperature between 20 to 150°C.

To record and analyze the scattered neutrons in accordance with their time of flight, the experimental equipment is provided with a measurement system having two detectors. The detectors are functionally identical but different by dimensions that assure their efficiency in observing the *Q* values at small and large ranges. Therefore, both detectors are provided with eight active ring detectors; one detector having the central opening of 80 mm while the second of 200 mm having outside diameters of 730 mm and 560 mm, respectively. Each detector can be settled in one of the eight positions having the sample detector distance fixed in the interval of 4.5 –12.5 m. The active part of each detector is divided into eight independent ring detectors by means of a concentric copper partition (the cathodes); at the center of each ring exists a tungsten wire of 25 µm thickness (the anode). This type of assembly provides eight independ‐ ent ring detectors that offer more accuracy of the measurements by partial overlapping of the detection range of each ring with the range of the previous ring. The detector efficiency for

The detector room is located after the samples and is made from a steel pipe of 1200 mm diameter and 12 m length having the inside surface covered with a cadmium layer of 0.5 mm thickness. Inside the room are mounted two rails for the movement of the electric-driven detectors trucks. The incident neutron beam is transmitted through the central hole of the detectors. At a distance of about 1.5 m from the reactor side, the detector is placed on a truck-

vacuum of about 10−2 Torr to avoid neutron absorption.

62 Neutron Scattering

neutrons with *λ* = 1.8Ǻ is about 70%.

To summarize, we can say that small angle scattering is the collective name given to the techniques of SANS and SAXS scattering. In each of these techniques, radiation is elastically scattered by a sample, and the resulting scattering pattern is analyzed to provide information on the size, shape, and orientation of some components of the sample. They offer the possibility to analyze particles without disturbing their natural environment. The type of sample that can be studied by SAS, the sample environment that can be applied, the actual length scales that can be probed, and the information that can ultimately be obtained all depend on the nature of the radiation used. For example, SAXS cannot be used to study thick samples or samples requiring complex containers, while SANS can penetrate deeper in the condensed matter. SANS is produced by heterogeneities in matter. If these are randomly oriented, every atom pair contributes to the scattering of a sample. Inhomogeneities of sizes larger than atomic distances (10–1000 Å) produce scattering patterns with *Q* ranges 1/*D* < *Q* < 10/*D*, if *D* is the dimension of the inhomogeneities.

A scattering experiment sees a scattering length density; in the case of X-rays, this is simply the electron density. The absolute density is not important but, in contrast, the difference between the particle and the surrounding medium.

The result of the experiment is the Fourier transformation of the contrast distribution. By comparing the experimental data with the theoretically calculated intensities or by a Fourier back transformation, we receive information about the contrast distribution, e.g., the mass density distribution of one particle. In the case of concentrated systems, we receive combined information about the single particle and the interaction between different particles.

In most cases, the sample and/or the sample environment are rather bulky.

Therefore, SANS instruments usually have to be large themselves in order to yield the desired resolution. Small SANS instruments can only serve a very limited number of applications. For reasons of intensity, a relatively large beam divergence, i.e., a beam cross section larger than the sample size is accepted as well as wavelength resolutions Δλ/λ of up to about 20%.

SANS and SAXS techniques are complementary; however, they share several similarities. Perhaps the most important of these is the fact that, with minor adjustments to account for the different types of radiation, the same basic equations and "laws" (for example, those due to Guinier and Porod) can be used to analyze data from any of the two techniques. This is a great advantage and one that has certainly eased the transition from one technique to another.

### **Author details**

Cristian A. Dragolici\*

Address all correspondence to: adrag@nipne.ro

Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH), Bucharest-Magurele, Romania

### **References**


## **Probing Giant Magnetism on Fe-N Thin Film by Polarized Neutron Reflectivity**

Xiaowei Zhang, Nian Ji, Valeria Lauter, Haile Ambaye and Jian-Ping Wang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62726

### **Abstract**

**Author details**

64 Neutron Scattering

**References**

Cristian A. Dragolici\*

Bucharest-Magurele, Romania

Sons Inc.; 1955. 268 p.

1817–1833.

Appl. Cryst. 1978; 11: 376–404.

Claredon Press; 1971. 599 p.

Academic Press; 1979. 523 p.

tron sources. J. Makromol. Chem. 1988; 15: 91–103.

Address all correspondence to: adrag@nipne.ro

Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH),

[1] Guinier A., Fournet G. Small-Angle Scattering of X-rays. New York: John Wiley &

[3] Gerold V., Kostorz G. Small-angle scattering applications to materials science. J.

[4] Allen A. J., Windsor C. G., Rainey V., Pearson D., Double D. D., Alford N. M. A small angle neutron scattering study of cement porosities. J. Phys. D: Appl. Phys. 1982; 15:

[5] Marshall W., Lovesey S. W. Theory of Thermal Neutron Scattering. 1st ed. Oxford:

[6] Kostorz G., editor. Treatise on Materials Science and Technology. 1st ed. New York:

[7] Ostanevich Y. M. Time-of-flight small-angle scattering spectrometers on pulsed neu‐

[2] Bacon G. E. Neutron Diffraction. 3rd ed. Oxford: Claredon Press; 1975. 636 p.

With the development in hard drive and permanent magnet industry, higher satu‐ ration magnetization (Ms) or magnetic induction (Bs) material is on high demand. According to the Slater-Pauling curve, the highest Bs value is ~2.45 T, which be‐ longs to FeCo alloy. However, in 1972, Kim and Takahashi [1] announced that the new material Fe-N thin film exhibited an increase in the Bs value of 18%. From then on, tons of research works [2, 3, 4] had been dedicated in this area with conclusions on both sides, achieving either high Bs or low Bs. Among those works, the Bs values were obtained from the measurement of the thin film magnetic moment and its vol‐ ume, which might cause a considerable amount of error depending on the accuracy of the film thickness and area measurement. Other concerns also include the uncer‐ tainty of Ms value due to the subtraction of Fe underlayer. Indeed, a direct meas‐ urement of Bs is the key to clarify the discrepancies between these results. Here, we are presenting the method of polarized neutron reflectometry (PNR) to measure the Bs of the partially ordered Fe16N2 thin film. PNR allows the interface magnetism study, [5] the *absolute* magnetization determination and magnetic depth profile in single thin films, [6, 7, 8, 9] and complicated superlattice structures. [10, 11] In the following scenarios, different partially ordered Fe16N2 thin films are fabricated and are measured using PNR subsequently. Their Bs values are obtained using the fit‐ ting results of nuclear scattering length density (NSLD) and magnetic scattering length density (MSLD). Also, a PNR application on FeN thin film will also help us to understand the switching picture of the FeN thin film with external applied field.

**Keywords:** Fe16N2, high saturation magnetization, polarized neutron reflectivity, thin film

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **1. Introduction**

### **1.1. PNR Introduction**

The interaction of neutrons with matter can be described in the optical formalism using the concept of reflective index for the medium. This can be obtained by Schrödinger equation, where the interaction of the neutron with medium is denoted by V:

$$\left[-\frac{\hbar^2}{2m}(\Delta + k\_0^2) + V\right] \Psi = 0\tag{1}$$

$$\frac{\hbar k\_0^2}{2m} = \alpha \tag{2}$$

where *m* is the neutron mass, *k0* is the wave vector in vacuum, *ћω* is its energy. So the refractive index can be written as:

$$m^2 = 1 - V / \hbar \alpha$$

Here, we can define the strong interaction potential with born approximation, which can be described by the Fermi pseudopotential: [12]

$$\left(V\right)\_F(r) = b \left(\frac{2\pi\hbar^2}{m}\right) \delta(r),\tag{4}$$

where *b* is the scattering length and *r* is neutron position. The mean potential of the interaction between neutron and medium is the given by the integration of the space:

$$V = \frac{1}{V} \int V(r)d^3r = \frac{2\pi\hbar^2}{m}Nb\_\prime \tag{5}$$

where *N* denotes the number density of the atoms. The multiplicity of number density and scattering length is called scattering length density (SLD) *ρ*.

### **2. Specular reflectivity of neutrons from interface**

If we compare the situation of neutron reflection to the light propagation at the interface of different mediums, the different interaction potential of neutron with medium can be seen as different mediums with different refraction indices. This follows Fresnel equations (Fig. 1).

**1. Introduction**

66 Neutron Scattering

**1.1. PNR Introduction**

index can be written as:

described by the Fermi pseudopotential: [12]

The interaction of neutrons with matter can be described in the optical formalism using the concept of reflective index for the medium. This can be obtained by Schrödinger equation,

(1)

(4)

2 <sup>0</sup> () 0 <sup>2</sup> *k V*

> 2 0 2 *k m* = w

<sup>2</sup> *n V* = -1 / h

<sup>2</sup> <sup>2</sup> ( ) ( ), *Vr b r <sup>F</sup> <sup>m</sup>* p

<sup>2</sup> 1 2 <sup>3</sup> *V V r d r Nb* ( ) , *V m*

between neutron and medium is the given by the integration of the space:

scattering length is called scattering length density (SLD) *ρ*.

**2. Specular reflectivity of neutrons from interface**

 æ ö <sup>=</sup> ç ÷ è ø h

where *m* is the neutron mass, *k0* is the wave vector in vacuum, *ћω* is its energy. So the refractive

Here, we can define the strong interaction potential with born approximation, which can be

where *b* is the scattering length and *r* is neutron position. The mean potential of the interaction

where *N* denotes the number density of the atoms. The multiplicity of number density and

If we compare the situation of neutron reflection to the light propagation at the interface of different mediums, the different interaction potential of neutron with medium can be seen as different mediums with different refraction indices. This follows Fresnel equations (Fig. 1).

w

d

p

<sup>=</sup> <sup>=</sup> ò <sup>h</sup> (5)

<sup>h</sup> (2)

(3)

é ù ê ú - D+ + Y= ë û

where the interaction of the neutron with medium is denoted by V:

2

h

*m*

**Figure 1.** A schematic illustration of neutron interacting with matters at the interface obeys the Fresnel equations in analogy to the optical formalism.

$$\begin{aligned} R &= \left| r \right|^2 = \left| \frac{k\_1 - k\_2}{k\_1 + k\_2} e^{i2k\_1 z} \right|^2 \\ T &= \left| t \right|^2 = \left| \frac{2k\_1}{k\_1 + k\_2} e^{i2(k\_1 - k\_2)z} \right|^2 \end{aligned} \tag{6}$$

For bulk materials, after applying Snell-Descartes' law, the reflection coefficient at a small incident angle can be written as a function of the wave-vector transfer *q*:

$$R(q) = \left| \frac{q\_z - \sqrt{q\_z^2 - q\_c^2}}{q\_z + \sqrt{q\_z^2 - q\_c^2}} \right|^2. \tag{7}$$

where *qc* is the wave vector, which corresponds to critical angle of total external reflection. It is also easy to show that at high *q* limit, the reflectivity *R* follows the *1/q*<sup>4</sup> power law (Fresnel decay).

Dynamic calculation can also be applied with Born approximation. The reflectivity is equal to the Fourier transformation of the SLD profile: [13]

$$R = \frac{1}{q\_z^2} \left| \int\_{-\alpha}^{\alpha} \mathbf{e}^{i q\_z y} \rho(y) \, \mathbf{d}y \right|^2. \tag{8}$$

It is worth mentioning that Fresnel decay originates from a planar surface and the decay will result in a extremely low intensity at high *q* value, which will be very difficult to collect with the mixture of background noise or incoherent scattering from the substrate.

The critical angle *θc* is usually small and can be written as follows at the total reflection condition:

$$
\cos \theta\_c = n = 1 - \frac{\lambda^2}{2\pi} N b\_\prime \tag{9}
$$

where *λ* is the wavelength. It is useful to use Taylor expansion. Therefore, the critical angle *θ<sup>c</sup>* and the corresponding critical wave vector *qc* are

$$
\theta\_c = \sqrt{\frac{Nb}{\pi}} \mathcal{X} \tag{10}
$$

$$q\_c = 4\sqrt{\pi N b}.\tag{11}$$

Usually the thickness of the thin film is larger than the wavelength of neutron, and so the reflecting beams of the neutron will give constructive/destructive patterns according to Bragg's diffraction law. Figure 2 is an example of a reflectivity from a bare bulk substrate and of a typical thin film:

**Figure 2.** Simulated neutron reflectivity curves of (a) a flat MgO substrate and (b) a flat 50-nm-thick Fe thin film grown on MgO substrate.

### **3. Neutron reflectivity from a magnetic interface**

In the presence of an external magnetic field, *B*, Zeeman energy will be present due to the spin of neutron itself:

$$\mathbf{V}\_{\mathbf{Z}} = -\mathbf{g}\_n \boldsymbol{\mu}\_n \mathbf{B}\_{\mathbf{z}} \tag{12}$$

where *gnμn* gives the neutron magnetic moment. Since *B* =*μ*0*H* + *M* , where *M* is the magneti‐ zation of the material, we can express the potential in terms of *M*. At the magnetic interface with the addition of this energy, the total interaction potential energy between the neutron and medium can be written as follows:

The critical angle *θc* is usually small and can be written as follows at the total reflection

cos 1 , <sup>2</sup> *<sup>c</sup> n Nb*

q

*c*

q

and the corresponding critical wave vector *qc* are

2

p

= = - (9)

<sup>=</sup> (10)

(11)

l

where *λ* is the wavelength. It is useful to use Taylor expansion. Therefore, the critical angle *θ<sup>c</sup>*

*Nb*

p

4 . *<sup>c</sup> q Nb* = p

 l

Usually the thickness of the thin film is larger than the wavelength of neutron, and so the reflecting beams of the neutron will give constructive/destructive patterns according to Bragg's diffraction law. Figure 2 is an example of a reflectivity from a bare bulk substrate and of a

**Figure 2.** Simulated neutron reflectivity curves of (a) a flat MgO substrate and (b) a flat 50-nm-thick Fe thin film grown

In the presence of an external magnetic field, *B*, Zeeman energy will be present due to the spin

**3. Neutron reflectivity from a magnetic interface**

condition:

68 Neutron Scattering

typical thin film:

on MgO substrate.

of neutron itself:

$$V = \frac{2\pi\hbar^2}{m}(\rho\_u \pm \rho\_w)\_\prime \tag{13}$$

where *ρ<sup>n</sup>* is the nuclear SLD of the medium and *ρ<sup>m</sup>* is the magnetic SLD of the medium, which can be defined by:

$$
\rho\_m = -\frac{m}{2\pi\hbar^2} \mu\_n \sum M.\tag{14}
$$

The sign ± represents spin-up and spin-down cases corresponding to the magnetization of the medium. In respect to neutron polarization, this shows that the magnetic SLD is directly related with the magnetization in the medium that the neutron is shining upon, which gives a direct measurement of the saturation magnetization when saturated in the external field.

Similarly, the critical angle *θc* and corresponding critical wave vector *qc* can be written as:

$$
\theta\_c = \sqrt{\frac{\rho\_u \pm \rho\_w}{\pi}} \lambda \tag{15}
$$

$$\eta\_c = 4\sqrt{\pi(\rho\_n \pm \rho\_w)}.\tag{16}$$

Therefore, spin-up reflectivity will have a large critical angle/momentum transfer compared to spin-down reflectivity (Fig. 3). If we present the potential in the matrix form with Pauli operator, then:

$$V = \frac{2\pi\hbar^2}{m} \begin{pmatrix} \rho\_u & 0\\ 0 & \rho\_u \end{pmatrix} \mp \mu\_n \begin{pmatrix} B\_z & B\_x - iB\_y\\ B\_x + iB\_y & -B\_z \end{pmatrix} = \frac{2\pi\hbar^2}{m} \begin{pmatrix} \rho\_u + \rho\_{uz} & \rho\_{uu} - i\rho\_{uy} \\ \rho\_{uu} - i\rho\_{uy} & \rho\_u - \rho\_{uz} \end{pmatrix}.\tag{17}$$

The potential we obtained above is the diagonal terms, which corresponds to the non-spin flip situation. The off diagonal terms [14]will represent the spin flip situation, which will not be discussed in this article.

**Figure 3.** Polarized neutron reflectivity with two different spin polarizations. The spin-up (µ+) and spin-down (µ–) neutrons are indexed as parallel and anti-parallel with respect to the external field direction.

**Figure 4.** (a) The simulated reflectivities of a 50-nm Fe thin film grown on MgO substrate after saturation in-plane with external field. (b) The depth-dependent SLD according to the reflectivity curves in (a).

As seen from the reflectivity curves (Fig. 4), the magnetic interaction splits the non-magnetic oscillation curve into two branches. Fitting the reflectivity will yield the SLD for each spin case:

$$
\begin{aligned}
\rho\_{\text{tot}\*} &= \rho\_u + \rho\_m \\
\rho\_{\text{tot}-} &= \rho\_u - \rho\_m.
\end{aligned}
\tag{18}
$$

We can solve for *ρm* and, therefore, retrieve the Ms of the sample.

### **4. Giant Bs induced by strain effect on epitaxial Fe16N2 thin film [15]**

Two sets of partially ordered Fe16N2 thin film samples are fabricated on MgO substrate with facing target sputtering system. After the deposition of Fe underlayer at 300°C, a Fe-N layer is grown subsequently on the top of Fe layer at room temperature with the optimum volume ratio of N2 and Ar mixture so that the stoichiometry in Fe-N layer is Fe/N=8:1. An in-situ annealing is carried out right after at 120°C for 20 hours in vacuum. To explore the in-plane tensile strain effect on the Bs value of Fe-N thin film, two samples are grown with different Fe underlayers, 2 nm and 20 nm, respectively.

**Figure 3.** Polarized neutron reflectivity with two different spin polarizations. The spin-up (µ+) and spin-down (µ–)

**Figure 4.** (a) The simulated reflectivities of a 50-nm Fe thin film grown on MgO substrate after saturation in-plane with

As seen from the reflectivity curves (Fig. 4), the magnetic interaction splits the non-magnetic oscillation curve into two branches. Fitting the reflectivity will yield the SLD for each spin case:

> *tot n m tot n m*

= +

 rr

 rr

**4. Giant Bs induced by strain effect on epitaxial Fe16N2 thin film [15]**

Two sets of partially ordered Fe16N2 thin film samples are fabricated on MgO substrate with facing target sputtering system. After the deposition of Fe underlayer at 300°C, a Fe-N layer

.

= - (18)

neutrons are indexed as parallel and anti-parallel with respect to the external field direction.

70 Neutron Scattering

external field. (b) The depth-dependent SLD according to the reflectivity curves in (a).

r

+ -

r

We can solve for *ρm* and, therefore, retrieve the Ms of the sample.

**Figure 5.** X-ray reflectivity characterization. (a) The fitted x-ray reflectivity curves measured on samples S1 and S2 (vertically offset by a factor of 10). (b) Calculated depth-dependent x-ray SLD profiles.

Both samples are investigated by PNR using the Magnetism Reflectometer at Spallation Neutron Source at Oak Ridge National Laboratory (SNS ORNL). [16] The reflectivities with the spin of the neutrons being either parallel (R+ ) or anti-parallel (R– ) to the applied magnetic field are collected simultaneously. From these data the depth profiles of the SLD of both NSLD and MSLD are obtained. The PNR experiments are performed at room temperature in the saturation external field of *H*=1.0 T applied in-plane of the sample. The R+ and R– reflectivity data are fitted simultaneously using a genetic algorithm with an exact recursive matrix calculation embedded in the *Simulreflec 1.0* package. [17] The experimental reflectivity and calculated curves with best chi-squared fit for samples S1 and S2 are shown on Fig. 6a and b, respectively. Their corresponding structural NSLD and magnetization depth profiles are plotted in Fig. 6c and d.

In data modeling process, the structural NSLD and layer choice were constrained to closely match X-ray Reflectivity (XRR) results (Fig. 5). To account for the possibility that the film possess homogeneous chemical composition but potentially different magnetization, the Fe-N layer was subdivided into three slabs where NSLD are forced to be identical but thickness, roughness, and MSLD were allowed to vary independently. When fitting the data to extract the physical depth profile of the film structure, the only free parameter that was allowed to vary was the magnetization as other parameters were already framed by the XRR result and only marginal improvement was possible. After the comparison of the XRR and PNR result, we can find a relatively large roughness at high Q range, which is most likely due to either not

**Figure 6.** Polarized neutron reflectivity characterization. (a) and (b) Experimental polarized neutron reflectivities to‐ gether with the fitted curves as functions of momentum transfer Q for samples S1 and S2 as labeled, respectively. The arrows at high Q region (>0.8nm–1) indicate the difference of the magnetic properties toward the bottom interface be‐ tween these two samples. (c) and (d) Structural (Brown) and magnetic (Green) depth profiles for samples S1 and S2 as labeled correspondingly.

enough data acquisition at this specific range or the chemical compound that is formed at the interface during the thin film growth or annealing process, such as MgO:N.

It is very obvious that an anomalously large magnetization is formed at the bottom of the film with a range of about 20 nm. The MSLD for this part of the film is 7.2~7.5 × 10–6Å–2, which in turn means the Ms value is 3.1~3.2 T. This value is significantly higher than bulk Fe (40~50%) and Fe65Co35 (20~30%). When the SLD profile extends to higher layer of the film, the MSLD values drops to 4.66 × 10–6Å–2 and represents a low Ms value of 2.01 T. This non-uniformity of the magnetization across the film cannot be replaced by single layer model, which will fail to resemble the reflectivity behavior of the experiment. For sample S2, the resulted MSLD is close to 5 × 10–6Å–2 for the Fe-N layer, corresponding to Ms of 2.15 T, which does not show the presence of giant Bs. The different Bs results of the two samples are due to the strain that is experienced by the epitaxial growth of the Fe-N layer. In sample S1, larger in-plane tensile strain contributes to high Bs.

Further justification for the different magnetic structure upon Fe buffer thickness change comes from the spin asymmetry (SA) (R+ –R– )/(R+ +R– ) plot shown in Fig. 7. SA of both samples were fitted with experiment data. Initially, the SA values of both samples are oscillating at a high value. However, the SA value of sample S2 drops rapidly close to zero, whereas that of sample S1 almost reaches unity. In this high scattering vector region (*q*>0.8nm–1), the interface of the bottom layer will dominate MSLD and NSLD. To account for the disparity of the different behaviors of two samples, |MSLD–NSLD| >> MSLD needs to be satisfied for sample S1. Since NSLD is similar for both samples, this observed feature directly proves the substantial enhancement of MSLD in the high Ms sample comparing to that of S2 at the bottom interface.

**Figure 7.** Spin asymmetry plotting of samples S1 (a) and S2 (b). It is clear as it approaches high q, the trend of the curves are different for these two samples.

### **5. High Bs obtained on partially ordered Fe16N2 multilayer thin film [16]**

enough data acquisition at this specific range or the chemical compound that is formed at the

**Figure 6.** Polarized neutron reflectivity characterization. (a) and (b) Experimental polarized neutron reflectivities to‐ gether with the fitted curves as functions of momentum transfer Q for samples S1 and S2 as labeled, respectively. The arrows at high Q region (>0.8nm–1) indicate the difference of the magnetic properties toward the bottom interface be‐ tween these two samples. (c) and (d) Structural (Brown) and magnetic (Green) depth profiles for samples S1 and S2 as

It is very obvious that an anomalously large magnetization is formed at the bottom of the film with a range of about 20 nm. The MSLD for this part of the film is 7.2~7.5 × 10–6Å–2, which in turn means the Ms value is 3.1~3.2 T. This value is significantly higher than bulk Fe (40~50%) and Fe65Co35 (20~30%). When the SLD profile extends to higher layer of the film, the MSLD values drops to 4.66 × 10–6Å–2 and represents a low Ms value of 2.01 T. This non-uniformity of the magnetization across the film cannot be replaced by single layer model, which will fail to resemble the reflectivity behavior of the experiment. For sample S2, the resulted MSLD is close to 5 × 10–6Å–2 for the Fe-N layer, corresponding to Ms of 2.15 T, which does not show the presence of giant Bs. The different Bs results of the two samples are due to the strain that is experienced by the epitaxial growth of the Fe-N layer. In sample S1, larger in-plane tensile

interface during the thin film growth or annealing process, such as MgO:N.

strain contributes to high Bs.

labeled correspondingly.

72 Neutron Scattering

Multilayer structures are very common in many areas that they can bring many new physical phenomena. To explore the possibility of applying Fe16N2 in multilayer structure, two samples are fabricated on GaAs substrate with Fe/Fe-N layer stack. Samples L1 and L3 are with 1 and 3 repetitions, respectively.

It is known that co-refining the x-ray and neutron reflectivity curves allows an unambiguous determination of magnetic depth profile. [19, 20] When modeling the chemical part of the film structure, consistent structure for fitting both samples is set the same for both PNR and XRR. In this case, only marginal adjustment is allowed during the fitting process. The two layer model for L1 and four layer model for L3 work well in both fitting, which illustrates the neutron scattering of Fe and N elements that accounts for the difference for each layer. It is also found out that the N concentration at the interface region is lower than that before annealing process.

PNR was conducted on these two samples, respectively, with field applied in-plane to saturate the magnetic moment. It is important to notice that these samples are different from the ones

**Figure 8.** (a) X-ray reflectivity of Fe16N2 samples L1 and L3. (b) Electron SLD of L3 was homogeneous through the whole sample. (c) SLD of L1 stayed the same after the first ~2 nm bump.

with optimum FeN atomic ratio (Fe/N=8/1), whose Ms values are homogeneous throughout the film according to PNR measurement. In Fig. 9, reflectivity curves showed different oscillatory periods due to the differences in film thickness. Green lines of both samples in Fig. 9 (b) and (d) indicate the giant Ms at the bottom of the film, with a thickness of 5 nm, respec‐ tively. Since all the other FeN phases possess low Ms values, the only option for this enhance‐ ment should come from Fe16N2. Considering the fact that Fe seed layer is deposited before FeN layer, N atoms diffusion is most likely to occur to form Fe16N2 at the bottom as N atoms are initially rich on the top of Fe underlayer. Fe16N2 is formed at this specific location near the substrate with the right stoichiometric amount of N atoms. On the top of this giant Ms layer, a N rich phase is expected since N atoms are initially rich and a lower Ms result is obtained by PNR. This transition of Ms from high to low also occurs on L3. A reduced but still "giant" magnetization resides at the bottom interface. Compared to sample L1, the transition from high Ms to low Ms is comparably slow. Considering the XRD analysis achieved above, α"- Fe16N2 and α'-Fe-N martensites are present in the film structure. Since high Ms behavior is obtained, these observation on PNR analysis can only be explained by the presence of high Ms Fe16N2 phase and N rich phase. The formation of these mixture phases is most likely due to the strain that the film experiences with thinner Fe underlayer on the substrate. Multilayer structure of Fe/FeN provides the same environment of single Fe/FeN layer, but the difference is that higher Fe/FeN layers, which are not next to substrate, do not exhibit giant Ms. This is a result of the lattice strain relaxation in the film norm direction. To better demonstrate the fitting of PNR, chemical structure extracted from NSLD was checked and the corresponding model also fitted XRR data. It is noticeable that the bottom part of samples L1 and L3 have different Probing Giant Magnetism in Fe-N Thin Film by Polarized Neutron Reflectometry http://dx.doi.org/10.5772/62726 75

**Figure 9.** (a) Experimental and fitted reflectivity of sample L1. (b) Sample L1 SLD and Ms in-depth profile after fitting. (c) Experimental and fitted reflectivity of sample L3. (d) Sample L3 SLD and Ms in-depth profile after fitting. Both (b) and (d) curves exhibit high Ms value at bottom part of the film for ~5 nm scale.

with optimum FeN atomic ratio (Fe/N=8/1), whose Ms values are homogeneous throughout the film according to PNR measurement. In Fig. 9, reflectivity curves showed different oscillatory periods due to the differences in film thickness. Green lines of both samples in Fig. 9 (b) and (d) indicate the giant Ms at the bottom of the film, with a thickness of 5 nm, respec‐ tively. Since all the other FeN phases possess low Ms values, the only option for this enhance‐ ment should come from Fe16N2. Considering the fact that Fe seed layer is deposited before FeN layer, N atoms diffusion is most likely to occur to form Fe16N2 at the bottom as N atoms are initially rich on the top of Fe underlayer. Fe16N2 is formed at this specific location near the substrate with the right stoichiometric amount of N atoms. On the top of this giant Ms layer, a N rich phase is expected since N atoms are initially rich and a lower Ms result is obtained by PNR. This transition of Ms from high to low also occurs on L3. A reduced but still "giant" magnetization resides at the bottom interface. Compared to sample L1, the transition from high Ms to low Ms is comparably slow. Considering the XRD analysis achieved above, α"- Fe16N2 and α'-Fe-N martensites are present in the film structure. Since high Ms behavior is obtained, these observation on PNR analysis can only be explained by the presence of high Ms Fe16N2 phase and N rich phase. The formation of these mixture phases is most likely due to the strain that the film experiences with thinner Fe underlayer on the substrate. Multilayer structure of Fe/FeN provides the same environment of single Fe/FeN layer, but the difference is that higher Fe/FeN layers, which are not next to substrate, do not exhibit giant Ms. This is a result of the lattice strain relaxation in the film norm direction. To better demonstrate the fitting of PNR, chemical structure extracted from NSLD was checked and the corresponding model also fitted XRR data. It is noticeable that the bottom part of samples L1 and L3 have different

**Figure 8.** (a) X-ray reflectivity of Fe16N2 samples L1 and L3. (b) Electron SLD of L3 was homogeneous through the

whole sample. (c) SLD of L1 stayed the same after the first ~2 nm bump.

74 Neutron Scattering

electron and NSLD from XRR and PNR result. This region is most likely Fe and GaAs com‐ pound due to the inter-diffusion of Fe and GaAs atoms at their interface. The addition of this layer actually improved our fitting to the experiment data and was confirmed by SLD depth profile.

### **6. Using PNR to visualize the high perpendicular crystalline anisotropy [19]**

The ordered Fe16N2 unit cell possesses tetragonality that gives rise to a high crystalline anisotropy, which is perpendicular to the film plane. This can be measured using vibrating sample magnetometer (VSM) in Fig. 10. However, VSM will not give detail information that will illustrate the FeN layer behavior in presence of the applied magnetic field.

During the PNR experiment, the external field is applied in the film plane direction. Both spin configuration signals are collected simultaneously. Fig. 11a–c show the reflectivity curves with fitting for spin-up and spin-down cases in three different external field intensities. At an external field of 20 kOe, the whole film magnetic moment will be saturated in the external field direction.

Since the same sample is tested in different fields, the chemical part, namely NSLD, stays the same during the fitting process. Three different MSLD depth profiles are generated after fitting in Fig. 11e. Since only one field intensity is strong enough to saturate the film in the film plane direction, the MSLD represent the in-plane component of the magnetization. It is clearly seen that the MSLD value drops rapidly for Fe-N layer, compared to Fe layer, when the external field is lowered. Since Fe layer is magnetically soft, all three fields will keep the Fe moment aligned in-plane. For Fe-N layer, there could be two scenarios to consider. First, the Fe-N film consists of antiferromagnetic domains in the film plane, which means the Fe-N layer has uniaxial in-plane anisotropy. This would directly contradict the fact that in-plane M-H loop does not depend on the in-plane orientation. Also, in this case, a strong off-specular signal should be observed, which is not our case either. Second, the magnetic moment turns out-ofplane due to perpendicular easy axis. This does explain our result since we do not observe off specular signal. When external field is lowered, the out-of-plane component of Fe-N film parallels the neutron beam momentum transfer and in-plane component contributes to the specular reflectivity. Therefore, the reduction of the magnetization in the in-plane direction after lowering external field can be explained by the out-of-plane easy axis of the Fe-N film.

Also, it is worth to mention that the oscillation amplitude of the reflectivity curves for R– curve is much smaller than that of R+ curve. This is most likely due to the modulation of the external field that the SLD is getting close to that of the substrate. This phenomenon is more robust when it approaches the Fe/GaAs interface where the reflectivity behavior is most dominated by this region at high qz.

**Figure 10.** In-plane (black) and out-of-plane (red) M-H loops measured on one partially ordered Fe16N2 sample. The Fe-N layer switches at Hc~5.7 kOe (black arrows).

**Figure 11.** (a)–(c) Fitted reflectivity curves with experiment data of Fe-N sample at external field of 5, 50, and 200 mT, respectively. (d) NSLD depth profile of Fe-N sample and (e) MSLD, the in-plane magnetization component, depth pro‐ file of Fe-N sample at above three different external fields.

### **7. Summary**

Since the same sample is tested in different fields, the chemical part, namely NSLD, stays the same during the fitting process. Three different MSLD depth profiles are generated after fitting in Fig. 11e. Since only one field intensity is strong enough to saturate the film in the film plane direction, the MSLD represent the in-plane component of the magnetization. It is clearly seen that the MSLD value drops rapidly for Fe-N layer, compared to Fe layer, when the external field is lowered. Since Fe layer is magnetically soft, all three fields will keep the Fe moment aligned in-plane. For Fe-N layer, there could be two scenarios to consider. First, the Fe-N film consists of antiferromagnetic domains in the film plane, which means the Fe-N layer has uniaxial in-plane anisotropy. This would directly contradict the fact that in-plane M-H loop does not depend on the in-plane orientation. Also, in this case, a strong off-specular signal should be observed, which is not our case either. Second, the magnetic moment turns out-ofplane due to perpendicular easy axis. This does explain our result since we do not observe off specular signal. When external field is lowered, the out-of-plane component of Fe-N film parallels the neutron beam momentum transfer and in-plane component contributes to the specular reflectivity. Therefore, the reduction of the magnetization in the in-plane direction after lowering external field can be explained by the out-of-plane easy axis of the Fe-N film.

Also, it is worth to mention that the oscillation amplitude of the reflectivity curves for R– curve is much smaller than that of R+ curve. This is most likely due to the modulation of the external field that the SLD is getting close to that of the substrate. This phenomenon is more robust when it approaches the Fe/GaAs interface where the reflectivity behavior is most dominated

**Figure 10.** In-plane (black) and out-of-plane (red) M-H loops measured on one partially ordered Fe16N2 sample. The Fe-

by this region at high qz.

76 Neutron Scattering

N layer switches at Hc~5.7 kOe (black arrows).

By using PNR, the Bs value can be accurately obtained directly in contrast to the conventional way that involves the thin film volume. In addition, we are also obtaining the Fe-N thin film NSLD and MSLD in-depth profile, which helps us to understand the physics behind the high Bs value. The above results are all derived from the specular reflections of the polarized neutron beam. This will facilitate the understanding of the giant Bs partially ordered Fe16N2 thin film in the film plane direction.

### **Acknowledgements**

This work was partially supported by ARPA-E (Advanced Research Projects Agency- Energy) project under Contract No. 0472-1595, Seagate Technology and Western Digital. Parts of this work were carried out in the Characterization Facility through NSF MRSEC program at University of Minnesota.

### **Author details**

Xiaowei Zhang1,2\*, Nian Ji1,2, Valeria Lauter3 , Haile Ambaye3 and Jian-Ping Wang1,2

\*Address all correspondence to: jpwang@umn.edu

1 Electrical and Computer Engineering Department and the Center for Micromagnetics and Information Technologies (MINT), University of Minnesota, Minneapolis, Minnesota, USA

2 Physics Department, University of Minnesota, Minneapolis, Minnesota, USA

3 Neutron Science Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA

### **References**


**Acknowledgements**

78 Neutron Scattering

University of Minnesota.

Xiaowei Zhang1,2\*, Nian Ji1,2, Valeria Lauter3

\*Address all correspondence to: jpwang@umn.edu

**Author details**

Tennessee, USA

**References**

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Phys. 70, 6427 (1991).

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This work was partially supported by ARPA-E (Advanced Research Projects Agency- Energy) project under Contract No. 0472-1595, Seagate Technology and Western Digital. Parts of this work were carried out in the Characterization Facility through NSF MRSEC program at

1 Electrical and Computer Engineering Department and the Center for Micromagnetics and Information Technologies (MINT), University of Minnesota, Minneapolis, Minnesota, USA

[2] Y. Sugita, H. Takahashi, M. Komuro, K. Mitsuoka and A. Sakuma, J. Appl. Phys. 76,

[3] M. A. Russac, C. V. Jahnes, E. Klokholm, J. Lee, M. E. Re, and B. C. Webb, J. Appl.

[4] J.-P. Wang, N. Ji, X. Liu, Y. Xu, C. Sánchez-Hanke, Y. Wu, F. M. F. de Groot, L. F. Al‐

[5] J. Chakhalian, J. W. Freeland, G. Srajer, J. Strempfer, G. Khaliullin, J. C. Cezar, T.

[7] M. A. M. Gijs, F. Petroff, Magnetic Ultra Thin Films, Multilayers and Surfaces Elsevi‐

[8] J. A. C. Bland, J. Lee, S. Hope, G. Lauhoff, J. Penfold, D. Bucknall, J. Magn. Magn.

[6] J. A. C. Bland, D. Pescia, and R. F. Willis, Phys. Rev. Lett. 58, 1244 (1987).

2 Physics Department, University of Minnesota, Minneapolis, Minnesota, USA

[1] T. K. Kim and Takahashi, Appl. Phys. Lett. 20, 492 (1972).

3 Neutron Science Scattering Division, Oak Ridge National Laboratory, Oak Ridge,

, Haile Ambaye3

and Jian-Ping Wang1,2


### **Thermophysical Properties and SANS Studies of Nanoemulsion Heat Transfer Fluids**

Bao Yang and Jiajun Xu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62313

### **Abstract**

Cooling is one of the most important technique challenges faced by a range of diverse in‐ dustries and military needs. There is an urgent need for innovative heat transfer fluids with improved thermal properties over currently available ones. This chapter discusses the development and characterization of nanoemulsion heat transfer fluids with phase changeable nanodroplets to increase the thermophysical properties and the heat transfer rate of the fluid. Nanoemulsion heat transfer fluids can be formed by dispersing one fluid into another immiscible fluid as nanosized structures such as droplets and tubes, in which those nanostructures are swollen reverse micelles with the dispersed phase and stabilized by the surfactant molecules. In addition to the enhancement of thermophysical properties such as thermal conductivity by mixing another liquid of higher thermal con‐ ductivity, an even larger amount of heat can be absorbed or released when these nano‐ droplets undergo phase transition from liquid to gas or vice versa, and thus enhancing the heat transfer rate. Three types of nanoemulsion heat transfer fluids are introduced: al‐ cohol-in-polyalphaolefin, water-in-FC-72, and water-in-polyalphaolefin. Structural and property characterizations of these nanoemulsion heat-transfer fluids are the two main aspects of this chapter. This chapter also identifies several critical issues in the nanoemul‐ sion heat transfer fluids to be solved in the future.

**Keywords:** Nanoemulsion, Thermophysical Property, Heat Transfer, Small Angle Neu‐ tron Scattering (SANS)

### **1. Introduction**

Cooling is one of the most important technical challenges faced by a range of diverse industries: microelectronics, optoelectronics, and, especially, power electronics [1–8]. This technology gap is the result of the higher currents, switching frequencies, and component densities of today's electronics and power electronics. The advances in semiconductor materials and more precise

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

fabrication techniques have the unfortunate side effect of generating higher amounts of waste heat within a smaller volume. Today, it is not unusual to see heat fluxes of 200 W/cm2 in a power module, a figure that is expected to increase over 1000 W/cm2 in the near future.

Thermal management of such high flux is becoming the bottleneck to improvements in electronics and power electronics. Existing cooling systems is striving to meet the everincreasing demand in higher computational power and smaller footprint. It is important that a cooling system with significantly improved heat transfer systems and their kernel compo‐ nents can be developed; in particular the cooling fluid used inside many heat transfer systems but yet has received little attention. The heat transfer fluids used in these heat transfer systems, including the coolants, lubricants, oils, and other fluids, limit the capacity and compactness of the heat exchangers that use these fluids due to their inherently poor heat transfer properties. The heat transfer capability of the heat exchangers can be easily amplified if fluids with better thermal properties are used. Therefore, development of innovative heat transfer fluids with improved thermal properties over those currently available is urgently needed.

The strategy of adding solid, highly conductive particles to improve thermal conductivity of fluids has been pursued since Maxwell's theoretical work was first published more than 100 years ago [9]. Early-stage studies have been confined to millimeter- or micrometer-sized solid particles dispersed in fluids. In the past decade, researchers have focused on suspensions of nanometer-sized solid particles, known as nanofluids [2, 8]. Many reviews and introductory reports on nanofluids have already been published [1, 2, 4, 10–14].

In this paper, it is intended to introduce some recent developments in another type of engi‐ neered heat transfer fluids, in which phase changeable nanodroplets are added to increase the thermophysical properties and heat transfer rate of the base fluids [15–22]. This chapter starts with the introduction of nanoemulsion fluids with potential application in thermal fluids. It is followed by the discussion on structural and thermophysical characterization techniques for nanoemulsion fluids [15–19, 21]. Then, three groups of nanoemulsion fluids and their prop‐ erties are discussed. This chapter is not intended to serve as a complete description of all nanoemulsion fluids available for heat transfer applications. The selection of the coverage was influenced by the research focus of the authors and reflects their assessment of the field.

### **2. Nanoemulsion heat transfer fluids**

One fluid is dispersed into another immiscible fluid as nanosized structures such as droplets and tubes to create a "nanoemulsion fluid." Those nanosized structures of the dispersed phase are micelles stabilized by the surfactant molecules on the outside. Nanoemulsion fluids are part of a broad class of multiphase colloidal dispersions [15–19, 21]. Different from the preparation of the nanofluids and emulsions [23–31], the nanoemulsion fluids are spontane‐ ously generated by self-assembly which does not require external shear force. Thus the nanoemulsion fluids are thermodynamically stable [16–19, 23, 32–44]. Table 1 is the compari‐ son between self-assembled nanoemulsion fluids and conventional emulsions. Nanoemulsion fluids made of specific fluids are suited for thermal management applications. Figure 1 shows a picture of ethanol-in-polyalphaolefin (PAO) nanoemulsion heat transfer fluids: both PAO and PAO-based nanoemulsion fluids are transparent but the nanoemulsion exhibits the Tyndall effect [15, 16, 19, 21, 45, 46].

**Figure 1.** Pictures of ethanol-in-PAO nanoemulsion fluids (Bottle A) and pure PAO (Bottle B). Liquids in both bottles are transparent. The Tyndall effect (i.e., a light beam can be seen when viewed from the side) can be observed only in Bottle A when a laser beam is passed through Bottles A and B.


**Table 1.** Comparison of nanoemulsion fluids and emulsions

fabrication techniques have the unfortunate side effect of generating higher amounts of waste heat within a smaller volume. Today, it is not unusual to see heat fluxes of 200 W/cm2

Thermal management of such high flux is becoming the bottleneck to improvements in electronics and power electronics. Existing cooling systems is striving to meet the everincreasing demand in higher computational power and smaller footprint. It is important that a cooling system with significantly improved heat transfer systems and their kernel compo‐ nents can be developed; in particular the cooling fluid used inside many heat transfer systems but yet has received little attention. The heat transfer fluids used in these heat transfer systems, including the coolants, lubricants, oils, and other fluids, limit the capacity and compactness of the heat exchangers that use these fluids due to their inherently poor heat transfer properties. The heat transfer capability of the heat exchangers can be easily amplified if fluids with better thermal properties are used. Therefore, development of innovative heat transfer fluids with

The strategy of adding solid, highly conductive particles to improve thermal conductivity of fluids has been pursued since Maxwell's theoretical work was first published more than 100 years ago [9]. Early-stage studies have been confined to millimeter- or micrometer-sized solid particles dispersed in fluids. In the past decade, researchers have focused on suspensions of nanometer-sized solid particles, known as nanofluids [2, 8]. Many reviews and introductory

In this paper, it is intended to introduce some recent developments in another type of engi‐ neered heat transfer fluids, in which phase changeable nanodroplets are added to increase the thermophysical properties and heat transfer rate of the base fluids [15–22]. This chapter starts with the introduction of nanoemulsion fluids with potential application in thermal fluids. It is followed by the discussion on structural and thermophysical characterization techniques for nanoemulsion fluids [15–19, 21]. Then, three groups of nanoemulsion fluids and their prop‐ erties are discussed. This chapter is not intended to serve as a complete description of all nanoemulsion fluids available for heat transfer applications. The selection of the coverage was influenced by the research focus of the authors and reflects their assessment of the field.

One fluid is dispersed into another immiscible fluid as nanosized structures such as droplets and tubes to create a "nanoemulsion fluid." Those nanosized structures of the dispersed phase are micelles stabilized by the surfactant molecules on the outside. Nanoemulsion fluids are part of a broad class of multiphase colloidal dispersions [15–19, 21]. Different from the preparation of the nanofluids and emulsions [23–31], the nanoemulsion fluids are spontane‐ ously generated by self-assembly which does not require external shear force. Thus the nanoemulsion fluids are thermodynamically stable [16–19, 23, 32–44]. Table 1 is the compari‐ son between self-assembled nanoemulsion fluids and conventional emulsions. Nanoemulsion fluids made of specific fluids are suited for thermal management applications. Figure 1 shows

power module, a figure that is expected to increase over 1000 W/cm2

82 Neutron Scattering

improved thermal properties over those currently available is urgently needed.

reports on nanofluids have already been published [1, 2, 4, 10–14].

**2. Nanoemulsion heat transfer fluids**

in a

in the near future.

### **2.1. Formation of self-assembled nanoemulsion fluids**

Self-assembled nanoemulsion fluids are thermodynamically stable, and the formation of these fluids can be explained using the classical thermodynamic theory [23, 34–39, 42, 43, 47–49]. The nanoemulsion fluid consists of one oil phase, one water phase, and certain surfactants. The adding of surfactant lowers the surface tension of the oil–water interface and the change in free energy of the system is given by Equation 1,

$$DG\_f = \gamma DA - T \cdot DS \tag{1}$$

where *DGf* is the free energy of formation, *γ* is the surface tension of the oil–waterinterface, *DA* is the change in interfacial area upon nanoemulsification, *DS* is the change in entropy of the system,and*T* is theabsolutetemperature.Thechangein*DA*isveryhighduetothelargenumber of nanosized droplets in the nanoemulsion fluids. However, the interfacial tension *γ* is very small (usually <<1 mN/m), and therefore the increase in the surface energy, *γ* DA, could be smaller than the entropy arising from the mixing. The favorable entropic contributions also arise fromdynamicprocesses suchas surfactantdiffusioninthe interfaciallayer andmonomer– micelle surfactant exchange. The entropy of mixing can be large enough to compensate for the positive interfacial free energy and to give the nanoemulsion system a free energy lower than that of the unmixed components. So the self-assembled nanoemulsion fluids can be formed through self-assembly and are in fact thermodynamically stable system [23, 43, 50]. There are mainlytwoapproachestopreparenanoemulsionfluids:thespontaneousemulsificationmethod (or the so-called phase titration method) and the phase-inversion method [23, 43].

Figure 2 shows the typical phase behavior diagram of a ternary system that contains two immiscible so-called oil and water phases and an amphiphilic surfactant component. The term "water" refers to a polar phase while "oil" is used for an apolar organic phase. When a system has a composition that lies in the shaded areas, a nanoemulsion fluid, either oil-in-water or water-in-oil, can be formed through self-assembly.

**Figure 2.** Schematic phase diagram of a water-oil-surfactant ternary system. The shaded areas are regions where selfassembled nanoemulsions are found.

### **2.2. Structure characterization methods**

Similar to other multiphase colloidal dispersions, the microstructure of nanoemulsion fluids is sensitive to many factors, including the different dispersed liquid, surfactants and base fluid, and molar ratio of dispersed liquid to surfactant [34]. In addition to that, temperature, pHvalue, and salinity also play an important role in the microstructure [23]. So accurate charac‐ terization of the microstructure of nanoemulsion fluids is important to understand the nanoemulsion fluids and yet challenging and costly to perform. In the past, small angle X-ray scattering (SAXS), small angle neutron scattering (SANS), dynamic or laser-light scattering (DLS), transmission electron microscopy (TEM), and nuclear magnetic resonance (NMR) have been used for the structure characterization [23]. In addition, the measurement of nanoemul‐ sion properties, such as viscosity, electric conductance, thermal conductivity, dielectric permittivity, electrophoretic birefringence, ultrasonic interferometry, and ultrasonic absorp‐ tion, can also provide information on the internal microstructure.

SANS allows the characterization of the structures inside the material on the nanometer (10-9 m) to micrometer (10-6 m) scale [51]. Many advanced engineering materials obtain unique performance because of their engineered structures on this length scale. For example, the toughness of high-impact plastics depends on the admixture of stiff and flexible segments of polymer molecules on the nano-to-micro scale, as well as, many biological processes in cells: from the storage of information on magnetic disks, to the hardness of steels and superalloys, to the conduction of current in superconductors, and many other materials properties.

Among all methods currently available for characterizing the microstructure of nanoemulsion fluids, small-angle neutron scattering (SANS) provides a unique approach to probe structure in liquids thanks to the distinctive penetrating power of neutron. Unlike the conventional dynamic light-scattering method using laser or X-rays, it can be applied to "concentrated" colloidal suspensions (e.g., >1 % volume fraction) and can penetrate through a container [51– 57]. Another advantage of SANS method is the deuteration method, in which deuterium labeled components in the sample in order to enhance their contrast that it can probe specific molecules or structure inside the sample with the deuteration technique. This unique method allows SANS to measure density fluctuations and composition (or concentration) fluctuations, which is very important to understand the structure inside nanoemulsion fluids [23].

### **2.3. Thermophysical properties characterization methods**

### *2.3.1. Thermal conductivity*

*DG DA T DS <sup>f</sup>* = -× g

(or the so-called phase titration method) and the phase-inversion method [23, 43].

water-in-oil, can be formed through self-assembly.

assembled nanoemulsions are found.

84 Neutron Scattering

**2.2. Structure characterization methods**

Figure 2 shows the typical phase behavior diagram of a ternary system that contains two immiscible so-called oil and water phases and an amphiphilic surfactant component. The term "water" refers to a polar phase while "oil" is used for an apolar organic phase. When a system has a composition that lies in the shaded areas, a nanoemulsion fluid, either oil-in-water or

**Figure 2.** Schematic phase diagram of a water-oil-surfactant ternary system. The shaded areas are regions where self-

Similar to other multiphase colloidal dispersions, the microstructure of nanoemulsion fluids is sensitive to many factors, including the different dispersed liquid, surfactants and base fluid, and molar ratio of dispersed liquid to surfactant [34]. In addition to that, temperature, pHvalue, and salinity also play an important role in the microstructure [23]. So accurate charac‐

where *DGf* is the free energy of formation, *γ* is the surface tension of the oil–waterinterface, *DA* is the change in interfacial area upon nanoemulsification, *DS* is the change in entropy of the system,and*T* is theabsolutetemperature.Thechangein*DA*isveryhighduetothelargenumber of nanosized droplets in the nanoemulsion fluids. However, the interfacial tension *γ* is very small (usually <<1 mN/m), and therefore the increase in the surface energy, *γ* DA, could be smaller than the entropy arising from the mixing. The favorable entropic contributions also arise fromdynamicprocesses suchas surfactantdiffusioninthe interfaciallayer andmonomer– micelle surfactant exchange. The entropy of mixing can be large enough to compensate for the positive interfacial free energy and to give the nanoemulsion system a free energy lower than that of the unmixed components. So the self-assembled nanoemulsion fluids can be formed through self-assembly and are in fact thermodynamically stable system [23, 43, 50]. There are mainlytwoapproachestopreparenanoemulsionfluids:thespontaneousemulsificationmethod

(1)

Low thermal conductivity is a primary limitation in the development of energy-efficient heat transfer fluids that are required in many industrial applications. Conventional heat transfer fluids have relatively poor thermal conductivity compared to metals [9]. It has been reported that the dispersed liquid nanodroplets could alter thermal conductivity of the base fluids [1, 2, 4, 58–61]. However, because of the absence of a theory for the thermal conductivity of nanoemulsion heat transfer fluids, an investigation of the effect of nanodroplets on the thermal conductivity will be conducted.

There are two widely used methods to measure the thermal conductivity of nanoemulsion fluids which includes (1) the transient hot-wire technique and (2) 3ω-wire method [62]. In the transient hot-wire method, thermal conductivity value is determined from the heating power and the slope of temperature change versus logarithmic time. The 3ω-wire method is used to measure the fluid thermal conductivity [19, 21, 62, 63]. This method is actually a combination of the transient hot-wire method and the 3ω-wire method, in which a metal wire is suspended to a liquid acting as both heater and thermometer. One advantage of this 3ω-wire method is that the temperature oscillation can be kept low enough: it is usually below 1 K as compared to about 5 K for the hot-wire method. It greatly helps to retain constant liquid properties of test liquid during measurement. Calibration experiments were performed for hydrocarbon (oil), fluorocarbon, and water at atmospheric pressure before each measurement.

### *2.3.2. Viscosity*

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. Viscosity is another macroscopically observable parameter that characterizes a nanoemulsion fluid, and it may range anywhere between a low viscous fluid and a gel state. It is an important quantity for many practical applications of nanoemulsion fluids, especially those used for hydraulic fluids. For instance, pumping such systems might be of interest in their application, and here viscosity plays an important role. Viscosity can be determined from the equation below:

$$F = \mu A \frac{\mu}{y} \tag{2}$$

where *F* is the stress force, *u* is the velocity, *A* is the area and *μ* is the proportionality factor called dynamic viscosity. The kinematic viscosity *ν* is related to the dynamic viscosity by dividing by the density of the fluid, *<sup>ν</sup>* <sup>=</sup> *<sup>μ</sup> ρ* .

Viscosity of a nanoemulsion fluid depends largely on its microstructure, that is, the type of aggregates that are present, on their interactions, and on the concentration of the system. So the viscosity can be used to monitor structural changes in the nanoemulsion system. In order to do so, one has to compare the experimental data to theoretical expressions that give the viscosity expected for certain model systems.

### *2.3.3. Specific heat measurement*

The specific heat is the amount of heat per unit mass required to raise the temperature by one degree Celsius. A differential scanning calorimeter (DSC) is usually used to measure a material's specific heat. In DSC measurement, it compares the differential heat flow (heat/time) between the measured material and the empty reference pan by adjusting the heat flux into a pan containing the sample with the heat flux into an empty pan while keeping both the measured sample and reference sample at nearly the same temperature. The difference in the amount of heat supplied to the sample and the reference is recorded as a function of temper‐ ature (or time), and the positive or negative peaks in the relationship correspond to exothermic or endothermic reactions in the sample, respectively. In order to determine the sample heat capacity, three measurements are usually carried out: for the sample, for the baseline, and for a standard. The baseline is subtracted from the sample measurement to obtain absolute values of the heat flow to the sample. The heat capacity is to be determined by the heat flow, the temperature rise, and the sample mass.

### **3. Nanoemulsion heat transfer fluids**

to a liquid acting as both heater and thermometer. One advantage of this 3ω-wire method is that the temperature oscillation can be kept low enough: it is usually below 1 K as compared to about 5 K for the hot-wire method. It greatly helps to retain constant liquid properties of test liquid during measurement. Calibration experiments were performed for hydrocarbon

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. Viscosity is another macroscopically observable parameter that characterizes a nanoemulsion fluid, and it may range anywhere between a low viscous fluid and a gel state. It is an important quantity for many practical applications of nanoemulsion fluids, especially those used for hydraulic fluids. For instance, pumping such systems might be of interest in their application, and here viscosity plays an important role. Viscosity can be determined from

> *<sup>u</sup> F A <sup>y</sup>* <sup>=</sup> m

> > *ρ* .

where *F* is the stress force, *u* is the velocity, *A* is the area and *μ* is the proportionality factor called dynamic viscosity. The kinematic viscosity *ν* is related to the dynamic viscosity by

Viscosity of a nanoemulsion fluid depends largely on its microstructure, that is, the type of aggregates that are present, on their interactions, and on the concentration of the system. So the viscosity can be used to monitor structural changes in the nanoemulsion system. In order to do so, one has to compare the experimental data to theoretical expressions that give the

The specific heat is the amount of heat per unit mass required to raise the temperature by one degree Celsius. A differential scanning calorimeter (DSC) is usually used to measure a material's specific heat. In DSC measurement, it compares the differential heat flow (heat/time) between the measured material and the empty reference pan by adjusting the heat flux into a pan containing the sample with the heat flux into an empty pan while keeping both the measured sample and reference sample at nearly the same temperature. The difference in the amount of heat supplied to the sample and the reference is recorded as a function of temper‐ ature (or time), and the positive or negative peaks in the relationship correspond to exothermic or endothermic reactions in the sample, respectively. In order to determine the sample heat capacity, three measurements are usually carried out: for the sample, for the baseline, and for a standard. The baseline is subtracted from the sample measurement to obtain absolute values of the heat flow to the sample. The heat capacity is to be determined by the heat flow, the

(2)

(oil), fluorocarbon, and water at atmospheric pressure before each measurement.

*2.3.2. Viscosity*

86 Neutron Scattering

the equation below:

dividing by the density of the fluid, *<sup>ν</sup>* <sup>=</sup> *<sup>μ</sup>*

viscosity expected for certain model systems.

temperature rise, and the sample mass.

*2.3.3. Specific heat measurement*

### **3.1. Ethanol-in-Polyalphaolefin (PAO) nanoemulsion fluids**

Polyalphaolefin (PAO) has been widely used as dielectric heat transfer fluids and lubricants due to its chemical stability within a wide temperature range. However, its thermal properties are relatively poor compared to other heat transfer fluids [64]. Ethanol-in-PAO is a liquid– vapor phase change nanoemulsion fluid, in which over 90 % consist of PAO to keep its chemical stability very close to pure PAO. Meanwhile, the ethanol nanodroplets could evaporate explosively and thus enhance the heat-transfer rate of the base fluid PAO [16]. The micro‐ structure and thermophysical properties of the ethanol-in-PAO nanoemulsion fluids are discussed below.

### *3.1.1. Microstructure of ethanol-in-PAO nanoemulsion fluids*

The microstructure of ethanol-in-PAO nanoemulsion fluids are on the NG-3 (30 m) beamline at the NIST Center for Neutron Research (NCNR) in Gaithersburg, MD. Samples are loaded into 2-mm quartz cells. Figure 3 shows the SANS data, the scattering intensity I versus the scattering vector q = 4π sin(θ/2)/λ, where λ is the wavelength of the incident neutrons, and θ is the scattering angle. The approximation q = 2πθ/λ is used for SANS (due to the small angle θ). The analysis of the SANS data suggests that the inner cores of the swollen micelles, that is, the ethanol droplets, are spherical and have a radius of about 0.8 nm for 9 vol. %.

**Figure 3.** Intensity I vs. Wave Vector q measured in ethanol-in-PAO nanoemulsion fluids.

### *3.1.2. Thermal conductivity of ethanol-in-PAO nanoemulsion fluids*

The relative thermal conductivity in ethanol-in-PAO nanoemulsion fluids measured using the 3ω-wire method along with the prediction by the Maxwell model (one model based on effective medium theory [EMT]) is shown in Figure 4. The relative thermal conductivity is defined as *k*eff/*k*o, where *k*o and *k*eff are thermal conductivities of the base fluid and nanoemulsion fluids, respectively. The effective medium theory reduces to Maxwell's equation for suspensions of well-dispersed, non-interacting spherical particles,

$$\frac{k\_{\rm eff}}{k\_{\rm o}} = \frac{k\_p + 2k\_{\rm o} + 2\phi \left(k\_p - k\_{\rm o}\right)}{k\_p + 2k\_{\rm o} - \phi \left(k\_p - k\_{\rm o}\right)}\tag{3}$$

where *ko* is the thermal conductivity of the base fluid, *k*p is the thermal conductivity of the particles, and ϕ is the particle volumetric fraction.

It can be seen in Figure 4 that the relative thermal conductivity of ethanol-in-PAO nanoemul‐ sion fluids is rather moderate (e.g. 2.3 % increase for 9 vol. % (*k*PAO = 0.143 W/mK and *k*alcohol = 0.171 W/mK [64, 65]), and no strong effects of Brownian motion on thermal transport are experimentally found in those fluids although the nanodroplets are extremely small, around 0.8 nm [16]. The thermal conductivity also appears to be linear with the loading of ethanol nanodroplets over the loading range from 0 to 9 vol. %. The Maxwell's equation, however, underestimates the viscosity increase in the ethanol-in-PAO nanoemulsion fluids, as can be seen in Figure 4. This is probably because Maxwell's equation does not take into account the shape transformation with different ethanol concentrations. It may suggest that the ethanol droplets form column-like with high aspect ratio of length to radius, which leads to a higher thermal conductivity enhancement in nanoemulsion fluids than the spherical droplets.

**Figure 4.** Relative thermal conductivity of ethanol-in-PAO nanoemulsions measured at different volume fractions of ethanol nanodroplets. The estimate from Maxwell's model is shown for comparison.

#### *3.1.3. Viscosity of ethanol-in-PAO nanoemulsion fluids*

Figure 5 shows the relative dynamic viscosity, *μ*eff/*μ*o, for the ethanol-in-PAO nanoemulsion fluids with varying alcohol loading, which is measured using a commercial viscometer (Brookfield DV-I Prime). The calibration is carried out using the pure PAO and its dynamic viscosity is found to be 7.3 cP which compares well with the literature value. Similar to thermal conductivity plotted in Figure 4, an approximately linear relationship is observed between the viscosity increase and the loading of alcohol nanodroplets in the range of 0–9 vol. %, but at a much larger percentage. For example, the measured viscosity increase is 31 % for 9 vol. %. ethanol loading, compared to a 2.3 % increase in thermal conductivity. The viscosity increase of dilute colloids can be predicted using the Einstein equation[12],

$$1\,\mu\_{\rm eff} \,\n\,\mu\_0 = 1\,\,+\,\text{2.5}\,\phi\tag{4}$$

where *μ*eff is the nanoemulsion viscosity, µ0 is the base fluid PAO viscosity and *φ* is the volumetric fraction of encapsulated ethanol nanodroplets. The viscosity increase in the ethanol-in-PAO nanoemulsion fluids is underestimated by the Einstein equation, as can be seen in Figure 5. This discrepancy is probably because the droplet volume fraction, ϕ, used in the viscosity calculation does not take into account the surfactant layer outside the alcohol core and Einstein equation used here is derived from dilute system.

**Figure 5.** Relative viscosities of ethanol-in-PAO nanoemulsions measured at different volume fractions of ethanol nanodroplets. The estimate from Einstein model is shown for comparison.

### **3.2. Water-in-FC-72 nanoemulsion fluids**

( ) ( )

(3)

o po eff o o po

++ - <sup>=</sup> +- -

*k k kk k k k k kk*

2

*p p*

particles, and ϕ is the particle volumetric fraction.

88 Neutron Scattering

2 2

f

f

where *ko* is the thermal conductivity of the base fluid, *k*p is the thermal conductivity of the

It can be seen in Figure 4 that the relative thermal conductivity of ethanol-in-PAO nanoemul‐ sion fluids is rather moderate (e.g. 2.3 % increase for 9 vol. % (*k*PAO = 0.143 W/mK and *k*alcohol = 0.171 W/mK [64, 65]), and no strong effects of Brownian motion on thermal transport are experimentally found in those fluids although the nanodroplets are extremely small, around 0.8 nm [16]. The thermal conductivity also appears to be linear with the loading of ethanol nanodroplets over the loading range from 0 to 9 vol. %. The Maxwell's equation, however, underestimates the viscosity increase in the ethanol-in-PAO nanoemulsion fluids, as can be seen in Figure 4. This is probably because Maxwell's equation does not take into account the shape transformation with different ethanol concentrations. It may suggest that the ethanol droplets form column-like with high aspect ratio of length to radius, which leads to a higher thermal conductivity enhancement in nanoemulsion fluids than the spherical droplets.

**Figure 4.** Relative thermal conductivity of ethanol-in-PAO nanoemulsions measured at different volume fractions of

Figure 5 shows the relative dynamic viscosity, *μ*eff/*μ*o, for the ethanol-in-PAO nanoemulsion fluids with varying alcohol loading, which is measured using a commercial viscometer (Brookfield DV-I Prime). The calibration is carried out using the pure PAO and its dynamic viscosity is found to be 7.3 cP which compares well with the literature value. Similar to thermal conductivity plotted in Figure 4, an approximately linear relationship is observed between the viscosity increase and the loading of alcohol nanodroplets in the range of 0–9 vol. %, but at a much larger percentage. For example, the measured viscosity increase is 31 % for 9 vol. %.

ethanol nanodroplets. The estimate from Maxwell's model is shown for comparison.

*3.1.3. Viscosity of ethanol-in-PAO nanoemulsion fluids*

Water-in-FC-72 nanoemulsion fluids are another group of nanoemulsion fluids designed for heat transfer purpose, in which water could undergo liquid–solid transition and thus increase heat transfer rate of the base fluid FC-72. FC-72 is one of the lines of Fluorinert™ Electronic Liquids developed by 3M™, which is used as the cooling fluids in liquid-cooled thermal management systems due to its low boiling point and excellent dielectric properties [66]. However, its heat transfer properties such as thermal conductivity and heat capacity are much inferior, compared to other fluids such as water.

### *3.2.1. Microstructure of water-in-FC-72 nanoemulsion fluids*

Water-in-FC-72 nanoemulsion fluids are generated by emulsifying deionized water into FC-72 with a small amount of perfluorinated amphiphiles. Figure 6 (a) shows the picture of the waterin-FC-72 nanoemulsion fluids and the pure FC-72. The autocorrelation function of the scattered light for the 12 vol. % water-in-FC-72 nanoemulsion fluids is plotted in Figure 6 (b), in which the curve shows a typical exponential decay of the correlation function versus time [19, 21]. The Brownian diffusivity and effective hydrodynamic radius of the nanodroplets are found to be 3.5×10-7 cm2 /s and 9.8 nm at *T* = 25o C, respectively.

*Correlation function of the scattered light for the 12 vol% Water‐in‐FC‐72 nanoemulsion fluids. Measurements taken by a Photocor‐Complex DLS instrument*  **Figure 6.** (a). Pictures of Water-in-FC-72 nanoemulsion fluids (Bottle A) and pure FC-72 (Bottle B). (b). Correlation function of the scattered light for the 12 vol.% Water-in-FC-72 nanoemulsion fluids. Measurements taken by a Photo‐ cor-Complex DLS instrument.

*Figure 6 (a) Pictures of Water‐in‐FC‐72 nanoemulsion fluids (Bottle A) and pure FC‐72 (Bottle B). (b)* 

### *3.2.2. Thermal conductivity of water-in-FC-72 nanoemulsion fluids*

Thermal conductivity of the water-in-FC-72 nanoemulsion is measured for different water loadings, and the results are shown in Figure 7. The 3ω -wire method is used to measure the fluid thermal conductivity. In water-in-FC-72 nanoemulsion fluids, the water phase has a thermal conductivity much higher than that of the base liquid FC-72. Water's thermal con‐ ductivity is 0.609 W/(mK) at 300 K and FC-72's thermal conductivity is much smaller, about 0.066 W/mK [66]. The addition of water is expected to improve the effective thermal conduc‐ tivity of FC-72.

A very large increase in thermal conductivity (up to 52 % for water-in-FC-72 nanoemulsion of 12 vol. % water) can be seen in Figure 7. The observed enhancement in thermal conductivity is much larger than that predicted by the EMT with assumption of spherical droplets [67]. This suggests that the water droplets are column-like with high aspect ratio of length to radius, which leads to a higher thermal conductivity enhancement in nanoemulsion fluids than the spherical droplets.

**Figure 7.** Relative thermal conductivity of water-in-FC-72 nanoemulsion fluids versus volume fraction of water. The estimate from EMT is shown for comparison.

### *3.2.3. Viscosity of water-in-FC-72 nanoemulsion fluids*

light for the 12 vol. % water-in-FC-72 nanoemulsion fluids is plotted in Figure 6 (b), in which the curve shows a typical exponential decay of the correlation function versus time [19, 21]. The Brownian diffusivity and effective hydrodynamic radius of the nanodroplets are found

C, respectively.

(a)

**Water-in-FC72 T=25<sup>o</sup> C**

(b)

**0**

*3.2.2. Thermal conductivity of water-in-FC-72 nanoemulsion fluids*

**0.1**

**0.2**

**Correlation Function**

**0.3**

**0.4**

*Figure 6 (a) Pictures of Water‐in‐FC‐72 nanoemulsion fluids (Bottle A) and pure FC‐72 (Bottle B). (b) Correlation function of the scattered light for the 12 vol% Water‐in‐FC‐72 nanoemulsion fluids. Measurements taken by a Photocor‐Complex DLS instrument* 

**Figure 6.** (a). Pictures of Water-in-FC-72 nanoemulsion fluids (Bottle A) and pure FC-72 (Bottle B). (b). Correlation function of the scattered light for the 12 vol.% Water-in-FC-72 nanoemulsion fluids. Measurements taken by a Photo‐

Thermal conductivity of the water-in-FC-72 nanoemulsion is measured for different water loadings, and the results are shown in Figure 7. The 3ω -wire method is used to measure the fluid thermal conductivity. In water-in-FC-72 nanoemulsion fluids, the water phase has a thermal conductivity much higher than that of the base liquid FC-72. Water's thermal con‐ ductivity is 0.609 W/(mK) at 300 K and FC-72's thermal conductivity is much smaller, about 0.066 W/mK [66]. The addition of water is expected to improve the effective thermal conduc‐

**10-7 10-6 10-5 0.0001 0.001 0.01**

**Delay Time (s)**

/s and 9.8 nm at *T* = 25o

to be 3.5×10-7 cm2

90 Neutron Scattering

cor-Complex DLS instrument.

tivity of FC-72.

The dynamic viscosity of water-in-FC-72 nanoemulsions of different nanodroplet concentra‐ tions is measured using a Brookfield viscometer at room temperature. The results have been normalized to the viscosity of pure FC-72 and are shown in Figure 8. The measured viscosity increase is nonlinear with the higher concentration of water added inside which agrees well with the nonlinear increase in thermal conductivity. This nonlinear increase in viscosity is common in colloidal systems, and has been interpreted by the aggregation of nanodroplets, that is, formation of column-like microstructure. Similar to the ethanol-in-PAO nanoemulsion fluids discussed previously, the Einstein equation significantly underpredicts the viscosity increase in the water-in-FC-72 nanoemulsion fluids at relatively high water loadings, as can be seen in Figure 8.

### *3.2.4. Effective heat capacity of water-in-FC-72 nanoemulsion fluids*

Another significant thermal property enhancement can be achieved here using the phase change of water nanodroplets formed inside the water-in-FC-72 nanoemulsion fluids. In water-in-FC-72 nanoemulsion fluids, the fluid's heat capacity can be increased by the high specific heat of water(the volumetric heat capacity of water is about 4.18 J/ml K, and is over two times the heat capacity of PAO (1.74 J/ml K) [65]) and/or the latent heat of water is the highest among common heat transfer fluids (Δ*H* =334 J / g), depending on the operating

**Figure 8.** Relative Viscosity of Water-in-FC-72 Nanoemulsions versus volume fractions of water nanodroplets. The es‐ timate from Einstein model is shown for comparison.

temperature of the fluids [66]. For example, the solid–liquid phase change of these droplets can increase the effective specific heat of the nanoemulsion fluid by a factor of 1 <sup>+</sup> *<sup>α</sup>* <sup>⋅</sup>*H*water Δ*T* ⋅*C*FC72 , where *α* is the water-volume fraction, *H*water is the latent heat of fusion of water per unit volume, and *∆T* is the temperature difference between the heat transfer surface and the bulk fluid.

The measured and calculated heat capacities of the water-in-FC-72 nanoemulsion fluids using a TA-CC100 DSC are shown in Figure 9. It can be seen that over 15 % increase in heat capacity can be achieved for a water volumetric fraction of 12 %, in which the measured *α* ⋅*H*water values of the water-in-FC-72 nanoemulsions for water loadings of 3, 6, 9, and 12 vol. % are 10.52, 15.44, 25.48, 39.78 J/ml, respectively. The effective heat capacity of FC-72 can be enhanced by more than 200 % for a 12 vol.% of water-in-FC-72 nanoemulsion, according to these experimental data. It agrees well with theoretical prediction using the model mentioned above.

The use of phase-changeable nanodroplets (e.g. water nanodroplets) provides another way to simultaneously increase the effective specific heat and thermal conductivity of conventional heat-transfer fluids.

### **3.3. Water-in-PAO nanoemulsion fluids**

### *3.3.1. Microstructure of water-in-PAO nanoemulsion fluids*

Figure 10 shows the SANS data for water-in-PAO nanoemulsion fluids with water volumetric concentration covering 1.8 vol. % to 10.3 vol.%.

It is clear from the SANS data that intensity curves of water-in-PAO nanoemulsion fluids gradually change the shape with increasing water loading. They can be further classified into three ranges (marked using three different colors): the 1.8–4.5 vol. % water-in-PAO nanoe‐ mulsion fluids with a smooth and gradually increasing scattering intensity for low q range

Thermophysical Properties and SANS Studies of Nanoemulsion Heat Transfer Fluids http://dx.doi.org/10.5772/62313 93

temperature of the fluids [66]. For example, the solid–liquid phase change of these droplets can increase the effective specific heat of the nanoemulsion fluid by a factor of 1 <sup>+</sup> *<sup>α</sup>* <sup>⋅</sup>*H*water

**Figure 8.** Relative Viscosity of Water-in-FC-72 Nanoemulsions versus volume fractions of water nanodroplets. The es‐

**0 2 4 6 8 10 12 14**

**Volmetric fraction of water (%)**

**Measured viscosity Einstein equation**

**Water-in-FC72 Nanoemulsions**

where *α* is the water-volume fraction, *H*water is the latent heat of fusion of water per unit volume, and *∆T* is the temperature difference between the heat transfer surface and the bulk fluid.

The measured and calculated heat capacities of the water-in-FC-72 nanoemulsion fluids using a TA-CC100 DSC are shown in Figure 9. It can be seen that over 15 % increase in heat capacity can be achieved for a water volumetric fraction of 12 %, in which the measured *α* ⋅*H*water values of the water-in-FC-72 nanoemulsions for water loadings of 3, 6, 9, and 12 vol. % are 10.52, 15.44, 25.48, 39.78 J/ml, respectively. The effective heat capacity of FC-72 can be enhanced by more than 200 % for a 12 vol.% of water-in-FC-72 nanoemulsion, according to these experimental

The use of phase-changeable nanodroplets (e.g. water nanodroplets) provides another way to simultaneously increase the effective specific heat and thermal conductivity of conventional

Figure 10 shows the SANS data for water-in-PAO nanoemulsion fluids with water volumetric

It is clear from the SANS data that intensity curves of water-in-PAO nanoemulsion fluids gradually change the shape with increasing water loading. They can be further classified into three ranges (marked using three different colors): the 1.8–4.5 vol. % water-in-PAO nanoe‐ mulsion fluids with a smooth and gradually increasing scattering intensity for low q range

data. It agrees well with theoretical prediction using the model mentioned above.

heat-transfer fluids.

**3.3. Water-in-PAO nanoemulsion fluids**

*3.3.1. Microstructure of water-in-PAO nanoemulsion fluids*

**1**

**1.5**

**2**

m**eff/**<sup>m</sup>

timate from Einstein model is shown for comparison.

92 Neutron Scattering

**2.5**

**3**

**3.5**

concentration covering 1.8 vol. % to 10.3 vol.%.

Δ*T* ⋅*C*FC72

,

fluids with different water loadings. (b) The measured and calculated heat capacity enhancement in the water‐in‐ FC‐72 nanoemulsion fluids when the dispersed water droplets undergo solid–liquid phase transition. **Figure 9.** (a) DSC curves and latent heat of fusion values of water of the water-in-FC-72 nanoemulsion fluids with dif‐ ferent water loadings. (b) The measured and calculated heat capacity enhancement in the water-in-FC-72 nanoemul‐ sion fluids when the dispersed water droplets undergo solid–liquid phase transition.

Figure 9. (a) DSC curves and latent heat of fusion values of water of the water‐in‐FC‐72 nanoemulsion

(less than 0.1 A˙ ); the 5.3–7.8 vol. % water-in-PAO nanoemulsion fluids with a sharper increase of q for the high q range (larger than 0.1 A˙ ) and a flatter intensity curve for low q (less than 0.03 A˙ ); and for even higher concentrations like 8.6 and 10.3 vol. % water-in-PAO nanoemulsion fluids tested here, the "hump" for high *q* is even more obvious and the intensity for low *q* increases more sharply which appears to have scattering curves made of three different sections. It indicates there is a dramatic microstructural change inside the water-in-PAO nanoemulsions of different water concentrations. Simple correlation length model cannot fit all the SANS curves well for both low and high *q* regions, as shown in Figure 11. The use of phase‐changeable nanodroplets (e.g. water nanodroplets) provides another way to simultaneously increase the effective specific heat and thermal conductivity of conventional heat‐ transfer fluids. *3.3 Water‐in‐PAO Nanoemulsion Fluids 3.3.1 Microstructure of water‐in‐PAO nanoemulsion fluids* 

The hard sphere model (one typical simple correlation length model) fits well for low water concentrations (i.e. 1.8 % to 4.5 % volume fractions), and nanodroplet radii are found to be 13.2

**Figure 10.** SANS curves for water-in-PAO nanoemulsion fluids for water volume concentration from 1.8 % to 10.3 %. Statistical error bars are smaller than the plotting symbols. 0.01 0.1 Wave Vector q(A-1)

3.6 vol%

7.8 vol%

3.6 vol% Hard Sphere Fit

100

**Figure 11.** Small-angle neutron scattering curves for water-in-PAO nanoemulsion fluids (solid symbols) and hard sphere model curve fittings (colored lines).

A˙ , 25.6 A˙ , and 96 A˙ for water loading 1.8 vol. %, 3.6 vol. %, and 4.5 vol. %, respectively. For higher water concentration (i.e. 7.8 vol. % to 10.3 vol.%), the hard-sphere model does not fit well especially for scattering variable q less than 0.1 A˙ region, which suggests that those nanodroplets are not simply spherical.

Thus, a more comprehensive fitting model must be used to take into account of the structure change inside the water-in-PAO nanoemulsion fluids. Here, the three-region Guinier–Porod empirical model is used to accommodate the structural changes inside the system by fitting curves that are shown in Figure 12 [51, 54, 56, 68].

Generally, the scattering intensity is given by two contributions in the Guinier–Porod model:

$$I(Q) = \frac{G\_{\circ}}{Q^{\circ\_{\circ}}} \exp(\frac{-Q^{\circ}R\_{\circ\_{\circ}}^{\circ}}{3 - s\_{\circ}}) \text{ for } Q \le Q\_{\circ}$$

$$I(Q) = \frac{G\_{\circ}}{Q^{\circ\_{\circ}}} \exp(\frac{-Q^{\circ}R\_{\circ\_{\circ}}^{\circ}}{3 - s\_{\circ}}) \text{ for } Q\_{\circ} \le Q \le Q\_{\circ} \tag{5}$$

$$I(Q) = \frac{D}{Q^{\circ}} \text{ for } Q \ge Q\_{\circ}$$

Here 3−s1 and 3−s2 are the dimensionality parameters, and *Rg*1 and *Rg*2 are the radii of gyration for the short and overall size of the scattering object. This generalized Guinier–Porod model can be used to analyze SANS patterns for nonspherical objects. In general, for scattering objects with spherical symmetry s1 =s2 =0 and for cylindrical objects s2 =0 and s1 =1. For lamellae with equal width and length, one has s2 =0 and s1 =2.

**Figure 12.** Small-angle neutron scattering curves for water-in-PAO nanoemulsion fluids (solid symbols) and three-re‐ gion Guinier–Porod model curve fittings (colored lines).

12

**0.17**

**0.14 0.15 0.16 0.0 3.0 6.0 9.0 Experimental Data Thermal Conductivity** *(W/mK)* **Water Volume Fraction (vol%)** 13 Based on the fitting curves using the three-region Guinier–Porod model, there are two dimensionality parameters s1 and s2, plus Rg2 and Rg1 are the radii of gyration for the short and overall sizes of the scattering object. The fitted curves give s2 =0.22 , s1 =1.4 and Rg2 =121A˙ , Rg1 =4.6A˙ for 10.3 vol. % water-in-PAO nanoemulsion fluids. For 7.8 % volume fraction water-in-PAO nanoemulsion fluid, s2 =0.18, s1 =0.97 and Rg2 =47.4A˙ , Rg1 =5.2A˙ . The dimensionality parameters suggest that those nanodroplets have a cylinder-like shape. So the SANS data confirm that there is an inner-structure variation with increasing water loading inside water-in-PAO nanoemulsion fluids. It is also noteworthy that the s2 =0.04 and s1 =0.25 is calculated for the 4.5 vol. % water-in-PAO nanoemulsion fluid and they can be approximated to s2 ≈0 and s1 ≈0 , which agrees well with the interpretation of being spherical shape based on Hard Sphere model fitting. Note that for cylindrical droplets Rg2 <sup>2</sup> <sup>=</sup> <sup>L</sup><sup>2</sup> <sup>12</sup> <sup>+</sup> <sup>R</sup><sup>2</sup> 2 , where L is the cylinder length and R is its radius and Rg1 <sup>2</sup> <sup>=</sup> <sup>R</sup><sup>2</sup> 2 .

A˙ , 25.6 A˙ , and 96 A˙ for water loading 1.8 vol. %, 3.6 vol. %, and 4.5 vol. %, respectively. For higher water concentration (i.e. 7.8 vol. % to 10.3 vol.%), the hard-sphere model does not fit well especially for scattering variable q less than 0.1 A˙ region, which suggests that those

**Figure 11.** Small-angle neutron scattering curves for water-in-PAO nanoemulsion fluids (solid symbols) and hard

Thus, a more comprehensive fitting model must be used to take into account of the structure change inside the water-in-PAO nanoemulsion fluids. Here, the three-region Guinier–Porod empirical model is used to accommodate the structural changes inside the system by fitting

Generally, the scattering intensity is given by two contributions in the Guinier–Porod model:

nanodroplets are not simply spherical.

sphere model curve fittings (colored lines).

curves that are shown in Figure 12 [51, 54, 56, 68].

1

1

1

10

1

10

100

Scattering Intensity I(cm-1)

Scattering Intensity I(cm-1)

100

Statistical error bars are smaller than the plotting symbols.

0.01 0.1

0.01 0.1

Wave vector q (A-1)

0.01 0.1

Wave vector q (A-1)

Wave Vector q(A-1)

3.6 vol%

**Figure 10.** SANS curves for water-in-PAO nanoemulsion fluids for water volume concentration from 1.8 % to 10.3 %.

0.01 0.1

7.8 vol%

3.6 vol% Hard Sphere Fit

Wave Vector q(A-1)

7.8 vol% Hard Sphere Fit 10.3 vol%

3.6 vol%

7.8 vol%

10.3 vol%

10.3 vol% Hard Sphere Fit

3.6 vol% Hard Sphere Fit

7.8 vol% Hard Sphere Fit

10.3 vol% Hard Sphere Fit

1.8 vol% 3.6 vol% 4.5 vol% 5.3 vol% 7.8 vol% 8.6 vol% 10.3 vol%

1.8 vol% 3.6 vol% 4.5 vol% 5.3 vol% 7.8 vol% 8.6 vol% 10.3 vol%

10

10

Intensity, I(Q) (cm-1)

94 Neutron Scattering

Intensity, I(Q) (cm-1)

100

100
