**Abstract**

The explicit form of the inelastic X-ray scattering, IXS, cross-section is derived within a time-dependent perturbative treatment of the scattering process. In this derivation, the double differential cross-section is obtained from the Fermi Golden Rule within a plane wave expansion of the vector potential. Furthermore, it is assumed throughout that the Thompson term of the perturbative Hamiltonian yields the overwhelming contribution to the scattering. The achievement of an explicit form for the double differential scattering cross-section rests on the validity of the adiabatic or Born-Oppenheimer approximation. As a result, it is here shown that that the IXS double differential cross-section is proportional to the spectrum of density fluctuations of the sample, which is thus the sample variable directly accessed by IXS measurements. Although the whole treatment is valid for monatomic systems only, under suitable approximations, it can be extended to molecular systems.

**Keywords:** inelastic X-ray scattering, theory of the scattering, theory of the line-shape, double differential scattering cross-section

#### **1. Introduction**

Inelastic scattering measurements are among the most powerful tools to investigate the collective terahertz dynamics of disordered systems [1, 2]. Although this subject has been the focus of intense scrutiny in the past few decades, it still presents many challenging aspects. In a spectroscopic measurement, the dynamic response of the target system is stimulated via the exchange of an energy ℏω and momentum ℏ*Q* where ℏ is the reduced Planck's constant. A suitable choice of the exchanged wavevector amplitude *Q* = |*Q*| and ω enables to tune the probe to dynamic events occurring over different scales. For infinitesimal *Q* and *ω* values, the measurement probes slowly decaying, hydrodynamic, density fluctuation modes either propagating or diffusing throughout the system, which resembles a continuous and homogeneous medium [3]. Upon increasing *Q's* and *ω's*, probed dynamic events become gradually faster and involve fewer atoms until the extreme, single-particle limit is reached. In this limit, the probe couples with the free recoil of the single atom after the collision with the photon and before any interaction with the first neighboring atomic cage [4].

initial 'plane wave'state and reradiate them into a spherical wave, which is

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations…*

in bold.

the sample.<sup>1</sup>

1

**5**

**Figure 1.**

using simple trigonometry.

to describe the scattering problem.

*DOI: http://dx.doi.org/10.5772/intechopen.93086*

consistent with well-known Huygens-Fresnel principle [10]. However, it is always safe to assume that the detector—which intercepts photons deviated by an angle <sup>2</sup>*θ*—has sensitive area *<sup>A</sup>* <sup>¼</sup> *<sup>r</sup>*2Δ*<sup>Ω</sup>* small enough to safely approximate the spherical wave impinging on it as a plane wave*.* As a consequence, the scattering event probed in a real experiment can be portrayed as a transition of the photon states between two different plane waves, as schematically shown in **Figure 2**. These are characterized by well-defined wavevector *k<sup>i</sup>*,*<sup>f</sup>* energy, ℏ*ωi*,*<sup>f</sup>* and polarization ^*εi*,*<sup>f</sup>* , with the indices '*i*' and '*f*' labelling the initial and final values, that is, the values before and after the scattering, respectively. Here all vector variables are indicated

As it appears from **Figure 1**, if one considers the plane defined by the two vectors *k<sup>i</sup>* and *k <sup>f</sup>* , only one angular coordinate, the scattering angle 2*θ*, is sufficient

To derive an expression of the intensity detected in high-resolution *IXS* measurements, it is useful to recognize that these measurements are typically executed in transmission geometry, that is, by detecting the scattering signal downstream of

For the sake of simplicity, a few more assumptions are here considered: (1) the sample has a straightforward shape: a slab of thickness *ts*; (2) such a slab is crossed by the incident beam orthogonally to its front area; (3) the beam cross-section *Σ<sup>B</sup>* is constant throughout the sample thickness, which implies that we are discarding the focusing of the incident beam; and (4) finally, for most *IXS* measurements, one can further assume that the beam only illuminates a limited portion of the whole crosssectional area of the sample. However, the detector has a sensitive area sufficiently

For simple sample shapes, the treatment can be easily extended to the case of finite scattering angle by

*A schematic rendering of the scattering process and the plane wave approximation (see text).*

Although the spectral profile is exactly known analytically in both hydrodynamic and single-particle limits, its evolution at the crossover in between them still eludes a firm understanding. Particularly insightful appears the study of the lineshape in the mesoscopic range, which corresponds to 2π/*Q* and 2π/ω values roughly matching nearest neighbor separations and 'in cage' rattling periods of atoms, respectively.

This range is the natural domain of high-resolution inelastic scattering, IXS [5], a spectroscopic method, which, since its development towards the end of the past millennium, has substantially improved the current understanding of the terahertz dynamics of condensed matter systems. This success partly owes to both inherent and practical advantages that this technique offers compared to the complementary terahertz spectroscopy, inelastic neutron scattering, INS. Intrinsic benefits include the virtual absence of kinematic limitations, the straightforward implementation of constant-*Q* energy scans, a mostly coherent cross-section and an often negligible multiple scattering contribution. More practical strengths are instead the substantially higher photon fluxes impinging on the sample and the smaller transversal size of the beam. However, these undoubted advantages can only be obtained at the cost of substantial count rate penalties. Indeed, the investigation of the collective dynamics in disordered systems imposes the access to energy transfer *E* =ℏ*ω* as low as a few meV. For IXS spectrometers, typically operated at 2.1 10<sup>4</sup> eV, resolving those energies imposes a resolving power of Δ*E/E* ≤ 10�<sup>7</sup> . The achievement of such a challenging performance has held back for long the development of high-resolution IXS, which was only made possible by the advent of high-brilliance third-generation synchrotron sources and by parallel advances in the X-ray optics [6, 7].

As an introduction to the field, this chapter is devoted to a derivation of the cross-section of IXS measurements, thus elucidating its direct connection with the Fourier transform of the atomic density fluctuations autocorrelation function. A similar treatment, which can also be found in Refs. [5, 8], is strictly valid for monatomic systems only, even though it can be easily generalized to the case of molecular systems.

#### **2. Generalities on an inelastic scattering measurement**

In a typical IXS measurement, a beam of particles–waves, as, for example, neutrons, X-rays or electrons, having well-defined energy, wavevector and polarization impinges on a sample and, after the impact, it is scattered all over the solid angle. A detector placed at a distance *r* from the sample is used to count the particles deviated by an angle 2θ within the small solid angle Δ*Ω* and intercepting its sensitive area *<sup>A</sup>* <sup>¼</sup> *<sup>r</sup>*<sup>2</sup>Δ*Ω*. Along the whole flight from the source to the detector, photons pass through optical elements filtering their energy both upstream and downstream of the sample, respectively referred to as monochromators and analysers. Other devices, such as collimators, mirrors, compound reflective lenses and so forth, are commonly used to shape the particle beam as required by experimental needs, and, specifically, they define its angular divergence and, whenever needed, its polarization.

At a long distance from the centre of the scattering, the electromagnetic wave generated by the scattering event is the sum of a plane and a spherical wave [9], that is, waves having respectively a planar and a spherical wavefront. In other terms, the ultimate effect of the scattering source is to remove a part of the photons from their

#### *High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations… DOI: http://dx.doi.org/10.5772/intechopen.93086*

initial 'plane wave'state and reradiate them into a spherical wave, which is consistent with well-known Huygens-Fresnel principle [10]. However, it is always safe to assume that the detector—which intercepts photons deviated by an angle <sup>2</sup>*θ*—has sensitive area *<sup>A</sup>* <sup>¼</sup> *<sup>r</sup>*2Δ*<sup>Ω</sup>* small enough to safely approximate the spherical wave impinging on it as a plane wave*.* As a consequence, the scattering event probed in a real experiment can be portrayed as a transition of the photon states between two different plane waves, as schematically shown in **Figure 2**. These are characterized by well-defined wavevector *k<sup>i</sup>*,*<sup>f</sup>* energy, ℏ*ωi*,*<sup>f</sup>* and polarization ^*εi*,*<sup>f</sup>* , with the indices '*i*' and '*f*' labelling the initial and final values, that is, the values before and after the scattering, respectively. Here all vector variables are indicated in bold.

As it appears from **Figure 1**, if one considers the plane defined by the two vectors *k<sup>i</sup>* and *k <sup>f</sup>* , only one angular coordinate, the scattering angle 2*θ*, is sufficient to describe the scattering problem.

To derive an expression of the intensity detected in high-resolution *IXS* measurements, it is useful to recognize that these measurements are typically executed in transmission geometry, that is, by detecting the scattering signal downstream of the sample.<sup>1</sup>

For the sake of simplicity, a few more assumptions are here considered: (1) the sample has a straightforward shape: a slab of thickness *ts*; (2) such a slab is crossed by the incident beam orthogonally to its front area; (3) the beam cross-section *Σ<sup>B</sup>* is constant throughout the sample thickness, which implies that we are discarding the focusing of the incident beam; and (4) finally, for most *IXS* measurements, one can further assume that the beam only illuminates a limited portion of the whole crosssectional area of the sample. However, the detector has a sensitive area sufficiently

<sup>1</sup> For simple sample shapes, the treatment can be easily extended to the case of finite scattering angle by using simple trigonometry.

the single atom after the collision with the photon and before any interaction with

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

Although the spectral profile is exactly known analytically in both hydrodynamic and single-particle limits, its evolution at the crossover in between them still eludes a firm understanding. Particularly insightful appears the study of the lineshape in the mesoscopic range, which corresponds to 2π/*Q* and 2π/ω values roughly matching nearest neighbor separations and 'in cage' rattling periods of atoms,

This range is the natural domain of high-resolution inelastic scattering, IXS [5], a spectroscopic method, which, since its development towards the end of the past millennium, has substantially improved the current understanding of the terahertz dynamics of condensed matter systems. This success partly owes to both inherent and practical advantages that this technique offers compared to the complementary terahertz spectroscopy, inelastic neutron scattering, INS. Intrinsic benefits include the virtual absence of kinematic limitations, the straightforward implementation of constant-*Q* energy scans, a mostly coherent cross-section and an often negligible multiple scattering contribution. More practical strengths are instead the substantially higher photon fluxes impinging on the sample and the smaller transversal size of the beam. However, these undoubted advantages can only be obtained at the cost of substantial count rate penalties. Indeed, the investigation of the collective dynamics in disordered systems imposes the access to energy transfer *E* =ℏ*ω* as low as a few meV. For IXS spectrometers, typically operated at 2.1 10<sup>4</sup> eV, resolving

. The achievement of

the first neighboring atomic cage [4].

those energies imposes a resolving power of Δ*E/E* ≤ 10�<sup>7</sup>

**2. Generalities on an inelastic scattering measurement**

such a challenging performance has held back for long the development of high-resolution IXS, which was only made possible by the advent of

high-brilliance third-generation synchrotron sources and by parallel advances in the

As an introduction to the field, this chapter is devoted to a derivation of the cross-section of IXS measurements, thus elucidating its direct connection with the Fourier transform of the atomic density fluctuations autocorrelation function. A similar treatment, which can also be found in Refs. [5, 8], is strictly valid for monatomic systems only, even though it can be easily generalized to the case of

In a typical IXS measurement, a beam of particles–waves, as, for example, neutrons, X-rays or electrons, having well-defined energy, wavevector and polarization impinges on a sample and, after the impact, it is scattered all over the solid angle. A detector placed at a distance *r* from the sample is used to count the particles

At a long distance from the centre of the scattering, the electromagnetic wave generated by the scattering event is the sum of a plane and a spherical wave [9], that is, waves having respectively a planar and a spherical wavefront. In other terms, the ultimate effect of the scattering source is to remove a part of the photons from their

deviated by an angle 2θ within the small solid angle Δ*Ω* and intercepting its sensitive area *<sup>A</sup>* <sup>¼</sup> *<sup>r</sup>*<sup>2</sup>Δ*Ω*. Along the whole flight from the source to the detector, photons pass through optical elements filtering their energy both upstream and downstream of the sample, respectively referred to as monochromators and analysers. Other devices, such as collimators, mirrors, compound reflective lenses and so forth, are commonly used to shape the particle beam as required by experimental needs, and, specifically, they define its angular divergence and, whenever

respectively.

X-ray optics [6, 7].

molecular systems.

needed, its polarization.

**4**

small, and a distance from the sample sufficiently large, that the scattered radiation impinging on it is schematizable as a plane wave, having wavevector *k <sup>f</sup>* and wavefront perpendicular to it.

Under these assumptions, we can write a general expression to estimate the number of photons per unit time impinging on the detector, which is given by:

$$I \propto \Phi n\_s \Sigma\_B t\_s \left(\frac{\partial^2 \sigma}{\partial \Omega \partial E\_f}\right) \Delta \Omega dE\_f,\tag{1}$$

**3. The interaction between impinging electromagnetic field and target**

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations…*

Given this preliminary discussion, the focus is now on the analytical derivation

where *r<sup>i</sup>* and *p<sup>i</sup>* are the position and the momentum of the *i*th electron, respectively, *V<sup>e</sup>*�*<sup>e</sup>* int is the electron–electron interaction potential averaged over the electron clouds of target atoms, while *V*(*ri*) is the potential acting on the *i*th electron. The

where the unperturbed Hamiltonian, associated with the multielectron system in

þ *V*ð Þ *r<sup>i</sup>* � �

þ<sup>X</sup> *i*

int <sup>þ</sup> *<sup>H</sup>*ð Þ<sup>2</sup>

requires, in the first place, a suitable expression for the Hamiltonian describing the interaction between the impinging photon beam and the electrons of the target sample. If one discards the relativistic nature of electron movements and neglects the usually weak contribution from the electron spin, such a Hamiltonian has the

*σ=dΩdEf* . An explicit analytical form

*<sup>V</sup>*ð Þþ *<sup>r</sup><sup>i</sup> <sup>V</sup>e*�*<sup>e</sup>* int , (5)

int, (6)

<sup>þ</sup> *<sup>V</sup><sup>e</sup>*�*<sup>e</sup>* int , (7)

� � (8)

*A r*ð Þ� *<sup>i</sup> A r*ð Þ*<sup>i</sup> :* (9)

**electrons**

following form [5]:

perturbation *H*ð Þ<sup>1</sup>

**7**

of the *IXS* double differential cross-section *d*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93086*

*<sup>H</sup>* <sup>¼</sup> <sup>1</sup> 2*Me*

the absence of the electromagnetic field, reads as:

impinging electromagnetic field, that is, respectively:

and the so-called Thomson scattering term:

X *i*

*<sup>p</sup><sup>i</sup>* � *<sup>e</sup> c A r*ð Þ*<sup>i</sup>* h i<sup>2</sup>

above Hamiltonian can be cast in the following perturbative form:

*<sup>H</sup>*el <sup>¼</sup> <sup>X</sup> *i*

> *H*ð Þ<sup>1</sup> int <sup>¼</sup> �*<sup>e</sup>* 2*Mec*

*H*ð Þ<sup>2</sup> int <sup>¼</sup> <sup>1</sup> 2 *r*0 X *i*

assumed to be entirely described by the Thomson term.

*<sup>H</sup>* <sup>¼</sup> *<sup>H</sup>*el <sup>þ</sup> *<sup>H</sup>*ð Þ<sup>1</sup>

*p*2 *i* 2*Me*

plus the other two terms accounting for the perturbation induced by the

X *i*

Here the symbol , f g denotes the anticommutator operator, while *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*<sup>=</sup> Mec*<sup>2</sup> ð Þ

As mentioned, in a typical scattering measurement, the X-ray photons undergo a transition between two different plane wave states. Therefore, one could, in principle, use the Fermi Golden Rule [11] to count all scattered photons emanating from a

absorption and emission, while two-photon processes, such as the scattering event, come into play to the second-order only. Conversely, the Thomson term (Eq. (9)), being quadratic in the vector potential, accounts to the first order for two photons interactions such as the scattering event. Away from an energy resonance, the latter term largely exceeds the second-order expansion of Eq. (8), thus providing an overwhelming contribution to the scattering process, which will be hereafter

int in Eq. (8) describes one-photon interactions with the sample as

is the classical electron radius expressed in cgs units. To its leading order, the

*A r*ð Þ*<sup>i</sup>* , *p<sup>i</sup>*

where *Φ* is the photon flux on the sample, defined as the number of photons impinging on the sample per unit time and unit area, while *ns* is the number of scattering units per unit volume, which is here assumed constant throughout the X-ray-illuminated sample.

The above formula introduces the double differential scattering cross-section:

$$\text{Rate of photons scattered into }d\Omega \text{ with}$$

$$\frac{d^2\sigma}{d\Omega \,\text{d}E\_f} = \frac{\text{final energy between }E\_f \text{ and } E\_f + dE\_f}{\Phi d\Omega \,\text{d}E\_f},\tag{2}$$

which is the only parameter of Eq. (1) conveying non-trivial information on the sample properties.

It can be recognized that the beam intensity across the sample thickness is not constant, as a part of it gets absorbed by the sample itself. This intensity reduction can be easily evaluated by expressing the attenuation caused by an elemental sample slice of thickness *dx* and located at a distance *x*. This intensity loss reads as:

$$dI = I(\mathbf{x} + d\mathbf{x}) - I(\mathbf{x}) = -I\mu d\mathbf{x},\tag{3}$$

where *μ* is the absorption coefficient at the energy of the incident beam. The integration of both members of the above equation leads to the conclusion that the intensity transmitted through the sample experiences an exponential decay. Assuming a forward scattering geometry, the attenuation factor can be simply obtained as *exp* ð Þ �*μts* and inserted in Eq. (1), thus obtaining:

$$I = \Phi \mathfrak{n}\_s \Sigma\_B \mathfrak{t}\_s \exp\left(-\mu \mathfrak{t}\_s\right) \left(\frac{\partial^2 \sigma}{\partial \Omega \partial E\_f}\right) \Delta \Omega dE\_{f.} \tag{4}$$

The above formula lends itself to a direct estimate of the ideal sample thickness, which is identified by the *<sup>∂</sup>I=∂ts* <sup>¼</sup> 0 condition, which yields *ts* <sup>¼</sup> <sup>1</sup>*=μ*. In summary, the optimal sample thickness should match the absorption length of the sample at the energy of the incident beam. For typical incident beam energies of most current *IXS* spectrometers, and for sample atomic species having electron number *Z* > 4, the extinction of the incident intensity is primarily caused by the photoelectric absorption process, which dominates over the Thomson scattering. The photoabsorption length typically decreases upon increasing *Z,* and this implies that *IXS* measurements on low Z materials require the use of relatively large samples, with a thickness in the *cm* range. However, this requirement becomes prohibitive for samples available in a small amount or that must be embedded in small volumes, as is typically the case of high-pressure experiments in Diamond Anvil Cells, *DACs*.

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations… DOI: http://dx.doi.org/10.5772/intechopen.93086*

### **3. The interaction between impinging electromagnetic field and target electrons**

Given this preliminary discussion, the focus is now on the analytical derivation of the *IXS* double differential cross-section *d*<sup>2</sup> *σ=dΩdEf* . An explicit analytical form requires, in the first place, a suitable expression for the Hamiltonian describing the interaction between the impinging photon beam and the electrons of the target sample. If one discards the relativistic nature of electron movements and neglects the usually weak contribution from the electron spin, such a Hamiltonian has the following form [5]:

$$H = \frac{1}{2M\_e} \sum\_{i} \left[ \mathbf{p}\_i - \frac{e}{c} \mathbf{A}(r\_i) \right]^2 + \sum\_{i} V(r\_i) + V\_{\text{int}}^{\epsilon - \epsilon},\tag{5}$$

where *r<sup>i</sup>* and *p<sup>i</sup>* are the position and the momentum of the *i*th electron, respectively, *V<sup>e</sup>*�*<sup>e</sup>* int is the electron–electron interaction potential averaged over the electron clouds of target atoms, while *V*(*ri*) is the potential acting on the *i*th electron. The above Hamiltonian can be cast in the following perturbative form:

$$H = H\_{\rm el} + H\_{\rm int}^{(1)} + H\_{\rm int}^{(2)},\tag{6}$$

where the unperturbed Hamiltonian, associated with the multielectron system in the absence of the electromagnetic field, reads as:

$$H\_{\rm el} = \sum\_{i} \left[ \frac{p\_i^2}{2\mathcal{M}\_{\epsilon}} + V(r\_i) \right] + V\_{\rm int}^{\epsilon - \epsilon},\tag{7}$$

plus the other two terms accounting for the perturbation induced by the impinging electromagnetic field, that is, respectively:

$$H\_{\rm int}^{(1)} = \frac{-e}{2M\_c\varepsilon} \sum\_{i} \{\mathcal{A}(r\_i), p\_i\} \tag{8}$$

and the so-called Thomson scattering term:

$$H\_{\rm int}^{(2)} = \frac{1}{2} r\_0 \sum\_i \mathbf{A}(r\_i) \cdot \mathbf{A}(r\_i). \tag{9}$$

Here the symbol , f g denotes the anticommutator operator, while *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*<sup>2</sup>*<sup>=</sup> Mec*<sup>2</sup> ð Þ is the classical electron radius expressed in cgs units. To its leading order, the perturbation *H*ð Þ<sup>1</sup> int in Eq. (8) describes one-photon interactions with the sample as absorption and emission, while two-photon processes, such as the scattering event, come into play to the second-order only. Conversely, the Thomson term (Eq. (9)), being quadratic in the vector potential, accounts to the first order for two photons interactions such as the scattering event. Away from an energy resonance, the latter term largely exceeds the second-order expansion of Eq. (8), thus providing an overwhelming contribution to the scattering process, which will be hereafter assumed to be entirely described by the Thomson term.

As mentioned, in a typical scattering measurement, the X-ray photons undergo a transition between two different plane wave states. Therefore, one could, in principle, use the Fermi Golden Rule [11] to count all scattered photons emanating from a

small, and a distance from the sample sufficiently large, that the scattered radiation

Under these assumptions, we can write a general expression to estimate the number of photons per unit time impinging on the detector, which is given by:

> *∂*2 *σ ∂Ω∂E <sup>f</sup>* !

where *Φ* is the photon flux on the sample, defined as the number of photons impinging on the sample per unit time and unit area, while *ns* is the number of scattering units per unit volume, which is here assumed constant throughout the

The above formula introduces the double differential scattering cross-section:

which is the only parameter of Eq. (1) conveying non-trivial information on the

It can be recognized that the beam intensity across the sample thickness is not constant, as a part of it gets absorbed by the sample itself. This intensity reduction can be easily evaluated by expressing the attenuation caused by an elemental sample

where *μ* is the absorption coefficient at the energy of the incident beam. The integration of both members of the above equation leads to the conclusion that the intensity transmitted through the sample experiences an exponential decay. Assuming a forward scattering geometry, the attenuation factor can be simply

> *∂*2 *σ ∂Ω∂E <sup>f</sup>* !

The above formula lends itself to a direct estimate of the ideal sample thickness, which is identified by the *<sup>∂</sup>I=∂ts* <sup>¼</sup> 0 condition, which yields *ts* <sup>¼</sup> <sup>1</sup>*=μ*. In summary, the optimal sample thickness should match the absorption length of the sample at the energy of the incident beam. For typical incident beam energies of most current *IXS* spectrometers, and for sample atomic species having electron number *Z* > 4, the extinction of the incident intensity is primarily caused by the photoelectric absorption process, which dominates over the Thomson scattering. The photoabsorption length typically decreases upon increasing *Z,* and this implies that *IXS* measurements on low Z materials require the use of relatively large samples, with a thickness in the *cm* range. However, this requirement becomes prohibitive for samples available in a small amount or that must be embedded in small volumes, as is typically the case of high-pressure experiments in Diamond Anvil

slice of thickness *dx* and located at a distance *x*. This intensity loss reads as:

obtained as *exp* ð Þ �*μts* and inserted in Eq. (1), thus obtaining:

*I* ¼ *ΦnsΣBts exp* ð Þ �*μts*

Rate of photons scattered into *dΩ* with

final energy between *Ef* and *Ef* þ *dEf ΦdΩ* d*Ef*

*dI* ¼ *I x*ð Þ� þ *dx I x*ð Þ¼�*Iμdx*, (3)

*ΔΩdEf* , (1)

, (2)

*ΔΩdE <sup>f</sup>:* (4)

impinging on it is schematizable as a plane wave, having wavevector *k <sup>f</sup>* and

*I* ∝ *ΦnsΣBts*

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

wavefront perpendicular to it.

X-ray-illuminated sample.

sample properties.

Cells, *DACs*.

**6**

*d*2 *σ dΩ* d*E <sup>f</sup>*

¼

single incident plane wave and having wavevector pointing to a 2*θ* direction to within a solid angle *ΔΩ*, thus deriving the double differential cross-section explicitly.

This strategy would require, in principle, a proper normalization of the photon wave functions, but, unfortunately, plane waves have normalization integral diverging for long distances. This difficulty is usually circumvented by confining the description of the scattering problem to a cubic box of size *L* and eventually considering the limit for large *L*. Within this *L*-sized cubic box, the vector potential becomes a linear combination of normalized plane waves which explicitly reads as [5]:

$$\mathbf{A}(\mathbf{r}) = \sum\_{k,a} \sqrt{\left(\frac{2\pi\hbar}{a\nu\_k L^3}\right)} c\hat{e}\_a \left[a\_{k,a} \exp\left(i\mathbf{k}\cdot\mathbf{r}\right) + a\_{k,a}^\dagger \exp\left(-i\mathbf{k}\cdot\mathbf{r}\right)\right].\tag{10}$$

Here the indexes '*k'* and 'α' label, respectively, the wavevector and the polarization states of the wave; *ak*,*<sup>α</sup>* and its Hermitian conjugate *a*† *<sup>k</sup>*,*<sup>α</sup>* are the annihilation and creation operators, respectively; *c* is the speed of light in vacuum and *ω<sup>k</sup>* is its angular frequency. Notice that the plus and minus signs in the phases of the exponential terms of Eq. (10) respectively define the upstream and downstream propagation of the photon plane wave.

Coming back to the double differential scattering cross-section, one can express it as:

$$\frac{d^2\sigma}{d\Omega dE\_f} = \frac{dP\_{i\to f}}{dt} \frac{1}{\Phi} \frac{d^2n}{d\Omega dE\_f},\tag{11}$$

**4. Counting the photon states**

*DOI: http://dx.doi.org/10.5772/intechopen.93086*

� � <sup>¼</sup> *<sup>d</sup>Ωk*<sup>2</sup>

where *nx*, *ny* and *nz* are generic integers.

can write *dV k <sup>f</sup>*

**Figure 2.**

**9**

*size Vmin* <sup>¼</sup> ð Þ <sup>2</sup>*π=<sup>L</sup>* <sup>3</sup> *(see text).*

It is worth noticing that the expedient of circumscribing the scattering within a *L*-sized cubic box, besides enabling a proper normalization of the plane waves, makes more straightforward the counting of the final state photon modes [11]. The

In the reciprocal space, the bandwidth *dEf* corresponds to the volume *dV k*ð Þ*<sup>F</sup>* of the spherical shell of infinitesimal thickness represented in **Figure 2**, for which one

The set of wavevectors defined in Eq. (15) identifies a lattice in *k-*space, whose

small enough—or, equivalently, if *L* is large enough—the number of lattice points

*The elemental volumes in the reciprocal space. Here the cube enclosing a lattice point represents the unit cell of*

*<sup>f</sup> dk <sup>f</sup>* . The wavevectors' components in the box *L*

*kx* ¼ ð Þ 2*π=L nx ky* ¼ ð Þ 2*π=L ny kz* ¼ ð Þ 2*π=L nz*, (15)

*dE <sup>f</sup>ΔΩ:* (14)

. If this volume is

� �*=V*min (**Figure 2**).

number of plane waves with energy included between *E <sup>f</sup>* and *Ef* þ *dEf* and

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations…*

*d*2 *n dΩdE <sup>f</sup>* !

pointing to a direction 2*θ* within a solid angle *ΔΩ* is given by:

representing the boundary of our scattering problem are:

simplest self-replicating unit cell has a volume *<sup>V</sup>*min <sup>¼</sup> ð Þ <sup>2</sup>*π=<sup>L</sup>* <sup>3</sup>

within the elemental volume is given by the ratio *dV k <sup>f</sup>*

where *dPi*!*<sup>f</sup> =dt* is the probability rate per sample and probe units that a photon experiences a transition between the initial and the final photon states, while the term *d*<sup>2</sup> *n=dΩdE <sup>f</sup>* represents the density of final photon states. The probability rate in Eq. (11) should be more appropriately written as a sum over all elementary excitations in the sample possibly coupling with the scattering event. Hence,

$$\frac{dP\_{i \to f}}{dt} = \sum\_{I,F} \frac{dP\_{I,i \to Ff}}{dt},\tag{12}$$

with *PI*,*i*!*F*,*<sup>f</sup>* denoting the probability of a transition j*I*, *i*⟩ ! j*F*, *f*⟩ between the combined states of the photon and the sample, labeled by lower case and capital fonts, respectively.

Eq. (12) is particularly useful as the term under summation can be derived explicitly using the Fermi Golden Rule, according to which:

$$\frac{dP\_{I,i \to Ff}}{dt} = \frac{2\pi}{\hbar} \left(\frac{d^2n}{d\Omega dE\_f}\right) \left| \langle F\_i f | H\_{int} | I, i \rangle \right|^2. \tag{13}$$

The last factor in the right-hand side of the above equation contains the perturbative part of the Hamiltonian computed between initial and final combined photon and sample states. As mentioned, we will assume that this term entirely coincides with the Thomson term in Eq. (9).

At this stage, the derivation of the double differential cross-section requires one to tackle the density of final states *d*<sup>2</sup> *n=dΩdE <sup>f</sup>* analytically, as discussed in the next paragraph.

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations… DOI: http://dx.doi.org/10.5772/intechopen.93086*

#### **4. Counting the photon states**

single incident plane wave and having wavevector pointing to a 2*θ* direction to within a solid angle *ΔΩ*, thus deriving the double differential cross-section

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

a linear combination of normalized plane waves which explicitly reads as [5]:

*<sup>c</sup>*^*εα ak*,*<sup>α</sup>* exp ð Þþ *<sup>i</sup><sup>k</sup>* � *<sup>r</sup> <sup>a</sup>*†

creation operators, respectively; *c* is the speed of light in vacuum and *ω<sup>k</sup>* is its angular frequency. Notice that the plus and minus signs in the phases of the exponential terms of Eq. (10) respectively define the upstream and downstream

Here the indexes '*k'* and 'α' label, respectively, the wavevector and the polariza-

Coming back to the double differential scattering cross-section, one can express

where *dPi*!*<sup>f</sup> =dt* is the probability rate per sample and probe units that a photon experiences a transition between the initial and the final photon states, while the

with *PI*,*i*!*F*,*<sup>f</sup>* denoting the probability of a transition j*I*, *i*⟩ ! j*F*, *f*⟩ between the combined states of the photon and the sample, labeled by lower case and capital

Eq. (12) is particularly useful as the term under summation can be derived

*d*2 *n dΩdE <sup>f</sup>* !

The last factor in the right-hand side of the above equation contains the perturbative part of the Hamiltonian computed between initial and final combined photon and sample states. As mentioned, we will assume that this term entirely

At this stage, the derivation of the double differential cross-section requires one

*n=dΩdE <sup>f</sup>* represents the density of final photon states. The probability rate

*dPI*,*i*!*F*,*<sup>f</sup>*

j⟨*F*, *f Hint* j j*I*, *i*⟩j

2

*n=dΩdE <sup>f</sup>* analytically, as discussed in the next

*:* (13)

1 *Φ*

*d*2 *n dΩdE <sup>f</sup>*

<sup>¼</sup> *dPi*!*<sup>f</sup> dt*

in Eq. (11) should be more appropriately written as a sum over all elementary excitations in the sample possibly coupling with the scattering event. Hence,

> *dPi*!*<sup>f</sup> dt* <sup>¼</sup> <sup>X</sup> *I*, *F*

explicitly using the Fermi Golden Rule, according to which:

*dPI*,*i*!*F*,*<sup>f</sup> dt* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>* ℏ

coincides with the Thomson term in Eq. (9).

to tackle the density of final states *d*<sup>2</sup>

h i

*<sup>k</sup>*,*<sup>α</sup>* exp ð Þ �*ik* � *r*

*:* (10)

*<sup>k</sup>*,*<sup>α</sup>* are the annihilation and

, (11)

*dt* , (12)

This strategy would require, in principle, a proper normalization of the photon wave functions, but, unfortunately, plane waves have normalization integral diverging for long distances. This difficulty is usually circumvented by confining the description of the scattering problem to a cubic box of size *L* and eventually considering the limit for large *L*. Within this *L*-sized cubic box, the vector potential becomes

explicitly.

it as:

term *d*<sup>2</sup>

fonts, respectively.

paragraph.

**8**

*A r*ð Þ¼ <sup>X</sup>

*k*, *α*

propagation of the photon plane wave.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*π*ℏ *ωkL*<sup>3</sup> s� �

tion states of the wave; *ak*,*<sup>α</sup>* and its Hermitian conjugate *a*†

*d*2 *σ dΩdEf*

It is worth noticing that the expedient of circumscribing the scattering within a *L*-sized cubic box, besides enabling a proper normalization of the plane waves, makes more straightforward the counting of the final state photon modes [11]. The number of plane waves with energy included between *E <sup>f</sup>* and *Ef* þ *dEf* and pointing to a direction 2*θ* within a solid angle *ΔΩ* is given by:

$$
\left(\frac{d^2n}{d\Omega dE\_f}\right)dE\_f\Delta\Omega.\tag{14}
$$

In the reciprocal space, the bandwidth *dEf* corresponds to the volume *dV k*ð Þ*<sup>F</sup>* of the spherical shell of infinitesimal thickness represented in **Figure 2**, for which one can write *dV k <sup>f</sup>* � � <sup>¼</sup> *<sup>d</sup>Ωk*<sup>2</sup> *<sup>f</sup> dk <sup>f</sup>* . The wavevectors' components in the box *L* representing the boundary of our scattering problem are:

$$k\_{\mathbf{x}} = (2\pi/L)n\_{\mathbf{x}}\ k\_{\mathbf{y}} = (2\pi/L)n\_{\mathbf{y}}\ k\_{\mathbf{z}} = (2\pi/L)n\_{\mathbf{z}},\tag{15}$$

where *nx*, *ny* and *nz* are generic integers.

The set of wavevectors defined in Eq. (15) identifies a lattice in *k-*space, whose simplest self-replicating unit cell has a volume *<sup>V</sup>*min <sup>¼</sup> ð Þ <sup>2</sup>*π=<sup>L</sup>* <sup>3</sup> . If this volume is small enough—or, equivalently, if *L* is large enough—the number of lattice points within the elemental volume is given by the ratio *dV k <sup>f</sup>* � �*=V*min (**Figure 2**).

#### **Figure 2.**

*The elemental volumes in the reciprocal space. Here the cube enclosing a lattice point represents the unit cell of size Vmin* <sup>¼</sup> ð Þ <sup>2</sup>*π=<sup>L</sup>* <sup>3</sup> *(see text).*

Therefore, one has

$$
\left(\frac{d^2n}{d\Omega dE\_f}\right)dE\_f\Delta\Omega = k\_f^2 \left(\frac{L}{2\pi}\right)^3 dk\_f d\Omega.\tag{16}
$$

latter can hardly be handled analytically, due to the complex interplay between electrons belonging to different atoms, which couples electronic and nuclear coordinates. However, it becomes treatable under the reasonable approximation that the centre of mass of the electronic cloud drifts following with no delay the slow nuclear motion. This assumption is customarily referred to as 'adiabatic', or Born-Oppenheimer, approximation [11]; its use justifies the factorization of the target system 'ket' as j*S*⟩ ¼ j*Sn*⟩ *Se* j ⟩, with nuclear and electronic states being labeled by the suffixes '*n*' and '*e*' respectively. The accuracy of this assumption ultimately owes to the substantially different nuclear and electronic masses and the correspondingly different timescales defining their dynamics. It holds validity when the energy exchange is smaller than all excitation energies of electrons in bound core states, which includes all cases of practical interest for this book. With *Se* j ⟩ being unaffected by the scattering process, the difference between the initial j*I*⟩ ¼ j*In*⟩ *Ie* j ⟩ and the final j*F*⟩ ¼ j*Fn*⟩ *Fe* j ⟩ states of the sample is uniquely due to excitations associated

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations…*

Within the validity of these assumptions, the double differential cross-section in

where *EIn* and *EFn* are the energies associated with the initial and final nuclear

X *Z*

*α*¼1

the centre of mass frame of the *j*th atom, while *Ie* j ⟩ coincides with the ground state of the electronic wave function of a given atomic nucleus. In practice, *f*ð Þ *Q* can be approximated by the value calculated for a free atom, that is, in the perfect gas phase, as the electronic cloud distribution is essentially unchanged upon phase transition. The primary contribution to this factor comes from core electrons whose

If a single atomic species is present in the sample, all atoms have the same form

*exp i*ð Þ *Q* � *R<sup>m</sup>* j*In*⟩

The above expression can be cast in a more compact form after the few additional manipulations as the use of an integral representation of the δ-function of energy, the Heisenberg representation of a time-dependent operator and the com-

In its initial state, the sample is usually a many-atoms system at equilibrium, and the sum over its initial state can be computed as an ordinary equilibrium average,

ð Þ� *Q f*ð Þ *Q* ; this further simplifies the expression of the double

� � � � �

2

orbits are more tightly bound to the much more massive atomic nucleus.

ð Þ *Q exp iQ* � *R<sup>j</sup>*

*exp i<sup>Q</sup>* � *<sup>r</sup> <sup>j</sup>*

*α*

� �j*In*⟩

� � � � �

2

� � *Ie* <sup>j</sup> ⟩ (23)

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

*δ* ℏ*ω* þ *EFn* � *EIn* ð Þ (24)

� *δ* ℏ*ω* þ *EFn* � *EIn* ð Þ,

(22)

with atomic density fluctuations.

*DOI: http://dx.doi.org/10.5772/intechopen.93086*

^*ε<sup>i</sup>* � ^*ε <sup>f</sup>* � �<sup>2</sup>X

*Fn*,*In*

*f j*

*PIn* ⟨*Fn*j

� � � � �

X *m*

j j *<sup>f</sup>*ð Þ *<sup>Q</sup>* <sup>2</sup> .

is the form factor of the *j*th atom. Here *r<sup>j</sup>*

ð Þ¼ *Q* ⟨*Fe*j

*PIn* ⟨*Fn*j

� � � � �

X *j f j*

Eq. (18) reduces to

¼ *r* 2 0 *k f ki*

states respectively, and

factor, that is, *f <sup>j</sup>*

differential cross-section

¼ *K* X *Fn*,*In*

> 0 *k f ki ε*´*<sup>i</sup>* � *ε*´ *<sup>f</sup>* � �<sup>2</sup>

pleteness of the final eigenstate.

which leads to the following identity:

*∂*2 *σ ∂Ω∂Ef*

where *<sup>K</sup>* <sup>¼</sup> *<sup>r</sup>*<sup>2</sup>

**11**

*∂*2 *σ ∂Ω∂E <sup>f</sup>*

For photons, the link between energy and wavevector is fixed by the linear law *E <sup>f</sup>* ¼ ℏ*ck <sup>f</sup>* , which can be differentiated to obtain *dE <sup>f</sup>* ¼ ℏ*cdk <sup>f</sup>* , thus eventually getting

$$
\left(\frac{d^2n}{d\Omega dE\_f}\right)dE\_f\Delta\Omega = \frac{k\_f^2}{\hbar c}\left(\frac{L}{2\pi}\right)^3dE\_fd\Omega.\tag{17}
$$

Therefore,

$$\frac{d^2n}{d\Omega dE\_f} = \frac{L^3}{8\pi^2} \frac{k\_f^2}{\hbar c},\tag{18}$$

which, combined with Eq. (11), yields

$$\frac{d^2\sigma}{d\Omega d E\_f} = \frac{L^3}{8\pi^2} \frac{k\_f^2}{\hbar c} \frac{dP\_{i \to f}}{dt} \,. \tag{19}$$

At this stage, the interaction term, that is, the squared matrix element appearing in the Fermi Golden Rule (Eq. (13)), can be made explicit by inserting in it the Thomson term in Eq. (9), while using the expression of the vector potential in Eq. (10), thus eventually obtaining

$$\sum\_{j,m} \langle F \vert \exp\left(-i\mathbf{Q} \cdot \mathbf{R}\_j\right) \vert I \rangle \langle I \vert \exp\left(i\mathbf{Q} \cdot \mathbf{R}\_m\right) \vert F \rangle,\tag{20}$$

where the vector *Rj* is the position of the *j*th atom. The above formula embodies the momentum conservation law as it was derived assuming the identity ℏ*Q* ¼ ℏ *k <sup>f</sup>* � *k<sup>i</sup>* � �. Furthermore, when using Eq. (10), it was considered that *<sup>ω</sup>*ð Þ*<sup>k</sup>* is equal to *cki* and *ckf* in the initial and the final photon states, respectively. Combining all analytical steps illustrated above, one eventually obtains the following expression for the double differential cross-section:

$$\frac{\partial^2 \sigma}{\partial \mathbf{Q} \partial \mathbf{E}\_f} = r\_0^2 \frac{k\_f}{k\_i} \left(\hat{\mathbf{e}}\_i \cdot \hat{\mathbf{e}}\_f\right)^2 \times \sum\_{\mathbf{F},I} \mathbf{P}\_I \left| \langle \mathbf{F} | \sum\_j \exp\left(i \mathbf{Q} \cdot \mathbf{R}\_j\right) | I \rangle \right|^2 \delta(\hbar \nu + E\_F - E\_I). \tag{21}$$

Here, ℏ*ω* is the energy gained by the photons in the scattering process, while the *δ-*function term accounts for the energy conservation in the scattering process, as it ensures that ℏ*ω* ¼ �ð Þ *EF* � *EI* with *EF* � *EI* being the energy gained by the sample. Notice that the cross-section defined above entails a sum over all states of the system, where the factor *PI* represents the statistical population of the initial states of the sample.

#### **5. From the adiabatic approximation to the dynamic structure factor**

The right-hand side of Eq. (21) contains three independent factors, the integral term being the only one directly relating to the properties of the target sample. The *High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations… DOI: http://dx.doi.org/10.5772/intechopen.93086*

latter can hardly be handled analytically, due to the complex interplay between electrons belonging to different atoms, which couples electronic and nuclear coordinates. However, it becomes treatable under the reasonable approximation that the centre of mass of the electronic cloud drifts following with no delay the slow nuclear motion. This assumption is customarily referred to as 'adiabatic', or Born-Oppenheimer, approximation [11]; its use justifies the factorization of the target system 'ket' as j*S*⟩ ¼ j*Sn*⟩ *Se* j ⟩, with nuclear and electronic states being labeled by the suffixes '*n*' and '*e*' respectively. The accuracy of this assumption ultimately owes to the substantially different nuclear and electronic masses and the correspondingly different timescales defining their dynamics. It holds validity when the energy exchange is smaller than all excitation energies of electrons in bound core states, which includes all cases of practical interest for this book. With *Se* j ⟩ being unaffected by the scattering process, the difference between the initial j*I*⟩ ¼ j*In*⟩ *Ie* j ⟩ and the final j*F*⟩ ¼ j*Fn*⟩ *Fe* j ⟩ states of the sample is uniquely due to excitations associated with atomic density fluctuations.

Within the validity of these assumptions, the double differential cross-section in Eq. (18) reduces to

$$\frac{\partial^2 \sigma}{\partial \mathbf{d} \partial \mathbf{E}\_f} = r\_0^2 \frac{k\_f}{k\_i} \left(\hat{\mathbf{e}}\_i \cdot \hat{\mathbf{e}}\_f\right)^2 \sum\_{\mathbf{F}\_n, I\_n} P\_{I\_n} \left| \langle \mathbf{F}\_n | \sum\_j f\_j(\mathbf{Q}) \exp\left(i \mathbf{Q} \cdot \mathbf{R}\_j\right) | I\_n\rangle \right|^2 \times \delta(\hbar \alpha + E\_{F\_n} - E\_{I\_n}), \tag{22}$$

where *EIn* and *EFn* are the energies associated with the initial and final nuclear states respectively, and

$$f\_j(\mathbf{Q}) = \langle F\_\epsilon | \sum\_{a=1}^{Z} \exp\left(i\mathbf{Q} \cdot r\_a^j\right) | I\_\epsilon \rangle \tag{23}$$

is the form factor of the *j*th atom. Here *r<sup>j</sup> <sup>α</sup>* is the coordinate of the *α*th electron in the centre of mass frame of the *j*th atom, while *Ie* j ⟩ coincides with the ground state of the electronic wave function of a given atomic nucleus. In practice, *f*ð Þ *Q* can be approximated by the value calculated for a free atom, that is, in the perfect gas phase, as the electronic cloud distribution is essentially unchanged upon phase transition. The primary contribution to this factor comes from core electrons whose orbits are more tightly bound to the much more massive atomic nucleus.

If a single atomic species is present in the sample, all atoms have the same form factor, that is, *f <sup>j</sup>* ð Þ� *Q f*ð Þ *Q* ; this further simplifies the expression of the double differential cross-section

$$\frac{\partial^2 \sigma}{\partial \Omega \partial E\_f} = K \sum\_{F\_n, I\_n} P\_{I\_n} \left| \langle F\_n | \sum\_m \exp \left( i \mathbf{Q} \cdot \mathbf{R}\_m \right) | I\_n \rangle \right|^2 \delta(\hbar \nu + E\_{F\_n} - E\_{I\_n}) \tag{24}$$

where *<sup>K</sup>* <sup>¼</sup> *<sup>r</sup>*<sup>2</sup> 0 *k f ki ε*´*<sup>i</sup>* � *ε*´ *<sup>f</sup>* � �<sup>2</sup> j j *<sup>f</sup>*ð Þ *<sup>Q</sup>* <sup>2</sup> .

The above expression can be cast in a more compact form after the few additional manipulations as the use of an integral representation of the δ-function of energy, the Heisenberg representation of a time-dependent operator and the completeness of the final eigenstate.

In its initial state, the sample is usually a many-atoms system at equilibrium, and the sum over its initial state can be computed as an ordinary equilibrium average, which leads to the following identity:

Therefore, one has

Therefore,

ℏ *k <sup>f</sup>* � *k<sup>i</sup>*

*∂*2 *σ ∂Ω∂Ef*

of the sample.

**10**

¼ *r* 2 0 *k f ki*

*d*2 *n dΩdE <sup>f</sup>* !

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

*d*2 *n dΩdE <sup>f</sup>* !

which, combined with Eq. (11), yields

Eq. (10), thus eventually obtaining

the double differential cross-section:

^*ε<sup>i</sup>* � ^*ε <sup>f</sup>* � �<sup>2</sup> �<sup>X</sup> *F*,*I*

*PI* ⟨*F*j

� � � � �

X *j*, *m*

*dE <sup>f</sup>ΔΩ* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

*dE <sup>f</sup>ΔΩ* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

*d*2 *n dΩdE <sup>f</sup>*

*d*2 *σ dΩdE <sup>f</sup>*

⟨*F*j*exp* �*iQ* � *R<sup>j</sup>*

For photons, the link between energy and wavevector is fixed by the linear law *E <sup>f</sup>* ¼ ℏ*ck <sup>f</sup>* , which can be differentiated to obtain *dE <sup>f</sup>* ¼ ℏ*cdk <sup>f</sup>* , thus eventually getting

*f L* 2*π* � �<sup>3</sup>

*f* ℏ*c*

<sup>¼</sup> *<sup>L</sup>*<sup>3</sup> 8*π*<sup>2</sup>

> *k*2 *f* ℏ*c*

At this stage, the interaction term, that is, the squared matrix element appearing

where the vector *Rj* is the position of the *j*th atom. The above formula embodies

� �. Furthermore, when using Eq. (10), it was considered that *<sup>ω</sup>*ð Þ*<sup>k</sup>* is equal to

*exp iQ* � *R<sup>j</sup>* � �j*I*⟩

Here, ℏ*ω* is the energy gained by the photons in the scattering process, while the *δ-*function term accounts for the energy conservation in the scattering process, as it ensures that ℏ*ω* ¼ �ð Þ *EF* � *EI* with *EF* � *EI* being the energy gained by the sample. Notice that the cross-section defined above entails a sum over all states of the system, where the factor *PI* represents the statistical population of the initial states

the momentum conservation law as it was derived assuming the identity ℏ*Q* ¼

*cki* and *ckf* in the initial and the final photon states, respectively. Combining all analytical steps illustrated above, one eventually obtains the following expression for

> X *j*

**5. From the adiabatic approximation to the dynamic structure factor**

The right-hand side of Eq. (21) contains three independent factors, the integral term being the only one directly relating to the properties of the target sample. The

*dPi*!*<sup>f</sup>*

� �j*I*⟩⟨*I*j*exp i*ð Þ *<sup>Q</sup>* � *<sup>R</sup><sup>m</sup>* <sup>j</sup>*F*⟩, (20)

� � � � �

2

*δ*ð Þ ℏ*ω* þ *EF* � *EI :* (21)

<sup>¼</sup> *<sup>L</sup>*<sup>3</sup> 8*π*<sup>2</sup>

in the Fermi Golden Rule (Eq. (13)), can be made explicit by inserting in it the Thomson term in Eq. (9), while using the expression of the vector potential in

*L* 2*π* � �<sup>3</sup>

*k*2 *f* ℏ*c* *dk <sup>f</sup> dΩ:* (16)

*dE <sup>f</sup> dΩ:* (17)

, (18)

*dt :* (19)

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

$$\sum\_{I\_k} P\_{I\_k} \langle I\_n | \exp \left[ -i \mathbf{Q} \cdot \mathbf{R}\_k(\mathbf{0}) \right] \exp \left[ i \mathbf{Q} \cdot \mathbf{R}\_j(t) \right] | I\_n \rangle = \sum\_{k,j} \langle \exp \left[ -i \mathbf{Q} \cdot \mathbf{R}\_k(\mathbf{0}) \right] \exp \left[ i \mathbf{Q} \cdot \mathbf{R}\_j(t) \right] | \rangle, \tag{25}$$

Again, the variable of direct pertinence for a spectroscopic measurement is

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations…*

in which it was considered that the Fourier transform of a constant function is a

We can now introduce the intermediate scattering function as the space Fourier

*exp i<sup>Q</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � � � *<sup>n</sup>δ*ð Þ *<sup>Q</sup>* ; (31)

*dr*⟨*δn*ð Þ *r*, *t δn*ð Þ *r*, 0 ⟩*exp i*ð Þ *Q* � *r* (32)

*dt F*ð Þ *Q*, *t exp* ð Þ �*iωt* , (33)

*Sn*ð Þ *Q*,*ω* , (34)

instead the Fourier transform of such a fluctuation:

*DOI: http://dx.doi.org/10.5772/intechopen.93086*

*F*ð Þ¼ *Q*, *t*

and its one-sided time Fourier transform

dynamic structure factor of the system.

*d*2 *σ dΩdE <sup>f</sup>*

*Sn*ð Þ¼ *<sup>Q</sup>*,*<sup>ω</sup>* <sup>1</sup>

δ-function.

**factor**

where

amplitude *Q* ¼ j j *Q* .

density as

**13**

*<sup>S</sup><sup>δ</sup>n*ð Þ¼ *<sup>Q</sup>*, *<sup>ω</sup>* <sup>1</sup>

2*π*ℏ*N*

<sup>¼</sup> <sup>1</sup> 2*π*ℏ*N* ð<sup>∞</sup> �∞

ð<sup>∞</sup> �∞

*<sup>δ</sup>n*ð Þ¼ *<sup>Q</sup>*, *<sup>t</sup>* <sup>X</sup>

*N*

*j*¼1

transform of the correlation function between density fluctuations:

2*π*ℏ

ð<sup>∞</sup> 0

**7. The double differential cross-section and the dynamic structure**

*k f ki* � � ^*ε<sup>i</sup>* � ^*<sup>ε</sup> <sup>f</sup>* � �<sup>2</sup>

ðþ<sup>∞</sup> �∞

which is customarily referred to as the spectrum of density fluctuations, or the

Given the dynamic variables introduced in the previous section, it can be readily

is the spectrum associated with the dynamic variable *n*ð Þ *Q*, *t* . Notice that for a homogeneous and isotropic system such as a liquid, such a variable does not depend on the direction of the exchanged wavevector, but uniquely on its

Let us discuss here how the spectrum in Eq. (35) relates to the variable density fluctuations as defined by Eq. (30). By definition, the spectrum of such a variable is

*dt*⟨*δn*ð Þ *Q*, *t δn*ð Þ *Q*, 0 ⟩*exp* ð Þ �*iωt*

with *<sup>C</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*=*ℏ*N*. At this stage, one can define the spectrum of the microscopic

*dt*⟨*n*ð Þ *Q*, *t n*ð Þ *Q*, 0 ⟩*exp* ð Þþ �*iωt Cδ ω*ð Þ*δ*ð Þ *Q* , (36)

j j *f Q*ð Þ <sup>2</sup>

*dt*⟨*n*ð Þ *Q*, 0 *n*ð Þ *Q*, *t* ⟩*exp* ð Þ �*iωt* (35)

1 *N* ð *V*

*<sup>S</sup>*ð Þ¼ *<sup>Q</sup>*,*<sup>ω</sup>* <sup>1</sup>

verified that Eq. (26) can be cast in the more compact form:

<sup>¼</sup> *<sup>N</sup> <sup>r</sup>*<sup>2</sup> 0 ℏ

2*π*ℏ*N*

the Fourier transform of the autocorrelation function. Explicitly

where as usual, the angle brackets ⟨ … ⟩ denote the thermal average on the system at equilibrium. The expression above is the time correlation function of the variable P *<sup>j</sup> exp* �*i<sup>Q</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � �, which involves the pair composed by the *<sup>j</sup>*th and *<sup>k</sup>*th atoms. The physical meaning of this variable will be discussed in the next section in further detail.

In summary, as a result of all manipulations mentioned above, the double differential cross-section in Eq. (24) eventually reduces to

$$\frac{\partial^2 \sigma}{\partial \Omega \partial E\_f} = \frac{K}{2\pi \hbar} \int\_{-\infty}^{\infty} dt \langle \sum\_{j=1}^{N} \sum\_{k=1}^{N} \exp\left\{i \mathbf{Q} \cdot \left[\mathbf{R}\_j(t) - \mathbf{R}\_k\right] \right\} \rangle \exp\left(-iat\right) \tag{26}$$

where *R<sup>k</sup>* is the shorthand notation for *Rk*ð Þ 0 .

## **6. Introducing a key stochastic variable: the microscopic density fluctuation**

The expression between angle brackets is the equilibrium autocorrelation function of the dynamic variable *<sup>n</sup>*ð Þ¼ *<sup>Q</sup>*, *<sup>t</sup>* <sup>P</sup> *<sup>j</sup>* exp *<sup>i</sup><sup>Q</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � �, which involves the positions of the generic *k*th and *k*th atom pair, that is, *Rk*ð Þ*t* and *Rj*ð Þ 0 respectively, evaluated at different times. The variable *n*ð Þ *Q*, *t* is the Fourier transform of the microscopic number density of the system, which, for a system of *N* atoms is defined as:

$$n(\mathbf{r}, t) = \sum\_{j=1}^{N} \delta \left[ \mathbf{r} - \mathbf{R}\_j(t) \right]. \tag{27}$$

The interpretation of this function as a microscopic density is perhaps more evident as one considers its average value over the whole sample volume:

$$n = \mathbf{1}/V \left[ \int\_{V} d\mathbf{r} n(\mathbf{r}, t) = \mathbf{1}/V \right]\_{V} d\mathbf{r} \sum\_{j=1}^{N} \delta[\mathbf{r} - \mathbf{R}\_{j}(t)] = N/V,\tag{28}$$

which is consistent with the macroscopic definition of number density. Notice that the δ-function is an extremely irregular discontinuous profile, which however adequately accounts for the atomistic, character of the system.

In the reciprocal space, one deals with the Fourier transform of the microscopic density, namely

$$m(\mathbf{Q}, t) = \int\_{V} d\mathbf{r} \left\{ \sum\_{j=1}^{N} \delta \left[ \mathbf{r} - \mathbf{R}\_{j}(t) \right] \right\} \exp\left( i \mathbf{Q} \cdot \mathbf{r} \right) = \sum\_{j=1}^{N} \exp\left[ i \mathbf{Q} \cdot \mathbf{R}\_{j}(t) \right]. \tag{29}$$

Furthermore, since scattering phenomena arise from inhomogeneities or fluctuation from equilibrium, we are here mainly interested in the microscopic density fluctuation:

$$\delta n(r, t) = \sum\_{j=1}^{N} \delta \left[ r - \mathbf{R}\_j(t) \right] - n,\tag{30}$$

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations… DOI: http://dx.doi.org/10.5772/intechopen.93086*

Again, the variable of direct pertinence for a spectroscopic measurement is instead the Fourier transform of such a fluctuation:

$$\delta n(\mathbf{Q}, t) = \sum\_{j=1}^{N} \exp\left[i\mathbf{Q} \cdot \mathbf{R}\_j(t)\right] - n\delta(\mathbf{Q});\tag{31}$$

in which it was considered that the Fourier transform of a constant function is a δ-function.

We can now introduce the intermediate scattering function as the space Fourier transform of the correlation function between density fluctuations:

$$F(\mathbf{Q}, t) = \frac{1}{N} \int\_{V} dr \langle \delta n(r, t) \delta n(r, \mathbf{0}) \rangle \exp\left(i \mathbf{Q} \cdot r\right) \tag{32}$$

and its one-sided time Fourier transform

$$S(\mathbf{Q}, \omega) = \frac{1}{2\pi\hbar} \int\_0^\infty dt \, F(\mathbf{Q}, t) \exp\left(-i\alpha t\right),\tag{33}$$

which is customarily referred to as the spectrum of density fluctuations, or the dynamic structure factor of the system.

### **7. The double differential cross-section and the dynamic structure factor**

Given the dynamic variables introduced in the previous section, it can be readily verified that Eq. (26) can be cast in the more compact form:

$$\frac{d^2\sigma}{d\Omega d E\_f} = N \frac{r\_0^2}{\hbar} \left(\frac{k\_f}{k\_i}\right) \left(\hat{e}\_i \cdot \hat{e}\_f\right)^2 |f(Q)|^2 \mathbb{S}\_n(Q, \omega o),\tag{34}$$

where

X *In*

P

*∂*2 *σ ∂Ω∂E <sup>f</sup>*

**fluctuation**

defined as:

density, namely

*n*ð Þ¼ *Q*, *t*

fluctuation:

**12**

ð *V* *dr* X *N*

*j*¼1

<sup>¼</sup> *<sup>K</sup>* 2*π*ℏ ð<sup>∞</sup> �∞ *dt*⟨ X *N*

tion of the dynamic variable *<sup>n</sup>*ð Þ¼ *<sup>Q</sup>*, *<sup>t</sup>* <sup>P</sup>

*n* ¼ 1*=V*

ð *V*

*PIn* ⟨*In*<sup>j</sup> exp ½ � �*i<sup>Q</sup>* � *<sup>R</sup>k*ð Þ <sup>0</sup> exp *<sup>i</sup><sup>Q</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � �j*In*⟩ <sup>¼</sup> <sup>X</sup>

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

differential cross-section in Eq. (24) eventually reduces to

*j*¼1

where *R<sup>k</sup>* is the shorthand notation for *Rk*ð Þ 0 .

X *N*

*k*¼1

**6. Introducing a key stochastic variable: the microscopic density**

The expression between angle brackets is the equilibrium autocorrelation func-

tions of the generic *k*th and *k*th atom pair, that is, *Rk*ð Þ*t* and *Rj*ð Þ 0 respectively, evaluated at different times. The variable *n*ð Þ *Q*, *t* is the Fourier transform of the microscopic number density of the system, which, for a system of *N* atoms is

*N*

*j*¼1

The interpretation of this function as a microscopic density is perhaps more

ð *V dr* X *N*

which is consistent with the macroscopic definition of number density. Notice that the δ-function is an extremely irregular discontinuous profile, which however

In the reciprocal space, one deals with the Fourier transform of the microscopic

Furthermore, since scattering phenomena arise from inhomogeneities or fluctuation from equilibrium, we are here mainly interested in the microscopic density

*j*¼1

*exp i*ð Þ¼ *<sup>Q</sup>* � *<sup>r</sup>* <sup>X</sup>

*N*

*j*¼1

*<sup>δ</sup> <sup>r</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � � � *<sup>n</sup>*, (30)

*<sup>n</sup>*ð Þ¼ *<sup>r</sup>*, *<sup>t</sup>* <sup>X</sup>

evident as one considers its average value over the whole sample volume:

*drn*ð Þ¼ *r*, *t* 1*=V*

adequately accounts for the atomistic, character of the system.

*<sup>δ</sup>n*ð Þ¼ *<sup>r</sup>*, *<sup>t</sup>* <sup>X</sup>

*N*

*j*¼1

*<sup>δ</sup> <sup>r</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � � ( )

*k*, *j*

where as usual, the angle brackets ⟨ … ⟩ denote the thermal average on the system at equilibrium. The expression above is the time correlation function of the variable

*<sup>j</sup> exp* �*i<sup>Q</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � �, which involves the pair composed by the *<sup>j</sup>*th and *<sup>k</sup>*th atoms. The physical meaning of this variable will be discussed in the next section in further detail. In summary, as a result of all manipulations mentioned above, the double

exp *iQ* � *Rj*ðÞ�*t R<sup>k</sup>*

⟨ exp ½ � �*i<sup>Q</sup>* � *<sup>R</sup>k*ð Þ <sup>0</sup> exp *<sup>i</sup><sup>Q</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � �⟩,

� � � � ⟩ exp ð Þ �*iω<sup>t</sup>* (26)

*<sup>j</sup>* exp *<sup>i</sup><sup>Q</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � �, which involves the posi-

*<sup>δ</sup> <sup>r</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � �*:* (27)

*<sup>δ</sup> <sup>r</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � � <sup>¼</sup> *<sup>N</sup>=V*, (28)

*exp i<sup>Q</sup>* � *<sup>R</sup>j*ð Þ*<sup>t</sup>* � �*:* (29)

(25)

$$\mathcal{S}\_n(Q,\omega) = \frac{1}{2\pi\hbar N} \int\_{-\infty}^{+\infty} dt \langle n(\mathbf{Q},0)n(\mathbf{Q},t) \rangle \exp\left(-i\alpha t\right) \tag{35}$$

is the spectrum associated with the dynamic variable *n*ð Þ *Q*, *t* . Notice that for a homogeneous and isotropic system such as a liquid, such a variable does not depend on the direction of the exchanged wavevector, but uniquely on its amplitude *Q* ¼ j j *Q* .

Let us discuss here how the spectrum in Eq. (35) relates to the variable density fluctuations as defined by Eq. (30). By definition, the spectrum of such a variable is the Fourier transform of the autocorrelation function. Explicitly

$$\begin{split} S\_{\delta n}(Q,\omega) &= \frac{1}{2\pi\hbar N} \Big|\_{-\infty}^{\infty} dt \langle \delta n(\mathbf{Q},t) \delta n(\mathbf{Q},0) \rangle \exp\left(-i\alpha t\right) \\ &= \frac{1}{2\pi\hbar N} \Big|\_{-\infty}^{\infty} dt \langle n(\mathbf{Q},t)n(\mathbf{Q},0) \rangle \exp\left(-i\alpha t\right) + \text{C}\delta(\omega)\delta(\mathbf{Q}), \end{split} \tag{36}$$

with *<sup>C</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>2</sup>*=*ℏ*N*. At this stage, one can define the spectrum of the microscopic density as

$$\mathcal{S}\_{\mathfrak{n}}(Q,\alpha) = \mathcal{S}\_{\mathfrak{dn}}(Q,\alpha) + n^2 \delta(\alpha) \delta(\mathbf{Q}).\tag{37}$$

**8. An estimate of the count rate**

*DOI: http://dx.doi.org/10.5772/intechopen.93086*

one has:

**Figure 3.**

**15**

*X-ray beam (courtesy of F. Sette).*

where *σ<sup>C</sup>* ¼ ð Þ *Zr*<sup>0</sup>

with the incident X-ray.

2

**8.1 The signal measured by a real instrument**

An estimate of the count rate achievable by an IXS measurement can be worked out starting from the expression of the total scattering cross-section, while assuming, for instance, a sample having an optimal thickness *ts* ¼ 1*=μ*. The flux of scattered photons in the solid angle *ΔΩ* and the energy interval *ΔEf* is thus given by

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations…*

*μ*

At this stage, both members of the equation can be integrated in time, and the double integration over both solid angle and final energy must also be performed to obtain the total cross-section of the IXS scattering. In the low *Q* limit, where

*d*2 *σ dΩdE <sup>f</sup>*

, while *σ<sup>A</sup>* ¼ *ns=μ* is the absorption cross-section. An idea of

*ΔΩΔEf :* (42)

^*ε<sup>i</sup>* � ^*<sup>ε</sup> <sup>f</sup>*

, (43)

<sup>2</sup> <sup>¼</sup> 1,

*dN*\_ <sup>¼</sup> *<sup>N</sup>*\_ <sup>0</sup> *exp* ð Þ �<sup>1</sup> *ns*

the atomic form factor *f Q*ð Þ≈*Z* and that the approximation *k <sup>f</sup> =ki*

*N N*<sup>0</sup> <sup>∝</sup> ð Þ *Zr*<sup>0</sup> 2 *ns <sup>μ</sup>* <sup>¼</sup> *<sup>σ</sup><sup>C</sup> σA*

the counting efficiency of IXS is provided by **Figure 3**, which displays the value of the *σC=σ<sup>A</sup>* ratio for an incident X-ray beam having 10 keV energy, as a function of the atomic number. The abrupt increase of this parameter can be readily appreciated at the absorption above the K-edge, that is, above the binding energy of the innermost electron shell; these innermost electrons are those primarily interacting

As a result of the previous treatment, it was demonstrated that the cross-section is proportional to *S Q*ð Þ , *ω* . However, such a treatment is entirely classical, insofar as

*The cross-sections ratio defined in Eq. (45) is reported as a function of the atomic number Z, for a 10 keV energy*

It appears that the spectra of either *n*ð Þ *Q*, *t* or *δn*ð Þ *Q*, *t* , which are labeled by the respective indexes *n* and *δn*, differ by a term proportional to the product *δ ω*ð Þ*δ*ð Þ *Q* .

This term accounts for the forward transmitted elastic scattering, which is of no relevance for a scattering experiment as it describes the signal from photons that have exchanged no energy or momentum with the target sample.

In practice, such a signal is never detected by scattering measurements, as it does not convey insight into non-trivial samples properties; furthermore, it fully overlaps with the forward transmitted beam, which is often so intense to burn or damage detectors. For these reasons, IXS measurements are always performed at finite scattering angles, where one has

$$\mathcal{S}\_n(Q,\alpha) = \mathcal{S}\_{\delta n}(Q,\alpha) \equiv \mathcal{S}(Q,\alpha),\tag{38}$$

with

$$S(Q,\omega) = \frac{1}{2\pi N} \int\_{-\infty}^{+\infty} dt \langle \delta n(\mathbf{Q}, \mathbf{0}) \delta n(\mathbf{Q}, t) \rangle \exp\left(-i\alpha t\right),\tag{39}$$

As discussed, the identity above entails the replacement of the microscopic density *n*ð Þ *Q*, *t* with its fluctuation from equilibrium *δn Q*ð Þ , *t*

$$\frac{d^2\sigma}{d\Omega dE\_f} = \text{KS}(Q, \alpha),\tag{40}$$

where *<sup>K</sup>* <sup>¼</sup> *N r*<sup>2</sup> <sup>0</sup>*=<sup>h</sup>* � � *<sup>k</sup> <sup>f</sup> <sup>=</sup>ki* � � ^*ε<sup>i</sup>* � ^*<sup>ε</sup> <sup>f</sup>* � �<sup>2</sup> j j *f Q*ð Þ <sup>2</sup> .

This expression of the cross-section above has been derived assuming a target sample composed by *N* identical atoms and within the Born-Oppenheimer approximation. When different atomic species are present in the sample, within the validity of the Born-Oppenheimer approximation, the derivation of the scattering cross-section is similar, provided the system is isotropic, that is, invariant under rotations, and a weak coupling exists between molecular rotations and centre of mass movements. The 'effective' form factor, in this case, results from the average value of the form factors of different atoms in the molecule. The general case, of a system composed of molecules with a pronounced anisotropy, that is, a markedly non-spherical shape, makes the computation of the cross-section slightly more complicated.

A more detailed treatment of this problem within the hypothesis of random molecular orientations and weak coupling between orientational and translational degrees of freedom leads to the conclusion that the spectrum splits into a coherent and an incoherent component. Consequently, the cross-section can be cast in the following general form:

$$\frac{\partial^2 \sigma}{\partial \Omega \partial \omega} = A \left\{ \langle F^2(Q) \rangle\_{\Omega} S\_C(Q, \alpha) + \delta \langle F^2(Q) \rangle\_{\Omega} S\_I(Q, \alpha) \right\} \tag{41}$$

where *<sup>δ</sup>*⟨*F Q*ð Þ<sup>2</sup> ⟩*<sup>Ω</sup>* <sup>¼</sup> ⟨*F Q*ð Þ<sup>2</sup> ⟩*<sup>Ω</sup>* � ⟨*F Q*ð Þ⟩ 2 *<sup>Ω</sup>*, where the suffix '*Ω*' indicates an average over molecular orientations, while the suffixes 'I' and 'C' label the incoherent and coherent parts of the dynamic structure factor.

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations… DOI: http://dx.doi.org/10.5772/intechopen.93086*

#### **8. An estimate of the count rate**

*Sn*ð Þ¼ *<sup>Q</sup>*,*<sup>ω</sup> <sup>S</sup><sup>δ</sup>n*ð Þþ *<sup>Q</sup>*,*<sup>ω</sup> <sup>n</sup>*<sup>2</sup>

have exchanged no energy or momentum with the target sample.

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

finite scattering angles, where one has

*S Q*ð Þ¼ , *<sup>ω</sup>* <sup>1</sup>

<sup>0</sup>*=<sup>h</sup>* � � *<sup>k</sup> <sup>f</sup> <sup>=</sup>ki*

2*πN*

density *n*ð Þ *Q*, *t* with its fluctuation from equilibrium *δn Q*ð Þ , *t*

� � ^*ε<sup>i</sup>* � ^*<sup>ε</sup> <sup>f</sup>*

ðþ<sup>∞</sup> �∞

*d*2 *σ dΩdE <sup>f</sup>*

� �<sup>2</sup>

sample composed by *N* identical atoms and within the Born-Oppenheimer

As discussed, the identity above entails the replacement of the microscopic

j j *f Q*ð Þ <sup>2</sup> .

This expression of the cross-section above has been derived assuming a target

approximation. When different atomic species are present in the sample, within the validity of the Born-Oppenheimer approximation, the derivation of the scattering cross-section is similar, provided the system is isotropic, that is, invariant under rotations, and a weak coupling exists between molecular rotations and centre of mass movements. The 'effective' form factor, in this case, results from the average value of the form factors of different atoms in the molecule. The general case, of a system composed of molecules with a pronounced anisotropy, that is, a markedly non-spherical shape, makes the computation of the cross-section slightly more

A more detailed treatment of this problem within the hypothesis of random molecular orientations and weak coupling between orientational and translational degrees of freedom leads to the conclusion that the spectrum splits into a coherent and an incoherent component. Consequently, the cross-section can be cast in the

ð Þ *<sup>Q</sup>* ⟩*ΩSC*ð Þþ *<sup>Q</sup>*, *<sup>ω</sup> <sup>δ</sup>*⟨*F*<sup>2</sup>

age over molecular orientations, while the suffixes 'I' and 'C' label the incoherent

2

⟩*<sup>Ω</sup>* � ⟨*F Q*ð Þ⟩

ð Þ *<sup>Q</sup>* ⟩*ΩSI*ð Þ *<sup>Q</sup>*,*<sup>ω</sup>* � � (41)

*<sup>Ω</sup>*, where the suffix '*Ω*' indicates an aver-

*δ ω*ð Þ*δ*ð Þ *Q* .

with

where *<sup>K</sup>* <sup>¼</sup> *N r*<sup>2</sup>

complicated.

following general form:

where *<sup>δ</sup>*⟨*F Q*ð Þ<sup>2</sup>

**14**

*∂*2 *σ <sup>∂</sup>Ω∂<sup>ω</sup>* <sup>¼</sup> *<sup>A</sup>* ⟨*F*<sup>2</sup>

⟩*<sup>Ω</sup>* <sup>¼</sup> ⟨*F Q*ð Þ<sup>2</sup>

and coherent parts of the dynamic structure factor.

It appears that the spectra of either *n*ð Þ *Q*, *t* or *δn*ð Þ *Q*, *t* , which are labeled by the respective indexes *n* and *δn*, differ by a term proportional to the product

This term accounts for the forward transmitted elastic scattering, which is of no relevance for a scattering experiment as it describes the signal from photons that

In practice, such a signal is never detected by scattering measurements, as it does not convey insight into non-trivial samples properties; furthermore, it fully overlaps with the forward transmitted beam, which is often so intense to burn or damage detectors. For these reasons, IXS measurements are always performed at

*Sn*ð Þ¼ *Q*,*ω S<sup>δ</sup>n*ð Þ� *Q*,*ω S Q*ð Þ , *ω* , (38)

*dt*⟨*δn*ð Þ *Q*, 0 *δn*ð Þ *Q*, *t* ⟩*exp* ð Þ �*iωt* , (39)

¼ *KS Q*ð Þ ,*ω* , (40)

*δ ω*ð Þ*δ*ð Þ *Q :* (37)

An estimate of the count rate achievable by an IXS measurement can be worked out starting from the expression of the total scattering cross-section, while assuming, for instance, a sample having an optimal thickness *ts* ¼ 1*=μ*. The flux of scattered photons in the solid angle *ΔΩ* and the energy interval *ΔEf* is thus given by

$$d\dot{N} = \dot{N}\_0 \exp\left(-\mathbf{1}\right) \frac{n\_s}{\mu} \frac{d^2 \sigma}{d\Omega dE\_f} \Delta\Omega \Delta E\_f. \tag{42}$$

At this stage, both members of the equation can be integrated in time, and the double integration over both solid angle and final energy must also be performed to obtain the total cross-section of the IXS scattering. In the low *Q* limit, where the atomic form factor *f Q*ð Þ≈*Z* and that the approximation *k <sup>f</sup> =ki* ^*ε<sup>i</sup>* � ^*<sup>ε</sup> <sup>f</sup>* <sup>2</sup> <sup>¼</sup> 1, one has:

$$\frac{N}{N\_0} \propto \frac{(Zr\_0)^2 n\_s}{\mu} = \frac{\sigma\_C}{\sigma\_A},\tag{43}$$

where *σ<sup>C</sup>* ¼ ð Þ *Zr*<sup>0</sup> 2 , while *σ<sup>A</sup>* ¼ *ns=μ* is the absorption cross-section. An idea of the counting efficiency of IXS is provided by **Figure 3**, which displays the value of the *σC=σ<sup>A</sup>* ratio for an incident X-ray beam having 10 keV energy, as a function of the atomic number. The abrupt increase of this parameter can be readily appreciated at the absorption above the K-edge, that is, above the binding energy of the innermost electron shell; these innermost electrons are those primarily interacting with the incident X-ray.

#### **8.1 The signal measured by a real instrument**

As a result of the previous treatment, it was demonstrated that the cross-section is proportional to *S Q*ð Þ , *ω* . However, such a treatment is entirely classical, insofar as

#### **Figure 3.**

*The cross-sections ratio defined in Eq. (45) is reported as a function of the atomic number Z, for a 10 keV energy X-ray beam (courtesy of F. Sette).*

all relevant observables are treated as commuting variables. Quantum effects are accounted for only through the so-called detailed balance principle, which takes into account the statistical population of the various ℏ*ω*-states of the sample. These effects ultimately result in an asymmetry of the spectrum respect to its elastic, ℏ*ω* ¼ 0, position. The most popular recipe for handling them is to assume that the true spectrum ~ *S Q*ð Þ ,*ω* can be obtained from the classic, symmetric, counterpart *S Q*ð Þ , *ω* by adding a suitable frequency-dependent factor. Explicitly,

$$\tilde{\mathcal{S}}(Q,\omega) = \frac{\hbar\omega}{k\_B T} \left[ \frac{1}{1 - \exp\left(-\hbar\omega/k\_B T\right)} \right] \mathcal{S}(Q,\omega). \tag{44}$$

and the dark counts of a detector; when modeling the line-shape, sometimes the

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations…*

The resolution profile represents the instrumental rendering of a spectral shape having zero energy width, that is, the δ(ω)–profile representing a perfectly elastic scattering. In a typical IXS measurement, such a resolution is estimated by measuring the scattering signal from an almost perfect elastic scatterer, often identified in a

In general, the kinematic laws ruling the scattering process impose some limitations to the dynamic Q,ω region explorable by the measurement. These kinematic constraints are especially severe for inelastic neutron scattering, INS [2]. Although these limitations are irrelevant for IXS, the portion of the dynamic plane explored by this technique is still limited in the low-energy, or low-frequency, side by the

**Figure 4** provides a clear example of how resolution and kinematic limitations differently affect IXS and INS. Indeed, the plot compares the spectra measured in a joint INS and IXS measurement on the same sample of heavy water [12], after normalization of the respective areas. The elastic peak in the INS spectrum has a spike-like shape. Such a sharp shape could be measured thanks to the 0.08 meV broad Gaussian resolution function. Which enables a superior definition of the spectral shape. However, this performance imposes an overall shrinkage of the spanned frequency range, which does not include the high-frequency shoulder in the IXS spectrum. On the other hand, the resolution of the IXS measurement is too coarse to enable a proper definition of the quasielastic portion of the scattering

In conclusion, we illustrated the main analytical steps leading to a derivation of the inelastic X-ray scattering, IXS signal, and demonstrated its direct link with the

Since its development in the mid-1990s, high-resolution IXS has rapidly transitioned to its mature age, nowadays representing an essential tool to characterize the terahertz dynamics of liquid and amorphous materials. Historically, the mainstream scientific interest of the IXS community was mostly limited to simple fluids and glass-forming materials. In recent years, such a focus has gradually shifted towards nanostructured metamaterials and biological systems. Since the high complexity of these systems often challenges a firm understanding of the measurement outcome, a firm theoretical modeling of the IXS signal from these

This work used resources of the National Synchrotron Light Source II, a U.S. Department of Energy Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.

latter coefficient is assumed either constant or linearly dependent on *ω*.

sample of plexiglas at the *Q*-position of the first sharp diffraction peak.

**8.2 A practical example: a comparison between an IXS and an INS**

**measurement**

profile.

**9. Conclusion**

**Acknowledgements**

**17**

finite instrumental energy resolution.

*DOI: http://dx.doi.org/10.5772/intechopen.93086*

terahertz spectrum of atomic density fluctuations.

highly heterogeneous systems would be highly beneficial.

Still, the above formula does not capture two essential aspects of the measured scattering signal, as the contribution of the instrumental resolution and the spectral background. These are explicitly accounted for by using the following general expression for the intensity profile:

$$I = I(Q, \boldsymbol{\alpha}) = A\left[\tilde{\mathcal{S}}(Q, \boldsymbol{\alpha}) \otimes \mathcal{R}(\boldsymbol{\alpha})\right] + B(\boldsymbol{\alpha})\tag{45}$$

where *A* is an overall intensity factor, while the usually mildly frequencydependent coefficient *B*ð Þ *ω* accounts in principle for both the spectral background

#### **Figure 4.**

*The spectral line-shapes measured by IXS (black line) and INS (shadowed blue line) on a D2O sample at ambient conditions. The spectra are reported after rescaling to the respective integrated intensities. Data are redrawn from ref. notice the remarkable difference in the explored* ω*-range. Data are redrawn from Ref. [13].*

*High-Resolution Inelastic X-Ray Scattering: A Probe of Microscopic Density Fluctuations… DOI: http://dx.doi.org/10.5772/intechopen.93086*

and the dark counts of a detector; when modeling the line-shape, sometimes the latter coefficient is assumed either constant or linearly dependent on *ω*.

The resolution profile represents the instrumental rendering of a spectral shape having zero energy width, that is, the δ(ω)–profile representing a perfectly elastic scattering. In a typical IXS measurement, such a resolution is estimated by measuring the scattering signal from an almost perfect elastic scatterer, often identified in a sample of plexiglas at the *Q*-position of the first sharp diffraction peak.

#### **8.2 A practical example: a comparison between an IXS and an INS measurement**

In general, the kinematic laws ruling the scattering process impose some limitations to the dynamic Q,ω region explorable by the measurement. These kinematic constraints are especially severe for inelastic neutron scattering, INS [2]. Although these limitations are irrelevant for IXS, the portion of the dynamic plane explored by this technique is still limited in the low-energy, or low-frequency, side by the finite instrumental energy resolution.

**Figure 4** provides a clear example of how resolution and kinematic limitations differently affect IXS and INS. Indeed, the plot compares the spectra measured in a joint INS and IXS measurement on the same sample of heavy water [12], after normalization of the respective areas. The elastic peak in the INS spectrum has a spike-like shape. Such a sharp shape could be measured thanks to the 0.08 meV broad Gaussian resolution function. Which enables a superior definition of the spectral shape. However, this performance imposes an overall shrinkage of the spanned frequency range, which does not include the high-frequency shoulder in the IXS spectrum. On the other hand, the resolution of the IXS measurement is too coarse to enable a proper definition of the quasielastic portion of the scattering profile.

#### **9. Conclusion**

all relevant observables are treated as commuting variables. Quantum effects are accounted for only through the so-called detailed balance principle, which takes into account the statistical population of the various ℏ*ω*-states of the sample. These effects ultimately result in an asymmetry of the spectrum respect to its elastic, ℏ*ω* ¼ 0, position. The most popular recipe for handling them is to assume that the

*S Q*ð Þ , *ω* by adding a suitable frequency-dependent factor. Explicitly,

*kBT*

*Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

*<sup>I</sup>* <sup>¼</sup> *I Q*ð Þ¼ ,*<sup>ω</sup> <sup>A</sup>* <sup>~</sup>

*S Q*ð Þ ,*ω* can be obtained from the classic, symmetric, counterpart

*S Q*ð Þ , *ω :* (44)

*S Q*ð Þ ,*<sup>ω</sup>* <sup>⊗</sup> *<sup>R</sup>*ð Þ *<sup>ω</sup>* <sup>þ</sup> *<sup>B</sup>*ð Þ *<sup>ω</sup>* (45)

1 1 � *exp* ð Þ �ℏ*ω=kBT* 

Still, the above formula does not capture two essential aspects of the measured scattering signal, as the contribution of the instrumental resolution and the spectral background. These are explicitly accounted for by using the following general

where *A* is an overall intensity factor, while the usually mildly frequencydependent coefficient *B*ð Þ *ω* accounts in principle for both the spectral background

*The spectral line-shapes measured by IXS (black line) and INS (shadowed blue line) on a D2O sample at ambient conditions. The spectra are reported after rescaling to the respective integrated intensities. Data are redrawn from ref. notice the remarkable difference in the explored* ω*-range. Data are redrawn from Ref. [13].*

true spectrum ~

**Figure 4.**

**16**

~

expression for the intensity profile:

*S Q*ð Þ¼ , *<sup>ω</sup>* <sup>ℏ</sup>*<sup>ω</sup>*

In conclusion, we illustrated the main analytical steps leading to a derivation of the inelastic X-ray scattering, IXS signal, and demonstrated its direct link with the terahertz spectrum of atomic density fluctuations.

Since its development in the mid-1990s, high-resolution IXS has rapidly transitioned to its mature age, nowadays representing an essential tool to characterize the terahertz dynamics of liquid and amorphous materials. Historically, the mainstream scientific interest of the IXS community was mostly limited to simple fluids and glass-forming materials. In recent years, such a focus has gradually shifted towards nanostructured metamaterials and biological systems. Since the high complexity of these systems often challenges a firm understanding of the measurement outcome, a firm theoretical modeling of the IXS signal from these highly heterogeneous systems would be highly beneficial.

#### **Acknowledgements**

This work used resources of the National Synchrotron Light Source II, a U.S. Department of Energy Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704. *Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications*

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