Atomic Dynamics in Real Space and Time

*Takeshi Egami*

#### **Abstract**

Atomic and molecular dynamics in strongly disordered matter, such as liquid, cannot be fully described in terms of phonons, because they are marginalized and often overdamped. Their dynamic and transport properties depend on local atomic rearrangements which are strongly correlated. To describe such local dynamics, the usual representation in momentum (*Q*) and energy (*E*) space in terms of the dynamic structure factor, *S*(*Q*, *E*), is not effective. We discuss an alternative approach in real space (*r*) and time (*t*), with the van Hove function, *G*(*r*, *t*), and show how this approach facilitates understanding of real-space local dynamics of liquids and other disordered systems in the length scale of Å and time scale of pico-second.

**Keywords:** local atomic dynamics, van Hove function, liquid, glass, correlated dynamics

#### **1. Introduction**

In crystalline solids, phonons are the elementary excitations of lattice dynamics. They can be observed with well-defined dispersions in the dynamic structure factor, *S*(*Q*, *E*), where *Q* is the momentum exchange and *E* (= ℏ*ω*) is the energy exchange in scattering. The *S*(*Q*, *E*) can be measured by inelastic X-ray scattering (IXS) as discussed in this book or by inelastic neutron scattering (INS). However, in strongly disordered materials, shortwave phonons have very short mean free path and lifetime. Only long-wave phonons, for which the atomic structure is irrelevant and the material acts as a continuum elastic body, propagate over some distance. However, the total spectral weight of long-wave phonons is small, because the phonon density of states increases as *E*<sup>2</sup> . For this reason, a majority of phonons are overdamped, and some of them are localized, particularly in liquid.

Atomic dynamics in liquid and soft matter is usually studied by quasi-elastic scattering (QES) at low *Q*, where the QES intensity is dominated by the selfcorrelation and the auto-correlation of the same atom. The energy width of QES is proportional to *DQ*<sup>2</sup> , where *D* is the diffusion constant. For hydrogen, which has a very large incoherent cross section, the value of *D* determined by QES at low *Q* is the isotopic diffusivity, *Di*, determined by the diffusion profile of isotopes measured by sectioning. The chemical diffusivity, *Dc*, which characterizes the flow of elements responding to concentration gradient, is usually different from *Di*, because of the backflow of diffusing atoms. The ratio, *f* = *Dc/Di*, is known as the correlation factor and describes how diffusive atomic jumps are correlated.

This example illustrates the importance of studying the distinct atomic correlation and the dynamic correlation among different atoms. The importance of knowing the distinct terms of the correlation function is beginning to be recognized [1, 2], but the difficulty of measurement delayed advances. Only recently the progress in IXS instrumentation [3, 4] and the advent of pulsed neutron sources [5] made it feasible to measure *S*(*Q*, *E*) over wide ranges of *Q* and *E* in a reasonable time and opened the possibility of garnering information on distinct atomic correlations.

*G r*ð Þ¼ *; t* 1 þ

*Atomic Dynamics in Real Space and Time DOI: http://dx.doi.org/10.5772/intechopen.88334*

measured is the energy-integrated intensity,

*g r*ð Þ¼ 1 þ

the snapshot correlation function, decays with time.

*Gs*ð Þ¼ *r; t*

1 *ρ*0

of the van Hove function. The double Fourier transform of Eq. (8) is

*Ss*ð Þ¼ *<sup>Q</sup>;<sup>ω</sup> DiQ*<sup>2</sup>

function in the diffusive regime should be

Fourier transform, the PDF,

**2.2 Evolution with time**

accurate determination of *Di*.

**49**

<sup>¼</sup> <sup>1</sup> 4*πρ*0*Nr*<sup>2</sup>

1 2*π*2*ρ*0*r*

*S Q*ð Þ¼ <sup>ð</sup>

1 2*π*<sup>2</sup>*ρ*0*r* ð

ð

X *i,j*

Again it is useful to divide it into the self-part, *Gs*(*r*, *t*), and the distinct-part, *Gd*(*r*, *t*). Although the van Hove function has been known for a long time, its experimental determination has rarely been done [13] because it requires *S*(*Q*, *E*) to be known over wide ranges of *Q* and *E*. Only recently it became practical [14, 15] because of the progress in instrumentation as noted above and discussed below. In the regular X-ray diffraction measurement, the energy resolution is of the order of 1 eV, far greater than the phonon excitation energies. Therefore what is

which leaves only the same-time (*t* = 0) contribution in Eq. (1). Therefore its

is equal to *G*(*r*, 0). In other words the van Hove function describes how the PDF,

At a short time scale (�0.1 ps), atomic motions are ballistic, but after atoms leave the neighbor cage, they become diffusive. Then the self-term of the van Hove

> 1 4*πDit* � �<sup>3</sup> 2

Therefore the self-diffusion coefficient, *Di*, can be determined from the self-part

which can be measured by QES and is routinely used for determining *Di*. In the case of quasi-elastic neutron scattering (QENS) from hydrogen, because the incoherent cross section of hydrogen is so large, the scattering intensity is totally dominated by *Ss*(*Q*, *ω*), and the measurement readily yields the value of *Di*. In general, however, there can be some contributions from the distinct-part to the low-angle QES, which can make the measurement inaccurate. On the other hand in the van Hove function the self-part is cleanly separated, at least initially, resulting in more

The decay of the first peak of the PDF with time describes how the nearest neighbor shell of an atom, known as the first-neighbor cage, disintegrates with

exp � *<sup>r</sup>*<sup>2</sup>

*=π*

4*Dit*

*<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *DiQ*<sup>2</sup> � �<sup>2</sup> *,* (9)

� �*:* (8)

*F Q*ð Þ *; t* sin ð Þ *Qr QdQ*

(5)

*<sup>δ</sup> <sup>r</sup>* � **<sup>r</sup>***i*ð Þ� <sup>0</sup> **rj**ð Þ*<sup>t</sup>* � � � � � �*:*

*S Q*ð Þ *; E dE,* (6)

½ � *S Q*ð Þ� 1 sin ð Þ *Qr QdQ ,* (7)

Conversely, even though the substantial progress in instrumentation allowed us to collect a much larger amount of data which contain vastly richer information, the IXS data are mostly processed in the same way, just by focusing on the phonon dispersion and its width [6]. Similarly, usually only the diffusivity is obtained from the QES data. What is missing is the analysis of the *diffuse* IXS intensity which is usually discarded as background. This situation is a perfect analog of the structural analysis of disordered crystals. The conventional methods of crystallographic analysis focus only on the Bragg peaks, and the diffuse scattering intensity is processed separately to determine the short-range order. However, by combining the Bragg and diffuse intensities as the total scattering intensity and Fourier-transforming it into the atomic pair distribution function (PDF), precise local structural information can be obtained [7]. In the same manner, by including the diffuse IXS intensity, valuable information on local dynamics can directly be obtained.

#### **2. Van Hove function**

#### **2.1 Definition**

After the correction for absorption and normalization, the IXS intensity, *I*(*Q*, *E*), is reduced to *S*(*Q*, *E*) [8],

$$S(\mathbf{Q}, E) = \frac{1}{2\pi\hbar N} \sum\_{i, j} \int \left< \exp\left[\mathbf{Q} \cdot \left(\mathbf{r}\_{\mathbf{i}}(\mathbf{0}) - \mathbf{r}\_{\mathbf{j}}(t)\right)\right] \right> \exp\left(-\frac{E}{\hbar}t\right) dt,\tag{1}$$

where *ri*(*t*) is the position of the atom *i* at time *t*. It is useful to separate it into the self-part (*i* = *j*), *Ss*(*Q*, *E*), and the distinct-part (*i* 6¼ *j*), *Sd*(*Q*, *E*).

$$\mathcal{S}(\mathbf{Q}, E) = \mathcal{S}\_{\mathfrak{r}}(\mathbf{Q}, E) + \mathcal{S}\_{d}(\mathbf{Q}, E). \tag{2}$$

For isotropic matter such as liquid, we use the spherical average,

$$S(Q,E) = \frac{1}{4\pi} \int S(\mathbf{Q},E)d\Omega,\tag{3}$$

where *Q* ¼ j j *Q* and Ω is the solid angle in *Q* space. Upon the Fourier transformation into the time-domain, we obtain the intermediate scattering function [8],

$$F(Q,t) = \int S(Q,E) \exp\left(i\frac{E}{\hbar}t\right) dE,\tag{4}$$

which has been widely used in the analysis of soft matter dynamics [9–11]. Another step of the Fourier transformation, this time from momentum space to real space, leads to the van Hove function [12],

*Atomic Dynamics in Real Space and Time DOI: http://dx.doi.org/10.5772/intechopen.88334*

This example illustrates the importance of studying the distinct atomic correlation and the dynamic correlation among different atoms. The importance of knowing the distinct terms of the correlation function is beginning to be recognized [1, 2], but the difficulty of measurement delayed advances. Only recently the progress in IXS instrumentation [3, 4] and the advent of pulsed neutron sources [5] made it feasible to measure *S*(*Q*, *E*) over wide ranges of *Q* and *E* in a reasonable time and opened the possibility of garnering information on distinct atomic

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

Conversely, even though the substantial progress in instrumentation allowed us to collect a much larger amount of data which contain vastly richer information, the IXS data are mostly processed in the same way, just by focusing on the phonon dispersion and its width [6]. Similarly, usually only the diffusivity is obtained from the QES data. What is missing is the analysis of the *diffuse* IXS intensity which is usually discarded as background. This situation is a perfect analog of the structural analysis of disordered crystals. The conventional methods of crystallographic analysis focus only on the Bragg peaks, and the diffuse scattering intensity is processed separately to determine the short-range order. However, by combining the Bragg and diffuse intensities as the total scattering intensity and Fourier-transforming it into the atomic pair distribution function (PDF), precise local structural information can be obtained [7]. In the same manner, by including the diffuse IXS intensity,

After the correction for absorption and normalization, the IXS intensity, *I*(*Q*,

where *ri*(*t*) is the position of the atom *i* at time *t*. It is useful to separate it into the

exp **<sup>Q</sup>** � **ri**ð Þ� <sup>0</sup> **rj**ð Þ*<sup>t</sup>* � � � � � � exp � *<sup>E</sup>*

*S*ð Þ¼ **Q***; E Ss*ð Þþ **Q***; E Sd*ð Þ **Q***; E :* (2)

*E* ℏ *t* � �

ℏ *t* � �

*S*ð Þ **Q***; E d*Ω*,* (3)

*dE,* (4)

*dt,* (1)

valuable information on local dynamics can directly be obtained.

correlations.

**2. Van Hove function**

*E*), is reduced to *S*(*Q*, *E*) [8],

*<sup>S</sup>*ð Þ¼ **<sup>Q</sup>***; <sup>E</sup>* <sup>1</sup>

2*π*ℏ*N*

X *i,j*

ð

self-part (*i* = *j*), *Ss*(*Q*, *E*), and the distinct-part (*i* 6¼ *j*), *Sd*(*Q*, *E*).

For isotropic matter such as liquid, we use the spherical average,

*S Q*ð Þ¼ *; <sup>E</sup>* <sup>1</sup>

ð

*F Q*ð Þ¼ *; t*

space, leads to the van Hove function [12],

**48**

4*π* ð

where *Q* ¼ j j *Q* and Ω is the solid angle in *Q* space. Upon the Fourier transformation into the time-domain, we obtain the intermediate scattering function [8],

*S Q*ð Þ *; E* exp *i*

which has been widely used in the analysis of soft matter dynamics [9–11]. Another step of the Fourier transformation, this time from momentum space to real

**2.1 Definition**

$$\begin{split} G(r,t) &= \mathbf{1} + \frac{\mathbf{1}}{2\pi^2 \rho\_0 r} \Big[ F(Q,t) \sin \left( Qr \right) \mathbf{Q} d\mathbf{Q} \\ &= \frac{\mathbf{1}}{4\pi \rho\_0 N r^2} \sum\_{i,j} \delta(r - \left| \mathbf{r}\_i(\mathbf{0}) - \mathbf{r}\_j(t) \right|). \end{split} \tag{5}$$

Again it is useful to divide it into the self-part, *Gs*(*r*, *t*), and the distinct-part, *Gd*(*r*, *t*). Although the van Hove function has been known for a long time, its experimental determination has rarely been done [13] because it requires *S*(*Q*, *E*) to be known over wide ranges of *Q* and *E*. Only recently it became practical [14, 15] because of the progress in instrumentation as noted above and discussed below.

In the regular X-ray diffraction measurement, the energy resolution is of the order of 1 eV, far greater than the phonon excitation energies. Therefore what is measured is the energy-integrated intensity,

$$\mathcal{S}(Q) = \int \mathcal{S}(Q, E) dE,\tag{6}$$

which leaves only the same-time (*t* = 0) contribution in Eq. (1). Therefore its Fourier transform, the PDF,

$$\mathbf{g}(r) = \mathbf{1} + \frac{\mathbf{1}}{2\pi^2 \rho\_0 r} \left[ [\mathbf{S}(\mathbf{Q}) - \mathbf{1}] \sin \left( \mathbf{Q}r \right) \mathbf{Q} d\mathbf{Q} \right] \tag{7}$$

is equal to *G*(*r*, 0). In other words the van Hove function describes how the PDF, the snapshot correlation function, decays with time.

#### **2.2 Evolution with time**

At a short time scale (�0.1 ps), atomic motions are ballistic, but after atoms leave the neighbor cage, they become diffusive. Then the self-term of the van Hove function in the diffusive regime should be

$$G\_{\boldsymbol{s}}(\boldsymbol{r},t) = \frac{1}{\rho\_0} \left(\frac{1}{4\pi D\_i t}\right)^{\frac{3}{2}} \exp\left(-\frac{r^2}{4D\_i t}\right). \tag{8}$$

Therefore the self-diffusion coefficient, *Di*, can be determined from the self-part of the van Hove function. The double Fourier transform of Eq. (8) is

$$\mathcal{S}\_i(Q, \alpha) = \frac{D\_i Q^2 / \pi}{\alpha^2 + \left(D\_i Q^2\right)^2},\tag{9}$$

which can be measured by QES and is routinely used for determining *Di*. In the case of quasi-elastic neutron scattering (QENS) from hydrogen, because the incoherent cross section of hydrogen is so large, the scattering intensity is totally dominated by *Ss*(*Q*, *ω*), and the measurement readily yields the value of *Di*. In general, however, there can be some contributions from the distinct-part to the low-angle QES, which can make the measurement inaccurate. On the other hand in the van Hove function the self-part is cleanly separated, at least initially, resulting in more accurate determination of *Di*.

The decay of the first peak of the PDF with time describes how the nearest neighbor shell of an atom, known as the first-neighbor cage, disintegrates with time. The van Hove function depicts this decay nicely and can relate the time scale of decay to the topological relaxation time and to viscosity as shown below.

The early prediction on the distinct-part of the van Hove function was that it could be expressed by the convolution of the PDF by the self-part (Eq. (8)) [16]. Then the QES width should be equal to *DQ*<sup>2</sup> , by Eq. (9). But de Gennes noted that QES becomes anomalously narrow in the vicinity of the first peak in *S*(*Q*) [17]. He suggested that this phenomenon, now known as the de Gennes narrowing, was due to the collective nature of the dynamics represented by the first peak in *S*(*Q*). Since then it became customary to equate the observation of the de Gennes narrowing to the confirmation of collective excitations. A recent study [18], however, showed otherwise. It was found that even in high-temperature liquid, in which atomic motions are uncorrelated, the decay time of *G*(*r*, *t*) depends linearly on distance. In general,

$$
\pi(r) = \pi\_0 + \pi\_r \left(\frac{r}{r\_1}\right)^\chi,\tag{10}
$$

**3. Local dynamics of water and aqueous solution of salt**

**Figure 2** shows the *S*(*Q*, *E*) of water at room temperature, determined by the IXS experiment at the beam line XL35 of the SPring-8 facility [14]. Earlier IXS experiments to observe phonons did not cover the *Q* space much beyond 1 Å<sup>1</sup> [6, 20, 21]. The *S*(*Q*, *E*) is dominated by QES, and as is given it is not easy to garner useful information without extensive modeling. Converting the data into the van Hove function makes local dynamics directly visible as shown in **Figure 3**. Because hydrogen is almost invisible to X-rays, the van Hove function is dominated by oxygen–oxygen correlation. To minimize the termination error for stopping the integration by Eq. (4) at a maximum *Q* value, *F*(*Q*, *t*) can be extended to large *Q*

*The S(Q , E) of water at room temperature, determined by the IXS experiment at the beam line XL35 of the*

**3.1 Van Hove function of water**

*Atomic Dynamics in Real Space and Time DOI: http://dx.doi.org/10.5772/intechopen.88334*

**Figure 2.**

**Figure 3.**

**51**

*The van Hove function of water [14].*

*SPring-8 facility [14].*

where *r*<sup>1</sup> is the position of the nearest neighbor peak of the PDF. As shown in **Figure 1**, the exponent *χ* depends on dimensionality *d* approximately as *χ* = (*d*-1)/2; thus *χ* = 1 for three dimensions. This is because at large *r,* each PDF peak describes not just one atomic distance but many. Therefore its decay with time does not correspond to the single atom dynamics. The number of pairs of atoms in each peak, *Nr*, is proportional to the surface area, 4*πr d*�1 . Then its fluctuation is proportional to ffiffiffiffiffiffi *Nr* <sup>p</sup> � *<sup>r</sup>*ð Þ *<sup>d</sup>*�<sup>1</sup> *<sup>=</sup>*2; therefore *<sup>χ</sup>* = (*d*-1)/2. Now the first peak of *<sup>S</sup>*(*Q*) represents the medium-range part of the PDF, beyond the first peak [19], so its decay is slow, reflecting the behavior of the PDF beyond the first peak. This argument proves that the de Gennes narrowing does not necessary imply collective excitations but can be just the natural consequence of geometry.

#### **Figure 1.**

*The r-dependent relaxation time τ(r) for model liquid iron in 2D (red triangle) χ = 0.66, 3D (black circle) χ = 1.04, and 4D χ = 1.45, beyond the first peak. The data points are shown in the form of log{[τ(r)-τ0]/τr} versus log(r/r1) to highlight χ from the expected power law dependence. The short dashed lines serve as guides to the eye [18].*
