## 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].

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 line-shape 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^{4} eV, resolving those energies imposes a resolving power of Δ*E/E* ≤ 10^{−7}. 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

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 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 2*θ*—has sensitive area *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

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.^{1}

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 *IXS* measurements, one can further assume that the beam only illuminates a limited portion of the whole cross-sectional area of the sample. However, the detector has a sensitive area sufficiently small, and a distance from the sample sufficiently large, that the scattered radiation impinging on it is schematizable as a plane wave, having wavevector

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:

where

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

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:

where

The above formula lends itself to a direct estimate of the ideal sample thickness, which is identified by the *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*.

## 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

where *i*th electron, respectively, *V*(**r**_{i}) is the potential acting on the *i*th electron. The above Hamiltonian can be cast in the following perturbative form:

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

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

and the so-called Thomson scattering term:

Here the symbol

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 single incident plane wave and having wavevector pointing to a

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]:

Here the indexes ‘*k’* and ‘α’ label, respectively, the wavevector and the polarization states of the wave; *c* is the speed of light in vacuum and

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

where

with

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

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

## 4. Counting the photon states

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 *θ* within a solid angle

In the reciprocal space, the bandwidth *L* representing the boundary of our scattering problem are:

where

The set of wavevectors defined in Eq. (15) identifies a lattice in *k-*space, whose simplest self-replicating unit cell has a volume *L* is large enough—the number of lattice points within the elemental volume is given by the ratio

Therefore, one has

For photons, the link between energy and wavevector is fixed by the linear law

Therefore,

which, combined with Eq. (11), yields

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

where the vector *j*th atom. The above formula embodies the momentum conservation law as it was derived assuming the identity *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:

Here, *δ-*function term accounts for the energy conservation in the scattering process, as it ensures that

## 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 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 *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

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

where

is the form factor of the *j*th atom. Here *α*th electron in the centre of mass frame of the *j*th atom, while

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

where

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:

where as usual, the angle brackets *j*th and *k*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

where

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

The expression between angle brackets is the equilibrium autocorrelation function of the dynamic variable *k*th and *k*th atom pair, that is, *N* atoms is defined as:

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

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

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

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

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:

and its one-sided time Fourier transform

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:

where

is the spectrum associated with the dynamic variable

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

with

It appears that the spectra of either

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

with

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

where

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:

where

## 8. An estimate of the count rate

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

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

where

### 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

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:

where

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

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.

## Acknowledgments

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.

## Notes

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