## Complex Refractive Index


Generally, optics based on the electromagnetics is described with the refractive index, while such index does not barely appear in X-ray diffraction (XRD) from the kinematical approach, presumably because of the different historical background; however, it is of great importance to see their relationship to understand the diffraction anomalous fine structure (DAFS) method. For example, the energy dependence of $$f\pprime$$ term is not completely equivalent to a linear absorption coefficient, $$\mu$$; $$f\pprime$$ should be divided by the photon energy in order to treat it as the absorption spectrum equivalent to $$\mu$$ in the theoretical framework and analysis of the XAFS field. Thus, the following section briefly describes the relationship between the complex atomic scattering factor derived before and the conventional complex refractive index.

The refractive index is defined as the ratio of the wave numbers in a material and a vacuum as follows:

\begin{align}
\tilde{n} \equiv k / K = 1 – \delta + i\beta = n + i\beta,
\label{Eq:refractive_index}
\tag{1}
\end{align}

where $$K$$ and $$k$$ are the wave numbers in the vacuum and the material, respectively, $$n$$ is the real part of the refractive index and $$\delta$$ corresponds to the its discrepancy from 1, and $$\beta$$ is the imaginary part of the refractive index. Note that the sign in the imaginary part depends on the definition of the wavefunction; the above description is based on the wavefunction of $$\exp \left\{ i \left( \vec{k}\cdot \vec{r} – \omega t \right) \right\}$$. Thus, the wave number in the materials is written as

\begin{align}
k = \tilde{n} K,
\label{Eq:wave_number_in_material}
\tag{2}
\end{align}

and consequently the electric field in the material is calculated as

\begin{align}
E &= E_{0} \exp \left\{ i (kx – \omega t)\right\} \notag \\
%&= E_{0} \exp \left\{ i(\tilde{n} K x – \omega t)\right\} \notag \\
%&= E_{0} \exp \left\{ i (nKx – \omega t) \right\} \exp (-\beta K x) \notag \\
&= E_{0} \exp \left\{ i(Kx-\omega t) \right\} \exp (-i \delta K x) \exp (- \beta K x) .
\label{Eq:wave_in_material}
\tag{3}
\end{align}

The second and the third terms in the last line of the above equation indicate the phase shift and the absorption, respectively. The absorption is further described with $$\beta$$ in the intensity (i.e., proportional to the square of the electric field ) as

\begin{align}
I(x) = I(0) \exp (- 2\beta K x).
\label{Eq:absorption_due_to_refractive_index}
\tag{4}
\end{align}

Since the absorption is also written as $$I(x) = I(0) \exp (- \mu x)$$ with the linear absorption coefficient, $$\mu$$, the relationship between $$\beta$$ and $$\mu$$ is obtained by comparing the exponential parts as

\begin{align}
\beta = \frac{\mu}{2K} = \frac{\lambda} {4\pi} \mu.
\label{Eq:absorption_coefficient}
\tag{5}
\end{align}

The complex refractive index is also written in the form of

\begin{align}
\tilde{n} = \sqrt{\frac{\epsilon \mu_{m}}{\epsilon_{0} \mu_{m0}}},
\label{Eq:full_refractive_index}
\tag{6}
\end{align}

where $$\epsilon$$ and $$\mu_{m}$$ are dielectric constants and magnetic permeability, respectively. A subscript 0 denotes the values in the vacuum. The material is magnetically equivalent to the vacuum. Thus, since $$\mu_{m} = \mu_{m0}$$, the above index can be reduced as

\begin{align}
\tilde{n} = \sqrt{\frac{\epsilon }{\epsilon_{0}}}.
\end{align}

The dielectric constant, $$\epsilon$$, is also related to an electric susceptibility, $$\chi$$, with an equation of

\begin{align}
\epsilon = \epsilon_{0} (1 + \chi_{e}).
\end{align}

### Connection between the refractive index and the scattering factors

The refractive index is related to the atomic scattering factor as follows. Electric dipole moment, $$P_{e}$$, is written as $$P_{e}= \epsilon_{0} \chi_{e} E$$. At the same time, $$P_{e}$$ is also described as $$-n_{s} ex$$, where $$n_{s}$$ is the volume density of the dipoles. Then,

\begin{align}
P_{e} = – n_{s} e x_{0} = \epsilon_{0} \chi_{e} E_{0}.
\end{align}

Therefore, $$\chi_{e}$$ is further calculated with the amplitude of the forced oscillator described in Resonant Scattering as follows:

\begin{align}
\chi_{e} &= \frac{-n_{s} e}{\epsilon_{0} E_{0}} x_{0} \notag \\
&= \frac{-n_{s} e}{\epsilon_{0} E_{0}} \left(- \frac{eE_{0}}{m} \frac{1}{(\omega_{s}^{2} – \omega^{2} -i\omega\Gamma)} \right) \notag \\
&=\frac{n_{s} e^{2} }{\epsilon_{0} m} \frac{1}{(\omega_{s}^{2} – \omega^{2} -i\omega\Gamma)} \notag \\
& = \frac{n_{s} e^{2} }{\epsilon_{0} m \omega^{2}} \frac{- \omega^{2}}{(\omega_{s}^{2} – \omega^{2} -i\omega\Gamma)}.
\end{align}

Furthermore, the last term can be replaced by the atomic scattering factor of the single oscillator. Then,

\begin{align}
\chi_{e} & = – \frac{n_{s} e^{2} \lambda^{2}}{\epsilon_{0} m (2\pi c)^{2}} f_{s} \notag \\
& = – \frac{r_{0} }{\pi} n_{s} \lambda^{2} f_{s}. \notag \\
\end{align}

Thus, we obtain the relationship between the electric susceptibility and the dispersion correction term of the atomic scattering factor.

On the other hand, the complex refractive index is written with the electric susceptibility, $$\chi_{e}$$, by assuming $$\chi_{e} \ll 1$$ and consequently with the dispersion correction term as follows:

\begin{align}
\tilde{n} &= \left( \frac{\epsilon}{\epsilon_{0}} \right)^{\frac{1}{2}} \notag \\
&= \left( 1 + \chi_{e} \right)^{\frac{1}{2}} \notag \\
&\sim 1 + \frac{1}{2} \chi_{e} \notag \\
&= 1- \frac{r_0}{2\pi} \lambda^{2} n_{s} f_{s}.
\label{Eq:refractive_index_with_fs}
\tag{7}
\end{align}

In the scope of the single forced oscillator mode, this refractive index is expressed in the form of

\begin{align}
\tilde{n} = 1- \frac{2\pi r_{0} n_{s} c^{2}}{\omega^{2} – \omega_{s}^{2} + i\Gamma \omega}.
\label{Eq:refractive_index_of_oscillator_model}
\tag{8}
\end{align}

Again, the refractive index is also affected by the bound of the electron to the nucleus as seen in the atomic scattering factor.

Practically, $$f_{s}$$ is replaced by the complex atomic scattering factor determined from experiments, i.e., $$f_j (\vec{Q}, E) = f^{0}_{j}(\vec{Q}) + f’_{j}(E) + i f\pprime_{j}(E)$$. When assuming the forward scattering, i.e., $$\vec{Q} = 0$$, $$f^{0}$$ value is identical to the atomic number, $$Z$$, and then

\begin{align}
\tilde{n}= 1- \frac{r_0}{2\pi} \lambda^{2} \sum_j n_{j} \left( Z_{j} + f’_{j} + if”_{j} \right),
\label{Eq:refractive_index_with_scattering_factor}
\tag{9}
\end{align}

where $$n_{j}$$ denotes the number of atoms of element $$j$$ in a unit volume. If the material is a crystal, Eq. \eqref{Eq:refractive_index_with_scattering_factor} is also rewritten with the unit cell volume, $$v_{c}$$ as

\begin{align}
\tilde{n}= 1- \frac{r_0}{2\pi v_{c}} \lambda^{2} F(Q=0, E) =1- \frac{r_0}{2\pi v_{c}} \lambda^{2} \sum_j \left( Z_{j} + f’_{j} + if\pprime_{j} \right).
\label{Eq:refractive_index_with_structure_factor}
\tag{10}
\end{align}

Thus, the real and imaginary pars of the complex refractive index are obtained by comparing each pat in the above equation and Eq. \eqref{Eq:refractive_index} as follows

\begin{align}
\delta = \frac{r_0}{2\pi v_{c}} \lambda^{2} \sum_j \left( Z_{j} + f’_{j}\right),\ \ \beta = -\frac{r_0}{2\pi v_{c}} \lambda^{2} \sum_j f\pprime_{j}.
\label{Eq:each_parts_of_refractive_index_with_structure_factor}
\tag{11}
\end{align}

Furthermore, $$\mu$$ is linked with $$f\pprime$$ by Eq. \eqref{Eq:absorption_coefficient} by

\begin{align}
\mu = \frac{4\pi}{\lambda} \beta = \frac{2\lambda r_{0}}{v_{c}} \sum_j \left( -f\pprime_{j} \right).
\label{Eq:mu_and_f_double_prime}
\tag{12}
\end{align}

Therefore, $$f\pprime$$ obtained from the DAFS method is completely equivalent to $$\mu$$ by multiplying wavelength, $$\lambda$$, to $$f\pprime$$ (or divided by the photon energy). Importantly, $$f\pprime$$ is a negative value because $$\mu$$ is positive from Eq. \eqref{Eq:mu_and_f_double_prime}. Conventionally, $$f\pprime$$ appears as a positive value in many textbooks and/or tables. It causes no problem as long as we discuss the diffraction intensity, where only the square of $$f\pprime$$ appears in the intensity; however, it is of importance for the DAFS method to distinguish the sign of $$f\pprime$$ values because we need to extract the site- and/or phase dependent $$f\pprime$$ value itself by directly solving the simultaneous equation of the weighted $$f\pprime$$ values.

###### For further study…(This article was written based on the following books)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
• 菊田惺志(S. Kikuta), X線散乱と放射光科学 基礎編 (X-ray scattering and synchrotron radiation science -basics- (English title was translated by T.K.)), 東京大学出版 (University of Tokyo Press), 2011.

## Resonant Scattering


### Resonant scattering term

In the previous article “scattering by one electron”, the classical Thomson scattering from an extended distribution of free electrons is derived as $$-r_{0}f^{0}(\vec{Q})$$, where $$-r_{0}$$ is the Thomson scattering length of a single element and $$f^{0}(\vec{Q})$$ is the atomic form factor. The atomic form factor is a Fourier transform of the electron distribution in an atom; therefore, it is a real number and independent of photon energy. In contrast, there exists an absorption edge and a fine structure in an absorption spectrum in the x-ray region. Thus, the absorption term should be included into the scattering length as an imaginary part, which is proportional to the absorption cross-section, by assuming a more elaborated model rather than that of a cloud of free electrons.

As energy dependent terms from the forced oscillator model will be derived in the following discussion, an atomic scattering factor consists of real and imaginary energy-dependent terms as well as the conventional atomic form factor as follows:

\begin{align}
f(\vec{Q}, E) = f^{0} (\vec{Q}) + f'(E) + i f\pprime(E),
\label{Eq:complex_scattering_factor}
\tag{1}
\end{align}

where $$f’$$ and $$f\pprime$$ are the real and imaginary parts of the dispersion corrections. These terms are called resonant scattering terms, while at one time it was conventionally referred to as anomalous scattering factors. Usually $$\vec{Q}$$ dependence of the resonant scattering terms are negligible, because the dispersion corrections are dominated by electrons in the core shell such as $$K$$ shell, which is spatially confined around an atomic nucleus.

Fig. 1 shows the theoretical curve of the resonant scattering terms of a bare Ni atom. $$f’$$ shows local minimum cusps at each absorption edge, while $$f\pprime$$, which should be , strictly speaking, negative values, has absorption edges as described in the XAFS section. These features of the resonant terms enable us to carry out various sophisticated x-ray scattering techniques, e.g., Multi-wavelength Anomalous Diffraction (MAD) for the determination of the unique crystalline structure without suffering from the phase problem [1]. Furthermore, by using the polarization and azimuthal dependences of the resonant terms, the resonant scattering techniques also contribute to the determination of spin and orbital orders seen in a strongly-correlated electron system and enantiomeric materials such as quartz [2, 3]. The structural analysis is usually carried out by hard x-ray (typically, >5 keV), where $$K$$ absorption edges of third-period elements locates as shown in Fig. 2.

Thus, we can evaluate the occupations of the similar elements such as Fe, Co and Ni in a crystal by the resonant scattering technique thanks to the characteristic of steep decrease in $$f’$$ at the each absorption edge, while the nonresonant x-ray diffraction technique hardly distinguishes the contributions from similar elements to a certain crystallographic site. This kind of approach to extract the structural information at a specific element is also applicable to the structural analysis of the amorphous materials by total scattering measurements [4, 5, 6]. Importantly, the diffraction anomalou fine structure (DAFS) method is also one of measurement techniques utilizing this resonant feature. In this method, we observe the $$f\pprime$$ and $$f’$$ spectra, which also reflect the fine structure as seen in a X-ray absorption fine structure (XAFS) spectrum of $$f\pprime$$, through the scattering channel.

In this article, a forced charged oscillator model is introduced to explain the basic principles behind how $$f’$$ and $$f\pprime$$ appear in the atomic scattering factor. This model is obviously classical and a crude approximation; however, it can help us to understand the relationship between $$f’$$ and $$f\pprime$$, i.e., scattering and absorption.

### The forced charged oscillator model

Suppose that an electron bound in an atom is subjected to the electric field of an incident x-ray beam, $$\vec{E}_{in} = \hat{\vec{x}} E_{0} \e^{-i\omega t}$$, which is linearly polarized along the $$x$$ axis with amplitude $$E_{0}$$ and frequency $$\omega$$. The motion equation for this electron is

\begin{align}
\ddot{x} + \Gamma \dot{x} + \omega_{s}^{2}x = – \left( \frac{eE_{0}}{m}\right) \e^{-i\omega t},
\label{Eq:forced_motion_electron}
\tag{2}
\end{align}

where $$\Gamma \dot{x}$$ is the velocity-dependent damping term corresponding to the dissipation of energy from the applied electric field due to the re-radiation, $$\omega_{s}$$ is the resonant frequency usually much larger than the damping constant, $$\Gamma$$. The solution for the differential equation is described as $$x(t) = x_{0}\e^{-i\omega t}$$ and consequently the amplitude of the forced oscillation is

\begin{align}
x_{0} = – \left( \frac{eE_{0}}{m}\right) \frac{1}{\omega_{s}^{2} – \omega^{2} -i\omega\Gamma}.
\label{Eq:coefficient_of_solution}
\tag{3}
\end{align}

The radiated field for an observer at a distance $$R$$ and at time $$t$$ is proportional to the $$\ddot{x}(t-R/c)$$ at the earlier time $$t’ = t -R/c$$; therefore,

\begin{align}
E_{\mathrm{rad}}(R,t) = \left( \frac{e^{2}}{4\pi \epsilon_{0} m c^{2} }\right) \ddot{x}(t-R/c),
\end{align}

where the polarization factor $$\hat{\vec{\epsilon}}\cdot \hat{\vec{\epsilon}}’$$ is assumed to be 1. By inserting the specific value of $$\ddot{x}(t-R/c)$$ calculated from $$x(t) = x_{0}\e^{-i\omega t}$$ and Eq. \eqref{Eq:coefficient_of_solution}, the above equation is expanded to be

\begin{align}
E_{\mathrm{rad}}(R,t) = \frac{\omega^{2}}{\omega_{s}^{2} – \omega^{2} -i\omega\Gamma} \left( \frac{e}{4\pi \epsilon_{0} R c^{2} }\right) E_{0} \e^{-i\omega t} \left( \frac{\e^{ikR}}{R} \right)
\end{align}

or equivalently

\begin{align}
\frac{E_{\mathrm{rad}}(R,t)}{E_{\mathrm{in}}} = -r_{0}\frac{\omega^{2}}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma} \left( \frac{\e^{ikR}}{R} \right).
\end{align}

The atomic scattering length, $$f_{s}$$, is defined to be the amplitude of the outgoing spherical wave, $$(\e^{ikR}/R)$$. (cf. scattering by one electron) Thus, $$f_{s}$$ in units of $$-r_{0}$$ is

\begin{align}
f_{s} = \frac{\omega^{2}}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma},
\label{Eq:resonant_terms_of_single_oscillator}
\tag{4}
\end{align}

where a subscript, $$s$$, denotes the “single oscillator”. For frequencies greatly larger than the resonant frequency, i.e., $$\omega \gg \omega_{s}$$, the value of $$f_{s}$$ should be approximated to be the Thomson scattering length of 1. Thus, the following reduction makes the equation clearer to understand the resonant scattering terms:

\begin{align}
f_{s} &= \frac{\omega^{2} – \omega_{s}^{2} + i\omega \Gamma + \omega_{s}^{2} – i\omega \Gamma}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma}
= 1 + \frac{ \omega_{s}^{2} – i\omega \Gamma}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma} \notag \\
&\sim 1 + \frac{ \omega_{s}^{2}}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma},
\label{Eq:fs_reduction}
\tag{5}
\end{align}

where the last line is derived based on the fact that $$\Gamma$$ is usually much less than $$\omega_{s}$$. Eq. \eqref{Eq:fs_reduction} clearly shows that the second term corresponds to the dispersion correction to the scattering factor. When the dispersion correction is written as $$\Delta f (\omega)$$, the value is described as

\begin{align}
\Delta f ( \omega) = f’_{s} + if\pprime_{s} = \frac{ \omega_{s}^{2}}{\omega^{2} – \omega_{s}^{2} + i\omega\Gamma}
\end{align}

with the real part given by

\begin{align}
f’_{s} = \frac{ \omega_{s}^{2} ( \omega^{2} – \omega_{s}^{2} )}{(\omega^{2} – \omega_{s}^{2} )^{2} + (\omega\Gamma)^{2}}
\label{Eq:dispersion_correction_real}
\tag{6}
\end{align}

and the imaginary part also given by

\begin{align}
f\pprime_{s} = – \frac{ \omega_{s}^{2} \omega \Gamma}{(\omega^{2} – \omega_{s}^{2})^{2} + (\omega\Gamma)^{2}}.
\label{Eq:dispersion_correction_imaginary}
\tag{7}
\end{align}

The dispersion correction terms calculated from the forced oscillator model are shown in Fig. 3. The imaginary part of the dispersion correction, $$f\pprime$$, corresponds to the absorption, showing the peak profile at $$\omega = \omega_{s}$$. In contrast, the absorption spectrum of a real material is like an edge rather than the peak; the discrepancy between two spectra shows the limitation of the single forced oscillator model. In order to model this behavior, we need to take into account so-called oscillator strength, $$g_{o}(\omega_{s})$$, to compensate the gap between the model and a real material, which gives the population of the single oscillators dependent on the photon energy; however, it still gives no explanation of XAFS (oscillation characteristic observed in the condensed material). Eventually, though the single oscillator model is not adequate for the quantitative understanding of the resonant scattering terms, it is helpful to understand the emergence of the dispersion correction terms due to the bound of the electron to the nucleus. The quantitative explanation and its evaluation require more sophisticated approach of quantum mechanics, where $$f’$$ and $$f\pprime$$ are derived from the 1st- and 2nd-order perturbation theory of the interaction Hamiltonian of $$(e \vec{A} \cdot \vec{p} / m)$$.

###### For further study…(This article was written based on the following books)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
• 菊田惺志(S. Kikuta), X線散乱と放射光科学 基礎編 (X-ray scattering and synchrotron radiation science -basics- (English title was translated by T.K.)), 東京大学出版 (University of Tokyo Press), 2011.
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## Calculation of a structure factor

A structure factor is calculated by summing up scattering factors of each atom with multiplying the phases at each atomic position in a unit cell as described previously in an equation. It is a facile approach to calculate the structure factor; however, the calculation becomes complicated when the unit cell includes the large number of atoms. Furthermore, it is frequently difficult to distinguish the equivalent and non-equivalent sites in this approach.

For example in body-centered cubic (BCC) metal, atom occupy (0, 0, 0), (1/2, 1/2, 1/2) sites, and all of them are equivalent. Thus, these atoms must have exactly the same local/electronic structure. In contrast, in cesium chloride structure, (0, 0, 0) and (1/2, 1/2, 1/2) are occupied by chloride and cesium atoms, respectively, being inequivalent each other; therefore, the local/electronic structure must be different. The “site-distinguished” analysis of the diffraction anomalous fine structure (DAFS) method provides individual XAFS spectra at the inequivalent sites in the material as seen in the latter case. These relations and concepts are comprehensively understood in the framework of Space Group, which is well-described in the International Tables for Crystallography vol. A[1]. Any crystalline materials except for a quasicrystal belong to a certain space group, and the structure factor of each space group is calculated and included in the international table [2].1 Thus, the calculation of the structure factor should be carried out based on the crystallographic site in space group, which is more versatile and convenient.

The table provides the structure factor as values of “$$A(\vec{G}\cdot \vec{r}_{m})$$” and “$$B(\vec{G}\cdot \vec{r}_{m})$$”, whose definition is as follows:

\begin{align}
A(\vec{G}\cdot \vec{r}_{m}) &= \sum_{e} \cos (\vec{G}\cdot \vec{r}_{m}) \\
B(\vec{G}\cdot \vec{r}_{m}) &= \sum_{e} \sin (\vec{G}\cdot \vec{r}_{m}),
\label{Eq:def_A_and_B}
\tag{1}
\end{align}

where $$\sum_{e}$$ denotes the summation of equivalent positions belonging the site in a unit cell. Then, the structure factor is described with these $$A$$ and $$B$$ values as

\begin{align}
F(\vec{G}) = \sum_{j} f_{j}A_{j} + i \sum_{j} f_{j}B_{j},
\label{Eq:A_B_based_structure_factor}
\tag{2}
\end{align}

where $$\sum_{j}$$ is the summation of independent sites, and $$f_{j}$$ is the atomic scattering factor of the atom at site $$j$$. Note that both $$A$$ and $$B$$ values should be multiplied by the ratio of the numbers of atoms at general and special positions. For example, when calculating the structure factor of $$32e$$ site (Wyckoff letter for a special position) in space group $$F d \bar{3} m$$ (No. 227), the factor of 32/192 ($$192i$$ is for general position in this space group) should be multiplied in order to take into account the overlap of the atoms at the special position. Also note that the value of $$B$$ disappears in the space group with centrosymmetry, because $$\sum_{e} \sin (\vec{G}\cdot \vec{r}_{m})$$ becomes 0 when the same element locates at $$\vec{r}_{m}$$ and $$-\vec{r}_{m}$$ due to the nature of $$\sin$$ function.

###### References
[1] International Tables for Crystallography Volume A, 5th ed., T. Hahn, Ed., Springer, 2006.
[Bibtex]
@book{Hahn2006,
edition = {5th},
editor = {Hahn, Th},
file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Unknown/Unknown\_Unknown\_2006.pdf:pdf},
isbn = {0792365909},
publisher = {Springer},
title = {{International Tables for Crystallography Volume A}},
year = {2006}
}
[2] International Tables for X-ray Crystallography Volume I, N. F. M. Henry and K. Lonsdale, Eds., Birmingham: Kynoch Press, 1952.
[Bibtex]
@book{Henry1952,
editor = {Henry, Norman F. M and Lonsdale, Kathleen},
file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Unknown/Unknown\_Unknown\_1952.pdf:pdf},
pages = {367},
publisher = {Kynoch Press},
title = {{International Tables for X-ray Crystallography Volume I}},
year = {1952}
}

## Lorentz factor

In the previous articles, the scattered intensity and related factors are derived. In an actual experiment of diffraction anomalous fine structure (DAFS) as well as the conventional X-ray diffraction (XRD), we need to “scan” a sample and/or a finite-size detector to obtain the whole shape of the diffraction peak, whose area is proportional to the structure factor described above. Thus, the factor concerning $$\vec{Q}$$ step2 and/or the fraction of detective diffraction, which are dependent on scattering angle and also energy, should be introduced for the evaluation of the scattered intensity, being called the Lorentz factor. This factor is essentially quite different from the other factors described above, i.e., atomic scattering factor, structure factor and Debye-Waller factor, because the Lorentz factor gives no structural information about the crystalline sample, being derived only from the experimental aspect. Thus, the effect of the Lorentz factor should be corrected for the subsequent structural and spectroscopic analyses using X-ray diffraction.

In the Lorentz factor appearing in a textbook and/or articles regarding the conventional XRD, only the angular dependence is focused and discussed, presumably because such a measurement is carried out by a single monochromatic x-ray. In contrast, the correction of the energy-dependence in the Lorentz factor is necessary in the spectroscopic analysis like the DAFS method. Thus, the following article briefly describes the derivation of the Lorentz factor including its energy dependence as well as that of the scattering angle.

### Single crystal diffraction

Let’s start from the case of a single crystal. A schematic of the experimental setup to measure the integrated intensity of diffraction from a single crystal particle is shown in Fig. 1. In the calculation, it is assumed that the incident and scatted beam are monochromatic, and that the incident is perfectly collimated while the scatted beam necessarily is not perfectly collimated, because the number of the lattice $$N$$ is finite and the beam will have some divergence. Left-hand side of Fig. 1 shows the schematic of the reciprocal space around a reciprocal lattice point, where only the portion of the point on the Ewald’s sphere, which is a sphere depicted by the possible terminal points of the outgoing wave number vector $$\vec{k’}$$, is observed at a certain incident angle (purple lines). Thus, the crystal has to be rotated (i.e., rocking scan) to obtain the integrated intensity from the reciprocal lattice points, which is drawn with watery, purple, pink lines, on an axis of $$\theta$$.

As shown in the previous article, the Laue function becomes the Dirac’s delta function when the number of the lattices is sufficiently large; therefore, the integrate intensity is described as follows:

\begin{align}
\int \diff \hat{\vec{k}}’ \delta (\vec{Q}-\vec{G} ) = \int \diff \hat{\vec{k}}’ \delta(\vec{k} – \vec{k}’ – \vec{G}),
\label{Eq:integration_of_k_LF}
\tag{1}
\end{align}

where $$\hat{\vec{k}}’$$ is a unit vector along $$\vec{k}’$$. The element of the solid angle $$\diff \hat{\vec{k}}’$$ is two-dimensional vector, which means the integration of the all scattered angles under a fixed incident angle. For the calculation, the vector, $$\vec{s} = k’ \hat{\vec{s}}$$, is introduced instead of $$\vec{k}’$$, where $$\hat{\vec{s}}$$ is a unit vector (see Fig.2). Then, the integration is transformed by adding the integration of the unit value2 into

\begin{align}
\tag{2}
\int \diff \hat{\vec{k}}’ \delta (\vec{k} – \vec{k}’ -\vec{G} ) = \overbrace{\frac{2}{k’} \int s^{2} \delta(s^{2} – k’^{2}) \diff s}^{1}\int \delta(\vec{k} – \vec{s} – \vec{G} ) \diff \hat{\vec{s}},
\end{align}

where $$\vec{k}’$$ is replaced by $$\vec{s}$$ in the second integration. The trick of adding the integration and the change of variable is to transform the integration of two-dimension into that of three-dimension.

Based on the schematic of the integration parameter in Fig. 2, the above equation is furthered transformed into

\begin{align}
\int \diff \hat{\vec{k}}’ \delta (\vec{k} – \vec{k}’ -\vec{G} ) &= \frac{2}{k’} \int \delta(s^{2} – k’^{2}) \delta(\vec{k} – \vec{s} – \vec{G} ) \diff \hat{\vec{s}}\diff s \notag \\
&= \frac{2}{k’} \int \delta(s^{2} – k’^{2}) \delta(\vec{k} – \vec{s} – \vec{G} ) \diff \vec{s}.
\label{Eq:int_para_change_LF}
\tag{3}
\end{align}

When $$\vec{s} = \vec{k} – \vec{G}$$, $$\delta(\vec{k} -\vec{s} – \vec{G}) = \delta (0)$$ and the integration is reduced to be

\begin{align}
\int \diff \hat{\vec{k}}’ \delta (\vec{k} – \vec{k}’ -\vec{G} ) &= \frac{2}{k’} \delta((\vec{k} – \vec{G})\cdot (\vec{k} – \vec{G}) – k’^{2}) \notag \\
&= \frac{2}{k} \delta(G^2 – 2kG \sin \theta ).
\label{Eq:int_result_LF}
\tag{4}
\end{align}

Finally, the preparation for the integration of $$\theta$$, which means the evaluation of the integrated intensity under the rocking scan, is completed.

The differential cross section of the diffraction is described by using the result above as

\begin{align}
\label{Eq:cross_section_LF}
\tag{5}
\left( \frac{\diff \sigma}{\diff \Omega}\right)_{\mathrm{int. \ over\ } \vec{k}’} = r_{0}^{2}P |F(\vec{Q})|^{2} N v_{c}^{*} \frac{2}{k}\delta(G^{2} -2kG \sin \theta).
\end{align}

Because the integration of the delta function by $$\theta$$ gives the following value

\begin{align}
\label{Eq:int_delta_func_LF}
\tag{6}
\int \delta(G^{2} -2kG \sin \theta) \diff \theta = \left[ \frac{-1}{2kG \cos \theta} \right] _{t = 0} = \frac{-1}{2k^{2} \sin 2 \theta},
\end{align}

The cross-section is further derived as

\begin{align}
\left( \frac{\diff \sigma}{\diff \Omega}\right)_{\mathrm{int. \ over}\ \vec{k}’, \theta} &= r_{0}^{2}P |F(\vec{Q})|^{2} N v_{c}^{*} \frac{2}{k} \frac{1}{2 k^{2} \sin 2 \theta} \notag \\
&= r_{0}^{2}P |F(\vec{Q})|^{2} N \frac{\lambda^{3}}{v_{c}} \frac{1}{\sin 2 \theta}
\label{Eq:cross_section_result_LF}
\tag{7}
\end{align}

and detective intensity is

\begin{align}
I_{\mathrm{SC}}\left( \mathrm{ \frac{photons}{sec}} \right) = \Phi_{0} \left( \mathrm{ \frac{photons}{unit\ area \times sec}} \right) r_{0}^{2}P |F(\vec{Q})|^{2} N \frac{\lambda^{3}}{v_{c}} \frac{1}{\sin 2 \theta}.
\label{Eq:scat_intensity_result_LF}
\tag{8}
\end{align}

Therefore, when we discuss the energy dependency of the DAFS spectrum, we need to correct the factor, $$\lambda^{3}/\sin 2 \theta$$ or $$1/(E^{3} \sin 2 \theta)$$, in a single crystal.

### Powder diffraction

In the powder diffraction, the Lorentz factor is decomposed into three parts as follows:

\begin{align}
\label{Eq:three_LF_in_XRPD}
\tag{9}
L(\theta, E) = L_{1}L_{2}L_{3},
\end{align}

where $$L_{1} = 1/ ( E^{3} \sin 2\theta )$$ same as that in single crystal, and both $$L_{2}$$ and $$L_{3}$$ are additional Lorentz factors for the powder diffraction, which will be subsequently introduced.

$$L_{2}$$ is derived from the angle dependence of the number of the observable crystalline particles. A sample for X-ray powder diffraction (XRPD) is randomly oriented crystallites, whose scattered intensity is described as a simple summation of the scattering intensity from each small crystalline particle. Thus, the scattered intensity is dependent on the number of the observable reciprocal lattice points at the same time. The sphere of the numerous reciprocal lattice point, whose radius is $$|\vec{G}| = 2\pi / d \equiv G$$, and the observable area on the “reciprocal sphere” drown as a ribbon are shown in Figure 3. When it is supposed that the number of the crystalline particle is $$N$$ and their reciprocal lattice points homogeneously distribute on the sphere, the angle dependence of the area of the ribbon corresponds to the $$L_{2}$$ value.

Straight line CP is a perpendicular to the lattice plane of a crystalline particle we observe, and $$\Delta \theta$$ is an acceptable angle of the diffraction derived from the divergence and the energy width of the incident beam. Namely, the particles whose reciprocal points locate among the range of $$\Delta \theta$$ satisfy the diffraction condition. Thus, from the geometric consideration of Fig. 3, the number, $$\Delta N$$, is described as

\begin{align}
\Delta N &= G \Delta \theta 2 \pi G \sin (\pi – \theta) \notag \\
&= 2\pi G^{2} \Delta \theta \cos \theta .
\label{Eq:number_of_particle_on_the_ribbon}
\tag{10}
\end{align}

Since the area of the whole sphere is $$4\pi G^{2}$$, the fraction of the observable number of the particles is

\begin{align}
\label{Eq:ratio_of_the_number_of_particles}
\tag{11}
\frac{\Delta N}{N} = \frac{\Delta \theta \cos \theta}{2}.
\end{align}

Therefore, the integrated intensity of XRPD is proportional to the factor:

\begin{align}
\label{Eq:L2_factor}
\tag{12}
L_{2} = \cos \theta.
\end{align}

The factor of $$L_{3}$$ is from the observation technique of XRPD, where we usually observe a portion of the Debye-Scherrer ring, by scanning a detector with a finite-size sensitive area. When the camera length, i.e., the distance between the detector and the sample, is $$R$$, the radius of the Debye-Sherrer ring at the detector position is $$R \sin 2 \theta$$, and consequently that of the length is $$2 \pi R \sin 2 \theta$$ as shown in Fig. 4.

If we observe the portion of the Debye-Scherrer ring of $$\delta R$$, the ratio of these lengths, i.e., $$\delta R/ 2\pi R \sin \theta$$, corresponds to the observable scattered intensity.
Thus, the integrated intensity is proportional to the factor:

\begin{align}
\label{Eq:L3_factor}
\tag{13}
L_{3} = \frac{1}{\sin 2 \theta}.
\end{align}

Finally, we obtain the complete Lorentz factor for XRPD and powder-DAFS as follows:
\begin{align}
L(\theta, E) &= L_{1} L_{2} L_{3} \notag \\
&= \frac{1}{E^{3}\sin 2\theta} \cos \theta \frac{1}{\sin 2\theta} \notag \\
&= \frac{1}{4 E^{3}\sin ^{2} \theta \cos \theta}.
\label{Eq:full_lorentz_factor_for_XRPD}
\tag{13}
\end{align}

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}

## Debye-Waller factor


The lattice has been assumed to be “perfectly rigid” in the evaluation of the scattering amplitude from the crystal in the previous article; however, the atoms vibrate due to two distinct causes in a real material. The first is from the uncertainty principle of the quantum mechanics, which is independent on temperature and observed even at 0 K, being called zero-point fluctuation. The second is from the elastic wave and/or phonon in the crystal, depending on the temperature. Whether the vibration is caused by above two mechanisms, the atomic vibration reduces the magnitude of the interference of the scattering wave from the different atoms due to the “ambiguity” of the atomic position, eventually decreasing the scattering amplitude. This attenuation factor is known as the Debye-Waller factor in x-ray diffraction.

The Debye-Waller factor is affected by some factors. The magnitude of the attenuation basically depends on the element; a heavier element shows small attenuation at a certain temperature. Furthermore, the attenuation magnitude is also dependent on the crystallographic site even when the same element is occupied. In addition, this vibration effect is enhanced in the higher scattering angle, i.e., small lattice spacing, because the scattering at the higher angle is more sensitive to the phase difference than that at the lower angle. Usually, the Debye-Waller factor is implemented into the structure factor by multiplying an exponential attenuation term, whose derivation will be given in the subsequent section. The site-selectivity of the diffraction anomalous fine structure (DAFS) originates from the difference in the contribution of the atoms to the a certain diffraction as described in the derivation of the structure factor. Thus, this factor should be included into the DAFS analysis to accurately separate each contribution of the atom in the different crystallographic sites.

For the sake of simplicity of the derivation, the scattering amplitude of a crystal consisting of a single element with some displacement from the average position is evaluated as follows:

\begin{align}
\label{Eq:DW_cal_displacement}
\tag{1}
F^{\mathrm{crystal}}(\vec{Q}) = \sum_{n} f(\vec{Q})\e^{i\vec{Q}\cdot (\vec{R}_{n} + \vec{u}_{n})},
\end{align}

where $$\vec{R}_{n} + \vec{u}_{n}$$ is the instantaneous position of the atom, $$\vec{R}_{n}$$ is the time-averaged mean position, and $$\vec{u}_{n}$$ is the displacement, which temporal average value, $$\left< \vec{u}_{n} \right>$$, is zero from the definition. Since the scattering intensity is calculated by taking the product of the scattering amplitude and its complex conjugate, the time-average scattering intensity is

\begin{align}
\label{Eq:time_average_intensity}
\tag{2}
I &= \left\langle \sum_{m} f(\vec{Q})\e^{i\vec{Q}\cdot (\vec{R}_{m} + \vec{u}_{m})} \sum_{n} f^{*}(\vec{Q})\e^{-i\vec{Q}\cdot (\vec{R}_{n} + \vec{u}_{n})} \right\rangle \notag \\
&= \sum_{m} \sum_{n} f(\vec{Q}) f^{*}(\vec{Q}) \e^{i\vec{Q}\cdot (\vec{R}_{m} -\vec{R}_{n})} \left\langle \e^{i\vec{Q}\cdot (\vec{u}_{m} -\vec{u}_{n})} \right\rangle.
\end{align}

For the further calculation, the last term of the second row in the equation is rewritten as

\begin{align}
\label{Eq:DW_dimension_reduction}
\tag{3}
\left\langle \e^{i\vec{Q}\cdot (\vec{u}_{m} -\vec{u}_{n})} \right\rangle = \left\langle \e^{iQ (u_{Qm} – u_{Qn})} \right\rangle,
\end{align}

where $$u_{Qn}$$ is the component of the $$\vec{u}_{n}$$ parallel to the vector, $$\vec{Q}$$, for the $$n$$-th atom. By using the Baker-Hausdorff theorem expressed as

\begin{align}
\label{Eq:BH_theory}
\tag{4}
\left\langle \e^{ix} \right\rangle = \e^{-\frac{1}{2} \left\langle x^{2} \right\rangle},
\end{align}

the right hand side in Eq. \eqref{Eq:DW_dimension_reduction} is reduced to be

\begin{align}
\left\langle \e^{iQ (u_{Qm} – u_{Qn})} \right\rangle &= \e^{-\frac{1}{2} \left\langle Q^{2}(u_{Qm}-u_{Qn})^{2} \right\rangle} \notag \\
&= \e^{-\frac{1}{2} Q^{2} \left\langle (u_{Qm}-u_{Qn})^{2} \right\rangle} \notag \\
&= \e^{-\frac{1}{2} Q^{2} \left\langle u_{Qm}^{2} \right\rangle} \e^{-\frac{1}{2} Q^{2} \left\langle u_{Qn}^{2} \right\rangle} \e^{Q^{2} \left\langle u_{Qm}u_{Qn} \right\rangle}.
\tag{5}
\label{Eq:reduction_of_u}
\end{align}

Because of the translation symmetry, $$u_{Qn}^{2} = u_{Qm}^{2}$$ and its value will be simply expressed as $$u_{Q}^{2}$$. In addition $$\e^{-Q^{2} \left\langle u_{Q}^{2} \right\rangle /2}$$ is also expressed as $$\e^{-M}$$ in the following derivation. In order to separate the scattering intensity into two terms, the last term of Eq. \eqref{Eq:reduction_of_u} is written as

\begin{align}
\label{Eq:correlated_vibration}
\tag{6}
\e^{Q^{2} \left\langle u_{Qm}u_{Qn} \right\rangle} = 1 + \left\{ \e^{Q^{2} \left\langle u_{Qm}u_{Qn} \right\rangle} – 1 \right\}.
\end{align}

Then, the scattered intensity is decomposed into two terms as

\begin{align}
I &= \sum_{m} \sum_{n} f(\vec{Q}) \e^{-M} \e^{i\vec{Q}\cdot \vec{R}_{m}} f^{*}(\vec{Q}) \e^{-M} \e^{-i\vec{Q}\cdot \vec{R}_{n}} \notag \\
&+ \sum_{m} \sum_{n} f(\vec{Q}) \e^{-M} \e^{i\vec{Q}\cdot \vec{R}_{m}} f^{*}(\vec{Q}) \e^{-M} \e^{-i\vec{Q}\cdot \vec{R}_{n}} \left\{ \e^{Q^{2} \left\langle u_{Qm}u_{Qn} \right\rangle} – 1 \right\}.
\label{Eq:decomposed_scattered_intensity}
\tag{7}
\end{align}

The first term is the elastic scattering from a lattice, i.e., x-ray diffraction; however the scatted intensity is weaken by the factor of $$\e^{-M} ( < 1)$$, which is known as the Debye-Waller factor. This factor can be generally introduced by replacing the atomic scattering factor by

\begin{align}
\label{Eq:introduction_of_DW_factor}
\tag{8}
f^{\mathrm{atom}} = f^{0}(\vec{Q}) \e^{- \frac{1}{2} Q^{2} \langle u_{Q}^{2} \rangle} \equiv f^{0}(\vec{Q}) \e^{-M}.
\end{align}

Conventionally, the magnitude of the Debye-Waller factor is given and discussed in the form of $$B_{T}$$ as

\begin{align}
\label{Eq:introduction_of_B}
\tag{9}
M = \frac{1}{2}Q^{2}\langle u_{Q}^{2} \rangle = \frac{1}{2} \left( \frac{4\pi \sin \theta}{\lambda} \right)^{2} \langle u_{Q}^{2} \rangle = B_{T} \left( \frac{\sin \theta}{\lambda} \right)^{2},
\end{align}

with

\begin{align}
\label{Eq:definition_of_B}
\tag{10}
B_{T} \equiv 8\pi^{2} \langle u_{Q}^{2} \rangle,
\end{align}

because of the traditional reason of the XRD description, where the angle dependence of a parameter is favorably expressed as a function of $$(\sin \theta / \lambda)$$ rather than $$Q$$ (for example, the atomic form factor is also given in the above form in equation in the previous article. If the atoms vibrate isotropically, $$\langle u^{2} \rangle = \langle u^{2}_{x} + u^{2}_{y} + u^{2}_{z} \rangle = 3 \langle u^{2}_{x} \rangle = 3 \langle u_{Q}^{2} \rangle$$, then

\begin{align}
\label{Eq:isotropic_DW_factor}
\tag{11}
B_{T, \mathrm{isotropic}} = \frac{8 \pi^{2}}{3}\langle u^{2} \rangle.
\end{align}

Though the deviation above proceeded on the assumption of the single element, the structure factor of plural elements is analogically derived as

\begin{align}
\label{Eq:structure_factor_with_DW_factor}
\tag{12}
F &= \sum_{m} f_{m} \exp \left( -M_{m}\right) \exp \left( i\vec{Q}\cdot \vec{r}_{m}\right) \\
&= \sum_{m} f_{m} \exp \left\{ -B_{T, m} \left( \frac{\sin \theta}{\lambda}\right)^{2} \right\} \exp \left( i\vec{Q}\cdot \vec{r}_{m}\right).
\end{align}

The magnitude of the Debye-Waller factor of each element can be evaluated by a preliminary XRD analysis such as the Rietveld analysis based on the XRPD. Typical values are available on International tables for x-ray crystallography vol. II, ranging from 0 to 2. The refined value should be used for the site-separation of the absorption spectrum obtained from the DAFS method.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}

## Scattering from a crystal

In a crystal, atoms or molecules form a periodic structure with the translational symmetry, which is frequently called long range order. Thus, scatterings from each atom are interfered and eventually create a diffraction pattern, which reflects the crystalline structure. Since each diffraction has a different contribution from each atom, the diffraction anomalous fine structure (DAFS) method is capable to distinguish the crystalline site-specific spectroscopic information by measuring an energy dependence of the diffraction intensity. This article will provide a brief review about the description of the conventional x-ray diffraction.

As seen in the discussion of the scattering from an atom, the scattering amplitude from plural atoms, i.e., a crystal, are similarly calculated from the summation of phases multiplied by atomic scattering factors of each atom. First, the position of an atom in the crystal is described by taking the translational symmetry into account as

\begin{align}
\label{Eq:position_vector}
\tag{1}
\vec{r}_{l}= \vec{R}_{n} + \vec{r}_{m},
\end{align}

with

\begin{align}
\label{Eq:atomic_position_vector_definition}
\tag{2}
\vec{R}_{n} &\equiv n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3} \\
\vec{r}_{m} &\equiv x_{m}\vec{a}_{1}+y_{m}\vec{a}_{2}+z_{m}\vec{a}_{3},
\end{align}

where $$\vec{R}_{n}$$ is a vector specifying the number $$n$$-th unit cell, $$\vec{r}_{m}$$ is a vector indicating $$m$$-th atomic position in a unit cell, $$\vec{a}_{1}$$, $$\vec{a}_{2}$$ and $$\vec{a}_{3}$$ are lattice vectors along with $$a$$, $$b$$ and $$c$$ axes, and $$x_{m}$$, $$y_{m}$$ and $$z_{m}$$ are the fractional coordinate of $$m$$-th atom. Thus, the scattering amplitude from the crystal can be described as

\begin{align}
\label{Eq:scat_amp_all_atom}
\tag{3}
F^{\mathrm{crystal}}(\vec{Q}) = \sum^{\mathrm{All\ atoms}}_{l}f_{l}(\vec{Q})\e^{i\vec{Q}\cdot \vec{r}_{l}},
\end{align}

where $$f_{l}(\vec{Q})$$ is the atomic form factor3 of the atom placed at position $$\vec{r}_{l}$$. The scattering amplitude is readily decomposed into two contributions from the lattice and that from the inside of the unit cell with Eq. \eqref{Eq:position_vector} as

\begin{align}
\label{Eq:decomposition_of_scattering_amplitude}
\tag{4}
F^{\mathrm{crystal}}(\vec{Q}) = \sum^{\mathrm{All\ atoms}}_{\vec{R}_{n}+\vec{r}_{j}} f_{j}(\vec{Q})\e^{i\vec{Q}\cdot(\vec{R}_{n}+\vec{r}_{j})} = \overbrace{\sum_{n}\e^{i\vec{Q}\cdot\vec{R}_{n}}}^{\mathrm{Lattice}} \overbrace{\sum_{m}f_{j}(\vec{Q})\e^{i\vec{Q}\cdot\vec{r}_{m}}}^{\mathrm{Unit\ cell}}
\end{align}

The first term of Eq. \eqref{Eq:decomposition_of_scattering_amplitude} is the sum of the scattering from the lattice, while the second term is the sum of atoms in the unit cell, which is known as structure factor.

The diffraction from a crystal is observed under the diffraction condition, which is conventionally described as $$2d\sin \theta = \lambda$$. The equivalent diffraction condition in the vector form is

\begin{align}
\label{Eq:laue_condition}
\tag{5}
\vec{Q} = \vec{G},
\end{align}

where $$\vec{G}$$ is a vector pointing reciprocal lattice defined as

\begin{align}
\label{Eq:def_G}
\tag{6}
\vec{G} = h\vec{a}^{*}_{1}+k\vec{a}^{*}_{2}+l\vec{a}^{*}_{3}.
\end{align}

$$h$$, $$k$$ and $$l$$ are all integers and called “Miller indices”. $$\vec{a}^{*}_{1}$$, $$\vec{a}^{*}_{2}$$ and $$\vec{a}^{*}_{3}$$ are basis vectors of reciprocal space defined as2

\begin{align}
\tag{7}
\vec{a}^{*}_{1} &= 2\pi \frac{\vec{a}_{2}\times \vec{a}_{3}}{v^{*}_{c}}\\
\vec{a}^{*}_{2} &= 2\pi \frac{\vec{a}_{3}\times \vec{a}_{1}}{v^{*}_{c}}\\
\vec{a}^{*}_{3} &= 2\pi \frac{\vec{a}_{1}\times \vec{a}_{2}}{v^{*}_{c}},
\end{align}

where the volume of the unit cell in the reciprocal space, $$v^{*}_{c}$$, is calculated as $$v^{*}_{c} = \vec{a}_{1}\cdot (\vec{a}_{2}\times \vec{a}_{3}) = \vec{a}_{2}\cdot (\vec{a}_{3}\times \vec{a}_{1}) = \vec{a}_{3}\cdot (\vec{a}_{1}\times \vec{a}_{2})$$.
These vectors fulfill a condition that

\begin{align}
\label{Eq:relation_between_real_and_reciprocal_vector}
\tag{8}
\vec{a}_{i} \cdot \vec{a}_{j} = 2\pi \delta_{ij},
\end{align}

where $$\delta_{ij}$$ is the Kronecker’s delta function. This diffraction condition is derived from the feature of the first term of right hand side in Eq. \eqref{Eq:decomposition_of_scattering_amplitude}, i.e., the scattering from the lattice. That is

\begin{align}
\label{Eq:laue_function}
\tag{9}
\left| \sum_{n}\e^{i\vec{Q}\cdot\vec{R}_{n}} \right|^{2} \to Nv^{*}_c \sum_{\vec{G}}\delta (\vec{Q}-\vec{G}) \qquad \text{as} \quad N \to \infty
\end{align}

where $$\delta$$ is the Dirac’s delta function, $$N$$ is the total number of unit cell, i.e., $$N_{1}\times N_{2}\times N_{3}$$. $$N_{1}$$, $$N_{2}$$ and $$N_{3}$$ are the number of unit cell along with $$\vec{a}_{1}$$, $$\vec{a}_{2}$$ and $$\vec{a}_{3}$$, respectively. Therefore, the diffraction intensity, $$\left| F^{\mathrm{crystal}}(\vec{Q}) \right|^{2}$$, is observed only when $$\vec{Q} = \vec{G}$$ as described in Eq. \eqref{Eq:laue_condition}, since the Dirac’s delta function, $$\delta(x)$$, shows the non-zero value only at $$x=0$$.

Under the diffraction condition, the structure factor is reduced as follows:

\begin{align}
\sum_{m}f_{m}(\vec{Q})\exp \left( i\vec{Q}\cdot\vec{r}_{m} \right)
&= \sum_{m}f_{m}(\vec{G})\exp \left( i\vec{G}\cdot\vec{r}_{m} \right) \notag \\
&= \sum_{m}f_{m}(\vec{G})\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\},
\end{align}

where $$\sum_{m}$$ is a summation of all atoms in a unit cell3 Therefore, the scattering amplitude from a crystal under diffraction condition is finally derived to be

\begin{align}
\label{Eq:derived_scattering_amplitude}
\tag{10}
I(\vec{Q} = \vec{G}) \propto \left| \sum_{m}f_{m}(\vec{G})\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\} \right|^{2}.
\end{align}

The essence of the site-distinguished analysis by the DAFS method is this structure factor, where the atomic scattering factor is multiplied by the phase factor, $$\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\}$$. This factor causes the difference in the contributions from a certain element at each diffraction line, consequently enabling us to site-selectively analyze the energy dependence of the atomic scattering factor4 of the same element at the different sites.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}

## Scattering by an atom


In the description of scattering by an atom, we need to take into account the interference of x-rays radiated from different positions in the atom since an electron spreads around the atomic nucleus in quantum-mechanical picture even when the number of electrons is one.

Figure 1 shows the configuration of the scattering process by an atom, whose atomic number is $$Z$$. $$\vec{k}$$ and $$\vec{k’}$$ are the wave number vectors, which lengths are the same, i.e., $$|\vec{k}| = |\vec{k’}| =k = 2\pi/\lambda$$, $$\vec{r}$$ is a position where we evaluate the interference of x-ray, $$\rho(\vec{r})$$ is an electron density at $$\vec{r}$$, and $$\diff V$$ is a volume element at $$\vec{r}$$. The phase difference, $$\Delta \phi(\vec{r})$$ between x-rays radiated at the origin and position $$\vec{r}$$ is described to be

\begin{align}
\label{Eq:phase_difference_in_an_atom}
\tag{1}
\Delta \phi(\vec{r}) = \left( \vec{k} – \vec{k}’ \right) \cdot \vec{r} = \vec{Q} \cdot \vec{r},
\end{align}

where

\begin{align}
\label{Eq:dif_Q_vector}
\tag{2}
\vec{Q} = \vec{k} – \vec{k}’.
\end{align}
This interference occurs in the whole atom; therefore, the scattering amplitude is as follows:

\begin{align}
\label{Eq:integral_of_phases}
\tag{3}
-r_{0}\int \rho(\vec{r}) \e^{i \Delta \phi (\vec{r})} \diff V =
-r_{0}\int \rho(\vec{r}) \e^{i \vec{Q}\cdot \vec{r}} \diff V
\equiv -r_{0}f^{0}(\vec{Q}),
\end{align}

where the integration is carried out in the whole atom and $$f^{0}(\vec{Q})$$ is known as the “atomic form factor“. From the definition, it is definitely expected that $$f^{0}(\vec{Q} = 0)$$ is identical to the atomic number, i.e., the number of electrons, $$Z$$. Thus, $$f^{0}(\vec{Q})$$ value is scattering power described with a unit of the number of electrons called “electron unit (eu), which is frequently used to discuss the absorption amplitude in the DAFS method as well as the conventional diffraction technique.

When we assume the spherical electron density, i.e., $$\rho(\vec{r}) \to \rho(r)$$, the atomic form factor is described and evaluated in a simpler form because the electric density is just a function of distance $$r$$. With the absolute value5 of $$|\vec{Q}| = 4\pi \sin \theta / \lambda$$, the atomic form factor can be reduced2 into

\begin{align}
\label{Eq:integration_f_zero}
\tag{4}
f^{0}(Q) = \int_{0}^{\infty} 4\pi r^{2} \rho(r) \frac{\sin Qr}{Qr} \diff r.
\end{align}

Thus, if one knows the electron density, $$\rho(r)$$, by some theoretical calculations, the atomic form factors of each element and ions can be evaluated and used for the structure analysis by x-ray diffraction technique. As seen in Eq. \eqref{Eq:integration_f_zero}, the atomic factor is a function of only $$Q$$; therefore, the values determined from the electron density calculated by the quantum mechanical approaches such as Hartree-Fock or Fermi-Thomas-Dirac are available as a function of $$(\sin \theta/ \lambda)$$ in International tables for crystallography; vol. C [1] as in the following form:

\begin{align}
\label{Eq:model_atomic_form_factor}
\tag{5}
f^{0}\left(\frac{\sin \theta}{\lambda}\right) = \sum_{i =1}^{4 \mathrm{\ or\ } 5} a_{i} \exp \left\{ -b_{i} \left( \frac{\sin \theta}{\lambda} \right)^{2} \right\} + c,
\end{align}

where $$a_{i}$$, $$b_{i}$$ and $$c$$ values of each element and ion are given in the book.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}
###### References
[1] International Tables For Crystallography Volume C, 3rd ed., E. Prince, Ed., Wiley, 2004.
[Bibtex]
@book{Prince2004,
doi = {10.1107/97809553602060000103},
edition = {3rd},
editor = {Prince, E},
publisher = {Wiley},
title = {{International Tables For Crystallography Volume C}},
year = {2004}
}

## Scattering by one electron


The ability of an electron to scatter an x-ray is described in terms of differential scattering length defined as follows:

\begin{align}
\label{Eq:dif_scattering_length}
\tag{1}
\left( \frac{\diff \sigma}{\diff \Omega}\right) \equiv \frac{I_{\rm{SC}}}{\Phi_0 \Delta \Omega},
\end{align}

where $$\Phi_0$$ is the strength of the incident beam (the number of photons passing through a unit area per second), $$I_{\rm{SC}}$$ is the number of scattered photons recorded per second in a detector positioned at a distance $$R$$ away from the object, $$\Delta \Omega$$ is a solid angle of the detector.

The values of the right-hand side in Eq. $$\eqref{Eq:dif_scattering_length}$$is also described by the electric fields of incoming and scattered x-ray with $$\Phi_0 = c \left| \vec{E}_{\rm{in}}\right|^2 /\hbar \omega$$ and $$I_{\rm{SC}} = cR^2\Delta \Omega \left| \vec{E}_{\rm{rad}}\right|^2 /\hbar \omega$$
as follows3:

\begin{align}
\label{Eq:dif_scattering_length_electric_field}
\tag{2}
\left( \frac{\diff \sigma}{\diff \Omega}\right) = \frac{\left| \vec{E}_{\rm{rad}}\right|^2 R^2}{\left| \vec{E}_{\rm{in}}\right|^2}.
\end{align}

In a classical model of the elastic scattering of the x-ray, the scattered x-ray is generated by the electron forcedly vibrated by the electric field of the incoming x-ray 2. The radiated field is proportional to the charge of the electron $$-e$$, and to the acceleration, $$a_X(t’)$$, cause by the electric filed of the incident x-ray, which is a linearly-polarized in $$x-z$$ plane, evaluated at a time $$t’$$ earlier than the observation time $$t$$ since the speed of light is finite value of $$c$$. Thus, the electron field of the radiated x-ray is written as

\begin{align}
\tag{3}
\end{align}

where $$t’ = t – R/c$$. The full acceleration from the force on the electron is evaluated with Newton’s equation of motion as

\begin{align}
\label{Eq:acceleration}
\tag{4}
a_{X}(t’) =\frac{-e E_{0}\e^{-i\omega t’}}{m}
= \frac{-e}{m}E_{\rm{in}}\e^{i\omega (R/c)}
= \frac{-e}{m}E_{\rm{in}}\e^{ikR},
\end{align}

where $$E_{\mathrm{in}} = E_{0}\e^{-i\omega t}$$ is the electric field of the incoming x-ray. Therefore, Eq. $$\eqref{Eq:rad_electric_field}$$ can be rearranged to be

\begin{align}
\label{Eq:ratio_of_electric_field}
\tag{5}
\frac{E_{\mathrm{rad}}(R,t)}{E_{\mathrm{in}}} \propto \left( \frac{e^2}{m} \right)\frac{\e^{ikR}}{R} \sin \Psi.
\end{align}

In order to complete the derivation of the differential cross section of the electron, it is necessary to check the dimension of both members of Eq. $$\eqref{Eq:ratio_of_electric_field}$$. First, the left-hand side is definitely dimensionless. On the other hand, the dimension of $$\e^{ikR}/R$$ is the inverse of the length. Therefore, the proportionality coefficient of Eq. $$\eqref{Eq:ratio_of_electric_field}$$ must have units of length. By noting that in SI units the Coulomb energy at distance $$r$$ from a point charge $$-e$$ is $$e^2/(4\pi \epsilon_{0}r)$$ while the dimensionally the energy is also described as the form of $$mc^2$$, the proportionality coefficient $$r_{0}$$ is then written as

\begin{align}
\tag{6}
r_{0} = \left( \frac{e^2}{4\pi \epsilon_{0} m c^{2} } \right) = 2.82 \times 10^{-5} \ \A.
\end{align}

This value is known as the Thomson scattering length or classical radius of the electron. By generalizing the relationship of the electric fields of incident and radiated x-rays, the ratio of the radiated electric field to the incident electric field described in Eq. $$\eqref{Eq:ratio_of_electric_field}$$ is reduced to

\begin{align}
\label{Eq:complete_ratio_of_electric_field}
\tag{7}
\frac{E_{\mathrm{rad}}(R,t)}{E_{\mathrm{in}}} = -r_{0} \frac{\e^{ikR}}{R} \left| \hat{\vec{\epsilon}} \cdot \hat{\vec{\epsilon}}’\right|,
\end{align}

where “$$-$$” indicates the radiated x-ray has a different phase from that of the incident x-ray by 180$$\deg$$ because the charge of the electron is negative, $$\hat{\vec{\epsilon}}, \hat{\vec{\epsilon}}’$$ are unit vectors for the electric field of the incident and radiated x-rays.
Therefore, the differential cross-section becomes

\begin{align}
\label{Eq:derived_cross_section}
\tag{8}
\left( \frac{\diff\sigma}{\diff\Omega} \right) = r_{0}^{2}\left| \hat{\vec{\epsilon}} \cdot \hat{\vec{\epsilon}}’\right|^{2}.
\end{align}

The factor of $$\left| \hat{\vec{\epsilon}} \cdot \hat{\vec{\epsilon}}’\right|^{2}$$ is called the Polarization factor and its value is dependent on the polarization of the incoming x-ray and experimental geometry:

\begin{align}
\label{Eq:polarization_factor}
\tag{9}
P = \left| \hat{\vec{\epsilon}} \cdot \hat{\vec{\epsilon}}’\right|^{2} =
\begin{cases}
1 & \text{horizontal linear polarization: vertical scattering plane} \\
\cos^{2} \Psi & \text{horizontal linear polarization: horizontal scattering plane}\\
\left( 1 + \cos^{2} \Psi \right)/2 & \text{unpolarized source: x-ray tube}
\end{cases}
\end{align}

The resultant equation predicts that the scattering intensity becomes very weak if $$\Psi$$ is around 90$$\deg$$ when a detector is scanned in the horizontal plane with the light source of horizontal linear polarization, which is definitely unfavorable to the usual scattering measurement3. This is the reason why a detector is vertically scanned in a synchrotron radiation facility, whose polarization is generally linear in the horizontal plane due to the electron orbit in a storage ring.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}
###### References
[1] J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}