# 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 $$\vec{Q}$$ step1 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}
}