Scattering factor

The scattering factor describes the amplitude of scattered rays from a single atom.

The presence or absence of an intensity peak depends on the interference of scattered X-rays at the detector of the powder X-ray diffractometer. As the interference of scattered X-rays is the sum of amplitudes of X-ray waves from many atoms in a sample, we have to begin with the analysis of the resultant amplitude of scattered rays from a single atom (scattering factor), which is in turn the sum of amplitudes of waves scattered by the electrons in the atom.

The amplitude of a travelling wave is expressed by the equation:

$y=Acos(kx-\omega t+\phi )$

We can also expressed the general wave equation in its complex form:

$y=Ae^{i(kx-\omega t+\phi )}\; \; \; \; \; \; \; (25)$

The complex form of the wave equation is equivalent to the travelling wave equation if we consider only the real part of it: Re(y), which is the value that is physically observed in an X-ray diffraction experiment. The reason eq25 is preferred over the travelling wave equation is that it is easier to manipulate mathematically to derive the scattering factor. Recall that ω = 2π/T (angular frequency), k = 2π/λ (wave number) and Φ is the phase difference. If we assume that the scattering of X-rays by an atom is elastic, i.e. there is no change in frequency of the X-ray between the incident and scattered rays, the term kx–ωt becomes a constant and we can simplify eq25 to:

$y=A'e^{i\phi }\; \; \; \; \; \; \; (26)$

where A’ = Ae^i(kx-ωt).

The amplitudes of the scattered X-rays not only differ in terms of their phase as suggested by eq26 but are also proportional to the number of electrons in the atom, ρdV, where ρ and V are electron density and volume of the atom respectively. If we define A’ as the number of electrons in the atom, eq26 becomes

$df=\rho e^{i\phi }dV\; \; \; \; \; \; \; (27)$

where the amplitude of the scattered X-rays from an atom is denoted by f, the scattering factor of an atom.

To determine an expression for Φ, we note that the ratio of path difference δ and wavelength is equal to the ratio of phase difference and 2π, i.e.

$\frac{\delta }{\lambda }=\frac{\phi }{2\pi }\; \; \; \; \; \; \; (28)$

Substitute eq19 in eq28,

$\phi =2\pi\textbf{\textit{a}}\cdot (\textbf{\textit{s}}-\textbf{\textit{s}}_0)\; \; \; \; \; \; \; (29)$

Since $\textbf{\textit{a}}\cdot (\textbf{\textit{s}}-\textbf{\textit{s}}_0)=\left |\textbf{\textit{a}} \right |\left | \textbf{\textit{s}}-\textbf{\textit{s}}_0\right |cos\mu$ and from eq23, $\left | \textbf{\textit{s}}-\textbf{\textit{s}}_0\right |=\frac{2sin\theta }{\lambda }$ , eq29 becomes:

$\phi =\frac{4\pi }{\lambda }\left | \textbf{\textit{a}}\right |sin\theta cos\mu$

In three dimensions, we let a = r and IaI = IrI = r,

$\phi =\frac{4\pi }{\lambda }rsin\theta cos\mu\; \; \; \; \; \; \; (30)$

Substitute eq30 in eq27

$df=\rho e^{ikrcos\mu }dV\; \; \; \; \; \; \; (31)$

where $k=\frac{4\pi }{\lambda }sin\theta$

Since ρ is spherically symmetrical, we can represent the volume element dV in eq31 in spherical coordinates where dV = r²sinμdrdμdΦ (see above diagram). Eq31 becomes:

$df=\rho e^{ikrcos\mu }r^2sin\mu drd\mu d\phi$

Integrating both sides,

$f=\int_{0}^{\infty }\int_{0}^{\pi }\int_{0}^{2\pi }\rho e^{ikrcos\mu }r^2sin\mu drd\mu d\phi$

$f=2\pi \int_{0}^{\infty }\rho r^2dr\int_{0}^{\pi }e^{ikrcos\mu }sin\mu d\mu \; \; \; \; \; \; \; (32)$

Note that d(cosμ) = -sinμdμ; when μ = π, cosμ = -1; when μ = 0, cosμ = 1. So the second integral in eq32 becomes

$\int_{0}^{\pi }e^{ikrcos\mu }sin\mu d\mu =-\int_{1}^{-1 }e^{ikrcos\mu } d(cos\mu)=\int_{-1}^{1 }e^{ikrcos\mu }d(cos\mu)$

$=\left [ \frac{1}{ikr}e^{ikrcos\mu } \right ]_{-1}^{1}=\frac{e^{ikr}-e^{-ikr}}{ikr}$

Eq32 now becomes

$f=2\pi \int_{0}^{\infty }\rho r^2 \frac{e^{ikr}-e^{-ikr}}{ikr}dr$

Since $e^{ikr}-e^{-ikr}=coskr+isinkr-(coskr-isinkr)=2isinkr$

$f=4\pi \int_{0}^{\infty }\rho \frac{sinkr}{kr}r^2dr\; \; \; \; \; \; \; (33)$

where $k=\frac{4\pi }{\lambda }sin\theta$ . ρ is a function of r and remains within the integral.

So far, we have developed an expression (eq33) that describes the resultant amplitude of scattered rays from a single atom. To further analyse the interference of scattered X-rays from multiple atoms in a sample, we have to derive another expression called the structure factor.

next article: structure factor

previous article: single crystal X-ray diffraction

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