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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.

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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