Debye-Waller factor

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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}
    }