The Effect of Thermal Vibration

The permanent state of thermal excitation experienced by the atoms of a real crystal is a further deviation from the ideal crystal structure. The fixed lattice positions are only ever average positions about which the atoms oscillate with a motion which is a function of temperature. Such dynamic fluctuations can be represented by the displacement $\Delta {\bf r}_{m,n}(t)$ being a function of time and the instantaneous position at a time $t$ can then be written

$${\bf R}( t ) = {\bf m} + {\bf r}_n + \Delta{\bf r}_{m,n}(t)$$ (4.14)

The experimentally observed intensity is a time average of the scattering; so correctly speaking, it is a thermal average of the atomic motions which will be observed.

The incorporation of thermal vibrations into the scattering equation was formulated by Debye. The Debye method is described below, with a simple approximation to the thermal vibrations made that each atom fluctuates independently about its own lattice position. Known more generally as the Einstein model of a solid, it denies the existence of phonon lattice excitations. It is a good approximation for elevated temperatures but shows stronger discrepancies at very low temperatures.

The result of Debye's formulation is achieved by a similar route to that for displacement disorder, only now the thermal average of equation 2.13 is required. A further simplification is to assume that the function $\Delta{\bf r}_m(t)$ has no correlation with the scattering direction. However, more realistically, a spherical displacement function can be used which allows for different vibration modes along the various crystallographic directions. It is then only the component of displacement along $({\bf k}_s - {\bf k}_o)$, the component normal to the diffraction plane, which will be of consequence to the scattering. The exponential can again be expanded, and the component of $\Delta{\bf r}_m(t)$ taken which is normal to k. If the displacements have equal probability of being +ve or -ve and are small, or if they follow a Gaussian distribution, then it follows that

$$<\exp(i{\bf k}.(\Delta{\bf r}_m - \Delta{\bf r}_{m'}))> =\exp(-\frac{1}{2}k^2<(\Delta r_m - \Delta r_{m'})^2>)$$ (4.15)

where $<$..$>$ denotes the thermal average. A complete proof is given by Warren [31]. The incorporation of this result into the expression for intensity leads to

$$\begin{eqnarray*} I({\bf k}) = \sum_m^N \sum_{m'}^{N'} &f^2& \exp(-2D) \exp(i{\b... ...&\times& \biggl[\exp(k^2 <\Delta r_m \Delta r_{m'}>) - 1 \biggr] \end{eqnarray*}$$

The result is to reduce the intensity of the fundamental reflections by a factor $\exp(-2D)$, which is known as the Debye-Waller factor, and where $D= \frac{1}{2}k^2<\Delta r_m^2>$. The intensity lost is redistributed throughout reciprocal space by the second term; similar to the disorder term, it is known as thermal diffuse scattering.

The term $<\Delta r^2>$ is the mean-square displacement of an atom and will obviously be a function of temperature. The value may be estimated most simply by relating the vibrations to that of a classical harmonic oscillator, the thermal average potential energy of which is

$$\frac{3}{2}k_B T = \frac{1}{2}M\omega^2<\Delta r^2>$$ (4.16)

Here the vibrational frequency is $\omega$, the mass of the atom is $M$, and the temperature $T$. From this the Debye-Waller factor now becomes

$$D=\frac{1}{2}k^2 \frac{k_B T}{M\omega^2}$$ (4.17)

It becomes clear that the scattered intensity will decrease exponentially with increasing temperature. Also that the Debye-Waller term affects reflections with large values of k, further out in reciprocal space, to a greater extent than those with smaller values closer to the reciprocal space origin.