Temperature Effects

Temperature Effects

Temperature will have an effect on the intensities of the powder diffraction pattern. However it is not an effect that can be expressed as a simple multiplier, as was the case for example with multiplicity, polarisation, Lorentz and absorption factors. This is because temperature expresses itself differently from one atom to another in the structure. The structure factor equation given earlier yields the diffracted intensities as though all atoms were stationary. As the temperature rises the atoms execute increased excursions from their average position. Such excursions can involve complex coupled motions of several atoms. However if one restricts consideration to just the independent motions of each atom there is a relatively simple trick that can be used which is to (artificially) modify the atomic scattering factors, f, of each atom to represent the net broadening of their electron clouds resulting from atomic motion. For example a given atom will have its "stationary" f changed to f_T according to:

f_T = f exp(−B sin²θ / λ²) where T is the temperature and B = 8π²<u²> (units of Å² ), <u²> being the root mean square displacement of the atom from its average position. The B values are often referred to as B factors (Debye-Waller is a another, similar, term often used) and increase with temperature (typical values are 0.2 to 0.8 Å² ). So clearly temperature will modify the diffraction intensities by changing the various values of f to f_T in the structure factor equation, but these changes are determined by structural refinement methods (rather than by simple multipliers) and these will be discussed further elsewhere in the course. Before that, one final point to be explained is that the above method assumes that the B factors and f_T values are the same in all directions. These are termed isotropic B factors but more challenging refinements allow each atom to vibrate differently in different directions. This is termed anisotropic B factor refinement and its result can be illustrated graphically using individual ellipsoids to indicate the angular profile of vibration for each atom; an example of this, for the tyrosine molecule in a structure, is shown below:

Complete Structure Factor Equation

If we now take into account the effect of temperature, the structure factor equation becomes:

F(S) =

Σ
n

f_n N_n exp{2π i (hx + ky + lz)} exp(−B sin²θ / λ)

for the case of isotropic thermal motion. There is now an additional multiplier, N, in this final equation for the structure factor, which takes into account atomic disorder in crystal structures. Thus an atomic site that is only half occupied will have this value set to 0.5, which is equivalent to reducing the scattering factor, f, by 50%.

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Author(s): Paul Barnes
Ian Tickle
Martin Vickers