Atomic Displacement Parameters

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Atomic Displacement Parameters

Atomic displacement parameters, often incorrectly called temperature factors, often present many problems to beginners in crystallography, and particularly for those working with powder diffraction data. There are several reasons for this: firstly, for historical reasons Rietveld programs often use a variety of parameters (i.e. β, B, U) to describe atomic displacements in marked contrast to their single-crystal equivalents, which now use just 1 isotropic or 6 anisotropic U values; secondly, the user has to make a scientific decision as to whether to use isotropic or anisotropic values. The latter is not just a consideration of whether the atoms are likely to be vibrating in an isotropic or anisotropic manner, but also whether the powder diffraction data can determine this reliably. Powder neutron diffraction data often permit the refinement of anisotropic values, whereas the equivalent X-ray data usually do not.

Computer programmers generally use β values since these optimise execution times of the computer code. However, their values are difficult to interpret quickly in contrast to B values, the latter being independent of the cell parameters. In addition, like the use of Ånstroms for bond lengths, B values are convenient in that typical values are around 1.0 Å²: for tightly bound atoms in a metal oxide, a typical value may be about 0.5 Å², while for molecular compounds, e.g. organic molecules, a typical value may be about 3 to 5 Å². Finally, many crystallographers prefer the use of U values since these are closely relate to the mean atomic displacements, i.e. U = <u²>, where u is an instantaneous atomic displacement. The square root of U provides an r.m.s. (root-mean-square) value for the atomic displacement.

Parameter Conversion

In practical method of testing whether a model can be refined with isotropic or anisotropic atomic displacement parameters is to try it out in practice and observe the result. A few Rietveld programs have the option of automatic conversion of, say, isotropic B values to anisotropic β values. This can be done readily by hand if such an option is not available:

Step 1.

Set B₁₁ = B₂₂ = B₃₃ = B(iso) and B₂₃ = B₃₁ = B₁₂ = 0.

Step 2.

Multiply the B values by the reciprocal space cell lengths (and divide by 4) to obtain β values, i.e.

β₁₁ = B₁₁ × a*² / 4
β₂₂ = B₂₂ × b*² / 4
β₃₃ = B₃₃ × c*² / 4
β₂₃ = B₁₂ × b* c* / 4
β₃₁ = B₃₁ × c* a* / 4
β₁₂ = B₁₂ × a* b* / 4

Step 3.

Insert them into the Rietveld program in the correct order! The logical order is the one given above, but is one that is rarely used in practice for the off-diagonal anisotropic values: two common orders are 11, 22, 33, 12, 13, 23 and 11, 22, 33, 12, 23, 31. (Remember that the anisotropic tensor matrix is symmetric so that B_ij equals B_ji.)

To convert the refined β values back to B values, simply divide the β values by the reciprocal space cell lengths, e.g. B₁₁ = β₁₁ × 4 / a*². Ideally, good software should do all this, but some does not.

Finally, it is easy to convert between B and U values (either isotropic or anisotropic) since this involves only the constant factor 8π², i.e. U = B / 8π².

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