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2.2 Equivalent isotropic displacement parameters next up previous
Next: 2.3 Graphical representations of the Gaussian mean-square displacement matrix Up: 2. Displacement Parameters Based on the Gaussian Approximation Previous: 2.1.2 Construction of Cartesian mean-square displacement tensors

2.2 Equivalent isotropic displacement parameters

It was pointed out by Hamilton (1959) and by Willis & Pryor (1975) that for minor departures from isotropic motion, or for anisotropic displacement parameters deemed to be physically insignificant, it may be worthwhile to replace the six-parameter description of anisotropic motion by a single quantity, which should describe an isotropic equivalent to the weakly or dubiously anisotropic case.

The IUCr Commission on Journals (1986) recommended that ``equivalent isotropic displacement parameters" be computed from the expressions proposed by Hamilton (1959) and by Willis & Pryor (1975). However, a number of different incorrect expressions have also been used (Fischer & Tillmanns, 1988), and this has led to considerable confusion. We first review the proper definitions and demonstrate their equivalence.

The first definition of the equivalent isotropic displacement parameter, as given by Hamilton (1959) and Willis & Pryor (1975), is

  equation1243

with tex2html_wrap_inline2708 an element of a mean-square displacement tensor, referred to a Cartesian basis [see eqs. (2.1.34) - (2.1.37)]. The trace of tex2html_wrap_inline2710 , as given on the right-hand side of eq. (2.2.51), is equivalent to the sum of the eigenvalues of this matrix. These eigenvalues are often computed, since an eigenvalue of the matrix tex2html_wrap_inline2710 represents the mean-square displacement along the corresponding eigenvector. The right-hand side of eq. (2.2.51) can then be interpreted as a mean-square displacement averaged over all directions.

Equation (2.2.51) can thus be applied to the computation of tex2html_wrap_inline2464 either by taking the trace of tex2html_wrap_inline2710 , which was obtained from eq. (2.1.44) or eq. (2.1.49), or by using the sum of the eigenvalues of tex2html_wrap_inline2710 . However, it is essential to note that eq. (2.2.51) holds only for the Cartesian displacement tensor tex2html_wrap_inline2710 . It will give incorrect values of tex2html_wrap_inline2464 if U is referred to oblique basis vectors and its trace taken instead of that of tex2html_wrap_inline2710 .

Since the basis vectors of the Cartesian system have the property tex2html_wrap_inline2726 , a consideration of eqs. (2.1.35), (2.1.37), and (2.2.51) readily leads to

  equation1287

This equation is a convenient starting point for testing the equivalence of various definitions of tex2html_wrap_inline2464 . The second definition by Willis & Pryor (1975) is the first line of the next equation

   eqnarray1300

with g the real space metric tensor. This shows that eqs. (2.2.51) and (2.2.53), the two recommended definitions of tex2html_wrap_inline2464 , are equivalent.

If we make use of eqs. (2.2.54) and (2.1.38), two additional expressions for tex2html_wrap_inline2464 can be obtained:

  equation1360

and

  equation1377

Thus, eqs. (2.2.51), (2.2.54), (2.2.55), and (2.2.56) are equivalent representations of the equivalent isotropic mean-square displacement parameter tex2html_wrap_inline2464 , obtainable from the commonly employed anisotropic displacement parameters.

We can also arrive at eq. (2.2.54) by directly combining eqs. (2.1.44) and (2.2.51), and making use of a known property of the matrix A. We have

  eqnarray1405

since

displaymath2706

(e.g., Prince, 1982). This derivation shows that the value of tex2html_wrap_inline2464 does not depend on the particular form of the matrix A, which transforms the components of u from the lattice to the Cartesian basis.

Acta Crystallographica requires that published values of tex2html_wrap_inline2464 be accompanied by an evaluation of the standard deviation (now standard uncertainty) in these quantities. The calculation of this estimate is described in detail by Schomaker and Marsh (1983). A useful measure of the anisotropy of the mean-square displacement tensor is the ratio of its minimum and maximum eigenvalues. We recommend that published or deposited values of tex2html_wrap_inline2464 be accompanied by both the standard uncertainties and the ratio of the minimum to the maximum eigenvalues of the corresponding anisotropic displacement tensors. Both the uncertainty of tex2html_wrap_inline2464 and the ratio may be helpful in judging the extent to which the use of tex2html_wrap_inline2464 is justified.


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Next: 2.3 Graphical representations of the Gaussian mean-square displacement matrix Up: 2. Displacement Parameters Based on the Gaussian Approximation Previous: 2.1.2 Construction of Cartesian mean-square displacement tensors

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