Link to IUCr Link to CCN
2.1.2 Construction of Cartesian mean-square displacement tensors next up previous
Next: 2.2 Equivalent isotropic displacement parameters Up: 2.1 Anisotropic displacement parameters Previous: 2.1.1 Relationships between the anisotropic displacement parameters

2.1.2 Construction of Cartesian mean-square displacement tensors

Referring an ADP tensor to a Cartesian basis is somewhat less simple. We proceed to show how this is done, both in order to illustrate the above-outlined transformation of tensors and to provide some background for the following section.

Construct a Cartesian system by taking, for example, tex2html_wrap_inline2660 along a, tex2html_wrap_inline2664 along tex2html_wrap_inline2686 , and tex2html_wrap_inline2662 along the vector product tex2html_wrap_inline2690 . The resulting vectors

  eqnarray959

comprise an appropriate and common orthonormal set of basis vectors.

Eqs. (2.1.35) and (2.1.17) are both expressions for u. Hence

  equation980

If we take the scalar products of the left-hand side and right-hand sides of eq. (2.1.40) with tex2html_wrap_inline2660 , tex2html_wrap_inline2662 and tex2html_wrap_inline2664 , we obtain three linear equations, or a matrix equation of the form

  equation1006

This is a transformation of the components of u, referred to the basis of the direct lattice, to its Cartesian components. The transformation matrix can be evaluated once the Cartesian basis vectors are defined (e.g., as above). If we adopt the index notation in the second line of eq. (2.1.17), eq. (2.1.41) can be written as

  equation1055

with

displaymath2680

We can similarly transform a product of components of u between the lattice and Cartesian bases, and finally an average of such a product:

  equation1073

If we now make use of eqs. (2.1.22) and (2.1.37) we obtain

  equation1090

which is the expression for the transformation of tex2html_wrap_inline2470 to Cartesian coordinates.

The orthogonalization of U proceeds along similar lines. The required version of eq. (2.1.40) is now [cf. eq. (2.1.23)]

  equation1109

and following the same procedure by which eq. (2.1.41) is obtained, we arrive at the transformation

  equation1132

which relates the components of u, referred to the tex2html_wrap_inline2700 basis, to its Cartesian components. Equation (2.1.46) can be written concisely as

  equation1188

with

equation1198

an element of the matrix product appearing in eq. (2.1.46). The desired transformation is obtained analogously to eq. (2.1.44) as

  equation1208

with tex2html_wrap_inline2702 and tex2html_wrap_inline2704 .

The explicit form of the transformation matrix appearing in eq. (2.1.41), for the specific Cartesian basis defined in eqs. (2.1.39), is:

  equation1232

Of course, a Cartesian basis associated with the direct and/or reciprocal bases can be chosen in an unlimited number of ways. A more general discussion of the construction of such Cartesian bases is given elsewhere (Shmueli, 1993).


next up previous
Next: 2.2 Equivalent isotropic displacement parameters Up: 2.1 Anisotropic displacement parameters Previous: 2.1.1 Relationships between the anisotropic displacement parameters

ADP Report Webmaster

Back to Table of Contents