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2.3.1 Ellipsoids of constant probability next up previous
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2.3.1 Ellipsoids of constant probability

In the absence of anharmonicity, the anisotropic mean-square displacement matrix U can be regarded as the variance-covariance matrix of a trivariate Gaussian probability distribution with probability density function

  equation1515

Here x is the vector of displacement of the atom from its mean position, and tex2html_wrap_inline2754 is the inverse of the quantity defined by eq. (2.1.25). If the eigenvalues of tex2html_wrap_inline2756 are all positive, then the surfaces of constant probability defined by the quadratic forms

  equation1533

are ellipsoids enclosing some definite probability for atomic displacement. This is the basis for the ORTEP ``vibration ellipsoids" (Johnson, 1965) that are used in so many illustrations of crystal structures. The lengths of the principal axes of the ellipsoids are proportional to the eigenvalues of the matrix tex2html_wrap_inline2710 expressed in the appropriate Cartesian system, and the directions of the principal axes correspond to the eigenvectors of this matrix. This representation cannot be used when tex2html_wrap_inline2756 has one or more negative eigenvalues, because the resulting non-closed surfaces are no longer interpretable in terms of the underlying physical model.


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