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