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.
