= Atomic displacement ellipsoid
An \b[atomic displacement ellipsoid] visualizes the directional positional spread represented by an <anisotropic atomic displacement parameter>. For a centered three-dimensional <normal distribution> with positive-definite Cartesian <covariance matrix> $U$, its probability surfaces have the form
$$
\mathbf u^TU^{-1}\mathbf u=c.
$$
The <eigenvectors> of $U$ give the principal directions; if its <eigenvalues> are $\lambda_i$, the semiaxis lengths are $\sqrt{c\lambda_i}$. The <atomic displacement probability level> specifies how much modeled probability is enclosed.<Atomic displacement parameter nomenclature (IUCr report)>[Section2.3.1, ellipsoids of constant probability]{ref}
The squared <Mahalanobis distance> $\mathbf u^TU^{-1}\mathbf u$ has a <chi-squared distribution> with3 degrees of freedom in this Gaussian model. A50% ellipsoid uses $c\simeq2.36597$. Taking $c=1$ instead encloses only about19.9%, so a three-dimensional probability ellipsoid cannot be read as a one-dimensional one-standard-deviation interval.
For an original example, $U=\operatorname{diag}(0.01,0.04,0.09)\,\mathrm{\mathring A^2}$ gives50% semiaxes approximately0.154,0.308 and0.461 angstroms. An isotropic matrix with the same <equivalent isotropic atomic displacement parameter>, $U_{\mathrm{eq}}=0.14/3\,\mathrm{\mathring A^2}$, instead gives a sphere of radius about0.332 angstroms.
\Image[/atomic-displacement-ellipsoid-comparison.png]
{title=Equal equivalent displacement does not imply equal ellipsoid shape}
{id=image-atomic-displacement-ellipsoid-comparison}
{description=Original Cartesian Gaussian models at50% enclosed probability. Both have the same <equivalent isotropic atomic displacement parameter>, but only the right-hand distribution is isotropic. Axes show displacements from the mean, not atomic radii.}
The ellipsoid represents modeled positional spread, including possible static disorder, rather than the atom's physical surface or a confidence region for its refined mean position. Its interpretation depends on the displacement model; the Gaussian probability construction requires a positive-definite covariance and does not turn a negative fitted eigenvalue into a physically meaningful radius.<Atomic displacement parameter nomenclature (IUCr report)>[Section2.3.1, positive eigenvalues and model interpretation]{ref}
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