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An anisotropic atomic displacement parameter describes direction-dependent positional spread about an atom's mean position in a crystal structure. In Cartesian coordinates a physical displacement model has a symmetric covariance matrixThe six independent entries have units of squared length. The crystallographic components stored in CIF use a cell-dependent convention, including reciprocal-cell lengths in the structure factor attenuation; they must be transformed before being interpreted as Cartesian covariance entries.^[ref]

For an original Cartesian example, letThe principal root-mean-square displacements are0.1,0.2 and0.3 angstroms. An atomic displacement ellipsoid displays those preferred directions; its displayed boundary also depends on the chosen probability level. Rotating the coordinate axes generally introduces off-diagonal entries without changing the physical spread or eigenvalues.

Under a centered multivariate normal distribution for $\mathbf{u}$, its contribution to the scattering amplitude is multiplied bywhere $\mathbf{s}$ is the Cartesian reciprocal vector in cycles per unit length. Thus reflections probing different directions can distinguish directional spread. A physically valid covariance is a positive semidefinite matrix.

The equivalent isotropic atomic displacement parameter is $U_{\mathrm{eq}}=\mathrm{tr}(U)/3$ in Cartesian coordinates. In this example it is $0.04667{\mathring{\mathrm{A}}}^2$; that average does not make the original distribution isotropic. Positional spread can include unresolved static variation as well as motion, and differs from uncertainty in the refined mean coordinate.^[ref]

An atomic displacement probability level is the fraction of modeled positional probability enclosed by an atomic displacement ellipsoid. It specifies the size of the drawn surface for a given anisotropic atomic displacement parameter; changing the level does not change the fitted displacement matrix.^[ref]

For a centered three-dimensional normal distribution with positive-definite Cartesian covariance matrix $U$, defineThe variable $Q$ has a chi-squared distribution with3 degrees of freedom. To enclose probability $p$, choose $c$ satisfying $\mathrm{Pr}(Q\le c)=p$ and draw $Q=c$. If ${\lambda }_i$ are the eigenvalues of $U$, its semiaxis lengths are $\sqrt{c{\lambda }_i}$.

An explicit calculation uses the cumulative distribution functionwhere $\mathrm{erf}$ is the error function. Solving $F_3(c)=0.5$ gives $c\simeq 2.36597$; for $p=0.9$, $c\simeq 6.25139$.

For an original comparison, keep $U=\mathrm{diag}(0.01,0.04,0.09){\mathring{\mathrm{A}}}^2$. The50% semiaxes are approximately0.154,0.308 and0.461angstroms. The90% semiaxes are approximately0.250,0.500 and0.750angstroms. Every semiaxis grows by the same factor, about1.625, although the underlying directional spread is unchanged.

This also explains why a surface drawn at one root-mean-square displacement in each principal direction, $Q=1$, encloses only about19.9% in three dimensions. The probability inside a three-dimensional ellipsoid differs from the probability inside a one-dimensional standard-deviation interval. Comparisons of displayed atomic sizes therefore need the stated probability levels as well as the displacement parameters.

The equivalent isotropic atomic displacement parameter summarizes an anisotropic atomic displacement parameter by the average mean-square displacement along three perpendicular Cartesian directions:It has units of squared length. The corresponding equivalent $B$ parameter is $B_{\mathrm{eq}}=8{\pi }^2{U}_{\mathrm{eq}}$. The CIF field for isotropic displacement may contain either a genuinely isotropic atomic displacement parameter or this anisotropic summary.^[ref]

For an original example, take a Cartesian displacement covariance matrixThen $U_{\mathrm{eq}}=0.020{\mathring{\mathrm{A}}}^2$ and $B_{\mathrm{eq}}\simeq 1.579{\mathring{\mathrm{A}}}^2$. The principal root-mean-square displacements are0.1,0.1 and0.2 angstroms. A genuinely isotropic matrix $0.020I$ has the same equivalent parameter but a different directional distribution.

The trace is invariant under Cartesian rotations, so the average does not depend on the laboratory orientation. In an oblique crystallographic basis, however, the raw diagonal entries of the stored displacement matrix do not in general have this Cartesian interpretation. The cell geometry must enter the conversion before taking the physical trace.^[ref]

This scalar is convenient for comparisons and tabulation, but discards directional information displayed by an atomic displacement ellipsoid. Equality of equivalent parameters therefore neither establishes isotropy nor proves that two sites have the same displacement distribution.