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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 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 formThe 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.^[ref]

The squared Mahalanobis distance ${\mathbf{u}}^T{U}^{-1}\mathbf{u}$ has a chi-squared distribution with3 degrees of freedom in this Gaussian model. A50% ellipsoid uses $c\simeq 2.36597$. 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, $U=\mathrm{diag}(0.01,0.04,0.09){\mathring{\mathrm{A}}}^2$ gives50% semiaxes approximately0.154,0.308 and0.461 angstroms. An isotropic matrix with the same equivalent isotropic atomic displacement parameter, $U_{\mathrm{eq}}=0.14/3{\mathring{\mathrm{A}}}^2$, instead gives a sphere of radius about0.332 angstroms.

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.^[ref]

An isotropic atomic displacement parameter describes the same mean-square positional displacement along every Cartesian direction in a crystal structure. Its Cartesian displacement covariance matrix isThus $U_{\mathrm{iso}}$ is a one-component variance, in squared angstroms, rather than a displacement length. The alternate conventional parameter is $B=8{\pi }^2{U}_{\mathrm{iso}}$.^[ref]

For an original example, $U_{\mathrm{iso}}=0.020{\mathring{\mathrm{A}}}^2$ givesThe radial root-mean-square displacement is instead $\sqrt{3U_{\mathrm{iso}}}\simeq 0.245\mathring{\mathrm{A}}$ because it includes three components.

For a centered isotropic normal distribution, the structure factor amplitude attenuation at reciprocal-vector magnitude $s$, measured in cycles per angstrom, isAt $s=1{\mathring{\mathrm{A}}}^{-1}$, the stated example gives $T\simeq 0.674$ and squared-amplitude attenuation $T^2\simeq 0.454$. Amplitude and intensity attenuation are therefore distinct, even in this simple Gaussian model.

An equivalent isotropic atomic displacement parameter can also summarize an anisotropic atomic displacement parameter by its Cartesian trace divided by3. Such an average can appear in the same CIF field as a genuinely isotropic parameter, but does not assert equal directional spread.^[ref]