= Anisotropic atomic displacement parameter
{title2=$U$}
An \b[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 matrix>
$$
U_{ij}=\mathbb E[u_i u_j],\qquad \mathbb E[\mathbf u]=0.
$$
The 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.<Anisotropic displacement components in the IUCr core CIF dictionary>[`_atom_site_aniso_U_`, Definition and structure-factor expression]{ref}
For an original Cartesian example, let
$$
U=\begin{pmatrix}0.01&0&0\\0&0.04&0\\0&0&0.09\end{pmatrix}
\,\mathrm{\mathring A^2}.
$$
The 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 by
$$
\mathbb E[e^{2\pi i\mathbf s\cdot\mathbf u}]
=\exp(-2\pi^2\mathbf s^TU\mathbf s),
$$
where $\mathbf s$ 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 $U_{\mathrm{eq}}=\operatorname{tr}(U)/3$ in Cartesian coordinates. In this example it is $0.04667\,\mathrm{\mathring A^2}$; 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.<Atomic displacement parameters in the IUCr core CIF dictionary>[`_atom_site_U_iso_or_equiv`, equivalent parameter calculated from anisotropic components]{ref}
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