= Equivalent isotropic atomic displacement parameter
The \b[equivalent isotropic atomic displacement parameter] summarizes an <anisotropic atomic displacement parameter> by the average mean-square displacement along three perpendicular Cartesian directions:
$$
U_{\mathrm{eq}}=\frac13\operatorname{tr}(U_{\mathrm{Cart}}).
$$
It has units of squared length. The corresponding equivalent $B$ parameter is $B_{\mathrm{eq}}=8\pi^2U_{\mathrm{eq}}$. The CIF field for isotropic displacement may contain either a genuinely <isotropic atomic displacement parameter> or this anisotropic summary.<Atomic displacement parameters in the IUCr core CIF dictionary>[`_atom_site_U_iso_or_equiv`, Definition and equivalent-parameter conversion]{ref}
For an original example, take a Cartesian displacement <covariance matrix>
$$
U_{\mathrm{Cart}}=\operatorname{diag}(0.010,0.010,0.040)
\,\mathrm{\mathring A^2}.
$$
Then $U_{\mathrm{eq}}=0.020\,\mathrm{\mathring A^2}$ and $B_{\mathrm{eq}}\simeq1.579\,\mathrm{\mathring A^2}$. 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.<Atomic displacement parameters in the IUCr core CIF dictionary>[`_atom_site_U_iso_or_equiv`, cell-dependent formula]{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.
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