= Isotropic atomic displacement parameter
{title2=$U_{\mathrm{iso}}$}
An \b[isotropic atomic displacement parameter] describes the same mean-square positional displacement along every Cartesian direction in a <crystal structure>. Its Cartesian displacement <covariance matrix> is
$$
U=U_{\mathrm{iso}}I.
$$
Thus $U_{\mathrm{iso}}$ is a one-component variance, in squared angstroms, rather than a displacement length. The alternate conventional parameter is $B=8\pi^2U_{\mathrm{iso}}$.<Atomic displacement parameters in the IUCr core CIF dictionary>[Definitions of isotropic U and B and their conversion]{ref}
For an original example, $U_{\mathrm{iso}}=0.020\,\mathrm{\mathring A^2}$ gives
$$
\sqrt{U_{\mathrm{iso}}}\simeq0.141\,\mathrm{\mathring A},
\qquad B\simeq1.579\,\mathrm{\mathring A^2}.
$$
The radial root-mean-square displacement is instead $\sqrt{3U_{\mathrm{iso}}}\simeq0.245\,\mathrm{\mathring A}$ 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, is
$$
T(s)=\exp(-2\pi^2U_{\mathrm{iso}}s^2).
$$
At $s=1\,\mathrm{\mathring A^{-1}}$, the stated example gives $T\simeq0.674$ and squared-amplitude attenuation $T^2\simeq0.454$. 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.<Atomic displacement parameters in the IUCr core CIF dictionary>[`_atom_site_U_iso_or_equiv`, isotropic or equivalent-isotropic meaning]{ref}
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