Displacement Parameters and NPD Atoms

A refined structure does not give an atom a position, it gives it a probability cloud — and the symmetry of the site decides what shape that cloud may have. Give a cell, a space group and the asymmetric unit: every site says how many of the six components of U a refinement could vary and what the rest are forced to be. Paste measured values in and it also says what they describe, and whether they describe anything possible.

You supply: A unit cell, a space group and one atom per line of the asymmetric unit — the same notation the geometry and structure factor pages read. Displacement parameters are optional: without them each site still shows what its symmetry allows.
Reading it: This site ships no measured displacement parameters for any structure, so with the box empty the values shown are a generic tensor projected onto each site — they demonstrate the checks and measure nothing. The principal axes are not shown either, only the displacements along them: a direction is only useful drawn.

Unit cell

Å

Space group

Number or symbol

Conditions belong to a setting, not to a space group number. Pnma, Pbnm and Pmcn are one space group with its axes labelled three ways, and the three tables differ: the mirror that empties hk0 in one of them empties 0kl in another. Every setting in the International Tables is here with its own operations, so P2₁/n and P2₁/a answer for themselves.

Atoms in the asymmetric unit

Label, x, y, z

The same notation the structure factor and geometry pages read, so a list works on any of the three. Only the coordinates matter here — a site is fixed by where it sits, not by what sits there.

Displacement parameters

Label, U₁₁ U₂₂ U₃₃ U₂₃ U₁₃ U₁₂

Optional, and left empty every site still shows what its symmetry allows. One atom per line in this order, which is the order a CIF’s _atom_site_aniso_U_* loop writes them — so rows paste straight out of a file, brackets and all. A line carrying one value instead of six is an isotropic atom.

What the symmetry allows

A site is fixed by the operations that carry it onto itself, and a displacement tensor is allowed only where those same operations leave it unchanged — U = W U W^T for every W that fixes the site. So the number of parameters a refinement can vary is a property of the position, not of the atom, and it is six only at a general position.

The independent components of U at each site.
Atom	Site symmetry	Multiplicity	Free	Independent	Forced by the site
Si	2	3	4	U₁₁, U₂₂, U₃₃, U₂₃	U₁₃ = U₂₃/2, U₁₂ = U₂₂/2
O	1	6	6	U₁₁, U₂₂, U₃₃, U₂₃, U₁₃, U₁₂	nothing — a general position

The tensors

These values are constructed, not measured. The site ships no displacement parameters for any structure, so each row here is a generic tensor projected onto what its own site allows — which is why every relation in the table above already holds. Paste your own into the box on the left and the checks below become about your structure.

The six components, and the equivalent isotropic parameter they average to.
Atom	U₁₁	U₂₂	U₃₃	U₂₃	U₁₃	U₁₂	U_eq
Si	0.01725	0.01970	0.02710	0.00890	0.00445	0.00985	0.02108
O	0.01310	0.01970	0.02710	0.00890	0.00230	0.00570	0.02108

Every value is in Å². U_eq is one third of the trace of the tensor in a Cartesian frame, which is what makes it independent of how the cell is set up.

The shape of the ellipsoid

The eigenvalues of U in a Cartesian frame are the mean-square displacements along its three principal axes, so their square roots are lengths a reader can picture. A tensor is physically possible only if all three are positive — the test is Sylvester’s criterion, and it has no tolerance in it.

Principal root-mean-square displacements, smallest first.
Atom	Smallest	Middle	Largest	Anisotropy	Possible?
Si	0.1173	0.1282	0.1818	2.40	yes
O	0.1088	0.1307	0.1852	2.90	yes

Displacements are in Å. The anisotropy is the ratio of the largest mean-square displacement to the smallest: 1 is a sphere. A row whose tensor is not possible shows no displacement along the offending axis, because the square root of a negative eigenvalue is not a length.

Sylvester’s criterion

A real symmetric matrix is positive definite exactly when its three leading principal minors are all greater than zero. That is the whole test — no eigenvalue solver, no threshold, and nothing to choose. An atom that fails it is the NPD atom a validation report flags: its ellipsoid has an imaginary axis, which is a sign that something in the refinement is wrong rather than merely uncertain.

The three leading principal minors.
Atom	U₁₁	U₁₁U₂₂ − U₁₂²	det U
Si	1.725e-2	2.428e-4	5.604e-6
O	1.310e-2	2.256e-4	5.205e-6

Where this comes from

International Tables for Crystallography
International Union of Crystallography · on the reading list under “The tables this site computes from”
The site-symmetry constraints on an anisotropic displacement parameter are Volume A's, and they are the external oracle this page is checked against — one free component at a cubic site, four on a trigonal two-fold. What is computed here rather than copied is the constraints themselves: they are derived from each site's own stabiliser, which is why the page can answer for a setting whose table nobody has typed in.
Atomic displacement parameter nomenclature. Report of a subcommittee on atomic displacement parameter nomenclature
K. N. Trueblood, H.-B. Bürgi, H. Burzlaff, J. D. Dunitz et al., Acta Cryst. A 1996, 52, 770–781 · doi:10.1107/S0108767396005697
The definitions this page works to: what U^ij means on the reciprocal basis, how U_eq is formed from it, and why a displacement parameter that is not positive definite describes nothing at all.