Displacement Parameters and NPD Atoms

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 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.

Every value is in Ų. 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 — and it is not the average of U₁₁, U₂₂ and U₃₃ unless the cell is orthogonal. The next panel has the formula and the size of the difference on this cell.

Why U_eq is not the average of the diagonal

U_eq is the mean-square displacement averaged over every direction, and the six U^ij are components on the reciprocal basis — so averaging the three diagonal ones treats a*, b* and c* as though they were three perpendicular unit vectors. The published closed form does not:

U_eq = ⅓ ∑_i ∑_j U^ij a*_i a*_j (a_i · a_j)

Everything in that sum except U^ij is fixed by the cell, so it collapses to six weights w_ij = a*_i a*_j (a_i · a_j), and U_eq is ⅓ of their weighted sum. Averaging the diagonal is the special case where every weight is 1 or 0. For the cell above they are:

They are not the identity, so the average of the diagonal is not U_eq on this cell. It is out by −5.27 % on O. The three diagonal weights are never below 1 — a*_i · a_i is 1 by definition, so the product of their lengths is 1/cos of the angle between them — while the cross weights carry the sign of the cosine of a cell angle and go either way. The error is therefore not a bias anyone could correct for: it lands on either side depending on which components are large.

The second basis is a’ = a, b’ = b, c’ = a + c — integer entries, determinant 1, so it spans the same lattice and describes the same atoms. Its constants are 4.9134 Å, 4.9134 Å, 7.3046 Å, 109.6529°, 47.7287°, 120.0000°. Nobody would choose it, and that is the point: being inconvenient is not a reason a physical quantity may change. The last column is unchanged to every digit printed. The one before it moves by +5.21 % on O, on the same atom, from the same measurement — because the trace of U^ij is not a property of the atom at all. It transforms as P^−T U P⁻¹, which preserves no trace; a linear map, which does, would transform as P U P⁻¹. Six numbers in Ų laid out as a symmetric matrix look like something whose diagonal can be averaged. They are not.

Both rows above are the same generic tensor projected onto each site’s own allowed subspace, which is why their U_eq agree exactly: the projection averages over the site symmetry, every operation of which is an isometry, and an average over directions cannot see a rotation. What the projection does change is the six components, and with them ⅓(U₁₁ + U₂₂ + U₃₃) — so on this page the wrong average disagrees between two atoms the right one calls identical.

The ellipsoid, drawn

A displacement parameter is a shape, and six numbers are not one. The surface below encloses a stated share of the atom’s probability: U₁₁, U₂₂ and U₃₃ stretch it along the cell axes, and U₂₃, U₁₃ and U₁₂ tilt it away from them. That is the whole distinction the table above cannot show — two tensors with the same three principal displacements can point in quite different directions.

Every picture is drawn at one scale, so a wide ellipsoid beside a narrow one is a real difference and not a framing one, and at 90 % probability — the surface 90 atoms in 100 would be found inside, which is 2.5003 times each root-mean-square displacement. An ellipsoid with no level stated is not a drawing anybody can read: the crystallographic convention is 50 %, and the same atom at one standard deviation looks identical while being a third smaller. The dashed circle is the sphere of the same U_eq — the atom this would be if it were isotropic. The three arcs are the principal sections and the straight segments the principal axes. The cell axes seen down b. b points at the reader and is drawn as a ring. The axis directions are drawn as directions only, at a fixed length that says nothing about the cell constants.

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, which is an exact statement about the matrix and has no tolerance in it. What it does not carry is the uncertainty of the numbers that went in: a refined U has standard uncertainties of its own, so a tensor that fails by a hair is a flag to go and look rather than a proven impossibility, while one that fails by a wide margin is a fault in the refinement.

The rest of this site works in B, and the two are one quantity in two conventions: B = 8π²U, so B = 1 Å² is U = 0.0127 Å² and the factor between them is about 79. A CIF writes whichever its refinement used, which is why _atom_site_aniso_U_* and _atom_site_B_iso_or_equiv both exist. The relation holds as written for the isotropic and the equivalent isotropic values, and it holds component by component for the anisotropic ones too: _atom_site_aniso_B_ij = 8π² U^ij. What does not scale that way is β^ij, the third convention a CIF may carry — β^ij = 2π²a*_ia*_jU^ij, which drags the reciprocal axis lengths in with it, so a β tensor multiplied through by 8π² is wrong by whatever those lengths are. Check which of the three a source used; the arithmetic itself is not in doubt.

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.