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Atomic displacement parameters

Displacement parameters describe the attenuation of coherent scattering by the distribution of positions around each site's average position. Their conventions and coordinate basis matter.

Isotropic forms

\[T_{\rm iso}(s)=\exp(-B_{\rm iso}s^2)=\exp(-8\pi^2U_{\rm iso}s^2),\qquad B_{\rm iso}=8\pi^2U_{\rm iso}.\]

Both Biso and Uiso have units Ų. An equal numerical value in the two forms does not represent equal displacement. adp_type chooses Biso or Uiso; adp_iso carries the corresponding value.

Anisotropic forms

With fractional reflection indices and reciprocal-cell lengths,

\[T_{\rm aniso}=\exp\left[-2\pi^2\sum_{i,j}U_{ij}h_i h_j a_i^*a_j^*\right] =\exp\left[-\tfrac14\sum_{i,j}B_{ij}h_i h_j a_i^*a_j^*\right].\]

The beta convention absorbs the reciprocal-length and \(2\pi^2\) factors:

\[T_{\rm aniso}=\exp[-\mathbf h^{\mathsf T}\boldsymbol\beta\mathbf h].\]

Bani, Uani and beta are distinct supported representations. The six independent symmetric components include cross terms: a full \(\sum_{i,j}\) counts \(ij\) and \(ji\), so an off-diagonal entry must not be added as if it were already doubled. Space-group symmetry can constrain the components.

Interpretation

Check physical plausibility and positive-definiteness where appropriate. A large displacement can compensate for another model error; its fitted uncertainty does not resolve that ambiguity. Link the value to the parameter index and the selected convention before discussing it with the assistant.

References

  1. International Union of Crystallography. Atomic displacement parameter nomenclature. Definitions and coordinate conventions.
  2. International Union of Crystallography. coreCIF definition of isotropic B.