Crystallography¶

Cell metric and spacing¶

Cell lengths \(a,b,c\) and angles \(\alpha,\beta,\gamma\) define the direct metric

\[ G=\begin{pmatrix}a^2&ab\cos\gamma&ac\cos\beta\\ab\cos\gamma&b^2&bc\cos\alpha\\ac\cos\beta&bc\cos\alpha&c^2\end{pmatrix},\qquad V=\sqrt{\det G},\qquad G^*=G^{-1}. \]

For reflection indices \(\mathbf h=(h,k,l)^{\mathsf T}\),

\[d_{hkl}^{-2}=\mathbf h^{\mathsf T}G^*\mathbf h.\]

Lengths are in Å, angles are entered in degrees, and the trigonometric functions use the corresponding angles. The reciprocal metric above excludes a \(2\pi\) factor. The scattering-vector magnitude is \(Q=2\pi/d\).

Bragg condition¶

\[2d\sin\theta=n\lambda,\qquad s=\sin\theta/\lambda=1/(2d).\]

For an indexed reflection the order can be incorporated in its Miller indices. The measured powder angle is \(2\theta\), not \(\theta\). A CW reflection is reachable only when \(\lambda/(2d)<1\) in the implemented open-angle domain. See calibration for how this becomes an experimental coordinate.

Symmetry and atom positions¶

Space-group operations map fractional positions as \(\mathbf r'=R\mathbf r+\mathbf t\) modulo lattice translations. The selected setting fixes the coordinate convention. Symmetry may relate cell parameters or fix atom coordinates; such values are not additional independent fitting directions.

The asymmetric unit and its symmetry-generated images define the unit cell. Wyckoff positions describe site symmetry and multiplicity. Do not count identical special-position images twice or confuse a site's multiplicity with the multiplicity of a reflection.

Reflection multiplicity and absences¶

Equivalent reciprocal-lattice vectors contribute a reflection multiplicity to a powder intensity. Centering, glide and screw translations can cause systematic absences through phase cancellation in the structure factor. A reflection's symmetry equivalence is a property of its space-group setting and radiation model; a count of atom sites is not its multiplicity.

For practical input, see Structure. Scattering connects the positions to intensities.

References¶

International Union of Crystallography. Bragg's law.