Crystal scattering

In the Born approximation, scattering from identical atomic potentials translated to lattice sites factors into one atomic form factor times a finite sum of phases over the crystal sites.

A three-dimensional Bravais lattice iswhere the primitive vectors $a_1,a_2,a_3$ are linearly independent.

A face-centered cubic lattice has lattice points at every corner and face center of a conventional cube. One primitive cell has one quarter of the conventional cube's volume.

A lattice point is a vector $l=n_1{a}_1+n_2{a}_2+n_3{a}_3$ whose coefficients in a chosen primitive basis are integers.

The reciprocal lattice isIf $\Omega =a_1\cdot (a_2\times a_3)$, its primitive vectors areand satisfy $a_i\cdot b_j=2\pi {\delta }_{ij}$.

The Wigner--Seitz cell of a lattice point consists of points at least as close to it as to every other lattice point. Its faces lie on perpendicular bisectors to neighbouring lattice points.

For direct primitive vectors $(1,0)$ and $(-1/2,\sqrt{3}/2)$, reciprocal primitive vectors areThe reciprocal Wigner--Seitz cell is a regular hexagon of area $8{\pi }^2/\sqrt{3}$ and circumradius $4\pi /3$.

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.

Fora reciprocal basis isThus ${\Lambda }^{\ast }=(2\pi /a)\{(h,k,l)\in {\mathbb{Z}}^3:h+k+l\text{ even}\}$ and its shortest nonzero vectors have length $2\sqrt{2}\pi /a$.

For a finite set $S$ of crystal sites, the lattice structure factor at momentum transfer $Q$ isIt multiplies the single-atom scattering amplitude.

For $S=\{{\sum }_i{l}_i{a}_i:-L_i/2\le l_i\le L_i/2\}$ and ${\alpha }_i=Q\cdot a_i$,For large $L_i$, each factor is sharply peaked when ${\alpha }_i\in 2\pi \mathbb{Z}$, exactly the condition $Q\in {\Lambda }^{\ast }$.

For elastic scattering with $k-k^{\prime }=q\in {\Lambda }^{\ast }$ and $|k|=|k^{\prime }|=k$,The smallest nonzero diffraction angle is determined by the shortest reciprocal-lattice vector.

Bragg scattering is coherent elastic scattering from a periodic lattice, enhanced when the wavevector transfer is a reciprocal lattice vector. Finite crystals have narrow rather than infinitely sharp reciprocal-space peaks.