## Structure factor for neutrons

The structure factor for neutrons can be calculated with the following formula,

$$F_{\vec{G}} = \sum\limits_j b_je^{- i\vec{G}\cdot\vec{r}_j} = \sum\limits_j b_j\left(\cos\left(\vec{G}\cdot\vec{r}_j\right)-i\sin\left(\vec{G}\cdot\vec{r}_j\right)\right).$$

where $\vec{r}_j$ defines the position of the atom $j$ and $\vec{G}$ is the reciprocal lattice vector. $\vec{b}_j$ is called the neutron scattering length, it depends on the spin-state of the neutron-nucleus system and the isotope the neutron is scattered from. The scattering lengths can be looked up at the NIST Center for Neutron Research.

The form below calculates the neutron structure factors. The script first calculates the reciprocal lattice vectors and from them calculates the reciprocal lattice vectors $\vec{G}_{hkl}=h\vec{b}_1+k\vec{b}_2+l\vec{b}_3$. The structure factors are calculated for a few reciprocal lattice vectors and listed in a table.

Primitive lattice vectors:

 $\vec{a}_1=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m] $\vec{a}_2=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m] $\vec{a}_3=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m]

Basis:
The positions of the atoms are given in fractional coodinates between -1 and 1.

 $\vec{a}_1+$ $\vec{a}_2+$ $\vec{a}_3$ $\vec{a}_1+$ $\vec{a}_2+$ $\vec{a}_3$ $\vec{a}_1+$ $\vec{a}_2+$ $\vec{a}_3$ $\vec{a}_1+$ $\vec{a}_2+$ $\vec{a}_3$ $\vec{a}_1+$ $\vec{a}_2+$ $\vec{a}_3$ $\vec{a}_1+$ $\vec{a}_2+$ $\vec{a}_3$ $\vec{a}_1+$ $\vec{a}_2+$ $\vec{a}_3$ $\vec{a}_1+$ $\vec{a}_2+$ $\vec{a}_3$

Primitive reciprocal lattice vectors

 $\vec{b}_1=2\pi\frac{\vec{a}_2\times\vec{a}_3}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1] $\vec{b}_2=2\pi\frac{\vec{a}_3\times\vec{a}_1}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1] $\vec{b}_3=2\pi\frac{\vec{a}_1\times\vec{a}_2}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1]

Structure factors

 $hkl$ $|\vec{G}|$ Å-1 $|F_{hkl}|$ fm $|F_{hkl}|^2$ fm² $\text{Re}\{F_{hkl}\}$ fm $\text{Im}\{F_{hkl}\}$ fm