Neutrons are quantum particles that move like waves and will diffract from a crystal. The wavelength of the neutrons is related to their kinetic energy $\frac{m_nv^2}{2}=\frac{p^2}{2m_n}=\frac{\hbar^2k^2}{2m_n} =\frac{h^2}{2m_n\lambda^2}.$ Here $m_n$ is the mass of a neutron, $v$ is the velocity, $p$ is the momentum, $k$ is the wave number, and $\lambda$ is the wavelength. Neutrons have a magnetic moment and scatter through a magnetic interaction. They are used in cases where it is difficult to distinguish between atoms in x-ray diffraction because the atoms have almost the same electron density, such as iron and nickel. Neutrons also detect hydrogen better than x-rays since hydrogen often loses its electron when it is bound in a crystal and there is little electron density for the x-rays to scatter from. A disadvantage of neutron diffraction is that it typically requires a nuclear reactor to produce the neutron beam. This makes neutron diffraction experiments significantly more expensive than x-ray diffraction. The diffraction condition for neutrons is the same as it is for x-rays and electrons, $\Delta\vec{k}=\vec{G}.$ The intensities of the diffraction peaks are proportional to the square of the structure factor,

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

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. A table of the scattering lengths can be found 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]

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Structure factors

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