# Bragg plane

Ray diagram of Von Laue formulation

In physics, a Bragg plane is a plane in reciprocal space which bisects one reciprocal lattice vector $\mathbf{K}$.[1] It is relevant to define this plane as part of the definition of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.

Considering the diagram at right, the arriving x-ray plane wave is defined by:

$e^{i\mathbf{k}\cdot\mathbf{r}}=\cos {(\mathbf{k}\cdot\mathbf{r})} +i\sin {(\mathbf{k}\cdot\mathbf{r})}$

Where $\mathbf{k}$ is the incident wave vector given by:

$\mathbf{k}=\frac{2\pi}{\lambda}\hat n$

where $\lambda$ is the wavelength of the incident photon. While the Bragg formulation assumes a unique choice of direct lattice planes and specular reflection of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as a source of secondary wavelets as described by the Huygens principle. Each scattered wave contributes to a new plane wave given by:

$\mathbf{k^\prime}=\frac{2\pi}{\lambda}\hat n^\prime$

The condition for constructive interference in the $\hat n^\prime$ direction is that the path difference between the photons is an integer multiple (m) of their wavelength. We know then that for constructive interference we have:

$|\mathbf{d}|\cos{\theta}+|\mathbf{d}|\cos{\theta^\prime}=\mathbf{d}\cdot(\hat n-\hat n^\prime)=m\lambda$

where $m\in\mathbb{Z}$. Multiplying the above by $2\pi/\lambda$ we formulate the condition in terms of the wave vectors $\mathbf{k}$ and $\mathbf{k^\prime}$:

$\mathbf{d}\cdot(\mathbf{k}-\mathbf{k^\prime})=2\pi m$
The Bragg plane in blue, with its associated reciprocal lattice vector K.

Now consider that a crystal is an array of scattering centres, each at a point in the Bravais lattice. We can set one of the scattering centres as the origin of an array. Since the lattice points are displaced by the Bravais lattice vectors $\mathbf{R}$, scattered waves interfere constructively when the above condition holds simultaneously for all values of $\mathbf{R}$ which are Bravais lattice vectors, the condition then becomes:

$\mathbf{R}\cdot(\mathbf{k}-\mathbf{k^\prime})=2\pi m$

An equivalent statement (see mathematical description of the reciprocal lattice) is to say that:

$e^{i(\mathbf{k}-\mathbf{k^\prime})\cdot\mathbf{R}}=1$

By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if $\mathbf{K}=\mathbf{k}-\mathbf{k^\prime}$ is a vector of the reciprocal lattice. We notice that $\mathbf{k}$ and $\mathbf{k^\prime}$ have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector $\mathbf{k}$ must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector $\mathbf{K}$. This reciprocal space plane is the Bragg plane.

## References

1. ^ Ashcroft, Neil W.; Mermin, David (January 2, 1976). Solid State Physics (1 ed.). Brooks Cole. pp. 96–100. ISBN 0-03-083993-9.