Bragg plane

Ray diagram of Von Laue formulation

In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, $\mathbf{𝐊}$, at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.

Considering the adjacent diagram, 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:

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

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

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

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:

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

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

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

By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if $\mathbf{𝐊}=\mathbf{𝐤}-{\mathbf{𝐤}}^{\mathit{\prime }}$ is a vector of the reciprocal lattice. We notice that $\mathbf{𝐤}$ and ${\mathbf{𝐤}}^{\mathit{\prime }}$ 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.

See also

• X-ray crystallography
• Reciprocal lattice
• Bravais lattice
• Powder diffraction
• Kikuchi line
• Brillouin zone