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, $\scriptstyle \mathbf{K}$, at right angles.^[1] The Bragg plane is defined as part 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 $\scriptstyle \mathbf{k}$ is the incident wave vector given by:

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

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

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

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, $\scriptstyle \mathbf{R}$, scattered waves interfere constructively when the above condition holds simultaneously for all values of $\scriptstyle \mathbf{R}$ which are Bravais lattice vectors, the condition then becomes:

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

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 $\scriptstyle \mathbf{K} ~=~ \mathbf{k} \,-\, \mathbf{k^\prime}$ is a vector of the reciprocal lattice. We notice that $\scriptstyle \mathbf{k}$ and $\scriptstyle \mathbf{k^\prime}$ have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector, $\scriptstyle \mathbf{k}$, must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector, $\scriptstyle \mathbf{K}$. This reciprocal space plane is the Bragg plane.

See also

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

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.