Bragg plane

In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, $\scriptstyle \mathbf$, 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:

:$e^ = \cos + i\sin$

Where $\scriptstyle \mathbf$ is the incident wave vector given by:

:$\mathbf = \frac\hat$

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:

:$\mathbf = \frac\hat^\prime$

The condition for constructive interference in the $\scriptstyle \hat^\prime$ 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:

:$, \mathbf, \cos + , \mathbf, \cos = \mathbf \cdot \left(\hat - \hat^\prime\right) = m\lambda$

where $\scriptstyle m ~\in~ \mathbb$. Multiplying the above by $\scriptstyle \frac$ we formulate the condition in terms of the wave vectors, $\scriptstyle \mathbf$ and $\scriptstyle \mathbf$:

:$\mathbf \cdot \left(\mathbf - \mathbf\right) = 2\pi m$

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

:$\mathbf \cdot \left(\mathbf - \mathbf\right) = 2\pi m$

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

:$e^ = 1$

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

See also

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

References

{{Crystallography
Crystallography
Geometry
Fourier analysis
Lattice points
Diffraction