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Bragg plane

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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.

See also

References

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