Bragg plane
In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, [math]\displaystyle{ \scriptstyle \mathbf{K} }[/math], 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 adjacent diagram, the arriving x-ray plane wave is defined by:
- [math]\displaystyle{ e^{i\mathbf{k} \cdot \mathbf{r}} = \cos {(\mathbf{k} \cdot \mathbf{r})} + i\sin {(\mathbf{k} \cdot \mathbf{r})} }[/math]
Where [math]\displaystyle{ \scriptstyle \mathbf{k} }[/math] is the incident wave vector given by:
- [math]\displaystyle{ \mathbf{k} = \frac{2\pi}{\lambda}\hat{n} }[/math]
where [math]\displaystyle{ \scriptstyle \lambda }[/math] 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:
- [math]\displaystyle{ \mathbf{k^\prime} = \frac{2\pi}{\lambda}\hat{n}^\prime }[/math]
The condition for constructive interference in the [math]\displaystyle{ \scriptstyle \hat{n}^\prime }[/math] 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:
- [math]\displaystyle{ |\mathbf{d}|\cos{\theta} + |\mathbf{d}|\cos{\theta^\prime} = \mathbf{d} \cdot \left(\hat{n} - \hat{n}^\prime\right) = m\lambda }[/math]
where [math]\displaystyle{ \scriptstyle m ~\in~ \mathbb{Z} }[/math]. Multiplying the above by [math]\displaystyle{ \scriptstyle \frac{2\pi}{\lambda} }[/math] we formulate the condition in terms of the wave vectors, [math]\displaystyle{ \scriptstyle \mathbf{k} }[/math] and [math]\displaystyle{ \scriptstyle \mathbf{k^\prime} }[/math]:
- [math]\displaystyle{ \mathbf{d} \cdot \left(\mathbf{k} - \mathbf{k^\prime}\right) = 2\pi m }[/math]
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, [math]\displaystyle{ \scriptstyle \mathbf{R} }[/math], scattered waves interfere constructively when the above condition holds simultaneously for all values of [math]\displaystyle{ \scriptstyle \mathbf{R} }[/math] which are Bravais lattice vectors, the condition then becomes:
- [math]\displaystyle{ \mathbf{R} \cdot \left(\mathbf{k} - \mathbf{k^\prime}\right) = 2\pi m }[/math]
An equivalent statement (see mathematical description of the reciprocal lattice) is to say that:
- [math]\displaystyle{ e^{i\left(\mathbf{k} - \mathbf{k^\prime}\right) \cdot \mathbf{R}} = 1 }[/math]
By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if [math]\displaystyle{ \scriptstyle \mathbf{K} ~=~ \mathbf{k} \,-\, \mathbf{k^\prime} }[/math] is a vector of the reciprocal lattice. We notice that [math]\displaystyle{ \scriptstyle \mathbf{k} }[/math] and [math]\displaystyle{ \scriptstyle \mathbf{k^\prime} }[/math] have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector, [math]\displaystyle{ \scriptstyle \mathbf{k} }[/math], must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector, [math]\displaystyle{ \scriptstyle \mathbf{K} }[/math]. This reciprocal space plane is the Bragg plane.
See also
- X-ray crystallography
- Reciprocal lattice
- Bravais lattice
- Powder diffraction
- Kikuchi line
- Brillouin zone
References
- ↑ Ashcroft, Neil W.; Mermin, David (January 2, 1976). Solid State Physics (1 ed.). Brooks Cole. pp. 96–100. ISBN 0-03-083993-9. https://archive.org/details/solidstatephysic00ashc/page/96.
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