Bragg plane

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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.

References

↑ Ashcroft, Neil W.; Mermin, David (January 2, 1976). Solid State Physics (1 ed.). Brooks Cole. pp. 96–100. ISBN 0-03-083993-9.

Bragg plane

References

See also