Bragg plane information

In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, ${\displaystyle \scriptstyle \mathbf {K} }$, 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 adjacent diagram, the arriving x-ray plane wave is defined by:

${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\cos {(\mathbf {k} \cdot \mathbf {r} )}+i\sin {(\mathbf {k} \cdot \mathbf {r} )}}$

Where ${\displaystyle \scriptstyle \mathbf {k} }$ is the incident wave vector given by:

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

where ${\displaystyle \scriptstyle \lambda }$ 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:

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

The condition for constructive interference in the ${\displaystyle \scriptstyle {\hat {n}}^{\prime }}$ 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:

${\displaystyle |\mathbf {d} |\cos {\theta }+|\mathbf {d} |\cos {\theta ^{\prime }}=\mathbf {d} \cdot \left({\hat {n}}-{\hat {n}}^{\prime }\right)=m\lambda }$

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

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

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

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

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

${\displaystyle e^{i\left(\mathbf {k} -\mathbf {k^{\prime }} \right)\cdot \mathbf {R} }=1}$

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