Bragg plane

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

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

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

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

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

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

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

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

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

\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:

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 $\scriptstyle \mathbf {K} ~=~\mathbf {k} \,-\,\mathbf {k^{\prime }}$ is a vector of the reciprocal lattice. We notice that $\scriptstyle \mathbf {k}$ and $\scriptstyle \mathbf {k^{\prime }}$ have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector, $\scriptstyle \mathbf {k}$ , must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector, $\scriptstyle \mathbf {K}$ . 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.

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

[1]

See also

References