Saddle-node bifurcation

Ray diagram of Von Laue formulation

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

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See also

• X-ray crystallography
• Reciprocal lattice
• Bravais lattice
• Powder diffraction
• Kikuchi line
• Brillouin zone