Bragg diffraction

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The Bragg formulation of X-ray diffraction (also referred to as Bragg diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid). They found that in these crystals, for certain specific wavelengths and incident angles, intense peaks of reflected radiation (known as Bragg peaks) were produced. The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes.

W. L. Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter d. It was proposed that the incident X-ray radiation would produce a Bragg peak if their reflections off the various planes interfered constructively

1b) Skewed Plane NaCl Crystal.

1a) Normal Plane NaCl Crystal.

1 Mechanics
2 Laue diffraction
3 Selection Rules and Practical X-Ray Crystallography
4 Nobel Prize for Bragg diffraction
5 See also
6 External links
7 References

[edit] Mechanics

As shown in the images on the right, a given crystal (in this case, NaCl) can be decomposed into any number of different Bragg plane configurations, due to the periodicity of the crystal lattice. The incident angle of the incoming wave and its wavelength determines which set of planes is relevant in the calculation.

As the wave enters the crystal, some portion of it will be reflected by the first layer, while the rest will continue through to the second layer, where the process continues. By the definition of constructive interference, the separately reflected waves will remain in phase if the difference in the path length of each wave is equal to an integer multiple of the wavelength.

In the figure 2 on the right, the path difference is given by $\begin{matrix}2d\sin\theta\end{matrix}\,$ , where d denotes the interplanar distance.

$2) Diffraction Calculation$

2) Diffraction Calculation

This gives the formula for what is known as the Bragg condition or Bragg's law:

$2 d\sin\theta = n\lambda\,$

Waves that satisfy this condition interfere constructively and result in a reflected wave of significant intensity.

[edit] Laue diffraction

The phenomenon of crystal diffraction can also be formulated in other equivalent ways. One such example is the von Laue formulation of X-ray and neutron diffraction. In this model, the crystal is instead seen in reciprocal space which describes the whole set of lattice plances of a real space crystal. It is equally regular and forms a related reciprocal lattice. The condition for constructive interference in this formulation is given by:

$\vec{R}\cdot(\vec{k} - \vec{k'}) = 2\pi n,$

where n is again an integer, $\vec k$ is the wave vector describing the incoming wave, $\vec{k}'$ is the wave vector describing the outgoing wave, and $\vec R$ is any reciprocal lattice lattice vector. This can be equivalently stated as

$e^{i(\vec{k'}-\vec{k})\cdot\vec{R}}=1,$

or, defining $\vec G$ to be a reciprocal lattice vector, and assuming that $|\vec k|=|\vec{k}'|$ ,

$\vec{k}\cdot\vec{G} = \frac{1}{2} G^2$

This final statement can be interpreted as saying that the Laue condition (for constructive interference) is satisfied if and only if the wave vector $\vec k$ lies in a plane that is the perpendicular bisector to a reciprocal lattice vector $\vec G$ lying at the origin of k-space. These planes are nothing other than the Bragg planes encountered earlier.

To further exemplify the equivalence between these two formulations (the Bragg formulation and the Van Laue formulation), note that the reciprocal lattice vector $\vec G$ must have a magnitude which is an integer multiple of $2π / d$ , where d is again the interplanar distance (this is a consequence of the definition of the reciprocal lattice). Therefore,

$|\vec{G}| = \frac{2 \pi n}{d}$

Furthermore, from the results of the Van Laue formulation, we know that in the case of constructive interference, we have

$|\vec{G}| = 2\vec{k}\cdot\frac{\vec{G}}{|\vec{G}|} = 2|\vec{k}|\sin \theta,$

where $θ$ is the angle between the incoming wave vector $\vec k$ and the plane perpendicular to the reciprocal lattice vector G.

Setting these two equations equal to each other, and recognizing the magnitude of the wave vector $\vec k$ is simply equal to $2π / λ$ , the Bragg condition is retrieved:

$2 d\sin\theta = n\lambda\,$

The Laue formalism is further exploited in the dynamical theory of diffraction.

[edit] Selection Rules and Practical X-Ray Crystallography

Bragg's Law, as stated above, can be used to obtain the lattice spacing of a particular cubic system through the following relation:

$d = \frac{a}{ \sqrt{h^2 + k^2 + l^2}}$

Where $a$ is the lattice spacing of the cubic crystal, and $h$ , $k$ , and $l$ are the Miller indices of the Bragg plane. Combining this relation with Bragg's Law:

$\bigg( \frac{ \lambda\ }{ 2a } \bigg)^2 = \frac{ \sin ^2 \theta\ }{ h^2 + k^2 + l^2 }$

One can derive selection rules for the Miller indices for different cubic Bravais lattices; here, selection rules for several will be given as is.

Selection Rules for the Miller Indices
Bravais Lattice	Example Compounds	Allowed Reflections	Forbidden Reflections
Simple Cubic	Simple Cubic	Any h,k,l	None
Body-Centered Cubic	Body-Centered Cubic	h+k+l even	h+k+l odd
Face-Centered Cubic	NaCl, KCl,Zinc Blende	h,k,l all odd or all even	h,k,l mixed odd or even
Hexagonal	Hexagonal close packed	l even, h+2k $\neq$ 3n	h + 2k = 3n for odd l

These selection rules can be used for any crystal with the given crystal structure. Selection rules for other structures can be referenced elsewhere, or derived: these were all the ones I could find.

[edit] Nobel Prize for Bragg diffraction

In 1915, William Henry Bragg and William Lawrence Bragg were awarded the Nobel Prize for their contributions to crystal structure analysis. They were the first and (so far) the only father-son team to have jointly won the prize. Other father/son laureates include Niels and Aage Bohr, Manne and Kai Siegbahn, J.J. and George Thomson, and Hans von Euler-Chelpin and Ulf von Euler all having been awarded the prize for separate contributions.