Diffraction Laws

Bragg’s law

Condition for constructive interference is given by $2d\sin\theta=n\lambda$ where $n$ is an integer.

• Most common applications to x-rays, neutrons, electrons, which have a wavelength comparable to interatomic spacing $∼ 1Å = 10^{−10}$m.

• Bragg’s law assumes specular reflection (equivalent angles of incidence and reflection) of waves from lattice planes.

Laue condition for diffraction

Laue condition for diffraction is $\vec Q=\vec G$, where $\vec Q$ is the scattering vector, $\vec Q=\vec k’-\vec k$.

• Laue approach assumes the crystal is made of identical (sets of) atoms at lattice sites $\vec R$ that scatter waves elastically.

• Constructive interference will occur if $\vec Q\cdot\vec R=2\pi\times$integer for all lattice vectors $\vec R$, and is only satisfied for all $\vec Q$ if the Laue condition is satisfied.

Positioning of $\vec Q$ relative to the scattering planes

When the Laue condition is satisfied, $\vec Q$ is normal to the scattering planes.

Ewald sphere

For a incident beam of certain wavelength 𝜆 on a crystal, the magnitude of $\vec k$ (and $\vec k'$) is fixed and therefore all possible scattering vectors $\vec Q$ lie on the surface of a sphere, the Ewald sphere, of radius $|\vec k|$.

Scattering amplitude $F(\vec Q)$

Is the Fourier transform of the electron density function $𝑛(\vec r)$. Experimentally, only intensity $I=|F|²$ can be measured, so information about phase is lost.

$$F (\vec Q) =\int_{solid} n(\vec r)e^{-i\vec Q\cdot\vec r}\,dV$$

Structure factor $S_G$

If the Laue condition is satisfied, all the points separated by lattice vector $\vec R$ scatter constructively, so $F(\vec Q)=NS_G$ where $N$ is the number of unit cells in a solid.

Atomic form factor $f_j(\vec Q)$

Is the amplitude scattered by the atom, given by the Fourier transform of the electron density of an individual atom, taken over a unit cell.

$$f_j(\vec Q)=\int_{cell} n_j(\vec \rho_j)e^{-i\vec G\cdot\vec \rho}\,dV$$

Atomic form factor value for $\vec Q$ limits

As $|\vec Q|\rightarrow 0$, we find $f_j= Z$, the atomic number.
As $|\vec Q|\rightarrow \infty$, we find $f_j= 0$.

Powder diffraction

A common methods for determining material structures, using x-rays.

Diffraction leads to Debye-Scherrer rings of intensity at specific values of $2\theta$.

Determining structure from powder diffraction

• Rearrange Bragg’s law and the distance between the planes, written $d=a/\sqrt{h^2+k^2+l^2}$ to give an equation in terms of $\sin²\theta$.

• Eliminate unknown lattice parameter $a$ by taking ratios from the $n^{th}$ to the 1st peak, giving $\sin²\theta_n/\sin²\theta_1=R$.

• Calculate the $(ℎ^2 + 𝑘^2 + 𝑙^2)$ ratios for all allowed ℎ𝑘𝑙 reflections for the two potential structures, and then compare to the experimental $\sin^2 𝜃$ ratios.

• Absence/presence of an extra $R$ determines which structure it is.

Intensities from powder diffraction

Variation of the structure factor and the form factor lead to peaks having different scattered intensities for a material.

Why is powder diffraction useful?

• All incident angles of the incoming waves to the Bragg planes are effectively sampled in a single experiment.

• Powdered (or polycrystalline) samples are generally easier to prepare than single-crystal samples.