Diffraction

3.1 Bragg”s Law

X-ray diffraction from crystal planes produces constructive interference when:

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

Where $d$ is the interplanar spacing, $\theta$ is the angle of incidence, and $n$ is the order.

Derivation. The path difference between waves scattered from adjacent planes is $2d\sin\theta$. Constructive interference requires this to be an integer multiple of $\lambda$. $\blacksquare$

For the $n$-th order reflection from $(hkl)$ planes, one can equivalently define it as the first-order Reflection from $(nh\ nk\ nl)$ planes with spacing $d/n$.

3.2 The Laue Condition

Diffraction occurs when the scattering vector equals a reciprocal lattice vector:

$\Delta\mathbf{k} = \mathbf{k}' - \mathbf{k} = \mathbf{G}$

This is equivalent to Bragg’s law. Since $\lvert\mathbf{k}\rvert = \lvert\mathbf{k}'\rvert$ (elastic Scattering), the Laue condition requires $\mathbf{k}$ to terminate on the Ewald sphere (a sphere Of radius $k$ centred at the tip of $\mathbf{k}$).

Equivalence with Bragg’s law. From $\lvert\mathbf{k}\rvert = \lvert\mathbf{k} + \mathbf{G}\rvert$:

$k^2 = \lvert\mathbf{k} + \mathbf{G}\rvert^2 = k^2 + G^2 + 2\mathbf{k}\cdot\mathbf{G}$

$\implies \mathbf{k}\cdot\mathbf{G} = -\frac{G^2}{2}$

Since $G = 2\pi/d$ and $\lvert\mathbf{k}\cdot\hat{\mathbf{G}}\rvert = k\sin\theta$:

$k\sin\theta = \frac{G}{2} = \frac{\pi}{d}$

Using $k = 2\pi/\lambda$: $2d\sin\theta = \lambda$ (first order). $\blacksquare$

3.3 Structure Factor

The structure factor determines the intensity of diffraction from planes $(hkl)$:

$S_{hkl} = \sum_j f_j e^{-i\mathbf{G}_{hkl}\cdot\mathbf{d}_j}$

Where $f_j$ is the atomic form factor of atom $j$ at position $\mathbf{d}_j$ in the basis.

Example: BCC. Two atoms at $(0,0,0)$ and $(a/2, a/2, a/2)$ in the conventional cell:

$S_{hkl} = f\left[1 + e^{-i\pi(h+k+l)}\right] = f\left[1 + (-1)^{h+k+l}\right]$

Reflections are present only when $h + k + l$ is even. When $h + k + l$ is odd, $S_{hkl} = 0$ (systematic absence).

Example: FCC. Atoms at $(0,0,0)$, $(a/2,a/2,0)$, $(a/2,0,a/2)$, $(0,a/2,a/2)$:

$S_{hkl} = f\left[1 + e^{-i\pi(h+k)} + e^{-i\pi(h+l)} + e^{-i\pi(k+l)}\right]$

Reflections present only when $h, k, l$ are all even or all odd.

3.4 Worked Examples: Structure Factor Calculations

3.5 Systematic Absences

Systematic absences arise from lattice centring and glide planes/screw axes, and are summarised by The structure factor:

Systematic absences allow unambiguous identification of the Bravais lattice from a diffraction pattern. The presence of a $(100)$ reflection rules out BCC; the presence of $(200)$ but absence of $(110)$ Identifies FCC.

3.6 Powder Diffraction

In a powder diffraction experiment, a polycrystalline sample with randomly oriented crystallites Is illuminated by a monochromatic X-ray beam. Each family of planes $(hkl)$ that satisfies Bragg’s Law produces a diffraction cone at angle $2\theta$ from the incident beam.

The Bragg—Brentano geometry uses a divergent beam and a focusing detector, recording intensity As a function of $2\theta$. Each peak position gives $d_{hkl}$ via Bragg’s law, and the peak Intensity is proportional to $\lvert S_{hkl}\rvert^2$ times multiplicity and geometric factors.

Scherrer equation. For crystallites of size $L$The diffraction peaks are broadened. The Full width at half maximum (FWHM) $\beta$ (in radians) relates to the crystallite size by:

$L = \frac{K\lambda}{\beta\cos\theta}$

Where $K \approx 0.89$ is the Scherrer constant. This provides a straightforward method for Estimating nanocrystallite sizes from powder diffraction data.

:::caution Common Pitfall Do not confuse the Laue condition $\Delta\mathbf{k} = \mathbf{G}$ with Bragg’s law $2d\sin\theta = n\lambda$. These are equivalent formulations of the same physics. The Laue condition is a vector equation in Reciprocal space, while Bragg’s law is a scalar equation in real space. Converting between them Requires careful geometry --- remember that $\theta$ in Bragg’s law is measured from the plane, Not from the normal.

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