X-Ray Crystallography Lab: Bragg, SC-XRD, PXRD, Size & Strain Analysis

Finite Thickness Interference

Bragg's Law & Finite Crystal Thickness ($N$ Planes)

Constructive interference occurs when path difference equals integer wavelengths $\lambda$. Adjust the number of atomic planes $N$ (thickness $t = N \cdot d$) to see how thin crystal plates cause diffraction peak broadening via the Laue interference function.

Bragg's Law $n\lambda = 2d\sin\theta$

Laue Factor $I(\theta) \propto \frac{\sin^2(N\phi/2)}{\sin^2(\phi/2)}$

Wavefronts & Lattice Planes ($N$ Layers)

Destructive Interference

Path Diff: 2.828 Å | Phase $\phi$: 360.0°

Crystal Thickness $t = N \cdot d$: 12.50 Å (1.25 nm)

Relative Intensity: 100.0% | FWHM $\beta$: 4.2°

Quick Bragg Angles:

Experimental Parameters

Planes $N$ (Thickness Effect) N = 5 planes (12.5 Å)

N = 2 (broad peak, thin) N = 30 (sharp peak, bulk)

Incident Angle ($\theta$) 17.9°

5.0° Diffraction Angle $2\theta$ = 35.8° 80.0°

Plane Spacing ($d$) 2.50 Å

X-ray Wavelength ($\lambda$) 1.5418 Å (Cu Kα)

Interference Function $I(\theta)$ vs $\theta$

Current θ marker

Thickness Effect: Increasing $N$ concentrates intensity into sharper peaks whose width is inversely proportional to $t = N \cdot d$. As $N \to \infty$, the function converges to the delta-function of ideal infinite Bragg reflection.

Single Crystal XRD & Ewald Sphere

Reciprocal Lattice, Ewald Sphere & 2D Area Detector

Because a single crystal has one unique orientation in 3D, constructive interference occurs only when a reciprocal lattice point intersects the Ewald sphere of radius $1/\lambda$, projecting sharp, discrete diffraction spots onto the detector.

$\Delta \mathbf{k} = \mathbf{k}_{\text{diff}} - \mathbf{k}_{\text{in}} = \mathbf{G}_{hkl}$

3D Goniometer & Reciprocal Lattice Drag to Orbit

• Cyan sphere: Ewald Sphere ($R = 1/\lambda$)

• Amber dots: Reciprocal Nodes $(hkl)$

• Green flash: Bragg condition met!

• Red beam: Incident X-ray vector $\mathbf{k}_{\text{in}}$

Crystal Rotation Angle ($\omega$): 0.0°

Lattice Parameter ($a$): 3.80 Å

2D Area Detector Frame

Sharply localized diffraction spots

0 Active Spots

Active Reflection $(hkl)$: Diffraction $2\theta$

Rotate crystal to bring reciprocal points to Ewald sphere...

Single Crystal Characteristic

In a single crystal, all unit cells have parallel lattice orientations. The condition $\mathbf{k}_{\text{diff}} - \mathbf{k}_{\text{in}} = \mathbf{G}_{hkl}$ is satisfied only at discrete rotational angles $\omega$. This produces isolated spots rather than continuous rings!

Powder XRD (PXRD)

Debye-Scherrer Cones & 1D Diffractogram: $I$ vs $2\theta$

Billions of randomly oriented crystallites project concentric Debye-Scherrer cones of aperture angle $4\theta$. Radially integrating these rings yields the 1D powder pattern plotted directly as Diffraction Intensity versus $2\theta$.

Sample:

3D Debye-Scherrer Diffraction Cones Aperture = 4θ

Capillary $\to$ Detector Plate

• Concentric cones: Each corresponds to an $(hkl)$ family

• Translucent plate: Intercepting detector plane

• Amber tube: Polycrystalline powder sample

Debye Rings on 2D Detector

$R = D \tan(2\theta)$

Cutting the cones with a flat camera produces concentric circular rings.

• 2D to 1D Reduction: Azimuthal integration along the circular rings converts the 2D pixel image into the standard 1D powder scan of Intensity vs $2\theta$ shown below.

Powder Diffractogram: Intensity vs $2\theta$ (Bragg-Brentano Geometry)

Horizontal axis is $2\theta$ (two times the Bragg angle) from $10^\circ$ to $90^\circ$. Broadening and shifts reflect both crystallite thickness and lattice strain parameters.

Instrument Noise

Diffraction Angle ($2\theta$) 28.44°

Miller Indices $(hkl)$ (1 1 1)

Interplanar Spacing $d$ 3.135 Å

Peak FWHM $\beta_{\text{total}}$ 0.36°

Strain & Peak Broadening Analysis

Microstrain, Macrostrain & Williamson-Hall Separation

Crystal thickness causes $1/\cos\theta$ broadening, microstrain causes $\tan\theta$ broadening, and macrostrain shifts peaks ($\Delta 2\theta = -2\varepsilon_{\text{macro}}\tan\theta$). The Williamson-Hall Plot ($\beta\cos\theta$ vs $4\sin\theta$) separates size (y-intercept) from strain (slope).

$\beta_{\text{total}}\cos\theta = \frac{K\lambda}{L} + 4\varepsilon_{\text{micro}}\sin\theta$

Crystallite Thickness ($L$) 25 nm

3 nm (Nanocrystal) 100 nm (Bulk)

Scherrer Broadening: $\beta_{\text{size}} = \frac{K\lambda}{L\cos\theta}$ (independent of strain)

Microstrain ($\varepsilon_{\text{micro}}$) 0.20 %

0.0% (Defect-free) 1.5% (High defect / milled)

Strain Broadening: $\beta_{\text{strain}} = 4\varepsilon\tan\theta$ (increases rapidly at high $2\theta$)

Uniform Macrostrain ($\varepsilon_{\text{macro}}$) 0.00 %

-1.0% (Compressive → Shift +) +1.0% (Tensile → Shift −)

Peak Position Shift: $\Delta 2\theta = -2\varepsilon_{\text{macro}}\tan\theta$ (no broadening)

Williamson-Hall Plot ($\beta \cos\theta$ vs $4 \sin\theta$)

Separates size from microstrain by linear regression

Extracted $L$: 25.0 nm
Extracted $\varepsilon$: 0.200 %

Intercept ($y = 0$): $\frac{K\lambda}{L} = $ 0.0055

Slope: $\varepsilon_{\text{micro}} = $ 0.0020

Microscopic Lattice State

Distortion of atomic lattice planes under chosen parameters

Why $\tan\theta$ vs $1/\cos\theta$? Differentiating Bragg's Law with respect to spacing $d$ gives $\frac{\Delta\theta}{\Delta d} = -\frac{\tan\theta}{d}$. Microstrain represents fluctuations in spacing $\Delta d/d$, causing broadening that diverges with $\tan\theta$ at high angles.

Theory & Quantitative Models

Physical Principles of X-Ray Diffraction, Size & Strain

Review the geometrical derivation of Bragg's Law, crystal thickness effects (Laue interference function and Scherrer broadening), lattice strain effects (microstrain vs macrostrain), and the Williamson-Hall deconvolution.

1. Bragg's Law & Geometric Path Difference

Parallel monochromatic X-ray beams striking adjacent planes separated by distance $d$ at grazing incident angle $\theta$ have an optical path difference equal to:

$$\Delta = AB + BC = d\sin\theta + d\sin\theta = 2d\sin\theta$$

$$\text{Constructive Interference Condition: } \Delta = n\lambda$$

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

Because $\sin\theta \le 1$, observable diffraction demands that $\lambda \le 2d$. X-rays possess wavelengths of $\approx 0.5 - 2.5\text{ \AA}$, making them ideally suited to probe crystal lattices.

2. Crystal Thickness & Scherrer Equation

For a finite crystal of $N$ planes (thickness $t = N \cdot d$), summing the electric fields yields the Laue interference function:

$$I(\theta) = I_0 \frac{\sin^2(N \phi / 2)}{\sin^2(\phi / 2)}, \quad \phi = \frac{4\pi d}{\lambda}\sin\theta$$

$$\text{Scherrer Crystallite Broadening: } \beta_{\text{size}} = \frac{K \lambda}{L \cos\theta}$$

Here, $\beta$ is the full width at half-maximum (FWHM) in radians, $K \approx 0.9$ is the Scherrer shape constant, and $L$ is the crystallite thickness. As $L \to \infty$, the function converges to an infinitely sharp delta-function.

3. Microstrain vs Macrostrain

Lattice strain exerts two distinctly different physical consequences on the diffraction pattern:

Uniform (Macro) Strain: Shifts peak positions:

$$\Delta 2\theta = -2 \varepsilon_{\text{macro}} \tan\theta$$

Non-uniform (Micro) Strain: Broadens peaks:

$$\beta_{\text{strain}} = 4 \varepsilon_{\text{micro}} \tan\theta$$

Tensile macrostrain increases $d$, shifting peaks to lower $2\theta$. Compressive macrostrain decreases $d$, shifting peaks to higher $2\theta$. Microstrain represents crystal defect fields (dislocations, vacancies) with distributed $\Delta d/d$.

4. Williamson-Hall Method

Assuming Lorentzian or Voigt sum of the size and strain contributions to the total peak integral breadth $\beta_{\text{total}}$:

$$\beta_{\text{total}} = \beta_{\text{size}} + \beta_{\text{strain}} = \frac{K\lambda}{L\cos\theta} + 4\varepsilon\tan\theta$$

$$\beta_{\text{total}}\cos\theta = \frac{K\lambda}{L} + 4\varepsilon\sin\theta$$

Plotting $\beta_{\text{total}}\cos\theta$ on the y-axis against $4\sin\theta$ on the x-axis yields a straight line with:
• Y-intercept: $K\lambda / L$ (inversely proportional to crystallite size)
• Slope: $\varepsilon$ (equal to the microstrain)

Comparison: Single Crystal XRD vs Powder XRD

Property	Single Crystal XRD (SC-XRD)	Powder XRD (PXRD)
Sample Form	One discrete, uniform crystal (~50–300 μm)	Ensemble of $10^7$ to $10^{10}$ microcrystallites
Diffraction Pattern	Discrete 3D spots in reciprocal space	Debye-Scherrer concentric rings $\to$ 1D trace ($I$ vs $2\theta$)
Strain Measurement	3D spot shape profile tensor & orientation matrix distortion	Peak broadening ($\tan\theta$ dependence) & position shifts
Thickness Effect	Reciprocal lattice nodes elongate into reciprocal rods ("rel-rods")	Diffraction peaks broaden into Cauchy/Voigt profiles (Scherrer effect)