Solid-State Chemistry/Part 4/X-ray Diffraction

4.2 X-ray Diffraction

X-ray diffraction is the most powerful technique for determining crystal structure. By analyzing the angles and intensities of diffracted X-ray beams, we can determine the positions of atoms within the unit cell with sub-angstrom precision.

X-ray Generation

X-rays are generated when high-energy electrons strike a metal target. Two types of radiation are produced:

- Bremsstrahlung: Continuous spectrum from deceleration of electrons. The short-wavelength limit is $\lambda_{\min} = hc/eV$.
- Characteristic radiation: Sharp peaks from electronic transitions in the target atom. Most useful for diffraction (e.g., Cu K$\alpha$ = 1.5406 A).

Moseley's Law

Moseley discovered that the frequency of characteristic X-rays varies systematically with atomic number:

$$\sqrt{\nu} = a(Z - b)$$

where $a$ and $b$ are constants for a given spectral series. For K$\alpha$ lines, $b \approx 1$. This law confirmed that atomic number, not atomic weight, is the fundamental ordering principle.

Bragg's Law

Constructive interference of X-rays reflected from parallel crystal planes occurs when:

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

where $n$ is the order of diffraction, $\lambda$ is the X-ray wavelength,$d$ is the interplanar spacing, and $\theta$ is the Bragg angle. For cubic crystals:

$$\sin^2\theta = \frac{\lambda^2}{4a^2}(h^2 + k^2 + l^2)$$

Structure Factor

The structure factor determines the intensity of each diffraction peak. For a unit cell with atoms at positions $(x_j, y_j, z_j)$:

$$F_{hkl} = \sum_j f_j\,e^{2\pi i(hx_j + ky_j + lz_j)}$$

where $f_j$ is the atomic scattering factor of atom $j$. The diffracted intensity is proportional to $|F_{hkl}|^2$.

Systematic Absences

When $F_{hkl} = 0$ for certain combinations of indices, those reflections are absent from the diffraction pattern. These selection rules reveal the lattice type:

FCC

Allowed: h, k, l all odd or all even.$F = 4f$ (allowed), $F = 0$ (mixed).

BCC

Allowed: $h + k + l$ = even.$F = 2f$ (allowed), $F = 0$ (odd sum).

Powder Diffraction

In powder (Debye-Scherrer) diffraction, a polycrystalline sample contains crystallites in all orientations. Each set of planes (hkl) produces a cone of diffracted beams at the appropriate Bragg angle. The resulting pattern of rings (or peaks on a detector) provides a fingerprint of the crystal structure.

Applications: phase identification, lattice parameter refinement, strain analysis, and crystallite size determination (via the Scherrer equation).

Video Lectures

20. X-ray Emission and Absorption

21. X-ray Diffraction Techniques I

22. X-ray Diffraction Techniques II

Python: Simulated Powder XRD Patterns

Simulated powder diffraction patterns for FCC (Cu) and BCC (Fe) structures showing peak positions, Miller indices, and selection rules.

Powder XRD Pattern Simulation

Python

Diffraction patterns for FCC Cu and BCC Fe with peak indexing

script.py133 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# ─────────────────────────────────────────────────
# Simulated Powder XRD Pattern for FCC and BCC
# ─────────────────────────────────────────────────

lambda_xray = 1.5406  # Cu K-alpha (angstroms)

def bragg_angle(d, lam):
    """Return 2theta in degrees, or None if not accessible."""
    sin_theta = lam / (2 * d)
    if sin_theta > 1:
        return None
    return 2 * np.degrees(np.arcsin(sin_theta))

def lorentzian_peak(x, x0, intensity, width=0.3):
    return intensity * width**2 / ((x - x0)**2 + width**2)

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(14, 8))
fig.patch.set_facecolor('#0a0a1a')
for ax in (ax1, ax2):
    ax.set_facecolor('#0a0a1a')
    for spine in ax.spines.values():
        spine.set_edgecolor('#334155')

two_theta = np.linspace(10, 140, 5000)

print("=" * 65)
print("Simulated Powder XRD Patterns")
print("=" * 65)

# --- FCC (Cu, a = 3.615 A) ---
a_fcc = 3.615
print(f"\nFCC: Cu (a = {a_fcc} A, Cu K-alpha)")
print(f"{'(hkl)':<10} {'h2+k2+l2':<12} {'d(A)':<10} {'2theta':<10} {'|F|2':<10}")
print("-" * 52)

fcc_peaks = []
hmax = 6
for h in range(hmax):
    for k in range(h+1):
        for l in range(k+1):
            if h == 0 and k == 0 and l == 0:
                continue
            # FCC selection rule: all odd or all even
            if not ((h%2 == k%2) and (k%2 == l%2)):
                continue
            N = h*h + k*k + l*l
            d = a_fcc / np.sqrt(N)
            tt = bragg_angle(d, lambda_xray)
            if tt is None:
                continue
            # Multiplicity (simplified)
            mult = 48  # approximate
            if h == k == l:
                mult = 8
            elif h == k or k == l:
                mult = 24
            # Structure factor for FCC: |F|^2 = 16*f^2 for allowed
            F2 = 16  # relative
            intensity = F2 * mult
            fcc_peaks.append((h, k, l, N, d, tt, intensity))
            print(f"({h}{k}{l})      {N:<12d} {d:<10.4f} {tt:<10.3f} {F2}")

# Plot FCC
pattern_fcc = np.zeros_like(two_theta)
for _, _, _, _, _, tt, intensity in fcc_peaks:
    pattern_fcc += lorentzian_peak(two_theta, tt, intensity / max(p[6] for p in fcc_peaks))

ax1.plot(two_theta, pattern_fcc, color='#22d3ee', linewidth=1.5)
ax1.fill_between(two_theta, 0, pattern_fcc, alpha=0.2, color='#22d3ee')
for h, k, l, N, d, tt, intensity in fcc_peaks:
    ax1.annotate(f'({h}{k}{l})', xy=(tt, intensity/max(p[6] for p in fcc_peaks)),
                fontsize=8, color='#f59e0b', ha='center', va='bottom')
ax1.set_ylabel('Relative Intensity', color='#94a3b8', fontsize=12)
ax1.set_title('FCC: Cu (a = 3.615 A)', color='white', fontsize=14)
ax1.tick_params(colors='#94a3b8')
ax1.grid(True, alpha=0.2)
ax1.set_xlim(10, 140)

# --- BCC (Fe, a = 2.8665 A) ---
a_bcc = 2.8665
print(f"\nBCC: Fe (a = {a_bcc} A, Cu K-alpha)")
print(f"{'(hkl)':<10} {'h2+k2+l2':<12} {'d(A)':<10} {'2theta':<10} {'|F|2':<10}")
print("-" * 52)

bcc_peaks = []
for h in range(hmax):
    for k in range(h+1):
        for l in range(k+1):
            if h == 0 and k == 0 and l == 0:
                continue
            # BCC selection rule: h+k+l = even
            if (h + k + l) % 2 != 0:
                continue
            N = h*h + k*k + l*l
            d = a_bcc / np.sqrt(N)
            tt = bragg_angle(d, lambda_xray)
            if tt is None:
                continue
            mult = 48
            if h == k == l:
                mult = 8
            elif h == k or k == l:
                mult = 24
            F2 = 4  # relative for BCC allowed
            intensity = F2 * mult
            bcc_peaks.append((h, k, l, N, d, tt, intensity))
            print(f"({h}{k}{l})      {N:<12d} {d:<10.4f} {tt:<10.3f} {F2}")

# Plot BCC
pattern_bcc = np.zeros_like(two_theta)
for _, _, _, _, _, tt, intensity in bcc_peaks:
    pattern_bcc += lorentzian_peak(two_theta, tt, intensity / max(p[6] for p in bcc_peaks))

ax2.plot(two_theta, pattern_bcc, color='#a78bfa', linewidth=1.5)
ax2.fill_between(two_theta, 0, pattern_bcc, alpha=0.2, color='#a78bfa')
for h, k, l, N, d, tt, intensity in bcc_peaks:
    ax2.annotate(f'({h}{k}{l})', xy=(tt, intensity/max(p[6] for p in bcc_peaks)),
                fontsize=8, color='#f59e0b', ha='center', va='bottom')
ax2.set_xlabel('2\\theta (degrees)', color='#94a3b8', fontsize=12)
ax2.set_ylabel('Relative Intensity', color='#94a3b8', fontsize=12)
ax2.set_title('BCC: Fe (a = 2.8665 A)', color='white', fontsize=14)
ax2.tick_params(colors='#94a3b8')
ax2.grid(True, alpha=0.2)
ax2.set_xlim(10, 140)

plt.tight_layout()
plt.savefig('xrd_patterns.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
print("\nPlot saved as 'xrd_patterns.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Structure Factor Calculator

Computes Bragg angles and structure factors for FCC and BCC cubic crystals, demonstrating systematic absences from the selection rules.

Bragg Angle & Structure Factor Calculator

Fortran

Structure factors and selection rules for FCC Al and BCC Fe

bragg_calc.f90121 lines

program bragg_structure_factor
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.14159265358979_dp

integer :: h, k, l, hmax, j, n_atoms
  real(dp) :: a, lambda, d_hkl, sin_theta, theta, two_theta
  real(dp) :: F_real, F_imag, F_mag2, f_atom
  real(dp) :: x, y, z, phase

! Atom positions for different structures
  real(dp) :: fcc_pos(4,3), bcc_pos(2,3), sc_pos(1,3), nacl_pos(8,3)
  real(dp) :: f_na, f_cl
  character(len=20) :: struct_name

! FCC atom positions (fractional)
  fcc_pos(1,:) = (/ 0.0_dp, 0.0_dp, 0.0_dp /)
  fcc_pos(2,:) = (/ 0.5_dp, 0.5_dp, 0.0_dp /)
  fcc_pos(3,:) = (/ 0.5_dp, 0.0_dp, 0.5_dp /)
  fcc_pos(4,:) = (/ 0.0_dp, 0.5_dp, 0.5_dp /)

! BCC atom positions
  bcc_pos(1,:) = (/ 0.0_dp, 0.0_dp, 0.0_dp /)
  bcc_pos(2,:) = (/ 0.5_dp, 0.5_dp, 0.5_dp /)

lambda = 1.5406_dp  ! Cu K-alpha
  hmax = 4
  f_atom = 1.0_dp     ! normalized atomic scattering factor

write(*,'(A)') '============================================================'
  write(*,'(A)') ' Bragg Angle & Structure Factor Calculator'
  write(*,'(A)') '============================================================'

! --- FCC (Al, a = 4.0495 A) ---
  a = 4.0495_dp
  n_atoms = 4
  write(*,'(A)') ''
  write(*,'(A,F7.4,A)') ' FCC: Al (a = ', a, ' A)'
  write(*,'(A)') ' (hkl)  d(A)      2theta    |F|^2/f^2  Allowed?'
  write(*,'(A)') ' -----  ------    ------    ---------  --------'

do h = 0, hmax
    do k = 0, h
      do l = 0, k
        if (h==0 .and. k==0 .and. l==0) cycle
        d_hkl = a / sqrt(real(h*h + k*k + l*l, dp))
        sin_theta = lambda / (2.0_dp * d_hkl)
        if (sin_theta > 1.0_dp) cycle
        theta = asin(sin_theta)
        two_theta = 2.0_dp * theta * 180.0_dp / pi

! Compute structure factor
        F_real = 0.0_dp
        F_imag = 0.0_dp
        do j = 1, n_atoms
          x = fcc_pos(j,1); y = fcc_pos(j,2); z = fcc_pos(j,3)
          phase = 2.0_dp * pi * (h*x + k*y + l*z)
          F_real = F_real + f_atom * cos(phase)
          F_imag = F_imag + f_atom * sin(phase)
        end do
        F_mag2 = F_real**2 + F_imag**2

if (F_mag2 > 0.01_dp) then
          write(*,'(2X,A1,I1,I1,I1,A1,2X,F7.4,3X,F7.3,5X,F6.1,6X,A)') &
            '(', h, k, l, ')', d_hkl, two_theta, F_mag2, 'YES'
        else
          write(*,'(2X,A1,I1,I1,I1,A1,2X,F7.4,3X,F7.3,5X,F6.1,6X,A)') &
            '(', h, k, l, ')', d_hkl, two_theta, F_mag2, 'no (extinct)'
        end if
      end do
    end do
  end do

! --- BCC (Fe, a = 2.8665 A) ---
  a = 2.8665_dp
  n_atoms = 2
  write(*,'(A)') ''
  write(*,'(A,F7.4,A)') ' BCC: Fe (a = ', a, ' A)'
  write(*,'(A)') ' (hkl)  d(A)      2theta    |F|^2/f^2  Allowed?'
  write(*,'(A)') ' -----  ------    ------    ---------  --------'

F_real = 0.0_dp
        F_imag = 0.0_dp
        do j = 1, n_atoms
          x = bcc_pos(j,1); y = bcc_pos(j,2); z = bcc_pos(j,3)
          phase = 2.0_dp * pi * (h*x + k*y + l*z)
          F_real = F_real + f_atom * cos(phase)
          F_imag = F_imag + f_atom * sin(phase)
        end do
        F_mag2 = F_real**2 + F_imag**2

write(*,'(A)') ''
  write(*,'(A)') ' Selection Rules Summary:'
  write(*,'(A)') '   FCC: h,k,l all odd or all even -> F = 4f'
  write(*,'(A)') '        mixed -> F = 0 (systematic absence)'
  write(*,'(A)') '   BCC: h+k+l = even -> F = 2f'
  write(*,'(A)') '        h+k+l = odd  -> F = 0 (systematic absence)'
  write(*,'(A)') ''
  write(*,'(A)') 'Done.'
end program bragg_structure_factor

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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