Plane-Crystal

Crystal Diffraction

We simulate X-ray diffraction using the dynamical theory in the Bragg-case geometry, assuming a perfect and thick crystal.

We compute the reflection coefficient based on this model. The reflected field is then determined from it. To run this simulation, the following user inputs are required:

Required Inputs

  • Photon energy (in eV)
  • lattice spacing*2 (in nm)
  • Unit cell volume (in nm³)
  • Structure factors:
    • F₀, F_H, F_H̄ with real/imaginary components
  • Crystal surface offset angle α (in radians)

The following sections explain how each physical quantity and formula is derived and used in the simulation. This section details the implementation of dynamical X-ray diffraction theory for perfect crystals, based on the foundational work of Batterman & Cole (1964).

Bragg Angle Calculation

\[ \theta_B = \arcsin\left(\frac{\lambda}{2d}\right) \]

  • Function: getBraggAngle(energy, dSpacing2)
  • Purpose: Calculate the Bragg diffraction angle
  • Parameters:
    • energy: Photon energy (eV)
    • dSpacing2: lattice spacing*2 (nm)
  • Returns: Bragg angle in radians

Asymmetry Factor

\[ b = \frac{\sin(\theta_B - \alpha)}{\sin(\theta_B + \alpha)} \]

  • Function: getAsymmetryFactor(braggAngle, offsetAngle)
  • Purpose: Account for crystal surface orientation effects
  • Parameters:
    • braggAngle: Calculated Bragg angle
    • offsetAngle: Surface tilt angle α

This definition corresponds to the asymmetry factor derived from direction cosines
and is consistent with the definition on page 690 in Batterman & Cole (1964).

Diffraction Prefactor

\[ \Gamma = \frac{r_e \lambda^2}{\pi V} \]

  • Function: getDiffractionPrefactor(wavelength, unitCellVolume)
  • Purpose: Calculate scaling factor for absorption/dispersion
  • Parameters:
    • wavelength: X-ray wavelength (nm)
    • unitCellVolume: Unit cell volume (nm³)

Gamma is defined in Batterman & Cole (1964) p. 685.

Eta Parameter (η)

\[ \eta = \frac{b\Delta\theta\sin 2\theta + \frac{1}{2}\Gamma F_0(1-b)}{\Gamma|P|\sqrt{|b|}\sqrt{F_H F_{\overline{H}}}} \]

  • Function: computeEta(theta, bragg, asymmetry, FH_re, FH_im, FHC_re, FHC_im, F0_re, F0_im, polFactor, gamma)
  • Purpose: Compute normalized angular deviation parameter
  • Parameters:
    • Structure factors (F₀, F_H, F_H̄) with real/imaginary components
    • polFactor: Polarization factor (1 or |cos2θ_B|)

Eta is defined as equation (32) in Batterman & Cole (1964) p. 690 .

Reflection Coefficient (R)

\[ R = \left(\eta \pm \sqrt{\eta^2 - 1}\right)\sqrt{\frac{F_H}{F_{\overline{H}}}} \]

  • Function: computeR(eta, FH_re, FH_im, FHC_re, FHC_im)
  • Purpose: Calculate complex reflection coefficient
  • Note: Sign selection based on real part of η

This function is based on Equation (103) from Batterman & Cole (1964) p. 706.
We applied the square to the structure factor terms in advance and omitted the leading (|b|) factor.

Polarization Factors

Two states considered:

  • \(\sigma\)-polarization: \(P = 1\)
  • \(\pi\)-polarization: \(P = |\cos 2\theta_B|\)

References