Atomic Scattering Factor Data¶

Understanding the atomic data sources and interpolation methods used in XRayLabTool.

Atomic Scattering Factors¶

Definition¶

Atomic scattering factors describe how X-rays scatter from atoms:

\[f = f_0 + f' + if''\]

Where:

f₀: Thomson scattering (classical, forward scattering)
f’: Dispersion correction (real part)
f’’: Absorption (imaginary part)

For X-ray optics calculations, we use:

f₁ = f₀ + f’: Total real part
f₂ = f’’: Imaginary part (absorption)

Physical Origin¶

Thomson Scattering (f₀): - Classical electron scattering - Energy-independent - Equals atomic number Z for forward scattering

Dispersion Correction (f’): - Quantum mechanical correction - Energy-dependent, especially near absorption edges - Can be positive or negative

Absorption (f’’): - Photoabsorption cross-section - Always positive - Shows sharp edges at absorption thresholds

Energy Dependence¶

Away from Absorption Edges¶

For energies well away from absorption edges:

\[ \begin{align}\begin{aligned}f' \approx -\frac{r_e mc^2}{2\pi} \sum_j \frac{\lambda^2 f_{j0}}{(\lambda^2 - \lambda_j^2)}\\f'' \approx \frac{Z^4 \text{const}}{E^3}\end{aligned}\end{align} \]

Where: - λⱼ: Absorption edge wavelengths - fⱼ₀: Oscillator strengths - The f’’ ∝ E⁻³ scaling is approximate

Near Absorption Edges¶

Near absorption edges, both f’ and f’’ show complex structure:

Pre-edge region: Smooth interpolation
Edge jump: Sharp discontinuity in f’’
Post-edge oscillations: XANES and EXAFS structure

Data Sources¶

Henke Tables¶

Coverage: - Elements: H (Z=1) to U (Z=92) - Energy range: 10 eV to 30 keV - Energy spacing: Variable, denser near edges

Method: - Combines experimental photoabsorption data - Theoretical calculations for f’ - Kramers-Kronig transformation ensures consistency

File Format: Standard .nff format with columns: - Energy (eV) - f₁ (real part) - f₂ (imaginary part)

CXRO Database¶

Extended Henke Tables: - Updated experimental data - Extended energy ranges for some elements - Web interface and downloadable files - Source: http://henke.lbl.gov/optical_constants/

Advantages: - Regular updates with new measurements - Quality control and validation - Widely accepted standard

NIST XCOM¶

Photoabsorption Data: - Primary source for absorption coefficients - Energy range: 1 keV to 100 GeV - Includes pair production and Compton scattering - Used to validate and extend other databases

Interpolation Methods¶

Linear Interpolation¶

XRayLabTool uses linear interpolation between tabulated values:

\[f(E) = f_1 + \frac{E - E_1}{E_2 - E_1}(f_2 - f_1)\]

This works well because: - Data points are closely spaced - Smooth variation between points - Computationally efficient

Logarithmic Interpolation¶

For some quantities, logarithmic interpolation may be more accurate:

\[\ln f(E) = \ln f_1 + \frac{\ln E - \ln E_1}{\ln E_2 - \ln E_1}(\ln f_2 - \ln f_1)\]

Used when: - Data spans many orders of magnitude - Exponential-like behavior expected - Higher accuracy needed

Spline Interpolation¶

For critical applications, spline interpolation provides: - Smooth derivatives - Better behavior near edges - Higher computational cost

Edge Handling¶

Absorption Edge Structure¶

Absorption edges create discontinuities in f’’:

K-edge (1s electron): - Largest jump in f’’ - Corresponding feature in f’ - Most prominent for light elements

L-edges (2s, 2p electrons): - Multiple edges (L₁, L₂, L₃) - Fine structure from chemical environment - Important for medium-Z elements

M-edges and higher: - Many closely spaced edges - Complex fine structure - Important for heavy elements

Pre-edge Features¶

Near absorption edges: - White lines: Sharp peaks just above edge - XANES: X-ray Absorption Near Edge Structure - Pre-edge peaks: Forbidden transitions

These features contain chemical information but complicate optical calculations.

Kramers-Kronig Relations¶

The real and imaginary parts are related by:

\[f'(E) = \frac{2}{\pi} P \int_0^{\infty} \frac{\omega f''(\omega)}{\omega^2 - E^2} d\omega\]

Where P denotes the principal value. This ensures physical consistency.

Quality and Accuracy¶

Experimental Uncertainties¶

Photoabsorption Measurements: - Systematic errors: 2-5% typical - Statistical errors: 1-2% for good measurements - Sample contamination affects results - Temperature and pressure effects

Transmission Measurements: - Sample thickness uncertainty - Multiple scattering corrections - Surface oxidation effects - Grain size and texture effects

Theoretical Limitations¶

Isolated Atom Approximation: - Ignores chemical bonding effects - Assumes spherical atoms - No crystal field effects - Limited accuracy for light elements

Relativistic Effects: - Important for inner shells of heavy elements - Affects edge positions and intensities - Modern calculations include these

Validation Methods¶

Cross-checks between databases: - NIST XCOM vs Henke tables - Independent measurements - Sum rule tests

Experimental validation: - Reflectometry measurements - Transmission measurements - Interferometry techniques

Data Processing in XRayLabTool¶

Caching Strategy¶

XRayLabTool uses a multi-level caching system:

Preloaded cache: 92 common elements loaded at startup
LRU cache: Recently used interpolations cached
Disk cache: Computed values saved for reuse
Memory management: Automatic cleanup of old entries

Performance Optimization¶

Vectorized operations: - NumPy arrays for energy ranges - Batch interpolation for efficiency - SIMD operations where available

Smart interpolation: - Adaptive mesh refinement near edges - Coarse grids away from features - Error estimation and mesh adaptation

Error Estimation¶

XRayLabTool provides error estimates based on:

Interpolation error: From data spacing
Experimental uncertainty: From literature values
Model limitations: Isolated atom approximation
Numerical precision: Machine epsilon effects

Usage Guidelines¶

Energy Range Selection¶

Recommended ranges: - 100 eV - 30 keV: Henke data most reliable - 30-100 keV: Extrapolation, larger uncertainties - Below 100 eV: Strong chemical bonding effects

Avoiding problematic regions: - Very close to absorption edges (±10 eV) - Regions with sparse data coverage - Energies requiring large extrapolations

Material Considerations¶

Light elements (Z < 10): - Large relative bonding effects - Limited experimental data - Consider molecular form factors

Heavy elements (Z > 80): - Complex edge structure - Relativistic effects important - Multiple absorption edges

Compounds vs Elements: - Additivity assumption generally good - Chemical shifts usually small - Exceptions: strongly bonded materials

Future Developments¶

Database Updates¶

New experimental measurements
Improved theoretical calculations
Extended energy ranges
Better uncertainty estimates

Computational Improvements¶

Machine learning interpolation
Quantum mechanical calculations
Many-body effects
Temperature-dependent data

Integration Features¶

Real-time database updates
Quality metrics and validation
User-contributed data
Community feedback mechanisms

References¶

Primary Sources: - Henke, B.L., et al. “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92”, Atomic Data and Nuclear Data Tables 54, 181-342 (1993) - NIST XCOM: Photon Cross Sections Database - CXRO X-ray interactions database

Theoretical Background: - Bethe, H.A. & Salpeter, E.E. “Quantum Mechanics of One- and Two-Electron Atoms” - Brown, G.S. et al. “X-ray absorption spectroscopy and its applications”