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Background Subtraction

The analysis.background module implements three physically motivated background models for removing inelastic scattering contributions from XPS spectra.

Mathematical Models

Shirley Background

The Shirley background models the inelastic background as proportional to the integrated peak intensity. It is computed iteratively:

\[ B(E) = k \int_{E}^{E_{\text{high}}} [I(E') - B(E')] \, dE' \]

where \(k\) is a scaling factor determined by the condition \(B(E_{\text{low}}) = I(E_{\text{low}})\). Convergence is reached when \(\max(|B_{n+1} - B_n|) < 10^{-5}\).

from xps_analyzer.analysis import shirley_background

result = shirley_background(spectrum, inplace=False)

Tougaard Background

The Tougaard model uses a universal cross-section parameterization:

\[ B(E) = \int_{E}^{\infty} K(E' - E) \, [I(E') - B(E')] \, dE' \]
\[ K(T) = \frac{B T}{(C + T^2)^2} + \frac{D T}{(C + T^2)^2} \]

Four variants are available, calibrated for different material classes:

Variant B (eV²) C (eV²) D (eV²)
Universal 2866 1643 0
Polymer 1122 1638 0
Sc元素 500 1500 500
Custom Configurable Configurable Configurable 
from xps_analyzer.analysis import tougaard_background

result = tougaard_background(spectrum, variant="universal", inplace=False)

Linear Background

A simple linear interpolation between two endpoints:

\[ B(E) = I(E_{\text{low}}) + \frac{I(E_{\text{high}}) - I(E_{\text{low}})}{E_{\text{high}} - E_{\text{low}}} (E - E_{\text{low}}) \]
from xps_analyzer.analysis import linear_background

result = linear_background(spectrum, inplace=False)

Fallback Strategy

The background_with_fallback function implements an automatic cascade:

  1. Try Shirley background
  2. If Shirley fails to converge (unusual lineshape), fall back to Tougaard
  3. If Tougaard produces unphysical results, fall back to Linear
from xps_analyzer.analysis import background_with_fallback

result = background_with_fallback(spectrum, inplace=False)

Visual Comparison

Background models comparison Comparison of Shirley, Tougaard, and Linear background models applied to an O 1s spectrum.