An approximation to the Mössbauer lineshape integral in terms of rational functions

Abstract

By replacing the exponential function e −ξ with a rational function R( ξ) whose numerator and denominator are each of degree 2, the Mössbauer lineshape integral can be evaluated analytically, and the result expressed in terms of rational functions. Expressions are given for the lineshape integral, and its first and second derivatives with respect to the parameters which define the line. Numerical values are given for the coefficients appearing in R( ξ) for 4 different R( ξ) which are the best (in the Chebyshev sense) approximations to e −ξ in the range 0≤ ξ≤ v, where v = 1, 2, 3, and 4. The maximum error ∣e − ξ − R( ξ)∣≈ 1.7 × 10 −6 for the v = 1 approximation, and ≈ 3.8 × 10 −4 for the v = 4 approximation.