Module pearson 

Pearson type IV and dual-Pearson representations of range profiles.

Pearson IV

The Pearson system (K. Pearson, “Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material”, Phil. Trans. R. Soc. Lond. A 186 (1895) 343) is the family of densities with d ln f / dx = -(x - c) / (b0 + b1 x + b2 x^2); its type IV member is fixed by the first four moments. W. K. Hofker, “Implantation of boron in silicon”, Philips Res. Repts. Suppl. No. 8 (1975), introduced it for ion implantation profiles, characterised by Rp, ΔRp, γ and β.

We use the explicit form and moment inversion of J. Heinrich, “A guide to the Pearson type IV distribution”, CDF/MEMO/STATISTICS/PUBLIC/6820 (2004):

f(x) = k [1 + ((x - λ)/a)^2]^(-m) exp(-ν atan((x - λ)/a)),
r = 6 (β - γ^2 - 1) / (2β - 3γ^2 - 6),   m = 1 + r/2,
D = 16 (r - 1) - γ^2 (r - 2)^2,
ν = -r (r - 2) γ / D^(1/2),   a = σ D^(1/2) / 4,   λ = μ - (r - 2) γ σ / 4,
k = |Γ(m + iν/2) / Γ(m)|^2 / (a B(m - 1/2, 1/2)).

Validity region. The inversion needs 2β - 3γ^2 - 6 > 0 (then r > 3, i.e. m > 5/2, so the fourth moment exists) and D > 0 (real ν). Together these are Pearson’s type IV criterion 0 < κ < 1; the boundary D = 0 (κ = 1, the type IV / VI border) is β_IV(γ) = (48 + 39γ^2 + 6 (γ^2 + 4)^(3/2)) / (32 - γ^2) for γ^2 < 32, which follows from solving D = 0 for β (a test checks the equivalence). The Gaussian point γ = 0, β = 3 lies on the boundary, so a nearly Gaussian sample can fall outside by noise; that is reported as PearsonError::OutsideTypeIv, never as NaN.

The normalisation uses the real part of the complex log-gamma function by the Stirling series (M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (1964), 6.1.40) after upward recurrence.

Dual Pearson

A. F. Tasch, H. Shin, C. Park, J. Alvis and S. Novak, “An improved approach to accurately model shallow B and BF2 implants in silicon”, J. Electrochem. Soc. 136 (1989) 810, represent a profile as R f_1(x) + (1 - R) f_2(x): two Pearson IV densities with their own four moments and a fraction R. DualPearson::fit fits the nine parameters to a depth histogram; the method is described there.

Structs

DualPearson

A dual-Pearson profile R f_head + (1 - R) f_tail (Tasch et al. 1989).

DualPearsonFit

The result of DualPearson::fit.

PearsonIv

A Pearson type IV density, stored both as the four moments it matches and as Heinrich’s parameters.

Enums

PearsonError

Errors from building a Pearson representation.

Functions

type_iv_min_kurtosis

The type IV / VI boundary β_IV(γ); None for γ^2 >= 32.