Module ikeda_carpenter 

Ikeda–Carpenter analytical moderator resolution model.

A third instrument-resolution model alongside the analytical Gaussian (crate::resolution::ResolutionParams) and the Monte-Carlo tabulated kernel (crate::resolution::TabulatedResolution). It exists to settle a methodological dispute about the VENUS instrument resolution: one camp trusts the MC-simulated tabulated kernel (UDR/FTS file); the instrument scientist distrusts the unproven MC and prefers an analytical Ikeda–Carpenter moderator model. NEREIDS implements IC as a first-class model so all three can be cross-validated against synthetic loop-closure and real VENUS data.

Physics — the Ikeda–Carpenter pulse

Reference: S. Ikeda & J. M. Carpenter, Nucl. Instrum. Methods A239 (1985) 536–544. The neutron emission-time distribution from a pulsed spallation moderator is

  I(τ) = (1−R)·g₃(τ;α)  +  R·[g₃(·;α) ⊛ β e^{−β·}](τ),    τ ≥ 0

where the prompt (slowing-down) term is a Gamma/Erlang density of shape 3:

  g₃(τ;α) = α³ τ² e^{−ατ} / 2          (mode 2/α, mean 3/α, ∫ = 1)

and the delayed (storage) term is g₃ convolved with an exponential of rate β. The convolution has the closed form

  g₃(·;α) ⊛ β e^{−β·} (τ) = β (α/γ)³ [ e^{−βτ} − e^{−ατ}(1 + γτ + ½γ²τ²) ],  γ = α−β

(re-derived and confirmed against the Codex independent derivation and the SAMMY-RPI χ²+double-exp moderator form). Both terms are individually unit-area, so I is unit-area for any α,β>0 and 0≤R≤1, with first moment ⟨τ⟩ = 3/α + R/β. The asymmetry — sharp rise, long tail toward larger τ (later TOF, lower apparent energy) — is the physical origin of the asymmetric MC kernel.

Parameters and their energy dependence

• α(E) [1/µs]: fast moderation/leakage rate; sets the prompt width. Leading epithermal scaling α ∝ √E (Mantid α = 1/(α₀+α₁λ), λ ∝ 1/√E).
• β [1/µs]: slow storage rate; sets the delayed tail. Energy-independent.
• R(E), 0 ≤ R ≤ 1: storage mixing fraction; R ≈ exp(−E_meV/κ) → R→0 in the 1–200 eV resonance regime, so IC there is dominated by the one-parameter prompt Gamma(3, α(E)) term.

Parameters are fixed when the instrument scientist provides them, or fit from a known calibration foil otherwise (general case).

Time → energy kernel and centering

For a flight path L, a resonance at E_r has nominal TOF t_r = TOF_FACTOR·L/√E_r. A neutron with emission delay τ arrives at TOF t_r+τ, apparent energy E' = (TOF_FACTOR·L/(t_r+τ))² — that is the kernel’s definition (its positive-τ tail is delayed emission). Sampling I(τ) on a τ-grid and mapping to TOF-offsets yields exactly the (offset, weight) kernel representation that crate::resolution::TabulatedResolution consumes — so IC rides the same verified broadening machinery, whose application is the convolution gather: the broadened value at measured TOF t reads theory at t−τ (a neutron measured at t with delay τ really flew t−τ; see resolution::broaden and SAMMY udr/mudr4.f90 Ud_Convolute). At apply time interpolated_kernel blends the two bracketing reference kernels as a width-normalized shape blend (offsets scaled to the geometrically interpolated width, shapes merged on the union grid) — unequal point counts from IC’s per-kernel tail trimming no longer trigger a nearest-reference fallback, so the between-reference width follows the physical power law smoothly. IC also synthesizes a dense reference grid (default 64 energies), so the between-reference error is negligible. The kernel is anchored with its mode at offset 0 (peak-centering), matching the UDR file convention (peak at offset 0); interpolated_kernel does not re-center it. Because the IC pulse is right-skewed, its mean lags its mode by ~1/α(E) in TOF, so the centroid — and even the minimum — of a broadened resonance shifts toward lower apparent energy by an α(E)-dependent amount (order 1e-2 eV, ~1e-3 relative, for α≈1.5 in the eV regime; larger toward lower energy). So it does move the broadened dip off the nominal energy, by a small amount.

For the prompt-only law α(E) = a0·√E + a1 with R≈0, the lag ~1/α(E) is exactly the 1/√E basis of a flight-path (L_scale) error iff a1 = 0: then the centroid offset scales as c/√E, and an L → L(1+δ) change shifts every TOF by δ·K·L/√E, also ∝ 1/√E, so the two are degenerate. With a1 ≠ 0 the lag 1/(a0√E + a1) only approximately follows that basis (and the storage term R/β, negligible in the eV regime, adds a further small departure). To leading order, then, the lag is confounded with the energy scale, and is handled by the SHARED (t0, L_scale) energy scale, not by any per-family knob:

• Run-time fitting fits the t0/L_scale energy-scale, which absorbs the constant-L part of the lag exactly (same basis); only the shape (skew/tail) of the asymmetry is not absorbable by position.
• Resolution calibration (the nereids-fitting calibrator) pins the energy scale by default — a pure shape/width fit. It can optionally fit a SHARED (t0, L_scale) under a metrology prior (with_position_prior) for joint energy-scale / identifiability work. A free, per-family position knob is deliberately NOT used: because the lag is the same basis as L_scale, a free position lets a wrong (symmetric) family imitate the asymmetric shift and erodes the model-selection χ² (the discriminator is then only the position-independent skew/tail). See nereids-fitting’s free_l_scale_absorbs_asymmetric_lag_and_erodes_discrimination.

ic_centering_shifts_broadened_symmetric_dip_with_alpha quantifies the bare mode→centroid shift; the calibrator’s fit_t0_recovers_injected_energy_scale_shift checks the energy-scale fit recovers an injected offset. Re-centering the kernel on its centroid (a future convention change spanning the UDR path too) would remove the shift at the source.

Optional instrument convolutions

The full instrument function folds the moderator with a proton-burst (Gaussian σ) and a chopper/channel (triangle, FWHM) term. Both are optional here (None ⇒ omitted).

Provenance of the triangle (channel_fwhm_us). SAMMY broadens for the accelerator burst either as a Gaussian of FWHM DELTAG (SAMMY Manual R8 Sec. III.C.1.a, eq. III C1 a.12) or as a square pulse of width BURST (Sec. III.C.2.a). At SNS the proton pulse delivered to the target is shaped by the accumulator ring (Proton Storage Ring, PSR) into an approximately triangular ~700 ns base — FWHM ≈ 350 ns — which is what the VENUS tabulated FTS kernel header records as “folded triang FWHM 350 ns PSR”. NEREIDS folds that PSR triangle via channel_fwhm_us (symmetric triangle, half-base = FWHM, triangle_kernel). Note: a tabulated file whose header says the triangle is already “folded” in must NOT be double-counted against an IC model that also applies it (the nereids-fitting calibrator therefore applies its psr_fwhm_ns fold to the IC family only, never to tabulated/UDR kernels).

Structs

IkedaCarpenter

Analytical Ikeda–Carpenter resolution model.

IkedaCarpenterParams

Parameters of the Ikeda–Carpenter resolution model.

SynthesisGrid

Configuration of the energy / time grids used to synthesize the IC kernel table. The energy grid spans the data range densely enough that interref interpolation error is negligible.

Enums

EnergyLaw

Energy-dependence law for an Ikeda–Carpenter parameter.

Constants

DEFAULT_N_ENERGIES

Default number of log-spaced reference energies in a synthesized table.

DEFAULT_N_TAU

Default number of τ-samples spanning the prompt core of each kernel.

Functions

ic_pulse

Ikeda–Carpenter moderator emission density I(τ).