# Mössbauer Recoilless Gamma Resonance Lab

Interactive Mössbauer effect: recoil-free fraction and the Debye–Waller factor, second-order Doppler shift, ⁵⁷Fe hyperfine spectra with Poisson counting, and the Pound–Rebka gravitational redshift.

> Canonical page: https://elysiatools.com/en/visualizations/mossbauer-recoilless-resonance

- **Category:** Physics

## Overview

Interactive nuclear + solid-state physics — distinct from the existing crystal-structures (Bravais lattices), hydrogen-wave-function / quantum-tunneling (single-particle quantum mechanics), and every spectroscopy-adjacent case; this is the only visualization of the coupling between a nuclear transition and quantized lattice vibrations: the Mössbauer effect, where a nucleus embedded in a crystal emits a 14.4 keV gamma photon with the entire lattice sharing the recoil momentum, giving the sharpest spectral line in physics (ΔE/E ~ 3×10⁻¹³). The model implements the full ⁵⁷Fe theory stack: free-nucleus recoil E_R = E_γ²/2Mc² ≈ 1.96 meV (~4×10⁵ natural linewidths, showing why free-atom resonance is impossible), natural width Γ = ħ/τ = 4.67 neV (τ = 141 ns, 0.0972 mm/s in velocity units), the Lamb–Mössbauer (Debye–Waller) recoil-free fraction f(T) = exp{−(3E_R/2k_BΘ_D)[1+4(T/Θ_D)²∫₀^{Θ_D/T}x/(eˣ−1)dx]} evaluated with exact Debye integrals (series + Simpson, verified against the π²/6 and high-T exp(−6E_R·T/k_BΘ_D²) limits and the f = exp(−k²⟨x²⟩) cross-route), the second-order Doppler thermal redshift δ_SOD = −⟨v²⟩/2c with ⟨v²⟩ = (9k_BΘ_D/8M)[1+8(T/Θ_D)⁴∫x³/(eˣ−1)dx] (−7.3×10⁻⁴ mm/s/K classical slope, −0.116 mm/s between 4 K and 300 K for Θ_D = 470 K, matching α-Fe), hyperfine patterns — isomer shift, quadrupole doublet, and the Zeeman sextet computed from nuclear moments μ_g = +0.0906 μ_N / μ_e = −0.15487 μ_N (α-Fe at 33 T: ±5.31/±3.08/±0.84 mm/s, intensities 3:2:1:1:2:3, verified to 0.03 mm/s), and the transmission spectrum with saturation: T(v) = exp(−t·Σ wᵢ/[1+((v−vᵢ)/(Γ_obs/2))²]) with t = σ₀·n·d·f(T), σ₀ = 2.4×10⁻¹⁸ cm², and Margulies–Ehrman broadening Γ_obs = Γ_nat(2+0.27t). A live Doppler drive sweeps a triangle wave while photons are counted channel-by-channel with seeded Poisson statistics (Knuth + normal branch), and a χ²-scan singlet-center fitter with parabolic refinement recovers line centers with σ ≈ Γ_obs/2√N precision — the statistical resolution that makes the Pound–Rebka gravitational redshift lab-observable (gΔh/c = 0.74 μm/s for 22.5 m = 0.0076 natural widths, detectable once ~10⁶–10⁷ counts accumulate; the fit is verified to recover imposed shifts within 5σ and shrink as 1/√N). Four visualization panels: (1) The transmission spectrum with the ideal model curve, Poisson-counted data points normalised to the off-line baseline, hyperfine line ticks, the moving pink drive-velocity marker, the fitted center (gold) and — in gravitational mode — the gΔh/c target (green) with live μm/s readouts. (2) Recoil-free fraction f(T) and its complement 1−f (phonon sideband fraction) over 4–600 K with the operating point. (3) Second-order Doppler shift δ(T) with the operating point. (4) A log₁₀ energy-scale bar from peV to keV placing Γ, the Doppler window, gΔh/c, the SOD shift, E_R, k_BΘ_D and E_γ — 17 orders of magnitude with the zero-phonon window and phonon-sideband zone shaded. Adjustable parameters: sample temperature (4–600 K), Debye temperature (250–600 K), Doppler scan range (±0.5–25 mm/s), absorber areal density (0.2–10 mg/cm² natural ⁵⁷Fe), hyperfine field (0–35 T), isomer shift (−0.5–+1.5 mm/s), Pound–Rebka height difference (−25–+25 m), drive pause/resume, clear counts, noise-seed re-roll; three absorber presets (stainless singlet, nitroprusside doublet Δ=1.71 mm/s, α-Fe sextet) and two experiment modes (transmission scan, gravitational redshift with a bright single-line source). Real-time diagnostics: f, 1−f, Γ (neV), observed FWHM (mm/s), effective thickness, E_R (meV), SOD shift, centroid δ+SOD, accumulated counts, fitted center ± σ (μm/s), and ΔE/E. Educational content covers why free-nucleus recoil kills resonance, the zero-phonon line vs the ~10⁵-linewidth-wide phonon sideband, the Debye–Waller factor's zero-point and high-T limits, SOD as a phonon thermometer, isomer/quadrupole/magnetic hyperfine interactions, linewidth saturation, and the Pound–Rebka gravitational-redshift test with statistics buying sub-linewidth resolution. Multi-language support (zh, en, es, fr, de, ru, pt).

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