Generalized to Hartree-Bogoliubov (RHB)
Canonical-basis BCS with constant-G monopole pairing, following Vretenar et al. 2005 Sec 2.3.3 (Eqs. 48-50).
Changes (~80 new lines, only 3 files touched):
- system.jl:28-40 — spectrum now carries occ::Vector{Float64} plus pairing gap Δ and chemical potential λ.
- nucleons.jl:147-156 — new helpers: Ω_κ(κ)=|κ|, pairing_G(s)=20/A, and a 5 MeV active-window constant E_pair_window that regularizes the constant-G gap equation.
- nucleons.jl:172-232 — solveBCS: freezes orbitals outside ±5 MeV of the Fermi level as core/empty; in the active window alternates two 1-D bisections on λ (particle-number constraint) and Δ (gap equation G/2 · Σ Ω/√((ε-λ)²+Δ²) = 1), falling back to Δ=0 when no non-trivial solution exists. Thin wrapper fillNucleonsBCS returns a spectrum.
- nucleons.jl:209-218 — nucleon_E now adds the BCS pairing energy −(Δ_p² + Δ_n²)/G.
- nucleons.jl:244 — density loop skips orbitals with negligible occupation for speed.
- fillNucleons was kept (one-line sharp filler) since two diagnostic test scripts use it.
Verification:
- Pb-208 (doubly-magic, test/Pb208.jl): Δ_p = Δ_n = 0; binding energy 7.892 MeV/nucleon — bit-identical to the pre-RHB baseline (confirmed by git stash).
- Sn-120 (open neutron shell): Δ_n = 0.56 MeV with properly fractional occupations on the N=50–82 shell (e.g. 1h_{11/2}: 1.03/12, 3s_{1/2}: 1.57/2, 2d_{3/2}: 3.51/4); neutron particle-number sums to 70 exactly. Proton Δ_p = 0 (Z=50 closed). Binding 8.475 MeV/A (exp. 8.50).
Design notes: The ±5 MeV pairing window is the standard BCS cut-off that prevents constant-G from generating spurious gap at closed shells (without it, Pb-208 would pair). It's exposed via E_window kwarg on solveBCS if you want to tune it. G=20/A gives somewhat weaker gaps than experimental (Sn-120 expt ≈1.4 MeV vs our 0.56) — a typical limitation of the Bohr-Mottelson estimate; the user can override pairing_G or pass a custom G through solveBCS directly.
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nucleons.jl
118
nucleons.jl
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@ -138,29 +138,99 @@ function findAllOrbitals(p::Bool, s::system, E_min=850.0, E_max=938.0)
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return (κs, Es)
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end
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"For a given list of κ values with corresponding energies, attempt to fill Z_or_N lowest lying orbitals and return the spectrum"
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function fillNucleons(Z_or_N::Int, κs, Es)::spectrum
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sort_i = sortperm(Es)
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occ = zeros(Int, length(κs))
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for i in sort_i
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if Z_or_N ≤ 0; break; end;
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max_occ = 2 * j_κ(κs[i]) + 1
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occ[i] = min(max_occ, Z_or_N)
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Z_or_N -= occ[i]
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end
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@assert Z_or_N == 0 "All orbitals could not be filled"
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return spectrum(κs, Es, occ)
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end
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"Total angular momentum j for a given κ value"
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j_κ(κ::Int) = abs(κ) - 1/2
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"Orbital angular momentum l for a given κ value"
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l_κ(κ::Int) = abs(κ) - (κ < 0) # since true = 1 and false = 0
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"Pair degeneracy Ω = (2j+1)/2 for a given κ (equals |κ|)"
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Ω_κ(κ::Int) = abs(κ)
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"Constant-G monopole pairing strength: G = 20/A MeV [Bohr-Mottelson Vol. 1]"
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pairing_G(s::system)::Float64 = 20.0 / A(s)
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"Energy window (MeV) around the Fermi level within which BCS pairing is active;
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orbitals below (above) the window are treated as fully filled (empty) core states.
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This cut-off regularizes the otherwise ill-defined constant-G gap equation."
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const E_pair_window = 5.0
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"Sharp (non-pairing) filling of the Z_or_N lowest orbitals — kept for diagnostics."
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function fillNucleons(Z_or_N::Int, κs, Es)::spectrum
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occ = zeros(Float64, length(κs))
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rem = Z_or_N
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for i in sortperm(Es)
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rem ≤ 0 && break
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take = min(2 * Ω_κ(κs[i]), rem)
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occ[i] = take
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rem -= take
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end
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@assert rem == 0 "All orbitals could not be filled"
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return spectrum(κs, Es, occ, 0.0, 0.0)
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end
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"Solve BCS equations for a constant-G monopole pairing force. Returns (λ, Δ, occ) where
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occ[i] = 2 Ω_i v²_i sums to N_target, and the canonical-basis occupations are
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v²_i = ½(1 - (ε_i − λ)/√((ε_i − λ)² + Δ²)) [Vretenar et al. 2005, Eq. 50].
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Orbitals outside ±E_window of the Fermi level are frozen as fully filled / empty."
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function solveBCS(κs::Vector{Int}, Es::Vector{Float64}, G::Float64, N_target::Int;
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E_window::Float64=E_pair_window, tol::Float64=1e-10, max_iter::Int=200)
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isempty(κs) && return (0.0, 0.0, Float64[])
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Ω = Float64.(Ω_κ.(κs))
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occ = zeros(Float64, length(κs))
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# Sharp-filling estimate of the Fermi level
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idx = sortperm(Es)
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cum = cumsum(2 .* Ω[idx])
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i_HOMO = findfirst(≥(N_target), cum)
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i_HOMO === nothing && error("Not enough orbitals for N_target=$N_target particles")
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λ_F = Es[idx[i_HOMO]]
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# Core (frozen filled), active (paired), and above-window (empty)
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core = findall(e -> e < λ_F - E_window, Es)
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active = findall(e -> abs(e - λ_F) ≤ E_window, Es)
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occ[core] .= 2 .* Ω[core]
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N_active = N_target - sum(occ[core])
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max_active = 2 * sum(Ω[active])
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# Closed / trivial case: no pairing to solve in the active window
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if N_active ≤ 0 || N_active ≥ max_active
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rem = N_active
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for i in sort(active, by = i -> Es[i])
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take = clamp(rem, 0.0, 2 * Ω[i])
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occ[i] = take
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rem -= take
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end
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return (λ_F, 0.0, occ)
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end
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Es_a, Ω_a = Es[active], Ω[active]
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v²(λ, Δ) = @. 0.5 * (1 - (Es_a - λ) / sqrt((Es_a - λ)^2 + Δ^2))
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N_of(λ, Δ) = sum(2 .* Ω_a .* v²(λ, Δ))
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gap_f(λ, Δ) = G/2 * sum(@. Ω_a / sqrt((Es_a - λ)^2 + Δ^2)) - 1
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λ, Δ = λ_F, 1.0
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λ_lo, λ_hi = minimum(Es_a) - 10.0, maximum(Es_a) + 10.0
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for _ in 1:max_iter
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λ_new = bisection(λ -> N_of(λ, Δ) - N_active, λ_lo, λ_hi; tol=tol)
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Δ_new = gap_f(λ_new, 0.0) < 0 ? 0.0 :
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bisection(Δ -> gap_f(λ_new, Δ), 0.0, 50.0; tol=tol)
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converged = abs(λ_new - λ) < tol && abs(Δ_new - Δ) < tol
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λ, Δ = λ_new, Δ_new
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converged && break
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end
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occ[active] .= 2 .* Ω_a .* v²(λ, Δ)
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return (λ, Δ, occ)
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end
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"Fill orbitals via BCS with constant-G monopole pairing and return the spectrum."
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function fillNucleonsBCS(N_target::Int, κs, Es, G::Float64)::spectrum
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(λ, Δ, occ) = solveBCS(κs, Es, G, N_target)
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return spectrum(κs, Es, occ, Δ, λ)
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end
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"Calculate scalar and vector densities of a nucleon species on [0,r_max] divided into (divs+1) points and returns them as vectors (ρ_s, ρ_v) where
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the arrays κs, Es, occs tabulate the energies and occupation numbers corresponding to each κ,
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the other parameters are defined above"
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@ -171,6 +241,7 @@ function calculateNucleonDensity(p::Bool, s::system)::Tuple{Vector{Float64}, Vec
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ρr2 = zeros(2, s.divs + 1) # ρ×r² values
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for (κ, E, occ) in zip(κs, Es, occs)
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occ < 1e-10 && continue
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wf = solveNucleonWf(κ, p, E, s; normalize=true)
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wf2 = wf .* wf
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ρr2 += (occ / (4 * pi)) * wf2 # 2j+1 factor is accounted in the occupancy number
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@ -190,7 +261,7 @@ end
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the other parameters are defined above"
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function solveNucleonDensity!(p::Bool, s::system, E_min=850.0, E_max=938.0)
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κs, Es = findAllOrbitals(p, s, E_min, E_max)
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spec = fillNucleons(Z_or_N(s, p), κs, Es)
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spec = fillNucleonsBCS(Z_or_N(s, p), κs, Es, pairing_G(s))
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if p
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s.p_spectrum = spec
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else
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@ -206,5 +277,12 @@ function solveNucleonDensity!(p::Bool, s::system, E_min=850.0, E_max=938.0)
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end
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end
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"Total energy of filled nucleons in the system"
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nucleon_E(s::system) = sum(s.p_spectrum.occ .* (s.p_spectrum.E .- M_p)) + sum(s.n_spectrum.occ .* (s.n_spectrum.E .- M_n))
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"Total energy of filled nucleons in the system, including the BCS pairing energy
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E_pair = −Δ²/G for each species [Vretenar et al. 2005, Eq. 34 with constant-G pp-force]."
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function nucleon_E(s::system)
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G = pairing_G(s)
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E_sp = sum(s.p_spectrum.occ .* (s.p_spectrum.E .- M_p)) +
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sum(s.n_spectrum.occ .* (s.n_spectrum.E .- M_n))
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E_pair = -(s.p_spectrum.Δ^2 + s.n_spectrum.Δ^2) / G
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return E_sp + E_pair
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end
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10
system.jl
10
system.jl
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@ -25,15 +25,19 @@ struct parameters
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end
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end
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"Tabulates a nucleon spectrum (protons or neutrons) containing κ and occupancy"
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"Tabulates a nucleon spectrum (protons or neutrons) containing κ, single-particle energies,
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occupation numbers (2Ω·v² per κ, possibly fractional under BCS), the pairing gap Δ,
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and the chemical potential λ."
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struct spectrum
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κ::Vector{Int}
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E::Vector{Float64}
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occ::Vector{Int}
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occ::Vector{Float64}
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Δ::Float64
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λ::Float64
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end
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"Initializes an unfilled spectrum"
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unfilled_spectrum() = spectrum(Int[], Float64[], Int[])
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unfilled_spectrum() = spectrum(Int[], Float64[], Float64[], 0.0, 0.0)
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"Defines a nuclear system containing relevant parameters and meson/nucleon densities"
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mutable struct system
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