43 lines
1.8 KiB
Julia
43 lines
1.8 KiB
Julia
using DifferentialEquations, Roots
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ħc = 197.327 # ħc in MeVfm
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M_n = 939.5654133 # Neutron mass in MeV/c2
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M_p = 938.2720813 # Proton mass in MeV/c2
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"The spherical Dirac equation that returns du=[dg, df] in-place where
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(g, f) are the reduced radial components evaluated at r,
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κ is the generalized angular momentum,
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M is the mass in MeV/c2,
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E in the energy in MeV,
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S(r) & V(r) are functions corresponding to scalar and vector potentials in MeV,
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r is the radius in fm.
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Reference: P. Giuliani, K. Godbey, E. Bonilla, F. Viens, and J. Piekarewicz, Frontiers in Physics 10, (2023)."
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function dirac!(du, (g, f), (κ, M, E, S, V), r)
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du[1] = -(κ/r) * g + (E + M - S(r) - V(r)) * f / ħc
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du[2] = (κ/r) * f - (E - M + S(r) - V(r)) * g / ħc
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end
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"Solve the Dirac equation and return g(r=r_max) for given scalar and vector potentials where
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κ is the generalized angular momentum,
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M is the mass in MeV/c2,
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E in the energy in MeV,
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S(r) & V(r) are functions corresponding to scalar and vector potentials in MeV,
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r_max is the outer boundary,
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r_min (=r_max/1000) is inside boundary which cannot be 0 due to the centrifugal term."
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function boundaryValue(κ, M, E, S, V, r_max, r_min=r_max/1000)
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prob = ODEProblem(dirac!, [0, 1], (r_min, r_max))
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sol = solve(prob, RK4(), p=(κ, M, E, S, V))
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return sol(r_max)[1]
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end
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"Find all bound energies between E_min (=0) and E_max (=M) where
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κ is the generalized angular momentum,
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M is the mass in MeV/c2,
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S(r) & V(r) are functions corresponding to scalar and vector potentials in MeV,
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r_max is the outer boundary,
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r_min (=r_max/1000) is inside boundary which cannot be 0 due to the centrifugal term."
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function findEs(κ, M, S, V, r_max, r_min=r_max/1000, E_min=0, E_max=M)
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f(E) = boundaryValue(κ, M, E, S, V, r_max, r_min)
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return find_zeros(f, (E_min, E_max))
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end
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