conflux.bsg.SpectralFunctions

Functions

phase_space(W, W0[, numass])

Phase space

fermi_function(W, Z, R, **kwargs)

Traditional Fermi Function

finite_size_L0_simple(W, Z, R, **kwargs)

Dominant electrostatic finite size correction

finite_size_L0(W, Z, R[, L0Const])

Dominant electrostatic finite size correction

finite_size_U_fermi(W, Z, **kwargs)

Higher-order electrostatic finite size correction

sirlin_g(W, W0, **kwargs)

Sirlin's g function for order alpha radiative corrections

radiative_correction_o1(W, W0, **kwargs)

Order alpha radiative correction to the beta spectrum shape

radiative_correction_o2(W, Z, R, **kwargs)

Order alpha^2 Z radiaive correction to the beta spectrum shape

radiative_correction_o3(W, Z, W0, R, **kwargs)

Radiative correction of order alpha^3 Z^2 to the beta spectrum shape

radiative_correction_L(W0, **kwargs)

Resummed order alpha^n radiative corrections

radiative_correction(W, Z, W0, R, **kwargs)

Total radiative correction up to order alpha^3 Z^2 to the beta spectrum shape

radiative_correction_neutrino(W, W0, **kwargs)

Radiative correction to the (anti)neutrino spectrum to order alpha

recoil_fermi(W, W0, A, **kwargs)

Kinematic recoil correction to the beta spectrum shape for a Fermi transition

recoil_gamow_teller(W, W0, A, **kwargs)

Kinematic recoil correction to the beta spectrum shape for a Gamow-Teller transition

recoil_Coulomb_fermi(W, Z, W0, A, **kwargs)

Coulomb-recoil correction to the beta spectrum shape for a Fermi transition

recoil_Coulomb_gamow_teller(W, Z, W0, A, **kwargs)

Coulomb-recoil correction to the beta spectrum shape for a Gamow-Teller transition

shape_factor_fermi(W, Z, W0, R, **kwargs)

Nuclear shape factor for an allowed Fermi transition

shape_factor_gamow_teller(W, Z, W0, R, A, b, c, d, ...)

Nuclear shape factor for an allowed Gamow-Teller transition

shape_factor_unique_forbidden(W, L, W0, Z, R, **kwargs)

Unique forbidden shape factor

atomic_screening(W, Z, R, l, **kwargs)

Screening correction due to atomic electrons in the final state

atomic_mismatch(W, Z, W0, A, **kwargs)

Correction due to non-orthogonality of initial and final electronic states, resulting in shake-up and shake-off

atomic_exchange(W, exPars)

atomic_exchange_simkovic(W, exPars)

Exchange correction due to Simkovic et al., https://journals.aps.org/prc/pdf/10.1103/PhysRevC.107.025501

Module Contents

conflux.bsg.SpectralFunctions.phase_space(W, W0, numass=0, **kwargs)

Phase space

Parameters:

  • W – Electron energy in iunits of me c^2

  • W0 – Electron endpoint energy in units of me c^2

conflux.bsg.SpectralFunctions.fermi_function(W, Z, R, **kwargs)

Traditional Fermi Function

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • R – Nuclear radius in units of the electron Compton wavelength

conflux.bsg.SpectralFunctions.finite_size_L0_simple(W, Z, R, **kwargs)

Dominant electrostatic finite size correction Correction to the traditional Fermi function to use a uniformly charged sphere rather than point charge

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • R – Nuclear radius in units of the electron Compton wavelength

conflux.bsg.SpectralFunctions.finite_size_L0(W, Z, R, L0Const=None, **kwargs)

Dominant electrostatic finite size correction Correction to the traditional Fermi function to use a uniformly charged sphere rather than point charge

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • R – Nuclear radius in units of the electron Compton wavelength

conflux.bsg.SpectralFunctions.finite_size_U_fermi(W, Z, **kwargs)

Higher-order electrostatic finite size correction Change from uniformly charged sphere to Fermi-type charge distribution

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

conflux.bsg.SpectralFunctions.sirlin_g(W, W0, **kwargs)

Sirlin’s g function for order alpha radiative corrections

Parameters:

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

conflux.bsg.SpectralFunctions.radiative_correction_o1(W, W0, **kwargs)

Order alpha radiative correction to the beta spectrum shape

Parameters:

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

conflux.bsg.SpectralFunctions.radiative_correction_o2(W, Z, R, **kwargs)

Order alpha^2 Z radiaive correction to the beta spectrum shape

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

conflux.bsg.SpectralFunctions.radiative_correction_o3(W, Z, W0, R, **kwargs)

Radiative correction of order alpha^3 Z^2 to the beta spectrum shape

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • R – Nuclear charge radius in units of the electron Compton wavelength

conflux.bsg.SpectralFunctions.radiative_correction_L(W0, **kwargs)

Resummed order alpha^n radiative corrections

Parameters:

W0 – Electron endpoint in units of me c^2

conflux.bsg.SpectralFunctions.radiative_correction(W, Z, W0, R, **kwargs)

Total radiative correction up to order alpha^3 Z^2 to the beta spectrum shape

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • R – Nuclear charge radius in units of the electron Compton wavelength

conflux.bsg.SpectralFunctions.radiative_correction_neutrino(W, W0, **kwargs)

Radiative correction to the (anti)neutrino spectrum to order alpha

Parameters:

  • Wv – (Anti)neutrino energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

conflux.bsg.SpectralFunctions.recoil_fermi(W, W0, A, **kwargs)

Kinematic recoil correction to the beta spectrum shape for a Fermi transition

Parameters:

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • A – Mass number (protons + neutrons) of the nuclear state

conflux.bsg.SpectralFunctions.recoil_gamow_teller(W, W0, A, **kwargs)

Kinematic recoil correction to the beta spectrum shape for a Gamow-Teller transition

Parameters:

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • A – Mass number (protons + neutrons) of the nuclear state

conflux.bsg.SpectralFunctions.recoil_Coulomb_fermi(W, Z, W0, A, **kwargs)

Coulomb-recoil correction to the beta spectrum shape for a Fermi transition

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • A – Mass number (protons + neutrons) of the nuclear state

conflux.bsg.SpectralFunctions.recoil_Coulomb_gamow_teller(W, Z, W0, A, **kwargs)

Coulomb-recoil correction to the beta spectrum shape for a Gamow-Teller transition

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • A – Mass number (protons + neutrons) of the nuclear state

conflux.bsg.SpectralFunctions.shape_factor_fermi(W, Z, W0, R, **kwargs)

Nuclear shape factor for an allowed Fermi transition

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • R – Nuclear charge radius in units of the electron Compton wavelength

conflux.bsg.SpectralFunctions.shape_factor_gamow_teller(W, Z, W0, R, A, b, c, d, Lambda, **kwargs)

Nuclear shape factor for an allowed Gamow-Teller transition

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • R – Nuclear charge radius in units of the electron Compton wavelength

  • A – Mass number (protons + neutrons) for the nuclear state

  • b – Weak magnetism form factor at q2=0

  • c – Gamow-Teller form factor at q2=0

  • d – Induced tensor form factor at q2=0

  • Lambda – Induced pseudoscalar form factor at q2=0

Form factors are in Holstein notation and that of Hayen et al., RMP 90 (2018) 015008

conflux.bsg.SpectralFunctions.shape_factor_unique_forbidden(W, L, W0, Z, R, **kwargs)

Unique forbidden shape factor

Parameters:

  • L – int Spin change

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • Z – Proton number of the final nuclear state

conflux.bsg.SpectralFunctions.atomic_screening(W, Z, R, l, **kwargs)

Screening correction due to atomic electrons in the final state

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Elecron energy in units of me c^2

  • R – Nuclear charge radius in units of the electron Compton wavelength

  • l – Shift in electric potential at the origin due to atomic electrons

conflux.bsg.SpectralFunctions.atomic_mismatch(W, Z, W0, A, **kwargs)

Correction due to non-orthogonality of initial and final electronic states, resulting in shake-up and shake-off

Parameters:

  • Z – Proton number of the final nuclear state

  • W – Electron energy in units of me c^2

  • W0 – Electron endpoint energy in units of me c^2

  • A – Nuclear mass number (protons + neutrons)

conflux.bsg.SpectralFunctions.atomic_exchange(W, exPars)
conflux.bsg.SpectralFunctions.atomic_exchange_simkovic(W, exPars)

Exchange correction due to Simkovic et al., https://journals.aps.org/prc/pdf/10.1103/PhysRevC.107.025501 Equation (34)