conflux.bsg.SpectralFunctions ============================= .. py:module:: conflux.bsg.SpectralFunctions Functions --------- .. autoapisummary:: conflux.bsg.SpectralFunctions.phase_space conflux.bsg.SpectralFunctions.fermi_function conflux.bsg.SpectralFunctions.finite_size_L0_simple conflux.bsg.SpectralFunctions.finite_size_L0 conflux.bsg.SpectralFunctions.finite_size_U_fermi conflux.bsg.SpectralFunctions.sirlin_g conflux.bsg.SpectralFunctions.radiative_correction_o1 conflux.bsg.SpectralFunctions.radiative_correction_o2 conflux.bsg.SpectralFunctions.radiative_correction_o3 conflux.bsg.SpectralFunctions.radiative_correction_L conflux.bsg.SpectralFunctions.radiative_correction conflux.bsg.SpectralFunctions.radiative_correction_neutrino conflux.bsg.SpectralFunctions.recoil_fermi conflux.bsg.SpectralFunctions.recoil_gamow_teller conflux.bsg.SpectralFunctions.recoil_Coulomb_fermi conflux.bsg.SpectralFunctions.recoil_Coulomb_gamow_teller conflux.bsg.SpectralFunctions.shape_factor_fermi conflux.bsg.SpectralFunctions.shape_factor_gamow_teller conflux.bsg.SpectralFunctions.shape_factor_unique_forbidden conflux.bsg.SpectralFunctions.atomic_screening conflux.bsg.SpectralFunctions.atomic_mismatch conflux.bsg.SpectralFunctions.atomic_exchange conflux.bsg.SpectralFunctions.atomic_exchange_simkovic Module Contents --------------- .. py:function:: phase_space(W, W0, numass=0, **kwargs) Phase space :param W: Electron energy in iunits of me c^2 :param W0: Electron endpoint energy in units of me c^2 .. py:function:: fermi_function(W, Z, R, **kwargs) Traditional Fermi Function :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param R: Nuclear radius in units of the electron Compton wavelength .. py:function:: 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 :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param R: Nuclear radius in units of the electron Compton wavelength .. py:function:: 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 :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param R: Nuclear radius in units of the electron Compton wavelength .. py:function:: finite_size_U_fermi(W, Z, **kwargs) Higher-order electrostatic finite size correction Change from uniformly charged sphere to Fermi-type charge distribution :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 .. py:function:: sirlin_g(W, W0, **kwargs) Sirlin's g function for order alpha radiative corrections :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 .. py:function:: radiative_correction_o1(W, W0, **kwargs) Order alpha radiative correction to the beta spectrum shape :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 .. py:function:: radiative_correction_o2(W, Z, R, **kwargs) Order alpha^2 Z radiaive correction to the beta spectrum shape :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 .. py:function:: radiative_correction_o3(W, Z, W0, R, **kwargs) Radiative correction of order alpha^3 Z^2 to the beta spectrum shape :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param R: Nuclear charge radius in units of the electron Compton wavelength .. py:function:: radiative_correction_L(W0, **kwargs) Resummed order alpha^n radiative corrections :param W0: Electron endpoint in units of me c^2 .. py:function:: radiative_correction(W, Z, W0, R, **kwargs) Total radiative correction up to order alpha^3 Z^2 to the beta spectrum shape :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param R: Nuclear charge radius in units of the electron Compton wavelength .. py:function:: radiative_correction_neutrino(W, W0, **kwargs) Radiative correction to the (anti)neutrino spectrum to order alpha :param Wv: (Anti)neutrino energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 .. py:function:: recoil_fermi(W, W0, A, **kwargs) Kinematic recoil correction to the beta spectrum shape for a Fermi transition :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param A: Mass number (protons + neutrons) of the nuclear state .. py:function:: recoil_gamow_teller(W, W0, A, **kwargs) Kinematic recoil correction to the beta spectrum shape for a Gamow-Teller transition :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param A: Mass number (protons + neutrons) of the nuclear state .. py:function:: recoil_Coulomb_fermi(W, Z, W0, A, **kwargs) Coulomb-recoil correction to the beta spectrum shape for a Fermi transition :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param A: Mass number (protons + neutrons) of the nuclear state .. py:function:: recoil_Coulomb_gamow_teller(W, Z, W0, A, **kwargs) Coulomb-recoil correction to the beta spectrum shape for a Gamow-Teller transition :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param A: Mass number (protons + neutrons) of the nuclear state .. py:function:: shape_factor_fermi(W, Z, W0, R, **kwargs) Nuclear shape factor for an allowed Fermi transition :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param R: Nuclear charge radius in units of the electron Compton wavelength .. py:function:: shape_factor_gamow_teller(W, Z, W0, R, A, b, c, d, Lambda, **kwargs) Nuclear shape factor for an allowed Gamow-Teller transition :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param R: Nuclear charge radius in units of the electron Compton wavelength :param A: Mass number (protons + neutrons) for the nuclear state :param b: Weak magnetism form factor at q2=0 :param c: Gamow-Teller form factor at q2=0 :param d: Induced tensor form factor at q2=0 :param Lambda: Induced pseudoscalar form factor at q2=0 Form factors are in Holstein notation and that of Hayen et al., RMP 90 (2018) 015008 .. py:function:: shape_factor_unique_forbidden(W, L, W0, Z, R, **kwargs) Unique forbidden shape factor :param L: int Spin change :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param Z: Proton number of the final nuclear state .. py:function:: atomic_screening(W, Z, R, l, **kwargs) Screening correction due to atomic electrons in the final state :param Z: Proton number of the final nuclear state :param W: Elecron energy in units of me c^2 :param R: Nuclear charge radius in units of the electron Compton wavelength :param l: Shift in electric potential at the origin due to atomic electrons .. py:function:: 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 :param Z: Proton number of the final nuclear state :param W: Electron energy in units of me c^2 :param W0: Electron endpoint energy in units of me c^2 :param A: Nuclear mass number (protons + neutrons) .. py:function:: atomic_exchange(W, exPars) .. py:function:: 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)