Geant4 Simulations for the Radon Electric Dipole Moment Search at

as [47]
I(E) =
G
2π 3 7 c 6 | M if | 2 (T 2 e +2T em e c 2 ) 1/2 (Q−T e ) 2 (T e +m e c 2 )F(Z,T e ) , (3.6)
where G is a constant representing the strength of the weak interaction, M if is the
transition matrix element, T e is the kinetic energy of the electron (or positron), Z is
the proton number of the daughter and F(Z,T e ) is called the Fermi function.
The Fermi function accounts for Coulomb effects between the emitted electron or
positron and the charge of the daughter nucleus; due to the opposite charges of the
electron and positron their spectral intensities differ. For example in a semi-classical
view of β − decay, the electron created in the decay of the neutron is held back by the
attractive Coulomb force with the daughter nucleus decreasing the average energy
of the emitted electrons. In β + decay, on the other hand, the positively-charged
positron created by the decay of the proton is repelled by the Coulomb force with the
daughter nucleus increasing the average energy of the emitted positrons.
The shape of the spectral intensity was calculated in the RnEDM simulations for
every β branch to accurately describe the energy of the emitted electrons from the
decay of 223 Rn into 223 Fr. An analytic approximation of the Fermi function was used
to calculate the shape of the spectrum, given by [48]
F(Z,E) ∼ = 4π(1+s)
[
[(2s)!] 2 (2pρ)2s−2 (s 2 +η 2 ) s−1/2 e
s
2φη−2s+
6(s 2 +η 2 )
]
, (3.7)
where s = [1−(Ze 2 /c) 2 ] 1/2 , ρ = R/(/mc), R is the nuclear radius approximated
as R = r ◦ A 1/3 with r ◦ = 1.2 fm and A the nuclear mass number, p is the electron (or
positron) momentum (in units of mc), η = ±Ze 2 /ν (positive sign for β − decay and
negativeforβ + decay)andν istheelectron(orpositron)velocity. Thisapproximation
does not account for electron screening, however this correction is important only at
very low energies (on the order of 100 keV or less) [48].
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