Geant4 Simulations for the Radon Electric Dipole Moment Search at

where (a b c d | e f) are Clebsch-Gordan coefficients and W is a Racah coefficient.
Racah coefficients can be expressed in terms of Wigner 6 j-symbols,
W(a,b,c,d,e,f) = (−1) (a+b+d+c) {
a b c
d e f
}
, (3.16)
and equivalently, combinations of Clebsch-Gordan coefficients. In the RnEDM simulationsClebsch-Gordan
coefficients werecalculatedusing adeveloped algorithmyielding
high computation efficiency and accuracy [54].
In more realistic scenarios, where the nuclei are not fully aligned, the angulardistribution
coefficients can be written as
A k (J i ,L 1 ,L 2 ,J f ) = α k (J i )A max
k (J i ,L 1 ,L 2 ,J f ) , (3.17)
where α k (J i ) is the attenuation coefficient of the alignment. This coefficient may be
represented as
α k (J i ) ≡ ρ k(J i )
B k (J i ) , (3.18)
where ρ k (J) is a statistical tensor defined as
ρ k (J) = (2J +1) 1/2∑ m
(−1) J−m (JmJ −m|k0)P m (J) . (3.19)
Yamazaki discusses the experimentally justifiable assumption that partial alignment
may be represented by a Gaussian of m-sates about m = 0 characterized by a parameter
σ which is the half-width of the Gaussian distribution, given by
P m (J) =
e −m2 /2σ 2
∑m ′ e −m′2 /2σ 2 , (3.20)
where σ = 0 defines the fully aligned state. As stated earlier, the nuclear m-states
are known at each step during the RnEDM Geant4 simulations. Therefore we may
describe any degree of orientation by explicitly inputting the m-state populations
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