Geant4 Simulations for the Radon Electric Dipole Moment Search at

for perfectly aligned nuclei with integer nuclear spin P(m = 0) = 100%, where P is
the population of the m-states. Polarization, on the other hand, has P(m = J) >>
P(m ≠ J) and P(m = J) = 100% for perfect polarization. Finally, orientation is the
completely general case, it can refer to any non-uniform population of m-states. In
the RnEDM simulations the m-states are tracked at each step, thus these populations
are known exactly. Given this information we can described any degree of orientation
using Yamazaki’s theory (see Appendix B for the MATLAB version of the γ-ray
angular distribution code).
InYamazaki’snotation,theangular-distributionfunctionforatransitionJ i → J f ,
where J is defined as the spin of the nuclear state, is expressed as
W(θ) = 1+A 2 P 2 (cosθ)+A 4 P 4 (cosθ) , (3.11)
where A k are the angular-distribution coefficients, P k are Legendre polynomials, and
θ is the angle of emission relative to the alignment axis. For fully aligned nuclei, the
angular-distribution coefficients are given the notation A max
k . For this ideal case,
A max
k = 1
1+δ 2 {
fk (J j ,L 1 ,L 1 ,J i )+2δf k (J j ,L 1 ,L 2 ,J i )+δ 2 f k (J j ,L 2 ,L 2 ,J i ) } , (3.12)
where δ is the mixing ratio (defined in Equation 3.3) and
f k ≡ B k (J i )F k (J f ,L 1 ,L 2 ,J i ) . (3.13)
The term f k can be broken up into a statistical tensor B k (J),
⎧
⎪⎨ (2J +1) 1/2 (−1) J (J0J0|k0) for integral spin,
B k (J) =
and (3.14)
⎪⎩ (2J +1) 1/2 (−1) J−1 2(J 1J 1|k0) for half-integral spin. 2 2
F k (J f ,L 1 ,L 2 ,J i ) ≡ (−1) J f−Ji−1 [ (2L 1 +1)(2L 2 +1)(2J i +1) ] 1/2
×(L 1 1L 2 −1|k0)W(J i J i L 1 L 2 ;kJ f ) , (3.15)
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