Exact computation of the 3-f and 6-f symbols - iucaa

S. -T. La,, Y -N. Chiu / Exact computation of the 3-f and 6-f symbols 351
and for the 6-f symbol as
{ } Ji J2 ~ = (_1)h+J2~4±J5([(f, —12 +f 3)!( f4 +J~ J3)!(J4 —15 +f3)!(—f4 ~f2 +j3)!
14 15 16
x (f ~f2 —f6)!(14 ~12 +16+ 1)!( —14 ~J2 +f6)!(jl +f~—j6)!(j1 —15 +16)!
x (—f’ ~15 +f~)!J/ [(f1+f~—f~)!(f1~f2 +13 + 1)!(_f~+12+13)!
x (14 ~15~J6 + 1)!(f4 12 +f6)!(fl +15+16 + 1)!] }1/2
L f~+12 —13 14 12 +16 —11 +J~+13 [ L + 1
x~(—1) L—f4—f5—f3 L—f1—f2—f3 L—f1—f5—f6 ~L—f4—j2—f6
where (~)‘s are the binomial coefficients. In eqs. (1) and (2), the summation is over all positive integral
values of L such that none of the factorials is negative. The main point is, how can we conveniently deal
with the large factorials exactly. We designed subroutines DECOM(L1,QK,R9) and PRIME(DIA,QK,R9)
which decompose each integer into a product of prime numbers with certain powers. In eqs. (1) and (2),
the value inside the square root is easy to compute, even if the jr’s (x = 1, 2, 3,.. . ,6) inside the square root
are very large. But there is a limitation on the f values that can be manipulated in the summation part.
Since each binomial coefficient is always an integer, the limitation of our computer requires that the
product of the three or four binomial coefficients cannot be beyond iO~~when we use Real * 1 6(H — floating)
on the Digital VAX computer. The value of H~floatingdata is in the approximate range 0.84 x 10 4932
to 0.59 x 104932. The precision of H_floating data is approximately one part in 2112, i.e. typically 33
decimal digits. According to our computation the 3-f symbols up to ~ ~ m3) and 6-f symbols up to
~ } can be handled on the VAX/VMS computer. 3g ~ That ~), ( means ~ _ ~) theand values ( ~ of~ 3-f }, and ( ~ 6-f ~} symbols and some are
exact approximate for thevalues abovefor values. larger We f’sgive in appendices examples of A ( and B.
(2)
3. Description of the program
The program contains five subroutines and one function which are named A3JSY (for computation of
3-f symbol), A6JSY (for computation of 6-f symbol), PRIM1(Q1) (for computation of selected prime
numbers below 602), DECOM(L1,QK,R9), PRIME(DIA,QK,R9) and FUNCTION DIFAC(N) (for
computation of factorials). For 3-f symbol calculation with integer f, we input angular momentum
numbers f~12’ 13’ and magnetic quantum numbers m
1, m2, m3. We give examples that are available in
Rotenberg’s book and can be checked against the latter. For example, for (1~1~~ we input 16.0, 16.0, 6.0,
0.0, 0.0, 0.0 from the WIG.DAT file, for half-integer 3-f symbol, for example, (~—9/2 —1/2)’ we input 8.0,
7.5, 6.5, 5.0, —4.5, —0.5. The procedures for computations of 6-f symbols are parallel to that for 3-f
symbols. The input WIG.DAT file format can be found in appendix C.
4. Output
The output shows the 3N-f symbols (N = 1, THRJ; N = 2, SIXJ) equal to a sign (+ 1.0 and —1.0)
multiplied by the prime numbers of given powers. The program output is as follows: