SO(5)⊃SO(3) Clebsch-Gordon coefficients and Spherical Harmonics

The data listed in these pages relate to the states |v,α,L,M⟩ and SO(5) spherical harmonics Υv,α,L,M which are labelled by SO(5) seniority v, SO(3) angular momentum L and magnetic component M, with α a "missing label" that distinguishes angular momentum L irreps in the SO(3) reduction of seniority v irreps of SO(5).

The seniority v representation of SO(5) is the "one-rowed" irrep of SO(5) having Dynkin label (v,0). It has dimension (v+1)(v+2)(2v+3)/6. On restricting this irrep to the particular subgroup SO(3) of SO(5) defined by the Bohr model [Bo1952] (see [RW2010] for a modern overview) of the atomic nucleus, the SO(3) irrep of angular momentum L occurs with a multiplicity dv,L given by dv,L= (⌊(v-b)/3⌋+1)θv-b -⌊(v-L+2)/3⌋θv-L+2, where b=L/2 for L even, and b=(L+3)/2 for L odd, and we define θk=1 for k≥0, and θk=0 for k<0. The missing label then takes the range 1≤α≤dv,L. Lists of valid labels (v,L;α) may be found here. Expressions for the SO(5) spherical harmonics Υv,α,L,M may be found here.

Each SO(5) Clebsch-Gordon coefficient (v1,&alpha1,L1,M1; v2,&alpha2,L2,M2 | v3,&alpha3,L3,M3) may be expressed
(v1,&alpha1,L1,M1; v2,&alpha2,L2,M2 | v3,&alpha3,L3,M3) = (v1,&alpha1,L1; v2,&alpha2,L2 || v3,&alpha3,L3) (L1,M1; L2,M2 | L3,M3),
via the Wigner-Eckart theorem. Here, (L1,m1; L2,m2 | L3,m3), is the usual SO(3) Clebsch-Gordon coefficient, and (v1,&alpha1,L1; v2,&alpha2,L2 || v3,&alpha3,L3) is a SO(5)⊃SO(3) reduced Clebsch-Gordon coefficient. Their use in the Algebraic Collective Model of atomic nuclei is expounded and illustrated in [RWC2009] and [RW2010].

For various v1, v2, v3, these reduced Clebsch-Gordon coefficients are given in the files listed below, with each file dealing with fixed values of v1,v2,v3. Thus for each file, the data may be interpreted as the entries in a matrix of coefficients, the rows of which correspond to pairs of vectors with fixed seniorities v1 and v2, and the columns of which correspond to vectors with fixed seniority v3. These data files have been compiled using the algorithm developed in [RTR2004], and refined in [CRW2009]. Note that the ordering employed in the generation of these components is that in the latter of these papers (this choice arises from the multiplicity in the spaces of constant v and L).

Each "raw data" file comprises a sequence of lines containing 10 values. The first value is the reduced CG coefficient (v1,&alpha1,L1; v2,&alpha2,L2 || v3,&alpha3,L3) in scientific format. Following it are the corresponding nine integers v1,&alpha1,L1, v2,&alpha2,L2, v3,&alpha3,L3. The data in this format is readily read using Maple's "readdata" command. Only those values for which the "triangle condition" |L1-L2|≤ L3≤ L1+L2 holds are given, for all other reduced CG coefficients are zero. A zipped archive containing all the raw data files listed below is available here.

The "annotated data" file contains the same information listed without the nine integers, and with sequences of 0s compressed (the triangle condition is not used here to exclude 0s). This file is also headed by other interesting information, such as the number of reduced CG coefficients in the file and the time taken to carry out the calculation.

If you have any questions, comments or complaints, please contact David Rowe or Trevor Welsh (David will handle the complaints).

Reduced CG coefficients with v2=1

v1v2v3
0 1 1 Raw Data Annotated Data
1 1 2 Raw Data Annotated Data
2 1 3 Raw Data Annotated Data
3 1 4 Raw Data Annotated Data
4 1 5 Raw Data Annotated Data
5 1 6 Raw Data Annotated Data
6 1 7 Raw Data Annotated Data
7 1 8 Raw Data Annotated Data
8 1 9 Raw Data Annotated Data
9 1 10 Raw Data Annotated Data
10 1 11 Raw Data Annotated Data
11 1 12 Raw Data Annotated Data
12 1 13 Raw Data Annotated Data
13 1 14 Raw Data Annotated Data
14 1 15 Raw Data Annotated Data
15 1 16 Raw Data Annotated Data
16 1 17 Raw Data Annotated Data
17 1 18 Raw Data Annotated Data
18 1 19 Raw Data Annotated Data
19 1 20 Raw Data Annotated Data
20 1 21 Raw Data Annotated Data
21 1 22 Raw Data Annotated Data
22 1 23 Raw Data Annotated Data
23 1 24 Raw Data Annotated Data
24 1 25 Raw Data Annotated Data
25 1 26 Raw Data Annotated Data
26 1 27 Raw Data Annotated Data
27 1 28 Raw Data Annotated Data
28 1 29 Raw Data Annotated Data
29 1 30 Raw Data Annotated Data
30 1 31 Raw Data Annotated Data
31 1 32 Raw Data Annotated Data
32 1 33 Raw Data Annotated Data
33 1 34 Raw Data Annotated Data
34 1 35 Raw Data Annotated Data
35 1 36 Raw Data Annotated Data
36 1 37 Raw Data Annotated Data
37 1 38 Raw Data Annotated Data
38 1 39 Raw Data Annotated Data
39 1 40 Raw Data Annotated Data
40 1 41 Raw Data Annotated Data
41 1 42 Raw Data Annotated Data
42 1 43 Raw Data Annotated Data

Reduced CG coefficients with v2=2

v1v2v3v1v2v3
0 2 2 Raw Data Annotated Data
1 2 3 Raw Data Annotated Data 1 2 1 Raw Data Annotated Data
2 2 4 Raw Data Annotated Data 2 2 2 Raw Data Annotated Data
3 2 5 Raw Data Annotated Data 3 2 3 Raw Data Annotated Data
4 2 6 Raw Data Annotated Data 4 2 4 Raw Data Annotated Data
5 2 7 Raw Data Annotated Data 5 2 5 Raw Data Annotated Data
6 2 8 Raw Data Annotated Data 6 2 6 Raw Data Annotated Data
7 2 9 Raw Data Annotated Data 7 2 7 Raw Data Annotated Data
8 2 10 Raw Data Annotated Data 8 2 8 Raw Data Annotated Data
9 2 11 Raw Data Annotated Data 9 2 9 Raw Data Annotated Data
10 2 12 Raw Data Annotated Data 10 2 10 Raw Data Annotated Data
11 2 13 Raw Data Annotated Data 11 2 11 Raw Data Annotated Data
12 2 14 Raw Data Annotated Data 12 2 12 Raw Data Annotated Data
13 2 15 Raw Data Annotated Data 13 2 13 Raw Data Annotated Data
14 2 16 Raw Data Annotated Data 14 2 14 Raw Data Annotated Data
15 2 17 Raw Data Annotated Data 15 2 15 Raw Data Annotated Data
16 2 18 Raw Data Annotated Data 16 2 16 Raw Data Annotated Data
17 2 19 Raw Data Annotated Data 17 2 17 Raw Data Annotated Data
18 2 20 Raw Data Annotated Data 18 2 18 Raw Data Annotated Data
19 2 21 Raw Data Annotated Data 19 2 19 Raw Data Annotated Data
20 2 22 Raw Data Annotated Data 20 2 20 Raw Data Annotated Data
21 2 23 Raw Data Annotated Data 21 2 21 Raw Data Annotated Data
22 2 24 Raw Data Annotated Data 22 2 22 Raw Data Annotated Data
23 2 25 Raw Data Annotated Data 23 2 23 Raw Data Annotated Data
24 2 26 Raw Data Annotated Data 24 2 24 Raw Data Annotated Data
25 2 27 Raw Data Annotated Data 25 2 25 Raw Data Annotated Data
26 2 28 Raw Data Annotated Data 26 2 26 Raw Data Annotated Data
27 2 29 Raw Data Annotated Data 27 2 27 Raw Data Annotated Data
28 2 30 Raw Data Annotated Data 28 2 28 Raw Data Annotated Data
29 2 31 Raw Data Annotated Data 29 2 29 Raw Data Annotated Data
30 2 32 Raw Data Annotated Data 30 2 30 Raw Data Annotated Data
31 2 33 Raw Data Annotated Data 31 2 31 Raw Data Annotated Data
32 2 34 Raw Data Annotated Data 32 2 32 Raw Data Annotated Data
33 2 35 Raw Data Annotated Data 33 2 33 Raw Data Annotated Data
34 2 36 Raw Data Annotated Data 34 2 34 Raw Data Annotated Data
35 2 37 Raw Data Annotated Data 35 2 35 Raw Data Annotated Data

Reduced CG coefficients with v2=3

v1v2v3v1v2v3
0 3 3 Raw Data Annotated Data
1 3 4 Raw Data Annotated Data 1 3 2 Raw Data Annotated Data
2 3 5 Raw Data Annotated Data 2 3 3 Raw Data Annotated Data
3 3 6 Raw Data Annotated Data 3 3 4 Raw Data Annotated Data
4 3 7 Raw Data Annotated Data 4 3 5 Raw Data Annotated Data
5 3 8 Raw Data Annotated Data 5 3 6 Raw Data Annotated Data
6 3 9 Raw Data Annotated Data 6 3 7 Raw Data Annotated Data
7 3 10 Raw Data Annotated Data 7 3 8 Raw Data Annotated Data
8 3 11 Raw Data Annotated Data 8 3 9 Raw Data Annotated Data
9 3 12 Raw Data Annotated Data 9 3 10 Raw Data Annotated Data
10 3 13 Raw Data Annotated Data 10 3 11 Raw Data Annotated Data
11 3 14 Raw Data Annotated Data 11 3 12 Raw Data Annotated Data
12 3 15 Raw Data Annotated Data 12 3 13 Raw Data Annotated Data
13 3 16 Raw Data Annotated Data 13 3 14 Raw Data Annotated Data
14 3 17 Raw Data Annotated Data 14 3 15 Raw Data Annotated Data
15 3 18 Raw Data Annotated Data 15 3 16 Raw Data Annotated Data
16 3 19 Raw Data Annotated Data 16 3 17 Raw Data Annotated Data
17 3 20 Raw Data Annotated Data 17 3 18 Raw Data Annotated Data
18 3 21 Raw Data Annotated Data 18 3 19 Raw Data Annotated Data
19 3 22 Raw Data Annotated Data 19 3 20 Raw Data Annotated Data
20 3 23 Raw Data Annotated Data 20 3 21 Raw Data Annotated Data
21 3 24 Raw Data Annotated Data 21 3 22 Raw Data Annotated Data
22 3 25 Raw Data Annotated Data 22 3 23 Raw Data Annotated Data
23 3 26 Raw Data Annotated Data 23 3 24 Raw Data Annotated Data
24 3 27 Raw Data Annotated Data 24 3 25 Raw Data Annotated Data
25 3 28 Raw Data Annotated Data 25 3 26 Raw Data Annotated Data
26 3 29 Raw Data Annotated Data 26 3 27 Raw Data Annotated Data
27 3 30 Raw Data Annotated Data 27 3 28 Raw Data Annotated Data
28 3 31 Raw Data Annotated Data 28 3 29 Raw Data Annotated Data
29 3 32 Raw Data Annotated Data 29 3 30 Raw Data Annotated Data
30 3 33 Raw Data Annotated Data 30 3 31 Raw Data Annotated Data
31 3 34 Raw Data Annotated Data 31 3 32 Raw Data Annotated Data
32 3 35 Raw Data Annotated Data 32 3 33 Raw Data Annotated Data
33 3 36 Raw Data Annotated Data 33 3 34 Raw Data Annotated Data
34 3 37 Raw Data Annotated Data 34 3 35 Raw Data Annotated Data
35 3 38 Raw Data Annotated Data 35 3 36 Raw Data Annotated Data

Reduced CG coefficients with v2=6

v1v2v3 v1v2v3 v1v2v3 v1v2v3
0 6 6 Raw Data Annotated Data
1 6 7 Raw Data Annotated Data 1 6 5 Raw Data Annotated Data
2 6 8 Raw Data Annotated Data 2 6 6 Raw Data Annotated Data 2 6 4 Raw Data Annotated Data
3 6 9 Raw Data Annotated Data 3 6 7 Raw Data Annotated Data 3 6 5 Raw Data Annotated Data 3 6 3 Raw Data Annotated Data
4 6 10 Raw Data Annotated Data 4 6 8 Raw Data Annotated Data 4 6 6 Raw Data Annotated Data 4 6 4 Raw Data Annotated Data
5 6 11 Raw Data Annotated Data 5 6 9 Raw Data Annotated Data 5 6 7 Raw Data Annotated Data 5 6 5 Raw Data Annotated Data
6 6 12 Raw Data Annotated Data 6 6 10 Raw Data Annotated Data 6 6 8 Raw Data Annotated Data 6 6 6 Raw Data Annotated Data
7 6 13 Raw Data Annotated Data 7 6 11 Raw Data Annotated Data 7 6 9 Raw Data Annotated Data 7 6 7 Raw Data Annotated Data
8 6 14 Raw Data Annotated Data 8 6 12 Raw Data Annotated Data 8 6 10 Raw Data Annotated Data 8 6 8 Raw Data Annotated Data
9 6 15 Raw Data Annotated Data 9 6 13 Raw Data Annotated Data 9 6 11 Raw Data Annotated Data 9 6 9 Raw Data Annotated Data
10 6 16 Raw Data Annotated Data 10 6 14 Raw Data Annotated Data 10 6 12 Raw Data Annotated Data 10 6 10 Raw Data Annotated Data
11 6 17 Raw Data Annotated Data 11 6 15 Raw Data Annotated Data 11 6 13 Raw Data Annotated Data 11 6 11 Raw Data Annotated Data
12 6 18 Raw Data Annotated Data 12 6 16 Raw Data Annotated Data 12 6 14 Raw Data Annotated Data 12 6 12 Raw Data Annotated Data
13 6 19 Raw Data Annotated Data 13 6 17 Raw Data Annotated Data 13 6 15 Raw Data Annotated Data 13 6 13 Raw Data Annotated Data
14 6 20 Raw Data Annotated Data 14 6 18 Raw Data Annotated Data 14 6 16 Raw Data Annotated Data 14 6 14 Raw Data Annotated Data
15 6 21 Raw Data Annotated Data 15 6 19 Raw Data Annotated Data 15 6 17 Raw Data Annotated Data 15 6 15 Raw Data Annotated Data
16 6 22 Raw Data Annotated Data 16 6 20 Raw Data Annotated Data 16 6 18 Raw Data Annotated Data 16 6 16 Raw Data Annotated Data
17 6 23 Raw Data Annotated Data 17 6 21 Raw Data Annotated Data 17 6 19 Raw Data Annotated Data 17 6 17 Raw Data Annotated Data
18 6 24 Raw Data Annotated Data 18 6 22 Raw Data Annotated Data 18 6 20 Raw Data Annotated Data 18 6 18 Raw Data Annotated Data
19 6 25 Raw Data Annotated Data 19 6 23 Raw Data Annotated Data 19 6 21 Raw Data Annotated Data 19 6 19 Raw Data Annotated Data
20 6 26 Raw Data Annotated Data 20 6 24 Raw Data Annotated Data 20 6 22 Raw Data Annotated Data 20 6 20 Raw Data Annotated Data
21 6 27 Raw Data Annotated Data 21 6 25 Raw Data Annotated Data 21 6 23 Raw Data Annotated Data 21 6 21 Raw Data Annotated Data
22 6 28 Raw Data Annotated Data 22 6 26 Raw Data Annotated Data 22 6 24 Raw Data Annotated Data 22 6 22 Raw Data Annotated Data
23 6 29 Raw Data Annotated Data 23 6 27 Raw Data Annotated Data 23 6 25 Raw Data Annotated Data 23 6 23 Raw Data Annotated Data
24 6 30 Raw Data Annotated Data 24 6 28 Raw Data Annotated Data 24 6 26 Raw Data Annotated Data 24 6 24 Raw Data Annotated Data
25 6 31 Raw Data Annotated Data 25 6 29 Raw Data Annotated Data 25 6 27 Raw Data Annotated Data 25 6 25 Raw Data Annotated Data

Last Updated: 04/02/2010

References

[Bo1952] "The coupling of nuclear surface oscillations to the motion of individual nucleons", by A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26:14 (1952).

[RTR2004] "Spherical harmonics and basic coupling coefficients for the group SO(5) in an SO(3) basis", by D.J. Rowe, P.S. Turner and J. Repka, J. Math. Phys. 45 (2004) 2761-2784.

[CRW2009] "Construction of SO(5) ⊃ SO(3) spherical harmonics and Clebsch-Gordan coefficients", by M.A. Caprio, D.J. Rowe and T.A. Welsh, Comp. Phys. Comm. 180 (2009) 1150-1163. (arXiv)

[RWC2009] "The Bohr Model as an algebraic collective model", by D.J. Rowe, T.A. Welsh and M.A. Caprio, Phys. Rev. C79 (2009) 054304.

[RW2010] "Fundamentals of Nuclear Models: I Foundational Models", by D.J. Rowe and J.L Wood, to appear 2010 (World Scientific, Singapore).