3b
Channels
The quantum numbers in the 3-body channels are coupled as follows
{$|p q (l s) j (\lambda 1/2) I_3 (I_3 j) J (t 1/2) T \rangle$}
Here p and q denote the Jacobi momenta. The angular momentum l and spin s of the two-body subsystem couple to total two-body channel angular momentum j. {$\lambda$} denotes the angular momentum of the spectator particle
Allowed two-body channels and ordering
We have two-body channels such as, where the channels are written in "spectroscopic" notation {${}^{2S+1}L_J$}
("1S0","3S1","1P1","3P0","3P1","3F3","3G4","3P2","3D2");
The construction and ordering of antisymmetric two-body channels {$ \alpha_2 $} is described in the following page.
The single cutoff parameter is {$ j_\mathrm{max} $}, which is the maximum total spin of the two-body subsystem. The number of allowed two-body channels for some values of {$ j_\mathrm{max} $} are given in the table
| {$ j_\mathrm{max} $} | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| # 2b-channels | 2 | 6 | 10 | 14 | 18 | 22 |
Allowed three-body channels and ordering
When you construct a 3-body channel with quantum numbers {$\alpha_3$} , you take a 2-body channel {$\alpha_2$} and check what the allowed quantum numbers of the spectator particle are with respect to the 2-particle subsystem.
The construction and ordering of partially symmetric three-body channels {$ \alpha_3 $} is described in the following page.
The following table gives the number of allowed three-body channels with total quantum numbers {$ (J, \Pi, T) = (1/2,+,1/2) $} for various two-body truncations {$ j_\mathrm{max} $}.
| {$ j_\mathrm{max} $} | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| # 3b-channels | 2 | 10 | 18 | 26 | 34 | 42 |
Sample output from reading of 3b-me from the Juelich group
Reading In the Channels
We want to be flexible about the 2-body channels that we include. I suggest therefore that we hardcode the order of the 2-body channels that we read in. We also want to be flexible whether we include a channel or not so it should be able to make this choice with a Bolean in an input file. Thus, the python code could create an input file with (when it comes to the channels) a well-defined number of lines corresponding to the maximal number of 2-body channels included in the calculation. Then there would be a number of columns where each column that defines which 3-body channel is taken into account. ""This number is finite!"" I will show later that the number of 3-body channels is maximally twice the number of included 2-body channels for the triton. These two 3-body channels arising from one 2-body channel have different {$\lambda$}s and can be ordered accordingly.
CF suggests that we:
- Specify in the input file what restrictions we want to have on the 2-body channels that we include. The minimu restriction is to give a {$ j_\mathrm{max} $} but we can also have additional truncations.
- We should also specify the q.n. of the three-body state. I.e. {$J, \Pi, T $}. Default values correspond to the triton (1/2,+1,1/2).
- I will then be able to create lists of two- and three-body channels {$ \alpha_2, \alpha_3 $} and print that information.
- From the files generated by the Juelich group I will be able to extract several ME files.
- Say that we have 10 three-body channels. I will then generate files: v3b_1_1.dat, v3b_1_2.dat, ..., v3b_1_10.dat, v3b_2_1.dat, ..., v3b_2_10.dat, ..., , v3b_10_1.dat, ..., v3b_10_10.dat
- Each such file will contain an array of floats corresponding to the NxN matrix elements, where N is the total number of (p,q) mesh points.
- The mesh is stored in one separate file pq_grid.dat (see below)
Structure of file pq.grid.dat
q' p' q p
# ---- ---- ---- ----
1.000000e-03 1.000000e-03 1.000000e-03 1.000000e-03
8.096000e-02 1.000000e-03 1.000000e-03 1.000000e-03
...
2.000000e+01 1.000000e-03 1.000000e-03 1.000000e-03
1.000000e-03 8.096000e-02 1.000000e-03 1.000000e-03
...
1.733330e+01 1.000000e+01 2.000000e+01 1.000000e+01
2.000000e+01 1.000000e+01 2.000000e+01 1.000000e+01