The convention is to use
so for a T-matrix with Nmax=3, we have
## n: 1..3
## m: -3..3
## q: 1..2
## 900 elements
for each block we introduce a combined p-index such that,
\[
p(n,m) = n(n+1)+m
\] which here varies as p: 1..15, as each block is of dimension pmax = nmax(nmax+1)+nmax = 3(3+1)+3 = 15
.
The whole T-matrix is indexed with a combined l-index such that,
\[ l(q,p) = (q-1)p_{max} + p \] which here gives us, l: 1..30. The total T-matrix dimension (along rows/columns) is thus \(l_{max}=2\times(nmax(nmax+1)+nmax)=30\).
In summary, for a given \(N_{max}\) the indices are given by:
Given \(l\) and \(N_{max}\) we can invert these indices as follows,
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
p | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
n | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
m | -1 | 0 | 1 | -2 | -1 | 0 | 1 | 2 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | -1 | 0 | 1 | -2 | -1 | 0 | 1 | 2 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
When combining multiple particles we introduce an additional index \(j=1:j_{max}\) tracking the particle number, and a combined index \(i\), \[ i = (j-1) l_{max} + l. \]
Knowing the number of particles, \(j_{max}\), and the maximum order \(n_{max}\), we can unpack the full set of indices with the following formulas,