Construction of physical channels

The wave functions considered so far have a definite total spin S and a definite total orbital angular momentum L, since only reduced matrix ele­ments in spin-space and orbital space were calculated, see eq. (3.21). If one wants to describe scattering, the various clusters have to be combined to two fragments with angular momentum ji and j2. This leads to new quan­tum numbers, the relative orbital angular momentum between the fragments Lrel, the channel spin Sc = j1 + j2, and the total angular momentum J = Lrel + Sc. Hence, a physical channel is characterized by the two fragments; their internal energy if one takes excited states into account, their spin j1

and j2, ^ Lreh and J.

So far the wave functions have been characterized by the spins of the frag­ments s1 and s2, the total spin S = s1 + s2, the internal orbital angular momenta l1 and l2 and their coupling to L3 = l1 + l2, and finally the total orbital angular momentum L = L3 + Lrei. This means till now we have
worked in an LS-coupling scheme, by calculating spin and orbital matrix el­ements separately, but for scattering reactions we have to use the jj-coupling scheme. This change requires a recoupling via standard procedures. In an obvious notation we find

Подпись: [Lrel[[h Sl]jl [l2S2]j2 ]Sc]J >= £ SL3 L Д? L2 (-1)Sc-Lre-J l1 l2 L3 I J S L 1 Si S2 S  Lrel L3 L  [ [ [ l1l2]L3Lrel]L[s1S2]S ]J > j1 j2 Sc J { Sc Sc 0 J

^ ^ (4-1)

where j = л/2j + 1 and the curly bracket symbols are 9 j-symbols [20].

If a fragment contains different orbital angular momenta, like the S — and D-wave component in the deuteron, then this linear combination can also be performed in eq. (4.1) with the appropriate coefficients. This is another meaning of the index a in eq. (3.3).

The generalized eigenvalue problem is then solved on the basis of physical channels, eq. (4.1). An essential point is that during solution no states ap­pear which have norm equal to zero, be they Pauli-forbidden states or states which cannot be coupled to the required quantum numbers. This is in con­trast to the treatment of ref. [3] where the Pauli-forbidden states are used as a test for a correct calculation. Here another test can be performed: Col­lecting all diagonal overlap matrix elements (and those of the Hamiltonian), which do not have permutation across fragment boundaries (nor interaction across fragment boundaries), the calculated norm is just the product of the internal norm of the fragments times the norm of the Gaussian function on the relative coordinate, which can easily be calculated. Hence, dividing by the norm of the relative motion, the calculated internal norms have to be independent of the width used for the relative motion. A further test are the matrix elements of the Hamiltonian described above, they contain all interactions of the fragments times the relative norm. Hence dividing these matrix elements by the corresponding norm matrix elements yields the in­ternal energy of the two fragments, the threshold energy, which again has to be independent of the width parameters used for the radial motion. In most cases these two checks are a very stringent test on the correctness of the calculation.

In order to calculate the reactance matrix amn eq. (2.31), the matrix elements for regular and irregular Coulomb functions are still missing.

Добавить комментарий

Ваш e-mail не будет опубликован. Обязательные поля помечены *