The RGM, however, usually deals with the much more complex case of the scattering of composite particles on each other. We will assume in the following, that the constituents interact via two-body forces, e. g. a short ranged nuclear force and the Coulomb force. An extension to three-body forces is straightforward and effects essentially only the treatment of the spin-isospin matrix elements. As alluded to in ref. [5], three body break-up channels pose a serious formal problem. Since for break-up channels the asymptotic wave function is not of the form of eq. (2.2), we have to neglect such channels. How they can be approximated is discussed in chapter 5.1.

With two-body forces alone, the Hamiltonian of an N-particle system can be split into

N 1

H(l,…,N)=’£Ti + -‘£/Vij (2-7)

i=1 i=j

where the centre of mass kinetic energy can be separated off by N 1 N

YTi = Tcm + Y (pi — pj)2 (2-8)

n=1 i<j

Here we assumed equal masses m for all the constituents, a restriction which can be removed, see ref. [7].

Due to our restriction we can decompose the translationally invariant part H’ of the Hamiltonian into the internal Hamiltonians of the two fragments, the relative motion one, and the interaction between nucleons being in different fragments

H (1,…,N) = Hi(l,…,Ni )+H2(Ni + 1,…,N)+Trei + Y, Vij

(2.9)

By adding and subtracting the point Coulomb interaction between the two fragments ZiZ2e2/Rrei the potential term becomes short ranged.

H'(1,…,N) = Hi(1,…,Ni)+H2(Ni + 1, … N) + Trel + ZZe2/Rrel + Vij — Zi Z2e2/Rrel (2.10)

ІЄ{ 1,…,N1}

^{N-l + W^N }

Here Rrel denotes the relative coordinate between the centres of mass of the two fragments. This decomposition of the Hamiltonian directs to an ansatz for the wave function in terms of an internal function of Hi and one of H2 and a relative motion function of type eq. (2.2). The total wave function is then a sum over channels formed out of the above functions properly antisymmetrised.

Nk

Фт = aY ф^ффз (2.11)

n=i

where A denotes the antisymmetriser, Nk the number of channels with chan­nel wave functions фсд described below and the relative motion wave function

«(Rrel) = $mn fm(Rrel) + amn gn(Rrel) + E bmnv Xnv (Rrel) (2.12)

V

The subscript m on фт indicates the boundary condition that only in channel m regular waves exist. The functions f and g are now regular and regularised

irregular Coulomb wavefunctions. How to use in — and outgoing waves and calculate the S-matrix directly is described in [8]. The sum n runs over physical channels, open or closed, and ’’distortion channels” without the standing wave terms. Such ’distortion channels” allow to take the distortion of the fragments in the interaction region into account, see the discussion in chapter 5.1. Sometimes they are called ’pseudo-inelastic” channels [3]. The coefficients amn and bmnv are variational parameters to be determined from

S(< ФтН’ — Ефт > ~amm) = 0 (2.13)

To simplify the notation we combine the channel functions and the relative motion part into one symbol and write in the obvious notation

The Hamiltonian H’ can be diagonalised in the space spanned by all the Xnv. Let us assume this diagonalisation to be done, then we can switch to new square integrable functions rv with

< rv ЛГ^ >— dv^ and	(2.15)
< rv^^ЛТ^ >— 6v ^	(2.16)

Since H’ commutes with the antisymmetrizer Л it suffices to apply Л on one side, see also chapter 3.2.

In eq. (2.14) the eigenfunctions Г of the Hamiltonian can be used as

фт — Л e( dmn Fn + amn + E dmv Г v (2.17)

^ n V J

where now the variational parameters amn and dmv have to be determined from the set of variational equations

< GnHAFm > + E < GnHAGn’ > amn’

n’

< rvHAFm > + ‘У ] < rvHAGni > amn’

n’

+ ^ ‘ < rvH ATv’ > dmv’ = 0

v’

with H = H’ — E. Since we prediagonalised the Hamiltonian only one term survives in the sum in eq. (2.19). Solving for dmv and taking eqs. (2.15, 2.16) into account we find

and inserting eq. (2.20) into eq. (2.18) yields

< Gn H Ag5 n’ > amn’ = — < <5n H AFm > (2.22)

n’

or in the obvious matrix notation

<GHG > aT = — <GHF > (2.23)

where aT denotes the transposed matrix a. This equation can easily be solved for a

a = — <GHF >T<GHG>-1 (2.24)

For known matrix elements of H, amn is known and via eq. (2.20) also dmv and hence the total wave function. Note that for a complete knowl­edge of the matrix a and the coefficients dmv the boundary condition of the total wavefunction фт has to run over all channels Nk. The expression for H (eq. (2.21)) indicates the close relationship of this approach to the quasiparticle method of Weinberg [9].

In general the reactance matrix amn in eq. (2.24) is not symmetric, therefore also the S-matrix given by the Caley transform

is not symmetric thus violating time-reversal invariance, even unitarity is not guaranteed. To enforce unitarity we have to have a symmetric reactance matrix a, which can be achieved by the so-called Kato correction [10]. In po­tential scattering the condition of stationarity leads to the same results [5]. For the scattering of composite systems, however, some integrals might di­verge, see the discussion below, so the more rigorous derivation [5] cannot be applied.

If we choose instead of eq. (2.17) another boundary condition as

Фт = A S (bmnFn + 5mnGn) + d! mvrv > (2.26)

^ n V J

then following along the lines of eqs. (2.17 — 2.24) we find

b = — <FHG >T< FHF>-1 (2.27)

again with an obviously unsymmetric matrix b. Since the boundary condi­tion should not affect observables, we should have

a = b-1 (2.28)

Therefore we can judge the quality of the calculation, by comparing the results of the two calculations. On the other hand we can follow the ideas of John [11] and insert the relation

< FHG >=< GHF >T + і 1 (2.29)

into eq. (2.28)

— < GHF >T< GHG >~l= — < FHF > GHF > + ^ l)

(2.30)

Multiplying by the transpose of the r. h.s of eq. (2.29) leads to [4]

a = —2(< FHiF > — < GHF >T < GHG >-1< GHF >) (2.31)

which is obviously symmetric. This expression has been derived as a second order correction in [2] and also in [5]. Analogously we find

Again from the comparison of the results for a and b we can judge the quality of the calculation. The most direct criterion of a ■ b being the unit matrix can easily fail near poles of a (resonances) or b without affecting physical observables. What remains to be done is the calculation of the matrix elements of H between F and G. For this purpose we need the channel wave function in eq. (2.11).

The ansatz for the internal wave functions is the most critical input. Because of the antisymmetrizer only two cases are realised in complicated systems: expansion in terms of harmonic oscillator wave functions or Gaussian func­tions and powers of r2, which can again be combined to harmonic oscillator functions. The difference of both expansions lies in the choice of parameters, a single oscillator frequency in one case, which allows to use the orthogonality of different functions, and a set of Gaussian width parameters, which allows to adjust the wave functions to different sizes of the fragments more easily. Therefore the harmonic oscillator expansion is well suited for the description of scattering of identical particles, or of the scattering of large nuclei on each other. In this case even algebraic methods can be used [12], [13]. Whereas the expansion in terms of Gaussians and powers of r2 can, in principle, be converted to harmonic oscillator functions, it becomes technically glumpy in more complicated cases, see the discussion in chapter 3. Since the sizes of light nuclei are quite different, we consider it, however, an advantage that different width parameters can be used.

To clarify our ansatz we consider just one term in eq. (2.11). Here the channel function has the structure

where the square brackets indicate angular momentum coupling of the trans­lationally invariant wave functions fiJ1 of the two fragments to channel spin Se and the coupling of the orbital angular momentum l and the channel spin Se to the total angular momentum J. In case of a bound state calculation the coupling to good channel spin is usually omitted. Since all the latter examples are nuclear physics ones, I will consider in the sequel wave func­tions of light nuclei for the fragment wave functions, but we could also use the technique described below for describing electron scattering off atomic or molecular systems [14].

The individual fragment wave function consists of a spatial part and a spin
(-isospin)-part, which may contain an arbitrary number of clusters. We use the expression ’’cluster” only for groups of particles without internal orbital angular momenta, that means that in nuclear physics a cluster can, at most, contain 4 nucleons, two protons and two neutrons with opposite spin projections. In hadron physics, a typical cluster would be a baryon containing 3 quarks or a meson containing a quark-antiquark pair.

The spatial wave function of a cluster h consists of a single Gaussian function

with nh the number of particles inside the cluster h and the width parameter Ph. Clusters containing only one particle are described by h = 1. In hadron physics the spin-isospin function is coupled to good total spin and isospin, in nuclear physics this coupling is not necessary in most cases, because the antisymmetriser projects onto total singlet states anyhow. The cluster rel­ative functions Xlkrel contain, in addition to the Gaussian function, a solid spherical harmonic Yik of angular momentum lk

Xk, rei = exp(-7fcРІ)Yik (pk) (2.35)

where pk denotes the Jacobi coordinate between the center-of-mass of cluster k + 1 and the center-of-mass of the clusters 1 to k, see fig. 1. The total wave function of a fragment is now a superposition of various combinations of internal and relative functions, e. g.

The spin-(-isospin) function HS,(T) is, in general, coupled to good spin (and may be coupled to good isospin). The set {lk} of orbital angular momenta between the clusters is denoted by lj, including the intermediate couplings. The sum a may run over different fragmentations, different sets of orbital angular momenta, e. g. ^-state admixtures, and different sets of width pa­rameters ph and Yk. The parameters (fah and Yak are determined from the Ritz variational principle together with the coefficients ClJs once the model space has been chosen. For this purpose one chooses the fragmentations and the set {lk} of angular momenta and the number of radial functions and asks for

Cluster 2 Figure 1: Schematic illustration of the intercluster coordinates p used in eq. (2.35).

5<fiJlH'(1,…,Ni) — EAifiJl >=0 (2.37)

where Ai is the antisymmetriser of the N1 particles in fragment 1. Therefore, we assume in the following, that fiJl and fij2 are bound states in the chosen model space and fulfill the equations

HiAifii >= EiAifii > i = 1,2 (2.38)

фі can be the lowest state but also an excited one, see the example below.

We can now demonstrate that the functional of eq. (2.13) exists and that all integrals exist in a Riemannian sense. Let us consider a fragmentation into Ni particles in fragment 1 and the rest in fragment 2. Then we can write the total antisymmetriser A in the form

A = A3A1A2 ‘У ‘ signPs P3A1A2 (2.39)

P3

where P3 permutes particles across the fragment boundaries including P3 = id. Choosing the kinetic energy Ek in the channel k to be

with Ei, k from eq. (2.38), we can then decompose the operator in eq. (2.10)

as

H'(1,…,N) — E = (Hi(1,…,Ni) — ElM) + (H2(Nl + 1,…,N) — E2,k)

+ Vij — Z1Z2 e2 / Rrel

je{N1 + l,…,N }

+ Trel + ZiZ2(?/Rrel — Ek (2.41)

All the integrals necessary for evaluating eqs. (2.18 — 2.19) are now well behaved, terms containing only square integrable functions in bra or ket cannot lead to divergent integrals. Because of the exponential fall off of the bound state functions, the same is true for all terms in which channels of different fragmentations are connected. Again, due to the properties of the bound state functions, integrals containing identical fragmentations but a genuine exchange of particles between the fragments, i. e. P3 = id, are of short range. Hence, the only possibly critical terms involve channels with identical fragmentations in bra and ket of eq. (2.13) with no exchange across the fragment boundaries.

In this case the first line of eq. (2.41) contributes zero, because according to eq. (2.38) the internal functions are solutions of the internal Hamiltonian Hi to just that energy Ei, k. The potential in the second line of eq. (2.41) is by construction short ranged, hence also this integral is short ranged. The re­maining line in eq. (2.41) is the (point-Coulomb) Hamiltonian of relative mo­tion Hrel whose solutions are the well-known Coulomb wave functions [15]. If and only if the functions Fk and Gk in eq. (2.12) are eigenfunctions of Hrel to the energy Ek, the related integrals are finite, to be precise they are zero. This choice, however, implies that the threshold energies are fixed by the energies of the fragments Ei ^. Besides choosing a different potential, the only possibility to vary the threshold energies is to modify the model space for the Ritz variation.

Since we have now shown that all integrals in eq. (2.13) and therefore also in eqs. (2.18, 2.19) are short ranged, we expand the regular and regularised irregular (Coulomb) functions in terms of square integrable functions, for simplicity those chosen in eq. (2.12). Hence, we have to calculate matrix elements of the Hamiltonian, or just overlap matrix elements, between anti­symmetrized translationally invariant wavefunctions where the spatial part consists of a superposition of multi-dimensional Gaussian functions and solid
spherical harmonics. In the next chapter we will describe how to calculate a typical matrix element.