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Imaginary potentials from many-body theory
NUCLEAR PHYSICS A
Nuclear Physics A567 (1994) 78-96 North-Holland
Imaginary potentials
from many-body theory
P. Danielewicz National Superconducting Cyclotron LaboratoT and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
P. Schuck Institut des Sciences Nucleaires, 53, Avenue des Martyrs, 38026 Grenoble Cedex, France Received (Revised
10 August 1990 2 June 1993)
Abstract The imaginary parts of the single-nucleon, particle-hole (phonon) and (deuteron) mass operators are considered. Expressions of a Fermi golden rule for each case. Special emphasis is laid on many-body aspects and collectivity amplitudes. Implications for the calculation of optical potentials for nuclei are
particle-particle form are derived in the transition discussed.
1. Introduction Nuclear optical potentials are of continued theoretical and experimental interest. On the theoretical side, the calculation of in-medium cross sections and the inclusion of collective phonon excitations is of prime interest. Though there exist “standard”
many-body
techniques
for calculation
of optical
potentials
(mass opera-
tors) [l-5], there have been recently new developments reported [6]. In this work, we continue along these lines, and derive new formulas for the optical potential of elastic nucleon and deuteron scattering. We treat the particle-hole (phonon) self-energy in a consistent manner. We concentrate on the formulation of the imaginary parts of the optical potentials. The corresponding real parts are in principle determined via dispersion relations. It should always be possible to write the diagonal elements of the imaginary potentials in a golden rule form; their definite sign is then ensured. Expressions that cannot be arranged in such a form had been used when attempting to include the phonon excitations. Due to antisymmetrization, overcounting problems occur which are partially cured by subtraction procedures [1,2,4,7]. As a consequence the imaginary potential can at least in principle acquire a wrong sign (flux is produced rather than absorbed). In this work we show that the collectivity may be put into the vertex functions without 0375-9474/94/$07.00
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P. Danielewicz, P. Schuck / Imaginary potentials
79
losing generality. The proper sign of the imaginary potential is then guaranteed, as the diagonal elements are indeed of the golden rule form. In sect. 2 we first treat the nucleon-nucleus potential quite in detail. In sect. 3 we generalize the results to the particle-particle and particle-hole self-energies. Finally in sect. 4 we confront our new formalism with commonly used procedures and give a critical assessment.
2. Single-particle
potential
The nucleon-nucleus optical potential can be identified with the mass operator appearing in the Dyson equation for the single-particle Green function G = Go + G’MG,
(2.1)
where G;,”
and Go suppose will also Being
= -i(O
1T[ $,( t)+@(t’)] IO>
(2.2)
is the Hartree-Fock propagator. In the remainder of the paper we to work, without loss of generality, in the Hartree-Fock basis, which we sometimes call the shell-model basis. concerned with the imaginary part of the potential, we write [8,9]
Im M[,” = Im - i(0 I T[ jl( t)j!p( t’)]
IO)irr
= -+] IO)irr,
(2.3)
with
(2.4) Explicitly the mass operator is
where u iZX4is the antisymmetrized
matrix element of the bare interaction,
From Eq. (2.5) we see that A4 is related to the irreducible function,
R 1231’2’3’
=
and
2plh (2hlp) Green
(2.7)
80
P. Danielewicz,
and we further
P. Schuck
/ Imaginary potentials
write
(2.8) where >I--f’ iR 1231’2’3 = co 1(~:IcI2~3),(~:‘~:‘~,‘)1,
(2.9a)
I’)iKY
and (2.9b)
Assuming a weak damping represented
of 2plh
(2hlp)
states,
the functions
(2.9) can be
as
>I--f’ = 2 C&,E:“,“,3E$53t e-iRL(t+r’), iR 1231,2,3,
(2.10a)
L
ePioLCf -“).
(2.10b)
L
Here BL are the wave functions and fiL are the energies of the correlated states. From general thermodynamic considerations [lo], FL = 0 for JJL < &r and NL = 0 for fiL > .5r, in the ground state of a system, where &r is Fermi energy. We can further assume that the states are normalized so that r\l, = 1 for 0, > &r and NL = 1 for 0, < or. Then Eqs. (2.9) and (2.10) yield the completeness relations
where n are occupations of the single-particle Upon introducing a dual system of functions g;-L * gL’ -123’123
and summing
states in the shell model, g such that
6LL’ =
(2.12)
9
Eqs. (2.11) side by side, we can obtain
E_1L;(n,Ti2ii3 + n,n,n,)E’,L;,= aLL’.
ii = 1 - n.
the normalization
condition (2.13)
P. Danielewicz, P. Schuck / Imaginary potentials
81
The three-body wave functions B and the energies of the states can be searched for by solving the three-body wave equation from ref. [ll] (see also ref. [6]),
- +(1 (flL+E1-&2 - E~)S:C~L~ - ( It1
-
EL It2 - n3)u232t3rw12f3r
n2)u21’12’s;2’3- (n1 - n3)u31r13r8~23’ = 0,
(2.14)
where E are the shell-model particle energies. On substituting Eqs. (2.10) into (2.8) we find for the diagonal elements of the imaginary potential in the energy representation Im M= -~~CI~,,,~~,I”(N,-N,)6(w-R,).
(2.15)
L
Some further insight into Eqs. (2.3), (2.6) and (2.15) may be gained by using the zero-temperature limit of finite-temperature theory to calculate the expectation values at the r.h.s. of (2.3) and (2.6). Two methods for calculating the expectation values exist. In appendix A we provide basic information on the method that yields chronological functions [such as in Eqs. (2.2), (2.5) and (2.711 in the amplitudes in the two-point expectation values. Details may be found e.g. in appendix C of ref. [12]. The other method 161utilizes retarded functions in the amplitudes, and it has some advantages, at finite temperatures and in nonequilibrium situations, over the first method, as will be indicated. Unless explicitly stated, we use the more common chronological functions. We find
where (2.17a) and (2.17b) The vertex function can be written in terms of R as
The time arguments of W are in general different from one another. The same holds for the right-hand time arguments of R. The dots at the r.h.s. of (2.16) indicate contributions from the 3p2h and 2h3p states that are products of singleparticle states, and such states with a still larger number of particle-hole pairs. The result of the form (2.16) is valid at finite temperature when the expectation values that determine the functions in (2.3), (2.5) and (2.17) are taken with respect
P. Danielewicz, P. Schuck / Imaginary potentials
82
to the equilibrium
density
operator
[6]. A complete
contribution
of the 2ph and
2hp states to the imaginary part of the mass operator is obtained, though, in that case, when the retarded rather than chronological amplitudes W are used [6]. The difference
vanishes
In the shell-model
at zero temperature. approximation
we have (2.19a)
and G<
II’
= in
1
6
11’
e-‘“,(‘~-‘;’
(2.19b)
>
where the ground-state occupation y1 is for a state below Fermi surface, and 0 above; E = 1 - n. On inserting (2.19) into (2.16), we find for the diagonal elements of imaginary potential in the energy representation, Im M = - $r
WI,,, I’( n,E,Z,
- E,n,n,)6(
w + 82 - Ed - .s4).
(2.20)
We should, of course, be aware of the fact that the shell-model states in (2.20) may themselves have a decay width and/or have a reduced strength (Z-factors less than 11. We want to ignore such refinements here and rather concentrate on the vertex function W. With the help of the Watson-Fadeev series expansion of the three-body propagator involved in (2.18) (refs. [9,11,6]) we can expand W in a series of pp and ph T-matrices. To lowest order W can be approximated by a pp T-matrix (a Brueckner T-matrix could be used, a Feynman-Galitski T-matrix, or any more sophisticated version thereof). Low-order terms of the expansion are shown in Fig. 1, and the second equality indicated in the figure follows from an integral equation satisfied by the T-matrix (the dot stands for cr&.
++2! +... TPh
=J?%+ *’ l
Fig. 1. Approximation
of the vertex function
W by a scattering
matrix
Tpp.
P. Danielewicz, P. Schuck / Imaginary potentials
Fig. 2. Approximation
to the vertex function
including
83
ph correlations.
A subject that we particularly want to dwell on in this work, is how to incorporate collectivity from low-lying collective states (including giant resonances> into the optical potential. For that purpose we add to the terms indicated in Fig. 1 the next terms in the series, see Fig. 2. The ph T-matrix satisfies an integral equation T1?& = Kf&
- iKP~~,,,G3,,3,G,,,,,Tph3”4”34
-
G&
that can be directly derived from the Bethe-Salpeter tor [ 13,141,
equation for the ph propaga-
IT = II” + III°KphIl,
(2.22)
where explicitly
n 121’2’
-i(O
=
I T(6p21&$1~)
IO>,
(2.23)
- (0 I e&b1 IO> and KPh is an irreducible kernel in the ph channel. With this we can now write the expression for the vertex function indicated in Fig. 2,
6p,, = $;$I
W1234(t
=
-
t’)
TIp2p348(
t -
t’)
+
TP2q3p4e-ia,(‘-t’)~12,3,2,,3,,( t - t’)K$‘!$,,,
+ T&Y,,, e--iE3(f--f )172f4f2fr4,,(t - t’) K$‘4,,2,.
(2.24)
P. Danielewicz, P. Schuck / Imaginary potentials
84
Here we assume that Tpp and KPh are energy-independent the relative times may be expressed with the &functions). nience tions
the time representation, may then
be written
since the forward-
concisely.
(the dependencies on We choose for conve-
and backward-going
Eq. (2.24) together
contribu-
with (2.20) is the basic
result of this work. Collectivity is accounted for via the density-density correlation function (2.23) which is available from extended RPA calculations [15,16]. For the based on Brueckner calcuinteractions T ** and K ph there exis t p arametrizations lations and corresponding Landau parameters, respectively. Skyrme-like forces might be used at low energies. It should then not be too difficult to evaluate (2.24) numerically. The definite sign of Im M and the correct second-order Born limit are guaranteed in this case (see discussion in sect. 4 on these points). We now try to give some connection of (2.20) and (2.24) with formulas more commonly in use when dealing with collectivity in the mass operator. We consider (2.24) in the energy representation and separate from IT possible low-lying isolated resonances in different channels,
= n;;;t2r +
fi,,,#2r)
(2.25)
where fi is the background. The amplitudes x” may e.g. be calculated from RPA theory. The resonance widths TV are supposed to come from the coupling to the continuum (escape width) as well as from coupling to more complicated states (spreading width). The resonances in (2.25) will lead to a pronounced peaking in the energy
dependence
of (2.24), whereas
I? will give rise to a renormalization
of
TPP.
= i=,pt4+ T,P23,411;:;s’2,,3t, K$‘3,,23 + T~?,4,n;e~,2,,4,,K~,~4,,24.
(2.26)
We now insert (2.26) into (2.20) and first discuss the case w > &_r. For narrow isolated resonances standing well above the background term T**, the direct product of resonances in 1W 12 will dominate in the resonance region over the interference terms resonance-background and between different resonances. Outside of the resonances the smooth part of the vertex function will be of impor-
85
tance. Following these considerations
XS(w+&*-Eg-Fq)
we write
+ .*. *
The dots in (2.27) indicate the interference matters when w = Ed = 0, and the quantity
(2.27) terms. A resonance
contribution
(2.28)
is recognized as the decay width of the state Y into the ph states. To the extent that these widths constitute most of the width of the states we finally get, including w < or that gives rise to an analogous expression,
(2.29)
This formula resembles the one in common use when dealing with collectivity [1,2]. It is, however, not equivalent, neither in spirit nor from a practical point of view. In (2.29) we sum only over a few resonances whereas in the usual formulation one sums, in principle, over a complete set of states. We also remark that (2.29), as (2.24), is of definite sign for either w > &r or o < &r (see sect. 4). It may be useful to generalize (2.29) to the finite-temperature case. The occupation numbers then become n = l/{exp[P(E - E.L)]+ 1) with p the inverse temperature. Starting from (2.20) and (2.24) one has to work, in the finite-temperature case, with retarded propagators instead of the chronoiogical ones we have been dealing with before. Also the quantity (2.27) does not represent the width of the state any longer, but instead the decay rate for fi, > 0, and the production rate
86
P. Danielewicz,
for 0, < 0. The exp( -pL?,)].
rates
P. Schuck
are related
/ Imaginary potentials
to the width
We then get for the imaginary
- ITl%X~; 12W>
with a factor
mV = sgn n,/[l
-
part of the mass operator:
L/257
(W-F4+t”)2++rv2
I+
. ..’
(2.30)
with NV = sgn R,/[exp(pR,,) - 11. Concluding this section we state that collectivity can either be put into the vertices (2.24) of the mass operator or, after certain approximations, more directly as intermediate states as in (2.29) and (2.30) and as is displayed in Fig. 3. The equivalence of (2.24) and (2.29) hinges on the ability of separating the spectrum of the response function L’ in a clear way into resonances and background. The detailed relation to the usual formulation will be given in sect. 4.
3. Particle-hole We now make
and particle-particle similar
cle-particle (deuteron) The imaginary part written as [14]
considerations
mass operators for the particle-hole
mass operators. of the particle-hole
mass operator
(phonon)
and parti-
in a state
v may be
Im Mph “(t - t’)
x ( cs’6’2’3’6*‘4’ Here
X is a wave function xL=
+(n,
-n,)i;;,
1’ ,‘5’7’4’~2’h’)X;:2’.
(3.1)
dual to ,Y, (3.2)
P. Danielewicz,
Fig. 3. Contribution
P. Schuck
87
/ Imaginary potentials
to the imaginary part of the optical potential
from the particle-hole
coupling.
where the upper sign refers to a,, > 0 and the lower one to 0, < 0, with the . . normahzatron set by i;;*x y’ = 1. When ignoring the damping of the 2p2h states, one can write
To every wave function rDJ there corresponds a wave function @‘, such that fiJ6 = a,, and @:;a4 = @<. For the imaginary part of Mph one then gets, in the energy representation,
The wave functions x might be searched for by solving a wave equation analogous to (2.14),
(3.5)
P. Danielewicz, P. Schuck / Imaginary potentials
88
The completeness
relation reads
and on introducing a system of dual wave functions @, the normalization can be written as nln2n3n4
condition
--
(3.7)
We can further expand diagrammatically the expectation value at the r.h.s. of (3.1) and apply a resummation as in the case of (2.6), to obtain
where the amplitude
with the 2p2h Green function
In the energy representation
we get the contribution
Im Mph ’ = - $r1&$34 I’( n,n,i?,Fi,
f
iyi2n3n4)fi(
from 2p2h states 0.J +
El
+
&2
-
83
-
F4),
(3.11) where, to lowest order in Tpp, l-%,4
=
%i*
( ci%
s52
-
T,p,P,3%4
-
G%4h
-
Tl%4W
(3.12)
*
The other important case is that of the deuteron ado-particle) mass operator MPP, see also ref. [17]. With Ik* being a two-particle wave function, the result analogous to (3.11) from retaining the transitions to 3ph states takes the form Im Mpp = Im P,:Mpp121’2’ 1v1’2’ = -
h! A
1234
I’(
---
n,n2n3n4
+
~,n2n3n4)S(w +
El
-
F2 -
E3 -
Eq). (3.13)
P. Danielewicz, P. Schuck / Imaginary potentials
89
Here the amplitude A is given by
G:,‘G-’ XD l,2,3,4,1,,2,,3,,4,,G,‘fG-’ 2”2 3 3 4”4?
(3.14)
where ?$ is a dual wave function, !P,2 = (1 - ~zi- n2>@,2, and iD 12341’2’3’4’ = (0 1T(
(cI:(cr2~3~41cr,lf~~f~~~~lf)
(3.15)
IO)irr.
It is important to note that into the vertices r, Eq. (3.12), and A, Eq. (3.14), enter the dual amplitudes with the result that integrations run over the whole rather than restricted phase space, even at zero temperature (see also ref. [14]).
4. Average mass operators of nuclei We now want to discuss in some detail the situation of nuclei. In nuclear physics essentially two different formulations are in use to calculate the imaginary part of the optical potential. The first one consists in calculating the expression (2.20) in infinite matter using the two-body scattering matrix for IV and going over to finite nuclei via the local density approximation [3,5,18]. The imaginary potential from the nuclear structure approach [1,2] takes the form Im M= --$rrCIU
12341y:4(2(n,NJ-ii2N,)G(w+F2-R,)
J
-
-atb234X;3~2(N,n,
+
--
%-I ul234
I*( n2n3n4
-
N,n,)6(
zi,n+,)a(
w
-
+
&2 -
0,
0”
-
E4)
&3 -
&4).
(4.1)
Here the sums run over all two-particle and particle-hole states and the last term at the r.h.s. is from subtraction of two second-order contributions to M that otherwise would be counted three times. When an effective interaction is used and the pp states are not collective, then the first term at the r.h.s. of (4.1) cancels one of the second-order contributions, leading to the result [2,7] Im M= -7rxIU
+
Y$rl
1234x:3
ul234
I’(
12(n2n3%4
xE4
-
-
Nvn4)S(
n2n3n4)6(
w
@
-
&
+
&2 -
-
‘4)
&3 -
&4)
3
(4.2)
P. Danielewicz, P. Schuck / Imaginary potentials
90
that is used in practice.
The second
half of the second-order difficulties
associated
and polarization
contributions
term at the r.h.s. of (4.2) still serves to cancel from
the first term.
There
with (4.1) and (4.2) that we want to discuss.
parts (cf. ref. 1191) of the imaginary
operator
of definite
sign, e.g. the correlation
operator
as representing
the particle
absorption.
part
potential should
are inherent
The correlation
should
each be an
be a positive
definite
This stems from the fact that the
imaginary parts of the single-particle Green function (2.21, i.e. the spectral functions, corresponding to poles either in the upper or lower half-planes, must be positive or negative definite operators, respectively. It follows then that the diagonal elements (in any basis) of the imaginary part of the mass operator must have a definite sign. Our expressions (2.20) with (2.24) [see also Eq. (2.2911 have this property since they are of the golden rule form. On the other hand this is obviously not the case with the quantities in Eqs. (4.1) and (4.2) (we explicitly demonstrate within two models in appendix B). This fact has apparently not caused problems in the practical calculations [4,20-221 up to now, but we find the situation uncomfortable. The fact that an expression such as (4.2) can give rise to problems has also been discussed in a slightly different context at the end of sect. 4.2.5 of ref. [23]. In short, we have shown in the preceeding sections the existence of an approximation scheme which allows the consideration of collective effects and at the same time ensures the correct sign and correct second-order Born limit of Im M.
5. Conclusion
In the paper we mainly addressed the very old question of how to put collectivity into the imaginary part of optical potentials. In our opinion this question was never really solved satisfactorily. We here give a novel prescription which assures that correlation and polarization parts have a definite sign, i.e. only flux absorption and no flux creation is possible for elastic scattering even in the presence of collectivity. Our formulation resides in the fact that we could show that all collectivity can be put into the vertices, and thus a Fermi’s golden rule form for the imaginary part of the optical potential can be applied under all circumstances. Our basic result is Eq. (2.24) which also holds at finite temperature and which has the correct second-order Born limit even in the presence of collectivity. It represents the first few terms when the vertex function is expanded in a Watson-Fadeev series. Eqs. (2.29) and (2.30) are approximations to (2.24) which preserve the definite sign property. They are valid when collective resonances are well separated from the background. The proposed formulation does not seem to be much more complicated than the ones in use and should be amenable to numerical application.
P. Danielewicz, P. Schuck / Imaginary potentials
91
P.S. expresses gratitude for the hospitality extended to him at the National Superconducting Cyclotron Laboratory during several visits. P.D. acknowledges partial support from the National Science Foundation under Grant Nos. PHY 8905933 and PHY 9017077.
Appendix A
We discuss here the computation of the expectation values such as at the r.h.s. of (2.3) and (2.6), or at the r.h.s. of (2.9). From (2.31, we have 2 Im
M:;“=
IO> -
10).
(‘4.1)
For the first term at the r.h.s. of (A.11, we get
(0 I j?(f’)j,(t>
IO)
= (OlU'( -m,
t')j[J(t')U'(t', m)U’(m, t)j:(t)U(t, co)IO)
= (0 I [ U’(m, t’)
= (0 I
j:,( t’)U’( t’,
-w)]+U’(m, f)j:(t)U(t,
03)IO>
(T[j:,(t')exP( -il_dr’ vl(T’))]}+ (A4
When we expand the exponentials at the r.h.s. of (A.21 and apply a Wick decomposition, we obtain contractions of the interaction-picture operators both of which are associated with j, or with jt, or one with j and the other with jt. For both field operators associated with j, the contraction coincides with the function iG. For both field operators associated with jt, the contraction coincides with the complex conjugate function (iG)*. The expansion of the exponential for jt in (A.21 further gives also conjugate potential factors (-iu)*. When one field operator is associated with jt and the other with j, then the contraction coincides either with iG> or iG<, cf. (2.17). Disconnected diagrams merely give an overall factor 1 = (01 U’( - ~0,m)U’(q - ~1 IO). Resummations can be carried out in a standard manner, so as to eliminate the expectation values of interaction-picture operators in favor of the expectation values of Heisenberg-picture operators,
P. Danielewicz, P. Schuck / Imaginary potentials
94
We shall write down the RPA equations in the original (“spherical”) basis used in (B.7) and (B.8), and solve for the density-density correlation function which for this model
reads
(B.9) The RPA equation
then becomes
(B.lO) The second term in the square bracket is an exchange term. Often that term is neglected. Keeping the term leads to noncollective roots. The solution of (B.lO) can be written as Iz m1m2m3m4
-2E
1
cl-_??
_
+(E+
-E -17
(B.ll)
95
P. Danielewicz, P. Schuck / Imaginary potentials
6-
0.0
Cl=4 x = 0.86 7)/E=0.14
0.5
1.0
1.5
2.0
-
2.5
Fig. B.l. Absorption rate - 2 Im M in the schematic model from Eq. (B.5).
and the energies
E = e/l
of the noncollective
- (V/e)’
and collective
states
are
(B.12)
,
and
(B.13)
P= qnUsing (B.ll)
(B.14)
1).
we obtain
for Im M, putting
e = 1,
We see that this expression correctly reduces to the second-order Born approximation for I/ -+ 0, i.e. 6, E + 1. In Fig. B.2 we show the absorption rate - 2 Im M from (B.15) for different values of LZ, n and I/. We see again in this model that
96
0.0
0.5
1.0
1.5
2.0
2.5
W/E Fig. B.2. Absorption
rate
-2
Im M in the Lipkin model from Eq. (B.15).
absorption rate may become negative. We therefore conclude that expression (4.2) does not, in general, ensure a proper sign of Im M. References [1] N. Vinh Mau, Theory of nuclear structure: Trieste lectures 1969 (IAEA, Vienna, 1970) p. 931 [Z] N. Vinh Mau, Microscopic optical potentials, Lecture notes in Physics, vol. 89, ed. H.V.V. Geramb (Springer, Berlin, 1978) p. 40 [3] C. Mahaux, Microscopic optical potentials, Lecture notes in physics, vol. 89, ed. H.V.V. Geramb (Springer, Berlin, 1978) p. 1 [4] G.F. Bertsch, P.F. Bortingnon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287 [5] R.W. Hasse and P. Schuck, Nucl. Phys. A438 (1985) 157 [6] P. Danielewicz, Ann. of Phys. 197 (1990) 154 [7] A. Bouyssy, H. Ngo and N. Vinh Mau, Nucl. Phys. A371 (1981) 173 [8] F. Villars, Fundamentals in nuclear theory, ed. A. de-Shalit (IAEA, Vienna, 1967) p. 269 [9] S. Ethofer and P. Schuck, Z. Phys. 228 (1969) 264 [lo] L.P. Kadanoff and G. Baym, Quantum statistical mechanics (Benjamin, New York, 1962) [I 11 P. Schuck, F. Villars and P. Ring, Nucl. Phys. A208 11973) 302 [12] P. Danielewicz, Ann. of Phys. 152 (1984) 239 113) A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems ~McGraw-Hill. New York, 1971) [14] S. Adachi and P. Schuck, Nucl. Phys. A496 (1989) 485 [15] S. Shlomo and G. Bertsch, Nucl. Phys. A243 (1975) SO7 [16] K.F. Liu and N. Van Giai, Phys. Lett. B65 (1976) 23 [17] P. Schuck, Z. Phys. 241 (1971) 395 [18] F. Brieva and J.R. Rook, Nucl. Phys. A291 (1977) 29Y, 317 [19] C. Mahaux and H. Ngo, Nucl. Phys. A378 (1982) 205 [20] V. Bernard and N. Van Giai, Nucl. Phys. A327 (1979) 397 [21] G.F. Bertsch, P.F. Bortignon, R.A. Broglia and C.H. Dasso, Phys. Lett. B80 (19791 161 [22] F. Osterfeld, J. Wambach and V.A. Madsen, Phys. Rev. C81 (19811 179 [23] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin. 1980) [24] C. Mahaux, P.F. Bortignon, R.A. Broglia and C.H. Dasso, Phys. Reports 120 (1985) 1
Nuclear Physics A567 (1994) 78-96 North-Holland
Imaginary potentials
from many-body theory
P. Danielewicz National Superconducting Cyclotron LaboratoT and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
P. Schuck Institut des Sciences Nucleaires, 53, Avenue des Martyrs, 38026 Grenoble Cedex, France Received (Revised
10 August 1990 2 June 1993)
Abstract The imaginary parts of the single-nucleon, particle-hole (phonon) and (deuteron) mass operators are considered. Expressions of a Fermi golden rule for each case. Special emphasis is laid on many-body aspects and collectivity amplitudes. Implications for the calculation of optical potentials for nuclei are
particle-particle form are derived in the transition discussed.
1. Introduction Nuclear optical potentials are of continued theoretical and experimental interest. On the theoretical side, the calculation of in-medium cross sections and the inclusion of collective phonon excitations is of prime interest. Though there exist “standard”
many-body
techniques
for calculation
of optical
potentials
(mass opera-
tors) [l-5], there have been recently new developments reported [6]. In this work, we continue along these lines, and derive new formulas for the optical potential of elastic nucleon and deuteron scattering. We treat the particle-hole (phonon) self-energy in a consistent manner. We concentrate on the formulation of the imaginary parts of the optical potentials. The corresponding real parts are in principle determined via dispersion relations. It should always be possible to write the diagonal elements of the imaginary potentials in a golden rule form; their definite sign is then ensured. Expressions that cannot be arranged in such a form had been used when attempting to include the phonon excitations. Due to antisymmetrization, overcounting problems occur which are partially cured by subtraction procedures [1,2,4,7]. As a consequence the imaginary potential can at least in principle acquire a wrong sign (flux is produced rather than absorbed). In this work we show that the collectivity may be put into the vertex functions without 0375-9474/94/$07.00
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P. Danielewicz, P. Schuck / Imaginary potentials
79
losing generality. The proper sign of the imaginary potential is then guaranteed, as the diagonal elements are indeed of the golden rule form. In sect. 2 we first treat the nucleon-nucleus potential quite in detail. In sect. 3 we generalize the results to the particle-particle and particle-hole self-energies. Finally in sect. 4 we confront our new formalism with commonly used procedures and give a critical assessment.
2. Single-particle
potential
The nucleon-nucleus optical potential can be identified with the mass operator appearing in the Dyson equation for the single-particle Green function G = Go + G’MG,
(2.1)
where G;,”
and Go suppose will also Being
= -i(O
1T[ $,( t)+@(t’)] IO>
(2.2)
is the Hartree-Fock propagator. In the remainder of the paper we to work, without loss of generality, in the Hartree-Fock basis, which we sometimes call the shell-model basis. concerned with the imaginary part of the potential, we write [8,9]
Im M[,” = Im - i(0 I T[ jl( t)j!p( t’)]
IO)irr
= -+
(2.3)
with
(2.4) Explicitly the mass operator is
where u iZX4is the antisymmetrized
matrix element of the bare interaction,
From Eq. (2.5) we see that A4 is related to the irreducible function,
R 1231’2’3’
=
and
2plh (2hlp) Green
(2.7)
80
P. Danielewicz,
and we further
P. Schuck
/ Imaginary potentials
write
(2.8) where >I--f’ iR 1231’2’3 = co 1(~:IcI2~3),(~:‘~:‘~,‘)1,
(2.9a)
I’)iKY
and (2.9b)
Assuming a weak damping represented
of 2plh
(2hlp)
states,
the functions
(2.9) can be
as
>I--f’ = 2 C&,E:“,“,3E$53t e-iRL(t+r’), iR 1231,2,3,
(2.10a)
L
ePioLCf -“).
(2.10b)
L
Here BL are the wave functions and fiL are the energies of the correlated states. From general thermodynamic considerations [lo], FL = 0 for JJL < &r and NL = 0 for fiL > .5r, in the ground state of a system, where &r is Fermi energy. We can further assume that the states are normalized so that r\l, = 1 for 0, > &r and NL = 1 for 0, < or. Then Eqs. (2.9) and (2.10) yield the completeness relations
where n are occupations of the single-particle Upon introducing a dual system of functions g;-L * gL’ -123’123
and summing
states in the shell model, g such that
6LL’ =
(2.12)
9
Eqs. (2.11) side by side, we can obtain
E_1L;(n,Ti2ii3 + n,n,n,)E’,L;,= aLL’.
ii = 1 - n.
the normalization
condition (2.13)
P. Danielewicz, P. Schuck / Imaginary potentials
81
The three-body wave functions B and the energies of the states can be searched for by solving the three-body wave equation from ref. [ll] (see also ref. [6]),
- +(1 (flL+E1-&2 - E~)S:C~L~ - ( It1
-
EL It2 - n3)u232t3rw12f3r
n2)u21’12’s;2’3- (n1 - n3)u31r13r8~23’ = 0,
(2.14)
where E are the shell-model particle energies. On substituting Eqs. (2.10) into (2.8) we find for the diagonal elements of the imaginary potential in the energy representation Im M= -~~CI~,,,~~,I”(N,-N,)6(w-R,).
(2.15)
L
Some further insight into Eqs. (2.3), (2.6) and (2.15) may be gained by using the zero-temperature limit of finite-temperature theory to calculate the expectation values at the r.h.s. of (2.3) and (2.6). Two methods for calculating the expectation values exist. In appendix A we provide basic information on the method that yields chronological functions [such as in Eqs. (2.2), (2.5) and (2.711 in the amplitudes in the two-point expectation values. Details may be found e.g. in appendix C of ref. [12]. The other method 161utilizes retarded functions in the amplitudes, and it has some advantages, at finite temperatures and in nonequilibrium situations, over the first method, as will be indicated. Unless explicitly stated, we use the more common chronological functions. We find
where (2.17a) and (2.17b) The vertex function can be written in terms of R as
The time arguments of W are in general different from one another. The same holds for the right-hand time arguments of R. The dots at the r.h.s. of (2.16) indicate contributions from the 3p2h and 2h3p states that are products of singleparticle states, and such states with a still larger number of particle-hole pairs. The result of the form (2.16) is valid at finite temperature when the expectation values that determine the functions in (2.3), (2.5) and (2.17) are taken with respect
P. Danielewicz, P. Schuck / Imaginary potentials
82
to the equilibrium
density
operator
[6]. A complete
contribution
of the 2ph and
2hp states to the imaginary part of the mass operator is obtained, though, in that case, when the retarded rather than chronological amplitudes W are used [6]. The difference
vanishes
In the shell-model
at zero temperature. approximation
we have (2.19a)
and G<
II’
= in
1
6
11’
e-‘“,(‘~-‘;’
(2.19b)
>
where the ground-state occupation y1 is for a state below Fermi surface, and 0 above; E = 1 - n. On inserting (2.19) into (2.16), we find for the diagonal elements of imaginary potential in the energy representation, Im M = - $r
WI,,, I’( n,E,Z,
- E,n,n,)6(
w + 82 - Ed - .s4).
(2.20)
We should, of course, be aware of the fact that the shell-model states in (2.20) may themselves have a decay width and/or have a reduced strength (Z-factors less than 11. We want to ignore such refinements here and rather concentrate on the vertex function W. With the help of the Watson-Fadeev series expansion of the three-body propagator involved in (2.18) (refs. [9,11,6]) we can expand W in a series of pp and ph T-matrices. To lowest order W can be approximated by a pp T-matrix (a Brueckner T-matrix could be used, a Feynman-Galitski T-matrix, or any more sophisticated version thereof). Low-order terms of the expansion are shown in Fig. 1, and the second equality indicated in the figure follows from an integral equation satisfied by the T-matrix (the dot stands for cr&.
++2! +... TPh
=J?%+ *’ l
Fig. 1. Approximation
of the vertex function
W by a scattering
matrix
Tpp.
P. Danielewicz, P. Schuck / Imaginary potentials
Fig. 2. Approximation
to the vertex function
including
83
ph correlations.
A subject that we particularly want to dwell on in this work, is how to incorporate collectivity from low-lying collective states (including giant resonances> into the optical potential. For that purpose we add to the terms indicated in Fig. 1 the next terms in the series, see Fig. 2. The ph T-matrix satisfies an integral equation T1?& = Kf&
- iKP~~,,,G3,,3,G,,,,,Tph3”4”34
-
G&
that can be directly derived from the Bethe-Salpeter tor [ 13,141,
equation for the ph propaga-
IT = II” + III°KphIl,
(2.22)
where explicitly
n 121’2’
-i(O
=
I T(6p21&$1~)
IO>,
(2.23)
- (0 I e&b1 IO> and KPh is an irreducible kernel in the ph channel. With this we can now write the expression for the vertex function indicated in Fig. 2,
6p,, = $;$I
W1234(t
=
-
t’)
TIp2p348(
t -
t’)
+
TP2q3p4e-ia,(‘-t’)~12,3,2,,3,,( t - t’)K$‘!$,,,
+ T&Y,,, e--iE3(f--f )172f4f2fr4,,(t - t’) K$‘4,,2,.
(2.24)
P. Danielewicz, P. Schuck / Imaginary potentials
84
Here we assume that Tpp and KPh are energy-independent the relative times may be expressed with the &functions). nience tions
the time representation, may then
be written
since the forward-
concisely.
(the dependencies on We choose for conve-
and backward-going
Eq. (2.24) together
contribu-
with (2.20) is the basic
result of this work. Collectivity is accounted for via the density-density correlation function (2.23) which is available from extended RPA calculations [15,16]. For the based on Brueckner calcuinteractions T ** and K ph there exis t p arametrizations lations and corresponding Landau parameters, respectively. Skyrme-like forces might be used at low energies. It should then not be too difficult to evaluate (2.24) numerically. The definite sign of Im M and the correct second-order Born limit are guaranteed in this case (see discussion in sect. 4 on these points). We now try to give some connection of (2.20) and (2.24) with formulas more commonly in use when dealing with collectivity in the mass operator. We consider (2.24) in the energy representation and separate from IT possible low-lying isolated resonances in different channels,
= n;;;t2r +
fi,,,#2r)
(2.25)
where fi is the background. The amplitudes x” may e.g. be calculated from RPA theory. The resonance widths TV are supposed to come from the coupling to the continuum (escape width) as well as from coupling to more complicated states (spreading width). The resonances in (2.25) will lead to a pronounced peaking in the energy
dependence
of (2.24), whereas
I? will give rise to a renormalization
of
TPP.
= i=,pt4+ T,P23,411;:;s’2,,3t, K$‘3,,23 + T~?,4,n;e~,2,,4,,K~,~4,,24.
(2.26)
We now insert (2.26) into (2.20) and first discuss the case w > &_r. For narrow isolated resonances standing well above the background term T**, the direct product of resonances in 1W 12 will dominate in the resonance region over the interference terms resonance-background and between different resonances. Outside of the resonances the smooth part of the vertex function will be of impor-
85
tance. Following these considerations
XS(w+&*-Eg-Fq)
we write
+ .*. *
The dots in (2.27) indicate the interference matters when w = Ed = 0, and the quantity
(2.27) terms. A resonance
contribution
(2.28)
is recognized as the decay width of the state Y into the ph states. To the extent that these widths constitute most of the width of the states we finally get, including w < or that gives rise to an analogous expression,
(2.29)
This formula resembles the one in common use when dealing with collectivity [1,2]. It is, however, not equivalent, neither in spirit nor from a practical point of view. In (2.29) we sum only over a few resonances whereas in the usual formulation one sums, in principle, over a complete set of states. We also remark that (2.29), as (2.24), is of definite sign for either w > &r or o < &r (see sect. 4). It may be useful to generalize (2.29) to the finite-temperature case. The occupation numbers then become n = l/{exp[P(E - E.L)]+ 1) with p the inverse temperature. Starting from (2.20) and (2.24) one has to work, in the finite-temperature case, with retarded propagators instead of the chronoiogical ones we have been dealing with before. Also the quantity (2.27) does not represent the width of the state any longer, but instead the decay rate for fi, > 0, and the production rate
86
P. Danielewicz,
for 0, < 0. The exp( -pL?,)].
rates
P. Schuck
are related
/ Imaginary potentials
to the width
We then get for the imaginary
- ITl%X~; 12W>
with a factor
mV = sgn n,/[l
-
part of the mass operator:
L/257
(W-F4+t”)2++rv2
I+
. ..’
(2.30)
with NV = sgn R,/[exp(pR,,) - 11. Concluding this section we state that collectivity can either be put into the vertices (2.24) of the mass operator or, after certain approximations, more directly as intermediate states as in (2.29) and (2.30) and as is displayed in Fig. 3. The equivalence of (2.24) and (2.29) hinges on the ability of separating the spectrum of the response function L’ in a clear way into resonances and background. The detailed relation to the usual formulation will be given in sect. 4.
3. Particle-hole We now make
and particle-particle similar
cle-particle (deuteron) The imaginary part written as [14]
considerations
mass operators for the particle-hole
mass operators. of the particle-hole
mass operator
(phonon)
and parti-
in a state
v may be
Im Mph “(t - t’)
x ( cs’6’2’3’6*‘4’ Here
X is a wave function xL=
+(n,
-n,)i;;,
1’ ,‘5’7’4’~2’h’)X;:2’.
(3.1)
dual to ,Y, (3.2)
P. Danielewicz,
Fig. 3. Contribution
P. Schuck
87
/ Imaginary potentials
to the imaginary part of the optical potential
from the particle-hole
coupling.
where the upper sign refers to a,, > 0 and the lower one to 0, < 0, with the . . normahzatron set by i;;*x y’ = 1. When ignoring the damping of the 2p2h states, one can write
To every wave function rDJ there corresponds a wave function @‘, such that fiJ6 = a,, and @:;a4 = @<. For the imaginary part of Mph one then gets, in the energy representation,
The wave functions x might be searched for by solving a wave equation analogous to (2.14),
(3.5)
P. Danielewicz, P. Schuck / Imaginary potentials
88
The completeness
relation reads
and on introducing a system of dual wave functions @, the normalization can be written as nln2n3n4
condition
--
(3.7)
We can further expand diagrammatically the expectation value at the r.h.s. of (3.1) and apply a resummation as in the case of (2.6), to obtain
where the amplitude
with the 2p2h Green function
In the energy representation
we get the contribution
Im Mph ’ = - $r1&$34 I’( n,n,i?,Fi,
f
iyi2n3n4)fi(
from 2p2h states 0.J +
El
+
&2
-
83
-
F4),
(3.11) where, to lowest order in Tpp, l-%,4
=
%i*
( ci%
s52
-
T,p,P,3%4
-
G%4h
-
Tl%4W
(3.12)
*
The other important case is that of the deuteron ado-particle) mass operator MPP, see also ref. [17]. With Ik* being a two-particle wave function, the result analogous to (3.11) from retaining the transitions to 3ph states takes the form Im Mpp = Im P,:Mpp121’2’ 1v1’2’ = -
h! A
1234
I’(
---
n,n2n3n4
+
~,n2n3n4)S(w +
El
-
F2 -
E3 -
Eq). (3.13)
P. Danielewicz, P. Schuck / Imaginary potentials
89
Here the amplitude A is given by
G:,‘G-’ XD l,2,3,4,1,,2,,3,,4,,G,‘fG-’ 2”2 3 3 4”4?
(3.14)
where ?$ is a dual wave function, !P,2 = (1 - ~zi- n2>@,2, and iD 12341’2’3’4’ = (0 1T(
(cI:(cr2~3~41cr,lf~~f~~~~lf)
(3.15)
IO)irr.
It is important to note that into the vertices r, Eq. (3.12), and A, Eq. (3.14), enter the dual amplitudes with the result that integrations run over the whole rather than restricted phase space, even at zero temperature (see also ref. [14]).
4. Average mass operators of nuclei We now want to discuss in some detail the situation of nuclei. In nuclear physics essentially two different formulations are in use to calculate the imaginary part of the optical potential. The first one consists in calculating the expression (2.20) in infinite matter using the two-body scattering matrix for IV and going over to finite nuclei via the local density approximation [3,5,18]. The imaginary potential from the nuclear structure approach [1,2] takes the form Im M= --$rrCIU
12341y:4(2(n,NJ-ii2N,)G(w+F2-R,)
J
-
-atb234X;3~2(N,n,
+
--
%-I ul234
I*( n2n3n4
-
N,n,)6(
zi,n+,)a(
w
-
+
&2 -
0,
0”
-
E4)
&3 -
&4).
(4.1)
Here the sums run over all two-particle and particle-hole states and the last term at the r.h.s. is from subtraction of two second-order contributions to M that otherwise would be counted three times. When an effective interaction is used and the pp states are not collective, then the first term at the r.h.s. of (4.1) cancels one of the second-order contributions, leading to the result [2,7] Im M= -7rxIU
+
Y$rl
1234x:3
ul234
I’(
12(n2n3%4
xE4
-
-
Nvn4)S(
n2n3n4)6(
w
@
-
&
+
&2 -
-
‘4)
&3 -
&4)
3
(4.2)
P. Danielewicz, P. Schuck / Imaginary potentials
90
that is used in practice.
The second
half of the second-order difficulties
associated
and polarization
contributions
term at the r.h.s. of (4.2) still serves to cancel from
the first term.
There
with (4.1) and (4.2) that we want to discuss.
parts (cf. ref. 1191) of the imaginary
operator
of definite
sign, e.g. the correlation
operator
as representing
the particle
absorption.
part
potential should
are inherent
The correlation
should
each be an
be a positive
definite
This stems from the fact that the
imaginary parts of the single-particle Green function (2.21, i.e. the spectral functions, corresponding to poles either in the upper or lower half-planes, must be positive or negative definite operators, respectively. It follows then that the diagonal elements (in any basis) of the imaginary part of the mass operator must have a definite sign. Our expressions (2.20) with (2.24) [see also Eq. (2.2911 have this property since they are of the golden rule form. On the other hand this is obviously not the case with the quantities in Eqs. (4.1) and (4.2) (we explicitly demonstrate within two models in appendix B). This fact has apparently not caused problems in the practical calculations [4,20-221 up to now, but we find the situation uncomfortable. The fact that an expression such as (4.2) can give rise to problems has also been discussed in a slightly different context at the end of sect. 4.2.5 of ref. [23]. In short, we have shown in the preceeding sections the existence of an approximation scheme which allows the consideration of collective effects and at the same time ensures the correct sign and correct second-order Born limit of Im M.
5. Conclusion
In the paper we mainly addressed the very old question of how to put collectivity into the imaginary part of optical potentials. In our opinion this question was never really solved satisfactorily. We here give a novel prescription which assures that correlation and polarization parts have a definite sign, i.e. only flux absorption and no flux creation is possible for elastic scattering even in the presence of collectivity. Our formulation resides in the fact that we could show that all collectivity can be put into the vertices, and thus a Fermi’s golden rule form for the imaginary part of the optical potential can be applied under all circumstances. Our basic result is Eq. (2.24) which also holds at finite temperature and which has the correct second-order Born limit even in the presence of collectivity. It represents the first few terms when the vertex function is expanded in a Watson-Fadeev series. Eqs. (2.29) and (2.30) are approximations to (2.24) which preserve the definite sign property. They are valid when collective resonances are well separated from the background. The proposed formulation does not seem to be much more complicated than the ones in use and should be amenable to numerical application.
P. Danielewicz, P. Schuck / Imaginary potentials
91
P.S. expresses gratitude for the hospitality extended to him at the National Superconducting Cyclotron Laboratory during several visits. P.D. acknowledges partial support from the National Science Foundation under Grant Nos. PHY 8905933 and PHY 9017077.
Appendix A
We discuss here the computation of the expectation values such as at the r.h.s. of (2.3) and (2.6), or at the r.h.s. of (2.9). From (2.31, we have 2 Im
M:;“=
IO> -
(‘4.1)
For the first term at the r.h.s. of (A.11, we get
(0 I j?(f’)j,(t>
IO)
= (OlU'( -m,
t')j[J(t')U'(t', m)U’(m, t)j:(t)U(t, co)IO)
= (0 I [ U’(m, t’)
= (0 I
j:,( t’)U’( t’,
-w)]+U’(m, f)j:(t)U(t,
03)IO>
(T[j:,(t')exP( -il_dr’ vl(T’))]}+ (A4
When we expand the exponentials at the r.h.s. of (A.21 and apply a Wick decomposition, we obtain contractions of the interaction-picture operators both of which are associated with j, or with jt, or one with j and the other with jt. For both field operators associated with j, the contraction coincides with the function iG. For both field operators associated with jt, the contraction coincides with the complex conjugate function (iG)*. The expansion of the exponential for jt in (A.21 further gives also conjugate potential factors (-iu)*. When one field operator is associated with jt and the other with j, then the contraction coincides either with iG> or iG<, cf. (2.17). Disconnected diagrams merely give an overall factor 1 = (01 U’( - ~0,m)U’(q - ~1 IO). Resummations can be carried out in a standard manner, so as to eliminate the expectation values of interaction-picture operators in favor of the expectation values of Heisenberg-picture operators,
P. Danielewicz, P. Schuck / Imaginary potentials
94
We shall write down the RPA equations in the original (“spherical”) basis used in (B.7) and (B.8), and solve for the density-density correlation function which for this model
reads
(B.9) The RPA equation
then becomes
(B.lO) The second term in the square bracket is an exchange term. Often that term is neglected. Keeping the term leads to noncollective roots. The solution of (B.lO) can be written as Iz m1m2m3m4
-2E
1
cl-_??
_
+(E+
-E -17
(B.ll)
95
P. Danielewicz, P. Schuck / Imaginary potentials
6-
0.0
Cl=4 x = 0.86 7)/E=0.14
0.5
1.0
1.5
2.0
-
2.5
Fig. B.l. Absorption rate - 2 Im M in the schematic model from Eq. (B.5).
and the energies
E = e/l
of the noncollective
- (V/e)’
and collective
states
are
(B.12)
,
and
(B.13)
P= qnUsing (B.ll)
(B.14)
1).
we obtain
for Im M, putting
e = 1,
We see that this expression correctly reduces to the second-order Born approximation for I/ -+ 0, i.e. 6, E + 1. In Fig. B.2 we show the absorption rate - 2 Im M from (B.15) for different values of LZ, n and I/. We see again in this model that
96
0.0
0.5
1.0
1.5
2.0
2.5
W/E Fig. B.2. Absorption
rate
-2
Im M in the Lipkin model from Eq. (B.15).
absorption rate may become negative. We therefore conclude that expression (4.2) does not, in general, ensure a proper sign of Im M. References [1] N. Vinh Mau, Theory of nuclear structure: Trieste lectures 1969 (IAEA, Vienna, 1970) p. 931 [Z] N. Vinh Mau, Microscopic optical potentials, Lecture notes in Physics, vol. 89, ed. H.V.V. Geramb (Springer, Berlin, 1978) p. 40 [3] C. Mahaux, Microscopic optical potentials, Lecture notes in physics, vol. 89, ed. H.V.V. Geramb (Springer, Berlin, 1978) p. 1 [4] G.F. Bertsch, P.F. Bortingnon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287 [5] R.W. Hasse and P. Schuck, Nucl. Phys. A438 (1985) 157 [6] P. Danielewicz, Ann. of Phys. 197 (1990) 154 [7] A. Bouyssy, H. Ngo and N. Vinh Mau, Nucl. Phys. A371 (1981) 173 [8] F. Villars, Fundamentals in nuclear theory, ed. A. de-Shalit (IAEA, Vienna, 1967) p. 269 [9] S. Ethofer and P. Schuck, Z. Phys. 228 (1969) 264 [lo] L.P. Kadanoff and G. Baym, Quantum statistical mechanics (Benjamin, New York, 1962) [I 11 P. Schuck, F. Villars and P. Ring, Nucl. Phys. A208 11973) 302 [12] P. Danielewicz, Ann. of Phys. 152 (1984) 239 113) A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems ~McGraw-Hill. New York, 1971) [14] S. Adachi and P. Schuck, Nucl. Phys. A496 (1989) 485 [15] S. Shlomo and G. Bertsch, Nucl. Phys. A243 (1975) SO7 [16] K.F. Liu and N. Van Giai, Phys. Lett. B65 (1976) 23 [17] P. Schuck, Z. Phys. 241 (1971) 395 [18] F. Brieva and J.R. Rook, Nucl. Phys. A291 (1977) 29Y, 317 [19] C. Mahaux and H. Ngo, Nucl. Phys. A378 (1982) 205 [20] V. Bernard and N. Van Giai, Nucl. Phys. A327 (1979) 397 [21] G.F. Bertsch, P.F. Bortignon, R.A. Broglia and C.H. Dasso, Phys. Lett. B80 (19791 161 [22] F. Osterfeld, J. Wambach and V.A. Madsen, Phys. Rev. C81 (19811 179 [23] P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin. 1980) [24] C. Mahaux, P.F. Bortignon, R.A. Broglia and C.H. Dasso, Phys. Reports 120 (1985) 1