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Compound resonances in the three-cluster system
Nuclear Physics A393 (1983) 158-172 North-Holland Publishing Company
COMPOUND RESONANCES IN THE THREE-CLUSTER SYSTEM*'** MARIUSZ ORLOWSKI
Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305, USA and Institut fuer Theoretische Physik der Universitaet Tuebingen, Federal Republic of Germany Received 14 October 1981 (Revised 11 June 1982)
Abstract: A model for resonances in a system of three composite particles is formulated, where sudden decay into three clusters is determinated by the knowledge of the microscopical compound state of the metastable decaying nucleus. This model is an extension of the Wildermuth-Ben6hr reaction theory from a two-body channel situation to the full three-body problem. The mathematical formalism used in the model is based on the study of the asymptotic behavior of the three-body Green function.
1. Introduction Resonances in three-particle systems have been a topic of great interest during the last decade, in particular in elementary particle physics. Most investigations, however, are based on the assumption that the three interacting particles are point particles without internal structure. In the present paper, we want to consider resonances which occur in systems of three composite particles, or clusters. The new feature will be the appearance of c o m p o u n d resonances and the difficulty will be that the c o m p o u n d resonances will interfere with subsystem resonances or resonance-like structures such as the final state peak. For two clusters, the theory of resonances is well-established in the context of the resonating group model 1.2). It has been shown that c o m p o u n d states give rise to resonances of Breit-Wigner type, in the single-channel case. Also, the coupled channel case has been investigated and the influence of a second open channel on the resonance behaviour in the first channel has been studied. W e shall try in this p a p e r to extend the theory of resonances from two to three clusters. T h e mathematical difficulty will be that, instead of the two-body G r e e n function, we will need the three-body G r e e n function. This means that we have to assume that a Faddeev equation has already been solved for the motion of the three clusters without the presence of a c o m p o u n d state. * Work supported in part by the Department of Energy under contract number DE-AC0376SF00515. ** Work supported in part by a grant from the Friedrich Naumann Foundation. 158
M.C.L. Orlowski / Compound resonances
159
In terms of physics, the difficulty is this: since there is a spectator cluster which can carry an arbitrary amount of energy, a subsystem resonance or a final-state p e a k will not only appear in a narrow energy range of the three-body system. It will appear at all energies above a certain minimum which corresponds to zero spectator energy. The c o m p o u n d resonance will thus never be an isolated resonance but will always find something to interfere with. This will be seen m o r e clearly from the relevant mathematical expressions. In sect. 2 a model of the three-cluster resonance is formulated in analogy to the work of Wildermuth and Ben6hr in the two-cluster case. A technique for solving this model is proposed in sect. 3. This technique is based on the asymptotic formulae of the full three-cluster resolvent. The corresponding formulae are discussed in the appendix. In sect. 4 the results are investigated and Breit-Wigner-like resonance formulae for the three-cluster case are discussed.
2. The model We consider a three-cluster system described by the following Schr6dinger equation: (h0+ V(1, 2 ) + V(1, 3 ) + V(2, 3 ) - E ) X = 0 ,
(2.1)
where h0 denotes the kinetic energy operator and V(i, j) are effective intercluster interactions. In addition to structures described by eq. (2.1) we like to consider an isolated three-cluster compound resonance. In order to describe such a c o m p o u n d resonance it is convenient to modify the structure of the ansatz for the solution of eq. (2.1) by introducing a single c o m p o u n d state F in the following ansatz:
X'(~, ~1) = X (6, ~1) + aF(~, ~i).
(2.2)
Here, ~ and ~1 are Jacobi coordinates and F is a three-cluster c o m p o u n d state which gives rise to a resonance. To be more specific, let us assume that F is a Feshbach projection 3) of a microscopic compound state [ which has been obtained by diagonalising a corresponding microscopic hamiltonian H of the system in a basis of Slater determinants. The Feshbach projection can be written as: F(~', lq') = I dt3A-lf(rl . . . . .
rA-1)~01q~2q~38(~--~')t~('tl --~i'),
(2.3)
where the ¢i are the internal cluster states corresponding to the microscopic interactions contained in H, and A is the number of all nucleons involved. A m o r e detailed discussion of the correlation between this model and the microscopic theory will be given in sect. 4. Note that eq. (2.1) can be written as a projection equation
(@lh - E I X ) -- 0 ,
(2.4)
where 6X denotes an arbitrary variation of X, and h = h0 +Y.j>i V(i, j). In ansatz
160
M.C.L. Orlowski / Compound resonances
(2.2) we assume the function F, whose expectation value exceeds the three-cluster breakup threshold but is still below the next breakup threshold, has only a small overlap with the first term on the right-hand side of (2.2). Using ansatz (2.2) and eq. (2.4) we get a set of equations:
(Sxlh - Elx ) + a (Sxlh - E L F ) = O, ( F l h - E I x ) + a (FIh -ELF) = 0 .
(2.5)
Formal elimination of the second equation leads to
(@lh _ E l x )
(@]h - E l F ) ( F l h - E l x ) (F]h - E L F )
- O.
(2.6)
While the first term of the last equation reproduces eq. (2.1) the second term introduces an additional separable potential Vc with a resonance denominator, 1
v c = ]P) E - (F]h ]F) (p]'
(2.7)
with IF)= (h -E)IF). The full effective potential of the three-cluster system can be written as: VCf61, 2, 3 ) = V(1, 2 ) + V(1, 3)+ V(2, 3)+ Vc.
(2.8)
In the next section we propose a method of solving the Schrbdinger equation (h0+ Veff(1, 2, 3))X = E x
(2.9)
with specific boundary condition for X and of investigating the scattering amplitudes in different channels.
3. Solution of the m o d e l
The aim is to obtain the solution
(ho + Ve,)X = ErX ,
(3.1)
with the boundary condition for bound-state scattering, i.e. for the scattering of a free particle 1 by the pair (1) = (2, 3), initially in a bound state of energy E l , . Here, Veff is given by eq. (2.8), ho represents the kinetic energy operator: ho = h2(A~ +An), and Er is the relative energy of the three-cluster system. [We work in the center-ofmass system of the three clusters and use Newton's notation of coordinates and momenta 4).] We define the following operators: h = h0+ Veff, g ( z ) = ( Z - h ) -1 ,
go(z) = (z - ho) -1 ,
/~ = h - Vc, i f ( z ) = (z - h )
-1 '
(3.2)
M.C.L. Orlowski / Compound resonances
161
with z = Er + ie. It holds that
g(z ) = ~(z ) + ~,(z ) Vcg(z ) .
(3.3)
It is assumed that the bound-state scattering problem, i.e. the equation lim ie~,(Er + i e ) ~ l = X~+) e~O
(3.4)
is already solved. Here, ~1 denotes the product of the plane wave tbl and the bound-state function of clusters 2 and 3, ~oln, where n denotes the nth bound state of clusters 2 and 3. This problem can be solved using some of the many techniques for solving the Faddeev equations. Using the operator identity (3.3) one obtains the Lippmann-Schwinger equation
limieg(Er+ie)~l
=
e-~0
X~+)= )~(1+)+g(Er +"l O +') V e ~ l (+} •
(3.5)
The last equation has a unique solution, since Vc has a finite Hilbert-Schmidt norm. Due to the separability of V¢ the solution X~÷) can be obtained immediately after a simple algebraic calculation. The result is
X ~+) = X~x+) + ~,(Er + iO +) (FIh IF) - E r -
(3.6)
In order to evaluate eq. (3.6) one needs the asymptotic formulae of the resolvent ~,+(Er)=~,(E~+iO +) in all channels. Such formulae, though not present in the literature, are well known. The derivation of these formulae are given in the appendix. The results are: -ilqinl bqil 1 ~e "~'(-)" ' E R'" ~,+(Er;R,R') I~,l-~ ' - 4 r r ( 2 ~ ) 3 / = L - - i ~ n t ~ ) x ~ " tq~,, ~n; ), (3.7) n I~1il IR'I.I~1<~ where q~, =
.q°lq,,,I = ( E r - E i , ) l / 2 ) , g (Er, R , R ' )
i = 1, 2, 3, for the two-body channels, and i~/4r.:.,3/4 elIKIIRI IRI-~ ' 2(2"rr)5/EIR[5/2 ;~°-'*(IKIR°'R') (3.8) IRq
for the breakup channel, where ~0-)* is the three-particle scattering function for three free incident particles. Here, the six-dimensional coordinates (~, "q~), i = 1, 2, 3, are denoted by a vector R, and the corresponding six-dimensional conjugate m o m e n t u m (p~, q~), i = 1, 2, 3, by a vector K. Then, the following formulae are valid 4):
IRI == I ,1 =+ln,I =, = IKI2=
Ip, l= + Iq, I= ,
K "R = ~i "Pl +'qi "qi,
(dR) = (d~,)(d'q~),
i = 1, 2, 3 .
M.C.L. Orlowski / Compound resonances
162
Using the formulae (3.7) and (3.8) and the known structure of the asymptotic form of the three-particle scattering function ~÷~ <5), one obtains all scattering amplitudes of the solution X~÷) of the eq. (3.6). One obtains (a) for the breakup channel, eil~l IRI I~l~oo > ~-7"~[ Rm(K'; ql, E1)+Rol. . . (K . ;qx, E~)] ,
X~+)(ql'E1;R)
(3.9)
(b) for the elastic channel,
x~÷~(q~, E~; R )
[,rll[~OO
) eiql"~ltPln(E1, ~1)
1~211
rn1(2/~1) w2 ["q1[
n
+ Rrly (q'l., El,,; ql, El)],
(3.10)
(c) for the rearrangement channels j # 1,
X~+)(ql'E1;R)
Y~ e'b'"Ft~'lq~j,(~i)[Ril(q ~n, E 1, ; qi, Ei) t~' t-'°~ > rnj(2/£i)~/= [~il . lejl
(3.11)
Here R o are the usual Faddeev scattering amplitudes in the absence of the threecluster potential 17"cand are given by
Ril(q'l., Exn ; qi, Ei)
; 1/2
= -16~.mitz3/2
I dR
e-iq'l"mqp 1".(SOl)i~lV(i)(~i)~l+)(qi, E i ; R ) ,
(3.12a)
l = 1, 2, 3, and
e' /4E r/4 f
g o l ( K ' ; ql, El) ~ 2 ( ~ ) 5
2 j dRtl~(1):~:(K ', R)i~1_ V(i)(~i)X(l+)(qx' El; R ) , (3.12b)
in which ~(1-) (K, R ) = e ~l''h& (1-) (pl, 61) and.& ~-)(~1) is a scattering state in subsystem (1).
The potentials V (i ) are defined by V ( i ) = V (f, k ), ( i, j, k) cyclic. The three-cluster resonance scattering amplitudes R ~ ~, i = 0, 1, 2, 3, which arise from the potential Vc can be found as
o3 >lp)(p[;,l+,> (PIg+(EOIR)'
R ~1" = - 16zrmilx 3/2 (F[h IF) - E ~ -
(3.13a)
M.C.L. Orlowski / Compound resonances
163
i = 1, 2, 3 and
ei•/4rr3/4 R~'f - 2(2zr)5/2
(FlhlF)_E,_(p[g+(Er)lP).
(3.13b)
Then the corresponding cross sections are given by (a) for the two-body channels i = 1, 2, 3:
d~
Iql°l m 1/.t- "i: --
dO
[ql[ mild, 1
-- res 2 -----: 1-7~-IRma+ R a I ,
(3.14a)
(b) for the breakup channel: dtr 41[q~]3 [Ro~ res2 , +R01 I (dq,) -- ,~5/2 2 3/2 dO z m l/xx IqlllK:
(3.14b)
It can be observed already here [formulae (3.14)] that the interference effects between the structures contained in Rii such as the final-state interaction and quasi-free scattering and R ~]~s are present in all channels. On the other side the decay of the three-cluster resonance itself is influenced by the existing structures produced by the two-body potentials [formulae (3.13)]. Thus formulae (3.12)-(3.14) give a qualitative insight into the variety of energy correlations in the case of a sudden decay of a three-cluster system. In the next section the scattering amplitudes R[~ ~,/" = 0, 1, 2, 3, will be studied in more detail.
4. Breit-Wigner resonance formulae for a three-cluster compound resonance
In this section the structure of the scattering amplitudes R[~ s, i = 0, 1, 2, 3, will be discussed. The terms of interest are:
07~°-)IP)(PI~?I+2)
(4.1a)
(F[h IF) - E r - (FIff+(Er)IP) '
(£12 IP)(FI£~+2)
(Flh IF) -Er- <#lff+(Er)IP> "
(4.1b)
The resolvent ff+(Er) can be represented in the following spectral decomposition:
ff+(Er) ~ ~
+jj
=~vEr-Evb?V) I ~<£°
fJ d-qiEr_Ei.O?f3I_lqJl )O?f3:+io+
-(+)~
~(+71
,XO )¢,Xo , I)~,,)O~K[ dqk apk E 2 Z+io--=S , r-- qkl --IPk[ ~ Er-E~ +i0 + _
(4.2)
where k = 1, 2, 3 is arbitrary. Here ~v is the three-body bound state, ~I~+) is a bound-state scattering function with initially bound state in subsystem (i), and ~(0+) is a scattering function for three free incident particles. We define a complex
M.C.L. Orlowski/ Compoundresonances
164 resonance shift by
A +(Er) = S ~I)~K)()~ IF) E~-E~ + i 0 ÷ "
(4.3)
Further, by using the identity
1
1
-.o EK -- Er - te CPVE~ Er+Drg(E,, - E r )
lim
(4.4)
. "=
one can separate the non-hermitian operator g+(Er) into hermitian and antihermitian parts:
g+(Er) = gh(Er) + ga'h'(Er) with
1~o)(~ol Xin "(+)")~(in "(+) ffh(Er)=~v Er-Ev "[-~ I CeWEr'q-lqi[2-Ein Er-lq, 12-lpil 2' (4.5)
~a'h'(Er) = ~
I
"(+) )(x,. ~(+) la(Er-E,.-Iq, I=) dqi Ix,.
;
÷ f f [;~(o+')~(o÷)lS(Er--[qkl2--lpk[E)dpkdqk. Then defining the following quantities:
(±)
]/in (Er)
= (FI)~l~)) ,
i = 1, 2, 3 ,
T(o±) (Er) = (Pl;(g)). F ,.±) (Er)--2~rl~.~)(Er)l
(4.6)
2
j--0. 1.2,3.
we can rewrite the formula (4.3) for the complex resonance shift as
A+(Er)=flc(Er)+il(~Fl+,)(Er-Ei,,)+f F(o+)(Er-lqk[zdqk),
(4.7)
where A c(Er) (Plff" (Er)lP). T h e magnitudes 3/(Er) describe the probability amplitude of the resonance clusterstate ff in the different scattering states at the same energy. The variety of the scattering states results from different initial conditions. Since ff is a square integrable function, the formulae (4.6) show that the resonance w i d t h s / ' are strongly influenced by the off-shell behaviour of the scattering states )~I~) and )~(0±). W e discuss the influence of the off-shell behavior on the resonance widths in terms of the Pauli principle and saturation properties of the nuclear force at the end of this section. =
165
M.C.L. Orlowski / Compound resonances
Due to the square integrability of P the resonance widths F are also very sensitive to subsystem resonances, since in that case the amplitude of this subsystem is large at small distances. Therefore the partial widths F are good candidates for probing the off-shell behavior of the three-cluster wave functions. With the definitions (4.6) and (4.7) we can transform (4.1a) and (4.1b) into
0?l-' IF)(PI~?<,+>) (F[h IF) - E r - (PIg(+)(G)IP) y~o-)*(Er)y ~+n)(Er)
E c - E r - Ac(Er)+ ~i(Yun FI +) (Er-
(4.8a)
Ein)+ f F(o+,(Er-1~12) d~)
and
O? IP) (PI '17 )
(Flh IF)- E,(PI~,+(E,)IP) •
=--i
(-),
lyj~
(+) (Er)yln (Er)
E ~ - Er-A~(E~) +½i ( x,orl;' ( E , - E , , ) +
I/'<°+'(E'-lq[2 dq) '
(4.8b)
respectively, where E¢ = (Flh IF). Obviously, these formulae reveal the Breit-Wigner structure. The resonance energy pole E~es-the pole at which the real part of the denominator in (4.8) becomes zero - is defined by E c - E r e s - "4c(Eres) = O .
The formulae (4.8) exhibit that even in the case of the three-body breakup channel the Breit-Wigner resonance formula is still valid. Moreover it allows one to study the rich energy correlations of the decay products. The only difference to this formula is that Ac(Er) and F(Er) are energy dependent. To eliminate this difference it is necessary to make certain approximations. We have to use the fact that we are considering an isolated, relatively sharp resonance. For such a resonance level all other resonances lie energetically so far away that F(Er) can be considered to be approximately energy-independent over the energy width of this level. Then we make the approximation that, in the neighbourhood of E .... ./d~c(Er)'~ Ac(Er) ~- ,4c(Eres) + (Er -- Eres) I T )
r
Er=Er¢s
'
F(E,) ~r(Er~). Then one arrives at the same formulae as (4.8) where Ac(Er) is replaced by '4c(Eres)
166
M.C.L Orlowski / Compound resonances
and the partial widths by: F(+)(E~)
F're+s)(Er¢~) = 1 + ~E~ Ac(Er)
~ Er=Eres
'
~//'tl
(Er) (Eres) = [1 + ~E~ A~(Er)]I/2 Tin
Er=Ere
s "
The main difference in the structure of formulae (4.8) compared with the resonance formulae for the two-body channel situation 6) is the appearance of an integral over resonance widths F~0+)(Er-lql2). This reflects the fact that the three-body break-up can be considered as a continuum of infinitely many two-body channels. Each distribution of the energy over the three bodies corresponds to one two-body channel. However, the interpretation of the resonance widths F according to the picture of a two-body channel situation is no longer possible. This can be seen in the following way: though )~0+) is a scattering function for three free incident particles, it apparently also contains contributions of all two-body channels. This aspect is lacking in the Breit-Wigner formula in the two-body channel situation and is obviously characteristic for (N~>3)-cluster systems. The analogy to the two-channel case is not possible because F(~-) cannot be interpreted as the transition amplitude from the resonance state ff whether to the breakup channel nor to some of the two-body channels; all these transition amplitudes are contained in Ft0+), but they altogether determine, nevertheless globally, the partial resonance width of the breakup channel. Therefore even in the case of a two-body channel no straightforward parametrization by phase shifts of the resonance amplitude can be given. Analogous considerations can be made for F ~ ). They show that vice-versa the transition amplitudes from the compound state P to the breakup channel influences the resonance width of a two-body channel. It should be pointed out that this many-cluster effect is qualitatively different from the mutual influence of the partial widths of the two-body coupled channels. The latter effect depends on the ratio of the partial width to the total resonance width. The origin of the new effect is a dynamical one. It leads to some interesting and important consequences. Let us assume that in addition to the three-cluster resonance there is at a lower energy a subsystem resonance in the subsystem (i), consisting of clusters ] and k. One considers now the three-cluster resonance behavior in the two-body channel, where clusters ] and k form just a bound state. In this case the three-cluster resonance width of this channel is strongly influenced by the resonance of the same subsystem (i), though in this channel just this subsystem is bound at each energy. Another difference is that the formulae given by Schranner 6) for two-body channels were derived under the strong approximation, namely that the direct coupling between the open two-body channels can be neglected and that the transition occurs through the formation of the resonance state. Such an approximation is valid only in specific situations, if for instance the resonance occurs in an energy region where all or all but one channels are shielded from the compound
M.C.L. Orlowski / Compound resonances
167
region by high potential barriers. The derivations presented here restricted to the two-body channel reactions are exact and cover all cases. The influence of the three-cluster breakup structures such as quasi-free scattering, final-state interaction, or two-body channel resonances on the three-cluster resonance is twofold. First, such structures strongly influence the three-cluster resonance widths F ( E ) , second, they interfere with the three-cluster resonance amplitude R ~s. The kinematic aspects of such interferences are extensively studied in ref. 7). In a contribution to come specific numerical examples will be presented. Finally we discuss to what extent our model is compatible with more microscopic theories, like for example the very successful resonating group method. This microscopic method was recently extended to the case of three - and more - cluster systems 8-10). It is known that microscopic models yield energy-dependent interactions and strong multi-cluster forces, which originate from the Pauli principle. In ref. 9) however, it has been shown that in the three-cluster system an appropriate off-shell transformation (i) removes the energy dependence of the interactions, (ii) changes the off-shell behavior of the two-cluster interactions while their on-shell properties remain unchanged, and (iii) it also reduces considerably the strength of the three-cluster potential. The crucial question is whether one is able to construct an appropriate off-shell behavior of the on-shell equivalent two-cluster forces, which would absorb the effect of the three-body force. This was studied in ref. 11) on specific examples. The numerical results in the case of a - ~ - a and A-o~-a binding problems have shown that the off-shell transformed three-body force is negligible. This is a very gratifying result if we think of the many phenomenological calculations on three-cluster systems. In most of these calculations, people have used - and with success - energyindependent two-body potentials and no three-body potential. In view of these results it is justified to say that our model, which is based on pairwise cluster-interactions, does not contradict microscopic models, provided that the two-cluster interactions have a proper off-shell behavior. The off-shell behavior should contain the Pauli principle, the saturation property of nuclear forces and the guarantee that no strong three-body potential is needed to describe the system correctly. It has been shown 12) that the off-shell behavior of two-body forces influences strongly the behavior of the scattering states XI±~ and Xto~ at short distances. This influence is reflected by the resonance widths F, as already mentioned. 5. Conclusion
A formal theory has been developed for the decay of a nuclear compound state into three clusters. The formalism shows clearly the interference of the sequential decay modes with the sudden decay mode. The differential cross section has a complicated structure, even when the compound resonance is sharp and isolated.
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M.C.L. Orlowski / Compound resonances
As compared with the standard Breit-Wigner formalism, the three-cluster decay of a compound state corresponds to a case with infinitely many two-body channels. Correspondingly the total decay width is a sum plus an integral over the partial widths. New effects of the mutual influence of the three-cluster resonance with subsystem resonances and other resonance-like structures have been discussed. For a practical application of the theory one has to solve the three-cluster Faddeev equation in absence of the three-cluster compound resonance potential and one has to calculate matrix elements of this potential with the obtained solutions. The evaluation of the resonance part of the three-cluster scattering amplitude is sensitive to the off-shell behavior of the three-cluster system. The author is pleased to acknowledge valuable suggestions and fruitful discussions with Prof. H. Pierre Noyes, Prof. R.G. Newton, Prof. S. Saito and especially with Prof. Erich W. Schmid.
Appendix THE ASYMPTOTIC FORMULAE OF THE FULL THREE-BODY GREEN FUNCTION IN COORDINATE RESPRESENTATION AS one knows from refs 13-18), the asymptotic'form of the three-body wave function depends on the direction in the six-dimensional coordinate space. The same, of course, must be true of the Green function. One can, therefore, expect that one should be able to use the various forms of the resolvent equation for the full resolvent ff in terms of the channel resolvent ffl, i = 0, 1, 2, 3, in order to find its asymptotic form in various directions. One has
~, = gi + (,,Q(i)~, ,
(A. 1)
where 17"(i)= (V(/) + V ( k ) ) , i, L k cyclic, and 9 ( 0 ) = Y.~=I V(i). We are considering a situation with pairwise interactions only. However in eq. (A.1) a term of the following form appears:
f dR"~, + (R, R") V(Z)(/~)~+(R ", R ' ) .
(A.2)
There is a convergence problem with alg integration. For this reason (A.1) is not the proper starting point for going to large IR I. One can expect that one has to start with Faddeev-like equations for the Green function ~. In order to obtain a Faddeev-like splitting of a resolvent identity, one considers eq. (A.1) for i = 0, 1, 2, 3 and obtains after an easy calculation 3
= Z ~i[M~+ V(i)(,oV(i)~,], i=1
(A.3)
M.C.L. Orlowski / Compound resonances
169
in which {V(i)~0 Mi =
"~x
i#h
i =A
Here A can be chosen arbitrarily, A = 1, 2, 3. One can show that the formula (A.3) corrresponds to the Weinberg decomposition of the three-particle wave function 19), and the choice of A corresponds to the determination of the entrance channel. The arbitrariness of A reflects the fact, that in the space of eigenstates of g~-~ the operator V(i)go is identical with identity Ui acting in this subspace. An equivalent formulation of (A.3) is 3
g = go+go ~ (V(i)~,~ + V(i)~,~V(i)~,).
(A.4)
i=1
We will show later that in eqs. (A,3) and (A.4) the disconnected parts in the kernel are avoided. In order to derive the asymptotic formulae for the full three-body Green function one needs corresponding asymptotic formulae for the channel Green functions gi. Such formulae have already been derived by Newton 4), who pointed out that gi can be split into two parts: ~- (E) = ff~c(E) + ff~b (E), where ~C(E) is responsible only for the asymptotic formulae of /~-(E) in the breakup channel and ff]-b(E) determines uniquely the asymptotic behaviour of g]-(E) in the two-body channel i: lim ~,~-(E;R,R')= IRl-,oo lim ~,.+,~(E;R,R')
IRl-,oo
Ia'l
IR'l
eiW/4E3/4 eilX[ IRI
= 2(2~.)s/21RIS/2 ~I -)*(K', R').
(A.5)
lim g~f(E;R,R')= lim ~,~b(E;R,R') Ind-,oo IR'l.14i~l
_ilqmIIn,I 1 t;; :g = 4,r(2tzl)3/2 ~ ~---~il ~o,,,(~,)q~, (q,,,, E,,~; R'), (A.6) where ql. = n°lq,.I. Iq,.I
= (E-E,.)
1/= .
E,.; R') -
The derivation of the asymptotic formula for the two-body channel is trivial. In this case one can start directly from the identity (A.1) i # 0. There is no trouble in the integration in the coordinates ~ and ~1~, since in the two body channel the
170
M.C.L. Orlowski / Compound resonances
leading contribution in the Green function comes only from g~+b (E). Thus the range of ~[ is limited by the bound-state function ¢i, and the range of ~ is limited by the potential V(/),/" ~ i. If the ranges of ~[ and of ~; ] ~ i are limited then the same holds true for R ' = (~, ~ ) . Starting from (A.1) and using (A.6) one obtains lira g + ( E ; R , R ' ) = - A ~,1 ~3/2Y. c~ g i , , In,I--,oo "t "n"( ~zt-ti ) n " IR'l,[~l
"(-)* .t ~. . . ) ,
(A.7)
where lim i e g ( E - ie ) ~ , = ~ i , + g - ~ ' , ) ~ , , -- ~I~ ) • ~"~0
In the case of the breakup channel the use of eq. (A.1) is no longer justified, since the corresponding kernels do not fulfil the requirement of the connectivity. In that case it is suitable to start from the identity (A.3) or (A.4). Studying eq. (A 3) one finds that the terms V(i)g0 cause no trouble because they represent compact two-body kernels embedded in the three-body space and the corresponding 6function can be factored out. But we have to ensure that ( R [ V ( i ) g o V ( / ) , i ~ i , decreases asymptotically sufficiently rapidly. For this purpose it is sufficient to show that V ( i ) g o ( z ) V ( / ) g o ( Z ) is square integrable. By taking the expectation value between m o m e n t u m states for (qkPk[ V ( k ) ] q k P k ) and the expectation value of the resolvent g0(z) we may work out explicitly that [ [ V ( i ) g o ( z ) V ( / ) g o ( z ) ] [ z is finite for z not on the real axis. Rubin, Sugar and Tiktopoulos 2o) have shown using Fredholm theory in the dispersion studies of the three-body amplitude that the square of Faddeev's kernel 21) is Fredholm even on the real axis. This strong result should be contrasted with that of Faddeev who was only able to show that the fifth power of the kernel was compact. The reason Faddeev could not obtain the stronger result was that he assumed only that the potentials involved satisfied smoothness conditions whereas here we will assume that the potential is analytic in its m o m e n t u m variables. Then the sufficient conditions for a Hilbert-Schmidt operator represented by a kernel K (x, y ; z) are (i) For each pair (x, y), K ( x , y; z) must be analytic for z ~D. (ii) There exists a uniform bound B
where ~o o-(/z) dtz is required to be of bounded variation.
M.C.L. Orlowski / Compound resonances
171
Thus we are allowed to use the formula (A.4) in order to derive the asymptotic formula of the resolvent go in the breakup channel. T o this purpose we need the corresponding formula for the free resolvent go. It holds ei~r/4E3/4 e l I K I R o - R '
lim go(E; R, R') I~l-,oo IR'l
=
e -ii/¢lR°'R'
2(2¢r)S/2[R{5/2
.
(A.8)
This formula is obtained using methods already applied by Newton 4). One obtains lim IRl~oo g ( E ; R , R ' ) I~'1
e
i-,r1,~,.-,314 i[KIRO.R' 1~ e !
=
2(2~r)5/2[Ri5/2
,
]
~"
[40 (K',R')+i=I (4oV(i)gf/(i)g)*iR')j, (A.9)
where 4 o ( K ' , R') = e ilrlR°'R', K ' = IKIR °. Considering that lim~_,o iegi (E - i e )40 = 41 -) one obtains 3
3
3
Y, (4oV(i)g~ + 4 o V ( i ) g ~ v ( i ) g ) = ~ (4~ -) + 4 1 - ) Q ( i ) g ) - ~ [ 4 0 + 4olP(i)~] i=1
i=1
i=1
and with this result finally lim g(E,s,R')= IRI-*~ Ia'l
e
irr/4~3/4 /:~
e
ilIKIIRI . 3 /
~ ( - ) * i rJ-t
i~ X,
t a , g ' ) - 2~(o-)* (K', R')
)
,
(AA0) in which ~I -) = lim ieg(E - ie)41 -) ,
(A. 11)
~(o-) = lim ieg(E - i e ) 4 o .
(A.12)
E~0
e-)0
One can easily show that the ~I -) as defined in (A.11) and (A.12) are identical for i = 0, 1,2, 3. Therefore one obtains eirr/gg3/4 e i l K I [RI
lim g(E" R, R') =
IRI-o~
IRq
'
2(2rr)S/21RI 5/2
;~(o-~* (K', R ' )
"
(A.13)
Finally, we mention another useful asymptotic formula for the breakup channel formulated not in terms of scattering functions but in terms of the scattering amplitude. This shows that the scattering amplitude Roo without the compound state ff is already contained in the resonance scattering amplitude R r~o~.It is known that the scattering amplitude Roo is obtained by quadrature from the scattering
M.C.L. Orlowski/ Compoundresonances
172
a m p l i t u d e s Roi, i = 1, 2, 3. T h e result is lira ~,(E;R,R') iRl~oo iR,l
ei~/4E3/4 eil~l lRl [ e-iK"w + (2 7r)3 I R~oo)(K,, K,,)(K,,,~oIR,) dK .] = 2(2rr)s/21RlS/2 (A.14) in which R~0o) d e n o t e s the f r e e - f r e e t r a n s i t i o n a m p l i t u d e in a t h r e e - b o d y system.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
H.C. Benoehr and K. Wildermuth, Nucl. Phys. A218 (1969) 1 K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977) H. Feshbach, Ann. of Phys. 19 (1962) 287 R.G. Newton, Ann. of Phys. 34 (1972) 324 S.P. Merkuriev, C. Gignoux and A. Laverne, Ann. of Phys. 99 (1976) 30 R. Schranner, Diploma Thesis, unpublished, University of Tuebingen M. Orlowski, Thesis, unpublished, University of Tuebingen E.W. Schmid, Phys. Rev. C21 (1980) 691 E.W. Schmid, Z. Phys. A302 (1981) 311 H. Horiuchi, Prog. Theor. Phys. 51 (1974) 1266; 53 (1975) 447; Prog. Theor. Phys. Suppl. 62 (1977) 90 E.W. Schmid, M. Orlowski and C.G. Bao, Importance of the three-body Pauli potential in three-cluster systems, Z. Phys. A (1982), to be published H. Leeb and E.W. Schmid, Talk given at the Liblice Conference, June 1981, Czechoslovakia W. Gloeckle, Z. Phys. A271 (1974) 31 C. Gignoux, A. Laverne and S.P. Merkuriev, Phys. Rev. Lett. 33 (1974) 1350 R.G. Newton, Ann. of Phys. 78 (1973) 561 A. Laverne and C. Gignoux, Nucl. Phys. A203 (1973) 597 J. Nuttall, Phys. Rev. Lett. 19 (1967) 473 J. Nutall and J.G. Webb, Phys. Rev. 178 (1968) 2226 S. Weinberg, Phys. Rev, 131 (1963) 440 M. Rubin, R. Sugar and G. Tiktopoulos, Phys. Rev. 159 (1967) 1348 L.D. Faddeev, Mathematical aspects of the three-body problem in quantum scatterings (Davey, New York, 1965)
COMPOUND RESONANCES IN THE THREE-CLUSTER SYSTEM*'** MARIUSZ ORLOWSKI
Stanford Linear Accelerator Center, Stanford University, Stanford, California 94305, USA and Institut fuer Theoretische Physik der Universitaet Tuebingen, Federal Republic of Germany Received 14 October 1981 (Revised 11 June 1982)
Abstract: A model for resonances in a system of three composite particles is formulated, where sudden decay into three clusters is determinated by the knowledge of the microscopical compound state of the metastable decaying nucleus. This model is an extension of the Wildermuth-Ben6hr reaction theory from a two-body channel situation to the full three-body problem. The mathematical formalism used in the model is based on the study of the asymptotic behavior of the three-body Green function.
1. Introduction Resonances in three-particle systems have been a topic of great interest during the last decade, in particular in elementary particle physics. Most investigations, however, are based on the assumption that the three interacting particles are point particles without internal structure. In the present paper, we want to consider resonances which occur in systems of three composite particles, or clusters. The new feature will be the appearance of c o m p o u n d resonances and the difficulty will be that the c o m p o u n d resonances will interfere with subsystem resonances or resonance-like structures such as the final state peak. For two clusters, the theory of resonances is well-established in the context of the resonating group model 1.2). It has been shown that c o m p o u n d states give rise to resonances of Breit-Wigner type, in the single-channel case. Also, the coupled channel case has been investigated and the influence of a second open channel on the resonance behaviour in the first channel has been studied. W e shall try in this p a p e r to extend the theory of resonances from two to three clusters. T h e mathematical difficulty will be that, instead of the two-body G r e e n function, we will need the three-body G r e e n function. This means that we have to assume that a Faddeev equation has already been solved for the motion of the three clusters without the presence of a c o m p o u n d state. * Work supported in part by the Department of Energy under contract number DE-AC0376SF00515. ** Work supported in part by a grant from the Friedrich Naumann Foundation. 158
M.C.L. Orlowski / Compound resonances
159
In terms of physics, the difficulty is this: since there is a spectator cluster which can carry an arbitrary amount of energy, a subsystem resonance or a final-state p e a k will not only appear in a narrow energy range of the three-body system. It will appear at all energies above a certain minimum which corresponds to zero spectator energy. The c o m p o u n d resonance will thus never be an isolated resonance but will always find something to interfere with. This will be seen m o r e clearly from the relevant mathematical expressions. In sect. 2 a model of the three-cluster resonance is formulated in analogy to the work of Wildermuth and Ben6hr in the two-cluster case. A technique for solving this model is proposed in sect. 3. This technique is based on the asymptotic formulae of the full three-cluster resolvent. The corresponding formulae are discussed in the appendix. In sect. 4 the results are investigated and Breit-Wigner-like resonance formulae for the three-cluster case are discussed.
2. The model We consider a three-cluster system described by the following Schr6dinger equation: (h0+ V(1, 2 ) + V(1, 3 ) + V(2, 3 ) - E ) X = 0 ,
(2.1)
where h0 denotes the kinetic energy operator and V(i, j) are effective intercluster interactions. In addition to structures described by eq. (2.1) we like to consider an isolated three-cluster compound resonance. In order to describe such a c o m p o u n d resonance it is convenient to modify the structure of the ansatz for the solution of eq. (2.1) by introducing a single c o m p o u n d state F in the following ansatz:
X'(~, ~1) = X (6, ~1) + aF(~, ~i).
(2.2)
Here, ~ and ~1 are Jacobi coordinates and F is a three-cluster c o m p o u n d state which gives rise to a resonance. To be more specific, let us assume that F is a Feshbach projection 3) of a microscopic compound state [ which has been obtained by diagonalising a corresponding microscopic hamiltonian H of the system in a basis of Slater determinants. The Feshbach projection can be written as: F(~', lq') = I dt3A-lf(rl . . . . .
rA-1)~01q~2q~38(~--~')t~('tl --~i'),
(2.3)
where the ¢i are the internal cluster states corresponding to the microscopic interactions contained in H, and A is the number of all nucleons involved. A m o r e detailed discussion of the correlation between this model and the microscopic theory will be given in sect. 4. Note that eq. (2.1) can be written as a projection equation
(@lh - E I X ) -- 0 ,
(2.4)
where 6X denotes an arbitrary variation of X, and h = h0 +Y.j>i V(i, j). In ansatz
160
M.C.L. Orlowski / Compound resonances
(2.2) we assume the function F, whose expectation value exceeds the three-cluster breakup threshold but is still below the next breakup threshold, has only a small overlap with the first term on the right-hand side of (2.2). Using ansatz (2.2) and eq. (2.4) we get a set of equations:
(Sxlh - Elx ) + a (Sxlh - E L F ) = O, ( F l h - E I x ) + a (FIh -ELF) = 0 .
(2.5)
Formal elimination of the second equation leads to
(@lh _ E l x )
(@]h - E l F ) ( F l h - E l x ) (F]h - E L F )
- O.
(2.6)
While the first term of the last equation reproduces eq. (2.1) the second term introduces an additional separable potential Vc with a resonance denominator, 1
v c = ]P) E - (F]h ]F) (p]'
(2.7)
with IF)= (h -E)IF). The full effective potential of the three-cluster system can be written as: VCf61, 2, 3 ) = V(1, 2 ) + V(1, 3)+ V(2, 3)+ Vc.
(2.8)
In the next section we propose a method of solving the Schrbdinger equation (h0+ Veff(1, 2, 3))X = E x
(2.9)
with specific boundary condition for X and of investigating the scattering amplitudes in different channels.
3. Solution of the m o d e l
The aim is to obtain the solution
(ho + Ve,)X = ErX ,
(3.1)
with the boundary condition for bound-state scattering, i.e. for the scattering of a free particle 1 by the pair (1) = (2, 3), initially in a bound state of energy E l , . Here, Veff is given by eq. (2.8), ho represents the kinetic energy operator: ho = h2(A~ +An), and Er is the relative energy of the three-cluster system. [We work in the center-ofmass system of the three clusters and use Newton's notation of coordinates and momenta 4).] We define the following operators: h = h0+ Veff, g ( z ) = ( Z - h ) -1 ,
go(z) = (z - ho) -1 ,
/~ = h - Vc, i f ( z ) = (z - h )
-1 '
(3.2)
M.C.L. Orlowski / Compound resonances
161
with z = Er + ie. It holds that
g(z ) = ~(z ) + ~,(z ) Vcg(z ) .
(3.3)
It is assumed that the bound-state scattering problem, i.e. the equation lim ie~,(Er + i e ) ~ l = X~+) e~O
(3.4)
is already solved. Here, ~1 denotes the product of the plane wave tbl and the bound-state function of clusters 2 and 3, ~oln, where n denotes the nth bound state of clusters 2 and 3. This problem can be solved using some of the many techniques for solving the Faddeev equations. Using the operator identity (3.3) one obtains the Lippmann-Schwinger equation
limieg(Er+ie)~l
=
e-~0
X~+)= )~(1+)+g(Er +"l O +') V e ~ l (+} •
(3.5)
The last equation has a unique solution, since Vc has a finite Hilbert-Schmidt norm. Due to the separability of V¢ the solution X~÷) can be obtained immediately after a simple algebraic calculation. The result is
X ~+) = X~x+) + ~,(Er + iO +) (FIh IF) - E r -
(3.6)
In order to evaluate eq. (3.6) one needs the asymptotic formulae of the resolvent ~,+(Er)=~,(E~+iO +) in all channels. Such formulae, though not present in the literature, are well known. The derivation of these formulae are given in the appendix. The results are: -ilqinl bqil 1 ~e "~'(-)" ' E R'" ~,+(Er;R,R') I~,l-~ ' - 4 r r ( 2 ~ ) 3 / = L - - i ~ n t ~ ) x ~ " tq~,, ~n; ), (3.7) n I~1il IR'I.I~1<~ where q~, =
.q°lq,,,I = ( E r - E i , ) l / 2 ) , g (Er, R , R ' )
i = 1, 2, 3, for the two-body channels, and i~/4r.:.,3/4 elIKIIRI IRI-~ ' 2(2"rr)5/EIR[5/2 ;~°-'*(IKIR°'R') (3.8) IRq
for the breakup channel, where ~0-)* is the three-particle scattering function for three free incident particles. Here, the six-dimensional coordinates (~, "q~), i = 1, 2, 3, are denoted by a vector R, and the corresponding six-dimensional conjugate m o m e n t u m (p~, q~), i = 1, 2, 3, by a vector K. Then, the following formulae are valid 4):
IRI == I ,1 =+ln,I =, = IKI2=
Ip, l= + Iq, I= ,
K "R = ~i "Pl +'qi "qi,
(dR) = (d~,)(d'q~),
i = 1, 2, 3 .
M.C.L. Orlowski / Compound resonances
162
Using the formulae (3.7) and (3.8) and the known structure of the asymptotic form of the three-particle scattering function ~÷~ <5), one obtains all scattering amplitudes of the solution X~÷) of the eq. (3.6). One obtains (a) for the breakup channel, eil~l IRI I~l~oo > ~-7"~[ Rm(K'; ql, E1)+Rol. . . (K . ;qx, E~)] ,
X~+)(ql'E1;R)
(3.9)
(b) for the elastic channel,
x~÷~(q~, E~; R )
[,rll[~OO
) eiql"~ltPln(E1, ~1)
1~211
rn1(2/~1) w2 ["q1[
n
+ Rrly (q'l., El,,; ql, El)],
(3.10)
(c) for the rearrangement channels j # 1,
X~+)(ql'E1;R)
Y~ e'b'"Ft~'lq~j,(~i)[Ril(q ~n, E 1, ; qi, Ei) t~' t-'°~ > rnj(2/£i)~/= [~il . lejl
(3.11)
Here R o are the usual Faddeev scattering amplitudes in the absence of the threecluster potential 17"cand are given by
Ril(q'l., Exn ; qi, Ei)
; 1/2
= -16~.mitz3/2
I dR
e-iq'l"mqp 1".(SOl)i~lV(i)(~i)~l+)(qi, E i ; R ) ,
(3.12a)
l = 1, 2, 3, and
e' /4E r/4 f
g o l ( K ' ; ql, El) ~ 2 ( ~ ) 5
2 j dRtl~(1):~:(K ', R)i~1_ V(i)(~i)X(l+)(qx' El; R ) , (3.12b)
in which ~(1-) (K, R ) = e ~l''h& (1-) (pl, 61) and.& ~-)(~1) is a scattering state in subsystem (1).
The potentials V (i ) are defined by V ( i ) = V (f, k ), ( i, j, k) cyclic. The three-cluster resonance scattering amplitudes R ~ ~, i = 0, 1, 2, 3, which arise from the potential Vc can be found as
o3 >lp)(p[;,l+,> (PIg+(EOIR)'
R ~1" = - 16zrmilx 3/2 (F[h IF) - E ~ -
(3.13a)
M.C.L. Orlowski / Compound resonances
163
i = 1, 2, 3 and
ei•/4rr3/4 R~'f - 2(2zr)5/2
(FlhlF)_E,_(p[g+(Er)lP).
(3.13b)
Then the corresponding cross sections are given by (a) for the two-body channels i = 1, 2, 3:
d~
Iql°l m 1/.t- "i: --
dO
[ql[ mild, 1
-- res 2 -----: 1-7~-IRma+ R a I ,
(3.14a)
(b) for the breakup channel: dtr 41[q~]3 [Ro~ res2 , +R01 I (dq,) -- ,~5/2 2 3/2 dO z m l/xx IqlllK:
(3.14b)
It can be observed already here [formulae (3.14)] that the interference effects between the structures contained in Rii such as the final-state interaction and quasi-free scattering and R ~]~s are present in all channels. On the other side the decay of the three-cluster resonance itself is influenced by the existing structures produced by the two-body potentials [formulae (3.13)]. Thus formulae (3.12)-(3.14) give a qualitative insight into the variety of energy correlations in the case of a sudden decay of a three-cluster system. In the next section the scattering amplitudes R[~ ~,/" = 0, 1, 2, 3, will be studied in more detail.
4. Breit-Wigner resonance formulae for a three-cluster compound resonance
In this section the structure of the scattering amplitudes R[~ s, i = 0, 1, 2, 3, will be discussed. The terms of interest are:
07~°-)IP)(PI~?I+2)
(4.1a)
(F[h IF) - E r - (FIff+(Er)IP) '
(£12 IP)(FI£~+2)
(Flh IF) -Er- <#lff+(Er)IP> "
(4.1b)
The resolvent ff+(Er) can be represented in the following spectral decomposition:
ff+(Er) ~ ~
+jj
=~vEr-Evb?V) I ~<£°
fJ d-qiEr_Ei.O?f3I_lqJl )O?f3:+io+
-(+)~
~(+71
,XO )¢,Xo , I)~,,)O~K[ dqk apk E 2 Z+io--=S , r-- qkl --IPk[ ~ Er-E~ +i0 + _
(4.2)
where k = 1, 2, 3 is arbitrary. Here ~v is the three-body bound state, ~I~+) is a bound-state scattering function with initially bound state in subsystem (i), and ~(0+) is a scattering function for three free incident particles. We define a complex
M.C.L. Orlowski/ Compoundresonances
164 resonance shift by
A +(Er) = S ~I)~K)()~ IF) E~-E~ + i 0 ÷ "
(4.3)
Further, by using the identity
1
1
-.o EK -- Er - te CPVE~ Er+Drg(E,, - E r )
lim
(4.4)
. "=
one can separate the non-hermitian operator g+(Er) into hermitian and antihermitian parts:
g+(Er) = gh(Er) + ga'h'(Er) with
1~o)(~ol Xin "(+)")~(in "(+) ffh(Er)=~v Er-Ev "[-~ I CeWEr'q-lqi[2-Ein Er-lq, 12-lpil 2' (4.5)
~a'h'(Er) = ~
I
"(+) )(x,. ~(+) la(Er-E,.-Iq, I=) dqi Ix,.
;
÷ f f [;~(o+')~(o÷)lS(Er--[qkl2--lpk[E)dpkdqk. Then defining the following quantities:
(±)
]/in (Er)
= (FI)~l~)) ,
i = 1, 2, 3 ,
T(o±) (Er) = (Pl;(g)). F ,.±) (Er)--2~rl~.~)(Er)l
(4.6)
2
j--0. 1.2,3.
we can rewrite the formula (4.3) for the complex resonance shift as
A+(Er)=flc(Er)+il(~Fl+,)(Er-Ei,,)+f F(o+)(Er-lqk[zdqk),
(4.7)
where A c(Er) (Plff" (Er)lP). T h e magnitudes 3/(Er) describe the probability amplitude of the resonance clusterstate ff in the different scattering states at the same energy. The variety of the scattering states results from different initial conditions. Since ff is a square integrable function, the formulae (4.6) show that the resonance w i d t h s / ' are strongly influenced by the off-shell behaviour of the scattering states )~I~) and )~(0±). W e discuss the influence of the off-shell behavior on the resonance widths in terms of the Pauli principle and saturation properties of the nuclear force at the end of this section. =
165
M.C.L. Orlowski / Compound resonances
Due to the square integrability of P the resonance widths F are also very sensitive to subsystem resonances, since in that case the amplitude of this subsystem is large at small distances. Therefore the partial widths F are good candidates for probing the off-shell behavior of the three-cluster wave functions. With the definitions (4.6) and (4.7) we can transform (4.1a) and (4.1b) into
0?l-' IF)(PI~?<,+>) (F[h IF) - E r - (PIg(+)(G)IP) y~o-)*(Er)y ~+n)(Er)
E c - E r - Ac(Er)+ ~i(Yun FI +) (Er-
(4.8a)
Ein)+ f F(o+,(Er-1~12) d~)
and
O? IP) (PI '17 )
(Flh IF)- E,(PI~,+(E,)IP) •
=--i
(-),
lyj~
(+) (Er)yln (Er)
E ~ - Er-A~(E~) +½i ( x,orl;' ( E , - E , , ) +
I/'<°+'(E'-lq[2 dq) '
(4.8b)
respectively, where E¢ = (Flh IF). Obviously, these formulae reveal the Breit-Wigner structure. The resonance energy pole E~es-the pole at which the real part of the denominator in (4.8) becomes zero - is defined by E c - E r e s - "4c(Eres) = O .
The formulae (4.8) exhibit that even in the case of the three-body breakup channel the Breit-Wigner resonance formula is still valid. Moreover it allows one to study the rich energy correlations of the decay products. The only difference to this formula is that Ac(Er) and F(Er) are energy dependent. To eliminate this difference it is necessary to make certain approximations. We have to use the fact that we are considering an isolated, relatively sharp resonance. For such a resonance level all other resonances lie energetically so far away that F(Er) can be considered to be approximately energy-independent over the energy width of this level. Then we make the approximation that, in the neighbourhood of E .... ./d~c(Er)'~ Ac(Er) ~- ,4c(Eres) + (Er -- Eres) I T )
r
Er=Er¢s
'
F(E,) ~r(Er~). Then one arrives at the same formulae as (4.8) where Ac(Er) is replaced by '4c(Eres)
166
M.C.L Orlowski / Compound resonances
and the partial widths by: F(+)(E~)
F're+s)(Er¢~) = 1 + ~E~ Ac(Er)
~ Er=Eres
'
~//'tl
(Er) (Eres) = [1 + ~E~ A~(Er)]I/2 Tin
Er=Ere
s "
The main difference in the structure of formulae (4.8) compared with the resonance formulae for the two-body channel situation 6) is the appearance of an integral over resonance widths F~0+)(Er-lql2). This reflects the fact that the three-body break-up can be considered as a continuum of infinitely many two-body channels. Each distribution of the energy over the three bodies corresponds to one two-body channel. However, the interpretation of the resonance widths F according to the picture of a two-body channel situation is no longer possible. This can be seen in the following way: though )~0+) is a scattering function for three free incident particles, it apparently also contains contributions of all two-body channels. This aspect is lacking in the Breit-Wigner formula in the two-body channel situation and is obviously characteristic for (N~>3)-cluster systems. The analogy to the two-channel case is not possible because F(~-) cannot be interpreted as the transition amplitude from the resonance state ff whether to the breakup channel nor to some of the two-body channels; all these transition amplitudes are contained in Ft0+), but they altogether determine, nevertheless globally, the partial resonance width of the breakup channel. Therefore even in the case of a two-body channel no straightforward parametrization by phase shifts of the resonance amplitude can be given. Analogous considerations can be made for F ~ ). They show that vice-versa the transition amplitudes from the compound state P to the breakup channel influences the resonance width of a two-body channel. It should be pointed out that this many-cluster effect is qualitatively different from the mutual influence of the partial widths of the two-body coupled channels. The latter effect depends on the ratio of the partial width to the total resonance width. The origin of the new effect is a dynamical one. It leads to some interesting and important consequences. Let us assume that in addition to the three-cluster resonance there is at a lower energy a subsystem resonance in the subsystem (i), consisting of clusters ] and k. One considers now the three-cluster resonance behavior in the two-body channel, where clusters ] and k form just a bound state. In this case the three-cluster resonance width of this channel is strongly influenced by the resonance of the same subsystem (i), though in this channel just this subsystem is bound at each energy. Another difference is that the formulae given by Schranner 6) for two-body channels were derived under the strong approximation, namely that the direct coupling between the open two-body channels can be neglected and that the transition occurs through the formation of the resonance state. Such an approximation is valid only in specific situations, if for instance the resonance occurs in an energy region where all or all but one channels are shielded from the compound
M.C.L. Orlowski / Compound resonances
167
region by high potential barriers. The derivations presented here restricted to the two-body channel reactions are exact and cover all cases. The influence of the three-cluster breakup structures such as quasi-free scattering, final-state interaction, or two-body channel resonances on the three-cluster resonance is twofold. First, such structures strongly influence the three-cluster resonance widths F ( E ) , second, they interfere with the three-cluster resonance amplitude R ~s. The kinematic aspects of such interferences are extensively studied in ref. 7). In a contribution to come specific numerical examples will be presented. Finally we discuss to what extent our model is compatible with more microscopic theories, like for example the very successful resonating group method. This microscopic method was recently extended to the case of three - and more - cluster systems 8-10). It is known that microscopic models yield energy-dependent interactions and strong multi-cluster forces, which originate from the Pauli principle. In ref. 9) however, it has been shown that in the three-cluster system an appropriate off-shell transformation (i) removes the energy dependence of the interactions, (ii) changes the off-shell behavior of the two-cluster interactions while their on-shell properties remain unchanged, and (iii) it also reduces considerably the strength of the three-cluster potential. The crucial question is whether one is able to construct an appropriate off-shell behavior of the on-shell equivalent two-cluster forces, which would absorb the effect of the three-body force. This was studied in ref. 11) on specific examples. The numerical results in the case of a - ~ - a and A-o~-a binding problems have shown that the off-shell transformed three-body force is negligible. This is a very gratifying result if we think of the many phenomenological calculations on three-cluster systems. In most of these calculations, people have used - and with success - energyindependent two-body potentials and no three-body potential. In view of these results it is justified to say that our model, which is based on pairwise cluster-interactions, does not contradict microscopic models, provided that the two-cluster interactions have a proper off-shell behavior. The off-shell behavior should contain the Pauli principle, the saturation property of nuclear forces and the guarantee that no strong three-body potential is needed to describe the system correctly. It has been shown 12) that the off-shell behavior of two-body forces influences strongly the behavior of the scattering states XI±~ and Xto~ at short distances. This influence is reflected by the resonance widths F, as already mentioned. 5. Conclusion
A formal theory has been developed for the decay of a nuclear compound state into three clusters. The formalism shows clearly the interference of the sequential decay modes with the sudden decay mode. The differential cross section has a complicated structure, even when the compound resonance is sharp and isolated.
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As compared with the standard Breit-Wigner formalism, the three-cluster decay of a compound state corresponds to a case with infinitely many two-body channels. Correspondingly the total decay width is a sum plus an integral over the partial widths. New effects of the mutual influence of the three-cluster resonance with subsystem resonances and other resonance-like structures have been discussed. For a practical application of the theory one has to solve the three-cluster Faddeev equation in absence of the three-cluster compound resonance potential and one has to calculate matrix elements of this potential with the obtained solutions. The evaluation of the resonance part of the three-cluster scattering amplitude is sensitive to the off-shell behavior of the three-cluster system. The author is pleased to acknowledge valuable suggestions and fruitful discussions with Prof. H. Pierre Noyes, Prof. R.G. Newton, Prof. S. Saito and especially with Prof. Erich W. Schmid.
Appendix THE ASYMPTOTIC FORMULAE OF THE FULL THREE-BODY GREEN FUNCTION IN COORDINATE RESPRESENTATION AS one knows from refs 13-18), the asymptotic'form of the three-body wave function depends on the direction in the six-dimensional coordinate space. The same, of course, must be true of the Green function. One can, therefore, expect that one should be able to use the various forms of the resolvent equation for the full resolvent ff in terms of the channel resolvent ffl, i = 0, 1, 2, 3, in order to find its asymptotic form in various directions. One has
~, = gi + (,,Q(i)~, ,
(A. 1)
where 17"(i)= (V(/) + V ( k ) ) , i, L k cyclic, and 9 ( 0 ) = Y.~=I V(i). We are considering a situation with pairwise interactions only. However in eq. (A.1) a term of the following form appears:
f dR"~, + (R, R") V(Z)(/~)~+(R ", R ' ) .
(A.2)
There is a convergence problem with alg integration. For this reason (A.1) is not the proper starting point for going to large IR I. One can expect that one has to start with Faddeev-like equations for the Green function ~. In order to obtain a Faddeev-like splitting of a resolvent identity, one considers eq. (A.1) for i = 0, 1, 2, 3 and obtains after an easy calculation 3
= Z ~i[M~+ V(i)(,oV(i)~,], i=1
(A.3)
M.C.L. Orlowski / Compound resonances
169
in which {V(i)~0 Mi =
"~x
i#h
i =A
Here A can be chosen arbitrarily, A = 1, 2, 3. One can show that the formula (A.3) corrresponds to the Weinberg decomposition of the three-particle wave function 19), and the choice of A corresponds to the determination of the entrance channel. The arbitrariness of A reflects the fact, that in the space of eigenstates of g~-~ the operator V(i)go is identical with identity Ui acting in this subspace. An equivalent formulation of (A.3) is 3
g = go+go ~ (V(i)~,~ + V(i)~,~V(i)~,).
(A.4)
i=1
We will show later that in eqs. (A,3) and (A.4) the disconnected parts in the kernel are avoided. In order to derive the asymptotic formulae for the full three-body Green function one needs corresponding asymptotic formulae for the channel Green functions gi. Such formulae have already been derived by Newton 4), who pointed out that gi can be split into two parts: ~- (E) = ff~c(E) + ff~b (E), where ~C(E) is responsible only for the asymptotic formulae of /~-(E) in the breakup channel and ff]-b(E) determines uniquely the asymptotic behaviour of g]-(E) in the two-body channel i: lim ~,~-(E;R,R')= IRl-,oo lim ~,.+,~(E;R,R')
IRl-,oo
Ia'l
IR'l
eiW/4E3/4 eilX[ IRI
= 2(2~.)s/21RIS/2 ~I -)*(K', R').
(A.5)
lim g~f(E;R,R')= lim ~,~b(E;R,R') Ind-,oo IR'l.14i~l
_ilqmIIn,I 1 t;; :g = 4,r(2tzl)3/2 ~ ~---~il ~o,,,(~,)q~, (q,,,, E,,~; R'), (A.6) where ql. = n°lq,.I. Iq,.I
= (E-E,.)
1/= .
E,.; R') -
The derivation of the asymptotic formula for the two-body channel is trivial. In this case one can start directly from the identity (A.1) i # 0. There is no trouble in the integration in the coordinates ~ and ~1~, since in the two body channel the
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M.C.L. Orlowski / Compound resonances
leading contribution in the Green function comes only from g~+b (E). Thus the range of ~[ is limited by the bound-state function ¢i, and the range of ~ is limited by the potential V(/),/" ~ i. If the ranges of ~[ and of ~; ] ~ i are limited then the same holds true for R ' = (~, ~ ) . Starting from (A.1) and using (A.6) one obtains lira g + ( E ; R , R ' ) = - A ~,1 ~3/2Y. c~ g i , , In,I--,oo "t "n"( ~zt-ti ) n " IR'l,[~l
"(-)* .t ~. . . ) ,
(A.7)
where lim i e g ( E - ie ) ~ , = ~ i , + g - ~ ' , ) ~ , , -- ~I~ ) • ~"~0
In the case of the breakup channel the use of eq. (A.1) is no longer justified, since the corresponding kernels do not fulfil the requirement of the connectivity. In that case it is suitable to start from the identity (A.3) or (A.4). Studying eq. (A 3) one finds that the terms V(i)g0 cause no trouble because they represent compact two-body kernels embedded in the three-body space and the corresponding 6function can be factored out. But we have to ensure that ( R [ V ( i ) g o V ( / ) , i ~ i , decreases asymptotically sufficiently rapidly. For this purpose it is sufficient to show that V ( i ) g o ( z ) V ( / ) g o ( Z ) is square integrable. By taking the expectation value between m o m e n t u m states for (qkPk[ V ( k ) ] q k P k ) and the expectation value of the resolvent g0(z) we may work out explicitly that [ [ V ( i ) g o ( z ) V ( / ) g o ( z ) ] [ z is finite for z not on the real axis. Rubin, Sugar and Tiktopoulos 2o) have shown using Fredholm theory in the dispersion studies of the three-body amplitude that the square of Faddeev's kernel 21) is Fredholm even on the real axis. This strong result should be contrasted with that of Faddeev who was only able to show that the fifth power of the kernel was compact. The reason Faddeev could not obtain the stronger result was that he assumed only that the potentials involved satisfied smoothness conditions whereas here we will assume that the potential is analytic in its m o m e n t u m variables. Then the sufficient conditions for a Hilbert-Schmidt operator represented by a kernel K (x, y ; z) are (i) For each pair (x, y), K ( x , y; z) must be analytic for z ~D. (ii) There exists a uniform bound B
where ~o o-(/z) dtz is required to be of bounded variation.
M.C.L. Orlowski / Compound resonances
171
Thus we are allowed to use the formula (A.4) in order to derive the asymptotic formula of the resolvent go in the breakup channel. T o this purpose we need the corresponding formula for the free resolvent go. It holds ei~r/4E3/4 e l I K I R o - R '
lim go(E; R, R') I~l-,oo IR'l
=
e -ii/¢lR°'R'
2(2¢r)S/2[R{5/2
.
(A.8)
This formula is obtained using methods already applied by Newton 4). One obtains lim IRl~oo g ( E ; R , R ' ) I~'1
e
i-,r1,~,.-,314 i[KIRO.R' 1~ e !
=
2(2~r)5/2[Ri5/2
,
]
~"
[40 (K',R')+i=I (4oV(i)gf/(i)g)*iR')j, (A.9)
where 4 o ( K ' , R') = e ilrlR°'R', K ' = IKIR °. Considering that lim~_,o iegi (E - i e )40 = 41 -) one obtains 3
3
3
Y, (4oV(i)g~ + 4 o V ( i ) g ~ v ( i ) g ) = ~ (4~ -) + 4 1 - ) Q ( i ) g ) - ~ [ 4 0 + 4olP(i)~] i=1
i=1
i=1
and with this result finally lim g(E,s,R')= IRI-*~ Ia'l
e
irr/4~3/4 /:~
e
ilIKIIRI . 3 /
~ ( - ) * i rJ-t
i~ X,
t a , g ' ) - 2~(o-)* (K', R')
)
,
(AA0) in which ~I -) = lim ieg(E - ie)41 -) ,
(A. 11)
~(o-) = lim ieg(E - i e ) 4 o .
(A.12)
E~0
e-)0
One can easily show that the ~I -) as defined in (A.11) and (A.12) are identical for i = 0, 1,2, 3. Therefore one obtains eirr/gg3/4 e i l K I [RI
lim g(E" R, R') =
IRI-o~
IRq
'
2(2rr)S/21RI 5/2
;~(o-~* (K', R ' )
"
(A.13)
Finally, we mention another useful asymptotic formula for the breakup channel formulated not in terms of scattering functions but in terms of the scattering amplitude. This shows that the scattering amplitude Roo without the compound state ff is already contained in the resonance scattering amplitude R r~o~.It is known that the scattering amplitude Roo is obtained by quadrature from the scattering
M.C.L. Orlowski/ Compoundresonances
172
a m p l i t u d e s Roi, i = 1, 2, 3. T h e result is lira ~,(E;R,R') iRl~oo iR,l
ei~/4E3/4 eil~l lRl [ e-iK"w + (2 7r)3 I R~oo)(K,, K,,)(K,,,~oIR,) dK .] = 2(2rr)s/21RlS/2 (A.14) in which R~0o) d e n o t e s the f r e e - f r e e t r a n s i t i o n a m p l i t u d e in a t h r e e - b o d y system.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
H.C. Benoehr and K. Wildermuth, Nucl. Phys. A218 (1969) 1 K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977) H. Feshbach, Ann. of Phys. 19 (1962) 287 R.G. Newton, Ann. of Phys. 34 (1972) 324 S.P. Merkuriev, C. Gignoux and A. Laverne, Ann. of Phys. 99 (1976) 30 R. Schranner, Diploma Thesis, unpublished, University of Tuebingen M. Orlowski, Thesis, unpublished, University of Tuebingen E.W. Schmid, Phys. Rev. C21 (1980) 691 E.W. Schmid, Z. Phys. A302 (1981) 311 H. Horiuchi, Prog. Theor. Phys. 51 (1974) 1266; 53 (1975) 447; Prog. Theor. Phys. Suppl. 62 (1977) 90 E.W. Schmid, M. Orlowski and C.G. Bao, Importance of the three-body Pauli potential in three-cluster systems, Z. Phys. A (1982), to be published H. Leeb and E.W. Schmid, Talk given at the Liblice Conference, June 1981, Czechoslovakia W. Gloeckle, Z. Phys. A271 (1974) 31 C. Gignoux, A. Laverne and S.P. Merkuriev, Phys. Rev. Lett. 33 (1974) 1350 R.G. Newton, Ann. of Phys. 78 (1973) 561 A. Laverne and C. Gignoux, Nucl. Phys. A203 (1973) 597 J. Nuttall, Phys. Rev. Lett. 19 (1967) 473 J. Nutall and J.G. Webb, Phys. Rev. 178 (1968) 2226 S. Weinberg, Phys. Rev, 131 (1963) 440 M. Rubin, R. Sugar and G. Tiktopoulos, Phys. Rev. 159 (1967) 1348 L.D. Faddeev, Mathematical aspects of the three-body problem in quantum scatterings (Davey, New York, 1965)