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G Y . B E N C Z E
N-BODY METHODS IN THE THEORY OF NUCLEAR REACTIONS
4Hungarian Academy o f Sciences
CENTRAL RESEARCH
INSTITUTE FOR PHYSICS
BUDAPEST
т т
KFKI-1980-52
N-BODY METHODS IN THE THEORY OF NUCLEAR REACTIONS
Gy. Bencze
Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary
Summary of the invited talk given at the 9th European Symposium on Few-Body Problems in Nuclear and Particle Physics, 3-6 June, 1980, Seeimbra, Portugal
HU ISSN 0368 5330 ISBN 963 371 684 5
1. Introduction
In the last decade m u l 11 part I с 1e scattering theory has developed into a separate bra n c h of mathematical physics. Unfortunately the increasing a c t i v
ity and rapid de v e l o p m e n t in the field of N-particle s cattering theory have mad e almost no impact on nuclear reaction theory. Traditional nuclear reaction
theory / f o l l o w i n g Kowalski " t r a d i t i o n a l " is used to mean p r e - F a d d e e v not n e cessarily c h r o n o l o g i c a l l y but in spirit/ bases its c o n siderations on the Vnul- tiparticle S c h r ö d i n g e r equat i o n and e m phasises the importance of numerical results as well as description of experimental data even if a large degree of p h e n o m e n o l o g y is involved. Such an a t t i t u d e is justified to some extent by the success of optical model, DWBA, CCBA, etc. However, in such circumstances it is easy to o v e r l o o k the limitations and possible inconsistencies of the basic formalism involved. H u l 1 1 part I cle scatt e r i n g theory on the o t h e r hand offers exact treatment and full unders t a n d i n g of the dynamics at the price of a rel
atively c o m p l i c a t e d formalism and e n o rmous numerical dif f i c u l t i e s in some of the a p p l ications. Of course Faddeev theory has also had s p e c t a c u l a r success in d e scribing the three-nucleon system and by now realistic four-nucleon c a l culations have a l s o become feasible. Due to the increasing amount of numerical work, however N - particle equations are clearly not practical for treating c o m
plex nuclear systems. Nevertheless the applicatiotr of concepts and methods developed by N - p a r t i c l e scattering theory may contribute to o u r better u n d e r standing of n u c l e a r reaction dynamics.
2. Traditional methods
Until Faddeev's w o r k only the two-body problem could be solved exactly.
Thus traditional nuclear reaction theory is necessarily b ased on two-body
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m e t h o d s even If mu)11 p a r t i c l e processes are stud i e d [1]. The Implications of this basic limitation are not always fully recognised.
Feshbach's e l e g a n t projection o p e rator formalism [2] provides a rig
orous J u s t ification for the optical model of elas t i c and inelastic scattering.
This theory of the optical potential can also be extended with some difficulty to account for e x c h a n g e effects. However, Feshbach's formalism is onl y for
m a l l y exact since in o r d e r to construct the optical potential one has to solve in principle the full mu 11ipart 1 cle scattering problem.
The treatment of rearrangement processes already presents some nontrivial difficulties. Since even the s i m plest case of rearrangement in
volves at least three c o n s t ituents or nuclear с 1 u s t e r s ,two-body m e t h o d s obvi
o u s l y cannot d e s c r i b e the dynamics in a conse q u e n t manner. The s t a ndard proce
dures, e.g. DWBA, C C B A treat the interactions respcnsible for the rearrangement by perturbation theory w i t h a suitable c o m b i n a t i o n of optical model and two- potential formalism. V a r i o u s attempts to ex t e n d the coupled channels method for rearrangement p r o c e s s e s In the frame of traditional theory suffer from serious mathematical and conceptual d i f f iculties, as is discussed in detail by Levin [3].
The r e sonating group me t h o d (RGM) is different in spirit s ince it is based on variational for m a l i s m [A], As a result it is flexible enough to treat all two cluster cha n n e l s , rearrangement and identical particle effe c t s w i t h out invoking perturbaticn t h e o r y . RGM In its present form cannot treat three- or m o r e cluster c h a n n e l s and some formal p roperties /e.g. n on-orthogonality e f f e c t s / reflect the b asic limitations of the method. Hovewer, if the known a s y m p t o t i c form of three-body scattering w a v e functions is made use of, it
is In p r inciple p o s s i b l e to extend RGM to treat three-body channels as well [5].
3. N - particle s c a t t e r i n g theory
These exist various exact formulations of N-particle scattering which are based on integral equations and det e r m i n e transition operators o r compo
nents of the s c a t t e r i n g o p e rator of the system.
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Details of N-oarticle scatterlrm theory can be found in recent review articles [6,7].
In the last year three Important developments occured w h i c h will be briefly dlscussed:
- the general algebraic theory of Identical particle scattering - Merkur i e v ' s theory of the Coulomb three body problem
- Combinatorial H amiltonian Unitary Connected Kernel /CHUCK/
theory.
The general algebraic theory of identical particle scattering d e v e l oped by Bencze and Redlsh [8] can be applied to a large class of N-partlcle scattering formalisms and results In c o n s i derable simplification, i.e. the maximal reduction of the number of coupled Integral equations. Thus the
theory can be f o rmulated in such a way as to involve only quantities relevant to physical processes [9].
The p r e sence of long range Coulomb interactions in nuclear systems leads to essential mathematical difficulties. As a consenquence the Faddeev and other N-pa r t i c l e approaches cannot be applied. Since in this case the a s ymptotic c o n dition has to be modified [10], correspondingly scattering
theory has to be reformulated as well. The problem of three charged particles has been recently solved by M e r k u r l e v [11] by introduclng suitably modified Faddeev equations. U n f o rtunately this theory is rather complicated and so far the nontrivial p r o b l e m of g e n e r alising it to an arbitrary number of particles has not been considered. Thus for partical applications simpler approximation schemes have to be introduced w h i c h acco m o d a t e the m o d ified asymptotic c o n d i tion [ 12].
As the n u m b e r of p a r ticles increases due to the large number of variables as well as coupled equations exact N-particle equations become
Impractical for describing múl ti particle collision processes. On the other hand In actual physical processes the m e c h a n i s m of the collision can be often
described In terms of only a small number of nuclear clusters, i.e. the system Is dominated by just a few channels. In such a case one intuitively expects to be able to give a simplified treatment in terms of the dominant channels or reaction mechanics. The CHUCK theory of Polyzou and Redish [13]
has been developed to handle such situations and combines the flexibility of the projection o p e r a t o r method w ith an exact múl ti particle scattering
formal ism.
Due to the above new developments it is nowefeasible to attack some of the problems of nuclear reaction theory w hich cannot be s y s tematically
treated by traditional methods.
4. Effective two-body problems
The simplest effective two-body prob l e m is the elastic /and inelastic/
scattering of two nuclear particles. Their interaction, the optical potential is given by a formal expression in Feshbach's formalism. N -particle
scattering theory on the other hand yields an explicit e x pression which can be evaluated by q u a d r atures provided the s u bsystem properties are known.
Indeed the elastic channel transition o p e rator can be shown ei t h e r by e l i m i nation or channel decoupling technique [14] to satisfy an L-S type equation.
By restriction to the channel subspace of the appropriate internal states of both p r ojectile and target one obtains the optical potential explicitly.
With a suitable rearrangement of the N-particle equations [15] the folding potential is o b t ained as a first a p p r oximation and the contribution of various processes to the imaginary part can be systematically studied.
However, there is a lot of practical problems to be solved be f o r e m i c r o s c o pic optical potentials can be actually calculated.
In traditional theory the coupled channel method cannot be s y s temat
ically extended to rearrangement processes. In N-particle scattering theory this presents no problem. Given a set of mi n i m a l l y coupled [6,7] N-particle equations the "p o l e " -or "bound state a p p r oximation" in the kernel imme
diately yields a set of coupled effective two-body equations w h i c h are
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suitable for numerical calculations. This a p p r o a c h has been extensively studied by Levin and c o l l aborators [16] . By ma k i n g use of the new d e v e l o p ments identical p a r ticle as well as Coulomb effects can be easy included and one arrives at an a l t e r n a t i v e to the RGM m e t h o d with the Important d i f ference that n o n - o r t h o g o n a l i t y terms are absent.
5. Effective few-body problems
A large number of multi p a r t i c l e c o llision processes can be intui- vely described as e f f e c t i v e few-body problems. If there are final states w h i c h involve more than two clusters two-body methods are clearly not
sufficient. However, o n e still expects to be able to treat these processes wit h few (<N)-body methods. Recently the RGM approach was extended by
Schmid [17] to obtain eff e c t i v e few-body Schröd i n g e r equation with effective interactions.
Starting w i t h exact N-particle f o r m a l i s m there are two ways of arriving at effective few-body problems. The " d ominant partition method" of Dixon and Redlsh [18] reduces the N-particle BRS-equations to a set of few-particle BRS-equations with effective / f e w - b o d y / interactions by a suitable elimination or truncation procedure. This remarkable property has been demonstrated so far only for the BRS-formalism.
C HUCK theory on the other hand offers a flexible wa y of keeping all the dominant channels irrespectively of their cluster structure. As a
result the few-body e q uations can be classified according to the number of vector variables rather than the cluster structure of channels. In this approach the " s p e c t r o s c o p i c factor problem" does not occur [19].
6. Conclusions
The above b rief summary gave only a g l i m p s e of the conceptual
clarity and promising new techniques N - particle methods can offer. However, there is clearly very m u c h to be done in o r d e r to develop practical methods of applications. In any case, as it was pointed out by Faddeev at the 1979
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Dubna meeting, it is time N-particel scattering theory was applied to nuclear reactions so as not to become an Isolated discipline alien to real physical p r o b l e m s .
So far only techniques relevant to so called direct reactions have been discussed. If there is a large number of degrees of freedom Involved in
the col l i s i o n process usually statistical considerations are used. These are based on the assumption that the number of particles is large. However, in real nucl e a r systems the number of reaction channels rather than that of the particles is very large. So that statistical or stochastic m e t h o d s are best to be applied to the set of channels rather than particles themselves. This idea has been elaborated by Baz and col 1aborators [20] and applied to heavy ion reactions with promising results. Such considerations can be easily extended to exact N - particle equations w ith the advantage that thtere channel structure is very suitably displayed.
The author Is indebted to Dr. J . Révai for several d iscussions and helpful comments.
R e f e r e n c e s :
1. P.E.Hpdgson, Nuclear Reactions and Nuclear Structure, Oxford, 1971.
2. H.Feshbach, Ann.Phys./M.Y./.
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/1958/ 357, J9. /1962/ 287 3. F.S.Levin, Invited paper to this conferencei*. K.WI Idermuth, E. J . Kanel lopoulos , Rep. P r o g . P h y s .
k2 /\Э 7Э /
1720 5. J . Révai, private communication6. V.Vanzani, in Few-Body Nuclear Physics, IAEA, Vienna 1978, p.57*
7. K . L . K o w a 1 s k i , in Few-Body Systems and Nuclear Forces, vol II. Springer, Berlin, 1978.
8. Gy.Bencze, in Few-Body Nuclear Physics, IAEA Vienna 1978, Gy. Bencze, E.F.Redish, J.M a t h . Phys. Jj} /1979/ 1909
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V
I
9. Gy.Bencze, C.Chandler, to be published 10. J.Dollard, J.Hath. Phys. W 1 9 6 V 729
11. S . P . Merkur i e v , Let t .M a t h . Phys . _3 /1979/ 1^1,
S . P .M e r k u r i e v , On the Three - B o d y Coulomb Scattering, Problem, preprint FUB/HEP 2/80.
12. Gy.Bencze, H . Zankel, Phys.Lett. 82B /1979/ 316, 13. W.P o l y z o u , E.F.Redish, J.Math.Phys. JJ_9 / 1979/ 1,.
]b. Gy.Bencze, Phys.Lett. 6 1 В /1976/ 139,
15. К . L . K o w a l s k i , Ann.Phys. /N.Y./ J_20 /1979/ 328, 16. J . Greben, F.S.Levin, Nucl.Phys. A325 /1979/ 1^5,
F.S.Levin, C.T.Li, to be published
1 7. E.W.Schmid, N-Cluster Dynamics and Effective Interaction of Composite Particles, preprint, Univ. of Tubingen, 1979-
18. R.M.Dixon, E.F.Redisn, preprint, University of M a r y l a n d 79-103 /1979/
1 9. W.N. Polyzou, Thesis, U n i v e r s i t y of Maryland, 1979
20. A.I.Baz, B.V. Danilin, Yad.Fiz. 29 /1979/ 1^89 /in Russian/
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Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Szegő Károly
Szakmai lektor: Szegő Károly Nyelvi lektor: Gombosi Tamás
Példányszám: 170 Törzsszám: 80-477 Készült a KFKI sokszorosító üzemében Budapest, 1980. augusztus hó
