REVI''TA MEXICANA DE FíSICA 48 SUPLEMENTO 2, 66-70

Quantal versions of Goldhaber's fragmentation model

p, Van Isacker, G. Auger, and C. Grosjean'Grand Accé/éralellr National d'/ons LOllrds, BP 55027, F-/4076 Caen Cedex 5, hance

Recibido el 20 de febrero de 2002; aceptado el I de mano de 2002

:O-;OVIE!\1BRE 2002

1\ quantal vcrsiUJl 01'Goldhabcr's fragrnt:ntation model is prcscnlcd. The basic assumption 01'sudden dichotorny is retained iniliaJ1y bul thedTcct 01' the nuclear mean field is taken into aceount by placing nucJeons in a harmonic osciUator potentia1. It is shown that. ir lhe aseillalarconstants of the initial nuclcus aod of lhe final fragmcnts are idcntical aod ir zcro-point motion is ncglcctcd, Goldhabcr's distribution isn.."Covered eX(l(:liy. An allcrnative to this simple model which ¡roposes conservalion of lhe number of oscillalor quanta is proposcd. Thiscondition is equivalent 10 conservation of energy only in the case of equal oscillator constanls bul can be readily generalil.cd lo unequal

unes.

Keywordc Fragmentation; Goldhaber model

Se presenta una versión cuántica del modelo de fragmentación de Goldhaber. La suposición básica de la dicotomía instanlanca se mantieneinicialmente pero el efecto del campo medio nuclear se toma en cuenta colocando a los nudeones en un potencial de oscilador armónico.Se muestra que, si las constantes del oscilador del núcleo original y de sus fragmentos finales son idénticos y que si el movimiento de puntocero se desprecia. la distribución de Goldhabcr se recupera de manera exacta. Se propone una alternativa a este modelo simple que imponeconservación de los cuantos del oscilador. Esta condición es equivalente a la conservación de la energía solo en el caso de constantes deoscilación iguales pero puede ser inmediatamente generalizada para el caso de constantes no iguales.

Descriptores: Fragmentación; modelo de Goldhaber

PACS: 21.60.$c; 21.60.Ev; 21.60.Fw

1. Introductionlo a elassic papcr 00 nucleus-nuclcus fragmcntation pro-cesses [IJ, Goldhaher showed !hat !he momentum dislribu-tion of fragmenls produced in a peripheral nuclear eoUisionfoUowcd !he rule

(p2 ) = (p2 ) = AlA2 1;;2) (1)Al A2 A _ 1\Y .This expression assumes lhe break-up of an initial compoundsystem with A nucleons into lwo fragmenl' with Al and A2nucleons with Al + A2 = A, and gives the momentum di s-tribution of lhe fragmenllólao;;a function of their rnass in terrnsof lhe mean squarcd nucleon momenlum (P2). (The expres-sion can be generalized to the calólCof fragmentation into kpieees). The derivation of Goldhaber's distribution rests onan assumption of "sudden dieholomy" whereby the nucle-ons of the initial system are randomly assigned to one or!heother fragment without a change in the individual nueleonmomenta. The derivation is entirely elassiea!. The nucleonsare non-interacting and no nuclear mean-field polential en-ters the discussion.

In !his eontribution a quantum analogue of Goldhaber'smodel is eonsidered, InitiaUy, !he quantal veesion is eon-strueted in closest possible analogy wi!h the elassieal modcl,and undcr such assurnptions il is shown lO givc idcntical re-sults. These poinls are developcd in Secs. 2 and 3. Althoughthere are no practieal advantages in using !he quantal model,lhe exercise nevertheless indica tes sorne conceptual diffieul-ties in lhe application of Goldhaber's mode!. Speeifically, lheassurnption of cqual oscillator constants for the initial nu-cleus and the final fragmento:;, neccssary to establish cquiv-

alenee wi!h classieal GoIdhaber, appears unrealistic, An al-lemative to Goldhaber's model whieh imposes eonservationof!he llumber of oscillator quanta is proposcd in Sec. 4 andin the final section sorne conclusions are offeced.

2. Expectation value oC the centre-oC-mass mo-mentum operator

We derive here lhe expression of the expeetation value of thecentre.of-rnao:;s operator (L¡ Pi) 2 in a Slater detenninaot ofhannonic oscillator wavefunctions of the form

where

QUANTAL VERSIONS OF GOLDHABER'S FRAGMENTATION MODEL

The expcetation value of the eentre-of-ma~s operator eontains a one-body and a two-body par!,

The expcetation value of the one-body par! follows from the virial thcorem for the harmonie oseillator:

(PAI LP¡lpA) = mnhwA L (n+ ~)knlm,J nlm

67

(3)

(4)

(5)

with mn !he ma" of a single particle and hw A the energy required for a single quantal exeitation in nucleus A. The summationin (4) extends over occ"pied states nlm only and requires the knowledge of the Slater configuration. A particular Slatereonfiguration is spceified by the oeeupation numbers {knlm} whieh can be ei!her O(empty) or I (oceupied).

The expcetation value of the two-body part between spherieal-ba~is states equals

(PA!2LPi'P;lpA)=-mnhwA L(n'I'llalllnl)' ¿ (~~,~)\n,mkn'l'm',i:.F{s. 4852 (2002) 66-70

68 P. VAN ISACKER, G. AUGER, AND C. GROSJEAN

(10)

juSI as il is in lhe elassieal ease. The symbol C(A, A,) slands for!he number of eonfiguralions whieh are possible if, from lheinitial A-partiele ground Slale, A, are retained and A - A, are removed. NOle !hal lo eaeh possible eonfiguration of fragmenlAl corrcsponds only onc configuration of fragrncnt A2 and vice versa. Thc morncntum distribution of fragrncnt Al 18thenfound by averaging over al! possible eonfigurations:

, 1 C(A,A,) ,

((A,i(~Pi) iA'))C(A,A,)= C(A,A,) ~ (~,I(LPi) I~,),where !he index q runs over all possible Slaler eonfiguralions ~,.

10A,

AN

'"V

In Fig. 1 lhe fragmenl momenlum distribution (10) isshown for mass numbers A = 4, 10 and 20 of lhe initialnucleus as a [unetion of lhe fragment's mass A" The massnumbers of lhe inilialnucleus eorrespond lo closed-shell eon-figuralions of identieal spinless particles in a harmonie oscil-lalor [5J. Thesc resulls are eompared lo Goldhaber's distri-bulion (1), showing a perfeet agreemenl. In facl, one maydeduce lhe relation (valid for elosed shells)

FIGURE 1. Fragmcnt morncntum dislributions (in units ffinhw) formass numbcrs A = 4, 10 and 20 of the initial nucleus as a functionof the fragment's mass Al_ The dots are calculated from (10) andthe curve is obtaincd from (11) with nmax = 1, 2 and 3, respec-tively.

A delieale problem arises wilh regard lo lOro-poinl mo-tion. It is elear !hal, in order lo reeover Goldhaber's elassicalresull, its eonlribulion should nol be ineluded sinee il is apurely quanlUm meehanieal effeel. Bul, even if quantum me-chanics is takcn iOlOaccount. there is the following argurncntnOllo eonsider zero-poinl mOlion. It can be shown [4] lhala ground-stale Slaler delerminant A (r" r" ... ,r A) can bewriltcn as the product of ellA (r~, f;, ... 1 fA)' depending 011the rc¡ative coordinatcs fi == Ti - R, and an oscillator wavc-funelion

QUANTAL VERSIONS OFGOLDHABER'S rRAGME.NTA'I10N MODEL 69

4. Conservation of energy

1= ¡¡(nmax + l)(nmax + 2)(nmax + 3). (13)

"m •." 1A= ¿ 2(n+l)(n+2)

n=O

(15)

(20,36) (28,28)

7.8 x 1014 7.6 X 1016

1.3 X 1041 8.1 X 1042

(10,46)

3.5 x 10'0

4.9 X 1032

(4,52)

3.6 x 10'

3.4 X lO"

(A" A,)

Goldhabcr

N,+N,=N

TABLEI. Numbcr of final-state configurations in the two-fragmentbreak-up oC an A = 56 nudeus.

N ,fA ,fA1 V A; + N, V A; = N,

whieh implies a number of final-state eonfigurations given by

¿ C3(A¡, N¡)C3(A" N,). (16)Nt..¡+N2.¡=N

lions, For a fragrnentation A -+ Al +A2 this number is givcnby

¿ C3(A"N¡)C3(A"N,), (14)Nt+N,=N

Preliminary eaieulations of the momentum distributions ob-taincd by avcraging over aH final-state configurations impliedby (14) or (16) indieate substantial deviations from Gold-haber's law (1). It should be noted, however, thal, even if all

where C3 (A, N) is !he number of Slater delerminanlS for Aparlieles in a harmonie oseillalor with a total of N oseil-lator quanta. This can be eomputed wi!h a recursive algo-ri!hm [61. Table 1 illustrates the number of final-state eon-figuralions in lhe specifie example of !he fragmentation ofa nueleus Wilh A = 56 partieles for a few eombinationsA¡ + A, = 56 and compares it to !he eorresponding num-ber found wi!h Goldhaber's sudden-dieholOmy hypo!hesis. Itis elear !hat many more eonfigurations are sampled wi!h theeondition NI +N, = N.

If the mean-field potentials of the initial nueleus A, and ofthefragmentsA¡ andA, are!hesame,wA = WA, = WA" theeondition of eonservation of !he number of quanta is equiv-alenl lo !hat of eonservation of energy sinee al! quanta havethe same energy. As a eorol!ary one finds that Goldhaber'shypothesis is eonsistent with eonservalion of energy (a~ itis a subease of N¡ + N, = N) but only if equal oseil!atoreonstal1l~ are taken, !hat is, !he sudden-diehotomy hypo!hesiseannot be generalized lOunequal oseil!ator eonstants wi!houtviolation of cnergy conscrvation.

This should be eonsidered a~ a serious drawbaek of Gold-haber's model. Indeed, a more realistie a~sumption is to sealethe harmonie oseil!ator eonstant with the size of!he nuelei asW ex A-¡/3 [7J. This immedialely leads lo unequal oseil!aloreonstanl~ for initial nueleus and final fragments and, underGoldhaber's hypothesis, to energy non-eonservation. On theOlher hand, unequal oseillators eonstanlS can be ea~ily ae-eommodated in the N¡ + N, = N seenario in whieh ea~ethis eondition should be replaeed by

A¡42

6

5

4A

0..3 A~8v2

1

where the upper result is exaet and the lower one is valid iflhe zero-point motion eontribulion to (p')n is neglected. Onlhe other hand, the partiele number A and nmax are relatedthrough

FIGURE2. Fmgrnent rnornenlum distributions (in units moñw) forrnass nurnbcrs A = 8 and 13 of the initial nucleus as a function oCthe fmgment's mass Al. The dots are calculated from (lO) and thecurve is obtaincd fmm (11) with nmax = 1.72594 and 2.35067,respccLiveJy.

Thisshows !hal (11) is obtained from the ela~sieal result (1)if the mean squared momenlum distribution of a nucleon ina hannonic oscillator is taken without it~zero-point motioneontribution. This eSk~blishes eomplele equivalenee betweenlhe ela"ieal and quantal versions of Goldhaber's fragmenta-tion model.

Figure 2 shows similar resull~ but now for the open-shelleonfigurations A = 8 and 13. Again, a Goldhaber distribu-tion is obtained (dots). The curve is ealculated from (11) Wilha (non-inleger) nmax derived from (13). The small deviationsare an indieation of a shell effect in the sense that an analytieeontinuation of !he relation (13) lowards non-integer valuesof nmax and its use in (11) does nOl coincide exaetly wi!h !henumerieally ealeulated distribution (lO). However,!he devia-tions are too small lo be of any eonsequenee in an applieationof Goldhaber's fragmentation law lo heavy-ion reaetions.

Goldhaber's hypothesis of sudden diehOlomy implies eonser-vation of number of quanta, N¡ + N, = N, where N is !hetotal number of quanla of the initia! nueleus and Ni are !hetotal numbcrs of quanla of the fragments. Thc converse is notneeessarily true in the sense that!he eondition N¡ + N, = Nis much wcakcr ami alIows many more final-state configura-

Rev. Me-

70 P. VAN ISACKER, G. AUGER, AND C. GROSJEAN

final-stale configurations in eilher (14) or (16) are consistentwith conservation of cnergy, they do nol necessarily conservemomcnlUm. It wiU !hus be necessary lO limil the final-stateconfigurations lo those thal conserve momenturn a~well, andlo make the corresponding averages of the momentum distri-bution.

5. Conclusion

The objective of this contribulion was mainly of a pcdago-logical kind. It was, first and foremost, the purpose to estab-lish an exact quanlum mechanical analogue of Goldhabcr'sfragmentation mode!' This was achieved by distributing par-licies, initiaUy confined by a harmonie oseillalor, over twoharmonic osciUators. It foUowed thal Goldhaber's momen-

Prescot address: CENBG, Le Haut- Vigncau, 33175 GradignanCcdcx, France.

1. A.S. Goldhaber, Phys. Lel/. B 53 (1974) 306.2. The n here does 1101 refee to the mdial quantum number (the

numbcr of nodes in the radial wavc funetion) but to the ma-jor oseillator quantum numbcr (the energy of the single-partielestate in units of hw). The notation N, usually reserved [or thisquantum numbcr, is here used [or the total number of quanta ofa many-particlc state.

3. M. Moshinky and Yu.E Smirnov, lfannonic OsciI1alor in Mod.em Physics (Harwood, Chur, Switzcrland, 1996).

tum distribution law is recovered exactly if zero-point motioois neglccted. Fur!hermore, it was shown !hat Goldhaber's hy-pothesis of sudden dichotomy is consistent with energy eon-servation if, and only if, the osciUalor constanls of !he ini-tial nucieus and of lhe final fragmenLs are aU cqua!. Depar-tures from !his assumption-which are eaUed for on lhe basisof standard nuclear physics considerations-eannot be madecompatible with lhe sudden-dicholomy hypo!hesis.

A possible alternative lo Goldhaber's modci was formu-¡ated which does allow fordifferent oseiUalor constants whilebeing consistent with cnergy conservation. However, morework is required lO iropase conscrvation of momcntum in lhiscase. Al !he moment it is too early to teU whe!her lhese ideascan be usefuUy applied in the analysis of heavy-ion reaetiondata.

4. L Talmi, Simple Models o/ Complex Nuclei (Harwood, Chur,Swit7.erland,1993).

5. The expressions (7) and (8) can be readi1y gcneralized to par-tieles with spin which wou1d 1ead to similar conc1usions. Forthe sake of limiting the number of possible configurationsC(A, At), this is not done hefe.

6. P. Van Isacker, unpublishcd.

7. A. Bohr and B.R. Motte1son, NuclearSlruclure. 1INuclear De-jormalions (Benjarnin, New York, 1975).

Re". Mex. Fls. 48 S2 (2002) 66-70