!C/jQ/lk9

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

A VIEW OH REACTIONS OP COMPLETE FUSION

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

I.I. Delohev

E.K. Marchev

and

I.J. Petkov

1978 MlRAMARE-TRIESTE

I. INTRODUCTION

International Atomic Energy Agency

andUnited Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHISICS

Inst i tute of

A VIEW ON FEACTI.OHS OF COMPLETE FUSION *

I . r . DelchevResearch and nuclear Energy, Bulgarian Academy of Sciences,

Sofia 1113. Bulgaria,

E.K. Marehev *•

International Centre for Theoretical Physics, Trieste, I ta ly ,

and

I . J , PetkovInst i tute of Nuclear Research and Xuclear Energy, Bulgarian Academy of Sciences

Sofia 1113, Bulgaria.

ABSTRACT

Complete fus ion r e a c t i o n s a r e analysed wi th in t h e frameyork of a t h e o r e t i c a l

model. Energy d e n s i t y i n t e r a c t i o n p o t e n t i a l s a r e made use of and a r e renormalized

for t h e purpose . A l a r g e number of he*vy ion r e a c t i o n s a r e s tudied and t h e c a l -

cu la t ed c r i t i c a l angular momenta a r e compared v l t h experimental da t a .

MTRAMftBE •> TRIESTE

November 1978

• To be submitted for p u b l i c a t i o n .

• • On l eave of absence from I n s t i t u t e of Nuclear Research and Nuclear Energy,Bulgarian Acadaay of Sc iences , Saf la 1113i Bu lga r i a .

Heavy ion physics has become the moat e x c i t i n g and dynamic "branch

of nuc lear phys ics . The primary impetus to t h e study of heavy ion r eac t ions

has been given by the poss ib le experimental syn thes i s of new heavy and super-

heavy elements . The experimental evidence has revealed a number of i n t e r e s t -

ing phenomena t h a t take place during the c o l l i s i o n between two heavy n u c l e i .

At k i n e t i c energ ies high above the Coulomb b a r r i e r , most of t h e r eac t ion e ros s -

section is due to complete fusion reactions and deep inelastic processes.

Experiment suggests the existence of a relation between the two types of

processes. Hence, the theoretical explanation and the quantitative descrip-

tion of complete fusion reactions becomes a necessary step towards the under-

standing of deep inelastic scattering. Most typical of both proceaees is the

considerable transfer of relative kinetic energy into intr insic degrees of

freedom-

The short reduced wavelength of the incident Ion allows for a classical

trajectory of relative motion [ l ] , i . e . one may attempt a classical descrip-

tion of the reaction process. The phenomenological dynamic models [2-5]

consider the energy loss by Introducing a dissipative friction force in the

classical equations of motion. Usually the former is taken proportional to

the velocity. These models do not provide a satisfactory quantitative

description of experimental data on complete fusion cross-sections,

though there is a wide choice of the strength and the form

factors of the friction farce, and the ion-ion interaction potential . The

failure of the dynamic models to obtaia an overall agreement betveen their

numerical predictions and experiments on complete fusion may be hidden in the

-2-

r o i i : i . i.foatiueii"L .if the- ^nur . -v r / ; i , . ^ l ior... i t .* x u l r ' o .rt, c ; . •. i * P :.' • r i ' ^ i o n

o f t i i a c o m p l e t e F u s i o n ?voiitf, i - f i . u i r e s rji,ii . 'V - i r r c a t i i o i c ?m:, f a c t o r s o f t h e

f r i c t i o n f o r c e | 4 1 . ipor.. f a c t o r s of El >.IUC'I >- o n e r r;!!-,,*1 a r o r'-Ti-rr^'M ".V^PT; ti;r-

sfviiE a p p r o a c h i p Hiyilie-i t o L1I;U.> i n e U i K L i c - i n a i r i ;> : j o j . . - i i . ' c t o r - .v>i <> "

nui-ioi'i cn.1 vi t lut ' i : a l /o j i . L .r. s i . i r . , l o s s e s an:i U'.ft a n . . n l : i r .T ;i i' i '..• , i nr.s - L--O

^r^^.ncvK i n d e i y i n o l a ^ t i c s o a U c r i i ^ :iii;- ».orr nii tainei1- i n [ 7 ] w i t h a r e a l i s t i c

rcti,"'p ill ' t h e i ' r j - t i o r . i o r t o 1.1 ini 11.11 i n - o u L i>-,,.::.,oLry ir . tiie ; ;ucl i ; : j . - - i ;ucl i -a» ' o L c a -

t i u l , iiif? ^ i i i iu l t iLneous .\e Lej'uu n a t i o JI o f t l i e c r i L i o ; i l aii-^"ular . u1.:oi.lfi,(tJi!e roi^T)iete

f u s i o n c r i i s s - s e c t i n a s ) r o [ ; u i r " 3 , * M i v e v e r 1 a f r e e aii jusUe-i.: I c - i r . i*t- 'r .

T ' r ip t io s t ^ e m T . i l .;i crt> . c u l i e a i i a l y n i s n i ; o . i i i>in " u i » : i . n - ' lioen j n i i i u s t i c

e v e n t s i s per fon , .K; l by t h e t i n e - u c j i e n - S ' i n t " n r t r : •• 1 —. 0 r -\ ( !.:DiF) n. i roac i i \ o j A ' T i f o r t u -

Ki i tEly i'l)l{f c a l c u l a t i o n ? i^rr ,.-.-]li»..i,](> o n l y t o l i g h t s y s t e m s f o r t h e p r e s e n t

I joci iuse o f t h e 1 ar[i:e Utao^rit o f L '> -i ,ui \ v-iiH. 1-i.,.o + o v ^ r t h c l o ^ ^ ? nicli c a l c u l a t i o n s

p r o v i d e v a l u a b l e i»nowl<v.!.;£e o f t h e t i m e e v o l u t i o n o f t h « syalr:.: o f t v o s l r o i v l v

i n t e r a c t i n g n u c l e i , T h i u ; , t i i c c o n t o u r " J u t s I 'nut - i vo t h e deu.-,ii.v 11 s t r i b i i t i o n s o f

Uif.-pl. and p r o j e c t i l e a s a f u n c t i o n o f f-irms i ! ! - „ r o r tiift f 0 i 1 o w i ti'i n u a H t t i t i v c

conclusions[fijt l.'ihe distor siotis of thp i< - -11y distributions o i' target niv!

projectile are iuaiynil'icaiit at tue first «U[-c of mutual overlap, i . e.Cnul o. t

barrier penetration is not accompanied \JY r* aovice^Lle deforiiiation of "both nuclei.

2 .The energy absorption is very sharp.'.jtAlter clutching,the system starts

oscillating around an equilibrium distance '.1 t-vci-vi the inerti;-.l co'.itres of the two

nuclei.

i such oscillations m*y passitly lead to fission of the compound

system. Thia stage of the reaction process is quite analogous to spontaneous

fission and a-decay phenomena. This probability cannot "be accounted for vithin the

frame-work of the classical dynamic models, where a certain trajectory is either

scattered or trapped in the pocket of the interaction potential.

- 3 -

;••• .]• :n t i.i-jor ui 'fers •» i o d d which descr ibes a.t)r>roxi;~ately the o s c i l l a -

t ion K "• i -•• '-•••' '•' ic le i ..^tli r e spec t lo f-ach o ther by a riuasimoleculhr s t a t e in the

>>!<--.lie;. I n . . 1 (iLeiiti.-l «el l of Liu- tooal ion-ion i n t e r a c t i o n p o t e n t i a l .

I I . THE MODEL OF COMPLETE FUSIOH

... • i.. :oli t Lor. i..ioii'-: Li.e nucl"ii.r . - t t r i ic t ionj tiie Couloiib repuls ion , anil the

<j' i. . r i i\: <v.\l ^rifcf a l lovs the formation >f an at t ract ivf? well in t'.ie

nuc!« ias-i".c]f.!5 inter . .c i ion p o t e n t i a l :

CR) ,

is the distance ljetT/(-Rn tile centres of target and pro ."jectile, V *^J\.) is the

i L.urn-,Lion >ot«ntial, V/" \_\\) ^ 3 the Coulomb repulsion>and \/p ( R ^ —

i s the effective centrifugal potential.

iletp fusion is considered as a tiro—stage pro cess-'I lie f i rs t step i s the

>enetratiou through the Coulomb barrier.A particle of reduced mass

• with an orbital angular momentum K and incident energy E , penetrates through the

interaction barrier with a probability '£ (_ EJ ) .The latter is calculated in the

qua si classical iKKB approximation.lt is assumed that energy losses are negligible

during the barrier penetration-Tile second stage of the process is the formation of

a nuclear quasimolecule vfcicli should be identified -iri-th the loirest possible quasi-

stationary state in the potential well of V / ^ R ) .The first oscillation of the

quo-Kimo] ecule(the first attempt for tunneling back into the entrance channel)

dct«naines the probability that the compound system will reuain in i t s quaai-

"ciolecular state.In case that no tunneling occurs,the corresponding partial ifave i«

considered contributing to the complete fusion cross-section.A sudden and strong

(lar'je) energy relaxation is assumed during the short time after the barrier

penetration and before the formation of the quasimolecule.The present model does

not specify and doea not treat the rather complicated intermediary s t a g e a t

• • ; • • • • - ' 3 . . :

the effective particle is trapped on the quasibound level.In the time interval after

the trapping o» the level and before the first collision on the potential barrier

(fv££ " iO~^\«)> t l l e l o 1 r e s t st^-te i n t l l e resultant potential well may be viewed

as the ground state of a spontaneously fissioning heavy nucleus.i'he level width

due to diffusion of nucleons ia negligible since diffusion is a slower process.

The higher lying states in the potential well are atrongly coupled to intrinsic

and other degrees of freedom,i.e. their level width is much larger and the energy

spectrum above the quaaimolecular ground state is very close to a continuous one.

Hence,trajectories of motion are meaningful only above the quasitiolecular ground

state. l t should be remarked that the forbidden appearance of the effective particle-

below the first level in the potential well is an essential result of the present

model and i t is a consequence of the quantum-mechanical treatment of the xi. —degree

of freedom.Classical dynamic models J_2—5J do not tulie this fact into consideration

and ao they are evidently forced to employ unrealistic friction fora factors vhen

trying to describe complete fusion events.

The probability k-Mf?) th.it I ho already forned tiuasi stationary state will

not decay hack into the direct fission channel is determined fay the penetration

probability la C£tf ) :

where is the of the lowest possible state in the potential

and \jJS lire calculated in the .flvJ] a;n)roxi;ar.tion. I''IUS, the partial coni.lctp fusion

probability Xe crI

a t a ; r i ve i l v a l u e o f L'le c c n t r r . i - o f - ; . ! : i , s s • i n n t i \-. i-n':imi\ C* ,nul F<> r

T h e co v - N - t ' 1 L u s i o n r r o w K - - i - 'c t iot i i.'i cr? L t?u 1 : L , .:

(3)

The infinite summation in (3) is terminated when:

i.The entrance channel kinetic energy becomes lower than the energy of the respect-

ive quasimolecular state.Obviously, for a given kinetic energy range the number of

fc -waves that contribute to the complete fusion cross-section Ogp * • (3) *1*1 *«•

larger as El increases.

2. In the case of high t -waves, the steeper centrifugal potential mftJceB the

corresponding potential wells shallower)while at some maximal value of the angular

momentum t the quasistationary state disappears.This maximal £ -val»e is not

identical to the critical angular momentum which follows from the sharp cutoff

fo rmula:

ZJi E(4)

'\urafrical calculations within the framework of the present model require

a <(ef'initr choice of the nucJ eus-nucleus interaction potential. In this, paper v« h«Ve

used ri:cr v Jruiaity iqtentiitls [_lOj,ihe l a t t e r make use of the ground stnte

(•(•i:::iL ili.; Iri !iutions .mi1 are cul cul.itud in the sudden collision npproximation.Tem-

:>i;r:iturr-i;f-r)Bn(!('nt Hartrnt—Fock nati.jiites JjtlJhave shown that density dis tr ibut ions

of' nuclei ur'; f.iior?illy inTiuenced b\ the temperature of the system though moderate

• ••. ciu t i n ' -: w uo u'..:t:'.nt i.-l nffi-ct on Clip density difitribution parameters, i-ence,

• •'•y-L-i; i i - r : 5 i l ' io t i ^ n t i a i .s ^ r o r - ' n l i < t i c i ' £ iouKh f o r n o t v^ry l i ^ r ^ ' e e x c i t a t i o n s o f

; •' '">• •''•'• "-"•.'t-rA i ! i-. ''•"• i - J T i ' d i v p ': ' .1 I ' - v i t l u p r i u i u s a m i t h e . s u r f a c e t h i c k n e s s

-•'*":•- "••'•••!• '' i ' . i i . i t . 1 ' i - t r i i . ' U t i o n i n t r c i u c ;ii )i i :iJi ^ e a n e r i i t u r e s . L a r g c - r J i n l f -

• • ; : ' i : ' : - ; < • • ' • •:) r ' ' i i : . ' : I . . ' •- . ' - ; iiirn-i-i-.se H I P i iuc l i . ' i i r i i U r u c t i o n

- 6 -

around the minimum of V w C R j a n d lo^er the value of the interaction barr ier

Let us consider the case when the incident energy in the entrance channel

ia close to both the interaction barr ier and the bottom of the potential well.That

happens for the high partju.1 t- waves which contribute to the complete fusion cro

cross-section can be obtained after a modification of the surface terwi

section V. p .The til lowed energy loss i s evidently sina.ll and "frozen" potentials

can be used.Note that these high f, -waves have the largest par t ia l weight in the

summation of the complete fusion formula (3).A deader interaction potential irell

due to internal excitations of the nuclei at lower intermediary £ —values will not

affect too much the direct fission probability of the qua.simolecule.Gne might be

obliged to follow the energy dissipation along a trajectory of motion for the lowest

(i —waves since a large amount of energy transfer is possible in that case. Further—

more,o consistent treatment of the excitation enerfiy effect on the ion-ion potential

Bay be neceasary.lt i s possible that for low values of the impact >.trimeter some of

the t ra jec tor ies mi^ht get scattered by t!ie repulsive core of the resultant

potential without the formation of u, iiuasiiiolccul".; .is nossibi I i f,y is nol accounted

for in our model.Nevertheless, the stuLurnd low £ -Mives will not influence too

much the value of the computed complete fusion cfo ?..->- section since their partial

weights in formula (3) are rather sciallo

I I I . THE IOH-I0H INTERACTION POTENTIAL

The present work is bused uinjn heavy ioi. Interaction pi, tential s calculated

in the sudden collision approximation ivibhin tlif- '•nertj density formalism [1C,13]

VCR) = (5)

i,s t h e ener ; rv d p i i s i L y , jetf t i 1 <i ,:bo"t. t l io n c i l ' l c c ' - u i c t > <• £ ( ? ) i'<j i- 11-

c a s e o l ' LN- r e r m i d e n s i t y d i s t r i b u t i o n u sed h e r e i n car . be found i r . I'.U) . The

.< t u d y o f t h e C O K L ' I 1 o t o T u ^ i o n c r o y f ^ i f c t i n n . s o : : 1.. . v | j j L ' M l L- ^ ^ i i . /

i l l s a g r n p i : u : r s t l i c t ^ c e n t l i r o r . v : i i s i i e x r i ' i i . . : : ' L L J ' i . j r i 1 ! . 1 \ l u c : 1 i ^ i 1 t ^ i n . > •'•••[,' f u s i o r *

(R) =\of the energy density potential.Such a modification can be carried out if one

replaces the'VJ parameter by an effective local variable Tj ( ,R,) o f t h e

following types

4,06^40 4.06 >S( 8 )

where

..Idle in react ions,where one of the two nuclei has a mass uumber j \ £iXiW j the

;>ura..ifti1rs i " (f>) take values £} "~"2.§0 ' C = 3*J 0 , and wiien the mass

numbers o t * arifet and project i le are greater than 20 J *-> —— ^ 6 9 > and

( =s 5^D^) .This difference in the parameter choice is due to the universal

KY!:vr;(-triz<!'] ^enai density dis t r ibut ion used in a l l ca lcula t ions . l t ia well

"sia'il isliud,ho jever, that density dis tr ibut ions of l ight nuclei are much closer to

,i "iuufMiii!i Liu- ii to a Ferrai di stribution.The different surface gradient of the

• W-iLsity -\L -irLbntioti ;Li'f't:cta mostly the surfo.ce |>art of the nuclear ion—ion

i) separate .'^Ls "I' n;ir:uetcr min t s are uaed in (6) when the

t i l " 'us" uuiiltir i.s :ii !icr or lover than2,0 .'I'he eorreeted

• n i i i - i T . i L L i M c t i i m l i t t l i t ' l r r i n h e r u l o v i T l i i p r e g i o n - V f i . r i o u s

>i<: !O'I i !' i <: • t . i o r it i' L.iie j i: I . e r f c t i u II >(i t e n t i n 1 • I t s h o u l d b e n o t e d ,

• '• - u r i -UM (•'!•!:; i i - n t ) I I T . : I i s t h e l iMi i i t ,•<:-'t i f ' i e d q i i i - n t i t y i n

'••r I i s o u . u r t l i e r l o r e , i t i . , is <<'ii s i m w n I 1 J 1 I l i n t a i i i i r e

-8 -

11.L1 i - : ' . t : L i u i i . c- c < : ,

1 - . M - . - . - 1 . o f L I i c '•[•:)

:! L i . - 1 i ; ; I . . ' i : L t ;i

•• j i t : . ' - . • f t

i i ' 1 • . ! : , i ' . . t

general definition of L'ie surface energy tera doct1 ;;ot change J-'c ,ir.diug en<?rf.vattraction

va lue of ., s epa ra t e mic l ras t u t b r ings about noticisc.'ol e piUtitini'.;:! • i r •*.!•<? 3iirTdc?

i j j r t of the nuc lea r i n t e r a c t i o n i o t c n t i . i l . The conclusion.? :~ir;i>rr i'- [15] are in

:;ualii t i . t ivo agreement u-it! '*•?• Present a.;<">roi£iriiatio_ri '.bout tiie 'urTnc"; i n t e r a c t i o n

energy* i i ' i s renoru.il i z a t i o n osn account also for the ''./niiaiic

deformation of tar '^nt and p r o j e c t i l e .

IV. RESULTS AND DISCUSSION

The inoiiel Ascr ibed in Sec . I I t a s been appl i f J to the study of c<i:..;ilete fusion

r e a c t i o n s between court lex nuc l e i ,The numerical r e s u l t s obtained re- repor ted in tl ;e

t a b l e . T h e r e i s s a t i s f a c t o r y agreement ')etween the ;:, l . - i l ' u f d :•'••.. :.'u>. ..-^.leri .• n t a l

va lues of the c r i t i c a l an:;-.: 1 u.r ;,io:i]:.i)ta for a l l , . . . irs 'i 1' nuc le i ::-iv<<i; in the t a b l e .

The simultaneous d e s c r i p t i o n of the •* Ar + ih and the o n T,r + * 3 E i

r e a c t i o n s should be noted, t i i ice a l l o ther u.ppro,\cni:s l£—5j i.^vn ."li-iied a t t h i s

ooint* i't e c i l cu i ; ; t ion.5? ire only" 'iuai i t^ tvivci^ in .:\ .i"o^;ji:iit i^itfi tlit? PXTicri ; e n t a l -

Ij found channel effect [16] vht;n d i i f c r e .t co ;il>inii. c ion« of tLtr(: f^ umi m-o,Tactile itre

fused, mud form one ;ind tne sa."ie cj:.:pou,' .i nucU'U-; ( B + Tb, C + (3d

0 + "*^Sta, ar;;i * " i e + Mo }. The crosf-i-tction •! viemlence on the

relat ive kinetic energy i s of ^rpat i,.:wriance .'nr ike study ;>f tin? co:uj>lete fu-

nion mechanism. I t is seen from the table that t • ''csent riui-erier-.l culcul^tioj.fl

risproduce fair ly well the experi.uenv.il energy C'i ! i t-aca of t'.o critic-il angular

i;ioKienttt. That becoaies evident in the ease of tin- " i r + ** 3b reaction where

the calculations are set against the values of wi*.- crit.ical an;..;!ii^r .jiOMonta oxr;e—

riiueatally deduced at six different V5.1ues oi' the d . r t r i - o f— cs -s inetic energy

ot"

1120, 266J MeV ) . Saturation of - y . in n-n.ctions Lutween 1 i .;!it ruclei

s tar t s tit lower values of £ as compered njtii t!it> i-x iori.i:eiit. At uresent the deter-

mination of f£r(*v* a*. very hig^i kinetic energies rsnains an

open problem. Reactions at ;l.i^li kinetic energies are ch:t.racteri?.orl Ly tbo large

amount of excitation energy which possibly leads to a different .,t?cliiuiisn of cca-

• c h ::.'; '.:•> t i - . i l l . - i l f j ' o i i,:i ,;ur .'.li.ul.tr :O,:]Ri: liurj.

i'1- I'C : . i - 3 ; OMiiiiin; ir. L.ie .u-f-.e'it, . o r i , . •:ois i:.-:t t w -ironosed t h e o r e t i -

!,] "Oiii'iur:: c o r r e c t l y accounta i.ir iiii- , . a in f e a t u r e s of L-'-D complete fusion

yocsKj 1 . • ;or . . ; i t ion oi' ti :.ur; ii-tir quasiroolecular syateuif t he .^iaxiiaa.1 energy

;=si-;r.tion and the Cou'o-Jb ijii,rri"r tunnel ing , «i:ich i s most ly r e s n o n s i b l e for

e cijaniicl :;-:<{ enori?y :ei>ent[ence oi1 t l ,e coap l c t e fusion c i o s s - s e c t i o n .

ACKHOWIiEDCMEBTS

One of t h e a u t h o r s (E.K.M.) would l i t e t o thank Profeasor AMua

t h e I n t e r n a t i o n a l Atomic Energy Agency and UNESCO fo r h o s p i t a l i t y a t the

International Centre for Theoretical Physics, Trieste. He also viahes to

thank Professor L. Fonda for careful reading of the manuscript.

-9-- 1 0 -

TABLE

&rp<?ri.;ient;il -Lnd t h e o r e t i c a l va lues of t'.'R c r i t i c a l . insular ::in;nei;tii For

d i f f e r e n t contr^-of-ni,-L^.s ! - inet ic energies^

TABLE ( c o n t . )

Centre-of-..i;iS3 i iaet ic

i ^ne r f i - ( i n i ^ e v )

12 c + *ttT i

c +6«Ni

31

44

G6

59

W

69

125

65

78

144

37

53

67

80

81

149

rre.-.ent remits .-. it-rimer.;. ! [lefcr^.ce

o n -. ••*? CV\L±- v a i n o s o f t h o

cal angular critical antiu—

£4

25

2-0

21

30

34

35

35

35

19

1'4

as

2S

2 s

29

us

29

32

49

± 2

± 2

+ 3

± 3

+_ 3

± 3

+ 3

± 3

+ 4

34

27 + 3

34 + 3

3 5 + 3

38 + 3

5 0 + 4

17

17

17

Centre-of-mass Present results Experimentalkinetic energy on the cri t ical values of the Eeferenc.(in MeV) angular momenta cr i t ical angular1 Qfflenta

63 ,,..

63o + Cu

37

54

SI

82

10(5

150

113

117

119

107

66

101

128

134

21

33

•i'o

35

35

35

35

40

47

53

43

35

37

43

45

21 i 3

£9 + 3

£5 + 3

35 + 3

39 + 4

45

50 + 3

40 + 4

46 + 4

53 +_ 8

40 +_ 3

34 + 3

27 +_ 3

41 + 3

44 + 5

18

I S

16

m

18

10?0 + Ag

0 +

146

124

57

53

46 + 5

58 + 4

13

1ft- 1 1 -

TABLE Ceont.)TABLE (oont.)

Experimentalvalues of the Heferencecr i t i ca l angular

Centre-of-mass Present results Experimentalkinetic energy on the crit ical values of the Reference

in (Mev) angular momenta critical angularjiqnenta

cen-tre-lof-mass present resultskinetic energy on the c r i t i ca l

in (He") angular norcenta

n . if*.0 + Au

2 0 N.

20 „ 63 „He + Cu

155

115

159

177

1-iti

192

70

60

76 + l i l

43 + o

o4 t 6

56 ± 6

39 + 1-1

13

21

18

ko Ar + oh

Ar +

135

148

168

195

226

190

254

2E

56

7 3

88

106

123

29

57

80

9 1

114

132

± 3

± 5

+ 3

± 9

i 12

+ 13

120

SO ± 5

1S9 + 7

23

161

102

71

107

7o +• 3

oii _r 8

C3 ^ o

IV + o

1 9 2 1 4

35T

370

81

102

•4C-50

0-20

92 + 6

127 + 7

35

18

24

24

96

132

138

EC j f 5

70 +_ ":

7C + 10

20

210

70

no

± •"'

i s

_+ 7

-, j , . , _

REFERENCESCURRENT ICTP PREPRINTS M B INTERNAL REPORTS

[1] R.A. Broglia and A. Winther, Pliys. Eep. Uc_, I55 (1972)-

[a] D.H.E. Gross and H. Ketlinovski, Phys. Letters k&B, 302 (1971*).

[3] J .P . Bondorf, M.I. Sobel and D. Sperber, Phys. Eep. 15C_, 84 (197M.

[it] E. Segl ie , V. Sperber and A. Sherman, Phys. Rev. Cll_, 1227 (1975)-

[5] R. Bass, Mucl. Phys. A231, 1*5 (197M.

[6] J .P . Bondorf, J.R. Huinenga, H.I. Sobel and D. Sperber, Phys. Rev,

Cl l , 1265 (1975) .

[7j K, Wylczynska and J. WylczynskI, JIucl. Phys. A26H, 115 (1976).

[a] P. Bonehe, S. Koonin and J.W. Megele, Phys. Rev. C13, 1226 (1976).

[9] P. Bonche, B. Gramnaticos and S. Koonin, Saclay prepr in t DPh-T/DOC/77/128.

[10] I . I . Delchev, E.K. Marchev and I . J . Petkov, Bulg. J . -'hys. j)_, 3^7 (1976).

[11] G. Sauer, H. Chandra and U. Mosel, Nucl. Phys. A26i», 221 (1976).

[12] I . I . Delchev, E.K, Marches and I . J . Petkov, unpublished.

[13] K.A.. Brueckner, J.R. Buchler and H.M, Kelly, Phys. Rev. 183., 9^ (1968).

[Ik] I . I . Delchev, E.K. Marchev and I . J . Petkov, Bulg. J . Phys. 111_, 238 (1976)

and JINR, Dubna, preprint Elt-9^56 (1976) .

[15] I . I . Delchev, E.K. Marchev and I . J . Petkov, JINR, Dubna, preprint E1*-101t20

(1977).

[16] A.M. Zebelmann and J . Miller , Phys. Rev. Letters 30_, 2T (1973);

A.M. Zebelmann et a l • , in Proceedings of the Third IAEA Symposium on the

Physics and Chemistry of Fission, Vol.11 (IAEA, Vienna 1973) p. 355 -

[17] J.B, Natowitz, E.T. Chulick and M.H. Nsmboodiri, Phys. Rev. c6_, 2133 {1972).

[18] J.B. Natowitz, Phys. Rev. Cl> 623 (1970).

[19] V.E, Viola J r . , R.G. Clark, W.G. Meyer, A.M. Zebelmann and R.G. Sextro,

Hucl. Phys. A.S61. 171* (1976).

[20] J . Galin, B. Gatty, D, Guerreau, C. P.ousset, U. Schlotthauer-Voos and

X. Tarago, Phys. Rev. C5, 1113 (197M-

[21] L. Kovalski, J.C. Jodogne and J.M. Miller , Phys. Rev. 169, 891* (196B).

[22] H.K. Guttrod, H.C. B r i t t , B. Erkkila, R,H. Stokes, Z. P l a s i l and M. Blann,

IAEA Report lAEA/'SM 17^/59 (1973), unpublished.

[23] M. Lefort, Y. Lebeyec and J. Peter, Riv. Huovo Cimen-fco _, 1 (I97I4).

[2U] M. Lefort, Y. Lebeyec and J . Peter, :'hejr.ie NiiCleaire,InstiT;ut physiq.ueHucles.ire, Orsay, Report IPNO-RC-73-C!1* .

IC/78/80

IC/78/81

IC/78/82

IC/78/83INT.REP.

Ic/78/84INT.REP.Ic/78/85

Ic/78/86INT.REP.

Ic/78/87INT.REP.

Ic/78/89

T. BARfTES and G. I . GHANDOTJRs ttriasmann functional Schrodinger " -na t ion ;An application to perturbation theory.

M.M. HAFIZ, A.A. AMMAfl and F.H. HAWAD: Trssmission oleotron mi: roscopjof thermally induced trangforroatians infilms.

J , IiQKIERSKIi Supercon formal PTOUC and curved fertnionio twis torspace.

* L. FO1IDA, C.C, OHIRAREI, C. OHERO, A. RIMINI and T. WBBEHi Quariidynamical semigroup descr ip t ion of sequent ia l decays.

162• H.R, PALAFI: Theore t i c" ! :rives-tif:ation of anomalies in Er,

C O . TIWACHUKU: ExpressionE for the eigenvalues of theopera tors of the 0(H) and ; ' . (2n) .

* H.R, DALAFI: Inves t iga t ion of Trast hand in Tb-nucls i .

* C. ARAOONB and A. RESTUCCIA; The Baker-Campbell- Hauadorff formulafor the o h i r a l SU(2) supergroup.

M. DE CRESCENZI, G. HARBEKE and E. TOSATTIifree eioiton polari tons in semiconducting filras.

CT diss ipat ion of

IC/78/92

IC/78/93

IC/78/94

IC/78/95

IC/78/96

lc/78/971ST,REP.

IC/78/100

IC/78/101

IC/78/102*

INT.REP.IC/78/1O3*IHT.REP,

IC/78/104*IHT.REP.

a, nltJLJANI, E. TOSATTI and H.P, TOSIt Quasi-one-dimenaicinal oitnttoinsu l s to r ,

T. SUZUKI, A.C. HIRSHFBLD and H. LESCHKEi The ro la of operatorordering in quantum f ie ld theory,

A. BALDEEE5CHI, R. CAR and E. TOSATTIi Local f i e lds in #roup IVnamironduotorn, MfO nnd W.il'1 .

U.K. l.EBi Mnnl I'eiitat Lnn-ntran^th of colour in two-photon procanrnB|below and above colour thrfluhold.

K. ABTA and S.5. JHA: Tifht-bindinp o rb i ta l model for third-ordernon-linear opt ical su scep t i b i l i t i e s in PTOUTJ I V c r y s t a l s ,

K. TABIR SHAH and H, YOURGRAO: On some aspects of themathematical foundations in physical theor i e s .

A. ADAMCZ1K and R. RACZKA: How r e l a t i v i a t i c wave equationsassocia ted with indecomposable r ep resen ta t ions of the PoincarS woup ,

R, R^CZKAi On the a i i s t a n c e of quantum i l e l d models in four-dimenaional apace-time.

mR angles of the vector and

A note on the Hall-Post theorem

C.V, SASTRY and B. MISRA:scalar meson, i sog ing le t s .

S.A. AFZAI, ana SHAM3HEB 'u0-oluster s t ructure of nuc le i .

SHAM3HER ALI and C. SCHIFr:.j:.j Theoretical aspects of the non-1folding model and the Persy-Buck po ten t i a l .

a l

* In te rna l Reports: Limited d i s t r i b u t i o n .THESZ PREPHINT3 ARE AVAILABLE FTOM THE PUBLICATIONS OFFICE, ICTP, P.O. "BOX

f ) * I.A. ELTATCS taA *„«.*. tftSSAffi On * • *M*4*««** evolution of sandINT.HJSI', dunaa.

IC/78/105 * E. RECAKI and K. TAHIH SHAHJ Multiply-connected space-time tiln. •'•I F T . R E P . holes and taehyonB,

ic/7fl/l°7 * R. MICHA3 and L. KOWALEWSKI: Dyson equation for transverse Iain/1ST,REP, «.»••••.-.«+

!C/7B/lj2 * G. GIULIANI and E. TOSATTIs Longitudinal phonon sueotrun of incom-INT.REP. manaurate one-dimsnaional oharpe-density-waves.

IC/78/109 * L.F. ABD ELAL: A Oalarkin algorithm for solving Cauehy-type sinfularIHT.REP. integral equations.

I c / 7 8 / l l l • M.A. ABOUZEIB, A. HABIB and A.A. EL-3HEIKHj Single neutron tramiferIHT.RBP. reaotions between heavy ionsi semiclaaaioal treatment.

Ic/78/]W * Second Latin-American workshop on self-consistent theories of condensedIKT.HEP. matter - 9-20 October 1976 (Contributions).

IC/78/113 * M.Y.A. TOUSSBFi Short note on the mixed KdV and MKdV aquation.IHT.RBP.IC/78/114 • 3.A. EL HAKIL and M.H. HAOGAO: Asymptotic solution to the reflectedIHT.RBP. c r i t i c a l s lab.IC/78/1J5. * K.A. 3AAD, M. SAWAS and 3. 3HEBLi Reaonanoe integral in fuel nithIJPT.RBP. frrain s t ruc ture .

IC/78/H6 L.SH, KHODJAEVt Basis of dynamical models of quantum th»ory loca l -isable interaotlons - Ii A model of exponential Lagranglans.

IC/78/117 • P. CAHBACZEWSKIi Almost Fermi, Boae dis t r ibut ions or sp in- l /aINT.fiE?« approiimation of Bose modes in quantum theory.

IC/78/141 * 3.K. PAWDBYI On oonformal invariance of gravi tat ional wave equations.INT.REP.Ic/78/l.i:1 « P. Btimwi, P. FURLAW and R. RJJCZKAs Poseibla origin of iaotopjo spinIlfT.REI', from extended conformal aymmetry.

IC/78/144 * L, BERTOCCHI and D. TRELEAIfli K° re^enaration in nuclei and the affectIHT.RBP, of the inelast ic- intermediate a ta tea .

10/78/1.18 • A. QADIHi Field equations in twistora.1ST.REP.

• K.H. MARCH and M.P. TOSIi Hydration of divalent ions in ice .

1C/78/122 •]:HT,REP,IC/78/123 •IST.RIP.IC/78/124 *IHT.SBP.IC/78/125

0. FUELAlf Mid B. CAVAi Heron solutions in a non-abelian conforms!invariant Higgs model.J . SAHTAMAHINAi SU(4) rang-Milla f ie ld solution.

H.B. CHASSIB and G, BASKARANs A He effective interaction for di lu tesolutions of ''He in liquid -%e at low temparaturea.0. SHfATOHB, M. PAHRIHBLLO and M.P. TOSIj Optical absorption ofdilute solutions of metals in molten aa l t a .

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