2.N Nuclear Physics A296 (1978) 177-188 ; ©North-Nollaisd PwWWÜna Co., AntatsrdantNot to be reproduced by photoprlnt or miceo8lm without wrhtm permioba 6tom the publhher

A MODEL FOR HIGH-ENERGY HEAVY-ION COLLISIONS

WILLIAM D. MYERS

Lawrence Berkeley Laboratory, Unioersity of CallJornia, Berkeley, Cal~/ornia 9720

Received 8 September 1977

A6strad: Amodel is developed for high-energy heavy-ion collisions thet treats the variation across theoverlap region ofthe target andprojectile in the amount ofenergy and momeatum chat is deposited.The expression for calculatinganyobservable takesthe formofasumover a series ofterms, each oneof which consists of a geometric, a kinematic, and a statistical factor. The geometrical fedora for anumber of target projedik systems are tabulated .

1. Ltr~otlaction

Heavy-ion collisions at high energies produce large showers of different kinds ofparticles. .Nucloons, light nuclei and pions are observed over a wide range of energyand angle. These various products seem to be associated with three relatively distinctsources. In the more peripherai collisions substantial fragments of both the targetand projectile may remain . These "spectators" may be highly excited leading toparticle emission (more or less isotropic in the rest frame of the fragment) and to adistribution of final products (evaporation residues) . In addition, some sort ofintermediate composite maybe formed from those parts ofthe nuclei that overlappedduring the collision . Particles from this region should be distributed in velocitybetween the target and projectile and if each particle experiences at least two orthree collisions on the average they should also be spread fairly widely in angle.[Refs. t " a) provide an excellent review of the experimental results.] This part of thecollision complex becomes more important with decreasing impact parameter.For central collisions where the entire projectile overlaps the target, no projectile"spectator" remains and most of the yield of the reaction is expected at intermediateenergies and widely spread in angle').The relative yields of the different products observed in this intermediate region

and their energy and angular distributions can be estimated from simple geometric,kinematic and statistical considerations. Such estimates serve to establish a "back-ground� against which the experimental results can be compared in order to searchfor anomalous behavior that might be associated with new phenomena. [Ref.')compares a number of different theoretical approaches to this problem with themeasured data for a particular experiment . It also contains extensive references toother work in the field.]

In the following section, a schematic model will be developed for predicting theIn

17 8

w. D. MYERS

distributions of products whose source seems to be the overlap region between thetarget and projectile . (Some of the geometrical features of this approach are similarto those employed in refs. s' s, e) .) Themodel itself is somewhat more general than theone used in earlier work along these lines 6). It includes a recognition of the fact thata velocity shear exists across the overlap region that depends locally on the amountof material coming from the projectile. In addition, the model is based on nucleardensity distributions with diffuse (rather than sharp) surfaces t . A special way ofsummarizing the geometrical aspects of the model will be introduced that allowsone to immediately assess the importance of its various features .

2. The model

In the approach described here the collision of a high-energy heavy ion with atarget nucleus is viewed as a totally inelastic process in the overlap region. Con-sequently, its range of applicability is limited. It is not expected to work at very lowenergies where the whole target and projectile may combine andsubsequently decay.At extremely high energies the target may become partially transparent to theprojectile due to the falloff of the nucleon-nucleon cross section.For any given collision the ratio of the amount of projectile matter to the amount

of target matter it is aimed at varies with position in the plane normal to the beamdirection . The diffuse nature of the nuclear surface region tends to smooth out thesevariations so that a continuous range ofq (the "projectilefraction") values is expected,where

_ number of particles from the projectilenumber of projectile plus target particles ~

(1)

The value of q can vary from zero (for the target spectator) smoothly through inter-mediate values (corresponding to the overlap region) to one (for the projectilespectator fragment). Between the target and projectile spectators (if the impactparameter is largeenough for any ofthe projectile to survive) a distribution ofmaterialis expected, which contains a large amount of internal energy andwhich is drawnoutinto a streak by the velocity shear across the overlap region .

Fig. 1 is a schematic two-dimensional representation of the model to illustratehow the nuclear density distributions are expected to evolve with time. In part(a) ofthe figure the material in the overlap region is seen to be moving with velocitiesspread uniformly between those of the target and projectile spectators. Some of thematerial has already "burned off' because of its high degrce of internal excitation .In part (b) the process has continued, and in part (c) the hottest part of the streakhas completely dissipated leaving behind the spectator fragments (that probably

f The importance of the nuclear difïuseness in these processes .was brought to my attention by J .Kapusta .

HEAVY-ION COLLTSIONS

179

~°Ne-~~U at 250 MeV/n

t " 10

t "20

t"30Fig. 1 . Schematic two~itpensïonal representation of a collision between a 250 MeV/n =°Ne projectilewith a sa°U target at an impact parameter of 8 fm. The three parts of the figure show how the system isexpected to have evolved at times t ~ 10, 20 end 30 fm/c. The intensity ofthe shading is proportional to

the degree of internal excitation generated by the collision .

begin to tumble because of the angular momentum imparted) which continue toburn off particles from the excited regions on their surfaces.

If the collision is completely inelastic the local values of the forward velocity vand the internal energy pcr particle t are givelp by the simple expressions :

v ° ~lvt,~ .m,

t = ~(1-~Iku"" m+

wherev~ is the velocity ofthe projectile beam, tt~ is the kinetic energy per particleof the projectile beam and >f (the "projectile fraction") is defined in eq . (1) .

For higher energies the simple classical expressions eq . (2) are no longer adequatebut the principle remains the same, as can be seen in fig. 2. In this figure the velocityofthecomposite system ß~ (or rather its ratio tothespced oflight, ß = v/c) is calculatedusing the correct relativistic expression for the case where the projectile initially has

180

W. D. MYERS

Projectile fraction,

Fig. 2. The velocity ßa and internal energy per particle to of a composite system (part target and partprojectile) are plotted against the projectile fractioq' q for the case where the initial projectile laboratorykinetic energy is 400 MeV per nucleon . The apprtipriate relativistic expressions were used to calculatethesecurves, and the to curve is shifted downwardby 8 MeV because ofthe binding energy ofthe nuclei that

must be overcome.

400 MeV/n kinetic energy . Its value is plotted against the projectile fraction q.In addition, the internal energy per particle generated by the collision is also plotted.Note that this latter curve has been reduced by 8 MeV/nbecause ofthe binding energyof the particles that must be overcome.

In fig. 3 the distribution in the x-y plane (perpendicular to the beam direction)of various quantities of interest is shown for the specific case of ~°Ne+23aU at400 MeV/n. The figure consists of three vertical columns corresponding to threedifferent values of the impact parameter, 6 = 2, 4 and 6 fm . Diffuse surface densitydistributions are used, but the location of the equivalent sharp surfaces is indicatedin part (a) (the uppermost entry in each column).

In part ro) of the figure the sum of the target and projectile density distributions isshown projected on to the x-y plane. Thecontour lines are at 0.05, 0.10, 0.15 and0.20.The maximumvalue is also indicated. (Actually the values shown must be multipliedby 12.5 in order to obtain the densities in units ofparticles/fm 2.) Not all the contours.are shown but only those that lie within the overlap region where the intennal energyper particle after the collision is positive.

In part (c) of the figure the local value of t, is indicated by contours at 0.1, 0.2, 0.3,eta, and the maximum values are indicated.

l:n part (d)thelocal value ofß, the velocity in the z-direction, is plotted with contoursat 0.1, 0.2, 0.3, etc. These plots are very similar to the ones for h since ß and n arenearly proportional (as can be seen in fig. 2).

In part (e) the local value of the internal energy (in .units of MeV/n) is plotted.rThe contours are at 10, 20, 30, etc. MeV. The maximum possible value of 87 MeV/n

HEAVY-ION COLLISIONS

l81

-4 0 +4 -4 0 +4 -4 0 +4

-4 0 +4 -4 0 +4 -4 0 +4Fig . 3 . Contour plots of p(x, y), q, ß and t, for the case of ~ °Ne+ =~~U at 400 MeV/n.

+a ~ +e

F4 I ~ #

-4b=2fm b=4fm b=6fm

-4+8

+4 ~

+8

+4

0 ~~ 0+(b)

-4 -4+8

'

~' +8

+4 1 + +4.28

0 +(c)

O

-4 -4+8

+4

r--

~

- ~--

~

+8

#J-

T_'1l

~~ ~ (d)0

-4 -4+8 +8

+4~,

~ I.,.U,~

r ~ ~ ~ +4

(e)

_a

18 2

W. D. MYERS

(see fig. 2) is only obtained in the last figure on the right; corresponding to an impactparameter of 6 fm.

3. The yield fa~ctioo, Y(iq)

Forthe actual case ofaparticular projectile and targetcombination, the geometricalaspects of the problem can be condensed into an especially simple form based onthe quantity ~. To see this we need only consider the origin of the functions F~(p~,which are the lab momentum space densities (in b) for particles of type j which areproduced in the collision. These are the quantities that are measured andconsequentlythey are the ones we address ourselves to in these model calculations. Such quantitiesmay be calculated from the expression,

F~(p) = J2~bdb

J Jwb(x, y~xdyJ'.~,(ß)f~' ; t~

(3)

where the first integral is over all possible values of the impact parameter b, and thesecond integrations are over the x-y plane normal to the beam direction. The quantitywb(x, y) (which is in units ofparticles/unit area) is obtained by projecting the combinedtarget and projectile density distributions on the to the x-y plane. The quantitiesf(p' ; t) are the c.m. frame momentum space distributions for particles of type jthat would be expected to emergefrom an excited composite system ôfinternal energyper particle t. (Normalized so as to correspond to the yield to be expected per nucleonof the original composite.)Some appropriate model must be chosen for distributing the internal energy t

among the available degrees of freedom in the composite system. For example,in appendix A the distribution in energy and angle of the protons arising from ahigh~nergy heavy-ion collision is calculated . The assumption is made there thatthe momentum distribution ofthe particles in the composite (with respect to their c.m.)is entirely random, i.e. a Maxwell distribution . Of course, this is an unnecessarilyoversimplified approach, and more realistic calculations using the other aspects ofthe model described here are possible, wherepion production is included and theyields of pions, protons, neutrons, deuterons, tritons, etc., are all predicted s " io) ,

In addition, it should be pointed out in passing, that nothing in the developmentwhich follows depends on an assumption that the particles have come to thermalequilibrium in their c.m. frame. Or, for that matter that they art even isotropicallydistributed in momentum . Even if a more microscopic analysis of the collisionprocess were to indicate that an auisotropic, non-Maxwell distribution is moreappropriate the same methods could be employed .The quantity Jr.~r(ß) in eq . (3) is the Jacobian transformation from the c.m .

frame (the primed frame) to the lab frame. It depends only on the lab velocity ß ofthe composite system .

HEAVY-ION COLLISIONS

183

Regardless of the actual form of the functions j~ the integrals in eq. (3) can beapproximated by the expression,

where~,+~

('('

Y(aJ = J-

dn'J2nbdb

J J

dxdywb(x, y~~1' -nb(x, y))~n~ ~

This kind of a separation can be employed, since both the Jacobian J and the c.m .distribution function jdepend only on ~ (through ßand t) . The rl-interval (from0 -" 1)can be divided into a number of equally spaced regions centered on the values qewith a width Sri which is small enough so that Jand j are well approximated by theirvalues at qr A substantial simplification results when this procedure is used sinceall the geometrical aspects of the problem are contained in the "yield function»Y(n). This function consists of a discrete set of numbers (19 in the examples given inappendix B) corresponding to the sequence of q-intervals that is chosen . The valuesof Y(q) (in units ofCTOSS section) can be determined, for any particular combination oftarget and projectile, by numerically integrating over impact parameter and over thex-y plane perpendicular to the beam .

4. Discoesion

The relative importance of different assumptions regarding geometrical aspects ofhigh-energy heavy-ion collisions can be understood most easily by considering theresulting functions Y(q) . Forexample consider the much studied case ofs°Ne+saeU.In fig. 4 the function Y(n) is plotted for three different models . In part (a) the nucleihave been assumed to be sharp surface spheres and the entire target and projectilecontributions to the overlap region are assumed to form a single composite (the"fireball model� ofref. 6)) with a single value for the velocity and local internal energy.The distribution in ~, which is quite narrow, is created solely by the sum over impactparameters. Part (b) shows the effect of including the dependence of q upon positionin the x-y plane. Finally, part (c) shows the result for the model being proposed here,where not only is the dependence of q upon position in the x-y plane included, butthe density distributions of both the target and projectile are taken to have diffuse(rather than sharp) surfaces . The implication of these differences for experimentallyobservable quantities can be scen in the comparison which is made in appendix A.Another dramatic illustration of the difference betwcen the three approaches

illustrated in fig. 4 can be seen in fig. 5. In this figure the functions Y(q) are shown forthree symmetric systems . In the fireball model these distributions would all bedelta functions at n = 0.5 . Ifthe variation ofq in the x-y plane is included they wouldbecome rather sharply peaked distributions at ~ = 0.5 with a width of about 0.2.

~sa

w. n. l~nrERs

Fig . 4. The yield function Y(q) for ~°Ne on

Fig. S . The yield function Y(q) for the three~3sU is plotted against q for three different

symmdric cases '=C+ ~'C, 2°Ne+'°Ne andgeometrical models .

`°A+ 4°A .

When thedifiuseness is also included, as it is in the cases plotted here, the distributionsspread out over the whole q interval .

In both figs. 4 and 5, and in the table of Y(r~) values given in appendix B, the valuesof Y(rl) for ~ = 0 and 1 are not given. These bins correspond, respectively, to con-tributions from the target and projectile spectators . The internal energy associatedwith this part of the cross section is quite low (see fig . 2) . Consequently, their con-tribution is negligible in the energy and angle region we are primarily interested inhere .When the diffuse outer tail of the projectile density distribution strikes a sub-

stantial amount of target material then n is close to zero . The velocity shear in thiscase is probably no longer sufficient to tear the excited material away from thematerial remaining at rest . If the internal energy deposited in this way is combinedwith the deformation energy of the fragment one has the basis for an ordinarycalculation of the (isotropic) energy distribution of the evaporated particles and theyields of the various evaporation residues, a calculation that procceds along slightlydifferent lines than the one described here, which seems to be appropriate over therest of the ~ interval . (Similar remarks apply to the projectile spectator fragments.)

HEAVY-ION COLLISIONS

185

In deriving eq. (4) we accomplished the goal we set for ourselves in the introduction .This expression shows how many of the results of .high~nergy heavy-ion collisionscan be approximated by a sum over a relatively small number of terms, each one ofwhich consists of a geometrical, a kinematical, and a statistical factor . All of thegeometry in this approach is contained in the function Y(q) which can be tabulatedto any target projectile combination (appendix B). Thekinematic considerations areall in the Jacobian Jn.~i, and all ofthe statistical considerations are contained in thefunctions jr

In appendix A the predictions of this approach (using a naive thermal model forf1) are compared with experiment and the general agreement is good . The samesort ofagreement (or better) is observed in calculations ofcomposite yields (deuterons,tritons, alphas, etc.) and pion yields that are currently being undertaken i'). Thereis a tendency (which can be seen in the comparison of appendix A) for the calculatedcross sections to be somewhat smaller than the measured ones, even though the shapeof the energy and angular distributions agree very well . This effect maybe associatedwith the way the geometrical considerations are handled. If the interaction regionwas allowed to spread during the collision and carry along adjacent material theoverall cross section would be increased. This kind of spreading could be includedby allowing neighboring regions to diffuse into each other (a sort of Boltzmannequation approach) or by introducing a viscosity term that would cause each regionto partially drag its neighbors along (a sort of hydrodynamic approach). In eithercase the factorization ofeq. (4) that gives this method its appealing simplicity, wouldbe destroyedThe ease with which this approach can be applied, and its flexibility with regard

to the incorporation of alternative forms for the functions f~ is its major strength.It is our hope that the predictions ofthis method will be compared with a wide rangeof experimental observations, and that it will be useful in helping to identify experi-mental results that suggest collective or otherwise anomalous effects.

Theauthor wishes toacknowledge contributions made to this work byM. Gyulassy,J. I. Kapusta, A. Mekjian and W. J. Swiatecki. This work was done with supportfrom the US Energy Research and Development Administration.

A COMPARISON V1+ITH EXPERIMENT

s. coal~

Appesidlx A

As an example of how the method is to be applied, the double differentialcross . section (d2a/dAdE) for proton emission will be calculated for the reactions2°Ne+238U at 250 and 400 MeV/n.

186

W. D. MYERS

The yield function Y(q) for this case is available from table 1 in appendix Band thekinematic transformation J~,~~ is determinedby the velocity ß~ for each value of q;.We need only to choose an appropriate form for the function fp,~�,(p' ; t). Calculationssimilar to the one being described here are underway where the function j is relativ-istically correct, where pion and deuteron, triton, etc. production are included ").However, for this introduction the protons will be treated as a classical ideal gas ofnon-interacting particles with a Maxwell distribution of velocities in the c.m . frameof the composite system,

.Îp~«oo(P' ; c) = Cn  +(1-n) ~` II (~Mc)- ~ exp (-p,2?cMJJp

t

3

The first term in eq . (A.1) is the ratio, locally, of the number of protons in thecomposite to .the total number of nucleons, where Z and A are the charge and massnumbers of the projectile (or target). The second factor is the normalized Maxwelldistribution for particles of mass M and average energy t.The cross section we seek can be obtained by summing, as in eq . (4), over the

different values of nt (and consequently different values of ß and t) each tenor beingweighted by Y(~1,). The result is shown as a solid line in fig . 6. The dashed line isthe result ofa fireball calculation s) where the yield function is the one shown in fig . 4a .The points in this figure were compiled by adding together the published proton

yields from tH, 2H, 3H, 3He, ~He, etc.') . If the function J we employed in eq . (6)had included provision for the formation of these other products we could simplyhave compared our predictions with the proton yields 6).

io"s

ios3E

IOs

400 AAeV/n =

Proton Ioboro~ kYwtle nmrgy (AAsV)

Fig. 6. The measured yield ofprotons as a fuadion oflab energy and angle for the reactions ~°Ne+=~sUat 250 aad 400 MeV is compared with the predictions oftwo models. The dashed line oorraponds to the

fireball model and the solid line to the model being proposed hero

HEAVY-ION COLLISIONS

187

NM ~? hIO 1~ OD OT O^ N M~ h ~O 1~ 00 T

00 00 ~°.= l+lô~`-°-Ô°~ô~ô°ôô~-°~ôen°..°nN ô~ e~,eneM+~~in°ôv ô °n~n~Mene~+~~ô

00 ~pv,v~ N- âN~+tONOO~O. ~~nrMNÔrOOMN ~NNNMe~+~~ .~+ .~r~,N_,O~~O V1v~~DOÔ~o

M Nl~~t~+1NOO7I~~OIh~~ÔÔM .~oDÔ~O ~D 1~ V1 ~ ~Vj V1 !~ Ô ~ h~M M e+1 e~1 ~ V1 fV

V~1 ~ f~+1 ONO ~~~ 0~0 ~n ~ O~O N ~ ~ ONO eQ+1 ~hO hMN NNM M V1 10 h MM N NN M hO

M N O~ N ~ ~ ~ ~ n M ~ ~ M N N n ~~rN ~~ O ~~ M N NN N N fV M ~ a0N NN

O,~~M .~+e0+.l~~~O,hN~ONNN~hO~ONNON DO ~h DO O~ ^ ~O ~MN (V N N NN N aa0

M N ~ ~ ~ r ~ ~ 00 00 OMO ~ n h M ~ n ~M ~O V M Ri ef vj V1 t"i N r. '. .~ ~ '+ .-~ N N ~D

C

NMOI OO OO v01 ? ~ ~O v~1 ~n RN~ N h eN~1 0N0

V1 fV -+ ~ r. '+ '+ N NN -. -~ '+ r. -+ '+ .-+ fV ~

NO N VI M V1 0l 0v~-.v,-oo-.enn .- ~~ .-.oo--v0~ 0 T V1 M v1 N 0

~ÔfV r .-. . ... .., .-+ .-. fV .-. .-. '. .-. .-+ .~ . .. fN ~

F.Q ~O IO ~p000nnO0âN~Nâ00nn0~00~p~C e+i .r~ OÔÔÔO~ "." '+ÔÔÔOÔ . . . . ..t~1RtdFO M ~ ~ O n ~ l~ 0~1 ~ ~ M ~ M M ~ ~t n ~p 0O

N viNa~OÔ N fV '. . .. ..+ .. . . .. ... ... .-i N ~. ..N0 MOR~~O:O~v~O-OM .~pI[~000~O~I~IO'w. V'1 M M f+1 M M N ^ ^~ O ~Ô O r. '+ .r N1

w Nl~ 00 N h~ h M NN OT MN00 ~~ CT o0 ~ O, N e+l ~ ~ ~ ~O, IO ~O, 1~ OO M 1~e~i .~ .-+O ÔO ~-+'+r+'+0000000 r+N

Q N ^~O ~Dh IO V1 ~ V1 VI 1~MIOOII~N~O~~OMV1 V1 V1 h ~O 1~fV ~-+0000ÔÔÔÔÔÔÔÔOOÔ'+fV

z~O~ONO. O, N~Ot~nh V]~b~O,00Nâ

Ô bMfJ'+ .~ N N .-+ .-.000OOOOÔ r+N.V

OI N. NO~~~~~bn~O,h~MM~~~OOMOlV ~ÔOOOOOOOOÔÔÔÔÔOÔ'+L~T M 00 n M ~M ~ ~ n~ ~~ ~ ~~ ~ V1 ~ 00

N v1N~vIN'+000OOO000Ô00

t` l~ V~0 T ~ â h ~ ~ N NN N~ N ~ eM+lN ÔÔÔ ÔÔÔ ÔÔÔ ÔÔ Ô ÔÔ Ô Ô

MNl~ V1 Ma r. ÔOO OO O OO O OO OOO OO

6.O ~~.~-" tV N e~I f~+1 ~~ VI ~D ODÔ

OOÔÔ ÔÔÔ ÔÔÔ Ô

~ ~O ~ l~ S~ 0;O Ô OO O ÔÔ O O

Ô

G

lgs

w.

D

.

MxERs

TABLE

OF VALUES FOR Y(q)

where

Appeadix

B

Table

1 contains a tabulation of Y(q) values for a number of different target and

projectile

combinations

.

Each column consists of nineteen entries each covering an

interval

ofbrl = 0

.05.

The values ofqi corresponding to the center of each ofthese~

intervals

are located at tt, = iSrl as is indicated in the second column of the

.table .

The fact that these is no entry for t)t = 0 or 1 has been discussed in the main

part

of the text

.The

integrals ofeq

.

(5) that are used to evaluate Y(qt) were done numerically using

a

grid point spacing of 0

.2

fen for the x-, y- and b-integkations

.

Tie diffuse density

distributions

were generated by folding a short-range function (a 7~ltlcawa) into the

equivalent

sharp spherical density distribution which results in an ptpression for

the

radial distribution of density that has the form

Be~tereacea

r5R

R

,e- "~ "

~~

cosh (R/a)-sinh (R/a)J r/a ,

r

Z R,

This

prooedttte has the advantage that the equivalent sharp radius R is simply

proportional

to A~ while the half-density radius of a Fermi function does not have

this

simple proportionality

.The

other advantage is that no special normalization is required to insure that

the

total number of particles is correct since the volume integral of this function is

independent

of the diffuseness

.

1)

H

.

Steirer, Proc

.

Int

.

Conf

.

on high energy collisions involving nuclei, Triste, Italy, September 9-13,

1974L.

S

.

Schroeder, Proc

.

XVI School ofTheoretical Physip, Zakopane

;

Poland, May 25-July 7, 1976,

Ads

Phys

.

PoL, to be published

2)

Nuclear Science Division 1975 Annual Report, Lawrence Herkeky Laboratory report m

.

LHL-5075

3)

J

.

D

.

Hôwman, W

.

J

.

Swiatccly and C

.

F

.

Tsaag, Lawrence Herkeky Laboratory reportno

.

LBL-2908,

July

1973,unpubtinhed

4)

A

.

A

.

Anreden et al

.,

Phys

.

Rev

.

Ldt

.

3B (1977) 1055

5)

J

.

Knoll, Proc

.

Int

.

worhahop on gross properties of nuclei and nuclear excitations V, Hirschegg,

Austria,

1977 (Technische Hochschule Darmstadt, Report no

.

AED - Conf

:

77-017-001, pp

.

44--48

;J .

Hüfner and J

.

Knoll, preprint, May, 1977

6)

G

.

D

.

Westfall er al

.,

Phys

.

Rev

.

Lett

.

37 (1976) 1202

7)

J

.

R Nix, private communication

8)

A

.

Mekjian, Phys

.

Rev

.

Ldt

.

3g (1977) 640

9)

R

.

Hoed, P

.

J

.

Johannen and S

.

E

.

Koonin, preprint NBI-77-9

10)

J

.

I

.

Kaposte, preprint, May, 1977

11)

I

.

Gosset and J

.

Kaposte, private communication