IL ~NUOVO CIMENTO VoL. 47 A, ~ . 1 1 Settembre 1978

Isobaric-Gas Model for High-Energy Nuclear Physics.

2K. LA CXM-ERA a n d A. WAT~GIffN

Isti tuto di Svienze .Fisiche dell' U~iversith - Geneva

Islitq, to Nazio~ale di _Fisica Nucleate - Sezione di Geneva

(ricevuto il 21 Aprile 1978)

Summary . - - We develop a model for high-energy inelastic processes. The model is pr imari ly intended for collisions of nuclei, however the equilibrium conditions permit its application also to hadron-hadron col- lisions. The model is constructed on a hydrodynamical basis and takes into account nuclear shock waves. Pionization is obtaincd in an in- direct way (production of Lsobaric resonances which subsequently decay or annihilate themselves). The nuclear temperature as a function of (, s is obtained and a comparison with experiment is made. Also an expres- sion for the mult ipl ici ty is obtained. Hadron mat ter density may become during the collision many times greater than the normal nuclear density.

1 . - I n t r o d u c t i o n .

This p a p e r will be c o n c e r n e d wi th some f e a t u r e s a p p e a r i n g in t h e col-

l is ion of r e l a t i v i s t i c h e a v y ions w i t h nucle i . I r speak ing , <( r e l a t i v i s t i c ~>

co r r e sponds to a k ine t i c e n e r g y p e r nuc l eon e q u a l to or g r e a t e r t h a n t h e nuc l e on

r e s t mass . A t t hese energies t h e co l l id ing nuc le i pas s t h r o u g h each o t h e r (1).

F o r h e a d - o n coll is ions of h e a v y nuc le i t h e m e a n free p a t h of h a d r o n s in

n u c l e a r m a t t e r is m u c h s h o r t e r t h a n t h e d i s t a n c e to be crossed. I n th i s case

shock waves m a y deve lop a n d l ead to dens i t i es h ighe r t h a n tw ice t h e n o r m a l

n u c l e a r d e n s i t y (.2).

(1) 1-~. [. SOBEL, P. J. SIEMENS, J. P. BO~DOI~F and II. A. BETnE: Nucl. Phys . , 251 A, 502 (1975). (2) W. SHEID, I t . MOI.I,):R and W. Gm,:I_~ER: Phys. Rev. Left., 32, 741 (1974).

35

36 ~. LA CAMERA a n d A. WATAGIIIN

The properties of such dense hot nuclear ma t t e r are proper ly described b y a many-phase isobaric gas, which allows for the presence of several nucleon excited states (s,~). As a simple example, we shall consider here only the A(1232) excited state of the nucleons, bu t clearly a more detailed calculation is needed to include formation of A-nuclei (~).

Even as it stands, however, our model allows us to obtain a first evaluat ion of the break-up tempera ture Tb, i.e. the t empera ture at which pionization can occur. Comparison with existing data will be made by taking account of the indirect mechanism of pion product ion which in this region is predom-

inant (8).

2. - Relativistic hydrodynamic equations.

In the case of head-on collision of a beam particle m~ with a target particle

m~, we m a y describe the process as follows. In the first stage, the projectile enters the target with supersonic speed

and thus produces a shock wave which will push nuclear ma t t e r to the side. In the nex t stage, shock waves propagate along the colliding particles,

leaving behind zones of high density and temperature . At the end of this stage, the creation of nuclear isobars makes it possible

to have a compressed hot nuclear ma t t e r in thermal equilibrium; with defined values for the tempera ture I ' and the compression n/no.

Subsequently~ there is the stage of the hydrodynamica l expansion (7), during which interaction decreases aud the mean free pa th increases. When n/no ~ 1 the mean free pa th becomes comparable with the linear dimensions of the system, and the la t ter disintegrates into separate particles. This last stage, the break-up, takes place at a tempera ture T b which we can calculate for given initial conditions and for given intermediate isobaric states.

Let us apply our model of nuclear mat te r as an ideal fluid to the compres-

sion stage of the collision. In the reference frame ~q in which the colliding particles have equal and

opposite velocities, with Lorentz factor

(1) 99 :1. V s - - (m~-- m.,) ~ -- 2 ~ m~m2

(a) l[. G. BAUMGARDT, J. U. SCHOTT, Y. SAKAMOTO, E. ScrroPPE~, ~. STeCKE•, J. HOFMANN, W. SCH}:ID and W. GRZIh'}:n: Zeits. Phys., 273 A ~, 359 (1975). (4) G. F. Cr~APLINE, .'~I. M. JORNSON, E. TELLER and M. S. WEISS: Phys. Rev. D, 8, 4302 (1973). (5) G. CtKAeI~INE trod M. WEISS: Bull. A?rber. Phys. Sos., 18, 18 (1973). (6) ~ . LA CAMERA and A. WATAGHIN: Left. 2~uovo Cimento 22, 463 (1978). (7) S. S. BELEN~KI and L. D. LANDAU: Suppl. Nuovo Cimento, 3, 15 (1956).

I80B&RIC-GXS 5~OD]~L FOR HIGII-]~N)]RGY NUOLE&Y~ PHYSICS 37

(s is the square of the avai lable e.m. energy), bo th projectile and ta rge t look

like thin disks, and the problem can be t rea ted as one-dimensional.

During the compression stage, shock waves p ropaga te in bo th directions

through hadronie m a t t e r with speed ft , . I n the ~- f rame the m a t t e r between the shock waves is a t rest. In order

to calculate the compression, we change to a p r imed f rame in which the shock

wave is a t rest (8). In the new sys tem the speed of m a t t e r behind the shock wave is ft , , while

ahead of the wave it is

( 2 ) ~ ' - +

.l_

with Lorentz fac tor

Because of the cont inui ty of energy, m o m e n t u m and n u m b e r of part icle

flow at the shock wave f ront (~), we Call write

(3) (4)

(5)

T P ) r D % = (tO + PO)r'*~' ,

(P + ~,~)r~, = (PO + ~,~O)r '~ '

n r , f l . = no r ' fl' ,

here the index (( o ~ refer to quanti t ies (pressure p, energy densi ty e and n u m b e r

of p~rticles n) measured in the rest f r ame of the (( normal )) nuclear ma t t e r ,

while the same quanti t ies wi thout index are measured in the rest f rame ,q of the shocked nuclear mat te r . F rom (3)-(5) it is possible to obta in the relativist ic l~ankine-IIugoniot equat ion (~0)

+ w) (6) Wo W 2 Wo n~ n 2 + ( p - p ~ ~ = 0 ,

where

(7) W = e + p

is the specific enthalpy.

I f in the shocked region we assume for the equat ion of s ta te of the fluid

(8) p := �89 (~-- nm) ,

(s) S. Z. B~]LI'iNKI and G-. k . MIL]~KIII.~: SOY. Phys. JETP, 2, 14 (1956). (~) L. LANDAU and E. LIFSHITZ: Mecanique des Fluides (~oscow, 1971), p. 621. (lo) A. TAUB: Phys. ]r 74, 328 (1948).

38 ~. LA CA~EI~A and x. WATAGIIIN

we obtain, for ~ not close to unity, the value of the compression in the presence

of shock waves:

~b (9) - - = 4 9 ~- 3 .

~t o

~ot ice tha t from (5) it follows tha t

(10) ~ = 3 \~ + 11 "

3. - The i sobaric gas.

At the end of the compression a situation of thermodynamical equilibrium

is reached, at a temperature T1. I t is then possible to use the statistical for-

mulae for an ideal quan tum gas (s,n):

(ii) s ----

(12) n ----

(13) ~ =

where

2z 2 . . . . 12 (--~)~-lexp ~7 ,

2~-" [ \ r ! ,-i ~

27~,2g--TS(T) 2~(~-I[~/T) K2(lm/T)|=I 12q-(lm/~)(gl(lm/T) (-- (~) z-1 exp ( ~ ) } ,

g is the statistical weight ,

m is the particle mass ,

tt is the chemical potent ia l ,

- - 1

~ 0

§

for bosons,

for bol tzmannions,

for fermions,

and K1(x), K2(x) are modified Bessel functions.

Let us now consider the collision of two nuclei of mass numbers A1 and A~.

In the H-frame their total energy is

(14)

(11) M. CH/tIOKIAN, R. I'[AGF, DORN and M. HAYASHI: Nucl. Phys., 92B, 445 (1975).

I S O B ~ R I G - G A S X0DEL FOR HIGH-ENERGY NUCLEAI~ I ' I I Y S I C S

and will be released, at the end of the compression, in a volume

(15) V(~) -- (A, J.-As)~(1/m~) _ (A, + As)Vo ~p + 3 4p + 3

Then conservation of energy, if one neglects free bosons, requires

(16) [e,~(1') + e,,(1') + ~ (so,o~(T) + ~v.(T))] V@) = E(~), t

where j~o stands for (( nucleons ~, J~'* for the i-th nuclear isobar and the bar indicates the corresponding antiparticle.

The chemical potential # is determined by the condition of nuclear-charge conservation:

39

(17) [~iv(T)- ns,(T) -[- ~, (n~v~(T) -- n~;(T))] V@) ---- A1 + A~. i

If we now consider the average entropy per particle, we lumw that this quantity remains constant during the next stage, which is the hydrodynamical expansion of an ideal fluid. Thus

(18) %(I,) + ~(T) + Z (%Z(V) + ~Z(T))

(

i

i

n~v(Tb) + n~'(Tb) -- X (nov~(Vb) -.'- n~(Tb)) "

i

Once again, to determine the chemical potential at the break-up temperature :Tb, we must add the condition of nuclear-charge conservation, remembering that at the break-up n/no = 1,

(19) [n,~o(Tb)- n.~(Tb) -k ~ (n.~o.(Tb)- nx-t(Tb)](A1 + A2) Vo = A1 -[- As . i

If we consider only the A(1232) excited state of the nucleons, we obtain, from (t6) to (19)~ after numerical evaluations, the dependence of T b (in the S-frame) on the e.m.s, energy s of the colliding particles:

(20) - - [" 4 m l m---~ -J "

See fig. 1.

40 ~I. LA CA)IERA and A. ~VATAGHIN

1 ~

A

: E

120[

I I I J I J I r ~ I 0 20 40 60 80 100 120

z,0

Fig. 1. - Final temperature T b as a function of 7~.

4 . - C o m p a r i s o n w i t h e x p e r i m e n t a l d a t a .

Exper imen ta l evidence (e,12,13) shows t h a t at high energies pion product ion occurs through the decay of one or more thermal ly excited centres (fireballs). In a previous le t ter (8) we showed tha t , due to this (( indirect ,~ mechanism,

the t runsver se -momentum distr ibution of emi t ted pions tukes, for sufficiently

high values of pT, the sealing form

(21) N PT; "

The average t ransverse m o m e n t u m of secondaries depends on the decay tem-

pera ture To of the fireball in its rest sys tem and on its t ransverse m o m e n t u m

Pr,~ as follows (eq. (11) or ref. (e)):

(22) (PT> _ To

2 1 - - pT,f/m~'

(12) k . ~kGNESE, 2~. WATAGHII~ ~, G. S. SHABRATOVA and K. D. TOLSTOV: .NUOVO Ci-

mento, 1 3 k , 144 (1973). (18) G. S. SH~RATOVA, K. D. TOLSTOY, P. CALVI~I and A. ~r .Sett. Nuovo Oimento, 18, 1511 (1977).

ISOBARIC-GAS MODEL FOR IIlGII-F~NERGY NUCLEAR I 'HYSICS 41

eq. (22) gives the apparent t empera ture from fit of cq. (21). This value is higher than 2'0. In this framework, what we called T b is nothing else than the proper temperaimre as seen from r ~-frame, i.e.

(23) T b To Yf

Thus the procedure for comparing the PT spectra with our T b (fig. 1) is the following. F rom a fit of (21) (or directly from <PT>) one obtains the ap- parent tempera ture <pT>/2. The11 from eq. (22) one obtains To, which, how- ever, is expected to be 110 MeV (see below). Then if ~f has been obtained (for equal colliding particles the S-frame is identical to the c.m.s, frame) by means of the methods of ref. (~3), use (23) to obtain T b.

~f is the Lorentz factor of the fireball in the ~-frame. We proceed now to compare this model with exper imental data in the case of hadron-hadron col- lisions (A -- 1). According to our estimate, at high energies this should be allo wed.

:From data published in (~3), which refer to pions produced in collisions of a 50 GeV/c 7:- bea.m in nuclear emulsions, and from (22) and (23) it follows that

<P~> ~ 160 ~1eV, To = (110 + 10) MeV, (24) - - ~ - . _

(25) 2'~ = ( s o . o • s . o ) M e V .

We expect the value of To not to depend oll s, at least in a very large range of energies.

I f we neglect leading-particle effects, eq. (20) can be used to obtain the theo- retical value of T b for the Dubna (~3) case:

(26) T b = 81 .2 . ~ i e V .

We think tha t the value I ' = 140 Meg, f requent ly found from fits of p~ dis- tr ibutions (14) is larger than our predicted value To =: 110 MeV, beca.use PT~ (see eq. (22)) has not been taken into account.

5. - Mult ipl ic i ty o f emi t ted pions.

The multiplicity of emit ted ,-:'s can be obtained, in our model, in the S-frame, by dividing the to ta l energy of the fireball by the average energy

(14) j . E1~wI.,,,', W. Ko, R. L. LAXDXR, D. E. PEI, Lt:T and t'. M. YAC_,VR: Phys. lgev. ]Lett., 27, 1534 (1971).

42 ~[. LA CA:Yl~I~A and a. WATAGHI,'ff

of each p ion (eqs. (11) a n d (12)):

f,,m,

This a s sumes t o t a l a n n i h i l a t i o n of t h e f i rebal l . E q u a t i o n (27) r e p r o d u c e s t h e

d a t a for energ ies up to s ~ 100 (GeV) ~, u s ing m ~ = 2 .6GeV. ~ o r h i g h e r energ ies

(ISI~ or c o s m i c - r a y region) , eq. (27) needs a new t y p e of f i reba l l to t i t t h e d a t a .

The n e w v a l u e of m~ shou ld be n e a r l y t w i c e t h e v a l u e of t h e f i rs t t y p e . E v i d e n c e

for t h i s s econd t y p e cou ld be t h e b e n d i n g of t h e p r s p e c t r u m in (~) a n d

fig. 14 of (x~).

W e t h a n k t h e J o i n t I n s t i t u t e for 1Wuclear l~esea.rch (D ubna ) for i t s

h o s p i t a l i t y in D e c e m b e r 1977 a n d in s eve ra l p r ev ious occas ions , a n d Prof . K .

D. TOLS~OV for v e r y use fu l d iscuss ions .

(x.~) CERN-CoLu~m~-ROCK):F~Lr,ym-ISR COLLABORATION: CERN Report (1973). (~) BRAZ~L-Ja~'~N E~TSLS~O~- COL~aBOI~aT~O-~: X V I I International Conference on High- Energy .Physics (London, 1974).

�9 R I ~ I S S U N T O

Si sviluppa un modcllo per processi inelas~ici ad al ta encrgia. I1 modello ~ primaria- mente costruito per urt i di nuclei. Cionondimeno le condizioni per l 'equilibrio pcrrnet- tono la sua applicazione anche ad urt i ndrono-adrone. I1 modcllo 6 costruito su basi idro- dinamichc c tiene conto delle onde di shock ncl nucleo. La pionizzazione 6 o t tcnuta in maniera indiret~a (produzione di risonanzc isobariche che in seguito decadono o s i annichilano). La tempera tura nuelcarc ~ calcolat~ in funzione di (~ s ~) e si effe~tua un confronto con i valori sperimentali . Si otticne anche un'cspressionc per la moltc- plicit~. La densit~ della mater ia adronica durante l 'ur to pub diventare molto su- pcriore alla densit'~ normale dei nuclei.

I/I3o6apaqecRa~ MO)Ie~b ra3a ~ym fl~epHofi ~H3HKH BI~ICOIgHX 3neprnfi.

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