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Dynamical aspects of the transition from nuclear matter to a quark-gluon plasma in heavy ion collisions
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
16 August 1984
DYNAMICAL ASPECTS OF THE TRANSITION FROM NUCLEAR MATTER T O A Q U A R K - G L U O N P L A S M A IN HEAVY ION C O L L I S I O N S H.W. B A R Z , B. K A M P F E R Central Institute for Nuclear Research, Rossendorf DDR-8051 Dresden, PF 19, GDR
L.P. C S E R N A I 1 School of Physics and Astronomy, University of Minnesota, 116 Church Street S.E., Minneapolis, MN 55455, USA
and B. L U K A C S Central Institute for Physics, Budapest, P.O.B. 49, H-1525 Budapest, Hungary
Received 20 February 1984 Revised manuscript received 14 May 1984
Taking into account a finite conversion rate the transition of nuclear matter to quark-gluon matter and vice versa is treated in a hydrodynamical model for heavy ion collisions. For fast conversion pure quark matter is formed for laboratory energies between 4 and 10 GeV per nucleon while below 4 GeV per nucleon matter stays in a mixed phase. Due to the large amount of latent heat the temperature does not exceed k T = 160 MeV.
Extrapolations of the present u n d e r s t a n d i n g of h a d r o n - n u c l e u s a n d n u c l e u s - n u c l e u s collisions s u p p o r t the idea that for b o m b a r d i n g energies between 5 a n d 10 G e V per n u c l e o n a w i n d o w exists for the creation of a hot state of nuclear m a t t e r with large b a r y o n density [1]. F o r a sufficiently large energy density the matter is expected to u n d e r g o a phase transition into a d e c o n f i n e d state c o n s u m i n g a rather large a m o u n t of latent heat [2]. I n the present p a p e r we investigate in a hydrod y n a m i c a l t r e a t m e n t the phase transition d u r i n g a heavy ion collision taking into a c c o u n t a finite conversion velocity. O n e often assumes that the transition proceeds in a shock-wave-like m a n n e r which can be described sufficiently by conservation laws (cf. ref. [3]). Such investigations are performed for the transition from h a d r o n m a t t e r 1 Permanent address: Central Institute for Physics, Budapest, P.O.B. 49, H-1525 Budapest, Hungary. 334
to quark matter [4] a n d vice versa [5]. F o r the particular case of the transition from nuclear m a t t e r to ideal q u a r k - g l u o n matter the R a n k i n e H u g o n i o t - T a u b adiabats are displayed in fig. 1 for different values of the bag c o n s t a n t B. F o r values B > W o / 4 = (127 MeV) 4 (w 0 being the enthalpy density of g r o u n d state nuclear matter) a C h a p m a n - J o u g u e t p o i n t does not exist a n d for B > 3w0/4 = (167 MeV) 4 the adiabats have a n u n usual p a t t e r n with positive slopes. T o such endotherm reactions the investigation of the stability of the relativistic shock waves [3,5] are not extended. Thus, the applicability of a stationary description is not obvious for bag c o n s t a n t s as large as ass u m e d in the majority of the literature [6]. If we further assume that the structural r e a r r a n g e m e n t d u r i n g the phase transition needs a finite time c o m p a r a b l e with the time of the fast relativistic collision, then a detailed d y n a m i c a l calculation is necessary to describe the behaviour of matter travelling through the transition zone.
0 3 7 0 - 2 6 9 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 143B, number 4, 5, 6 r
06
i
i
PHYSICS LETTERS
i
i
v
n 0.4
0.2-
2
z,
6 X[GeV fm 3]
Fig. 1. Rankine-Hugoniot-Taubadiabats for the transition of nuclear matter to ideal quark matter in a plot of the pressure p versus the generalized specific volume x = (e + p ) / n 2 (e: energy density, n: baryon number density). The curves are calculated with the equations of state explained in the text for different bag constants B 1/4. Point A indicates the initial state being normal nuclear matter.
If there is an i n t e r n a l conversion of the m e d i u m the e q u a t i o n of state d e p e n d s o n three intensive variables, e.g. the density of b a r y o n s n, the temperature T a n d in a d d i t i o n the so-called progress variable [7]. F o r this variable we employ c~ = N 1 / N , where N 1 is the b a r y o n n u m b e r in the nuclear m a t t e r phase a n d N is the total n u m b e r of b a r y o n s in a fluid element. Following ref. [8] we assume that d u r i n g the d y n a m i c a l process both phases are in thermal a n d mechanical e q u i l i b r i u m b u t n o t in chemical equilibrium. T o complete the hydrodyn a m i c a l equations which follow from the vanishing 4-divergence of the stress tensor T ik a n d the b a r y o n i c flux n u ~, V , T 'k = O,
V,(nu')
= 0,
(1)
a n a d d i t i o n a l d y n a m i c a l equation for the progress variable a is necessary. W e use the linearized form of the relaxation e q u a t i o n (cf, ref. [7]) = - - q ( a -- C%q),
(2)
where q is the conversion rate factor a n d the dot m e a n s the time derivation in the local rest frame. The conversion rate is p r o p o r t i o n a l to the devia t i o n from the phase e q u i l i b r i u m a e q = n l ( n n 2 ) / n ( n 1 - n 2 ), where the lower a n d u p p e r b o u n d s n~.z of the b a r y o n n u m b e r density of the coexis-
16 August 1984
tence region d e p e n d on the temperature. There is a possibility that the transition from quark matter to h a d r o n i c matter is hindered because of the necessity to form colour singlet states from the rand o m l y distributed colour q u a n t u m n u m b e r s in the q u a r k phase [9]. T o model this case we use two different values for q in eq. (2) for & ~ 0 . The Q C D scale h / B 1/4 suggests a value of q - a in the order of 1 f m / c . Since a reliable estimate is n o t available we vary q in the range between 0.1 a n d 10 c / f m . F o r the energy per b a r y o n e / n in the nuclear m a t t e r phase we employ a quadratic form for the cold compression energy a n d the thermal energy of a B o l t z m a n n gas: e/n
= Wo/n o + ~K(n/n
K = 280 MeV,
o - 1) 2 + 3T,
(3)
where n o denotes the density of g r o u n d state nuclear matter a n d T is the temperature. The q u a r k - g l u o n p l a s m a is described b y s t a n d a r d Q C D [10] without i n t e r a c t i o n a n d with the bag c o n s t a n t B 1/4 = 235 MeV. We present the results for slab o n slab collisions with each slab having a thickness of 11 fm corres p o n d i n g to a collision of U o n U. The o n e - d i m e n sional relativistic h y d r o d y n a m i c calculation is performed with a m e t h o d which is new in the field of relativistic heavy ion physics. The matter is described as a perfect fluid. Thus, as usual in supersonic problems an artificial viscosity must be added to treat shock p h e n o m e n a . This viscosity causes a small b u t finite width of a shock front in which e n t r o p y is p r o d u c e d in the same a m o u n t as in an ideal shock. The n o n - e q u i l i b r i u m phase transition is a further source of entropy. Eqs. (1) are solved in a comoving, s y n c h r o n o u s curvilinear coordinate system. This m e t h o d applied till now for stellar p r o b l e m s [11] is modified for our slab geometry. The difference scheme o b t a i n e d from eqs. (1) corresponds to that derived in ref. [11]. I n fig. 2 typical paths of different fluid elements through the d e n s i t y - t e m p e r a t u r e p l a n e are shown. I n the b e g i n n i n g the temperature rises steeply in the h a d r o n i c phase b u t as soon as the phase border line of the f o r m a t i o n for the q u a r k - g l u o n phase is reached the temperature starts to decrease due to 335
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
16 A u g u s t 1984
2 ~150 I-.........
Etob/A = 6 GeV
125
0'.5
1
1.5
100 0
19
28 n [fm'~
Fig. 2. Typical paths through the n - T plane for different bombarding energies per nucleon. The conversion rate factors q are 10 c/fm and 1 c Yfm for compression and expansion, respectively. The splitting of the paths for compression and expansion is mainly caused by the different composition of the matter due to the delay in the phase transition. The hatched area indicates the coexistence region of both phases calculated via the Gibbs condition.
the c o n s u m p t i o n of the latent heat. The temperature increases again if the q u a r k - g l u o n matter is compressed till the t u r n i n g p o i n t is reached. F o r energies below El,4 = 4 GeV the position of the t u r n i n g p o i n t lies in the coexistence region a n d varies sensitively c h a n g i n g the energy or the conversion rate factor whereas above this energy the variations are moderate. The striking p o i n t is that the temperature never exceeds the limit of 160 MeV for b o m b a r d i n g energies below E/A = 10 G e V provided that q >__1 c/fm. This fact occurs although the linear fluid model overestimates systematically the density a n d the temperature by a b o u t 20% [12] because of the absence of the transverse flow a n d the t r a n s p a r e n c y effect. Also other authors using stationary models [4] f o u n d this limiting temperature. F o r energies above El.4 = 4 GeV we find a stable shock front which separates the two phases provided that q >__1 c/fm. Otherwise there is a weaker shock which compresses the nuclear m a t t e r in a mixed state followed by a region, where nuclear matter transforms slowly in plasma. F o r illustration we show in fig. 3 the density profiles for two q values. The velocity of the shock front is f o u n d to be i n d e p e n d e n t of q for q > 1 c/fm, b u t increases strongly for small q because of the larger stiffness of n u c l e a r m a t t e r present in this case. 336
×
[fmJ
2
Fig. 3. S n a p s h o t s o f d e n s i t y profiles of a slab as f u n c t i o n s of the C M time t a n d p o s i t i o n x. F o r q = 10 c/fm (full lines) a p u r e h i g h d e n s i t y p l a s m a ( a = 0) is f o r m e d while for q = 0.1
c/fm (dashed line) only partial conversion ( a = 0.75) is reached. The velocities of the shock fronts are 0.2 c and 0.5 c, respectively. I n fig. 4 the time d e v e l o p m e n t of the progress variable a averaged over the central part of the slabs as a f u n c t i o n of the rate factor q for the b o m b a r d i n g energy of 4 G e V per n u c l e o n is displayed. I n the expansion stage the quark matter cools until the phase b o r d e r line is reached. T h e n the nucleonic r e c o m b i n a t i o n process starts a n d the latent heat is released. I n d e p e n d e n t l y of the b o m b a r d i n g energy a n d the conversion rate factors the matter reaches at very low density a temperature of 155 MeV which corresponds to the critical temperature of the phase transition. W h e t h e r this fact offers possible experimental implications deserves further investigations since in this region pionic c o n t r i b u t i o n s to the e q u a t i o n of state (3) become i m p o r t a n t . (~):
,
10/
Etob/A=4GeV
'..
-0
lO/lO
/ ....."
, nL-
lOllO,
!
"'~//
,
10
,
I
20 t/fmcq
Fig. 4. The time development of the ratio (et) of baryons in hadronic phase to the total number of baryons averaged over the inner halves of the slabs corresponding to a U on U collision at E/A = 4 GeV. The curves are labelled by the conversion rate factors for compression/expansion. The time t is the centre of mass time.
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
In summary we present the results of a proper dynamical treatment of the transition from nuclear m a t t e r to q u a r k - g l u o n m a t t e r i n h e a v y i o n collis i o n s a t i n t e r m e d i a t e e n e r g i e s u p to 10 G e V p e r n u c l e o n . I n o r d e r to f o r m a p u r e q u a r k g l u o n p l a s m a a c o n v e r s i o n r a t e is n e c e s s a r y w h i c h is f a s t e r t h a n t h a t g i v e n b y eq. (2) w i t h q = 1 c/fm. We find a transition to a relatively cool quark-gluon matter (T_<160 MeV) and a constant recombination temperature at low matter density. Although the threshold for complete quark m a t t e r p r o d u c t i o n is E]ab/A = 4 G e V w e s h o w t h a t f r a c t i o n a l q u a r k m a t t e r p r o d u c t i o n is p o s s i b l e at lower energies. At the energy of 2 GeV in a U + U collision according to our calculation some percent of the matter can be converted. This fraction increases with increasing projectile energy, h o w e v e r , it is s e n s i t i v e o f t h e c o n v e r s i o n r a t e fact o r q, e s p e c i a l l y if q_< 1 c/fm. T h i s m a k e s it d i f f i c u l t to o b s e r v e a d e f i n i t e t h r e s h o l d b e h a v i o u r for the onset of a phase transition because the a p p e a r a n c e o f t h e q u a r k - g l u o n p l a s m a is a g r a d u a l p r o c e s s . A d e l a y in t h e p h a s e t r a n s i t i o n y i e l d s a n e x t r a e n t r o p y p r o d u c t i o n [8]. T h i s e x t r a e n tropy might be a good signature for finite conversion rate factors. This work was supported in part by the US Department of Energy under contract number DE-AC-02-79ER-10364.
16 August 1984
References [1] M. Gyulassy, Proc. Intern. Conf. on Nucleus-nucleus collisions (Michigan State University, September 1982), Nucl. Phys. A400 (1983) 31c; Proc. Intern. Conf. on High energy nuclear physics (Balatonfured, Hungary, June 1983) (KFKI, Budapest, 1983) p. 489. [2] M. Jacob and H. Satz, Proc. Bielefeld Workshop on Quark matter formation and heavy ion collisions (May 1982) (World Scientific, Singapore, 1982) p.1; T. (~elik, J. Engels and H. Satz, Phys. Lett. 125B (1983) 411; 133B (1983) 427. J. Kogut et al., Phys. Rev. Lett. 50 (1983) 393. [3] P.J. Steinhardt, Phys. Rev. D25 (1982) 2074. [4] A.C. Chin, Phys. Lett. 78B (1978) 552; H. St6cker et al., Phys. Lett. 95B (1980) 192; B. Kampfer, J. Phys. G9 (1983) 1487; T. Bir6 and J. Zimhnyi, Nucl. Phys. A395 (1983) 525. [5] B.L. Friman, G. Baym and J.P. Blaizot, Phys. Lett. 132B (1983) 291; G. Baym, B.L. Friman, J.P. Blaizot and M. Soyer, Nucl. Phys. A407 (1983) 541; M. Gyulassy, K. Kajantie, H. Kurki-Suonio and L. McLerran, Nucl. Phys. B237 (1984) 477. [6] P. Hasenfratz et al., Phys. Lett. 95B (1980) 299; H. Satz, Phys. Lett. l13B (1982) 245. [7] P.P. Wegener and B.J.C. Wu, in: Nucleation phenomena, ed. A.C. Zettlemoyer (Elsevier, Amsterdam, 1977) p. 325. [8] L.P. Csernai and B. Lukhcs, Phys. Lett. 132B (1983) 295. [9] B. Mi~ller and J. Rafelski, Phys. Lett. 101B (1981) 111. [10] J.I. Kapusta, Nucl. Phys. B148 (1979) 461. [11] K.A. Van Riper, Astrophys. J. 232 (1979) 558. [12] H. St~Scker et al., Nucl. Phys., to be published.
337
PHYSICS LETTERS
16 August 1984
DYNAMICAL ASPECTS OF THE TRANSITION FROM NUCLEAR MATTER T O A Q U A R K - G L U O N P L A S M A IN HEAVY ION C O L L I S I O N S H.W. B A R Z , B. K A M P F E R Central Institute for Nuclear Research, Rossendorf DDR-8051 Dresden, PF 19, GDR
L.P. C S E R N A I 1 School of Physics and Astronomy, University of Minnesota, 116 Church Street S.E., Minneapolis, MN 55455, USA
and B. L U K A C S Central Institute for Physics, Budapest, P.O.B. 49, H-1525 Budapest, Hungary
Received 20 February 1984 Revised manuscript received 14 May 1984
Taking into account a finite conversion rate the transition of nuclear matter to quark-gluon matter and vice versa is treated in a hydrodynamical model for heavy ion collisions. For fast conversion pure quark matter is formed for laboratory energies between 4 and 10 GeV per nucleon while below 4 GeV per nucleon matter stays in a mixed phase. Due to the large amount of latent heat the temperature does not exceed k T = 160 MeV.
Extrapolations of the present u n d e r s t a n d i n g of h a d r o n - n u c l e u s a n d n u c l e u s - n u c l e u s collisions s u p p o r t the idea that for b o m b a r d i n g energies between 5 a n d 10 G e V per n u c l e o n a w i n d o w exists for the creation of a hot state of nuclear m a t t e r with large b a r y o n density [1]. F o r a sufficiently large energy density the matter is expected to u n d e r g o a phase transition into a d e c o n f i n e d state c o n s u m i n g a rather large a m o u n t of latent heat [2]. I n the present p a p e r we investigate in a hydrod y n a m i c a l t r e a t m e n t the phase transition d u r i n g a heavy ion collision taking into a c c o u n t a finite conversion velocity. O n e often assumes that the transition proceeds in a shock-wave-like m a n n e r which can be described sufficiently by conservation laws (cf. ref. [3]). Such investigations are performed for the transition from h a d r o n m a t t e r 1 Permanent address: Central Institute for Physics, Budapest, P.O.B. 49, H-1525 Budapest, Hungary. 334
to quark matter [4] a n d vice versa [5]. F o r the particular case of the transition from nuclear m a t t e r to ideal q u a r k - g l u o n matter the R a n k i n e H u g o n i o t - T a u b adiabats are displayed in fig. 1 for different values of the bag c o n s t a n t B. F o r values B > W o / 4 = (127 MeV) 4 (w 0 being the enthalpy density of g r o u n d state nuclear matter) a C h a p m a n - J o u g u e t p o i n t does not exist a n d for B > 3w0/4 = (167 MeV) 4 the adiabats have a n u n usual p a t t e r n with positive slopes. T o such endotherm reactions the investigation of the stability of the relativistic shock waves [3,5] are not extended. Thus, the applicability of a stationary description is not obvious for bag c o n s t a n t s as large as ass u m e d in the majority of the literature [6]. If we further assume that the structural r e a r r a n g e m e n t d u r i n g the phase transition needs a finite time c o m p a r a b l e with the time of the fast relativistic collision, then a detailed d y n a m i c a l calculation is necessary to describe the behaviour of matter travelling through the transition zone.
0 3 7 0 - 2 6 9 3 / 8 4 / $ 0 3 . 0 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 143B, number 4, 5, 6 r
06
i
i
PHYSICS LETTERS
i
i
v
n 0.4
0.2-
2
z,
6 X[GeV fm 3]
Fig. 1. Rankine-Hugoniot-Taubadiabats for the transition of nuclear matter to ideal quark matter in a plot of the pressure p versus the generalized specific volume x = (e + p ) / n 2 (e: energy density, n: baryon number density). The curves are calculated with the equations of state explained in the text for different bag constants B 1/4. Point A indicates the initial state being normal nuclear matter.
If there is an i n t e r n a l conversion of the m e d i u m the e q u a t i o n of state d e p e n d s o n three intensive variables, e.g. the density of b a r y o n s n, the temperature T a n d in a d d i t i o n the so-called progress variable [7]. F o r this variable we employ c~ = N 1 / N , where N 1 is the b a r y o n n u m b e r in the nuclear m a t t e r phase a n d N is the total n u m b e r of b a r y o n s in a fluid element. Following ref. [8] we assume that d u r i n g the d y n a m i c a l process both phases are in thermal a n d mechanical e q u i l i b r i u m b u t n o t in chemical equilibrium. T o complete the hydrodyn a m i c a l equations which follow from the vanishing 4-divergence of the stress tensor T ik a n d the b a r y o n i c flux n u ~, V , T 'k = O,
V,(nu')
= 0,
(1)
a n a d d i t i o n a l d y n a m i c a l equation for the progress variable a is necessary. W e use the linearized form of the relaxation e q u a t i o n (cf, ref. [7]) = - - q ( a -- C%q),
(2)
where q is the conversion rate factor a n d the dot m e a n s the time derivation in the local rest frame. The conversion rate is p r o p o r t i o n a l to the devia t i o n from the phase e q u i l i b r i u m a e q = n l ( n n 2 ) / n ( n 1 - n 2 ), where the lower a n d u p p e r b o u n d s n~.z of the b a r y o n n u m b e r density of the coexis-
16 August 1984
tence region d e p e n d on the temperature. There is a possibility that the transition from quark matter to h a d r o n i c matter is hindered because of the necessity to form colour singlet states from the rand o m l y distributed colour q u a n t u m n u m b e r s in the q u a r k phase [9]. T o model this case we use two different values for q in eq. (2) for & ~ 0 . The Q C D scale h / B 1/4 suggests a value of q - a in the order of 1 f m / c . Since a reliable estimate is n o t available we vary q in the range between 0.1 a n d 10 c / f m . F o r the energy per b a r y o n e / n in the nuclear m a t t e r phase we employ a quadratic form for the cold compression energy a n d the thermal energy of a B o l t z m a n n gas: e/n
= Wo/n o + ~K(n/n
K = 280 MeV,
o - 1) 2 + 3T,
(3)
where n o denotes the density of g r o u n d state nuclear matter a n d T is the temperature. The q u a r k - g l u o n p l a s m a is described b y s t a n d a r d Q C D [10] without i n t e r a c t i o n a n d with the bag c o n s t a n t B 1/4 = 235 MeV. We present the results for slab o n slab collisions with each slab having a thickness of 11 fm corres p o n d i n g to a collision of U o n U. The o n e - d i m e n sional relativistic h y d r o d y n a m i c calculation is performed with a m e t h o d which is new in the field of relativistic heavy ion physics. The matter is described as a perfect fluid. Thus, as usual in supersonic problems an artificial viscosity must be added to treat shock p h e n o m e n a . This viscosity causes a small b u t finite width of a shock front in which e n t r o p y is p r o d u c e d in the same a m o u n t as in an ideal shock. The n o n - e q u i l i b r i u m phase transition is a further source of entropy. Eqs. (1) are solved in a comoving, s y n c h r o n o u s curvilinear coordinate system. This m e t h o d applied till now for stellar p r o b l e m s [11] is modified for our slab geometry. The difference scheme o b t a i n e d from eqs. (1) corresponds to that derived in ref. [11]. I n fig. 2 typical paths of different fluid elements through the d e n s i t y - t e m p e r a t u r e p l a n e are shown. I n the b e g i n n i n g the temperature rises steeply in the h a d r o n i c phase b u t as soon as the phase border line of the f o r m a t i o n for the q u a r k - g l u o n phase is reached the temperature starts to decrease due to 335
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
16 A u g u s t 1984
2 ~150 I-.........
Etob/A = 6 GeV
125
0'.5
1
1.5
100 0
19
28 n [fm'~
Fig. 2. Typical paths through the n - T plane for different bombarding energies per nucleon. The conversion rate factors q are 10 c/fm and 1 c Yfm for compression and expansion, respectively. The splitting of the paths for compression and expansion is mainly caused by the different composition of the matter due to the delay in the phase transition. The hatched area indicates the coexistence region of both phases calculated via the Gibbs condition.
the c o n s u m p t i o n of the latent heat. The temperature increases again if the q u a r k - g l u o n matter is compressed till the t u r n i n g p o i n t is reached. F o r energies below El,4 = 4 GeV the position of the t u r n i n g p o i n t lies in the coexistence region a n d varies sensitively c h a n g i n g the energy or the conversion rate factor whereas above this energy the variations are moderate. The striking p o i n t is that the temperature never exceeds the limit of 160 MeV for b o m b a r d i n g energies below E/A = 10 G e V provided that q >__1 c/fm. This fact occurs although the linear fluid model overestimates systematically the density a n d the temperature by a b o u t 20% [12] because of the absence of the transverse flow a n d the t r a n s p a r e n c y effect. Also other authors using stationary models [4] f o u n d this limiting temperature. F o r energies above El.4 = 4 GeV we find a stable shock front which separates the two phases provided that q >__1 c/fm. Otherwise there is a weaker shock which compresses the nuclear m a t t e r in a mixed state followed by a region, where nuclear matter transforms slowly in plasma. F o r illustration we show in fig. 3 the density profiles for two q values. The velocity of the shock front is f o u n d to be i n d e p e n d e n t of q for q > 1 c/fm, b u t increases strongly for small q because of the larger stiffness of n u c l e a r m a t t e r present in this case. 336
×
[fmJ
2
Fig. 3. S n a p s h o t s o f d e n s i t y profiles of a slab as f u n c t i o n s of the C M time t a n d p o s i t i o n x. F o r q = 10 c/fm (full lines) a p u r e h i g h d e n s i t y p l a s m a ( a = 0) is f o r m e d while for q = 0.1
c/fm (dashed line) only partial conversion ( a = 0.75) is reached. The velocities of the shock fronts are 0.2 c and 0.5 c, respectively. I n fig. 4 the time d e v e l o p m e n t of the progress variable a averaged over the central part of the slabs as a f u n c t i o n of the rate factor q for the b o m b a r d i n g energy of 4 G e V per n u c l e o n is displayed. I n the expansion stage the quark matter cools until the phase b o r d e r line is reached. T h e n the nucleonic r e c o m b i n a t i o n process starts a n d the latent heat is released. I n d e p e n d e n t l y of the b o m b a r d i n g energy a n d the conversion rate factors the matter reaches at very low density a temperature of 155 MeV which corresponds to the critical temperature of the phase transition. W h e t h e r this fact offers possible experimental implications deserves further investigations since in this region pionic c o n t r i b u t i o n s to the e q u a t i o n of state (3) become i m p o r t a n t . (~):
,
10/
Etob/A=4GeV
'..
-0
lO/lO
/ ....."
, nL-
lOllO,
!
"'~//
,
10
,
I
20 t/fmcq
Fig. 4. The time development of the ratio (et) of baryons in hadronic phase to the total number of baryons averaged over the inner halves of the slabs corresponding to a U on U collision at E/A = 4 GeV. The curves are labelled by the conversion rate factors for compression/expansion. The time t is the centre of mass time.
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
In summary we present the results of a proper dynamical treatment of the transition from nuclear m a t t e r to q u a r k - g l u o n m a t t e r i n h e a v y i o n collis i o n s a t i n t e r m e d i a t e e n e r g i e s u p to 10 G e V p e r n u c l e o n . I n o r d e r to f o r m a p u r e q u a r k g l u o n p l a s m a a c o n v e r s i o n r a t e is n e c e s s a r y w h i c h is f a s t e r t h a n t h a t g i v e n b y eq. (2) w i t h q = 1 c/fm. We find a transition to a relatively cool quark-gluon matter (T_<160 MeV) and a constant recombination temperature at low matter density. Although the threshold for complete quark m a t t e r p r o d u c t i o n is E]ab/A = 4 G e V w e s h o w t h a t f r a c t i o n a l q u a r k m a t t e r p r o d u c t i o n is p o s s i b l e at lower energies. At the energy of 2 GeV in a U + U collision according to our calculation some percent of the matter can be converted. This fraction increases with increasing projectile energy, h o w e v e r , it is s e n s i t i v e o f t h e c o n v e r s i o n r a t e fact o r q, e s p e c i a l l y if q_< 1 c/fm. T h i s m a k e s it d i f f i c u l t to o b s e r v e a d e f i n i t e t h r e s h o l d b e h a v i o u r for the onset of a phase transition because the a p p e a r a n c e o f t h e q u a r k - g l u o n p l a s m a is a g r a d u a l p r o c e s s . A d e l a y in t h e p h a s e t r a n s i t i o n y i e l d s a n e x t r a e n t r o p y p r o d u c t i o n [8]. T h i s e x t r a e n tropy might be a good signature for finite conversion rate factors. This work was supported in part by the US Department of Energy under contract number DE-AC-02-79ER-10364.
16 August 1984
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