close

Вход

Забыли?

вход по аккаунту

?

Entropy of a Two-phase Mixture.

код для вставкиСкачать
Annalen der Physik. 7. Folge, Band 45, Heft 4, 1988, S. 258-264
VEB J. A. Barth, Leipzig
Entropy of a Two-phase Mixture
By H. SCHULZ,
H. REINHARDT
The Niels Bohr Institute, Copenhagen (Denmark)
and
Central Institute for Nuclear Research Rossendorf, Dresden (GDR)
B. KAMPFER
Central Institute for Nuclear Research Rossendorf, Dresden (GDR)
Abstract. The isentropic expansion of a blob of nuclear matter towards the two-phase instability
region is studied by considering two extreme conditions: the instantaneous development of the phase
transition and the spinodal decomposition. We show that the experimentally observed entropy values
of light clusters give evidence for the onset of a liquid-gas phase transition in heavy ion and proton
induced reactions, but to be more conclusive the disassembly of relatively cold nuclear systems has
experimentally to be investigated in more detail.
Entropie einer nuklearen Zweiphasenmischung
Inhaltsubersicht. Die isentrope Expansion eines nuklearen Feuerballs und das folgende Eindringen in die Region der Fliissigkeit-Gas-Phaseninstabilitatwird in zwei NLherungen untersucht :
es wird entweder eine schnelle Phasemeparation oder die spinodale Entmischung angenommen. Es
kann gezeigt werden, daB die experimentell bestimmten Entropiewerte der leichten Cluster Hinweise
auf eine Zweiphasenentmischung in Schwerionen- und Protonen-induzierten Reaktionen geben. Die
detailliertere experimentelle Untersuchung der Disintegration relativ kalter nuklearer Systeme ist
notig, um zu weitergehenden Aussagen zu gelangen.
One of the intriguing aspects of heavy ion and proton induced reactions would be
the proof (or disproof) of the existence of a liquid-gas like phase transition in hot nuclear
matter. Such a phase transition is predicted by different theoretical approaches for
infinite nuclear matter (see [l--61 and references quoted therein). At present it is not
fully understood what are the experimental evidence for such a phase transition. Following Fisher's droplet model [7] critical exponents z of a power law fit A-" to medium
fragment mass spectra has been extracted [S], whereas the light cluster abundance8
have been fitted by considering the onset of a phase transition [3, 91.
Additional informat,ion may be gained from entropy measurements because the
entropy is the interesting quantity which provides a window into the earlier stage of
the disassembly process. The total entropy of the system will grow rapidly when the
collision process begins and remains nearly constant when the expansion of t h e entire
system sets in. The subsequent multifragmentation process is very complicated and
its time evolution proceeds through thermodynamical and hydrodynamical instabilities
[5, 13-16, 23, 241. Since the total entropy is approximately conserved one can expect
H. SCIIULZ
et al., Entropy of n Two-phase Mixture
259
that the specific entropy values associated with the yield of light or medium clusters
can give valuable information and signal the onset of the phase traiisition process. Such
specific entropy values have recently been inferred from data on light fragment cross
sections. I n ref. [lo] the entropy was extracted by using a quantum statistical motlel
which assumes that a t nuclear disassembly chemical arid thermal equilibrium is established. The average value of the specific entropy for fragments with 1 2 A 2 4 was
found to be s = 3.60 f 0.3 2 and turned out to be nearly independent of the beam energy
ant1 projectile nucleus.
A previously suggested model [ 1I ] has been generalized [ 121 to consider deuteron
like pairs contained in larger composites and to infer entropy values from data on light
fragment cross sections. It was found that the entropy decreased slightly with decreasing beam energy but never dropped below its value a t the critical point of about SIN
= 3.
I n what follouw we show how these results can be interpreted in the physical picture
of the onset of a phase transition familar from the van der M'aals gas.
I
T
Fig. 1. The nuclear phase diagram with the liquid and gas border lines pL and pG and the spinodnls
s, and sl. The isentropes labelled by the specific entropy value are calculated either by considering
an instantaneous phase transition (dashed lines) or by ignoring the phase separation. The numbers
on the spinodal s2 and on the phase border lines give the specific entropy values. The break-up rcgion
and the region where predominantly compound nuclei are formed are hatched.
Fig. 1 shows a typical nuclear matter phase diagram [13] with the border lines pL
and eGfor the liquid and gaseous phase and the spinodals s1 and s2 defined by the locus
where the compressibility vanishes. Within the region bounded by the spinodals the
, ~
respectively,
nuclear matter is unstable, whereas in the region confined by s ~ and
the nuclear matter is in a metastable state.
As usual we assume that the intermediate hot nuclear system formed by a heavy
ion or proton induced reaction expands along an isentrope (see Fig. I ) and finally reaches
a stage where the system becomes metastable or unstable. The conditions for the initial
heating-up and compression t o arrive a t these situations have been studied recently
[13-161.
Ann. Physik Leipzig 46 (1988) 4
2(iO
Following the isentropic expansion towards the unstable region one can distinguish
the following two extreme scenarios for developing the phase transition :
(i) The system expands slowly enough so that, when reaching the metastable region,
bubbles of the new phase are formed due to thermodynamic and quantum fluctuations.
At the spinodal s1 the entire system becomes unstable and decays.
(ii) The initially homogeneous system is driven rather rapidly along the isentrope
into a situation where it becomes thermodynamically unstable and the spinodal decomposition occwrs. Given the fact that the spinodal decomposition represents a break-up
of the initially homogeneous unstable system, it is natural to assume that this process
happens in the vicinity of the break-up density Q w ~ , / 3(hatched area in Pig. 1).
Let us first consider the case (i),i. e. a phase separation process which is fast compared
to the expansion. The total entropy of the two-phase mixture is then given by
=
A
(eL
-
e)
(2)
@G ( c L - - P f 2 )
is the mixing coefficient determined by the leverage rule. This coefficient. is nothing
else than the ratio of the gaseous particles (N,) t o the total particle number ( N ) . At
given temperature T the specific entropy values S, and SL are fixed and the mixing coefficient has to be adjusted to conserve the total entropy. The resulting path of the system
in the T,e plane is shown in Fig. 1 by dashed lines. Fig. 2 illustrates how the total entropy
values are composed by those of the two separated phases. The corresponding mixing
coefficients are displayed in Fig. 3. Sote that only within the break-up region the fraction of the gaseous particles becomes noticeable. According to this picture the specific
entropy value of the gaseous (liquid) phases are determined through their corresponding
values along the phase border line. Kote that S, increases for decreasing total entropy
values or temperature.
-_-_ ___ .
_-_
I
1
i
05
x
I
I
3
hm-31
0
0 05
01
3 [fm-?;
Fig. 2
Fig, 3
Pig. 2. The sharing of the total entropy S into the entropy values S , and 8, d b function of the density g. The dotted line is calculated by considering the bubble formation explicitly. The numbers on
this curve denote the partirle number of the critical bubbles. The break-up region is hatched.
Fig. 3. The mixing coefficient x as function of the density at given total entropy values.
H. SCHULZ
e t al., Entropy of a Two-phase Mixture
261
Till i ~ o wwe hare considered the phase separation as an instantaneous process by
ignoring the underlying microscopic nature of the two-phase formation via the growth
of critical bubbles in the metastable region. The bubble formation probability has been
estimated in ref. [16] by using a droplet model and in ref. [16] within a field theoretical
approach. Following these results we illustrate for the SIN = 2 isentrope iri Fig. 2
the corresponding path of the system (dotted line). The figures on this path denote the
number of nucleons in the critical bubbles. It can be seen that only close to the spinodal
sl, wert' the size of the bubbles is much smaller than that of the system, the phase
transition can develop. I n other words, only if the system has almost passed the metastable region the application of the picture of a n instantaneous phase transition (using
the Maxwell construction) is justified because then the critical bubbles are so small
that the phase transition is not anymore hindred. Compared to the picture of an instantaneous phase transition described above we have now the situation that the liquid
becomes first superheated and afterwards the phase separation occurs. Furthermore,
i t has been shown [13, 1.5, 161 that the phase transition may predominantly develop when
the total entropy S is confined t o the interval Smi,,
< S < S,,. For entropies S > S,,,
the system remains always in a gaseous phase and, hence, is not influenced by the onset
,
,
, m 2.81) (see also Pig. 1).
of the phase transition a t all. The calculation show that S
The lower limit Smin
N 1.2 results from two effects. Due to the occurence of a negttive
pressure in the system the collective expansion is slowed down and eventually comes t o
halt before reaching the spinotlal s1 except for extreme initial compression energies
(Ieposited in the system, which experimentally can hardly be achieved. Furthermore,
the two-phase separation is strongly hindred because the critical bubbles exceed the
size of the system. This qualitative picture is in good agreement with the findings of
ref. [17], where the fragmentation process has been investigated by means of a statistical
model including finite size effects. There it has been found that for total entropy values
S < 1.3 only compound nuclei are formed which can lower their excitation energy by
particle emission, garnnia rays and fission.
SIN
Fig. 4. The specific values of the gaseous phase calculated in different approximations as functioii
of the total entropy. The curve labelled by 8LP.de'/N represents the entropy values when assuming
spinodal decomposition, whereas SF9t/N is calculated by considering a n instantaneous phase
transition. The curve Ssp gives the entropy values of the gaseous phase along the spinodal s2 (cf.
Pig. I). The dashed line would give the results of the gaseous phase when there is no phase separation.
Above the critical entropy, 8, coincides with the total entropy 8.
I)
Kapusta [ I d ] has obtained S,,,,/N m 3.3 with a slightly different equation of state.
268
Ann. Physik Leipzig 45 (1988) 4
Now let us turn t o the case (ii). We consider the other extreme situation, where the
system is isentropically driven into the break-up stage (full lines in Fig. 1).Once the
system is in such an unstable state it decays immediately. This is because fluctuations
grow here exponentially, since there is no longer an energy barrier for forming critical
clusters (bubbles). This process called spinodal decomposition is familiar from nucleation
theory in statistical and fluid mechanics [18, 191. During the nucleation process in the
vicinity of the growing clusters the available matter is exhausted for forming them and,
therefore, no other clusters can be built up. This effect acts among the clusters like an
excluded volume interaction introduced explicitly in statistical nuclear fragmentation
models [20, 211.
Compared t o the processes discussed before (case (i))the multifragmentatiori proceeding through an instantaneous spinodal decomposition in the break-up region would
mean that the expanding fluid blob is more cooled down. The situation is illustrated in
Fig. 4, where the specific entropy of the gaseous phase, S,, has been extracted by using
the results of Fig. 1 for the different processes characterized by the items (i) and (ii)
and drawn as a function of the total initial entropy S. It can clearly be seen that S,
increases as the total entropy decreases. I n other words, one gets the result that the
specific entropy associated with the gaseous phase increases as the beam energy (or
deposited energy) decreases. Such an effect is a t present not seen from the existing experimental data and seems to be an artifact of retaining the thermodynamic equilibrium
conditions for the whole separation process.
I n ref. [22] the relatively strong increase of the values for 8, is obscured by considering an unphysically small break-up density, @bu -N eo/lO. (Note that in ref. [lo] a
break-up density pbu = e0/3 has been used to infer the entropy values from the experimental data).
The average values for the specific entropy for fragments with 1 5 A 5 4 were
determined in ref. [lo] to be SIN = 3.60 f 0.12 and turned out to be independent of
the incident energy and the projectile nucleus. I n ref. [22] these entropy values has
been tried to fit by assuming pbu = po/10 and by invoking the two-phase picture, i.e.
the specific entropy of the light clusters has been associated with that of the gaseous
phase of the two-phase region (cf. Figs. 1, 2 and 4).It seems to us that this interpretation
can not be applied, because one has to analyze simultaneously light and heavier composites which are emitted from the s a m e source. Usually, heavier clusters detected experimentally stem from the decay of spectator matter, which is often characterized by a
much lower temperature than the participant matter source, which predominantly
emitts the light particles with A 5 4. The experimentally found constancy of the entropy
associated with the light particle production seems to indicate that in the participant
matter the disorder has already achieved such a degree that it does not depend very
much on how it was initially formed. This tendency is in agreement with recent results
for entropy values extracted from 4n measurements of the light cluster production
Ar a t energies 400 MeV/
yields [25]. Also cascade calculations [26] for the reaction Ar
u 5 Elab5 2 100 MeV/u show that SIN increases only slightly in this large energy
interval.
It seems t o us that the rapidity dependent entropy values as extracted in ref. [ 101
are not conclusive for gaining safe information on the onset of the phase transition of
the liquid - gas type. This is because the uncertainties coming from the experimental
conditions are a t present too large t o be sure that light and heavier clusters are emitted
from the same source which can be characterized by a common temperature and total
entropy value. Experimental studies of light and medium cluster production with a 4n
detector allowing for triggering central collision events would give us much more valuable
information on the two-phase separation onset.
+
H. SCHULZ
et al., Entropy of a Two-phase Mixture
263
Analyzing the experimental data within the static two-phase picture oue has to be
aware that in a kinetic treatment the equilibrium situation will never be reached and
therefore the final stage of the evolution will be characterized by a lower entropy value
of the gaseous phase as, for instance, predicted by the leverage rule. The final values for
S, and their precise values can only be calculated by solving kinetic equations which
govern the evolution of a system which consists of droplets immersed in a gaseous
nuclear medium and moves towards the final state. From Fig. 1 and especially from
Fig. 4 it can be seen that along the spinodal s2 the specific entropy values remain fairly
constant, S,JN = 3.2. This almost constant value represents a better estimate for the
entropy of the gaseous phase and considers that some non-equilibrium aspects of the
separation process are accounted for.
Therefore, we would expect that due t o the existence of the two-phase separation
the specific entropy of the gaseous phase should remain almost constant with S, ‘v 3.2
if the experimental conditions are such that the disassembling intermediate nuclear
system can be characterized a t break-up by a temperature T 5 Tcrit.The average
value found in ref. [lo] SIN = 3.60 f- 0.12 is very close to our predictions but to be
more conclusive one has to consider the disassembly of even colder systems with T 5
8 MeV. The method of ref. [22] would predict then entropy values of SIN N 6 unless
extremely low break-up densities are taken into account.
To summarize, the interesting conclusion which can be drawn from our present analysis is that the behaviour of the measured specific entropy values of the light clusters
gives some evidence for the development of a first order phase transition. But to be more
conclusive one has to treat relatively cold systems. The phase transition manifests then
itself by the fact that with decreasing beam energy (temperature) the specific entropy
values inferred from light cluster yields does not drop below some critical value of SIN m 3
in contrast to the total entropy. The development of a theory which includes nonequilibrium aspects but having a t the same time predictive power seems to be a major
challenge to the theorists. A first attempt in this direction has been undertaken in ref.
~41.
Two of the authors (H. S. and H. R.) are grateful to the Niels Bohr Institute for the
kind hospitality extended to them.
References
[l] LAMB,D. Q.; LATTIMER,
J. M. ; PETHICK,
C. J.; RAVENHALL,
D. G.: Phys. Rev. Lett. 41 (1978)
1623; Nucl. Phya. A 360 (1981) 459.
IS] DANIELEWICZ,
P.: Nucl. Phys. A 314 (1979) 465.
[3] SCHULZ,
H.; MUNCHOW,
L.; ROPICE,
G.; SCHMIDT,
M.: Phys. Lett. 119B (1982) 12.
[4] CURTIN,
M. W.; TOKI,H.; SCOTT,
D. K.: Phys. Lett. 123 B (1983) 289.
[5] BERTSCH,
G.; SIEMENS,
P. J.: Phys. Lett. 126 B (1983) 9.
[GI STOCKER,
H. ;BUCHWALD,
G. ; GRAEBNER,
G. ;SUBRAMANIAN,
P.; MARUHN,
J. A. ; GREINER,
W. ;
JACAK,
B. V.; WESTFALL,
G. D.: Nucl. Phys. A 4.00 (1983) G3.
171 FISHER,M. E.: Physics 3 (1967) 255.
[8] PANAOIOTOU,
A. D.; CURTIN,
&I. W.; TOKI,H.; SCOTT,D. K.; SIEMENS,
P. J.: Phys. Rev.
Lett. 68 (1984) 496.
[9] SC~HULZ,
H.; ROPKE,Q.; SCHMIDT,
M.: Z. Phys. A313 (1983) 369.
[lo] JACAK,B. V.; STOCKER,
H.; WESTFALL,
G. D.: Phys. Rev. C 29 (1984) 1744.
1111 SIEMENS,
P. J.; KAPUSTA,
J. 1.:Phys. Rev. Lett. 43 (197G) 1486.
L12] KAPWSTA,
J. I.: Phys. Rev. C 29 (1933) 1733.
264
Ann. Physik Leipzig 45 (1988) 4
[13] SCRULZ,
H.; KLMPFER,
B.; BARZ,H. W.; ROPKE,G.; BONDORF,
J.: Phys. Lett. 147 B (1984) 17.
KAMPFER,B.; SCHULZ,
H.; LIJKACS,
B.: J. Phys. G 11 (1985) L 47.
[14] CUGNON,J.: Phys. Lett. 136 B (1984) 374.
H.; VOSKRESENSKY,
D. N.; BONDORF,
J.: Phys. Lett. 110B (1983) 141.
[15] SCHULZ,
H.; SCHULZ,
H.: Nucl. Phys. A 432 (1985) 630.
[16] REINRARDT,
1171 BONDORF,
J.; DONANGELO,
R.; MISNUSTIN,
I. N.; SCHULZ,
H.: Nucl. Phys. A, 444 (1985) 460.
[IS] BINDER,K.; STAUFFER,
D.: Adv. Phys. 25 (1976) 343.
[19] LANCER,
J. S.: Lecture Notes in Physics 132 (1980) 12.
CZO] RANDRUP,
J.; KOONIN,
S. E.: Nucl. Phys. A366 (1981) 223.
[21] BONDORF,
J.; DONANGELO,
R.; MISHUSTIN,
1.N.; PETHICK,
C. J.; SCHULZ,
H.; SNEPPEN,
K.:
Nucl. Pliys. A 443 (1985) 321.
[22] CSERNAI,
L. P.: Phys. Rev. Lett. 54 (1985) 639.
[ 2 3 ] LOPEZ,J. A.; SIEMENS,
P. J.: Nucl. Phys. A 431 (1984) 728.
[24] STRACK,
B.; KNOLL,
J.: Z. Phys. A315 (1984) 249.
[25] GUTBROD,
H. H.; LOHNER,
H.; POSKANZER,
A. M.; RENNER,
T.; RIEDESEL,
H.; RITTER,H. G.;
WARWICK,
A.; WIEMAN,
H.: Phys. Rev. C 32 (1986) 116.
[26] GUDIMA,
K. K.; TONEEV,
V. B.; ROPKE,G.; SCHULZ,
H.: Phys. Rev. C 32 (1985), 1605.
Bei der Redaktion eingegangen am 31. MXLrz 1986.
Anschr. d. Verf.: Dr. H. SCHULZ
Dr. H. REINHARDT
Dr. B. KAMPFER
Central Institute for Nuclear Research
Rossendorf
Postfach 19
Dresden
DDR-8051
Документ
Категория
Без категории
Просмотров
2
Размер файла
476 Кб
Теги
two, mixtures, entropy, phase
1/--страниц
Пожаловаться на содержимое документа