Phase diagram of alpha matter with Skyrme-like scalar interaction
Leonid M. Satarov, Roman V. Poberezhnyuk, Igor N. Mishustin, Horst Stoecker
PPhase diagram of alpha matterwith Skyrme-like scalar interaction
L. M. Satarov, R. V. Poberezhnyuk,
1, 2
I. N. Mishustin,
1, 3 and H. Stoecker
1, 4, 5 Frankfurt Institute for Advanced Studies,D-60438 Frankfurt am Main, Germany Bogolyubov Institute for Theoretical Physics, 03680 Kiev, Ukraine National Research Center ”Kurchatov Institute” 123182 Moscow, Russia Institut f¨ur Theoretische Physik, Goethe Universit¨at Frankfurt,D-60438 Frankfurt am Main, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
Abstract
The equation of state and phase diagram of strongly interacting matter composed of α particlesare studied in the mean-field approximation. The particle interactions are included via a Skyrme-like mean field, containing both attractive and repulsive terms. The model parameters are found byfitting the values of binding energy and baryon density in the ground state of α matter, obtainedfrom microscopic calculations by Clark and Wang. Thermodynamic quantities of α matter arecalculated in the broad domains of temperature and baryon density, which can be reached inheavy-ion collisions at intermediate energies. The model predicts both first-order liquid-gas phasetransition and Bose-Einstein condensation of α particles. We present the profiles of scaled variance,sound velocity and isochoric heat capacity along the isentropic trajectories of α matter. Strongdensity fluctuations are predicted in the vicinity of the critical point at temperature T c ≈
14 MeVand density n c ≈ .
012 fm − . a r X i v : . [ nu c l - t h ] O c t . INTRODUCTION It is well known from experimental observations [1–3] and theoretical studies [4–7] thatsymmetric nuclear matter at subsaturation densities has a tendency for clusterization. And α particles are the most abundant clusters, as observed in many experiments [8, 9]. One mayconclude that α -particle is the next most important building block of nuclear matter afterthe nucleon. The fact that α particles are bosons makes α matter even more interesting tostudy, because of the possibility of Bose-Einstein condensation. In our previous studies of α matter, we have used the Skyrme-like model [10], where the effective mean-field potentialis a function of α particle density. In the present work, we propose another approach whereself-interaction of α ’s is introduced in terms of a scalar field φ . The system of α particles isdescribed by an effective Lagrangian which contains the attractive ( φ ) and repulsive ( φ )terms. A similar scalar potential was first introduced by Boguta et al. in the relativisticmean-field model of nuclear matter [11, 12]. In Ref. [13] the same type of self-interactionwas used in the mean-field model for cold α matter. Recently, a similar model was used [14]for bosonic particles without any conserved charge, i.e., with zero chemical potential (e.g.,pions). In the present paper, we further develop this approach for α matter which carriesthe conserved baryon charge.A special role of α particles is well established in nuclear physics. The examples include:2 α structure of Be which decays into two α ’s in about 10 − s and the famous 3 α Hoyle statein C with width of only 8 . α -clustered final states, analogous to the Hoylestate, have been identified in O and Mg [15]. In Ref. [16] an evidence for 7 α resonancehas been reported. All these states can be interpreted as Bose condensates of α particlesin finite systems [17]. This interpretation is justified by microscopic calculations, see, e.g.,recent review [18] and references therein.In our recent publications [19, 20], we have proposed several Skyrme-like models to de-scribe α condensation in nuclear matter. We have considered two possibilities regarding theinteraction between α particles and nucleons. In the first scenario [19], pure α matter andpure nucleonic matter have their own ground states separated by a potential barrier. In thesecond scenario [20], there is only one ground state with coexisting nucleons and α ’s. Inthis case the nuclear matter at moderate temperatures and densities contains only a smalladmixture of α particles. Having in mind that experimental data [15, 16] show enhanced2ields of α particles and α -conjugate nuclei, we are tempted to conclude that the first sce-nario is more realistic. In this case, there is a chance that the initial fireball will be createdon a metastable branch of the phase diagram where α clusters are more abundant and mayeven form a condensate. In the present paper, we study this scenario within a mean-fieldSkyrme-type model for pure α matter.The paper is organized as follows. In Section II we formulate the model. In Sec. IIIwe analyze the phase diagram of α matter which contains both the liquid-gas phase tran-sition (LGPT) and the Bose-Einstein condensation (BEC). We also discuss isentropic andisothermal trajectories of α matter in different thermodynamic variables. In Sec. IV A wepresent a fluctuation observable in terms of the scaled variance ω , and show that it hasa strong peak at the critical point of liquid-gas phase transition. In Sec. IV B and IV C wecalculate isentropic profiles of the sound velocity and the isohoric heat capacity. It is shownthat both these quantities exhibit jumps at the mixed phase boundary. II. THE MODELA. Bosonic matter in the mean-field approximation
Following [14] we describe the system of scalar bosons with zero spin by a scalar fieldoperator φ ( x ) with the Lagrangian ( (cid:126) = c = 1) L = 12 (cid:0) ∂ µ φ ∂ µ φ − m φ (cid:1) + L int (cid:0) φ (cid:1) . (1)Here m is the boson mass in the vacuum and L int describes particles’ interactions. In themean-field approximation (MFA) one expands L int in the lowest order in φ − σ , where σ = (cid:104) φ (cid:105) (2)is the scalar density ( c -number) and angular brackets denote the quantum-statistical ave-raging in the grand-canonical ensemble (see below).One gets the following result L ≈ L
MFA = 12 (cid:2) ∂ µ φ ∂ µ φ − M ( σ ) φ (cid:3) + p ex ( σ ) , (3) All numerical calculations below are carried out for the specific case of α particles. M ( σ ) = m − L (cid:48) int ( σ ) (4)is the effective mass squared, and p ex ( σ ) = L int ( σ ) − σ L (cid:48) int ( σ ) (5)is the so-called excess pressure [14] (hereinafter primes denote derivatives over σ ). Theterms containing L int in these two equations describe deviations from the ideal gas. Lateron we apply the Skyrme-like parametrization for L int ( σ ) containing both the attractive andrepulsive parts. From Eqs. (4), (5) one gets the differential relation between the effectivemass and excess pressure p (cid:48) ex ( σ ) = σM M (cid:48) ( σ ) , (6)which guaranties the thermodynamic consistency of the present model (see below). In fact,the appearance of the effective mass makes the main difference between the present approachand our previous studies of α [10] and α − N [19, 20] matter.Using a plane-wave decomposition of φ in terms of creation ( a + k ) and annihilation ( a k )operators φ ( x ) = g (2 π ) (cid:90) d k √ E k (cid:0) a k e − ikx + a + k e ikx (cid:1) , (7)where k = E k = √ M + k , g is the statistical weight of a boson particle, one gets [14] thefollowing equations for the particle number and Hamiltonian density operators in the MFA NV = g (2 π ) (cid:90) d k a + k a k , HV = g (2 π ) (cid:90) d k E k a + k a k − p ex . (8)In the grand canonical ensemble one can find the particle momentum distribution by aver-aging a + k a k over the statistical operator ρ ∝ exp [( µN − H ) /T ] at given chemical potential µ and temperature T . Within the MFA one has n k = (cid:104) a + k a k (cid:105) = (cid:20) exp (cid:18) E k − µT (cid:19) − (cid:21) − . (9)One can see that this distribution coincides with the ideal-gas (Bose-Einstein) distributionof quasiparticles with mass M . The latter is in general not equal to m and should be found For states with BEC one should add [14] an additional condensate component φ c to the right handside (r.h.s.) of this equation. T, µ . According to Eq. (9) possible values of µ are bound fromabove, namely µ (cid:54) M . In the particular case µ = M the distribution n k is singular at k = 0 .Similarly to the ideal-gas case, this implies the appearance of BEC with macroscopic numberof zero momentum particles. Therefore, in our model the condensate appears at µ = M ( σ ).We will see that this takes place either at low enough temperatures or at sufficiently largedensities.Using Eqs. (2), (7), (9) one can calculate the scalar density σ : σ = σ th [ T, µ, M ( σ )] , (10)where σ th ( T, µ, M ) = g (2 π ) (cid:90) d k n k E k = g π ∞ (cid:90) M dE √ E − M exp (cid:16) E − µT (cid:17) − . (11)In fact, Eq. (10) plays a role of self-consistent gap equation which determines σ and M asfunctions of T, µ .The pressure p is determined by spatial components of the energy-momentum tensor T αα which in turn can be calculated from the Lagrangian. Within the MFA one has p = 13 (cid:104) T xx + T yy + T zz (cid:105) = 16 (cid:10) ( ∇ φ ) (cid:11) + p ex ( σ ) . (12)Using Eqs. (7), (9) one can easily calculate the kinetic term in the second equality (belowwe denote it by p th ). Finally one gets the equation p = p th ( T, µ, M ) + p ex ( σ ) , (13)where p th = g (2 π ) (cid:90) d k k E k n k = g π ∞ (cid:90) M dE ( E − M ) / exp (cid:16) E − µT (cid:17) − . (14)One can also obtain explicit expressions for the number density of α particles n = (cid:104) N (cid:105) /V and the internal energy density ε = (cid:104) H (cid:105) /V (note that the baryon density n B = 4 n ). UsingEqs. (8), (9) one has n = n th ( T, µ, M ) = g (2 π ) (cid:90) d k n k , (15)and ε = ε th ( T, µ, M ) − p ex ( σ ) , ε th = g (2 π ) (cid:90) d k E k n k . (16)5n the case of α matter, the temperatures of interest are lower or of the order of α -particlebinding energy ( ≈
28 MeV). To a good accuracy, one can apply the non-relativistic approx-imation (NRA) by taking the lowest-order terms in
T /M in Eqs. (11), (14)–(16). This leadsto approximate relations σ th ≈ n th M ≈ gM λ T g / ( z ) , (17) p th ≈
23 ( ε th − M n th ) ≈ gTλ T g / ( z ) , (18)where z (cid:54) λ T = λ T ( T, M ) is the thermal wave length: z ≡ exp (cid:18) µ − MT (cid:19) , λ T = (cid:114) πM T . (19)The dimensionless function (polylogarithm) g β ( z ) is defined as g β ( z ) ≡ β ) ∞ (cid:90) dx x β − z − ex − ∞ (cid:88) k =1 z k k − β , (20)where Γ( β ) is the gamma function.The following properties of the polylogarithm will be used below: z g (cid:48) β ( z ) = g β − ( z ) , g β ( z ) = z, z (cid:28) ,ξ ( β ) , z → , (21)where ξ ( β ) = (cid:80) ∞ k =1 k − β is the Riemann function . The function g β ( z ) diverges at z → β (cid:54) z . According to Eqs. (17)and (21), in this case z ≈ nλ T /g (cid:28) p th ≈ nT .On the other hand, for states where the degeneracy parameter nλ T (cid:38)
1, the effects ofquantum statistics are important. As discussed above, the BEC starts when µ → M , i.e.,at z → T < T
BEC where T BEC ≈ πµ (cid:18) ng ξ (3 / (cid:19) / (22)is the threshold temperature of BEC. The chemical potential in the r.h.s. of this equationis determined from the relation µ = M ( σ ). At not too high densities one can take µ ≈ m with a good accuracy. The resulting value of T BEC coincides with that for the ideal Bosegas. Therefore, for nonrelativistic bosons (like α -particles) the BEC onset line in the ( n, T )plane is the same as in the ideal gas [10, 21].6t T < T
BEC additional condensate terms should be added to the r.h.s. of Eqs. (10), (15)and (16) : σ ( µ ) = σ c + σ th ( T, µ, µ ) , n = n c + n th ( T, µ, µ ) , ε = ε c + ε th ( T, µ, µ ) − p ex ( σ ) , (23)where n c = µσ c , ε c = µn c and σ ( µ ) is determined by solving the equation µ = M ( σ ). Notethat ( µ, T ) dependence of the condensate terms is fully determined by the first equalityin (23) .Let us consider now the entropy of a bosonic system S . The entropy density s = S/V canbe calculated by using the general relation s = ( ε + p − µn ) /T . One can see that the excess-and (possible) condensate terms in ε, p and n are cancelled and we arrive at the equation: s = ε th + p th − µn th T . (24)Formally, we get the same expression as for the ideal Bose gas [14]. However, the interactioneffects enter via the effective mass M . Substituting µ = M + T ln z and using Eqs. (17), (18)one obtains the following expression for the entropy per particle (cid:101) s = s/n (cid:101) s = SN ≈ g / ( z ) g / ( z ) − ln z , T > T BEC , ξ (5 / nλ T ( T, µ ) , T < T BEC . (25)Again, we arrive at the relations for the ideal Bose gas [22], but with the modified (state-dependent) particle mass. These results show that the specific entropy (cid:101) s is constant at theBEC boundary ( z = 1): (cid:101) s ≈ ξ (5 / ξ (3 / ≈ .
284 ( T = T BEC ) . (26)This means that this boundary is an isentrope and the bosonic matter can not cross itduring the isentropic evolution. For example, it is not possible to reach the BEC regionby an adiabatic expansion of α matter from non-condensed initial states . Note that thethreshold value (26) does not depend on the particle mass and the interaction parameters. By presence of BEC the pressure is modified only indirectly, via the condensate scalar density σ c in theterm p ex ( σ ) . However, this is possible if the bosonic matter contains also the LGPT (see Sec. III). dp = sdT + ndµ , (27) dε = T ds + µdn . (28)To prove (27), we directly calculate the derivatives entering the pressure differential dp = dT ∂ p/∂T + dµ ∂ p/∂µ + dM ∂ p/∂M . Using Eqs. (13)–(16), (24) one can check that ∂p/∂T = ∂ p th /∂T = s, ∂ p/∂µ = ∂ p th /∂µ = n th and [14] ∂ p∂M = ∂ ( p th + p ex ) ∂M = − M σ th + M σ = , T > T BEC ,µσ c , T < T BEC , (29)where we have used Eqs. (6) and (23) (in the second and third equalities, respectively).Finally we arrive at Eq. (27). Note that in the BEC region we substitute M = µ and µσ c = n − n th . Equation (28) can be obtained from (27) by differentiating the relation ε = T s + µn − p . B. Skyrme-like parametrization of particle interactions in α matter The above results are obtained in the MFA, and they do not depend on a specific form ofthe interaction Lagrangian. The following calculations are carried out with the Skyrme-like( φ − φ ) parametrization, L int ( σ ) = a σ − b σ , (30)where a, b are positive constants. The first and the second terms in (30) describe, respec-tively, the attractive and the repulsive interactions between the scalar bosons. It will beshown below that the presence of attraction leads to the appearance of the first-order (liquid-gas) phase transition of bosonic matter. On the other hand, the repulsive term stabilizes thismatter at high densities . Substituting (30) into Eqs. (4) and (5) gives the explicit relationsfor the effective mass and excess pressure M ( σ ) = √ m − aσ + bσ , (31) p ex ( σ ) = − a σ + b σ . (32) A similar quasiparticle model for scalar bosons with repulsive interaction was constructed in Ref. [24]. IG. 1: Effective mass of particles in α matter as the function of scalar density σ . Full dot showsposition of GS . Note that p ex vanishes at σ = 3 a b and the minimum value of the effective mass M min = (cid:114) m − a b is reached at σ = a b . A typical plot of M ( σ ) for the case of α mat-ter ( m = 3727 . M from its vacuum value at σ → a, b by fitting known propertiesof cold α matter. According to microscopic calculations of Ref. [25] this matter has thefollowing values of the binding energy per α particle W and the equilibrium density n : W = m − (cid:16) εn (cid:17) min = 19 . , n = 0 .
036 fm − . (33)The bosonic matter at T = 0 consists of condensed quasiparticles with µ = M ( σ ), n = n c = µσ , and p = p ex ( σ ). For the GS one has d (cid:16) εn (cid:17) /dn = p/n = 0 and we obtain thefollowing equations connecting equilibrium parameters n , W and σ : p ex ( σ ) = 0 , m − W = M ( σ ) = n σ . (34) The corresponding binding energy per nucleon, W = m N + ( W − m ) / ≈
12 MeV is slightly less thanthe binding energy of isosymmetric nuclear matter (about 16 MeV/nucleon).
9s already mentioned, the first equation holds at σ = 3 a b . Solving two remaining equalities,one obtains the following values of parameters a, b : a ≈ , b ≈ .
94 MeV − . (35)Note, that the position of GS is indicated by dot in Fig. 1. The GS value σ ≈
75 MeV is significantly lower than the close packing density σ p ∼
200 MeV (it corresponds to thevector density n p ≈ mσ p ∼ . − [19]).It is instructive to calculate the incompressibility modulus: K = 9 dpdn ≈ p (cid:48) ex ( σ ) m = 2716 a bm ≈
354 MeV . (36)This agrees with the value obtained in Ref. [19] for a pure α matter with a mean-field vectorpotential U ( n ) . The latter is defined as the shift of chemical potential with respect tothe ideal gas. One can show that within the NRA such an approach gives results which aresimilar to the present model with scalar interaction. Indeed, at T = 0 the chemical potentialin the scalar theory equals M ( σ ) where σ ≈ n/m . Decomposing the r.h.s. of Eq. (31) inpowers of σ one obtains the Skyrme-like expression for the equivalent vector potential: U ( n ) ≈ M ( σ ) − m ≈ − An + B n , where A = a m , B = b m . (37)Substituting the values of a and b from (35), we get the values of A, B close to those obtainedin Ref. [19]. However, one can show that the approach with vector potential (37) leads tosuperluminal sound velocities c s at sufficiently large n . On the other hand, the presentmodel does not violate the relativistic causality condition c s < III. PHASE DIAGRAM OF α MATTER
By using the model with vector mean-field interaction we have calculated in Refs. [10, 19]the phase diagram of the bosonic matter. It was demonstrated that if this interactionincludes both attractive and repulsive term, the resulting phase diagram contains bothregions of LGPT and BEC. Below we show that a similar phase diagram takes place in thepresent model with scalar interaction. More exactly, one gets the ’stiff’ ( γ = 1) version of the potential introduced in Ref. [19]. . The BEC boundary As explained above, the BEC boundary in the ( µ, T ) plane, T = T BEC ( µ ) , is determinedby simultaneous solving the equations M ( σ ) = µ, σ = σ th ( T, µ, µ ) where M and σ th are givenby Eqs. (31) and (11), respectively. The equation M ( σ ) = µ can be solved analytically : σ = σ ( µ ) = a + (cid:112) a + 4 b ( µ − m )2 b . (38)As shown in Sec. II A, within the NRA one can use the approximate relations σ th ≈ gξ (3 / µλ T ( T, µ ) and n ≈ µσ ( µ ) . Finally we get Eq. (22) for the temperature T BEC ( µ ). B. Liquid-gas phase transition
Let us consider first the states without BEC. At given
T, µ one can solve Eqs. (10) and (31)with respect to σ . At sufficiently low temperatures there is a region of µ with severalsolutions σ i ( µ, T ). The solution with the largest (lowest) pressure is thermodynamicallystable (unstable) [23]. The critical line of the LGPT µ = µ ( T ) is found from the Gibbscondition of phase equilibrium. The latter implies equality of pressure in coexisting liquid-like ( i = l ) and gas-like ( i = g ) domains of the mixed phase (MP) . Within our model onegets three coupled equations for µ ( T ) , σ g ( T ), and σ l ( T ) : p th [ T, µ, M ( σ g )] + p ex ( σ g ) = p th [ T, µ, M ( σ l )] + p ex ( σ l ) , (39) σ i = σ th [ T, µ, M ( σ i )] ( i = g, l ) . (40)Solving these equations gives the critical lines µ = µ ( T ) , p = p ( T ) of the LGPT andthe values σ i ( T ) , M i ( T ) , n i ( T ) , s i ( T ) . . . at the MP boundaries (’binodals’) i = g, l . Notethat within the MP region, µ and p are functions of temperature only and do not de-pend, e.g., on density. The latter is connected with the volume fraction of the gasphase λ ≡ V g / ( V g + V l ) ∈ [0 ,
1] : n = λn g ( T ) + (1 − λ ) n l ( T ) . (41)At given T, n one gets the relation λ = n l ( T ) − nn l ( T ) − n g ( T ) . (42) We have checked that the second root of this equation corresponds to unstable states. We neglect surface effects associated with finite sizes of domains. IG. 2: (a) The isotherm T = 5 MeV of α matter on the ( µ, σ ) plane. The solid and short-dashedlines show equilibrium states without and with BEC. The filled dot marks the boundary of the BECstates. The vertical section GL corresponds to MP states of the LGPT. The dashed and dottedlines represent the metastable and unstable states. The thin dashed-dotted curve corresponds tothe ideal-gas limit. The cross shows the boundary of BEC states in the ideal gas. (b) Isothermsof α matter on the ( µ, n ) plane. The filled square, triangle and circle correspond, respectively, tothe GS, TP and CP. The region of MP is shown by shading. Our calculations show that at T = T TP ≈ .
67 MeV (so-called triple point temperature)the BEC boundary reaches the MP region. At
T < T TP , Bose condensate appears inthe liquid-phase domains. In this case the conditions of phase equilibrium are obtainedfrom Eqs. (39), (40) after replacing M ( σ l ) by µ , and σ l by σ ( µ ) from Eq. (38).A typical example for T = 5 MeV is shown in Fig. 2 (a). The solid and short-dashed linesshow equilibrium states on the ( µ, σ ) plane. At the considered temperature the BEC stateslie outside the MP region (the vertical line GL). One can see that at large scalar densitiesthe results strongly deviate from the ideal gas of bosons with the vacuum mass m . We alsoshow positions of metastable and unstable (spinodal) states. They correspond to solutionsof the gap equation with lower values of pressure as compared to equilibrium state at thesame T and µ .In Fig. 2 (b) we compare different isotherms of α matter on the ( µ, n ) plane. The lowerand upper boundaries of the MP (shaded region) correspond, respectively to the gas-likeand liquid-like binodals. Note that at T = 0 the liquid-like binodal state coincides with12 ABLE I: Characteristics of critical point of α matter T (MeV) µ − m (MeV) M − m (MeV) σ (MeV ) n (cid:0) fm − (cid:1) S/N p/ ( nT ) nλ T . − . . . . .
54 0 .
334 0 . the GS of equilibrium α matter. One can see that the LGPT disappears when T exceedsthe critical point (CP) temperature T CP ≈ . ∂ n/∂µ ) T diverges. Table I shows characteristics of CP obtained within the present model. FIG. 3: Phase diagram of α matter on the ( µ, T ) plane. The dashed curves are adiabatic trajec-tories, i.e., lines of constant entropy per particle. The values of S/N are given in boxes. The BECregion is shown by shading.
Figure 3 represents the phase diagram on the ( µ, T ) plane. The MP region correspondsto the thick solid line between the GS and the CP. The BEC states lie below the thin curvewhich crosses the LGPT critical line at the triple point TP. The domain of BEC states isshown in Fig. 3 by the shaded area. At the same plot we show the behavior of adiabatictrajectories (isentropes), i.e., the lines of constant specific entropy
S/N = const.The isentropes play an important role in a fluid-dynamical evolution of excited matter . In absence of dissipation and shock waves, the total entropy is conserved for thermally equilibrated mattereven in presence of collective flow.
13n particular, we would like to mention the hydrodynamic [26] and thermal [27, 28] modelsof heavy-ion collisions which successfully describe particle production at high energies. Theypostulate that a hot and compressed ’fireball’ is formed at some intermediate stage of a nu-clear collision. Due to the presence of internal pressure, the fireball expands at later stages,producing secondary particles. Fitting the observed data confirmed that the specific entropyof fireball is fully determined by the initial c.m. energy of colliding nuclei.As one can see in Fig. 3, the isentropes enter the MP region at any
S/N . Especiallyimportant is the isentrope
S/N = (
S/N ) CP ≈ .
54 which goes through the CP. As will beshown later, the trajectories with the specific entropy close to (
S/N ) CP go through stateswith anomalously large fluctuations of the particle density. Note, that isentropes with S/N larger (smaller) than (
S/N ) CP enter the MP region at the gas-like (liquid-like) side. Asmentioned above, the isentropes do not cross the BEC boundary outside the MP region. Weshall come back to discussing this phase diagram in Sec. IV A (see Fig. 9). (cid:2) (cid:2) (cid:1) (cid:2) (cid:3) (cid:2) (cid:1) (cid:2) (cid:4) (cid:2) (cid:1) (cid:2) (cid:5) (cid:2) (cid:1) (cid:2) (cid:6) (cid:2) (cid:1) (cid:2) (cid:7)(cid:2)(cid:4)(cid:6)(cid:8)(cid:9)(cid:3) (cid:2)(cid:3) (cid:4)(cid:3) (cid:6)(cid:3) (cid:8)(cid:3) (cid:9)(cid:4) (cid:2) (cid:1) (cid:10) (cid:6) (cid:18) (cid:2) (cid:13) (cid:11) (cid:12) (cid:9) (cid:1) (cid:7) (cid:11) (cid:14) (cid:15) (cid:8) (cid:6) (cid:13)(cid:1) (cid:13) (cid:11) (cid:16) (cid:19) (cid:11) (cid:8) (cid:2) (cid:13) (cid:11) (cid:12) (cid:9) (cid:1) (cid:7) (cid:11) (cid:14) (cid:15) (cid:8) (cid:6) (cid:13)(cid:1) (cid:3) (cid:5) (cid:4) (cid:1) (cid:7) (cid:15) (cid:19) (cid:14) (cid:8) (cid:6) (cid:17) (cid:20) (cid:2) (cid:4)(cid:5) (cid:3) (cid:5) (cid:7) (cid:3) (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:2) (cid:6)(cid:2) (cid:1) (cid:2) (cid:5)(cid:2) (cid:1) (cid:2) (cid:4)(cid:2) (cid:1) (cid:2) (cid:3) (cid:5) (cid:1) (cid:2) (cid:4) (cid:7) (cid:6) (cid:3) (cid:6) (cid:1) (cid:2) (cid:4) (cid:5) (cid:1) (cid:2) (cid:3) (cid:2)(cid:2) (cid:1) (cid:2) (cid:3)(cid:2) (cid:1) (cid:2) (cid:4)(cid:2) (cid:1) (cid:2) (cid:5)(cid:2) (cid:1) (cid:2) (cid:6)(cid:2) (cid:1) (cid:2) (cid:7) (cid:6) (cid:1) (cid:1) (cid:2) (cid:4) (cid:5) (cid:1) (cid:2) (cid:3) (cid:1) (cid:3) FIG. 4: Phase diagram of α matter on the ( n, T ) plane. The values of the condensate density n c are shown by different shades of blue color (see the color map on r.h.s.). The dotted lines arecontours of equal n c in the BEC region. The square, circle and star correspond, respectively, tothe GS, TP, and CP. n c at fixed n, T , first, for statesoutside the MP, i.e., for n > n l ( T ) and T < T
BEC . As explained in Sec. II A this can bedone by solving the equations n c = µ [ σ ( µ ) − σ th ( T, µ, µ )] = n − n th ( T, µ, µ ) , (43)where σ ( µ ) is given by Eq. (38). One can see that n c increases from zero to n whentemperature drops from T = T BEC to T = 0 . FIG. 5: Phase diagrams of α matter on the ( T, M ) (a) and (
T, S/N ) (b) planes. The regionsof MP (a) and BEC (b) are shown by shading.
Inside the MP the condensate appears at
T < T TP . At such temperatures the densitiesof the gas- and liquid-like domains satisfy the relation n g ( T ) (cid:28) n l ( T ) . Unless the tem-perature is extremely low, one can disregard presence of BEC in the gaseous phase. Thenone can write the relation (cid:104) n c (cid:105) ≈ (1 − λ ) n cl ( T ) for the density of BEC averaged over theensemble of coexisting domains. Here n cl ( T ) is the condensate density in the liquid phase at T < T TP . Substituting further Eq. (42) and neglecting terms ∼ n g /n l one gets the relation (cid:104) n c (cid:105) /n ≈ n cl ( T ) /n l ( T ). More detailed information is given in Fig. 4 where we show lines ofequal condensate densities on the ( n, T ) plane. The obtained phase diagrams are similar tothose derived in Ref. [10]. It is interesting that they qualitatively agree with phase diagramsobserved [29] for atomic He.Figure 5(a) shows the phase diagram on the (
T, M ) plane. One can seethat M g ( T ) > M l ( T ) . At low temperatures M g ( T ) ≈ m and M l ( T ) = µ ( T ) ≈ m − W IG. 6: The isentropes
S/N = 3 (the dashed line) and
S/N = 5 (the dash-dotted line) onthe ( n, T ) plane. The dots mark their intersection with the MP boundary. Thin dashed anddash-dotted lines represent the ideal-gas calculation. [see Eq. (34)]. In Fig. 5(b) we show the values of specific entropy (cid:101) s i = ( S/N ) i at the MPboundaries i = g, l . One can see that (cid:101) s l ( T ) < (cid:101) s CP < (cid:101) s g ( T ) . It is possible to derive a simpleanalytic formula for (cid:101) s g ( T ) at T →
0. Indeed, in this limit the density n = n g ( T ) is smalland one can apply the Boltzmann approximation in calculating the specific entropy on thegas-like binodal. One gets approximate relations (cid:101) s g ( T ) ≈ − ln z ≈
52 + m − µ ( T ) T ≈
52 + W T . (44)In the first equality we have used the upper line of Eq. (25). Comparison of the dashed anddash-dotted lines in Fig. 5 (b) confirms good accuracy of these relations at low T .In Fig. 6 we compare the behavior of isentropes S/N = 3 and
S/N = 5 on the ( n, T )plane. Within the MP region we use the relation for the entropy density s = λs g ( T ) + (1 − λ ) s l ( T ) , (45)where λ is given by Eq. (42) . One can see jumps in slopes of isentropes at the MP boundary(a similar behavior takes place also on the ( n, p ) plane). This in turn leads to jumps of thesound velocity and the heat capacity at these boundaries (see Secs. IV B and IV C). Note16hat at small n and T isentropes go near the gas-like binodal. The calculation shows thatoutside the MP region the adiabatic trajectories are close to those in the ideal Bose-gas. IV. POSSIBLE SIGNATURES OF PHASE TRANSITION IN α MATTERA. Strong density fluctuations
Statistical fluctuations of conserved charges (e.g., baryon number) are important ob-servables for experimental studies of the phase diagrams of interacting systems. In thegrand-canonical ensemble such fluctuations are expressed via ’susceptibilities’, i.e., higher-order derivatives of pressure with respect to
T, µ . Below we calculate the scaled variance ω defined as the second moment of the particle number in a given volume: ω = (cid:104) N (cid:105) − (cid:104) N (cid:105) (cid:104) N (cid:105) , (46)where averaging is performed at fixed temperature. Outside the MP region one gets [23] therelation ω = Tn (cid:18) ∂ n∂µ (cid:19) T = Tn (cid:18) ∂ p∂µ (cid:19) T . (47)In calculating the derivatives in the r.h.s. of (47) one should explicitly take into account thedependence M = M ( T, µ ) .At the region
T > T
BEC , using Eqs. (15), (31) and (10), one has ω = Tn (cid:20) ∂ n th ( T, µ ; M ) ∂µ + ∂ n th ( T, µ ; M ) ∂M M (cid:48) ( σ ) (cid:18) ∂σ∂µ (cid:19) T (cid:21) , (48)where M (cid:48) ( σ ) = (cid:16) bσ − a (cid:17) M − , (cid:18) ∂σ∂µ (cid:19) T = ∂σ th ∂µ (cid:20) − ∂σ th ∂M M (cid:48) ( σ ) (cid:21) − . (49)At µ → µ CP , T → T CP both ( ∂ n/∂µ ) T and ( ∂ σ/∂µ ) T diverge which leads to the relation ∂σ th ∂M M (cid:48) ( σ ) = 1 at CP . (50)As one can see from Eq. (11) the density σ th decreases with M at fixed T, µ . Therefore,the CP (and LGPT) may exist if M ( σ ) contain regions with M (cid:48) ( σ ) <
0. In accordance withfirst equality in Eq. (49), this is possible only if the interaction has an attractive term withnonzero a . 17y using Eqs. (17) and (19) one can calculate the density derivatives entering Eq. (48)in the NRA. Then one obtains (cid:18) ∂σ∂µ (cid:19) T ≈ σg / ( z ) σM (cid:48) ( σ ) g / ( z ) + T g / ( z ) . (51)At given n, T , the values z and σ are determined from the approximate rela-tions n ≈ mσ ≈ g g / ( z ) /λ T ( T, m ) . Substituting (51) into (48) gives ω ≈ (cid:20) g / ( z ) g / ( z ) + σM (cid:48) ( σ ) T (cid:21) − . (52)The second term in denominator vanishes in the limiting case of the ideal Bose gas ( M = m ).Note, that at T → T BEC the first term in brackets goes to zero.At
T < T
BEC using Eq. (23) one has: ω = Tn (cid:20) ∂ n th ( T, µ ; µ ) ∂µ + ∂ n c ( T, µ ) ∂µ (cid:21) , (53)where, n c ( T, µ ) is defined by the first equality in (43). Within the NRA one has approxi-mately n ≈ µσ ( µ ), where σ ( µ ) is given by Eq. (38) . Using further Eq. (47) one gets therelations ω ≈ T σ (cid:48) ( µ ) σ ( µ ) = µTσ ( bσ − . a ) . (54)One can see that ω → T → a = 0 , b = 0) where ω is infinite [23] in the BEC region.In the MP region one should take into account not only fluctuations of particle numbersinside the coexisting domains of matter, but also fluctuations of the interphase boundaries,which change the relative fraction of domain volumes. The latter corresponds to fluctuationsof the parameter λ around the equilibrium value given by Eq. (42). One can include bothtypes of fluctuations by calculating ω directly from Eq. (46) . In Ref. [30] such a calculationwas made for a particular case of the classical van der Waals model. Below we apply a similarapproach for an arbitrary LGPT.We obtain the following result at given n, T in the MP region: ω = 1 n (cid:20) λn g ω g + (1 − λ ) n l ω l + ( n l − n g ) λ (1 − λ ) ω g ω l λn l ω g + (1 − λ ) n g ω l (cid:21) . (55) A similar result can be obtained in the vector model of Ref. [10] with the replacement σM (cid:48) ( σ ) → nU (cid:48) ( n )where U ( n ) is the potential introduced in Eq. (37) . IG. 7: Scaled variance of α matter as the function of temperature along the isentropes S/N = 3 (a) and
S/N = 4 .
54 (b). The dashed lines are calculated without the third term inEq. (55). The MP regions are shown by shading. The open and full dots correspond, respectively,to boundaries of MP and BEC states.FIG. 8: Scaled variance of α matter as the function of temperature along the isentropes with S/N = 3 −
7. Circles mark points where the isentropes cross the MP boundary. The verticaldotted line shows the temperature T = T CP . Here λ is defined in Eq. (42), n i and ω i are, respectively, the particle densities and scaledvariances inside the domains i = g, l . These quantities are equal to their values at the gas-19 (cid:10) (cid:8) (cid:1) (cid:8) (cid:3) (cid:1) (cid:5) (cid:8) (cid:3)(cid:3)(cid:5)(cid:7)(cid:9)(cid:11)(cid:4) (cid:3)(cid:4) (cid:5)(cid:4) (cid:7)(cid:4) (cid:9)(cid:4) (cid:11)(cid:5) (cid:3) (cid:2) (cid:7) (cid:3) (cid:4) (cid:3) (cid:2) (cid:1) (cid:5) (cid:2) (cid:1) (cid:2) (cid:5) (cid:3)(cid:4)(cid:5)(cid:6)(cid:8) (cid:7)(cid:11) (cid:1) (cid:10) (cid:9) (cid:3) (cid:2) (cid:5) (cid:1) (cid:2) (cid:4) (cid:7) (cid:6) (cid:3) m (cid:4) (cid:1) (cid:11) (cid:1) (cid:1) (cid:5) (cid:9) (cid:6) (cid:1) (cid:3) (cid:3) (cid:2) (cid:3) (cid:4) (cid:3) (cid:2) (cid:3) (cid:5) (cid:3) (cid:2) (cid:3) (cid:6) (cid:3) (cid:2) (cid:3) (cid:7) (cid:3) (cid:2) (cid:3) (cid:8)(cid:3)(cid:5)(cid:7)(cid:9)(cid:11)(cid:4) (cid:3)(cid:4) (cid:5)(cid:4) (cid:7)(cid:4) (cid:9)(cid:4) (cid:11)(cid:5) (cid:3) (cid:2) (cid:8) (cid:3) (cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:11) (cid:1) (cid:10) (cid:9) (cid:8) (cid:2) (cid:1) (cid:2) (cid:5) (cid:2) (cid:1) (cid:3) (cid:3)(cid:4) (cid:12) (cid:1) (cid:2) (cid:10) (cid:11) (cid:1) (cid:2) (cid:3) (cid:3)(cid:3) (cid:2) (cid:8)(cid:4)(cid:6)(cid:4) (cid:3)(cid:8) (cid:3) w FIG. 9: (a) The dashed lines: isentropic trajectories with different
S/N (shown by black numbersin boxes) on the ( µ, T ) plane. Colors show values of scaled variance ω . Thin white lines arecontours of equal ω (their values are given by white numbers). The thick green and white linesrepresent the LGPT line and the BEC boundary, respectively. The star marks position of the CP.(b) Same as (a) but on the ( n, T ) plane. Note the strong ω peak near the critical point. and liquid-like binodals. The contribution of λ fluctuations is described by the third termin the r.h.s. of (55). This term vanishes near the MP boundary where λ (1 − λ ) → ω calculation for states along the isentropes with different S/N are shownin Figs. 7–8. The calculations predict strong peaks of scaled variance for
S/N = 4 . ± ω to particle interactions. Figures 9 (a) and (b) show the density plots of ω in the ( µ, T ) and ( n, T ) planes, respectively. One can see a very narrow peak of ω ( T, µ ) at T ≈ T CP . This can be used as a clear signal of the CP in experimental searches of LGPT. B. Softening of the equation of state
An important characteristics of the equation of state is the sound velocity c s whichcharacterizes propagation of small perturbation in the local rest frame of matter . In the20deal fluid dynamics the sound velocity squared is equal to [23] c s = (cid:18) ∂ p∂ε (cid:19) (cid:101) s = (cid:18) ∂ p∂ε (cid:19) n + nε + p (cid:18) ∂ p∂n (cid:19) ε , (56)where (cid:101) s = s/n is the entropy per particle . Using Eqs. (27), (28) one can directly calculate c s by expressing the derivatives entering Eq. (56) via the derivatives of n, s over T, µ . Forthe MP states one can use Eqs. (41), (45) and rewrite these derivatives via the binodalquantities s (cid:48) i ( T ) and n (cid:48) i ( T ) ( i = g, l ) .Within the NRA one can get much simpler expressions (outside the MP). Indeed, fromthe first equality in (56), substituting ε ≈ mn one has c s ≈ m (cid:20)(cid:18) ∂ p th ∂n (cid:19) (cid:101) s + (cid:18) ∂ p ex ∂n (cid:19) (cid:101) s (cid:21) . (57)According to Eq. (25), in this approximation the specific entropy (cid:101) s depends in the NRAonly on fugacity z at T > T
BEC or on nλ T at T < T
BEC . From Eqs. (17) and (18) onecan see that in both cases nT − / and p th T − / are approximately constant at (cid:101) s = const.Using Eq. (6) one gets c s ≈ σM (cid:48) ( σ ) m + 5 T m g / ( z ) g / ( z ) , T > T BEC ,ξ (5 / nλ T ( T, m ) , T < T BEC , (58)where σ ≈ n/m and z is found by solving the equation nλ T ( T, m ) /g = g / ( z ). The secondterm in (58) is the same as for ideal Bose-gas , and the first one takes into account theinteraction effects.The results of calculation are presented in Figs. 10 (a) and (b) for states along the isen-tropes S/N = 3 and
S/N = 4 .
54 , respectively. As expected, the sound velocity is stronglysuppressed inside the MP region. On the other hand, it becomes much larger at higher den-sities where the repulsive interaction dominates. Outside the MP region c s increases with T faster as compared to the ideal-gas calculation. One can see discontinuities of sound veloci-ties at the MP boundary: c s jumps down during the adiabatic expansion. It is known [31] Equation (56) is derived for a continuous matter without large gradients of density. On the other hand, thisis not true for MP with different densities of liquid and gas domains. Nevertheless, one can approximatelyconsider the MP region as a homogeneous matter if the wavelength of a sound wave exceeds typical domainsizes (i.e., at low enough frequencies). In the Boltzmann approximation c s = (cid:112) T / (3 m ). IG. 10: Sound velocity as the function of temperature along the isentropes
S/N = 3 (a) and
S/N = 4 .
54 (b) (the solid lines). The dashed curves correspond to ideal gas. The MP states areshown by shading. Full dots mark the BEC boundary at T = T TP . that such a behavior leads to the formation of a rarefaction shock. The latter may beregarded as a signature of the LGPT. C. Enhanced heat capacity
The isohoric heat capacity is another observable which is sensitive to the LGPT and BECeffects. This quantity (per particle) is defined as [23] c v = Tn (cid:18) ∂ s∂T (cid:19) n = 1 n (cid:18) ∂ ε∂T (cid:19) n . (59)We calculate c v outside the MP by using formulae similar to Eqs. (48), (49). For stateswithin the MP we perform the direct calculation based on Eqs. (41) and (45).Within the NRA one can use approximate relations (17), (25) with M ≈ m . Outsidethe MP, we obtain the result coinciding with that for the ideal Bose gas [22] c v ≈ g / ( z ) g / ( z ) − g / ( z ) g / ( z ) , T > T BEC , ξ (5 / nλ T ( T, m ) , T < T BEC . (60)In the Boltzmann approximation one gets the well-known value c v = 3 /
2. At the BECboundary c v = 3 . ξ (5 / /ξ (3 / ≈ .
925 .22
IG. 11: Isochoric heat capacity per particle as the function of temperature along the isentropes
S/N = 3 (dashed) and
S/N = 4 .
54 (solid). Full dots correspond to T = T TP . Open dots mark theMP boundary. Thin dotted and dash-dotted lines represent the ideal gas calculation. Figure 11 shows the temperature dependence of c v along the isentropes with same S/N asin Fig. 10. Again one can see jumps of the heat capacity at the MP boundary. The predicted c v -values are much larger in the MP region as compared to the ideal gas calculation. Thesevalues increase roughly as T − at T →
0. Such a behavior can be qualitatively understood.Indeed, as mentioned above, at low temperatures the isentrope (cid:101) s = const becomes close tothe gas-like binodal, i.e., λ ≈ n ≈ n g ( T ). Substituting s ≈ (cid:101) sn g ( T ) into Eq. (59) gives c v (cid:101) s ≈ T n (cid:48) g ( T ) n g ( T ) ≈ m − µ ( T ) T ≈ W T . (61)In the second equality we neglect deviations from the Boltzmann statistics having in mindthat the density n g is small at low temperatures. V. CONCLUSIONS
We have proposed a field-theoretical model to describe α -clustered nuclear matter atfinite temperatures. The system of interacting α particles is represented by a scalar field φ with the Lagrangian containing the attractive ( φ ) and repulsive ( φ ) self-interaction terms.23he calculations are done within the mean-field approach which obeys a self-consistencyrelation between the scalar mean field and the particle effective mass. The model has twofree parameters which are fixed by fitting properties of the ground state of cold α matterknown from microscopic calculations of Clark and Wang [25].Our main results are as follows: 1) α matter exhibits a liquid-gas phase transition withthe critical point at T c ≈
14 MeV , n c ≈ .
012 fm − ; 2) at low temperatures the α mattercontains the Bose-Einstein condensate, which appears in the liquid phase; 3) all isentropictrajectories, S/N = const, terminate in the mixed-phase region; 4) the BEC boundaryoutside the mixed phase coincides with the isentrope
S/N ≈ .
28 ; 5) the scaled varianceof density fluctuations has a strong peak at the critical point which lies on the isentrope
S/N ≈ . α -clusteredmatter. We believe that enhanced yields of α particles and α -conjugate nuclei observed inRefs. [2, 15] are associated with the mixed phase of α matter formed at a late stage ofthe nuclear matter evolution. Since at this stage the expansion is approximately isentropic,one may hope to select events which correspond to the critical point of α matter around S/N ≈ .
5. In the vicinity of this point we expect a wide (power-law) mass distribution ofproduced fragments, like Y ( A ) ∝ A − τ where τ ≈ α clustering can be expected in heavy and superheavy nuclei.As demonstrated in Ref. [32], due to strong Coulomb repulsion such nuclei may develop ahollow structure where α ’s are condensed in the outer shell but neutrons fill the centralregion. We hope that future experiments with heavy-ion beams at intermediate energieswill bring new evidences for α clustering and α condensation in nuclear systems. Acknowledgments
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