Critical Temperature for α -Particle Condensation within a Momentum Projected Mean Field Approach
aa r X i v : . [ nu c l - t h ] J a n Critical Temperature for α -Particle Condensation within a Momentum ProjectedMean Field Approach T. Sogo, G. R¨opke
Institut f¨ur Physik, Universit¨at Rostock, D-18051 Rostock, Germany
P. Schuck
Institut de Physique Nucl´eaire, CNRS, UMR 8608, Orsay F-91406, FranceUniversit´e Paris-Sud, Orsay F-91505, France andLaboratoire de Physique et Mod´elisation des Milieux Condens´es,CNRS and Universit´e Joseph Fourier, 25 Avenue des Martyrs,Boˆıte Postale 166, F-38042 Grenoble Cedex 9, France
Alpha-particle (quartet) condensation in homogeneous spin-isospin symmetric nuclear matter isinvestigated. The usual Thouless criterion for the critical temperature is extended to the quartetcase. The in-medium four-body problem is strongly simplified by the use of a momentum projectedmean field ansatz for the quartet. The self-consistent single particle wave functions are shown anddiscussed for various values of the density at the critical temperature.
PACS numbers: 03.75.Nt, 03.75.Ss, 21.65.-f, 74.20.Fg
I. INTRODUCTION
Investigation of pairing in different Fermi systems isstill on the forefront of active research. Examples are nu-clear physics [1] and the physics of cold fermionic atoms[2]. The formation and condensation of heavier clustersin Fermi systems is much less studied.In cold atom physics the recent advent of trappingthree different species of fermions [3] has opened up thepossibility of creating gases of heavier clusters. For thetime being those may be trions (bound state of three dif-ferent fermions) but in the future one also can think ofquartets (bound state of four different fermions). Thelatter are specially interesting because of their bosonicnature and the possibility of Bose-Einstein condensation(BEC) of quartets. The description of quartet conden-sation to occur has been attempted with an extension ofthe so-called Cooper problem to the four body case in[4]. In [5] a variational procedure for the condensation of(2 s + 1)-component fermion clusters, with s the fermionspin, has been proposed. A quartet phase has been foundin a one dimensional model with four different fermions[6].On the other hand, nuclear physics, because it isa four-component fermion-system (proton/neutron spinup/down), all fermions attracting one another, leadingto the very strongly bound α -particle, is a proto-typesystem for quartetting. There, the formation of clustershas been an object of study almost since the beginningof nuclear physics [7]. Of course, pairing also exists innuclei from where it is concluded that neutron stars aresuperfluid. Nuclei are very small quantum objects withonly a (slowly) fluctuating phase (the conjugate variableto particle number N ). Actually the number of Cooperpairs in nuclei generally does not exceed about a dozen(often much less) and yet clear signs of superfluidity areobserved in nuclei (e.g., moments of inertia strongly re- duced from their classical value), implying that no criticalsize exists from where signatures of superfluidity abruptlydisappear. One, thus, can safely extrapolate from finitenuclei to superfluidity in neutron stars. On the otherhand, as already mentioned, in nuclear physics the ex-istence of quartets ( α -particles) as subclusters of nucleiis omnipresent. As well known, many lighter nuclei withequal proton and neutron numbers ( Z = N ) show, for in-stance in excited states, strong α clustering. The conceptthat these α -particles may form a condensate in certainlow density states of nuclei and that this may, in analogyto the pairing case, be a precursor sign of α -particle con-densation in infinite matter [8], has come up only recently[9]. Also heavy nuclei seem to have preformed α -clustersin the surface because of their well known spontaneous α decay properties.Symmetric nuclear matter does not exist in nature be-cause of the too strong Coulomb repulsion. However, incollapsing stars, so-called proto-neutron stars, the frac-tion of protons is still high [10] and the formation of α -particles and, at sufficiently low temperature, their con-densation may eventually be possible. At any rate, itseems evident that nuclear matter at various degrees ofasymmetry is unstable with respect to cluster formationin the low density regime. At zero temperature, the moststable nucleus is Fe but as a function of temperature,density, asymmetry, other cluster compositions of infi-nite baryonic matter may be formed. Several theoreticalstudies predict α -phases to exist in certain temperature-density- asymmetry domains [11].In view of the complexity of the task, the objectiveof the present work is quite modest. We want to studythe critical temperature of α -particle condensation as afunction of density and temperature in symmetric nuclearmatter. Still, even this task will not be carried out downto the BEC limit. We will study the critical temperature T αc for the onset of formation of α -particles in a thermal FIG. 1: Sketch of α -particle configuration, indicating that thetwo protons and two neutrons occupy the lowest 0 S level inthe mean field potential of harmonic oscillator shape. gas of nucleons. This shall be done with a theory anal-ogous to the famous Thouless criterion for the onset offormation of Cooper pairs in a superconductor. On themicroscopic level the problem is still very challenging,since it amounts to solve an in-medium four-body prob-lem. In spite of that, solutions have already been workedout in the past, either solving the Faddeev-Yakubovskyequations [12] or with an approximate procedure [8].In this work, we will continue along those lines. Thefinal objective is to reach the BEC regime in a treatmentsimilar to the one of Nozi`eres and Schmitt-Rink (NSR)theory [13], but for quartets. Needless to say that thisonly will be possible if the whole formalism can radicallybe simplified. Actually, as we will show in this work,such a procedure may well exist. In any case, it is notconceivable that one treats condensation of bosonic clus-ters built out of N fermions on the level of non-linearin medium N -body equations for N >
2. On the otherhand, it is well known, that nuclei can satisfactorily bedescribed, in mean field approximation [14]. Projectingthese mean field (Hartree-Fock) type of solutions on zerototal momentum ( K = 0) will then allow these meanfield clusters to Bose condense. Actually it is well knownamong the nuclear physics community that even for sucha small nucleus as the α -particle a momentum projectedmean field approach yields a very reasonable description[15]. The reason for this stems, as already mentioned,from the presence of four different fermions, all attract-ing one another with about the same force.In Fig. 1 we sketch the situation, indicating that thetwo protons and two neutrons occupy the lowest 0 S levelof the mean field potential. Actually calculations showthat the 0 S orbital of the self-consistent mean field resem-bles very much an oscillator wave function of Gaussianshape. In this respect the cartoon in Fig. 1 is not sofar from reality. We suspect that the situation is genericfor all strongly bound quartets which may be producedin the future and, therefore, our present study is of quitegeneral interest. We will adopt this momentum projectedmean field procedure in this work. II. THE IN-MEDIUM FOUR BODY EQUATION
In-medium four body equations are well documentedin the literature since long [16]. In the present case of an in-medium quartet, the corresponding equation reads asfollows [8]:( E − ε )Ψ = (1 − f − f ) X ′ ′ v , ′ ′ Ψ ′ ′ + (1 − f − f ) X ′ ′ v , ′ ′ Ψ ′ ′ + permutations , (1)where f i = f ( ε i ) = [ e ( ε i − µ ) /T + 1] − with ε i = ε ( k i ) = k i / (2 m ) is the Fermi-Dirac distribution and ε = ε + ε + ε + ε (¯ h = c = k B = 1: natural units). The matrixelement of the interaction is v , ′ ′ with the numbers 1,2, 3, · · · standing for all quantum numbers as momenta,spin, isospin, etc., as also in all other quantities in (1).In Eq. (1), when E = 4 µ , this signals quartet conden-sation in very much the same manner as in the two bodyequation( E − ε )Ψ = (1 − f − f ) X ′ ′ v , ′ ′ Ψ ′ ′ , (2)where ε = ε + ε , the approach of T → T c such that E → µ signals the transition to a superconducting orsuperfluid state (the well known Thouless criterion [17]).Of course, as already stated several times, the determi-nation of T αc needs the heavy solution of the in-mediummodified four particle equation (1).Following the discussion in the introduction, we, there-fore, make the following ’projected’ mean field ansatz forthe quartet wave function [4, 5, 18],Ψ = (2 π ) δ (3) ( k + k + k + k ) Y i =1 ϕ ( k i ) χ ST , (3)where χ ST is the spin-isospin function which we supposeto be the one of a scalar ( S = T = 0). We will notfurther mention it from now on. We work in momen-tum space and ϕ ( k ) is the as-yet unknown single par-ticle 0 S wave function. In position space, this leads tothe usual formula [14] Ψ → R d R Q i =1 ˜ ϕ ( r i − R )where ˜ ϕ ( r i ) is the Fourier transform of ϕ ( k i ). If wetake for ϕ ( k i ) Gaussian shape, this gives: Ψ → exp[ − c P ≤ i 71 fm) of the free ( f i = 0) α -particle come outright. The adjusted values are: λ = − 992 MeV fm , and b = 1 . 43 fm − . The results of the calculation are shownin Fig. 2.In Fig. 2, the maximum of critical temperature T αc, max is at µ = 5 . α -condensation can exist up to µ max = 11 MeV. It is very remarkable that the obtainedresults for T αc well agree with a direct solution of (1)[12]. These results for T αc are by about 25 percent higherthan the ones of our earlier publication [8]. We, however,checked that the underlying radius of the α -particle inthat work is larger than the experimental value and that T αc decreases with increasing radius of α -particle. Fur-thermore a different variational wave function was usedin [8].In Fig. 2 we also show the critical temperature fordeuteron condensation derived from Eq. (11). In thiscase, we take λ = − and k = 1 . 46 fm − to get experimental energy ( − . . α -particle.The latter breaks down rather abruptly at a critical pos-itive value of the chemical potential. Roughly speaking,this corresponds to the point where the α -particles startto overlap. This behavior stems from the fact that Fermi-Dirac distributions in the four body case, see (1), cannever become step-like, as in the two body case, even notat zero temperature, since the pairs in an α -particle arealways in motion. As a consequence, α -condensation gen-erally only exists as a BEC phase and the weak couplingregime is absent.Fig. 3 shows the normalized self-consistent solutionof the wave function in momentum space derived fromEq. (8) and the wave function in position space definedby its Fourier transform ˜ ϕ ( r ). Fig. 3-(a1) and (b1) arethe wave functions of the free α -particle. As discussed inthe Introduction, the wave function resembles a Gaussianand this shape is approximately maintained as long as µ is negative, see Fig. 3-(a2). On the contrary, the wavefunction of Fig. 3-(a3), where the chemical potential ispositive, has a dip around k = 0 which is due to the Pauliblocking effect. For the even larger positive chemical po-tential of Fig. 3-(a4) the wave function develops a node.This is because of the structure of the wave function de-rived in Eq. (4) from where one can realize that again thisstems from the Pauli blocking factor. The maximum ofthe wave function shifts to higher momenta and followsthe increase of the Fermi momentum k F , as indicatedon Fig. 3. From a certain point on the denominator in(8) develops a zero and no self-consistent solution can befound any longer.On the other hand, the wave functions in position spacein Figs. 3-(b2), (b3) and (b4) develop an oscillatory be-havior, as the chemical potential increases. This is remi-niscent to what happens in BCS theory for the pair wavefunction in position space [19]. IV. DISCUSSION AND CONCLUSIONS In this work we again took up the study of the criti-cal temperature of α -particle (quartet) condensation inhomogeneous symmetric nuclear matter. We essentiallyconfirm the behavior of two previous studies [8, 12]. Theobjective of the paper was to show that practically sameresults as before can be obtained with a strongly simpli-fying ansatz for the four particle wave function. Namely,this time, we used a momentum projected mean fieldvariational wave function. This is based on the fact thatthe four different fermions of the quartet can occupy thesame single particle 0 S -wave function in the mean field.The latter is to be determined from a self-consistent nonlinear HF-type of equation as a function of chemical po-tential or density. The relation between the chemical (a1) µ = − . 08 (MeV) T c = 0 (MeV) (a2) µ = − . 22 (MeV) T c = 6 . 61 (MeV) (a3) µ = 6 . 17 (MeV) T c = 8 . 45 (MeV) (a4) µ = 10 . T c = 5 . 54 (MeV) (b1)(b2)(b3)(b4) k (fm − ) ϕ ( k )( f m / ) r (fm) r ˜ ϕ ( r )( f m − / ) k (fm − ) ϕ ( k )( f m / ) r (fm) r ˜ ϕ ( r )( f m − / ) k (fm − ) ϕ ( k )( f m / ) r (fm) r ˜ ϕ ( r )( f m − / ) k (fm − ) ϕ ( k )( f m / ) r (fm) r ˜ ϕ ( r )( f m − / ) FIG. 3: Single particle wave functions in momentum space ϕ ( k ) (a), and in position space r ˜ ϕ ( r ) (b) at critical temper-ature, Eq. (8). From top to bottom: (1) µ = − . 08 MeV, T c = 0 MeV, n = 0 fm − (2) µ = − . 22 MeV, T c = 6 . n = 9 . × − fm − , (3) µ = 6 . 17 MeV, T c = 8 . n = 3 . × − fm − , and (4) µ = 10 . T c = 5 . n = 3 . × − fm − . Figs. (a1) and (b1) correspondto the wave functions for free α -particle. The vertical lines in(a3) and (a4) are at the Fermi wave length k F = √ mµ . potential and density is taken from the free Fermi gasrelation, Eq. (12). However, the total nucleon densityof the system must be calculated from the mean singlenucleon state occupation number taking into account cor-relations, so that the contribution of bound states to thetotal nucleon density is taken into account, see Ref. [20].To calculate the critical temperature not as function ofthe free nucleon density, see Fig. 2b, but of the totalnucleon density, a generalization `a la NSR [13] must beperformed, that is we have at least to incorporate thecontribution of the α -particle density including the con-densate to the single particle occupation numbers. Thisshall be investigated in future work.Besides, in this work, we used the isospin-independentseparable potential, Eq. (9), for the two-body interactionas a simplification. Comparison with a realistic two-bodyinteraction is interesting. This study also shall be donein the future.The self-consistent wave function has been studied inmomentum and position space. For negative chemicalpotential the single particle wave function behaves likea Gaussian. However, once the chemical potential turnspositive, then the single particle wave function in r -spacestarts to oscillate. This is a well known feature fromordinary pairing.We, therefore, have demonstrated that a very simpli-fying momentum projected mean field ansatz suffices toaccount for the salient features of quartet condensation.This is very helpful for the next step which is more com-plicated, i.e. the incorporation of quartet condensationself-consistently into the Equation of State (EOS).We should, however, be aware of the fact that our pro-jected mean field ansatz for the quartet wave functioncan only be a valid approximation so long as well de-fined quartets exist. In the break down region seen onFig. 2, this is certainly no longer the case. How the quar-tet phase evolves into a superfluid phase of pairs is anopen question. A possibility to study this very inter-esting problem could be to write down the in-mediumfour body equation (1) directly in the BCS formalism,i.e. with the corresponding BCS coherence factors. Itmay be foreseen that the latter only catch on close to the transition region. Another interesting problem forthe future is how the present results are modified in theasymmetric case, that is in the case of neutron excess.The success of our study to employ a very simplifyingansatz of the mean field type for the quartet wave func-tion, may open wide perspectives. Besides to push thedescription of quartet condensation much further, theremight exist the possibility that even for the case of a gasof trions such a projected mean field ansatz is a quitevalid approach. In the case of three colors, like quarksin the constituent quark model for nucleons, a harmonicconfining potential is frequently assumed and the threequarks can occupy the lowest 0 S state, analogously to thecase of quartets treated in the present paper. Of course,trions are composite fermions and cannot be treated inthe same way as bosonic composites, since they shall forma new Fermi gas with their own new Fermi level. Howthis situation can eventually be treated has recently beenoutlined in [21]. 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