Critical Temperature for $α$-Particle Condensation within a Momentum Projected Mean Field Approach

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by  T. Sogo

Critical Temperature for α-Particle Condensation within a Momentum Projected
Mean Field Approach
T. Sogo, G. Ro¨pke
Institut fu¨r Physik, Universit¨at Rostock, D-18051 Rostock, Germany
P. Schuck
Institut de Physique Nucl´eaire, CNRS, UMR 8608, Orsay F-91406, France
Universit´e Paris-Sud, Orsay F-91505, France and
Laboratoire 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 is
9 investigated. The usual Thouless criterion for the critical temperature is extended to the quartet
0
case. Thein-mediumfour-bodyproblemisstrongly simplifiedbytheuseofamomentumprojected
0
mean field ansatz for the quartet. The self-consistent single particle wave functions are shown and
2
discussed for various values of the density at the critical temperature.

n
a PACSnumbers: 03.75.Nt,03.75.Ss,21.65.-f,74.20.Fg
J

6
I. INTRODUCTION ducedfromtheirclassicalvalue),implyingthatnocritical

] sizeexistsfromwheresignaturesofsuperfluidityabruptly
h
disappear. One, thus, can safely extrapolate from finite
t Investigation of pairing in different Fermi systems is
- nuclei to superfluidity in neutron stars. On the other
l stillontheforefrontofactiveresearch. Examplesarenu-
c hand, as already mentioned, in nuclear physics the ex-
clear physics [1] and the physics of cold fermionic atoms
u istence of quartets (α-particles) as subclusters of nuclei
n [2]. The formation and condensation of heavier clusters
is omnipresent. As well known, many lighter nuclei with
[ in Fermi systems is much less studied.
equalprotonandneutronnumbers(Z =N)show,forin-
In cold atom physics the recent advent of trapping
1 stanceinexcitedstates,strongαclustering. Theconcept
v three different species of fermions [3] has opened up the that these α-particles may form a condensate in certain
5 possibility of creating gases of heavier clusters. For the lowdensitystatesofnucleiandthatthismay,inanalogy
7 timebeingthosemaybetrions(boundstateofthreedif-
tothe pairingcase,be a precursorsignofα-particlecon-
6 ferent fermions) but in the future one also can think of
densationininfinitematter[8],hascomeuponlyrecently
0
quartets (bound state of four different fermions). The
. [9]. Also heavy nuclei seemto have preformedα-clusters
1 latter are specially interesting because of their bosonic
inthesurfacebecauseoftheirwellknownspontaneousα
0 nature and the possibility of Bose-Einsteincondensation
decay properties.
9
(BEC) of quartets. The description of quartet conden-
0 sation to occur has been attempted with an extensionof Symmetric nuclearmatterdoes notexistinnaturebe-
:
v the so-called Cooper problem to the four body case in cause of the too strong Coulomb repulsion. However, in
i [4]. In[5]avariationalprocedureforthecondensationof collapsing stars, so-called proto-neutron stars, the frac-
X
(2s+1)-component fermion clusters, with s the fermion tion of protons is still high [10] and the formation of α-
ar spin,hasbeenproposed. Aquartetphasehasbeenfound particles and, at sufficiently low temperature, their con-
in a one dimensional model with four different fermions densation may eventually be possible. At any rate, it
[6]. seems evident that nuclear matter at various degrees of
asymmetry is unstable with respect to cluster formation
On the other hand, nuclear physics, because it is
inthelowdensityregime. Atzerotemperature,themost
a four-component fermion-system (proton/neutron spin
stable nucleus is 56Fe but as a function of temperature,
up/down), all fermions attracting one another, leading
density, asymmetry, other cluster compositions of infi-
to the very strongly bound α-particle, is a proto-type
nite baryonic matter may be formed. Severaltheoretical
system for quartetting. There, the formation of clusters
studies predict α-phases to exist in certain temperature-
has been an object of study almost since the beginning
density- asymmetry domains [11].
of nuclear physics [7]. Of course, pairing also exists in
nuclei from where it is concluded that neutron stars are In view of the complexity of the task, the objective
superfluid. Nuclei are very small quantum objects with of the present work is quite modest. We want to study
only a (slowly) fluctuating phase (the conjugate variable the critical temperature of α-particle condensation as a
to particle number N). Actually the number of Cooper functionofdensityandtemperatureinsymmetricnuclear
pairs in nuclei generally does not exceed about a dozen matter. Still, eventhis taskwill notbe carriedoutdown
(often much less) and yet clear signs of superfluidity are totheBEClimit. Wewillstudythecriticaltemperature
observed in nuclei (e.g., moments of inertia strongly re- Tα for the onset of formation of α-particles in a thermal
c
2
in-medium quartet, the corresponding equation reads as
follows [8]:
(E−ε1234)Ψ1234 = (1−f1−f2) v12,1′2′Ψ1′2′34
1′2′
X
+ (1−f1−f3) v13,1′3′Ψ1′23′4
1′3′
FIG.1: Sketchofα-particleconfiguration,indicatingthatthe X
+ permutations, (1)
two protons and two neutrons occupy the lowest 0S level in
themean field potential of harmonic oscillator shape. where f = f(ε ) = [e(εi−µ)/T +1]−1 with ε = ε(k ) =
i i i i
k2/(2m)istheFermi-Diracdistributionandε =ε +
i 1234 1
ε +ε +ε (h¯ =c=k =1: naturalunits). Thematrix
2 3 4 B
gas of nucleons. This shall be done with a theory anal- element of the interaction is v12,1′2′ with the numbers 1,
ogous to the famous Thouless criterion for the onset of 2, 3, ··· standing for all quantum numbers as momenta,
formation of Cooper pairs in a superconductor. On the spin, isospin, etc., as also in all other quantities in (1).
microscopic level the problem is still very challenging, In Eq. (1), when E =4µ, this signals quartet conden-
since it amounts to solve an in-medium four-body prob- sationinverymuchthesamemannerasinthetwobody
lem. Inspiteofthat,solutionshavealreadybeenworked equation
out in the past, either solving the Faddeev-Yakubovsky
equations [12] or with an approximate procedure [8]. (E−ε12)Ψ12 =(1−f1−f2) v12,1′2′Ψ1′2′, (2)
In this work, we will continue along those lines. The 1′2′
X
finalobjectiveistoreachthe BECregimeinatreatment where ε = ε +ε , the approach of T → T such that
12 1 2 c
similar to the one of Nozi`eres and Schmitt-Rink (NSR) E → 2µ signals the transition to a superconducting or
theory [13], but for quartets. Needless to say that this superfluidstate (the wellknownThouless criterion[17]).
onlywillbe possible ifthe wholeformalismcanradically Ofcourse,asalreadystatedseveraltimes,thedetermi-
be simplified. Actually, as we will show in this work, nation of Tα needs the heavy solution of the in-medium
c
such a procedure may well exist. In any case, it is not modified four particle equation (1).
conceivablethatone treatscondensationofbosonicclus- Followingthediscussionintheintroduction,we,there-
ters built out of N fermions on the level of non-linear fore,makethe following’projected’meanfieldansatzfor
in medium N-body equations for N > 2. On the other the quartet wave function [4, 5, 18],
hand, it is well known, that nuclei can satisfactorily be
described, in mean field approximation [14]. Projecting 4
Ψ =(2π)3δ(3)(k +k +k +k ) ϕ(k )χST, (3)
thesemeanfield(Hartree-Fock)typeofsolutionsonzero 1234 1 2 3 4 i
total momentum (K = 0) will then allow these mean iY=1
fieldclusterstoBosecondense. Actuallyitiswellknown where χST is the spin-isospinfunction which we suppose
amongthenuclearphysicscommunitythatevenforsuch to be the one of a scalar (S = T = 0). We will not
a small nucleus as the α-particle a momentum projected further mention it from now on. We work in momen-
mean field approachyields a very reasonable description tum space and ϕ(k) is the as-yet unknown single par-
[15]. The reason for this stems, as already mentioned, ticle 0S wave function. In position space, this leads to
from the presence of four different fermions, all attract- the usual formula [14] Ψ → d3R 4 ϕ˜(r − R)
1234 i=1 i
ing one another with about the same force. where ϕ˜(r ) is the Fourier transform of ϕ(k ). If we
i i
In Fig. 1 we sketch the situation, indicating that the take for ϕ(k ) Gaussian shape, Rthis gQives: Ψ →
i 1234
twoprotonsandtwoneutronsoccupythelowest0S level exp[−c (r −r )2] which is the translationally
1≤i<k≤4 i k
of the mean field potential. Actually calculations show invariant ansatz often used to describe α-clusters in nu-
thatthe0Sorbitaloftheself-consistentmeanfieldresem- clei. FoPr instance, it is also employed in the α-particle
bles very much an oscillator wave function of Gaussian condensate wave function of Tohsaki, Horiuchi, Schuck,
shape. In this respect the cartoon in Fig. 1 is not so Ro¨pke (THSR) in [9].
far from reality. We suspect that the situation is generic Inserting the ansatz (3) into (1) and integrating over
for all strongly bound quartets which may be produced superfluousvariables,orminimizingtheenergy,wearrive
inthe future and,therefore,ourpresentstudy isofquite atthefollowingnon-linear,Hartree-Focktypeofequation
generalinterest. Wewilladoptthismomentumprojected for the single particle 0S wave function ϕ(k)=ϕ(|k|)
mean field procedure in this work.
A(k)ϕ(k)+3B(k)+3C(k)ϕ(k)=0, (4)
II. THE IN-MEDIUM FOUR BODY EQUATION where A(k), B(k), and C(k) are given by:
4 d3k 4 k2
In-medium four body equations are well documented A(k ) = i i −4µ
inthe literaturesince long[16]. Inthe presentcaseofan 1 (2π)3 " 2m #
Z i=2 i=1
Y X
3
4 12
× |ϕ(k2)|2|ϕ(k3)|2|ϕ(k4)|2(2π)3δ(3)( ki) (5) 10 (a) deuatelprohna
i=1
X
8
4 d3k d3k′ d3k′ V)
B(k1) = (2π)i3(2π)13(2π)23(1−f(ε1)−f(ε2)) (Me 6
Z Yi=2 Tc 4
× vk1k2,k′1k′2ϕ(k1′)ϕ(k2′)ϕ(k2)|ϕ(k3)|2|ϕ(k4)|2 2
4
× (2π)3δ(3)( k ) (6) 0
i -10 0 10 20 30 40 50 60 70
Xi=1 µ(MeV)
4 d3k d3k′ d3k′
C(k1) = (2π)i3(2π)23(2π)33(1−f(ε2)−f(ε3)) 12 (b) alpha
Z Yi=2 10 deuteron
× vk2k3,k′2k′3ϕ(k2)ϕ(k2′)ϕ(k3)ϕ(k3′)|ϕ(k4)|2 8
4 V)
× (2π)3δ(3)( k ), (7) Me 6
i (
Xi=1 Tc 4
where, in this pilot study, we neglect mean field shifts 2
and effective mass contributions.
0
From Eq. (4), we obtain the single particle wave func- 0.001 0.01 0.1 1
tion in momentum space as n(fm−3)
−3B(k)
ϕ(k)= . (8)
FIG. 2: Critical temperature of alpha and deuteron conden-
A(k)+3C(k)
sationsasfunctionsofchemicalpotential(a)anddensity(b),
derived from Eq. (4) for the α-particle and Eq. (11) for the
As seen in Eqs. (5), (6), and (7), since A(k), B(k), and
deuteron,respectively.
C(k) depend on the wave function of ϕ(k), Eq. (8) is
strongly non-linear. Its solution can be found by itera-
tion.
III. RESULTS FOR THE CRITICAL
For a general two body force vk1k2,k′1k′2, the equation TEMPERATURE Tcα
to be solved is still rather complicated. We, therefore,
proceed to the last simplification and replace the two
In order to determine the critical temperature for α-
body force by a unique separable one, that is
condensation as a function of density n, we need to de-
vk1k2,k′1k′2 =λe−k2/k02e−k′2/k02(2π)3δ(3)(K−K′), (9) termine the chemical potential µ via
d3k
where k =(k −k )/2, k′ = (k′ −k′)/2, K = k +k , n=4 f(ε) (12)
and K′ = k′1+ k′2. This mean1s th2at we take a1 spin2- Z (2π)3
1 2
isospinaveragedtwobodyinteractionanddisregardthat
and adjust the temperature so that the eigenvalue of (1)
in principle the force may be somewhat different in the
hits 4µ. The two open constants λ and k in Eq. (9)
0
S,T =0,1 or 1,0 channels.
aredetermined sothat binding energy (−28.3MeV) and
We are now ready to study the solution of (1) for the
radius (1.71fm) ofthe free ( f =0) α-particle come out
criticaltemperatureTcα whentheeigenvaluehits4µ. For right. The adjusted values are:i λ=−992MeV fm3, and
later comparison, the deuteron (pair) wave function at b = 1.43 fm−1. The results of the calculation are shown
thecriticaltemperatureisalsorepresentedfromEqs.(2)
in Fig. 2.
and (9) to be InFig. 2, the maximumof criticaltemperature Tα
c,max
isatµ=5.5MeV,andtheα-condensationcanexistupto
φ(k)=− 1−2f(ε) λe−k2/k02 d3k′ e−k2/k02φ(k′), µmax =11 MeV. It is very remarkable that the obtained
k2/m−2µ (2π)3 results for Tα well agree with a direct solution of (1)
Z (10) [12]. TheserecsultsforTα arebyabout25percenthigher
where φ(k) is a relative wave function of two parti- c
thantheonesofourearlierpublication[8]. We,however,
cles given by Ψ → φ(|k1−k2|)δ(3)(k +k ), and ε =
12 2 1 2 checked that the underlying radius of the α-particle in
k2/(2m). FromEq.(10),the criticaltemperatureofpair thatworkis largerthanthe experimentalvalue andthat
condensation is obtained with the following equation: Tα decreases with increasing radius of α-particle. Fur-
c
thermore a different variational wave function was used
1=−λ d3k 1−2f(ε) e−2k2/k02. (11) in [8].
(2π)3k2/m−2µ In Fig. 2 we also show the critical temperature for
Z
4
deuteron condensation derived from Eq. (11). In this
12 0.3
case, we take λ = −1305 MeV fm3 and k0 = 1.46 fm−1 10 (a1) 0.25 (b1)
tfdTmoehu)geteoeltarfotettxnheprecebordniremeduaeetknnestrsaoadlntoi.ewonnnIetrwrgaiiystnhss(e−ereo2nva.e2btrrhMutahpetetVlay)otnaahetnigdaohfrcearαridt-diipuceaasnrls(tp1iitc.oi9lese5s-. 32/ϕ(k)(fm) 864 µ=T−c7=.080((MMeeVV)) −12/˜ϕ(r)(fm) 00..0010..05521
itive value of the chemical potential. Roughly speaking, 2 r -0.05
0 -0.1
this corresponds to the point where the α-particles start 0 1 2 3 4 5 0 2 4 6 8 10 12 14
k(fm−1) r(fm)
tooverlap. ThisbehaviorstemsfromthefactthatFermi-
8 0.3
Dirac distributions in the four body case, see (1), can
7 (a2) 0.25 (b2)
naatlewvzaeeryrsobietnceommmopeteirsoatnet.uprA-leis,kaes,icnaocsneisntehqteuheepnatcwierso,αibn-ocdaonyndcαea-nspseaa,rtetivoicenlnegnaeonret- 32/(fm) 654 µT=c=−62..2621((MMeeVV)) −12/(fm) 0.001..521
erally only exists as a BEC phase andthe weak coupling ϕ(k) 32 ˜ϕ(r) 0.005
r
regime is absent. 1 -0.05
Fig. 3 shows the normalized self-consistent solution 0 -0.1
0 1 2 3 4 5 0 2 4 6 8 10 12 14
of the wave function in momentum space derived from k(fm−1) r(fm)
Eq. (8) and the wave function in position space defined 6 0.3
by its Fourier transform ϕ˜(r). Fig. 3-(a1) and (b1) are 5 (a3) 0.25 (b3)
tthheeIwnatvroedfuunctcitoinon,tshoefwthaevefrfeuenαct-ipoanrtriecsleem. bAlsesdaisGcuasusessdiainn 32/(fm) 43 Tµc==86..4157((MMeeVV)) −12/(fm) 0.001..521
and this shape is approximately maintained as long as µ ϕ(k) 2 ˜ϕ(r) 0.005
is negative, see Fig. 3-(a2). On the contrary, the wave 1 r -0.05
function of Fig. 3-(a3), where the chemical potential is 0 -0.1
0 1 2 3 4 5 0 2 4 6 8 10 12 14
positive,hasadiparoundk =0whichisduetothePauli k(fm−1) r(fm)
blockingeffect. Forthe evenlargerpositivechemicalpo- 0.3
6
tential of Fig. 3-(a4) the wave function develops a node. (a4) 0.25 (b4)
TrivheisdiisnbEeqc.a(u4s)eforofmthwehsterruecotnuerecaonf trheealwizaevtehafutnacgtaioinntdheis- 32/(fm) 420 −12/(fm) 0.001..521
sthteemwsafvreomfunthcteioPnaushliifbtslotcokihnigghfaecrtmoro.mTehnetamaanxdimfoulmlowosf ϕ(k)--24 µ=10.6(MeV) r˜ϕ(r) 0.005
the increase of the Fermi momentum kF, as indicated -6 Tc=5.54(MeV) -0-0.0.15
on Fig. 3. From a certain point on the denominator in 0 1 2 3 4 5 0 2 4 6 8 10 12 14
k(fm−1) r(fm)
(8)developsazeroandno self-consistentsolutioncanbe
found any longer.
Ontheotherhand,thewavefunctionsinpositionspace FIG. 3: Single particle wave functions in momentum space
in Figs. 3-(b2), (b3) and (b4) develop an oscillatory be- ϕ(k) (a), and in position space rϕ˜(r) (b) at critical temper-
havior,as the chemicalpotential increases. This is remi- ature, Eq. (8). From top to bottom: (1) µ = 7.08 MeV,
niscenttowhathappensinBCStheoryforthepairwave Tc = 0 MeV, n = 0 fm−3 (2) µ = 2.22 MeV−, Tc = 6.61
function in position space [19]. MMeeVV,,nn==39.0.471×101−02−3fmf−m3−,3a,n(d3()4µ)µ=−=6.1107.6MMeeVV,,TTcc==85..4554
MeV, n=3.34× 10−2 fm−3. Figs. (a1) and (b1) correspond
×
tothewavefunctionsforfreeα-particle. Theverticallinesin
IV. DISCUSSION AND CONCLUSIONS (a3) and (a4) are at the Fermi wavelength kF =√2mµ.
In this work we again took up the study of the criti-
cal temperature of α-particle (quartet) condensation in potential and density is taken from the free Fermi gas
homogeneous symmetric nuclear matter. We essentially relation, Eq. (12). However, the total nucleon density
confirmthe behavioroftwopreviousstudies[8,12]. The of the system must be calculated from the mean single
objective of the paper was to showthat practically same nucleonstateoccupationnumbertakingintoaccountcor-
results as before can be obtained with a strongly simpli- relations,so that the contributionof bound states to the
fying ansatzforthe four particlewavefunction. Namely, total nucleondensity is taken into account,see Ref. [20].
this time, we used a momentum projected mean field To calculate the critical temperature not as function of
variationalwave function. This is based on the fact that the free nucleon density, see Fig. 2b, but of the total
the four different fermions of the quartet can occupy the nucleon density, a generalization `a la NSR [13] must be
same single particle 0S-wave function in the mean field. performed, that is we have at least to incorporate the
The latter is to be determined froma self-consistentnon contribution of the α-particle density including the con-
linear HF-type of equation as a function of chemical po- densate to the single particle occupation numbers. This
tential or density. The relation between the chemical shall be investigated in future work.
5
Besides,in this work,we used the isospin-independent the transition region. Another interesting problem for
separablepotential,Eq.(9),forthetwo-bodyinteraction the future is how the present results are modified in the
asasimplification. Comparisonwitharealistictwo-body asymmetric case, that is in the case of neutron excess.
interaction is interesting. This study also shall be done The success of our study to employ a very simplifying
in the future. ansatz of the mean field type for the quartet wave func-
The self-consistent wave function has been studied in tion, may open wide perspectives. Besides to push the
momentum and position space. For negative chemical description of quartet condensation much further, there
potential the single particle wave function behaves like mightexistthe possibility that evenfor the case ofa gas
a Gaussian. However, once the chemical potential turns of trions such a projected mean field ansatz is a quite
positive,thenthesingleparticlewavefunctioninr-space valid approach. In the case of three colors, like quarks
starts to oscillate. This is a well known feature from in the constituent quark model for nucleons, a harmonic
ordinary pairing. confining potential is frequently assumed and the three
We, therefore, have demonstrated that a very simpli- quarkscanoccupythelowest0Sstate,analogouslytothe
fying momentum projected mean field ansatz suffices to case of quartets treated in the present paper. Of course,
account for the salient features of quartet condensation. trions are composite fermions and cannot be treated in
This is veryhelpful for the nextstep whichis more com- thesamewayasbosoniccomposites,sincetheyshallform
plicated, i.e. the incorporation of quartet condensation a new Fermi gas with their own new Fermi level. How
self-consistently into the Equation of State (EOS). thissituationcaneventuallybetreatedhasrecentlybeen
Weshould,however,beawareofthe factthatourpro- outlined in [21].
jected mean field ansatz for the quartet wave function
can only be a valid approximation so long as well de-
fined quartets exist. In the break down region seen on
Acknowledgments
Fig.2,thisiscertainlynolongerthecase. Howthequar-
tet phase evolves into a superfluid phase of pairs is an
open question. A possibility to study this very inter- This work is part of an ongoing collaboration with Y.
esting problem could be to write down the in-medium Funaki,H.Horiuchi,A.Tohsaki,andT.Yamada. Useful
four body equation (1) directly in the BCS formalism, discussions are gratefully acknowledged. We thank P.
i.e. with the corresponding BCS coherence factors. It Nozi`eres for his interest in quartet condensation. This
may be foreseen that the latter only catch on close to work is supported by the DFG grant No. RO905/29-1.
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