Spin polarized states in strongly asymmetric nuclear matter

0.29 MB·English

by  A. A. Isayev

Spin polarized states
in strongly asymmetric nuclear matter
A. A. Isayev
Kharkov Institute of Physics and Technology, Academicheskaya Str. 1, Kharkov, 61108, Ukraine
J. Yang
Dept. of Physics and Center for Space Science and Technology,
Ewha Womans University, Seoul 120-750, Korea
and
Center for High Energy Physics,
Kyungbook National University, Daegu 702-701, Korea
4
(Dated: February 8, 2008)
0
0 The possibility of appearance of spin polarized states in strongly asymmetric nuclear matter is
2 analyzed within theframework of a Fermiliquid theory with theSkyrmeeffectiveinteraction. The
n zero temperature dependence of the neutron and proton spin polarization parameters as functions
a of density is found for SLy4 and SLy5 effective forces. It is shown that at some critical density
J strongly asymmetric nuclear matter undergoes a phase transition to the state with the oppositely
6 directed spins of neutrons and protons while the state with the same direction of spins does not
appear. In comparison with neutron matter, even small admixture of protons strongly decreases

2 thethreshold densityof spin instability. Itis clarified that protonsbecome totally polarized within
v very narrow density domain while the density profile of the neutron spin polarization parameter is
3 characterized by theappearance of long tails near thetransition density.
1
1 PACSnumbers: 21.65.+f;75.25.+z;71.10.Ay
7
0
3 I. INTRODUCTION density. The general conditions on the parameters of
0 neutron–neutron interaction, which result in a magnet-
/ ically ordered state of neutron matter, were formulated
h The spontaneous appearance of spin polarized states
t in Ref. [6]. Spin correlations in dense neutron matter
- in nuclear matter is the topic of a great current interest
l were studied within the relativistic Dirac–Hartree–Fock
c duetorelevanceinastrophysics. Inparticular,theeffects
approachwiththeeffectivenucleon–mesonLagrangianin
u ofspin correlationsin the medium stronglyinfluence the
Ref. [7], predicting the ferromagnetic transition at sev-
n neutrino cross section and neutrino luminosity. Hence,
v: depending on whether nuclear matter is spin polarized eraltimesnuclearmattersaturationdensity. Theimpor-
tanceoftheFockexchangetermintherelativisticmean–
i ornot, drasticallydifferentscenariosofsupernovaexplo-
X field approach for the occurrence of ferromagnetism in
sion and cooling of neutron stars can be realized. An-
nuclear matter was established in Ref. [8]. The stabil-
r other aspect relates to pulsars, which are considered to
a ity of strongly asymmetric nuclear matter with respect
be rapidly rotating neutron stars, surrounded by strong
to spin fluctuations was investigatedin Ref. [9], where it
magneticfield. Thereisstillnogeneralconsensusregard-
wasshownthatthesystemwithlocalizedprotonscande-
ingthemechanismtogeneratesuchstrongmagneticfield
velop a spontaneous polarization, if the neutron–proton
ofaneutronstar. Oneofthehypothesesisthatmagnetic
spin interactionexceeds some threshold value. This con-
fieldcanbe producedbya spontaneousorderingofspins
clusionwasconfirmedalsobycalculationswithintherel-
in the dense stellar core.
ativisticDirac–Hartree–Fockapproachtostronglyasym-
The possibility ofa phase transitionofnormalnuclear metric nuclear matter [10]. Competition between ferro-
matter to the ferromagnetic state was studied by many magnetic (FM) and antiferromagnetic (AFM) spin or-
authors. In the gas model of hard spheres, neutron mat- dering in symmetric nuclear matter with the Skyrme ef-
ter becomes ferromagnetic at ̺ ≈ 0.41fm−3 [1]. It was fective interaction was studied in Ref. [11], where it was
found in Refs. [2, 3] that the inclusion of long–range at- clarifiedthatFMspinstateisthermodynamicallyprefer-
traction significantly increases the ferromagnetic transi- able over AFM one for all relevant densities.
tion density (e.g., up to ̺ ≈ 2.3fm−3 in the Brueck-
ner theory with a simple central potential and hard core Forthemodelswithrealisticnucleon–nucleon(NN)in-
only for singlet spin states [3]). By determining mag- teraction,theferromagneticphasetransitionseemstobe
netic susceptibility with Skyrme effective forces, it was suppressed up to densities well above ̺0 [12]–[14]. In
shown in Ref. [4] that the ferromagnetic transition oc- particular, no evidence of ferromagnetic instability has
curs at ̺ ≈ 0.18–0.26fm−3. The Fermi liquid crite- been found in recent studies of neutron matter [15] and
rion for the ferromagnetic instability in neutron matter asymmetric nuclear matter [16] within the Brueckner–
with the Skyrme interaction is reached at ̺ ≈2–4̺0 [5], Hartree–Fock approximation with realistic Nijmegen II,
where ̺0 = 0.16fm−3 is the nuclear matter saturation Reid93 and Nijmegen NSC97e NN interactions. The
2
same conclusion was obtained in Ref. [17], where mag- it is assumed to consider a nuclear system with an ex-
neticsusceptibilityofneutronmatterwascalculatedwith cess of neutrons, the positive isospin projection is as-
the use of the Argonne v18 two–body potential and Ur- signed to neutrons. This is different from the formalism
bana IX three–body potential. ofRef.[11],aimedtoinvestigatesymmetricnuclearmat-
Here we continue the study of spin polarizability of ter. Further we shall study the possibility of formation
nuclear matter with the use of an effective NN interac- of various types of spin ordering (ferromagnetic and an-
tion. As a framework of consideration, a Fermi liquid tiferromagnetic) in nuclear matter.
(FL) description of nuclear matter is chosen [18, 19]. As The normal distribution function can be expanded in
a potential of NN interaction, we use the Skyrme effec- the Pauli matrices σ and τ in spin and isospin spaces
i k
tive interaction, utilized earlier in a number of contexts
for nuclear matter calculations [20]–[23]. The main em- f(p)=f00(p)σ0τ0+f30(p)σ3τ0 (3)
phasis will be laid on strongly asymmetric nuclear mat- +f03(p)σ0τ3+f33(p)σ3τ3.
ter and neutron matter as its limiting case. We explore
the possibility of FM and AFM phase transitions in nu- For the energy functional invariant with respect to ro-
clearmatter,whenthe spinsofprotonsandneutronsare tations in spin and isospin spaces, the structure of the
aligned in the same direction or in the opposite direc- single particle energy is similar to the structure of the
tion, respectively. In contrast to the approach,based on distribution function f:
the calculation of magnetic susceptibility, we obtain the
self–consistentequationsfortheFMandAFMspinorder ε(p)=ε00(p)σ0τ0+ε30(p)σ3τ0 (4)
parameters and find their solutions at zero temperature. +ε03(p)σ0τ3+ε33(p)σ3τ3.
This allows us to determine not only the critical density
ofinstability with respectto spinfluctuations,but to es- Using Eqs. (2), (4), one can express evidently the distri-
tablish the density dependence of the order parameters bution functions f00,f30,f03,f33 in terms of the quanti-
andtoclarifythe questionofthermodynamicstabilityof ties ε:
various phases.
1
Note that we consider the thermodynamic properties f00 = {n(ω+,+)+n(ω+,−)+n(ω−,+)+n(ω−,−)},
4
of spin polarized states in nuclear matter up to the high
1
densityregionrelevantforastrophysics. Nevertheless,we f30 = {n(ω+,+)+n(ω+,−)−n(ω−,+)−n(ω−,−)},
4
take into account the nucleon degrees of freedom only,
1
although other degrees of freedom, such as pions, hyper- f03 = {n(ω+,+)−n(ω+,−)+n(ω−,+)−n(ω−,−)},
ons, kaons, or quarks could be important at such high 4
1
densities.
f33 = {n(ω+,+)−n(ω+,−)−n(ω−,+)+n(ω−,−)}.
4
(5)
II. BASIC EQUATIONS
Here n(ω)={exp(Y0ω)+1}−1 and
The normal states of nuclear matter are described by
ω+,+ =ξ00+ξ30+ξ03+ξ33,
the normal distribution function of nucleons f =
Tr̺a+ a , where κ ≡ (p,σ,τ), p is momentumκ1,κσ2(τ) ω+,− =ξ00+ξ30−ξ03−ξ33,
κ2 κ1
is the projection of spin (isospin) on the third axis, and ω−,+ =ξ00−ξ30+ξ03−ξ33,
̺ is the density matrix of the system. The energy of ω−,− =ξ00−ξ30−ξ03+ξ33,
the systemis specified as a functional ofthe distribution
functionf,E =E(f), anddetermines the singleparticle where
energy
ξ00 =ε00−µ000, ξ30 =ε30,
∂E(f)
εκ1κ2(f)= ∂f . (1) ξ03 =ε03−µ003, ξ33 =ε33,
κ2κ1 µ0 +µ0 µ0 −µ0
µ0 = n p, µ0 = n p.
The self-consistent matrix equation for determining the 00 03
2 2
distributionfunction f followsfromthe minimum condi-
tion of the thermodynamic potential [18] and is Asfollowsfromthestructureofthedistributionfunctions
f, the quantity ω , being the exponent in the Fermi
±,±
f ={exp(Y0ε+Y4)+1}−1 ≡{exp(Y0ξ)+1}−1. (2) distributionfunctionn,playstheroleofthequasiparticle
spectrum. We consider the case when the spectrum is
Here the quantities ε and Y4 are matrices in the space four–foldsplit due to the spin and isospindependence of
of κ variables, with Y4κ1κ2 = Y4τ1δκ1κ2 (τ1 = n,p), the single particle energy ε(p) in Eq. (4). The branches
Y0 = 1/T, Y4n = −µ0n/T and Y4p = −µ0p/T the La- ω±,+correspondtoneutronswithspinupandspindown,
grange multipliers, µ0 and µ0 the chemical potentials and the branches ω to protons with spin up and spin
n p ±,−
of neutrons and protons, and T the temperature. Since down.
3
1 p−q
lizTathioendcisotnridbiutitoinons functionsf shouldsatisfythe norma- ε˜00(p)= 2V XU0(k)f00(q), k= 2 ,
q
4
V Xf00(p)=̺, (6) ε˜30(p)= 21V XU1(k)f30(q),
p
q
4
V Xf03(p)=̺n−̺p ≡α̺, (7) ε˜03(p)= 21V XU2(k)f03(q),
p
q
4
V Xf30(p)=̺↑−̺↓ ≡∆̺↑↑, (8) ε˜33(p)= 21V XU3(k)f33(q).
p
q
4
V Xf33(p)=(̺n↑+̺p↓)−(̺n↓+̺p↑)≡∆̺↑↓. (9) Here m0 is the bare mass of a nucleon, U0(k),...,U3(k)
p are the normal FL amplitudes, and ε˜00,ε˜30,ε˜03,ε˜33 are
Hereαisthe isospinasymmetryparameter,̺ ,̺ and the FL corrections to the free single particle spectrum.
n↑ n↓
̺ ,̺ are the neutron and proton number densities Further we do not take into account the effective tensor
p↑ p↓
withspinup andspindown,respectively;̺ =̺ +̺ forces,whichleadtocouplingofthemomentumandspin
↑ n↑ p↑
and ̺ = ̺ +̺ are the nucleon densities with spin degrees of freedom [24, 25, 26], and, correspondingly, to
↓ n↓ p↓
up and spin down. The quantities ∆̺ and ∆̺ may anisotropy in the momentum dependence of FL ampli-
↑↑ ↑↓
be regardedas FM and AFM spin order parameters. In- tudes with respect to the spin polarization axis. Using
deed, in symmetric nuclear matter, if all nucleon spins Eqs.(1)and(11),we getthe self–consistentequationsin
are aligned in one direction (totally polarized FM spin the form
state), then ∆̺ =̺ and ∆̺ =0; if all neutron spins
are aligned in o↑n↑e direction a↑n↓d all proton spins in the ξ00(p)=ε0(p)+ε˜00(p)−µ000, ξ30(p)=ε˜30(p), (12)
opposite one (totally polarized AFM spin state), then ξ03(p)=ε˜03(p)−µ003, ξ33(p)=ε˜33(p).
∆̺ = ̺ and ∆̺ = 0. In turn, from Eqs. (6)–(9) one
↑↓ ↑↑
To obtain numerical results, we use the Skyrme effective
can find the neutron and proton number densities with
interaction. In the case of Skyrme forces the normal FL
spin up and spin down as functions of the total density
amplitudes read [19]
̺, isospin excess δ̺ ≡ α̺, and FM and AFM order pa-
rameters ∆̺ and ∆̺ :
↑↑ ↑↓ 2
1 U0(k)=6t0+t3̺β + ¯h2[3t1+t2(5+4x2)]k2, (13)
̺ = (̺+δ̺+∆̺ +∆̺ ),
n↑ 4 ↑↑ ↑↓ 1
1 U1(k)=−2t0(1−2x0)− 3t3̺β(1−2x3)
̺ = (̺+δ̺−∆̺ −∆̺ ),
n↓ 4 ↑↑ ↑↓ 2
1 − 2[t1(1−2x1)−t2(1+2x2)]k2 ≡a+bk2,
¯h
̺ = (̺−δ̺+∆̺ −∆̺ ),
p↑ ↑↑ ↑↓
4 1
1 U2(k)=−2t0(1+2x0)− t3̺β(1+2x3)
3
̺ = (̺−δ̺−∆̺ +∆̺ ).
p↓ ↑↑ ↑↓
4 2
In order to characterize spin ordering in the neutron − ¯h2[t1(1+2x1)−t2(1+2x2)]k2,
andprotonsubsystems,itisconvenienttointroduceneu- 1 2
tron and proton spin polarization parameters U3(k)=−2t0− 3t3̺β − ¯h2(t1−t2)k2 ≡c+dk2,
̺ −̺ ̺ −̺
Πn = n↑ n↓, Πp = p↑ p↓. (10) where ti,xi,β are the phenomenologicalconstants, char-
̺ ̺
n p acterizingagivenparameterizationoftheSkyrmeforces.
The expressions for the spin order parameters ∆̺ and InthenumericalcalculationsweshalluseSLy4andSLy5
↑↑
∆̺ through the spin polarization parameters read potentials[27], developedto fitthe propertiesofsystems
↑↓
with large isospin asymmetry. With account of the ev-
∆̺ =̺ Π +̺ Π , ∆̺ =̺ Π −̺ Π .
↑↑ n n p p ↑↓ n n p p ident form of FL amplitudes and Eqs. (6)–(9), one can
To obtainthe self–consistentequations,we specify the obtain
energy functional of the system in the form
p2
E(f)=E0(f)+Eint(f), (11) ξ00 = 2m00 −µ00, (14)
p2 p2
E0(f)=4Xp ε0(p)f00(p), ε0(p)= 2m0, ξ03 = 2m03 −µ03, (15)
p2 ∆̺ b
Eint(f)=2X{ε˜00(p)f00(p)+ε˜30(p)f30(p) ξ30 =(a+b 4 ) 8↑↑ + 32hq2i30, (16)
p p2 ∆̺ d
+ε˜03(p)f03(p)+ε˜33(p)f33(p)}, ξ33 =(c+d 4 ) 8↑↓ + 32hq2i33, (17)
4
wheretheeffectivenucleonmassm00andeffectiveisovec- With account of Eqs. (22) and (23), the normalization
tor mass m03 are defined by the formulae: conditions for the distribution functions can be written
2 2 in the form
¯h ¯h ̺
= + [3t1+t2(5+4x2)], (18)
2m00 2m0 16 2
¯h2 α̺ V Xf0(p)=̺, (24)
= [t2(1+2x2)−t1(1+2x1)], p
2m03 16 2
and the renormalized chemical potentials µ00 and µ03 V Xf3(p)=̺↑−̺↓ ≡∆̺↑↑. (25)
p
should be determined from Eqs. (6), (7). In Eqs. (16)
and(17),hq2i30andhq2i33arethesecondordermoments Here ̺ and ̺ are the neutron number densities with
↑ ↓
of the corresponding distribution functions
spin up and spin down and
4
hq2i30 = V Xq2f30(q), (19) 1
q f0 = {n(ω+)+n(ω−)}, ω± =ξ0±ξ3, (26)
2
4
hq2i33 = V Xq2f33(q). (20) f3 = 1{n(ω+)−n(ω−)}, (27)
q 2
p2
Thus, with account of the expressions (5) for the dis- ξ0 = −µn, (28)
tributionfunctionsf,weobtaintheself–consistentequa- 2mn
tions (6)–(9), (19), and (20) for the effective chemical p2 ∆̺ b
potentials µ00,µ03, FM and AFM spin order parameters ξ3 =(an+bn 4 ) 4↑↑ + 1n6hq2i3. (29)
∆̺↑↑,∆̺↑↓,andsecondordermomentshq2i30,hq2i33. It
is easy to see, that the self–consistent equations remain The effective neutron mass mn is defined by the formula
invariable under a global flip of spins, when neutrons
2 2
(protons) with spin up and spin down are interchanged, ¯h ¯h ̺
and under a global flip of isospins, when neutrons and 2mn = 2m0 + 8[t1(1−x1)+3t2(1+x2)], (30)
protons with the same spin projection are interchanged.
Let us consider, what differences will be in the case and the quantity hq2i3 in Eq. (29) is the second order
of neutron matter. Neutron matter is an infinite nuclear moment of the distribution function f3:
system, consisting of nucleons of one species, i.e., neu-
trons,and,hence,theformalismofone–componentFermi 2
liquidshouldbe appliedforthe descriptionofits proper- hq2i3 = V Xq2f3(q). (31)
q
ties. Formally neutron matter can be considered as the
limiting case of asymmetric nuclear matter, correspond-
Thus, with account of the expressions (26) and (27) for
ingtotheisospinasymmetryα=1. Theindividualstate
thedistributionfunctionsf,weobtaintheself–consistent
of a neutron is characterized by momentum p and spin
equations (24), (25), and (31) for the effective chemical
projection σ. The self–consistent equation has the form
potential µ , spin order parameter ∆̺ , and secondor-
of Eq. (2), where all quantities are matrices in the space n ↑↑
ofκ≡(p,σ)variables. Thenormaldistributionfunction der moment hq2i3.
and single particle energy can be expanded in the Pauli
matrices in spin space
III. PHASE TRANSITIONS IN STRONGLY
f(p)=f0(p)σ0+f3(p)σ3, (21) ASYMMETRIC NUCLEAR MATTER
ε(p)=ε0(p)σ0+ε3(p)σ3.
The energy functional of neutron matter is character- Early researcheson spin polarizability of nuclear mat-
ized by two normal FL amplitudes Un(k) and Un(k). ter with the Skyrme effective interaction were based on
0 1
Applying the sameprocedure,asinRef.[19], thenormal the calculation of magnetic susceptibility and finding its
FLamplitudescanbefoundintermsoftheSkyrmeforce pole structure [4, 5], determining the onset of instabil-
parameters t ,x ,β: ity with respect to spin fluctuations. Here we shall find
i i
directly solutions of the self–consistent equations for the
U0n(k)=2t0(1−x0)+ t33̺β(1−x3) (22) FMandAFMspinorderparametersasfunctionsofden-
sity at zero temperature. A special emphasis will be
2
+ ¯h2[t1(1−x1)+3t2(1+x2)]k2, l(aαid<∼on1)thwehislteuidnysoyfmsmtroetnrgiclynauscylmeamremtraitctenrucFlMearspminatoterr-
U1n(k)=−2t0(1−x0)− t3̺β(1−x3) (23) dering is thermodynamically more preferable than AFM
3 one [11].
2
+ 2[t2(1+x2)−t1(1−x1)]k2 ≡an+bnk2. If all neutron and proton spins are aligned in one di-
¯h rection,thenfornontrivialsolutionsoftheself–consistent
5
equations we have consequence,thespinpolarizedstateisformedmuchear-
lierindensitythaninpureneutronmatter. Forexample,
∆̺↑↑ =̺, ∆̺↑↓ =α̺, (32) thecriticaldensityinneutronmatteris̺≈0.59fm−3 for
3 SLy4 potential and ̺ ≈0.58fm−3 for SLy5 potential; in
hq2i30 = 10̺kF2[(1+α)5/3+(1−α)5/3], asymmetricnuclearmatterwithα=0.95thespinpolar-
3 ized state arisesat ̺≈0.38fm−3 for SLy4 and SLy5 po-
hq2i33 = 10̺kF2[(1+α)5/3−(1−α)5/3], tentials. Hence, even small quantity of protons strongly
favorsspininstability ofhighly asymmetricnuclearmat-
where k =(3π2̺)1/3 is the Fermi momentum of totally ter, leading to the appearance of states with the oppo-
F
polarized symmetric nuclear matter. Therefore, for the sitely directed spins of neutrons and protons. As follows
partial number densities of nucleons with spin up and fromFig.1,protonsbecometotallyspinpolarizedwithin
spin down one can get very narrow density domain (e.g., if α=0.95,full polar-
ization occurs at ̺ ≈ 0.41fm−3 for SLy4 force and at
̺ = 1+α̺, ̺ = 1−α̺, ̺ =̺ =0. (33) ̺≈0.40fm−3 for SLy5 force) while the threshold densi-
n↑ p↑ p↓ n↓
2 2 tiesfortheappearanceandsaturationoftheneutronspin
order parameter are substantially different (if α = 0.95,
Ifallneutronspins arealignedin onedirectionandall
neutrons become totally polarized at ̺ ≈ 1.05fm−3 for
proton spins in the opposite one, then
SLy4 force and at ̺≈1.02fm−3 for SLy5 force).
∆̺ =α̺, ∆̺ =̺, (34)
↑↑ ↑↓
3
hq2i30 = 10̺kF2[(1+α)5/3−(1−α)5/3],
3
hq2i33 = 10̺kF2[(1+α)5/3+(1−α)5/3],
and, hence, 1.0
0.8 n SLy4
1+α 1−α
̺ = ̺, ̺ = ̺, ̺ =̺ =0. (35) 0.6
n↑ p↓ p↑ n↓
2 2 0.9
0.4
Now we present the results of numerical solution of 0.8 a =1
0.2
the self–consistent equations with the effective SLy4 0.95
p
0.0
0a.n9d5,S0L.9y,50.f8o)rcaensdfnoreusttrroonng(lαy=as1y)mmmaetttreicr.nTuchleeanreu(αtro=n P, n-0.2
P
and proton spin polarization parameters Πn and Πp are -0.4
shown in Fig. 1 as functions of density at zero temper-
-0.6
ature. Since in totally polarized state the signs of spin p
-0.8 (a)
polarizations are opposite (Π = 1,Π = −1), consider-
n p
ing solutions correspond to the case, when spins of neu- -1.0
trons and protons are aligned in the opposite direction. 1.0
Note that for SLy4 and SLy5 forces, being relevant for
0.8 n
the description of strongly asymmetric nuclear matter, SLy5
0.6
there are no solutions, corresponding to the same direc- 0.90
tion of neutron and proton spins. The reasonis that the 0.4 0.80
sign of the multiplier t3(−1+2x3) in the density depen- p 0.2 a =1
dent term of the FL amplitude U1, determining spin– P, n 0.0 0.95
spin correlations, is positive, and, hence, corresponding P
-0.2
term increases with the increase of nuclear matter den-
-0.4
sity, preventing instability with respect to spin fluctua-
tions. Contrarily, the density dependent term −t3̺β/3 -0.6 p
in the FL amplitude U3, describing spin–isospin correla- -0.8 (b)
tions,is negative,leading to spininstability with the op-
-1.0
positely directedspins ofneutronsandprotonsathigher
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
densities. r [fm-3]
Another nontrivial feature relates to the density be-
haviorofthespinpolarizationparametersatlargeisospin
asymmetry. AsseenfromFig.1,evensmalladmixtureof
protons leads to the appearance of long tails in the den- FIG. 1: Neutron and proton spin polarization parameters as
sity profiles of the neutron spin polarization parameter functions of density at zero temperature for (a) SLy4 force
and (b) SLy5force.
near the transition point to a spin ordered state. As a
6
0
1.0 SLy4
-20
p SLy4 a =1
0.8
0.8 cl]
0.9 n nu -40
>00.6 0.95 eV/ 0.95
22<q>/<q 0.4 0.08.9 dE/A [M --8600 0.90.8
a =1
0.2 0.95 -100
(a)
0.0 0
1.0 SLy5
p -20 a =1
0.8 SLy5
0.8 0.9 ucl]
0.95 n V/n -40 0.95
>00.6 e
2q M 0.9
2q>/< 0.4 0.08.9 E/A [ -60 0.8
< d -80
0.2
0.95
a =1 (b) -100
0.0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
r [fm-3] r [fm-3]
FIG.2: Sameas inFig. 1,butfor thesecond ordermoments FIG.3: Totalenergypernucleon,measuredfrom itsvaluein
hq2ip andhq2in,normalized totheirvaluesinthetotally po- the normal state, for the state with the oppositely directed
larized state. spinsofneutronsandprotonsasafunctionofdensityatzero
temperature for (a) SLy4force and (b) SLy5force.
Note that the second order moments
hq2i ≡hq2i −hq2i (36)
n n↑ n↓
1
= V Xq2(cid:16)n(ω+,+)−n(ω−,+)(cid:17), onIdnoFrdige.r2mowmeepnltosthtqh2ei daenndsithyq2diep,ennodremnacleizoefdtthoethseecir-
q n p
values in the totally polarized state, for different asym-
hq2i ≡hq2i −hq2i
p p↑ p↓ metries at zero temperature. These quantities behave
1
= V Xq2(cid:16)n(ω+,−)−n(ω−,−)(cid:17) sthimerileaerxtiosttlhoengsptianilpsoinlarthizeatdieonnsiptayrpamroefitleerssoifnthFeign.e1u,tir.oe.n,
q
spin order parameter and the proton spin order parame-
also characterize spin polarization of the neutron and ter is saturated within very narrow density interval.
proton subsystems. If the solutions hq2i30 and hq2i33 of
the self–consistent equations are known, then To check thermodynamic stability of the spin ordered
state with the oppositely directed spins of neutrons and
1
hq2in = hq2i30+hq2i33 , protons, it is necessary to compare the free energies of
2(cid:0) (cid:1)
this state and the normal state. In Fig. 3 the difference
1
hq2ip = hq2i30−hq2i33 . of the total energies per nucleon of the spin ordered and
2(cid:0) (cid:1)
normal states is shown as a function of density at zero
The values ofhq2in andhq2ip forthe totallypolarized temperature. One can see that nuclear matter under-
state are goes a phase transition to the state with the oppositely
3 3 directed spins of neutrons and protons at some critical
hq2in0 = 10̺kF2(1+α)5/3, hq2ip0 =−10̺kF2(1−α)5/3. density, depending on the isospin asymmetry.
7
IV. DISCUSSION AND CONCLUSIONS be constrained. Probably, these constraints will be ob-
tainedfrom the data onthe time decay of magnetic field
Spin instability is a common feature, associated with of isolated neutron stars [33]. In spite of this shortcom-
a large class of Skyrme models, but is not realized in ing, SLy4 and SLy5 effective forces hold one of the most
more microscopic calculations. The Skyrme interaction competing Skyrme parameterizationsat presenttime for
hasbeen successfulindescribingnucleiandtheirexcited descriptionofisospinasymmetricnuclear matter athigh
states. In addition, various authors have exploited its densitywhileaFermiliquidapproachwithSkyrmeeffec-
applicability to describe bulk matter at densities of rel- tive forces provides a consistent and transparent frame-
evance to neutron stars [28]. The force parameters are work for studying spin instabilities in a nucleon system.
determined empirically by calculating the ground state
In summary, we have considered the possibility of
in the Hartree–Fock approximation and by fitting the
phase transitions into spin ordered states of strongly
observed ground state properties of nuclei and nuclear
asymmetric nuclear matter within the Fermi liquid for-
matteratthesaturationdensity. Inparticular,theinter-
malism,whereNNinteractionisdescribedbytheSkyrme
actionparameters,describingspin–spinandspin–isospin
effective forces (SLy4 and SLy5 potentials). In contrast
correlations, are constrained from the data on isoscalar
to the previous considerations, where the possibility of
[29,30]andisovector(giantGamow–Teller)[31,32]spin–
formation of FM spin polarized states was studied on
flip resonances.
the base of calculationof magnetic susceptibility, we ob-
In a microscopic approach, one starts with the bare
tain the self–consistent equations for the FM and AFM
interaction and obtains an effective particle–hole inter-
spin polarization parameters and solve them in the case
action by solving iteratively the Bethe–Goldstone equa-
ofzerotemperature. Ithasbeenshowninthemodelwith
tion. IncontrasttotheSkyrmemodels,calculationswith
SLy4 and SLy5 effective forces, that strongly asymmet-
realistic NN potentials predict more repulsive total en-
ric nuclear matter undergoes a phase transition to the
ergy per particle for a polarized state comparing to the
spin polarized state with the oppositely directed spins
nonpolarized one for all relevant densities, and, hence,
of neutrons and protons, while the state with the same
give no indication of a phase transition to spin ordered
direction of the neutron and proton spins does not ap-
state. It must be emphasized that the interaction in the
pear. An important peculiarity of this phase transition
spin–andisospin–dependentchannelsisacrucialingredi-
is the existence of long tails in the density profile of the
ent in calculating spin properties of isospin asymmetric
neutron spin polarization parameter near the transition
nuclear matter and different behavior at high densities
point. Thismeans,thatevensmalladmixtureofprotons
of the interaction amplitudes, describing spin–spin and
to neutron matter leads to the considerable shift of the
spin–isospin correlations, lays behind this divergence in
critical density of spin instability in the direction of low
calculations with the effective and realistic potentials.
densities. In the model with SLy4 effective interaction
InthisstudyasapotentialofNNinteractionwechoose
this displacement is from the critical density ̺ ≈ 3.7̺0
SLy4 and SLy5 Skyrme effective forces, which were con-
for neutron matter to ̺ ≈ 2.4̺0 for asymmetric nuclear
strained originally to reproduce the results of micro-
matter at the isospin asymmetry α= 0.95, i.e. for 2.5%
scopic neutron matter calculations (pressure versus den-
of protons only. As a result, the state with the oppo-
sity curve) [27]. Besides, in the recent publication [28]
sitely directed spins of neutrons and protons appears,
it was shown that the density dependence of the nuclear
whereprotons become totally polarizedina verynarrow
symmetryenergy,calculateduptodensities̺<∼3̺0with density domain. This picture is different from the case
SLy4andSLy5effectiveforces(togetherwithsomeother
of symmetric nuclear matter, where the FM spin config-
sets of parameters among the total 87 Skyrme force pa-
uration is thermodynamically more preferable, than the
rameterizations checked) gives the neutron star models
AFM one [11]. Obtained results may be of importance
in a broad agreement with the observables, such as the
for the description of thermal and magnetic evolution of
minimum rotation period, gravitational mass–radius re-
pulsars, whose core represents strongly asymmetric nu-
lation,thebindingenergy,releasedinsupernovacollapse,
clear matter.
etc. This is important check for using these parameter-
izations in high density region of strongly asymmetric A.I. is grateful for support of Topical Program of
nuclear matter. However, it is necessary to note, that APCTP during his stay at Seoul. J.Y. is partially sup-
the spin–dependent part of the Skyrme interaction at portedbyKoreaResearchFoundationGrant(KRF-2001-
densities of relevance to neutron stars still remains to 041-D00052).
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