Spin polarized neutron matter within the Dirac-Brueckner-Hartree-Fock approach

0.18 MB·English

by  F. Sammarruca

Spin polarized neutron matter
within the Dirac-Brueckner-Hartree-Fock approach
F. Sammarruca
Physics Department, University of Idaho, Moscow, ID 83844, U.S.A
P. G. Krastev
Physics Department, Texas A&M University – Commerce, Commerce, TX 75429-3011, U.S.A
(Dated: February 9, 2008)
The relation between energy and density (known as thenuclear equation of state) plays a major
role in a variety of nuclear and astrophysical systems. Spin and isospin asymmetries can have a
7 dramatic impact on the equation of state and possibly alter its stability conditions. An example
0 is the possible manifestation of ferromagnetic instabilities, which would indicate the existence, at
0 a certain density, of a spin-polarized state with lower energy than the unpolarized one. This issue
2 isbeingdiscussed extensively intheliteratureand theconclusions are presentlyverymodeldepen-
dent. Wewill report anddiscuss ourrecentprogress in thestudyof spin-polarized neutronmatter.
n
The approach we take is microscopic and relativistic. The calculated neutron matter properties
a
are derived from realistic nucleon-nucleon interactions. This makes it possible to understand the
J
properties of the equation of state in termsof specific features of thenuclear force model.
7
1

1. INTRODUCTION Reid hard-core potential as well as a non-local separa-
2
ble potentials [10]. Relativistic calculations based on ef-
v
9 The propertiesof dense and/orhighly asymmetric nu- fectivemeson-nucleonLagrangians[13]predictthe ferro-
2 clear matter, where asymmetric may refer to isospin or magnetictransitiontotakeplaceatseveraltimesnuclear
0 matterdensity, withits onsetbeing cruciallydetermined
spinasymmetries,areofgreatcurrentinterestinnuclear
7 by the inclusion of the isovector mesons. Clearly, the
physics and astrophysics. This topic is broad-scoped
0 existence of such phase transitiondepends sensitively on
sinceitreachesouttoexoticsystemsonthenuclearchart
6
the modeling of the spin-dependent part of the nuclear
0 aswellas,onadramaticallydifferentscale,exoticobjects
force and its behavior in the medium. Thus, this unset-
/ in the universe such as compact stars.
h tled issue goes to the very core of nuclear physics.
In this paper, we investigate the bulk and single-
t
- particlepropertiesofspin-polarizedneutronmatter. The Our calculationis microscopic and treats the nucleons
l
c study of the magnetic properties of dense matter is of relativistically. A parameter-free and internally consis-
u tentapproachisimportantifwearetointerpretourcon-
considerable interest in conjunction with the physics of
n clusions in terms of the underlying nuclear force. This is
pulsars, which are believed to be rapidly rotating neu-
:
v tron stars with strong surface magnetic fields. The po- preciselyourfocus,namelytounderstandthein-medium
i behavior of specific components of the nuclear force (in
X larizability of nuclear matter can have strong effects on
this case, the spin dependence). Different NN potentials
neutrinodiffusionand,inturn,variationsoftheneutrino
r can have comparable quality as seen from their global
a mean free path due to changes in the magnetic suscepti-
descriptionofNNdataandyetdiffer inspecific features.
bility ofneutronmattercanimpactthe physicsofsuper-
Thus, it will be interesting to explore how, for a given
novae and proton-neutron stars.
many-bodyapproach,predictionsforspin-polarizedneu-
The magnetic properties of neutron/nuclear matter
tronmatter depend upon specific features of the NN po-
have been studied extensively since a long time by
tential. Second,itwillbeinsightfultocomparewithpre-
many authors and with a variety of theoretical meth-
dictions based on a realistic NN potential and the BHF
ods [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
method[18],especiallyatthe higherdensities,wherethe
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Never-
repulsiveDirac effect canhavea dramatic impact onthe
theless,conclusionsaboutthepossibilityofaphasetran-
short-range nature of the force.
sition to a ferromagnetic state at some critical density
This work is organized in the following way: after the
are still contradictory. For instance, calculations based
introductory notes in this section, we briefly review our
on Skyrme-type interactions [23] predict that such in-
theoretical framework (section 2); our results are pre-
stabilities will occur with increasing density. In partic-
sented and discussedin section3; we conclude in section
ular, currently used Skyrme forces show a ferromagnetic
4 with a short summary and outlook.
transition for neutron matter at densities between 1.1ρ0
and 3.5ρ0 [24]. On the other hand, more recent pre-
dictions based on Monte Carlo simulations [16] and the
Brueckner-Hartree-Fock (BHF) approach with realistic 2. BRIEF DESCRIPTION OF THE
CALCULATION
nucleon-nucleon (NN) interactions [18, 19] exclude these
instabilities, atleastatdensities upto severaltimes nor-
malnucleardensity. Similarly,noevidenceofatransition Thestartingpointofanymicroscopiccalculationofnu-
to a ferromagnetic state was found in older calculations clearstructureorreactionsisarealisticfree-spaceNNin-
based on the Brueckner-Hartree-Fockapproachwith the teraction. A realistic and quantitative model for the nu-
2
clear force with reasonabletheoreticalfoundations is the where u and d refer to “up” and “down” polarizations,
one-boson-exchange(OBE)model[30]. Unlessotherwise respectively, and where each Uσσ′ term contains the ap-
specified, our standard framework consists of the Bonn propriate(spin-dependent) partofthe interaction,Gσσ′.
B potential together with the Dirac-Brueckner-Hartree- More specifically,
Fock (DBHF) approach to nuclear matter. A detailed
description of our application of the DBHF method to
nuclear, neutron, and asymmetric matter can be found
in our earlier works [31, 32, 33]. U (p~)= <σ,σ′|G(p~,~q)|σ,σ′ >, (3)
σ
Similarly to what we have done to describe isospin
σ′X=u,dqX≤kσ′
asymmetriesofnuclearmatter,thesingle-particlepoten- F
tial is the solution of a set of coupled equations
U =U +U (1)
u ud uu
where the second summation indicates integration over
the two Fermi seas of spin-up and spin-down neutrons,
U =U +U (2) and
d du dd
1 1 1 1
<σ,σ′|G(p~,~q)|σ,σ′ > = < σ; σ′|S(σ+σ′)>< σ; σ′|S(σ+σ′)>
2 2 2 2
L,L′,SX,J,M,ML
× <LM ;S(σ+σ′)|JM ><L′M ;S(σ+σ′)|JM >
L L
× iL′−LYL∗′,ML(kˆrel)YL,ML(kˆrel)<LSJ|G(krel,Kc.m.)|L′SJ > (4)
The notation < j1m1;j2m2|j3m3 > is used for the dure is applied to the Pauli operator (when expressed in
Clebsh-Gordon coefficients. Clearly, the need to sepa- terms of k and K ) and to the two-particle propa-
rel c.m.
ratetheinteractionbyindividualspincomponentsbrings gator in the kernel of the G-matrix equation.
alongangulardependence,withtheresultthatthesingle- Once a self-consistent solution is obtained for the
particle potential depends also on the direction of the single-particlespectrum,theaveragepotentialenergyfor
momentum. Notice that the G-matrixequationissolved particles with spin polarization σ is obtained as
using partial wave decomposition and the matrix ele-
ments are then summed as in Eq. (4) to provide the 1 1 1 kFσ π 2π 2
<U >= U (p~)p dpdΩ
new matrix elements in the uncoupled-spin representa- σ 2(2π)3ρσ Zp=0Zθp=0Zφp=0 σ p
tion needed for Eq. (3). The three-dimensional integra- (6)
tion in Eq. (3) is performed in terms of the spherical co- The average potential energy per particle is then
ordinates of~q, (q,θ ,φ ), with the final resultdepending
q q ρ <U >+ρ <U >
upon both magnitude and direction of ~p. On the other <U >= u u d d (7)
ρ
hand,thescatteringequationissolvedusingrelativeand
center-of-mass coordinates, krel and Kc.m.. These are The kinetic energy (or, rather,the free-particle opera-
easily related to the momenta of the two particles in the tor in the Dirac equation when using the DBHF frame-
nuclear matter rest frame through the standard defini- work), is also averaged over magnitude and direction of
tionsK~ =~p+~q,and~k = p~−q~. (The latterdisplays the momentum. In particular, we calculate the average
c.m. rel 2
thedependenceoftheargumentofthesphericalharmon- free-particle energy for spin-up(down) neutrons as
ics upon p~ and ~q.)
dΩT¯(m∗(θ))
Solving the G-matrix equation requires knowledge <T >= σ (8)
of the single-particle potential, which in turn requires σ R dΩ
knowledge of the interaction. Hence, Eqs. (1-2) together R
where T¯ is the average over the magnitude of the mo-
with the G-matrix equation constitute a self-consistency
mentum. Notice that the angular dependence comes
problem, which is handled, technically, exactly the same
in through the effective masses, which, being part of
wayaspreviouslydoneforthecaseofisospinasymmetry
the parametrization of the single-particle potential, are
[31]. The Pauli operator for scattering of two particles
themselves direction dependent (and of course different
with unequal Fermi momenta, contained in the kernelof
for spin-up or spin-down neutrons).
the G-matrix equation, is also defined in perfect analogy
Finally
with the isospin-asymmetric one [31],
ρ <T >+ρ <T >
Qσσ′(p,q,kFσ,kFσ′)=(cid:26) 10 iofthpe>rwkiFsσe.and q >kFσ′ (5) <T >= u u ρ d d (9)
and the average energy per neutron is
Noticethat,althoughafullthree-dimensionalintegration
is performed in Eq. (3), the usual angle-average proce- e¯=<U >+<T > (10)
3
80 DBHF, Bonn B 60 DBHF, Bonn B 80 DBHF, Bonn B 50 DBHF, Bonn B
60 40
β = +0.4, kn = 1.764 fm-1 60 β = +0.8, kFn = 1.764 fm-1
F 20
0
40
) ) ) )
V V V V
0
e k = 5.292 e e 40 e
M k = 3.528 M M M
( 20 k = 1.764 ( ( (
Uu kk == 11..144171 Ud -20 Uu Ud
k = 0.882 -50
k = 0.529 20
0 k = 0.088 -40
-60
-20 0
-80 -100
0.0 π/4 π/2 0.0 π/4 π/2 0.0 π/4 π/2 0.0 π/4 π/2
θ(rad) θ(rad) θ(rad) θ(rad)
FIG. 1: (Color online) Angular dependence of the single- FIG. 2: (Color online) As in the previous figure but for a
particlepotentialforspin-upandspin-downneutronsatfixed larger valueof thespin asymmetry.
spin asymmetry and Fermi momentum and for different val-
ues of the neutron momentum. The momenta are in unitsof
fm−1. The angle is definedrelative tothe polarization axis. For the most generalcase,it will be necessaryto com-
bine isospin and spin asymmetry. With twice as many
degrees of freedom, the coupled self-consistency problem
Asinthecaseofisospinasymmetry,itcanbeexpected
schematically displayed in Eqs.(1-2) is numerically more
that the dependence of the average energy per particle
involved but straightforward. This is left to a future
upon the degree of polarization [18] will follow the law
work.
2
e¯(ρ,β)=e¯(ρ,β =0)+S(ρ)β (11)
where β is the spin asymmetry, defined by β = ρu−ρd. 3. RESULTS AND DISCUSSION
ρ
A negative value of S(ρ) would signify that a polarized
We begin by showing the angular and momentum de-
system is more stable than unpolarized neutron matter.
pendence of the single-neutron potential, see Figs. 1-2.
From the energy shift,
The angular dependence is rather mild, especially at the
lowest momenta. As can be reasonably expected, it be-
S(ρ)=e¯(ρ,β =1)−e¯(ρ,β =0), (12)
comes stronger at larger values of the asymmetry, see
the magnetic susceptibility can be easily calculated. If Fig.2. InFig.3,theasymmetrydependenceisdisplayed
the parabolicdependenceis assumed,thenonecanwrite for fixed density and momentum (here the angular de-
the magnetic susceptibility as [18] pendence is averaged out). As the density of u particles
µ2ρ
χ= , (13)
2S(ρ)
20
DBHF, Bonn B
where µis the neutronmagnetic moment. The magnetic u
susceptibility in oftenexpressedin units ofχF, the mag- V) 0 knF = k = 1.764 fm-1
netic susceptibility of a free Fermi gas, Me
( −20
χF = ¯hµ22πm2kF, (14) θ > u / d −40
U
wherek denotestheaverageFermimomentumwhichis < d
F −60
related to the total density by
k =(3π2ρ)1/3. (15) 0.0 0.2 0.4 0.6 0.8
F β
The Fermi momenta for up and down neutrons are
ku =k (1+β)1/3 FIG. 3: (Color online) Asymmetry dependenceof the single-
F F particlepotentialforspin-upandspin-downneutronsatfixed
kd =k (1−β)1/3. (16) density and momentum. The angular dependence is inte-
F F
grated out.
4
BHF, Bonn B
40
800
DBHF, Bonn B V) 20
MeV) 750 knF = 1.764 fm-1 u <U>(Me 0
-20
(
> 700
d -40
*< mu / 660500 d eV) 11802000 ρ230 .ρρ0500ρ0
M
0.0 0.2 0.4 0.6 0.8 T>( 60
<
β
40
20
150
FIG.4: (Coloronline)Asymmetrydependenceoftheeffective
V)
masses for upward and downward polarized neutrons under e
M
thesame conditions as in Fig. 3. e ( 100
cl
arti
p
120 DBHF, Bonn B gy/ 50
er
100 en
V) 80 0
Me 60 −1.0−0.5 0β.0 0.5 1.0
>( 40 FIG.6: (Coloronline)Samelegendasforthepreviousfigure,
U
< 20 but thepredictions are obtained with theBHF calculation.
0
-20
100
80 ρ0.5ρ0 120 Bonn B
MeV) 60 23ρρ000 100
>(
T ) 80
< 40 V
e
M 60
20 S (
40 DBHF
V) 200 parabolic approx. BHF
e 20
M
cle ( 150 0
arti 100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
p
y/ ρ/ρ
erg 50 0
n
e
0
−1.0−0.5 0.0 0.5 1.0 FIG. 7: (Color online) Energy difference between the polar-
β
FIG. 5: (Color online) Average potential, kinetic, and to- ized and the unpolarized states corresponding to Fig. 5 and
tal energy per particle at various densities as a function of 6.
thespin asymmetry. Predictions obtained with our standard
DBHFcalculation.
the average kinetic energy, respectively. The parabolic
dependence on β, or linear on β2, is obviously verified.
goes up, the total density remaining constant, the most InFig.6weshowthecorrespondingpredictionsobtained
likelykindofinteractionforuneutronsisoftheuutype. withtheconventionalBrueckner-Harteee-Fockapproach.
Similarly, the largest contribution to the d-particle po- This comparison may be quite insightful, as we further
tential is of the du type, see Eqs. (1-2), with the latter discussnext. We notice thatthe Diracenergiesareover-
being apparently more attractive, as can be inferred by all more repulsive, but the parabolas predicted with the
the spin splitting of the potential shown in Fig. 3. Be- BHF prescription appear to become steeper, relative to
fore we move on to discuss this issue in greater details, each other, as density grows. The energy difference be-
we also show the effective masses of u and d neutrons, tweenthetotallypolarizedstateandtheunpolarizedone
see Fig. 4, and observe that they display a qualitatively for both the relativistic and the non-relativistic calcula-
similar behavior as the one of the corresponding single- tion is shown in Fig. 7. Although initially higher, the
particle potentials. growth of the DBHF curve shows a tendency to slow
The average energy per particle at various densities downandthetwosetsofpredictionscrossoverjustabove
and as a function of the asymmetry parameter is shown 3ρ0.
in the third frame ofFig. 5. The firsttwo framesdisplay Before leaving this detour into the non-relativistic
the contribution from the average potential energy and model, we observe that the predictions shown in Fig. 6
5
concerned,wewouldexpectthemtobemorepronounced
1000 for nuclear matter than for pure neutron matter, since
DBHF the largest variations among modern realistic potentials
800 are typically found in the strength of the tensor force,
V) β = +1.0 whichisstrongerinT=0partialwaves(obviouslyabsent
e 600
M β = 0.0 in the nn system). This point will be explored in a later
> ( 400 investigation.
*m In the remainder of this paper, we will focus on the
< DBHF model, which is our standard operational ap-
200
proach. To further explore the possibility of a ferro-
0 magnetic transition, we have extended the DBHF cal-
0 1 2 3 4 5 6 culation to densities as high as 10ρ0. The same method
ρ/ρ as described in Ref. [33] is applied to obtain the energy
0
per particle where a self-consistent solution cannot be
obtained (see Section III of Ref. [33] for details). The
FIG. 8: (Color online) Neutron effective masses used in the (angle-averaged) neutron effective masses for both the
DBHF calculations of the EOS. The angular dependence is unpolarized and the fully polarized case are shown in
averaged out. Fig. 8 as a function of density.
DBHF predictions for the average energy per particle
are shownin Fig. 9 at densities rangingfrom ρ=0.5ρ0 to
10ρ0. What we observe is best seen through the spin-
DBHF, Bonn B symmetry energy, which we calculate from Eq. (12) and
1000
show in Fig. 10. We see that at high density the energy
)
V shiftbetweenpolarizedandunpolarizedmattercontinues
Me 800 to grow, but at a smaller rate, and eventually appear to
( saturate. Similar observations already made in conjunc-

le 600 tion with isospin asymmetry were explained in terms of
c
i stronger short-range repulsion in the Dirac model [33].
t
r
a It must be kept in mind that some large contributions,
p
y/ 400 such as the one from the 1S0 state, are not allowed in
g
r the fully polarizedcase. Now,if suchcontributions(typ-
e
n 200 ically attractive at normal densities) become more and
e
morerepulsivewithdensity(duetotheincreasingimpor-
tance of short-range repulsive effects), their absence will
0
amounttolessrepulsiveenergiesathighdensity. Onthe
-1.0 -0.5 0.0 0.5 1.0
other hand, if large and attractive singlet partial waves
β
remain attractive up to high densities, their suppression
(demanded in the totally polarized case) will effectively
amount to increased repulsion.
FIG. 9: (Color online) Average energy per particle at densi-
In conclusion, although the curvature of the spin-
ties equal to 0.5,1,2,3,5,7,9, and 10 times ρ0 (from lowest to symmetry energy may suggest that ferromagnetic insta-
highest curve). Predictions obtained with our DBHF calcu-
bilities are in principle possible within the Dirac model,
lation.
inspection of Fig. 10 reveals that such transition does
are reasonablyconsistentwith those frompreviousstud-
ieswhichusedtheBrueckner-Hartree-Fockapproachand
200
the Nijmegen II and Reid93 NN potentials [18]. In fact,
DBHF, Bonn B
comparisonwiththatworkallowsustomakesomeuseful
observationsconcerningthechoiceofaparticularNNpo- 150
tential, for a similar many-body approach (in this case,
)
V
BHF). We must keep in mind that off-shell differences Me 100
exist among NN potentials (even if nearly equivalent in
(
theirfitofNNscatteringdata)andthosewillimpactthe S
G-matrix(which,unliketheT-matrix,isnotconstrained 50
bythetwo-bodydata). Furthermore,off-shelldifferences
will have a larger impact at high Fermi momenta, where 0
the higher momentum components of the NN potential, 0 2 4 6 8 10
(usually also the most model dependent), play a larger ρ/ρ
0
role in the calculation. Accordingly, the best agreement
between our BHF predictions and those of Ref. [18] is
seenatlowtomoderatedensities. Furthermore,asfaras FIG.10: (Color online) Densitydependenceof thespin sym-
differences based on the choice of the NN potential are metry energy obtained with theDBHF model.
6
change of S(ρ), in case of a ferromagnetic instability.)
3.0
DBHF, Bonn B
4. CONCLUSIONS
χ
/ F 2.5 We have calculated bulk and single-particle proper-
χ ties of spin-polarized neutron matter. The EOSs we ob-
tainwiththeDBHFmodelaregenerallyratherrepulsive
at the larger densities. The energy of the unpolarized
system (where all nn partial waves are allowed), grows
2.0
rapidly at high density with the result that the energy
0 2 4 6 8 10
differencebetweentotallypolarizedandunpolarizedneu-
ρ/ρ
0 tron matter tends to slow down with density. This may
be interpreted as a precursor of spin-separationinstabil-
ities, although no such transition is actually seen up to
FIG. 11: (Color online) Density dependence of the ratio
10ρ0. Our analysis allowedus to locate the originof this
χF/χ. Predictions are obtained with theDBHF model.
behavior in the contributions to the energy from specific
partial waves and their behavior in the medium, partic-
ularly increased repulsion in the singlet states.
not take place at least up to 10ρ0. Clearly it would not
In future work, the impact of further extensions will
be appropriate to explore even higher densities without
be considered, such as: examining the effects of con-
additional considerations, such as transition to a quark
tributions that soften the EOS (especially at high den-
phase. In fact, even on the high side of the densities
sity);extending ourframeworktoincorporatebothspin-
considered here, softening of the equation of state from
and isospin-asymmetries; examining the temperature
additionaldegreesoffreedomnotincludedinthepresent
dependence of our observations for spin- and isospin-
modelmaybenecessaryinorderto drawamoredefinite
asymmetries of neutron and nuclear matter.
conclusion.
Acknowledgements
Finally, in Fig. 11 we show the ratio χ /χ, whose be-
F
havior is directly related to the spin-symmetry energy,
see Eq. (13). Clearly, similar observations apply to both TheauthorsacknowledgefinancialsupportfromtheU.S.
Fig. 11 and Fig. 10. (The magnetic susceptibility would Department of Energy under grant number DE-FG02-
show an infinite discontinuity, corresponding to a sign 03ER41270.
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