Landau parameters for asymmetric nuclear matter with a strong magnetic field

0.31 MB·English

by  M. A. Perez-Garcia

Landau parameters for asymmetric nuclear matter with a strong magnetic field
M. A´ngeles P´erez-Garc´ıa,1,∗ C. Providˆencia,2,† and A. Rabhi2,3,‡
1Departamento de F´ısica Fundamental and IUFFyM,
Universidad de Salamanca, E-37008 Salamanca
2Centro de F´ısica Computacional, Department of Physics,
University of Coimbra, 3004-516 Coimbra, Portugal
3Laboratoire de Physique de la Mati`ere Condens´ee,
1
1 Facult´e des Sciences de Tunis, Campus Universitaire, Le Belv´ed`ere-1060, Tunisia
0
2 (Dated: January 11, 2011)

n
a TheLandauFermiLiquidparametersarecalculatedforchargeneutralasymmetricnuclear
J
matter in beta equilibrium at zero temperature in the presence of a very strong magnetic
9

field with relativistic mean-field models. Due to the isospin structure of the system, with
]
h
different populations of protons and neutrons and spin alignment to the field, we find non-
t
-
l vanishing Landau mixing parameters. The existence of quantized Landau levels for the
c
u
charged sector has some impact on the Landau parameters with the presence of discretized
n
[ features in those involving the proton sector. Using the Fermi liquid formalism singlet and

1
triplet excited quasiparticle states are analyzed, and we find that in-medium effects and
v
6 magnetic fields are competing, however, the former are more important in the interaction
5
6 energy range considered. It is found that for magnetic field strengths Log10 B (G) 17
≤
1
. the relative lowpolarizationof the systemproduces mild changesin the generalizedLandau
1
0 parameters with respect to the unmagnetized case, while for larger strengths there is a
1
1 resolution of the degeneracy of the interaction energies of quasiparticles in the system.
:
v
i PACS numbers: 97.60.Jd 24.10.Jv 26.60.-c 21.65.-f 26.60.Kp
X
r
a
∗ [email protected]
† cp@teor.fis.uc.pt
‡ rabhi@teor.fis.uc.pt
2
I. INTRODUCTION
Asymmetricnuclearmatterispresentlyanimportanttopicproposedforstudyinexperimentsat
radioactive beam facilities such as FAIR [1] at GSI, SPIRAL2 [2] at GANIL, ISAC-IIIat TRIUMF
[3] or FRIB [4] at MSU, among others [5]. These will allow the investigation of nuclei in regions of
the nuclear chart far from the stability line. These nuclear regions, where the isospin asymmetry
ratio, given by the ratio of the proton vector number density, nv, to baryonic vector number
p
density, n , defined as Y = nv/n , largely departs from the 0.5 value, are of interest in the study
B P p B
of stability of exotic nuclei asin theneutronrichnuclei. Recently studiedexamples of thosearethe
isotopes of Ca [6] with Z = 20 or the isotones N = 30 of Ca and Sc with CLARA experiment [7] in
Legnaro. The study of observables related to nuclei far from the isospin stability line allows for an
improved description of neutron rich environments as those of interest in astrophysical scenarios
of neutron star matter. They are relevant to interiors of compact stellar objects like neutron stars
(NS) arising in the aftermath of a supernova event. In this context, the rapid deleptonization of
the NS following the gravitational collapse of the inner regions with the proton-electron capture
process makes matter more neutron rich since (anti) neutrinos diffuse out of the star [8, 9]. Also
supernova matter can be considered to be constitued by a set of nuclei, in the nuclear statistical
equilibrium(NSE)approximation, andpresentsadistributionofnucleishiftedtotheN > Z region
[10, 11].
On Earth, the high temperature (T) and small baryonic chemical potential (µ ) region of
B
matter phase space has been somewhat tested [12] at RHIC, and it will be possible to further
test with other heavy-ion experiments like ALICE [13] at CERN. Although this improvement of
our knowledge of the phases of matter is certainly valuable, the phase diagram of nuclear matter
relevant for the equation of state (EOS) of NS is that of cold asymmetric nuclear matter in the low
temperature range.
Historically, most of the existing literature on the nuclear matter EOS characterization [14]
has neglected the input from external fields, in particular, in the case of magnetic fields, this is
claimed due to the tiny value of the nuclear magnetic moment [15]. On terrestrial experiments
recent estimations of the magnetic field strength that could be produced dynamically at CERN
or BNL energies [16] are of the order B 1017 1019 G. In nature, we have another indication
≈ −
of sources of intense magnetic fields of astrophysical origin such as magnetars [17]. Magnetars are
neutron stars which may have surface magnetic fields B 1015 G [18–20] discovered in the X-ray
≈
andγ-ray electromagnetic spectrum(for areview see [21]). Theyare identified with theanomalous
3
X-ray pulsars (AXP) and soft γ-ray repeaters [22].
Taking as reference the critical field, Be, at which the electron cyclotron energy is equal to
c
the electron mass, Be = 4.414 1013 G, we define B∗ = B/Be. It has been shown by several
c × c
authors [23–26] that magnetic fields larger than B∗ = 105 will affect the EOS of compact stars. In
particular,field-theoreticaldescriptionsbasedonseveralparametrizationsofthenon-linearWalecka
model (NLWM) [27] show an overall similar behavior. According to the scalar virial theorem [28]
the interior magnetic field strength could be as large as B = 1018 G so, in principle, this is the
maximum field strength that is meaningful to consider.
In 1959 the formal theory for treatment of low temperature (non-superfluid) fermion systems,
known as normal Fermi Liquids, was developed by Landau [29, 30] to describe the behavior of 3He
below100mK.WiththisFermiLiquidTheory(FLT)(seearecentreference[31])theexcited states
in the system could bedescribed as quasiparticles (qp) as long as these states have sufficiently long
lifetimes. At low temperature, the small excitation energy (compared to the chemical potential)
will assure this fact. In the context of the FLT, the so-called Landau parameters, can parametrize
the interaction energy between a pair of qp in the medium. Previous works have attempted to
partially study the behavior of Landau parameters for non-magnetized symmetric nuclear matter
or neutron matter [32–34] or in magnetized matter under the presence of a magnetic field either in
a non-relativistic formalism [35–37] or in magnetized matter without considering B field including
exchange terms in a relativistic way [38].
In this work we will be interested in calculating the Landau Fermi Liquid parameters for an
isospin asymmetric nuclear system in beta equilibrium and in charge neutrality under the effect
of an intense magnetic field. The FLT used in this case must describe relativistically the more
general condition of a magnetized non-pure isospin system to account for the fact that the intense
magnetic field can modify isospin populations and partially align nucleon magnetic moments with
respect to the case of vanishing magnetic field. In addition, the different dynamics of proton and
neutronsectorsunderthepresenceofamagneticfield(includingtheexistenceofanomalousnucleon
magnetic moments) will have effects in the Landau parameter computation showing discretized or
continue features for protons and neutrons, respectively. In section II we introduce the relativistic
lagrangian model used in this work and the generalized formalism of the FLT for charge neutral
isospin asymmetric hadronic systems under the presence of a magnetic field. In section III we
discuss the explicit form of the matricial structure of the coefficients describing the interaction of
qpinthemagnetizedsystemthroughtheLandauparameters. InsectionIVweanalyzetheobtained
coefficients for either individual spin quantum numbers or total spin (singlet or triplet) for the qp
4
excitations for the electrically neutral system configurations calculated underbeta equilibrium and
the Landau parameter behavior in presence of a strong magnetic field and, finally, in section V, we
summarize and draw some conclusions.
5
II. THE FORMALISM
For the description of the EOS of neutron star matter, we employ a relativistic field-theoretical
approach in which thebaryons, neutrons(n) andprotons (p), interact via theexchange of σ ω ρ
− −
mesons in the presence of a uniform magnetic field B along the z-axis. The Lagrangian density for
the TM1 parametrization [39] of the non-linear Walecka model (NLWM) reads [27]
= + + . (1)
b m l
L L L L
b=n,p m=σ,ω,ρ l=e
X X X
The baryon (b=n, p), meson (m = σ, ω and ρ) and lepton (l = e) lagrangians are given by
(c = ~=1),
1 1
= Ψ¯ iγ ∂µ q γ Aµ m +g σ g γ ωµ g τ γ ρµ µ κ σ Fµν Ψ , (2)
b b µ b µ b σ ω µ ρ 3b µ N b µν b
L − − − − 2 − 2
(cid:18) (cid:19)
1 1 1 1 1 1 1
= ∂ σ∂µσ m2σ2 κσ3 λσ4+ m2ω ωµ+ ξg4(ω ωµ)2 ΩµνΩ
Lm 2 µ − 2 σ − 3! − 4! 2 ω µ 4! ω µ − 4 µν
1 1 1
FµνF + m2ρ ρµ PµνP , (3)
−4 µν 2 ρ µ − 4 µν
= Ψ¯ (iγ ∂µ q γ Aµ m )Ψ , (4)
l l µ l µ l l
L − −
where Ψ , Ψ are the baryon and lepton Dirac fields respectively.
b l
The nucleon isospin z-projection for the proton (neutron) is denoted by τ = 1 ( τ = 1).
3p 3n
−
The nucleon mass is m (m = m = m = 938 MeV), its charge is q and the baryonic anomalous
b b n p b
magnetic moments (AMM) areintroduced via thecouplingto theelectromagnetic fieldtensor with
σ = i [γ ,γ ] and strength κ . In particular, κ = 1.91315 for the neutron and κ = 1.79285
µν 2 µ ν b n − p
for the proton. m and q are the mass and charge of the lepton. We will consider the simplest
l l
model where the leptonic sector is formed just by electrons (l = e), with no anomalous magnetic
moment, providing charge neutrality in this astrophysical scenario. Despite there is a non-zero
electron AMM its value [15] is tiny when compared to that in the hadronic sector and it was shown
that this contribution is negligible for the magnetic fields of interest in astrophysics if properly
introduced [40]. The mesonic and electromagnetic field strength tensors are given by their usual
expressions: Ω = ∂ ω ∂ ω , P = ∂ ρ ∂ ρ , and F = ∂ A ∂ A .
µν µ ν ν µ µν µ ν ν µ µν µ ν ν µ
− − −
The electromagnetic field is assumed to be externally generated (and thus has no associated
fieldequation), andonlyfrozen-fieldconfigurationswillbeconsideredinthiswork. Tocalculatethe
thermodynamic conditions in this charge neutral asymmetric nuclear system in beta equilibrium
with an intense magnetic field, isoscalar and isovector current conservation must be imposed.
6
Explicitly, the following conditions are fulfilled: i) electrical charge neutrality ii) conservation of
baryonic charge iii) mesonic field equations selfconsistency. In addition, we will assume thorought
this work that neutrinos scape freely and therefore there is no neutrino trapping.
The field equations of motion are determined from the Euler-Lagrange equations arising from
the lagrangian density in Eq.(1). Under the conditions of the present calculation, a relativistic
mean field (RMF) approximation will be used so that the space-time varying fields are replaced by
a homogeneous value, φ(x ) φ. In this way, the Dirac equation for a nucleon is given by,
µ
→
(iγ ∂µ q γ Aµ (m g σ) g γ ωµ
µ b µ b σ ω µ
− − − −
1 1
g τ γ ρµ µ κ σ Fµν)Ψ = 0, (5)
ρ 3b µ N b µν b
−2 − 2
where the effective baryon mass is m∗ = m g σ. For leptons,
b σ
−
(iγ ∂µ q γ Aµ m )Ψ = 0. (6)
µ l µ l l
− −
For meson fields we obtain,
1 1
m2σ+ κσ2+ λσ3 = g ns +ns = g ns, (7)
σ 2 3! σ p n σ
m2ω0+ 1ξg4 ω02 ω2 ω0 = g (cid:0)nv +nv(cid:1) = g n , (8)
ω 3! ω − ω p n ω B
m2ω+ 1ξg4(cid:16) ω02 ω2(cid:17) ω = g j(cid:0) , (cid:1) (9)
ω 3! ω − ω B
(cid:16) (cid:17) 1 1
m2ρ0 = g nv nv = g n , (10)
ρ 2 ρ p− n 2 ρ 3
m2ρ = 1g j(cid:0) , (cid:1) (11)
ρ 2 ρ 3
where we use the notation as in the work of Matsui [34] and n is the baryonic (vector) particle
B
number density constructed as the sum of (vector) particle number density of protons (nv) and
p
neutrons (nv), n = nv +nv .
n B p n
The baryon current j = j +j is also the sum of the proton (j ) and neutron (j ) currents.
B p n p n
ns = ns + ns is the scalar density constructed from that of protons (ns) and neutrons (ns).
p n p n
n = nv nv and j = j j are the isoscalar particle number density and isovector baryon
3 p − n 3 p − n
current, respectively. When solving for equilibrium conditions in the nuclear system governed by
Eqs.(5)-(11) we impose j = 0 and j = 0. Then, we have for the σ field,
B 3
g2
g σ = σ ns, (12)
σ m′2
σ
with m′2 = m2 + 1κσ+ 1λσ2. For the ω0 field we get
σ σ 2 3!
g2
g ω0 = ω n , (13)
ω m′2 B
ω
7
with m′2 = m2 + 1ξg4ω02.
ω ω 3! ω
When the Dirac equation for nucleons Eq.(5) is solved, a magnetic field B in the z-direction
given by B = Bkˆ is used. The energies for the quasi-protons and quasi-neutrons in the medium
with spin z-projection, s, are given by the following expressions [23],
2 1
ǫp = k2+ m∗2+2νq B sµ κ B +g ω0+ g ρ0, (14)
ν,s z p p − N p ω 2 ρ
r
(cid:16)q (cid:17)
2
1
ǫn = k2+ m∗2+k2 sµ κ B +g ω0 g ρ0, (15)
s s z n ⊥ − N n ω − 2 ρ
(cid:18)q (cid:19)
where ν = n+ 1 sgn(q )s = 0,1,2,... enumerates the quantized Landau levels for protons with
2 − b 2
electric charge q . The quantum number s is +1 for spin up and 1 for spin down quasipar-
p
−
ticles. Due to the fact that the magnetic field is taken in the z-direction, it is useful to define
three-momentum (k) components along parallel (k ) and perpendicular (k ) directions. Then, for
z ⊥
neutrons, the surface of constant energy is an ellipsoid while, for protons, constant energy surfaces
are formed by circumferences on nested cylinders with radius labeled by the Landau level, see next
section. Inthisworkwearemainlyinterested inhadronicproperties,andelectron dynamicswillbe
such that at high B field strengths, they will mostly be in the low Landau levels. For completeness
we write the expressions of the scalar and vector densities for protons and neutrons for both spin
polarizations as follows [23]
ns = qpBm∗p νmax m∗p2+2νqpB −sµNκpB ln kFp,ν,s+EFp ,
p 2π2 Xν=0 Xs q m∗p2+2νqpB (cid:12)(cid:12)(cid:12) m∗p2+2νqpB−sµNκpB(cid:12)(cid:12)(cid:12)
m∗ q kn +En (cid:12)q (cid:12)
ns = n Enkn m¯2 ln F,s F , (cid:12) (cid:12)
n 4π2 F F,s− n m¯ (cid:12) (cid:12)
s (cid:20) (cid:12) n (cid:12)(cid:21)
Xνmax (cid:12) (cid:12)
nv = qpB kp , (cid:12)(cid:12) (cid:12)(cid:12)
p 2π2 F,ν,s
ν=0 s
X X
1 1 1 m¯ π
nv = kn 3 sµ κ B m¯ kn +En2 arcsin n , (16)
n 2π2 3 F,s − 2 N n n F,s F En − 2
s (cid:20) (cid:18) (cid:18) (cid:18) F(cid:19) (cid:19)(cid:19)(cid:21)
X (cid:0) (cid:1)
wherekp ,kn aretheFermimomentaofprotonsandneutronsrelatedtotheprotonandneutron
F,ν,s F,s
Fermi energies, Ep and En, respectively, by
F F
2
kp2 = Ep2 m∗2+2νq B sµ κ B ,
F,ν,s F − p p − N p
kn2 = En2 hm¯q2, i (17)
F,s F − n
and m¯ = m∗ sµ κ B m¯s. The summation in ν in the above expressions terminates at
n n − N n ≡ n
ν , the largest value of ν for which the square of Fermi momentum of the charged particle is
max
8
still positive and corresponds to the closest integer from below defined by the ratio
(Ep +sµ κ B)2 m∗2
ν = F N p − p . (18)
max
2q B
" p #
| |
For electrons ke = Ee2 (m2+2νq B) and ν does not have the same value as that for
F,ν,s F − e e max
q
protons.
III. LANDAU FERMI LIQUID PARAMETERS
We now calculate the Landau parameters using a generalized formulation of Landau Theory of
Fermi Liquids [31], from the variations of the energy density of the system, , given by [32]
E
1g2m2 1g2m2 1 ξg8 1 ξg8 2 ξg8
= ω ωn2 ω ωj2 + ωn4 + ωj4 ωj2n2
E 2 m′4 B − 2 m′4 B 4!m′8 B 4!m′8 B − 4!m′8 B B
ω ω ω ω ω
1 g2 1 g2
+ ρ n2 ρ j2+ npEp + nnEn + nlEl
8m2 3− 8m2 3 i iν,s i is i iν,s
ρ ρ i,ν,s i,s i,ν,s
X X X
1m2 1 κ 1 λ B2
+ σ(m m∗)2+ (m m∗)3+ (m m∗)4+ , (19)
2 g2 b− 3!g2 b− 4!g2 b− 2
σ σ σ
where we have defined np = np(k ,ν,s ) as the occupation number of the quasiprotons and nn =
i i i i
nn(k ,s ) for the quasineutrons. For leptons the occupation number is nl = nl(k ,ν,s ). Also we
i i i i i
have defined the energies,
2
Ep = p2+ m∗2+2νq B sµ κ B , (20)
iν,s Kzi p p − N p
r
(cid:16)q (cid:17)
2
En = n2+ m∗2+ 2 sµ κ B , (21)
is sKzi n K⊥i − N n
(cid:18)q (cid:19)
and analogous for electrons, El . We use generalized three-momenta depending on isospin
iν,s
Kj = k Vj, j = p, n (22)
i i− i
with Vp = g ω+ 1g ρ and Vn = g ω 1g ρ. The equation for the nucleon effective mass can be
ω 2 ρ ω − 2 ρ
written as
g2
m∗ = m g σ = m σ (ns +ns), (23)
b− σ b− m′2 p n
σ
where
nnm∗ sµ κ B
ns = is 1 N n , (24)
n Xi,s Eins  − m∗n2+K⊥2i
 q 
9
np m∗ m¯p
ns = i,ν,s νs , (25)
p Ep m¯p +sµ κ B
i,ν,s iνs νs N p
X
where the following definition has been used
m¯p = m∗2+2q νB sκ µ B = ǫ˜p sκ µ B. (26)
νs p − p N νs− p N
q
The vector current for quasiprotons is written as
Kpnp
j = i i , (27)
p [K2p +(m¯p )2]1/2
i,ν,s i νs
X
and for quasineutrons,
Kn sµ κ B Kn
j = nn ⊥i 1 N n + zi , (28)
n Xi,s is Eins  − m∗n2+Kn⊥2i Eins
  q  
so that j and j can be constructed.
B 3
Accordingto theFLT [31]thefirstvariation of theenergy density of thesystem, , withrespect
E
j j
to the ocupation numberfor qp with isospin of jth-type, n , defines theqp energy, ǫ . Let us notice
i i
that in reduced notation the index i means (i,ν,s) for quasi-protons and (i,s) for quasi-neutrons:
j j
δ = ǫ δn . (29)
E i i
i,j
X
and
g2 g2
ǫp = Ep + ω n + ρ n , (30)
i i,ν,s m′2 B 4m2 3
ω ρ
g2 g2
ǫn = En + ω n ρ n . (31)
i is m′2 B − 4m2 3
ω ρ
The single quasiparticle energy has, each, two explicit contributions, one due to the motion under
the influence of a strong quantizing magnetic fiel and another due to the motion in a medium
with mesonic self-interacting fields. To calculate the Landau parameters we use the standard
approach [31] but generalizing to the case when there are external fields. These will allow to
extract information on the interaction energies of quasiparticles of spin z-projection (s,s′) and
isospin (i,j) and, if they are protons, different Landau levels (ν,ν′) in the system. In a generalized
system with aqpstate theLandauparameters arecalculated as thesecond derivative of theenergy
j
densityofthesystem, ,withrespecttotheqpstatewithoccupationnumbern ,thatisj-isospin,
E l,s′
10
spin s′ and momentum j (if they are quasiprotons they include the additional quantum number
Kl
j
ν, in the way n ). Using Eq. (29) it is defined,
lνs′
∂ǫi
ij r,s
f = . (32)
rlss′ j
∂n
l,s′
From the original formulation by Landau of the pure neutron system without considering spin
degrees calculated in [29, 30] the above expression generalizes to a (2ν + 2) (2ν + 2)
max max
×
matrix, f , in isospin and spin space. In this way we have a characterization of the nuclear system
L
when an external quantizing magnetic field is considered,
pp pn
f f
ilss′νν′ ilss′ν
f = . (33)
L
 fnp fnn 
ilss′ν′ ilss′
 
The detailed calculation of the matrix elements is given in the appendix in section VI.
In order to obtain the relativistic Landau parameters we must consider in the context of the
FLT that the interactions of the effective quasiparticles in the system will take place close to the
Fermi surfaces, since the lifetime of these excitations varies inversely with the departure of its
energy, E, from the Fermi energy τ 1 .
≈ (E−EF)2
Since the matrix elements in Eq.(33) have dimensions of energy divided by number density
it seems convenient to define new dimensionless coefficients by multiplying them by the density
of states at each Fermi level for quasiprotons, Ns,ν and quasineutrons, Ns, at the Fermi surface
p n
with a given spin projection. In the case of protons the quantized level filling must be carefully
considered and the definition of density of states at the Fermi level and in a given Landau level
with polarization s is
1 q B
Nps,ν = V δ(ǫkzp,ν,s−µ)= |2πp|L δ(ǫkzp,ν,s−µ)
Xk Xkzp
→ |q2pπ|B2 ∞dkzpδ(ǫkzp,ν,s−µ) = |q2pπ|B2 ∞dǫkzp,ν,sǫkkzp,pν,sδ(ǫkzp,ν,s−µ)
Z0 Z0 z
q B µ
p
= | | , (34)
2π2 kp
F,ν,s
p
where µ = E is the proton Fermi energy. Summing over all possible levels we have,
F
Ns = ∂nps = Ns,ν = qpB EFp , (35)
p (cid:18)∂ǫps (cid:19)EFp Xν p 2π2 Xν kFp,ν,s
p p2 p 2
where k = E (m¯ ) . For neutrons,
F,ν,s F − νs
q
∂nn En m¯s π
Ns = s = F kn sµ κ B arcsin n . (36)
n ∂ǫn 2π2 F,s− N n En − 2
(cid:18) s (cid:19)EFn (cid:26) (cid:20) (cid:18) F(cid:19) (cid:21)(cid:27)