1307.2534

Compact Stars in the QCD Phase Diagram III (CSQCD III)
December 12-15, 2012, Guarujá, SP, Brazil
http://www.astro.iag.usp.br/~foton/CSQCD3
arXiv:1307.2534v1 [nucl-th] 9 Jul 2013
Nuclear matter EOS with light clusters within the
mean-field approximation
Márcio Ferreira and Constança Providência
Centro de Fı́sica Computacional
Department of Physics, University of Coimbra
P-3004-516 Coimbra
Portugal
Email: [email protected], [email protected]
1
Introduction
The crust of a neutron star is essentially determined by the low-density region (ρ <
ρ0 ≈ 0.15 − 0.16 fm−3 ) of the equation of state. At the bottom of the inner crust,
where the density is ρ . 0.1ρ0 , the formation of light clusters in nuclear matter will
be energetically favorable at finite temperature. At very low densities and moderate
temperatures, the few body correlations are expected to become important and light
nuclei like deuterons (d ≡ 2 H), tritons (t ≡ 3 H), helions (h ≡ 3 He) and α-particles
(4 He) will form. Due to Pauli blocking, these clusters will dissolve at higher densities
ρ & 0.1ρ0 . The presence of these clusters influences the cooling process and quantities,
such as the neutrino emissivity and gravitational waves emission. The dissolution
density of these light clusters, treated as point-like particles, will be studied within
the Relativistic Mean Field approximation. In particular, the dependence of the
dissolution density on the clusters-meson couplings is studied [1].
2
The model
The Lagrangian density of a system of nucleons and light clusters (d, h, t, and α
particles) is given by
X
L=
Lj + Lα + Ld + Lσ + Lω + Lρ + Lωρ ,
j=p,n,t,h
where the Lagrangian density Lj is
Lj = ψ j γµ iDjµ − Mj∗ ψj ,
1
(1)
with
gρj
τ · bµ ,
2
j = p, n, t, h
iDjµ = i∂ µ − gvj ω µ −
(2)
Mj∗ = Mj − gsj σ,
(3)
where Mp,n = M = 938.918695 MeV is the nucleon mass and Mt,h = At,h M − Bt,h
are the triton and helion masses. The binding energies of the clusters Bi are in [1].
The couplings between the cluster i and the meson fields ω, σ and ρ are given by gvj ,
gsj and gρj , respectively. The α particles and the deuterons are described as in [2]
1
1
(iDαµ φα )∗ (iDµα φα ) − φ∗α (Mα∗ )2 φα ,
2
2
1
1
(iDdµ φνd − iDdν φµd )∗ (iDdµ φdν − iDdν φdµ ) − φµ∗
(Md∗ )2 φdµ ,
=
4
2 d
Lα =
(4)
Ld
(5)
where iDjµ = i∂ µ − gvj ω µ and Mj∗ = Mj − gsj σ with j = α, d.
The meson Lagrangian densities are
1
1 3 3
1 4 4
µ
2 2
Lσ =
∂µ σ∂ σ − ms σ − κgs σ − λgs σ
2
3
12
1
1 4
1
µν
2
µ
µ 2
− Ωµν Ω + mv ωµ ω + ξgv (ωµ ω )
Lω =
2
2
12
1
1
µν
2
µ
Lρ =
− Bµν · B + mρ bµ · b
2
2
2 2
Lωρ = Λv gv gρ ωµ ω µ bµ · bµ
where Ωµν = ∂µ ων − ∂ν ωµ , Bµν = ∂µ bν − ∂ν bµ − gρ (bµ × bν ).
The equations of motion in the relativistic mean field approximation for the meson
fields are given by
X
X
λ
κ
m2s σ + gs3 σ 2 + gs4 σ 3 =
gsi ρis +
gsi ρi
2
6
i=p,n,t,h
i=d,α
X
1
3
2
m2v ω 0 + ξgv4 ω 0 + 2Λv gv2 gρ2ω 0 b03 =
gvi ρi
6
i=p,n,t,h,d,α
X
2
gρi I3i ρi
m2ρ b03 + 2Λv gv2 gρ2 ω 0 b03 =
i=p,n,t,h
where ρis =
2S i +1
2π 2
R kfi
0
k 2 dk √
Mi −gsi σ
k 2 +(Mi −gsi σ)2
is the scalar density, ρi is the vector density,
and I3i (S i ) is the isospin (spin) of the specie i for i = n, p, t, h. At zero temperature all
the α and deuteron populations will condense in a state of zero momentum, therefore
ρα and ρd correspond to the condensate density of each specie.
2
0.012
ρdis fm−3
0.0014
Yp = 0.5
gρt
Yp = 0.1
0.001
0
0.5
1
1.5
δt
= δt gρ
2
2.5
3
FSU
Yp = 0.1
0.008
0.006
gst = 0
gvt = ηtgv
0.002
gρt = gρ
0.004
0
0.012
0
0.5
1
NL3
FSU
Yp = 0.1
gst = βtgs
gvt = 3gv
gρt = gρ
0.008
FSU
0.0018
Yp = 0.5
Yp = 0.5
NL3
(a)
(b)
0.01
ρdis fm−3
(a)
(a)
NL3
ρdis fm−3
0.0022
0.004
(c) (a)
1.5
ηt
2
2.5
3
0
0
0.5
1
βt
1.5
2
Figure
= 0 for
tritons
Yp =lines)
0.5 (solid
Figure1:1:The
Thedissolution
dissolution density
density at
for Ttritons
with
= with
0 5 (solid
and lines)
= 0 and
t
t
t
Yp(dashed
= 0.1 (dashed
as aof:
function
of: a) δ ),(gb)ρ = δt gρ ), b) η) tand
(gv c)
= ηt gv ) and c)βt
lines) as lines)
a function
a)
(gst = βt gs )
3 Results
3 Results
The dissolution density
of each light cluster is studied separately. We assume the
coupling
constants
as ρd of each light cluster is studied
and separately.
= 0 with We assume
t, h, d, αthe
The
dissolution
density
t, hconstants
and
d,
Then,
coupling
asαgvjrespectively.
= ηj gv , gsj =
βj gs ,wegρi determine
= δi gρ andthe
gρl dissolution
= 0 with jdensity
= t, h, d, α,
and we
for determine
different global
proton fractions
i dependence
= t, h and on
l =thed,parameters
α respectively. Then,
the dissolution
density ρd
The
results
for
triton
at
=
0
are
shown
in
Fig.
1
(the
results
for
all
clusters
can beYP .
dependence on the parameters ηj , βi and δi for different global proton fractions
seen
in
[1]).
The results for triton at T = 0 are shown in Fig. 1 (the results for all clusters can be
cluster formation and dissolution are insensitive to the -meson-cluster couseen The
in [1]).
pling
constants
when compared
with andare insensitive
[1]. Therefore,
we ρ-meson-cluster
fix the for triton
The cluster formation
and dissolution
to the
couand
helion
as
.
The
decreases
with
in
-meson-cluster
coupling
coni an increase
i
i
pling constants when compared with gs and gv [1]. Therefore, we fix the gρ for triton
stant
and increases
with
increasewith
in -meson-cluster
constant. Forcoupling
some
and
helion
as gρ . The
ρd an
decreases
an increase of coupling
the ω-meson-cluster
value of below the mass number , the
increases abruptly and the clusters will
constant and increases with an increase of the σ-meson-cluster coupling constant. For
not dissolve.
some value of βi below the mass number Ai , ρd increases abruptly and the clusters
To fix and
for each cluster two constraints are required. We consider the
will not dissolve.
for symmetric
nuclear
matter ( = 0 5) at = 0 MeV [3], the in-medium binding
i
i
To
fix
g
and
g
for
each[2],
cluster
two constraints are required. We consider
the ρ
s
v
energies
and recent experimentally Mott points at
5 MeV d
for
nuclear
matter
(Yp = /g
0.5)=at0,TA= 0 MeV4 [3],
binding
[4].symmetric
We contruct
several
sets with
and the
4 in-medium
5, and calculate
∗
∗
energies
B
=
A
M
−
M
[2],
and
recent
experimentally
Mott
points
at
T
≈
MeV
i
the /g i valuei of each set
that reproduces
the
for
= 0 5 at = 0 MeV5 [3].
i
[4].
Wewe
contruct
sets with
gs /g
Ai /2,
and with
4Ai /5,
calculate
s = 0, sets
i /4
After,
compareseveral
the behavior
of the
different
at 3A=
5 MeV
theand
in-medium
i
value of [2]
each
that
reproduces
for Y
= 0 parameter
MeV [3]. After,
the
gv /gv energies
p = 0.5
binding
andset
the
Mott
points [4].ρdThe
results
forat
theT best
set
we
compare
the
behavior
of
the
different
sets
at
T
=
5
MeV
with
the
in-medium
[1] are shown in Figure 2. To test the choosen parameter set, we calculate the particle
binding
and the Mott
[4]. The
results for the best
set
fractionenergies
at finite[2]
temperature
and points
in chemical
equilibrium,
+ parameter
(
[1]with
are shown
in
Figure
2.
To
test
the
parametrization
chosen,
we
calculate
the
particle
α, h, t, d [5, 6]. The effect of
at = 5 is shown in Fig. 3 (a). In Fig.
fraction
temperature
equilibrium,
µi = Ztwo
+ (Ai −NL3
Zi )µn
3 (b) – at
(d),finite
we always
set
=and
3 in chemical
. In Fig.3
(b) we compare
i µp models,
i
with
i = α,
h, t,and
d [5,
6]. (full
Thelines).
effect of
gseffect
at T =
MeV is shown
Fig. 3asymmetry
(a). In Fig.
(dashed
lines)
FSU
The
of 5
temperature
and in
isopsin
shown
in we
Fig.3
(c) and
Fig.3
(c)
particle
fractions
in
symmetric
matter
are
3 is(b)
– (d),
always
set(d).
gsi =In3A
/4
g
.
In
Fig.3
(b)
we
compare
two
models,
NL3
i
s
compared
for and
= FSU
10 MeV
lines)
and
5 MeV
(dashed and
lines).
(dashed
lines)
(full(full
lines).
The
effect=of
temperature
isopsin asymmetry
is shown in Fig.3 (c) and (d). In Fig.3 (c) particle fractions in symmetric matter are
compared for T = 10 MeV (full lines) and T = 5 MeV (dashed lines).
3
2.5
25
Bi ( MeV)
10
(b)
•(a)
•
α
20
(b)
Typel 2010
gsi /gs
8
= 3Ai /4
T ( MeV)
30
15
10
t
0
0.002
0.004
ρ fm−3
0.006
6
4
0
0.008
α
t
h
Hagel 2012
Typel 2010
2
5
h
0 d
d
gsi /gs
0
0.003
0.006
ρ fm−3
= 3Ai /4
0.009
0.012
Figure2:2: The
The in-medium
in-medium binding
binding energy in our model is
Figure
is compared
compared in
in (a),
(a), for
forall
allthe
the
clusters
(dashed
lines),
with
the
corresponding
results
of
Typel
2010
[2]
at
T
=
clusters (dashed lines), with the
results of Typel 2010 [2] at
= 55
MeV(full
(fulllines).
lines). In
In (b)
(b) we
we compare
compare our Mott points with
MeV
with the
the ones
ones from
from Typel
Typel2010
2010
[2](dashed
(dashed lines)
lines) and
and the
the experimental
experimental prediction by Hagel
[2]
Hagel 2012
2012 [4]
[4] (full
(full dots
dots with
with
errorbars).
The
FSU
parametrization
was
considered.
errorbars). The FSU parametrization
Forlarger.
a largerFor
coupling
gsi /gs ,nuclear
both the
fraction
andthe
thedeuteron
dissolution
densityis
are
symmetric
matter
at of particles
= 10 MeV,
fraction
are larger.
For symmetric
at T the
= 10
MeV, the
deuteronrich
fraction
is
already
the largest
fraction nuclear
and thematter
fraction
smallest.
In neutron
matter,
already
the
largest
fraction
and
the
α
fraction
the
smallest.
In
neutron
rich
matter,
the deuteron fraction is the largest and the tritons come in second both at = 5 and
theMeV.
deuteron
is the
theon
tritons
come inparametrization
second both at Tchoosen.
= 5 and
10
The fraction
results do
not largest
dependand
much
the nuclear
10 MeV. The results do not depend much on the nuclear parametrization chosen.
4 Conclusion
4 Conclusion
Light clusters have been included in the EOS of nuclear matter as point-like particles
Light
clusterscoupling
have been
includedwithin
in therelativistic
EOS of nuclear
as point-like
particles
with
constant
constants
meanmatter
field approach.
It was
shown
with
constant
coupling
constants
within
relativistic
mean
field
approach.
Results
that the dissolution of clusters is mainly determined by the isoscalar part of the EOS.
demonstrate
that theresults
dissolution
clusters is binding
mainly energy
determined
by clusters
the isoscalar
Recent
experimental
for theofin-medium
of light
[4] at
part of the EOS. Recent experimental results for the in-medium binding energy of
5 MeV and results from a quantum statistical approach [2] were used to constraint
light clusters [4] at T ∼ 5 MeV and results from a quantum statistical approach
the clusters couplings constants. It was shown that a larger -cluster coupling gives
[2] were used to constraint the clusters couplings constants. It was shown that a
rise to larger dissolution densities and larger particle fractions in chemical equilibrium
larger σ-cluster coupling gives rise to larger dissolution densities and larger particle
at = 5 MeV and = 10 MeV for symmetric and asymmetric nuclear matter with
fractions in chemical equilibrium at T = 5 MeV and T = 10 MeV for symmetric and
light clusters.
asymmetric nuclear matter with light clusters.
The in-medium binding energies proposed in [2] may be reproduced within our
The in-medium binding energies proposed in [2] may be reproduced within our
model with a temperature dependent meson-cluster couplings. More experimental
model with a temperature dependent meson-cluster coupling constants. More exMott points at more temperatures would allow the determination of the temperature
perimental Mott points at more temperatures would allow the determination of the
dependence.
temperature dependence.
4
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K+ '!" )(--(3,+4
N*',(+ DO℄ () '!"
-,4!' -%&'"$& ,+
/"$,+4 "* ! -%&0
-%&'"$& (%9-,+4
6
α6
t, h *+/
" /"'"$1,+" '!"
" (+ '!" 9*$*10
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; K+ 728 3" (19*$" '3( 1(/"-&6 CIS 7/*&!"/ -,+"&8 *+/
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@"R 7/*&!"/ -,+"&8;
T = 5 MeV
Yp = 0.5
0.1
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T = 10 MeV
T = 5 MeV
(b)
0.1
0.01
(c)
0.1
0.01
0.0001
0.001
ρ fm−3
Yp = 0.5
0.001
gsi /gs = 3A/4
gsi /gs = 0
h α
T = 10 MeV
T = 5 MeV
Yp = 0.3
FSU
(d)
0.01
T = 5 MeV
FSU
0.001
0.001
FSU
0.0001
0.01 10−5
0.0001
0.001
ρ fm−3
0.0001
0.01 10−5
0.0001
0.001
ρ fm−3
0.0001
0.01 10−5
0.0001
0.001
0.01
ρ fm−3
A($ * -*$4"$
(%9-,+4of light
/g 6 2('!
'!" )$*
',(+ () 9*$', with
-"& *+/nuclear
'!" /,&&(-%',(+
-*$4"$
Figure
3: Fraction
clusters
in equilibrium
matter: /"+&,'.
(a) for*$"
FSU
i
with!"Y$"&%-'&
T = 5 1%
MeV,
lines) and
gsi = 3 Ai /4 gs (full
p = 0.5
s =
/( and
+('
/"9"+/
! (+and
'!" g+%
-"*$0 (dashed
9*$*1"'$,N*',(+
!((&"+
i
lines); (b) for gs = 3 Ai /4 gs with Yp = 0.5 and T = 5 MeV and FSU (full lines)
Q' NL3
= 10
@"R '!"lines);
)$* (c)
',(+for
,& *-$"*/.
'!" -*$4"&'
',(+
',(+and
'!"T&1*--"&'
and
(dashed
FSU with
gsi = 3)$*
Ai /4
gs *+/
and'!"
Yp =)$*
0.5,
= 10
i
MeV
(full lines)! 1*''"$
and T =
(dashed
with,+ g&"
3 A2('!
i /4 g*'
s and= 5
K+ +"%'$(+0$,
'!"5 MeV
)$* ',(+
,& '!"lines);
-*$4"&'(d)
*+/for'!"FSU(1"
(+/
s =
Yp*+/
= 0.3,
and T = 10 MeV (full lines) and T = 5 MeV (dashed lines).
10 @"R
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FSU
NL3
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0.01
0.0001 −5
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Acknowledgments
This work was partially supported by QREN/FEDER, through the Programa Operacional Factores de Competitividade - COMPETE and by National funds through FCT
- Fundação para a Ciência e Tecnologia under Project No. PTDC/FIS/ 113292/2009
and Grant No. SFRH/BD/51717/2011.
References
[1] M. Ferreira and C. Providencia, Phys. Rev. C 85, 055811 (2012).
[2] S. Typel, G. Röpke, T. Klahn, D. Blaschke and H.H. Wolter, Phys. Rev. C 81,
015803 (2010).
[3] G. Röpke, Phys. Rev. C 79, 014002 (2009).
[4] K. Hagel et al., Phys. Rev. Lett. 108, 062702 (2012)
[5] S. S. Avancini, C. C. Barros, Jr., D. P. Menezes, and C. Providência, Phys. Rev.
C 82, 025808 (2010)
[6] S. S. Avancini, C. C. Barros, Jr., L. Brito, S. Chiacchiera, D. P. Menezes, and
C. Providência, Phys. Rev. C 85, 035806 (2012).
5