Nuclear structure and dynamics in the neutron star crust

Nuclear structure and
dynamics in
the neutron star crust
Piotr Magierski (Seattle & Warsaw)
Collaborators:
Aurel Bulgac (Seattle)
Paul-Henri Heenen (Brussels)
Andreas Wirzba (Bonn)
Neutron star discovery
-The existence of neutr on star s was pr edicted by Landau (1932), Baade & Zwicky (1934) and
Oppenheimer & Volkoff (1939).
- On November 28, 1967, Cambr idge gr aduate student J ocelyn Bell (now Bur nell) and her advisor ,
Anthony Hewish discover ed a sour ce with an exceptionally r egular patter n of r adio flashes. These
r adio flashes occur r ed ever y 1 1/3 seconds like clockwor k. After a few weeks, however , thr ee mor e
r apidly pulsating sour ces wer e detected , all with differ ent per iods. They wer e dubbed " pulsar s."
Natur e of the pulsar s
Pulsar in the Cr ab Nebula
pulse r ate = 30/second
slowing down r ate = 38 nanoseconds/day
Calculated ener gy loss
due to r otation of a possible
neutr on star

Ener gy r adiated
Conclusion: the pulses are produced by rotation!
Basic facts about neutr on star s:


Radius:
10 km
Mass:
1-2 solar masses
14
3
10
g
/
cm

Aver age density:
8
12
10

10

Magnetic field:
G
 1015 G
Magnetar s:
Rotation per iod: 1.5 msec. – 5 sec.
Number of known pulsar s: > 1000
Number of pulsar s in our Galaxy:
Gr avitational ener gy
of a nucleon at the sur face
of neutr on star

 10
8
100 MeV
Binding ener gy per nucleon in an atomic nucleus:
 8 MeV
Neutron star is bound by gravitational force
Bir th of a neutr on star
Super nova explosion and for mation of pr oto-neutr on star
Ther mal evolution of a neutr on star :
Temper atur e: 50 MeV 0.1 MeV (t  min.)
URCA pr ocess: p  e  n  
e
n  p  e  e
Temper atur e: 0.1 MeV
MURCA pr ocess:


Cr ust
100eV (t  10 yr.)
5
p  p  e  p  n e
e
n  p  e  n  n e
p  n  p  p  e e
n  n  n  p  e e
e
Ener gy tr ansfer between cor e
and sur face:
T

 D2T; D 
t
Cv
For  < 100 years:
T cor e < T sur f
URCA &
MURCA
Tcor e
Cor e
 1km
e
e

Tsur f
Cooling wave

No URCA
Relaxation time: a typical
time for the cooling wave
to r each the sur face
Structure of a neutron star
~ 0.01 - 0.02 M NS
Cr ystalline solid
.
.. . .. .
.
. .. . .. ..
.
Electrons
. .
Nuclei
.
.
. ..
.
.
.
.
.
.
Neutrons
Nuclei
∼
56
Fe
ρ 106 g/cm3
ρ ≈ 4×1011 g/cm3
ρ ≈1014 g/cm3
Outer
cr ust Inner
cr ust
Cor e
Unifor m
nuclear matter
Exotic nuclear shapes
„pasta” phase
QuarkQuarkgluon
plasma?
plasma?
Neutr on density distr ibution
Str uctur e of the inner cr ust
Electr ons
.
.
..
.
.
.
.
.
.
Unbound
neutr ons
Nuclei
r (fm)
ρtot (fm -3 )
Pr oton density distr ibution
Neutr on Cooper pair s
s-wave pair ing gap in infinite neutr on matter
with r ealistic NN-inter actions
r (fm)
ρtot (fm -3 )
Pr oton fr action
ρp
ρtot
K.Klimaszewski, diploma thesis, 2004
ρtot (fm -3 )
Dynamics of a nucleus immer sed
in a neutr on super fluid
Neutr ons Cooper pair s
 | ˆ | 2 C

H    m  | ˆ |2 
m 

2
M
2
m


( -1) 2
5
M  m
R
;   out
in 2(   1) N

in
sur f
C C
 C coul
RN - nuclear radius
ρin - nuclear density
ρout - density of unbound neutrons
Vibrating nucleus
Dynamics of a nucleus immer sed in a neutr on super fluid:
cor r ection due to the coupling to the lattice
Vibrational energy depends on the
orientation with respect to the lattice
Splitting of a quadr upole
vibr ational multiplet:
1
tot  VCoul VWS 
VCoul -Coulomb interaction energy of the nucleus with the lattice
Total volume
-Wigner - Seitz cell volume
Number of nuclei
α - amplitude of vibration
VWS =
.
. ..
.
. .
Spherical symmetry breaking
due to the coupling between lattice and
nuclear vibrations
spher ical nuclei
.
.
. .
.
defor med nuclei
Nuclear quadr upole excitation
ener gy in the inner cr ust
Structure of the bottom of the inner crust: „pasta” phases
Nuclear Pasta
At densities of 1013 -1014 g/cm 3 competition between
nuclear attr action and Coulomb r epulsion leads
to a ver y complex gr ound state that involve r ound
(meat ball), r od (spaghetti), plate (lasagna), and
other shapes.
This „nuclear pasta” is expected to have unusual
pr oper ties and dynamics. It may be impor tant for
r adio, x-r ay, and neutr ino r adiation.
Simple semiclassical model pr edict a sequence of
phase tr ansitions:
Lorenz, Ravenhall and Pethick, Phys. Rev. Lett. 70, 379 (1993)
First fully microscopic 3D calculations of pasta phases:
Hartree-Fock calcs. In
coordinate space for
1000 nucleons in
the Wigner-Seitz (WS)
cell!
Pr oton density
distr ibution
Neutr on density
distr ibution
Coulomb interaction
treated beyond the WS
approximation!
‘Spaghetti’
phase
P. Magierski and P.-H. Heenen, Phys.Rev.C65,045804 (2002)
‘Lasagna’
phase
scc lattice
bcc lattice
P. Magierski and P.-H. Heenen, Phys.Rev.C65,045804 (2002)
Ener gy differ ence between the spher ical phase and the ‘spaghetti’ phase:
Ener gy differ ence between the spher ical phase and the ‘lasagna’ phase: ........
lasagna
spaghetti
Spher ical
phase
What is the or igin of these oscillations?
Skyrme HF with SLy4, P.Magierski and P.-H.Heenen, Phys.Rev.C 65, 045804 (2002)
Quantum fluctuation effects
Let me create a caricature of a the “pasta phase” in the crust of a neutr on star .
Neutr on gas
Question:
What is the most favor able ar r angement of these two spher es?
Casimir Inter action among Objects Immer sed in a Fer mionic Envir onment
EC
 2kF a 2

c o s ( 2 k F ( r  2 a ))
3
8 m r
A.Bulgac, P. Magierski,
Nucl. Phys. A683, 695 (2001),
A.Bulgac, A. Wirzba,
Phys.Rev.Lett.87,120404(2001)
The Casimir energy for the displacement of a single void in the lattice
Rod phase
Slab phase
y
x
Bubble phase
A. Bulgac and P. Magierski
Nucl. Phys. 683, 695 (2001)
Energy transfer between core and surface:
∂T
κ
2
= D∇ T; D =
∂t
CV
CV = ?
What are the basic degrees of freedom of
the neutron-proton-electron matter at subnuclear densities?
Estimates of various contributions
to the specific heat in the crust
Char acter istic temper atur e:
Electr ons:
T  0.1 MeV
CV  T/ε F
Super fluid unifor m nuclear liquid: CV  Δ/T 3/2 Δexp -Δ/T 
Lattice vibr ations:
Nuclear shape vibr ations:
CV  3N
CV ≈ (2 l +1)N
Contributions to the specific heat of the inner crust
Conclusions
•
•
•
•
•
•
Ther e is a substantial r enor malization effect of a nuclear /ion mass in the
inner cr ust of a neutr on star , due to the pr esence of a super fluid neutr on
liquid.
Ther mal and electr ic conductivities of the inner cr ust ar e expected to be
modified. In par ticular , the contr ibutions coming fr om Umklapp pr ocesses
have to be r ecalculated using the r enor malized ion masses.
Due to the coupling between the nuclear sur face vibr ations and the ion latt ice
par t of the cr ust is filled with non-spher ical nuclei. The phase tr ansition
takes place at densities far lower than the pr edicted density for the tr ansition
to the exotic „pasta phases”.
The contr ibution to the specific heat associated with nuclear shape vibr ations
seems to be impor tant at densities ar ound 0.02 fm -3 wher e the pair ing
cor r elations ar e pr edicted to r each their maximum.
Quantum cor r ections (Casimir ener gy) to the gr ound state ener gy of an`
inhomogeneous neutr on matter at the bottom of the cr ust ar e of the same
magnitude or lar ger than the ener gy differ encies between spher ical,
„spaghetti”, and „lasagna” phases.
The “pasta phase” might have a r ather complex str uctur e, var ious shape can
coexist, and at the same time significant lattice distor tions ar e likely and the
bottom of the neutr on star cr ust could be on the ver ge of a disor der ed phase.
Open questions:
• Basic degrees of freedom of the „pasta phase”?
• Influence on the cooling curve of neutron stars?
• The role of isovector nuclear modes?
• Mechanical properties of the crust? Liquid crystal?
• The physics of superfluid vortices in the inner crust?