Nuclear structure and Nuclear structure and dynamics in dynamics in the the neutron star neutron star crustcrust

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Nuclear structure and Nuclear structure and

dynamics in dynamics in

the the neutron star neutron star crust crust

Piotr Magierski (Seattle & Warsaw) Collaborators:

Aurel Bulgac (Seattle)

Paul-Henri Heenen (Brussels) Andreas Wirzba (Bonn)

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Neutron star

Neutron star discovery 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

Natur e of the pulsar s Pulsar in the Cr ab Nebula



Conclusion: the pulses are produced by rotation!

Calculated ener gy loss due to r otation of a possible

neutr on star

Ener gy r adiated pulse r ate = 30/second

slowing down r ate = 38 nanoseconds/day

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Basic

Basic facts about facts about neutr on neutr on star s star s : : Radius: 10 km

Mass: 1-2 solar masses Aver age density:

Magnetic field: G Magnetar s: G

Rotation per iod: 1.5 msec. – 5 sec.

8 12

10  10



  14 3

10 g cm /

 10 15



Gr avitational ener gy of a nucleon at the sur face

of neutr on star  ^{100 MeV}

Binding ener gy per nucleon in an atomic nucleus: 8 MeV 

Neutron star is bound by gravitational force

Number of known pulsar s: > 1000

Number of pulsar s in our Galaxy:  10 ⁸

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Bir th of a neutr on star

Super nova explosion and for mation of pr oto-neutr on star

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Cr ust

Cor e

Cor e ^

^e



^e



^e



^e

T T

cor e cor e

sur f sur f



T ^{cor e} < T ^{sur f}

For  < 100 years:

Cooling wave Cooling wave Ther mal evolution of

Ther mal evolution of a neutr on star : a neutr on star :

Temper atur e: 50 MeV 0.1 MeV URCA pr ocess:

e e

p e n

n p e



  

   Temper atur e: 0.1 MeV 100eV

MURCA pr ocess:

e e

p p e p n

n p e n n

p n p p e

n n n p e



    

    

Ener gy tr ansfer between cor e and sur face:

2

;

v

T D T D

t C

    



URCA &

MURCA

 1km ( t  min.)

( t  10 5 yr .)

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Relaxation time: a typical time for the cooling wave to r each the sur face

No URCA

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Structure of a neutron star

~ 0.01 - 0.02 M

NS

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Outer

cr ust Inner cr ust

Nuclei

. . . . . . .

.

. . . .

.

. .

.

Electrons

Neutrons

Exotic nuclear shapes

„pasta” phase

. . . .

. .

.

. .

.

Cor e

Unifor m

nuclear matter

Quark Quark - - gluon gluon plasma plasma ? ?

14 3

ρ ≈10 g/cm

∼ Fe 56

 ⁶ ³

ρ 10 g/cm

Nuclei

Cr ystalline solid

11 3

ρ ≈ 4×10 g/cm

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Str uctur e of the inner cr ust

Neutr on density distr ibution

Pr oton density distr ibution

Pr oton fr action

p tot

ρ ρ

-3

ρ (fm )tot -3

ρ (fm )tot

r (fm)

-3

ρ (fm )tot

. . . .

. .

Nuclei Electr ons

Neutr on Cooper pair s

s- s -wave pair ing gap in infinite wave pair ing gap in infinite neutr on matter neutr on matter with r ealistic

with r ealistic NN- NN -inter actions inter actions

Unbound neutr ons

K.Klimaszewski, diploma thesis, 2004

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Dynamics of a nucleus immer sed in a neutr on super fluid

2 ˆ 2

| |

ˆ

| |

2 2

( - 1) 2 5 ; 2( 1)

m C

H m M m

M m R out

in N

sur f coul in

C C C

 

 

 

 

 

 

  

 

 

 



 

N in out

R - nuclear radius ρ - nuclear density

ρ - density of unbound neutrons

Neutr ons Cooper pair s

Vibrating nucleus

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Dynamics of a nucleus immer sed in a neutr on super fluid:

cor r ection due to the coupling to the lattice

1 tot V Coul V WS ^ 

   

Vibrational energy depends on the orientation with respect to the lattice

Splitting of a quadr upole vibr ational multiplet:

Coul

WS

V -Coulomb interaction energy of the nucleus with the lattice Total volume

V = -Wigner - Seitz cell volume

Number of nuclei α - amplitude of vibration

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. .

.

. .

. . .

. .

spher ical nuclei

defor med nuclei

Nuclear quadr upole excitation ener gy in the inner cr ust

Spherical symmetry breaking

due to the coupling between lattice and nuclear vibrations

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Lorenz, Ravenhall and Pethick, Phys. Rev. Lett. 70, 379 (1993)

Structure of the bottom of the inner crust : „pasta” phases

Nuclear Pasta

At densities of 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:

13 14 3

10 -10 g/cm

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P. Magierski and P.-H. Heenen, Phys.Rev.C65,045804 (2002)

‘Spaghetti’

phase

Pr oton density distr ibution

First fully microscopic 3D calculations of pasta phases:

Hartree-Fock calcs. In coordinate space for 1000 nucleons in

the Wigner-Seitz (WS) cell!

Coulomb interaction treated beyond the WS approximation!

Neutr on density distr ibution

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bcc lattice

‘Lasagna’

phase

scc lattice

P. Magierski and P.-H. Heenen, Phys.Rev.C65,045804 (2002)

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Skyrme HF with SLy4, P.Magierski and P.-H.Heenen, Phys.Rev.C 65, 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: ...

spaghetti

Spher ical phase

lasagna

What is the or igin of these oscillations?

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Let me create a caricature of a the “pasta phase” in the crust of a neutron star.

Question: What is the most favor able ar r angement of these two spher es?

Quantum fluctuation effects

Casimir Inter action among Objects Immer sed in a Fer mionic Envir onment

2 2

3 c o s ( 2 ( 2 ) ) 8

F

C F

k a

E k r a

 m r

  

A.Bulgac, P. Magierski, Nucl. Phys. A683, 695 (2001), A.Bulgac, A. Wirzba,

Phys.Rev.Lett.87,120404(2001)

Neutr on gas

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The Casimir The Casimir energy for the displacement of a single void in the lattice energy for the displacement of a single void in the lattice

Slab phase

Bubble phase Rod phase

A. Bulgac and P. Magierski Nucl. Phys. 683, 695 (2001)

x y

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What are the basic degrees of freedom of

the neutron-proton-electron matter at subnuclear densities?

Estimates of various contributions Estimates of various contributions

to to the specific heat in the crust the specific heat in the crust

Char acter istic temper atur e: T  0.1 MeV Electr ons:

Super fluid unifor m nuclear liquid:

Lattice vibr ations:

Nuclear shape vibr ations:

V  F

C T/ ε

   

 ^3/2

C V Δ/T Δexp -Δ/T

V 

C 3N

V l

C ≈ (2 +1)N

2 V

∂T κ

= D ∇ T; D =

∂t C

Energy transfer between core and surface:

C = ? V

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Contributions

Contributions to to the specific heat of the inner crust the specific heat of the inner crust

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• There is a substantial renormalization 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.

• Thermal and electric conductivities of the inner crust are 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 surface vibrations 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 contribution to the specific heat associated with nuclear shape vibrations seems to be impor tant at densities ar ound 0.02 fm wher e the pair ing

cor r elations ar e pr edicted to r each their maximum.

• Quantum corrections (Casimir energy) to the ground state energy 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 rather complex structure, various 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.

Nuclear structure and Nuclear structure and dynamics in dynamics in the the neutron star neutron star crustcrust