Origin of Magnetars — induced magnetic field

of anisotropic neutron superfluid

涡丝核心 (正常中子流体 )

Neutron superfluid vortex

Qiu-he Peng(Department of Astronomy, Nanjing Universit

y)

2

Main works of Mine I. On Explosion Mechanism of SN(1990-2006) 1.Neutrino flux by phase transition from nuclear matter into quarks in the collapsed core of SNII (1995, Dai, Peng and Lu) 2.Effect of electron screening on electron capture in presupernova (1997-2002) 。 3. New mechanism of core collapse of SNII and tested simulation with 1-D (2003-2006)II. Nucleosynthesis of 26Al and origin of interstellar 26Al from SN (1990-1997; 2005-2006) (Supported by a experiment in nuclear physics )III. Determination of scale-height of the disk for Face-on galaxies and effect of the thickness on structure of disk galaxies(1978-2006) (Two independent methods for determination of scale-hei

ght of the disks for Face-on galaxies)

3

IV. On Pulsars ( and Neutron stars) (1979-1985; 2002-2006)Two kinds of radiation emitted from neutron superfluid v

ortices (1979-1985):1 ) A spin down mechanism by neutrino cyclotron radiat1 ) A spin down mechanism by neutrino cyclotron radiat

ion of ion of neutron superfluid vortices (Peng, Huang & Huang (Peng, Huang & Huang 1982, A & A)1982, A & A)

2) A2) A heating mechanism by magnetic dipole radiation of heating mechanism by magnetic dipole radiation of anisotropic (anisotropic (33PP22 ) neutron superfluid vortecies ) neutron superfluid vortecies

(Peng, Huang & Huang, 1980 ; Huang, Lingenfelter, Peng and H(Peng, Huang & Huang, 1980 ; Huang, Lingenfelter, Peng and Huang, 1982, A & A).uang, 1982, A & A).

3) A rocked model of a neutrino jet for NS kicks (2003)3) A rocked model of a neutrino jet for NS kicks (2003)V. On origin of “extra high energy cosmic ray” by a AGN

model with magnetic monopoles (1986-2003)

4

Question

Observation: for majority of pulsars

B ~ (1011 – 1013 ) Gauss• magnetars: B ~ (1014 – 1015) Gauss

The fossil magnetic field from collapsed core of supernova

B(collapse) ~ (109 – 1011 ) Gauss

Origin of the Strong magnetic field of neutron stars?

Origin of the Magnetars

5

2R B const ( ) 1310BackgroundB Gauss

Initial magnetic field of neutron stars originates from collapsed core of SNII by the conservation of magnetic flux

Proposed models of the magnetars :Ferrario & Wickrammasinghe(2005)suggest that the extra-strong magnetic field of the magnetars is descended from their stellar progenitor with high magnetic field core. Iwazaki(2005)proposed the huge magnetic field of the magnetars is some color ferromagnetism of quark matter. Vink & Kuiper (2006) suggest that the magnetars originate from rapid ratating proto-neutron stars. My new model: The extra-strong magnetic field of the magnetars originates due to the induced magnetic moment of the anisotropic (3P2 ) neutron superfluid

6

Structure of the Neutron star

Core(1km)

3P2(isotropic)(isotropic)

NSVNSV

11SS00 (isotropic)NSV(isotropic)NSV

Protons (5-8)% ( Type II superconductor?) (normal) electron Fermi gas

= (g/cm3)

1014

1011 107

Inner crust

nuclei with Extra-

neutron rich

outer crust

(crystal of heavy metal)

Quarks ??

5×1014

104

NSV: NSV: Neutron superfluid vorticesNeutron superfluid vortices

III. Induced magnetic fields of anisotropic neutron superfluid

涡丝核心 (正常中子流体 )

Neutron superfluid vortex

8

1S0 &3PF2 Neutron superfluid1S0 neutron superfluid1S0 neutron Cooper pair: S=0, isotropic Energy gap :△(1S0) ≥ 0, 1011 < ρ(g/cm3) < 1.4×1014

△(1S0)≥2MeV 7×1012 <ρ(g/cm3)< 5×1013

3PF2 neutron superfluid3PF2 neutron Cooper pair: S =1, anisotropic, abnormal magnetic moment ~10-23 c.g.s.Energy gap:△ n(

3PF2) ～ 0.05MeV

(3.31014 < (g/cm3) < 5.21014)314 /108.2 cmgnuc

9

A Taskabnormal magnetic moment of a neutron

gaussergn /10966.0~ 23

(Bohr’s magnetic moment))/10927.0~)(( 20 gaussergeB

Behavior of 3P2 neutron superfluid is similar to that of the B phase of the liquid 3He with the very low temperature

A 3P2 neutron Cooper pair possesses a magnetic moment1) To calculate the magnetic field produced by the Pauli paramagnetic moment of highly degenerate Fermi (neutron, prot

on or electron) system .2) To calculate the magnetic field produced by the induce magnetic

moment of the 3P2 neutron pairs in the anisotropic neutron superfluid.

10

II. Pauli (induced) paramagnetic moment of a highly degenerate neutron system

The spin of a neutron for some momentum state by Fermi statistics)

11, 1, 1

2 ZS h

Its projection on z- ( or magnetic field) direction SZ = -h/2, +h/2 。The projection of its magnetic moment on direction of the magnetic field : zμ0 = μ0 , -μ0 。Its energy under the magnetic field: zμ0 B。

11

Method of Statistical PhysicsThe induced magnetic moment of a Fermi system may be found by a relation of

thermal dynamics

( ) lninBk T B

Ξ: grand partition function of the system.Ｂ : external magnetic field; kB: Boltzmann constantΨ: chemical potential of the Fermi gas/2: the projection of the spin(quantum number) = -1, +1N(ε): The ( level) state density of energy.

, 0,

01, 1 0

ln ln(1 exp{ ( )}

( ) ln(1 exp{ ( )}

k kk

B

N B d

1

Bk T

23

8( )

VN d p dp

h

12

Calculation of lnΞ

μnB (100eV) <<EF(n), it may be expanded a series of μnB that

( ) 2

ln(1 exp{ ( )})

1ln(1 ) ( ) ( ) ( )[1 ( )]

2

n

n n

B

e Bn B n n

( )

1( )

1n

e

It is the average number of neutrons occupied at the quantum state with energyε.

( ) 1n

But at the case ε > ψ ( ) 0n

At the deep of the Fermi see (ε<< ψ),

13

The sum of the second term for (=-1/2, +1/2) equal to zero. Both the sum of the first and third term for (= -1/2, +1/2) are just to double.

( ) 2

0 0

ln 2 ( ) ln(1 ) ( ) ( ) ( )[(1 ( )]nd N e B d N n n

The first term has no contribution to calculating the magnetic moment due to no relation with magnetic field. Thus it may be neglected for the calculation of the magnetic moment .

22

0

( )( ) ( ) ( ) ( )

6

dnd N N kT N

d

( )( )[1 ( )]

dnn n kT

d

2

( ) 2 ( ) 2ln ( )2 ( ){1 ( ) }

6 ( )in tot

n

NkT B N kT

B N

2

2

( )( )

d NN

d

14

The state density of energy N(ε)

2

1/ 2 1/ 22 3

2

( ) (2 )

n

n n

p

m

VN m m

For neutron system with highly non-relativistic degeneracy

V : Volume of the system

23

8( )

VN d p dp

h

For electron system with highly relativistic degeneracy

23

8( )

( )

cp

VN

hc

15

Total induced paramagnetic momentμ(in) for the neutrons

1/ 2 1/ 22 3

( ) (2 )2 n n

VN m m

2 1

2

( ) 1(4 )

( ) 4 F

NE

N

2( ) 2 (0) 22 ( ){1 ( ) )}

24in

nF

kTB N

E

3 / 2NS p NSB R Using a relation of magnetic field with the magnetic moment

(RNS radius of the NS) The induced magnetic field is

2( )( ) (0) (0)

3 3

2

3

42( )

4( )

inin n

FNS NS

nF

NS

B N E B ABR R

A N ER

16

Numerical estimation

22 / 3 2 2 / 3

5 / 3

3( ) ( ) 2

8F nn

E nm

2 / 331/ 3 3 1/ 3 22

2

33 3 1/ 32

( )16 3( ) ( )

3 8

( )2.1 10 ( ) ( )

nn n

NS

n

NS nuc

mR PA

R

R P

R

( ) (0) 6 (0) (0)10inB AB B B

32( )

0.1 1NS

R PA

R

For protons: there is only (5-8)% protons in a neutron star, so the induced magnetic field of the proton system is even smaller compared with the neutron system

17

Induced paramagnetic moment of the electron gas

2( ) 2 (0) 2ln ( )

2 ( ){1 ( ) }6 ( )

inn

NkT B N kT

B N

23

8( )

( )

cp

VN

hc

2( )( ) (0) (0)

3 3

2 21/ 3 2

3 3

42( ( ))

4 8( ( )) 16 ( ) ( )

3 ( )

inin e

FNS NS

e eF F

NS

B N E e B ABR R

A N E e E eR hc

The electron gas is in a highly relativistic degeneracy in NS

2

( ) 2

( )F

F F

N E

N E E

18

( ) (0)inB AB

33

8, ,

3F

e F F e e A

En p p n Y N

ch

Ｙ e the fraction of electrons;

( ) (0)( )inB e B

21/ 3 2 / 3 2 / 3

2 2 / 3

816 ( ) ( )

3

1.8 10 [ ]0.05

eA e

e

nuc

A N YhcY

( ) (0)( )inB n B

Conclusion:

B(in)(e) is no relative with temperature

19

3P2 neutron superfluid phase A

• interior in the neutron superfluid, in the vicinity of the Fermi surface, two neutrons with the opposite momentum cooperate into a 3P2 neutron cooper pair.

• every 3P2 neutron Cooper pair has spin =1, its projection onto the Z-direction has three values:

Z = -1, 0, +1 with the corresponding magnetic moment projection:

(μB)Z = 2 μn , 0, -2 μn 1. when the external magnetic field is very weak, the s

pin projection of every 3P2 neutron Cooper pair is in equal spin pair(ESP). Even deep in the neutron Fermi sea, the neutrons are also in ESP. Therefore, there is no net magnetic moment on the macro-scale, almost isotropic. We call it phase A of 3P2 neutron superfluid (similar to the liquid 3He-A).

20

3P2 netron superfluid phase B

2. under strong external magnetic field, the neutron superfluid in not in ESP. This is the case for both the 3

P2 netron Cooper pair and neutrons deep in the neut

ron Fermi sea. Therefore, the 3P2 netron superfluid o

btains a net magnetic momentum, anisotropic. We call it phase B of 3P2 netron superfluid (similarly to th

e liquid 3He-B).

Behavior of 3P2 neutron superfluid is similar to that of the B phase of the liquid 3He at very low temperature.

21

II. Pauli (induced) paramagnetic moment of a highly degenerate neutron system

22

statistical consideration• the 3P2 neutron Cooper pair system is a Bose system, it will condensa

te onto the ground state at low temperature. Every 3P2 neutron Cooper pair has magnetic momentum:

μB = 2 μn=1.9 ×10-23 ergs/gauss• Its projection onto the Z-direction is: [-Z2n] (Z=1,0,-1), where Z= -1 corresponds to paramagnetic. For a magnetic dipole, it will lie along the direction of the external

magnetic field. That is, the Z= -1 state has lower energy than the Z=1, 0 state.

(1) (1)2 / 2 /1 1

0 0

31 0 1 2

,

with ( ) (normalization)

n nB kT B kT

n

n ne e

n n

n n n n P

(1) (0) ( ) ( )inB B B e

23

Total number of 3P2 netron Cooper pair• number difference between the paramagnetic and

diamagnetic 3P2 neutron Cooper pairs is:(1)

31 1 1 2

4( )

3n

n

Bn n n n P

kT

322 ( )nm P

2 3 1/ 2 31/ 22 2

3

4 ( )[2 ( ( ))] ( ( ))3[ ]

( )(4 / 3)F n

FF

p n m P n P nq

E np

EF(n) ~ 200 MeV, (3P2(n)) ~ 0.05 MeV, q ~ 4.7% total number of 3P2 neutron Copper pair: N(3P2(n)) = q NA× m(3P2(n))/2

The 3P2 neutron Cooper pairs only exist in a small shell of the Fermi sphere with thickness of momentum

the paired neutrons is only a fraction of the total neutrons:

24

Induced magnetic moment• Total number difference between the paramagnetic

and diamagnetic 3P2 neutron Cooper is:

(1) (1)3 3

1 2 2

4 2( ) ( )

3 3n n

A

B BN N P qN m P

kT kT

(1)( ) 3 3

2 1 2

(1)

4( ) 2 ( )

3

conditions:

tot npair n A n

n

BP N qN m P

kT

B kT

The induced magnetic moment:

25

physical scenario• the magnetic projection of the 3P2 neutron Cooper pa

ir is almost stochastic distributed. The paramagnetic pairs is a little more than the diamagnetic pairs. And it is this small difference that caused the magnetic moment and anisotropic of the 3P2 neutron superfluid.

• in the case of a neutron star, we can say that, the strong surface magnetic field is caused by the 3P2 neutron superfluid which is not in ESP state (about 1 per 103). Finally, the total number of 3P2 neutron Cooper pairs is approximately 4.7% of the total number of neutrons.

26

Induced magnetic fields

Employing the dipole model of pulsars:

3 / 2NS p NSB R

3( ) 2

3

(1)(1) ( ) (0)

3( ) (1) 3212 12 ,6

7

2( )( )( , )

4( , ) ( )

3

( )1[ ]0.3

in Sun A n

Sun NS

inn

inNS

sun

m Nm PB q f T B Gauss

m R

Bf T B B B e B

kT

m PB AB A R

T m

27

Final result3

32,6

7

(in) 3 (1) ( ) (0)2

7 7

( ),

0.3

B ( ) ( ( ) )

NSSun

in

m PA R

T m

P B B e BT T

( ) 2 (0)( ) 1.8 10inB e B

( ) (0) ( ) ( ) 32( ) ( )tot in inB B B e B P

28

My ideaB(0): The fossil magnetic field from collapsed core of SN

B(0) is weaker than 1011 guass

B(1)(e) :

The induced magnetic field caused by the paramagnetic moment of highly relativistic degenerate electron gas of the neutron star

B(1)(e) 180 B(0) no relative with temperature

(in) 3 ( ) (0)2

7

332

,6

B ( ) ( ( ) )

( )

0.3

in

NSSun

P B e BT

m PR

m

29

ConclusionWhen T7>>1,

( ) 3 ( )2( ) ( )in inB P B e

( ) 3 ( )2( ) ( )in inB P B e

If η1, When T7<1

When T7<<1 ( ) 3 ( )2( ) ( )in inB P B e

Magnetars

30

conclosion

• The strong magnetic field of neutron stars originates from the induced field by the paramagnetic moment of electrons. It is much higher than the original one from collapsed progenitors.

• for a nascent neutron star, the interior temperature is very high, the parameter A is very small, so the induced magnetic fields by 3P2 is far less than the induced field by the paramagnetic moment of electronsas a neutron star cools, its temperature decreases, the parameter A will increase, and the induced magnetic fields will increase gradually.

31

magnetic field evolution of neutron star

• as soon as the interior temperature cools down to Tλ

=2.8×108K, there is a phase transition from normal Fermi liquid to 3P2 superfluid. at the same time, there is a change in the magnetic field, since the 3P2 neutron Cooper pair tends to lie along the direction of the fossil field.

• at high temperatures, the 3P2 Cooper pairs are in a chaos conditions. there is only a small fraction lie paramagneticly. as a neutron star cools, its interior temperature decrease, the 3P2 Cooper pairs that lie paramagneticly will increase exponentially.

32

magnetic field evolution of neutron star(2)

• as a neutron star cools down, two factors may affect the growth of the magnetic field

1. more and more 3P2 Cooper pairs tend to lie paramagneticly. It makes an increase in the induced magnetic moment , hence an increase in the induced magnetic field.

2. enlargement of the 3P2 superfluid region. total mass of the 3P2 neutron superfluid increase (see picture below).

33

3P2 neutron energy gap(Elgagøy et al.1996, PRL, 77, 1428-1431)

34

magnetic field evolution of neutron star(3)

• in the vicinity of original 3P2 superfluid region, as the temperature decreases, the phase transiton from normal Fermi liquid to 3P2 superfluid continues. The aniostropic superfluid region grows gradually, hence the induced magnetic moment will increase continuously, accompanied by a continuous increase of the induced magnetic field

• that’s the way to magnetars!

35

upper limit of a neutron star’s surface magnetic field

we consider here the zero temperatrue limit. In this case, the 3

P2 neutron Cooper pairs will all lie along the direction of the fossil field. Thus, we obtain an upper limit of the induced magnetic moment of the 3P2 nuetron superfluid region

The upper limit of the induced magnetic moment:

3( ) 3 3 32 2

2 2

( )( ) ( ) 1.63 10 [ ] ( . . .)

0.3uptot A n

Sun

m PP qN m P c g s

m

( ) 14max 3.26 10inB gauss

The corresponding upper limit of the induced magnetic moment field

36

upper limit of a neutron star’s surface magnetic field(2)

η 1 2 3 4 5

Bmax(in)(gauss) 3.3×1014 6.5×1014 0.98×1015 1.3×1015 1.7×1015

37

IV. pulsar’s spin down and age of pulsars

38

3PF2 neutron superfluid affects pulsar’s

spin down

3PF2 neutron superfluid normal neutron Fermi liquid phase tr

anstion， a dramatic change of a pulsar’s magnetic field:normal neutron Fermi liquid: B = B(0)

2 30 ,12

78 2

( ( ) )

( ) , 3.01 10 sin

p SF

SF

PP A B C G n P

nG n C

n

standard model standard model (magnetic dipole model)(magnetic dipole model) ,12 160.32pB PP Gauss

hybrid modelhybrid model(our model)(our model)

2 312 160.10 ( )SFB PP C G n P

3 (0)2 2( ) ( )SF SFC G n P B P C G n P 3PF2 nuetron superfluid(temperature belown 1.1×107K): B = B(induced) >> B(0)

39

Age of pulsars

0

2 (3 )0 12 0 0( ( ) )

P

SFP

PdPt

A B C G n P P

different from the standard dipole model!evolution of pulsar’s magnetic field：1. The fossil field B(0) < (109 -1011 ) gauss2. The strong magnetic field of neutron stars

originates from the induced field by the paramagnetic moment of electrons. It is much higher than the original one from collapsed progenitors.

40

3. when the temperature is less than Tλ,max(3P2)=2.78×108K,

3P2 neutron superfluid appears. when T7 >η, the induced magnetic field issmaller than the induced field by the paramagnetic moment of electrons. It may be neglectable. But when T7 <η , the induced magnetic fields exceed B(in)(e). we have to make modifications to calculate th

e age of pulsars, (see below).

41

Age of pulsars(2)

• where P1 is the pulsar’s spin period when its temperature T7 <η. B12(t) is time depending function, increase sharply as t increase.• conclusion: the standard dipole model should not be used to caculate the age of pulsars.

1

0 1

(0)2 (3 ) 2 (3 )0 12 0 0 0 12 0 0( ( ) ) ( ( ) ( ) )

P P

SF SFP P

PdP PdPt

A B C G n P P A B t C G n P P

0

20 12

P

P

PdPt

A B

42

Goal:

I wish to set up a united model of the neutron superfluid to explain many important observed phenomena of pulsars:

1)Origin of PSR kick? (2003)

2) Origin of Glitch for PSR(2006)

3) Origin of Magnetars?(2006)

43

4) On Some times pulsar and Null Pulse5) Slowing glitch pulsar6)Millisecond PSR: low magnetic field, no

glitch and lower space velocity. 7) LMXB with low magnetic field HMXB with high magnetic field Why?

44

Discussion

21( )

2 e

eh p A

m c

The electron system in a metal use to possesses a diamagnetic moment due to the electric charge of the electrons. The Harmiton of a electron in an external electro-magnetic field is

A

the electro-magnetic vector potential.

It is rather complicated , for we have to solve the Schödinger equation of the system. But I guess it is less than the paramagnetic moment of the electron system 。

45

Thanks!

46

B(in)/B(0) – T7(from left to right corresponds to =1,2,3,4,5,6

respectively)