1. Essential of Ultra Strong Magnetic field and Activity For Magnetars
Qiu-he Peng
(Dept. of Astronomy, Nanjing University, China)
< CSQCD II>, KIAA-PKU,Bejing, China, May 20-24, 2009

2. Observation Ultra strong magnetic field
For AXPs (Anormalous X-ray Pulsars)
and SGRs (Soft Gamma Repeaters )
 Long spin period and spin-down rate
 Absorption line at 10 keV

3. Activity of Magnetars Abnormal high X-ray luminosity For AXP
Decaying pulsating tails of giant flares with energy 1044 ergs
It can not be explained by loss of the rotational energy.
It is guessed by transformed from the energy of the ultra strong magnetic field of the magnetars.
Surface temperature (by observation):
T > 107 K for AXP
T~ K (for usual pulsars)
Why??

4. Proposed Models for the ultra strong B
Duncan & Thompson (1992, 1993): α-Ωdynamo with initial spin period less than 3ms
Ferrario & Wickrammasinghe(2005)suggest that the extra-strong magnetic field of the magnetars descends from their stellar progenitor with high magnetic field core.
Vink & Kuiper (2006) suggest that the magnetars originate from rapid rotating proto-neutron stars.
Iwazaki(2005)proposed the huge magnetic field of the magnetars is some color ferromagnetism of quark matter.
The question is still open!

5. Idea of mine: 1) Origin of Ultra strong B:
 Ferromagnetism of Anisotropic ( 3P2 ) neutron superfluid
2) Activity:
Instability due to high Fermi energy of electrons under the ultra strong B

6. Part I Origin of Ultra strong Magnetic Field Is from Ferromagnetism
of Anisotropic ( 3P2 ) neutron superfluid

7. Structure of a neutron star
= (g/cm3)
107
104
1011
Inner crust
nuclei with Extra-neutron rich
1014
5×1014
1S0 NSV (isotropic)
Core
(1km)
3P2NSV (anisotropic)
Protons (5-8)% ( Type II superconductor?)
(normal) electron Fermi gas
Quarks ??
NSV: Neutron superfluid vortices
outer crust
(crystal of heavy metal)

8. Two kinds of Neutron Cooper pairs
(Pauli matrix)
A neutron has a spin :
Vector addition :
Total spin of a Cooper pair of neutrons:
(1S0-Cooper pair of neutrons)
3P2-Cooper pair of neutrons:
(Projection along the external magnetic field)

9. 1S0 &3PF2 Neutron superfluid
(BEC based superfluidity )
1S0 neutron Cooper pair: S=0, isotropic
Energy gap :
△(1S0) ≥ 0, < ρ(g/cm3) < 1.4×1014
△(1S0)≥2MeV 7×1012 <ρ(g/cm3)< 5×1013
3P2 neutron Cooper pair: S =1, anisotropic,
abnormal magnetic moment ~10-23 c.g.s.
Energy gap:
△ n(3P2) ～0.05MeV
(3.31014 <  (g/cm3) < 5.21014)

10. Anisotropic 3P2 neutron superfluid
Critical temperature:
 Magnetic moment:
A magnetic moment tends to point at the direction of applied magnetic field with lower energy due to the interaction of the magnetic field with the magnetic moment of the 3P2 neutron Cooper pair.

11. Energy distribution of neutrons in the presence of a magnetic field

12. Statistics
Hamiltonian:
Ensemble average:
The Brillouin function .

13. Energy gap --- Combing energy of a Cooper pair
A key idea: The energy gap, Δ, is a combining energy of couple of of neutrons (the Cooper pair). It is a real energy, rather than the variation of the Fermi energy due to the variation of neutron number density.
Corresponding momentum of the combing energy of the neutron Cooper pair is (in non-relativity)

14. q Value
How many neutrons have been combined into the 3P2 Cooper pairs?
Since only particles in the vicinity of the Fermi surface contribute (Lifshitz et al. 1999), there is a finite probability q for two neutrons being combined into a Cooper pair.

15. Total induced magnetic field by the 3P2 superfluidor

16. From paramagnetism to ferromagnetism
(B < 1013 gauss , T > 107 K )
Set 
(Curea Temparature)
Paramagnetism is not important when T>Tc
Phase Transition From paramagnetism to ferromagnetism
When T down to T→Tc and the induced magnetic is very
strong 

18. Increase of magnetic field of NS
The induced magnetic field for the anisotropic neutron superfluid increases with decreasing temperature due to More and more neutron 3P2 Copper pairs transfer into paramagnetic states.
b) The region and then mass of anisotropic neutron superfluid is increasing with decreasing temperature

19. Energy gap of the 3P2 neutron pair (Elgagøy et al
Energy gap of the 3P2 neutron pair (Elgagøy et al.1996, PRL, 77, )

20. The up limit of the magnetic field for magnetars
Conclusion:
All assumptions with B > 31015 gauss are unrealistic.

21. Part II: are Caused by The increase of Fermi energy of electrons
Instability and activity of magnetars
are Caused by
The increase of Fermi energy of electrons
under ultra strong magnetic field

23. Landau quantization pz
n=5 p
n=6
n=3
n=2
n=1 
n=4
n=0

24. The radius of its cross section is p .pz p
The overwhelming majority of neutrons congregate in the lowest levels n=0 or n=1,
When
The Landau column is a rather long cylinder along the magnetic filed, but it is very narrow.
The radius of its cross section is p .

25. Under the ultra strong magnetic field
The Landau energy level is quantized when B  Bcr
( Bcr =4.4141013 gauss)
Landau column pz p
n: quantum number of the Landau energy level
n=0, 1,2,3……

26. Total Occupied number of electrons(1)
We may calculate the total occupied number by two different ways:
1) Total electron number density:
Ye: electron fraction
The total number of state occupied by the electrons in a complete degenerate electron gas should be equal to the electron number due to the Pauli’s Exclusion Principle

27. Total Occupied number of electrons(2)
2) Method in Statistical physics:
State number in a volume element of phase space:
Landau quantization

29. Cont.

30. Result

31. Fermi energy of electrons in ultra strong magnetic field

32. B (1014 Gauss) b
EF (MeV)
1.0
2.415
49.86
3.0
7.246
65.63
5.0
12.077
74.57
10.0
24.155
88.68
15.0
36.232
98.14
20.0
48.309
105.45

33. Magnetars are unstable due to the ultra high Fermi energy of electrons

34. Basic idea
When the energy of the electron near the Fermi surface is rather high (EF>60 MeV)
Energy of the resulting neutrons will be rather high and they will react with the neutrons in the 3P2 Cooper pairs and will destroy these 3P2 Cooper pairs .
It will cause the anisotropic superfluid disappear and then the magnetic field induced by the 3P2 Cooper pairs will also disappear.

35. Total thermal energy
Average energy of outgoing neutrons after breakdown of the 3P2 Cooper pairs
It will be transfer into thermal energy.
Total thermal energy will be released after all 3P2 Cooper being breaked up

36. Life time of magnetar activity
X ray luminosity of AXPs
It may be maintained ~ yr

37. Phase Oscillation Afterwards,
Revive to the previous state just before formation of the 3P2 neutron superfluid.
 Phase Oscillation .

38. Questions? Detail process: The rate of the process Time scale ??
2. What is the evolution of the previous instable process
of the magnetars?
3. What is the real maximum magnetic field of the magnetars?
4. How long is the period of oscillation above?
5. How to compare with observational data
6. Estimating the appearance frequency of AXP and SGR ?
A further work of mine:
“Heating by 3P2 Neutron Superfluid Votexes and Glitch of Pulsars Due to the Oscillation between A and B phase of Anisotropic superfluid”. (It Will be contributed to the conference on <Pulsars> next year)

39. Thanks