Relation of Electron Fermi Energy With Magnetic Field in Magnetars and their x-ray Luminosity

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Relation of Electron Fermi Energy With Magnetic
Field in Magnetars and their x-ray Luminosity
Qiu-he Peng (Nanjing University )
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Some published relative works
1) Qiu-he Peng and Hao Tong, 2007, The Physics
of Strong magnetic fields in neutron stars,
Mon. Not. R. Astron. Soc. 378, 159-162(2007) Obser
ved strong magnetic fields (1011-1013 gauss) is
actually come from one induced by Pauli
paramagnetic moment for relativistic degenerate
electron gas
2) Qiu-he Peng Hao Tong, 2009,
The Physics of Strong Magnetic Fields and
Activity of Magnetars Procedings of Science
(Nuclei in the Cosmos X) 189, 10th Symposium On
Nuclei in the Cosmos, 27 July 1 August 2008,
Mackinac Island, Michigan, USA It is found that
super strong magnetic fields of magnetars is come
from one induced by paramagnetic moment for
anisotropic 3P2 neutron Cooper pairs
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Behavior of electron gas under strong magnetic
field

• Question How is the relation of electron Fermi
  energy with magnetic field?

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Landau Column under strong magnetic field
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(No Transcript)
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Landau quantization
pz
n5
p?
n3
n2
n6
n1
?
n0
n4
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Landau Column
Landau column
pz
p?
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In the case B gt Bcr
pz
p?
(Landau coulumn)

• The overwhelming majority of neutrons
  congregates in the lowest levels n0 or n1,
• When

The Landau column is a very long cylinder along
the magnetic filed, but it is very narrow. The
radius of its cross section is p? .
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Fermi sphere in strong magnetic field

• Fermi sphere without magnetic field
• Both dpz and dp? chang continuously.
• the microscopic state number in a volume element
  of phase space d3x d3p is d3x d3p /h3.

Fermi sphere in strong magnetic field
along the z-direction dpz changes continuously.
In the x-y plane, electrons are populated on
discrete Landau levels with n0,1,2,3 For a
given pz (pz is still continuous),there is a
maximum orbital quantum number nmax(pz,b,s)nmax(p
z,b). In strong magnetic fields, an envelope of
these Landau cycles with maximum orbital quantum
number nmax(pz,b,s) (0 ? pz ? pF ) will
approximately form a spherical sphere, i.e. Fermi
sphere.
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Behavior of the envelope Fermi sphere under
ultra strong magnetic field

• In strong magnetic fields, things are different
• along the z-direction dpz changes continuously.
• In the x-y plane, however, electrons are
  populated on discrete Landau levels with
  n0,1,2,3nmax (see expression below).
• The number of states in the x-y plane will be
  much less than one without the magnetic field.
  For a given electron number density with a highly
  degenerate state in a neutron star, however, the
  maximum of pz will increase according to the
  Paulis exclusion principle (each microscopic
  state is occupied by an electron only). That
  means the radius of the Fermi sphere pF being
  expanded. It means that the Fermi energy EF also
  increases.
• For stronger field,nmax(pz,b) is lower,there will
  be less electron in the x-y plane. The
  expansion of Fermi sphere is more obvious along
  with a higher Fermi energy EF.

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• Majority of the Fermi sphere is empty, without
  electron occupied,
• In the x-y plane, the perpendicular momentum of
  electrons is not
• continue, it obeys the Landau relation .

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Landau Column
pz
p?

• The overwhelming majority of neutrons
  congregates
• in the lowest levels n0 or n1, when

The Landau column is a very long cylinder along
the magnetic filed, but it is very narrow. The
radius of its cross section is p? . More the
magnetic filed is, more long and more narrow the
Landau column is .
i.e. EF(e)is increasing with increase of
magnetic field in strong magnetic field What is
the relation of EF(e) with B ? We may find it by
the Pauli principle Ne (number density of
state) ne (number density of electron)
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An another popular theory

• Main idea electron Fermi Energy decreases with
  increasing magnetic field
• Typical papers referenced by many papers in
  common
• a) Dong Lai, S.L. Shapiro, ApJ., 383(1991)
  745-761
• b) Dong Lai, Matter in Strong Magnetic Fields
  (Reviews of Modern Physicsgt, 2001, 73629-661)
• c) Harding Lai , Physics of Strongly
  Magnetized Neutron Stars.
• (Rep. Prog. Phys. 69 (2006) 2631-2708)

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Paper a) (Dong Lai, S.L. Shapiro,ApJ.,383(1991)
745-761 )

• Main idea (p.746)
• a) In a case no magnetic field ( for an unit
  volume )

(2.4)
b) For the case with strong magnetic field
(2.5)
nL Quantum number of Landau energy
Compton wave length of an electron
g? degeneracy for spin g? 1, when ? 0
g? 2 when ? ? 1
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Paper a) continue

• Number density of electrons in the case T ?0

(2.6)
It is the maximum of momentum of the electron
along z-direction With a given quantum number
(2.7)
?e chemical potential of electrons ( i.e.
Fermi energy). The up limit, ?m , of the sum is
given by the condition following
(2.8)
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Paper b) Matter in Strong Magnetic Fields
(Reviews of Modern Physicsgt, 2001, 73629-661)

• VI. Free-Electron Gas in Strong Magnetic Fields
  (p.647)?

The pressure of free electrons is isotropic. ?0
is the radius of gyration of the electron in
magnetic field
Note 1 nL in paper b) is ? in paper a) really
Note 2 in the paper c) (Harding _at_Lai ,
2006Rrp. Prog. Phys.69 2631-2708), (108)-(110)
in 6.2 (p.2669) are the same as (6.1)-(6.3)
above
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Results in these papers

• For non-relativistic degenerate electron gas with
  lower density

????,???Fermi????????????????? ??gtgt?B
???,?????????? ????????????Landau???????????! ???
?
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Query on these formula and looking into the
causes
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Landau theory (non -relativity)

• By solving non-relativistic Schrödinger equation
  with magnetic field
• ( Landau Lifshitz , lt Quantum Mechanismgt 112
  (pp. 458-460 ))
• 1)electron energy (Landau quantum)

?B Larmor gyration frequency of a non
relativistic electron in magnetic field
2)The state number of electrons in the interval
pz?pzdpz is
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Relativistic Landau Energy in strong magnetic
field
In the case EF gtgtmec2 , Landau energy is by
solving the relativistic Dirac equation in
magnetic field
(Bohr magnetic moment of the electron)
Landau quantum in strong magnetic field
n quantum number of the Landau energy level
n0, 1,2,3(?n 0 ?, ??s -1)
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For the non-relativistic case
The state number of electrons in the interval
pz?pzdpz
(A)
(Due to Pauli Principle)
?
It is contrary with our idea that the Fermi
energy will increases with increasing magnetic
field of super strong magnetic field
The expression (A) is derived by solving the non
relativistic gyration movement
It should be revised for the case of super
strong magnetic field
(It is relativistic gyration movement)
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Result in some text-book(Pathria R.K., 2003,
Statistical Mechanics, 2nd edn.
lsevier,Singapore)
n1
The state number of electrons in the interval
pz?pzdpz along the direction of magnetic field
n
(B)
The result is the same with previous one for the
non relativistic case It is usually referenced by
many papers in common.
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My opinion

• There is no any state between p?(n) - p?(n1)
  according to the idea
• of Landau quantization. It is inconsistent with
  Landau idea.
• In my opinion, we should use the Dirac ? -
  function to represent Landau quantization of
  electron energy

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More discussion

• The eq. (2.5) in the previous paper a) (in strong
  magnetic field)

And some eq. in paper b)
The authors quote eq. (B) above . Besides, the
order 1) to integral firstly 2) to sum then It
is not right order. In fact, to get the Landau
quantum number nL , we have to give the momentum
pz first, rather than giving nL first. The two
different orders are different idea in physics.
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Our method
The microscopic stae number (in an unit volume) is
g0 is degeneracy of energy
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continue
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In super strong magnetic field
The state density of electrons in super strong
magnetic field
??I????????
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Principle of Paulis incompatibility

• Pauli principle
• The total number states ( per unite volume)
  occupied by the electrons in the complete
  degenerate electron gas should be equal to the
  number density of the electrons.

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Relation between Fermi energy of electrons and
magnetic field
?
For the case with lower magnetic field in NS
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IVActivity of magnetars and their high x-ray
luninocity
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Question

• What is the mechanism for very high x-ray
  luminosity of magnetars ?

What is the reason of x-ray flare or
of x-ray Burst for some magnetars ?
(???)
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Basic idea

• Electron capture by protons will happen

When the magnetic field is more strong than Bcr
and then
Energy of the outgoing neutrons is high far more
than the Binding energy of a 3P2 Cooper. Then the
3P2 Cooper pairs will be broken by nuclear
interact with the outgoing neutrons.
It makes the induced magnetic field by the
magnetic moment of the 3P2 Cooper pairs
disappearing , and then the magnetic energy the
magnetic moment of the 3P2 Cooper pairs
Will be released and then will be transferred
into x-ray radiation
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Total Energy may be released
It may take 104 -106 yr for x-ray luminocity
of AXPs
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??????

• ?1???,?????Ee?????????Ep?????,???
• ???????E? (????????En)??????(???)?

??,f??????Fermi????? ????????Q??????????????
??En ?Ep ????????????????CV, CA ???Wemberg-Salam
????????????????????
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?????????x-??
????????????????????????????Fermi?(?????3P2
Cooper?,???????????????3P2 Cooper?(????, ?
ltlt1),?????3P2 Cooper??????????,?????,?x-ray???????
??????????????x-ray???
x-ray ????
??,? ???????????? (? ltlt1) lt?gt?x-ray????????????
??????(lt?gt ltlt1)
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?
(?j ???j ????)
??????,?????????????????????? ?,????
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?????

• d3nj ???????? j ???????

?j ??j ??????????
??????,???????????
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?????

• ????????(??????????????????)?? ?lt?gt???,????,
  ?ltlt1????????????????

??????,????????,????B?????LX ???????????
?????????????????B????LX ,????????
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???????????

• ??????SGR(?????), ??????AXP(??X-ray?
• ??????????????3?AXP?????????(??????)
• ??????3?????(a0,0.5,1.0)???????

(????????310-17)
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Phase Oscillation

• Afterwards,

Revive to the previous state just before
formation of the 3P2 neutron superfluid. ? Phase
Oscillation .
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Questions?

• Detail process
• The rate of the process

• Time scale ??

2. What is the real maximum magnetic field of the
magnetars?

1. How long is the period of oscillation above?

4. How to compare with observational data
5. Estimating the appearance frequency of AXP and
SGR ?
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??Flare?Burst????

• 1)???(2010)????3P2??????????????????3P2?????A?-B?
  ????????Glitch
• 2)????????????????????????????????????????????????
  ????????Flare?Burst????(????????)

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???(???)???????

• 1)???????????(2003)
• 2)??????(1011-13 gauss)????(2006)
• 3) ??(1014-15 gauss)???????????(2009-2010)
• 4)?????????(Glitch)????????(2010)
• 5)???(Null-pulse)?Some times pulsars??
• 6)???X-??(LMXB)?????????
• ???X-??(HMXB)??????????????
• 7)?????????
• ???, ?Glitch, ??????, ?????
• ????? ???????????????
• 8) ????? (X-ray, ?-ray)????? ???????
• 9)????(?)??(??)??

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????