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Relation of Electron Fermi Energy With Magnetic

Field in Magnetars and their x-ray Luminosity

Qiu-he Peng (Nanjing University )

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

Behavior of electron gas under strong magnetic

field

- Question How is the relation of electron Fermi

energy with magnetic field?

Landau Column under strong magnetic field

(No Transcript)

Landau quantization

pz

n5

p?

n3

n2

n6

n1

?

n0

n4

Landau Column

Landau column

pz

p?

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? .

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.

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.

- 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 .

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)

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)

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

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)

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

Results in these papers

- For non-relativistic degenerate electron gas with

lower density

????,???Fermi????????????????? ??gtgt?B

???,?????????? ????????????Landau???????????! ???

?

Query on these formula and looking into the

causes

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

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)

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)

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.

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

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.

Our method

The microscopic stae number (in an unit volume) is

g0 is degeneracy of energy

continue

In super strong magnetic field

The state density of electrons in super strong

magnetic field

??I????????

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.

Relation between Fermi energy of electrons and

magnetic field

?

For the case with lower magnetic field in NS

IVActivity of magnetars and their high x-ray

luninocity

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 ?

(???)

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

Total Energy may be released

It may take 104 -106 yr for x-ray luminocity

of AXPs

??????

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

??,f??????Fermi????? ????????Q??????????????

??En ?Ep ????????????????CV, CA ???Wemberg-Salam

????????????????????

?????????x-??

????????????????????????????Fermi?(?????3P2

Cooper?,???????????????3P2 Cooper?(????, ?

ltlt1),?????3P2 Cooper??????????,?????,?x-ray???????

??????????????x-ray???

x-ray ????

??,? ???????????? (? ltlt1) lt?gt?x-ray????????????

??????(lt?gt ltlt1)

?

(?j ???j ????)

??????,?????????????????????? ?,????

?????

- d3nj ???????? j ???????

?j ??j ??????????

??????,???????????

?????

- ????????(??????????????????)?? ?lt?gt???,????,

?ltlt1????????????????

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

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

???????????

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

(????????310-17)

Phase Oscillation

- Afterwards,

Revive to the previous state just before

formation of the 3P2 neutron superfluid. ? Phase

Oscillation .

Questions?

- Detail process
- The rate of the process

- Time scale ??

2. What is the real maximum magnetic field of the

magnetars?

- How long is the period of oscillation above?

4. How to compare with observational data

5. Estimating the appearance frequency of AXP and

SGR ?

??Flare?Burst????

- 1)???(2010)????3P2??????????????????3P2?????A?-B?

????????Glitch - 2)????????????????????????????????????????????????

????????Flare?Burst????(????????)

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

- 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)????(?)??(??)??

????