Relation of Electron Fermi Energy With Magnetic Field in Magneta

rs and their x-ray Luminosity

Qiu-he Peng(Nanjing University )

Some published relative works

gaussergn /10966.0~ 23

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)Observed 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, USAIt is found that super strong magnetic fields of magnetars is come from one induced by paramagnetic moment for anisotropic 3P2 neutron Cooper pairs

20( ) ~ 0.927 10 /B e erg gauss

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

Landau quantization 2 2 4 2 2 2 2

zE m c p c p c

2( ) (2 1 )e

p n bm c

n=0n=1n=4

n=3 n=2n=5

n=6

pz

p

/ crb B B

Landau Column

2( ) (2 1 )e

p n bm c

2 2 4 2 2 2 2zE m c p c p c

/ crb B B

pz

p

Landau column

In the case B > Bcr

The overwhelming majority of neutrons congregates in the lowest levels n=0 or n=1,

When

crB B

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 .

p

pz

(Landau coulumn)

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 n=0,1,2,3… For a given pz (pz is still continuous) ， there is a maximum orbital quantum number nmax(pz,b,σ)≈nmax(pz,b). In strong magnetic fields, an envelope of these Landau cycles with maximum orbital quantum number nmax(pz,b,σ) (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 n=0,1,2,3…nmax (see expression below). The number of states in the x-y plane will be much less than on

e 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 Pauli’s 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 .

2 2max 2

1( , , 1) { [( ) 1 ( ) ]}2

F zz

e e

E pn p b Intb m c m c

2max( ) (2 1 ) 0,1,2,3..... ( , , )z

e

p n b n n p bm c

2 2max 2

1( , , 1) { [( ) 1 ( ) ] 1}2

F zz

e e

E pn p b Intb m c m c

max max max

2 2max 2

( , , 1) ( , , 1) ( , )1( , ) [( ) 1 ( ) ]2

z z z

F zz

e e

n p b n p b n p bE pn p b

b m c m c

Landau ColumnThe overwhelming majority of neutrons congregates in the lowest levels n=0 or n=1, when

( 1)crB B b

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 .

p

pz

i.e. EF （ e ） is increasing with increase of magnetic field in strong magnetic fieldWhat 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 commona) Dong Lai, S.L. Shapiro, ApJ., 383(1991) 745-761 b): Dong Lai, Matter in Strong Magnetic Fields

(Reviews of Modern Physics>, 2001, 73:629-661)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 )

)()(12 232

33 cm

pdcm

ppdh eeee

(2.4

)b) For the case with strong magnetic field:

)()2(/

322 cmpdgBBdpg

cheB

e

z

e

cz

e

12Ln

(2.5)

g : degeneracy for spin g = 1, when =0

g = 2 when 1

/e em c Compton wave length of an electron

nL : Quantum number of Landau energy

）（ h

dphceBdp

hdpdp

hpd

hz

zyx111

23

3

Paper a): continue

Number density of electrons in the case T →0

0

2 ( )m

Fe e

eBn g phc h

(2.6)

( ) 0Fep （ ）

It is the maximum of momentum of the electron along z-directionWith a given quantum number

2 2 2 2 2( ) (1 2 )Fe e e

c

Bp v c m cB

e : chemical potential of electrons ( i.e. Fermi energy).The up limit, m , of the sum is given by the condition following

2[ ( )] 0Fep

2 2 4 (1 2 )e ec

Bm cB

(2.7)

(2.8)

Paper b): Matter in Strong Magnetic Fields (Reviews of Modern Physics>, 2001, 73:629-661)

§VI. Free-Electron Gas in Strong Magnetic Fields (p.647) 中 :

200

1 (6.1)(2 ) L

L

e n zn

n g fdp

1[1 exp( )] 6.2eE

fkT

（ ）

2 2

200

1 6.3(2 ) L

L

ze n z

n

p cP g f dpE

（ ）

2 2 2 4 1/ 2[ (1 2 )] (2.12)z e Lc

BE c p m c nB

1/ 20 ) 2.5c

eB

（ （ ） 2 2

0

1 22

eBh c

（ ）（ ）

The pressure of free electrons is isotropic.ρ0 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 @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

2 2122.67 ( ) ( ) (6.14)F

F e BET B Y K fork

27 3/ 2 3122 3

0

1 4.24 102B A BN n B cm

1/ 2 10 1/ 20 12) 2.5656 10c B cm

eB

（ （ ）

3 3/ 2 3127.04 10 /B B g cm

磁场增强，电子的 Fermi 能降低。磁场降低了电子的简并性质。当 ρ>>ρB 情形下，磁场对电子影响很小。 这个结论同我们对强磁场下 Landau 能级量子化的图象不一致！为什么 ?

Query on these formula and

looking into the causes

Landau theory (non -relativity) By solving non-relativistic Schrödinger equation with magnetic field( Landau & Lifshitz , < Quantum Mechanism> §112 (pp. 458-460 ))1 ） electron energy (Landau quantum) ：

2( 1/ 2 ) / 2B z eE n p m

2 ） The state number of electrons in the interval pzpz+dpz is

24zdpeB

c

/B ee B m c

ωB : Larmor gyration frequency of a non relativistic electron in magnetic field

2B e B 2e

e

em c

Relativistic Landau Energy in strong magnetic field

2 22 2

2

2( ) ( , , , ) 1 ( ) (2 1 )

1 ( ) (2 1 )

ezz

e e e

z

e

BpE p B n nm c m c m c

p n bm c

n: quantum number of the Landau energy level n=0, 1,2,3……( 当 n = 0 时 , 只有 σ = -1)

20e ~ 0.927 10 / serg gaus

2

2 1e cr

e

Bm c

/ crb B B

(Bohr magnetic moment of the electron)

Landau quantum in strong magnetic field

In the case EF >>mec2 , Landau energy is by solving the relativistic Dirac equation in magnetic field

2134, 414 10

2e

cre

m cB gauss

For the non-relativistic case

2( )4

zphase z z

dpeBN p dpc

The expression (A) is derived by solving the non relativistic gyration movement

2 2 3 ( ) 1 when )Bcr

cre e

eB B B BBm c m c

（

The state number of electrons in the interval pzpz+dpz

1( )FE e B

It is contrary with our idea that the Fermi energy will increases with increasing magnetic field of super strong magnetic field

It should be revised for the case of super strong magnetic field

(A)

eAeF

p

zzphasephase YNncEeBdppNN

F

20

24)(

(Due to Pauli Principle)

(It is relativistic gyration movement)

Result in some text-book(Pathria R.K., 2003, Statistical

Mechanics, 2nd edn. lsevier,Singapore)

2 12 2 2

41 1 n Bx y n

m Bdp dp ph h h

n

n+1

The result is the same with previous one for the non relativistic caseIt is usually referenced by many papers in common.

cme

eB 2

The state number of electrons in the interval pzpz+dpz along the direction of magnetic field

(B)

My opinion There is no any state between p(n) - p(n+1) according to the idea of Landau quantization. It is inconsistent with Landau idea. In my opinion, we should use the Dirac’ - function to represent Lan

dau quantization of electron energy

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

）（ 5.2)()2(/

322

cm

pdgBBdpgch

eB

e

z

e

cz

e

200

1 (6.1)(2 ) L

L

e n zn

n g fdp

2 2

200

1 6.3(2 ) L

L

ze n z

n

p cP g f dpE

（ ）

And some eq. in paper b):

1/ 2 10 1/ 20 12) 2.5656 10c B cm

eB

（ （ ）

The authors quote eq. (B) above .Besides, the order : 1) to integral firstly 2) to sum thenIt 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

3

1phase x y zN dxdydzdp dp dp

h

max

max

/ ( , , 1)3

000

( , , 1)

01

2 ( ) ( ){ ( 2 )( ) ( )

( 2( 1) )( ) ( )}

F e z

z

p m c n p be z

phasene e e e

n p b

n e e e

m c p p p pN d g nb dh m c m c m c m c

p p pg n b dm c m c m c

The microscopic stae number (in an unit volume) is

g0 is degeneracy of energy

continue

max max

30

/ ( , , 1) ( , , 1)

0 10

2 ( )

( ){ 2 2( 1) }F e z z

ephase

p m c n p b n p bz

n ne

m cN gh

pd nb n bm c

2 2max 2

1( , , 1) { [( ) 1 ( ) ]}2

F zz

e e

E pn p b Intb m c m c

2 2max 2

1( , , 1) { [( ) 1 ( ) ] 1}2

F zz

e e

E pn p b Intb m c m c

max max max

2 2max 2

( , , 1) ( , , 1) ( , )1( , ) [( ) 1 ( ) ]2

z z z

F zz

e e

n p b n p b n p bE pn p b

b m c m c

In super strong magnetic field/7 / 2

1/ 2 3 3/ 2 2 2 3/ 20 2

0

2 1( ) ( ) [( ) 1 ( ) ] ( )3 2

F ep m ce F z z

phasee e e

m c E p pN b g dh b m c m c m c

3 40 2

4 ( ) ( )3

e Fphase

e

m c EN g Ib h m c

其中 I 为一个具体数值。1

2 3/ 2

0

(1 )I t dt

The state density of electrons in super strong magnetic field

3 2 2 3/ 2e 0 2 2 2

4 1( ) [( ) ( ) ]3

e F

e e e

m c E Egb h m c m c m c

Principle of Pauli’s 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.

phase A eN N Y

Relation between Fermi energy of electrons and magnetic field

( ) 60FE e MeV

3 40 2

4 ( ) ( )3

e Fphase e A e

e

m c EN g I N N Yb h m c

1 14 4

2 [ ]0.05

eF

e nuc

YE C bm c

（b>1)）1/ 4

076.69C g

For the case with lower magnetic field in NS

1/ 4( ) 60( ) ( )F crcr

BE e MeV B BB

IVActivity of magnetars and their high x-ray luni

nocity

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 ?

( 短时标 )sergsLx /1010~ 4342

sec/1010 3634 ergsLx

Basic ideaElectron capture by protons will happen

Bn

2

ee p n

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.

( , )n n n n n n It makes the induced magnetic field by the magnetic moment of the 3

P2 Cooper pairs disappearing , and then the magnetic energy the magnetic moment of the 3P2 Cooper pairs

When the magnetic field is more strong than Bcr and then

MeVnEeE FF 60)()(

Will be released and then will be transferred into x-ray radiation

keVBBkT n 1510

Total Energy may be released

ergs1.0

)(1012)( 23

472

3

SunnA m

PmBPmqNE

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

34 3610 10 / secxL ergs

电子俘获速率在 1 秒钟內 , 一个能量为 Ee的电子被一个能量为 Ep的质子俘获 , 出射中乀微子的能量为 Eν (出射中子的能量为En)的事件的几率 ( 即速率 ) 为 :

enpe

)()1)(31(C2 222eVF EQEdEfaG

hd

32

2

)(2])([

hccmmEEE pnnpe

其中， fν 为中微子的 Fermi 分布函数。电子俘获的能阈值 Q 和中微子的能级密度 ρν 分別为2)( cmmEEQ pnpn

其中 En 、 Ep 分别为中子与质子的非相对论能量。 CV, CA 分别是 Wemberg-Salam 弱电统一理论中的矢量耦合与轴矢量耦合系数

0.9737; 1.253AV

V

CC aC

电子俘获过程产生的 x-光度

dBdL nx 2

由于每一次电子俘获过程的出射自由中子能量明显超过了中子的 Fermi 能 ( 它远远超过 3P2 Cooper 对 , 这个出射的高能中子立即摧毁一个 3P2 Cooper 对 ( 几率为η, η <<1), 同时将这个 3P2 Cooper 对的磁矩能量释放出来，转化成热能，以 x-ray 形式发射出来。上述每一次电子俘获过程产生的 x-ray 光度为x-ray 总光度为 :

《 》

其中， 为热能转化为辐射能的效率 ( <<1); <θ>为 x-ray 从中子星内部转移到表面的辐射透射系数 (<θ> <<1)

43 2 2 2

21

3 3 3 3 3

(2 )( ) (1 3 )

( 0.61 ) ( ) 2

x F V

e p n n e f i n

L V P G G aV

d n d n d n d n E E MeV E k k S B

续( ) ( )[1 ( )][1 ( )]e e p p n nS f E f E f E f E

1( ) [exp( ) / 1]j jf E E kT （ j 是粒子 j 的化学势 )

在中子星内部，能量不太高的中微子几乎透明地不受任何阻拦而逸出，可近似取 : 1 ( ) 1f E ) 1 ( )) 0 ( )

e e e F

e e e F

f E when E E ef E when E E e

（（

1 ( ) 0 ( )1 ( ) 1 ( )

n n n F

n n n F

f E when E E nf E when E E n

) 1 ( )

) 0 ( )p p p F

p p p F

f E when E E p

f E when E E p

（

（

14( ) 60( / ) MeVF crE e B B

能级状态数d3nj 为单位体积内粒子 j 的微观状态数。 3

31 3

jj j j j

d pd n V g dE

h

在超强磁场下 ,电子气体的能级态密度为3 2 2 3/ 2

e 0 2 2 2

4 1( ) [( ) ( ) ]3

e F

e e e

m c E Egb h m c m c m c

ρj 为第 j 种粒子的能级态密度。

2

2 3 3

( )12

eQ Ec

3/ 2 3/ 2 3/ 2 3/ 23 3

8 8( ) ( )n n F p p Fm E n m E ph h

2( )n p n pQ E E m m c

关于参量 ζ目前我们不知道η(它应该由凝聚态物理和核物理联合来计算 )、

和 <θ> 的数值，仅仅知道 , ζ<<1 。在这项工作中只能当作可调参量。在实际计算中，将 ζ 当作待定参量，由对某个 B 值计算出的 LX 同观测值比较后来估计 ζ 的大小。再由此确定的 ζ 值来计算其它 B值对应的 LX , 再去同观测对比。

理论计算结果与观测对比

红色圆圈代表 SGR( 软 γ 重复暴 ), 兰色方块代表 AXP( 反常 X-ray 脉冲星。最左边远高于理论曲线的 3个 AXP 己发现有明显的吸积 (密近双星系统 )在磁场较高时 3个理论模型 (α=0,0.5,1.0) 曲线趋于一致。(计算中 ξ 值选取为 3×10-17)

Phase Oscillation

Afterwards,

en p e Revive to the previous state just before formation of the 3P2 neutron superfluid. Phase Oscillation .

Questions?1. Detail process: The rate of the process ee p n

Time scale ??

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

3. How long is the period of oscillation above?

4. How to compare with observational data

5. Estimating the appearance frequency of AXP and SGR ?

磁星 Flare 与 Burst 的活动性1) 彭秋和 (2010): 中子星内 3P2中子超流涡旋的磁偶极辐射的加热机制与 3P2中子超流体 A相 -B 相震荡触发脉冲星的 Glitch2) 内部超流体带动中子星壳层物质突然加快引起物质较差自转、导致磁力线扭曲和磁重联将磁能释放转化为突然能量释放引起磁星 Flare与 Burst 的活动性 (正在构思的探讨中 )

中子星 (脉冲星 )的主要疑难问题1) 高速中子星的物理原因 ?(2003)2) 中子星强磁场 (1011-13 gauss) 的起源 ?(2006)3) 磁星 (1014-15 gauss) 及其活动性的物理本质 ?(2009-2010)4) 年轻脉冲星周期突变 (Glitch)现象的物理本质 ?(2010)5)缺脉冲 (Null-pulse) 和 Some times pulsars现象6) 低质量 X-双星 (LMXB) 内的中子星磁场很低 ; 高质量 X-双星 (HMXB) 内的中子星磁场很强。为什么 ?7)毫秒脉冲星重要特性 : 低磁场 , 无 Glitch, 空间速度不高 , 物理原因 ?我们的目标 : 统一解释的脉冲星的主要观测现象8) 脉冲星射电 (X-ray, -ray) 辐射机制 ? 辐射产生区域 ?9) 是否存在 (裸 )奇异 (夸克 ) 星 ?

谢谢大家