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)
Observed strong magnetic fields (1011-1013 gauss) is actually
come from one induced by Pauli paramagnetic moment
for relativistic degenerate electron gas
 B (e) ~ 0.927  1020
erg / gauss
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
3P neutron Cooper pairs
2
n ~ 0.966  1023
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
pz
E 2  m 2 c 4  pz2 c 2  p2 c 2
n=6
n=5
n=4
p 2
(
)  (2n  1   )b
me c
b  B / Bcr
n=3 n=2
n=1
n=0
Landau
quantization
p
Landau Column
E m c  p c  p c
2
2 4
2 2
z
2 2

p 2
(
)  (2n  1   )b
me c
b  B / Bcr
Landau column
pz
p
In the case B > Bcr
(Landau coulumn)
The overwhelming majority of
neutrons congregates in the lowest
levels n=0 or n=1,
When
B  Bcr
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 .
pz
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 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
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 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 .
p 2
(
)  (2n  1   )b
me c
nmax ( pz , b,   1)  Int{
n  0,1, 2,3.....nmax ( pz , b,  )
E
p
1
[( F 2 ) 2  1  ( z ) 2 ]}
2b me c
me c
EF 2
pz 2
1
nmax ( pz , b,   1)  Int{ [(
) 1 (
) ]  1}
2
2b me c
me c
nmax ( pz , b,   1)  nmax ( pz , b,   1)  nmax ( pz , b)
EF 2
pz 2
1
nmax ( pz , b)  [(
) 1  (
) ]
2
2b me c
me c
pz
Landau Column
The overwhelming majority of neutrons congregates
in the lowest levels n=0 or n=1, when
B  Bcr
(b  1)
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)
p
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 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 )

e

2
1
p 2
p
3
d
p

(
)
d
(
)
h3 
 2 3e  me c
me c
(2.4)
b) For the case with strong magnetic field:
eB
g  dp
 
h
c

2
z
e

B / Bc
p
g d( z )
2 3  
(2 ) e 
me c
(2.5)
1
1
1
eB dp z
3
（ 3  d p  2  dp x dp y  dp z 
）

h
h
h
hc h
1
2
  nL   
nL : Quantum number of Landau energy
e  / mec
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
m
eB 2 F
g pe ( )
h
  0 hc
ne  
(2.6)
peF ( ) （  0）
It is the maximum of momentum of the electron along z-direction
With a given quantum number
peF (v) 2 c 2  me2 c 2 (1  2
B
)  e2
Bc
(2.7)
e : chemical potential of electrons ( i.e. Fermi energy).
The up limit, m , of the sum is given by the condition following
[ peF ( )]2  0
B
  m c (1  2 )
Bc
2
e
2 4
e
(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:

1
ne 
(20 )
Pe 
2
1
(20 ) 2
 gnL 


nL  0

g 
nL  0
nL
fdpz


pz2 c 2
f
dpz
E
(6.1)
f  [1  exp(
E  e 1
)]
kT
（6.3）
The pressure of free electrons is isotropic.
ρ0 is the radius of gyration of the electron in magnetic field
0 （
c 1/ 2
)
eB
E  [c 2 pz2  me2 c 4 (1  2nL
（2.5）
B 1/ 2
)]
Bc
1
2 eB
（

）
2
2
（20）
hc
(2.12)
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
（6.2）
Results in these papers
For non-relativistic degenerate electron gas with lower density
EF
 2.67 B122 (Ye  ) 2 K
( for
  B )
k
1
27 3/ 2
3
B N A  nB 

4.24

10
B
cm
12
2 2 03
TF 
(6.14)
 B  7.04 103 B123/ 2 g / cm3
c 1/ 2
（0 （ )  2.5656 1010 B121/ 2 cm）
eB
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)：
E  (n  1/ 2   ) B  pz2 / 2me
ωB : Larmor gyration frequency of a non relativistic electron in magnetic field
B  e B / mec
 B  2 e B
e 
e
2me c
2）The state number of electrons in the interval pzpz+dpz is
eB dpz
4 2 c
Relativistic Landau Energy in strong magnetic field
In the case EF >>mec2 , Landau energy is by solving the relativistic
Dirac equation in magnetic field
(
2 e B
pz 2
E 2
)
(
p
,
B
,
n
,

)

1

(
)

(2
n

1


)
z
me c 2
me c
me c 2
pz 2
 1 (
)  (2n  1   )b
me c
b  B / Bcr
2  e Bcr
1
2
me c
me c 2
Bcr 
 4, 414 1013 gauss
2 e
e ~ 0.927 1020 erg / gauss
(Bohr magnetic moment of
the electron)
Landau quantum in strong magnetic field
n: quantum number of the Landau energy level
n=0, 1,2,3……(当n = 0 时, 只有σ= -1)
For the non-relativistic case
The state number of electrons in the interval pzpz+dpz
N phase ( pz )dpz 
N phase 
eB dpz
4 2 c
pF
 N phase( pz )dpz 
0

(A)
eB EF
 ne  N A Ye
2
2
4 c
(Due to Pauli Principle)
EF (e)  B 1
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
B
me c
2

eB
B

(
)  1 （when
2 3
Bcr
me c
B  Bcr )
(It is relativistic gyration movement)
Result in some text-book
(Pathria R.K., 2003, Statistical
Mechanics, 2nd edn. lsevier,Singapore)
The state number of electrons
in the interval pzpz+dpz along
the direction of magnetic field
1
1
2
dp
dp


p
x
y

h2 
h2
n 1
n
4 m B B

h2
n+1
n
(B)
e
B 
2me c
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(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
Landau quantization of electron energy
More discussion
The eq. (2.5) in the previous paper a) (in strong magnetic field)
eB
g  dp
 
 h c
2
z

e
B / Bc
p
g d( z )
2 3  
(2 ) e 
me c
（2.5）
And some eq. in paper b):
ne 
Pe 
1
(20 )2
1
(20 ) 2
（0 （

g 
nL  0
nL



g 
nL  0
nL
fdpz


(6.1)
pz2 c 2
f
dpz
E
（6.3）
c 1/ 2
)  2.5656 1010 B121/ 2 cm）
eB
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
 N phase
1
 3 dxdydzdpx dp y dpz
h
The microscopic stae number (in an unit volume) is
N phase
mc
 2 ( e )3
h

pF / me c

0
nmax ( pz ,b , 1)

n 1
pz nmax ( pz ,b , 1)
p
p
p
d(
){
g

(

2
nb
)(
)
d
(
)

0
me c
me c
me c
me c
n 0
p
p
p
g0   (
 2(n  1)b )(
)d (
)}
me c
me c
me c
g0 is degeneracy of energy
continue
N phase
me c 3
 2 (
) g0 
h
pF / me c

0
nmax ( pz ,b , 1)
pz nmax ( pz ,b , 1)
d(
){
2nb 
2(n  1)b}


me c
n 0
n 1
EF 2
pz 2
1
nmax ( pz , b,   1)  Int{ [(
) 1  (
) ]}
2
2b me c
me c
EF 2
pz 2
1
nmax ( pz , b,   1)  Int{ [(
) 1 (
) ]  1}
2
2b me c
me c
nmax ( pz , b,   1)  nmax ( pz , b,   1)  nmax ( pz , b)
EF 2
pz 2
1
nmax ( pz , b)  [(
) 1  (
) ]
2
2b me c
me c
In super strong magnetic field
N phase
27 / 2 1/ 2 me c 3 1 3/ 2

b (
) ( ) g0
3
h
2b
pF / me c

0
[(
EF 2
pz 2 3/ 2 pz
)

1

(
) ] d(
)
2
mec
mec
mec
The state density of electrons in super strong magnetic field
me c 3 1
EF 2
4
E 2 3/ 2
e 
g0 (
)
[(
) (
) ]
2
2
2
3b
h
me c me c
me c
N phase
me c 3 EF 4
4

g0 I (
)(
)
2
3b
h
me c
1
其中I为一个具体数值。
I   (1 t 2 )3/ 2 dt
0
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.
N phase  N AYe 
Relation between Fermi energy of electrons and magnetic field
N phase

me c 3 EF 4
4

g0 I (
)(
)  N e  N AYe 
2
3b
h
me c
Ye
EF
 14 14
 C[

] b
2
me c
0.05 nuc
C  76.69 g01/ 4
（b>1) ）
For the case with lower magnetic field in NS
B 1/ 4
EF (e)  60( ) MeV
Bcr
EF (e)  60MeV
( B  Bcr )
IV
Activity of magnetars
and their high x-ray
luninocity
Question
What is the mechanism for very high x-ray luminosity
of magnetars ?
Lx  10  10 ergs / sec
34
36
What is the reason of x-ray flare or
of x-ray Burst for some magnetars ?
Lx ~ 10  10 ergs / s
42
43
(短时标)
Basic idea
Electron capture by protons will happen
e  p  n   e
When the magnetic field is more strong than Bcr and then
EF (e)  EF (n)  60MeV
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
3P Cooper pairs disappearing , and then the magnetic energy the
2
magnetic moment of the 3P2 Cooper pairs 2  B
n
Will be released and then will be transferred into x-ray radiation
kT  n B  10 B15
keV
Total Energy may be released
m(3P2 )
E  qN A m( P2 )  2 n B  110
0.1mSun
3
47
ergs
It may take ~ 104 -106 yr for x-ray luminocity of
AXPs
Lx  10  10 ergs / sec
34
36
电子俘获速率
e   p  n  e
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:
2 2 2
d 
GF CV (1  3a 2 )(1  f )  dE  ( E  Q  Ee )
h
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
Q  En  E p  (mn  mp )c 2
 
[ E e  E p  E n  ( mn  m p )c 2 ]
CV  0.9737;
2 2 (hc) 3
a
CA
 1.253
CV
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
电子俘获过程产生的x-光度
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
dLx  2n B  d
x-ray 总光度为:
(2 ) 4 2 2
Lx   V ( P2 )
GF GV (1  3a 2 ) 
V1
3
3
3
3
3
3
d
n
d
n
d
n
d
n

(
E

E

0.61
MeV

E
)

(k f  k i ) S  2  n B
e
 e p n   n
  《 》
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)
续
S  f e ( Ee ) f p ( E p )[1  f n ( En )][1  f ( E )]
（j 是粒子j 的化学势)
f j ( E)  [exp( E   j ) / kT  1]1
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  f ( E )  1


f（
e Ee )  1
when
Ee  EF (e)
f（
e Ee )  0
when
Ee  EF (e)
1  f n ( En )  0
when
En  EF (n)
f（
p Ep )  1
when
E p  EF ( p)
1  f n ( En )  1
when
En  EF (n)
f（
p Ep )  0
when
E p  EF ( p)
EF (e)  60( B / Bcr )
1
4
MeV
能级状态数
j 为单位体积内粒子
d3n
j 的微观状态数。
d 3n j  V1 g j
ρj 为第j 种粒子的能级态密度。
在超强磁场下,电子气体的能级态密度为
me c 3 1
EF 2
4
E 2 3/ 2
e 
g0 (
)
[(
) (
) ]
2
2
2
3b
h
me c me c
me c
 
1
2 2 3c3
(Q  Ee )2
8 3/ 2 3/ 2
 n  3 mn EF (n)
h
8 3/ 2 3/ 2
 p  3 m p EF ( p )
h
Q  En  E p  (mn  mp )c 2
d3 pj
h
3
  j dE j
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,
n  p  e  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
e  p  n  e

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Glitch
2)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