A numerical model of non-equilibrium thermal plasmas, Part I:
Transport properties
Xiao-Ning Zhang,1,2 He-Ping Li,2,1) A. B. Murphy,3 and Wei-Dong Xia1
1
Department of Thermal Science and Energy Engineering, University of Science and
Technology of China, Hefei, Anhui Province 230026, P. R. China
2
Department of Engineering Physics, Tsinghua University, Beijing 100084, P. R. China
3
CSIRO Materials Science and Engineering, PO Box 218, Lindfield NSW 2070, Australia
Supplementary Materials:
SOLUTION OF THE BOLTZMANN EQUATION
A. Modified Chapman-Enskog perturbation method
Using the perturbation technique of Chapman and Enskog,1,2 the distribution function
f  r , ci , t  of species i can be given from the solution of the Boltzmann equation (S-1):2
N
fi
X
 ci fi  i ci fi     fi ' f j'  fi f j  g ij dΩdc j .
t
mi
j 1
(S-1)
For the sake of convenience, Eq. (S-1) can be re-expressed as
Dfi N
  J ij  fi , f j ,
Dt j 1
(S-2)
Dfi fi
X

 ci fi  i  ci fi ,
Dt t
mi
(S-3)
J ij  fi , f j     fi ' f j'  fi f j  g ij dΩ dc j ,
(S-4)
with
1)
Author to whom correspondence should be addressed. Electronic mail: liheping@tsinghua.edu.cn.
1
where ci , X i and mi are the velocity, external force and mass of species i, r and t are
the space vector and time, f i ' is the distribution function after collision of species i, g is the
relative speed of the species i and j,  ij is the differential collision cross section and  is the
solid angle.
Enskog obtained a series solution by introducing a perturbation parameter  to the
Boltzmann equation as
Dfi 1 N
  J ij .
Dt  j 1
(S-5)
In fact, Eq. (S-5) corresponds to the Boltzmann equation for the distribution function
1
f , so  is merely a scaling factor. Since we eventually return  to the value of unity, all
 i
the results are in formal agreement with the original Boltzmann equation (S-2).
If the distribution function, fi, is expanded in a series of ε, i.e.,
fi  fi 0   fi 1 
(S-6)
,
where the superscripts indicate the degree of approximation, by substituting the series (S-6)
into the modified Boltzmann equation (S-5) and equating the coefficients with the equal
powers of  , the following set of equations can be obtained
N


0   J ij fi  0 , f j 0 ,
j 1
(S-7)
N
Dfi  0
   J ij fi  0 , f j1  J ij f i 1 , f j 0  J ij f i  0 , f j 0 .


Dt
j 1






(S-8)
Equations (S-7) and (S-8) are the zero-order and first-order approximation of the
Boltzmann equation, respectively. These equations allow us to determine the distribution
function uniquely.
B. Zero-order and first-order solutions
2
In the following derivations, the subscripts “1” or “e” denotes the electrons, while “i
(i ≥ 2)” or “h” denotes the heavy-species, and “i (∀i)” represents an arbitrary species for
simplicity. For the zero-order solution, the electron–heavy-particle collision terms are
expected to have little effect on the distribution functions of heavy-species and electrons due
to the small size of the ratio me/mh. Equation (S-7) can be re-written for the electrons
  f
 0 '  0 '
f
1

 f1 0 f  0 g11 dΩ dc  0,
(S-9)
and for the heavy-species
  f   f    f   f    g
N
0 '
i
0 '
0
j
i
0
j
ij
dΩ dc j  0,
(S-10)
j 2
where f  0  and c (without any subscript) in Equation (S-9) denote collision partner
electrons. Thus, the Maxwellian distribution functions with Te for electrons and Th for
heavy-species are obtained by solving the foregoing zero-order approximation equations (S-9)
and (S-10) as
3
 0
1
f
 m1  2
 m1C12 
 n1 
exp


,
 2 k BTe 
 2k BTe 
(S-11)
3
fi  0
 mi  2
 mi Ci2 
 ni 
 exp  

 2 k BTh 
 2k BTh 
i  2 ,
(S-12)
where the peculiar particle velocity of species i is defined as Ci  ci  v0 , and kB represents
the Boltzmann constant. Consequently, the species number densities ( ni ), the mass-averaged
velocity ( v0 ) and the temperatures (Te, Th) can be defined as
ni   f i   dci ,
0
(S-13)
N
N
i 1
i 1
 v0  mi ci f i  0 dci  ni mi vi ,
(S-14)
3
1
n1k BTe  m1 C12 f1 0 dc1 ,
2
2
(S-15)
3
3
1 N
 nt  n1  kBTh  mi Ci2 fi 0dci ,
2
2 i 2
N
N
i 1
i 1
(S-16)
where nt   ni is the total number density and    ni mi is the total mass density.
Because of the isotropy of the zero-order distribution function in velocity phase space,
all the transport fluxes except for the momentum flux are zero. To obtain the non-zero
transport fluxes, it is necessary to derive the first-order perturbation solution.2 The equation
for the fi 1 may be written in terms of a perturbation function i , defined by
fi 1  fi  0i  r , ci , t  .
(S-17)
It is worth noting that the definitions of Eqs. (S-13)-(S-16) have placed certain
restrictions on i , namely,
f
 0
i dci  0
i
N
m c f   dc
0
i
i i
i
i
(S-18)
0
(S-19)
i 1
2  0
1 1
1
C
f  dc1  0
N
m C
(S-20)
2  0
i i
i
i
f  dci  0
(S-21)
i 2
Thus, Eq. (S-8) may be re-written in terms of this function (S-17) to give
Dfi  0 N   0 '  0 ' ' '
    fi f j i   j   fi  0 f j 0 i   j g ij ddc j
Dt
j 1

N
  fi
j 1
 0 '  0 '
fj
 0  0
 fi f j
g ddc ,
ij
(S-22)
j
The equations for electrons and heavy-species can then be written, respectively, as
Df1 0
  f1 0 f  0 1'   '  1    g 11 dΩ dc
Dt
    f1 0 ' f j 0 ' 1'   j'   f1 0 f j 0 1   j   g 1 j dΩ dc j
N
j 2
N


   f1 0 ' f j 0 '  f1 0 f j 0 g 1 j dΩ dc j ,
j 2
4
(S-23)
Dfi  0
   fi  0 ' f1 0 ' i'  1'   fi  0 f1 0 i  1   g i1dΩ dc1
Dt
   fi  0 f j 0 i'   j'  i   j  g ij dΩ dc j
N
j 2

, ( i  2 ) (S-24)

  fi  0 ' f1 0 '  fi  0 f1 0 g i1dΩ dc1.
Based
on
the
form
of
Maxwellian
f1 0 ' f j 0 '  f1 0 f j 0 1  O  m1 m j 
relationship
can
distribution
be
given;3
functions,
thus,
the
we
have
f1 0 ' f j 0 '  f10 f j 0 due to the fact m1 m j  1 . On the other hand, there exists the
relationship i'  i   O  m1 mi  1'  1  for the electron–heavy-particle collisions ( i  2 );
and therefore, the change in the first order perturbation function of heavy species may be
neglected compared with that of electrons. According to the foregoing analysis, the
zero-order approximation integral terms in Eqs. (S-23) and (S-24) are negligible.
The validity of the simplifications can be verified by the very minor differences between
the simplified theory and complete theory in comparisons presented in recent works.4-6
Equations (S-23) and (S-24) can be re-written as
Df1 0
  f1 0 f  0 1'   '  1    g 11dΩ dc
Dt
N
   f1 f j
 0
 0
j 2

'
1
 1  g 1 j dΩ dc j ,
Dfi  0
  fi  0 f1 0 1'  1  g i1dΩ dc1
Dt
N
   fi
j 2
 0
 0
fj
  
'
i
'
j
 i   j  g ij dΩ dc j .
(S-25)
(S-26)
It should be emphasized that the first integration term of Eq. (S-26), i.e.,
 f
 0
i
f1
0
    g
'
1
1
dΩ dc1 , is retained, unlike in the simplified theory of Devoto.3
i1
The left-hand sides of Eqs. (S-25) and (S-26) can be reduced using Enskog’s equation of
change.1,2 We adopt the definition of the diffusion driving force ( di ) introduced by Rat et al.7
5
Df1 0
to take into account the coupling between electrons and heavy species. Thus,
and
Dt
Dfi  0
( i  2 ) become
Dt

Df1 0
C1
5
 f1 0  W12   C1  ln Th  2W1 W1 : v0 
 d1
Dt
2
n1k BTe

Q 0  2
5



C1  ln    W12   C1  ln   1  W12  1  ,
n1k BTh
2
n1k BTe  3


1

Dfi  0
Ci
5
 fi  0  Wi 2   Ci  ln Th  2Wi Wi : v0 
 di
Dt
2
ni k BTh

Q1 0
 2 2 

Ci  ln  
1 W 
ni k BTh
 nt  n1  kBTh  3 i 
i
(S-27)
(S-28)
i  2 ,
where
N
1
Q1 0   m1  C1'2  C12  f1 0 f j 0 g 1 j dΩ dc j dc1 ,
j 2 2
(S-29)
represents the change of average kinetic energy of electrons via elastic collisions with the
heavy particles, and
d1 
di 
i

N
n X
j
j 1
j
 x  
1 N
p
n j X j  n1 X 1   1  1  p  2 x1 ,

 j 1
D
 D 
x 
 ni X i   i  i
D 
xi   1 p

p
x1
 p  xi 
D
D2

where xi  ni nt is the mass fraction of species i ( 1,
(S-30)
 i  2  , (S-31)
, N ) . The thermal non-equilibrium
coefficient is defined as   Te Th . For an arbitrary species i, the partial pressure is
N
pi  ni k BTi , p   pi represents the total pressure, Wi 
i 1
mi
Ci corresponds to the
2kBTi
1 2
reduced velocity and Wi Wi is the symmetric traceless tensor WW
i i  Wi I , with I as the
3
6
unit tensor. In addition, D , 1 and i ( i  2 ) are given by D  1  x1   1 ,
1  x1 p 1  x1  D2 and i   xi x1 p D 2 , respectively.
C. Expressions for the integral equations
Comparing the form of Eq. (S-25) with that of Eq. (S-27), the perturbation function 1
for electrons can be expressed as
N
N
j 1
j 1
1   A1  ln Th  B1 : v0  C1j  d j  D1Q1 0  E1j j  ln   F1  ln  .
(S-32)
It is important to note that the form of the third term on the right-hand side of Eq. (S-32),
N
C
j
1
 d j , is equivalent to taking into account the diffusion process between electrons and
j 1
heavy-species, which makes the calculation of the complete set of diffusion coefficients
available. This differs from the simplified theory presented in Refs. 3, 8-9.
Similarly, the expression of i for heavy-species ( i  2 ) can be written as
N
N
i   Ai  ln Th  Bi : v0  Ci  d j  Di Q1  Ei j j  ln   Fi  ln  . (S-33)
j
j 1
 0
j 1
Substituting 1 and i into Eqs. (S-25) and (S-26), we obtain the expressions of the
tensors  1 s ,k  and  i s ,k  ( i  2 ) for the electrons and heavy species, respectively, as follows:
R1
s ,k 
s ,k
s ,k
s ,k
s ,k
0
0
   1      1      f1  f   g 11dΩdc


'
'
N
s ,k
s ,k
0
0
    1   1   f1  f j  g 1 j dΩdc j ,


j 2
Ri
s ,k 
'
(S-34)
s ,k
s ,k
0
0
   1   1   fi   f1  g i1dΩdc1


'
N
s ,k
s ,k
s ,k
s ,k
0
0
    i   j   i   j   fi   f j  g ij dΩdc j .


j 2
'
'
(S-35)
The expressions for R1 s ,k  ,  1 s ,k  , Ri s ,k  and  i s ,k  are listed in Table I. The term Di
(∀i) is, in fact, not taken into account in the following derivations of the transport
7
coefficients, since the corresponding integrals representing the transport fluxes vanish
identically. Thus, Eqs. (S-34) and (S-35) and the auxiliary equation
N

miTi   i s ,k  Wi  fi  0 dci  0,
i 1
(S-36)
which is equivalent to Eqs. (S-18)-(S-21), serve to specify  i s ,k  uniquely.
Solutions to the integral equations (S-34) and (S-35) are obtained with a similar method
to that presented in Refs. 1 and 2, in which the unknown quantity  i s ,k  are expanded in a
series of Sonine polynomials
 s ,k 
i
 1
 i tip
s ,k 
p 0
  Sn p  Wi 2  ,
(S-37)
where  is the so-called level of approximation to the transport coefficients. The value of the
index n, the meaning of tensor i and the unknown coefficient tip
s ,k 
are given in Table II.
 s ,k 
For simplicity, Rim
is defined as


Rim s ,k    Ri s ,k  : i Sn m Wi 2  dci ,
(S-38)
Eventually, it is shown that tip  ( ) can be obtained by solving the following equations:
s ,k
 1
 R1m   Q1mp t1 p
s ,k
s ,k 
  ,
(S-39)
p 0
and

 Rim
s ,k 
 1 N
 Qijmp t jp
s ,k 
p 0 j  2
 1
*  s ,k 
   Qimp
  ,
1 t1 p
(i  2),
(S-40)
p 0
with
Q1mp  n1n j 1Sn m W12  ; 1Sn p  W12 
N
1j
j 1
 m
 n 1Sn
2
1
W  ; S  W  ,
2
1
8
 p
n
2
1
(S-41)
when t jps ,k   b jp

Qijmp

Qijmp   mp n j m jT j mp
Qii  m 0 p 0
( i  2, j  2 )
Qij 
ni miTi

when t jps ,k   a jp , c jps ,k  , ejps ,k  or f ip
, (S-42)
and
*

Qimp
1

*
Qimp
  mp* n1 m1T1 mp
1
( i  2)
Qii  m 0 p 0
Qi1 
n
m
T
i
i
i

when t1 ps ,k   b1 p
when t1 ps ,k   a1 p , c1 sp,k  , ejps ,k  or f1 p
. (S-43)
The second terms in the lower expressions of Eqs. (S-42) and (S-43) arise from the
auxiliary equation (S-36), which can be written as
N
n
miTi ti0s ,k   0,
i
(S-44)
i 1
*
and Qijmp and Qimp
are defined as
1

Qijmp   ni nk  ij i Sn m  Wi 2  ; i Sn p  Wi 2  
ik
k 2
N
W  ;  S W  ,
 n n  S   W  ;  S   W  
 m
 jk i Sn
*
Qimp
1
 p
2
i
k
m
i 1
The expressions for R1ms ,k  and Rim
s ,k 
i
n
n
p
2
i
2
k
1 n
(S-45)
ik
2
1
i1
.
(S-46)
 i  2  can then be obtained as listed in Table III. In
Eqs. (S-45) and (S-46), the terms of form Gij ; Hij  are the bracket integrals defined by
ij
Chapman and Cowling,8 which may be written as a linear combination of a set of collision
*
integrals defined. Expressions for some of the bracket integrals, Q1mp , Qijmp and Qimp
in
1
terms of the collision integrals are given in Refs. 1-3, and 10.
D. COEFFICIENTS OF THE SONINE POLYNOMIALS

The coefficients tip
s ,k 
listed in Table II can be derived from Eqs. (S-39) and (S-40) as
follows.
9
c1 sp,k    

4
Q100
Q101 Q102
Q103
Q110
Q120
Q130
Q111 Q112
Q121 Q122
Q131 Q132
Q113
Q123
Q133
 p0
 p1
 p3
 p2
k BTe 1s  1k 
2me
k BTe
3
0
0
0
0
Q100
Q110
Q120
Q130
Q101 Q102
Q111 Q112
Q121 Q122
Q131 Q132
3
,
Q103
Q113
Q123
Q133
(S-47)
k BTh  ls   lk  3 0 p*  s ,k 
  Ql1 c1 p
2ml
k BTh
p 0
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
 Ql11p*c1 p
Qlj20
Qlj21 Qlj22
Qlj23
 Ql21 p*c1 sp,k 
Qlj30
Qlj31 Qlj32
Qlj33
 Ql31p*c1 sp,k 
0
0
3
s ,k 
p 0
3
p 0
3
ci0s ,k    

4
 ij
0
0
p 0
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
Qlj20
Qlj21 Qlj22
Qlj23
Qlj30
Qlj31 Qlj32
Qlj33
, (S-48)
where
Q22mp
Q2mpN
Qljmp 
,
QNmp2
10
mp
QNN
(S-49)
e1 sp,k    

4
Q100
Q101 Q102
Q103
Q110
Q120
Q130
Q111 Q112
Q121 Q122
Q131 Q132
Q113
Q123
Q133
 p0
 p1
 p3
 p2
3
k BTe 1s  1k 
2me
k BTh
0
0
0
0
Q100
Q110
Q120
Q130
Q101 Q102
Q111 Q112
Q121 Q122
Q131 Q132
3
,
Q103
Q113
Q123
Q133
(S-50)
k BTh   ls   lk  3 0 p*  s ,k 
  Ql1 e1 p
2ml
k BTh
p 0
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
 Ql11p*e1 p
Qlj20
Qlj21 Qlj22
Qlj23
 Ql21 p*e1 sp,k 
Qlj30
Qlj31 Qlj32
Qlj33
 Ql31p*e1 sp,k 
0
0
3
s ,k 
p 0
3
p 0
3
ei0s ,k    

4
 ij
 a1 p   
4
0
0
p 0
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
Qlj20
Qlj21 Qlj22
Qlj23
Qlj30
Qlj31 Qlj32
Qlj33
Q100
Q101 Q102
Q103
Q110
Q111
Q112
Q113
Q120
Q130
Q121 Q122
Q131 Q132
Q123
Q133
0
0
 p0
 p1
 p3
0
 p2
0

kT
15
ne B e
2
2me
Q100
Q110
Q101 Q102
Q111 Q112
Q103
Q113
Q120
Q130
Q121 Q122
Q131 Q132
Q123
Q133
11
, (S-51)
,
(S-52)
3
 Ql01 p*a1 p
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
Qlj20
Qlj21 Qlj22
Qlj23
 Ql21 p*a1 p
Qlj30
Qlj31 Qlj32
Qlj33
 Ql31p*a1 p
0
0
p 0

3
kT
15
nl B h   Ql11p*a1 p
2
2ml p 0
3
p 0
3
 ai 0 4  
 ij
0
0
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
Qlj20
Qlj21 Qlj22
Qlj23
Qlj30
Qlj31 Qlj32
Qlj33
0

kT
15
ne B e
2
2me
0
0
 f10 4 
Q100
c11 s ,k   

4
3
p 0
Q101 Q102
Q103
Q111
Q112
Q113
Q121 Q122
Q131 Q132
Q123
Q133
Q100
Q110
Q120
Q101 Q102
Q111 Q112
Q121 Q122
Q103
Q113
Q123
Q130
Q131 Q132
Q133
,
,
k BTe 1s  1k 
Q102
2me
k BTe
Q103
Q110
0
Q112
Q113
Q120
0
Q122
Q123
Q130
0
Q132
Q133
Q100
Q101 Q102
Q103
Q110
Q111
Q112
Q113
Q120
Q121 Q122
Q123
Q130
Q131 Q132
Q133
12
(S-53)
(S-54)
,
(S-55)
3
k BTh  ls   lk  3 0 p*  s ,k 
  Ql1 c1 p
2ml
k BTh
p 0
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
 Ql11p*c1 p
Qlj20
Qlj21 Qlj22
Qlj23
 Ql21 p*c1 sp,k 
Qlj30
Qlj31 Qlj32
Qlj33
 Ql31p*c1 sp,k 
0
0
3
s ,k 
p 0
3
p 0
3
ci1s ,k    

4
0
 ij
0
Q100

Q110
 f11 4 
Q100
e11 s ,k   

4
p 0
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
Qlj20
Qlj21 Qlj22
Qlj23
Qlj30
Qlj31 Qlj32
Qlj33
0
Q102
Q103
kT
15
ne B e
2
2me
Q112
Q113
0
0
Q122
Q132
Q123
Q133
Q120
Q130
Q100
Q110
Q120
Q101 Q102
Q111 Q112
Q121 Q122
Q103
Q113
Q123
Q130
Q131 Q132
Q133
3
,
,
k BTe 1s  1k 
Q102
2me
kTh
Q103
Q110
0
Q112
Q113
Q120
0
Q122
Q123
Q130
0
Q132
Q133
Q100
Q101 Q102
Q103
Q110
Q111
Q112
Q113
Q120
Q121 Q122
Q123
Q130
Q131 Q132
Q133
13
(S-56)
(S-57)
,
(S-58)
k BTh   ls   lk  3 0 p*  s ,k 
  Ql1 e1 p
2ml
k BTh
p 0
3
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
 Ql11p*e1 p
Qlj20
Qlj21 Ql22
j
Qlj23
 Ql21 p*e1 sp,k 
Qlj30
Qlj31 Qlj32
Qlj33
 Ql31p*e1 sp,k 
0
0
3
s ,k 
p 0
3
p 0
3
ei1s ,k    

4
 ij
0
0
p 0
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
Qlj20
Qlj21 Qlj22
Qlj23
Qlj30
Qlj31 Qlj32
Qlj33
Q100

Q110
 a11 4 
0
Q102
Q103
kT
15
ne B e
2
2me
Q112
Q113
0
0
Q122
Q132
Q123
Q133
Q120
Q130
Q100
Q110
Q120
Q101 Q102
Q111 Q112
Q121 Q122
Q103
Q113
Q123
Q130
Q131 Q132
Q133
, (S-59)
,
(S-60)
3
 Ql01 p*a1 p
Qlj00
Qlj01 Qlj02
Qlj03
Qlj10
Qlj11
Qlj12
Qlj13
Qlj20
Qlj21 Qlj22
Qlj23
 Ql21 p*a1 p
Qlj30
Qlj31 Qlj32
Qlj33
 Ql31p*a1 p
p 0

3
kT
15
nl B h   Ql11p*a1 p
2
2ml p 0
3
p 0
3
 ai1 4  
0
 ij
0
p 0
0
00
lj
10
lj
20
lj
30
lj
0
01
lj
11
lj
21
lj
31
lj
02
lj
12
lj
22
lj
32
lj
03
lj
13
lj
23
lj
33
lj
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
14
.
(S-61)
REFERENCES
1
J. O. Hirschfelder, C. F. Curtis, and R. B. Bird, Molecular Theory of Gases and Liquids, 2nd
ed. (Wiley, New York, 1964).
2
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd ed.
(Cambridge University Press, Cambridge, England, 1970).
3
R. S. Devoto, Ph. D. Thesis, Stanford University (1965).
4
V. Colombo, E. Ghedini, and P. Sanibondi, Prog. Nucl. Energy 50, 921 (2008).
5
V. Colombo, E. Ghedini, and P. Sanibondi, J. Phys. D: Appl. Phys. 42, 055213 (2009).
6
V. Colombo, E. Ghedini, and P. Sanibondi, Plasma Sources Sci. Technol. 20, 035003 (2011).
7
V. Rat, P. Andre, J. Aubreton, M. F. Elchinger, P. Fauchais, and A. Lefort, Phys. Rev. E 64,
026409 (2001).
8
C. Bonnefoi, Ph. D. Thesis, University of Limoges, France (1983).
9
J. Aubreton, C. Bonnefoi, and J. M. Mexmain, Rev. Phys. Appl. 21, 365 (1986).
10
R. S. Devoto, Phys. Fluids 10, 2105 (1967).
15
TABLE I. Expressions of R1 s ,k  ,  1 s ,k  , Ri s ,k  and  i s ,k  .
R1 s ,k 
 1 s ,k 
Ri s ,k  ( i  2 )
 i s ,k  ( i  2 )
5

f1 0   W12  C1
2

A1
5

fi  0   Wi 2  Ci
2

Ai
2 f1 0W1 W1
B1
2 fi 0Wi Wi
Bi
f1 0
C1
1s  1k 
n1k BTe
C1s  C1k
fi  0
f1 0
C1
1s  1k 
n1k BTh
E1s  E1k
fi  0
5

f1 0   W12  C1
2

Ci
 is   ik 
ni k BTh
 Ci
ni k BTh
 is   ik 
Fi 0
F1
Here,  ij is Kronecker delta.  ij  1 if i  j ; otherwise  ij  0 .
16
Cis  Cik
Eis  Eik
0
TABLE II. Expressions of i , tip
s ,k 
and n.
 i s ,k 
n
tip s ,k 
i
Ai
3
2
aip
Wi
Bi
5
2
bip
Wi Wi
Cis  Cik
3
2
cip s ,k 
Wi
Eis  Eik
3
2
eip s ,k 
Wi
Fi
3
2
f ip
Wi
17
 s ,k 
TABLE III. Expressions of R1ms ,k  and Rim
.
tip s ,k 
R1ms ,k 
 s ,k 
(i  2 )
Rim
aip
k T
15
n1 B 1  m1
2
2m1
kT
15
ni B i  m1
2
2mi
bip
5n1 m0
5ni m0
cip s ,k 
3
kBT1 1s  1k 
 m0
2m1
kBT1
eip s ,k 
3
kBT1 1s  1k 
 m0
2m1
kBTh
f ip
k T
15
n1 B 1  m1
2
2m1
18
3
3
kBTi  is   ik 
 m0
2mi
kBTi
kBTi   is   ik 
 m0
2mi
kBTh
0