Boyd T.J.M., Sanderson J.J. The Physics of Plasmas
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Appendix 1
Numerical values of physical constants and plasma
parameters
Physical constants
Velocity of light c 2.998 × 10
8
ms
−1
Electron charge e 1.602 × 10
−19
C
Electron mass m
e
9.109 × 10
−31
kg
Proton mass m
p
1.673 × 10
−27
kg
Proton/electron mass ratio m
p
/m
e
1.836 × 10
3
Classical electron radius r
e
2.818 × 10
−15
m
Thomson scattering cross-section σ
T
6.652 × 10
−29
m
2
Planck’s constant h 6.626 ×10
−34
Js
Electron volt eV 1.602 × 10
−19
J
Ionization potential of hydrogen 2.180 × 10
−18
J
Boltzmann’s constant k
B
1.381 × 10
−23
JK
−1
Vacuum permittivity ε
0
8.854 × 10
−12
Fm
−1
Vacuum permeability µ
0
1.257 × 10
−6
Hm
−1
Gravitational constant G 6.673 × 10
−11
m
3
kg
−1
s
−2
Solar mass M
1.989 × 10
30
kg
Solar radius R
6.960 × 10
8
m
Sun–Earth distance 1 AU 1.496 × 10
11
m
Earth’s gravitational acceleration g 9.807 m s
−2
Temperature equivalent to 1 eV 1.160 × 10
4
K
Plasma parameters
Electron plasma frequency
n
e
e
2
/m
e
ε
0
1/2
56.4 n
1/2
e
rad s
−1
Ion plasma frequency
n
i
(Ze)
2
/m
i
ε
0
1/2
1.32 Z(n
i
/A)
1/2
rad s
−1
Electron cyclotron frequency eB/m
e
1.76 × 10
11
B rad s
−1
Ion cyclotron frequency ZeB/m
i
9.58 × 10
7
(ZB/ A) rad s
−1
Debye length
ε
0
k
B
T
e
/n
e
e
2
1/2
69.1 (T
e
/n
e
)
1/2
m
507
508 Numerical values of physical constants and plasma parameters
Electron Larmor radius m
e
v
⊥
/eB 5.68 × 10
−12
v
⊥
B
−1
m
Ion Larmor radius m
i
v
⊥
/ZeB 1.04 × 10
−8
v
⊥
(A/ZB) m
Electron thermal speed (k
B
T
e
/m
e
)
1/2
3.89 × 10
3
T
1/2
e
ms
−1
Ion thermal speed (k
B
T
i
/m
i
)
1/2
90.9 (T
i
/A)
1/2
ms
−1
In the above formulae densities are expressed in m
−3
, B in teslas, T
e
, T
i
in K and
v
⊥
(ms
−1
) is the component of particle velocity perpendicular to B.
Appendix 2
List of symbols
The following is a list of symbols which appear in the text beyond the location of
their definition or first introduction and indicates the page on which this may be
found. Some of these symbols also appear with subscript e(i) indicating electron
(ion) equivalents. Likewise, symbols with subscript (⊥) indicate the component
parallel (perpendicular) to the magnetic field.
Roman alphabet
Symbol Description Page
A magnetic vector potential 32
A atomic number 9
a minor radius of torus 95
a
s
(m
s
/2k
B
T
s
)
1/2
313
B magnetic field intensity 13
˜
B vacuum magnetic field 75
B
ind
magnetic field induced by magnetization current 16
B magnetic field magnitude 11
B
p
poloidal magnetic field 33
B
t
toroidal magnetic field 33
b unit vector in direction of B 43
b impact parameter 309
b
0
impact parameter for π/2 scattering 10
C
v
specific heat at constant volume 184
C
α
‘collision term’ for wave–particle interactions 409
c
α
random or thermal velocity for species α 481
c speed of light 18
c
s
ion acoustic speed 65
D component of cold plasma dielectric tensor 205
509
510 List of symbols
D diffusion coefficient 299
D(k, p) plasma dielectric function 258
D
a
ambipolar diffusion coefficient 301
E electric field 13
ˆ
E effective electric field 494
E
s
scattered electric field 355
E
T
electric field of transverse wave 395
E energy 3
e magnitude of electronic charge 7
e
j
charge on particle j 13
E internal energy per unit mass 55
E(k, t) spectral energy density of electrostatic field 380
F force per unit mass 52
F
i
(t) force on electron i at time t 465
F field reversal parameter 144
F fast wave 218
F
0
(u) one-dimensional distribution function 258
f (1) one-particle distribution function 469
f (r, v, t) particle distribution function 252
f
K
(r, v, t) Klimontovich distribution function 252
f
N
N -particle distribution function 466
f
s
distribution function of scattering particles 309
f
s
generic s-particle distribution function 467
f
α0
local Maxwellian distribution for species α 489
G(v) Rosenbluth potential 312
G(ωb
0
/v) Gaunt factor 336
g relative velocity of colliding particles 309
g plasma parameter 8
g 1 −n·β
β
β 326
g(1, 2) pair correlation function 469
¯g(ω, T ) Maxwellian-averaged Gaunt factor 338
H(v) Rosenbluth potential 312
I unit dyadic 78
I total plasma current 87
I internal energy 183
I
p
poloidal current 98
I
ω
(s) radiation intensity 330
J longitudinal invariant 28
j current density 13
List of symbols 511
j
G
current density due to grad B drift 23
j
g
guiding centre current density 70
j
M
magnetization current density 16
j
P
polarization current density 37
j
p(t)
poloidal (toroidal) current density 99
K magnetic helicity 103
K common coupling coefficient 391
k
L
wavenumber of Langmuir wave 394
k
s
wavenumber of ion acoustic wave 394
k
s
wavenumber of scattered wave 356
k
T
wavenumber of transverse wave 394
k
B
Boltzmann’s constant 3
k
c
critical wavenumber 240
L scale length 12
L component of cold plasma dielectric tensor 205
L left circularly polarized wave 210
L Langmuir wave 394
L
H
hydrodynamic length scale 49
L
s
shock thickness 408
L
, L
⊥
, L
∧
transport coefficients 495
˜
L
dimensionless L
497
˜
L
⊥
,
˜
L
∧
dimensionless L
⊥
, L
∧
498
ln ζ Euler’s constant 338
ln Coulomb logarithm 10
M magnetization per unit volume due to Larmor motion 16
M Mach number 187
M
A
Alfv
´
en Mach number 174
M
solar mass 170
m particle mass 14
m
j
mass of particle j 13
m
s
mass of scattering particle 309
N total particle number density 57
n unit vector normal to surface 74
n dimensionless wave propagation vector 203
ˆ
n unit vector in direction of shock propagation 184
n particle number density 3
n refractive index 331
n quantum state 339
n
b
number density of electron beam 241
512 List of symbols
O ordinary wave 214
P pressure 54
P component of cold plasma dielectric tensor 205
P power radiated by accelerated charge 326
P total scalar pressure 486
P
l
power in harmonic line l 345
P wave polarization 199
P polarization dyadic 355
Q heat energy per unit mass 55
Q 1 + k
2
c
2
/ω
2
pe
232
Q
α
rate of thermal energy transfer to species α 484
q heat flux 297
q
α
heat flux for species α 482
q charge density 13
q safety factor 34
q
T
test particle charge 287
R
α
rate of momentum transfer to species α 484
R hydrodynamic Reynolds number 61
R pressure ratio across shock 185
R component of cold plasma dielectric tensor 205
R right circularly polarized wave 210
R |r − r
0
| 325
R reflection coefficient 428
R
E
Earth’s radius 7
R
M
magnetic Reynolds number 61
R
y
Rydberg constant 340
R
0
tokamak major radius 33
R
0
gas constant 58
R
solar radius 173
r
g
guiding centre position vector 15
r
j
position vector of particle j 13
r tokamak minor radius 33
r density ratio across shock 185
r
A
Alfv
´
en radius 172
r
e
classical electron radius 328
r
L
Larmor radius 15
S Poynting vector 200
S Lundquist number 141
S entropy 184
List of symbols 513
S component of cold plasma dielectric tensor 205
S slow wave 218
S sound wave 394
S(k,ω) form factor 358
T Maxwell stress tensor 78
T temperature 3
T characteristic time scale 12
T transverse wave 395
T tunnelling (transmission) coefficient 434
T
scattered transverse wave 394
T
α
temperature of species α (in energy units) 481
t
retarded time 325
t
F
time scale for magnetic field changes 29
U total energy density 78
u velocity of fluid element 49
V(t) test particle velocity 313
V warm plasma thermal speed 227
V wave phase velocity 269
V (x) electrostatic potential in plasma sheath 454
V
b
electron beam thermal speed 273
V
p(s)
group velocity of Langmuir (Raman scattered) wave 444
V
th
thermal speed 10
v
B
combined grad B and curvature drift velocity 22
v
b
flow velocity of electron beam 241
v
c
curvature drift velocity 21
v
E
E × B drift velocity 17
v
G
grad B drift velocity 20
v
g
guiding centre velocity 43
v
P
polarization drift velocity 37
v
p
poloidal velocity 34
v
s
velocity of scattering particle 309
v
A
Alfv
´
en speed 130
v
d
electron–ion drift velocity 243
v
g
group velocity 201
v
p
phase velocity 72
v
0
quiver velocity 441
W particle kinetic energy 14
W total potential energy 120
W
p(s,v)
plasma (surface, vacuum) potential energy 122
514 List of symbols
X extraordinary wave 214
X
α
α
τ
α
498
Z charge state 9
Z(ζ ) plasma dispersion function 265
z atomic number of scattered particle 310
z
c
cut-off point in WKBJ model 428
z
s
atomic number of scattering particle 310
Greek alphabet
α Clebsch variable 32
α
2
ω
2
p
/ω
2
222
α
kl
thermoelectric coefficient 494
α
ω
absorption coefficient 331
β
β
β v/c 47
β Clebsch variable 32
β ratio of plasma to magnetic pressure 61
β
2
|
i
e
|/ω
2
222
β
e
|
e
|/ω 222
β
i
i
/ω 222
β
p(t)
poloidal (toroidal) β 89
particle flux 299
z
2
e
2
ln /4πε
2
0
m
2
311
γ(1 − v
2
/c
2
)
−1/2
38
γ ratio of specific heats 57
γ instability growth rate 121
γα
2
/β
2
= c
2
/v
2
A
222
γ Landau damping decrement 261
discontinuity in magnetic field gradient 152
inverse aspect ratio 33
ratio of collision time to hydrodynamic time 493
ω
emission coefficient 331
ε cold plasma dielectric tensor 203
ε
0
vacuum permittivity 8
η plasma resistivity 140
θ pitch angle 27
θ poloidal angle 33
θ wave propagation angle relative to B 205
θ scattering angle 309
θ
res
resonant angle 207
List of symbols 515
ι rotational transform 88
κ curvature of magnetic field 122
κ coefficient of thermal conductivity 56
κ
α
kl
element of thermal conductivity tensor for species α 494
λ k
2
k
B
T/m
2
e
280
λ
c
collisional mean free path 63
λ
D
Debye length 8
µ
B
magnetic moment 16
µ coefficient of viscosity 54
µ
0
vacuum magnetic permeability 13
µ
α
klmn
fourth-order viscosity tensor for species α 495
µ
α
1,2,3,4,
independent elements of viscosity tensor 495
ν kinematic viscosity 192
ν
c
collision frequency 10
ν
ei
electron–ion collision frequency 10
ξ(r, t) displacement vector 109
α
ij
traceless pressure tensor for species α 481
ρ mass density 51
σ electrical conductivity tensor 203
σ collisional cross-section 3
σ electrical conductivity 56
σ(|g|,θ) differential scattering cross-section 309
σ
kl
element of electrical conductivity tensor 494
τ t − x/c 38
τ optical depth 332
τ
A
Alfv
´
en transit time 140
τ
B
bounce time 306
τ
c
collision time 62
τ
e(i)
electron (ion) collision time 62
τ
G
gravitational time scale 155
τ
H
hydrodynamic time 58
τ
L
Larmor period 32
τ
P
precession time 32
τ
R
resistive diffusion time 140
τ
longitudinal invariant transit time 29
magnetic flux 31
ij
element of stress tensor 52
φ magnetic scalar potential 113
φ(x) error function 314
516 List of symbols
φ(z) wave phase or eikonal 427
φ
ij
electrostatic potential between electrons i and j 468
χ
α
susceptibility of species α 362
χ
α
normalized deviation from local Maxwellian 489
(x) [φ(x) − xφ
(x)]/2x
2
314
ψ flux function 96
Larmor frequency 15
solid angle 326
relativistic Larmor frequency (
0
/γ ) 344
ω
L
L-mode cut-off frequency 210
ω
L
Langmuir wave frequency 394
ω
LH
lower hybrid resonance frequency 209
ω
p
plasma frequency 8
ω
R
R-mode cut-off frequency 210
ω
s
ion acoustic wave frequency 394
ω
T
transverse electromagnetic pump wave frequency 394
ω
UH
upper hybrid resonance frequency 209