2007

NRL PLASMA FORMULARY

J.D. Huba

Beam Physics Branch

Plasma Physics Division

Naval Research Laboratory

Washington, DC 20375

Supported by

The Office of Naval Research

1
CONTENTS

Numerical and Algebraic . . . . . . . . . . . . . . . . . . . . . 3

Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . 4

Differential Operators in Curvilinear Coordinates . . . . . . . . . . . 6

Dimensions and Units . . . . . . . . . . . . . . . . . . . . . . . 10

International System (SI) Nomenclature . . . . . . . . . . . . . . . 13

Metric Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Physical Constants (SI) . . . . . . . . . . . . . . . . . . . . . . 14

Physical Constants (cgs) . . . . . . . . . . . . . . . . . . . . . 16

Formula Conversion . . . . . . . . . . . . . . . . . . . . . . . 18

Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . 19

Electricity and Magnetism . . . . . . . . . . . . . . . . . . . . . 20

Electromagnetic Frequency/Wavelength Bands . . . . . . . . . . . . 21

AC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Dimensionless Numbers of Fluid Mechanics . . . . . . . . . . . . . 23

Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Fundamental Plasma Parameters . . . . . . . . . . . . . . . . . . 28

Plasma Dispersion Function . . . . . . . . . . . . . . . . . . . . 30

Collisions and Transport . . . . . . . . . . . . . . . . . . . . . 31

Approximate Magnitudes in Some Typical Plasmas . . . . . . . . . . 40

Ionospheric Parameters . . . . . . . . . . . . . . . . . . . . . . 42

Solar Physics Parameters . . . . . . . . . . . . . . . . . . . . . 43

Thermonuclear Fusion . . . . . . . . . . . . . . . . . . . . . . 44

Relativistic Electron Beams . . . . . . . . . . . . . . . . . . . . 46

Beam Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 48

Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Atomic Physics and Radiation . . . . . . . . . . . . . . . . . . . 53

Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 59

Complex (Dusty) Plasmas . . . . . . . . . . . . . . . . . . . . . 62

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2
NUMERICAL AND ALGEBRAIC

Gain in decibels of P2 relative to P1

G = 10 log10(P2/P1).

To within two percent

(2π)

1/2 ≈ 2.5; π2 ≈ 10; e3 ≈ 20; 210 ≈ 103

.

Euler-Mascheroni constant1 γ = 0.57722

Gamma Function Γ(x + 1) = xΓ(x):

Γ(1/6) = 5.5663 Γ(3/5) = 1.4892

Γ(1/5) = 4.5908 Γ(2/3) = 1.3541

Γ(1/4) = 3.6256 Γ(3/4) = 1.2254

Γ(1/3) = 2.6789 Γ(4/5) = 1.1642

Γ(2/5) = 2.2182 Γ(5/6) = 1.1288

Γ(1/2) = 1.7725 = √

π Γ(1) = 1.0

Binomial Theorem (good for | x |< 1 or α = positive integer):

(1 + x)

α =

X∞

k=0

α

k



x

k ≡ 1 + αx +

α(α − 1)

2!

x

2 +

α(α − 1)(α − 2)

3!

x

3 + . . . .

Rothe-Hagen identity2(good for all complex x, y, z except when singular):

Xn

k=0

x

x + kz

x + kz

k

 y

y + (n − k)z

y + (n − k)z

n − k



=

x + y

x + y + nz

x + y + nz

n



.

Newberger’s summation formula3[good for μ nonintegral, Re (α + β) > −1]:

X∞

n=−∞

(−1)nJα−γn(z)Jβ+γn(z)

n + μ

=

π

sin μπ

Jα+γμ(z)Jβ−γμ(z).

3
VECTOR IDENTITIES4

Notation: f, g, are scalars; A, B, etc., are vectors; T is a tensor; I is the unit

dyad.

(1) A·B×C = A×B·C = B·C×A = B×C·A = C·A×B = C×A·B

(2) A × (B × C) = (C × B) × A = (A · C)B − (A · B)C

(3) A × (B × C) + B × (C × A) + C × (A × B) = 0

(4) (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C)

(5) (A × B) × (C × D) = (A × B · D)C − (A × B · C)D

(6) ∇(fg) = ∇(gf) = f∇g + g∇f

(7) ∇ · (fA) = f∇ · A + A · ∇f

(8) ∇ × (fA) = f∇ × A + ∇f × A

(9) ∇ · (A × B) = B · ∇ × A − A · ∇ × B

(10) ∇ × (A × B) = A(∇ · B) − B(∇ · A) + (B · ∇)A − (A · ∇)B

(11) A × (∇ × B) = (∇B) · A − (A · ∇)B

(12) ∇(A · B) = A × (∇ × B) + B × (∇ × A) + (A · ∇)B + (B · ∇)A

(13) ∇2f = ∇ · ∇f

(14) ∇2A = ∇(∇ · A) − ∇ × ∇ × A

(15) ∇ × ∇f = 0

(16) ∇ · ∇ × A = 0

If e1, e2, e3 are orthonormal unit vectors, a second-order tensor T can be

written in the dyadic form

(17) T =

P

i,j

Tijeiej

In cartesian coordinates the divergence of a tensor is a vector with components

(18) (∇·T)i =

P

j

(∂Tji/∂xj )

[This definition is required for consistency with Eq. (29)]. In general

(19) ∇ · (AB) = (∇ · A)B + (A · ∇)B

(20) ∇ · (fT) = ∇f·T+f∇·T

4
Let r = ix + jy + kz be the radius vector of magnitude r, from the origin to

the point x, y, z. Then

(21) ∇ · r = 3

(22) ∇ × r = 0

(23) ∇r = r/r

(24) ∇(1/r) = −r/r3

(25) ∇ · (r/r3) = 4πδ(r)

(26) ∇r = I

If V is a volume enclosed by a surface S and dS = ndS, where n is the unit

normal outward from V,

(27) Z

V

dV ∇f =

Z

S

dSf

(28) Z

V

dV ∇ · A =

Z

S

dS · A

(29) Z

V

dV ∇·T =

Z

S

dS ·T

(30) Z

V

dV ∇ × A =

Z

S

dS × A

(31) Z

V

dV (f∇

2

g − g∇

2

f) = Z

S

dS · (f∇g − g∇f)

(32) Z

V

dV (A · ∇ × ∇ × B − B · ∇ × ∇ × A)

=

Z

S

dS · (B × ∇ × A − A × ∇ × B)

If S is an open surface bounded by the contour C, of which the line element is

dl,

(33) Z

S

dS × ∇f =

I

C

dlf

5
(34) Z

S

dS · ∇ × A =

I

C

dl · A

(35) Z

S

(dS × ∇) × A =

I

C

dl × A

(36) Z

S

dS · (∇f × ∇g) = I

C

fdg = −

I

C

gdf

DIFFERENTIAL OPERATORS IN

CURVILINEAR COORDINATES5

Cylindrical Coordinates

Divergence

∇ · A =

1

r

∂

∂r

(rAr) + 1

r

∂Aφ

∂φ

+

∂Az

∂z

Gradient

(∇f)r =

∂f

∂r

; (∇f)φ =

1

r

∂f

∂φ

; (∇f)z =

∂f

∂z

Curl

(∇ × A)r =

1

r

∂Az

∂φ

−

∂Aφ

∂z

(∇ × A)φ =

∂Ar

∂z

−

∂Az

∂r

(∇ × A)z =

1

r

∂

∂r

(rAφ) −

1

r

∂Ar

∂φ

Laplacian

∇

2

f =

1

r

∂

∂r 

r

∂f

∂r 

+

1

r

2

∂

2

f

∂φ2

+

∂

2

f

∂z2

6
Laplacian of a vector

(∇

2A)r = ∇2Ar −

2

r

2

∂Aφ

∂φ

−

Ar

r

2

(∇

2A)φ = ∇2Aφ +

2

r

2

∂Ar

∂φ

−

Aφ

r

2

(∇

2A)z = ∇2Az

Components of (A · ∇)B

(A · ∇B)r = Ar

∂Br

∂r

+

Aφ

r

∂Br

∂φ

+ Az

∂Br

∂z

−

AφBφ

r

(A · ∇B)φ = Ar

∂Bφ

∂r

+

Aφ

r

∂Bφ

∂φ

+ Az

∂Bφ

∂z

+

AφBr

r

(A · ∇B)z = Ar

∂Bz

∂r

+

Aφ

r

∂Bz

∂φ

+ Az

∂Bz

∂z

Divergence of a tensor

(∇ · T)r =

1

r

∂

∂r

(rTrr) + 1

r

∂Tφr

∂φ

+

∂Tzr

∂z

−

Tφφ

r

(∇ · T)φ =

1

r

∂

∂r

(rTrφ) + 1

r

∂Tφφ

∂φ

+

∂Tzφ

∂z

+

Tφr

r

(∇ · T)z =

1

r

∂

∂r

(rTrz) + 1

r

∂Tφz

∂φ

+

∂Tzz

∂z

7
Spherical Coordinates

Divergence

∇ · A =

1

r

2

∂

∂r

(r

2Ar) + 1

r sin θ

∂

∂θ

(sin θAθ) + 1

r sin θ

∂Aφ

∂φ

Gradient

(∇f)r =

∂f

∂r

; (∇f)θ =

1

r

∂f

∂θ

; (∇f)φ =

1

r sin θ

∂f

∂φ

Curl

(∇ × A)r =

1

r sin θ

∂

∂θ

(sin θAφ) −

1

r sin θ

∂Aθ

∂φ

(∇ × A)θ =

1

r sin θ

∂Ar

∂φ

−

1

r

∂

∂r

(rAφ)

(∇ × A)φ =

1

r

∂

∂r

(rAθ) −

1

r

∂Ar

∂θ

Laplacian

∇

2

f =

1

r

2

∂

∂r 

r

2 ∂f

∂r 

+

1

r

2 sin θ

∂

∂θ 

sin θ

∂f

∂θ 

+

1

r

2 sin2 θ

∂

2

f

∂φ2

Laplacian of a vector

(∇

2A)r = ∇2Ar −

2Ar

r

2

−

2

r

2

∂Aθ

∂θ

−

2 cot θAθ

r

2

−

2

r

2 sin θ

∂Aφ

∂φ

(∇

2A)θ = ∇2Aθ +

2

r

2

∂Ar

∂θ

−

Aθ

r

2 sin2 θ

−

2 cos θ

r

2 sin2 θ

∂Aφ

∂φ

(∇

2A)φ = ∇2Aφ −

Aφ

r

2 sin2 θ

+

2

r

2 sin θ

∂Ar

∂φ

+

2 cos θ

r

2 sin2 θ

∂Aθ

∂φ

8
Components of (A · ∇)B

(A · ∇B)r = Ar

∂Br

∂r

+

Aθ

r

∂Br

∂θ

+

Aφ

r sin θ

∂Br

∂φ

−

AθBθ + AφBφ

r

(A · ∇B)θ = Ar

∂Bθ

∂r

+

Aθ

r

∂Bθ

∂θ

+

Aφ

r sin θ

∂Bθ

∂φ

+

AθBr

r

−

cot θAφBφ

r

(A · ∇B)φ = Ar

∂Bφ

∂r

+

Aθ

r

∂Bφ

∂θ

+

Aφ

r sin θ

∂Bφ

∂φ

+

AφBr

r

+

cot θAφBθ

r

Divergence of a tensor

(∇ · T)r =

1

r

2

∂

∂r

(r

2Trr) + 1

r sin θ

∂

∂θ

(sin θTθr)

+

1

r sin θ

∂Tφr

∂φ

−

Tθθ + Tφφ

r

(∇ · T)θ =

1

r

2

∂

∂r

(r

2Trθ) + 1

r sin θ

∂

∂θ

(sin θTθθ)

+

1

r sin θ

∂Tφθ

∂φ

+

Tθr

r

−

cot θTφφ

r

(∇ · T)φ =

1

r

2

∂

∂r

(r

2Trφ) + 1

r sin θ

∂

∂θ

(sin θTθφ)

+

1

r sin θ

∂Tφφ

∂φ

+

Tφr

r

+

cot θTφθ

r

9
DIMENSIONS AND UNITS

To get the value of a quantity in Gaussian units, multiply the value expressed in SI units by the conversion factor. Multiples of 3 in the conversion

factors result from approximating the speed of light c = 2.9979 × 1010 cm/sec

≈ 3 × 1010 cm/sec.

Dimensions

Physical Sym- SI Conversion Gaussian

Quantity bol SI Gaussian Units Factor Units

Capacitance C

t

2

q

2

ml2

l farad 9 × 1011 cm

Charge q q

m1/2l

3/2

t

coulomb 3 × 109statcoulomb

Charge ρ

q

l

3

m1/2

l

3/2t

coulomb 3 × 103statcoulomb

density /m3/cm3

Conductance

tq2

ml2

l

t

siemens 9 × 1011 cm/sec

Conductivity σ

tq2

ml3

1

t

siemens 9 × 109sec−1

/m

Current I, i

q

t

m1/2l

3/2

t

2

ampere 3 × 109statampere

Current J,j

q

l

2t

m1/2

l

1/2t2

ampere 3 × 105statampere

density /m2/cm2

Density ρ

m

l

3

m

l

3

kg/m310−3g/cm3

Displacement D

q

l

2

m1/2

l

1/2t

coulomb 12π × 105statcoulomb

/m2/cm2

Electric field E

ml

t

2q

m1/2

l

1/2t

volt/m 1

3

× 10−4statvolt/cm

Electro- E,

ml2

t

2q

m1/2l

1/2

t

volt

1

3

× 10−2statvolt

motance Emf

Energy U, W

ml2

t

2

ml2

t

2

joule 107erg

Energy w, ǫ

m

lt2

m

lt2

joule/m310 erg/cm3

density

10
Dimensions

Physical Sym- SI Conversion Gaussian

Quantity bol SI Gaussian Units Factor Units

Force F

ml

t

2

ml

t

2

newton 105 dyne

Frequency f, ν

1

t

1

t

hertz 1 hertz

Impedance Z

ml2

tq2

t

l

ohm

1

9

× 10−11 sec/cm

Inductance L

ml2

q

2

t

2

l

henry

1

9

× 10−11

sec2/cm

Length l l l meter (m) 102centimeter

(cm)

Magnetic H

q

lt

m1/2

l

1/2t

ampere– 4π × 10−3oersted

intensity turn/m

Magnetic flux Φ

ml2

tq

m1/2l

3/2

t

weber 108 maxwell

Magnetic B

m

tq

m1/2

l

1/2t

tesla 104gauss

induction

Magnetic m, μ

l

2

q

t

m1/2l

5/2

t

ampere–m2103oersted–

moment cm3

Magnetization M

q

lt

m1/2

l

1/2t

ampere– 4π × 10−3oersted

turn/m

Magneto- M,

q

t

m1/2l

1/2

t

2

ampere–

4π

10

gilbert

motance Mmf turn

Mass m, M m m kilogram 103gram (g)

(kg)

Momentum p, P

ml

t

ml

t

kg–m/s 105g–cm/sec

Momentum

m

l

2t

m

l

2t

kg/m2–s 10−1g/cm2–sec

density

Permeability μ

ml

q

2

1 henry/m 1

4π

× 107 —

11
Dimensions

Physical Sym- SI Conversion Gaussian

Quantity bol SI Gaussian Units Factor Units

Permittivity ǫ

t

2

q

2

ml3

1 farad/m 36π × 109 —

Polarization P

q

l

2

m1/2

l

1/2t

coulomb/m23 × 105statcoulomb

/cm2

Potential V, φ

ml2

t

2q

m1/2l

1/2

t

volt

1

3

× 10−2

statvolt

Power P

ml2

t

3

ml2

t

3

watt 107erg/sec

Power

m

lt3

m

lt3

watt/m310 erg/cm3–sec

density

Pressure p, P

m

lt2

m

lt2

pascal 10 dyne/cm2

Reluctance R

q

2

ml2

1

l

ampere–turn 4π × 10−9cm−1

/weber

Resistance R

ml2

tq2

t

l

ohm

1

9

× 10−11 sec/cm

Resistivity η, ρ

ml3

tq2

t ohm–m

1

9

× 10−9sec

Thermal con- κ, k

ml

t

3

ml

t

3

watt/m– 105erg/cm–sec–

ductivity deg (K) deg (K)

Time t t t second (s) 1 second (sec)

Vector A

ml

tq

m1/2l

1/2

t

weber/m 106gauss–cm

potential

Velocity v

l

t

l

t

m/s 102cm/sec

Viscosity η, μ

m

lt

m

lt

kg/m–s 10 poise

Vorticity ζ

1

t

1

t

s

−1

1 sec−1

Work W

ml2

t

2

ml2

t

2

joule 107erg

12
INTERNATIONAL SYSTEM (SI) NOMENCLATURE6

Physical Name Symbol Physical Name Symbol

Quantity of Unit for Unit Quantity of Unit for Unit

*length meter m electric volt V

potential

*mass kilogram kg

electric ohm Ω

*time second s resistance

*current ampere A electric siemens S

conductance

*temperature kelvin K

electric farad F

*amount of mole mol capacitance

substance

magnetic flux weber Wb

*luminous candela cd

intensity magnetic henry H

inductance

†plane angle radian rad

magnetic tesla T

†solid angle steradian sr intensity

frequency hertz Hz luminous flux lumen lm

energy joule J illuminance lux lx

force newton N activity (of a becquerel Bq

radioactive

pressure pascal Pa source)

power watt W absorbed dose gray Gy

(of ionizing

electric charge coulomb C radiation)

*SI base unit †Supplementary unit

METRIC PREFIXES

Multiple Prefix Symbol Multiple Prefix Symbol

10−1 deci d 10 deca da

10−2centi c 102 hecto h

10−3 milli m 103 kilo k

10−6 micro μ 106 mega M

10−9 nano n 109giga G

10−12 pico p 1012 tera T

10−15 femto f 1015 peta P

10−18 atto a 1018 exa E

13
PHYSICAL CONSTANTS (SI)7

Physical Quantity Symbol Value Units

Boltzmann constant k 1.3807 × 10−23 J K−1

Elementary charge e 1.6022 × 10−19 C

Electron mass me 9.1094 × 10−31 kg

Proton mass mp 1.6726 × 10−27 kg

Gravitational constant G 6.6726 × 10−11 m3s

−2kg−1

Planck constant h 6.6261 × 10−34 J s

h ̄ = h/2π 1.0546 × 10−34 J s

Speed of light in vacuum c 2.9979 × 108 m s−1

Permittivity of ǫ0 8.8542 × 10−12 F m−1

free space

Permeability of μ0 4π × 10−7 H m−1

free space

Proton/electron mass mp/me 1.8362 × 103

ratio

Electron charge/mass e/me 1.7588 × 1011 C kg−1

ratio

Rydberg constant R∞ =

me4

8ǫ0

2ch3

1.0974 × 107 m−1

Bohr radius a0 = ǫ0h

2/πme2

5.2918 × 10−11 m

Atomic cross section πa0

2

8.7974 × 10−21 m2

Classical electron radius re = e

2/4πǫ0mc2

2.8179 × 10−15 m

Thomson cross section (8π/3)re

2

6.6525 × 10−29 m2

Compton wavelength of h/mec 2.4263 × 10−12 m

electron h/m  ̄ ec 3.8616 × 10−13 m

Fine-structure constant α = e

2/2ǫ0hc 7.2974 × 10−3

α

−1

137.04

First radiation constant c1 = 2πhc23.7418 × 10−16 W m2

Second radiation c2 = hc/k 1.4388 × 10−2 m K

constant

Stefan-Boltzmann σ 5.6705 × 10−8 W m−2K−4

constant

14
Physical Quantity Symbol Value Units

Wavelength associated λ0 = hc/e 1.2398 × 10−6 m

with 1 eV

Frequency associated ν0 = e/h 2.4180 × 1014 Hz

with 1 eV

Wave number associated k0 = e/hc 8.0655 × 105 m−1

with 1 eV

Energy associated with hν0 1.6022 × 10−19 J

1 eV

Energy associated with hc 1.9864 × 10−25 J

1 m−1

Energy associated with me3/8ǫ0

2h2

13.606 eV

1 Rydberg

Energy associated with k/e 8.6174 × 10−5eV

1 Kelvin

Temperature associated e/k 1.1604 × 104 K

with 1 eV

Avogadro number NA 6.0221 × 1023 mol−1

Faraday constant F = NAe 9.6485 × 104 C mol−1

Gas constant R = NAk 8.3145 J K−1mol−1

Loschmidt’s number n0 2.6868 × 1025 m−3

(no. density at STP)

Atomic mass unit mu 1.6605 × 10−27 kg

Standard temperature T0 273.15 K

Atmospheric pressure p0 = n0kT0 1.0133 × 105 Pa

Pressure of 1 mm Hg 1.3332 × 102 Pa

(1 torr)

Molar volume at STP V0 = RT0/p0 2.2414 × 10−2 m3

Molar weight of air Mair 2.8971 × 10−2 kg

calorie (cal) 4.1868 J

Gravitational g 9.8067 m s−2

acceleration

15
PHYSICAL CONSTANTS (cgs)7

Physical Quantity Symbol Value Units

Boltzmann constant k 1.3807 × 10−16 erg/deg (K)

Elementary charge e 4.8032 × 10−10 statcoulomb

(statcoul)

Electron mass me 9.1094 × 10−28 g

Proton mass mp 1.6726 × 10−24 g

Gravitational constant G 6.6726 × 10−8 dyne-cm2/g

2

Planck constant h 6.6261 × 10−27 erg-sec

h ̄ = h/2π 1.0546 × 10−27 erg-sec

Speed of light in vacuum c 2.9979 × 1010 cm/sec

Proton/electron mass mp/me 1.8362 × 103

ratio

Electron charge/mass e/me 5.2728 × 1017 statcoul/g

ratio

Rydberg constant R∞ =

2π

2me4

ch3

1.0974 × 105cm−1

Bohr radius a0 =  ̄h

2/me2

5.2918 × 10−9cm

Atomic cross section πa0

2

8.7974 × 10−17 cm2

Classical electron radius re = e

2/mc2

2.8179 × 10−13 cm

Thomson cross section (8π/3)re

2

6.6525 × 10−25 cm2

Compton wavelength of h/mec 2.4263 × 10−10 cm

electron h/m  ̄ ec 3.8616 × 10−11 cm

Fine-structure constant α = e

2/hc  ̄ 7.2974 × 10−3

α

−1

137.04

First radiation constant c1 = 2πhc23.7418 × 10−5erg-cm2/sec

Second radiation c2 = hc/k 1.4388 cm-deg (K)

constant

Stefan-Boltzmann σ 5.6705 × 10−5erg/cm2-

constant sec-deg4

Wavelength associated λ0 1.2398 × 10−4cm

with 1 eV

16
Physical Quantity Symbol Value Units

Frequency associated ν0 2.4180 × 1014 Hz

with 1 eV

Wave number associated k0 8.0655 × 103cm−1

with 1 eV

Energy associated with 1.6022 × 10−12 erg

1 eV

Energy associated with 1.9864 × 10−16 erg

1 cm−1

Energy associated with 13.606 eV

1 Rydberg

Energy associated with 8.6174 × 10−5eV

1 deg Kelvin

Temperature associated 1.1604 × 104 deg (K)

with 1 eV

Avogadro number NA 6.0221 × 1023 mol−1

Faraday constant F = NAe 2.8925 × 1014 statcoul/mol

Gas constant R = NAk 8.3145 × 107erg/deg-mol

Loschmidt’s number n0 2.6868 × 1019 cm−3

(no. density at STP)

Atomic mass unit mu 1.6605 × 10−24 g

Standard temperature T0 273.15 deg (K)

Atmospheric pressure p0 = n0kT0 1.0133 × 106 dyne/cm2

Pressure of 1 mm Hg 1.3332 × 103 dyne/cm2

(1 torr)

Molar volume at STP V0 = RT0/p0 2.2414 × 104cm3

Molar weight of air Mair 28.971 g

calorie (cal) 4.1868 × 107erg

Gravitational g 980.67 cm/sec2

acceleration

17
FORMULA CONVERSION8

Here α = 102cm m−1, β = 107erg J−1, ǫ0 = 8.8542 × 10−12 F m−1,

μ0 = 4π×10−7 H m−1, c = (ǫ0μ0)

−1/2 = 2.9979×108 m s−1

, and  ̄h = 1.0546×

10−34 J s. To derive a dimensionally correct SI formula from one expressed in

Gaussian units, substitute for each quantity according to Q ̄ = kQ ̄ , where k ̄ is

the coefficient in the second column of the table corresponding to Q (overbars

denote variables expressed in Gaussian units). Thus, the formula  ̄a0 = h ̄ ̄2/m ̄ e ̄

2

for the Bohr radius becomes αa0 = ( ̄hβ)

2/[(mβ/α2

)(e

2αβ/4πǫ0)], or a0 =

ǫ0h

2/πme2

. To go from SI to natural units in which  ̄h = c = 1 (distinguished

by a circumflex), use Q = kˆ−1Qˆ, where kˆ is the coefficient corresponding to

Q in the third column. Thus ˆa0 = 4πǫ0h ̄

2/[( ˆmh/c  ̄ )(ˆe2

ǫ0hc  ̄ )] = 4π/mˆ eˆ

2

. (In

transforming from SI units, do not substitute for ǫ0, μ0, or c.)

Physical Quantity Gaussian Units to SI Natural Units to SI

Capacitance α/4πǫ0 ǫ0

−1

Charge (αβ/4πǫ0)

1/2

(ǫ0hc ̄ )

−1/2

Charge density (β/4πα5ǫ0)

1/2

(ǫ0hc ̄ )

−1/2

Current (αβ/4πǫ0)

1/2

(μ0/hc ̄ )

1/2

Current density (β/4πα3ǫ0)

1/2

(μ0/hc ̄ )

1/2

Electric field (4πβǫ0/α3)

1/2

(ǫ0/hc ̄ )

1/2

Electric potential (4πβǫ0/α)

1/2

(ǫ0/hc ̄ )

1/2

Electric conductivity (4πǫ0)

−1

ǫ0

−1

Energy β ( ̄hc)

−1

Energy density β/α3( ̄hc)

−1

Force β/α ( ̄hc)

−1

Frequency 1 c

−1

Inductance 4πǫ0/α μ0

−1

Length α 1

Magnetic induction (4πβ/α3μ0)

1/2

(μ0hc ̄ )

−1/2

Magnetic intensity (4πμ0β/α3)

1/2

(μ0/hc ̄ )

1/2

Mass β/α2c/h ̄

Momentum β/α h ̄

−1

Power β ( ̄hc2)

−1

Pressure β/α3( ̄hc)

−1

Resistance 4πǫ0/α (ǫ0/μ0)

1/2

Time 1 c

Velocity α c

−1

18
MAXWELL’S EQUATIONS

Name or Description SI Gaussian

Faraday’s law ∇ × E = −

∂B

∂t

∇ × E = −

1

c

∂B

∂t

Ampere’s law ∇ × H =

∂D

∂t

+ J ∇ × H =

1

c

∂D

∂t

+

4π

c

J

Poisson equation ∇ · D = ρ ∇ · D = 4πρ

[Absence of magnetic ∇ · B = 0 ∇ · B = 0

monopoles]

Lorentz force on q (E + v × B) q



E +

1

c

v × B



charge q

Constitutive D = ǫE D = ǫE

relations B = μH B = μH

In a plasma, μ ≈ μ0 = 4π × 10−7 H m−1(Gaussian units: μ ≈ 1). The

permittivity satisfies ǫ ≈ ǫ0 = 8.8542 × 10−12 F m−1(Gaussian: ǫ ≈ 1)

provided that all charge is regarded as free. Using the drift approximation

v⊥ = E × B/B2to calculate polarization charge density gives rise to a dielectric constant K ≡ ǫ/ǫ0 = 1 + 36π × 109ρ/B2

(SI) = 1 + 4πρc2/B2(Gaussian),

where ρ is the mass density.

The electromagnetic energy in volume V is given by

W =

1

2

Z

V

dV (H · B + E · D) (SI)

=

1

8π

Z

V

dV (H · B + E · D) (Gaussian).

Poynting’s theorem is

∂W

∂t

+

Z

S

N · dS = −

Z

V

dV J · E,

where S is the closed surface bounding V and the Poynting vector (energy flux

across S) is given by N = E × H (SI) or N = cE × H/4π (Gaussian).

19
ELECTRICITY AND MAGNETISM

In the following, ǫ = dielectric permittivity, μ = permeability of conductor, μ

′ = permeability of surrounding medium, σ = conductivity, f = ω/2π =

radiation frequency, κm = μ/μ0 and κe = ǫ/ǫ0. Where subscripts are used,

‘1’ denotes a conducting medium and ‘2’ a propagating (lossless dielectric)

medium. All units are SI unless otherwise specified.

Permittivity of free space ǫ0 = 8.8542 × 10−12 F m−1

Permeability of free space μ0 = 4π × 10−7 H m−1

= 1.2566 × 10−6 H m−1

Resistance of free space R0 = (μ0/ǫ0)

1/2 = 376.73 Ω

Capacity of parallel plates of area C = ǫA/d

A, separated by distance d

Capacity of concentric cylinders C = 2πǫl/ ln(b/a)

of length l, radii a, b

Capacity of concentric spheres of C = 4πǫab/(b − a)

radii a, b

Self-inductance of wire of length L = μl

l, carrying uniform current

Mutual inductance of parallel wires L = (μ

′

l/4π) [1 + 4 ln(d/a)]

of length l, radius a, separated

by distance d

Inductance of circular loop of radius L = b



μ

′

[ln(8b/a) − 2] + μ/4

	

b, made of wire of radius a,

carrying uniform current

Relaxation time in a lossy medium τ = ǫ/σ

Skin depth in a lossy medium δ = (2/ωμσ)

1/2 = (πfμσ)−1/2

Wave impedance in a lossy medium Z = [μ/(ǫ + iσ/ω)]1/2

Transmission coefficient at T = 4.22 × 10−4(fκm1κe2/σ)

1/2

conducting surface9

(good only for T ≪ 1)

Field at distance r from straight wire Bθ = μI/2πr tesla

carrying current I (amperes) = 0.2I/r gauss (r in cm)

Field at distance z along axis from Bz = μa2I/[2(a

2 + z2

)

3/2

]

circular loop of radius a

carrying current I

20
ELECTROMAGNETIC FREQUENCY/

WAVELENGTH BANDS10

Frequency Range Wavelength Range

Designation

Lower Upper Lower Upper

ULF* 30 Hz 10 Mm

VF* 30 Hz 300 Hz 1 Mm 10 Mm

ELF 300 Hz 3 kHz 100 km 1 Mm

VLF 3 kHz 30 kHz 10 km 100 km

LF 30 kHz 300 kHz 1 km 10 km

MF 300 kHz 3 MHz 100 m 1 km

HF 3 MHz 30 MHz 10 m 100 m

VHF 30 MHz 300 MHz 1 m 10 m

UHF 300 MHz 3 GHz 10 cm 1 m

SHF† 3 GHz 30 GHz 1 cm 10 cm

S 2.6 3.95 7.6 11.5

G 3.95 5.85 5.1 7.6

J 5.3 8.2 3.7 5.7

H 7.05 10.0 3.0 4.25

X 8.2 12.4 2.4 3.7

M 10.0 15.0 2.0 3.0

P 12.4 18.0 1.67 2.4

K 18.0 26.5 1.1 1.67

R 26.5 40.0 0.75 1.1

EHF 30 GHz 300 GHz 1 mm 1 cm

Submillimeter 300 GHz 3 THz 100 μm 1 mm

Infrared 3 THz 430 THz 700 nm 100 μm

Visible 430 THz 750 THz 400 nm 700 nm

Ultraviolet 750 THz 30 PHz 10 nm 400 nm

X Ray 30 PHz 3 EHz 100 pm 10 nm

Gamma Ray 3 EHz 100 pm

In spectroscopy the angstrom is sometimes used (1 ̊A = 10−8cm = 0.1 nm).

*The boundary between ULF and VF (voice frequencies) is variously defined.

†The SHF (microwave) band is further subdivided approximately as shown.11

21
AC CIRCUITS

For a resistance R, inductance L, and capacitance C in series with

a voltage source V = V0 exp(iωt) (here i =

√

−1), the current is given

by I = dq/dt, where q satisfies

L

d

2

q

dt2

+ R

dq

dt

+

q

C

= V.

Solutions are q(t) = qs + qt, I(t) = Is + It, where the steady state is

Is = iωqs = V/Z in terms of the impedance Z = R + i(ωL − 1/ωC) and

It = dqt/dt. For initial conditions q(0) ≡ q0 =  ̄q0 + qs, I(0) ≡ I0, the

transients can be of three types, depending on ∆ = R2 − 4L/C:

(a) Overdamped, ∆ > 0

qt =

I0 + γ+q ̄0

γ+ − γ−

exp(−γ−t) −

I0 + γ−q ̄0

γ+ − γ−

exp(−γ+t),

It =

γ+(I0 + γ−q ̄0)

γ+ − γ−

exp(−γ+t) −

γ−(I0 + γ+q ̄0)

γ+ − γ−

exp(−γ−t),

where γ± = (R ± ∆1/2)/2L;

(b) Critically damped, ∆ = 0

qt = [ ̄q0 + (I0 + γRq ̄0)t] exp(−γRt),

It = [I0 − (I0 + γRq ̄0)γRt] exp(−γRt),

where γR = R/2L;

(c) Underdamped, ∆ < 0

qt =

h

γRq ̄0 + I0

ω1

sin ω1t +  ̄q0 cos ω1t

i

exp(−γRt),

It =

h

I0 cos ω1t −

(ω1

2 + γR

2

) ̄q0 + γRI0

ω1

sin(ω1t)

i

exp(−γRt),

Here ω1 = ω0(1 − R

2C/4L)1/2

, where ω0 = (LC)

−1/2

is the resonant

frequency. At ω = ω0, Z = R. The quality of the circuit is Q = ω0L/R.

Instability results when L, R, C are not all of the same sign.

22
DIMENSIONLESS NUMBERS OF FLUID MECHANICS12

Name(s) Symbol Definition Significance

Alfv ́en, Al, Ka VA/V *(Magnetic force/

K ́arm ́an inertial force)1/2

Bond Bd (ρ

′ − ρ)L2

g/Σ Gravitational force/

surface tension

Boussinesq B V/(2gR)

1/2

(Inertial force/

gravitational force)1/2

Brinkman Br μV 2/k∆T Viscous heat/conducted heat

Capillary Cp μV/Σ Viscous force/surface tension

Carnot Ca (T2 − T1)/T2 Theoretical Carnot cycle

efficiency

Cauchy, Cy, Hk ρV 2/Γ = M2Inertial force/

Hooke compressibility force

Chandra- Ch B

2L2/ρνη Magnetic force/dissipative

sekhar forces

Clausius Cl LV 3ρ/k∆T Kinetic energy flow rate/heat

conduction rate

Cowling C (VA/V )

2 = Al2 Magnetic force/inertial force

Crispation Cr μκ/ΣL Effect of diffusion/effect of

surface tension

Dean D D3/2V/ν(2r)

1/2 Transverse flow due to

curvature/longitudinal flow

[Drag CD (ρ

′ − ρ)Lg/ Drag force/inertial force

coefficient] ρ

′V2

Eckert E V

2/cp∆T Kinetic energy/change in

thermal energy

Ekman Ek (ν/2ΩL

2

)

1/2 = (Viscous force/Coriolis force)1/2

(Ro/Re)1/2

Euler Eu ∆p/ρV 2 Pressure drop due to friction/

dynamic pressure

Froude Fr V/(gL)

1/2

†(Inertial force/gravitational or

V/NL buoyancy force)1/2

Gay–Lussac Ga 1/β∆T Inverse of relative change in

volume during heating

Grashof Gr gL3β∆T/ν2 Buoyancy force/viscous force

[Hall CH λ/rL Gyrofrequency/

coefficient] collision frequency

*(†) Also defined as the inverse (square) of the quantity shown.

23
Name(s) Symbol Definition Significance

Hartmann H BL/(μη)

1/2 = (Magnetic force/

(Rm Re C)1/2 dissipative force)1/2

Knudsen Kn λ/L Hydrodynamic time/

collision time

Lewis Le κ/D *Thermal conduction/molecular

diffusion

Lorentz Lo V/c Magnitude of relativistic effects

Lundquist Lu μ0LVA/η = J × B force/resistive magnetic

Al Rm diffusion force

Mach M V/CS Magnitude of compressibility

effects

Magnetic Mm V/VA = Al−1(Inertial force/magnetic force)1/2

Mach

Magnetic Rm μ0LV/η Flow velocity/magnetic diffusion

Reynolds velocity

Newton Nt F/ρL2V

2

Imposed force/inertial force

Nusselt N αL/k Total heat transfer/thermal

conduction

P ́eclet Pe LV/κ Heat convection/heat conduction

Poisseuille Po D2∆p/μLV Pressure force/viscous force

Prandtl Pr ν/κ Momentum diffusion/

heat diffusion

Rayleigh Ra gH3β∆T/νκ Buoyancy force/diffusion force

Reynolds Re LV/ν Inertial force/viscous force

Richardson Ri (NH/∆V )

2 Buoyancy effects/

vertical shear effects

Rossby Ro V/2ΩL sin Λ Inertial force/Coriolis force

Schmidt Sc ν/D Momentum diffusion/

molecular diffusion

Stanton St α/ρcpV Thermal conduction loss/

heat capacity

Stefan Sf σLT 3/k Radiated heat/conducted heat

Stokes S ν/L2f Viscous damping rate/

vibration frequency

Strouhal Sr fL/V Vibration speed/flow velocity

Taylor Ta (2ΩL

2/ν)2 Centrifugal force/viscous force

R

1/2

(∆R)

3/2

(Centrifugal force/

·(Ω/ν) viscous force)1/2

Thring, Th, Bo ρcpV/ǫσT 3 Convective heat transport/

Boltzmann radiative heat transport

Weber W ρLV 2/Σ Inertial force/surface tension

24
Nomenclature:

B Magnetic induction

Cs, c Speeds of sound, light

cp Specific heat at constant pressure (units m2s

−2 K−1

)

D = 2R Pipe diameter

F Imposed force

f Vibration frequency

g Gravitational acceleration

H, L Vertical, horizontal length scales

k = ρcpκ Thermal conductivity (units kg m−1s

−2

)

N = (g/H)

1/2 Brunt–V ̈ais ̈al ̈a frequency

R Radius of pipe or channel

r Radius of curvature of pipe or channel

rL Larmor radius

T Temperature

V Characteristic flow velocity

VA = B/(μ0ρ)

1/2 Alfv ́en speed

α Newton’s-law heat coefficient, k

∂T

∂x

= α∆T

β Volumetric expansion coefficient, dV/V = βdT

Γ Bulk modulus (units kg m−1s

−2

)

∆R, ∆V, ∆p, ∆T Imposed differences in two radii, velocities,

pressures, or temperatures

ǫ Surface emissivity

η Electrical resistivity

κ, D Thermal, molecular diffusivities (units m2s

−1

)

Λ Latitude of point on earth’s surface

λ Collisional mean free path

μ = ρν Viscosity

μ0 Permeability of free space

ν Kinematic viscosity (units m2s

−1

)

ρ Mass density of fluid medium

ρ

′ Mass density of bubble, droplet, or moving object

Σ Surface tension (units kg s−2)

σ Stefan–Boltzmann constant

Ω Solid-body rotational angular velocity

25
SHOCKS

At a shock front propagating in a magnetized fluid at an angle θ with

respect to the magnetic induction B, the jump conditions are 13,14

(1) ρU =  ̄ρU ̄ ≡ q;

(2) ρU2 + p + B 2

⊥ /2μ =  ̄ρU ̄ 2 +  ̄p + B ̄ 2⊥ /2μ;

(3) ρUV − BkB⊥/μ =  ̄ρU ̄V ̄ − B ̄

kB ̄⊥/μ;

(4) Bk = B ̄

k;

(5) UB⊥ − V Bk = U ̄B ̄⊥ − V ̄ B ̄

k;

(6) 1

2

(U

2 + V2

) + w + (UB 2

⊥ − V BkB⊥)/μρU

= 1

2

(U ̄ 2 + V ̄ 2) +  ̄w + (U ̄B ̄ 2

⊥ − V ̄ B ̄

kB ̄⊥)/μρ ̄U ̄.

Here U and V are components of the fluid velocity normal and tangential to

the front in the shock frame; ρ = 1/υ is the mass density; p is the pressure;

B⊥ = B sin θ, Bk = B cos θ; μ is the magnetic permeability (μ = 4π in cgs

units); and the specific enthalpy is w = e + pυ, where the specific internal

energy e satisfies de = Tds − pdυ in terms of the temperature T and the

specific entropy s. Quantities in the region behind (downstream from) the

front are distinguished by a bar. If B = 0, then15

(7) U − U ̄ = [( ̄p − p)(υ − υ ̄)]1/2;

(8) ( ̄p − p)(υ − υ ̄)

−1 = q2

;

(9)  ̄w − w = 1

2

( ̄p − p)(υ +  ̄υ);

(10)  ̄e − e = 1

2

( ̄p + p)(υ − υ ̄).

In what follows we assume that the fluid is a perfect gas with adiabatic index

γ = 1 + 2/n, where n is the number of degrees of freedom. Then p = ρRT/m,

where R is the universal gas constant and m is the molar weight; the sound

speed is given by Cs

2 = (∂p/∂ρ)s = γpυ; and w = γe = γpυ/(γ − 1). For a

general oblique shock in a perfect gas the quantity X = r

−1

(U/VA)

2

satisfies14

(11) (X−β/α)(X−cos2θ)

2 = X sin2

θ



[1 + (r − 1)/2α] X − cos2θ

	

, where

r =  ̄ρ/ρ, α = 1

2

[γ + 1 − (γ − 1)r], and β = Cs

2/VA

2 = 4πγp/B2

.

The density ratio is bounded by

(12) 1 < r < (γ + 1)/(γ − 1).

If the shock is normal to B (i.e., if θ = π/2), then

(13) U

2 = (r/α)



Cs

2 + VA2

[1 + (1 − γ/2)(r − 1)]	;

(14) U/U ̄ = B/B  ̄ = r;

26
(15) V ̄ = V ;

(16)  ̄p = p + (1 − r

−1

)ρU2 + (1 − r

2

)B2/2μ.

If θ = 0, there are two possibilities: switch-on shocks, which require β < 1 and

for which

(17) U

2 = rVA2

;

(18) U ̄ = VA

2/U;

(19) B ̄ 2

⊥ = 2B

2

k

(r − 1)(α − β);

(20) V ̄ = U ̄B ̄⊥/Bk;

(21)  ̄p = p + ρU2(1 − α + β)(1 − r

−1

),

and acoustic (hydrodynamic) shocks, for which

(22) U

2 = (r/α)Cs

2

;

(23) U ̄ = U/r;

(24) V ̄ = B ̄⊥ = 0;

(25)  ̄p = p + ρU2(1 − r

−1

).

For acoustic shocks the specific volume and pressure are related by

(26)  ̄υ/υ = [(γ + 1)p + (γ − 1) ̄p] / [(γ − 1)p + (γ + 1) ̄p].

In terms of the upstream Mach number M = U/Cs,

(27)  ̄ρ/ρ = υ/υ ̄ = U/U ̄ = (γ + 1)M2/[(γ − 1)M2 + 2];

(28)  ̄p/p = (2γM2 − γ + 1)/(γ + 1);

(29) T/T  ̄ = [(γ − 1)M2 + 2](2γM2 − γ + 1)/(γ + 1)2M2;

(30) M ̄ 2 = [(γ − 1)M2 + 2]/[2γM2 − γ + 1].

The entropy change across the shock is

(31) ∆s ≡ s ̄ − s = cυ ln[( ̄p/p)(ρ/ρ ̄)

γ

],

where cυ = R/(γ − 1)m is the specific heat at constant volume; here R is the

gas constant. In the weak-shock limit (M → 1),

(32) ∆s → cυ

2γ(γ − 1)

3(γ + 1)

(M2 − 1)3 ≈

16γR

3(γ + 1)m

(M − 1)3.

The radius at time t of a strong spherical blast wave resulting from the explosive release of energy E in a medium with uniform density ρ is

(33) RS = C0(Et2/ρ)

1/5

,

where C0 is a constant depending on γ. For γ = 7/5, C0 = 1.033.

27
FUNDAMENTAL PLASMA PARAMETERS

All quantities are in Gaussian cgs units except temperature (T, Te, Ti)

expressed in eV and ion mass (mi) expressed in units of the proton mass,

μ = mi/mp; Z is charge state; k is Boltzmann’s constant; K is wavenumber;

γ is the adiabatic index; ln Λ is the Coulomb logarithm.

Frequencies

electron gyrofrequency fce = ωce/2π = 2.80 × 106B Hz

ωce = eB/mec = 1.76 × 107B rad/sec

ion gyrofrequency fci = ωci/2π = 1.52 × 103Zμ−1B Hz

ωci = ZeB/mic = 9.58 × 103Zμ−1B rad/sec

electron plasma frequency fpe = ωpe/2π = 8.98 × 103ne

1/2 Hz

ωpe = (4πnee

2/me)1/2

= 5.64 × 104ne

1/2

rad/sec

ion plasma frequency fpi = ωpi/2π

= 2.10 × 102Zμ−1/2ni

1/2 Hz

ωpi = (4πniZ

2

e

2/mi)1/2

= 1.32 × 103Zμ−1/2ni

1/2

rad/sec

electron trapping rate νT e = (eKE/me)

1/2

= 7.26 × 108K1/2E

1/2

sec−1

ion trapping rate νT i = (ZeKE/mi)

1/2

= 1.69 × 107Z

1/2K1/2E1/2μ−1/2

sec−1

electron collision rate νe = 2.91 × 10−6ne ln ΛTe

−3/2

sec−1

ion collision rate νi = 4.80 × 10−8Z

4μ−1/2ni ln ΛTi−3/2

sec−1

Lengths

electron deBroglie length  ̄λ =  ̄h/(mekTe)

1/2 = 2.76 × 10−8Te−1/2

cm

classical distance of e

2/kT = 1.44 × 10−7T−1

cm

minimum approach

electron gyroradius re = vT e/ωce = 2.38Te

1/2B−1

cm

ion gyroradius ri = vT i/ωci

= 1.02 × 102μ

1/2Z−1Ti1/2B−1

cm

electron inertial length c/ωpe = 5.31 × 105ne

−1/2

cm

ion inertial length c/ωpi = 2.28 × 107Z

−1

(μ/ni)

1/2

cm

Debye length λD = (kT/4πne2)

1/2 = 7.43 × 102T1/2n−1/2

cm

28
Velocities

electron thermal velocity vT e = (kTe/me)

1/2

= 4.19 × 107Te

1/2

cm/sec

ion thermal velocity vT i = (kTi/mi)

1/2

= 9.79 × 105μ

−1/2Ti1/2

cm/sec

ion sound velocity Cs = (γZkTe/mi)

1/2

= 9.79 × 105(γZTe/μ)

1/2

cm/sec

Alfv ́en velocity vA = B/(4πnimi)

1/2

= 2.18 × 1011μ

−1/2ni

−1/2B cm/sec

Dimensionless

(electron/proton mass ratio)1/2(me/mp)

1/2 = 2.33 × 10−2 = 1/42.9

number of particles in (4π/3)nλD

3 = 1.72 × 109T3/2n−1/2

Debye sphere

Alfv ́en velocity/speed of light vA/c = 7.28μ

−1/2ni

−1/2B

electron plasma/gyrofrequency ωpe/ωce = 3.21 × 10−3ne

1/2B−1

ratio

ion plasma/gyrofrequency ratio ωpi/ωci = 0.137μ

1/2ni1/2B−1

thermal/magnetic energy ratio β = 8πnkT/B2 = 4.03 × 10−11nTB−2

magnetic/ion rest energy ratio B2/8πnimic

2 = 26.5μ−1ni−1B2

Miscellaneous

Bohm diffusion coefficient DB = (ckT/16eB)

= 6.25 × 106TB−1cm2/sec

transverse Spitzer resistivity η⊥ = 1.15 × 10−14Z ln ΛT

−3/2

sec

= 1.03 × 10−2Z ln ΛT

−3/2 Ω cm

The anomalous collision rate due to low-frequency ion-sound turbulence is

ν* ≈ ωpeW/kT e = 5.64 × 104ne

1/2W/kT e sec−1

,

where We is the total energy of waves with ω/K < vT i.

Magnetic pressure is given by

Pmag = B

2

/8π = 3.98 × 106(B/B0)

2

dynes/cm

2 = 3.93(B/B0)2

atm,

where B0 = 10 kG = 1 T.

Detonation energy of 1 kiloton of high explosive is

WkT = 1012 cal = 4.2 × 1019erg.

29
PLASMA DISPERSION FUNCTION

Definition16 (first form valid only for Im ζ > 0):

Z(ζ) = π

−1/2

Z +∞

−∞

dt exp −t

2



t − ζ

= 2i exp −ζ

2



Z iζ

−∞

dt exp −t

2



.

Physically, ζ = x + iy is the ratio of wave phase velocity to thermal velocity.

Differential equation:

dZ

dζ

= −2 (1 + ζZ) , Z(0) = iπ1/2;

d

2Z

dζ2

+ 2ζ

dZ

dζ

+ 2Z = 0.

Real argument (y = 0):

Z(x) = exp −x

2





iπ1/2 − 2

Z x

0

dt exp t

2





.

Imaginary argument (x = 0):

Z(iy) = iπ1/2exp y

2



[1 − erf(y)] .

Power series (small argument):

Z(ζ) = iπ1/2exp −ζ

2



− 2ζ



1 − 2ζ

2

/3 + 4ζ

4

/15 − 8ζ

6

/105 + · · ·.

Asymptotic series, |ζ| ≫ 1 (Ref. 17):

Z(ζ) = iπ1/2σ exp −ζ

2



− ζ

−1



1 + 1/2ζ

2 + 3/4ζ4 + 15/8ζ6 + · · ·

,

where

σ =

0 y > |x|

−1

1 |y| < |x|

−1

2 y < −|x|

−1

Symmetry properties (the asterisk denotes complex conjugation):

Z(ζ*) = − [Z(−ζ)]*;

Z(ζ*) = [Z(ζ)] * + 2iπ1/2exp[−(ζ*)2] (y > 0).

Two-pole approximations18 (good for ζ in upper half plane except when y <

π

1/2x2

exp(−x

2

), x ≫ 1):

Z(ζ) ≈

0.50 + 0.81i

a − ζ

−

0.50 − 0.81i

a* + ζ

, a = 0.51 − 0.81i;

Z

′

(ζ) ≈

0.50 + 0.96i

(b − ζ)

2

+

0.50 − 0.96i

(b* + ζ)

2

, b = 0.48 − 0.91i.

30
COLLISIONS AND TRANSPORT

Temperatures are in eV; the corresponding value of Boltzmann’s constant

is k = 1.60 × 10−12 erg/eV; masses μ, μ

′

are in units of the proton mass;

eα = Zαe is the charge of species α. All other units are cgs except where

noted.

Relaxation Rates

Rates are associated with four relaxation processes arising from the interaction of test particles (labeled α) streaming with velocity vα through a

background of field particles (labeled β):

slowing down

dvα

dt

= −ν

α\β

s vα

transverse diffusion

d

dt

(vα −  ̄vα)

2

⊥ = ν

α\β

⊥ vα

2

parallel diffusion

d

dt

(vα −  ̄vα)

2

k = ν

α\β

k

vα

2

energy loss

d

dt

vα

2 = −να\β

ǫ

vα

2

,

where vα = |vα| and the averages are performed over an ensemble of test

particles and a Maxwellian field particle distribution. The exact formulas may

be written19

ν

α\β

s = (1 + mα/mβ)ψ(x

α\β

)ν

α\β

0

;

ν

α\β

⊥ = 2 

(1 − 1/2x

α\β

)ψ(x

α\β

) + ψ

′

(x

α\β

)



ν

α\β

0

;

ν

α\β

k =



ψ(x

α\β

)/xα\β



ν

α\β

0

;

ν

α\β

ǫ = 2 

(mα/mβ)ψ(x

α\β

) − ψ

′

(x

α\β

)



ν

α\β

0

,

where

ν

α\β

0 = 4πeα

2

eβ

2λαβnβ/mα2

vα

3

; x

α\β = mβvα2

/2kTβ;

ψ(x) = 2

√

π

Z x

0

dt t1/2e

−t

; ψ

′

(x) = dψ

dx

,

and λαβ = ln Λαβ is the Coulomb logarithm (see below). Limiting forms of

νs, ν⊥ and νk are given in the following table. All the expressions shown

31
have units cm3

sec−1. Test particle energy ǫ and field particle temperature T

are both in eV; μ = mi/mp where mp is the proton mass; Z is ion charge

state; in electron–electron and ion–ion encounters, field particle quantities are

distinguished by a prime. The two expressions given below for each rate hold

for very slow (x

α\β ≪ 1) and very fast (xα\β ≫ 1) test particles, respectively.

Slow Fast

Electron–electron

ν

e|e

s /neλee ≈ 5.8 × 10−6T

−3/2 −→ 7.7 × 10−6

ǫ

−3/2

ν

e|e

⊥

/neλee ≈ 5.8 × 10−6T

−1/2

ǫ

−1 −→ 7.7 × 10−6

ǫ

−3/2

ν

e|e

k

/neλee ≈ 2.9 × 10−6T

−1/2

ǫ

−1 −→ 3.9 × 10−6Tǫ−5/2

Electron–ion

ν

e|i

s /niZ

2λei ≈ 0.23μ3/2T−3/2 −→ 3.9 × 10−6

ǫ

−3/2

ν

e|i

⊥

/niZ

2λei ≈ 2.5 × 10−4

μ

1/2T−1/2

ǫ

−1−→ 7.7 × 10−6

ǫ

−3/2

ν

e|i

k

/niZ

2λei ≈ 1.2 × 10−4

μ

1/2T−1/2

ǫ

−1−→ 2.1 × 10−9

μ

−1Tǫ−5/2

Ion–electron

ν

i|e

s /neZ

2λie ≈ 1.6 × 10−9

μ

−1T−3/2 −→ 1.7 × 10−4

μ

1/2

ǫ

−3/2

ν

i|e

⊥ /neZ

2λie ≈ 3.2 × 10−9

μ

−1

T

−1/2

ǫ

−1 −→ 1.8 × 10−7

μ

−1/2

ǫ

−3/2

ν

i|e

k

/neZ

2λie ≈ 1.6 × 10−9

μ

−1T−1/2

ǫ

−1 −→ 1.7 × 10−4

μ

1/2Tǫ−5/2

Ion–ion

ν

i|i

′

s

ni

′Z2Z′2λii′

≈ 6.8 × 10−8 μ

′1/2

μ



1 +

μ

′

μ

−1/2

T

−3/2

−→ 9.0 × 10−8



1

μ

+

1

μ′



μ

1/2

ǫ

3/2

ν

i|i

′

⊥

ni

′Z2Z′2λii′

≈ 1.4 × 10−7μ

′1/2

μ

−1T−1/2

ǫ

−1

−→ 1.8 × 10−7μ

−1/2

ǫ

−3/2

ν

i|i

′

k

ni

′Z2Z′2λii′

≈ 6.8 × 10−8μ

′1/2

μ

−1

T

−1/2

ǫ

−1

−→ 9.0 × 10−8μ

1/2

μ

′−1

Tǫ−5/2

In the same limits, the energy transfer rate follows from the identity

νǫ = 2νs − ν⊥ − νk,

except for the case of fast electrons or fast ions scattered by ions, where the

leading terms cancel. Then the appropriate forms are

ν

e|i

ǫ −→ 4.2 × 10−9

niZ

2λei



ǫ

−3/2

μ

−1 − 8.9 × 104

(μ/T)

1/2

ǫ

−1

exp(−1836μǫ/T)



sec

−1

32
and

ν

i|i

′

ǫ −→ 1.8 × 10−7

ni

′Z

2Z′2λii′



ǫ

−3/2

μ

1/2

/μ

′

− 1.1[(μ + μ

′

)/μμ′](μ

′

/T′)

1/2

ǫ

−1

exp(−μ

′

ǫ/μT′)



sec

−1

.

In general, the energy transfer rate ν

α\β

ǫ

is positive for ǫ > ǫα* and negative for ǫ < ǫα*, where x* = (mβ/mα)ǫα*/Tβ is the solution of ψ

′

(x*) =

(mα|mβ)ψ(x*). The ratio ǫα*/Tβ is given for a number of specific α, β in the

following table:

α\β i|e e|e, i|i e|p e|D e|T, e|He3e|He4

ǫα*

Tβ

1.5 0.98 4.8 × 10−32.6 × 10−31.8 × 10−31.4 × 10−3

When both species are near Maxwellian, with Ti <∼ Te, there are just

two characteristic collision rates. For Z = 1,

νe = 2.9 × 10−6nλTe

−3/2

sec

−1

;

νi = 4.8 × 10−8nλTi

−3/2

μ

−1/2

sec

−1

.

Temperature Isotropization

Isotropization is described by

dT⊥

dt

= −

1

2

dTk

dt

= −ν

α

T (T⊥ − Tk),

where, if A ≡ T⊥/Tk − 1 > 0,

ν

α

T =

2

√

πeα

2

eβ

2nαλαβ

mα

1/2(kTk)3/2

A

−2



−3 + (A + 3) tan−1

(A

1/2

)

A1/2



.

If A < 0, tan−1(A

1/2

)/A1/2is replaced by tanh−1(−A)

1/2/(−A)1/2

. For

T⊥ ≈ Tk ≡ T,

ν

e

T = 8.2 × 10−7nλT −3/2

sec

−1

;

ν

i

T = 1.9 × 10−8nλZ2

μ

−1/2

T

−3/2

sec

−1

.

33
Thermal Equilibration

If the components of a plasma have different temperatures, but no relative drift, equilibration is described by

dTα

dt

=

X

β

ν ̄

α\β

ǫ

(Tβ − Tα),

where

ν ̄

α\β

ǫ = 1.8 × 10−19 (mαmβ)

1/2Zα2Zβ2nβλαβ

(mαTβ + mβTα)

3/2

sec

−1

.

For electrons and ions with Te ≈ Ti ≡ T, this implies

ν ̄

e|i

ǫ /ni =  ̄ν

i|e

ǫ /ne = 3.2 × 10−9Z

2λ/μT 3/2

cm

3

sec

−1

.

Coulomb Logarithm

For test particles of mass mα and charge eα = Zαe scattering off field

particles of mass mβ and charge eβ = Zβe, the Coulomb logarithm is defined

as λ = ln Λ ≡ ln(rmax/rmin). Here rmin is the larger of eαeβ/mαβu ̄

2

and

h/ ̄ 2mαβu ̄, averaged over both particle velocity distributions, where mαβ =

mαmβ/(mα + mβ) and u = vα − vβ; rmax = (4π

Pnγ eγ

2/kTγ )−1/2

, where

the summation extends over all species γ for which  ̄u

2 < vT γ2 = kTγ /mγ . If

this inequality cannot be satisfied, or if either  ̄uωcα

−1 < rmax or  ̄uωcβ−1 <

rmax, the theory breaks down. Typically λ ≈ 10–20. Corrections to the transport coefficients are O(λ

−1

); hence the theory is good only to ∼ 10% and fails

when λ ∼ 1.

The following cases are of particular interest:

(a) Thermal electron–electron collisions

λee = 23.5 − ln(ne

1/2Te−5/4

) − [10−5 + (ln Te − 2)2/16]1/2

(b) Electron–ion collisions

λei = λie = 23 − ln ne

1/2ZT −3/2

e



, Time/mi < Te < 10Z

2

eV;

= 24 − ln ne

1/2T−1

e



, Time/mi < 10Z

2

eV < Te

= 30 − ln ni

1/2

Ti

−3/2Z2

μ

−1



, Te < TiZme/mi.

(c) Mixed ion–ion collisions

λii′ = λi

′i = 23 − ln 

ZZ′(μ + μ

′

)

μTi

′ + μ′Ti



niZ

2

Ti

+

ni

′Z

′2

Ti

′

1/2



.

34
(d) Counterstreaming ions (relative velocity vD = βDc) in the presence of

warm electrons, kTi/mi, kTi

′ /mi′ < vD

2 < kTe/me

λii′ = λi

′i = 35 − ln 

ZZ′(μ + μ

′

)

μμ′βD

2



ne

Te

1/2



.

Fokker-Planck Equation

Df α

Dt

≡

∂f α

∂t

+ v · ∇f

α + F · ∇vfα =



∂f α

∂t 

coll

,

where F is an external force field. The general form of the collision integral is

(∂f α/∂t)coll = −

P

β

∇v · J

α\β

, with

J

α\β = 2πλαβ

eα

2

eβ

2

mα

Z

d

3

v

′

(u

2

I − uu)u

−3

·

n

1

mβ

f

α

(v)∇v′f

β

(v

′

) −

1

mα

f

β

(v

′

)∇vf

α

(v)

o

(Landau form) where u = v

′ − v and I is the unit dyad, or alternatively,

J

α\β = 4πλαβ

eα

2

eβ

2

mα

2

n

f

α

(v)∇vH(v) −

1

2

∇v ·



f

α

(v)∇v∇vG(v)

o

,

where the Rosenbluth potentials are

G(v) = Zf

β

(v

′

)ud3v

′

H(v) = 1 +

mα

mβ

Z

f

β

(v

′

)u

−1

d

3

v

′

.

If species α is a weak beam (number and energy density small compared with

background) streaming through a Maxwellian plasma, then

J

α\β = −

mα

mα + mβ

ν

α\β

s vf

α −

1

2

ν

α\β

k

vv · ∇vf

α

−

1

4

ν

α\β

⊥



v

2

I − vv· ∇vf

α

.

35
B-G-K Collision Operator

For distribution functions with no large gradients in velocity space, the

Fokker-Planck collision terms can be approximated according to

Dfe

Dt

= νee(Fe − fe) + νei(F ̄e − fe);

Dfi

Dt

= νie(F ̄i − fi) + νii(Fi − fi).

The respective slowing-down rates ν

α\β

s

given in the Relaxation Rate section

above can be used for ναβ, assuming slow ions and fast electrons, with ǫ replaced by Tα. (For νee and νii, one can equally well use ν⊥, and the result

is insensitive to whether the slow- or fast-test-particle limit is employed.) The

Maxwellians Fα and F ̄α are given by

Fα = nα

 mα

2πkTα

3/2

expn−

h mα(v − vα)

2

2kTα

io;

F ̄α = nα

 mα

2πkT ̄α

3/2

expn−

h mα(v −  ̄vα)

2

2kT ̄α

io,

where nα, vα and Tα are the number density, mean drift velocity, and effective

temperature obtained by taking moments of fα. Some latitude in the definition

of T ̄α and  ̄vα is possible;20 one choice is T ̄e = Ti, T ̄i = Te,  ̄ve = vi,  ̄vi = ve.

Transport Coefficients

Transport equations for a multispecies plasma:

d

αnα

dt

+ nα∇ · vα = 0;

mαnα

d

αvα

dt

= −∇pα − ∇ · Pα + Zαenα

h

E +

1

c

vα × B

i

+ Rα;

3

2

nα

d

αkTα

dt

+ pα∇ · vα = −∇ · qα − Pα : ∇vα + Qα.

Here d

α/dt ≡ ∂/∂t + vα · ∇; pα = nαkTα, where k is Boltzmann’s constant;

Rα =

P

β

Rαβ and Qα =

P

β

Qαβ, where Rαβ and Qαβ are respectively

the momentum and energy gained by the αth species through collisions with

the βth; Pα is the stress tensor; and qα is the heat flow.

36
The transport coefficients in a simple two-component plasma (electrons

and singly charged ions) are tabulated below. Here k and ⊥ refer to the direction of the magnetic field B = bB; u = ve − vi is the relative streaming

velocity; ne = ni ≡ n; j = −neu is the current; ωce = 1.76 × 107B sec−1and

ωci = (me/mi)ωce are the electron and ion gyrofrequencies, respectively; and

the basic collisional times are taken to be

τe =

3

√me(kTe)

3/2

4

√

2π nλe4

= 3.44 × 105 Te

3/2

nλ

sec,

where λ is the Coulomb logarithm, and

τi =

3

√mi(kTi)

3/2

4

√

πn λe4

= 2.09 × 107 Ti

3/2

nλ

μ

1/2

sec.

In the limit of large fields (ωcατα ≫ 1, α = i, e) the transport processes may

be summarized as follows:21

momentum transfer Rei= −Rie ≡ R = Ru + RT ;

frictional force Ru = ne(jk/σk + j⊥/σ⊥);

electrical σk = 1.96σ⊥; σ⊥ = ne

2

τe/me;

conductivities

thermal force RT = −0.71n∇k(kTe) −

3n

2ωceτe

b × ∇⊥(kTe);

ion heating Qi =

3me

mi

nk

τe

(Te − Ti);

electron heating Qe = −Qi − R · u;

ion heat flux qi = −κ

i

k∇k(kTi) − κ

i

⊥∇⊥(kTi) + κ

i

∧b × ∇⊥(kTi);

ion thermal κ

i

k = 3.9

nkTiτi

mi

; κ

i

⊥ =

2nkTi

miω 2

ci τi

; κ

i

∧ =

5nkTi

2miωci

;

conductivities

electron heat flux qe = q

e

u + q

e

T ;

frictional heat flux q

e

u = 0.71nkTeuk +

3nkTe

2ωceτe

b × u⊥;

thermal gradient q

e

T = −κ

e

k∇k(kTe) − κ

e

⊥∇⊥(kTe) − κ

e

∧b × ∇⊥(kTe);

heat flux

electron thermal κ

e

k = 3.2

nkTeτe

me

; κ

e

⊥ = 4.7

nkTe

meω 2

ce τe

; κ

e

∧ =

5nkTe

2meωce

;

conductivities

37
stress tensor (either Pxx= −

η0

2

(Wxx + Wyy) −

η1

2

(Wxx − Wyy) − η3Wxy;

species)

Pyy= −

η0

2

(Wxx + Wyy) + η1

2

(Wxx − Wyy) + η3Wxy;

Pxy= Pyx = −η1Wxy +

η3

2

(Wxx − Wyy);

Pxz= Pzx = −η2Wxz − η4Wyz;

Pyz = Pzy = −η2Wyz + η4Wxz;

Pzz = −η0Wzz

(here the z axis is defined parallel to B);

ion viscosity η

i

0 = 0.96nkTiτi; η

i

1 =

3nkTi

10ω 2

ci τi

; η

i

2 =

6nkTi

5ω 2

ci τi

;

η

i

3 =

nkTi

2ωci

; η

i

4 =

nkTi

ωci

;

electron viscosity η

e

0 = 0.73nkTeτe; η

e

1 = 0.51

nkTe

ω 2

ce τe

; η

e

2 = 2.0

nkTe

ω 2

ce τe

;

η

e

3 = −

nkTe

2ωce

; η

e

4 = −

nkTe

ωce

.

For both species the rate-of-strain tensor is defined as

Wjk =

∂vj

∂xk

+

∂vk

∂xj

−

2

3

δjk∇ · v.

When B = 0 the following simplifications occur:

Ru = nej/σk; RT = −0.71n∇(kTe); qi = −κ

i

k∇(kTi);

q

e

u = 0.71nkTeu; q

e

T = −κ

e

k∇(kTe); Pjk = −η0Wjk.

For ωceτe ≫ 1 ≫ ωciτi, the electrons obey the high-field expressions and the

ions obey the zero-field expressions.

Collisional transport theory is applicable when (1) macroscopic time rates

of change satisfy d/dt ≪ 1/τ, where τ is the longest collisional time scale, and

(in the absence of a magnetic field) (2) macroscopic length scales L satisfy L ≫

l, where l =  ̄vτ is the mean free path. In a strong field, ωceτ ≫ 1, condition

(2) is replaced by Lk ≫ l and L⊥ ≫

√

lre (L⊥ ≫ re in a uniform field),

where Lk is a macroscopic scale parallel to the field B and L⊥ is the smaller

of B/|∇⊥B| and the transverse plasma dimension. In addition, the standard

transport coefficients are valid only when (3) the Coulomb logarithm satisfies

λ ≫ 1; (4) the electron gyroradius satisfies re ≫ λD, or 8πnemec

2 ≫ B2

; (5)

relative drifts u = vα − vβ between two species are small compared with the

38
thermal velocities, i.e., u

2 ≪ kTα/mα, kTβ/mβ; and (6) anomalous transport

processes owing to microinstabilities are negligible.

Weakly Ionized Plasmas

Collision frequency for scattering of charged particles of species α by

neutrals is

να = n0σ

α|0

s

(kTα/mα)

1/2

,

where n0 is the neutral density and σ

α\0

s

is the cross section, typically ∼

5 × 10−15 cm2and weakly dependent on temperature.

When the system is small compared with a Debye length, L ≪ λD, the

charged particle diffusion coefficients are

Dα = kTα/mανα,

In the opposite limit, both species diffuse at the ambipolar rate

DA =

μiDe − μeDi

μi − μe

=

(Ti + Te)DiDe

TiDe + TeDi

,

where μα = eα/mανα is the mobility. The conductivity σα satisfies σα =

nαeαμα.

In the presence of a magnetic field B the scalars μ and σ become tensors,

J

α = σα

· E = σ

α

k Ek + σ

α

⊥E⊥ + σ

α

∧E × b,

where b = B/B and

σ

α

k = nαeα

2

/mανα;

σ

α

⊥ = σ

α

k

να

2

/(να

2 + ω2

cα);

σ

α

∧ = σ

α

k

ναωcα/(να

2 + ω2

cα).

Here σ⊥ and σ∧ are the Pedersen and Hall conductivities, respectively.

39
APPROXIMATE MAGNITUDES

IN SOME TYPICAL PLASMAS

Plasma Type n cm−3 T eV ωpe sec−1 λD cm nλD

3 νei sec−1

Interstellar gas 1 1 6 × 1047 × 1024 × 1087 × 10−5

Gaseous nebula 1031 2 × 10620 8 × 1066 × 10−2

Solar Corona 1091022 × 1092 × 10−18 × 10660

Diffuse hot plasma 1012 1026 × 1010 7 × 10−34 × 10540

Solar atmosphere, 1014 1 6 × 1011 7 × 10−540 2 × 109

gas discharge

Warm plasma 1014 10 6 × 1011 2 × 10−48 × 102107

Hot plasma 1014 1026 × 1011 7 × 10−44 × 1044 × 106

Thermonuclear 1015 1042 × 1012 2 × 10−38 × 1065 × 104

plasma

Theta pinch 1016 1026 × 1012 7 × 10−54 × 1033 × 108

Dense hot plasma 1018 1026 × 1013 7 × 10−64 × 1022 × 1010

Laser Plasma 1020 1026 × 1014 7 × 10−740 2 × 1012

The diagram (facing) gives comparable information in graphical form.22

40
41
IONOSPHERIC PARAMETERS23

The following tables give average nighttime values. Where two numbers

are entered, the first refers to the lower and the second to the upper portion

of the layer.

Quantity E Region F Region

Altitude (km) 90–160 160–500

Number density (m−3) 1.5 × 1010–3.0 × 1010 5 × 1010–2 × 1011

Height-integrated number 9 × 1014 4.5 × 1015

density (m−2)

Ion-neutral collision 2 × 103–1020.5–0.05

frequency (sec−1)

Ion gyro-/collision 0.09–2.0 4.6 × 102–5.0 × 103

frequency ratio κi

Ion Pederson factor 0.09–0.5 2.2 × 10−3–2 × 10−4

κi/(1 + κi

2

)

Ion Hall factor 8 × 10−4–0.8 1.0

κi

2/(1 + κi2

)

Electron-neutral collision 1.5 × 104–9.0 × 10280–10

frequency

Electron gyro-/collision 4.1 × 102–6.9 × 1037.8 × 104–6.2 × 105

frequency ratio κe

Electron Pedersen factor 2.7 × 10−3–1.5 × 10−410−5–1.5 × 10−6

κe/(1 + κe

2

)

Electron Hall factor 1.0 1.0

κe

2/(1 + κe

2

)

Mean molecular weight 28–26 22–16

Ion gyrofrequency (sec−1) 180–190 230–300

Neutral diffusion 30–5 × 103105

coefficient (m2sec−1)

The terrestrial magnetic field in the lower ionosphere at equatorial lattitudes is approximately B0 = 0.35×10−4

tesla. The earth’s radius is RE = 6371

km.

42
SOLAR PHYSICS PARAMETERS24

Parameter Symbol Value Units

Total mass M⊙ 1.99 × 1033 g

Radius R⊙ 6.96 × 1010 cm

Surface gravity g⊙ 2.74 × 104cm s−2

Escape speed v∞ 6.18 × 107cm s−1

Upward mass flux in spicules — 1.6 × 10−9g cm−2s

−1

Vertically integrated atmospheric density — 4.28 g cm−2

Sunspot magnetic field strength Bmax 2500–3500 G

Surface effective temperature T0 5770 K

Radiant power L⊙ 3.83 × 1033 erg s−1

Radiant flux density F 6.28 × 1010 erg cm−2s

−1

Optical depth at 500 nm, measured τ5 0.99 —

from photosphere

Astronomical unit (radius of earth’s orbit) AU 1.50 × 1013 cm

Solar constant (intensity at 1 AU) f 1.36 × 106erg cm−2s

−1

Chromosphere and Corona25

Quiet Coronal Active Parameter (Units) Sun Hole Region

Chromospheric radiation losses

(erg cm−2s

−1

)

Low chromosphere 2 × 1062 × 106 >∼ 107

Middle chromosphere 2 × 1062 × 106107

Upper chromosphere 3 × 1053 × 1052 × 106

Total 4 × 1064 × 106 >∼ 2 × 107

Transition layer pressure (dyne cm−2) 0.2 0.07 2

Coronal temperature (K) at 1.1 R⊙ 1.1–1.6 × 1061062.5 × 106

Coronal energy losses (erg cm−2s

−1

)

Conduction 2 × 1056 × 104105–107

Radiation 1051045 × 106

Solar Wind <∼ 5 × 1047 × 105 < 105

Total 3 × 1058 × 105107

Solar wind mass loss (g cm−2s

−1

) <∼ 2 × 10−11 2 × 10−10 < 4 × 10−11

43
THERMONUCLEAR FUSION26

Natural abundance of isotopes:

hydrogen nD/nH = 1.5 × 10−4

helium nHe3 /nHe4 = 1.3 × 10−6

lithium nLi6 /nLi7 = 0.08

Mass ratios: me/mD = 2.72 × 10−4 = 1/3670

(me/mD)

1/2= 1.65 × 10−2 = 1/60.6

me/mT = 1.82 × 10−4 = 1/5496

(me/mT )

1/2 = 1.35 × 10−2 = 1/74.1

Absorbed radiation dose is measured in rads: 1 rad = 102erg g−1. The curie

(abbreviated Ci) is a measure of radioactivity: 1 curie = 3.7×1010 counts sec−1.

Fusion reactions (branching ratios are correct for energies near the cross section

peaks; a negative yield means the reaction is endothermic):27

(1a) D + D −−−−→ 50% T(1.01 MeV) + p(3.02 MeV)

(1b) −−−−→ 50% He3(0.82 MeV) + n(2.45 MeV)

(2) D + T −−−−→He4(3.5 MeV) + n(14.1 MeV)

(3) D + He3−−−−→He4(3.6 MeV) + p(14.7 MeV)

(4) T + T −−−−→He4 + 2n + 11.3 MeV

(5a) He3 + T−−−−→ 51% He4 + p + n + 12.1 MeV

(5b) −−−−→ 43% He4(4.8 MeV) + D(9.5 MeV)

(5c) −−−−→6% He5(2.4 MeV) + p(11.9 MeV)

(6) p + Li6 −−−−→He4(1.7 MeV) + He3(2.3 MeV)

(7a) p + Li7 −−−−→ 20% 2 He4 + 17.3 MeV

(7b) −−−−→ 80% Be7 + n − 1.6 MeV

(8) D + Li6 −−−−→2He4 + 22.4 MeV

(9) p + B11 −−−−→3 He4 + 8.7 MeV

(10) n + Li6 −−−−→He4(2.1 MeV) + T(2.7 MeV)

The total cross section in barns (1 barn = 10−24 cm2) as a function of E, the

energy in keV of the incident particle [the first ion on the left side of Eqs.

(1)–(5)], assuming the target ion at rest, can be fitted by28

σT (E) =

A5 +



(A4 − A3E)

2 + 1−1

A2

E



exp(A1E−1/2) − 1



44
where the Duane coefficients Aj for the principle fusion reactions are as follows:

D–D D–D D–T D–He3 T–T T–He3

(1a) (1b) (2) (3) (4) (5a–c)

A1 46.097 47.88 45.95 89.27 38.39 123.1

A2 372 482 50200 25900 448 11250

A3 4.36 × 10−43.08 × 10−41.368 × 10−23.98 × 10−31.02 × 10−30

A4 1.220 1.177 1.076 1.297 2.09 0

A5 0 0 409 647 0 0

Reaction rates σv (in cm3sec−1), averaged over Maxwellian distributions:

Temperature D–D D–T D–He3 T–T T–He3

(keV) (1a + 1b) (2) (3) (4) (5a–c)

1.0 1.5 × 10−22 5.5 × 10−21 10−26 3.3 × 10−22 10−28

2.0 5.4 × 10−21 2.6 × 10−19 1.4 × 10−23 7.1 × 10−21 10−25

5.0 1.8 × 10−19 1.3 × 10−17 6.7 × 10−21 1.4 × 10−19 2.1 × 10−22

10.0 1.2 × 10−18 1.1 × 10−16 2.3 × 10−19 7.2 × 10−19 1.2 × 10−20

20.0 5.2 × 10−18 4.2 × 10−16 3.8 × 10−18 2.5 × 10−18 2.6 × 10−19

50.0 2.1 × 10−17 8.7 × 10−16 5.4 × 10−17 8.7 × 10−18 5.3 × 10−18

100.0 4.5 × 10−17 8.5 × 10−16 1.6 × 10−16 1.9 × 10−17 2.7 × 10−17

200.0 8.8 × 10−17 6.3 × 10−16 2.4 × 10−16 4.2 × 10−17 9.2 × 10−17

500.0 1.8 × 10−16 3.7 × 10−16 2.3 × 10−16 8.4 × 10−17 2.9 × 10−16

1000.0 2.2 × 10−16 2.7 × 10−16 1.8 × 10−16 8.0 × 10−17 5.2 × 10−16

For low energies (T <∼ 25 keV) the data may be represented by

(σv)DD = 2.33 × 10−14T

−2/3

exp(−18.76T

−1/3

) cm3sec

−1

;

(σv)DT = 3.68 × 10−12T

−2/3

exp(−19.94T

−1/3

) cm3sec

−1

,

where T is measured in keV.

The power density released in the form of charged particles is

PDD = 3.3 × 10−13nD

2

(σv)DD watt cm−3

(including the subsequent

D–T reaction);

PDT = 5.6 × 10−13nDnT (σv)DT watt cm−3

;

PDHe3 = 2.9 × 10−12nD nHe3 (σv)DHe3 watt cm−3

.

45
RELATIVISTIC ELECTRON BEAMS

Here γ = (1 − β

2

)

−1/2

is the relativistic scaling factor; quantities in

analytic formulas are expressed in SI or cgs units, as indicated; in numerical

formulas, I is in amperes (A), B is in gauss (G), electron linear density N is

in cm−1, and temperature, voltage and energy are in MeV; βz = vz/c; k is

Boltzmann’s constant.

Relativistic electron gyroradius:

re =

mc2

eB

(γ

2 − 1)1/2

(cgs) = 1.70 × 103(γ

2 − 1)1/2B−1

cm.

Relativistic electron energy:

W = mc

2

γ = 0.511γ MeV.

Bennett pinch condition:

I

2 = 2Nk(Te + Ti)c2

(cgs) = 3.20 × 10−4N(Te + Ti) A2.

Alfv ́en-Lawson limit:

IA = (mc

3

/e)βzγ (cgs) = (4πmc/μ0e)βzγ (SI) = 1.70 × 104βzγ A.

The ratio of net current to IA is

I

IA

=

ν

γ

.

Here ν = Nre is the Budker number, where re = e

2/mc2 = 2.82 × 10−13 cm

is the classical electron radius. Beam electron number density is

nb = 2.08 × 108Jβ−1cm

−3

,

where J is the current density in A cm−2. For a uniform beam of radius a (in

cm),

nb = 6.63 × 107Ia−2β

−1

cm

−3

,

and

2re

a

=

ν

γ

.

46
Child’s law: (non-relativistic) space-charge-limited current density between

parallel plates with voltage drop V (in MV) and separation d (in cm) is

J = 2.34 × 103V

3/2

d

−2 A cm−2

.

The saturated parapotential current (magnetically self-limited flow along equipotentials in pinched diodes and transmission lines) is29

Ip = 8.5 × 103Gγ ln γ + (γ

2 − 1)1/2



A,

where G is a geometrical factor depending on the diode structure:

G =

w

2πd

for parallel plane cathode and anode

of width w, separation d;

G =



ln

R2

R1

−1

for cylinders of radii R1 (inner) and R2 (outer);

G =

Rc

d0

for conical cathode of radius Rc, maximum

separation d0 (at r = Rc) from plane anode.

For β → 0 (γ → 1), both IA and Ip vanish.

The condition for a longitudinal magnetic field Bz to suppress filamentation

in a beam of current density J (in A cm−2) is

Bz > 47βz(γJ)

1/2 G.

Voltage registered by Rogowski coil of minor cross-sectional area A, n turns,

major radius a, inductance L, external resistance R and capacitance C (all in

SI):

externally integrated V = (1/RC)(nAμ0I/2πa);

self-integrating V = (R/L)(nAμ0I/2πa) = RI/n.

X-ray production, target with average atomic number Z (V <∼ 5 MeV):

η ≡ x-ray power/beam power = 7 × 10−4ZV.

X-ray dose at 1 meter generated by an e-beam depositing total charge Q

coulombs while V ≥ 0.84Vmax in material with charge state Z:

D = 150V

2.8

max QZ1/2

rads.

47
BEAM INSTABILITIES30

Name Conditions Saturation Mechanism

Electron- Vd > V ̄ej , j = 1, 2 Electron trapping until

electron V ̄ej ∼ Vd

Buneman Vd > (M/m)

1/3V ̄i, Electron trapping until

Vd > V ̄e V ̄e ∼ Vd

Beam-plasma Vb > (np/nb)

1/3V ̄b Trapping of beam electrons

Weak beam- Vb < (np/nb)

1/3V ̄b Quasilinear or nonlinear

plasma (mode coupling)

Beam-plasma V ̄e > Vb > V ̄b Quasilinear or nonlinear

(hot-electron)

Ion acoustic Te ≫ Ti, Vd ≫ Cs Quasilinear, ion tail formation, nonlinear scattering,

or resonance broadening.

Anisotropic Te⊥ > 2Tek Isotropization

temperature

(hydro)

Ion cyclotron Vd > 20V ̄i (for Ion heating

Te ≈ Ti)

Beam-cyclotron Vd > Cs Resonance broadening

(hydro)

Modified two- Vd < (1 + β)

1/2VA, Trapping

stream (hydro) Vd > Cs

Ion-ion (equal U < 2(1 + β)

1/2VA Ion trapping

beams)

Ion-ion (equal U < 2Cs Ion trapping

beams)

For nomenclature, see p. 50.

48
Parameters of Most Unstable Mode

Name Wave Group

Growth Rate Frequency Number Velocity

Electron1

2

ωe 0 0.9

ωe

Vd

0

electron

Buneman 0.7

 m

M

1/3

ωe 0.4

 m

M

1/3

ωe

ωe

Vd

2

3

Vd

Beam-plasma 0.7



nb

np

1/3

ωe ωe−

ωe

Vb

2

3

Vb

0.4



nb

np

1/3

ωe

Weak beamnb

2np



Vb

V ̄b

2

ωe ωe

ωe

Vb

3V ̄ 2

e

plasma Vb

Beam-plasma 

nb

np

1/2 V ̄e

Vb

ωe

Vb

V ̄e

ωe λ

−1

D Vb

(hot-electron)

Ion acoustic  m

M

1/2

ωi ωi λ

−1

D Cs

Anisotropic Ωe ωe cos θ ∼ Ωe r

−1

e V ̄e⊥

temperature

(hydro)

Ion cyclotron 0.1Ωi 1.2Ωi r

−1

i

1

3

V ̄i

Beam-cyclotron 0.7Ωe nΩe 0.7λ

−1

D

>∼ Vd;

(hydro) <∼ Cs

Modified two1

2

ΩH 0.9ΩH 1.7

ΩH

Vd

1

2

Vd

stream

(hydro)

Ion-ion (equal 0.4ΩH 0 1.2

ΩH

U

0

beams)

Ion-ion (equal 0.4ωi 0 1.2

ωi

U

0

beams)

For nomenclature, see p. 50.

49
In the preceding tables, subscripts e, i, d, b, p stand for “electron,” “ion,”

“drift,” “beam,” and “plasma,” respectively. Thermal velocities are denoted

by a bar. In addition, the following are used:

m electron mass re, ri gyroradius

M ion mass β plasma/magnetic energy

V velocity density ratio

T temperature VA Alfv ́en speed

ne, ni number density Ωe, Ωi gyrofrequency

n harmonic number ΩH hybrid gyrofrequency,

Cs = (Te/M)

1/2

ion sound speed ΩH

2 = ΩeΩi

ωe, ωi plasma frequency U relative drift velocity of

λD Debye length two ion species

50
LASERS

System Parameters

Efficiencies and power levels are approximate.31

Power levels available (W)

Type

Wavelength

(μm) Efficiency

Pulsed CW

CO2 10.6 0.01–0.02 > 2 × 1013 > 105

(pulsed)

CO 5 0.4 > 109 > 100

Holmium 2.06 0.03†–0.1‡ > 10780

Iodine 1.315 0.003 3 × 1012–

Nd-glass 1.06 – 1.25 × 1015–

Nd:YAG 1.064 – 109 > 104

Nd:YLF 1.045, – 4 × 10880

1.54,1.313

Nd:YVO4 1.064 – – > 20

Er:YAG 2.94 – 1.5 × 105–

*Color center 1–4 10−35 × 1081

*Ti:Sapphire 0.7–1.5 0.4 × ηp 1014 150

Ruby 0.6943 < 10−31010 1

He-Ne 0.6328 10−4– 1–50×10−3

*Argon ion 0.45–0.60 10−35 × 104150

*OPO 0.3–10 > 0.1 × ηp 1010 5

N2 0.3371 0.001–0.05 106–

*Dye 0.3–1.1 10−35 × 107 > 100

Kr-F 0.26 0.08 1012 500

Xenon 0.175 0.02 > 108–

Ytterbium fiber 1.05–1.1 0.55 5 × 107104

Erbium fiber 1.534 – 7 × 106100

Semiconductor 0.375–1.9 > 0.5 3 × 109 > 103

*Tunable sources †lamp-driven ‡diode-driven

Nd stands for Neodymium; Er stands for Erbium; Ti stands for Titanium;

YAG stands for Yttrium–Aluminum Garnet; YLF stands for Yttrium Lithium

Fluoride; YVO5 stands for Yttrium Vanadate; OPO for Optical Parametric

Oscillator; ηp is pump laser efficiency.

51
Formulas

An e-m wave with k k B has an index of refraction given by

n± = [1 − ω

2

pe/ω(ω ∓ ωce)]1/2

,

where ± refers to the helicity. The rate of change of polarization angle θ as a

function of displacement s (Faraday rotation) is given by

dθ/ds = (k/2)(n− − n+) = 2.36 × 104NBf−2cm

−1

,

where N is the electron number density, B is the field strength, and f is the

wave frequency, all in cgs.

The quiver velocity of an electron in an e-m field of angular frequency ω

is

v0 = eEmax/mω = 25.6I

1/2λ0 cm sec−1

in terms of the laser flux I = cE 2

max/8π, with I in watt/cm2

, laser wavelength

λ0 in μm. The ratio of quiver energy to thermal energy is

Wqu/Wth = mev0

2

/2kT = 1.81 × 10−13λ0

2

I/T,

where T is given in eV. For example, if I = 1015 W cm−2, λ0 = 1 μm, T =

2 keV, then Wqu/Wth ≈ 0.1.

Pondermotive force:

F = N∇hE

2

i/8πNc,

where

Nc = 1.1 × 1021λ0

−2

cm

−3

.

For uniform illumination of a lens with f-number F , the diameter d at

focus (85% of the energy) and the depth of focus l (distance to first zero in

intensity) are given by

d ≈ 2.44Fλθ/θDL and l ≈ ±2F

2λθ/θDL.

Here θ is the beam divergence containing 85% of energy and θDL is the

diffraction-limited divergence:

θDL = 2.44λ/b,

where b is the aperture. These formulas are modified for nonuniform (such as

Gaussian) illumination of the lens or for pathological laser profiles.

52
ATOMIC PHYSICS AND RADIATION

Energies and temperatures are in eV; all other units are cgs except where

noted. Z is the charge state (Z = 0 refers to a neutral atom); the subscript e

labels electrons. N refers to number density, n to principal quantum number.

Asterisk superscripts on level population densities denote local thermodynamic

equilibrium (LTE) values. Thus Nn* is the LTE number density of atoms (or

ions) in level n.

Characteristic atomic collision cross section:

(1) πa0

2 = 8.80 × 10−17

cm

2

.

Binding energy of outer electron in level labelled by quantum numbers n, l:

(2) E

Z

∞(n, l) = −

Z

2EH∞

(n − ∆l)

2

,

where E

H∞ = 13.6 eV is the hydrogen ionization energy and ∆l = 0.75l−5

,

l >∼ 5, is the quantum defect.

Excitation and Decay

Cross section (Bethe approximation) for electron excitation by dipole

allowed transition m → n (Refs. 32, 33):

(3) σmn = 2.36 × 10−13 fmng(n, m)

ǫ∆Enm

cm

2

,

where fmn is the oscillator strength, g(n, m) is the Gaunt factor, ǫ is the

incident electron energy, and ∆Enm = En − Em.

Electron excitation rate averaged over Maxwellian velocity distribution, Xmn

= Nehσmnvi (Refs. 34, 35):

(4) Xmn = 1.6 × 10−5 fmnhg(n, m)iNe

∆EnmT

1/2

e

exp −

∆Enm

Te



sec

−1

,

where hg(n, m)i denotes the thermal averaged Gaunt factor (generally ∼ 1 for

atoms, ∼ 0.2 for ions).

53
Rate for electron collisional deexcitation:

(5) Ynm = (Nm*/Nn*)Xmn.

Here Nm*/Nn* = (gm/gn) exp(∆Enm/Te) is the Boltzmann relation for level

population densities, where gn is the statistical weight of level n.

Rate for spontaneous decay n → m (Einstein A coefficient)34

(6) Anm = 4.3 × 107(gm/gn)fmn(∆Enm)

2

sec

−1

.

Intensity emitted per unit volume from the transition n → m in an optically

thin plasma:

(7) Inm = 1.6 × 10−19AnmNn∆Enm watt/cm

3

.

Condition for steady state in a corona model:

(8) N0Nehσ0nvi = NnAn0,

where the ground state is labelled by a zero subscript.

Hence for a transition n → m in ions, where hg(n, 0)i ≈ 0.2,

(9) Inm = 5.1 × 10−25 fnmgmNeN0

g0T

1/2

e



∆Enm

∆En0

3

exp −

∆En0

Te



watt

cm3

.

Ionization and Recombination

In a general time-dependent situation the number density of the charge

state Z satisfies

(10) dN(Z)

dt

= Ne

h

− S(Z)N(Z) − α(Z)N(Z)

+S(Z − 1)N(Z − 1) + α(Z + 1)N(Z + 1)i.

Here S(oZ) is the ionization rate. The recombination rate α(Z) has the form

α(Z) = αr(Z) + Neα3(Z), where αr and α3 are the radiative and three-body

recombination rates, respectively.

54
Classical ionization cross-section36 for any atomic shell j

(11) σi = 6 × 10−14bj gj (x)/Uj

2

cm

2

.

Here bj is the number of shell electrons; Uj is the binding energy of the ejected

electron; x = ǫ/Uj , where ǫ is the incident electron energy; and g is a universal

function with a minimum value gmin ≈ 0.2 at x ≈ 4.

Ionization from ion ground state, averaged over Maxwellian electron distribution, for 0.02 <∼ Te/EZ∞

<∼ 100 (Ref. 35):

(12) S(Z) = 10−5 (Te/EZ∞)

1/2

(EZ∞)

3/2(6.0 + Te/EZ∞)

exp −

E

Z∞

Te



cm

3

/sec,

where E

Z∞ is the ionization energy.

Electron-ion radiative recombination rate (e + N(Z) → N(Z − 1) + hν)

for Te/Z2 <∼ 400 eV (Ref. 37):

(13) αr(Z) = 5.2 × 10−14Z



E

Z∞

Te

1/2 h

0.43 +

1

2

ln(E

Z

∞/Te)

+0.469(E

Z

∞/Te)

−1/3

i

cm

3

/sec.

For 1 eV < Te/Z2 < 15 eV, this becomes approximately35

(14) αr(Z) = 2.7 × 10−13Z

2

Te

−1/2

cm

3

/sec.

Collisional (three-body) recombination rate for singly ionized plasma:38

(15) α3 = 8.75 × 10−27Te

−4.5

cm

6

/sec.

Photoionization cross section for ions in level n, l (short-wavelength limit):

(16) σph(n, l) = 1.64 × 10−16Z

5

/n3K

7+2l

cm

2

,

where K is the wavenumber in Rydbergs (1 Rydberg = 1.0974 × 105cm−1).

55
Ionization Equilibrium Models

Saha equilibrium:39

(17) NeN1*(Z)

Nn*(Z − 1) = 6.0 × 1021 g

Z

1 Te

3/2

g

Z−1

n

exp −

E

Z∞(n, l)

Te



cm

−3

,

where g

Z

n is the statistical weight for level n of charge state Z and E

Z∞(n, l)

is the ionization energy of the neutral atom initially in level (n, l), given by

Eq. (2).

In a steady state at high electron density,

(18) NeN*(Z)

N*(Z − 1) =

S(Z − 1)

α3

,

a function only of T.

Conditions for LTE:39

(a) Collisional and radiative excitation rates for a level n must satisfy

(19) Ynm >∼ 10Anm.

(b) Electron density must satisfy

(20) Ne >∼ 7 × 1018Z

7

n

−17/2

(T/EZ

∞)

1/2

cm

−3

.

Steady state condition in corona model:

(21) N(Z − 1)

N(Z)

=

αr

S(Z − 1) .

Corona model is applicable if40

(22) 1012tI

−1 < Ne < 1016Te7/2

cm

−3

,

where tI is the ionization time.

56
Radiation

N. B. Energies and temperatures are in eV; all other quantities are in

cgs units except where noted. Z is the charge state (Z = 0 refers to a neutral

atom); the subscript e labels electrons. N is number density.

Average radiative decay rate of a state with principal quantum number n is

(23) An =

X

m<n

Anm = 1.6 × 1010Z

4

n

−9/2

sec.

Natural linewidth (∆E in eV):

(24) ∆E ∆t = h = 4.14 × 10−15 eV sec,

where ∆t is the lifetime of the line.

Doppler width:

(25) ∆λ/λ = 7.7 × 10−5(T/μ)

1/2

,

where μ is the mass of the emitting atom or ion scaled by the proton mass.

Optical depth for a Doppler-broadened line:39

(26) τ = 3.52×10−13fnmλ(Mc2/kT)

1/2NL = 5.4×10−9

fmnλ(μ/T)

1/2NL,

where fnm is the absorption oscillator strength, λ is the wavelength, and L is

the physical depth of the plasma; M, N, and T are the mass, number density,

and temperature of the absorber; μ is M divided by the proton mass. Optically

thin means τ < 1.

Resonance absorption cross section at center of line:

(27) σλ=λc = 5.6 × 10−13λ

2

/∆λ cm

2

.

Wien displacement law (wavelength of maximum black-body emission):

(28) λmax = 2.50 × 10−5T

−1

cm.

Radiation from the surface of a black body at temperature T:

(29) W = 1.03 × 105T

4 watt/cm2

.

57
Bremsstrahlung from hydrogen-like plasma:26

(30) PBr = 1.69 × 10−32NeTe

1/2X

Z

2N(Z)



watt/cm

3

,

where the sum is over all ionization states Z.

Bremsstrahlung optical depth:41

(31) τ = 5.0 × 10−38NeNiZ

2

gLT −7/2,

where g ≈ 1.2 is an average Gaunt factor and L is the physical path length.

Inverse bremsstrahlung absorption coefficient42 for radiation of angular frequency ω:

(32) κ = 3.1 × 10−7Zne

2

ln Λ T

−3/2

ω

−2

(1 − ω

2

p/ω2

)

−1/2

cm

−1

;

here Λ is the electron thermal velocity divided by V , where V is the larger of

ω and ωp multiplied by the larger of Ze2/kT and  ̄h/(mkT)

1/2

.

Recombination (free-bound) radiation:

(33) Pr = 1.69 × 10−32NeTe

1/2Xh

Z

2N(Z)



E

Z−1 ∞

Te

i watt/cm

3

.

Cyclotron radiation26 in magnetic field B:

(34) Pc = 6.21 × 10−28B

2NeTe watt/cm3

.

For NekTe = NikTi = B2/16π (β = 1, isothermal plasma),26

(35) Pc = 5.00 × 10−38N

2

e T

2

e watt/cm

3

.

Cyclotron radiation energy loss e-folding time for a single electron:41

(36) tc ≈

9.0 × 108B−2

2.5 + γ

sec,

where γ is the kinetic plus rest energy divided by the rest energy mc2.

Number of cyclotron harmonics41 trapped in a medium of finite depth L:

(37) mtr = (57βBL)

1/6

,

where β = 8πNkT/B2

.

Line radiation is given by summing Eq. (9) over all species in the plasma.

58
ATOMIC SPECTROSCOPY

Spectroscopic notation combines observational and theoretical elements.

Observationally, spectral lines are grouped in series with line spacings which

decrease toward the series limit. Every line can be related theoretically to a

transition between two atomic states, each identified by its quantum numbers.

Ionization levels are indicated by roman numerals. Thus C I is unionized

carbon, C II is singly ionized, etc. The state of a one-electron atom (hydrogen)

or ion (He II, Li III, etc.) is specified by identifying the principal quantum

number n = 1, 2, . . . , the orbital angular momentum l = 0, 1, . . . , n − 1, and

the spin angular momentum s = ± 1

2

. The total angular momentum j is the

magnitude of the vector sum of l and s, j = l ± 1

2

(j ≥ 1

2

). The letters s,

p, d, f, g, h, i, k, l, . . . , respectively, are associated with angular momenta

l = 0, 1, 2, 3, 4, 5, 6, 7, 8, . . . . The atomic states of hydrogen and hydrogenic

ions are degenerate: neglecting fine structure, their energies depend only on n

according to

En = −

R∞hcZ2n

−2

1 + m/M

= −

RyZ

2

n2

,

where h is Planck’s constant, c is the velocity of light, m is the electron mass,

M and Z are the mass and charge state of the nucleus, and

R∞ = 109, 737 cm−1

is the Rydberg constant. If En is divided by hc, the result is in wavenumber

units. The energy associated with a transition m → n is given by

∆Emn = Ry(1/m2 − 1/n2),

with m < n (m > n) for absorption (emission) lines.

For hydrogen and hydrogenic ions the series of lines belonging to the

transitions m → n have conventional names:

Transition 1 → n 2 → n 3 → n 4 → n 5 → n 6 → n

Name Lyman Balmer Paschen Brackett Pfund Humphreys

Successive lines in any series are denoted α, β, γ, etc. Thus the transition 1 →

3 gives rise to the Lyman-β line. Relativistic effects, quantum electrodynamic

effects (e.g., the Lamb shift), and interactions between the nuclear magnetic

59
moment and the magnetic field due to the electron produce small shifts and

splittings, <∼ 10−2cm−1; these last are called “hyperfine structure.”

In many-electron atoms the electrons are grouped in closed and open

shells, with spectroscopic properties determined mainly by the outer shell.

Shell energies depend primarily on n; the shells corresponding to n = 1, 2,

3, . . . are called K, L, M, etc. A shell is made up of subshells of different

angular momenta, each labeled according to the values of n, l, and the number

of electrons it contains out of the maximum possible number, 2(2l + 1). For

example, 2p5indicates that there are 5 electrons in the subshell corresponding

to l = 1 (denoted by p) and n = 2.

In the lighter elements the electrons fill up subshells within each shell

in the order s, p, d, etc., and no shell acquires electrons until the lower shells

are full. In the heavier elements this rule does not always hold. But if a

particular subshell is filled in a noble gas, then the same subshell is filled in

the atoms of all elements that come later in the periodic table. The ground

state configurations of the noble gases are as follows:

He 1s2

Ne 1s22s22p6

Ar 1s22s22p63s23p6

Kr 1s22s22p63s23p63d104s24p6

Xe 1s22s22p63s23p63d104s24p64d105s25p6

Rn 1s22s22p63s23p63d104s24p64d104f145s25p65d106s26p6

Alkali metals (Li, Na, K, etc.) resemble hydrogen; their transitions are described by giving n and l in the initial and final states for the single outer

(valence) electron.

For general transitions in most atoms the atomic states are specified in

terms of the parity (−1)Σli and the magnitudes of the orbital angular momentum L = Σli, the spin S = Σsi, and the total angular momentum J = L + S,

where all sums are carried out over the unfilled subshells (the filled ones sum

to zero). If a magnetic field is present the projections ML, MS , and M of

L, S, and J along the field are also needed. The quantum numbers satisfy

|ML| ≤ L ≤ νl, |MS | ≤ S ≤ ν/2, and |M| ≤ J ≤ L + S, where ν is the

number of electrons in the unfilled subshell. Upper-case letters S, P, D, etc.,

stand for L = 0, 1, 2, etc., in analogy with the notation for a single electron.

For example, the ground state of Cl is described by 3p5 2P

o

3/2

. The first part

indicates that there are 5 electrons in the subshell corresponding to n = 3 and

l = 1. (The closed inner subshells 1s22s22p63s2, identical with the configuration of Mg, are usually omitted.) The symbol ‘P’ indicates that the angular

momenta of the outer electrons combine to give L = 1. The prefix ‘2’ represents the value of the multiplicity 2S + 1 (the number of states with nearly the

same energy), which is equivalent to specifying S = 1

2

. The subscript 3/2 is

60
the value of J. The superscript ‘o’ indicates that the state has odd parity; it

would be omitted if the state were even.

The notation for excited states is similar. For example, helium has a state

1s2s 3S1 which lies 19.72 eV (159, 856 cm−1) above the ground state 1s2 1S0.

But the two “terms” do not “combine” (transitions between them do not occur)

because this would violate, e.g., the quantum-mechanical selection rule that

the parity must change from odd to even or from even to odd. For electric

dipole transitions (the only ones possible in the long-wavelength limit), other

selection rules are that the value of l of only one electron can change, and only

by ∆l = ±1; ∆S = 0; ∆L = ±1 or 0; and ∆J = ±1 or 0 (but L = 0 does not

combine with L = 0 and J = 0 does not combine with J = 0). Transitions

are possible between the helium ground state (which has S = 0, L = 0, J = 0,

and even parity) and, e.g., the state 1s2p 1P

o

1

(with S = 0, L = 1, J = 1,

odd parity, excitation energy 21.22 eV). These rules hold accurately only for

light atoms in the absence of strong electric or magnetic fields. Transitions

that obey the selection rules are called “allowed”; those that do not are called

“forbidden.”

The amount of information needed to adequately characterize a state increases with the number of electrons; this is reflected in the notation. Thus43

O II has an allowed transition between the states 2p23p′

2Fo

7/2

and 2p2(

1D)3d′ 2F7/2 (and between the states obtained by changing

J from 7/2 to 5/2 in either or both terms). Here both states have two electrons with n = 2 and l = 1; the closed subshells 1s2

2s2are not shown. The

outer (n = 3) electron has l = 1 in the first state and l = 2 in the second.

The prime indicates that if the outermost electron were removed by ionization,

the resulting ion would not be in its lowest energy state. The expression (1D)

give the multiplicity and total angular momentum of the “parent” term, i.e.,

the subshell immediately below the valence subshell; this is understood to be

the same in both states. (Grandparents, etc., sometimes have to be specified

in heavier atoms and ions.) Another example43 is the allowed transition from

2p2(

3P)3p 2Po

1/2

(or 2P

o

3/2

) to 2p2(

1D)3d′ 2S1/2, in which there is a “spin

flip” (from antiparallel to parallel) in the n = 2, l = 1 subshell, as well as

changes from one state to the other in the value of l for the valence electron

and in L.

The description of fine structure, Stark and Zeeman effects, spectra of

highly ionized or heavy atoms, etc., is more complicated. The most important

difference between optical and X-ray spectra is that the latter involve energy

changes of the inner electrons rather than the outer ones; often several electrons

participate.

61
COMPLEX (DUSTY) PLASMAS

Complex (dusty) plasmas (CDPs) may be regarded as a new and unusual

state of matter. CDPs contain charged microparticles (dust grains) in addition

to electrons, ions, and neutral gas. Electrostatic coupling between the grains

can vary over a wide range, so that the states of CDPs can change from weakly

coupled (gaseous) to crystalline. CDPs can be investigated at the kinetic level

(individual particles are easily visualized and relevant time scales are accessible). CDPs are of interest as a non-Hamiltonian system of interacting particles

and as a means to study generic fundamental physics of self-organization, pattern formation, phase transitions, and scaling. Their discovery has therefore

opened new ways of precision investigations in many-particle physics.

Typical experimental dust properties

grain size (radius) a ≃ 0.3−30 μm, mass md ∼ 3×10−7 −3×10−13 g, number

density (in terms of the interparticle distance) nd ∼ ∆−3 ∼ 103 − 107cm−3,

temperature Td ∼ 3 × 10−2 − 102eV.

Typical discharge (bulk) plasmas

gas pressure p ∼ 10−2 − 1 Torr, Ti ≃ Tn ≃ 3 × 10−2eV, vTi ≃ 7 × 104

cm/s

(Ar), Te ∼ 0.3 − 3 eV, ni ≃ ne ∼ 108 − 1010 cm−3, screening length λD ≃

λDi ∼ 20 − 200 μm, ωpi ≃ 2 × 106 − 2 × 107s

−1

(Ar). B fields up to B ∼ 3 T.

Dimensionless

Havnes parameter P = |Z|nd/ne

normalized charge z = |Z|e

2/kTea

dust-dust scattering parameter βd = Z

2

e

2/kTdλD

dust-plasma scattering parameter βe,i = |Z|e

2/kTe,iλD

coupling parameter Γ = (Z

2

e

2/kTd∆) exp(−∆/λD)

lattice parameter κ = ∆/λD

particle parameter α = a/∆

lattice magnetization parameter μ = ∆/rd

Typical experimental values: P ∼ 10−4 −102,z ≃ 2−4 (Z ∼ 103 −105electron

charges), Γ < 103, κ ∼ 0.3 − 10, α ∼ 10−4 − 3 × 10−2, μ < 1

Frequencies

dust plasma frequency ωpd = (4πZ2e

2nd/md)1/2

≃ (|Z|

P

1+P mi/md)

1/2ωpi

charge fluctuation frequency ωch ≃ √

1+z

2π

(a/λD)ωpi

62
dust-gas friction rate νnd ∼ 10a

2p/mdvTn

dust gyrofrequency ωcd = ZeB/mdc

Velocities

dust thermal velocity vTd = (kTd/md)

1/2 ≡ [

Td

Ti

mi

md

]

1/2vTi

dust acoustic wave velocity CDA = ωpdλD

≃ (|Z|

P

1+P mi/md)

1/2vTi

dust Alfv ́en wave velocity vAd = B/(4πndmd)

1/2

dust-acoustic Mach number V/CDA

dust magnetic Mach number V/vAd

dust lattice (acoustic) wave velocity C

l,t

DL = ωpdλDFl,t(κ)

The range of the dust-lattice wavenumbers is K∆ < π The functions Fl,t(κ)

for longitudinal and transverse waves can be approximated44,45 with accuracy

< 1% in the range κ ≤ 5:

Fl ≃ 2.70κ

1/2

(1 − 0.096κ − 0.004κ

2

), Ft ≃ 0.51κ(1 − 0.039κ

2

),

Lengths

frictional dissipation length Lν = vTd/νnd

dust Coulomb radius RCe,i = |Z|e

2/kTe,i

dust gyroradius rd = vTd/ωcd

Grain Charging

The charge evolution equation is d|Z|/dt = Ii − Ie. From orbital motion

limited (OML) theory46 in the collisionless limit len(in) ≫ λD ≫ a:

Ie =

√

8πa

2

nevTeexp(−z), Ii =

√

8πa

2

nivTi



1 +

Te

Ti

z



.

Grains are charged negatively. The grain charge can vary in response to spatial

and temporal variations of the plasma. Charge fluctuations are always present,

with frequency ωch. Other charging mechanisms are photoemission, secondary

emission, thermionic emission, field emission, etc. Charged dust grains change

the plasma composition, keeping quasineutrality. A measure of this is the

Havnes parameter P = |Z|nd/ne. The balance of Ie and Ii yields

exp(−z) =  mi

me

Ti

Te

1/2 

1 +

Te

Ti

z



[1 + P (z)]

63
When the relative charge density of dust is large, P ≫ 1, the grain charge Z

monotonically decreases.

Forces and momentum transfer

In addition to the usual electromagnetic forces, grains in complex plasmas are

also subject to: gravity force Fg = mdg; thermophoretic force

Fth = −

4

√

2π

15

(a

2

/vTn )κn∇Tn

(where κn is the coefficient of gas thermal conductivity); forces associated

with the momentum transfer from other species, Fα = −mdναdVαd, i.e.,

neutral, ion, and electron drag. For collisions between charged particles, two

limiting cases are distinguished by the magnitude of the scattering parameter

βα. When βα ≪ 1 the result is independent of the sign of the potential. When

βα ≫ 1, the results for repulsive and attractive interaction potentials are

different. For typical complex plasmas the hierarchy of scattering parameters

is βe(∼ 0.01 − 0.3) ≪ βi(∼ 1 − 30) ≪ βd(∼ 103 − 3 × 104). The generic

expressions for different types of collisions are47

ναd = (4√

2π/3)(mα/md)a

2

nαvTα Φαd

Electron-dust collisions

Φed ≃

1

2

z

2Λed βe ≪ 1

Ion-dust collisions

Φid =

n 1

2

z

2

(Te/Ti)

2Λid βi < 5

2(λD/a)

2

(ln2 βi + 2 ln βi + 2), βi > 13

Dust-dust collisons

Φdd =

n

z

2

dΛdd βd ≪ 1

(λD/a)

2

[ln 4βd − ln ln 4βd], βd ≫ 1

where zd ≡ Z

2

e

2/akTd.

For νdd ∼ νnd the complex plasma is in a two-phase state, and for νnd ≫ νdd

we have merely tracer particles (dust-neutral gas interaction dominates). The

momentum transfer cross section is proportional to the Coulomb logarithm

Λαd when the Coulomb scattering theory is applicable. It is determined by

integration over the impact parameters, from ρmin to ρmax. ρmin is due to finite

grain size and is given by OML theory. ρmax = λD for repulsive interaction

(applicable for βα ≪ 1), and ρmax = λD(1 + 2βα)

1/2

for attractive interaction

(applicable up to βα < 5).

64
For repulsive interaction (electron-dust and dust-dust)

Λαd = zα

Z ∞

0

e

−zαx

ln[1 + 4(λD/aα)

2

x

2

]dx − 2zα

Z ∞

1

e

−zαx

ln(2x − 1)dx,

where ze = z, ae = a, and ad = 2a.

For ion-dust (attraction)

Λid ≃ z

Z ∞

0

e

−zx ln h

1 + 2(Ti/Te)(λD/a)x

1 + 2(Ti/Te)x

i

dx.

For νdd ≫ νnd the complex plasma behaves like a one phase system (dust-dust

interaction dominates).

Phase Diagram of Complex Plasmas

The figure below represents different “phase states” of CDPs as functions of

the electrostatic coupling parameter Γ and κ or α, respectively. The vertical dashed line at κ = 1 conditionally divides the system into Coulomb and

Yukawa parts. With respect to the usual plasma phase, in the diagram below the complex plasmas are “located” mostly in the strong coupling regime

(equivalent to the top left corner).

Regions I (V) represent Coulomb (Yukawa) crystals, the crystallization condition is48 Γ > 106(1 + κ + κ

2/2)−1

. Regions II (VI) are for Coulomb (Yukawa)

non-ideal plasmas – the characteristic range of dust-dust interaction (in terms

of the momentum transfer) is larger than the intergrain distance (in terms of

the Wigner-Seitz radius), (σ/π)

1/2 > (4π/3)−1/3∆, which implies that the

interaction is essentially multiparticle.

Regions III (VII and VIII) correspond to

Coulomb (Yukawa) ideal gases. The range

of dust-dust interaction is smaller than the

intergrain distance and only pair collisions

are important. In addition, in the region

VIII the pair Yukawa interaction asymptotically reduces to the hard sphere limit,

forming a “Yukawa granular medium”. In

region IV the electrostatic interaction is

unimportant and the system is like a uaual

granular medium.

0.1 1 10 10-4

10-2

100

102

104

101102103

α

-1

=∆/a

VII VIII

VI

V

IV

III

II

I

Γ

κ=∆/λ

65
REFERENCES

When any of the formulas and data in this collection are referenced

in research publications, it is suggested that the original source be cited rather

than the Formulary. Most of this material is well known and, for all practical

purposes, is in the “public domain.” Numerous colleagues and readers, too

numerous to list by name, have helped in collecting and shaping the Formulary

into its present form; they are sincerely thanked for their efforts.

Several book-length compilations of data relevant to plasma physics

are available. The following are particularly useful:

C. W. Allen, Astrophysical Quantities, 3rd edition (Athlone Press, London, 1976).

A. Anders, A Formulary for Plasma Physics (Akademie-Verlag, Berlin,

1990).

H. L. Anderson (Ed.), A Physicist’s Desk Reference, 2nd edition (American Institute of Physics, New York, 1989).

K. R. Lang, Astrophysical Formulae, 2nd edition (Springer, New York,

1980).

The books and articles cited below are intended primarily not for the purpose

of giving credit to the original workers, but (1) to guide the reader to sources

containing related material and (2) to indicate where to find derivations, explanations, examples, etc., which have been omitted from this compilation.

Additional material can also be found in D. L. Book, NRL Memorandum Report No. 3332 (1977).

1. See M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical

Functions (Dover, New York, 1968), pp. 1–3, for a tabulation of some

mathematical constants not available on pocket calculators.

2. H. W. Gould, “Note on Some Binomial Coefficient Identities of Rosenbaum,” J. Math. Phys. 10, 49 (1969); H. W. Gould and J. Kaucky, “Evaluation of a Class of Binomial Coefficient Summations,” J. Comb. Theory

1, 233 (1966).

3. B. S. Newberger, “New Sum Rule for Products of Bessel Functions with

Application to Plasma Physics,” J. Math. Phys. 23, 1278 (1982); 24,

2250 (1983).

4. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGrawHill Book Co., New York, 1953), Vol. I, pp. 47–52 and pp. 656–666.

66
5. W. D. Hayes, “A Collection of Vector Formulas,” Princeton University,

Princeton, NJ, 1956 (unpublished), and personal communication (1977).

6. See Quantities, Units and Symbols, report of the Symbols Committee

of the Royal Society, 2nd edition (Royal Society, London, 1975) for a

discussion of nomenclature in SI units.

7. E. R. Cohen and B. N. Taylor, “The 1986 Adjustment of the Fundamental

Physical Constants,” CODATA Bulletin No. 63 (Pergamon Press, New

York, 1986); J. Res. Natl. Bur. Stand. 92, 85 (1987); J. Phys. Chem. Ref.

Data 17, 1795 (1988).

8. E. S. Weibel, “Dimensionally Correct Transformations between Different

Systems of Units,” Amer. J. Phys. 36, 1130 (1968).

9. J. Stratton, Electromagnetic Theory (McGraw-Hill Book Co., New York,

1941), p. 508.

10. Reference Data for Engineers: Radio, Electronics, Computer, and Communication, 7th edition, E. C. Jordan, Ed. (Sams and Co., Indianapolis,

IN, 1985), Chapt. 1. These definitions are International Telecommunications Union (ITU) Standards.

11. H. E. Thomas, Handbook of Microwave Techniques and Equipment

(Prentice-Hall, Englewood Cliffs, NJ, 1972), p. 9. Further subdivisions

are defined in Ref. 10, p. I–3.

12. J. P. Catchpole and G. Fulford, Ind. and Eng. Chem. 58, 47 (1966);

reprinted in recent editions of the Handbook of Chemistry and Physics

(Chemical Rubber Co., Cleveland, OH) on pp. F306–323.

13. W. D. Hayes, “The Basic Theory of Gasdynamic Discontinuities,” in Fundamentals of Gas Dynamics, Vol. III, High Speed Aerodynamics and Jet

Propulsion, H. W. Emmons, Ed. (Princeton University Press, Princeton,

NJ, 1958).

14. W. B. Thompson, An Introduction to Plasma Physics (Addison-Wesley

Publishing Co., Reading, MA, 1962), pp. 86–95.

15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd edition (AddisonWesley Publishing Co., Reading, MA, 1987), pp. 320–336.

16. The Z function is tabulated in B. D. Fried and S. D. Conte, The Plasma

Dispersion Function (Academic Press, New York, 1961).

17. R. W. Landau and S. Cuperman, “Stability of Anisotropic Plasmas to

Almost-Perpendicular Magnetosonic Waves,” J. Plasma Phys. 6, 495

(1971).

67
18. B. D. Fried, C. L. Hedrick, J. McCune, “Two-Pole Approximation for the

Plasma Dispersion Function,” Phys. Fluids 11, 249 (1968).

19. B. A. Trubnikov, “Particle Interactions in a Fully Ionized Plasma,” Reviews of Plasma Physics, Vol. 1 (Consultants Bureau, New York, 1965),

p. 105.

20. J. M. Greene, “Improved Bhatnagar–Gross–Krook Model of Electron-Ion

Collisions,” Phys. Fluids 16, 2022 (1973).

21. S. I. Braginskii, “Transport Processes in a Plasma,” Reviews of Plasma

Physics, Vol. 1 (Consultants Bureau, New York, 1965), p. 205.

22. J. Sheffield, Plasma Scattering of Electromagnetic Radiation (Academic

Press, New York, 1975), p. 6 (after J. W. Paul).

23. K. H. Lloyd and G. H ̈arendel, “Numerical Modeling of the Drift and Deformation of Ionospheric Plasma Clouds and of their Interaction with

Other Layers of the Ionosphere,” J. Geophys. Res. 78, 7389 (1973).

24. C. W. Allen, Astrophysical Quantities, 3rd edition (Athlone Press, London, 1976), Chapt. 9.

25. G. L. Withbroe and R. W. Noyes, “Mass and Energy Flow in the Solar

Chromosphere and Corona,” Ann. Rev. Astrophys. 15, 363 (1977).

26. S. Glasstone and R. H. Lovberg, Controlled Thermonuclear Reactions

(Van Nostrand, New York, 1960), Chapt. 2.

27. References to experimental measurements of branching ratios and cross

sections are listed in F. K. McGowan, et al., Nucl. Data Tables A6,

353 (1969); A8, 199 (1970). The yields listed in the table are calculated

directly from the mass defect.

28. G. H. Miley, H. Towner and N. Ivich, Fusion Cross Section and Reactivities, Rept. COO-2218-17 (University of Illinois, Urbana, IL, 1974); B. H.

Duane, Fusion Cross Section Theory, Rept. BNWL-1685 (Brookhaven

National Laboratory, 1972).

29. J. M. Creedon, “Relativistic Brillouin Flow in the High ν/γ Limit,”

J. Appl. Phys. 46, 2946 (1975).

30. See, for example, A. B. Mikhailovskii, Theory of Plasma Instabilities

Vol. I (Consultants Bureau, New York, 1974). The table on pp. 48–49

was compiled by K. Papadopoulos.

68
31. Table prepared from data compiled by J. M. McMahon (personal communication, D. Book, 1990) and A. Ting (personal communication, J.D.

Huba, 2004).

32. M. J. Seaton, “The Theory of Excitation and Ionization by Electron Impact,” in Atomic and Molecular Processes, D. R. Bates, Ed. (New York,

Academic Press, 1962), Chapt. 11.

33. H. Van Regemorter, “Rate of Collisional Excitation in Stellar Atmospheres,” Astrophys. J. 136, 906 (1962).

34. A. C. Kolb and R. W. P. McWhirter, “Ionization Rates and Power Loss

from θ-Pinches by Impurity Radiation,” Phys. Fluids 7, 519 (1964).

35. R. W. P. McWhirter, “Spectral Intensities,” in Plasma Diagnostic Techniques, R. H. Huddlestone and S. L. Leonard, Eds. (Academic Press, New

York, 1965).

36. M. Gryzinski, “Classical Theory of Atomic Collisions I. Theory of Inelastic

Collision,” Phys. Rev. 138A, 336 (1965).

37. M. J. Seaton, “Radiative Recombination of Hydrogen Ions,” Mon. Not.

Roy. Astron. Soc. 119, 81 (1959).

38. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena (Academic Press, New York,

1966), Vol. I, p. 407.

39. H. R. Griem, Plasma Spectroscopy (Academic Press, New York, 1966).

40. T. F. Stratton, “X-Ray Spectroscopy,” in Plasma Diagnostic Techniques,

R. H. Huddlestone and S. L. Leonard, Eds. (Academic Press, New York,

1965).

41. G. Bekefi, Radiation Processes in Plasmas (Wiley, New York, 1966).

42. T. W. Johnston and J. M. Dawson, “Correct Values for High-Frequency

Power Absorption by Inverse Bremsstrahlung in Plasmas,” Phys. Fluids

16, 722 (1973).

43. W. L. Wiese, M. W. Smith, and B. M. Glennon, Atomic Transition Probabilities, NSRDS-NBS 4, Vol. 1 (U.S. Govt. Printing Office, Washington,

1966).

44. F. M. Peeters and X. Wu, “Wigner crystal of a screened-Coulombinteraction colloidal system in two dimensions”, Phys. Rev. A 35, 3109

(1987)

69
45. S. Zhdanov, R. A. Quinn, D. Samsonov, and G. E. Morfill, “Large-scale

steady-state structure of a 2D plasma crystal”, New J. Phys. 5, 74 (2003).

46. J. E. Allen, “Probe theory – the orbital motion approach”, Phys. Scripta

45, 497 (1992).

47. S. A. Khrapak, A. V. Ivlev, and G. E. Morfill, “Momentum transfer in

complex plasmas”, Phys. Rev. E (2004).

48. V. E. Fortov et al., “Dusty plasmas”, Phys. Usp. 47, 447 (2004).

70
AFTERWORD

The NRL Plasma Formulary originated over twenty five years ago

and has been revised several times during this period. The guiding spirit and

person primarily responsible for its existence is Dr. David Book. I am indebted

to Dave for providing me with the TEX files for the Formulary and his continued suggestions for improvement. The Formulary has been set in TEX by Dave

Book, Todd Brun, and Robert Scott. Finally, I thank readers for communicating typographical errors to me as well as suggestions for improvements.

71