anon - naval research laboratory plasma formulary - nrl - 2009

8/20/2019 Anon - Naval Research Laboratory Plasma Formulary - NRL - 2009

1/73


2/73


3/73

2009

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


4/73

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) . . . . . . . . . . . . . . . . . . . . . . 14Physical 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 . . . . . . . . . . . . . . . . . . . . . . 44Relativistic 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


5/73

NUMERICAL AND ALGEBRAIC

Gain in decibels of P 2 relative to P 1

G = 10 log10(P 2/P 1).

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) = 1.4892Γ(1/5) = 4.5908 Γ(2/3) = 1.3541Γ(1/4) = 3.6256 Γ(3/4) = 1.2254Γ(1/3) = 2.6789 Γ(4/5) = 1.1642Γ(2/5) = 2.2182 Γ(5/6) = 1.1288Γ(1/2) = 1.7725 =

√ π Γ(1) = 1.0

Binomial Theorem (good for | x | −1]:∞

n=−∞(−1)nJ α−γn(z)J β+γn(z)

n + µ

= π

sin µπ

J α+γµ(z)J β−γµ(z).

3


6/73

VECTOR IDENTITIES4

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

(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) ∇(f g) = ∇(gf ) = f ∇g + g∇f (7) ∇ · (f A) = f ∇ · A + A · ∇f (8) ∇ × (f A) = 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 bewritten in the dyadic form

(17) T =

i,jT ijeiej

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

(18) (∇·T )i =

j(∂T ji/∂xj)

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

(19) ∇ · (AB) = (∇ · A)B + (A · ∇)B(20) ∇ · (f T ) = ∇f ·T +f ∇·T

4


7/73

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 unitnormal outward from V,

(27)

V

dV ∇f = S

dSf

(28)

V

dV ∇ · A = S

dS · A

(29)

V

dV ∇·T = S

dS ·T

(30)

V

dV ∇ × A = S

dS × A

(31) V dV (f ∇2g − g∇2f ) = S dS · (f ∇g − g∇f )

(32)

V

dV (A · ∇ × ∇ × B − B · ∇ × ∇ × A)

=

S

dS · (B × ∇ × A − A × ∇ × B)

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

(33) S

dS × ∇f = C

dlf

5


8/73

(34) S dS · ∇ × A = C dl · A(35)

S

(dS × ∇) × A = C

dl × A

(36)

S

dS · (∇f × ∇g) = C

f dg = − C

gdf

DIFFERENTIAL OPERATORS INCURVILINEAR COORDINATES5

Cylindrical Coordinates

Divergence

∇ · A = 1r

∂

∂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

∇2f =

1

r

∂

∂r r ∂f ∂r + 1r2 ∂ 2f

∂φ2 +

∂ 2f

∂z 2

6


9/73

Laplacian of a vector

(∇2A)r = ∇2Ar − 2

r2∂Aφ

∂φ − Ar

r2

(∇2A)φ = ∇2Aφ + 2

r2∂Ar

∂φ− Aφ

r2

(

∇2A)z =

∇2

Az

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(rT rr) +

1

r

∂T φr

∂φ +

∂T zr

∂z − T φφ

r

(∇ · T )φ = 1

r

∂

∂r(rT rφ) +

1

r

∂T φφ

∂φ+

∂T zφ

∂z+

T φr

r

(∇ · T )z = 1

r

∂

∂r(rT rz) +

1

r

∂T φz

∂φ+

∂T zz

∂z

7


10/73

Spherical Coordinates

Divergence

∇ · A = 1r2

∂

∂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 = 1r

∂ ∂r

(rAθ) − 1r

∂Ar∂θ

Laplacian

∇2f = 1r2

∂

∂r

r

2 ∂f

∂r

+

1

r2 sin θ

∂

∂θ

sin θ

∂f

∂θ

+

1

r2 sin2 θ

∂ 2f

∂φ2

Laplacian of a vector

(∇2A)r = ∇2Ar − 2Ar

r2 − 2

r2∂Aθ

∂θ− 2cot θAθ

r2 − 2

r2 sin θ

∂Aφ

∂φ

(∇2A)θ = ∇2Aθ + 2

r2∂Ar

∂θ − Aθ

r2 sin2 θ − 2 cos θ

r2 sin2 θ

∂Aφ

∂φ

(∇

2A)φ =∇

2Aφ−

Aφ

r2 sin2 θ+

2

r2 sin θ

∂Ar

∂φ+

2 cos θ

r2 sin2 θ

∂Aθ

∂φ

8


11/73

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

r2∂

∂r(r2T rr) +

1

r sin θ

∂

∂θ(sin θT θr)

+ 1

r sin θ

∂T φr

∂φ− T θθ + T φφ

r

(∇ · T )θ = 1

r2∂

∂r(r2T rθ) +

1

r sin θ

∂

∂θ(sin θT θθ)

+ 1

r sin θ

∂T φθ

∂φ+

T θr

r− cot θT φφ

r

(∇ · T )φ = 1

r2∂

∂r(r2T rφ) +

1

r sin θ

∂

∂θ(sin θT θφ)

+ 1

r sin θ

∂T φφ

∂φ+

T φr

r+

cot θT φθ

r

9


12/73


13/73


14/73


15/73


16/73


17/73


18/73


19/73


20/73


21/73

MAXWELL’S EQUATIONS

Name or Description SI Gaussian

Faraday’s law ∇ × E = − ∂ B∂t

∇ × E = − 1c

∂ B

∂t

Ampere’s law ∇ × H = ∂ D∂t

+ J ∇ × H = 1c

∂ D

∂t+

4π

cJ

Poisson equation ∇ · D = ρ ∇ · D = 4πρ[Absence of magnetic ∇ · B = 0 ∇ · B = 0monopoles]Lorentz force on q (E + v × B) q

E +

1

cv × B

charge q

Constitutive D = ǫE D = ǫErelations B = µH B = µH

In a plasma, µ ≈ µ0 = 4π × 10−7 H m−1 (Gaussian units: µ ≈ 1). Thepermittivity satisfies ǫ ≈ ǫ0 = 8.8542 × 10−12 F m−1 (Gaussian: ǫ ≈ 1)provided that all charge is regarded as free. Using the drift approximationv⊥ = E × B/B2 to calculate polarization charge density gives rise to a dielec-tric 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

V

dV (H · B + E · D) (SI)

= 1

8π V dV (H · B + E · D) (Gaussian).Poynting’s theorem is

∂W

∂t +

S

N · dS = − 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


22/73

ELECTRICITY AND MAGNETISM

In the following, ǫ = dielectric permittivity, µ = permeability of conduc-tor, 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−1Permeability 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 ΩCapacity 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/8πl, carrying uniform current

Mutual inductance of parallel wires L = (µ′l/4π) [1 + 4 ln(d/a)]of length l, radius a, separatedby 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(f κm1κe2/σ)1/2conducting 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(a2 + z2)3/2]

circular loop of radius acarrying current I

20


23/73


24/73

AC CIRCUITS

For a resistance R, inductance L, and capacitance C in series witha voltage source V = V 0 exp(iωt) (here i =

√ −1), the current is givenby I = dq/dt, where q satisfies

Ld2q

dt2 + R

dq

dt+

q

C = V.

Solutions are q(t) = qs + qt, I (t) = I s + I t, where the steady state isI s = iωqs = V /Z in terms of the impedance Z = R + i(ωL − 1/ωC ) andI t = dqt/dt. For initial conditions q(0)

≡ q0 = q̄0 + qs, I (0)

≡ I 0, the

transients can be of three types, depending on ∆ = R2 − 4L/C :(a) Overdamped, ∆ > 0

qt = I 0 + γ +q̄0

γ + − γ −exp(−γ −t) −

I 0 + γ −q̄0γ + − γ −

exp(−γ +t),

I t = γ +(I 0 + γ −q̄0)

γ + − γ −exp(−γ +t) −

γ −(I 0 + γ +q̄0)γ + − γ −

exp(−γ −t),

where γ ± = (R ± ∆1/2)/2L;(b) Critically damped, ∆ = 0

qt = [q̄0 + (I 0 + γ Rq̄0)t] exp(−γ Rt),I t = [I 0 − (I 0 + γ Rq̄0)γ Rt] exp(−γ Rt),

where γ R = R/2L;

(c) Underdamped, ∆ < 0

qt =

γ Rq̄0 + I 0

ω1sin ω1t + q̄0 cos ω1t

exp(−γ Rt),

I t =

I 0 cos ω1t −

(ω12 + γ R

2)q̄0 + γ RI 0

ω1sin(ω1t)

exp(−γ Rt),

Here ω1 = ω0(1 − R2C/4L)1/2, where ω0 = (LC )−1/2 is the resonantfrequency. 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


25/73


26/73


27/73

Nomenclature:

B Magnetic induction

C s, c Speeds of sound, light

cp Specific heat at constant pressure (units m2 s−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−1

s−2

)N = (g/H )1/2 Brunt–Väisälä frequency

R Radius of pipe or channel

r Radius of curvature of pipe or channel

rL Larmor radiusT Temperature

V Characteristic flow velocity

V A = B/(µ0ρ)1/2 Alfvén 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−1 s−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 m2 s−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 m2 s−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


28/73

SHOCKS

At a shock front propagating in a magnetized fluid at an angle θ withrespect to the magnetic induction B, the jump conditions are 13,14

(1) ρU = ρ̄Ū ≡ q;(2) ρU 2 + p + B 2⊥ /2µ = ρ̄Ū

2 + ¯ p + B̄ 2⊥ /2µ;

(3) ρU V − BB⊥/µ = ρ̄Ū V̄ − B̄ B̄⊥/µ;(4) B = B̄;

(5) U B⊥ − V B = ¯U

¯B⊥ −

¯V

¯B;

(6) 12 (U 2 + V 2) + w + (U B 2⊥ − V BB⊥)/µρU

= 12 (Ū 2 + V̄ 2) + w̄ + (Ū B̄ 2⊥ − V̄ B̄ B̄⊥)/µρ̄Ū .

Here U and V are components of the fluid velocity normal and tangential tothe front in the shock frame; ρ = 1/υ is the mass density; p is the pressure;B⊥ = B sin θ, B = B cos θ; µ is the magnetic permeability (µ = 4π in cgsunits); and the specific enthalpy is w = e + pυ, where the specific internalenergy e satisfies de = T ds − pdυ in terms of the temperature T and thespecific entropy s. Quantities in the region behind (downstream from) the

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

(7) U − Ū = [(¯ p − p)(υ − ῡ)]1/2;(8) (¯ p − p)(υ − ῡ)−1 = q2;(9) w̄ − w = 12 ( ¯ p − p)(υ + ῡ);

(10) ē − e = 12 ( ¯ p + p)(υ − ῡ).In what follows we assume that the fluid is a perfect gas with adiabatic indexγ = 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 C s2

= (∂p/∂ρ)s = γpυ; and w = γe = γpυ/(γ − 1). For ageneral oblique shock in a perfect gas the quantity X = r−1(U/V A)2 satisfies14

(11) (X−β/α)(X−cos2 θ)2 = X sin2 θ

[1 + (r − 1)/2α] X − cos2 θ

, where

r = ρ̄/ρ, α = 12 [γ + 1 − (γ − 1)r], and β = C s2/V A2 = 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)C s2 + V A2 [1 + (1 − γ/2)(r − 1)];(14) U/Ū = B̄/B = r;

26


29/73

(15) V̄ = V ;

(16) ¯ p = p + (1 − r−1)ρU 2 + (1 − r2)B2/2µ.If θ = 0, there are two possibilities: switch-on shocks, which require β


30/73

FUNDAMENTAL PLASMA PARAMETERS

All quantities are in Gaussian cgs units except temperature (T , T e, T i)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 f ce = ωce/2π = 2.80 × 106B Hzωce = eB/mec = 1.76 × 107B rad/sec

ion gyrofrequency f ci = ωci/2π = 1.52 × 103Zµ−1B Hzωci = ZeB/mic = 9.58 × 103Zµ−1B rad/sec

electron plasma frequency f pe = ωpe/2π = 8.98 × 103ne1/2 Hzωpe = (4πnee

2/me)1/2

= 5.64 × 104ne1/2 rad/secion plasma frequency f pi = ωpi/2π

= 2.10 × 102Zµ−1/2ni1/2 Hzωpi = (4πniZ

2e2/mi)1/2

= 1.32 × 103

Zµ−1/2

ni1/2

rad/secelectron trapping rate ν Te = (eKE/me)

1/2

= 7.26 × 108K 1/2E 1/2 sec−1ion trapping rate ν Ti = (ZeKE/mi)

1/2

= 1.69 × 107Z 1/2K 1/2E 1/2µ−1/2 sec−1electron collision rate ν e = 2.91 × 10−6ne ln ΛT e−3/2 sec−1ion collision rate ν i = 4.80 × 10−8Z 4µ−1/2ni ln ΛT i−3/2 sec−1

Lengths

electron deBroglie length λ̄ = h̄/(mekT e)1/2 = 2.76 × 10−8T e−1/2 cm

classical distance of e2/kT = 1.44 × 10−7T −1 cmminimum approach

electron gyroradius re = vTe/ωce = 2.38T e1/2B−1 cm

ion gyroradius ri = vTi/ωci

= 1.02 × 102µ1/2Z −1T i1/2B−1 cmelectron inertial length c/ωpe = 5.31 × 105ne−1/2 cm

ion inertial length c/ωpi = 2.28 × 107

Z −1

(µ/ni)

1/2

cmDebye length λD = (kT /4πne

2)1/2 = 7.43 × 102T 1/2n−1/2 cm

28


31/73

Velocities

electron thermal velocity vTe = (kT e/me)1/2

= 4.19 × 107T e1/2 cm/secion thermal velocity vTi = (kT i/mi)

1/2

= 9.79 × 105µ−1/2T i1/2 cm/secion sound velocity C s = (γZkT e/mi)

1/2

= 9.79 × 105(γZ T e/µ)1/2 cm/secAlfvén velocity vA = B/(4πnimi)

1/2

= 2.18 × 1011µ−1/2ni−1/2B cm/secDimensionless

(electron/proton mass ratio)1/2 (me/mp)1/2 = 2.33 × 10−2 = 1/42.9

number of particles in (4π/3)nλD3 = 1.72 × 109T 3/2n−1/2

Debye sphere

Alfvén velocity/speed of light vA/c = 7.28µ−1/2ni−1/2B

electron plasma/gyrofrequency ωpe/ωce = 3.21 × 10−3ne1/2B−1ratio

ion plasma/gyrofrequency ratio ωpi/ωci = 0.137µ1/2ni

1/2B−1

thermal/magnetic energy ratio β = 8πnkT/B2 = 4.03 × 10−11nT B−2magnetic/ion rest energy ratio B2/8πnimic

2 = 26.5µ−1ni−1B2

MiscellaneousBohm diffusion coefficient DB = (ckT/16eB)

= 6.25 × 106T B−1 cm2/sectransverse Spitzer resistivity η⊥ = 1.15 × 10−14Z ln ΛT −3/2 sec

= 1.03 × 10−2Z ln ΛT −3/2 Ω cmThe anomalous collision rate due to low-frequency ion-sound turbulence is

ν * ≈ ωpe W/kT = 5.64 × 104ne1/2 W/kT sec−1,where W is the total energy of waves with ω/K < vTi.Magnetic pressure is given by

P mag = B2/8π = 3.98 × 106(B/B0)2 dynes/cm2 = 3.93(B/B0)2 atm,

where B0 = 10 kG = 1 T.Detonation energy of 1 kiloton of high explosive is

W kT = 1012

cal = 4.2 × 1019 erg.

29


32/73

PLASMA DISPERSION FUNCTION

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

Z (ζ ) = π−1/2

+∞

−∞

dt exp−t2

t − ζ = 2i exp

−ζ 2

iζ−∞

dt exp−t2

.

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;

d2Z

dζ 2 + 2ζ

dZ

dζ

+ 2Z = 0.

Real argument (y = 0):

Z (x) = exp−x2

iπ1/2 − 2

x0

dt exp

t2

.

Imaginary argument (x = 0):

Z (iy) = iπ1/2 exp

y2

[1 − erf(y)] .Power series (small argument):

Z (ζ ) = iπ1/2 exp−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 + 3/4ζ 4 + 15/8ζ 6 + · · ·

,

where

σ =

0 y > |x|−11 |y| < |x|−12 y <

−|x

|−1

Symmetry properties (the asterisk denotes complex conjugation):

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

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

π1/2x2 exp(−x2), x ≫ 1):

Z (ζ ) ≈ 0.50 + 0.81ia − ζ −

0.50 − 0.81ia* + 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


33/73

COLLISIONS AND TRANSPORT

Temperatures are in eV; the corresponding value of Boltzmann’s constantis k = 1.60 × 10−12 erg/eV; masses µ, µ′ are in units of the proton mass;eα = Z αe is the charge of species α. All other units are cgs except wherenoted.

Relaxation RatesRates are associated with four relaxation processes arising from the in-

teraction of test particles (labeled α) streaming with velocity vα through abackground of field particles (labeled β):

slowing down dvα

dt= −ν α\βs vα

transverse diffusion d

dt(vα − v̄α)2⊥ = ν α\β⊥ vα

2

parallel diffusion d

dt(vα − v̄α)2 = ν α\β vα

2

energy loss d

dtvα

2= −ν α\βǫ vα2,

where vα = |vα| and the averages are performed over an ensemble of testparticles and a Maxwellian field particle distribution. The exact formulas maybe written19

ν α\βs = (1 + mα/mβ)ψ(xα\β)ν α\β0 ;

ν α\β⊥ = 2

(1 − 1/2xα\β)ψ(xα\β) + ψ′(xα\β)

ν α\β0 ;

ν α\β =

ψ(xα\β)/xα\β

ν α\β0 ;

ν α\β

ǫ = 2 (mα/mβ)ψ(xα\β) − ψ′(xα\β) ν α\β0 ,where

ν α\β0 = 4πeα

2eβ2λαβnβ/mα

2vα3; xα\β = mβvα

2/2kT β ;

ψ(x) = 2√

π

x0

dt t1/2e−t; ψ′(x) = dψ

dx,

and λαβ = l n Λαβ is the Coulomb logarithm (see below). Limiting forms of ν s, ν ⊥ and ν are given in the following table. All the expressions shown

31


34/73

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 chargestate; in electron–electron and ion–ion encounters, field particle quantities aredistinguished by a prime. The two expressions given below for each rate hold

for very slow (xα\β ≪ 1) and very fast (xα\β ≫ 1) test particles, respectively.Slow Fast

Electron–electronν e|es /neλee ≈ 5.8 × 10−6T −3/2 −→ 7.7 × 10−6ǫ−3/2

ν e|e⊥ /neλee ≈ 5.8 × 10

−6T −1/2ǫ−1 −→ 7.7 × 10−6ǫ−3/2ν e|e /neλee ≈ 2.9 × 10

−6T −1/2ǫ−1 −→ 3.9 × 10−6T ǫ−5/2

Electron–ionν e|is /niZ

2λei ≈ 0.23µ3/2T −3/2 −→ 3.9 × 10−6ǫ−3/2ν e|i⊥ /niZ

2λei ≈ 2.5 × 10−4µ1/2T −1/2ǫ−1−→ 7.7 × 10−6ǫ−3/2ν e|i /niZ

2λei ≈ 1.2 × 10−4µ1/2T −1/2ǫ−1−→ 2.1 × 10−9µ−1T ǫ−5/2

Ion–electron

ν i|es /neZ 2λie ≈ 1.6 × 10−9µ−1T −3/2 −→ 1.7 × 10−4µ1/2ǫ−3/2

ν i|e⊥ /neZ

2λie ≈ 3.2 × 10−9µ−1T −1/2ǫ−1 −→ 1.8 × 10−7µ−1/2ǫ−3/2ν i|e /neZ

2λie ≈ 1.6 × 10−9µ−1T −1/2ǫ−1 −→ 1.7 × 10−4µ1/2T ǫ−5/2

Ion–ion

ν i|i′

s

ni′Z 2Z ′2λii′≈ 6.8 × 10−8 µ

′1/2

µ

1 +

µ′

µ

−1/2T −3/2

−→ 9.0 × 10−8

1

µ+

1

µ′

µ1/2

ǫ3/2

ν i|i′⊥

ni′Z 2Z ′2λii′≈ 1.4 × 10−7µ′1/2µ−1T −1/2ǫ−1

−→ 1.8 × 10−7µ−1/2ǫ−3/2

ν i

|i′

ni′Z 2Z ′2λii′

≈ 6.8 × 10−8µ′1/2µ−1T −1/2ǫ−1

−→ 9.0 × 10−8µ1/2µ′−1T ǫ−5/2

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

ν ǫ = 2ν s − ν ⊥ − ν ,

except for the case of fast electrons or fast ions scattered by ions, where theleading terms cancel. Then the appropriate forms are

ν e|i

ǫ −→ 4.2 × 10−9

niZ 2

λeiǫ−3/2µ−1 − 8.9 × 104(µ/T )1/2ǫ−1 exp(−1836µǫ/T )

sec−1

32


35/73

and

ν i|i′ǫ −→ 1.8 × 10−7ni′Z 2Z ′2λii′ǫ−3/2

µ1/2

/µ′ − 1.1[(µ + µ′)/µµ′](µ′/T ′)1/2ǫ−1 exp(−µ′ǫ/µT ′)

sec−1.

In general, the energy transfer rate ν α\βǫ is positive for ǫ > ǫα* and nega-tive for ǫ < ǫα*, where x* = (mβ/mα)ǫα*/T β is the solution of ψ

′(x*) =(mα|mβ)ψ(x*). The ratio ǫα*/T β is given for a number of specific α, β in thefollowing table:

α\β i|e e|e, i|i e| p e|D e|T, e|He3 e|He4ǫα*

T β1.5 0.98 4.8 × 10−3 2.6 × 10−3 1.8 × 10−3 1.4 × 10−3

When both species are near Maxwellian, with T i 0,

ν αT = 2

√ πeα

2eβ2nαλαβ

mα1/2(kT )3/2 A−2

−3 + (A + 3) tan

−1(A1/2)A1/2

.

If A < 0, tan−1(A1/2)/A1/2 is replaced by tanh−1(−A)1/2/(−A)1/2. ForT ⊥ ≈ T ≡ T ,

ν eT = 8.2 × 10−

7

nλT −3/2

sec−1

;

ν iT = 1.9 × 10−8nλZ 2µ−1/2T −3/2 sec−1.

33


36/73

Thermal Equilibration

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

dT α

dt=β

ν̄ α\βǫ (T β − T α),

where

ν̄ α\βǫ = 1.8 × 10−19 (mαmβ)

1/2Z α2Z β

2nβλαβ

(mαT β + mβT α)3/2 sec−1.

For electrons and ions with T e ≈ T i ≡ T , this implies

ν̄ e|iǫ /ni = ν̄

i|eǫ /ne = 3.2 × 10−9Z 2λ/µT 3/2cm3 sec−1.

Coulomb Logarithm

For test particles of mass mα and charge eα = Z αe scattering off fieldparticles of mass mβ and charge eβ = Z βe, the Coulomb logarithm is defined

as λ = l n Λ ≡ ln(rmax/rmin). Here rmin is the larger of eαeβ/mαβū2 andh̄/2mαβū, averaged over both particle velocity distributions, where mαβ =

mαmβ/(mα + mβ) and u = vα − vβ ; rmax = (4πnγeγ2/kT γ)−1/2, wherethe summation extends over all species γ for which ū2 < vTγ2 = kT γ/mγ . If this inequality cannot be satisfied, or if either ūωcα

−1 < rmax or ūωcβ−1 <rmax, the theory breaks down. Typically λ ≈ 10–20. Corrections to the trans-port coefficients are O(λ−1); hence the theory is good only to ∼ 10% and failswhen λ ∼ 1.

The following cases are of particular interest:

(a) Thermal electron–electron collisions

λee = 23.5 − ln(ne1/2T e−5/4) − [10−5 + (ln T e − 2)2/16]1/2

(b) Electron–ion collisions

λei = λie = 23 − ln

ne1/2ZT −3/2e

, T ime/mi < T e


37/73

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

warm electrons, kT i/mi, kT i′/mi′ < vD2 < kT e/me

λii′ = λi′i = 35 − ln

ZZ ′(µ + µ′)µµ′βD2

ne

T e

1/2.

Fokker-Planck Equation

Df α

Dt≡ ∂f

α

∂t+ v · ∇f α + F · ∇vf α =

∂f α

∂t coll ,where F is an external force field. The general form of the collision integral is

(∂f α/∂t)coll = −

β∇v · Jα\β , with

Jα\β = 2πλαβeα

2eβ2

mα

d

3v′(u2I − uu)u−3

·

1

mβf α(v)∇

v′f β(v′) − 1

mαf β(v′)∇vf α(v)

(Landau form) where u = v′ − v and I is the unit dyad, or alternatively,Jα\β = 4πλαβ

eα2eβ

2

mα2

f α

(v)∇vH (v) − 1

2∇v ·

f α

(v)∇v∇vG(v)

,

where the Rosenbluth potentials are

G(v) = f β(v′)ud3v′H (v) =

1 +

mα

mβ

f

β(v′)u−1d3v′.

If species α is a weak beam (number and energy density small compared withbackground) streaming through a Maxwellian plasma, then

Jα\β =

−

mα

mα + mβ

ν α\βs vf α

−

1

2

ν α\β

vv

· ∇vf

α

− 14

ν α\β⊥

v

2I − vv

· ∇vf α.

35


38/73

B-G-K Collision Operator

For distribution functions with no large gradients in velocity space, theFokker-Planck collision terms can be approximated according to

Df e

Dt = ν ee(F e − f e) + ν ei( F̄ e − f e);

Df i

Dt = ν ie( F̄ i − f i) + ν ii(F i − f i).

The respective slowing-down rates ν

α

\β

s given in the Relaxation Rate section

above can be used for ν αβ, assuming slow ions and fast electrons, with ǫ re-placed by T α. (For ν ee and ν ii, one can equally well use ν ⊥, and the resultis insensitive to whether the slow- or fast-test-particle limit is employed.) TheMaxwellians F α and F̄ α are given by

F α = nα

mα

2πkT α

3/2exp

−

mα(v − vα)22kT α

;

F̄ α = nα mα2πkT̄ α3/2 exp− mα(v − v̄α)22kT̄ α ,where nα, vα and T α are the number density, mean drift velocity, and effectivetemperature obtained by taking moments of f α. Some latitude in the definitionof T̄ α and v̄α is possible;

20 one choice is T̄ e = T i, T̄ i = T e, v̄e = vi, v̄i = ve.

Transport Coefficients

Transport equations for a multispecies plasma:

dαnα

dt + nα∇ · vα = 0;

mαnαdαvα

dt= −∇ pα − ∇ · P α + Z αenα

E +

1

cvα × B

+ Rα;

3

2nα

dαkT α

dt + pα∇ · vα = −∇ · qα − P α : ∇vα + Qα.

Here dα/dt ≡ ∂/∂t + vα · ∇; pα = nαkT α, where k is Boltzmann’s constant;Rα = β Rαβ and Qα = β Qαβ , where Rαβ and Qαβ are respectivelythe momentum and energy gained by the αth species through collisions withthe βth; P α is the stress tensor; and qα is the heat flow.

36


39/73

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

and singly charged ions) are tabulated below. Here and ⊥ refer to the di-rection of the magnetic field B = bB; u = ve − vi is the relative streamingvelocity; ne = ni ≡ n; j = −neu is the current; ωce = 1.76 × 107B sec−1 andωci = (me/mi)ωce are the electron and ion gyrofrequencies, respectively; andthe basic collisional times are taken to be

τ e = 3

√ me(kT e)

3/2

4√

2π nλe4= 3.44 × 105 T e

3/2

nλ sec,

where λ is the Coulomb logarithm, and

τ i = 3

√ mi(kT i)

3/2

4√

πnλe4 = 2.09 × 107 T i

3/2

nλ µ

1/2sec.

In the limit of large fields (ωcατ α ≫ 1, α = i, e) the transport processes maybe summarized as follows:21

momentum transfer Rei= −Rie ≡ R = Ru + RT ;frictional force Ru = ne( j

/σ

+ j

⊥/σ

⊥);

electrical σ = 1.96σ⊥; σ⊥ = ne2

τ e/me;conductivities

thermal force RT = −0.71n∇(kT e) − 3n

2ωceτ eb × ∇⊥(kT e);

ion heating Qi = 3me

mi

nk

τ e(T e − T i);

electron heating Qe = −Qi − R · u;ion heat flux qi =

−κi

∇(kT i)

−κi

⊥∇⊥(kT i) + κ

i

∧b

× ∇⊥(kT i);

ion thermal κi = 3.9

nkT iτ i

mi; κ

i⊥ =

2nkT i

miω 2ci τ i; κ

i∧ =

5nkT i

2miωci;

conductivities

electron heat flux qe = qeu

+ qeT ;

frictional heat flux qeu

= 0.71nkT eu + 3nkT e

2ωceτ eb × u⊥;

thermal gradient qeT = −κe∇(kT e) − κe⊥∇⊥(kT e) − κe∧b × ∇⊥(kT e);heat flux

electron thermal κe

= 3.2

nkT eτ e

me; κ

e

⊥ = 4.7

nkT e

meω 2ceτ e; κ

e

∧ =

5nkT e

2meωce;

conductivities

37


40/73

stress tensor (either P xx=

−

η0

2

(W xx + W yy)

−

η1

2

(W xx

−W yy)

−η3W xy;

species)

P yy= −η0

2(W xx + W yy) +

η1

2(W xx − W yy) + η3W xy;

P xy= P yx = −η1W xy + η3

2(W xx − W yy);

P xz= P zx = −η2W xz − η4W yz ;P yz = P zy = −η2W yz + η4W xz;P zz = −η0W zz

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

ion viscosity ηi0 = 0.96nkT iτ i; η

i1 =

3nkT i

10ω 2ci

τ i; η

i2 =

6nkT i

5ω 2ci

τ i;

ηi3 = nkT i

2ωci; ηi4 =

nkT i

ωci;

electron viscosity ηe0 = 0.73nkT eτ e; ηe1 = 0.51

nkT e

ω 2ceτ e; ηe2 = 2.0

nkT e

ω 2ceτ e;

ηe3 = −nkT e

2ωce; ηe4 = −

nkT e

ωce.

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

W jk = ∂vj

∂xk+

∂vk

∂xj− 2

3δjk∇ · v.

When B = 0 the following simplifications occur:

Ru = ne j/σ; RT = −0.71n∇(kT e); qi = −κi∇(kT i);

qeu

= 0.71nkT eu; qe

T =

−κe

∇(kT e); P

jk =

−η0W

jk.

For ωceτ e ≫ 1 ≫ ωciτ i, the electrons obey the high-field expressions and theions obey the zero-field expressions.

Collisional transport theory is applicable when (1) macroscopic time ratesof 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 = v̄τ is the mean free path. In a strong field, ωceτ ≫ 1, condition(2) is replaced by L ≫ l and L⊥ ≫

√ lre (L⊥ ≫ re in a uniform field),

where L is a macroscopic scale parallel to the field B and L⊥ is the smallerof B/|∇⊥B| and the transverse plasma dimension. In addition, the standardtransport coefficients are valid only when (3) the Coulomb logarithm satisfiesλ ≫ 1; (4) the electron gyroradius satisfies re ≫ λD, or 8πnemec2 ≫ B2; (5)relative drifts u = vα − vβ between two species are small compared with the

38


41/73

thermal velocities, i.e., u2

≪kT α/mα, kT β/mβ ; and (6) anomalous transport

processes owing to microinstabilities are negligible.

Weakly Ionized Plasmas

Collision frequency for scattering of charged particles of species α byneutrals is

ν α = n0σα|0s (kT α/mα)

1/2,

where n0 is the neutral density, σα\0s is the cross section, typically ∼ 5×10−15

cm2 and weakly dependent on temperature, and (T 0/m0)1/2 < (T α/mα)

1/2

where T 0 and m0 are the temperature and mass of the neutrals.When the system is small compared with a Debye length, L

≪ λD, the

charged particle diffusion coefficients are

Dα = kT α/mαν α,

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

DA = µiDe − µeDi

µi − µe=

(T i + T e)DiDe

T iDe + T eDi,

where µα = eα/mαν α is the mobility. The conductivity σα satisfies σα =nαeαµα.

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

Jα = σσα · E = σα E + σα⊥E⊥ + σα∧E × b,

where b = B/B and

σ

α

= nαeα2

/mαν α;

σα⊥ = σα ν α

2/(ν α2

+ ω 2cα);

σα∧ = σα ν αωcα/(ν α

2+ ω 2cα).

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

39


42/73


43/73

41


44/73


45/73


46/73

THERMONUCLEAR FUSION26

Natural abundance of isotopes:

hydrogen nD/nH = 1.5 × 10−4helium nHe3 /nHe4 = 1.3 × 10−6lithium nLi6 /nLi7 = 0.08

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

1/2= 1.65 × 10−2 = 1/60.6me/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 = 102 ergg−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 sectionpeaks; 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 , theenergy 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 by 28a

σT (E ) =

A5 + (A4 − A3E )2 + 1

−1A2

E exp(A1E −1/2) − 144


47/73


48/73

RELATIVISTIC ELECTRON BEAMS

Here γ = (1 − β2)−1/2 is the relativistic scaling factor; quantities inanalytic formulas are expressed in SI or cgs units, as indicated; in numericalformulas, 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 isBoltzmann’s constant.

Relativistic electron gyroradius:

re = mc2

eB

(γ 2

−1)

1/2(cgs) = 1.70

×10

3(γ

2

−1)

1/2B−1

cm.

Relativistic electron energy:

W = mc2γ = 0.511γ MeV.

Bennett pinch condition:

I 2 = 2N k(T e + T i)c2

(cgs) = 3.20 × 10−4N (T e + T i) A2.

Alfvén-Lawson limit:

I A = (mc3

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

The ratio of net current to I A is

I

I A=

ν

γ .

Here ν = Nre is the Budker number, where re = e2/mc2 = 2.82 × 10−13 cm

is the classical electron radius. Beam electron number density is

nb = 2.08 × 108Jβ−1 cm−3,

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

nb = 6.63 × 107Ia−2β−1 cm−3,

and 2re

a=

ν

γ .

46


49/73

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 3/2d−2 A cm−2.

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

I p = 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

−1for 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 I A and I p vanish.

The condition for a longitudinal magnetic field Bz to suppress filamentationin 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 inSI):

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


50/73


51/73


52/73

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

“drift,” “beam,” and “plasma,” respectively. Thermal velocities are denotedby a bar. In addition, the following are used:

m electron mass re, ri gyroradiusM ion mass β plasma/magnetic energyV velocity density ratioT temperature V A Alfvén speedne, ni number density Ωe, Ωi gyrofrequencyn harmonic number ΩH hybrid gyrofrequency,

C s = (T e/M )1/2 ion sound speed ΩH

2 = ΩeΩiωe, ωi plasma frequency U relative drift velocity of

λD Debye length two ion species

50


53/73


54/73

Formulas

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

n± = [1 − ω 2pe/ω(ω ∓ ωce)]1/2,

where ± refers to the helicity. The rate of change of polarization angle θ as afunction of displacement s (Faraday rotation) is given by

dθ/ds = (k/2)(n− − n+) = 2.36 × 104N Bf −2 cm−1,

where N is the electron number density, B is the field strength, and f is thewave frequency, all in cgs.

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

v0 = eE max/mω = 25.6I 1/2λ0 cm sec

−1

in terms of the laser flux I = cE 2max/8π, with I in watt/cm2, laser wavelength

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

W qu/W th = mev02/2kT = 1.81 × 10−13λ02I/T,

where T is given in eV. For example, if I = 1015 W cm−2, λ0 = 1 µm, T =2 keV, then W qu/W th ≈ 0.1.

Pondermotive force:

FF = N ∇E 2/8πN c,

whereN c = 1.1 × 1021λ0−2cm−3.

For uniform illumination of a lens with f -number F , the diameter d atfocus (85% of the energy) and the depth of focus l (distance to first zero inintensity) 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 thediffraction-limited divergence:

θDL = 2.44λ/b,

where b is the aperture. These formulas are modified for nonuniform (such asGaussian) illumination of the lens or for pathological laser profiles.

52


55/73

ATOMIC PHYSICS AND RADIATION

Energies and temperatures are in eV; all other units are cgs except wherenoted. Z is the charge state (Z = 0 refers to a neutral atom); the subscript elabels electrons. N refers to number density, n to principal quantum number.Asterisk superscripts on level population densities denote local thermodynamicequilibrium (LTE) values. Thus N n* is the LTE number density of atoms (orions) in level n.

Characteristic atomic collision cross section:

(1) πa02 = 8.80 × 10−17 cm2.

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

(2) E Z∞(n, l) = −

Z 2E H∞(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 dipoleallowed transition m → n (Refs. 32, 33):

(3) σmn = 2.36 × 10−13f mng(n, m)

ǫ∆E nmcm2,

where f mn is the oscillator strength, g(n, m) is the Gaunt factor, ǫ is theincident electron energy, and ∆E nm = E n − E m.Electron excitation rate averaged over Maxwellian velocity distribution, Xmn= N eσmnv (Refs. 34, 35):

(4) Xmn = 1.6 × 10−5f mng(n, m)N e

∆E nmT 1/2e

exp

− ∆E nm

T e

sec

−1,

where g(n, m) denotes the thermal averaged Gaunt factor (generally ∼ 1 foratoms, ∼ 0.2 for ions).

53


56/73

Rate for electron collisional deexcitation:

(5) Y nm = (N m*/N n*)Xmn.

Here N m*/N n* = (gm/gn) exp(∆E nm/T e) is the Boltzmann relation for levelpopulation 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)f mn(∆E nm)2 sec−1.

Intensity emitted per unit volume from the transition n → m in an opticallythin plasma:

(7) I nm = 1.6 × 10−19AnmN n∆E nm watt/cm3.

Condition for steady state in a corona model:

(8) N 0N e

σ0nv

= N nAn0,

where the ground state is labelled by a zero subscript.

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

(9) I nm = 5.1 × 10−25f nmgmN eN 0

g0T 1/2e

∆E nm

∆E n0

3exp

− ∆E n0

T e

watt

cm3 .

Ionization and RecombinationIn a general time-dependent situation the number density of the charge

state Z satisfies

(10) dN (Z )

dt= N e

− S (Z )N (Z ) − α(Z )N (Z )

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

.

Here S (oZ ) is the ionization rate. The recombination rate α(Z ) has the formα(Z ) = αr(Z ) + N eα3(Z ), where αr and α3 are the radiative and three-bodyrecombination rates, respectively.

54


57/73

Classical ionization cross-section36 for any atomic shell j

(11) σi = 6 × 10−14bjgj(x)/U j2 cm2.

Here bj is the number of shell electrons; U j is the binding energy of the ejectedelectron; x = ǫ/U j , where ǫ is the incident electron energy; and g is a universalfunction with a minimum value gmin ≈ 0.2 at x ≈ 4.Ionization from ion ground state, averaged over Maxwellian electron distribu-tion, for 0.02


58/73

Ionization Equilibrium Models

Saha equilibrium:39

(17) N eN 1*(Z )

N n*(Z − 1) = 6.0 × 1021 g

Z1 T e

3/2

gZ−1nexp

− E

Z∞(n, l)

T e

cm−3,

where gZn 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 byEq. (2).

In a steady state at high electron density,

(18) N eN *(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) Y nm >∼ 10Anm.

(b) Electron density must satisfy

(20) N e >∼ 7 × 1018Z 7n−17/2(T /E Z∞)1/2cm−3.

Steady state condition in corona model:

(21) N (Z − 1)

N (Z ) =

αr

S (Z − 1) .

Corona model is applicable if 40

(22) 1012tI−1 < N e


59/73

Radiation

N. B. Energies and temperatures are in eV; all other quantities are incgs units except where noted. Z is the charge state (Z = 0 refers to a neutralatom); the subscript e labels electrons. N is number density.

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

(23) An =m


60/73

Bremsstrahlung from hydrogen-like plasma:26

(30) P Br = 1.69 × 10−32N eT e1/2

Z 2

N (Z )

watt/cm3

,

where the sum is over all ionization states Z .

Bremsstrahlung optical depth:41

(31) τ = 5.0 × 10−38N eN iZ 2gLT −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 fre-quency ω:

(32) κ = 3.1 × 10−7Zne2 ln Λ T −3/2ω−2(1 − ω2p/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 h̄/(mkT )1/2.

Recombination (free-bound) radiation:

(33) P r = 1.69

×10−32

N eT e1/2

Z 2

N (Z )E Z−1∞

T e watt/cm3

.

Cyclotron radiation26 in magnetic field B:

(34) P c = 6.21 × 10−28B2N eT e watt/cm3.

For N ekT e = N ikT i = B2/16π (β = 1, isothermal plasma),26

(35) P c = 5.00 × 10−38N 2e T 2e watt/cm3.

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


61/73

ATOMIC SPECTROSCOPY

Spectroscopic notation combines observational and theoretical elements.Observationally, spectral lines are grouped in series with line spacings whichdecrease toward the series limit. Every line can be related theoretically to atransition between two atomic states, each identified by its quantum numbers.

Ionization levels are indicated by roman numerals. Thus C I is unionizedcarbon, 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 quantumnumber n = 1, 2, . . . , the orbital angular momentum l = 0, 1, . . . , n

−1, and

the spin angular momentum s = ± 12 . The total angular momentum j is themagnitude of the vector sum of l and s, j = l ± 12 (j ≥ 12 ). The letters s,p, d, f, g, h, i, k, l, . . . , respectively, are associated with angular momental = 0, 1, 2, 3, 4, 5, 6, 7, 8, . . . . The atomic states of hydrogen and hydrogenicions are degenerate: neglecting fine structure, their energies depend only on naccording to

E n = −R∞hcZ 2n−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 E n is divided by hc, the result is in wavenumberunits. The energy associated with a transition m → n is given by

∆E mn = 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 thetransitions m → n have conventional names:

Transition 1 → n 2 → n 3 → n 4 → n 5 → n 6 → nName 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 electrodynamiceffects (e.g., the Lamb shift), and interactions between the nuclear magnetic

59


62/73

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

splittings,


63/73

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 state1s2s 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 thatthe parity must change from odd to even or from even to odd. For electricdipole transitions (the only ones possible in the long-wavelength limit), otherselection rules are that the value of l of only one electron can change, and onlyby ∆l = ±1; ∆S = 0; ∆L = ±1 or 0; and ∆J = ±1 or 0 (but L = 0 does notcombine 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 1Po1 (with S = 0, L = 1, J = 1,odd parity, excitation energy 21.22 eV). These rules hold accurately only forlight atoms in the absence of strong electric or magnetic fields. Transitionsthat obey the selection rules are called “allowed”; those that do not are called“forbidden.”

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

O I I has an allowed transition between the states 2p23p′2F o7/2 and 2p

2(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 elec-trons with n = 2 and l = 1; the closed subshells 1s22s2 are not shown. Theouter (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 bethe same in both states. (Grandparents, etc., sometimes have to be specifiedin heavier atoms and ions.) Another example43 is the allowed transition from2p2(3P)3p 2Po1/2 (or

2Po3/2) 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 aschanges from one state to the other in the value of l for the valence electronand in L.

The description of fine structure, Stark and Zeeman effects, spectra of highly ionized or heavy atoms, etc., is more complicated. The most importantdifference between optical and X-ray spectra is that the latter involve energychanges of the inner electrons rather than the outer ones; often several electronsparticipate.

61


64/73

COMPLEX (DUSTY) PLASMAS

Complex (dusty) plasmas (CDPs) may be regarded as a new and unusualstate of matter. CDPs contain charged microparticles (dust grains) in additionto electrons, ions, and neutral gas. Electrostatic coupling between the grainscan vary over a wide range, so that the states of CDPs can change from weaklycoupled (gaseous) to crystalline. CDPs can be investigated at the kinetic level(individual particles are easily visualized and relevant time scales are accessi-ble). CDPs are of interest as a non-Hamiltonian system of interacting particlesand as a means to study generic fundamental physics of self-organization, pat-tern formation, phase transitions, and scaling. Their discovery has thereforeopened 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, numberdensity (in terms of the interparticle distance) nd ∼ ∆−3 ∼ 103 − 107 cm−3,temperature T d ∼ 3 × 10−2 − 102 eV.Typical discharge (bulk) plasmas

gas pressure p ∼ 10−2 − 1 Torr, T i ≃ T n ≃ 3 × 10−2 eV, vT i ≃ 7 × 104 cm/s(Ar), T

e ∼ 0.3

−3 eV, n

i ≃ n

e ∼ 108

−1010 cm−3, screening length λ

D ≃λDi ∼ 20 − 200 µm, ωpi ≃ 2 × 106 − 2 × 107 s−1 (Ar). B fields up to B ∼ 3 T.

Dimensionless

Havnes parameter P = |Z |nd/nenormalized charge z = |Z |e2/kT eadust-dust scattering parameter βd = Z

2e2/kT dλD

dust-plasma scattering parameter βe,i = |Z |e2/kT e,iλDcoupling parameter Γ = (Z 2e2/kT d∆) 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 −105 electroncharges), Γ


65/73

dust-gas friction rate ν nd

∼10a2 p/mdvT n

dust gyrofrequency ωcd = ZeB/mdc

Velocities

dust thermal velocity vT d = (kT d/md)1/2 ≡ [T dT i

mimd

]1/2vT i

dust acoustic wave velocity C DA = ωpdλD

≃ (|Z | P 1+P mi/md)1/2vT idust Alfvén wave velocity vAd = B/(4πndmd)

1/2

dust-acoustic Mach number V /C DA

dust magnetic Mach number V /vAd

dust lattice (acoustic) wave velocity C l,tDL = ωpdλDF l,t(κ)

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

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


66/73

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

Fth = −4√

2π

15(a

2/vT n)κn∇T n

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

with the momentum transfer from other species, Fα = −mdν αdVαd, i.e.,neutral, ion, and electron drag. For collisions between charged particles, twolimiting cases are distinguished by the magnitude of the scattering parameterβα. When βα ≪ 1 the result is independent of the sign of the potential. Whenβα ≫ 1, the results for repulsive and attractive interaction potentials aredifferent. For typical complex plasmas the hierarchy of scattering parametersis βe(∼ 0.01 − 0.3) ≪ βi(∼ 1 − 30) ≪ βd(∼ 103 − 3 × 104). The genericexpressions for different types of collisions are47

ν αd = (4√

2π/3)(mα/md)a2

nαvT αΦαd

Electron-dust collisions

Φed ≃ 1

2z2Λed βe ≪ 1

Ion-dust collisions

Φid =

12 z

2(T e/T i)2Λid βi 13

Dust-dust collisons

Φdd = z2dΛdd βd ≪ 1(λD/a)2[ln 4βd − ln ln 4βd], βd ≫ 1where zd ≡ Z 2e2/akT d.

For ν dd ∼ ν nd the complex plasma is in a two-phase state, and for ν nd ≫ ν ddwe have merely tracer particles (dust-neutral gas interaction dominates). Themomentum transfer cross section is proportional to the Coulomb logarithmΛαd when the Coulomb scattering theory is applicable. It is determined byintegration over the impact parameters, from ρmin to ρmax. ρmin is due to finitegrain 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


67/73

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

Λαd = zα

∞0

e−zαx ln[1 + 4(λD/aα)2x2]dx − 2zα

∞1

e−zαx ln(2x − 1)dx,

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

For ion-dust (attraction)

Λid

≃z

∞

0

e−zx ln1 + 2(T i/T e)(λD/a)x

1 + 2(T i/T e)x dx.For ν dd ≫ ν nd the complex plasma behaves like a one phase system (dust-dustinteraction 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 verti-cal dashed line at κ = 1 conditionally divides the system into Coulomb and

Yukawa parts. With respect to the usual plasma phase, in the diagram be-low 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 condi-

tion is48 Γ > 106(1 + κ + κ2/2)−1. Regions II (VI) are for Coulomb (Yukawa)non-ideal plasmas – the characteristic range of dust-dust interaction (in termsof 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 theinteraction is essentially multiparticle.

Regions III (VII and VIII) correspond toCoulomb (Yukawa) ideal gases. The rangeof dust-dust interaction is smaller than theintergrain distance and only pair collisionsare important. In addition, in the regionVIII the pair Yukawa interaction asymp-totically reduces to the hard sphere limit,forming a “Yukawa granular medium”. Inregion IV the electrostatic interaction isunimportant and the system is like a uaualgranular medium.

0.1 1 1010

-4

10-2

100

102

104

101

102

103

α-1

=∆/a

VIIIVII

VI

V

IV

III

II

I

Γ

κ=∆/λ

65


68/73

REFERENCES

When any of the formulas and data in this collection are referencedin research publications, it is suggested that the original source be cited ratherthan the Formulary . Most of this material is well known and, for all practicalpurposes, is in the “public domain.” Numerous colleagues and readers, toonumerous 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 physicsare available. The following are particularly useful:

C. W. Allen, Astrophysical Quantities, 3rd edition (Athlone Press, Lon-don, 1976).

A. Anders, A Formulary for Plasma Physics (Akademie-Verlag, Berlin,1990).

H. L. Anderson (Ed.), A Physicist’s Desk Reference , 2nd edition (Amer-ican 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 purposeof giving credit to the original workers, but (1) to guide the reader to sourcescontaining related material and (2) to indicate where to find derivations, ex-planations, examples, etc., which have been omitted from this compilation.Additional material can also be found in D. L. Book, NRL Memorandum Re-port 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 somemathematical constants not available on pocket calculators.

2. H. W. Gould, “Note on Some Binomial Coefficient Identities of Rosen-baum,” J. Math. Phys. 10, 49 (1969); H. W. Gould and J. Kaucky, “Eval-uation of a Class of Binomial Coefficient Summations,” J. Comb. Theory1, 233 (1966).

3. B. S. Newberger, “New Sum Rule for Products of Bessel Functions withApplication to Plasma Physics,” J. Math. Phys. 23, 1278 (1982); 24,2250 (1983).

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

66


69/73

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 Committeeof the Royal Society, 2nd edition (Royal Society, London, 1975) for adiscussion of nomenclature in SI units.

7. E. R. Cohen and B. N. Taylor, “The 1986 Adjustment of the FundamentalPhysical Constants,” CODATA Bulletin No. 63 (Pergamon Press, NewYork, 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 DifferentSystems 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 Com-munication , 7th edition, E. C. Jordan, Ed. (Sams and Co., Indianapolis,IN, 1985), Chapt. 1. These definitions are International Telecommunica-tions Union (ITU) Standards.

11. H. E. Thomas, Handbook of Microwave Techniques and Equipment (Prentice-Hall, Englewood Cliffs, NJ, 1972), p. 9. Further subdivisionsare 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 Fun-damentals 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-WesleyPublishing Co., Reading, MA, 1962), pp. 86–95.

15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd edition (Addison-Wesley 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 toAlmost-Perpendicular Magnetosonic Waves,” J. Plasma Phys. 6, 495(1971).

67


70/73

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,” Re-views of Plasma Physics, Vol. 1 (Consultants Bureau, New York, 1965),p. 105.

20. J. M. Greene, “Improved Bhatnagar–Gross–Krook Model of Electron-IonCollisions,” 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 (AcademicPress, New York, 1975), p. 6 (after J. W. Paul).

23. K. H. Lloyd and G. Härendel, “Numerical Modeling of the Drift and De-formation of Ionospheric Plasma Clouds and of their Interaction withOther Layers of the Ionosphere,” J. Geophys. Res. 78, 7389 (1973).

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

25. G. L. Withbroe and R. W. Noyes, “Mass and Energy Flow in the SolarChromosphere 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 crosssections are listed in F. K. McGowan, et al., Nucl. Data Tables A6,353 (1969); A8, 199 (1970). The yields listed in the table are calculateddirectly from the mass defect.

28. (a) 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); (b) X.Z. Li, Q.M. Wei, andB. Liu, “A new simple formula for fusion cross-sections of light nuclei,”Nucl. Fusion 48, 125003 (2008).

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 InstabilitiesVol. I (Consultants Bureau, New York, 1974). The table on pp. 48–49was compiled by K. Papadopoulos.

68


71/73

31. Table prepared from data compiled by J. M. McMahon (personal com-

munication, D. Book, 1990) and A. Ting (personal communication, J.D.Huba, 2004).

32. M. J. Seaton, “The Theory of Excitation and Ionization by Electron Im-pact,” 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 Atmo-spheres,” 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 Tech-niques, R. H. Huddlestone and S. L. Leonard, Eds. (Academic Press, NewYork, 1965).

36. M. Gryzinski, “Classical Theory of Atomic Collisions I. Theory of InelasticCollision,” 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 High-Temperature 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-FrequencyPower Absorption by Inverse Bremsstrahlung in Plasmas,” Phys. Fluids16, 722 (1973).

43. W. L. Wiese, M. W. Smith, and B. M. Glennon, Atomic Transition Prob-abilities, NSRDS-NBS 4, Vol. 1 (U.S. Govt. Printing Office, Washington,1966).

44. F. M. Peeters and X. Wu, “Wigner crystal of a screened-Coulomb-interaction colloidal system in two dimensions”, Phys. Rev. A 35, 3109(1987)

69


72/73

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. Scripta45, 497 (1992).

47. S. A. Khrapak, A. V. Ivlev, and G. E. Morfill, “Momentum transfer incomplex plasmas”, Phys. Rev. E (2004).

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

70


73/73

AFTERWORD

The NRL Plasma Formulary originated over twenty five years agoand has been revised several times during this period. The guiding spirit andperson primarily responsible for its existence is Dr. David Book. I am indebtedto Dave for providing me with the TEX files for the Formulary and his con-tinued suggestions for improvement. The Formulary has been set in TEX byDave Book, Todd Brun, and Robert Scott. I thank readers for communicatingtypographical errors to me as well as suggestions for improvements.

Finally, I thank Dr. Sidney Ossakow for his support of the NRL Plasma

Formulary during his tenure as Superintendent of the Plasma Physics Division.He was a steadfast advocate of this important project at the Naval ResearchLaboratory.

anon - naval research laboratory plasma formulary - nrl - 2009

Documents