NRL Plasma Formulary

7/31/2019 NRL Plasma Formulary

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2011

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

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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

Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . 19Electricity 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

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NUMERICAL AND ALGEBRAIC

Gain in decibels of P2 relative to P1

G = 10 log10(P2/P1).

To within two percent

(2)1/2 2.5; 2 10; e3 20; 210 103.

Euler-Mascheroni constant1 = 0.57722

Gamma Function (x + 1) = x(x):

(1/6) = 5.5663 (3/5) = 1.4892(1/5) = 4.5908 (2/3) = 1.3541(1/4) = 3.6256 (3/4) = 1.2254(1/3) = 2.6789 (4/5) = 1.1642(2/5) = 2.2182 (5/6) = 1.1288(1/2) = 1.7725 =

(1) = 1.0

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

(1 + x) =

k=0

kxk

1 + x +

( 1)

2!

x2 +( 1)( 2)

3!

x3 + . . . .

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

nk=0

x

x + kz

x + kzk

yy + (n k)z

y + (n k)zn k

=

x + y

x + y + nz x + y + nz

n .

Newbergers summation formula3 [good for nonintegral, Re ( + ) > 1]:

n=

(1)nJn(z)J+n(z)n +

=

sin J+(z)J(z).

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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) = fg + gf

(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 bewritten in the dyadic form

(17) T =

i,j

Tijeiej

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

(18) (T)i =

j(Tji/xj)

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

(19) (AB) = ( A)B + (A )B(20) (fT) = fT+fT

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Let r = ix + jy + kz be the radius vector of magnitude r, from the origin tothe point x ,y ,z. Then

(21) r = 3

(22) r = 0(23) r = r/r

(24) (1/r) = r/r3

(25) (r/r3) = 4(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

dVf = S

dSf

(28)

V

dV A =S

dS A

(29)

V

dVT =S

dS T

(30)V

dV A = S

dS A

(31)

V

dV(f2g g2f) =S

dS (fg gf)

(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

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(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 CoordinatesDivergence

A = 1r

r(rAr) +

1

r

A

+

Az

z

Gradient

(f)r =f

r; (f) =

1

r

f

; (f)z =

f

z

Curl

( A)r =1

r

Az

A

z

( A) =Ar

z Az

r

( A)z =1

r

r (rA) 1

r

Ar

Laplacian

2f = 1r

r

r

f

r

+

1

r22f

2+

2f

z 2

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Laplacian of a vector

(2A)r = 2Ar 2

r2A

Ar

r2

(2A) = 2A +2

r2Ar

A

r2

(2A)z = 2Az

Components of (A )B

(A B)r = Ar Brr

+ Ar

Br

+ Az Brz

ABr

(A B) = ArB

r+

A

r

B

+ Az

B

z+

ABr

r

(A B)z = ArBz

r+

A

r

Bz

+ Az

Bz

z

Divergence of a tensor

( T)r =1

r

r(rTrr) +

1

r

Tr

+

Tzr

z T

r

( T) =1

r

r(rTr) +

1

r

T

+

Tz

z+

Tr

r

( T)z =1

r

r(rTrz) +

1

r

Tz

+

Tzz

z

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Spherical Coordinates

Divergence

A =

1

r2

r

(r2

Ar) +1

r sin

(sin A) +1

r sin

A

Gradient

(f)r =f

r; (f) =

1

r

f

; (f) =

1

r sin

f

Curl

( A)r =1

r sin

(sin A)

1

r sin

A

( A) =1

r sin

Ar

1

r

r(rA)

( A) =1

r

r(rA)

1

r

Ar

Laplacian

2f = 1r2

r

r

2 f

r

+

1

r2 sin

sin

f

+

1

r2 sin2

2f

2

Laplacian of a vector

(2A)r = 2Ar 2Ar

r2 2

r2A

2cot A

r2 2

r2 sin

A

(2A) = 2A + 2r2

Ar

A

r2 sin2 2 cos

r2 sin2

A

(2A) = 2A A

r2 sin2 +

2

r2 sin

Ar

+

2 cos

r2 sin2

A

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Components of (A )B

(A B)r = ArBr

r+

A

r

Br

+

A

r sin

Br

AB + AB

r

(A B) = ArB

r+

A

r

B

+

A

r sin

B

+

ABr

r cot AB

r

(A B) = ArB

r+

A

r

B

+

A

r sin

B

+

ABr

r+

cot AB

r

Divergence of a tensor

( T)r = 1r2

r

(r2Trr) + 1r sin

(sin Tr)

+1

r sin

Tr

T + T

r

( T) =1

r2

r(r

2Tr) +1

r sin

(sin T)

+ 1r sin

T

+ Trr

cot Tr

( T) =1

r2

r(r

2Tr) +1

r sin

(sin T)

+1

r sin

T

+

Tr

r+

cot T

r

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MAXWELLS EQUATIONS

Name or Description SI Gaussian

Faradays law E = Bt

E = 1c

B

t

Amperes law H = Dt

+ J H = 1c

D

t+

4

cJ

Poisson equation D = D = 4[Absence of magnetic B = 0 B = 0

monopoles]

Lorentz force on q (E + v B) q

E +

1

cv B

charge qConstitutive D = E D = Erelations B = H B = H

In a plasma, 0 = 4 107 H m1 (Gaussian units: 1). Thepermittivity satisfies 0 = 8.8542 1012 F m1 (Gaussian: 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 /0 = 1+ 36 109/B2 (SI) = 1+4c2/B2 (Gaussian),where is the mass density.

The electromagnetic energy in volume V is given by

W =1

2

V

dV(H B + E D) (SI)

=1

8

V

dV(H B + E D) (Gaussian).

Poyntings theorem is

W

t+S

N dS = V

dVJ E,

where S is the closed surface bounding V and the Poynting vector (energy fluxacross S) is given by N = E H (SI) or N = cE H/4 (Gaussian).

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ELECTRICITY AND MAGNETISM

In the following, = dielectric permittivity, = permeability of conduc-tor, = permeability of surrounding medium, = conductivity, f = /2 =radiation frequency, m = /0 and e = /0. 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 0 = 8.8542 1012 F m1Permeability of free space 0 = 4 107 H m1

= 1.2566 106 H m1Resistance of free space R0 = (0/0)

1/2 = 376.73

Capacity of parallel plates of area C = A/dA, separated by distance d

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

of length l, radii a, bCapacity of concentric spheres of C = 4ab/(b a)

radii a, b

Self-inductance of wire of length L = l/8l, carrying uniform current

Mutual inductance of parallel wires L = (l/4) [1 + 4 ln(d/a)]of length l, radius a, separatedby distance d

Inductance of circular loop of radius L = b

[ln(8b/a) 2] + /4

b, made of wire of radius a,carrying uniform currentRelaxation time in a lossy medium = /

Skin depth in a lossy medium = (2/)1/2 = (f)1/2

Wave impedance in a lossy medium Z = [/( + i/)]1/2

Transmission coefficient at T = 4.22 104(f m1e2/)1/2conducting surface9

(good only for T 1)Field at distance r from straight wire B = I/2r tesla

carrying current I (amperes) = 0.2I/r gauss (r in cm)Field at distance z along axis from Bz = a

2I/[2(a2 + z2)3/2]circular loop of radius acarrying current I

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AC CIRCUITS

For a resistance R, inductance L, and capacitance C in series witha voltage source V = V0 exp(it) (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) = Is + It, where the steady state isIs = iqs = V /Z in terms of the impedance Z = R + i(L 1/C) andIt = dqt/dt. For initial conditions q(0) q0 = q0 + qs, I(0) I0, thetransients can be of three types, depending on = R2 4L/C:(a) Overdamped, > 0

qt =I0 + +q0

+ exp(t) I0 + q0

+ exp(+t),

It =+(I0 + q0)

+ exp(+t)

(I0 + +q0)+

exp(t),

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

qt = [q0 + (I0 + Rq0)t] exp(Rt),It = [I0 (I0 + Rq0)Rt] exp(Rt),

where R = R/2L;

(c) Underdamped, < 0

qt =

R q0 + I0

1sin 1t + q0 cos 1t

exp(Rt),

It = I0 cos 1t (12 + R2)q0 + RI01 sin(1t) exp(Rt),Here 1 = 0(1 R2C/4L)1/2, where 0 = (LC)1/2 is the resonantfrequency. At = 0, Z = R. The quality of the circuit is Q = 0L/R.Instability results when L, R, C are not all of the same sign.

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Nomenclature:

B Magnetic induction

Cs, c Speeds of sound, light

cp Specific heat at constant pressure (units m2 s2 K1)

D = 2R Pipe diameter

F Imposed force

f Vibration frequency

g Gravitational acceleration

H, L Vertical, horizontal length scales

k = cp Thermal conductivity (units kg m1 s2)

N = (g/H)1/2 BruntVaisala frequency

R Radius of pipe or channel

r Radius of curvature of pipe or channelrL Larmor radiusT Temperature

V Characteristic flow velocity

VA = B/(0)1/2 Alfven speed

Newtons-law heat coefficient, kT

x= T

Volumetric expansion coefficient, dV/V = dT

Bulk modulus (units kg m1 s2)

R, V, p, T Imposed differences in two radii, velocities,pressures, or temperatures

Surface emissivity

Electrical resistivity

, D Thermal, molecular diffusivities (units m2 s1) Latitude of point on earths surface

Collisional mean free path

= Viscosity

0 Permeability of free space

Kinematic viscosity (units m2 s1) Mass density of fluid medium

Mass density of bubble, droplet, or moving object

Surface tension (units kg s2) StefanBoltzmann constant Solid-body rotational angular velocity

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SHOCKS

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

(1) U = U q;(2) U2 +p + B 2 /2 = U

2 + p + B 2 /2;

(3) U V BB/ = UV BB/;(4) B = B;

(5) U B V B = UB VB;(6) 12 (U

2 + V2) + w + (UB 2 V BB)/U= 12 (U

2 + V2) + w + (UB 2 VBB)/U.Here U and V are components of the fluid velocity normal and tangential tothe front in the shock frame; = 1/ is the mass density; p is the pressure;B = B sin , B = B cos ; is the magnetic permeability ( = 4 in cgsunits); and the specific enthalpy is w = e + p, where the specific internalenergy e satisfies de = T ds pd 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 U = [(p p)( )]1/2;(8) (p p)( )1 = q2;

(9) w w = 12 (p p)( + );(10) e e = 12 ( p +p)( ).

In what follows we assume that the fluid is a perfect gas with adiabatic index = 1 + 2/n, where n is the number of degrees of freedom. Then p = RT/m,where R is the universal gas constant and m is the molar weight; the soundspeed is given by Cs

2 = (p/)s = p; and w = e = p/( 1). For ageneral oblique shock in a perfect gas the quantity X = r1(U/VA)2 satisfies14

(11) (X/)(Xcos2 )2 = X sin2

[1 + (r 1)/2] X cos2

, where

r = /, = 12

[ + 1

(

1)r], and = Cs2/VA

2 = 4p/B2.

The density ratio is bounded by

(12) 1 < r < ( + 1)/( 1).If the shock is normal to B (i.e., if = /2), then

(13) U2 = (r/)

Cs2 + VA

2 [1 + (1 /2)(r 1)]

;

(14) U/U = B/B = r;

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(15) V = V;

(16) p = p + (1 r1)U2 + (1 r2)B2/2.If = 0, there are two possibilities: switch-on shocks, which require < 1 andfor which

(17) U2 = rVA2;

(18) U = VA2/U;

(19) B 2 = 2B2 (r 1)( );

(20) V = UB/B;

(21) p = p + U2(1 + )(1 r1),and acoustic (hydrodynamic) shocks, for which

(22) U2 = (r/)Cs2;

(23) U = U/r;

(24) V = B = 0;

(25) p = p + U2(1 r1).For acoustic shocks the specific volume and pressure are related by

(26) / = [( + 1)p + ( 1)p] / [( 1)p + ( + 1)p].In terms of the upstream Mach number M = U/Cs,

(27) / = / = U/U = ( + 1)M2/[(

1)M2 + 2];

(28) p/p = (2M2 + 1)/( + 1);(29) T /T = [( 1)M2 + 2](2M2 + 1)/( + 1)2M2;(30) M2 = [( 1)M2 + 2]/[2M2 + 1].

The entropy change across the shock is

(31) s s s = c ln[(p/p)(/) ],where c = R/( 1)m is the specific heat at constant volume; here R is thegas constant. In the weak-shock limit (M 1),

(32) s c 2( 1)3( + 1)

(M2 1)3 16R3( + 1)m

(M 1)3.

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

(33) RS = C0(Et2/)1/5,

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

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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, = mi/mp; Z is charge state; k is Boltzmanns constant; K is wavenumber;

is the adiabatic index; ln is the Coulomb logarithm.Frequencies

electron gyrofrequency fce = ce/2 = 2.80 106B Hzce = eB/mec = 1.76 107B rad/sec

ion gyrofrequency fci = ci/2 = 1.52 103Z1B Hzci = ZeB/mic = 9.58 103Z1B rad/sec

electron plasma frequency fpe = pe/2 = 8.98 103ne1/2 Hzpe = (4nee

2/me)1/2

= 5.64 104ne1/2 rad/secion plasma frequency fpi = pi/2

= 2.10 102Z1/2ni1/2 Hzpi = (4niZ

2e2/mi)1/2

= 1.32 103Z1/2ni1/2rad/secelectron trapping rate Te = (eKE/me)

1/2

= 7.26 108K1/2E1/2 sec1

ion trapping rate Ti = (ZeKE/mi)1/2

= 1.69 107Z1/2K1/2E1/21/2 sec1electron collision rate e = 2.91 106ne ln Te3/2 sec1ion collision rate i = 4.80 108Z41/2ni ln Ti3/2 sec1

Lengths

electron deBroglie length = h/(mekTe)1/2 = 2.76 108Te1/2 cm

classical distance of e2/kT = 1.44 107T1 cmminimum approach

electron gyroradius re = vTe/ce = 2.38Te1/2

B1

cmion gyroradius ri = vTi/ci

= 1.02 1021/2Z1Ti1/2B1 cmelectron inertial length c/pe = 5.31 105ne1/2 cmion inertial length c/pi = 2.28 107Z1(/ni)1/2 cmDebye length D = (kT /4ne

2)1/2 = 7.43 102T1/2n1/2 cm

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Velocities

electron thermal velocity vTe = (kTe/me)1/2

= 4.19 107Te1/2 cm/secion thermal velocity vTi = (kTi/mi)

1/2

= 9.79 1051/2Ti1/2 cm/secion sound velocity Cs = (ZkTe/mi)

1/2

= 9.79 105(Z Te/)1/2 cm/secAlfven velocity vA = B/(4nimi)

1/2

= 2.18 10111/2ni1/2B cm/secDimensionless

(electron/proton mass ratio)1/2 (me/mp)1/2 = 2.33 102 = 1/42.9

number of particles in (4/3)nD3

= 1.72 109

T3/2

n1/2

Debye sphere

Alfven velocity/speed of light vA/c = 7.281/2ni1/2B

electron plasma/gyrofrequency pe/ce = 3.21 103ne1/2B1ratio

ion plasma/gyrofrequency ratio pi/ci = 0.1371/2ni

1/2B1

thermal/magnetic energy ratio = 8nkT/B2 = 4.03 1011nT B2magnetic/ion rest energy ratio B2/8nimic

2 = 26.51ni1B2

MiscellaneousBohm diffusion coefficient DB = (ckT/16eB)

= 6.25 106T B1 cm2/sectransverse Spitzer resistivity = 1.15 1014Z ln T3/2 sec

= 1.03 102Z ln T3/2 cmThe anomalous collision rate due to low-frequency ion-sound turbulence is

* pe

W/kT = 5.64 104ne1/2

W/kTsec

1,

where W is the total energy of waves with /K < vTi.Magnetic pressure is given byPmag = B

2/8 = 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

WkT = 1012

cal = 4.2 1019 erg.

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PLASMA DISPERSION FUNCTION

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

Z() = 1/2

+

dt expt2

t = 2i exp2i

dt expt2 .

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

Differential equation:

dZ

d= 2 (1 + Z) , Z(0) = i1/2; d

2Z

d2+ 2

dZ

d+ 2Z = 0.

Real argument (y = 0):

Z(x) = exp x2i1/2 2x

0dt expt2 .

Imaginary argument (x = 0):

Z(iy) = i1/2

exp

y2

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

Z() = i1/2

exp2

2

1 22/3 + 44/15 86/105 +

.

Asymptotic series, || 1 (Ref. 17):

Z() = i1/2 exp2 1 1 + 1/22 + 3/44 + 15/86 + ,where

=

0 y > |x|11 |y| < |x|12 y < |x|1

Symmetry properties (the asterisk denotes complex conjugation):

Z(*) = [Z()]*;Z(*) = [Z()] * + 2i

1/2exp[(*)2] (y > 0).

Two-pole approximations18 (good for in upper half plane except when y * and nega-tive for < *, where x* = (m/m)*/T is the solution of

(x*) =(m|m)(x*). The ratio */T is given for a number of specific , in thefollowing table:

\ i|e e|e, i|i e|p e|D e|T, e|He3 e|He4*

T1.5 0.98 4.8 103 2.6 103 1.8 103 1.4 103

When both species are near Maxwellian, with Ti 0,

T =2

e2e

2n

m1/2(kT

)3/2

A2

3 + (A + 3) tan1(A1/2)A1/2

.

If A < 0, tan1(A1/2)/A1/2 is replaced by tanh1(A)1/2/(A)1/2. ForT T T,

eT = 8.2 107nT3/2 sec1;

iT = 1.9 108nZ21/2T3/2 sec1.

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Thermal Equilibration

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

dT

dt =

\

(T T),

where

\ = 1.8 1019(mm)

1/2Z2Z

2n

(mT + mT)3/2sec

1.

For electrons and ions with Te Ti T, this implies

e|i /ni =

i|e /ne = 3.2 109Z2/T3/2cm3 sec1.

Coulomb Logarithm

For test particles of mass m and charge e = Ze scattering off fieldparticles of mass m and charge e = Ze, the Coulomb logarithm is defined

as = ln ln(rmax/rmin). Here rmin is the larger of ee/mu2 andh/2mu, averaged over both particle velocity distributions, where m =

mm/(m + m) and u = v v ; rmax = (4

ne2/kT)

1/2, wherethe summation extends over all species for which u2 < vT

2 = kT/m . If

this inequality cannot be satisfied, or if either uc1 < rmax or uc1 47z(J)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)(nA0I/2a);

self-integrating V = (R/L)(nA0I/2a) = RI/n.

X-ray production, target with average atomic number Z (V


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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 plasma/magnetic energyV velocity density ratioT temperature VA Alfven speedne, ni number density e, i gyrofrequencyn harmonic number H hybrid gyrofrequency,

Cs = (Te/M)1/2 ion sound speed H

2 = eie, i plasma frequency U relative drift velocity ofD Debye length two ion species

50


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Formulas

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

n

= [1

2pe/(

ce)]

1/2,

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

d/ds = (k/2)(n n+) = 2.36 104N Bf2 cm1,

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 is

v0 = eEmax/m = 25.6I1/20 cm sec1

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

0 in m. The ratio of quiver energy to thermal energy is

Wqu/Wth = mev02/2kT = 1.81 10130 2I/T,

where T is given in eV. For example, if I = 1015 W cm2, 0 = 1 m, T =2 keV, then Wqu/Wth 0.1.

Pondermotive force:

FF= NE2/8Nc,

whereNc = 1.1 102102cm3.

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

2.44F/DL

and l

2F2

/DL

.

Here is the beam divergence containing 85% of energy and DL is thediffraction-limited divergence:

DL = 2.44/b,

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

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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 e

labels electrons. N refers to number density, n to principal quantum number.Asterisk superscripts on level population densities denote local thermodynamicequilibrium (LTE) values. Thus Nn* is the LTE number density of atoms (orions) in level n.

Characteristic atomic collision cross section:

(1) a02

= 8.80 1017 cm2.

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

(2) EZ(n, l) =

Z2EH(n l)2

,

where EH = 13.6 eV is the hydrogen ionization energy and l = 0.75l5,

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) mn = 2.36 1013fmng(n, m)

Enmcm2,

where fmn is the oscillator strength, g(n, m) is the Gaunt factor, is theincident electron energy, and Enm = En Em.Electron excitation rate averaged over Maxwellian velocity distribution, Xmn= Ne

mnv

(Refs. 34, 35):

(4) Xmn = 1.6 105fmng(n, m)Ne

EnmT1/2e

exp

Enm

Te

sec

1,

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

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Rate for electron collisional deexcitation:

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

Here Nm*/Nn* = (gm/gn) exp(Enm/Te) 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)fmn(Enm)2 sec1.

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

(7) Inm = 1.6 1019AnmNnEnm watt/cm3.

Condition for steady state in a corona model:

(8) N0Ne0nv = NnAn0,

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) Inm = 5.1 1025fnmgmNeN0

g0T1/2e

EnmEn0

3exp

En0

Te

wattcm3

.

Ionization and Recombination

In a general time-dependent situation the number density of the chargestate Z satisfies

(10)dN(Z)

dt = Ne S(Z)N(Z) (Z)N(Z)

+S(Z 1)N(Z 1) + (Z + 1)N(Z + 1)

.

Here S(oZ) is the ionization rate. The recombination rate (Z) has the form(Z) = r(Z) + Ne3(Z), where r and 3 are the radiative and three-bodyrecombination rates, respectively.

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Classical ionization cross-section36 for any atomic shell j

(11) i = 6 1014bjgj(x)/Uj2 cm2.

Here bj is the number of shell electrons; Uj is the binding energy of the ejectedelectron; x = /Uj , where 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


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Ionization Equilibrium Models

Saha equilibrium:39

(17)NeN1*(Z)

Nn*(Z 1) = 6.0 1021gZ

1

Te3/2

gZ1nexp EZ(n, l)Te cm3,

where gZn is the statistical weight for level n of charge state Z and EZ(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)NeN*(Z)

N*(Z 1)=

S(Z 1)

3

,

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 1018Z7n17/2(T /EZ)1/2cm3.

Steady state condition in corona model:

(21)N(Z 1)

N(Z)=

r

S(Z 1) .

Corona model is applicable if40

(22) 1012

tI1

< Ne < 1016

Te7/2

cm3

,

where tI is the ionization time.

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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


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Bremsstrahlung from hydrogen-like plasma:26

(30) PBr = 1.69 1032NeTe1/2

Z2

N(Z)

watt/cm3

,

where the sum is over all ionization states Z.Bremsstrahlung optical depth:41

(31) = 5.0 1038NeNiZ2gLT7/2,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 :

(32) = 3.1 107Zne2 ln T3/22(1 2p/2)1/2 cm1;

here is the electron thermal velocity divided by V, where V is the larger of and p multiplied by the larger of Ze

2/kT and h/(mkT)1/2.

Recombination (free-bound) radiation:

(33) Pr = 1.69 1032NeTe1/2

Z2

N(Z)

EZ1

Te

watt/cm

3.

Cyclotron radiation26 in magnetic field B:

(34) Pc = 6.21 1028

B

2

NeTe watt/cm

3

.

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

(35) Pc = 5.00 1038N2e T2e watt/cm3.

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

(36) tc 9.0 108B2

2.5 + sec,

where 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 = (57BL)1/6

,

where = 8NkT/B2.

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

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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 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, andthe 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 hydrogenic

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

En = RhcZ2n2

1 + m/M= RyZ

2

n2,

where h is Plancks 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 cm1

is the Rydberg constant. If En is divided by hc, the result is in wavenumberunits. 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 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 , , , etc. Thus the transition 1 3 gives rise to the Lyman- line. Relativistic effects, quantum electrodynamiceffects (e.g., the Lamb shift), and interactions between the nuclear magnetic

59


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


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the value of J. The superscript o indicates that the state has odd parity; itwould 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). Transitionsare 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. Transitions

that obey the selection rules are called allowed; those that do not are calledforbidden.

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 II has an allowed transition between the states 2p23p2Fo7/2 and 2p

2(1D)3d 2F7/2 (and between the states obtained by changingJ 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 ofhighly ionized or heavy atoms, etc., is more complicated. The most important

difference between optical and X-ray spectra is that the latter involve energychanges of the inner electrons rather than the outer ones; often several electronsparticipate.

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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 m, mass md

3

107

3

1013 g, number

density (in terms of the interparticle distance) nd 3 103 107 cm3,temperature Td 3 102 102 eV.Typical discharge (bulk) plasmas

gas pressure p 102 1 Torr, Ti Tn 3 102 eV, vTi 7 104 cm/s(Ar), Te 0.3 3 eV, ni ne 108 1010 cm3, screening length D Di 20 200 m, pi 2 106 2 107 s1 (Ar). B fields up to B 3 T.

Dimensionless

Havnes parameter P = |Z|nd/nenormalized charge z = |Z|e2/kTeadust-dust scattering parameter d = Z

2e2/kTdD

dust-plasma scattering parameter e,i = |Z|e2/kTe,iDcoupling parameter = (Z2e2/kTd) exp(/D)lattice parameter = /D

particle parameter = a/

lattice magnetization parameter = /rd

Typical experimental values: P 104

102

,z 24 (Z 103

105

electroncharges), < 103, 0.3 10, 104 3 102, < 1

Frequencies

dust plasma frequency pd = (4Z2e2nd/md)

1/2

(|Z| P1+Pmi/md)1/2picharge fluctuation frequency ch 1+z

2(a/D)pi

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dust-gas friction rate nd 10a2p/mdvTndust gyrofrequency cd = ZeB/mdc

Velocities

dust thermal velocity vTd = (kTd/md)1/2 [TdTi

mimd ]1/2vTi

dust acoustic wave velocity CDA = pdD

(|Z| P1+Pmi/md)1/2vTidust Alfven wave velocity vAd = B/(4ndmd)

1/2

dust-acoustic Mach number V /CDA

dust magnetic Mach number V /vAd

dust lattice (acoustic) wave velocity Cl,tDL

= pdDFl,t()

The range of the dust-lattice wavenumbers is K < The functions Fl,t()for longitudinal and transverse waves can be approximated44,45 with accuracy< 1% in the range 5:

Fl 2.701/2(1 0.096 0.0042), Ft 0.51(1 0.0392),

Lengths

frictional dissipation length L = vTd/nd

dust Coulomb radius RCe,i = |Z|e2/kTe,idust gyroradius rd = vTd/cd

Grain Charging

The charge evolution equation is d|Z|/dt = Ii Ie. From orbital motionlimited (OML) theory46 in the collisionless limit len(in) D a:

Ie =

8a2nevTe exp(z), Ii =

8a

2nivTi

1 +

Te

Tiz

.

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 ch. Other charging mechanisms are photoemission, secondaryemission, thermionic emission, field emission, etc. Charged dust grains changethe plasma composition, keeping quasineutrality. A measure of this is theHavnes parameter P = |Z|nd/ne. The balance of Ie and Ii yields

exp(z) =

mi

me

Ti

Te

1/21 +

Te

Tiz

[1 + P(z)]

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When the relative charge density of dust is large, P 1, the grain charge Zmonotonically 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

2

15(a

2/vTn)nTn

(where n is the coefficient of gas thermal conductivity); forces associatedwith the momentum transfer from other species, F = mddVd, i.e.,neutral, ion, and electron drag. For collisions between charged particles, twolimiting cases are distinguished by the magnitude of the scattering parameter. When 1 the result is independent of the sign of the potential. When

1, the results for repulsive and attractive interaction potentials aredifferent. For typical complex plasmas the hierarchy of scattering parametersis e( 0.01 0.3) i( 1 30) d( 103 3 104). The genericexpressions for different types of collisions are47

d = (4

2/3)(m/md)a2

nvTd

Electron-dust collisions

ed 1

2z2ed e 1

Ion-dust collisions

id =

12 z

2(Te/Ti)2id i < 5

2(D/a)2(ln2 i + 2 ln i + 2), i > 13

Dust-dust collisons

dd =

z2ddd d 1(D/a)

2[ln 4d l nl n4d], d 1

where zd Z2e2/akTd.

For dd nd the complex plasma is in a two-phase state, and for nd ddwe have merely tracer particles (dust-neutral gas interaction dominates). Themomentum transfer cross section is proportional to the Coulomb logarithmd when the Coulomb scattering theory is applicable. It is determined byintegration over the impact parameters, from min to max. min is due to finitegrain size and is given by OML theory. max = D for repulsive interaction

(applicable for 1), and max = D( 1 +2)1/2 for attractive interaction(applicable up to < 5).

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

d = z

0

ezx ln[1 + 4(D/a)

2x2]dx 2z

1

ezx ln(2x 1)dx,

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

For ion-dust (attraction)

id z

0

ezx

ln

1 + 2(Ti/Te)(D/a)x

1 + 2(Ti/Te)x

dx.

For dd 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 ofthe electrostatic coupling parameter and or , respectively. The verti-cal dashed line at = 1 conditionally divides the system into Coulomb andYukawa 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 > 106(1 + + 2/2)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), (/)1/2 > (4/3)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 region

VIII 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

-1

=/a

VIIIVII

VI

V

IV

III

II

I

=/

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REFERENCES

When any of the formulas and data in this collection are referencedin 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 practicalpurposes, is in the public domain. Numerous colleagues and readers, toonumerous to list by name, have helped in collecting and shaping the Formularyinto 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 Physicists 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 MathematicalFunctions (Dover, New York, 1968), pp. 13, 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. 4752 and pp. 656666.

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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 a

discussion 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. I3.

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. F306323.

13. W. D. Hayes, The Basic Theory of Gasdynamic Discontinuities, in Fun-damentals of Gas Dynamics, Vol. III, High Speed Aerodynamics and JetPropulsion, 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. 8695.

15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd edition (Addison-Wesley Publishing Co., Reading, MA, 1987), pp. 320336.

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

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18. B. D. Fried, C. L. Hedrick, J. McCune, Two-Pole Approximation for thePlasma 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 BhatnagarGrossKrook Model of Electron-IonCollisions, Phys. Fluids 16, 2022 (1973).

21. S. I. Braginskii, Transport Processes in a Plasma, Reviews of PlasmaPhysics, 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. Harendel, 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 andReactivities, 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, and

B. 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 / 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. 4849was compiled by K. Papadopoulos.

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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 Lossfrom -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, New

York, 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 Hydrogenic Ions, Mon. Not.Roy. Astron. Soc. 119, 81 (1959).

38. Ya. B. Zeldovich 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. Fluids

16, 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)

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45. S. Zhdanov, R. A. Quinn, D. Samsonov, and G. E. Morfill, Large-scalesteady-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).

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AFTERWORD

The NRL Plasma Formulary originated over twenty five years agoand has been revised several times during this period. The guiding spirit and

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

NRL Plasma Formulary

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