huba j.d. (ed.) the natl. plasma physics formulary

DIMENSIONS AND UNITSTo get the value of a quantity in Gaussian units, multiply the value pressed in SI units by the conversion factor... INTERNATIONAL SYSTEM SI NOMENCLATURE6Physical Nam

2004 REVISED

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

The NRL Plasma Formulary originated over twenty five years ago andhas been revised several times during this period The guiding spirit and per-son primarily responsible for its existence is Dr David Book I am indebted toDave for providing me with the TEX files for the Formulary and his continuedsuggestions for improvement The Formulary has been set in TEX by DaveBook, Todd Brun, and Robert Scott Finally, I thank readers for communicat-ing typographical errors to me

Numerical and Algebraic 4

Vector Identities 5

Differential Operators in Curvilinear Coordinates 7

Dimensions and Units 11

International System (SI) Nomenclature 14

Metric Prefixes 14

Physical Constants (SI) 15

Physical Constants (cgs) 17

Formula Conversion 19

Maxwell’s Equations 20

Electricity and Magnetism 21

Electromagnetic Frequency/Wavelength Bands 22

AC Circuits 23

Dimensionless Numbers of Fluid Mechanics 24

Shocks 27

Fundamental Plasma Parameters 29

Plasma Dispersion Function 31

Collisions and Transport 32

Ionospheric Parameters 41

Solar Physics Parameters 42

Thermonuclear Fusion 43

Relativistic Electron Beams 45

Beam Instabilities 47

Approximate Magnitudes in Some Typical Plasmas 49

Lasers 51

Atomic Physics and Radiation 53

Atomic Spectroscopy 59

Complex (Dusty) Plasmas 62

References 66

NUMERICAL AND ALGEBRAICGain in decibels of P2 relative to P1

G = 10 log10(P2/P1)

To within two percent

(2π)1/2 ≈ 2.5; π2 ≈ 10; e3 ≈ 20; 210 ≈ 103.Euler-Mascheroni constant1 γ = 0.57722

(1 + x)α =

∞Xk=0

αk

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

(19) ∇ · (AB) = (∇ · A)B + (A · ∇)B

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

f dg = −

IC

gdf

DIFFERENTIAL OPERATORS INCURVILINEAR COORDINATES5Cylindrical Coordinates

∂Aφ

∂φ +

∂Az

∂zGradient

∂φ +

∂Tzz

∂z

DIMENSIONS AND UNITS

To get the value of a quantity in Gaussian units, multiply the value pressed in SI units by the conversion factor Multiples of 3 in the conversionfactors result from approximating the speed of light c = 2.9979 × 1010cm/sec

ex-≈ 3 × 1010cm/sec

DimensionsPhysical Sym- SI Conversion GaussianQuantity bol SI Gaussian Units Factor UnitsCapacitance C t

2q2

ml2 l farad 9 × 1011 cmCharge q q m

1/2l3/2

t coulomb 3 × 109 statcoulombCharge ρ q

t

m1/2l3/2

t2 ampere 3 × 109 statampereCurrent J, j q

l2

m1/2

l1/2t coulomb 12π × 105 statcoulomb

/m2 /cm2Electric field E ml

lt2

m

lt2 joule/m3 10 erg/cm3density

DimensionsPhysical Sym- SI Conversion GaussianQuantity bol SI Gaussian Units Factor Units

Force F ml

t2

ml

t2 newton 105 dyneFrequency f, ν 1

t

1

t hertz 1 hertzImpedance Z ml

(cm)Magnetic H q

lt

m1/2

l1/2t ampere– 4π × 10−3 oerstedintensity turn/m

Magnetic m, µ l

2qt

t

m1/2l1/2

t2 ampere– 4π

10 gilbertmotance Mmf turn

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

(kg)Momentum p, P ml

Permeability µ ml

q2 1 henry/m 1

4π × 107 —

DimensionsPhysical Sym- SI Conversion GaussianQuantity bol SI Gaussian Units Factor UnitsPermittivity  t

2q2

ml3 1 farad/m 36π × 109 —Polarization P q

2

t3

ml2

t3 watt 107 erg/secPower m

3

tq2 t ohm–m 1

9 × 10−9 secThermal con- κ, k ml

t3

ml

t3 watt/m– 105 erg/cm–sec–ductivity deg (K) deg (K)Time t t t second (s) 1 second (sec)Vector A ml

tq

m1/2l1/2

t weber/m 10

6 gauss–cmpotential

INTERNATIONAL SYSTEM (SI) NOMENCLATURE6

Physical Name Symbol Physical Name SymbolQuantity of Unit for Unit Quantity of Unit for Unit

*length meter m electric volt V

potential

*mass kilogram kg

electric ohm Ω

*time second s resistance

*current ampere A electric siemens S

†solid angle steradian sr intensity

frequency hertz Hz luminous flux lumen lmenergy joule J illuminance lux lxforce newton N activity (of a becquerel Bq

radioactivepressure pascal Pa source)

power watt W absorbed dose gray Gy

(of ionizingelectric charge coulomb C radiation)

*SI base unit †Supplementary unit

METRIC PREFIXESMultiple Prefix Symbol Multiple Prefix Symbol

PHYSICAL CONSTANTS (SI)7

Physical Quantity Symbol Value UnitsBoltzmann constant k 1.3807 × 10−23 J K−1

Elementary charge e 1.6022 × 10−19 C

Electron mass me 9.1094 × 10−31 kg

Proton mass mp 1.6726 × 10−27 kg

Gravitational constant G 6.6726 × 10−11 m3s−2kg−1Planck constant h 6.6261 × 10−34 J s

¯

h = h/2π 1.0546 × 10−34 J sSpeed of light in vacuum c 2.9979 × 108 m s−1

Permittivity of 0 8.8542 × 10−12 F m−1free space

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

Atomic cross section πa02 8.7974 × 10−21 m2

Classical electron radius re = e2/4π0mc2 2.8179 × 10−15 m

Thomson cross section (8π/3)re2 6.6525 × 10−29 m2

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

electron ¯h/mec 3.8616 × 10−13 m

Fine-structure constant α = e2/20hc 7.2974 × 10−3

α−1 137.04First radiation constant c1 = 2πhc2 3.7418 × 10−16 W m2

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

constant

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

Physical Quantity Symbol Value UnitsWavelength associated λ0 = hc/e 1.2398 × 10−6 m

Avogadro number NA 6.0221 × 1023 mol−1

Faraday constant F = NAe 9.6485 × 104 C mol−1Gas constant R = NAk 8.3145 J K−1mol−1Loschmidt’s number n0 2.6868 × 1025 m−3

PHYSICAL CONSTANTS (cgs)7

Physical Quantity Symbol Value UnitsBoltzmann constant k 1.3807 × 10−16 erg/deg (K)Elementary charge e 4.8032 × 10−10 statcoulomb

(statcoul)Electron mass me 9.1094 × 10−28 g

Atomic cross section πa02 8.7974 × 10−17 cm2

Classical electron radius re = e2/mc2 2.8179 × 10−13 cm

Thomson cross section (8π/3)re2 6.6525 × 10−25 cm2

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

electron ¯h/mec 3.8616 × 10−11 cm

Fine-structure constant α = e2/¯hc 7.2974 × 10−3

α−1 137.04First radiation constant c1 = 2πhc2 3.7418 × 10−5 erg-cm2/secSecond radiation c2 = hc/k 1.4388 cm-deg (K)constant

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

Wavelength associated λ0 1.2398 × 10−4 cm

with 1 eV

Physical Quantity Symbol Value UnitsFrequency associated ν0 2.4180 × 1014 Hz

Avogadro number NA 6.0221 × 1023 mol−1

Faraday constant F = NAe 2.8925 × 1014 statcoul/molGas constant R = NAk 8.3145 × 107 erg/deg-molLoschmidt’s number n0 2.6868 × 1019 cm−3

(no density at STP)

Atomic mass unit mu 1.6605 × 10−24 g

Standard temperature T0 273.15 deg (K)Atmospheric pressure p0 = n0kT0 1.0133 × 106 dyne/cm2Pressure of 1 mm Hg 1.3332 × 103 dyne/cm2(1 torr)

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

Molar weight of air Mair 28.971 g

calorie (cal) 4.1868 × 107 erg

Gravitational g 980.67 cm/sec2acceleration

FORMULA CONVERSION8Here α = 102cm m−1, β = 107erg J−1, 0 = 8.8542 × 10−12F m−1,

µ0 = 4π×10−7H m−1, c = (0µ0)−1/2 = 2.9979×108m s−1, and ¯h = 1.0546×

10−34J s To derive a dimensionally correct SI formula from one expressed inGaussian units, substitute for each quantity according to ¯Q = ¯kQ, where ¯k isthe coefficient in the second column of the table corresponding to Q (overbarsdenote variables expressed in Gaussian units) Thus, the formula ¯a0 = ¯¯h2/ ¯m ¯e2for the Bohr radius becomes αa0 = (¯hβ)2/[(mβ/α2)(e2αβ/4π0)], or a0 =

0h2/πme2 To go from SI to natural units in which ¯h = c = 1 (distinguished

by a circumflex), use Q = ˆk−1Q, where ˆˆ k is the coefficient corresponding to

Q in the third column Thus ˆa0 = 4π0¯h2/[( ˆm¯h/c)(ˆe20¯hc)] = 4π/ ˆm ˆe2 (Intransforming from SI units, do not substitute for 0, µ0, or c.)

Physical Quantity Gaussian Units to SI Natural Units to SICapacitance α/4π0 0−1

v⊥ = E × B/B2 to calculate polarization charge density gives rise to a tric constant K ≡ /0 = 1 + 36π × 109ρ/B2 (SI) = 1 + 4πρc2/B2 (Gaussian),where ρ is the mass density.

dielec-The electromagnetic energy in volume V is given by

W = 1

2

ZV

dV (H · B + E · D) (SI)

= 18π

ZV

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

∂W

∂t +

ZS

N · dS = −

ZV

dV J · 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/4π (Gaussian)

ELECTRICITY AND MAGNETISM

In the following,  = dielectric permittivity, µ = permeability of tor, µ0 = permeability of surrounding medium, σ = conductivity, f = ω/2π =radiation frequency, κm = µ/µ0 and κe = /0 Where subscripts are used,

conduc-‘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 × 10−12F m−1

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

= 1.2566 × 10−6H m−1Resistance of free space R0= (µ0/0)1/2 = 376.73 Ω

Capacity of parallel plates of area C = A/d

Self-inductance of wire of length L = µl

l, carrying uniform current

Mutual inductance of parallel wires L = (µ0l/4π) [1 + 4 ln(d/a)]

of length l, radius a, separated

by distance d

Inductance of circular loop of radius L = b

µ0[ln(8b/a) − 2] + µ/4

b, made of wire of radius a,

carrying uniform current

Relaxation time in a lossy medium τ = /σ

Skin depth in a lossy medium δ = (2/ωµσ)1/2 = (πf µσ)−1/2Wave impedance in a lossy medium Z = [µ/( + 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 a

carrying current I

ELECTROMAGNETIC FREQUENCY/

WAVELENGTH BANDS10Frequency Range Wavelength RangeDesignation

Lower Upper Lower UpperULF* 30 Hz 10 Mm

VF* 30 Hz 300 Hz 1 Mm 10 MmELF 300 Hz 3 kHz 100 km 1 MmVLF 3 kHz 30 kHz 10 km 100 km

X Ray 30 PHz 3 EHz 100 pm 10 nmGamma Ray 3 EHz 100 pm

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

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

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

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

a voltage source V = V0exp(iωt) (here i = √

−1), the current is given

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

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

It = dqt/dt For initial conditions q(0) ≡ q0 = ¯q0 + qs, I(0) ≡ I0, thetransients can be of three types, depending on ∆ = R2 − 4L/C:

(b) Critically damped, ∆ = 0

qt = [¯q0 + (I0+ γR¯0)t] exp(−γRt),

It = [I0 − (I0+ γR¯0)γRt] exp(−γRt),where γR = R/2L;

DIMENSIONLESS NUMBERS OF FLUID MECHANICS12Name(s) Symbol Definition Significance

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

K´arm´an inertial force)1/2

Bond Bd (ρ0− ρ)L2g/Σ Gravitational force/

surface tensionBoussinesq B V /(2gR)1/2 (Inertial force/

gravitational force)1/2Brinkman Br µV2/k∆T Viscous heat/conducted heatCapillary Cp µV /Σ Viscous force/surface tensionCarnot Ca (T2 − T1)/T2 Theoretical Carnot cycle

efficiencyCauchy, Cy, Hk ρV2/Γ = M2 Inertial force/

Hooke compressibility force

Chandra- Ch B2L2/ρνη Magnetic force/dissipative

Clausius Cl LV3ρ/k∆T Kinetic energy flow rate/heat

conduction rateCowling C (VA/V )2 = Al2 Magnetic force/inertial forceCrispation Cr µκ/ΣL Effect of diffusion/effect of

surface tensionDean D D3/2V /ν(2r)1/2 Transverse flow due to

curvature/longitudinal flow[Drag CD (ρ0− ρ)Lg/ Drag force/inertial force

coefficient] ρ0V2

Eckert E V2/cp∆T Kinetic energy/change in

thermal energyEkman Ek (ν/2ΩL2)1/2 = (Viscous force/Coriolis force)1/2

(Ro/Re)1/2Euler Eu ∆p/ρV2 Pressure drop due to friction/

dynamic pressureFroude Fr V /(gL)1/2 †(Inertial force/gravitational or

V /N L buoyancy force)1/2Gay–Lussac Ga 1/β∆T Inverse of relative change in

volume during heatingGrashof Gr gL3β∆T /ν2 Buoyancy force/viscous force[Hall CH λ/rL Gyrofrequency/

coefficient] collision frequency

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

Name(s) Symbol Definition Significance

Hartmann H BL/(µη)1/2 = (Magnetic force/

(Rm Re C)1/2 dissipative force)1/2Knudsen Kn λ/L Hydrodynamic time/

collision timeLewis Le κ/D *Thermal conduction/molecular

diffusionLorentz Lo V /c Magnitude of relativistic effectsLundquist Lu µ0LVA/η = J × B force/resistive magnetic

Al Rm diffusion forceMach M V /CS Magnitude of compressibility

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

Magnetic Rm µ0LV /η Flow velocity/magnetic diffusionReynolds velocity

Newton Nt F/ρL2V2 Imposed force/inertial force

Nusselt N αL/k Total heat transfer/thermal

conductionP´eclet Pe LV /κ Heat convection/heat conductionPoisseuille Po D2∆p/µLV Pressure force/viscous forcePrandtl Pr ν/κ Momentum diffusion/

heat diffusionRayleigh Ra gH3β∆T /νκ Buoyancy force/diffusion forceReynolds Re LV /ν Inertial force/viscous force

Richardson Ri (N H/∆V )2 Buoyancy effects/

vertical shear effectsRossby Ro V /2ΩL sin Λ Inertial force/Coriolis force

Schmidt Sc ν/D Momentum diffusion/

molecular diffusionStanton St α/ρcpV Thermal conduction loss/

heat capacityStefan Sf σLT3/k Radiated heat/conducted heatStokes S ν/L2f Viscous damping rate/

vibration frequencyStrouhal Sr f L/V Vibration speed/flow velocityTaylor Ta (2ΩL2/ν)2 Centrifugal force/viscous force

R1/2(∆R)3/2 (Centrifugal force/

·(Ω/ν) viscous force)1/2Thring, Th, Bo ρcpV /σT3 Convective heat transport/

B Magnetic induction

Cs, c Speeds of sound, light

cp Specific heat at constant pressure (units m2s−2K−1)

D = 2R Pipe diameter

F Imposed force

f Vibration frequency

g Gravitational acceleration

H, L Vertical, horizontal length scales

k = ρcpκ Thermal conductivity (units kg m−1s−2)

N = (g/H)1/2 Brunt–V¨ais¨al¨a frequency

R Radius of pipe or channel

r Radius of curvature of pipe or channel

β Volumetric expansion coefficient, dV /V = βdT

Γ Bulk modulus (units kg m−1s−2)

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

pressures, or temperatures

 Surface emissivity

η Electrical resistivity

κ, D Thermal, molecular diffusivities (units m2s−1)

Λ Latitude of point on earth’s surface

λ Collisional mean free path

µ = ρν Viscosity

µ0 Permeability of free space

ν Kinematic viscosity (units m2s−1)

ρ Mass density of fluid medium

ρ0 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

B⊥ = B sin θ, Bk = 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) thefront are distinguished by a bar If B = 0, then15

(7) U − ¯U = [(¯p − p)(υ − ¯υ)]1/2;

(8) (¯p − p)(υ − ¯υ)−1 = q2;

(9) ¯w − w = 12(¯p − p)(υ + ¯υ);

(10) ¯e − e = 12(¯p + p)(υ − ¯υ)

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

γ = 1 + 2/n, where n is the number of degrees of freedom Then p = ρRT /m,where R is the universal gas constant and m is the molar weight; the soundspeed is given by Cs2 = (∂p/∂ρ)s = γpυ; and w = γe = γpυ/(γ − 1) For ageneral oblique shock in a perfect gas the quantity X = r−1(U/VA)2 satisfies14(11) (X−β/α)(X−cos2θ)2 = X sin2θ

(33) RS = C0(Et2/ρ)1/5,

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

FUNDAMENTAL PLASMA PARAMETERSAll quantities are in Gaussian cgs units except temperature (T , Te, Ti)expressed in eV and ion mass (mi) expressed in units of the proton mass,

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

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

Frequencies

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

ωce = eB/mec = 1.76 × 107B rad/secion gyrofrequency fci = ωci/2π = 1.52 × 103Zµ−1B Hz

ωci = ZeB/mic = 9.58 × 103Zµ−1B rad/secelectron plasma frequency fpe = ωpe/2π = 8.98 × 103ne1/2Hz

ωpe = (4πnee2/me)1/2

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

= 2.10 × 102Zµ−1/2ni1/2Hz

ωpi = (4πniZ2e2/mi)1/2

= 1.32 × 103Zµ−1/2ni1/2rad/secelectron trapping rate νT e = (eKE/me)1/2

= 7.26 × 108K1/2E1/2sec−1ion trapping rate νT i = (ZeKE/mi)1/2

= 1.69 × 107Z1/2K1/2E1/2µ−1/2sec−1electron collision rate νe = 2.91 × 10−6neln ΛTe −3/2sec−1

ion collision rate νi = 4.80 × 10−8Z4µ−1/2niln ΛTi −3/2sec−1Lengths

electron deBroglie length λ = ¯¯ h/(mekTe)1/2 = 2.76 × 10−8Te−1/2cmclassical distance of e2/kT = 1.44 × 10−7T−1cm

minimum approach

electron gyroradius re = vT e/ωce = 2.38Te1/2B−1cm

ion gyroradius ri = vT i/ωci

= 1.02 × 102µ1/2Z−1Ti1/2B−1cmelectron inertial length c/ωpe = 5.31 × 105ne−1/2cm

ion inertial length c/ωpi = 2.28 × 107(µ/ni)1/2cm

electron thermal velocity vT e = (kTe/me)1/2

= 4.19 × 107Te1/2cm/secion thermal velocity vT i = (kTi/mi)1/2

= 9.79 × 105µ−1/2Ti1/2cm/secion sound velocity Cs = (γZkTe/mi)1/2

= 9.79 × 105(γZTe/µ)1/2cm/secAlfv´en 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 = 1/42.9number of particles in (4π/3)nλD3 = 1.72 × 109T3/2n−1/2Debye sphere

Alfv´en velocity/speed of light vA/c = 7.28µ−1/2ni −1/2B

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

ratio

ion plasma/gyrofrequency ratio ωpi/ωci = 0.137µ1/2ni1/2B−1

thermal/magnetic energy ratio β = 8πnkT /B2 = 4.03 × 10−11nT B−2magnetic/ion rest energy ratio B2/8πnimic2 = 26.5µ−1ni −1B2

Miscellaneous

Bohm diffusion coefficient DB = (ckT /16eB)

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

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

ν* ≈ ωpeW /kT = 5.64 × 10e 4ne1/2W /kT sece −1,where W is the total energy of waves with ω/K < ve T i

Magnetic pressure is given by

Pmag = B2/8π = 3.98 × 106(B/B0)2dynes/cm2 = 3.93(B/B0)2atm,where B0 = 10 kG = 1 T

Detonation energy of 1 kiloton of high explosive is

WkT = 1012cal = 4.2 × 1019erg

PLASMA DISPERSION FUNCTIONDefinition16 (first form valid only for Im ζ > 0):

dt exp t2


.Imaginary argument (x = 0):

Z(iy) = iπ1/2exp y2

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

Z(ζ) = iπ1/2exp −ζ2

− 2ζ 1 − 2ζ2/3 + 4ζ4/15 − 8ζ6/105 + · · ·

.Asymptotic series, |ζ|  1 (Ref 17):

Z(ζ) = iπ1/2σ exp −ζ2

− ζ−1 1 + 1/2ζ2 + 3/4ζ4 + 15/8ζ6 + · · ·

,where

σ =

0 y > |x|−1

1 |y| < |x|−1

2 y < −|x|−1Symmetry properties (the asterisk denotes complex conjugation):

Z0(ζ) ≈ 0.50 + 0.96i + 0.50 − 0.96i

, b = 0.48 − 0.91i

COLLISIONS AND TRANSPORT

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

is k = 1.60 × 10−12erg/eV; masses µ, µ0 are in units of the proton mass;

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

Relaxation Rates

Rates are associated with four relaxation processes arising from the teraction of test particles (labeled α) streaming with velocity vα through abackground of field particles (labeled β):

in-slowing down dvα

dt = −νsα|βvαtransverse diffusion d

dt(vα − ¯vα)2⊥ = ν⊥α|βvα2parallel diffusion d

ν0α|β = 4πeα2eβ2λαβnβ/mα2vα3; xα|β = mβvα2/2kTβ;

ψ(x) = √2

π

Z x 0

dt t1/2e−t; ψ0(x) = dψ

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

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

have units cm3sec−1 Test particle energy  and field particle temperature Tare 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 holdfor very slow (xα|β  1) and very fast (xα|β  1) test particles, respectively.

νe|i −→ 4.2 × 10−9niZ2λei

nega-α|β i|e e|e, i|i e|p e|D e|T, e|He3 e|He4

α*

Tβ 1.5 0.98 4.8 × 10−3 2.6 × 10−3 1.8 × 10−3 1.4 × 10−3When both species are near Maxwellian, with Ti <

∼ Te, there are justtwo characteristic collision rates For Z = 1,

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

trans-The following cases are of particular interest:

(a) Thermal electron–electron collisions