2006

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 FOREWARD

The NRL Plasma Formulary originated over twenty five years ago and has 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 to Dave for providing me with the TEX files for the Formulary and his continued suggestions for improvement. The Formulary has been set in TEX by Dave Book, Todd Brun, and Robert Scott. Finally, I thank readers for communicat- ing typographical errors to me.

2 CONTENTS

Numerical and Algebraic ...... 4 VectorIdentities ...... 5 Differential Operators in Curvilinear Coordinates ...... 7 Dimensions and Units ...... 11 International System (SI) Nomenclature ...... 14 MetricPrefixes ...... 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 ACCircuits ...... 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 AtomicSpectroscopy ...... 59 Complex (Dusty) Plasmas ...... 62 References ...... 66

3 NUMERICAL AND ALGEBRAIC

Gain in decibels of P2 relative to P1

G = 10 log10(P2/P1).

To within two percent

(2π)1/2 2.5; π2 10; e3 20; 210 103. ≈ ≈ ≈ ≈

Euler-Mascheroni constant1 γ = 0.57722

Gamma Function Γ(x + 1) = xΓ(x):

Γ(1/6) = 5.5663 Γ(3/5) = 1.4892 Γ(1/5) = 4.5908 Γ(2/3) = 1.3541 Γ(1/4) = 3.6256 Γ(3/4) = 1.2254 Γ(1/3) = 2.6789 Γ(4/5) = 1.1642 Γ(2/5) = 2.2182 Γ(5/6) = 1.1288 Γ(1/2) = 1.7725 = √π Γ(1) = 1.0

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

∞ α α(α 1) α(α 1)(α 2) (1 + x)α = xk 1 + αx + − x2 + − − x3 + .... k ≡ 2! 3! Xk=0

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

n x x + kz y y +(n k)z − x + kz k y +(n k)z n k − − Xk=0

x + y x + y + nz = . x + y + nz n
Newberger’s summation formula3 [good for µ nonintegral, Re (α + β) > 1]: −

∞ n ( 1) Jα γn(z)Jβ+γn(z) π − − = Jα+γµ(z)Jβ γµ(z). n + µ sin µπ − n= X−∞

4 VECTOR IDENTITIES4

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

(1) A B C = A B C = B C A = B C A = C A B = C A B · × × · · × × · · × × · (2) A (B C)=(C B) A = (A C)B (A B)C × × × × · − · (3) A (B C) + B (C A) + C (A B) = 0 × × × × × × (4) (A B) (C D)=(A C)(B D) (A D)(B C) × · × · · − · · (5) (A B) (C D)=(A B D)C (A B C)D × × × × · − × · (6) (fg) = (gf) = f g + g f ∇ ∇ ∇ ∇ (7) (fA) = f A + A f ∇ · ∇ · ·∇ (8) (fA) = f A + f A ∇× ∇× ∇ × (9) (A B) = B A A B ∇ · × ·∇× − ·∇× (10) (A B) = A( B) B( A)+(B )A (A )B ∇× × ∇ · − ∇ · ·∇ − ·∇ (11) A ( B)=( B) A (A )B × ∇× ∇ · − ·∇ (12) (A B) = A ( B) + B ( A)+(A )B +(B )A ∇ · × ∇× × ∇× ·∇ ·∇ (13) 2f = f ∇ ∇·∇ (14) 2A = ( A) A ∇ ∇ ∇ · −∇×∇× (15) f = 0 ∇×∇ (16) A = 0 ∇·∇× If e1, e2, e3 are orthonormal unit vectors, a second-order tensor T can be written in the dyadic form

(17) T = Tij eiej i,j In cartesianP coordinates the divergence of a tensor is a vector with components

(18) ( T )i = (∂Tji/∂xj ) ∇· j [This definition isP required for consistency with Eq. (29)]. In general (19) (AB)=( A)B +(A )B ∇ · ∇ · ·∇ (20) (fT ) = f T +f T ∇ · ∇ · ∇· 5 Let r = ix + jy + kz be the radius vector of magnitude r, from the origin to the point x,y,z. Then (21) r = 3 ∇ · (22) r = 0 ∇× (23) r = r/r ∇ (24) (1/r) = r/r3 ∇ − (25) (r/r3) = 4πδ(r) ∇ · (26) r = I ∇ If V is a volume enclosed by a surface S and dS = ndS, where n is the unit normal outward from V,

(27) dV f = dSf ∇ ZV ZS

(28) dV A = dS A ∇ · · ZV ZS

(29) dV T = dS T ∇· · ZV ZS

(30) dV A = dS A ∇× × ZV ZS

(31) dV (f 2g g 2f) = dS (f g g f) ∇ − ∇ · ∇ − ∇ ZV ZS

(32) dV (A B B A) ·∇×∇× − ·∇×∇× ZV

= dS (B A A B) · ×∇× − ×∇× ZS If S is an open surface bounded by the contour C, of which the line element is dl,

(33) dS f = dlf ×∇ ZS IC 6 (34) dS A = dl A ·∇× · ZS IC

(35) (dS ) A = dl A ×∇ × × ZS IC

(36) dS ( f g) = fdg = gdf · ∇ ×∇ − ZS IC IC

DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES5

Cylindrical Coordinates Divergence

1 ∂ 1 ∂Aφ ∂Az A = (rAr ) + + ∇ · r ∂r r ∂φ ∂z

Gradient ∂f 1 ∂f ∂f ( f)r = ; ( f)φ = ; ( f)z = ∇ ∂r ∇ r ∂φ ∇ ∂z

Curl

1 ∂Az ∂Aφ ( A)r = ∇× r ∂φ − ∂z

∂Ar ∂Az ( A)φ = ∇× ∂z − ∂r

1 ∂ 1 ∂Ar ( A)z = (rAφ) ∇× r ∂r − r ∂φ

Laplacian

1 ∂ ∂f 1 ∂2f ∂2f 2f = r + + ∇ 2 2 2 r ∂r  ∂r  r ∂φ ∂z

7 Laplacian of a vector

2 2 2 ∂Aφ Ar ( A)r = Ar ∇ ∇ − r2 ∂φ − r2

2 2 2 ∂Ar Aφ ( A)φ = Aφ + ∇ ∇ r2 ∂φ − r2

( 2A) = 2A ∇ z ∇ z

Components of (A )B ·∇ ∂Br Aφ ∂Br ∂Br AφBφ (A B)r = Ar + + Az ·∇ ∂r r ∂φ ∂z − r

∂Bφ Aφ ∂Bφ ∂Bφ AφBr (A B)φ = Ar + + Az + ·∇ ∂r r ∂φ ∂z r

∂Bz Aφ ∂Bz ∂Bz (A B)z = Ar + + Az ·∇ ∂r r ∂φ ∂z

Divergence of a tensor

1 ∂ 1 ∂Tφr ∂Tzr Tφφ ( T )r = (rTrr ) + + ∇ · r ∂r r ∂φ ∂z − r

1 ∂ 1 ∂Tφφ ∂Tzφ Tφr ( T )φ = (rTrφ) + + + ∇ · r ∂r r ∂φ ∂z r

1 ∂ 1 ∂Tφz ∂Tzz ( T )z = (rTrz ) + + ∇ · r ∂r r ∂φ ∂z

8 Spherical Coordinates Divergence

1 ∂ 2 1 ∂ 1 ∂Aφ A = (r Ar ) + (sin θAθ ) + ∇ · r2 ∂r r sin θ ∂θ r sin θ ∂φ

Gradient ∂f 1 ∂f 1 ∂f ( f)r = ; ( f)θ = ; ( f)φ = ∇ ∂r ∇ r ∂θ ∇ r sin θ ∂φ

Curl

1 ∂ 1 ∂Aθ ( A)r = (sin θAφ) ∇× r sin θ ∂θ − r sin θ ∂φ

1 ∂Ar 1 ∂ ( A)θ = (rAφ) ∇× r sin θ ∂φ − r ∂r

1 ∂ 1 ∂Ar ( A)φ = (rAθ ) ∇× r ∂r − r ∂θ

Laplacian

1 ∂ ∂f 1 ∂ ∂f 1 ∂2f 2f = r2 + sin θ + ∇ 2 2 2 2 2 r ∂r  ∂r  r sin θ ∂θ  ∂θ  r sin θ ∂φ Laplacian of a vector

2 2 2Ar 2 ∂Aθ 2 cot θAθ 2 ∂Aφ ( A)r = Ar ∇ ∇ − r2 − r2 ∂θ − r2 − r2 sin θ ∂φ

2 2 2 ∂Ar Aθ 2 cos θ ∂Aφ ( A)θ = Aθ + ∇ ∇ r2 ∂θ − r2 sin2 θ − r2 sin2 θ ∂φ

2 2 Aφ 2 ∂Ar 2 cos θ ∂Aθ ( A)φ = Aφ + + ∇ ∇ − r2 sin2 θ r2 sin θ ∂φ r2 sin2 θ ∂φ

9 Components of (A )B ·∇ ∂Br Aθ ∂Br Aφ ∂Br AθBθ + AφBφ (A B)r = Ar + + ·∇ ∂r r ∂θ r sin θ ∂φ − r

∂Bθ Aθ ∂Bθ Aφ ∂Bθ AθBr cot θAφBφ (A B)θ = Ar + + + ·∇ ∂r r ∂θ r sin θ ∂φ r − r

∂Bφ Aθ ∂Bφ Aφ ∂Bφ AφBr cot θAφBθ (A B)φ = Ar + + + + ·∇ ∂r r ∂θ r sin θ ∂φ r r

Divergence of a tensor

1 ∂ 2 1 ∂ ( T )r = (r Trr ) + (sin θTθr ) ∇ · r2 ∂r r sin θ ∂θ

1 ∂Tφr Tθθ + Tφφ + r sin θ ∂φ − r

1 ∂ 2 1 ∂ ( T )θ = (r Trθ ) + (sin θTθθ) ∇ · r2 ∂r r sin θ ∂θ

1 ∂Tφθ Tθr cot θTφφ + + r sin θ ∂φ r − r

1 ∂ 2 1 ∂ ( T )φ = (r Trφ) + (sin θTθφ) ∇ · r2 ∂r r sin θ ∂θ

1 ∂Tφφ Tφr cot θTφθ + + + r sin θ ∂φ r r

10 DIMENSIONS AND UNITS To get the value of a quantity in Gaussian units, multiply the value ex- pressed in SI units by the conversion factor. Multiples of 3 in the conversion factors result from approximating the speed of light c = 2.9979 1010 cm/sec × 3 1010 cm/sec. ≈ × Dimensions Physical Sym- SI Conversion Gaussian Quantity bol SI Gaussian Units Factor Units

t2q2 Capacitance C l farad 9 1011 cm ml2 × m1/2l3/2 Charge q q coulomb 3 109 statcoulomb t × q m1/2 Charge ρ coulomb 3 103 statcoulomb 3 3/2 density l l t /m3 × /cm3 tq2 l Conductance siemens 9 1011 cm/sec ml2 t × 2 tq 1 9 1 Conductivity σ siemens 9 10 sec− 3 ml t /m × q m1/2l3/2 Current I,i ampere 3 109 statampere t t2 × q m1/2 Current J, j ampere 3 105 statampere 2 1/2 2 density l t l t /m2 × /cm2 m m 3 3 3 Density ρ kg/m 10− g/cm l3 l3 q m1/2 Displacement D coulomb 12π 105 statcoulomb l2 l1/2t /m2 × /cm2 1/2 ml m 1 4 Electric field E volt/m 10− statvolt/cm t2q l1/2t 3 × 2 1/2 1/2 ml m l 1 2 Electro- , volt 10− statvolt 2 motance EmfE t q t 3 × ml2 ml2 Energy U, W joule 107 erg t2 t2 m m Energy w,ǫ joule/m3 10 erg/cm3 2 2 density lt lt

11 Dimensions Physical Sym- SI Conversion Gaussian Quantity bol SI Gaussian Units Factor Units

ml ml Force F newton 105 dyne t2 t2 1 1 Frequency f,ν hertz 1 hertz t t 2 ml t 1 11 Impedance Z ohm 10− sec/cm tq2 l 9 × 2 2 ml t 1 11 2 Inductance L henry 10− sec /cm q2 l 9 × Length l l l meter (m) 102 centimeter (cm) 1/2 q m 3 Magnetic H ampere– 4π 10− oersted 1/2 intensity lt l t turn/m × ml2 m1/2l3/2 Magnetic flux Φ weber 108 maxwell tq t m m1/2 Magnetic B tesla 104 gauss tq l1/2t induction l2q m1/2l5/2 Magnetic m,µ ampere–m2 103 oersted– t t moment cm3 1/2 q m 3 Magnetization M ampere– 4π 10− oersted lt l1/2t turn/m × q m1/2l1/2 4π Magneto- , ampere– gilbert 2 motance MmfM t t turn 10 Mass m, M m m kilogram 103 gram (g) (kg) ml ml Momentum p, P kg–m/s 105 g–cm/sec t t m m 2 1 2 Momentum kg/m –s 10− g/cm –sec l2t l2t density ml 1 Permeability µ 1 henry/m 107 — q2 4π ×

12 Dimensions Physical Sym- SI Conversion Gaussian Quantity bol SI Gaussian Units Factor Units

t2q2 Permittivity ǫ 1 farad/m 36π 109 — ml3 × q m1/2 Polarization P coulomb/m2 3 105 statcoulomb l2 l1/2t × /cm2 2 1/2 1/2 ml m l 1 2 Potential V,φ volt 10− statvolt t2q t 3 × ml2 ml2 Power P watt 107 erg/sec t3 t3 m m Power watt/m3 10 erg/cm3–sec lt3 lt3 density m m Pressure p, P pascal 10 dyne/cm2 lt2 lt2 2 q 1 9 1 Reluctance ampere–turn 4π 10− cm− 2 R ml l /weber × 2 ml t 1 11 Resistance R ohm 10− sec/cm tq2 l 9 × 3 ml 1 9 Resistivity η,ρ t ohm–m 10− sec tq2 9 × ml ml Thermal con- κ, k watt/m– 105 erg/cm–sec– 3 3 ductivity t t deg (K) deg (K) Time t t t second (s) 1 second (sec) ml m1/2l1/2 Vector A weber/m 106 gauss–cm potential tq t l l Velocity v m/s 102 cm/sec t t m m Viscosity η,µ kg/m–s 10 poise lt lt

1 1 1 1 Vorticity ζ s− 1 sec− t t ml2 ml2 Work W joule 107 erg t2 t2

13 INTERNATIONAL SYSTEM (SI) NOMENCLATURE6

Physical Name Symbol Physical Name Symbol Quantity of Unit for Unit Quantity of Unit for Unit *length meter m electric volt V potential *mass kilogram kg electric ohm Ω *time second s resistance *current ampere A electric siemens S conductance *temperature kelvin K electric farad F *amount of mole mol capacitance substance magnetic flux weber Wb *luminous candela cd intensity magnetic henry H inductance plane angle radian rad † magnetic tesla T solid angle steradian sr intensity † frequency hertz Hz luminous flux lumen lm energy joule J illuminance lux lx force newton N activity (of a becquerel Bq radioactive pressure pascal Pa source) power watt W absorbed dose gray Gy (of ionizing electric charge coulomb C radiation) *SI base unit Supplementary unit † METRIC PREFIXES

Multiple Prefix Symbol Multiple Prefix Symbol

1 10− deci d 10 deca da 2 2 10− centi c 10 hecto h 3 3 10− milli m 10 kilo k 6 6 10− micro µ 10 mega M 9 9 10− nano n 10 giga G 12 12 10− pico p 10 tera T 15 15 10− femto f 10 peta P 18 18 10− atto a 10 exa E

14 PHYSICAL CONSTANTS (SI)7

Physical Quantity Symbol Value Units

23 1 Boltzmann constant k 1.3807 10− JK− × 19 Elementary charge e 1.6022 10− C × 31 Electron mass m 9.1094 10− kg e × 27 Proton mass m 1.6726 10− kg p × 11 3 2 1 Gravitational constant G 6.6726 10− m s− kg− × 34 Planck constant h 6.6261 10− J s × 34 h¯ = h/2π 1.0546 10− J s × 8 1 Speed of light in vacuum c 2.9979 10 m s− × 12 1 Permittivity of ǫ 8.8542 10− F m− 0 × free space 7 1 Permeability of µ 4π 10− H m− 0 × free space Proton/electron mass m /m 1.8362 103 p e × ratio 11 1 Electron charge/mass e/m 1.7588 10 C kg− e × ratio 4 me 7 1 Rydberg constant R = 1.0974 10 m− 2 3 ∞ 8ǫ0 ch × 2 2 11 Bohr radius a = ǫ h /πme 5.2918 10− m 0 0 × 2 21 2 Atomic cross section πa 8.7974 10− m 0 × 2 2 15 Classical electron radius r = e /4πǫ mc 2.8179 10− m e 0 × 2 29 2 Thomson cross section (8π/3)r 6.6525 10− m e × 12 Compton wavelength of h/mec 2.4263 10− m × 13 electron h/m¯ c 3.8616 10− m e × 2 3 Fine-structure constant α = e /2ǫ0hc 7.2974 10− 1 × α− 137.04 2 16 2 First radiation constant c = 2πhc 3.7418 10− W m 1 × 2 Second radiation c = hc/k 1.4388 10− mK 2 × constant 8 2 4 Stefan-Boltzmann σ 5.6705 10− W m− K− × constant

15 Physical Quantity Symbol Value Units

6 Wavelength associated λ = hc/e 1.2398 10− m 0 × with 1 eV Frequency associated ν = e/h 2.4180 1014 Hz 0 × with 1 eV 5 1 Wave number associated k = e/hc 8.0655 10 m− 0 × with 1 eV 19 Energy associated with hν 1.6022 10− J 0 × 1 eV 25 Energy associated with hc 1.9864 10− J 1 × 1 m− 3 2 2 Energy associated with me /8ǫ0 h 13.606 eV 1 Rydberg 5 Energy associated with k/e 8.6174 10− eV × 1 Kelvin Temperature associated e/k 1.1604 104 K × with 1 eV 23 1 Avogadro number N 6.0221 10 mol− A × 4 1 Faraday constant F = N e 9.6485 10 C mol− A × 1 1 Gas constant R = NAk 8.3145 JK− mol− 25 3 Loschmidt’s number n 2.6868 10 m− 0 × (no. density at STP) 27 Atomic mass unit m 1.6605 10− kg u × Standard temperature T0 273.15 K Atmospheric pressure p = n kT 1.0133 105 Pa 0 0 0 × Pressure of 1 mm Hg 1.3332 102 Pa × (1 torr) 2 3 Molar volume at STP V = RT /p 2.2414 10− m 0 0 0 × 2 Molar weight of air M 2.8971 10− kg air × calorie (cal) 4.1868 J 2 Gravitational g 9.8067 m s− acceleration

16 PHYSICAL CONSTANTS (cgs)7

Physical Quantity Symbol Value Units

16 Boltzmann constant k 1.3807 10− erg/deg (K) × 10 Elementary charge e 4.8032 10− statcoulomb × (statcoul) 28 Electron mass m 9.1094 10− g e × 24 Proton mass m 1.6726 10− g p × 8 2 2 Gravitational constant G 6.6726 10− dyne-cm /g × 27 Planck constant h 6.6261 10− erg-sec × 27 h¯ = h/2π 1.0546 10− erg-sec × Speed of light in vacuum c 2.9979 1010 cm/sec × Proton/electron mass m /m 1.8362 103 p e × ratio Electron charge/mass e/m 5.2728 1017 statcoul/g e × ratio 2 4 2π me 5 1 Rydberg constant R = 1.0974 10 cm− ∞ ch3 × 2 2 9 Bohr radius a =h ¯ /me 5.2918 10− cm 0 × 2 17 2 Atomic cross section πa 8.7974 10− cm 0 × 2 2 13 Classical electron radius r = e /mc 2.8179 10− cm e × 2 25 2 Thomson cross section (8π/3)r 6.6525 10− cm e × 10 Compton wavelength of h/mec 2.4263 10− cm × 11 electron h/m¯ c 3.8616 10− cm e × 2 3 Fine-structure constant α = e /hc¯ 7.2974 10− 1 × α− 137.04 2 5 2 First radiation constant c = 2πhc 3.7418 10− erg-cm /sec 1 × Second radiation c2 = hc/k 1.4388 cm-deg (K) constant 5 2 Stefan-Boltzmann σ 5.6705 10− erg/cm - × constant sec-deg4 4 Wavelength associated λ 1.2398 10− cm 0 × with 1 eV

17 Physical Quantity Symbol Value Units

Frequency associated ν 2.4180 1014 Hz 0 × with 1 eV 3 1 Wave number associated k 8.0655 10 cm− 0 × with 1 eV 12 Energy associated with 1.6022 10− erg × 1 eV 16 Energy associated with 1.9864 10− erg 1 × 1 cm− Energy associated with 13.606 eV 1 Rydberg 5 Energy associated with 8.6174 10− eV × 1 deg Kelvin Temperature associated 1.1604 104 deg (K) × with 1 eV 23 1 Avogadro number N 6.0221 10 mol− A × Faraday constant F = N e 2.8925 1014 statcoul/mol A × Gas constant R = N k 8.3145 107 erg/deg-mol A × 19 3 Loschmidt’s number n 2.6868 10 cm− 0 × (no. density at STP) 24 Atomic mass unit m 1.6605 10− g u × Standard temperature T0 273.15 deg (K) Atmospheric pressure p = n kT 1.0133 106 dyne/cm2 0 0 0 × Pressure of 1 mm Hg 1.3332 103 dyne/cm2 × (1 torr) Molar volume at STP V = RT /p 2.2414 104 cm3 0 0 0 × Molar weight of air Mair 28.971 g calorie (cal) 4.1868 107 erg × Gravitational g 980.67 cm/sec2 acceleration

18 FORMULA CONVERSION8 2 1 7 1 12 1 Here α = 10 cm m− , β = 10 erg J− , ǫ0 = 8.8542 10− F m− , 7 1 1/2 8 1 × µ0 = 4π 10− H m− , c = (ǫ0µ0)− = 2.9979 10 m s− , andh ¯ = 1.0546 34 × × × 10− J s. To derive a dimensionally correct SI formula from one expressed in Gaussian units, substitute for each quantity according to Q¯ = kQ¯ , where k¯ is the coefficient in the second column of the table corresponding to Q (overbars 2 2 denote variables expressed in Gaussian units). Thus, the formulaa ¯0 = h¯ /m¯ e¯ 2 2 2 for the Bohr radius becomes αa0 = (¯hβ) /[(mβ/α )(e αβ/4πǫ0)], or a0 = 2 2 ǫ0h /πme . To go from SI to natural units in whichh ¯ = c = 1 (distinguished 1 by a circumflex), use Q = kˆ− Qˆ, where kˆ is the coefficient corresponding to 2 2 2 Q in the third column. Thusa ˆ0 = 4πǫ0h¯ /[(m ˆ h/c¯ )(ˆe ǫ0hc¯ )] = 4π/mˆ eˆ . (In transforming from SI units, do not substitute for ǫ0, µ0, or c.)

Physical Quantity Gaussian Units to SI Natural Units to SI

1 Capacitance α/4πǫ0 ǫ0− 1/2 1/2 Charge (αβ/4πǫ0) (ǫ0hc¯ )− 5 1/2 1/2 Charge density (β/4πα ǫ0) (ǫ0hc¯ )− 1/2 1/2 Current (αβ/4πǫ0) (µ0/hc¯ ) 3 1/2 1/2 Current density (β/4πα ǫ0) (µ0/hc¯ ) 3 1/2 1/2 Electric field (4πβǫ0/α ) (ǫ0/hc¯ ) 1/2 1/2 Electric potential (4πβǫ0/α) (ǫ0/hc¯ ) 1 1 Electric conductivity (4πǫ0)− ǫ0− 1 Energy β (¯hc)− 3 1 Energy density β/α (¯hc)− 1 Force β/α (¯hc)− 1 Frequency 1 c− 1 Inductance 4πǫ0/α µ0− Length α 1 3 1/2 1/2 Magnetic induction (4πβ/α µ0) (µ0hc¯ )− 3 1/2 1/2 Magnetic intensity (4πµ0β/α ) (µ0/hc¯ ) Mass β/α2 c/h¯ 1 Momentum β/α h¯− 2 1 Power β (¯hc )− 3 1 Pressure β/α (¯hc)− 1/2 Resistance 4πǫ0/α (ǫ0/µ0) Time 1 c 1 Velocity α c−

19 MAXWELL’S EQUATIONS

Name or Description SI Gaussian ∂B 1 ∂B Faraday’s law E = E = ∇× − ∂t ∇× − c ∂t ∂D 1 ∂D 4π Ampere’s law H = + J H = + J ∇× ∂t ∇× c ∂t c Poisson equation D = ρ D = 4πρ ∇ · ∇ · [Absence of magnetic B = 0 B = 0 monopoles] ∇ · ∇ · 1 Lorentz force on q (E + v B) q E + v B × × charge q  c  Constitutive D = ǫE D = ǫE relations B = µH B = µH

7 1 In a plasma, µ µ0 = 4π 10− H m− (Gaussian units: µ 1). The ≈ × 12 1 ≈ permittivity satisfies ǫ ǫ0 = 8.8542 10− F m− (Gaussian: ǫ 1) provided that all charge≈ is regarded as free.× Using the drift approximation≈ v = E B/B2 to calculate polarization charge density gives rise to a dielec- ⊥ × 9 2 2 2 tric constant K ǫ/ǫ0 = 1 + 36π 10 ρ/B (SI) = 1 + 4πρc /B (Gaussian), where ρ is the mass≡ density. × The electromagnetic energy in volume V is given by

1 W = dV (H B + E D) (SI) 2 · · ZV 1 = dV (H B + E D) (Gaussian). 8π · · ZV

Poynting’s theorem is

∂W + N dS = dV J E, ∂t · − · ZS ZV 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). × ×

20 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.

12 1 Permittivity of free space ǫ = 8.8542 10− F m− 0 × 7 1 Permeability of free space µ0 = 4π 10− H m− × 6 1 = 1.2566 10− H m− × 1/2 Resistance of free space R0 = (µ0/ǫ0) = 376.73 Ω Capacity of parallel plates of area C = ǫA/d A, separated by distance d Capacity of concentric cylinders C = 2πǫl/ ln(b/a) of length l, radii a, b Capacity of concentric spheres of C = 4πǫab/(b a) radii a, b − Self-inductance of wire of length L = µl l, carrying uniform current

Mutual inductance of parallel wires L = (µ′l/4π)[1 +4ln(d/a)] of length l, radius a, separated by distance d

Inductance of circular loop of radius L = b µ′ [ln(8b/a) 2] + µ/4 b, made of wire of radius a, − carrying uniform current  Relaxation time in a lossy medium τ = ǫ/σ 1/2 1/2 Skin depth in a lossy medium δ = (2/ωµσ) = (πfµσ)− Wave impedance in a lossy medium Z = [µ/(ǫ + iσ/ω)]1/2 4 1/2 Transmission coefficient at T = 4.22 10− (fκ κ /σ) × m1 e2 conducting surface9 (good only for T 1) ≪ Field at distance r from straight wire Bθ = µI/2πr tesla carrying current I (amperes) = 0.2I/r gauss (r in cm) 2 2 2 3/2 Field at distance z along axis from Bz = µa I/[2(a + z ) ] circular loop of radius a carrying current I

21 ELECTROMAGNETIC FREQUENCY/ WAVELENGTH BANDS10

Frequency Range Wavelength Range Designation Lower Upper Lower Upper

ULF* 30 Hz 10 Mm VF* 30 Hz 300 Hz 1 Mm 10 Mm ELF 300 Hz 3 kHz 100 km 1 Mm VLF 3 kHz 30 kHz 10 km 100 km LF 30 kHz 300 kHz 1 km 10 km MF 300 kHz 3 MHz 100 m 1 km HF 3 MHz 30 MHz 10 m 100 m VHF 30 MHz 300 MHz 1 m 10 m UHF 300 MHz 3 GHz 10 cm 1 m SHF 3 GHz 30 GHz 1 cm 10 cm † S 2.6 3.95 7.6 11.5 G 3.95 5.85 5.1 7.6 J 5.3 8.2 3.7 5.7 H 7.05 10.0 3.0 4.25 X 8.2 12.4 2.4 3.7 M 10.0 15.0 2.0 3.0 P 12.4 18.0 1.67 2.4 K 18.0 26.5 1.1 1.67 R 26.5 40.0 0.75 1.1 EHF 30 GHz 300 GHz 1 mm 1 cm Submillimeter 300 GHz 3 THz 100 µm 1 mm Infrared 3 THz 430 THz 700 nm 100 µm Visible 430 THz 750 THz 400 nm 700 nm Ultraviolet 750 THz 30 PHz 10 nm 400 nm X Ray 30 PHz 3 EHz 100 pm 10 nm Gamma Ray 3 EHz 100 pm 8 In spectroscopy the angstrom is sometimes used (1A˚ = 10− cm = 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 † 22 AC CIRCUITS For a resistance R, inductance L, and capacitance C in series with a voltage source V = V0 exp(iωt) (here i = √ 1), the current is given by I = dq/dt, where q satisfies −

d2q dq q L + R + = V. dt2 dt C

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 I = dq /dt. For initial conditions q(0) q =q ¯ + q , I(0)− I , the t t ≡ 0 0 s ≡ 0 transients can be of three types, depending on ∆ = R2 4L/C: − (a) Overdamped, ∆ > 0

I0 + γ+q¯0 I0 + γ q¯0 qt = exp( γ t) − exp( γ+t), γ+ γ − − − γ+ γ − − − − − γ+(I0 + γ q¯0) γ (I0 + γ+q¯0) It = − exp( γ+t) − exp( γ t), γ+ γ − − γ+ γ − − − − − − where γ = (R ∆1/2)/2L; ± ± (b) Critically damped, ∆ = 0

q = [¯q +(I + γ q¯ )t] exp( γ t), t 0 0 R 0 − R I = [I (I + γ q¯ )γ t] exp( γ t), t 0 − 0 R 0 R − R where γR = R/2L; (c) Underdamped, ∆ < 0

γRq¯0 + I0 qt = sin ω1t +q ¯0 cos ω1t exp( γRt), ω1 − h 2 2 i (ω1 + γR )¯q0 + γRI0 I = I cos ω t sin(ω t) exp( γ t), t 0 1 − 1 − R h ω1 i 2 1/2 1/2 Here ω = ω (1 R C/4L) , where ω = (LC)− is the resonant 1 0 − 0 frequency. At ω = ω0, Z = R. The quality of the circuit is Q = ω0L/R. Instability results when L, R, C are not all of the same sign.

23 DIMENSIONLESS NUMBERS OF FLUID MECHANICS12

Name(s) Symbol Definition Significance

Alfv´en, Al, Ka VA/V *(Magnetic force/ K´arm´an inertial force)1/2 2 Bond Bd (ρ′ ρ)L g/Σ Gravitational force/ − surface tension Boussinesq B V/(2gR)1/2 (Inertial force/ gravitational force)1/2 Brinkman Br µV 2/k∆T Viscous heat/conducted heat Capillary Cp µV/Σ Viscous force/surface tension

Carnot Ca (T2 T1)/T2 Theoretical Carnot cycle − efficiency Cauchy, Cy, Hk ρV 2/Γ = M2 Inertial force/ Hooke compressibility force Chandra- Ch B2L2/ρνη Magnetic force/dissipative sekhar forces Clausius Cl LV 3ρ/k∆T Kinetic energy flow rate/heat conduction rate 2 2 Cowling C (VA/V ) = Al Magnetic force/inertial force Crispation Cr µκ/ΣL Effect of diffusion/effect of surface tension Dean D D3/2V/ν(2r)1/2 Transverse flow due to curvature/longitudinal flow

[Drag CD (ρ′ ρ)Lg/ Drag force/inertial force − 2 coefficient] ρ′V 2 Eckert E V /cp∆T Kinetic energy/change in thermal energy Ekman Ek (ν/2ΩL2)1/2 = (Viscous force/Coriolis force)1/2 (Ro/Re)1/2 Euler Eu ∆p/ρV 2 Pressure drop due to friction/ dynamic pressure Froude Fr V/(gL)1/2 (Inertial force/gravitational or † V/NL buoyancy force)1/2 Gay–Lussac Ga 1/β∆T Inverse of relative change in volume during heating Grashof Gr gL3β∆T/ν2 Buoyancy force/viscous force

[Hall CH λ/rL Gyrofrequency/ coefficient] collision frequency *( ) Also defined as the inverse (square) of the quantity shown. †

24 Name(s) Symbol Definition Significance

Hartmann H BL/(µη)1/2 = (Magnetic force/ (Rm ReC)1/2 dissipative force)1/2 Knudsen Kn λ/L Hydrodynamic time/ collision time Lewis Le κ/ *Thermal conduction/molecular D diffusion Lorentz Lo V/c Magnitude of relativistic effects

Lundquist Lu µ0LVA/η = J B force/resistive magnetic Al Rm ×diffusion force Mach M V/CS Magnitude of compressibility effects 1 1/2 Magnetic Mm V/VA = Al− (Inertial force/magnetic force) Mach Magnetic Rm µ0LV/η Flow velocity/magnetic diffusion Reynolds velocity Newton Nt F/ρL2V 2 Imposed force/inertial force Nusselt N αL/k Total heat transfer/thermal conduction P´eclet Pe LV/κ Heat convection/heat conduction Poisseuille Po D2∆p/µLV Pressure force/viscous force Prandtl Pr ν/κ Momentum diffusion/ heat diffusion Rayleigh Ra gH3β∆T/νκ Buoyancy force/diffusion force Reynolds Re LV/ν Inertial force/viscous force Richardson Ri (NH/∆V )2 Buoyancy effects/ vertical shear effects Rossby Ro V/2ΩL sin Λ Inertial force/Coriolis force Schmidt Sc ν/ Momentum diffusion/ D molecular diffusion Stanton St α/ρcpV Thermal conduction loss/ heat capacity Stefan Sf σLT 3/k Radiated heat/conducted heat Stokes S ν/L2f Viscous damping rate/ vibration frequency Strouhal Sr fL/V Vibration speed/flow velocity Taylor Ta (2ΩL2/ν)2 Centrifugal force/viscous force R1/2(∆R)3/2 (Centrifugal force/ (Ω/ν) viscous force)1/2 · 3 Thring, Th, Bo ρcpV/ǫσT Convective heat transport/ Boltzmann radiative heat transport Weber W ρLV 2/Σ Inertial force/surface tension 25 Nomenclature: B Magnetic induction

Cs, c Speeds of sound, light 2 2 1 cp Specific heat at constant pressure (units m s− K− ) D = 2R Pipe diameter F Imposed force f Vibration frequency g Gravitational acceleration H,L Vertical, horizontal length scales 1 2 k = ρcpκ Thermal conductivity (units kg m− s− ) N = (g/H)1/2 Brunt–V¨ais¨al¨afrequency R Radius of pipe or channel r Radius of curvature of pipe or channel rL Larmor radius T Temperature V Characteristic flow velocity 1/2 VA = B/(µ0ρ) Alfv´en speed ∂T α Newton’s-law heat coefficient, k = α∆T ∂x β Volumetric expansion coefficient, dV/V = βdT 1 2 Γ Bulk modulus (units kg m− s− ) ∆R, ∆V, ∆p, ∆T Imposed differences in two radii, velocities, pressures, or temperatures ǫ Surface emissivity η Electrical resistivity 2 1 κ, Thermal, molecular diffusivities (units m s− ) D Λ Latitude of point on earth’s surface λ Collisional mean free path µ = ρν Viscosity

µ0 Permeability of free space 2 1 ν Kinematic viscosity (units m s− ) ρ Mass density of fluid medium

ρ′ Mass density of bubble, droplet, or moving object 2 Σ Surface tension (units kg s− ) σ Stefan–Boltzmann constant Ω Solid-body rotational angular velocity

26 SHOCKS

At a shock front propagating in a magnetized fluid at an angle θ with respect to the magnetic induction B, the jump conditions are 13,14 (1) ρU =ρ ¯U¯ q; ≡ (2) ρU 2 + p + B 2/2µ =ρ ¯U¯ 2 +p ¯ + B¯ 2/2µ; ⊥ ⊥ (3) ρUV B B /µ =ρ ¯U¯V¯ B¯ B¯ /µ; − k ⊥ − k ⊥ (4) B = B¯ ; k k (5) UB VB = U¯B¯ V¯ B¯ ; ⊥ − k ⊥ − k 1 2 2 2 (6) 2 (U + V ) + w +(UB VB B )/µρU ⊥ − k ⊥ 1 ¯ 2 ¯ 2 ¯ ¯ 2 ¯ ¯ ¯ ¯ = 2 (U + V ) +w ¯ +(UB V B B )/µρ¯U. ⊥ − k ⊥ Here U and V are components of the fluid velocity normal and tangential to the front in the shock frame; ρ = 1/υ is the mass density; p is the pressure; B = B sin θ, B = B cos θ; µ is the magnetic permeability (µ = 4π in cgs ⊥ k units); and the specific enthalpy is w = e + pυ, where the specific internal energy e satisfies de = T ds pdυ in terms of the temperature T and the specific entropy s. Quantities− in the region behind (downstream from) the front are distinguished by a bar. If B = 0, then15 (7) U U¯ = [(¯p p)(υ υ¯)]1/2; − − − 1 2 (8) (¯p p)(υ υ¯)− = q ; − − (9)w ¯ w = 1 (¯p p)(υ +υ ¯); − 2 − (10)e ¯ e = 1 (¯p + p)(υ υ¯). − 2 − In what follows we assume that the fluid is a perfect gas with adiabatic index γ =1+2/n, where n is the number of degrees of freedom. Then p = ρRT/m, where R is the universal gas constant and m is the molar weight; the sound 2 speed is given by Cs = (∂p/∂ρ)s = γpυ; and w = γe = γpυ/(γ 1). For a 1 −2 14 general oblique shock in a perfect gas the quantity X = r− (U/VA) satisfies

(11) (X β/α)(X cos2 θ)2 = X sin2 θ [1+(r 1)/2α] X cos2 θ , where − − − − r =ρ/ρ ¯ , α = 1 [γ + 1 (γ 1)r], and β = C 2/V 2 = 4πγp/B2. 2 − −  s A The density ratio is bounded by (12) 1

(13) U 2 = (r/α) C 2 + V 2 [1 + (1 γ/2)(r 1)] ; s A − − (14) U/U¯ = B/B¯ = r;

27 (15) V¯ = V ; 1 2 2 2 (16)p ¯ = p + (1 r− )ρU + (1 r )B /2µ. − − If θ = 0, there are two possibilities: switch-on shocks, which require β < 1 and for which 2 2 (17) U = rVA ; 2 (18) U¯ = VA /U; (19) B¯ 2 = 2B 2(r 1)(α β); ⊥ k − − (20) V¯ = U¯B¯ /B ; ⊥ k 2 1 (21)p ¯ = p + ρU (1 α + β)(1 r− ), − − and acoustic (hydrodynamic) shocks, for which 2 2 (22) U = (r/α)Cs ; (23) U¯ = U/r; (24) V¯ = B¯ = 0; ⊥ 2 1 (25)p ¯ = p + ρU (1 r− ). − For acoustic shocks the specific volume and pressure are related by (26)υ/υ ¯ = [(γ + 1)p +(γ 1)¯p] / [(γ 1)p +(γ + 1)¯p]. − − In terms of the upstream Mach number M = U/Cs, (27)ρ/ρ ¯ = υ/υ¯ = U/U¯ = (γ + 1)M 2/[(γ 1)M 2 + 2]; − (28)p/p ¯ = (2γM 2 γ + 1)/(γ + 1); − (29) T/T¯ = [(γ 1)M 2 + 2](2γM 2 γ + 1)/(γ + 1)2M 2; − − (30) M¯ 2 = [(γ 1)M 2 + 2]/[2γM 2 γ + 1]. − − The entropy change across the shock is (31) ∆s s¯ s = c ln[(¯p/p)(ρ/ρ¯)γ ], ≡ − υ where cυ = R/(γ 1)m is the specific heat at constant volume; here R is the gas constant. In the− weak-shock limit (M 1), → 2γ(γ 1) 2 3 16γR 3 (32) ∆s cυ − (M 1) (M 1) . → 3(γ + 1) − ≈ 3(γ + 1)m − 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 ρ is 2 1/5 (33) RS = C0(Et /ρ) , where C0 is a constant depending on γ. For γ = 7/5, C0 = 1.033.

28 FUNDAMENTAL PLASMA PARAMETERS

All quantities are in Gaussian cgs units except temperature (T , Te, Ti) expressed in eV and ion mass (mi) expressed in units of the proton mass, µ = mi/mp; Z is charge state; k is Boltzmann’s constant; K is wavenumber; γ is the adiabatic index; ln Λ is the Coulomb logarithm. Frequencies electron gyrofrequency f = ω /2π = 2.80 106B Hz ce ce × ω = eB/m c = 1.76 107B rad/sec ce e × 3 1 ion gyrofrequency f = ω /2π = 1.52 10 Zµ− B Hz ci ci × 3 1 ω = ZeB/m c = 9.58 10 Zµ− B rad/sec ci i × electron plasma frequency f = ω /2π = 8.98 103n 1/2 Hz pe pe × e 2 1/2 ωpe = (4πnee /me) = 5.64 104n 1/2 rad/sec × e ion plasma frequency fpi = ωpi/2π 2 1/2 1/2 = 2.10 10 Zµ− n Hz × i 2 2 1/2 ωpi = (4πniZ e /mi) 3 1/2 1/2 = 1.32 10 Zµ− n rad/sec × i 1/2 electron trapping rate νTe = (eKE/me) 8 1/2 1/2 1 = 7.26 10 K E sec− × 1/2 ion trapping rate νT i = (ZeKE/mi) 7 1/2 1/2 1/2 1/2 1 = 1.69 10 Z K E µ− sec− × 6 3/2 1 electron collision rate ν = 2.91 10− n ln ΛT − sec− e × e e 8 4 1/2 3/2 1 ion collision rate ν = 4.80 10− Z µ− n ln ΛT − sec− i × i i Lengths 1/2 8 1/2 electron deBroglie lengthλ ¯ =h/ ¯ (m kT ) = 2.76 10− T − cm e e × e 2 7 1 classical distance of e /kT = 1.44 10− T − cm minimum approach × 1/2 1 electron gyroradius re = vTe/ωce = 2.38Te B− cm

ion gyroradius ri = vT i/ωci 2 1/2 1 1/2 1 = 1.02 10 µ Z− T B− cm × i 5 1/2 electron inertial length c/ω = 5.31 10 n − cm pe × e ion inertial length c/ω = 2.28 107(µ/n )1/2 cm pi × i 2 1/2 2 1/2 1/2 Debye length λ = (kT/4πne ) = 7.43 10 T n− cm D ×

29 Velocities 1/2 electron thermal velocity vTe = (kTe/me) = 4.19 107T 1/2 cm/sec × e 1/2 ion thermal velocity vT i = (kTi/mi) 5 1/2 1/2 = 9.79 10 µ− T cm/sec × i 1/2 ion sound velocity Cs = (γZkTe/mi) = 9.79 105(γZT /µ)1/2 cm/sec × e 1/2 Alfv´en velocity vA = B/(4πnimi) 11 1/2 1/2 = 2.18 10 µ− n − B cm/sec × i Dimensionless 1/2 1/2 2 (electron/proton mass ratio) (m /m ) = 2.33 10− = 1/42.9 e p × 3 9 3/2 1/2 number of particles in (4π/3)nλD = 1.72 10 T n− Debye sphere × 1/2 1/2 Alfv´en velocity/speed of light vA/c = 7.28µ− ni− B 3 1/2 1 electron plasma/gyrofrequency ωpe/ωce = 3.21 10− ne B− ratio × 1/2 1/2 1 ion plasma/gyrofrequency ratio ωpi/ωci = 0.137µ ni B− 2 11 2 thermal/magnetic energy ratio β = 8πnkT/B = 4.03 10− nTB− × 2 2 1 1 2 magnetic/ion rest energy ratio B /8πnimic = 26.5µ− ni− B Miscellaneous Bohm diffusion coefficient DB = (ckT/16eB) 6 1 2 = 6.25 10 TB− cm /sec × 14 3/2 transverse Spitzer resistivity η = 1.15 10− Z ln ΛT − sec ⊥ × 2 3/2 = 1.03 10− Z ln ΛT − Ω cm × The anomalous collision rate due to low-frequency ion-sound turbulence is

4 1/2 1 ν* ω W/kT = 5.64 10 n W/kT sec− , ≈ pe × e where W is the total energye of waves with ω/K

P e = B2/8π = 3.98 106(B/B )2 dynes/cm2 = 3.93(B/B )2 atm, mag × 0 0 where B0 = 10 kG = 1 T. Detonation energy of 1 kiloton of high explosive is

W = 1012 cal = 4.2 1019 erg. kT ×

30 PLASMA DISPERSION FUNCTION

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

+ 2 iζ ∞ dt exp t 1/2 2 2 Z(ζ) = π− − = 2i exp ζ dt exp t . t ζ − − Z −
Z −∞
−∞
Physically, ζ = x + iy is the ratio of wave phase velocity to thermal velocity. Differential equation: dZ d2Z dZ = 2(1+ ζZ) , Z(0) = iπ1/2; + 2ζ + 2Z = 0. dζ − dζ2 dζ Real argument (y = 0): x Z(x) = exp x2 iπ1/2 2 dt exp t2 . −  − Z 
0
Imaginary argument (x = 0):

Z(iy) = iπ1/2 exp y2 [1 erf(y)] . − Power series (small argument):

Z(ζ) = iπ1/2 exp ζ2 2ζ 1 2ζ2/3 + 4ζ4/15 8ζ6/105 + . − − − − · · · Asymptotic series, ζ 1
(Ref. 17):
| | ≫ 1/2 2 1 2 4 6 Z(ζ) = iπ σ exp ζ ζ− 1 + 1/2ζ + 3/4ζ + 15/8ζ + , − − · · · where

1 0 y > x − | | 1 σ = 1 y < x − | | | | 1  2 y < x − −| | 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): − ≫ 0.50 + 0.81i 0.50 0.81i Z(ζ) − , a = 0.51 0.81i; ≈ a ζ − a* + ζ − − 0.50 + 0.96i 0.50 0.96i Z′(ζ) + − , b = 0.48 0.91i. ≈ (b ζ)2 (b* + ζ)2 − −

31 COLLISIONS AND TRANSPORT

Temperatures are in eV; the corresponding value of Boltzmann’s constant 12 is k = 1.60 10− erg/eV; masses µ, µ′ are in units of the proton mass; × eα = Zαe is the charge of species α. All other units are cgs except where noted. Relaxation Rates Rates are associated with four relaxation processes arising from the in- teraction of test particles (labeled α) streaming with velocity vα through a background of field particles (labeled β):

dvα α β slowing down = ν \ vα dt − s d 2 α β 2 transverse diffusion (vα ¯vα) = ν \ vα dt − ⊥ ⊥ d 2 α β 2 parallel diffusion (vα ¯vα) = ν \ vα dt − k k d 2 α β 2 energy loss vα = ν \ vα , dt − ǫ where vα = vα and the averages are performed over an ensemble of test particles and a| Maxwellian| field particle distribution. The exact formulas may be written19

α β α β α β \ νs \ = (1+ mα/mβ )ψ(x \ )ν0 ; α β α β α β α β α β \ \ ν = 2 (1 1/2x \ )ψ(x \ ) + ψ′(x \ ) ν0 ; ⊥ − α β α β α β α β \  \  ν = ψ(x \ )/x \ ν0 ; k α β α β α β α β ν \ = 2 (m /m )ψ(x \ ) ψ′(x \ ) ν \ , ǫ α β − 0   where

α β 2 2 2 3 α β 2 \ ν0 = 4πeα eβ λαβ nβ /mα vα ; x \ = mβ vα /2kTβ ;

x 2 1/2 t dψ ψ(x) = dt t e− ; ψ′(x) = , √π dx Z0 and λαβ = lnΛαβ is the Coulomb logarithm (see below). Limiting forms of νs, ν and ν are given in the following table. All the expressions shown ⊥ k 32 3 1 have units cm sec− . Test particle energy ǫ and field particle temperature T are both in eV; µ = mi/mp where mp is the proton mass; Z is ion charge state; in electron–electron and ion–ion encounters, field particle quantities are distinguished by a prime. The two expressions given below for each rate hold α β α β for very slow (x \ 1) and very fast (x \ 1) test particles, respectively. ≪ ≫ Slow Fast Electron–electron e e 6 3/2 6 3/2 νs| /neλee 5.8 10− T − 7.7 10− ǫ− e e ≈ × 6 1/2 1 −→ × 6 3/2 ν | /neλee 5.8 10− T − ǫ− 7.7 10− ǫ− ⊥e e ≈ × 6 1/2 1 −→ × 6 5/2 ν | /neλee 2.9 10− T − ǫ− 3.9 10− Tǫ− k ≈ × −→ × Electron–ion e i 2 3/2 3/2 6 3/2 νs| /niZ λei 0.23µ T − 3.9 10− ǫ− e i 2 ≈ 4 1/2 1/2 1−→ × 6 3/2 ν | /niZ λei 2.5 10− µ T − ǫ− 7.7 10− ǫ− ⊥e i 2 ≈ × 4 1/2 1/2 1−→ × 9 1 5/2 ν | /niZ λei 1.2 10− µ T − ǫ− 2.1 10− µ− Tǫ− k ≈ × −→ × Ion–electron i e 2 9 1 3/2 4 1/2 3/2 νs| /neZ λie 1.6 10− µ− T − 1.7 10− µ ǫ− i e 2 ≈ × 9 1 1/2 1 −→ × 7 1/2 3/2 ν | /neZ λie 3.2 10− µ− T − ǫ− 1.8 10− µ− ǫ− ⊥i e 2 ≈ × 9 1 1/2 1 −→ × 4 1/2 5/2 ν | /neZ λie 1.6 10− µ− T − ǫ− 1.7 10− µ Tǫ− k ≈ × −→ × Ion–ion i i′ 1/2 1/2 νs| 8 µ′ µ′ − 3/2 6.8 10− 1 + T − 2 2 ni Z Z′ λii ≈ × µ µ ′ ′   1/2 8 1 1 µ 9.0 10− + 3/2 −→ × µ µ′ ǫ i i   | ′ ν 7 1/2 1 1/2 1 ⊥ 1.4 10− µ′ µ− T − ǫ− 2 2 ni Z Z′ λii ≈ × ′ ′ 7 1/2 3/2 1.8 10− µ− ǫ− i i −→ × ν | ′ 8 1/2 1 1/2 1 k 6.8 10− µ′ µ− T − ǫ− 2 2 ni Z Z′ λii ≈ × ′ ′ 8 1/2 1 5/2 9.0 10− µ µ′− Tǫ− −→ × In the same limits, the energy transfer rate follows from the identity

νǫ = 2νs ν ν , − ⊥ − k except for the case of fast electrons or fast ions scattered by ions, where the leading terms cancel. Then the appropriate forms are

e i 9 2 ν | 4.2 10− n Z λ ǫ −→ × i ei 3/2 1 4 1/2 1 1 ǫ− µ− 8.9 10 (µ/T ) ǫ− exp( 1836µǫ/T ) sec− − × −   33 and i i 7 2 2 ν | ′ 1.8 10− n Z Z′ λ ǫ −→ × i′ ii′ 3/2 1/2 1/2 1 1 ǫ− µ /µ′ 1.1[(µ + µ′)/µµ′](µ′/T ′) ǫ− exp( µ′ǫ/µT ′) sec− . − −

 α β  In general, the energy transfer rate νǫ \ is positive for ǫ>ǫα* and nega- tive for ǫ<ǫα*, where x*=(mβ /mα)ǫα*/Tβ is the solution of ψ′(x*) = (mα mβ )ψ(x*). The ratio ǫα*/Tβ is given for a number of specific α, β in the following| table:

α β i e e e, i i e p e D e T, e He3 e He4 \ | | | | | | | | ǫα* 3 3 3 3 1.5 0.98 4.8 10− 2.6 10− 1.8 10− 1.4 10− Tβ × × × ×

When both species are near Maxwellian, with Ti < Te, there are just two characteristic collision rates. For Z = 1, ∼

6 3/2 1 ν = 2.9 10− nλT − sec− ; e × e 8 3/2 1/2 1 ν = 4.8 10− nλT − µ− sec− . i × i

Temperature Isotropization Isotropization is described by

dT 1 dT α ⊥ k = = νT (T T ), dt − 2 dt − ⊥ − k where, if A T /T 1 > 0, ≡ ⊥ k −

2 2 1 1/2 α 2√πeα eβ nαλαβ 2 tan− (A ) ν = A− 3+(A + 3) . T 1/2 3/2 1/2 mα (kT ) − A k  

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

e 7 3/2 1 ν = 8.2 10− nλT − sec− ; T × i 8 2 1/2 3/2 1 ν = 1.9 10− nλZ µ− T − sec− . T × 34 Thermal Equilibration If the components of a plasma have different temperatures, but no rela- tive drift, equilibration is described by

dTα α β = ν¯ \ (Tβ Tα), dt ǫ − Xβ where 1/2 2 2 α β 19 (mαmβ ) Zα Zβ nβ λαβ 1 ν¯ \ = 1.8 10− sec− . ǫ 3/2 × (mαTβ + mβ Tα) For electrons and ions with T T T , this implies e ≈ i ≡ e i i e 9 2 3/2 3 1 ν¯ | /n =ν ¯ | /n = 3.2 10− Z λ/µT cm sec− . ǫ i ǫ e × Coulomb Logarithm

For test particles of mass mα and charge eα = Zαe scattering off field particles of mass mβ and charge eβ = Zβ e, the Coulomb logarithm is defined as λ = lnΛ ln(r /r ). Here r is the larger of e e /m u¯2 and ≡ max min min α β αβ h/¯ 2mαβu¯, averaged over both particle velocity distributions, where mαβ = 2 1/2 mαmβ /(mα + mβ ) and u = vα vβ ; rmax = (4π nγ eγ /kTγ )− , where − 2 2 the summation extends over all species γ for whichu ¯ < vTγ = kTγ /mγ . If P 1 1 this inequality cannot be satisfied, or if eitheruω ¯ cα− < rmax oruω ¯ cβ − < rmax, the theory breaks down. Typically λ 10–20. Corrections to the trans- 1 ≈ port coefficients are O(λ− ); hence the theory is good only to 10% and fails when λ 1. ∼ The∼ following cases are of particular interest: (a) Thermal electron–electron collisions

1/2 5/4 5 2 1/2 λ = 23.5 ln(n T − ) [10− + (ln T 2) /16] ee − e e − e − (b) Electron–ion collisions

1/2 3/2 2 λ = λ = 23 ln n ZT − , T m /m < T < 10Z eV; ei ie − e e i e i e 1/2 1 2 = 24 ln n T − ,
T m /m < 10Z eV < T − e e i e i e 1/2 3/2 2 1 = 30 ln n T −
Z µ− , T < T Zm /m . − i i e i e i
(c) Mixed ion–ion collisions

2 2 1/2 ZZ′(µ + µ′) niZ n Z′ i′ λii = λi i = 23 ln + . ′ ′ − µT + µ Ti Ti T  i′ ′  i′   35 (d) Counterstreaming ions (relative velocity vD = βDc) in the presence of 2 warm electrons, kTi/mi,kT /m < vD < kTe/me i′ i′

1/2 ZZ′(µ + µ′) ne λ = λ = 35 ln . ii′ i′i − 2  µµ′βD  Te  

Fokker-Planck Equation

α α α Df ∂f α α ∂f + v f + F vf = , Dt ≡ ∂t ·∇ ·∇ ∂t  coll where F is an external force field. The general form of the collision integral is α α β (∂f /∂t)coll = v J \ , with − β ∇ · P 2 2 α β eα eβ 3 2 3 J \ = 2πλαβ d v′(u I uu)u− mα Z −

1 α β 1 β α f (v) v f (v′) f (v′) vf (v) · ∇ ′ − ∇ n mβ mα o

(Landau form) where u = v′ v and I is the unit dyad, or alternatively, −

2 2 α β eα eβ α 1 α J \ = 4πλ f (v) vH(v) v f (v) v vG(v) , αβ 2 mα ∇ − 2 ∇ · ∇ ∇ n  o where the Rosenbluth potentials are

β 3 G(v) = f (v′)ud v′ Z

mα β 1 3 H(v) = 1 + f (v′)u− d v′.  mβ  Z If species α is a weak beam (number and energy density small compared with background) streaming through a Maxwellian plasma, then

α β mα α β α 1 α β α J \ = νs \ vf ν \ vv vf − mα + mβ − 2 k ·∇

1 α β 2 α ν \ v I vv vf . − 4 ⊥ − ·∇
36 B-G-K Collision Operator For distribution functions with no large gradients in velocity space, the Fokker-Planck collision terms can be approximated according to

Dfe = νee(Fe fe) + νei(F¯e fe); Dt − −

Dfi = νie(F¯i fi) + νii(Fi fi). Dt − −

α β The respective slowing-down rates νs \ given in the Relaxation Rate section above can be used for ναβ, assuming slow ions and fast electrons, with ǫ re- placed by Tα. (For νee and νii, one can equally well use ν , and the result is insensitive to whether the slow- or fast-test-particle limit is⊥ employed.) The Maxwellians Fα and F¯α are given by

3/2 2 mα mα(v vα) F = n exp − ; α α −  2πkTα  n h 2kTα io

3/2 2 mα mα(v ¯vα) F¯ = n exp − , α α ¯ − ¯  2πkTα  n h 2kTα io where nα, vα and Tα are the number density, mean drift velocity, and effective temperature obtained by taking moments of fα. Some latitude in the definition 20 of T¯α and ¯vα is possible; one choice is T¯e = Ti, T¯i = Te, ¯ve = vi, ¯vi = ve. Transport Coefficients Transport equations for a multispecies plasma:

α d nα + nα vα = 0; dt ∇ ·

α d vα 1 m n = p P + Z en E + v B + R ; α α −∇ α − ∇ · α α α α × α dt h c i α 3 d kTα nα + pα vα = qα Pα : vα + Qα. 2 dt ∇ · −∇ · − ∇ Here dα/dt ∂/∂t + v ; p = n kT , where k is Boltzmann’s constant; ≡ α ·∇ α α α Rα = Rαβ and Qα = Qαβ , where Rαβ and Qαβ are respectively β β the momentumP and energy gainedP by the αth species through collisions with the βth; Pα is the stress tensor; and qα is the heat flow.

37 The transport coefficients in a simple two-component plasma (electrons and singly charged ions) are tabulated below. Here and refer to the di- k ⊥ rection of the magnetic field B = bB; u = ve vi is the relative streaming − 7 1 velocity; n = n n; j = neu is the current; ω = 1.76 10 B sec− and e i ≡ − ce × ωci = (me/mi)ωce are the electron and ion gyrofrequencies, respectively; and the basic collisional times are taken to be

3/2 3/2 3√me(kTe) 5 Te τe = = 3.44 10 sec, 4√2π nλe4 × nλ where λ is the Coulomb logarithm, and

3/2 3/2 3√mi(kTi) 7 Ti 1/2 τi = = 2.09 10 µ sec. 4√πn λe4 × nλ

In the limit of large fields (ω τ 1, α = i, e) the transport processes may cα α ≫ be summarized as follows:21

momentum transfer R = R R = Ru + R ; ei − ie ≡ T frictional force Ru = ne(j /σ + j /σ ); k k ⊥ ⊥ 2 electrical σ = 1.96σ ; σ = ne τe/me; conductivities k ⊥ ⊥ 3n thermal force RT = 0.71n (kTe) b (kTe); − ∇k − 2ωceτe ×∇⊥ 3me nk ion heating Qi = (Te Ti); mi τe − electron heating Q = Q R u; e − i − · i i i ion heat flux qi = κ (kTi) κ (kTi) + κ b (kTi); − k∇k − ⊥∇⊥ ∧ ×∇⊥ i nkTiτi i 2nkTi i 5nkTi ion thermal κ = 3.9 ; κ = 2 ; κ = ; conductivities k mi ⊥ miωci τi ∧ 2miωci e e electron heat flux qe = qu + qT ;

e 3nkTe frictional heat flux qu = 0.71nkTeu + b u ; k 2ωceτe × ⊥ e e e e thermal gradient qT = κ (kTe) κ (kTe) κ b (kTe); heat flux − k∇k − ⊥∇⊥ − ∧ ×∇⊥ nkTeτe nkTe 5nkTe electron thermal κe = 3.2 ; κe = 4.7 ; κe = ; 2 conductivities k me ⊥ meωce τe ∧ 2meωce η0 η1 stress tensor (either Pxx= (Wxx + Wyy ) (Wxx Wyy ) η3Wxy ; species) − 2 − 2 − −

38 η0 η1 Pyy = (Wxx + Wyy ) + (Wxx Wyy ) + η3Wxy; − 2 2 − η3 Pxy = Pyx = η1Wxy + (Wxx Wyy ); − 2 − P = P = η W η W ; xz zx − 2 xz − 4 yz P = P = η W + η W ; yz zy − 2 yz 4 xz Pzz = η0Wzz (here the z axis is defined parallel− to B);

i i 3nkTi i 6nkTi ion viscosity η0 = 0.96nkTiτi; η1 = 2 ; η2 = 2 ; 10ωci τi 5ωci τi i nkTi i nkTi η3 = ; η4 = ; 2ωci ωci nkTe nkTe electron viscosity ηe = 0.73nkT τ ; ηe = 0.51 ; ηe = 2.0 ; 0 e e 1 2 2 2 ωce τe ωce τe e nkTe e nkTe η3 = ; η4 = . − 2ωce − ωce For both species the rate-of-strain tensor is defined as

∂vj ∂vk 2 Wjk = + δjk v. ∂xk ∂xj − 3 ∇ ·

When B = 0 the following simplifications occur:

i Ru = nej/σ ; RT = 0.71n (kTe); qi = κ (kTi); k − ∇ − k∇

e e e qu = 0.71nkTeu; qT = κ (kTe); Pjk = η0Wjk. − k∇ −

For ωceτe 1 ωciτi, the electrons obey the high-field expressions and the ions obey the≫ zero-field≫ expressions. Collisional transport theory is applicable when (1) macroscopic time rates of change satisfy d/dt 1/τ, where τ is the longest collisional time scale, and (in the absence of a magnetic≪ field) (2) macroscopic length scales L satisfy L l, where l =vτ ¯ is the mean free path. In a strong field, ω τ 1, condition≫ ce ≫ (2) is replaced by L l and L √lre (L re in a uniform field), k ≫ ⊥ ≫ ⊥ ≫ where L is a macroscopic scale parallel to the field B and L is the smaller k ⊥ of B/ B and the transverse plasma dimension. In addition, the standard transport|∇⊥ coefficients| are valid only when (3) the Coulomb logarithm satisfies λ 1; (4) the electron gyroradius satisfies r λ , or 8πn m c2 B2; (5) ≫ e ≫ D e e ≫ relative drifts u = vα vβ between two species are small compared with the − 2 thermal velocities, i.e., u kTα/mα, kTβ /mβ ; and (6) anomalous transport processes owing to microinstabilities≪ are negligible.

39 Weakly Ionized Plasmas Collision frequency for scattering of charged particles of species α by neutrals is α 0 1/2 να = n0σs | (kTα/mα) ,

α 0 where n0 is the neutral density and σs \ is the cross section, typically 15 2 ∼ 5 10− cm and weakly dependent on temperature. × When the system is small compared with a Debye length, L λD , the charged particle diffusion coefficients are ≪

Dα = kTα/mανα,

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

µiDe µeDi (Ti + Te)DiDe DA = − = , µ µ T D + T D i − e i e e i where µα = eα/mανα is the mobility. The conductivity σα satisfies σα = nαeαµα. In the presence of a magnetic field B the scalars µ and σ become tensors,

Jα = σσα E = σαE + σα E + σαE b, · k k ⊥ ⊥ ∧ × where b = B/B and

α 2 σ = nαeα /mανα; k α α 2 2 2 σ = σ να /(να + ωcα); ⊥ k α α 2 2 σ = σ ναωcα/(να + ωcα). ∧ k

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

40 IONOSPHERIC PARAMETERS23

The following tables give average nighttime values. Where two numbers are entered, the first refers to the lower and the second to the upper portion of the layer.

Quantity E Region F Region

Altitude (km) 90–160 160–500 3 10 10 10 11 Number density (m− ) 1.5 10 –3.0 10 5 10 –2 10 × × × × Height-integrated number 9 1014 4.5 1015 2 × × density (m− ) Ion-neutral collision 2 103–102 0.5–0.05 1 × frequency (sec− ) Ion gyro-/collision 0.09–2.0 4.6 102–5.0 103 × × frequency ratio κi 3 4 Ion Pederson factor 0.09–0.5 2.2 10− –2 10− 2 × × κi/(1 + κi ) 4 Ion Hall factor 8 10− –0.8 1.0 2 2 × κi /(1 + κi ) Electron-neutral collision 1.5 104–9.0 102 80–10 × × frequency Electron gyro-/collision 4.1 102–6.9 103 7.8 104–6.2 105 × × × × frequency ratio κe 3 4 5 6 Electron Pedersen factor 2.7 10− –1.5 10− 10− –1.5 10− 2 × × × κe/(1 + κe ) Electron Hall factor 1.0 1.0 2 2 κe /(1 + κe ) Mean molecular weight 28–26 22–16 1 Ion gyrofrequency (sec− ) 180–190 230–300 Neutral diffusion 30–5 103 105 2 1 × coefficient (m sec− )

The terrestrial magnetic field in the lower ionosphere at equatorial latti- 4 tudes is approximately B0 = 0.35 10− tesla. The earth’s radius is RE = 6371 km. ×

41 SOLAR PHYSICS PARAMETERS24

Parameter Symbol Value Units

Total mass M 1.99 1033 g ⊙ × Radius R 6.96 1010 cm ⊙ × 4 2 Surface gravity g 2.74 10 cm s− ⊙ × 7 1 Escape speed v 6.18 10 cm s− ∞ × 9 2 1 Upward mass flux in spicules — 1.6 10− g cm− s− × 2 Vertically integrated atmospheric density — 4.28 g cm−

Sunspot magnetic field strength Bmax 2500–3500 G

Surface effective temperature T0 5770 K 33 1 Radiant power 3.83 10 erg s− ⊙ L × 10 2 1 Radiant flux density 6.28 10 erg cm− s− F × Optical depth at 500 nm, measured τ5 0.99 — from photosphere Astronomical unit (radius of earth’s orbit) AU 1.50 1013 cm × 6 2 1 Solar constant (intensity at 1 AU) f 1.36 10 erg cm− s− × Chromosphere and Corona25 Quiet Coronal Active Parameter (Units) Sun Hole Region

Chromospheric radiation losses 2 1 (erg cm− s− ) Low chromosphere 2 106 2 106 > 107 × × Middle chromosphere 2 106 2 106 ∼107 × × Upper chromosphere 3 105 3 105 2 106 × × × Total 4 106 4 106 > 2 107 2 × × ∼ × Transition layer pressure (dyne cm− ) 0.2 0.07 2 Coronal temperature (K) at 1.1 R 1.1–1.6 106 106 2.5 106 ⊙ 2 1 × × Coronal energy losses (erg cm− s− ) Conduction 2 105 6 104 105–107 × × Radiation 105 104 5 106 × Solar Wind < 5 104 7 105 < 105 × × Total ∼3 105 8 105 107 2 1 × 11 × 10 11 Solar wind mass loss (g cm− s− ) < 2 10− 2 10− < 4 10− ∼ × × ×

42 THERMONUCLEAR FUSION26

Natural abundance of isotopes: 4 hydrogen nD/nH = 1.5 10− × 6 helium n 3 /n 4 = 1.3 10− He He × lithium nLi6 /nLi7 = 0.08 4 Mass ratios: me/mD = 2.72 10− = 1/3670 1/2 × 2 (me/mD) = 1.65 10− = 1/60.6 × 4 me/mT = 1.82 10− = 1/5496 1/2 × 2 (m /m ) = 1.35 10− = 1/74.1 e T × 2 1 Absorbed radiation dose is measured in rads: 1 rad = 10 erg g− . The curie 10 1 (abbreviated Ci) is a measure of radioactivity: 1 curie = 3.7 10 counts sec− . × Fusion reactions (branching ratios are correct for energies near the cross section peaks; a negative yield means the reaction is endothermic):27 (1a) D + D T(1.01 MeV) + p(3.02 MeV) −−−−→50% (1b) He3(0.82 MeV) + n(2.45 MeV) −−−−→50% (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 He4 + p + n + 12.1 MeV −−−−→51% (5b) He4(4.8 MeV) + D(9.5 MeV) −−−−→43% (5c) He5(2.4 MeV) + p(11.9 MeV) −−−−→6% (6) p+Li6 He4(1.7 MeV) + He3(2.3 MeV) −−−−→ (7a) p + Li7 2 He4 + 17.3 MeV −−−−→20% (7b) Be7 + n 1.6 MeV −−−−→80% − (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) −−−−→ 24 2 The total cross section in barns (1 barn = 10− cm ) as a function of E, the energy in keV of the incident particle [the first ion on the left side of Eqs. (1)–(5)], assuming the target ion at rest, can be fitted by28 1 A + (A A E)2 + 1 − A 5 4 − 3 2 σT (E) = E exp(A E 1/2) 1 1 − −   43 where the Duane coefficients Aj for the principle fusion reactions are as follows:

D–D D–D D–T D–He3 T–T T–He3 (1a) (1b) (2) (3) (4) (5a–c)

A1 46.097 47.88 45.95 89.27 38.39 123.1

A2 372 482 50200 25900 448 11250 4 4 2 3 3 A 4.36 10− 3.08 10− 1.368 10− 3.98 10− 1.02 10− 0 3 × × × × × A4 1.220 1.177 1.076 1.297 2.09 0 A5 0 0 409 647 0 0

3 1 Reaction rates σv (in cm sec− ), averaged over Maxwellian distributions:

Temperature D–D D–T D–He3 T–T T–He3 (keV) (1a + 1b) (2) (3) (4) (5a–c)

22 21 26 22 28 1.0 1.5 10− 5.5 10− 10− 3.3 10− 10− × 21 × 19 23 × 21 25 2.0 5.4 10− 2.6 10− 1.4 10− 7.1 10− 10− × 19 × 17 × 21 × 19 22 5.0 1.8 10− 1.3 10− 6.7 10− 1.4 10− 2.1 10− × 18 × 16 × 19 × 19 × 20 10.0 1.2 10− 1.1 10− 2.3 10− 7.2 10− 1.2 10− × 18 × 16 × 18 × 18 × 19 20.0 5.2 10− 4.2 10− 3.8 10− 2.5 10− 2.6 10− × 17 × 16 × 17 × 18 × 18 50.0 2.1 10− 8.7 10− 5.4 10− 8.7 10− 5.3 10− × 17 × 16 × 16 × 17 × 17 100.0 4.5 10− 8.5 10− 1.6 10− 1.9 10− 2.7 10− × 17 × 16 × 16 × 17 × 17 200.0 8.8 10− 6.3 10− 2.4 10− 4.2 10− 9.2 10− × 16 × 16 × 16 × 17 × 16 500.0 1.8 10− 3.7 10− 2.3 10− 8.4 10− 2.9 10− × 16 × 16 × 16 × 17 × 16 1000.0 2.2 10− 2.7 10− 1.8 10− 8.0 10− 5.2 10− × × × × × For low energies (T < 25 keV) the data may be represented by ∼ 14 2/3 1/3 3 1 (σv) = 2.33 10− T − exp( 18.76T − ) cm sec− ; DD × − 12 2/3 1/3 3 1 (σv) = 3.68 10− T − exp( 19.94T − ) cm sec− , DT × − where T is measured in keV.

The power density released in the form of charged particles is 13 2 3 PDD = 3.3 10− nD (σv)DD watt cm− (including the subsequent × D–T reaction); 13 3 P = 5.6 10− n n (σv) watt cm− ; DT × D T DT 12 3 P 3 = 2.9 10− n n 3 (σv) 3 watt cm− . DHe × D He DHe

44 RELATIVISTIC ELECTRON BEAMS

2 1/2 Here γ = (1 β )− is the relativistic scaling factor; quantities in analytic formulas are− expressed in SI or cgs units, as indicated; in numerical formulas, I is in amperes (A), B is in gauss (G), electron linear density N is 1 in cm− , and temperature, voltage and energy are in MeV; βz = vz /c; k is Boltzmann’s constant. Relativistic electron gyroradius:

2 mc 2 1/2 3 2 1/2 1 re = (γ 1) (cgs) = 1.70 10 (γ 1) B− cm. eB − × −

Relativistic electron energy:

W = mc2γ = 0.511γ MeV.

Bennett pinch condition:

2 2 4 2 I = 2Nk(T + T )c (cgs) = 3.20 10− N(T + T )A . e i × e i

Alfv´en-Lawson limit:

I = (mc3/e)β γ (cgs) = (4πmc/µ e)β γ (SI) = 1.70 104β γ A. A z 0 z × z

The ratio of net current to IA is

I ν = . IA γ

2 2 13 Here ν = Nre is the Budker number, where re = e /mc = 2.82 10− cm is the classical electron radius. Beam electron number density is ×

8 1 3 n = 2.08 10 Jβ− cm− , b ×

2 where J is the current density in A cm− . For a uniform beam of radius a (in cm), 7 2 1 3 n = 6.63 10 Ia− β− cm− , b × and 2re ν = . a γ 45 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

3 3/2 2 2 J = 2.34 10 V d− A cm− . ×

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

I = 8.5 103Gγ ln γ +(γ2 1)1/2 A, p × −   where G is a geometrical factor depending on the diode structure: w for parallel plane cathode and anode G = 2πd of width w, separation d; 1 R2 − G = ln for cylinders of radii R1 (inner) and R2 (outer);  R1  Rc for conical cathode of radius R , maximum G = c d0 separation d0 (at r = Rc) from plane anode. For β 0 (γ 1), both I and I vanish. → → A p The condition for a longitudinal magnetic field Bz to suppress filamentation 2 in a beam of current density J (in A cm− ) is 1/2 Bz > 47βz (γJ) G.

Voltage registered by Rogowski coil of minor cross-sectional area A, n turns, major radius a, inductance L, external resistance R and capacitance C (all in SI):

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

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

X-ray production, target with average atomic number Z (V < 5 MeV): 4 ∼ η x-ray power/beam power = 7 10− ZV. ≡ × X-ray dose at 1 meter generated by an e-beam depositing total charge Q coulombs while V 0.84V in material with charge state Z: ≥ max

2.8 1/2 D = 150Vmax QZ rads.

46 BEAM INSTABILITIES30

Name Conditions Saturation Mechanism

Electron- Vd > V¯ej , j = 1, 2 Electron trapping until electron V¯ V ej ∼ d 1/3 Buneman Vd > (M/m) V¯i, Electron trapping until V > V¯ V¯ V d e e ∼ d 1/3 Beam-plasma Vb > (np/nb) V¯b Trapping of beam electrons

1/3 Weak beam- Vb < (np/nb) V¯b Quasilinear or nonlinear plasma (mode coupling)

Beam-plasma V¯e > Vb > V¯b Quasilinear or nonlinear (hot-electron)

Ion acoustic T T , V C Quasilinear, ion tail form- e ≫ i d ≫ s ation, nonlinear scattering, or resonance broadening.

Anisotropic Te > 2Te Isotropization ⊥ temperature k (hydro)

Ion cyclotron Vd > 20V¯i (for Ion heating T T ) e ≈ i

Beam-cyclotron Vd > Cs Resonance broadening (hydro)

1/2 Modified two- Vd < (1 + β) VA, Trapping stream (hydro) Vd > Cs

1/2 Ion-ion (equal U < 2(1 + β) VA Ion trapping beams)

Ion-ion (equal U < 2Cs Ion trapping beams) For nomenclature, see p. 50.

47 Parameters of Most Unstable Mode Name Wave Group Growth Rate Frequency Number Velocity

1 ωe Electron- ωe 0 0.9 0 electron 2 Vd 1/3 1/3 m m ωe 2 Buneman 0.7 ωe 0.4 ωe Vd M M Vd 3  1/3   nb ωe 2 Beam-plasma 0.7 ωe ωe Vb np − Vb 3   1/3 nb 0.4 ωe np 2   2 nb Vb ωe 3V¯ Weak beam- ω ω e ¯ e e plasma 2np  Vb  Vb Vb 1/2 ¯ nb Ve Vb 1 Beam-plasma ωe ωe λ− Vb n V V¯ D (hot-electron)  p  b e 1/2 m 1 Ion acoustic ωi ωi λ− Cs M D   1 ¯ Anisotropic Ωe ωe cos θ Ωe re− Ve temperature ∼ ⊥ (hydro) 1 1 Ion cyclotron 0.1Ωi 1.2Ωi r− V¯i i 3 1 > Beam-cyclotron 0.7Ωe nΩe 0.7λD− Vd; (hydro) <∼ Cs ∼ 1 ΩH 1 Modified two- ΩH 0.9ΩH 1.7 Vd stream 2 Vd 2 (hydro) ΩH Ion-ion (equal 0.4ΩH 0 1.2 0 beams) U ωi Ion-ion (equal 0.4ωi 0 1.2 0 beams) U For nomenclature, see p. 50.

48 In the preceding tables, subscripts e, i, d, b, p stand for “electron,” “ion,” “drift,” “beam,” and “plasma,” respectively. Thermal velocities are denoted by a bar. In addition, the following are used: m electron mass re, ri gyroradius M ion mass β plasma/magnetic energy V velocity density ratio T temperature VA Alfv´en speed ne,ni number density Ωe, Ωi gyrofrequency n harmonic number ΩH hybrid gyrofrequency, 1/2 2 Cs = (Te/M) ion sound speed ΩH = ΩeΩi ωe, ωi plasma frequency U relative drift velocity of λD Debye length two ion species

APPROXIMATE MAGNITUDES IN SOME TYPICAL PLASMAS

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

4 2 8 5 Interstellar gas 1 1 6 10 7 10 4 10 7 10− × × × × 3 6 6 2 Gaseous nebula 10 1 2 10 20 8 10 6 10− × × × 9 2 9 1 6 Solar Corona 10 10 2 10 2 10− 8 10 60 × × × 12 2 10 3 5 Diffuse hot plasma 10 10 6 10 7 10− 4 10 40 × × × 14 11 5 9 Solar atmosphere, 10 1 6 10 7 10− 40 2 10 × × × gas discharge 14 11 4 2 7 Warm plasma 10 10 6 10 2 10− 8 10 10 × × × 14 2 11 4 4 6 Hot plasma 10 10 6 10 7 10− 4 10 4 10 × × × × 15 4 12 3 6 4 Thermonuclear 10 10 2 10 2 10− 8 10 5 10 × × × × plasma 16 2 12 5 3 8 Theta pinch 10 10 6 10 7 10− 4 10 3 10 × × × × 18 2 13 6 2 10 Dense hot plasma 10 10 6 10 7 10− 4 10 2 10 × × × × 20 2 14 7 12 Laser Plasma 10 10 6 10 7 10− 40 2 10 × × ×

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

49 50 LASERS

System Parameters Efficiencies and power levels are approximate.31 Wavelength Power levels available (W) Type Efficiency (µm) Pulsed CW

13 5 CO2 10.6 0.01–0.02 > 2 10 > 10 (pulsed) × CO 5 0.4 > 109 > 100 Holmium 2.06 0.03 –0.1 > 107 80 † ‡ Iodine 1.315 0.003 3 1012 – × Nd-glass 1.06 – 1.25 1015 – × Nd:YAG 1.064 – 109 > 104 Nd:YLF 1.045, – 4 108 80 1.54,1.313 × Nd:YVO4 1.064 – – > 20 Er:YAG 2.94 – 1.5 105 – 3 × 8 *Color center 1–4 10− 5 10 1 × 14 *Ti:Sapphire 0.7–1.5 0.4 ηp 10 150 × 3 10 Ruby 0.6943 < 10− 10 1 4 3 He-Ne 0.6328 10− – 1–50 10− 3 4 × *Argon ion 0.45–0.60 10− 5 10 150 × 10 *OPO 0.3–10 > 0.1 ηp 10 5 × 6 N2 0.3371 0.001–0.05 10 – 3 7 *Dye 0.3–1.1 10− 5 10 > 100 × Kr-F 0.26 0.08 1012 500 Xenon 0.175 0.02 > 108 – Ytterbium fiber 1.05–1.1 0.55 5 107 104 × Erbium fiber 1.534 – 7 106 100 × Semiconductor 0.375–1.9 > 0.5 3 109 > 103 × *Tunable sources lamp-driven diode-driven † ‡ Nd stands for Neodymium; Er stands for Erbium; Ti stands for Titanium; YAG stands for Yttrium–Aluminum Garnet; YLF stands for Yttrium Lithium Fluoride; YVO5 stands for Yttrium Vanadate; OPO for Optical Parametric Oscillator; ηp is pump laser efficiency.

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

2 1/2 n = [1 ω /ω(ω ωce)] , ± − pe ∓ where refers to the helicity. The rate of change of polarization angle θ as a function± of displacement s (Faraday rotation) is given by

4 2 1 dθ/ds = (k/2)(n n+) = 2.36 10 NBf − cm− , − − × where N is the electron number density, B is the field strength, and f is the wave frequency, all in cgs. The quiver velocity of an electron in an e-m field of angular frequency ω is 1/2 1 v0 = eEmax/mω = 25.6I λ0 cm sec− 2 2 in terms of the laser flux I = cEmax/8π, with I in watt/cm , laser wavelength λ0 in µm. The ratio of quiver energy to thermal energy is

2 13 2 W /W = m v /2kT = 1.81 10− λ I/T, qu th e 0 × 0

15 2 where T is given in eV. For example, if I = 10 W cm− , λ0 = 1 µm, T = 2 keV, then W /W 0.1. qu th ≈ Pondermotive force:

= N E2 /8πN , FF ∇h i c where 21 2 3 N = 1.1 10 λ − cm− . c × 0 For uniform illumination of a lens with f-number F , the diameter d at focus (85% of the energy) and the depth of focus l (distance to first zero in intensity) are given by

d 2.44Fλθ/θ and l 2F 2λθ/θ . ≈ DL ≈ ± DL

Here θ is the beam divergence containing 85% of energy and θDL is the diffraction-limited divergence:

θDL = 2.44λ/b, where b is the aperture. These formulas are modified for nonuniform (such as Gaussian) illumination of the lens or for pathological laser profiles.

52 ATOMIC PHYSICS AND RADIATION

Energies and temperatures are in eV; all other units are cgs except where noted. Z is the charge state (Z = 0 refers to a neutral atom); the subscript e labels electrons. N refers to number density, n to principal quantum number. Asterisk superscripts on level population densities denote local thermodynamic equilibrium (LTE) values. Thus Nn* is the LTE number density of atoms (or ions) in level n. Characteristic atomic collision cross section:

2 17 2 (1) πa = 8.80 10− cm . 0 ×

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

2 H Z Z E (2) E (n, l) = ∞ , ∞ − (n ∆ )2 − l

H 5 where E = 13.6 eV is the hydrogen ionization energy and ∆l = 0.75l− , 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): →

13 fmng(n,m) 2 (3) σmn = 2.36 10− cm , × ǫ∆Enm where fmn is the oscillator strength, g(n,m) is the Gaunt factor, ǫ is the incident electron energy, and ∆E = E E . nm n − m Electron excitation rate averaged over Maxwellian velocity distribution, Xmn = N σ v (Refs. 34, 35): eh mn i

5 fmn g(n,m) Ne ∆Enm 1 (4) Xmn = 1.6 10− h i exp sec− , × 1/2 − T ∆EnmTe  e  where g(n,m) denotes the thermal averaged Gaunt factor (generally 1 for atoms,h 0.2 fori ions). ∼ ∼

53 Rate for electron collisional deexcitation:

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

Here Nm*/Nn*=(gm/gn) exp(∆Enm/Te) is the Boltzmann relation for level population densities, where gn is the statistical weight of level n. Rate for spontaneous decay n m (Einstein A coefficient)34 →

7 2 1 (6) A = 4.3 10 (g /g )f (∆E ) sec− . nm × m n mn nm

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

19 3 (7) I = 1.6 10− A N ∆E watt/cm . nm × nm n nm

Condition for steady state in a corona model:

(8) N N σ v = N A , 0 eh 0n i n n0 where the ground state is labelled by a zero subscript. Hence for a transition n m in ions, where g(n, 0) 0.2, → h i ≈

3 25 fnmgmNeN0 ∆Enm ∆En0 watt (9) Inm = 5.1 10− exp . × 1/2 ∆E − T cm3 g0Te  n0   e 

Ionization and Recombination In a general time-dependent situation the number density of the charge state Z satisfies

dN(Z) (10) = N S(Z)N(Z) α(Z)N(Z) e − − dt h +S(Z 1)N(Z 1) + α(Z + 1)N(Z + 1) . − − i Here S(oZ) is the ionization rate. The recombination rate α(Z) has the form α(Z) = αr(Z) + Neα3(Z), where αr and α3 are the radiative and three-body recombination rates, respectively.

54 Classical ionization cross-section36 for any atomic shell j

14 2 2 (11) σ = 6 10− b g (x)/U cm . i × j j j

Here bj is the number of shell electrons; Uj is the binding energy of the ejected electron; x = ǫ/Uj , where ǫ is the incident electron energy; and g is a universal function with a minimum value g 0.2 at x 4. min ≈ ≈ Ionization from ion ground state, averaged over Maxwellian electron distribu- Z tion, for 0.02 < Te/E < 100 (Ref. 35): ∼ ∞ ∼

Z 1/2 Z 5 (Te/E ) E 3 (12) S(Z) = 10− ∞ exp ∞ cm /sec, Z 3/2 Z (E ) (6.0 + Te/E ) − Te ∞ ∞   where EZ is the ionization energy. ∞ Electron-ion radiative recombination rate (e + N(Z) N(Z 1) + hν) 2 → − for Te/Z < 400 eV (Ref. 37): ∼

Z 1/2 14 E 1 Z (13) α (Z) = 5.2 10− Z ∞ 0.43 + ln(E /T ) r × e  Te  h 2 ∞

Z 1/3 3 +0.469(E /Te)− cm /sec. ∞ i

2 35 For 1 eV < Te/Z < 15 eV, this becomes approximately

13 2 1/2 3 (14) α (Z) = 2.7 10− Z T − cm /sec. r × e

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

27 4.5 6 (15) α = 8.75 10− T − cm /sec. 3 × e

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

16 5 3 7+2l 2 (16) σ (n, l) = 1.64 10− Z /n K cm , ph ×

5 1 where K is the wavenumber in Rydbergs (1 Rydberg = 1.0974 10 cm− ). × 55 Ionization Equilibrium Models Saha equilibrium:39

Z 3/2 Z NeN1*(Z) 21 g1 Te E (n, l) 3 (17) = 6.0 10 exp ∞ cm− , Z 1 Nn*(Z 1) × g − − Te − n  

Z Z where gn is the statistical weight for level n of charge state Z and E (n, l) is the ionization energy of the neutral atom initially in level (n, l), given∞ by Eq. (2). In a steady state at high electron density,

NeN*(Z) S(Z 1) (18) = − , N*(Z 1) α − 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

18 7 17/2 Z 1/2 3 (20) Ne > 7 10 Z n− (T/E ) cm− . ∼ × ∞

Steady state condition in corona model:

N(Z 1) αr (21) − = . N(Z) S(Z 1) −

Corona model is applicable if40

12 1 16 7/2 3 (22) 10 tI − < Ne < 10 Te cm− ,

where tI is the ionization time.

56 Radiation N. B. Energies and temperatures are in eV; all other quantities are in cgs units except where noted. Z is the charge state (Z = 0 refers to a neutral atom); the subscript e labels electrons. N is number density. Average radiative decay rate of a state with principal quantum number n is

10 4 9/2 (23) A = A = 1.6 10 Z n− sec. n nm × m

Natural linewidth (∆E in eV):

15 (24) ∆E ∆t = h = 4.14 10− eV sec, × where ∆t is the lifetime of the line. Doppler width:

5 1/2 (25) ∆λ/λ = 7.7 10− (T/µ) , × where µ is the mass of the emitting atom or ion scaled by the proton mass. Optical depth for a Doppler-broadened line:39

13 2 1/2 9 1/2 (26) τ = 3.52 10− f λ(Mc /kT ) NL = 5.4 10− f λ(µ/T ) NL, × nm × mn where fnm is the absorption oscillator strength, λ is the wavelength, and L is the physical depth of the plasma; M, N, and T are the mass, number density, and temperature of the absorber; µ is M divided by the proton mass. Optically thin means τ < 1. Resonance absorption cross section at center of line:

13 2 2 (27) σ = 5.6 10− λ /∆λ cm . λ=λc ×

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

5 1 (28) λ = 2.50 10− T − cm. max ×

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

(29) W = 1.03 105T 4 watt/cm2. ×

57 Bremsstrahlung from hydrogen-like plasma:26

32 1/2 2 3 (30) P = 1.69 10− N T Z N(Z) watt/cm , Br × e e X   where the sum is over all ionization states Z. Bremsstrahlung optical depth:41

38 2 7/2 (31) τ = 5.0 10− N N Z gLT − , × e i 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:

7 2 3/2 2 2 2 1/2 1 (32) κ = 3.1 10− Zn ln Λ T − ω− (1 ω /ω )− cm− ; × e − p here Λ is the electron thermal velocity divided by V , where V is the larger of 2 1/2 ω and ωp multiplied by the larger of Ze /kT andh/ ¯ (mkT ) . Recombination (free-bound) radiation:

Z 1 32 1/2 2 E − 3 (33) Pr = 1.69 10− NeTe Z N(Z) ∞ watt/cm . × T X h  e i Cyclotron radiation26 in magnetic field B:

28 2 3 (34) P = 6.21 10− B N T watt/cm . c × e e

2 26 For NekTe = NikTi = B /16π (β = 1, isothermal plasma),

38 2 2 3 (35) P = 5.00 10− N T watt/cm . c × e e

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

8 2 9.0 10 B− (36) tc × sec, ≈ 2.5 + γ 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:

1/6 (37) mtr = (57βBL) , where β = 8πNkT/B2. Line radiation is given by summing Eq. (9) over all species in the plasma.

58 ATOMIC SPECTROSCOPY

Spectroscopic notation combines observational and theoretical elements. Observationally, spectral lines are grouped in series with line spacings which decrease toward the series limit. Every line can be related theoretically to a transition between two atomic states, each identified by its quantum numbers. Ionization levels are indicated by roman numerals. Thus C I is unionized carbon, C II is singly ionized, etc. The state of a one-electron atom (hydrogen) or ion (He II, Li III, etc.) is specified by identifying the principal quantum number n = 1, 2,..., the orbital angular momentum l = 0, 1,...,n 1, and 1 − the spin angular momentum s = 2 . The total angular momentum j is the ± 1 1 magnitude of the vector sum of l and s, j = l 2 (j 2 ). The letters s, p, d, f, g, h, i, k, l, . . . , respectively, are associated± with≥ angular momenta l = 0, 1, 2, 3, 4, 5, 6, 7, 8, . . . . The atomic states of hydrogen and hydrogenic ions are degenerate: neglecting fine structure, their energies depend only on n according to 2 2 2 R hcZ n− RyZ En = ∞ = , − 1 + m/M − 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

1 R = 109, 737 cm− ∞ is the Rydberg constant. If En is divided by hc, the result is in wavenumber units. The energy associated with a transition m n is given by →

∆E = Ry(1/m2 1/n2), mn − with mn) for absorption (emission) lines. For hydrogen and hydrogenic ions the series of lines belonging to the transitions m n have conventional names: →

Transition 1 n 2 n 3 n 4 n 5 n 6 n → → → → → → Name Lyman Balmer Paschen Brackett Pfund Humphreys

Successive lines in any series are denoted α, β, γ, etc. Thus the transition 1 3 gives rise to the Lyman-β line. Relativistic effects, quantum electrodynamic→ effects (e.g., the Lamb shift), and interactions between the nuclear magnetic

59 moment and the magnetic field due to the electron produce small shifts and 2 1 splittings, < 10− cm− ; these last are called “hyperfine structure.” ∼ In many-electron atoms the electrons are grouped in closed and open shells, with spectroscopic properties determined mainly by the outer shell. Shell energies depend primarily on n; the shells corresponding to n = 1, 2, 3,... are called K, L, M, etc. A shell is made up of subshells of different angular momenta, each labeled according to the values of n, l, and the number of electrons it contains out of the maximum possible number, 2(2l + 1). For example, 2p5 indicates that there are 5 electrons in the subshell corresponding to l = 1 (denoted by p) and n = 2. In the lighter elements the electrons fill up subshells within each shell in the order s, p, d, etc., and no shell acquires electrons until the lower shells are full. In the heavier elements this rule does not always hold. But if a particular subshell is filled in a noble gas, then the same subshell is filled in the atoms of all elements that come later in the periodic table. The ground state configurations of the noble gases are as follows:

He 1s2 Ne 1s22s22p6 Ar 1s22s22p63s23p6 Kr 1s22s22p63s23p63d104s24p6 Xe 1s22s22p63s23p63d104s24p64d105s25p6 Rn 1s22s22p63s23p63d104s24p64d104f145s25p65d106s26p6

Alkali metals (Li, Na, K, etc.) resemble hydrogen; their transitions are de- scribed by giving n and l in the initial and final states for the single outer (valence) electron. For general transitions in most atoms the atomic states are specified in terms of the parity ( 1)Σli and the magnitudes of the orbital angular momen- − tum L = Σli, the spin S = Σsi, and the total angular momentum J = L + S, where all sums are carried out over the unfilled subshells (the filled ones sum to zero). If a magnetic field is present the projections ML, MS , and M of L, S, and J along the field are also needed. The quantum numbers satisfy ML L νl, MS S ν/2, and M J L + S, where ν is the |number| ≤ of electrons≤ | in| the ≤ unfilled≤ subshell.| | Upper-case ≤ ≤ letters S, P, D, etc., stand for L = 0, 1, 2, etc., in analogy with the notation for a single electron. 5 2 o For example, the ground state of Cl is described by 3p P3/2. The first part indicates that there are 5 electrons in the subshell corresponding to n = 3 and l = 1. (The closed inner subshells 1s22s22p63s2, identical with the configura- tion of Mg, are usually omitted.) The symbol ‘P’ indicates that the angular momenta of the outer electrons combine to give L = 1. The prefix ‘2’ repre- sents the value of the multiplicity 2S + 1 (the number of states with nearly the 1 same energy), which is equivalent to specifying S = 2 . The subscript 3/2 is

60 the value of J. The superscript ‘o’ indicates that the state has odd parity; it would be omitted if the state were even. The notation for excited states is similar. For example, helium has a state 3 1 2 1 1s2s S1 which lies 19.72 eV (159, 856 cm− ) above the ground state 1s S0. But the two “terms” do not “combine” (transitions between them do not occur) because this would violate, e.g., the quantum-mechanical selection rule that the parity must change from odd to even or from even to odd. For electric dipole transitions (the only ones possible in the long-wavelength limit), other selection rules are that the value of l of only one electron can change, and only by ∆l = 1; ∆S = 0; ∆L = 1 or 0; and ∆J = 1 or 0 (but L = 0 does not combine± with L = 0 and J =± 0 does not combine± with J = 0). Transitions are possible between the helium ground state (which has S = 0, L = 0, J = 0, 1 o and even parity) and, e.g., the state 1s2p P1 (with S = 0, L = 1, J = 1, odd parity, excitation energy 21.22 eV). These rules hold accurately only for light atoms in the absence of strong electric or magnetic fields. Transitions that obey the selection rules are called “allowed”; those that do not are called “forbidden.” The amount of information needed to adequately characterize a state in- creases with the number of electrons; this is reflected in the notation. Thus43 2 O II has an allowed transition between the states 2p 3p′ 2 o 2 1 2 F7/2 and 2p ( D)3d′ F7/2 (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. The outer (n = 3) electron has l = 1 in the first state and l = 2 in the second. The prime indicates that if the outermost electron were removed by ionization, the resulting ion would not be in its lowest energy state. The expression (1D) give the multiplicity and total angular momentum of the “parent” term, i.e., the subshell immediately below the valence subshell; this is understood to be the same in both states. (Grandparents, etc., sometimes have to be specified in heavier atoms and ions.) Another example43 is the allowed transition from 2 3 2 o 2 o 2 1 2 2p ( P)3p P1/2 (or P3/2) to 2p ( D)3d′ S1/2, in which there is a “spin flip” (from antiparallel to parallel) in the n = 2, l = 1 subshell, as well as changes from one state to the other in the value of l for the valence electron and in L. The description of fine structure, Stark and Zeeman effects, spectra of highly ionized or heavy atoms, etc., is more complicated. The most important difference between optical and X-ray spectra is that the latter involve energy changes of the inner electrons rather than the outer ones; often several electrons participate.

61 COMPLEX (DUSTY) PLASMAS

Complex (dusty) plasmas (CDPs) may be regarded as a new and unusual state of matter. CDPs contain charged microparticles (dust grains) in addition to electrons, ions, and neutral gas. Electrostatic coupling between the grains can vary over a wide range, so that the states of CDPs can change from weakly coupled (gaseous) to crystalline. CDPs can be investigated at the kinetic level (individual particles are easily visualized and relevant time scales are accessi- ble). CDPs are of interest as a non-Hamiltonian system of interacting particles and as a means to study generic fundamental physics of self-organization, pat- tern formation, phase transitions, and scaling. Their discovery has therefore opened new ways of precision investigations in many-particle physics.

Typical experimental dust properties 7 13 grain size (radius) a 0.3 30 µm, mass md 3 10− 3 10− g, number ≃ − ∼ × −3 × 3 7 3 density (in terms of the interparticle distance) nd ∆− 10 10 cm− , 2 2 ∼ ∼ − temperature T 3 10− 10 eV. d ∼ × − Typical discharge (bulk) plasmas

2 2 4 gas pressure p 10− 1 Torr, Ti Tn 3 10− eV, vTi 7 10 cm/s ∼ − ≃ 8 ≃ 10× 3 ≃ × (Ar), Te 0.3 3 eV, ni ne 10 10 cm− , screening length λD ∼ − ≃ 6∼ − 7 1 ≃ λ 20 200 µm, ω 2 10 2 10 s− (Ar). B fields up to B 3 T. Di ∼ − pi ≃ × − × ∼ Dimensionless

Havnes parameter P = Z nd/ne | | 2 normalized charge z = Z e /kTea | |2 2 dust-dust scattering parameter βd = Z e /kTdλD dust-plasma scattering parameter β = Z e2/kT λ e,i | | e,i D coupling parameter Γ = (Z2e2/kT ∆) exp( ∆/λ ) d − D lattice parameter κ = ∆/λD particle parameter α = a/∆

lattice magnetization parameter µ = ∆/rd

4 2 3 5 Typical experimental values: P 10− 10 ,z 2 4(Z 10 10 electron 3 ∼ −4 ≃ −2 ∼ − charges), Γ < 10 , κ 0.3 10, α 10− 3 10− , µ < 1 ∼ − ∼ − × Frequencies 2 2 1/2 dust plasma frequency ωpd = (4πZ e nd/md) ( Z P m /m )1/2ω ≃ | | 1+P i d pi 1+z charge fluctuation frequency ωch (a/λD )ωpi ≃ √2π 62 dust-gas friction rate ν 10a2p/m v nd ∼ d Tn dust gyrofrequency ωcd = ZeB/mdc

Velocities 1/2 Td mi 1/2 dust thermal velocity vT = (kTd/md) [ ] vT d ≡ Ti md i dust acoustic wave velocity CDA = ωpdλD ( Z P m /m )1/2v ≃ | | 1+P i d Ti 1/2 dust Alfv´en wave velocity vAd = B/(4πndmd)

dust-acoustic Mach number V/CDA

dust magnetic Mach number V/vAd l,t dust lattice (acoustic) wave velocity CDL = ωpdλD Fl,t(κ)

The range of the dust-lattice wavenumbers is K∆ < π The functions Fl,t(κ) for longitudinal and transverse waves can be approximated44,45 with accuracy < 1% in the range κ 5: ≤ F 2.70κ1/2(1 0.096κ 0.004κ2), F 0.51κ(1 0.039κ2), l ≃ − − t ≃ − Lengths

frictional dissipation length Lν = vTd /νnd dust Coulomb radius R = Z e2/kT Ce,i | | e,i dust gyroradius rd = vTd /ωcd

Grain Charging

The charge evolution equation is d Z /dt = Ii Ie. From orbital motion limited (OML) theory46 in the collisionless| | limit l− λ a: en(in) ≫ D ≫

Te I = √8πa2n v exp( z), I = √8πa2n v 1 + z . e e Te − i i Ti  Ti  Grains are charged negatively. The grain charge can vary in response to spatial and temporal variations of the plasma. Charge fluctuations are always present, with frequency ωch. Other charging mechanisms are photoemission, secondary emission, thermionic emission, field emission, etc. Charged dust grains change the plasma composition, keeping quasineutrality. A measure of this is the Havnes parameter P = Z n /n . The balance of I and I yields | | d e e i

1/2 mi Ti Te exp( z) = 1 + z [1 + P (z)] −  me Te   Ti  63 When the relative charge density of dust is large, P 1, the grain charge Z monotonically decreases. ≫ Forces and momentum transfer

In addition to the usual electromagnetic forces, grains in complex plasmas are also subject to: gravity force Fg = mdg; thermophoretic force

4√2π 2 Fth = (a /vT )κn Tn − 15 n ∇

(where κn is the coefficient of gas thermal conductivity); forces associated with the momentum transfer from other species, Fα = mdναdVαd, i.e., neutral, ion, and electron drag. For collisions between charged− particles, two limiting cases are distinguished by the magnitude of the scattering parameter β . When β 1 the result is independent of the sign of the potential. When α α ≪ βα 1, the results for repulsive and attractive interaction potentials are different.≫ For typical complex plasmas the hierarchy of scattering parameters 3 4 is βe( 0.01 0.3) βi( 1 30) βd( 10 3 10 ). The generic expressions∼ for− different≪ types∼ of− collisions≪ are∼47 − ×

√ 2 ναd = (4 2π/3)(mα/md)a nαvTα Φαd Electron-dust collisions

1 2 Φed z Λed βe 1 ≃ 2 ≪ Ion-dust collisions

1 2 2 2 z (Te/Ti) Λid βi < 5 Φid = 2 2 n 2(λD/a) (ln βi + 2 ln βi + 2), βi > 13 Dust-dust collisons

2 zdΛdd βd 1 Φdd = ≪ (λ /a)2[ln 4β ln ln 4β ], β 1 n D d − d d ≫ where z Z2e2/akT . d ≡ d

For νdd νnd the complex plasma is in a two-phase state, and for νnd νdd we have∼ merely tracer particles (dust-neutral gas interaction dominates).≫ The momentum transfer cross section is proportional to the Coulomb logarithm Λαd when the Coulomb scattering theory is applicable. It is determined by integration over the impact parameters, from ρmin to ρmax. ρmin is due to finite grain size and is given by OML theory. ρmax = λD for repulsive interaction (applicable for β 1), and ρ = λ (1+2β )1/2 for attractive interaction α ≪ max D α (applicable up to βα < 5).

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

∞ ∞ z x 2 2 z x Λ = z e− α ln[1 + 4(λ /a ) x ]dx 2z e− α ln(2x 1)dx, αd α D α − α − Z0 Z1 where ze = z, ae = a, and ad = 2a.

For ion-dust (attraction)

∞ zx 1+2(Ti/Te)(λD/a)x Λid z e− ln dx. ≃ 1+2(Ti/Te)x Z0 h i

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

Phase Diagram of Complex Plasmas

The figure below represents different “phase states” of CDPs as functions of the electrostatic coupling parameter Γ and κ or α, respectively. The 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- 48 2 1 tion is Γ > 106(1 + κ + κ /2)− . Regions II (VI) are for Coulomb (Yukawa) non-ideal plasmas – the characteristic range of dust-dust interaction (in terms of the momentum transfer) is larger than the intergrain distance (in terms of 1/2 1/3 the Wigner-Seitz radius), (σ/π) > (4π/3)− ∆, which implies that the interaction is essentially multiparticle. α-1= /a 101 102 103 Regions III (VII and VIII) correspond to 4 10 Coulomb (Yukawa) ideal gases. The range I V of dust-dust interaction is smaller than the 102 intergrain distance and only pair collisions are important. In addition, in the region II VIII the pair Yukawa interaction asymp- 100 VI Γ totically reduces to the hard sphere limit, forming a “Yukawa granular medium”. In 10-2 III region IV the electrostatic interaction is IV VII VIII unimportant and the system is like a uaual granular medium. 10-4 0.1 1 10 κ=/λ

65 REFERENCES

When any of the formulas and data in this collection are referenced in research publications, it is suggested that the original source be cited rather than the Formulary. Most of this material is well known and, for all practical purposes, is in the “public domain.” Numerous colleagues and readers, too numerous to list by name, have helped in collecting and shaping the Formulary into its present form; they are sincerely thanked for their efforts. Several book-length compilations of data relevant to plasma physics are available. The following are particularly useful:

C. W. Allen, Astrophysical Quantities, 3rd edition (Athlone Press, 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 purpose of giving credit to the original workers, but (1) to guide the reader to sources containing related material and (2) to indicate where to find derivations, 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 some mathematical constants not available on pocket calculators.

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22. J. Sheffield, Plasma Scattering of Electromagnetic Radiation (Academic Press, New York, 1975), p. 6 (after J. W. Paul).

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

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

25. G. L. Withbroe and R. W. Noyes, “Mass and Energy Flow in the Solar Chromosphere and Corona,” Ann. Rev. Astrophys. 15, 363 (1977).

26. S. Glasstone and R. H. Lovberg, Controlled Thermonuclear Reactions (Van Nostrand, New York, 1960), Chapt. 2.

27. References to experimental measurements of branching ratios and cross sections are listed in F. K. McGowan, et al., Nucl. Data Tables A6, 353 (1969); A8, 199 (1970). The yields listed in the table are calculated directly from the mass defect.

28. G. H. Miley, H. Towner and N. Ivich, Fusion Cross Section and Reactivi- ties, Rept. COO-2218-17 (University of Illinois, Urbana, IL, 1974); B. H. Duane, Fusion Cross Section Theory, Rept. BNWL-1685 (Brookhaven National Laboratory, 1972).

29. J. M. Creedon, “Relativistic Brillouin Flow in the High ν/γ Limit,” J. Appl. Phys. 46, 2946 (1975).

30. See, for example, A. B. Mikhailovskii, Theory of Plasma Instabilities Vol. I (Consultants Bureau, New York, 1974). The table on pp. 48–49 was compiled by K. Papadopoulos.

68 31. Table prepared from data compiled by J. M. McMahon (personal 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, New York, 1965).

36. M. Gryzinski, “Classical Theory of Atomic Collisions I. Theory of Inelastic Collision,” Phys. Rev. 138A, 336 (1965).

37. M. J. Seaton, “Radiative Recombination of Hydrogen Ions,” Mon. Not. Roy. Astron. Soc. 119, 81 (1959).

38. Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and 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-Frequency Power Absorption by Inverse Bremsstrahlung in Plasmas,” Phys. Fluids 16, 722 (1973).

43. W. L. Wiese, M. W. Smith, and B. M. Glennon, Atomic Transition 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 45. S. Zhdanov, R. A. Quinn, D. Samsonov, and G. E. Morfill, “Large-scale steady-state structure of a 2D plasma crystal”, New J. Phys. 5, 74 (2003).

46. J. E. Allen, “Probe theory – the orbital motion approach”, Phys. Scripta 45, 497 (1992).

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

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

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