Advanced Mathematics and Physics: Vector Calculus, Laplacian, and Collision Integrals, Lab Reports of Electrical and Electronics Engineering

Download Advanced Mathematics and Physics: Vector Calculus, Laplacian, and Collision Integrals and more Lab Reports Electrical and Electronics Engineering in PDF only on Docsity! 2000 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 1 CONTENTS Numerical and Algebraic . . . . . . . . . . . . . . . . . . . . . 3 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . 4 Differential Operators in Curvilinear Coordinates . . . . . . . . . . . 6 Dimensions and Units . . . . . . . . . . . . . . . . . . . . . . . 10 International System (SI) Nomenclature . . . . . . . . . . . . . . . 13 Metric Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Physical Constants (SI) . . . . . . . . . . . . . . . . . . . . . . 14 Physical Constants (cgs) . . . . . . . . . . . . . . . . . . . . . 16 Formula Conversion . . . . . . . . . . . . . . . . . . . . . . . 18 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . 19 Electricity and Magnetism . . . . . . . . . . . . . . . . . . . . . 20 Electromagnetic Frequency/Wavelength Bands . . . . . . . . . . . . 21 AC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Dimensionless Numbers of Fluid Mechanics . . . . . . . . . . . . . 23 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Fundamental Plasma Parameters . . . . . . . . . . . . . . . . . . 28 Plasma Dispersion Function . . . . . . . . . . . . . . . . . . . . 30 Collisions and Transport . . . . . . . . . . . . . . . . . . . . . 31 Approximate Magnitudes in Some Typical Plasmas . . . . . . . . . . 40 Ionospheric Parameters . . . . . . . . . . . . . . . . . . . . . . 42 Solar Physics Parameters . . . . . . . . . . . . . . . . . . . . . 43 Thermonuclear Fusion . . . . . . . . . . . . . . . . . . . . . . 44 Relativistic Electron Beams . . . . . . . . . . . . . . . . . . . . 46 Beam Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 48 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Atomic Physics and Radiation . . . . . . . . . . . . . . . . . . . 52 Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 58 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2 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) ∫ V dV∇f = ∫ S dSf (28) ∫ V dV∇ ·A = ∫ S dS ·A (29) ∫ V dV∇·T = ∫ S dS ·T (30) ∫ V dV∇×A = ∫ S dS×A (31) ∫ V dV (f∇2g − g∇2f) = ∫ S dS · (f∇g − g∇f) (32) ∫ V dV (A · ∇ × ∇×B−B · ∇ × ∇×A) = ∫ S dS · (B×∇×A−A×∇×B) If S is an open surface bounded by the contour C, of which the line element is dl, (33) ∫ S dS×∇f = ∮ C dlf 5 (34) ∫ S dS · ∇ ×A = ∮ C dl ·A (35) ∫ S (dS×∇)×A = ∮ C dl×A (36) ∫ S dS · (∇f ×∇g) = ∮ C fdg = − ∮ C gdf DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES5 Cylindrical Coordinates Divergence ∇ ·A = 1 r ∂ ∂r (rAr) + 1 r ∂Aφ ∂φ + ∂Az ∂z Gradient (∇f)r = ∂f ∂r ; (∇f)φ = 1 r ∂f ∂φ ; (∇f)z = ∂f ∂z Curl (∇×A)r = 1 r ∂Az ∂φ − ∂Aφ ∂z (∇×A)φ = ∂Ar ∂z − ∂Az ∂r (∇×A)z = 1 r ∂ ∂r (rAφ)− 1 r ∂Ar ∂φ Laplacian ∇2f = 1 r ∂ ∂r ( r ∂f ∂r ) + 1 r2 ∂2f ∂φ2 + ∂2f ∂z2 6 Laplacian of a vector (∇2A)r = ∇2Ar − 2 r2 ∂Aφ ∂φ − Ar r2 (∇2A)φ = ∇2Aφ + 2 r2 ∂Ar ∂φ − Aφ r2 (∇2A)z = ∇2Az Components of (A · ∇)B (A · ∇B)r = Ar ∂Br ∂r + Aφ r ∂Br ∂φ + Az ∂Br ∂z − AφBφ r (A · ∇B)φ = Ar ∂Bφ ∂r + Aφ r ∂Bφ ∂φ + Az ∂Bφ ∂z + AφBr r (A · ∇B)z = Ar ∂Bz ∂r + Aφ r ∂Bz ∂φ + Az ∂Bz ∂z Divergence of a tensor (∇ · T )r = 1 r ∂ ∂r (rTrr) + 1 r ∂Tφr ∂φ + ∂Tzr ∂z − Tφφ r (∇ · T )φ = 1 r ∂ ∂r (rTrφ) + 1 r ∂Tφφ ∂φ + ∂Tzφ ∂z + Tφr r (∇ · T )z = 1 r ∂ ∂r (rTrz) + 1 r ∂Tφz ∂φ + ∂Tzz ∂z 7 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 Capacitance C t2q2 ml2 l farad 9× 1011 cm Charge q q m1/2l3/2 t coulomb 3× 109 statcoulomb Charge ρ q l3 m1/2 l3/2t coulomb 3× 103 statcoulomb density /m3 /cm3 Conductance tq2 ml2 l t siemens 9× 1011 cm/sec Conductivity σ tq2 ml3 1 t siemens 9× 109 sec−1 /m Current I, i q t m1/2l3/2 t2 ampere 3× 109 statampere Current J, j q l2t m1/2 l1/2t2 ampere 3× 105 statampere density /m2 /cm2 Density ρ m l3 m l3 kg/m3 10−3 g/cm3 Displacement D q l2 m1/2 l1/2t coulomb 12π × 105 statcoulomb /m2 /cm2 Electric field E ml t2q m1/2 l1/2t volt/m 1 3 × 10−4 statvolt/cm Electro- E, ml2 t2q m1/2l1/2 t volt 1 3 × 10−2 statvolt motance Emf Energy U,W ml2 t2 ml2 t2 joule 107 erg Energy w,  m lt2 m lt2 joule/m3 10 erg/cm3 density 10 Dimensions Physical Sym- SI Conversion Gaussian Quantity bol SI Gaussian Units Factor Units Force F ml t2 ml t2 newton 105 dyne Frequency f, ν 1 t 1 t hertz 1 hertz Impedance Z ml2 tq2 t l ohm 1 9 × 10−11 sec/cm Inductance L ml2 q2 t2 l henry 1 9 × 10−11 sec2/cm Length l l l meter (m) 102 centimeter (cm) Magnetic H q lt m1/2 l1/2t ampere– 4π × 10−3 oersted intensity turn/m Magnetic flux Φ ml2 tq m1/2l3/2 t weber 108 maxwell Magnetic B m tq m1/2 l1/2t tesla 104 gauss induction Magnetic m,µ l2q t m1/2l5/2 t ampere–m2 103 oersted– moment cm3 Magnetization M q lt m1/2 l1/2t ampere– 10−3 oersted turn/m Magneto- M, q t m1/2l1/2 t2 ampere– 4π 10 gilbert motance Mmf turn Mass m,M m m kilogram 103 gram (g) (kg) Momentum p,P ml t ml t kg–m/s 105 g–cm/sec Momentum m l2t m l2t kg/m2–s 10−1 g/cm2–sec density Permeability µ ml q2 1 henry/m 1 4π × 107 — 11 Dimensions Physical Sym- SI Conversion Gaussian Quantity bol SI Gaussian Units Factor Units Permittivity  t2q2 ml3 1 farad/m 36π × 109 — Polarization P q l2 m1/2 l1/2t coulomb/m2 3× 105 statcoulomb /cm2 Potential V, φ ml2 t2q m1/2l1/2 t volt 1 3 × 10−2 statvolt Power P ml2 t3 ml2 t3 watt 107 erg/sec Power m lt3 m lt3 watt/m3 10 erg/cm3–sec density Pressure p, P m lt2 m lt2 pascal 10 dyne/cm2 Reluctance R q2 ml2 1 l ampere–turn 4π × 10−9 cm−1 /weber Resistance R ml2 tq2 t l ohm 1 9 × 10−11 sec/cm Resistivity η, ρ ml3 tq2 t ohm–m 1 9 × 10−9 sec Thermal 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 106 gauss–cm potential Velocity v l t l t m/s 102 cm/sec Viscosity η, µ m lt m lt kg/m–s 10 poise Vorticity ζ 1 t 1 t s−1 1 sec−1 Work W ml2 t2 ml2 t2 joule 107 erg 12 Physical Quantity Symbol Value Units Wavelength associated λ0 = hc/e 1.2398× 10−6 m with 1 eV Frequency associated ν0 = e/h 2.4180× 1014 Hz with 1 eV Wave number associated k0 = e/hc 8.0655× 105 m−1 with 1 eV Energy associated with hν0 1.6022× 10−19 J 1 eV Energy associated with hc 1.9864× 10−25 J 1 m−1 Energy associated with me3/80 2h2 13.606 eV 1 Rydberg Energy associated with k/e 8.6174× 10−5 eV 1 Kelvin Temperature associated e/k 1.1604× 104 K with 1 eV Avogadro number NA 6.0221× 1023 mol−1 Faraday constant F = NAe 9.6485× 104 C mol−1 Gas constant R = NAk 8.3145 J K −1mol−1 Loschmidt’s number n0 2.6868× 1025 m−3 (no. density at STP) Atomic mass unit mu 1.6605× 10−27 kg Standard temperature T0 273.15 K Atmospheric pressure p0 = n0kT0 1.0133× 105 Pa Pressure of 1 mm Hg 1.3332× 102 Pa (1 torr) Molar volume at STP V0 = RT0/p0 2.2414× 10−2 m3 Molar weight of air Mair 2.8971× 10−2 kg calorie (cal) 4.1868 J Gravitational g 9.8067 m s−2 acceleration 15 PHYSICAL CONSTANTS (cgs)7 Physical Quantity Symbol Value Units Boltzmann constant k 1.3807× 10−16 erg/deg (K) Elementary charge e 4.8032× 10−10 statcoulomb (statcoul) Electron mass me 9.1094× 10−28 g Proton mass mp 1.6726× 10−24 g Gravitational constant G 6.6726× 10−8 dyne-cm2/g2 Planck constant h 6.6261× 10−27 erg-sec h̄ = h/2π 1.0546× 10−27 erg-sec Speed of light in vacuum c 2.9979× 1010 cm/sec Proton/electron mass mp/me 1.8362× 103 ratio Electron charge/mass e/me 5.2728× 1017 statcoul/g ratio Rydberg constant R∞ = 2π2me4 ch3 1.0974× 105 cm−1 Bohr radius a0 = h̄ 2/me2 5.2918× 10−9 cm Atomic cross section πa0 2 8.7974× 10−17 cm2 Classical electron radius re = e 2/mc2 2.8179× 10−13 cm Thomson cross section (8π/3)re 2 6.6525× 10−25 cm2 Compton wavelength of h/mec 2.4263× 10−10 cm electron h̄/mec 3.8616× 10−11 cm Fine-structure constant α = e2/h̄c 7.2974× 10−3 α−1 137.04 First radiation constant c1 = 2πhc 2 3.7418× 10−5 erg-cm2/sec Second radiation c2 = hc/k 1.4388 cm-deg (K) constant Stefan-Boltzmann σ 5.6705× 10−5 erg/cm2- constant sec-deg4 Wavelength associated λ0 1.2398× 10−4 cm with 1 eV 16 Physical Quantity Symbol Value Units Frequency associated ν0 2.4180× 1014 Hz with 1 eV Wave number associated k0 8.0655× 103 cm−1 with 1 eV Energy associated with 1.6022× 10−12 erg 1 eV Energy associated with 1.9864× 10−16 erg 1 cm−1 Energy associated with 13.606 eV 1 Rydberg Energy associated with 8.6174× 10−5 eV 1 deg Kelvin Temperature associated 1.1604× 104 deg (K) with 1 eV Avogadro number NA 6.0221× 1023 mol−1 Faraday constant F = NAe 2.8925× 1014 statcoul/mol Gas constant R = NAk 8.3145× 107 erg/deg-mol Loschmidt’s number n0 2.6868× 1019 cm−3 (no. density at STP) Atomic mass unit mu 1.6605× 10−24 g Standard temperature T0 273.15 deg (K) Atmospheric pressure p0 = n0kT0 1.0133× 106 dyne/cm2 Pressure of 1 mm Hg 1.3332× 103 dyne/cm2 (1 torr) Molar volume at STP V0 = RT0/p0 2.2414× 104 cm3 Molar weight of air Mair 28.971 g calorie (cal) 4.1868× 107 erg Gravitational g 980.67 cm/sec2 acceleration 17 ELECTRICITY AND MAGNETISM In the following,  = dielectric permittivity, µ = permeability of conduc- tor, µ′ = permeability of surrounding medium, σ = conductivity, f = ω/2π = radiation frequency, κm = µ/µ0 and κe = /0. Where subscripts are used, ‘1’ denotes a conducting medium and ‘2’ a propagating (lossless dielectric) medium. All units are SI unless otherwise specified. Permittivity of free space 0 = 8.8542× 10−12 F m−1 Permeability of free space µ0 = 4π × 10−7 H m−1 = 1.2566× 10−6 H m−1 Resistance of free space R0 = (µ0/0) 1/2 = 376.73 Ω Capacity of parallel plates of area C = A/d A, separated by distance d Capacity of concentric cylinders C = 2πl/ ln(b/a) of length l, radii a, b Capacity of concentric spheres of C = 4πab/(b− a) radii a, b Self-inductance of wire of length L = µl l, carrying uniform current Mutual inductance of parallel wires L = (µ′l/4π) [1 + 4 ln(d/a)] of length l, radius a, separated by distance d Inductance of circular loop of radius L = b { µ′ [ln(8b/a)− 2] + µ/4 } b, made of wire of radius a, carrying uniform current Relaxation time in a lossy medium τ = /σ Skin depth in a lossy medium δ = (2/ωµσ)1/2 = (πfµσ)−1/2 Wave impedance in a lossy medium Z = [µ/(+ iσ/ω)]1/2 Transmission coefficient at T = 4.22× 10−4(fκm1κe2/σ)1/2 conducting surface9 (good only for T  1) Field at distance r from straight wire Bθ = µI/2πr tesla carrying current I (amperes) = 0.2I/r gauss (r in cm) Field at distance z along axis from Bz = µa 2I/[2(a2 + z2)3/2] circular loop of radius a carrying current I 20 ELECTROMAGNETIC FREQUENCY/ WAVELENGTH BANDS10 Frequency Range Wavelength Range Designation Lower Upper Lower Upper ULF* 30 Hz 10 Mm VF* 30 Hz 300 Hz 1 Mm 10 Mm ELF 300 Hz 3 kHz 100 km 1 Mm VLF 3 kHz 30 kHz 10 km 100 km LF 30 kHz 300 kHz 1 km 10 km MF 300 kHz 3 MHz 100 m 1 km HF 3 MHz 30 MHz 10 m 100 m VHF 30 MHz 300 MHz 1 m 10 m UHF 300 MHz 3 GHz 10 cm 1 m SHF† 3 GHz 30 GHz 1 cm 10 cm S 2.6 3.95 7.6 11.5 G 3.95 5.85 5.1 7.6 J 5.3 8.2 3.7 5.7 H 7.05 10.0 3.0 4.25 X 8.2 12.4 2.4 3.7 M 10.0 15.0 2.0 3.0 P 12.4 18.0 1.67 2.4 K 18.0 26.5 1.1 1.67 R 26.5 40.0 0.75 1.1 EHF 30 GHz 300 GHz 1 mm 1 cm Submillimeter 300 GHz 3 THz 100µm 1 mm Infrared 3 THz 430 THz 700 nm 100µm Visible 430 THz 750 THz 400 nm 700 nm Ultraviolet 750 THz 30 PHz 10 nm 400 nm X Ray 30 PHz 3 EHz 100 pm 10 nm Gamma Ray 3 EHz 100 pm In spectroscopy the angstrom is sometimes used (1Å = 10−8 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 21 AC CIRCUITS For a resistance R, inductance L, and capacitance C in series with a voltage source V = V0 exp(iωt) (here i = √ −1), the current is given by I = dq/dt, where q satisfies L d2q dt2 + R dq dt + q C = V. Solutions are q(t) = qs + qt, I(t) = Is + It, where the steady state is Is = iωqs = V/Z in terms of the impedance Z = R+ i(ωL− 1/ωC) and It = dqt/dt. For initial conditions q(0) ≡ q0 = q̄0 + qs, I(0) ≡ I0, the transients can be of three types, depending on ∆ = R2 − 4L/C: (a) Overdamped, ∆ > 0 qt = I0 + γ+q̄0 γ+ − γ− exp(−γ−t)− I0 + γ−q̄0 γ+ − γ− exp(−γ+t), It = γ+(I0 + γ−q̄0) γ+ − γ− exp(−γ+t)− γ−(I0 + γ+q̄0) γ+ − γ− exp(−γ−t), where γ± = (R±∆1/2)/2L; (b) Critically damped, ∆ = 0 qt = [q̄0 + (I0 + γRq̄0)t] exp(−γRt), It = [I0 − (I0 + γRq̄0)γRt] exp(−γRt), where γR = R/2L; (c) Underdamped, ∆ < 0 qt = [ γRq̄0 + I0 ω1 sinω1t+ q̄0 cosω1t ] exp(−γRt), It = [ I0 cosω1t− (ω1 2 + γR 2)q̄0 + γRI0 ω1 sin(ω1t) ] exp(−γRt), Here ω1 = ω0(1 − R2C/4L)1/2, where ω0 = (LC)−1/2 is the resonant frequency. At ω = ω0, Z = R. The quality of the circuit is Q = ω0L/R. Instability results when L, R, C are not all of the same sign. 22 Nomenclature: B Magnetic induction Cs, c Speeds of sound, light cp Specific heat at constant pressure (units m 2 s−2 K−1) D = 2R Pipe diameter F Imposed force f Vibration frequency g Gravitational acceleration H,L Vertical, horizontal length scales k = ρcpκ Thermal conductivity (units kg m −1 s−2) N = (g/H)1/2 Brunt–Väisälä frequency R Radius of pipe or channel r Radius of curvature of pipe or channel rL Larmor radius T Temperature V Characteristic flow velocity VA = B/(µ0ρ) 1/2 Alfvén speed α Newton’s-law heat coefficient, k ∂T ∂x = α∆T β Volumetric expansion coefficient, dV/V = βdT Γ Bulk modulus (units kg m−1 s−2) ∆R,∆V,∆p,∆T Imposed differences in two radii, velocities, pressures, or temperatures  Surface emissivity η Electrical resistivity κ,D Thermal, molecular diffusivities (units m2 s−1) Λ Latitude of point on earth’s surface λ Collisional mean free path µ = ρν Viscosity µ0 Permeability of free space ν Kinematic viscosity (units m2 s−1) ρ Mass density of fluid medium ρ′ Mass density of bubble, droplet, or moving object Σ Surface tension (units kg s−2) σ Stefan–Boltzmann constant Ω Solid-body rotational angular velocity 25 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 = ρ̄Ū ≡ q; (2) ρU2 + p+ B 2⊥ /2µ = ρ̄Ū 2 + p̄+ B̄ 2⊥ /2µ; (3) ρUV − B‖B⊥/µ = ρ̄Ū V̄ − B̄‖B̄⊥/µ; (4) B‖ = B̄‖; (5) UB⊥ − V B‖ = ŪB̄⊥ − V̄ B̄‖; (6) 12 (U 2 + V 2) + w + (UB 2⊥ − V B‖B⊥)/µρU = 12 (Ū 2 + V̄ 2) + w̄ + (ŪB̄ 2⊥ − V̄ B̄‖B̄⊥)/µρ̄Ū . 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 units); and the specific enthalpy is w = e + pυ, where the specific internal energy e satisfies de = Tds − pdυ in terms of the temperature T and the specific entropy s. Quantities in the region behind (downstream from) the front are distinguished by a bar. If B = 0, then15 (7) U − Ū = [(p̄− p)(υ − ῡ)]1/2; (8) (p̄− p)(υ − ῡ)−1 = q2; (9) w̄ − w = 12 (p̄− p)(υ + ῡ); (10) ē− e = 12 (p̄+ p)(υ − ῡ). In what follows we assume that the fluid is a perfect gas with adiabatic index γ = 1 + 2/n, where n is the number of degrees of freedom. Then p = ρRT/m, where R is the universal gas constant and m is the molar weight; the sound speed is given by Cs 2 = (∂p/∂ρ)s = γpυ; and w = γe = γpυ/(γ − 1). For a general oblique shock in a perfect gas the quantity X = r−1(U/VA) 2 satisfies14 (11) (X−β/α)(X−cos2 θ)2 = X sin2 θ { [1 + (r − 1)/2α]X − cos2 θ } , where r = ρ̄/ρ, α = 12 [γ + 1− (γ − 1)r], and β = Cs 2/VA 2 = 4πγp/B2. The density ratio is bounded by (12) 1 < r < (γ + 1)/(γ − 1). If the shock is normal to B (i.e., if θ = π/2), then (13) U2 = (r/α) { Cs 2 + VA 2 [1 + (1− γ/2)(r − 1)] } ; (14) U/Ū = B̄/B = r; 26 (15) V̄ = V ; (16) p̄ = p+ (1− r−1)ρU2 + (1− r2)B2/2µ. If θ = 0, there are two possibilities: switch-on shocks, which require β < 1 and for which (17) U2 = rVA 2; (18) Ū = VA 2/U ; (19) B̄ 2⊥ = 2B 2 ‖ (r − 1)(α− β); (20) V̄ = ŪB̄⊥/B‖; (21) p̄ = p+ ρU2(1− α+ β)(1− r−1), and acoustic (hydrodynamic) shocks, for which (22) U2 = (r/α)Cs 2; (23) Ū = U/r; (24) V̄ = B̄⊥ = 0; (25) p̄ = p+ ρU2(1− r−1). For acoustic shocks the specific volume and pressure are related by (26) ῡ/υ = [(γ + 1)p+ (γ − 1)p̄] / [(γ − 1)p+ (γ + 1)p̄]. In terms of the upstream Mach number M = U/Cs, (27) ρ̄/ρ = υ/ῡ = U/Ū = (γ + 1)M2/[(γ − 1)M2 + 2]; (28) p̄/p = (2γM2 − γ + 1)/(γ + 1); (29) T̄ /T = [(γ − 1)M2 + 2](2γM2 − γ + 1)/(γ + 1)2M2; (30) M̄2 = [(γ − 1)M2 + 2]/[2γM2 − γ + 1]. The entropy change across the shock is (31) ∆s ≡ s̄− s = cυ ln[(p̄/p)(ρ/ρ̄)γ ], where cυ = R/(γ − 1)m is the specific heat at constant volume; here R is the gas constant. In the weak-shock limit (M → 1), (32) ∆s→ cυ 2γ(γ − 1) 3(γ + 1) (M 2 − 1)3 ≈ 16γR 3(γ + 1)m (M − 1)3. The radius at time t of a strong spherical blast wave resulting from the explo- sive release of energy E in a medium with uniform density ρ is (33) RS = C0(Et 2/ρ)1/5, where C0 is a constant depending on γ. For γ = 7/5, C0 = 1.033. 27 PLASMA DISPERSION FUNCTION Definition16 (first form valid only for Im ζ > 0): Z(ζ) = π −1/2 ∫ +∞ −∞ dt exp ( −t2 ) t− ζ = 2i exp ( −ζ2 )∫ iζ −∞ dt exp ( −t2 ) . Physically, ζ = x+ iy is the ratio of wave phase velocity to thermal velocity. Differential equation: dZ dζ = −2 (1 + ζZ) , Z(0) = iπ1/2; d2Z dζ2 + 2ζ dZ dζ + 2Z = 0. Real argument (y = 0): Z(x) = exp ( −x2 )( iπ 1/2 − 2 ∫ x 0 dt exp ( t 2 )) . Imaginary argument (x = 0): Z(iy) = iπ 1/2 exp ( y 2 ) [1− erf(y)] . Power series (small argument): Z(ζ) = iπ 1/2 exp ( −ζ2 ) − 2ζ ( 1− 2ζ2/3 + 4ζ4/15− 8ζ6/105 + · · · ) . Asymptotic series, |ζ|  1 (Ref. 17): Z(ζ) = iπ 1/2 σ exp ( −ζ2 ) − ζ−1 ( 1 + 1/2ζ 2 + 3/4ζ 4 + 15/8ζ 6 + · · · ) , where σ = { 0 y > |x|−1 1 |y| < |x|−1 2 y < −|x|−1 Symmetry properties (the asterisk denotes complex conjugation): Z(ζ*) = − [Z(−ζ)]*; Z(ζ*) = [Z(ζ)] * + 2iπ 1/2 exp[−(ζ*)2] (y > 0). Two-pole approximations18 (good for ζ in upper half plane except when y < π1/2x2 exp(−x2), x 1): Z(ζ) ≈ 0.50 + 0.81i a− ζ − 0.50− 0.81i a* + ζ , a = 0.51− 0.81i; Z ′ (ζ) ≈ 0.50 + 0.96i (b− ζ)2 + 0.50− 0.96i (b* + ζ)2 , b = 0.48− 0.91i. 30 COLLISIONS AND TRANSPORT Temperatures are in eV; the corresponding value of Boltzmann’s constant is k = 1.60 × 10−12 erg/eV; masses µ, µ′ are in units of the proton mass; eα = Zαe is the charge of species α. All other units are cgs except where noted. Relaxation Rates Rates are associated with four relaxation processes arising from the in- teraction of test particles (labeled α) streaming with velocity vα through a background of field particles (labeled β): slowing down dvα dt = −να\βs vα transverse diffusion d dt (vα − v̄α)2⊥ = ν α\β ⊥ vα 2 parallel diffusion d dt (vα − v̄α)2‖ = ν α\β ‖ vα 2 energy loss d dt vα 2 = −να\β vα 2 , where the averages are performed over an ensemble of test particles and a Maxwellian field particle distribution. The exact formulas may be written19 ν α\β s = (1 +mα/mβ)ψ(x α\β )ν α\β 0 ; ν α\β ⊥ = 2 [ (1− 1/2xα\β)ψ(xα\β) + ψ′(xα\β) ] ν α\β 0 ; ν α\β ‖ = [ ψ(x α\β )/x α\β ] ν α\β 0 ; ν α\β  = 2 [ (mα/mβ)ψ(x α\β )− ψ′(xα\β) ] ν α\β 0 , where ν α\β 0 = 4πeα 2 eβ 2 λαβnβ/mα 2 vα 3 ; x α\β = mβvα 2 /2kTβ ; ψ(x) = 2 √ π ∫ x 0 dt t 1/2 e −t ; ψ ′ (x) = dψ dx , and λαβ = ln Λαβ is the Coulomb logarithm (see below). Limiting forms of νs, ν⊥ and ν‖ are given in the following table. All the expressions shown have units cm3 sec−1. Test particle energy  and field particle temperature T 31 are both in eV; µ = mi/mp where mp is the proton mass; Z is ion charge state; in electron–electron and ion–ion encounters, field particle quantities are distinguished by a prime. The two expressions given below for each rate hold for very slow (xα\β  1) and very fast (xα\β  1) test particles, respectively. Slow Fast Electron–electron ν e\e′ s /ne′λee′ ≈ 5.8× 10 −6 T −3/2 −→ 7.7× 10−6−3/2 ν e\e′ ⊥ /ne′λee′ ≈ 5.8× 10 −6 T −1/2  −1 −→ 7.7× 10−6−3/2 ν e\e′ ‖ /ne′λee′ ≈ 2.9× 10 −6 T −1/2  −1 −→ 3.9× 10−6T−5/2 Electron–ion ν e\i s /niZ 2 λei ≈ 0.23µ3/2T−3/2 −→ 3.9× 10−6−3/2 ν e\i ⊥ /niZ 2 λei ≈ 2.5× 10−4µ1/2T−1/2−1−→ 7.7× 10−6−3/2 ν e\i ‖ /niZ 2 λei ≈ 1.2× 10−4µ1/2T−1/2−1−→ 2.1× 10−9µ−1T−5/2 Ion–electron ν i\e s /neZ 2 λie ≈ 1.6× 10−9µ−1T−3/2 −→ 1.7× 10−4µ1/2−3/2 ν i\e ⊥ /neZ 2 λie ≈ 3.2× 10−9µ−1T−1/2−1 −→ 1.8× 10−7µ−1/2−3/2 ν i\e ‖ /neZ 2 λie ≈ 1.6× 10−9µ−1T−1/2−1 −→ 1.7× 10−4µ1/2T−5/2 Ion–ion νi\i ′ s ni′Z 2Z′2λii′ ≈ 6.8× 10−8 µ′1/2 µ ( 1 + µ′ µ )−1/2 T −3/2 −→ 9.0× 10−8 ( 1 µ + 1 µ′ ) µ1/2 3/2 ν i\i′ ⊥ ni′Z 2Z′2λii′ ≈ 1.4× 10−7µ′1/2µ−1T−1/2−1 −→ 1.8× 10−7µ−1/2−3/2 ν i\i′ ‖ ni′Z 2Z′2λii′ ≈ 6.8× 10−8µ′1/2µ−1T−1/2−1 −→ 9.0× 10−8µ1/2µ′−1T−5/2 In the same limits, the energy transfer rate follows from the identity ν = 2νs − ν⊥ − ν‖, except for the case of fast electrons or fast ions scattered by ions, where the leading terms cancel. Then the appropriate forms are ν e\i  −→ 4.2× 10 −9 niZ 2 λei[  −3/2 µ −1 − 8.9× 104(µ/T )1/2−1 exp(−1836µ/T ) ] sec −1 32 (c) Mixed ion–ion collisions λii′ = λi′i = 23− ln [ ZZ′(µ+ µ′) µTi′ + µ ′Ti ( niZ 2 Ti + ni′Z ′2 Ti′ )1/2] . (d) Counterstreaming ions (relative velocity vD = βDc) in the presence of warm electrons, kTi/mi, kTi′/mi′ < vD 2 < kTe/me λii′ = λi′i = 35− ln [ ZZ′(µ+ µ′) µµ′βD2 ( ne Te )1/2] . Fokker-Planck Equation Dfα Dt ≡ ∂fα ∂t + v · ∇fα + F · ∇vfα = ( ∂fα ∂t ) coll , where F is an external force field. The general form of the collision integral is (∂fα/∂t)coll = − ∑ β ∇v · Jα\β , with J α\β = 2πλαβ eα 2eβ 2 mα ∫ d 3 v ′ (u 2I− uu)u−3 · { 1 mβ f α (v)∇v′f β (v ′ )− 1 mα f β (v ′ )∇vfα(v) } (Landau form) where u = v′ − v and I is the unit dyad, or alternatively, J α\β = 4πλαβ eα 2eβ 2 mα2 { f α (v)∇vH(v)− 1 2 ∇v · [ f α (v)∇v∇vG(v) ]} , where the Rosenbluth potentials are G(v) = ∫ f β (v ′ )ud 3 v ′ H(v) = ( 1 + mα mβ )∫ f β (v ′ )u −1 d 3 v ′ . 35 If species α is a weak beam (number and energy density small compared with background) streaming through a Maxwellian plasma, then J α\β =− mα mα +mβ ν α\β s vf α − 1 2 ν α\β ‖ vv · ∇vf α − 1 4 ν α\β ⊥ ( v 2I− vv ) · ∇vfα. B-G-K Collision Operator For distribution functions with no large gradients in velocity space, the Fokker-Planck collision terms can be approximated according to Dfe Dt = νee(Fe − fe) + νei(F̄e − fe); Dfi Dt = νie(F̄i − fi) + νii(Fi − fi). The respective slowing-down rates να\βs given in the Relaxation Rate section above can be used for ναβ , assuming slow ions and fast electrons, with  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 F̄α are given by Fα = nα ( mα 2πkTα )3/2 exp { − [ mα(v − vα)2 2kTα ]} ; F̄α = nα ( mα 2πkT̄α )3/2 exp { − [ mα(v − v̄α)2 2kT̄α ]} , where nα, vα and Tα are the number density, mean drift velocity, and effective temperature obtained by taking moments of fα. Some latitude in the definition of T̄α and v̄α is possible; 20 one choice is T̄e = Ti, T̄i = Te, v̄e = vi, v̄i = ve. Transport Coefficients Transport equations for a multispecies plasma: dαnα dt + nα∇ · vα = 0; mαnα dαvα dt = −∇pα −∇ · Pα + Zαenα [ E + 1 c vα ×B ] + Rα; 36 3 2 nα dαkTα dt + pα∇ · vα = −∇ · qα − Pα : ∇vα +Qα. Here dα/dt ≡ ∂/∂t + vα · ∇; pα = nαkTα, where k is Boltzmann’s constant; Rα = ∑ β Rαβ and Qα = ∑ β Qαβ , where Rαβ and Qαβ are respectively the momentum and energy gained by the αth species through collisions with the βth; Pα is the stress tensor; and qα is the heat flow. The transport coefficients in a simple two-component plasma (electrons and singly charged ions) are tabulated below. Here ‖ and ⊥ refer to the di- rection of the magnetic field B = bB; u = ve − vi is the relative streaming velocity; ne = ni ≡ n; j = −neu is the current; ωce = 1.76× 107B sec−1 and ωci = (me/mi)ωce are the electron and ion gyrofrequencies, respectively; and the basic collisional times are taken to be τe = 3 √ me(kTe) 3/2 4 √ 2π nλe4 = 3.44× 105 Te 3/2 nλ sec, where λ is the Coulomb logarithm, and τi = 3 √ mi(kTi) 3/2 4 √ πnλe4 = 2.09× 107 Ti 3/2 nλ µ 1/2 sec. In the limit of large fields (ωcατα  1, α = i, e) the transport processes may be summarized as follows:21 momentum transfer Rei= −Rie ≡ R = Ru + RT ; frictional force Ru = ne(j‖/σ‖ + j⊥/σ⊥); electrical σ‖ = 1.96σ⊥; σ⊥ = ne 2 τe/me; conductivities thermal force RT = −0.71n∇‖(kTe)− 3n 2ωceτe b×∇⊥(kTe); ion heating Qi = 3me mi nk τe (Te − Ti); electron heating Qe = −Qi −R · u; ion heat flux qi = −κi‖∇‖(kTi)− κ i ⊥∇⊥(kTi) + κ i ∧b×∇⊥(kTi); ion thermal κ i ‖ = 3.9 nkTiτi mi ; κ i ⊥ = 2nkTi miω 2ci τi ; κ i ∧ = 5nkTi 2miωci ; conductivities electron heat flux qe = q e u + q e T ; frictional heat flux q e u = 0.71nkTeu‖ + 3nkTe 2ωceτe b× u⊥; 37 APPROXIMATE MAGNITUDES IN SOME TYPICAL PLASMAS Plasma Type n cm−3 T eV ωpe sec −1 λD cm nλD 3 νei sec −1 Interstellar gas 1 1 6× 104 7× 102 4× 108 7× 10−5 Gaseous nebula 103 1 2× 106 20 107 6× 10−2 Solar Corona 109 102 2× 109 2× 10−1 8× 106 60 Diffuse hot plasma 1012 102 6× 1010 7× 10−3 4× 105 40 Solar atmosphere, 1014 1 6× 1011 7× 10−5 40 2× 109 gas discharge Warm plasma 1014 10 6× 1011 2× 10−4 103 107 Hot plasma 1014 102 6× 1011 7× 10−4 4× 104 4× 106 Thermonuclear 1015 104 2× 1012 2× 10−3 107 5× 104 plasma Theta pinch 1016 102 6× 1012 7× 10−5 4× 103 3× 108 Dense hot plasma 1018 102 6× 1013 7× 10−6 4× 102 2× 1010 Laser Plasma 1020 102 6× 1014 7× 10−7 40 2× 1012 The diagram (facing) gives comparable information in graphical form.22 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 Number density (m−3) 1.5× 1010–3.0× 1010 5× 1010–2× 1011 Height-integrated number 9× 1014 4.5× 1015 density (m−2) Ion-neutral collision 2× 103–102 0.5–0.05 frequency (sec−1) Ion gyro-/collision 0.09–2.0 4.6× 102–5.0× 103 frequency ratio κi Ion Pederson factor 0.09–0.5 2.2× 10−3–2× 10−4 κi/(1 + κi 2) Ion Hall factor 8× 10−4–0.8 1.0 κi 2/(1 + κi 2) 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 Electron Pedersen factor 2.7× 10−3–1.5× 10−4 10−5–1.5× 10−6 κe/(1 + κe 2) Electron Hall factor 1.0 1.0 κe 2/(1 + κe 2) Mean molecular weight 28–26 22–16 Ion gyrofrequency (sec−1) 180–190 230–300 Neutral diffusion 30–5× 103 105 coefficient (m2 sec−1) The terrestrial magnetic field in the lower ionosphere at equatorial latti- tudes is approximately B0 = 0.35×10−4 tesla. The earth’s radius is RE = 6371 km. 42 SOLAR PHYSICS PARAMETERS24 Parameter Symbol Value Units Total mass M 1.99× 1033 g Radius R 6.96× 1010 cm Surface gravity g 2.74× 104 cm s−2 Escape speed v∞ 6.18× 107 cm s−1 Upward mass flux in spicules — 1.6× 10−9 g cm−2 s−1 Vertically integrated atmospheric density — 4.28 g cm−2 Sunspot magnetic field strength Bmax 2500–3500 G Surface effective temperature T0 5770 K Radiant power L 3.83× 1033 erg s−1 Radiant flux density F 6.28× 1010 erg cm−2s−1 Optical depth at 500 nm, measured τ5 0.99 — from photosphere Astronomical unit (radius of earth’s orbit) AU 1.50× 1013 cm Solar constant (intensity at 1 AU) f 1.36× 106 erg cm−2 s−1 Chromosphere and Corona25 Quiet Coronal Active Parameter (Units) Sun Hole Region Chromospheric radiation losses (erg cm−2 s−1) Low chromosphere 2× 106 2× 106 >∼ 10 7 Middle chromosphere 2× 106 2× 106 107 Upper chromosphere 3× 105 3× 105 2× 106 Total 4× 106 4× 106 >∼ 2× 10 7 Transition layer pressure (dyne cm−2) 0.2 0.07 2 Coronal temperature (K) at 1.1 R 1.1–1.6× 106 106 2.5× 106 Coronal energy losses (erg cm−2 s−1) Conduction 2× 105 6× 104 105–107 Radiation 105 104 5× 106 Solar Wind <∼ 5× 10 4 7× 105 < 105 Total 3× 105 8× 105 107 Solar wind mass loss (g cm−2 s−1) <∼ 2× 10 −11 2× 10−10 < 4× 10−11 43 RELATIVISTIC ELECTRON BEAMS Here γ = (1 − β2)−1/2 is the relativistic scaling factor; quantities in analytic formulas are expressed in SI or cgs units, as indicated; in numerical formulas, I is in amperes (A), B is in gauss (G), electron linear density N is in cm−1, and temperature, voltage and energy are in MeV; βz = vz/c; k is Boltzmann’s constant. Relativistic electron gyroradius: re = mc2 eB (γ 2 − 1)1/2 (cgs) = 1.70× 103(γ2 − 1)1/2B−1 cm. Relativistic electron energy: W = mc 2 γ = 0.511γ MeV. Bennett pinch condition: I 2 = 2Nk(Te + Ti)c 2 (cgs) = 3.20× 10−4N(Te + Ti) A2. Alfvén-Lawson limit: IA = (mc 3 /e)βzγ (cgs) = (4πmc/µ0e)βzγ (SI) = 1.70× 104βzγ A. The ratio of net current to IA is I IA = ν γ . Here ν = Nre is the Budker number, where re = e 2/mc2 = 2.82 × 10−13 cm is the classical electron radius. Beam electron number density is nb = 2.08× 108Jβ−1 cm−3, where J is the current density in A cm−2. For a uniform beam of radius a (in cm), nb = 6.63× 107Ia−2β−1 cm−3, and 2re a = ν γ . 46 Child’s law: (non-relativistic) space-charge-limited current density between parallel plates with voltage drop V (in MV) and separation d (in cm) is J = 2.34× 103V 3/2d−2 A cm−2. The saturated parapotential current (magnetically self-limited flow along equi- potentials in pinched diodes and transmission lines) is29 Ip = 8.5× 103Gγ ln [ γ + (γ 2 − 1)1/2 ] A, where G is a geometrical factor depending on the diode structure: G = w 2πd for parallel plane cathode and anode of width w, separation d; G = ( ln R2 R1 )−1 for cylinders of radii R1 (inner) and R2 (outer); G = Rc d0 for conical cathode of radius Rc, maximum separation d0 (at r = Rc) from plane anode. For β → 0 (γ → 1), both IA and Ip vanish. The condition for a longitudinal magnetic field Bz to suppress filamentation in a beam of current density J (in A cm−2) is Bz > 47βz(γJ) 1/2 G. Voltage registered by Rogowski coil of minor cross-sectional area A, n turns, major radius a, inductance L, external resistance R and capacitance C (all in SI): externally integrated V = (1/RC)(nAµ0I/2πa); self-integrating V = (R/L)(nAµ0I/2πa) = RI/n. X-ray production, target with average atomic number Z (V <∼ 5 MeV): η ≡ x-ray power/beam power = 7× 10−4ZV. X-ray dose at 1 meter generated by an e-beam depositing total charge Q coulombs while V ≥ 0.84Vmax in material with charge state Z: D = 150V 2.8 max QZ 1/2 rads. 47 BEAM INSTABILITIES30 Name Conditions Saturation Mechanism Electron- Vd > V̄ej , j = 1, 2 Electron trapping until electron V̄ej ∼ Vd Buneman Vd > (M/m) 1/3V̄i, Electron trapping until Vd > V̄e V̄e ∼ Vd Beam-plasma Vb > (np/nb) 1/3V̄b Trapping of beam electrons Weak beam- Vb < (np/nb) 1/3V̄b Quasilinear or nonlinear plasma (mode coupling) Beam-plasma V̄e > Vb > V̄b Quasilinear or nonlinear (hot-electron) Ion acoustic Te  Ti, Vd  Cs Quasilinear, ion tail form- ation, nonlinear scattering, or resonance broadening. Anisotropic Te⊥ > 2Te‖ Isotropization temperature (hydro) Ion cyclotron Vd > 20V̄i (for Ion heating Te ≈ Ti) Beam-cyclotron Vd > Cs Resonance broadening (hydro) Modified two- Vd < (1 + β) 1/2VA, Trapping stream (hydro) Vd > Cs Ion-ion (equal U < 2(1 + β)1/2VA Ion trapping beams) Ion-ion (equal U < 2Cs Ion trapping beams) For nomenclature, see p. 50. 48 Formulas An e-m wave with k ‖ B has an index of refraction given by n± = [1− ω 2pe/ω(ω ∓ ωce)] 1/2 , where ± refers to the helicity. The rate of change of polarization angle θ as a function of displacement s (Faraday rotation) is given by dθ/ds = (k/2)(n− − n+) = 2.36× 104NBf−2 cm−1, where N is the electron number density, B is the field strength, and f is the wave frequency, all in cgs. The quiver velocity of an electron in an e-m field of angular frequency ω is v0 = eEmax/mω = 25.6I 1/2 λ0 cm sec −1 in terms of the laser flux I = cE 2max/8π, with I in watt/cm 2, laser wavelength λ0 in µm. The ratio of quiver energy to thermal energy is Wqu/Wth = mev0 2 /2kT = 1.81× 10−13λ02I/T, where T is given in eV. For example, if I = 1015 W cm−2, λ0 = 1µm, T = 2 keV, then Wqu/Wth ≈ 0.1. Pondermotive force: F = N∇〈E2〉/8πNc, where Nc = 1.1× 1021λ0−2cm−3. For uniform illumination of a lens with f-number F , the diameter d at focus (85% of the energy) and the depth of focus l (distance to first zero in intensity) are given by d ≈ 2.44Fλθ/θDL and l ≈ ±2F 2λθ/θDL. Here θ is the beam divergence containing 85% of energy and θDL is the diffraction-limited divergence: θDL = 2.44λ/b, where b is the aperture. These formulas are modified for nonuniform (such as Gaussian) illumination of the lens or for pathological laser profiles. 51 ATOMIC PHYSICS AND RADIATION Energies and temperatures are in eV; all other units are cgs except where noted. Z is the charge state (Z = 0 refers to a neutral atom); the subscript e labels electrons. N refers to number density, n to principal quantum number. Asterisk superscripts on level population densities denote local thermodynamic equilibrium (LTE) values. Thus Nn* is the LTE number density of atoms (or ions) in level n. Characteristic atomic collision cross section: (1) πa0 2 = 8.80× 10−17 cm2. Binding energy of outer electron in level labelled by quantum numbers n, l: (2) E Z ∞(n, l) = − Z2EH∞ (n−∆l)2 , where EH∞ = 13.6 eV is the hydrogen ionization energy and ∆l = 0.75l −5, l >∼ 5, is the quantum defect. Excitation and Decay Cross section (Bethe approximation) for electron excitation by dipole allowed transition m→ n (Refs. 32, 33): (3) σmn = 2.36× 10−13 fnmg(n,m) ∆Enm cm 2 , where fnm is the oscillator strength, g(n,m) is the Gaunt factor,  is the incident electron energy, and ∆Enm = En − Em. Electron excitation rate averaged over Maxwellian velocity distribution, Xmn = Ne〈σmnv〉 (Refs. 34, 35): (4) Xmn = 1.6× 10−5 fnm〈g(n,m)〉Ne ∆EnmT 1/2 e exp ( − ∆Enm Te ) sec −1 , where 〈g(n,m)〉 denotes the thermal averaged Gaunt factor (generally ∼ 1 for atoms, ∼ 0.2 for ions). 52 Rate for electron collisional deexcitation: (5) Ynm = (Nm*/Nn*)Xmn. Here Nm*/Nn* = (gm/gn) exp(∆Enm/Te) is the Boltzmann relation for level population densities, where gn is the statistical weight of level n. Rate for spontaneous decay n→ m (Einstein A coefficient)34 (6) Anm = 4.3× 107(gm/gn)fmn(∆Enm)2 sec−1. Intensity emitted per unit volume from the transition n → m in an optically thin plasma: (7) Inm = 1.6× 10−19AnmNn∆Enm watt/cm3. Condition for steady state in a corona model: (8) N0Ne〈σ0nv〉 = NnAn0, where the ground state is labelled by a zero subscript. Hence for a transition n→ m in ions, where 〈g(n, 0)〉 ≈ 0.2, (9) Inm = 5.1× 10−25 fnmg0NeN0 gmT 1/2 e ( ∆Enm ∆En0 )3 exp ( − ∆En0 Te ) watt cm3 . Ionization and Recombination In a general time-dependent situation the number density of the charge state Z satisfies (10) dN(Z) dt = Ne [ − S(Z)N(Z)− α(Z)N(Z) +S(Z − 1)N(Z − 1) + α(Z + 1)N(Z + 1) ] . Here S(Z) 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. 53 Radiation N. B. Energies and temperatures are in eV; all other quantities are in cgs units except where noted. Z is the charge state (Z = 0 refers to a neutral atom); the subscript e labels electrons. N is number density. Average radiative decay rate of a state with principal quantum number n is (23) An = ∑ m<n Anm = 1.6× 1010Z4n−9/2 sec. Natural linewidth (∆E in eV): (24) ∆E∆t = h = 4.14× 10−15 eV sec, where ∆t is the lifetime of the line. Doppler width: (25) ∆λ/λ = 7.7× 10−5(T/µ)1/2, where µ is the mass of the emitting atom or ion scaled by the proton mass. Optical depth for a Doppler-broadened line:39 (26) τ = 3.52× 10−13fnmλ(Mc2/kT )1/2NL = 5.4× 10−9λ(µ/T )1/2NL, where fnm is the absorption oscillator strength, λ is the wavelength, and L is the physical depth of the plasma; M , N , and T are the mass, number density, and temperature of the absorber; µ is M divided by the proton mass. Optically thin means τ < 1. Resonance absorption cross section at center of line: (27) σλ=λc = 5.6× 10 −13 λ 2 /∆λ cm 2 . Wien displacement law (wavelength of maximum black-body emission): (28) λmax = 2.50× 10−5T−1 cm. Radiation from the surface of a black body at temperature T : (29) W = 1.03× 105T 4 watt/cm2. 56 Bremsstrahlung from hydrogen-like plasma:26 (30) PBr = 1.69× 10−32NeTe1/2 ∑[ Z 2 N(Z) ] watt/cm 3 , where the sum is over all ionization states Z. Bremsstrahlung optical depth:41 (31) τ = 5.0× 10−38NeNiZ2gLT−7/2, where g ≈ 1.2 is an average Gaunt factor and L is the physical path length. Inverse bremsstrahlung absorption coefficient42 for radiation of angular fre- quency ω: (32) κ = 3.1× 10−7Zne2 ln ΛT−3/2ω−2(1− ω2p/ω 2 ) −1/2 cm −1 ; here Λ is the electron thermal velocity divided by V , where V is the larger of ω and ωp multiplied by the larger of Ze 2/kT and h̄/(mkT )1/2. Recombination (free-bound) radiation: (33) Pr = 1.69× 10−32NeTe1/2 ∑[ Z 2 N(Z) ( EZ−1∞ Te )] watt/cm 3 . Cyclotron radiation26 in magnetic field B: (34) Pc = 6.21× 10−28B2NeTe watt/cm3. For NekTe = NikTi = B 2/16π (β = 1, isothermal plasma),26 (35) Pc = 5.00× 10−38N2eT 2 e watt/cm 3 . Cyclotron radiation energy loss e-folding time for a single electron:41 (36) tc ≈ 9.0× 108B−2 2.5 + γ sec, where γ is the kinetic plus rest energy divided by the rest energy mc2. Number of cyclotron harmonics41 trapped in a medium of finite depth L: (37) mtr = (57βBL) 1/6 , where β = 8πNkT/B2. Line radiation is given by summing Eq. (9) over all species in the plasma. 57 ATOMIC SPECTROSCOPY Spectroscopic notation combines observational and theoretical elements. Observationally, spectral lines are grouped in series with line spacings which decrease toward the series limit. Every line can be related theoretically to a transition between two atomic states, each identified by its quantum numbers. Ionization levels are indicated by roman numerals. Thus C I is unionized carbon, C II is singly ionized, etc. The state of a one-electron atom (hydrogen) or ion (He II, Li III, etc.) is specified by identifying the principal quantum number n = 1, 2, . . . , the orbital angular momentum l = 0, 1, . . . , n − 1, and the spin angular momentum s = ± 12 . The total angular momentum j is the magnitude of the vector sum of l and s, j = l ± 12 (j ≥ 1 2 ). The letters s, p, d, f, g, h, i, k, l, . . . , respectively, are associated with angular momenta l = 0, 1, 2, 3, 4, 5, 6, 7, 8, . . . . The atomic states of hydrogen and hydrogenic ions are degenerate: neglecting fine structure, their energies depend only on n according to En = − R∞hcZ 2n−2 1 +m/M = − RyZ2 n2 , where h is Planck’s constant, c is the velocity of light, m is the electron mass, M and Z are the mass and charge state of the nucleus, and R∞ = 109, 737 cm −1 is the Rydberg constant. If En is divided by hc, the result is in wavenumber units. The energy associated with a transition m→ n is given by ∆Emn = Ry(1/m 2 − 1/n2), with m < n (m > n) for absorption (emission) lines. For hydrogen and hydrogenic ions the series of lines belonging to the transitions m→ n have conventional names: Transition 1→ n 2→ n 3→ n 4→ n 5→ n 6→ n Name Lyman Balmer Paschen Brackett Pfund Humphreys Successive lines in any series are denoted α, β, γ, etc. Thus the transition 1→ 3 gives rise to the Lyman-β line. Relativistic effects, quantum electrodynamic effects (e.g., the Lamb shift), and interactions between the nuclear magnetic 58 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. 2. H. W. Gould, “Note on Some Binomial Coefficient Identities of Rosen- baum,” J. Math. Phys. 10, 49 (1969); H. W. Gould and J. Kaucky, “Eval- uation of a Class of Binomial Coefficient Summations,” J. Comb. Theory 1, 233 (1966). 3. B. S. Newberger, “New Sum Rule for Products of Bessel Functions with Application to Plasma Physics,” J. Math. Phys. 23, 1278 (1982); 24, 2250 (1983). 4. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw- Hill Book Co., New York, 1953), Vol. I, pp. 47–52 and pp. 656–666. 61 5. W. D. Hayes, “A Collection of Vector Formulas,” Princeton University, Princeton, NJ, 1956 (unpublished), and personal communication (1977). 6. See Quantities, Units and Symbols, report of the Symbols Committee of the Royal Society, 2nd edition (Royal Society, London, 1975) for a discussion of nomenclature in SI units. 7. E. R. Cohen and B. N. Taylor, “The 1986 Adjustment of the Fundamental Physical Constants,” CODATA Bulletin No. 63 (Pergamon Press, New York, 1986); J. Res. Natl. Bur. Stand. 92, 85 (1987); J. Phys. Chem. Ref. Data 17, 1795 (1988). 8. E. S. Weibel, “Dimensionally Correct Transformations between Different Systems of Units,” Amer. J. Phys. 36, 1130 (1968). 9. J. Stratton, Electromagnetic Theory (McGraw-Hill Book Co., New York, 1941), p. 508. 10. Reference Data for Engineers: Radio, Electronics, Computer, and Com- munication, 7th edition, E. C. Jordan, Ed. (Sams and Co., Indianapolis, IN, 1985), Chapt. 1. These definitions are International Telecommunica- tions Union (ITU) Standards. 11. H. E. Thomas, Handbook of Microwave Techniques and Equipment (Prentice-Hall, Englewood Cliffs, NJ, 1972), p. 9. Further subdivisions are defined in Ref. 10, p. I–3. 12. J. P. Catchpole and G. Fulford, Ind. and Eng. Chem. 58, 47 (1966); reprinted in recent editions of the Handbook of Chemistry and Physics (Chemical Rubber Co., Cleveland, OH) on pp. F306–323. 13. W. D. Hayes, “The Basic Theory of Gasdynamic Discontinuities,” in Fun- damentals of Gas Dynamics, Vol. III, High Speed Aerodynamics and Jet Propulsion, H. W. Emmons, Ed. (Princeton University Press, Princeton, NJ, 1958). 14. W. B. Thompson, An Introduction to Plasma Physics (Addison-Wesley Publishing Co., Reading, MA, 1962), pp. 86–95. 15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd edition (Addison- Wesley Publishing Co., Reading, MA, 1987), pp. 320–336. 16. The Z function is tabulated in B. D. Fried and S. D. Conte, The Plasma Dispersion Function (Academic Press, New York, 1961). 17. R. W. Landau and S. Cuperman, “Stability of Anisotropic Plasmas to Almost-Perpendicular Magnetosonic Waves,” J. Plasma Phys. 6, 495 (1971). 62 18. B. D. Fried, C. L. Hedrick, J. McCune, “Two-Pole Approximation for the Plasma Dispersion Function,” Phys. Fluids 11, 249 (1968). 19. B. A. Trubnikov, “Particle Interactions in a Fully Ionized Plasma,” Re- views of Plasma Physics, Vol. 1 (Consultants Bureau, New York, 1965), p. 105. 20. J. M. Greene, “Improved Bhatnagar–Gross–Krook Model of Electron-Ion Collisions,” Phys. Fluids 16, 2022 (1973). 21. S. I. Braginskii, “Transport Processes in a Plasma,” Reviews of Plasma Physics, Vol. 1 (Consultants Bureau, New York, 1965), p. 205. 22. J. Sheffield, Plasma Scattering of Electromagnetic Radiation (Academic Press, New York, 1975), p. 6 (after J. W. Paul). 23. K. H. Lloyd and G. Härendel, “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. 31. Table prepared from data compiled by J. M. McMahon (personal commu- nication, 1990). 63