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NRL PLASMA FORMULARY Supported by the Office of Naval 2013 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 NumericalandAlgebraic . 3 VectorIdentities . 4 Differential Operators in Curvilinear Coordinates . .... 6 DimensionsandUnits . .10 International System (SI) Nomenclature . 13 MetricPrefixes . .13 Physical Constants (SI) . 14 Physical Constants (cgs) . 16 FormulaConversion . .18 Maxwell’s Equations . 19 ElectricityandMagnetism . .20 Electromagnetic Frequency/Wavelength Bands . 21 ACCircuits . ... ... .. ... ... ... .. ... ... .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 IonosphericParameters . .42 Solar Physics Parameters . 43 ThermonuclearFusion . .44 Relativistic Electron Beams . 46 BeamInstabilities . .48 Lasers .............................51 Atomic Physics and Radiation . 53 AtomicSpectroscopy . .59 Complex (Dusty) Plasmas . 62 References . .66 Afterword ............................71 2 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−∞ 3 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 ∇ · ∇ · ∇· 4 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 5 (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 6 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 7 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 8 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 9 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 10 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 × 11 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 12 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
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