Naval Research Laboratory Washington, DC 20375-5320 NRL/PU/6790--18-640 NRL Plasma Formulary Est. 1966 52 Years 2018 Approved for public release; distribution is unlimited. 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