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

2004 REVISED NRL PLASMA FORMULARY J.D. Huba Beam Physics Branch Plasma Physics Division Naval Research Laboratory Washington, DC 20375 Supported by The Office of Naval Research 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 T E X files for the Formulary and his continued suggestions for improvement. The Formulary has been set in T E X 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 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . 5 Differential Operators in Curvilinear Coordinates . . . . . . . . . . . 7 Dimensions and Units . . . . . . . . . . . . . . . . . . . . . . . 11 International System (SI) Nomenclature . . . . . . . . . . . . . . . 14 Metric Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Physical Constants (SI) . . . . . . . . . . . . . . . . . . . . . . 15 Physical Constants (cgs) . . . . . . . . . . . . . . . . . . . . . 17 Formula Conversion . . . . . . . . . . . . . . . . . . . . . . . 19 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . 20 Electricity and Magnetism . . . . . . . . . . . . . . . . . . . . . 21 Electromagnetic Frequency/Wavelength Bands . . . . . . . . . . . . 22 AC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Dimensionless Numbers of Fluid Mechanics . . . . . . . . . . . . . 24 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Fundamental Plasma Parameters . . . . . . . . . . . . . . . . . . 29 Plasma Dispersion Function . . . . . . . . . . . . . . . . . . . . 31 Collisions and Transport . . . . . . . . . . . . . . . . . . . . . 32 Ionospheric Parameters . . . . . . . . . . . . . . . . . . . . . . 41 Solar Physics Parameters . . . . . . . . . . . . . . . . . . . . . 42 Thermonuclear Fusion . . . . . . . . . . . . . . . . . . . . . . 43 Relativistic Electron Beams . . . . . . . . . . . . . . . . . . . . 45 Beam Instabilities . . . . . . . . . . . . . . . . . . . . . . . . 47 Approximate Magnitudes in Some Typical Plasmas . . . . . . . . . . 49 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Atomic Physics and Radiation . . . . . . . . . . . . . . . . . . . 53 Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 59 Complex (Dusty) Plasmas . . . . . . . . . . . . . . . . . . . . . 62 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3 NUMERICAL AND ALGEBRAIC Gain in decibels of P 2 relative to P 1 G = 10 log 10 (P 2 /P 1 ). To within two percent (2π) 1/2 ≈ 2.5; π 2 ≈ 10; e 3 ≈ 20; 2 10 ≈ 10 3 . Euler-Mascheroni constant 1 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): (1 + x) α = ∞  k=0  α k  x k ≡ 1 + αx + α(α − 1) 2! x 2 + α(α − 1)(α − 2) 3! x 3 + . . . . Rothe-Hagen identity 2 (good for all complex x, y, z except when singular): n  k=0 x x + kz  x + kz k  y y + (n − k)z  y + (n − k)z n − k  = x + y x + y + nz  x + y + nz n  . Newberger’s summation formula 3 [good for µ nonintegral, Re (α + β) > −1]: ∞  n=−∞ (−1) n J α−γn (z)J β+γn (z) n + µ = π sin µπ J α+γµ (z)J β−γµ (z). 4 VECTOR IDENTITIES 4 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) ∇ 2 f = ∇ · ∇f (14) ∇ 2 A = ∇(∇ · A) − ∇ × ∇ × A (15) ∇ × ∇f = 0 (16) ∇ · ∇ × A = 0 If e 1 , e 2 , e 3 are orthonormal unit vectors, a second-order tensor T can be written in the dyadic form (17) T =  i,j T ij e i e j In cartesian coordinates the divergence of a tensor is a vector with components (18) (∇·T ) i =  j (∂T ji /∂x j ) [This definition is 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/r 3 (25) ∇ · (r/r 3 ) = 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∇ 2 g − g∇ 2 f) =  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 6 (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 COORDINATES 5 Cylindrical Coordinates Divergence ∇ · A = 1 r ∂ ∂r (rA r ) + 1 r ∂A φ ∂φ + ∂A z ∂z Gradient (∇f) r = ∂f ∂r ; (∇f) φ = 1 r ∂f ∂φ ; (∇f) z = ∂f ∂z Curl (∇ × A) r = 1 r ∂A z ∂φ − ∂A φ ∂z (∇ × A) φ = ∂A r ∂z − ∂A z ∂r (∇ × A) z = 1 r ∂ ∂r (rA φ ) − 1 r ∂A r ∂φ Laplacian ∇ 2 f = 1 r ∂ ∂r  r ∂f ∂r  + 1 r 2 ∂ 2 f ∂φ 2 + ∂ 2 f ∂z 2 7 Laplacian of a vector (∇ 2 A) r = ∇ 2 A r − 2 r 2 ∂A φ ∂φ − A r r 2 (∇ 2 A) φ = ∇ 2 A φ + 2 r 2 ∂A r ∂φ − A φ r 2 (∇ 2 A) z = ∇ 2 A z Components of (A · ∇)B (A · ∇B) r = A r ∂B r ∂r + A φ r ∂B r ∂φ + A z ∂B r ∂z − A φ B φ r (A · ∇B) φ = A r ∂B φ ∂r + A φ r ∂B φ ∂φ + A z ∂B φ ∂z + A φ B r r (A · ∇B) z = A r ∂B z ∂r + A φ r ∂B z ∂φ + A z ∂B z ∂z Divergence of a tensor (∇ · T ) r = 1 r ∂ ∂r (rT rr ) + 1 r ∂T φr ∂φ + ∂T zr ∂z − T φφ r (∇ · T ) φ = 1 r ∂ ∂r (rT rφ ) + 1 r ∂T φφ ∂φ + ∂T zφ ∂z + T φr r (∇ · T ) z = 1 r ∂ ∂r (rT rz ) + 1 r ∂T φz ∂φ + ∂T zz ∂z 8 Spherical Coordinates Divergence ∇ · A = 1 r 2 ∂ ∂r (r 2 A r ) + 1 r sin θ ∂ ∂θ (sin θA θ ) + 1 r sin θ ∂A φ ∂φ Gradient (∇f) r = ∂f ∂r ; (∇f) θ = 1 r ∂f ∂θ ; (∇f) φ = 1 r sin θ ∂f ∂φ Curl (∇ × A) r = 1 r sin θ ∂ ∂θ (sin θA φ ) − 1 r sin θ ∂A θ ∂φ (∇ × A) θ = 1 r sin θ ∂A r ∂φ − 1 r ∂ ∂r (rA φ ) (∇ × A) φ = 1 r ∂ ∂r (rA θ ) − 1 r ∂A r ∂θ Laplacian ∇ 2 f = 1 r 2 ∂ ∂r  r 2 ∂f ∂r  + 1 r 2 sin θ ∂ ∂θ  sin θ ∂f ∂θ  + 1 r 2 sin 2 θ ∂ 2 f ∂φ 2 Laplacian of a vector (∇ 2 A) r = ∇ 2 A r − 2A r r 2 − 2 r 2 ∂A θ ∂θ − 2 cot θA θ r 2 − 2 r 2 sin θ ∂A φ ∂φ (∇ 2 A) θ = ∇ 2 A θ + 2 r 2 ∂A r ∂θ − A θ r 2 sin 2 θ − 2 cos θ r 2 sin 2 θ ∂A φ ∂φ (∇ 2 A) φ = ∇ 2 A φ − A φ r 2 sin 2 θ + 2 r 2 sin θ ∂A r ∂φ + 2 cos θ r 2 sin 2 θ ∂A θ ∂φ 9 Components of (A · ∇)B (A · ∇B) r = A r ∂B r ∂r + A θ r ∂B r ∂θ + A φ r sin θ ∂B r ∂φ − A θ B θ + A φ B φ r (A · ∇B) θ = A r ∂B θ ∂r + A θ r ∂B θ ∂θ + A φ r sin θ ∂B θ ∂φ + A θ B r r − cot θA φ B φ r (A · ∇B) φ = A r ∂B φ ∂r + A θ r ∂B φ ∂θ + A φ r sin θ ∂B φ ∂φ + A φ B r r + cot θA φ B θ r Divergence of a tensor (∇ · T ) r = 1 r 2 ∂ ∂r (r 2 T rr ) + 1 r sin θ ∂ ∂θ (sin θT θr ) + 1 r sin θ ∂T φr ∂φ − T θθ + T φφ r (∇ · T ) θ = 1 r 2 ∂ ∂r (r 2 T rθ ) + 1 r sin θ ∂ ∂θ (sin θT θθ ) + 1 r sin θ ∂T φθ ∂φ + T θr r − cot θT φφ r (∇ · T ) φ = 1 r 2 ∂ ∂r (r 2 T rφ ) + 1 r sin θ ∂ ∂θ (sin θT θφ ) + 1 r sin θ ∂T φφ ∂φ + T φr r + cot θT φθ r 10 [...]... 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 = 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... bar If B = 0, then15 ¯ (7) U − U = [(¯ − p)(υ − υ )]1/2 ; p ¯ (8) (¯ − p)(υ − υ)−1 = q 2 ; p ¯ (9) w − w = ¯ (10) e − e = ¯ 1 p 2 (¯ − 1 p 2 (¯ + p)(υ + υ); ¯ p)(υ − υ) ¯ In what follows we assume that the fluid is a perfect gas with adiabatic index γ = 1 + 2/n, where n is the number of degrees of freedom Then p = ρRT /m, where R is the universal gas constant and m is the molar weight; the sound speed... −1/2 − 1.8 × 10−7 µ−1/2 → i|i Z2Z 2λ −8 µ −1 1/2 µ −1 T −3/2 −1/2 −1 − 9.0 × 10 → −8 µ 1/2 µ −1 T −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 ni Z 2 λei → −3/2 µ−1 − 8.9 × 104 (µ/T )1/2 33 −1 exp(−1836µ... /mγ If ¯ ¯ this inequality cannot be satisfied, or if either uωcα −1 < rmax or uωcβ −1 < ¯ rmax , the theory breaks down Typically λ ≈ 10–20 Corrections to the transport coefficients are O(λ−1 ); 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 λee = 23 − ln(ne 1/2 Te −3/2 ), = 24 − ln(ne 1/2 Te −1 ), Te... kQ, where k is the coefficient in the second column of the table corresponding to Q (overbars ¯ denote variables expressed in Gaussian units) Thus, the formula a0 = h2 /m e2 ¯ ¯ ¯¯ 2 2 2 for the Bohr radius becomes αa0 = (¯ β) /[(mβ/α )(e αβ/4π 0 )], or a0 = h h2 /πme2 To go from SI to natural units in which h = c = 1 (distinguished ¯ 0 −1 ˆ ˆ ˆ by a circumflex), use Q = k Q, where k is the coefficient... µH In a plasma, µ ≈ µ0 = 4π × 10−7 H m−1 (Gaussian units: µ ≈ 1) The permittivity satisfies ≈ 0 = 8.8542 × 10−12 F m−1 (Gaussian: ≈ 1) provided that all charge is regarded as free Using the drift approximation v⊥ = E × B/B 2 to calculate polarization charge density gives rise to a dielectric constant K ≡ / 0 = 1 + 36π × 109 ρ/B 2 (SI) = 1 + 4πρc2 /B 2 (Gaussian), where ρ is the mass density The electromagnetic... of the upstream Mach number M = U/Cs , ¯ (27) ρ/ρ = υ/¯ = U/U = (γ + 1)M 2 /[(γ − 1)M 2 + 2]; ¯ υ (28) p/p = (2γM 2 − γ + 1)/(γ + 1); ¯ ¯ (29) T /T = [(γ − 1)M 2 + 2](2γM 2 − γ + 1)/(γ + 1)2 M 2 ; ¯ (30) M 2 = [(γ − 1)M 2 + 2]/[2γM 2 − γ + 1] The entropy change across the shock is (31) ∆s ≡ s − s = cυ ln[(¯ ¯ p/p)(ρ/¯ γ ], ρ) where cυ = R/(γ − 1)m is the specific heat at constant volume; here R is the. .. Z(ζ) ≈ 31 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 = Zα e 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 interaction of test particles (labeled α) streaming with... 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 32 have units cm3 sec−1 Test particle energy 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... Infrared 100 pm In spectroscopy the angstrom is sometimes used (1˚ = 10−8 cm = 0.1 nm) A *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, . ×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. surface S and dS = ndS, where n is the unit normal outward from V, (2 7)  V dV ∇f =  S dSf (2 8)  V dV ∇ · A =  S dS · A (2 9)  V dV ∇·T =  S dS ·T (3 0)  V dV ∇ × A =  S dS × A (3 1)  V dV (f∇ 2 g. second-order tensor T can be written in the dyadic form (1 7) T =  i ,j T ij e i e j In cartesian coordinates the divergence of a tensor is a vector with components (1 8) (∇·T ) i =  j (∂T ji /∂x j ) [This