Table Of Content

2007 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 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. 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THIS PAGE 71 unclassified unclassified unclassified Report (SAR) Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Atomic Physics and Radiation . . . . . . . . . . . . . . . . . . . 53 Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 59 Complex (Dusty) Plasmas . . . . . . . . . . . . . . . . . . . . . 62 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2 NUMERICAL AND ALGEBRAIC Gain in decibels of P relative to P 2 1 G = 10log (P /P ). 10 2 1 To within two percent 1/2 2 3 10 3 (2π) 2.5; π 10; e 20; 2 10 . ≈ ≈ ≈ ≈ 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) α k 2 3 (1 + x) = x 1 + αx + − x + − − x + .... k ≡ 2! 3! X(cid:0) (cid:1) k=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 X (cid:0) (cid:1) − (cid:0) − (cid:1) k=0 x + y x + y + nz = . x + y + nz n (cid:0) (cid:1) Newberger’s summation formula3 [good for µ nonintegral, Re(α + β) > 1]: − ∞ ( 1)nJ (z)J (z) π α γn β+γn − − = J (z)J (z). α+γµ β γµ n + µ sinµπ − X n= −∞ 3 VECTOR IDENTITIES4 T I Notation: f, g, are scalars; A, B, etc., are vectors; is a tensor; 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 ∇ · ∇ × T If e , e , e are orthonormal unit vectors, a second-order tensor can be 1 2 3 written in the dyadic form T (17) = T e e ij i j i,j P In cartesian coordinates the divergence of a tensor is a vector with components T (18) ( ) = (∂T /∂x ) i ji j ∇· j P [This definition is required for consistency with Eq. (29)]. In general (19) (AB) = ( A)B + (A )B ∇ · ∇ · · ∇ T T T (20) (f ) = f +f ∇ · ∇ · ∇· 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) ∇ · I (26) r = ∇ 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 Z ∇ Z V S (28) dV A = dS A Z ∇ · Z · V S T T (29) dV = dS Z ∇· Z · V S (30) dV A = dS A Z ∇ × Z × V S 2 2 (31) dV(f g g f) = dS (f g g f) Z ∇ − ∇ Z · ∇ − ∇ V S (32) dV(A B B A) Z · ∇ × ∇ × − · ∇ × ∇ × V = dS (B A A B) Z · × ∇ × − × ∇ × S If S is an open surface bounded by the contour C, of which the line element is dl, (33) dS f = dlf Z × ∇ I S C 5 (34) dS A = dl A Z · ∇ × I · S C (35) (dS ) A = dl A Z × ∇ × I × S C (36) dS ( f g) = fdg = gdf Z · ∇ × ∇ I −I S C C DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES5 Cylindrical Coordinates Divergence 1 ∂ 1 ∂A ∂A φ z A = (rA ) + + r ∇ · r ∂r r ∂φ ∂z Gradient ∂f 1 ∂f ∂f ( f) = ; ( f) = ; ( f) = r φ z ∇ ∂r ∇ r ∂φ ∇ ∂z Curl 1 ∂A ∂A z φ ( A) = r ∇ × r ∂φ − ∂z ∂A ∂A r z ( A) = φ ∇ × ∂z − ∂r 1 ∂ 1 ∂A r ( A) = (rA ) z φ ∇ × r ∂r − r ∂φ Laplacian 1 ∂ ∂f 1 ∂2f ∂2f 2 f = r + + ∇ r ∂r ∂r r2 ∂φ2 ∂z2 (cid:16) (cid:17) 6 Laplacian of a vector 2 ∂A A 2 2 φ r ( A) = A r r ∇ ∇ − r2 ∂φ − r2 2 ∂A A 2 2 r φ ( A) = A + φ φ ∇ ∇ r2 ∂φ − r2 2 2 ( A) = A z z ∇ ∇ Components of (A )B · ∇ ∂B A ∂B ∂B A B r φ r r φ φ (A B) = A + + A r r z · ∇ ∂r r ∂φ ∂z − r ∂B A ∂B ∂B A B φ φ φ φ φ r (A B) = A + + A + φ r z · ∇ ∂r r ∂φ ∂z r ∂B A ∂B ∂B z φ z z (A B) = A + + A z r z · ∇ ∂r r ∂φ ∂z Divergence of a tensor 1 ∂ 1 ∂T ∂T T T φr zr φφ ( ) = (rT ) + + r rr ∇ · r ∂r r ∂φ ∂z − r 1 ∂ 1 ∂T ∂T T T φφ zφ φr ( ) = (rT ) + + + φ rφ ∇ · r ∂r r ∂φ ∂z r 1 ∂ 1 ∂T ∂T T φz zz ( ) = (rT ) + + z rz ∇ · r ∂r r ∂φ ∂z 7 Spherical Coordinates Divergence 1 ∂ 1 ∂ 1 ∂A 2 φ A = (r A ) + (sinθA ) + r θ ∇ · r2 ∂r rsinθ ∂θ rsinθ ∂φ Gradient ∂f 1 ∂f 1 ∂f ( f) = ; ( f) = ; ( f) = r θ φ ∇ ∂r ∇ r ∂θ ∇ rsinθ ∂φ Curl 1 ∂ 1 ∂A θ ( A) = (sinθA ) r φ ∇ × rsinθ ∂θ − rsinθ ∂φ 1 ∂A 1 ∂ r ( A) = (rA ) θ φ ∇ × rsinθ ∂φ − r ∂r 1 ∂ 1 ∂A r ( A) = (rA ) φ θ ∇ × r ∂r − r ∂θ Laplacian 1 ∂ ∂f 1 ∂ ∂f 1 ∂2f 2 2 f = r + sinθ + ∇ r2 ∂r ∂r r2 sinθ ∂θ ∂θ r2 sin2 θ ∂φ2 (cid:16) (cid:17) (cid:16) (cid:17) Laplacian of a vector 2A 2 ∂A 2cotθA 2 ∂A 2 2 r θ θ φ ( A) = A r r ∇ ∇ − r2 − r2 ∂θ − r2 − r2 sinθ ∂φ 2 ∂A A 2cosθ ∂A 2 2 r θ φ ( A) = A + θ θ ∇ ∇ r2 ∂θ − r2 sin2 θ − r2 sin2 θ ∂φ A 2 ∂A 2cosθ ∂A 2 2 φ r θ ( A) = A + + φ φ ∇ ∇ − r2 sin2 θ r2 sinθ ∂φ r2 sin2 θ ∂φ 8 Components of (A )B · ∇ ∂B A ∂B A ∂B A B + A B r θ r φ r θ θ φ φ (A B) = A + + r r · ∇ ∂r r ∂θ rsinθ ∂φ − r ∂B A ∂B A ∂B A B cotθA B θ θ θ φ θ θ r φ φ (A B) = A + + + θ r · ∇ ∂r r ∂θ rsinθ ∂φ r − r ∂B A ∂B A ∂B A B cotθA B φ θ φ φ φ φ r φ θ (A B) = A + + + + φ r · ∇ ∂r r ∂θ rsinθ ∂φ r r Divergence of a tensor 1 ∂ 1 ∂ T 2 ( ) = (r T ) + (sinθT ) r rr θr ∇ · r2 ∂r rsinθ ∂θ 1 ∂T T + T φr θθ φφ + rsinθ ∂φ − r 1 ∂ 1 ∂ T 2 ( ) = (r T ) + (sinθT ) θ rθ θθ ∇ · r2 ∂r rsinθ ∂θ 1 ∂T T cotθT φθ θr φφ + + rsinθ ∂φ r − r 1 ∂ 1 ∂ T 2 ( ) = (r T ) + (sinθT ) φ rφ θφ ∇ · r2 ∂r rsinθ ∂θ 1 ∂T T cotθT φφ φr φθ + + + rsinθ ∂φ r r 9