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NRL: Plasma Formulary 5B Naval Research Laboratory Washington, DC 20375-5320 NRL/PU/6790--04-477 NRL Plasma Formulary Revised 2004 Approved for public release; distribution is unlimited. Form Approved Report Documentation Page 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. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 2. REPORT TYPE 3. DATES COVERED 2004 N/A - 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER NRL: Plasma Formulary 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER Naval Research Laboratory Washington, DC 20375-5320 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE UU 72 unclassified unclassified unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 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 TEX files for the Formulary and his continued suggestions for improvement. The Formulary has been set in TEX 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 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 = pπ Γ(1) = 1.0 Binomial Theorem (good for x < 1 or α = positive integer): j j 1 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]: − 1 n ( 1) Jα γn(z)Jβ+γn(z) π − − = Jα+γµ(z)Jβ γµ(z): n + µ sin µπ − n= X−∞ 4 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 r r r r (7) (fA) = f A + A f r · r · · r (8) (fA) = f A + f A r × r × r × (9) (A B) = B A A B r · × · r × − · r × (10) (A B) = A( B) B( A) + (B )A (A )B r × × r · − r · · r − · r (11) A ( B) = ( B) A (A )B × r × r · − · r (12) (A B) = A ( B) + B ( A) + (A )B + (B )A r · × r × × r × · r · r (13) 2f = f r r · r (14) 2A = ( A) A r r r · − r × r × (15) f = 0 r × r (16) A = 0 r · r × 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 cartesian Pcoordinates the divergence of a tensor is a vector with components (18) ( T )i = (@Tji=@xj ) ∇· j [This definition isPrequired for consistency with Eq. (29)]. In general (19) (AB) = ( A)B + (A )B r · r · · r (20) (fT ) = f T +f T r · r · ∇· 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 r · (22) r = 0 r × (23) r = r=r r (24) (1=r) = r=r3 r − (25) (r=r3) = 4πδ(r) r · (26) r = I 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 r ZV ZS (28) dV A = dS A r · · ZV ZS (29) dV T = dS T ∇· · ZV ZS (30) dV A = dS A r × × ZV ZS (31) dV (f 2g g 2f) = dS (f g g f) r − r · r − r ZV ZS (32) dV (A B B A) · r × r × − · r × r × ZV = dS (B A A B) · × r × − × r × ZS If S is an open surface bounded by the contour C, of which the line element is dl, (33) dS f = dlf × r ZS IC 6 (34) dS A = dl A · r × · ZS IC (35) (dS ) A = dl A × r × × ZS IC (36) dS ( f g) = fdg = gdf · r × r − ZS IC IC DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES5 Cylindrical Coordinates Divergence 1 @ 1 @Aφ @Az A = (rAr ) + + r · r @r r @φ @z Gradient @f 1 @f @f ( f)r = ; ( f)φ = ; ( f)z = r @r r r @φ r @z Curl 1 @Az @Aφ ( A)r = r × r @φ − @z @Ar @Az ( A)φ = r × @z − @r 1 @ 1 @Ar ( A)z = (rAφ) r × r @r − r @φ Laplacian 1 @ @f 1 @2f @2f 2f = r + + r 2 2 2 r @r @r r @φ @z 7 Laplacian of a vector 2 2 2 @Aφ Ar ( A)r = Ar r r − r2 @φ − r2 2 2 2 @Ar Aφ ( A)φ = Aφ + r r r2 @φ − r2 ( 2A) = 2A r z r z Components of (A )B · r @Br Aφ @Br @Br AφBφ (A B)r = Ar + + Az · r @r r @φ @z − r @Bφ Aφ @Bφ @Bφ AφBr (A B)φ = Ar + + Az + · r @r r @φ @z r @Bz Aφ @Bz @Bz (A B)z = Ar + + Az · r @r r @φ @z Divergence of a tensor 1 @ 1 @Tφr @Tzr Tφφ ( T )r = (rTrr ) + + r · r @r r @φ @z − r 1 @ 1 @Tφφ @Tzφ Tφr ( T )φ = (rTrφ) + + + r · r @r r @φ @z r 1 @ 1 @Tφz @Tzz ( T )z = (rTrz ) + + r · r @r r @φ @z 8 Spherical Coordinates Divergence 1 @ 2 1 @ 1 @Aφ A = (r Ar ) + (sin θAθ ) + r · r2 @r r sin θ @θ r sin θ @φ Gradient @f 1 @f 1 @f ( f)r = ; ( f)θ = ; ( f)φ = r @r r r @θ r r sin θ @φ Curl 1 @ 1 @Aθ ( A)r = (sin θAφ) r × r sin θ @θ − r sin θ @φ 1 @Ar 1 @ ( A)θ = (rAφ) r × r sin θ @φ − r @r 1 @ 1 @Ar ( A)φ = (rAθ ) r × r @r − r @θ Laplacian 1 @ @f 1 @ @f 1 @2f 2f = r2 + sin θ + r 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 r r − r2 − r2 @θ − r2 − r2 sin θ @φ 2 2 2 @Ar Aθ 2 cos θ @Aφ ( A)θ = Aθ + r r r2 @θ − r2 sin2 θ − r2 sin2 θ @φ 2 2 Aφ 2 @Ar 2 cos θ @Aθ ( A)φ = Aφ + + r r − r2 sin2 θ r2 sin θ @φ r2 sin2 θ @φ 9 Components of (A )B · r @Br Aθ @Br Aφ @Br AθBθ + AφBφ (A B)r = Ar + + · r @r r @θ r sin θ @φ − r @Bθ Aθ @Bθ Aφ @Bθ AθBr cot θAφBφ (A B)θ = Ar + + + · r @r r @θ r sin θ @φ r − r @Bφ Aθ @Bφ Aφ @Bφ AφBr cot θAφBθ (A B)φ = Ar + + + + · r @r r @θ r sin θ @φ r r Divergence of a tensor 1 @ 2 1 @ ( T )r = (r Trr ) + (sin θTθr ) r · r2 @r r sin θ @θ 1 @Tφr Tθθ + Tφφ + r sin θ @φ − r 1 @ 2 1 @ ( T )θ = (r Trθ ) + (sin θTθθ) r · 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) r · r2 @r r sin θ @θ 1 @Tφφ Tφr cot θTφθ + + + r sin θ @φ r r 10 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.
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