NRL_FORMULARY_07 - 2007 NRL PLASMA FORMULARY J.D Huba Beam...

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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
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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
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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 (2 π ) 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) = x Γ( 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 = summationdisplay 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 summationdisplay 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]: summationdisplay n = −∞ ( 1) n J α γn ( z ) J β + γn ( z ) n + μ = π sin μπ J α + γμ ( z ) J β γμ ( z ) . 3
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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) ∇ · ( f A ) = f ∇ · A + A · ∇ f (8) ∇ × ( f A ) = 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) ∇ · ( f T ) = f · T + f ∇· T 4
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Let r = i x + j y + k z
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