The Natl. Plasma physics formulary

70 Pages·2004·0.396 MB·English

by  Huba J.D. (ed.)

2004
REVISED
NRL PLASMA FORMULARY
J.D. Huba
Beam Physics Branch
Plasma Physics Division
Naval Research Laboratory
Washington, DC 20375
Supported by
The O(cid:14)ce of Naval Research
1
FOREWARD
The NRL Plasma Formulary originated over twenty (cid:12)ve 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 (cid:12)les 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
Di(cid:11)erential Operators in Curvilinear Coordinates . . . . . . . . . . . 7
Dimensions and Units . . . . . . . . . . . . . . . . . . . . . . . 11
International System (SI) Nomenclature . . . . . . . . . . . . . . . 14
Metric Pre(cid:12)xes . . . . . . . . . . . . . . . . . . . . . . . . . . 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 relative to P
2 1
G = 10log (P =P ):
10 2 1
To within two percent
1=2 2 3 10 3
(2(cid:25)) 2:5; (cid:25) 10; e 20; 2 10 :
(cid:25) (cid:25) (cid:25) (cid:25)
Euler-Mascheroni constant1 (cid:13) = 0:57722
Gamma Function (cid:0)(x + 1) = x(cid:0)(x):
(cid:0)(1=6) = 5.5663 (cid:0)(3=5) = 1.4892
(cid:0)(1=5) = 4.5908 (cid:0)(2=3) = 1.3541
(cid:0)(1=4) = 3.6256 (cid:0)(3=4) = 1.2254
(cid:0)(1=3) = 2.6789 (cid:0)(4=5) = 1.1642
(cid:0)(2=5) = 2.2182 (cid:0)(5=6) = 1.1288
(cid:0)(1=2) = 1:7725 = p(cid:25) (cid:0)(1) = 1.0
Binomial Theorem (good for x < 1 or (cid:11) = positive integer):
j j
1 (cid:11) (cid:11)((cid:11) 1) (cid:11)((cid:11) 1)((cid:11) 2)
(cid:11) k 2 3
(1 + x) = x 1 + (cid:11)x + (cid:0) x + (cid:0) (cid:0) x + ::::
k (cid:17) 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
(cid:0)
x + kz k y + (n k)z n k
X (cid:0) (cid:1) (cid:0) (cid:0) (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 (cid:22) nonintegral, Re((cid:11) + (cid:12)) > 1]:
(cid:0)
1 ( 1)nJ (z)J (z) (cid:25)
(cid:11) (cid:13)n (cid:12)+(cid:13)n
(cid:0) (cid:0) = J (z)J (z):
(cid:11)+(cid:13)(cid:22) (cid:12) (cid:13)(cid:22)
n + (cid:22) sin(cid:22)(cid:25) (cid:0)
X
n=
(cid:0)1
4
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
(cid:1) (cid:2) (cid:2) (cid:1) (cid:1) (cid:2) (cid:2) (cid:1) (cid:1) (cid:2) (cid:2) (cid:1)
(2) A (B C) = (C B) A = (A C)B (A B)C
(cid:2) (cid:2) (cid:2) (cid:2) (cid:1) (cid:0) (cid:1)
(3) A (B C)+ B (C A) + C (A B) = 0
(cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)
(4) (A B) (C D) = (A C)(B D) (A D)(B C)
(cid:2) (cid:1) (cid:2) (cid:1) (cid:1) (cid:0) (cid:1) (cid:1)
(5) (A B) (C D) = (A B D)C (A B C)D
(cid:2) (cid:2) (cid:2) (cid:2) (cid:1) (cid:0) (cid:2) (cid:1)
(6) (fg) = (gf) = f g + g f
r r r r
(7) (fA) = f A + A f
r (cid:1) r (cid:1) (cid:1) r
(8) (fA) = f A + f A
r (cid:2) r (cid:2) r (cid:2)
(9) (A B) = B A A B
r (cid:1) (cid:2) (cid:1) r (cid:2) (cid:0) (cid:1) r (cid:2)
(10) (A B) = A( B) B( A) + (B )A (A )B
r (cid:2) (cid:2) r (cid:1) (cid:0) r (cid:1) (cid:1) r (cid:0) (cid:1) r
(11) A ( B) = ( B) A (A )B
(cid:2) r (cid:2) r (cid:1) (cid:0) (cid:1) r
(12) (A B) = A ( B) + B ( A) + (A )B+ (B )A
r (cid:1) (cid:2) r (cid:2) (cid:2) r(cid:2) (cid:1) r (cid:1) r
(13) 2f = f
r r (cid:1) r
(14) 2A = ( A) A
r r r (cid:1) (cid:0) r (cid:2) r (cid:2)
(15) f = 0
r (cid:2) r
(16) A = 0
r (cid:1) r (cid:2)
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
r(cid:1) j
P
[This de(cid:12)nition is required for consistency with Eq. (29)]. In general
(19) (AB) = ( A)B + (A )B
r (cid:1) r (cid:1) (cid:1) r
T T T
(20) (f ) = f +f
r (cid:1) r (cid:1) r(cid:1)
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 (cid:1)
(22) r = 0
r (cid:2)
(23) r = r=r
r
(24) (1=r) = r=r3
r (cid:0)
(25) (r=r3) = 4(cid:25)(cid:14)(r)
r (cid:1)
I
(26) r =
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 r Z
V S
(28) dV A = dS A
Z r (cid:1) Z (cid:1)
V S
T T
(29) dV = dS
Z r(cid:1) Z (cid:1)
V S
(30) dV A = dS A
Z r (cid:2) Z (cid:2)
V S
2 2
(31) dV(f g g f) = dS (f g g f)
Z r (cid:0) r Z (cid:1) r (cid:0) r
V S
(32) dV(A B B A)
Z (cid:1) r (cid:2) r (cid:2) (cid:0) (cid:1) r (cid:2)r (cid:2)
V
= dS (B A A B)
Z (cid:1) (cid:2) r (cid:2) (cid:0) (cid:2) r (cid:2)
S
If S is an open surface bounded by the contour C, of which the line element is
dl,
(33) dS f = dlf
Z (cid:2)r I
S C
6
(34) dS A = dl A
Z (cid:1) r (cid:2) I (cid:1)
S C
(35) (dS ) A = dl A
Z (cid:2) r (cid:2) I (cid:2)
S C
(36) dS ( f g) = fdg = gdf
Z (cid:1) r (cid:2) r I (cid:0)I
S C C
DIFFERENTIAL OPERATORS IN
CURVILINEAR COORDINATES5
Cylindrical Coordinates
Divergence
1 @ 1 @A @A
(cid:30) z
A = (rA ) + +
r
r (cid:1) r @r r @(cid:30) @z
Gradient
@f 1 @f @f
( f) = ; ( f) = ; ( f) =
r (cid:30) z
r @r r r @(cid:30) r @z
Curl
1 @A @A
z (cid:30)
( A) =
r
r (cid:2) r @(cid:30) (cid:0) @z
@A @A
r z
( A) =
(cid:30)
r (cid:2) @z (cid:0) @r
1 @ 1 @A
r
( A) = (rA )
z (cid:30)
r (cid:2) r @r (cid:0) r @(cid:30)
Laplacian
1 @ @f 1 @2f @2f
2
f = r + +
r r @r @r r2 @(cid:30)2 @z2
(cid:16) (cid:17)
7
Laplacian of a vector
2 @A A
2 2 (cid:30) r
( A) = A
r r
r r (cid:0) r2 @(cid:30) (cid:0) r2
2 @A A
2 2 r (cid:30)
( A) = A +
(cid:30) (cid:30)
r r r2 @(cid:30) (cid:0) r2
2 2
( A) = A
z z
r r
Components of (A )B
(cid:1) r
@B A @B @B A B
r (cid:30) r r (cid:30) (cid:30)
(A B) = A + + A
r r z
(cid:1) r @r r @(cid:30) @z (cid:0) r
@B A @B @B A B
(cid:30) (cid:30) (cid:30) (cid:30) (cid:30) r
(A B) = A + + A +
(cid:30) r z
(cid:1) r @r r @(cid:30) @z r
@B A @B @B
z (cid:30) z z
(A B) = A + + A
z r z
(cid:1) r @r r @(cid:30) @z
Divergence of a tensor
1 @ 1 @T @T T
T (cid:30)r zr (cid:30)(cid:30)
( ) = (rT ) + +
r rr
r (cid:1) r @r r @(cid:30) @z (cid:0) r
1 @ 1 @T @T T
T (cid:30)(cid:30) z(cid:30) (cid:30)r
( ) = (rT ) + + +
(cid:30) r(cid:30)
r (cid:1) r @r r @(cid:30) @z r
1 @ 1 @T @T
T (cid:30)z zz
( ) = (rT ) + +
z rz
r (cid:1) r @r r @(cid:30) @z
8
Spherical Coordinates
Divergence
1 @ 1 @ 1 @A
2 (cid:30)
A = (r A ) + (sin(cid:18)A ) +
r (cid:18)
r (cid:1) r2 @r rsin(cid:18) @(cid:18) rsin(cid:18) @(cid:30)
Gradient
@f 1 @f 1 @f
( f) = ; ( f) = ; ( f) =
r (cid:18) (cid:30)
r @r r r @(cid:18) r rsin(cid:18) @(cid:30)
Curl
1 @ 1 @A
(cid:18)
( A) = (sin(cid:18)A )
r (cid:30)
r (cid:2) rsin(cid:18) @(cid:18) (cid:0) rsin(cid:18) @(cid:30)
1 @A 1 @
r
( A) = (rA )
(cid:18) (cid:30)
r (cid:2) rsin(cid:18) @(cid:30) (cid:0) r @r
1 @ 1 @A
r
( A) = (rA )
(cid:30) (cid:18)
r (cid:2) r @r (cid:0) r @(cid:18)
Laplacian
1 @ @f 1 @ @f 1 @2f
2 2
f = r + sin(cid:18) +
r r2 @r @r r2 sin(cid:18) @(cid:18) @(cid:18) r2 sin2 (cid:18) @(cid:30)2
(cid:16) (cid:17) (cid:16) (cid:17)
Laplacian of a vector
2A 2 @A 2cot(cid:18)A 2 @A
2 2 r (cid:18) (cid:18) (cid:30)
( A) = A
r r
r r (cid:0) r2 (cid:0) r2 @(cid:18) (cid:0) r2 (cid:0) r2 sin(cid:18) @(cid:30)
2 @A A 2cos(cid:18) @A
2 2 r (cid:18) (cid:30)
( A) = A +
(cid:18) (cid:18)
r r r2 @(cid:18) (cid:0) r2 sin2 (cid:18) (cid:0) r2 sin2 (cid:18) @(cid:30)
A 2 @A 2cos(cid:18) @A
2 2 (cid:30) r (cid:18)
( A) = A + +
(cid:30) (cid:30)
r r (cid:0) r2 sin2 (cid:18) r2 sin(cid:18) @(cid:30) r2 sin2 (cid:18) @(cid:30)
9
Components of (A )B
(cid:1) r
@B A @B A @B A B + A B
r (cid:18) r (cid:30) r (cid:18) (cid:18) (cid:30) (cid:30)
(A B) = A + +
r r
(cid:1) r @r r @(cid:18) rsin(cid:18) @(cid:30) (cid:0) r
@B A @B A @B A B cot(cid:18)A B
(cid:18) (cid:18) (cid:18) (cid:30) (cid:18) (cid:18) r (cid:30) (cid:30)
(A B) = A + + +
(cid:18) r
(cid:1) r @r r @(cid:18) rsin(cid:18) @(cid:30) r (cid:0) r
@B A @B A @B A B cot(cid:18)A B
(cid:30) (cid:18) (cid:30) (cid:30) (cid:30) (cid:30) r (cid:30) (cid:18)
(A B) = A + + + +
(cid:30) r
(cid:1) r @r r @(cid:18) rsin(cid:18) @(cid:30) r r
Divergence of a tensor
1 @ 1 @
T 2
( ) = (r T ) + (sin(cid:18)T )
r rr (cid:18)r
r (cid:1) r2 @r rsin(cid:18) @(cid:18)
1 @T T + T
(cid:30)r (cid:18)(cid:18) (cid:30)(cid:30)
+
rsin(cid:18) @(cid:30) (cid:0) r
1 @ 1 @
T 2
( ) = (r T ) + (sin(cid:18)T )
(cid:18) r(cid:18) (cid:18)(cid:18)
r (cid:1) r2 @r rsin(cid:18) @(cid:18)
1 @T T cot(cid:18)T
(cid:30)(cid:18) (cid:18)r (cid:30)(cid:30)
+ +
rsin(cid:18) @(cid:30) r (cid:0) r
1 @ 1 @
T 2
( ) = (r T ) + (sin(cid:18)T )
(cid:30) r(cid:30) (cid:18)(cid:30)
r (cid:1) r2 @r rsin(cid:18) @(cid:18)
1 @T T cot(cid:18)T
(cid:30)(cid:30) (cid:30)r (cid:30)(cid:18)
+ + +
rsin(cid:18) @(cid:30) r r
10