NRL PLASMA FORMULARY

64 Pages·1994·0.4 MB·English

1994
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
CONTENTS
Numerical and Algebraic . . . . . . . . . . . . . . . . . . . . . 3
Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . 4
Di(cid:11)erential Operators in Curvilinear Coordinates . . . . . . . . . . . 6
Dimensions and Units . . . . . . . . . . . . . . . . . . . . . . . 10
International System (SI) Nomenclature . . . . . . . . . . . . . . . 13
Metric Pre(cid:12)xes . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Atomic Physics and Radiation . . . . . . . . . . . . . . . . . . . 52
Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2
NUMERICAL AND ALGEBRAIC
Gain in decibels of P2 relative to P1
G = 10log10(P2=P1):
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)
1
Euler-Mascheroni constant (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) k (cid:11)((cid:11) 1) 2 (cid:11)((cid:11) 1)((cid:11) 2) 3
(1 + x) = x 1+ (cid:11)x + (cid:0) x + (cid:0) (cid:0) x + ::::
k (cid:17) 2! 3!
Xk=0 (cid:0) (cid:1)
2
Rothe-Hagen identity (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
Xk=0 (cid:0) (cid:1) (cid:0) (cid:0) (cid:0) (cid:1)
x + y x+ y + nz
= :
x+ y + nz n
(cid:0) (cid:1)
3
Newberger’s summation formula [good for (cid:22) nonintegral, Re((cid:11) + (cid:12)) > 1]:
(cid:0)
1
n
( 1) J(cid:11)(cid:0)(cid:13)n(z)J(cid:12)+(cid:13)n(z) (cid:25)
(cid:0) = J(cid:11)+(cid:13)(cid:22)(z)J(cid:12)(cid:0)(cid:13)(cid:22)(z):
n + (cid:22) sin(cid:22)(cid:25)
nX=(cid:0)1
3
4
VECTOR IDENTITIES
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
2
(13) f = f
r r (cid:1) r
2
(14) A = ( 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 e1, e2, e3 are orthonormal unit vectors, a second-order tensor can be
written in the dyadic form
T
(17) = Tijeiej
i;j
In cartesianPcoordinates the divergence of a tensor is a vector with components
T
(18) ( )i = (@Tji=@xj)
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)
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
r (cid:1)
(22) r = 0
r (cid:2)
(23) r = r=r
r
3
(24) (1=r) = r=r
r (cid:0)
3
(25) (r=r ) = 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
r
ZV ZS
(28) dV A = dS A
r(cid:1) (cid:1)
ZV ZS
T T
(29) dV = dS
r(cid:1) (cid:1)
ZV ZS
(30) dV A = dS A
r(cid:2) (cid:2)
ZV ZS
2 2
(31) dV (f g g f) = dS (f g g f)
r (cid:0) r (cid:1) r (cid:0) r
ZV ZS
(32) dV (A B B A)
(cid:1) r (cid:2) r (cid:2) (cid:0) (cid:1) r (cid:2) r (cid:2)
ZV
= dS (B A A B)
(cid:1) (cid:2) r (cid:2) (cid:0) (cid:2) r (cid:2)
ZS
If S is an open surface bounded by the contour C, of which the line element is
dl,
(33) dS f = dlf
(cid:2) r
ZS IC
5
(34) dS A = dl A
(cid:1) r (cid:2) (cid:1)
ZS IC
(35) (dS ) A = dl A
(cid:2) r (cid:2) (cid:2)
ZS IC
(36) dS ( f g) = fdg = gdf
(cid:1) r (cid:2) r (cid:0)
ZS IC IC
DIFFERENTIAL OPERATORS IN
5
CURVILINEAR COORDINATES
Cylindrical Coordinates
Divergence
1 @ 1 @A(cid:30) @Az
A = (rAr) + +
r (cid:1) r @r r @(cid:30) @z
Gradient
@f 1 @f @f
( f)r = ; ( f)(cid:30) = ; ( f)z =
r @r r r @(cid:30) r @z
Curl
1 @Az @A(cid:30)
( A)r =
r (cid:2) r @(cid:30) (cid:0) @z
@Ar @Az
( A)(cid:30) =
r (cid:2) @z (cid:0) @r
1 @ 1 @Ar
( A)z = (rA(cid:30))
r (cid:2) r @r (cid:0) r @(cid:30)
Laplacian
2 2
2 1 @ @f 1 @ f @ f
f = r + +
2 2 2
r r @r @r r @(cid:30) @z
(cid:16) (cid:17)
6
Laplacian of a vector
2 2 2 @A(cid:30) Ar
( A)r = Ar
2 2
r r (cid:0) r @(cid:30) (cid:0) r
2 2 2 @Ar A(cid:30)
( A)(cid:30) = A(cid:30) +
2 2
r r r @(cid:30) (cid:0) r
2 2
( A)z = Az
r r
Components of (A )B
(cid:1) r
@Br A(cid:30) @Br @Br A(cid:30)B(cid:30)
(A B)r = Ar + + Az
(cid:1) r @r r @(cid:30) @z (cid:0) r
@B(cid:30) A(cid:30) @B(cid:30) @B(cid:30) A(cid:30)Br
(A B)(cid:30) = Ar + + Az +
(cid:1) r @r r @(cid:30) @z r
@Bz A(cid:30) @Bz @Bz
(A B)z = Ar + + Az
(cid:1) r @r r @(cid:30) @z
Divergence of a tensor
1 @ 1 @T(cid:30)r @Tzr T(cid:30)(cid:30)
T
( )r = (rTrr) + +
r (cid:1) r @r r @(cid:30) @z (cid:0) r
1 @ 1 @T(cid:30)(cid:30) @Tz(cid:30) T(cid:30)r
T
( )(cid:30) = (rTr(cid:30)) + + +
r (cid:1) r @r r @(cid:30) @z r
1 @ 1 @T(cid:30)z @Tzz
T
( )z = (rTrz) + +
r (cid:1) r @r r @(cid:30) @z
7
Spherical Coordinates
Divergence
1 @ 2 1 @ 1 @A(cid:30)
A = (r Ar) + (sin(cid:18)A(cid:18)) +
2
r (cid:1) r @r rsin(cid:18) @(cid:18) rsin(cid:18) @(cid:30)
Gradient
@f 1 @f 1 @f
( f)r = ; ( f)(cid:18) = ; ( f)(cid:30) =
r @r r r @(cid:18) r rsin(cid:18) @(cid:30)
Curl
1 @ 1 @A(cid:18)
( A)r = (sin(cid:18)A(cid:30))
r (cid:2) rsin(cid:18) @(cid:18) (cid:0) rsin(cid:18) @(cid:30)
1 @Ar 1 @
( A)(cid:18) = (rA(cid:30))
r (cid:2) rsin(cid:18) @(cid:30) (cid:0) r @r
1 @ 1 @Ar
( A)(cid:30) = (rA(cid:18))
r (cid:2) r @r (cid:0) r @(cid:18)
Laplacian
2
2 1 @ 2@f 1 @ @f 1 @ f
f = r + sin(cid:18) +
2 2 2 2 2
r r @r @r r sin(cid:18) @(cid:18) @(cid:18) r sin (cid:18) @(cid:30)
(cid:16) (cid:17) (cid:16) (cid:17)
Laplacian of a vector
2 2 2Ar 2 @A(cid:18) 2cot(cid:18)A(cid:18) 2 @A(cid:30)
( A)r = Ar
2 2 2 2
r r (cid:0) r (cid:0) r @(cid:18) (cid:0) r (cid:0) r sin(cid:18) @(cid:30)
2 2 2 @Ar A(cid:18) 2cos(cid:18) @A(cid:30)
( A)(cid:18) = A(cid:18) +
2 2 2 2 2
r r r @(cid:18) (cid:0) r sin (cid:18) (cid:0) r sin (cid:18) @(cid:30)
2 2 A(cid:30) 2 @Ar 2cos(cid:18) @A(cid:18)
( A)(cid:30) = A(cid:30) + +
2 2 2 2 2
r r (cid:0) r sin (cid:18) r sin(cid:18) @(cid:30) r sin (cid:18) @(cid:30)
8
Components of (A )B
(cid:1) r
@Br A(cid:18) @Br A(cid:30) @Br A(cid:18)B(cid:18) + A(cid:30)B(cid:30)
(A B)r = Ar + +
(cid:1) r @r r @(cid:18) rsin(cid:18) @(cid:30) (cid:0) r
@B(cid:18) A(cid:18) @B(cid:18) A(cid:30) @B(cid:18) A(cid:18)Br cot(cid:18)A(cid:30)B(cid:30)
(A B)(cid:18) = Ar + + +
(cid:1) r @r r @(cid:18) rsin(cid:18) @(cid:30) r (cid:0) r
@B(cid:30) A(cid:18) @B(cid:30) A(cid:30) @B(cid:30) A(cid:30)Br cot(cid:18)A(cid:30)B(cid:18)
(A B)(cid:30) = Ar + + + +
(cid:1) r @r r @(cid:18) rsin(cid:18) @(cid:30) r r
Divergence of a tensor
1 @ 2 1 @
T
( )r = (r Trr) + (sin(cid:18)T(cid:18)r)
2
r (cid:1) r @r rsin(cid:18) @(cid:18)
1 @T(cid:30)r T(cid:18)(cid:18) + T(cid:30)(cid:30)
+
rsin(cid:18) @(cid:30) (cid:0) r
1 @ 2 1 @
T
( )(cid:18) = (r Tr(cid:18)) + (sin(cid:18)T(cid:18)(cid:18))
2
r (cid:1) r @r rsin(cid:18) @(cid:18)
1 @T(cid:30)(cid:18) T(cid:18)r cot(cid:18)T(cid:30)(cid:30)
+ +
rsin(cid:18) @(cid:30) r (cid:0) r
1 @ 2 1 @
T
( )(cid:30) = (r Tr(cid:30)) + (sin(cid:18)T(cid:18)(cid:30))
2
r (cid:1) r @r rsin(cid:18) @(cid:18)
1 @T(cid:30)(cid:30) T(cid:30)r cot(cid:18)T(cid:30)(cid:18)
+ + +
rsin(cid:18) @(cid:30) r r
9
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. Multiples of 3 in the conversion
10
factors result from approximating the speed of light c = 2:9979 10 cm/sec
10 (cid:2)
3 10 cm/sec.
(cid:25) (cid:2)
Dimensions
Physical Sym- SI Conversion Gaussian
Quantity bol SI Gaussian Units Factor Units
2 2
t q 11
Capacitance C l farad 9 10 cm
2
ml (cid:2)
1=2 3=2
m l 9
Charge q q coulomb 3 10 statcoulomb
t (cid:2)
1=2
q m 3
Charge (cid:26) coulomb 3 10 statcoulomb
l3 l3=2t 3 (cid:2) 3
density /m /cm
2
tq l 11
Conductance siemens 9 10 cm/sec
2
ml t (cid:2)
2
tq 1 9 (cid:0)1
Conductivity (cid:27) siemens 9 10 sec
3
ml t (cid:2)
/m
1=2 3=2
q m l 9
Current I;i ampere 3 10 statampere
2
t t (cid:2)
1=2
q m 5
Current J;j ampere 3 10 statampere
l2t l1=2t2 2 (cid:2) 2
density /m /cm
m m 3 (cid:0)3 3
Density (cid:26) kg/m 10 g/cm
3 3
l l
1=2
q m 5
Displacement D coulomb 12(cid:25) 10 statcoulomb
l2 l1=2t 2 (cid:2) 2
/m /cm
1=2
ml m 1 (cid:0)4
Electric (cid:12)eld E volt/m 10 statvolt/cm
2 1=2
t q l t 3 (cid:2)
2 1=2 1=2
ml m l 1 (cid:0)2
Electro- , volt 10 statvolt
2
E t q t 3 (cid:2)
motance Emf
2 2
ml ml 7
Energy U;W joule 10 erg
2 2
t t
m m 3 3
Energy w;(cid:15) joule/m 10 erg/cm
2 2
lt lt
density
10