Plasma Formulary 2016 PDF

71 Pages·2016·0.45 MB·English

2016
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
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
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
factors result from approximating the speed of light c = 2.9979 1010 cm/sec
×
3 1010 cm/sec.
≈ ×
Dimensions
Physical Sym- SI Conversion Gaussian
Quantity bol SI Gaussian Units Factor Units
t2q2
Capacitance C l farad 9 1011 cm
ml2 ×
m1/2l3/2
Charge q q coulomb 3 109 statcoulomb
t ×
q m1/2
Charge ρ coulomb 3 103 statcoulomb
density l3 l3/2t /m3 × /cm3
tq2 l
Conductance siemens 9 1011 cm/sec
ml2 t ×
tq2 1
Conductivity σ siemens 9 109 sec 1
−
ml3 t ×
/m
q m1/2l3/2
Current I,i ampere 3 109 statampere
t t2 ×
q m1/2
Current J,j ampere 3 105 statampere
density l2t l1/2t2 /m2 × /cm2
m m
Density ρ kg/m3 10 3 g/cm3
−
l3 l3
q m1/2
Displacement D coulomb 12π 105 statcoulomb
l2 l1/2t /m2 × /cm2
ml m1/2 1
4
Electric field E volt/m 10− statvolt/cm
t2q l1/2t 3 ×
ml2 m1/2l1/2 1
2
Electro- , volt 10− statvolt
E t2q t 3 ×
motance Emf
ml2 ml2
Energy U,W joule 107 erg
t2 t2
m m
Energy w,ǫ joule/m3 10 erg/cm3
lt2 lt2
density
10