NRLPLASMAFORMULARY
J.D.HubaBeamPhysicsBranchPlasmaPhysicsDivisionNavalResearchLaboratoryWashington,DC20375
Supportedby
TheOfficeofNavalResearch
1
FOREWARD
TheNRLPlasmaFormularyoriginatedovertwentyfiveyearsagoandhasbeenrevisedseveraltimesduringthisperiod.Theguidingspiritandper-sonprimarilyresponsibleforitsexistenceisDr.DavidBook.IamindebtedtoDaveforprovidingmewiththeTEXfilesfortheFormularyandhiscontinuedsuggestionsforimprovement.TheFormularyhasbeensetinTEXbyDaveBook,ToddBrun,andRobertScott.Finally,Ithankreadersforcommunicat-ingtypographicalerrorstome.
2
CONTENTS
NumericalandAlgebraicVectorIdentities
.....................
457
.........................
DifferentialOperatorsinCurvilinearCoordinates...........
DimensionsandUnits.............InternationalSystem(SI)Nomenclature.....MetricPrefixes................PhysicalConstants(SI)............PhysicalConstants(cgs)...........FormulaConversion.............Maxwell’sEquations
.............ElectricityandMagnetism...........ElectromagneticFrequency/WavelengthBands..ACCircuits.................DimensionlessNumbersofFluidMechanics
...Shocks
...................FundamentalPlasmaParameters........PlasmaDispersionFunction..........CollisionsandTransport...........IonosphericParameters............SolarPhysicsParameters...........ThermonuclearFusion............RelativisticElectronBeams
..........BeamInstabilities..............ApproximateMagnitudesinSomeTypicalPlasmasLasers
...................AtomicPhysicsandRadiation.........AtomicSpectroscopy.............Complex(Dusty)Plasmas
...........References..................3
........................................................................................................................................................................................11..14..14..15..17
..19..20
..21..22..23
..24
..27
..29..31..32..41..42..43
..45
..47
..49
..51
..53
..59
..62
..66
..........................NUMERICALANDALGEBRAIC
GainindecibelsofP2relativetoP1
G=10log10(P2/P1).
Towithintwopercent
(2π)
1/2
≈2.5;π≈10;e≈20;2
2310
≈10.
3
Euler-Mascheroniconstant1γ=0.57722GammaFunctionΓ(x+1)=xΓ(x):
Γ(1/6)Γ(1/5)Γ(1/4)Γ(1/3)Γ(2/5)Γ(1/2)
======
5.56634.59083.62562.67892.2182√1.7725=
Γ(3/5)Γ(2/3)Γ(3/4)Γ(4/5)Γ(5/6)
=====1.48921.35411.22541.16421.1288
2!
x2+
α(α−1)(α−2)
x+kz
x+kz
k
y
x+y+nz
x+y+nz
n
.
Newberger’ssummationformula3[goodforµnonintegral,Re(α+β)>−1]:
∞
(−1)nJα−γn(z)Jβ+γn(z)
sinµπ
Jα+γµ(z)Jβ−γµ(z).
n=−∞
4
VECTORIDENTITIES4
Notation:f,g,arescalars;A,B,etc.,arevectors;Tisatensor;Iistheunitdyad.
(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
Ife1,e2,e3areorthonormalunitvectors,asecond-ordertensorTcanbewritteninthedyadicform(17)T=
Incartesiancoordinatesthedivergenceofatensorisavectorwithcomponents(18)(∇·T)i=
i,j
Tijeiej
[ThisdefinitionisrequiredforconsistencywithEq.(29)].Ingeneral(19)∇·(AB)=(∇·A)B+(A·∇)B(20)∇·(fT)=∇f·T+f∇·T
j
(∂Tji/∂xj)
5
Letr=ix+jy+kzbetheradiusvectorofmagnituder,fromtheorigintothepointx,y,z.Then(21)∇·r=3(22)∇×r=0(23)∇r=r/r(24)∇(1/r)=−r/r3(25)∇·(r/r3)=4πδ(r)(26)∇r=I
IfVisavolumeenclosedbyasurfaceSanddS=ndS,wherenistheunitnormaloutwardfromV,(27)
(28)
(29)
(30)
dV∇f=
V
dSf
S
dV∇·A=
V
dV∇·T=
V
dS·A
S
dS·T
S
dV∇×A=
V
(31)
(32)
dS×A
S
dV(f∇2g−g∇2f)=
V
dS·(f∇g−g∇f)
S
dV(A·∇×∇×B−B·∇×∇×A)
V
=
dS·(B×∇×A−A×∇×B)
S
IfSisanopensurfaceboundedbythecontourC,ofwhichthelineelementisdl,(33)
dS×∇f=
S
dlf
C
6
(34)
(35)
dS·∇×A=
S
dl·A
C
(dS×∇)×A=
S
(36)
dl×A
C
dS·(∇f×∇g)=
S
fdg=−
C
gdf
C
DIFFERENTIALOPERATORSINCURVILINEARCOORDINATES5
CylindricalCoordinatesDivergence
∇·A=
1∂r
(rAr)+
1∂φ
+∂Az
∂r
;
(∇f)φ=
1∂φ
;(∇f)z=
∂f
∂Azr∂Ar
∂r
1∂r
1∂φ∂z
(∇×A)φ=
(∇×A)z=(rAφ)−
Laplacian
∇f=
2
1∂r
r
∂f
r2
∂2f
∂z2
7
Laplacianofavector
(∇A)r=∇Ar−
2
2
2
∂φ∂Arr2
−
Ar
r2
(∇A)z=∇Az
22
Componentsof(A·∇)B
(A·∇B)r=Ar
∂Br
rAφ
∂φ∂Bzr
∂z
∂Br
∂z∂Bφ
r−AφBφ
∂r∂Bz
++Az
(A·∇B)z=Ar
Divergenceofatensor
(∇·T)r=
1∂r
(rTrr)+
1∂φ
+∂Tzr
r
∂Tzφ
r
(∇·T)φ=
1∂r
(rTrφ)+
1
∂φ
+
(∇·T)z=
1∂r
(rTrz)+
1
∂φ
+
∂Tzz
SphericalCoordinatesDivergence
∇·A=Gradient
(∇f)r=
∂f
r∂f
rsinθ
∂f
1∂r
(rAr)+
2
1
∂θ
(sinθAθ)+
1
∂φ
∂
rsinθ
∂Ar
rsinθ∂r
rr∂Ar∂
rsinθ
∂Aθ
∂r2
∂r
+
1
∂θ
sinθ
∂f
r2sin2θ
∂2f
r2
−
2
∂θ
−
2cotθAθ
r2sinθ
2cosθ
∂φ
∂Aφ
∂Arr2
Aφ
r2sinθr2sin2θ
−
(∇2A)φ=∇2Aφ−
∂Ar
r2sin2θ
∂Aθ
Componentsof(A·∇)B
(A·∇B)r=Ar
∂Br
r
∂Bθ
rAθ
∂θ∂Bθ
rsinθAφ
∂φ
∂Br
rsinθ
∂Bθ
rAφBr
r
∂Br
r
cotθAφBφ
(A·∇B)θ=Ar
−
∂r
Divergenceofatensor
(∇·T)r=
1∂r
+++
(rTrr)+
2
1
∂θ
(sinθTθr)1
∂φ
Tθθ+Tφφ
+
∂r2
rsinθ
∂
−
∂Tφθ
rsinθ
∂r2
rsinθ
∂Tφφ
rsinθ
r
∂
r
−
cotθTφφ
+
cotθTφθ
DIMENSIONSANDUNITS
TogetthevalueofaquantityinGaussianunits,multiplythevalueex-pressedinSIunitsbytheconversionfactor.Multiplesof3intheconversionfactorsresultfromapproximatingthespeedoflightc=2.9979×1010cm/sec≈3×1010cm/sec.
PhysicalQuantityCapacitance
q
ChargedensityConductanceConductivityCurrentCurrentdensityDensityDisplacementElectricfield
E,
qtq2tq2qqmqml
Conversion
SIt2q2
l
m1/2l3/2m1/2l1m1/2l3/2m1/2mm1/2m1/2
coulombcoulomb/m3siemenssiemens/mampereampere/m2kg/m3coulomb/m2volt/m
31
t2qml2m
ml2m
t
joulejoule/m3
×10
−4
Units
9×1011
Units
statcoulombstatcoulomb/cm3cm/secsec−1statamperestatampere/cm2g/cm3
statcoulomb/cm2
statvoltergerg/cm3
motanceEnergyEnergydensity
11
PhysicalQuantityForceFrequencyImpedance
L
LengthMagneticintensityMagneticfluxMagneticinductionMagneticmomentMagnetizationMagneto-Mmfm,Mp,P
l
Conversion
SIml1ml2
ml1t
Unitsnewtonhertzohm
91
×10−11
sec2/cmcentimeter(cm)oerstedmaxwellgaussoersted–cm3oerstedUnitsdynehertz
q2l
meter(m)
m1/2m1/2l3/2m1/2m1/2l5/2m1/2m1/2l1/2
ampere–turn/mweberteslaampere–m2ampere–turn/mampere–
1010310510−1
henry/m
×10
7
qml2ml2qqq
m
tl2t
tl2t
µ
q24π
PhysicalQuantityPermittivity
PV,φ
PowerPowerdensityPressureReluctanceResistance
η,ρκ,ktAvη,µζW
Conversion
SIt2q2
1l1/2t
tml2mm1t
wattwatt/m3pascalampere–turn/weberohmohm–m
t3ttqtlttt2
tlttt2
t
99×10×10
−11
Units
36π×1093×1051
Units
l2t2qml2mmq2ml2
statvolterg/secerg/cm3–secdyne/cm2cm−1
tq2t3
−9
1051106102101107
INTERNATIONALSYSTEM(SI)NOMENCLATURE6PhysicalQuantity*length*mass*time*current*temperature*amountofsubstance*luminousintensity†planeangle†solidanglefrequencyenergyforcepressurepower
SymbolforUnit
mkgsAKmolcdradHzJNPaWsr
PhysicalQuantityelectricpotentialelectricresistanceelectric
conductanceelectric
capacitancemagneticfluxmagneticinductancemagneticintensity
SymbolforUnit
VΩSFWbHTlmlxBqGy
luminousfluxilluminanceactivity(ofaradioactivesource)
electricchargeC*SIbaseunit†Supplementaryunit
absorbeddose
(ofionizingradiation)
METRICPREFIXES
Multiple10−110−210−310−610−910−1210−1510−18
Symboldcmµnpfa
Multiple
10102103106109101210151018
SymboldahkMGTPE
14
PHYSICALCONSTANTS(SI)7
SymbolkemempGh
SpeedoflightinvacuumPermittivityof
µ0
freespace
Proton/electronmass
e/me
ratio
Rydbergconstant
802ch3
a0=0h2/πme2πa02
re=e2/4π0mc2(8π/3)re2h/mec
electron
Fine-structureconstant
α−1
c1=2πhc2c2=hc/k
constantStefan-Boltzmann
5.6705×10−83.8616×10−137.2974×10−31.8362×1031.0546×10−342.9979×1088.8542×10−12
UnitsJK−1Ckgkg
m3s−2kg−1Js
Hm−1
Ckg−1m−1mm2mm2m
Wm2mK
with1eV
Frequencyassociated
with1eV
Energyassociatedwith
1m−1
Energyassociatedwith
1Kelvin
Temperatureassociated
(no.densityatSTP)AtomicmassunitStandardtemperatureAtmosphericpressurePressureof1mmHg
acceleration
SymbolUnits
λ0=hc/e
m
2.4180×1014
k0=e/hc
m−1
1.6022×10−19
hc
J
13.606
k/e
eV
1.1604×104
NA
mol−1F=NAeCmol−1R=NAkJK−1mol−1n0
m−3
1.6605×10−27
273.151.0133×1051.3332×102
V0=RT0/p0m3MairkgJ
g
ms−2
16
PHYSICALCONSTANTS(cgs)7
Symbolke
ElectronmassProtonmass
GravitationalconstantPlanckconstant
h¯=h/2πcmp/me
ratio
Electroncharge/mass
R∞=
Bohrradius
AtomiccrosssectionClassicalelectronradiusThomsoncrosssectionComptonwavelengthof
h¯/mecα=e2/h¯c
137.04
FirstradiationconstantSecondradiation
σ
constant
Wavelengthassociated
1.2398×10−43.7418×10−5
1.4388
2π2me4
5.2728×10171.0974×1055.2918×10−98.7974×10−172.8179×10−136.6525×10−252.4263×10−10
cm
9.1094×10−281.6726×10−246.6726×10−86.6261×10−27
Unitserg/deg(K)statcoulomb
erg-seccm/sec
erg/cm2-
Symbolν0
with1eV
Wavenumberassociated
8.0655×103
erg
1eV
Energyassociatedwith
1.9864×10−16
eV
1Rydberg
Energyassociatedwith
8.6174×10−5
Hz
Units
deg(K)
with1eVAvogadronumberFaradayconstantGasconstantLoschmidt’snumber
muT0
p0=n0kT0
(1torr)
MolarvolumeatSTPMolarweightofaircalorie(cal)Gravitational
2.2414×104
28.9714.1868×107
980.676.0221×10232.8925×10148.3145×1072.6868×1019
gdeg(K)dyne/cm2dyne/cm2
FORMULACONVERSION8
Hereα=102cmm−1,β=107ergJ−1,0=8.8542×10−12Fm−1,µ0=4π×10−7Hm−1,c=(0µ0)−1/2=2.9979×108ms−1,andh¯=1.0546×10−34Js.ToderiveadimensionallycorrectSIformulafromoneexpressedin
¯,wherek¯is¯=kQGaussianunits,substituteforeachquantityaccordingtoQ
thecoefficientinthesecondcolumnofthetablecorrespondingtoQ(overbars
¯2/mdenotevariablesexpressedinGaussianunits).Thus,theformulaa¯0=h¯¯e¯2
fortheBohrradiusbecomesαa0=(¯hβ)2/[(mβ/α2)(e2αβ/4π0)],ora0=0h2/πme2.TogofromSItonaturalunitsinwhichh¯=c=1(distinguished
ˆ−1Qˆisthecoefficientcorrespondingtoˆ,wherekbyacircumflex),useQ=k
Qinthethirdcolumn.Thusaˆ0=4π0h¯2/[(mˆh¯/c)(ˆe20h¯c)]=4π/mˆeˆ2.(IntransformingfromSIunits,donotsubstitutefor0,µ0,orc.)
PhysicalQuantityCapacitanceCharge
ChargedensityCurrent
CurrentdensityElectricfield
ElectricpotentialElectricconductivityEnergy
EnergydensityForce
FrequencyInductanceLength
MagneticinductionMagneticintensityMass
MomentumPowerPressureResistanceTimeVelocity
NaturalUnitstoSI0−1(0h¯c)−1/2(0h¯c)−1/2(µ0/h¯c)1/2(µ0/h¯c)1/2(0/h¯c)1/2(0/h¯c)1/20−1(¯hc)−1(¯hc)−1(¯hc)−1c−1µ0−11(µ0h¯c)−1/2(µ0/h¯c)1/2c/h¯h¯−1(¯hc2)−1(¯hc)−1
(0/µ0)1/2cc−1
19
MAXWELL’SEQUATIONS
NameorDescriptionFaraday’slaw
∇×H=∇·B=0
chargeqConstitutiverelations
∇·D=ρ
∂t∂D
Gaussian
∂Bc
∇×H=
1∂t
+4π
q(E+v×B)
c
D=EB=µH
v×B
Inaplasma,µ≈µ0=4π×10−7Hm−1(Gaussianunits:µ≈1).Thepermittivitysatisfies≈0=8.8542×10−12Fm−1(Gaussian:≈1)providedthatallchargeisregardedasfree.Usingthedriftapproximationv⊥=E×B/B2tocalculatepolarizationchargedensitygivesrisetoadielec-tricconstantK≡/0=1+36π×109ρ/B2(SI)=1+4πρc2/B2(Gaussian),whereρisthemassdensity.
TheelectromagneticenergyinvolumeVisgivenby
1
W=
8π
Poynting’stheoremis
∂W
dV(H·B+E·D)
V
(Gaussian).
ELECTRICITYANDMAGNETISM
Inthefollowing,=dielectricpermittivity,µ=permeabilityofconduc-tor,µ=permeabilityofsurroundingmedium,σ=conductivity,f=ω/2π=radiationfrequency,κm=µ/µ0andκe=/0.Wheresubscriptsareused,‘1’denotesaconductingmediumand‘2’apropagating(losslessdielectric)medium.AllunitsareSIunlessotherwisespecified.PermittivityoffreespacePermeabilityoffreespaceResistanceoffreespace
CapacityofparallelplatesofareaA,separatedbydistancedCapacityofconcentriccylindersoflengthl,radiia,bCapacityofconcentricspheresofradiia,b
Self-inductanceofwireoflengthl,carryinguniformcurrentMutualinductanceofparallelwiresoflengthl,radiusa,separatedbydistancedInductanceofcircularloopofradiusb,madeofwireofradiusa,carryinguniformcurrentRelaxationtimeinalossymediumSkindepthinalossymediumWaveimpedanceinalossymediumTransmissioncoefficientatconductingsurface9(goodonlyforT1)
FieldatdistancerfromstraightwirecarryingcurrentI(amperes)FieldatdistancezalongaxisfromcircularloopofradiusacarryingcurrentI
0=8.8542×10−12Fm−1µ0=4π×10−7Hm−1
=1.2566×10−6Hm−1R0=(µ0/0)1/2=376.73ΩC=A/dC=2πl/ln(b/a)C=4πab/(b−a)L=µl
L=(µl/4π)[1+4ln(d/a)]
L=b
µ[ln(8b/a)−2]+µ/4
τδ
=/σ
=(2/ωµσ)1/2=(πfµσ)−1/2
Z=[µ/(+iσ/ω)]1/2
T=4.22×10−4(fκm1κe2/σ)1/2
Bθ=µI/2πrtesla
=0.2I/rgauss(rincm)Bz=µa2I/[2(a2+z2)3/2]
21
ELECTROMAGNETICFREQUENCY/
WAVELENGTHBANDS10
WavelengthRange
Designation
Lower
Lower10Mm
30Hz300Hz3kHz30kHz300kHz3MHz30MHz300MHz3GHz2.63.955.37.058.210.012.418.026.530GHz300GHz3THz430THz750THz30PHz3EHz
1Mm100km10km1km100m10m1m10cm1cm7.65.13.73.02.42.01.671.10.751mm100µm700nm400nm10nm100pm
ACCIRCUITS
ForaresistanceR,inductanceL,andcapacitanceCinserieswith√avoltagesourceV=V0exp(iωt)(herei=
dt2
+R
dq
C
=V.
Solutionsareq(t)=qs+qt,I(t)=Is+It,wherethesteadystateisIs=iωqs=V/ZintermsoftheimpedanceZ=R+i(ωL−1/ωC)andIt=dqt/dt.Forinitialconditionsq(0)≡q0=q¯0+qs,I(0)≡I0,thetransientscanbeofthreetypes,dependingon∆=R2−4L/C:(a)Overdamped,∆>0
qt=It=
I0+γ+q¯0γ+(I0+γ−q¯0)
γ+−γ−
exp(−γ+t),
exp(−γ−t),
γ+−γ−
whereγ±=(R±∆1/2)/2L;(b)Criticallydamped,∆=0
qt=[¯q0+(I0+γRq¯0)t]exp(−γRt),It=[I0−(I0+γRq¯0)γRt]exp(−γRt),
whereγR=R/2L;(c)Underdamped,∆<0
qt=
γRq¯0+I0
ω1
sin(ω1t)exp(−γRt),
Hereω1=ω0(1−R2C/4L)1/2,whereω0=(LC)−1/2istheresonant
frequency.Atω=ω0,Z=R.ThequalityofthecircuitisQ=ω0L/R.InstabilityresultswhenL,R,Carenotallofthesamesign.
23
DIMENSIONLESSNUMBERSOFFLUIDMECHANICS12Name(s)Alfv´en,
Bd
Boussinesq
BrCpCa
Cauchy,sekharClausius
Ch
LV3ρ/k∆T
CCr
Dean
CD
coefficient]Eckert
Ek
Euler
Fr
V/NL
Gay–Lussac
GrCH
1/β∆T
volumeduringheating
Buoyancyforce/viscousforceGyrofrequency/
D3/2V/ν(2r)1/2
2
DefinitionVA/V
inertialforce)1/2Gravitationalforce/
V/(2gR)1/2
gravitationalforce)1/2Viscousheat/conductedheatViscousforce/surfacetensionTheoreticalCarnotcycle
ρV2/Γ=M2
compressibilityforceMagneticforce/dissipativeconductionrate
Magneticforce/inertialforceEffectofdiffusion/effectofcurvature/longitudinalflowDragforce/inertialforce
ρV
V2/cp∆T(Ro/Re)1/2∆p/ρV2
thermalenergy
(Viscousforce/Coriolisforce)1/2
dynamicpressure
†(Inertialforce/gravitationalor
coefficient]
*(†)Alsodefinedastheinverse(square)ofthequantityshown.
24
Name(s)Hartmann
DefinitionBL/(µη)1/2=
Kn
Lewis
LoLu
MachMachMm
Magnetic
NtN
P´ecletPoisseuillePrandtl
RaReRi
RossbySchmidt
St
StefanStokes
SrTa
BoltzmannTh,Bo
Weber
κ/D
AlRmV/CS
µ0LV/η
LV/κD2∆p/µLVν/κ
V/2ΩLsinΛν/D
σLT3/kν/L2f
R
1/2
(∆R)
3/2
ρLV2/Σ
25
dissipativeforce)1/2Hydrodynamictime/
diffusion
MagnitudeofrelativisticeffectsJ×Bforce/resistivemagneticeffects
(Inertialforce/magneticforce)1/2velocity
Imposedforce/inertialforceTotalheattransfer/thermal
heatdiffusion
Buoyancyforce/diffusionforceInertialforce/viscousforceBuoyancyeffects/
moleculardiffusion
Thermalconductionloss/
vibrationfrequency
Vibrationspeed/flowvelocity
Centrifugalforce/viscousforceviscousforce)1/2
Convectiveheattransport/
Nomenclature:BCs,ccp
D=2RFfgH,Lk=ρcpκN=(g/H)1/2RrrLTV
VA=B/(µ0ρ)1/2α
MagneticinductionSpeedsofsound,light
Specificheatatconstantpressure(unitsm2s−2K−1)PipediameterImposedforce
Vibrationfrequency
Gravitationalacceleration
Vertical,horizontallengthscales
Thermalconductivity(unitskgm−1s−2)Brunt–V¨ais¨al¨afrequencyRadiusofpipeorchannel
RadiusofcurvatureofpipeorchannelLarmorradiusTemperature
CharacteristicflowvelocityAlfv´enspeed
Newton’s-lawheatcoefficient,k
∂T
SHOCKS
AtashockfrontpropagatinginamagnetizedfluidatanangleθwithrespecttothemagneticinductionB,thejumpconditionsare13,14
¯≡q;(1)ρU=ρ¯U
¯V¯−B¯B¯⊥/µ;(3)ρUV−BB⊥/µ=ρ¯U¯;(4)B=B
1
2(U
2¯2+p¯2/2µ;(2)ρU2+p+B⊥/2µ=ρ¯U¯+B⊥
¯B¯⊥−V¯B¯;(5)UB⊥−VB=U
¯2+V¯2)+w¯B¯2−V¯B¯B¯⊥)/µρ¯.¯+(U¯U⊥
(6)
HereUandVarecomponentsofthefluidvelocitynormalandtangentialto
thefrontintheshockframe;ρ=1/υisthemassdensity;pisthepressure;B⊥=Bsinθ,B=Bcosθ;µisthemagneticpermeability(µ=4πincgsunits);andthespecificenthalpyisw=e+pυ,wherethespecificinternalenergyesatisfiesde=Tds−pdυintermsofthetemperatureTandthespecificentropys.Quantitiesintheregionbehind(downstreamfrom)thefrontaredistinguishedbyabar.IfB=0,then15
¯=[(¯(7)U−Up−p)(υ−υ¯)]1/2;(8)(¯p−p)(υ−υ¯)−1=q2;(9)w¯−w=
1
p+2(¯
p)(υ−υ¯).
Inwhatfollowsweassumethatthefluidisaperfectgaswithadiabaticindex
γ=1+2/n,wherenisthenumberofdegreesoffreedom.Thenp=ρRT/m,whereRistheuniversalgasconstantandmisthemolarweight;thesoundspeedisgivenbyCs2=(∂p/∂ρ)s=γpυ;andw=γe=γpυ/(γ−1).ForageneralobliqueshockinaperfectgasthequantityX=r−1(U/VA)2satisfies14
(11)(X−β/α)(X−cosθ)=Xsinθr=ρ/ρ¯,α=
1
2
2
2
[1+(r−1)/2α]X−cosθ,where
2
¯=V;(15)V
(16)p¯=p+(1−r−1)ρU2+(1−r2)B2/2µ.
Ifθ=0,therearetwopossibilities:switch-onshocks,whichrequireβ<1and
forwhich
(17)U2=rVA2;¯=VA2/U;(18)U
¯2=2B2(r−1)(α−β);(19)B⊥¯=U¯B¯⊥/B;(20)V
(21)p¯=p+ρU2(1−α+β)(1−r−1),andacoustic(hydrodynamic)shocks,forwhich(22)U2=(r/α)Cs2;¯=U/r;(23)U
¯=B¯⊥=0;(24)V
(25)p¯=p+ρU2(1−r−1).
Foracousticshocksthespecificvolumeandpressurearerelatedby(26)υ¯/υ=[(γ+1)p+(γ−1)¯p]/[(γ−1)p+(γ+1)¯p].IntermsoftheupstreamMachnumberM=U/Cs,¯=(γ+1)M2/[(γ−1)M2+2];(27)ρ/ρ¯=υ/υ¯=U/U¯/T=[(γ−1)M2+2](2γM2−γ+1)/(γ+1)2M2;(29)T
(28)p/p¯=(2γM2−γ+1)/(γ+1);
Theentropychangeacrosstheshockis
¯2=[(γ−1)M2+2]/[2γM2−γ+1].(30)M
(31)∆s≡s¯−s=cυln[(¯p/p)(ρ/ρ¯)γ],
wherecυ=R/(γ−1)misthespecificheatatconstantvolume;hereRisthe
gasconstant.Intheweak-shocklimit(M→1),(32)∆s→cυ
2γ(γ−1)
3(γ+1)m
(M−1)3.
Theradiusattimetofastrongsphericalblastwaveresultingfromtheexplo-sivereleaseofenergyEinamediumwithuniformdensityρis(33)RS=C0(Et2/ρ)1/5,
whereC0isaconstantdependingonγ.Forγ=7/5,C0=1.033.
28
FUNDAMENTALPLASMAPARAMETERS
AllquantitiesareinGaussiancgsunitsexcepttemperature(T,Te,Ti)expressedineVandionmass(mi)expressedinunitsoftheprotonmass,µ=mi/mp;Zischargestate;kisBoltzmann’sconstant;Kiswavenumber;γistheadiabaticindex;lnΛistheCoulomblogarithm.Frequencies
electrongyrofrequencyiongyrofrequency
electronplasmafrequency
ionplasmafrequency
electrontrappingrateiontrappingrateelectroncollisionrateioncollisionrateLengths
electrondeBroglielengthclassicalminimumdistanceapproachofelectrongyroradiusiongyroradius
electroninertiallengthioninertiallengthDebyelength
fce=ωce/2π=2.80×106BHzωce=eB/mec=1.76×107Brad/secfci=ωci/2π=1.52×103Zµ−1BHzωci=ZeB/mic=9.58×103Zµ−1Brad/secfpe=ωpe/2π=8.98×103ne1/2Hzωpe=(4πnee2/me)1/2
=5.64×104ne1/2rad/sec
fpi=ωpi/2π
=2.10×102Zµ−1/2ni1/2Hzωpi=(4πniZ2e2/mi)1/2
=1.32×103Zµ−1/2ni1/2rad/sec
νTe=(eKE/me)1/2
=7.26×108K1/2E1/2sec−1
νTi=(ZeKE/mi)1/2
=1.69×107Z1/2K1/2E1/2µ−1/2sec−1νe=2.91×10−6nelnΛTe−3/2sec−1
νi=4.80×10−8Z4µ−1/2nilnΛTi−3/2sec−1
¯=h¯/(mekTe)1/2=2.76×10−8Te−1/2cme2/kT=1.44×10−7T−1cmre=vTe/ωce=2.38Te1/2B−1cmri=vTi/ωci
=1.02×102µ1/2Z−1Ti1/2B−1cmc/ωpe=5.31×105ne−1/2cmc/ωpi=2.28×107(µ/ni)1/2cm
λD=(kT/4πne2)1/2=7.43×102T1/2n−1/2cm29
λVelocities
electronthermalvelocityionthermalvelocityionsoundvelocityAlfv´envelocity
Dimensionless
(electron/protonmassratio)1/2
numberofparticlesinDebyesphereAlfv´envelocity/speedoflightelectronplasma/gyrofrequencyratio
ionplasma/gyrofrequencyratiothermal/magneticenergyratiomagnetic/ionrestenergyratioMiscellaneous
BohmdiffusioncoefficienttransverseSpitzerresistivity
vTe=(kTe/me)1/2
=4.19×107Te1/2cm/secvTi=(kTi/mi)1/2
=9.79×105µ−1/2Ti1/2cm/secCs=(γZkTe/mi)1/2
=9.79×105(γZTe/µ)1/2cm/secvA=B/(4πnimi)1/2
=2.18×1011µ−1/2ni−1/2Bcm/sec(me/mp)1/2=2.33×10−2=1/42.9(4π/3)nλD3=1.72×109T3/2n−1/2vA/c=7.28µ−1/2ni−1/2Bωpe/ωce=3.21×10−3ne1/2B−1ωpi/ωci=0.137µ1/2ni1/2B−1
β=8πnkT/B2=4.03×10−11nTB−2B2/8πnimic2=26.5µ−1ni−1B2DB=(ckT/16eB)
=6.25×106TB−1cm2/secη⊥=1.15×10−14ZlnΛT−3/2sec
=1.03×10−2ZlnΛT−3/2Ωcm
Theanomalouscollisionrateduetolow-frequencyion-soundturbulenceis
isthetotalenergyofwaveswithω/K whereB0=10kG=1T. Detonationenergyof1kilotonofhighexplosiveis WkT=10 12 /kT=5.64×10neν*≈ωpeW 4 1/2 /kTsecW −1 , Pmag=B2/8π=3.98×106(B/B0)2dynes/cm2=3.93(B/B0)2atm, cal=4.2×10 19 erg. 30 PLASMADISPERSIONFUNCTION Definition16(firstformvalidonlyforImζ>0): Z(ζ)=π −1/2 +∞ dtexp−t −∞ 2 dζ =−2(1+ζZ),Z(0)=iπ 1/2 ; d2Z dζ +2Z=0. Realargument(y=0): Z(x)=exp−x Imaginaryargument(x=0): 2 iπ 1/2 −2 x dtexpt 0 2 . Z(iy)=iπ Powerseries(smallargument): Z(ζ)=iπ 1/2 1/2 expy exp−ζ Asymptoticseries,|ζ|1(Ref.17): Z(ζ)=iπwhere 1/2 2 σexp−ζ −2ζ1−2ζ/3+4ζ/15−8ζ/105+···.−ζ −1 2 2 2 [1−erf(y)]. 46 σ= 0 y>|x|−11|y|<|x|−12y<−|x|−1 1+1/2ζ+3/4ζ+15/8ζ+···, 246 Symmetryproperties(theasteriskdenotescomplexconjugation): Z(ζ*)=−[Z(−ζ)]*; Z(ζ*)=[Z(ζ)]*+2iπ1/2exp[−(ζ*)2] (y>0). Two-poleapproximations18(goodforζinupperhalfplaneexceptwheny<π1/2x2exp(−x2),x1): Z(ζ)≈Z(ζ)≈ 0.50+0.81i a*+ζ 0.50+0.96i (b*+ζ)2 ,a=0.51−0.81i;,b=0.48−0.91i. 31 COLLISIONSANDTRANSPORT TemperaturesareineV;thecorrespondingvalueofBoltzmann’sconstantisk=1.60×10−12erg/eV;massesµ,µareinunitsoftheprotonmass;eα=Zαeisthechargeofspeciesα.Allotherunitsarecgsexceptwherenoted. RelaxationRates Ratesareassociatedwithfourrelaxationprocessesarisingfromthein-teractionoftestparticles(labeledα)streamingwithvelocityvαthroughabackgroundoffieldparticles(labeledβ): slowingdown dvα (vα−v¯α)⊥=ν⊥vα 2 α|β 2 paralleldiffusion dtd dt vα=−ν 2α|β vα, 2 wherevα=|vα|andtheaveragesareperformedoveranensembleoftestparticlesandaMaxwellianfieldparticledistribution.Theexactformulasmaybewritten19 α|βνs=(1+mα/mβ)ψ(xα|β)ν0α|βν⊥α|βνα|βν α|β =2(1−1/2x=ψ(x =2(mα/mβ)ψ(x where ν0 α|β ; α|β α|β α|β )/x α|β )ψ(xν0 α|β )+ψ(x )ν0 α|β ; α|β α|β )−ψ(x )ν0 α|β ; α|β , =4πeα2eβ2λαβnβ/mα2vα3; 2π xα|β=mβvα2/2kTβ; dψ ψ(x)= x dtt1/2e−t;ψ(x)= 0 haveunitscm3sec−1.TestparticleenergyandfieldparticletemperatureTarebothineV;µ=mi/mpwherempistheprotonmass;Zisionchargestate;inelectron–electronandion–ionencounters,fieldparticlequantitiesaredistinguishedbyaprime.Thetwoexpressionsgivenbelowforeachrateholdforveryslow(xα|β1)andveryfast(xα|β1)testparticles,respectively. Slow Electron–electron e|eνs/neλee≈5.8×10−6T−3/2e|e ≈5.8×10−6T−1/2−1ν⊥/neλeeElectron–ionν/neλee e|e ≈2.9×10−6T−1/2−1 −→7.7×10−6−3/2 −→7.7×10−6−3/2 −→3.9×10−6T−5/2 e|i νs/niZ2λei≈0.23µ3/2T−3/2−→3.9×10−6−3/2e|i →7.7×10−6−3/2ν⊥/niZ2λei≈2.5×10−4µ1/2T−1/2−1−e|i Ion–electron ν/niZ2λei≈1.2×10−4µ1/2T−1/2−1−→2.1×10−9µ−1T−5/2 i|e νs/neZ2λie≈1.6×10−9µ−1T−3/2−→1.7×10−4µ1/2−3/2i|e2−9−1−1/2−1−7−1/2−3/2 −→1.8×10µν⊥/neZλie≈3.2×10µTi|e Ion–ion ν/neZ2λie≈1.6×10−9µ−1T−1/2−1−→1.7×10−4µ1/2T−5/2 i|i νs µ 1+ µ + 1 3/2 i|iν⊥ µ ni Z2Z2λ ii ≈6.8×10 −8 µ 1/2 µ −1 T −1/2−1 Inthesamelimits,theenergytransferratefollowsfromtheidentity ν=2νs−ν⊥−ν, exceptforthecaseoffastelectronsorfastionsscatteredbyions,wheretheleadingtermscancel.Thentheappropriateformsare e|iν−→4.2×10−9niZ2λei −→9.0×10 −8 µ 1/2 µ −1 T −5/2 −3/2 µ −1 −8.9×10(µ/T) 4 1/2−1 exp(−1836µ/T)sec−1 33 and i|iν −→1.8×10 α|β Ingeneral,theenergytransferrateνispositivefor>α*andnega-tivefor<α*,wherex*=(mβ/mα)α*/Tβisthesolutionofψ(x*)=(mα|mβ)ψ(x*).Theratioα*/Tβisgivenforanumberofspecificα,βinthefollowingtable: −7 −3/2 niZZλii 1/2 2 2 µ /µ−1.1(µ/T) 1/2−1 exp(−µ/T)sec −1 . i|e Tβ e|e,i|ie|pe|D e|T,e|He3e|He4 WhenbothspeciesarenearMaxwellian,withTi<∼Te,therearejusttwocharacteristiccollisionrates.ForZ=1, νe=2.9×10−6nλTe−3/2sec−1;νi=4.8×10 −8 nλTi −3/2 µ −1/2 sec −1 . TemperatureIsotropization Isotropizationisdescribedby dT⊥ dT 2 πeαeβnαλαβ A1/2 22 . IfA<0,tan−1(A1/2)/A1/2isreplacedbytanh−1(−A)1/2/(−A)1/2.ForT⊥≈T≡T, eνT=8.2×10−7nλT−3/2sec−1; νT=1.9×10 i −8 nλZµ 2 −1/2 T −3/2 sec −1 . 34 ThermalEquilibration Ifthecomponentsofaplasmahavedifferenttemperatures,butnorela-tivedrift,equilibrationisdescribedby dTα (mαTβ+mβTα)3/2 ForelectronsandionswithTe≈Ti≡T,thisimplies ν¯/ni=ν¯/ne=3.2×10 e|i i|e −9 sec −1 . Zλ/µT 23/2 cmsec 3 −1 . CoulombLogarithm Fortestparticlesofmassmαandchargeeα=Zαescatteringofffieldparticlesofmassmβandchargeeβ=Zβe,theCoulomblogarithmisdefinedasλ=lnΛ≡ln(rmax/rmin).Hererministhelargerofeαeβ/mαβu¯2andh¯/2mαβu¯,averagedoverbothparticlevelocitydistributions,wheremαβ=mαmβ/(mα+mβ)andu=vα−vβ;rmax=(4πnγeγ2/kTγ)−1/2,where thesummationextendsoverallspeciesγforwhichu¯2 Thefollowingcasesareofparticularinterest:(a)Thermalelectron–electroncollisions λee=23−ln(ne1/2Te−3/2), =24−ln(ne (b)Electron–ioncollisionsλei=λie=23−lnne =24−lnne=30−lnni 1/2 Te −1 ), Te<∼10eV;Te>∼10eV. 1/21/2 −3/2 ZTe−1Te 1/2 Ti −3/2 , , Time/mi 2 −1 35 Time/mi<10ZeV (c)Mixedion–ioncollisions λii=λii=23−ln ZZ(µ+µ) Ti + niZ2 µµβD2 ne Dt ≡ ∂fα ∂t , coll whereFisanexternalforcefield.Thegeneralformofthecollisionintegralis(∂fα/∂t)coll=−∇v·Jα|β,with β J α|β =2πλαβ eα2eβ2 1 mβ fα(v)∇vfβ(v)− mα2 f(v)∇vH(v)− α 1 mβ 36 fβ(v)u−1d3v. Ifspeciesαisaweakbeam(numberandenergydensitysmallcomparedwithbackground)streamingthroughaMaxwellianplasma,then J α|β =−mα να|β vv·∇vf α mdα vαnα α 2 −1 Dt=νee(Fe−fe)+νei(F ¯e−fe);Dfi 2πkTα 3/2 exp − mα(v−vα)2 3/2 2πkT ¯αexp − mα(v−v¯α)2 dt +nα∇·vα=0; c vα×B +Rα; 37 3 dt +pα∇·vα=−∇·qα−Pα:∇vα+Qα. themomentumandenergygainedbytheαthspeciesthroughcollisionswiththeβth;Pαisthestresstensor;andqαistheheatflow. Thetransportcoefficientsinasimpletwo-componentplasma(electronsandsinglychargedions)aretabulatedbelow.Hereand⊥refertothedi-rectionofthemagneticfieldB=bB;u=ve−viistherelativestreamingvelocity;ne=ni≡n;j=−neuisthecurrent;ωce=1.76×107Bsec−1andωci=(me/mi)ωcearetheelectronandiongyrofrequencies,respectively;andthebasiccollisionaltimesaretakentobe 3√4 Heredα/dt≡∂/∂t+vα·∇;pα=nαkTα,wherekisBoltzmann’sconstant;Rα=RαβandQα=Qαβ,whereRαβandQαβarerespectively β β τe= √ nλ sec, whereλistheCoulomblogarithm,and 3√4 1/2 τi= √ nλ µsec. Inthelimitoflargefields(ωcατα1,α=i,e)thetransportprocessesmaybesummarizedasfollows:21 momentumtransferfrictionalforceelectrical conductivitiesthermalforce Ru=ne(j/σ+j⊥/σ⊥);Rei=−Rie≡R=Ru+RT; σ=1.96σ⊥;σ⊥=ne2τe/me;RT=−0.71n∇(kTe)− nkmi 3n conductivitieselectronheatfluxfrictionalheatflux mi qe=qu+qT; equ e e ; iκ⊥ = 2nkTi 2miωci ; =0.71nkTeu+ 3nkTe thermalgradientheatfluxelectronthermal eee qe=−κ∇(kT)−κ∇(kT)−κee⊥T⊥∧b×∇⊥(kTe);eκ =3.2 nkTeτe (Wxx+Wyy)−(Wxx+Wyy)+ η1η1 2τmeωcee ; e κ∧ = 5nkTe 22 Pxz=Pzx=−η2Wxz Pzz=−η0Wzz (herethezaxisisdefinedparalleltoB);ionviscosity iη0iη3 Pyz=Pzy=−η2Wyz+η4Wxz; (Wxx−Wyy);2 −η4Wyz; =0.96nkTiτi;=nkTi iη1 =; 3nkTi 2τ5ωcii ; ωci e η1 electronviscosity eη0eη3 =0.73nkTeτe;=− nkTe =0.51ωce . nkTe 2τωcee ; Forbothspeciestherate-of-straintensorisdefinedas Wjk= ∂vj ∂xj −2 lre(L⊥reinauniformfield), 39 whereLisamacroscopicscaleparalleltothefieldBandL⊥isthesmallerofB/|∇⊥B|andthetransverseplasmadimension.Inaddition,thestandardtransportcoefficientsarevalidonlywhen(3)theCoulomblogarithmsatisfiesλ1;(4)theelectrongyroradiussatisfiesreλD,or8πnemec2B2;(5)relativedriftsu=vα−vβbetweentwospeciesaresmallcomparedwiththethermalvelocities,i.e.,u2kTα/mα,kTβ/mβ;and(6)anomaloustransportprocessesowingtomicroinstabilitiesarenegligible. WeaklyIonizedPlasmas Collisionfrequencyforscatteringofchargedparticlesofspeciesαbyneutralsis α|0 να=n0σs(kTα/mα)1/2, α\\0 wheren0istheneutraldensityandσsisthecrosssection,typically∼5×10−15cm2andweaklydependentontemperature. WhenthesystemissmallcomparedwithaDebyelength,LλD,thechargedparticlediffusioncoefficientsare Dα=kTα/mανα, Intheoppositelimit,bothspeciesdiffuseattheambipolarrate DA= µiDe−µeDi , TiDe+TeDi whereµα=eα/mαναisthemobility.Theconductivityσαsatisfiesσα=nαeαµα. InthepresenceofamagneticfieldBthescalarsµandσbecometensors, J α =σ α ·E=σE+σ⊥E⊥+σ∧E×b, ααα whereb=B/Band ασ=nαeα2/mανα; σ⊥=σνα/(να+ωcα); αα2σ∧=σναωcα/(να2+ωcα). αα222 Hereσ⊥andσ∧arethePedersenandHallconductivities,respectively. 40 IONOSPHERICPARAMETERS23 Thefollowingtablesgiveaveragenighttimevalues.Wheretwonumbersareentered,thefirstreferstothelowerandthesecondtotheupperportionofthelayer. Quantity Altitude(km) Numberdensity(m−3)Height-integratednumberdensity(m−2)Ion-neutralcollisionfrequency(sec−1)Iongyro-/collisionfrequencyratioκiIonPedersonfactorκi/(1+κi2)IonHallfactorκi2/(1+κi2) Electron-neutralcollisionfrequencyElectrongyro-/collisionfrequencyratioκeElectronPedersenfactorκe/(1+κe2)ElectronHallfactorκe2/(1+κe2)MeanmolecularweightIongyrofrequency(sec−1)Neutraldiffusion coefficient(m2sec−1) FRegion160–5005×1010–2×1011 4.5×10150.5–0.054.6×102–5.0×1032.2×10−3–2×10−4 1.080–10 7.8×104–6.2×10510−5–1.5×10−6 1.022–16230–300105 Theterrestrialmagneticfieldinthelowerionosphereatequatoriallatti-tudesisapproximatelyB0=0.35×10−4tesla.Theearth’sradiusisRE=6371km. 41 SOLARPHYSICSPARAMETERS24 SymbolMRgv∞——BmaxT0LFτ5 fromphotosphere Astronomicalunit(radiusofearth’sorbit)Solarconstant(intensityat1AU)ChromosphereandCorona25 Parameter(Units) (ergcm−2s−1) LowchromosphereMiddlechromosphereUpperchromosphereTotal Transitionlayerpressure(dynecm−2)Coronaltemperature(K)at1.1RCoronalenergylosses(ergcm−2s−1) ConductionRadiationSolarWindTotalSolarwindmassloss(gcm−2s−1) QuietSun ActiveRegion 1.50×10131.36×106 Unitsgcmcms−2cms−1gcm−2s−1 gcm−2 GKergs−1ergcm−2s−1 — 2×1062×1063×1054×1060.071066×1041047×1058×1052×10−10 42 THERMONUCLEARFUSION26 Naturalabundanceofisotopes: hydrogenheliumlithium Massratios: nD/nH=1.5×10−4 nHe3/nHe4=1.3×10−6nLi6/nLi7=0.08 2.72×10−41.65×10−21.82×10−41.35×10−2 ====1/36701/60.61/54961/74.1 me/mD=(me/mD)1/2=me/mT=(me/mT)1/2= Absorbedradiationdoseismeasuredinrads:1rad=102ergg−1.Thecurie(abbreviatedCi)isameasureofradioactivity:1curie=3.7×1010countssec−1.Fusionreactions(branchingratiosarecorrectforenergiesnearthecrosssectionpeaks;anegativeyieldmeansthereactionisendothermic):27 (1a)D+D−−−−→T(1.01MeV)+p(3.02MeV) 50% (1b)−−−−→He3(0.82MeV)+n(2.45MeV) 50% (2)D+T−−−−→He4(3.5MeV)+n(14.1MeV) (3)(4)(5a)(5b)(5c)(6)(7a)(7b)(8)(9)(10) D+He3−−−−→He4(3.6MeV)+p(14.7MeV)T+T −−−−→He4+2n+11.3MeV He3+T−−−−→He4+p+n+12.1MeV 51%−−−−→He4(4.8MeV)+D(9.5MeV)43%−−−−→He5(2.4MeV)+p(11.9MeV)6%6 p+Li−−−−→He4(1.7MeV)+He3(2.3MeV)p+Li7−−−−→2He4+17.3MeV 20%−−−−→Be7+n−1.6MeV80%6 D+Li−−−−→2He4+22.4MeVp+B11−−−−→3He4+8.7MeV n+Li6−−−−→He4(2.1MeV)+T(2.7MeV) Thetotalcrosssectioninbarns(1barn=10−24cm2)asafunctionofE,the energyinkeVoftheincidentparticle[thefirstionontheleftsideofEqs.(1)–(5)],assumingthetargetionatrest,canbefittedby28 σT(E)= A5+(A4−A3E)+1 2 −1 A2 wheretheDuanecoefficientsAjfortheprinciplefusionreactionsareasfollows: D–D(1b) A1A2A3A4A5 ReactionratesTemperature (1a+1b)1.5×10−225.4×10−211.8×10−191.2×10−185.2×10−182.1×10−174.5×10−178.8×10−171.8×10−162.2×10−16 D–T (3)10−26 1.4×10−236.7×10−212.3×10−193.8×10−185.4×10−171.6×10−162.4×10−162.3×10−161.8×10−16 T–T (5a–c)10−2810−25 2.1×10−221.2×10−202.6×10−195.3×10−182.7×10−179.2×10−172.9×10−165.2×10−16 47.88482 3.08×10−4 1.1770 D–He3(3)89.27259003.98×10−3 1.297647 T–He3(5a–c)123.111250000 σv)DD=2.33×10( −14 T −2/3 exp(−18.76T −1/3 )cmsec 3 −1 ; PDT σv)DDwattcm−3(includingthesubsequent D–Treaction); =5.6×10−13nDnT( σv)DHe3wattcm−3. 44 RELATIVISTICELECTRONBEAMS Hereγ=(1−β2)−1/2istherelativisticscalingfactor;quantitiesinanalyticformulasareexpressedinSIorcgsunits,asindicated;innumericalformulas,Iisinamperes(A),Bisingauss(G),electronlineardensityNisincm−1,andtemperature,voltageandenergyareinMeV;βz=vz/c;kisBoltzmann’sconstant. Relativisticelectrongyroradius: re= mc2 ν I= A a = ν Child’slaw:(non-relativistic)space-charge-limitedcurrentdensitybetweenparallelplateswithvoltagedropV(inMV)andseparationd(incm)is J=2.34×103V3/2d−2Acm−2. Thesaturatedparapotentialcurrent(magneticallyself-limitedflowalongequi-potentialsinpincheddiodesandtransmissionlines)is29 Ip=8.5×10Gγlnγ+(γ−1) 3 whereGisageometricalfactordependingonthediodestructure:G= w 21/2 A, R1 G= Rc −1 forcylindersofradiiR1(inner)andR2(outer); BEAMINSTABILITIES30 NameElectron-electronBunemanBeam-plasmaWeakbeam-plasmaBeam-plasma(hot-electron)Ionacoustic Anisotropictemperature(hydro)IoncyclotronBeam-cyclotron(hydro)Modifiedtwo-stream(hydro)Ion-ion(equalbeams)Ion-ion(equalbeams) Fornomenclature,seep.50. SaturationMechanismElectronV¯trappinguntilej∼VdElectrontrappingV¯untile∼VdTrappingofbeamelectronsQuasilinearornonlinear(modecoupling)QuasilinearornonlinearQuasilinear,iontailform-ation,nonlinearscattering,orresonancebroadening.Isotropization Ionheating ResonancebroadeningTrappingIontrappingIontrapping 47 ParametersofMostUnstableMode GrowthRate 1 0 0.4 WaveNumber Vdωe 2 0.70.7 mnb m ωe−0.4 Vb nb ωe 3 Vb nb (hot-electron)Ionacoustic ¯bVnb 2 Vb ωeωe ωe ¯23Ve M 1/2 Ωe ¯eV Vb 1 λ−D−1re ωi 0.1Ωi Beam-cyclotron(hydro)Modifiedtwo-2ΩH nΩe −1ri 1.7 ΩH 3>Vd;∼ 0.4ΩH beams)Ion-ion(equal 0 1.2 ΩH 0 U Intheprecedingtables,subscriptse,i,d,b,pstandfor“electron,”“ion,”“drift,”“beam,”and“plasma,”respectively.Thermalvelocitiesaredenotedbyabar.Inaddition,thefollowingareused:mMVT ne,nin Cs=(Te/M)1/2ωe,ωiλD electronmassionmassvelocity temperaturenumberdensityharmonicnumberionsoundspeedplasmafrequencyDebyelength re,riβVA Ωe,ΩiΩHU gyroradius plasma/magneticenergydensityratioAlfv´enspeedgyrofrequency hybridgyrofrequency,ΩH2=ΩeΩi relativedriftvelocityoftwoionspecies APPROXIMATEMAGNITUDESINSOMETYPICALPLASMAS PlasmaTypeInterstellargasGaseousnebulaSolarCoronaDiffusehotplasmaSolaratmosphere,gasdischargeWarmplasmaHotplasmaThermonuclearplasmaThetapinchDensehotplasmaLaserPlasma TeV11102102110102104102102102 λDcm7×102 202×10−17×10−37×10−52×10−47×10−42×10−37×10−57×10−67×10−7 νeisec−17×10−56×10−2 60402×109 1074×1065×1043×1082×10102×1012 Thediagram(facing)givescomparableinformationingraphicalform.22 49 50 LASERS SystemParameters Efficienciesandpowerlevelsareapproximate.31 Type 10.652.061.3151.061.0641.045,1.54,1.3131.0642.941–40.7–1.50.69430.63280.45–0.600.3–100.33710.3–1.10.260.1751.05–1.11.5340.375–1.9 Efficiency Powerlevelsavailable(W)Pulsed>2×1013>109>1073×10121.25×1015 1094×108–1.5×1055×10810141010–5×10410101065×1071012>1085×1077×1063×109 Formulas Ane-mwavewithkBhasanindexofrefractiongivenby n±=[1−ωpe/ω(ω∓ωce)] 2 1/2 , where±referstothehelicity.Therateofchangeofpolarizationangleθasafunctionofdisplacements(Faradayrotation)isgivenby dθ/ds=(k/2)(n−−n+)=2.36×10NBf 4 −2 cm −1 , whereNistheelectronnumberdensity,Bisthefieldstrength,andfisthewavefrequency,allincgs. Thequivervelocityofanelectroninane-mfieldofangularfrequencyωis v0=eEmax/mω=25.6I1/2λ0cmsec−1 2 intermsofthelaserfluxI=cEmax/8π,withIinwatt/cm2,laserwavelengthλ0inµm.Theratioofquiverenergytothermalenergyis Wqu/Wth=mev02/2kT=1.81×10−13λ02I/T, whereTisgivenineV.Forexample,ifI=1015Wcm−2,λ0=1µm,T=2keV,thenWqu/Wth≈0.1. Pondermotiveforce: F=N∇E/8πNc, where Nc=1.1×1021λ0−2cm−3. 2 Foruniformilluminationofalenswithf-numberF,thediameterdatfocus(85%oftheenergy)andthedepthoffocusl(distancetofirstzeroinintensity)aregivenby d≈2.44Fλθ/θDL and l≈±2Fλθ/θDL. 2 Hereθisthebeamdivergencecontaining85%ofenergyandθDListhediffraction-limiteddivergence: θDL=2.44λ/b, wherebistheaperture.Theseformulasaremodifiedfornonuniform(suchasGaussian)illuminationofthelensorforpathologicallaserprofiles. 52 ATOMICPHYSICSANDRADIATION EnergiesandtemperaturesareineV;allotherunitsarecgsexceptwherenoted.Zisthechargestate(Z=0referstoaneutralatom);thesubscriptelabelselectrons.Nreferstonumberdensity,ntoprincipalquantumnumber.Asterisksuperscriptsonlevelpopulationdensitiesdenotelocalthermodynamicequilibrium(LTE)values.ThusNn*istheLTEnumberdensityofatoms(orions)inleveln. Characteristicatomiccollisioncrosssection:(1) πa02=8.80×10−17cm2. Bindingenergyofouterelectroninlevellabelledbyquantumnumbersn,l: Z E∞(n,l) H Z2E∞ (2)=− ∆Enm cm2, wherefmnistheoscillatorstrength,g(n,m)istheGauntfactor,istheincidentelectronenergy,and∆Enm=En−Em. ElectronexcitationrateaveragedoverMaxwellianvelocitydistribution,Xmn=Neσmnv(Refs.34,35):(4) Xmn=1.6×10 −5 fmng(n,m)Ne Te sec −1 , whereg(n,m)denotesthethermalaveragedGauntfactor(generally∼1foratoms,∼0.2forions). 53 Rateforelectroncollisionaldeexcitation:(5) Ynm=(Nm*/Nn*)Xmn. HereNm*/Nn*=(gm/gn)exp(∆Enm/Te)istheBoltzmannrelationforlevelpopulationdensities,wheregnisthestatisticalweightofleveln.Rateforspontaneousdecayn→m(EinsteinAcoefficient)34(6) Anm=4.3×10(gm/gn)fmn(∆Enm)sec 7 2 −1 . Intensityemittedperunitvolumefromthetransitionn→minanopticallythinplasma:(7) Inm=1.6×10−19AnmNn∆Enmwatt/cm3. Conditionforsteadystateinacoronamodel:(8) N0Neσ0nv=NnAn0, wherethegroundstateislabelledbyazerosubscript.Henceforatransitionn→minions,whereg(n,0)≈0.2,(9) Inm=5.1×10 −25fnmgmNeN0 ∆En0 3 exp − ∆En0 cm3 . IonizationandRecombination Inageneraltime-dependentsituationthenumberdensityofthechargestateZsatisfies(10) dN(Z) Classicalionizationcross-section36foranyatomicshellj(11) σi=6×10−14bjgj(x)/Uj2cm2. Herebjisthenumberofshellelectrons;Ujisthebindingenergyoftheejectedelectron;x=/Uj,whereistheincidentelectronenergy;andgisauniversalfunctionwithaminimumvaluegmin≈0.2atx≈4. Ionizationfromiongroundstate,averagedoverMaxwellianelectrondistribu-Z Z1/2 (Te/E∞) Te cm3/sec, Z whereE∞istheionizationenergy. Electron-ionradiativerecombinationrate(e+N(Z)→N(Z−1)+hν)forTe/Z2<∼400eV(Ref.37):(13) αr(Z)=5.2×10 −14 Z Z E∞ 2 Z ln(E∞/Te) Z +0.469(E∞/Te)−1/3cm3/sec. For1eV −13 ZTe 2 −1/2 cm/sec. 3 Collisional(three-body)recombinationrateforsinglyionizedplasma:38(15) α3=8.75×10 −27 Te −4.5 cm/sec. 6 Photoionizationcrosssectionforionsinleveln,l(short-wavelengthlimit):(16) σph(n,l)=1.64×10 −16 Z/nK 537+2l cm, 2 whereKisthewavenumberinRydbergs(1Rydberg=1.0974×105cm−1). 55 IonizationEquilibriumModels Sahaequilibrium:39 NeN1*(Z) Z−1gn (17)exp − ZE∞(n,l) N*(Z−1) = S(Z−1) N(Z) = αr Radiation N.B.EnergiesandtemperaturesareineV;allotherquantitiesareincgsunitsexceptwherenoted.Zisthechargestate(Z=0referstoaneutralatom);thesubscriptelabelselectrons.Nisnumberdensity. Averageradiativedecayrateofastatewithprincipalquantumnumbernis(23) An= m Anm=1.6×1010Z4n−9/2sec. Naturallinewidth(∆EineV):(24) ∆E∆t=h=4.14×10 −15 eVsec, where∆tisthelifetimeoftheline.Dopplerwidth:(25) ∆λ/λ=7.7×10−5(T/µ)1/2, whereµisthemassoftheemittingatomorionscaledbytheprotonmass.OpticaldepthforaDoppler-broadenedline:39(26)τ=3.52×10 −13 fnmλ(Mc/kT) 21/2 NL=5.4×10 −9 fmnλ(µ/T) 1/2 NL, wherefnmistheabsorptionoscillatorstrength,λisthewavelength,andListhephysicaldepthoftheplasma;M,N,andTarethemass,numberdensity,andtemperatureoftheabsorber;µisMdividedbytheprotonmass.Opticallythinmeansτ<1. Resonanceabsorptioncrosssectionatcenterofline:(27) σλ=λc=5.6×10−13λ2/∆λcm2. Wiendisplacementlaw(wavelengthofmaximumblack-bodyemission):(28) λmax=2.50×10−5T−1cm. RadiationfromthesurfaceofablackbodyattemperatureT:(29) W=1.03×105T4watt/cm2. 57 Bremsstrahlungfromhydrogen-likeplasma:26(30) PBr=1.69×10 −32 NeTe 1/2 wherethesumisoverallionizationstatesZ.Bremsstrahlungopticaldepth:41(31) τ=5.0×10 −38 ZN(Z)watt/cm, 2 3 NeNiZ 2 g≈1.2isanaverageGauntfactorandListhephysicalpathlength. Inversebremsstrahlungabsorptioncoefficient42forradiationofangularfre-quencyω:(32) 2 κ=3.1×10−7Zne2lnΛT−3/2ω−2(1−ωp/ω2)−1/2cm−1; hereΛistheelectronthermalvelocitydividedbyV,whereVisthelargerof ωandωpmultipliedbythelargerofZe2/kTandh¯/(mkT)1/2.Recombination(free-bound)radiation:(33) Pr=1.69×10 −32 NeTe 1/2 ZN(Z) 2 Z−1 E∞ 2.5+γ sec, whereγisthekineticplusrestenergydividedbytherestenergymc2.Numberofcyclotronharmonics41trappedinamediumoffinitedepthL:(37) whereβ=8πNkT/B2. LineradiationisgivenbysummingEq.(9)overallspeciesintheplasma. mtr=(57βBL) 1/6 , 58 ATOMICSPECTROSCOPY Spectroscopicnotationcombinesobservationalandtheoreticalelements.Observationally,spectrallinesaregroupedinserieswithlinespacingswhichdecreasetowardtheserieslimit.Everylinecanberelatedtheoreticallytoatransitionbetweentwoatomicstates,eachidentifiedbyitsquantumnumbers.Ionizationlevelsareindicatedbyromannumerals.ThusCIisunionizedcarbon,CIIissinglyionized,etc.Thestateofaone-electronatom(hydrogen)orion(HeII,LiIII,etc.)isspecifiedbyidentifyingtheprincipalquantumnumbern=1,2,...,theorbitalangularmomentuml=0,1,...,n−1,andthespinangularmomentums=±1 1 2(j≥ 1+m/M =− RyZ2 TransitionName Successivelinesinanyseriesaredenotedα,β,γ,etc.Thusthetransition1→3givesrisetotheLyman-βline.Relativisticeffects,quantumelectrodynamiceffects(e.g.,theLambshift),andinteractionsbetweenthenuclearmagnetic 59 momentandthemagneticfieldduetotheelectronproducesmallshiftsand −2 splittings, Inthelighterelementstheelectronsfillupsubshellswithineachshellintheorders,p,d,etc.,andnoshellacquireselectronsuntilthelowershellsarefull.Intheheavierelementsthisruledoesnotalwayshold.Butifaparticularsubshellisfilledinanoblegas,thenthesamesubshellisfilledintheatomsofallelementsthatcomelaterintheperiodictable.Thegroundstateconfigurationsofthenoblegasesareasfollows: HeNeArKrXeRn 1s2 1s22s22p6 1s22s22p63s23p6 1s22s22p63s23p63d104s24p6 1s22s22p63s23p63d104s24p64d105s25p6 1s22s22p63s23p63d104s24p64d104f145s25p65d106s26p6 Alkalimetals(Li,Na,K,etc.)resemblehydrogen;theirtransitionsarede-scribedbygivingnandlintheinitialandfinalstatesforthesingleouter(valence)electron. Forgeneraltransitionsinmostatomstheatomicstatesarespecifiedintermsoftheparity(−1)Σliandthemagnitudesoftheorbitalangularmomen-tumL=Σli,thespinS=Σsi,andthetotalangularmomentumJ=L+S,whereallsumsarecarriedoutovertheunfilledsubshells(thefilledonessumtozero).IfamagneticfieldispresenttheprojectionsML,MS,andMofL,S,andJalongthefieldarealsoneeded.Thequantumnumberssatisfy|ML|≤L≤νl,|MS|≤S≤ν/2,and|M|≤J≤L+S,whereνisthenumberofelectronsintheunfilledsubshell.Upper-caselettersS,P,D,etc.,standforL=0,1,2,etc.,inanalogywiththenotationforasingleelectron.Forexample,thegroundstateofClisdescribedby3p52Po3/2.Thefirstpartindicatesthatthereare5electronsinthesubshellcorrespondington=3andl=1.(Theclosedinnersubshells1s22s22p63s2,identicalwiththeconfigura-tionofMg,areusuallyomitted.)Thesymbol‘P’indicatesthattheangularmomentaoftheouterelectronscombinetogiveL=1.Theprefix‘2’repre-sentsthevalueofthemultiplicity2S+1(thenumberofstateswithnearlythesameenergy),whichisequivalenttospecifyingS=1 thevalueofJ.Thesuperscript‘o’indicatesthatthestatehasoddparity;itwouldbeomittedifthestatewereeven. Thenotationforexcitedstatesissimilar.Forexample,heliumhasastate1s2sS1whichlies19.72eV(159,856cm−1)abovethegroundstate1s21S0.Butthetwo“terms”donot“combine”(transitionsbetweenthemdonotoccur)becausethiswouldviolate,e.g.,thequantum-mechanicalselectionrulethattheparitymustchangefromoddtoevenorfromeventoodd.Forelectricdipoletransitions(theonlyonespossibleinthelong-wavelengthlimit),otherselectionrulesarethatthevalueoflofonlyoneelectroncanchange,andonlyby∆l=±1;∆S=0;∆L=±1or0;and∆J=±1or0(butL=0doesnotcombinewithL=0andJ=0doesnotcombinewithJ=0).Transitionsarepossiblebetweentheheliumgroundstate(whichhasS=0,L=0,J=0,andevenparity)and,e.g.,thestate1s2p1Po1(withS=0,L=1,J=1,oddparity,excitationenergy21.22eV).Theserulesholdaccuratelyonlyforlightatomsintheabsenceofstrongelectricormagneticfields.Transitionsthatobeytheselectionrulesarecalled“allowed”;thosethatdonotarecalled“forbidden.” 3 Theamountofinformationneededtoadequatelycharacterizeastatein-creaseswiththenumberofelectrons;thisisreflectedinthenotation.Thus43OIIhasanallowedtransitionbetweenthestates2p23p2o F7/2and2p2(1D)3d2F7/2(andbetweenthestatesobtainedbychangingJfrom7/2to5/2ineitherorbothterms).Herebothstateshavetwoelec-tronswithn=2andl=1;theclosedsubshells1s22s2arenotshown.Theouter(n=3)electronhasl=1inthefirststateandl=2inthesecond.Theprimeindicatesthatiftheoutermostelectronwereremovedbyionization,theresultingionwouldnotbeinitslowestenergystate.Theexpression(1D)givethemultiplicityandtotalangularmomentumofthe“parent”term,i.e.,thesubshellimmediatelybelowthevalencesubshell;thisisunderstoodtobethesameinbothstates.(Grandparents,etc.,sometimeshavetobespecifiedinheavieratomsandions.)Anotherexample43istheallowedtransitionfrom 2o212 2p2(3P)3p2Po(orP)to2p(D)3dS1/2,inwhichthereisa“spin1/23/2 flip”(fromantiparalleltoparallel)inthen=2,l=1subshell,aswellaschangesfromonestatetotheotherinthevalueoflforthevalenceelectronandinL. Thedescriptionoffinestructure,StarkandZeemaneffects,spectraofhighlyionizedorheavyatoms,etc.,ismorecomplicated.ThemostimportantdifferencebetweenopticalandX-rayspectraisthatthelatterinvolveenergychangesoftheinnerelectronsratherthantheouterones;oftenseveralelectronsparticipate. 61 COMPLEX(DUSTY)PLASMAS Complex(dusty)plasmas(CDPs)mayberegardedasanewandunusualstateofmatter.CDPscontainchargedmicroparticles(dustgrains)inadditiontoelectrons,ions,andneutralgas.Electrostaticcouplingbetweenthegrainscanvaryoverawiderange,sothatthestatesofCDPscanchangefromweaklycoupled(gaseous)tocrystalline.CDPscanbeinvestigatedatthekineticlevel(individualparticlesareeasilyvisualizedandrelevanttimescalesareaccessi-ble).CDPsareofinterestasanon-Hamiltoniansystemofinteractingparticlesandasameanstostudygenericfundamentalphysicsofself-organization,pat-ternformation,phasetransitions,andscaling.Theirdiscoveryhasthereforeopenednewwaysofprecisioninvestigationsinmany-particlephysics.Typicalexperimentaldustproperties grainsize(radius)a0.3−30µm,massmd∼3×10−7−3×10−13g,numberdensity(intermsoftheinterparticledistance)nd∼∆−3∼103−107cm−3,temperatureTd∼3×10−2−102eV.Typicaldischarge(bulk)plasmas gaspressurep∼10−2−1Torr,TiTn3×10−2eV,vTi7×104cm/s(Ar),Te∼0.3−3eV,nine∼108−1010cm−3,screeninglengthλDλDi∼20−200µm,ωpi2×106−2×107s−1(Ar).BfieldsuptoB∼3T.Dimensionless Havnesparameter normalizedcharge dust-dustscatteringparameterdust-plasmascatteringparametercouplingparameterlatticeparameterparticleparameter latticemagnetizationparameter P=|Z|nd/nez=|Z|e2/kTeaβd=Z2e2/kTdλDβe,i=|Z|e2/kTe,iλD Γ=(Z2e2/kTd∆)exp(−∆/λD)κ=∆/λDα=a/∆µ=∆/rd Typicalexperimentalvalues:P∼10−4−102,z2−4(Z∼103−105electroncharges),Γ<103,κ∼0.3−10,α∼10−4−3×10−2,µ<1Frequencies dustplasmafrequency ωpd=(4πZ2e2nd/md)1/2 (|Z| √ P dust-gasfrictionratedustgyrofrequencyVelocities dustthermalvelocityνnd∼10a2p/mdvTnωcd=ZeB/mdc vTd=(kTd/md)1/2≡[ Td md ]1/2vTi dustacousticwavevelocity CDA=ωpdλD (|Z| P 8πa2nevTeexp(−z), Ii= √ Tz i . Grainsarechargednegatively.Thegrainchargecanvaryinresponsetospatialandtemporalvariationsoftheplasma.Chargefluctuationsarealwayspresent,withfrequencyωch.Otherchargingmechanismsarephotoemission,secondaryemission,thermionicemission,fieldemission,etc.Chargeddustgrainschangetheplasmacomposition,keepingquasineutrality.AmeasureofthisistheHavnesparameterP=|Z|nd/ne.ThebalanceofIeandIiyields exp(−z)= mi Te 1/2 1+ Te Whentherelativechargedensityofdustislarge,P1,thegrainchargeZmonotonicallydecreases. Forcesandmomentumtransfer Inadditiontotheusualelectromagneticforces,grainsincomplexplasmasarealsosubjectto:gravityforceFg=mdg;thermophoreticforce √4 (a2/vTn)κn∇TnFth=− 15(whereκnisthecoefficientofgasthermalconductivity);forcesassociatedwiththemomentumtransferfromotherspecies,Fα=−mdναdVαd,i.e.,neutral,ion,andelectrondrag.Forcollisionsbetweenchargedparticles,twolimitingcasesaredistinguishedbythemagnitudeofthescatteringparameterβα.Whenβα1theresultisindependentofthesignofthepotential.Whenβα1,theresultsforrepulsiveandattractiveinteractionpotentialsaredifferent.Fortypicalcomplexplasmasthehierarchyofscatteringparametersisβe(∼0.01−0.3)βi(∼1−30)βd(∼103−3×104).Thegenericexpressionsfordifferenttypesofcollisionsare47 √ ναd=(4 2 Ion-dustcollisions Φid= z2Λed βe1 1 Forrepulsiveinteraction(electron-dustanddust-dust)Λαd=zα ∞0 e−zαxln[1+4(λD/aα)2x2]dx−2zα ∞ e−zαxln(2x−1)dx, 1 whereze=z,ae=a,andad=2a.Forion-dust(attraction) Λidz ∞0 e−zxln 1+2(Ti/Te)(λD/a)x REFERENCES Whenanyoftheformulasanddatainthiscollectionarereferenced inresearchpublications,itissuggestedthattheoriginalsourcebecitedratherthantheFormulary.Mostofthismaterialiswellknownand,forallpracticalpurposes,isinthe“publicdomain.”Numerouscolleaguesandreaders,toonumeroustolistbyname,havehelpedincollectingandshapingtheFormularyintoitspresentform;theyaresincerelythankedfortheirefforts. Severalbook-lengthcompilationsofdatarelevanttoplasmaphysics areavailable.Thefollowingareparticularlyuseful: C.W.Allen,AstrophysicalQuantities,3rdedition(AthlonePress,Lon-don,1976). A.Anders,AFormularyforPlasmaPhysics(Akademie-Verlag,Berlin,1990). H.L.Anderson(Ed.),APhysicist’sDeskReference,2ndedition(Amer-icanInstituteofPhysics,NewYork,1989). K.R.Lang,AstrophysicalFormulae,2ndedition(Springer,NewYork,1980). Thebooksandarticlescitedbelowareintendedprimarilynotforthepurposeofgivingcredittotheoriginalworkers,but(1)toguidethereadertosourcescontainingrelatedmaterialand(2)toindicatewheretofindderivations,ex-planations,examples,etc.,whichhavebeenomittedfromthiscompilation.AdditionalmaterialcanalsobefoundinD.L.Book,NRLMemorandumRe-portNo.3332(1977). 1.SeeM.AbramowitzandI.A.Stegun,Eds.,HandbookofMathematical Functions(Dover,NewYork,1968),pp.1–3,foratabulationofsomemathematicalconstantsnotavailableonpocketcalculators.2.H.W.Gould,“NoteonSomeBinomialCoefficientIdentitiesofRosen-baum,”J.Math.Phys.10,49(1969);H.W.GouldandJ.Kaucky,“Eval-uationofaClassofBinomialCoefficientSummations,”J.Comb.Theory1,233(1966).3.B.S.Newberger,“NewSumRuleforProductsofBesselFunctionswith ApplicationtoPlasmaPhysics,”J.Math.Phys.23,1278(1982);24,2250(1983).4.P.M.MorseandH.Feshbach,MethodsofTheoreticalPhysics(McGraw-HillBookCo.,NewYork,1953),Vol.I,pp.47–52andpp.656–666. 66 5.W.D.Hayes,“ACollectionofVectorFormulas,”PrincetonUniversity, Princeton,NJ,1956(unpublished),andpersonalcommunication(1977).6.SeeQuantities,UnitsandSymbols,reportoftheSymbolsCommittee oftheRoyalSociety,2ndedition(RoyalSociety,London,1975)foradiscussionofnomenclatureinSIunits.7.E.R.CohenandB.N.Taylor,“The1986AdjustmentoftheFundamental PhysicalConstants,”CODATABulletinNo.63(PergamonPress,NewYork,1986);J.Res.Natl.Bur.Stand.92,85(1987);J.Phys.Chem.Ref.Data17,1795(1988).8.E.S.Weibel,“DimensionallyCorrectTransformationsbetweenDifferent SystemsofUnits,”Amer.J.Phys.36,1130(1968).9.J.Stratton,ElectromagneticTheory(McGraw-HillBookCo.,NewYork, 1941),p.508.10.ReferenceDataforEngineers:Radio,Electronics,Computer,andCom-munication,7thedition,E.C.Jordan,Ed.(SamsandCo.,Indianapolis,IN,1985),Chapt.1.ThesedefinitionsareInternationalTelecommunica-tionsUnion(ITU)Standards.11.H.E.Thomas,HandbookofMicrowaveTechniquesandEquipment (Prentice-Hall,EnglewoodCliffs,NJ,1972),p.9.FurthersubdivisionsaredefinedinRef.10,p.I–3.12.J.P.CatchpoleandG.Fulford,Ind.andEng.Chem.58,47(1966); reprintedinrecenteditionsoftheHandbookofChemistryandPhysics(ChemicalRubberCo.,Cleveland,OH)onpp.F306–323.13.W.D.Hayes,“TheBasicTheoryofGasdynamicDiscontinuities,”inFun-damentalsofGasDynamics,Vol.III,HighSpeedAerodynamicsandJetPropulsion,H.W.Emmons,Ed.(PrincetonUniversityPress,Princeton,NJ,1958).14.W.B.Thompson,AnIntroductiontoPlasmaPhysics(Addison-Wesley PublishingCo.,Reading,MA,1962),pp.86–95.15.L.D.LandauandE.M.Lifshitz,FluidMechanics,2ndedition(Addison-WesleyPublishingCo.,Reading,MA,1987),pp.320–336.16.TheZfunctionistabulatedinB.D.FriedandS.D.Conte,ThePlasma DispersionFunction(AcademicPress,NewYork,1961).17.R.W.LandauandS.Cuperman,“StabilityofAnisotropicPlasmasto Almost-PerpendicularMagnetosonicWaves,”J.PlasmaPhys.6,495(1971). 67 18.B.D.Fried,C.L.Hedrick,J.McCune,“Two-PoleApproximationforthe PlasmaDispersionFunction,”Phys.Fluids11,249(1968).19.B.A.Trubnikov,“ParticleInteractionsinaFullyIonizedPlasma,”Re-viewsofPlasmaPhysics,Vol.1(ConsultantsBureau,NewYork,1965),p.105.20.J.M.Greene,“ImprovedBhatnagar–Gross–KrookModelofElectron-Ion Collisions,”Phys.Fluids16,2022(1973).21.S.I.Braginskii,“TransportProcessesinaPlasma,”ReviewsofPlasma Physics,Vol.1(ConsultantsBureau,NewYork,1965),p.205.22.J.Sheffield,PlasmaScatteringofElectromagneticRadiation(Academic Press,NewYork,1975),p.6(afterJ.W.Paul).23.K.H.LloydandG.H¨arendel,“NumericalModelingoftheDriftandDe-formationofIonosphericPlasmaCloudsandoftheirInteractionwith OtherLayersoftheIonosphere,”J.Geophys.Res.78,7389(1973).24.C.W.Allen,AstrophysicalQuantities,3rdedition(AthlonePress,Lon-don,1976),Chapt.9.25.G.L.WithbroeandR.W.Noyes,“MassandEnergyFlowintheSolar ChromosphereandCorona,”Ann.Rev.Astrophys.15,363(1977).26.S.GlasstoneandR.H.Lovberg,ControlledThermonuclearReactions (VanNostrand,NewYork,1960),Chapt.2.27.Referencestoexperimentalmeasurementsofbranchingratiosandcross sectionsarelistedinF.K.McGowan,etal.,Nucl.DataTablesA6,353(1969);A8,199(1970).Theyieldslistedinthetablearecalculateddirectlyfromthemassdefect.28.G.H.Miley,H.TownerandN.Ivich,FusionCrossSectionandReactivi-ties,Rept.COO-2218-17(UniversityofIllinois,Urbana,IL,1974);B.H.Duane,FusionCrossSectionTheory,Rept.BNWL-1685(BrookhavenNationalLaboratory,1972).29.J.M.Creedon,“RelativisticBrillouinFlowintheHighν/γLimit,” J.Appl.Phys.46,2946(1975).30.See,forexample,A.B.Mikhailovskii,TheoryofPlasmaInstabilities Vol.I(ConsultantsBureau,NewYork,1974).Thetableonpp.48–49wascompiledbyK.Papadopoulos. 68 31.TablepreparedfromdatacompiledbyJ.M.McMahon(personalcom-munication,D.Book,1990)andA.Ting(personalcommunication,J.D.Huba,2004).32.M.J.Seaton,“TheTheoryofExcitationandIonizationbyElectronIm-pact,”inAtomicandMolecularProcesses,D.R.Bates,Ed.(NewYork,AcademicPress,1962),Chapt.11.33.H.VanRegemorter,“RateofCollisionalExcitationinStellarAtmo-spheres,”Astrophys.J.136,906(1962).34.A.C.KolbandR.W.P.McWhirter,“IonizationRatesandPowerLoss fromθ-PinchesbyImpurityRadiation,”Phys.Fluids7,519(1964).35.R.W.P.McWhirter,“SpectralIntensities,”inPlasmaDiagnosticTech-niques,R.H.HuddlestoneandS.L.Leonard,Eds.(AcademicPress,NewYork,1965).36.M.Gryzinski,“ClassicalTheoryofAtomicCollisionsI.TheoryofInelastic Collision,”Phys.Rev.138A,336(1965).37.M.J.Seaton,“RadiativeRecombinationofHydrogenIons,”Mon.Not. Roy.Astron.Soc.119,81(1959).38.Ya.B.Zel’dovichandYu.P.Raizer,PhysicsofShockWavesandHigh-TemperatureHydrodynamicPhenomena(AcademicPress,NewYork,1966),Vol.I,p.407.39.H.R.Griem,PlasmaSpectroscopy(AcademicPress,NewYork,1966).40.T.F.Stratton,“X-RaySpectroscopy,”inPlasmaDiagnosticTechniques, R.H.HuddlestoneandS.L.Leonard,Eds.(AcademicPress,NewYork,1965).41.G.Bekefi,RadiationProcessesinPlasmas(Wiley,NewYork,1966).42.T.W.JohnstonandJ.M.Dawson,“CorrectValuesforHigh-Frequency PowerAbsorptionbyInverseBremsstrahlunginPlasmas,”Phys.Fluids16,722(1973).43.W.L.Wiese,M.W.Smith,andB.M.Glennon,AtomicTransitionProb-abilities,NSRDS-NBS4,Vol.1(U.S.Govt.PrintingOffice,Washington,1966).44.F.M.PeetersandX.Wu,“Wignercrystalofascreened-Coulomb-interactioncolloidalsystemintwodimensions”,Phys.Rev.A35,3109(1987) 69 45.S.Zhdanov,R.A.Quinn,D.Samsonov,andG.E.Morfill,“Large-scale steady-statestructureofa2Dplasmacrystal”,NewJ.Phys.5,74(2003).46.J.E.Allen,“Probetheory–theorbitalmotionapproach”,Phys.Scripta 45,497(1992).47.S.A.Khrapak,A.V.Ivlev,andG.E.Morfill,“Momentumtransferin complexplasmas”,Phys.Rev.E(2004).48.V.E.Fortovetal.,“Dustyplasmas”,Phys.Usp.47,447(2004). 70 因篇幅问题不能全部显示，请点此查看更多更全内容