Ion heating due to ionisation and recombination - CLF€¦ · · 2016-11-18Ion heating due to...
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Ion heating due to ionisation and recombination R. G. Evans Central Laser Facility, CCLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon., OX11 0QX, UK High Power Laser Science ■ Theory and Computation 95 Central Laser Facility Annual Report ■ 2005/2006 Main contact email address [email protected] 3 Introduction It is very common in plasma physics, particularly in plasmas of moderately high atomic number for the average charge state <Z> = n e /n i to be somewhat less than the nuclear charge Z nuc . For instance Neon is only fully ionised for temperatures of more than 500eV or so and Xenon is only fully ionised above around 50keV. It is usual to distinguish between Z nuc and <Z> in the description of the plasma but much less common [1,2] to recognise that plasmas contain ions of many different charge states which each have different dynamics and collision frequencies. In reality any one ion will not have a constant charge state but its charge will fluctuate as a result of ionisation and recombination events and each charge state will have slightly different dynamics in the plasma electromagnetic fields. In many ways the charge fluctuations are analogous to the electron-ion collisions that result in inverse bremsstrahlung absorption of electromagnetic wave energy by electrons. If the electrons are initially hotter than the ions or there is an external energy source driving plasma waves then it is possible for the ions to steadily gain energy as a result of the random changes of charge state. Somewhat similar effects occur in dusty plasmas [3] where the electric charge on dust particles fluctuates, energy is not always conserved since the Hamiltonian H = p 2 /2M + ZeΦ is time dependent through the fluctuations of Z. Motion in an Ion-acoustic wave We consider an idealised case where the plasma ions of mass M may exist in two charge states Z 1 and Z 2 (Z 2 >Z 1 ) and for simplicity we consider that the ionisation rate ν ion from Z 1 to Z 2 is the same as the recombination rate ν rec from Z 2 to Z 1 so that the equilibrium densities of Z 1 and Z 2 are equal. We also define ν = ν ion /2 = ν rec /2 which is the rate for a complete cycle of ionisation and recombination. Consider the motion of the ion in charge state Z 1 in a monochromatic ion acoustic wave described by its longitudinal electric field E = E 0 cos(ωt - kx). Since the wave is longitudinal and assumed to be small amplitude then the density fluctuation is (δn/n) = v o /v ph where v o = Z 1 eE 0 /Mω is the oscillation velocity and v ph = ω/k = (Z 1 k B T e /M) 1/2 is the ion acoustic speed, k B being Boltzmann’s constant. Initially the ion oscillation is around the point x = 0 and has no net drift so that v = (Z 1 eE 0 /Mω) sin(ωt). We now consider what happens when the ion of charge state Z 1 is ionised to charge state Z 2 at some phase φ 1 of the electric field. Ionisation generally occurs either via electron impact or by photoionisation and in either case there is little change in the ion momentum. So immediately after the ionisation event we may write the ion velocity as v = (Z 2 eE 0 /Mω) sin ωt + (Z 1 -Z 2 )(eE 0 /Mω) sin φ 1 The first term corresponds to the oscillatory motion in the new charge state and the second to an average drift velocity. If the ion of charge Z 2 now recombines with an electron at phase φ 2 to produce an ion of charge Z 1 again with little change in the ion momentum then the drift velocity after one ionisation - recombination cycle is v D = (Z 1 -Z 2 )(eE o /Mω)sinφ 1 + (Z 2 -Z 1 )(eE/Mω) sin φ 2 The velocity increments do not cancel out since they occur at different phases of the wave. We rewrite this last equation as (1) where φ =(φ 1 + φ 2 )/2 and δφ = φ 1 - φ 2 After many uncorrelated cycles of this process the drift velocity will perform a random walk so that the expectation value of the energy increases linearly with time. The absolute phase φ is random so <cos 2 φ> = 1/2 and we evaluate the term in δφ = ωτ by taking the time delay τ between ionisation and recombination to be exponentially distributed so that we need to evaluate giving our final result for the rate of increase of ion energy (2) For small values of ν/ω the heating rate is proportional to ν while for ν >> ω the heating rate falls as ν -1 since the successive ionisation and recombination events are strongly correlated. This calculation is strictly valid only for small ion wave amplitudes (v o /v ph )=(δn/n) << 1 since we assume that the ionisation and recombination rates are independent of the phase of the waves [4] . Motion In a Static Potential Well We consider the motion of an ion initially of charge Z 1 in a potential well eΦ = ax 2 with x = x 1 sinωt and initial energy E 1 =Z 1 ax 1 2 . If the ion changes its charge state to Z 2 at phase φ 1 with no change in momentum then the new 2 2 2 2 o 2 1 2 1 2 v 2 1 v 2 1 ω ν ω ν + - = M Z Z Z t M D + = - ∫ ∞ 2 2 2 0 2 2 1 2 sin ) exp( ω ν ω τ ωτ ντ ν d - = 2 sin cos v 4 v 2 2 2 o 2 1 1 2 2 δφ φ ν Z Z Z t D ) 2 sin( ) cos( v 2 v 1 2 1 δφ φ o D Z Z Z - =
Transcript of Ion heating due to ionisation and recombination - CLF€¦ · · 2016-11-18Ion heating due to...
Ion heating due to ionisation and recombinationR. G. EvansCentral Laser Facility, CCLRC Rutherford Appleton Laboratory, Chilton, Didcot, Oxon., OX11 0QX, UK
High Power Laser Science n Theory and Computation
95Central Laser Facility Annual Report n 2005/2006
Main contact email address [email protected]
3
IntroductionIt is very common in plasma physics, particularly inplasmas of moderately high atomic number for the averagecharge state <Z> = ne/ni to be somewhat less than thenuclear charge Znuc. For instance Neon is only fullyionised for temperatures of more than 500eV or so andXenon is only fully ionised above around 50keV. It isusual to distinguish between Znuc and <Z> in thedescription of the plasma but much less common [1,2] torecognise that plasmas contain ions of many differentcharge states which each have different dynamics andcollision frequencies.
In reality any one ion will not have a constant charge statebut its charge will fluctuate as a result of ionisation andrecombination events and each charge state will haveslightly different dynamics in the plasma electromagneticfields. In many ways the charge fluctuations are analogousto the electron-ion collisions that result in inversebremsstrahlung absorption of electromagnetic wave energyby electrons. If the electrons are initially hotter than theions or there is an external energy source driving plasmawaves then it is possible for the ions to steadily gain energyas a result of the random changes of charge state.
Somewhat similar effects occur in dusty plasmas [3] wherethe electric charge on dust particles fluctuates, energy isnot always conserved since the Hamiltonian H = p2/2M +ZeΦ is time dependent through the fluctuations of Z.
Motion in an Ion-acoustic waveWe consider an idealised case where the plasma ions ofmass M may exist in two charge states Z1 and Z2 (Z2 > Z1)and for simplicity we consider that the ionisation rate νionfrom Z1 to Z2 is the same as the recombination rate νrecfrom Z2 to Z1 so that the equilibrium densities of Z1 andZ2 are equal. We also define ν = νion/2 = νrec/2 which isthe rate for a complete cycle of ionisation andrecombination.
Consider the motion of the ion in charge state Z1 in amonochromatic ion acoustic wave described by itslongitudinal electric field E = E0 cos(ωt - kx). Since thewave is longitudinal and assumed to be small amplitudethen the density fluctuation is (δn/n) = vo/vph where vo =Z1eE0/Mω is the oscillation velocity and vph = ω/k =(Z1kBTe/M)1/2 is the ion acoustic speed, kB beingBoltzmann’s constant. Initially the ion oscillation isaround the point x = 0 and has no net drift so that
v = (Z1eE0/Mω) sin(ωt).
We now consider what happens when the ion of chargestate Z1 is ionised to charge state Z2 at some phase φ1 ofthe electric field. Ionisation generally occurs either viaelectron impact or by photoionisation and in either casethere is little change in the ion momentum. So immediatelyafter the ionisation event we may write the ion velocity as
v = (Z2eE0/Mω) sin ωt + (Z1 - Z2)(eE0/Mω) sin φ1
The first term corresponds to the oscillatory motion in thenew charge state and the second to an average driftvelocity. If the ion of charge Z2 now recombines with anelectron at phase φ2 to produce an ion of charge Z1 againwith little change in the ion momentum then the driftvelocity after one ionisation - recombination cycle is
vD = (Z1 - Z2)(eEo/Mω)sinφ1 + (Z2-Z1)(eE/Mω) sin φ2
The velocity increments do not cancel out since they occurat different phases of the wave. We rewrite this lastequation as
(1)
where φ = (φ1 + φ2)/2 and δφ = φ1 - φ2
After many uncorrelated cycles of this process the driftvelocity will perform a random walk so that theexpectation value of the energy increases linearly withtime.
The absolute phase φ is random so <cos2φ> = 1/2 and weevaluate the term in δφ = ωτ by taking the time delay τbetween ionisation and recombination to be exponentiallydistributed so that we need to evaluate
giving our final result for the rate of increase of ion energy
(2)
For small values of ν/ω the heating rate is proportional toν while for ν >> ω the heating rate falls as ν-1 since thesuccessive ionisation and recombination events are stronglycorrelated. This calculation is strictly valid only for smallion wave amplitudes (vo/vph) = (δn/n) << 1 since weassume that the ionisation and recombination rates areindependent of the phase of the waves [4].
Motion In a Static Potential WellWe consider the motion of an ion initially of charge Z1 ina potential well eΦ = ax2 with x = x1 sinωt and initialenergy E1 = Z1ax1
2. If the ion changes its charge state toZ2 at phase φ1 with no change in momentum then the new
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High Power Laser Science n Theory and Computation
Central Laser Facility Annual Report n 2005/200696
3energy is E2 = E1 + (Z2 - Z1)ax1
2sin2φ1. The frequency andphase of the oscillation in the new charge state aredifferent to their previous values and the exact algebrabecomes lengthy. If we restrict the analysis to ionisationand recombination rates smaller than the oscillation periodwe can assume all phases to be randomly distributed andneglect the phase change at ionisation and recombination.If the ion now returns to charge Z1 at phase φ2 of its newoscillation and its new energy is E3, then we find
so that with randomly distributed φ1 and φ2:
This is an exponential increase of the mean energy witheach ionisation recombination cycle. It is possible becausethe motion of a particle with time dependent chargecorresponds to a time dependent Hamiltonian and energyis in general not conserved. Truly electrostatic structuresin plasmas are rare but the late stage evolution of manyplasma instabilities such as the two stream instability givesrise to long lived density structures. In most cases thedensity structures correspond to the expulsion of electronsby the ponderomotive force associated with wave energyand so do not provide confining potentials for the ions.Quasi resonant particles in a wave see an almost staticpotential and will experience this exponential growth inenergy until they are shifted away from resonance.
Electron Ion EquilibrationNormally electron ion temperature equilibration proceedsvia the mechanism of screened binary collisions giving theSpitzer rate for Te >> Ti
where µ = M/mp and mp is the proton mass
The process is slower than other collision rates because thelarge electron ion mass ratio gives only a small energytransfer in each collision.
If ZTe >> Ti the thermal plasma will have an equilibriumexcitation of ion-acoustic waves with dispersion relationωia = kcs. The energy in the ion-acoustic modes iscalculated by a procedure analogous to the simplederivation of the ratio of kinetic to electrostatic energy inthe Langmuir oscillation of electrons.
If the plasma is considered to be in a box of side L thenmodes exist for k = 2πN/L and the maximum value of k isgiven by the Landau damping limit kcs = ωpi where ωpi
2 = 4πniZ2e2/M. The energy driving the ion waves is
due to the electron thermal energy so we now ascribe anenergy of kBTe to each ion mode giving after a littlemanipulation exactly the same result as for the Langmuirmodes:
(Electron Kinetic Energy / Ion Acoustic wave energy) =neλD
3 where λD is the electron Debye length
It is only when plasmas become non-ideal (few particlesper Debye sphere) that there is significant energy in theelectrostatic modes. To use this result with (2) we need anexpression for the average oscillatory velocity of the ion-acoustic modes and we note that for a mode ofwavenumber k, vosc
2 ~ k-2 while the density of modes ~k2
so that the mean square oscillation velocity is
M<vosc2> = ZkBTe / (neλD
3) or <vosc2> / cs
2 = 1 / (neλD3)
Using very approximate ionisation rates [5] we find that forstrongly coupled plasmas of high Z, around solid densityand at temperatures of hundreds of eV the ionisationheating can be important but it is in a regime where theSpitzer theory is already of limited applicability.
ConclusionThe ion dynamics driven by stochastic changes of ioncharge lead to a novel ion heating mechanism which mayneed to be included in the analysis of some high Z plasmaexperiments. The simple model presented here would needto be extended for very large amplitude waves whereionisation and recombination rates may change with thedensity (and in some cases temperature) changesassociated with the waves. Also a more general treatmentwould allow for a wider distribution of simultaneous ioncharge states.
References1. S. J. Gitomer et al., 1986 Phys Fluids 29, 2679 2. E. M. Epperlein, R. W. Short, A. Simon (1984) Phys
Rev E 49, 24803. M. R. Jana, A. Sen and P. K. Kaw (1993) Phys Rev E
48, 4, 39304. R. W. P. McWhirter and R. Wilson (1974) J Phys B
Atom Molec Phys 7, 15885. R. W. P. McWhirter (1965) “Spectral Intensities” in
Plasma Diagnostic Techniques ed R. H. Huddlestoneand S. L. Leonard (Academic Press, New York 1965)
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