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2 VOLUmn 22, NumaSnS8 PHYSICAL P EVIEW LETTERS 24 Fsuu~ay 1969 t I I FREE-FALL DECAY OF A MERCURY VAPOR PLASMA IN THE EARLY AFTERGLOW* B. C. Gregory Physics Department, Trent University, Peterborough, Canada (Received 6 January 1969) "A free-fall theory is developed for the decay of the average electron number density and of the electron temperature in a pulsed cylindrical mercury vapor discharge plasma using moment eqations for the Ions with a collision term for momentum loss, and &asumritained with Maxwellisn experimental electrons. results The on plasma four tubes assumption of varying was diameter.1 used. Good agreement is ob- As the mercury vapor pressure is lowered, the tion, an ion-neutral momentum-lose collision afterglow decay profile of the average electron term, the assumption that eiectrons are Maxweldensity (ne(t)) in a cylindrical discharge tube be- lian, and an equation describing the energy decay comes quite nonexponential in time (Fig. 1). This of the electrons. Letter is concerned with the nonexponential be- The theory is based upon the good agreement havior of (xe(t)) which occurs path when of the the charged mean free particles is greater than the between the simplified moment treatment and of Kino Shawl and the more exact treatment of Self' transverse dimensions of the tube. Experimental and Parker.$ We have used the steady-state iniresults are explained by using the first two mo- tial conditions for electron density (= ion density) ment equations for the ions, the plasma assump- and drift velocity worked out in Ref. 1 for cylin- Remoduced by the CLEARINGHOUSE 335 I- Federal Scientific & Technical 33 Information Springfield Vs 22151

3 VOLUNM22, NUMoSi8 PHYSICAL REVIEW LETTERS 24FaasuAAvI969 'Nre where ne is the electron number density, neo its axial value, o the space potential, -e the electronic charge, k Boltzmann's constant, and Te the electron temperature (assumed a function of time only and not of space). With these assumpmi 0tort tions, the first two moment equations for the ions e.. are ort.aty.. an 1 8, S\ 57 + = 0, (2) tenrn3) = and DyO ' a 8t 8i e a o,+8r+,+ + i?0. (3) K " -,.~ Here v is the ion drift velocity, v an ion-neutral -. N, collision frequency, and M the ion mass. I T - P A third equation is necessary to describe ener- " C ]J gy changes. This states that the rate of decrease &OR of average electron energy equals the electron rewall R 0potential: flux times the energy equivalent of the wall d a 3 2nkTe2rrdr= -2va(vn) e p " (4) '09! 1 where a is the tube radius and Vow is the tube t fo )--,wall potential. This equation implies that each :i. G. Measured average electron number-density electron e~fo arriving h lcrngs at the wall transports hc sr-awlenergy profiles in the afterglow. Tube C with pressure as ei~w from the electron gas, which is re-maxwelparameter. lianized immediately. The electron arrival rate at the wall equals that of the ions. In reality drical geometry, and (W) constant ionization with most electrons (energy less than e~pw) will undergo times. numerous wall collisions during their lifenlo ion kinetic pressure term, (ii) ionization proportional to electron density with no pressure times. term, and (iii) ionization proportional to electron Since Te is only a function of time, Eqs. (4) and density with an ion kinetic pressure term. Case (2) may be combined to give (iii) was shown in Ref. 1 to give results very sim- dt T dn ilar to the more exact treatment of Refs. 2 and 3, --- (--)-N-", (5) even though the fluid method assumes that all the ions have the same velocity at any point, where N= fo'21nrdr and 3 =epw,,/kte. Our theoretical results for the afterglow indi- The fact that 03 is almost constant for a very cate that in case (i) above the electron density de- wide variety of discharge parameters 2 enables cays about 20-30% more slowly than cases (ii) us to integrate numerically the three coupled nonand (iii), which agree quite closely. Since case linear equations (2), (3), and (5) starting with the (iii) seems to be the most realistic here we use initial n and r, spatial distributions, case (iii) its results for the initial conditions in what fol- above. Constant collision frequency gives results lows. which do not agree with experiment, and so a In the afterglow we assume that the ion and constant cross section for ion-atom collisions, electron densities are equal at all points in space a, was assumed. The collision frequency is then and at all times. The ion kinetic pressure tensor v = naom t, where na is the atom density. A typical term is dropped. We also assume that no further set of curves with kteo = 3.0 ev and parameter ionization occurs after i =0. The electrons are nsa is shown in Fig. 2. assumed Maxwellian: 1i en exp(ew/kte, () The hot-cathode discharge tubes were described earlier. 4 Current pulses were of 0.2 to 2.0 A and 336 e e 0 e from 5 to 50 Msec duration. Most results were

4 VOLUaI22, NUMvIuS8 PHYSICAL REVIEW LETTERS 24 PissuABy taken at 40 at sec, which time steady-state con- Electron density profiles were extremely reditions prevailed. The tubes were run in an oven producible, even after periods of months. The at (100* 5)*C with a finger of liquid mercury pro- shape of the profile did not depend on the distruding below into a water bath whose tempera- charge current pulse, but only on pressure and ture was controlled to within ±0.2 Ca. initial electron temperature. For tubes of differ- The average electron number density was mea- ent radii but with roughly similar initial condisured with a dipole resonance cavity" and a per- tions, approximate time constants of the (ne(t)) turbation TM 010 mode cavity. The two methods profiles scaled directly as the tube radii, as agree very well (see Fig. 1) when care is taken might t expected for a free-fall regime. to account for the cavity end holes and the glass To compare theory and experiment we proceed tube envelope.' The former method was used as follows: (i) Choose an initial electron tempermost of the time because the Q of the dipole- ature (usually measured) and select initial densimode cavity is higher, yielding superior time ty and velocity space profiles. We have used resolution. those corresponding to free-fall conditions as de- Electron temperature decay curves in these scribed above, but collision effects could be acrapidly changing plasmas are exceedingly diffi- counted for here using the theories of Self and cult to measure. We have used a pulsed Lang- Ewald' and Forrest and Franklin.' (ii) Calculate muir probe technique, a swept probe technique, %au. Alternatively this may be selected to best and a time-resolving (1-2 gsec) 3.0-GHz radiom- fit experimental data. (iii) Solve Eqs. (2), (3), eter.6 However, at the pressures of interest and (5) to obtain (n(w)) and kte(t). here the absorptivity of the plasma is so low that Figure 3 shows such a comparison for two the last method is rather insensitive. We chose tubes. Agreement for (n(t)) is excellent, but for to use the radiometer to verify the probe results kte(t) somewhat poorer. As stated above, we at higher pressures. Even so, we do not place are not very confident of the experimental elecmuch confidence in our probe results at very low tron temperature points, especially later in the pressures, afterglow. S~(=j) 7 pww 0 St _ 1 I:-4oIllFIG. 3. Comparins, FIG. 2. I Theoretical f j of experimental norma t j electron lized average electron J 0 den n u m b e r d e n s t. ty P a ra ty and m e te temperature r,i., A T * profile. 0 ff3.o. in ev the afterglow ti for be (ra two d ii 1.13 an d 1.85 c m ). si- Sd o

5 VOLums22, NumsaS8 PHYSICAL REVIEW LETTERS 24 hmnuusa 1969 The parameter oa used to best fit theory with nical assistance and to B. F. George for program. experiment yields an 5x 10' ml, a value of the ming the computer and for many useful discusorder of the elastic ion-atom cross section but sions. at least an order of magnitude smaller than the average charge-transfer cross section.9 This is *Work supported by a National Research Council of not yet understood fully. Possibly the introduc- Canada Grant No. A-3157, by a Defence Research tion of the ion kinetic pressure term in the after- Board of Canada Grant No , and by an Ontario glow theory will clarify this point. Department of University Affairs Grant. Using this theory with experimentally deter- (1966). 'G. S. Kino and E. K. Shaw, Phys. Fluids., 587 mined initial electron temperatures, we were 'S. A. Self, Phys. Fluids 6, 1762 (1963). able to obtain excellent agreement with electron 3 G. V. Parker, Phys. Fluids 6, 1657 (1963). density measurements made on four discharge 4 B. C. Gregory, Can. J. Phys. 44, 2615 (196). tubes of radii 0.55, 0.85, 1.13, and 1.85 cm. 'B. C. Gregory, Can. J. Phys. 4, 2281 (1968). More complete results will be presented else- 'J. C. Ingraham and S. C. Brown, Ph.s. Rev. 138 where. This work tends to support the validity A1015 (1965). 7 of using fluid theory for this sort of problem. It S. A. Self and H. N. Ewald, Phys. Fluids 9, 2486 also predicts supercooling of the electrons 1 0 ; (1966). pj. R. Forrest and R. N. Franklin, Brit. J. Appl. whether or not this is real or a phenomenon in- Phys (1966). troduced by approximations in the theory will be 9S. C. Brown, Basic Data of Plasma Physics (John explored. Wiley & Sees, Inc., New York. 1959), p. 38. We are grateful to R. S. Daniel for expert tech- 10 M. A. Biondi, Phys. Rev (1964).!8 I 338

One dimensional hybrid Maxwell-Boltzmann model of shearth evolution

Technical collection One dimensional hybrid Maxwell-Boltzmann model of shearth evolution 27 - Conferences publications P. Sarrailh L. Garrigues G. J. M. Hagelaar J. P. Boeuf G. Sandolache S. Rowe B. Jusselin