Diffusion of Arc Plasmas across a Magnetic Field

Transcription

1 P366 USA Diffusion of Arc Plasmas across a Magnetic Field By Albert Simon* The effect of a magnetic field В is to reduce the coefficients of diffusion, ) p, across the magnetic field to the values equations for the ion and electron densities, щ and n e are: «i тл -о d 2 %i та d, >+ p = 2, x 2? D - T = l + U T_)2 ^) where D+ denotes the coefficient for В = 0, tü-t = eb ni±c and т± = mean free time between collisions for ions or electrons. Diffusion in the direction of the magnetic field, D\ is unchanged from its field-free value. Hence At 1 = D ±. (2) In many arc experiments, (OJ-T-) > {OJ+T+) > 1 and, as a result, DJ > D+? > Z)_ p. In this case the plasma has a highly anisotropic conductivity, since currents may flow in the direction of the field far more easily than in the perpendicular direction. In the analysis which follows, it will be shown that this anisotropy produces a diffusion across the magnetic field which is not ambipolar. 1 The ions and electrons, instead, diffuse at their own intrinsic rates. Space-charge neutralization is maintained by slight adjustments of the currents in the direction of the field lines. These results will then be compared with the experiments of Bohm, Neidigh, and Bickerton. The analysis below will be valid only for the case of a weakly ionized gas. This means that the dominant mechanisms of diffusion are ion-neutral atom and electron-neutral atom collisions. In typical arc plasmas (ion and electron energies of the order of a few electron volts) this requires the degree of ionization to be smaller than about 1%. In the final section, the effects of providing magnetic mirrors at the ends of the arc and the effect of increasing the tube length in the direction of the field lines will be considered. DIFFUSION IN A UNIFORM FIELD OF FINITE LENGTH Theory Consider a two-dimensional problem in which the magnetic field is in the jy-direction and the ^-direction is perpendicular to the field. The time dependent * Oak Ridge National Laboratory, operated by Union Carbide Corporation for the U.S. Atomic Energy Commission. The report covers work by R. V. Neidigh and A. Simon, Oak Ridge National Laboratory. 143 ^1 dt = T) v d 2 n e dx 2 d 2 n e p d_ dx d Here x denotes the mobility and E x and E y are electric field components in the plasma. The relation between the mobility across a magnetic field and that parallel to the field is the same as that for the diffusion coefficients in Eq. (1). Hence x p <^ x. In addition, let us consider a finite system bounded by conducting walls. If L is the dimension of the system in the ^-direction and R the dimension in the ^-direction, then one would expect that E y \E x ~ RL. The reason for this is that the electric fields are the derivatives of the scalar potential, which varies monotonically from a central positive value to zero at the walls. Thus, except for the case of a very long and thin container, one may safely neglect the second term on the right side of Eq. (3) compared to the last term. This is the formal way of stating that the essential currents which adjust for space charge neutrality are those which flow in the direction of the magnetic field. One may now proceed in the usual fashion to eliminate the explicit dependence on E y by the assumption of approximate charge neutrality m = n e = n. By combining the ion and electron expressions of Eq. (3), there results: dn (L-D + -fl+d-\ dt \ Note that the effective diffusion coefficient in the ^-direction has the usual ambipolar value, (4) D eit = (5) while the coefficient in the ^-direction is that of the ions since x_ > fjl+ and D+ p > Z)_ p. Thus = = D p

2 344 SESSION A-10 P366 ALBERT SIMON e-folding Length Suppose now that ions and electrons are being produced equally and uniformly along the^-axis (x = 0). The steady state solution of Eqs. (4), (5) and (6), with inclusion of the source term and boundary conditions, will be of the form n(x } y) = N(x)G(y). With the principal solution having the value G(y) = sin (тгуl), and with N(x) = Ае~ х \1я. where A is a constant and where -»- Г2П.Р b V + (10) The corresponding result for the case of free streaming to the end walls is VACUUM MANIFOLD T Figure 1. Apparatus used by Neidigh In many cases the dimensions of the system in the field direction are small compared to the collision mean free paths. In this case, the streaming of ions and electrons in the ^-direction may not be represented by the diffusion terms. It may be shown that the new equations corresponding to those of Eq. (3) are now, dni _ P^i p + dt dx 2 t dt щ\е\е у where M is the ion mass, m the electron mass, and Vi and v e are the average thermal velocities. Note that there is now free streaming to the end walls as weh as electric field accelerations of the ions and electrons through a distance of the order of half the arc length. Once again, it may be shown that the mobility term in the ^-direction is usually negligible. In the same fashion as before, one finds dn It \ -*\d4 n )dx 2 L\ The effective streaming velocity to the end walls (for comparable electron and ion energy) is 2v\ while the effective diffusion coefficient across the field is D+ p, as before. It should be noted that the effective diffusion coefficient across the magnetic field, D+ p, is much larger than the " ambipolar " coefficient, D P AMB> which would have been obtained by requiring that the electron and ion currents balance precisely to zero in the x- and ^-directions separately. This coefficient, by analogy to Eq. (5), has the value -D P AMB = 2P_g. (9) (7) Note that the ion density decreases exponentially with an -folding length q. Note also that the -folding length varies as PoB in the diffusion case of Eq. (10) and as PQ%B in the free streaming case of Eq. (11). Here Ро(~т-!) is the gas pressure. These results go over exactly in the case of cylindrical symmetry except that N{x) -> r-ie-rq. (12) Experiment A measurement of the ion diffusion coefficient across a magnetic field was obtained by Bohm et al. at Berkeley. 2 A complete description of the experiments may be found in Ref. 2. In essence, an arc was struck parallel to the magnetic field and along the long axis of a rectangular arc chamber made of graphite. The resultant ion density in the median plane was measured with a cold shielded probe. In most measurements, the chamber was filled with argon gas at a pressure of about 10~ 3 mm Hg and the magnetic field strength was 3700 gauss. These observations established that the ion density decreased exponentially from the center outwards with an -folding length of about 0.3 cm. If one assumes that the ion energy in the plasma is about 2 ev, the resultant mean free path at a pressure of 1.4xlO~ 3 mm is then Л# 5 cm. The thermal velocity is t>i # 3 x 10 5 cmsec and the arc length L = 12 cm. From Eq. (11), using the measured value of the -folding length, one finds that D+ p л 4.5 x 10 3 cm 2 sec. This is to be compared with the theoretical value of Z)+ p obtained from Eq. (1). Since си+т+ = 14.4, this value is 2.5 x 10 3 cm 2 sec, which is in good agreement with the measured value. Unfortunately, at that time, it was assumed that the diffusion across the magnetic field was ambipolar, and the experimental value was compared with the theoretical value of D P AMB which is given in Eq. (9). This value is D P AMB #10 cm 2 sec in complete disagreement with the experiment. As a result of this apparent anomaly, Bohm postulated that the principal mechanism of diffusion was by plasma oscillations rather than by the classical collision mechanism. The

3 DIFFUSION OF ARC PLASMAS TRARY U»IITS! DEPj NDEI ВО ч ICE V.as- Î f N; V V л ее < e S ы 5 3 i fq m Á о f i 4 у у 4J. DEPE NDEh СЕ m ARC CENTER AT (UNITS OF >24 in ) у у У о 2 * i MAGNETIC FIELD STRENGTH ( kilo - oersteds) < Figure 2. Radial variation of ion density Figure 3. Variation of e-folding length with В details of this theory have not been published. However, the resultant diffusion coefficient is stated to be 2 D 16B ' (13) where kt e is in electron volts and В in kilogauss. Note that this coefficient varies as B~ l and is pressure independent, whereas the classical coefficient varies as B~ 2 and is directly proportional to the pressure. In order to test these theories, Neidigh 3 has carried out a series of experiments at Oak Ridge. A sketch of his apparatus is shown in Fig. 1. The ion chamber is cylindrical and most experiments were carried out with nitrogen gas at a pressure of about 10~ 3 mm. The magnetic field could be varied from 2000 to 14,000 gauss. The carbon probe is biased 20 volts negative, which is on the flat portion of its characteristic curve. Measured ion currents varied from 10 xa near the cylinder wall to 10 ma near the arc. It is assumed that ion density is proportional to probe current. Further details of the apparatus may be found in Ref. 3. Some typical measurements of the -folding length as a function of В are shown in Fig. 2. The resulting values of the reciprocal of the -folding length are then plotted as a function of В in Fig. 3. The plot should be linear if Z>+ p varies as B~ 2 [see Eqs. (1) and (11)] and should vary as the square root if <D+ P ~ В~ г. The results clearly favor the B~ 2 behavior. Incidentally, the experimental magnitude of Z)+ p (#4xlO 3 cm 2 sec) is in good agreement with that determined in the experiment of Ref. 2. In addition to a study of the dependence on B, Neidigh has also investigated the variation with pressure. 4 For a case in which free streaming to the end wall should prevail (Л # 6 cm, L 6 cm), the results of Fig. 4 indicate a square-root dependence, in agreement with the discussion following Eq. (11). Another case in which diffusion to the end walls has set in (Л # 6 cm, L = 26 cm) is shown in Fig. 5. Here a Hnear behavior is observed, again in agreement with the theory. These last experiments on the pressure dependence are only preliminary; more careful investigations are planned. Further confirmation of the B~ 2 dependence of the diffusion coefficient is provided, for electrons, by the experiments of Bickerton. 5 In this case the spreading parameter R was found to vary linearly with B 2 in agreement with the classical theory. DIFFUSION IN A MAGNETIC MIRROR GEOMETRY The theory presented above indicates that the essential cause of the non-adiabatic diffusion across the magnetic fieldis the very large conductivity of the system in the direction of the field lines. It is of interest to inquire as to how this conductivity might be reduced and ambipolar diffusion restored. One

4 346 SESSION A-10 P366 ALBERT SIMON 10 s UNIFORM MA3NETIC FIELD: 58 О UNIFORM MA( 3NETIC FIELD.* 35( X) gaussj D U S S W<^ f A Pressure (mm Hg X 10 3 ) Figure 4. Variation of e-folding length with streaming case pressure. Free possible way is to put magnetic "mirrors" on the ends of the arc. This effect is discussed below. Theory The theory of the confinement of charged particles by a magnetic mirror 6 indicates that a particle will be reflected from the mirror if its velocity angle 9 is greater than 9 C where e G = sin-i (1Й). (14) Here 9 is the angle between the velocity vector of the particle and the mirror axis. The ratio of the magnetic field strength in the mirror region to that in the central region is denoted by R. Let us assume that the mean free path Л is large compared to the arc length L. Those ions with 9 < 9 C escape immediately. The remaining ions are confined until a collision produces a new 9 which is less than 9 C. If P is the probability that an ion will be scattered into the escape cone (9 > 9 C ) the rate of production of such ions per unit volume per sec is then nvpx and the total current streaming to the end walls is J = nvplx. (15) This is to be compared with the corresponding expression for free streaming without mirrors, which is = nv. (16) If one assumed isotropic scattering, the probability P of scattering into the escape cone, by use of Eq. (14) becomes: 366 PRESSURE ( mm Hg x 1O~ 3 ) Figure 5. Variation of e-folding length with pressure. End diffusion case which has the value P # (2R)- 1 for large R. It may be shown that the effect of an electric field potential ф is to increase the escape cone for the ions and decrease it for the electrons in such a way that P «P(l ±еф!щ (18) where W is the average kinetic energy of the particles in the region between the mirrors. The combined effect of Eqs. (15) and (18) may be seen to be equivalent to changing the streaming terms in Eq. (7) by the factor LPX, since фl & E y. This in turn yields a new expression for the e-folding length, q 2 = \D+*l2viP. l (19) It should be noted that the addition of the mirrors reduced the current in the direction of the field by the factor LPX. This will have no effect on the non-ambipolar diffusion, as the discussion in the final section will indicate. However, there is an immediate effect in the behavior of q 2 with pressure. Equation (19), together with Eq. (1), indicates that q 2 should be pressure independent and that q # ro2p% where ro is the Larmor radius of the ions. The case of diffusion flow to the end walls is of no interest for a mirror field, since the mirrors will then have only a minor effect on the current flowing in this direction.

5 DIFFUSION OF ARC PLASMAS 347 N 2 PRESSURE (Ю mm Hg) r 0 (cm) Figure 6. Radial variation of ion density. Magnetic mirrors used Experiment Again there has been a preliminary set of measurements by Neidigh. 4 The mirror field used has a 2:1 ratio. The field was 3500 gauss in the central region and 7000 gauss in the mirror region. The results are plotted in Fig. 6. The -folding length seems independent of pressure over the range studied. However, the magnitude of q is about a factor of 4 larger than the Larmor radius. Further experiments are planned. LONG TUBE LENGTH It seems clear that if the arc chamber is made long enough the effect of the electric fields parallel to the tube axis will finally become comparable with the radial electric fields; ambipolar diffusion will then be restored. The necessary length may be estimated. From the first part of Eq. (3) the order of magnitude of the perpendicular current is (<x) + T+) 2 q (20) where q is the -folding length. The magnitude of the parallel current is I t * р+щеуь. (21) Finally, since E x «фr and E y «фl, one has (22) Hence, for ambipolar diffusion to be restored, one must have LjR «œ+r + (qr)i. (23) Finally, since it is electron current in the magnetic field direction which neutralizes space charge, it is necessary that this relation be satisfied for the electrons as well. This is the worst case, hence the condition is LR S* a>-t-{q R)K (24) This condition may be difficult to obtain in practice. For example, in Neidigh's experiments а)+т+ # 14.4 and œ-r-» 3 x Since qr must be appreciably larger than unity, this requires that LjR > It should be kept in mind that ео+т+ must be kept appreciably larger than unity if the magnetic field is to play a dominant role in the behavior of the gas. In the case of free streaming to the end walls, Eq. (7), the corresponding condition is easily shown to be LjR > (a)t) 2 qx. This condition is incompatible, however, with the necessary requirement for free streaming, A > L. It might be thought that the use of insulating end walls would require exact ambipolar diffusion in the field direction and hence produce ambipolar diffusion radially. However, there is no perfect insulator (for example, surface collisions allow electrons to walk radially along the wall) and this probably prevents this scheme from compensating for the huge anisotropy-of the plasma conductivity. Some unpublished experiments by Neidigh have shown that an insulating end wall had only a small effect on the radial ion diffusion.

6 348 SESSION A-10 P366 ALBERT SIMON SUMMARY It is shown that experimental results on diffusion of ions and electrons in a weakly ionized plasma across a magnetic field are in agreement with the classical collision-diffusion mechanism. Hence, no additional mechanisms, such as plasma oscillations, need be postulated. The resultant diffusion rate is not ambipolar, however. The ions and electrons diffuse at their own intrinsic rates. REFERENCES 1. A. Simon, Ambipolar Diffusion in a Magnetic Field, Phys. Rev. 98, 317 (1955). 2. A. Guthrie and R. K. Wakerling, The Characteristics of Electrical Discharges in Magnetic Fields, Vol. 5, Div. 1, p McGraw-Hill Co., Inc., New York (1949). 3. A. Simon and R. V. Neidigh, Diffusion of Ions in a Plasma Across a Magnetic Field, Oak Ridge National Laboratory Report, ORNL-1890 (1955). 4. R. V. Neidigh, Some Experiments Relating Ion Diffusion In a Plasma to the Neutral Gas Density in the Presence of a Magnetic Field, Oak Ridge National Laboratory Report, ORNL-2024 (1956). 5. R. J. Bickerton, The Lateral Diffusion of an Electron Swarm in a Magnetic Field, Proc. Phys. Soc. (London), 70B, 305 (1957). 6. E. Fermi, On the Origin of the Cosmic Radiation, Phys. Rev. 75, 1169 (1949).