Diffusion Processes in the Positive Column in a Longitudinal Magnetic Field

Transcription

1 P/146 Sweden Diffusion Processes in the Positive Column in a Longitudinal Magnetic Field By B. Lehnert* The motion of a charged particle in a magnetic field may be described as the sum of a gyration around thefieldlines, a drift motion across the same lines a motion parallel to the field. 1 External force fields, magnetic field inhomogeneities inertia forces are responsible for the drift motion in this single-particle picture. In an ionized gas the picture has to be modified by particle interactions, such as collisions phenomena caused by space charges. Only encounters between nonidentical particles produce a diffusion in first order across the magnetic field. 2» 3 The macroscopic correspondence to a situation the transverse drift due to particle interactions inertia forces can be neglected the motions are relatively slow is an ionized gas nearly "frozen to" the magnetic field lines. This strong coupling between ionized matter magnetic fields is of great importance, not only in astrophysics but also for the problem of producing controlled thermonuclear energy in a plasma confined by a magnetic field. However, it has been emphasized by Bohm, Burhop, Massey Williams 4 that rom fluctuations of charge density plasma oscillations may produce electric fields which in their turn give rise to drift motions across the magnetic field lines. This ' "drain" diffusion provides an additional mechanism for ionized matter to "slip" across a magnetic field may, when it dominates over collision diffusion, introduce considerable difficulties into the physics of a plasma in a magnetic field. Bohm collaborators performed experiments with an arc plasma in a magnetic field came to the conclusion that the diffusion was not consistent with collision phenomena but could be explained by "drain". This interpretation was criticized by Simon, 5 who pointed out that the transverse diffusion of the plasma is not necessarily ambipolar. Thus, in the arc experiment, space-charge neutralization can be maintained by the conducting end walls which produce an electron short-circuit across the magnetic field. With this new interpretation Simon was able to show that the arc experiments did not conflict with the binary collision theory. The purpose of the present investigation is to study diffusion across a magnetic field in a configuration * Royal Institute of Technology, Stockholm, Sweden. 349 which is free from short-circuiting effects such as those described by Simon. It provides the possibility of deciding whether collision or "drain" diffusion is operative. For the purpose a long cylindrical plasma column with a homogeneous magnetic field along the axis has been chosen as described in Section 5. The theoretical treatment is given in Sections 2 to 4. On the basis of the collision diffusion theory Tonks, 6 Rokhlin, 7 Cummings Tonks 8 Fataliev 9 have pointed out that a longitudinal magnetic field will reduce the losses of particles to the walls. Consequently, when the magneticfieldis present, a lower electron temperature a smaller potential drop along the plasma column should be required to sustain a certain ion density. The same conclusions have recently been drawn by Bickerton 10 f by Bickerton von Engel 11 in theoretical experimental investigations which also include probe measurements in a magnetic field. In the range of the apparatus used by these authors the positive column was found to behave in good agreement with the collision theory. In the absence of a magnetic field the behaviour of the column was consistent with the theory of Tonks Langmuir, 12 when thefieldwas present a modified Schottky theory was applicable (next section). The present experiment forms an extension of that of Bickerton von Engel into a range the Schottky theory 13 is applicable in the absence of a magnetic field the applied magnetic field is still made strong enough to influence the electron temperature. THEORY OF THE POSITIVE COLUMN IN A LONGITUDINAL MAGNETIC FIELD A cylindrically symmetric, stationary, ionized gas column is assumed to be situated in a constant, homogeneous, longitudinal magnetic field B. The basic macroscopic equations for the ion, electron neutral gases are assumed to be valid as well as the following conditions: (a) The mean free paths of ions electrons are small compared to the tube radius. (b) The production rate %n e of charged particles is proportional to the electron density n e, % is a f The author is indebted to Dr. Bickerton Professor von Engel for placing this manuscript at his disposal.

2 350 SESSION A-10 P/146 B. LEHNERT function of the electron temperature the neutral gas density. Thus, two-stage ionization is neglected as well as volume recombination. These processes introduce a variation in the longitudinal potential drop with the discharge current. 11» 14 However, since the potential drop varies slowly with the discharge current in the present experiments these approximations do not introduce serious errors. (c) The ionization degree is low the frictional coupling between charged neutral particles does not set the neutral gas into motion. (d) Electron attachment can be neglected. This is true for a rare gas such as helium which has been used in the present experiment (see Loeb 15 ). (e) The electron ion gases have Maxwellian distributions with constant temperatures T e Ti across the column. In reality, striated columns often appear with periodically varying conditions in the longitudinal direction. However, Eméleus Burns 16 have pointed out that this is not likely to change the theory of the positive column fundamentally. In the present theory the temperatures T e Ti can be regarded as symbols for the mean energies of electrons ions. Not until the next section does the special form of the particle distribution affect the results. (/) The column is extended very far in the axial direction so as to make end effects deviations from cylindrical symmetry negligible. (g) The velocities Vi v e of the ion electron fluids are small nonlinear terms can be neglected. Basic Equations With щ indicating the ion density, E the electric field, b the magnetic field from the current in the plasma, v\ v e the collision frequencies between the charged particles the neutral gas, the basic equations become div (^Vi) = div (n e V e ) = Z n *> W 0 = ещ (E + Vi x B) k TiVni ^m^vi, (2) 0= -en e (E+v e xb)-kt e Vn e -v e tnenev e, (3) curlb = [ е(щу1-п е Уе)> (4) _ curle = 0, E = -VF, (5) dive = е(пх-п е )/е 0} (6) Eq. (1) expresses the conservation of mass charge, (2) (3) the conservation of momentum. Cylindrical symmetry is assumed in a coordinate system (г, ф, z) with the z axis along the column. Equations (5) give E r = -dv/dr, Е ф = 0, E z = const = E (7) if it is assumed that the component Е ф should be finite at the axis r = 0. Equations (2) (3) now become 0 = e 0 = -еп е (-^ -v ir B, E^- ^, 0, o) ty, v iz ) ^> 0,, V ф ez ). (9) From the ф components, Ч Ф = i Vir, У еф = V er, (Ю) П Ve OÜÍ = Б/mi œ e = eb/m e are the gyrofrequencies of ions electrons. The macroscopic motions Viф Veф around the axis of symmetry arise partly from the radial density gradients, partly from drift motion in the crossed fields E r В = B z. In the axial direction Transverse Diffusion CÜQ vente E. (11) The radial components of Eqs. (8) (9) combine with Eq. (10) to give n e V er = f$en Q -D e -^, we have introduced the notations D e = (12) (13) (14) (15) Di D e may be regarded as transverse diffusion coefficients, /?i /? e are the corresponding mobilities. Further, the symmetry condition imposed on the radial component of Eq. (4) gives mvir = n e v er, (16) which is an exact expression, even if the eigenfield b cannot be neglected. Ecker 17 obtains the relation (16) from Eq. (1) by use of the condition that the fluxes nivir n e v er have to be finite at r = 0. The notations П = w e, n' = fli-fle (17) are now introduced Eqs. (12), (13), (1), (16) (6) become nv er = - d 1 от/ j \ ) -Dij- (n + ri) -*. -!>.& (18) eo(d 1\3V ir (19) (20) In a subnormal positive column, the axial current density is very small, considerable deviations from electrical neutrality may arise, i.e., the net space charge en' may be comparable to en. This has been discussed by Ecker 17 in the absence of an external

3 POSITIVE COLUMN IN LONGITUDINAL MAGNETIC FIELD 351 magnetic field with the assumption that ri is nearly proportional to n. For experiments on the normal positive column, however, ri\n <^ 1 is usually a good approximation in the major part of its cross section. Dropping terms containing ri in Eqs. (18) gives c& (21) dr dn Pipe + PePi (22) (23) is the transverse ambipolar diffusion coefficient. We observe that, when P e /Pi < 1, ions will have a stronger tendency to diffuse towards the walls, corresponding to a reversal of the radial potential distribution, compared to the nonmagnetic case. This point is discussed later. Applying d/dr+l/r to Eq. (22) gives, with Eq. (19), dr 2 r dr D a with the solution which is finite at r = 0: n{r) = n O Jo{ar), (24) (25) The results (23) (25) have been obtained in a different way by Bickerton, 10 who starts with the assumption that the diffusion coefficients of ions electrons are reduced by the factors 1/(1 +O)Í 2 /VÍ 2 ) 1/(1+cü 2 e /v 2 e ) i n a magnetic field. In the present treatment Eqs. (23) (25) have been deduced directly from the basic equations (1) to (5). From Eq. (21) the radial potential distribution becomes Special Case V = De-Dt, ( 26 ) A theory for the positive column in an axial magnetic field has been developed earlier by Tonks 6 under the assumption that the motion of the electrons is affected by the magnetic field, but not that of the ions. This is a special case which is obtained by putting coi 2 /vi 2 <^ 1 in all equations of this paper. In such a situation Eq. (26) reduces to V-Vo= (kter/e) log [JoMJ, (27) e Pe-Pi kt e (28) is a parameter used by Tonks. From expressions (27) (28), with ап 2 М 2 <^ 1, Tonks concludes that a strong magnetic field may flatten the radial potential distribution or even reverse its sign, i.e., the sign of т. However, it has been pointed out by Bickerton 10 that the reverse field may be very small because an introduction of the mean free paths of ions electrons, Ai = {3kTilfni)i vu gives the ratio Pe Ae (29) It should be observed that T e is a slowly varying function of the magnetic field as given in Section 3. A lower limit of P e /Pi is obtained by putting into Eq. (30) the smallest value of T e the largest value of Ti in the range of variation. The function (PelPi)min thus obtained varies monotonically with the magnetic field B. The electron temperature in discharge experiments is usually of the order of 10 4 to 10 5 K., as the ion temperature is considerably lower at moderate ionization degrees power inputs. Since the mean free paths Ai A e are of the same order of magnitude, the ratio P e /Pi does not necessarily go below unity, a reversal does not always take place, even in very strong magnetic fields, as shown by Eqs. (30) (26). Space Charge Returning to the solutions (25) (26), the space charge is calculated from Eq. (20) in the quasi-neutral case: (a2) are the Debye distances corresponding to the thermal energies of the ions electrons. Since the electron density is a positive quantity, Eq. (25) shows that ar must be less than or equal to the first zero of Jo, R is the radius of the bounding wall. From this condition from Eqs. (31) (32) it is seen that the quasi-neutral approximation rijn <^ 1 is valid throughout the major part of the column as soon as the Debye distances are much smaller than the tube radius R. Only in the immediate neighbourhood of the wall do Eqs. (18) (19) lose their validity. The density no at the axis may be calculated from Eqs. (11) (25) the total axial current, /, which becomes {cf. von Engel Steenbeck 14 ) 7-Г* JO Consequently, no = ai (33) r = R' forms the ''boundary' ' between the quasi-neutral plasma the wall sheath. Situations (R R f )/R <^ 1 are of special interest here.

4 352 SESSION A-10 P/146 B. LEHNERT The discussion is now restricted to a nonconducting wall the boundary condition at r = R' is given by the balance between the particles produced per unit volume, jjn, the particles absorbed by the wall. Per unit length time there are created -ZV = [ Rf 2irr^ndr = 27Tn o %R'J 1 {ar')la (34) Jo particles inside r = R'. On the other h, ions leave the plasma edge r = R' with a mean velocity towards the wall which may be defined by an equivalent ' 'temperature" Ti'. Thus, the wall absorbs N = far'n(r')(3kti'lmi)* (35) particles per unit length time. Eqs. (34) (35) combine to give n(r')/n 0 = 4feZ)^fn/3Ar 1 / )*/i(a2î / ). (36) Since n(r) ^ 0, Eqs. (23), (25), (36) (29) give the condition Л = 1.66 n(r')/no = Jo(aR') < X/R f «XjR, (37) (38) can be regarded as a mean value of the mean free paths is equivalent to a mean value used earlier by von Engel Steenbeck. 14 In order to estimate A the magnitude of 7Y has to be determined. For the nonmagnetic case Bickerton 10 Bickerton von Engel 11 assume that T± = T\. On the other h, Bohm 18 has pointed out that a stable sheath cannot exist unless the ions leave the plasma edge with a velocity corresponding to T\ «T e /2. This is caused by an acceleration in a weak electric field which penetrates from the wall region through the plasma edge. When a magnetic field is present, in cases of interest in this connection, the Larmor radius of ions is larger than the Debye distances (32) the thickness of the wall sheath. This implies that the collection of ions by the wall across the sheath is largely unaffected by the magnetic field (Bohm, 18 Bohm, Burhop Massey, 4 Bickerton 10 Bickerton von Engel 11 ). According to Bohm collaborators 7Y should still be of the order of T e /2 when a magnetic field is present T e > Ti. In Section 3 the condition \/R <^ 1 is assumed to be valid. It is seen that a strong reduction of X/R takes place according to Eq. (38) when the magnetic field is increased (cf. Bickerton 10 ). DETERMINATION OF ELECTRON The Plasma Balance Equation TEMPERATURE When X/R <^ 1 the balance between charge production wall losses is given by = Xi 2, (39) Ki = is the first zero of /o. Combination of Eqs. (36) (37) gives a corresponding balance equation which leads to a "modified Schottky theory' ' applicable also to situations when X/R <fé 1 (cf. Bickerton 10 Bickerton von Engel 11 ). Assuming a Maxwellian distribution, von Engel Steenbeck 14 have deduced the number % of ion pairs produced per unit time per electron: x (1+eVil2hTe)er*r,lw. t (Щ p is the neutral gas pressure (in newtons/m 2 ; 1 mm Hg = n/m 2 ), T a the neutral gas temperature, V\ the ionization potential a, is a constant with a numerical value times that given by von Engel Steenbeck when Eq. (40) is expressed in MKSA units. For helium, a = amp sec 5 /kg 2 m 2 V = volts. The notations x = Ae = Ae/p, Л е = Л в0 (Г п /273), Xi = evilkti, (41) Ai = Л#, (42) Ai = Лю(Г /273) (43) are introduced, Л е о is a function of the electron ion temperatures given by von Engel Steenbeck 14 Maier-Leibnitz. 19 Eq. (39) now becomes F(x) = A (Rp)*[L{x, Xi) +y*m{x, Xi )], (44) F(x) =**e*/(l+2/*),' (45) A = 2{6l^a(273IT n )Vi{milm e )ikr 2, (46) Ц Х, XÍ) =. **., [l +% f\ > (47) v v v ' Ai(l+*i/*)L А е \пцх) J ' У =, (48) M(x, Xi ) = (49) With helium A, equals 31.7 sec 2 /kg when T n = 300 K. Relation between Ion. Electron Temperatures Von Engel Steenbeck 14 obtain the energies da da (50) which are transferred to the neutral gas by collisions with ions electrons, respectively. In Eqs. (50) (51) KÍ к е are the fractions of the total energy of a particle being lost in a collision. The energy is supplied by the electric field E which is parallel to the magnetic field B. The energy supply is given by ( ) = -

5 POSITIVE COLUMN IN LONGITUDINAL MAGNETIC FIELD 353 both in the presence in the absence of the magnetic field, from Eqs. (11), (50), (51) (52) or (*l/*) 2 = -^n), (53) The temperature of the ions is not very much greater than that of the neutral gas, ions may be assumed to lose their energy by elastic collisions, i.e., /q «0.5. For electrons, however, inelastic collisions will be taken into account к е becomes a function of T e as shown below. The electron temperature is determined by the root x of Eqs. (44) to (49) (53). LONGITUDINAL ELECTRIC FIELD IN A HELIUM DISCHARGE The total fractional loss of energy of an electron in a collision with a neutral gas particle is /c e i = 2m e /m n is the energy loss from elastic collisions, *ion = (evfál{ikt 9 v e ) = (55) is the ionization loss, the rate % is given by Eq. (40) "exc = 2 =2f*r?: ( 56 ) Figure 1. Square root of the total fractional loss к е of energy of the electrons as a function of T e. Full line indicates results according to Eq. (60) broken line experimental results of Bickerton von Engel assumed, the number of exciting collisions per unit time electron becomes for "level" s: 8 is the total excitation loss, V s s s are the excitation potential excitation rate of the sth level. Since the electrons diffuse in a radial direction there is a wall loss as well. The plasma balance not only requires the ionization work to be done at the rate %, but the electron which is produced must also be '"heated" to the temperature T e. Since ^ electrons of this temperature are lost to the walls per unit time electron, the fractional wall loss becomes "wall = T dl(iktelve) = = f In order to calculate the excitation loss in helium the experimentally determined excitation cross section (Maier-Leibnitz 19 ) is used as a starting point. As shown by Fabrikant 20 Karelina 21 a good approximation to the experimental results is given by an empirical expression for the total excitation free path 273р у 1 V-V s V ms -V 132.8Г -f A, V ms -V e x p V ms -V s ' (58) V is the potential corresponding to the particle energy. In this approximation two "levels", s = l, 2, are used with Ai = 7.7xlO" 2 m, A 2 = 2.7xlO" 2 m, V x = volts, V 2 = 20.2 volts, V ml = 20 volts V m 2 = 28 volts. If a Maxwellian distribution is _ /273\ Г 00 4 v* v ~ P \TJ Jev V^A S y-v s v is the total velocity of an electron, w = {2kTelfn e )i A s = 132.8A S. The total loss к е is given by the integrated Eq. (59) Eqs. (54) to (57), (60) (40) to (43): (57) *e = ^ + (^ [ A s V ms -V s xvm.-v.rn (60) The result is shown in Fig. 1 agrees fairly well with experimental results of Bickerton 10 Bickerton von Engel. 11 Finally, the longitudinal electric field is obtained from Eqs. (51), (52) (11) as a function of x: E = (61) The factor in front of the right-h member of Eq. (61) is somewhat uncertain because it is based upon

6 354 SESSION A-10 P/146 B. LEHNERT discharge tube ' de л [cathode condition y 2 ym > L. In the present measurements this is the case only for у > 0.1 when the strong fields of the right-h sides of Figs. 3a b are considered. Figure 2. Outline of the experimental arrangement. The discharge tube has an inner radius of 1 cm elementary kinetic methods in the calculation of the frictional coefficients in Eqs. (2) (3). A more rigorous kinetic theory gives a factor (64/тг) in front of Eq. (61) instead of (96/тг)^ (von Engel Steenbeck 14 ). However, this source of error should play a minor role in the relative magnitude в = E(B ф 0)/E{B = 0) (62) of the electric field at varying magnetic fieldstrengths. Theoretical results for Rp = \S\n\m (=1.46 cm mm Hg) 4.52n/m ( = 3.40 cm mm Hg) with T n = 300 K are shown by the full curves in Figs. 3a b, as calculated from Eqs. (44) to (49), (60) (62). For the mean free path of ions defined by Eqs. (42) (43) a value д/2 times that of neutral particles given by Kennard 22 has been taken. A source of error may be introduced by this value but it does not change the position of the theoretical curve fundamentally; appreciable changes in the rate of diffusion are required to cause a noticeable change in the electron temperature since the production rate ^ is a sensitive function of T e according to Eq. (40). In Figs. 3a b a change of the diffusion coefficient has also been simulated by substituting the function M of Eq. (49) by ym. The value у = 1 corresponds to the actual situation, as у = 0.5, (broken lines) give diffusion coefficients which are about twice, ten one hundred times larger than the present in a magnetic field,which is strong enough to satisfy the EXPERIMENT Apparatus The experimental arrangement is shown in Fig. 2. A discharge tube of inner radius R = 1 cm has been placed inside a magnetic coil of four meters length. In one of the runs being made (Fig. 3b) the magnetic field was varied up to В = 0.53 v sec/m 2 ( = 5300 gauss), corresponding to a power input of about 3 x 10 5 watts. Since every measurement required only a few seconds' time it was sufficient to cool the coil with fans. The tube ends with anode cathode were both extended far outside of the coil. The diverging field at the coil ends, the large ratio between the tube length tube radius, the relatively high pressure used in the experiments (^1.46 mm Hg) make impossible short-circuiting effects of the type discussed by Simon. 5 Measurements of the longitudinal electric fieldwere made with an electrostatic voltmeter connected between two floating platinum wire probes, one meter apart both far from the tube ends. The probes had the shape of circular sectors which followed the tube wall closely. Their connections were screened electrostatically all the way out to the voltmeter. The discharge current was closed through an inductance-free resistor over which the voltage could be examined with an oscilloscope a wave analyser. Before the measurements the electrodes were outgassed the tube run with 180 ma current. Impurities in the helium discharge were easily detected with a small spectroscope. Narrow slits in the coil made observations possible over the entire length of the tube. It was found that the most rapid way to clean the tube of impurities was to place the cathode at the opposite side of the vapor trap filledwith carbon e,46 B c Vs/m* Bc Vs/m 2 1- Y-ОЛ 0.5- a 60 ma * 160 ma co /p Figure 3a С 10* Figure 3b ' 2*10*' Figure 3. The ratio 0 = E(B)/E(0), (fi) is the longitudinal electric field when the magnetic field is present (0) is the corresponding value without magnetic field, щ ев/гщ isthegyrofrequency of the ions. Experimental results are indicated by marks the theoretical by the full curve (y = 1). A double, a tenfold a hundredfold reduction of the magnetic contribution to the diffusion coefficient is simulated by the broken lines with у = 0.5, , respectively. (a) Rp = 1.96n/m = 1.47 cm mm Hg. (b) Rp = 4.33n/m = 3.26 cm mm Hg 3^10*'

7 POSITIVE COLUMN IN LONGITUDINAL MAGNETIC FIELD x 5*10 3 c/s B c =0.26 Vs/m «10 3 c/s 10*c/s B c =0.2U Vs/m 2 ^ x 6 о 5s- 10* 2*10* 3«10 A A*10* Figure 4o 5*10* 10* 2*10* Figure 4b Figure 4. The ratio в between the noise voltage in the presence of the magnetic field the voitage in the absence of the same field, (a) Rp = 1.94n/m = 1.46 cm mm Hg. (b) Rp = 4.52n/m = 3.40 cm mm Hg u>i/p (mvns) 3-10* (Fig. 2). With a strong current the discharge acted as an ion pump the gradual disappearance of the impurities could be followed from the anode end to the cathode by means of the spectroscope. Measurements were not performed until the tube was observed to be clean over its entire length. The pressure was measured with a McLeod gauge. Results Two series of electric fieldmeasurements were made at the pressures mmhg as shown by Figs. 3a b. The results at zero magnetic field are given in Table 1 together with theoretical results calculated from Eqs. (44) to (47), (53) (61), which are also compared to results by Kareh'na 21 for Rp = 1.47 cm mm Hg by Klarfeld 23 for Rp = 3.40 cm mm Hg. The agreement between the experimental results of these authors the present experiments is as good as can be expected. The present theoretical value of E/p for Rp = 1.47 cm mm Hg is closer to the experimental than the value calculated by Karelina, but there still remains a discrepancy which may be due partly to an erroneous factor in front of the expression for the mobility. Figures 4a b show the corresponding measurements at Rp = cm mm Hg of the relative magnitude в of the noise voltage over the resistor in the discharge circuit. The measurements have been made with a discharge current of 160 ma at RP (cm. mm Hg) Table 1. Values of E/p in volts/cm mm Hg x {ma) Present results Exptl Theor Earlier results Exptl Theor x cycles per second with the wave analyser adjusted to a b width of 145 cycles per second. Measurements at lower frequencies (10 3 c/sec) were difficult to interpret; even in the absence of a magnetic fieldthere was a strong rapidly changing noise voltage. It is seen from Figs. 3 4 that the experimental points form a "knee", which indicates that the state of the discharge is changing at a magnetic field B c « v sec/m 2 for Rp ~ cm mm Hg, respectively. DISCUSSION Experimental Conditions In the absence of the magnetic field at the pressures of the present experiments the electron temperature is about 4 x 10 4 K the ion temperature about 1000 K. From Eq. (38) the values X/R « are obtained at Rp = cm mm Hg when T{ is put equal to T e /2 according to Bohm. 18 With 7Y ~ T\ according to Bickerton von Engel 11 the values X/R» are obtained, which can be regarded as upper limits since the stability^ of the wall sheath requires T\ «T e /2. Even with X/R = 0.3 the change in the theoretical value of the electron temperature is only a few per cent. At the lowest pressure current density used in the experiments the charge density is calculated from Eq. (33) to 4 x m~ 3, corresponding to the Debye distances h e = 1.4 x 10~ 4 m hi = 3x 10~ 5 m as given by Eq. (32). Equation (31) gives n'/n < 0.2 at a distance greater than 1 mm ( = 0.1 R) from the wall. Consequently, the quasi-neutral approximation is valid throughout the major part of the tube. Conditions are improved with increasing current density pressure; for / = 0.16 amp Rp = 4.33 n/m the results are h e = 1.0xl0~ 5 m n'/n < 1.0 xlo" 3 sir = 0.1Я. Finally, the smallest radius of gyration of the ions in the present experiments has been 2.8 x 10~ 4 m (at JB = 0.53 v sec/m 2 ), which is greater than the largest occurring Debye distance h e = 1.4 x 10~ 4 m. From these considerations it is seen that the basic

8 356 SESSION A-10 P/146 B. LEHNERT conditions underlying the theory given earlier have their correspondence in the present experiments. Conclusions The left-h parts of the theoretical curves in Figs. 3a b show a satisfactory agreement between theory experiment; the small discrepancies may be due to an error in the mean free paths used in the calculations. An increase in the diffusion coefficient by a factor of ten (y = 0.1) or more is, in any case, too large not to be distinguished from the present results. Consequently, the left-h parts of Figs. 3a b show that the diffusion is taking place according to the binary collision theory that a strong reduction in the diffusion coefficient is caused by the magnetic field which traps the particles effectively in its transverse direction. There seems to be no difference between the measurements at varying current densities from this it is concluded that deviations from electrical neutrality as well as two-stage processes recombination do not influence the results noticeably. Correspondingly, the left-h parts of Figs. Aa b show no increase in the noise level caused by the magnetic field. However, when the magnetic field reaches a certain critical value which is about B c = v- sec/m 2 at Rp = n/m the diffusion coefficient starts to increase rapidly the noise level is suddenly increased at the same time. The deviations between the binary collision theory the experiments are considerable in this region as shown by the right-h parts of Figs. 3a b. In Fig. 3a the diffusion coefficient exceeds its value due to the collision theory by more than a factor of ten when the magnetic field exceeds about 0.3 v-sec/m 2 in Fig. 3b this occurs at a field strength of about 0.4 v-sec/m 2. These results support strongly the existence of the "drain" diffusion mechanism suggested by Bohm, Burhop, Massey Williams. 4 It should also be observed that the right-h part of Fig. 3a clearly indicates an increase in the diffusion rate at increasing current densities, i.e., at increasing charge densities, n. This also speaks in favour of such a mechanism which is based upon the effect of local electric fieldscaused by deviations from electric neutrality. The values of в somewhat above unity at strong fields in Fig. 3b are not unimaginable if the same mechanism is operative, i.e., the diffusion may even be accelerated by the magnetic field. Whether the oscillations consist only of a broad spectrum of "electromagnetic turbulence" or include plasma oscillations concentrated around some special frequencies cannot be judged at this stage requires further investigation. It should be pointed out that noise oscillations in the presence of a magnetic field have also been observed by Âstrôm 24 Webster 25 in an electron gas by Block 26 in model experiments on the auroral discharge. Finally, the conclusions drawn here are also consistent with the results obtained by Bostick Levine 27 for a toroidal discharge in a toroidal magnetic field. These authors find a considerable decrease in the diffusion time as well as the presence of oscillations in a Hmited range of magnetic field strengths. However, as pointed out by Bostick Levine 27 later by Biermann Schlüter 28 Lehnert, 29 particle losses are also caused by the gradient in a toroidal magnetic field no definite conclusions can be drawn about the magnitude of the diffusion coefficient in this experiment. The diffusion caused by oscillations "electromagnetic turbulence" provides a mechanism for ionized matter to slip across a magnetic field. When this mechanism is acting, modifications of the present theories of an ionized gas have to be undertaken, both in astrophysics when the motions of magnetic fields matter are considered, in the thermonuclear problem when the confinement of a hot gas in a magnetic field is discussed. It is desirable to extend the present investigations to a fully ionized gas in stronger magnetic fields than those being used here. ACKNOWLEDGEMENTS The author is indebted to Mr. Bertil Andersson for building the apparatus for this experiment for valuable assistance during the measurements. A grant from Statens Naturvetenskapliga Forskningsrâd for covering the costs of the experimental equipment is gratefully acknowledged. 1. H. Alfvén, Cosmical Electrodynamics, Clarendon Press, Oxford (1950). 2. L. Spitzer, Physics of Fully Ionized Gases, Interscience Publ. Inc., New York London (1956). 3. C. L. Longmire M. N. Rosenbluth, Phys. Rev., 103, 507 (1956). 4. D. Bohm, E. H. S. Burhop, H. S. W. Massey R. M. 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