Roy, M

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Indian J. Phys. 73B ( 1), 83-91 ( 1999)I J P  B— an international journalD e p e n d e n c e  o f  th e  r a d ia l d is tr ib u tio n  fu n c t io n  fo r  a z im u th a l  c o n d u c t iv ity  o f  a r c  p la sm a  o n  tu b e  ra d iiM RoyD epartm ent o f  Physics, Pachhunga U niversity  C ollege, N orth -E astern  Hill U niversity , A i/aw l-7 9 6  001. M izoram . IndiaR e c e i v e d  24  S ep tem b er  1998,  a c c e p t e d  3  N o v e m b e r  199 8A b s tr a c t  The d istribution  function for the radial variation o f  a /im u thal conductivityo f an arc plasm a as proposed by G hosal e t a l  (1978) has been studied in four arc tubes of d ifferen t radii The ca lcu lation  o f  half-w id ths fio in  the d istribu tion  I unction  ind icates that the p lasm a becom es less and less concentrated  along the axis w ith the increase o f  the tube radius The value o f  axial conductivity has been calculated m lour arc tubes and an analytical expression  presen ted  to represent the variation  o f axial conductiv ity  w ith arc tube radiusK e y w o rd s  R a d ia l d is t r ib u t io n  fu n c tio n , a / im u th a l  c o n d u c tiv ity , a rc  p la sm a , rad io frequency  currentPA C S  N o. 52 70 Ds1. IntroductionThe radiofrequency non-immersive probe method has been utilised to study the variation of radial distribution profile of conductivity in a mercury arc by Ghosal et al 11. 2]. It has been shown that radio frequency coil probe method in conjunction with the usual probe method may jointly provide useful information in connection with the structural behaviour of conductivity and electron density of the plasma. They have concluded, however, that the average azimuthal electrical conductivity of an arc plasma can be estimated by determining the change in band­width of a coil wound around the arc tube due to the presence of the plasma column within it. The method used in the present investigation as utilised by Ghosal et al[ 1 ] needs no calibration in contrast to the methods used by other authors like Mikoshiba and Smy [3] and Ciampi and Talini [4],Ciampi and Talini [4] deduced expressions and measured two distinct average values of conductivity but did not, however, explore the structural behaviour of conductivity or electron density. With the help of a method based on the relation between the plasma parameters and the impedence of a radiofrcqucncy coil placed co-axially around the plasma they obtained electron concentration and electron atom collision frequency for momentum transfer, assuming©19991ACS84 M RoyBessel type electron distribution along radius of the tube. Ghosal et al [2] questioned the Bessel distribution of charge density for an arc in the usual pressure range 1 O' 2 to 1 O' 1 torr and presented the new technique for obtaining the conductivity profile for a mercury arc plasma. The object of the present work is to find out and explain the dependence of the proposed distribution function for conductivity profile and their half-widths on tube radius.2. Theoretical considerationAccording to Ghosal et a\ [2J, the azimuthal conductivity C70(r) of the plasma at a distance r from the axis can be expressed as a polynomial expansion around /* = /?, and is written in the approximate form<T(r) = <T0 [ l- ( r /# ? )2 r  . (I)where a 0 and n are to be determined and R is the radius of the tube. The quantity n can be determined experimentally and is given asn = R2/a -  2, (2)where a = [ (a - 1 ) / f 2k 2l ] (Ei  /) RQ .In cq. (2) a  denotes the ratio of the radiofrequency current without and with the plasma of interest,/the frequency of the radiofrequency coil probe, / the length of the coil, k a constant depending upon the number of turns of the primary coil, RQ the radiofrequency resistance, E the axial electric field strength per unit length and / is the discharge current.SinceJ*<y„ (1 — ( r /  R)2 ]" rdr = l /2 j tE ,  after integration, it has been deduced (Ghosal et al [2]) asCT0 = < //£ )  2(n + \)l 2 n R 1. 0 )Though an expression has been provided for n in the radial distribution formula for conductivity, it is worthwhile to investigate whether the factor n depends on the radius of the arc tube. It is also to examine whether the distribution formula is valid irrespective of the radius of the arc tube and whether the axial conductivity becomes a function of the radius of the arc tube for the same input energy.3. Experimental arrangementThe experimental arrangement for this work has been already reported by Ghosal et al [I], Measurements have been made for a mercury arc plasma formed within the four arc lubes of approximately constant length 31-32 cm in which anode-cathode spacing is around 24 cm. The oscillator coil was placed closed to the work-coil wound around the discharge tube and the induced radiofrequency voltage was tuned with a variable condenser which forms a secondary tank circuit with the work-coil. The tuned currents were measured with the help of a radiofrequency milliammeter. The main discharge current has been changed by the high current capacity rheostats inserted in series with the DC supply. Any change in the tuned radiofrequency current should provide an indication of the change of resistive impedance of the resonantDependence o f the radial distribution function etc 85circuit. A number of measurements of tuned radiofrequency current, both in the presence and in the absence of plasma have been made for different discharge currents but at fixed exciting frequency 3.69 MHz. The circuit constants are given in Table 1.Table 1. C ircu it constan ts.P a ra m e te r Set I Set II Set III Set IVO uter d iam eter o f  tube in cm 1.22 1.85 2 15 2 6 4Inner d iam eter o f  tube in cm 0 98 1.56 1 75 2 35C oil length in cm 9 .0 0 5 .6 0 5 9 0 4 0 0Coil d iam eter in cm 1.22 1 85 2 15 2 64W ire d iam eter in cm 0 20 0 20 0 20 0 .2 0P ro b e -to -p ro b e  sep ara tio n  in cm 8 .5 0 8 .5 0 6 6 0 9 30N um ber o f  turns in the coil 77 4 4 5 0 33R adiofrequency  resistance o f  the coil in O 14 1 1 19 15Inductance o f the coil in mH 10 10 19 16The two tungsten probes within glass capsules with a bare tip of 0.1 cm have been utilised to measure the axial electric field strength in the plasma column.4. Results and discussionThe values of ( a -  1) as determined experimentally have been plotted against different arc currents (in the range of 2.25 A -  5A) for different tube radii in Figure 1. The variation is linear in nature for different values of tube radii.F ig u re  1. V aria tio n  o f  ( a  -  1) w ith  a rc  c u r re n t, I (R  = 0  49  cm ), II (/? = 0 .7 8  cm ), III (R -  0.875 cm), IV (I? = 1.185 cm).86 M  RoyAs has been shown by Ghosal et al f 1 ] a - \  = (o2 M 2 / R tRn ,where co is the angular frequency of the applied radiofrequency field, M the mutual inductance, R() is the radio frequency resistance of the coil and R} is the axial resistance of the plasma. For a particular tube radius, co, M and R0 arc constants and hence ( a -  1) is proportional to I/Ri and hence to the arc current. As the values of M and RQ will be different for different tube radii, the dependence of ( a -  1) with arc current will vary with tube radius.Values of/t calculated from cq. (2) have been plotted in Figure 2 for different HE values against different tube radii. It is observed that n has a tendency towards saturation for higher //£, values and it becomes a function of R, the radius of the arc tube.F ig u re  2. Van at ion o f  the value o f  n with / /£  ; R as a param eter.Dependence of the radial distribution function etc 87In order to find how the axial conductivity a0 varies with tube radius the values of a0 have been calculated from eq. (3) and plotted against HE in Figure 3. In all the cases a0 increases linearly with l/E but with increasing slope as tube radius is reduced.Figure 3. Variation of o;, with 1/L at different tube radii ! ( /? — () 40 cm), 11 ( R -  0 78 cm). Ill (R = 0 875 cm ), IV <K ^  1.185 cm)T h e  ra d ia l v a r ia tio n  o f a ( r )  f ro m  r =  0  to  r = R h a s  b een  c a lc u la te d  fo r  d if fe re n t tu b e  rad ii fo r so m e  s p e c if ic  v a lu e s  o f  I!E o n e  o f  w h ic h  is s h o w n  in  F ig u re  4 . F ro m  th e  n a tu re  o f  th eFigure 4. Electrical conductivity distribution as function of r for HE values (amp.cm/volt) 4.4, 3.6, 2.8, 2.36 and 2.186 respectively from top downwards in the arc tube of radius 049 cm.M Royrepresentative curve it is evident that not only the conductivity at the axis shows a rapid increase with the increase of the arc current but at the same time the nature of the distribution of azimuthal conductivity undergoes a remarkable change indicating that the discharge becomes more and more constricted with the increase of the arc current. The nature of variation is, however, the same in arc tubes of different radii indicating that the proposed distribution formula (cq. 1) is valid for tube of any radius.To study the variation of radial conductivity profile for different tube radii the value of C(r) has been calculated against tube radii for some values of I/E. In this case also, one of the representative curves is shown in Figure 5. It is evident from the figure that for each value of HE the axial conductivity is largest for tube with smallest radius and gradually decreases with the increase in tube radius. The radial conductivity falls much more rapidly for the tube with small radius whereas the rate of fall is much less for the tubes with larger radii. The case is almost analogous with the selectivity of radio circuit with smaller and smaller resistance. The rate of dimunition of the radial conductivity thus depends upon the tube radius which indicates that n in eq. ( 1) is an explicit function of the tube radius as has been observed here independently.F ig u re  5. E lectrical conductivity distribution as function o f  r  for R  values (cm ) 0  49, 0  78, 0  875 and 1.185 respectively from  top dow nw ards at fixed l /E  value (4.5 am p.cm /volt).If r, a particular value of r, denotes the half-width of the conductivity distribution curve, then we getr = R [1-(1 /2 ) l/n]l/2Taking the value of n from Figure 2 for I/E = 4.5 amp.cm/volt, half-widths for different tube radii have been plotted in Figure 6 . The half-width construed as a measure of the constriction of the plasma column increases rapidly with the increase of the tube radius for a fixed value of I/E.Dependence o f the radial distribution function etc 89The values of a0, the axial conductivity have been plotted against the tube radii for tubes used in the present investigation for 1/E values namely 4.5,4.0 and 3.5 amp.cm/volt andFigure 6. V ariation o f  ha lf-w id lhs o f  the conductiv ity  d istribu tion  function  w ith  A\ tube radius at fixed I /h  value (4 5 am p cm /v o lt)are shown in Figure 7. The curves show that o{) decreases exponentially with the increase of/? Alternatively, values of log cx0 have been plotted against R in Figure 8 and the graph shows a linear variation with/?. Thus the variation of cr0 with R can be represented by an equation ol the form<r0(/?) = cr0<K_)0>exp(-p«),Figure 7. Variation of o tt with tube radius I(//£ = 4.5 ampcm/volt) 11 (//£ = 4.0 amp.cm/ volt), III (//£ * 3.5 amp.cm/volt).90 M Roywhere P is a constant and cro<y?_^o> the axial conductivity at R  —> (). We can calculate the value of P by a statistical method, which is shown here for I/E = 4.5 amp.cm/volt as :cx0 = <*-,<» CXP H W ).lOgc(V/?)=IOge(T0<fi_ ()> ~ P K -s  = l l  logcd 0(/? )-lo g e(T0<^ ()> +PR}2,dSI dp = 2 S  R [ loge<T()(R) -  loge<r((<if_>0> + pR ] = 0,I / f l o gi;<r0(/f) = logcCT()< ^ ()> I R - p i R 2 ,^  = l°ge(To<*-*» r / f - r / ? ] o g e<r0( /? ) /X /r  ,=3.65 r / ?  = 3.33 Z  R 1 =3.01835,£ /?  logger„(/?) = 5.80589 p  = 2.103 .: ' r * y r T ° f '°g'CT" W1,h ,ube radlus Ul/E = 4 5  ampxm/volt) ll<//£ = 40 amp cm/ volt). Ill (//£ = 3.5 amp cm/volt)The calculatedvaluc ol 0 for//£=4.0 and 3.5 amp.cm/volt has been found to be 2.091 and 2.079 respectively. Thus, it is observed that /J decreases with decrease of I/E i.e. input energy.5, ConclusionsIn a mercury vapour tube, the arc completely occupies the tube for low currents but as the current is increased, the arc column contracts and the light becomes more intense at the axis ofD ependence o f  the radial d istribu tion  fu n c tio n  etc 91the tube. Besides this, with the reduction of the lube radius the plasma column also contracts. It is also established that the value of on-axis conductivity decreases exponentially with tube radius. It has also been observed that the conductivity at the axis increases with the decrease of the tube radius for the same value of HE. This may be attributed to the fact that the gas density is higher in a tube of smaller areas for the same value of pressure and consequently the rate of collision becomes higher which results in greater ionization and hence to increased conductivity.R e fe re n c e s[1] S K Ghosal. G P Nandi and S N Sen Ini .1 Election 41 509 (1976)[2] S K Ghosal, G P Nandi and S N Sen h i t  J  E l e c t r o n  44 409 (1978)131 S Mikoshiba and P R Silly Rev Si i lustrum 40 1187 (1969)14 1 M Ciampi and N Talmi /. A p p l .  P h y s .  38 3771 (1967)

Dependence of the Radial Distribution Function for Azimuthal Conductivity of Arc Plasma on Tube Radii

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Dependence of the Radial Distribution Function for Azimuthal Conductivity of Arc Plasma on Tube Radii

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