Jana, D C

Sen, S N

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Momentum transfer collision cross section for slow electrons in magnetic field from radio frequency conductivity measurementsS. N. Sen and  D. C. Jan aDepartment of Physics, North Bengal University, DarjeelingMoniontum tranKfor uolliHion (jroas Section for slow electrons in hydrogen and oxygen in a transverse magnetic field varying from zero to 1850 gauss have b(^ en obtained by studying the variation of radio frequency conducti­vity t>f the ionized gas within the pressure range of a few microns to 6 torr. The momentum transfer cross section is higher than that without field for the same range of electron energies. The results have been discussed in the light of the experimental breakdown voltage measure­ments in these gases. It is further pointed out that the methcKl is capable o f providing accurate values for the electron density in these gases.Indian J. Phya, 52B, 288-295 (1978)1. I ntrodu ctionIn previous communications (Sen and Ghosh 1966, Gupta and Mandal 1967, Sen and Gupta ] 969) it has been shown that tlie measurement of radio frequency conductivity o f an ionistKl ga»s and its variation with pressure enables us to calcu­late the various parameters of tire ionisrsi gas such as the electron density, collision frequency and electron temperature. The measurement in presence of a mag­netic field enables us to find the variation of electron temperature with magnetic field. The only effecd of magnetic field that has been taken into consideration in the previous papers is the introduction of the concept of equivalent pressure and it has betm found that the experimental results can be satisfactorily ex­plained quantitatively for small values of (HjP) only where H is the magnetic field and P  is the pressure; this is due to the fact that the equivalent pressure concept is valid only for small values of {HjP). The measurement of momentum transfer cross Section of electrons for elastic scattering has been carried out in a large nmnber of atomic and molecular gases for a wide range of electron energies by various standard methods such as swarm experiment, microwave after-glow method and the cyclotron resonance method and a com­prehensive review of these methods and the analysis o f the results obtained has been provided by Massy and Burhop (1969). The effect of a magnetic field on the collision cross section of electrons with atoms afid molecules is important not only for understanding the natiire of interaction o f magnetic field with ionised gases but is absolutely necessary for explaining the phenomena of breakdown of gases in magnetic field. As the measurement of radio frequency conductivity of ionised gases and its variation with pressure and magnetic field enables us to calculate the electron density, collision frequency and electron288Momentum transfer collision cross section 289temperature, ao the same method is proposed to be adopted in the prosent investi­gation to calculate the momentum transfer cross section of eliKjtrons iind its variation with electron energy (slow electrons) in a transverse magnetic field. A programme for the measurement of momentum transfer cross scKjtion in mag­netic field, spexjially for low energy electrons has therefore been undertaken and the present paper reports the results in case of hydrogen and oxygen.2. T h e o r y  of Calculation  of Momentum  T ran sfer  Cross Section FROM R. F Con du ctivity  M easurem entIn the present experimental setup It is assumed that the dlschaige current is flowing along the a;-axLs, tl)e radio frj^pumey field uaotl for measurement of r.f. conductivity along //-axis and the n|agnetic field along the ;:-axis. Then the equations of motion aredvydt VVy-\where v is the frequency for momentum transfer, a> is the frequency ol the mea­suring radio frequency field and wb {pB\w) the cyclotron frequency and simi­larly-  0tlionand m[(v+ J 1Henco which is tJio real part of r.f. conductivity in a dirwtion porpondicularto the transverse magtietic fichinehi{i^+ to*+ ‘•'B*} . . .  (1)which is the same expression as deduced previously by Appleton and Boohariwalla (1935) and later by Gilardini (1959). The energy of the electron m the magnetic field can be calculated and the energye. = 1 (2)2m [v®+“»B*]as the value of a. is amaUer than either v or wb by two orders of magnitude. Differ-290 S. N. Sen and D. C. Janaontiating oq. (1) with roapoct to v, which in effect means the change with respect to pressure, we getlOo t^cB*]—(<u*— <uj*) =  0As the magnetic field used in the experiment is o f the order o f a few kil(jga\isa, and the measuring field has the frequency of a few megacycles i.e.(JjJi ^  COwe get,(v'‘ —co^ )^(v2 fcojj^ 0or  ^ ^B-In deriving the above exprosHion it has been assumed that n, th^ ‘ <^ hrcdron density romaias constant with the changes of pressure. To justify this it is notcnl tJiat«  ne^E ne^E% =  nefiE ^ ------mv moFwhere h is a constant. It is observed that with the cJiango of pros>sur(^  the di»s- charge ciirrent changes but in our experiment we havi  ^ k*ipt the cliscJiarge current constant by changing the value of E. To tost this point an experiment has been performed in which the pressure is varied and the corresponding value of E is notcKl so that the discharge current is kept constant and the results aie j)lotte^d in figure 1. The linear relation observed betwinui E atul P  indi(?ated that for constant discharge current n the elecjtron density remains constant with pressure.$P « E S S U f t £  I N  T O A DFig. 1. Variation of applied voltage at different pressures to keep constant discharge current.(T^ rf)B will become a maximum when v =  wjb from which the collision froqvenc}^ at the particular pressure can be calculated and as vc =  Vrj?^ e ~ VfPjLwe get r^Pfnam _L mor H - :  constant,maxMomentum transfer collision cross section 291wlu^re Pmax ^  prosKuro at which tht^  radio fn^quoncy (conductivity luvconios a maximum whon tho magnotic hold H  is applitMl.As N(Tc \ / SkTewhero N  is the number o f molecules pet c.c. at the pressure considered, 0*^ is the momentum transb^r cross scncticm, Tf, is the electron temperature, w is the mass o f the electron and K  is the Boltzman constant. The electron temperature has b(Hm measured by single probe method and its variation 'with magmdic field li,as been found to satisfy thi^  tlveorotieal dfiiiution (Sen, T)as and Gupta 1972) for small (HjP) valiK s^ namelyT eH ^ T e[l-\ n ,  ]wiiereI e L' m ?vywlion; L  is the in<‘aii frw> path o f tJio electron in the gas at. one torr. Tl>e value o f has heeii foiiml to h(» 5.62 < 10 " in c1f)Se agreement with that obtained by Bleviii and Haydon (1958) and that o f oxygen is 17.14 x'lO-''*.Hence from t'q. (3) (Tc the momentum transfer cross swtion in presence of magnetic field is giv<*n bya-e m \/H KTe\P } \ mnivlP)V[ 1-1 c, ~ w11/4... (4)3. Expebttmbntal ArrangementThe present investigation reports tho results regarding the variation of r.f. conductivity with pressure in a transverse magnetic field in case of hydrogen and oxygen. The measuring field has a frequency of 2.45 MHz and the variation of pressure is in the millimeter range and magnetic field employed is 1 kilogauss to 2 kilogauss so that the aasranption is justified.Tho meth(Kl of measurement is the .same as has been us.kI in tlm previous paper (Sen and Ghosh 1966); hydrogen and oxygen have been prepwedby the olertrolyais of a saturated solution of warm barium hydroxide and dried by potassium hydroxide and phosphorous pontoxide.The pressure has been accurately measured by a calibrated MacLeod gauge and the magnetic field by a calibrated gauss meter.292 S. N. Sen and D. C. Jana4. R esults an d  D iscu ssionThe vajiation of r.f. conductivity with pressure in transverse magnetic field 1150G, 13506 and 18506 in case of hydrogen and oxygen have been plotted in figures 2 and 3. In each case it is observed that the r.f. conductivity becomes a maximum at a certain pressure and the pressure at which the conductivity becomes a maximum always shifts towards the higher pressure with the increase o f magnetic field and the absolute value of conductivity diminishes with mag- metio field for all values o f pressure. The experimental results have been entered in table 1.Pig. 2. Variation of r.f.conductivity with pressure for different values of magnetic field; Hydrogen.It is evident from column VI that the theoretical deduction HIPmam =  constant for maximum conductivity is well satisfied. From the expression v =  for maximum (r^ rf)H the value of v can be obtained and the results are entered in the fifth column in table 1. We have then calculated utilizing eq. (4) the values o f momentum transfer cross section for various values o f magnetic field rangingMomentum transfer collision cross section 2B3Table 1. Frequency of meaain'ement 2.45 MHzGas Magnetic field in gauss(P/r)ma« iu mm X  10+is o.m.u.VtifP maxnH50 1.85 1.16 .3222x1010 621.6 1.668X10®Hydrogen 1350 2.15 .985 .3791 X 10^ 0 626.9 1.662x10®1850 3.05 .72 .5184 X 10»o 618.0 1.666x10“1150 .9 1.18 .3222x 101° 1277 1.693X 10“Oxygen 1350 1.1 .985 .3791 X  lO^ o 1228 1.629 X  10“1850 1.5 .70 .5184 X  10^ 0 1236 1.619x10“Oxygen.294 S. N. Sen and B. C. Janafroju 50 to 1850 gauSK fot corresponding energies calcTilatecl from eq. (4) for both hydrogen and oxygon and the results are plotted in figure 4; for c()nij)ari»S()u, the values of momentum transfer ci*oss st c^tion without magnetic field by previous workers are also shown in th(> figun^ . Tt is noUnl that for range of ek^dron energies investigated here the momentum transfer (.toss section when magnetic field is present is always greatcT tlian in th(^  absences of the field with a tendemey to incrcasi  ^for lUgher electron energies. TJie variat ion of monientuni transftT cross section with electron energy without the juagnetic field l»,as Ixxvn explaimxl by yorst and Phelps (1952) and llu^  (^xperiimuital results arc^  found to be in good agrtxunent. Ill thojiaid deduction the distribution of (electron energies lia»s been assunuHl to be Maxwellian but as no adequate  ^ imergy distribution fujicdioii h,as boon found for electrons in a magnetic fiedd, no attempt has been made Imre to dcsluce any quanti­tative expression for the nionKuituiu transfer cross Section in magnetic field. The increase of momentum transfer cross section in proSimce of niagnotic fi(dd indicates more loss of energy by the (doctrons and eonsequentlj’^ higher breakdown voltages will be reqidr(Kl in prcSeiice of magmdic field. T]\o results are corrobo­rated by breakdown field measurements.lliese nieasuronients also jirovide us with a metliod of calculating the id(X)tron density; when v we l)ave(^rf)max ~ 2mo>nMomentum, transfer collision cross section 295TJi(» valu(i of n  oaloulatwl bv this mothod Uan hoc'ii oiitoi’rtl hi t)u‘ ,s(ivoiit]>, coluiiiii of table 1. Tho near coiiKiKtonciy of tlio raluort obtainwl iii due to tlio fact tliat the discliarge curroih lias bix'n kepi couatanl for tiie magnetic fields studied here. Tllo results are also corroborated by probe measurements.Tt is thus obsei vwl that the nu^asureinent of tins radio frequency conductivity of an ionised gas and its variation u’itli pres.sure in a magnetic field enables ns to calculate the plasma paranuttm-s such as electron densit}' and collision frequency. This measurement combined with, measurement ol' electron tcmjx'i-ature enables US to calculate! the momentum transfer cross section and its variation with electron energy.R e f e e Hn o b sAppleton E .  V .  & Boohariwalla D . B. Proe. Phi/s Hoc. 47, 1374.B lovin H .  A . &  Ila yd o u  S. C. JO.'IS Jour. Phys. 11, IS.Frost L. S. & Phelps A . V . 1962 Phys. Rev. 127, 1621.(fila rd in i A . 1959 N imvo Cimntto Siipplm oil. Hermon 13. in.OuptH R . X . & Mu ik IhI S. K . 1967 Jiul. J. Phys. 41, 251.Massey I f .  S. W . and Hurhop E . IT. S. FAcclromc and ionic impact phenomena. Vo l. 1, ClwcndonI’ross, Oxford.Son S. N . & Ghosh A . Iv. 1906 I ml. Jour. Pure & .-\ppl. Phys. 4, 79.Son S. N . & Gupta N . 1969 hid. Jour. Pure & Appl. Phys. 7, 462.Son S. N ., Bas R . P . & Gupta R . N . 1972 Jour. Phys. D. Appl. Phys. 5.1260.

Momentum Transfer Collision Cross Section for Slow Electrons in magnetic Field from Radio Frequency Conductivity Measurements

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Momentum Transfer Collision Cross Section for Slow Electrons in magnetic Field from Radio Frequency Conductivity Measurements

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