I t has been observed th a t the temperature o f hydrogen atoms in the anode region of the d isch arge is much higher than th at o f the argon atoms (se c tio n 4 ,4 ) . This may be expected sin c e a hydrogen atom b ein g lig h te r gains more energy from each c o llis io n w ith an e le c tr o n than does an argon atom. This p rocess w i l l be examined in more d e ta il u sin g elem entary k in e t ic theory.
Only e la s t ic c o llis io n s are co n sid ered , and the e le c tr o n s and atoms are trea ted as smooth, hard sp h eres. The plasma i s assumed to c o n s is t of 5 components; heavy atoms (argon) mass m^, l ig h t atoms
(hydrogen) mass m^, e le c tr o n s mass m^, heavy ion s mass m^ and lig h t ion s mass m^. Heavy p a r tic le s w i l l be d esign ated type (1) and lig h t p a r tic le s by type (2 ), The co n cen tra tio n o f lig h t atoms is much le s s
than th at o f the heavy atoms and the fr a c tio n a l io n iz a tio n i s assumed to be sm all ( < 1%), The co n cen tra tio n o f lig h t io n s is assumed to be very sm all and w i l l be ign ored .
The energy gained by a p a r tic le of mass c o llid in g w ith a p a r t ic le of mass m^ is given by
2 m^m^
= --- {m u ^ - m u_^ 4- m u-u„ - m«u p } (A9)
(m^+m^)^ ^ ^ 1 1 2 2 1 2
where u^ and u^ are the components o f the v e lo c it ie s o f p a r t ic le s (1)
and (2) along the lin e o f c e n tr e s ,
Although the system i s n ot in thermodynamic eq u ilib riu m s i t i s convenien t to assume a Max^vell Boltzmann d is tr ib u tio n o f en ergy, The Langmuir probe measurements (appendix A3) in d ic a te th a t th is is q u ite ,a good approxim ation fo r .th e e le c tr o n energy d is tr ib u tio n .
In p r in c ip le i t i s p o s s ib le to deduce the v e lo c ity d is tr ib u tio n of the atoms from the D oppler broadened p r o f ile o f the sp e c tr a l lin e s ,
although in p r a c tic e th is method i s lik e ly to be rath er in accu rate due to n o ise on the measured lin e p r o f ile .
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Using a Maxwell Boltzmann d is tr ib u tio n (A9) can be averaged over the v e lo c it ie s o f p a r t ic le s type (1) and (2)„ The average energy tra n sferred from (2) to (.1) per c o llis io n is given by
2 m^m^ k
A2 = - T. ) , (AlO)
where T^ and T^ are the r e sp e c tiv e tem peratures and k is Boltzmann's con stan t,
For the chosen model o f the plasm a, the ra te a t which energy i s gained by type (.2) p a r t ic le s from the e le c tr o n s must be equal to the ra te at which i t i s lo s t to the type (1) p a r t ic le s , (The e le c tr o n s are trea ted as an i n f in it e h ea t source and the heavy p a r tic le s as an in f i n i t e h ea t s in k ). Equating the r a te s o f gain and lo ss o f energy g iv es j
, 2 m„m k , 2 .m,.m k , . 2 im.m^
^ ^ ^ ^ ^ ^ (A ll) where v^^ the c o ll is i o n r,ate between lig h t and heavy atoms
v^^ i s the c o llis io n ra te between e le c tr o n s and lig h t atoms v^^ i s the c o llis io n ra te between lig h t atoms and heavy iqns T^;T2ÿTg and T^ are the tem peratures o f the heavy atoms, lig h t atoms e le c tr o n s , and heavy ion s r e s p e c tiv e ly .
Since m^ >> m^ >> m^ (A ll) may be s im p lifie d to g iv e , m.m V v „ , m_m v v ,
T _(l + — + — ) = T. + — T_ + — T, (A12)
m^^ ^12 ^12 ^ m^' ^12 ^ ^12 ^
E xperim entally i t is found T^ i s c lo s e to T^(T^ = 2 ,4 0 0 K, T^ = 2 0 0 0 K) and sin c e the fr a c tio n a l io n iz a tio n i s sm all we have ^24 ^12* (A12) may be s im p lifie d to give
m m . V mm v
T ( 1 + _ji£) = T. + - L A T (A13)
2 ''12 ^ mg' ''12 ^
■66*
^ 1 2 ” ^1^2^1 2 ^ 2 c o llis io n s cm"^ se c ”'^ (Â14)
\ ■
where N. and N. are the number d e n s itie s o f the heavy and lig h t atom s,
1 2 ‘
0 ^ 2 the c o llis io n cross s e c tio n and U2 i s the average v e lo c ity o f
the lig h t, atoms, (The heavy atoms move much more slo w ly than the lig h t atoms so U2 must be u sed ).
S im ila rly the c o ll is i o n rate between the lig h t atoms and the
e le c tr o n s i s given by - ^
^23 “ ^2^3^23^3 '
where i s the e le c tr o n number d e n sity , O2 3 the cross s e c tio n for the c o ll is i o n , and u^ i s the average v e lo c it y .o f the e le c tr o n s .
From (A14) and (A15) the r a tio of the c o llis io n r a te s i s given by
The e a s ie s t way to compare t h is theory w ith experim en tal r e s u lts i s to use the measured v a lu es o f T^;T2 , and T^ to c a lc u la te '^23^^12 from (A13) and then compare th is w ith the valu e c a lc u la te d from (A16),
From (A13) the r a tio ''^23^^12 given by
V .. T^-* T m^
I f the lig h t atoms are hydrogen and the heavy atoms argon -m^ = 40 m2 and m2 - 1840 T yp ical v a lu es o f T^, T2 and T^ are 2000 K, 5 ,5 0 0 K, and 20,000 K, Tlius V2 2/ v^ 2 i s equal to 90, ■
To c a lc u la te '^23^^12 (A16) req u ires a knowledge ■‘^23^‘^12* If the atoms and e le c tr o n s are tre a te d as hard sp h eres w ith the i e le c tr o n s much sm a ller than the o th er p a r t ic le s which are assumed to have the same d iam eter, then ^ 2 3 ^ °\1 ^ 1 /4 , T yp ical valu es o f
and N are 10‘ ^ cmT^ and 2 x 10^® cm“^ . S u b stitu tin g th ese va lu es
1 ^ . i ■
in (A16) g iv es ^23^"^\2 " 1/10» very much le s s than the valu e (90) •
” 67”
I t i s c le a r th erefo re th a t th is theory i s t o t a lly inadequate to ex p la in the la rg e d iffe r e n c e between the temperature o f the argon and the temperature o f the hydrogen. Using th is model the expected d iffe r e n c e in temperature i s o n ly 70 K,
To ex p la in the observed tem peratures, we pow consid er the e f f e c t s
o f in e la s t ic c o llis io n s . These c o llis io n s g e n e r a lly tr a n sfe r le s s tr a n sla tio n a l energy from the e le c tr o n s to the atoms than e la s t ic c o ll is i o n s , however there is an im portant ex cep tio n . A hydrogen m olecule may be d is so c ia te d by an e le c tr o n c o llis io n i f the m olecule i s e x c ite d to the 1^2^ s t a t e . This s ta t e is r e p u lsiv e and the atoms f ly a p a rt, each carryin g o f f 2 .1 eV o f k in e tic energy (4 4 ). This p rocess provides a source o f h igh energy hydrogen atoms which only lo s e th e ir energy to the h ea v ier argon atoms a fte r many c o llis io n s . The th resh old fo r e x c ita tio n of the s ta te by e le c tr o n c o llis io n i s 8 . 8 eV.
The high tem perature o f the hydrogen atoms i s probably due to the energy gained by the hydrogen atoms in the d is s o c ia tio n o f a
hydrogen m olecu le. The average energy of the hydrogen w i l l be determ ined by the ra te a t which energy i s lo s t by the atoms to the h ea v ier argon atoms and by the d is s o c ia tio n r a te . Re com bination o f
hydrogen atoms (probably at the w a lls ) w ill provide a continuous source of hydrogen m olecules to su sta in the d is s o c ia tio n r a te . This p rocess i s d i f f i c u l t to analyse sin c e recom bination ra tes and
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APPENDIX A5