Io n isin g rea ctio n s by electro n a n d ion im p a ct a re of th e form
e + X <=> X + + e'+ e , ...(R4) a n d
X + + Y <^> X + + Y ++ e , ...(R 5)
w h e re X a n d Y r e p r e s e n t a n y n e u tr a l a to m or m o lecu le a n d e" re p re s e n ts free e le ctro n s. F o r e le c tro n im p a c t io n is a tio n to proceed, in c id e n t e le c tro n s m u s t h a v e e n e rg ie s a t le a s t e q u a l to th e f ir s t io n is a tio n p o te n tia l E j. T h e cross sectio n for th is re a c tio n ra p id ly in c r e a s e s ab o v e t h r e s h o ld to a m a x im u m w h e re i t f a lls off ap p ro x im ate ly lin e a rly w ith h ig h en erg ies. I f th e ato m in th e collision h a s m ore th a n one su b sh ell, m ore th a n one th re s h o ld occurs a t h ig h e n e rg ie s re s u ltin g in som e s tr u c tu r e in th e io n is a tio n fu n ctio n . The e le c tro n im p a c t io n is a tio n p ro ce ss is a n e n d o th e rm ic re a c tio n a n d h en ce th e r e e x is ts a th re s h o ld v a lu e for th e e le c tro n e n e rg y below w h ich th e re a c tio n w ill n o t proceed. T he cross sectio n for e le ctro n collisional io n is a tio n v a rie s as E ’MogE a t h ig h en erg ies as p red ic te d by th e B eth e-B o rn a p p ro x im atio n . T h ere is a n (E-Ej)11 dep en d en ce on th e cross section n e a r th resh o ld . W a n n ie r (1953) observed th e dependence of th e yield on th e electron energy j u s t above th resh o ld . A v alu e for n of 1.127 w as o b ta in e d b y W a n n ie r u s in g a sem i-classical tr e a tm e n t (for ions th e ex p o n en t lie s b e tw ee n 1.127 a n d 1.0), w hile v a rio u s q u a n tu m m ec h an ica l c a lc u la tio n s h av e g iv en v a lu e s for n of 1.0, 1.13 a n d 1.5 (Rudd, 1981).
Chapter 2 : Ionisation p ro c e sse s and g a s emissivities 54
F o r electro n s w ith en erg ies less th a n th e en erg y difference b e tw ee n th e lo w e st ex cited s ta te a n d th e g ro u n d s ta te , th e to ta l collision cross section is sim ply eq u al to th e e lastic collision cross section (M assey a n d B u rh o p , 1952). H ow ever as th e electro n en erg ies in c re a se beyond th is difference, e x cita tio n a n d io n isatio n of ato m s w ill proceed a n d th e to ta l cross section w ill now becom e eq u al to th e su m of th e cross sectio n for e la s tic co llisio n s to g e th e r w ith th o se for e x c ita tio n a n d io n is a tio n . M assey a n d B u rh o p give
E
max
Q - Qo + + JQe d E ...(2.9)
S 0
for th e to ta l cross section for electro n im p a ct w ith ato m s or ions. Q0 is th e e la stic cross section, Qg is th e cross section for ex citatio n a n d QE dE is th e cross sectio n for io n isin g collisions for w hich th e e n e rg y of th e lib e ra te d electron lies b etw een E an d E +dE .
L in a n d T ea re (1963) rep ro d u ce th e to ta l io n isatio n cross section QiX(e) for collisions b e tw ee n electro n s w ith en erg y e a n d ato m s or m olecules of ty p e X, in te rm s of th e local electron en erg y d istrib u tio n fu n ctio n f(e). T he r a te of ion p a ir p ro d u c tio n p e r u n it volum e by e le c tro n im p a c t b eh in d a shock fro n t is given as
^ = ne J u Q v (e)f(e) de , ...(2.10)
at v J lX
x wiX
w here n e is th e local electron n u m b e r density; n x is th e n u m b e r d e n sity of m olecules of ty p e X; wiX is th e th re sh o ld energy; a n d v is th e electro n
Chapter 2 : Ionisation processes and gas emissivities 55
v elo city w ith a n e n e rg y e. T h e e le c tro n v elo city d is tr ib u tio n w ill a p p ro x im a te th e M ax w ellian fu n ctio n a t th e ele ctro n te m p e r a tu re T e w hich is s ta te d as b ein g com parable to, b u t n o t id en tica l w ith, th e heav y p a rtic le tr a n s la tio n a l te m p e r a tu re T. As a f ir s t a p p ro x im a tio n , th e te m p e ra tu re s T a n d Te a re assu m e d by L in an d T eare to be e q u al w hen e s tim a tin g th e e le ctro n im p a c t io n is a tio n ra te . W ith th is a s s u m p tio n eq u atio n 2.10 is re -w ritte n as
3/2
Y , n x
J
e d e •- (2-1:L)
* wiX
w h ere m e is th e electro n m ass.
T a te a n d S m ith (1932) m e a s u re d th e io n is a tio n cro ss se c tio n s of e le c tro n im p a c t w ith th e d iato m ic m o lecu les N 2, 0 2, a n d N O . T h e ir m e a su re d cross sections ex h ib it th e ra p id rise to th re s h o ld follow ed by a m o n o to n ic d e c re a se w ith in c re a s in g e le c tro n en erg y . T h e e le c tro n io n is a tio n cross sections for atom ic N a n d 0 a re n o t as w ell k n o w n as th e ir m o lecu lar c o u n te rp a rts. T he cross section v e rs u s electro n en erg y curve for d iato m ic m olecules a n d noble g a ses a re show n to be q u ite s im ila r e x c e p t for th e d iffe rin g th r e s h o ld e n e rg ie s in e a c h case. M a ssey a n d B u rh o p h a v e com piled th e a v a ila b le d a ta on co llisio n al io n is a tio n b y n e u tr a l a to m s , io n s a n d m o le c u le s. L o o k in g for c o rre la tio n s in th is d a ta , it is ev id e n t th a t, w ith th e ex cep tio n of th e io n is a tio n p o te n tia l, th e r e is no th r e s h o ld e n e rg y b elo w w h ic h io n is a tio n is n o t p o ssib le, a n d for p o sitiv e io n s, th e cross se ctio n is e x c e e d in g ly sm a ll u p to e n e rg ie s c o n s id e ra b ly g r e a t e r t h a n th e io n isatio n p o ten tial. In com parison w ith th e electro n im p a ct io n isatio n p ro cesses, ion im p a c t io n isa tio n will be in sig n ific an t. Since th e cross
Chapter 2 : Ionisation processes and gas emissivities 56
sectio n v e rs u s e lectro n en erg y curve for d iatom ic m olecules a n d noble g a se s a re show n to be sim ilar, L in a n d T ea re s ta te t h a t it is rea so n a b le to a ssu m e t h a t th e cross section for io n isatio n b y electro n im p a c t for N a n d O follow th is p a tte r n a n d ex h ib its sim ila r curves b u t w ith d ifferen t th re s h o ld e n erg ies.