ON THE EVALUATION OF ELASTIC AND
INELASTIC COLLISION FREQUENCIES
FOR HYDROGENIC-LIKE PLASMAS
E. L. BYDDER
September, 1967
Department of Engineering Physics
Research School of Physical Sciences
THE AUSTRALIAN NATIONAL UNIVERSITY
HANCOCK
f T J 16 3
. A87
EP - RR1 7
erra, A.C.T., Australia.
TJ163.A87 EP-RR17.
1924139
A.N.U.
LIBRARY
EP-RR 17
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ON THE EVALUATION OF ELASTIC AND INELASTIC
COLLISION FREQUENCIES FOR HYDROGENIC-LIKE PLASMAS
by
E . L. BYDDER
S eptem ber, 1967
Publication E P -R R 17
D epartm ent of E ngineering P h y sics
R e se a rc h School of P h y sical Sciences
THE AUSTRALIAN NATIONAL UNIVERSITY
C an b e rra, A . C . T .
R S P H Y S .S
A u stra lia
. 8 FEö 1968
CONTENTS
page
SUMMARY
iii
1.
Introduction
1
2.
The D ifferential C ro ss Sections
2 .1
E lastic C ollisions Between E lec tro n s
2 .2
E lastic C ollisions Between E lectro n s and
Hydrogen Atoms
2. 3
E lastic C ollisions Between E lec tro n s and P ro to n s
2 .4
E lastic C ollisiops Between Hydrogen Atoms
2 .5
E lastic C ollisions Between Hydrogen Atom s
and P rotons
2 .6
E lastic C ollisions Between P ro to n s
2 .7
I s - 2p Excitation C ollisions Between E lec tro n s
and Hydrogen Atoms
2 .8
Ionisation C ollisions Between E lectro n s and
Hydrogen Atoms
2 .9
C harge Exchange C ollisions Between Hydrogen
A tom s and P ro to n s
3
4
4
4
4
5
5
5
5
6
3.
E valuation of the <j) 's
3 .1
E lastic C ollisions
3 .2
Inelastic C ollisions
6
6
10
4.
E valuation of the jTL 's
4 .1
E lastic C ollisions
4 .2
E lectron-H ydrogen Atom I s - 2p E xcitation
C ollisions
4 .3
E lectron-H ydrogen Atom Ionisation C ollisions
4 .4
Hydrogen A tom -P roton C harge Exchange C ollisions
23
24
29
31
34
APPENDIX A
36
APPENDIX B
57
APPENDIX C
66
REFERENCES
70
ii
SUMMARY
D etailed calculations of the g en eralised
c o llisio n freq u e n cies fo r the p rin cip al e la stic , e x cita ­
tion, ionisation, and charge exchange co llisio n s in the
fo rm atio n of a hydrogenic p lasm a a re given.
A m odi­
fied fo rm of B o m ’s approxim ation is used fo r the
d iffe re n tia l c ro s s sectio n s, and approxim ations c o r ­
responding to a high te m p e ra tu re gas a re used.
The
re s u lts enable c e rta in se ts of 13-m om ent equations
d e sc rib in g the p la sm a form atio n s to be com pleted.
iii
Introduction
.
1
R ecently m ethods have been devised to enable full account to be taken of
in elastic as w ell a s e lastic collisio n p ro c e s s e s in the application of G ra d 's 13-m om ent
1
2
equations to p a rtia lly ionised g a se s (Bydder, B ydder and Liley ). The re su ltin g c o l­
lisio n in te g ra ls a re e x p re ssed in te rm s of c e rta in g e n eralise d co llisio n "freq u en cies ”
(but including a num ber density fa c to r), j f l ( V )
a
Or)
, which a re s im ila r to functions
used by Chapm an and Cowling3 fo r e la stic c o llisio n s.
The final fo rm s of
th ese co llisio n te rm s a re , how ever, of little p ra c tic a l value w ithout the a sso ciated
iT . 's being d eterm in ed explicitly through the use of the re le v an t co llisio n d ifferen tial
c ro s s sec tio n s.
In the lim iting c a s e of e la stic co llisio n s th e re a re only five such p a r a ­
m e te rs involved, and th ese have alread y been obtained fo r many d ifferen t types of in te ra c tio n s (e .g . re fe re n c e 3).
However, since the
± L
(.W
a re p e c u lia r to
re fe re n c e s 1 and 2, obviously no o th e r equivalent calculations have been c a rrie d out
fo r the in e la stic c a s e .
F o r th is re a so n the following m ore g en eral calcu latio n s have
been m ade, e sse n tia lly com pleting the equations of re fe re n c e 2 and also of a p a p er to
4
be published (Bydder and Liley ), fo r a p a rtic u la r form of hydrogenic p lasm a.
In g en eral th e re a re 20 J T L 's to be d eterm in ed fo r each p a rtic le type
p re se n t.
T h e re fo re a t f ir s t sight the ta sk is a form idable one.
However, as w ill be
seen, reaso n ab le and in te r-r e la te d ex p re ssio n s can be obtained fo r c e rta in specific
c a s e s.
In this re p o rt the
SI
J ( r ) a re obtained explicitly fo r p ro c e ss e s applicable
to the fo rm atio n of a la b o ra to ry h ydrogenic-lik e p lasm a.
In attem pting to follow the collisio n p ro c e ss e s d u rin g the developm ent of
a fully ionised p la sm a from m o lecu lar hydrogen, co llisio n s involving (Goodyear and
5
*
*
+ +
Von Engel ) H , H, H , H , H , II
and e le c tro n s should be taken into account.
&
Li
Z
B ecause of the la rg e num ber of p o ssib le types of c o llisio n s betw een th ese p a rtic le s ,
*
+
a sim plified m odel, re s tric te d to H, H , H
and e lec tro n s, is co n sid ered .
In all
p ro b ab ility th is is a re a lis tic m odel fo r sufficiently high in itial tem p eratu res* and in
g en eral should give re s u lts suitable fo r co m p arisio n w ith ex perim ent.
even fo r th is sim ple c a se , the re le v a n t ex p re ssio n s fo r the
Düs a re
U nfortunately,
not p a rtic u la rly
am enable to integ ratio n , and in c e rta in c a s e s v a rio u s approxim ations a re n e c e ssa ry .
G en erally speaking, re la tiv e ly ex act e x p re ssio n s can be obtained fo r e la stic co llisio n s
(Chapman and Cowling, loc cit, and also Appendix C), the m a jo r approxim ations
Introduction
2
applying to the in elastic p ro c e s s e s .
These approxim ations a re such that the
CL ' s ten<
to be in accu rate n e a r the th resh o ld energy, which is unfortunate, but they do lead to
reaso n ab ly a c c u ra te re s u lts fo r appreciably h ig h er e n erg ies.
The n iTi (°
a re defined in re fe re n c e s 1, 2 and 4 by
-Y\ *r+z
O
’ =
w here
(J)j^ ^
In these form ulae,
>V )
is given by
Y =
9
*
re la tiv e velocity of the colliding p a rtic le s , and
9
being the
<T SA#\ % d y d £ ~
is the d ifferen tial c ro s s section fo r the co llisio n betw een p a rtic le s of types j
involving p a rtic le s of type
t
, &
. The rem ain in g p a ra m e te rs , also defined in r e f e r ­
en ces 1, 2 and 4, a re given as they subsequently o ccu r in th is re p o rt.
F o r conven­
ien ce, how ever, the following may be listed :
is the re la tiv e velocity of the colliding
p a rtic le s j ft b efo re the collision;
0
—
\
is the re la tiv e velocity of the p a rtic le s
j / ^
produced in the collision;
— ( 9 1 / 1 9 1
=
w here
(see a^so Appendix B);
$ ; S y/ + J. sck' - fifcj - i - . t i k,
is som e function of the p a rtic le p a ra m e te rs , and the
S t j a re o rd in a iy
K ronnecker deltas;
At
is the kinetic energy "gain" in the collisio n .
3
The D ifferential C ro ss Sections
2.
Since th e re a re not suitable analytical ex p re ssio n s fo r the d ifferen tial (or
fo r th at m a tte r, total) c ro s s sections fo r all of the p ro c e s s e s being co n sid ered , two
m ain ap p roaches a re p o ssib le.
The f ir s t would be to obtain em p iric a l re la tio n s fo r
the d ifferen tia l c ro s s section fo r each p ro c e ss , in the relev an t energy range, based
on the use of a cc u ra te calcu latio n s and ex p erim en tal re s u lts .
U nfortunately th e re a re
re la tiv e ly few a cc u ra te calculations fo r the d ifferen tial c ro s s sectio n s, and few er ex ­
p e rim e n ta l re s u lts .
More data is available on to tal c ro s s sectio n s, and e m p iric a l
re la tio n s fo r the d ifferen tial c ro s s sectio n s w hich in te g ra te to the known total values
could have been chosen.
This would, how ever, be p a rtic u la rly tedious.
The second
approach is to use a c o n siste n t m athem atical approxim ation fo r th e o re tic a l evaluations
of the d ifferen tia l c ro s s sectio n s, fo r exam ple, the B om approxim ation o r c e rta in m o­
d ificatio n s of this approxim ation.
quent calcu latio n s.
It is the la tte r approach which is used in the su b se ­
It m ust be stated, how ever, th at the in v e rse fifth pow er law and
o th e r such c la s s ic a l m odels provide a much b e tte r d e sc rip tio n of c e rta in of the e la stic
c o llisio n s than any form of the B om approxim ation, job fs calcu lated fo r such c la s s i3
cal c r o s s sectio n s a re given in Chapman and Cowling, and fo r convenience th ese c a l­
cu latio n s a re b rie fly su m m a rised in Appendix C.
F o r the p re s e n t re p o rt, the B om
approxim ation has been applied to all c o llisio n s, being m ultiplied by a n u m erical fa c ­
to r to produce "re aso n a b le" approxim ations to the total c ro s s sectio n s in the energy
ran g e 0 - 100 ev.
The detailed deriv atio n of the v ario u s c a s e s and a co m p ariso n w ith
known v alues of total c ro s s section is given in Appendix A.
F o r notational convenience,
is the e le c tro n ch arg e, and
units being used).
€0
°f the colliding p a rtic le s is deflected with
is the reduced m a ss of the colliding p a rtic le s;
co n stan t divided by 2 tT
; and
p / T?
.
fe
p
K
=
-
£o
“6
is P la n k ’s
is the (reduced) wave v e cto r of the colliding p a r ­
is the reduced m om entum
is the B ohr ra d iu s, and the change in wave v e c to r in a co llisio n , K
(l)
m kS
Except w here o th erw ise designated, d C i is the solid angle into
; mr
tic le s , given by
w here
is the fre e -s p a c e p e rm ittiv ity (ratio n alised
w hich the asym ptotic re la tiv e velocity ^
p o la r angle X
/ 4 T T € 0 is rep laced by e 2
,
(J
. Finally,
Q,0
, is defined by
4
ko
and
fef being the wave v e c to rs b efo re and a fte r the co llisio n .
th e se wave v e c to rs,
c o llisio n s.
and
The m oduli of
(the wave n u m b ers), a re of co u rse equal fo r e lastic
The d ifferen tia l c ro s s sectio n s given in the following su b -se c tio n s a re
d e riv e d in Appendix A, and include the " c o rre c tio n " fa c to rs.
2.1
E la stic C ollisions Between E lectro n s
The d ifferen tia l c ro s s sectio n fo r these Coulomb co llisio n s is
(2)
w ith
CT d -0. =
% >/
Xo
2 .2
e
.
> 0
fc/a
rl 0
is the Debye c u t-o ff angle (see Appendix A).
E lastic C ollisions Between E lectro n s and Hydrogen A tom s
The adjusted B om d ifferen tia l c ro s s sectio n is
(3)
O' d f l
=
(-43")
4 nV
+
8 )X dA
* 4 ( K l a* + 4 ) 4
2. 3
E lastic C ollisions Between E lec tro n s and P ro to n s
T his Coulomb d ifferen tia l c ro s s section has the sam e fo rm as (2)
(4)
w ith
cr
X
dfl
=
^
*Xo
2 .4
n)reh o
4 W kt
c
dD-
(Debye cut-off).
E lastic C ollisions Between Hydrogen Atom s
The adjusted B o m c ro s s sectio n is
(5)
<T d f l
rr
( 2 -ox io
z ) 4
n v e^cto
K
* * C a* K1 + 4 ) 9
(aj
K
+* %)4 d_Q
5
The D ifferen tial C ro ss Sections
2. 5
E lastic C ollisions Between Hydrogen Atom s and P ro to n s
Of the sam e gen eral fo rm as (3), the d ifferen tial c ro s s section is
(6)
<rcUl
=
U-Qx
2 .6
io'z) 4«ifV fl.*(K ^%
8
)z d.il .
E lastic C ollisions Between P ro to n s
F o rm ally the sam e as (2) and (4),
(7)
dfl
<r
w ith
=
X
i«f e*
X 0
2.7
X / x dfl
(Debye cut-off).
I s - 2p E xcitation C ollisions Between E lec tro n s and Hydrogen
Atom s
The adjusted B orn approxim ation gives
(8)
O'
dQ
=
(-4.S) 31 215
kot* K 1 ( A
2. 8
d il
K1+ 9 ) 6
Ionisation C ollisions Between E lec tro n s and Hydrogen A tom s
A sim plified fo rm of the B orn approxim ation used fo r th is c a se
gives
(9)
<T
w h ere
dfl, dAi dck = (-I4X IQ?) Z6tta« nye*
dll, d-Q1dj<
k, K4 ( a * K1 + I ) 4
d - 0 .j
w ave v e c to r
is the solid angle into w hich the incident e le c tro n is s c a tte re d (with
k-f
), and
ejec te d w ith wave v e c to r "J<
the solid angle into w hich the atom ic e le c tro n is
. D etails a re given in Appendix A.
6
2. 9
C harge Exchange C ollisions Between H ydrogen A tom s and P ro to n s
A sim plified B orn approxim ation gives
(10)
5- djfl
= (2-1X 10'") 28 mr e V o
d fi
xc £ + O 6
w here, fo r this c ase ,
K *
Evaluation of the
3 .1
fe-o + i - f
•
<^> T!
E lastic C ollisions
The g en eral
4)
defined p rev io u sly
a r e sim plified fo r e lastic c o llisio n s, since X — | . Hence the
pendent of ytL
:
(j x \>) =
( 11 )
a re inde­
J ( I - 003,v;t ) q cr s A
a
4 M
dy
,
and th ese (j> (, V) a re the sam e as those used by Chapm an and Cowling.
(a)
^
E le c tro n -e le c tro n co llisio n s
T hese re s u lts a re given in re fe re n c e 3.
(12) <J>ee( ') =
f j
( |-
C ^ , y . ) e * C] ^ e c ^ / z S w X d y - =se^^
X o
:
4
_
i
Wr* 9
__ ^
^
i
w here
(13)
(
(14)
=
A
XD
Xo g 1
/e 1
being the Debye length).
(j>eeC2.)
( A n (| + A \e)
^ r9 3
since A . gg
S im ilarly ,
is la rg e .
A ce
I + A ee
)
^
t 1) ,
?
E v alu ation of the
(j> ’s
(b)
E lectro n -h y d ro g en atom co llisio n s
A ccording to (3),
<15> <J>ea0 ) =
I ( I~
°
Since fo r e la stic co llisio n s
(16)
fC
=
2 C
CI-
UrsX )Cj (-4.5)41Tye^CU (K ^o + 8 )^ ~Xft*
<UT5 X -
using the substitu tio n vj ^=. | — Co^
(17)
<j)pa( i ) =
w here
( K ^do +
)
4 )*
,
, a fte r som e m anipulation the re s u lt is
(-4..T) 4 m r e V t ^ (-in; ct * 2 4
=
X ko &o
=
ct & ~ ^
- 2 4- <1 ^
2 Mr g 2 Clo / A* .
S im ilarly
/ t TT
(18) <J>e a ( z )
—
J0C
x
I - c o ^ x x ) q ( 45~)4mr e ^ a o ( K V t -*-8) S ^ » X d X
fc* ( 4 - + K1 ^ ) 4
=
2
<j>ea0 ) -
(^ s-K n ^ e ^ o 9
+ '<>)
3 fc* ( c i . 1 - 2 ) 3
A ltern ativ ely , fo r the sam e collisio n s co n sid ered according to an in v e rse pow er fo rce
law, the
a re e sse n tia lly given in Appendix C.
(c)
E lec tro n -p ro to n co llisio n s
T hese collisio n s give the sam e e x p re ssio n s as fo r e le c tro n -
e le c tro n c o llisio n s.
(19) <j>e O) = - S l _ "V93
^j(M -Aep),
8
<20>
^ c,+AV>-
<fcpC O
m r3 *
l+ A e p
) - ^epO),
'
^ d g1mr / e 1.
w here
(d)
H ydrogen atom -hydrogen atom co llisio n s
Using equation (5)
f
(j>a a 0 ) = .
( 21 )
CI - < ^ s X ) g
1 0 ^ ) 4 i t v e ^ a ® K * ( q I K % 8 ' ) s ^ i n X d X -.
°
On substituting fo r K
(22)
I
=
j
( a * « 1 - *- * ) ®
(from (16)) the p a rt of (21) to be in teg rated is
°
(T -
= 2 ko d* ;
w here
P-
C I - C47SX.)3 (
f
<1 < * * % )* S ^ 1 > C d x
c y * s )i)8
=
+ 8
,
r
=
+ 4
.
E lem entary application of stan d ard in te g ra ls p ro v id es the re su lt:
T
_ _L [~d^57C,
_
— <?'V
<* LL?(q,ij+4)7
(23) X
I 92
>
y
(24) 1
I
y
.
_
J(n U
+A) 7
( ( O U t -(iJ
).ij
1 -l-v)3
'
3072
2 16
3 ( q ,y + 4 ) 3
the lim its being on
143340
iC ayt^)1
■^3
F o r the g re a te r p a rt of the energy range being co n sid ered ,
<£
w ill be used.
Xav
_L
J
2
In this c ase ,
(25) (j^O) =
VCiy^4)v
C ^y + 4 )
; taking th is into account, the approxim ation
-
768
g ( l x lo'b 16 mre^ Ca®
C
kU l
E valuation of the
9
Ts
S im ilarly, with the sam e approxim ation,
<26> <j>aaCO =
q ( 2 . x IO x ) 3X mr
X o g itll
Again, possibly b e tte r a lte rn a tiv e re s u lts , based on in v e rse pow er fo rce law s, a re
given in re fe re n c e 3 (see also Appendix C).
(e)
Hydrogen atom -p ro to n co llisio n s
Using the differen tial c ro s s sectio n (6), fo r th is c ase
f
<27> «tapCO =
C l - < ^ % ) q ( 2 x i o ' 2) 4 m r e * a t ( K a o + 9 ) ' t v w i * - d X .
** ( K2a | + 4 )*
°
The in tegration is s im ila r to the previous c a s e s .
(28) «Kp (*) =
( 2 x io' 1) 4. cj mr
Taking
M rq
=
2 feo&o
I
y
%±1
Again, it may be shown that
(29) <}>ap(l)
=
C2 x lo- i ) 8 g m ^ e* <io
2. 4*ap C 1) •
(As m entioned with c e rta in o th e r c a se s, re fe re n c e may also be made to Chapm an and
3
Cowling .)
(f)
P ro to n -p ro to n c o llisio n s
The calculation fo r th is case is fo rm ally the sam e as fo r e le c tro n -
e le c tro n co llisio n s, the re s u lts being
10
(30)
<J>ppC0
=
-4 -3 -
^
0
9
w here -A -p p
(31)
~
X o C A n r y / e 1 ; and sim ila rly ,
“ -Mr
(^ (!+^ p) “
"V*9
^
z)
C
3 .2
I") ~
-A^PP
t
~
I +-Y C,
to produce p a rtic le s
Xx
$„< ■')■
In elastic C ollisions
F o r in elastic c o llisio n s involving p a rtic le s j
(32)
z
=
j 7
noj
^
colliding
;
_
rnj / mk/
, ft
1 (t h j + m h ^
"V " V 91
T herefore it is n e c e s s a ry to use the g en eral fo rm fo r the
<j) Ts
which, fro m the
Introduction, is
^7T
(33)
= J
( I - A C O h ’x ) S'
X ^X .
F o r excitation c o llisio n s w here the n a tu re of the colliding p a rtic le s does not change
(the energy being c a r rie d away by photons, fo r exam ple), the following (J> Ts a re
re q u ire d to evaluate the collisio n te rm s : < ^ (- 00, 0),
(j)(2 >X)
<pC lj l)
, (|) ( 2. o )
and
. In m ore gen eral in e la stic c o llisio n s it is also n e c e s sa ry to calcu late
(a)
E lectro n -h y d ro g en atom I s - 2p excitation collisions
Using the d ifferen tia l c ro s s sectio n (8),
(34 > <|>(yu.,v) = f (1- )Acoivx) g (jus) 3*2*
** fco K*(4K*ai + 9 ) 6
S^X dx
11
E valuation of the
and since
(35)
K 1 =.
ko + kf
Z ko fe-f CX5X;
-
ignoring the constant te rm s in (34), th is becom es
(36)
I ( A v)= /
(i- \
( p-<j,cosx)(r-sors>)6
w here
p = feo *+
(37)
cj^ =- Z fe0 fef ; r = ? + 4 do (feo + fef) i
*,
S = 8a0
H 0k( • t =
r ± s
•= 7 + 4
Using the substitution
(38)
l (
=s
) =
and
R
-
T — S C<rs X
V
can
P = Jr
r"s (t + <iy) ^
rs-s
(40)
Q
S
(41)
R
=
- ^ r > i a
e x p re sse d in te rm s of in te g ra ls
takes the values
r+s
(39)
j
s
X (yy)
, w here (since
= - i8
( fe0 i kf)
r r*i o - ^ c - A
r
and it follows that
ps - <jr
C t + q, y) ys
[
(t^ d ) r
O
,
I
and
X
):
P , Q
12
T hese in te g ra ls a re read ily evaluated.
C onsidering (39), it is found that
T*+ S
A fter in se rtin g the lim its of integratio n in the g en eral te rm
n
^ _ ■y* + 5
+
j r-s
and su b stitu tin g fo r
T +S
and
V - S
,
except n e a r thresh o ld it is found th at th is can be approxim ated by
<«, [ 1f n(t_Li3)',J r ',_S
i ~
%
= (ZkA,f.
S im ilarly, con sid erin g the te rm in (42) involving the lo g arith m , away fro m the th re s h ­
old,
,44,
~
On the o th er hand, n e a r th reshold, it is found th at both of th ese te rm s (43) and (44)
a re of the sam e o rd e r.
F rom a physical point of view, it is the n e a r th resh o ld region
th a t is usually of p a rtic u la r im portance d esp ite the sm all total c r o s s section, since
during the excitation and ionisation stag es the m ean p a rtic le e n e rg ie s a re w ithin th is
region.
However, it is only with the sim p lificatio n achieved by d isc a rd in g all te rm s
in (42) except the lo garithm te rm that a co m p reh en sib le form fo r the jf}. ’s can be
obtained.
In th is c ase, the in teg ral (42) becom es
<45> P = l l
iu m -l
= 3'1 feolf
------^
?
E valuation of the
13
<t> ’s
<j) ’ I
while s im ila r approxim ations in calculating
(46)
Q
=
?
P
5
(47)
R
=
81 P
-
Q
,
Taking the re le v a n t p a rtic u la r values of
(48)
I ( A
o)
=
give
R\
V
,
( I- A^)P
X ( A ,) = ( I - \ mtA ) P ^ C X ^ ) Q
X ( / ^ z ) = ( I - X*V/s)P + ( 2 XV s 1) Q - ( x " V ) R .
W riting
(49) A = (• 45) 31
and using (46), (47), the
nrip
<D ’s
e^ao /
ka
becom e
(50)
(p(Mo)
Ad-OP
(51)
<j) ( / * l)
A ( I-
(52)
(j) ( /* , 1 )
AO - xM( ^ £ f ) P
feo feL) P
2 fc0 fcf 7
Again it is n e c e s s a ry to m ake assu m p tio n s co rresp o n d in g to a n o n -th re sh ­
old region.
Since fo r the ex citation co llisio n s being co n sid ered ,
(53)
A2 —
I -
(54)
A
I
~
Z A t/rh rq'
- mrA ? /^ k o
—
I
=
-
2 mr
<*
/ fco
,
14
Also fro m the energy co n serv atio n (since
SS
(55)
fcf
+
A 6/
feo +
-
k(•
/ ko
is the ex citatio n energy)
k1 3
2 (lo -
/ r
— 2 <* 3
.
Using th ese re la tio n s in equations (50 to (52), giving the
(56)
(57)
<j>(-®,o) - AP
=
4>C»,0
A O -
it follows that
,
|SA)P
~ (mr Ag/fi‘ fci) AP,
(58)
<j>(2,2)
=
(59)
4>(2,0)
Z<p(bU ,
= A
=
Hence fo r the
(z m rAf/fcH o ) A P
= A(l - ( H ) ^ ) P
n
( I -A1) P
4> ( z ; z h
the following rela tio n sh ip s apply:
(60)
Again, using (55) in (45),
/\
P sim p lifies
to
(6i) A P = ('^ O 2 cj m ? X n j
3 10 V fcj-
(
3 t feo ^
2mr / i f /
E valuation of the
<|> ’s
15
Equations (56) to (59) enable the
(b)
<j)Ts
to be e x p re sse d in te rm s of (61).
E lectro n -h y d ro g en atom ionising c o llisio n s
An ionisation co llisio n is e sse n tia lly an encounter in which the
atom ic e lec tro n is excited to the continuum sta te .
F o r th is re a so n , fo r the incident
e le c tro n s c a tte re d in the co llision, the p ro c e ss may be tre a te d as ex citatio n w ith
v a ria b le
A S • On the o th e r hand, owing to the strip p in g of the atom and the p ro d u c ­
tion of new p a rtic le s , fo r the atom , the proton produced, and the atom ic e lec tro n
e jec te d , the c o llisio n s need to be co n sid ered in m ore d etail.
p h y sics of these collisio n s is contained w ithin c e rta in
en ces 1, 2 and 4; n e v e rth e le ss the -O . Ts
To som e extent the
-functions used in r e f e r ­
m u st be calcu lated fo r each d ifferen t
p a rtic le type fo r which the 13-m om ent equations a re re q u ire d .
p a rtic le co n sid ered , the A ? a re d ifferen t, and the
te rm s of these fo r in teg ratio n o v er the range of
Let
m ust be e x p re sse d in
A P •
d -0 . i be the solid angle into w hich the incident e le c tro n
(with incident (reduced) wave v e c to r
d /1 -2
X's
In g en eral, fo r each
k 0 ) is s c a tte re d with wave v e c to r
, and
the solid angle into which the atom ic e le c tro n is ejected with wave v e c to r X
It is shown in Appendix A that it is s a tisfa c to ry to re fe r all th ese p a ra m e te rs to a
fra m e moving with the c en tre of m a ss of the two colliding p a rtic le s .
Q
,
€
x
»
CL >
The s u b sc rip ts
w ill r e f e r re sp ec tiv e ly to the s c a tte re d elec tro n , atom ic
p
e lec tro n , atom , and proton produced.
F o r the co llisio n dynam ics (but not the g en eral
equations fo r the e le c tro n component) the s c a tte re d and atom ic e le c tro n s a re tre a te d
se p a ra te ly .
(i)
S cattered e le c tro n , € ,
Using equation (33) and the d ifferen tial c ro s s sectio n (9),
( X
being the p o la r angle of
(62) <t )e ,a.,e .(/u,v ) =
kf
w ith re s p e c t to
\'
Iko
)
(I - A '£ a j i 'x ) q ( - I 4 x l ö 3) z V m J e V t f e f
K * ( a 0* K %
I)*
dx
16
w here
K
(63)
=
(if
+
fco
2
-
feo
fcf C^rsX
.
It is a lm o st im possible to c alcu late the jO . ts
e v er, by approxim ating the te rm
(a *
I
fro m (62) as such; how­
in the denom inator by (&*
0
it becom es p o ssib le to evaluate the sequence of in te g ra ls re q u ire d to obtain the D i's .
Ij
Although by no m eans an a c c u ra te approxim ation, the f t
te rm , also in the denom i­
n a to r, n u llifies the effect of th is approxim ation.
F ro m the co n serv atio n of energy, it follows that
(64;
•+ 3^
w here T
fa c to r
■+
2. W rT / t\
is the ionisation energy of the atom .
In teg ratio n o v er
4 TT . The lim its of in teg ratio n o v e r
the lim its on X
a re
( O
(65) <f>eoJe , ( / i v) =
a re ( o
, (
).
4X 1 ^
y ield s a
, rr ), w hile fro m (64)
The equation (62) b eco m es
^rroiv.io
J,
(^)
** ko ( a? k? +f
w here
,-IT , a ? - 2 m rI / # j i ) V i
- \
(66)
I (I
- b/UfU
ef S'UlX d%
CC +
th is being evaluated in Appendix B.
(67)
\ X =
I -
fef -
z k A - f O n x ) 3'
F o r th is c a se ,
X
is given by
( tK*+ 2 m r l / ^ )
U sing the values from th is appendix fo r the a p p ro p ria te (yU , V ) ,
(68)
« *■ =
ko
-
2
m
rl/ f>
on putting
3c
,
E valuation of the
17
(h ’s
f*
X
•
ii
(69)
>
and
io 3) z 6tt g a *
e * 4 tt
+ i )*
K * ko ( a *
2 <x3
3 ( 2 m r / f i 1) 1
the re s u lts a re :
(70)
(71)
(72)
i >eo.Je l ( - ao'°'>=
A >
4*eaj e, ( 1j 0
=
A (
^ e a e , ( 2 ,2 .)
=
A
1-
( 1 -
C + < *N
Ckl
’
( k \ 4 o**)1 )
4 ft«,*
(73) (ftea^e, ( 2 , o )
=
(74)
=
4>ea e, C 3 j )
A
A
CI -
( i
-
’
HP
’
3 cK2- ( feo -4* °<x ) \
IO feo“
Finally, 2 . ^ 1
/ft*
is neglected com pared w ith
a fte r substituting fo r <* fro m (59).
valid away from th re sh o ld .
fe0
in the p a re n th e sise d te rm s
(Again th is c o rre sp o n d s to an approxim ation
However, a p a rt from being c o n sisten t with p rev io u s
approxim ations taken, th is type of approxim ation in the p re s e n t calcu latio n s is
n e c e s sa ry to obtain w orkable r e s u l t s .) The re s u lts then becom e
(75)
4 > e ^ e . C ‘ > *)
=
2
3
<ba
Y e a >e. ( - ° ° j ° )
(76)
^ e a e, ( 2 , 2 )
_
Z m rT
>
^ e a , e , C '00.»0 )
ki
(77)
4*ea e, U , ° )
=
f
4 e a e!, C - ° ° ; ° )
35
y
18
(ii)
The atom ic e le c tro n , 6 ^
S im ilarly to (62), but fo r th is c ase w ith
angle of
(79)
w ith re s p e c t to
^ea e^C/V) =
, and
/(> -
,
x
x (*f
being the p o la r
the deflection angle of
^ f
»
) g (-14- * I03) IT mp e ' V a *
X dx cJiT,
K4(a*Kx+ IT
w here now K
is in te rm s of % i (being defined in fa c t fro m the d ifferen tial c ro s s
section):
(80)
ft1
=
feo +
- 2 f e 0 ^ f UTS >6,
.
F or th is c a se (Appendix B)
(81)
A1 =
I
-
(fco-ddi/fc»
In tegration o v er "X
independence of this angle.
(82)
T ^ C /^ v O
=
is stra ig h tfo rw a rd , owing to the c ro s s sectio n al
Defining
J
( I ~ x
>
e lem en ta ry in teg ratio n gives
(83)
T2 (yUV)
=
2
-
XA ( l + ( - 0 V) / C V +
In teg ratio n of (79) o v er the solid angle
(84)
M
-
f
d X t d fc ,
K^(at K3+ I
< J 0 -|
I)
.
involves only te rm s in f^l
w h ere
E valuation of the
< j>
f !
K “►l )
As in the p revious c a se , the te rm
In teg ratio n o v e r the azim uthal angle gives
is rep laced by ( c io ^ o ■+ I
)
2 TT , and with the above approxim ation,
in teg ratio n o v e r ")C, m ay be effected, giving
(85) M =
4Tr/((atf£+
R eturning to (79),
(j[)(yU^V) becom es
r <*
(86)
_3
<J>e a „ C /t.V ) = I TTIDp M
9 ('IV x JO ) f
fcf * - d *
J°
_
4TT ß
|_ C y U ,V )
w here
(87) 6 =
(•i4*iö3) 2 <aiTTm?efc
Wk o
o<
of
is defined by (68) and
L
is calcu lated in Appendix B.
from th is Appendix, and since
A
> alread y defined by (69), is
(88^ A — 4 TT B
(aH l +i f 3UmrI / t f )
the
are given by
,
<»« ^ ( 2 , 2 )
~ (§ + s k i } >
V)
<90> ^eae.C1»1)
II
>
(89) te a e /- “ ^0) = A
' RS PHYS.S
Ur*. PAr ^
Taking the values
*V
20
(92)
(93)
^ e a e2
i 2LXA ,
=
k a e x^ ° )
r 1C
A •
=
0
’
Since
+
(94)
(2.
the
<J> ( y ^ v )
U +
'
,
7
may be expressed as follows in term s of < f > ( - C O , o ) :
C 0
(95) (h
17
T e a e ,.
■
= £
^ e a , e 2C2' ^ )
_
O
_
(97)
<T>
n>
»*
/^s
_
(96)
115-1
~
°<2 )
s k i
(98)
( -C O 0 )
=
=
ryj
3
<j
h
a
^ 0 0 ' 0 ) ,
f e a C - 00' 0 ) ■
In any c a s e it is obvious from (33) th at
Cpea
tion fo r the co llision m ultiplied by <0 / 2TT
is sim ply the to tal c ro s s s e c ­
and th e re fo re fo r all p a rtic le s involved
in a p a rtic u la r (inelastic) c o llisio n , the <j> (-0 0 ^ 6 ) a re id en tical
Thus (89) is id en ti­
cal w ith (70).
(iii)
The atom s
In the p a r t of Appendix B dealing w ith the calcu latio n of the
X
fs it w as pointed out that fo r ato m s as a p a rtic le type in th is type of ionising c o l­
lisio n , the only . O / s
involved a r e the
(V )
, w hich a re independent of
Hence fo r the atom s,
(99)
=•
J’yGSMny.dy. —
X
•
J
E valuation of the
w here
X
-
Again rep lacin g
(100,
(101)
Cc7^ '( ( “
(d j
*( “
0 ^
x,
<j>
21
<j) 's
(~ o o ,o )
) / ko
by
fef )
=
(d© feo + I
=
KdK/fe»fef
=.
C
/
X |
•
» since on d ifferen tiatin g (63),
?
^
T
K3
w here
(102) C
=
g C i^ io W e ^ T iV .
The u p p er and low er lim its on
(103)
—
tK l
a re
<j>e<A~°°’ 0 )
(?f
. Using (64) and (68),
J
and th e re fo re , on integ ratin g o v er
(104)
fec t
_
K
%6x i k 0k{
~ ( k o - k V z
L
The rem aining in te g ra tio n is stra ig h tfo rw a rd also :
(105)
<J)e a C - ® > ° )
Z c
feo °< 3
3 ( a m . I / f i 1)1
It is ap p aren t on com paring (105) w ith (70), fo r exam ple, th a t the
obtained
fo r the d ifferen t c a s e s a re equal; th is su g g ests th a t the d ifferen t approxim ations made
a re c o n sisten t.
22
(iv)
The protons
As d iscu ssed in Appendix B the D l ' s fo r the p ro to n s p ro ­
duced in the ionising co llisio n s a re read ily obtained fro m those fo r the ato m s.
tic u la r, when te rm s of o rd e r
In p a r ­
a re neglected, the -Q -'s a re the sam e fo r the p ro ­
bet
tons as fo r the atom s.
T here is, th e re fo re , no need to calcu late the p a rtic u la r
<J)ea(yU v ) fo r these protons.
(c)
C harge exchange co llisio n s
As d iscu ssed in Appendix A, only reso n an t ch arg e exchange c o lli­
sio n s a re co n sid ered .
T hese ch arg e exchange co llisio n s a re e la stic in the sen se that
th e re is no kinetic energy absorbed in the collision, although in the s tr ic t sen se of a
change in the n a tu re of the colliding p a rtic le s the c o llisio n s a re in e la stic .
N ev erth e­
le s s , the m ost convenient way of con sid erin g this c a se is to calcu late the
and hence the
(j) ’s fo r the
-functions of the atom (or proton) p re s e n t a fte r the
co llisio n m inus the
-functions of the atom (or proton), involving the o th e r nucleus,
b e fo re the collision.
F rom th is point of view, the co llisio n is e la stic , w ith d ifferen tial
c r o s s sectio n (10).
Since
need to co n sid e r
= <t>(0 say.
(106)
?>(l) = £>(.2) = O
(^ (-O O ^ O )
, while
A lso le t
( } ) ( 2 ,o )
=. <J> (2*)
°
-
o
; then
rI (-
(J>Cv) = D
fo r e la stic co llisio n s, th e re is no
dx
( a J K ^ l )6
w here
(107)
D =•
Wr
T*
) Q
0*0
2, 6
Substituting fo r r \
(defined fo r equation (10))
dos) < L p(v) =
D / " " Cl - c « » » x ) s * . x d *
J °
CI •+
and
(j) ( J ?
»
E valuation of the
23
s
4
from which is obtained
<109
< J> ap C l,0
=
D (—- - - - --------- !- - - - ' lo f e ja ?
=
D / 16 fe ta !
=
D (
=
D / s ko a ;
<110> 4>aPW
I
i t f e ta !
I
io f eo t a 1
X X
i o f\o
fe:
al )
.
F o r th ese calcu latio n s, it has been assum ed th at
fe„ &0
)
E valuation of the A - ,
4.
The n - s , defined in the introduction and re fe re n c e s 1, 2 and 4, a re
e sse n tia lly the re s u lt of integrating the
the definition, (w here
an)
w here
Y0
-Q jfe’ t ( r )
Y
^ 9
=
=
T ri
?
ftij
4
e
Yo"
Y0
/ Z fa (.
"
Y
o r sin ce
is the th resh o ld value of Y
co llisio n s
(112)
“
S’
<j)’s o v er the im pact velocity.
Y
Z
T
+
R estatin g
m^ i } )
Z
fi fe0 — rn r <J
d
?
Y
fo r a p a rtic u la r collisio n .
Y
=
S ^ fe o /^ r
•
Thus fo r e la stic
is zero, w hile fo r a collisio n w ith excitation energy
A € 0
»
=■
This is taken account of, in a sen se, in the
(J) ’s, in that being functions of the d iffe r­
en tial c ro s s sectio n , below the th re sh o ld energy they take the value z ero .
H ow ever,
the approxim ate c ro s s sections used a r e not zero fo r all e n erg ies below the c o llisio n
th resh o ld , and it is n e c e s sa ry to specify the p a rtic u la r low er lim it as in (112).
24
4. 1
E lastic C ollisions
F o r th ese co llisio n s, w ith the j Q js as w ith the
(J) Ts, the d efi3
nitions a re form ally the sam e as those given by Chapman and Cowling.
The re s u lts
fo r Coulomb co llisio n s a re given below a re in fact the sam e as in re fe re n c e 3; how­
e v e r, as already m entioned, the use of "ad ju sted " Born approxim ations fo r the
d ifferen tia l c ro s s sectio n s fo r o th e r e lastic co llisio n s lead s to d ifferen t re s u lts from
those of re fe re n c e 3 based on in v e rse pow er fo rc e law s.
Since
<j) (
)
=E <J) (v)
fo r elastic co llisio n s, it is evident that
(113)
“
- ° j * (T )
(using this equation to define
(a)
. 0 . ^ C ^)
)
E le c tro n -e le c tro n co llisio n s
(j) 0
Equation (12) fo r
n.i4i _Q c v ) =
u1/
)
, with (111) gives
y x r*
e
r
Although W e e
.
l+Aee)dY
Y
i s a function of Y
, owing to its o c c u rre n c e only in the lo g a rith 0
m ic te rm , it is g en erally adequate to re p lace it by an "av erag ed " value (Liley ),
re g a rd in g it as constant fo r the p u rp o ses of in teg ratio n .
In th is c a se the in teg ral is
e a sily reduced to an ex p re ssio n involving the fa c to ria l function:
(115)
=
S im ilarly, since
(j>e e ( x )
(116 )
n
ee
Cr)
=
TTZ & e ^ i ^ g ( A e e )
=
2
^ec C‘ )
z f l eAr
)
c
^
I
/ Z YX)r
Evaluation of the
25
s
The ’’av eraged" value of
(117)
w h ere
W e e is
_A_,. = 3feTe\D/e a =
Xo
3 (R TeM e^T rnee^) ,
is the Debye length.
(b)
E lectro n -h y d ro g en atom co llisio n s
Using equation (17) fo r Cj)(l) gives
(118)
n ' e^ r ) =
e Y Yar^^(-45)4 ni?e*ao s
ftA ( 2 feo a t ) 1
( ioo(a£fco + 0 +
~
\
R etaining only the lo g arith m te rm , th is equation becom es
00
(119)
rijr)
=
tt1
(*45)
e^_g3
y
v .x
e~Y
Yr
-»■
dY•
v * ft-
F o r the vario u s
T
, the following e x p re ssio n s fo r the j f l ( T )
(120)
^
I*2«
n '
a)
(J ) -
r
a re obtained:
i* ( - « ‘' kEiC-'/W) >
Tr>c^y)e*ft3 i (, + Lzb e,/bEU-i/b)),
i j j . - , (-ZbVlb-.)eW.Ei(. , ^
26
Ei (-l/b )
argum ent
(123)
is the exponential-in te g ra l function (G radshteyn and Ryzhik4), and the
b
is given by
b =
.
The s im ila r approxim ation of d iscard in g te rm s o th er than the
lo g arith m te rm in
^^0 0
(equation (18)) gives
<124> <ka(X) ® Z
}
and th e re fo re
(125)
f l e(LCr)
=
(c)
2 . n ea(r)
.
E lec tro n -p ro to n co llisio n s
The calculation of the . 0 , 's is fo rm ally the sam e as fo r e le c tro n -
e lec tro n c o llisio n s.
(126)
G ^ p( r )
=
G
=
e p (>")
w here in th is case
(127)
The re s u lts a re
A
ep
=•
T T ^ y " -i«g( i f^ep") P ( r ) /
XG
W e p is given by
I
Xo (
n v / e 2
oC g <X p
~
3
( feTe )3/a
€ a (^rrne ea)t/a
e p ^ 1-
X m rx
)
-}
E v a lu a tio n o f the
n
(d)
H y d ro g e n a to m -h y d ro g e n a to m c o llis io n s
I
0)
(128)
cj> (V)
The re le v a n t
,
a re g iv e n in (25), (26). U sin g (25) f o r
is th e in te g r a l
=
n ' aM
( 2x iq' 1) y 3evTr1/1
JUm 0 + b Y l )dY,
Jo
w h e re
b=
J
as b e fo re .
F ro m the r e s u lts in v o lv in g s im i la r in te g ra ls f o r the e le c tro n -h y d ro g e n a to m c o llis io n s ,
in the sam e n o ta tio n , the
<129> A
x<
l0
)
=
jT l
(t )
beco m e
(z * l 6l ) y Je * Tr'/a i t - p i/h F; /-l/fa ))
m rx
1 V
<130> r i c J 1)
= U < l 5' ) y 3e V A I ( , + t b e,/b E l ( - >/b))
nor *L
^
b
(131) t T a a O )
=
( Z x l ö b y V T r'A
,
(t(3~ j 4 t-2lf * 2b- ' )e>/l’Ei(-|/b)).
0
A g a in , s in c e
<j>a a ( 2 .)
—
^ 4aa
)
i t fo llo w s th a t
(132)
(e)
H y d ro g e n a to m - p r o to n c o llis io n s
F ro m the e q u a tio n (28) f o r
(133) i 7 ap(r) = ( 2.<»o ) e^y
r
*V X
4 *d p ^1 )
/* e’ ^ Yzr
Vo
( I + bYz) dY
28
with
b
as in (123).
The in te g ra ls have the sam e form as for the prev io u s c ase s
involving atom s, being e x p re ssib le in te rm s of the ex p o n en tial-in teg ral functions.
The re s u lts a re
(134)
f l a p Cl)
=
( Z x l ö b e V T T ' 71 i
rvV1
a ss » 1 1 '
(:o
=
z
( z , i ö 1) e V i r l/l i p
mrz
«■>«>
4-
b ^ e ^ E iC -i/b ));
b
- t * f c ^ V * E iC - ./b j ) .
Again, since
(137)
( _ e '/ b E i ( - ! / b) ) ,
^ a p (^ ) =
i T Q p (T")
(f)
=
^-^apCO
» it follows that
z Q a p iT)
.
P ro to n -p ro to n co llisio n s
F o rm ally the sam e as fo r e le c tro n -e le c tro n co llisio n s and
e le c tro n -p ro to n c o llisio n s, the re s u lts a re
(138)
("Ip p C r)
(1 3 9 ) i T p p ( r )
ss
=
( I
tt
z O p p C r )
+■ -A
p p ) T '( y')
.
F o r th ese c o llisio n s, how ever,
(140) A pp =
3 \ Ditv /o ^ e*
s
(3f?TP/e a) ( ^TeAirnee1) 1'1.
E valuation of the
4 .2
I I 's
29
E lectron-H ydrogen Atom I s - 2p E xcitation C ollisions
The A i-
( r j given by (111) a re functions of, am ongst o th er
v a ria b le s, the low er lim it of in teg ratio n Y© =
function of the te m p e ra tu re , the A L
tro n and atom te m p e ra tu re s .
• Since
Y
is a
Ct ) w ill e sse n tially be functions of the e le c -
As w ill be seen, in g en eral the a L
( t *)
can only be
reduced to e x p re ssio n s in te rm s of incom plete Gam m a functions, which in th is case
a re d im en sio n less functions of the te m p e ra tu re s .
The incom plete G am m a functions
a re , how ever, tabulated functions, and in c e rta in ex trem e c a se s they can be w ritten
in te rm s of analytical functions.
iV ^C r)
, -O
, f l
* (r)
The following .O - ’s a re req u ired : Cl.
J C*r) ,
with r
and 3 ; but since re la tio n s (60) e x p re ss -fi. jl(jr) and
—C O j O
it is only n e c e s sa ry to evaluate A 1
and
r\~c°? o
The calculation of A
o
,
,
| ,%
,
r ) in te rm s of jQ . J(r),
/n ), I
(T )
=
(/V )
1L
( t ) , fo r th ese value of T .
involves using (56) fo r (J
This gives
(141)
w here b
Cl
(r)=
( ^ y ) e 4‘ Z l5a-o Y tt1'*
has the value
^e ' Y y i r + l ^
(bearing no re la tio n to the te rm
b
, de­
fined in (123) ).
C onsider the function
. oo
I
(T j
Y0)
defined by
J e“Y Y'11"^ 1
(142) I( r , Y,) =•
Vo
(bY) dY .
A change of v a ria b le yields
oo
(143) I(r,Yo) -
i
*o
fe_JCx r
d x,
30
y* ■ Let
and this can be in teg rated by p a rts fo r the v a rio u s values of
V C ^ Z ) be an
incom plete Gamma function, defined by
(144)
7\r,z) =
£°V XXr-' dx
.
(Some auth o rs define the incom plete Gam m a function in the in v e rse way, that is, by
(
—
PCY, z)
) in the above notation, but the definition (144) is m ore suitable
fo r the p u rp o se s of this re p o rt)
fo r all
T
P rovided
X.
O , the
T^Cy^ z )
a re defined
.
In te rm s of th ese functions (144), in teg ratio n of (143) gives
(145)
I(o ,Y .)
(146) I ( |,Y .)
(147) I
=
- ■
C e~Yo
yo2 + r(o,Y»a) ) J
= 1 ( e ’Y,a , 14 .)i9 lfXl + r0,Y.i ) + r(o>X1)),
( a , Y .)
=
^
( e " Yo ( Y / + 2 Y0a +
2) Xoj bx Y*
+
V ( 2,
Y .x)
+ 2 T (i,Y .a) + 2 r<o,Y.A) ) ,
= ^-(e"Yo(Y.4 -
(148) I C 3 , Y 0 )
+3 T u x ' )
-
It is evident that the
.
T" !
c Y.l + o ^ g
+ c r ( i,Y o b
+
6 r c o .Y .b )
•
00,0
(r )
a re obtained using th ese re s u lts fo r
I
)
in equation (141), that is,
(149)
r T ~ '°
11
ea
(r)y = U iO e V ^ Y ir1
*- I(r,Y.) .
3' ° fc1
a re calcu lated using equation (57) fo r 4 w o , i )
E valuation of the
31
j O - 's
T his gives
(iso) n Pf l ( r )
w here again
XX 7 ( t )
(-4-0 TT
3
-
b =
wrA j y 3
YTir
ft
3
can also be e x p re ssed in te rm s of the
Cl
l(-r )
=
3
• It follows fro m th is equation th at the
X O ^ Y o ) , defined in (142), and
hence in te rm s of the incom plete Gamma functions.
(151)
J^Yoe'YYZr~'i<g(bY)dY
r°°
(•*
O
TT
In p a rtic u la r,
e * l'sg0 r r y Y3A"to I ( r - |, Y 0)
.
ea
X
The in teg ral
! j Yo )
, not p reviously given, has the value
(152) I ( - i ,Y 0) = + (r«V».2)igb* + j
cb ) .
\o
The la s t in te g ra l on the rig h t hand side of (152) e x ists fo r
To
^
O
; being of a
s im ila r n atu re to the incom plete G am m a functions, it may also re ad ily be evaluated
n u m erically .
Finally, the rem aining X l Ts a re re la te d to (149) and (151) by
(153) , Q e * C r )
(154)
Cl e a
(r)
4. 3
Clpa
=
1
=•
1 0 - ea
(r)
Cr )
•
E lectron-H ydrogen Atom Ionisation C ollisions
F o r th ese co llisio n s the XX ’s a re defined by (111).
lim it of integration is given by Y> 58 (
energy.
As with excitation co llisio n s, since
w here
X
The lo w er
is the ionisation
Y is a function of the e le c tro n and
atom te m p e ra tu re s , the jTX's a re also functions of th ese te m p e ra tu re s .
When
32
Ta /T e
<£< ^ a / m e » then X 2^ C X e —
, in which case the O ' s a re
sim ply functions of the e lec tro n te m p e ra tu re .
c o llisio n s of c o u rse .
This also applies to the excitation
F or each of the p a rtic le s (incident electro n , atom , atom ic
e lec tro n , and proton produced), the follow ing - T
0 ,v's a re req u ired : . 0 . C T )
h i
Q
7(t )
’
C**) , - O
( Y“)
with T taking the values
0, 1, 2, and 3.
Fortunately, it is not n e c e s s a ry to calcu late all 80 - 0 , 's .
As shown
in Appendix B, the -O .’s fo r the pro to n s a re e sse n tia lly equal to the _ 0 - 's fo r the
atom s.
F u rth e rm o re , fo r the individual p a rtic le s , re la tio n s have been given betw een
v ario u s
<J> (yu3V) , enabling many jC ' L ' s to be e x p re ssed in te rm s of o th e rs .
In p a rtic u la r it h as been shown that fo r the incident electro n , all
<J>ea e
can be e x p re ssed in te rm s of <j>e<^ e<( y ° ° 9 o ) o r ( p e a e ( 2 ^ 2 )
fo r the atom ic e lec tro n , all (pe< l e (yU } l> )
and fo r the protons and atom s, the only
finally th a t the d)
.
(p
involved is
o ) ; and
a
( M,V)
1 L /
( T ) being given in equation (111), it is evident th at it is a
function of CD ( M y ) • Yo , and c e rta in o th er constants of the collisio n .
all p a rtic le s in the collision have the sam e
A
£T )
T h erefo re
. It follows th at fo r the
/ N
io n isation c o llisio n s, only 8 of the i i ' s need be calculated, nam ely
2 1
and f l ’ (T*) w ith T = 0 ^ 1 , 2 and 3 .
e j ei
Since
(155)
Y
o1
-=
X V 2-
I /" V
,
using th is re la tio n and also those a fte r (111),
(.56)
£
a 1
-
£
l m rI / ^
=
mir1 ( Y 1 - Y o 1 ) /
+
=
a el M (? ( Y 1 4 Y ; i )
I
e ( r 00!,0)
o o o ) a re equal fo r all p a rtic le s involved in a p a rtic u la r collision.
Te0,L
The g en eral fo rm of
can be e x p re ssed in te rm s of
;
/ ^
1
- O
° ; 0
1 JL
(t)
^a
E valuation of the j f L 's
w here
Y ,i= r Y Zfi
<h (-C30jO)
T ea
(157)
3
n
, and, like Y0
, is a constant of the co llisio n .
With
* given in equation (70) fo r exam ple, from (111),
-3 V
(r ) -
( - I » x l 0 ) t t‘a * X T ^ * y V
2 ( r Y Y)
3 a ‘ mr* (Z m rI / f , 1)i
w here
I
(t j
Yo ?Yi )
is defined by
f
ass» Jcr^Yo^X) =•
e Y Ya 4a(Y2-X»2) 7 <jy •
( Y z +
lt^ Y ijY , )
X x )¥
m ay v,e reduced on m aking a p p ro p ria te approxim ations, o r it may be
n u m erically evaluated.
As d iscu ssed previo u sly , X ( t jYo,Y| )
is e sse n tia lly a
function of the e lec tro n te m p e ra tu re .
S im ilarly , using equation (72) fo r
, it may be
v e rifie d that
(159)
n
M
(r) =
(•<Vx<ö)TT,/^ e V ° T T 1 y 7mr I
ea'e'
Y#>> ; )
7TF
3 mP‘ a.*(amrT
K r - Y0, Y, ) being again given by (158).
Finally, using equations (70) to (78), (95) to (98), and (10$, the
s e t of 48 X jL ' s fo r the co llisio n (as alre ad y m entioned,
32 of the X I 's asso c iate d
w ith the atom and proton p a rtic le types need not be co n sid ered ), in te rm s of the 8
independent X X 's a re as follow s.
F o r convenience, X I ^
CT)
and X I ea ^ (.***)
a re also rep eated .
-co ,
o
(160) Clel \ r ) =
( • li, x io3) f j * e Y S I ( r ) Y o ,Y ,)
3
do2mrI
(
34
with X ( y' ; X ?Y. ) defined in equation (158),
(161)
^ 1 ,1
- O . ea,c,N
oM
= 33 A , « (r)
(162) l l ea
e (r)
e o j« ,
(163)
/■r00' 0
=
f
H e«
C*0
=
( - i i , iö 3) tt 4 ^ e v t ' ° y T» ^ r I
33,1
,'
eea,
a ,ee,'
,^ T )
D
<164> -^ e a e, t r )
3 m? «.‘ ( i m p l / f i 1 ) 1
1,1
=
2 ,X
=•
ea,e2^r ^
Ji D
(r)
15 AZe a ^ ! /
- o o ,°
ii? A w
(168)
n
—
-fle a
=•
-^ e a C r)
(r)
A ’ “ .“
‘ea,ez ^
-00,0
-00,0
-C e a
^ ea^a ^
-COjO
_ —
oo o
*
e a , p (t )
4 .4
< »
-^eo
Hydrogen A to m -P ro to n C harge Exchange C ollisions
The n atu re of the c h arg e exchange co llisio n s has already been
d isc u sse d .
The ( f r a p C / U j V ) a re given b y ^109) and (110), being independent of yU
w hile the X l 's re q u ire d a re
-0 (**), -Q^*)with
I7 X
and since the collisio n s a re e sse n tia lly e la stic , from (111),
(i.7i)
- Q l pc r ) =
(
v
X
u
i
r<r)
and
3
,
• Using (109)
Evaluation of the
a
35
s
Similarly with (j)^
(172)
D . ap 0*) -
( x*ixio2)
zB
10 f t *
given by (110), it follows that
y
P(r4i)
36
APPENDIX A
The D ifferential C ro ss Sections
A l. 1
The B om A pproxim ation
In th is Appendix, quantum -m echanical ex p re ssio n s a re obtained fo r the
d ifferen tia l c ro s s sectio n s fo r c e rta in p ro c e ss e s involving e le c tro n s, hydrogen atom s
and p ro to n s, which a re re q u ire d to calcu late the
's and
's .
A B om type of
approxim ation has been chosen to give analy tical ex p re ssio n s fo r the d ifferen tial c ro s s
sectio n , and a n u m erical adjustm ent fa c to r is included which is sele c ted to obtain a
"re a so n a b le " approxim ation to the known total c ro s s sectio n s in the energy range
0 - 100 ev.
P roceeding fo rm ally , co n sid e r a co llisio n involving atom ic p a rtic le s J
and
fe re fe rre d to a fram e moving with the c e n tre of m a ss.
Suppose th at jTj , JTj,
re p re s e n t the e lec tro n position v e cto rs of the p a rtic le s re la tiv e to th e ir resp ectiv e
n u clei (these being zero of c o u rse if the p a rtic le s a re e lec tro n s), and
n u c le a r distance, with
( Tfe )
the in te r-
the in te ra ctio n p otential.
,
a re the in te rn a l (unperturbed) H am iltonians of the two p a rtic le s,
re f e r r e d to th e ir re sp ec tiv e nuclei,
and
T
( t ? Tj ^
)
g
is the in itial asym ptotic re la tiv e velocity,
is the com plete wave function fo r both p a rtic le s .
The Schrö­
d in g er equation fo r the sy stem is
(Al)
fiL1 a?
~
L^mr brx
w h ere
mr =
/
( Wj *+* Wfc)
is the reduced m a ss, and
tu rb ed ground state energy of the sy stem a sso ciated with
te rm
O
j
-j
x
E l0 the u n p er­
jo L j and
• The
is the asym ptotic kinetic energy of the sy stem .
N eglecting the p o ssib ility of elec tro n exchange,
tjHt.tj.r,) - ( I * /)
^
,t„) F.ci)
can be expanded as
Appendix A
37
w here
is the s e t of all p o ssib le
eigenfunctions fo r the p a rtic le s
J
(j^CTj )
and
which a re
is e sse n tially a s c a tte r ­
ing function, to be d eterm in ed by solving the S chrödinger equation.
^
E<^ =
+
is the sum of the eigenvalues of the two p a rtic le s co rresp o n d in g to the eigen­
functions Cj>^ C
)
and ( p q C Y ' f a )
• The sum m ation is o v e r d is c re te s ta te s and
th e integration o v e r continuum s ta te s .
Substituting this expansion in the Schrödinger
equation (Al) gives
<A3>(I +-/ ) ^ C r jlrfe) [ ^ irg ^ x rnf3"fe + Eo-Ei]f“^ r )
S'
= V(r,rJ,rfe) J(r,rj,rO M ultiplying (Al) by
<A4>
( r ^ T ^ ) d f j c lr^
and in teg ratin g , th is becom es
4 T mr3jh + Eo ” Ec^J *V X)
= /vd.rj.rfc) J(T,rj,rfc)i|^(rj,r(i)drj <lr(, •
p0 ( t )
( Cj^
re p re s e n ts the sum of an incident and a s c a tte re d w ave, w hile
o)
m ust re p re s e n t s c a tte re d w aves only.
g
fo rm s can be shown to be (Mott and M assey ):
(As) F. ~
(A6>
w h ere
p
F^
jl©
F<^ OX )
Hence the re sp ec tiv e asym ptotic
ec^* ~ + i e i ‘ *w 'I i ( x . e )
-
a
f^ ( x , e)
is the (reduced) incident wave v ecto r such th a t the reduced m om entum
, is given by
f\
. S im ilarly
of the s c a tte re d p a rtic le fo r the state
fcf C f y )
is the red u ced wave v e cto r of
. Then the sc a tte re d c u rre n t density fo r
38
The D ifferential C ro ss Sections
the
state of excitation is
<a 7> fcfcfQj,) 1 1
Iy
mr
t
x
y t )|a .
^
The d ifferen tial c ro s s section fo r the deflection of the final wave vecto r,
kf
, into a solid angle
(XL
X I 4 djX"jL
) with excitation sta te
is the
ra tio of the s c a tte re d c u rre n t p e r unit solid angle about X T. to the incident c u rre n t
density, giving
(as) o ^ d n
=
if
.
Ko
Subject to the plane wave B orn approxim ation,
/Am .T
vj; =
(A9)
e
t£o-r
,
(Xj , If.)
Substituting th is e x p ressio n fo r T
.
in the rig h t hand side of (A4), th is equation
becom es
(Aio) [*j»L
L 2mr öT1
+.
=
p (r)
zw>P J ^
J
V(r,tj,rfc) eL-° rii0(rj,rb)>^(Tj,rfc)dxj<Jrft
with asym ptotic solution
<A1U
- V
/ v e‘- -
or
(A12)
fl(x,o = I2X_ J v el - r vj/04>* djj drfa dr .
< drj dift
39
Appendix A
In these equations,
K
=
kp -
kf ( CJ,) j ftf -
<jjh -+ i l t l r ( E 0" Eq) .
“ F
~V
“
In the c a se of excitation to the continuum lev el, vj/^ is the wave function
w hose m a trix elem ent, taken with the ground sta te wave function and in te ra ctio n po­
te n tia l, gives the p robability of a p a rtic u la r d ire c tio n and energy of the atom ic e le c ­
tro n (as w ell as the d ire c tio n and energy of the s c a tte re d p a rtic le ).
With the H am il­
tonians
and
of the elec tro n s of the p a rtic le s
j
k
re f e r r e d to th e ir re sp ec tiv e c e n tre s of m a ss, the eigenfunctions <J)^(Y*j) ^
grouped under the te rm
>
, th e re fo re re la te the continuum state e lec tro n p a r a ­
m e te rs to the c e n tre of m a ss of the p a rtic le , say
k
, fro m which it w as d isplaced.
Consequently the angle and energy of the ejected e lec tro n is not re fe rre d to the c en tre
of m a ss of the sy stem j
and ^
, but to p a rtic le
k
.
In o rd e r to use the
d ifferen tia l c ro s s sections in the c en tre of m a ss sy stem , it becom es n e c e s sa ry to
co n v ert th ese p a ra m e te rs to th is sy stem .
give, fo rm ally , fo r ionisation (with tK
(A13) < r < u i , d n z d *
w here the su b sc rip ts
B earing th is in mind, the above re s u lts
the atom ic e lec tro n wave vector):
=
I 2.
re fe r to the s c a tte re d (incident) p a rtic le and the atom ic
(ejected) e lectro n , resp ectiv ely .
R egarding the fra m e s of re fe re n c e , the d ifferen tial c ro s s sectio n as
obtained above has to be tra n sfo rm e d to the fram e moving with the c en tre of m ass of
the colliding p a rtic le s (which w ill be called the "c. m. fram e ").
It is also n e c e s sa ry
to obtain an e x p re ssio n fo r the in tern al energy of the "com p o site" p a rtic le produced
in the co llision, as noted previously.
The types of co llisio n s being co n sid ered
a re those in which one of the colliding p a rtic le s , say J
o th e r p a rtic le , say
, is s c a tte re d , w hile the
k , on collision b re a k s up into two " su b -p a rtic le s " ,
nr
and S
The form (A13) of the d ifferen tial c ro s s section r e fe rs the angle and velocity of
to the c. m . fram e , but (owing to the re fe re n c e fram e of the H am iltonians) it re fe rs
40
The D ifferential C ro ss Sections
the v elo cities and solid angles of t
m a ss (the " k " fram e).
and
Using v
S
, X
solid angles in the c. m. fra m e , and
to a fram e moving with th e ir c en tre of
and -O-
U.
,
to denote v elo cities, angles and
©
and to
v e lo cities, angles, and
solid angles in the M fe " fram e , the problem is to e x p re ss
U>s
(A14)
in te rm s of -O.
and
M jV j -t*
Ur
, Us
, to r
and
v . C learly,
—°
or
=
V^
_
V-
Then
(A15)
Vr
—
U-r -+■ ¥ fa
—
Ur
—
üüi
y
k
Vs
u&
=
-
Ü Ü
Y ;
The azim uthal angle, that is the angle with the plane of the co llisio n , is the sam e fo r
both fra m e s.
Hence the elem en ts of solid angle in the two fra m e s a re resp ectiv ely
d -O . — S/i*\ % d% d i.
and
du) =
& d©d€
Taking com ponents
of velocity of (A15) in a c a rte s ia n fram e gives
(A16)
V rx
—
Vr CCS X r
=
U r OCTS (9r
-
rv\j y ^ CCS
x
j
rv\ k
Vry
Vr
Sxxo)C r
—
u r
<9 r —
V j S^r> X ,
F ro m th ese re la tio n s it follows that
(A17)
U rX
=•
v*
(ÜÜf
-1-
v j
+
2
V jV r
r*v c < n s ( X j - ) - r )
m k
(A 18)
(A19)
—
er
v/
-
4
(ÜÜ f v ]
ve s*n%r
C
+
4
4
1
( m j / w K ) Vj s^w x j _____________
4
z vr Vj (»*>j/mk) c o 5 ( ^ -y-r))Vl
41
(A20)
Appendix A
s w ie , =
y-s
+ (n»j /m fe) v, a**»y-j_______________
( v i :L+
,
v j1■+ a vä Vj ( ' " j / m h) c o ' 5 ( > j - y J ) ) l/l
When I is an e lec tro n and k
an atom , on neglecting
com pared with unity,
J
rv)a
the c. m . and " fe " fra m e s becom e identical, a s would be expected.
The
te rm is d iscu ssed in re fe re n c e s 1, 2 and 4.
E ssen tially it is
the sum of the ionisation energy and the in tern al kinetic energy of the com posite
" p a r tic le ” . Suppose of th re e p a rtic le s ,
(j^ ; S and t
,
and S
form a com ­
p o site " p a rtic le " , then fo r th is case
(A21)
Ai
J_ n'lq.tns
z na«* + m$
X
=
T his is given la te r (in Appendix B) fo r the p a rtic u la r c ase of electro n -h y d ro g en atom
ionising co llisio n s.
A l. 2
The B orn D ifferential C ro ss Sections
Al.2.1
E lastic C ollisions
a.
E le c tro n -e le c tro n c o llisio n s
W ith the neglect of the e le c tro n spin, the c la s s ic a l method
the B orn approxim ation, and the ex act p a rtia l wave calcu latio n all yield the w ell known
Coulomb co llisio n re s u lt fo r the d ifferen tial c ro s s section:
(A22)
<T d -O - =
* C<JSACV X -A
<j.Q-
Since the n a tu re of the co llisio n te rm s in the m om ent equations involves tre a tin g the
colliding p a rtic le s as distinguishable (when they a re identical), it is doubtful w hether
it would be adequate to take account of exchange effects p u rely in the d ifferential
c r o s s sectio n .
Although the energy range 0 - 100 ev is that in which exchange effects
m ake th e ir g re a te s t contribution to the c ro s s sec tio n s, they a re not considered in th is
re p o rt.
42
The D ifferential C ro ss Sections
b.
E lectro n -h y d ro g en atom co llisio n s
Using (A8), (A12) with the ground state wave functions fo r
hydrogen, since the in teractio n potential is
I
(A23)
V = e* ( r " r-Ti
w here T\
is the atom ic e lec tro n position v e cto r with re s p e c t to the nucleus, the B orn
d ifferen tial c ro s s section is
(A24) <y d O
=
4 Q.O
e
( a0 K +
8)
J .Q
.
**(aox Ka + 4 V*
In th is equation (as defined previously),
c.
K = (ko kf) — 2,fcoC^
.
E lec tro n -p ro to n c o llisio n s
The Coulomb re s u lt applies fo r th is c a se , the d ifferen tial
c ro s s section being
(A25)
c
d .0
=
mr
c w ic + jL /2
4 * * k*
d.
Hydrogen atom -hydrogen atom co llisio n s
Using “p ,
>Tx
fo r the position v e c to rs of the atom ic
e le c tro n s re fe rre d to th e ir re sp ec tiv e nuclei, and
fo r the in te m u c le a r d istan ce,
the in teractio n potential is
(A26)
V = e*(-A
It - t-,1
ir+ rj
l T + f j .- r J
The colliding atom s a re reg ard ed as d isting u ish ab le, in which Case, using the ground
s ta te wave functions, from (A8) and (A12), the d ifferen tia l c ro s s sectio n fo r th ese
43
Appendix A
c o llisio n s is
(A27)
CT" d f l
=
4m?e4aSK*Ca;K^8 )*
■ f c * ( a * K 2 -*-4
e.
.
) 8
Hydrogen ato m -p ro to n co llisio n s
A p art from the d ifference in the reduced m a ss e s, these
c o llisio n s a re identical with the electro n -h y d ro g en atom c o llisio n s, of c a se (b).
The
d ifferen tia l c ro s s section is
(A28) (T d-0. = 4 wre^tt<> ( ö l K +8)
.
fi*(aJK*+4r
f.
P ro to n -p ro to n co llisio n s
T hese Coulomb co llisio n s a re fo rm ally the sam e as case
(a).
The d ifferen tial c ro s s section is th e re fo re
(A29)
<rdQ = Urg-Eg?r*x /z
A2.2.2
da.
Inelastic C ollisions
It is reaso n ab le to suppose th at in an in elastic co llisio n with an
atom , owing to the scree n in g effect of the atom ic e lec tro n , atom s w ill have a s m a lle r
c ro s s section than p ro to n s.
Again, using the B orn approxim ation, an e stim a te of the
re la tiv e m agnitudes of the c ro s s sectio n s of protons and e le c tro n s in in e la stic c o lli­
sions w ith atom s can be obtained as follows (Bates and G riffing^).
be total c ro s s sections fo r a given e le c tro n o r proton energy
Then if
A§
E
L et
Q ^C E^Q pC E]
resp ectiv ely .
is the kinetic energy defect in the co llisio n the B orn approxim ation
gives
(A30)
Q e (J T E ) = ^ [ Q p ( m p E A O - ( £ ! ) 1 Q p ( < " p 4 f / l < ' n > e E ) J
44
T he D iffe r e n tia l C ro s s S e ctio n s
w h e re
(A31)
S' =
and
(A32)
( \ +At / uE) X ~
I
a re the e le c tr o n and p ro to n m a s s e s .
V e ry a p p ro x im a te ly , th is g iv e s
Qp(E) =
A c c o rd in g to (A 32), the n , the p ro to n -a to m c ro s s s e c tio n f o r a g iv e n in c id e n t e n e rg y
is o f the sam e o r d e r as the e le c tr o n -a to m c ro s s s e c tio n f o r an in c id e n t e n e rg y
Mp
o f the p ro to n e n e rg y .
M e/
S ince f o r lo w e n e rg ie s the e le c tro n c ro s s s e c tio n is s m a ll,
th e p ro to n -a to m c ro s s s e c tio n (and the a to m -a to m c ro s s s e c tio n a ls o ) w i l l be v e ry
s m a ll in th e e n e rg y ra n g e 0 - 100 ev.
I t is re a s o n a b le , th e re fo re , to n e g le c t p ro to n s
and ato m s as in e la s tic c o llis io n p r o je c t ile s f o r the c a lc u la tio n s o f the r e p o r t.
a.
E le c tr o n -h y d ro g e n a to m ls - 2 p e x c ita tio n c o llis io n s
T he o n ly e x c ita tio n p ro c e s s th a t w i l l be c o n s id e re d is
ls - 2 p e x c ita tio n , th is p ro c e s s h a v in g a m u ch la r g e r to ta l c ro s s s e c tio n tha n th a t to
any o th e r le v e l f r o m the g ro u n d s ta te .
T h e in te r a c tio n p o te n tia l is g iv e n in (A23).
U s in g th e g ro u n d s ta te h y d ro g e n a to m w a ve fu n c tio n and the Z p ( m - O ) w ave fu n c tio n
in ( A l l ) to c a lc u la te
(A33)
f 0 & -Z p )
, f r o m (A8) the d if fe r e n t ia l c ro s s s e c tio n is
df)..
or d H .
+4aJKV
b.
E le c tro n -h y d ro g e n a to m io n is in g c o llis io n s
F o r th e se c o llis io n s , u s in g the e x a c t c o n tin u u m w ave
g
fu n c tio n (M o tt and M a sse y,
p 233) in th e c a lc u la tio n o f the B o rn a p p ro x im a tio n
f o r the d iffe r e n tia l c ro s s s e c tio n le a d s to an e x c e e d in g ly c o m p lic a te d e x p re s s io n .
Let
d -Q |
be the s o lid angle in to w h ic h the in c o m in g e le c tr o n is s c a tte re d ;
th e s o lid angle in to w h ic h the a to m ic e le c tr o n is e je c te d w ith w ave v e c to r
tK
d -O -j
; and
A ppendix A
45
^
y
— t 0 —
fco
in c id en t e le c tro n , a s b e fo re .
(A34)
j
kf
b ein g in itia l and fin al w ave v e c to r s of the
T hen th e d iffe re n tia l c r o s s s e c tio n f o r th is c o llis io n is
o- dfl, d fu* = *Va.*e*fcf *e
J-
W
"5^ 0^
L
x (a *
In th is eq u atio n ,
S
is the an g le b etw een
M + a£(Kx-;x2))
005^) ctfl, d-0.a d ^
K
and
%
.
dO-|
and 7K
a r e in
fa c t r e f e r r e d to a fra m e m oving w ith th e c e n tr e of m a s s of th e ato m a f te r th e c o llis io n ,
b u t it is a p p a re n t fro m eq u atio n s (A17) to (A20) th a t th e e r r o r involved in ta k in g th e s e
v a ria b le s a s r e f e r r e d to th e c. m . fo r th e c o llis io n is only of o r d e r
YY\e
J m «.
H ow ever, it is obvious th a t th is e x p re s s io n (A34) is u n­
su ita b le fo r c a lc u la tio n of th e C[> ’s and . 0 . Ts , w h e re s u c c e s s iv e in te g ra tio n s o v e r
fu n ctio n s involving th e d iffe re n tia l c r o s s s e c tio n a r e n e c e s s a r y .
E ven w hen th is d if­
fe r e n tia l c r o s s s e c tio n is a v e ra g e d o v e r the so lid an g le of e je c tio n of th e a to m ic e le c ­
tro n , th e re s u ltin g e x p re s s io n is too c o m p lic a te d to b e u sed in ev alu atin g th e .O - 's .
The p ro b le m m ay, h o w ev er, be ap p ro ac h ed in a s im p le r
w ay.
T he m a in re a s o n f o r the c o m p lic a te d fo rm of (A34) is th e co m p lex n a tu re of the
continuum w ave fu n ctio n fo r the h y d ro g en ato m ic e le c tro n .
It is th e re f o r e re a s o n a b le
to c o n s id e r re p la c in g it by a p lan e w ave fu nction, in a s im ila r m a n n e r to th a t in w hich
the re le v a n t B o rn ap p ro x im a tio n w as ob tain ed by u sin g a p la n e w ave (A9) in eq u ation
(A 10).
Such a p lan e w ave is
(A3 5)
e
T*,
L*-r;
j
b ein g th e p o sitio n v e c to r of the ato m ic e le c tro n r e la tiv e to tiie c. m .
(But, s im i­
la r ly to th e B o m a p p ro x im atio n b eing v alid fo r la r g e e n e rg ie s of th e in c id e n t e le c tro n ,
so th is ap p ro x im a tio n is a c tu a lly v a lid only f o r high e n e rg ie s of the e je c te d e l e c t r o n . )
46
The D ifferential C ro ss Sections
The ap p ro p riate n o rm alisin g fa c to r is not e n tire ly c le a r, but to obtain the c o rre c t
fo rm of the d ifferen tial c ro s s section, (j> <J> d;fc
should re p re s e n t a p ro b ab ility d en ­
sity of p a rtic le s with wave nu m b ers "X in the range (*J< ,
dim ensions (volume) \
t-d * )
, and so have
A com parison of the final fo rm obtained, using th is wave
function, w ith (A34), obtained by using the c o rre c t wave function, su g g ests that the
c o r r e c t form for
(A36)
,
( =
cj)(
is
/
( * / a °)
£
l£*r,
The use of an adjusting fa c to r with the to tal c ro s s section obtained using th is wave
function w ill m ean that the neglect of n u m e ric al fa c to rs is of no consequence.
With
(A36) fo r the final wave function and the no rm al ground state hydrogen wave function
as the initial wave function, the B orn approxim ation, equation (A13), may be used to
calc u la te f (j<) .
F rom (A8) the d ifferen tia l c ro s s sectio n is found to be
»dfl.dft.J* = 2 V . ; » , y t t » d n . d f l , d »
K*(aHKl- **> + ')*
In g en eral, j f tI
7 7 1^1
_
and th e re fo re it is convenient to sim plify (A37) to
(A38) a d n , d n 2 <u = iWmye+fcf* d a ,d ü xdK
k'+ko
Again, since
<
K 4 ( & o Kz+ 0*
, (A37) is e asily re a rra n g e d to give the p ro b a ­
b ility of ejection of the atom ic e le c tro n in the energy range
c.
( £
, E-^ +
d E .^
P ro to n -h y d ro g en atom ch arg e exchange co llisio n s
R esonant ch arg e exchange, w here the e le c tro n energy
re la tiv e to the nucleus is the sam e fo r the final "a to m s” as fo r the in itial "ato m s",
is in g en eral the only type of charge exchange co llisio n w ith an ap p reciab le c ro s s
)•
47
Appendix A
section. This can occur between hydrogen atoms and protons. It has been shown
(Bates‘S ) that the correct interaction potential for such collisions is not the simple
"post” or "prior" interaction (identical for the symmetrical proton hydrogen atom
system), but a much more complicated function. Furthermore, in calculating the
collision cross section, account should be taken of the change in translational motion
of the electron that accompanies the jump between the nuclei. However, in order to
achieve reasonable analytical expressions, both of these considerations will be ig­
nored, it being assumed that the inclusion of a numerical adjustment factor will bring
the Born calculation under these simplifications to an acceptable approximation for
this process.
The "prior" interaction potential is
(A39)
e1 (
V
where V
ir+r.i - ± )
is the internuclear separation, and
the position vector of the elec­
tron relative to the nucleus that it has before collision.
The Born approximation for
this case gives
(A40)
*
=
Zirfc
—
f V e1^
vj/0cr,')
,
but, somewhat differently from the previous cases, feo is the (reduced) wave vector
of the incident nucleus, fef the wave vector of the "scattered" nucleus; dl# ,
are the wave functions of the electron in the initial and final atoms; and also
(A41)
K =
feo + fef ;
K2’ ==ZfeoL( | - h<^r5 X ) .
Using ground state hydrogen atom wave functions for vj;o ,
differential cross section follows from (A8):
to obtain -f
, the
48
The D ifferential C ro ss Sections
(A*2 ) a~ dO — 2. mr e flo
dQ.
fi*(aJKx+ 0 6
A l. 3
The Total C ro ss Sections
In this section, the e x p re ssio n s obtained fo r the d ifferen tial c ro s s sections
in the p revious section a re in teg rated o v er the sc a tte rin g angles to obtain the total
co llisio n c ro s s sections as a function of incident energy and o th e r p a ra m e te rs .
These
to tal c ro s s sections a re com pared with ex p erim en tal re s u lts o r exact n u m e ric al c a l­
cu latio n s to give a n u m erical adjustm ent fa c to r.
T his fa c to r is used to b rin g the "B orn"
approxim ations into reaso n ab le a g reem en t w ith the probable c ro s s sectio n s in a region
fo r which the plane wave assum ption is e n tire ly inadequate.
However since the energy
ran g e of in te re s t is lim ited, the approxim ations a re co n sid ered sufficient to be used in
the calculations of the - Q .’s to obtain an e stim a te of the co llisio n te rm s in the m om ent
equations fo r the form ation of a p la sm a.
As d iscu ssed p rev io u sly , it is likely that
c e rta in c la s s ic a l approxim ations b e tte r d e sc rib e the non-Coulom b e la stic co llisio n s
involved.
A l.3 .1
E lastic C ollisions
a.
Coulomb co llisio n s (e le c tro n -e le c tro n , elec tro n -p ro to n
proton-proton)
With Coulomb co llisio n s the in te g ra l of the d ifferen tial
c r o s s section d iv e rg e s.
F o r this re a so n the w ell known Debye cut-o ff is introduced,
the Debye length re p re se n tin g a d is ta n c e from a p a rtic le beyond which the e le c tro ­
sta tic shielding of the o th er p a rtic le s is ’’com plete".
Usually put fo rw ard as an
u p p er lim it on an im pact p a ra m e te r, it may equally well be e x p re ssed as a m inim um
angle of deflection in a collision.
Since the d e ta ils of Coulomb co llisio n s a re d iscu ssed in
m any books, only the b a re fa c ts need be given.
the s c a tte rin g angle X
(A4 3)
X
z
is
vuyT1(
i/c I +y\?)‘A)
In te rm s of an im pact p a ra m e te r
b ,
A p p e n d ix A
49
w h e re
(A44)
TV
=
W rb s y e 1 •
T h e D ebye le n g th ,
(A45)
\ D
, being
kTe
Xp
--------------- —
ut
r ne e a
V''1
)
">
'
it- fo llo w s a t once th a t th e m in im u m d e fle c tio n
W =• _A-\ L
(A4 6)
A jk =
<Ft
is g iv e n b y (A43) on ta k in g
w h e re
W m '3 ik A ‘
T h e to ta l c ro s s s e c tio n , in te r m s o f
(A4 7)
%o
Z TT
e*'
J»A‘
*X-0
JL
is
.
No c o r r e c tio n fa c to rs a re re q u ire d f o r th is w e ll know n r e s u lt.
b.
E le c tro n -h y d ro g e n a to m c o llis io n s
Since
(A 48)
K dK =
K * — 2.
A
( i “ Co*>)L) , on d iffe r e n tia tin g ,
•
T he u p p e r and lo w e r l i m it s o f in te g ra tio n o v e r
K
b eco m e
'Zko
and
O
.
U sin g
th e d if fe r e n t ia l c ro s s s e c tio n (A 24), w ith the s u b s titu tio n (A48) the to ta l c ro s s s e c tio n
is
50
The D ifferen tial C ro ss Sections
(A49)
c rT
r i e f l n
/
I 4 a .* m r
_
°
TTHlr
°
( a° K C s ) 2 KdKdfc
V k U a l K 2 +u r
( 7
k o < U
3 fc* C
c.
18 C a t + I 2f e oao )
■*■
( I * a ^ tc ) 3
H ydrogen atom -hydrogen atom co llisio n s
Using (A48) in the d ifferen tial c ro s s sectio n (A27), the
application of stan d ard in te g ra ls gives the to tal c ro s s section:
(A50)
c ry
=
l
- g'±
8 u m r e'*a
M C al" + £
,^(1
+a
?
k
:
)
7
V'vo
FTk
+ IS C C
-+ it Cat ^ .
3
d.
K
3
/
Hydrogen ato m -p ro to n c o llisio n s
T his is a lg eb raic a lly the sam e re s u lt as fo r e le c tro n -
hydrogen atom collisio n s:
(A5i)
cry. =
Trmrevao ( 7 C at + '8 C a t -+ iz C a t )
3 C
Al.3.2
C
( I *+
fcjaj-y
In elastic C ollisions
Since
(A52)
K 1
=
C
+
fef
-
J
d ifferen tia tio n yields
(A53)
K <jK
C (if
>■ d y
Appendix A
51
The lim its of integration with re s p e c t to
Ka re k0—
fe f
; fro m the energy equa-
(l ?
II
7T*
(A 54)
° P
tion fo r the type of co llision being considered,
4.
ft*
giving
kf
(A55)
ft.;-
=
* ■ » ; ? ' ~
V*
Ü.O _
w h ere A £ is the energy tra n s fe rre d in elastic ally to the atom .
lim its of
K
a re taken as
2
and fYV
/n r A ¥ \
The upper and low er
resp ectiv ely .
The approxim ate
lim its cannot be expected to be reaso n ab le n e a r the th resh o ld energy, and they a re
th e re fo re " o v e r-ru le d ” in th is region by taking
a.
O
E lectro n -h y d ro g en atom ls - 2 p excitation
F or these co llisio n s A ?
en erg y (10. 2 ev).
at th resh o ld .
is the (d iscrete) excitation
F rom (A33) and using (A53) the total c ro s s section is
- 2TT
(A 56)
<?T -
l
f
fef mr
KdK
JJ°
2
kf
X
2 "Trmr e 'a .o
7r
fo r sufficiently la rg e
a<,m r A £
$ 4 (u
ko
b.
•
E lectro n -h y d ro g en atom ionisation collisions
W ith ionisation c o llisio n s, in equation (A54), A £ becom es
(A57)
AE = I
+
52
The D ifferential C ro ss Sections
w here 1X)q is the m a ss of the ejected electro n , and I
th is case the upper and low er lim its on K
and
(I
is the ionisation energy.
a re m o re conveniently w ritten as
The lim its of in teg ratio n o v er %
( kj*’—
ixr r
.
(A58) <rT =
w here d
a re
O
For
2 k0
and
F rom (A38) and using (A53) the total c ro s s sectio n is given by
f
f
y
K m r e 4 2 6 T r a 2 - d < d 3 c c lK < lfe d -^ > .
is the solid angle of the ejected e lec tro n .
(A59) (fco + R f ) a „
~
Z k 0 <lo “
j^° (
( k 0 - k f ) a 0 ~ if
With feo sufficiently la rg e ,
)
> /’
^
*
"J
\ .
Z.
fe n
Using th ese approxim ations (invalid n e a r thresh o ld ), the re s u lt is
(A60)
<rr
=
rn<?
17 a °
^ fiV2meI
~ I
)
fi *
(since, on neglecting the te rm
is equal to the e lec tro n m a ss,
c.
com pared with unity, the reduced m a ss, tmr
).
C harge exchange c o llisio n s
As d iscu ssed e a r lie r fo r this c a se ,
, yT.
is given by
Z fe* ( I f CCTsX) and th e re fo re
(A6i)
svn)cdx
=
— Kd K/ k o .
T his co llisio n being effectively e la stic , using th is re la tio n in (A42) and in teg ratin g
over
K
with lim its
O ,
2fe0
gives the to tal c ro s s section.
(owing to the la rg e reduced m ass in this collision), the re s u lt is
F o r 6 0 la rg e
,
53
(A62)
Appendix A
<rT =
'n~Wr ^ ^ a°
5- 6 * f £
A l. 3 .3
C om parison w ith Known V alues of C ro ss Sections
Since the Coulomb calcu latio n s a re exact (ap art fro m neglect of
exchange) these a re not d iscu ssed .
F o r all of the o th e r c a se s co n sid ered except e la s ­
tic hydrogen atom -hydrogen atom co llisio n s (for which no d ata ap p ears to be available,
p erhaps owing to the problem of recom bination to form m o lecu lar hydrogen), c u rv e s
showing a com parison of num erically adjusted B o m -ty p e total c ro s s sectio n s with
known values a re given. Most of the "known" d ata is taken fro m B arn ett, Ray and
11
12
Thom pson.
The ch arg e exchange c ro s s sectio n s a re fro m B ates.
The n u m erical fa c to rs w hich the calcu lated B orn-type total
c ro s s sections have been m ultiplied by to give the re s u lts in the fig u res a re as follows:
Cross section curves for elastic e -H collisions
adj. Born
impact energy (ev)
The Differential Cross Sections
54
Cross section curves for elastic H -II collisions
a d j . Born
impact energy (ev)
Cross section curves for ls-2p e -H excitation collisions
a dj. Born
0
20
80
impact energy (ev)
100
120
Appendix A
55
1.2
cross section x 10
—16
™
(cm
Cross section curves for e -H ionisation collisions
adj .
Born
0
20
40
60
80
100
120
impact energy (ev)
cross section x .873 10 ^
(cm*')
Cross section curves for H + -H charge exchange collisions
expt.
adj .
Born
impact energy (kev)
56
The D ifferential C ro ss Sections
a.
E lectron-hydrog en atom elastic co llisio n s.
B orn calculation m ultiplied by 0. 45.
b.
Hydrogen atom -proton e lastic co llisio n s.
B om calculation m ultiplied by 2. 0 x 10
c.
-2
Hydrogen atom -hydrogen atom e lastic co llisio n s.
No com pariso n with "known” re s u lts being possible,
the sam e fa c to r as with (b) above is taken: 2. 0 x 10
d.
-2
E lectron-hydrog en atom ls -2 p excitation.
B orn calculation m ultiplied by 0.45.
e.
E lectron-hydrog en atom ionisation.
_3
B orn calculation (sim plified) m ultiplied by 0.14 x 10
f.
Hydrogen atom -p ro to n ch arg e exchange.
B om calculation (sim plified) m ultiplied by 2 .1 x 10
^.
57
APPENDIX B
S ubsidiary R elations for E lectron-H ydrogen
Atom Ionisation C ollisions
B l.
The
X 's
and
A§
Ts
In co nsidering the dynam ics of in elastic b in ary co llisio n s, the
X
fs a re
defined by
(Bl)
X2* s-
w here Zl f
m
j
/ mj /
/
my
is the ap p aren t kinetic energy lo ss in the collisio n .
co llisio n fo r exam ple, A ?
F o r an ionisation
is the sum of the ionisation energy and the change in
. /
in te rn a l energy of the p a rtic le s J
/
, R
/
.
com pared with J
,
/
. With the p a r ­
tic u la r ionisation co llisio n s being considered , co n sid erab le sim p lificatio n is achieved
by neglecting te rm s of o rd e r
Y Y \e /
com pared with unity; it has been shown in
Appendix A th at this p e rm its the fram e of re fe re n c e fo r all d ifferen tial c ro s s section
p a ra m e te rs to be taken as the c . m . fram e .
This is not n e c e s sa rity so fo r the "com ­
p o site " p a rtic le in tern al e n erg ies, how ever.
a.
The incident electro n , C t
F or the collision dynam ics of th is p a rtic le , the two " p a rtic le s "
a fte r the c o llisio n a re the incident (and sca tte re d ) electro n , and the proton + atom ic
e lec tro n sy ste m .
(B2)
The ap p aren t energy lo ss in the collisio n ,
A
i? =I +
§
is, quite sim ply,
•
The la s t te rm in (B2) being the in tern al kinetic energy of the proton + e lec tro n s y s ­
tem ; as b efo re
is the wave v e c to r of the atom ic electro n , and
neglected com pared with unity.
X
is the ionisation energy.
is given by
(B3)
=
I
-
( 2 .J +
92
•
me / Wa
Using (Bl),
is
X
58
A p p e n d ix B
b.
The A to m ic E le c tro n ,
0a.
C o rre s p o n d in g to th is p a r tic le b e in g one o f the p a r tic le s a fte r
the c o llis io n , the re m a in in g " p a r t ic le ” is the s y s te m p ro to n + in c id e n t e le c tro n .
the c . m . s y s te m , th is la t t e r " p a r t ic le " has in te r n a l e n e rg y
te r m s o f o r d e r
W e / W a ).
(B4)
X
A
j*
=
/ZfY \r
In
(n e g le c tin g
T he a p p a re n t e n e rg y lo s s is th e re fo re
.
■+
F ro m th is i t fo llo w s tn a t
(B5)
X1
=
I
-
c.
( 2
l
+
The P ro to n ,
^ » V w r V m
r g '
•
p
T he c o m p o s ite " p a r t ic le " a fte r th is c o llis io n is the s y s te m o f
the tw o e le c tro n s .
In the c. m . s y s te m , the re s p e c tiv e v e lo c itie s o f the s c a tte re d and
a to m ic e le c tro n s a re
£1 k f / l ^ e
and
.
e n e rg y r e la tiv e to t h e ir c e n tre o f m a ss is th e r e fo r e
M
The in te r n a l k in e tic
^ — j< ) 2*
q
and so
(B6)
A g
=
I
+
h i.
4 me
T h e re fo re
(B7)
X a
=
I
-
2 - ( l
+
In g e n e ra l, away fr o m th re s h o ld
|^K |
^
^
, and so in th is r e -
I jk f I
g io n i t is p o s s ib le to w r it e
(B 8 )
A1 =
I -
'
,
Subsidiary R elations fo r E lectron-H ydrogen
Atom Ionisation C ollision
d.
The atom ,
59
Q.
T his p a rtic le is not p re se n t a fte r the collisio n , and th e re fo re ,
using the
fa -functions defined in re fs . 1, 2, it may be v erified that the collision
in te g ra ls fo r the atom s only involve
a re independent of
.X
.0 .
(t )
. Since by definition th ese 0 . ’s
, they a re also independent of
A£
. The
A£
, how ever,
a re im p licit in the in tegration of the d ifferen tial c ro s s section o v er all p o ssib le e n e r­
gies of the ejected e lec tro n in the calculation of the . 0 . ' s fo r th is case.
It is w orthw hile noting that, with the neglect of te rm s of o rd e r
me
/ W q. , th e re is a sim ple re la tio n sh ip betw een the X ^ i's fo r the atom s and
those fo r the protons in an ionisation collisio n .
T his may be seen by co n sid erin g the
fa -functions (refs. 1, 2) and neglecting all te rm s of o rd e r m e / fY)a . It can also
be seen in another sim ple way.
N eglect of te rm s involving the e lec tro n m ass is equi­
v alen t to regarding the collision as an event which s trip s the e lec tro n from the atom,
form ing a proton with the sam e m a ss and velocity as that of the atom before the c o lli­
sion.
In the notation of re fe re n c e 2, th is m eans th at
(B9)
AeJ^p) =
Since only the
r\ - oo, o
1 Z Lt )
= ~ 4a
a re involved, it follows that (the
£ -functions con­
taining the sign re le v an t to (B9) ):
(BIO)
B2.
-c o ^ o
/ V 00' 0
eq
cl
O )
In teg rals O ccurring in the C alculation of the (|) Ts and - Q j s
a.
The Integral
C onsider
J| (
V
)
Appendix B
00
(B ll)
J ( (yu .,v) =
/ I T , ( C " 2 m r I / 6 * ) /2
J
/ ( I-
° °
SvnX
C kt + kf - 2 f».
X )x
with
(B12)
X1 =
I ”
*-r *
o
2 -(I +
" v /^ k
Zm e y
This integral (Bll) occurs for the case of an ionisation collision where the particle
being considered is the incident (and scattered) electron. Conservation of energy in
the collision gives
(B13)
J
+
2m r
^ Xj<‘X
Zmr
and since
^
Z Wr
the reduced mass for the collision, it follows from (Bll)
that
(B14)
A2
=
The following values of
(2,2
U
are required: ( —OO 5 o
2 , 0 ), ( 3 J
i.
X
In equation (B ll), integration over
with the use of a substitution of the form
(B15)
(/IjV) =
=
2
x
J Otze C ~
/(C
-
k f )z
"X
is readily effected
■= 00-5 X . This gives
( £ - 2 « V l/^ ) 'A
where
(B16) I ( o )
| ? |
).
Integration over
r
), (
^
),
61
S ubsidiary R elatio n s fo r E lectron-H ydrogen
Atom Ionisation C ollision
(B18)
J(2)
4 kl kf
0
( feo + fcf V
ko k{
k o + fcf '
Ignoring the lo g a rith m te rm and, in addition, retain in g only the te rm s which a re
la rg e s t away fro m the thresh o ld , these form u lae a re re p re se n te d by
(B 1 9 )
TO) =.
2 ( feo 4 f e f )
A ccordingly,
becom es
2. ‘/a
(fe0-2mrT/f> )
(B20)
i ki - kV
ii.
Integration ov er
Using CX
(B21) CX x
U
and
(B22)
X1
=
ko
as defined previously, i. e.
- 2mrI A
X
a re
=
cx - X
=
ko
F ro m equation (B20)? J* (yU^V) can
(B23)
( /S V)
=
X.
P (/S V3 *“
.
w ritten as
62
Appendix B
w here
(B24)
P (y U ,\> )
f XkfX(i±
=
J°
(B25)
Q C /^ V )
(fe ^ -fe f)1
f* Zh*d*
=
°
v * (feN<W)v .
( f e o - f e f )1
In g e n eral, fo r ionisation collisio n s of th is type, the atom ic elec tro n s a re ejected
with low energy.
(B26)
In view of th is, the e r r o r introduced by rep lacin g the te rm s
(C -
C f ) 0-
in the denom inator by
X \ 2.
=
a m j / * 1)
( 2 P1r I / fi )
is not unduly la rg e .
This sim p lificatio n
gives
(B27)
pCuv) =
2 o<
f 2
^
-4
Again, in calculating
( 2 n rl / $ * ) z
,
3 ( 2mrX / Ä1)1
in the te rm involving ( 6 0
)
,
&
neglected so that
(B28)
-+
kf
— 2 ke — 2mrJ./f>
~
2 C - 2 m rl / ^
=
kt
+ °<
In th is c ase ,
(B29)
Q (yU ,V )
=1
I
Z(ko+<x)
(2m r I
/ m
v fc f*
(ot3- * 1)
*
fccl*
,
is
63
S u b sid iary R e la tio n s fo r E le c tro n -H y d ro g e n
A tom Io n isatio n C o llisio n
z U.1- « 1)'
2\Z
( M+V
UmrI/*TZvfc
$ ( /,* )
o
where the S ( V , v )
(B30)
are given by
S C -o o ,o )
==
O
S o ,n
=
5(2,2)
=
S (3 ,0
=
s
=
( 2, 0)
°<3 / 3
•
Finally, therefore,
3
=
(B31)
Z °<
3 ( 2 m rX / ^ ) 2
(B32)
T, ( i , l)
=
2c<3
3 ( 2 m rI / f i Jf
(B33)
J, ( 2, 2)
j
_
C fe* +<**)
( ,
_
( fei - + « 1)
-
3<*x \
S ■k l
J
f
-
Z * *
V
)
3 O m r I / f r 1) 2 ^
(B34)
Ll, 0)
(B35) J t ( 3 , 0
b.
=
2o<3
( \
3 ( 2 m rX / ft1)*- ^
2°<3
/ | _
3 ( i m r l / f i 1) 1 '
The In te g ra l
io
ft*
)
•
I—
C o n sid e r
(B36)
L C /jV )
- /° (2mrX/ft1+ **)
( i - f o h£ I) .
\
V -M
64
Appendix B
T his in teg ral a ris e s when the p a rtic le being co n sid ered is the ejected atom ic electro n .
F o r this c a se ,
(B37)
XX =
I -
>
the notation being as p reviously used in th is section.
into two p a rts , the f ir s t being independent of
(B38)
(B39)
L (y u ,y ) = W
+ X
V)
. Taking
)
v = z
(2n K1)*
w ith
(B36) is conveniently sep a ra te d
•)
as in (B21). Although it is p o ssib le to evaluate the in teg ral in this form
(B39), usable ex p re ssio n s fo r calculatin g the jO - 's a re only obtained if the denom ina­
to r is rep la ce d by ( 2 m r X / ^
th is substitution,
(B40)
W
as in e a r lie r in te g ra ls of th is section.
is e asily obtained:
2
W
)
*
3 ( 2 mrI /
*
The second p a rt of in teg ral (B36), with (B38) and using (B37), is
(B41)
X (yU ,v) =
'V'
ko
(v .ti)U n r l/f i* ) 2
F o r the individual c a se s,
(B42)
X (-oo.o )
=
X ( 1,1)
X C 3, I)
O
With
S u b s id ia ry R e la tio n s f o r E le c tro n -H y d ro g e n
A to m Io n is a tio n C o llis io n
65
w h ile
(B43)
X(2-,2)
=
-
I
^ kf x d fr
3 ( 2 ™ rI / V
/ I_
) a
fep - 3<X\
^
ß
and th is b ecom es
(B44)
X ( a ,i)
=
-
____2
3 (z m
_________
(« :5 _
rl / V ) ' \ 3
S&J’
w h ile la s tly ,
(B45)
X C ^O )
-
-
fI
Jo
U s in g (B 38),
(B46)
L C yU ^v)
L ( 3 , i)
=
-
fe»-*1)
(2*ri m 2 '
= 3X£i,J)
>
has the v a lu e s
L C '.O
-
L c - o o jo )
=
3 (2 n T 1 J .I/V )1
(B47)
L(2,2)
2<x
«
-
2 <* 3
(B 48)
L (2 ,° )
a U m . I A 1) i
(
*
*
3 « 1
5 fef-
#
£
)
66
APPENDIX C
In v erse P ow er and A ttractiv e-R ep u lsiv e Type In teractio n s
F o r the sake of com pleteness, and to provide a com p ariso n with the B o rn type calculations p re sen te d in this re p o rt, the in v e rse pow er and a ttra c tiv e -re p u lsiv e
type in teractio n c la s s ic a l calculations of the j f l j s fo r e lastic co llisio n s a re given.
3
T hese a re as d iscu ssed by Chapm an and Cowling, and th is Appendix is a sum m ary
of the relev an t p a rts of th e ir tenth ch ap ter.
C l.
In v erse P ow er R epulsive F o rce
Let
d istan ce
T
(C l)
P =
P
be the fo rce betw een two m olecules of m a sse s
rrij
, nr)^ at a
, satisfying the relatio n
/ tV
•
By considering the equations of m otion of the two p a rtic le s , it may read ily be shown
that, in p o la r coordinates ( T , © ),
(C2)
T 1©
=
(C3)
i ^
r 1 © 2) +
b
(C4)
S
C o n st.;
=
mo
gb
/ mj
is the im pact p a ra m e te r of the collisio n , such that
<T S ^ n
x
dX
—
kdb
W ritin g
(C5)
V*
=
Vo
=
b /r
,
_L
=
c o n s t \> -
\
9
•
67
Appendix C
at the apse of the o rb it
(C6)
I
-
\ r 2- -
^ r r ( v o )
”
°
L et V 00 denote the re a l positive ro o t of (C6); then fro m the geom etry of the co llisio n
it is apparent that the angle betw een the assy m p to tes is tw ice the value of
ponding to \ T s U oo , and th e re fo re
(C7)
^). ( ^ )
vJ"
has the value
can be tra n sfo rm e d as follows:
C^) = /
C I - <***%)
30*
2
5"
= / m°
=
with
At (v)
^^
Z"00
qv^r I
\~ 1
/ yv>°
g “ i
( I
Ajy)
a pure num ber depending only on
c e rta in values of
and
^
A,
-
v
r
0<iu-c
and
V
, and tabulated fo r
i
in Chapman and Cowling.
Using th is ex p ressio n fo r
($(<()
, -fi- (_t )
is read ily evaluated.
Since
“e'YV r+1 4>C^)ctY ,
(C 9)
it follows that
(CIO)
c o rre s­
X = TT
Using (C7),
(c8>
pC
0
1 2 j ,(.»*)
^
=
-------
^
2.6*) m */m o V /l
m
- ^
)
68
In v e rse P o w er and A ttra c tiv e
R epulsive Type In teractio n s
C2.
A ttra c tiv e -R ep u lsiv e Interaction
The fo rc e s betw een sev e ra l types of m olecules can be w ell approxim ated
to by an equation of the form
(cii)
p =
*jfe/Tv -
(where the fo rce
P
/r v^
is taken as positive when rep u lsiv e, and
V
^
V
).
Using a s im ila r m ethod to th at indicated above, it may be shown that the angle "X- is
given by
(C12)
X
w here
(C13)
V0
=
b(
v"‘
.
V«/ =
_ !__
b ( Wj mfag2/m 0
v/'*
and ~\^oo is the (least) positive root of the equation
V->
(C14)
v-1 v xr0 i
The evaluation of
\> -
*
(f)(£) and
(•£-)
0
.
I
= o .
£
-
w hat difficult in the g en eral c a se .
C y' )
a re , fo r th is potential, so m e­
If, how ever, the a ttra c tiv e p a rt of the field is weak,
/
fa irly sim ple approxim ations can be m ade. When
is sm all,
can be
w ritte n approxim ately as
v-w'
69
Appendix C
w here "T
is a m ean te m p e ra tu re of the colliding p a rtic le types, and
value of X
sid ered .
Using th is approxim ation (C15),
/
~ / ( I-
-» X^jb/T v^7)) S'5"
/
°
=
w here
ß(Jt)
is the
obtained when only the rep u lsiv e p a rt of the in teractio n fo rce is con­
TT
(C16>
“y ,0
( <t>j f c ^ ) ) o ( 1 + /* C i) / T ^
is identical w ith the
is indepe: lent of T
^ jf c C 'O
^ 7
°f
Pr evious c ase , and
. Hence finally,
/
(ci7) 0 * C r ) =
w
h
e r e )o
p a rt of this Appendix.
V /j
( ü / hCr))ö ( l + sJ k ( b ) / T ^
is identical w ith the co rresp o n d in g
S j^
r )
is a function of J??
)
of the previous
^
and
and is tabulated (in a com ponent form ) fo r the Sutherland and L e n n ard -Jo n es
m o lecu lar m odels in re fe re n c e 3.
70
REFERENCES
BYDDER, E. L . : ’’The Theory of P a rtia lly Ionised and N on-U niform P la sm a , "
Ph. D. T h e sis, A u stralian National U niversity, June, 1967.
BYDDER, E. L. and B .S . LILEY: "On the Integration o f'B o ltz m a n n -L ik e ’
C ollision In te g ra ls, " D epartm ent of Engineering P h y sic s, A u stralian
National U niversity P ublication E P -R R 15, M arch, 1967.
CHAPMAN, S. and T .G . COWLING: "The M athem atical Theory of NonUniform G ases, " 2nd edition, C am bridge, 1960.
BYDDER, E .L . and B .S . LILEY: "The F orm al N on-E quilibrium Theory
of P a rtia lly Ionised a n d /o r N on-U niform Gas M ix tu re s," (to be
published).
GOODYEAR, C .C . and A. VON ENGEL: P ro c . P hys. S o c ., 79, 732, 1962.
LILEY, B. S. : "The N on-E quilibrium T heory of M ono-Atomic Gas M ixtures
and P la sm a , " A. E . I . , R eport A, 1392, July 1963 and Ph. D. T h esis,
Univ. of Reading, May, 1963.
GRADSHTEYN, I. S. and I.M . RYZHIK: "T ab les of In te g ra ls, S eries, and
P ro d u cts, " (tran sl. from 4th R ussian edition), A cadem ic, 1965.
MOTT, N. F. and H .S .W . MASSEY:
2nd edition, Oxford, 1949.
"The Theory of Atomic C ollisions, "
BATES, D .R . and G .W . GRIFFING:
P ro c . P h y s .S o c ., A66, 961, 1953.
BATES, D .R .:
P ro c . Roy. S oc., A247, 294, 1958.
BARNETT, C .F ., J .A . RAY and J .C . THOMPSON: "Atomic and M olecular
C ro ss Sections of In te re s t in C ontrolled T h erm o n u clear R esearch , "
Oak Ridge N ational L ab o rato ry , R eport ORNL-3113. R evised, August,
1964.
BATES, D . R . : "T h e o re tic al T reatm en t of C ollisions Between Atomic Sys­
tem s, " in B ates, D. R. (ed), "Atomic and M olecular P ro c e s s e s , "
A cadem ic, 1962.
RI
P u b lic a tio n s by D e p a rtm e n t of E n g in e e rin g P h y s ic s
No._________ A u th o r
T itle
F irs t
P u b lish e d
R e -is s u e d
E P -R R 1
H ib b ard , L. U.
C em en tin g R o to rs fo r the
C a n b e rra H o m o p o lar
G e n e ra to r
May, 1959
A p ril, 1967
E P -R R 2
C a rd e n , P . O.
L im ita tio n s of R ate of R ise
of P u ls e C u r r e n t Im posed
by Skin E ffec t in R o to rs
S ept., 1962
A p ril, 1967
E P -R R .3
M a rs h a ll, R. A.
T he D esign of B ru s h e s fo r
th e C a n b e rra H om opolar
G e n e ra to r
J a n ., 1964
A p ril, 1967
E P -R R 4
M a rs h a ll, R. A.
The E le c tro ly tic V a ria b le
R e s is ta n c e T e s t L o ad /S w itch
fo r the C a n b e rra H om opolar
G e n e ra to r
May, 1964
A p ril, 1967
E P -R R 5
In a ll, E . K.
The M ark II C oupling and
R o to r C e n te rin g R e g is te r s
fo r the C a n b e r r a H om opol a r G e n e ra to r
O ct. , 1964
A p ril, 1967
E P -R R 6
In a ll, E . K.
A R eview of th e S p e c ific a ­
tio n s and D esign of the
M ark II Oil L u b ric a te d
T h ru s t and C e n te rin g
B e a rin g s of th e C a n b e rra
H o m o p o lar G e n e ra to r
Nov. , 1964
A p ril, 1967
E P -R R 7
In a ll, E . K.
P ro v in g T e s ts on the
C a n b e rra H o m o p o lar G en­
e r a t o r w ith th e Two R o to rs
C onnected in S e rie s
F e b ., 1966
A p ril, 1967
E P -R R 8
B rad y , T. W.
N otes on Speed B alan ce
C o n tro ls on the C a n b e rra
H o m o p o lar G e n e ra to r
M ar. ,1966
A p ril, 1967
E P -R R 9
In a ll, E . K.
T e s ts on th e C a n b e rra
H o m o p o lar G e n e ra to r
A rra n g e d to Supply the
5 M egaw att M agnet
May, 1966
A p ril, 1967
R2
P u b lic a tio n s by D e p a rtm e n t of E n g in eerin g P h y s ic s (C o n t.)
No.
A u th o r
T itle
A Study of the P e r f o r m a n c e
of the 1000 kW M o to r G en­
e r a t o r Set Supplying th e
C a n b e rra H o m o p o lar G en­
e r a t o r F ield
F irs t
P u b lish ed
R e -is s u e d
Ju n e, 1966
A p ril, 1967
A p ril, 1967
E P -R R 10
B rad y , T .W .
E P -R R 11
M acleod, I.D .G . In s tru m e n ta tio n and C o n tro l
of the C a n b e rra H o m o p o lar
G e n e ra to r by O n -L in e C o m ­
p u te r
O c t., 1966
E P -R R 12
C a rd e n , P . O.
M echanical S tr e s s e s in an
In fin itely Long H om ogeneous
B itte r Solenoid w ith F in ite
E x te rn a l F ield
J a n . , 1967
E P -R R 13
M acleod, I.D .G . A Survey of Iso la tio n A m p lif i e r C irc u its
F eb. , 1967
E P -R R 14
In a ll, E. K.
T he M ark III C oupling fo r
th e R o to rs of th e C a n b e rra
H o m o p o lar G e n e ra to r
F eb. , 1967
E P -R R 15
B y d d er, E . L .
L iley , B. S.
On th e In te g ra tio n of
" B o ltz m a n n -L ik e "
C o llisio n In te g ra ls
M ar. ,1967
E P -R R 16
V ance, C . F .
S im ple T h y r is to r C irc u its
to P u l s e - F i r e Ig n itro n s
f o r C a p a c ito r D isc h a rg e
M ar. ,1967
E P -R R 17
B y d d er, E . L .
On the E v alu atio n of E la s tic
and In e la s tic C o llisio n F r e ­
q u e n c ie s fo r H y d ro g e n ic -L ik e
P la s m a s
Sept. ,1967
E P -R R 18
S teb b e n s, A.
W ard , H.
T he D esig n of B ru s h e s fo r
th e H o m o p o lar G e n e ra to r a t
The A u s tra lia n N ational
U n iv e rsity
M ar. ,1964
S e p t., 1967
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