Louisiana State University
LSU Digital Commons
LSU Historical Dissertations and Theses
Graduate School
1971
The Photoionization of Lithium, Sodium, and
Potassium and the Photodetachment of the
Negative Hydrogen Ion.
Richard Lee Smith
Louisiana State University and Agricultural & Mechanical College
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Recommended Citation
Smith, Richard Lee, "The Photoionization of Lithium, Sodium, and Potassium and the Photodetachment of the Negative Hydrogen
Ion." (1971). LSU Historical Dissertations and Theses. 2090.
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« a4 L w '• 'fit* - t ■**J te n r
1!
72-3526
SMITH, R ich ard Lee, 1943THE PHOTOIONIZATION OF LITHIUM, SODIUM,
• AND POTASSIUM AND THE PHOTODETACHMENT
OF THE NEGATIVE HYDROGEN ION.
The L o u isia n a S ta te U n iv e rs ity and
A g r ic u ltu r a l and M echanical C o lle g e ,
P h .D ., 1971
P h y s ic s , atom ic
| University Microfilms, A XEROXCom pany, Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED
THE PHOTOIONIZATION OF LITHIUM, SODIUM, AND POTASSIUM
AND THE PHOTODETACHMENT OF THE NEGATIVE HYDROGEN ION
A D is s e rta tio n
S u b m itted t o th e G rad u ate F a c u lty o f th e
L o u is ia n a S t a t e U n iv e r s ity and
A g r i c u l tu r a l and M echanical C o lleg e
i n p a r t i a l f u l f i l l m e n t o f th e
re q u ire m e n ts f o r th e d e g re e o f
D octor o f P h ilo so p h y
in
The D epartm ent o f P h y sic s and Astronomy
by
R ic h a rd Lee Sm ith
B .S ., B ay lo r U n iv e r s ity , 1966
A u g u st, 1971
EXAMINATION AND THESIS REPORT
Candidate:
R ic h a r d L ee S m ith
Major Field:
P h y s ic s
Title of Thesis: The P h o t o i o n i z a t i o n o f L i t h i u m , S o d iu m ,
and th e P h o to d e ta c h m e n t o f th e N e g a tiv e
an d P o ta s s iu m
H y d ro g e n I o n
Approved:
1 ? .U A
Major Professor and Chairman
Dean o( the Graduate School
EXAMINING COMMITTEE:
Date of Examination:
J u ly
16,
1971
PLEASE NOTE:
Some P a g e s h a v e i n d i s t i n c t
p rin t.
Filmed as r e c e iv e d .
UNIVERSITY MICROFILMS
ACKNOWLEDGMENTS
The a u th o r w ish es to e x p re ss h i s g r a t i t u d e t o D rs. R. W.
LaBahn and J . J . M atese f o r t h e i r a s s i s t a n c e th ro u g h o u t th e
co m p letio n o f t h i s d i s s e r t a t i o n .
The a u th o r i s in d e b te d to D r. J . N. B a rd s le y f o r communicating
h i s p s e u d o p o te n tia l p a ra m e te rs p r i o r t o p u b l i c a t i o n , and to D r. R. S .
O beroi f o r th e u s e o f h i s com puter program s.
Thanks a r e a l s o due to D rs. R. J . W. H enry and J . C allaw ay
f o r h e l p f u l d is c u s s io n s and comments.
F in a n c ia l a s s i s t a n c e r e c e iv e d from th e "D r. C h a rle s E. C oates
M emorial Fund o f th e LSU F oundation" f o r th e p u b l i c a t i o n o f t h i s
work i s g r a t e f u l l y acknow ledged.
T h is d i s s e r t a t i o n i s d e d ic a te d to my w if e , E l iz a b e th , who
ty p e d th e m a n u sc rip t and was a c o n s ta n t so u rc e o f encouragem ent.
ii
TABLE OF CONTENTS
Page
A ck n o w led g m en ts................................................................................................................. i i
L i s t o f T a b l e s ................................................................................................................. i v
L i s t o f F i g u r e s .....................................................................................................................
A b s t r a c t ........................................................................................................................
P a rt I ;
v ii
HIOTOIONIZATION OF LITHIUM,
SODIUM, POTASSIUM
i
1 -1
I n t r o d u c t i o n ....................................................................................................... 2
1 -2
T h e o r y .....................................................................................................................^
1 -3
E v a lu a tio n o f P s e u d o p a r a m e te r s
I-U
*
8
R e s u lts an d D is c u s s io n ...................................................................... •
10
A.
B.
C.
1 -5
S o d i u m ................................................................................................... 10
L i t h i u m ...................................................................................................i 2
P o ta s s iu m ................................................................................................... i 1*
C o n c l u s i o n s ..................................................................................................... i®
P art I I :
HIOTOBETACHMENT OF NEGATIVE
HYDROGEN ION
28
I I -1
I n t r o d u c t i o n ..................................................................................................... 29
I I -2
Wave F u n c t i o n s ...............................................................................................32
A.
B.
Bound S t a t e ............................................................................................ 32
Continuum S t a t e ........................................................................................3 °
1.
2.
S h o rt Range C o e f f i c i e n t s .................................................... 38
A sy m p to tic C o e f f i c i e n t s ........................................................... ^2
I I -3
M u ltic h a n n e l P h o to d e ta c h m e n t....................................................................^9
Il-k
P hotodetachm ent R e s u lts and C onclusions Below
t h e I n e l a s t i c T h r e s h o l d .........................................................................52
I I -5
R e s u lts an d C o n clu sio n s i n t h e ^P Resonance R egion
•
•
55
R e f e r e n c e s .............................................................................................................................. 88
V ita -
........................................
69
iii
LIST OF TABLES
T ab le
I
C a lc u la te d non-coulom b an d quantum d e f e c t p-wave
p h a se s h i f t s i n r a d ia n s
II
The e f f e c t t h a t t h e b a s is s e t h a s on th e c o n tr ib u tio n s t o
t h e p h o to d etach m en t c ro s s s e c tio n s f o r a f i n a l s t a t e i n
c h a n n e l f o u r.
LIST OF FIGURES
F ig u re
I
Page
P h o to io n iz a tio n c ro s s s e c tio n s o f sodium , u s in g
X -in d ep en d en t p s e u d o p o te n tia l and in c lu d in g
p o la riz a tio n e f f e c ts .
II
21
P h o to io n iz a tio n c r o s s s e c tio n o f sodium u s in g th e
X -dependent p s e u d o p o te n tia l o f B a rd s le y .
III
Com parison o f t h e p h o to io n iz a tio n r e s u l t s o f
sodium t o o th e r sodium r e s u l t s .
IV
23
P h o to io n iz a tio n c ro s s s e c tio n s o f li th iu m , u s in g
th e Yukawa form o f B a r d s le y 's p s e u d o p o te n tia l.
V
22
2k
Com parison o f o u r r e s u l t s f o r lith iu m t o o th e r
c a l c u l a t i o n s o f p h o to io n iz a tio n c ro s s s e c tio n
o f li t h i u m .
VI
25
P h o to io n iz a tio n c r o s s s e c tio n o f p o ta ss iu m , u s in g
th e p s e u d o p o te n tia l o f B a rd s le y .
V II
V III
A d ju s te d p h o to io n iz a tio n c ro s s s e c tio n o f p o ta ss iu m .
26
27
P hotodetachm ent c ro s s s e c tio n s i n th e d ip o le
v e l o c i t y a p p ro x im a tio n f o r H~ u s in g t h r e e d i f f e r e n t
hound s t a t e s .
IX
60
A com parison o f p h o to d etach m en t c ro s s s e c tio n s i n
d ip o le v e l o c i t y a p p ro x im a tio n o b ta in e d u s in g th e
S chw artz bound s t a t e o f H*" t o th e r e s u l t s u s in g
th e bound s t a t e o f M atese and O b ero i.
v
6l
A com parison o f p h o to d etach m en t c r o s s s e c tio n s i n
d ip o le le n g th a p p ro x im a tio n o b ta in e d u s in g th e
Schw artz bound s t a t e o f H" t o th e r e s u l t s u s in g
th e bound s t a t e o f M atese and O b e ro i.
XI
Photodetachm ent r e s u l t s f o r H~ o b ta in e d w ith th e
s ix s ta te
X II
e x p a n sio n o f M atese and O b e ro i.
63
( l s - 2 s ) and ( ls - 2 p ) c r o s s s e c tio n s i n th e en erg y
r e g io n o f th e ^P re s o n a n c e .
X III
62
The ^P shape re so n a n c e p h o to d etach m en t c ro s s s e c t io n s .
vi
6^
65
ABSTRACT
The d i s s e r t a t i o n i s a two p a r t w ork.
e n t i t y from P a r t I I .
P a r t I i s a s e p a r a b le
I n P a r t I a p s e u d o p o te n tia l fo rm alism i s u sed
t o c a l c u l a t e th e c r o s s s e c tio n f o r p h o to io n iz a tio n o f sodium ,
l i t h i u m , and p o ta s s iu m f o r e j e c t e d e l e c t r o n e n e r g ie s from th r e s h o ld
t o a b o u t 15 eV.
B oth .th e d ip o le le n g th and d ip o le v e l o c i t y m a trix
form s a r e com puted.
F or sodium th e d ip o le le n g th r e s u l t s a r e i n
good agreem ent w ith ex p erim en t away from th e lo w e s t th r e s h o l d .
The d ip o le v e l o c i t y r e s u l t s f o r sodium , on t h e o th e r h an d , s e v e r e ly
u n d e re s tim a te t h e c r o s s s e c tio n e x c e p t v e r y n e a r t h r e s h o ld .
For
li t h i u m th e d ip o le le n g th and v e l o c i t y r e s u l t s a r e l e s s th a n th e
e x p e rim e n ta l r e s u l t s b u t compare f a v o r a b ly w ith o th e r t h e o r e t i c a l
re s u lts .
F o r p o ta s s iu m t h e c r o s s s e c t io n e x h i b i t s th e g e n e r a l
shape o f e x p e rim e n ta l c u rv e s.
I n P a r t I I th e v a r i a t i o n a l m ethod p ro p o se d b y F .E . H a r r is and
ex ten d ed by R.K . N esb et i s employed t o o b ta in wave f u n c tio n s f o r
t h e s i n g l e t p-w ave continuum f u n c tio n s o f th e n e g a tiv e hydrogen
i o n , u s in g a l s - 2 s - 2 p c lo s e c o u p lin g a p p ro a c h .
The b o u n d -fre e
a b s o r p tio n c o e f f i c i e n t i s c a lc u la te d u s in g l e n g t h , v e l o c i t y , and
a c c e l e r a t i o n form s o f th e d ip o le m a trix e le m e n t.
t h e s e c a l c u l a t i o n s a r e p r e s e n te d .
The r e s u l t s o f
P a r t i c u l a r a t t e n t i o n i s g iv e n
t o th e e n e rg y r e g io n n e a r 0 .7 5 1 ry d w here th e ^P shape re so n a n c e
e x is ts .
Com parison o f th e r e s u l t s from t h e l e n g th , v e l o c i t y and
a c c e l e r a t i o n o p e r a to r m ethods i n d i c a t e t h a t th e wave f u n c tio n s
o b ta in e d by t h i s v a r i a t i o n a l m ethod a r e s u f f i c i e n t l y a c c u r a te f o r
c a l c u l a t i n g t h e p h o to d etach m en t o f H’ .
vii
PART I - PHOTOIONIZATION OF LITHIUM, SODIUM, POTASSIUM
1
SECTION 1-1
INTRODUCTION
I n p r i n c i p l e th e c ro s s s e c tio n s f o r p h o to io n iz a tio n can b e
c a l c u l a t e d when a c c u r a te wave f u n c tio n s a r e known f o r th e s t a t e s o f
th e atom s and i o n s .
I n p r a c t i c e , assu m p tio n s have t o be made i n
o r d e r t o o b ta in t h e n e c e s s a r y wave f u n c tio n s .
I n a p re v io u s p u b l i ­
c a tio n by t h e au th o r,'* ' a p s e u d o p o te n tia l form alism was u sed to
c a l c u l a t e t h e p h o to io n iz a tio n c r o s s s e c tio n o f sodium.
The p h o to ­
i o n i z a t i o n c ro s s s e c t io n r e s u l t s u s in g th e d ip o le le n g th form o f th e
m a trix w ere i n v e ry good agreem ent w ith ex p erim en t.
The o b je c t o f
t h i s work i s to e x te n d th e work on sodium to in c lu d e p o l a r i z a t i o n
e f f e c t s and to a p p ly t h e p s e u d o p o te n tia l method t o o b ta in p h o to i o n i­
z a tio n c r o s s s e c tio n s o f lith iu m and p o ta ssiu m .
The p s e u d o p o te n tia l method
2
h a s b een used e x te n s iv e ly i n s o li d
s t a t e p h y s ic s and h a s o n ly r e c e n t l y b een a p p lie d to ato m ic c o l l i s i o n
p ro cesses.
3 "6
The g e n e r a l co n cep t o f th e p s e u d o p o te n tia l m ethod i s
t h a t a v a le n c e e l e c t r o n i n an atom o r a s o lid sees a weak n e t e f f e c t i v e
p o te n tia l.
I n s id e th e co re o f th e atom th e n u c le a r p o t e n t i a l a c t in g
on a v a le n c e e l e c t r o n i s v e ry s tr o n g and a t t r a c t i v e .
A lso i n t h i s
r e g io n th e P a u li p r i n c i p l e r e q u ir e s t h a t th e v ale n c e wave f u n c tio n be
o rth o g o n a l t o th e o r b i t a l s o f th e c o re e le c tr o n s .
Thus th e v a le n c e
e l e c t r o n wave f u n c tio n o s c i l l a t e s r a p i d l y , co rresp o n d in g t o a h ig h
k i n e t i c e n e rg y .
The l a r g e n e g a tiv e p o t e n t i a l en erg y i n s i d e t h e co re
o f th e atom and th e l a r g e p o s i t i v e k i n e t i c energy which th e v a le n c e
2
3
e le c tr o n h a s t h e r e c a n c e l to g iv e a weak n e t e f f e c t i v e p o te n tia l.
Thun
a system o f ( Z - l ) co re e le c tr o n s and a v a le n c e e l e c t r o n can he a p p ro x i­
mated a s a o n e e l e c t r o n system w ith th e e f f e c t o f th e core e le c tr o n s
and ( Z - l) p r o to n s b e in g r e p la c e d by a weak (and u s u a ll y re p u ls iv e )
p o t e n t i a l c a l l e d th e " p s e u d o p o te n tia l" .
The p s e u d o p o te n tia l used i n
2
t h i s p a p e r i s n o t d ev elo p ed a lo n g th e l i n e o f P h i l i p s and Kleinman o r
7
A u s tin , H e in e , and Sham, b u t alo n g th e model p o t e n t i a l method o f
g
Abarenkov an d H eine . The o n ly req u irem en t on t h e p s e u d o p o te n tia l i s
t h a t i t g iv e s th e same en erg y e ig e n s ta te s a s th e r e a l problem .
The p s e u d o p o te n tia l form alism i s e s p e c i a l l y s u ite d to th e a l k a l i
atom s.
The v a le n c e e l e c t r o n and th e c lo s e d - s h e ll c o re a re known t o
i n t e r a c t r a t h e r w eakly an d th u s th e e f f e c t s o f t h e c o re upon th e v a le n c e
e le c tr o n may b e r e p r e s e n te d to a good ap p ro x im a tio n b y some e f f e c t iv e
c e n tr a l p o t e n t i a l .
The a l k a l i atoms a r e a ls o w e ll s u ite d f o r com pari­
son o f th e o r y t o e x p e rim e n t, due to th e e a se w ith w hich th e y may b e
o b ta in e d as a p p ro x im a te ly monatomic v ap o rs and t o th e co n v en ien tly low
i o n iz a tio n p o t e n t i a l .
The p l a n o f P a r t I i s as fo llo w s : S e c tio n 1 -2 p r e s e n ts th e p s e u ­
d o p o te n tia l ...method a s a p p lie d t o c a lc u la te p h o to io n iz a tio n cross s e c ­
t i o n s ; i n S e c tio n 1 -3 t h e method o f d eterm in in g t h e p s e u d o p o te n tia l i s
d is c u s s e d ; i n S e c tio n 1-1+ th e r e s u l t s a r e compared t o experim ental
r e s u l t s and o t h e r t h e o r e t i c a l w ork; th e c o n c lu s io n s a r e p resen ted i n
S e c tio n 1-5*
SECTION 1-2
THEORY
The i o n i z a t i o n o f a n ato m ic system by a n e x t e r n a l e l e c t r o ­
m ag n etic f i e l d i s r e a d i l y t r e a t e d b y p e r t u r b a t i o n t h e o r y u s in g a
s e m i - c l a s s i c a l m odel f o r t h e i n t e r a c t i o n betw een bound e le c tr o n s
and th e r a d i a t i o n f i e l d . ^
I n th e d ip o le a p p ro x im a tio n an e le c tr o n
e j e c t e d from t h e v a le n c e s - s t a t e o f an a l k a l i atom w i l l go in to a
continuum p - s t a t e .
U sin g a " s i n g le ch an n el" a p p ro x im a tio n and
assum ing t h a t t h e d i f f e r e n c e s i n t h e c o re wave f u n c tio n s f o r th e
atom and th e io n a r e n e g l i g i b l e , t h e d ip o le le n g th (L) and th e
d ip o le v e l o c i t y (V) fo rm s o f th e p h o to io n iz a tio n c r o s s s e c tio n
a re
ct(L5V) = | T raao2 ( I + k 2 )|M ^L,V^ |2
w here I i s th e f i r s t i o n i z a t i o n p o t e n t i a l , k
2
I.
i s t h e e n e rg y o f th e
e je c t e d e l e c t r o n ,
m (i) = J V
r>r W
r)a r
and
M(v ) .
J > s{ r)[l V r )
+ A
V p )]a r.
!.
h
2
—IQ 2
I n t h e above fo rm u la s — Traa^ = 8 .5 6 x 10
cm and th e rem aining
q u a n t i t i e s a r e ta k e n i n th e system o f ato m ic u n i t s
= 1, e
2
1
= —= 2
5
The wave f u n c tio n s 7
and X a r e n o rm a liz e d red u ced r a d i a l wave
ns
Kp
f u n c tio n s f o r th e v a le n c e e l e c t r o n .
The two a l t e r n a t i v e fo rm s,
and M^, a r e i d e n t i c a l when th e wave f u n c tio n s V ( r ) and \ ( r )
'
ns
Kp
a r e e x a c t e i g e n s ta te s o f t h e same H a m ilto n ia n .
One i n t e r e s t i n g f e a tu r e a b o u t v a le n c e e le c tr o n s o f a l k a l i
atom s i s th e e x te n t to w hich t h e i r o b se rv e d p r o p e r t i e s p a r a l l e l
th o s e t o be ex p e c te d from a p p a r e n tly cru d e m o d els.
i s th e p s e u d o p o te n tia l fo rm a lism .
One su ch model
I n in tr o d u c in g th e p seu d o p o te n ­
t i a l fo rm a lism , th e a c t u a l sy stem i s r e p la c e d b y a one e l e c tr o n
sy stem .
The e f f e c t o f th e c o re e l e c t r o n s i s r e p r e s e n te d b y th e
p s e u d o p o te n tia l
where we choose t h e model form
e
I.b
Vp = ^
The pseudo wave f u n c tio n s s a t i s f y t h e S c h ro d in g e r e q u a tio n :
fo r
t h e bound s - s t a t e
2
[-%■ + § " V + e ]$ ( r ) = 0
dr
p
1 .5
and f o r th e continuum p-wave
1.6
dr
r
The bound s t a t e pseudo wave f u n c tio n i s n o rm a liz e d so t h a t
1 .7
The continuum pseudo wave f u n c tio n h a s th e a s y m p to tic form
§ ^ ( r) ~ k 2 s i n [ k r - ^
^ l n ( 2k r ) + cr^ + 61 (k 2 ) ] .
2
61 (k ) i s th e non-coulom b p h ase s h i f t and
1 .8
i
= a r g T (2 + £•).
The pseudo wave f u n c tio n $ i s a sm oothly v a ry in g f u n c tio n
in s id e th e c o re .
Tlje a c t u a l wave f u n c tio n o f th e v a le n c e e l e c t r o n
m ust be o rth o g o n a l t o th e c o re s t a t e s .
Y (r) i s th e pseudo wave
f u n c tio n p r o p e r ly o rth o g o n a l! z e d t o th e c o re s t a t e wave f u n c tio n s
1 -9
F o r th e bound s t a t e , c o re s t a t e s o f th e atom a r e u s e d and f o r th e
continuum s t a t e co re s t a t e s o f th e io n sh o u ld be u s e d .
However f o r
t h e a l k a l i s th e d i f f e r e n c e s i n th e c o re s t a t e s o f th e atom and th e
s in g l y io n iz e d io n a r e n e g l i g i b l e .
A c c u ra te c o re s t a t e s have been
c a l c u l a t e d w ith in th e H a rtre e -F o c k ap p ro x im a tio n b y C lem en ti and
o t h e r s . T h e o r th o g o n a liz e d bound s t a t e wave f u n c tio n Yn s i s r e ­
n o rm a liz e d so t h a t
1 .10
The continuum wave f u n c tio n ^
Eq. ( 1 .8 ) .
i s a s y m p to tic a lly n o rm a liz e d b y
The pseudo d ip o le le n g th c ro s s s e c tio n m en tio n ed i n
t h i s paper i s
1 .11
7
w here t h e m a trix e lem en t i s c a lc u l a te d w ith th e n o n -o rth o g o n a liz e d
pseudo wave f u n c tio n s ,
= / V 1*) r \ ( r )d*.
1.12
SECTION 1-3
EVALUATION OF PSEUDO PARAMETERS
The p s e u d o p o te n tia l V i s d e fin e d by E q. ( i A ) . V alues f o r
P
th e d ip o le p o l a r i z a b i l i t y , o^ a r e ta k e n from th e b e s t e s tim a te s
a v a i l a b l e i n th e l i t e r a t u r e . ^
The v a lu e s o f th e a
and th e s c re e n q
in g c o n s ta n t, d , axe chosen so t h a t th e p s e u d o p o te n tia l re p ro d u c e s ,
a s c l o s e ly a s p o s s i b l e , t h e spectrum o f f - s t a t e i e v e l s f o r th e
v a le n c e e l e c t r o n .
12
The f - l e v e l s a r e u sed t o d eterm in e or ,d
q
b ecau se f o r a l a r g e A v a lu e t h e c e n t r i f u g a l te rm w i l l dom inate
o v er th e f i r s t te rm i n Eq. (I A ) and one can s e t
= 0.
The
r e s u l t i n g v a lu e s o f d a r e on th e o rd e r o f th e r a d iu s o f th e c o re .
The v a lu e s o f ^
a r e u s u a ll y s m a lle r th a n th e co rre sp o n d in g quad-
ru p o le p o l a r i z a b i l i t i e s .
The p a ra m e te rs
and 0^ a r e e v a lu a te d by
r e q u ir in g t h a t th e pseudo wave fu n c tio n s have th e same e n erg y spectrum
a s th e lo w er A s t a t e s o f th e v a le n c e e l e c t r o n o f th e a l k a l i atom .
E x p e rim e n ta l e n e r g ie s w ith s p i n - o r b i t s p l i t t i n g s u b tr a c te d o f f
13
a re
u sed to e v a lu a te Q and 0.
Two m ethods a r e u sed t o e v a lu a te Q and 0. The f i r s t m ethod,
5
su g g e ste d by C allaw ay and Laghos , r e q u ir e s t h a t Q and 0 be chosen
so t h a t th e e ig e n - e n e r g ie s o f th e f i r s t s - and p-pseudo wave fu n c tio n s
a g re e e x a c t l y w ith th e e x p e rim e n ta l e n e r g ie s o f th e n s - and n p - s ta te s
o f th e v a le n c e e l e c t r o n o f th e a l k a l i atom , where n = 2 ,3 , and U f o r
l ith iu m , sodium , and p o ta s s iu m , r e s p e c t i v e l y .
8
Thus i n th e f i r s t
9
method th e p a ra m e te rs Q and 0 a r e in d e p e n d e n t o f t h e o r b i t a l a n g u la r
momentum X.
A lth o u g h th e ^ -in d e p e n d e n t p s e u d o p o te n tia ls g iv e good
r e s u l t s f o r sodium , i n th e g e n e r a l c a se Q, and 0 sh o u ld be fu n c tio n s
o f Si.
The e f f e c t o f exchange can be c o n s id e re d a s g iv in g r i s e t o
a d i f f e r e n t p o t e n t i a l f o r s t a t e s o f d i f f e r e n t a n g u la r momentum.
The f a c t t h a t t h i s e f f e c t i s s m a ll i n sodium i s ev id en ced b y th e
su c c e ss o f P ro k o fje w
lU
i n a c c o u n tin g f o r a l l th e s p e c t r a l l e v e l s
o f sodium on t h e b a s i s o f a s in g le p o t e n t i a l .
F o r L i and K th e
e f f e c t s o f exchange can n o t be n e g le c te d .
B a rd s le y
12
h a s found v a lu e s o f Q, and 0 f o r n = 1 (Yukawa form )
f o r L i , Na, an d K w hich a re X -dependent.
He h a s a l s o d eterm in ed
th e s e w ith n = 0 (e x p o n e n tia l form ) f o r L i and Na.
F or a s t a t e
w ith H - 0 , th e pseudo p a ra m e te rs Q and 0 a r e chosen so t h a t th e
s
s
f i r s t two s - s t a t e e n erg y l e v e l s o f t h e pseudo system a g re e e x a c tly
w ith th e e n e r g ie s o f th e"g ro u n d and f i r s t e x c ite d s - s t a t e o f th e
v a le n c e e l e c t r o n .
S im ila r l y f o r a s t a t e w ith X = 1 ,
and 0^ a r e
chosen so a s t o rep ro d u c e th e c o r r e c t e n e r g ie s f o r t h e f i r s t and
second e x c i t e d p - s t a t e s o f th e v a le n c e e l e c t r o n .
F o r bound s t a t e s , th e o n ly re q u ire m e n t o f th e p o t e n t i a l i s
t h a t i t y i e l d c o r r e c t e n e rg ie s o v e r th e e n e rg y ra n g e i n which th e
p o t e n t i a l i s b e in g u s e d .
However, f o r t h e continuum s t a t e s , th e
re q u ire m e n t o f t h e p s e u d o - p o te n tia l i s t h a t t h e p o t e n t i a l y i e l d
Q
a c c u r a te p h a s e s h i f t s .
B a r d s le y 's v a lu e s o f
and 0^ le d t o non­
coulomb p-w ave p h a se s h i f t s which a r e shown i n T a b le I t o b e i n good
agreem ent t o t h e quantum d e f e c t v a lu e s .
SECTION 1-1+
RESULTS AND DISCUSSION
E q u a tio n s ( 1 .5 ) and ( 1 .6 ) f o r th e pseudo wave f u n c tio n s were
s o lv e d n u m e r ic a lly u s in g N um erov's m ethod.
A ll i n t e g r a l s were
e v a lu a te d n u m e ric a lly u s in g a f i v e - p o i n t fo rm u la .
The r e s u l t s o f
th e c a l c u l a t i o n s a r e p r e s e n te d i n F ig u re s 1 th ro u g h 7 where th e y
a r e com pared w ith o th e r c a l c u l a t i o n s and e x p e rim e n ta l d a t a ( th e
o r d in a te a x is i n th e s e f ig u r e s i s la b e le d i n term s o f th e e n erg y
2
o f t h e e j e c t e d e l e c t r o n i n u n i t s o f e l e c tr o n v o l t s ; e(eV) = 13*6 k ) .
A - SODIUM
I n an e a r l i e r p a p e r1 t h e a u th o r h a s a p p lie d a p s e u d o p o te n tia l
method f o r sodium and o b ta in e d good agreem ent betw een th e d ip o le
le n g th and ex p erim en t away from t h e lo w e st th r e s h o ld .
The p seu d o ­
p o t e n t i a l i n t h a t work was o f t h e form
Vp M
= |
e ' Pr
w here Q, = 20.1+3 and 3 = 2.01+75*
T h is p o t e n t i a l e x a c t l y re p ro d u c e s
th e e x p e rim e n ta l 3 s and 3p e n e rg y l e v e l s o f th e v a le n c e e l e c tr o n b u t
n e g le c ts p o la r iz a tio n e f f e c ts .
B e tte r agreem ent betw een th e o r y and
e x p erim e n t i s o b ta in e d by in c lu d in g p o l a r i z a t i o n te rm s i n th e pseudop o te n tia l.
Upon u s in g th e same p o l a r i z a t i o n te rm s a s i n Eq. (1.1+)
10
11
and s e t t i n g a\^ = 0 .9 4 5 a ^ , ar^ = 1.5 2 3 a^ and d = 1 . 1 , th e r e s u l t i n g
p o t e n t i a l e x a c tly re p ro d u c e s t h e e x p e rim e n ta l 3s and 3P en erg y
l e v e l s o f th e v a le n c e e l e c tr o n f o r Q, = 16.1*37 and P = 1 .7 7 4 4 8 .
The
p h o to io n iz a tio n c ro s s s e c tio n f o r th e p s e u d o p o te n tia l w ith p o l a r i ­
z a tio n in c lu d e d i s compared t o t h e e x p e rim e n ta l r e s u l t s o f Hudson
and C a r te r
17
i n F ig u re 1 .
L and V i n d i c a t e th e d ip o le le n g th and
v e l o c i t y r e s u l t s from th e f u l l p s e u d o p o te n tia l fo rm alism w h ile P
i s th e p s e u d o -d ip o le le n g th c ro s s s e c t io n d e fin e d by Eq. ( i . l l ) .
A more g e n e r a l fo rm alism i s t o make th e p s e u d o p o te n tia l
p a ra m e te rs Q and P X -dependent a s i n d ic a te d i n Eq. ( 1 . 4 ) .
When
t h i s i s d o n e, B a rd s le y o b ta in s th e fo llo w in g p a ra m e te rs f o r th e
Yukawa form ( n = l ) ; Q = 329. 792, P
S
Pp = 2 .4 0 3 .
ab o v e.
12
S
= 3*858 and 0
'jp
= 5 2 .7 3 6 ,
The v a lu e s o f or^, or^, an d d a r e t h e same a s s t a t e d
The r e s u l t i n g p h o to io n iz a tio n c ro s s s e c tio n s a r e compared
t o ex p erim en t i n F ig u re 2 .
A lth o u g h t h e ^-d ep en d en t p se u d o p o te n ­
t i a l c r o s s s e c tio n s do n o t seem to a g re e as w e ll w ith e x p e rim e n t,
th e r e s u l t s a r e s t i l l good when compared to o th e r t h e o r e t i c a l
re s u lts .
A s l i g h t im provem ent o f th e c ro s s s e c tio n can b e o b ta in e d b y
changing from th e Yukawa form t o an e x p o n e n tia l form (n = 0 ).
The
c o rre sp o n d in g p s e u d o p o te n tia l p a ra m e te rs a r e ; Q = 635*024, p = 4 .5 3 4
s
s
and Op = 79*542, P^ = 2 .9 2 2 .
The r e s u l t i n g d ip o le l e n g th p h o to io n iz a ­
t i o n c ro s s s e c t i o n s , la b e le d L ', a r e shown i n F ig u re 2 .
The p seu d o ­
p o t e n t i a l o f th e Yukawa form seems t o g iv e b e t t e r r e s u l t s n e a r th e
th r e s h o ld w hereas th e e x p o n e n tia l form te n d s to g iv e b e t t e r r e s u l t s
12
f o r h ig h e r p h o to n e n e r g i e s .
However th e d if f e r e n c e does n o t seem to
be s i g n i f i c a n t f o r sodium .
The t h e o r e t i c a l w orks o f S e a to n , B urgess and S e a to n , Cooper,
Boyd, S h eld o n , and McGuire a r e compared to ex p erim en t i n F ig u re 3«
S ea to n
18
com putes t h e c r o s s s e c t io n u s in g H a rtre e -F o c k wave f u n c tio n s .
19
Cooper ^ u s e s a H a rtre e -F o c k bound s t a t e and a l o c a l i z e d form o f
th e same H a rtre e -F o c k p o t e n t i a l a s found i n th e bound s t a t e c a lc u ­
l a t i o n s t o compute t h e continuum o r b i t a l .
Boyd
20
The c a l c u l a t i o n s o f
c o n s i s t o f a c e n t r a l H a r tr e e f i e l d a p p ro x im a tio n m o d ifie d
t o in c lu d e some c o r r e l a t i o n and p o l a r i z a t i o n e f f e c t s .
S eato n
B u rg ess and
u s e th e quantum d e f e c t m ethod t o o b ta in th e c ro s s s e c t io n s .
S heldon
21
a ls o u s e d t h e quantum d e f e c t m ethod b u t a d ju s te d th e
p a ra m e te rs t o o b t a i n ag reem ent w ith th e e x p e rim e n ta l c r o s s s e c tio n
a t th r e s h o l d .
McGuire
22
u s e s a c e n t r a l p o t e n t i a l fo rm alism w ith a
model p o t e n t i a l o f th e form V (r) = 2 z / r - A,, r < r ^ ; V( r ) = 2 / r ,
r > r ^ ; r^ = 2 ( z - l) /A ,
where z and A a r e chosen t o f i t o b se rv e d
te rm v a lu e s f o r t h e v a r io u s ato m s.
ap p ro x im a te d th e Herman and S k illm a n
s tra ig h t lin e s .
I n a l a t e r c a l c u l a t i o n , McGuire
2b
23
p o te n tia l by a s e rie s o f
The p a ra m e te rs w ere th e n a d ju s te d so t h a t th e model
e ig e n v a lu e s and th o s e o f Herman and S k illm a n w ere i n r e a s o n a b le
ag reem en t.
B - LITHIUM
The e f f e c t o f r e p la c in g th e co re b y a p s e u d o p o te n tia l u s u a lly
le a d s t o a r e p u l s i v e p o t e n t i a l .
T h is can be a t t r i b u t e d t o th e P a u li
13
e x c lu s io n p r i n c i p l e , w hich i n h i b i t s an e l e c tr o n from p e n e tr a tin g
i n t o a r e g io n a lr e a d y o c c u p ie d by e l e c t r o n s o f th e same symmetry.
F or li t h i u m an e l e c t r o n i n a p - s t a t e i s n o t ex clu d ed from th e co re
b y t h e P a u li p r i n c i p l e .
As a r e s u l t , th e p - s t a t e e n erg y l e v e l s f o r
l i th iu m a r e lo w er th a n th e hydrogen l e v e l s .
The p s e u d o p o te n tia l
f o r th e p - s t a t e s o f l i t h i u m m ust th u s b e a t t r a c t i v e and so a s in g le
p s e u d o p o te n tia l w i l l n o t g e n e r a te b o th t h e s - s t a t e s and p - s t a t e s .
B a rd s le y h a s a ls o o b ta in e d th e p a ra m e te r f o r th e ^ -d e p e n d e n t
p s e u d o p o te n tia l ( 1 .4 ) f o r li t h i u m .
12
The p o l a r i z a t i o n te rm s i n th e
p s e u d o p o te n tia l a r e , aa = 0 .1 9 2 5 a ^ ,
= 0>11£, a 5 > ^
The rem a in in g p a ra m e te rs a r e a s fo llo w s :
d . 0 _?5>
f o r th e Yukawa form ( n = l) ,
Q = 53*524, 0 = 2 .8 9 6 and © = - 3 . 710, 0 = 2 . 676; f o r t h e expos
s
tp
p
n e n t i a l form (n=0 ) Q = 1 1 3 . 0 1 0, 0
S
S
= 3 »6l 6 and Q = - 1 0 . 3 0 ,
jp
0p = 3 . 569.
The p h o to io n iz a tio n c r o s s s e c tio n s f o r lith iu m o b ta in e d f o r
t h e Yukawa form and th e e x p o n e n tia l form o f th e p s e u d o p o te n tia l a re
t h e same to t h r e e s i g n i f i c a n t f i g u r e s .
Thus o n ly one p l o t o f c ro s s
s e c t io n v e rs u s p h o to n e n e rg y i s shown and t h i s i s g iv e n i n F ig . 4 i n
17
co m parison w ith th e e x p e rim e n ta l r e s u l t s o f Hudson and C a r te r .
The
r e s u l t s o f th e c a l c u l a t i o n s f o r li t h i u m a r e i n good agreem ent w ith
e x p erim e n t n e a r th e t h r e s h o l d .
However, away from t h r e s h o ld th e
c a l c u l a t i o n s f a l l o f f f a s t e r th a n e x p e rim e n t.
Some p re v io u s t h e o r e t i c a l r e s u l t s a r e g iv e n i n F ig u re 5S te w a rt
25
, S ew ell
26
, and Chang and McDowell
27
have each found th e
p h o to io n iz a tio n c ro s s s e c t io n u s in g wave f u n c tio n s c a lc u la te d w ith in
Ik
t h e H a rtre e -F o c k a p p ro x im a tio n .
S e w e ll's r e s u l t s a r e i n good a g r e e ­
m ent w ith e x p e rim e n t, b u t d is a g r e e w ith t h e r e s u l t s o f S te w a rt and
o f Chang and McDowell w hich a r e i n a g reem en t.
M atese and LaBahn
28
have a l s o done a H a rtre e -F o c k c a l c u l a t i o n , b u t d id n o t r e p o r t t h e i r
r e s u l t s w hich w ere f o r a l l p u rp o se s th e same a s Chang and M cD ow ell's
re s u lts .
Thus i t would seem t h a t t h e r e s u l t s o f S te w a rt g iv e n i n
F ig u re 5 a r e a good i n d i c a t i o n o f th e p h o to io n iz a tio n c ro s s s e c tio n
o f li t h i u m w ith in th e H a rtre e -F o c k a p p ro x im a tio n .
why t h e r e s u l t s o f S ew ell a r e d i f f e r e n t .
a r e t h e r e s u l t s Chang and McDowell
27
I t i s n o t known
A lso shown i n F ig u re 5
o b ta in e d b y u s in g th e B ru eck n er
G o ld sto n e many-body p e r t u r b a t i o n th e o r y , and th e r e s u l t s o f M atese
and LaBahn
28
who o b ta in e d t h e p h o to io n iz a tio n c ro s s s e c tio n o f
l i t h i u m by m ethod o f p o l a r iz e d o r b i t a l s .
C - POTASSIUM
A p s e u d o p o te n tia l w hich i s in d e p e n d e n t o f a n g u la r momentum X
can b e found f o r p o ta s s iu m .
The p a ra m e te rs a r e :
0 = .8 8 715, <*d = 5 .^ 0 a ^ , of = 1 7 .6
s?q9
Q, = 7 . 5656,
and d = 1 .5 .
The d ip o le
l e n g th form o f th e c r o s s s e c tio n a t t h r e s h o l d was 2 . 3U x 10
b u t t h e c ro s s s e c t io n in c r e a s e d m o n o to n ic a lly w ith e n e rg y .
-20
2
cm ,
T h is i s
29
s i m i l a r t o th e r e s u l t s B a te s
o b ta in e d f o r a d ip o le p o l a r i z a b i l i t y
■3
g r e a t e r th a n 11 a ^ . B a te s u s e d s o lu tio n s o f t h e H a rtre e -F o c k
e q u a tio n s so lv e d by H a r tr e e and H a r tr e e
30
f o r th e bound s t a t e s and
f o r t h e continuum s t a t e h e n e g le c te d exchange i n t h e H a rtre e -F o c k
e q u a tio n s b u t in c lu d e d a p o l a r i z a t i o n p o t e n t i a l o f th e form
15
V (r) = - P / ( r ^ + d ^ )^ .
He found t h a t t h e p o l a r i z a b i l i t y P had a
g r e a t I n f lu e n c e on th e b e h a v io r o f t h e c r o s s s e c tio n v e rs u s e n e rg y .
H is b e s t ag reem en t w ith ex p erim en t was o b ta in e d f o r P = 10.1+6 a ^ .
I n th e p r e s e n t work c a l c u la t io n s w ere made u s in g th e p s e u d o p o te n tia l
Q
method and v a ry in g th e d ip o le p o l a r i z a t i o n betw een z e ro and 10.1+6 a ^ .
But f o r t h e d ip o le le n g th form a minimum was n e v e r o b ta in e d u s in g
th e X -in d ep en d en t p s e u d o p o te n tia l.
I n t h e c a se o f t h e ^ -d ep en d en t p s e u d o p o te n tia l, th e e n erg y
dependence o f th e p h o to io n iz a tio n c r o s s s e c t io n c l o s e ly re sem b les
th e g e n e r a l shape o f th e e x p e rim e n ta l d a t a .
from B a r d s le y
12
c*d = 5.1+7 a 3 ,
The p a ra m e te rs o b ta in e d
f o r th e Yukawa form ( n = l) o f t h e p s e u d o p o te n tia l a r e :
= 12.
ajj, d = 1.1+, Qg = 1337.0721+, 0g = 3 . ^ 1+831+,
= 115.37211+ and 0^ = 1 . 877117.
The q u a d ru p o le p o l a r i z a b i l i t y was
chosen so a s t o rep ro d u ce th e c o r r e c t f - s t a t e e n e rg y l e v e l s o f
p o ta ss iu m .
I t i s somewhat s m a lle r i n m ag n itu d e th a n th e b e s t th e o ­
r e t i c a l c a l c u l a t i o n s w hich g iv e
01
1
=16.2.
The ^ -d e p e n d e n t p seu d o -
p o t e n t i a l r e s u l t s a r e compared t o t h e e x p e rim e n ta l r e s u l t s o f Hudson
and C a r te r
31
i n F ig u re 6 .
A lthough t h e cu rv e does have a minimum,
th e ag reem en t betw een th e o r y and e x p e rim e n t i s n o t a s good f o r
p o ta ss iu m a s i t i s f o r sodium an d l i t h i u m .
C a lc u la tio n s w ere a ls o made u s in g d i f f e r e n t v a lu e s f o r th e
p o la riz a b ility .
Somewhat b e t t e r ag reem en t betw een th e o r y and e x p e r iO
ment i s o b ta in e d b y in c r e a s in g t h e p o l a r i z a b i l i t y . F o r
= 10.1+6 a Q,
a
C[
= 0 . 0 , d = 1.1+, n = 1 , Q = 1 2 9 .1 0 1 , 0e = 2 .1 1 7 2 5 ,
S
S
T?
= 1+0.025,
and 0 = 1 . 3921+, th e a d ju s te d d ip o le l e n g t h cu rv e f a l l s ab o u t midway
Jtr
betw een t h e d ip o le le n g th curve and t h e e x p e rim e n ta l p o in ts i n F ig u re 6 .
16
However, th e p o s i t i o n o f t h e minimum does n o t seem t o be ex trem ely
s e n s i t i v e to changes i n th e p o l a r i z a t i o n p a r a m e te rs , c o n tr a r y to
th e r e s u l t s o b ta in e d b y B a te s .
29
The c r o s s s e c t i o n i s s e n s i t i v e ,
th o u g h , t o s m a ll changes i n th e en erg y sp ectru m u s e d t o compute
th e p s e u d o p o te n tia l ( c f . t h e b e s t f i t cu rv e L ' i n F ig u re 7 ) .
O th er t h e o r e t i c a l r e s u l t s f o r p o ta ss iu m a r e i l l u s t r a t e d i n
F ig u re 7*
The r e s u l t s o f S heldon
21
and M cGuire
22
a r e o b ta in e d
by a d j u s t i n g p a ra m e te rs t o g iv e a b e s t f i t t o e x p e rim e n ta l r e s u l t s .
A lso in c lu d e d i n F ig u re 7 i s a b e s t f i t c u rv e o b ta in e d u s in g th e
p s e u d o p o te n tia l m ethod.
th e p a ra m e te rs
in g p a ra m e te rs .
The b e s t f i t cu rv e was o b ta in e d b y v a ry in g
and 3^ and u s in g B a r d s le y 's v a lu e s f o r t h e re m a in ­
F o r a f ix e d Q^, 3p i s ch o sen so t h a t th e e n e rg y o f
th e f i r s t p s t a t e o f t h e pseudo system e q u a ls t h e Up e n e rg y l e v e l o f
p o ta s s iu m .
F o r eac h s e t o f v a lu e s
and 3pj t h e c r o s s s e c tio n i s
c a l c u l a t e d and com pared t o e x p e rim e n ta l r e s u l t s a t t h r e s h o l d .
F or
Qp = 3 0 . and 3p = 1.37117> th e c a lc u la te d r e s u l t i s a p p ro x im a te ly
e q u a l t o th e e x p e rim e n ta l r e s u l t o f 1 .2 x 10
and 3
-20
2
cm .
For t h i s Qp
th e e n e rg y o f t h e ’ second p - s t a t e o f t h e pseudo system i s
Xr
- . 09U2076 r y , com pared t o -.0938238 r y t h e e n e r g y ^ o f th e 5p l e v e l
o f p o ta ss iu m .
F o r e n e r g ie s above 2 eV t h e c r o s s s e c tio n o b ta in e d by
a d j u s t i n g t h e p s e u d o p o te n tia l a r e i n much b e t t e r agreem ent w ith
ex p erim en t th a n p r e v io u s t h e o r e t i c a l c a l c u l a t i o n s .
C a lc u la tio n s f o r t h e p h o to io n iz a tio n o f p o ta ss iu m a r e co m p li­
c a te d b y many f a c t o r s .
n e a r th r e s h o l d .
The minimum i n t h e c r o s s s e c tio n o c c u rs v e ry
The s p i n - o r b i t e f f e c t f o r p o ta s s iu m i s much l a r g e r
c
th a n f o r sodium .
A lso t h e (3p)
p
(4 s)
t r a n s i t i o n s i n p o tassiu m a r e
v e ry im p o rta n t i n th e a b s o rp tio n sp ectru m and p ro b a b ly e f f e c t s th e
p h o to io n iz a tio n c ro s s s e c tio n f o r th e e n e rg y ran g e d is c u s s e d i n
t h i s p a p er.
Thus i t i s n o t to o s u r p r i s i n g t h a t t h e r e s u l t s f o r
p o ta ssiu m do n o t seem t o be a s good a s th e r e s u l t s o f sodium and
li th iu m .
SECTION 1-5
CONCLUSION
The p h o to io n iz a tio n c r o s s s e c tio n s f o r th e a l k a l i s a r e verys e n s i t i v e to th e wave f u n c tio n s due t o t h e h ig h d eg ree o f c a n c e l l a ­
t i o n w hich o c c u rs i n th e m a trix e le m e n ts , Eq. ( 1 .5 - 1 . 6 ) .
s e n s i t i v i t y seems t o he g r e a t e s t i n p o ta ss iu m .
T h is
P re v io u s t h e o r e ­
t i c a l c a l c u l a t i o n s a r e i n good agreem ent w ith experim ent n e a r
th r e s h o ld h u t f o r e n e r g ie s ahove a few e l e c tr o n v o l t s th e c a lc u la t e d
c ro s s s e c tio n s d e c re a s e w ith in c r e a s in g e n erg y much f a s t e r th a n th e
e x p e rim e n ta l r e s u l t s would i n d i c a t e .
The p r e s e n t p s e u d o p o te n tia l
c a l c u l a t i o n s a ls o f a l l o f f f a s t e r th a n ex perim ent f o r h ig h e r
e n e r g ie s h u t much l e s s so th a n p r e v io u s t h e o r e t i c a l w orks.
Thus
i t i s fo u n d t h a t p s e u d o p o te n tia ls , e v a lu a te d from e x p e rim e n ta lly
d e te rm in e d e n e rg y s p e c t r a , y i e l d q u a l i t a t i v e l y good a p p ro x im a tio n s
f o r th e p h o to io n iz a tio n c r o s s s e c t i o n s .
A com parison o f th e p s e u d o p o te n tia l method and th e quantum
d e f e c t method
shows many s i m i l a r i t i e s .
I n h o th m ethods, model
p a ra m e te rs a r e chosen t o f i t s e le c t e d e x p e rim e n ta l in fo rm a tio n and
t h e r e s u l t s u s e d t o p r e d i c t a d d i t i o n a l phenomena.
The quantum d e f e c t
method h a s th e a d v a n ta g e s o f h e in g more a n a l y t i c a l w ith e s s e n t i a l l y
a l l th e a n a l y t i c a l a n a l y s is a lr e a d y done h y S eato n and co w o rk ers.
The p s e u d o p o te n tia l m ethod e n t a i l s s p e c i f i c c a lc u la tio n s f o r each
system h e in g s tu d ie d .
However t h e p s e u d o p o te n tia l method m ig h t h e
18
e x p ec te d t o g iv e " b e tte r r e s u l t s th a n th e quantum d e f e c t m ethod,
s in c e th e quantum d e f e c t m ethod i s e q u iv a le n t to s e t t i n g t h e p seu d o p o t e n t i a l e q u a l t o z e ro an d t r u n c a t in g th e wave f u n c tio n s a t some
f i n i t e r a d iu s t o a v o id t h e s i n g u l a r i t y a t th e o r i g i n .
I n r e g a r d t o p r e d i c t i n g p h o to io n iz a tio n c r o s s s e c t i o n s , a
r e l a t i v e l y cru d e m odel p s e u d o p o te n tia l y i e l d s s u p e r io r r e s u l t s o v er
th e quantum d e f e c t m ethod f o r a t l e a s t sim p le h y d r o g e n -lik e system s
such as th e a l k a l i m e ta ls .
T a b le I .
C a lc u la te d non-coulom b and quantum d e f e c t p-w ave p h ase
s h i f t s i n r a d ia n s
E lem ent
Na
Li
K
K2
Exp.
Ttp. (k 2 )
Quantum D e fe c t
^ ( k 2)
Yukawa
0 .0
2 .6 8 k
2.6 8 5
2 .6 8 7 a
0 .1
2 .6 5 0
2.6 5 0
2.6 5 6
0 .2
2 .6 1 8
2.6 1 9
2.632
0 .0
0 . 11+82
0 . 11+81
O.H+88b
0 .1
0 . 151+9
0.151+7
0 .1 5 6 8
0 .2
0 . 1611+
0.1611
0.1652
0 .0
2 .2 3 5
2 . 23!+°
0 .1
2 . 161+
2 .1 5 2
0 .2
2 .1 0 0
2.0 7 0
R e f e r e n c e 15
^ C a lc u la te d from th e t a b l e s i n R e f. 13
cR e fe re n c e 16
21
24
SODIUM
22
20
o Hudson 8 Carter
(Experiment) -
F ig u re 1 F h o to io n iz a tio n c ro s s s e c tio n s o f sodium , u s in g A -independent
p s e u d o p o te n tia l and in c lu d in g p o l a r i z a t i o n e f f e c t s . L and V i n d i c a t e th e
le n g th and v e l o c i t y r e s u l t s from t h e f u l l p s e u d o p o te n tia l fo rm alism
w h ile P i s th e p se u d o d ip o le le n g th c ro s s s e c tio n d e f in e d b y Eq. ( I . 1 1 ).
E x p e rim e n ta l r e s u l t s o f Hudson and C a r te r (R e f. 17) a r e g iv e n b y th e
c irc le s .
22
24
SODIUM
Hudson & Carter
22
20
OJ
E
o
o
CM
O
X
b
2.
F ig u re 2 F h o to io n iz a tio n c ro ss s e c tio n o f sodium u s in g th e 4 -d ep en d en t
p s e u d o p o te n tia l o f B a rd s le y . L an d V i n d i c a t e th e le n g th and v e l o c i t y
forms o f t h e m a trix ele m e n ts f o r t h e Yukawa form o f p s e u d o p o te n tia l.
The d a sh e d cu rv e L ' i n d i c a t e s th e le n g th form o f th e m a trix elem en t f o r
th e e x p o n e n tia l form o f th e p s e u d o p o te n tia l. E x p e rim en tal r e s u l t s o f
Hudson a n d C a r te r (R e f. 17) a r e g iv e n by t h e c i r c l e s .
I
h
I
I
I
I
i
i
i
i
i
i
r
SODIUM
o Hudson 8 Carter
[Burgess a Seaton °_° 0 °
S
' ' x
'
CM 16
\C o o p e r
s\
i
Sheldon
i
f
Seaton
Burgess 8 Seaton
1
2
1,
4
I
I
6
I
I
8
10
e(eV)
F ig u re 3 The c i r c l e s i n d i c a t e t h e e x p e rim e n ta l r e s u l t s o f Hudson
and C a r te r (R e f. 17) • The o th e r c u rv e s a r e th e t h e o r e t i c a l
c a lc u la tio n s o f S e a to n (R ef. 1 8 ) , B u rg ess and S eato n (R e f. 1 6 ),
Cooper (R ef. 19)> Boyd (R ef. 2 0 ) , an d S heldon (R e f. 2 1 ) . M -l
i n d i c a t e s th e c u rv e f o r McGuire (R e f. 22) w ith A=l8 .6 and M-2
i n d i c a t e s th e l a t e r r e s u l t s o f McGuire (R ef. 23) u s in g th e HermanS k illm an p o t e n t i a l .
2k
LITHIUM
180
—o
160
o Hudson 8 Carter
100
80
0
2
4
6
8
10
12
14
16
18
e(eV)
F ig u re k F h o to io n iz a tio n c r o s s s e c tio n s o f li t h i u m , u s in g th e Yukawa
form o f B a r d s le y 's p s e u d o p o te n tia l. L and V i n d i c a t e th e le n g th and
v e l o c i t y forms o f th e m a tr ix e le m e n ts . E x p e rim e n ta l r e s u l t s o f Hudson
and C a r te r (R ef. 17) a r e g iv e n by th e c i r c l e s .
T"
I T I
\
° /^ \
180 l - o/
/
J
160
E
•
140
o
o
X
r
I1 "T
V m'
C M
\
^
i
o
\
\
\
■
\
1
/
\
° \
°
1
\
\
*
a
rl
T
I
CM
I 'I
LITHIUM
.
'
/Stew art
O
I
O \
I
CM
|
I
° \
^
i •
I
\
n
«i-
I
'
\
\0
\
o
120
A
°
V
°
\
\\
*
100
°
\
\\
\
°
o
Hudson 8 C arter
\
\Sewell
* ML-V
° ML-L
o
o
80
i
0
2
l
4
l
I
6
i
l
8
I
10
12
14
16
18
e(eV)
F ig u re 5 O th er c a l c u l a t i o n s o f p h o to io n iz a tio n c r o s s s e c t io n o f
l i t h i u m . The d ash ed c u rv e i s t h e le n g th form o f th e H a r tr e e Fock
r e s u l t o f S te w a rt (R e f. 2 5 ) . The s h o r t d ash es r e p r e s e n t t h e le n g th
form o f th e H a r tr e e Fock r e s u l t o f S ew ell (R e f. 2 6 ). The v e r t i c a l
b a r s m arked CM i n d i c a t e th e e x te n t o f th e le n g th and v e l o c i t y c a l ­
c u la tio n s o f Chang and McDowell (R ef. 27) > u s in g many body p e r t u r b a ­
t i o n th e o r y . ML-L and ML-V i n d i c a t e th e le n g th and v e l o c i t y r e s u l t s
o f M atese and LaBahn (R e f. 2 8 ) , u s in g th e m ethod o f p o la r iz e d o r b i ­
t a l s . E x p e rim e n ta l r e s u l t s o f Hudson and C a r te r (R e f. 17) a r e g iv e n
by c ir c l e s .
26
POTASSIUM
80
E
o
Hudson 8 Carter
cvi
O
20
o //
0
2
4
6
8
e(eV)
F ig u re 6 F h o to io n iz a tio n c r o s s s e c t io n o f p o ta ss iu m , u s in g th e p seu d o ­
p o t e n t i a l o f B a rd s le y . L and V i n d i c a t e th e le n g th and v e l o c i t y forms
o f t h e m a trix e le m e n t. The c i r c l e s i n d i c a t e th e e x p e rim e n ta l r e s u l t s o f
Hudson and C a r te r (R e f. 31) u s in g t h e v ap o r p r e s s u r e d a ta o f Nesmeyanov.
27
POTASSIUM
80
CM 60
£
o
o
Hudson ft Carter “
CM
o
b 40
' u
20
Sheldon
McGuire
0
2
4
6
8
10
12
e(eV)
F ig u re 7 F h o to io n iz a tio n c r o s s s e c tio n o f p o ta ssiu m . The s h o r t dash
cu rv e i n d i c a t e s th e a d ju s te d quantum d e f e c t r e s u l t s o f S heldon (R e f. 2 1 ).
The d a sh cu rv e i n d i c a t e s th e r e s u l t s o f McGuire (R ef. 22) f o r &=6.20.
L ' i s a b e s t f i t cu rv e o b ta in e d i n t h i s p a p e r by a d ju s tin g th e p seu d o ­
p o t e n t i a l p a r a m e te rs . The e x p e rim e n ta l r e s u l t s o f Hudson and C a r te r
(R e f. 31) a r e g iv e n b y t h e c i r c l e s .
PART II - PHOTODETACHMENT OF NEGATIVE HYDROGEN ION
28
SECTION II-1
INTRODUCTION
The n e g a tiv e hydrogen io n , H~ i s th e p r i n c i p l e so u rc e o f
o p a c ity i n th e s o l a r atm osphere.
T h is a s tr o p h y s ic a l im p o rtan ce
o f n e g a tiv e io n s was f i r s t n o te d b y W ild t
32
i n 1939*
W ild t p o in te d
o u t t h a t i n an atm o sp h ere c o n ta in in g b o th m e ta l and hydrogen atoms
th e i o n i z a t i o n o f t h e m e ta l atoms can s u p p ly th e e l e c t r o n s n e c e s s a r y
f o r th e fo rm a tio n o f n e g a tiv e hydrogen i o n s .
The e l e c t r o n a f f i n i t y o f h y d ro g en i s due t o in c o m p le te
s c re e n in g o f th e n u c le u s and to th e p o l a r i z a t i o n o f th e h y d ro g en ic
c o re .
A lth o u g h a n e u t r a l atom e x e r t s an a t t r a c t i v e f o r c e on an
e l e c t r o n , t h e r e a r e a l im i t e d number o f s t a t i o n a r y s t a t e s f o r
a tta c h e d e l e c t r o n s .
T h is i s due t o th e s h o r t ra n g e o f th e e f f e c t i v e
a t t r a c t i v e f i e l d i n w hich th e e l e c t r o n moves.
p le l i m i t s th e number o f a v a i l a b l e s t a t e s .
A lso t h e P a u li P r i n c i ­
F or th e n e g a tiv e hydrogen
io n t h e r e i s o n ly th e one bound s t a t e ; e x c it e d bound s t a t e s do n o t
e x is t.
Thus t h e ato m ic a b s o rp tio n c o e f f i c i e n t i s i d e n t i c a l t o th e
c r o s s - s e c t i o n f o r th e p hotodetachm ent o f an e l e c t r o n .
I n 19^0, u s in g th e pho to d etach m en t r e s u l t s o f M assey and B a te s ,
S tro m g ien
3I4.
was a b le to produce a th e o r y o f th e s o l a r atm osphere w ith
no p r i n c i p a l d i s c r e p a n c ie s .
However W ild t* s
35
i n v e s t i g a t i o n on th e
t h e o r e t i c a l r e l a t i o n betw een th e c o lo r -te m p e r a tu re and th e e f f e c t i v e te m p e ra tu re o f s t a r s d id have d is c r e p a n c ie s .
29
The maximum o f th e
33
p h o to d etach m en t cu rv e needed t o "be moved from 4000A tow ard l a r g e r
w a v e le n g th s.
By 1959 th e p h o to d etach m en t cu rv e was known p r e c i s e enough
f o r a s tro n o m ic a l p u r p o s e s .
The maximum o f t h e p h o to d etach m en t
curve was a p p ro x im a te ly 4 .5 x 10
-17
2
cm and l o c a te d n e a r 8200A.
The im provem ent i n t h e photodetachm ent r e s u l t s was m ain ly due to
C handrasekhar and h i s c o -w o rk ers.
C h andrasekhar was th e f i r s t
t o p o in t o u t t h e need f o r more v a r i a t i o n a l p a ra m e te rs i n th e
ex p an sio n o f t h e bound s t a t e wave f u n c tio n o f H .
He a ls o n o te d
t h a t th e d ip o le v e l o c i t y photodetachm ent r e s u l t s a r e more s ta b l e
t o d i f f e r e n t bound s t a t e s th a n th e le n g th r e s u l t s and th u s a r e
p ro b a b ly more a c c u r a t e .
The m ain m o tiv a tio n o f t h e work a f t e r
i 960 was n o t n e c e s s a r i l y t o o b ta in more a c c u r a te p h o to d etach m en t
r e s u l t s , b u t r a t h e r t o show th e c a p a b i l i t i e s o f o t h e r t h e o r ie s i n
p r e d i c t in g a c c u r a te p h otodetachm ent r e s u l t s .
T h is i s one re a s o n
t h a t th e p r e s e n t work was u n d e rta k e n .
N esb et
h a s fo rm u la te d a p ro c e d u re t h a t i s r e a d i l y a p p lie d
t o atom ic s c a t t e r i n g p ro b le m s.
S e i l e r , O b e ro i, and C a lla w a y ^ have
a p p lie d t h i s m ethod t o t h e slow c o l l i s i o n s o f e l e c t r o n s and p o s i ­
t r o n s w ith ato m ic h y d ro g en .
They c a lc u la t e d th e a p p r o p r ia te phase
s h i f t s and c r o s s s e c t i o n s .
U sing th e N esbet p ro c e d u re o r a lg e b r a ic
c lo s e c o u p lin g m ethod one can a ls o d e te rm in e th e wave f u n c tio n o f
t h e s c a t t e r e d e l e c t r o n i n an a n a ly tic form .
I f an a c c u r a te wave
f u n c tio n i s known many o th e r p r o p e r tie s o f th e s c a t t e r e d e le c tr o n
can be e a s i l y o b ta in e d .
Thus i t i s o f i n t e r e s t t o i n v e s t i g a t e th e
31
a c c u r a c y o f a wave f u n c tio n g e n e r a te d by t h e a l g e b r a i c c lo se
c o u p lin g m ethod.
U sing th e knowledge t h a t th e p h otodetachm ent
r e s u l t s a r e e q u a l i n th e l e n g th , v e l o c i t y , and a c c e l e r a t i o n d ip o le
a p p ro x im a tio n f o r e x a c t wave f u n c tio n s , one can u s e photodetachm ent
c a lc u la tio n s fo r H
t o d e te rm in e th e a c c u ra c y o f t h e wave fu n c tio n s
g e n e r a te d b y th e a l g e b r a ic c lo s e co u p lin g m ethod.
Once th e wave f u n c tio n s were found t o be a c c u r a t e , th e work
was e x te n d e d i n t o t h e e n e rg y r e g io n where i n e l a s t i c p ro c e s s e s a re
p o s s i b l e an d a m u ltic h a n n e l fo rm alism i s need ed .
n=2 l e v e l t h e r e e x i s t s a
re s o n a n c e .
J u s t above th e
I t s c o n tr ib u ti o n to th e
p h o to a b s o r p tio n c r o s s s e c tio n i s i n v e s t i g a t e d .
The p l a n o f P a r t I I i s a s fo llo w s :
i n S e c tio n I I -2 th e
a n a l y s i s f o r d e te rm in in g th e wave f u n c tio n s i s g iv e n ; i n S e c tio n I I -3
t h e m ethod o f d e te rm in in g pho to d etach m en t c r o s s - s e c t i o n s i s d is c u s s e d ;
i n S e c tio n II-U t h e p h otodetachm ent r e s u l t s i n t h e n o n -reso n an ce
r e g i o n a r e d is c u s s e d ; i n S e c tio n I I -5 th e p h o to d etach m en t r e s u l t s
f o r t h e ^P re so n a n c e a r e d is c u s s e d and t h e c o n c lu s io n s a r e p re s e n te d .
SECTION II-2
WAVE FUNCTIONS
A - Bound S t a t e
I n , t h i s s e c t i o n , th e wave f u n c tio n s w hich a r e needed f o r c a l ­
c u l a t i n g th e p h o to d etach m en t c r o s s s e c t io n o f H
a re
d is c u s s e d . The
p ro c e d u re f o r c a l c u l a t i n g th e bound s t a t e wave f u n c tio n o f H" i s
d is c u s s e d i n p a r t A and i n p a r t B t h e N esb et m ethod f o r o b ta in in g
th e s c a t t e r e d wave f u n c tio n i s d e s c r ib e d .
T h ree d i f f e r e n t bound s t a t e s a r e u s e d i n t h i s w ork.
The
a u th o r o b ta in e d two bound s t a t e s , one w hich in c lu d e s th e th r e e
h y d ro g en c h a n n e ls I s , 2 s , and 2p and one w hich in c lu d e s f o u r
c h a n n e ls I s , 2 s , 2 p , and 2p.
The 2p i s t h e pseudo s t a t e o f B urke,
G a lla h e r , and G e ltm a n ^ w hich was chosen t o re p ro d u c e t h e f u l l
p o l a r i z a b i l i t y o f th e g ro u n d s t a t e o f H.
R |= = - 0 .9 6 6 r ( l + '| r ) e r + .3^0 r e
I I.1
^
A s i x c h a n n el bound s t a t e o b ta in e d by M atese and O beroi
Uo
w hich
c o n s i s t s o f t h r e e hy d rogen s t a t e s I s , 2 s , and 2p and t h r e e pseudo
s t a t e s 3 s , 3P> and 3d was a l s o u s e d .
t o b e o f th e form
32
The p s e u d o - s ta te s were chosen
33
R3 s = Ns ( l+ a sr ) e " V
Rr— = N r ( l + a r ) e
3P
P
P
+ ^
-TLr
P + X0 R_
2p 2p
I I.2
p
®3d ' V
where th e
and
m u tu a lly o rth o n o rm a l.
<1+V > e
a r e such t h a t th e s i x b a s is f u n c tio n s axe
The p s e u d o - s ta te p a ra m e te rs w ere o b ta in e d ; * y
u s in g c o n v e n tio n a l v a r i a t i o n a l te c h n iq u e s t o m inim ize th e en erg y o f
H“ .
The tw o - e le c tr o n H a m ilto n ia n i s g iv e n b y
H ( l ,2 ) =
I I.3
-V22 - 2/ r i - 2/ r 2 +2/ r 12 .
A tom ic u n i t s a r e t o b e u sed c o n s i s t e n t l y th ro u g h o u t t h i s work;
th e e n e rg y b e in g m easured i n R ydbergs and $ = 1 , m =
and e2= t
The t r i a l wave f u n c tio n i s o f th e form
I 1” ( 1 ,2 ) = [1+P12] p
r ( l ) F r (2 )
hS
^ O
ls, ) .
II. k
H ere H and T axe c h an n el in d ic e s and d e s ig n a te s e t s o f quantum
numbers
r — (n p , A p,S pj^jL ,S
jtr)
I I.5
3b
w here:
1.
rip i s t h e p r i n c i p a l quantum number o f th e ato m ic s t a t e ;
2.
j£p>Sp, s p e c if y th e o r b i t a l a n g u la r momentum an d s p in o f
th e ato m ic s t a t e ;
3.
4 i s th e o r b i t a l a n g u la r momentum o f th e s c a t t e r e d
p a rtic le ;
U.
L, S , M^, Mg,
it
s p e c if y t h e t o t a l o r b i t a l and s p in a n g u la r
momentum o f th e two p a r t i c l e sy stem , t h e i r components on t h e a x is o f
q u a n t i z a t i o n , and t h e t o t a l p a r i t y r e s p e c tiv e ly .
T hese q u a n t i t i e s
a r e c o n se rv e d i n t h e c o l l i s i o n .
P12 i s t h e sp ace exchange o p e r a to r .
The f u n c tio n Y c o u p le s th e
a n g u la r momentum o f th e e l e c t r o n and th e atom
I I .6
The s in g le p a r t i c l e wave f u n c tio n F p ( r) i s expanded i n th e
form
Fr (r) = S t>rnf rn(r)
n
II.7
w here t h e f r _ a r e n o rm a liz e d S l a t e r o r b i t a l s
f
I I .8
The in d e x n d e s ig n a te s th e number o f b a s i s f u n c tio n ( S l a t e r o r b i t a l s )
c o n s id e re d .
CKr<100a .
The c o e f f i c i e n t s §n a r e chosen so a s t o span th e ran g e
35
The R i t z v a r i a t i o n a l method i s u s e d t o s o lv e d th e e ig e n v a lu e
problem
HB = ESB.
I I.9
H ere H i s t h e H a m ilto n ia n m a trix and S i s th e o v e r la p m a trix . B
r>J i s
a v e c to r o f t h e ele m e n ts b ^ .
The s m a lle s t e ig e n v a lu e o f th e m a tr ix
e q u a tio n i s t h e e n e rg y o f H~ and th e c o rre s p o n d in g e ig e n v e c to r i s th e
bound s t a t e wave f u n c t io n .
A l o c a l minimum f o r th e energy o f H- i s
o b ta in e d b y t r e a t i n g t h e s m a lle s t e ig e n v a lu e a s a n o n - lin e a r f u n c tio n
o f t h e p s e u d o - s t a t e p a ra m e te rs a ^ ,
U sing t e n b a s is fu n c tio n s f o r
each o f t h e co u p le d c h a n n e ls M atese and O beroi o b ta in e d -1 .0 5 3 7 Ryd.
f o r th e e n e rg y o f H .
The e x a c t v a lu e i s -1 .0 5 5 5 Ryd.
k2
36
B - Continuum S t a t e
The v a r i a t i o n m ethod p ro p o se d b y N esbet i s u se d to c a lc u la t e
s i n g l e t continuum p-w ave e l e c t r o n hydrogen wave f u n c tio n s .
t r i a l wave f u n c t io n i s ta k e n t o b e o f th e form
The
37
pi
N pi
T (1,2) = 2 Yr (1,2)
11.10
r=i 1
T' b e in g th e i n c i d e n t c h a n n e l, and T a f i n a l c h a n n e l.
i f ' (1 , 2) i s
*fcll
th e component o f t h e t o t a l wave f u n c tio n i n th e T
c h a n n el.
T h is
f u n c tio n may b e e x p re s s e d as
i f ’ (1,2) = x f (1,2) + ^ ^ ( 1 , 2 ) + qT jA ^ I ^ )
11.11
where Xp i s a n o rm a liz a b le f u n c tio n and AQp and A1 p a r e f u n c tio n s
h a v in g a s p e c i f i e d a s y m p to tic form .
T1
The f u n c tio n Xp rosy
e x p re s s e d a s
r ' „ _\
7 /,
j t
yL
(1,2) = r Jef
(1 ,2)0
“
b=l
1
11.12
D
w here b i s t h e b a s i s in d e x and
r
%
p
=
p
Mp
i f <2 )
r
Ila 3
fp c
The n o rm a liz a b le b a s i s f u n c tio n s 'IL (r) = [*
jl o
] 2r
p „
e” b .
r(2Je+3)
And above t h e th r e s h o l d f o r th e e x c i t a t i o n o f th e n=2 s t a t e s , two
a d d i t i o n a l b a s i s f u n c tio n s a r e in c lu d e d o f th e form
( l - e "^r )^+2 s i n ( k p r ) / r 2
II. i h
( l - e "^**)^*2 c o s ( k p r ) / r 2 .
The f u n c tio n Rp i s t h e h y d ro g en !c r a d i a l f u n c tio n .
I n t h i s work
t h r e e s t a t e s a r e in c lu d e d t h e I s , 2s , an d 2p s t a t e s o f hyd ro g en .
The o p e r a to r P^2 i s t h e two p a r t i c l e exchange o p e r a to r .
The a s y m p to tic f u n c tio n s may h e e x p re s s e d a s
1+P1?
Mr
Ai r ~ <-7§ ^ Rr^r l^ Si ^ r , r 2^ YL,J&p,J&
The fu n c tio n s SQ a n d
* “ 0 ,1 *
a r e p r o p o r ti o n a l to r~"Ls in ( k p r -
r e s p e c t i v e l y , a t l a r g e d is ta n c e .
SQ(r,r) = kp(L- e~
-Gtt/
2),
The s p e c i f i c form s u s e d were
^ (k p r)
I I .1 6
S1 ( r , r ) = k r ( l - e “ P r) 2‘e+1Nje(k p r)
1 1 .1 7
and
i n w hich
an d N^ a r e s p h e r ic a l B e s s e l and Neumann f u n c tio n s .
q u a n tity 3 i s an a r b i t r a r y p a ra m e te r.
in tro d u c e d so t h a t t h e f u n c tio n
o r ig in .
The f a c t o r ( l - e
w i l l behave a s r
••Si* 2^*1
)
is
c lo s e t o th e
F o r SQ t h e f a c t o r ( l - e ” ^r ) 2'^+'1' i s in c lu d e d s o le l y f o r
convenience i n c a l c u l a t i n g th e n e c e s s a r y i n t e g r a l s .
The
38
SHORT RANGE COEFFICIENTS - C ^ '
P
The wave f u n c tio n a s i t s ta n d s h as th r e e unknowns, a^p .
P
,
FP
TP
P
and C^ . However th e C^ can h e e x p re s s e d i n term s o f c^p and
r*
P
P
q^P . And e i t h e r QfQp o r c^p can h e d e f in e d t o h e a d e l t a f u n c tio n
6pp, le a v in g o n ly one unknown.
I n t h e Kohn V a r ia tio n a l Method
P
P
orQp = 6^ , , a n d i n th e I n v e rs e Kohn V a r i a ti o n a l Method a£p = 6p p , .
The t r i a l wave f u n c tio n vF
m ust s a t i s f y th e S c h r o d in g e r
e q u a tio n i n th e suhspace o f H i l h e r t space spanned hy th e h a s is
r
f u n c tio n s T]^. T h is c o n d itio n i s u s e d t o d e te rm in e th e s h o r t ran g e
TP
p
c o e f f i c i e n t s C^ i n term s o f th e a s y m p to tic c o e f f i c i e n t s Q^p . The
i n i t i a l c h a n n e l in d ex P w i l l he s u p p re s s e d i n th e rem ain d er o f
t h i s d is c u s s io n .
The demand t h a t th e t r i a l wave f u n c tio n Y s a t i s f y th e
S c h ro d in g e r
e q u a tio n i n th e su h sp ace spanned h y th e s h o r t ran g e
r
f u n c tio n s 7]^ le a d s t o th e system o f l i n e a r e q u a tio n s .
E < © ^ (l,2 )|H -E |x p (l,2 )+ Q ropAo p ( l , 2 ) + a Lp A1 p ( l ,2 ) > = 0 ,
I I.1 8
f o r a c h a n n e l in d e x v = 1 , . . . , N and h a s i s in d e x a = l , . . . . , n v .
An e q u iv a le n t e x p re s s io n f o r Eq. I I . 1 8 i s
£ E MP® c f = -E ( a MPJ + a. MP^ ) ,
. ah
v oq aS
jLq aC ’
s h
q
I I . 19
3l)
where
Mg =
(« P |H -s |e g ),
n .a o
MS = ( ^ l H-E|Ao q ) ,
I n th e re m a in in g d is c u s s io n p , q , and s a r e c h an n el i n d i c e s .
a and b a r e b a s i s i n d i c e s .
t h a t th e c o e f f i c i e n t s
And
The p ro b lem can be s im p lif ie d by n o tin g
can b e e x p re s s e d a s l i n e a r com bination o f
a 's ,
cf = E
(cv c f 1 + a. a ? ? )
Hd
q v oq bo
lq b l
where f o r p = 1 , . . . , N and a = l , . . . n
S £
1 1 .2 1
Jfcr
= -< ^ |H -E |A o q >
I I . 22
E E < # |H - E |S g > C ^ = -< # |H - E |A 1(1>.
These 2N inhom ogeneous l i n e a r e q u a tio n s o f Eq. (1 1 .2 2 ) may b e s o lv e d
by two m eth o d s.
One method i s an e ig e n v a lu e problem and th e o th e r
method c o n s i s t s o f m a trix in v e r s io n .
ko
I n th e e ig e n v a lu e m ethod one f i r s t o b ta in s t h e homogeneous
s o lu tio n .
The homogeneous e q u a tio n may b e e x p re s s e d a s
f jjj (“ S
- V X X *
" °-
1 1 -2 3
Eq. (1 1 .2 3 ) i s an e ig e n v a lu e problem o f d im en sio n En_ .
P =
The e ig e n f u n c tio n s ,
y i.2 )
- 2 £ ( 1 ,2 ) = E ! £ ( 1 ,2 )
1 1 . Zk
r
span t h e same su b sp ace o f H i l b e r t sp ace a s th e b a s i s f u n c tio n s T)^
T hus-E q. 1 1 .2 2 may a l s o b e e x p re s s e d a s ,
E E < £ |H - e | £ > £ | - - < ^ M | A o q >
1 1 .2 5
E E < £ | h - e | £ > £ J = -< ’^ |H -E |A l q >.
g
I t i s c o n v e n ie n t t o in tr o d u c e th e f u n c tio n s
4i q = £
,
°bi
an d t o e x p re ss them i n te rm s o f t h e e ig e n f u n c tio n s
11-26
T h is le a d s
t o t h e e x p re s s io n
| E < £ Ik-E | £ , >Kjt i . - < £ |h - E |Al q >
X I . 27
kl
f o r i = 0 ,1 where
n -28
S in c e t h e e ig e n f u n c tio n s ¥a a r e ta k e n t o h e o rth o n o rm al
Kc k “ ( E - E ^ E ^ I h - E ^
iq >
II> 2 9
and
‘i q = §
Mcd
1 = °>1 '
H -3 0
Sf ( 1 , 2 ) ] .
lq lq '
J
1 1 .3 1
V
The f u n c tio n Xg may "be e x p re s s e d a s
XV( 1 ,2 ) = 2[cvV $s ( 1 ,2 ) +
s' ’
q oq o q '
ol
av
Thus t h e s h o r t ra n g e c o e f f i c i e n t s
a r e e x p re s s e d i n term s o f t h e
\)
a s y m p to tic c o e f f i c i e n t s
an d th e e ig e n f u n c tio n s
and e ig e n e n e r g ie s Ea hy Eq. I I . 30 and E q. 11.31*
E q. 1 1 .2 1 can a l s o e a s i l y b e s o lv e d b y u s in g m a tr ix in v e r s io n
i n Eq. 1 1 .2 2 .
H ere
I Z -3S
<£? = - S
n il
ra '
'b a
Mr «
aC
1+2
ASYMPTOTIC COEFFICIENTS - or?
iq
The asy m p to tic c o e f f i c i e n t s a r e d e te rm in e d v a r i a t i o n a l l y .
The
v a r i a t i o n a l f u n c tio n a l i s ta k e n t o he
i CTV - <V '|h - e | t v>
=
1 1 < ^ ( i , 2 ) |h - e |
1 1 .3 3
»q( i , a ) > .
The t r i a l f u n c tio n was e x p re s s e d a s
f v( i , a ) = s ^ ( i , a )
I I . 3^
* > , 2 ) = X > , 2 ) + % SA0S(1»2) + “ L V
1 ’25-
F o r th e v a r i a t i o n a l c a l c u l a t i o n i t i s c o n v e n ie n t t o e x p re ss th e
t r i a l f u n c tio n Y w ith component i n ch an n el s a s
* > , 2) = f E
1 1 .3 5
where th e fu n c tio n s Y? a r e d e f in e d h y
iq
l j q ( l , 2 ) = * Jq ( l , 2 ) + V i q ( 1 ’ 2 ) -
1 1 .3 6
U3
Then th e v a r i a t i o n a l f u n c tio n a l can b e e x p re s s e d as
1 = 2
2 cF *MP? o£
i j pq iP
i d dq
1 1 .3 7
I
1 1 .3 8
or
crv
= 2 2 o? * IP ,
i p iP i v
w here
Xi v
Mi j “iq »
1 ,d = 0,15 P,<1 =
1 1 ,3 9
The m a tr ix elem en ts MPjjj- a r e d e f in e d b y
MPq = 2 2
|H - E 6
)
00
r s v op1 rs
r s 1 oq7
= 2 (S |H - E 6 1 ^ ) ,
s ' p 1 ps
p s 1 o q 77
MPq- = 2 2 { ' f jH - E 6 1 ^ )
ol
r s ' op' rs
r s 1 lq 7
= 2 (S |H - E 6 \ t ) ,
s v p ' ps
p s ' l q 7’
MPq = 2 2
|h - E 6
)
lo
r s ' l p 1 rs
r s 1 oq7
= 2 (C |H - E 6
),
s
P ps
p s 1 oq ’
MPq = 2 2
|H - E 6 l ^ 5 )
11
r s ' l p 1 rs
r s 1 xq7
= 2 (C |H - E 6 K ) .
s ' p 1 ps
p s 1 lq 7
II.U O
The m a tr ix elem en ts MPq can b e d e te rm in e d by e i t h e r o f two m ethods.
I f th e e ig e n v a lu e problem h a s b e e n s o lv e d th e n
And i f t h e e ig e n v a lu e p roblem h a s n o t b een s o lv e d , th e M fr a r e
e a s i l y d e te rm in e d u s in g th e fo llo w in g e q u a tio n s ,
MPq = m*§ - 2 s.
00
bb r s ab <
II
"2-
•s II
V
U sing t h e i d e n t i t y
4 bb
f t - r s2 ab
\
•e *
MPq - 2 2 M1"
CS r s ab Ca
"S -
Mpq - 2 2
CC r s ab
*2
Ca
where
k 2 = E-E
P
P
II
th e v a r i a t i o n a l f u n c tio n a l may b e e x p re s s e d a l t e r n a t i v e l y a s
I
aV
= E (ij
dq
I f th e c o e ffic ie n ts a
(J
Vdq
+
k ( 6,-Of0 * - 6, o? * )o % )
S d l oq.
do l q
jq
II
y
and or a r e in d e p e n d e n t, from E q. 1 1 .3 8
Si
~
a*
Sa
ip
II
xi v
and from E q . I I . ^5
Si (TV
_0
*
rr-= I j
d aV
dq
dCT
+
Thus th e f i r s t o r d e r v a r i a t i o n o f I
61
av
=SZ
ip
O *
k (6 a
q d l oq
<•
CT*\
- 6 ar
),
do l q
II
i s g iv e n i n g e n e r a l b y
6a°* i f + E E i j * 6c^
ip iv
i q d^
dq
II
_
+ E
q
, / 0 * c V
O' * «. V N
k (or
6a . - a
6a ) .
q ' oq
iq
lq
oq
U sing th e Kohn V a r i a t i o n a l M ethod th e R m a trix i s d e f in e d as
V
where
■ C ^ /k P )4 Ypq
n
U6
an d
c? = 6
oq
qp
q = 1 , . . . ,N
1 1 .5 1
where p i s th e i n i t i a l c h a n n e l in d e x .
F o r th e above d e f i n i t i o n o f th e R m a tr ix , th e f i r s t v a r i a t i o n
o f th e v a r i a t i o n a l f u n c tio n a l s im p l i f i e s c o n s id e ra b ly .
c i e n t s o! a r e r e a l and th e v a r i a t i o n s o f 0P v a n is h .
oq
A ll c o e ffi­
E q. I I . U 8
re d u c e s t o
61 k = E l j %, 6y
+ S I ? 6y%i +
ov
p I V 'o p
q
l o Vq
k 6yvw
o vo
I I . 52
The i n t e g r a l s I ^ v t h a t o c c u r i n Eq. 1 1 .5 2 can a l l s im u lta n e o u s ly
b e re d u c e d t o z e ro b y an a p p r o p r ia te c h o ic e o f c o e f f i c i e n t s
.
U sing E q s. I I . 1+3, I I . 5 0 , 1 1 .5 1
■ <
+ S '®
Thus th e c o e f f i c i e n t s Y^°
Yvq
s a t i s f y t h e s e t o f l i n e a r e q u a tio n s
2 M?? v l o ) = - M?V .
q
11 Vq
lo
I f 1^
H - 53
1 1 .5 5
= 0 f o r a l l p , q th e n E q. 1 1 .5 2 becomes
•tto v -
V w P
■ °-
I T -56
T h is g iv e s a p p ro x im a te ly s t a t i o n a r y v a lu e s o f th e c o e f f i c i e n t s
I I - 57
YVq - ^
- f 1 A q ) K o + ? M® Y^ 5 >•
T h is co m p letes th e c a l c u l a t i o n o f
v
= y
.
However t h e r e a r e
d i f f i c u l t i e s w ith s p u rio u s s i n g u l a r i t i e s f o r th e Kohn method i f
|m®J( e ) | sh o u ld have i s o l a t e d z e r o e s .
com puting e lem e n ts o f th e
r ”'L
These can "be a v o id e d by
m a tr ix f o r th e s e i s o l a t e d en erg y
re g io n s .
A m ethod, sim ilar* t o t h a t u s e d f o r th e R m a tr ix , can be u s e d
f o r com puting e le m en ts o f R~^ m a tr ix .
I n v e r s e Kohn b y N e s b e t.
t o second H u lth e n m ethod.
f o llo w s .
T h is m ethod i s c a l l e d th e
I n t h e s in g le - c h a n n e l problem i t re d u c e s
A b r i e f ’ e q u a tio n o u t l i n e o f th e m ethod
1 1 • 58
1(8
‘^ov + 1 V v o ) * 0
\
^
PVq “ ^
+
+ |
»S
S p u rio u s s i n g u l a r i t i e s a l s o o c c u r i n t h e I n v e r s e Kohn M ethod,
if
|M ^ (E ) | h a s i s o l a t e d z e r o e s .
However a p o in t o f s i n g u l a r i t y i n
t h e Kohn m ethod i s a lm o st n e v e r a s i n g u l a r i t y i n t h e I n v e r s e Kohn
m ethod an d v ic e v e r s a .
a v o id e d .
Thus th e s p u rio u s s i n g u l a r i t i e s can he
N e sb et s u g g e s ts u s in g t h e Kohn fo rm u la when th e r a t i o
o f d e te rm in a n ts
i s l e s s th a n u n i t y , and u s in g th e I n v e r s e Kohn fo rm u la when t h i s
r a t i o i s g r e a t e r th a n u n i t y .
T h is ty p e o f s p u rio u s s i n g u l a r i t y d id
n o t c r e a t e an y d i f f i c u l t i e s i n th e p r e s e n t work.
SECTION I I -3
MULTICHANNEL JHOTOBETA.CHMENT
H aving d e te rm in e d th e n eed ed wave f u n c tio n s i n t h e p re v io u s
s e c t i o n , i n t h i s s e c t io n we w i l l d e s c r ib e how th e s e wave f u n c tio n s
can b e u s e d to d e te rm in e t h e p h o to d etach m en t c r o s s s e c tio n .
B ecause
i n e l a s t i c p r o c e s s e s a r e to b e a llo w e d , a m u ltic h a n n e l fo rm alism i s
n eed ed .
I4.Q
H enry and L ip sk y J have d e s c r ib e d a th e o r y f o r m u ltic h a n n e l
p h o to io n iz a tio n w hich in c lu d e s c o u p lin g betw een f i n a l - s t a t e c h a n n e ls .
They a p p lie d t h e fo rm alism t o t h e p h o to io n iz a tio n o f neon.
M atese
and O beroi a l s o u se d t h i s fo rm alism i n com puting th e p hotodetachm ent
o f H~ below th e n=2 t h r e s h o ld .
The d e s c r i p t i o n w hich fo llo w s i s
t h a t fo u n d i n t h e p a p e r b y M atese and O b e ro i.
1+0
The t o t a l c ro s s s e c t i o n , w ith c o n tr ib u tio n s from a l l open
c h a n n e ls i s g iv e n by
= E CTr ^
r 1
1 1 .6 1
where
*r(3) ■f “%
4
and
Ap.p, —(l-iR)pp, -1 (l+iR)p,p -1
k9
-^ A
rr" 4 %
H ere j = 1 ,2 ,3 i n d i c a t e s th e le n g th , v e l o c i t y o r a c c e l e r a t i o n d ip o le
a p p ro x im a tio n .
1% P
and P
i n d i c a t e open f i n a l s t a t e c h a n n e ls .
R i s t h e r e a c ta n c e m a tr ix and i s d e te rm in e d b y Eq. ( I I . ^9) •
f i
= N 2
2
(
Yf Yi or=+l
+ h
lf,£
r i s a ch a n n el in d e x ,
h
2f,je+CT
1 1 .6 2
C<FYf l ° ( j ) lF Yi > < u Y K
Yf
jla
Yi
Yf
Yi
>
u i s e i t h e r a I s , 2s , 2p re d u c e d h y d ro g en ic
s t a t e o r a re d u c e d pseudo s t a t e wave f u n c tio n .
U sing th e n o ta tio n
o f S e c tio n I I - 2 , t h e red u ced f i n a l s t a t e s in g le p a r t i c l e wave
P
f u n c tio n Fp i s
Fr'“
+ “or (i-e 'Pr)2*+1J.e(V >
11.6 3
+ o [p ( l - e ' ^ J ^ N ^ C k p r ) ] .
51
N i s t h e bound s t a t e n o r m a liz a tio n , ^=\ ;f=^ 2 i, an^
o p e r a to r o j ^
^-P 0-1-0
i s g iv e n by
0=1
^
- <t[ 4 + £ (l+ ff)]/r
j= 2
I I . 6k
3=3
F o r e n e r g ie s below th e n=2 t h r e s h o ld , o n ly th e I s c h an n el
i s o p en .
I n t h i s c a s e Eq. ( I I . 6 l ) becomes
^ ) = | 2 f f ao 2 ( ^ ) 3 - 2 J
w here 6 i s th e
P p h ase s h i f t and
„(J_) Cos2 6
f i
i s th e I s c h a n n e l.
1 1 .6 5
SECTION n -k
PHOTODETACHMENT RESULTS AND CONCLUSIONS BELOW THE INELASTIC THRESHOLD
I n t h i s s e c t io n t h e p h o to d etach m en t c ro s s s e c tio n i s c a lc u la t e d
u s in g E q . ( I I . 6 5 ) an d t h e wave f u n c tio n s d e s c r ib e d i n S e c tio n I I - 2 .
The p h o to n e n e r g ie s c o n s id e re d i n t h i s s e c t io n (A>113l) a r e n o t l a r g e
enough t o p ro d u ce a n i n e l a s t i c c o l l i s i o n , th u s o n ly a s in g l e ch an n el
fo rm alism i s n eed ed .
The p u rp o se o f t h i s s e c t io n i s t o d e m o n stra te
t h a t t h e wave f u n c tio n s o b ta in e d u s in g t h e a l g e b r a i c c lo s e c o u p lin g
m ethod a r e s u f f i c i e n t l y a c c u r a te t o compute th e p h otodetachm ent c r o s s
s e c tio n o f H .
F i r s t th e p h o to d etach m en t c ro s s s e c tio n i s computed f o r th e
same continuum wave f u n c tio n and t h r e e d i f f e r e n t bound s t a t e s .
The
continuum s t a t e i s a t h r e e s t a t e ex p an sio n composed o f t h e I s , 2 s ,
2p h y d ro g en s t a t e s .
e a rlie r.
The t h r e e bound s t a t e s a r e th o s e d e s c rib e d
I n H g u r e 8 th e d ip o le v e l o c i t y p h o to d etach m en t r e s u l t s
a r e shown.
Curve 1 i n F ig u re 8 r e p r e s e n ts t h e r e s u l t s u s in g th e
s i x c h an n e l bound s t a t e o f M atese and O b e ro i.
I n F ig u re 9 t h i s same
cu rv e i s shown t o b e i n good agreem ent t o a s i m i l a r c a l c u l a t i o n by
I4L.
D oughty and F r a s e r .
D oughty and F r a s e r computed th e p h o to d e ta c h ­
ment c r o s s s e c t io n o f H" u s in g th e s e v e n ty p a ra m e te r S chw artz bound
s ta te .
They u s e d a t h r e e s t a t e c lo s e c o u p lin g c a lc u la tio n f o r th e
continuum s t a t e .
N ote i n F ig u re 8 t h a t t h e f o u r s t a t e ex p an sio n
which in c lu d e s th e 2p pseudo s t a t e y i e l d s r e s u l t s which a r e
52
53
s i g n i f i c a n t l y b e t t e r th a n t h e r e s u l t s o b ta in e d u s in g th e th r e e s t a t e
ex p a n sio n .
The p h o to d etach m en t r e s u l t s i n th e d ip o le le n g th a p p ro x i-
i
m atio n a r e shown i n F ig u re 10.
I n F ig u re 11 th e s i x s t a t e e x p an sio n o f M atese and O beroi i s
u se d t o compute b o th th e bound and th e continuum s t a t e .
A lso shown
i n F ig u re 11 a r e th e r e s u l t s o f D oughty, F r a s i e r , and McEachran.
They u s e d t h e S chw artz bound s t a t e o f H ".
They c a lc u la t e d th e
continuum s t a t e u s in g th e c lo s e c o u p lin g m ethod w ith th e f i r s t s i x
h y d ro g e n ic s t a t e s in c lu d e d i n th e e x p an sio n o f th e wave f u n c tio n .
F ig u re s 8 -11 i l l u s t r a t e th e a lr e a d y known f a c t s t h a t th e d ip o le
v e l o c i t y a p p ro x im a tio n y i e l d s th e b e s t p h o to d etach m en t c ro s s s e c tio n s ,
and t h a t th e p h o to d etach m en t r e s u l t s a r e n o t a s s e n s i t i v e to th e con­
tinuum s t a t e a s th e y a r e th e bound s t a t e .
F ig u re s 8-11 sh o u ld a ls o
e s t a b l i s h t h a t a c c u r a te wave f u n c tio n s can be o b ta in e d by u sin g th e .
a l g e b r a i c c lo s e c o u p lin g m ethod.
However t h e r e i s one d i f f i c u l t y
t h a t a r i s e s i n u s in g t h e a lg e b r a ic c lo s e co u p lin g m ethod.
If. th e
t o t a l e n e rg y o f th e sy stem i s v e ry c lo s e t o one o f th e e ig e n v a lu e s
o f E q. (1 1 .2 3 ) th e n th e wave f u n c tio n s e x h i b i t s p u rio u s reso n an ce
s tru c tu re ,
s p u rio u s i n t h e se n se t h a t t h i s e i g e n s t a te d o e s n 't e x i s t
i n th e r e a l o r a c t u a l p h y s ic a l problem an d th u s th e reso n an ce s tr u c t u r e
a l s o d o e s n 't e x i s t i n t h e r e a l p ro b lem .
T h e re fo re th e en erg y r e g io n
o f th e n o n - p h y s ic a l e ig e n v a lu e s sh o u ld be a v o id e d .
I f an energy
r e g io n c o n ta in in g a n o n -p h y s ic a l e ig e n v a lu e n eeds t o be in v e s tig a te d ,
th e e ig e n v a lu e sp ectru m i s e a s i l y s h i f t e d by changing th e e x p o n e n tia l
c o e f f i c i e n t s o f th e b a s i s s e t .
A d e s c rip tio n o f t h i s d if f ic u lty in
r e g a r d t o th e p h otodetachm ent o f H
an d O b ero i.
1*0
i s g iv e n i n th e p a p e r by Mate
SECTION I I -5
1P RESONANCE
I n th e e l e c t r o n hydrogen problem t h e r e e x i s t s a
j u s t above th e n=2 t h r e s h o ld .
reso n a n c e
B ecause th e n=2 c h a n n e ls a r e open, th e
p roblem o f d e te rm in in g t h e s i n g l e t p-wave f u n c tio n i s more d i f f i c u l t
i n t h i s e n e rg y r e g io n .
The s o u rc e o f th e d i f f i c u l t y i s th e c o u p lin g
o f th e d e g e n e ra te 2s and 2p s t a t e s o f ato m ic h y d ro g en , which g iv e s
r i s e t o a lo n g -ra n g e o f f - d ia g o n a l d ip o le p o t e n t i a l .
b6
E. R. S m ith , R. S . O b e ro i, and R .J.W . H enry have i n v e s ti g a te d
t h i s problem b y c o n s id e rin g a two ch an n el model problem which con­
t a i n e d an o f f d ia g o n a l d ip o le p o t e n t i a l .
They found t h a t i n th e
model p ro b lem , f o r d e g e n e ra te e n e r g ie s , t h e a s y m p to tic s o lu ti o n o f
t h e wave f u n c tio n co u ld n o t b e a c c u r a te ly ap p ro x im ated u s in g s p h e r ic a l
B e s s e l and Neumann f u n c tio n s .
fo r th is .
They su g g e ste d two means o f c o r r e c tin g
One method was t o in c lu d e e n e rg y -d e p e n d e n t s in u s o id a l term s
in th e b a s is s e t.
Two such a d d i t i o n a l te rm s w ere added i n t h e p r e s e n t
work (Eq. I I . 1 4 ).
The a d d itio n o f th e en erg y -d e p e n d e n t s in u s o id a l
te rm s means t h a t th e e ig e n v a lu e e q u a tio n (Eq. 1 1 .2 3 ) m ust be so lv e d
f o r e v e ry e n e rg y , w hereas b e f o r e i t needed t o be s o lv e d o n ly once.
S in c e th e a l t e r n a t e method o f m a trix in v e r s io n i s much q u ic k e r , one
would n a t u r a l l y want t o s o lv e th e problem u s in g t h a t p ro c e d u re .
However, one would l i k e t o know th e e ig e n v a lu e sp ectru m , a s m entioned
i n th e l a s t s e c t io n .
Knowing th e e ig e n v a lu e sp ectru m i s even more
55
im p o rta n t i n th e i n e l a s t i c w ork, b e c a u se t h e en erg y -d ep en d en t
s in u s o id a l term s seem t o cause a c l u s t e r i n g o f e ig e n v a lu e s a b o u t
th e en erg y b e in g c o n s id e re d .
The p ro b lem o f n o n -p h y s ic a l e ig e n e n e r g ie s does n o t seem t o
a r i s e i n c a l c u l a t i n g t h e A=1 c ro s s s e c tio n s o f th e l s - 2 s and l s - 2 p
tra n s itio n s .
T hese c r o s s s e c tio n s a r e g iv e n i n F ig u re 12 a lo n g w ith
lj.7
th e r e s u l t s o f T a y lo r an d B urke.
T a y lo r and Burke u se d th e c l o s e c o u p lin g m ethod f o r two e x p a n s io n s, one in c lu d in g t h r e e s t a t e s and
t h e o th e r t h r e e s t a t e s p lu s tw e n ty c o r r e l a t i o n te rm s .
A lth o u g h th e
maximums i n t h e c r o s s s e c tio n o c c u rre d a t t h e same e n e r g ie s , t h e
v a lu e s o f th e maximums a r e s l i g h t l y l e s s i n th e p r e s e n t work.
T h is
m ight be c au sed by n o t in c lu d in g enough s in u s o i d a l term s i n th e
b a s i s e x p a n sio n .
U8
The agreem ent betw een t h e p r e s e n t work and t h a t
o f T a y lo r and Burke seems t o i n d i c a t e t h a t t h e p r e s e n t work h a s b e e n
ex ten d ed above n=2 th r e s h o l d c o r r e c t l y .
Thus t h e wave f u n c tio n s a r e
u se d t o compute t h e c o n tr ib u tio n o f th e "4? re so n a n c e t o th e p h o to detachm ent c r o s s s e c t i o n .
The p h o to d etach m en t c ro s s s e c tio n f o r th e
reso n a n c e was
d e term in e d u s in g th e bound s t a t e o f M atese an d O b ero i.
The f i n a l
s t a t e was o b ta in e d u s in g t h e a lg e b r a ic c lo s e c o u p lin g method w ith
t h r e e s t a t e s ( l s - 2 s - 2 p ) in c lu d e d i n t h e e x p a n sio n .
S ix te e n b a s is
f u n c tio n s w ere u s e d i n th e e x p an sio n o f th e s in g le p a r t i c l e f u n c tio n
F ^ , f o u r te e n S l a t e r ty p e and two h arm o n ic.
The ph o to d etach m en t c r o s s
s e c tio n was com puted u s in g th r e e d i f f e r e n t s e t s o f s ix te e n b a s is
f u n c tio n s .
F ig u re 13 r e p r e s e n ts a b e s t f i t cu rv e f o r th e t h r e e s e t s
o f ph o to d etach m en t d a t a .
The d a ta p o in t s i n th e energy r e g io n
57
( .7 5 - • 76 R y d .) do n o t f i t a smooth c u rv e .
However t h e same smooth
curve g iv e s a good b e s t f i t cu rv e t o a l l t h r e e s e t s o f d a t a .
The
a u th o r f e e l s t h a t t h e g iv e n cu rv e i s a c c u r a te t o w ith in 10$.
A lso
hg
in c lu d e d i n F ig u re 13 a r e th e v e l o c i t y r e s u l t s o b ta in e d b y Macek
u s in g th e bound s t a t e 2 0 -p a ra m e te r H y lle r a a s - ty p e wave f u n c tio n f o r
H o f H a rt an d H e rz b e rg and a l s - 2 s - 2 p c lo s e c o u p lin g f i n a l s t a t e .
The d i f f i c u l t y i n o b ta in in g a sm ooth curve i s a t t r i b u t e d to two
re a s o n s .
One i s t h e d i f f i c u l t y w ith a n e ig e n v a lu e b e in g to o c lo s e
to th e t o t a l e n e rg y E .
o f f p a ra m e te r P.
The o th e r s o u rc e o f d i f f i c u l t y i s th e c u t
A good c u t o f f p a ra m e te r f o r c h a n n e l one i s P = k^
o r a p p ro x im a te ly = 0 .8 a~ ^.
W hereas a good c u t o f f p a ra m e te r f o r
channel fo u r i s P = kg o r .05 a Q^*
u sed f o r a l l c h a n n e ls .
The same c u t o f f p a ra m e te r was
B e ta was ch o sen to be th e a v e ra g e o f k^ and
kg o r P e q u a l a p p ro x im a te ly .^ 5 - .5 0 a ”1 .
T h is P i s as sm a ll a s
p o s s ib le i f one i s t o o b ta in r e a s o n a b le r e s u l t s f o r th e I s c h a n n e l.
However i t i s much to o l a r g e , t o a c c u r a te ly c u t o f f t h e s p h e r ic a l
Neumann f u n c tio n 7]g i n channel f o u r .
T h is i s i l l u s t r a t e d i n T a b le I I ,
which shows t h e c o n t r ib u t i o n t o th e p h otodetachm ent c r o s s s e c t io n o f
a c o l l i s i o n w here t h e h y d ro g e n ic e l e c t r o n i s l e f t i n th e e x c ite d 2p
s t a t e o f hy d ro g en an d t h e o th e r e l e c t r o n i s s c a t t e r e d w ith a n g u la r
momentum j6=2 .
I n summary, a c c u r a te wave f u n c tio n s can be o b ta in e d u s in g t h e
A lg e b ra ic C lo se C o u p lin g M ethod.
However th e ap p e a ra n c e o f s p u rio u s
OQ
s i n g u l a r i t i e s a s n o te d by M atese and O beroi
i n t h e i r p a p e r seems t o
be even a l a r g e r s o u rc e o f d i f f i c u l t y f o r e n e r g ie s above th e n=2
58
th r e s h o ld .
A lso a s in g le c u t - o f f p a ra m e te r f o r a l l open ch an n els
sh o u ld n o t b e u sed f o r e n e r g ie s v e r y n e a r th e th r e s h o ld .
Work i s p r e s e n t l y underw ay to make th e s e two changes in th e
program :
t o make th e c u t o f f p a ra m e te r e q u a l to t h e wave v e c to r o f
th e c h a n n e l an d t o in c lu d e two a d d i t i o n a l harm onic t e r n s i n th e b a s is
s e t.
When t h i s i s com pleted th e work w i l l be e x ten d ed t o s ix
c h a n n e ls .
TABLE II
PHOTODETACHMENT CROSS SECTIONS: FINAL STATE CHANNEL FOUR
(n^=2 , / = 1 , X=2 )
CVI
LENGTH a x 1 0 ~^cm ^
BASIS
A
B
0 .7 5 1
0 .0 5
0 .0 9
0 .7 5 2
0 .7 9
0 .2 9
5.55
0 .7 5 3
0 .3 ^
0 .3 2
0 .3 8
0 .7 5 ^
0 .6 5
0 .1 1
0 .5 2
0 .7 5 5
0 .6 0
0 .9 0
0 .6 6
0 .7 5 6
0 .7 7
0.7^-
1 .2 2
0 .7 5 7
0 .3 9
0.1*7
0 .3 7
0 .7 5 8
0 .3 7
o.M*
0 .3 6
.0 .7 5 9
0 .3 3
0 .3 7
3 .2 0
c
B a s is A: §. = [ .0 0 5 , .0 1 5 , . 0 3 , .0 7 5> «1> *3* »6 ,
.8, 1 .1 , 1 .3 , 1 . 6 , 2 . , 3 . , 5.]
B a s is B: §. = [ .0 0 5 , .0 1 , .OU, .0 7 , .2 , .4 , . 6 ,
. 8 , 1 . 2 , l.k, 1 . 8 , 2 . 5 , 3 . 5, ^. 5]
B a s is C: §. = [ .0 0 5 , .0 1 5 , «03> »075> *1> *3> »^5>
. 6 , . 8 , 1 .3 , 1 . 6 , 2 . , 3 . , 5 .]
DIPOLE VELOCITY
6
BOUND STATE
ls -2 s-2 p _
ls -2 s -2 p -2 p
ls - 2 s - 2p-s-p-3 "
CONTINUUM STATE
Is - 2 s —2p
5
4
3
2
0
2000
4000
6000
8000
10000
12000
14000
16000
18000 20000
X (A)
F ig u re 8 F hotodetachm ent .c ro ss s e c tio n s i n th e d ip o le v e l o c i t y
a p p ro x im a tio n f o r H u s in g an a lg e b r a ic c lo s e co u p lin g t h r e e s t a t e
continuum wave f u n c tio n and t h r e e d i f f e r e n t hound s t a t e s .
ON
o
DIPOLE VELOCITY
CONTINUUMSTATE
ls-2s-2p
6
BOUND STATE
Is - 2s - 2p- 5p
Is-2s-2p-5-p-J
o o SCHWARTZ (70)
5
4
3
2
2000
4000
6000
8000
10000
12000 14000
16000
18000
X (A°)
F ig u re 9 F hotodetachm ent c ro s s s e c tio n s o f H i n th e d ip o le v e l o c ity
ap p ro x im a tio n . The dashed curve i s th e r e s u l t u s in g a f o u r channel
"bound s t a t e . The s o l i d curve i s th e r e s u l t u s in g th e s i x channel
"bound s t a t e . The c i r c l e s a r e th e r e s u l t o f Doughty and F r a s e r u s in g
th e Schw artz "bound s t a t e .
DIPOLE LENGTH
CONTINUUM STATE
Is -2s-2p
6
BOUND STATE _
Is-2s-2p- 2p
Is-2s-2p-s-p-3
° ° SCHWARTZ (70)
cm
4
(10
5
3
2
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
X(A°)
F ig u re 10 Photodetachm ent c ro s s s e c tio n s o f H i n th e d ip o le le n g th
a p p ro x im a tio n . The dashed curve i s th e r e s u l t u s in g a f o u r channel
hound s t a t e . The s o l i d curve i s th e r e s u l t u s in g th e s ix channel
hound s t a t e . The c i r c l e s a r e th e r e s u l t s o f Doughty and F r a s e r u s in g
th e Schw artz hound s t a t e .
ON
ro
DiPOLE LENGTH AND VELOCITY
CONTINUUM STATE
BOUND STATE
Is- 2s-2p-*-p-<J
Is-2 s-2p-s-p-(J
Is - 2*-2p-3s-3p-3d
SCHWARTZ (70)
“ 3
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
X(A°)
F ig u re 11 S ix ch an n el photodetachm ent c ro s s s e c tio n s o f H~ i n th e d ip o le
le n g th and v e l o c i t y ap p ro x im a tio n . The s o l i d cu rv es la b e le d L and V a r e
th e le n g th and v e l o c i t y r e s u l t s u s in g a s ix ch an n el hound s t a t e and a s ix
ch an n el continuum s t a t e . The dashed cu rv es la b e le d L and V a r e t h e le n g th
and v e l o c i t y r e s u l t s o f D oughty, F r a s e r , and M cEachran. They u se d a d o s e
co u p lin g continuum wave f u n c tio n w ith s i x hydrogen s t a t e s in c lu d e d , and
th e y u se d th e Schw artz bound s t a t e .
6H
0.6
Is, 2 s , 2 p CLOSE COUPLING
Is, 2 s , 2 p + 20 CORRELATION TERMS
• • • • I s , 2 s , 2 p ALGEBRAIC CLOSE COUPLING
0.5
0.4
0.3
o
a
0.2
s - 2p)
X
0.75
0.755
0.76
0.765
ELECTRON ENERGY IN RYDBERGS
Figure 12 Partial P Cross Sections of the ls-2s and ls-2p Transition
above the n=2 Threshold: the dots are this work and the dashed and
solid curves are the results of Taylor and Burke.
CVJ
I
I
I
I
I
I
3 3.0
CO
s
111
CO
CO
CO
o 2.0
oc
o
o
im
£C
o
co
m
1.0
<
g
o
X
Q.
.70
.72
.74 .75 .76
.78
.80
k 2 ( Ryd)
F ig u re 13 The photodetachm ent c ro s s s e c tio n o f H i n th e energ y r e g io n
o f th e l p re so n a n c e . The s o l i d cu rv e i s th e b e s t f i t curve o f t h i s
w ork. The dashed curve i s th e r e s u l t o f Macek. Macek u se d a th r e e
s t a t e c lo s e co u p lin g continuum s t a t e and th e tw en ty p a ra m e te r hound
s t a t e wave f u n c tio n o f H a rt and H erzb erg .
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69
VITA
R ich a rd Lee S m ith , son o f B roadus P. and L ouise S m ith , was
h o rn i n H ouston, Texas on O ctober 6 , 19^3•
H ouston H igh S chool i n P asad en a, T ex as.
He a tte n d e d South
Upon g ra d u a tio n he e n r o lle d
a t B ay lo r U n iv e r s ity and re c e iv e d h i s B .S . from B ay lo r U n iv e r s ity
i n th e S p rin g o f 1966 where he r e c e iv e d a d o u b le m ajor i n math and
p h y s ic s .
An a d d i t i o n a l y e a r o f g ra d u a te s tu d y i n p h y s ic s was s p e n t
a t B a y lo r.
On May 2 7 , 19^7 h e m a rrie d E liz a b e th Ann V a r n e ll.
In
th e F a l l o f 1967, he e n te r e d th e g ra d u a te sch o o l o f L o u is ia n a S t a t e
U n iv e r s ity .
On May 7 , E liz a b e th and R ich ard were b le s s e d w ith a
baby g i r l , Carey Ann.
R ich ard Lee Sm ith i s p r e s e n t l y a c a n d id a te
f o r th e d eg ree o f D o cto r o f P h ilo so p h y i n th e D epartm ent o f P h y sic s
and Astronomy a t th e 1971 Summer Commencement.