Calculations of photo-ionization cross sections for diatomic molecules

b-l-6

â

.q.4Js_u!4[QNÐ_03-!_8.0åq-JoNrz-4rI0t{p-Ro$.ÞsEcrIqN$F03

!ruLg-¡IQtllculÐg.

H. C. ÎUCKV{EIL¡ B.Sc.

A thesis subnittecl in accorda¡rce with ^tho

recluirenents of the Degree of ^Master of ^Science'

by

,

Departrent of Mathematical Physicst

University of Ad.elaidet South Australia.

February, 1969. ^I

I give consent to this copy of my thesis, whon d.eposited. il the university Library, being available for loan and photocopying.

Date aaaaaaaaaaa Si-gned

SU}ÍMARÏ.

SI'ATEI,EM.

PREFACE.

CH¡.PTER ¹ 1.1

1.2

1.,

1,5

CI{APTER 2

2.1 2.2

'2.1

2.4

CHAPTER

'

5"1

7.2

7.7 1.4

coJ!ru

-IIIIIR0DUCTIOI{ . . . . ' ' ' ' ' ' ' Interactions of ^Photons with ^{Bound Eleetronic}

Systems . . . .. . ' ' ' ' ' ' ' '

Tirne Dependent Perturbation ^Theory.

Formulae for Photoionization ^Cross Sections '

Many Particle ^{tr'Iave Functions ancl the One} ElectronÂpproximation. . . ' '

(

1.4 ^Atoroic PhotoÍonization Cross Sections

a

a

i

Prevíous l{ork ^{on lufole cule} s c . c '

-

EtECTRoNTc srÀTES OF N2,

*r*,

^{02, 02+ -}

ElectronicTerurs . . . c . '

S.C.F. - ^L.C.4.0. - llolecular Orbitals

aa

aa

aa

a T2

a 74

9

B:

77

5'

24

a

Photoionization ^{Processes from} o, ⁽ Xtlt ⁾

ancl Nr( X'Zi ) ror ^{Fixed Nucrei a 44}

Expressions for ^{Bou¡d' State M.0.} lfave Functions 4B

in Prolate Spheroid-a1 Co-ordinates ' ' '

EIECTRONIC STATES OF THE CONTINUUI'I.

Discussion of ^{the Final State} l{oclel . . c ^.

The Schrorlinger Equation in Prolate

SpheroidalCo-orclinates . . . ' ' ' Solutions of the ârtgular Equations . . . . ^e TheRadialSolutioflg ¡ . . . t. . ' ' '

61 64 66

CHi.PTER 4 - EVAIUATION OF TI{E EI'ECTRONIC ^I'IATRIX ELF]]\MNTS.

4.1 Tfy

^(o"

.iV) _rnitiar

_statos

' ' '

^'

4.2 Tfn (ot lr* ) ^rnitiar

^states

' ' '

^'

4.t Úl fnitttf ^States . ' ' ' ' '

^'

4.+ t; Initialstates . . . . ' "'

4.5 ^{High Energy} Calctrlation for ff¡ ^Orbitals'

oHAPTER5.RESuLTSANDDISCUSSIONFORFIXÐDI{.[-ÐLEI.

a

.84

.BB .92

.96

a a 98

7t

B1

5.1 5.2

5.7

5.4 5.5

6.1

6.2 6.3

6.4

Details of tre l{urnerical- Methods

Partial ^{Cross Sections} for Ind'iviclual

Orbitals ^ard. Their Behaviour llear

Thresholils o . . . . ' ' '

The Effects of Changing the L'C'A'0' Coefficients of { 0rbitals ' ' Effects of ^{Varying the B-ourd State}

Pararneter gz for Tl ^Orbital-s ' ' '

Bigh Enerry ^Behaviour of the Cross Sections

a

a

. ^lo9

.112

.118

.122

. ¹²⁷

CHAPTER 6 - INCTUSION OF THE VIBRATIO\TAL A}ID ROTATIONA]J EIGEI{STATES.

The Born-Oppenheimer Approxination'

Expressions for tre ^Cross Section ^¡

Results for Partial ^{Cross Sections} Calculatetl

with Franck-Cord on Factors ard ^Comparison

with Experinent . . . ' ^'157 Inclusion of the DePendence of the

Electronic Transition ^Moment on Internt¡cIear

Distance . . ' ' ' ^.179

6,5 Totat Cross Sect$ons for ^N,

*

l, lt"i .

^.14,'

Thresholds to 5OA. . '

cr{APlER 7

-

7.1 7.2

7.'

'1.4 CHÀPI'ER 8 -

CHÄPTER 9 -

9,1

9.2

9.t

REFERENCES.

PHOTOIONIZATIOIT OF ^THE 2]f EI,ECTRON OF

NITRIC OXIDE.

Bound.statel{aveFunction . . . . ^. I,tatrixEÌenents . . . .'. ^.

Approximate Calcqlation of the Cross Section

¡rãn 1t4O - ^10229' . . . . o . . . ^. Resultsard.Discussion . . . ...

CONCLUSIONEi.

APPENDICES.

Norroalization of the Bounil Molecular

OrbitalsofChaPter2 . . . . i . ^. Angularlntegrals . . . ^. Brief Description of Programmes . . . ^.

a

a

a

a

150 152

156 158 160

17'

181

lBB

ì

"s.ulfll¿BI.

In this thesis ue present calculations of ^photo- ionization cross sections for the diatomic rnolecules

N^, O^ and. N0. These are expected'to have irnportant

¿¿

as trophysical applicati ons.

The necessary formulae and. the justification of

the one electron approach fcr evaluating electronic matrix elements for many-electron s¡rstems are ^given in ^CÌ:.epter 1, where an attempt is also ^matle to survey

briefly the relatively well ^known fielcl of atomic

calculations. ^The Chapter concludes with a resume

of ^previous theoretical work on photo-ionization cross secti-ons for molecules.

fn ^Chapters 2 and.7 the electronic wave functions

need.ed. for the evaluation of the natrix elernents are

presented and discussed. It is ^{apparent that} approximate

ne thods ¡nust be r:sed for both the initial and final states.

S.C.F.-t.C.À.0.-l'{.0. uave functions are f ou¡d most

convenient for the bourd states and. analytic expressions

for these I1.0.ts are derived for the case cf Slater type 4.0. ^I s, using prolate spheroidal co-orclinates. For the

fj-rst attempt at calculatiôns of the ct'oss sectÍons,

!'lannery and. Opikts final state ^rnodel is chosen by

analory with ^{coulonb rvaves which} are a first approximation

for atomic calculatÍons.

The first ^parts of chapter 4 are ^concerned with the evaluation of the electronic netrix ^elements with ^the inÍtial ^and. final states of ^{Chapters 2 ard} 1' In tlæ

final part, a calculation of the cross section using plane wave finaÌ states is ^presented for a ^ÍTY eleettont this approach not being ^pursued in subseqrænt numerical v¡ork.

The results of ^the calculations fcn fixed' nuclei reveal several interesting features. ^{Each bourd state} orbital type ^has a characteristic ^cross section curve and

this is explained.: In ^{the high energy behaviour} of ths cross sections for ^{lTly and. 3l¿} orbitals, ^{we observe}

peaks which are interpretecl as the basis of ^{Cohen and'}

Fanors trshoulderrr effect. ^{lle also} incorporate ^changes in the parameters fc¡r fr¡- tl.O.rs and. a]so in the ^arnorrnt of hybrid.ization for ^{Cf. I,t.O. rs and d.iscuss} their effects.

The inclusion of the vibrational eigenstates is consid.ered necessary in the evaluation of the cross

section,s near threshold. and in ^{Chapter 6 we develop}

formul-ae for the cross section using the Franck-Cordon

factor approximation. comparison with experiÍent ^reveals that our ^apploach is reliable for the llTã, l'ffn *'a 3{t orbítals of ^{N, and} o, (if occupied.)" ^For the transitions to ^or+( d,"i'r*, A'Trn)

"n "rrotalous eff ect is fourd near

threshold. but for al-l ?-fy ^{anð' 2fa} orbitalsr ^{the nodel meets} with a radical failure ^r+hich is d.iscussed in d.etail. ^Before

the total cross sections for ^{N, ard} o, are discusseil, ^we investigate the vatid.ity of ^the Franck-cond.on factor

approxirnation in the case of photo-ionization of the lIî, electron (of zero kinetic ^ettergy) in Or. ^We find the

variation of ^the electronic transÍtion ^moment with inter- mrclear separation ard reach the conclusion that the Franck- cord.on factor approxination is valicì. to within ^{a f ew percent.}

In ^Chapter ? ^{we evaluate} the electronic rstrix ^elenents for the 27f ^electron of nitric oxíde. ^\{e fird that ^nuch

computational work can be avoidecl with an approximate approach which pro\res reliable near thresholcl'

tr'inallyinChapterSrgeneralconclusionsaregiven with a d.iscussion of ^{possi.ble nethods} for fr¡rther work

in this field.

STATEME_U

This thesi-s contains no naterial ^{which has been}

acceptecl for the ^award. cf any other degree and to the best of ny-knowlectge and. belief , ^{contains no rnaterial}

previously published. or written by another ^persoa except where tlue reference is nade in the text.

E. C. flICKI,{EtL

ACICSoIII,EDGnIENTS.

The author would llke to thank Professor C.A. ^Euret

for his supervlsion ancl encouragement throughout this ^work

and aÌso for his critieaL

"eading of the nanuscript. ^{Thanks are} also due to ^{Professor J.H. Carver and. other nernbers} of the

.A,tie1airte University ^Physics De¡nrtnent for nany useful tliscussions. In partieular, Dr. AoJ. Blake for ^provlding cross sections for 0, ^{ancl N,} prior to publlcation, llfr. J.L.

Bahr for ^providing Franck4ondon factors for ionization ^to

02+

(zi

_), antt Mr. Vijay ^Kumar for ^providing experinental results for nitric ^oxide.

Fu¡ther thanks are ^d.ue to Dr. G"J. Iverson for val-uable

critlcisns ^and. help during the earl-ier stages of this ^{work and} also to Dr. Fomest R. Gilnore for ^providing spectroscopic constants and potential energy curve d.ata for various'states of ^{Nr+ ancl} Or+. Finally it is a ^pleasure to aeknowletlge the

r:atiring efforts of ^{Miss N. Ad.ams} in her excellent typing of the nanuscript.