UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Positron-atom interaction Pai, David Mieng 1975

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1975_A6_7 P03.pdf [ 5.34MB ]
Metadata
JSON: 831-1.0085260.json
JSON-LD: 831-1.0085260-ld.json
RDF/XML (Pretty): 831-1.0085260-rdf.xml
RDF/JSON: 831-1.0085260-rdf.json
Turtle: 831-1.0085260-turtle.txt
N-Triples: 831-1.0085260-rdf-ntriples.txt
Original Record: 831-1.0085260-source.json
Full Text
831-1.0085260-fulltext.txt
Citation
831-1.0085260.ris

Full Text

P O S I T R O N - A T O M I N T E R A C T I O by DAVID MIENG PAI B.Sc, University of Bri t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE In the Department of Physics • i ' ' We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1975 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of irhys < c$ The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date SO /75~ i i Abstract Any attempt to describe accurately the l i f e time of positrons in atoms or the low energy phase shifts for positron-at<bm scattering relies on an adequate treatment of electron-positron correlations. In this thesis e an interaction model which treats such correlations is presented. This model has other desirable features. It is non-phenomenologlcal, amenable to calculation even for complicated atoms, and applicable to calculations of both bound and scattering states of positron. Following the presentation of the theoretical aspect of this model* methods which can be used to carry out the prescribed calculations are discussed. At the end of this thesis, results obtained from applying this model to positron-helium system axe presented. In particular, the calculated total elastic scattering cross sections are compared with experimental measurements. i i i Table of Contents Page Abstract H Table of Contents i i i Acknowledgement iv Chapter 1 Introduction 1 Chapter 2 Theory and Model of Interaction 4 2 . 1 . Physical Consideration of Interaction 4 2 . 2 . Model of Interaction 7 Chapter 3 Methods of Calculation 19 3 . 1 . Solution of Electron Equation 19 3 . 2 . Calculation of V(rp) and 24 3 . 2 . 1 . Reductions of Integrals in VC^p) and Mz to Sums of Matrix Elements between Basis Functions 24 3 . 2 . 2 . Evaluation of Matrix Elements between Basis Functions 28 Chapter 4 Results 35 Bibliography 4? Appendix A Derivation of ( 3 . 6 ) 49 Appendix B Derivation of 7p£ a n d 5 2 Appendix C Derivation of 7pt a n ( i Vpl 56 Appendix D Alternative Method on Deriving Vf \ and vp% 58 Appendix E Electron Wavefunction 61 Appendix F Matrix Elements "between Molecular Orbital $ 69 Appendix G V/(rP,&)and S(<>,$,U) 72 Appendix H S @) as a Function of Positron Energy 75 ) Acknowledgement I wish to take this opportunity to express my sincere gratitude to my research supervisor, Dr. E.W. Vogt, for his continuous patience and encouragement in guiding me during the entire course of this research. It would have been difficult to complete the project had he not given me the help that I received. I am also grateful to Dr.D.M. Schrader of Marquette University for his many Invaluable contributions. The project would have been a difficult task without his participation. Finally I want to thank Dr. P.H.R. Orth who set up the computer programme BISON for us. The authors* of BISON are also appreciated. * A.G. Wahl, P.J. Bertoncini, and R.H. Land of Argonne National Lab. 1 Chapter 1 Introduction The i n t e r a c t i o n of positrons with matter i s an i n t e r e s t i n g and important phenomenon because the manner i n which a positron annihilate with an electron can often reveal the e l e c t r o n i c structure of the material. I t i s no wonder that positron has been used successfully to probe s o l i d s and 1 l i q u i d s . Of more fundamental i n t e r e s t , however, i s the i n t e r a c t i o n of positrons with atomic gases because t h i s kind of process i s dominated by a positron i n t e r a c t i n g with one gas atom at a time, instead of an assembly of atoms, i . e . assuming the s c a t t e r i n g length i s much l e s s than the average interatomic spacing in gas. Studies of t h i s kind can reveal the i n t e r a c t i o n mechanisms which are valuable i n understanding the more complicated s i t u a t i o n s . Experimental studies on positron one-atom i n t e r a c t i o n s are mainly of two kinds. One i s the l i f e time spectrum measurement of positrons i n gases, usually as a function of temperature and applied e l e c t r i c f i e l d . S i g n i f i c a n t amount of 2-5 measurement has been made on rare gases . I t i s d i f f i c u l t however to use l i f e time spectrum measurements to deduce without ambiguity the scattering or the a n n i h i l a t i o n cross sections which characterize the i n t e r a c t i o n . A d i f f e r e n t kind of experiment i s to measure the attenuation of positrons of a 6-7 d e f i n i t e energy i n gases . This experiment was not possible unt'il recently a method was found on how to obtain mono energetic positrons. The attenuation measurement i s more d i r e c t than the l i f e time measurement i n y i e l d i n g the t o t a l cross sections, but uncertainty s t i l l e x i s t s . At the moment, there are some unresolved discrepancies among the measurements. Most t h e o r e t i c a l i n v e s t i g a t i o n s on positron one-atom system are r e s t r i c t e d to e l a s t i c s c a t t e r i n g or bound state c a l c u l a t i o n s . I n e l a s t i c s c a t t e r i n g , which means that the i n t e r a c t i o n involves e x c i t a t i o n of the atom or r e a l positronium formation, i s d i f f i c u l t to treat- So f a r , most of the meaningful e l a s t i c scattering c a l c u l a t i o n s were f o r 3-10 simple atoms. Helium atom , i n p a r t i c u l a r , received much attention because experimental data were r e a d i l y a v a i l a b l e . Agreements between ca l c u l a t i o n s and measurements are s t i l l debatable. Some of the ca l c u l a t i o n s are phenomenological, others are v a r i a t i o n a l . Regrettablly, none shows promise i n giving an accurate and solvable method for more complicated atoms. For bound state c a l c u l a t i o n s , Schrader constructed an elaborate s e l f - c o n s i s t e n t - f i e l d theory**. However complete and correct, the theory i s d i f f i c u l t to be applied to complicated systems. This t h e s i s presents a model f o r the i n t e r a c t i o n between a positron and a single atom. This model has several desirable features. F i r s t of a l l , i t i s applicable to both e l a s t i c scattering and bound state c a l c u l a t i o n s , though not 3 to i n e l a s t i c s c a t t e r i n g . Secondly, i t i s not phenomenological. There i s no parameter to be varied against experimental data. T h i r d l y , i t retains the s i n g l e p a r t i c l e picture of the positron. This treatment enables s c a t t e r i n g phase s h i f t s or binding energy of the positron ( i f applicable) to be e a s i l y c a l c u l a t e d . Fourthly, i t accounts f o r the c r u c i a l short range s p a t i a l c o r r e l a t i o n between the positron and the electrons,, This i s important because process liJte a n n i h i l a t i o n i s very s e n s i t i v e to the behavior of electrons near the positron. L a s t l y , and most importantly, the model i s amenable to c a l c u l a t i o n , even f o r complicated atoms. I t was decided to use helium atom as a t e s t i n g ground f o r the model. The obtained r e s u l t s are compared i n t h i s t h e s i s with presently available experimental data. I t should be pointed out that the idea behind t h i s model i s applicable to any system of p a r t i c l e s where s p a t i a l c o r r e l a t i o n s between the p a r t i c l e s are important. In f a c t , t h i s idea was used by Vogt to c a l c u l a t e the e f f e c t of 12 c o r r e l a t i o n s between nucleons i n a nucleus Chapter 2 Theory and Model of Interaction 4 2»/\ Physical Consideration of Interaction I t i s worthwhile to describe i n words the i n t e r a c t i o n between a positron and an atom before gi v i n g the mathematical treatment of the model. F i r s t , consider the s i t u a t i o n when the positron i s f a r enough from the atom that the only e f f e c t the aton can f e e l i s a slowly varying e l e c t r i c f i e l d as the positron changes i t s position,. S p e c i f i c a l l y , the e l e c t r i c f i e l d v a r i a t i o n i s so slow that the electrons have ample time to rearrange themselves as the positron moves about. This s i t u a t i o n i s analogous to the case of a molecule i n which the electrons can almost follow up the nucleus motion instantaneously. Therefore when the positron i s s u f f i c i e n t l y f a r away from the atom, i t i s j u s t i f i e d to use the 'adiabatic approximation', i . e . calculate the electron motion as i f the positron i s stationary. This approximation greatly s i m p l i f i e s the problem. The r e s u l t i s that the positron i n t e r a c t s with a polarized atom. The degree of p o l a r i z a t i o n depends on the positron's distance from the atom. In f a c t , the i n t e r a c t i o n becomes predominantly a dipole-charge type because at large distance from the atom other multipoles of the atom give l i t t l e e f f e c t . I t i s therefore expected that, the positron z experiences a po t e n t i a l having the asymtotic form - ^  where oC i s the dipole p o l a r i z a b i l i t y of the atom. This i s the well known long range p o l a r i z a t i o n p o t e n t i a l . Now what i f the positron i s not far away enough from the atom f o r the adiabatic approximation to be va l i d ? In other words, consider the s i t u a t i o n when the e f f e c t of the positron on the atom varies so s w i f t l y that the electrons can no longer follow up the positron's motion instantaneously. This i s the case when the positron i s close by or i n f a c t i n s i d e the electron cloud* Consider for the moment a simpler but s i m i l a r problem, the i n t e r a c t i o n between a positron and a si n g l e electron. As can be found i n any mechanics text that a two body i n t e r a c t i o n problem can always be s i m p l i f i e d using the concept of reduced mass i n t o a one body problem. S p e c i f i c a l l y , the wavefunction ^(C) which describes the r e l a t i v e motion between the positron and the ele c t r o n s a t i s f i e s a one body Schrodinger equation. ahere / x i s the reduced mass of the system. I t equals 1/2 of the electron mass for the system being considered. The the positron and the electron, gives the p r o b a b i l i t y density f o r finding the electron Y apart from the positron. In other words, one can solve for the electron wavefunction i n the coordinate r e l a t i v e to the positron as i f the positron i s (2.1) where £ i s the r e l a t i v e coordinate between i n f i n i t e l y heavy and the e l ec t ron mass i s reduced by h a l f . I f only in teres ted i n the correct r e l a t i v e wavefunction ^(C) # one can always solve the a l t e r n a t i v e equation where m g i s the e lec t ron mass. In words, t h i s means that the wavefunction which describes the e l ec t ron motion r e l a t i v e to the pos i t ron can be obtained as i f the pos i t ron s t a t i o n a r y , but with a charge of 1/2 e instead of 1 e. Now come back to the o r i g i n a l problem. I t was mentioned that the ad iaba t i c approximation i s not v a l i d when the pos i t ron i s c lose by the atom because the e lec t rons can no longer rearrange themselves quick ly to the p o s i t r o n ' s motion. However fast the pos i t ron may seem to the e l e c t r o n s , one knows by using the idea described for a pos i t ron i n t e r a c t i n g with a s ing le e lec t ron that the e lec t ron wavefunction, at l ea s t the part near by the po s i t ron , can be e a s i l y c a l c u l a t e d t r ea t ing the pos i t ron s ta t ionary and using the e f f e c t i v e pos i t ron charge 1/2 e. For e lec t rons not so c lose to the p o s i t r o n , however, because of i n t e r a c t i o n s with the nucleus and other e lec t rons the e f f e c t i v e charge may range anywhere from 0 to 1 e. Therefore, one can perhaps approximate the e l ec t ron wavefunction by t rea t ing the i n t e r a c t i o n with the pos i t ron as such that the pos i t ron i s s ta t ionary but with some e f f e c t i v e charge l e s s than 1 e. The value of t h i s charge depends on the positron's distance from the atom. This way, as far as the electron motion i s concerned, the positron-atom system resembles a s p e c i a l diatomic molecule f o r which the electron wavefunction has been well studied. This i s the main idea of our model. I t i s appropriate now to present the mathematics of the model. Some of the ideas can then be more p r e c i s e l y stated. A l l g u a n t i t i e s from here on are given i n Atomic Unit («v=e=n=1) unless s p e c i f i e d otherwise. 2-.2 Model of Interaction The approximate wavefunction used to describe the positron-atom system i s taken to be the product form ^((tv] •> •, £0rp))U(rp). The electron wavefunction ^ i s expressed e x p l i c i t l y by the electron coordinates Ifi (i=1,2,... #Z) where Z i s the atomic number. It also depends on the positron coordinate Tp ,although parametrically. This dependence together with @(rp) builds into ^ the important e l e c t r o n -positron c o r r e l a t i o n s . The positron wavefunction l i (Tp) , though a function of only Ifp , does depend on the e l e c t r o n motion because i t i s determined from a p o t e n t i a l which depends on the electron configuration. The electron wavefunction ; Pp > (lOp) ^ s defined by the equation 8 \ u \«t i*< '» tat j>i MJ js, t i p / ^V; : k i n e t i c energy operator of the i - t h e l e c t r o n ^js. : i n t e r a c t i o n between the nucleus and the i — t h electron -~ : i n t e r a c t i o n between the i - t h and the j - t h e l e c t r o n , u n j « l r i - r i | 6(Vo)—-•: i n t e r a c t i o n between the i - t h electron and the r v Tip positron i n the context of present method, Hp-| £V"Tp| The nucleus, as usual, i s treated as stationary i n the process of interaction., I t can conveniently be used as the o r i g i n for ITi and £p „ The r a t i o n a l behind the term @(rp) i s already given. I t was stated that i t i s the e f f e c t i v e charge of the positron seen from the electrons i f the positron i s treated as quasi stationary when the electrons move about. I t should be determined by the extent to which the electrons can follow up the positron's motion. The behavior of electrons near the positron, the:so c a l l e d short range c o r r e l a t i o n , i s very s e n s i t i v e to t h i s parameter. S p e c i f i c a l l y , at fi = £p, ^ s a t i s f i e s the electron-positron cusp condition which can be derived e a s i l y from (2.3) by ignoring a l l i n t e r a c t i o n terms except the electron-positron one. To describe i d e a l l y the short range c o r r e l a t i o n , according to (2 . 1 ) or (2.2), @ should be set to 1/2 i r r e s p e c t i v e the positron's distance from the atom. More importantly, however, Q should describe an accurate electron wavefunction not only i n the region close to the positron, but everywhere. In other words, @«p) i s intended to account f o r c o r r e l a t i o n s i n an average way i n the sense that at a p a r t i c u l a r positron p o s i t i o n ifp both the long and the short range c o r r e l a t i o n s are described by t h i s single parameter. When the positron i s i n s i d e the electron cloud, £ should be close to 1/2 because short range c o r r e l a t i o n dominates there. On the other hand, when the positron i s s u f f i c i e n t l y f a r away from the atom, (3 should equal to 1 since only long range c o r r e l a t i o n e x i s t s there. In between these two extreme cases, (3 i s some s o r t of average between long and short range c o r r e l a t i o n s . I t i s hoped that the important short range c o r r e l a t i o n i s not too badly represented t h i s way. The c o r r e l a t i o n s would of course be better treated i f one could construct a theory using (3 as not only a function of Tp , but also of H p Now consider the positron wavefunction U(rp) . I t i s defined by ( I T V P + \ + Vcrp))U(rp)=ePUccp) ( 2.5) r 10 . 2 -^-V p:the positron k i n e t i c energy operator :the positron nucleus i n t e r a c t i o n 'P y(fp):th.e e f f e c t i v e p o t e n t i a l experienced by the positron due to the electrons 6p:the positron energy This equation shows that the positron i s treated as a s i n g l e p a r t i c l e moving i n an e f f e c t i v e p o t e n t i a l . This treatment has the advantage of c a l c u l a t i n g positron-atom s c a t t e r i n g phase s h i f t s e a s i l y . In case one i s interested i n bound s t a t e , binding energy of the positron can also be r e a d i l y c a l c u l a t e d provided VOtf i s negative enough. There i s no preset boundary condition ( K^-^oo) on the positron wavefunction. The model i s general enough for either e l a s t i c s cattering or bound state c a l c u l a t i o n s . The roles played by @(rP) and V(rp) are now explained. But how exactly are they determined? Consider f i r s t the case of \J(Yp) . The i d e a l wavefunction which describes an i n t e r a c t i o n should have a d e f i n i t e energy. In the present case, the t r i a l product wavefunction ^/Ul does not s a t i s f y t h i s condition since i t i s not an eigenstate of the t o t a l Hamiltonian H. S p e c i f i c a l l y , 11 - He + HP+0-?M.£^-p-V(rp) (2.6) where He and Hp are respectively the Halmiitonian i n (2.3) and (2.5). Therefore H(^U)=£CrP)¥u + ePW+^^^^ (2.7) Unfortunate i t i s that does not give an energy independent of positron and electron coordinates ( i f possible at a l l to f i n d such a wavefunction). An immediate question one can ask i s whether i t i s possible to formulate the treatment such that energy of the system averaged over a l l the electron coordinates i s independent of the positron coordinate. In other words, one asks ^ * m ^ U)^*Yz...^l AUOrpj (2.8) where A , a constant, i s the energy of the system averaged over a l l the electron coordinates. This condition i s equivalent as demanding the energy of the system, averaged over a l l the electron coordinates, to remain constant as the 12 positron moves about. I t i s indeed a ph y s i c a l l y reasonable condition and should be incorporated, i f possible, i n t o the treatment. One sees from (2.7) that JVH (vpU)dV, - «*S = { £ C r P ) + e p + (v-0<rp))<^|^|*> * u J (2 .9 ) Hhere the electron wavefunction i s taken to be properly normalized, i . e . 0 ^ 1 ^ ; = 1 • Condition (2.8) can therefore be s a t i s f i e d i f X.£CrP> + e PH»-P< R P>><¥l£^^^ (2.10) One r e a l i z e s immediately that t h i s equation can be used f o r determining K/i^p) , assuming at the moment that A can be s p e c i f i e d . In other words, the e f f e c t i v e p o t e n t i a l experienced by the positron due to the electrons can be chosen via the desirable condition that the t o t a l energy, although i n t h i s case not a d e f i n i t e eigenvalue but an average over the electron coordinates, remains constant no matter where the positron i s . Such scheme i s i n f a c t equivalent as assuming that the positron i n t e r a c t s with the electron d i s t r i b u t i o n , rather than with the i n d i v i d u a l electrons. This i s indeed a reasonable approximation as i t has been used well i n other s i t u a t i o n s . For example, Hartree's s e l f - c o n s i s t e n t - f i e l d equation for electrons i n an atom can be derived by using the s i m i l a r condition that the t o t a l energy averaged over a l l electron coordinates, except the one under consideration, i s a constant. Consider now the value of A - I t i s defined as the t o t a l energy, averaged over a l l the electron coordinates, of the positron-atom system. It i s impossible to c a l c u l a t e A f o r any positron p o s i t i o n other than for the simple s i t u a t i o n when the positron i s far away from the atom. In that case, there i s almost no i n t e r a c t i o n , the t o t a l energy simply equals £ 0 + 6 p where So i s the atom energy. Mathematically, i f one prefers, the product R^J i s an exact eigenfunction at large Tp since there i s no i n t e r a c t i o n . One can see that energy of the system as given by (2.7), or the average energy as given by (2 .9) i n keeping with the d e f i n i t i o n of A , becomes fcico^-tQ-p since the l a s t four terms, -—r-^ - » Vp^S? , Vp^ e and V , which describe i n t e r a c t i o n s , a l l vanish at large To . Notice that 1'^ £(rp)-*£,0 . Therefore the t o t a l energy of the system, averaged over a l l the elec t r o n coordinates, i s independent of the positron's p o s i t i o n i f \jQTp) i s calcula t e d by (2.10) with A = £ o + e p , i . e . V(rp)= 8Crp)-eo+(t-<?(rP))<^|r^|^>+f (2.11) 14 For c a l c u l a t i o n f e a s i b i l i t y one may wonder, about the term containing U which, a f t e r a l l , i s determined from VC p^) . Fortunately t h i s term vanishes i d e n t i c a l l y because the electron wavefunction ^ i s r e a l (see p. 23) and normalized. <v£'iYP^YpU>=<^i7p^>-vPu=^7p(<:vi'i^>)-ypa=o (2.12) For s i m p l i c i t y , define three terms: A - £ ^ j , J B V ^ ^ ; C 3 - - ^ Y p ^ U l J (2.13) where they a l l are functions of JTi and £p . Equation (2.11) can then be conveniently written as \J(rP)- i c r p ) - £ 0 + (i-^(rP))A(rp')+ § (2.14) where the symbol •—• (bar) means to average over a l l the electron coordinates. For example, A ( R P ^ ^ f t — d V . ^ - ^ r 2 (2.15) To show the consequence of choosing V^p) according to (2.8), one can rewrite equation (2.7) using V^p) as given by (2.14). H ^ U ) = (£• + e P ) *(i-^c.r l))(A- A ) ^ U + ( 8 - 8 ) ^  U (2.16) This equation c l e a r l y shows that the t r i a l product wavefunction ^ f U i s an eigenfunction i n the sense of averaging over the electron coordinates. This way the t o t a l energy in the same average sense i s kept at the constant value 8o +^P • How should @Cp) be determined? I t was introduced i n the f i r s t place to describe as an accurate electron wavefunction as possible when the positron i s treated as guasi stationary r e l a t i v e to the electron motion. In other words, §(rp) should be determined such that the t r i a l wavefunction ^ U i resembles the true energy eigenstate as much as possible. One guantity which can measure the extent to which ^ L i approximates the true eigensate i s H 2 ( P<*P>) S < ¥11 |(H - Eo) 21 ^ U > (2.17) where Eo i s the true energy of the system, already shown to be the value £ o + ^ P . The value ( M z ) ' 2 measures the energy spread makes about the true energy. S p e c i f i c a l l y M2 can be written as co M 2 = Z K E i l ^ U > | 2 ( E i - E 0 ) 2 + { KEIvyU>|2(E-Eb)'dE (2.18) where [Ei) 's are the bound s t a t e s , i f t h e r e i s any, o f energy £; ( E; < E 0 ) ; and |E> 's are the continuum s t a t e s o f energy E . This expression c l e a r l y shows that non-vanishing 2 M2 i s due to the undesirable c o e f f i c i e n t s I^EtlSPU^ and UEl^U)! where Ei,E * Eo . It i s e a s i l y seen that M 2 ought to be minimized i f (MAJVs component oh lEo^ , i . e . (CEol^ UM > i s to be maximized. The smaller M 2 i s , the better l^ PU) approximates the true eigenstate |E(^ > . I t i s therefore reasonable to determine @("p) by minimizing M a • To write down H2 e x p l i c i t l y i n terms of and Ut , one can use Eq. (2.16) which shows how H operates on ^Ll • M 2 =<^UI H 2-2HEo + Eo I^U> = <^u iH 2 | ^ u > - E 0 2 <u ia> = <H(^U)|H(^U)>-Eo<U\U> = ^U(rP,^rp))|2S(rPf (^rP))dVP (2.19) where S ( ^ = 0 - ^ 2 ( A 2 ) + 2 ( i - ^ ( A 6 - A B ) + ( ^ - B 2 ) - r 2 ( l -^ ) (^ -AC) + 2(6C-8C) + (C5--C2) (2.20) The terms A 7 , A B , Bz, A C , B£, and C~* are not defined yet. Their symbols however should indicate c l e a r l y what they are. For example, re f e r i n g to (2.13) and (2.15) one can see that 17 AB*$mXiifpXTi^)^' (2.21, Equation (2.19) states that i n order to minimize H 2 one has to know the positron's wavefunction Ui , which supposedly i s determined from Wp) . But V W i s dep endent on 0(fp) which comes from minimizing M 2 « Therefore there seems to be a loop process which suggests d i f f i c u l t y i n actual computation. Fortunately, f o r the case of e l a s t i c s c a t t e r i n g of positrons from helium atom, the values obtained f o r \Afp,(3) and SCrpj^ jU) f o r a s e r i e s of (3 at each Tp indicate that no integration over Tp i s necessary to f i n d minimum M 2 «. In other words, (3(rp) can be determined from minimizing $ at each Tp „ Furthermore, for finding mimimum S , i t was found that no extensive i t e r a t i o n i s reguired to take care of the f a c t that 6 depends on U . The idea that @(*p0 depends on Q.p (since c* depends on 14 ) i s p h y s i c a l l y reasonable. The f a c t o r @ , as described, measures the extent to which the electrons can follow up the positron's motion. Therefore, i n case of sca t t e r i n g the higher the i n i t i a l positron energy Qp i s , the lower (30Tp) everywhere i s expected. The following f i g u r e gives a rough idea for how (3ttp) should vary with . 18 Figure 1 The energy range f o r e l a s t i c p o s i t r o n - h e l i u m s c a t t e r i n g i s 0 t o 17.7 eV. Over t h i s s m a l l range, no v a r i a t i o n o f $(fp) with Sp was found. A l l the c l a i m s made here w i l l he s u b s t a n t i a t e d i n the f i n a l c h a p t e r u s i n g a c t u a l numbers o b t a i n e d from c a l c u l a t i o n s . 19 Chapter 3 Methods of Calcula t i o n The methods which were used to carry out the necessary c a l c u l a t i o n s prescribed by the model f o r helium atom are presented i n t h i s chapter. Most techniques used are also applicable to the more complicated atoms. For c l a r i t y , many detailed derivations are given i n the appendices. 3. ,1 Solution of Electron Equation The f i r s t tasX i s to solve f o r the electron wavefunction ^({tt}-, rf; §(rr\) from Eg. (2.3) f o r a p a r t i c u l a r TP and (3 . There e x i s t well documented computer programmes to solve t h i s p a r t i c u l a r eguation since, as mentioned, i t i s i d e n t i c a l to the electron wave equation of a diatomic molecule except the 13 f a c t o r ^ . A programme by the name BISON was used in t h i s study af t e r a few modifications were made on i t to accomodate The method BISON uses to solve (2.3) i s the Hartree-Fock-Bdothaan 1s SCF ( s e l f - c o n s i s t e n t - f i e l d ) LCAO ( l i n e a r 14 combination of atomic o r b i t a l s ) method . S p e c i f i c a l l y , f o r the ground state of a close s h e l l system, the wavefunction i s approximated as a Slater determinant of one electron wavefunctions (commonly c a l l e d molecular o r b i t a l s ) , i . e . 20 (3.1) where i s the i - t h o r b i t a l , each being doubly occupied by electrons of spin t and spin V . It should be pointed out that m has a more complicated form for an open s h e l l system which, although can s t i l l be handled by BISON, does make other c a l c u l a t i o n s d i f f i c u l t . Fortunately, rare gases which are of main i n t e r e s t do form close s h e l l systems. The method of approximating ^ i n terms of o r b i t a l s does not tr e a t electron-electron c o r r e l a t i o n s properly. This i s however not a serious defect since present i n t e r e s t s , which are associated with the positron motion, are not expected to be strongly dependent on electron-electron c o r r e l a t i o n s , but indeed on electron-positron c o r r e l a t i o n s . Subjecting ^ of the assumed form shown by (3.1) to the v a r i a t i o n a l p r i n c i p l e that &(vp,p) be minimum, one finds that the Cj? *s s a t i s f y an eigenvalue equation, the so c a l l e d Hartree-Fock Equation. This equation can be approximately §,(r.)t(i) $,fr.)i(l) 4Vr,)t(i) CZ!) solved by the Roothaan Expansion Method in which the molecular orbitals are expressed as l inear combinations of some basis functions (usually in the form of atomic orb i ta l s ) , i . e . § i = i:C;t,F* (3 .2) where the F » s are the basis functions; and the c ' s are the associated coeff ic ients . Using this approximation, the Hartree-Fock Eguation can be reduced to a set of f i r s t order l inear equations. The coefficients are then solved by i terat ion unt i l reaching self-consistency. The basis functions BISON uses are the normalized Slater type orbitals centered either on the nucleus or on the positron and defined as F „ , 1 . , , 1 ( r ,9 .? )=- | | I p r n"'e <3.3> where Vjlm ^e>^) i s t n e usual spherical harmonics. The coordinate (r, i s defined according to which center the function resides on as shown below. 22 Figure 2 One can see that (fa.Qa.^a) r e f e r s to the coordinate of basis functions on the nucleus, and (Y~b,9b,*Pb) i s f o r those on the positron. The degree of accuracy f o r obtained t h i s way d e f i n i t e l y depends on the number of basis functions used to expand the molecular o r b i t a l s . Fortunately, one can obtain reasonable accuracy with p r a c t i c a l computing cost as BISON has been used s u c c e s s f u l l y to many diatomic systems. The feature that basis functions can reside on e i t h e r center has one great advantage. I t i s that the electron wavefunction near the positron, i n p a r t i c u l a r the e l e c t r o n -positron cusp condition stated i n (2.4), can be e a s i l y represented by basis functions on the positron, but not by those on the nucleus. In other words, i t can be d i f f i c u l t to describe s a t i s f a c t o r i l y the important short range e l e c t r o n -positron c o r r e l a t i o n i f r e s t r i c t e d i n using basis functions 9 only on the nucleus. If so, as in some previous work , many 23 basis functions have to be then used. It should therefore be recognized that this model not only accounts for the c ruc ia l short range electron-positron correlation in theory as by (3(*"p) t but also in actual calculation by using basis functions on the positron. However, for evaluating V(*p) and H2 f two center basis functions give r ise to two center integrals which can not be done analytical ly as in the case of one center integrals . Fortunately, as w i l l be shown, this problem can be le f t to BISON to take care of. For helium atom, the situation is part icularly simple since there needs be only one molecular orbi ta l s to accomodate the two electrons. According to (3 .1) , the electron wavefunction i s then expressed as m= ^ <$(rO<$(r2)(t(i)U2)-Ul)t(2)) (3.4) Ac this i s a singlet state since the electron spins are ant ipara l le l . &s in a diatomic molecule, the molecular orb i ta l ^ should have rotational symmetry about the nucleus-positron axis. In other words, g5 *s dependence on the angle _ _ _ im<p Y ( = T a = Trb) t shown in F ig . 2, should be of the form Q. . The energy is |m| dependent, with m=0 being the lowest. Assuming the positron interacts with a helium atom i n i t i a l l y at i t s ground state (thus tota l orbi ta l angular momentura=0), i t i s then necessary (but not sufficient) to choose m=0 for to 24 approach a ground state atom at large Tp . The s t a t e m=0 i s non-degenerate. One has therefore a close s h e l l system. I n c i d e n t a l l y , other rare gases which w i l l be investigated i n the future form close s h e l l also. To ensure <3? having the correct dependence on <P , the basis functions (3.3) should then be the Fn*o$ o r b i t a l s . The ones chosen f o r t h i s study are given i n the next chapter together with the obtained c o e f f i c i e n t s for various Ifp and <> 3.2 C a l c u l a t i o n of \J(rp) and Mz. As mentioned i n Chapter 2, there are a s e r i e s of i n t e g r a l s to be evaluated from the electron wavefunction ^ i n order to c a l c u l a t e (3(n>) and V(*p) , namely A, A5 ,B, E*,AB,C^,AC, and BC. Since ^ i s expressed by molecular o r b i t a l s which are l i n e a r combinations of basis functions, these i n t e g r a l s can be reduced to sums of matrix elements between the basis functions. This section describes f o r each i n t e g r a l what these matrix elements are and how they are evaluated. -3.2. 1 Reduction of Integrals i n V(Vp) and Ms to Sums of Matrix Elements Between-Basis Functions The f i r s t step i s to s i m p l i f y the i n t e g r a l s to be i n terms of molecular o r b i t a l s . For helium atom, contains only one molecular o r b i t a l as i n (3.4). For convenience. 25 .1 denote by <£> , and <£(!f2) by $ . One then has •2 where Vb » as i n F i g . 2, i s the distance from the in t e g r a t i o n point to the positron* By using the same technigue, i t i s shown i n Appendix A that A2=e<$i^j$w<$i-^i$>2 ^ " £ ^T7p~l ->rp 7KT>rp ik' L v rP 1 rP *ep Ar P ^  u ' . 9 / w _ r i I I - a l l i \ Z , / , / <)$ i i ^ <&y-»U_L?U _ ! _ ' v rP^eP afy 1 rps«hep afy ^sme^ty u J ^ J^p_ _*p >ep/-8*p rp IT , A / -a^ i i -al>v -aU i ~>U i A / i ~d& i I -a^v i au i *U 2. AC= 2<« ,l 1g^p> TFp"0" + * _JJ>_!L!f* u 7pti^ pi»<Pp XT A l l the underlined terms above w i l l be shown to be equal to zero. Since the molecular o r b i t a l ^ i s a l i n e a r combination of basis functions as i n (3.2), the terms i n (3.6) can be s i m p l i f i e d further i n t o sums of matrix elements between the basis functions. I t i s simple to see that <$i^-»$>=z:'z : c i C j < F « ^ i F j > < $ l ^ $ > s T H C r C j < F l l - f e l F j > (3-7) However, the ones involving derivatives with respect to the positron coordinate, i . e . those with B or C present, are not so st r a i g h t forward because the c o e f f i c i e n t s c; 's are also functions of positron's coordinate. For example It i s simpler , however, for or since the c o e f f i c i e n t s , by symmetry (no preferred nucleus-positron a x i s ) , are independent of the positron's angular p o s i t i o n . One simply has ^$__rc-25- ^ - r c ' ^ ' 13 9) 27 The case for Vp?<£ i s a l i t t l e more complicated. V P 2 $ = £ Ifrp^Fi + Ci VP2Fi * Z vPC- • 7PFi] (3 . i o ) Again, because * 0 # one has V P C i V p F i = ^ ^ (3.11) Therefore ^ J ^ l f c f R + C i ^ i + e C i ' ^ ) (3.12) where C* , c" denote , Vp 2C; r e s p e c t i v e l y . By using (3.8), (3.9), and (3.12), i t i s simple (but tedious) to reduce a l l the terms i n (3.6) int o sums of matrix elements between the basis functions. One has <*i -k = C i CJ' ( F ; ' i 1 Fj > +£ fCi<;i <Fi '-^  ^ I -4- w l > • ?T C i C J < F ( 1 -re- w 8 > ,_!__?$_, j _ 2 $ _ \ T T r.r• /_!_l£l I . L 2 f L \ \ irp *36p • rp "^6ip~ 2- 4- 1 J fp ">6p rp •jBp ' J 28 / _ ! _ ? _ & I _ ! C-C • (—! l H i _ J l £ k VpSwfcp^V ^pSi.&p ^p/-4-4- ^ IrpSthfrp * V rpt»6p * V _ !?_S??L*i » t J v p ^ V ^ > * X J ^ V 5 ? ^ ^vp^V rp<u* V V " i j ^ *p>dr vptt»»i»*V «B <FiIFj >*Z^CiCj <Ff IVp l Fj>-2Z ICCjVF,-1^ *b i j * i J b ' j * "'P +EEc I Cj<^|2g > + z r tc,^Fii |B> + ? £ t Ci ,e ,<2Bi-2&> ^1^4?* ^ F CC3'<ft 1 ^ * CiCj <<Fi 1 . + r i f e < i < ^ i V B > + 4 r ^ H t . i | s > + 4 r p f v ^ i ^ ) ( 3 . i 3 j where the underlined terms i n (3.6) are again underlined. The guestion now i s how to evaluate these matrix elements. 3..2..2 Evaluation of Matrix Elements Between Basis Functions 29 An inspection of the terms i n (3.13) shows that most of the matrix elements involve two types of d i f f e r e n t i a l operators with respect to the positron coordinate, namely Yp and vp . i t i s therefore necessary to f i n d out how these operators act on basis functions. This i n f a c t was found to be the major task i n carrying t h i s model to completion. The d i f f i c u l t y can be anticipated from the f a c t that the dependence of basis functions on the positron coordinate i s i m p l i c i t , rather than e x p l i c i t . Conceivably, basis functions reside on the nucleus react d i f f e r e n t l y from those on the positron under operations by Yp and Vp . To d i s t i n g u i s h them, l e t 4- denotes the ones on the nucleus, and ^ the ones on the positron. Consider f i r s t the simpler case, Vp-f . I t can be expressed as rP -&6p ~ rp\-aiQ» aeP >ea »<fe >eP ) rp$*ep»fy - rpsio©pV^ ra *«PD "** -aGa a<?P -a«pa *«PP / Refering to F i g . 1, one sees that Ifa , the r a d i a l coordinate of f , does not depend on any positron coordinate, i . e . = 0 -^r-=0 ( 3 . 1 5 ) 30 However, the angular coordinate of £ , ( &a,9ai ) # i s defined by the nucleus-positron axis. I t i s therefore dependent on the angular p o s i t i o n of the positron (Op.^pJr but not the r a d i a l position of the positron Vp . Hence Cos 6a Sin ©a 3 % oos ©a •aCpp ~ Sine* cosqsias^ &p+cosep (3.16) where the dependence of (0a, % ) on ( ©pf <PP ) i s derived i n Appendix B. Eg. (3.14) can therefore be written as - 0 * 9 a "a ©a (3.17) where the fa c t that ( i s independent of 9a (since m=0) has been used. I t i s now straight forward to f i n d . By d e f i n i t i o n ^ T " r P t r ? ^ "*• r p l pn&p r P a ?ep* v^ stfep *<pf* (3.18) Using (3.16), i t i s found that = ( 3 . 1 9 ) How Vp and Vp operates on ^ , however, i s more d i f f i c u l t to f i n d . The complication a r i s e s from the f a c t that the o r i g i n of q^s coordinate ( fi^Sb, Vb ) i s the positron i t s e l f . Now one has the s i t u a t i o n that a l l three of \%, 6b t and *Pb are functions of rp , Qp , and ... By going through s i m i l a r but c e r t a i n l y more d i f f i c u l t considerations as f o r £ , as shown i n Appendix C, one f i n d s that 4- = co S e b a i -S ^ 4 . r^p Tb ^eb » ri « * f l ^ a COS 6b I ri \ -i ^9 ^ o . ^ cos61, 3 $ i ri.\ ^ p l W = ^ n ^ ( ^ " - ^ ^ b ' - ^ W b ) ^ ^ p^? - vvJ +• + V b ( 3 . 2 0 ) where, again, m=0 i s assumed. I t is" apparent that most d i f f i c u l t i e s associated with the above method of treating Vp and Vp1 l i e i n the f a c t that basis functions are defined with respect to coordinates which are functions of the positron coordinate. In view of t h i s weakness, Vogt proposed an a l t e r n a t i v e approach i n which the basis functions are transformed (by r o t a t i o n of coordinate) 3 2 i n t o a new c o o r d i n a t e system whose o r i e n t a t i o n •• i s f i x e d i n space, i . e . independent of the p o s i t r o n ' s angular p o s i t i o n . Such t r a n s f o r m a t i o n , understandably, depends on the p o s i t r o n ' s angular c o o r d i n a t e . The r e s u l t i s t h a t t h e b a s i s f u n c t i o n s can be expressed i n the form where the dependence on the p o s i t r o n ' s angular c o o r d i n a t e (9p,<2Pp) i s e x p l i c i t , and t h a t |T ( p o s i t i o n v e c t o r i n new c o o r d i n a t e system) i s completely independent of the p o s i t r o n ' s c o o r d i n a t e i n the case of £ , and a simple a d d i t i v e dependence ( r - t ' - l f p r where r' i s independent of )Tp ) i n the case of ^ . T h i s way, d i f f e r e n t i a l o p e r a t o r s on F with r e s p e c t to the p o s i t r o n c o o r d i n a t e can be d i r e c t l y c a r r i e d out. T h i s approach proved to be u s e f u l i n t r e a t i n g the o p e r a t o r V p . i t i s shown i n Appendix D how Vp + and v*p ^ can be q u i c k l y s i m p l i f i e d using t h i s method. For , the same e x p r e s s i o n as (3.19) i s o b t a i n e d . However, the one o b t a i n e d f o r V p 2^ , (0.12 ) , i s s i m i l a r but not i d e n t i c a l to (3.20). I t i s expected t h a t (D..12) w i l l be reduced to (3.20) i f the l a s t term i n i t can be s i m p l i f i e d f u r t h e r . I t i s now s t r a i g h t forward to show t h a t the u n d e r l i n e d terms i n (3 . 6 ) or (3.13) 'are indeed zero as c l a i m e d . N o t i c e t h a t they are sums of the f o l l o w i n g matrix elements, -arp i rp »e P'' ^ >rf 1 rpsin9P afy ' ' <• rp *e P >rpli^ p 7? P> <• ? r'> r P * e P 7 ' / T 7 X P I — ! ^S- S / which a l l vanish since, t h e i r i n t e g r a n d s 33 depend on the angle (= <Pa= %) in the form of sin<P, cos<P, or sintycosCp as can be seen from (3.17), (3.19), and (3.20). How are the other matrix elements i n (3.13) evaluated? The only problem remaining now i s the actual i n t e g r a t i o n i t s e l f . In t h i s study, since the basis functions used can reside either on the nucleus or on the positron, some of the matrix elements are two center i n t e g r a l s which can only be evaluated numerically. Fortunately, i t i s not necessary f o r us to do any actual integration f o r the following reason. I t turns out that a Sl a t e r type o r b i t a l a f t e r acted on by the operations s p e c i f i e d i n (3.17), (3.19), and (3.20) can always be expressed as a sum of Slater type o r b i t a l s , or o r b i t a l s m u l t i p l i e d by ^ • o r « In other words, a l l matrix elements i n (3.13) can be f i n a l l y broken down i n t o sums of < F i \ F j > , (Fi l-^r- lFj> and/or < F i l -J jr lFp which are the kinds of matrix element that BISON can evaluate. To v e r i f y the correctness of the c a l c u l a t i o n scheme described i n t h i s section, i t i s appropriate to note that the following two r e l a t i o n s h i p s are indeed s a t i s f i e d as can be seen from the r e s u l t s presented i n Chapter 4. < * l ^ > - - ° <4 I V p ' * > - - - < ^ > - < ^ ' i ^ - < 7 ^ 4 l ^ > j | > ( 3 - 2 1 > these two equations are obtained from -^ -<<£l<£> and Vjf{4?|<|>> * P \ using the fact that i s r e a l and normalized. 35 Chapter 4 Results Presented i n t h i s chapter are r e s u l t s obtained from a l l the c a l c u l a t i o n s prescribed by the model f o r positron-helium i n t e r a c t i o n . The obtained W P ) i s then used to c a l c u l a t e e l a s t i c scattering cross sections which are then compared with experimental r e s u l t s and other c a l c u l a t i o n s . Before proceeding , r e c a l l that a l l quantities are given i n Atomic Unit unless s p e c i f i e d otherwise. In t h i s study, c a l c u l a t i o n s were c a r r i e d out f o r eighteen positron positions ranging from .3 to 6.5 . Outside t h i s range, c a l c u l a t i o n i s not necessary. For Yp smaller than .3 , the i n t e r a c t i o n i s dominated by nucleus-positron repulsion. As f o r Tp larger than 6.5 ,one simply has a well known dipole-charge i n t e r a c t i o n as described i n Chapter 2. At each positron p o s i t i o n , a s e r i e s of @ were investigated i n order to f i n d (i(rp) and thus \/(*p) . F i r s t set of r e s u l t s to be presented i s the e l e c t r o n wavefunction for various jfp and (3 * Recall that i s expressed by a molecular o r b i t a l <|> which i s a l i n e a r combination of S l a t e r type o r b i t a l s FnAo^  centered e i t h e r on the nucleus *s) or on the positron ($'s). The basis functions used are the following ; -fl0o i.83tf/"f»o» 3.18*7 * £a»» i.6oa7 # 9ioe 1.133 » a n <* fcioi.*33» This set was chosen on a semi t r i a l and error basis. S p e c i f i c a l l y , the basis functions used i n a study performed on the (He H) system J were adopted as the : 36 i n i t i a l set since the electron wave equation of (He H) d i f f e r s from (2.3) only by the f a c t o r Ci . To accomodate t h i s d i f f erence, the parameter % of the <ft 's were then varied u n t i l reaching minimum This procedure was c a r r i e d out at *p = .5 using (3 assumed to be 1/2. The obtained set, admittedly, may not be the best one f o r other choices of fp and $ . Given i n Appendix E are the obtained £( rp,§) and c o e f f i c i e n t s f o r each choice of lr*p and . Notice that f o r each positron position Yp under consideration, wavefunctions at fp + STp and lp-**"p also have to be evaluated i n order to f i n d C' and C* » By using the scheme described i n Section 3.2, various i n t e g r a l s prescribed by the model were cal c u l a t e d from the electron wavefunction. Presented i n Appendix F are the values of matrix elements (between the molecular o r b i t a l $ ) s p e c i f i e d i n (3.13) f o r each choice of lTp and ^ . Note that the two conditions s p e c i f i e d i n (3.21) are indeed s a t i s f i e d . This establishes c e r t a i n c r e d i b i l i t y to the methods used i n c a l c u l a t i o n . The values of \ / ( r p,^) and o O ^ ^ J j ) , given i n Appendix G, can be e a s i l y c a l c u l a t e d from the matrix elements i n Appendix F. It i s now necessary to extract ^(^p) from these two sets of values according to the c r i t e r i o n that Mz » as defined i n (2.19), i s a minimum. As mentioned at the end of Chapter 2, i t i s d i f f i c u l t to carry out t h i s process because 37 of -the inter-dependence among the quantities SC^p,^, U) , V ( ^ P , ^ ) , UU p^) , P ( rp) # and Mz • A f i r s t step one can take, however, i s to obtain an approximate (3(Yp) from the minimum of (I - A* - ** ) + 2 ( I-@KAB - A8 ) + (B*-S2) since i t v a r i e s most s i g n i f i c a n t l y with (3 than any other term i n & . This gives r i s e to an approximate V(fp) which enables to be evaluated for various Qp . The obtained \Jl can then be used to f i n d the l e a s t $(^,(3) which w i l l probably give a (£(rp) a l i t t l e d i f f e r e n t from the one started with. This procedure can then be i t e r a t e d u n t i l reaching s e l f -consistency. In f a c t , a f t e r only two i t e r a t i o n s , i t was found that ($(rp) stayed constant. The values of i C f p , ^ ) as a function of £p are presented i n Appendix H. I n c i d e n t a l l y , during the i t e r a t i o n procedure, the terms invo l v i n g T-r^r- r rand TVT^I^y i° c Z were neglected since s-wave sc a t t e r i n g dominates f o r the e l a s t i c s c a t t e r i n g energy range (0-17.7 eV). How does one, however, j u s t i f y i n using S(rp,(i) to determine (^Vp") which i n f a c t should be determined from M i • Notice from G & Appendix AB that, at each rp , the minimum of \/(p) and % (@) occurs nearly at the same (5 . In other words, = 0 i f = 0 everywhere. I t i s therefore highly probable that = 0 when —"STpT" = everywhere, since h (4.1) It i s plotted i n F i g . 3 how & varies with at three 38 positron positions. The e s s e n t i a l idea of the model i s well i l l u s t r a t e d by t h i s graph. Notice that, for each Tp , there i s a d e f i n i t e value of ($ f o r which S i s a minimum. As expected, t h i s value i s nearly 1/2 when positron i s close by or i n s i d e the atom, and 1 when far away. I t i s i n t e r e s t i n g to see that $ f o r the choice of ^=1 i s higher than ^=0 at rp=.5 and 2.?-.. In other words, using s t r a i g h t a d i a b a t i c approximation (^=1) i s probably even worse than t r e a t i n g no p o l a r i z a t i o n at a l l ( Q=0) when positron i s close by or in s i d e the atom. Plotted i n F i g . 4 i s the curve §(*p) extracted from Appendix H, Notice that over the e l a s t i c s c a t t e r i n g energy range (0-17.7 eV) , no v a r i a t i o n of ^ (r p) with Gp i s necessary. By using ^C«p) and Appendix 6, the po t e n t i a l ^J(Vp) can be s p e c i f i e d . Plotted i n F i g . 5 i s the net e f f e c t i v e 2 p o t e n t i a l ( V(V p) ) experienced by a positron i n in t e r a c t i n g with a helium atom. In agreement with previous fin d i n g s , the pot e n t i a l i s found to be not deep enough to support any bound state. I t should be noted, however, that t h i s p o t e n t i a l does not have the expected dipole p o t e n t i a l form --^--^ (oc =1.38 f o r helium atom) at large ITp . According to (2.14),, V(rp,@»0 a t • large Tp becomes Sirfi-Zo which t h e o r e c t i c a l l y gives a l l multipoles of a long range 16 i n t e r a c t i o n . The net p o t e n t i a l should therefore approach the expected dipole form -4r-br since other multipoles give 39 l i t t l e e f f e c t at large Tp . In the present case, however, because the •£ • s used were a l l s - o r b i t a l s which are incapable of expanding long range polarized electron wavefunction. The obtained £(rp) consequently f a i l e d to account fo r any long range p o l a r i z a t i o n e f f e c t . For comparison, the p o t e n t i a l (V^p,^-0")4 "f^ ) ^ s s i l O W I i i n the same graph. I t would be the p o t e n t i a l experienced by the positron i f the electrons were not influenced at a l l by the positron. Difference between the two curves can be a t t r i b u t e d to p o l a r i z a t i o n s caused by the positron. For reference, the s i z e of helium atom i s shown i n F i g . 6. Osing the obtained N/Op) (long range part corrected to i 1.38 ^ a --^-jpf), s c a t t e r i n g phase s h i f t s Aup to p a r t i a l wave X =5 were calc u l a t e d . The r e s u l t s are plotted i n F i g . 7. Humberston's r e s u l t 1 0 which i s considered the most r e l i a b l e c a l c u l a t i o n i s shown on the same graph fo r comparison. T o t a l e l a s t i c s c attering cross section (J can be calculated from (T = J s " Z(2fc+0 Sirf&jt (4 .2 ) In our case, contributions to (J* from higher p a r t i a l waves (JL>5) are not included since they are n e g l i g i b l e . Plotted i n F i g . 8 i s the c a l c u l a t e d (J" as a function of positron energy below the f i r s t i n e l a s t i c threshold (17.7 eV) • On the same graph, r e s u l t s from two experimental groups are p r e s e n t e d ^ . ttO Discrepancies among t h e i r measurements are s t i l l unresolved at the moment. Results obtained from t h i s model for positron-helium i n t e r a c t i o n i s encouraging. With only f i v e b a s i s functions, the c a l c u l a t e d e l a s t i c s c attering cross sections are i n f a i r agreement with experimental r e s u l t s . C a l c u l a t i o n s can. d e f i n i t e l y be improved by using more and better basis functions. More extensive c a l c u l a t i o n s f o r helium w i l l be i n progress. Argon atom w i l l be investigated afterwards. io l .ooolL Fig. 3 &)at three positron positions for 6p«" 7 eV. Fig. 4 (30^ ) obtained from minimizing &(l"p,(3) Fig. 6 Size of helium atom, I.e. electron radial probability density as a function of radius. Radius in Bohr 45 and f part i a l waves. Dotted lines are Humberston's results (flef. 10). Fig, 8 Total elastic positron-helium scattering cross section. Experimental points are from Ref. 6 (dotted error bars) and Ref, 7 (solid error bars). BIBLIOGRAPHY 47 1) Stewart, A.T. and Roellig, L.O., Eds., 1967 f "Positron Annihilation". 2) Lee, G.F., Orth, P.H.R., and Jones, G. 1969. Phys. Lett., A, 28, 674. 3) Orth, P.H.R. and Jones, G. 1969. Phys. Rev. 183, 7. 4) Leung, C.Y. and Paul, D.A.L. I969. J. Phys. B, 2, 1278. 5) Coleman, P.G., Griffith, T.G., Heyland, G.R., and Killeen, T.L. 1975. J. Phys. Bi Atom. Molec Phys., Vol. 8, No. 10e 173^» 6.a) Jaduszliwer, B. and Paul, D.A.L. 1973« Canadian Journal Physics, 51. 1565. 6. b) Jaduszliwer, B. and Paul, D.A.L. 1974. Canadian Journal Physics, 52, 1C47. 7. a) Canter, K.F., Coleman, P.G., Griffith, T.C., and Heyland, G.R. 1972. J. Phys. B» Atom. Molec. Phys., Vol. 5» LI67. 7.b) Canter, K.F., Coleman, P.G., Griffith, T.C., and Heyland, G.R. 1973. J. Phys. Bi Atom. Molec. Phys., Vol. 6, L201. 8) Houston, S.K. and Drachman, R.J. 1971. Phys., Rev., A, 3, 1335-42. 9) Aulenkamp, H., Heiss, P., and Wichmann, E. 1974. Z. Phys., 268, 213-5. lO.a) Humberston, J.W., 1973* J. Phys. Bi Atom. Molec. Phys., Vol. 6, L305-8. lO.b) Humberston, J.W. 1973* J« Phys. B» Atom. Molec. Phys., Vol. 7, L286-9. 10.c) Campeanu, R.I. and Humberston, J.W. 1975* J. Phys. Bt Atom. Molec. Phys., Vol. 8, No. 11, L244. 11) Schrader, D.M. 1969. Phys. Rev., Vol. 1, No. 4, 1070. 12) Vogt, E.W. and Lascoux, J. 1957. Phys. Rev. 107, 1028. 13) Wahl, A.C., Bertpncini, P.J., and Land, R.H. 1968. "BISONt A FORTRAN Computer System for the Calculation of Analytic Self-consistent-field Wavefunctions, Properties, and Charge Densities for Diatomic Molecules", published as an ANL report, ANL-7271. / 48 14) Roothaan, C.G.J. 1951 Revs. Mod. Phys. 23, 69. 15) Peyerimhoff, S. 1965 J. Chem. Phys. 43, 998. 16) Kleinman, C.J., Hahn, Y., and Spruch, L. 1968 Phys. Rev. l65» 53. 49 Appendix A Derivation of (3.6) =4<<£l 4- <$' i - <$'i **">•< **ift*'> = -<<£|V*$> = • f <V*q?| Vp^>+<$'|<$ ,><^Vp^>+2<^|Vp$y^l7p$ ?> SO + 2<?P$,|$'>-<?p$VPl*2>+2<7p l^^ ><7p$,l?p^> 7 ° » . * = ° T^N>r p arp ^ r^ eb rp>ep",'rsJntv^rpO^ I * / ] = ^ r ( 2 < V p 2 $ | V p i $ > + 2 ^ l V p 2 $ > % c ^ *rP I rpSj-^ Hp7^ >ifp 'ipSihepjep' HplVrp~*§p'MpBp»p"*V + / _ ! I _ ! _ _ > # v / ' vFt » N rPsih6p 1 rpsi«ep ^ rpsjnbpx(pl rPs»eP xp,/J J 50 , v»<& 13$ J , , « i ' ^ \ , / 1 ilL i_j " ^ r p 1 -atp ' v r pa6 P> rP->ep7 Vpsin8p *<PP V rpSin&p->>9P' C ^ >r p 1 r P>©p ' T * v -»rp ' ^ e ^ f y ' v i ^ V ip^y**?.. =^f£Vip^ + < T ^ * * * * ' W < # ™ > ( i S B t ^ 1 , 9 / _ i V | i _ J *®\( 1 * u 1 f 51 A8 S<$ ,^l(-^--t p)TJVp^V> I < * ' I Vp' * <$ 4 | Vpl$*> * 2 < * * l vp f > . v pd5'>] AC- < ^ ' < ! > l | ( ^ ^ ) | - ? p ^ ) > . y p U 1 j . - <* ,*'ll | r , p + r>pJI ^ p « + »rp * . ? >ITp u + < $ $ I (-r7p - -7-)| TPIe>^ + * U + 2 < $ | - ! — 3 L ^ > _J l U J L * rp*»*p »fy V»»&p *<PP Ll g a s < 4 v p W ) | -Vp (^J> . v P a-[x = ^ ( $ ' V P 1 **+ $V P '$'+ * I #yP$V £*7P$'>-7pU-Q-= ^ ^ V p ^ ' l Y p ^ + K v p ^ ' l y p ^ ^ - V p U - n - < V P * I •j>*-p>>Kp u ^VP*I r p ^ep ? r p »e P u 5 2 Appendix B Derivation of and VpV Let (Tp, ©p^p) denote the positron coordinate. It i s defined with respect to a system centered on the nucleus and fixed i n orientation. The coordinate systems of -f and ^ , ( r a t ©a,T) and ( r b , © b J < P ) , are defined with respect to the positron-nucleus axis as shown below. For purpose of i l l u s t r a t i o n , i t i s convenient, but with no loss of generality, to orient x a (or ) i n the 9 P direction. The diagram below defines the symbols used in this and the following appendix. The present aim i s to find out how ©a and <P vary with changes of 0p and Qpp . Consider f i r s t ©a . Since I - fa Sin & a , racos©a ~ | . But dJN r& cos ©a C«eP)(- cos <*>) - ^ = -cos<p. Similarly, SmQp $.n e P Consider now the case of <P which i s much more complicated. Since (ra - Ta cos 6a fp ) - e P = £ cos <p d © p u ^ - • - " ^ f j - - d i p - - H.-»Y - j ^ p where • has already shown to be - fa cos 6 a CoS<p At © P + d 8 P , (ra-raCos0aCp)-©p becomes [jTa - ra cos (Oa - cos q> deP) C fp + d9P eP)] • f 9 P - dQP fp] Keep f i r s t order in d©p , the above expression can be simplified to ( Ta - r a cos9a f P) • e P - fa cos 9 a dQp Therefore - ^ f (fa-fa C*>S9a CP ) • | P ] = - Ta cos ©a d <D _ cos Can use similar technique for -d_5L. . Since dQpp (ra - Ta cos e a r P ) • Gp = 2 coscp -|^[cra-r3coSoa rP).eP7^ cos T _ 2 - OP where has already shown to be *a cos ©a sin<p $<" ©p ClQPp r At <Pp + d<pp , (Ha - T a cos6a fp) - eP becomes [ra - ra C05(ea+ sf'»epsin<pd<pp)Crp+ sinepd<pp£ip)]. [eP + cosepdov^ p] 54 Keep f i r s t order in d<pp , the above expression can be simplified to C Ta - fa cosQa r p ) . Q p - Pa sin ©a Sin<p Cos Sp d <Pp Therefore, ( fa - cos 0a Tp) • <§P] = - ra sin 0 a sin <p cos Op For -f , the operator V p can therefore be written as «* = o ctrP rP dQp~ rP I <jea+ ?«.ea 2<p J The above expression can be simply reduced to Eq. (3 .17) by using the fact that -p i s independent of 9 since m=0. "2 C It i s now straight forward to derive the expression for V p i- . By definition, But = s i " ^ - f£i! t ^ f p ">*<P ^ _ f o r m = 0 o r M t a l s . efore 2 Appendix G Derivation of V p $ and "Vp ^  Consider f i r s t how 1^  varies with ^ , © P , and % . Since rb*=ra2 + r P 2 - 2 ra rP cos ©a where and &a stay constant as Y~P changes, one has 2 T b ^ ^ 2 r p - 2 r a cos©a -rjpp- = -7=-frP - fa cos 9a) = cos 9b For , since Va a*id Yp stay constant as ©p changes, one has where ' . ^ =. -cosSP a s shown i n Appendix B. Therefore 7) Dp = - r a r p s i n e a e o s < P _ r p S | r i e b C o s ( p Similarly, -^5- = r p Sin 0 b sio 9 Sin Qp Consider now how Qb changes Yp , Op , and <Pp . Since H N rP2 - 2 r b r p cos 9b where stays constant as Tp changes, one has o - 2 r b ^ + 2 r p - 2 r p c o s e b ^ + 2 r b r p s . n e b ^ b It i s shown above that - J * ; a Cos 9h , therefore 36b _ -sin 8b ? rp - rb 56 For ^ • , since fa stays constant as 0p changes, one has where ^ = - r p sin0b tos<p as shown above. Therefore = -coscpC-^-cxjseb - 0 Similarly, Z Q b r . The case for <P has already been considered in Appendix B. The results are r b Sin ©b 'b Sm t?b For $ , the operator Vp can therefore be written as J _ J L _ =-coS<pfsin© b- (-^i'2 • j>_-i ^.(pf ' a I Cos9b "a 1 _j 3 c - . n t o f < ? , n 9 K 2 , C^ Q^b -a Ll_l+cos(pf—J ^ _ ' c°sfoB 1 » ' c a s 9 p * rP5i^ ©p ^q?p" Y »• 0 2 r b rb aeb rp -j©bJ T l r b s i n 9 b aop " rp s;«e b aq?J ^ rp s i * e P The above expression is reduced to Eq. (3.2l) for m=0 orbitals. To 2 derive V p using the above expression is a straight forward but very tedious process. It is not necessary to go through the detailed steps. The result for m=0 orbitals is Vp %~ ^ + $ - -pr LSl,0b - ^ b - t y ^ - H 7 ^ - ¥ - _ J 57 2r 2 Appendix D Alternative Method on Deriving VP * and VP ^ -Consider f i r s t the case of . Since -f i s defined with respect to a coordinate system which rotates with positron's angular position, i t i s inconvenient to apply any operator with respect to positron's coordinate on f . Now we wish to express I In a coordinate system fixed in space. It i s convenient to choose one with the same orientation as (x ,y ,z ). The thing to do therefore i s to rotate (x ,y ,z ) to p *-p p a -a -a coincide with (x ,y ,z ). How does -f transform under such a rotation? -P -P "P In the present case, since the -f 's are a l l m=0: orbitals, one can use the well known Addition Theorem for spherical harmonics to express £ in the new coordinate system. Specifically, 1 ft ( c o s e , ) - ^ £ Y , m ( e , < D ) Y , * c e p , ^ (D .1) where, as shown in the diagram below, (9,<P) refers to the new coordinate system (x,y,z) which has the same orientation as (x^.y^z^) and i s completely independent of the positron position. itnin Since Y{0 ( 9a") = (cos 8a) 58 one has ? n J l 0 ( ra,©a) = R„(0( « ) £ Y ? m(9,<P) Y 8 m(e P )<P P) (D.2) where Rn stands for the radial part of -f . The. operator V p now can be simply applied to -f since only the term Yem^ ep,^  i n (D.2) needs be operated, i.e. ' ^ m - r which i s identical to the one derived i n Appendix B. A similar transformation for <j can be carried out. Now l e t (x,y,z) stand for the new coordinate system which has the same orientation as (Xp,yp,Zp) but centered on the positron, as shown in the diagram below. | % In the new coordinate system, again by using Addition Theorem, ^ can can be expressed as 9n f l 0Crb,eb)= Rn(r)^_) £ Ve,<P)Y2m(ir-ep,n+<pp) (D.L) L" m---f. Unlike the case of "f , however, i t i s s t i l l d i f f i c u l t to apply the 59 operator Vp on $ because the new coordinate ( T, & , 9 ) s t i l l depends on the positron position. Because of the simple relationship (x=x'-x , etc.) between the two coordinate systems, (x ,y ,z ) and * p ' x_p'-p*—p' (x,y,z) , i t i s convenient to consider V p 2 in Cartesian Coordinate. Since <fy now i s of the form 9 = % ( v. X >^ *p,Vp.,Zp) where the dependence on x^ p etc. i s e x p l i c i t , one has d * p " 7 * a^p + - 3 7 -»x P -a? * * P + -3*p v ^ o ; where -4xr-- a n d should be distinguished. Since =. — O «*p 0*p ^ X p X p and ? < = -1 , Eq. (D .5) i s reduced to ~i Tip d^p ~ -ay -ayp Operate by again, one gets (D .6) (D .7) dVp1 ^y 1 ^yp >*p -a-/p1 Similar expressions can be obtained for y^and z^ components. Since <^  can be written in the form 6= ZI [9.^ .) 9 a(i2p)] M (D.8) as indicated by Eq. (D.4), vp2^ becomes '-I »(p i y aVp"1" "a* Hp' (D .9) 60 The operator^ Vr i s invariant under rotation, i.e. ^7r - ^rb , hence L[h ^J>}m = I>rb2t«,L = (D.10) m Furthermore, since $ 2 - \Qp)(n-9P Jt(pp), one has Therefore, VP2# i s simplified to ^ P 2 ^ = If2)[SV9,-?p90i» ' (3.12) P rr> This expression i s identical to the one i n Appendix G except the last term which i s d i f f i c u l t to be simplified further. Appendix E. Electron Wavefunction r 0,3 0,00 r 0,20 0,50 0,00 o.as 0 , 2 9 9 0,100 — o , 3 o r ~ 0 ,299 0,100 0.301 0 ,299 0,100 ~ 0, 3 0 1 ' 0 ,299 0,300 "0,301 •— 0 ,299 0.300 -0 ,301 0 ,299 0,300 " " © . S O I — 0 ,299 0,300 0,301 0,60 0 , 2 9 9 0,300 0,301 — 0,70 0 , 2 9 9 0,300 ' 0,301 ' ' 0,50 0,53 02 ,660099763 •2,860100431 '•2,860100699" •3,485136043 " 3 , 4 8 4 6 9 7 0 7 1 • 3 , 4 6 4 2 5 7 6 1 9 -•3,816117943 •S.B156II2271 •3,814946129~ • 4 , 160015183 •4 , 159038671 •4,158061459 " •4,116516681 •4,115190411 • 4 , 1 3 4 2 6 1 5 9 9 " • 4 ,516122523 •4,514640281 •4,511557709 " •4 ,698834971 •4,697390631 • 0 , 6 4 5 9 « 5 9 7 9 ~ » 4 , 6 ' 8 4 6 6 4 1 7 1 - 4 , 8 8 3 0 5 1 2 5 1 • 4 , 8 8 1 4 1 7 8 1 9 " •5,265774673 •5,263800961 •S,261626789 ~ 0 0,«5" 0 0 0 O.Sff a 0 0 0,55" « * i •> v tf , S 0 2 "oTio--«;75~ 0,300 0,502 "O.OOS -0,500 0,502 "0,498" • 2 . 8 6 0 1 0 0 4 4 3 -•2,860101010 •2,860100413 • 3 , 5 3 7 9 1 9 6 5 0 " •3,536846946 •3,535774456 • 3,679808608"-•3,678496945 =1,677185882 ' " • 3 , 8 2 3 8 2 4 4 7 4 — •3,822265965 •3,820708483 " ' • 3 , 9 6 9 9 7 0 8 2 « -•3,968157729 •3,966346125 - • 4 , I | 6 2 « 9 J 5 5 -•4,116174091 • 4 , l | 4 | 0 0 8 2 1 • 4 , 2 6 8 6 5 9 8 5 7 ~ •4,266315019 •4,263972708 ' "•4 ,421200188" -•4,418578555 •4,415960019 " "•4,575866252— •4,572960800 •4,570059063 " • 5 I 0 5 2 5 « B « « 0 ~ •0,967360546 •0,967360546 — • 0 , 9 6 7 3 8 0 5 4 6 -•0,951751561 •0,951777240 - • 0 , 9 5 1 8 0 0 3 3 5 -•0,947878432 •0,947838222 - • 0 ; 9 4 7 7 9 7 B 9 2 " •0,938152156 •0,938066575 " - 0 , 9 1 7 9 8 1 6 7 5 -•0,930892312 •0,910795735 - • 0 , 9 1 0 7 0 0 4 4 6 -•0,921864450 •0,921768823 • 0 , 9 2 1 6 7 5 2 0 2 -•0,910881165 •0,910801114 - S 0 , 9 1 0 7 2 8 5 7 2 _ •0,897681861 •0,897645946 • 0 ,897614416 " •0,862970514 •0,663145225 0,097699510 00 ,150174021 0,097699530 •0,154174021 0.097699S30 » 0 7 1 5 4 l 74021" 0 . o o n 7 Q ? < i n o 0 ,140159661 0.101S1690S " 0 ,103265199 0,113752679 _ - . 0 , 1 11436524 •0,863327338 0,113120233 ,090 92509 30,090740188 0,119752085 " 0 ,090687965 0 ,139345513" 0 ,090034985 0,279116729 0,089942266 0,278600611 ~070B9S49831 07278085186~ 0 ,091929751 0 ,425288912 0,091790014 0 , 4 2 4 6 1 8 0 8 ] " 0 , 0 9 1 6 5 0 7 0 9 0 , 4 2 1 9 8 6 9 2 1 -0,091881006 0,501119878 0,091715188 fl,5005«4S84 - "0,091550250 0 , 4 9 9 8 4 8 0 9 2 " 0,096501781 0 .579S61422 0.096310798 0,578725721 " 0 , 0 9 6 1 1 8 1 0 9 0,577885555" 0,099799130 0 ,660118125 0 .099S77171 0,659352177 - 0 . 0 9 9 3 5 6 0 7 B - 0 , 6 5 8 3 8 2 0 1 6 " -0 ,103768775 • » « " « " —SO-,967380546"-•0,967380546 •0,967380546 • 0 , 9 5 0 4 0 0 5 7 1 -•0,950416249 •0,950431865 —i0,9«528322r-•0,945293250 •0,945303370 - i " 0 , - 9 3 9 1 5 S 7 6 9 -•0,919163711 •0,919171869 " " • 0 , 9 3 2 6 3 5 0 1 6 " •0,912644165 •0,912651666 " • 0 , 9 2 5 1 4 0 2 5 7 " •0,925151578 •0,925167356 " S O , 9 1 6 8 9 3 1 1 1 — •0,9169112S1 •0,916933922 - • 0 , 9 0 7 9 1 7 5 0 6 — •0,907946667 •0,907976511 "~ i0 ,B9823912B— •0.S982794S6 •0,898320320 - • 0 7 8 6 5 2 7 S S 9 T -— 0 , 0 9 7 6 9 9 5 3 0 -0,097699530 0,097699530 0 , 0 7 6 7 1 8 0 8 0 -0 ,078629323 0,078541106 - 0 , 0 7 5 7 5 2 0 6 1 — 0,075643622. 0 ,075535878 _ 0 , 0 7 3 1 3 3 7 1 9 — 0,073001112 0,072871167 " " 0 , 0 7 0 8 6 3 1 8 6 -0,070708177 0,070554012 _ 0 , 0 6 6 9 4 0 2 1 9 — 0,068758040 0,068577094 "-OV06736220r~ 0,067150756 0,066940745 - 0 , 0 6 6 1 2 6 1 5 8 — 0,065883232 0.065641954 " 0 , 0 6 5 2 2 7 7 7 3 — 0,064951249 0,064676596 0 , 0 6 4 4 9 9 2 1 2 — 0,744014811 0.742882B01 " 0 , 7 4 1 7 4 4 2 7 1 " 0 ,923054415 0,921387451 ~ 0 ; 9 1 9 7 0 7 8 0 2 " •0;154174023" •0.1S417U023 •0,154174021 0 , 121009893" 0 ,120181516 0,119355148 — 0 , 1 7 7 3 1 0 0 9 1 -0 ,176134838 0,175161814 "O";234457070~ 0,233330232 0,212205887 " 0 , 2 9 2 3 8 1 2 4 1 " 0,291098752 0,269819046 ~ 0 , 1 5 1 0 f 0 5 0 6 -0,149566956 0,148110504 - 0 . 4 1 0 2 7 0 5 6 0 -0,408667228 0,407067317 ""07070085219-0,468318104 0,466554721 - 0 , 5 3 0 1 7 6 8 9 7 — 0,528444651 0,526516488 0 .7J1323359 0,000000000 0,000000000 0,-000000000" •0,271170174 •0,270935482 — • 0 . 2 7 0 5 0 2 4 2 5 " •0,396396609 •0,39589585} - » 0 , 3 9 5 3 9 6 U J -•0,535197538 •0,534581839 - • 0 , 5 3 3 9 6 5 6 2 8 " •0,610226378 •0,609521577 - . 0 , 6 0 8 8 1 8 9 1 0 " " •0,689421057 •0,68860135} •0,687777869 •0,773248332 •0,772270014 • 0 , 7 7 1 2 8 5 3 2 2 -•0,662164181 •0,66116754} " • 0 , 8 5 9 9 6 1 1 3 4 " •1,060679420 •1,056733080 - » r , 0 5 & 7 6 8 i n r -0 ,000000000 ~ 0,000000000 0,000000000 — • 0 , 2 4 5 3 5 5 3 7 3 — •0.24454929B •0,243745795 -•0 ,297213684 "" •0,296269614 •0,29S32B32S " •0,350709786""" •0,349617850 •0,346526975 " • 0 , 4 0 5 7 7 2 1 4 9 — •0,404523291 •0,403277637 - i 0 7 f l 6 2 3 2 3 9 0 8 — •0,460909954 •0,459499795 " • 0 , 5 2 0 2 8 3 2 2 1 — ' •0,518696929 •0,517114860 "•"0757956167?— 0,000000000 0,000000000 0"iO000O00OO •0,031836640 •0,031872044 —•0,031905318 "0 ,046161584 •0,046218182 — r ,04631 48Bl" •0,063686793 •0,063606052 - "•0,061924583 o0,073737419 •0,073875622 - • 0 , 0 7 4 0 1 2 7 0 6 " •0,084721731 oO,084877565 •0,065029817 oO,096712507 •0,096896740 •0iO9705B639 •0,109066243 •0,110052460 "•0,110215401 •0,140421673 •0,140543205 "oO-iT4065B7V5 •0,577798781 " 0 , 5 7 6 0 3 8 5 6 8 ~ V 0 " , 6 O 0 0 7 « 7 2 2 -•0,638125999 •0,636182411 •0,828735287" 0,000000000 ' 0 ,000000000 0,000000000 - . 0 7 0 4 1 6 9 7 6 7 5 •0,041b59106 •0,0416)9732 "•0,051431011" -" 0 , 0 5 1 3 9 7 5 5 0 " 0 , 0 5 1 1 6 3 0 7 2 " * 0 , 0 6 1 7 8 2 8 5 4 •0,061752769 •0,061721490 "•0,072712845 •0,072704667 •0,072675069 "10,084258727 • 0 . 0 6 4 2 H 2 2 2 •0,064202022 •0,096316469 •0,096308611 •0,096278812 "voTToaoaoffOTi •0,108911444 •0,108680261 " • 0 7 1 2 2 0 4 3 3 8 1 — •0,122012834 •0,121979810 "3V75"6"JI«6"491 O — CO OO rt <0 <0 3 3 3 • • t n rt co •*» o> r » o> • n o j ^ s »/» AI -*» rt o> AJ ftl IP (A 9 « CO —» • * o tft A n > N O o S S o AI * * 1 «•* o o o 9 IA o- IA 9 t w a> N 3 o I O >0 O <\| (O l 9 i t 9 (O OO <*o « o -o <o < « O O O O O «*Oo3AIAl| - * GO AJ 00 < (O (O « 4 rt 9 - * r t 9 4) A » » » OO CD N K K ' 1D3 B3» K O O l O - O 3 O O * 4 o r r t i M n j H o m w n j •-. o eo o co O O O fl O O O 9 I _ O O O i n i A i n f l 4 ) J ) N N K f f 0> :o — L. » O O O Q O O O O O O O —• 3 - « « l / < | / t ^ « * 0 ( f l o o o r » r\j — ru rt 3 o so o-O O O LH J~i o A ^ I N N N C O o o o o o o o o o o o o o O O O O O O O O O O O O O " 9{acortftjcortb> o i r n i i / i n o s a ^ 9 0 H ^ r * M . 4 > o v * N ^ n t o o - a -o m » d i f l d m m t o a > c * o » o * o o o o p O O D o o o o r t 7 w m r . MI s> o> . « m u i m o o o x> — » o i v m ™ « u j „ n i o >» — » [ I l l — — — U » « o » « o ( . « i n » . , m o ~ ~ t\~\ ~ ~ " m i m n n n n n n o o o a k o o a a o ' o o o a ' a s a o M M O O o m » « - A O K i n O 4 » ^ f « . 4 > 0 > - « 4 l 9 < t n t n i n « - w r t M r o « * * ~ ^ e ? » £ e o f u N N N O O O O ^ * ( \ | ( M r u » 0 ' C > U , I L f l U » O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o i n 9 9 9 O O O A j t f l o } 0 « o o o r t ~ « * « c 0 n j O O O J i f l i / U H ) O O O f t t f t l A I W U t O O O A I M N N n i to »*- tn ro HI HI O 9 3 y i ^ > K N O l/l * —• rt r» o o 9 « co —• — r - —• o o m m e> o o> AJ 1/t 49 M iA O -A A i d • « 3 HI o* m n i m rt HI rt > o o o o o o o o o < 0 » » • 8 HI _ _ > o c i N r u o - 4 r t r o> » n j - o « 3 »»» O f\J r 9 4> m 3 O o •> i n o» o> o* HI ru • —• o o o - * v N m o o =x o o o r». 9 ru HI m co n j u> « -a o* O O O O O O O O O O D O O -O 9 r - ^> 4> I O O « S l> N r*» >o r*\ O* O* rt — rti • I • 0 o o o p -*» i*- •»» *s »• 4crgao>B AI so r* — w rtj tO 0> >^ KT Al IM K f» co N M t r u IA|A<M>C> r  rt p o o o o b 1 j l I C II AI •>* co r*. to f\ **\ -a ~o VQ n i r a n i f t i f M O O O O O O O < ~ " I B . •A , ! i n i <o • > D 9 IA ft! 9 <A 1 rt SO 4 OO IO 1/1 i i n -o co 9 HI HI * O"- r - r » r>- .43 »/> » f- N M s ft) <y > D" i n tn i n i n m > o o o o o o i n co o> m HI so « N cr 4 O C > O f \ | N i n o o ru r»» K> ««"t rg o fU eo ao'co Hi i n 3 3 3 3 o o o o o o o o o o o p o o p L =o 3 L r u >o o 3 In c* c * -o M M C> 3 3 HI O O O >inMonini 0 | > oO o o <o o* to co m HI i n i n r t H i M O* O 1 U l t / t u t O O O O o > p o o o o o o o o o « o o o o o o o o HlWWfcf <0 rt 0> 9 r>» 'O m i f l i n l w o n i o o M i n o*- o> o> i n co " co •< 3 cv « -O « ^ V ( M H > « N N M N N N O Cf 3 O* 0> O* U"i O O O - i f l i n ^ o 3 3 C f — _ i j o o o o >jo O C o «o O O CO o 3 eo o tn o i n i n -o co co o* O 3 £ f ) N a 3 O O K m m i n m m ' o o o> o o i n m 3 m m I o o o o o j O O O O O O O O l K 0 3 « « n o 3 m 4} " l e > 0 3 » B 3 3 0 > B 3 C > ( D 3 « < o n i A < i t H ) i n c f n i i n N < A K i I •) M J) * • 9 rf| ' 9 -O O 9 41 <n > i n -o o> Kt o> • o co -o m tn 1 03 3 <4ft O > rt 0 co so co o* K X 4 4 0 0 •o rt rt rt rt H V O C* 0> 0> O" o* o o o o o o i l l I * • m 3 O 9 co <o co i n CO « A I O p« rt o cr oo o> 3 ifl i n N O 3 3 3 t— rt ru ra ru o o o « o> » r tn rt 9 tn HI ru 9o co ru t n o o m co rt o o rt m * • * co ru O i ^ — < r y flow K 3 l > 4 3 0 * 0 a £ I V O N C f l } 4 ) 9 N N O O O f t i r M M * o o o o o o o o o o -o as 9 9 in tn * x <o o o CO CO CO SO CO r u ru o > < u r u ( M i o m c r a ) K K » * o i r * o» o» O ' O O O M ^ ^ ^ K I I ru rt rt HI m HI rt393333333 o « O J l sO X CO X 3 rt —• o* ec A <o K 4 >a f s O H I M O N M U I Q O ; o 3 i n N o ^ o r < - ~ « o . 9 N O N r U o 3 0 s O > 0 | i O* rt CO I O - * H I O O * - * f i m 3 r u < n r t * « u K K C 0 4 } c O O > 0 0 > ( «OCOJ-O>O» o o o r in rt i n co o co i o a> i n 9 ru —• 3 3 3 3 3 3 » r u c o o A t c o o r u c o < o r u a > o r u e o o r t « o c> o o o > o o o > o o » o o » ru co > o o> • o* CO o ru co o r u o o o » o o o c * ID o> e> > o o o o o o o o o o c o o r u c o o r u c o o r u c o ^ o o c o o o ^ o o c * C * » a ) » 0 ' < o o ' o e I M K 4 Q O O I V n i < o eg o i n eo o rt i n x co 9 c r t> rt co rt c o K eo co o* rt 3 3 i n •3 3 3 O C* C* HI rt rt rt n j n j m n i o> o» o> o> o> o> o* Q O O O O O O O O O K e> n i h i n i n h i OB i n r u c o r u o u a t n H i o H i m o Cf ( V r t O > O D > I A O A i M * « o o r f i o r u f u « t f i « « o o o p K i p c o > * u i a ) - • - * - * - * 0 > s 0 9 AlO>r-> o o o i A H i r u x m r t e o •O -B 3 3 3 HI HI HI A * v O O A J t O O A j a O O A I O O j 0 > O O D > 0 0 0 > 0 0 9 o o o o o o o o o o < o o o o o o o < L 1. • — * * - * o o o o > o * o » c o a i e o 0 0 0 0 0 0 0 0 * « > * * « o o o o o in c* i ru — i cn in i AI r u i i - o 01/1 c i n i o r t ' r t o ^ < x > 4 > r u < D 9 o o o o o o O O o o O O o o O - * U * A I - « - « A I » < » 0 9 ^ * 9 - * o r - r - « . o a t n v o N i n N o o o o o o o o AI t> O Al <M - * O- 4> 9 co N r u • « o n O O <C D d l/t •O O -O 4 O *•» o o o o o o o* a a1 at j% r u o u i M o -O 0> r » i/t H I rt rt o o o o b o o o o o i • • 6 0 B k O O O O O O O O O O » 0> r •43 Al P > -* In e it) n o A o 0 r l n i l f l 0 3 3 0 o f f « o w m n j o M f l * • * - * i A 4 > < o o o * * * « - « * a 9 9 9 9 9 r t r t r t O O O > o o r t o 4 > « - o D > o t n > o o p » ^ r > > o o f \ i r » » O O O I > < 0 0 0 0 « « > o o i P t n m a > N r » o k O O O O O O O O O * rt tn o m o m o o r> 9 9 * • m o ^ o n r > i n - * o - * * t tn o o ru AI AI 9 ru AI ru ru r u ru MI . © ! » — > r u » a> i n rt-9 AMI/IK A I r u ^ 9 9 • 0 B ,0 Q 0 mowif lKoor t J A i > - < t - 0 9 0 r t r t - 0 o v « H n i o 9 m o t n r t r o t o v r x M o o ( t n o o o r u r u r u t n { < s t n o f V ) « n j r> o o a> r u 9 •o ru ^ _ x IA o o 9 co AI 9 9 ^ - -O O-O O O O O O 00 i * CO - o a « r > o tn 9 ao ru rt O 4) 9 9 4> i n o o c •o o to r» N 0» O « » a a O O A l A i A l O O O O O O O O O O O O O O O O O i o o m i n O* o> 4> 4> M « K t 9 9 N I V 9 l A ^ O t r t A l A I r t * * » T H » O ^ A i l M ^ N n j - o m o i f l o • O A I r t r t O > - O O > 0 o o o o o o o o O I A 9 « 0 4 ff t o i A i n o 9 9 9 9 9 9 r At tO tA 4 m W CO AI 9 O H « H I 9 K t n o»- o* r-- <D o o o* o o 9 9 O O O O O O O O O O O c O c O O O O O O O ^ N J I N N O O O - 0 - 0 c 0 < D 0 » 0 0 0 - 0 ^ 1 2 - ^ - 0 . 0 O O O O K l h IMl/l O O O t r O O O M O O O O - O - C y - L n i n O O O r t r t r t 9 9 O O O O O O O O to Al « o o> rt 9 Al I*- tf » « » • * 63-i ~ * r t 9 t O O r t r > 9 * * k « rt—AJrteO<Of0O>-*Kl c o r u - » o ' j i 9 r t j n 9 i r u " « » w a 9 9 rt rt ~rt i n i n i A i n . a . O H > - * - « ^ « o o o o o o p - * - * u * • o o p [ I r t i n rt Jt tt I O O 0 N I S 9 I / I I > o o A J A i « * i r » r t < i o o o co a) i / i 3 i » O O Al Al AI 9 9 = ao -o co i n 9 p i n 9 Al O* - * rt «© CO O* CO ^ •o ru o* o rt .o rt o A • * r * rt ru AI - * o o* 1 0 9 0 < 0 0 c > »n 9 rt rt 9 = r » <o ru 9 ru c ru « » » JS -o 4) ao r» o io :c rttn p o 9> « o> « r* <o m b> eo 4> t* rt f*. in o lO 9 rt 9 Al A * w+ 9 At JO 9 0 > O S > 4 ( A a> a> K . A I A I w* <*• O* i> 9 9 » * i - » rt rt rt • O O Q O O O O O O O O O O O I N -< » N U CO • irt to ru o» rt r • o» o rt 9 « t ru AI rt — . •> o o o o o < IA IA CO IA • r* m rt 9 o m ru iA w j t i > h > A i n i o 41 9 O 9 CO » 9 O —• -O -« *« x» r -o o o ^ * co «t to r» co co o CO o 0* A * rt r- 9 9 »n rt ru h N S I M 3 >^ •o «« 4 -0 co b 9 9 rt O O* O* rt rt rt -o IA m o o rt rt rt lA IA IA O. o» O o> r> o* •O -O •£ o o p o o o o o o o o o o o p o o p o o l o " [ " [ " i ' i c r [ o o 4 > o r » 9 ^ > c o t n o > 9 > » » r t c o 9 0 r t 4 i M 9 0> »^ O co tA rt 41  AI m co o i o 9 r> 9 -o « « rt rt 9 9 m rt rt rt o» <o *o i n 4A - • 9 o s i n o o h t f i M A I A I C 0 t A C 0 O 9 - 0 r t -* f y a N « 9 - « r t i n » f ^ r t O 1 in to ru o> rt o r ininm<oiA<otoiA4)>j 9 - i A i A i n » « « ~ « < - « r « f f ^ i i n m m t n i n i A i A 9 9 c O O O O O O O O O * i IA in i - * ru > eo co rt99cOD-«rt~«rt9 0 > 9 — O i r t O * * - * 9 r t rtlAr*C009f«-««94> l A f f O f f ^ . o ^ r t r t r t o> o r ^ 9 ( 0 4 9 ' O n i i o < 0 9 9 i A c o r > o o ««ni ^ N N N O O — 3 U M / 1 A I * « * « - « — i l A L A l A C> » O1 » » r> O CO IS CO o o o o o o o o o o o p o o o o o o o o o o o o o o I ! ! . I. < O « 0 - O O K > < O t A O r t - 4 > t A 9 f U t A O m r t - 4 > A f 9 < > O 9 9 9 r t - « 0 » t a . | A A J ( A 0 9 0 k - O A I i A < 0 9 r ^ 9 0 t n L A i A o r - > 9 c o 4 > 9 A i i A r o o - * r u 1 o t o o A i r t r t O O O 9 ( 0 n M 3 N 4 N O ( D C 0 O E 0 K N 9 r U O » n a 9 r t N A i * « . £ < « a } 9 0 9 « ( O C O ' 0 9 4 ^ 4 ) rtrtrtAtrurtA4>r«4i^c09iA^icoo>ortin4) O* O* C 9 9 9 0 > 9 9 ' 9 9 9 C O C O O O O O •O -O • O r t r t r t r t r t r t r u A l A i r u r u A i - « - ^ * « » ^ t « * ^ .  o o o < I— o o o o o o o o o o o o o , > o o o o o o o • I A m o m rt O* O 9 o» i O CD I Al • o o ru CO IA AJ CO IA in AI rt 9 o* o • ru rt 9 m * * r t > o o <o r»» o : 9 « P > 9 C > 4 | ru i n to — rt 41; *0 9 Al lA Al O* • ~« _« o o o * : o o o »>» 4 i ; A I O A I 9 « - * M > - M \ I M O O O r t 3 l > f m O I > m A J A I A l i n o - « r » O k * « 9 9 « J — i K t N A I O" « • « Ai O O O r t r t 4 > r u A I I A M O A l c O l A 9 9 4 > O t « - r t l A o o o ^ s O m * * i A r t o r * 9 « 9 0 4 > o i A r t r t r t O O O r t A l — * A l « « O A I * « O r t ^ 0 9 r t * * e O < < 0 9 4><0419 9 9 A J A l A I O O O c 0 X C 0 - 0 4 1 < 0 ; A l A l A COCDCOinmiA 4 ) 4 4 > N S K K N K c o n c 0 9 a 9 c O P A i e o O A I c D O A I c o O A I d O O A i e o O A I c o O A l 9 0 0 0 * 0 0 0 0 0 0 * 0 0 0 * 0 0 0 ^ 0 0 0 * 0 0 > 0 4 ) 4 > r t l A r t ^ r u r U ^ r ^ r > 0 > 0 - 9 r t « « ( A A l A f l 9 9 9 9 0 m m r u S M ^ 9 H | M M N O ' > 1 - « 0 « M I A l A l A * « r ^ r t » - ^ A J O > » * I A A i * « » 4 > 0 * - * c 0 * » ( 9 o o o a N M » o * « j i 9 ) o n i » i n i A 0 4 c a « i < \ O C O C D K m A n t e S K l ^ K t n A j C O t f t r t ^ C O f O N M N K M ' J M M O ' f f O k n m t A * * > w M 4 ) 4 1 « 0 9 9 9 • 0 < 0 - 0 9 9 9 r t r t r t r t r t r t r t r t r t A I A I A I C O c O R > 0 > 0 > 0 * 0 * 0 > C > O k O * 0 * O t O i O * 0 " - 0 » O k O > 0 * O k e O c O C O < M > < A 9 ( D o a « M A K i i n 9 N h > o r t n o « « « O 4 ) r t « * o k 9 o r u r u 9 -o h» ru o> co -o -o o m o ru o o o « o n i x i K n i « o i / i « « n i H ) x i 9 » 9 u O O O ^ - r t D O * 4 3 9 C O r « I A C O t , ^ l ^ > A f A i A I 9 4 l i O ) o o o « o o o o > i > o o t R ) o 9 > g } « * o D ' r > a ) t a 4 > - 0 4 ) < 0 4 > ' 4 ) r t A I A J O 0 k 0 > r » 4 l 4 > 9 9 r t i n i A l / t c o « a > 9 9 D i A t n i n 4 i i / t u v « « « N N K A ) n i n i coorucoolujeoonicooLcBoiucooi^  * Ai • o* o o > 9 m IA 0 * 0 0 0 > 0 0 0 > 0 0 0 ' 0 9 1 A I A 9 1 A I A 9 I A I A 9 I A — IA o IA 4> • • O o o o.as ~<r.sv--OT60 — -0765 "T700— - 1 , 9 'fig. I « . - 5 I •" I I. 0.50 - 0 . 5 - L 0 ,60 0,»5 1,00 1,700 1.702 1.696 1,700 1,702 1 . 6 9 8 -1,700 1,702 l , 6 9 « -1,700 1,702 1,698 " I .700 1.702 i . 6 9 8 — ,700 .702 , 6 9 8 " ,700 ,702 1.896 1 . 9 0 0 " 1 . 9 0 4 1,896 1.900 -1,904 1 ,896 1.900 " , 9 0 4 . 8 9 6 , 9 0 0 " " , 9 0 4 , 8 9 6 .OOO" -, 9 0 4 , 8 9 6 , 9 0 0 — ,904 ,896 900~ 904 •2,860100224 •2.860100807 •J .194612012 •1,194028791 1 9 1 4 2 7 H 9 • 1 . 4 5 5 8 2 9 2 2 2 -- 1 . 4 5 5 1 5 1 7 5 3 •1,454480179 • 3 , 5 1 7 4 7 7 0 6 2 — •1.516728203 " 3 . 5 1 5 9 8 1 3 8 9 •3.579604692 — •1.578781293 •3,577960089 • 3 . 6 4 2 2 4 H J 2 — " 3 , 6 4 1 3 4 4 2 2 1 •3,640447459 •4 ,098092992 — •4 ,096632003 "0 .095174449 _"2 .860100221 • 2 . 6 6 0 1 0 0 8 0 9 " •2 ,660100648 • 3 . 3 4 0 2 4 2 6 1 2 • 3 , 3 3 9 2 4 6 8 6 8 -•3 ,338254097 •3 ,394917862 •3 ,393821248 " • 3 . 3 9 2 7 1 2 U 7 "3 , 4 * 9 9 8 1 152 • 3 . 4 4 8 7 4 5 8 4 8 — • 3 , 4 4 7 5 1 4 4 4 7 _ " 3 , 5 0 5 3 9 9 3 6 2 •1 ,504041418 — •3 ,502687857 • 3 . 5 6 1 2 2 | 8 5 2 • 3 . 5 5 9 7 3 9 2 B 8 — •3 .558261587 _^1 ,966161242 •1 .963949098 •1 .06154S267 '•I 0,00 2 .096 •2.860101119 2 j -• too ""I mr~ •2,660100833 • 1 V If •2,8601 00952— 0,05 2 .096 •3 ,294667459 j" : 2 '2 .100 J flfl"~ 293817226 t * VH •3 ,291010165: 0 ,50 ,096 •3 ,343919679 i — — — 100 ' l n n •1 .141010986 _ • I 104 • 3 , 3 4 2 0 8 6 0 2 5 0,55 I. 096 •1 ,193475579 2. 100 •3 ,392446916 «. 104 "17191422345 0 ,60 1. 096 •3 .4432986^9 V _2. — * — 100 • — — » w • u u j ' f •3 ,442168446 104" •17"44 TO 4 2705" "0 ,967380546 •0 ,967380546 •0 ,947881764 •0 .9479219S4 •0 ,947962101 • 0 V 9 4 4 7 0 5 1 B 8 -•0 ,944792083 •0 ,944838927 • 0 , 9 4 1 3 7 2 6 0 8 -•0 ,941426529 •0 ,941480186 •0 ,917709289— •0 ,917810510 •0 ,917871704 • 0 , 9 1 1 8 5 9 9 1 3 — •0 ,931928746 •0 ,913997510 •0 ,897831228"— •0 ,897957440 •0 ,898083516 0 , 0 . 0 . 0 . 0 . 0, 0. - o , o . o . -07 o . o . "0 , o , - f t o, o . •0 ,967360546 •0 ,967180546™ •0 ,967380546 "0 ,951635645 • 0 , 9 5 ( 7 0 6 7 0 7 ~ •0 .9S1777363 "0 .949121S14 •0 ,949206137 ~ •0 ,949288519 •0 ,946406783 •0 ,946501806 •0 ,946596554 •0 ,943469465 •0,941577511 " — 0,943685247 0,940294222 0.9.40415B83 0,940537201 0,909739959 0,909967937 0,910195399 097699530 » 0 , 1 5 4 1 7 4 0 2 1 097699510 "0,154174021 068619596 •0,059066111 068671612 « 0 , 0 5 9 1 2 1 0 6 6 068721S99 •0,059579108 065271040 40;0471B5042-065330629 "0,047474101 065388192. >0,047762390 061866966 *0,034987507" 061930040 •0,015109136 061993091 "0,035629917 058198351! • 0,TJ2245926«T ~ • 056466799 "0,022813648 "0 058535228 "0,023167112 >0 054864281 "0,009586302- 50 054937975 •0,009971572 « 0 055011652 "0,010359844 -0 028108450 07591104623 SO 028212903 0,090490746 >0 028317389 0,089678334 »0 .000000000 ,000000000 ,104681623 ,104471207 ,104261292 V I18272745" ,118035420 ,117798657 . 1 3 2 4 2 6 8 2 9 -•132162630 ,131899053 7147164529— ,146873599 .146583349 .T6250512J— .162167718 .161871050 7287B57542"— .287373053 .286869682 0 0 •0 • 0 • 0 ~«"0 •0 • 0 "•0 0,000000000 0,000000000 •0 ,034601943 •0 .0345S1640 _*0 ,034505441 •0 .039607836 •0 ,039551042 •0 ,039494366 •0^044732426 •0 ,044666945 •0 ,044601596 "0,049979761 •0 ,049905196 •0 ,049811176 •0 ,055151830 •0 ,055270384 ^0 ,055187102 -070968TSS1J _ 0.097699S30 0,097699530 0,097699530 0,073588485 0,073684115 0,073779937 0,070796107 0 ,070903017 -0,071009708 0,067919000 0,068056956 "* 0,068174686 0,065012199 0.065141337— 0,065270044 0.062011386 0 .062151208— 0,062290796 0.038513172 "0.038722029 — 0,036930735 "0 ,096662450 •0 ,096510579 _ ° 0 . • 0 . " 0 , • 0 , —*o, • 0 , •0 •o, •o. _-o. • 0 , " 0 , • 0 , • 0 . •o, ^ 0 , • o , _ o , o. 0 154174023 o 154174021 o 154174021 o 081411986 o0 081807196 — a0 08219631J . g 072378891 . n 072822761 i"0 071264286 . 0 061056764 . 0 063551999 i 0 064044620 » 0 051428871 » 0 051976050 . 0 054520191 . o 041475860 . 0 044075537 . 0 044672138 . 0 036877605 . 0 035907400 50 034941745 " 0 ,000000000 ,000000000"-,000000000 ,066050476 .085715444" ,085381881 .097243756 .096865299— .096468496 .108997146 .108575066™ .106154815 .12134145s 0,000000000 o ;ooooooooo~ 0,000000000 •0,030279061 • 0 , 0 3 0 2 0 0 4 8 8 " •0,030122230 _^0^034509654 • O , 0 1 4 « l " 6 9 2 0 — • 0 , 0 3 4 1 2 « 1 5 j •0,018842768 •0,038735190 " •0,018628028 •0,043281736 0,967380546 s 0,967380546 0 » ; 9 6 7 3 B 0 5 4 6 - o .954921029 0 .954981307 0 .955041371 o .952953658 0 .953023930 0 ,953093951 0 .9508 IJ76S 0 ,950894663 0 i9S097527S ff ,948484845 6 .948577005 0 "9486688"4"3r ,097699530 .097699530 ,097699530" .078107413 .078191799 . 0 7 8 2 7 5 9 4 1 -.075850640 .075945252 .076039597— .073530282 • 0 73635195 . 0 7 3 7 3 9 8 1 — -•071140428 .071255701 ,0T117066"7— H l 0 B 7 ^ ^ ^ » f | | » J . .120412040 . 0 0 4 j o 3 7 l f 2 .131307417 . 0 0478307?! :m ^ ^ ' ! 3^ oJ~" i^ o 4^' ^''o 244f?i«I "°."<I755S784 ; | 23%^ J ' i | * __ rO , 083064905 .243169?ff 'F082«r2«8-t<41 |6916J •0 ,082561371 •0 •0 ""•o; " 0 , •0, " • o , •o, •o. - * o •0 •0 "V0 •o, •o, ^07 .154174023 ,154174023 154174021-0,000000000 0,000000000 -07000000000" 098718550 » 0 , 0 7 0 9 5 0 8 4 0 ,099019841 *0,070680200 ,099119516 SO,070410677" 091929694 « 0 , 0 8 0 1 7 5 1 0 5 092270157 •0,079869149 ;09260922I iO " 7 0 7 9 S 6 « 2 4 i r ,084891112 •0,089942760 ,085271961 •0,089600601 ,085651)791 i T , 0892598110— ,077581930 •0,100292881 078003700 >0.099914387 078423273 S07TJ99537SITS ~V0 • 0 • 0 ,000000000 ,000000000 70000 0"000O— ,026704586 ,026639798 . . 0 2 6 5 7 5 2 4 8 — .030274330 .010197448 ,"o:sffr2o"B~i— ,033933869 ^013844677 . 0 3 1 7 5 5 8 0 8 — " ,017686971 .037585256 7 0 T r « T 3 W 4 — 0„6S 1,00 ? !~r,T~o;oo-2,096 2.100 2,104 2,096 2,tOO "2,104-•5,095(135109 • 3 , 4 9 2 2 0 ( 6 6 6 •3,490973115 •3,656060739 •3,854058006 ~«3;e5205B90S -"0750" 0,55 T;OO— "2,695 " 2,700 2,70S 2.695 2,700 2,705 2,695 2.700 2,705 "2 ,695— 2,700 2,705 "2.695 -2,700 ' 2.705 "2,695 2.700 2,705 2,695 ~-2,700 2,705 • 2 , 8 6 0 1 0 1 0 6 2 " •2 , 8 6 0 1 0 1 0 0 5 •2 , 8 6 0 1 0 1 0 1 3 •3,196902160 •3 ,196266296 •3 , 1 9 5 6 3 2 4 7 8 •3 , 2 3 0 6 8 1 7 7 0 •3 . 2 3 3 9 7 1 7 0 6 •3 . 2 3 3 2 6 0 2 7 8 • 3 , 2 7 2 5 6 5 0 1 0 " •3 , 2 7 1 7 8 0 1 8 6 •3 , 2 7 0 9 9 8 2 8 8 • 3 , 3 1 0 5 6 0 3 5 0 " •3,309703816 • 3 , 3 0 « 6 « b S l 8 •3 .3O8690060"" •3 , 3 0 7 7 5 6 7 7 6 •3 , 3 0 6 8 2 3 0 3 8 • 3 . 6 2 1 126780 " •3 , 6 1 9 6 1 3 3 9 6 •3,618105908 2,9 0,00 0,05 0,50 0,55 2,895 •2,860100868 2,900 " -2,660101081' 2,905 •2,860101069 2,895 •3,173182805 "2,900 53,17263205a -2,905 •3,172083708 2,695 >3,208216505 "2,900 53,207602659" 2,905 "3,206990698 2,695 •3.203320165 _ 2 ,900—sj ;2a2606 ia i r -2,905 •3,201970298 2.B9J -3,278510805 •0,905908712 •0,906052766 •0,906156065 •0,920090907 •0,920696351 f0,920B97250" "•0,967380596" •0,967380506 •0,967380506 •0,961729610 •0,961769606 •0,961809363 •0,960882080" •0,960928932 •0,960975100 ~50,959947285" •0,960000775 •0,960053900 •0,958911993 •0,958973157 •0,959033908 "50,957762397" •0,957831929 •0,957901002 •0,900622981 •0,904770934 •0,904926120 0,068674715 0,068800386 0,068925734 0,048681990 0,006877965 "07049073585" "" 0,097699530-0,097699530 0,097699530 "0,086044492" 0,088105954 0,088167096 "0,087013844 " 0,087062816 0,087151025 "0,085942193 " 0,08b01B898 0,086095197 "0,084823993" 0,080908675 0.084992907 "0,083652757" 0,063705677 0,083838104 "0,073136404" 0,073295135 0,073453110 •0,069979697 •0,070443143 •0,070904197 •0,006460575 •0,007227777 -=070*1799X380-•0,111267066 •0,110852102 •0,110438771 •0,209098295 •0,208450230 -5TJ7207809425-«0,041537481 •0,001423019 •0,041308967 •0,071530113 •0,071320282 -iTT7071TTT20r-"~*07150174023-•0,154170023 •0,154170023 ""•0,129948244" •0,130116561 •0,130287752 "-"•0,127229827" •0,127421832 •0,127612554 ""•0,124378889" •0,124593038 •0,120806541 •0.121378S29 " •0,121616548 •0,121852961 "•0,118209075" •0,118471564 •0,118732280 "•0,088806802 " •0,089272595 •0,089735481 0^000000000" 0,000000000 0,000000000 ""•0,000322267" •0,040133280 •0,039905169 "•0,045478109" •0,045263768 •0,045050019 "•0,051057226" •0,050816277 •0,05057643S "•0,057110206 •0,056841380 •0,056573773 "•0,063694809" •0,06339685] •0,063100222 "•0,131422240" •0,130901621 •0,130382689 0^000000000" 0,000000000 0,000000000 SO,018991807" •0,018939752 •0,016887843 50,021145403" •0,021080318 •0,021023016 50,023363902" •0,023293612 •0,023223502 ""•0,025650859 " •0,025571187 •0,025091774 •0,026010052" •0,027920823 •0,027831891 ""•0.0068934S9 " •0,046732976 •0,006573066 •0,967380546 " •0,967180546 ' •0,967380506 •0,963152613 •"5079 6 318 3 817" •0,963210822 •0.962533138 0,96  0,9625-•0,962569278-•0,962605185 •0 , 961806850 "V0.961BB840r 0,097699530 0,097699530" 0,097699530 0,090264699 ""07090314202" 0,090363511 0,089501929 ""0,0895S73S3_ 0,089612063 0 . 0 8 8 7 0 6 7 6 7 6S , 4   0,088706767 -V , 6——0,088766295" •0,961929692 0,088829052 •0,961083972 0,087874587 "•0,961131980 — " 0,087902408" 8 •0,961178677 0,088009838 5 •0,960232762 0,086999746 0 50, 960286817 0,087079133" i •0,960300521 0,087148089 5 •0,950121242 0,078908099 1 - i O , ' ) S 0 2 4 S 9 9 r 0 , 0 7 9 0 J 7 6 J 6 ~ •0,950366056 0,079166035 •0,154174023 -"0,154174023-•0,154174023 •0,135950955 "SB;136081489" "0,136211154 •0,133986950 -•0,134133617-•0,134279293 •0.131921974 "•0,132085490" •0,132247890 •0,129742109 "•0,129923265" •0,130103179 •0,127430642 "•0,127630515-•0,127628813 •0,105318682 -»0,10S67965r" •0,106038284 0,000000000 ""0,000000000" 0,000000000 •0,033405393 ^07033248293" 0,000000000 "0,000000000 0,000000000 •0,017013049 i0y01696606O-• 0,033091912 •0,0169191*81 •0,037632727 •0,018833147 •0,037454537 •0,018778504 •0,037277164 •0,018724002 •0,04223235} "0,020711333 "0,"092 031789 •0,041832143 •0,047253164 ~ . 0 . 0 4 7 0 2 8 8 7 9 •0,046BOS607 •0,052752134 -5070525027237 •0,052254419 •0,111850759 -5o-.TnJ"»2Be5f - . •0,110916474 . 0 , 0 4 0 6 0 9 2 0 2 50,020648870" •0,020586579 •0,022651074 "•0,022580621" •0,022510372 •0,020656120 "507020577506-•0,020499127 •0,000686823 ~0",«407il878B-• . , . ! « » . » , . •0,967380506 0,097699530 •0.154174023 •2,860100968 »0.96738QS46 0,097699530 •0.154174023 • 2,860100895 50,967380546 0,097699530 -07154174023-•3.I34S0341S .0,965096137 0,093446081 •0.144051280 —J4J2-?£25I5 -0,965114300 0,093476944 .0,144127189 • J.T3365B8 T1~— »0,96 515 2 J HI 0709« 076TP SOfTTlW"2 "07575" 0,000000000 0,000000000 —07000000000" •0,022860682 •0,022751813 0,000000000 0,000000000 "0,000000000" •0,013557856 •0.013518122 !26HTiT«l ^70TJ4784*JT Js 0 t f i r « l v U t > K t - - ' 9 K » 4 ) 4 > 0 ) 0 o * o ^ r t r t 9 r t 4 i r t r u k - « i , - » c o r t O ' ' M - * t > r u i \ i K i e r i A o - i k n s o o o > r u ino4>«>9o>coru4tAij4>9-9rtru 9 9 -|-0|04NKS»iooiOQb o o o o o o o o o r -o o o o o o o o o o i o o o o o 0>«Sl>l/)OI-}c0 >«!9 0 O O O O O O O O 9 lr> «-•A M> • 9 1/11 ut in o o 9 4) AI ao kit 9 9 r > A l l O i n - » 4 > - e 09.nino>N«4> aNnitoinojN 9 9 9 r t 9 9 r t 9 fti w r\j »«i « • O O O O O O O O O O A » » « K rv p- so o o* rvi n i K N r o o 9 rt O f> « 9 rt ru ru — r— W r\i l\i O O O O O 66-<Ort « « r>»<-iri (T* i n - o b r t V t Iri o rkrtMK0Do < M > i>>4O4 ) ( M 9 r \ i n i i-.tftrt4>mo>N»<«4i9Fo(>inoin «j i9r*« a - « b 3 } o > r - t f > i n n i c e inu*incoeocoAiru--a-.Ainj-»f>f>:o < u r u r u r u r u r u r t r t r t r t r t r t ^ - r * * » o o o o o o o o o o p o o p o o o o o o o o ' o o o o o o i a o • > » i ^ r ^ l » r u o i o « « r ^ -> •<-9 o > u " » 9 * - o i n 9 4 » i / » * o e o t n e o o * o V)eOBOr«rtrt-«»*l(OrtNrtN9t» 9 9 o r w r u ; 4 0 k f f l « * r t o r t 4 > i n t / t r t i / i r t m o i x o e o e p o o ^ n j A N ' O 0'«NOO^a)0'OMSO>M> —• rtrtr»»rururuoo — t> o> o> r » I-* co 999999999rtrtrtrurMrU o o o o o o p o o o o o 0 0 0 0:0 0 0 O* r*- 9 0> rt fU O* o p o o o o o i o o o r ^ r t c o » w o 9 c o 4 > r - > ( « - i r t r « > « > 9 c o n e D o o i o - t o 9 « o > - > r « i n t M 0 r t u i « a C , p o o a o c o c o t o a t c o o ^ o o ^ o o o o o o C O 0 ° 0 O 0 O 0 O O 0 0 ) » « > « * ^ w « « « * « t o o o o o o o o o o o o o o o o o o o o o> o o o o o» o o o t ^ r u r o r u p ^ r t c b r t O O O 1/t c~ -o o o m • o o o i n i o o o o o o « 4> ru P»-- * n i 9 rt o i o • 1» 9 flO IA eo <D ru o* o 9 o *o m 0 r > - « r > ~ * f < » r > ' » 0 ~ » N N f f -n ia ru9rtrui*»4>iAru ~ ~ »»—•**•-<• 9 —» - * AI AI AI AI o o o o o o O O O O I ! 9 CO rt 9 0* •o co o ru 9 9 9 o o* a-rt - * r u m r> o ao m • 3 9 1 /19 9 ru r u i n tn IA o o o o a >oN.<iN^a>'rih.ui<tf«i>i«ico9rv >Of>03-^N 9 r » 9 4) U> CO U> rt lA 9 >ocor<>oifla)rtM-«i/iK0^(V44 >O-«0kC0AlAfrtrt941i94l9'9(A>>-»0rt4>09t>»0c00rui'->c00>999 » o o o o — * - « - « r t r t r u 9 9 9 4 > 4 > 4 > o o o o o « • 0 |0 0 8 i l 0 o o o o o o o o o o o o o p o '•••'•"rrr A t 9 t n < O O C O A J M r t 9 C O r t * « A i « « 94«moi / IO9tAB)OOr0K4 » i n o c o t n A i r t 9 D 0 9 0 * i n i A i n 9 c o « « r t 4 > o » rt h> o i | > - r t i-« r> rtrtrtAiAjruAiAiru^—**-*-N.F*»>. (>o>c>»r>c>»o>i>c>r>»iO(fi(o o o o o o o o < o o o o o o p o t > o n » r t a » o > r M K r t n j u n a L _ ««*4i/)>«ni«irirtoAtON«i-*j) 9 r t o - o m A i o > A i r t r t r t - * r > i o * « N«3f>iM<Aomnoo*4rur>90' N O - r y cjsruinojio-a 9 r u o> * * * - I ' , » S 0 9 9 9 0 0 0 l A 4 1 0 0 > 0 0 9 9 9 9 9 9 9 9 9 r t r t r t r > > c O C 0 4 > o - o o < o 4 < i o « « > o 4 i n i r ) i A (\icgrycoffo-oNcorvj-onj(Dt>Nrt 9cort9tn OOOOffUJU'0(\J«INN(>M/loo<lW- w o o 999 4>0-fMD4>rtAJ-J^)9-««00ru990in 999 -.«-Nr-(o I\J rv ^  K N O O xrvjrvii/i^su'i 4 K . « - * - « • « 9 9 9 0 i S ( a a c O ( O N N N K S N « « 4 o o o t n i A i n 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 o o o p o o o o o o — _ - - - IA 9 9 9 9 9 , 9 AlAI Al o o o o o o o o o o o o o o o o o o o o o T t in ) o 0» » CO O* O O O O P O 4 I o o o o o O r ^ i n i ^ i n i n B ^ i n i n i - * i n t n ( - « t n i n t « . N ^ K r t r t i r i M K M p . s n i ' } ' o i n —• 9 r*. • o ~ - * « - * 4 ) r u 4 ) - * i A ' o r v i i n < - o 4)99rt941Al9l>-C0O90>£DO Ort-0 9. AiOAimc09rt—3>Ort 4> — N rvi r*. a to m-*ino>rt99 i n 9 a t n i n a 4 ir* i n s 4 m a rtru 4 « 4 > o e f r g r u r v i / t i n i r < i s r » K • - « M n M M f u r i i r u i \ i r u n ( 9 9 9 » • • » j» t » • ft • ;a • » j» » t0044p«rtt7rVK4tMArVI4K •*ifl«Bfti!>o-«<0 4 r \ i i / K y i n o N**oa^a«a j i f i i r t -K io o>o"Ai9'mr>»ortotnoo<oruco o*«9in-o<otDOkr-«coo94)cort Airuooos-oco4)or*.999tn 4j)4jj>otni/\ini^iriinirtinirru -0'0 4>4)4>04>414>4>4>O4)414) o > o k o > o k o > o > o > o > o > o > o > » o > r > o » k O O O O O O O O O O O O O O O O O O VA 4> ru AI o> o-... t o i n LA o i n o t n i n P i n t A o i n i n b t n i n o i A O* O O O k O O O > 0 0 0 * 0 0 0 > 0 0 tn i n to i n o i n t n « t n t ^ 4 i i n i n 4 i i n i A 4 > i n t n 4 i i n Oe3*«K{>OAj^io»K>ipiOAJNI>tto«rUrO o o - o K n i i o f t j o i M r i ^ N - « « M N 3 > o r > ^ i f t *« O —• CO 9 0 * 0 4 ) 9 99 ( n o A l t A f*-.~.4)r > 4 l 9 o o o * w o l / n o - ' j l f l - i r > 9o>rtm -oru4>o rtM^noNtn^-oKrtO1 OinorvNnioniin — <-»»«^ininiArtAiAiiA9rt 4J o •ooooKixtMi/tinincotua-M-Mooo * % * - - - • " " " • ' " M w r u a a c r AiAiAirtrtrtrtrtrtrtrtrtrtrtHirt rt rt rt rt rt . . j . . i n o i ^ i n o t n u ' t o t n i n o t n i n o t n i n o i n i n c e> o o o> o o o > o o a o o o > o o o - o o o > < O r ^ r ^ o r ^ r ^ 4 ) r ^ r > . - o r ^ ^ > o * > . r ^ 4> **» i * . 4> r rtrtirtrtrtrt rtrtrtrtrtrtrtrtrtrtrtrtrtr o o o p o o o o o j o o o o o o ............... Ai-OAiru»r>r>9rtosooo k rtrt in A - r t 0 i s r U i - > . « M ^ 4 n ) p * « - ) 9 9 o j i r t « - > r t 9 - / i oo r \ i i / i K 9 « - > 0 ' 0 > r > N r \ J i A t D r > A | t r t - T E0_c0AJ4)O9C0 • « r t r t r t o « - . - « a o c o o > i n - 0 4 > r t r t r t 9 * - . * * ~ - > > - « - * - « o o o o o o o o o u ^ i A i n i n i r i i A i A i n k / t i / i L n u - i i r t i r t i A i A OOOtD9rt44««O«4e04KniB9N03 P o 0 0 0 0 0 0 0 0 0 0 ft « • • • • « i 1 1 I I 4»<-o—or».—*rurto rtrt<omrt9.omr>co9 ini/ ir>«*ivin , u>4comi / i r> okco«-«rtin3>*->4>w«i/) X M O O x t h o n i f f M i M 4 > 4 ) 9 I A i A r t 9 9 A J r t r t I-**r— 4 > 4 1 4 } 4 > 4 > 4 1 4 1 4 > 4 } O O O O O O O O O O O p O 0 o o o r t r> I CO o I o o > -O 4> ' r> r> > o o <o I n o p o o o o o o o o o o o o o o c > rt 9 r 9 9 » ru i n > rt 0> I o- o> I 1 [ • 0 4 > o r > « * > o K . i n r g i n i « . Kt ut K m ; n m AI AI rt ) A l : 4> 41 41 4) o> o> 4> 41 4> 4> 9AJOO0>O9IAt rt9IA99lArt9l 4> A - o i A i n i n a 9 t 4) 4)4>4>4S4)4)4>« ~0-04>4>4)4>4>4>< ift • 0 0 0 ft » t CO O o- t n o» o> o o co co i tn co . o» m r O 4> « O O ( L * - * rt r> O O 4 SO 4> -to co a c o c o o c o c o o c o c o o c o c o o c o t o IA4fUOIAOtOOrt-r\|*.NO 0> 9*N.v«o-(il,*-9inocOAIOO o rt — on t> ru—>4)rt9rtHtir> tnt>rtNCOCOe>l7>IAAIOrt4*W 4 > r * > 9 o r u s 9 i « - > r t r > r t c o r t c o o* 9 9 9 o* c o c o r t r t A i c o r - » r - > A i l^>0 O O A l l V ) A l i A i n i A N K h > 0 — — ru l» O 41 9> p — i I > 4 > 9 0 4 > 9 0 4 > 9 0 4 > k > 0 0 * - O 0 0 * ' 0 0 0 ^ 0 0 t > 0 0 0 k 9 9 4 9 9 "9 9 I u L L 1 • 4 . 1 0 0 • ! , 2 0 2 3 2 ( 4 2 0 •0,966326160 4 , 1 0 6 •3,201814530 - 0 , 9 * 6 3 3 7 7 4 4 0 , 7 5 4 ,094 •3,2273686/8 •0,966176487 4 , 1 0 0 •3.226B23IS0 •0,966189744 4 , 1 0 6 - 3 , 2 2 6 2 7 9 1 3 8 - . 0 , 9 6 6 2 0 2 8 6 4 "17 1TB 0,094" •3,350391728 i 0 , 9 6 5 0 1 5 0 3 5 4 , 1 0 0 - 3 , 3 4 9 6 5 7 1 7 0 •0,965041149 4 , 1 0 6 •3,348924968 •0,965066995 8.5 0,00_ 0.50 0.55 0,60 0,65 0 , 7 0 1.00 0 , 6 0 4 .493 " 4 , 5 0 0 — 4 ,507 4 , 4 9 3 " 4 , 5 0 0 — 4 , 5 0 7 4 , 4 9 3 " 4 , 5 0 0 4 ,507 4 ,493 - 4 . 5 0 0 — 4 , 5 0 7 4 , 4 9 3 " 4 , 5 0 0 4 ,507 4 , 4 9 1 _ 4 ,500 — 4 ,507 4 , 4 9 3 ~ " , 5 0 0 4 , 5 0 7 0 . 7 5 4 , 4 9 3 — 4 , 5 0 0 " 4 , 5 0 7 4 , 4 9 3 " 4 , 5 0 0 " 4 ,507 •2,860100975 • 2 , 8 6 0 1 0 0 9 8 3 -•2,860100986 •3,060605175 •3 ,060291318" •3,059978457 •3,082890715 « 3 , 0 8 2 5 4 IB68" •3,08219(1117 •3,105179685 • 1 , 1 0 0 7 9 5 7 7 8 -•3,104411087 - 1 , 1 2 7 4 7 2 5 5 5 • 3 . 1 2 7 0 5 1 5 3 8 -- 3 , 1 2 6 6 1 5 8 4 7 •3,149769895 • 3 , 1 4 9 3 1 5 7 0 5 -•3,148862957 •3,172072425 • 1 , 1 7 1 5 8 2 9 8 8 " •3,171095107 •3,19438|04S • 3 , 193856278"" •1,193313177 •3,306072285 • 3 , 3 0 5 1 6 8 7 4 8 " •3,304667487 0 ,967180546 0 ,967380546 0 ,967180546 0 ,967098564 0 ,967102249 0 ,967105888 0 ,967061837 07 9 6 7 0 6 7 9 9 4 0 ,967072099 0 ,967025053 0 ,967029738 0 ,967034364 0.9669B152B 0 ,966986806 0,966992017 0 ,966932400 0 . 9 6 6 9 3 8 3 4 7 " 0 ,966944218 0,966876561 0 ,966883269 0,966889891 0 ,966812565 0,966820145" 0 .966B27630 0,966269661 0,966284613 0 ,966299372 •0,967380546 •0,967380546 ~~iD, 967380546" •0,967335784 •0,967116682 " > 0 , 9 6 7 3 3 7 5 6 2 •0,967330589 •0,967331592 " • 0 , 9 6 7 3 3 2 5 7 6 •0,967324772 •0,967325895 ,967326995 ,967310798 ,967312208 ,967313590" ,967292511 ,967294297 ,967296048 ,967267445 - 9 f c 7 ? h 0 7 U 9 0,095983443 0,096001440 0,095838312 0,095857946 0,095877170 "07094960049" 0,094989458 0,095018576 •0,150052957 •0,150095943 " 0 , 1 4 9 6 8 1 8 0 5 •0,149710905 •0,149777541 •"07147176700" •0,147449585 •0,147521701 - 0 , 0 1 8 0 1 4 8 5 1 •0,017905112 •0,020259928 •0,020136350 •0,020013487 • 07037364283" •0,037141097 •0,036919105 •0,011508297 •0,011455360 •0,012340326 " 0 , 0 1 2 2 8 2 9 3 9 •0,012225763 "5"0;016879851 •0,016797995 •0,016716458 0,097699530 "07097699530-0,097699510 0,097094718 0 ,097102274 -0,097109723 0,097051923 "070^70601)5—" 0,097068095 0,097006959 0 . 0 9 7 0 1 5 7 1 8 " 0 ,097024375 0,096959482 0,096968902"" 0,096978211 0,096909036 0 , 0 9 6 9 1 9 1 5 8 " 0 ,096929159 0,096855038 " 0 , 0 9 6 8 6 5 9 1 0 " 0,096876651 0,096796733 0 , 0 9 6 8 0 8 4 1 3 " 0,096819952 0,096388970 0 , 0 9 6 4 0 6 2 4 3 " 0 ,096423301 •0,154174021 —SO,"! 5*174021-• 0 , 154174023 •0,152737460 —SO", 152755045" •0,152772429 =0,152630631 —S07152649685" •0,152668519 •0,152517728 — • 0 , 1 5 2 5 3 8 3 3 5 " •0,152558700 " 0 , 1 5 2 3 9 7 7 4 5 " ' • 0 , 1 5 2 4 2 0 0 0 1 " •0. I5244199S •0.152269419 " " • 0 , 1 5 2 2 9 3 4 4 0 " •0,152317174 •0,152131144 " S O , 1 5 2 1 5 7 0 6 0 " "0 ,152182674 •0,151980835 ""•0,152008821 " •0,152036468 •0,150907384 • 0 , 1 5 0 9 5 0 0 0 9 " •0,150992107 0,000000000 070 00 OOO 0 0 0 " 0,000000000 •0,007085224 " S 07 0 0 7 0 1 5 7 4 5"" •0,006986599 •0,007869700 D7007B14225" •0,007759127 •0,008739921 "•0,008677821" •0,008616147 •0,009710653 •0 ,009641195" •0,009572218 •0,010800237 • 0 , 0 1 0 7 2 2 5 6 3 -•0,010645428 •0,012031737 —;o";ori944B3o •0,011658529 •0,013434549 "•0,013337198 •0,013240524 •0,024900907 " • 0 , 0 2 4 7 2 1 3 3 1 •0,024542966 0,000000000 07O0O000000 0,000000000 •0,006060518 —•0,006028227 •0,005996044 •0,006496596 ~SO;00646T352 •0,006426254 •0,006949739 — » 0 ; 0 0 6 9 1 1 4 5 1 •0,006873328 •0,007421231 ""•070073797B1 •0,007318515 •0,007912517 " • 0 , 0 0 7 8 6 7 7 7 7 •0,007823240 •0,008425267 SD7O0B377096 •0,006329150 •0,008961430 ""•0,006909677 •0,006858169 •0,012096032 "•0 ,012023405 " 0 , 0 1 1 9 5 1 1 4 4 0,097699530 0,097699530 0 ,097699530" 0,097603402 0,097605316 0,097607194 0,097597803 0,097599837 0 , 0 9 7 6 0 1 8 3 2 " 0,097591876 0,097594037 0 , 0 9 7 5 9 6 1 5 6 " 0 ,097578782 0,097581221 0 , 0 9 7 5 8 3 6 T B -0,097563390 0,097566156 0 , 0 9 7 5 6 8 8 6 9 " 0 ,097544389 0,097547557 , v975506"6r" •0,154174021 " 0 , 1 5 4 1 7 4 0 2 3 " i iO ,154174023" - 0 . 1 5 3 9 3 4 7 9 6 •0,153939438 " • 0 , 1 5 3 9 4 3 9 9 4 •0,153920202 •0,15392S1S0 " ^ 0 , 1 5 3 9 3 0 0 0 6 " •0,153904664 •0,153909958 - • 0 , 1 5 3 9 1 5 1 3 2 -•0,153870169 •0,153876165 "S0" ,153882007~ •0,153629245 •0,153836096 " • 0 , 1 5 3 8 4 2 8 1 5 " •0,153776314 •0,153786226 - 5 0 , l 5 3 7 9 T 9 f f S — 0,000000000 0,000000000 0,000000000-•0,002828014 •0,002798729 " • 0 , 0 0 2 7 6 9 7 4 1 " •0,003116601 •0,003084106 — . 0 7 0 0 3 0 5 1 9 4 2 " •0,003439539 •0,003403457 " • 0 , 0 0 3 3 6 7 7 4 7 " - 0 , 0 0 4 2 1 6 1 7 3 •0,004171496 "~0—J04T272BJ— •0,005234788 •0,005176900 " - 0 , 0 0 5 1 2 3 5 9 ; -•0,006626588 •0,006557460 "V0"7o-Tj6TJB7i20" 0,000000000 0,000000000 0,000000000 •0,002803989 •0,002778761 •0,002753729 •0,002970325 •0,002943368 " S O ; 0 0 2 9 1 6 6 2 5 " 0 , 0 0 3 1 4 3 5 2 5 " 0 , 0 0 3 ( 1 4 7 6 9 " • 0 , 0 0 3 0 8 6 2 4 5 •0.003S1266S •0,003480077 -~)700J4TI7757 •0,003916569 •0,003879786 "•0,003643313 •0,004162456 •0.004121039 "TTOrO—>T9 975 t.00 5,490 -J,221515781 •0,967210562 0,09751914* « 0 , 1 5 1 7 1 0 1 8 0 •0.0086S0121 .0.004861049 5,500 -1.223851055 -0,967211631 0,097522842 -0,153719507 -0.008557195 -0,004814823 5,510 -3,223188822 -0,967236637 0,097526466 -0,153728652 -0,008465244 -0,004768601 f-575-"OTBfl 6 T 0 7 0 -6,500 6,530 "T770 6,470-6,500 6,530 "O7B0 6,470" 6,500 6,510_ * • 6)500 6,510 T700 5747O— 6,500 6.S30 "•1,045578966" •3.044722559 •3,043874046 -"•3,076492286" •3,075493119 •3,074503166 "•3,107405956-•3,106264009 •3,105132586 "•I,118320126--1,137015359 •3,135762446 --37I59235096 -•3,167807459 -3,166393006 "•0,967373228-•0,967373676 •0,967374097 —•0,96737146J" •0,967372021 •0,967372545 -•0,967369168" •0.967369869 •0,967370528 "50,967366043-•0,967166940 •0,967367783 -•"079 6 736146^-•0,967362651 •0,967363765 -0709T685765-0,097686629 0,097667441 -07097680250-0,097685214 0,097686119 -0,097682453" 0.097681S33 0,097684546 "0,097680206-0,097681430 0,097682579 "07097677173-0,097678591 0,097679921 -STJ7154137750" •0,154139976 •0,154142067 - » 0 , 1 5 4 1 3 3 6 0 2 -•0,154136091 •0,154138433 "•0,154128654" •0,154131460 •0,154134096 "•0,154122450" •0,154125649 •0,154128654 -*07O01212182 5"<M>01?2Tff39 •0,001173789 -0,001185485 •0,001136596 -0,001150140 "50;00|471159 SO",OOI3S0661 •0,001424159 •0,001310121 •0,001378644 •0,001270715 -•0,001812446 -0,001491612 •0,001754142 .0,001446486 •0,001697694 •0,001402637 ~tfO", 002282517 ~ " - 0 7 1 5 4 n T 0 6 7 " •0,154117796 •0,154121297 •0,002206713 _ » 0 , 0 0 2 1 3 7 2 7 3 • 070 0297081. •0,002874535 •0,002781281 •0,001647025 •0,001596639 •0,001548089 5TJ70"OT8T03T2 •0,001764539 •0,001710109 Appendix F, Matrix Elements between Molecular Orbital § O.J 0,00 0,20 0,30 0,40 0 .4S 0,50 J>.S5_ 0,b0 0.70 0.5 0,00 0.25 0,30 6 ,35" 0,40 0.45 0.50 0.55 o , * o _ "6,75 1,00 Oo» 0,00 0,40 _0j45 0,56" 0,55 0,60_ 0,9 0,00 0,40_ 0,45 0,50 0j55_ 0,60 0,65 1.1 0,00 0,45 0,50 0.55 0,60 0,65 • t .Og-1.5 i f~ 0,00 _0.45_ 0,50 0.55 0.60 1,4948 1.623S 1,6859 1.7480 1,7790 1.8100 1,8410 "1,8722 1,9356 3,9304 4,6474 5,0044 5,3736_ 5.5632 5,7565 5,9542 _ 6 , 1 5 7 t ' 6,5836 0,0000 •0,0303 •0.0654 _»0jU>93_ •6,1381 • 0,1714 •0.2101 • 0,2514 " •0,3654 0,0000 •1,2274 •1.8819 =2 ._6 7 0 3^  •3,1480 •3,7013 •4,3541 •5,1258" •7,1703 0,0000 0,7705 1,5991 2,9249 3,9295 5,3436 10,6111 23,0488 0,0000 0,0138 0,0305 0,0537 0,0678 0,0835 _0,I012 6.1211 0,1713 0,0000 0,0074 0,0161 0.0282 0.0358 0,0444 _0,0542 0,0653" 0,0924 1,2995 1,4059 J . ^ ' l 1,4483" 1,4695 _1_,4 90B_ 1.5120 1.5332 J .5544 1.6174" 1.7196 2,7134 3,2211 _3,3268_ 3,4347 3,5445 3,6563 3,7699 3,8852 _4,0022_ 4,3615 4,9805 0,0000 •0,0327 •0,0465 •6,0624" •0,0822 ^0, 1028_ •0.1272 •0,1532 •0.1833 • 0,2656 •0,5097 0,"66"00 0,0000 n o u t A C C A A 0,0000 0.0074 0,0161  0,0282 6,0356 0,0444 _0,0542 6,0653 0,0924 0,0000 0,0420 0,0923 _0..1S66_ 0,"1969 0,2405 0,2929 0,3618 0,6314 0,0000 •0,0676 •0,0974 •0.1266 •0,1417 •0,1577 •0,1753 "•6,1957 " •0,2531 1,0000 1,0000 1 ,0000 1 ,0000 - i .0060"-1,0000 J .O00O 1 ,0006" 1,0000 0,0000 • 0,0000 •0,0000 •0,0000_ •6,60oi •0,0001 •0,000l_ •0,0001 •0,8161 ^1 .0039 "•I ,"2024" • 1,4148 ^ U 6 3 6 6 •1,8 7 "20 •2,1182 •2,3783 •3,2257 •4,8730 1,1190 1,2359 J j * s 1 2 _ 1.2667 1.2822 -L.A?L8_ 1,8797 2,3844 2,4556_ "2,5286 2,6033 2,6797 0,0000 •0,0546 •0,0688 •6,0858" -0,1038 •0.1241 0,0000 •0,7606 _=0,9093_ •1,0465 •1,1903 -1.3422 V g V V V V 0,5590 0,7933_ 1,0748 1.4068 1.7914 2.2310 2.7282 3,2841 5.3286 10,0503 0,0000 • 0,0000 0,0159 0,0080 _0.0226 0,0115 6,0305 0,0i56 0,0396 0,0204 0.0499 0 t0259_ 0,0613 0,0322 0,0739 0,0392 0,0875 0,0469 0.1347 " 0,0746" 0,2330 0,1361 0,0000 0,0080 _0,0115 0.01S6 0,0204 0,0259 0,0 322— 0,0392 0,0469 "0,0746"'"' 0,1361 0,0000 0,0254 0,0349 "0,6456 0.0573 0,6S3f 0,0966 0,1104 "0.1S16 0,204V 0,0000 •0,0566 •0,0665 ""•6 , 0762" •0,0857 •0,0949 •6,1040 •0, |127 -0,1212 •0,1446 •0,1759 —ITOJoo" 1,0000 1,0000 " 1,0000 1,0000 1,0000 t ,6000 1,0000 _ 1 , 0 0 0 0 1 ,0000 1,0000 •0,0001 "0,6000 •0,0001 •0,0001 •0,0001 •0,0001 •0 ,0001 • 0,0 O i l -0,0001 • 0,0002 •0,0002" •0,0002 0,9649 l_t t0465_ 1,0599 1,0716 I,0635 1,6956 1,1078 1.3217 _lj6458_ 1,6937 1,7433 JL»7."»£_ 1,8471 1,9015 0,0000 •0,015S_ -6,0456 •0,0570 •J>,0692_ •0,0836 •0,0990 0,00 0,8375 0,94 80 0,45 0,9079 1,1983 0.S0 0.9169 LL23J2_ 0,55 0,9263 1,2695 0,60 . 0,9359 1,3074 J . ,65 0.9457 1.3469 1,00 1,0215 1,6672" 0,0000 •0,4620 "•"0.5424 •0,6278 _«0j,7l82_ -6,8146 • 0,9162 0,0000 0,7148 _0.919t 1,1532" 1,4189 1.7172 O.OOOO 0,0000 0,0256 0,0141 Oj.0324 " "6 "" 0,0000 0,0141 0,0000 0,0199 0,0000 0.9028 0,5217 0,6585 _0,8139 0,9889 1,1836 _0,7334 0,6953_ 0,7863 0,7935 0,8678 0,8929 _0,8009 0,9J9J_ 0,8066 0,9472 0,8166 0,9765 0.8814 1.2264 6 ,6 o"6 6" •0,0306 •0,0386 •0,6469 •0,0568 j .0,0682 •0.1797 _0,0000_ •0,0209 •0,0261 ^ 0 , 0 3 3 l _ _ A A "f A A. •0.0396 •0,0472 •0.1325 0,6465 _0,6B85_ 6,6941 0,7000 0.7062 0,0000 -0,3382 _-0.1944 •0.4540" •0,5160 _^0,5868_ •1,1916 _0,0000_ -0,2161 •0,2537 _M>_,2952_ •0,3390 •0,3664 •0.8297 0,0000 0.3177 0.4030 0,5007 0,6107 __0,7342 2,6613"" 0.0000 0.2017 0,2574 0.3216 0,3948 0. 4771 1. » 3 4 0,0000 0,0000 0,0160 0,0096 ""0,020 4 6,0123" 0,0253 0,0155 _010308 0,0190 0,6369 0.0229 0,0437 0,0272 ""676566 0 .606T" 0,0130 0,0085 0.0163 0.0107 0,0200 0,0132 0,0242 0,0161 0,0288 0,0000 0,0096 0.0123" 0,0155 J>j0190 0,0229 0,0272 0,0000 0,0080 ~ 6 ,009r 0,0114 0.0133 0,0152 0,0171 -0,0540 0,0000 •0,0543 =0,0601 •0,0657" •0,0712 •0.0765 0,0000 •0.0359 •0,0198" '0,0435 0,0471 0,0506 1,0000 1,0000 1,0000 "1 .0000" 1,0000 1,0000 0,00"06~ •0 ,0001 •0 ,0001 ••0,0001" •0 ,0001 • 0,0001 1,0000 1,0000 "l .OuOO 1,0000 _I A 00 00 1,0006 1,0000 0,0006""^ 6,6606" 0,0085 0,0051 0,0107 0^00 61 0.0192 0,0132 6,0674 0,0161 0,0086 _0,0192 * - | v f c v w v j ^ v * T I v ,ju, TC 0 ^ 0098 • 0«0 5 S 6 1, uuoo 0,0759 0,0514 0,0514 0,0189 "•0,0511 1,6666" " 0,0660 TToTioo •0,0282 1,0000 •0.0109 1,0000 •0,0116 1,0000 0,0162 0,0000 •0 ,0001 •0 ,0001 • 0,0001 • 0,0001 •o.oOoT" • 0,0001 "T,"6So"S" •0,0001 •0,0001 1,0000 _1,0000_ 0,0000 0,0000 0,0000 0,0000 . . . 0,0067 0,0060 0,0060 O.OOle" 0,0110 0.0075 0,0075 0,0047 0.0135 0.0093 0,0093 0.0056 0,0165 0,0114 — ' 0,0198 " 0.0548 w • v • « -1 0.0137 0.0384 0,01140,0137 0.0384 0,0000 •0,0211 •0,0233 •0,0255 •0,0276 1.0000 1,0000 1,0000 1,0000 • O.OOTf" •0 ,0001 •0 ,0001 • 6 , 6 6 6 ? " 0,0000 0,5221 0.6424 0,6605 0,6798 0,0000 _»0j,0146_ •0,0180 0,0233 0.7004 •0.0285 0,0000 0,0000 0,0000 0,0000 _i0j14 0 2 p ,1318 0.0061 0.0042 •0,1655 0,1690 0,0077 * -'---"" -0,1944 0,2123 0,0096 •0.2255 0.2625 0.0118 0,0066 1,0000 0,0077 -0,0296 1,0000 0.0156 -0,0396 1,0000 A AAAfl A AA AA « A A A rt ,  0,0000 0,0000 0,0000 1,0000 JJL? P 4 2 0,0 042 0.0 0 30 - 0 . 0162 1.000 0 0,0053 0,0053 0,0018 -O.OIBl 1,0000" 0,0066 0,0066 0,0047 -0,0200 1,0000 0,0081 0,0081 0.0056 -0.0219 1.0000 » I V V W ft •» o ,T6o l •0 ,0001 •0 ,0001 •0 ,6061 •0 ,0001 • 0 , 0 0 0 2 0,0000 •0,0001 •0,0001 • 0,0001 • 0,0001 0 I" 1.7 0,65" 1,00 0,00 0,45 _0j50_ 0,55 0. 60 _0,65_ 1, bo -0 ,0 0_ 0,<15 0,50 -0.55.... 0,60 0,65 _ K 0 0 _ 2.1 0,00 o««s_ 0,50 0.55 _o.*o_ 6,65 1.00 2,7 J.J 0,00 0,a5 _o_.so_ 0,55 0,60 _0..65_ 1,00 _o.oo_ 0,45 0,50 _0-.55_ 0,60 0,65 0,00 _ 0 J « 5 _ 0,50 0.55 0,60_ 6,65 1,00 J.7 0,00 0,45 _0,50_ 0.55 0,60 _0,65_ 1,00 0","7l27~ 0,7684 0,5789 0,6091 _0,61 J4_ 0,6161 0,6230 _0,6283_ 6,6759 0.5215 0,5441 0,5475 0,5510_ 0,5549 0.5591 _0,5994_ 0,4737 0,4905 0,4930 0,4957 0,4967 "6,5026" 0,5355 ^ 7 2 2 4 " 0,9207 0,4014 0,4655 0,4965 0.5126 " 0,5279 -°-o5444_ 0,7023 0.3158 0,3744 0,3837 _0,39J9 6,4051 0,4173 - J . . 5»28_ 0,2539 0,2944 0,3010 " 0,3063 _0,1164_ 0,3254 0,4242 •0,0144" •0,0997 0,0000 •0,0106 _?0j0133_ •6,0181 •0,0202 • 0,0246 •6,0766 0X000 0_ •0,0078 •0.0098 •0,0124 •0.0155 " •0,0188 _TP-t-?.*04_ 0,0000 •0,0057 •0,0074" •0,0092 •0,0115 •6,0140 •0,0477 "V0"~!51»S-•0,5934 0,0000 •0.0917 j^0,1091_ •0,1297 •0,1506 "0,1752_ •0,4314 _ 0,0000 •0,0601 •0,0721 _^0,0858_ •6.1014" •0,1186 •.0j3171_ 0,0000 •0.039S •0,0477 " •0,0571 =0,0680_ •0,0604 •0,2337 "673T95 V~,WH~ 0,9544 0,0412 0,0290 0,0290 0,0000 0,0000 0,0000 0,0000 0,0676 0,0044 0,0029 0,0029 0,1129 0j,0057 0,0037 0,0037 0,1426 0,6671 0,0047 o.ooiir 0,1774 0,0086 0,0058 0,0058 0,2176 0,0107 0,0070 0,0070 0,6914 0,0321 0,0222 0,0222 0.0000 0,0590 0.0763 _0,0969 " 6.1212" 0.1497 _0j5096_ 0,0000 _0,0402 0,0520" 0,0663 _0,0833_ 6,1036 0,3795 _Ot0OOO 0,0033 0,0043 0,0054 ""6,0067" 0,0082 0,0000 0.0021 0,0027 _0,0033_ 6,0041 0,0051 J>.,0000_ 0,0021 0,0027 0,0033_ 0.0041 0,0051 w ** ~ » • •» » •» a W f f v w ^ t JL«0?_57 0,0170 0,01701 0,0000 0,0026 6,0033" 0,0042 _0,00S3_ 6,0065 0 .02H 0,0000 0,0015 0,0019" 0,0024 0,0030_ "0,6036 0,0131 0,0000 _0,0015 6,0019" 0,0024 _0,0030 0.0036" 0,0131 0,3700 0,3766 -0,3775.. 0,3786 0,3799 _0,3812_ 0.3983 _0,3446_ '6,3494 0,3501 _P,3508_ 0,3517 0,3527 0_,3657 0,1469 0,0000 0,1599 .0,0024 _0,J62J • 0,0 0 31 0,1646 •0.0040 0,1674 .0,0049 .P.J707 .0,0061 0,2148 •0.0240" _0,1260 0,1349 0,1363 J>j 1 J80 0,1400 0,1422 0,0000 '0,0019 0,0023 0,0030 6,6637" >0,0045 JbJ 74 7 -0.0189 0,0000 •0,0112 _;0,0J37_ •6,0166 •0,0201 _-0,0243 •6,0910 0,0000_ •6,6674 •0,0090 _«0 L 0110 •0,6)33 •0,0161 _-0,0649 0,0000 0,0134 -OiPJIL. 0,0219 0,0276 0,0346 6 , i 547 _ _0,0000_ 0.0095 0,0121 _?.0153_ 0,0193 0,0241 0,1124 6 ,006r 0,0144 0,0000 0,0024 _0 ,003 l_ 0.66J9 0,0047 0.0056 0,6134 _0 , 000 p_ o",ooi9 0,0024 0,0031 6,0038" 0,0046 -JLnOlfL. 0,0000 0,0014 0,0018 " 0,0023 _0,0029_ 6,6636 0,0108 ^676«r •0,0329 0,0000 •0,0126 •0,0143 •0,0159 "0,0176 _"_0,0192 •6 ,0284" _0,0000_ •6,6698 •0,0112 •0,0126 •0,0141 •O.OtSS •0,0248 1,0000 1,0000 1,0000 1,0000 1,0000 1,0000 _1,0000_ 1,6666 X.oooo "1,666o" 1,0000 1,0000 1,0000 1,0000 1,0000 "VOTiTO^r •0,0001 0,0000 •0,0001 J^0,0001_ •6,0001 •0,0001 -0 ,000l_ •6,6ooi 0,0000 •0,0001 " •0,0001 •0,0002 •0,0002 •0,0002 -_0,0003_ 0,0000 •0,0076 •0,0087' •0,0099 J;O,OIU__ •6,0124 •0,0216 0,0000 0,0013 0 '0° l*_ 0,0021 0,0026 _0,0033 6,6i23" __0,0000 0,0010 0,0013 _0,0016 0,0621 0,0026 0,0102 0,3030 _P,3054 0,3057" 0,3061 0,3065 "6 ,3070-0,3141 "T,"2 7 03" 0,2714 0,2718"" 0,2720 _0 J2722_ 6,2758 0,0958 _0,0998 6,1005" 0,1013 0 , l 0 2 l _ 6,1032 0,1199 0,0000 0,0005 0.0007 0,0009" 0,0011 _0,0013_ 0,0057 0,0000 0,0004 0,0005 . 0.0006 0,0008 0,0009 0.0042 0,0000 0,0005 _0j0007_ 0,0009 0,0011 _0,0013_ 6,0057 0,0000 0,0005 0.0006 0,0008 0,0010 J .0013 6,66«3" o.oooo 0,0004 0,0005 0,0006 0,0008 0,0009 0,0042 0„0000 0,0003 0,0004 0.0005 0,00 06 0,0008 0.0047 0,0000 "0,0033 _^ 0_,J0 038_ •0,0044 •0,0051 »0,0058__ •6.0131 0.0000 •0,0024 •0,0028 JJ0,0033_ •0,0036 •0,0043 •0.0107 1,0000 1,0000 "1,0000 1,0000 _1,0000_ 1,00 00 1,0000 "i76666~ 1,0000 _lxOOjOO_ 0,0000 •0,0001 •0,0001 •0,0001 •0,0001_ •0,0001 •0,0002 1,0000 1,0000 _1,0000 1,0000 1.0000 1,0000 1,0000 __1,0000 1,0000" 1 ,0000 1.0000 0,0000 •0,0001 •0,0001 •0,0001" •0,0001 •0,0001_ -0,0002 0,0000 • 0,0000 • 0,0001 ;0,0001_ • 6,6001 •0,0001 • 0,0001 0,0755 0,0773 0,0775 0,0779 0,0783 0.0787 0.0867 0,0000 0,0000 0,0000 •0,0011 •0,0032 0,0049 •0,0013 •0,0039 6,0061 •0,0017 •0,0047 0,0076 •0,0020 •0,0057 0.0095 •0,0026 •0,0070 0,0118 •0,0113 •0,0315 0,0570 "o"V6666~ 0,0000 0,0000 •0,0006 •0,0014 0,0025 0,0000 _0,0006_ 0,00 08 0,0010 _it?0J2 o.ooiT" 0,0066 0,0000 _0,0002_ 6,0002 0,0003 0.0004 O.OOOS 0,0022 0,0000 _0j0002_ 0,0662 0,0003 0.0004 0,0065" 0,0022 0,0000 _0.0O0J_ 0,666| 0,0002 0,0002 ~6"75OTJ-0,0020 0,0000 •0,0013 •o'.ooTS" •0,0018 •0,0020 •0,0023" •0,0065 •0,0009 •0,0010 _ » O , O O I J L •0,0062 0.00 0.2439 0.0610 0,0000 • 0 , - - . •0,6020 •0,0024 _?0 t0030_ • 6,6143 0,0000 0.0000 0,0004 O.OOOS v j v v j w 0,0276" 0,0000 0,0006" 0,0007 0.0009 0.0000 0,0001 0,0001 0,0002 0,0002 0.0002 0,0000 0,0001 0.0001 0,0002" 0,0002 0.0002 0,0000 0,0000 OjOOOO 0,0000 0,0000 0.0000 0,0005 0.0000 0,0000 •0,0007 •0,0008 •0,0009 •0,0010 •0.0012 •0,0036 1.0000 J_,0000_ 1,0066 1,0000 _1.0000 I TSooo 1,0000 1,0666" 1,0000 1 .0000 0.0000 •j>,0000_ •6,6600 "0,0000 •0,0000 •o,6"0oo -0,0001 1,0000 1,0000 1 .0000 0,0000 •0,0000 •0,0000_ •0,0000 •0,0000 0,0000 1,0000 • O.O'OOl 0.0000 1,0000 0.0000 I1 O O O O O O O O O O O O O O O O p o O O O O O O j O O O O O O O O o o o o o o o o o O O O O O O O O O o o o o o O O O O O O O O O O O O O I ? 0 0 0 0 0 ! 0 0 o o o i o O O O O O » O O O O < * o o o o < • O O O i o ' g * » m m <t c o . O O O O O . O -< • O O O O O O O O > O O O O O o o > • • • • * { • » j o o o o o o p o I'T o o o b ( > o o o o o o o o o o o o o o o o o o o o o o o o o o i o o o o o o <* Al ; o o o , 0 0 0 0 0 0 i - • -! ; o o o 0 0 O O O O O O O * J O O O O O O O O o o o o o o o o • ; — » - * ! o o o o o o o o o o o o o o o o o o o f O O O O O O J O O O O O O O j O o » o o o o j o o 1 rt 9 f>| o o o o o o l 0 0 0 0 0 0 O O O O O O ) • O O O O O O • •••61 o o 0-0 ru • O O O O O O O O O O O O [ 0 0 0 . 0 0 0 • O O O O ' . O O O O O O O O O O O O O O O O O O * " * * o o O O O O O O , 0 0 0 0 0 0 ) 0 0 O O O O O O O O O o o o o o o o > o o o o j o o O c f O < o < t: 0 0 0 <o o o 0 0 0 p o o p u rt rt 9 i n 411 O O O O O O JOAI O O O O O O O O f o o o o o o p o o < o • I o — — 1000 0 0 0 o o j o O O O O > o H - « * « r u t M r u r t ! 9 r t O o o o o o ; o * * J O O O O O O O O • o o o p o o i [ o o — « - » - «AI o o o o o o o o o 0 0 o o l o o o o o o o o o o o o o o o o j o O O O O O O OJO o o • o o o l o o o | 1 O O O :0 O O ; • O 0;0 O O j > 0 0 o o ! o c » O O O O l O C > 0,0 o o i o « B «r. • o o > O - O O O j O CL _ 0,0 0 00 0 B I O O O O O O • O O O O O o o • 0 0 0 0 0 0 a » . o « i > o < ! o o o » j o O O i ' O O O » ; 0 O O • 0 : 0 O O i O O O O O O O O » o o o o • o o o o 3 0 0 0 . O O O O O O O O O O O O O O ; O O O O O O O O O O O O O O : O O O O O O O O j o o 0 0 o o i 0 0 0 0 o i o o o k u A I A I rt 9 —! o o 0.0 o —«i O O O O O O ; O O O O O O . O O O O O O t o o o U * » r u » 0 o 0 0 o o | » o o o : o O o » Q O O O O O | o -p* co o m a W o o j o « - * r u r v tn-sf i n oo O O O U ^ M ^ i M ^ ! O O O O O O O O 0 0 0 , 0 0 0 0 0 0 O O O O O O o o O O O | 0 O 0 0 O O i O O O O O O O o MINO> O H i m < 000—•«-•>** j . * ,b e o O O O O o o 0 0 0 0 0 0 0 0 0 0 0 0 0 ; o o 0 0 0 M I I I ( l 1 j •» • f l t J i n - O c o c o U r v j i o r u r u o o o o o O O O O O O O O O 1 > ' o » . o > i o - -»V- -1 s 1I1 • e, o o l o o 0! o o j o o o l o 0 0 o o 0 o j o o o | 0 o ; o o o o o i o o o j o o o o  o , . 1 o ° o ; « 0 o l o 0 o o o o o o o o i o o o | o o ! o o o O O JO O O o o o o o o I o O O ; 0 O O ' |9 9 in'4> co r - l 0 0 0 0 « » • ' * * r u * o o o o o rvi • 0 0 0 0 0 0 o « 0 0 0 0 0 0 ' o o o o o o o < O O O O O O ' 0 0 0 0 0 0 0 4 t ' O O O O O O p o o o o o • O O O O O O * rt 9 i9 in 41' I 0 i o O O O O «**> o o o j o O O O O O , 0 0 0 , 0 0 0 O O O O O O j O O O O O O ' 4> 4) ! o 0 I o o rt i n tr* t o - • r u r u ru r u irt 4» 4) 4) 4> 4) 4) 4) o o o 0;0 o o o o o i o o O O O • * wm rU| O O O O O O O Ol O O O O 0,0 o o O O O O O O O O > O O O O O o t 9 9 ; 9 9 : (ru r u c [9 CO COt» 0> o ' * - * r t 41 j o o o i o o — — rvi •in in m ;t/> on in .in in in O O O l O o 0.0 o o | » » [ o o o | o o O O o o I ! k n o i n o m o v i o » co r> j . - . in r 9 9 rtn 41 r 9 9 9 9 •1 r u r u Ai r u i • i r u i/l m>41 4> 4J P— co 9 A i r u r u r u r u r u r u r u r t ^ r u r v i r u r u r u . - u r u r u . r u r u r u r u r u r u r u r u r u tn i n rt rt rt rt o o CO 9 4) » 41 41 4) rt rt rt rt rt rt: O O O o o c 41 CO 00: O O O O O Oi 0 0 © — - * ru I o o o j , 0 0 0 : o o o o o 0 0 . 0 0 0 ' 0 0 , 0 0 0 o o o i o o o l O O O o o j o o o o j o o o j • o O O O tO O jO 0 4 A O L f i O m ' O i O O O 1 9 in in 41 41 I*.II-. o jO 9 iA tA 4) « K O O O O O o — o o O O O i n eft » o*jo> *> o>! I rt rt rt rt t m m m m l ! C 3 O-Appendix G, V (r p, lj) and £ ( Vp. @, U) f 0 1 A AH - §*) *2<st-ec) i il rill. • \ V t UO 0,20 0,30 0.40 •2,9b96 •3.1935 •3.2532 •3 2664 3,6771 3,4732 _3,41 J5 3,3920 1,0708 0.43U o.oooo •CIP.J •0,1324 ' •0,0880 - o T o ^ ^ m ^ 0,0277 0,0149 0,0149 0,0609 0,0321 0.0321 i _ ' : w , " 0,45 0.50 0.55 •3J2928 •3.2924 * 1 p H li i •J , 36 01 3,3738 3,3743 0,1566 0,2231 0,5188 0,0131 0,0820 0,1655 0,1074 0,1355 0.1671 0,056$ 0,0716 0,0668 0,0565 0,0716 0,0888 • 0,60 0,70 ™ j f eo*» / •3,2691 •3,2090 3,3620 3,3976 3,4577 1,1774 2,4219 8.1295 0,2703 0,4105 0,9591 0,2020 0,2421 0,3427 0,1084 0,1305 0.1848 0.1084 0,1305 0 >1848 1 0,5 0,00 0,25 0 iO •2,599| •2,7538 .3 l i n o 1,4009 1,2462' 2,0492 0.5224 0,0000 •0,1191 0,0000 0.0317 0,0000 0.0160 0,0000 0,0160 > 0J35 0,4 0 0 45 • c | ( f U O •2.7632 •2,7910 _ 3 j n i i i " 1.2292 1,2168 lj2090 0,1461 0,2199 0.1419 •0,1163 -0,1069 •0,0909 0,0452 0,0611 0,0793 0,0229 0,0312 0,0406 0.0229 0.0312 0,0408 o.so 0.55 •2,7925 •2,7662 1,2059 1,2075 1,2138 0,1286 0,1734 0.2661 •0,0690 •0,0418 •0,0094 0,0998 0.1227 0,1477 0,0519 0,0644 0,0783 0,0519 • 0,0644 0,0783 0 f O ! -0,60 0.75 1.00 •e.»/31 •2.7135 •2.5180 1,2249 1,2865 1,4820 0,4680 1,4683 4.9869 0,0269 0.1586 0,4089 0,1750 0,2694 0,4659 0,0937 0,1491 0.2721 0.091/ 0,1491 0.2721 0.T 0.00 0.40 0 n't •2,2379 -2,3707 m"3 I 7 t tt 0,6192 _ 0,4864 1.2553 0.1169 0,0000 •0,0905 0,0000 0,0512 0,0000 0.0281 0.0000 0.0281 V , M3 0,50 _.0.S5 0,60 •2,3736 •2,3701 .a TK la 0,4813 0,4835 _ P.4871 0,0911 0,0984 0,1440 •0,0819 •0,0744 •0,0620 0,0647 0,0799 0,0966 0,0159 0,0447 0.0546 0,0159 0,0447 0.0546 wC|3DJc 0,4940 0,2290 •0,0467 0,1150 0,0656 0,0656 " " 0,9 0,00 0,40 •1,9298 .) A 3 T a 0,2925 0.781S 0,0000 0,0000 0,0000 0,0000 i 0,45 0,50 0 S5. mc f e •2,0306 -2.0317 a > hlA.l 0,1IS I 0,1917 0.1905 0,0647 0,0615 0,0566 •0,6702 •0,0682 •0,0642 0,6326 " 0,0408 0,0506 0,0193 0,0247 0,0309 0.01n 0,0247 0,0309 » i 3 ' 0,60 0.65 •C , U i U It•2.0267 •2.0205 0,1918 0,1955 0,2017 0,0762 0.1212 0,1903 •0,0583 •0,0507 •0,0413 0,0616 0,0739 0,0674 0,0379 •" 0,0457 0,0544 6,6179— . 0,0457 0,0544 1 III 0,00 0,45 0 50 •1,6750 •Ij7524 0,1431 _0,0658 0,4931 0,0410 0,0000 •0,0515 0,0000 0,0261 0,0000 0,0171 0,0000 0,0171 0^55 0,60 0.65 • 1 ,7543 •1,7544 •1,7528 mi 1 UQ I 0,0639 0,0637 0,0654 0,0176 0,0469 0,0707 •0,0492 •0,0457 •0,0406 6,0326 0,0400 0,0484 0,0215 0,0265 0,0321 0,0215 0,0265 0,0321 *•> 1.1 1,00 A A A 9l 1 ' *• » J •1,6635 0,0669 0,1547 0,1110 0,9957 -0,0344 0,0176 0,0576 0,1519 0,0184 0,1029 0,03 6* : 0,1029 • I * U f uu 0,45 0.50 a 5*. •1,4669 •1.S270 •1.5291 0,0716 0,0115 0,0094 0,1146 0,0318 0,0271 0,0000 •0,0187 •0,0173 0,0000 0,0174 0,0219 0,0000 0,0119 0.0150 0,0066 1 -0.0119 0,0150 " i .-• i1 a • u , " 0,60 0.65 1 00 • 1 | DCT*J •1.5295 •1.5276 0,0085 0,0090 0.0109 0,0107 0,0452 0.070S -0,0147 •0,0309 •0,0261 0,0271 0,0330 0,0395 0,0186 0,0227 0,0273 0,0186 0.0227 0.027) -0 r , 1 | V U •I,4648 0,0736 0,6740 o.o iu ' 6,169} " 0,076? 6.6767 — 1.5 0.00 •1.2970 0.0363 0.2030 0,0000 0,0000 0,0000 0.0000 ""o7*r 0,50 0,55 0,«0 0,65 1.00 0,00 0,50 0.5$ 0,60_ 0,65 1,00 0,00 0,45 _0,50 0,55" 0,60 0,65_ 1,00 OjOO 0,50 •1**. 0,60 0.65 J,M. 0,00 0,«S_ 0,50 0,55 0,*0_ 0,65 1,00 0,00 0,45 0,50 0.55 0,60 •»-•*_ 1,00 OjOO. 6,45 0,50 0,60 0,65 1.00.. 0.00 0,50 0,55 ».*9. * T ; J * J * -B.oior •0,0122 •0,0134 •0,0135 •0,0125 0.0345 •1,345* -1.3467 •1,3466 -I.3450 •1.2960 •1,1570 .•1,1936 •1,195 3" 0,0166 ^OfOITl •6,016*"" •1,1964 •0.0200 _?1,I966 -0,0203 •1,1961 -0,0199" •1,1601 0,0164 •1.041T •1,0702 _-l,671» 0,0096 •0,0175 r0,0169 •1.0725 .676.9T" •1,0729 -0.0202 _-l,0726_-0,0200 •1,0441 "0,0085" •0,9«7« ~=6 ,9 i7r •0,9606 _"0,969S -6.9696" •0,9697 _-0,9466 •0,0164 _iOt017l •6,6l7S~ •0,0171 _0,0056 •0,7400 0,0007 _rO,74«l -0,0071 »0,7*«*" •0,7467 •0,7467 •0,7466 •0,7156 -0,0077 •0,0079 •0,0060 •0,0076 0,0049 •0,669"r •0,6951 _-0,6953 •0,6954 •0,695* j-0,695i •0 ,660$" 0,0004 •0,0050 .•OjOOS* •0,0056" •0,0058 •0,0057 0,0046 _£P,*060_ 0,0001 •0,6084 •0,0028 •0,6090 •0,0029 —6-» 60,JO_iO, 002* •0,60*0 -0,002*" •0,6069 -0,0028 •0.6022 0.0019 •0,5405 _S0,S420_ •0,5420 •0,5420 0,0006 -0.0014 •0,0014 •0,0014 •0.5410. .0 flfl|fl " 0,0208" 0,0210 0,0224 0,0110 0,0471 0,4756 0,1125 0,0192 "5,01*5" 0,0166 _0,02)0 0,6337"" 0,3446 0,0677" 0,0151 0,013) "T,onj-0.0172 0,0245 "0,25*1" .0589 "o.otar 0,0106 0,0111 0,0134"" 0,018) 0,1894 0,01*9 _0,0063 6,0059" 0,0059 0.0066 0,0081 0,077) 0,0145 0,0050 0.0047 "67664 r 0.0052 0,0062 "olossr __0.0081 0,0032" 0,0031 .0.00)1 0,003r 0.00)7 0.0265 0,00*8 0.0021 4,0020 0,0019 0.0020 "•TOW •0,0266 •0,0267 •0,0237 -0,0196 0.0269 0,0000 J?0j02)0 •6,622*" •0,0209 •0,0166 •0,0156 0,0266 0,0000 •0,0179 •0,0176 « . O T * T " •0,0149 •0,0125 "0,02*3" 0,0000 •0,01)9" •0.01)6 •0,01)2 •6,0126" •0,0102 0.0217 0,0000 -0,0062 •0,00*4" • 0,0061 .•0,00*1 •o.oorr 0,0125 •0*0047 •0.00*9 •0,0049 •0,0048 •0,004* 0,009T" 0,0000 •6,0627" •0,0028 •0,0026 • 0 , 5 0 2 6 " • 0 , 0 0 2 6 0 .00*0 0,0000 •0.0015 •0.0015 •0,001* •0.0016 "6,0121-0.0154 0,0192 0,021* 0,02*4 .6,6624 0,0000 0,0069 l . o i i r 0,0142 0.017* 0.62TJ" 0,0**2 oToOoi" 0,00*7 0,00a* "ovoio*" 0,0115 0.0165 "0,0514" JljOOOO 0.0052" 0,00*7 0,0085 "6,616*-0,0110 0.0422 0,0000 _0,002S 0,0031" 0,0042 0.0051 0,66*1" 0,024* o.oonr 0,0020 0.0026 0,0013" 0,0041 0.0051 0,020* 0,0000 "6,0012"" 0,001* 0,0020 6.0031 6.01)) 6,0006 _6j6067 6,6609 6.6611 6.6614 "B.00B1" 6,616* 6,6112 6,61*2 6,019* 6,6566 6,6006 6,6059 "6.607r 0.0094 _6,0115_ 6,0141 0,0443 T Y S 6 6 6 " 0,0042 6,005) "670^ *7" 0,0063 _6,610t 0,0)46' _e_,oooo 6,60)9" 0,60)6 6,6046 ~ 5 . 6 o v r -6,007) 0.02*1 0,6666 _0j00l| 0,0014 0,0017 0.0021 o.oosr 0,0114 O.OOoT" 0,0006 0,00|0 "T.ooir 0.0015 6,001* "676085" 6,0066 "4,0004" 6,6005 _6.000* O^SOT" 0,6669 6.6645 6,6060 0,0002 O.OOOl" 6,6063 6.666* " 0 7 W 6 1 0 , 0 1 0 * 0 , 01 )2 0 , 0 1 * 2 0 , 0 1 9 * 0,0560 0,0006 0,005* "070075 0,009* . 0.0115 TTeiil 0,0*43 "9,0066 0,00*2 0,0053 0.0067 0,0063 0,0101 0 ,03*0 0,0006 "6,0636 8,0036 0 ,00*6 T.60S* 0,0073 0,0261 0,0006 0.0011 0,0014 6.0017 0,0021 6,662f 0,011* "676056 6,0006 6.0016 "T760I2 6.0015 _6.00l» "o7008"5 0,0000 O.OOO* 0,0005 JjOOO* 0,0 008 6,600* 6,0045 0.0000 6,0002 O.OOOl 6.0003 6.6604 Appendix H. oOp,p) as a r IL _ f i _ e^3«V \ 0j_0 0 3.391984 3.391984 0,20 0,914897 0.915113 0.J0 0,514798 0,512627 0,10 0.656665 0,650653 0,45 0,991508 0,963069 0,50 1.601421 1,590178 0.55 2.635296 2.620768 0,60 4,348691 4,330179 0,70 11,649500 11,618390 8,5 0,00 2,049150 2,049150 0,25 0,423108 0,024054 0.J0 0.271605 0.271706 0,35 0,181668 0,180632 0,40 0.154155 . 0,151692 0.45 0.198309 0.194147 0,50 0,312794 0,306676 0,55 0,505278 0,496953 0.60 0.775121 0.760372 0,75 2,107690 2,088003 1.00 6,228328 6,191834 0,7 0,00 1,255282 1,255282 0.40 0,077032 0,077271 0.45 0.066153 0.065001 0.50 0,091914 0,069978 0,55 0,159453 0,156142 0.60 0.270062 0 . ? h 5 1 8 1 0,9 0,00 0,781509 0,761509 0.40 0.054789 0,056389 0,45 0,035686 0,036746 0.50 0,038435 0,038801 0.55 0.065139 0,064661 0,60 0,116629 0,115168 0,65 0, |9S7|8 0,193126 111 0,00 0,493092 0,093092 0,45 0,026258 0,028114 0,50 0.023047 0,024077 0,55 0,034875 0.035763 0,60 0,062193 0,062011' 0,65 0.106602 0.106039 1,00 1,036950 1,027897 1.1 0.00 0.314643 0.31064] 0,45 0,021832 0,023980 0,50 0,018040 0,019910 0.55 0.022815 0,024302 0,60 0,038905 0,039903 0,65 0,066212 0,066615 1.00 0,692765 0.686011 1.5 0,00 0,202972 0,202972 0.45 0.018612 0,020814 function of positron energy ep eP*7«y 3.191964 0,915015 0,509762 0,602690 0.971883 1.575269 2.601496_ 4.305612 11,577060 2,049150 0,425374 _0^271?18_ 0,179350 0,106533 0„18875!_ 6,298703 0,086071 _017S0294_ 2,063067 6,103821 1,255262 0,077785 _0_,064635_ 0,087678 0,152059 _0,259063_ 0,781509 _0A058 770_ 0.038456 0,039641 _0j064437 0,113699 0,190219 0,493092 0,030936 _0.026791 0", 637422" 0,063209 0,105914 ep»i?«v 3,391984 0,915817 0,506226 0,632612 0,957995 1,556746 2_,5_7750_1 4,275063 11,525620 2,049150 0,427118 0,272309 0,177910 0,144786 0,182255 6,289036 0,472625 0,733112 2,032621" 6,064859 1.255282 0,078749 0,064066 0,065268 0,147502 0.252047 0,781509 0X062165 0,041094 0,041285 0,060851 0,112668 0 . I675U 0,093092 0,035065 0,030391 0,000317 0,065245 0.106606 17506417 0,314643 T,?3'2o2J 0,027385 0.030921 —67015171 0.070644 0.673793 0,202972 0.029282 3.191964 0.916158""-0,503439 0,624975_ 0 , 9 4 6 9 6 6 S,542026 2,558496 4,250764" 11,080660 27609150 0,426539 0,272723 0,176890 0,101957 __0,177260 6.2BiS05 0,462514 _ 0 , 7 19702 2,007680 6,038539 ~"77255282 0,079784 0.063931 0,083721 0,100305 0.206958 0,781509 0.065232 0,041627" 0,003100 0.065768 0,112521 0.186126 0,491092 0,018690 0,011885 - 0 , 0 Oil09 0,067669 0.106500 ~ 0 7 9 9 9 9 2 T " 0j3l0643 "0,6 3 6492"-0,031754 0,035075 •,017145" „ 0.314643 0,027238 0,022864 0,026818 0,041844 0,067847 0.679521 07649194 0,074027 0,671861 ~0 0.202972 0,024202 0,202972 0,014152 0,"5T" 0 , 5 5 0 , 6 0 0 , 6 5 1 ,00 0,00 0,45 0.S5 0,60 0.65 1.00 OjO0_ 0,45 0,50 0,60 0,65 >i-°.P_ 0,00 0.45 0,50 0.55 0,60_ 0,65 1.00 0,00 0,45 0,50_ 0.5S 0,60 1,00 0j00_ 0,45 0,50 0.55_ 0,60 0,65 0,00 0,45 0,50 0,55 0 t60_ 0,65 1,00 0,00 0,45 0 , 5 0_ 0 , 5 5 0 , 6 0 0 . «>5_ TToimr 0,017575 0,027026 0,044320 0,485608 0,132490 0,015777 _0,0l323l_ 0,013690 0,020479 - P - l l 1 ' ! * ! . 0,350537 _0j087660_ 0,013310 0,011418 _0j0_12007_ 0,015674 0,023235 _0.,257767_. 0,058903 _P,011105_ 0,009710 0,010072 _0,012470_ 0,017561 0,019942 0,006063 _0,005646_ 0,005667 0,006369 _0,0079LL 0,077746 _0,01446|_ 0,004875 0,004603 _0,0046M_ 0,005089 0,006125 .0,056443 0,008067 _0,003182 6,003027~ 0,003011 _0|0_03213 0,004835 0,002081 0.001962 0,001921 0,002018 0.002177 0,617549" 0,019310 0,028381 0,045193 0,480677 "6713 FSW" 0,017919 _0,015257_ 0,615508 0,021992 0.032886 0,346438 _0,087660_ 0,0153.24 0,013371 _0j0i3816_ 0,017249 0,024488 _0,.25422?_ 0,058903 _0j_pl2943_ 0,011527" 0.0U797 _0,014025_ • i > w « i 0,018660 0,191599 0,188474 0,019942 0,007299 _0j006906 6,666938"-0,007611 0.009Q7S_ 0,075945 _0,014461 0,005916 0,005694 _0,005729_ 0,06626"5 0,007203 _0 |055l5a_ 0,008067 _0,003919_ 6,003812 0,003835 0L004065 0,004835 0,002569 _0,002509_ 0,002502 0,002634 0.002822 0,026760" 0,022220 0.030B68 0,047128 0,475842 0,132"490 0,021288 _0,016548 0,018606 0,024776 _0,J>3S245 0,342864 .0,087660 0,016592 0,016643 —Oj_OL6990 0.020213 0,027128 _0,251615 0,058903 0.016046 0,01470! 0,014956 0,017068 "0,621663" 0,186638 0,019942 0,009704 _0,009497 0,669686 0,010477 0.012009 0,076715 0.014461 0,008158 0 ,008143 _ ° - i 0 0 | 3 7 3 _ 0,009024 0,010167 _0. 057121 0,003649 0,664513 u . g u m f a 0,028554 0,028270 0,032906 0,008067 _O,005893_ 0.006018 0,006281 A l i o 67 8 3_ 0,007475 0,004835 0,004465 0.0046S6_ 0,004928 0,005377 0.005919 D70"2"S755 : 6 7 0 3 0 5 8 ^ 0,026987 0,031683 0,035267 0,040027 0,051012 0,055534 0,473149 0,474272 ~o",Tsr49^ om"2'4~j 0,026518 0,031741 _0.023842 0,029287 0,023831 0,0294 05 0,029794 0,035402 _0,039936 0.045504 0,342386 0,346164 _0, 087660 _ 0 , 087660 0,023913 6,029549 0,022167 0,028194 _0j 022602 0L028949 0,025789 0,632179 0,032547 0,039306 _9,251095 0. .25990 J 0,056901 0,058901 _PJ021438 0,027644 0,020428 0;o27210 0,020929 0,026287 Jb 021184 0,031047 0.027 8 31 0|6161 52 0,190225 0,201122 0.019942 676T99 42 0,015290 0,025256 0,01578? 0,027419 0,01670| 6,010214 0,018236 0,011811 JLLP 20 519 0jJ>38171 0,090082 0 ,11220?" Jj .0 |446l 0,014461 0,014474 0,011020 0,015182 0,015154 0,016610 OjJ 4 00 80 0,018142 0,046054 0,020651 0,051226 078396 0,171864 0,008067 0,008067 -?j0l6793_^ 1,288423 0,018913 1.62626T 0,021470 2,035139 0j_0246 07 2^529 9 21 0,028408 3,128915 0,096747 13,043730 0,004635 0,004815 0,226117 0,002944 0.278420 0j_0 03798 0,341455 0,005026" 0.417601 0,006740 0.510325 0,008971 i . I, O 0 0 P 5 |LA K» •- - rvj ru r^ . to o ru | « o t < i N j t i / t ; ^ o i M 0 0 > 0 0 j 0 « « 9 ' 0 > 9 D — - • . * o o o o o o o o o o O O O O O O O ' O O •o to r » ;^> -O 4> r*> i _ _ kuooo O O I O O K I : o o o o o o io o o; o o o o o o-o o oj O O O O O O O rt O* O f - O O O ? rt A I A I rt ,9 4 > r » o o o o o o o o o ! « * r t •«IVO!««<f tf>>9 9 l\J 4>o»9rt41rt'*>»c04) O A 4 ) l 4 > « M S O N r u o o o o o o - « 9 o o o o o o o o o o o o o o o o o o o » m i / t o e 9 9 r u rt rt 9 9 t r » m 4> »•«• o " O O O O O O O O * * O O O O IO O O O O o o o oio o oio o » » 3 rt |tn t-> f LA rt CO 9 IMIA Al o 0 o -< » r\j j co rt rt rt AI Irt 9 4k o o o o o j o o o o o o o o o o o o o o o o l o o o o o i o o o !o o o IA U V * At * * - | o O - * | <OcOiAAtcOtQ>IA-* « ru A I Al » 00000:00 o o o o o o o o o o o o o o o o o O O J O 4 IA 41 'CO • * (A 41 O 41 I"- O ;Ai O iA '0 O A I ; O O* 0> O S J O ; — 4t A l ! AI41^ *iO> O Al '9 41 9 1 000 o —• 9 j 00 O I O O O t O O O : o o o j o O O I O o o i O O O IA O CF> O 9l<0 Al CO 41 ; !T rt rtlO 4t *«| CO Al Al Al Al Al 9 CO o 00 o ojo o o o 0:0 o olo O O o 00 0 olo O O ' > 9 V* f>- Al A l O 1-* 4> 9|o> rt O* J 9 AJ —' • OO ( V l O | ? f t l - • O* :41 O LA Al 9 :fU 9 iA' > 4 1 4 1 4 ) - 4 1 l * > 0 > - ' « « 0 | o * 4 0^ 0 0 0i>«nil f l I — — - « — f\J K Al - * * • * « m Kl • O O O jO O O O O O 9 O O O O O O O » o o o o ° o o ° * o o o o o ' o o o 0000(0000 ° jO O O jO O O iO O I »:rt rt rt -rt 9 41 • o o ojo o o i j O O O I O O O J ''•> ••> O 0 i -j o j j ' l ' I I 1 1 1 >L rt r- L A 4> i r » » r» 4> i Al Al rt 9 . O O i O O ( O o o o O l O* 9<AI (A < 0> r«. ;4) 41 C o oio o < o o o o < O O O O < 77 O O O O O i o o o o o o o o o o o o o o o » • * iA A l 0> 9 rt rt 41 A l — -+ - * Al o o o o o o o o o o o o o o o o o o o o 0vArt9lr^04>'lAtA O rt Al A1:AJ Al 9 .41 * o o o o io O O O O O O O O O O O O O !«•* co r*:Ai I A r*lr* o* o! |4» r»- o I A rt rt!<o f » ol >0 CO OO N r-if- co O'-Al o o o o olo Ort' o o o o o ojo O O O O O O O o o o o. io o oio o olo o o:. IA Aj!**».9!4> 4> •»( W 33 «A **cO|CO 9 Oi CO At Ai Al Al 9| o o o o olo o o! O O O O O O o o -o o o o ojo o oj O OJO O OJO C rt (A 0> IA •« <o i n rt rt IA O O O O O O O O O O o o o o o o o o o o o o o o o O IA O lAO iA OllA O O Lfi O'lA O lA'O IA O' O 9 IA LAJ4) 4> r- O I— o o o oio o oio —« O 9 IA lA 4» 4>!»*> f » Oi O 0:IA O O O O O O IA:tA 41 F*ICO 0> oi » - * ; O O O O O ' O O - * o o o o H 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0085260/manifest

Comment

Related Items