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The infinite order sudden approximation and the delta-shell potential 1992

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THE INFINITE ORDER SUDDEN APPROXIMATION AND THE DELTA-SHELL POTENTIAL by STEPHEN JOHN DANCHO B.Sc., The University of Winnipeg, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE ill THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CHEMISTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1992 © Stephen John Dancho 1992 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of  Chemistry The University of British Columbia Vancouver, Canada Date DE-6 (2/88) The Infinite Order Sudden Approximation and the Delta-Shell Potential Abstract The Infinite Order Sudden (IOS) approximation is applied to the collision of an atom with a diatom where the intermolecular potential is given by a delta-shell. It is shown that modelling the potential as such allows for a simpler calculation of the close-coupled equations, and using the MS results in even further savings in calculations. Ex- act and IOS calculations at 300K and 1000K are compared and it is found that the WS overestimates inelastic cross sections for both temperatures. A variety of corrections to the IOS are considered and the Energy Corrected MS (ECIOS) approximation is shown to be the best of those studied. Other possible improvements to the IOS are proposed.ii Contents Abstract 	 ii List of Tables 	 vi List of Figures 	 viii Acknowledgement 	 ix 1 INTRODUCTION 	 1 2 ATOM-DIATOM COLLISION THEORY 	 8 2.1 Uncoupled Angular Momentum Representation 	 8 2.2 The Total,/ Representation 	  15 2.3 Cross Sections in the Total-.I Representation 	  22 2.4 The IOS Approximation 	  25 2.4.1 IOS Cross Sections 	  28 2.4.2 Energy Corrected IOS Cross Sections 	  31 2.4.3 General S Matrix Cross Sections 	  32 2.4.4 Accessible States Scaling Law 	  33 3 THE DELTA-SHELL POTENTIAL 	 36 3.1 Scattering From a Spherical Delta-Shell Potential 	  36 3.2 Scattering From a Non-Spherical Delta-Shell Potential 	  40 3.3 Simplification Using Only Open States 	  46 4	3.4	 Inclusion of Closed States  3.5 	 IOS T-Matrix Calculation CALCULATIONS AND RESULTS  4.1 	 Parameter Determination 51 56 60 60 4.1.1 Atom and Diatom Parameters 	 60 4.1.2 Choice of Energy 	 60 4.1.3 Range of Partial Waves 	 61 4.1.4 Inverse Power Potential Comparisons 	 62 4.2 Cross Sections at 300K 	 67 4.2.1 Exact Cross Sections Including Only Open States 	 67 4.2.2 Exact Cross Sections With Inclusion of Closed States 	 . 69 4.2.3 IOS 0 --* L Cross Sections 	 71 4.2.4 IOS Scaling Relations 	 74 4.2.5 Energy-Corrected Scaling Relation 	 87 4.2.6 General S-Matrix Scaling Relation 	 97 4.2.7 Accessible States Scaling Relation 	 104 4.3 Cross Sections at 1000K 	 112 4.3.1 The Exact Cross Sections 	 112 4.3.2 The IOS crL, 0 Cross Sections 	 114 4.3.3 ECIOS UL,-0 Cross Sections 	 117 4.4 Changing Parameter Cv,  119 4.5 Changing Parameter a 	 121 iv 5 DISCUSSION 124 5.1 Time Savings of the IOS 	 124 5.2 Possible Improvements to the IOS 	 125 5.3 Applications of the 105 	 126 5.4 Molecular Potentials 	 128 5.4.1 	 Time Savings of the Delta-Shell 	 128 5.4.2 	 Comparison of Potential Parameters 	 129 5.5 Calculations on a PC 	 132 6 CONCLUSIONS 133 References 136 List of Tables 1 	 A Comparison of the Delta-Shell and 7. -12 Potentials 	  66 2	 Exact Cross Sections 	  67 3	 Effect of Including Closed States 	  70 4 	 IOS o-L,0 Cross Sections 	  72 5 	 IOS Cross Sections at 300K Using k o =   75 6 	 IOS Cross Sections at 300K Using k o = kfinal 	  77 7 	 Comparison of k i„ itial and kfilial IOS Cross Sections 	  78 8 	 Comparison of k,„,,„ and krni„ IOS Cross Sections 	  79 9 	 IOS Cross Sections at 300K Using ko = 14.27 A-1 	  81 10 IOS Cross Sections At 300K Using Exact o-L,_0 Values 	  83 11 	 Effect Of Using Different k Values In Exact o -L-0 Values on the Cross Sections 	  85 12 IOS Cross Sections At 300K Using k=16.92 A -1 	  86 13 	 ECIOS Cross Sections At 300K Using T = 71- a/(2v inin ) and Exact O•L—o(Ek,„,,,,) Values 	  89 14 	 ECIOS Cross Sections At 300K Using 7 = fa/v,,,;,, and 105 aL—o(Ek,,,,.) Values 	  91 15 	 ECIOS Cross Sections at 300K Using T = 0.16a/Vmin and 105 01-0 Ek.„“„) Values 	  92 16 0i,_0 k Values and ECIOS T Values Used In Table 14 	  94 v i 17 	 How f Varies According To the k Used in the Calculation of the crL4-0 Cross Sections 	  96 18 GSMSR Cross Sections At 300K Using IOS o -L,0 Values 	  98 19 Input crL,o (Ek EL ) Values for the GSMSR at 300K for f=10 	  101 20 GSMSR at 300K Using Exact crL-0(Ek EL) Values 	  102 21 7 Values Required to Match the GSMSR with Exact Results at 300K 	  103 22 ASSR Cross Sections at 300K Using IOS 01_ 0 Values 	  106 23 	 Input o-o_L(Ek 	 L) Values for the ASSR at 300K for j' =6 . 	  108 24 ASSR Cross Sections Using Exact au—L(Ek (L) Values for 300K 109 25 	 Exact Cross Sections at Energy=1000K 	  113 26 	 IOS Cross Sections at 1000K Using ko =   115 27 ECIOS Cross Sections at 1000K 	  118 28 Exact and IOS Cross Sections at 300K for Cv, = 1000 	  120 29 Exact and IOS Cross Sections at 300K for a =0.55 A-1 	  122 30 	 Computer Time Required for IOS and Exact Calculations 	  124 31 	 Computer Time Required for IOS Calculations for a Continuous and Delta-Shell Potential 	  129 vii List of Figures 1 	 Coordinates used for diatom-atom collision problems 	  10 2 	 Impact parameter b 	  63 viii Acknowledgement This project is actually the completion of some work that was originally started in 198 and worked on until 1985. The work was resumed last year and it has only been with the help of some very exceptional and special people that I have been able to finish the thesis. I would like to express thanks: To my parents Joan and John for their love and support, to Sheryl and Vince for helping me through the toughest time, to Cathy and Nigel and Connie and Dave for the phone calls and letters, to Heather and Jennifer for their drawings which I have on the wall at the office, to my Grandmother Stella who taught for j1 years in the Manitoba public school system and to my other Grandmother Catherine who I wish could be with me for this graduation. To my friends Daniel and Jaleel for helping me keep perspective while I worked on this research, and to Valerie and Colleen. To my Aunt Rita and Uncle Don for helping me on my move to Van- couver, to my godparents Aunt Margaret and Uncle Ed for their hospitality on my many trips between Winnipeg and Vancouver, and to my cousins and Toni and my godchild Marie. To my students Vanessa, Leora, Nicol, Julie, Mark, Linda, Arlann, Ian, Jaye, Michael, Dave, Bron, Arnie, Cathy, Christina and Carol-Anne who showed me that Science can explain a lot more than I thought it could. To my teachers Dr. R. Wasylishen, Dr. C. Campbell, Dr. W. Mabb, Dr. H. Hutton, Dr. H. E. Duckworth, Mr. M. Selby, Mr. J. Dobrovolny, Dr. Kerr, Dr. D. Topper, Dr. C. Ridd, Dr. E. A. Ogryzlo, Dr. D. G. Fleming, Dr. M. C. L. Gerry, to my references Mr. P. K. Bingham, Mr. T. Kostynyk and Mr. C. Buffie, to my customers at Eatons', Mr. D. Feinberg, Mr. H. Delorme and Mr. D. Sloan, to my colleagues Mr. Guowei Wei, Mr. Pat Duffy and Mr. Dan Berard for help on Quantum Mechanics, DOS real mode and Unix, and to the professors in the UBC Theoretical Chemistry group, ix Dr. D. P. Chong, Dr. J.A.R. Coope, Dr. G.N. Patey and Dr. B. Shizgal, for their examples of excellence in teaching and research. To the doctors who have kept me healthy, Dr. B. Jones and Dr. G. Laws and to all the researchers in neurochemistry. To the administration at UBC for assisting me this year, Dr. Legzdins, Head, Dept. of Chemistry, Ms. Tilly Schreinders, Graduate Secretary, Mr. Alnoor Aziz, Finance Department, Ms. Anne Grierson, Graduate Studies. For the generous financial support from NSERC. To Dr. R. Pincock, Graduate Admissions, Dept. of Chemistry, for giving me the chance to finish this work and for his constant assistance throughout the year. And finally to my supervisor Dr. R. F. Snider for suggesting this fasci- nating topic, a topic that has kept me intrigued and confused for the last 8 years and probably will for the rest of my life. I owe a great deal of thanks to him for all the encouragement, patience and enlightening conversations throughout the years. And especially for giving me the freedom to discover this part of the world in my own way. It is indeed a pleasure to work with such a gifted educator and scientist. 1. INTRODUCTION The study of molecular collisions is the basis for understanding a large number of chemical phenomena such as chemical reactions, gas viscosity and pressure broadening. Specifically, the equation used to describe molecular collisions is the SchrOdinger equation. The aim of this thesis is to further investigate an approx- imation used to numerically solve the SchrOdinger equation — the Infinite Order Sudden (10S) approximation — on the scattering problem where the interaction potential between an atom and diatom is given by a delta-shell. This chapter gives a brief introduction to and describes the development of both the IOS and the delta-shell potential. The description of the collision of an atom and a diatom requires consideration of the process before, during and after the collision. Before the collision, it is necessary to set the initial conditions, that is, to describe the state of the free atom and free diatom. The collision itself involves a choice of intermolecular potential and a set of coordinates suitable for the mathematical description of the collision process. Finally one must identify the amplitude and relative probability for the products of the collision as they are separating at infinite distance from each other. Here the latter are reported only in terms of the total degeneracy averaged cross section into each of the allowed states. In setting up the problem, attention is paid to how the calculation can be performed with computational efficiency. [1]. The system to be studied is the collision of a homonuclear diatom, treated as a rigid rotor, with an atom. Later on, the description will be more specific and the molecular parameters will be chosen to model the argon (Ar) - nitrogen (N2) system. Until that is done, the description is for the general collision of an atom with a rigid rotor. A detailed description of rotational excitation caused by the collision of two molecules requires the solution of a set of close coupled equations [2]. Approxi- mation techniques have been developed over the last 20 years in order to reduce the amount of computation required to solve these equations. The approximation technique of interest in this thesis is the 10S. The close coupled equations are equivalent to the SchrOdinger equation. The solution of the SchrOdinger equation for inelastic processes involves the proper treatment of both the angular and radial motion. Depending on the basis set used, the equations are coupled via angular momentum operators or the inter- action potential. The first type of operator, responsible for directional coupling, is the interaction potential, which is generally diagonal in orientation representa- tion but non-diagonal in angular momentum representation. The second type of operator includes the orbital angular momentum operator and the translational energy operator, which, in contrast to the potential, are diagonal in angular mo- 2 mentum representation and non-diagonal in orientation representation. Hence, in either representation — angular momentum or orientation representation — the SchrOdinger equation will couple direction dependent states. The IOS approximation replaces the quantum numbers for rotational and cen- trifugal angular momentum with constant values, decoupling the set of equations in an orientational basis. This has the computational advantage of decoupling the angular and radial motion, treating the former as a constant and the SchrOdinger equation then reduces to motion in one dimension (the radial motion). A stan- dard further simplification is to use the WKB approximation [3]. An alternate approach is to look for a potential that simplifies the radial motion. In partic- ular, the SchrOdinger equation for the delta-shell potential can be reduced to treating the radial motion by matching inner and outer solutions. For the exact close coupled equations, this leaves the angular motion to be treated by matrix methods. In the 105 approximation the solution is obtained analytically so that not even matrix methods are required. The IOS is a combination of two approximations – the Energy Sudden (ES) [4, 5] and Centrifugal Sudden (CS) [6, 7, 8, 9, 10, 11] approximations. The Energy Sudden approximation is equivalent to assuming the rotor's ori- entation is fixed for the duration of the collision, but exact solutions to collision problems involving a homonuclear diatom (rotor) and an atom take into account the fact that the rotor's orientation changes during the collision process. The ES, which involves treating all rotational states as degenerate, was first used 3 by Drozdov [4] and Chase [5], according to an IOS history given by Parker and Pack [12]. The CS approximation involves treating the centrifugal potentials as degen- erate. There were two approaches taken leading to this approximation method. One approach, taken by Pack and co-workers [6, 7, 8, 9, 10], was to treat all centrifugal potentials as degenerate as well as incorporating the ES in a space fixed frame of reference. The other approach, that of McGuire and Kouri [11], was to treat all centrifugal potentials as degenerate in a body fixed frame of reference. The approach taken by Pack and collaborators was first proposed in 1963 by Takayanagi [13]. In 1970 Tsien and Pack [6] applied Takayanagi's approximation and tested it numerically on an He — N2 system. The results proved encouraging and further work by Pack and co-workers [7, 8, 9, 10] led to what is now called the IOS approximation. In 1974 Pack [3] then extended his work to a body-fixed coordinate system and chose to replace the centrifugal potential operator with a single centrifugal potential which he identified as the total angular momentum. He termed this the CS,/ approximation. In 1972 Rabitz [14] developed an effective Hamiltonian method which suc- ceeded in decoupling orbital and rotor angular momentum. An alternative way of achieving this decoupling was presented in 1974 by McGuire and Kouri [11]. In their work they fixed the centrifugal potential as a final orbital angular mo- mentum and termed it the Coupled States (CS) approximation. 4 The CS of McGuire and Kouri and the CS-J of Pack were shown to be equiv- alent in 1977 by Parker and Pack [15] when they identified the single centrifugal potential in the CS-J as labelled with a final orbital, rather than total, angular momentum. Simultaneously Shimoni and Kouri [16, 17, 18] found that the cross sections for Pack's CS-J approximation were greatly improved when a similar substitution was made. An equivalent approximation to the IOS is assuming constant orientations for the duration of the collision. This idea had been used as early as 1961 by Monchick and Mason [19] when it was applied as an approximation for classical scattering. In 1975 Secrest [20] and Hunter [21] simultaneously succeeded in formalizing the IOS from a fixed-angle approach. Hunter [21] further pointed out that this approach was similar to an approach taken by Curtiss in 1968 [22] who, while developing a formalism describing molecule-molecule collisions, presented an approximation where all orientations of both molecules are fixed. The IOS is expected to be a valid approximation when the collision is sudden, that is, when the collision process occurs over such a short time duration that only negligible rotations of the molecular system can take place. To enhance its range of applicability, a variety of corrections have been considered. In this thesis, the IOS and some of the methods of correcting it are applied to a system which has a delta-shell potential — a potential having an infinite height extending over an infinitesimal width at some distance from the origin. As a basis for comparison, the degeneracy averaged cross sections of Ar — N2 collisions of the exact, IOS and 10S-corrected solutions are studied. It would be expected that a delta-shell potential would make for conditions where the IOS approximation is valid. An interaction described by a delta-shell potential would have only an infinitesimal distance in which to act, which implies an interaction taking very little time — the basic concept of a sudden collision. The delta-shell is first investigated in its simplest form — as a spherical potential. Then an angular dependence for the potential is introduced. The resulting exact and IOS degeneracy averaged cross sections for this potential are then derived and studied. The savings in calculation noted above merits further investigation into whether more accurate modelling of potentials with one or more delta-shells can be accomplished. While this thesis uses the delta shell as an interaction potential between two molecules, most work with the delta-shell has been in nuclear physics. One study found that two terms of a spin-angle expansion of an effective neutron- proton potential for deuteron-proton reactions to be well approximated using delta-shells [23]. Kok et. al. have found the delta-shell potential useful in calcu- lating phase shifts for proton-proton and N —a scattering [24]. Other applications for the delta-shell have been in molecular physics [25] (studying hyperfine inter- actions) and in solid state physics [26] (calculating band structures). Chapter 2 outlines the theory for the exact and IOS calculations of cross sections. Chapter 3 describes the calculational details required to obtain these 6 cross sections with first a spherical delta-shell and then a non-spherical delta- shell. Chapter 4 describes the parameters used, lists the cross sections obtained, and compares the values calculated for the exact, 10S, and 10S-corrected cross sections. A brief discussion of computer time saved by using the 10S, current work on the 10S, and possible future work for the delta-shell potential then follows in Chapter 5. Finally Chapter 6 concludes with a summary of the main points of this thesis. 7 2. ATOM-DIATOM COLLISION THEORY 2.1 Uncoupled Angular Momentum Repre- sentation The dynamics of the collision process are governed by the SchrOdinger equation. Since the collision process starts and ends with two free particles (in this case, an atom and a diatom) a positive energy solution for the SchrOdinger equation is required. As well, a centre of mass co-ordinate system is chosen so only relative motion between the particles need be considered. The SchrOdinger equation is HT(P, = ET R) (1) where H is the Hamiltonian, kli(7•,11) is the wave function of the system, and E is the energy. This section solves (1) in an uncoupled representation — both the rotational angular momentum of the rotor and orbital angular momentum of the relative motion between the atom and diatom are treated independently. In the next section a coupled representation is introduced, allowing for some simplifications in both notation and in manipulation of terms. 8 To set up the mathematical description of an atom colliding with a diatom it is necessary to first define a set of coordinates. For this study, a space fixed centre of mass coordinate system is chosen. With reference to Figure 1 on page 10 , R is the position vector describing the distance from the diatom (rotor) centre of mass to the atom centre of mass, r is the vector between atoms in the diatom and 0 is the relative angle between R and r. In this paper, three-dimensional vectors will be designated in bold and the following convention will be used: r = 7'7' R = Ri? 	 (2) where r, R denote the magnitude of vectors r, R and r" , j denote the direction unit vectors of r, R respectively. In order to solve this equation, a specification of the Hamiltonian H is needed. We are interested only in the rotational aspects of the diatom so the rigid rotor approximation for the diatom is assumed, with both electronic and vibrational contributions being neglected. Since the rotor is rigid, the magnitude of r will not change so the potential will be only a function of R and the direction f-. There are thus two types of motion to be considered for the Hamiltonian. The first is that of the spinning rotor and the second is the relative motion of the atom with respect to the diatom centre of mass. The Hamiltonian H1 for the rotor is jr2 Hi _ 2I (3) 9 R FIGURE 1: Coordinates used for diatom-atom collision problem. 1 0 where Jr is the angular momentum operator of the rotor and I is the moment of inertia of the rotor. Next, an appropriate complete set of states for the expansion of IP is chosen. Since in free motion, the spinning rotor motion is independent of that of the orbital motion of the atom around the diatom, 'P can be expanded in a complete set of rotational eigenfunctions, each expressed as a product of eigenfunctions of the two rotational motions. These comprise a discrete, as opposed to continuous, basis. The eigenvectors of Jr are the spherical harmonics Y,„(• and so JiYpn( 7') =	 + 1 )h 2 rini(r')  (4) where j is the quantum number for magnitude of rotor angular momenta and m is the quantum number representing its component along the z-axis. The Hamiltonian H2 for the atom-diatom relative motion is h2 r_72 H2 = - - v R 2/..t where it is the reduced mass. In the Ar-N2 system the reduced mass t is MAr MN2 - MAr MN2 ) MAr being the atomic mass of Ar, m N2 the molecular mass of N2. The eigenvectors of H2 are e ik 'R so that u ik•R	 h 2 k2 „ik•R II 2 e	 — 	 (7)2/2 where eik 'R is a plane wave of momentum hk. For the scattering problem to be studied, one assumes an incoming plane wave in initial momentum state hk' and rotor state j'm'. (5) (6) 11 The expression for the total Hamiltonian H can be written H = Hi + H2 + 	 R) 	 (8) where V(71 , R) is the atom-diatom potential. This potential can be written in the form  R) = E vi,(R)PL (cos o) 	 (9) L where Pi, is the Lth Legendre function and 17L (R) its expansion coefficient. Converting Vila to polar co-ordinates R, co and 0, the total Hamiltonian H given by (8) becomes: —h 2 { 1 a ,,,  a 	 A' 	 2H 	 7;	 R) 	 (10) 2/2 R2 aR aR n2R2 where A2 is the operator for the square of the orbital angular momentum, which is, in orientation representation: 2 _ _ n2 [ 1 —a sin y a +  1  '2A 	sin yoap 	 ac,o 	 sin2 e a02 • (11) The eigenvectors of A 2 are the spherical harmonics YA,V), A being the quantum number of angular momentum magnitude for the orbital angular momenta and s is its component along the z-axis. Hence  A 2 Y,\ ,(R) = 	 + 1)K.s(1).  (12) In the orientation representation it is required to solve (1) for T. With the  expansion as given by (12) and (4), 	 can now be written:  R) = > YimMYAs(I) ,,,„,(R) 	 (13) jniAs 12 where a is an abbreviation for the set of quantum numbers j, m, and s, a' the initial set of quantum numbers and tk,„,(R) is the coefficient for this expansion for a given a and a'. Putting (13) and (10) into (1) and operating on the left of each equation by f f Y2*.(0YA*s(f)ch'di 	 (14) gives the set of close coupled equations in an uncoupled angular momentum representation: ti 2 1 d2 	 	 d z A(A + 1) 71) 0 ( a / 	 = E vaa4,„„,(R) 27 PR' 	 lidR 4- 3 R2 	 I a" where k'? .±2 [E 	+1)01 3 	 h2 	 2 1 and = I f ri,n (i9YA*8 (i?)V(1., R)Yinno (P)YA, s , (ft)df-dil. 	 (17) Next the boundary conditions are set for and its expansion coefficients 0,„,„,• As well, qi and all the 	 must be regular at the origin. Finally the must be chosen so that satisfies the incoming wave condition, ie., the solution of (1) starting with planar motion in rotational state j', m', k' (eigenstates of H 1 and H2 ). So eile•Ryi,m,(0 scatt • 	 (18) where W scatt is the scattered part of 	 The experimental observable to be examined in this thesis is the cross section (which will be defined in Section 2.3). (15) (16) 13 In order to calculate the cross section it is necessary to determine the amplitude fin, ;j im i(i?) of the spherical wave in the final state jm. (18) is rewritten as e R iki R kP eiki .RYjni 	+ E Yjm(il f im; 	 (R) 	 R 	 oo.	 (19) j?n We make a spherical harmonic expansion of the initial plane wave, = E omeRwAs (iT)YA-s (P) 	 (20) As where jA (k'R) is a Bessel function of order A, and k' is shorthand for kj ,. Thus the wave function with appropriate initial conditions is expanded as in (13) with expansion coefficients 0„„,(R) = 47r E 	 (21) A' .5' The x„,(R) functions at large R satisfy the coupled equations (15) with boundary conditions 1x,„„,(R) 	 (eR)(5.' (22) where 	 is  the Kronecker symbol, to match the incoming plane wave and the outgoing spherical wave of equation (18) in terms of the transition matrix element h A (k R) is the Hankel function and its asymptotic solution as R 	 oo is given by: R 	 oo . 	 (23)—kjRei(kiR-A7/2) , Using (23) in (22) yields R—Ar12) Xaa i ( R) 	 j (k i R)6acv , (24) 2 11 kik' 14 so that comparison of (19) and (24) leads to the following relationship between and 27riJae(' 	 (25) In conclusion it is noted that to attain the goal of a calculation of the atom- diatom collision cross section, it is necessary to solve the set of differential equa- tions (15) which are coupled by potential matrix elements subject to the boundary conditions of being regular at the origin and asymptotically of the form of (22). 2.2 The Total-J Representation Since H is invariant with respect to a rotation of the co-ordinate system, the total angular momentum is conserved. Hence states having a total angular momentum J are not coupled to states with a different total angular momentum J' and the SchrOdinger equation is decoupled into different total angular momentum components. The total angular momentum operator J = A + Jr has eigenfunctions n'Am 	 EE(jmAs jAJ M)Y3 ,„(ilY),,(1) 	 (26) where (jmAs I j AJ M) is a Clebsch-Gordan coefficient[27}. The expansion coefficients v3T,,A ,(R) of the scattering wave function (22) in this angular momentum representation satisfy 15 R) 47r E iAriAmv,ibx;JA;i , A ,(R) ,A3M A' s' X (j i 	M I j im')'s')YA*,,,(1;'). (27) Performing a similar set of operations as outlined in the previous section only this time operating on the left of each equation by I Y3;;T,(i',1-=?)di. d1 	 (28) gives the set of close coupled equations in the coupled angular momentum rep- resentation: d2 	  d 	 2 	 Ak. 2u dR2 R dR 	 R2 [ P■	 1 )] AjA ; ii,v(R) 	 jA; j".X" 	 ;3',X'Vj 	Vj	 • ( R) (29) where VIA;3 ,A, = JI Yi*A1 M V, i?)V(r,R)Y3'ZI, , 	 (30) For the potential of (9) we are interested in determining an angular momen- tum representation. First the expression for the uncoupled representation, that is, for the V-matrix element (jnas Vijim! s') is determined. This quantity is then transformed into a total-J representation. It is then shown that rotational invariance of the potential allows certain simplifications in the total-J represen- tation. The potential given by (9) is in an orientation representation. The angular momentum representation of V ,given in terms of the orientation representation, 16 is: (jmAsIVI,j'712' A's') = f f f f (Pl) sh 1- )(1'; flIV (R)11'' ; ir)(f-'; 	 77-1' s i)di'dkdf- idk Using 4.6.6 of Edmonds[27], ie., 47r PL(cos 0) = 2L 1 E YLAMYL'A (9) can be written as mv(R)Ip ; fr) E vL (R),L47+ 	 4,(0y;,;,(t)so 	 — fi). (33) Lti Using the following defined phase conventions [28] and substituting (jmAsIV > Lp 771As (33) into j'nt i	= 47r 1?) = (-i)'Y3*,„(o(-i)AYas(R) (31), one obtains the following equation: (—i)jyr,„(1')(—i)AYA*.s(f) YLA(')Yi„(f)i ji Yi'.1(0(i) Al YA's'( 11)th'dk (34) (35)vi(R) 2L + 1 By using the following two properties of the spherical harmonics, (equations 2.5.6 and 4.6.3 of Edmonds [27]) YiKti(f) = (-)AYL_„(fi)  (36) (31) (32) 17 and = (2j + 1)(2L + 1)(2j' + 1)  (	 j L j'L j' 47r 	 —rn ( m ) 0 0 0 , (37) (35) can be written as (jmAsIV (R)1j'm' s') =	 (2j + 1)(2j' + 1)(2A + 1)(2A' + 1) 	L '	 A L Ax O o o) ( ) o o ' ) j L j') (A	 LA' ) ). (38 -771 it 711 / 	-8 -	 8/ Equation (38) is the expression for the V-matrix element in the uncoupled representation. Eight quantum numbers, j, rn, A, s, j', rn', A' and s' are required to label each element. The total-J representation, however, can reduce the number of labels needed to five, ie., J, j, Ad and .\', as will be shown. In order to arrive at an expression for the total,/ V-matrix element, the uncoupled V- matrix expression is converted to the coupled expression through the following transformation: (jAJMIV(R)Ij i)'J'M') = E (jAJM I jrrr,\.․ )(j1n\sIV (R)Ij"m" A" s")(j"rn"A"s"Ij'A'J'M'). (39) msm"3" Using the phase convention given by equation 3.7.3 of Edmonds [27] which relates the Clebsch-Gordan coefficients with the 3-j symbols through the following: A 	 J V,2J + 1	 (40) s — MAJ M jmAs) = ( 77/ 18 and using (38) for (jinAsIV(R)I j'n'A's'), (39) can be written A 	 J /AVM') = E (—)3 -- A+m,\/2J + 	 1 ( 7n s —M ) 711S711' (jAJM1V (R ) xi3i+A1-3-A \I(2j + 1)(2j' + 1)(2A + 1)(2A' + 1) x E 000 ( oo o) — 	 m' L j' 	 A L 771 —s A 	 L A' s' x(—)-A-m-s(—)ji—v+miV2J' + 	 1 ( 	 Ji )m' 	 —M' ) • Two symmetry property of the 3-j symbols, given by equations 3.7.6 and 3.7.5 of Edmonds [27], ie., j' A' 	 J' 	 = 	 j' 	 A' J' in , .s' —M' 	 —m ' —s '	 ) (42) and j L j' ) ) — 	 —m m' L iu ) (43) allows (41) to be written as (jAJM1V(R) = \I(2j + 1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1) x E ij i+A -3-AvL(R) (j L j' ) (A L A' L 	 0 0 0 	 0 0 0 x ` (_ ) i-i-, , -Fx+m+m , ( 	 i' 	 A' J' in,,s, ) —m' —s M' x E(—)m—s+A-Fj+A-EL ( A' 	 A 	 L ) ( 	 i inns 	 s —s —it 	 -77t 771 , A 	 J s —M (44) Further simplification in (44) is possible by considering equation 6.2.8 from Lg (41) 19 Edmonds [27], where E (4., i+t,2+A3+i1+12+13 1 31 l2 	 /3 1 iL2 113 1711 P2 — /13 X 	 _( 	 11 	 j2 	 13 ) 	 /1 	 12 	 j3 	 = 3 _{ 31 	 2 	 3 — 111 111 2 ,u3 	 /Li — ,u2 7n3 	ti 12 13 (  31 	 j2 mi m2 m3 (45) where j l , j 2 , j3 , 177 1 , 771 2 , 7173 , / 1 , / 2 and 13 are the arbitrary indices used in Ed- monds [27] and j1 j2 /1 	/2	l3 is the 6-j symbol defined by Edmonds [27]. (44) can be written j'A'I'M') = V(2j 1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1)(jAJM V(R) x E 	 VL(R) j 	 L A /0 x E(_)i+i,+x+m+m, A' j' A, j A L 	 —m' —s ' J' ( 	 j' 	 J ) s' m' —M ) • (47) Using (42) and equation 3.7.8 from Edmonds [27], ie. , ( 31 	 32 	 ) 	 31 	 j2 = (2:73+1 r i bm,„i t 	 (5 • 	 (48)3 3333 31 2 3177 1 m 2 1713 	mi m2 m3m i M2 where b i1h73 = 1 if j 1 , j2 and j3 satisfy the triangular conditions and 6171j2j3 = 0 otherwise, (47) is further reduced to (46) 20 (jAJMIV(R)Ij'AVM') = 1/(2j + 1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1) x E i3 17/, (R)  L	 ( A L A' 0 0 0 )0) 0 	 ) x(—) 3 +31 +i+m+mi (2J 1) -1 8j,p6mm , I A'jij LI Finally, using the symmetry of the 6-j symbols as given by equation 6.2.4 of Edmonds [27], .12 :13 l = 1 .13 	 J2 j4 js j6 j 	 1  j6 ,14 j5 the following expression is obtained: ( 5 0 ) j 1 J i M i ) =	 02j + 1)(2j' + 1)(2A + 1)(2A' + 1)jAJMIV(R) L j' ( A L A' f J A' j'E vL(R)(-Y+3+3'  . 	 (51)x j 0 0)0 0 	 L j A Note that the V matrix element is independent of M, the projection of the total angular momentum on the z-axis. This is expected of any rotationally invariant (scalar) quantity such as the interaction between atom and diatom. Secondly it is noted that the total angular momentum J is the same in the initial and final state. For this reason J need not appear in both initial and final states but can be denoted as a superscript on the operator V, ie., 1/ J . Hence the V-matrix element (j)JMIV(R)Ij'A'J'M') may be written as (jAIVJ Ij')'). This new representation, having taken advantage of the simplifi- cations made possible by the rotational invariance of the interaction potential, now requires only five indices instead of the original eight. (49) 21 As done in (22), an incoming plane wave of constant flux in initial momentum state Pik' and rotor state j'71-1 1 is assumed for the boundary conditions for T. It is also required that the solution be regular at the origin and have the asymptotic form eikR X, ; j , A , ( R) Mk/ R) 8 j":5 AA' + 	 R 	 oo. 	 (52) 2\c' R 3Au' A" 2.3 Cross Sections in the Total-J Representa- tion The cross section is defined as the ratio of outgoing spherical flux to incoming planar flux. Planar flux is defined as the number of incident particles crossing a unit surface area placed perpendicular to the direction of propagation per unit time while spherical flux is defined as the number of outgoing particles scattering through solid angle cll2 per unit time. The incoming plane wave e iki * R has flux - ,Pncorning [29] h k' Fincoming = IL (53) Putting (22) in (21) and then into (13), and substituting k for R results in the following expression for the spherical flux Foutgoing for the outgoing spherical wave e ikR/R [29] : 2 Ai 	 (54) t k' Foutgoing = (471-) 	 YA.5(k)TamAs ; j /nt i s i YA 1 si(ki )lkj kJ , As A' .9' 22 From (53) and (54) the differential cross section o-i„,,i,„,,i(k) is: .7r. 	 4 2 a- . 	 (k)m4---2 1 111 1 2 E Ks(k)TjtnAs ; jim'AisiYA1 3/(k i ) (55 ) In this work only the total degeneracy averaged cross section, 	 ie., the differential cross section integrated over all angles and summed over all m's, TDA  =- 2j' + 1 	 0.3„3,7n,(k)dk is to be calculated. It is possible to simplify the resulting total cross section by choosing the incoming direction k' as the z-axis. Since (56) m m' yAs ( ki) 	 y\s (z) 2A + 1 oso 47r (57) the expression for o-TD1- becomes: TDA — E E TimAs;?rnivo(E)YAs(ic)ko(2j , + 1) AsA' 47.2 2A' + 1 47r x E YA„.,„(k)77„,,,„,„ ;i,„„ A „,o (E) „ s„ ,\” , 2A"' + 1 dk 	 (58) 47r Using the orthonormality of the spherical harmonics, I }I s ( k ) YA*,,„ ( k )dk 	 (59) the expression for the total cross section now is 1"TDA = k , 2j + 1) Asv„,„,' 	„ 	 , 	Tjm  ; jimiA 10(. 	2.)■• + 1) 	 (60) 23 A 	 J" ) ( j' x s —M" ) in' A' 	 J" s /	 Mil/ JIIMI/j/IIMIll (62) Finally, then, we seek the appropriate form for o-TW; in the total-J represen- tation. Using the identity (jm)sT(E)Vin'A's') = E (iniAskmiA)(JmiAlT(E)Ifivricv)(fm -iyAilfm')'s') (61) JMJ'M' and (40), (60) in the total J representation is: 71TDA 	 E E— 	 + 	 770)11 AsAisi 4_ 1) JMJIMI A 	 J x —M ) ( in' s' —M' ) E (2] +1) 2 (jAT J JMJ'M' X j A 	 J	 ( j' A' 	 J' —M )	 s' —M' ) nimissi 2 where the notation = (JMjAIT(E)1J'MT)') Smm, 	 (63) has been introduced since J = J' and M = M' by conservation of angular momentum. Using 3.7.8 from Edmonds[27], ie., 77/ A 	 J ( j A s —M ) —M' J'	 1 = 2J +18"'SmAli	 (64)SS 24 the following expression for the degeneracy averaged cross section for the j 4-- j' transition in the total-J representation is obtained: aTDA V2(2j' + 1) EA jEm (JA rimliiy)1 2 	E E(2J + 1)1(j A1TJ j' A') k 12 (2j/ + 1) AA, j 2 (65) It is this quantity which will be used as the basis for both the exact and IOS results. 2.4 The IOS Approximation The close coupled equations (29) consist of a set of coupled differential equa- tions which must be simultaneously integrated to obtain a solution. For example, expanding (29) to 100 partial waves and ten rotor states leaves over a thousand coupled equations to solve. The IOS approximation is a drastic simplification since it decouples these equations (but in an orientational basis). This approx- imation assumes the collision to be sudden, allowing for a diagonalization in orientation representation. This is equivalent to replacing all and all A with the constants ko2 and A c, respectively in the coupled set. Actually, the IOS is a combination of two approximations, the Energy Sudden approximation (ES) and Centrifugal Sudden approximation (CS). These will be examined in turn. The Energy Sudden approximation replaces the j dependent wave number k3 with a j independent effective value ko . Since this approximation no longer treats the angular momentum of A and Jr equally, the uncoupled representation 25 (15) is used. Putting (21) in (15) and converting to Dirac notation gives the following equation: 	d 2 	9 d 	 2 	2/1 idR2  t. 	 A(A  R2 1)  Kalxc',(R)) = E(a1V(R) la")(a"IX'(R))* (66) Replacing vi with q, leaves (66) in the form (a x.,,(R)) = E(a1V(R)Ia"\Ka" IX.i(R)). (67) Operating on the left by Eim (djm) gives 	E(dini) [ddR22 ddR ko2 A(R2  1)] (Asjmxa,(R)) 3m h 2 2µ [ d2 	 	 d 2 AO 4- 1) [c/R2 	dR R2 . 	 \ 2 dm) 1-7 E(a co V(R) all) ( a" lx,,, , ( R)). (68) Using Eljm)kgj  (69) and E E(dint)(a a" jm V( R) cr")(a"lx,,(R)) = E (AsIV (P , R)1A" s")(A" 	 , R)) A"s" (70) we finally get the ES equation: I (12 	 2 d 	 1.2 	 A( 	( Aslx,vs, 	 R))[dR2 	-fidR n'") 	R2 =	 E 081 V(r, R) A"s")(A"slxv,,,(f-, R)). 	 (71) 26 We note that the effect of replacing 	 with the constant parameter 10.2 allows the SchrOdinger equation to be diagonalized in the rotor orientation. In other words, the collision process occurs with fixed rotor orientation. A collision process which is fixed for a particular rotor orientation corresponds to a situation where the incoming atom has a velocity relative to the diatom centre of mass much larger than the angular velocity of the rotating diatom. Hence the ES approximation is expected to get better with larger translational kinetic energy of the incoming atom or slower angular velocity of the rotor. In the CS approximation, .\ is replaced with the constant parameter A., so (29) now becomes: n2 [ d2 	 2 d 	 A (A + 1 k, .j2 	° ° 1 '](alx„,()R  = Vall)(a"a"lx„,( ))R. 	dR2 ridR 	 R2 (72) As) and transforming into the orientation repre-Operating on the left by EA, sentation by carrying out operations analogous to those discussed above for the ES approximation, we get [ d2 2 d 1.2 A.(A. + . {dR2 dR n'j R2 (R)) 2 — h2 E (im V(R)ljuni")(j"ni"lx,,(R)). 	 (73) Note that (73) has a fixed R orientation-the sudden approximation for large centrifugal energy. One limitation of the CS approximation is that it breaks down in the region of the turning point [30]. Classically, the turning point is defined as the point 27 where the radial kinetic energy vanishes [31]. The rotor, however, is still rotating and thus collisions at the turning point could not be considered sudden. As the energy of the collision increases, the CS improves [30]. An increase of energy would correspond to a small rotation of the rotor relative to the distance covered by an incoming atom with high velocity. Finally the IOS combines both of the above approximations, replacing k? with ko2 and replacing A with Ao . We operate on the left by > c,(/; da) and since the angular momentum operators are now merely parameters we can use closure of the la) states to give the IOS equation: 	d2 	  d	 o(A. + 1) 	 2,a 	I/R2 	RdR n'°2	 R2	 R)	 R)x„,(i, R). (74) Note the effect of replacing operators by numbers and the subsequent change of basis leaves the equations without any sum over intermediate states and all equations are decoupled. 2.4.1 IOS Cross Sections The T-matrix was introduced in (22) in an uncoupled representation. As was noted in the previous section, the IOS decouples the SchrOdinger equation in a basis with fixed orientation. Hence all quantities calculated from the IOS SchrOdinger equation, such as the T-matrix, are diagonal in the orientation rep- resentation. The only rotationally invariant quantity that depends on the different possible orientations is the scalar product of the rotor orientation 7 and atom- diatom orientation h. Thus T is a function of the angle 0 between r and R. 28 This can be expanded in Legendre polynomials: v0k0(7•, 1?) 	 E '/2L + 1 Ti;° k0 PL,(1' • 11). 	 (75) where &(7 • .fi) is the Legendre polynomial of order L. Equation (55) can be expressed in a different representation as: crini.iim , 	= -2C4 L m ; k T 	 kl) 1 2 . (76) For the IOS cross sections cr.Tis_j , n,,(k) we have: 	0 .19 S — 47x2  3 m	 111 	 k4 .7 1 2 IOm ; kV , 17)T A ° k° 	 1?j'rn'; k')ci•df I I E YA3(00 4-3 's1(k)YiniMYA*3(R) 2 47r 2 k ?.7 1 (77) Using (75) and the expansion given by (32), the IOS expression for the degeneracy averaged differential cross sections, or a i°s (h,k',:// —> k, j) is 2a/ sok' , ji 	 j) = (2j +0 YA.,(00'4-3A mm' AsAisI xV(2L• + 1)(2A + 1)(2)x' + 1)TE° k°(—) 7"'+8 Ye s ,(P) E E X ( iLn(L j' (j L 	 \ a L0 0 0 ) 	 —ill —Cr 77/ 	 0 0 0 ) 	 s cr s (78) Equation (78) may also be written as cr ic's(hk'd 	 k, j) = (2j + 1)E 0 0 0	( 	 L 	 2 cr (Me , 0 —4 k, L)	 (79) 2 29 where 47r2cr(hk' , 0 	 k, L) =  E 01-A 02A + 1)(2A' + 1) AsAisi (AL A') (AL Ax YAs (k)TE°"(—) 3 0 0 	 ) 	 a s' Y ,(P)) 	 s 2 . 	 (80) If the initial momentum direction k' is chosen as the z axis, there will not be any z projections of angular momentum for the initial state, so (57) holds. Therefore ams (hk' ,j' --+ k, j) 7r 	 (= 	 (2j + 1) 0 L 0 0 ) 2 ) AA' A L ) ( 	 A L ) 2 X 71 ,A°k° 	— ) s 0 0 0 ) —a a 0 ) + 1)(2A' + 1) (81) To get the total cross section 	 (81) is integrated over all angles to give the following aL4—o 27 fir = 	 o-(hk' , 0	 k, L) sin OdOdO o 	 o A L A' ) 2 0 (82) 7r „ = 	 2_, 2_,(2A + 1)(2\' + )71\°"iA'-A A L A' ) as 0 0 0 ) ir =  2E(2A + 1)(2A' + 1) nokoiA 1 --A 1 2 A Lkit  I 	 0 0 0 ) If the initial A paramaterization for the 105 is assumed, ie., A = A', (82) simplifies to , 	 1 01—o k'2= —7r E(2L + 1) IT17°1 2 . (83) 30 For total cross section am', , (79) is integrated over all angles to give the following IOS scaling law: 	, 	 2 ros 	 j= (2j + 1) E o o o 	L j. 	 0./0s L4-0* 2.4.2 Energy Corrected IOS Cross Sections The scaling law given by (84) can be modified in a manner such that it takes into account a collision time T. (Recall the IOS assumes a sudden collision, ie, T 0.) DePristo et. al. [32] have proposed a scaling law called the Energy Corrected Sudden (ECS) approximation which incorporates the collision time T as a correction to the IOS approximation in the form of a simple multiplicative factor. DePristo [32] corrects for the S-matrix, but since the S- and T-matrices differ only by the identity matrix and some phase and the correction factor does not change the identity matrix we relate the T-matrices by the same correction factor: TECros( 	6+  — os t 6 + 	 — €i)27-2/(202 3j 1 k-Lik Ejt). (85) where Ek is the translational energy, E ., is the rotor energy of the jth rotor state and T is the collision time. In this study the centrifugal sudden approximation is also assumed so that the correction factor is applied to the IOS T-matrices, giving an "Energy Corrected 105" T-matrix, or g?los: (84) L 31 Since by (76) cross sections are proportional to the square of the T-matrices, and assuming that T is independent of A and A', the following simple scaling rule is obtained: 24h 2 2 °Jos.,.ECIOS„. 24h2 	 (e 	 6)272 	 34-3 (86) T shall be treated as a parameter to help fit the cross sections from this scaling to the cross sections from the exact calculations. DePristo [32] gives a method of estimating T and this is then compared in a later section to the T value which gives the best agreement with the exact results. 2.4.3 General S-Matrix Cross Sections One of the interesting results arising from the IOS treatment of cross sections is that all j 4— j' transitions can be calculated from a single set of cross section values using (84). DePristo et. al. [32] have taken this property of the IOS to investigate whether the exact cross sections follow the same scaling laws. The General S-Matrix (GSM) scaling relationship is a way of calculating a set of cross sections from another set of cross sections with different energies and transitions. The relationship makes use of some IOS properties which allow these connections, but the GSM scaling relationship is actually a relationship between exact cross section values. The IOS implies that the L 	 0 and j	 j' S-matrices are related by [32] (js. 	 j' L s ios ( E 	)	 (87).ri o.s;, ( Ek 	ei,) = E o 0 0  L4-0 k 	 L • 39 Using the factor in (85) to relate S -?' 	 with S°s 4.act 	 with, and to relate Swsp—Ji 	 1.--.7 	 L4—o SYxatjt and then using (87) allows for the L 4-- 0 and j 4-- j' exact S-matrices to be related by _re j,) r 24h 2 + (EL — co)272 j 	 EcGsm 	 , 02T2  Srxac t (Ek+EL). (88)(L 24h2 	 f_ 	 0 0 03 	 3 Here EL is the rotational energy associated with the Lth rotor state. Then, in order to get a cross section relation, DePristo et. al. [32] make an assumption similar to that made in the previous section, ie., that 7 is independent of A and .\'. They arrive at the following scaling law for the GSM cr3 ,_j i: CSM 	 24h2+ (EL — fo)2T2 2 j (Ek -Fc :p) (2j+1) r 24h 2 + ( 6j, — Ej )2 7-2 	 0 2 j' L Exact 0 0 ) °-L4---0 (Ek+CL)• (89) The above scaling law will be used with of,"6' values to calculate o-2 ,_is values and these will be compared with the actual crExa;t values to determine whether 1 4-3 the above scaling law is valid. As well, the scaling law shall be used with o-Pso to determine which of the scaling laws, (86) or (89) leads to a more useful manner of correcting the IOS cross sections. 2.4.4 Accessible States Scaling Law DePristo and Rabitz [33] arrive at an expression for the cross section which does not involve 3-j symbols. To get this expression, they make the following 3 assumptions: 33 • angular coupling coefficients are mainly a function of difference jj' — ji only • quantum tunneling is unimportant, so the main energy dependence corre- sponds to the criterion of determining the outermost turning point. • the transition probability is inversely proportional to the number of acces- sible states The result of the first assumption is that cross sections are a function of — ji. The result of the second assumption, that tunneling is not important, allows the cross section to be evaluated at the kinetic energy corresponding to the highest rotor level (where the kinetic energy is the least and where tunneling is least effective). Combining these two assumptions with the third allows the Accessible States (AS) cross section o-).7‘4_si,(E) to be written as criA,si,(E) = f 	 — § 1, E —  (90) where f is a function of — j1 and E — ;EJmax • tjmax is the energy of the rotor for the maximum allowed j value, and N(E) is the number of available rotor states for a given energy E. The assumed functional properties of the f function implies a relationship between cross sections which have the same If —j1 and E — €3,.. values, namely [2j + 1][2(j' — j) + 1] N(Ek Ea , —j) [2:1' + 1] 	 N(Ek 	 cji) > J. (91) 34 (The form of (91) is equation (4.9) of [32] which was first presented in a different form in [331.) The scaling law given by (91) is here referred to as the AS scaling law, referring to the N(Ek + ci ,-3 )/N(Ek + 63 , term, which actually describes only one of the three approximations made. Note that in contrast to the previous scaling laws, (91) relates cross sections of different total energies, ie, cr.)P,(Ek + c.3 ,) with ao.._3 , _3 (Ek + ci,_3). A set of exact crj ,,, and (3-0,3 ,_3 values are related by a given total energy. Although there is a question in the exact case as to which total energy to use, this study will show that this choice is not a significant factor determining cross section values. However, in this study we will use (91) to determine its validity as a scaling law for the IOS approximation. Since the IOS treats all rotor states as having the same energy, cr.f° :,(Ek + e3 ,) and cri°1,_i (Ek cii_j) should actually be writ- ten as c .f° ,(Ekc,) and o-4° ,_3 (Eq) where ko and ko are those wave numbers corresponding to the parameters chosen to replace the k3 value and E kc, is the energy associated with the ko wave number. It is this aspect of the IOS approx- imation - the incorporation of rotor and translational energies into one effective wave number - which makes (91) lose some of its specificity when applied to the IOS results. It will be shown later that this loss of energy dependence does not significantly affect the accuracy of the scaling law given by (91). 35 3. THE DELTA-SHELL POTENTIAL 3.1 Scattering From a Spherical Delta Shell Potential In its simplest form the delta shell potential is V(R) = V8 8(R — a) (92) where V8 is a potential strength parameter having units of energy times distance, a is the distance from the origin at which the potential acts and 8(R — a) is the Dirac delta function. If the collision involves only a spherical potential, rotor states are unchanged in the collision process and need not be considered further. The form for the Hamiltonian H is thus h 2 H — 2p where A 2 is given by (11). 1 	 a R2 + v(R) 	(93)—IR2 aR 	 OR 	 R2 ] The SchrOdinger h 2 equation 1	 a pt2 a now takes the form A2	 2p 2/1 [R2 OR A(' OR R2 + h2 ^(R)R)	 V(R)^(R)=	 (R)	 (94) 36 where kli(R) is the wave function for a spherical delta shell. Using the partial wave expansion of the wave function t11(R) 	tP(R) = E YA,5(f)oAs(R) 	 (95) as where 	7PAs(R) = 47ri A xA(R)YA*.,(P) 	 (96) and using equation (92) for V (R) and introducing the dimensionless variable x = kR where k2 = 21LE Ih 2 we now can write (94) as h2 0 	 02 	 AO + 1) 	VO +1iXA = k 8(x — ka)XA2 2 1 	rxOx ax2 	 x2 (97) where the following property of the delta function [34] has been used: 8(R — a) = kS(x — ka) 	 (98) For x 	 ka, there is no potential and the partial wave solutions are the well known spherical Bessel and Hankel functions [35). With the constraint that x), must be well behaved at the origin, the solution of (97) is given up to a constant by XA 	 jA(x) - ji-TAh),(x), x > ka XA 	 BAjA(X), 	 x < ka 	 (99) where the jA's are the regular Bessel functions and the hA's are the Hankel func- tions. The expansion coefficients TA and BA are to be determined by appropriate matching conditions at x = ka. :37 Defining ka- 	ka — e ka+ 	 ka + 6,	 0 	 (100) and integrating (97) over x from ka- to ka+ and requiring x), to be continuous at ka gives 2 , k vsXA• Since x A is continuous, a 	 2/1 OX x),(ka+ ) — OX lxA(ka- ) = h2k V5 )00 Using (99) in (102) gives Mka+)— 7TAhVka + )— BAA(ka  h,2k ) = — 2p lissjA(ka)BA 	 (103) 2 where A(x) = cclix :7,\(x) h a (x) 	 (x hA (x). 	 (104) Since x ), is continuous at ka, xA(ka+ ) = x),(ka - ) 	 (105) and so 1  jA(ka)— .-T),h,\(ka) = BA:7),(ka). 	 (106) ka+ 2xA ka — 	 ka ka+ ka - axA ax (101) (102) 38 Allowing E —> 0 and multiplying (103) by h A (ka), (106) by hjka) and then using the following property of the Wronskian W of the Bessel and Hankel func- tions [35]: W = A(x)h A (x)— jA (x)//,',\ (x) 	 ix1 2 	(107) an expression for BA is obtained: BA = 1 — i0jAhA where jA is shorthand for jA (ka), hA for h A (ka), = ka 	 (109) and g 	 2iiVa a h2 	 (110) Finally, multiplying (103) by jA, (106) by j'A , and using (107) and (108) gives the following expression for TA: Ta =  	 (111) 1 — igjAhA . With TA and BA solved, we are interested in the behaviour of XA at R> a. From the asymptotic behaviour of the Bessels and Hankels given in (23) the expression for x ), as R oo is now 	— 	-i(ka-A1r/2)	 0 	 Toei(ka-A7/2)1 	 (112) XA 	2i1ka [e Using (60) and assuming j' = 0 (ie, no rotor states) the total cross section is .total = -;75- 	 (2A + 1) ITAl 2 • 	 (113) A 1 (108) 39 Hence a calculation of the cross section for a spherical delta shell potential requires the wave number k, the parameter strength V5 and the distance a of the delta shell from the origin. After evaluating the j A 's and h A 's at ka, the total cross section for the spherical delta shell potential is obtained from (111) and (11:3). 3.2 Scattering From a Non-Spherical Delta Shell Potential For the system being studied — an atom-diatom — the fact that the scatter- ing centre is non-spherical (ie., the potential between the atom and diatom is 0 dependent) means that the angular momentum of the diatom can couple with the orbital angular momentum of the atom. It is the 0 dependence in the poten- tial which is responsible for angular momentum transfer between rotor angular momentum and orbital angular momentum. A 0 dependent potential can be expanded in Legendre functions. Since we are studying a homonuclear system, only even Legendre functions appear in the expansion. Here only the zeroth and second order Legendre functions are retained to give a potential in the form V(0) = 17,5 [1 b2 P2 (cos 0)] (R — a) (114) where b2 is a constant. Explicitly, the second order Legendre is 0 , 	 1 /0 	2 0 	 1 )P2 (cos 11) = — 2 COS Of — (115) 40 With a spherical potential there is no coupling between orbital and rotor angular momentum and therefore it is not necessary to designate initial and final angular momenta — they are always the same. With a non-spherical potential, transfer of angular momentum may occur between the two types of angular momentum. This implies that the initial state, designated by quantum numbers j', m', A' and s', can be different from the final state, designated by quantum numbers j, m, A and s. Since must then be described by an initial and final state, a notation is introduced where the vector wave function x is represented as a matrix x. (Matrices will be indicated with bold type.) As well, since the rotor operator k, orbital angular momentum operator A and potential operator V all must be parameterized by indices corresponding to the ket and bra states, they too will be represented by a similar matrix notation, ie., k, A and V respectively. In a previous section it was shown how a total-J coupling scheme can separate the V-matrix elements into blocks having the same total-J. Since by (133) the only non-diagonal matrix that T depends on is V it is possible to separate the calculation of the T-matrix as well into blocks with the same total-J. With this in mind, and using the above notation, (29) is actually one matrix element of a matrix equation and can be generalized as h 2 d2 	  d 2/1 tdR2 	fidR k2 h 212R21_1 Xj VjXj (116) where k 2 and A 2 are diagonal matrices whose elements in the angular momentum representation are defined by 41 j(j 	 1 h 2 2)1 	c kjj = kj = () [E 21 (117) and ALI = A(A 	 1)h 2 6), ),183j ,	 (118) The form of (116) corresponding to matrix element (jAlx -i lj'A') is then h 2 d2 	  d 	 2 2 dR2 TtdR k3 A(h2R+21)] (iA l x j (R) IjA) /2[ E (jA VJ ICA")(5(R — a)(j"A" x J(R)VA') 	 (119) This is similar in form to (97). In the same manner as we solved (97), we choose solutions jA(kiR)— A/)k;71 ' R > a (jA Blj'A)j),(kiR), R < a (120) where the coefficients T and B are no longer simply parameterized by A but must be matrices themselves corresponding to the representation of X. If the matrices j and h are now introduced, their matrix elements are given by (jA 1j(kiR)Ij')')  j), (kiR)Sij 1 5AA I (jA li(kjR)Ij'A') = 11,A(kiR)8,w8AA ,  (121) (120) can be written in matrix notation as 1 X= j — 5-hk2Tk -2, R> a = jB, 	 R < a 	 (122) Ki A = = 42 	j represents the prepared incoming state; 	 represents the scattered outgoing wave and jB represents the part of the incoming wave that has tunneled through the potential barrier. Next (119) is integrated with respect to R from ka- to ka+ where ka+ = ka + e ka - = ka - e, e -4 0	 (123) The requirement of continuity of x, ie., x (ki a+ ) = x (k3 a- ), 	 (124) is used to get 	dy X(kj a+ ) - dy— d x(k3 a- ) = 2  Vx  (125) where 	y = kR. 	 (126) Again the requirement of continuity for the wave function x at R = a implies 1  jB = j- - 2 hk 7 Tk - 2, 	 R = a 	 (127) Inserting the expression for x given by (122) into (125) gives dj B + 2,a k _ ilf I_ 	 _dj 	 dh k 	2 hkiTk - f) 	 (128)dy 	 dy 2 dy Operating on the left of (127) by -(d dy)j, on the left of (128) by j and then adding the two resulting equations gives 1 	 i 	 1 	 dj 	 .dh) , —h2 jV - 	 = - —n j— K 7 K - 1 	(129) 2 dy 	 dy 	 2 43 Using the matrix version of the Wronskian given by (107), ie., dj 	 .dh 	 1 W = —h — j — .  dy 	 dy i(kR) 2 allows (129) to be written as 42.ift kiV 	 --1 hkiTk — i) = k 2 Tk -4 2 	- 2 or, rearranging, 4ipa 2 kNj = [2iya2 h 2 	h2 ViVhk2 + 1] Tk — i so that finally an expression for T is obtained: ita 2 	4ipa2 1. 	T = [1 + 2i  kIjVhk2 	 k NA  h2 J 	 h2 Another important point to mention regarding the separation of the T matrix into smaller blocks comes from the form of the potential used. The potential is expanded so that there are only two parameters for the VL(R)'s, ie., Vo and V2 . Hence L = 0 or L = 2. Equation (51) reveals that the potential matrix element is expanded in 3-j symbols ( L j' 0 0 0 ( A 0 0 0 L A' ) which are 0 if j+ L-F j' or A L A' is odd by equation (3.7.14) of Edmonds [27]. This means that j and j' must both be even or they must both be odd. The same is true for A and A'. Hence the V-matrix can be broken down into blocks categorized by: (130) (131) (132) (133) 44 1) total-J 2) even or odd A 3) even or odd j. And since T is a function of V then T as well can be broken down into these blocks. One important difference between T and V is in the range of parameter off- diagonality that is allowed, that is, the difference allowed between j and j' and A L 'between ). and V. The V matrix, because of the ( 	 AL 0 0 0 	 0 0 0 ) terms, can only couple A and A' states that differ by 0 or by 2. The T-matrix, however, allows for any even transitions. The reason for this is that the T-matrix depends on V not only linearly but on arbitrary products of V as well since (133) has an inverse of V in it. With the above in mind the study was further restricted to even j states. The calculations were done by choosing a total-J value, then calculating cross sections for even A and then odd A. This was done for J ranging from 0 to its maximum value, A max -I- max • Though one wave number and an initial j' value are chosen, (and a total energy value Etotal is obtained from these values) other matrix elements having the same Eto tai but different j' (and therefore different wavelengths as well) must be calculated. That is, there is no way in the total-J coupling scheme to separate the various initial j states. Hence if the T-matrix elements are calculated 45 for Etotai = 5.820 x 10' 1 J., so that the initial translational state corresponds to a temperature of 300K and the initial rotor state corresponds to a value of j' = 6, T-matrix elements corresponding to this same total energy but having different initial conditions are also calculated. For example, the initial conditions of j' = 0 and k = 16.92 x 10 10m. -1 correspond as well to a total energy of 5.820 x 10 -21 J. So too do j' = 2 and k = 16.57 x 1010 m. -1 , etc., all the way to j' = 10 and k = 8.37 x 10 10m. -1 Thus the sum of appropriate T-matrix elements will give cross sections not only for the intended initial conditions but for all other combinations of initial conditions having the same total energy. 3.3 Simplification Using Only Open States Generally, in the solution of the T-matrix, the wave function must be expanded in a complete set of basis functions. This implies that internal states with energy greater than the total energy must also be considered in the expansion which leads to imaginary k values. These closed channel states are coupled to the initial and final scattering states by the intermolecular potential. In a physical sense, what this means is that the interaction perturbs the internal states and some of these energy forbidden internal states are necessary to represent the eigenstates of the perturbed system [36]. As a first approximation only real k values were used in the calculation. That is, only the energetically accessible (open) channels were used in the calculation of the T-matrix. The theory to include closed states is developed in the next 46 section. For ease in computation, it is desirable to avoid inversion of a complex matrix. With this in mind, we define h = in + j 	 (134) where n is the matrix of Neumann functions. Equation (133) may be written as 2i T — 	 J 	 (135) 1 — N + iJ where 2,ua 2 1 . . iJ — 	 k7jVjk 2 h 2 2pa 2 1. 	 1 N _7.. 	 kTivnk2 h2 (136) (137) and so that J and N are always real. Equation (135) can be expressed in another form: T = [N — 1 + 4 -1 [N — 1] [N — 1] -1 [-24 = [[N — 1] -1 [N — 1 — ii]] 1 [ N 2i 	 1 [J]] = [1 N 1 i 	 AI ' { N 2i 1 [J]]— 	 — = [1 — iK] -1 [-2iK] 	 (138) where 1 K N — 1 iii (139) 47 and K is always real. The following expression is obtained upon further expanding (139): 1 1 — iK [1 + iK] — ' [1 + iK] [-2iK] — 1 	[ 2 iK + 2K 2 ] 1 + K 2 1 	i2K21 	2i = 1 + K 2 I 	 -I 	 1 + K 2 [K]. 	 (140) Since 1 + K 2 is real, it is now possible to calculate both the real and imaginary part of T without using complex inversion. For further ease of computation, it was investigated whether a symmetric form for the calculation of K could be found. Letting 2,a0 1 	 1 W - - — h 2  kTVICT allows K to be expanded as follows: (141) 1 K 1 = N — 1 Pi  jWn — 1 [Pi] 1  = 	 {jWn] [1] (j/n) [nWn — (n/j)] 	 n 1 	= nWn — (n/j) [j [iifti  n — 1 + 1] [i] n . i 	 1 n + nWn — (n/j) • (142) Since the j's and n's are diagonal matrices and W is a symmetric matrix, only a symmetric matrix inversion routine is needed in order to evaluate K. T = 48 Problems, however, arose with this procedure. Neumann functions for suffi- ciently small values of ka at large A approach oo as given by [35], ie., n,\ (2A — 1)!! (ka)A+1 (143) Using Stirling's approximation for large N N! = ( N ) N —	 (144) and (2A — 1)! (2A — 1)!!(145)2A-1(A _ 1)! and approximating 2A-1 as 2A and A-1 as A for large A, the Neumann functions become, as ka —> 0, nA ti  1 ( 2A A kakae) ka --> 0.  (146) Treating the limiting behaviour of the Bessel functions in an analogous manner as ka ---> 0 gives the limit A(ka)' 	e	 kae) ka —> 0. 	 (147) (2A + 1)!! — 2A 2A ) n 	 kla k2aAe ) A 	2A )2A+1 e (Lfy (kae 2A 2A When the actual run was made, exponential overflow occurred in the inversion routine for (142) for initial rotor state j'=8, A=74, j=12, ka=34 so that n 74 (34)  [1481 149 J74(34) — 	 3 x 103° . 	 (149) Hence the (n/j) term in (142) is of the order of , ka —> 0. 	 (148) 49 Solving for an inverse of a matrix requires the multiplication of matrix elements; three terms of the order of (149) would be sufficient for an exponential over- flow [37] on the Amdahl. Thus another form for the K matrix was needed., Equation (142) can be rewritten in the form 	1 	[K 	 1 n + jWn — 1 1 . ni 1 n j [W — (1/jn)] n [ni l  1 1 1 	  	111 Ltd W — (1/jn) Lni ( 150) In this rewritten form, the magnitude of the numbers were much more manage- able. The (1/jn) term was, for the parameters previously mentioned, of the order of 1 	 2)ka 	 2(74)34 103 , 	 ka 	 0. 	 (151) ..1AnA 	 e 	 2.303 The W term 1 ranged from 4 x 10 4 to 2 x 108 and an exponential overflow was avoided using this form. (The only possibility of an overflow is for W = (1/jn) but this did not occur as (1/jn) was evaluated at a small number of specific ka points; for the cases studied here W — (1/jn) never got close enough to 0 to cause overflow problems.) 1 The range for W was calculated by recognizing that the term (2/ta/h 2 )V6 is of the order of 100 while the actual V-matrix element, as given by (51) depends on 3-j and 6-j symbols, which, by their unitary nature [27], take on a maximum value of 1. Finally, the k values used in this study ranged from 5 x 10 10 Tn. -1 to 30 x 10 10 rn. -1 50 3.4 Inclusion of Closed States One approximation made in the previous section was limiting the basis set to energetically allowed states. We seek now a more general solution that includes "closed" channels, those channels (states) which are not physically allowed as final states for the collision process but nonetheless may affect the calculations. Treatment of the problem with the inclusion of closed states is similar to that presented in (116) to (133), the only difference is that the incoming prepared state is designated Poj where P o is the projection operator onto open chan- nels. The resulting equation is similar in form to (133) except for the projection operator at the end: 1 4 2 T = [1 h 2itta2 V 	ipa 	  iVhk 	 [ h 2 k 2 iVjk 2 Vol . 	 (152) With the following definitions H 2 	k7jVhk 2 = JVVhpa2 1. 	 . h 2 	 (153) and 2//a 2 (154)J 	 k-1,jVjk 2 = JWJ h 2 (where W is defined in (141)), the T-matrix can be written T =  [1 -Fl iF11 2ijP°• 	 (155) For the calculation of the scattering cross section, only the open state part Too of the T-matrix is required: Too = PoT = Po	1[ 1 iH 2i0o• 	 (156) 51 The term P o [1/(1 + iH)] can be broken up into open, closed and open-closed coupling contributions  Po[ 1 4- 1 iH -= Po [ 1 + 1 iHo ] [1 i [H H1 + 1 iH ]	 (157) where Ho P0HPo,  (158) H1 PoHPc PcHPo,  (159) Pc being the projection operator on closed states, defined by P c = 1 — Po 	(160) and finally,  H e 	P c FIP c •	 (161) The following calculations can be performed: (non-contributing terms are marked with slashes) [ 1 I 	 1 	1	 D  iH = P ° [1 + iHo 	° [1 -FliHo ] 	 [1 :Po ill] PO [ 1 	 i  +1itio l poHipc [ 1 +1iFic ] x [1 	 i [1-1 — 	 {i P _1 	  	 1 	 J 	1 	1P 14 IP u 1. 1 	 1[1 + &Li °"1. c [1 +1iFi c l r 	r 	1 	r 1+ •zl-l c 	1 + iH[1 + iH0.1 13°H1Pc I. 	•	 i P c H I P°I. 	• j. (162) 52 So [ 1 1 poHipc 1 1 	1 Fic H i poi Po [ 1 	 = -I iH l[ 	 [1 + iF10 .1 	 {1 + ihI c i Pa [1 +iiHo l 	 iP°H1Pc (1 +1 H e  • If the above is multiplied on the left by [1 + iH o], the following is obtained: Po[ 1 1+ iH [1 + iHo  1 PoHiPc [ 1 + iHc PcHiPoll 1 x Po [1 — iPoHiPc ( 1 iFic )] • (164) The T-matrix can be written: TOO iHo PoHiPc [1  +1 iHi= 2i [1 + 	 PcHiPol x [P0 -113 , — iPo HiPc [ 1 +laic ] PciPo] (165) 	= PO — [Po + iHo + PoH1Pc1 +1 	 PcH1P0 x [P o + i [Ho — 2J0] PoHiPc  1 [ 1 + iHc I Pc (H1 — 2-1) Pc] (166) where 	PoiWiPo•	 (1 67) (1 63) Or TOO 53 H — 2J = = Using = 2, h n 2 k iiv [h _ 2j] ki 2fin 2 k fiv [i n _ j] ki h 2 2/./a 2 0 jvh( 2 )k i h 2 (168) or since for open channels h (2) = h* [H — 2J] P o = 2"2 1cijVh*IciP o , h 2 (169) the following expression for the T oo matrix is obtained: Too = P o — P 0 [1 + iHo + PoHiPc [1 + iHd PciliP° 1 x [Po — i (J 0 — iNo ) — P0 H1P c [ 1 + 1 ill] P C (J — iN) P o l (170) where N o F.-_,- PojWnP o . 	 (171) We define Ro -P.. P oWP c h [ 1 + 1 &lc liP cWP 0 	(172) where W has been defined in (141). Using (159) and (153) we can now write 1-1 54 [ii PoHiPc [ 1 + 1  &lc ] PcH1P0 = Po [PoFIPc -I- PcHPo] pc 1 +l He x P c [P 0 HP c + PcHPo] Po = PoHP4 1 +1iFicl  PcHPo = P ojWhP c [ 1 + 1 iHcl  PcjWhP0 = jR0 h = jRoj + ijR o n  = i [Ji + 	 iNi.]  (173) where ii 	 — iiRoj 	 (174) and 	N1 E.--  —ijRo n.	 (175) With the following definitions H o E Jo + iNo, 	 (176) 	.1-E. Jo + ii 	 (177) and 	ITI F- No + N1, 	 (178) the Too matrix can be written 5500 TOO Po P2 	tri _ ij _ K ] = 1 +LI — N Or and, finally where [1 — Ki ] [1+ [i/ (1 — IV -.J] [1 —1 IV 	 1 — KI j	I [1 	i	  —1 Po Po 	 	 —I J (179) 1 + r 	 , 	 .__\ 1ii/ O. — NA [1 	 -J 	 1 —N 	 i PoToo = PO (180)iR]1 - iR [1 + 2iPoRToo (181)— 1 — iK 1 K -.L- J. (182)— N 	 1 = Po = Po 3.5 IOS T-Matrix Calculation It is useful to note here that the method which will be used to obtain the 105 solution has already been used in the section where the central 6-shell potential was considered. To understand why this is so it is necessary to consider the two types of operators in the SchrOdinger equation (ie., (29)) that bring about the directional coupling. If the interaction potential is diagonal in orbital and rotor angular momentum representation, ie., it no longer couples different angular momentum states ) and A' (and this is the case when there is only a central potential V(R)) then all 56 operators in the SchrOdinger equation can be thought of as diagonal in angular momentum representation which leads to radial solutions parameterized by only one A, ie.: h 2 1 d 2/1 [R2 dff R2  RR2  A( + 1) k2] 2P),(kR) 	 V(R)1PA(kR) 	 (183) where k 2 = (2y1h 2 )E. Conversely, if the angular momentum operators are replaced by quantities which are diagonal in orientation representation (and this is the case in the IOS approximation where operator A 2 is replaced by the parameter A o (to + 1) so that all operators in the SchrOdinger equation can be thought of as diagonal in orientation representation which leads to a set of uncoupled differential equations (as opposed to matrix radial solutions in the exact case) which is similar in form to (183) , ie.: h2 1 d R2 d	 Ao (A0 + 1) + 	 7PA,(koR, 0) = V(R, 0)71)A 0 (koR, 0)2µ[R2 dR dR 	 R2 (184) except that there is the extra constant parameter 0 classifying the solution OA° (koR, 0). (Here 11 (2p/h 2 ) {E — jo (jo + 1)h 2 /21] where jo is some cho- sen parameter.) Further, there is no longer the restriction that A o be an integer, as in (183). Hence the methods of solution of the radial part of (183) and (184) are similar — (183) gives solutions parameterized by A and k while (184) gives solutions parameterized by A o , ko and 0. To demonstrate the above, the potential given by (114) is inserted in the IOS 57 and 211 g(0) = [1 — b2 P2 (cos 0)] . V52 (188) equation (ie., equation (74)) and the following expression is obtained: n2 21.i [ [ R2 dR 1 d R2 dR d 	 Ao(A  R2 0 + 1) + 41 11) Ao(koR, 0) = V5 [1 + b2P2(cos 0)] S(R a)60 (koR, 0). (185) In terms of the above discussion, the radial solution to (185) is dependent only on 0 and hence the directional couplings which required matrix manipulations are avoided and (185) is solved for in the same manner as was (97). Since the only integration done in solving (97) is over R, the addition of the 0 term in the potential and the 0 parameter dependence of the wave function makes no difference to the method of solution. So following the same steps as were used to solve for (97), a 0 dependent T-matrix is obtained: —2iGg(0)jt(ko a) TAok0 (0) =  	 (186) 1 — iGiAo (koa)hA o (koa) where = ko a 	 (187) As was discussed in a previous section on IOS cross sections, the quantity Mo ko must be calculated in order to obtain the o-j ,_j , quantities. By inversion of (75), no ko is given by: itko = V2L + 112 TAoko ( 6) ) PL (COS 0) sin Oa. 	 (189) 58 Thus to get 01j._3/ cross sections, TAok„(0) is calculated as given by (186) for appropriate values of A 0 , ko and 0. Next, using numerical methods this TA 0 k0 (0) is integrated over all 0 as given by (189) in order to get values for the noka 's. Finally (80) is used to get the resulting cross sections. 59 4 CALCULATIONS AND RESULTS 4.1 Parameter Determination 4.1.1 Atom and Diatom Parameters As mentioned in an earlier section, the molecular parameters are chosen so that the diatom is a model of nitrogen and so that the atom is a model of argon. Hence the reduced mass y is set to 2.734 x 10 -26kg. For the rotor, the atom separation is chosen to be 1.094 x 10 -10 m. [38] which gives a moment of inertia I for nitrogen as 1.392 x 10 -"kg — ni. 2 , or a characteristic rotational temperature h 2 /2/kB for nitrogen of 2.894K where kB is the Boltzmann constant. 4.1.2 Choice of Energy At temperature T the most probable energy for each of translational and rota- tional motion is kBT. In this thesis the major computation has been carried out at 300K. At this temperature j = 6 is the j-even state closest to a rotational energy of kB T while the wave number 2itkB T (190) 60 has the value 14.27 x 10 -10 7n. -1 For this choice of rotor state (j =6) and trans- lational energy kBT, the total energy is 5.82 x 10 -21 J. This is the total energy used for the cross section calculations in the following section. Corresponding to this energy, rotor states up to j = 10 are open and j > 11 closed. An analogous choice of energy parameters is used in section 4.3 for a calculation at 1000K. Note that the potential, (51), does not couple j-even and j-odd states, so the choice in this thesis is to restrict the calculation to j-even states. 4.1.3 Range of Partial Waves In both of eqs. (81) (the IOS cross section) and (65) (the exact cross section) it is necessary to set an upper limit on A, the maximum orbital angular momentum that significantly contributes to the sums over partial waves. The largest A value considered to contribute significantly to the scattering cross sections is that which corresponds to a particle just passing by the outer edge of the delta-shell, ie., the A where the incoming particle approached a distance a from the scattering centre. To determine this A, it is necessary to associate the angular momentum as expressed in the quantum mechanical equation L2 = A(A + 1)h 2 	(191) which for large A can be approximated as 1 L r-:::', (A + -)h (192) 61 with the angular momentum as expressed in the classical expression L = ybv 	 (193) where b is the impact parameter (see Figure 2 on page 63) and v is the particle velocity given by hk v = —. Et Equating (192) and (193) gives ,abv = (A + )h. Since the maximum range of the potential is a, this gives ka = (A + 2— I ) A. (194) (195) (196) For a given translational and rotational energy the calculation for maximum A was done as follows: the total energy (the kinetic and translational) was converted to a k value, then multiplied by a to arrive at a given A. All A beyond approximately twice this value did not contribute significantly to the cross sections. 4.1.4 Inverse Power Potential Comparisons Since the aim of this study is to compare exact and 10S results using a delta- shell potential, the choice of potential parameters is arbitrary — just as long as the same parameters are used in both the exact and IOS calculations. However arbitrary this choice of parameters may be, it is nonetheless desirable that the parameters chosen yield cross sections which are comparable to cross sections 62 FIGURE 2: Impact parameter b is "the distance of the asymptotic path of the particle from the line of head on collision" [311 63 obtained for a realistic system. In order to accomplish this, an IOS calculation us- ing a realistic smooth repulsive potential was performed and then several choices of the delta-shell potential parameters were tried in equation (114) to see if these results could approximately reproduce the realistic results. The potential chosen for comparison was the repulsive part of the Pattengill et. al. [39] potential 2.2 x 10' 2 V (R, 0) = R12 	[1 + 0.5P2 (cos 0)] 1.2 x 10 -21 [1 + 0.13P2 (cos 0)] R6 (197) expressed in SI units, specifically the inverse power potential corresponding to the positive term 2.2 x 10' 2 V (R, 0) = R12 [1 	 0.5P2 (cos 0)] . (198) Cross sections were calculated for this potential using the procedure reported by Snider and Coombe [40]. Phase shifts 77,\„(cos 0) were calculated using the WKB approximation (\.  R2 V (R 	r2—riAk (cos 0) = k f , 1  1 R  2,a 1 — —Rc2 dR 1  k(b — rc)rh 2 k2 , 0) (199) where b is the impact parameter (A + Wti,- (see (196) ) and re is the largest classical turning point. By definition, the phase shifts are related to the S matrix by sAk(o) = exp 	 (cos 0)] . 	 (200) 64 Since the potential is even in cos 0, it follows that the S-matrix can be expanded in even Legendre functions, sAk (0) = E V2L 1,9j/A,(cos 0). L(even) The expansion coefficients Si\," are given by the inverse of (201), ie. (201)  	 1 SLR = V2L 11 PL(cos 0) exp [2i7/Ak(cos 0)] d(cos 0). 	 (202) To carry out this integration the phase shifts are fitted to an expansion in even Chebyshev polynomials: 1/Ak(cos 0) = A + BT2(cos 0) + CT4 (cos 0) DT6 (cos 0) 	 (203) where the T„'s are Chebyshev polynomials and A, B, C and D are the fitting coefficients. A 60 point Gauss-Chebyshev integration scheme was then used to evaluate the t.9 , 6 values. The T-matrix expansion coefficients are then obtained according to Ti k = 5OL — SL R 	(204) and these were then substituted into (83) to obtain the aL ,_.0 cross sections. The results are shown in Table 1. For various values of the delta-shell potential parameters, ie., V6, b2 and a, the IOS cross sections UL- 0 were calculated, giving a wide range of results. For each choice of parameters, the 0 dependent matrices were calculated using (186) and then using (189) integrated via a 40 point Gauss Legendre integration to 65 give the Ti`K's. The aro were calculated using (80). After comparison with the results of the repulsive r -12 potential, parameters were chosen to give a reasonable fit. The best fit delta-shell potential is V (R, 0) = 3.697 x 10 -32 J7n [1 + 1.50P2 (cos 0)] S(R — a) 	 (205) where a = 5.5 x 10 -1° 77z. The leading coefficient for the potential is not directly comparable to the leading coefficient for the r -12 repulsive potential in (198) since the units and form are different. In the delta-shell V8 = 3.697 x 10 -3 J. — 7n. has units of Joule-meters, whereas for the r -12 repulsive potential the coefficient has units of Joule-meters 12 . Nonetheless, a form of comparison for these two parameters will be offered later in the discussion. Table 1 gives a comparison of the cross sections for the two potentials: Table 1: A Comparison of the Delta-Shell and r -12 Potentials All cross sections in A 2 ic = 14.2717 A , Amax 	 120 L value IOS with repulsive l' -12 IOS with delta shell L=0 115.15 113.75 L=2 25.32 23.30 L=4 12.09 6.41 L=6 3.16 1.98 L=8 0.50 0.68 L=10 0.06 0.25 66 It is noted that the delta shell has a much higher anisotropy parameter but still gives smaller inelastic cross sections. 4.2 Cross Sections at 300K 4.2.1 Exact Cross Sections Including Only Open States An exact calculation was done using as initial state, the thermally most proba- ble rotor state and velocity corresponding to a temperature of 300K (equation (190)) . For this temperature, j'=6, k' = 14.27 x 10 -10 m. -1 , and the maximum rotor state into which the molecule can scatter is j = 10. The present calculation was restricted to including only the open rotor states in solving the SchrOdinger equation, thus ,max = 10. k'a 	 77 and )'max was chosen to be 120. Since all the T-matrix elements are obtained from the calculation, all cross sections at this total energy are readily available. These are reported in Table 2. Table 2: Exact Cross Sections All cross sections in A 2 , Total Energy=5.82 x 10' J. T.E. 1 j'=0 421K j'=2 404K j'=4 364K j'=6 300K j'=8 213K j'=10 103K j =0 117.52 4.78 0.35 0.03 0.00 0.00 j =2 22.93 126.24 6.06 0.55 0.05 0.00 j =4 2.71 9.83 130.25 7.73 0.76 0.05 j =6 .32 1.06 9.21 143.19 8.75 0.66 j =8 .03 0.09 0.84 8.13 159.03 6.45 j =10 .00 0.00 0.03 0.36 3.86 197.47 1. TE = Translational Energy expressed as an equivalent temperature (E = kBT). 67 From the detailed calculations it was found that significant contributions to the cross sections fell off (ie., were of the order of 10 -6 A 2 ) for the j'=6 column at J=93 which corresponded to A contributions ranging from A=83 to A=103. From J=93 to J=130 the contributions to the cross sections strictly decreased. Checks were maintained on the accuracy of the matrix inversion by a calculation of the condition numbers for both the K 2 +1 and W-1/jn inversions as required by equations (140) and (141). The worst condition numbers found were of the order of about 2 x 10 4 for W — 1/jn and 2 x 10 7 for K 2 + 1. A further check as to the reliability of both inverses was performed by an actual multiplication of the matrix by its inverse and determining how close the result was to the unit matrix. For the product of the W — 1/jn matrix (the matrix having condition number of 2 x 10 4 ) and its inverse, the largest off-diagonal term was of the order of 10 -11 and the largest off-diagonal term for the product of the K 2 + 1 matrix (the matrix having condition number of 2 x 10 7 ) and its inverse were of the order of 10-8 . Hence the inversion proved reliable for this set of initial conditions. One trend is noted: as the kinetic energy decreases (and j' goes up), the elastic j 	 j' cross sections all increase. 1 The condition number was calculated automatically by the inversion routine INV available as part of the support software at the 1.113C computing centre. According to ref- erence [41] the condition number is a form of Turing's N-condition number[42]. Generally, the larger the condition number, the poorer the inverse. Well conditioned matrices will have a condition number of the order of N where N is the dimension of the matrix. At the other end of the scale, one of the most poorly conditioned matrices is the Hilbert matrix which has a condition number of order EXP(3.5N) [41]. For this run N=36. 68 4.2.2 Exact Cross Sections With Inclusion of Closed States A calculation was run using the closed state calculation of (181) and then com- pared with a calculation done under similar conditions using the open state cal- culation of (133). Close-state calculations require the use of modified spherical Bessel functions (MSBF), and the values used in these calculations were obtained from a fitting of a table of values for MSBF for a given argument to a polyno- mial third order in A. Two closed states, j' =10 and j' =12 were included in the calculations which are displayed in Table 3. 69 Table 3: Effect of Including Closed States Part A: Using only open states. Total energy corresponds to translational energy of 300K and rotational state of O. Units are in A 2 . j' denotes the initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j=0 126.95 5.00 0.49 0.06 0.01 j=2 23.56 142.98 7.39 0.80 0.09 j=4 3.59 11.39 145.24 8.08 0.84 j=6 0.43 1.31 8.60 164.04 7.76 j=8 0.04 0.10 0.61 5.21 202.36 Part B: Inclusion of closed states. Total energy corresponds to translational energy of 300K and rotational state of 0. Units are in A 2 . j' denotes the initial state, j the final state. j 1 =0 j'=2 j'=4 j'=6 j'=8 j=0 126.94 5.00 0.49 0.06 0.01 j=2 23.57 142.97 7.39 0.80 0.07 j=4 3.58 11.40 145.31 8.09 0.69 j=6 0.43 1.31 8.61 166.11 6.41 j=8 0.03 0.08 0.49 4.31 208.20 From the above table it is concluded that closed states only affect the calcula- tions for those cross sections involving high rotor states (104-8, 84-10, 104-10). These differences can be of the order of about 20% but are confined to cross sections in the lower right hand corner of the above table of values (ie., high rotor states). 70 4.2.3 IOS 0 ---+ L Cross Sections The IOS calculations are carried out in two steps. First, the o-L,0 cross sections are calculated as given by (83) and then used along with the scaling relations as given by (84) to give the particular j 4- j' transition cross section. The first of these steps, the results of the calculation of the (1_0 cross sections, are given and compared with the exact results in this section. Before any calculations using the IOS approximation can be performed, it is necessary to choose values for the parameters Ao and k0 . In equation (83), A o was chosen to be an average of the ). and A' values associated with the particular T-matrix element calculated. Hence for each combination of A and A' that gives a different ) value, another T-matrix element was calculated. The same choices for k0 are available — one can choose k 0 as being equal to the k value corresponding to the j' state or to the k value corresponding to the j state or as being an average of both. For the L 4-  0 calculations, both k ip corresponding to the j' state and ko corresponding to the j state were used. A Gauss Legendre angular integration scheme using 40 points was performed for the integration of the T-matrix as given by (189). Sums were done up to )=120 and L=30 with contributions trailing off in significance (ie. giving cross section contributions of less than 10' A 2 ) at A=87 for ko = 14.27 x 10 -10m.-1 The unitarity of the S-matrix summing from L=0 to L=30, ie., L=30 2E 1.00000 (206) L=0 was verified to 5 significant figures. The following results were obtained: 71 Table 4: IOS o- L,0 Cross Sections All cross sections in A 2 Final State L Final k in A -1 Trans. Energy (T .E.) Using ko = kiiiit (16.92A ') T.E.=421.5K Using ko = kfinal Using ko = 14.27 A-1 T.E.=300K Exact Results, k = 16.92 A -1 T.E.=421.5K L =0 16.92 421.5K 101.90 101.90 113.75 117.52 L =2 16.57 404.2K 23.58 23.09 23.30 22.93 L =4 15.71 363.7K 5.85 5.50 6.41 2.71 L =6 14.27 300.0K 1.64 1.98 1.98 0.32 L =8 12.03 213.2K 0.52 0.80 0.68 0.03 L =10 8.37 103.2K 0.18 0.38 0.25 0.01 L =12 0.07 0.10 L =14 0.03 0.04 L =16 0.01 0.02 L =18 0.01 0.01 The general trend to be noted is that the IOS calculations underestimate the elastic 04-0 transitions and overestimate the inelastic cross sections. This can be explained by recognizing that a difference in energy between two states serves to hinder the excitation of the higher state. Hence with an approximation such as the 10S, which treats all rotational states as degenerate, it would be expected that the IOS would give higher inelastic cross sections than the exact results. Another more classical explanation of these results is that at high L the rotor is moving more quickly with respect to the incoming atom than at say L =0. Hence the collision is less sudden at high L and it would be expected the IOS approximation to be less accurate for large angular momentum transfer. This 79 trend would be expected for transitions for j' 	 since all other cross sections are dependent on these 0 1, 4-0 values by (84). One other effect arises because the IOS treats all rotational states as degener- ate. This is that energetically inaccessible states are allowed, in particular, in the present calculation, transitions are allowed from L=0 to rotational states higher than L=10. The cross sections for these transitions however are so low that they may be effectively neglected. The next section contains some IOS results where energetically forbidden transitions are fairly significant. Finally, the results above show that choosing the k 0 parameter to be either the initial or final k value does not significantly change the results. The only difference the k0 choice seems to make is in the elastic cross section, where a choice of k0 =14.27 A -1 gives a value closer to the exact value than choosing k0 =16.92 A -1 , the actual initial k value for this transition. The above trends apply to the 0L-0 calculations. Since the cross sections are calculated from these aL,0 cross sections, the differences between exact and IOS calculations noted here will be further examined after the scaling relations are applied to better determine how the IOS and exact calculations compare. 7:3 4.2.4 IOS Scaling Relations Using IOS 0L 4_0 Cross Sections Since the 105 T-matrix was 0 dependent it was expanded in Legendre polynomials in equation (75). This expansion subsequently led to a scaling relationship (84) which expresses j j' cross sections in terms of L 0 cross sections. This relationship is used in this section to calculate the IOS cross sections. The previous section compared the 105 0 L ,_0 and exact cross sections. Whether their differences and similarities carry over to the crj„, cross sections is examined here in order to determine how successful the scaling relation (84) is in predicting j 4- I cross sections, once given the L 0 cross sections. As well, different ko choices are examined in light of the scaling relations . For each of the three choices of 4, ko = ko = kfinai and 14 ) =-14.92 A -1 , the IOS o-3,j, cross sections were calculated using the 0L __0 cross sections from Table 4 and equation (84). The sum in (84) was taken up to the maximum allowed L value. For the choice of ko = ki„itiai , the cross sections are given in Table 5: 74 Table 5: IOS Cross Sections at 300K Using k o  = kinitial Part A: Units are in A 2 . j' denotes the initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 101.90 4.62 0.61 0.15 0.05 0.02 j=2 23.58 111.03 7.48 1.40 0.37 0.13 j=4 5.85 13.86 113.22 8.54 1.63 0.46 j=6 1.64 3.17 11.65 120.97 8.43 1.70 j=8 0.52 0.89 2.47 11.60 132.19 7.41 j=10 0.18 0.29 0.64 2.70 10.65 155.32 j=12 0.07 0.10 0.19 0.78 2.64 8.96 j=14 0.03 0.04 0.06 0.26 0.81 2.50 Part B: Ratio of IOS to exact cross sections. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 0.87 0.97 1.75 4.48 15.9 154 j=2 1.03 0.88 1.23 2.55 6.92 46.7 j=4 2.15 1.41 0.87 1.11 2.14 9.80 j=6 5.21 2.99 1.27 0.85 0.96 2.59 j=8 20.4 9.36 2.93 1.43 0.83 1.15 j=10 297 98.3 20.6 7.40 2.76 0.79 The same trend is observed for cross sections with higher initial j-state as was observed for the j'=0 initial state cross sections - 105 inelastic cross sections are larger than the exact values while the elastic cross sections are lower. The results actually seem to get better for larger j' on using the scaling relationship but this is basically due to the nature of the summation. For example, the j'=6 column depends less on the large L-valued o-L-0 cross section than the j' =0 columns. That is, to get the 104-0 cross section, only for L=10 is used, 75 which is 297 times the exact o-L,0 value. To get the 104-6 term, the o-L 4-0 cross sections for the L=4, L=6, . . L=16 terms are used. And for L=4, the IOS aL,0 is only 2.15 times the exact 471_0 value. Not only are the lower-L 0L,0 values more accurate, but they also contribute more in equation (84) relative to the other terms which also leads to better results for cross sections in the mid-table region. With these points in mind, the L 0 results should be compared with the j j' results in terms of how many rotor states the initial state is from the elastic transition. For instance, the 0+-4 should be compared with the 24-6 transition. On this basis, the scaling law given by (84) preserves the ratios of the 10S to exact cross sections for the L 0 transitions when it is used to calculate the j 4— j' transitions. The scaling relation also allows energetically forbidden cross sections and these may be of significant size. For example, the 105 124-10 transition is calculated at 8.96 A 2 but is 0 (ie., not allowed) for the exact calculation. The cross sections using the choice ko = kfinm are given in Table 6. 76 Table 6: IOS Cross Sections at 300K Using k0 = kfinal Part A: Units are in A 2 . j' denotes the initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j'=12 j'=14 j=0 101.90 4.72 0.65 0.13 0.03 0.01 0.00 0.00 j=2 23.09 111.03 7.70 1.22 0.26 0.07 0.02 0.01 j=4 5.50 13.46 1 13.22 8.06 1.31 0.27 0.07 0.02 j=6 1.98 3.64 12.33 120.97 8.87 1.67 0.40 0.12 j=8 0.80 1.24 3.07 11.02 132.19 8.62 1.79 0.47 j=10 0.38 0.53 1.07 2.75 9.16 155.32 7.53 1.81 Part B: Ratio of IOS to exact cross sections. j 1 =0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 0.87 0.99 1.86 3.71 10.3 72.7 j=2 1.01 0.88 1.27 2.22 4.94 25.1 j=4 2.03 1.37 0.87 1.04 1.72 5.86 j=6 6.29 3.44 1.34 0.85 1.01 2.55 j=8 31.4 13.1 3.65 1.36 0.83 1.34 j=10 627 183 34.5 7.53 2.37 0.79 Compared with the k0 = knn t,1„1 choice the ko = kfinal choice was better (although only of the order of about 5% for values about 1.0 times the exact value) in 17 cases and the k0 = ',Initial was better in 13 cases with the two being the same for the elastic cross sections. The following table shows which of the two choices is best for each transition: 77 Table 7: Comparison of kmitial and kfi„ai IOS Cross Sections The best agreement with the exact cross sections is : I if Initial state, F if Final state or E if equal. j' notes initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 j=2 j=4 j=6 j=8 j=10 E F F I I I F E F I I I I I E I I I F F F E F I F F F F E F F F F F I E Rather than analyzing the above trend in terms of initial or final state pa- rameter ko , it is useful to analyze the results in terms of choice of kmax or kmm . Table 8 demonstrates the trend for the best ko value being ko = kmax for large IAA transitions and ko = kmi„ generally being the best choice for transitions where Ion I=2. 78 Table 8: Comparison of kmax and 	 IOS Cross Sections The best agreement with the exact cross sections is: Max if ko  = kinax, Min if ko = kinin or E if equal. j' notes initial state, j the final state j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 j=2 j=4 j=6 j=8 j=10 E Min Min Max Max Max Max E Min Max Max Max Min Min E Max Max Max Max Max Max E Min Max Max Max Max Max E Min Max Max Max Max Min E The above two tables can be rationalized as follows: Since the total energy in a collision is conserved, a downward transition (going from a high rotor state to a low rotor state) corresponds to a km;„ to kmax transition while an upward transition corresponds to a kmax to a k11m transition. Now for the a-L„._0 cross sections the k1111, value is always the final k o choice and the kmax value is always the initial ko choice. Regarding Table 4 it is noted that for o•2,0 and a4.._0 the kfi„al or k11111 choice is best while for cr6-0, (3-8 4_0 and o-10,_0, kiiijtiai or kmax is best. By the scaling relationship in (84), transitions are governed mainly by the o-L _.0 value corresponding to L = 1j/ — j since this is the largest term. Assuming that more accurate aL,o values will give more accurate aj,j, values, then downward transitions (ie. going from a low ko to a high ko) with IAA greater than or equal to 4 would tend to favour a k final choice or kmax . Upwards transitions (going from a high ko to a low ko ) with 16,j1 greater than or equal 79 to 4 would favor the kmax choice as well but this now corresponds to the k initial choice. For transitions with downward transitions would favour Ic i„itiai or kniax (as is the case in the 04-2, 44-6 and 44-8 transitions) and upward transitions would favour kfi„al or kmm (as is the case in 24-0, 44-2, 84-6, and 104-8 ). Finally, to complete the study of k o values, one fixed energy was chosen, k0 =14.27 	 . The cross sections using this choice of ko are given in Table 9: $0 Table 9: IOS Cross Sections At 300K Using ko = 14.27 A -1 Part A: Units are in A 2  . j' denotes the initial state, j the final state. j'=-0 j'=2 j'=4 j'=6 j'=8 j'=10 j'=12 j'=14 j=0 113.75 4.66 0.71 0.15 0.04 0.01 0.00 0.00 j=2 23.30 122.24 7.93 1.40 0.33 0.09 0.03 0.01 j=4 6.41 14.27 121.25 8.54 1.57 0.38 0.11 0.03 j=6 1.98 3.64 12.33 120.97 8.87 1.67 0.40 0.12 j=8 0.68 1.11 2.97 11.60 120.87 9.08 1.74 0.43 j=10 0.25 0.38 0.88 2.70 11.22 120.82 9.23 1.79 j=12 0.10 0.14 0.30 0.78 2.55 10.98 120.80 9.33 j=14 0.04 0.06 0.11 0.26 0.73 2.47 10.83 120.78 Part B: Ratio of 105 to exact cross sections :1=0 j'=2 j'=4 j'=6 j'=8 f=10 j=0 0.97 0.97 2.04 4.48 15.6 103 1=2 1.02 0.97 1.31 2.55 6.20 33.4 j=4 2.36 1.45 0.93 1.11 2.07 8.02 j=6 6.29 3.44 1.34 0.85 1.01 2.55 j=8 26.8 11.7 3.53 1.43 0.76 1.41 j=10 419 131 28.3 7.40 2.91 0.61 In comparison with the previous results, the k o =14.27A -1 choice works best in the elastic cross sections. This can be accounted for in that the largest term in the sum in (84) is the term involving the 00-0 value and this is estimated better when kb is chosen as ko = 14.27 A -1 rather than kinitiai or kfinai. In the inelastic transitions, however, the kinitial and kfilim choices are better. A further 81 initial 	 kfinal kaverage 2 (207) investigation could study a kaverage value where but since the results in the above three choices of k o differ much less among themselves than they do with the exact results, little change from what has already been given would be expected. The main point to note about the scaling relations is that the discrepancies noted in the cri,–.0 cross sections are carried over into the 0-.7,3 , cross sections with the results getting neither better nor worse. This is important to note since it suggests these scaling laws can be applied to the exact results which is the topic of investigation in the next section. Using Exact Cross Sections There are two parts to the prediction of by means of the scaling law in (84) — the values of the 0L,0 cross sections and the way these al.„_0 cross sections are combined with the 3-j symbols — each affects the result. In particular, the question arises if the 3-j symbols predict the correct j, j' dependence of This can easily be tested with the delta-shell potential by putting in the exact 01.–o values in (84) to determine how the resulting j 	 j' cross sections compare with their corresponding exact cross sections. The cross sections obtained using the exact 0 L,_0 values of Table 4 are reported in Table 10. 82 Table 10: IOS Cross Sections At 300K Using Exact aL, 0 Values Part A: Units are in A 2 . j' denotes the initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j'=12 j'=14 j=0 117.52 4.59 0.30 0.02 0.00 0.00 0.00 0.00 j=2 22.93 124.85 7.00 0.51 0.04 0.00 0.00 0.00 j=4 2.71 12.60 123.97 7.63 0.58 0.05 0.00 0.00 j=6 0.32 1.32 11.02 123.80 7.97 0.62 0.05 0.00 j=8 0.03 0.14 1.10 10.43 123.75 8.18 0.65 0.06 j=10 0.00 0.01 0.12 1.01 10.11 123.72 8.32 0.67 j=12 0.00 0.00 0.01 0.10 0.96 9.91 123.70 8.43 j=14 0.00 0.00 0.00 0.01 0.10 0.92 9.77 123.70 Part B: Ratio of IOS to exact cross sections. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 1.00 0.96 0.86 0.71 0.51 0.25 j=2 1.00 0.99 1.15 0.93 0.79 0.94 j=4 1.00 1.28 0.95 0.99 0.76 1.05 j=6 1.00 1.25 1.20 0.87 0.91 0.95 j=8 1.00 1.51 1.30 1.28 0.78 1.27 j=10 1.00 3.67 3.71 2.76 2.62 0.63 Two trends in the above table can be noted: downward transitions are calculated as somewhat lower than the actual value, and upward transitions are calculated as being higher. As well, the elastic cross sections become progressively less than the actual values as j' increases. That the downward transitions are lower than upward transitions can be accounted for by considering that the IOS has replaced wave number ki with 83 one fixed parameter ko . One implication of this approximation can be seen in the detailed balance equation: Exact 	 [j] k 2 Ell) = 	 0-3E'ix,__va3c't(-Elk 	 (j)1.) 	 ji 	 (208) where [A = 2j + 1 and kj is the k value corresponding to the state with rotor quantum number equal to j. For the IOS case, with lq = (208) becomes: (7.1)(234E) = 	 (E). 	 (209) (The above could also be derived using (84) for each of cri_j, and a- i, and equating the two expressions.) For downward transitions kj, < k j so ki /ki, > 1 but by (209) the 105 approximates this term as 1. As well, in upward transitions the term k3 /k3 , < 1 is also approximated by 1. If for j' > j (ie., a downward transition) the cr3 ,,_ i term is very close to the exact term, then the resulting (73,3 , cross section would then be lower than the exact value. Conversely, for an upward transition the o -34_3 , cross section would be higher than the exact value, as is the case in the above table. For downward transitions with final state 0, the trend for this type of study will always lead to lower than exact values since the o 3,, values are the exact values. For downward transitions with final state other than 0, this trend would not always be expected. For example, the a 3 1,_i term may be very much higher than the exact o 3 _3 , term which, in a downward transition could compensate for the lack of the k3 /kl , > 1 term and give an accurate 105 downward transition. 84 The other trend — that elastic 105 cross sections become progressively less than the exact value for larger initial rotor state — can also be attributed partly to the 105 approximating with k;i. The ko value used for the cross section calculations was 16.92A -1 . These in turn were used to calculate the 0,;,j, cross sections for j' = 6 where k=14.27A -1 . If the exact 01.0 values corresponding to ki„i t iai =14.27A -1 are used, the following values (shown in comparison to the kinitia1=16.92A -1 values) are obtained: Table 11: Effect Of Using Different k Values In Exact o- L, 0 Values on the 	 Cross Sections Exact a-L,_ 0 C7'088 sections calculated at k=14.92 A -1 and substituted into (84).Results are in A 2  and are shown in comparison to the values given in Table 10: Transition k=14.92A -1 Ratio to exact results k=16.92A -1 Ratio to exact results 04— 6 0.03 0.96 0.02 0.71 24— 6 0.67 1.23 0.51 0.93 44— 6 7.96 1.03 7.63 0.99 64— 6 133.54 0.93 123.80 0.87 84— 6 10.87 1.34 10.43 1.28 The elastic cross section does get better on using the proper energy but most of the other transitions do not. Hence while a proper accounting for different wave numbers among transitions may allow the scaling relation in (84) to better replicate elastic cross sections, inelastic cross sections may require even further types of corrections. 85 The results in Table 10 are now compared to results obtained using scaling relation (84) and 105 01,0 values for an energy corresponding to 16.92A -1 ; see Table 12. Table 12: IOS Cross Sections At 300K Using k=16.92 A -1 Part A: Sums were done to maximum L using (84) and the crL,_0 values given for k=16.92 A -1 . Units are in A 2 . j' denotes the initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j'=12 j'=14 j=0 101.90 4.72 0.65 0.13 0.03 0.01 0.00 0.00 j=2 23.58 110.30 7.87 1.25 0.26 0.06 0.02 0.01 j=4 5.85 14.16 109.30 8.48 1.40 0.30 0.08 0.02 j=6 1.64 3.25 12.26 109.04 8.82 1.49 0.33 0.09 j=8 0.52 0.90 2.65 11.54 108.94 9.04 1.55 0.35 j=10 0.18 0.28 0.71 2.41 11.16 108.90 9.19 1.60 j=12 0.07 0.10 0.22 0.63 2.28 10.94 108.88 9.29 j=14 0.03 0.04 0.08 0.19 0.59 2.20 10.78 108.87 Part B: Ratio of 108 to exact cross sections. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 0.87 0.99 1.86 3.71 10.3 72.7 j=2 1.03 0.87 1.30 2.27 5.01 24.8 j=4 2.15 1.44 0.84 1.10 1.84 6.49 j=6 5.21 3.06 1.33 0.76 1.01 2.27 j=8 20.4 9.49 3.14 1.42 0.69 1.40 j=10 297 97.1 22.9 6.61 2.89 0.55 The major finding to note is that in all but the 6<-8 and 04-2 transitions the cross sections obtained using the exact ol_o values fared better than those 86 obtained using the IOS 01-0 values. This demonstrates that the scaling law works best with a1,,..0 values that are closer to the exact results. (It could have been that the scaling law corrected for inaccurate trends in the 0i„_.0 cross sections — in which case using exact cri,,0 cross sections would give worse results.) Further, the errors due to the scaling laws are generally less than those at- tributed to the o-L4. 0 values. For example, in the 44-6 transition, the use of an exact ol,,0 value and the scaling law gives a 1% error but the use of the IOS 0 L,0 value and the scaling law gives a 10% error. The derivation of the scaling law involved only the angular momentum cou- pling made possible through a 0 parameterized T-matrix which was made possible by the assumption of suddenness. The calculation of the IOS oL*—o values, involve not only angular momentum simplifications but linear momentum simplifications associated with the assumption of suddenness. Since the scaling law at 300K is more accurate than the crL„_.0 values, it appears that the concept of suddenness for this collision may be better suited to studying the angular, rather than energy, or combination of energy and angular aspects of the collision. 4.2.5 Energy-Corrected Scaling Relation As derived in (86), the Energy Corrected Scaling Relation requires a collision time T. Since the shell of the potential is of negligible width (it takes no time to pass through the potential) 7 might be considered to be 0. Nonetheless two 87 proposals for a finite value of T are presented and then various T values are tested to determine whether the Energy-Corrected Sudden scaling relation can improve the IOS results. One possibility for the calculation of T is to take the average time for a straight line trajectory through the sphere inside the Delta Shell potential at a given impact parameter b. Assuming the velocity v during the collision is constant, T is calculated to be (1 a  = — — Va2 — b2db —7a (210) v a fo 2v where a is the radius of the delta-shell. The value for v could correspond to the final or initial collision velocity or some average of the two. Table 13 gives the result of choosing the lowest velocity to calculate a r value for the correction. The IOS 01 4_0 cross sections, on the other hand, were calculated using the initial state k value, whether this corresponded to the lowest velocity or not. 88 Table 13: ECIOS Cross Sections At 300K Using T = 71( (21) min) and Exact aL_0(Ek,,,,,,,,) Values Part A: Units are in A 2 . j' denotes initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j 1 =10 j=0 101.90 2.38 0.02 0.00 0.00 0.00 j=2 12.14 111.03 0.65 0.00 0.00 0.00 j=4 0.17 1.21 113.22 0.13 0.00 0.00 j=6 0.00 0.01 0.18 120.97 0.02 0.00 j=8 0.00 0.00 0.00 0.03 132.19 0.00 j=10 0.00 0.00 0.00 0.00 0.00 155.32 Part B: Ratio of ECIOS to exact cross sections. j'=0 j'=2 j'=4 f=6 j'=8 j'=10 j=0 0.87 0.50 0.05 0.01 0.00 0.00 j=2 0.53 0.88 0.11 0.01 0.00 0.00 j=4 0.06 0.12 0.87 0.02 0.00 0.00 j=6 0.01 0.01 0.02 0.85 0.00 0.00 j=8 0.00 0.00 0.00 0.00 0.83 0.00 j=10 0.00 0.00 0.00 0.00 0.00 0.79 Aside from the elastic cross sections, which are not affected by the ECIOS scaling relation, all cross sections have been practically reduced to 0 by this choice of T. Clearly this choice of T is too large. This does however demonstrate how the ECIOS scaling relation works. The 10S, by assuming a 0 collision time, overestimates inelastic cross sections. The worse this assumption (ie., the less sudden the collision) the more the IOS will overestimate the inelastic cross sections. Introducing a correction term inversely proportional to a collision time, as is done in the ECIOS scaling relation, reduces inelastic collision cross sections. 89 Further studies were done on finding a better T value. If a factor f is defined so that a T . f. — v (211) then Table 13 displays the results obtained using f=1.57. It was found that an f value of 100 completely reduces all inelastic cross sections to 0.00 A 2 while an f value of 1x10 -5 does not change any of the IOS cross sections. In the investigation as to which T value worked best for the ECIOS scaling relation it was found that while one choice of f was sufficient to get a column (ie. a set of cross sections with the same initial state) of IOS upwards transitions within 55% of exact values, this same value of f was not suitable for any other column. Table 14 reports the ECIOS cross sections for upward transitions which best fit the exact results as well as the f value used to obtain these results. Upward transitions can be calculated using detailed balance and would then give the same ratio to exact cross sections as their respective downward transition. 90 Table 14: ECIOS Cross Sections At 300K Using T = fa/vmin and 'OS aL,0 (Ekma.) Values Part A: Cross sections are in A 2 . j' denotes initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 101.90 j=2 22.91 111.03 j=4 4.22 12.60 113.22 j=6 0.43 1.64 10.01 120.97 .i=8 0.02 0.11 0.94 8.63 132.19 j=10 0.00 0.00 0.03 0.37 4.57 155.32 f value 0.300 0.225 0.165 0.145 0.145 0.79 Part B: Ratio of ECIOS to exact cross sections. 0 2 4 6 8 10 j=0 0.87 j=2 1.00 0.88 j=4 1.55 1.28 0.87 j=6 1.37 1.54 1.09 0.85 j=8 0.84 1.15 1.12 1.06 0.83 j=10 0.75 0.89 0.89 1.03 1.18 0.79 It can be seen from Table 14 that the actual time that is needed to make the scaling relation work best is actually about 1/10th to 1/5th that calculated when it was assumed that the interaction lasts for the whole time it takes for the atom to traverse the diatom potential shell. Another result brought to light by this study is the fact that the correction, which uses T corresponding to the least velocity works best on an lOS result that calculates the 01-0 values based on the highest velocity and hence highest 91 energy. DePristo et. al. [32] recommend the minimum k value for the calculation of o-L-0 . This is based on the assumption that if the collision is sudden at the minimum kinetic energy value, it will be sudden at the maximum value as well. However a study of rate constants by Chapman and Green (43], essentially a Boltzmann average of cross sections, found that using the initial energy for upward j 4-- 0 transitions and the initial energy for downward 0 4-  j' transitions gave best results. The present study comes to the same conclusion. If, however, the minimum k value is used in both the calculation of the ol,_0 cross sections and the ECIOS scaling relation then it is not possible to get the results to all agree within a 50% deviation of the exact results, as is demonstrated in Table 15, where the cross sections for initial rotor state equal to 10 are reported. The f value used was that which gave results that deviated least from the exact cross sections. Table 15: ECIOS Cross Sections at 300K Using T = 0.16a/vrnin and IOS 	 (Ekr,„,i) Values Transition Cross Section in A 2 Ratio to Exact Results 04-10 0.00 3.8 24-10 0.00 1.4 44-10 0.02 0.46 6+-10 0.18 0.28 84-10 2.75 0.43 104-10 155.32 0.79 9 9 The resulting cr -1() ,(Ek,„,„) cross sections with scaling could not be fitted to the exact results as closely as were the or -'3',(Ekm,„) cross sections. The ratio of results:exact ranged by an order of 10 using minimum k value for the crL,o cross sections and ranged by an order of about 1.5 using the maximum k value. A possible interpretation of why different k values work best in the scaling relation and 01,0 cross sections is that the IOS calculates the sudden part of the collision where the kmax value would be dominant while the ECIOS scaling relation calculates the non-sudden part of the collision where the km'. value would be dominant. Table 14 may be further understood by considering the actual collision time, once T is calculated using the f value which worked best. Table 16 shows both the k value used to calculate the o -L,0 cross sections and the actual T value used (obtained by multiplying f times a, divided by the least velocity) that gave the results in Table 14. 9:3 Table 16: o- L-0 k Values and ECIOS 7- Values Used In Table 14 For each entry, the upper number is the k value in A -1 used to calculate the o L_O cross sections while the lower number is the collision time r in fs (10-15 s.). Elastic cross sections are not corrected by the ECIOS scaling relations and so there are no collision times for these values. j' denotes ini- tial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 16.92 j=2 16.92 16.57 258.3 j=4 16.92 16.57 15.71 272.3 204.2 j=6 16.92 16.57 15.71 14.27 299.8 224.8 164.9 j=8 16.92 16.57 15.71 14.27 12.03 355.6 266.7 195.6 171.9 j=10 16.92 16.57 15.71 14.27 12.03 8.37 511.0 383.3 281.1 247.0 247.0 Factor 0.300 0.225 0.165 0.145 0.145 If the T value is regarded as a reliable measure of the time of interaction then the collision time seems to be proportional to the difference in rotor states between initial and final states. For example, transitions where jAjj=2 seem to require about 200 fs. whereas those with 0,j1=8 require about 370 fs. This could explain why using the least T value in the correction works so well, as well as why a different factor must be used for each j' value. If the same r value was used for the j'=0 and f=2 columns then the 44-0 and 44-2 would have the same T value. But because the j'.0 column requires a larger f value than the 94 j'=2 column, the T values for the 44-0 and 44-2 transitions are 272 and 204 fs. respectively. Hence the outcome of calculating 7 based on final velocity and f based on initial state gives a set of actual values that increase with IAA. Note that the same sort of effect could not be achieved by using a T value dependent on the average of initial and final velocities. For example, using an average T velocity for the 104-0 transition (lAj1=10) would give a T value much less than using an average velocity for the 104-8 transition ( 1= 2 )• One final point was investigated — whether using the different k values in the calculation of the ci-L_ o cross sections affected the choice of collision time. It turns out that while certain combinations seem to work best (ie., largest k value for the calculation of the (31_ 0 cross sections and least velocity value for the calculation of 7 in the ECIOS scaling relation) there does not appear to be a direct relationship between the two possible choices. Table 17 demonstrates that if the 0L- 0 cross sections are calculated at a lower k value, then a higher T term in the ECIOS scaling relation is not always needed to compensate for this. 95 Table 17: How f Varies According To the k Used in the Calculation of the o - L, 0 Cross Sections The f values chosen so that the 64-4, 84-4 and 104-4 ECIOS cross sections are within 15% of the exact values. k Used to Calculate 0L,0 in A -1 f Value Giving Best Results 16.92 15.71 8.37 0.170 0.165 0.190 The lack of correlation between the 0L-0 value and ECIOS scaling relation velocity can be accounted for in that the 10S, regardless of choice of k value, assumes an instantaneous collision. Any correction accounting for collision time, such as the ECIOS scaling relation, would not be directly correlated to the k value used in the IOS o-L,0 cross section calculations. In summary, then, the results in this section recommend for the calculation of IOS o-L_o cross sections the use of the highest k value and then the use of these cross sections to calculate upward transitions. These cross sections are then to be used with the ECIOS scaling relations, where best results are obtained when the least velocity weighted by a factor dependent on the initial state is used for the r value. The net effect of this method of calculating cross sections is to allow for a collision time in the 10S, and further, for this collision time to be dependent on the magnitude of lAjl for the transition. Finding that r increases as does lAjlor .AE appears to violate the uncertainty principle. This seems to imply that interpreting r as only a collision time may 96 be too narrow a definition of T. Since E ., — 	 is also dependent on lAjl via h 2 — =(J — i i )(3 + 3 + 1 ) 27 it could well be that instead of interpreting (E3 — Ej i)7. in the ECIOS correction factor as an energy-collision time term it is found out that an (energy) 2 term gives a better approximation to the exact cross sections. 4.2.6 General S-Matrix Scaling Relation Using IOS 0L-0 Cross Sections In this section the General S-Matrix Scaling Relation (GSMSR, Equation (89)) is studied with regards to how well it can correct the IOS 0 L-0 cross sections. That is, even though exact o-j,  values are intended to be used in the GSMSR, it is investigated whether the GSMSR can improve the regular 105 results. In replacing the exact aL,o(Ek + EL) with IOS 01.0(Ek0 ), there is some arbitrariness as to the choice of Eko since in the 105 all rotational states are degenerate. The choice is made to interpret the [L]o-PanEk + EL) term as airl(Ek0 ). That is, the choice of the k'd value which will replace operator k 2 will correspond to the kinetic energy Ek with the EL term ignored. With this substitution Equation (89) becomes 2j j' L	 r6 +RE', — COT /2hi 2 ] 2 IOS 	i(Ek+Cp) = ( 2j+1) E 	 01,,,o(Ek).0 0 ) [6 + [ (( EL  — cj)7/2h1 2 	L 	 ° (213) As well, the optimal choice of 7 was involved using for each j', an f value and the minimum collision velocity. This resulted in the cross sections listed in Table 18. (212) 97 The cross sections were calculated using (213) and the values where Ely is the translational energy associated with rotor state j' from Table 5. The T value is calculated using the minimum collision velocity, the value for f reported below and equation (211). Table 18: GSMSR Cross Sections At 300K Using IOS o- L,0 Values Part A: j' denotes initial state and j the final state. All cross sections are in A 2 . j' =0 j' =2 j' =4 j' =6 j' =8 j' =10 j=0 101.90 4.62 0.61 0.15 0.05 0.02 j=2 23.58 111.55 7.25 1.27 0.32 0.11 j=4 5.85 13.53 115.11 7.92 1.27 0.32 j=6 1.64 2.73 9.98 122.88 7.52 1.14 j=8 0.52 0.72 1.44 9.64 134.71 6.33 j=10 0.18 0.23 0.30 1.48 9.30 158.25 f Value 1 0.25 0.30 0.20 0.15 0.10 1 The r value cancels out for these transitions Part B: Ratio of G.SAISR to exact cross sections. j' =0 j' =4 j' =6 j' =8 j' =10 j=0 0.87 0.97 1.7 4.5 16 1.5x102 j=2 1.0 0.88 1.2 2.3 6.1 40 j=4 2.2 1.4 0.88 1.0 1.7 6.9 j=6 5.2 2.6 1.1 0.86 0.86 1.7 j=8 20 7.6 1.7 1.2 0.85 1.0 j=10 3.0x102 79 9.7 4.0 2.4 0.80 98 For transitions with j'=0, (213) reduces to 0.9SM/ 	 E 	 0.0S 34-0 kJ-1k + 3 = 4---o(ik) and for those with j=0, (213) reduces to csm E k 	 — a.081i-0 k(E )ao—p  r 	 3 13 1 so there is no change in these values from Table 5. The other values report a marginal improvement over Table 5 (ie., thelOS Scaling Relation with IOS o-L 4_0 values) in that elastic cross sections are increased and inelastic cross sections are decreased, more in keeping with exact results. The 7 values which worked best tend to decrease as j' increases but this is not a definite trend as the j'=2 column demonstrates. An alternative for these calculations is to convert (89) using the equation for detailed balance (equation (208)) to _ 3,3 C;S.1.11 1 (Ek 	 Ea' ) = (2i + i)E J 3 L	o o L ) 2 r6 + [(EL — E0)7/2h] 2  2 (Ek + EL) Exact 0 ) [6 + Rej, — ti)7/2/42 .1 Ek (216) and replace o-Exa: with o-Pso . This results in crGS.111 ( E , + ci , ) 3 4-3 k 	 i'' ( i j' L ) 2 16 + [(CL — f0) 71 2h] 2 1 2 (Ek + EL) = (2j + 1)E 	 r,—o- 	 (217) L 0 0 0 ) [6 + [(ei ' — fi )-r/2/42 ] 	 E—k Comparing (217) with (216) reveals that the effect of this treatment would be to multiply all cross sections by a factor of (Ek + EL)/Ek > 1 which would (214) (215) 99 lead to even poorer agreement with the exact cross sections. Other ways of approximating the ofx o(Ek eL ) term by a cri 0 (Eko ) value would not be expected to significantly improve these results owing to the fact that the values of (71,T0(Ek„) change only slightly with respect to a change in ko (Table 4). It may be concluded that the GSMSR, even though intended for use with exact 0l_0 values does have an advantage over the ECIOS (one intended for IOS o-L, 4_0 values) in increasing elastic cross sections. With regards to inelastic cross sections, however, the ECIOS does a better job than the GSMSR of scaling down the high an, values to match exact cross sections. Using Exact o-L-0 Cross Sections The GSMSR was tested as to how well it applied to reproducing the exact re- sults. In order to calculate the column of cross sections with f=10, the GSMSR required the calculation of o-L,_.0 (Ek + ti.,) cross sections for Ek correspond- ing to 103K and ( L, for L=0,2,4,6,8,12,14 in order to get enough terms in (89) to have the resulting o-3,j , cross sections converge to two significant figures. Table 19 lists the details of this calculation, ie., the input values required for the calculation of cross sections for the GSMSR. Ek is the energy according to the most probable translational energy at 103K. J = A j is the total angular momentum. The last contributing .1 value is that value of J for which there was a contribution of at least 1 x 10' A 2 to the cross section. 100 Table 19: Input o-L-0(Ek + EL ) Values for the GSMSR at 300K for Y=10 Largest  Max. Max. Max. Contri- Calc. 	 Cross  Cross 	 condition rotor 	 J 	 buting 	 time 	 Section  Section 	 number' state 	 J  (sec) 2 	 in A2 ao_o(Ek + 6 0) 2.2x10 3 4 80 84 55 3.3 1.77x10 2 0-04-2(Ek + 6 2) 1.4x10 5 4 80 84 59 3.3 4.71x100 0-0■-4(Ek 	 64) 8.5x10 5 6 90 96 67 6.7 4.32x10 -1 0-0■-6(Ek + 6 6) 9.5x10 6 8 100 108 78 14.5 6.50x10 -2 70 4_8(Ek 	 E8) 1.1x10 7 8 110 118 90 16.2 3.55x10 -3 Cro,m(Ek + 610) 1.9x10 7 10 120 130 104 68.9 1.17x10 -4 Cr0■-12(Ek 	 €12) 6.1x10 5 12 140 152 118 81.2 3.11x10 -6 0-0.-14(Ek 	 614) 3.5x10 6 14 150 164 132 162.1 1.50x10 -7 1. Not all the conditions numbers checked for the 0,-10 transition (Calculations for the 04-10 transition done on an Amdahl 470 V8) 2. The calculation time is that required for the Amdahl 5840. The GSMSR also requires the calculation of a collision time T. Using an average 7 value with minimum v resulted in inelastic cross sections that were too low and an elastic cross section that was too high. If, however, 7 is decreased, it is found that this scaling relationship decreases elastic cross sections since the coefficient in each term in the sum in (89) becomes [6 [L(L OhT/(442  2 (218)6 Not only does a decrease in T decrease elastic cross sections, but it also increases inelastic cross sections since the coefficient in each term for these cross sections 101 in (89) becomes 6 + [L(L 1)/i7/(41)? 6 + [[ji(ji + 1) — j(j + 1)] tir/(4/)1 2 (219) and for L small (where most of the contribution occurs) and j 1 =10 and j=2,4,6,8 Pi' +1) — j(j +1) > L(L + 1). 	 (220) A T value of 1.28 x 10' seconds was then found to bring all results to within 25% of the exact results as is shown in Table 20. Table 20: GSMSR at 300K Using Exact aL4_0(Ek + EL) Values The a-L,_0 (Ek + L) values used are from Table 19. 7 = 1.28 x 10' s. Transition Cross Section in A 2 Ratio to Exact Results 0 4— 10 0.00 1.00 2 <— 10 0.00 1.25 4 <— 10 0.06 1.24 6 4— 10 0.49 0.75 8 4— 10 6.73 1.04 10 <— 10 184 0.93 The results in Table 20 demonstrate that the GSMSR, with its allowance of a collision time, is about as accurate in predicting cross sections as the regular IOS Scaling Relation (Table 10), where no collision times were taken into account. (The j'=0 is a special case for the GSMSR in which (89) reduces to Qj •/(Ek + eji ) = 0-1.,o (Ek 	 ejl)  (221) 102 with j = L). The success of the GSMSR then seems to come not so much in accounting for a collision time but in that it makes use of the IOS scaling relationship. One further result from Table 20 is that the 7 value was found to have the same magnitude as in the ECIOS case, again suggesting the actual time of interaction to be about 1/10th the time it takes for a particle going at minimum collision velocity to traverse a shell with radius a=5.5A. . There are, however, some limitations to interpreting this T value as an actual time of collision. This is demonstrated by adjusting the 7 value for each transition in order for the cross section to fit with exact results. Table 21 lists the T values required to get cross sections that are within 1% of the value of the exact results. Table 21: -r- Values Required to Match the GSMSR with Exact Results at 300K The T reported below is that value required by the GSMSR to give cross sec- tions to within 1% of the exact values. Transition 7 value in seconds 2 4- 10 1.79 x 10' 4 4-  10 1.53 x 10' 64-1010 0.89 x 10' 8 4-  10 1.36 x 10–n 104-1010 8.52 x 10-13 The 7 value required to fit the elastic cross section to exact results is 5- 10 times higher than the 7 value required for inelastic cross sections but there 103 does not appear to be any physical reason why this should be so. As well, no direct relationship appears between the 7 value and the type of transition (eg. decreasing r with increasing j) which cautions one to not make too literal an interpretation of the 7 value as being an accurate measure of the collision time. 4.2.7 Accessible States Scaling Relation Using IOS ol_ o Cross Sections In the previous section, the GSMSR used the assumptions of the IOS to relate o-i,j,(Ek+ej,) values with cro —L(Ek+cL) cross sections. In the Accessible States Scaling Relation (ASSR), the assumptions involve a simplification of angular coupling coefficients, neglect of quantum tunnelling and a statistical treatment of transition probabilities [33]. This leads to an alternative relationship between cro—L(Ek + €L) values with o- , , (Ek + t,i) cross sections. In this section several cross sections are calculated using the ASSR in order to determine how well it can reproduce exact cross sections. The difference of the ASSR with the GSMSR is that the ASSR uses only one ao.L(Ek+cL) value for each o-j-_, , (Ek+€3 , ) cross section calculated whereas the GSMSR involves a sum of (30,__L(Ek+EL) values for calculation of the cr,i(Ek+ ej ,) cross section. For this reason the ASSR calculations are easier and less costly to obtain than the GSMSR calculations. As mentioned above, since the ASSR is similar to the GSMSR in that it relates o-j _j i(Ek (J O cross sections with cross sections of a different energy, namely the o-o—L(Ek + td cross section. 104 Although the ASSR is intended to be used with exact o-L-0(Ek + EL) values, a study is made in this section as to how good a scaling relation the ASSR is with the IOS values. Comparing these results with exact results, it will be determined how compatible the assumptions that lead to the 105 cross sections are with the assumptions that lead to the ASSR. As in the GSMSR when IOS .91_ 0 values were used, the choice is made to 	(E 	 . .interpret the quantity [j' — j]o 	 k. 	 (- 13 ,4 as ar,_so(Eko)• substitution, equation (91) becomes  [j] N ( Ek 	 p--EJ) loscy,,s P—.71 	 D (Ek  (ii) = •  CfulLal.-0(Eko)• ii N(Ek With this (222) Table 22 lists the cross sections calculated from (222) using the IOS cross sections of Table 5. 105 Table 22: ASSR Cross Sections at 300K Using IOS crL,0 Values Part A: Accessible States Cross Sections using 01-0(Ek j,). j' denotes initial state and j the final state. All cross sections are in A 2 . j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 101.90 4.62 0.61 0.15 0.05 0.02 j=2 117.88 102.81 11.35 2.24 0.58 0.18 j=4 57.39 41.56 96.54 13.20 2.82 0.72 j=6 25.24 16.15 29.51 93.07 12.13 2.11 j=8 11.18 5.97 10.38 24.93 91.29 6.71 j=10 5.13 2.76 3.82 9.42 19.59 67.94 Part B: Ratio of above to the exact cross sections j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 0.87 0.97 1.7 4.5 16 1.5 x 102 j=2 5.1 0.81 1.9 4.1 11 66 j=4 21 4.2 0.74 1.7 3.7 15 j=6 80 15 3.2 0.65 1.4 3.2 j=8 4.4x10 2 63 12 3.1 0.57 1.0 j=10 8.5x103 9.4x10 2 1.2x10 2 26 5.1 0.34 Downward transitions appear more accurate than upward transitions at the same j' and IAjl. Elastic cross sections underestimate exact results and become increasingly lower as j' increases. For transitions with j=0, (222) reduces to the expression 1 	 s ( EA 	(3') =	 -C'ko	 (223) and so for these transitions the ASSR using IOS 01_0 values is the same as the IOS scaling relation using or so values. For all other transitions, however, these 106 ASSR cross sections prove much more inaccurate than the IOS scaling relation using aro values in that the ASSR reports higher inelastic and lower elastic cross sections than the IOS results. An alternative calculation, ie., use in (91) the exact relation ) (224)r •/ .1 Exact t Ek = [Ek Exact 1.1 — Ek , 3Lik and then approximate atLa .c7 1_0 (Ek cw_ ji ) by arso (Eko ) as outlined in (216) to (217) for the General S-Matrix Scaling relation, leads to all cross sections being multiplied by a factor of (Ek tb,_,1)/Ek > 1. This would increase the already high inelastic cross sections. From the results in this section it appears important to retain the angular coupling coefficients (the 3-j symbols) when relating or:, cross sections with cri°50' values since the scaling relations based on the 3-j symbols (the regular IOS and ECIOS scaling relations) prove better than the ASSR. The ASSR, by assuming these coefficients to be functions of differences in j and j' only, is not compatible with the ar so values since it does not correct for and in fact amplifies the inaccuracies resulting from using the aro values, namely, low elastic and high inelastic cross sections. Using Exact a-L_o Cross Sections Accessible states cross sections have been calculated for j'=10 and j'=6. For j'=10 the input cross sections are given in Table 19 while the j'=6 while the j'=6 input cross sections appear in Table 23. The same ao—L(Ek + EL) values 107 given in Table 19 are then used for cross sections with initial rotor state of 10. As well, the o-o—L(Ek (L) values given in Table 23 are used to calculate cross sections with initial rotor state of 6. In Table 23 E 1, is the energy according to the most probable translational energy at 300K, and fL = L(L 1)h 2121. J = A j is the total angular momentum. The last contributing J value is that value of J for which there was a contribution of at least 1 x 10' A 2 to the cross section. The time for calculations is that required for the Amdahl 5840. Table 23: Input ao,L(Ek + E L ) Values for the ASSR at 300K for j' =6 Cross Section Largest Condition Number' Max. Rotor State Max. ) Max. J Contri- buting J Calc. Time2 (sec.) Cross Section in A 2 as—o(Ek + (0) 1.2x10 7 8 110 118 89 20.138 0.126952 x 103 ao-2(Ek + (2) 1.8x10 5 8 110 118 91 19.737 0.507163x10 1 ao_4(Ek + ( 4 ) 2.1x10 5 10 120 130 96 39.148 0.404343x10° uo,-6(Ek + (6) 1.9x10 7 10 120 130 104 68.934 0.340958x10-1 1. Not all the conditions numbers checked for the 04-6 transition 2. Calculations for the 04-6 transition done on an Amdahl 470 V8 The values from Tables 19 and 23 were then substituted into equation (91) and the resulting cross sections are displayed in Table 24. 108 Table 24: ASSR Cross Sections Using Exact ao<—L(Ek + EL) Values for 300K The o0,L(Ek eL) values used are from Tables 19 and 23 . Transition Cross Section in A2 Ratio to Exact Results Transition Cross Section in A 2 Ratio to Exact Results 04-6 0.03 1.00 04-10 0.00 1.00 24-6 1.27 2.32 24-10 0.00 4.32 4+-6 14.36 1.86 44-10 0.03 5.62 64-6 126.95 0.73 64-10 1.31 2.00 84-6 25.36 3.34 84-10 8.67 1.34 104-6 3.64 14.7 104-10 80.58 0.41 The calculations in Table 22 are also more inaccurate than the ASSR using ak—xV—ii(Ek (13 ,_ 3 1) values (Table 24). On the basis of these results, the ASSR calculates elastic cross sections that are lower and inelastic cross sections that are higher than the exact cross sections. (The special case where j=0 reduces (91) to 0.04—ji(Ek e .1 ) = 0-0*L(Ek + EL) (225) with j' = L and so the 04-10 and 0+-6 ASSR cross sections are exact). As well, as Ij —ft increases, so too does the error. In the f=6 column the cross section for a downward transition with a given — is much closer to the exact value than is the upward transition cross section for the same Jj — ft. In the paper where the ASSR is introduced some calculations [32] using the ASSR for Ar-N2 using the potential of Pattengill, LaBudde, Bernstein and Curtiss [39], it was found that error increases as Ij ft increases. In these calculations 109 it was also found that the error for transitions for the Ar-N2 system increases as j' increases but this is not the case for the results in Table 24 . For example, the ratio of the 44-6 transition cross section to exact results is 1.9 but for the 84-10 transition is more accurate, with a ratio to exact results of 1.3. It was also found in the calculations by DePristo [33] that the ASSR for some transitions where lj — j'1 , 4 can overpredict by a factor of 2 (as is the case in this study for the 24-6 and 64-10 transitions) and the ASSR for Ij — j'I=6 can overpredict by as much as 11 times (here, for the 44-10 transition, the ASSR overpredicts by only about 5.5 times). In comparison to the ASSR cross sections, the IOS cross sections which use the IOS ol_ o values (Table 5) are better for all the j'=6 transitions and for the j'=10 transitions for lj — j'I=0 and 2 (ie. the 104-10 and 84-10 transitions). The two are similar in that they become worse with increasing Ij — j'I and this error in both scaling relations grows to about the same magnitude as the transition becomes more inelastic. For example, the errors in the IOS for the 84-10, 64-10 and 44-10 transitions are 15%, 160% and 880% and for the ASSR are 34%, 100% and 460% respectively. DePristo further notes [33] that using pure statistical theory (ie. treating all rotational states as degenerate) to relate the ao—L(Ek EL) values with the o-j _j , (Ek f 4) cross sections leads to even greater inelastic cross sections and hence even poorer results. It seems possible then that the reason for the overestimation of the inelastic collisions comes from the statistical assumptions made in each scaling relation, the ASSR when it 110 assumes that the transition probability is inversely proportional to the number of accessible states, and the 10S, when it replaces operator with one parameter k?, and in doing so treats all rotational states as energetically equivalent. One further note on DePristo's work [33] which may have some relevance on this work is that DePristo found that his scaling theory worked best for systems with a lower reduced mass, such as a He-CO system, for which the reduced mass is about 3.5 atomic mass units (amu). Less accurate results were found by DePristo for the Ar-N2 system, which has a reduced mass of 16.5 amu. Finally, in comparing the ASSR cross sections (which vary by as much as 1500% from the exact results) with the GSMSR cross sections (which vary by at most only 25% from the exact results (Table 20)), it may be concluded that the manner in which the GSMSR relates the o-o:___L(Ek EL ) values with the cri,j,(Ek ef ) cross sections appears to be more reliable. In the ASSR, the combined assumptions of degeneracy, the Effective Hamiltonian having angular coupling coefficients dependent on energy differences and neglect of quantum tunnelling lead to errors of the same magnitude found in IOS scaling relations with IOS 01,0 values even though exact input cross sections are used. One possibility for decreasing this error of the ASSR is to modify or remove its assumption that on statistical grounds that the transition probability is proportional to the number of accessible states. 1 l 1 4.3 Cross Sections at 1000K 4.3.1 The Exact Cross Sections An exact calculation for 1000K was carried out. This requires changing certain parameters from the values used for the 300K calculation. Specifically, at 1000K, the thermally most probable rotor state is 12.64 (which was taken to be 12 since this study considers only the even rotor states) and the maximum rotor state for N2 with a translational energy of 1000K (ie. using (190)) and initial rotational state of 12 is 20. This gives a total energy of E = kBT E12 = 2.004 x 10 -20 J. If all the energy from the initial conditions is converted to kinetic energy then the maximum k value possible is :3.14 x 10 11 m.. -1 Using (196) the maximum partial wave contributing is estimated at 173 so contributions from 190 partial waves were kept. All other parameters — those for the potential and N—N distance — were kept at the same values as used in the 300K calculations. As this study is focussed on the utility of the IOS approximation and its variants to calculate rotational cross sections, any aspects of vibrational motion have been ignored. The calculated cross sections retaining only open states are listed in Table 25. 112 Table 25: Exact Cross Sections at Energy=1000K j' denotes initial state and j the final state. All cross sections are in A2 . j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j=0 70.07 3.13 0.19 0.01 0.00 0.00 j=2 15.44 73.92 5.24 0.27 0.02 0.00 j=4 1.68 9.16 75.64 4.96 0.30 0.02 j=6 0.16 0.66 6.83 80.03 5.52 0.32 j=8 0.01 0.06 0.51 6.75 79.69 6.01 j=10 0.00 0.00 0.03 0.44 6.76 85.11 j=12 0.00 0.00 0.00 0.03 0.44 6.70 j=14 0.00 0.00 0.00 0.00 0.03 0.36 j=16 0.00 0.00 0.00 0.00 0.00 0.02 j=18 0.00 0.00 0.00 0.00 0.00 0.00 j=20 0.00 0.00 0.00 0.00 0.00 0.00 j'=12 j'=14 j'=16 j'=18 j'=20 j=0 0.00 0.00 0.00 0.00 0.00 j=2 0.00 0.00 0.00 0.00 0.00 j=4 0.00 0.00 0.00 0.00 0.00 j=6 0.02 0.00 0.00 0.00 0.00 j=8 0.37 0.02 0.00 0.00 0.00 j=10 6.38 0.35 0.02 0.00 0.00 j=12 90.09 6.63 0.39 0.03 0.00 j=14 6.49 98.79 7.00 0.50 0.03 j=16 0.34 6.28 108.63 8.05 0.49 j=18 0.02 0.35 6.28 127.48 7.05 j=20 0.00 0.01 0.22 4.00 171.52 For the inversion of matrix K 2 + 1 the largest condition number was found to be of the order of 1 x 10'. For the inversion of matrix W - 1/nj the largest condition number was found also to be about 1 x 10 7 . 11:3 A comparison of the 300K cross sections (Table 2) with the 1000K cross sections reveals that all inelastic and elastic cross sections are reduced as the temperature increases. 4.3.2 The IOS aL,0 Cross Sections As noted in the literature, the Energy Sudden [30] and Centrifugal Sudden [44] approximations improve for higher collision energies. Thus it is expected that the 10S, a combination of the ES and CS, should improve at a higher collision energy. The IOS calculation was carried out for 1000K keeping all the parameters the same as described for the 300K calculation except: contributions from 190 instead of 120 partial waves were included and the ko = ki „it value was calculated as that value which would give a total energy corresponding to 	h 2	,;	 h2 1  J- 	 I initial(J initial + 1)- = 004X10 -20 J f (226) where 2.004x10 -20 J. is the total energy appropriate for 1000K. Again the Ao value was chosen to be the average of ) and A'. The calculated IOS cross sections are listed in Table 26. 114 Table 26: IOS Cross Sections at 1000K Using ko = Part A: j' denotes initial state and j the final state. All cross sections are in A 2 . j'=0 j'=2 j'=4 j'=6 j'=8 f=10 j=0 62.40 3.97 0.34 0.05 0.01 0.00 j=2 19.79 69.14 6.16 0.62 0.10 0.02 j=4 3.05 11.18 69.04 6.77 0.73 0.13 j=6 0.59 1.59 9.67 70.28 7.15 0.84 j=8 0.14 0.30 1.29 9.24 71.98 7.52 j=10 0.04 0.07 0.23 1.22 9.07 75.00 j=12 0.01 0.02 0.05 0.22 1.20 9.11 j=14 0.00 0.01 0.01 0.05 0.22 1.25 j=16 0.00 0.00 0.00 0.01 0.05 0.24 j=18 0.00 0.00 0.00 0.00 0.01 0.06 j=20 0.00 0.00 0.00 0.00 0.00 0.02 j'=12 j'=14 j'=16 j'=18 j'=20 j=0 0.00 0.00 0.00 0.00 0.00 j=2 0.01 0.00 0.00 0.00 0.00 j=4 0.03 0.01 0.00 0.00 0.00 j=6 0.16 0.04 0.01 0.01 0.01 j=8 0.95 0.19 0.06 0.03 0.02 j=10 7.80 1.10 0.24 0.09 0.05 j=12 78.95 8.40 1.25 0.33 0.14 j=14 9.16 85.18 8.62 1.52 0.47 j=16 1.32 9.65 92.63 9.00 1.81 j=18 0.26 1.46 9.74 105.03 8.75 j=20 0.06 0.29 1.60 10.03 127.47 115 Table 26 - Continued Part B: Ratio of 108 to exact cross sections j'=0 j'=2 j'=4 f=6 j'=8 f=10 j=0 0.89 1.3 1.7 3.6 9.8 52 j=2 1.3 0.94 1.2 2.3 4.9 20 j=4 1.8 1.2 0.91 1.4 2.4 7.6 j=6 3.8 2.4 1.4 0.88 1.3 2.7 j=8 10 5.2 2.5 1.4 0.90 1.3 j=10 48 19 7.4 2.8 1.3 0.88 j=12 3.1x10 2 1.0x102 30 8.5 2.7 1.4 j=14 1.3x103 4.5x10 2 1.2x10 2 33 8.4 3.4 j=16 1.2x104 4.3x103 1.0x10 3 2.0x10 2 48 14 j=18 1.3x10 5 3.7x104 6.1x103 1.2x103 2.4x102 71 j=20 1.9x10 7 2.2x10 6 3.3x10 5 3.6x104 5.4x10 3 1.3x103 j'=12 j'=14 j'=16 j'=18 j'=20 j=0 4.1x102 1.7x10 3 2.5x104 4.8x10 5 1.6x106 j=2 1.3x10 2 5.5x10 2 8.0x10 3 1.2x10 5 1.4x10 7 j=4 36 1.5x102 1.8x103 1.8x104 1.8x106 j=6 8.8 36 2.7x102 2.4x10 3 1.1x10 5 j=8 2.6 8.4 55 3.7x102 1.1x104 j=10 1.2 3.1 13 78 1.6x10 3 j=12 0.88 1.3 3.2 13 1.6x102 j=14 1.4 0.86 1.2 3.1 18 j=16 3.9 1.5 0.85 1.1 3.7 j=18 15 4.2 1.6 0.82 1.2 j=20 1.9x10 2 29 7.4 2.5 0.74 Contributions for A > 184 for j=0 were found to be negligible (ie., affecting only the sixth significant figure). The unitarity of the S matrix (equation (206)) again was verified to 6 sig- nificant figures for (206) summed up to L = :30. As well, a check was done 116 on the S-matrix integration procedure, using 96 as well as 40 points for Gauss- Legendre integration. The 96 point integration procedure agreed with the 40 point procedure to 6 significant figures. Table 26 demonstrates that in 28 of the 36 transitions for j and j' ranging from 0 to 10 the ratios of the IOS cross sections to exact cross sections at 1000K are closer to one than the corresponding ratios at 300K (Table 5). 4.3.3 ECIOS Cross Sections One of the most attractive features of the ECIOS scaling law is that it corrects for the large IOS values for highly inelastic collisions. This warranted investigation at higher temperatures, where there is the possibility for even higher energy inelasticity. Will the ECIOS be able to correct for this as well? Table 27 gives a comparison between exact, 10S, and ECIOS values at 1000K. The f value chosen for the ECIOS calculation is 0.5. 117 Table 27: ECIOS Cross Sections at 1000K Transition Calculation Cross Section Ratio to Exact Exact 70.07 04-0 IOS 62.40 0.89 ECIOS 62.40 0.89 Exact 15.4 24-0 lOS 19.8 1.3 ECIOS 19.4 1.3 Exact 1.68 4+-0 IOS 3.05 1.8 ECIOS 2.40 1.4 Exact 0.157 6+-0 lOS 0.589 3.8 ECIOS 0.232 1.5 Exact 0.0137 84-0 IOS 0.138 10 ECIOS 0.0167 1.2 Exact 7.73 x 10 -4 10+-0 lOS 3.72 x 10 -2 48 ECIOS 1.11 x 10 -3 1.4 Exact 3.57 x 10 -5 124-0 IOS 1.11 x 10 -2 310 ECIOS 7.86 x 10 -5 2.2 Exact 2.80 x 10 -6 144-0 IOS 3.56 x 10 -3 1300 EGOS 5.97 x 10 -6 2.1 Exact 1.01 x 10 -7 164-0 IOS 1.19 x 10 -3 12000 ECIOS 4.60 x 10 -7 4.6 Exact 3.20 x 10 -9 18+-0 IOS 4.12 x 10-4 130000 ECIOS 3.15 x 10 -8 10 Exact 7.74 x 10 -12 204-0 IOS 1.46 x 10 -4 19000000 ECIOS 1.29 x 10 -9 167 115 The results in Table 27 seem to indicate that the corrections invoked by the ECIOS are mathematically valid; the ECIOS follows the behaviour of the exact calculations even to values as low as 1 x 10' A 2 . 4.4 Changing Parameter Cv6 In the calculations parameter Cy, is the unitless constant 2 a C = =2-ti a v (227) and so an increase in Cy, corresponds to increasing value V5, or "height" of the shell. It was investigated whether an increase in this value would make the collision more "sudden" and hence bring about better agreement between the IOS and exact calculations. Table 28 displays the results of this calculation. Initial conditions were set with ji=0 and translational energy equivalent to 300K 119 Table 28: Exact and IOS Cross Sections at 300K for Cy, =1000 Part A: Exact Results. Units are in A 2 . 1' denotes initial state, j the final state. j'=0 j'=2 j'=4 j'=6 j'=8 j=0 198.63 0.59 0.07 0.01 0.001 j=2 2.80 203.92 0.70 0.09 0.01 j=4 0.52 1.08 202.36 0.35 0.04 j=6 0.08 0.15 0.37 202.12 0.17 j=8 0.006 0.01 0.03 0.12 205.62 Part B: IOS Results. j'=0 j'=2 j'=4 j'=6 j'=8 j=0 199.87 0.54 0.16 0.022 0.0029 j=2 2.19 201.74 1.21 0.16 0.021 j=4 0.92 1.91 202.42 0.64 0.087 j=6 0.47 0.86 1.93 201.97 0.39 j=8 0.29 0.46 0.94 0.86 202.46 Part C: Ratio of IOS to exact cross sections. j'=0 j'=2 j' =4 j'=6 j'=8 j=0 1.006 0.90 2.36 2.06 2.50 j=2 0.78 0.989 1.73 1.77 2.17 j=4 1.76 1.77 1.000 1.81 2.46 j=6 5.65 5.89 5.17 0.999 2.22 j=8 48.40 42.04 37.40 7.36 0.985 120 While the values in Table 28 indicate that changing parameter Cy, does not improve agreement for inelastic cross sections, it does show that the 105 elastic cross sections do improve in their agreement with exact results. In Table 5 with Cy, =100, elastic cross sections are out by about 20%, whereas in Table 28, with Cy, =1000, the 105 elastic cross sections agree to within about 1% with exact values. Note also that increasing parameter C y, increases elastic cross sections and decreases inelastic cross sections in general. 4.5 Changing Parameter a It was investigated whether reducing the parameter a, the delta-shell radius, would increase the agreement between 105 and exact results. The motivation behind this investigation is that a smaller radius would correspond to a shorter time of interaction between the atom and diatom and make for a more sudden collision. The initial conditions were chosen with j'=0 and the translational energy corresponding to 300K. The results are shown in Table 29. 121 Table 29: Exact and IOS Cross Sections at 300K for a=0.55A. -1 Part A: Exact Results. Units are in A 2 . j' denotes initial state, j the final state. j 1=0 j'=2 j'=4 j'=6 j'=8 j=0 2.29 0.0063 0.00076 0.00011 0.0000051 j= 0.030 2.33 0.0091 0.0013 0.000068 j=4 0.0056 0.014 2.29 0.0026 0.00025 j=6 0.00082 0.0021 0.0028 2.35 0.0018 j=8 0.000026 0.000075 0.00018 0.0012 2.47 Part B: 103 Results. j '=0 j'=2 j'=4 j'=6 j'=8 j=0 2.21 0.0068 0.00089 0.00062 0.000064 j=2 0.018 2.22 0.0081 0.0040 0.00046 j=4 0.0076 0.027 2.28 0.014 0.0017 j=6 0.0044 0.016 0.013 2.31 0.0069 j=8 0.0033 0.010 0.0039 0.018 2.35 Part C: Ratio of 108 to exact cross sections. :1=0 j'=2 j'=4 j'=6 j'=8 j=0 0.96 1.1 1.16 5.82 13 j=2 0.61 0.95 0.89 3.08 6.8 j=4 1.4 1.9 0.99 5.23 6.9 j=6 5.4 7.2 4.5 0.98 3.72 j=8 126 139 22 15 0.95 1 99 The first observation to be made from Table 29 is that cross sections in both the exact and IOS calculations are drastically reduced, about 2 orders of magnitude for both elastic and inelastic collisions from their values when a =5.5 A -1 . Yet even at these reduced values the IOS still overestimates inelastic collisions. Elastic collisions agree to within about 5%. The results of the last two tables indicate that something more complex than reaction time considerations may be required to further improve the IOS relia- bility for inelastic cross sections. Another useful investigation involving adjusting a collision parameter would be to decrease the anisotropy parameter b 2 . Yet another would be to calculate cross sections for a lighter system, eg., He-N 2 . 123 5 DISCUSSION 5.1 Time Savings of the IOS One of the most attractive features of the IOS approximation is its computational efficiency. Matrix manipulation is greatly reduced and the programs required for calculations are shorter and much quicker than those for the exact calculations. Table 30 lists the computer time required for the 105 and exact calculations presented in this study. It lists the times in seconds required for program com- pilation and execution for the cross sections calculated in the previous chapter. The 300K calculation was done on an Amdahl 470V8 and the 1000K calculation on an Amdahl 5840. Table 30: Computer Time Required for IOS and Exact Calculations Temperature IOS Calculation Exact Calculation 300K 1000K 30.0 64.0 67.3 1000.0 As previously noted, the IOS becomes more accurate at higher collision en- ergies [30], [44]. This feature of the 105, coupled with the fact that at higher 191 energies the exact solution requires a far greater amount of calculation make the IOS a very attractive alternative to close-coupled calculations at high energies. 5.2 Possible Improvements to the IOS In replacing values kJ and A by parameters ko and A o in (29), the IOS allows for an easier method of calculation which, depending on the conditions of the collision process, may or may not be an accurate reflection of the exact results. Certain modifications to the IOS are proposed in this section which may serve to enhance the accuracy of the approximation. One promising result is that the ECIOS proved to correct high IOS inelastic cross sections. A possible extension to this study could be to: 1) improve the calculation of the T value, and, 2) extend the ECIOS correction to include a Centrifugal Sudden correction where 7 would be dependent on A as well as j. A correction to the ECS proposed by Richard and DePristo [45] does not improve the agreement with exact cross sections. Even if it did improve the agreement, applying such a correction to the values presented in this work may not be as effective as correcting for the CS approximation, as was done by McLenithan and Secrest [46]: ,-cs = 	 [0,,,,(R)]T (21 t. 1) 6 ,- 11,,cs( f)dR, 	(scs	 siscs) 2z 0 	 9 (228) where S cs is the CS 8-matrix, 1 is A o and fiJ is a null matrix, save for its jth 125 CCIOS Cr 	 7 24 2 	 „ios 24h 2 	(( A , — c.\ ) 2 7- 2 ' (230) diagonal element, which equals unity. The above is essentially a first order pertur- bative correction to the CS scattering matrix, where that part of the Hamiltonian neglected by the IOS is calculated and then applied as a correction to the ap- proximation. A first order correction has been shown to improve the CS results for cross sections involving m transitions [46]. Another way of correcting for the CS approximation could be to split the free motion Hamiltonian into a radial and angular part and from this get Hfree = Hradial	 Hangular 	 (229) in order to get a correction term something like Equations (229) and (230) are at best very sketchy and the details remain to be worked out. 5.3 Applications of the IOS Since its introduction in 1974 [3, 6] the IOS has been used in a variety of chemical systems and processes. Examples of its usefulness can be found in papers deal- ing with the Senftleben-Beenakker effect [47], calculation of pressure broadening cross sections [48], calculation of molecular fragmentation [49], angular momen- tum alignment due to collisions [50], and modeling potential parameters [51]. Two particular areas of study are mentioned, namely using the IOS for collisions 126 involving vibrational transitions and using the lOS in the calculation of reaction cross sections. The Vibrational Infinite Order Sudden Approximation (VIOSA) is an approx- imation which deals with the vibrational quantum number 71 the same way the rotational quantum numbers j and ) are dealt with in the 105. This idea was suggested by Pack in 1974 [3] and then formally derived by Pfeffer in 1985 [52]. Pfeffer obtains the following equation for the VIOSA: 21t v(r, 	 illxnjA(r, R)) d2 	  d	 k2 	A(A + 1)1 / 	 , n)/ —h2R2	idR2+ "1-:?,dR 	 "3 (231) The only difference between (231) and (74) is that the parameter k2i j has an extra subscript — n — that comes about by replacing operator k 2 with the parameter k72L3 . This is equivalent to assuming that the duration of interaction is much less than the time required for a vibration of the diatom. Results are encouraging and give reasonable agreement with exact quantum results [52] . The Reactive Infinite Order Sudden (RIOS) approximation was developed in 1980 by Bowman and Lee [53] and by Khare, Kouri and Baer [54]. Most work so far is on the atom — diatom system, eg. H and H2. As an A-FBC system, there are three arrangement channels to consider, A-I-BC (cr arrangement), ACI-B (/3 arrangement) and ABH-C (-y arrangement), each with its own set of coupled equations and potential function. The goal in reaction theory is to solve for these three sets of coupled equations while matching the wave functions 0,„ 00 and so that the functions and first derivatives are continuous at the boundaries. 127 The IOS is used in decoupling each of the three separate blocks of equations and since the angular momentum operators have been replaced by parameters, the matching conditions are simpler as well (eg. setting A 02„ = Ao on the matching surface) where AL is the parameter replacing the operator A 2 for the a set of coupled equations and AL for the /1 set. Work is continuing on the RIOS theory, which is proving to be a valuable approach to study chemical reactions [55]. Work on the 105 approximation and the exploration of new areas where it may be applied is attractive not only because of the substantial saving in calculation time that the IOS affords but also because it is through comparison of exact and IOS results that a further understanding of the underlying details which make up the final results is achieved. Through work such as this it is hoped that a fuller knowledge of the dynamics of chemical systems may be developed. 5.4 Molecular Potentials 5.4.1 Time Savings of the Delta-Shell Since use of the delta-shell potential allows an R-integration step to be avoided that would otherwise be needed if a continuous potential was used, computations using the delta-shell potential are much simpler and faster than those using a continuous potential. Table 31 compares the computer time needed in the calculation of the elastic cross sections reported in Table 1. 128 Table 31: Computer Time Required for IOS Calculations for a Continuous and Delta-Shell Potential The calculations were done on an Amdahl 5840. Type of Potential Time in Seconds Delta-Shell Inverse Power 2.446 3.562 5.4.2 Comparison of Potential Parameters For purposes of comparing the effect of modelling an inverse power potential with a delta-shell potential, a way of comparing the inverse power potential parameters and the delta-shell potential parameters used in this study is presented in this section. There are two factors which must be taken into account in comparing the relative strengths of the potentials. Firstly the parameters are of different units, the inverse power potential having a parameter in units of J-m 12 and the delta- shell potential having a parameter in units of J-m -1 . Further, the potentials go to infinity at different points, the delta-shell at a = 5.5x10 -10 m. and the inverse power at R = Om. One method of comparison is to integrate the delta- shell potential over the a region of space from 1 -1 = Om. to R = 5.5x10 -10 m. Then the inverse power potential will be assigned a con.stant value, namely, that value it has at R = 5.5x10 -1O m. and then integrated over the same region of space. The two resulting energy-distance values obtained will then be compared. 129 Equation (198) gives the following value for the inverse power potential at R = 5.5x10 -10 m.: vinverse P0w"(5.5 X 10 -1°7n., 0) = 2.911 x 10 -23J [1 + 0.5P2 (cos 0)] 	 (232) Integration of this value over the region R = Om. to R = 5.5x10 -10 m. leads to the following energy-distance value: r=a vmverse  P"er (5.5 X 10 -10 7n., 0) = 1.6 x 10 -32 [1 + 0.5P2 (cos 0)] J — m r=o (233) Integrating the delta-shell potential over the same region of space gives the following value for comparison: fr=a r=avdelta 	 = 	 3.697 x 10 -32 [1 + 1.5132 (COS 0)] 6(r — a)dr L=o 	 f=o = 3.70 x 10 -32 [1 + 1.5P2 (cos 0)] J — m 	 (234) Hence the area under the delta-shell used in this investigation is roughly twice what one would obtain with a square well potential extending from R = Om. to R = 5.5x10 -10 m. having a height that is given by the inverse power potential at R = 5.5x10 -10 m. Another parameter to consider is the value for a, the position of the delta- shell. This value will be compared with the point where the inverse power po- tential becomes equal to the kinetic energy of the relative motion of the atom- diatom. Equating 2.2 x 10 -14 X [10— 12vinverse power = R12 77 112 [1 + 0.5 P2 (cos 0)] = h 2 k2 2/1 (235) 130 where k = 14.27x10 10 m -1 and = 2.73x10 -26 kg. gives a value for R of 3.63x10 -10 m. This is to be compared with the value of a= 5.5x10 -10 m. A possible algorithm to determine magnitudes for delta-shell parameters in order to model an inverse power potential with a delta-shell potential could be: • choose a to be twice the distance where the inverse power potential is equal to the relative kinetic energy of the atom-diatom • choose the strength parameter 1 76 to be twice the product of the strength of the inverse power potential at a and the distance a. The final comparison is that for the anisotropy parameter. The inverse power potential uses 0.5 while the delta-shell requires a parameter of 1.5 to give equiv- alent 04-0 and 24-0 cross sections. The comparison presented offers a quick method of comparing the two types of potentials. As for future delta-shell potential modelling, many possibilities ex- ist. Another repulsive shell could be added to determine if the higher transitions, eg. 04--4, 04-6, ... can be matched with the inverse power potential cross sec- tions. One could try to model the attractive part of a molecular potential with a delta-shell. Finally, an attractive and repulsive delta-shell could be combined in order to give a reasonable approximation to an actual molecular potential. The fact that 04-0 and 24-0 cross sections from the inverse power and delta-shell po- tentials can be matched fairly closely (see Table 1) is a promising note for future development of how to make a delta-shell to replace a continuous potential. 131 5.5 Calculations on a PC The advent of a world wide standard PC in 1981 has resulted in improved porta- bility of computer programs and availability of computing resources. Once a program is debugged and running successfully it is no longer necessary to change the source code to conform to the standards of the mainframe of the institution that the scientist is working at or visiting. One other advantage is in the area of numerics. Overflows of the order of 10' that could not be handled by a mainframe are handled easily by Microsoft Fortran 5.1 (which can handle values up to 10'87 ). There are however still disadvantages with using a PC. Two of the main concerns are limited memory and a slower CPU (depending on machines used for comparison). 132 6 CONCLUSIONS Many useful and interesting results have come about from this investigation of the IOS and delta-shell potential. The following is a summary of the most important results, as well as a summary of what further investigations may be performed in light of these results. It was found that cross sections from a 0 dependent delta-shell can be com- puted that are comparable to a more realistic potential, such as an inverse-power potential. Further studies may include various combinations of repulsive and attractive delta shell potentials. This work suggests future work in many areas may prove beneficial, such as further investigating how to model a continuous potential (such as a Lennard- Jones) with a delta-shell potential. Perhaps a hard sphere repulsive (since it is non-penetrable) and non-spherical attractive delta-shell may offer a combination of ease of computing and a fairly realistic model of certain systems. Expressions for the T-matrices have been derived from the exact and IOS solutions for an atom-diatom system with a 0-dependent delta-shell potential. As well, the rotor transition cross sections have been calculated for a variety of energies. The exact and IOS cross sections were then compared. At translational 1 :3 3 energies corresponding to 300K and 1000K the cross sections were found to be reasonable but at high rotational energy transfer the IOS consistently overesti- mated the values. Several scaling laws aimed at improving the 105 cross sections were inves- tigated. These all are based on the IOS ol_ o cross sections. The Accessible States Scaling Relation, an approximate scaling law based on angular momentum transfer of the rotor states was found not to be a significant improvement over regular IOS scaling laws at 300K. The General S-Matrix Scaling Law, based on a scaling based on reaction times and rotor energy separation was found to be useful in some cases in correcting for the large inelastic cross sections given by the IOS approximation. The Energy Corrected Scaling Law, based entirely on correcting for the IOS S matrix with a reaction time parameter T was found to correct very well for IOS differences from exact results for 300K. The T values that worked best were found to be 1/10th to 1/5th that calculated for the time it takes an atom to traverse the diatom potential shell. The scaling laws also were investigated as to how well they performed using exact 01_ 0 cross-sections. All four scaling laws performed very well at 300K. 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