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The infinite order sudden approximation and the delta-shell potential Dancho, Stephen John 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. Exact 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 i proposed.  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 3.1  36  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    	3.4  Inclusion of Closed States  3.5  IOS T-Matrix Calculation  4  51   56   CALCULATIONS AND RESULTS  60  4.1  60  4.2  4.3  Parameter Determination   4.1.1  Atom and Diatom Parameters  60  4.1.2  Choice of Energy  60  4.1.3  Range of Partial Waves  4.1.4  Inverse Power Potential Comparisons  Cross Sections at 300K  61   62   67   4.2.1  Exact Cross Sections Including Only Open States  4.2.2  Exact Cross Sections With Inclusion of Closed States  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    Cross Sections at 1000K  67 .  69  112  4.3.1  The Exact Cross Sections  4.3.2  The IOS crL, 0 Cross Sections  114  4.3.3  ECIOS  117  UL,-0    Cross Sections  112  4.4  Changing Parameter Cv,  119  4.5  Changing Parameter a  121  iv  5  DISCUSSION 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 6  124  Calculations on a PC    132  CONCLUSIONS  133  References  136  List of Tables 1  A Comparison of the Delta-Shell and  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  6  IOS Cross Sections at 300K Using k o = kfinal  77  7  Comparison of k i „ itial and k filial IOS Cross Sections  78  8  Comparison of k,„,,„ and k rni „ IOS Cross Sections  79  9  IOS Cross Sections at 300K Using ko  81  7. -12  Potentials  66  =  =  75  14.27 A -1  10 IOS Cross Sections At 300K Using Exact o L,_ 0 Values -  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 13  ECIOS Cross Sections At 300K Using T =  71- a/(2v ini n )  86  and Exact  O•L—o(Ek,„,,,,) Values 14  89  ECIOS Cross Sections At 300K Using 7 = fa/v,,,;,, and 105 aL—o(Ek,,,,.) Values  15  83  91  ECIOS Cross Sections at 300K Using T = 0.16a/V mi n and 105 01 - 0 Ek.„“„) Values  16 0i,_ 0 k Values and ECIOS  92 T  Values Used In Table 14  vi  94  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 cr L , o (E k E L ) Values for the GSMSR at 300K for f=10 	 101 20 GSMSR at 300K Using Exact crL- 0 (Ek EL) Values 21  7  102  Values Required to Match the GSMSR with Exact Results at  300K  103  22 ASSR Cross Sections at 300K Using IOS 01_ 0 Values 23  Input o o_L(Ek -  L)  106  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  26  IOS Cross Sections at 1000K Using k o  113 =  115  27 ECIOS Cross Sections at 1000K  118  28 Exact and IOS Cross Sections at 300K for C v, = 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  2  Impact parameter b  10 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 Vancouver, 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 fascinating 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 approximation 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]. Approximation 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 interaction potential. The first type of operator, responsible for directional coupling, is the interaction potential, which is generally diagonal in orientation representation 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 centrifugal 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 standard further simplification is to use the WKB approximation [3]. An alternate approach is to look for a potential that simplifies the radial motion. In particular, 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 orientation 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 degenerate. 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 succeeded 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 equivalent 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 neutronproton potential for deuteron-proton reactions to be well approximated using delta-shells [23]. Kok et. al. have found the delta-shell potential useful in calculating phase shifts for proton-proton and N —a scattering [24]. Other applications for the delta-shell have been in molecular physics [25] (studying hyperfine interactions) 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 deltashell. 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 Representation 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, where  =  ET  R)  (1)  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 =  where r,  Ri?  (2)  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 H 1 for the rotor is jr2  Hi _ 9  2I  (3)  R  FIGURE 1: Coordinates used for diatom-atom collision problem.  10  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 h 2 r_72  H2 = - - v R 2/..t  (5)  where it is the reduced mass. In the Ar-N2 system the reduced mass t is MAr MN2 -  (6)  MAr M N2 ) MA r  being the atomic mass of Ar, m N2 the molecular mass of N2.  The eigenvectors of  H2 are e ik ' R so that u  II  ik•R  2 e  —  h 2 k2 „ik•R  2/2  (7)  where e ik ' 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'.  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  H  a ,,, a  A'  2/2 R 2 aR aR n2R2  7;	2  R)  (10)  where A 2 is the operator for the square of the orbital angular momentum, which is, in orientation representation:  A2 _ 	sin  _ n2 [  1  a sin y ac,o a + —  yoap  1 '2 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:  >  YimMYAs(I) ,,,„,(R)  R) =  jniAs  12  (13)  where a is an abbreviation for the set of quantum numbers initial set of quantum numbers and  j, m,  and  s, a' the  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:  d lidR  ti 2 1 d 2	2 27 PR'  z 4-  A(A + 1) R2  3  I 71) 0 ( a /  =Ev a  "  aa  4,„„,(R)  (15)  where  k'? 3  .± 2 [E  	+1)01  (16)  21  h2  and =  I f  ri, n (i9YA*8 (i?)V(1., R)Yin no (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  where  W scatt  is the scattered part of  scatt •    (18)  The experimental observable to be  examined in this thesis is the cross section (which will be defined in Section 2.3). 13  In order to calculate the cross section it is necessary to determine the amplitude  fin, ;j i m i(i?) of the spherical wave in the final state jm. (18) is rewritten as kP eiki  .R Yj ni	+  e iki R  E Yjm(il f im;  (R)  j?n  R  R  oo.  (19)  We make a spherical harmonic expansion of the initial plane wave,  = E omeRw  As  As  (iT)YA-s (P)    (20)  where j A (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  x,„„,(R) where  (e R )(5.'  1  (22)  is the Kronecker symbol, to match the incoming plane wave and the is  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: —kjRei(kiR-A7/2)  ,  R  oo .  (23)  Using (23) in (22) yields R—Ar12) Xaa i ( R)  j (k i R)6 acv ,  1  2 1 kik' 14  ( 24)  so that comparison of (19) and (24) leads to the following relationship between and  Jae('  27ri  (25)  In conclusion it is noted that to attain the goal of a calculation of the atomdiatom collision cross section, it is necessary to solve the set of differential equations (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  n'Am where  J = A + Jr has eigenfunctions  EE(jmAs jAJ M)Y3 ,„(ilY),,(1)  (26)  (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  E ,A3M  R) 47r  iAriAmv,ib x;JA;i , A ,(R)  A' s'  X  (j i	M I j i m')'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 representation:  [ d 2	2 2u  dR 2  R  d  k.2  dR  A P■ R2  1 )]  VjA; j	V j".X"	 j  AjA ii,v(R) ;  • ;3',X'  ( R) (2 9)  where  VIA;3 , A  ,  = JI Yi*A1 M V, i?)V(r,R)Y3'Z, , I  (30)  For the potential of (9) we are interested in determining an angular momentum 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 representation. 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- i dk  -  (31) Using 4.6.6 of Edmonds[27], ie.,  47r PL(cos 0) = 2L 1  E  (32)  YLAM YL'A  (9) can be written as  mv(R)Ip ; fr)  E v (R), L + 47  L  4,(0y;,;,(t)so  — fi). (33)  Lti  Using the following defined phase conventions [28]  771As  1?) = (-i)'Y *,„(o(-i)AYas(R) 3  (34)  and substituting (33) into (31), one obtains the following equation:  (jmAsIV j'nt i	=  (—i)jyr,„(1')(—i)AYA*.s(f)  > vi(R) 2L47r+ 1 YLA(')Yi„(f)i  ji Yi'.1(0(i) Al YA's'( 11 )th'dk  (35)  Lp  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)  17  (36)  and  = (2j + 1)(2L + 1)(2j' + 1) ( 47r  j L j'L j' ( , (37) —rn m ) 0 0 0  (35) can be written as  (2j + 1)(2j' + 1)(2A + 1)(2A' + 1) j' 	 ( A L A '  (jmAsIV (R)1j'm' s') = 	L  x  O o o)  j L j') (A -771 it 711 /	-8  ) o o )  LA'	 ). -  (38 )  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 Vmatrix expression is converted to the coupled expression through the following transformation:  (jAJMIV(R)Ij i )'J'M') =  E (j AJM  msm"3"  I jrrr,\.․ )(j1n\sIV (R)Ij"m" A" s")(j"rn"A"s"Ij'A'J'M').  (39)  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:  AJ M jmAs) = (  77/  18  A J , V 2J + 1 s—M  (40)  and using (38) for (jinAsIV(R)I j'n'A's'), (39) can be written  (jA JM1V (R) /AVM') =  E  A  (—)3 A+m,\/2J + 	 1 (  J  7n s —M )  --  711S711'  xi 3i + A1-3-A \I(2j + 1)(2j' + 1)(2A + 1)(2A' + 1) x  L j' ( A L 000  E  oo o)  Lg  x(—)-A-m-s(—)ji—v+miV2J' +	 1 (	 m'  —	 771  A —s  m'  L A' s'  Ji )  (41)  —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' in .s ' —M'  j' A' J' —m ' —s ' )  =  ,  (42)  and  j  L  iu  j'  ))  )   (43)  —m m' L  —  allows (41) to be written as  (jA JM1V (R) x  = \I(2j + 1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1)  L j' E ij +A -3-AvL(R) 0(j 0 0 i  L  x  ) (A L A' 0 0 0    ` ( _ )i-i-, -Fx+m+m ( ,  ,  in,,s, x E(—)m—s+A-Fj+A-EL ( inns  i' —m'  A' J' ) —s  M'  L ) ( s —s —it  A'  A  -77t  J s —M  A  i 771  ,  (44) Further simplification in (44) is possible by considering equation 6.2.8 from 19  Edmonds [27], where  (4.  E  1 iL2  ,  i+t, 2+A3+i1+12+13  113  11  X	 ( —  j2  13 ) 11 1 111 2 ,u3  /3 31 l2 1711 P2 — /13 1 /1  12  j3  3 _ 2	 _ 3  =	 { 31  /Li — ,u2 7n 3	ti  12 13  31 (  j2  mi m2 m3 (45)  where j l , j 2 , j 3 , 177 1 , 771 2 , 717 3 , / 1 , / 2 and 1 3 are the arbitrary indices used in Edmonds [27] and j1 j2  (46)  / 1	/ 2	l3  is the 6-j symbol defined by Edmonds [27]. (44) can be written  (jAJM V(R) j'A'I'M') x  E  = V(2j  1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1)  VL(R) j  L  x E(_)i+i,+x+m+m, A'  j  A0/  A,  j'  A L  —m' —s '  J j' J' ( ) s' m' —M ) • (47)  Using (42) and equation 3.7.8 from Edmonds [27], ie. , (  m i M2  31  177 1  32  )  31  j2  m 2 171 3	mi m2 m3  = (2:73+1 r i b m ,„ i 3t	 3333(531 •	 2 3 (48)  where b i1 h 73 = 1 if j 1 , j 2 and j 3 satisfy the triangular conditions and 6171j2j3 = 0 otherwise, (47) is further reduced to  20  (jAJMIV(R)Ij'AVM') = 1/(2j + 1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1)  x E i3  L ( A L A' 0 0 0 )0) 0 )  17/, (R)  x(—) 3 + 31 + i + m + mi (2J 1) -1 8j ,p6mm ,  IijALI  (49)  A'j  Finally, using the symmetry of the 6-j symbols as given by equation 6.2.4 of Edmonds [27],  .12 :13  l  = 1 .13  j4 j s j6 j  J2  (50)  11 j6 ,14 j5  the following expression is obtained:  jAJMIV(R) j 1 J i M i ) =  x E vL(R)(-Y+3+3'  02j + 1)(2j' + 1)(2A + 1)(2A' + 1) j L j' ( A L A' f J A' j' L j A 0 0)0 0  .  (51) 1)  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 and final state. For this reason  J  need not appear in  J  is the same in the initial  both initial and final states  but can be denoted as a superscript on the operator V, ie., 1/ J . Hence the V-matrix element  (jAIV J Ij')').  (j)JMIV(R)Ij'A'J'M')  may be written as  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.  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' +  2\c' R  3A u' A "  R  oo.  (52)  2.3 Cross Sections in the Total-J Representation 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 =  (53)  IL  Putting (22) in (21) and then into (13), and substituting  k for R results in  the following expression for the spherical flux F outgoi ng for the outgoing spherical wave e ikR/R [29] :  t k' Foutgoing =  2  (471-)  l    kj kJ , As A' .9'  YA.5(k)TamAs ;  22  j / nt i  s i YA 1 si( ki )  Ai  (54)    	„  From (53) and (54) the differential cross section o i„,, i ,„,,i(k) is: -  a- .	 m4---2  . (k)  4.7r 2  2  E  Ks(k)TjtnAs ; jim'AisiYA1 3 /(k i )  (55 )  In this work only the total degeneracy averaged cross section,  ie., the  1 111 1  differential cross section integrated over all angles and summed over all m's, TDA =-  (56)  0.3„3,7n,(k)dk  2j' + 1  m m'  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 yAs ki)  2A + 1  y\s (z)  (  (57)  oso  47r  the expression for o - T D 1- becomes: 47.2  TDA —  ko(2j + 1)  EE  ,  x  E „  s „ ,\” ,  2A' + 1  TimAs;?rnivo(E)YAs(ic)  47r  AsA'  YA„.,„(k)77„,,,„,„ ;i ,„„ A „, o (E)  2A"' + 1 47r  dk  (58)  Using the orthonormality of the spherical harmonics,  I }I ( k Y*,, ( k ) k s  )  A  „  d  (59)  the expression for the total cross section now is TDA  1"  = k 	TjmAs , 2j + 1) Asv„,„,' 23    ,  ; jimiA 1 0( 	2.)■• .  + 1)  (60)  Finally, then, we seek the appropriate form for o-T W; in the total-J representation. 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  EE  71  TDA  —  +  770)11  A  x  J representation is:  4_ 1)  AsAisi  JMJIMI 2  J ) —M ) ( in' s' —M'  E  (2] +1) 2 (jAT J  JMJ'M' j  X  A  nimissi  x JIIMI/j/IIMIll  J ( j' A' J' —M ) s' —M' ) J" A J" ) ( j' A' s / Mil/ s —M" ) in'  (62)  where the notation = has been introduced since  (JMjAIT(E)1J'MT)') Smm,  (63)  J = J' and M = M' by conservation of angular  momentum. Using 3.7.8 from Edmonds[27], ie., A SS  77/  J ( j A s —M ) 24  J' —M'  1 = 2J +18"'SmAli  (64)  the following expression for the degeneracy averaged cross section for the j 4-- j' transition in the total-J representation is obtained:  a  TDA  V2(2j' + 1)  	E k (2j/ + 1) 12  E Em (JA rimliiy)1 A  j  2  E(2J + 1)1(j A1T J j' A')  2  AA , j  (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 equations 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 approximation 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 k o2 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  d  2	9  t.	 2 A(A  idR2  	2/1  R2  1)  Kalxc',(R)) = E(a1V(R)  la")(a"IX'(R))* (66)  Replacing  vi with q, leaves (66) in the form  h 2 [ d 2	2	 d  AO 4- 1) R2  2  2µ [c/R 2	dR  (a x.,,(R)) = E(a1V(R)Ia"\Ka" IX.i(R)). (67)  E im (djm) gives  Operating on the left by  ddR ko2 	E( 3m d ini) [ddR22  .	 \ 2 dm) -7 1  A(R2  1)]  E(a V(R) co  (Asjmx a ,(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  A( 	( Aslx,vs, )	R2  1.2  [dR 2	-fidR =  n'"  R))  E 081 V(r, R) A"s")(A"slxv,,,(f-, R)). 26  (71)  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:  n 2 [ d2  2 d  	dR 2 ridR  A (A + 1 k, .j2	° ° 1 '](alx„,() R) = Vall)( a"a"lx„,( R. )) R2  (72) Operating on the left by EA,  As) and transforming into the orientation repre-  sentation by carrying out operations analogous to those discussed above for the ES approximation, we get [ d2 {dR 2  2  d dR  1.2  A.(A. +  n'j  . (R))  R2 2 — h2  E (im V(R)lj  u ni")(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 A o . 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:  	d 	2  o(A.  d  I /R 2	RdR n'°2 2  + 1)  R)  R2  2,a  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 representation. The only rotationally invariant quantity that depends on the different possible orientations is the scalar product of the rotor orientation 7 and atomdiatom orientation  h.  Thus T is a function of the angle 0 between r and 28  R.    	0  	(  This can be expanded in Legendre polynomials: v0k0(7•,  E '/2L + 1 Ti;°  1?)  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:  C  crini.iim ,	= -42L m k ;  (76)  kl) 1 2 .  T  For the IOS cross sections cr.Tis_ j , n ,,(k) we have: .19 S  47x2  3 m  —  111  k4.7  1  47r 2 k.7  ?  1  I  Om  II E  ;  kV , 17)T ° °  1?j'rn'; k')ci•df  A k  2  YA3(00 4-3 's1(k)YiniMYA*3(R) 2  (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  0 sok' , ji j) = (2j +  2  a/  EE  YA.,(00'4-3A  mm' AsAisI  xV(2L • + 1)(2A + 1)(2)x' + 1)TE° k°(—) 7 "' +8 Ye s ,(P) X  Equation  (  iLn(L j' (j L 0 0 0 )  a  0 0 0 )  —ill —Cr 77/  2  L  \ s cr s  (78)  (78) may also be written as  cr ic' s (hk'd  k, j) = (2j + 1)E 29    L  0 0 0  2 cr (Me , 0  4  —  k, L)  (79)  where  k, L) =  cr(hk' , 0  47r2  E  01-A  02A + 1)(2A' + 1)  AsAisi 2  x YAs (k)TE°"(—) 3  (AL A') (AL A Y ,(P) . 0 0 ) a s')	 s  If the initial momentum direction  (80)  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)  1  0  X 71 A°k°	— ) s ,  To get the  (  L )2 0 0 )  A L 0 0  total cross section  + 1)(2A' + 1) AA'  ) (	 A L 0 ) —a a (81)  2  )  (81)  0 )  is integrated over all angles to give  the following  aL4—o  27 fir  =  = =  o-(hk' , 0 k, L) sin OdOdO o o 7r „ L A' ) 2_, 2_,(2A + 1)(2\' + ) 71\°"iA'-A A 0 0 0 ) as  A L A' ) 0  ir L 2 + 1) nokoiA 1 --A 1 2 A kit E(2A + 1)(2A' 0 0 0 ) I   If the initial  A  paramaterization for the 105 is assumed, ie.,  (82) A = A', (82)  simplifies to  01—o=k'2 — 7r E(2L + 1) 30  ,  12  IT17°1  .  2  (83)  For total cross section am',  ,  (79) is integrated over all angles to give the  following IOS scaling law: 	,  j. ros = (2j + 1) E  	L  L  j    ooo    2  0./0s  (84)  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  T.  (Recall the IOS assumes a sudden collision, ie,  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: + 	6 TE Cros( os t —  3j 1 k-Lik i)27-2/(202 — € 6 +  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: 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: „. ECIOS  T  2  24h	 2 24h2  (e  6)272  °Jos.,.  (86)  34- 3  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  s.	i.r o.s;, ( Ek	ei ,) =  0 and j  E  (j  j' S-matrices are related by [32]  j' L  o 0 0  39  s ios ( E	) L4-0 k  L •  (87)  Using the factor in (85) to relate S p—Ji -?'	 ° s wit4.act with S1.--.7	 , 	 and to relate Sws L4—o with SYxatjt and then using (87) allows for the L 4-- 0 and j 4-- j' exact S-matrices to be related by c Gsm  _re , j,)  r 24h L  + (EL — co)272 j ( f_	 02T2 0 0 0 24h2 3 3 2  E  t  Srxac (Ek+EL).  (88)  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  (Ek -Fc :p) (2j+1)  r  24h2+ (E L — 24h 2  + ( 6j , —  fo)2T2 2  j  Ej )2 7-2  0  j' L 0 0  )2  Exact  °-L4---0 (Ek+CL)• (89)  The above scaling law will be used with  of,"6' values to calculate o-2 ,_i s values  and these will be compared with the actual cr 1E xa; t values to determine whether 3 4-  the above scaling law is valid. As well, the scaling law shall be used with o-P so 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 corresponds to the criterion of determining the outermost turning point. • the transition probability is inversely proportional to the number of accessible 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  criA,si,(E) = where  f  f  is a function of — j1 and  the maximum allowed for a given energy  j  value, and  o-).7‘4_s i ,(E) to be written as — § 1, E —    (90)  is the energy of the rotor for E —EJmax ; • tjmax  N(E)  is the number of available rotor states  E.  The assumed functional properties of the  f  function implies a relationship  between cross sections which have the same If —j1 and  [2j + 1][2(j' — j) + 1] N(Ek Ea —j) N(Ek	 cji) [2:1' + 1] ,  E — €3 ,..  values, namely  > 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 + 6 3 , 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 of  total  exact crj ,,,  energies, ie, cr.P ) ,(Ek + c. 3 ,) with ao.._ 3 , _ 3 (Ek + c i ,_3). A set  and (3- 0 , 3 ,_ 3 values are related by a given  there is a question in the exact case as to which  total energy. Although  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 + e 3 ,) and  cri ° 1,_ i (Ek ci i_j) should actually be writ-  ten as c .f ° ,(Ek c,) and o- 4 ° ,_ 3 (Eq) where k o and  ko are those wave numbers  corresponding to the parameters chosen to replace the k3 value and  E k c,  is the  energy associated with the ko wave number. It is this aspect of the IOS approximation - 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 h2  1  + v(R)  a R2  H — 2p IR2 aR  OR	 — R2 ]  	(93)  where A 2 is given by (11). The SchrOdinger equation now takes the form  h2  1	 a  pt2  a  2/1 [R 2 OR A(' OR  2p R 2 + h2  A2  36  ^(R) V(R)^(R) R)	 = (R)  (94)  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)  = 47ri A xA(R)YA*.,(P)  (96)  as  where 	7PAs(R)  and using equation (92) for V (R) and introducing the dimensionless variable  x = kR where k 2 = 21LE Ih 2 we now can write (94) as  1  AO + 1)	VO  2 h2 0  	r 2  02 xOx ax2  +1iXA =  x2  k  8(x — ka)XA  (97)  where the following property of the delta function [34] has been used:  8(R — a) = kS(x — ka) For x  (98)  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 functions. 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,  (100)  0  and integrating (97) over x from ka - to ka+ and requiring x), to be continuous at ka gives  axA ax  ka+    2xA   ka  —  ka  ka+ ka  ,  2  k vsXA•  -  (101)  Since x A is continuous,  a OX  x),(ka + ) — lxA(ka - ) =  OX  2/1 h2k  V5 )00  (102)  Using (99) in (102) gives 2p Mka+)— 7TAhVka + )— BAA(ka 	 ) = — lis sjA(ka)BA 2 h,2k  (103)  where  A(x) = ci clx :7,\(x)  h A (x).  (104)  xA(ka + ) = x),(ka - )  (105)  jA(ka)— .-T),h,\(ka) 1 = BA:7),(ka).  (106)  h a (x)  (x  Since x ), is continuous at ka,  and so  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 functions [35]: W = A(x)h A (x)— j A (x)//,',\ (x)  ix1 2	(107)  an expression for BA is obtained: BA =  1  (108)  1 — i0jAhA  where j A is shorthand for j A (ka), h A for h A (ka),  = ka  (109)  and  2iiVa	 a h2   g  (110)  Finally, multiplying (103) by jA, (106) by j'A , and using (107) and (108) gives the following expression for TA:  Ta =  1 — igjAhA  (111)  .  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  XA	2i1ka [e  (112)  Using (60) and assuming j' = 0 (ie, no rotor states) the total cross section is .total =  -;75- A  (2A + 1) ITAl 2 • 39    (113)  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 scattering 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 potential 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 b 2 P2 (cos 0)] (R — a) (114) where b 2 is a constant. Explicitly, the second order Legendre is 0  ,  1 /0	2  P2 (cos 11) = —  2  40  0  COS Of —  11))  (115)  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	2  d 2/1 tdR 2	fidR  k2  h 212 R 21_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  k jj = kj = () [E  1)1 h 2 	c 2  (117)  21  and  ALI = A(A  1)h 2 6 ), ),183j ,  (118)  The form of (116) corresponding to matrix element (jAlx -i lj'A') is then  2 A(h2R+21)] ( iA l x j (R) I jA ) d 2 /2 [ dR 2 TtdR k3 h 2 d 2	2  E  (jA V J 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  Ki A =  jA(kiR)—  =  (jA Blj'A)j),(kiR),  A/)k;71 '  R  >a  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  j), (kiR) Sij 5AA  (jA 1j(ki R)Ij')')  1  (jA li(kj R)Ij'A') = 11,A(kiR)8,w8AA  I   ,  (121)  (120) can be written in matrix notation as 1 X= j — 5-hk2Tk  = jB,  -  2,  R> a R < a  42  (122)    j represents the prepared incoming state;  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  	2  4 0  (123)  -  The requirement of continuity of x, ie.,  x (ki a+ ) = x (k3 a ),    (124)  -  is used to get 	dy  	y  represents the scattered  d X(kj a + ) - dy — x(k3 a - ) =  2  Vx    (125)  where kR.  =  (126)  Again the requirement of continuity for the wave function x at R = a implies 1 jB = j- - hk 7 Tk - 2,  R = a  2  (127)  Inserting the expression for x given by (122) into (125) gives dj B + 2,a k _ ilf  I_  hkiTk - f)  dy  dh k  _dj  (128)  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 — h2 jV -  1  i  1 	 dj = n 2 dy —  43  .dh) ,  j  —  dy  K7 K  -  1	(129)  2    	T  Using the matrix version of the Wronskian given by (107), ie., dj .dh 1 W = —h — j — . dy dy ( i kR) 2  (130)  allows (129) to be written as 42.ift  kiV  2	-  --1 hkiTk — i) = k 2 Tk -4  2  (131)  or, rearranging, 4ipa 2  h2  [2iya2  = 	hkNj 2 ViVhk2 + 1] Tk — i  (132)  so that finally an expression for T is obtained:  =  [1  +  2i ita 2	4ipa2 1.  k NA h2 kIjVhk2 h2  J  (133)  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  L j'  is expanded in 3-j symbols ( 0 0 0  ( A  L A'  0 0 0  ) 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:  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 offdiagonality that is allowed, that is, the difference allowed between  between ). and V. The V matrix, because of the (  L  0 0 0  j and j' and A L A' 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 maximum value, A max -I-  J ranging from 0 to its  max •  Though one wave number and an initial  j' value are chosen, (and a total  energy value E total is obtained from these values) other matrix elements having the same  Et o t ai  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 10 m. -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 10 10 m. -1 , etc., all the way to j' = 10 and k = 8.37 x 10 10 m. -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 T—  2i  J  (135)  J — 2,ua 2 1 . . i  k7jVjk 2  (136)  2pa 2 1. 1 kTivnk2 h2  (137)  1 — N + iJ  where  h2  N _7..  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 [  = [1  i  N—	 1  N 2i	 1 [J]]  AI { N —1 [J]] '  = [1 — iK] -1 [-2iK]  2i  (138)  where K  1 N — 1 iii  47  (139)    and K is always real. The following expression is obtained upon further expanding (139): 1 1 — iK [1 + iK] — ' [1 + iK] [-2iK]  T =  1  —  	[ 2  1 + K2  1  =  iK + 2K 2 ] 	2i  	i2K21  1 + K 2 I  -I  1 + K2  [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 1 2,a0 1 W - —kTVICT - h2  (141)  allows K to be expanded as follows:  K  1  1 jWn — 1 [Pi] 1 {jWn] [ 1 ] (j/n) [nWn — (n/j)] n N — 1 Pi  = =  	=    .  1 nWn — (n/j) [j  [iifti  n  — 1 + 1] [i] n  1 i+ 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.  48  Problems, however, arose with this procedure. Neumann functions for sufficiently small values of ka at large A approach oo as given by [35], ie., n,\  (2A — 1)!!  (143)  (ka)A+1  Using Stirling's approximation for large N N! = (  N) N  (144)  —  and  (2A — 1)! (2A 2A-1(A — _ 1)! 1)!!(145) 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  ka --> 0.  kakae)  (146)  Treating the limiting behaviour of the Bessel functions in an analogous manner as ka ---> 0 gives the limit (ka)'	e  kae)  A  > 0.  (147)  , ka —> 0.  (148)  ka  (2A + 1)!! — 2A 2A )  —  Hence the (n/j) term in (142) is of the order of 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 J74(34) — 49  149  3 x 103° .  (149)    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 overflow [37] on the Amdahl. Thus another form for the K matrix was needed., Equation (142) can be rewritten in the form  	1  K  n n  +  	[ 1  jWn — 1 1 ni .  1 j [W — (1/jn)] n [ni l 	111 1 1 1 	1 Ltd W — (1/jn) Lni  (  150)  In this rewritten form, the magnitude of the numbers were much more manageable. The (1/jn) term was, for the parameters previously mentioned, of the order  of  The W term  1  1  2)ka  2(74)34  ..1AnA  e  2.303  10 3 ,  ka  0.  (151)  ranged from 4 x 10 4 to 2 x 10 8 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  P o j 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:  	ipa 1 4 2 2itta2 T = [1  V iVhk [ 2 k 2 iVjk 2 Vol .  h  h  (152)  With the following definitions 2 pa2 	 1. H 	k7jVhk  h2  .  2  = JVVh  (153)  and  J (where  2//a 2  h2  k-1,jVjk 2 = JWJ  (154)  W is defined in (141)), the T-matrix can be written T=  (155)  [1 -Fl iF11 2ijP°•  For the calculation of the scattering cross section, only the open state part  T oo  of the T-matrix is required: T oo = P o T = P o	1 [  1 iH  51  2i0o•    (156)    The term P o [1/(1 + iH)] can be broken up into open, closed and open-closed coupling contributions Po[  1  1 4- 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 — P o(  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 = P iH ° [1 + iH o	° [1 -FliH o ] Po 1ill] [1 :  [1 i +1itio l poHipc [ 1 +1iFic PO ] [  x [1  i [1-1 —  P _1 	1 u 1 1  r  .  	 1  {i J 	1  	 1P 14 IP °"1. c [1 +1iFi c l  1[1 + &Li  	r  	1  	r  I P°I.	• i P c H+ 	 c	1 [1 + iH 0.1 13°H1Pc I.	• 1+ •zl-l iH j. 52  (162)  	=  So [ 1  [  1 	1 Fic H i p o i {1 + ihI c i  1 poHipc 1  [1 + iF10 .1  Pa [1 +iiH o l  Po 1 [  -I iH  	l =  iP°H1Pc (1 +1 H e  •  (1 6 3)  If the above is multiplied on the left by [1 + iH o ], the following is obtained: Po[  1 1+ iH  [1 + iHo  PoHiPc [  1 PcHiPoll 1 + iHc  1 x Po [1 — iPoHiPc ( 1 iFic )] •  (164)  The T-matrix can be written: TOO  = 2i [1 +	 iH o PoHiPc [1  iHi PcHiPol +1  x [P 0 -113 , — iP o HiP c [ 1 +laic ] PciPo]  (165)  Or  P O — [Po + iHo + PoH1Pc1 TOO  +1  o  x [P + i [H o — 2J0] PoHiPc	 [  PcH1P0  1  1 + iHc  I Pc (H1 — 2-1) Pc]  (166) where 	PoiWiPo•  53    (1 67)  Using H — 2J =  = =  2, n 2 k i iv  [h _ 2j] ki h 2 fin 2 k f iv [i n _ j] ki h2 2/./a 2 0 jvh( 2 )k i  (168)  h2  or since for open channels h (2) = h* 2"2  [H — 2J] P o =  h2  1cijVh*IciP o ,  the following expression for the T oo matrix is obtained:  1-1  1  T oo = P o — P 0 [1 + iHo + PoHiPc [1 + iHd  (169)  PciliP°  1  x [P o — i (J 0 — iN o ) — P 0 H1P c [ 1 + ill ] P C (J — iN) P o l (170) where N o F.-_,- P o jWnP o .  (171)  We define R o -P.. P o WP c h [  1 1 + &lc  liP c WP 0	(172)  where W has been defined in (141). Using (159) and (153) we can now write  54    PoHiPc [  1 1 + &lc  ] PcH1P0 = Po [PoFIPc -I- PcHPo]  [ii He  pc 1 +l  x P c [P 0 HP c + PcHPo] Po = PoHP4  1 +1iFicl  = P o jWhP c [  PcHPo  1 1 + iHcl  PcjWhP0  = jR 0 h = jRoj + ijR o n  = i [Ji +	 iNi.]  (173)  where ii  —  iiRoj  (174)  and  	N1  E.- —ijR o n. -  (175)  With the following definitions  H o E J o + iNo, 	.1-E.  Jo  +  ii    (176) (177)  and  	ITI  F N o + N1, -  the T oo matrix can be written  00 55  (178)  TOO  =  Po  = Po  P2 	tr i _ ij _ K ] 1 +LI — N Po —1 IV [1 — Ki ] [1+ [i/ (1 — IV .J] [1	I -  = Po  r  [1	i 1 —	 KI j 1 —  	 —I .__\ 1 [1	 1 —N - Ji ,	P o	i  (179)  1 + ii/ O. — NA J  Or  Po  Too = PO  1  (180)  - iR [1 + iR]  and, finally 2iP o  Too —  R  (181)  1 — iK  where  K  .L -  -  1  — J. N 	1  (182)  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  56  V(R))  then all  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 R2 A( + 1) 2/1 [R 2 dff RR2 where k 2 =  (2y1h 2 )E.  k2] 2P),(kR)  V(R)1PA(kR)  (183)  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  parameter A o (to +  operator  A 2 is replaced by the  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.:  d R2 d 2µ[R 2 dR dR h2 1  A o (A 0 + 1) R2  +  7PA,(koR, 0) = V(R, 0)71)A 0 (koR, 0) (184)  except that there is the extra constant parameter  OA° (koR, 0). (Here  11  (2p/h 2 )  0  classifying the solution  {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 solutions parameterized by A o , ko and  k while (184) gives  0.  To demonstrate the above, the potential given by (114) is inserted in the IOS  57  equation (ie., equation (74)) and the following expression is obtained: n 2 [ 1 d R2 d 21.i [ R 2 dR  Ao(A 0 + 1)  dR  R2  +  41 1  1)  Ao(koR, 0) =  V5 [1 + b2P2(cos 0)] S(R a)6 0 (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:  TA ok0 (0) =  —2iGg(0)jt(k o a) 1 — iGiAo (koa)hA o (koa)    (186)  where  = k o a  (187)  [1 — b 2 P2 (cos 0)] .  (188)  and  g(0) =  211 2  V5  As was discussed in a previous section on IOS cross sections, the quantity Mo k o must be calculated in order to obtain the o j ,_ j , quantities. By inversion  of (75),  n  o ko  -  is given by: itk o = V2L + 112 TAoko ( 6) ) PL (COS 0) sin Oa.  58  (189)  Thus to get  01j._ 3 /  cross sections,  appropriate values of A 0 , k o and is integrated over all  TAok„(0)  is calculated as given by (186) for  0. Next, using numerical methods this TA 0 k0 (0)  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 -26 kg. 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 k B is the Boltzmann constant.  4.1.2 Choice of Energy At temperature T the most probable energy for each of translational and rotational motion is k B T. 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 k B T while the wave number  2itk B T  60  (190)  has the value 14.27 x 10 -10 7n. -1 For this choice of rotor state (j =6) and translational energy k B T, 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  L 2 = A(A + 1)h 2	(191) which for large A can be approximated as 1 L r-:::', (A + -)h  61  (192)  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  v=  hk —.  Et  (194)  Equating (192) and (193) gives  ,abv = (A + )h.  (195)  Since the maximum range of the potential is a, this gives  ka =  I  (A + 2— ) A.  (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 deltashell 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 using 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  V (R, 0)  =  2.2 x 10' 2 	[1 + 0.5P2 (cos  R 12  1.2 x 10 -21 R6  0)] (197)  [1 + 0.13P2 (cos 0)]  expressed in SI units, specifically the inverse power potential corresponding to the positive term  V (R, 0) =  2.2 x 10' 2 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 WKB approximation ri Ak (cos  0) = k f  ,  1  (\.  1 	R — R2  0) were calculated using the  2,a	 V (R	r , 0) 2 1—— Rc2 dR h 2 k2  1  k(b — r c )r (199)  where  b is  the impact parameter (A + Wti,- (see (196) ) and r e is the largest  classical turning point. By definition, the phase shifts are related to the sAk(o) = exp   64  (cos  S matrix by  0)] .  (200)  Since the potential is even in cos  0, it follows that the S-matrix can be expanded  in even Legendre functions,  s Ak (0) =  E  V2L 1,9j/A,(cos 0).  (201)  L(even)  The expansion coefficients  SL R = V2L  Si\," are given by the inverse of (201), ie. 	 1 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)  where the T„'s are Chebyshev polynomials and  (203)  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 = 5O L — SL R 	(204) and these were then substituted into (83) to obtain the  a L ,_. 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  L value L=0 L=2 L=4 L=6 L=8 L=10  A2  ic = 14.2717  IOS with repulsive l' -12 115.15 25.32 12.09 3.16 0.50 0.06  66  A  ,  Amax  120  IOS with delta shell 113.75 23.30 6.41 1.98 0.68 0.25  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 probable 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 j j j j j  =0 =2 =4 =6 =8 =10  j'=0 421K 117.52 22.93 2.71 .32 .03 .00  j'=2 404K  j'=4 364K  j'=6 300K  4.78 126.24 9.83 1.06 0.09 0.00  0.35 6.06 130.25 9.21 0.84 0.03  0.03 0.55 7.73 143.19 8.13 0.36  j'=8 213K 0.00 0.05 0.76 8.75 159.03 3.86  j'=10 103K 0.00 0.00 0.05 0.66 6.45 197.47  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 elastic  j  j' goes up), the  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 reference [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 compared with a calculation done under similar conditions using the open state calculation 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 polynomial 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 23.56 3.59 0.43 0.04  j'=2 5.00 142.98 11.39 1.31 0.10  j'=4 0.49 7.39 145.24 8.60 0.61  j'=6 0.06 0.80 8.08 164.04 5.21  j'=8 0.01 0.09 0.84 7.76 202.36  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=0 j=2 j=4 j=6 j=8  j 1 =0 126.94 23.57 3.58 0.43 0.03  j'=2 5.00 142.97 11.40 1.31 0.08  j'=4 0.49 7.39 145.31 8.61 0.49  j'=6 0.06 0.80 8.09 166.11 4.31  j'=8 0.01 0.07 0.69 6.41 208.20  From the above table it is concluded that closed states only affect the calculations 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 A o and k 0 . 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 k o 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 k o = 14.27 x 10 -10m.-1 The unitarity of the S-matrix summing from L=0 to L=30, ie., L=30  E  2  1.00000  L=0  was verified to 5 significant figures. The following results were obtained: 71  (206)  Table 4: IOS o L , 0 Cross Sections -  All cross sections in A Final State  Final k in A -1  Trans. Energy  (T .E.)  L L =0 L =2 L =4 L =6 L =8 L =10 L =12 L =14 L =16 L =18  16.92 16.57 15.71 14.27 12.03 8.37  421.5K 404.2K 363.7K 300.0K 213.2K 103.2K  2  Using  Using  ko = kiiiit (16.92A ') T.E.=421.5K 101.90 23.58 5.85 1.64 0.52 0.18 0.07 0.03 0.01 0.01  ko = kfinal  101.90 23.09 5.50 1.98 0.80 0.38  Using ko = 14.27 A -1 T.E.=300K 113.75 23.30 6.41 1.98 0.68 0.25 0.10 0.04 0.02 0.01  Exact Results, k = 16.92 A -1 T.E.=421.5K 117.52 22.93 2.71 0.32 0.03 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 Hence the collision is less sudden at high  L  =0.  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' are dependent on these  0 1 , 4- 0  since all other cross sections  values by (84).  One other effect arises because the IOS treats all rotational states as degenerate. 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 k 0 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 k 0 =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 4.2.4.1 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 k o choices are examined in light of the scaling relations . For each of the three choices of 4, k o  =  ko = kfinai and 14 ) =-14.92 A -1 ,  the IOS o- 3 , j , cross sections were calculated using the 0 L __ 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 k o = 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=12 j=14  j'=0 101.90 23.58 5.85 1.64 0.52 0.18 0.07 0.03  j'=2 4.62 111.03 13.86 3.17 0.89 0.29 0.10 0.04  j'=4 0.61 7.48 113.22 11.65 2.47 0.64 0.19 0.06  j'=6 0.15 1.40 8.54 120.97 11.60 2.70 0.78 0.26  j'=8 0.05 0.37 1.63 8.43 132.19 10.65 2.64 0.81  j'=10 0.02 0.13 0.46 1.70 7.41 155.32 8.96 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 1.03 2.15 5.21 20.4 297  j'=2 0.97 0.88 1.41 2.99 9.36 98.3  j'=4 1.75 1.23 0.87 1.27 2.93 20.6  j'=6 4.48 2.55 1.11 0.85 1.43 7.40  j'=8 15.9 6.92 2.14 0.96 0.83 2.76  j'=10 154 46.7 9.80 2.59 1.15 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 a L , 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 j  L 0 results should be compared with the  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 k o = 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'=0 101.90 23.09 5.50 1.98 0.80 0.38  j'=2 4.72 111.03 13.46 3.64 1.24 0.53  j'=4 0.65 7.70 1 13.22 12.33 3.07 1.07  j'=6 0.13 1.22 8.06 120.97 11.02 2.75  j'=8 0.03 0.26 1.31 8.87 132.19 9.16  j'=10 0.01 0.07 0.27 1.67 8.62 155.32  j'=12 0.00 0.02 0.07 0.40 1.79 7.53  j'=14 0.00 0.01 0.02 0.12 0.47 1.81  Part B: Ratio of IOS to exact cross sections. j=0 j=2 j=4 j=6 j=8 j=10  j 1 =0 0.87 1.01 2.03 6.29 31.4 627  j'=2 0.99 0.88 1.37 3.44 13.1 183  j'=4 1.86 1.27 0.87 1.34 3.65 34.5  j'=6 3.71 2.22 1.04 0.85 1.36 7.53  j'=8 10.3 4.94 1.72 1.01 0.83 2.37  j'=10 72.7 25.1 5.86 2.55 1.34 0.79  Compared with the k 0 = knn t,1„1 choice the k o = kfi nal 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 k mitial 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 E F F I I I  j'=2 F E F I I I  j'=4 I I E I I I  j'=6 F F F E F I  j'=8 F F F F E F  j'=10 F F F F I E  Rather than analyzing the above trend in terms of initial or final state parameter 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 k o = kmax for large IAA transitions and ko = kmi „ generally being the best choice for transitions where  Ion I=2.  78  Table 8: Comparison of k max and Sections  IOS Cross  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 E Min Min Max Max Max  j'=2 Max E Min Max Max Max  j'=4 Min Min E Max Max Max  j'=6 Max Max Max E Min Max  j'=8 Max Max Max Max E Min  j'=10 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 transition corresponds to a k max to a sections the  k1111,  km; „ to k max transition while an upward k11m  transition. Now for the  a L„._ 0 cross -  value is always the final k o choice and the k max value is always  the initial k o choice. Regarding Table 4 it is noted that for o•2,0 and a4.._0 the kfi„ al or k 11111 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 a L , o values will give more accurate aj, j , values, then downward transitions (ie. going from a low k o 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 k o to a low k o ) 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 23.30 6.41 1.98 0.68 0.25 0.10 0.04  j'=2 4.66 122.24 14.27 3.64 1.11 0.38 0.14 0.06  j'=4  j'=6  j'=8  j'=10  0.71 7.93 121.25 12.33 2.97 0.88 0.30 0.11  0.15 1.40 8.54 120.97 11.60 2.70 0.78 0.26  0.04 0.33 1.57 8.87 120.87 11.22 2.55 0.73  0.01 0.09 0.38 1.67 9.08 120.82 10.98 2.47  j'=12 0.00 0.03 0.11 0.40 1.74 9.23 120.80 10.83  j'=14 0.00 0.01 0.03 0.12 0.43 1.79 9.33 120.78  Part B: Ratio of 105 to exact cross sections j=0  1=2 j=4 j=6 j=8 j=10  :1=0 0.97 1.02 2.36 6.29 26.8 419  j'=2 0.97 0.97 1.45 3.44 11.7 131  j'=4 2.04 1.31 0.93 1.34 3.53 28.3  j'=6 4.48 2.55 1.11 0.85 1.43 7.40  j'=8 15.6 6.20 2.07 1.01 0.76 2.91  f=10 103 33.4 8.02 2.55 1.41 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 k o = 14.27 A -1 rather than kinitiai or kfinai. In the inelastic transitions, however, the kinitial and kfilim choices are better. A further  81  investigation could study a  kaverage  value where  kavera g e  initial  kfinal  (207)  2  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.  4.2.4.2 Using Exact Cross Sections There are two parts to the prediction of by means of the scaling law in (84) — the values of the 0 L , 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 a L , 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 22.93 2.71 0.32 0.03 0.00 0.00 0.00  j'=2 4.59 124.85 12.60 1.32 0.14 0.01 0.00 0.00  j'=4 0.30 7.00 123.97 11.02 1.10 0.12 0.01 0.00  j'=6 0.02 0.51 7.63 123.80 10.43 1.01 0.10 0.01  j'=8 0.00 0.04 0.58 7.97 123.75 10.11 0.96 0.10  j'=10 0.00 0.00 0.05 0.62 8.18 123.72 9.91 0.92  j'=12 0.00 0.00 0.00 0.05 0.65 8.32 123.70 9.77  j'=14 0.00 0.00 0.00 0.00 0.06 0.67 8.43 123.70  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 1.00 1.00 1.00 1.00 1.00  j'=2 0.96 0.99 1.28 1.25 1.51 3.67  j'=4 0.86 1.15 0.95 1.20 1.30 3.71  j'=6 0.71 0.93 0.99 0.87 1.28 2.76  j'=8 0.51 0.79 0.76 0.91 0.78 2.62  j'=10 0.25 0.94 1.05 0.95 1.27 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 k o . One implication of this approximation can be seen in the detailed balance equation:  [j] k 2  Exact Ell)  where  [A = 2j + 1 and kj is the k  quantum number equal to  j.    =  1.)  (208)  (j)  0-3E'ix,__ va3c't(-Elk  ji  value corresponding to the state with rotor  For the IOS case, with  lq = (208)  becomes:  (E).  (7.1)(234E) =  (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 transition) the  j' > j  (ie., a downward  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 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 accurate 105 downward transition.  84  a 3 1,_i  k3 /kl ,  > 1 term and give an  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 i ai =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: Ratio to  Ratio to  Transition  k=14.92A -1  exact results  k=16.92A -1  exact results  04— 6 24— 6  0.03  0.02  0.71  0.67  0.96 1.23  0.51  0.93  44— 6  7.96  1.03  7.63  0.99  64— 6  133.54  0.93  123.80  84— 6  10.87  1.34  10.43  0.87 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 cr L,_ 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 23.58 5.85 1.64 0.52 0.18 0.07 0.03  j'=2 4.72 110.30 14.16 3.25 0.90 0.28 0.10 0.04  j'=4 0.65 7.87 109.30 12.26 2.65 0.71 0.22 0.08  j'=6 0.13 1.25 8.48 109.04 11.54 2.41 0.63 0.19  j'=8 0.03 0.26 1.40 8.82 108.94 11.16 2.28 0.59  j'=10 0.01 0.06 0.30 1.49 9.04 108.90 10.94 2.20  j'=12 0.00 0.02 0.08 0.33 1.55 9.19 108.88 10.78  j'=14 0.00 0.01 0.02 0.09 0.35 1.60 9.29 108.87  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 1.03 2.15 5.21 20.4 297  j'=2 0.99 0.87 1.44 3.06 9.49 97.1  j'=4 1.86 1.30 0.84 1.33 3.14 22.9  j'=6 3.71 2.27 1.10 0.76 1.42 6.61  j'=8 10.3 5.01 1.84 1.01 0.69 2.89  j'=10 72.7 24.8 6.49 2.27 1.40 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 a 1,,.. 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 attributed 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 coupling 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 — b 2 db  7a —  v a fo 2v  (210)  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=10  j'=0  j'=2  j'=4  j'=6  j'=8  j 1 =10  101.90 12.14 0.17 0.00 0.00 0.00  2.38 111.03 1.21 0.01 0.00 0.00  0.02 0.65 113.22 0.18 0.00 0.00  0.00 0.00 0.13 120.97 0.03 0.00  0.00 0.00 0.00 0.02 132.19 0.00  0.00 0.00 0.00 0.00 0.00 155.32  Part B: Ratio of ECIOS to exact cross sections. j'=0 j=0 j=2 j=4 j=6 j=8 j=10  0.87 0.53 0.06 0.01 0.00 0.00  j'=2 0.50 0.88 0.12 0.01 0.00 0.00  j'=4 0.05 0.11 0.87 0.02 0.00 0.00  f=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 T  .a  . 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 (Ek ma .) 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 .i= 8 j=10 f value  j'=0 101.90 22.91 4.22 0.43 0.02 0.00 0.300  j'=2  j'=4  j'=6  j'=8  j'=10  111.03 12.60 1.64 0.11 0.00 0.225  113.22 10.01 0.94 0.03 0.165  120.97 8.63 0.37 0.145  132.19 4.57 0.145  155.32 0.79  Part B: Ratio of ECIOS to exact cross sections. j=0 j=2 j=4 j=6 j=8 j=10  0 0.87 1.00 1.55 1.37 0.84 0.75  2  4  6  8  10  0.88 1.28 1.54 1.15 0.89  0.87 1.09 1.12 0.89  0.85 1.06 1.03  0.83 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  Ratio to  in A 2  Exact Results  04-10  0.00  3.8  24-10  0.00  1.4  44-10  0.02  0.46  6+-10  0.28  84-10  0.18 2.75  104-10  155.32  99  0.43 0.79  The resulting  cr -1()  ,(Ek,„,„) cross sections with scaling could not be fitted  to the exact results as closely as were the  or 3'',(E km ,„) -  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 Table 14 -  -  7  -  Values Used In  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 initial state, j the final state. j'=0 j=0 j=2 j=4 j=6 j=8 j=10 Factor  If the  T  16.92 16.92 258.3 16.92 272.3 16.92 299.8 16.92 355.6 16.92 511.0 0.300  j'=2  j'=4  j'=6  j'=8  j'=10  16.57 16.57 204.2 16.57 224.8 16.57 266.7 16.57 383.3 0.225  15.71 15.71 164.9 15.71 195.6 15.71 281.1 0.165  14.27 14.27 171.9 14.27 247.0 0.145  12.03 12.03 247.0 0.145  8.37  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 94  f value than the  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 velocity for the 104-0 transition (lAj1=10) would give a  T  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  f Value  Calculate 0L, 0 in A -1  Giving Best Results  16.92  0.170  15.71  0.165  8.37  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 96  only a collision time may  be too narrow a definition of  T.  —  Since E ., —  =(J  is also dependent on lAjl via h2  —  (212)  i i )(3 + 3 + 1 ) 27  it could well be that instead of interpreting  (E 3 — 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 4.2.6.1 Using IOS  Cross Sections  0L- 0  In this section the General S-Matrix Scaling Relation (GSMSR, Equation (89)) is studied with regards to how well it can correct the IOS That is, even though exact  o-j,  0 L -0  cross sections.  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(Ek 0 ), there is some  arbitrariness as to the choice of E ko since in the 105 all rotational states are degenerate. The choice is made to interpret the [L]o PanE k + -  EL)  term as  airl(Ek 0 ). That is, the choice of the k'd value which will replace operator k 2 will correspond to the kinetic energy Ek with the substitution Equation (89) becomes 	i(Ek+Cp)  	L  =(  E 2j+1)   2  EL  term ignored. With this  r6 [ (  j j' L +RE', EL — COT /2hi 2 ] — cj)7/2h1 2 ° 0 0 ) [6 +  201,,,o(Ek). IOS  (  (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.  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 f Value 1 The  j' =0  j' =2  j' =4  j' =6  j' =8  j' =10  101.90 23.58 5.85 1.64 0.52 0.18 1  4.62 111.55 13.53 2.73 0.72 0.23 0.25  0.61 7.25 115.11 9.98 1.44 0.30 0.30  0.15 1.27 7.92 122.88 9.64 1.48 0.20  0.05 0.32 1.27 7.52 134.71 9.30 0.15  0.02 0.11 0.32 1.14 6.33 158.25 0.10  r value cancels out for these transitions  Part B: Ratio of G.SAISR to exact cross sections. j' =0 j=0 j=2 j=4 j=6 j=8 j=10  0.87 1.0 2.2 5.2 20 3.0x10 2  0.97 0.88 1.4 2.6 7.6 79  j' =4  j' =6  j' =8  1.7 1.2 0.88 1.1 1.7 9.7  4.5 2.3 1.0 0.86 1.2 4.0  16 6.1 1.7 0.86 0.85 2.4  98  j' =10 1.5x10 2 40 6.9 1.7 1.0 0.80  For transitions with j'=0, (213) reduces to 0.9SM/  E 34-0 kJ-1 k + 3  =	 0.0S 4---o(ik)  (214)  and for those with j=0, (213) reduces to  csm E  ao—p  — r  k  13 1  a.08 (Ek ) 3 1i-0  (215)  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 best tend to decrease as  7  values which worked  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 _ C;S.1.11  3,3 1 (Ek  Ea' )  = (2i + i)E J 3 L	o o  L ) 2 r6 + [(EL — E0)7/2h] 2 2 (Ek + 0 ) [6 + Rej, — ti)7/2/42 1 Ek  EL) Exact  (216)  .  and replace o- Exa: with o- P so . This results in cr GS.111 ( 3 4- 3 k  E , + ci , ) i''  = (2j + 1)E L  (  i  j'  L  )  2  16 + [(CL — f0) 71 2 h] 2 1 2 (Ek  0 0 0 ) [6 +  [(e i ' — f i )-r/2/42 ]  + EL)  Ek —  r,—o-  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 99  (217)  lead to even poorer agreement with the exact cross sections. Other ways of approximating the ofx o (Ek e L ) 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.  4.2.6.2 Using Exact o L- 0 Cross Sections -  The GSMSR was tested as to how well it applied to reproducing the exact results. 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   Cross  Section  + 6 0) 0- 04-2(Ek + 6 2) ao_o(Ek 0- 0■-4(Ek  64)  0- 0■-6(Ek  + 6 6)  70 4_8(Ek  E8)  Values for the GSMSR at 300K for Y=10  o L-0(Ek + E L ) -   Largest Max. Max. Max. Contri- Calc.   condition rotor J buting time   number' state (sec) 2 J 2.2x10 3 4 80 84 55 3.3 1.4x10 5 4 80 84 59 3.3 8.5x10 5 9.5x10 6 1.1x10 7  6 8 8  90 100  10 12 14  120  Cro,m(Ek  + 6 10)  Cr 0■-12(Ek  €12)  1.9x10 7 6.1x10 5  614)  3.5x10 6  0- 0.-14(Ek  110 140 150  96 108 118  67 78  130 152 164  Cross Section in A2 1.77x10 2 4.71x10 0  6.7  4.32x10 -1  14.5 16.2  6.50x10 -2  90 104 118 132  68.9 81.2 162.1  1.17x10 -4 3.11x10 -6  3.55x10 -3  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 average  7  T.  Using an  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 6  Not only does a decrease in  T  2  (218)  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  (219) 2  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 Values The a L,_ 0 (Ek + L) values used are from Table 19.  7  -  Transition  Cross Section in A 2 0.00 0.00 0.06 0.49 6.73 184  0 4— 10 2 <— 10 4 <— 10 6 4— 10 8 4 10 10 <— 10 —  aL4_0(Ek  +  EL)  = 1.28 x 10' s.  Ratio to Exact Results 1.00 1.25 1.24 0.75 1.04 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  •/(E  k + eji )  = 0 1., o (Ek  102  -  ejl)    (221)  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 is demonstrated by adjusting the  7  T  value as an actual time of collision. This  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 sections to within 1% of the exact values. Transition  7  value  in seconds 2  4-  10  4  4-  10  1.53 x 10'  64-10 10  0.89 x 10' 1.36 x 10 –n  8  4-  10  104-10 10  The  7  1.79 x 10'  8.52 x 10-13  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 4.2.7.1 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+ e j ,) 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)• -  With this  substitution, equation (91) becomes cy,,s P—.71  (Ek  [j] N Ek (ii) = D •i   i  (  N(Ek  p-EJ) l os  CfulLal.-0(Eko)•  (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 cr L , 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'=0 101.90 117.88 57.39 25.24 11.18 5.13  j'=2 4.62 102.81 41.56 16.15 5.97 2.76  j'=4 0.61 11.35 96.54 29.51 10.38 3.82  j'=6 0.15 2.24 13.20 93.07 24.93 9.42  j'=8 0.05 0.58 2.82 12.13 91.29 19.59  j'=10 0.02 0.18 0.72 2.11 6.71 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 5.1 21 80 4.4x10 2 8.5x10 3  j'=2 0.97 0.81 4.2 15 63 9.4x10 2  j'=4 1.7 1.9 0.74 3.2 12 1.2x10 2  j'=6 4.5 4.1 1.7 0.65 3.1 26  j'=8 16 11 3.7 1.4 0.57 5.1  j'=10 1.5 x 10 2 66 15 3.2 1.0 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 ( E A 	(3')  1 =  s -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 r •/ 1.1  .1 Exact t Ek  =  —  [Ek  ,  Exact  Ek  and then approximate atLa .c7 1_ 0 (E k c w _ ji ) by  ar so (Ek o )  Lik  3  ) (224)  as outlined in (216)  to (217) for the General S-Matrix Scaling relation, leads to all cross sections being multiplied by a factor of  (E k  tb,_,1)/E k > 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.  4.2.7.2 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 107  ao—L(Ek + EL) values  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 f L = L(L 1)h 2 121. 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 Largest Cross Section as—o(Ek + (0) ao-2(Ek + (2) ao_4(Ek + ( 4 ) uo,-6(Ek + (6)  Condition Number' 1.2x10 7 1.8x10 5 2.1x10 5 1.9x10 7  Max. Rotor State 8 8 10 10  Max. )  Max. J  Contributing  J 110 110 120 120  118 118 130 130  89 91 96 104  Calc. Time 2 (sec.) 20.138 19.737 39.148 68.934  Cross Section in A 2 0.126952 x 10 3 0.507163x10 1 0.404343x10 ° 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 04-6 24-6 4+-6  Cross  Ratio to  Section  Exact  in A 2  Results  0.03 1.27  1.00 2.32 1.86  64-6  14.36 126.95  84-6  25.36  104-6  3.64  Cross Section  Ratio to  Transition  in A 2  Results  04-10  0.00  24-10 44-10  0.00 0.03 1.31  1.00 4.32  0.73  64-10 84-10  3.34 14.7  104-10  Exact  5.62 2.00  8.67  1.34  80.58  0.41  The calculations in Table 22 are also more inaccurate than the ASSR using ak— x V—ii(Ek (1 3 ,_ 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  with  e .1  ) = 0- 0*L(Ek + EL)  (225)  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  a o —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 E L ) values with the -  cri , j ,(Ek e f ) 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.  1l1  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 = kB T  E 12  = 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 A 2 . j=0 j=2 j=4 j=6 j=8 j=10 j=12 j=14 j=16 j=18 j=20  j=0 j=2 j=4 j=6 j=8 j=10 j=12 j=14 j=16 j=18 j=20  j'=0 70.07 15.44 1.68 0.16 0.01 0.00 0.00 0.00 0.00 0.00 0.00 j'=12 0.00 0.00 0.00 0.02 0.37 6.38 90.09 6.49 0.34 0.02 0.00  j'=2 3.13 73.92 9.16 0.66 0.06 0.00 0.00 0.00 0.00 0.00 0.00  j'=4 0.19 5.24 75.64 6.83 0.51 0.03 0.00 0.00 0.00 0.00 0.00  j'=14 0.00 0.00 0.00 0.00 0.02 0.35 6.63 98.79 6.28 0.35 0.01  j'=6 0.01 0.27 4.96 80.03 6.75 0.44 0.03 0.00 0.00 0.00 0.00  j'=16 0.00 0.00 0.00 0.00 0.00 0.02 0.39 7.00 108.63 6.28 0.22  j'=8 0.00 0.02 0.30 5.52 79.69 6.76 0.44 0.03 0.00 0.00 0.00  j'=18 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.50 8.05 127.48 4.00  j'=10 0.00 0.00 0.02 0.32 6.01 85.11 6.70 0.36 0.02 0.00 0.00 j'=20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.49 7.05 171.52  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 a L , 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	,; I J initial(J initial -  f  h2  + 1) - = 004X10 -20 J 1  (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 A2 .  j=0 j=2 j=4 j=6 j=8 j=10 j=12 j=14 j=16 j=18 j=20 j=0 j=2 j=4 j=6 j=8 j=10 j=12 j=14 j=16 j=18 j=20  j'=0 62.40 19.79 3.05 0.59 0.14 0.04 0.01 0.00 0.00 0.00 0.00  j'=2 3.97 69.14 11.18 1.59 0.30 0.07 0.02 0.01 0.00 0.00 0.00  j'=12 0.00 0.01 0.03 0.16 0.95 7.80 78.95 9.16 1.32 0.26 0.06  j'=4 0.34 6.16 69.04 9.67 1.29 0.23 0.05 0.01 0.00 0.00 0.00  j'=6 0.05 0.62 6.77 70.28 9.24 1.22 0.22 0.05 0.01 0.00 0.00  j'=14 j'=16 0.00 0.00 0.00 0.00 0.01 0.00 0.04 0.01 0.19 0.06 1.10 0.24 8.40 1.25 85.18 8.62 9.65 92.63 1.46 9.74 0.29 1.60  115  j'=8 0.01 0.10 0.73 7.15 71.98 9.07 1.20 0.22 0.05 0.01 0.00  j'=18 0.00 0.00 0.00 0.01 0.03 0.09 0.33 1.52 9.00 105.03 10.03  f=10 0.00 0.02 0.13 0.84 7.52 75.00 9.11 1.25 0.24 0.06 0.02  j'=20 0.00 0.00 0.00 0.01 0.02 0.05 0.14 0.47 1.81 8.75 127.47  Table 26 - Continued Part B: Ratio of 108 to exact cross sections j=0 j=2 j=4 j=6 j=8 j=10 j=12 j=14 j=16 j=18 j=20  j'=0 0.89 1.3 1.8 3.8 10 48 3.1x10 2 1.3x10 3 1.2x10 4 1.3x10 5 1.9x10 7  j=0 j=2 j=4 j=6 j=8 j=10 j=12 j=14 j=16 j=18 j=20  j'=2 1.3 0.94 1.2 2.4 5.2 19 1.0x10 2 4.5x10 2 4.3x10 3 3.7x10 4 2.2x10 6  j'=12 4.1x10 2 1.3x10 2 36 8.8 2.6 1.2 0.88 1.4 3.9 15 1.9x10 2  j'=4 1.7 1.2 0.91 1.4 2.5 7.4 30 1.2x10 2 1.0x10 3 6.1x10 3 3.3x10 5  f=6 3.6 2.3 1.4 0.88 1.4 2.8 8.5 33 2.0x10 2 1.2x10 3 3.6x10 4  j'=14  j'=16  1.7x10 3 5.5x10 2 1.5x10 2 36 8.4 3.1 1.3 0.86 1.5 4.2 29  2.5x10 4 8.0x10 3 1.8x10 3 2.7x10 2 55 13 3.2 1.2 0.85 1.6 7.4  j'=8 9.8 4.9 2.4 1.3 0.90 1.3 2.7 8.4 48 2.4x10 2 5.4x10 3  j'=18 4.8x10 5 1.2x10 5 1.8x10 4 2.4x10 3 3.7x10 2 78 13 3.1 1.1 0.82 2.5  f=10 52 20 7.6 2.7 1.3 0.88 1.4 3.4 14 71 1.3x10 3  j'=20 1.6x10 6 1.4x10 7 1.8x10 6 1.1x10 5 1.1x10 4 1.6x10 3 1.6x10 2 18 3.7 1.2 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 significant 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 GaussLegendre 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 04-0  24-0  4+-0  6+-0  84-0  10+-0  124-0  144-0  164-0  18+-0  204-0  Calculation Exact IOS ECIOS Exact lOS ECIOS Exact IOS ECIOS Exact lOS ECIOS Exact IOS ECIOS Exact lOS ECIOS Exact IOS ECIOS Exact IOS EGOS Exact IOS ECIOS Exact IOS ECIOS Exact IOS ECIOS  Cross Section 70.07 62.40 62.40 15.4 19.8 19.4 1.68 3.05 2.40 0.157 0.589 0.232 0.0137 0.138 0.0167 7.73 x 10 -4 3.72 x 10 -2 1.11 x 10 -3 3.57 x 10 -5 1.11 x 10 -2 7.86 x 10 -5 2.80 x 10 -6 3.56 x 10 -3 5.97 x 10 -6 1.01 x 10 -7 1.19 x 10 -3 4.60 x 10 -7 3.20 x 10 -9 4.12 x 10 -4 3.15 x 10 -8 7.74 x 10 -12 1.46 x 10 -4 1.29 x 10 -9 115  Ratio to Exact 0.89 0.89 1.3 1.3 1.8 1.4 3.8 1.5 10 1.2 48 1.4 310 2.2 1300 2.1 12000 4.6 130000 10 19000000 167  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  C  2a  == - v ti 2a  (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 2.80 0.52 0.08 0.006  j'=2 0.59 203.92 1.08 0.15 0.01  j'=4 0.07 0.70 202.36 0.37 0.03  j'=6 0.01 0.09 0.35 202.12 0.12  j'=8 0.001 0.01 0.04 0.17 205.62  Part B: IOS Results. j=0 j=2 j=4 j=6 j=8  j'=0 199.87 2.19 0.92 0.47 0.29  j'=2 0.54 201.74 1.91 0.86 0.46  j'=4 0.16 1.21 202.42 1.93 0.94  j'=6 0.022 0.16 0.64 201.97 0.86  j'=8 0.0029 0.021 0.087 0.39 202.46  Part C: Ratio of IOS to exact cross sections. j=0 j=2 j=4 j=6 j=8  j'=0 1.006 0.78 1.76 5.65 48.40  j'=2 0.90 0.989 1.77 5.89 42.04  j' =4 2.36 1.73 1.000 5.17 37.40  120  j'=6 2.06 1.77 1.81 0.999 7.36  j'=8 2.50 2.17 2.46 2.22 0.985  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=0 j= j=4 j=6 j=8  j 1 =0 2.29 0.030 0.0056 0.00082 0.000026  j'=2  j'=4  j'=6  j'=8  0.0063 2.33 0.014 0.0021 0.000075  0.00076 0.0091 2.29 0.0028 0.00018  0.00011 0.0013 0.0026 2.35 0.0012  0.0000051 0.000068 0.00025 0.0018 2.47  Part B: 103 Results. j=0 j=2 j=4 j=6 j=8  j '=0 2.21 0.018 0.0076 0.0044 0.0033  j'=2 0.0068 2.22 0.027 0.016 0.010  j'=4 0.00089 0.0081 2.28 0.013 0.0039  j'=6 0.00062 0.0040 0.014 2.31 0.018  j'=8 0.000064 0.00046 0.0017 0.0069 2.35  Part C: Ratio of 108 to exact cross sections. j=0 j=2 j=4 j=6 j=8  :1=0  j'=2  j'=4  j'=6  j'=8  0.96 0.61 1.4 5.4 126  1.1 0.95 1.9 7.2 139  1.16 0.89 0.99 4.5 22  5.82 3.08 5.23 0.98 15  13 6.8 6.9 3.72 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 reliability 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 compilation 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 IOS  Exact  Calculation  Calculation  300K  30.0  67.3  1000K  64.0  1000.0  Temperature  As previously noted, the IOS becomes more accurate at higher collision energies [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 k o 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]:  [0,,,,(R)]T (21  ,-cs =  t 1) .  6 ,- 11, ,cs( f)dR,	(scs  siscs)  9  2z 0  (228) where S cs is the CS 8-matrix, 1 is A o and fiJ is a null matrix, save for its jth 125  diagonal element, which equals unity. The above is essentially a first order perturbative 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 approximation. 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  Cr  CCIOS 7  24  2  „ios 24h 2	(( A , — c.\ ) 2 7- 2 '  (230)  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 dealing with the Senftleben-Beenakker effect [47], calculation of pressure broadening cross sections [48], calculation of molecular fragmentation [49], angular momentum 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 approximation 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: d 2	2 d  	idR 2+ "1-?: ,dR  k2	A(A  "3  + 1)1 /  , n) /  R2  21t v(r,  — h2  illxnjA(r, (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 k 72L3 . 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,„ 0 0 and  so that the functions and first derivatives are continuous at the boundaries.  127  R))  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  2.446  Inverse Power  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 deltashell potential having a parameter in units of J-m -1 . Further, the potentials go to infinity at different points, the delta-shell at inverse power at  R=  a=  5.5x10 -10 m. and the  Om. One method of comparison is to integrate the delta-  - = Om. to shell potential over the a region of space from 1 1  Then the inverse power potential will be assigned a value it has at  R=  R = 5.5x10 -10  m.  con.stant value, namely, that  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 -23 J [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  r=o  P " er (5.5 X 10 -10 7n.,  0) = 1.6 x 10 -32 [1 + 0.5P2 (cos 0)] J — m (233)  Integrating the delta-shell potential over the same region of space gives the following value for comparison:  f r=a vdelta  L=o  =  r=a  f=o  3.697  x 10 -32 [1 + 1.5132 (COS  0)] 6(r — a)dr  = 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 deltashell. This value will be compared with the point where the inverse power potential becomes equal to the kinetic energy of the relative motion of the atomdiatom. Equating vinverse power = 	 2.2 x 10 -14 X [10— 12  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 equivalent 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 exist. 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 sections. 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 potentials 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 portability 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 computed 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 LennardJones) 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 overestimated the values. Several scaling laws aimed at improving the 105 cross sections were investigated. 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. This suggests that there exists for the delta-shell potential a quick way of calculating other transitions from the a L , 0 cross sections. Various choices for the k o parameter were investigated as to whether they affected the IOS results, but it was found that changes in its value did not  131  significantly improve agreement with exact values. Inclusion of closed channels was not found to significantly affect the results. Increasing the parameter C y, improved both elastic and downwards transitions but upwards transitions became worse. Decreasing the parameter a brought about very good agreement between exact and 105 results on elastic cross sections, but not for inelastic cross sections. The savings in computational time was found to be very significant in each of the approximations considered. Using a delta-shell was far simpler and quicker than using the standard Lennard Jones potential. And using the 105 approximation on this delta-shell potential was found to be much easier than using the exact calculations for the same situation. This suggests that investigations into further improvements to the delta-shell potential and 105 approximation may prove beneficial to shedding light on problems that have been impossible to solve with conventional potentials and computing procedures.  135  References [1] A. M. Arthurs and A. Dalgarno. Proc. R. Soc. London, Ser. A, 256:540, 1960. [2] W.A.Lester, Jr. The N coupled-channel problem. In W. H. Miller, editor,  Dynamics of Molecular Collisions, Part A, pages 1-32, Plenum Press, New York, 1976. [3] R. T. Pack. J. Chem. Phys., 60:633, 1974. [4] S. I. Drozdov. Soy. Phys. JETP, 1:591, 1955. [5] D. M. Chase. Phys. Rev., 104:838, 1956. [6] T. P. Tsien and R. T. Pack. Chem,. Phys. Lett., 6:54, 1970. [7] T. P. Tsien and R. T. Pack. Chem. Phys. Lett., 6:400, 1970. [8] T. P. Tsien and R. T. Pack. Chem. Phys. Lett., 8:579, 1971. [9] R. T. 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