THE INFINITE ORDER SUDDENAPPROXIMATION AND THEDELTA-SHELL POTENTIALbySTEPHEN JOHN DANCHOB.Sc., The University of Winnipeg, 1981A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEillTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CHEMISTRYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1992© Stephen John Dancho 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of ChemistryThe University of British ColumbiaVancouver, CanadaDate DE-6 (2/88)The Infinite Order Sudden Approximationand the Delta-Shell PotentialAbstractThe Infinite Order Sudden (IOS) approximation is applied to thecollision of an atom with a diatom where the intermolecular potentialis given by a delta-shell. It is shown that modelling the potential assuch allows for a simpler calculation of the close-coupled equations,and using the MS results in even further savings in calculations. Ex-act and IOS calculations at 300K and 1000K are compared and itis found that the WS overestimates inelastic cross sections for bothtemperatures. A variety of corrections to the IOS are considered andthe Energy Corrected MS (ECIOS) approximation is shown to be thebest of those studied. Other possible improvements to the IOS areproposed.iiContentsAbstract iiList of Tables viList of Figures viiiAcknowledgement ix1 INTRODUCTION 12 ATOM-DIATOM COLLISION THEORY 82.1 Uncoupled Angular Momentum Representation 82.2 The Total,/ Representation 152.3 Cross Sections in the Total-.I Representation 222.4 The IOS Approximation 252.4.1 IOS Cross Sections 282.4.2 Energy Corrected IOS Cross Sections 312.4.3 General S Matrix Cross Sections 322.4.4 Accessible States Scaling Law 333 THE DELTA-SHELL POTENTIAL 363.1 Scattering From a Spherical Delta-Shell Potential 363.2 Scattering From a Non-Spherical Delta-Shell Potential 403.3 Simplification Using Only Open States 464 3.4 Inclusion of Closed States 3.5 IOS T-Matrix Calculation CALCULATIONS AND RESULTS 4.1 Parameter Determination 515660604.1.1 Atom and Diatom Parameters 604.1.2 Choice of Energy 604.1.3 Range of Partial Waves 614.1.4 Inverse Power Potential Comparisons 624.2 Cross Sections at 300K 674.2.1 Exact Cross Sections Including Only Open States 674.2.2 Exact Cross Sections With Inclusion of Closed States . 694.2.3 IOS 0 --* L Cross Sections 714.2.4 IOS Scaling Relations 744.2.5 Energy-Corrected Scaling Relation 874.2.6 General S-Matrix Scaling Relation 974.2.7 Accessible States Scaling Relation 1044.3 Cross Sections at 1000K 1124.3.1 The Exact Cross Sections 1124.3.2 The IOS crL, 0 Cross Sections 1144.3.3 ECIOS UL,-0 Cross Sections 1174.4 Changing Parameter Cv, 1194.5 Changing Parameter a 121iv5 DISCUSSION 1245.1 Time Savings of the IOS 1245.2 Possible Improvements to the IOS 1255.3 Applications of the 105 1265.4 Molecular Potentials 1285.4.1 Time Savings of the Delta-Shell 1285.4.2 Comparison of Potential Parameters 1295.5 Calculations on a PC 1326 CONCLUSIONS 133References 136List of Tables1 A Comparison of the Delta-Shell and 7. -12 Potentials 662 Exact Cross Sections 673 Effect of Including Closed States 704 IOS o-L,0 Cross Sections 725 IOS Cross Sections at 300K Using k o = 756 IOS Cross Sections at 300K Using k o = kfinal 777 Comparison of k i„ itial and kfilial IOS Cross Sections 788 Comparison of k,„,,„ and krni„ IOS Cross Sections 799 IOS Cross Sections at 300K Using ko = 14.27 A-1 8110 IOS Cross Sections At 300K Using Exact o-L,_0 Values 8311 Effect Of Using Different k Values In Exact o -L-0 Values on theCross Sections 8512 IOS Cross Sections At 300K Using k=16.92 A -1 8613 ECIOS Cross Sections At 300K Using T = 71- a/(2v inin ) and ExactO•L—o(Ek,„,,,,) Values 8914 ECIOS Cross Sections At 300K Using 7 = fa/v,,,;,, and 105aL—o(Ek,,,,.) Values 9115 ECIOS Cross Sections at 300K Using T = 0.16a/Vmin and 10501-0 Ek.„“„) Values 9216 0i,_0 k Values and ECIOS T Values Used In Table 14 94v i17 How f Varies According To the k Used in the Calculation of thecrL4-0 Cross Sections 9618 GSMSR Cross Sections At 300K Using IOS o -L,0 Values 9819 Input crL,o (Ek EL ) Values for the GSMSR at 300K for f=10 10120 GSMSR at 300K Using Exact crL-0(Ek EL) Values 10221 7 Values Required to Match the GSMSR with Exact Results at300K 10322 ASSR Cross Sections at 300K Using IOS 01_ 0 Values 10623 Input o-o_L(Ek L) Values for the ASSR at 300K for j' =6 . 10824 ASSR Cross Sections Using Exact au—L(Ek (L) Values for 300K 10925 Exact Cross Sections at Energy=1000K 11326 IOS Cross Sections at 1000K Using ko = 11527 ECIOS Cross Sections at 1000K 11828 Exact and IOS Cross Sections at 300K for Cv, = 1000 12029 Exact and IOS Cross Sections at 300K for a =0.55 A-1 12230 Computer Time Required for IOS and Exact Calculations 12431 Computer Time Required for IOS Calculations for a Continuousand Delta-Shell Potential 129viiList of Figures1 Coordinates used for diatom-atom collision problems 102 Impact parameter b 63viiiAcknowledgementThis project is actually the completion of some work that was originallystarted in 198 and worked on until 1985. The work was resumed last yearand it has only been with the help of some very exceptional and special peoplethat 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 andVince for helping me through the toughest time, to Cathy and Nigel andConnie and Dave for the phone calls and letters, to Heather and Jenniferfor their drawings which I have on the wall at the office, to my GrandmotherStella who taught for j1 years in the Manitoba public school system andto my other Grandmother Catherine who I wish could be with me for thisgraduation.To my friends Daniel and Jaleel for helping me keep perspective while Iworked on this research, and to Valerie and Colleen.To my Aunt Rita and Uncle Don for helping me on my move to Van-couver, to my godparents Aunt Margaret and Uncle Ed for their hospitalityon my many trips between Winnipeg and Vancouver, and to my cousins andToni 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 whoshowed 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. Kostynykand 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. PatDuffy and Mr. Dan Berard for help on Quantum Mechanics, DOS real modeand Unix, and to the professors in the UBC Theoretical Chemistry group,ixDr. 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. Lawsand 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 givingme the chance to finish this work and for his constant assistance throughoutthe year.And finally to my supervisor Dr. R. F. Snider for suggesting this fasci-nating topic, a topic that has kept me intrigued and confused for the last 8years and probably will for the rest of my life. I owe a great deal of thanksto him for all the encouragement, patience and enlightening conversationsthroughout the years. And especially for giving me the freedom to discoverthis part of the world in my own way. It is indeed a pleasure to work withsuch a gifted educator and scientist.1. INTRODUCTIONThe study of molecular collisions is the basis for understanding a large numberof chemical phenomena such as chemical reactions, gas viscosity and pressurebroadening. Specifically, the equation used to describe molecular collisions is theSchrOdinger equation. The aim of this thesis is to further investigate an approx-imation used to numerically solve the SchrOdinger equation — the Infinite OrderSudden (10S) approximation — on the scattering problem where the interactionpotential between an atom and diatom is given by a delta-shell. This chaptergives a brief introduction to and describes the development of both the IOS andthe delta-shell potential.The description of the collision of an atom and a diatom requires considerationof the process before, during and after the collision. Before the collision, it isnecessary to set the initial conditions, that is, to describe the state of the freeatom and free diatom. The collision itself involves a choice of intermolecularpotential and a set of coordinates suitable for the mathematical description of thecollision process. Finally one must identify the amplitude and relative probabilityfor the products of the collision as they are separating at infinite distance fromeach other. Here the latter are reported only in terms of the total degeneracyaveraged cross section into each of the allowed states. In setting up the problem,attention is paid to how the calculation can be performed with computationalefficiency. [1].The system to be studied is the collision of a homonuclear diatom, treatedas a rigid rotor, with an atom. Later on, the description will be more specificand 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 anatom with a rigid rotor.A detailed description of rotational excitation caused by the collision of twomolecules requires the solution of a set of close coupled equations [2]. Approxi-mation techniques have been developed over the last 20 years in order to reducethe amount of computation required to solve these equations. The approximationtechnique of interest in this thesis is the 10S.The close coupled equations are equivalent to the SchrOdinger equation. Thesolution of the SchrOdinger equation for inelastic processes involves the propertreatment of both the angular and radial motion. Depending on the basis setused, the equations are coupled via angular momentum operators or the inter-action potential. The first type of operator, responsible for directional coupling,is the interaction potential, which is generally diagonal in orientation representa-tion but non-diagonal in angular momentum representation. The second type ofoperator includes the orbital angular momentum operator and the translationalenergy operator, which, in contrast to the potential, are diagonal in angular mo-2mentum representation and non-diagonal in orientation representation. Hence,in either representation — angular momentum or orientation representation —the SchrOdinger equation will couple direction dependent states.The IOS approximation replaces the quantum numbers for rotational and cen-trifugal angular momentum with constant values, decoupling the set of equationsin an orientational basis. This has the computational advantage of decoupling theangular and radial motion, treating the former as a constant and the SchrOdingerequation then reduces to motion in one dimension (the radial motion). A stan-dard further simplification is to use the WKB approximation [3]. An alternateapproach is to look for a potential that simplifies the radial motion. In partic-ular, the SchrOdinger equation for the delta-shell potential can be reduced totreating the radial motion by matching inner and outer solutions. For the exactclose coupled equations, this leaves the angular motion to be treated by matrixmethods. In the 105 approximation the solution is obtained analytically so thatnot even matrix methods are required.The IOS is a combination of two approximations – the Energy Sudden(ES) [4, 5] and Centrifugal Sudden (CS) [6, 7, 8, 9, 10, 11] approximations.The Energy Sudden approximation is equivalent to assuming the rotor's ori-entation is fixed for the duration of the collision, but exact solutions to collisionproblems involving a homonuclear diatom (rotor) and an atom take into accountthe fact that the rotor's orientation changes during the collision process. TheES, which involves treating all rotational states as degenerate, was first used3by Drozdov [4] and Chase [5], according to an IOS history given by Parker andPack [12].The CS approximation involves treating the centrifugal potentials as degen-erate. There were two approaches taken leading to this approximation method.One approach, taken by Pack and co-workers [6, 7, 8, 9, 10], was to treat allcentrifugal potentials as degenerate as well as incorporating the ES in a spacefixed 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 ofreference.The approach taken by Pack and collaborators was first proposed in 1963 byTakayanagi [13]. In 1970 Tsien and Pack [6] applied Takayanagi's approximationand tested it numerically on an He — N2 system. The results proved encouragingand further work by Pack and co-workers [7, 8, 9, 10] led to what is now calledthe IOS approximation. In 1974 Pack [3] then extended his work to a body-fixedcoordinate system and chose to replace the centrifugal potential operator with asingle centrifugal potential which he identified as the total angular momentum.He termed this the CS,/ approximation.In 1972 Rabitz [14] developed an effective Hamiltonian method which suc-ceeded in decoupling orbital and rotor angular momentum. An alternative wayof 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.4The CS of McGuire and Kouri and the CS-J of Pack were shown to be equiv-alent in 1977 by Parker and Pack [15] when they identified the single centrifugalpotential in the CS-J as labelled with a final orbital, rather than total, angularmomentum. Simultaneously Shimoni and Kouri [16, 17, 18] found that the crosssections for Pack's CS-J approximation were greatly improved when a similarsubstitution was made.An equivalent approximation to the IOS is assuming constant orientationsfor the duration of the collision. This idea had been used as early as 1961 byMonchick and Mason [19] when it was applied as an approximation for classicalscattering. In 1975 Secrest [20] and Hunter [21] simultaneously succeeded informalizing the IOS from a fixed-angle approach. Hunter [21] further pointed outthat this approach was similar to an approach taken by Curtiss in 1968 [22] who,while developing a formalism describing molecule-molecule collisions, presentedan 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 thatonly negligible rotations of the molecular system can take place. To enhanceits range of applicability, a variety of corrections have been considered. In thisthesis, the IOS and some of the methods of correcting it are applied to a systemwhich has a delta-shell potential — a potential having an infinite height extendingover an infinitesimal width at some distance from the origin. As a basis forcomparison, the degeneracy averaged cross sections of Ar — N2 collisions of theexact, IOS and 10S-corrected solutions are studied.It would be expected that a delta-shell potential would make for conditionswhere the IOS approximation is valid. An interaction described by a delta-shellpotential would have only an infinitesimal distance in which to act, which impliesan interaction taking very little time — the basic concept of a sudden collision.The delta-shell is first investigated in its simplest form — as a sphericalpotential. Then an angular dependence for the potential is introduced. Theresulting exact and IOS degeneracy averaged cross sections for this potential arethen derived and studied.The savings in calculation noted above merits further investigation intowhether more accurate modelling of potentials with one or more delta-shellscan be accomplished.While this thesis uses the delta shell as an interaction potential between twomolecules, most work with the delta-shell has been in nuclear physics. Onestudy found that two terms of a spin-angle expansion of an effective neutron-proton potential for deuteron-proton reactions to be well approximated usingdelta-shells [23]. Kok et. al. have found the delta-shell potential useful in calcu-lating phase shifts for proton-proton and N —a scattering [24]. Other applicationsfor the delta-shell have been in molecular physics [25] (studying hyperfine inter-actions) and in solid state physics [26] (calculating band structures).Chapter 2 outlines the theory for the exact and IOS calculations of crosssections. Chapter 3 describes the calculational details required to obtain these6cross sections with first a spherical delta-shell and then a non-spherical delta-shell. Chapter 4 describes the parameters used, lists the cross sections obtained,and compares the values calculated for the exact, 10S, and 10S-corrected crosssections. A brief discussion of computer time saved by using the 10S, currentwork on the 10S, and possible future work for the delta-shell potential thenfollows in Chapter 5. Finally Chapter 6 concludes with a summary of the mainpoints of this thesis.72. ATOM-DIATOM COLLISIONTHEORY2.1 Uncoupled Angular Momentum Repre-sentationThe 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 isrequired. As well, a centre of mass co-ordinate system is chosen so only relativemotion between the particles need be considered. The SchrOdinger equation isHT(P, = ET R) (1)where H is the Hamiltonian, kli(7•,11) is the wave function of the system, and Eis the energy.This section solves (1) in an uncoupled representation — both the rotationalangular momentum of the rotor and orbital angular momentum of the relativemotion between the atom and diatom are treated independently. In the nextsection a coupled representation is introduced, allowing for some simplificationsin both notation and in manipulation of terms.8To set up the mathematical description of an atom colliding with a diatom itis necessary to first define a set of coordinates. For this study, a space fixed centreof mass coordinate system is chosen. With reference to Figure 1 on page 10 , Ris the position vector describing the distance from the diatom (rotor) centre ofmass to the atom centre of mass, r is the vector between atoms in the diatomand 0 is the relative angle between R and r. In this paper, three-dimensionalvectors will be designated in bold and the following convention will be used:r = 7'7'R = Ri? (2)where r, R denote the magnitude of vectors r, R and r" , j denote the directionunit 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 rotorapproximation for the diatom is assumed, with both electronic and vibrationalcontributions being neglected. Since the rotor is rigid, the magnitude of r willnot 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 ofthe atom with respect to the diatom centre of mass. The Hamiltonian H1 forthe rotor isjr2Hi _ 2I (3)9R FIGURE 1: Coordinates used for diatom-atom collision problem.1 0where Jr is the angular momentum operator of the rotor and I is the momentof inertia of the rotor. Next, an appropriate complete set of states for theexpansion of IP is chosen. Since in free motion, the spinning rotor motion isindependent of that of the orbital motion of the atom around the diatom, 'Pcan be expanded in a complete set of rotational eigenfunctions, each expressedas a product of eigenfunctions of the two rotational motions. These comprisea discrete, as opposed to continuous, basis. The eigenvectors of Jr are thespherical harmonics Y,„(• and soJiYpn( 7') = + 1 )h 2 rini(r') (4)where j is the quantum number for magnitude of rotor angular momenta and mis the quantum number representing its component along the z-axis.The Hamiltonian H2 for the atom-diatom relative motion ish2 r_72H2 = - - v R2/..twhere it is the reduced mass. In the Ar-N2 system the reduced mass t isMAr MN2-MAr MN2 )MAr being the atomic mass of Ar, m N2 the molecular mass of N2.The eigenvectors of H2 are e ik 'R so thatu ik•R h 2 k2 „ik•RII 2 e — (7)2/2where eik 'R is a plane wave of momentum hk. For the scattering problem to bestudied, one assumes an incoming plane wave in initial momentum state hk' androtor state j'm'.(5)(6)11The expression for the total Hamiltonian H can be writtenH = Hi + H2 + R) (8)where V(71 , R) is the atom-diatom potential. This potential can be written inthe form R) = E vi,(R)PL (cos o) (9)Lwhere 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 Hgiven by (8) becomes:—h 2 { 1 a ,,, a A' 2H 7; R) (10)2/2 R2 aR aR n2R2where A2 is the operator for the square of the orbital angular momentum, whichis, in orientation representation:2 _ _ n2 [ 1 —a sin y a + 1 '2A sin yoap ac,o sin2 e a02 • (11)The eigenvectors of A 2 are the spherical harmonics YA,V), A being the quantumnumber of angular momentum magnitude for the orbital angular momenta ands is its component along the z-axis. Hence A 2 Y,\ ,(R) = + 1)K.s(1). (12)In the orientation representation it is required to solve (1) for T. With the expansion as given by (12) and (4), can now be written: R) = > YimMYAs(I) ,,,„,(R) (13)jniAs12where a is an abbreviation for the set of quantum numbers j, m, and s, a' theinitial set of quantum numbers and tk,„,(R) is the coefficient for this expansionfor a given a and a'.Putting (13) and (10) into (1) and operating on the left of each equation byf f Y2*.(0YA*s(f)ch'di (14)gives the set of close coupled equations in an uncoupled angular momentumrepresentation:ti 2 1 d2 d z A(A + 1)71) 0 ( a / = E vaa4,„„,(R)27 PR' lidR 4- 3 R2 I a"wherek'? .±2 [E +1)013 h2 2 1and= I f ri,n (i9YA*8 (i?)V(1., R)Yinno (P)YA, s , (ft)df-dil. (17)Next the boundary conditions are set for and its expansion coefficients 0,„,„,•As well, qi and all the must be regular at the origin. Finally themust be chosen so that satisfies the incoming wave condition, ie., the solutionof (1) starting with planar motion in rotational state j', m', k' (eigenstates of H 1and H2 ). Soeile•Ryi,m,(0 scatt • (18)where W scatt is the scattered part of The experimental observable to beexamined in this thesis is the cross section (which will be defined in Section 2.3).(15)(16)13In order to calculate the cross section it is necessary to determine the amplitudefin, ;j im i(i?) of the spherical wave in the final state jm. (18) is rewritten aseRiki RkP eiki .RYjni + E Yjm(il f im; (R) R oo. (19)j?nWe make a spherical harmonic expansion of the initial plane wave,= E omeRwAs (iT)YA-s (P) (20)Aswhere jA (k'R) is a Bessel function of order A, and k' is shorthand for kj ,.Thus the wave function with appropriate initial conditions is expanded as in(13) with expansion coefficients0„„,(R) = 47r E (21)A' .5'The x„,(R) functions at large R satisfy the coupled equations (15) withboundary conditions1x,„„,(R) (eR)(5.' (22)where is the Kronecker symbol, to match the incoming plane wave and theoutgoing spherical wave of equation (18) in terms of the transition matrix elementh A (k R) is the Hankel function and its asymptotic solution as R oo isgiven by:R oo . (23)—kjRei(kiR-A7/2) ,Using (23) in (22) yieldsR—Ar12)Xaa i ( R) j (k i R)6acv , (24)2 11 kik'14so that comparison of (19) and (24) leads to the following relationship betweenand27riJae(' (25)In conclusion it is noted that to attain the goal of a calculation of the atom-diatom collision cross section, it is necessary to solve the set of differential equa-tions (15) which are coupled by potential matrix elements subject to theboundary conditions of being regular at the origin and asymptotically of the formof (22).2.2 The Total-J RepresentationSince H is invariant with respect to a rotation of the co-ordinate system, the totalangular momentum is conserved. Hence states having a total angular momentumJ are not coupled to states with a different total angular momentum J' andthe SchrOdinger equation is decoupled into different total angular momentumcomponents.The total angular momentum operator J = A + Jr has eigenfunctionsn'Am EE(jmAs jAJ M)Y3 ,„(ilY),,(1) (26)where (jmAs I j AJ M) is a Clebsch-Gordan coefficient[27}.The expansion coefficients v3T,,A ,(R) of the scattering wave function (22)in this angular momentum representation satisfy15R) 47r E iAriAmv,ibx;JA;i , A ,(R),A3MA' s'X (j i M I j im')'s')YA*,,,(1;'). (27)Performing a similar set of operations as outlined in the previous section onlythis time operating on the left of each equation byI Y3;;T,(i',1-=?)di. d1 (28)gives the set of close coupled equations in the coupled angular momentum rep-resentation:d2 d 2 Ak.2u dR2 R dR R2[ P■ 1 )]AjA ; ii,v(R) jA; j".X" ;3',X'Vj Vj • ( R)(29)whereVIA;3 ,A, = JI Yi*A1 M V, i?)V(r,R)Y3'ZI, , (30)For the potential of (9) we are interested in determining an angular momen-tum representation. First the expression for the uncoupled representation, thatis, 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 rotationalinvariance of the potential allows certain simplifications in the total-J represen-tation.The potential given by (9) is in an orientation representation. The angularmomentum representation of V ,given in terms of the orientation representation,16is:(jmAsIVI,j'712' A's') =f f f f (Pl) sh 1- )(1'; flIV (R)11'' ; ir)(f-'; 77-1' s i)di'dkdf- idkUsing 4.6.6 of Edmonds[27], ie.,47rPL(cos 0) = 2L 1 E YLAMYL'A(9) can be written asmv(R)Ip ; fr) E vL (R),L47+ 4,(0y;,;,(t)so — fi). (33)LtiUsing the following defined phase conventions [28]and substituting(jmAsIV>Lp771As(33) intoj'nt i =47r1?) = (-i)'Y3*,„(o(-i)AYas(R)(31), one obtains the following equation:(—i)jyr,„(1')(—i)AYA*.s(f)YLA(')Yi„(f)i ji Yi'.1(0(i) Al YA's'( 11)th'dk(34)(35)vi(R) 2L + 1By using the following two properties of the spherical harmonics, (equations2.5.6 and 4.6.3 of Edmonds [27])YiKti(f) = (-)AYL_„(fi) (36)(31)(32)17and=(2j + 1)(2L + 1)(2j' + 1) ( j L j'L j'47r —rn (m ) 0 0 0 , (37)(35) can be written as(jmAsIV (R)1j'm' s') = (2j + 1)(2j' + 1)(2A + 1)(2A' + 1) L ' A L Ax O o o) () o o' )j L j') (A LA' )).(38-771 it 711 / -8 - 8/Equation (38) is the expression for the V-matrix element in the uncoupledrepresentation. Eight quantum numbers, j, rn, A, s, j', rn', A' and s' are requiredto label each element. The total-J representation, however, can reduce thenumber of labels needed to five, ie., J, j, Ad and .\', as will be shown. In orderto arrive at an expression for the total,/ V-matrix element, the uncoupled V-matrix expression is converted to the coupled expression through the followingtransformation:(jAJMIV(R)Ij i)'J'M') =E (jAJM I jrrr,\.․ )(j1n\sIV (R)Ij"m" A" s")(j"rn"A"s"Ij'A'J'M'). (39)msm"3"Using the phase convention given by equation 3.7.3 of Edmonds [27] which relatesthe Clebsch-Gordan coefficients with the 3-j symbols through the following: A J V,2J + 1 (40)s — MAJ M jmAs) = ( 77/ 18and using (38) for (jinAsIV(R)I j'n'A's'), (39) can be written A J/AVM') = E (—)3 -- A+m,\/2J + 1 ( 7n s —M )711S711'(jAJM1V (R ) xi3i+A1-3-A \I(2j + 1)(2j' + 1)(2A + 1)(2A' + 1)x E 000 (oo o) — m'L j' A L 771 —sA L A's'x(—)-A-m-s(—)ji—v+miV2J' + 1 ( Ji )m' —M' ) •Two symmetry property of the 3-j symbols, given by equations 3.7.6 and3.7.5 of Edmonds [27], ie.,j' A' J' = j' A' J'in , .s' —M' —m ' —s ' ) (42)andj L j' ) ) — —m m' Liu ) (43)allows (41) to be written as(jAJM1V(R) = \I(2j + 1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1)x E ij i+A -3-AvL(R) (j L j' ) (A L A'L 0 0 0 0 0 0x ` (_ ) i-i-, , -Fx+m+m , ( i' A' J'in,,s, )—m' —s M'x E(—)m—s+A-Fj+A-EL ( A' A L ) ( iinns s —s —it -77t 771,A Js —M(44)Further simplification in (44) is possible by considering equation 6.2.8 fromLg(41)19Edmonds [27], whereE (4., i+t,2+A3+i1+12+13 131 l2 /31 iL2 113 1711 P2 — /13X _( 11 j2 13 ) /1 12 j3 = 3 _{ 31 2 3— 111 111 2 ,u3 /Li — ,u2 7n3 ti 12 13 ( 31 j2mi m2 m3(45)where j l , j 2 , j3 , 177 1 , 771 2 , 7173 , / 1 , / 2 and 13 are the arbitrary indices used in Ed-monds [27] andj1 j2/1 /2 l3is the 6-j symbol defined by Edmonds [27]. (44) can be written j'A'I'M') = V(2j 1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1)(jAJM V(R) x E VL(R) j L A /0x E(_)i+i,+x+m+m, A' j' A,j A L —m' —s 'J' ( j' J) s' m' —M ) •(47)Using (42) and equation 3.7.8 from Edmonds [27], ie. ,( 31 32 ) 31 j2 = (2:73+1 r i bm,„i t (5 • (48)3 3333 31 2 3177 1 m 2 1713 mi m2 m3m i M2where b i1h73 = 1 if j 1 , j2 and j3 satisfy the triangular conditions and 6171j2j3 = 0otherwise, (47) is further reduced to(46)20(jAJMIV(R)Ij'AVM') = 1/(2j + 1)(2j' + 1)(2A + 1)(2A' + 1)(2J + 1)(2J' + 1)x E i3 17/, (R) L ( A L A'0 0 0 )0) 0 )x(—) 3 +31 +i+m+mi (2J 1) -1 8j,p6mm , I A'jij LIFinally, using the symmetry of the 6-j symbols as given by equation 6.2.4 ofEdmonds [27],.12 :13 l = 1 .13 J2j4 js j6 j 1 j6 ,14 j5the following expression is obtained:( 5 0 ) j 1 J i M i ) = 02j + 1)(2j' + 1)(2A + 1)(2A' + 1)jAJMIV(R)L j' ( A L A' f J A' j'E vL(R)(-Y+3+3' . (51)x j 0 0)0 0 L j A Note that the V matrix element is independent of M, the projection of thetotal angular momentum on the z-axis. This is expected of any rotationallyinvariant (scalar) quantity such as the interaction between atom and diatom.Secondly it is noted that the total angular momentum J is the same in the initialand final state. For this reason J need not appear in both initial and final statesbut can be denoted as a superscript on the operator V, ie., 1/ J .Hence the V-matrix element (j)JMIV(R)Ij'A'J'M') may be written as(jAIVJ Ij')'). This new representation, having taken advantage of the simplifi-cations made possible by the rotational invariance of the interaction potential,now requires only five indices instead of the original eight.(49)21As done in (22), an incoming plane wave of constant flux in initial momentumstate Pik' and rotor state j'71-1 1 is assumed for the boundary conditions for T. It isalso required that the solution be regular at the origin and have the asymptoticformeikRX, ; j , A , ( R) Mk/ R) 8 j":5 AA' + R oo. (52)2\c' R 3Au' A"2.3 Cross Sections in the Total-J Representa-tionThe cross section is defined as the ratio of outgoing spherical flux to incomingplanar flux. Planar flux is defined as the number of incident particles crossing aunit surface area placed perpendicular to the direction of propagation per unittime while spherical flux is defined as the number of outgoing particles scatteringthrough solid angle cll2 per unit time.The incoming plane wave e iki * R has flux - ,Pncorning [29]h k'Fincoming =IL(53)Putting (22) in (21) and then into (13), and substituting k for R results inthe following expression for the spherical flux Foutgoing for the outgoing sphericalwave e ikR/R [29] : 2Ai (54)t k'Foutgoing = (471-) YA.5(k)TamAs ; j /nt i s i YA 1 si(ki )lkj kJ , As A' .9' 22From (53) and (54) the differential cross section o-i„,,i,„,,i(k) is:.7r. 4 2a- . (k)m4---2 1 111 1 2 E Ks(k)TjtnAs ; jim'AisiYA1 3/(k i ) (55 ) In this work only the total degeneracy averaged cross section, ie., thedifferential cross section integrated over all angles and summed over all m's,TDA =- 2j' + 1 0.3„3,7n,(k)dkis to be calculated.It is possible to simplify the resulting total cross section by choosing theincoming direction k' as the z-axis. Since(56)m m'yAs ( ki) y\s (z) 2A + 1 oso47r(57)the expression for o-TD1- becomes:TDA— E E TimAs;?rnivo(E)YAs(ic)ko(2j , + 1) AsA'47.2 2A' + 147rx E YA„.,„(k)77„,,,„,„ ;i,„„ A „,o (E)„ s„ ,\” ,2A"' + 1 dk (58)47rUsing the orthonormality of the spherical harmonics,I }I s ( k ) YA*,,„ ( k )dk (59)the expression for the total cross section now is1"TDA= k, 2j + 1) Asv„,„,' „ , Tjm ; jimiA 10(. 2.)■• + 1) (60) 23A J" ) ( j'xs —M" ) in'A' J"s / Mil/JIIMI/j/IIMIll(62)Finally, then, we seek the appropriate form for o-TW; in the total-J represen-tation. Using the identity(jm)sT(E)Vin'A's') =E (iniAskmiA)(JmiAlT(E)Ifivricv)(fm -iyAilfm')'s') (61)JMJ'M'and (40), (60) in the total J representation is:71TDA E E— + 770)11 AsAisi4_ 1)JMJIMIA Jx—M ) ( in' s' —M' )E (2] +1) 2 (jAT JJMJ'M'X j A J ( j' A' J'—M ) s' —M' )nimissi2where the notation= (JMjAIT(E)1J'MT)') Smm, (63)has been introduced since J = J' and M = M' by conservation of angularmomentum.Using 3.7.8 from Edmonds[27], ie.,77/ A J ( j As —M ) —M'J' 1= 2J +18"'SmAli (64)SS24the following expression for the degeneracy averaged cross section for the j 4-- j'transition in the total-J representation is obtained:aTDAV2(2j' + 1) EA jEm (JA rimliiy)1 2 E E(2J + 1)1(j A1TJ j' A')k 12 (2j/ + 1) AA, j2 (65) It is this quantity which will be used as the basis for both the exact and IOSresults.2.4 The IOS ApproximationThe close coupled equations (29) consist of a set of coupled differential equa-tions which must be simultaneously integrated to obtain a solution. For example,expanding (29) to 100 partial waves and ten rotor states leaves over a thousandcoupled equations to solve. The IOS approximation is a drastic simplificationsince it decouples these equations (but in an orientational basis). This approx-imation assumes the collision to be sudden, allowing for a diagonalization inorientation representation. This is equivalent to replacing all and all A withthe constants ko2 and A c, respectively in the coupled set. Actually, the IOS is acombination of two approximations, the Energy Sudden approximation (ES) andCentrifugal Sudden approximation (CS). These will be examined in turn.The Energy Sudden approximation replaces the j dependent wave numberk3 with a j independent effective value ko . Since this approximation no longertreats the angular momentum of A and Jr equally, the uncoupled representation25(15) is used. Putting (21) in (15) and converting to Dirac notation gives thefollowing equation: d 2 9 d 2 2/1 idR2 t. A(A R2 1) Kalxc',(R)) = E(a1V(R)la")(a"IX'(R))*(66)Replacing vi with q, leaves (66) in the form(a x.,,(R)) = E(a1V(R)Ia"\Ka" IX.i(R)).(67)Operating on the left by Eim (djm) gives E(dini) [ddR22 ddR ko2 A(R2 1)](Asjmxa,(R))3mh 22µ[ d2 d 2 AO 4- 1)[c/R2 dR R2 . \ 2dm) 1-7 E(aco V(R) all) ( a" lx,,, , ( R)). (68) UsingEljm)kgj (69)andE E(dint)(aa" jm V( R) cr")(a"lx,,(R)) = E (AsIV (P , R)1A" s")(A" , R))A"s" (70)we finally get the ES equation:I (12 2 d 1.2 A( (Aslx,vs, R))[dR2 -fidR n'") R2= E 081 V(r, R) A"s")(A"slxv,,,(f-, R)). (71)26We note that the effect of replacing with the constant parameter 10.2 allowsthe SchrOdinger equation to be diagonalized in the rotor orientation. In otherwords, the collision process occurs with fixed rotor orientation.A collision process which is fixed for a particular rotor orientation correspondsto a situation where the incoming atom has a velocity relative to the diatomcentre of mass much larger than the angular velocity of the rotating diatom.Hence the ES approximation is expected to get better with larger translationalkinetic energy of the incoming atom or slower angular velocity of the rotor.In the CS approximation, .\ is replaced with the constant parameter A., so(29) now becomes:n2 [ d2 2 d A (A + 1k, .j2 ° ° 1 '](alx„,()R = Vall)(a"a"lx„,( ))R. dR2 ridR R2 (72)As) and transforming into the orientation repre-Operating on the left by EA, sentation by carrying out operations analogous to those discussed above for theES approximation, we get[ d2 2 d 1.2 A.(A. + .{dR2 dR n'j R2 (R))2—h2 E (im V(R)ljuni")(j"ni"lx,,(R)). (73)Note that (73) has a fixed R orientation-the sudden approximation for largecentrifugal energy.One limitation of the CS approximation is that it breaks down in the regionof the turning point [30]. Classically, the turning point is defined as the point27where the radial kinetic energy vanishes [31]. The rotor, however, is still rotatingand thus collisions at the turning point could not be considered sudden.As the energy of the collision increases, the CS improves [30]. An increase ofenergy would correspond to a small rotation of the rotor relative to the distancecovered by an incoming atom with high velocity.Finally the IOS combines both of the above approximations, replacing k? withko2 and replacing A with Ao . We operate on the left by > c,(/; da) and sincethe angular momentum operators are now merely parameters we can use closureof the la) states to give the IOS equation: d2 d o(A. + 1) 2,a I/R2 RdR n'°2 R2 R) R)x„,(i, R). (74)Note the effect of replacing operators by numbers and the subsequent changeof basis leaves the equations without any sum over intermediate states and allequations are decoupled.2.4.1 IOS Cross SectionsThe T-matrix was introduced in (22) in an uncoupled representation. As wasnoted in the previous section, the IOS decouples the SchrOdinger equation ina basis with fixed orientation. Hence all quantities calculated from the IOSSchrOdinger equation, such as the T-matrix, are diagonal in the orientation rep-resentation. The only rotationally invariant quantity that depends on the differentpossible orientations is the scalar product of the rotor orientation 7 and atom-diatom orientation h. Thus T is a function of the angle 0 between r and R.28This can be expanded in Legendre polynomials:v0k0(7•, 1?) E '/2L + 1 Ti;° k0 PL,(1' • 11). (75)where &(7 • .fi) is the Legendre polynomial of order L.Equation (55) can be expressed in a different representation as:crini.iim , = -2C4 L m ; k T kl) 1 2 . (76) For the IOS cross sections cr.Tis_j , n,,(k) we have: 0 .19 S — 47x2 3 m 111 k4.7 1 2 IOm ; kV , 17)T A ° k° 1?j'rn'; k')ci•dfI I E YA3(00 4-3 's1(k)YiniMYA*3(R)2 47r 2k ?.7 1 (77) Using (75) and the expansion given by (32), the IOS expression for the degeneracyaveraged differential cross sections, or a i°s (h,k',:// —> k, j) is2a/ sok' , ji j) = (2j +0 YA.,(00'4-3Amm' AsAisIxV(2L• + 1)(2A + 1)(2)x' + 1)TE° k°(—) 7"'+8 Ye s ,(P)E E X ( iLn(L j' (j L \ a L0 0 0 ) —ill —Cr 77/ 0 0 0 ) s cr s(78)Equation (78) may also be written ascr ic's(hk'd k, j) = (2j + 1)E 0 0 0 ( L 2 cr (Me , 0 —4 k, L) (79)229where47r2cr(hk' , 0 k, L) = E 01-A 02A + 1)(2A' + 1)AsAisi (AL A') (AL Ax YAs (k)TE°"(—) 3 0 0 ) a s' Y ,(P)) s2 . (80) If the initial momentum direction k' is chosen as the z axis, there will not beany z projections of angular momentum for the initial state, so (57) holds.Thereforeams (hk' ,j' --+ k, j)7r (= (2j + 1) 0L0 0) 2)AA'A L ) ( A L ) 2X 71 ,A°k° — ) s 0 0 0 ) —a a 0 )+ 1)(2A' + 1)(81)To get the total cross section (81) is integrated over all angles to givethe followingaL4—o27 fir= o-(hk' , 0 k, L) sin OdOdOo oA L A' ) 20(82)7r „= 2_, 2_,(2A + 1)(2\' + )71\°"iA'-A A L A' )as 0 0 0 )ir= 2E(2A + 1)(2A' + 1) nokoiA 1 --A 1 2 A Lkit I 0 0 0 )If the initial A paramaterization for the 105 is assumed, ie., A = A', (82)simplifies to, 101—o k'2= —7r E(2L + 1) IT17°1 2 . (83)30For total cross section am', , (79) is integrated over all angles to give thefollowing IOS scaling law: , 2ros j= (2j + 1) E o o o L j. 0./0sL4-0*2.4.2 Energy Corrected IOS Cross SectionsThe scaling law given by (84) can be modified in a manner such that it takesinto account a collision time T. (Recall the IOS assumes a sudden collision, ie,T 0.) DePristo et. al. [32] have proposed a scaling law called the EnergyCorrected Sudden (ECS) approximation which incorporates the collision time Tas a correction to the IOS approximation in the form of a simple multiplicativefactor.DePristo [32] corrects for the S-matrix, but since the S- and T-matricesdiffer only by the identity matrix and some phase and the correction factor doesnot change the identity matrix we relate the T-matrices by the same correctionfactor:TECros( 6+ — os t6 + — €i)27-2/(202 3j 1 k-Lik Ejt). (85)where Ek is the translational energy, E ., is the rotor energy of the jth rotor stateand T is the collision time.In this study the centrifugal sudden approximation is also assumed so that thecorrection factor is applied to the IOS T-matrices, giving an "Energy Corrected105" T-matrix, or g?los:(84)L31Since 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 ruleis obtained:24h 22 °Jos.,.ECIOS„.24h2 (e 6)272 34-3 (86)T shall be treated as a parameter to help fit the cross sections from thisscaling to the cross sections from the exact calculations. DePristo [32] gives amethod of estimating T and this is then compared in a later section to the Tvalue which gives the best agreement with the exact results.2.4.3 General S-Matrix Cross SectionsOne of the interesting results arising from the IOS treatment of cross sections isthat all j 4— j' transitions can be calculated from a single set of cross sectionvalues using (84). DePristo et. al. [32] have taken this property of the IOS toinvestigate whether the exact cross sections follow the same scaling laws.The General S-Matrix (GSM) scaling relationship is a way of calculating a setof cross sections from another set of cross sections with different energies andtransitions. The relationship makes use of some IOS properties which allow theseconnections, but the GSM scaling relationship is actually a relationship betweenexact cross section values.The IOS implies that the L 0 and j j' S-matrices are related by [32](js. j' L s ios ( E ) (87).ri o.s;, ( Ek ei,) = E o 0 0 L4-0 k L •39Using the factor in (85) to relate S -?' with S°s 4.act with, and to relate Swsp—Ji 1.--.7 L4—oSYxatjt and then using (87) allows for the L 4-- 0 and j 4-- j' exact S-matricesto be related by_re j,) r 24h 2 + (EL — co)272 j EcGsm ,02T2 Srxact(Ek+EL). (88)(L 24h2 f_ 0 0 03 3Here EL is the rotational energy associated with the Lth rotor state. Then, inorder to get a cross section relation, DePristo et. al. [32] make an assumptionsimilar to that made in the previous section, ie., that 7 is independent of A and.\'. They arrive at the following scaling law for the GSM cr3 ,_j i:CSM 24h2+ (EL — fo)2T2 2 j(Ek -Fc :p) (2j+1) r24h 2 + ( 6j, — Ej )2 7-2 0 2j' L Exact0 0 ) °-L4---0 (Ek+CL)•(89)The above scaling law will be used with of,"6' values to calculate o-2 ,_is valuesand these will be compared with the actual crExa;t values to determine whether1 4-3the above scaling law is valid.As well, the scaling law shall be used with o-Pso to determine which of thescaling laws, (86) or (89) leads to a more useful manner of correcting the IOScross sections.2.4.4 Accessible States Scaling LawDePristo and Rabitz [33] arrive at an expression for the cross section whichdoes not involve 3-j symbols. To get this expression, they make the following 3assumptions:33• angular coupling coefficients are mainly a function of difference jj' — jionly• quantum tunneling is unimportant, so the main energy dependence corre-sponds to the criterion of determining the outermost turning point.• the transition probability is inversely proportional to the number of acces-sible statesThe 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 tothe highest rotor level (where the kinetic energy is the least and where tunnelingis least effective). Combining these two assumptions with the third allows theAccessible States (AS) cross section o-).7‘4_si,(E) to be written ascriA,si,(E) = f — § 1, E — (90)where f is a function of — j1 and E — ;EJmax • tjmax is the energy of the rotor forthe maximum allowed j value, and N(E) is the number of available rotor statesfor a given energy E.The assumed functional properties of the f function implies a relationshipbetween cross sections which have the same If —j1 and E — €3,.. values, namely[2j + 1][2(j' — j) + 1] N(Ek Ea , —j) [2:1' + 1] N(Ek cji) > J.(91)34(The form of (91) is equation (4.9) of [32] which was first presented in a differentform in [331.)The scaling law given by (91) is here referred to as the AS scaling law,referring to the N(Ek + ci ,-3 )/N(Ek + 63 , term, which actually describes onlyone of the three approximations made.Note that in contrast to the previous scaling laws, (91) relates cross sectionsof different total energies, ie, cr.)P,(Ek + c.3 ,) with ao.._3 , _3 (Ek + ci,_3). A setof exact crj ,,, and (3-0,3 ,_3 values are related by a given total energy. Althoughthere is a question in the exact case as to which total energy to use, this studywill show that this choice is not a significant factor determining cross sectionvalues.However, in this study we will use (91) to determine its validity as a scalinglaw for the IOS approximation. Since the IOS treats all rotor states as having thesame energy, cr.f° :,(Ek + e3 ,) and cri°1,_i (Ek cii_j) should actually be writ-ten as c .f° ,(Ekc,) and o-4° ,_3 (Eq) where ko and ko are those wave numberscorresponding to the parameters chosen to replace the k3 value and E kc, is theenergy associated with the ko wave number. It is this aspect of the IOS approx-imation - the incorporation of rotor and translational energies into one effectivewave number - which makes (91) lose some of its specificity when applied to theIOS results. It will be shown later that this loss of energy dependence does notsignificantly affect the accuracy of the scaling law given by (91).353. THE DELTA-SHELL POTENTIAL3.1 Scattering From a Spherical Delta ShellPotentialIn its simplest form the delta shell potential isV(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 theDirac delta function.If the collision involves only a spherical potential, rotor states are unchangedin the collision process and need not be considered further. The form for theHamiltonian H is thush 2H —2pwhere A 2 is given by (11).1 a R2 + v(R) (93)—IR2 aR OR R2 ]The SchrOdingerh 2equation1 a pt2 anow takes the formA2 2p2/1 [R2 OR A(' OR R2 + h2^(R)R) V(R)^(R)= (R) (94)36where 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)aswhere 7PAs(R) = 47ri A xA(R)YA*.,(P) (96)and using equation (92) for V (R) and introducing the dimensionless variablex = kR where k2 = 21LE Ih 2 we now can write (94) ash2 0 02 AO + 1) VO+1iXA =k 8(x — ka)XA22 1 rxOx ax2 x2 (97)where the following property of the delta function [34] has been used:8(R — a) = kS(x — ka) (98)For x ka, there is no potential and the partial wave solutions are the wellknown 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 constantbyXA jA(x) - ji-TAh),(x), x > kaXA BAjA(X), x < ka (99)where the jA's are the regular Bessel functions and the hA's are the Hankel func-tions. The expansion coefficients TA and BA are to be determined by appropriatematching conditions at x = ka.:37Definingka- ka — eka+ ka + 6, 0 (100)and integrating (97) over x from ka- to ka+ and requiring x), to be continuousat ka gives2 ,k vsXA•Since x A is continuous,a 2/1OXx),(ka+ ) — OXlxA(ka- ) = h2k V5 )00Using (99) in (102) givesMka+)— 7TAhVka + )— BAA(ka h,2k) = —2p lissjA(ka)BA (103)2 whereA(x) = cclix :7,\(x)h a (x) (x hA (x). (104)Since x ), is continuous at ka,xA(ka+ ) = x),(ka - ) (105)and so1 jA(ka)— .-T),h,\(ka) = BA:7),(ka). (106) ka+ 2xAka — kaka+ka -axAax (101)(102)38Allowing E —> 0 and multiplying (103) by h A (ka), (106) by hjka) and thenusing the following property of the Wronskian W of the Bessel and Hankel func-tions [35]:W = A(x)h A (x)— jA (x)//,',\ (x) ix1 2 (107)an expression for BA is obtained:BA =1 — i0jAhAwhere jA is shorthand for jA (ka), hA for h A (ka),= ka (109)andg 2iiVa ah2 (110)Finally, multiplying (103) by jA, (106) by j'A , and using (107) and (108) givesthe following expression for TA:Ta = (111)1 — igjAhA .With TA and BA solved, we are interested in the behaviour of XA at R> a.From the asymptotic behaviour of the Bessels and Hankels given in (23) theexpression for x ), as R oo is now — -i(ka-A1r/2) 0 Toei(ka-A7/2)1 (112)XA 2i1ka [eUsing (60) and assuming j' = 0 (ie, no rotor states) the total cross sectionis.total = -;75- (2A + 1) ITAl 2 • (113)A1(108)39Hence a calculation of the cross section for a spherical delta shell potentialrequires the wave number k, the parameter strength V5 and the distance a of thedelta shell from the origin. After evaluating the j A 's and h A 's at ka, the totalcross section for the spherical delta shell potential is obtained from (111) and(11:3).3.2 Scattering From a Non-Spherical DeltaShell PotentialFor the system being studied — an atom-diatom — the fact that the scatter-ing centre is non-spherical (ie., the potential between the atom and diatom is 0dependent) means that the angular momentum of the diatom can couple withthe orbital angular momentum of the atom. It is the 0 dependence in the poten-tial which is responsible for angular momentum transfer between rotor angularmomentum and orbital angular momentum.A 0 dependent potential can be expanded in Legendre functions. Since weare studying a homonuclear system, only even Legendre functions appear in theexpansion. Here only the zeroth and second order Legendre functions are retainedto give a potential in the formV(0) = 17,5 [1 b2 P2 (cos 0)] (R — a) (114)where b2 is a constant. Explicitly, the second order Legendre is0 , 1 /0 2 0 1 )P2 (cos 11) = —2 COS Of — (115)40With a spherical potential there is no coupling between orbital and rotorangular momentum and therefore it is not necessary to designate initial and finalangular momenta — they are always the same. With a non-spherical potential,transfer of angular momentum may occur between the two types of angularmomentum. This implies that the initial state, designated by quantum numbersj', m', A' and s', can be different from the final state, designated by quantumnumbers j, m, A and s. Since must then be described by an initial and finalstate, a notation is introduced where the vector wave function x is represented asa matrix x. (Matrices will be indicated with bold type.) As well, since the rotoroperator k, orbital angular momentum operator A and potential operator V allmust be parameterized by indices corresponding to the ket and bra states, theytoo 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 separatethe V-matrix elements into blocks having the same total-J. Since by (133) theonly non-diagonal matrix that T depends on is V it is possible to separate thecalculation of the T-matrix as well into blocks with the same total-J. With thisin mind, and using the above notation, (29) is actually one matrix element of amatrix equation and can be generalized ash 2 d2 d2/1 tdR2 fidR k2 h 212R21_1 Xj VjXj (116)where k 2 and A 2 are diagonal matrices whose elements in the angular momentumrepresentation are defined by41j(j 1 h 2 2)1 ckjj = kj = () [E 21 (117)andALI = A(A 1)h 2 6), ),183j , (118)The form of (116) corresponding to matrix element (jAlx -i lj'A') is thenh 2 d2 d 22 dR2 TtdR k3 A(h2R+21)] (iA l x j (R) IjA)/2[ E (jA VJ ICA")(5(R — a)(j"A" x J(R)VA') (119)This is similar in form to (97). In the same manner as we solved (97), wechoose solutionsjA(kiR)— A/)k;71 ' R > a(jA Blj'A)j),(kiR), R < a (120)where the coefficients T and B are no longer simply parameterized by A butmust be matrices themselves corresponding to the representation of X. If thematrices j and h are now introduced, their matrix elements are given by(jA 1j(kiR)Ij')') j), (kiR)Sij 1 5AA I(jA li(kjR)Ij'A') = 11,A(kiR)8,w8AA , (121)(120) can be written in matrix notation as1X= j — 5-hk2Tk -2, R> a= jB, R < a (122)Ki A==42 j represents the prepared incoming state; represents the scatteredoutgoing wave and jB represents the part of the incoming wave that has tunneledthrough the potential barrier.Next (119) is integrated with respect to R from ka- to ka+ whereka+ = ka + eka - = ka - e, e -4 0 (123)The requirement of continuity of x, ie.,x (ki a+ ) = x (k3 a- ), (124)is used to get dyX(kj a+ ) - dy—d x(k3 a- ) =2 Vx (125)where y = kR. (126)Again the requirement of continuity for the wave function x at R = a implies1 jB = j- -2hk 7 Tk - 2, R = a (127)Inserting the expression for x given by (122) into (125) givesdj B + 2,a k _ ilf I_ _dj dh k 2 hkiTk - f) (128)dy dy 2 dyOperating on the left of (127) by -(d dy)j, on the left of (128) by j and thenadding the two resulting equations gives1 i 1 dj .dh) ,—h2 jV - = - —n j— K 7 K - 1 (129)2 dy dy 243Using the matrix version of the Wronskian given by (107), ie.,dj .dh 1W = —h — j — . dy dy i(kR) 2allows (129) to be written as42.iftkiV --1 hkiTk — i) = k 2 Tk -42 - 2or, rearranging,4ipa 2 kNj = [2iya2 h 2 h2 ViVhk2 + 1] Tk — iso that finally an expression for T is obtained:ita 2 4ipa2 1. T = [1 + 2i kIjVhk2 k NA h2 J h2Another important point to mention regarding the separation of the T matrixinto smaller blocks comes from the form of the potential used. The potential isexpanded 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 elementis expanded in 3-j symbols ( L j'0 0 0 ( A0 0 0L A') which are 0 if j+ L-F j'or A L A' is odd by equation (3.7.14) of Edmonds [27]. This means that jand j' must both be even or they must both be odd. The same is true for A andA'. Hence the V-matrix can be broken down into blocks categorized by:(130)(131)(132)(133)441) total-J2) even or odd A3) even or odd j.And since T is a function of V then T as well can be broken down into theseblocks.One important difference between T and V is in the range of parameter off-diagonality that is allowed, that is, the difference allowed between j and j' andA L 'between ). and V. The V matrix, because of the ( AL0 0 0 0 0 0 )terms, can only couple A and A' states that differ by 0 or by 2. The T-matrix,however, allows for any even transitions. The reason for this is that the T-matrixdepends 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 crosssections for even A and then odd A. This was done for J ranging from 0 to itsmaximum value, A max -I- max •Though one wave number and an initial j' value are chosen, (and a totalenergy value Etotal is obtained from these values) other matrix elements havingthe same Eto tai but different j' (and therefore different wavelengths as well)must be calculated. That is, there is no way in the total-J coupling scheme toseparate the various initial j states. Hence if the T-matrix elements are calculated45for Etotai = 5.820 x 10' 1 J., so that the initial translational state correspondsto a temperature of 300K and the initial rotor state corresponds to a value ofj' = 6, T-matrix elements corresponding to this same total energy but havingdifferent initial conditions are also calculated. For example, the initial conditionsof j' = 0 and k = 16.92 x 10 10m. -1 correspond as well to a total energy of5.820 x 10 -21 J. So too do j' = 2 and k = 16.57 x 1010 m. -1 , etc., all the wayto j' = 10 and k = 8.37 x 10 10m. -1 Thus the sum of appropriate T-matrixelements will give cross sections not only for the intended initial conditions butfor all other combinations of initial conditions having the same total energy.3.3 Simplification Using Only Open StatesGenerally, in the solution of the T-matrix, the wave function must be expandedin a complete set of basis functions. This implies that internal states with energygreater than the total energy must also be considered in the expansion whichleads to imaginary k values. These closed channel states are coupled to theinitial and final scattering states by the intermolecular potential. In a physicalsense, what this means is that the interaction perturbs the internal states andsome of these energy forbidden internal states are necessary to represent theeigenstates of the perturbed system [36].As a first approximation only real k values were used in the calculation. Thatis, only the energetically accessible (open) channels were used in the calculationof the T-matrix. The theory to include closed states is developed in the next46section.For ease in computation, it is desirable to avoid inversion of a complex matrix.With this in mind, we defineh = in + j (134)where n is the matrix of Neumann functions. Equation (133) may be written as2iT — J (135)1 — N + iJwhere2,ua 2 1 . . iJ — k7jVjk 2h 22pa 2 1. 1N _7.. kTivnk2h2(136)(137)and so that J and N are always real. Equation (135) can be expressed in anotherform:T = [N — 1 + 4 -1 [N — 1] [N — 1] -1 [-24= [[N — 1] -1 [N — 1 — ii]] 1 [ N 2i 1 [J]]= [1 N 1i AI ' { N2i 1 [J]]— — = [1 — iK] -1 [-2iK] (138)where1 KN — 1 iii (139)47and K is always real. The following expression is obtained upon further expanding(139):11 — iK [1 + iK] — ' [1 + iK] [-2iK]— 1 [ 2iK + 2K 2 ]1 + K 21 i2K21 2i=1 + K 2 I -I 1 + K 2[K]. (140)Since 1 + K 2 is real, it is now possible to calculate both the real and imaginarypart of T without using complex inversion.For further ease of computation, it was investigated whether a symmetricform for the calculation of K could be found. Letting2,a0 1 1W -- —h 2 kTVICTallows K to be expanded as follows:(141)1 K 1= N — 1 Pi jWn — 1 [Pi]1 = {jWn] [1](j/n) [nWn — (n/j)] n1 = nWn — (n/j) [j [iifti n— 1 + 1] [i]n. i 1n + nWn — (n/j) •(142)Since the j's and n's are diagonal matrices and W is a symmetric matrix,only a symmetric matrix inversion routine is needed in order to evaluate K.T =48Problems, however, arose with this procedure. Neumann functions for suffi-ciently small values of ka at large A approach oo as given by [35], ie.,n,\(2A — 1)!! (ka)A+1 (143)Using Stirling's approximation for large NN! = (N ) N— (144)and(2A — 1)! (2A — 1)!!(145)2A-1(A _ 1)!and approximating 2A-1 as 2A and A-1 as A for large A, the Neumann functionsbecome, as ka —> 0,nAti 1 ( 2A Akakae)ka --> 0. (146)Treating the limiting behaviour of the Bessel functions in an analogous manneras ka ---> 0 gives the limitA(ka)' e kae)ka —> 0. (147)(2A + 1)!! — 2A 2A )n kla k2aAe ) A 2A )2A+1e (Lfy (kae2A 2AWhen the actual run was made, exponential overflow occurred in the inversionroutine for (142) for initial rotor state j'=8, A=74, j=12, ka=34 so thatn 74 (34) [1481 149J74(34) — 3 x 103° . (149)Hence the (n/j) term in (142) is of the order of, ka —> 0. (148)49Solving for an inverse of a matrix requires the multiplication of matrix elements;three terms of the order of (149) would be sufficient for an exponential over-flow [37] on the Amdahl.Thus another form for the K matrix was needed., Equation (142) can berewritten in the form 1 [K 1n+ jWn — 1 1 . ni1 n j [W — (1/jn)] n [ni l 1 1 1 111Ltd W — (1/jn) Lni( 150)In this rewritten form, the magnitude of the numbers were much more manage-able. The (1/jn) term was, for the parameters previously mentioned, of the orderof1 2)ka 2(74)34 103 , ka 0. (151)..1AnA e 2.303The W term 1 ranged from 4 x 10 4 to 2 x 108 and an exponential overflow wasavoided 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 kapoints; for the cases studied here W — (1/jn) never got close enough to 0 tocause overflow problems.)1 The range for W was calculated by recognizing that the term (2/ta/h 2 )V6 is of theorder of 100 while the actual V-matrix element, as given by (51) depends on 3-j and 6-jsymbols, which, by their unitary nature [27], take on a maximum value of 1. Finally, thek values used in this study ranged from 5 x 10 10 Tn. -1 to 30 x 10 10 rn. -1503.4 Inclusion of Closed StatesOne approximation made in the previous section was limiting the basis set toenergetically allowed states. We seek now a more general solution that includes"closed" channels, those channels (states) which are not physically allowed asfinal states for the collision process but nonetheless may affect the calculations.Treatment of the problem with the inclusion of closed states is similar to thatpresented in (116) to (133), the only difference is that the incoming preparedstate is designated Poj where P o is the projection operator onto open chan-nels. The resulting equation is similar in form to (133) except for the projectionoperator at the end:1 4 2T = [1 h2itta2 V ipa iVhk [ h2 k 2 iVjk 2 Vol . (152)With the following definitionsH 2 k7jVhk 2 = JVVhpa2 1. .h 2 (153)and2//a 2(154)J k-1,jVjk 2 = JWJh 2(where W is defined in (141)), the T-matrix can be writtenT = [1 -Fl iF11 2ijP°• (155)For the calculation of the scattering cross section, only the open state part Tooof the T-matrix is required:Too = PoT = Po 1[1 iH 2i0o• (156)51The term P o [1/(1 + iH)] can be broken up into open, closed and open-closedcoupling contributions Po[1 4-1iH -= Po [1 +1iHo] [1 i [H H1+1iH] (157)whereHo P0HPo, (158)H1 PoHPc PcHPo, (159)Pc being the projection operator on closed states, defined byP c = 1 — Po (160)and finally, H e P c FIP c • (161)The following calculations can be performed: (non-contributing terms are markedwith slashes)[ 1 I 1 1 D iH = P° [1 + iHo ° [1 -FliHo ] [1 :Po ill]PO [ 1 i +1itio l poHipc [ 1 +1iFic]x [1 i [1-1 — {i P _1 1 J 1 1P 14 IP u 1. 1 1[1 + &Li °"1. c [1 +1iFi c lr r 1 r 1+ •zl-l c 1 + iH[1 + iH0.1 13°H1Pc I. • i P c H I P°I. • j.(162)52So[ 1 1 poHipc 1 1 1 Fic H i poi Po [ 1 =-I iHl[ [1 + iF10 .1 {1 + ihI c iPa [1 +iiHo l iP°H1Pc (1 +1 H e •If the above is multiplied on the left by [1 + iH o], the following is obtained:Po[1 1+ iH[1 + iHo 1PoHiPc [ 1 + iHc PcHiPoll1 x Po [1 — iPoHiPc ( 1 iFic )] • (164)The T-matrix can be written:TOO iHo PoHiPc [1 +1iHi= 2i [1 + PcHiPolx [P0 -113 , — iPo HiPc [ 1 +laic ] PciPo] (165) = PO — [Po + iHo + PoH1Pc1 +1 PcH1P0x [P o + i [Ho — 2J0] PoHiPc 1[ 1 + iHc I Pc (H1 — 2-1) Pc](166)where PoiWiPo• (1 67)(1 63)OrTOO53H — 2J ==Using=2,hn 2 k iiv [h _ 2j] ki2fin 2 k fiv [i n _ j] kih 22/./a 2 0 jvh( 2 )k ih 2(168)or since for open channels h (2) = h*[H — 2J] P o =2"21cijVh*IciP o ,h 2(169)the following expression for the T oo matrix is obtained:Too = P o — P 0 [1 + iHo + PoHiPc [1 + iHd PciliP°1 x [Po — i (J 0 — iNo ) — P0 H1P c [ 1 +1ill] P C (J — iN) P o l(170)whereN o F.-_,- PojWnP o . (171)We defineRo -P.. P oWP c h [1 +1&lcliP cWP 0 (172)where W has been defined in (141). Using (159) and (153) we can now write1-154[iiPoHiPc [1 + 1 &lc ] PcH1P0 = Po [PoFIPc -I- PcHPo] pc 1 +lHex P c [P 0 HP c + PcHPo] Po= PoHP41 +1iFicl PcHPo= P ojWhP c [1 +1iHcl PcjWhP0= jR0 h= jRoj + ijR o n = i [Ji + iNi.] (173)whereii — iiRoj (174)and N1 E.-- —ijRo n. (175)With the following definitionsH o E Jo + iNo, (176) .1-E. Jo + ii (177)and ITI F- No + N1, (178)the Too matrix can be written5500TOO PoP2 tri _ ij _ K ]= 1 +LI — NOrand, finallywhere[1 — Ki ] [1+ [i/ (1 — IV -.J] [1 —1 IV 1 — KI j I [1 i —1Po Po —IJ (179)1 + r , .__\ 1ii/ O. — NA [1 -J 1 —N iPoToo = PO (180)iR]1 - iR [1 +2iPoRToo (181)—1 — iK1K -.L- J. (182)—N 1= Po= Po3.5 IOS T-Matrix CalculationIt is useful to note here that the method which will be used to obtain the 105solution has already been used in the section where the central 6-shell potentialwas considered. To understand why this is so it is necessary to consider the twotypes of operators in the SchrOdinger equation (ie., (29)) that bring about thedirectional coupling.If the interaction potential is diagonal in orbital and rotor angular momentumrepresentation, ie., it no longer couples different angular momentum states ) andA' (and this is the case when there is only a central potential V(R)) then all56operators in the SchrOdinger equation can be thought of as diagonal in angularmomentum representation which leads to radial solutions parameterized by onlyone A, ie.:h 2 1 d2/1 [R2 dffR2 RR2 A( + 1) k2] 2P),(kR) V(R)1PA(kR) (183)where k 2 = (2y1h 2 )E. Conversely, if the angular momentum operators arereplaced by quantities which are diagonal in orientation representation (and thisis the case in the IOS approximation where operator A 2 is replaced by theparameter A o (to + 1) so that all operators in the SchrOdinger equation canbe thought of as diagonal in orientation representation which leads to a set ofuncoupled differential equations (as opposed to matrix radial solutions in theexact case) which is similar in form to (183) , ie.:h2 1 d R2 d Ao (A0 + 1) + 7PA,(koR, 0) = V(R, 0)71)A 0 (koR, 0)2µ[R2 dR dR R2(184)except that there is the extra constant parameter 0 classifying the solutionOA° (koR, 0). (Here 11 (2p/h 2 ) {E — jo (jo + 1)h 2 /21] where jo is some cho-sen parameter.) Further, there is no longer the restriction that A o be an integer,as in (183). Hence the methods of solution of the radial part of (183) and (184)are similar — (183) gives solutions parameterized by A and k while (184) givessolutions parameterized by A o , ko and 0.To demonstrate the above, the potential given by (114) is inserted in the IOS57and211g(0) = [1 — b2 P2 (cos 0)] .V52(188)equation (ie., equation (74)) and the following expression is obtained:n221.i [[ R2 dR1 d R2 dRd Ao(A R20 + 1) + 41 11) Ao(koR, 0) =V5 [1 + b2P2(cos 0)] S(R a)60 (koR, 0). (185)In terms of the above discussion, the radial solution to (185) is dependent onlyon 0 and hence the directional couplings which required matrix manipulationsare 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 the0 term in the potential and the 0 parameter dependence of the wave functionmakes no difference to the method of solution. So following the same steps aswere used to solve for (97), a 0 dependent T-matrix is obtained:—2iGg(0)jt(ko a)TAok0 (0) = (186)1 — iGiAo (koa)hA o (koa)where= ko a (187)As was discussed in a previous section on IOS cross sections, the quantityMo ko must be calculated in order to obtain the o-j ,_j , quantities. By inversionof (75), no ko is given by:itko = V2L + 112 TAoko ( 6) ) PL (COS 0) sin Oa. (189)58Thus to get 01j._3/ cross sections, TAok„(0) is calculated as given by (186) forappropriate values of A 0 , ko and 0. Next, using numerical methods this TA 0 k0 (0)is integrated over all 0 as given by (189) in order to get values for the noka 's.Finally (80) is used to get the resulting cross sections.594 CALCULATIONS AND RESULTS4.1 Parameter Determination4.1.1 Atom and Diatom ParametersAs mentioned in an earlier section, the molecular parameters are chosen so thatthe diatom is a model of nitrogen and so that the atom is a model of argon.Hence the reduced mass y is set to 2.734 x 10 -26kg. For the rotor, the atomseparation is chosen to be 1.094 x 10 -10 m. [38] which gives a moment of inertia Ifor nitrogen as 1.392 x 10 -"kg — ni. 2 , or a characteristic rotational temperatureh 2 /2/kB for nitrogen of 2.894K where kB is the Boltzmann constant.4.1.2 Choice of EnergyAt temperature T the most probable energy for each of translational and rota-tional motion is kBT. In this thesis the major computation has been carried outat 300K. At this temperature j = 6 is the j-even state closest to a rotationalenergy of kB T while the wave number2itkB T (190)60has the value 14.27 x 10 -10 7n. -1 For this choice of rotor state (j =6) and trans-lational energy kBT, the total energy is 5.82 x 10 -21 J. This is the total energyused for the cross section calculations in the following section. Corresponding tothis energy, rotor states up to j = 10 are open and j > 11 closed. An analogouschoice 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 thechoice in this thesis is to restrict the calculation to j-even states.4.1.3 Range of Partial WavesIn both of eqs. (81) (the IOS cross section) and (65) (the exact cross section) itis necessary to set an upper limit on A, the maximum orbital angular momentumthat significantly contributes to the sums over partial waves.The largest A value considered to contribute significantly to the scatteringcross sections is that which corresponds to a particle just passing by the outeredge of the delta-shell, ie., the A where the incoming particle approached adistance a from the scattering centre.To determine this A, it is necessary to associate the angular momentum asexpressed in the quantum mechanical equationL2 = A(A + 1)h 2 (191)which for large A can be approximated as1L r-:::', (A + -)h (192)61with the angular momentum as expressed in the classical expressionL = ybv (193)where b is the impact parameter (see Figure 2 on page 63) and v is the particlevelocity given byhkv = —.EtEquating (192) and (193) gives,abv = (A + )h.Since the maximum range of the potential is a, this giveska = (A + 2—I) A.(194)(195)(196)For a given translational and rotational energy the calculation for maximum A wasdone as follows: the total energy (the kinetic and translational) was converted toa k value, then multiplied by a to arrive at a given A. All A beyond approximatelytwice this value did not contribute significantly to the cross sections.4.1.4 Inverse Power Potential ComparisonsSince the aim of this study is to compare exact and 10S results using a delta-shell potential, the choice of potential parameters is arbitrary — just as long asthe same parameters are used in both the exact and IOS calculations. Howeverarbitrary this choice of parameters may be, it is nonetheless desirable that theparameters chosen yield cross sections which are comparable to cross sections62FIGURE 2: Impact parameter b is "the distance of the asymptotic pathof the particle from the line of head on collision" [31163obtained for a realistic system. In order to accomplish this, an IOS calculation us-ing a realistic smooth repulsive potential was performed and then several choicesof the delta-shell potential parameters were tried in equation (114) to see if theseresults could approximately reproduce the realistic results.The potential chosen for comparison was the repulsive part of the Pattengillet. al. [39] potential2.2 x 10' 2 V (R, 0) =R12 [1 + 0.5P2(cos 0)]1.2 x 10 -21[1 + 0.13P2 (cos 0)]R6(197)expressed in SI units, specifically the inverse power potential corresponding tothe positive term2.2 x 10' 2 V (R, 0) =R12 [1 0.5P2 (cos 0)] . (198)Cross sections were calculated for this potential using the procedure reportedby Snider and Coombe [40]. Phase shifts 77,\„(cos 0) were calculated using theWKB approximation(\. R2 V (R r2—riAk (cos 0) = k f , 1 1 R 2,a 1 — —Rc2 dR 1 k(b — rc)rh 2 k2, 0)(199)where b is the impact parameter (A + Wti,- (see (196) ) and re is the largestclassical turning point.By definition, the phase shifts are related to the S matrix bysAk(o) = exp (cos 0)] . (200)64Since the potential is even in cos 0, it follows that the S-matrix can be expandedin even Legendre functions,sAk (0) = E V2L 1,9j/A,(cos 0).L(even)The expansion coefficients Si\," are given by the inverse of (201), ie.(201) 1SLR = V2L 11 PL(cos 0) exp [2i7/Ak(cos 0)] d(cos 0). (202)To carry out this integration the phase shifts are fitted to an expansion in evenChebyshev polynomials:1/Ak(cos 0) = A + BT2(cos 0) + CT4 (cos 0) DT6 (cos 0) (203)where the T„'s are Chebyshev polynomials and A, B, C and D are the fittingcoefficients. A 60 point Gauss-Chebyshev integration scheme was then used toevaluate the t.9 , 6 values.The T-matrix expansion coefficients are then obtained according toTi k = 5OL — SL R (204)and these were then substituted into (83) to obtain the aL ,_.0 cross sections. Theresults 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. Foreach choice of parameters, the 0 dependent matrices were calculated using (186)and then using (189) integrated via a 40 point Gauss Legendre integration to65give the Ti`K's. The aro were calculated using (80). After comparison withthe results of the repulsive r -12 potential, parameters were chosen to give areasonable fit. The best fit delta-shell potential isV (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 theleading coefficient for the r -12 repulsive potential in (198) since the units andform are different. In the delta-shell V8 = 3.697 x 10 -3 J. — 7n. has units ofJoule-meters, whereas for the r -12 repulsive potential the coefficient has unitsof Joule-meters 12 . Nonetheless, a form of comparison for these two parameterswill 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 -12PotentialsAll cross sections in A 2 ic = 14.2717 A , Amax 120L value IOS with repulsive l' -12 IOS with delta shellL=0 115.15 113.75L=2 25.32 23.30L=4 12.09 6.41L=6 3.16 1.98L=8 0.50 0.68L=10 0.06 0.2566It is noted that the delta shell has a much higher anisotropy parameter butstill gives smaller inelastic cross sections.4.2 Cross Sections at 300K4.2.1 Exact Cross Sections Including Only Open StatesAn exact calculation was done using as initial state, the thermally most proba-ble rotor state and velocity corresponding to a temperature of 300K (equation(190)) . For this temperature, j'=6, k' = 14.27 x 10 -10 m. -1 , and the maximumrotor state into which the molecule can scatter is j = 10. The present calculationwas restricted to including only the open rotor states in solving the SchrOdingerequation, thus ,max = 10. k'a 77 and )'max was chosen to be 120. Since allthe T-matrix elements are obtained from the calculation, all cross sections atthis total energy are readily available. These are reported in Table 2.Table 2: Exact Cross SectionsAll cross sections in A 2 , Total Energy=5.82 x 10' J.T.E. 1j'=0421Kj'=2404Kj'=4364Kj'=6300Kj'=8213Kj'=10103Kj =0 117.52 4.78 0.35 0.03 0.00 0.00j =2 22.93 126.24 6.06 0.55 0.05 0.00j =4 2.71 9.83 130.25 7.73 0.76 0.05j =6 .32 1.06 9.21 143.19 8.75 0.66j =8 .03 0.09 0.84 8.13 159.03 6.45j =10 .00 0.00 0.03 0.36 3.86 197.471. TE = Translational Energy expressed as an equivalent temperature(E = kBT).67From the detailed calculations it was found that significant contributions tothe cross sections fell off (ie., were of the order of 10 -6 A 2 ) for the j'=6 columnat 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 calculationof the condition numbers for both the K 2 +1 and W-1/jn inversions as requiredby equations (140) and (141). The worst condition numbers found were of theorder of about 2 x 10 4 for W — 1/jn and 2 x 10 7 for K 2 + 1. A further checkas to the reliability of both inverses was performed by an actual multiplicationof the matrix by its inverse and determining how close the result was to the unitmatrix. For the product of the W — 1/jn matrix (the matrix having conditionnumber of 2 x 10 4 ) and its inverse, the largest off-diagonal term was of the orderof 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 orderof 10-8 . Hence the inversion proved reliable for this set of initial conditions.One trend is noted: as the kinetic energy decreases (and j' goes up), theelastic j j' cross sections all increase.1 The condition number was calculated automatically by the inversion routine INVavailable as part of the support software at the 1.113C computing centre. According to ref-erence [41] the condition number is a form of Turing's N-condition number[42]. Generally,the larger the condition number, the poorer the inverse. Well conditioned matrices willhave a condition number of the order of N where N is the dimension of the matrix. At theother end of the scale, one of the most poorly conditioned matrices is the Hilbert matrixwhich has a condition number of order EXP(3.5N) [41]. For this run N=36.684.2.2 Exact Cross Sections With Inclusion of ClosedStatesA calculation was run using the closed state calculation of (181) and then com-pared with a calculation done under similar conditions using the open state cal-culation of (133). Close-state calculations require the use of modified sphericalBessel functions (MSBF), and the values used in these calculations were obtainedfrom a fitting of a table of values for MSBF for a given argument to a polyno-mial third order in A. Two closed states, j' =10 and j' =12 were included in thecalculations which are displayed in Table 3.69Table 3: Effect of Including Closed StatesPart A: Using only open states. Total energy corresponds to translationalenergy of 300K and rotational state of O. Units are in A 2 . j' denotes theinitial state, j the final state.j'=0 j'=2 j'=4 j'=6 j'=8j=0 126.95 5.00 0.49 0.06 0.01j=2 23.56 142.98 7.39 0.80 0.09j=4 3.59 11.39 145.24 8.08 0.84j=6 0.43 1.31 8.60 164.04 7.76j=8 0.04 0.10 0.61 5.21 202.36Part B: Inclusion of closed states. Total energy corresponds to translationalenergy of 300K and rotational state of 0. Units are in A 2 . j' denotes theinitial state, j the final state.j 1 =0 j'=2 j'=4 j'=6 j'=8j=0 126.94 5.00 0.49 0.06 0.01j=2 23.57 142.97 7.39 0.80 0.07j=4 3.58 11.40 145.31 8.09 0.69j=6 0.43 1.31 8.61 166.11 6.41j=8 0.03 0.08 0.49 4.31 208.20From the above table it is concluded that closed states only affect the calcula-tions for those cross sections involving high rotor states (104-8, 84-10, 104-10).These differences can be of the order of about 20% but are confined to crosssections in the lower right hand corner of the above table of values (ie., highrotor states).704.2.3 IOS 0 ---+ L Cross SectionsThe IOS calculations are carried out in two steps. First, the o-L,0 cross sectionsare calculated as given by (83) and then used along with the scaling relations asgiven by (84) to give the particular j 4- j' transition cross section. The first ofthese steps, the results of the calculation of the (1_0 cross sections, are givenand compared with the exact results in this section.Before any calculations using the IOS approximation can be performed, it isnecessary to choose values for the parameters Ao and k0 . In equation (83), A owas chosen to be an average of the ). and A' values associated with the particularT-matrix element calculated. Hence for each combination of A and A' that givesa different ) value, another T-matrix element was calculated.The same choices for k0 are available — one can choose k 0 as being equalto the k value corresponding to the j' state or to the k value corresponding tothe j state or as being an average of both. For the L 4- 0 calculations, both k ipcorresponding to the j' state and ko corresponding to the j state were used.A Gauss Legendre angular integration scheme using 40 points was performedfor 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 crosssection contributions of less than 10' A 2 ) at A=87 for ko = 14.27 x 10 -10m.-1The unitarity of the S-matrix summing from L=0 to L=30, ie.,L=30 2E 1.00000 (206)L=0was verified to 5 significant figures. The following results were obtained:71Table 4: IOS o- L,0 Cross SectionsAll cross sections in A 2FinalStateLFinal kin A -1Trans.Energy(T .E.)Usingko = kiiiit(16.92A ')T.E.=421.5KUsingko = kfinalUsingko =14.27 A-1T.E.=300KExactResults, k =16.92 A -1T.E.=421.5KL =0 16.92 421.5K 101.90 101.90 113.75 117.52L =2 16.57 404.2K 23.58 23.09 23.30 22.93L =4 15.71 363.7K 5.85 5.50 6.41 2.71L =6 14.27 300.0K 1.64 1.98 1.98 0.32L =8 12.03 213.2K 0.52 0.80 0.68 0.03L =10 8.37 103.2K 0.18 0.38 0.25 0.01L =12 0.07 0.10L =14 0.03 0.04L =16 0.01 0.02L =18 0.01 0.01The general trend to be noted is that the IOS calculations underestimate theelastic 04-0 transitions and overestimate the inelastic cross sections. This canbe explained by recognizing that a difference in energy between two states servesto hinder the excitation of the higher state. Hence with an approximation suchas the 10S, which treats all rotational states as degenerate, it would be expectedthat 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 rotoris moving more quickly with respect to the incoming atom than at say L =0.Hence the collision is less sudden at high L and it would be expected the IOSapproximation to be less accurate for large angular momentum transfer. This79trend would be expected for transitions for j' since all other cross sectionsare dependent on these 0 1, 4-0 values by (84).One other effect arises because the IOS treats all rotational states as degener-ate. This is that energetically inaccessible states are allowed, in particular, in thepresent calculation, transitions are allowed from L=0 to rotational states higherthan L=10. The cross sections for these transitions however are so low that theymay be effectively neglected. The next section contains some IOS results whereenergetically forbidden transitions are fairly significant.Finally, the results above show that choosing the k 0 parameter to be eitherthe initial or final k value does not significantly change the results. The onlydifference the k0 choice seems to make is in the elastic cross section, where achoice of k0 =14.27 A -1 gives a value closer to the exact value than choosingk0 =16.92 A -1 , the actual initial k value for this transition.The above trends apply to the 0L-0 calculations. Since the crosssections are calculated from these aL,0 cross sections, the differences betweenexact and IOS calculations noted here will be further examined after the scalingrelations are applied to better determine how the IOS and exact calculationscompare.7:34.2.4 IOS Scaling Relations4.2.4.1 Using IOS 0L 4_0 Cross SectionsSince the 105 T-matrix was 0 dependent it was expanded in Legendre polynomialsin equation (75). This expansion subsequently led to a scaling relationship (84)which expresses j j' cross sections in terms of L 0 cross sections. Thisrelationship 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 sectionsis 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. Aswell, different ko choices are examined in light of the scaling relations .For each of the three choices of 4, ko = ko = kfinai and 14 ) =-14.92 A -1 ,the IOS o-3,j, cross sections were calculated using the 0L __0 cross sections fromTable 4 and equation (84). The sum in (84) was taken up to the maximumallowed L value.For the choice of ko = ki„itiai , the cross sections are given in Table 5:74Table 5: IOS Cross Sections at 300K Using k o = kinitialPart 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'=10j=0 101.90 4.62 0.61 0.15 0.05 0.02j=2 23.58 111.03 7.48 1.40 0.37 0.13j=4 5.85 13.86 113.22 8.54 1.63 0.46j=6 1.64 3.17 11.65 120.97 8.43 1.70j=8 0.52 0.89 2.47 11.60 132.19 7.41j=10 0.18 0.29 0.64 2.70 10.65 155.32j=12 0.07 0.10 0.19 0.78 2.64 8.96j=14 0.03 0.04 0.06 0.26 0.81 2.50Part B: Ratio of IOS to exact cross sections.j'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0 0.87 0.97 1.75 4.48 15.9 154j=2 1.03 0.88 1.23 2.55 6.92 46.7j=4 2.15 1.41 0.87 1.11 2.14 9.80j=6 5.21 2.99 1.27 0.85 0.96 2.59j=8 20.4 9.36 2.93 1.43 0.83 1.15j=10 297 98.3 20.6 7.40 2.76 0.79The same trend is observed for cross sections with higher initial j-state as wasobserved for the j'=0 initial state cross sections - 105 inelastic cross sectionsare larger than the exact values while the elastic cross sections are lower. Theresults actually seem to get better for larger j' on using the scaling relationshipbut this is basically due to the nature of the summation. For example, the j'=6column depends less on the large L-valued o-L-0 cross section than the j' =0columns. That is, to get the 104-0 cross section, only for L=10 is used,75which is 297 times the exact o-L,0 value. To get the 104-6 term, the o-L 4-0 crosssections for the L=4, L=6, . . L=16 terms are used. And for L=4, the IOSaL,0 is only 2.15 times the exact 471_0 value. Not only are the lower-L 0L,0values more accurate, but they also contribute more in equation (84) relativeto the other terms which also leads to better results for cross sections in themid-table region.With these points in mind, the L 0 results should be compared with thej j' results in terms of how many rotor states the initial state is from theelastic transition. For instance, the 0+-4 should be compared with the 24-6transition. On this basis, the scaling law given by (84) preserves the ratios of the10S to exact cross sections for the L 0 transitions when it is used to calculatethe j 4— j' transitions.The scaling relation also allows energetically forbidden cross sections andthese may be of significant size. For example, the 105 124-10 transition iscalculated at 8.96 A 2 but is 0 (ie., not allowed) for the exact calculation.The cross sections using the choice ko = kfinm are given in Table 6.76Table 6: IOS Cross Sections at 300K Using k0 = kfinalPart 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'=14j=0 101.90 4.72 0.65 0.13 0.03 0.01 0.00 0.00j=2 23.09 111.03 7.70 1.22 0.26 0.07 0.02 0.01j=4 5.50 13.46 1 13.22 8.06 1.31 0.27 0.07 0.02j=6 1.98 3.64 12.33 120.97 8.87 1.67 0.40 0.12j=8 0.80 1.24 3.07 11.02 132.19 8.62 1.79 0.47j=10 0.38 0.53 1.07 2.75 9.16 155.32 7.53 1.81Part B: Ratio of IOS to exact cross sections.j 1 =0 j'=2 j'=4 j'=6 j'=8 j'=10j=0 0.87 0.99 1.86 3.71 10.3 72.7j=2 1.01 0.88 1.27 2.22 4.94 25.1j=4 2.03 1.37 0.87 1.04 1.72 5.86j=6 6.29 3.44 1.34 0.85 1.01 2.55j=8 31.4 13.1 3.65 1.36 0.83 1.34j=10 627 183 34.5 7.53 2.37 0.79Compared with the k0 = knn t,1„1 choice the ko = kfinal choice was better (althoughonly of the order of about 5% for values about 1.0 times the exact value) in 17cases and the k0 = ',Initial was better in 13 cases with the two being the same forthe elastic cross sections. The following table shows which of the two choices isbest for each transition:77Table 7: Comparison of kmitial and kfi„ai IOS CrossSectionsThe best agreement with the exact cross sections is : I if Initial state, F ifFinal state or E if equal. j' notes initial state, j the final state.j'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0j=2j=4j=6j=8j=10EFFIIIFEFIIIIIEIIIFFFEFIFFFFEFFFFFIERather than analyzing the above trend in terms of initial or final state pa-rameter ko , it is useful to analyze the results in terms of choice of kmax or kmm .Table 8 demonstrates the trend for the best ko value being ko = kmax for largeIAA transitions and ko = kmi„ generally being the best choice for transitionswhere Ion I=2.78Table 8: Comparison of kmax and IOS CrossSectionsThe best agreement with the exact cross sections is: Max if ko = kinax, Min ifko = kinin or E if equal. j' notes initial state, j the final statej'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0j=2j=4j=6j=8j=10EMinMinMaxMaxMaxMaxEMinMaxMaxMaxMinMinEMaxMaxMaxMaxMaxMaxEMinMaxMaxMaxMaxMaxEMinMaxMaxMaxMaxMinEThe above two tables can be rationalized as follows: Since the total energyin a collision is conserved, a downward transition (going from a high rotor stateto a low rotor state) corresponds to a km;„ to kmax transition while an upwardtransition corresponds to a kmax to a k11m transition. Now for the a-L„._0 crosssections the k1111, value is always the final k o choice and the kmax value is alwaysthe initial ko choice. Regarding Table 4 it is noted that for o•2,0 and a4.._0the kfi„al or k11111 choice is best while for cr6-0, (3-8 4_0 and o-10,_0, kiiijtiai or kmaxis best. By the scaling relationship in (84), transitions are governed mainly bythe o-L _.0 value corresponding to L = 1j/ — j since this is the largest term.Assuming that more accurate aL,o values will give more accurate aj,j, values,then downward transitions (ie. going from a low ko to a high ko) with IAAgreater than or equal to 4 would tend to favour a k final choice or kmax . Upwardstransitions (going from a high ko to a low ko ) with 16,j1 greater than or equal79to 4 would favor the kmax choice as well but this now corresponds to the k initialchoice. For transitions with downward transitions would favour Ic i„itiaior kniax (as is the case in the 04-2, 44-6 and 44-8 transitions) and upwardtransitions would favour kfi„al or kmm (as is the case in 24-0, 44-2, 84-6, and104-8 ).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:$0Table 9: IOS Cross Sections At 300K Usingko = 14.27 A -1Part 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'=14j=0 113.75 4.66 0.71 0.15 0.04 0.01 0.00 0.00j=2 23.30 122.24 7.93 1.40 0.33 0.09 0.03 0.01j=4 6.41 14.27 121.25 8.54 1.57 0.38 0.11 0.03j=6 1.98 3.64 12.33 120.97 8.87 1.67 0.40 0.12j=8 0.68 1.11 2.97 11.60 120.87 9.08 1.74 0.43j=10 0.25 0.38 0.88 2.70 11.22 120.82 9.23 1.79j=12 0.10 0.14 0.30 0.78 2.55 10.98 120.80 9.33j=14 0.04 0.06 0.11 0.26 0.73 2.47 10.83 120.78Part B: Ratio of 105 to exact cross sections:1=0 j'=2 j'=4 j'=6 j'=8 f=10j=0 0.97 0.97 2.04 4.48 15.6 1031=2 1.02 0.97 1.31 2.55 6.20 33.4j=4 2.36 1.45 0.93 1.11 2.07 8.02j=6 6.29 3.44 1.34 0.85 1.01 2.55j=8 26.8 11.7 3.53 1.43 0.76 1.41j=10 419 131 28.3 7.40 2.91 0.61In comparison with the previous results, the k o =14.27A -1 choice works bestin the elastic cross sections. This can be accounted for in that the largest termin the sum in (84) is the term involving the 00-0 value and this is estimatedbetter when kb is chosen as ko = 14.27 A -1 rather than kinitiai or kfinai. In theinelastic transitions, however, the kinitial and kfilim choices are better. A further81initial kfinalkaverage2(207)investigation could study a kaverage value wherebut since the results in the above three choices of k o differ much less amongthemselves than they do with the exact results, little change from what hasalready been given would be expected.The main point to note about the scaling relations is that the discrepanciesnoted in the cri,–.0 cross sections are carried over into the 0-.7,3 , cross sectionswith the results getting neither better nor worse. This is important to note sinceit suggests these scaling laws can be applied to the exact results which is thetopic of investigation in the next section.4.2.4.2 Using Exact Cross SectionsThere are two parts to the prediction of by means of the scaling law in (84)— the values of the 0L,0 cross sections and the way these al.„_0 cross sectionsare combined with the 3-j symbols — each affects the result. In particular, thequestion arises if the 3-j symbols predict the correct j, j' dependence ofThis can easily be tested with the delta-shell potential by putting in the exact01.–o values in (84) to determine how the resulting j j' cross sections comparewith their corresponding exact cross sections. The cross sections obtained usingthe exact 0 L,_0 values of Table 4 are reported in Table 10.82Table 10: IOS Cross Sections At 300K Using ExactaL, 0 ValuesPart 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'=14j=0 117.52 4.59 0.30 0.02 0.00 0.00 0.00 0.00j=2 22.93 124.85 7.00 0.51 0.04 0.00 0.00 0.00j=4 2.71 12.60 123.97 7.63 0.58 0.05 0.00 0.00j=6 0.32 1.32 11.02 123.80 7.97 0.62 0.05 0.00j=8 0.03 0.14 1.10 10.43 123.75 8.18 0.65 0.06j=10 0.00 0.01 0.12 1.01 10.11 123.72 8.32 0.67j=12 0.00 0.00 0.01 0.10 0.96 9.91 123.70 8.43j=14 0.00 0.00 0.00 0.01 0.10 0.92 9.77 123.70Part B: Ratio of IOS to exact cross sections.j'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0 1.00 0.96 0.86 0.71 0.51 0.25j=2 1.00 0.99 1.15 0.93 0.79 0.94j=4 1.00 1.28 0.95 0.99 0.76 1.05j=6 1.00 1.25 1.20 0.87 0.91 0.95j=8 1.00 1.51 1.30 1.28 0.78 1.27j=10 1.00 3.67 3.71 2.76 2.62 0.63Two trends in the above table can be noted: downward transitions are calculatedas somewhat lower than the actual value, and upward transitions are calculatedas being higher. As well, the elastic cross sections become progressively less thanthe actual values as j' increases.That the downward transitions are lower than upward transitions can beaccounted for by considering that the IOS has replaced wave number ki with83one fixed parameter ko . One implication of this approximation can be seen inthe detailed balance equation:Exact [j] k 2Ell) = 0-3E'ix,__va3c't(-Elk (j)1.) ji (208)where [A = 2j + 1 and kj is the k value corresponding to the state with rotorquantum number equal to j. For the IOS case, with lq = (208)becomes:(7.1)(234E) = (E). (209)(The above could also be derived using (84) for each of cri_j, and a- i, andequating the two expressions.) For downward transitions kj, < k j so ki /ki, > 1but by (209) the 105 approximates this term as 1. As well, in upward transitionsthe term k3 /k3 , < 1 is also approximated by 1. If for j' > j (ie., a downwardtransition) the cr3 ,,_ i term is very close to the exact term, then the resulting(73,3 , cross section would then be lower than the exact value. Conversely, for anupward 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 sincethe o 3,, values are the exact values. For downward transitions with final stateother than 0, this trend would not always be expected. For example, the a 3 1,_iterm may be very much higher than the exact o 3 _3 , term which, in a downwardtransition could compensate for the lack of the k3 /kl , > 1 term and give anaccurate 105 downward transition.84The other trend — that elastic 105 cross sections become progressively lessthan the exact value for larger initial rotor state — can also be attributed partlyto the 105 approximating with k;i. The ko value used for the cross sectioncalculations was 16.92A -1 . These in turn were used to calculate the 0,;,j, crosssections for j' = 6 where k=14.27A -1 . If the exact 01.0 values correspondingto ki„i t iai =14.27A -1 are used, the following values (shown in comparison to thekinitia1=16.92A -1 values) are obtained:Table 11: Effect Of Using Different k Values In Exacto- L, 0 Values on the Cross SectionsExact 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 inTable 10:Transition k=14.92A -1Ratio toexact results k=16.92A -1Ratio toexact results04— 6 0.03 0.96 0.02 0.7124— 6 0.67 1.23 0.51 0.9344— 6 7.96 1.03 7.63 0.9964— 6 133.54 0.93 123.80 0.8784— 6 10.87 1.34 10.43 1.28The elastic cross section does get better on using the proper energy but mostof the other transitions do not. Hence while a proper accounting for differentwave numbers among transitions may allow the scaling relation in (84) to betterreplicate elastic cross sections, inelastic cross sections may require even furthertypes of corrections.85The results in Table 10 are now compared to results obtained using scalingrelation (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=16.92A -1Part A: Sums were done to maximum L using (84) and the crL,_0 values givenfor k=16.92 A -1 . Units are in A 2 . j' denotes the initial state, j the finalstate.j'=0 j'=2 j'=4 j'=6 j'=8 j'=10 j'=12 j'=14j=0 101.90 4.72 0.65 0.13 0.03 0.01 0.00 0.00j=2 23.58 110.30 7.87 1.25 0.26 0.06 0.02 0.01j=4 5.85 14.16 109.30 8.48 1.40 0.30 0.08 0.02j=6 1.64 3.25 12.26 109.04 8.82 1.49 0.33 0.09j=8 0.52 0.90 2.65 11.54 108.94 9.04 1.55 0.35j=10 0.18 0.28 0.71 2.41 11.16 108.90 9.19 1.60j=12 0.07 0.10 0.22 0.63 2.28 10.94 108.88 9.29j=14 0.03 0.04 0.08 0.19 0.59 2.20 10.78 108.87Part B: Ratio of 108 to exact cross sections.j'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0 0.87 0.99 1.86 3.71 10.3 72.7j=2 1.03 0.87 1.30 2.27 5.01 24.8j=4 2.15 1.44 0.84 1.10 1.84 6.49j=6 5.21 3.06 1.33 0.76 1.01 2.27j=8 20.4 9.49 3.14 1.42 0.69 1.40j=10 297 97.1 22.9 6.61 2.89 0.55The major finding to note is that in all but the 6<-8 and 04-2 transitionsthe cross sections obtained using the exact ol_o values fared better than those86obtained using the IOS 01-0 values. This demonstrates that the scaling lawworks best with a1,,..0 values that are closer to the exact results. (It couldhave been that the scaling law corrected for inaccurate trends in the 0i„_.0 crosssections — in which case using exact cri,,0 cross sections would give worseresults.)Further, the errors due to the scaling laws are generally less than those at-tributed to the o-L4. 0 values. For example, in the 44-6 transition, the use of anexact ol,,0 value and the scaling law gives a 1% error but the use of the IOS0 L,0 value and the scaling law gives a 10% error.The derivation of the scaling law involved only the angular momentum cou-pling made possible through a 0 parameterized T-matrix which was made possibleby the assumption of suddenness. The calculation of the IOS oL*—o values, involvenot only angular momentum simplifications but linear momentum simplificationsassociated with the assumption of suddenness. Since the scaling law at 300K ismore accurate than the crL„_.0 values, it appears that the concept of suddennessfor 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 RelationAs derived in (86), the Energy Corrected Scaling Relation requires a collisiontime T. Since the shell of the potential is of negligible width (it takes no timeto pass through the potential) 7 might be considered to be 0. Nonetheless two87proposals for a finite value of T are presented and then various T values are testedto determine whether the Energy-Corrected Sudden scaling relation can improvethe IOS results.One possibility for the calculation of T is to take the average time for astraight line trajectory through the sphere inside the Delta Shell potential ata given impact parameter b. Assuming the velocity v during the collision isconstant, T is calculated to be(1 a = — — Va2 — b2db —7a (210)v a fo 2vwhere a is the radius of the delta-shell. The value for v could correspond to thefinal or initial collision velocity or some average of the two. Table 13 gives theresult 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 initialstate k value, whether this corresponded to the lowest velocity or not.88Table 13: ECIOS Cross Sections At 300K UsingT = 71( (21) min) and Exact aL_0(Ek,,,,,,,,) ValuesPart A: Units are in A 2 . j' denotes initial state, j the final state.j'=0 j'=2 j'=4 j'=6 j'=8 j 1 =10j=0 101.90 2.38 0.02 0.00 0.00 0.00j=2 12.14 111.03 0.65 0.00 0.00 0.00j=4 0.17 1.21 113.22 0.13 0.00 0.00j=6 0.00 0.01 0.18 120.97 0.02 0.00j=8 0.00 0.00 0.00 0.03 132.19 0.00j=10 0.00 0.00 0.00 0.00 0.00 155.32Part B: Ratio of ECIOS to exact cross sections.j'=0 j'=2 j'=4 f=6 j'=8 j'=10j=0 0.87 0.50 0.05 0.01 0.00 0.00j=2 0.53 0.88 0.11 0.01 0.00 0.00j=4 0.06 0.12 0.87 0.02 0.00 0.00j=6 0.01 0.01 0.02 0.85 0.00 0.00j=8 0.00 0.00 0.00 0.00 0.83 0.00j=10 0.00 0.00 0.00 0.00 0.00 0.79Aside from the elastic cross sections, which are not affected by the ECIOSscaling relation, all cross sections have been practically reduced to 0 by thischoice of T. Clearly this choice of T is too large. This does however demonstratehow the ECIOS scaling relation works. The 10S, by assuming a 0 collisiontime, overestimates inelastic cross sections. The worse this assumption (ie., theless sudden the collision) the more the IOS will overestimate the inelastic crosssections. Introducing a correction term inversely proportional to a collision time,as is done in the ECIOS scaling relation, reduces inelastic collision cross sections.89Further studies were done on finding a better T value. If a factor f is definedso thataT . f. —v(211)then Table 13 displays the results obtained using f=1.57. It was found that anf value of 100 completely reduces all inelastic cross sections to 0.00 A 2 whilean f value of 1x10 -5 does not change any of the IOS cross sections. In theinvestigation as to which T value worked best for the ECIOS scaling relation itwas found that while one choice of f was sufficient to get a column (ie. a setof cross sections with the same initial state) of IOS upwards transitions within55% 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 fitthe exact results as well as the f value used to obtain these results. Upwardtransitions can be calculated using detailed balance and would then give thesame ratio to exact cross sections as their respective downward transition.90Table 14: ECIOS Cross Sections At 300K UsingT = fa/vmin and 'OS aL,0 (Ekma.) ValuesPart A: Cross sections are in A 2 . j' denotes initial state, j the final state.j'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0 101.90j=2 22.91 111.03j=4 4.22 12.60 113.22j=6 0.43 1.64 10.01 120.97.i=8 0.02 0.11 0.94 8.63 132.19j=10 0.00 0.00 0.03 0.37 4.57 155.32f value 0.300 0.225 0.165 0.145 0.145 0.79Part B: Ratio of ECIOS to exact cross sections.0 2 4 6 8 10j=0 0.87j=2 1.00 0.88j=4 1.55 1.28 0.87j=6 1.37 1.54 1.09 0.85j=8 0.84 1.15 1.12 1.06 0.83j=10 0.75 0.89 0.89 1.03 1.18 0.79It can be seen from Table 14 that the actual time that is needed to makethe scaling relation work best is actually about 1/10th to 1/5th that calculatedwhen it was assumed that the interaction lasts for the whole time it takes for theatom 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 resultthat calculates the 01-0 values based on the highest velocity and hence highest91energy. DePristo et. al. [32] recommend the minimum k value for the calculationof o-L-0 . This is based on the assumption that if the collision is sudden at theminimum 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 aBoltzmann average of cross sections, found that using the initial energy forupward j 4-- 0 transitions and the initial energy for downward 0 4- j' transitionsgave best results. The present study comes to the same conclusion.If, however, the minimum k value is used in both the calculation of theol,_0 cross sections and the ECIOS scaling relation then it is not possible toget the results to all agree within a 50% deviation of the exact results, as isdemonstrated in Table 15, where the cross sections for initial rotor state equalto 10 are reported. The f value used was that which gave results that deviatedleast from the exact cross sections.Table 15: ECIOS Cross Sections at 300K UsingT = 0.16a/vrnin and IOS (Ekr,„,i) ValuesTransitionCross Sectionin A 2Ratio toExact Results04-10 0.00 3.824-10 0.00 1.444-10 0.02 0.466+-10 0.18 0.2884-10 2.75 0.43104-10 155.32 0.799 9The resulting cr -1() ,(Ek,„,„) cross sections with scaling could not be fittedto the exact results as closely as were the or -'3',(Ekm,„) cross sections. Theratio of results:exact ranged by an order of 10 using minimum k value for thecrL,o cross sections and ranged by an order of about 1.5 using the maximumk value. A possible interpretation of why different k values work best in thescaling relation and 01,0 cross sections is that the IOS calculates the suddenpart of the collision where the kmax value would be dominant while the ECIOSscaling 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 boththe 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 theresults in Table 14.9:3Table 16: o- L-0 k Values and ECIOS 7- Values Used InTable 14For each entry, the upper number is the k value in A -1 used to calculatethe o L_O cross sections while the lower number is the collision time r infs (10-15 s.). Elastic cross sections are not corrected by the ECIOS scalingrelations and so there are no collision times for these values. j' denotes ini-tial state, j the final state.j'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0 16.92j=2 16.92 16.57258.3j=4 16.92 16.57 15.71272.3 204.2j=6 16.92 16.57 15.71 14.27299.8 224.8 164.9j=8 16.92 16.57 15.71 14.27 12.03355.6 266.7 195.6 171.9j=10 16.92 16.57 15.71 14.27 12.03 8.37511.0 383.3 281.1 247.0 247.0Factor 0.300 0.225 0.165 0.145 0.145If the T value is regarded as a reliable measure of the time of interactionthen the collision time seems to be proportional to the difference in rotor statesbetween initial and final states. For example, transitions where jAjj=2 seem torequire about 200 fs. whereas those with 0,j1=8 require about 370 fs. Thiscould explain why using the least T value in the correction works so well, as wellas why a different factor must be used for each j' value. If the same r value wasused for the j'=0 and f=2 columns then the 44-0 and 44-2 would have thesame T value. But because the j'.0 column requires a larger f value than the94j'=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 fbased on initial state gives a set of actual values that increase with IAA. Notethat the same sort of effect could not be achieved by using a T value dependenton the average of initial and final velocities. For example, using an average Tvelocity for the 104-0 transition (lAj1=10) would give a T value much less thanusing an average velocity for the 104-8 transition ( 1= 2 )•One final point was investigated — whether using the different k values inthe 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 kvalue for the calculation of the (31_ 0 cross sections and least velocity value forthe calculation of 7 in the ECIOS scaling relation) there does not appear to bea direct relationship between the two possible choices. Table 17 demonstratesthat if the 0L- 0 cross sections are calculated at a lower k value, then a higher Tterm in the ECIOS scaling relation is not always needed to compensate for this.95Table 17: How f Varies According To the k Used inthe Calculation of the o - L, 0 Cross SectionsThe f values chosen so that the 64-4, 84-4 and 104-4 ECIOS cross sectionsare within 15% of the exact values.k Used toCalculate 0L,0 in A -1f ValueGiving Best Results16.9215.718.370.1700.1650.190The lack of correlation between the 0L-0 value and ECIOS scaling relationvelocity 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 kvalue used in the IOS o-L,0 cross section calculations.In summary, then, the results in this section recommend for the calculation ofIOS o-L_o cross sections the use of the highest k value and then the use of thesecross sections to calculate upward transitions. These cross sections are then tobe used with the ECIOS scaling relations, where best results are obtained whenthe least velocity weighted by a factor dependent on the initial state is used forthe r value. The net effect of this method of calculating cross sections is toallow for a collision time in the 10S, and further, for this collision time to bedependent on the magnitude of lAjl for the transition.Finding that r increases as does lAjlor .AE appears to violate the uncertaintyprinciple. This seems to imply that interpreting r as only a collision time may96be too narrow a definition of T. Since E ., — is also dependent on lAjl viah 2— =(J — i i )(3 + 3 + 1 ) 27it could well be that instead of interpreting (E3 — Ej i)7. in the ECIOS correctionfactor as an energy-collision time term it is found out that an (energy) 2 termgives a better approximation to the exact cross sections.4.2.6 General S-Matrix Scaling Relation4.2.6.1 Using IOS 0L-0 Cross SectionsIn this section the General S-Matrix Scaling Relation (GSMSR, Equation (89))is studied with regards to how well it can correct the IOS 0 L-0 cross sections.That is, even though exact o-j, values are intended to be used in the GSMSR,it is investigated whether the GSMSR can improve the regular 105 results.In replacing the exact aL,o(Ek + EL) with IOS 01.0(Ek0 ), there is somearbitrariness as to the choice of Eko since in the 105 all rotational states aredegenerate. The choice is made to interpret the [L]o-PanEk + EL) term asairl(Ek0 ). That is, the choice of the k'd value which will replace operator k 2will correspond to the kinetic energy Ek with the EL term ignored. With thissubstitution Equation (89) becomes2j j' L r6 +RE', — COT /2hi 2 ] 2 IOS i(Ek+Cp) = ( 2j+1) E 01,,,o(Ek).0 0 ) [6 + [ (( EL — cj)7/2h1 2 L °(213)As well, the optimal choice of 7 was involved using for each j', an f value and theminimum collision velocity. This resulted in the cross sections listed in Table 18.(212)97The cross sections were calculated using (213) and the values where Ely is thetranslational energy associated with rotor state j' from Table 5. The T value iscalculated using the minimum collision velocity, the value for f reported belowand equation (211).Table 18: GSMSR Cross Sections At 300K Using IOSo- L,0 ValuesPart A: j' denotes initial state and j the final state. All cross sections arein A 2 .j' =0 j' =2 j' =4 j' =6 j' =8 j' =10j=0 101.90 4.62 0.61 0.15 0.05 0.02j=2 23.58 111.55 7.25 1.27 0.32 0.11j=4 5.85 13.53 115.11 7.92 1.27 0.32j=6 1.64 2.73 9.98 122.88 7.52 1.14j=8 0.52 0.72 1.44 9.64 134.71 6.33j=10 0.18 0.23 0.30 1.48 9.30 158.25f Value 1 0.25 0.30 0.20 0.15 0.101 The r value cancels out for these transitionsPart B: Ratio of G.SAISR to exact cross sections.j' =0 j' =4 j' =6 j' =8 j' =10j=0 0.87 0.97 1.7 4.5 16 1.5x102j=2 1.0 0.88 1.2 2.3 6.1 40j=4 2.2 1.4 0.88 1.0 1.7 6.9j=6 5.2 2.6 1.1 0.86 0.86 1.7j=8 20 7.6 1.7 1.2 0.85 1.0j=10 3.0x102 79 9.7 4.0 2.4 0.8098For transitions with j'=0, (213) reduces to0.9SM/ E 0.0S34-0 kJ-1k + 3 = 4---o(ik)and for those with j=0, (213) reduces tocsm E k —a.081i-0 k(E )ao—p r 313 1so there is no change in these values from Table 5. The other values report amarginal improvement over Table 5 (ie., thelOS Scaling Relation with IOS o-L 4_0values) in that elastic cross sections are increased and inelastic cross sectionsare decreased, more in keeping with exact results. The 7 values which workedbest tend to decrease as j' increases but this is not a definite trend as the j'=2column demonstrates.An alternative for these calculations is to convert (89) using the equation fordetailed balance (equation (208)) to_3,3C;S.1.111 (Ek Ea' )= (2i + i)E J 3L o oL ) 2 r6 + [(EL — E0)7/2h] 2 2 (Ek + EL) Exact0 ) [6 + Rej, — ti)7/2/42 .1 Ek(216)and replace o-Exa: with o-Pso . This results incrGS.111 ( E , + ci , )3 4-3 k i''( i j' L ) 2 16 + [(CL — f0) 71 2h] 2 1 2 (Ek + EL) = (2j + 1)E r,—o- (217)L 0 0 0 ) [6 + [(ei ' — fi )-r/2/42 ] E—kComparing (217) with (216) reveals that the effect of this treatment would beto multiply all cross sections by a factor of (Ek + EL)/Ek > 1 which would(214)(215)99lead to even poorer agreement with the exact cross sections. Other ways ofapproximating the ofx o(Ek eL ) term by a cri 0 (Eko ) value would not beexpected to significantly improve these results owing to the fact that the valuesof (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 withexact 0l_0 values does have an advantage over the ECIOS (one intended forIOS o-L, 4_0 values) in increasing elastic cross sections. With regards to inelasticcross sections, however, the ECIOS does a better job than the GSMSR of scalingdown the high an, values to match exact cross sections.4.2.6.2 Using Exact o-L-0 Cross SectionsThe GSMSR was tested as to how well it applied to reproducing the exact re-sults. In order to calculate the column of cross sections with f=10, the GSMSRrequired 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 forthe calculation of cross sections for the GSMSR. Ek is the energy according tothe most probable translational energy at 103K. J = A j is the total angularmomentum. The last contributing .1 value is that value of J for which there wasa contribution of at least 1 x 10' A 2 to the cross section.100Table 19: Input o-L-0(Ek + EL ) Values for the GSMSRat 300K for Y=10Largest Max. Max. Max. Contri- Calc. Cross Cross condition rotor J buting time Section Section number' state J (sec) 2 in A2ao_o(Ek + 6 0) 2.2x10 3 4 80 84 55 3.3 1.77x10 20-04-2(Ek + 6 2) 1.4x10 5 4 80 84 59 3.3 4.71x1000-0■-4(Ek 64) 8.5x10 5 6 90 96 67 6.7 4.32x10 -10-0■-6(Ek + 6 6) 9.5x10 6 8 100 108 78 14.5 6.50x10 -270 4_8(Ek E8) 1.1x10 7 8 110 118 90 16.2 3.55x10 -3Cro,m(Ek + 610) 1.9x10 7 10 120 130 104 68.9 1.17x10 -4Cr0■-12(Ek €12) 6.1x10 5 12 140 152 118 81.2 3.11x10 -60-0.-14(Ek 614) 3.5x10 6 14 150 164 132 162.1 1.50x10 -71. Not all the conditions numbers checked for the 0,-10 transition(Calculations for the 04-10 transition done on an Amdahl 470 V8)2. The calculation time is that required for the Amdahl 5840.The GSMSR also requires the calculation of a collision time T. Using anaverage 7 value with minimum v resulted in inelastic cross sections that weretoo 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 thecoefficient in each term in the sum in (89) becomes[6 [L(L OhT/(442 2(218)6Not only does a decrease in T decrease elastic cross sections, but it also increasesinelastic cross sections since the coefficient in each term for these cross sections101in (89) becomes6 + [L(L 1)/i7/(41)? 6 + [[ji(ji + 1) — j(j + 1)] tir/(4/)1 2(219)and for L small (where most of the contribution occurs) and j 1 =10 and j=2,4,6,8Pi' +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 within25% of the exact results as is shown in Table 20.Table 20: GSMSR at 300K Using Exact aL4_0(Ek + EL)ValuesThe a-L,_0 (Ek + L) values used are from Table 19. 7 = 1.28 x 10' s.Transition CrossSectionin A 2Ratio toExactResults0 4— 10 0.00 1.002 <— 10 0.00 1.254 <— 10 0.06 1.246 4— 10 0.49 0.758 4— 10 6.73 1.0410 <— 10 184 0.93The results in Table 20 demonstrate that the GSMSR, with its allowance of acollision time, is about as accurate in predicting cross sections as the regular IOSScaling 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 toQj •/(Ek + eji ) = 0-1.,o (Ek ejl) (221)102with j = L). The success of the GSMSR then seems to come not so muchin accounting for a collision time but in that it makes use of the IOS scalingrelationship.One further result from Table 20 is that the 7 value was found to havethe same magnitude as in the ECIOS case, again suggesting the actual time ofinteraction to be about 1/10th the time it takes for a particle going at minimumcollision velocity to traverse a shell with radius a=5.5A. . There are, however,some limitations to interpreting this T value as an actual time of collision. Thisis demonstrated by adjusting the 7 value for each transition in order for the crosssection to fit with exact results. Table 21 lists the T values required to get crosssections that are within 1% of the value of the exact results.Table 21: -r- Values Required to Match the GSMSRwith Exact Results at 300KThe T reported below is that value required by the GSMSR to give cross sec-tions to within 1% of the exact values.Transition 7 valuein seconds2 4- 10 1.79 x 10'4 4- 10 1.53 x 10'64-1010 0.89 x 10'8 4- 10 1.36 x 10–n104-1010 8.52 x 10-13The 7 value required to fit the elastic cross section to exact results is 5-10 times higher than the 7 value required for inelastic cross sections but there103does not appear to be any physical reason why this should be so. As well, nodirect 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 aninterpretation of the 7 value as being an accurate measure of the collision time.4.2.7 Accessible States Scaling Relation4.2.7.1 Using IOS ol_ o Cross SectionsIn the previous section, the GSMSR used the assumptions of the IOS to relateo-i,j,(Ek+ej,) values with cro —L(Ek+cL) cross sections. In the Accessible StatesScaling Relation (ASSR), the assumptions involve a simplification of angularcoupling coefficients, neglect of quantum tunnelling and a statistical treatmentof transition probabilities [33]. This leads to an alternative relationship betweencro—L(Ek + €L) values with o- , , (Ek + t,i) cross sections. In this section severalcross sections are calculated using the ASSR in order to determine how well itcan reproduce exact cross sections.The difference of the ASSR with the GSMSR is that the ASSR uses only oneao.L(Ek+cL) value for each o-j-_, , (Ek+€3 , ) cross section calculated whereas theGSMSR involves a sum of (30,__L(Ek+EL) values for calculation of the cr,i(Ek+ej ,) cross section. For this reason the ASSR calculations are easier and less costlyto obtain than the GSMSR calculations. As mentioned above, since the ASSR issimilar to the GSMSR in that it relates o-j _j i(Ek (J O cross sections with crosssections of a different energy, namely the o-o—L(Ek + td cross section.104Although 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 iswith the IOS values. Comparing these results with exact results, it will bedetermined how compatible the assumptions that lead to the 105 cross sectionsare with the assumptions that lead to the ASSR.As in the GSMSR when IOS .91_ 0 values were used, the choice is made to (E . .interpret the quantity [j' — j]o k. (- 13 ,4 as ar,_so(Eko)•substitution, equation (91) becomes [j] N ( Ek p--EJ) loscy,,sP—.71 D(Ek (ii) = • CfulLal.-0(Eko)•ii N(EkWith this(222)Table 22 lists the cross sections calculated from (222) using the IOS crosssections of Table 5.105Table 22: ASSR Cross Sections at 300K Using IOScrL,0 ValuesPart A: Accessible States Cross Sections using 01-0(Ek j,). j' denotes initialstate and j the final state. All cross sections are in A 2 .j'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0 101.90 4.62 0.61 0.15 0.05 0.02j=2 117.88 102.81 11.35 2.24 0.58 0.18j=4 57.39 41.56 96.54 13.20 2.82 0.72j=6 25.24 16.15 29.51 93.07 12.13 2.11j=8 11.18 5.97 10.38 24.93 91.29 6.71j=10 5.13 2.76 3.82 9.42 19.59 67.94Part B: Ratio of above to the exact cross sectionsj'=0 j'=2 j'=4 j'=6 j'=8 j'=10j=0 0.87 0.97 1.7 4.5 16 1.5 x 102j=2 5.1 0.81 1.9 4.1 11 66j=4 21 4.2 0.74 1.7 3.7 15j=6 80 15 3.2 0.65 1.4 3.2j=8 4.4x10 2 63 12 3.1 0.57 1.0j=10 8.5x103 9.4x10 2 1.2x10 2 26 5.1 0.34Downward transitions appear more accurate than upward transitions at thesame j' and IAjl. Elastic cross sections underestimate exact results and becomeincreasingly lower as j' increases.For transitions with j=0, (222) reduces to the expression1 s( EA (3') = -C'ko (223)and so for these transitions the ASSR using IOS 01_0 values is the same as theIOS scaling relation using or so values. For all other transitions, however, these106ASSR cross sections prove much more inaccurate than the IOS scaling relationusing aro values in that the ASSR reports higher inelastic and lower elasticcross sections than the IOS results. An alternative calculation, ie., use in (91)the exact relation) (224)r •/ .1 Exact tEk = [Ek Exact1.1 — Ek ,3Lik and then approximate atLa .c7 1_0 (Ek cw_ ji ) by arso (Eko ) as outlined in (216)to (217) for the General S-Matrix Scaling relation, leads to all cross sectionsbeing multiplied by a factor of (Ek tb,_,1)/Ek > 1. This would increase thealready high inelastic cross sections.From the results in this section it appears important to retain the angularcoupling coefficients (the 3-j symbols) when relating or:, cross sections withcri°50' values since the scaling relations based on the 3-j symbols (the regularIOS and ECIOS scaling relations) prove better than the ASSR. The ASSR, byassuming these coefficients to be functions of differences in j and j' only, is notcompatible with the ar so values since it does not correct for and in fact amplifiesthe inaccuracies resulting from using the aro values, namely, low elastic andhigh inelastic cross sections.4.2.7.2 Using Exact a-L_o Cross SectionsAccessible states cross sections have been calculated for j'=10 and j'=6. Forj'=10 the input cross sections are given in Table 19 while the j'=6 while thej'=6 input cross sections appear in Table 23. The same ao—L(Ek + EL) values107given 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 crosssections with initial rotor state of 6.In Table 23 E 1, is the energy according to the most probable translationalenergy at 300K, and fL = L(L 1)h 2121. J = A j is the total angularmomentum. The last contributing J value is that value of J for which therewas a contribution of at least 1 x 10' A 2 to the cross section. The time forcalculations is that required for the Amdahl 5840.Table 23: Input ao,L(Ek + E L ) Values for the ASSR at300K for j' =6Cross SectionLargestConditionNumber'Max.RotorStateMax.)Max.JContri-butingJCalc.Time2(sec.)Cross Sectionin A 2as—o(Ek + (0) 1.2x10 7 8 110 118 89 20.138 0.126952 x 103ao-2(Ek + (2) 1.8x10 5 8 110 118 91 19.737 0.507163x10 1ao_4(Ek + ( 4 ) 2.1x10 5 10 120 130 96 39.148 0.404343x10°uo,-6(Ek + (6) 1.9x10 7 10 120 130 104 68.934 0.340958x10-11. Not all the conditions numbers checked for the 04-6 transition2. Calculations for the 04-6 transition done on an Amdahl 470 V8The values from Tables 19 and 23 were then substituted into equation (91)and the resulting cross sections are displayed in Table 24.108Table 24: ASSR Cross Sections Using Exactao<—L(Ek + EL) Values for 300KThe o0,L(Ek eL) values used are from Tables 19 and 23 .TransitionCrossSectionin A2Ratio toExactResults TransitionCrossSectionin A 2Ratio toExactResults04-6 0.03 1.00 04-10 0.00 1.0024-6 1.27 2.32 24-10 0.00 4.324+-6 14.36 1.86 44-10 0.03 5.6264-6 126.95 0.73 64-10 1.31 2.0084-6 25.36 3.34 84-10 8.67 1.34104-6 3.64 14.7 104-10 80.58 0.41The calculations in Table 22 are also more inaccurate than the ASSR usingak—xV—ii(Ek (13 ,_ 3 1) values (Table 24).On the basis of these results, the ASSR calculates elastic cross sections thatare lower and inelastic cross sections that are higher than the exact cross sections.(The special case where j=0 reduces (91) to0.04—ji(Ek e .1 ) = 0-0*L(Ek + EL) (225)with j' = L and so the 04-10 and 0+-6 ASSR cross sections are exact). As well,as Ij —ft increases, so too does the error. In the f=6 column the cross sectionfor a downward transition with a given — is much closer to the exact valuethan is the upward transition cross section for the same Jj — ft.In the paper where the ASSR is introduced some calculations [32] using theASSR 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 calculations109it was also found that the error for transitions for the Ar-N2 system increases asj' 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 the84-10 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 sometransitions where lj — j'1 , 4 can overpredict by a factor of 2 (as is the case inthis study for the 24-6 and 64-10 transitions) and the ASSR for Ij — j'I=6 canoverpredict by as much as 11 times (here, for the 44-10 transition, the ASSRoverpredicts by only about 5.5 times).In comparison to the ASSR cross sections, the IOS cross sections which usethe IOS ol_ o values (Table 5) are better for all the j'=6 transitions and for thej'=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 andthis error in both scaling relations grows to about the same magnitude as thetransition becomes more inelastic. For example, the errors in the IOS for the84-10, 64-10 and 44-10 transitions are 15%, 160% and 880% and for the ASSRare 34%, 100% and 460% respectively. DePristo further notes [33] that usingpure statistical theory (ie. treating all rotational states as degenerate) to relatethe ao—L(Ek EL) values with the o-j _j , (Ek f 4) cross sections leads to evengreater inelastic cross sections and hence even poorer results. It seems possiblethen that the reason for the overestimation of the inelastic collisions comes fromthe statistical assumptions made in each scaling relation, the ASSR when it110assumes that the transition probability is inversely proportional to the number ofaccessible states, and the 10S, when it replaces operator with one parameterk?, and in doing so treats all rotational states as energetically equivalent.One further note on DePristo's work [33] which may have some relevance onthis work is that DePristo found that his scaling theory worked best for systemswith a lower reduced mass, such as a He-CO system, for which the reducedmass is about 3.5 atomic mass units (amu). Less accurate results were found byDePristo 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 as1500% from the exact results) with the GSMSR cross sections (which vary byat most only 25% from the exact results (Table 20)), it may be concluded thatthe manner in which the GSMSR relates the o-o:___L(Ek EL ) values with thecri,j,(Ek ef ) cross sections appears to be more reliable. In the ASSR, thecombined assumptions of degeneracy, the Effective Hamiltonian having angularcoupling coefficients dependent on energy differences and neglect of quantumtunnelling lead to errors of the same magnitude found in IOS scaling relations withIOS 01,0 values even though exact input cross sections are used. One possibilityfor decreasing this error of the ASSR is to modify or remove its assumption thaton statistical grounds that the transition probability is proportional to the numberof accessible states.1 l 14.3 Cross Sections at 1000K4.3.1 The Exact Cross SectionsAn exact calculation for 1000K was carried out. This requires changing certainparameters 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 sincethis study considers only the even rotor states) and the maximum rotor state forN2 with a translational energy of 1000K (ie. using (190)) and initial rotationalstate of 12 is 20. This gives a total energy of E = kBT E12 = 2.004 x 10 -20 J.If all the energy from the initial conditions is converted to kinetic energy then themaximum k value possible is :3.14 x 10 11 m.. -1 Using (196) the maximum partialwave contributing is estimated at 173 so contributions from 190 partial waveswere 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 studyis focussed on the utility of the IOS approximation and its variants to calculaterotational cross sections, any aspects of vibrational motion have been ignored.The calculated cross sections retaining only open states are listed in Table 25.112Table 25: Exact Cross Sections at Energy=1000Kj' 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'=10j=0 70.07 3.13 0.19 0.01 0.00 0.00j=2 15.44 73.92 5.24 0.27 0.02 0.00j=4 1.68 9.16 75.64 4.96 0.30 0.02j=6 0.16 0.66 6.83 80.03 5.52 0.32j=8 0.01 0.06 0.51 6.75 79.69 6.01j=10 0.00 0.00 0.03 0.44 6.76 85.11j=12 0.00 0.00 0.00 0.03 0.44 6.70j=14 0.00 0.00 0.00 0.00 0.03 0.36j=16 0.00 0.00 0.00 0.00 0.00 0.02j=18 0.00 0.00 0.00 0.00 0.00 0.00j=20 0.00 0.00 0.00 0.00 0.00 0.00j'=12 j'=14 j'=16 j'=18 j'=20j=0 0.00 0.00 0.00 0.00 0.00j=2 0.00 0.00 0.00 0.00 0.00j=4 0.00 0.00 0.00 0.00 0.00j=6 0.02 0.00 0.00 0.00 0.00j=8 0.37 0.02 0.00 0.00 0.00j=10 6.38 0.35 0.02 0.00 0.00j=12 90.09 6.63 0.39 0.03 0.00j=14 6.49 98.79 7.00 0.50 0.03j=16 0.34 6.28 108.63 8.05 0.49j=18 0.02 0.35 6.28 127.48 7.05j=20 0.00 0.01 0.22 4.00 171.52For the inversion of matrix K 2 + 1 the largest condition number was foundto be of the order of 1 x 10'. For the inversion of matrix W - 1/nj the largestcondition number was found also to be about 1 x 10 7 .11:3A comparison of the 300K cross sections (Table 2) with the 1000K crosssections reveals that all inelastic and elastic cross sections are reduced as thetemperature increases.4.3.2 The IOS aL,0 Cross SectionsAs noted in the literature, the Energy Sudden [30] and Centrifugal Sudden [44]approximations improve for higher collision energies. Thus it is expected thatthe 10S, a combination of the ES and CS, should improve at a higher collisionenergy.The IOS calculation was carried out for 1000K keeping all the parametersthe same as described for the 300K calculation except: contributions from 190instead of 120 partial waves were included and the ko = ki „it value was calculatedas that value which would give a total energy corresponding to h 2 ,; h21 J- I initial(J initial + 1)- = 004X10 -20 Jf(226)where 2.004x10 -20 J. is the total energy appropriate for 1000K. Again the Aovalue was chosen to be the average of ) and A'. The calculated IOS cross sectionsare listed in Table 26.114Table 26: IOS Cross Sections at 1000K Usingko =Part A: j' denotes initial state and j the final state. All cross sections are inA 2 .j'=0 j'=2 j'=4 j'=6 j'=8 f=10j=0 62.40 3.97 0.34 0.05 0.01 0.00j=2 19.79 69.14 6.16 0.62 0.10 0.02j=4 3.05 11.18 69.04 6.77 0.73 0.13j=6 0.59 1.59 9.67 70.28 7.15 0.84j=8 0.14 0.30 1.29 9.24 71.98 7.52j=10 0.04 0.07 0.23 1.22 9.07 75.00j=12 0.01 0.02 0.05 0.22 1.20 9.11j=14 0.00 0.01 0.01 0.05 0.22 1.25j=16 0.00 0.00 0.00 0.01 0.05 0.24j=18 0.00 0.00 0.00 0.00 0.01 0.06j=20 0.00 0.00 0.00 0.00 0.00 0.02j'=12 j'=14 j'=16 j'=18 j'=20j=0 0.00 0.00 0.00 0.00 0.00j=2 0.01 0.00 0.00 0.00 0.00j=4 0.03 0.01 0.00 0.00 0.00j=6 0.16 0.04 0.01 0.01 0.01j=8 0.95 0.19 0.06 0.03 0.02j=10 7.80 1.10 0.24 0.09 0.05j=12 78.95 8.40 1.25 0.33 0.14j=14 9.16 85.18 8.62 1.52 0.47j=16 1.32 9.65 92.63 9.00 1.81j=18 0.26 1.46 9.74 105.03 8.75j=20 0.06 0.29 1.60 10.03 127.47115Table 26 - ContinuedPart B: Ratio of 108 to exact cross sectionsj'=0 j'=2 j'=4 f=6 j'=8 f=10j=0 0.89 1.3 1.7 3.6 9.8 52j=2 1.3 0.94 1.2 2.3 4.9 20j=4 1.8 1.2 0.91 1.4 2.4 7.6j=6 3.8 2.4 1.4 0.88 1.3 2.7j=8 10 5.2 2.5 1.4 0.90 1.3j=10 48 19 7.4 2.8 1.3 0.88j=12 3.1x10 2 1.0x102 30 8.5 2.7 1.4j=14 1.3x103 4.5x10 2 1.2x10 2 33 8.4 3.4j=16 1.2x104 4.3x103 1.0x10 3 2.0x10 2 48 14j=18 1.3x10 5 3.7x104 6.1x103 1.2x103 2.4x102 71j=20 1.9x10 7 2.2x10 6 3.3x10 5 3.6x104 5.4x10 3 1.3x103j'=12 j'=14 j'=16 j'=18 j'=20j=0 4.1x102 1.7x10 3 2.5x104 4.8x10 5 1.6x106j=2 1.3x10 2 5.5x10 2 8.0x10 3 1.2x10 5 1.4x10 7j=4 36 1.5x102 1.8x103 1.8x104 1.8x106j=6 8.8 36 2.7x102 2.4x10 3 1.1x10 5j=8 2.6 8.4 55 3.7x102 1.1x104j=10 1.2 3.1 13 78 1.6x10 3j=12 0.88 1.3 3.2 13 1.6x102j=14 1.4 0.86 1.2 3.1 18j=16 3.9 1.5 0.85 1.1 3.7j=18 15 4.2 1.6 0.82 1.2j=20 1.9x10 2 29 7.4 2.5 0.74Contributions for A > 184 for j=0 were found to be negligible (ie., affectingonly the sixth significant figure).The unitarity of the S matrix (equation (206)) again was verified to 6 sig-nificant figures for (206) summed up to L = :30. As well, a check was done116on the S-matrix integration procedure, using 96 as well as 40 points for Gauss-Legendre integration. The 96 point integration procedure agreed with the 40point procedure to 6 significant figures.Table 26 demonstrates that in 28 of the 36 transitions for j and j' rangingfrom 0 to 10 the ratios of the IOS cross sections to exact cross sections at 1000Kare closer to one than the corresponding ratios at 300K (Table 5).4.3.3 ECIOS Cross SectionsOne of the most attractive features of the ECIOS scaling law is that it corrects forthe large IOS values for highly inelastic collisions. This warranted investigationat higher temperatures, where there is the possibility for even higher energyinelasticity. 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 0.5.117Table 27: ECIOS Cross Sections at 1000KTransition Calculation Cross Section Ratio to ExactExact 70.0704-0 IOS 62.40 0.89ECIOS 62.40 0.89Exact 15.424-0 lOS 19.8 1.3ECIOS 19.4 1.3Exact 1.684+-0 IOS 3.05 1.8ECIOS 2.40 1.4Exact 0.1576+-0 lOS 0.589 3.8ECIOS 0.232 1.5Exact 0.013784-0 IOS 0.138 10ECIOS 0.0167 1.2Exact 7.73 x 10 -410+-0 lOS 3.72 x 10 -2 48ECIOS 1.11 x 10 -3 1.4Exact 3.57 x 10 -5124-0 IOS 1.11 x 10 -2 310ECIOS 7.86 x 10 -5 2.2Exact 2.80 x 10 -6144-0 IOS 3.56 x 10 -3 1300EGOS 5.97 x 10 -6 2.1Exact 1.01 x 10 -7164-0 IOS 1.19 x 10 -3 12000ECIOS 4.60 x 10 -7 4.6Exact 3.20 x 10 -918+-0 IOS 4.12 x 10-4 130000ECIOS 3.15 x 10 -8 10Exact 7.74 x 10 -12204-0 IOS 1.46 x 10 -4 19000000ECIOS 1.29 x 10 -9 167115The results in Table 27 seem to indicate that the corrections invoked by theECIOS are mathematically valid; the ECIOS follows the behaviour of the exactcalculations even to values as low as 1 x 10' A 2 .4.4 Changing Parameter Cv6In the calculations parameter Cy, is the unitless constant2 aC = =2-ti a v (227)and so an increase in Cy, corresponds to increasing value V5, or "height" ofthe shell. It was investigated whether an increase in this value would make thecollision more "sudden" and hence bring about better agreement between theIOS and exact calculations.Table 28 displays the results of this calculation. Initial conditions were setwith ji=0 and translational energy equivalent to 300K119Table 28: Exact and IOS Cross Sections at 300K forCy, =1000Part A: Exact Results. Units are in A 2 . 1' denotes initial state, j the finalstate.j'=0 j'=2 j'=4 j'=6 j'=8j=0 198.63 0.59 0.07 0.01 0.001j=2 2.80 203.92 0.70 0.09 0.01j=4 0.52 1.08 202.36 0.35 0.04j=6 0.08 0.15 0.37 202.12 0.17j=8 0.006 0.01 0.03 0.12 205.62Part B: IOS Results.j'=0 j'=2 j'=4 j'=6 j'=8j=0 199.87 0.54 0.16 0.022 0.0029j=2 2.19 201.74 1.21 0.16 0.021j=4 0.92 1.91 202.42 0.64 0.087j=6 0.47 0.86 1.93 201.97 0.39j=8 0.29 0.46 0.94 0.86 202.46Part C: Ratio of IOS to exact cross sections.j'=0 j'=2 j' =4 j'=6 j'=8j=0 1.006 0.90 2.36 2.06 2.50j=2 0.78 0.989 1.73 1.77 2.17j=4 1.76 1.77 1.000 1.81 2.46j=6 5.65 5.89 5.17 0.999 2.22j=8 48.40 42.04 37.40 7.36 0.985120While the values in Table 28 indicate that changing parameter Cy, does notimprove agreement for inelastic cross sections, it does show that the 105 elasticcross sections do improve in their agreement with exact results. In Table 5 withCy, =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% withexact values.Note also that increasing parameter C y, increases elastic cross sections anddecreases inelastic cross sections in general.4.5 Changing Parameter aIt was investigated whether reducing the parameter a, the delta-shell radius,would increase the agreement between 105 and exact results. The motivationbehind this investigation is that a smaller radius would correspond to a shortertime of interaction between the atom and diatom and make for a more suddencollision.The initial conditions were chosen with j'=0 and the translational energycorresponding to 300K. The results are shown in Table 29.121Table 29: Exact and IOS Cross Sections at 300K fora=0.55A. -1Part A: Exact Results. Units are in A 2 . j' denotes initial state, j the finalstate.j 1=0 j'=2 j'=4 j'=6 j'=8j=0 2.29 0.0063 0.00076 0.00011 0.0000051j= 0.030 2.33 0.0091 0.0013 0.000068j=4 0.0056 0.014 2.29 0.0026 0.00025j=6 0.00082 0.0021 0.0028 2.35 0.0018j=8 0.000026 0.000075 0.00018 0.0012 2.47Part B: 103 Results.j '=0 j'=2 j'=4 j'=6 j'=8j=0 2.21 0.0068 0.00089 0.00062 0.000064j=2 0.018 2.22 0.0081 0.0040 0.00046j=4 0.0076 0.027 2.28 0.014 0.0017j=6 0.0044 0.016 0.013 2.31 0.0069j=8 0.0033 0.010 0.0039 0.018 2.35Part C: Ratio of 108 to exact cross sections.:1=0 j'=2 j'=4 j'=6 j'=8j=0 0.96 1.1 1.16 5.82 13j=2 0.61 0.95 0.89 3.08 6.8j=4 1.4 1.9 0.99 5.23 6.9j=6 5.4 7.2 4.5 0.98 3.72j=8 126 139 22 15 0.951 99The first observation to be made from Table 29 is that cross sections inboth the exact and IOS calculations are drastically reduced, about 2 orders ofmagnitude for both elastic and inelastic collisions from their values when a =5.5A -1 . Yet even at these reduced values the IOS still overestimates inelasticcollisions. Elastic collisions agree to within about 5%.The results of the last two tables indicate that something more complex thanreaction time considerations may be required to further improve the IOS relia-bility for inelastic cross sections. Another useful investigation involving adjustinga collision parameter would be to decrease the anisotropy parameter b 2 . Yetanother would be to calculate cross sections for a lighter system, eg., He-N 2 .1235 DISCUSSION5.1 Time Savings of the IOSOne of the most attractive features of the IOS approximation is its computationalefficiency. Matrix manipulation is greatly reduced and the programs required forcalculations are shorter and much quicker than those for the exact calculations.Table 30 lists the computer time required for the 105 and exact calculationspresented in this study. It lists the times in seconds required for program com-pilation and execution for the cross sections calculated in the previous chapter.The 300K calculation was done on an Amdahl 470V8 and the 1000K calculationon an Amdahl 5840.Table 30: Computer Time Required for IOS andExact CalculationsTemperatureIOSCalculationExactCalculation300K1000K30.064.067.31000.0As previously noted, the IOS becomes more accurate at higher collision en-ergies [30], [44]. This feature of the 105, coupled with the fact that at higher191energies the exact solution requires a far greater amount of calculation make theIOS a very attractive alternative to close-coupled calculations at high energies.5.2 Possible Improvements to the IOSIn replacing values kJ and A by parameters ko and A o in (29), the IOS allowsfor an easier method of calculation which, depending on the conditions of thecollision 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 toenhance the accuracy of the approximation.One promising result is that the ECIOS proved to correct high IOS inelasticcross 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 Suddencorrection where 7 would be dependent on A as well as j.A correction to the ECS proposed by Richard and DePristo [45] does notimprove the agreement with exact cross sections. Even if it did improve theagreement, applying such a correction to the values presented in this work maynot be as effective as correcting for the CS approximation, as was done byMcLenithan and Secrest [46]:,-cs = [0,,,,(R)]T (21 t. 1) 6 ,- 11,,cs( f)dR, (scs siscs)2z 0 9(228)where S cs is the CS 8-matrix, 1 is A o and fiJ is a null matrix, save for its jth125CCIOSCr 724 2 „ios24h 2 (( A , — c.\ ) 2 7- 2 '(230)diagonal element, which equals unity. The above is essentially a first order pertur-bative correction to the CS scattering matrix, where that part of the Hamiltonianneglected by the IOS is calculated and then applied as a correction to the ap-proximation. A first order correction has been shown to improve the CS resultsfor cross sections involving m transitions [46].Another way of correcting for the CS approximation could be to split the freemotion Hamiltonian into a radial and angular part and from this getHfree = Hradial Hangular (229)in order to get a correction term something likeEquations (229) and (230) are at best very sketchy and the details remain to beworked out.5.3 Applications of the IOSSince its introduction in 1974 [3, 6] the IOS has been used in a variety of chemicalsystems and processes. Examples of its usefulness can be found in papers deal-ing with the Senftleben-Beenakker effect [47], calculation of pressure broadeningcross sections [48], calculation of molecular fragmentation [49], angular momen-tum alignment due to collisions [50], and modeling potential parameters [51].Two particular areas of study are mentioned, namely using the IOS for collisions126involving vibrational transitions and using the lOS in the calculation of reactioncross sections.The Vibrational Infinite Order Sudden Approximation (VIOSA) is an approx-imation which deals with the vibrational quantum number 71 the same way therotational quantum numbers j and ) are dealt with in the 105. This idea wassuggested by Pack in 1974 [3] and then formally derived by Pfeffer in 1985 [52].Pfeffer obtains the following equation for the VIOSA:21t v(r, illxnjA(r, R))d2 d k2 A(A + 1)1 / ,n)/ —h2R2 idR2+ "1-:?,dR "3(231)The only difference between (231) and (74) is that the parameter k2i j hasan extra subscript — n — that comes about by replacing operator k 2 with theparameter k72L3 . This is equivalent to assuming that the duration of interactionis much less than the time required for a vibration of the diatom. Results areencouraging and give reasonable agreement with exact quantum results [52] .The Reactive Infinite Order Sudden (RIOS) approximation was developed in1980 by Bowman and Lee [53] and by Khare, Kouri and Baer [54]. Most work sofar is on the atom — diatom system, eg. H and H2. As an A-FBC system, thereare 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 coupledequations and potential function. The goal in reaction theory is to solve for thesethree sets of coupled equations while matching the wave functions 0,„ 00 andso that the functions and first derivatives are continuous at the boundaries.127The IOS is used in decoupling each of the three separate blocks of equations andsince the angular momentum operators have been replaced by parameters, thematching conditions are simpler as well (eg. setting A 02„ = Ao on the matchingsurface) where AL is the parameter replacing the operator A 2 for the a set ofcoupled 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 maybe applied is attractive not only because of the substantial saving in calculationtime that the IOS affords but also because it is through comparison of exact andIOS results that a further understanding of the underlying details which make upthe final results is achieved. Through work such as this it is hoped that a fullerknowledge of the dynamics of chemical systems may be developed.5.4 Molecular Potentials5.4.1 Time Savings of the Delta-ShellSince use of the delta-shell potential allows an R-integration step to be avoidedthat would otherwise be needed if a continuous potential was used, computationsusing the delta-shell potential are much simpler and faster than those usinga continuous potential. Table 31 compares the computer time needed in thecalculation of the elastic cross sections reported in Table 1.128Table 31: Computer Time Required for IOSCalculations for a Continuous and Delta-ShellPotentialThe calculations were done on an Amdahl 5840.Type of Potential Time in SecondsDelta-ShellInverse Power2.4463.5625.4.2 Comparison of Potential ParametersFor purposes of comparing the effect of modelling an inverse power potential witha delta-shell potential, a way of comparing the inverse power potential parametersand the delta-shell potential parameters used in this study is presented in thissection.There are two factors which must be taken into account in comparing therelative strengths of the potentials. Firstly the parameters are of different units,the inverse power potential having a parameter in units of J-m 12 and the delta-shell potential having a parameter in units of J-m -1 . Further, the potentialsgo to infinity at different points, the delta-shell at a = 5.5x10 -10 m. and theinverse power at R = Om. One method of comparison is to integrate the delta-shell potential over the a region of space from 1 -1 = Om. to R = 5.5x10 -10 m.Then the inverse power potential will be assigned a con.stant value, namely, thatvalue it has at R = 5.5x10 -1O m. and then integrated over the same region ofspace. The two resulting energy-distance values obtained will then be compared.129Equation (198) gives the following value for the inverse power potential at R =5.5x10 -10 m.:vinverse P0w"(5.5 X 10 -1°7n., 0) = 2.911 x 10 -23J [1 + 0.5P2 (cos 0)] (232)Integration of this value over the region R = Om. to R = 5.5x10 -10 m. leads tothe following energy-distance value:r=avmverse P"er (5.5 X 10 -10 7n., 0) = 1.6 x 10 -32 [1 + 0.5P2 (cos 0)] J — mr=o(233)Integrating the delta-shell potential over the same region of space gives thefollowing value for comparison:fr=a r=avdelta = 3.697 x 10 -32 [1 + 1.5132 (COS 0)] 6(r — a)drL=o f=o= 3.70 x 10 -32 [1 + 1.5P2 (cos 0)] J — m (234)Hence the area under the delta-shell used in this investigation is roughly twicewhat one would obtain with a square well potential extending from R = Om. toR = 5.5x10 -10 m. having a height that is given by the inverse power potentialat R = 5.5x10 -10 m.Another parameter to consider is the value for a, the position of the delta-shell. This value will be compared with the point where the inverse power po-tential becomes equal to the kinetic energy of the relative motion of the atom-diatom. Equating2.2 x 10 -14 X [10— 12vinverse power = R1277 112 [1 + 0.5 P2 (cos 0)] =h 2 k2 2/1(235)130where k = 14.27x10 10 m -1 and = 2.73x10 -26 kg. gives a value for R of3.63x10 -10 m. This is to be compared with the value of a= 5.5x10 -10 m. Apossible algorithm to determine magnitudes for delta-shell parameters in orderto model an inverse power potential with a delta-shell potential could be:• choose a to be twice the distance where the inverse power potential isequal to the relative kinetic energy of the atom-diatom• choose the strength parameter 1 76 to be twice the product of the strengthof the inverse power potential at a and the distance a.The final comparison is that for the anisotropy parameter. The inverse powerpotential uses 0.5 while the delta-shell requires a parameter of 1.5 to give equiv-alent 04-0 and 24-0 cross sections.The comparison presented offers a quick method of comparing the two typesof potentials. As for future delta-shell potential modelling, many possibilities ex-ist. Another repulsive shell could be added to determine if the higher transitions,eg. 04--4, 04-6, ... can be matched with the inverse power potential cross sec-tions. One could try to model the attractive part of a molecular potential with adelta-shell. Finally, an attractive and repulsive delta-shell could be combined inorder to give a reasonable approximation to an actual molecular potential. Thefact that 04-0 and 24-0 cross sections from the inverse power and delta-shell po-tentials can be matched fairly closely (see Table 1) is a promising note for futuredevelopment of how to make a delta-shell to replace a continuous potential.1315.5 Calculations on a PCThe advent of a world wide standard PC in 1981 has resulted in improved porta-bility of computer programs and availability of computing resources. Once aprogram is debugged and running successfully it is no longer necessary to changethe source code to conform to the standards of the mainframe of the institutionthat the scientist is working at or visiting.One other advantage is in the area of numerics. Overflows of the order of10' that could not be handled by a mainframe are handled easily by MicrosoftFortran 5.1 (which can handle values up to 10'87 ).There are however still disadvantages with using a PC. Two of the mainconcerns are limited memory and a slower CPU (depending on machines used forcomparison).1326 CONCLUSIONSMany useful and interesting results have come about from this investigationof the IOS and delta-shell potential. The following is a summary of the mostimportant results, as well as a summary of what further investigations may beperformed in light of these results.It was found that cross sections from a 0 dependent delta-shell can be com-puted that are comparable to a more realistic potential, such as an inverse-powerpotential. Further studies may include various combinations of repulsive andattractive delta shell potentials.This work suggests future work in many areas may prove beneficial, such asfurther investigating how to model a continuous potential (such as a Lennard-Jones) with a delta-shell potential. Perhaps a hard sphere repulsive (since it isnon-penetrable) and non-spherical attractive delta-shell may offer a combinationof ease of computing and a fairly realistic model of certain systems.Expressions for the T-matrices have been derived from the exact and IOSsolutions 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 ofenergies. The exact and IOS cross sections were then compared. At translational1 :3 3energies corresponding to 300K and 1000K the cross sections were found to bereasonable but at high rotational energy transfer the IOS consistently overesti-mated the values.Several scaling laws aimed at improving the 105 cross sections were inves-tigated. These all are based on the IOS ol_ o cross sections. The AccessibleStates Scaling Relation, an approximate scaling law based on angular momentumtransfer of the rotor states was found not to be a significant improvement overregular IOS scaling laws at 300K. The General S-Matrix Scaling Law, based ona scaling based on reaction times and rotor energy separation was found to beuseful in some cases in correcting for the large inelastic cross sections given bythe IOS approximation.The Energy Corrected Scaling Law, based entirely on correcting for the IOSS matrix with a reaction time parameter T was found to correct very well forIOS differences from exact results for 300K. The T values that worked best werefound to be 1/10th to 1/5th that calculated for the time it takes an atom totraverse the diatom potential shell.The scaling laws also were investigated as to how well they performed usingexact 01_ 0 cross-sections. All four scaling laws performed very well at 300K. Thissuggests that there exists for the delta-shell potential a quick way of calculatingother transitions from the aL,0 cross sections.Various choices for the k o parameter were investigated as to whether theyaffected the IOS results, but it was found that changes in its value did not131significantly improve agreement with exact values.Inclusion of closed channels was not found to significantly affect the results.Increasing the parameter Cy, improved both elastic and downwards transi-tions but upwards transitions became worse.Decreasing the parameter a brought about very good agreement betweenexact 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 eachof the approximations considered. Using a delta-shell was far simpler and quickerthan using the standard Lennard Jones potential. And using the 105 approxi-mation on this delta-shell potential was found to be much easier than using theexact calculations for the same situation. This suggests that investigations intofurther improvements to the delta-shell potential and 105 approximation mayprove beneficial to shedding light on problems that have been impossible to solvewith conventional potentials and computing procedures.135References[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. Pack. Chem. Phys. 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The infinite order sudden approximation and the delta-shell potential Dancho, Stephen John 1992
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Title | The infinite order sudden approximation and the delta-shell potential |
Creator |
Dancho, Stephen John |
Date Issued | 1992 |
Description | 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 proposed. |
Extent | 3694002 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-12-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0061747 |
URI | http://hdl.handle.net/2429/2947 |
Degree |
Master of Science - MSc |
Program |
Chemistry |
Affiliation |
Science, Faculty of Chemistry, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1992-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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