THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGtE OF ENGINEERING SLOW-NEUTRON DIFFRACTION IN NORMAL-LIQUIDS Arnulfo Morales A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1960 June, 1960 IP-438

Doctoral Committee Professor Richard K. Osborn, Chairman Professor Kenneth M. Case Professor Henry J. Gomberg Professor William Kerr Associate Professor John S. King Associate Professor Paul F. Zweifel

ACKNOWLEDGMENT The author wishes to express his gratitude to the members of his Doctoral Committee for their interest and their understanding, and in particular to Professor Richard K. Osborn for his invaluable guidance, help, and encouragement throughout the course of this work, and to Professor Kenneth M. Case for fruitful discussions and suggestions. The use of The University of Michigan Computing Center facilities and the help from Mr0 Larry B. Evans and several members of the staff is gratefully acknowledged. The financial support received from the Institute of Physics of the National Autonomous University of Mexico, the Mexican Nuclear Energy Commission and the Mexican Light and Power Company made possible the completion of this work and is also gratefully acknowledged. Finally the author wishes to express his gratitude to the Industry Program of the College of Engineering for the reproduction of this thesis. iii

TABLE OF CONTENTS Page AC KNOWLEDG1ENTS. +................e. eo —e............................. iii LIST OF TABLESo........... e. * o..o* —-*o**** vi LIST OF FIGURES,...a....... e e o e O.... ****.. edo e e e * Vii LIST OF APPEINDICES.................. viii INTRODUCTION................ 1 CHAPTER I. QUANTUM MECHANICAL THEORY OF SCATTERING,..................... 3 II. NORMAL AND ASSOCIATED LIQUIDS,...,........... 8 i. Lennard-Jones and Devonshire Theory Review............. 2. Approximation for Potential.... 11 3. Equation of State and Derived Thermodynamic Functions, 13 4. Trouton's Rule.......................... 14 5, Entropy of Melting~.....~ D. e..............15 III. DIFFERENTIAL CROSS-SECTION ACCORDING TO THE APPROXIMATION TO TEE LENNARD-JONES AND DEVONSHIRE LIQUID MODEL POTENTIAL*....O.. a O,......... 0 *-...*... 16 1, Inner Efect,....................... 16 2. Outer Effect............................ 20 3. Incident Spectrum.............................e 22 IV, CALCULATIONAL PROCEDURES AND DATA25.................. 25 1. Energy Transfers.............................. 25 2, Nearest-Neighbor Distance and Hard-SphereDiameter... 26 3. Calculational Procedure,....,, e......,. o....... 26 V, COMPARISON OF AVAILABLE THEORIES WITH "SMEARED" POTENTIAL MODEL,...........*....................... 36 1. Zernike-Prins Approximation..................... 36 2, Convolution Approximation,............... 36 3. Summary...............39 iv

TABLE OF CONTENTS CONTT'D CHAPTER Page VI.I NORMAL DIATOMIC LIQUIDSI............................... 41 1. Diatomic Molecule,................................. 41 2. Spin Dependence,................................... 43 3. Thermal Average...................................... 45 4. Static Approximation...................... 46 5. Results,................................................ 48 APPEINDICES0.. a.....1........1.... BI:BLIOGRAPHY.... *. e e * + e e X * + @ @ -. w v v * 111

LIST OF TABLES Table Page I CALCULATIONAL PARAMETERS FOR MONATOMIC ELEMENTS *..... 28 II LIQUID LEAD FINAL EINERGY SPECTRUM WIDTHS6...........e 40 III NITROGEN MOLECULE CALCULATIONAL PARAMETERS........... 49 vi

LIST OF FIGURES Figure Page 1 Scattering Collision Diagram..................... 4 2 Lennard-Jones and Devonshire Cell.......... 10 3 Lennard.-Jones and Devonshire Potential.. *.......... 12 4 Nuclear Coordinates......... i...... 18 5 Normalized Cross-Section ( 4,.7) for Liquid Argon at 840K as a Function of Scattering Angle,........ 31 6 Normalized Cross-Section ( 4,7) for Liquid Lead at 620~K as a Function of Scattering Angle,........ 32 7 Normalized Cross.Section ( 4,8) for Liquid Neon at 24.570K as a Function of Scattering Angle,........... 33 8 Normalized Cross-Section ((4.8) for Liquid Krypton at 116~K as a Function of ScatteriAnAngle,.............. 34 9 Normalized Cross-Section ( 4.8) for Liquid Xenon at 163~K as a Function of Scattering Angle,.......,... 35 10 Diatomic Molecule Coordinates.............................. 42 11 Normalized Cross-Section for Liquid Nitrogen at 77.40K as a Function of Scattering Angle............. 50 vii

LIST OF APPENDICES Appendix Page A. 1 Transition Cross -Section,....,........... o.. 8 e. 52 A. 2 Integral Equation..*.O... o.......Q....... 53 A. 3 Green' s Function.......,.,..... 0.........a a 55 A. 4 Energy Conservation............ *......, 57 A. 5 Parabolic Range............... a * * * a * w 60 A. 6 Harmonic Oscillator Free Energy............... 62 A. 7 Thermodynamical Functions................................. 64 A. 8 Heisenberg's Operator.... *.... w *......... 66 A. 9 Thermal Average.....e 68 A. 10 Irving-Zwanzig Theorem and Radial Distribution Function.... 78 A. 11 Time Expansiona.ni.oen*. a a aa a. 0B.&. 4 a 84 A. 12 Hard Sphere Radial Distribution and Density Expansion...,,. 89 A. 13 Incident Spectrum Calculation,......o o o *. * 9 * a 94 A, 14 Outer Effect Calculation,...*...,,.,..... 98 A. 15 Spin Dependence Calculation............. 100 A. 16 Time Expansion for Internal Degrees of Freedom............ 103 viii

INTRODUCTION Extensive studies on the scattering of slow neutrons by a system of atoms have been carried out(12,3)). However, only for gases and solids has it been possible to make accurate detailed calculations. For the case of liquids the complexity of the interatomic interactions and their time variations make these calculations very difficult and only through the use of several approximations has it been possible to make any progress. Two of the approximations commonly used are the static approximation(4) in which the change of energy of the neutron upon scattering is assumed to be negligi:-.le compared with the initial energy of the neutron, and the convolution approximation(5), for which it is difficult to find a physical interpretation, although mathematically it is equivalent to setting the expectation value of a product equal to the product of expectation values. The static approximation gives no energy spread produced by the scattering; and the diffusion model, which in conjunction with the convolution approximation, predicts too great an energy spread as compared with experiment. In this work a new approach is attempted. The method consists of introducing a well known liquid potential function, limited to normal liquids, due to Lennard-Jones and Devonshire(6) into the scattering formulae. The energy spread predicted by this model is in better agreement with experiment for the cases studied than the previously mentioned theories -1

-2In Chapter I we have used the quantum mechanical theory of scattering in order to obtain the basic formula (114) for the differential cross-section (in angle and energy) of slow-neutrons, The LennardJones and Devonshire liquid theory together with some thermodynamic properties of normal liquids are reviewed in Chapter II, In Chapter III an approximation to the Lennard-Jones and Devonshire potential function (2.9) is used in order to obtain the differential cross-section for monatomic normal liquids (3.20). The parameters used for several monatomic normal liquids are discussed in Chapter IV and exhibited in Table I. The results obtained for the angular cross-section, (4~7) and (4,8), for these liquids together with experimental data available are shown in Figures 5 through 9 at the end of Chapter IV, Some of the existent theories are discussed in Chapter V. The resulting formulae, (5.18) through (5.20), are compared with the model introduced in this dissertation, Table II, at the end of Chapter V, shows experimental as well as theoretical results obtained for the broadening of the neutron final energy spectrum width at half-height for liquid Lead at different temperatures, A generalization to diatomic normal liquids and the introduction of spin dependence into the scattering formulae are made in Chapter VI, The results are applied to liquid Nitrogen, and the angular cross-section thus obtained for liquid Nitrogen is compared to the experimental cross-section in Figure 11,

CHAPTER I QUANTUM MECHANICAL THEORY OF SCATTERING In what follows we will derive the basic scattering formulae to which all the foregoing theories are applied. We want to consider a system of N bodies being bombarded by monoenergetic slow neutrons of incident energy Con and we want to calculate the number of neutrons scattered by this system into 0, cp and E (See Figure 1). We can show that the number of particles scattered per unit time into the solid angle d2 by atoms undergoing a transition from initial state a to final state r, per unit incident flux, is given by (Appendix A.1) ao ()d - _k I: 2() (1.1) ko'' where k and k are the neutron wave vectors for the final and the -'q - -o initial state respectively, and fr(e) is the amplitude of the 1- th spherically scattered wave. If we now define pa as the probability of finding the system in the initial state aa (Z pa = 1) a(e)da = Z E pug (0)da (1.2) would give us the angular cross-section due to transitions between all final and initial states of the system. -3

-4k,m ikr e ~~~~~~~~~~~~~kz r ko'mn~~~~~~~~~~ rI 1~~~~~~~~~~~.1 I ___________ ________Y_ Figure 1. Scattering Collision IDiagram. Z Ir J~~~~~~~~~~~~~~~~~~~~~~~~~ i~~~~~ I - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I, -I, e o,, I I I I Figure 1. Scattering Collision Diagram.

-5Let 0 (r) be the incident neutron wave function, (Figure 1), GE (r,r') the Green's function for the appropriate Schrodinger equation (Appendix A.2), and re (r) = (rir (r, I) Ic4) - f d3 0A% (A ) r (r, i) Acx), then (r) = _(r)+ f'd3rElc (r, r) r Ad(r')(r ()r') where A~(j) is an energy eigenfunction for the scattering system, will be an integral representation of the time independent Schrodinger equation [H(Q) + H(r) + r (r, i) - et] 0. 1) = (1.4) In (1.4) i = -, 2, * - N represent the position vectors of N bodies in the system, H(_) the system Hamiltonian, H(r) the neutron Hamiltonian, r(r, _ ) the neutron-system interaction potential, and et the total neutron-system energy. In order to solve Equation (1.3) we make use of the well known Born's approximation(7)which may be obtained by simply taking inside the integral sign in (1.3) tY (r') ~ @ (r) bYu (1.5) which leaves us with r) (r) + S d3r' GE (r, r) (r') (1.6) r' (.) The boundary conditions that the solution of our problem must satisfy are that there is an incoming wave for the system initial state a, and asymptotically spherically outgoing waves in every state

r->oo ik z C ikr (r) - ike + f(0)e /r; i = (1.7) rI_ fa(T)eik r/r; otherwise, Thus, if we introduce the asymptotic expression for GE (r,r') (Appendix A,j3) in (1i6),. we get f (r) r-o (r.). m-, eik r d3r -ik (r)(r) (1,8) T(r 2d" r r"rl where mn is the neutron's mass, Therefore, f() =. -. f d rI e-ik. r' (r') $ (r' (1.9) 2..I2 r' and since $ (r) = exp (iko. r) (1.,9) becomes fa(0) mn d3r' eir' (ko-k ) (r.') TI(e) r (-r (r) (1.10) amE62 r - r= a (rMaking use of this result and of Equation (1,2) we may write man )7C 2~ tC e (1.1T) ir K ( P mn kp. | f d3r ei_ r ~ K | (r')-2 () ( 2 Za ko r' where K. = k -k is the momentum transfer vector, Thus far, we have said nothing with respect to F(r,), We will now assume it to be given by what is known as the Fermi pseudopotential(7, and refer the reader elsewhere for justification of this

-7assumption. Thus, let 2 N r (r, i) 2 m gbs (r- s) (1.12) mn s=1 s where bs is the scattering length of the s-th atom and in general may be spin dependent, and 6(r) is the Dirac delta function. Introducing in (1.11) the energy conservation relation'(-'9), we get a(e,O)de = k, )2 p k d3r 2eir( a) 5 which with the help of Equation (1,12) and of the integral representation of Dirac's delta becomes (Appendix A.4),0 eN N -iK os (t) iK e) = k 1 dteitq Z pa(cl, b b e e J i-a) (1,14) ko 2~d~ -, =1 j=l j where q = e0 - c (1.15) is the energy transfer, and _is(t) = exp (itE/fi) 4s exp (-itH/-) (1.16) the Heisenberg's operator. In order to proceed further we have to decide which system we are going to treat and, therefore, which Hamiltonian we are going to use. Having done this, we could formally calculate (1.16) and could go on with the operations indicated in (1,. 14).

CHAPTER II NORMAL AND ASSOCIATED LIQUIDS It has been customary to classify liquids into two categories, those that followed certain empirical rules, such as Trouton's Rule, and had similar physical properties like vapor pressure, surface tension, etc,; and that minority that-did not show such correlations, As more data were accumulated and knowledge of the chemical structure, of the molecules that formed these liquids increased, a better understanding of the empirical correlations was obtained. These empirical correlations were only explainable if it was assumed that the internal degrees of freedom of the molecule were not seriously disturbed by the closely packed molecules in the liquid state; and that the molecule in the gaseous state was essentially spherically symmetric, It.is thus implied that in the passage from the gas phase to the liquid phase the internal structure of the individual molecules was little modified, These liquids were called Normal, On the other hand, the liquids that deviated from these "normal" conditions were believed to form in the liquid phase links among molecules causing disturbances in the internal degrees of freedom (rotational, -..vibrat-ional., and electronical)o These liquids were called Associated, Lennard-Jone.s and Devonshire Theory Peview Several attempts 0'l have been made trying to explain the thermodynamic properties of liquids. Lennard-Jones and Devonshir4 6)(LJD) made a significant contribution in this respect when they introduced a -8

-9potential energy, alrqady found empirically to fit the properties of slightly imperfect gases, in their liquid model, They found that this particular choice could predict with satisfactory accuracy the temperature dependence of the virial coefficient B(PV/kBT = 1 + B/V + C/V2 +.-.) of the slightly imperfect gas, the critical temperature, the boiling point, and the heat of evaporation. In their work it is assumed that each molecule moves in it's' own cell and a calculation is performed in which the actual time varying field to which a molecule in the liquid is subjected is replaced by a suitable average. This average field for a molecule moving.in its cell is then obtained by fixing the.immediate neighbors to their respective equilibrium positions. All these "smeared" nearest neighbors are assumed to be arranged in a spherically symmetric fashion. That is, arranged with equal probability on the surface -of the cell. In Figure 2 the central molecule is at P, a distance r from the center of the cell. The center of the cells are separated a distance a. The probability of finding x "smeared" nearest neighbors in the area encircled by the rings is x(2ta2 sinG dO / 4ca2) = (x/2)sinO de; (2.1) the interaction potential between the particle at P and the particle at Q is given by V(R). Therefore, the average potential energy within the cell (r < a) will be given by V(r) = fo/2V(Jr2 + a2 - 2ar co ) sin 0 dO (22) 2

-10p a Figure 2. Lennard-Jones and Devonshire Cell It was mentioned before that the interaction potential chosen will be that which predicts with reasonable accuracy the equilibrium properties of the slightly imperfect gases, namely V(R) =-+ R1 (23) the well-known Lennard-Jones six-twelve potential function. Therefore, after trivial integration and rather lengthy algebraic manipulations, we obtain: V(r) =V(o) + A*(V*/V)2 [(V*/V)2 a l(r2/a2) 2- m(r2/a2)] (2.4) where V(o) = A*(V*/V)2 [(V*/V)2 - 2] (2.5) in which if e* denotes the minimum value of V(R), and r* is the value of R at which this minimum occurs1 then

A* XE- xC* /4v (2.6) V*/V = (r*/a)3 = (2v/). 2a,3( where V denotes the volume per molecule and l(y) and m(y) are functions defined by I.(y) = (1 + 12y + 25.2y2 +12y3 y4)(1 y)10 _ 1 (2.7) m(y) = (1 + y)(l - y)4 1 ApProximation for Pqtential Equation (2.4) tells us how the potential energy of a given molecule changes as it moves in its cell, while (2.5) gives the value of this potential at the center of the cell, The zero of-energy is, as usual, that of infinite separation, The right hand si.d.e (RHS) of (2.5) is a function of A* and V*/V only; that of' (2.4) depends on these two quantities and also, through a, on the numerical constant y (a3 = y' V) which is determined by the geometrical arrangement of the molecules, Some representati.ve curves for V(r) for certain val.ues of V*/V are shown in Figure 3,. Curves 1 and 2 have a potential hump at the center, while 3 does not, It is:found that -the field ceases to have this central hump when (Appendix A,5)...a aa >.67( For this range it is easy to see that if one expands V(r) in a series of even powers of r, the main contribution to V(r) comes from the r2 term. The contribution from the rest of the terms being neglected,

*Tvstuo cT aJTqsuoAa'put sauo r-p.ZUUa'. aJmty g. \' ~~~o/i~i D/J 7 r [ l l' I~~(~h

-13Therefore, the potential can be written V(r) = V(o) + sr2 (2~9) where Mw2 A* V* V )( 44v 2'2 K. 2 a12 2 a1 V V-' Equation of State and Derived Thermodynamic Functions It has ba.en shown elsewhere(l0) that the free energy of an assembly of N molecules in a volume V may be exhibited as the sum of three independent terms, the translational or kinetic motion, the potential, and the internal degrees of freedom contribution. This is possible if the partition functions of the system is given by the product of partition functions for the kinetic, potential, and internal degrees of freedom, respectively. This is satisfied by dilute gases and presumed to be a good approximation for highly compressed gases and for normal liquids. To these approximations the only distinctions among normal liquids, compressed gas, and a dilute gas are entirely due to differences in the potential part in the free energy. Under this assumption one can show (Appendix A.6) that for our parabolic approximation the free energy per molecule is given by f = V(o) - 3kBT In (kgT/hw) - kBT in j(T) (2.11)

-14where j(T) stands for the internal degrees of freedom partition function, From this relation one can derive related thermodynamic functions such as the partial potential Y, the molecular heat content H, and the equation of state (Appendix A.7)o Trouton's Rule Trouton's rule can be formulated as follows: all normal liquids have the same entropy of evaporation at their boiling points. It is shown in Appendix A,7 that the heat content HL, for the harmonic oscillator model is given by LH =V(o) + 3k T[10 - 33 (V*/V)2][5 - 11 (V*/V)2]- 1 + L H VB~o) + 5kinT~lO 55 mt. ~(2.12) v4A*(V*/v)2[l - (v*/v)2] and for the gas-phase(l0) HG = (5/2)kBT + EGt (2.13) Therefore, the molecular heat of evaporation Ae would be Ae = HGH (5/2)kLT - V(o) - G(V) + Eint - EiLnt (214) G(V) = 3kBT[10 - 33 (V*/V)2][5 - 11 (V*/V)2] — 4A*(V*/V)2 (2.15) *[1 - (V*/V)2] and the entropy at boiling is then MS Ae 5 "V(o) G(V) -kg=~~~~ kg~~~- - -~~3 - kg(2,16) kB kBTB 2 kBTB kBT(1 where we have assumed G EL int. int,

For Argon, for example, (v*/v) - l, (A*/kBT) - 16,6 o A s/kB - 7,65 The experimental value is approximately 8. (See Table 2 Chapter VI of reference 10). Entropy.. of Melting In reference 10 it is shown that using the harmonic oscillator liquid model, one can exhibit the entropy of melting as a function, although approximated, only of the frequency of vibration of the crystal and the liquid state, The value obtained for the entropy of melting is A Sm/kB = m/kBTm = 3'.ln(w/wL) + 1 (2o17) Formula (2.17) has been used in order to obtain eL for Lead, In order to do this, we have assumed that to was determined by the characteristic Debye temperature (wC = (kB8/T) = 12.371 x 1012 sec-1 for a = 94,5~K12) and that the entropy of melting!.Am/kBTm =.95 10,135 These values lead to wL = 1257 x 1012sec-l We will use this value L of et in Chapter V,

CHAPTER III DIFFERENTIAL CROSS-SECTION ACCORDING TO THE APPROXIMATION TO THE LENNARD-JONES AND DEVONSHIRE LIQUID MODEL POTENTIAL In Chapter I (1.14) we have established an expression for the differential cross-section, a (e, 0), which we may write in the following form af (E,' ) = inner(E e0) + aouter(", 0) (3.1) where k N 20 tJ ~ iK'4(t) iK- (sja (e) k. dteit(a bse (5.2) inner K(,t)...c) a 2a"k 2 sl and k 1 CO teitqj ~outei O) k - dte" a Er(6 bsbj b -ieK~_s(t)eiK'-jil) (K.3) ko 2j-tfa-c s=lj=l The reason for this break-up of the cross-section is that for a monoatomic system a. (E0 ) represents the non-interference cross-section, inner sometimes referred to as the direct-scattering part of the cross-section and gives the contribution to the cross-section from those waves scattered by each atom individually. The aouter(e, 0) represents the part due to the scattering by different atoms giving, thus, the interference part or outer effect. Inner Effect We will proceed first to calculate ainner( C 0). In Chapter II we have established that when the range (2v/p)~ a-3.67 -16

-17is satisfied a suitable approximation to the LJD model potential was given by V (r) = V (o) + s r2;, r < a with V (o) and s given by (2.5) and (2.10), respectively. With the help of Figures 1 and 2 we construct Figure 4, and we define is(t) = Bs + rs(t). (3.5) The vector ~s(t) is then the position vector of the s-th nucleus at time t, and it is given by the vector addition of a fixed vector Bs to the center of the cell, and a vector rs(t) which gives the actual position of the s-th nucleus with respect to its equilibrium position at time t. We may write the true Hamiltonian H' i=l 2i X>(R) + + v (rR) for n electrons, N nuclei, and V the sum of electrostatic interactions between all of them, In this section an approximate Hamiltoni.an has been used in which we have replaced the potential V (r, R) by our approximation to the LJD potential, Thus N H' - H = Hs (3.6) s 1 and H = (ps/2Ms) + (M w /2)r + V(o) (537) for:. monatomic systems Ms = Mo

-18z Figure 4. Nuclear Coordinates.

-19-. Let Qs(q, t) = exp [-iKo4s(t)] (308) Then 2s(l, t)s(q,o) = 2s(rt)s(r, where it is the hermitean conjugate of Q. Therefore, ainner(EO) = (k/2Nko)oJ dteitq< Z b2g (rt)i+(ro)>T (310). -' S S ~ where the thermal average indicated by < >T is evaluated through the use of a corollary to Blocht's theorem2' 14, and (Appendix A.9) < s(r't)2+(r,0) >T: C(lb 2e-iK~ rs(t)ei-K"rs 1) (3 11) - b0 enlintw-A.In(A2'K2),. n=-o The Heisenberg's operator is given by (Appendix Ao8) rs(t) = exp (iHt/~-)rs exp(-iHt/fi) = cos wtrS(o) + (sin wt/Mw)ps(o) -(3,12) where K2 = (2mn/y2)(e + o - 21 o cos e) A = (/2kT) (13) A' = (~/2Mw)csch A 2 1 A' = A' cosh A 3 2 1 and In(x) is the modified Bessel's function of order n, which can be written in terms of the ordinary Besselts function, Jn(x), of order n (In(x)= i-nJ (ix)) Upon substitution of (3.11) into (3,10) we obtain

0inner I' o) 0o b2 N ~ (k/ko) e- A3K + nA1- (.14) n= —oo *In (AiK2) 5 (Go -c- n- w) Outer Effect As has been mentioned before, the interference between the amplitudes scattered by independent atoms of the system produces changes in the intensity which have been observed in X-ray diffraction and lately also in Neutron diffraction. In order to calculate this intensity exactly, it is necessary to have a detailed knowledge of the interparticle interactions and of the position correlations to which they lead. In order to determine the inelastic part of the outer effect, a knowledge of the time dependence of these position correlations is necessary as well. The exact calculation of these effects is extremely difficult. This fact forces us to make some approximations as described below, In what follows we will make use of the familiar Heisenberg's operator time expansion technique(2l'15) to calculate the thermal average in aouter We will then show, in the Appendices A,10 and A.11, that the

-21zero-th order term of this expansion may be related to the well-known radial distribution function, g(r), through the use of the Irving Zwanzig theorem. The function g(r)d3r is defined to be the probability that, given an atom at the origin, a distinct one will be found simultaneously in d3r about r, The first order term is shown to be zero(l) if the Wigner distribution functions are assumed to be isotropic in the momenta, (Appendix A.lO). In this way it is shown as has been pointed out elsewhere(17), that inelastic effects are much less prominent in the interference terms than in the direct scattering. Therefore, we can write N N router (-") = (k/ko)b(Eo-E) < Z b bje ( ) s j (3.15) iKr (Nb (h/ko)(ot.~) p S d3rei g(r) where p is the number of atoms per unit volume, (See Spin Dependence Chapter VI). Much theoretical and experimental work has been done with regard to the radial distribution function in liquids. On the theoretical side Kirkwood(18), Ivon(19), Green-Born(20) have derived integro-differential equations which through some approximations have been solved for certain choices of potentials. These cases have been reviewed in a book by T. L,. Hill on Statistical Mechanics. () On the experimental side the determinations of the radial distribution function, using X-ray diffraction, for several elements have been reviewed to great extent by Gingrich(21); and for those cases in which neutron diffraction techniques have been used Henshaw, Pope, Hurst, and Brokhouse, to mention a few, have measured g(r) for Argon, Nitrogen molecule, Oxygen molecule, and Lead(22'23)

For our particular problem the choice of the theoretical radial distribution function, which we will use in our calculation of outer will be governed by the consistency in the interaction potential used to calculate on the one hand, the g(r) function and on the other our Ginner calculation. Namely, we would have to calculate the g(r) function by using the six-twelve Lennard-Jones potential, But, since this calculation has been done elsewhere using a modified Lennard-Jones and Hard-sphere potential because the calculation of the Lennard-Jones potential alone turned out to be extremely difficultl')) there is no need to repeat it here. Instead we will use the hardsphere potential, U (r) = O; r > d (3.16) co; r <d and an expansion to first order in the density(11,29) ( O; r > d gr 1; r < 2d (3517) = 1 + (4kpd3/3)[1 - (3r/4d) + (r/d)3/16]d < r < 2d of the radial distribution function (Appendix A.12) for simplicity's sake, since the results may be exhibited in closed form, Incident Spectrum Although we have assumed in the theory that we had monoenergetic neutrons of incident energy co, this is not achieved in practice, and therefore, in order to compare theoretical results with experiment, we have to introduce into the theory the fact that the incident beam has a finite energy width, about an. average Ec) rather than

-23being of the ideal 5(Eo0-o) type, Assuming this energy spectrum is given by a Gaussian distribution (which is approximately the case for most crystal spectrometers) as G(cb,6=o) L exp[-(co-Eo)2/ 2] (3,18) we maythen calculate 00 a(E';8; Co0,) - fodEoG(e oEo) f(E:, ) (3619) and obtain (Appendix A,13) a(e,6;o,oP)/N b L = aouter(eY;EoEp) +'o(E:,;oP) + +( Ep) (3~20) + U_(,e; E% ) where we have written ainner(~,e;Eo,) ='go + 9+ + a and.2(5o21) Qn = c + n(5w/2) - (+nh) cos (21 O(6;6 —o) ~= exp[- (e —0o)2'2-2eA3sin2 8 ]1i0(2eA2sin2 ) (522) 00 a_+(~ (; E,) = e(~+e-nhw) 1In(A )exp[-A3 Q2+nA1 (E-Lo+nh w)2 2] n=l f(323).a (6, 8;Q~o,~)2= ~ ~ 6( 2-n) -exp [-A3Q'n.~nA~-_ -,.nG/n=l n(A2Qn( (3,,24) and for.96'ter (Appendix A.14) aouter(E,; CEo) = exp.[ -(-Eo0)22]{B} (3,25) In expression (3,25) the bracket is given by {B} = -DA12(y) + (3D2/16)[24(y2-l)ocos 2y - 48y sin 2y + (5y4 (3+ 5y + 24) 4 COS y + (4y2 + 24y) si.26) + (5y4 + 12y2 + 24) cos y + (4y2 + 24y) sin y]

-24where y = Kd = (4Td/X) sin (0/2) (3 27) and X is the wavelength corresponding to the energy -, D = (4td3p/3); A2 = (4mn/~2)A (3.28) A3/2(Y) = 3(sin y - y cos y)y53; A3 = A2cosh A1 In formula (3.26) we can immediately recognize the first term as the familiar Debye approximation to the outer effect, while the rest is closely related to a similar expression derived in the classical approximation (high temperature) by Mazo and Zemach(l in connection with neutron diffraction by imperfect gases.

CHAPTER IV CALCULATIONAL PROCEDURES AND DATA Energy Transfers In Chapter III, (3.20), we have expressed the cross-section as the sum of four terms, the second of which, oo(e,&; Co, ), represents the elastic contribution to the direct scattering of the neutron final energy spectrum. This is seen to be the case because for a narrow width incident spectrum the main contribution to ao comes from those values of e close to eo, as given by the Gaussian centered at about eo? In the same way one can see that the Gaussians in the third and fourth terms (3.23) and (3.24) are centered ~aboiit co - nEw and Ec + n~w, respectively, which indicates that the neutron has lost or gained n~w units of energy in collision. Thus A+ and v_ give the inelastic contribution to the cross-section resulting from the loss and gain of energy by the neutron. The first term, aouter, in the approximation considered is formed only of elastic scattering. Therefore, we can see that the summations in a+ and a_ are limited, essentially by the amount of energy the neutron can lose to or gain from the system, We can then say that if in the avera;ge the system is in thermal equilibrium, and thus has kBT units of energy the neutron can approximately gain this same amount. Then the upper limit in the summation in the v_ expression would be given by n_ = (kgT / ico) (4.1) On the other hand if the neutron has co units of energy with which it started out, it can only lose this much; and the upper limit in the summation in the a+ expression is -25

-26n (/ Aw) (4.2) Nearest Neighbor Distance and Hard Sphere Diameter We have seen (Chapter II) that the average nearest neighbor distance or the distance between the center of the cells is given by "a'' and that "a" enters into the expression for w, the oscillator frequency. We have also seen (Chapter III) that in the calculation of g(r) another parameter entered, d, the hard-sphere diameter. These two parameters are not equal. However, one may obtain an approximate relationship between them by noticing that if one would use the LennardJones six-twelve potential to calculate the first peak in the g(r) function, it would be given by the minimum of the Lennard-Jones potential. The value at which the g(r) function has its first maximum is then rlst max = r* for LJ potential (453) For the hard-sphere potential this maximum appears at "d" rlst max = d for hard-sphere potential (4o4) Therefore, using (2.6) we get d = r* = a ( 3 4)1/3 (45) Calculational Procedure We can summarize our results by writing for the differential cross-section of normal liquids [E~eG; Co,p,T; w(i,v,a),p] (4.6) which tells us that if the inequality 1(v/ (2v/~)32 5 67

-27is satisfied, and if we also know a) the experimental conditions i. average incident neutron energy, co, ii. incident energy spectrum width, P, (which may easily be related to the half-height width), and iii. the temperature, T. at which the liquid was kept during the experiment, and b) the liquid under investigation, defined by i. the number of atoms per unit volume in the liquid, p, and ii. the oscillator frequency, w, which is related to the six-twelve Lennard-Jones gas-phase potential parameter p. and v, and the nearest-neighbor's distance, a, we can obtain the intensity of neutrons scattered through angle 0 and into final energy E. Table I shows those elements for which there is gas-phase data13 available and for which the cross-section has been calculated according to the parameters also shown, In the Case of Lead the p and v parameters are not known. However, one can estimate the oscillator frequency for the liquid state by making use of an approximate expression for the entropy of melting (Chapter II, (.2.o.17))o The value of w thus obtained was used in our calculation The re'sults for the angular cross-section are shown in Figures 5 through 9. Figures 5 and 6 show the angular cross-section

TABLE I CALCUIATIONAL PARAMETERS FOR MONATOMIC ELEMENTS r* A* V* T kBT A*/kT V*/V M p Elements ergs/moIecule (A) ev/molecule (A3) (OK) ev (gram ) (Atoms/cm3) ev sec-1 (x10 ) (x1-3). (x10-2 ) (x1022) (x102) (x1012) Neon 48.2 3.12.0361 21.2 24.57 2 16,9.975 33.463 3.597 7.00 6.669 Argon 165.0 3.84.1240 40.0 84.00 7,22 17.1.970 66.370 2.130 7.57 8,198 Krypton 238.4 4.03.11i90 46.4 11i6.-oo 10,00 17.9.990 138.940 1.765 7.00 5.855 Xenon 309.9 4.56.2320 67.1 1163.00 14J0 16.5.960 217.695 1A00 7.00 4.500 Lead 609.00 5.25 344,864 3.067 6.85 12.600 lead 620.00 5,34 344.864 3.060 4.42 12.600 Lead 823,00 7.09 344.864 2.995 6.85 12.600

-29for Argon and Lead, respectively. Experimental data ("22'23 )is available for these two elements, and we have shown it in the same graphs. The theoretical curves were obtained by integrating the final energy spectra at each angle, i.e., o(O) = f da(e,O;,) (4..7) and the normalization of the experimental points is that which gives closest fit at large angles. Notice, however, that in the last three figures 7, 8, and 9 (Neon, Krypton, and Xenon) we have only shown a few integrated points at large angles. The integrated points are properly indicated on the curves. The angular distribution shown is a(e) = f dE ( - o) a(E,0; Eo, ) (4.8) which is equivalent to counting particles with a counter that only detects those particles with final energy co* This was done due to the fact that there is not any experimental data available for these elements. Therefore, the nearest neighbor distance "a" is not known. We have guessed this parameter for Ne, Kr, and Xe from crystal data and calculated r*, Thus it was felt that the complete energy distributions were not necessary. Notice, also that the asymptote (see diagram) about which the intensity fluctuates is a function of temperature. Although,. not a very strong function of temperature, this feature may be detected experimentally. The final energy spectra widths at half-maximum are given in Table II, Chapter V.

-30az 1I.o b 0 WINNER 9Q0 0.0 00 90~ 0 OUTER 2.0 00T_ 0 0.0 o 00 90 C0 = INNER + C OUTER Diagram. Intensity versAs Scattering Angle as a Function of Temperature.

2.0 LIQUID ARGON 2~~~~~~~~~~~~~~~~2 I z I L w~~~~~~~~~~~~~~~~~ T:84~ K (I) u1.0 tVO" Q o I.'.0. o) I H U4 I o / I W-0 0 0 10 20 30 40 50 60 70 80 90 SCATTERING ANGLE Figure 5. Normalized Cross-Section (4.7) for Liquid Argon at 840K as a Function of Scattering Angle. Also shown is the experimental cross-section(22) (broken line).

I'\ I' 2.0 Z~~~~~~~~~~~~ zo LIQUID LEAD 0 I Io~~~~ E:=~4.42 x IO2ev. L3~ I ~0O ~~~~~~~'I i62 K cc Cl) C.)0 ~~~~i,0o of I i II: I I 0 I i Z: I /, ~/~ 0 10 20 30 40 50 60 70 SCATTERING ANGLE e Figure 6. Normalized Cross-Section (4.7) for Liquid Lead at 6200K as a Function of Sc ttering Angle. Also shown is the experimental cross-section 23) (broken line).

2.0 LIQUID NEON -2 E = 7.0 x 10 ev. o_ 0o I i I iiFgur T = 24.570 K IU. 1.0 0 0 0 10 20 30 40 50 60 70 80 90 SCATTERING ANGLE Figure 7. Normalized Cross-Section (4.8) for Liquid Neon at 24.570K as a Function of Scattering Angle. Also shown are a few integrated points (4.7) (broken line).

LIQUID KRYPTON 2.0 Eo: 7.0 x IO6' ev. 0 z o T=116 K 1 — o0 w cn I Cl) C) 0 o~ ~ Ix~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ LUJ 0. 0 0 10 20 30 40 50 60 70 80 90 SCATTERING ANGLE9 Figure 8.Nornmalized Cross-Section (4t.8) for Liquid Krypton at 1160K as a Function of Scattering Angle. Also shown are a few integrated points (4.7) (broken line).

LIQUID XENON -2 2.0 E 7.0 x 10 ev. 0 T=163 K z o I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ U) W, o -2.0 4 u) 4~ ~~~~~~~~~~~~~ I SCATTERING ANGLE 8 Figure 9. Normalized Cross-Section (4-8) for Liquid Xenon at 1630K as a Function of Scattering Angle. Also shown are a few integrated points Q4.7) (broken line). V)~~~~~~~~~~~~~~ z;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I ~ or z ~ 0 20 30 90 50 60 70 80 90~~~~~~~~~~~~~~~~~~~~ SCA~~~~fERING ANGLE ~~~~~~~~ Figure 9. Normalized Cross-Section (4.8) for Liquid Xenon a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 163"K~~~~~~ F~~intgur.Nrmalied Cronss-eto (4.8) fbor iuienoin a 13~

CHAPTER V COMPARISON OF AVAILABLE THEORIES WITH "SMEARED" POTENTIAL MODEL In what follows we will try to outline briefly some of the available theories and to compare the results obtained with these theories to those obtained with the model investigated in this work. Zernike-Prins Static Approximation'(4 25) Although originally the Zernike-Prins formula (5.3) was not derived in the manner we will describe below, it is equivalent to assuming the time-dependent operator is(t) to be slowly varying in time. Thus, setting t = 0 (static approximation).% (ft) s+ (5,o) - 2s(so) 4 (~,0) - 1 (541) and as before (3.15) N N < Z Z Qs (ft) j (,0)>= Np dr g(r) exp (iKr) exp (5.2) s=l j=l T Therefore a(e,e) = b2N(E-)(k/kok/ko)(l+p d3rg(r) exp(iK-r)) (5.3) which with an incident spectrum of the form (3.18) gives a(e,0e;-,-o)/Nb2L = exp[-(E-eo)2/p2](l+pfd3rg(r)exp(iK'r)) (5.4) Convolution Approximation This approximation due to George Vineyard(5) is essentially described in the following manner: define g(r)d3r = the radial -36

-37distribution function = the probability that, if an atom is at the origin, a distinct atom will be found simultaneously in d3r about r, G(r,t)d3r the time-displaced pair distribution function, the probability that, if an atom is at the origin at time zero, an atom will be found in d3r about r, at time t, Therefore, we can write G(r,t) = Gs(r,t) + Gd(r,t) (5.5) where Gs(r,t) = the probability of finding at r and time t the atonm, that was at the origin at t = 0, and Gd(r,t) = the probability of finding at r and t an atom distinct from the one that was at the origin at t = 0. These two functions are formally defined by N GS(r,t) = (2e)3N-l fd3Ke-iK.r < E ( ) (5.6)t(,o)> and Gd(r,t) = (2=)-3-1 S d3Kei-Kr< Z (,t)Q*(,O)> (5.7) K =lj=l J T If now in (5,7) we let (convolution approximation) N N N N < C 7. n( i,t) +.(,o )> -< C ~. (it)a+(i~0) Q$( ) j( 0) >T ~e-i jl e=lj- l j (.8) N 1, — vv^'~^'

-38(5~7) becomes Gd(r,t) = (2t)-3N-lfd3Ke-iKr'. [N-1< Z A(t)s(qo)>T (5.9) N N N N *< Z Z ~(S0o)n+(Q O)>T= fd3r'[fd3Ke - r K Z Z (0,o)OJ (i O)>T] R=1 j=l j T =l j=l -' fj fj.[d3KTe iK (r-I ) < (t)NQ(,0)>T Gd(r,t) = f d3r' g(r') Gs (r-r; t) where g(r) is formally defined by N N g(r) = (2) 3Nfd3KeiK r < E E (),O)()Q,0)>T (510) K a=1 j=l-1 Finally, by applying the convolution theorem, Vineyard showed that k 20 itq N N + (C) - k b2 00 dte itq < ( Y. t)j(2 O)>T Nb2-kor (Kq)[l+y(K)] k ( 2 Tct 0 Y=1 j=l -) ko swhere rs (K, q) = f dt fd3r Gs(rt)ei(K1r-qt) (5o12) -00 r' and 7(K) = f g(r) eiKl-d3r (5o13) r By definition the Gs(r,t) function is interpreted to give the wandering away of an atom from the origin, In one of the models Vineyard used, he assumed that this distribution was governed by the diffusion equation (DV + t) Gs(r,t) = 0 (5o14)

-39which when solved for a point source at the origin gives Gs(rt) = (4kDt) 3/2exp(-r2/4Dt) (5.15) This:.res.ult:leads. to rs(K,q) = 2DK2[(DK2)2 + q2]-i (5o16) and thus to a(e,O) = (Nb2k/ko)2DK2[(DK2)2 + q2]-l(1 + pfeiKEg(r)d3r) (5Q17) Summary Let al(Ee), a2(,eO), and a3(e,O), represent the Zernikeprins formula, the Vineyard diffusion model, and the "smeared' potential liquid model cross-sections, respectively as given below al(E,9)/b2N =k 5(-Eo)(l + P eiK r, g(r)d3r) (5,18) ko ~2(E,e)/b2N k 2DK2 (1 + p J efiK g(r)d3r) (5.19) kb (DK2)2 + q2, G3(e_)_/b 2N kZ 5(~o_ -E-nTw)[e-A3Q~+nAl n(A2Q2) (5.20) + P8no I eiKr g(r)d3r]. We, therefore, have that (a) the static approximation, al(Ce,), gives no final energy spectrum broadening, (b) the diffusion model, a2(C,e), gives an energy broadening given by the half-height width equal to A3 2fDK2, and (c) the "smeared" potential, o3(E, 0), although difficult to write in closed form, gives an energy broadening, calculated by a 704 IBM machine, usually smaller than —that predicted by (b).

The half-height widths for liquid Lead, at two different temperatures, have been measured and reported in the literature(26) in an attempt to check the diffusion model, We have calculated these widths using the'smeared" potential model, The results of the experiment, the diffusion model and the "smeared" potential model are shown in Table II. The incident halfheight width is given by Aco, the "smeared" potential half-height width by Asc, the experimental half-height widths by AE2, and the diffusion model by Ac53 It is observed that the width at half-maximum increases with temperature. This effect is also observed in the other elements studied. It is also observed that the energy spectra are more sensitive to these changes in temperature than the angular distributions (See end of Chap. IV). TABLE II LIQUID LEAD FINAL ENERGY SPECTRUM WIDTHS T | Aeo Ac| 1 Ae2 ] Ae3 ~K (x 10-2) (x 10-2) (x 10-2) (x 10-2) Element ev ev ev ev Lead 609 1,06 1.78 1o,86 +.11 2,30 Lead 823 1.06 2.07 1,98 + 11 4.160 NOTE: Comparison for Liquid Lead of Experimental Widths at Half-Maximum, AE2 26), with the Widths at Half Maximum for the "Smeared" Potential, ACj, Calculated for an Oscillator Frequency, o = 12.6 x 1012 sec-1 (See Chapter II, pa 15) and the Diffusion Model Ac3(5,26).

CHAPTER VI NORMAL DIATOMIC LIQUIDS Diatomic Molecule The time independent Schrodinger equation for a molecule may be written as 52 N 2 N r2 2 (- 2me V - Z 2j V. + V) = E (6.1) i-l j=1 J There are n electrons and N nuclei and V is the sum of electrostatic interactions between all pairs of them. It can be shown(27) that the above equation may be separated into two equations. One of them describing the electrons motion and the other an approximate equation for the nuclear motion. Born and Oppenheimer(28) have shown formally that this approximation is justified so long as not too high vibrational and rotational modes are excited. In Figure 10 we show a diatomic molecule, the center of mass position vector R, and the relative position vector r. We will use the following notation. The actual position vector of the vth atom in the s-th molecule will be denoted by e. In terms of the center of mass -S position vector and the relative vector it is given by V R + f r (6.2) -S -S -s where M1; v = 2 (6.3) M

-42ZI (1/X2 >M2 Figure 10. Diatomic Molecule Coordinates.

Turning our attention to the diatomic molecule Hamiltonian, we can show that the approximate nucleax motion Hamiltonian, HDM, may be written as the sum of the relative, Hr, and the center of mass Hamiltonian, HCM, as follows DM HCM +E ( 6*4) where HCiM = _6 V2(R); MM1 + M ( 65) and h2.12L2 Hr =r 2 2 6 2 r_2 a r r + 21tr2 where [i is the reduced mass ( = M1M2/M), L the angular momentum operator and U(r) the potential energy, In order'to describe the motion of the -diatomic molecule in the liquid phase, we introduce our approximation to the LJD liquid model potential as an additional potential to which the center of mass is subjected. For normal liquids this is believed to be a good assumption (see discussion in Chapter II), We may thus write as an approximation for the diatomic liquid Hamiltonian,.2 HDL = HCM + V(O) + 2 R + Hr ( 67) Spin Dependence Spin dependence may be introduced through'the neutron-system interaction potential (1.12) if one writes for the scattering length b. =.+~ V S ( 6, 8)

where b is the scattering length associated with the vth atom in the I-th molecule, a is the neutron spin operator (Pauli spin matrices), SV and SV are the nuclear spin operator and the nuclear spin of the vth atom in the Ath molecule, respectively; V and e are given by a n a= [(SV + l)AV + Se AV](2SV + 1)1 (6.9) = 2 (A+ - Al") (2S + 1) These are the customary definitions of coherent and incoherent scattering amplitudes, respectively. The scattering,,amplitudes for the two values of the total angular momentum 7e + 1/2 are given by A+* For unpolarized neutron beams and nuclear spins we want to calculate the average over neutron and nuclear spin direction of the thermal average, We will indicate these averages by { }S, Therefore, we want N N 2 s < c b A (a,' QJt*) > (6.10) 2=1 J v=l ~=1 where S', (t,t) = exp [-x k V] I(t)] which in Appendix A.15 is shown to be equal to { < bSbi nV Sla (t9,) a,:+ (if) > Irk (.1 v S(S + 1) 5 Ijvj,< i + I2 %jvll < sVe ~t n4,+ t) >n (e) >

-45where 6L = 1, if atoms v and p. in molecule 2 are alike, = 0, otherwise. For spinless systems this reduces to < bV bo(t)Q(0) =i < (,t)Q,o)> (6.12) For monatomic systems we have + S 2 2222 T t < A.y b b j 2(0 t t) 0j ( n 0) >;TV=' q Z [0aj+4 S (S +1) ]<2Q(., t) ( i,0 )>' [IV, J (6.13) N N + Z Z C c <.=1 j=l j Q j Thermal Average We can write for the Thermal average with the help of (6.2), < V (It) j (0,o)>T = < e iKT R(t)+f1:_(]eiK.Rj+fr j)> (6.14) - < cl e- iK'R(t)eiK'-R j >I< r e- ifVK'r2(t)eif[pKr > The last step is easy to obtain once one notices that the diatomic liquid Hamiltonian is separated in two, the center of mass hamiltonian, He = HcM + V(o) + Mw2R2/2, and the relative motion hamiltonian, Hr, and that they commute. Also because of this fact, the total wave function may be written as a product (Inj' 2) =c(B)_r(r). (6.15) In (6.14) we recognize the thermal average of the center of mass motion as the one we encountered in the monatomic liquid treatment,

and, of course, the same remarks apply. That is, when e = j we use (6.19) for the Hamiltonian, and for R / j we use the Heisenberg's operator time expansion method, which we have seen lead to the radial distribution function, Static Approximation The relative motion thermal average will be treated in this section in the static approximation (See Appendix A.16 for discussion of further correction terms), Therefore, setting t = O we have -ifVK.rj(t) if'Ko r< < frli ee Irj T r lexp[iKo(f"r -fVr)]rT (6.16) At this point it is necessary to make an approximation in the relative motion Hamiltonian. As it is standard practice in treating diatomic molecules, we will expand the potential U(r) about its minimum rM. Thus Hr = Hrot + Hvib (6~17) where Hrot -i2L2/22M (6,18) and Hvib = 2 r2 a r - + U(rM) + JM (r-rM)2 (6.19) The relative motion wave function *r separates into a product of the rigid rotator wave function, trot) and the vibrational wave function vibe Averaging over directions of the molecule axis, we obtain

-47<r I exEiiK- (f lJr -fvr ]I *r> 1r < Vib io[Kr (f -f )"fvi>; i 4=vib ljO(Krjif) jo(Krf~f) fVvib>T; f i. (6.20) Expanding rj about the minimum rM of the potential U(r) we finally obtain <~ rIexp[iKQ(fLrj-fvr2)] Ir > r jo[KrMJ(ff)> = i (6.21) = o(KrMjfL)Jo(KrrMfV);) / j. The cross-section may then be written, after integrating over incident energy spectrum'a( e;, )/2A.NL e= 3 hn e A nA E-,O+nhw)2/p2 (6.22) n (A2Qn) [j (alrMn)+Bo2/Ao2 ]+2j o2 (alrMo/2) p drg( r) 4r2 [s in(alor)/alr ] r where a1 = 48m1J2 Ao = coherent scattering amplitude, 4 B2 = the bound-scattering cross-section = a2 + -S(S+l1) and Qn is given by (3.21). In the aforementioned we have assumed equal mass atoms in the molecule and neglected the spin correlation term that appears when the atoms are alike (last term (6.11)). In Reference 2 the spin correlation term is discussed, and it was found to be negligible except for very low mass number atoms. Equation (6.22) has been computed in a 704 IBM machine for the Nitrogen molecule. The results are shown in Figure 11 together with the experimental data(22). In Table 3 we have given the numerical constants that entered in this calculation.

Results In Figures 5 through 9 we have shown the calculated and the experimental angular cross-sections (when available) for the monatomic elements studied. In Figure 11 we have shown the nitrogen diatomic molecule angular cross-section. The agreement observed is believed to be satisfactory. However, a radial distribution function more accurate than the one employed in this paper for the outer effect would probably improve the agreement. In Table II we have shown the broadening of the neutron final energy spectrum as calculated with the "smeared potential" liquid model and compared it with the experimental data as well as with the diffusion model. It is seen that the "smeared potential" model is in better agreement with experiment than the diffusion model. It is felt, however, that more experiments on the broadening of the energy spectrum should be performed, using monatomic elements as well as polyatomic elements, in order to further test the accuracy of the model introduced in this dissertation. No experiments on the energy distribution of normal polyatomic liquids have been reported in the literature.

TABLE III NITROGEN MOLECULE CALCULATIONAL PARAMETERS * r"~* A V* JT A^X*/kT * 0 0') (K Element (ergs/molec.) (A) (ev/molec.) (A3) (OK) (x 10o"16) Nitrogen 132,40 4.15.0994 51.5 7740 14,9.99,p ~ 0 p Bound C ohl M Element (grams) (atoms/cm5) (ev) (sec-1) barns barns () (x lo-2 (x 1022) (x o102) (x o1012) rNitrogen 46*48 35,480 7o0 7.897 11.4 11.0 1,094 - - - - -- - -, -,,__

3.0 I I LIQUID NITROGEN I-2 II [~~~~~ E~ 7.0 x 10 ev. 0 T:77.4 K 2. 1 Z 0 t-' UJ i, 0 I' 0 o 1.0 IL)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ cr~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ OL z 1.0 Z~~~~~~ I / / /_ / 0 10 20 30 40 50 60 70 80 90 SCATTERING ANGLE Figure 11. Normalized Cross-Section for Liquid Nitrogen at 77.40K as a Function of Scattering Aigle. Also shown is the experimental cross-section ~2) (broken line).

APPENDICES -51

APPENDIX AM1 TRANSITION CROSS-SECTION(31, 32) Let Cs (r) represent the scattered wave function from initial state a to final state., We determine the number Pa (e)dQ of particles scattered rs per second into da by equating it to the flux through a surface element of the sphere of radius R, and since # (r) has the form nS (r\ i ( I\; fir / r (A.1l2) we obtain for the transition cross-section per unit incident flux in the z direction, jOz' C5(2- P c9l~n_ (9 /(A^1.3) o

APPENDFIX A.2 INTEG1RAL EQUATION The time independent Schrodinger equation is given by [ H~~~ A~~~~~~(r +A( r(<Ji) (rip =0~ O(A421) Expand'Y(r,_) in terms of the complete set of stationary state wave function Qf the system; (rY a\ 5 = X, 1 ( r5 1\' R (A,222) where W ( % ) < ( R =~ S < ( G ) (A.235) Then (Ao2,1) becomes after substitution of (A.2,2) and (A.2,3) E lW. t ++( ( "5 Ih-' = ~ (A,2 4) Define -53

-54and multiply Equation (A.2.4) on the left by A1*(Q) and integrate over._space, then we obtain [ l l ( r) _ 6 19 A) ( or) =- E A) ( f B (A.2.6) where =- f\ (Z ( (N) A I, (A.2.7) Introducing the Green's Function, Gc (r,r') defined in Appendix A.3 7 we can write Equation (A.2.6) in the form where (r ) is an incident plane wave where O(r) is an incident plane wave,

APPENDIX A.3 GREEN'S FUNCTION Define Gk(r,r') by the following equation ( VI ( (r% (A.3 1) and take the Fourier transform of both sides of this equation -'f -'\ l I tt k V CT( 7') 4 (A,3o2) 4 5g i do3r I, C C 4- = and since (V - k (A5303) __ _,, we obtain F (IJ 1 (zni j = (z7 | r Q/ t ( \\l j (A.3-4) -55

-56Therefore, by inverse transformation, we obtain that Gr~ I A-(r 14_( ^274s/lh~k\ Fl~. (r.% This integral has been evaluated elsewhere( 7)and the result is Cw (_,_ 5, \ \ r \C1 | (A,3.6) That takes the asymptotic form \t — \r )r\ (A 3,7) since - — > o0 I-_'r -- =(,r- ol' _ _X = [t o/ 71kA (A 3,8)

APPENDIX A,4 ENERGY CONSERVAT1N (2) In Chapter I we obtained (1,11) &r \< (Aq4I1) which may be written.e(- (l - - )I in which &l 1 &.' I' ((A.oCE -E )) -57

-58represents the differential cross-section (in angle and energy), and where 6(E - E) takes care of energy conservation. Consider t > (, r (,. (A.,44) MVu\!'I l l s CE 64 (Ao4K5) _ ] - o ~ \\ sgg ~ lCG

-59where we have used the hermiticity property of H(q) and the relation 0 itt l tt I /AC -I A.CI t\,'~ t(A,4.6) easily proved by expansion, and According to (1.12) we have that -(<lR A, E b~ therefore (Us r*( (r'n" i)C9rNh 5(A.407) which when substituted baick into (A.4. -5) gives II,V l i(A.4V8) tT~~~~uS~ ~ - |& ( fiSla

APPENDIX A35 PARABOLIC RANGE10 In order to show that in the range V*/V >.67 the LJD potential V(r) has a minimum at r = 0 and no other, one would have to take the derivative with respect to r of V(r), equate it to zero, and then show that r = O is a root; and that there are no other real roots for the above indicated range. Also the second derivative at r = 0 must be positive. Thus, using (204) and y = r2/a2 aV~r —A-r (Vi/V I(9 /V) - (A,5o1) Therefore, r = 0, and b al] V ~ m = V / V A l 2 (A 5/2) which leads to (A,5.3) -60o

-61For the range 2b = 5/11 and for y < 1 there are no positive rootso The second derivative evaluated at r = 0 is o~ t(X) - K (Oh = 40o. 4 c > C (A.5.4) for 2b = 5./11

APPENDIX A.6 HARMONIC OSCILLATOR FREE ENERGY It is shown in Reference 10 that the free. energy of an imperfect gas or normal liquid is given by F=-; \ N (2K e T / h 4 1 (A.6.1) 0'V where the first term represents the contribution from the translational motion, the second the internal degrees of freedom, j(T) the partition function for the internal degrees of freedom, and Q(T) the potential energy W contribution, _- W / i? ~-,Q (A.6,2) dwi the volume element, to which the center of the i-th molecule in 2 the liquid belongs. For a harmonic oscillator W = V(o) + sr _a C \ = ( N \, ~p -1 Vt c T'1 cc (A,653) -62

-63ZL (T) —-N _, -T (aid) la(A.6.4) (A.6. 4) -il n \ 9 (0 _ \is k j (T)

APPENDIX A,7 THERMODYNAMICAL FUNCTIONS10 By definition we have that p = Pressure=- t` r (A.7 1) E = Molecular energy = - \ - / T) (A.7.2) Y = Partial potential (A-703) and H = Heat content = E - V (A.7.4) which for the harmonic oscillator potential, and with the help of (A.6,4) become?= \/ j40 (V / \18 t(9 + \ (A*7*5 (A.7.5) - 3; T (+- llsL (V /V )I7 ~L.~2 (A.7.6) 9\t Zt na Wj (T)

'- - L] ~ (L.LV) + V z A - " lt"/A t h rii -~ t I pi + tQ ) A = 1

APPENDIX A.8 HEISENBERG'S OPERATOR32 Making use of the expansion (A 8.1) co where )HBe O I i A,& 2 HR- e;) ) ) )eai = At t ~(A.82),.nd.. with U; Ip~~/2 m~ 5t s ~r'~ i~ t. ~ (A.8,3) we can show that [ g r]:Ch~ A = (_ (i 8 z ( S / z m) \r C O B (Ae8,4) and where which when substituted back into (Ao8o1) gives

-67(A.8.6) [whic(h or s - - gives which for s = ma gives 2 (t = \?&Dr ) Or'.c0sat 4- A -X / $ mu) (A.8/) It is interesting to notice that this result is just what one would get by solving'S I ~~ =_ k r (-) (A..8)

APPENDIX A.9 THERMAL AVERAGE We will show that, making use of Bloch's theorem14 jT q~'r (Ag.l) where Q is an oscillator coordinate, and of the Baker-Hausdorff theorem A C (A.9,2) where C-A N +3 \D (A.9.3) the thermal average, operationally defined by (A,9.4) is equal to z I`i \r i~(oi\ eCA" \< i ) (A9 4) -68

whe re A- t = Aik) / k (A.9-5) From (A.9.2) we get _< QK e (A.9.6) The last term may be factored out of the thermal average because the commutator is a constant. Using (A.9.1) \, \r Cte (A 9\7 <q) TD

-70((A.9o7) cont'd) -,,,p- <e Ct r1- t' sz[c rs > 0< and since we will show, at the end of this appendix, that < rat (t )r fs et) fTs= <. 9.1) T T and that for z = exp (hw/kBT) S: r t s-, -, (i. A.9.7) < %, r =_~ S r (ii. A.9,3)

-71where we have used the expans ion We can show that for the harmonic oscillator the following relations hold (io A9.l) 0)( Y\ O( 0lt fl \ ( VW + i God -t (g ~t-\ zlien-;?\t)v~ I

-72WV' \ A(i A.9.2) Therefore, 1G C \ t t ) - ( \ A) \ (i. A9.3): ~ \ ~V'A \V'? ) ~ since ~(%.,b tQV\ ( \ AA\V\) (X k\Eiv )- 0 _

Cons ider (i. A9.k4):_ (+~~(Q4~~~~~~L~~~ /l The thermal average thus becomes < = -tU)K~ w(~~ r ~ 1 W 38(i. A.9,5) /\ )es/\~~t W

-74 (i.,A,9.5 con't) EAl = 9) 254- -t i/(\-Qt \n)v "er e' (i't(i A.9.6) where i = taut ( i a) | I, T > (i, A.9.8)

-75-" =*l t \% _ (\ V \vV = i (ii; Aogo.) since (A P)\ \t)Xd\V\(> (y \\\= t)\ \ Consider 4 akSM8 eV 8\(fiC t X4 k:,~~~o~";M~~~~~x ~ + \ At-' VQ/2..

(ii, A.9,2 cont'd) Therefore, the thermal average <P t > = t i% Mu3 (s \\ /. C'ti 1) is ((ii A.903) Similarly, (ii, A.9g4) MM k = Ai < (iie A.9o5) e i. T 7

-77< rOit+) vb t) C = 4 o': (iii,) We had in Appendix 8 (A.8,7) I~. T - to' at ~ \r,\ >t I t/,5 < f0,S (iii. A.9.1) _:*xCeAr U) C'Mi j i<81 And e 7. C+ 1~ uj ~147

APPENDIX A.10 IRVING-ZWANZIG THEOREM AND RADIAL DISTRIBUTION FUNCTION The Irving-Zwanzig theoreA 6)states that the expectation value over a state 4, of an operator depending on the position and momentum operators of N bodies, is identical with the integral of the corresponding classical function of position and momentum, over the so-called Wigner distribution function of the state 4o The Wigner distribution function is the quantum mechanical analogue of the classical phase distribution function, and depends on the position and momenta of the N bodies. It is defined as in which and has the following properties and has the following properties low l. I - A..iA,' | P - ~ ~ ( +' (A. 102) -78

-79|r~~~i~ ei~~~~i~ d=~ ~~ iii~ W -(A,10 3) 5 * W l t j ) A ~i ( )x} )wi3~ ( XA(Ao10,4) We will also define the reduced Wigner distribution functions (AlO.5) and the reduced spatial distribution functions jsh l jc| *,ee(Ao6) ~- l \Mr~,l Ijr jvj 4

-80The theorem states that l tle rllp9~I ~I"~"e) X W 8 0 1 C,r \X (A,10,7) where the classical function g(r, S1) may be obtained30 from the function of operators by substituting the operators R and P in G (R, P) by the variables r, P, and by applying the operator exp [~ ih Vp.] on the left, Therefore, E S U ~t t &i PIv q\ t vs A cw ( T(A*10,8) NOTE.o The sign in the exponential is negative if in the function G(R, _P) the operator R appears on the right, Consider \ (k \ Q (>t t * (\ ( ( k (A,10.9) Look at dp( ~ ~ A \CtZ a. = 8Saa t (f\t

(AolO.10) Let r =r and r2 =rl dk i~~~ t ( \82 j 9 ( % (v \ A (A.*lo_11)' \ If and dropping the primes in (A,10.11) we get a, at, 0(Ao10 12) ov z

-82The thermal average of n2 is equal to _~ /4t zT -\ Z r C< I......... r A-% |2;5\r 3 ( The function P2 (r1, r2) is just the quantum mechanical analogue of the one defined in Chapter VI of Hill's booll). This is seen to be the case after recognizing Slater's sum inside the integral sign, It is shown in Reference 11, p. 185, that for a fluid this function is related to the experimental radial distribution function g (r) by w tr of ato'ms pr( LC~ (A.10.L4) where p = number of atoms per unit volume = N/V.

-83Therefore, our thermal average becomes (Ao 1. 15) since there are N(N - 1) terms in the summation,

APPENDIX A,11 TIME EXPANSION In Chapter III we encountered the aouter thermal average, and we established the fact that in order to calculate it one was forced into using the Heisenberg's operator time expansionl2) It was also mentioned that in order to calculate g(r) there was no need to go to the averaged LJD potential, since calculations for g(r) have been performed using the Lennard-Jones six-twelve potential. The thermal average was of the form In Appendix 10 we have shown that the zero-th order term is related to theT g(r) function in which we have only shown a few term of the expansion (Ao8o1). In Appendix 10 we have shown that the zero-th order term is related to the g(r) function. -84

For the first and second order terms, we have that for a Hamiltonian of the form H = E tz / C~2 K A 2) V the commutators take the following form [ A Z L ES = [ (' WA) / 2e5 1s + kI 5 / 2' C 0 (A.ll, 3) and C A - = (i; /'S -.4) -;2i' <.. V ( / -?...

-86Consider C - - -sz a -j j In order to evaluate the second term in the RHS of (All,5) we made use of the Irving-Zwanzig theorem and of the Weyl correspondence rule to evaluate the classical function that enters in the theorem (see Appendix 10). which applies when the operators R appear on the left in the function G (p, r). Therefore, 4 (nj?) % ( t ( -' ~ V Q7 Q s Ts(A.ll.7)

-87Since there are only terms linear in pso We obtain (A...8) Therefore, the thermal average may be written as follows: =- _ S&xi.( i KC'ielwo 7 i~~~~~~~~~~~~~~m MWcc~T'5,5'1 ~ ~ ~ t'c-i,"c- 3 ~~~~~I Q MZ 1~~~~~-~~~~~~2 f152~~~~(I119

-88The last term vanishes on the assumption that the f2 distribution is isotropic in the momenta. Substituting this result in (A.115) we see that the first order term in the expansion is zero, <[ | -H] ) -G ) >-= o (A.1.)10 Proceeding in the same fashion, we obtain' 0 (A.llll) L and t ~ - -'~ > Z ('

APPENDIX Ao12 HARD SPHERE RADIAL DISTRIBUTION AND DENSITY EXPANSION All thermodynamical functions of a system of N bodies can be expressed in term of two very simple distribution functions, the P1 (r1) and the P2 (rl, r2). This fact alone makes distribution functions very useful tools. However, things get complicated because in order to calculate the P2 (rl, -L2) one has to know the P3 (El, _2, r3) etc. To this effect several approaches have been tried. Most outstanding are those of Kirkwood(A and Ivon(l9)-Green-Born(20)who have obtained integrodifferential equations linking these distribution functions. In this appendix we will limit ourselves to briefly outlining the Kirkwood method, since it is the one we have used in order to obtain the radial distribution function. In order to do this we define a potential energy LU (,cy,-; \ L... \..C ( _ (A. 12.] 1 where the a's represent the intermolecular couplings. They range from zero to unity. Full coupling (real fluid) aj = 1, for all j's, and \ 2 )_X). _ z ( -89

-9o0the probability that a molecule (not necessarily molecule 1) will be found in the element of volume d3rl at l, a second in d3r2 at r2, ~-, another in d3rn at rn = ij,3r / where Z7 = I |. 3r (A.12.3) By taking the derivative with respect to one of the a,'s of Equation (A.12.2) the Kirkwood integro.-differential equation is obtained (Ivon-Green-Born take derivative with respect to one of the r's). The result may be written aS11) (A. 1i24) A(- /'/ I \ L ~k~'r u(, ( \r,r o,. +P y:' 3

-91(A.12.4 cont'd) _ C which in terms of the correlation function, g2 (-1, r2; a), p tT r. ~~ - - pa\ ( T 02) C ( 9 @ C ) (A.12.5). q ( r' k) can be written in the following form after integration over a':3 It can be shown that~ for p-4 0 the gn ( r1' *'', rn; a) equation (neglecting terms of order 1/N) is

-92(A.12 7) that, when substituted in Equation (A.12.6), as a first approximation for g3 and g2 inside the integral sign, give:(1l, 29) % ( \r Ij -U,, / TA ] QJ2 (A. 12.8) For a fluid (a = 1) %2q~~~~~ ut~ r ~a~ r. ) = Gk (A.12,9) -UL|~ \333

-93(A.12.9' cont'.d)'3 Hard Sphere Potential U(r = 0 r d (4.12.10) ) Using cylindrical coordinates.it is easy to show that for the above potential I cQf\I QJ ( z _~~ 83r/t % -~ _ 1 [-'I 1 (A.12. 11) =o; rcd Therefore, (A.12.9) becomes4l, 29) (A.-2.l2):. ~,\r 24

APPENDIX A.13 INCIDENT SPECTRUM CALCULATION We have shown that (3*14 and 3.15) ~ttO)/b = ( /k9) Gg _E6';\L~\~~~~Td ~(A13,2): C \&3r Q.1r, These are the results one would get for the theoretical cross-sections when a monochromatic beam of neutrons of energy Eo hits the target, Howe.ver, this is not accomplished in practice. The true energy distribution out of a crys;t.al spectrometer is of the Gaussian type rather than of the.ideal Delta type. We will here assume that the incident energy spectrum distribution is given by (3.18) -9t4 -L - (- o /r (Ael373) -94

Therefore the theoretical cross-section would become 00 c( j L c -\ - o r7_~~~ ) j |c3Cri. k+ ~~) ct;o 0 _________ -te-e -V ( u_ ( Q =-o,,0 A' (2s,>/ C, C Ci-+:\tM W)O4b 4Ll S VI 4vLO /WO 9d o

_96therefore T( }-;60-)D / tb K - (0 CKA.15) 00PI y hReqY 3 7 n(\,, I2) + where (4 / ) (A.13.6)

-97The summation over n may be divided into three parts as follows a0c 0, ~.; 0,- - (_6ZaI~ _ - _(A,13,7) -r+ I,;(Az13.8) (A.13.9) 00 X o7-64tilY/ j. - Vj 2

APPENDIX A.14 OUTER EFFECT CALCULATION We have shown with the h.eip of Appendices A.10 and A1 that the outer effect may be exhibited as follows, Also in Appendix A*12 we have shown that Therefore r 0 0 j 0 0 ~ +98-,

-99leads after trivial integration and somewhat lengthy manipulation to To (e l@ N ~b (k / h) i6t (Oe E (A 14'4) where VR1 } o 3/2A(C) + ( %~/,6t )[ 24 (x- A) Gos at - 4z (A. 14.5) i (5Xi4 +\2yi 424)4 C@o~x + 4 ( TC3 -tx)'Vv\ ] in which D = (4r 7A> /3 ) m- = - dx (A.14.6) 3/(3 / >. Integrating (Ao14,4) over incident energies, after introducing the incident energy spectrum we get (3o25).

APPENDIX A.1 5 SPIN DEPENDENCY1) We want to calculate the average over neutron and nucleus spin directions { }s of the thermal average < >T given by The molecular wave functions may be written as a product of a spatial function jr and a spin function X.provided the molecule does not have like nuclei. In this section we will assume this to be the case and refer the reader elsewhere) for the opposite case. We will assume Xn.~~~ =0(~~~~ ~C~ it(A.15.2) This allows us to write the average (A.15.1) as a product. The neutron and nuclear spin average times the usual thermal average. Consider the former /, /- Si-~S iI } -(A l5-3) -100

-101(A.15.3 cont'd) j( t4- ( j _ + ( c.$ ~) (.S Making use of the well-known relation between the Pauli matrices and any vector A and B, ( f. A B ( f. 3 % = e A 3 + 2 -e C ( A x 3 8 (A.15.4) we obtain for (A.15.3) \o \bj + I c.S sj, + ~, j -c ~j- 0j g(5x *

-102For unpolarized neutrons the average over neutron spin directions. of the operator a vanishes. The average of S7-S.3 over nuclear spin directions is just 1 i-e/ [,' i 0 - S ( 5 +1~6 R3 > i(A,15.6) These two results simplify expression (A.15.5) considerably X4 Ct S -)iio

APPENDIX A.16 TIME EXPANSION FOR INTERNAL DEGREES OF FREEDOM In Chapter VI we made use of the static approximation in order to calculate the internal degrees of freedom thermal average. Here we will outline the time expansion technique used in order to obtain further correction terms, However, due to the fact that the thermal average under discussion reduces to that discussed in great length in reference 2, we will limit ourselves to illustrate the calculation for the direct non-interference scattering contribution. The thermal average was given by (6.16) <Q1~~~ i<- Sl~~~~ S i b j l(A16.61) Expanding the time operator in the usual form (A.8.1) - _ - -Ei)~ ( / i ) [ fl \ S -0O ) b SI(~ 4 / i > t; z Z 1 + (A.16.2) -103

It is observed that for diatomic molecules(3) the rotational energy level separation is in general very much smaller than the corresponding vibrational level. The vibrational energy separation is usually greater than the neutron energy used.in neutron diffraction experiments, On the other hand, the rotational energy level separation is smaller than the incident neutron energy. In particular for the Nitrogen molecule the rotational level separation is approximately.3 ev.,and the vibrational level separation is approximately 4.9 x 10-4 ev. The incident neutron energy used in diffraction experiments is of the order 7 x 10-2 ev, This implies that the neutron will excite very many rotational levels but none of the vibrational levels, In order to go further in the calculation of the thermal average, we are forced to make the following approximations i) the Hamiltonian Hc is written in the approximation used in Chapter VI (6017 to 6.19), i.e., r ot A\ In lieu of the aforementioned discussion we will take as the second approximation ii) the static approximation for the vibrational degrees of freedom These approximations enable us to write for (A.16,2)

-105-~~~~Q~~j~~; Sott I1 e i wt /+ - K f - H ~It i A Q) _ot X[ Q1 1 Let >( =s l S / 4 f (A.16,4)

-106and write the angular momentum operator in the form L I=% _ k~ \5 V9\ (A.16.5) After substitution of (A.16.3) and (A.16.5) into (A.16.1) we obtain where E_ ( 2I -d -J + Zorw% p~I- uec~ii~ (?rizT3 b (A.16.7) +~tl~bE b-~_ r~'~, 4 14 ).e

-107Consider the coefficient of the term linear in the time for s = j and kt ( \V W\ and using the recursion relations for the associated Legendre functions, we get (A.16.9) ~A~v Qg3'

-108The matrix element becomes ('Ak rvA IA NA (A.l6 zlo) ILC~ sR C-IA ) v - -_' /i I (A~~~~~~~~~~~~~~~~~~~.610 c h \C ~~~~~~~~~,$ i~~~~

-109This term reduces to the equivalent one in reference 2 when we expand rs about the minimum rM of the potential U(r). That is -i _ i.y j*. 2(A,16.11) The contribution of the term linear in time to the cross-section is (A.16.12) For purposes of comparing the order of magnitude of this term with respect to the static approximation term, we will assume that the center of mass thermal average is unity. We will then integrate (A.16o12) over incident energies after multiplying by the usual Gaussian, The result obtained is

110lbo Qu1( gcet\et/ 4 Yfi (, 6c66,aa e

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