THE SCATTERING OF SLOW NEUTRONS BY POLAR LIQUIDS by Sidney Yip of Nuclear Engineering Department A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan June 1962

ACKNOWLEDGMENTS The members of the doctoral committee have been very helpful and cooperative during the course of the research. In particular I wish to acknowledge stimulating discussions with Professors A. G. DeRocco, K. Hecht, C. Kikuchi, J. S. King, and C. E. Nordman. I should like to express my gratitude to Professor K. M. Siegel, then the director of the Radiation Laboratory, for the kindness extended to me when I was a part-time member of that laboratory, and to Professors W. Kerr, C. Kikuchi, P. F. Zweifel for constant support, encouragement and advice during my days as a graduate student in the Department of Nuclear Engineering. The financial support of the present investigation has been provided by the Rackham Graduate School in the form of a scholarship and by the Michigan Memorial-Phoenix Project in the form of two fellowships, one of these being the Owens-Corning Fiberglas Fellowship. In addition, the Phoenix Project has provided funds for travel as well as funds for the preparation of this dissertation. To the Project and acting director, Professor W. Kerr, I remain deeply grateful. The kind assistance of D. Danford of the Industry Program of the College of Engineering and of G. Prosser and staff of the Office of Research Administration has made the preparation of the manuscript a relatively painless task. ii

The use of the IBM-709 digital computer at The University of Michigan Computing Center is hereby acknowledged, Inasmuch as it is difficult to properly acknowledge the invaluable guidance of Professor R. K. Osborn, chairman of the doctoral committee, nevertheless, I wish to express my deepest appreciation to my teacher for his kind interest and thoughtful attention in all phases of my activities during these past four, highly important years of my formal education, iii

TABLE OF CONTENTS LIST OF FIGURES ABSTRACT CHAPTER Page vi vii I. II. III. IV. V. VI, VII. APPENDIX INTRODUCTION GENERAL FORMULATION OF NEUTRON SCATTERING BY MACROSCOPIC SYSTEMS MOLECULAR TRANSLATIONS MOLECULAR ROTATIONS HINDERED MOLECULAR ROTATIONS SLOW-NEUTRON SCATTERING BY DIATOMIC LIQUIDS CONCLUDING REMARKS 1 8 24 41 6o 87 107 Ao A NON-SPHERICAL LENNARD-JONES-DEVONSHIRE "SMEARING" PROCEDURE Bo "CREATION" AND "ANNIHILATION" OPERATORS C. NEUTRON SCATTERING ACCORDING TO THE SQUARE-WELL APPROXIMATION D. ANGULAR MOMENTUM OPERATORS AND HAMILTONIAN FOR RIGID MOLECULES E. THE ASYMMETRIC MOLECULE IN AN ELECTRIC FIELD Fo THE INTEGRAL S ^,( > ^) Go THE HINDERED ROTATOR-OSCILLATOR APPROXIMATION H. EFFECTS OF FREE ROTATIO N IN NEUTRON SCATTERING BY HINDERED ROTATORS 115 121 125 127 133 136 14o 143 iv

TABLE OF CONTENTS (Concluded) APPENDIX Page Io SEPARATION OF ROTATION AND VIBRATION IN DIATOMIC POLAR MOLECULES 148 J. THE THERMAL AVERAGE FOR INTERNAL DEGREES OF FREEDOM OF A DIATOMIC MOLECULE 152 REFERENCES 156 v

LIST OF FIGURES Figure Page 1. Schematic diagram of neutron scattering experiment, 9 2. Position vectors for the Lennard-Jones-Devonshire "smearing" procedure. 28 3. Curves of the effective potential V(u) in units of Y. 32 4I Eulerian angles ( ) ()) defining orientation of body axes I; with respect to space axes.44 5. Energy distributions normalized at 0.062 ev of an initially "narrow" beam of neutrons scattered by hydrogen chloride according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown is the incident spectrum. 97 6. Energy distributions normalized at 0.058 ev of an initially "broad" beam of neutrons scattered by water (center-of-mass degrees of freedom only) according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown are the experimental points14 normalized at 0.052 ev and the incident spectrum. 101 7. Energy distributions normalized at 0.064 ev of an initially "narrow" beam of neutrons scattered by water (center-of-mass degrees of freedom only) according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown is the incident spectrum, 103 8. Energy distributions of an initially "broad" beam of neutrons scattered by water according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown are Nelkin's calculation23 (dotted curve), the experimental points14 normalized at 0.052 ev, and the incident spectrum. All theoretical curves are normalized at 0.058 ev. 105 vi

ABSTRACT A quasi-crystalline model of polar liquids for the analysis of slowneutron scattering is proposed. Effects of the macroscopic medium are considered in terms of specific potentials which hinder molecular translations and rotations, By smearing the neighboring molecules over a spherical distribution and choosing a two-particle interaction which includes dipole-dipole forces an effective potential for the center-of-mass degrees of freedom is derivedo A sufficient condition for the parabolic approximation to this potential is established, Hindrance in the rotational motion is treated by means of a coupling between the permanent electric dipole moment of the molecule and a local electric field which represents orientation-dependent effects of intermolecular forces. Under the condition of strong coupling this hindering potential leads to useful wave functions for symmetric and linear molecules which describe free as well as hindered rotations, On the basis of the present treatment of molecular translations and rotations the energy and angular differential scattering cross section is calculated without introducing further approximations, Particularly simple expressions are obtained when freerotation energies are ignoredo Energy distributions of neutrons scattered by hydrogen chloride and water are calculatedo It is shown that hindered rotation effects are experimentally observable whereas the structure associated with hindered translations can be observed only vii

by means of high-resolution measurements. The present results provide a suitable basis for systematic studies of medium effects in polar liquids by inelastic slow-neutron scatteringo Specific experiments are suggested, viii * * Vlll

CHAPTER I INTRODUCTION In recent years considerable attention has been directed toward the -3 -^use of inelastic scattering of slow ( - 10 -o10 ) neutrons for the study of atomic motions in macroscopic systems. Of the various modes of molecular motions, translation of the center-of-mass and rotation are of particular interest because of their more sensitive dependence upon the physical state of the medium. The characteristic energies associated with these motions are of the order of 103-10-2ev, so in the event of an inelastic collision with a neutron the energy transfer is of the same order as the incident neutron energy. The energy spectrum of the scattered neutrons therefore provides a means with which the details of molecular translations and rotations, and the effects on these motions due to surroundings in a liquid or a crystal, can be directly examined. In this respect, slow neutrons are very suitable microscopic probes for the investigation of intermolecular forceso The general problem of low-energy neutron scattering by free molecules has been considered by Zemach and Glauber in treating those systems which can be characterized by the degrees of freedom appropriate to gas molecules. Krieger and Nelkin,2 in a somewhat different approach, used the Sachs-Teller maas tensor approximation5 to describe the combined 2,4 effects of translations and rotation. Both methods give results for gaseous methane in good agreement with measurements,5 1

2 In crystals the translational degrees of freedom can be appropriately characterized by a system of oscillators (if diffusion in crystals is ignored) according to a dynamical description such as that of Debye or w 6 Born-Von Karman. The first general treatment of neutron scattering by crystals was given by Weinstock;7 subsequently the theory has been refined and valuable information about interatomic forces in crystals, namely, frequency distribution and dispersion relations, have been ob8,9 tained by means of neutron diffraction studies. 9 The analysis of neutron scattering by liquids is considerably more difficult since the dynamics of molecular translations and rotations are much more complicated. In the absence of detailed knowledge of intermolecular forces it has become necessary, in cases where a specific model is used to interpret measurements, to introduce greatly simplified descriptions. Commonly either a gas-like or crystal-like description is employed for the translational degrees of freedom. A number of slow10 l11 12-14 neutron experiments using liquid tin, ammonium halides, and water have been reported where the results tend to support the point of view that liquids behave much like disordered crystals with short range ordering and that the molecules spend appreciable amount of time in a localized region before undergoing diffusion. The energy distributions of 12,13 neutrons scattered by water at very low energies12 showed considerable structure in the region of small energy transfer and this structure is regarded as direct evidence of hindered translational motions.

Morales15 has studied hindered translations in normal liquids and has found that an Einstein-crystal model of liquid lead leads to energy broadenings in better agreement with observed valuesl6 than those predicted by a continuous diffusion model proposed by Vineyardo17 Singwi and Sjolanderl8 have shown that the previously mentioned structure in the water results2 can be calculated by treating the liquid as a Debye crystal. They have also suggested a model in which each molecule intermittently undergoes oscillations and diffusions. On the basis of an extensive series of measurements Larsson and coworkers13 concluded that on the average the water molecule performs about ten oscillations during a period of /V 2 x 10-12 sec before diffusing a distance of at least 1.5A. Since the time interval in which a slow neutron (v, 105 cm/sec) is in the immediate vicinity of a molecule is approximately 10-13 sec, then the supposition that dominant aspects of translational motions are crystal-like seems reasonable~ at least in so far as slow-neutron scattering is concerned, Although quasi-crystalline models have been employed with some success in correlating with experimental data, the only attempt to justify such descriptions on theoretical basis has been made by Morales in the case of normal liquids. In addition to translations, molecular rotations in liquids can also become hindered as a result of intermolecular interactions. The phenomenon of hindered rotation has been observed in waterl2-14 and in ammonium halides using the neutron technique, and also in Raman spectra19 and infrared absorption20 studieso in early papers, Pauling21

has considered the rotational motion in crystals in connection with its influence on the dielectric constant, and Magat22 has carried out an electrostatic calculation to determine the amount of energy a water molecule must acquire in order to have complete freedom of rotation in the potential field of four nearest neighbors. Aside from these considerations the study of hindered rotation in liquids and crystals has received little theoretical attention. Thus far, the effect of this type of motion upon neutron scattering has been treated only recently by Nelkin,23 who, in constructing a model for water, assumed that the hindered molecular rotations can be replaced by a single mode of torsional oscillation. Nelkin's approach is phenomenological, and the model, although providing an adequate basis for calculations of neutron thermalization in nuclear reactors, does not lead to much physical insight as to the nature of intermolecular forces in liquids. In the present work we propose a dynamical model for polar liquids for the analysis of slow-neutron scattering. The atomic motions are considered in terms of molecular translations, rotations, and internuclear vibrations. The model consists of descriptions, derived in a systematic and consistent approach, of translations and rotations in the presence of appreciable intermolecular interactions. The method used by Morales in treating normal liquids is extended in order to study molecular translations in polar liquids where noncentral forces arising from dipole-dipole couplings are taken into accounto, For those liquids where the existence of short range ordering

5 can be expected, an averaged, effective single-particle potential for the center-of-mass degrees of freedom is derived. This potential provides, in principle, at least a qualitative representation of the interactions generated by neighboring molecules in the liquido To the extent that a certain condition can be satisfied by the parameters characterizing the two-particle potential employed, some theoretical justification for the use of oscillator models in treating translational motions is obtained. A description of the rotational motions of polar molecules is proposed in which the permanent electric dipole moment of the molecule is coupled to a local electric field which represents the orientationdependent effects produced by the ordered near neighbors. In the case of strong coupling, physically meaningful solutions for symmetric and linear molecules are obtained. The model treats free as well as hindered rotations because in this instance the potential contains only one angular variableo On the basis of the proposed model an energy and angular differential cross section is calculated, New formulae for the scattering by rotational degrees of freedom are obtained. The results of the present investigation are considerably less empirical than Nelkin's treatment since the molecular rotational degrees of freedom, the polar nature of the molecule, and internuclear separations appear explicitly in the expression for the intensity of the scatteringo Moreover, the approach allows the magnitudes of the energy transfer to be directly estimated

6 in terms of the dipole moment of the molecule, the moment of inertia, and the crystalline electric field in the corresponding solid phase. Numerical computations are presented in which the calculated energy distributions are compared with the available water data and future scattering measurements are suggested. In Chapter II the formalism of Zemach and Glauber is derived and a general expression for the energy and angular differential cross section is obtained.* The single-particle model is then adopted in which couplings between translation, rotation, and vibration are ignored so that these motions can be studied separately. Hindered molecular translation is treated in Chapter III in terms of a potential obtained by appropriate position and orientation averages of the two-particle Stockmayer potential. A sufficient condition for an Einstein-crystal approximation is established, and the cross section for a system of oscillators is calculatedo The dynamics of free molecular rotation are reviewed in Chapter IV where the known symmetry properties of the wave functions are utilized in computing the cross sections for asymmetric, symmetric and linear molecules rigorously, A rotator model of hindered rotation is proposed in Chapter V. A formal solution of the molecular problem is first discussed, then strong-field solutions for symmetric and linear molecules are obtained and used in cross section calculationso Effects *Van Hove24 has discussed neutron scattering in tems o space and time correlation functions. This approach is quite general and has been extensively used by Egelstaff25 and others,26 Inasmuch as specific models are proposed and employed in the present work there is no advantage to consider Van Hove's approach,

7 of free and hindered motions are examined, and using the method of massratio expansion the angular differential cross section in the limit of high incident energy is derived, Numerical computations for hydrogen chloride and water are discussed in Chapter VI, In the case of water, the molecule is approximated by a diatomic molecule of equal mass and having the same dipole moment, Concluding remarks regarding the present model are given in Chapter VII where specific scattering experiments are also suggested.

CHAPTER II GENERAL FORMULATION OF NEUTRON SCATTERING BY MACROSCOPIC SYSTEMS In this section a general description of neutron scattering by molecules in an arbitrary macroscopic system is developedo The formalism is suitable for the analysis of low-energy inelastic scattering by liquids on the basis of specific dynamical models. In order to calculate the effects characteristic of atomic motions in the liquid state an independentparticle model is adopted in which molecular translations, rotations, and vibrations are approximated by independent motions. Each motion can then be treated separately. We consider an idealized experiment to illustrate the type of measurements with which subsequent calculations are to be correlated. A beam of monoenergetic neutrons impinges upon a scattering system as shown in Figo 1, and collisions between the neutrons and target molecules result in some of the neutrons being deflected from the original path with energies greater or less than that before scattering. By counting neutrons of a given energy scattered in a given direction, an energy distribution is obtainedo If neutrons of all energies are counted for a given direction, then by repeating this measurement at different scattering angles an angular distribution is obtained. Finally, if all scattered neutrons are counted regardless of their directions or energies then the measurement is said to be a total scattering measurement and the result depends only upon the incident energy. We observe that of the three types of 8

9 -NEUTRON DETECTOR z r2 d SCATTERED NEUTRON WAVE VECTOR Q.MY N- PARTICLE SCATTERING SYSTEM X WAVE VECTOR Fig. 1. Schematic diagram of neutron scattering experiment.

10 measurement, the energy distribution experiment is the most fundamental and should provide maximum information about the scattering process, and that with respect to it the other two are integral measurements. The scattering event under consideration is a single collision process, and the resolution of the neutron detector shown in Figo 1 is infinitely sharpo In practice, however, the thickness of the sample is dictated by multiple scattering and effects of finite resolution, which is inherent in any energy measurement device, must be taken into account in specific applications o The energy distribution measurement can be analyzed in terms of a fundamental quantity which characterizes the interaction, the probability that an incident neutron with initial energy ii will be scattered into direction Q with final energy CS o When appropriately normalized to unit scatterer, this probability becomes a differential cross section t,,S) which, we emphasize, is a distribution in final neutron energy and direction. In order to derive an expression for ((J i. 4) we return to the idealized experiment of Figo 1o The number of neutrons scattered into an element of solid angle Ad about ~Q (denoted by r ) per unit incident neutron with energy.i and per unit scatterer is (tO; aya = - d (2 01) N(i3i(2 where T3 and _ are initial and final neutron currents and q is the total number of scattering centers. From the well-known relation

11 3 %L 2 -Wf ) (2e2) where = - -r Y being the Planck's constant, W is the neutron mass, and \) is the neutron wave function, the cross section can therefore be calculated once the initial and final neutron wave functions are determined. The initial wave function may be taken as a plane wave with wave vector __ The final neutron wave function necessarily depends upon the particular scattering process involved, and can be derived from the appropriate stationary Schrodinger equation, - q'J 2... ~.... (2.5) where ~ is the total energy of the system, molecules plus neutron, whose wave function is s, The appropriate Hamiltonian is ~-\ 2 +- i s+ (2.4) where is the kinetic energy of the neutron, V the Hamiltonian of the scattering system, and V the neutron-nuclear interaction potential. The Hamiltonian \-\ represents only the various molecular energies of the scattering system. Because in the experiments under consideration the neutron energies are much too low to excite any electronic levels, and also electron scattering is negligible, the electronic energies are of no interest and have been separated by means of the Born and Oppenheimer approximation.27 It is expected that given -, there exists a complete, orthonormal set of functions -':^ which satisfy the equation, ItS,\'-I,.\'

12 where ~[i[is the energy eigenvalue of the system associated with the ht eigenstate. Then we can consider the expansion \CtRj{4t)4) (2.5) where V' represents neutron position and'i?., -?.: <, represents the set of.-:\i coordinates of a system containing i( particles. The function \), is seen to be that neutron state which corresponds to the V1t4l state of the system. Inserting the above expansion in (2.3) we obtain +- l h 2 ll 2. (V ) (2.6) where c N \o ~_ 2i E C- = - ~ Equation (2.6) represents a set of integro-differential equations and is conveniently converted to a set of integral equations. If we define a Green's function by C^+b^^-Z ^^ 1(2.7) then because (2.6) is a linear equation, where4='htZ'rz" saisie Qd,()2.) )stsis QI~(%Y (2.8) where g satisfies (. 9) ~ ~"\ \a%.. (2,9)

13 Physically, if there were no interaction between the neutron and system we would expect to be just the product of incident neutron and initial system wave functions. Hence,'A(1j= l to (2.10) where V\ denotes the initial state of the scattering system. The solution of (2,7) is8,\ t 4$\r \r= P ^ i ~crg~4- YKV^ ), (2.11) where terms of order t and higher can be ignored as the region of deD 1 tection is such that g b, In fact, in the limit r'70 contribution to neutron current from higher-order terms vanish identically. An approximate solution to the integral equation (2.8), according to the "first Born approximation," is obtained by iterating once, <LG^r^ (27 + z L6 (2.12) The first term in (2.12) represents the unscattered incident beam and the second term can be interpreted as the spherically outgoing scattered wave whose amplitude is SVthe) dff eet cs se<ti>o (2.13) Using the results just derived we arrive at a formal expression for the differential cross section of (2.1), w(s;,,)dQ = rN IP i/d L. (2V14) ~~~5~ ViA0~~~~~~~~~~~

14 where we average over initial system states and sum over all possible final system states because they cannot be observed. The probability that the system is in initial state V\0 is denoted by, which becomes a Kronecker delta should the scattering system be initially prepared. For each collision process in which the scattering system undergoes a transition from state VlI to state V), the associated condition of energy conservation, + vAE cV\< j (2.15) must be satisfied. It will be convenient to incorporate this statement 29 into the cross section. By defining [ 5(E;,Q^aQd = ct, aQs (2.16) and noting the integral representation of the Dirac delta function, CO )1 we obtain an expression for the energy and angular differential cross section,* 1 ~ (&,,Q,)d- 2I It 2 -A,@(Ev-E (2.17) where on account of the delta function K and CA can be written as l *In the present notation the variable t has the unit of reciprocal energy, that is to say, J='T/[ where % has the unit of time.

15 and (-; this is to avoid confusion and to indicate that the scattered neutron energy is an observed quantity. The manipulation also conveniently removes the factor -1 from the summand. It is noted that the amplitude of the scattered wave (2.13) contains the neutron-nuclear interaction potential W/ which has yet to be specified. For neutron energy sufficiently low (i,4e}) such that its wavelength is large compared to the range of nuclear forces, neutronnuclear collisions can be represented quite accurately as a "localized impact" for which the interaction potential may be taken as the "Fermi pseudopotential, "*30 <v- X0 a t\ I, &) (2.18) where li(W is the scattering length of U-l4 nucleus in the 4-l' molecule, whose position is Yiand summation is over all nuclei and molecules in the system. This particular form of the potential is so chosen that when used with the "first Born approximation" the correct total scattering cross section for a free atom is guaranteed in the high-energy limit (ae.) i.e., i —.>.idr where A- is the experimentally determined free-atom scattering length and is related to the so-called bound-atom scattering length by l^bMC i S M being the mass of the scatterer. With this choice of the potential the scattered amplitude takes the form *The use of pseudopotential is appropriate so long as the incident neutron is in the energy range for which nuclear scattering is isotropic in the center-of-mass coordinate system and independent of neutron energy.

16 - -n - - / M - t'1/, (2.19) where t<" eJ- a.: is the neutron momentum transfer. The differential m_.. cross section thus becomes K(*;- t\ ~ 2 K < (2.20) where <:;/' -' 1tcs;i%-'a' -;tt -*K-Q^, (2 21) -= <V(Di^t t 4 AT and where. -- -,;. In arriving at (2.21) we have used the closure property of the wave function to perform the sum over final states. The subscript "' is used to denote the average of the expectation value of the indicated operator in a given initial eigenstate VI (previously t, ) over all initial states; because the scattering system is usually taken to be in thermodynamic equilibrium the quantity < > is often called a thermal average. Expressions (2.20) and (2.21) represent the basic equations used by Zemach and Glauber1 in treating neutron scattering by gases. Since the Hamiltonian for the scattering system has not yet been specified these expressions can be applied to crystals and liquids as well. We have purposely placed CQoj3k, between the initial system wave functions. The reason for this is that (2.21) can be generalized to include spin effects which we have not considered thus far and which can be taken into account by introducing a spin-dependent expression for the

17 scattering length, GQ A~l d w )d) (2.22) where'S and S are respectively the neutron and nuclear spins, and. )~ are nuclear constants to be determined by experiment. The necessary modification in (2.21) merely requires that the thermal average to include the matrix elements of spin-dependent factors. The system eigenfunction now appears as tD where nuclear spin function 3 is a column vector specified by V, the projection of the nuclear spin along the E axis. Separation of spin and coordinate dependences is not always allowed; however, it has been shown4 that spin correlation effects may be neglected for all but the lightest diatomic molecules containing like nuclei (eogo,H2 and D2) at very low temperatures. The thermal average (2.21) becomes t~;rpd -t^ -;, / e S< ='\- gl$QiA H<- ( ~' ( >T ) (2.23) where S specifies the neutron spin function represented by a two-component column vector. The spin thermal average in (2.23) can be readily performed by noting the following: a - <z l S;) >- ~. 7 (2.24) (2.25)

18 Hence -^^A.^ -N^^^ ^( (lI: ( * (2.26) This result leads to a convenienit and meaningful separation of the cross section into various contributions, (2.27) dG-' 1 i*t, iX' Kt where the prime over the summation indicates that terms for which the two indices being equal are to be omitted. In this notation, ^^ A\4-^ ^K-^, 1)/4 and /'t" -' 2 -' Y (2.28) tin (2 The three terms in (2,27) represent contributions due to direct, "inner" (interference withini the same molecule) and "outer" (interference from different molecules) scatteringso The nuclear constants A and O.[19<. ~A. /2~. are conventionally known as the coherent and incoherent scattering amplitudes respectivelyo They are related to the "parallel" and "anti-parallel" (total spin ='i!it/,. respectively) scattering lengths by t' (< B st u C( ) E t,^S ^(2.29)

19 Thus we see that direct scattering is both coherent and incoherent while interference scattering is coherent; moreover, only incoherent scattering is spin-dependent. When more than one isotope is present in the system the scattering amplitudes should be replaced by appropriate isotopic average s. We will now briefly indicate the essential connection between the 24 present formalism and Van Hove's approach. The differential cross section for a monatomic system is* where the symbol 2 is used to denote the appropriate scattering length. If a function C(_ is defined as forsa Foutitr <tr a C e >, then the cross section can be calculated in terms of ba(,t) by means of a four-dimensional Fourier transform, One advantage of this approach lies in the fact that it is possible *Vineyard has discussed the generalization to polyatomic systems.17

20 to assign physical interpretation to the function (ts) in the classical limit and obtain different methods of calculation.* For then ~(-tdake represents the probability that given a molecule at the origin at t-=0 there will be a molecule ind about _ and dt about t. The presentation of experimental data in terms of t(_j) have also been proposed.25'26 The general expression for the differential cross section (2.27) provides a basis for a direct calculation of the quantity measured in a scattering experiment. The thermal average (%SQ' contains effects of neutron interaction with all possible degrees of freedom of the molecular system, and in order to study characteristic atomic motions in liquids it will be necessary to obtain an appropriate decomposition of (2.27). It is anticipated that because of the relatively small energies ( 10 - lO eV\ associated with translations of the centers of mass and molecular rotations these motions are apt to be influenced to the greatest extent by existing intermolecular forces in the liquid.** We now make the fundamental assumptions that the energy of the liquid can be represented as a sum of individual molecular energies and that for times long compared to neutron-nuclear interaction times each molecule moves in a potential generated by interactions with its neighbors, which is approximated by a *The transition between classical and quantum mechanicfl (A2) functions has been recently discussed by Rosenbaum and Zweifelo **The internuclear vibrational motions, which involve energies C(i 11 eV) large compared to translations and rotations, are less affected by surroundings. It is known that the frequencies of normal vibrations are different in the liquid phase from those in the corresponding vapor; however, even for water where considerable amount of hydrogen bonding exists in the liquid state the changes in the frequencies are no more than 5-10%.31

21 function depending upon the coordinates of that molecule only and is taken to be the sum of two terms. One of these is assumed to be a function only of the center-of-mass coordinates and the other a function of molecular orientation. Thus A-AaA A (2.29) and A Tnd ^v. (2.30) According to (2.29) the physical many-body problem is reduced to a singleparticle description, and in (2.30) the energy of each polyatomic molecule is further separated into energies of translation, rotation, and vibration. The present treatment will ignore couplings between different degrees of freedom. The dynamical correlation effects among molecules are treated in terms of effective interaction potentials which are singleparticle potentials, and evidently, such a description of the liquid state emphasizes aspects characteristic of individual particle motions. Because the model does not take into account explicit correlations in positions and orienitations of the molecules it is not expected to be suitable for direct studies of molecular symmetries. The form of the Hamiltonian (2.30) suggests a decomposition for the position of a nucleus measured in the laboratory system, ^'^v — rag- oaf Ural(2.31)

22 where R is the position of center of mass of the t4V molecule, bW the equilibrium position of the (~ nucleus in the Q-V molecule and V its instantaneous displacement. When the molecule consists of more than two atoms nuclear vibrations can be treated in terms of normal coordinates, which are obtained by a transformation of the form, Ux(o- t CQbj, \ (2.32) where C is the \4\ polarization vector associated with normal coordinate qu. Among the components in (2.31) b., 9 will not commute with Hq, At,Q respectively, and in addition C9, a vector which rotates with the molecule, also does not commute with \/;4 otherwise all commutators vanish. The thermal average (2.28) is therefore factored into a product, ~' y T Tix T a- 2 i\k$i. 4 i kab -;Wt -Xt7 — 4<, -i% o< = < t e e e T 6 e C )> (2.34) The nonseparability of rotation and vibration in ( G T results from the commutation relation noted above and also follows directly from the decomposition (2.31) since only three coordinates are needed to specify each \^ and the coordinates of and L are clearly not independent sets. Because the operators in each thermal average involve only the i —

23 and; molecules, the matrix elements therefore are taken with respect to the appropriate coordinates of these two molecules. It is seen that 9-.Q, will be time-dependent only when 3 =', so energy transfer is'",' possible only by direct and "inner" scatterings. The fact that intermolecular interference is a purely elastic effect arises from the assumption of no explicit dynamical correlations among the molecules. In the following chapters we shall be concerned with neutron scattering by associated or polar liquidso Specific dynamical models describing molecular translations and rotations will be proposed and the corresponding thermal averages evaluated. It will be assumed that average over molecular orientations in the rotational and vibrational thermal averages can be performed separately, although the coupling effect can be formally treated by the methods developed later.* This assumption corresponds to ignoring the effect of rotation of the axes of vibration during a scattering event. *-It has been shown in the calculation of angular distribution of neutrons scattered by CH4 vapor that little error is introduced by ignoring the nonseparability of rotational and vibrational thermal averages.2

CHAPTER III MOLECULAR TRANSLATIONS The translational motion of a molecule is influenced only by its interactions with the surroundings. In a gas the molecules tend to behave as free particles whereas in a crystal they would tend to oscillate about their respective lattice sites. Unfortunately, for molecules in the liquid state no simple description of their translational degrees of freedom is available primarily because these molecules neither diffuse continuously nor remain in any oscillatory state over a long period of time. It is however reasonable for some class of experiments to suppose that the molecules would vibrate in a semistable structure for a certain length of time after which diffusion takes place by activation through potential barriers created by intermolecular interactions, and the entire process then repeats itself. In recent neutron experiments with water12-14 and liquid tin10 solidlike behavior has been observed which suggests the existence of a certain degree of local order in the liquid. For the description of neutron scattering by polar molecules we feel it is appropriate, particularly in view of the appreciable molecular associations, to adopt a quasi-crystalline model in which it is assumed that the neutron will encounter, on the average, a molecule oscillating about a position of temporary equilibrium rather than one undergoing diffusive motions. Similar description has been used by Morales15 in analyzing neutron diffraction by liquid lead, 24

25 and by Singwi and SjBlanderl8 in calculating the differential cross section for water. In both cases the results are in better agreement with measurements than those obtained assuming continuous diffusion. The present chapter is devoted to a discussion of attempts to justify the assumption of hindered molecular translations. We shall derive an approximate effective potential in terms of the Stockmayer two-particle interaction where, under a certain condition to be satisfied by the associated parameters, the potential may be approximated by a threedimensional isotropic harmonic oscillator potential. Such an approximation would effectively correspond to an Einstein-crystal description of the liquid in which liquid parameters are used to determine the force constant. The oscillator model thus obtained is then used to calculate the direct and interference scattering contributions to the differential cross section from center-of-mass motions. Similar calculations for a square-well potential, an alternative to the parabolic approximation, have also been considered and are discussed in Appendix C. THE EFFECTIVE POTENTIAL We consider a molecule in a pure liquid moving in the field of its neighbors. The energy of interaction for this molecule, which, according to independent-particle assumption, is the same for any molecule in the system, can be written as

26't -\' is' >C-'^'iA'? a,'v (5. 3 1) where ( r is the two-particle interaction for a molecule at R with dipole moment j and another at R' and dipole moment L', and tR XI).i.^' is the probability that given a molecule at R its dipole orientation is in:V..k') about a and that another molecule is in 3 n va The inega d R' about R' with dipole orientation in ct&1 ) about'. The integrations in (3.1) extend over all configuration space excluding the volume occupied by the molecule in question and over all orientations of both dipoles. The frequency may be decomposed into a product of two frequencies, i(@t'e23&i')dV Jd^ ) ( 32 where T('s\?'. ja' is the probability that given a molecule at R there will be another in d3RT about', and v jU(J~? is the probability that given a molecule at R and another at R' the former will have dipole orientation in Aai\ about {I and the latter will have dipole orientation in Ij\ about A. A useful approximation for treating V QR), first used by Lennard-Jones and Devonshire32 in attempts to calculate thermodynamic properties of dense gases and more recently applied by Morales15 to neutron scattering by normal liquids, is to assume that all the neighboring molecules which interact with the molecule under consideration to an

27 appreciable extent are uniformly distributed on the surface of a sphere centered about the "equilibrium" position of the molecule and whose radius is of the order of intermolecular separation.* In applying such a "smearing" procedure we first write where the various position vectors are defined in Fig. 2. Now according to the Lennard-Jones and Devonshire approximation, HFX)it>=-, k). ^ -~v), (3.4) where: is the number of neighboring molecules involved in the interaction and s is the mean separation between molecules. It is expected that \ should be of the order of coordination number for the corresponding crystal and s may be estimated from the radial distribution function studied in X-ray scattering. In choosing a two-particle interaction L ]) ) it is evident that any reasonable potential should be attractive at large molecular separations and repulsive when the molecules are close together. In ad*The assumption of the existence of "equilibrium" positions is central to the quasi-crystalline model which we have adopted. At least for neutron scattering this is believed to be reasonable since neutronnuclear interaction times are very short compared to times required for molecules to change their positions appreciably.

28 Z N / 7/ I/ ) Fig. 2. Position vectors for the Lennard-Jones-Devonshire "smearing" procedure.

29 dition, the potential should be orientation-dependent to account for electrostatic interactions between dipoleso Such a potential can be constructed by taking a central potential widely known as the LennardJones 6-12 potential and the interaction between two rigid, point dipoles. The resulting potential is the Stockmayer potential,35 U-^^ -^f]^ Ci. \br (3.5) U - 4-dT\;AL\jd), (535) _ 3( _ __ ____) (3.6) with r = R'-R. In (305) constants G and CT are characteristic of the chemical species of the molecule; E is the maximum energy of attraction Y6 of the two molecules which occurs at a separation of 2 (, and G is the distance of closest approach. ^ A Now it remains to obtain an explicit expression for R_,J |!_). From the form of the two-particle potential just chosen it follows that at a given intermolecular separation there will be relative orientations for which Ud is a minimum or maximumo The fact that statistically molecules prefer those orientations for which the interaction energy is -bUd small can be expressed by means of the Boltzmann factor t, where tre th), k i being the Boltzmasln constant and T the absolute temperature of the liquid, The effective potential (5,1) now becomes V -tC re'li T i\'-r)(s — > ^ d\. (35.7) V v J~~~~

30 where cXNk^ &nad~l (3.8) It has been assumed in (358) that any variation in r during a period of rotation can be neglected. At small separations or low temperatures Ud~(Ud)min because the molecules will tend to oscillate only about the orientation corresponding to minimum energy, For large separations and high temperatures the Boltzmann factor can be expanded in powers of (kT). The first nonvanishing contribution to Ud is3 * \ _ 2JI d ST^s (3.9) The dipole-dipole interaction averaged in the above manner is seen to be attractive and vanishes as (kT)-1 for large temperatures. Using (3.4) and (3.9) we obtain a - o \, (-> ( \s i\(C (35.10) where E0, Gi are constants chosen so the integrand has the same form as the Lennard-Jones 6-12 potential, ( 2 t3 0~ ( with =- -. After a trivial integration, W (u = 0i -f0- ( 1 + )(-8) (3.12)

31 where &(_,,(()' and Thus V(u) is only a function of the displacement of the molecule from its "equilibrium" position, and the shape of this isotropic potential is characterized entirely by the parameter. At the origin u = 0, V(u) has the value-8k It is necessary to introduce further approximations of V(u). This is because the Schrodinger equation with (3.12) as the potential energy cannot be solved analytically and numerical solutions for the wave functions are not suitable for later calculations of the differential cross section. In Fig. 3 we show a number of typical curves for V(u) with different values of I. As we would expect, the associative nature of the dipole-dipole interaction tends to cause the molecule to move away from its "equilibrium" position. However, beyond a certain point the repulsive part of the potential becomes dominant quickly and the molecule is thus always bound. Moreover, for T b5xi the potential can be reasonably approximated by either a square well or a parabola. In either case the approximation results in one parameter for the present model, the width of the well or the strength of the restoring force, By considering the expansion of V(u) in a power series about u = 0, we find that in order for the potential to have a minimum only at the origin in the range L<l, the condition WkK)( ~) > 1 (3.15)

52 i.5x 10I' V (u),5x10-2 12.5 XIO-2 IL/iI - x i I 1. I i ~~~~~~U/S g. Curves of the effective potential V(u) in u Y=ts of.5x10-2 Fig. 3. Curves of the effective potential V(u) in units of X.

33 must be satisfied,* For liquids whose parameters are such that (3.13) holds,** a first estimate of the oscillator frequency in the parabolic approximation is given by [\ ~) We suspect that even when the above condition is not met the use of an oscillator potential is et least consistent with the fundamental assumption of quasi-crystalline behavior of polar molecules in the liquid state. The present description ignores diffusive motion which certainly must exist: for this reason usefulness of the model is questionable in considerations of long-term phenomena such as those encountered in thermodynamic measurements, For special cases where it is felt that dipole-dipole interactions should result in a significant anisotropic character of the effective potential, a slightly different method may be used, We may assume a fixed relative orientation between the molecules and relegate any directional effects to an asymmetrical distribution of the neighbors. For simplicity the dipole orientation of the neighbors can be taken to be the same as that of the central molecule, a choice that is actually quite reasonable except when k$t -, in which case the molecular dipoles will tend to point in opposite directions. This particular approximation leads to an expression for the effective potential, *This condition withJ = 0 has been noted by Morales15 in connection with deriving an effective potential for normal liquids. **Using the parameters given by Hirschfelder Curtiss, and Bird,33 p. 214, we have found that the oscillator approximation is justified, on the basis of condition (3.13), in the cases of CHC12F (dichlorofluoromethane, tf 0O403), C2H5C1 (ethyl chloride, s 0.654), and possibly CH3C1 (methyl chloride, h = 0.0795).

7-L \V /= ua ><+W~ (3-14) where \2 U) and P(V) specifies the nonspherical distribution. Calculations have been made assuming the neighbors are "smeared" over the surface of a spheroid and are discussed in Appendix A. THE THERMAL AVERAGE The preceding discussions indicate that a tractable description of molecular translations in polar liquids is obtained by treating the molecules as a system of independent oscillatorso Using such a model we shall proceed to calculate the thermal average appropriate to the translational degrees of freedom. The analysis involved can be considerably simplified by introducing time-dependent operator formalism, in fact, it will be seen that explicit form of the molecular wave function is never needed in obtaining the necessary matrix elementso The thermal average is first written as < C 1 > c i > (5.15) T T where Ci-' ~ ~t~~ (35.16)

35 is the Heisenberg operator. The "equilibrium" position vector x is not a dynamical variable and therefore appears only in the multiplicative phase factor. Both time-dependent and time-independent forms of the instantaneous displacement vector are used, the time-independent operator u being the same as u(t) at t = O. The case of Q.-Q corresponds to direct and "inner" scattering while that of W/ corresponds to interference scattering. In the former case, it is evident that corresponding components of *_[) and t_ will not commute, and in the latter case all components commute with each other regardless of time-dependence. We shall consider direct scattering first. Because the Hamiltonian \ is separable into three independent components the thermal average (3.15) for =QI factors into a product of three thermal averages, one for each component, <A, Tl~ V the subscript Q being suppressed and ni is the quantum number specifying the ith component of the molecular translational wave function. In arriving at (3.17) use has been made of the fact that Pn is similarly expressible as a product of Pni. In thermodynamic equilibrium we have explicitly -1 M, -. e (r Q- zv~vl. 2~~-V2~I _ ~j~(3.18) - ( CA - t ) )

36 where z = 2.-,U) being the frequency which characterizes the oscillator potential, Because in the present instance the effective potential describing translations is a three-dimensional isotropic oscillator potential each thermal average in the product can be treated identically. By virtue of the operator identity9 e e'\ (3-19) which holds whenever A and B commute with their commutator [A,B], W)</( -/ > ( <( > c (3.20) T where we have suppressed index i, and the commutator [u(t),u] is just a number. Next we make use of Bloch's theoreml Q zEQ > <i) = i T r (3.21) Q being a multiple, or some linear combination, of commuting oscillator coordinates and their conjugate momenta. Thus ilul~t)e -i l) c-K iit)T T T?n)T (3.22) T A more detailed discussion of the time-dependent operator u(t) is given in Appendix B. It is shown that u(t) can be expressed in terms of the "creation" and "annihilation" operators a(t) and a+(t). The thermal averages in (3.22), evaluated in terms of thermal averages of a(t), a (t) and their time-independent analogues, are

37 2 (UT'T ^^l4^ +^ it (3.23) where <KM>=V P = Cl e ) (5.24) After some rearrangements we arrive at the expression for direct scattering 2 ~<Ai't fc e - s, T^^Ve, (3.25) where D P - and we have used the generating function of the modified Bessel function of the first kind,^3 t In a very similar manner the corresponding result for interference scattering is found to be I' T (3526) The time-dependence of the thermal average determines the manner in which energy may be exchanged with the neutron in a collision. To

38 illustrate this remark more explicitly we imagine the differential cross section (2.20) to depend only upon the translational degrees of freedom of the system. In this case the "time" integration yields delta functions which obviously represent statements of energy conservation. For direct scattering we would obtain terms containing >(-iT-Il) which, depending on the sign of n, correspond to neutron energy gain or loss in multiplies of the oscillator energy spacing. Of course not all negative values of n are allowed since the neutron cannot give more than its incident energy to the liquid. The energy distribution of the scattered neutrons, according to the oscillator model, therefore exhibits a certain uniform and discrete structure consisting of peaks whose amplitudes decrease with increasing amount of energy transfer. This structure is in marked contrast to the smooth distribution of energies predicted by the gas model. In fact, the comparison of the calculated distribution with measurements provides a direct means with which it is possible to determine whether a gas or an oscillator model offers the more appropriate description of the translational degrees of freedom, At any finite temperature neutron energy loss is always more probable then energy gain since the lower system states are more heavily populated This fact is expressed by the factor.e In particular, at T = 0 energy gain is no longer possible. When the molecules are more tightly bound, the frequency is will be large and in the limit all inelastic term vanish by virtue of the small-argument representation, rvlC) f.4!) (X/2,) I This is to be expected on physical grounds as

39 a system of rigidly-fixed molecules cannot possibly exchange energy with the neutron. Elastic scattering (n = 0), however, is still allowed. It is seen that the thermal average for interference scattering is time-independent and thus leads to no energy transfer, The reason for interference scattering being a purely elastic effect lies in the assumption of no dynamical correlations among molecules, a situation in which it is highly improbable for neutron waves to be scattered from two uncorrelated scattering centers with identical energy exchangeso The phase factor } is ultimately to be summed over all 9i and Q' molecules (Q-/- K) To do so explicitly would require the determination of these "equilibrium" positions relative to each other. This difficulty can be avoided if we follow the earlier method of smearing the neighbors over the surface of a sphere, then -/ /) (Yq 4> VcJ v,\............. ni ) ( w 2) where s is the mean intermolecular separation. Alternatively, we may adopt a more elaborate treatment of the summation in terms of the radial distribution function15 or simply apply Debye's outer-effect approximation well known in X-ray diffraction,35 The exponential - which appears in both (3.25) and (3.26) is the Debye-Waller factor originally derived in X-ray scattering to account for temperature effects. This factor attenuates the intensity of all scattering processes particularly at high temperature or small 7, In addition to being temperature-dependent it also varies with the angle

4o of scattering. For sufficiently small i,Pk will be large so ^f\) -'l/ X "('(.2~X) (, and the Debye-Waller factor is thus canceled. This implies that at high temperatures or large momentum transfer only direct scattering will be appreciable. We have just used a quasi-crystalline model to study neutron scattering by the translational degrees of freedom of molecules in the liquid state. The general expressions derived here will be used to calculate energy distributions of neutrons scattered by water and hydrogen chloride, and the results will be compared to corresponding distributions based on the gas model.1 In the case where observable differences can be shown to exist in distributions according to the two models a scattering experiment will then be suggestedo Because the present thermal average calculation is actually applicable to any bound system in which interparticle interactions may be approximated by harmonic forces, similar cross section results can also be used to study neutron scattering by vibrational degrees of freedom of molecules. In fact, the treatment applies equally well to analysis of neutron diffraction by crystals,* the significant difference being the appearance of the well-known Bragg condition for interference which would not be obtained for liquids. *Neutron diffraction by crystals have been discussed by a number of authorso For an excellent review readers may see Kothari and Singwi; a number of fundamental aspects have also been examined in detail by Yip, Osborn, and Kikuchio9

CHAPTER IV MOLECULAR ROTATIONS _.3 -2 When the energy of incident neutrons is very low ( 6. o\ -lo Iv ) the scattered neutron energy distribution can be significantly affected by angular momentum exchanges resulting from neutron interaction with the rotational degrees of freedom of the scattering molecule. The problem of inelastic scattering by rigid molecules rotating freely in space has been treated in terms of the Sachs-Teller mass tensor2 which combines the effects of molecular translations with rotations, and also from the point of view of computing inelastic corrections1 6' to the static approximation which ignores energy transfero In this section a formal calculation is described in which matrix elements in the thermal average are evaluated rigorously. The present approach has been used independently by Rahman37* following the work of Gorgunovo38 Because of the assumption of free rotational motions the results are applicable primarily to gases and normal liquids; the effects of intermolecular forces will be considered in. the following chapter in terms of a specific model for polar molecules, We shall begin with a review of the dynamics of molecular rotations in free space. The purpose for the review is to present a complete development of those aspects of the molecular problem relevant to the *We wish to thank Dr. KoSo Singwi for bringing this work to our attention, in

42 analysis of neutron scattering. In addition, certain features of the free-rotator problem which are also pertinent to later development of a description of hindered molecular rotation can be brought forth. DYNAMICS OF FREE MOLECULAR ROTATIONS* Consider an arbitrary polyatomic molecule rotating freely in space as a rigid body with energy pi p2 pZ = -2( J + -) 5 (4.1) where Pi and Ii are the ith components of the total angular momentum and moment of inertia respectively in the "body" system, i.e., that system of axes fixed in the molecule which coincide with the principal axes, It is conventional to express this energy as i -Q P\ P+ + bCtP>-P X ) (4.2) with a = I,- ^u 1I If Iy is assigned the intermediate value of the three moments, then — I b. ~. In (4,2) we also have the Hamiltonians for symmetric mol*A large amount of literature is available; for review reader may see Nielsen39 and Van Wintero 4

43 ecules (Ix = Iy) and spherical molecules (Ix = Iy = Iz). The Eulerian angles (0Jt) which specify the orientation of the body axes relative to a set of axes fixed in space are defined in Fig. 4. In terms of these dynamical variables explicit expressions of the angular momentum operators have been derived and are given in Appendix D. It is.convenient to treat the symmetric molecule first and then use the result to obtain solutions for other cases. The eigenvalue problem is (4 ) ( (4.3) where from (4.2) u P2 ( KLI ) 2 P'jHSRZ= Z ^ + ^;(4.4) with Ix = Iy = I, and from Appendix D, FH^ - \e) Y-2Xt D49.5.S'), (4-5) with B = -2/2I. In (4,5) the coefficient of every term is independent of ~ and (, the equation is therefore separable. The solution becomes <^. t.J',l (4.6) where M, K are integers, and R(4-) is defined by the equation Rl- & lO^ \ e' ^ -X; — @- -- AA -O, (4.7)..... x' "Sie...

t==, -3 -3 I' = e' 3 -, 2 2 -3- 4' Fig. 4. Eulerian angles ( (7 ) defining orientation of body axes. with respect to space axes (. L - L

45. E 2. with = - ~. This equation can be solved by studying its singular points. A variable transformation 2.t-cc~G leads to l-t R'.t) I O-2(- KA-t - t2 i) - (4.8) with regular singular points at t = 0, 1 and another regular singular point at infinity which is nonphysical since t is defined in the range (0, 1). In the neighborhood of the first two singular points the wellbehaved solutions are lk4Ah -tKM1l/S2 (4.9) ic(-t) The transformation is therefore suggested) tt-11/2 I k+ /2 =(t}= t C-vt) f t) e (4.10) After some manipulations, an equation for F(t) is obtained, tl-tf"- t -.- ((+4 t$ -F 0(- = o, (4.11) with i=ttM + i v = &-MI | +1 I ~1&~~4A v(4.12) i&=I(iYZMl )0^+M\+lt-Kl4+2)/4j- VC-X. Equation (4.11) is known as the hypergeometric equation which is the standard equation for three regular singular points The anaytic standard equation for three regular singular points. The analytic

solution at t = 0 is called the hypergeometric function, expressible as a convergent series for [j\ < /, U2lrfto (4. 13) Since the wave function must be finite everywhere (including t = 1) the series must terminate. This will be the case if ( is a negative integer or zero. Thus it follows that the energy eigenvalues are given by E= = 6T(Jcy- - i( <. ( (4.14) (4. 4) where - I+ (1K+Mt +lK-Mt) is denoted by J,J=0, 1,...; K,M = +1,.,. The corresponding eigenfunction is, lx A',\,, )-4A i >-JC, ^ } X et t a(" At Wme %, (4.15) and39 N _ = (t~.~.'.lo',ai-!'~_~,-.. - ll -2.C~.~(il~+,~At+ l -~xl-.!. (!t-Ml- )! L - i —— q —— K —--- (^k ~(4016) In the solutions just obtained J, K, M are quantum numbers appropriate to the total angular momentum, its component along the body-z axis and its component along the space-z axis. Since P commutes with Pz in both systems then the total angular momentum and its two z components are all conserved. From this standpoint the energy EJKM could have been written down directly. The fact that the energy is independent of M is not surprising because the molecule has no preferred orientations in

47 space. Associated with this randomness is a degeneracy of 2J+1. The energy is also independent of the sign of K, corresponding to clockwise or counter clockwise rotations about its symmetry axis, hence each state JKM> is 2(2J+l)-fold degenerate. A convenient representation of the symmetric-molecule wave function has been given by Wigner4 4 > (. ) O^-(H)', (4.17) 44 where if we follow the notation used by Rose43 -yTi =ta, &8V = X OW) t V,V (4.18).2:T(C-1- 2S - -K S (4.19) Summation over s is such that arguments in all the factorials are never negative. The function DK is known as the rotation matrix, or the MKth matrix element of the Jth irreducible unitary representation of the rotation group. Because of its well-known symmetry and transformation 43 properties this function is very useful in the calculation of maxtrix elements. Solutions for the spherical molecule are obtained by simply setting Iz = I in the above results. The complete degeneracy in K leads to a total degeneracy of (2J+1). The wave functions and energies of a linear molecule can also be derived as a special case in which there is no

48 rotation about the molecular axis. Thus J^ ~=\CSce n eg- Y, ~titsh~ t P iy( (4.20) ZT* /2 Y 24 + o T (4.21) -M- = 2(21 ) (4.22) where PM is an associated Legendre polynomial. Degeneracy for this system with only two degrees of freedom is 2J+1, We next consider the general case of the asymmetrical molecule whose energy is given by (4.2) or 2AR 1+ s/. 2(4.25) The eigenvalue problem l Se - p ^-0^ (4.24) with, = f;, may be treated by noting that the states l JKI> form a complete set in the space of the Eulerian angles and thus provide a basis for the expansion of the present solution =M> K ^ I 7<U.>2 l d /. (4. 25) The coefficients Y depend only upon K because J and M are still good quantum numbers (P and the space-z component still commute with HAR).

49 Now quantum number E is anticipated here as a replacement for K. Equation (4.24) becomes -~7K~' ~ ck t, ~,: =(4.26) where we have suppressed the label M. As usual there will be no nontrivial solution for 0K unless the secular determinant vanishes, i.e., CM | 2 )) 7 b < T~l \\^ \ =- c) (4v27) Since K ranges from -J to J the determinant yields 2J+1 values of ), each of which will be labeled by the index E. The energy of an asymmetric molecule is therefore given by C)E/, where for a given J, E may range from zero to 2J in integral steps. For a given J and E, the appropriate set of coefficients of expansion are then obtained from the set of simultaneous equations (4.26). The present method of solution corresponds to an orthogonal transformation from an "old" basis {)JKM>\ to a "new" basis { JEM> in which HAR is diagonal. In order to evaluate the matrix element <JK|Hl\JK'> the "raising" and "lowering" angular momentum operators P+ = PxiPy can be introduced. Thus:L X l 2 2. -\= P- P - P ) (4.28) In terms of P+ the necessary computation is greatly simplified because the symmetric-molecule wave function, denoted here as JKb, transforms according to the Jth irreducible representation of the rotation group,

50 and under the operation of infinitesimal rotation operators P+ yields48 P?^y<> - 9 (t+<)(is\<+l) \^^ < (4.29) In other words, la l = ~ ij,;9,Kt{)|:l*+^^F —atT (4,0) with Q a,^)\44 = tts'-(a-<+ (T4S (4.5-31) Thus Hi couples only adjacent states K' = It2.* The determinant (4.27) has the tridiagonal form Z - O <T3T(HlJ-^)> 0 \ b(<TT -2 i 4A,: ~> K 0 o K \ b<K -1\H Il > *The same result is obtained by using the differential-operator form of HAR and deriving recursion relations for the wave functions (4.15).44

51 where Q-K M E - ^J(j+). The energy of the molecule is therefore E^ - 2|[a<T(+l))+QR,, (4.33) and the associated eigenfunctions are given by (4.25). The (2J+l)-fold degenerary due to the arbitrary orientation of the space-z axis still remains; however, previous degeneracy in the sign of K is removed by the asymmetry. THE THERMAL AVERAGE The wave functions and energies of rigid molecules obtained in the preceding section can now be used to investigate neutron scattering by a system with only rotational degrees of freedom. From the standpoint of energy transfer the case of interest is the scattering from a single molecule since "outer" scattering is necessarily elastic in the present model. The appropriate thermal average will be evaluated first for symmetric molecules, then the asymmetrical case can be derived as a generalization. * For molecules with axial symmetry the matrix elements under consideration are >tHs i' \ — i~ti- iC b (dd' j3X4>=(CK4\6 M a a taKe> (4.54),7'K~s, / / *Rahman37 has independently obtained results similar to those derived here.

52 the index % being suppressed since -='. In the product only one matrix element needs to be obtained, the other is merely its complex conjugate with appropriate change of nuclear subscript. The equilibrium position b is a constant vector in the body system while neutron momentum transfer K, is fixed in space; in the scalar product both vectors must be expressed in the same system of coordinates. A useful expansion for the purpose of calculating the indicated matrix elements is Ci = An d~it4Zfr (b)) (4) (4.35) where I/ is the Qth order spherical Bessel function and the prime indicates that arguments of the spherical harmonics are measured with respect to the body axes. Because KY is specified in the laboratory, a rotation yields -: () - t -', -~ )A (4.36) A Q where KJ has components in the space system. The rotation matrix is an explicit function of the Eulerian angles describing rotations of the molecule with respect to the space axes. The desired matrix element now becomes (S ^ _ | K >. The expectation value of li in an eigenstate of the symmetric molecule is easily obtained since the wave functions are themselves rotation matrices. The product of rotation matrices can be reduced by means of the tabulated Clebsch-Gordon coefficients Ci G; the relevant reduction formula as well as other useful properties of the

53 Clebsch-Gordon coefficients are summarized below:* q C-~~vvM K T J ~)M'-(: M+4 ikt IC i2 S V I ^tl vc 1+ L i iti2 - c- ^ - Ci YV I - z o 2, C -12 1 I I lrm i/2 1 1* C,) IQ^V ^-w, C-)ii2 1t >itl e ml -M! r1 \z k 1 2.+ 2 t I/2 3 j 2 it 12 - C' i\ (4,57) M, V T M,-K'm K.. C T -T' 9, Ilp v-KA CQ cm y-I T cQ C - a)IZ *See Rose4i for a thorough discussion of the matrix and Clebsch-Gordon coefficients. properties of rotation

Using these results we obtain rc b ASJ~zAK X U, s1' <i at 7T i (4.38) where use has been made of the orthogonality property i~ e~,SQ:~ediR, $I ki,-la S<;e ^ ^ ^(4-39) In arriving at (4.38) the space-z axis is chosen to be along \4 so that The energy eigenvalues of the symmetric molecule have been shown to be degenerate with respect to M and the sign of K. The matrix element (4,34) therefore should be averaged over K, -K and the 2J+1 values of M. Summation over M and M' gives atn n o-_ (-s M (4.40) KA7 - i' Q,. and we have <J i / ) I L <^J4 I 7 /<\Q\ s'- 7, "/, 4 ) t ~t,47d;' y o(tia C-K K

55 The thermal average for scattering by a single symmetric molecule thus becomes iftc i, by -its^ _ -_ilcb T (4.42) = vil-t ) t<^ Tk>l Inelastic neutron interaction with rotational degree of freedom results in angular momentum exchanges which are governed by selection rules for J' and K' for fixed values of J and K. The energy transfer depends upon I, which from the expansion (4.35) is interpretable as the amount of angular momentum exchanged in the collision. When 9. = O, we would expect only elastic scattering; this is precisely the case since K' = K and the Clebsch-Gordon coefficients require J' = J. Explicitly, =O T (4e43 ) and so <(dd';T) = l<' 4) ), (4.44) As b — 0 only the term with = 0 survives in (4.41) because dl() d —i~(X<; the vanishing of the "moment arm" therefore renders the scattering process purely elastic Equation (4.44) is, however, not the sole elastic contribution to the cross section since for finite angular momentum exchanges there will also be selection rules for K' = K and

56 J' = J, and each term in the 9-sum contributes to elastic scattering. The Clebsch-Gordon coefficient C restricts J' to range from It7-l to J+Q, then for small Ki the spherical Bessel function ^([) r ~Y/(9Wl)K! limits inelastic processes to small energy transfer. Spherical and linear molecules are special cases of some interest in which rotational energy is the same and depends only upon quantum number J. The difference in these two cases lies in system symmetries and hence the associated degeneracies as noted previously. The matrix elements in the thermal average must be appropriately averaged over all orientations with respect to which the energy is degenerate. The results are <,> Qi=.C-@5 —,) —12 i'j^\(4.45) T v v where for spherical molecules, --......i':.~k -. (in By1' (I< -, / <', where - is the angle between and. Similarly for linear molecules, K ecules, -; - M e-^-. 0(02t) X ^('.j ~ =i^ d ^*^ba^ijkc'

57 The absence of the degree of freedom characterized by K results in (275 +1)/1 i f) being replaced by (Ctl) I C QT The present method of obtaining the thermal average can be generalized to the case of an asymmetric molecule in a straightforward manner, We recall from the discussion of the molecular problem that the wave function of the asymmetric molecule is expressible as a linear combination of IJKM, and that the energy is still degenerate with respect to M, The eigenstates are K k7^-M -J~ (4[,48) where the coefficients -rT are real and the energy eigenvalues are labeled by J and Eo Following the same procedure as before, we obtain the formal expression ~ T.ITe ) X i E ~TElit. & ( Y4.49A)) Vk A 4' T -T' (( Ti,'T d2E t -ik' iC k( C v-k In the results derived thus far the scattering process is treated by explicit summation of individual rotational transitionso While the approach is rigorous and accounts for all the inelastic processes at once, the amount of labor involved in an actual calculation may be pro

58 hibitively large if the incident neutron energy is sufficiently high as to induce a great number of transitions. On the other hand the results obtained here are in a form suitable for exhibiting the low energy limit. By virtue of the small argument representation for the spherical Bessel function, &dYW P Q(*QQjL"', the cross section may be readily expressed as series in powers of. When the incident energy is large compared to rotational level spacings the mass-ratio expansion method developed by Zemach and Glauberl and 56 extended by Volkin can be applied. The static approximation, or the high-energy approximation which completely ignores inelastic scattering, may be readily deduced from the present expressions for spherical and linear molecules.37 In this case energy transfer is a small fraction of the incident energy so KY is essentially only a function of ki and the angle of scattering; moreover, the time-dependent factor will be ignored and the sum over J' extended to all possible final states. Then from (4.46) w- z ) ICKY -- 2 244 S2L V (4.50) - -K ~_ Q and r>T j,0,%Q( i ( (4-51) g W ^i-b,\).

59 The last step in (4*51) can be shown by considering expressions similar IK(bu'g);t ^ -it bW to (4.35) for ~ and e, and then integrate over all directions of K. The same result can be derived for linear molecules using (4.47), thus implying that the scattering of high-energy neutrons is not sensitive to system symmetries. In the cases of spherical and linear molecules the scattering is also not influenced by the particular state in which the molecule is found initially.

CHAPTER V HINDERED MOLECULAR ROTATIONS The preceding analysis of neutron scattering by systems with only rotational degrees of freedom assumes the molecules are freely rotating, The assumption would be erroneous if existing intermolecular electrostatic effects in the liquid are sufficiently strong as to cause the molecules to spend considerable period of time in certain preferred orientations of minimum energy. This type of motion is then said to be hindered in the sense that one or more of the rotational degrees of freedom is partially or completely restrained. Recent slow-neutron experiments with water12'l9 have revealed pronounced inelastic scattering involving energies larger than any characteristic energies associated with molecular translations or free rotations and smaller than the vibrational energies. As in early interpre22 tations of similar observations in Raman spectra,2 the specific atomic motions responsible for this type of energy transfer are taken to be hindered rotations, or torsional oscillations of the molecule in the 12 potential field of its neighbors, In this section we propose a dynamical model for the description of hindered as well as free rotational motions. The basis of the model lies in the assumption that the hindrances are predominantly produced by electrostatic interactions among molecular dipoles and because of the appreciable amount of crystalline symmetry still persisting in the 6o

61 liquid strong direction-dependent forces prevent the molecules from having complete freedom in rotation. It is evident that this viewpoint is consistent with the model developed earlier for molecular translations. The present approach treats the interactions of a molecular dipole with its neighbors in terms of the coupling between the dipole moment and a local electric field which represents the effects of the surrounding. As in the previous determination of an effective potential for translations, only one parameter appears in the model to account for the dynamical characteristics of different liquids. The electric field in this connection will be taken as uniform in the immediate neighborhood of the molecule and constant in time, at least during a period of rotation; its magnitude is presumably of the same order as the crystalline field in the corresponding solid phase.* In the single-particle description the potential for rotations appropriate to our model is just -J.E, the interaction of a dipole J in a uniform electric field E. The Hamiltonian now becomes A/ C a- A (5.1) where HR is the free-rotator Hamiltonian used in the preceding chapter. The direction E is fixed for the molecular problem; however, in evaluating the corresponding thermal average its direction must be averaged, *Spatial and time dependences of such a field can be assigned physical meanings and may be introduced as modifications of the model.

62 in which case E will be assumed to have random orientation. In the presence of the potential the total angular momentum is not conserved although with proper choice of coordinates its two z components will still commute with HR. We shall first consider a rigorous solution of the molecular problem for symmetric molecules and then indicate a possible generalization to the asymmetric case in Appendix E. It will be seen that the rigorous approach is not convenient for the calculation of thermal average because the determination of rotator energy and wave function requires substantial amount of numerical work. Moreover, as the existence of hindered rotations implies large local electric fields, strong-field solutions should be physically meaningful and capable of providing a suitable means for the analysis of slow-neutron scattering. A FORMAL SOLUTION The eigenvalue problem for a symmetric polar molecule rotating in a uniform electric field is* ( -* = \. hJi> )> (5.2) where HSR is given by (4.4), \.-Y\., and i., (.KJ i [see Eq. (4.21)]. The dipole moment a has been chosen along the body-z or C, axis and electric field E along the space-z or e3 axis so the potential is simply...C\ ci Since the other two Eulerian angles are not affected by E, M and K re*The same equation is encountered in the study of Stark effects in polar molecules. An application of the present approach to linear molecules has been described by Kusch and Hughes 45

63 main as good quantum numbers while > is anticipated as a replacement for J. Again making use of the fact that the free-rotator eigenstates form a complete set, we seek an expression for i1t<k of the form i > — 2 C;Q) l'_ T.V;\ (5 3) The coefficients of expansion are determined from the coupled set of equations 5 C- at CABI z SNT' 5 /I jW z )\ c' )~ (5 4) where EJ is the free-rotator energy, indices M and K being temporarily suppressed, and E2 has yet to be found. The interaction matrix element UjJ, is readily obtained by means of the reduction formula (4.37) and orthogonality integral (4.39), \ ^J~^C^S+c^i^3 /-,,+ (5-5) with..^ S a l T C r T T ~<......t'\.j ^ _.. C. o:i.+}. I CT L:r L C T-:; (5.6) The next step is usually to demand that the determinant formed by the coefficients of CJ,(C) in (5.4) vanish, a condition giving rise to allowed values of E, each of which is then used to obtain a corresponding set of Cj(<,). From (5.5) the matrix Ujji is seen to be an infinite tridiagonal matrix since J is unrestricted, and for this reason it is not possible to exhibit the solutions in closed form. Nevertheless, by using

64 a method first discussed by Feenberg46 in connection with perturbation theory the solutions can be given in continued-fraction form which will be suitable for weak-field approximations or numerical computation. We shall write the energy as* F= 7 7 7^ E7.T - e-7 An argument is given to ES for identification purposes since in the event the electric field vanishes the energy given by (5.7) must coincide with the corresponding free-rotator energy. The effect of the potential can therefore be studied at any field strength. The close resemblence of (5-7) to ordinary perturbation theory should not imply that the expression is approximate. Due to the fact that DoO couples only adjacent levels in 3 addition to having nonzero diagonal matrix elements all terms in k and higher do not appear. To show this we only need to consider the term in \, which would be ordinarily present, For a fixed J Ujp requires p = J~l and Upq requires q = J~2. Since Ujj' = 0 for J' / J, J~l then Uqj must vanish. In view of (5.5) the energy is rigorously given by Eg(T)-E )(I+ A 1i t —ETlc (5.8) ^-^ ^{^i — -^}.~F E ^!S.J- 7. f^ ~ 7- *See Morse and Feshbach41 Chapter 9.

65 E'a -,:' 7/^) + A i, E^) K (5.9) EJIT"I L33'" r'/ with J' = J~l, J'' = J~2, etc. It is more convenient to cast H into the form of a continued fraction,41 Z V r^PO~q.+(T) i Eg(T)- Ei,+ X\(2lAl) - \XZ W t5+ ) (5.10) E^~Ls?- ^7-t 4C-^ 4(T) _ftq (1- xxi1+ tts1)-^ e o) -,, The coefficients of expansion corresponding to this energy are similarly obtained, CSt>= 1, taz1+ c. (5.1) Gil Eg23)-E^^ I + } 6(XTil- Thus the expansion (5.5) consists of three terms, and it is obvious from these results that as \ —6 6) i1M-)t->l and ~S3 —.. The present formulation is particularly useful in studying the splitting of the energy levels as the electric field varies.* Since the wave function is expressed *The total number of splittings is more easily determined from the strongfield solution. For some numerical results in the case of linear molecules, see Kusch and Hughes.45

66 as a linear combination of the rotation matrices the method used previously can be applied in computing the matrix elements for the thermal average. There is little advantage in this approach, however, as it would be very difficult to treat the physically interesting case of strong electric field to exhibit the nature of hindered rotations. In the following we shall therefore directly seek analytic solutions in the high-field limit. THE HINDERED ROTATOR The eigenvalue problem for symmetric polar molecules in an electric field expressed in the form of a differential equation is obtained from (5.1) and (4.5). By choosing4l along el and E along e_ the equation is still separable in the three Eulerian angles, so the wave function is of the form 44' (5.12) and Y(9) satisfies |\~J,ot<- (T&~ ~ -> ~( a (5.15) It is expected that under the influence of strong field the molecular dipole will tend to be aligned in the direction of E, thus angle e is restricted to having small values. A meaningful first approximation is then obtained by a power expansion in 9. To second order, 1r (43 3 )i HL r( 3 )-~i l ((-lfAt, tjI(Y)=o (5.14) B I II-M)-I: 2.22h/

67 The $i term can be eliminated by a change of variable Y =l p(i ) A further transformation y= — yields 1i Mh (X+X-o )1 |,P& = o ) (5 15) where 0 G E (- ce- iy -^ 3 (4+84 ~(5.16) an equation with singularieties at -=o 0, o In the neighborhood of these points the well-behaved solutions are X In the small-angle approximation the range of e is now (O,oo) instead of (0Or). The asymptotic behavior of P suggests the dependent-variable transformation.X).) e L(X) which leads to an equation for L(x), xL'' l - L'( - )L + V. (5017) Equation (5.17) is in the form of confluent hypergeometric equation with 4l regular singular point at x = 0 and an irregular point at x = c. The solution, like that of the hypergeometric equation, can be expressed as a convergent series for Ix < |. If we require the wave function to be square integrable over the range (O0,o) it is necessary that I-_ - - be a positive integer or zero, in which case the solution is the associated Laguerre polynomial The energy and wave function according Laguerre polynomial \_ ()<. The energy and wave function according to

68 this mode of quantization are 1^> 4Ni i x (X (5.18) tcak- ^ <^' - -. (5.19) Ik z. In writing these results we have used, to a very good approximation, 2p -12. and 1 The normalization constant is determined from a special case of a more general integral discussed in Appendix F, tN -ii ---— (5.20) The wave function and energy of a linear molecule are derivable from (5.18) and (5.19) by simply removing the q degree of freedom and seting K = 0. ^ M -S- 1 \C-^ X % L x) (5.21) g- i;(2 s,+ i I)- ^ ~ - )\.3 (5.22) Normalization constant here is that given by (5.20) with K = 0 and less -1/2 a factor of (2T) -/2 The rotational energies of the symmetric molecule are seen to consist of contributions from both hindered and free motions, the latter being the terms proportional to B. This consequence could have been predicted from the form of the potential wch onais y the e variable. Although each of the three degrees of freedom is involved in hindered rotation, the energy of this type of motion actually depends upon only

69 two numbers. This implies that if the energies of free rotation were ignored then the small-angle approximation leads to a description of the symmetric molecule as a system with effectively two degrees of freedom, and in this sense there is no difference in the dynamical behavior between the symmetric and linear molecules. Moreover, through the form of the energy of hindered motion the nature of the small-angle approximation is revealed; it is not surprising to find that under strong fields the system behaves as if it were a two-dimensional isotropic oscillator. In Appendix G we discuss a somewhat different approach to the strongfield solution which leads to the same results as those obtained here. The presence of free rotations gives rise to a rather complicated pattern of energy levels superimposed upon the equally spaced levels of hindered rotationso It is interesting to note that by changing the quantum numbers K and M such that IK-MI remains the same the symmetric molecule can undergo relatively small amounts of energy exchange. Physically this is to be expected since the rotational frequency about the axis of symmetry can be altered without affecting the relative orientation of 4_ and _. Since there is no rotation about the et axis in linear molecules this type of low-energy transitions is not possible as can be observed in (5.22). The strong-field solutions which we have obtained represent a meaningful description of rotations of polar molecules whenever \ is sufficiently large so the use of small-angle approximation is justified. However, not all solutions are admissible since

70 <X) = -(~2%,zr \t-Al A l ),(5.23) and there will be values of Sand IK-M| for which <)S is no longer small. The physical breakdown of the model occurs when the energy available for rotation is so large that the effect of the electric field becomes negligible. Such a situation may be achieved by increasing the system temperature to the extent that the molecules can no longer "feel" the influence of the potential. In this limit one would expect the molecules to again resume completely free rotations. Unfortunately, the use of small-angle approximation at the outset prevents the dynamical behavior during this transition to be examined in the present formulation."* The description of hindered rotations has not been satisfactorily generalized to the case of asymmetric molecule. We have considered the method of orthogonal transformation employed by Wang44 in treating the free asymmetrical top and have found that the terms arising from the asymmetry [terms proportional to b in Eq. (D.5)] couple all states of the symmetric molecule. Hence numerical methods are required to diagonalize the resulting infinite matrix. On the other hand, we have observed that the energies of hindered rotation for symmetric and linear molecules are of the same form. Then it may be argued that in cases of strong coupling the symmetric molecule provides a reasonable first approximation to the asymmetrical molecule. The supposition that hindered *The use of a temperature-dependent field offers an interesting possible modification of the model through which a high-temperature limit may be derived.

71 rotations are not sensitive to system symmetries will be tested in a specific calculation of water to be discussed in the next chapter. THE THERMAL AVERAGE 25 Recently Nelkin23 has treated hindered rotations in water by means of an oscillator whose frequency was taken directly from neutron experiment and whose mass was an adjusted parameter fitted to the free-atom cross section in the Sachs-Teller mass tensor approximation. The reduction of rotations in a potential field to torsional oscillations has just been explicitly derived and in this respect it is possible to justify Nelkin's approach. The model proposed in this investigation provides a reasonably simple but realistic description of rotations of polar molecules in the liquid or even solid state, and is suitable for use in studying neutron scattering. We shall follow essentially the same approach developed in Chapter IV in evaluating the thermal average. It will be seen that the main features of Nelkin's calculation are contained in the present results although the corresponding expressions for the intensity of the scattering are markedly different. Moreover, the present treatment allows the effects of free rotations to be included in the general analysis. We consider first the matrix element appropriate to neutron scattering by a single symmetric molecule, W ^U>-<^ t i4 = <kI~ to e, i, lK>IC 1(5.24)

72 where re''b i wQ<he ~i > ~>KI'K'M'' \ ^ (5.25) The expansion (4o35) now gives ^-^ _ 4T tD^^A(')r(4i m4te^d? l(d&^) (5.26) where under the transformation of successive rotations the spherical harmonic YQ(i^ whose arguments are measured in the ej system is expressed as a linear combination of _ with components of t measured in the laboratory or Q system. The orientation of the space of ei axes with respect to the laboratory system is specified by the Eulerian angles (C&). The laboratory axes are introduced because the direction of E, which is along e39 must be averaged over all possible directions. By means of the orthogonality integral (4.59), tr2. IpQc t)D M Be, I/= (Q -\ and the relation we obtain an expression for Q averaged over all the random directions of E, The indicated matrix elements can be reduced in view of (5.l8), <4SMl|..l M > = ( K i^<N tup $ 42SK F., (5.28)

73 <k'id, ig >=2 A ^' it eC F (5.29) where normalization constant N, is given by (5.20) and integral F is defined by kq/~ F=|xX x s 4s d ^^5KV (5,30) with =-(3 It is seen that for fixed K and K' the product of the two matrix elements requires k = k'. Thus ~ = ~,j 7(, ( i (db 2" lu'i,(^)(i' b i id i( F (5 31) In the integral F the matrix element _ is according to (4.19), ri' Q = G C4-FAZIQ' S (.s-f)(S8(^ \2- (- ^ -2\ (5.32) where 1 t- A F= ~, 9t9X_\w! -s4,-^c (-Mv. i In keeping with the small-angle approximation used to obtain the rotator wave function we may write 2 ~2-W~t2, (- -W+ 3-! n_~ -~-~ ~dwihleads to anexplii tWiS)) (5 33) which leads to an explicit expression for F,'F -Q^[ t S ) C @- 20 Mk / 2tX(S(40-8 XttIp~gS) J (5.5)

<-(S I. P — cJ t P-j i -I J1 —i (2Q+CM-i- as) (5.36) 2i = Q- l K-M w -tel s2^ -Mt k 2q - iK-M\ - -i — *v ~-k\ S- 2 -M k. We note that p, j, q are always positive integers or zero. The integral S can be evaluated in terms of the generating function of the associated Laguerre polynomial (see Appendix F), ^^ ^fd. ^M CptV.Cj+-ff-^ ccq^^-d^7:'' d — d!((-d)!cS'4!, — d ^l-a —^! eqtn-g where the summations are such that arguments of the factorials in the denominator must never be negative. For a given b, j, q, and p Eq. (5.57) can be used to derive the correponding selection rules for'/. The Eqs. (5.24), (5.31), (5.34), and (5.37) comprise a formal description of neutron scattering by a hindered symmetric rotator. As with the earlier case of free rotators, the present approach results in an explicit summation of all possible transitions, and consequently requires considerable reductions before a particular process can be investigated in detail. It is noted that in (5.34) a natural and convenient expression for F can be developed as a power series in (i4a ) where =( The leading terms are

75 F= F-+ (+:w F, + ilF +( (5.38) C lp-Ml -x V- F1 = -m (4)(+ 1 x Ik-MI 2l+l- _ - l ( - ( L Le ), Z 2 \' r ~K-M9lt -\-X \ K\-Ml K-lA F =-' S-4-l^^ 6 LL, i. The integrals in Fi are special cases of (5.37) with J=O, the form %~%'P A X tiS' =(>- ct,^->'! (c^-X>! ql they are of (53.9) T2 ~:y CS-l cg~! (tr~ ((5*~-~) in which if \ )<<M\ ( V y I,. If the condition on Q is not satisfied then the integral is zero, indicating all transitions are forbidden. Since F appears in the thermal average as F2 it will be sufficient for most cases to retain only the first two terms in (.538). By ignoring the term 1. in F we are effectively neglecting a correction to Fo which represents elastic scattering, of order and inelastic terms of order (id) which are different from those given by F1. In order to exhibit the nature of the energy transfer with hindered modes of rotation we shall, for the present, ignore free-rotation energies in Eat. The matrix element (5.24) can now be written as

76!_:1~', (5.40) where j f ^ 2 =kccio) (5.42) l _. l 4K~2lt- 2+1tl-Q /+1-irCtI l 5hil~ilb-lk7(24s~ 2.a i T2 ). iI In the above notation ( l-k( 2 l1 and in (5.42) the second term is to be regarded as a sum of two terms corresponding to upper and lower signs respectively. The energy becomes -: K2.(9..-. — ) —\ as a result of ignoring free rotations, and the matrix element is only a function of, instead of K and M separately. The simplification thus resulted is more apparent in performing average over initial states, for we have 4;iY)(' ^ ^ —.^t S.'f-4i,> (5.43) where. > and L [~-cmb i i()l -{((c) l( rt 4 k e) E; The calculation of W(VK +.)K) requires explicit determination of the integrals Zk. pi. ) and associated selection rules for (.' After a series of straightforward manipulations with (5.42) and (5.39) we find

77 WQ t=.I+ ~lM ls; (1) 4-] ) (5.45) = + Thus for Cf b, 4 VheiW)t h l1i ~ ^stL + (c i1 t (5.46) + Aet-i+ -7 Q\k ~ it~~' -?i~ T n, - t- 2 1 + K - lti v(Q \ +(a 2)e ] The indicated summations over I and 4 can now be carried out, (5.47) Lo 0-o XC=-C t -^ _ <2 2 (t ) -i) (5.48) ) (2,++ -2 P cl-e l- 1) where - )= /\k The thermal average becomes simply,(K (bi> (6i b d^^s^^Z.^^^' T =u M r) I/,, -. i + -I x I';t -tt Q 2aS In arriving at (5.49) average over initial states has been performed using all possible states. However, this procedure is not entirely consistent

78 with the rotator model since large E and t values are not allowed by virtue of (5.23). For cases with P \ we anticipate no significant error due to the rapid attenuation provided by the Boltzmann factor. If only a smaller set of quantum states are to be used in the average the appropriate partial sums may still be performed without difficulty. The first term in K(d > represents the contribution to the cross section from elastic scattering. It is in fact expressible as,(K_-b <,l) the result obtained in the static approximation for free molecules. As to be expected, elastic processes only depend upon system symmetries and is not influenced by dynamical details of a particular model. The time-dependent terms in (5.49) correspond to "one-quantum" transitions in which neutron energy is increased and decreased respectively by an amount Aif the spacing of hindered rotation levels. This relatively large energy transfer may be interpreted as the hindered rotation peak similar to those observed in scattering from water and ammonium halides. The same nature of inelasticity can be obtained by treating the rotator as an ordinary oscillator with frequency \ and in this respect the nature of approximation in Nelkin's model for water is demonstrated. Higher-order inelastic scatterings do not appear in the thermal average because terms of order and higher have been neglected in computing F. It can be readily shown that the term F2 will give rise to "two-quanta" transitions whose intensity is lower by a factor of tX compared to the "one-quantum" processes. The index Q plays essentially the same role as in the free-rotator

79 problem since identical expansions have been employed. Inelastic scattering is seen to be possible only if Q is non-zero. Because each term in (5.49) represents all the contributions to that particular process, all values of y greater than or equal to one will contribute to the "onequantum" transitions. It is observed that elastic scattering is temperature independent. The absence of a Debye-Waller factor is a direct consequence of the assumption of rigid molecular bondso Nevertheless, temperature effects introduced by the averaging over initial states indicate that neutron energy loss at finite temperatures is always more probable than energy gain, the latter process is completely restricted at T = O. In the limit of high temperatures the intensity of both processes varies like )- due to the fact that many more states are able to participate in inelastic scattering. The equal probability for scattering up or down suggests in these results a certain degree of classical phenomenon. As noted previously, the high-temperature limit in which free rotations are obtained is not available in the present model by virtue of the small-angle approximation. Corresponding result for linear molecules may again be derived directly from (5.49). By noting that ~ (6h =S-~ we have T t l ( ar)e vaid fr The thermal averages given by (5.49) and (5.*50) are valid for any

80 incident neutron energy. In the case of very low energy further simplification is available. The small arguments expression for di() mentioned in Chapter IV leads to expressions in power series of Kb. For instance, for linear molecules we would find (29 +I Vq o(I Xi T 2(Q- 3 2cQt\ An appropriate high energy limit will be discussed separately in the following section. For scattering from different molecules, we need to evaluate <cca'j^^=kM -<< l M1><CM|^ C L A) We shall be content with averaging the product over orientations of the electric field separately. This is equivalent to assuming no directional correlation from one molecule to another. Thus <0W0{ i By> =. jo(l~io|a^^^) *(5.51) Again, as a consequence of the single-particle model, "outer" effects are purely elastic and independent of initial state of the molecule. Thus far we have not considered the effects of free rotations. It is to be expected that these effects will not significantly alter the essential features of the preceding results. On the other hand, there

81 are two interesting aspects associated with the results which take into account of free rotation energies, namely, that there will be inelastic scatterings associated with low-energy transfer and that many transitions involving energies of order \A will be allowed. A detailed consideration of these effects is discussed in Appendix H, we shall mention here some of the conclusions. The selection rules for., is independent of the form of., therefore only the case m = k, which previously gave rise to elastic scattering, needs to be examined for transitions involving only the terms proportional to B. The cases m = k~l will contribute to a number of transitions involving an exchange of \2X; and some freerotation energies; because of the dependence of l-. on K2, M, and KM a rich spectrum of inelasticities is produced. The resulting energy distribution will thus exhibit certain peaks in the low energy region corresponding to excitations of free-rotation levels and the somewhat broadened hindered rotation peaks centered roughly about il.- ii"'1N Ki For linear molecules the low-energy peaks do not appear as previously noted. A HIGH-ENERGY APPROXIMATION The formalism in the preceding section becomes cumbersome whenever the incident neutron energy is high enough to induce a great number of transitionso In this case the quantity of interest is usually the angular differential cross section for which the mass-ratio expansion developed by Zemach and Glauber can be applied. This method, previously discussed only in connection with free molecules, can be used with the hindered

82 rotator model. We will consider the application to linear molecules and show that for very high energy neutrons the distinction between hindered and free rotations vanishes, a limit one would expect on purely physical grounds. We return to the matrix element for direct scattering by a linear molecule in a given initial state, W X;SM> -- < t e l east (5.52) where t-:,'l -j U'E, with WlUkb, and we have defined \. — if-b -I_,b e — ~ i, e According to the Lie expansion ^t Q bpl+^^t^Q i^-". (5-53) we have ( ) - _ -V l t-r l G ( x- L )- t t.rbk b (5.55) In arriving at - the different components may be taken with respect to either body or space axes; in the former case the commutators involve components of Li and Ki, and will have a minus sign associated with the commutation relations.36'47 For the moment we shall proceed formally. Since H is diagonal in the representation y, Eq. (5.52) can be written as <io jW\=>=<< l ( \<ik, (5.56)

83 where 2A i - ( A + 2f,~-=. ^<tbxlb_0/ b t = cK /Wibx?%% l For the purpose of present discussion we shall ignore the translational and vibrational degrees of freedom of the molecule. Then for a given initial state the contribution to the cross section from direct scattering is ~it(-,\Xti4= (i,0&j7<<b(9li9 BC A>.(5.57) where the scattering length is denoted as a. Now consider the Taylor expansion A=0 (5.58) which when inserted into (5.57) gives i(G;,Q,^f - = tX(2 Q K.C (5-59) The angular distribution is obtained by integrating over all energy transfer, (5.60) 37= M a (hi(fC r =^zc^(^y^^ ^~~~~~~~~~~

84 Thus the time expansion (5.58) results in an expression in power series of the mass ratio for the angular differential cross section. In order to determine the coefficients of the time expansion we examine (5.56) in the form For the expectation values of these terms we introduce the hindered rotator model in which three vectors need to be specified. As before the neutron momentum transfer I will be chosen along the laboratory-z axis, the electric field E along the space-z axis, and dipole momentQ. along the molecular axis. Since E does ot appear in (5.61) the average over all.61) diretions of which obviously ver, among the products of various hav componen omitted. to the laborexpectatory system for the averaging the introdame the hindere obtained model in which three vectors need to be specified. As before the neutron momeby integratingsfer thes will be chosen along with respect laboratory-z a set of axes oft electric field E along the space-z axis is along b One readily findst molecular axis. Since E does not appear in ( ) the average over all electric field orientations may be conveniently replaced by average over directions of V4. Moreover, among the products of various components of K and b it is not necessary to rotate the momentum transfer vector back to the laboratory system for the averaging, the same results are obtained by integrating these components of kq with respect to a set of axes of which the z axis is along b. One readily finds, f -- A A \, (5.62)

85 where < >\V denotes the average over electric field orientations. To order t the only expectation value in (5.61) that requires further attention is the term 6> ) for which explicit representation of the angular momentum operators Pi are given in Appendix D. In this case L3 = - A Of the nine terms in the summation only six have to be evaluated. After some manipulation it is found that <v _1 1 + I,\ X3 ) ce~~~.l~ si~;3~ iuze~j (5.63) The last two terms in the bracket are seen to be precisely the kinetic energy operator so the expectation of < >ha can be readily computed. The matrix element thus has the following time expansion KL~ScA) - rN)~ Ic~ 1 r (V.6 ) <^ - i-Af^^iK2K -A-~ For comparison we quote the free molecule result derived by Zemach and Glauber, ~' (. ik^^(i.6,) <,iA-JM>= 1 -(, 3 + t ~'' F' b ~ j(5.65) Since J(J+l) is the energy of a free linear molecule there exists complete analogy between these two results except for the somewhat puzzling reversal

86 of sign of the last term. However, both terms in are always positive even in the ground state. The coefficients of (t) in (5.64) are the required CS. According to (5.60) the angular differential cross section for direct scattering by a hindered rotator in the eigenstate I~M> becomes ~I ~0(fi= 1-^ - 3-1 3@S ( tX I ((2-Ak41())1-' 2i (5.66) This result is useful for the study of angular distributions at relatively high incident energy (ib > )and small mass ratio. The "inner" scattering can be similarly treated. In this result dependence on system dynamics first appears in the correction term of order ) C Therefore at large energies effects of the hindrance become increasingly insignificant and the scattering is expected to be insensitive to the choice of models.

CHAPTER VI SLOW-NEUTRON SCATTERING BY DIATOMIC LIQUIDS We consider specific calculations based on the dynamical models developed in Chapters III and V. It is our main purpose in this chapter to determine what aspects of the differential cross section are sensitive to details of molecular translations and rotations, and to what extend does the incident neutron spectrum influence the final energy distribution in studying hindrances in molecular motions. The calculations are made for diatomic polar liquids since it is the simplest system that embodies all the essential features of molecular dynamics in the present description. The differential cross section (C(E- Gc) is computed for liquid HC1 at 188~K. As there are no reported measurements using any diatomic polar liquid the results obtained here can well serve as a guide for future experiments. Because water is an important system from the standpoint of liquid theory as wellas neutron moderation in nuclear reactors it has received considerable experimental attention as previously mentioned. In this investigation the differential cross section for water is computed in an approximation where the asymmetric molecule is replaced by a diatomic molecule of equivalent mass and permanent electric dipole moment. The results are then compared with the measurements of Brockhouse and Nelkin's calculation. 87

88 ENERGY AND ANGULAR DIFFERENTIAL CROSS SECTION FOR POLAR DIATOMIC MOLECULES According to the formalism derived in Chapter II the differential cross section 9(C,) ))C) (the cross section is azimthally symmetric) is determined once the thermal averages appropriate to the various degrees of freedom are obtained. When the center-of-mass degrees of freedom are treated by the methods of Chapter III the translational thermal averages are given by Eqs. (3.25) and (3.26). By ignoring the vibrationrotation coupling it is shown in Appendix I that the description of vibrational degree of freedom is that appropriate to an oscillator, but on account of the non-commutation between HR and CQy vibrational and rotational thermal averages are still not separable. The thermal average for the internal degrees of freedom, in the case of _= /,can be taken from Eq. (2.33) a -^ V^<g _, (6.1) Muitt -2lby, -, (6.2) where,,; -t' — -- ^? i-^v -^v The eigenstates /| are just oscillator eigenstates, so

89 __b h j P- (6.5) L YUij:j Co4 2 ( where for diatonic molecules [= +, and hM The internuclear vibrational frequency and reduced mass are denoted here as L and et respectively. Note the ordering of time-dependent and timeindependent operators, b[ and _, are respected. This expression for yd'/ is to be inserted in (6.1) when averaging over-all molecular orient. a i-a S ons i ~~C A ientations. A formal method which is applicable when < Since the effect of rotation of axis of vibration on neutron scattering has been shown to be small we shall proceed under the approxi~~A A~~~~ ~ A mation that the effect can be ignored entirely in which case ~._ - (t) A z and (K-i k)) will be replaced by V,. Compared to the Krieger and Nelkin approach of averaging the exponent the present approximation allows greater attenuation of the intensity of scattering due to vibrational motions. Thus

90 I'."P " 4 ", NI ~ ~~Z~~~~~ *~'\\ - / K>)~~~~~~~~~~~ i,(~~~~~~~~1 A/ II-~~~) V,. ~ ~ _.i. i (6.4) For the range of neutron energies T... ~n... CD ) ) peratures (T, 373~K) of interest excitations of vibrational highly unlikely. We may then adopt the low-temperature limit only elastic scattering, and temlevels are and admit - / - ( y _ ~ -2 _ Vk j) 1'~, ~ l! -/L V.,, (6.5) The effect of internuclear vibration is therefore characterized entirely by an exponential factor, and in this approximation the same results applies in the case of _L. X When the vibrational and rotational thermal averages are evaluated independently as we have done here, the formulae derived in Chapter V can be used directly. The thermal average (6.1) becomes' I-' 1 Il)~ (6.6) t- I 9 (6.7)

91 where <(i,> is given by (5.50). The appropriate thermal averages can then be inserted into the expression for the differential cross section (2.27). Performing the indicated integration we find in discrete energy transfers as noted in Chapter III. The first term in ( 6.8) consists of contributions from direct and v; B — D<S l o^/iA - S The use of oscillator approximation for molecular translations results in discrete energy transfers as noted in Chapter III. The first term (\-9j') in (6.8) consists of contributions from direct ( Ot-J) and "inner" scatterings. The second term represents "outer" scattering (I jt )O and since it is purely elastic this term will not be considered in computing the energy distribution of scattered neutrons. An alternative to the oscillator description of the center-of-mass is the description in which the molecules move in a square well (see

92 Appendix C). In the limit of very large well width or high temperature where it is justified to replace the summation over initial states by an integral the translational thermal average is that appropriate to gas molecules, 1. I _'. r;, -V t I <'i e 5, I (6.9) the "outer" scattering being negligible. The differential cross section corresponding to free translations is then readily derived, Li/. d P\ ( C C i' J ^' -uc^^/ cycy (2\)~~ bi -K> c~~i 2~ b~".X O*'i (6.10) -) -2 -V i~~~i)Qv~~~-Q~~) ~~ C, C,. t 72- (<2Kr )I with C(-:- il. The distribution of energy transfer is seen to be continuous. In the calculations we will also use this expression as a comparison with Eq. (6.8). It is of some interest to note that the differential cross sections exhibited in Eqs. (6.8) and (6,10) both satisfy the detailed balance condition which can be stated as23

93 This property of the cross section is useful in neutron thermalization studies. THE MEASURED DIFFERENTIAL CROSS SECTION The measured differential cross section G(t @ ~ CJ ) is actually an appropriate average of (SC~ i lE C) since due to the inherent finite resolution associated with any energy-measuring device the initial and final neutron energies cannot be specified precisely. If we let \(,. and,.f j-a. be the energy distributions of the incident neutrons and the neutrons actually detected respectively then 5(&he D1, (6.11) We shall approximate the incident spectrum by a Gaussian, ARE:SZ~~;;4=4(6.12) where is the full width at half maximum and N is the normalization constant. The distribution B will be approximated by a step function so that ~f, -' for Ca-' -.. A. a + _-: and zero otherwise. Then we may write

94 f _ t-2 %;~ 2. ( -^^r^)^s~ Q G(6to>,N &sW>=9oGJ(G~aF d, U(6.13) For the following calculations we will use a scattering angle of 90~ in order to obtain maximum energy transfer and a value of 0.065 ev for G -o. In applying Eq. (6.8) the final energy space is divided into equal intervals of LEi, then all contributions to an interval are summed and assigned a value at the midpoint of the interval. The energy distribution thus appears in the form of a histogram. The energy distribution computed with Eq. (6.10) is necessarily a smooth function so the results are presented as a curve drawn through the discrete points of computation. To compare the results obtained with Eqs. (6.8) and (6.10) the distributions are normalized at the energy in the vicinity of the incident spectrum where the scattering intensity according to (6.8) is a maximum. SLOW-NEUTRON SCATTERING BY HYDROGEN CHLORIDE Among the hydrogen halides, HF, HC1, HBr and HI are in the order of increasing mass and decreasing permanent electric dipole moment. From the standpoint of obtaining maximum energy transfer and applying the strong field description of hindered rotations HF should be the most appropriate scattering system. But because of the practical problem of finding suitable substance to contain HF and of the availability of optical and thermodynamic studies of HC1 in the literature we have chosen HC1 to

95 perform sample calculations, Two parameters in the present description, the oscillation frequency of the centers of mass and the magnitude of the local electric field, are to be estimated. In the absence of Stockmayer potential parameters for HC1 the method developed in Chapter III is not applicable; we shall therefore follow an approach previously employed by Morales.15 On the basis of an oscillator description of the liquid an approximate expression has been derived relating the entropy of melting S to the ratio of oscillation frequencies in the solid and liquid states,49, 5 I =~S O A + (6014) where Tw, denotes the melting temperature. In this expression we have replaced the frequency of oscillation in the solid phase by an average frequency corresponding to a Debye temperature E3. Using the values51 T. = 162~K, %S = 476 cal/mol and lm= 128~K we obtain C = 1.112 x 1013 sec"-l This is the value of Z used in (6.8) to compute the energy distribution for HC1. The local electric field produced by the neighboring molecules, in principle, can be estimated from the known crystalline structure. On the other hand, in infrared studies of crystalline HC1 Hornig and Osberg52 observed a line corresponding to an energy of about 0,033 ev interpreted to be a torsional lattice vibration.* We will interpret this energy to corA weaker and oader boerand corresponding to 466 cml (, 0.0575 ev) has also been observed,

96 respond to the energy of hindered rotation, \j2. From the H-C1 separation of 1.2747A 53 we find = 1.2395A, b= 0.0352A, and the rotational constant B = 0.00131 ev. The dipole moment of HC1 in liquid phase (162188~K) is about 1.15 debye54 (1 debye = 1018 _) a slight increase over the gas-phase value of 1.0.55 In order that 92XA b' 0't3 Z\/ a field strength of about 5.77 x 105 dyne/cm will be used. This value is not unreasonable since in water the estimated field is of the same order of magnitude. The frequency of internuclear vibration in the ground state of HC1 has been reported at 5.65 x 104 sec1.3 This frequency corresponds to an energy of 0.535 ev so that at 1880K the population of the excited levels should be negligible. For 0.065 ev neutrons the assumption of no energy transfer with the vibrational mode is also justified. Thus our approximation of the vibrational thermal average is not expected to introduce any appreciable error. The appropriate nuclear scattering amplitudes of hydrogen and chlo2 2- 2 rine have been measured56 at LiT\t -, A_ 2 0a. 2 - and L4TiA = iQt in units of barn (1 barn = 1024 cm2). The scattering by hydrogen is seen to be predominantly incoherent, while the scattering by chlorine is mostly coherent, The energy distributions of neutrons scattered by HC1 at 188~K as predicted by the hindered and free translation descriptions are shown in Fig. 5o Since it is anticipated that such an experiment may be carried out here at The University of Michigan the incident spectrum used is a

97 1.4 HCL T = 188~K 1.2 8 = 90~ 1.0 - (I z 0.6 0.4 I'.02.04.06.08.10 i2 N \ I- \ o.0.; -r ----- ^f ENERGY (EV) Fig. 5. Energy distributions normalized at 0.062 ev of an initially "narrow" beam of neutrons scattered by hydrogen chloride according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown is the incident spectrum.

98 narrow Gaussian as indicated by the dotted curve where O- ~ 5-,, This spectrum represents the optimum that can be achieved with the crystal spectrometer in the Phoenix Memorial Laboratory.57 The result obtained according to Eq. (6.8) is presented in the form of a histogram at a uniform width of 0.002 ev, a value of LE that may be somewhat optimistic even for high-resolution measurements. The incident spectrum is sufficiently narrow so the hindered translation result shows considerable structure. Both distributions indicate that the two peaks associated with hindered rotation scattering should be easily measurable; consequently, an experiment measuring the intensity of scattering in these vicinities is strongly suggested as a test of the proposed model. An identical calculation using a broad ( r, O.IC-o ) Gaussian incident spectrum shows little or no evidence of the structure due to hindered translational motions, a result not at all surprising since there now exists considerable overlap of contributions from the various direct scattering terms in (6.8). The contribution from "inner" scattering is found to be less than 1% and hence is ignored. SLOW-NEUTRON SCATTERING BY WATER In calculating the differential cross section for water we introduce an approximation in which the asymmetric water molecule is replaced by a fictitious diatomic molecule of equivalent mass and electric dipole moment ( J = l.8 debye55). This approximation greatly simpli

99 fies the calculation of rotational thermal average and does not alter any essential feature of hindered rotations. Although the appropriate moment of inertia to be used in computing the rotational constant is now somewhat ambiguous, B can be expected to be of the order of 10-3 ev. The internuclear vibrational frequency of this diatomic molecule is taken to correspond to the first measured vibrational level, -Q 3.1Q_ X'$ -1 58 0o %( sec.5 Using an 0-H distance of 0.96A and H-0-H angle of 105~59 we have estimated b4, to be 0.905A. In obtaining the oscillation frequency (I, the Stockmayer potential for water33 gives a value for ) v 0.023, thereby indicating that the potential \/(A) is not parabolic. Nevertheless we shall proceed to approximate the actual\ (A) by a parabola. If we apply Eq. (6.14) using Do 250~K60 and 3 ~ 1400 cal/mol61 then IO) - 1.5 x 1013 sec-1. According to recent calculations of Singwi and Sjolander18 and highresolution neutron experiments13 the Debye temperature for water, assuming such a concept is meaningful, is about 135 ~Ko The frequency appropriate to the present hindered translation description is then three fourth of the frequency corresponding to this value of G, or CO C 1.3 x 1013 sec- We shall use this latter value of l) in the calculations. An attempt is made to estimate the magnitude of the local electric field by directly computing the field produced at the central molecule which is surrounded by four dipoles at the corners of a tetrahedron. The orientations ofthe dipoles are chosen to the are he extent that there is a hydrogen between two oxygens, otherwise they are arbitrary. The value

100 of E computed in this manner is about 3 x 105 dyne/cm, thus giving \ as, 0.35 ev which may be compared to the height of a hindering potential of r 0.7 ev from an early calculation by Magat.22 Since the rotational constant B is not known in the diatomic approximation we shall choose B such that \QA- corresponds roughly to the observed energy of hindered rotation N 0.06 ev,12,15,19 then F - 5o3 x 10-3 ev. We have mentioned that scattering by hydrogen is mainly incoherent. The scattering by oxygen however is entirely coherent 56.ry -llW't barns. On the basis of the scattering amplitudes, hydrogen effects constitute about 95% of total scattering so we will ignore the small contribution from oxygen in the following calculations. In order to determine the nature of translational motions in water we compute the energy distributions according to the oscillator and free gas descriptions of the center-of-mass degrees of freedom and neglect, for the moment, the effects of internal degrees of freedom. The two distributions are compared in Fig, 6 where the experimental points of Brockhouse14 are also shown. The dotted curve represents a relatively broad incident spectrum ( 0. O\5eo ) corresponding to the actual spectrum employed in the experiment. We observe that the two descriptions give essentially the same distribution, The discrete structure associated with the hindered motions is almost completely smeared into a smooth distribution by virtue of the relatively large spread of the incident spectrum which causes considerable overlap of different transitions. On the basis of this comparison we can conclude that measure

0. 8 L vy z w o 0.6 - W 0,, 0! 0 0.4 - z I' I'I) 0.2 ----- i \ -~ 0.02.04.06.08.10.12.14.16 ENERGY(EV) Fig. 6. Energy distributions normalized at 0.058 ev of an initially "broad" beam of neutrons scattered by water (center-of-mass degrees of freedom only) according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown are the experimental points14 normalized at 0.052 ev and the incident spectrum. points omlz a.5 a t icdn pcr.

102 ments with the present spectrum are not sensitive to details of molecular translations and therefore cannot yield any information regarding any possible hindrances which may exist in water. This observation also explains why gas-like descriptions have been used with success in computing neutron distributions in reactors. The peaks of the two distributions in Fig. 6 are seen to be shifted to a lower energy. The observation that the broadening predicted by the hindered translation model is slightly smaller can be explained by the fact that a hindered molecule develops a thermal cloud of smaller extent during the scattering process than the free molecule. Except for the regions of large energy transfer both models give reasonable fit to the data thus indicating that the differential cross section for small energy transfer is influenced mainly by interactions with the molecule as a point masso A similar comparison is given in Fig. 7 in which the narrow incident spectrum of Fig. 5 is used. In this case the discrete structure associated with the oscillator model is clearly observable. There is essentially no shift of the elastic peak in the hindered translation distribution where the intensity of low order (n) inelastic scattering is greater when neutron loses energy than when energy is gained. The attenuation of the higher order processes is not the same for energy gain and loss, hence the distribution becomes asymmetrical as is apparent in Fig. 6. It is noted in Fig. 7 that transitions for which w >5 need not be considered since their intensities are of the order 1% or less compared to the intensity of elastic scattering.

103 H20 8=90~ > 0.8 CE 0 bLJ 0 bCL~~~~~~~~J ~ </~~ ~ \.02.04.06.08.10.12 ENERGY (EV) Fig. 7. Energy distributions normalized at 0.064 ev of an initially "narrow" beam of neutrons scattered by water (center-of-mass degrees of freedom only) according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown is the incident spectrum.

The energy distributions for water calculated according to Eqso (6.8) and (6.10) are shown in Fig. 8 along with the measurements of Brockhouse and the theoretical curve (dotted line) of Nelkin. The effects of zero point vibration and hindered rotation are readily observed by comparing these results with those in Fig. 6, As can be expected with a broad incident spectrum little difference exists between free and hindered translation descriptions. A value of E = 4 x 105 dyne/cm is used in the computation which corresponds to \2.\ = 0.069 ev. It is observed that within the large experimental uncertainties the theoretical curves are in reasonable agreement with the measurements. The essential difference between Nelkin's theory and the present treatment manifests mainly in the predicted intensity of scattering associated with hindered rotations, the calculation in this investigation showing a considerably more pronounced effect than Nelkin's distribution. Because of experimental uncertainties and the scarcity of data in the region around 0.12 ev it is not possible to arrive at any definite conclusion regarding relative accuracy of the two models. In cold-neutron ( o- 00.005 ev) measurements of water12,13 the observed intensity of the hindered rotation peak is about twice that at the incident energies. The intensity calculated according to (6.8) is found to be low by a factor of' 10. The discrepancy may indicate that symmetry effects are important in this case, hence the breakdown of the diatomic approximation, On the basis of the comparison in Fig. 8, it

z HwZ OG0-'6 ^ ^ <>1 \ 0.2 ENERGY (EV) Fig. 8. Energy distributions of an initially "broad" beam of neutrons scattered by water according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown are Nelkin's calculation23 (dotted curve), the experimental points14 normalized at 0.052 ev, and the incident spectrum. A theoretical curves are normalized at 58 ev. I \ 0 / \ 0.02.04.06.08.10.12.14. 16 ENERGY (EV) Fig. 8. Energy distributions of an initially "broad" beam of neutrons scattered by water according to descriptions of hindered (histogram) and free (solid curve) translations. Also shown are Nelkin's calculation23 (dotted curve), the experimental points14 normalized at 0.052 ev, and the incident spectrum. All theoretical curves are normalized at 0.058 ev.

106 is likely that even lower intensity will be predicted by Nelkin's model although such a calculation has not been reported. In these experiments a smaller peak corresponding to energy transfer of ev 0.008 ev is also observed. The position of this peak is predicted by the present model as the primary hindered translation peak although the calculated intensity is lower than the observed value by a factor - 2.

CHAPTER VII CONCLUDING REMARKS By considering the effects of intermolecular forces on molecular translation and rotation the model proposed in this investigation reduces the description of slow-neutron scattering to a consideration of two parameters which characterize the scattering system, a frequency of oscillation of the center of mass and a local electric field which hinders rotation. The oscillation frequency arises from a parabolic approximation to an effective potential derived on the basis of assigning the neighboring molecules to a spherical distribution and choosing an appropriate two-particle interaction, Such a potential will always tend to confine the central molecule to one, or at most two, small local regions, The validity of the parabolic approximation then depends on the fact that the localized region of minimum interaction be necessarily at the center of the distribution. This sufficient condition can be stated essentially in terms of the ratio of the distance of closest approach, a parameter appropriate to the two-particle potential, and the radius of the spherical distribution. When the ratio is of order unity repulsive forces dominate in the interaction between the central molecule and its symmetrically-located neighbors in which case a justification for the use of an oscillator description of translational motions is obtainedo 107

108 The polar nature of the molecules can be taken into account in estimating this oscillation frequency. It is found that the specific effect of dipole-dipole interaction suitably averaged over orientations produces attraction among molecules, and hence leads to a more stringent condition for the oscillator approximation, There is no aposteriori reason for choosing a spherical distribution for the neighbors; in fact, the orientation-dependent dipole-dipole effects suggest that the tendency for molecules to associate is probably greater in some directions than in others. The results of smearing the neighbors according to a prolate or oblate spheroidal distribution are complicated but are simplified considerably in. the case of small deformation from spherical symmetry. The oscillator frequency estimated in this manner is now directionally dependent and contains the additional parameter characterizing the deformation. The estimation of an oscillator frequency according to the smearing procedure is a useful method provided that in the case where the sufficient condition is not satisfied a parabola can still be found by numerical means which is a reasonable fit to the calculated effective potential. This method is, nevertheless, limited since the necessary parameters of the two-particle interaction may not be known for a specific liquid. On the other hand, such a frequency may also be estimated from a simple expression which requires only a knowledge of the entropy of melting, the melting temperature, and a characteristic frequency of oscillation in the corresponding solid, This somewhat less direct ap

109 proach has a much wider range of applicability and is found to be satisfactory in the cases of lead and water. A means by which the rotational motions of polar molecules are hindered is obtained by assigning the hindrance to purely electrostatic effects, specifically, the coupling between the dipole moment and a uniform and constant electric field. In principle, the magnitude of the field is expected to be roughly the same as the crystalline field in the corresponding solid. In practice, the field estimated from the structure of ice is found to be of the proper order of magnitude expected for water on the basis of measurements. The particular type of hindrance considered here affects only one angular displacement, so there will still be free rotational motions associated with the other two rotational degrees of freedom of the molecule. The present description of hindered rotation leads to useful wave functions for symmetric and linear molecules under the condition of strong coupling. The energies associated with hindered rotation of both types of molecules are those of a two-dimensional isotropic oscillator with level spacing 2\@, where A and 3 are the coupling and rotational constants respectively. This fact is actually to be expected since for small angular displacements any molecule behaves like a torsional oscillator. The energies of this oscillator can then depend on system symmetries only through its level spacing, hence the definition of (s. This observation implies that the hindered rotational energy spectrum of an asymmetric molecule should also consist of an equally

110 spaced set of levels. It then appears that in the sense of similar energy spectrum the effects of hindered rotation on neutron scattering by an asymmetric molecule may be approximated by those calculated using the wave functions of the symmetric, or even the diatomic, molecule. However, the asymmetry which is ignored in such an approximation may have significant influence on the scattering intensity in which case substantial error can be introduced in the calculated energy distribution of the scattered neutrons. The condition of strong coupling requires that the mean square of the angular displacement between the dipole moment and the field be small. For this reason, only a subset of the totality of the available eigenstates corresponds to physically meaningful solutions and, in principle, only these states are involved in the scattering process, Using the present descriptions of molecular translation and rotation the differential cross section for slow-neutron scattering can be calculated without introducing further approximations. As a consequence of the assumption of no dynamical coupling between molecules inelastic processes are possible only by direct and "inner" scatterings. These effects therefore determine the energy distribution of the scattered neutrons. The purely elastic "outer" effect, which can be important in considerations of angular distributions, has not been dealt with to any appreciable extent in this investigation. In the parabolic approximation the scattering by the center-of-mass degrees of freedom is described by the well-known results of neutron

11l scattering by a system of independent oscillatorso The method employed in treating the scattering by rotational degrees of freedom can be applied to both free and hindered molecules. In the former case the evaluation of the matrix elements is considerably facilitated by making use of known symmetry properties of the wave functions. The results obtained are particularly useful when the incident neutron energy is sufficiently low so that only the lowest order excitations contribute significantly to the cross section. In the case of hindered rotators consideration of the integrals involved leads to a separation of the different contributions to the scattering in powers of (i2x) o The zeroth-order terms in the cross section therefore consist of elastic processes and inelastic processes involving only changes in free-rotation energies. The firstorder terms consist of transitions involving the exchange of \ in the presence of free rotation excitations. These latter terms can be interpreted to give rise to the hindered rotation transitions observed in neutron as well as optical measurements. Higher-order terms in the cross section correspond to excitations involving multiples of 92XA and also free-rotation energies; these are of much lower intensity and may be ignored (for water the second-order terms are of an order of magnitude lower in intensity as compared to the first-order terms), When the small free-rotation energies are ignored the total scattering consists of only elastic and ~ -QANB inelastic contributions and the cross section can be exhibited in a particularly simple form, The fact that only a limited number of the hindered rotator eigenstates

112 can be used does not impose any significant restriction on the calculation, This is because in averaging over initial states the summation? 4+ involves terms in powers of Q which is a small quantity ( @ 8 and 20 for HC1 and H20 respectively), and hence the inclusion of all powers of e introduces little error. The expression for the cross section may be compared to Nelkin's model of hindered rotation.23 The present treatment is seen to be less empirical in that all the rotational degrees of freedom are considered and the polar nature of the molecule is explicitly taken intc account. In addition to the parameters which characterize the particular model the difference in the two descriptions manifests itself mainly in intensity factors. It is recommended that the model proposed in this investigation be subjected to thorough experimental test. The calculations for hydrogen chloride and water both indicate that effects of hindered rotational motions can be observed by measuring the energy distribution of scattered neutrons. On the other hand) relatively high-resolution experiments are necessary in order to resolve the distribution in the region of small energy transfer for structure associated with hindered translational motions. Results of the water calculation are in good agreement with lowresolution measurements.4 However, the experiment is not sufficiently precise as to provide a basis for quantitive comparison of Nelkin's model and the description presented here, even though the two calcula tions deviate significantly in the region where hindered rotation ef

fects are prominent. In view of the calculations a number of scattering experiments are suggested. It will be desirable to repeat Brockhouse's water experiment to examine with greater precision the region where prominent hindered rotation excitation rv0.06 ev is predicted. If sufficiently good resolution (, 5% resolution) can be obtained the region of small energy transfer should be studied to determine if any structure exists in the distribution. In this respect, it is perhaps more appropriate to use incident neutrons of lower energy than - 0.065 ev. Similar types of measurements are also highly recommended for hydrogen chloride. If energy exchanges -- 0~035 ev of appreciable intensity are indeed observed then the neutron method promises to be a very powerful tool for the study of hindered rotations in polar liquids, particularly those liquids, such as hydrogen flouride, which may not be suitable for optical studies. The present investigation provides a basis which can be modified and extended in subsequent work, The hindered rotation descriptions can be refined by introducing temporal and spatial dependences in the local electric field, the latter effect thus leading to directional correlation between moleculeso Some consideration should be given to a description in which all three rotational degrees of freedom are hindered for the analysis of neutron scattering by systems, such as perhaps ammonium halides, which do not seem to exhibit any free rotations. Thus far Nelkin's model has been used to study neutron thermalization and diffusion in water. It is suggested that some of the calculations be

114 repeated using the results derived in this investigation. Since the quantities to be calculated are integral measurements of the differential cross section, no major deviation in the two models can be expected; nevertheless, second-order differences can prove to be very interesting and informative. Only qualitative agreement is obtained with the coldneutron water data 12'l where the calculated intensities in the diatomic approximation are considerably lower than the measured values, It appears that a more refined computation, where the molecular symmetries are treated more appropriately, is required to determine if the present model is adequate in obtaining quantitative agreement.

APPENDIX A A NON-SPHERICAL LENNARD-JONES-DEVONSHIRE "SMEARING" PROCEDURE We consider the derivation of an effective potential for molecular translational degrees of freedom in which the distribution of neighbors about the "equilibrium position" of the central molecule is assumed to be non-spherical. By taking all molecular dipoles to be along the same direction, say the z axis, the effective potential is given by (3.14), V=|d^ P(< \(.,'r (A.1) UVf) = ^ A r3- 5 12 G with L=4-6- b=.U' The surface of distribution will be taken to be a spheroid for which we introduce ellipsoidal coordinates,51 p b ic cam c. on The probability of finding a molecule on a given surface {( = 0o, a constant) becomes -1 In the present approach, a direction, that of the dipoles, has been defined for the problem, therefore it is convenient to exhibit V in terms of its components, VZ and Vx, corresponding to the cases in which displacements 115

of the central molecule are parallel and perpendicular to the dipole orientation respectively. We find 1r~ ~ ~ i^2~~~~ ~ (A.5) where > 2- 1/2 following approximation. For small displacements, inie a r tbu (A.l4). yz where indices i,,j represent different components of the vector and repeated indices are to be summede After some manipulation it can be shown that j ^3 )( ) -) = ~ O for all j 1u)

117 and sat~~~F~~~i~C~~3CI ~ ~ l~iAS, tB(A.5) In this notation denotes the or component and the X component only; moreover do12 does not imply summation over. We note that if \.Z>C> then V may be approximated by an oscillator potential with frequency in the X and t directions different from that in the i direction. Explicit expressions for (A.5) have been obtained after a great deal of manipulations. We shall merely record the results here, |p X=otel andi thX o d (A.6) where we have separated the contributions from the Lennard-Jones 6-12 potential and those from dipole-dipole interactions. *t_0>Lz t\ L(2\ - Nf -M1 p 110 z - ~; + l i ~ - l ((24 w- (A.7) ~+ ^ ^8 jM~t(v-^ ^ ^"^7-7^ D - i31e)(A —A 4

cO (n -f)... -!c,-_ L? —rI L - -C- e,/^ -f -- 5 0 + ^^ ^- 4, i t -n'^ ~-l 1 c| rat"L L CY. ^^ — -i- LII ~JI,col 5LL oi fS- * JJ V-, [ f N — L. f-'^ S S'\I i~ ZS 5~0 -t4r/C f 113 i:D, i~ j` nz -tor3,

119 where * _ in the case o sall deoration ro s rical s rin F- - C~i C; other hand, if the elongation is in the x and y directions (oblate other hand, if the elongation is are always negative, becoming more so spheroid ) then the coefficients are always negative, becoming more so

120 for larger:. The second case is believed to be more reasonable since for the present choice of dipole orientations the interaction is more attractive along the z direction than the x and y directions. According to the approach described in Chapter III the effective potential for water does not have a minimum around the origin since ( ) (see Figure 3).

APPENDIX B "CREATION" AND "ANNIHILATION" OPERATORS tA -;tA In Chapter III the time-dependent position operator, ()(t)= C t 6 was introduced in (3.16). When the Hamiltonian H is that of a system of independent oscillators it is most convenient to express Lk[(t) in terms of certain operators of which the matrix elements in any eigenstate of the oscillator are trivially obtained. As defined U(t) obviously satisfies the Heisenberg equation of motion c do)_= t _u{t), [a[ (B.I) where the is omitted because here is actually the conventional t/l Since the oscillator is isotropic we need only to consider one of its cartesian components, 2 2 k-2( (B.2) and we have the usual commutation rule t = According to (B.1), These results combine to give an equation for lU(), namely,,( t)=-l l(t) Thus Constant -i and can be determinedt -we find, for uG^)= a e it+. p d= - (e - ) /(B.3) Constants C and? can be determined from W(O) and P(`); we find, for 121

122 instance, t - (c- AC t t (B.4) which is in fact the result one would obtain for a classical harmonic oscillator. ittV) 6 \ t -- tt + t Ct-l A u -~t) jl-,C^ -.] 1 (B.5) mutation relation for and saotisfy fC a) 0 a- =t -sy j moreover, since A and p are Hermitian =-, It is convenient to replace the dynamical variables! and p by so-called "creation" and "annihilation" operators _ _ _ \_- 0 ( 6(B.6 ) wAcA acZ eo a a Ce t me r -nCepenCent counterparts wnen 7 (3. Yihls choice produces the commutation relation La(t ] t(4,).-_- L(~ ^(6 | I

123 In terms of these operators, ^( M \ (! \ ) (B.T) and similarly for A[0). The effect of the operation of 0, or on an oscillator eigenstate i can be found by the following. The Hamiltonian becomes 5- ~,. 7> \. u. \,....... —-.,. -..,,. G I. -:? < t) t (B.8) which is also valid at \j/'." since H itself is time-independent. We can write (B.8) as (H/ 2,) - 0i o^ doB, ^'- - - a(o)a () ) Itt _V. (a ( o ( V/-i- / ) I or ~'o a(b V> -- (^+ C)A > o which shows that (tVl> must be p Similarly we find CO()\/b to be p To determine A and B, we simply c <n a()( O V//- \-I' Hence, a(o4\> - ^ -i aotcu-^'l l(It^ iroportional to t\t()'v>t ->?roportional to \jV-l>, or o (ro6\K\:!\\v>Q>:onsider K, <iV\A(3o(o( l> kV1., (t% u> — sn 1-'i)

124 This shows clearly why a\ and ( are called "creation" and "annihilation" operators.

APPENDIX C NEUTRON SCATTERING ACCORDING TO THE SQUARE-WELL APPROXIMATION In this appendix we shall consider the thermal average <e e > /T appropriate to molecules each moving in a three-dimensional square well. Since the well width is taken to be the same in all three directions we only need to treat one component in the calculation. The molecular problem is simply | id- FC 823~ ~ 41(\;n}_~ &(C.1) where we have defined 5= A QA being the width of the well, and z 2 Wa A. *The origin is chosen at - a with, o. By taking the barrier to be infinite at t — c2A the solutions to (C.1) are 4 A) X 4t i>\- gg9-ffi5~ ~ (C.2) where Q, - VTh/2 The required matrix elements can be readily computed. For direct scattering, Iwelt) — a, a inte ge, th whele < S. = aCm i net )g - n (2;' - tCw >v/4 (t -f-,-^ /4 a If we let -Y\t'+S, S an integer, then 125

126 it it -; (c.4) with' 4Z ( 2t Hence the thermal average becomes, in the case of direct scattering, - n,-> (c.5) ~ / 1<^|6 l;t For interference scattering,'')'\Wt' -- /k-,-' - ~ "wl M') <i b <~=Q _ ) 1f 1<"1&~(ntZ i>\ (c.6) In an actual calculation, the required thermal average is a product of three terms like (C.5) or (C.6) since we have only considered one component here.

APPENDIX D ANGULAR MOMENTUM OPERATORS AND HAMILTONIAN FOR RIGID MOLECULES The orientation of a body in space can be specified by the Eulerian angles (I 4., By this we mean that the set of axes fixed in space and another set fixed in the body can be brought into coincidence by three successive rotations; explicitly, where and are respectively the space and body axes. Since a rotation is an orthogonal transformation then matrix $ic) is orthogonal, i.e., XQ, [2 _ \=Qi(I"). When the body is undergoing arbitrary rotation, rotation about any axis gives rise to an angular velocity 3 along that axis, and for infinitesimal rotation the angular momentum generated may be treated like vectors.63 The angular velocity 0 for the arbitrary rotation can be expressed in terms of the Eulerian angles, where the three rotations are about -3 and __ respectively (see Figure 4). Expressing the angular velocity either in the space system or the body system we have 127

128 1 I 2 ( + ~~~ i +' (D'2) where these relations will be used in deriving angular momentum operators. Consider an infinitesimal rotation of a scalar field () about the space axis e by an amount 0i, (^`- 1^(l - ( (D.e) the change tin,due to the rotation being a Since t is small the expansion is appropriate. Now define a generator of infinitesimal rotation about II~. In this rotation the summation over repeated indices is implied. By using Em as48 -l >1^ 1 / In this rotation the summation over repeated indices is implied. By using u instead of (.) (D.4) yields i a generator of infinitesimal rotation about the body axis. The operator Li is actually related to the / angular momentum operator t< by merely adding a factor " ==,i 1' i'';

129 ANGULAR MOMENTUM OPERATORS IN "SPACE" SYSTEM We may regard (Dol) as a matrix equation 0j-i $ from which we need to determine -, -'. In the present order of rotations ned i /CZ o -5S \i C, iS \ R1(= m I < -S, C, 0 ) Sz o C2 \ 0 / C2C cs C -s S25S Cl / ) where CowCti S-sA^G;. From (D.1) matrix B is seen to be - L J -* ~~~~~~ O sI 0 s-rC, sz`51 and inverse* 0 -.S( Sz S2. Cl I1 0 o Hence the angular momentum operators in the space system are:** (^\to ),'P I Q~vi -41A I with n: E 2. S2 I -S, CI 0 Cl S2\ Si S_2. 0 0 / ANGULAR MOMENTUM OPERATORS IN "BODY" SYSTEM We could follow the same procedure in obtaining expressions for ~. 1 * E9 =( t/t( ), where t is the adjoint matrix. **This method of deriving t. has been used by Osborn and Klema. ~I

130 However, once Q is determined a more direct method is available. Note that where rotation matrix R is given by Q7 = = Q(BQi~ / C2CC -3 I ( -S3C2CI C3SI SzCi -S \2 C a Thus i- i~Q I ~VI ) with C.-C -R II 0 C2S3 S2 1. 48= ~ according to Figure 4 Summary: If we designate ~1 At_ ) then

131 R.= He - < i { W^- The preceding formulae for / are in agreement with Eq. (II.29) of 39 Nielsen. The present Eulerian angles are defined with respect to a 11/ right-hand coordinate system and the second rotation is about the 2 axis rather than the ( axis as used by Goldstein,63 Margenau and Murphy, etc. If we follow Van Winter4 who has adopted the convention of using the t axis for the second rotation, then I/ which can be obtained from the previous results by letting G — (>'-'/ THE HAMILTONIAN The Hamiltonian is given by (4.2), By straightforward differentia

132 / tion (using ) we obtain - > i... -'z i (ii )\ ^ -:~?)'''' r i t..... (D.5) where i n iic,8"I < —)- 1 j lj)1:+Co0f-~~ - -- — p CC + i - 2 1~~~ Ili\ A, C)~~CUYI-~ 7-:1:1" —, -7,:';* (. 1 c:; i' k~~~~~1 12I > - + SV4 —."8a -11.4%6`~. 2; ~3 LAA2- 6c= "I —; 1~~~~~~ ec 4 ti In the literature the sign of b often appears as minus4 and is due to the use of axis instead of the use of I/, axis instead of ^^;,it

APPENDIX E THE ASYMMETRIC MOLECULE IN AN ELECTRIC FIELD We record here a procedure whereby, at least in principle, rigorous solutions or solutions to any order of accuracy to the problem of the hindered asymmetrical rotator can be obtained. The method is a straightforward generalization of the formal approach described in Chapter V, and, as before, involves considerable numerical efforts in any application. The eigenvalue problem is C RH tod ^- )\I11, (E.1) where ris the free-rotator Hamiltonian given by (4.1). From Chapter IV recall the solution Az TEM> = E T EM T=-: Now we seek solutions to (E.1) of the form.,\'-. )Ti'\)>, (E.2) T=O Ean= since M remainss as good quantum number, is used to replace J and E. Henceforth we shall suppress index M. (E.l) then reads`533'4'-6^ ^-E3Et(5a ~i<5'='~0TyE0_ O (E.3) (E E- ) =< -T0 133

where < C'> =lK,(fo > ) t K K'E -bD cc)K' (E.3) becomes o(S K =/ — Qg4 E ) 0 (E.4) T21 2) S, CB / Jt/ \)(\5ti )cJJSE = ~ ) v2(31i,;)' E ^=^1:y)&&<(0iDC E'o where _ ) ^^E )- - ^te^E^^^C^.(-5) In (E.5) K ranges from -N to N, N being J or J whichever is the smaller. For instance, 2 The first equation in (E.4) corresponds to J = 0 where ) has to be 1, and the second equation is for J I, We can simplify the notations somewhat by introducing a single index q for J and E. The following correspondence is noted: N A; = o C >424 0 — 4+ = 123 4,'' NZ C(N+2)

135 When J = N, the highest that V can assume is 2 - Therefore in this group V starts at (0-)(-)+1 N and runs through the 2N+\ values of E up to N(N2). It is necessary to devise a scheme by which a given ), say l- X can be related uniquely to a set of (J,E). Consider the test: (1) If X= N then the corresponding (2E) is A= K ET) 0 Z2 (2) If X-* N, then find that N such that N < X < (Ntl) and the corresponding (TE) is = K N X -, With this substitution (E.4) becomes tp-^-)(K 2-x L W ^|t-#)X(|1 b O U (E.6) 1 ^ where it being understood that N i 9 4(&H-! Equation (E.6) is in the same form as (54) so e and p[f) can be formally expressed in similar continued-fraction forms as (5o10) and (5.11).

APPENDIX F THE INTEGRAL ie. tP i Jn1) We shall evaluate the integral r t S a = - x x I 0~~ p -Vc+Ox p j -ip A II by using the generating function of associate Laguerre polynomial -,.. ". i -*t t) i" -/ " —,X — t ) AII \'- 0 Consider n-5 _t" rm A ~ ^ *P-/V +. 1t&tm1 1) tu.iq) AA-A(' \\i L I x d x. 0 (F.1) - /~ j. C 9 + where Q -. t,-'i - i.- ~ _We next write The integral in (F.1) is simply p! ir A i - I P+lI Y S1;~~~~ -~ -~ so i [ - t. -t 0- ^i - -([^t -- xUOl-t5(l-V%\ 1 (F.2) The following expansions are useful: --.-).. C [ - iC-t - -t (- X A- I. (-t. j -— __. _t r a I ( -t)( -- t), 136

137 Q- "a dU r 4 tj ~(t- i ==) - S)t y-dy. d! * —tX)-) i Thus I - iJt-(g+lcfl-tYI-)^a d p (IA) - -t) ( )(Q) ) Since j and q can be positive, negative or zero we need to consider three cases separately. (1) and q either positive or zero _ 0t —(!' ( x) From this we want coefficient t so we set k — =- C —/ thus implying - O(3~ Y<_ Now we obtain

138 sc= (9 ) <Pt<)! <ik( -rF) (q+d)' ) X (p~qa)= t C^^ ^pql (ec-i)l L!(4-dM') ^!(-d)!(it^s- c 9(tf-S (F,5) with a < hmly \ lt I and C > lmX ( x'^- ( When 4JL=O, only non-vanishing term is C =d;with max. (! ) < 6 ( with max./~"~-f m/s4-^\ (2) negative, positive or zero In this case ) q~ ( a ~ (S-0)~ ( T G —~) I',(I ~ 6- ~, (F.4) f c, (-) O d r-a) ( Cl~-t %I (rt) if ~ ~ 1t ~ We can still expand -\A m3E. f -. O Thus, —iI~` V.)1 -kl\ 21A I. - <.) tt 1= ^')! (Ct p- )! (F.5) S-d fiscd, fl, ~ d -', a (p ts-)'.a) t, (i i -- i ) di"' r \i~~~~~~. ~~, -' I 1.-7 i' t- g1, C I: ( iLq i V. 3 L Y~.

139 Again when.0X, =cik' then only the term with = — o-G survives in the summation. Also note symmetry in ( i ) and (.l\). (3) 4 and Q negative Similar procedure leads to S;I -" I 4 -,9)I t-^ -^ ^ t — W ^~~~~ (F.6) _ -d =f -A A ~y 4 = Bs-. (Y = S!-' j - CA( ~-8 )! ( 6 A ~ ( I i A- -1! C ( ( ( + l ) When W= 0 set (I — (= 6-< =T) (.

APPENDIX G THE HINDERED ROTATOR-OSCILLATOR APPROXIMATION* We shall consider an approximation in which Eq. (5.13) can be reduced to the oscillator equation. Equation (5o13) is Id + co 6:I d 4 2 (M- o - Xl+ AToso + IU I J -,= zli~v\^V^o (5.135) x where A -=- eliminated by t with E TsI, The first-order derivative can be the transformation (-(- = C()v- ( ) CX k Y0 () Then z d I ff ) (t+ 4) ~ 62. ei 4 O Q(e - C (G.1) - +(M - Ycx^B -4 - X CG - + V(^) For \i>1 and Mv K, efR(' has a minimum at e=:(W which is near -= O. We can therefore expandV L8) in a Taylor series, e f where by definiti r\11 419 1 "A -F 0'-,7- W,, 4 -1^ _,'dn t abA6 (G.2) It is found that for small I -: \ -. 0 * It is found that for small 6 6(& A. O. ^, / P -_ J<' - /u / X 4-! (B~is So v,($ _-A' ^ _'-44_ ~~~~~~~~~~~~~~~ / I\ \/eq ) t~1 -! - ~ I2 ~, + k (M-^ V- + (k- // j) 2 ( d( # o\t~\/s a;di rY, + (:z2tV *We thank Professor K. Hecht for recently been used by Maker.65 a discussion of this approach which has 14o

141 We have kept terms independent of XI in fl(9 because they represent the energies of free rotation which are of some interest. On the other hand, the leading terms which are ignored in (,t when multiplied by a term a, As, are of the same order, but they are neglected becuase their presence introduce undesirable coupling effects between the different degrees of freedom characterized by q J) ) and O e In the present approximation (Gol) thus becomes an oscillator equation. The energy eigenvalues are i~"S,<, -,\t 16' (2e.+\; (k\-kS5-, + \) t 6i1\ (G. 3) For l\< we can ignore the 1/4 so (G.3) agrees with (519) to within a constant. The result (Go3) actually is not valid for M = K in which case cV(j) has no true minimum. The reason for this behavior may be seen from the fact that J=0 is a simple pole in the differential equation and physically the molecule can never be in the position B- 0O ('\ and A colinear) unless M = K. It is therefore not surprising that the M = K case leads to difficulty in the present approximation. If we insist on ignoring the!/4lv2g term and proceed with the oscillator approximation then essentially we are neglecting effects of irregular behavior near the origin whenever M = K. For large \ this irregularity is confined to a very small region about - =6 hence the oscillator approach may still be useful in cases where one is primarily interested in the over-all behavior of the wave function.

142 The eigenfunctions of the hindered rotator are -. -. t. il! H e' -1-/,..... 77 >- )(my d)C 4 e s Id) (G.4) where -k is the Hermite polynomial of order f. In this case, has the range (a9-CO ) )

APPENDIX H EFFECTS OF FREE ROTATION IN NEUTRON SCATTERING BY HINDERED ROTATORS In this appendix we shall examine the influence of free-rotation energies upon the scattered neutron energy distribution. It has been 1i shown in Chapter V that to order [. in the cross section only the cases Y-\- A and W\= /1 need to be considered. For these cases the selection rules for! as given by >g> show that ^=\= corresponds to no excitation of the hindered modes, therefore when free-rotation energies are considered this particular case will give rise to energy transfer involving terms proportional to B. On the other hand, the cases M- t' correspond to "one-quantum" excitations, and in the presence of freerotation excitations the amount of energy exchanged will not be precisely \I\2 thus resulting in a broadening of the inelastic peaks at:=- a Cii_\ s. Since the two effects just mentioned arise from different relative values of m and k they can be discussed separately. (1) SMALL ENERGY TRANSFER Z. In (5042) (2 F) is given as a sum of three terms, the first of which is <), C O6) The contribution of this term in (5.41) will be denoted as, t( C<-) w(H.1) where now E is given by (5.18), Stl<M

l44 I~~~y(2~i-R BI~~~~~(Mt rri~~~~~~~~~i-i~~~ (H.2) Thus elastic scattering is obtained only when k = 0. It is interesting to note that in the case of linear molecules all terms vanish except those with k = 0 [see (5.50)] and so no small energy transfer is possible. The corresponding thermal average may be written as 0 T _o K iV\ A ~,r< ( b T (H.3) with V I tE(_s) Is<jE %!~~~K ~ ~ ti)g _ (e\2l\ (Y+!l-) + 3FSK: i ) 3 e In (Ho3) the summand actually consists of two terms corresponding to upper and lower signs of n, and similarly in. The elastic term is still 0( W ).I-by,)

145 (2) LARGE ENERGY TRANSFER The preceding discussion shows how free rotational motions can lead to low-energy inelastic scattering. Now we shall consider the effects of free rotations on large energy transfer processeso The inelastic terms in (5.45) and (5.46) become \, 1/ ( i? m), I, -- -- -y~~~~~~~~~~~~~~~~~~~~~~~~~~~ i^+ kh ^ V 1 j (ji. k-,, b A i \' (t si) (t i i)(l-lc2e`t- S'e l' )\~ (-~; ) -.-.^ (.1 qt9 c a-yi i-)T,, c,,^, i 1"~~,, - i - ^ - - ^s s i ^i _.\ i E. (-^i) JiF_ K( t) J Ai-O (T i -,. c4 * i+ — 4 --—.^0i^ - eS^. f- (2, Wi t<^ m ",-bI~~~t ll-'/l~~~~~~~~~~~~ ~~~~~~~~~ ~F71 _> i/^ c i~( k 4I -I + - f _ c ve -I tc (i-2)(t~~k+1) i~t l-)'C-| /fln \/(1 I \ - ^E; lo l) A, Z., (,-, —+ " 7 -^ (C,- 1) " ~ ^ ~ ~ ~ ~ ----- Z [' S t tly S. L^ ^, L-' /^ ^~~tE ^)i

146 where +6,(i< 2= Q(dl-'g - - A*t 6iQ<-^K-K 4 \ E+(rxa=~~~~1Xt; I~~i, t..~ )+ it c OA4- -K.' (i k i 7 t$( - 12 I - -. A r-%o.[-| (.),. 5 I 3(\ r I/l ~, m ~ -(~- + j- ( A- L"K-ii with %r - ji: -)b(Qi-k). Hence WI ),= M~b = 6 I t! (2K-a, 1'a -~lt~c~~ 1t;iz csl~~~~P;~~~i~~~~ht~~~ e~ isw A4 (4A)AZ- tt * ~4-A ) ~ tqi^t,^<,)+@,(Sk54) e,. I i E iem (N L<2-^^) 6 + c,, eje (MiV)- +. la> il-~ age E~~~-t i il v 5 a, GuWiQ) e i Ii~c~-~e fttk-h 24)

These results may be compared to (5.45) and (5.47). The major effects due to free-rotations appear in the time dependence of the matrix elements and greatly complicates the inelastic spectrumo Because the energy now depends separately in M and K the thermalaverage can no longer be given in closed form as (5.49), nevertheless, the expression can be written down without difficulty.

APPENDIX I SEPARATION OF ROTATION AND VIBRATION IN DIATOMIC POLAR MOLECULES The procedure for treating rotational and vibrational motions of a diatomic molecule is well known, The method is equally applicable when the molecule is in a potential field, In the following we give a brief description to illustrate the necessary approximations in separating rotation and vibration, The six coordinates of a diatomic molecule are chosen to be the coordinates of center-of-mass, the interatomic separation, and the two angles specifying the orientation of the molecular axis. The molecular energy is therefore just the sum of energies associated with center-ofmass and internal, motionso Explicitly'\ i 4 t A (I.l) with U~ crz)~ ~ ^i^~I (1.2) -SVtr 2M T'Io2) H - -2zG 1 t'((^)\'<r E DT Y 4 f l^_W a^'5+t\ ) (I,5) 148

149 where ki') is the effective potential governing translations and therefore depending on the center-of-mass coordinate R only, and \ A ), as discussed in Chapter V, is a function only of the angle ~ under appropriate choice of axes. The interatomic potential \)(() governs the vibrational motion and hence depends on the interatomic separation. The reduced mass,-2,where -^.-t-.Liz, is denoted here as It is apparent from the form of (1.3) that the wave function describing the internal degrees of freedom is separable, 41' ^( t) _ - k'0 (ti i) l d (1.4) where (it Ktti ) ( ('.5) and ~ and s satisfy respectively, svi ~ (__At)3 _ R (' (1.6) In arriving at (Io7) we have employed the approximation ( \jL/ ( \V~C(),,V being the separation when both atoms are at rest. Rotational constant is defined as before, " -)Z/ QT _, and E represents the energies of internal motion. The rotational energy E of a diatomic molecule under the influence of a strong local electric field

150 is given by Eq. (5.22) as e and the associated wave function )( ) is given by (5.21) as / ). The potential for vibrational motion, to a fair approximation, can be given by a Hooke s-law interaction of the form \|(F (r- r where b is the restoring force constant. To take advantage of this quadratic form we introduce the transformation, (CV)= CQ) I /r and p = t-t, so that U^~t-[^ - e-_ bQi (E (I.8) The choice of \(t) presupposes ) to be small compared to i. Then to order 0 (1.8) becomes (a)_ ti^& %s l2 t + --. =0, (I.9) where = L E We observe that (I.9) resembles the oscillator equation if the term proportional to p can be eliminatedo This is accomplished by letting =\-t~, and Q'( l t ( "- -tt) 0 _ o, (I.10) with \i 2, L hy~ y t = a (I - _ ) and )=.E o It is seen that is always negative because the ratio b/foK A f / b^ ) is expected to be considerably greater than unity. Equation (I.10) is the oscillator equation provided we can ignore the fact that the actual problem requires 4=o for Y — O, 0 whereas the oscillator solution vanishes at 1 =_ -- j 00,

151 If the oscillator approximation is adopted /. can then be exhibited as \=it t,(Ski) +ta ere is t.n1/. where -i=..( / j) is the vibrational frequency. The three terms in this equation obviously represent the energies of vibration, rotation, and rotation-vibration coupling. For most cases (_.<, and since E) & we may ignore the stretching of the molecule due to rotation in which case vibration and rotation become separable. We thus find li,':,. -'t --'-1. and Ie, l? >, where /~ is the wave function of an oscillator of mass'A and associated frequency Q.

APPENDIX J THE THERMAL AVERAGE FOR INTERNAL DEGREES OF FREEDOM OF A DIATOMIC MOLECULE We consider a formal method of evaluating the thermal average "','X >.. *The vibrational thermal average is given by Eq. (6.3). ~, < I In general 4'Y.)>2>\ iand.\,'. (,, the small argument expression of the modified Bessel function can therefore be used, Vyl') r. (>Y' / d'!. We find itti -0(44 ^>Z -i;tt -OD(<c 5' Aa -,Q -,'t' -vi - h ) where ^' -nuE ^l/2bT / ~b/ The first two terms in \7, correspond to elastic and first-order ( = -+- 1) inelastic scatterings. Dependence upon Hr is explicit so that each term may be inserted into Eq. (6.1) to perform the average 152

153 over molecular orientation. We consider the first term of Va l which represents the effects of zero vibration in (6.1). The corresponding thermal average eA - i, l(J.2) <^>~Ah^l/:. exponential ^ can be expanded according to Eq. (5.26), similarly A (J.5); 0 _ i(. ^ Cbe) (}) ) C D L v' c - I - -.: -1 —' The leading term in (J.3) leads to the result given in Chapter V and the second term when inserted in (J.2) can be reduced using the relations (4.37) and the explicit form of IA. After a series of manipulations we obtain to order UlJ <0 Add for - Add > D~~(J. 4<() M' T T I

154 where <~(/ T is given by Eq. (5.50) and nt4m z 2 1 <^ - ^ 2-^(^~\ Cl)2+ c)jh P( IQSM7 ^'*A-o^ Ztc~le1,' tle o - ~ + -t - \ (X (0 o6 6 O (M+tt -S - m -t Q') SM|d h^JS'~Al S e,'W(J.4) ~+ e ^ J — m u S' c 0 1' C cM'1 (, (-) C-o ) I, C.') t' d k (*s t) (XMth\da-tMisl t) \( eM _-I -9' I\tt' /+S' j.

155 The integral V, is ( <SM' i'e,,), like the integral F in Chapter.-1'7 IPAc I i ioz (/ C! Z.iB ":l 7, (J. 5) where i is known (see (5.37)) and gives the selection rule for in a rather complicated way. The higher order terms in \ may be treated in a similar manner although the amount of labor involved is expected to rapidly become prohibitive.

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157 REFERENCES (Continued) 14. B. N. Brockhouse, "Structural Dynamics of Water by Neutron Spectrometry," Nuovo Cimento Suppl., 9, 45 (1958). 15. A. Morales, Slow-Neutron Diffraction in Normal Liquids, thesis, The University of Michigan, Ann Arbor, 1960. 16. I. Pelah, W, L. Whittemore, and A. W. McReynolds, "Energy Distribution of Neutrons Scattered by Liquid Lead," Phys. Rev. 113, 767 (1959). 17. G. H. Vineyard, "Scattering of Slow Neutrons by a Liquid," Phys. Rev, 110, 999 (1958). 18. K. S. Singwi and Alf SjOlander, "Diffusive Motions in Water and Cold Neutron Scattering," Phys. Rev. 119, 863 (1960). 19. P. C. Cross, J. Burnham, and P. A. Leighton, "The Raman Spectrum and the Structure of Water," Jo Am. Chem. Soc. 59, 1134 (1937). 20. R. C. Plumb and D. F. Hornig, "Infrared Spectrum, X-Ray Diffraction Pattern, and Structure of Ammonium Flouride," Jo Chem. Phys. 23, 947 (1955). 21. L. Pauling, "The Rotional Motion of Molecules in Crystals," Phys. Rev. 36, 430 (1930). 22. M. Magat, "Recherches sur le Spectra Raman et la Constitution de leau Ligue," Ann, d. Physique 6, 109 (1936). 25. Mark Nelkin, "Scattering of Slow Neutrons by Water," Phys. Rev. 119, 741 (1960). 24. L. Van Hove, "Correlation in Space and Time and Born Approximation Scattering in Systems of Interacting Particles," Phys. Rev. 95, 249 (1954). 25. P. A. Egelstaff, "The Theory of Thermal-Neutron Scattering Law," Proc. Symposium on Inelastic Scattering of Neutrons in Solids and Liquids, I.A.E.Ao, Vienna (1960); see also "The Treatment of Thermal Neutron Scattering Law Data," AERE Report R-3622 (1961). 26. See the Proceedings of Symposium on Inelastic Scattering of Neutrons in Solids and Liquids, I.AoEA., Vienna (1960)

158 REFERENCES (Continued) 27. M. Born and J. R. Oppenheimer, "Zur Quantenthearie der Molekeln," Ann. d. Physik 84, 457 (1927). 28. L. I. Schiff, Quantum Mechanics, McGraw Hill, New York (1955). 29. W. E. Lamb, "Capture of Neutrons by Atoms in a Crystal," Phys. Rev. 55, 190 (1939). 30. E. Fermi, Ricerca Sci. 7, 13 (1936); English translation available as USAEC Rept. NP-2385. 31. A. G. DeRocco, private communication. 32. J. E. Lennard-Jones and Ao F. Devonshire, "Critical Phenomena in Gases-I," Proc. Roy. Soc. (London), 163, 53 (1937); "Critical Phenomena in Gases II Vapour Pressures and Boiling Points," 165A, 1 (1938). 33. J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, New York (1954) 34. Go N. Watson, Bessel Functions, Macmillan Company, New York (1948). 35. P. Debye, "IUber dio Zerstreuang von Rontgenstrahlen an amorphen Korpen," Physik. Z. 28, 135 (1927); see also "Note on the Scattering of X-Rays," J, Math, and Phys. 4, 133 (1925) or The Collected Papers of Peter J. Wo Debye, Interscience, New York (1954). 36. A. Rahman, "Scattering of Slow Neutrons by Molecules," J. Nucl. Energy 13, 128 (1961). 38. A. F. Gorgunov, "The Scattering of Slow Neutrons by Water Molecules," J. Nucl. Energy 4, 109 (1957). 39. H. H. Nielsen, "The Vibration-Rotation Energies of Molecules," Rev, Mod. Phys. 23, 90 (1951); "The Vibration-Rotation Energies of Molecules," Handbuch der Physik 37, 173 (1959) 40. C. Van Winter, "The Asymmetric Rotator in Quantum Mechanics," Physica XX, 274 (1954). 41. P. M. Morse and Ho Feshbach, Methods of Theoretical Physics, Vols. I and II, McGraw-Hill, New York (1953).

159 REFERENCES (Continued) 42. E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York (1959). 43. M. E. Rose, Elementary Theory of Angular Momentum, Wiley, New York (1957). 44. S. C. Wang, "On the Asymmetric Top in Quantum Mechanics," Phys. Rev. 34, 243 (1929). 45. P. Kusch and V.. Hughes, "Atomic and Molecular Beam Spectroscopy," Handbuch der Physik 37, 1 (1959) o 46. E. Feenberg, "A Note on Perturbation Theory," Phys. Rev. 74, 206 (1948). 47. J. H. Van Vleck, "The Coupling of Angular Momentum Vectors in Molecules," Rev. Mod. Phys. 23, 213 (1951). 48. V. Heine, Group Theory in Quantum Mechanics, Pergamon, New York (1960). 49. R. H. Fowler and E. A, Guggenheim, Statistical Thermodynamics, Cambridge, London (1949). 50. N. F. Mott, "The Resistance of Liquid Metals," Proc, Roy. Soc., London, 146, 465 (1934). 51. W, F. Giaugue and R. Wiebe, "The Entropy of Hydrogen Chloride. Heat Capacity from 16~K to Boiling Pointo Heat of Vaporization. Vapor Pressures of Solid and Liquid," J. Am. Chem. Soc. 50, 101 (1928). 52. D. F. Hornig and W. E, Osberg, "Infrared Spectrum and the Structure of the Low-Temperature Phases of Crystalline HC1, HBr, and HI," J. Chem. Physo 23, 662 (1955)o 53. G. Herzberg, Molecular Spectra and Molecular Structure I. Diatomic Molecule, Prentice-Hall, New York (1939). 54. Re W. Swenson and R. H. Cole, "Dielectric Properties of Hydrogen Halides II. Hydrogen Chloride," J. Chem. Phys. 22, 284 (1954). 55. Po Debye, Polar Molecules, Dover, New York (1949).

160 REFERENCES (Concluded) 56. C. G. Shull and E. 0. Wollan, "Coherent Scattering Amplitudes as Determined by Neutron Diffraction," Phys. Rev. 81, 527 (1951). 57. J. S. King, private communicationo 58. G. Herzberg, Molecular Spectra and Molecular Structure, II. D. Van Nostrand, New Jersey (1950). 59. J. D. Bernal and Ro H. Fowler, "A Theory of Water and Ionic Solution, with Particular Reference to Hydrogen and Hydroxyl Ions," J. Chem. Phys. 1, 515 (1933) 60. N. E. Dorsey, Properties of Ordinary Water Substance, Reinhold, New York (1940). 61. S. Glasstone, Elements of Physical Chemistry, D. Van Nostrand, New York (1955). 62. H, Margenau and G. Murphy, The Mathematics of Physics and Chemistry, D. Van Nostrand, New York (1957) 63. H. Goldstein, Classical Mechanics, Addison-Wesley, New York (1950). 64. R. Ko Osborn and E. D. Klema, "Dynamics of an Elastic Ellipsoid," Nuovo Cimento 9, 791 (1958). 65. P. D. Maker, Stark Effects in the Near Infrared Spectra of Simple Polyatomic Molecules, thesis, The University of Michigan, Ann Arbor, 1961. 66. L. Pauling and E. Bo Wilson, Introduction to Quantum Mechanics, McGraw-Hill, New York (1935). 67. M. Rosenbaum and P. F. Zweifel, "Comparison of'Classical' and'Quasi-Classicl' Cross Sections for Some Simple Systems," Transactions of International Conference on Neutron Thermalization, Brookhaven National Laboratory (1962); see also R. Aamodt, K. M. Case, M. Rosenbaum, and P. F. Zweifel, "Quasi-Classical Treatment of Neutron Scattering," to be published in The Physical Review.

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