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Transport properties of gases with rotational states McCourt, Frederick Richard Wayne 1966

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TRANSPORT PROPERTIES OF GASES WITH ROTATIONAL STATES by FREDERICK RICHARD WAYNE McCOURT B. Sc. (Hons.), University of British Columbia, 1963 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of CHEMISTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1966 iv In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C HEMfSTEY The University of British Columbia Vancouver 8, Canada The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of FREDERICK RICHARD McCOURT B,Sc = (Hons.)j The Uni v e r s i t y of B r i t i s h Columbia, 1963 MONDAY, AUGUST 15, 1966 AT 3:30 P.M. IN ROOM 261, CHEMISTRY BUILDING COMMITTEE IN CHARGE Chairman: I. McT. Cowan Research Supervisor: R. F. Snider External Examiner: C. F. Curtiss T h e o r e t i c a l Chemistry I n s t i t u t e U n i v e r s i t y of Wisconsin Madison s Wisconsin R. Barrie J. A. R. Coope B. A, Dunell C. A. McDowell R. F. Snider L. Sobrino TRANSJPOftT PROPERTIES OF GASES WITH ROTATIONAL STATES ABSTRACT Theo r e t i c a l expressions for the transport c o e f f i c i e n t s of a single component gas with a nonzero but small l o c a l angular momentum density are obtained from a modified Boltzmann equation which takes into account the presence of degenerate i n t e r n a l states ( s p e c i f i c a l l y , r o t a t i o n a l s t a t e s ) . As i s to be expected, a number of Onsager r e c i p r o c a l r e l a t i o n s are found connecting the transport c o e f f i c i e n t s . A l i n e a r i z a t i o n of the Boltzmann E l a t i o n i s car r i e d out by means of a perturbation expansion about a l o c a l e q uilibrium state which i s characterized by a l o c a l temperature, stream v e l o c i t y , and angular momentum density. This perturbation i s expressed as a line a r combination of the macroscopic gradients of the system s whose c o e f f i c i e n t s being tensors s are expanded in terms of i r r e d u c i b l e Cartesian tensors made up of the angular momentum pseudovector operator J and the reduced v e l o c i t y vector W. The transport c o e f f i c i e n t s are then given by combinations of c e r t a i n s c a l a r expansion c o e f f i c i e n t s . Expressions for these expansion c o e f f i c i e n t s i n terms of square bracket i n t e g r a l s are obtained with the a i d of an i t e r a t i v e v a r i a t i o n a l procedure based on a scalar product which allows for the lack of time reversal, symmetry of the Boltzmann c o l l i s i o n operator. F i n a l l y ^ the square bracket integr a l s are reduced to r e l a t i v e and center-of-mass coordinates and expressed i n terms of generalized c o l l i s i o n cross sections. The techniques developed for the ro t a t i n g gas with a nonzero l o c a l angular momentum density are u t i l i z e d to obtain an expression for the change i n the thermal, conductivity of a gas when placed i n a magnetic f i e l d , It i s shown that at saturation the r a t i o of the changes in the thermal conductivity with the magnetic f i e l d (a) p a r a l l e l t o s and (b) perpendicular t o s the temperature gradient i s 2/3, This value agrees with the experimental r e s u l t for paramagnetic gases. GRADUATE STUDIES F i e l d of Study: Theoretical Chemistry Topics i n Physical Chemistry J. A. R, Coope A, V. Bree Seminar i n Physical Chemistry N. B a r t l e t t Quantum Chemistry J. A. R. Coope Advanced Theoretical Chemistry R„ F. Snider Transport Properties of Gases R. F. Snider Topics i n Chemical Physics C. B. A. Dune11 A. McDowell Chemical Thermodynamics J. N. Butler Topics i n Inorganic Chemistry W„ R. Cullen H. C, Clark N. B a r t l e t t J. T. Kwon Spectroscopy and Molecular Structure A. V. Bree C Reid B. A. Dunell Chemistry of the S o l i d State L. , G. Harrison Related Studies: Elementary Quantum Mechanics F, A, Kaempffer Plasma Physics L= Sobrino Introduction to Low Temperature Physics J. B. Brown P, W, Matthews PUBLICATIONS F. R, McCourt, An Infrared Study of S o l i d Gas Hydrates at 77°K, B. Sc. Thesis, U.B.C., (1962). R. F. Snider and F. R. McCourt s K i n e t i c Theory of Moderately Dense Gases; Inverse Power Pot e n t i a l s , Phys. F l u i d s 63 1020 (1963)= K, B. Harvey, F. R. McCourt and H, F. Shur v e l l , Infrared Absorption of the S02 Clathrate-Hydrate °, Motion of the S02 Molecule, Can, J, Chem. 42, 960 (1964). F. R, McCourt and R. F. Snider, Thermal Conductivity of a Gas with Rotational States, J. Chem. Phys. 41, 3185 (1964). J. A. R. Coope, R, F. Snider and F. R. McCourt, Irreducible Cartesian Tensors, J. Chem, Phys. 43, 2269 (1965). F. R. McCourt and R. F. Snider, Transport Properties of Gases with Rotational States, I I , J. Chem. Phys. 43, 2276 (1965). F. R. McCourt and R. F. Snider, "Transport Properties of Gases with Rotational States, I I I " (to be published). F„ R. McCourt and R, F. Snider, "The Thermal Conductivit a Gas of Rotating Molecules i n an External Magnetic F i e l d " (to be published) FREDERICK RICHARD WAYNE McCOURT. i i TRANSPORT PROPERTIES OF GASES WITH ROTATIONAL STATES. Supervisor. R. F. Snider. ABSTRACT Theoretical expressions for the transport coefficients of a single component gas with a nonzero but small local angular momentum den-sity are obtained from a modified Boltzmann equation which takes into account the presence pf degenerate internal states (specifically, ro-tational states). As is to be expected, a number of Onsager reciprocal relations are found connecting the transport coefficients. A linearization of the Boltzmann equation is carried out by means of a perturbation expansion about a local equilibrium state which is characterized by a local temperature, stream velocity and angular momentum density. This perturbation is expressed as a linear combi-nation of the macroscopic gradients of the system, whose coefficients, being tensors, are expanded in terms of irreducible Cartesian tensors made up of the angular momentum pseudovector operator J and the re-duced velocity vector \ y . The transport coefficients are then given by combinations of certain scalar expansion coefficients. Expressions for these expansion coefficients in terms of square bracket integrals are obtained with the aid of an iterative variational procedure based on a scalar product which allows for the lack of time reversal symmetry of the Boltzmann coll is ion operator. Finally, the square bracket i n -i i i tegrals are reduced to relative and center-of-mass coordinates and expressed in terms of generalized col l is ion cross sections. The techniques developed for the rotating gas with a nonzero local angular momentum density are utilized to obtain an expression for the change in the thermal conductivity of a gas when placed in a magnetic f ie ld . It is shown that at saturation the ratio of the changes in the thermal conductivity with the magnetic field (a) parallel to, and (b) perpendicular to, the temperature gradient is 2/3. This value agrees with the experimental result for paramagnetic gases. V TABLE OF CONTENTS Page Abstract i i List of Tables and Figures v i i Acknowledgment v i i i CHAPTER I INTRODUCTION 1 1.1 Treatments Based on Class ica l Mechanics . 2 1.2 Treatments Based on Quantum Mechanics . . 6 1.3 Effect of a Magnetic Field on the Transport Properties 10 CHAPTER II THE BOLTZMANN AND HYDRODYNAMIC EQUATIONS 14 2.1 The Boltzmann Equation 14 2.2 The General Equation of Change 18 2.3 The Hydrodynamic Equations 21 CHAPTER III LINEARIZED BOLTZMANN EQUATION FOR SMALL J Q 27 3.1 The Equilibrium Distribution Function 27 3.2 Linearization of the Boltzmann Equation for Small J G 31 CHAPTER IV THE FLUX TENSORS 40 4.1 Fluxes and Forces 40 4.2 The Pressure Tensor 44 4.3 The Angular Momentum Flux Tensor 53 4.4 The Heat Flux Vector 61 4.5 Discussion 65 vi CHAPTER V THE VARIATIONAL PROCEDURE . 69 5.1 The Variational Principle 69 5.2 The Iterative Equations 7 3 5.3 The Onsager Reciprocal Relations 7 8 CHAPTER VI THE FIRST ITERATION—ZEROTH ORDER IN £ 89 6.1 Method of Approximation 89 6. 2 Coefficient of Shear Viscosity 92 6.3 Coefficient of Bulk Viscosity 98 6.4 Coefficient of Thermal Conductivity 101 6.5 The Rotational Diffusion Coefficients. 104 6.6 The Coefficients ^ and V 108 CHAPTER VII EFFECTS WITH LINEAR 'CL-DEPENDENCE . . I l l 7.1 Preliminary Remarks I l l 7.2 The Anisotropic Viscosity Coefficients 112 7.3 Other "Linear-in- oL » Effects 120 CHAPTER VIII THE SENFTLEBEN-BEENAKKER EFFECT FOR THE THERMAL CONDUCTIVITY, , , . 122 8.1 Introduction 122 8.2 Form of the Boltzmann Equation . 123 8.3 The Tensor Equations Determining the Anisotropy 128 8.4 The Anisotropic Thermal Conductivity 138 8.5 Reduction of the Square Bracket Integrals . . 145 8.6 Discussion 148 BIBLIOGRAPHY 151 APPENDIX.I 157 APPENDIX II 161 APPENDIX III 164 APPENDIX IV 170 APPENDIX V 174 v i i LIST OF TABLES AND FIGURES following page TABLE I A complete set of irreducible tensors (at most linear in ) which are linear in the macroscopic gradients 43 TABLE II Complete sets of independent irreducible isotropic tensors of ranks two to five . . . . . 49 TABLE III Identification of the /6 coefficients for the iterative solutions 112 FIGURE I An invariant representation of isotropic tensors 49 v i i i ACKNOWLEDGMENT It is a pleasure to acknowledge the able and enthusiastic super-vision of Dr. R. F . Snider, without whom this work would not have progressed. CHAPTER I INTRODUCTION In statistical mechanics, an attempt is made to predict or explain the properties and behaviour of a macroscopically small but micro-* scopically large number of particles when the laws governing the interaction between the constituent particles are known. This sub-ject has been subdivided into two main branches, the first dealing with the properties of systems in equilibrium and knownas "equilibrium statistical mechanics"; the second dealing with those properties of systems which depend on deviations from equilibrium, known as "non-equilibrium statistical mechanics" or, alternatively, as "the statistical mechanicsof irreversible processes". The determination of the trans-port properties of a system is a central problem of nonequilibrium statistical mechanics and in particular for gases, is usually known as the "kinetic theory of gases". 2 1.1 Treatments Based on Class ica l Mechanics. A mathematically rigorous approach to the problems of the kinetic theory of gases was not made until the mid-nineteenth century when Maxwell* in 1866 derived the equations of change for a non-uniform 2 3 gas. Shortly thereafter, Boltzmann ' published in 187 2 a derivation of the integro-differential equation, now bearing his name, upon which the rigorous kinetic theory of gases is founded. Since the Boltzmann equation is a non-linear integro-differential equation, attempts to 4 5 solve it met with little success until Chapman and Enskog indepen-dently published their solutions. Their methods basically consist of restricting the gas to being always near local equilibrium so that the velocity distribution function can be obtained through a series of successive approximations, the first of which w i l l give the local equilibrium distribution function. This restriction allows the linear-ization of the Boltzmann equation, thus permitting the techniques of linear analysis to be employed. For gases not deviating too far from local equilibrium a second approximation should give quite good re-sults for the transport coefficients when compared with experiment. Two standard methods of solution of the above-mentioned linear-ized Boltzmann equation may be ut i l ized. The first of these is the 3 method used by Chapman and Cowling in which an approximate sol -ution for the linearized integral equation is obtained by expanding the solution in a specially chosen complete set of functions, expressing each of the expansion coefficients as a ratio of infinite determinants (each element of which is a complicated integral) and then approx-imating this ratio by the ratio of finite determinants. This procedure has now largely been replaced by the variational method introduced by 7 Hellund and Uehling and extensively discussed in the well-known 8 9 treatises by Hirschfelder, Curtiss and Bird and by Waldmann. It must be made clear that certain assumptions are inherent in the Boltzmann equation approach to the kinetic theory of gases. Occurring in Boltzmann's original derivation of his equation and in the Chapman-Enskog solution are the assumptions that: c lass ical mechanics is valid for describing molecular interactions; the gas is sufficiently dilute that three-body collisions may be ignored; the molecules possess no inter-nal structure, that i s , that they are point particles surrounded by spherically symmetric fields .of force. Much of the research in kinetic theory since Chapman and Enskog has been directed towards the remo-val of these three restrictions. Jeans^ considered the effect of internal degrees of freedom on the transport coefficients for a gas which could be described by the special molecular model of the loaded sphere, that i s , for spheres in 4 which the center of mass is slightly displaced from the geometrical center. He derived, using the mean free path approach, an expression for the rate of equilibration of rotational and translational kinetic energy. A later, more exact c lass ical treatment of the transport prop-erties for the loaded sphere model has been given by Dahler and Sather^ while a quantum mechanical treatment has recently been 12 carried out for this model by Mueller. Using the Chapman-Enskog 13 method, Pidduck determined the transport coefficients for a gas of perfectly rough spheres (the simplest molecular model possessing energy of rotation which is interconvertible with energy of translation). A rigorous c lass ical treatment of the transport properties (with expl i -cit consideration of the angular momentum variables) for a single component gas consisting of nonspherical molecules was first given 14 by Curtiss in 1956, including a consideration of the hitherto neglec-ted transport of angular momentum through the gas. His treatment is quite ingenious. He considers the gas to be, in effect, a multicom-ponent mixture in configuration space thus enabling him to utilize techniques developed for multicomponent gas mixtures. Using this theory and the spherocylinder model, expressions were obtained for the thermal conductivity and the shear and bulk viscosities by Curtiss 15 and Muckenfuss, while a determination of the angular momentum 5 transport coefficients for the same model was carried out by Livingston 16 and Curtiss. Curtiss 1 method was further generalized to multi-17 18 component gas mixtures by Muckenfuss ' who also carried out a calculation of the coefficients of diffusion, thermal diffusion, shear viscosity, bulk viscosity and thermal conductivity for a binary gas mixture. A slightly more detailed treatment of the derivation of the Boltzmann equation for gases composed of molecules with arbitrary internal de-grees of freedom and which interact according to the laws of c lass ical mechanics through arbitrary non-central pairwise-additive forces was 19 presented in 1963 by Curtiss and Dahler, and as an application of their theory, expressions were obtained for the transport coefficients of a gas of symmetric top molecules. 20 It was first notedby Chapman and Cowling and later emphasized 21 by Kagan and Afanas'ev that in a system of rotating molecules there are in fact, two vector quantities, namely the reduced velocity vector W and the angular momentum vector J , which must be employed in obtaining the correct tensorial character of the coefficients used in writing the perturbation function <^> as a linear combination of the macroscopic gradients of the system. The treatment given by Kagan and Afanas'ev was class ical in nature and can be considered as a 6 generalization of the method employed earlier by Curtiss but with the inclusion of the angular momentum dependence of the solutions. A similar calculation has been carried out for rough spheres by Condiff, 22 23 Lu and Dahler and for loaded spheres by Sandler and Dahler. 24 Waldmann has also taken the J-dependence of the solutions into account in treating the partial polarization in a Lorentz gas of rotating molecules. 1.2 Treatments Based on Quantum Mechanics. Quantum mechanics was introduced successfully into the kinetic theory description of transport processes in 1933 by Uehling and 25 Uhlenbeck. The main result of their work was the replacement of the c lass ical cross section by its quantum analogue (without internal 9 states). Theseresultshavebeenextensivelydiscussed byWaldmann , 26 27 by Mor i , Oppenheim and Ross and by Hoffmann (see also Chapman 6 8 and Cowling and Hirschfelder, Curtiss and Bird ) . The first serious consideration of the general kinetic theory of gases for molecules with internal states was put forward by Wang 28 Chang and Uhlenbeck (see also Wang Chang, Uhlenbeck and de 29 Boer ). In their treatment, Wang Chang and Uhlenbeck assumed a 7 condition on the quantum mechanical cross section which is analogous to the existence of inverse collisions in c lass ical mechanics. Since this "detailed balance" criterion is only valid if the molecular states are nondegenerate, the validity of their equations for most real gases was in serious question. Under the same assumptions, Mason and 30 Monchick have applied the formal theory of Wang Chang and Uhlenbeck to derive expressions for the transport coefficients of polyatomic gases and have arrived at fair agreement between theory 9 and experiment. This work and that of Waldmann have shown that although the treatment employed by Wang Chang and Uhlenbeck was at fault in the assumption of detailed balance, their results are valid for a l l molecules provided their quantum mechanical cross sections are replaced by the corresponding degeneracy-averaged quantum mechanical cross sections. The required symmetry property of the degeneracy-averaged cross sections arises naturally from considera-tions of space inversion and time reversal invariance although the 31 principle of detailed balance does not generally hold. The Wang Chang and Uhlenbeck method was generalized to mixtures of poly-32 .. 33 atomic gases by Snider, by Waldmann and Trubenbacher and by 34 Monchick, Yun and Mason. A derivation of the quantum mechanical Boltzmann equation taking 8 into account the presence in a gas of rotating molecules was first 35 given by Waldmann and later an independent derivation starting from the quantum Liouville equation and employing the formal Lippmann-36 Schwinger scattering theory was given by Snider. For the case of rotationally degenerate states the Wigner distribution function-density matrix is employed rather than a simple Wigner distribution function. Inclusion of the off-diagonal terms in the Wigner distribution function-density matrix necessitates a modified col l is ion term in which the coll is ion is described in terms of combinations'of the transition opera-tor rather than simply in terms of the col l is ion cross section. This modified coll is ion integral reduces to the usual quantum mechanical Boltzmann equation for the case of nondegenerate states. Waldmann has also derived a Boltzmann equation for a gas composed of spin one-37 half particles and as an application considered the diffusion of these 38 spin-particles in a magnetic f ield. This thesis i s principally concerned with obtaining expressions, for the transport coefficients for a gas of (diatomic) rotating molecules starting from the Boltzmann equation of Waldmann and Snider while taking into account the dependence of the solutions on the angular momentum. However, in any quantum mechanical treatment following the Kagan and Afanas'ev method, the commutation problems arising 9 from the noncummutativity of the internal angular momentum pseudo-vector operator 2 must be treated explici t ly . In particular, serious complications arise in an evaluation of the pressure tensor P and the angular momentum flux tensor L .^ In order to avoid such difficulties, it is convenient to utilize an alternative procedure suggested by Kagan 39 and Maksimov. These authors employed a tensorial expansion of the perturbation function in terms of products of irreducible tensors, one formed from W and the other from2« In such a manner, they de-termined the thermal conductivity of a c lass ical gas of rotating para-magnetic molecules in the case of zero local angular momentum density. The first part of this thesis considers the form for the perturbation function <^> and the resulting forms for £ , L and the heat flux vector q when there is a nonzero but small local angular momentum density J^. In Chapter; II the Boltzmann equation of Waldmann and Snider is discussed and the resulting hydrodynamic equations obtained, while in Chapter III the linearized form of this equation for a nonzero but small local angular momentum density is derived. P ,L and q are ex-pressed in Chapter IV as linear combinations of generalized forces (gradients) whose coefficients (the transport coefficients) are obtained in terms of the expansion coefficients of the perturbation function. Treatment of the problem using the Kagan and Maksimov method 10 involves tensorial coupling coefficients which are, in general, not unique. These are most conveniently expressed in terms of a set of independent irreducible isotropic tensors with scalar expansion coef-ficients. A variational procedure resulting in an iterative method is introduced in Chapter V and the Onsager reciprocal relations connecting some of the transport cross effects are examined. Chapters Viand VII are devoted to the variational calculation of the transport coefficients in the J -independent and J, -linear approximations respectively, -o —o The J^-independent coefficients correspond to the ordinary transport coefficients. In the expressions for the angular momentum flux trans-port coefficients, the introduction of new J-dependent col l is ion cross sections is required. 1.3 Effect of a Magnetic Field on the Transport Properties. That magnetic fields affect the transport properties of a gas of charged particles has long been known, but until 1930 when Senftleben in Germany reported the experimental observation of the effect of an 40 external magnetic field on the thermal conductivity of paramagnetic diatomic gases (oxygen and nitric oxide) it had been assumed that a magnetic field would have no effect on the transport properties of a 11 gas of neutral particles. Senftleben's results were a l l the more start-ling in that the thermal conductivity was found to depend on the ratio of the magnitude of the magnetic f ield, H , to the equilibrium pressure, p, of the gas. A saturation effect was found, for which at the satura-tion value of H/p 100-200 Oe/mm Hg, the thermal conductivity of the gas was decreased by about 2% from its field-free value. In the fo l -41 lowing ten years, a whole series of experimental papers was devoted to this effect. It was at the same time found that the shear viscosity of a paramagnetic gas was also affected by a magnetic field in much 42 the same way as the thermal conductivity. A qualitative explana-43 44 45 tion was proposed by Laue ' in 1935-36, by Gorter in 1938, fo l -lowed closely by a more quantitative explanation by Zernike and van 46 Lier in 1939, a l l based on the mean free path approach to kinetic theory. These workers proposed that the coll is ion probability of a fast rotating diatomic molecule depends on the angle between its axis of rotation and the direction of its motion. Quite good agreement be-tween theory and experiment was attained. No further work, either experimental or theoretical, appears to have been undertaken until 39 Kagan and Maksimov presented their work on transport phenomena in paramagnetic gases from the point of view of the Boltzmann equation in 1961. Treating the rotational motion c lass ica l ly , they set up a 12 Boltzmann equation with a term on the left hand side (called for ob-vious reasons a magnetic operator term) containing the magnetic mom-ent due to the unpaired electron spin. With the inclusion of this term they were able to relate the Senftleben effect to the degree of non-sphericity of the molecules. It has been known for some time now from the theoretical.work of 47 Wick , and the molecular beam experiments of Ramsey and co-48 workers, that many diatomic molecules possess a magnetic moment (of the order of a nuclear magneton) due to the nonzero angular mo-mentum of their rotational states. From this result and the work of Kagan and Maksimov, it can be surmised, then, that it is possible for a magnetic field to affect the transport properties of such molecules. However, as this rotational magnetic moment is approximately a nuclear magneton while that due to the unpaired electron spin in paramagnetic molecules is approximately a Bohr magneton, the magnetic field effects cannot be expected to show up except for very high H/p values* This problem was investigated for a number of diatomic diamagnetic gases 49 by Beenakker and coworkers in the Netherlands for the magnetic 50 field dependence of the shear viscosity and by Gorelik and Sinitsyn in the Soviet Union for the corresponding dependence of the thermal 51 conductivity. Recently, Korving et a l . have even observed this 13 magnetic field effect on the viscosity of C H ^ and CF^ (rough spherical molecules). The final part of this thesis is concerned with obtaining an ex-pressionfor this "Senftleben-Beenakker effect" for the thermal conduc-tivity of a gas of rotating molecules. Thus, in Chapter VIII, a linear-ized Boltzmann equation allowing for the presence of a magnetic field is obtained and, as an application, an expression is derived giving the change in the thermal conductivity of a gas of rotating molecules in a magnetic f ield. 14 CHAPTER II THE BOLTZMANN AND HYDRODYNAMIC EQUATIONS 2.1 The Boltzmann Equation. Transport of particles through a region of phase-space can take place in two ways. The first of these is through particles "drifting" into and out of the region of phase space under consideration, and the second is through particles entering and leaving the region via the mech-anism of col l is ions . The number of such particles entering and leaving the given region is governed, in the limit of low density and the binary coll is ion approximation, by the Boltzmann equation. This equation gives the total time-rate-of-change of the phase space distribution function in terms of a "streaming" or "drift" term and a "col l is ion" term. The use of a distribution function i s , of course, valid for particles whose states can be described c lass ica l ly . Unfortunately, the situationis not so simple for particles whose states have to be described quantum mechanically, since there is. no complete analogue of the phase space distribution function in quantum mechanics. In this case the so-called 52 Wigner distribution function is ut i l ized. The Wigner distribution function w i l l then satisfy a quantum mechanical Boltzmann equation in the low density-binary coll is ion approximation. So far, the absence of internal states in the particles has been 15 assumed. The internal states of a particle are described by a density matrixin internal state space. Should the internal states be nondegen-erate, the internal state wave function for a free particle with specified energy contains only one possible state so that the corresponding density matrix w i l l be diagonal in internal state space. Hence for the quantum mechanical treatment of the transport properties of a gas whose particles possess only nondegenerate internal states, it is sufficient to employ a set of Wigner distribution functions (one for each internal state) in the Boltzmann equation. On the other hand, if the particle possesses degen-erate internal states, it must be described by a density matrix which is no longer diagonal in internal state space. This, combined with the c lass ical description of translational motion, requires that a Wigner distribution function-density matrix be employed. The existence of particles with nondegenerate internal states has been considered quantum mechanically by Wang Chang, Uhlenbeck and 30 31 de Boer, by Mason and Monchick and has been generalized to mix-32 33 tures by Snider, by Waldmann and Trubenbacher and by Monchick, 34 53 Yun and Mason. Dahler considered a diatomic gas possessing de-generate internal states, but in so doing employed only the diagonal elements of the Wigner distribution function-density matrix. The inclusion of off-diagonal elements of the Wigner distribution function-density 16 matrix requires a modification of the Boltzmann coll ision term which involves combinations of the transition operator* t and its adjoint t+ 35,36 rather than the usual coll is ion cross section. The Boltzmann equation for particle s with degenerate internal states is where f< :^(r ,P,t) is a matrix element of the Wigner distribution function-density matrix diagonal in the quantum number j (quantum number denoting energy shells) and non-diagonal in the quantum number * A discussion of this operator and some of its properties is giv^n in Appendix I. 17 m (quantum number for degenerate states), p is the linear momentum, iik is the mass of the particle in state j , and ( £ , 2 ^ is the relative linear momentum of the two colliding particles given by ft b\- £.-"•)? - b - m< p > (2-2) where P_ is the center of mass momentum. £(E) i s a Dirac delta function of the energy expressing conservation of energy during a co l l i s ion . The first of the two terms on the right hand side of Eq. (2-1) gives the number of particles which, due to a col l i s ion, end up in a state contri-buting to fj^.^v,/(£_'£'*) while the second gives the number of particles ' which are removed by coll isions from state s contributing to f j ^ . ^ / (L ' l l ' t ) • It i s , however, more convenient to utilize the Boltzmann equation in operator form in internal state space while maintaining the matrix element form in momentum space. Thus the operator Boltzmann equation is - ^ h > f [ i j f f 1 . f f 1 ^ ' ] d l | ! > (2-3) 18 where the quantities and are defined by # - < ( h . o l t l ^ ' - t > , , ) > < 2 - 4 ) and in which j ^ ' i s understood to mean £ + £ 1 - £ ' • The symbol f1 denotes the adjoint in internal state space, g and g/ denote the relative v e l -ocit iesof the two particles corresponding to the relative momenta (£,£^) and (£' ,£^) respectively, and the prime s on f and f designate the functional dependence on £ ' and £^ respectively, ^hj represents a trace over the internal states of the second particle. 2.2 The General Equation of Change. Certain quantities such as the mass, the total energy, the three components of the total angular momentum and those of the linear mo-mentum are conserved during a col l i s ion . These eight quantities are the sums of one-particle attributes. Such conserved quantities with this 19 summational property, are called "summational invariants", and are of special interest in kinetictheory since their equations of change are the familiar hydrodynamic equations of the gas. It is best to consider first anequation of change for a general sum-mational invariant and from this equation obtain the individual hydro-dynamic equations. Since it is easier to follow such a derivation in operator notation, the operator form of the Boltzmann equation is used as the starting point. Letting (r_,£) stand for a summational invariant, the equation of change for the quantity is given by - i ^ b ^ ^ l ^ . ^ ' ] ^ ^ (2-6, during a col l i s ion . Symmetrizing with respect to particle labels and integrating over in the first term gives for the right hand side of Eq. (2-6) function expressing conservation of momentum 20 (2-7) I f In the second term of Eq. (2-7), since Lq i s diagonal in momentum space, "5f + and 'tf' commute. Ut i l iz ing this fact along with the cycl ic property of the trace and the operator identity (2-8) or r - t - 2k < 1 "? (2-9) Eq. (2-7) becomes " ^ M P . ^ ^ ^ ' ^ . V P ' < 2 - 1 0 ) 21 where it is of course clear that the primed andunprimed variables in the first term have been interchanged. The commutation of a summational invariant with the adjoint of the transition operator (in matrix element form in momentum space) can be shown to be With this result, the right hand side of the generalized equation of change for a summational invariant, as given by Eq. (2-10), is easily (2-11) seen to vanish. Hence the equation of change for a summational i n -variant *>Jr (r_,jo) is 2.3 The Hydrodynamic Equations. Before carrying out the derivation of these equations, it is useful to define several average quantities. The Wigner distribution function-22 density matrix is normalized to give the number density n(r_,t), where signifies a trace over the internal state space of the first particle. With this normalization, the mass density ^(L'$ i s , of course, simply given by / ° ( t 4 i ) - M (2-14) The brackets ^ ^ are used to denote an average over the momentum and internal state spaces, i . e . , n<A> = tJiJAf ( 2 ' 1 5 ) Using this mass density, a mass average velocity or "stream" velocity v (r,t) can in turn be defined by —o ' (2-16) 23 The mass average velocity is used as a reference velocity for the motion of the whole gas (and hence the terminology "stream velocity") so that the motion of a particular particle in the gas relative to the mass motion is given by V < r ^ , 0 = t - V . t c . - l 5 * (2-17> and is called the "peculiar velocity". Two other commonly occurring average quantities in kinetic theory are the pressure tensor* and the heat flux vector, defined respectively by P(r , i ) s n<mVl/> <2-18> and ?<c.O ^ n<(i*il/\H')y>. (2-19) The operator H ' appearing in Eq. (2-19) is the internal state Hamil-tonian, assumed to have orthonormal eigenfunctions S such that jm jm j jm * The pressure tensor is symmetric throughout this thesis since the treatment is concerned only with a dilute gas. t 24 The equation of change, Eq. (2-12), can be rewritten in the form With each of the summational invariants, Eq. (2-20) gives a hydro-dynamic equation. Thus, for ^$~ = m, the equation of continuity for the mass density is obtained, while = £ gives the equation of motion of the gas, p23 (£ . + p V , J L V - - - S i . P . (2-22) Defining the average total energy by U = n < ^ V % H'> ? (2-23) 25 the energy balance equation for this quantity is n<^_ + V W . 3 R - p v . (2-24) When the summational invariant is the total angular momentum, = M , the derivation is not quite so straight-forward as for the pre-vious cases. A breakdown of the total angular momentum into two con-tributions, the so-called "external" angular momentum term, irxp_, and the "internal" angular momentum operator J is made, i . e . , (2-25) Substituting this into the general equation of change and employing the equation of continuity gives + & « H c r x p ) > + £ ' < " V J > = o ( 2 - 2 6 ) By algebraic manipulation, this equation can be converted into the form 26 r x ^ ^ + n w ^ v . + ^ . P } + n ^ f > Since the cross-product term is just r_ crossed into the equation of motion , it vanishes, and by defining an "angular momentum flux tensor" L as L = n < V J > , (2-28) the equation of angular momentum balance in final form is ( " O L where J Q = ( J ) (2-29) 27 CHAPTER III LINEARIZED BOLTZMANN EQUATION FOR SMALL J 0 3 .1 The Equilibrium Distribution Function. The Boltzmann equation is a nonlinear integro-differential equation of a form for which at present there are no known techniques available for its general solution. It is thus desirable from-a mathematical point of view to replace this difficult nonlinear equation by a linearized equation so that the powerful techniques of linear analysis can be ap-plied. The linearized Boltzmann equation obtained below can be expec-ted to describe processes in which the distribution is always close to local equilibrium. The linearization of the Boltzmann equation is accomplished by an expansion about local equilibrium, hence the local equilibrium distr i-bution function-density matrix has first to be determined. Waldmann^ has shown that l n f ^ ° \ where f ^ is the local distr i-bution function-density matrix, is a summational invariant. It is a we l l -known result of mechanics that there are eight and only eight indepen-dent functions of the dynamical variables of a particle involved in a 28 coll is ion with another particle which are summational invariants and that a l l other summational invariants must be linear combinations of 54 these eight. The eight independent summational invariants are just those eight mentioned in Chapter II, i . e . , the mass, total energy, three components of the linear momentum and the three components of the total angular momentum. Hence, l n f ^ can be written as where £ , M_, H 1 and m are as defined previously. It is worth noting that the first serious consideration of the total angular momentum as a summational invariant seems to have been made 20 by Chapman and Cowling in considering the rough spherical molecule. In their work, they considered very briefly the possibility that l n f ^ may be proportional to M and even pointed out the nonexistance of i n -verse collisions in the c lass ical sense for rough spheres. They then eliminated the dependence of l n f ^ on _M by invoking an argument which does not allow for the partial polarization of the molecules in a gas of 24 rotating molecules as has been demonstrated by Waldmann. A local angular momentum density can, in general, arise should the principle of detailed balance not hold. It hasby now, however, been conclusively 29 shown that detailed balance does not hold in general. The four constants a^, a^, a_ and a^ of Eq. (3-1) can now be ob-tained in terms of physical parameters of the gas. By introducing the constraint <y> > .0-2) a can be shown to take the form ~2 a * - - a 3 x r - a 4 y 0 • ( 3 _ 3 ) Setting a ' - a 1 m - i a 4 v 0 ; , 0-4) f can be written as = ^ p { a ; + a 4 ( ^ V % H 0 + ^ - ? } ' ( 3 _ 5 ) Temperature is defined as for the canonical distribution in statistical mechanics, so that a^ = - l/W£. The constant a^ is evaluated through the normalization n = t a ( f ( ° ^ d £ . Having evaluated a' and replaced 30 §_2 by 2s/kT, the local equilibrium distribution function-density matrix f is seen to be I " 0 ) - D / t>» \ (Q (3-6) where (P is the density matrix for internal states defined by the relation < p . < p - ^ P ( - y i ) « p ( ^ ) <3-7) and Q is the internal state partition function defined by ^M-£t)MW) <3-8) The parameter o(. occurring in Eq. (3-7) is related to the local angular momentum density J Q by the relation - r - L ~ r (3-9) and corresponds to what has been called the "local average angular 20 14 velocity" ' in the c lass ical treatment of the problem. 31 3. 2 Linearization of the Boltzmann Equation for Small • In carrying out the linearization of the Boltzmann equation, it is assumed that the distribution of particles is never far from the local equilibrium distribution so that f can be written as* -p = f ' % ± c r + + - 4 » r * > (3-10 This form has been chosen so that the perturbation of f from f ^ is 56 I (o) 1 Hermitian for Hermitian <p since in general f and <P do not commute. Since the only assumption made on f is that the magnitude of the deviation from f ^ is small, situations can be treated for which <^> (r_,£,t) does not change appreciably over distances of the order of a mean free path or times of the order of a col l is ion time. Using this form of the perturbation and neglecting terms quadratic in ^ , the Boltzmann equation, Eq. (2-3), becomes * In this treatment, method "b" of Reference 56 is ut i l ized. It should be noted that in the approximation applied here ( oL small), 4>,b>= <£«># 32 -it + V ^ r - t N I V t, i d Clearly, the terms not involving the perturbation <j? w i l l cancel, since the transition operator t and commute, allowing Eq. (2-9) to be uti l ized. Consistent with the Enskog expansion, the derivative terms involving ^»on the left hand side of the equation can be dropped. Thus the linearized Boltzmann equation is + l i s . 5 ^ . W (3-12) e(o) Because of the form of f , Eq. (3-6), the left hand side of Eq. (3-12) can be written in the form 3 T + y ^ T a t 3£ J •f 3f a n 3n" —- + *v • •at - s r ^ 0 :o r —-1 + v . — v 4- 2* to) r (3-13) 33 The time derivatives in this equation may be eliminated by util izing the hydrodynamic equations in the local equilibrium approximation. In this approximation, the fluxes appearing in the hydrodynamic equations become L - o j % - <v> -o • p -.(^y. (3-14) where p = nkT and U is the unit tensor. Thus, at equilibrium, the equation of angular momentum balance reduces to while the energy balance equation becomes where the substantial derivative D/Dt is defined as = ^ _ + v . ^ - - 0-17) The local angular momentum density, J , is considered in this the sis to be nonzero but small, An explicit relation connecting J q and 34 the parameter cL in v°' allows the exponential term of f con-taining oC to be expanded in a power series in . Since J is — —o given by T _ k \ T*t?(-"'/kT)*wU-Z/kT)} (3.18) ih{**?(r H 7 f c T > K P (±7/kT)] an expansion of exp( rA« J/kT) to terms linear in gives - < j V > . 0-19) ' 31eT * ' where Q0--ih^?(~H'/kT)' (3-20) To terms quadratic in 0^ . , the internal partition function Q for this system is given by 35 which shows that the total energy U and the total heat capacity C y differ from their = 0 values by terms at least quadratic in oL . Hence, to terms linear in 0^  D t v D t \*Llr O - t =. C v D I _ - kTV-Uo > (3"22) where C v is simply given by C v = [^ j^ and use has been made of Eqs. (3-15) and (3-16). Considering (L as a function of J Q and T £ s e e Eq. (3-19)"], it follows that <H,J1>-<32><H/> kT<3l> (3-23) 36 so that and •at a t D t - v -(3-25) (o) Considering these results and evaluating the derivatives of f with respect t o n , v , ot. and T, the linearized Boltzmann equation becomes 37 i l - e leT 2V: V * 0 -< J 1 H ' > - < T l X H < > (3-26) where in calculating the derivative of f ^ with respect to oL, the 57 Fre'chet derivative has been used since oL'Tand J do not commute; ^ is the commutator superoperator of Reference 56, here understood to mean defined as 4 = T - 7 - - e - 3 leT (3-27) and a l l terms quadratic in d. have been dropped. (o) Defining f by o Qo U T r k T (3-28) 38 the distribution function f ^ in this approximation is given by (3-29) and Eq. (3-26) becomes, again to terms linear in cL 7 - ^ W 2. k - r J — + + U K ( U ) K - l ) - ^ + ^ ] i y ) : C v , J * } ^ ^ 4 ) , ( 3 - 30) where W is the dimensionless or "reduced" velocity given by 39 (3-31) S is the symmetric traceless "rate of shear" tensor defined as* § = £ [ v * . +cvv„) t] - $v-v..y > ( 3 - 3 2 ) while b i s given by m b = 2 ~ | w ' y > ( 3 - 3 3 ) (2) 58 and [ j j j is defined as 59 where the superscript "2" designates the weight of the tensor. * ( ) denotes the vector transpose of a second rank tensor, formed by interchanging indices. The distinction between (ab) andba, for example, is retained since in general a and b may not commute. 40 CHAPTER IV THE FLUX TENSORS 4.1 Fluxes and Forces. For a single-component gas in a nonequilibrium state, each of the transports of energy, linear momentum and angular momentum is due to gradients in the macroscopic quantities T, v^, and ^ . The flux tensors arising through the transport of these quantities are the heat flux vector, q, the pressure tensor, P, and the angular momentum flux tensor, L . These flux tensors have been defined in Chapter II and their equilibrium values were given in Chapter III [see Eq. (3-14)]. As is usual in kinetic theory, the transport coefficients w i l l be obtained through an evaluation of the corresponding fluxtensor in terms of the perturbation on the local equilibrium distribution. It w i l l be sufficient to carry out an expansion of the perturbation function ^ t o terms linear in the macroscopic gradients (thermodynamic forces) of the system since, as was seen in Chapter III, V l n T , S, \/QL and the contracted form V , v o w e r e t n e only macroscopic gradients which arose naturally in the formulation of the left hand side of the linearized Boltzmann equation. 41 The nonequilibrium expressions for the flux tensors q, P and L are given by (4-1) (4-2) and L - Lt>.f(+bf*)V!<*•?> (4"3) respectively. The expressions for q and P have somewhat simpler forms than that for L since H ' and £ commute with f ^ while J does not. As the local values for the densities of mass, linear momentum, energy and angular momentum must be given by the local equilibrium distribution function, this places a number of auxiliary conditions on the perturbation term. That i s , (4-4) 42 fe [ pf% <Af = o » ( 4 " 5 ) and Since q is a first rank tensor, P a second rank tensor, and L a second rank pseudotensor, a l l tensors of corresponding rank and i n -version symmetry which can be formed from v , , V^HTT V y o a n d and which are linear in one of the three gradients are of interest in the expansion of the perturbation function. Al l of the tensors i n -volving v can be omitted since Galilean invariance of the phenomen-"~o ological equations implies that they must be independent of V q . How-ever, this is not so for the corresponding tensors involving p(. , since no corresponding symmetry argument can be invoked. 43 From these considerations, the fluxes q, P and L can be ex-pressed as linear combinations of the tensors of corresponding rank 59 and inversion symmetry. The particular set of independent tensors given in Table I shows which phenomenological tensors have the same tensorial behaviour as the corresponding fluxes so that only these need be included in an expansion of the perturbation function <p . Definitions of the various S, and K tensors of Table I are given later in this chapter. The perturbation function dp when expanded in terms of the l i n -early independent macroscopic gradients w i l l have the general form* where the expansion tensors are a l l independent of d. . For a sys-tem in which the particles possess angular momentum, there are two independent vector quantities (or, more precisely, one vector and one pseudovector operator) upon which the expansion tensors may depend, * A convention in contracting tensors of arbitrary rank is used, namely, that a pair of indices which are physically closest to one another are contracted together, then the next pair, and so forth. (4-8) TABLE I. A complete set of irreducible tensors (at most linear in<* ) which are linear in the macroscopic gradients. Scalars Pseudoscalars Vectors Pseudovectors V x y e %'* (**.>*] Tensors ? s . Pseudotensors 44 namely, the reduced velocity W and the internal angular momentum 20 operator J. It was first pointed out by Chapman and Cowling and 21 later by Kagan and Afanas'ev that the solutions for the expansion of the perturbation function must contain a l l tensors of corresponding rank and inversion symmetry which can be constructed from these two vector quantities. 4.2 The Pressure Tensor. Due to the independence of the macroscopic gradients, certain of the terms in the expansion of cj?,Eq. (4-8), w i l l not contribute to the pressure tensor; upon examination of Table I it is immediately apparent that the relevant terms in this case are (to terms linear in oL) 4> - - B:v*. - <g;(v04- ( 4- 9 ) 39 A technique first introduced by Kagan and Maksimov for ex-panding the tensors B and (£> in terms of irreducible Cartesian ten-sors formed from W and J can now be ut i l ized. These tensors can be 45 written (in component form) as (4-10) and 7 « . . . = L <e£..wW'>t33<t> > where [ > 3 ( p > = C W - W D ^ u and C J j ' ? ) S C ? ' ' ' J 3 ^ th th denote the irreducible p - and q -rank tensors formed from W and B , J respectively.* The tensorial coefficients D ^ , . . c • and <Db,... Wp f^^  are functions only of W and J . It is useful to expand these tensorial coupling coefficients in terms of sets of associated Laguerre polynomials**, L_ s (w ; , and sets of discrete orthogonal th th * By "irreducible p - and q -rank tensors" is meant tensors in natural form, i . e . , tensors having the same weight and rank. This concept is fully discussed in Reference 59. ** These associated Laguerre polynomials are the same as the Sonine polynomials found in most treatises on kinetic theory. See, for example, References 6 and 7. 46 polynomials, (n) The associated Laguerre polynomials L g (x), having weight functions w(n) = x 1 1 exp(-x), are given b y ^ \ ^ \ ^ = ^ C-pTftt+S+QX* (4-13) A - O rW<:+i)<s-;)U! with the well-known property that o _ ^ ) The discrete polynomials having weight functions w(q) = Q Q 1 (2j + l)g(q) exp(-H'AT) with g(0) = 1, g(l) = j( j+l)and g(2) = (20)~ 1j(j+l)(4j 2 + 4j - 3) (the polynomials for q = 0 are those 47 28 introduced by Wang Chang and Uhlenbeck ) are chosen so that they satisfy the relations £ j=kT£^- is an eigenvalue of the internal energy operator H ' . Explici t ly, the (£j/feT) polynomials which are required in this thesis are R';\ vw; = R: C * . ) l - I J feT<T*> " (4-16) (4-17) and (4-18) 48 The pressure tensor P is given by (4-19) Equation (4-19) has been written in the above form to facilitate the evaluation of Writing a l l tensors in terms of irreducible tensors and carrying out the summations over p and q by util izing the fact that the various irreducible tensors are orthogonal when integrated over the angles of W and/or when the trace is taken over the degen-erate angular momentum subspaces, it is found that + f 0 O , o ) } : (V*. )* ( 4 " 2 1 )  and (4-22) 49 59 where the invariant notation for the fourth rank isotropic tensors is explained in Figure 1.* 2000 Since the tensorial coupling coefficients such as B . are 2 2 constants, ( i .e . , independent of both W and J ), they can be ex-pressed as linear combinations of the isotropic tensors of correspon-ding rank. This is given in Table II.** Chopsing the coupling con-«. + D2000 D00s0 ^OOsO _01s0 <o. 2000 , D2100 . stants B , B , (g , B , and B in accor-dance with Table II, Eqs. (4-20), (4-21) and (4-22) reduce to p =- M feT[B; o °°S +(Be"'-B"")V-v.y~}> <4-23) + ( 6 " 6 0 - © " " ) « } . - V x y . g ] (4-24) * The symbolic representation of these tensors in intersecting and overlapping LJ's as given in Figure 1 has been found to be very use-ful in practice and is a technique developed by Snider and Curtiss. [ R . F. Snider and C . F. Curtiss, Phys. Fluids 1, 122 (1958)J. ** The tensors appearing in this table have been chosen in accor-dance with Ref. 59. Two sets of isotropic fifth rank tensors are given, the first set corresponding to a reduction of a third rank ten-sor, e .g . (Vgi)oC, to a second rank tensor and the second set being an indicial permutation of the first which is useful in the reduction of the term containing c ' z o ° of Eq. (4-48). to follow page 49 TABLE II. Complete sets of independent irreducible isotropic tensors for ranks two to five Rank Isotropic tensors 2 3 4 5 to follow page 49 FIGURE 1. An Invariant Representation of Isotropic Tensors Rank Invariant Form 2 T 2 y 3 • T < v * = - s 4 5 § 1 4 L i | i ? J - t — (s'ify Y = - « [ UB- & "] 50 and E - H - B T S . + B T S . . + CB°'°°- B°"° y ] > <4-2S> where S is as defined in Eq. (3-32) and S^, § 2 are defined by § , = ^ [ ^ C V x y , ) + ( 7 x y 0 ) o i ] - j o C ' V K y e ( J (4-26) and ^ 1 = "1 L " " S ^ J * (4-27) Now, defining the phenomenological coefficients V | , 7^  ) ^ ^ ; ^ and ^ by the equation P - (nkT- x v - y 8 - X o L - V x O ^ - 2 > ? | -2>?( S -iy[x\iU-and comparing it with Eqs. (4-23) to (4-25), it is clear that 28) 51 and , ^1000 + _ l _ ( B ° ' 0 O - B 0 " ° ) ]• < 4- 3 3 > The auxiliary conditions imposed by the local values of the number, linear momentum, energy and angular momentum densities do not influence ^] , , or For X and Kj , the auxiliary conditions imposed by the densities of linear and angular momenta have no effect on the final result, while those of the number and energy densities lead to the auxiliary conditions (4-29) X- n l e T 1-&T--B,"'0! (4-32) 52 B -o (4-34a) . e o o i •a ^ o o t o C> 4. and ^ < 8 " ~ ^ . ( u . - u o f l " " " " - | (8' (4-3 4b) where b i s defined by (4-35) The final expressions for K and K( are thus 53 X - - n k T B OOI 0 (4-36) and B°"° ] • (4-37) 4.3 The Angular Momentum Flux Tensor. 55 Grad was apparently the first to introduce the concept of the angular momentum flux tensor, but did not attempt to determine the 14 transport coefficients. Curtiss was the first to derive a formal expression for L using the Boltzmann equation to setup equations for the evaluation of the transport coefficients for a c lass ica l gas of ro-tating molecules. An actual evaluation of these coefficients for the 16 spherocylinder model was laterobtained by Livingston and Curtiss . For the angular momentum flux tensor, the relevant terms in the expansion of the perturbation function can be written as ' % : ( V * ) * - Q.'&4«T)d - Ai-VAT, (4"38) 54 so that the angular momentum flux tensor L is given by n 7.TT (4-39) where X is independent of <k . The tensorial coefficients in Eq. (4-38) are expanded as in the calculation of the pressure tensor, v i z . , C < < t e K w v M ^ ' ' ( 4 " 4 1 ) 1st and where LwD ( P and C j ] ^ a r e defined as in Eq. (4-11). Thus 55 Using Table II to write A 1 1 ^ ^ and writing C 1 1 ^ as (4-44) X becomes £ = " r l w ' -& + c«i , 4 - k - V v * + a f .wf 4 4 5 ) where i L (the symmetric traceless part of the macroscopic tensor VoO and _T)!^ (the antisymmetric part of V £ ) are defined by - Q ^ - ±[v<6 - i V ' * y ( 4 _ 4 6 ) and •Q- 0 > - i [ w - t v * ) * ] - ( 4 - 4 7 ) 56 It can be shown that* where the identity ^ i l r ^ ( 4 r 4 - 3 T ' } + ¥ ) - 3 w] ( 4- 4 9 ) has been used. Thus, choosing a l l isotropic tensors in Eq. (4-48) in compliance with Table II, Y( ol ) is obtained as * The tensors C * ^ * and A * ^ * arise since for a diatomic molecule J is related to the reduced internal energy £j by the relation J = 2IkT£., where I is the moment of inertia of the molecule. 57 c,l0° A c l ) 3teT 1 « ' kT 1 - t ^ 3 ~ k ^ + 3 k . T ' 100 6i 4 [a1 1 , 0 0 , Z I c -+ It A I 00 I 3leT A' -1 -[ ^ "0 0 + U c , « A 1 0 0 1 " A 1 A K CO 10 ^ 3 + 3 k h q i e T ^ i A llOO 3fc OL-Vx <* (J (4-50) (2) (1) The macroscopic tensors -W-, , _1_Z. , / K and K are defined by means of the relations J u L ^ = i - [ _ C V x * ) * + c^Cv 1**)] - ^ - V x o ^ U ^ (4-51) 58 and J L M = ^ [ ( V « - i(.V*i)) , (4-52) = X [ * * 4 f ( 4 - 5 3 ) ft-) K = ^ [<* CViv^T) 4- (VJUT) V ^ T (J (4-54) ,C0 K r ^ [ ^ I W v J O 4 V ^ 7 ) < T ) . (4-55) The linear momentum auxiliary condition resulting in C 1 ( ^ = 0, A 1 0 0 0 = 0 and (X - (3kT) - 1 A 1 1 0 0 = 0 has been taken into account in the above expression for Y( oL ) so that the full expression for L is obtained simply by combining Eqs. (4-45) and (4-50). In the same way as the coefficients for the transport of linear momentum were defined, the angular momentum transport coefficients A .4 can be defined by means of the relation 59 Thus, comparing Eq. (4-56) with Eqs. (4-45) and (4-50), it is seen that the angular momentum transport coefficients are given by* T - - H iihl\"'L r"°° ^ ~ /Z [ M J I' > r - n 12kxV^ f 1 1 0 0  J i " IZ V v" ) 11 > T - nflkX \ " y " o c ? M ^ v o o + •> { 4" 6 o ) (4-56) (4-57) (4-58) (4-59) * A change in notation has been made from that used in Reference 58 where the results of this chapter have been previously reported. Several errors occurring in the expressions for the transport coeffici-ents of L have been corrected. 60 5 = - - I" 1 j " < " ' ! _ £ ' " " . - 1 , (4-63) and 61 4.4 The Heat Flux Vector. As is suggested in Table I, cross effects between the heat flux and the driving forces producing the angular momentum flux should be expected in the presence of a nonzero local angular momentum density. The reciprocal coupling was indeed obtained above in the expression for L , where certain contributions are proportional to the temperature gradient. G r a d , ^ 5 in a nonequilibrium thermodynamic study of such a system, first pointed out that such a coupling could 14 be expected. Curtiss has obtained an expression for the heat flux of a c lass ical gas of nonspherical molecules which explici t ly deter-mines the form of this coupling. The form of the perturbation function required for the determina-tion of the heat flux vector q is given by Eq. (4-38). Now, as the heat flux vector is defined by n T 3 ' T Q , 62 using Eq. (4-38); the result is + Upon integration over the angles of W , taking the trace over the degenerate subspaces, replacing the tensorial parts of the tensorial r f l . , A10st ^ lOs t l i s t ^10st ~ l l s t , ^10st coupling coefficients A i (A. # A , C , C and by isotropic tensors, and util izing the definitions of the polynomials R ^ ( H ' A T ) of Eqs. (4-16) and (4-17), q becomes t -63 - [ ( £ t u . ) a " - f a ' ° ' V ^ a ' M ' ( A " 0 0 - A"10) (bA"%u,A"")]^ ' . + LCf*"-)C""-f C' A . = I +zfr <C- O + 3 E T + < " ) ] • ( 4 - 7 2 ) where Q = i . i • cL , Q = i l -ot and Q = £(V'#) • Hence, de-1 2 3 fining the phenomenological coefficients (including the well-known coefficient of thermal conductivity "X ) by - X i j - 2 ^ - ^ - 2 ^ , (4-73) a comparison of Eq. (4-73) with Eq. (4-72) requires that 64 i I- C„x n " * " a b A " ° ' 1 , <«-"> „ = *&\gLfSf±c™-f C'°'°], (4"76> and *• 4 I / - I O B U _ 1 0 1 0 X - -In writing down these expressions, the auxiliary condition im-posed by the linear momentum density has been ut i l ized. This aux-il iary condition may easily be shown to lead to the conditions 65 A1 0 0 0 _ ^1000 I AllOO > (4-78) As w i l l be shown in a later chapter, because of the aforementioned cross effects, certain Onsager reciprocal relations can be expected to hold in this system. Suffice it to say at this point that some con-siderable simplification arises in the coupling phenomena on account of these reciprocal relations. 4.5 Discussion. At this point it seems useful to clarify what, exactly, has been attained so far in the development of a linear theory of transport properties of a gas with a nonzero local angular momentum density. Starting from the usual Boltzmann equation-one particle picture of a dilute gas consisting of nonspherical molecules, expressions have been obtained for the fluxes of linear momentum, angular momentum and energy. The local equilibrium state of the gas is parametrized by a mass density, stream velocity .temperature and angular momen-66 turn density. Near equilibrium, the thermodynamic fluxes are given as linear combinations of the gradients of the last three of these quantities. Because of the Galileaninvariance of the phenomenolog-ica l equations, fluxes formed from v^ and these gradients can be omitted, whereas fluxes formed from the angular momentum density J , or equivalently, its conjugate variable ^ with the gradients can -o not. In this way both the fluxes and the perturbation ^ depend, in general, on the vector cL though not on the vector v^ . This i n -creasesthe possible number of tensors on which the individual fluxes may depend; a tabulation of those possibilities which are at most linear in cL has been given in Table I. The phenomenological coef-ficients of the fluxes relative to these tensors are expressed in terms of certain tensorial expansion coefficients of the perturbation function A quantum mechanical approach has been stressed throughout this thesis. In the quantum mechanical approach, the translational motion is given by a Wigner distribution function, entirely analogously to the c lass ical treatment, whereas the rotational motion is treated in terms of a density matrix in rotational states. This is quite un-55 14 like the c lass ical treatments of Grad and Curtiss, in which the rotational state of a molecule is described in terms of its orientation 67 and angular velocity. As a result of this , Curtiss has based his dis-cussion on an equation of change for an average angular velocity rather than on an equation of change for an angular momentum den-sity as has been done here. Another point of departure is that Curtiss has given the rotational energy as a quadratic form in angular veloc-ity (or equivalently, angular momentum) with an orientation dependent moment of inertia. This has naturally led to an equation of change for an average moment of inertia as well as allowing one to eliminate the average angular velocity cd0 from the rotational energy expres-sion. Thus, in analogy to the peculiar velocity usually found in kinetic theory, Curtiss uses the peculiar angular velocity to define the fluxes of angular momentum and energy. A similar treatment could be accomplished in quantum mechanics, and such considerations suffice to identify Curtiss ' c*J0 with the present o£ if the Hamiltonian is quadratic in J . It would seem, however, to be more reasonable to base the treatment on a peculiar angular momentum rather than on a peculiar angular velocity. Further-more, the commutation relations for the peculiar angular momentum are not the same as those of the usual, angular momentum operators. For these reasons and the fact that for a spin in a magnetic field the energy i s not quadratic in J, the rotational Hamiltonian has been left 68 in general form with the consequence that both the hydrodynamic equations and the fluxes of energy and angular momentum are defined somewhat differently here from those given by Curtiss. An important consequence of the method of expansion in terms of irreducible ten-sors is that this technique leads naturally to all possible tensors upon which <$> may depend and also elucidates all of the possible tensors which can contribute to the various flux tensors. 69 CHAPTER V THE VARIATIONAL PROCEDURE 5.1 The Variational Principle. A variational principle for solving the Boltzmann equation in or-der to obtain the transport coefficients for gases was introduced by 7 Hellund and Uehling in 1939 and has since become the most popu-lar such method available. Its use in obtaining solutions for the Boltzmann equation when the principle of detailed balance is applic-able has been considered in detail by Hirschfelder, Curtiss and o Bird, while a modified variational procedure has been outlined by 61 Ziman in the case where a magnetic field is present (certainly causing the breakdown of detailed balance or "microscopic reversi-62 bility"). Snider has recently reviewed the variational methods as appliedto gases for which the principle of detailed balance does not hold. For a gas with zero local angular momentum density, the pertur-bation is considered to be an element of a vector space over the real field. A scalar product defined in this space which is approp-70 riate for the following variational procedure is <4I^>„ =iA.iC+rV*t'' l5"1) where 4^  is obtained from by time reversing the variables and the parameter v^. The linearized Boltzmann equation is obtained as an integral equation of the form where >^ is the true solution; the explicit form of this equation will be given later in this section. If, now, ^ is any function for which then the variational principle says that of all functions satisfy-ing this relation, <^> istheone which makes ^ Vl(R\ 0^anextremum. The inner product of the functions G and <R.H, ( G \<R \ VV)0 ' 71 is the analog of the square bracket integral of G and H employed by 8 6 Hirschfelder, Curtiss and Bird and by Chapman and Cowling, namely, [GvH] - «?I<RH>„ -- -^j"<V^,CH)<*? > ( 5 - 4 ) where the double dot product implies that the proper tensorial con-traction between G and H has been used. This assures that [G;H"3 is always a scalar. From the time reversal properties of the Hamil-56 tonian of the system, Snider has shown that [ S j H J - C H j = L H T ; G T 3 - < 5 " 5 ' The following discussion of the variational principle as applied to a gas with nonzero local angular momentum density follows closely Snider's work on perturbation variation methods for a quantum 56 Boltzmann equation. He has shown that if i = f&\r= 'f%id+«*xw+» <5-6) 72 then a simple scalar product for purposes of the variational procedure is <rI* >, = M f w i % r 1 x)=< * i n T , (s-7) in which is the time reversed operator corresponding to |^*r" where both system and surroundings (represented by and ot ) 56 are time reversed. As has been shown, when using this scalar product, the linearized Boltzmann equation must be modified into the form iM=cV r _ i = * • <5-8) Up to and including terms linear in J (or oL ), the nonequilibrium o expectation value* of an operator O can also be written in terms of the scalar product ^ \ /. as * The only expectation values of interest in this thesis are those of mVY, VI and Y(^mVa+H' -«<•]), whose equilibrium expectation values are nkTU, 0 and 0 respectively. 73 = i t f 4 o r { ( e ' ' ' l * e 4 ' I ) + If f ^ is expressed in the form of Eq. (3-29) where f ^ commutes o with <ty while, in general exp( oi• J/kT) does not, the scalar product of Eq. (5-7) can be obtained for small <^  from the scalar product ^ \ ^ 0 of Eq. (5-1) by means of the formula ^ <^iyX^TUx<t*'I,«+|y> e • (5-10) 5.2 The Iterative Equations. An iterative procedure can be utilized in solving the Boltzmann equation if a separation is made into oC -independent and cL-depen-dent parts on each side of the equation and the corresponding parts 74 equated. To this end, f ^ is expanded in accordance with Eq. (3-29), so that X ^ is given by J ^ T U W - < ^ H ' > . ( y ) < ) (5-11) while (/< <f> is (5-12) if quadratic and higher terms in ci are neglected. Now, writing 75 x(d) ^ ^ a n d ^ a s * X " *o + X , y (5-13) <R - <£0 +<R. (5-14) and where the subscripts o and 1 refer to terms independent of and linear in oL respectively, substituting these equations into Eq. (5-8) and equating the corresponding -independent and q(. -linear terms, a set of two equations is obtained, v i z . , X 0 = (R0 + . (5-16) and * The superscript "d" is understood in a l l that follows. 76 X, - <R, 4>. ^ C • (5-17) Here, Eq. (5-16) is just the usual Boltzmann equation for a system with zero local angular momentum density, while Eq. (5-17) is new. The existance of nontrivial solutions of the inhomogeneous equations requires that the solutions of the adjoint homogeneous equation (both equations have the same homogeneous equation whose adjoint equation has the summational invariants as solutions) must be 63 orthogonal to the inhomogeneous terms. That this condition is sat-isfied by X Q i s at once obvious; it may easily be shown to hold for - ^>0 also since the scalar product of a summational invariant with $,4j is just a modified square bracket integral involving a summational invariant, and this clearly vanishes. Thus, nontrivial solutions of Eqs. (5-16) and (5-17) exist and their solutions can be approximated using an iterative procedure. To achieve this, an approximation for 4? * s obtainedfrom Eq. (5-16) using the standard variational procedure,^ 2 ' ^ 4 and this value of 4? is substituted into Eq. (5-17) which is then used to obtain an approxi-mation for in an analogous manner. By writing <X> as and uti l izing the linear independence of the macroscopic forces, two sets of Boltzmann equations are obtained. The first set of equations i s b= <J?0B„, <5-20» and 78 These equations are referred to hereafter as the equations of the first iteration. The second set of equations is + ( i e T ) 1 ' # ( £ - - ( ( e l f <£0 (5-24) and CVT<1X> - kT<Rt B c - W T ( £ t o ^6 > ( 5 " 2 5 ) and is called the equations of the second iteration. 5.3 The Onsager Reciprocal Relations. There are two ways of looking at the transport coefficients for a gas with nonzero local angular momentum density. The first of these i s to use 0^ -dependent macroscopic forces (as has been done in 79 Chapter IV) while treating the resulting system as isotropic; the sec-ond is to consider the transport coefficients as -dependent while retaining the original set of macroscopic forces, treating the result-ing system as anisotropic. Whichever interpretation is used, the final results must be the same. The latter interpretation, that i s , considering the transport coefficients themselves as dependent on the local angular momentum density is similar to a technique which has often been used in the evaluation of expressions for the transport properties of an electron gas in the presence of an external magnetic f i e l d ^ (e.g. the H a l l , ^ Righi-Leduc, ^ ' ^ ' ^ and magneto-67, 68, 69 resistance effects in solids). The principal advantage offered by the first-mentioned interpre-tation of the transport coefficients is that it allows direct application of the well-known Curie principle to an isotropic system. This principle in effect states that the Cartesian components of the fluxes do not depend on a l l the Cartesian components of the thermodynamic 70 forces. One of the results of the Curie principle is that fluxes and thermodynamic forces of different tensprial character do not couple in anisotropic system. Thus, a classification of the thermodynamic forces and fluxes by their behaviour under space inversion makes it 80 possible to tel l upon which thermodynamic forces a particular flux can depend. This has already been done in setting up Table I. Con-sultation of Table I leads, then, to the conclusion that there can be no cross effects between the pressure tensor and either the heat flux or the angular momentum flux. However, since a second rank pseudo-tensor has the same tensorial behaviour under space inversion as a third rank tensor, cross effects can be expected between the angular momentum and heat fluxes. Having eliminated the possibility of any cross effects occurring between P and either of L or q, the system to be considered has been somewhat simplified. The nature of the cross effects between L and q may be most conveniently treated by reverting to the anisotropic description of the system, for which case only the four thermodynamic forces V ' ^ . £ V ^ ] ( ° , LVr f®, ^ T a n d the four fluxes it L , , U and q need be considered. Thus T r"bl L £ H V i + | ( , ' V i v i T + | ( 3 : L 7 * ] t , V ^ ; , C 7 ^ T (5-26) 81 or, in matrix notation, (5-27) are the equations giving the fluxes as linear functions of the indepen-dent thermodynamic forces.* The Curie principle can give no further information regarding the off-diagonal components of this set of equations. However, a set of nal components of . These reciprocal relations are obtained through considerations of time reversal symmetry. Along this same vein, a de-tailed discussion of the irreversible thermodynamics of a fluid with a nonzero local angular momentum density stressing a l l the possible tensorial properties of the phenomenological coefficients has been car-72 ried out by Snider and Lewchuk. An immediate deduction can be made regarding the phenomenological coefficients *C,4 and from a cursory examination of Eqs. (5-26). Onsager reciprocal relations can be expected coupling the off-diago-* It is important to bear in mind that these phenomenological (or kinetic) coefficients are functions of the pseudovector parameter ol . 82 To terms linear in 0^ there is no way to construct a symmetric trace-less second rank tensor from cL , U and £ and so <^,4= =0. Afurther examination of Eqs. (5-26) reveals that four Onsager relations can be expected connecting the phenomenological coefficients of q and L: specifically, one from with £ , two from < ^ 2 3 with and one from with 4^2. • • ^ w o f u r t n e r reciprocal relations connecting four of the phenomenological coefficients of L should be expected, namely, from with and with These relations are obtained through the two Boltzmann equations 3 V = kT(RC (5-28) and feT k T with the aid of the scalar product ^ \ ^ . The heat flux vector is defined here as (5-30) 83 where the minus sign comes from time reversing V(^mV + H ' - cL*]) as required in the scalar product K \ ^ . A definition of q including the 55 53 ^.•J term has also been employed by Grad and Dahler. Since no contribution to q can arise from the terms ^7v and ^7«v , the pertur-— —o v o bation function can be written as ^ - _ A - v - ^ t - C : V * <5-31> for the purposes of these calculations, so that from which the appropriate coefficients can be identified. Similarly, the angular momentum flux tensor is 84 The cross terms between L and q are denoted by € | l ( * ) = - < v- l i " v * + H ' - I i \ (5"34) and t^W - <YJ!A>, (5-35) Recognizing that constant multiples ofV vanish in these phenomenolog-ica l coefficients due to the linear momentum auxiliary condition on c£>, and X . q C ^ ) can be related through the following sequence of steps: - -feT<<*A 1C>, =-leT<A|«C>| = - <A|xv> ( =-<yr|A>* T = - ( ^ ( * ) ) T - (5-36> 85 The superscript "t" on £^(9?-) signifies a transpose of the first and third indices of ^^<t)> Thus, the Onsager reciprocal relation ex-pressing the reciprocity of the cross terms between q and L is itLLi) = (5-37) The change of sign of d± follows from the fact that this is the only effect of time reversal on Is L<J ^  ^  • Since these third rank tensors can be at most of weight one in oL , they can be expressed as linear combinations of the tensors In accordance with the definitions of the phenomenological coefficients in Chapter IV, the scalar coefficients of these tensors in and e C are identified through the following two equations M U ^ - W O +^ *y < s" 3 8 > and 86 + A 1 i - [ ^ - y f l 4- A 3 i U - (5-39) The Onsager reciprocal relation of Eq. (5-37) then implies that i? = - A 4 < 5 - 4 0 ) and tf. = A . U - J . 2 , 3 ) . (5-4D Four of the six expected Onsager reciprocal relations have now been shown. The final two reciprocal relations are obtained from an examina-tion of the symmetry of - kT<ct|d?C>. > ( 5- 4 2 ) 87 which has used Eq. (5-28) and the self-adjointness property of in the scalar product Hence, again using Eq. (5-28), j £ ( gj. ) becomes - <yj|c>1T <5"43> where the superscript " t ( l ,4; 2,3)" denotes an interchange of the first and fourth indices simultaneously with an interchange of the second and third indices. With this result, the Onsager reciprocal relation expressing the internal symmetry of i s Since this tensor is fourth rank and at most of weight one in , it can be written as a linear combination of the fourth rank tensors T/* ' 0 * = 1, 2, 3) and t ^ | T ( , f , 0 ) ( X = 1, 2, . . . , 6) as u «--1 v T T r ; s ' ° ' > ( s - 4 5 ) 88 where the scalar coefficients T^ . and are synonymous with those defined in Section 4 .3 . Substitution of Eq. (5-45) into the Onsager reciprocal relation (5-44) results in the following two relations amongst the % 's: and (5-46) 89 CHAPTER VI THE FIRST ITERATION — ZEROTH ORDER IN oL . 6.1 Method of Approximation. The iterative equations are set up, naturally enough, so that the first iteration determines the transport coefficients for vanishing J . Aprocedure of approximating the expansion of the perturbation function for this iteration is employed, in which, if a certain transport coef-ficient, as expressed in Chapter IV, involves a set of i expansion coefficients with corresponding tensors T ^  , it is convenient to approximate ^ by 4> - ZL 3 s*Tv •7n ( 6 _ 1 ) where "TT] i s the appropriate thermodynamic force. With this approx-imation, a set of i equations in the i unknown coefficients is obtained by substituting the approximation of Eq. (6-1) into the cor-responding linearized Boltzmann equation and then dotting in each of the approximation tensors in turn. The set of equations so obtained 90 may be represented by ch ~-t (>"^-.*) (6-2) in which the h ^ are square bracket integrals defined by V - - ( U - - C r , ; r j ( 6 - 3 ) while the c^ are constants given by (6-4) and X is the left hand side of the appropriate linearized Boltzmann equation. Since a l l expansion tensors T., that are used in this the sis ~r satisfy the restricted commutation 7~:T~ = ~f > T / it follows that the square bracket integrals of Eq. (6-3) can be written as V - - - c « « * , J ] j r c : ^ ' [ < ( r ; + r ; ) C ' 91 where the prime on T' denotes, once again, that W is to be replaced by W ' . This expression can be further simplified by introducing the reduced relative velocity ^ and the reduced center of mass peculiar velocity ^ defined implicitly by (6-6) The Jacobian of the transformation to relative and center of mass co-ordinates implies that J J J • * p f c 4 - T * (H'+HD/krlr'dMl<LSL , (6-7) where dSL is an element of solid angle for the variable ,^ ' . Hence h ^ m a y be written in the form V > = a ^ M e T h \ \ **? [ - A a (H V H ; )/kT ] V C ( T ^ £ c i r a a . ( 6 - 8 , Clearly, any additive part of which is a summational invariant w i l l give no contribution to . This is likewise true for T^due to the symmetry of the square bracket, Eq. (5-5). 6.2 Coefficient of Shear Viscosity. Since the coefficient of shear viscosity, lr\ , is determined by one expansion coefficient from the expansion of the perturbation func-tion, cf. Eq. (4-29), it is sufficient to choose* which, when using the Boltzmann equation (5-20) and the method of 2000 Section 6 .1 , results in an equation determining B . Thus _ -7D0O / f / 0 —ro) , u ' \ (6-9) (6-10a) * The breakdown of § into B 0and B ( = £- • $ corresponds to the breakdown of c|> into 4^ and <4?( (Section 5.2). 93 or 1O00 JL B - (6-10b) where the constant , as determined from the left hand side of Eq. (5-20), is r ( | ) - S H (6-11) and D. , the corresponding square bracket integral of Eq. (6-8), is I given by 94 f 3 7 .iia/1U)C %le)Et\^ciJk di (6-12) Since W , W' commute with and t<£ this can be further simpli-fied to \ . . . W A V - ^ 2 / feT^r- x f f K e " ' H W V f c r ( w 4 w l v . ) : f <^«' '-jw'V + ^ ^ i ^ - i i s V K i M ^ V i o A | u > J U A . (6-i3) 95 This integral can now be expressed in terms of the degeneracy-averaged cross section defined as ~ 4*rt,t (6"14) where - X and Cp are the polar and azimuthal angles describing the orientation of \ relative to V and is the degeneracy of the energy shell denoted by j ( i . e . , 2j+l). In the last form for the cross section, t is an abbreviated form for a matrix element of t* , namely, I H < i ' j , ' H : { | n > < 6 - 1 5 ' which is s t i l l an operator in the degenerate subspaces labeled by m, m., and t * i s the adjoint of X in this space. t a # J de notes a trace over a l l the degenerate subspaces. Combined time reversal and space inversion invariance of the Hamiltonian implies that - <-vn rrn, \V w / ) j (6-16) 96 where t ' is identical to t with the primed and unprimed indices i n -terchanged, i . e . , = ^ j j ^ \ t | ' | ' ' ^ ] _ ^ * F r o m Eq. (6-16) it fo l -lows that ^ j ^ a - j ' ' ' " ^ . ^ ^ - <r-';(Y,K,f) (6-17) and thus D ( becomes f f ( w l ^ J ( / } Vcjj-o^ ^ ' ( ^ ^ S t E ) ^ ^ ^ - ^ - ( 6 " 1 8 ) Now, utilizing the definitions of ^ and ^ given in Eq. (6-6), inte-grating over the angles and magnitude of and using the fact that conservation of energy during a coll is ion requires that y - y ' 1 - n A £ (6-19) 97 where (6-20) it is not difficult to reduce b, to I — J ^ 3 3, (6-21) I. A further simplification of D, can be attained by utilization of the relation E Y e *"ajcv j v^'* 4^fH&<^ JUT whence 98 "I H,11 J ' ' so that from Eq. (4-29), the. coefficient of shear viscosity 'Vj is seen to be >1 = 5" ( T i ^ l e T ) 2 L (6-24) which is in agreement with the results of Wang Chang, Uhlenbeck, 29 30 and de Boer and Mason and Monchick with, of course, the intro-duction of degeneracy-averaged coll is ion cross sections. 6. 3 Coefficient of Bulk Viscosity. An expression similar to that for the shear viscosity, but with one important difference, relates the coefficient of bulk viscosity, 99 X . - to an expansion coefficient of Cp [see, e . g . , Eq. (4-36) J . This difference arises from the auxiliary condition of Eq. (4-34a) which links a second non-vanishing coefficient to the one given in Eq. (4-36) and which must, therefore, be included in the expansion of cp • Hence, the approximation involved in this calculation is Co) + I (6-25) which, together with Eq. (5-21) leads to (6-26) where _ vj\ _ H'/hrr e e V-|VU (6-27) 100 and -^(^) zT~^&6 ]<kJbdl*^ic<Lxdf *e" € r £ j | (ke)%A^%*%dy A). (6-28) Thus, from Eqs. (4-36), (6-27) and (6-28), the coefficient of bulk viscosity is 6 • ••/•/ JJ.jJ, ^ I'I" (6-29) 101 6. 4 Coefficient of Thermal Conductivity. In the case of the coefficient of thermal conductivity, there are two expansion coefficients to be calculated, [^see Eq. (4-74)~\ , hence two simultaneous equations are to be solved. The approximation con-sidered here is then 30) so that the two equations to be solved are* i ,1010 , i i o o l N a. ^ h „ A +- h L X A ^ i o ( o i o o | (6-31) The constants and a^ are given by * In actual fact, the expansion should include the term in A determining A 4 , so that three equations are obtained, the third of which is homogeneous, allowing A"° to be obtained in terms of A and A 1 ° ° Xsee Eq. (6-53)3. To terms independent of the nonspheri-ci ty, the two equations (6-31) are a l l that is required for the deter-mination of "X . 102 and - i r h S±*. (1^L\//V ( 6 - 3 3 ) while by the methods employed in Sections 6.2 and 6.3, the h ^ square bracket integrals may be obtained in.the form with X - 4 L J T U . V + Htef( $ - ^ 1 ( 6 " 3 5 > . T , z - | C A £ f ( 6 " 3 6 ) and 103 30 in agreement with the results of Mason and Monchick. It should be noted that particle symmetrization and symmetrization with respect to primed and unprimed variables have been used to simplify these a 1 0 1 0 ^ aIOOI formulae. The expansion coefficients A and A are therefore given by A i bio a , a . (6-38a) and A too where Sb i (.6-3 8b) s the determinant of the square bracket integrals (6-39) Substitution of these expressions into Eq. (4-74) results in an ex-pression giving the coefficient of thermal conductivity X / again in 30 agreement with the form found by Mason and Monchick. 104 6.5 The Rotational Diffusion Coefficients. The coefficients , and are called in this thesis the first, second and third coefficients of rotational diffusion re spectively. Since, as can be seen from Eqs. (4-57), (4-58) and (4-59), these coefficients each involve only one expansion coefficient of ^ , relatively simple expressions can be obtained for them. From Eq. (5-22), the integral equation for CQl it would appear from first i n -spection that the three coefficients are in effect coupled; on closer inspection, however, since e .g . °^ Eq. ( 4 _57) carries with it a weight two tensor made up of J and W, it w i l l pick out only the weight two parts of C Q and JV upon contraction with them. Similarly, -,1100 , „1100 + , . , + , + . . C ^ a n c * C-^2 carry tensors of weight one and zero, respectively. Because of these tensorial properties, a separation of the three sim-ultaneous equations in three unknowns into three separate equations, (each containing only one unknown) is accomplished. In particular, the equation for C ^ is < ^ [+.+; f ' p w r * ^ _ . . . , e i V ( - j 9 - / r _ . . , ^ r , ...>,V\l*t' X - ( t T ^ + B " , ^ j ^ O ^ V f i f ( - p ) 4 ( % ^ p . ( 6 - 4 0 ) 105 Afurther evaluation of this expression requires the introduction of the 73 generalized coll ision cross sections mCD^ C T ^ ' c o l o l =7fT-?lt* ^ (6-41) a j : £ D . - ^ 5 : f J ' L 0 1 1 1 E W T ' T , , (6-42) .^•uJj. ^ ^ [ ^ r v ' l T l r Tfl'lJjt*! (6-43) and eo'.u;: o~^' 11 = yy i-i^'itV (6-44) in which 7 ^ is the operator defined by 7^=: L(xTi)4tl'LYn'i'7l'ti,. ' (6-45) and which a l l satisfy the time-reversal, space inversion symmetry property ^ j / V ^ C A l B D = ^y^;^^:tK\K^ (6-46) 106 With these generalized cross sections, Eq. (6-40) becomes c uoo 5- /a^ 3 n k>T ^ ( T t o 103 - ^ L H o 3 ) + ^ ( 5= D ' t r ] - ^ ( c r [ o o | 3 _ 0=-Loi| 3)y<*SL<ftJ ' (6-47) so that, £see Eq. (4-57)"] V is given by b / 7C i x (6-48) 107 = i'i! All of the cf 's used above are labeled 0~3~ ' ( ^-')-In a similar manner, the second and third rotational diffusion co-efficients Xj, and are determined to be - <r -1 (6-49) and 3 7zl^/ery L Q a o i y J J\ 4- ^ [ } W X d ^ f ^ . (6-50) 108 6.6 The Coefficients Z*4 and The only remaining coefficients defined for the flux tensors inde-pendent of d- are in the heat flux vector and A4 in the angular momentum flux tensor. Because of the Onsager relation (5-40), it is sufficient to determine only (which is simpler than 1) ) . can be approximated by (see footnote on p. 101) a a 1 0 1 0 ( J M A i o o l O c o } a I ' 0 0 (,S Ao - A wL,(wl; + A wK,^) + A R0^ > (6-51) for the determination of A 1 1 ^ . The three equations obtained by the variational procedure are a. , k A'0,%k,A'0°' + K,A"00 ' 1 ' l | ' * 1 "(2. IOIO , A 1 0 0 1 &x - h 2 ) K°° + h^A vk l 3A 0 n 0 0 1 0 0 1 it© 0 (6-52) Since the third of these equations is homogeneous, it may be solved 1100 r 1010 J 1001 to give A in terms of A and A , v i z . , i HOO I . . , 1 0 1 0 , . 1 oe?l \ h 3 3 (6-53) 109 The constants a, and a . and the square bracket integrals h , h 1 9 1 1 11 and h are as determined in Section 6.4, while h ^ X - W l ^ , ~ 0 ? (6-54) /J/J J/ J and x (T»y'fl^'Xcr[o-i|o-i2 - o" [>M2 - V D ) ^ / ^ f d ) ( 6 _ 5 6 ) where T » ( £ * $ ' ) = ^ t f W i ^ (6-57) and the cross section defined by 110 (6-58) have been used in obtaining Eq. (6-55). With these equations and Eq. (4-69) defining , an expression for A .^ (and hence for ) is obtained, v i z . , n. (i\er\<i/\"00 H 2kT\ h *3 cxMu - aM .(6-59) I l l CHAPTER VII EFFECTS WITH LINEAR (X_-DEPENDENCE 7 .1 Preliminary Remarks. As has been stated previously in Chapter V, the transport coef-ficients for the macroscopic gradients which depend linearly on can be obtained from the second iteration of an iterative solution of the Boltzmann equation. It is possible to eliminate completely from the equations of the second iteration j^Eqs. (5-23) to (5-25) ]^ when the explicit form of the integral operator (R. | is considered. Unfortunately, a higher approximation for is required for the evaluation of the transport coefficients in the linear cL-dependent effects than in the oC -independent effects (that i s , it has more terms in.the expansion). Thus, a new first iteration must be performed to the variational method in these two iterations can be represented as higher approximation. The equations obtained by 112 = S ^ r ( ^ l , l , S - l , V " J) (7-1) where the c r are constants, the h are modified square bracket inte-rs grals and the coefficients p^ are identified in Table III.* In this chapter, a reasonably detailed calculation of the transport coefficient V ^ i s presented and the procedure outlined for obtaining expressions for the other "linear-in- qL " transport coefficients. 7 . 2 The Anisotropic Viscosity Coefficients. The simplest transport coefficients to calculate for the linear oC-dependent effects are the three anisotropic viscosity coefficients "^ |,Tj 1_and X ( . Of these three, Y[x can immediately be seen to vanish since from its definition, Eq. (4-28), it is a coefficient for a macroscopic gradient which is essentially antisymmetric in V y 0 and there is no corresponding term on the left hand side of the Boltzmann * Note that in Eq. (7-1), a summation convention has not been used. The superscript " 1 " distinguishes between the first and sec-ond iterations. to follow page 112 TABLE III. Identification of the /S coefficients for the iterative solutions. 1. First Iteration. Coefficients Identification Expansion Tensor IT-D2000 B l A 1 0 1 0 2 w w - w y (1/2) 0 L ( W ) W A 1 0 0 1 RJ°* ( H ' A T ) W „ 1110 A 1101 A i > 2 ) J W : T ( 3 ' 0 , R ( 1 ) 1 - - = o R ( 1 ) ( H ' A T ) J W : T ( 3 ' 0 ) i3 1200 A -1100 C l 1200 ° 1 1200 ° 2 (2) R V ' ( H ' A T ) W o R ^ ^ ( H ' A T ) W • o R ^ ( H ' A T ) W ' o R ^ ( H ' A T ) W • l i 4 ' ° ' - l T ( 1 S , ' 0 , : C i n ( 2 > 14 /*-'4 c i i o o 2 • c i o o i R ^ ( H ' A T ) W • R | ° ' ( H ' A T ) W • T(3,0) to follow page 112 TABLE III. Identification of the /S coefficients for the Iterative solutions. 2. Second Iteration. Coefficient Identification Expansion Tensor I 8- . W W : T( 5 ' 0 ) 12 R ( 1 ) ( H ' A T ) J W : T ! 4 ' 0 ) o '- - =1 a T (1) (4,0) R ( H ' A T ) J W : T o - — =2 A a '0 0 0 (3,0) w • t\ - =3 23 ft L ' ; 1 W2 ) W . T ( 3 ' 0 ) ^ 1 00 1 (A R ^ H ' A T j W . f ' ^ lido a 3 R^ (H'AT) JW^Tg . i s -R ^ ( H ' A T ) J W : T ( 2 5 ' 0 ) as" c (1), „ x (5,0) R (H'AT) JW : T o - - §4 (1) (5,0) R L ' ( H ' A T ) J W : T o - _ §3 (1) (5,0) R Q ( H ' A T ) J W : T 5 113 equation. A detailed analysis tr ivial ly confirms this symmetry argu-ment, as it leads to two simultaneous homogeneous equations with the determinant of the coefficients non-vanishing. Similarly, X\ also vanishes, again because terms proportional to do not arise on the left hand side of the Boltzmann equation. For the calculation of ")] , must first be determined from the integral equation b_ = # 0 3 6 (7-2) with B q approximated by* BO = B ^ W - y y ) 4- <T> $ T z ; , r o > . (7-3) The two simultaneous equations so obtained are * Only the first term of this approximation was used in Chapter VI because, as w i l l be seen, B T is linear in the nonsphericity and is hence small compared with Bf00 . For this same reason, the term in the left hand side of Eq. (5-25) arising from B^' ° w i l l be neglected in the second iteration as it w i l l give rise to a term proportional to the square of the nonsphericity. 114 E B, fc„ + (7-4a) and ^ i 0 0 ° ^ 6r,o) -, x - L _ - R T , 0 ° O ^ T 6 ^ ' t ^ : T M ) ] ^ B , 4-13, b a x (7-4b) where is given in Eq. (6-11) and b ( | is clearly just the of Eq. (6-23). The other square bracket integrals and O^have yet 2000 , 2100 to be evaluated. Thus, and a r e and g I , 0 ° _ ^ £ x ( ^ Jr, bu . (7-6) 115 The evaluation of the square bracket integrals and t ^ i s straight-forward but not simple. Explici t ly, 1D1( is given by * S C j g ' + p ' ^ F / d p ^ J ; o l ( D 5 (7-7) which, upon converting to relative and center of mass coordinates via the transformation of Eqs. (6-6) and (6-7) becomes * ^ 2 " * ^ *l ? (7-8) where the terms arising from WW with WW and W^W^ in the square bracket integral have vanished due to tensorial contraction and/or integration over the angles of . Similarly, the terms involving and J h in the first bracket of Eq. (7-8), being odd in J van-ish upon integration over the angles of ^ while the terms containing A%%M and i i i w i l l vanish upon contraction with T leaving only the term containing tfjf^'^ contributing to the integral. Hence, 116 after performing the integration overdo and the angles of 1 . K, be-comes 21 (7-9) By utilizing the relations (7-10) and defining the cross section (7-11) where % is the angle of deflection in the center of mass system, simplifies to JJ|J J, (7-12) 117 In an entirely analogous manner, D 2 2 ^ s f ° u n c ' t o be * c r (o + i i -o+i ) + a - ( o i l t o + i ) 4- % i o x % < r (nJ ,U - ^ ( r C l H , ^ ) , ) ] ^ ^ ^ ^ ^ - ( 7 " 1 3 ) 2000 2100 ^ 2 0 0 0 Having determined B and B , only QV, has yet to be 1 2 ^ obtained for the evaluation of ^ [.see Eq. (4 -31)~ | . The equation of /Q 2000 the second iteration which determines IQ is (7-14) which, with the approximation 6 ^ ( 8 " ° ° w:T^'0> (7"15) for the expansion tensor leads to an equation giving in terms of and two square bracket integrals, v i z . , 118 1.000 1 I / ? & • H p , = — i [ y jy . -^^^j j f jyy .y iv ' t / ) ] ( 7 _ 1 6 ) Simplification of these square bracket integrals into expressions in terms of cross sections, although not t r iv ia l , is straightforward. The 21 21 square bracket integrals of c^ and h are thus 100O _ . , /» je-rcj- v ^ ^ y T u t J J J' ^ [ ^ V j u | ] CM'XM^J tfdif d) (7-17) and Substituting these equations into Eq. (7-16) results in an expression r rt. 2 0 0 0 • for ko ^ , v i z . , 119 (7-19) As is expressed in terms of B 2 and Co a s I _ n o o X (4-31) it can be obtained as proportional to Vj through the equation i * (7-20) when t can be neglected compared with Vjn D 120 7.3 The Transport Coefficients for the Other "Linear-in- £ " Effects. Only an outline of the procedure to be used for obtaining these transport coefficients is presented. For determining the coefficients I"i =~^5> , ^ , 5 3 ' % = "£& / the expansion coefficients are again de-termined by an iterative procedure. By means of their tensorial weights and symmetries, they can be split into four sets, each of which may be determined independently (see the argument tendered in Section 6.3, p. 103). Thus, two* sets of expansion coefficients are to be obtained 1100 1200 1200 1100 1001 in the first iteration, v i z . : C^ , C^ , Cg ; and C^ , C ; _s> II O O and similarly two sets in the second iteration, these being: "fe^ ^ I I O O \\00 jpUOO teq ; and toj , . These expansion coefficients and their corresponding tensors together with their assignment as the ^ 's of Eq. (7-1) are found in Table III. Analogously, the expansion coefficients used in determining the transport coefficients A , , A x , A 3 a n d ~X , the A's and ( X ' s , can 1010 1001 be divided into four sets. The first set of coefficients is A , A * Two sets only, since Cg is the only expansion^coefficient which carries with it a weight zero tensor, i . e . , J»W R o U , and is therefore as determined previously in Chapter VI. 121 . 1 1 1 0 . . 1 1 0 1 ' , A 1 2 0 0 . . . u , t . , , .• ,. A , A and A which can be determined from the first ltera-n l l O O ~ It O O ^ Ioeo , o i o tion; the second set, LL. ; the third, CA, , 6 ^ , (a - and l l 0 0 ; and the fourth, CX^ . Each of the last three sets of coef-ficients can be obtained from the equations of the second iteration. ^ : and the fourth. LA. 122 CHAPTER VIII THE SENFTLEBEN-BEENAKKER EFFECT FOR THE THERMAL CONDUCTIVITY 8.1 Introduction. The effect of a magnetic field on the transport properties of a dia-tomic gas is by now well-known experimentally. Recently, a number of investigations have been undertaken studying the coefficients of 50 49 51 thermal conductivity and viscosity ' for a number of diatomic gases. Magnetic field effects were first reported for diamagnetic dia-49 tomic gases by Beenakker e t a l . as recently as 1962.' This same type of magnetic field dependence had (under the name "Senftleben effect") been known for paramagnetic gases for many years. ^ A magnetic field decreases the measured values of the coefficients of thermal conductivity and viscosity by around 1% at saturation and results in a dependence of these transport coefficients on the variable H / p , where p is the equilibrium gas pressure. This decrease is quali-tatively seen to be reasonable, since the class ical col l is ion cross 123 section depends on the (relative) orientation of the molecules and, since, a magnetic field causes a precession, the orientation of a molecule relative to its velocity direction varies between col l is ions . This pre-cession effectively results in an extra averaging process caused by randomization of orientations between col l is ions, which in turn causes an increased cross section and a consequent decrease of the transport coefficients. 8. 2 Form of the Boltzmann Equation. Since the Boltzmann equation employed in the earlier part of this thesis does not take into account the presence of a magnetic field, the first task is to determine the appropriate form of the Boltzmann equation necessary for treating the magnetic field dependent effects. Following 38 74 39 Waldmann ' and Kagan and Maksimov, it is assumed in this thesis that the effect of a magnetic field on the Boltzmann coll is ion term can be ignored. Thus the magnetic field enters only in the stream-ing terms of the Boltzmann equation and the form of these are ascer-tained. To this end, the equation of motion for the singlet density namely, 124 where ^ is the two particle density matrix. The subscripts "1" and "2" refer, respectively, to particles 1 and 2. The Hamiltonian for the system, , is written as a sum of one particle Hamiltonians V and « A and an interaction term V-^ a s w x°'+K° + (8-2) In Eq. (8-1), the term containing V is . re sponsible for coll is ion effects and, as magnetic field effects in the coll is ion term are ignored, it 35 36 w i l l be identically given bythat derived by Waldmann and Snider. Thus the coll is ion term is as given in Eq. (2-1). Since translational states are treated c lass ical ly , the singlet density matrix can be converted by means of the Weyl correspondence 26 7 5 to a Wigner distribution function in position-momentum space ' while retaining its density matrix properties in the space of internal states. Before carrying out the Weyl correspondence, the Hamiltonian ft ^ ) i s expanded in a Taylor series about r, keeping only the first two terms, 125 7/0)(e) ^ rf'ir) *(<*-r> ^ * (8-3) so that Eq. (8-1) can be written as - ( Coll. W v ) . (8-4) It is convenient to split the single particle Hamiltonian into two parts, the ordinary momentum term for the translational states and a term which acts on the internal states only. Thus, J% is written as (8-5) Now, the Wigner distribution function is defined in terms of the singlet density matrix jO0^ by * (? , & are the position and momentum operators, respectively. No confusion should arise between this (? and the density matrix (p of Eq. (3-7). - f ( - r ^ ) 126 (8-6) JO, when jO is in position representation, or by (8-7) when is in momentum representation. Thus, under the Weyl cor-respondence df/dt becomes (8-8) and using Eq. (8-5), — [[^(V) j ^ 0 ) J splits into two terms, (8-9) and 127 ft 1 .§£ (8-10) In a similar manner, using the position representation of Eq. (8-6), the final term arising from the commutator is KJ L 2>± J + (8-11) Hence, the appropriate form of the Boltzmann equation to be employed in the presence of a magnetic field is 128 - * f t * j * A V fe, f E ^ f f , - ^ ^ ' j ^ . (8-12) The form of the left hand side of this equation has also been obtained 7 6 by Emery although he gives no derivation. In the absence of exter-nal forces, the anti-commutator term of Eq. (8-12) vanishes. In par-ticular, for a homogeneous magnetic f ield, there w i l l be no contribu-tion from this term. 8. 3 The Tensor Equations Determining the Anisotropy. Just as Senftleben was the first to observe that a magnetic field 40 affects the thermal conductivity of a paramagnetic gas, so Beenakker et a l . were the first to observe the effect of a magnetic field on the 129 transport properties,' in particular, the viscosity, of diamagnetic gases. He reasoned that any effect produced by a magnetic field on a diamagnetic gas would have to occur at much higher H/p values than for a paramagnetic gas since the gyromagnetic ratio Y of the former is much smaller. On this basis, the effect of a magnetic field on the transport properties of (diatomic) gases is herein calledthe Senftleben-Beenakker effect. Since, as has been already discussed in the previous section, the magnetic field appears only on the left hand side of the Boltzmann equation, the linearized equation used in treating the thermal conduc-tivity takes the form Certain assumptions have been made in obtaining this equation from Eq. (8-12). Firstly, 7/.^ is given as the sum of the magnetic field independent Hamiltonian H 1 and the 2eeman Hamiltonian -"tfH'J, thus %T = H'-*H-T. (8"14) Secondly, J Q is assumed to be zero and thirdly, H ' is assumed to 130 commute with <^=> (hence with A). The thermal conductivity is given by a formula similar to Eq. (4-7 4). However, in this case the expansion coefficients are tensors, made up in part of an anisotropic contribution from the magnetic field H and so the resulting thermal conductivity tensor is (8-15) The auxiliary condition arising from the linear momentum density, Aiooo r O , <8-16> has been used in obtaining this equation. 39 Following Kagan and Maksimov., the expansion coefficient A is expanded in irreducible Cartesian tensors of W and J. The first term contributing to the commutator of Eq. (8-13) and hence giving rise to a magnetic field effect is JW:A^ . An analysis of the contribution arising from this term shows that, there is no magnetic field effect on the heat flux if H is perpendicular to the temperature gradient.* * See Appendix II for an outline of the tensorial analysis employed in arriving at this conclusion. 131 Since this cannot explain the experimentally observed dependence of the heat flux on the magnetic f ield, the next commutator-contributing term should also be included. This second term is W v A 1 2 ^ : [ n ] ^ ^ -For the sake of simplicity, the A . 1 1 ^ term w i l l be ignored in the re-maining discussion. Thus, A is expanded as Now, substituting Eq. (8-17) into the integral equation (8-13), it be-comes It is convenient to split A into two parts, one independent of the magnetic field and the other dependent on the magnetic field, v i z . , A - A +niB) (8-19) with^?lO) = o. Hence, in the absence of a magnetic field, A = A , 132 the corresponding tensors of Eq. (8-17) are isotropic. Therefore they can be written as A'°'\ A'"\i A'c°'--A'°c'u • A'^--A'20° T'*-°> (8-20) where the f\ are scalars. For the magnetic field free case, use of the variational procedure with Eq. (8-18) leads to the three equations i n . i o i o i v . | 0 0 | . 13 , u o o (4,,,) - n 0. = h . . A * l v * , A T ; \ -J. , T \ » (010 , t V ,(06/ , 7 l , | 7 0 0 _ r _ K o ) a - = U .A ir hr A \Vl. , 4 7 • '1 J (8-21) for the three unknowns A ° . A and 12 oo Since a l l the ten-1 2 sor square bracket analogues and the constants a , a are always independent of H , they are isotropic and may be written as scalar multiples of the isotropic tensors of appropriate indicial symmetry. Making these substitutions, equations (8-21) reduce to 133 2 CO - , i<Alo<o 2% k ioo i e~ I 73 j t i c o (8-22) 1 2 The constants a , a are equal to a^, a 2 respectively of Section 6.4 11 12 21 22 while h = h _ , h = h = h and h = h where h , h and h were also evaluated in Section 6.4. The scalar square bracket 22 13 31 23 32 33 integrals h =I.h, , h = h and h w i l l be evaluated in Section 8.5. I2O0 As the third of Eqs. (8-22) is homogeneous, >4 can be ex-aIo'° J t 0 ° ' pressed in terms of A and-A by A n ^ h A + k ^ (8-23) 1.33 Employing this result in the first two of Eqs. (8-22) results in a set A ' c ' ° Jl'ooi of two simultaneous equations in s\ and A , whose solutions, 134 to terms linear in the nonsphericity, are given by Eqs. (6-38a) and (6-38b). For nonzero H , there are twice as manyterms in each equation de-termined by the variational procedure since both >4 and Hi t O are nonvanishing. A simplification is achieved, however, by subtracting away the field-free equations as.given by Eqs. (8-21). The resulting three equations determining the expansion tensors are thus: „ ^ 1010 1 7 _ ^ , 0 0 , i3 MIZOO i X Ate. f * >le. v>p.V Xle^.V & J s ^ \ t - i > ^ ' " x x / a V ^ | 11.0 o V 1X0 O (8-24) J where the constant a . , , arises from the commutator term. This constant is defined by 135 and involves, in particular, the factor ^ ) ( ^ 3 ^ f3*, ^ " ^ " V ] ) The latter is given by* + •£*^v.S M v> ^ } > ( 8 " 2 6 ) from which, with H = H \) , + 6w^v §^v> <~ ^n^L/ S^ vrv) J ^ v (8-27) or „ 3 •» C r 0 ) * See Appendix III for the derivation of the expression for this re-lation. 136 3 a is defined to be 4 p ^ H _ 4o ttVleT W_ ( 8 _ 2 9 ) r-0) where p = nkT is the equilibrium gas pressure and n. is a fourth rank anisotropic tensor of weight one in \) defined by ti° = ^ l ( £ f teUtiif $ L ? - b - (8-30) r\ is symmetric traceless in both the front and back pairs of indices, The fact that the first two equations of (8-24) are homogeneous allows Ji - ^ and to be determined from an internally contracted form of / " [ ^ ^ , v i z . , (8-31) and (8-32) 137 It should be noted that the isotropy of the h 1 J square bracket tensors and the inherent indicial symmetry of H. , i . e . , r ( . = , have been utilized in arriving at the above two equations. When the results of Eqs. (8-31) and (8-32) are substituted back into the third of Eqs. (8-24), it is immediately apparent that the terms containing ...0 . , 0 0 1 / l ^ and w i l l be quadratic in the nonsphericity (due to the pro-i3 3j ducts h h ) and hence w i l l be small compared to the other terms. rf1200 Neglecting these two terms allows a closed equation for H to be set down, namely,* * There are three sixth rank isotropic tensors which can be written down immediately from inspection of the tensorial symmetry of In so that I % i f ^ 1 f Since the last two of these tensors imply mixing amongst the J-tensors by the transition operator, the scalar coefficients h ' 5 3 and h " ^ can be expected to depend at least linearly on the nonsphericity. These last two terms can, in fact, be neglected since r| t-te and n l k are already l i n -ear in the nonsphericity and so only the first of these three terms need be retained. 138 a H. ri h - - W H . (8-33) r 0) Since 7"7. has the symmetry property ti" ' • ' = - J , " ' , (8-34) Eq. (8-33) can be rewritten.in the form This equation must be solved for / t , hence determining the aniso-tropic contribution to ~X which arises through the terms M and [see Eqs. (8-15), (8-31) and (8-32)"] . 8. 4 The Anisotropic Thermal Conductivity. Al l of the expansion tensors in have a definite character un-der time reversal. They are, in fact, a l l odd. For this reason, a separation of the anisotropic parts of , i . e . , those arising from 139 ry as determined in Section 8.3, into contributions which are even and those which are odd under time reversal is made. These contri-butions w i l l , by the nature of the anisotropy, be simply either even or odd in )^ , respectively. For notational simplicity*, the part of d I S " ° C " t (+) r1,z°° r\ even in y w i l l be denoted by Y and the part of K odd in 7 (-) h w i l l be denoted by Y . Thus, a separation of Eq. (8-35) into two parts is achieved, these being, \ I k - O ( 8 " 3 6 ) which is even in a n < 3 Y H a L Y * ' - ^ -OLln°°L0) (8-37) which is oddin . Using Eq. (8-37) to eliminate Y ^ from Eq. (8-36) yields W " * \ ^ ~ % A = 0 (8-38) This notation is the same as that used by Kagan and Maksimov. 140 as a closed equation for Y ! + ^ , where* y CO r - fO r - CD " K C ^ h ^ ^ ( 8 " 3 9 ) is a weight two in ^ fourth rank tensor. Similarly J^ T. is defined as th r'0-* ~ the i power of fa where multiplication is double dot contraction. - V r a) r «-) Since a l l terms in Eq. (8-38) are even in </ and since fa and fa form a complete set of fourth rank tensors even in ^ (see Appendix IV), Y V ' can be written as a linear combination of fa and fa , thus Y , " = C 7 l & ) + C • (8-40) Replacing Y V ; , in Eq. (8-38) by Eq. (8-40) and substituting for fa the relation - 4 h J U ^ - f A l ^ W , (8-41) p ft) r then equating the resulting coefficients of fa and rl to zero * See Appendix IV for the derivation and properties of these fourth rank,weight " i " in ^5 J ^ t o tensors. 141 gives a set of two equations determining c^ and c . These equations are C, + jlc^ - (8-42a) and ( S f 4- 4 . ) C V t c (8-42b) 4 s from which C - 4-f S c (8-43) and C - 4 * i ' 2 0 0 . (8-44) An expression giving c wholly in terms of j? and A> can be ob-tained simply from Eqs. (8-43) and (8-44) and is c . r i 4 + s y > ( 8 . 4 5 ) 142 Thus, Y^) is _ l Z i A h • 8"46 However, in order to determine the form of the anisotropic part of X / the internally contracted form /-/.. .~ fsee Eqs. (8-31), (8-32)j a a o o Ale*. A L is required.* Thus, is given by J 4 , 1 2 0 0 ^ 1 i > D L t 31*<3 + 1 " ) / * 0 0 o (8-47) * The odd in p part of g has been neglected since it is not difficult to show that these terms vanish for infinite magnetic fields and are thus very small in the ordinary experimental situation. This is shown in Appendix V. 143 Since A \^e is given by A. \ V* ' J loo I (8-15) the anisotropic part is s » k » / 2 i e T \ " * r n , 0 ' ° z c ^ n I 0 0 I (8-48) In order to get Eq. (8-48) into a more tractable form, it is convenient to define s, , s . w, and w„ by 1 2 1 2 h 1 1 lo" - w 1 ^ ' (8-49) and i . r - f ~ ( l f D , r r . ._. 3 ? \ H - V ) (8-50) so that r | . and " ( ^ are given by 144 (8-51) and K°l -'^Jn0"b:k^s^^°°^k- (8-52) With these equations, becomes ( 2 0 0 ,2 CO Now, replacing A by Eq. (8-23) which expresses A in terms of , A A^ve i s given by A \ and l L* 1 >l ( < 3 n J , \ "^^  >| too I (8-54) 145 8. 5 Reduction of the Square Bracket Integrals, 1 1 1 2 21 22 The square bracket integrals h , h = h and h have been reduced to expressions involving the cross section 0~ in Chapter VI and are given by Eqs. (6-34) to (6-3.7). The remaining scalar square bracket integrals to be reduced to relative coordinate expressions are u 13 .31 u 23 ,32 , . 3 3 h = h , h = h and h 13 13 For the reduction of h ,. the scalar is formed from h by a quad-ruple dot product with T ^ ' ^ in the manner (8-55) 13 In this form, h is just a standard square bracket integral as defined by Eq. (6-5) and so is given by - H ^ r f * . ^ ' } ^ ^ f i ' 4 ! 5 - S ' f VpUp'elpdp (8-56) 146 which, after changing to relative and center-of-mass coordinates, integrating over ^ and the angles of ^ becomes ^[-(rt^+crcwli 4- c tV i l 1 -<?t\m]<*si J)- ( 8 _ 5 7 ) The cross sections O^L | "3 have been introduced and are defined by cj.fi>; Y V . ! ' 1 ' C wi'-tuift*! (s-ss) and *>:H>; ^0° J J ' L^l 3 =- Wl'tiZ-P-l tif . (8-59) It is worth noting that these 0°L I cross sections are tr ivial ly re-lated to the.<J-[ \ ~\ cross sections of Section 6.5 by the relation (8-60) 147 In an entirely analogous manner, the square bracket integrals 23 33 h and h are determined to be <4 ( i L . y - r L v * v . JU)i - <^C*,Vl 1 icTl U,y,D }o(iIe(> j (8-61) and -iiij j ( 4 VfrWC^ 0 1 0)"^ (H0)3-* a IX IOC) -<J°( |Ol)]]dJlA(8-62) 64 33 where the new cross sections I ) used in h are defined by *j^^'(0\l) ^ [ J j f ; ( LZl^t* (8-63) 148 and ^.•Oj V ^ V ^ ' C O U ) = I T [ J j ] 6 3 : C ^ J , ] ^ . (8-64)' 8.6 Discussion. 39 As in the case of a paramagnetic gas, the thermal conductivity tensor for a gas of rotating (diatomic) molecules has been shown to have an anisotropic contribution. This anisotropy of A causes the heat flux to have different values when the magnetic field is applied in different directions to the temperature gradient. In particular, if the magnetic field is parallel to the temperature gradient, ^ h L ^ _ c L U A + h A i u r ^ t (8-65) 5-le while if the magnetic field is perpendicular to the temperature gradi-ent, 149 A X , _ * W ° + W ' 0 ' ^ (8.66) For large values of H , saturation sets in and the terms w and w X Li attain the following asymptotic values: ; - A < ~ i(\ 4 - 2 f 2 - 4 \ - 1 (8-67a) ' ' n — ^ * ~ x _ \ r •> . and (8-67b) Hence, the ratio AX// / A X J . at saturation is _L ^  3 - * 1 _ "5 2 (8-68) which is. in agreement with the result for a c lass ical paramagnetic gas 150 39 as obtained by Kagan and Maksimov. This result holds true, in this case even for a (diatomic) diamagnetic gas since the derivation as-sumes no more than the presence of a constant gyromagnetic ratio as-sociated with the angular momentum J. Kagan and Maksimov, on the other hand, resorted to a treatment in which the gyromagnetic ratio is not constant and depends on a spin projection quantum number. The angular momentum J treated in this thesis can, of course, contain more than one contribution. In particular, in the case of para-magnetic gases, the angular momentum resulting from the electronic spin of the unpaired electron w i l l be the predominant factor as the cor-responding gyromagnetic ratio w i l l be many orders of magnitude larger 47 48 than that for the rotational angular momentum. ' Unfortunately, no experimental measurement seems to have been undertaken yet in which the thermal conductivity has been determined for a magnetic field parallel to the temperature gradient so that no experimental test of the ratio / A ^ j ^ h a s yet been obtained for non-paramagnetic gases. 151 BIBLIOGRAPHY 1 J. C . Maxwell , Ph i l . Trans. Roy. Soc. .157, 49 (1867); see also The Scientific Papers of James Clerk Maxwell , (Dover Publications, New York, 1952), Vol . 2, p . 26. 2 L . Boltzmann, Wien. Ber. 66, 275 (1872). 3 L . Boltzmann, Lectures on Gas Theory, translated by S. G . Brush (University of California Press, Berkeley and Los Angeles, 1964). 4 S. Chapman, Ph i l . Trans. Roy. Soc. A216, 279 (1916). 5 D. Enskog, Klnetlsche Theorle der Vorgange in massig verdiinnten  Gasen, Dissertation, Uppsala, Almqvist and W i k s e l l , 1917. 6 S. Chapman and T. G . Cowling, The Mathematical Theory of Non- Uniform Gases , (Cambridge University Press, London, 1952). 7 E. J. 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Phys. 29, 1257 (1958). 19 C . F. Curtiss and J. S. Dahler, J . Chem. Phys. .38, 2352 (1963). 20 S. Chapman and T. G . Cowling, op. c i t . , Chapter 11. 21 Y. Kagan and A. M . Afanas'ev, Zh. Eksperim. i Teor. F i z . 41, 1536 (1961) [English transl . : Soviet Phys.—JETPL4, 1096 (1962)]. 22 D. W . Condiff, W . - K . Lu and J. S. Dahler, J. Chem. Phys. 42, 3445 (1965). 23 S. I. Sandler and J. S. Dahler, J. Chem. Phys. 43, 1750 (1965). 24 L . Waldmann, Z. Naturforsch. 18a, 1033 (1963). 25 E. A. Uehling and G . E. Uhlenbeck, Phys. Rev. 43, 552 (1933). 26 H . Mori , I. Oppenheim and J. Ross, "Some Topics in Quantum Statistics: The Wigner Function and Transport Theory" in Studies  in Statistical Mechanics, edited by J. deBoer and G . E. Uhlenbeck (North Holland Publishing Company, Inc. Amsterdam, 1962). Vol . 1. 27 D. K. Hoffmann, J. Chem. Phys. 44, 2644 (1966). 28 C . S. Wang Chang and G . E. Uhlenbeck, "Transport Phenomena in Polyatomic Gases", University of Michigan Report, CM-681 (1951). 29 C . S. Wang Chang, G . E. Uhlenbeck and J. de Boer, "The Heat Conductivity and Viscosity of Polyatomic Gases" in Studies in  Statistical Mechanics, edited by J. deBoer and G . E. Uhlenbeck (North Holland Publishing Company, Inc. Amsterdam, 1964) Vol . 2. 153 30 E. A. Mason and L . Monchick, J. Chem. Phys. 36_, 1622 (1962). 31 G . Gioumousis and R. F. Snider, to be published. 32 R. F. Snider, The Quantum Mechanical Kinetic Theory of Non- spherical Molecules , Dissertation, Madison, Wisconsin, 1958. 33 L . Waldmann and E. Triibenbacher, Z. Naturforsch. 17a, 363 (1962). 34 L . Monchick, K. S. Yun and E. A. Mason, J. Chem. Phys. 39_, 654 (1963). 35 L . Waldmann, Z. Naturforsch. 12a, 660 (1957). 36 R. F. Snider, J. Chem. Phys. 32^ , 1051 (1960). 37 L . Waldmann, Z. Naturforsch. _13a, 609 (1958); Nuovo Cim. 14, 893 (1959). 38 L . Waldmann and H . - D . Kupatt, Z. Naturforsch. l_8a, 86 (1963); H . . - D . Kupatt, Z. Naturforsch. 19a, 303 (1964). 39 Y. Kagan and L . Maksimov, Zh. Eksperim. i Teor. F i z . 41_, 842 (1961) [English transl . : Soviet Phys.--JETP L4, 604 (1962)3. 40 H . Senftleben, Physik. Z. 31., 961 (1930). 41 H . Senftleben and J. Piezner, Ann. Physik _16, 907 (1933); ibid 27, 108, 117 (1936); ibid 30, 541 (1937). E. Reiger, Ann. Physik .31, 4 5 3 (1938). H . Torwegge, Ann. Physik 33, 459 (1938). 42 H . Senftleben and H . Gladisch, Ann. Physik 30, 713 (1937); ibid 33., 471 (1938). H . Engelhardt and H . Sach, Physik. Z. 33., 724 (1932). M . TrantzandE. Froschel, Physik. Z. 33., 947 (1932). 43 M . Laue, Ann. Physik 23., 1 (1935). 44 M . Laue, Ann. Physik 2^, 373 (1936). 45 C . J. Gorter, Naturwiss. 26, 140 (1938). 46 F. Zernike and C . van Lier, Physica 6, 961 (1939). 154 47 G . C . Wick, Z. Physik 85, 25 (1933); Nuovo Cim. JJD, 118 (1933), in Italian; Phys. Rev. 73, 51 (1948). 48 N . F. Ramsey, Jr. , Phys. Rev. 78, 699 (1950); ibid 58, 226 (1940); ibid 87, 1075 (1952). N . J. Harrick, R. - G . Barnes, P. J. Bray and N . F . Ramsey, Phys. Rev. 90_, 260 (1953); R. - G . Barnes, P. J. Bray and N . F. Ramsey, Phys. Rev. 94, 893 (1954). See also H . F.. Hameka, Advanced Quantum Chemistry, (Addison-Wesley Publishing Company, Inc. , Reading, Massachusetts, 1965), p. 175. 49 J. J. M . Beenakker, G . Scoles, H . F. P. Knaap and R. M . Jonkman, Phys. Letts. 2, 5 (1962); J . J . . M . Beenakker, H . Hulsman, H . F . P. Knaap, J. Korving and G . Scoles, Advances in Thermo- physlcal Properties at Extreme High Temperatures and Pressures, ASME (1965), p. 216. 50 L. ; L . Gorelik and V. V. Sinitsyn, Zh. Eksperim. i Teor. F i z . 46, 401 (1964) [English transl . : Soviet Phys.--JETP _19 , 272 (19643; see also L . L . Gorelik, Y. N . Redkoborodyi and V. V. Sinitsyn, Zh. Eksperim. i Teor. F i z . 48, 761 (1965) [ English transl. : Soviet Phys.—JETP 2J., 503 (1965)] . 51 J. Korving, H . Hulsman, H . F. P. Knaap and J. J. M . Beenakker, Phys. Letts. \7_, 33 (1965); ibid 21, 5 (1966). 52 E. Wigner, Phys. Rev. 40, 479 (1932). 53 J. S. Dahler, J. Chem. Phys. 30, 1447 (1959); Phys. Rev. 129, 1464 (1963). 54 E. H . Kennard, Kinetic Theory of Gases, (McGraw-Hil l Book Company, Inc. , New York, 1938), pp .42-45 ;H. Grad, Commun. Pure and Applied Math. 2, 331 (1949). Grad presents an interest-ing geometric proof that the only two linearly independent sum-mational invariants which are functions of the particle velocity are the kinetic energy and the linear momentum in Appendix I of his paper. This does not, of course, allow for the existance of internal states. 55 H . Grad, Commun. Pure and Applied Math. _5, 455 (1952). 155 56 R. F . Snider, J. Math. Phys. 5, 1580 (1964). 57 E. Hil le and R. S. Phi l l ips , Functional Analysis and Semigroups, (American Mathematical Society Colloquium Publications, Provi-dence, Rhode Island, 1957), Vol . 31. 58 F . R. McCourt and R. F . Snider, J. Chem. Phys. 43, 227 6 (1965). 59 J. A. R. Coope, R. F . Snider and F . R. McCourt, J. Chem. Phys. , 43, 2269 (1965). 60 See, for example, Higher Transcendental Functions , edited by A. Erd&lyi (McGraw-Hil l Book Company, Inc. , New York, 1953) V o l . 2 . 61 J. M . Ziman, Can. J. Phys. 34, 1256 (1956). 62 R. F. Snider, J. Chem. Phys. 41, 591 (1964). 63 R. Courant and D. Hilbert, Methods of Mathematical Physics, (Interscience Publishers, Inc. , New York, 1953), Vol . 1. 64 F . R. McCourt and R. F. Snider, J. Chem. Phys. 4J., 1385 (1964). 65 S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics, (North Holland Publishing Company, Inc. , Amsterdam, 1962). 66 A. H . Wilson, Theory of Metals , (Cambridge University Press, London, 1965). 67 W . Zawadzki and J. Kotodziejczak, Phys. Stat. Sol. j i , 419 (1964). W . Zawadzki, ibid 2, 385 (1962); Phys. Stat. Sol. 8, 739 (1965). J. KoiodziejczakandS. Zukotynski, Phys. Stat. Sol . 5, 145 (1964). 68 B. Fogarassy, Phys. Stat. Sol . 3., 1646 (1963). 69 H . Gabriel and R. Klein, Z. Naturforsch. 19a, 524 (1964). 70 P. Curie, Oeuvres, (Gauthier-Villars, Paris, 1908), p. 118; see also reference 65, p. 57. 71 L . Onsager, Phys. Rev. 37, 405 (1931); ibid 38., 2 2 6 5 (1931). H . B.. G . Casimir, Revs. Mod. Phys. J 7 , 343 (1945). 156 72 R. F. Snider and K. S. Lewchuk, to be published. 73 The first definition of such generalized col l is ion cross sections seems to have been made by L . Waldmann, Z. Naturforsch. 15a, 19 (1960). 74 L . Waldmann, "Dilute Polyatomic Gases. Accuracy and Limits of Applicability of Transport Equations" in Proceedings of the Inter- national Seminar on the Transport Properties of Gases, organized by J. Kestin and J. Ross (Brown University, Providence, Rhode Island, 1964), p. 59. 75 K. SchramandB. R. A. Nijboer, Physica 2j>, 733 (1959). 76 V. J. Emery, Phys. Rev. _133, A661 (1964). 157 APPENDIX I SYMMETRY PROPERTIES OF GENERALIZED COLLISION CROSS SECTIONS. The transition operator t for a binary coll is ion is determined from the integral equation where ~H i s the total Hamiltonian for the pair of molecules and V the interaction term or potential energy operator. Generalized coll is ion cross sections are expressed in terms of this transition operator and its adjoint t* through the matrix element forms t and .£* (which are s t i l l operators in the degenerate angular momentum subspaces) through the relation r2^{]'VAIis) s ^ ) 4 A V n ' h ^ A t B t * d-2) where A and B are, in general, functions of J , J^ , V and X • When A and B are tensors, the appropriate dot product between them is implied in the definition (1-2). 158 The behaviour of these generalized coll is ion cross sections under combined time reversal and space inversion is most easily seen by writing down the matrix element form of t. The behaviour of the full operator t under combined time reversal and space inversion is easily ascertained from Eq* (1-1) where, since the Hamiltonians ~H and V are time reversal-space inversion invariant, the only effect of the operators Q (time reversal operator) and f~\ (parity operator) is to change L to - L in the resolvent. Thus the relation tBn = e - ' r u n e = tf a-3> is obtained connecting the time reversed and space inverted transition operator with.t* . The corresponding matrix representation is a,—vv\. ¥ , where the * represents complex conjugation. Physically, this equation is interpreted as declaring that the matrix element of t for a forward coll ision ( e . g . , <^  mm^ > ^! m ' m p is equal to the complex 159 conjugate of the matrix element of t for the time reversed, space i n -verted coll is ion ( i . e . , ^/ - m' - m ^  > ^ -m-m^) . In order to make any statement regarding the generalized coll is ion cross sections, the quantity of interest is which, since the m's are summed over, can be replaced by a summation over - i l l ' s , so that Now, Eq. (1-4) can be used, giving (1-7) Since the quantities A ( <^ ,-m,-m^) and B(<|, , - m ' , - m p are matrix el-ements of the time reversed, space inverted operators AQ^  and Eq. (1-7) can be written as 160 This implies that the generalized cross sections as defined in Eq. (1-2) satisfy the time reversal-space inversion symmetry property ; ( A I B > = 2 " ^ ' ( B 9 0 I A W ) . (i-9) 161 APPENDIX II TENS OKI AL ANALYSIS FOR THE J W: A 1 1 0 0 TERM. By splitting the A expansion vector into two parts as was done in Chapter VIII, two sets of equations are obtained for the A terms. The first set, for the isotropic part, A , is 100 I 1001 *5 | l 00 0 - n l 34 -v V i 3 3 A y (ii-1) where the fact that h vanishes has been used \_see Eq. (6-54)1 and 13 . J (2) the superscript "(2)" on h-jg in the third equation means that the tensor contributor was that corresponding to T ^ ' ^ . From Eqs. (II-1), / 1 1 0 0 i • • * u A 1 0 0 1 jB^ i s determined by / \ as IIOO (II-2) '33 162 The second set of equations obtained via the variational procedure for the F{ 's is 1010 tool "A ||0 o 1(00 tool iioo (II-3) in which and the subscript "t" on signifies a transpose of the first two indices. <T~ is a constant arising from f-,1010 the. commutator term. From the first of these equations, /-f is determined by simply as io/o (II-4) in — n 1001 and hence the second equation results in f~\ being determined 1100 as e-.f) lioo (II-5) 163 Since, by definition, n a s no isotropic component, the con-tracted tensor § • a l s o has no isotropic component and so can give no term in ^ proportional to U so that there is no change in the heat flux parallel to the temperature gradient and perpendicular to H . This does not agree with experiment. 164 APPENDIX III ISOTROPIC PART OF FIFTH RANK TENSORS IN J . In general, there are six linearly independent fifth rank tensors although it is possible to write down, by inspection, a total of ten 59 such tensors each made up of one § and one y . Here the form of the isotropic part of the tensor J J J J J is obtained. This may be given in terms of the overcomplete set often fifth rank isotropic tensors according to the following method suggested by Coope.* The isotropic part of J J J J J is represented by ^ ^ J J J J J ) where ^(d^)denotes a trace over the angular momentum space. &-^)(JJ JJJ) can be written down as a linear combination of the ten aforementioned isotropic tensors. Due to the cycl ic symmetry of the trace (and hence of the tensorial indices) only two independent coefficients are ob-tained. Each of these two coefficients multiplies five of the original ten isotropic tensors. These two tensors are denoted by 4p( and <^ >x and are characterized by the diagrammatic representations J. A. R. Coope, private communication. 165 ( I I I - D where the lines are considered to connect the various vertices of a regular pentagon. These pentagonal vertices represent the five free indices of the tensor. Thus, for example, the bent line i-"^*>">k represents an £ connecting the three indices i , j , k, v i z . , Similarly, a straight line 1 m represents a U connecting the two indices 1, m, v i z . , § . The sum over " i " in Eq. (Ill— 1) is taken lm over a l l cycl ic permutations of the diagrams. Thus, the two tensors and a r e (III-2) and (III-3) with 166 4> I + x - ° ( I I I _ 4 ) expressing the orthogonality of <fc>( and , and their normalization. Since ^ ^ ( J J J J J) is the isotropic part of JJJJ_J_, Eqs. (III-4) and (III-5) imply that a and b are easily seen to be a - t ft. , T< (m-7) and The determination of k ( d U ^( [ JJ 3 ^ J [ JJ ] ^ ) is quite straight-167 forward and is simplified by the fact that it must be symmetric trace-less in the front and back pairs of indices and w i l l thus vanish unless each of the three parts is connected to an £ tensor. Thus, it can be written as a scalar multiple of a special fifth rank isotropic tensor combination, namely, By utilizing the cycl ic property of the trace for the operators J (while retaining, of course, the correct tensorial order) the constant c can be determined from the equation C . i ' - f c ^ \&\ C n J ' V t j T D * } . d i i - i o ) (III-9) This equation is easily reduced to the form - 3 o c {(%)\[JI + I Li? I} which, in component form becomes (III-12) 168 59 which, with the relation (III-13) gives C- ~- % ^ ) ( 4 T 4 - 3 T ^ (111-14) and %. K i t $ L ? g 3 W T t n 3 w ^ ± | hti^ ( 4 T M ^ ) ^ ( | ) + | L 4 f +4j}-(m-i5) In an analogous manner, fe-^j(fJJ 3 [JJ 3 " c a n b e shown to be - - % ^ } ( a j 4 - 3 ^ ) | ^ & -h l |+ & + f } ( m - 1 6 ) It should be noted that if one set of second weight J's is dist in-guished by primes fromthe second, tensorially fo-^^J J 3^[j'J'3 ^ ' J is quite different from the cycl ic permutation of [ j j ] J [jj3 given by ( ^ ^ ( { J T ~\ C T J3 J ) although these two are the same 169 from the operator point of view. Subtacting fe^trjj] ^ [ U ^ J , ) of Eq. (Ill-16) from £ w ^ ( [ j n ( 2 ) J [ J J J* 2 ? ) o f E c*- (H.I-15) gives the relation of Eq. (8-26) 170 APPENDIX IV r-U) THE ANISOTROPIC rL TENSORS. A closed set of fourth rank anisotropic tensors with weights from one to four in the magnetic field direction )^ can be constructed be-ginning with the weight one tensor which arises naturally from the commutator term in the Boltzmann equation. Thus, if h. is given by . • ' & 0 i = ± \ (ft +(SL f ti+M (IV"1} then h. can be generated from h. by double dotting h, into itself, then double dotting h- into the resulting fourth rank tensor and continuing in this manner (i-1) times. However, this method is very cumbersome and so a more elegant and simple method of genera-tion is employed here. In a notation developed by Snider* using over-lapping and intersecting n's , a technique identical to that used by R. F. Snider, private communication. 171 39 Kagan and Maksimov can be utilized to generate these anisotropic tensors. Denoting by the fourth rank tensor F*" - f?iv +- ff?\* +- tfTV1 * n * * ( I V " 2 ) where the n's are second rank tensors given by n ' ° = V av-3) and the superscripts and (3 correspond to the weights of the n tensors. From this it is obvious that o 2 L F +- F J (IV-4) 172 in which two superscripts appearing together, as for example, <x V" («*) m implies that the corresponding second rank n , n tensors are to be dotted together. Thus, since 4 l 4 (IV-5) these double dot products of the F tensors can easily be evaluated <|V> oi r ( t > r 0 ) r ^ s clearly just F , while h. , rt and n. can be shown to be ft) / T r F ° ' ' . F 0 1 - 2 ( F ° ° ' " +- F0,''° ) ^ 2 ( F " % F " ) -2 0^ +.4."):bb } ' ( I V " 6 ) i r-0< + 3F ) which in component form is 173 (IV-7) and is obtained from the expression ro<i) r o) = - i r -^ i 4 12 r Q : > r 0 ) which, since F can be determined in terms of h, and ri by becomes r 6 0 5* r & ) / r 0 ) § = -•££ -H • ^ closes under double dot contraction at i = 4. 174 APPENDIX V THE ANISOTROPIC TERMS ODD IN }) . The odd part of f\ * ^ ^ # v i z . , Y / ~ ^ / w a s ignored in Chapter VIII since, as was stated there, it vanishes as H * for H large. In this appendix, the results leading to such a conclusion are presented. A closed equation, similar in form to Eq. (8-38) for , can be written for Y^ ^ and is (V-l) Now, considering Y^ ^ to consist of a linear combination of those tensors (as given in Appendix IV) which are odd in h and therefore odd under time reversal, namely, Y ~ C^K 4- <T4 k > (V-2) Eq. (V-l) for ^ becomes 175 r-Or) r (0 Relation (IV-10) has to be used in replacing H. in terms of h. _ or> = • ~ and ri • Equating the coefficients of rt and At to zero results in two equations determining c^ and c , which are J 4 and - s*c 3 f H i s * - o < <v-5> From Eqs. (V-4) and (V-5), c^ and c^ are given as and C - 4 | 3 (V-7) Thus, is Y M - r•> , « s ' , 4 " " r » (v-8) For large magnetic fields (I.e. , for 5 —* °° ) / is seen to make no contribution to the saturation value and, indeed, is inversely pro-portional to the magnitude of the field strength. 

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