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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 i s 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 U n i v e r s i t y  o f 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 R. B a r r i e J . A. R. Coope B. A, D u n e l l Research  C. A. McDowell R. F. S n i d e r L. S o b r i n o  Supervisor:  External  Examiner:  Theoretical  R. F. S n i d e r C. F. C u r t i s s  Chemistry  University Madison  s  Institute  of Wisconsin Wisconsin  TRANSJPOftT PROPERTIES OF GASES WITH ROTATIONAL STATES  ABSTRACT  T h e o r e t i c a l e x p r e s s i o n s f o r the t r a n s p o r t c o e f f i c i e n t s of a s i n g l e component gas w i t h a nonzero but s m a l l l o c a l a n g u l a r momentum d e n s i t y a r e o b t a i n e d from a m o d i f i e d Boltzmann e q u a t i o n which takes i n t o account the presence o f degenerate i n t e r n a l s t a t e s (specifically, rotational states). As i s t o be expected, a number o f Onsager r e c i p r o c a l r e l a t i o n s a r e found c o n n e c t i n g the t r a n s p o r t 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 o f the Boltzmann E l a t i o n i s c a r r i e d out by means o f a p e r t u r b a t i o n expansion about a l o c a l e q u i l i b r i u m s t a t e which i s c h a r a c t e r i z e d by a l o c a l temperature, stream v e l o c i t y , and a n g u l a r momentum d e n s i t y . T h i s p e r t u r b a t i o n i s e x p r e s s e d as a l i n e a r combination o f the macroscopic g r a d i e n t s o f the system whose c o e f f i c i e n t s b e i n g t e n s o r s a r e expanded i n terms o f i r r e d u c i b l e C a r t e s i a n t e n s o r s made up o f the a n g u l a r momentum pseudovector o p e r a t o r J and the reduced v e l o c i t y v e c t o r W. The t r a n s p o r t c o e f f i c i e n t s are then g i v e n by combinations o f 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 . E x p r e s s i o n s f o r these expansion c o e f f i c i e n t s i n terms o f square b r a c k e t i n t e g r a l s a r e o b t a i n e d w i t h the a i d o f 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 s c a l a r product which a l l o w s f o r the l a c k o f time reversal, symmetry o f the Boltzmann c o l l i s i o n o p e r a t o r . F i n a l l y ^ the square b r a c k e t i n t e g r a l s a r e reduced to r e l a t i v e and center-of-mass c o o r d i n a t e s and expressed i n terms of g e n e r a l i z e d c o l l i s i o n cross sections. s  s  The t e c h n i q u e s developed f o r the r o t a t i n g gas w i t h a nonzero l o c a l a n g u l a r momentum d e n s i t y a r e u t i l i z e d t o o b t a i n an e x p r e s s i o n f o r the change i n the thermal, c o n d u c t i v i t y o f a gas when p l a c e d i n a magnetic f i e l d , I t i s shown that a t s a t u r a t i o n the r a t i o of the changes i n the thermal c o n d u c t i v i t y w i t h the magnetic f i e l d  (a) p a r a l l e l t o and (b) p e r p e n d i c u l a r t o the temperature g r a d i e n t i s 2/3, T h i s v a l u e agrees w i t h the e x p e r i m e n t a l r e s u l t f o r paramagnetic gases. s  s  GRADUATE STUDIES F i e l d of Study: Topics  Theoretical  i n Physical  Chemistry  Seminar i n P h y s i c a l Quantum  J . A. R, Coope A, V. Bree  Chemistry  N. B a r t l e t t  Chemistry  J . A. R. Coope  Advanced T h e o r e t i c a l Transport Topics  Chemistry  Chemistry  R„ F. Snider  P r o p e r t i e s of Gases  i n Chemical  R. F. Snider B. A. Dune11 C. A. McDowell  Physics  Chemical Thermodynamics  J . N. B u t l e r  Topics  W„ R. C u l l e n H. C, C l a r k N. B a r t l e t t J . T. Kwon  i n Inorganic  Spectroscopy  Chemistry  and M o l e c u l a r  Structure  A. V. Bree C Reid B. A. D u n e l l  State  L., G. H a r r i s o n  Elementary Quantum Mechanics  F, A, Kaempffer  Chemistry Related  of the S o l i d  Studies:  Plasma P h y s i c s I n t r o d u c t i o n to Low Temperature Physics  L=  Sobrino  J . B. Brown P, W, Matthews  PUBLICATIONS F. R, McCourt, An I n f r a r e d Study o f S o l i d Gas Hydrates at 77°K, B. Sc. T h e s i s , U.B.C., (1962). R. F. Snider and F. R. M c C o u r t K i n e t i c Theory o f M o d e r a t e l y Dense Gases; Inverse Power P o t e n t i a l s , Phys. F l u i d s 6 1020 (1963)= s  3  K, B. Harvey, F. R. McCourt and H, F. S h u r v e l l , I n f r a r e d A b s o r p t i o n of the S02 C l a t h r a t e - H y d r a t e °, M o t i o n o f the S02 M o l e c u l e , Can, J , Chem. 42, 960 (1964). F. R, McCourt and R. F. S n i d e r , Thermal C o n d u c t i v i t y o f a Gas w i t h R o t a t i o n a l S t a t e s , J . Chem. Phys. 41, 3185 (1964). J . A. R. Coope, R, F. Snider and F. R. McCourt, I r r e d u c i b l e C a r t e s i a n T e n s o r s , J . Chem, Phys. 43, 2269 (1965). F. R. McCourt and R. F. S n i d e r , T r a n s p o r t P r o p e r t i e s o f Gases w i t h R o t a t i o n a l S t a t e s , I I , J . Chem. Phys. 43, 2276 (1965). F. R. McCourt and R. F. Snider, " T r a n s p o r t P r o p e r t i e s of Gases w i t h R o t a t i o n a l S t a t e s , I I I " ( t o be published). F„ R. McCourt and R, F. Snider, "The Thermal C o n d u c t i v i t a Gas of R o t a t i n g M o l e c u l e s i n an E x t e r n a l Magnetic F i e l d " ( t o be p u b l i s h e d )  FREDERICK RICHARD WAYNE McCOURT. TRANSPORT PROPERTIES OF GASES WITH ROTATIONAL STATES.  ii  S u p e r v i s o r . R. F. S n i d e r .  ABSTRACT  Theoretical expressions for the transport coefficients of a single component gas with a nonzero but small local angular momentum density are obtained from a modified Boltzmann equation which takes into account the presence pf degenerate internal states (specifically, rotational 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 i s expressed as a linear combination of the macroscopic gradients of the system, whose coefficients, being tensors, are expanded i n terms of irreducible Cartesian tensors made up of the angular momentum pseudovector operator J and the reduced 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 collision operator.  Finally, the square bracket i n -  iii  tegrals are reduced to relative and center-of-mass coordinates and expressed in terms of generalized collision 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 field.  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  ii  List of Tables and Figures  vii  Acknowledgment  viii  CHAPTER I 1.1 1.2 1.3 CHAPTER II 2.1 2.2 2.3 CHAPTER III  INTRODUCTION Treatments Based on C l a s s i c a l Mechanics . Treatments Based on Quantum Mechanics . . Effect of a Magnetic Field on the Transport Properties  10  THE BOLTZMANN AND HYDRODYNAMIC EQUATIONS  14  The Boltzmann Equation The General Equation of Change The Hydrodynamic Equations  14 18 21  LINEARIZED BOLTZMANN EQUATION FOR SMALL J  27  Q  3.1 3.2  The Equilibrium Distribution Function Linearization of the Boltzmann Equation for Small J  27  THE FLUX TENSORS  40  Fluxes and Forces The Pressure Tensor The Angular Momentum Flux Tensor The Heat Flux Vector Discussion  40 44 53 61 65  G  CHAPTER IV 4.1 4.2 4.3 4.4 4.5  1 2 6  31  vi CHAPTER V 5.1 5.2 5.3 CHAPTER VI 6.1 6. 2 6.3 6.4 6.5 6.6 CHAPTER VII 7.1 7.2 7.3 CHAPTER VIII 8.1 8.2 8.3  THE VARIATIONAL PROCEDURE .  69  The Variational Principle The Iterative Equations The Onsager Reciprocal Relations  69 73 78  THE FIRST ITERATION—ZEROTH ORDER IN £  89  Method of Approximation Coefficient of Shear Viscosity Coefficient of Bulk Viscosity Coefficient of Thermal Conductivity The Rotational Diffusion Coefficients. The Coefficients ^ and V EFFECTS WITH LINEAR 'CL-DEPENDENCE  89 92 98 101 104 108 ..  Ill  Preliminary Remarks The Anisotropic Viscosity Coefficients Other "Linear-in- oL » Effects  Ill 112 120  THE SENFTLEBEN-BEENAKKER EFFECT FOR THE THERMAL CONDUCTIVITY, , , .  122 122 123  8.4 8.5  Introduction Form of the Boltzmann Equation . The Tensor Equations Determining the Anisotropy The Anisotropic Thermal Conductivity Reduction of the Square Bracket Integrals . .  8.6  Discussion  148  128 138 145  BIBLIOGRAPHY  151  APPENDIX.I  157  APPENDIX II  161  APPENDIX III  164  APPENDIX IV  170  APPENDIX V  174  vii  LIST OF TABLES AND FIGURES  following page  TABLE I  TABLE II TABLE III  FIGURE I  A complete set of irreducible tensors (at most linear in ) which are linear in the macroscopic gradients  43  Complete sets of independent irreducible isotropic tensors of ranks two to five . . . . .  49  Identification of the /6 coefficients for the iterative solutions  112  An invariant representation of isotropic tensors  49  viii  ACKNOWLEDGMENT  It is a pleasure to acknowledge the able and enthusiastic supervision 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 "nonequilibrium statistical mechanics" or, alternatively, as "the statistical mechanicsof irreversible processes". The determination of the transport properties of a system is a central problem of nonequilibrium statistical mechanics and in particular for gases, i s usually known as the "kinetic theory of gases".  2  1.1  Treatments Based on C l a s s i c a 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. equation is a non-linear integro-differential solve it met with little success until Chapman dently published their solutions.  Since the Boltzmann  equation, 4  attempts to 5  and Enskog  indepen-  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  equilibrium distribution function.  give the local  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 results for the transport coefficients when compared with experiment. Two standard methods of solution of the above-mentioned linearized Boltzmann equation may be u t i l i z e d .  The first of these is the  3  method used by Chapman and Cowling  in which an approximate s o l -  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 approximating 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 ChapmanEnskog solution are the assumptions that: c l a s s i c a l mechanics is valid for describing molecular interactions; the gas is sufficiently dilute that three-body collisions may be ignored; the molecules possess no internal 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 removal 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 l a s s i c a l treatment of the transport properties 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 i s interconvertible with energy of translation). A rigorous c l a s s i c a l treatment of the transport properties (with e x p l i cit consideration of the angular momentum variables) for a single component gas consisting of nonspherical molecules was first given 14 by Curtiss  i n 1956, including a consideration of the hitherto neglec-  ted transport of angular momentum through the gas. quite ingenious.  His treatment is  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 method was further generalized to multi17 18 1  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 degrees of freedom and which interact according to the laws of c l a s s i c a l 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 21 by Kagan and Afanas'ev  and later emphasized  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 macroscopic gradients of the system.  the  The treatment given by Kagan  and Afanas'ev was c l a s s i c a l 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 l a s s i c a l cross section by its quantum analogue (without internal 9 states). Theseresultshavebeenextensivelydiscussed byWaldmann , 26 27 by M o r i , Oppenheim and Ross and by Hoffmann (see also Chapman and Cowling  6  8 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 i s analogous to the existence of inverse collisions in c l a s s i c a l mechanics. Since this "detailed balance" criterion i s 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 mechanical cross sections.  quantum  The required symmetry property of the  degeneracy-averaged cross sections arises naturally from considerations 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 poly32 .. 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 Lippmann36 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 functiondensity matrix necessitates a modified collision term in which the collision is described in terms of combinations'of the transition operator rather than simply in terms of the collision cross section.  This  modified collision 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 one37 half particles  and as an application considered the diffusion of these 38  spin-particles in a magnetic field. 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 pseudovector operator 2 must be treated explicitly.  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 i s 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 l a s s i c a l gas of rotating paramagnetic 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 i s derived. P , L and q are expressed 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, unique.  in general, not  These are most conveniently expressed in terms of a set of  independent irreducible isotropic tensors with scalar expansion coefficients.  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 -o  and J, -linear approximations respectively, —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 collision 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 startling in that the thermal conductivity was found to depend on the ratio of the magnitude of the magnetic field, 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 H g , the thermal conductivity of the  gas was decreased by about 2% from its field-free value. In the f o l 41 lowing ten years, a whole series of experimental papers to this effect.  was devoted  It was at the same time found that the shear viscosity  of a paramagnetic gas was also affected by a magnetic field i n much 42 the same way as the thermal conductivity. A qualitative explana43 44 45 tion was proposed by Laue  '  in 1935-36, by Gorter  in 1938, f o l -  lowed closely by a more quantitative explanation by Zernike and van 46 Lier theory.  in 1939, a l l based on the mean free path approach to kinetic These workers proposed that the collision 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 between 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 l a s s i c a l l y , they set up a  12  Boltzmann equation with a term on the left hand side (called for obvious reasons a magnetic operator term) containing the magnetic moment due to the unpaired electron spin. With the inclusion of this term they were able to relate the Senftleben effect to the degree of nonsphericity 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 c o 48  workers,  that many diatomic molecules possess a magnetic moment  (of the order of a nuclear magneton) due to the nonzero angular momentum 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 i n 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 C F ^ (rough spherical molecules). The final part of this thesis is concerned with obtaining an expressionfor this "Senftleben-Beenakker effect" for the thermal conductivity of a gas of rotating molecules. Thus, in Chapter VIII, a linearized Boltzmann equation allowing for the presence of a magnetic field i s 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 field.  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 i n two ways. The first of these is through particles "drifting" into and out of the region of phase space under consideration, and the second i s through particles entering and leaving the region via the mechanism of c o l l i s i o n s .  The number of such particles entering and leaving  the given region is governed, i n the limit of low density and the binary collision 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 "collision" term. The use of a distribution function i s , of course, valid for particles whose states can be described c l a s s i c a l l y . 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 utilized.  The Wigner distribution  function w i l l then satisfy a quantum mechanical Boltzmann equation in the low density-binary collision 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 i s 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 l a s s i c a l 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 de Boer,  30  by Mason and Monchick  tures by Snider,  32  and has been generalized to mix-  by Waldmann and Trubenbacher  34 Yun and Mason.  31  33  and by Monchick,  53 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 collision term which involves combinations of the transition operator* t and its adjoint  t  +  35,36 rather than the usual collision 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), iik is the mass of the particle in state  p is the linear momentum,  j , and ( £ , 2 ^ i s the relative  linear momentum of the two colliding particles given by  ft b\-  £.-"•)?  - b -  <  p >  m  where P_ i s the center of mass momentum. £(E)  i s  a  (2-2)  Dirac delta function  of the energy expressing conservation of energy during a c o l l i s i o n . first of the two terms on the right hand side of Eq. (2-1)  The  gives the  number of particles which, due to a c o l l i s i o n , end up in a state contributing to  fj^.^v,/(£_'£'*) while the second gives the number of particles '  which are removed by collisions from state s contributing to  fj ^ . ^ / ( 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[ijff .ff ^']dl| 1  1  ! >  (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 £ + £ the adjoint in internal state space,  - £ ' • The symbol f denotes 1  1  g and  g/ denote the relative v e l -  ocitiesof the two particles corresponding to the relative momenta (£,£^) and (£',£^) respectively, and the prime s on f and f functional dependence on £ '  and £^  designate the  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 momentum are conserved during a c o l l i s i o n . sums of one-particle attributes.  These eight quantities are the  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 i s best to consider first anequation of change for a general summational invariant and from this equation obtain the individual hydrodynamic equations.  Since it i s easier to follow such a derivation i n  operator notation, the operator form of the Boltzmann equation i s used as the starting point. Letting  (r_,£) stand for a summational invariant,  the equation of change for the quantity  -  i  ^  b  ^  ^  l  ^  .  ^  i s given by  '  ]  ^  ^  (2-6,  function expressing conservation of momentum during a c o l l i s i o n .  Symmetrizing with respect to particle labels and  integrating over  in the first term gives for the right hand side of  Eq. (2-6)  20  (2-7)  If In the second term of Eq. (2-7), since space, "5f +  and  Lq  i s diagonal in momentum  'tf' commute. Utilizing this fact along with  the c y c l i c property of the trace and the operator identity (2-8)  or  r 1  -  t "?  -  2k <  (2-9)  Eq. (2-7) becomes  " ^ M P . ^ ^  ^ ' ^ . V P '  <2  -  10)  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  (2-11)  With this result, the right hand side of the generalized equation of change for a summational invariant, as given by Eq. (2-10), i s easily 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 i s 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 i) 4  The brackets  ^  ^  -  M  (2-14)  are used to denote an average over the momentum  and internal state spaces, i . e . ,  n<A> = tJiJAf  (2  '  15)  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 i s 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 i s given by  V<r^,0 = t-V.tc.-l5* and i s called the "peculiar velocity".  ->  (2  17  Two other commonly occurring  average quantities i n kinetic theory are the pressure tensor* and the heat flux vector, defined respectively by  P ( r , i ) s n<mVl/>  <- >  ?<c.O ^ n<(i*il/\H')y>.  (2-19)  2 18  and  The operator  H ' appearing i n Eq. (2-19) is the internal state Hamil-  tonian, assumed to have orthonormal eigenfunctions jm  S such that jm  j jm  * The pressure tensor i s 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 hydrodynamic equation. Thus, for ^$~ = m, the equation of continuity for the mass density  is obtained, while  p23(£.  +  p  V  = £ gives the equation of motion of the gas,  , 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 i s  <^_  n  +  V W  .3R  -  p  v  .  (2-24)  When the summational invariant is the total angular momentum, = M , the derivation i s not quite so straight-forward as for the previous cases. A breakdown of the total angular momentum into two contributions, the so-called "external" angular momentum term, irxp_, and the "internal" angular momentum operator  J i s made, i . e . ,  (2-25)  Substituting this into the general equation of change and employing the equation of continuity gives  + &«Hcrxp)>+£'<"VJ> = o  (2  -  26)  By algebraic manipulation, this equation can be converted into the form  26  r  x  ^  ^  +  n  w  ^  v  .  +  ^ . P }  Since the cross-product term is just  +  n ^ f >  r_ crossed into the equation of  motion , it vanishes, and by defining an "angular momentum flux tensor" L  as  L  =  n<VJ>  (2-28)  ,  the equation of angular momentum balance in final form is  ("O where J = ( J ) Q  L  (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 applied. The linearized Boltzmann equation obtained below can be expected 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 distribution function-density matrix has first to be determined. Waldmann^ has shown that l n f ^ ° \  where f ^  is the local distri-  bution function-density matrix, is a summational invariant. It is a w e l l known result of mechanics that there are eight and only eight independent functions of the dynamical variables of a particle involved in a  28  collision 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 and m are as defined previously. 1  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 l a s s i c a l 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'  -  - a xr  - a y  3  4  •  0  ( 3 _ 3 )  Setting  f  a  1  m - i a  4  v  ; 0  ,  0-4)  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 normalization  n=ta(f(°^d£.  The constant a^ is evaluated through 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 i s the density matrix for internal states defined by the relation  <p.  and  < p - ^  P  ( - y i ) « p ( ^ )  <3- ) 7  Q is the internal state partition function defined by  ^M-£t)MW)  -  <3 8)  The parameter o(. occurring in Eq. (3-7) i s related to the local angular momentum density  J  -r  -  Q  by the relation  L~r  (3-9)  and corresponds to what has been called the "local average angular 20 14 velocity" ' in the c l a s s i c a l 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 Hermitian  56  I for Hermitian <p since in general  f  (o)  and  f^  is  1  <P do not  commute. Since the only assumption made on f is that the magnitude of the deviation from f ^  i s 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 collision 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 utilized. It should be noted that in the approximation applied here ( oL small), 4> >= <£«> ,b  #  32  -it  +  V  ^ r -  t  N  I  V  t,  id Clearly, the terms not involving the perturbation <j? w i l l cancel, since the transition operator t and utilized.  commute, allowing Eq. (2-9) to be  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  + lis. (3-12)  ^ . W  5  (o) Because of the form of f , Eq. (3-6), the left hand side of Eq. (3-12) e  can be written in the form  3  at  —-1 + v . — v  4-  T  3£ J  +  to) r  :o r ^0  y ^  T  2*  •f  3f  an 3n" —- + *v • •at sr  (3-13)  33  The time derivatives in this equation may be eliminated by utilizing 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. where  p = nkT and  (3-14)  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  =  ^ _  +  v . ^ -  D/Dt i s defined as  -  The local angular momentum density,  0-17)  J , is considered in this  the sis to be nonzero but small, An explicit relation connecting J  q  and  34  the parameter cL i n v°' allows the exponential term of f containing oC to be expanded i n a power series i n . Since J is — —o given by  T  _ k \ T*t?(-"'/kT)*wU-Z/kT)} ih{** (r H 7 f c T > K (±7/kT)] ?  P  an expansion of exp( rA« J/kT) to terms linear i n  '  <  j  V  .  (3 18)  >  31eT  gives  . * '  0-19)  where  Q --ih^ (~ '/k )' H  0  ?  T  To terms quadratic i n 0^. , the internal partition function system i s given by  -  (3 20)  Q for this  35  which shows that the total energy differ from their  U and the total heat capacity C  = 0 values by terms at least quadratic in  y  oL .  Hence, to terms linear in 0^  Dt  v  =. where  C  v  D t  \*Ll  r  O-t  C D I _ - kTV-Uo >  (3  v  i s simply given by C  v  "  22)  = [^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  <HJ >-<3 ><H> kT<3> ,  1  2  /  l  (3-23)  36  so that  •at  at  D t  -  v  -  and  (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 leT  l - e  2V: V *  <J H'>-<T XH > 1  l  <  0(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 =  (3-27)  T-7 -  -  e  -  3  leT  and a l l terms quadratic i n d. have been dropped. Defining  (o) f by o  (3-28)  Qo U T r k T  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  —  + + + where  U K ( U ) K - l ) - ^ ^]iy):Cv,J*}^^4),(3W is the dimensionless or "reduced" velocity given by  30)  39  (3-31)  S i s the symmetric traceless "rate of shear" tensor defined as*  § = £ [ v * . +cvv„) ] - $v-v..y > t  while  (3  -  32)  (3  -  33)  b i s given by m  b = 2 and [ j j j  (2)  ~ |w'y >  is defined as  58  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) a n d b a , 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 i s 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 , the contracted form V  , v  o  w  e  r  e  t  n  e  S, \/QL and  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 respectively.  - Lt>.f(+bf*)V!<*•?>  "  (4  3)  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 a n d o  and which are linear in one of the three gradients are of interest in the expansion of the perturbation function.  All 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*  (4-8)  where the expansion tensors are a l l independent of d. . For a system 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 i s used, namely, that a pair of indices which are physically closest to one another are contracted together, then the next pair, and so forth.  TABLE I. A complete set of irreducible tensors (at most linear in<* ) which are linear in the macroscopic gradients.  Scalars Pseudoscalars Vectors  Pseudovectors  Tensors  Pseudotensors  Vxy  e  %'* (**.>*]  ?  s.  44  namely, the reduced velocity W and the internal angular momentum 20 operator J.  It was first pointed out by Chapman and Cowling 21  later by Kagan and Afanas'ev  and  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 tensors formed from W and J can now be utilized. These tensors can be  45  written (in component form) as  (4-10)  and  where  «... 7  = L  [>3  = CW-WD^  ( p >  <e£..wW'>t33<t>  u  and C J j '  >  ? )  S C ?''' J 3 ^  th th denote the irreducible p - and q -rank tensors formed from W and J respectively.*  The tensorial coefficients  <Db,... Wp^f^ are functions only of W and J .  B D, ^ ,..  c  •  and  It i s 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 exp(-x), are given b y ^ 11  \^\^ = ^ A-O  C-pTftt+S+QX*  rW<:+i)<s-;)U!  (4-13)  with the well-known property that  o  _ ^ ) The discrete polynomials  having weight functions  w(q) = Q Q (2j + l)g(q) exp(-H'AT) 1  g(2) = (20)~ j(j+l)(4j + 4j - 3) 1  2  with g(0) = 1, g(l) = j(j+l)and  (the polynomials for  q = 0 are those  47  28 introduced by Wang Chang and Uhlenbeck ) are chosen so that they satisfy the relations  £ j = k T £ ^ - is an eigenvalue of the internal energy operator H ' . Explicitly, the  (£j/feT) polynomials which are required in this  thesis are  R';\ vw;  = R:  C*.  )  l (4-16)  - I  (4-17)  J  feT<T*>  "  and  (4-18)  48  The pressure tensor P i s given by  (4-19)  Equation (4-19) has been written i n the above form to facilitate the evaluation of  Writing a l l tensors i n terms of irreducible tensors  and carrying out the summations over p and q by utilizing 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 degenerate angular momentum subspaces, it i s found that  +  f  0O,o  )}:(V*.)*  (4  "  21)  and  (4-22)  49  59  where the invariant notation for the fourth rank isotropic tensors is explained in Figure 1.* Since the tensorial coupling coefficients such as 2 2  2000 B. are  constants, ( i . e . , independent of both W and J ), they can be expressed as linear combinations of the isotropic tensors of corresponding rank.  This is given in Table II.** Chopsing the coupling con-  «. 2000 00s0 ^OOsO _01s0 <o. 2000 , 2100 . stants B , B , (g , B , and B in accordance with Table II, Eqs. (4-20), (4-21) and (4-22) reduce to +  D  D  D  p = - f e T [ B ; ° ° S +(B "'-B"")V-v.y~}> o  e  M  + (6" -©"")«}.-Vxy.g 60  ]  -  <4 23)  (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 useful in practice and i s a technique developed by Snider and C u r t i s s . [ R . F. Snider and C . F . C u r t i s s , Phys. Fluids 1, 122 (1958)J. ** The tensors appearing in this table have been chosen in accordance with Ref. 59. Two sets of isotropic fifth rank tensors are given, the first set corresponding to a reduction of a third rank tensor, 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 ' ° of Eq. (4-48). z o  to follow page 49  TABLE II. Complete sets of independent irreducible isotropic tensors for ranks two to five Rank 2 3 4  5  Isotropic tensors  to follow page 49  FIGURE 1. An Invariant Representation of Isotropic Tensors Rank  Invariant Form  2  T  3  • T  2 <  v  *  y  = -s  4  5  §1  4  - t — (s'ify Y =-  L  « [  i  |  i  UB- & "]  ?  J  50  and  E - H - B T S .  + BTS..  + CB°'°°- B°"°  y] >  <- > 4 2S  where S i s as defined i n Eq. (3-32) and S^, § are defined by 2  §,  =  (4- 26)  ^ [ ^ C V x y , ) + (7xy )oi]-joC'VKy (J 0  e  and  ^1  =  "1 L "  " S ^ J *  (4-27)  Now, defining the phenomenological coefficients ^  and ^  V | , 7^ ) ^ ^  by the equation  P - (nkT- x v - y  8  - X o L - V x O ^ - 2 > ? | -2>? S (  -iy[ \iU- 28)  and comparing it with Eqs. (4-23) to (4-25), it i s clear that  x  ;  51  X-  ,  ^1000  nleT  1-&T--B,"' !  (4-29)  -  0  (4 32)  and  _ l _ ( B ° ' - B " ° ) ]• 0 O  +  0  <4  -  33>  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  52  -o  B  (4-34a) •a  ^ o o t o  C>  4.  .eoo i  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- -nkT  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 l a s s i c a l gas of rotating 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 i n 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  ^ ' '  1st  and  where LwD and C j ] ^ a r e defined as i n Eq. (4-11). Thus (P  (4  "  41)  55  (4-44)  Using Table II to write A  1 1  ^ ^ and writing C  1  1  ^ as  X becomes  £ =" r l w where i L  '  -&  +  c  «i , 4  -k-V v*+a  f.wf  4  4 5 )  (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 - tv*)*]-  ( 4  -  4 7 )  56  It can be shown that*  where the identity  ^ i l r ^ has been used.  (  4  r  4  -  3  T  '  }  +  ¥ ) - 3w]  (4  -  49)  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° A ,l0  3teT  kT  4  ^3  [a  -[  ~  ^ "  0  ^3  k  ^  ZIc+ It  0 +  +  )  1  ,  1 , 0 0  1  l  «'  1  100  -t  c  A  +  I 00 I  3leT  Uc,« A  "A  3k  3k.T  1 0 0 1  1  h  llOO  ' -1  A'  A  The macroscopic tensors -W-, , _1_Z. ,  CO K  10 A qieT^i  OL-Vx <* (J  3fc  6i  (4-50)  (2) (1) / K and K are defined  by means of the relations  J u L ^ = i - [ _ C V x * ) * + c^Cv **)] - ^ - V x o ^ U ^ 1  (4-51)  58  JL  M  = ^ [(V«  =  -  i(.V*i)) ,  (4-52)  X [**4f  ( 4  -  5 3 )  ft-) = ^ [<* CViv^T) 4- (VJUT)  K  V ^ T (J  (4-54)  and ,C0 K  r  ^ [^IWvJO  4V^7)<T).  The linear momentum auxiliary condition resulting in C A  1 0 0 0  = 0 and (X - ( 3 k T ) A -1  1 1 0 0  (4-55)  1  (  ^ = 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  (4-56)  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\"' r"°° ^ ~ /Z [ M J I' L  r - n 12kxV^ f i " IZ V v" )  nflkX  M ^ v  o  \"y"  o  +  >  1 1 0 0  J  T -  (4-57)  11  (4-58)  >  o c ?  (4-59)  •>  {4  "  6o)  * 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 coefficients of L have been corrected.  60  5  and  =  -  - I" j " " ' <  1  !_ £'"".-1 ,  (4-63)  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.  Grad,^  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 l a s s i c a l gas of nonspherical molecules which explicitly determines the form of this coupling. The form of the perturbation function required for the determination of the heat flux vector q i s given by Eq. (4-38). heat flux vector i s defined by  n T ' Q, 3  T  Now, as the  62  using Eq. (4-38); the result i s  + Upon integration over the angles of W , taking the trace over the degenerate subspaces, replacing the tensorial parts of the tensorial . , 10st ^ l O s t l i s t ^10st ~ l l s t , ^10st coupling coefficients A i (A. #A , C , C and r f l  A  by isotropic tensors, and utilizing the definitions of the polynomials R ^ ( H ' A T ) of Eqs. (4-16) and (4-17), q becomes t -  63  -[(£tu.)a"  - f a ' ° ' V ^ a '  M  ' ( A "  0  -  0  A" ) 10  (bA"%u,A"")]^' . + LCf*"-)C""-f  C'  A. = I  zfr <C- O +  +  3ET  where Q = i . i • cL , Q = i l -ot 1 2  + < " ) ]  and Q = £(V'#) 3  fining the phenomenological coefficients  •  •  Hence,  (  4  -  7  2  de-  (including the well-known  coefficient of thermal conductivity "X ) by  -Xij  - 2 ^ - ^ - 2 ^  ,  a comparison of Eq. (4-73) with Eq. (4-72) requires that  (4-73)  )  64  i  „  =  I- C„x n " * "  a  b  A"°' 1  ,  <«-">  C'°'°],  *&\gLfSf±c™-f  "  (4  76>  and  *• 4  -  I  I O B U  _  1010  X-  -  /  In writing down these expressions, the auxiliary condition i m posed by the linear momentum density has been u t i l i z e d .  This aux-  iliary condition may easily be shown to lead to the conditions  65  A  1  0  0  0  _  ^1000  I  AllOO  >  (4-78)  As w i l l be shown i n 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 considerable 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 i s 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 phenomenologi c a 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 coefficients 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 i s given by a Wigner distribution function, entirely analogously to the c l a s s i c a l treatment, whereas the rotational motion is treated in terms of a density matrix in rotational states. like the c l a s s i c a l treatments of Grad  55  This is quite un-  and Curtiss,  14  i n which the  rotational state of a molecule i s described in terms of its orientation  67  and angular velocity. As a result of t h i s , Curtiss has based his d i s cussion on an equation of change for an average angular velocity rather than on an equation of change for an angular momentum density as has been done here. Another point of departure i s that Curtiss has given the rotational energy as a quadratic form in angular velocity (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 cd sion.  0  from the rotational energy expres-  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' present o£ if the Hamiltonian i s quadratic in J .  c*J with the 0  It would seem,  however, to be more reasonable to base the treatment on a peculiar angular momentum rather than on a peculiar angular velocity. Furthermore, 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 i n a magnetic field the energy i s not quadratic i n 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 tensors 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 order 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 applicable 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 reversi62 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 perturbation real field.  is considered to be an element of a vector space over the A scalar product defined in this space which is approp-  70  riate for the following variational procedure is  <4I^>„  =iA.iC+rV*t''  where 4^ is obtained from  l5  "  1)  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\^0anextremum. The inner product of the functions G and <R.H, ( G \<R \ VV) ' 0  71  i s the analog of the square bracket integral of G and H employed by Hirschfelder, Curtiss and Bird  8  and by Chapman and Cowling,  6  namely,  [GvH] - «?I<RH>„ -- -^j"<V^,CH)<*? >  (5  -  4)  where the double dot product implies that the proper tensorial contraction between G and H has been used. i s always a scalar.  This assures that [G;H"3  From the time reversal properties of the H a m i l 56  tonian of the system, Snider  [ S j HJ -  CHj  has shown that  = LH ;G 3T  <"' 5  T  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+«*x+» <&  w  5 6)  72  then a simple scalar product for purposes of the variational procedure is  <rI* >, = M f in which  w i  % r  1  x)=< * i n  T  , (s-7)  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  i =cV M  r  _i  =  *  • <-  58 )  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(^mV +H' -«<•]), whose equilibrium expectation values are nkTU, 0 and 0 respectively. a  73  = i t f  4  o r  {  ( e ' ' '  l  * e  4  '  I  If f ^ i s expressed in the form of Eq. (3-29) where f ^ o  )  +  commutes  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 ^ i s expanded in accordance with Eq. (3-  29), so that X ^ is given by  J ^  T  U  W - < ^ H ' >  .  (  y  ) <  )  (5-11)  while (/< <f> i s  (5-12)  if quadratic and higher terms in ci are neglected.  Now, writing  75  x  (d) ^ ^  a  n  d  ^  a s  *  X " *o + X ,  <R - <£  0  (5-13)  y  +<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  =  (R  0  +.  and  *  The superscript "d" is understood in a l l that follows.  (5-16)  76  X, - <R,  4>.  ^  C  •  (5-17)  Here, Eq. (5-16) i s 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 i s at once obvious; it may easily be shown to hold for Q  -  ^> also since the scalar product of a summational invariant 0  with $,4j  i s 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? * obtainedfrom Eq. (5-16) using s  the standard variational procedure,^ ' ^ 2  4  and this value of 4? is  substituted into Eq. (5-17) which i s then used to obtain an approximation for  in an analogous manner.  By writing <X> as  and utilizing the linear independence of the macroscopic forces, two sets of Boltzmann equations are obtained.  The first set of equations  is  b = <J? B„, 0  and  <- » 5 20  78  These equations are referred to hereafter as the equations of the first iteration. The second set of equations is  + ( i e T ) ' # £ - - ( ( e l f <£  (5-24)  1  (  0  and  C T<1 > X  V  -  kT<R  t  B  c  -  WT(£  to  ^6  >  (5  "  25)  and i s 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 second is to consider the transport coefficients as  -dependent while  retaining the original set of macroscopic forces, treating the resulting system as anisotropic.  Whichever interpretation is used,  final results must be the same.  the  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 , resistance  67, 68, 69  ^  Righi-Leduc, ^ ' ^ ' ^  and magneto-  effects in solids).  The principal advantage offered by the first-mentioned interpretation of the transport coefficients i s 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 tell upon which thermodynamic forces a particular flux can depend. This has already been done i n setting up Table I. C o n 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 pseudotensor 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 ^ ] ° , L V r f ® , ^ T a n d the four fluxes it L , (  U  ,  and q need be considered. Thus  T r"bl L  £  H  V i  +| ,'ViviT+| (  :L7*] V^;,C7^ t ,  ( 3  T (5-26)  81  or, in matrix notation,  (5-27)  are the equations giving the fluxes as linear functions of the independent thermodynamic forces.* The Curie principle can give no further information regarding the off-diagonal components of this set of equations. Onsager reciprocal relations nal components of  However, a set of  can be expected coupling the off-diago-  . These reciprocal relations are obtained through  considerations of time reversal symmetry. Along this same vein, a detailed 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 car72 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).  * It is important to bear i n 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 traceless second rank tensor from cL , U and  £  and so  <^, =  =0.  4  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 and one from  with £  with ^42.  • •^  wo  , two from < ^ f  u r t n e r  2 3  with  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  kT  with the aid of the scalar product ^ \ ^ . The heat flux vector i s 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  ^.•J term has also been employed by Grad  53 and Dahler.  Since no  contribution to q can arise from the terms ^7v and ^7«v , the pertur— —o o v  bation function can be written as  ^  -  _ A-v-^t -  C  :  V *  for the purposes of these calculations, so that  from which the appropriate  coefficients can be identified.  Similarly, the angular momentum flux tensor i s  <- > 5  31  84  The cross terms between L and q are denoted by  € | l  (  * )  =  -  <- li" v  v  *+  H  '-  Ii  \  (5  "  34)  and  t^W  -  (5-35)  <YJ!A>,  Recognizing that constant multiples ofV vanish i n these phenomenologi c a 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  i Li) tL  =  -  (5 37)  The change of sign of d± follows from the fact that this i s the only effect of time reversal on Is L<J ^ ^ • Since these third rank in  tensors can be at most of weight one  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 i n and  eC  are identified through the following two equations  M U ^ - W O +^ *y and  <s  "  38>  86  +  A  1  i - [ ^  - y f l 4- A  3  i U -  (5-39)  The Onsager reciprocal relation of E q . (5-37) then implies that  i? = - A  < 5  4  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 examination of the symmetry of  - kT<c|d?C>. > t  (5  -  42)  87  which has used Eq. (5-28) and the self-adjointness property of the scalar product  in  Hence, again using Eq. (5-28), j £  ( gj. )  becomes  - <yj|c>  <"> 5 3 4  1T  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 i s fourth rank and at most of weight one in be written as a linear combination of the fourth rank tensors 1, 2, 3) and t ^ | T , f (  u  «--1  v  , 0 )  , it can T/*' *  =  0  ( X = 1, 2, . . . , 6) as  TT  r; '°'> s  ( s  -  4 5 )  88  where the scalar coefficients T^. and defined in Section 4 . 3 .  are synonymous with those  Substitution of Eq. (5-45) into the Onsager  reciprocal relation (5-44) results in the following two relations amongst the %  's:  (5-46)  and  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 coefficient, as expressed in Chapter IV, involves a set of i expansion coefficients  with corresponding tensors T ^ , it i s convenient to  approximate ^ by  4> - ZL 3 *Tv •7n s  ( 6 _ 1 )  where "TT] i s the appropriate thermodynamic force. With this approximation, a set of i equations in the i unknown coefficients  is  obtained by substituting the approximation of Eq. (6-1) into the corresponding 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  c  h  ~-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]jrc:^'[<(r; r;)C' +  91  where the prime on T' denotes, once again, that W i s 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 coordinates implies that  J JJ  •  *pfc4-T*  (H'+HD/krlr'dMl<LSL  where dSL is an element of solid angle for the variable ,^ ' .  ,  (6-7)  Hence  h ^ m a y be written in the form  V> =  a ^ M e T  V  C(T^  h  \ \ **  £  c  ?  i  (H V H ; )/kT  [-A a  r  a  a  .  (  6  -  ]  8  ,  Clearly, any additive part of give no contribution to  .  which is a summational invariant w i l l This is likewise true for T ^ d u e to the  symmetry of the square bracket, Eq. (5-5).  6.2 Coefficient of Shear Viscosity.  Since the coefficient of shear v i s c o s i t y , lr\ , is determined by one expansion coefficient from the expansion of the perturbation function, cf. Eq. (4-29), it is sufficient to choose*  _  -7D0O  / /0  —ro)  f  , u ' \  (6-9)  which, when using the Boltzmann equation (5-20) and the method of 2000 Section 6 . 1 , results in an equation determining B . Thus  (6-10a)  * The breakdown of § into B and B = £- • $ corresponds to the breakdown of c|> into 4^ and <4? (Section 5.2). 0  (  (  93  or  JL  1O00  B  where the constant  -  (6-10b)  , as determined from the left hand side of  Eq. (5-20), is  r(|)  and  -  S H  (6-11)  D. , the corresponding square bracket integral of Eq. (6-8), is I  given by  94  f  7  3  .iia/ U)C  %le)E \^ciJk di  1  (6-12)  t  Since W , W ' commute with  and t<£ this can be further simpli-  fied to  \  ...WAV-  K e"'  ^ /feT^r- x  H W V f c r  2  (w4w v.) l  :  ^ ^ i ^ - i i s V K i M ^ V i o A  f  f  f <^«''-jw'V +  |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  which is s t i l l an operator in the degenerate  1 5  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 rn, \V r  w  / ) j  (6-16)  96  where t ' is identical to t with the primed and unprimed indices i n terchanged, i . e . ,  = ^jj^ \ | ' | ' ' ^ ] _ ^ * F t  r o m  Eq. (6-16) it f o l -  lows that  ^ j ^ a - j ' ' ' " ^ . ^ ^  and thus  <r-';(Y,K,f)  -  (6-17)  D becomes (  f f(w ^J(/} l  Vcjj-o^ ^ ' ( ^ ^ S t E ) ^ ^ ^ - ^ -  Now, utilizing the definitions of ^  ( 6  "  1 8 )  and ^ given i n Eq. (6-6), inte-  grating over the angles and magnitude of  and using the fact that  conservation of energy during a collision requires that  y -y' - n A £ 1  (6-19)  97  where  (6-20)  it i s not difficult to reduce  I  b, to  —  J ^  3 3,  (6-21)  A further simplification of I.D, can be attained by utilization of the relation  E  Y  JUT whence  e  *"ajcvj  v^'*  4^fH&<^  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, 29  Uhlenbeck,  30  and de Boer and Mason and Monchick with, of course, the introduction of degeneracy-averaged collision 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)  (6-25)  + I which, together with Eq. (5-21) leads to  (6-26)  where  e  _ vj\ _ H'/hrr  e  (6-27) -|VU  V  100  and  -^(^)  *e"  zT~^& ]<kJbdl*^ic<Lxdf 6  €r£j|  (ke) A^%*%dy A). %  (6-28)  Thus, from Eqs. (4-36), (6-27) and (6-28), the coefficient of bulk viscosity i s  6  ^  I'I"  • ••/•/ JJ.jJ,  (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 considered here is then  30)  so that the two equations to be solved are* i  ,1010  a. ^ h „ A  ,  i i ool  +- h L X A  N  ^ (6-31)  io(o  The constants  ioo|  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 ° ° Xsee Eq. (6-53)3. To terms independent of the nonsphericity, the two equations (6-31) are a l l that is required for the determination of "X . 1  102  and  -  irh  S±*. (1^L\  //V  (6-33)  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  T, and  4 LJTU.V +  -  z  -  | C A £ f  Htef( $  - ^  1  ( 6  ( 6  "  "  3 5 >  .  3 6 )  103  in agreement with the results of Mason and Monchick.  30  It should  be noted that particle symmetrization and symmetrization with respect to primed and unprimed variables have been used to simplify these a  formulae.  The expansion coefficients A  1 0 1  0  ^ aIOOI  and A  are therefore  given by  A  i bio  a,  (6-38a)  a.  and  A where  Sb i s  too  (.6-3 8b)  the determinant of the square bracket integrals  (6-39)  Substitution of these expressions into Eq. (4-74) results in an expression 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. (5-22), the integral equation for C  Ql  ^  ,  From Eq.  it would appear from first i n -  spection that the three coefficients are in effect coupled; on closer inspection, however, since e . g .  °^ Eq. ( 57) carries with it 4_  a weight two tensor made up of J and W , it w i l l pick out only the weight two parts of C and JV upon contraction with them. Q  -,1100 C^  a n c  Similarly,  , „1100 , . , , . . * C-^2 carry tensors of weight one and zero, respectively. +  +  +  Because of these tensorial properties, a separation of the three simultaneous equations in three unknowns into three separate equations, (each containing only one unknown) is accomplished. the equation for C ^  X  <  ^  In particular,  is  [+.+;  - ( t T ^ + B " , ^  f'p r*^_...,eiV(-j9-/r_..,^ w  r , ...>,V\l*t'  j ^O^Vfif -p)4 %^p.(6-40) (  (  105  Afurther evaluation of this expression requires the introduction of the generalized collision cross sections  73  mCD^ C T ^ ' c o l o l =7fT-?lt*  ^  (6-41)  aj:£D.-^5:f 'L011  ,  (6-42)  J  .^•uJj.  1  E  ^^[^rv'lTl  r  W  T'T,  Tfl'lJjt*!  (6-43)  yy i-i^'itV  (6-44)  and  eo'.u;:  o~^'  11 =  in which 7 ^ i s the operator defined by  7^=: L(xTi) t ' Yn' '7l'ti,. 4  L  i  l  '  (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  53n  /a^ k>T  ^  - ^(  ( T t o 103 - ^ L H o 3  c r [ o o | 3 _ 0=-Loi|  )  +  ^ ( 5= D ' t r ]  3)y<*SL<ftJ  '  (6-47)  so that, £ s e e 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 coefficients  Xj, and  are determined to be  -1 -  (6-49)  <r  and  3  7zl^/ery  4- ^ [  LQ o y a  J J\  i  } W X d ^ f ^  .  (6-50)  108  6.6 The Coefficients Z*4 and  The only remaining coefficients defined for the flux tensors independent 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 i s simpler than 1) ).  can  be approximated by (see footnote on p. 101)  a  a 1010  A - A o  (JM  A  iool  O  a I '  co}  0  0  wL,(w; + A wK,^) + A l  for the determination of A  1  1  ^.  (,S  R ^> 0  (6-51)  The three equations obtained by the  variational procedure are  a. , k A' %k,A' °' K,A" 0,  'l| ' *  '1  0  1  IOIO  +  "(2. ,  A1001  & - h K°° + h^A x  00  2)  n  vk A  00  (6-52)  l3  it© 0  1001  0  Since the third of these equations is homogeneous, it may be solved 1100 1010 1001 to give A in terms of A and A , viz. , r  i HOO  I h  33  . .  J  ,1010 ,  . 1 oe?l \  (6-53)  109  The constants a, and a . and the square bracket integrals h 1  and h  1  11  , h  1 9  are as determined in Section 6.4, while  h^X-Wl^,  ~ 0  (6-54)  ?  /J/J J/ J  and  x  (T»y'fl^'Xcr[o-i|o-i2 - o" [ > M 2 - V D ) ^ / ^ f d )  ( 6 _ 5 6 )  where  T»(£*$')  =  ^ t f W i ^  and the cross section defined by  (6-57)  110  (6-58)  have been used i n obtaining Eq. (6-55). Eq. (4-69) defining  With these equations and  , an expression for A^. (and hence for  ) is  obtained, v i z . ,  n.  H  (i\er\<i/\"  00  cxMu -  2kT\ h  *3  M  a  .(6-59)  Ill  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 higher approximation.  The equations obtained by  the variational method in these two iterations can be represented as  112  =  where the c  r  S  ^  ( ^ l , l , S - l , V " J)  r  are constants, the h rs  grals and the coefficients  (7-1)  are modified square bracket inte-  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 _and X . |  1  (  Of these three, Y[ can immediately be seen to x  vanish since from its definition, Eq. (4-28), it is a coefficient for a macroscopic gradient which i s 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 second iterations.  to follow page 112 TABLE III. Identification of the /S coefficients for the iterative solutions. 1. First Iteration.  Coefficients  Identification  D B  2000 l  Expansion Tensor  2 ww -  IT-  A  1 0 1 0  A  1 0 0 1  „ 1110 A 1101 A 1200 i3  y  (1/2) 0  (W ) W  RJ°* ( H ' A T ) W  i> )JW T ' 1 - - = 2  ( 3  R  ( 1 )  R^^(H'AT) W •  1200 °1  R ^ ( H ' A T ) W '  1200 °2  R^  /*'4 c  iooi  ( 1 )  o  ( 3  '  0 )  (2)' ( H ' A T ) W  -1100 l  iioo 2 •  R  (H'AT)JW:T  R  c  0 ,  :  A  C  14  L  w  V  o  o o  li '°'-l 4  T  (  S , 1  ' :Cin > 0 ,  (H'AT) W •  R^(H'AT) W •  R|°'(H'AT) W •  T  (3,0)  ( 2  to follow page 112 TABLE III. Identification of the /S coefficients for the Iterative solutions. 2. Second Iteration.  Coefficient  Identification  Expansion Tensor WW:T ' 12 ( 5  I-  .  8  R  (  1  )  0 )  ( H ' A T ) J W : T ! ' 4  o  A  '- -  0  )  =1  aT  (1) (4,0) R (H'AT)JW:T o =2  a'  (3,0) w • t\ =3  —  0 0 0  23 L';1W )W.T 2  ft ^  1  00  ( 3  '  0 )  1  (A  R^H'ATjW.f'^ lido  a . is  3  R^ (H'AT) JW^Tg  -  R ^ ( H ' A T )  as"  c  JW:T  (  5 2  '  0 )  (1), „ (5,0) R (H'AT) JW : T o - - §4 x  R  (1) '(H'AT) o L  (5,0) §3  JW:T  _  (1) (5,0) R (H'AT)JW:T Q  5  113  equation.  A detailed analysis trivially 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_  =  #  3  0  (7-2)  6  with B approximated by* q  B  O  =  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 00 is linear in the nonsphericity and is hence small compared with Bf . 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. T  114  E  B,  fc„  +  (7-4a)  and ^  i  0  0  °  ^  x-L_-R  ^  where  6r,o)  T , 0  B,  °  -,  O ^ T  4-13,  b  ^ '  6  , and  )  ]  (7-4b)  is clearly just the  (|  Eq. (6-23). The other square bracket integrals 2000  M  a x  is given in Eq. (6-11) and b  to be evaluated. Thus,  t ^ : T  2100  a  r  of  and O ^ h a v e yet  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. Explicitly, 1D is given by 1(  * S C j g ' + p ' ^ F / d p ^ J ; o l ( D  (7-7)  5  which, upon converting to relative and center of mass coordinates via the transformation of Eqs. (6-6) and (6-7) becomes  *l  * ^2"* ^  (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 and  Jh  .  Similarly, the terms involving  in the first bracket of Eq. (7-8), being odd in J  ish upon integration over the angles of ^ A%%M  while the terms containing  and i i i w i l l vanish upon contraction with T  only the term containing  tfjf^'^  van-  contributing to the integral.  leaving Hence,  116  after performing the integration overdo and the angles of 1 . K, becomes  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 ^ f ° s  u n c  2 2  * cr(o + i i-o+i)  +  a - ( o i l  '  t o  be  t o + i ) 4-  - ^ ( r C l H , ^ ) , ) ] ^ ^ ^ ^ ^ 2000  2100  Having determined B 1  and B 2  obtained for the evaluation of ^  , only  [.see Eq.  <r(nJ,U  % i o x %  ( 7  ^2000  QV, ^  (4-31)~|  .  "  1 3 )  has yet to be The equation of  /Q 2 0 0 0  the second iteration which determines  IQ  is (7-14)  which, with the approximation  6 ^ ( 8 " ° ° w:T^'  0>  for the expansion tensor terms of  leads to an equation giving  and two square bracket integrals, v i z . ,  "  (7  15)  in  118 1.000  1  &•  I  /?  H p ,  =  —  i  [yjy.-^^^jjfjyy.yiv't/)]  ( 7  _  1 6 )  Simplification of these square bracket integrals into expressions in terms of cross sections, although not t r i v i a l , is straightforward. The 21 21 square bracket integrals of c^ and h are thus 100O  je-rcj-  _  .  ,  /»  v^^y ^ [ ^  V  Tut j  u  |  J  ] CM'XM^J  J  tfdif  J  ' d)  (7-17)  and  Substituting these equations into Eq. (7-16) results in an expression rfor rt.ko ^ 2  0  0  0  , v•i z . ,  119  (7-19)  As  is expressed in terms of B  2  I  and Co  _  no  a  s  (4-31)  o  X it can be obtained as proportional to Vj through the equation  i *  (7-20)  when t  can be neglected compared with Vj D n  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 ' % "£& / the expansion coefficients are again de>  =  3  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: ^  teq  IIOO  \\00  ; and  toj  "fe^  jpUOO  ,  . 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  , A 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 x  3  * Two sets only, since Cg is the only expansion^coefficient which carries with it a weight zero tensor, i . e . , J»W R U , and is therefore as determined previously in Chapter VI. o  121  .1110..1101'  A  , A  , 1200  tion; the second set, ^  . . .  A  and A  u  ,  .  t  ,,  .•  ,.  which can be determined from the first lteran  llOO  LL.  ~  It O O  ; the third, CA,  ll 0 0  : and the fourth. LA. ; and the fourth, CX^ .  ^  ,6^  Ioeo  ,oio  , (a-  and  Each of the last three sets of coef-  ficients can be obtained from the equations of the second iteration.  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 diatomic 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 d i a 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 qualitatively seen to be reasonable,  since the c l a s s i c a l collision 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 c o l l i s i o n s .  This pre-  cession effectively results in an extra averaging process caused by randomization of orientations between c o l l i s i o n s , 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. Waldmann  Following  38 74 39 ' and Kagan and Maksimov, it is assumed in this  thesis that the effect of a magnetic field on the Boltzmann collision term can be ignored. Thus the magnetic field enters only in the streaming terms of the Boltzmann equation and the form of these are ascertained.  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, and «  A  , is written as a sum of one particle Hamiltonians and an interaction term V - ^  w  x°'+K°  a  s  +  In Eq. (8-1), the term containing V  V  -  (8  2)  is. re sponsible for collision effects  and, as magnetic field effects in the collision term are ignored, it 35 w i l l be identically given bythat derived by Waldmann  36 and Snider.  Thus the collision term is as given in Eq. (2-1). Since translational  states are treated c l a s s i c a l l y , 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/ (e) ^ rf'ir) *(<*-r> 0)  ^  *  (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 jO ^ by 0  * (? , & are the position and momentum operators, respectively. No confusion should arise between this (? and the density matrix (p of Eq. (3-7).  126  -f(- ^) r  (8-6)  when jOJO,is in position representation, or by  (8-7)  when  is in momentum representation.  respondence  df/dt  Thus, under the Weyl cor-  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  -*ft*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 76 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 field, there w i l l be no contribution 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 SenftlebenBeenakker 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 conductivity 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 and the 2eeman Hamiltonian -"tfH'J, thus 1  %  T  Secondly, J  Q  = H'-*H-T.  "  (8 14)  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,  A iooo  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 i s 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 field, the next commutator-contributing term should also be included. This second term is W v A For the sake of simplicity, the A .  1 1  1 2  ^: [n]^^-  ^ 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 becomes  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 where the f\  A' °'--A'° 'u • A'^--A' ° T'*-°> c  c  20  (8-20)  are scalars. For the magnetic field free case, use of  the variational procedure with Eq. (8-18) leads to the three equations  i  0. -J.  a-  n  = h..  A  , T \ » (010  = U .A '1  iv  . i o i o  * l v  ,tV  .  *  ,(06/  ir h A  for the three unknowns  r  J  | 0 0  |  . 13  ,uoo  ,  ;  \  r  ,4  and  T  -n  ,|700_ _Ko)  , 7l  \Vl.  A ° . A  A  (4,,,)  (8-21)  7 •  12 oo  Since a l l the ten1  sor square bracket analogues and the constants a  2 , 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  e~ I 73  i  j tic  o  (8-22)  1 2 The constants a , a are equal to a^, a respectively of Section 6.4 11 12 21 22 while h = h _ , h =h = h and h = h where h , h and 2  h  22  were also evaluated in Section 6.4.  integrals h  13  The scalar square bracket  31 23 32 33 =I.h, , h = h and h w i l l be evaluated in Section  8.5. As the third of Eqs. (8-22) is homogeneous, >4 pressed in terms of A  A  n  ^  aIo'°  J  and-A  h  A  t  0  I2O0  can be ex-  ° '  by  +  ^  k  (8-23)  1.33  Employing this result in the first two of Eqs. (8-22) results in a set A '  of two simultaneous equations in s\  c  ' °  Jl'ooi  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 determined by the variational procedure since both >4 nonvanishing.  and Hi t O are  A simplification is achieved, however, by subtracting  away the field-free equations as.given by Eqs. (8-21). three equations determining the „  ^  iX  1010  Ate.  1 7  f  ^  _  *  ,  0  0  ,  >le.  The resulting  expansion tensors are thus: MIZOO  i3  v>p.V  Xle^.V  | 11.0 o  V & J s ^ \  t  ^ ' " x x / a V  - i >  (8-24)  ^  1X0 O J  where the constant a . constant is defined by  ,  , arises from the commutator term.  This  135  and involves, in particular, the factor ^ ) ( ^  3  ^  f *, ^ " ^ " V ] ) 3  The latter is given by*  + •£*^v.S v>  ^  M  } >  ( 8  "  2 6 )  from which, with H = H \) ,  + 6 w ^ 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 relation.  136  a  3  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  1J  and the inherent indicial symmetry of H.  square bracket tensors  , 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  / l ^  .,001  and  w i l l be quadratic in the nonsphericity (due to the pro-  i3 ducts h  3j h  ) and hence w i l l be small compared to the other terms.  rf  1200  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 ' and h " ^ can be expected to depend at least linearly on the nonsphericity. These last two terms can, in fact, be neglected since r| - and n are already l i n ear in the nonsphericity and so only the first of these three terms need be retained. 5 3  t te  l k  138  a  H.  ri  h  --W H .  (8-33)  ,  (8-34)  r 0)  Since 7"7.  has the symmetry property  ti"  '• ' = -  J,"'  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.  All of the expansion tensors in der time reversal.  have a definite character un-  They are, i n 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 "° " t (+) r1 °° r\ even in y w i l l be denoted by Y and the part of K odd in 7 (-) I S  C  ,z  h w i l l be denoted by Y  . Thus, a separation of Eq. (8-35) into two  parts is achieved, these being, \  I  which is even in  Y  an<  which is oddin  -  O  ( 8  "  3 6 )  3  aLY*'-^  H  k  -OLl °°L n  0)  (8-37)  . 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  fO  r-  "  r-  C ^  K  CD ^ ^  h  ( 8  "  3 9 )  is a weight two in ^ fourth rank tensor. Similarly J^T. is defined as the i  ~  r' -* 0  th  power of  fa where multiplication is double dot contraction. V a) r «-) Since a l l terms in Eq. (8-38) are even in </ and since fa and fa r  form a complete set of fourth rank tensors even in ^ (see Appendix IV), Y ' can be written as a linear combination of V  fa  and  fa  ,  thus  Y,"  Replacing Y  V ;  =  C 7l  &  ) +  C  •  (8-40)  , in Eq. (8-38) by Eq. (8-40) and substituting for fa  the relation  -4 JU^  -  h  f  then equating the resulting coefficients of  A l ^ W, p fa  ft) and  (8-41) r rl  to zero  * See Appendix IV for the derivation and properties of these fourth rank,weight " i " in ^ 5 J ^ tensors. t o  141  gives a set of two equations determining c^ and c . These equations are  C,  + jlc^  (Sf  4- 4 . ) C  -  (8-42a)  and  t  V  4  (8-42b)  c  s  from which  -  C  4-f S  (8-43)  c  and  C  -  4  An expression giving c  *  i '  2  0  0  .  wholly in terms of j? and A>  (8-44)  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  A h  Z i  •  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 o o Ale*. A is required.* Thus, is given by L  a  J  ^  1  4  i>D  31*<3 + 1 " )  ,1200  L t  /*  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  loo I A.  \  '  V*  J  (8-15)  the anisotropic part is  s»k»/2ieT\"*r  n  , 0  '° z  c  ^  n  I 00  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  (8-49) h lo" - w ^ ' 11  1  and  i.r -  f~(lf D  so that r | .  , r r  . ._. 3 ? \ H - V )  and " ( ^ are given by  (8-50)  144  (8-51)  and  K°l  -'^J "b:k^ ^^°°^kn0  s  With these equations,  (8-52)  becomes  (200  Now, replacing A A \  and  ,2 CO by Eq. (8-23) which expresses A  , A A^ve  l  L* >l 1  i  s  (<3nJ  in terms of  given by  , \ ^"^ >| too I  (8-54)  145  8. 5 Reduction of the Square Bracket Integrals,  The square bracket integrals h  1112 21 22 , 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 13 .31 2 3 ,32 ,.33 h = h , h = h and h u  u  13  13  For the reduction of h ,. the scalar i s formed from h ruple dot product with T ^ ' ^ in the manner  by a quad-  (8-55)  13 In this form, h  i s just a standard square bracket integral as defined  by Eq. (6-5) and so i s given by  -  H ^ r f * .  ^  ' } ^ ^ f i ' ! - S ' f VpUp'elpdp 4  5  (8-56)  146  which, after changing to relative and center-of-mass coordinates, integrating over ^ and the angles of ^ becomes  ^[-(rt^+crcwli 4- c t V 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 ° ' L^l 3 =- Wl'tiZ-P-l JJ  It i s worth noting that these 0°L I  ti  f  .  (8-59)  cross sections are trivially 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 h  23  and h  33  are determined to be  (iL.y-r  <4  L v * v .  JU)i  - <^C*,Vl  1 icTl  U,y,D } o ( i I e ( > j  (8-61)  and  -iiij j (  4 VfrWC^ )"^ (H0)3-* I X 010  a  where the new cross sections  *j^^'(0\l)  64  IOC) -<J°( |Ol)]]dJlA(8-62)  I ) used in h  ^ [ J j f ;  33  are defined by  ( LZl^t*  (8-63)  148  and  ^.•Oj  V ^ V ^ ' C O U  )=  [Jj]  I T  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 gradient,  149  AX,  _  *  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<~ ' '  n  — ^ *~  i(\  4- f 2  x_ \  r  2  4  \  -  1  (8-67a)  •> .  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 l a s s i c a l paramagnetic gas  150  as obtained by Kagan and Maksimov. case even for a  39  This result holds true, in this  (diatomic) diamagnetic gas since the derivation as-  sumes no more than the presence of a constant gyromagnetic ratio associated 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, i n the case of paramagnetic gases, the angular momentum resulting from the electronic spin of the unpaired electron w i l l be the predominant factor as the corresponding gyromagnetic ratio w i l l be many orders of magnitude larger 47 than that for the rotational angular momentum.  48 '  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 gases.  / A ^ j ^ h a s yet been obtained for non-paramagnetic  151  BIBLIOGRAPHY  1  J. C . Maxwell, P h i l . Trans. Roy. Soc. .157, 49 (1867); see also The Scientific Papers of James Clerk Maxwell, (Dover Publications, New York, 1952), V o l . 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. 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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). V o l . 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) V o l . 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 Nonspherical 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 C i m . 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 t r a n s l . : 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, (1938). H . Torwegge, Ann. Physik 33, 459 (1938). 4  5  3  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 . T r a n t z a n d E . 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 . W i c k , Z. Physik 85, 25 (1933); Nuovo C i m . 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, (AddisonWesley 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 Thermophyslcal 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-Hill Book Company, Inc., New York, 1938), p p . 4 2 - 4 5 ; H . Grad, Commun. Pure and Applied Math. 2, 331 (1949). Grad presents an interesting geometric proof that the only two linearly independent summational 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. Hille and R. S. Phillips, Functional Analysis and Semigroups, (American Mathematical Society Colloquium Publications, Providence, Rhode Island, 1957), V o l . 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-Hill Book Company, Inc. , New York, 1953) Vol.2.  61  J. M . Ziman, C a n . 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), V o l . 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 . W i l s o n , 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. S o l . 5, 145 (1964).  68  B. Fogarassy, Phys. Stat. S o l . 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., H . B.. G . Casimir, Revs. M o d . Phys. J 7 , 343 (1945). 2  2  6  5  (1931).  156  72  R. F . Snider and K. S. Lewchuk, to be published.  73  The first definition of such generalized collision 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 International 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 collision 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 collision  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  r ^{ 'VAIis) 2  ]  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 collision 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.  t  Bn  = e-'rune  Thus the relation  = t  a-3>  f  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 collision ( 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 collision ( i . e . , ^/ - m' - m ^  > ^  -m-m^).  In order to make any statement regarding the generalized collision 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 elements 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  ;  (AIB>=  2 " ^ ' (B  9 0  IA  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  |l  00  (ii-1)  *5  0  n 4  -  -v  l3  Vi A 3 3  y  where the fact that h  vanishes has been used \_see Eq. (6-54)1 and 13 . (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), J  jB^/  1  1  0  0  ii s determined • • * u by /A \  1  0  IIOO  0  1  as  (II-2) '33  162  The second set of equations obtained via the variational procedure for the  F{  's is 1010  tool  "A ||0 o  (II-3)  1(00  tool  in which  iioo  and the subscript "t" on  a transpose of the first two indices.  signifies  <T~ is a constant arising from f-,1010  the. commutator term. determined by  From the first of these equations,  /-f  is  simply as io/o  (II-4) in —  and hence the second equation results in 1100 as  n 1001 f~\ being determined  lioo e-.f)  (II-5)  163  Since, by definition, tracted tensor  §•  can give no term in ^  n  also  a  s  no isotropic component, the con-  has no isotropic component and so  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 i s 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 ) ^(d^)denotes  a trace over the angular momentum space.  where  &-^)(JJ JJJ)  can be written down as a linear combination of the ten aforementioned isotropic tensors. Due to the cyclic symmetry of the trace (and hence of the tensorial indices) only two independent coefficients are obtained.  Each of these two coefficients multiplies five of the original  ten isotropic tensors.  These two tensors are denoted by 4p and <^>  and are characterized by the diagrammatic representations  J. A. R. Coope, private communication.  (  x  165  (III-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. represents an £  Thus, for example, the bent line i-"^* ">k >  connecting the three indices i , j , k, v i z . ,  Similarly, a straight line 1 indices 1, m, v i z . , §  .  m represents a U connecting the two The sum over " i " in Eq. (Ill— 1) is taken  lm over a l l cyclic permutations of the diagrams. and  Thus, the two tensors  a r e  (III-2)  and  (III-3)  with  166  4> I +  - °  x  expressing the orthogonality of <fc> and (  their normalization. Since  ( I I I _ 4 )  , and  ^ ^ ( J J J J J) i s 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  (dU  ^ ( [ JJ 3 ^ J [ JJ ] ^ )  is quite straight-  167  forward and is simplified by the fact that it must be symmetric traceless 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,  (III-9)  By utilizing the cyclic 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'-fc^  \&\  CnJ'VtjTD*}.  dii-io)  This equation is easily reduced to the form  -3oc  {(%)\[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  %. $ L ? g 3 T t n 3 W  ^  w  Kit  In an analogous manner,  ±|  h^ ti  fe-^j(fJJ  ( 4 T M ^ ) ^  3  [JJ 3  ( | ) + | L 4 f +4j}-(m-i5)  "  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 d i s t i n guished by primes fromthe second, tensorially fo-^^J J 3^[j'J'3 ^ ' J is quite different from the cyclic permutation of [ j j ] given by  ( ^ ^ ( { J T  ~\  C T J3  J)  J [jj3  although these two are the same  169  from the operator point of view. Subtacting  £w^([jn Eq. (8-26)  ( 2 )  fe^trjj] J [JJJ* ?) 2  ^ o f  [ U ^ J , ) of Eq. (Ill-16) from Ec  *- (H.I-15) gives the relation of  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 beginning with the weight one tensor which arises naturally from the commutator term in the Boltzmann equation.  Thus, if  h.  is given  by  .•'&  then  h.  0 i  = ± \  ti+M  (ft +(SL f  can be generated from  itself, then double dotting h-  h.  by double dotting h,  (IV  "  1}  into  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 generation is employed here. In a notation developed by Snider* using overlapping and intersecting n's, a technique identical to that used by  R. F . Snider, private communication.  171  Kagan and Maksimov tensors.  39  can be utilized to generate these anisotropic  Denoting by  the fourth rank tensor  F*" - f?iv +-ff?\*+- tfTV  1  * n**  ( I V  "  2 )  where the n's are second rank tensors given by  n'°=  and the superscripts tensors.  V  av-3)  and (3 correspond to the weights of the n o  From this it is obvious that  2 LF  +- F  J  (IV-4)  172  in which two superscripts appearing together, as for example, <x V" («*) implies that the corresponding second rank n be dotted together.  m , n  tensors are to  Thus, since 4  l  4  (IV-5)  these double dot products of the F tensors can easily be evaluated  <>  oi  V|  s clearly just F  r , while  h.  ( t  r  >  , rt  r ^  0 )  and  n.  can be  shown to be ft) /T r  F°''. F  0 1  - 2 0^  - 2 ( F ° ° ' " +- F ''° ) ^ 2 ( F " % F " ) 0,  +.4."):bb } ' i  which in component form i s  r-0<  + 3F  ( I V  )  "  6 )  173  (IV-7)  and  is obtained from the expression r  o<i)  =  -  which, since F  o)  r  i  ^ i  12  r  -  4 r h,  Q : >  can be determined in terms of  and  r ri  0 )  by  becomes  r  6 0  §  5* r  & )  / r  0 )  = -•££ -H • ^  under double dot contraction at i = 4.  closes  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 i s  (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 Eq. (V-l) for  ~  C^K  ^ becomes  4- <T k 4  >  (V-2)  175  r-Or) H. in terms of  Relation (IV-10) has to be used in replacing  _ or> ri •  and  =  Equating the coefficients of rt  and  At  •  r  h.  (0  ~  to zero results in two  equations determining c^ and c , which are J 4  and  - s*c  3  f  His*  -  o  <  <-> v  5  From Eqs. (V-4) and (V-5), c^ and c^ are given as  and C Thus,  Y  -  4|  (V-7)  3  is  M  -  r•> ,  For large magnetic fields (I.e. , for  «s',4""  5 —* ° ° ) /  r»  (v-8)  i s seen to make  no contribution to the saturation value and, indeed, i s inversely proportional to the magnitude of the field strength.  


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