27407Quantum mechanical calculations of the transport cross https://ntrs.nasa.gov/search.jsp?R=19650019806 2018-05-12T03:28:42+00:00Z.The t h e o r y of t r a n s p o r t phenomena i n a low d e n s i t y g a s of s t r u c t u r e l e s s s p h e r i c a l molecules i s w e l l developed . The t r a n s p o r t 1 c o e f f i c i e n t s a r e g i v e n i n terms o f t h e t e m p e r a t u r e dependent r e d u c e d , which i n t u r n are i n t e g r a l s o f t h e e n e r g y i s used t o d e s c r i b e t h e b i n a r y c o l l i s i o n s between t h e molecules, t h e s e c r o s s s e c t i o n s may be expressed i n terms o f t h e phase s h i f t s , 1. Cross S e c t i o n s 71 * I n t h e p r e s e n t p a p e r , we c o n s i d e r t h e quantum mechanical d e s c r i p t i o n o f t r a n s p o r t phenomena i n a g a s made up o f m o l e c u l e s which i n t e r a c t a c c o r d i n g t o a Lennard-Jones(l2,6)potentialA s u s u a l , t h e c o n s t a n t E i s t h e d e p t h o f t h e p o t e n t i a l minimum, t h e c o n s t a n t b i s t h e s e p a r a t i o n d i s t a n c e a t which t h e p o t e n t i a l i s z e r o , and R*=n/ris t h e reduced s e p a r a t i o n . E* , t h e quantum p a r a m e t e r , A " , and t h e Jr(*,t)YThe reduced t e m p e r a t u r e dependent c r o s s s e c t i o n s ,, are s i m p l y i n t e g r a l s o v e r t h e energy o f t h e a p p r o p r i a t e c r o s s s e c t i o n s ,. is the reduced temperature. These cross sections are functions of the reduced temperature, T'' , the quantum parameter, A9', and the s t a t i s t i c s .The expressions for the transport coefficients of single component systems and mixtures depend simply on the reduced cross sect ions,. These expressions are given elsewhere' and n(fi,tJ 4are not repeated here. Numerical ProceduresThe phase shifts 7t were computed by direct numerical integration (RKG method) of the radial wave equation using a program previously developed , but now improved by the introduction 2 of a modification which continuously adjusted the integration interval depending upon the curvature of the wave function.For convenience, the equivalences among the notation of references 1, 2, and the present discussion are summarized as follows :The method of d e t e r m i n a t i o n o f t h e a p p r o p r i a t e i n t e r v a l s i z e i s as f o l l o w s . F i r s t , a n i n i t i a l i n t e r v a l s i z e was chosen a s t h e smaller o f two q u a n t i t i e s : ' ) . During t h e c o u r s e o f t h e i n t e g r a t i o n , t h e i n t e r v a l s i z e w a s v a r i e d so as t o g i v e a p p r o x i m a t e l y 40i n t e g r a t i o n i n t e r v a l s between two s u c c e s s i v e nodes.The r a d i a l f n t e g r a t i o n was s t o p p e d when t h e v a l u e s of t h e " a p p a r e n t phase s h i f t s ' ' c a l c u l a t e d a t f o u r s u c c e s s i v e nodes d i f f e r e d from e a c h o t h e r by l e s s t h a n 10 r a d i a n . The p h a s e s t h u s o b t a i n e d -4w...
Quantum mechanical c a l c u l a t i o n s , based on t h e Lennard-Jones (12,6) p o t e n t i a l , a r e presented showing t h e dependence of t h e reduced phase s h i f t , 7* , on t h e quantum parameter, h * , f o r a f i x e d reduced e f f e c t i v e p o t e n t i a l and v a r i o u s reduced e n e r g i e s , E". The observed o s c i l l a t o r y behavior of ?*[.') i s due p r i m a r i l y t o t h e i n c l u s i o n of t h e p h y s i c a l l y unimportant c o n t r i b u t i o n of wv t o t h e phase s h i f t , where i s t h e number of quasi-bound ( v i r t u a l ) s t a t e sof energy l e s s than E*. A modified reduced phase s h i f t , ?* 9 d e f i n e d by e x c l u d i n g t h i s c o n t r i b u t i o n , d i s p l a y s only t h e s h a r p i n f l e c t i o n s a s s o c i a t e d w i t h b a r r i e r p e n e t r a t i o n under resonance c o n d i t i o n s . Except f o r t h e resonance c o n t r i b u t i o n , the phase s h i f t s may be a c c u r a t e l y reproduced by a second-order JWKB procedure. T h i s method a l s o a c c u r a t e l y p r e d i c t s t h e r e s o n a n t e n e r g i e s ( i . e . t h e e n e r g i e s of t h e quasi-bound s t a t e s ) . The f i r s t -o r d e r JWKB treatmentof t h e b a r r i e r p e n e t r a t i o n problem by Ford, H i l l , Wakano, and Wheeler s u f f i c e s f o r t h e purpose of e s t i m a t i n g t h e l e v e l widths and l i f e t i m e s of t h e v i r t u a l s t a t e s a s w e l l a s t h e main f e a t u r e s of t h e r e s o n a n t phase s h i f t s , b u t does n o t a c c u r a t e l y reproduce t h e q u a n t a l c a l c u l a t i o n s .W T h i s r e s e a r c h r e c e i v e d f i n a n c i a l support from t h e N a t i o n a l Aeronautics fi t t i h f r-----e and Space A d m i n i s t r a t i o n Grant . W P r e s e n t a d d r e s s : N a t i o n a l Bureau of Standard https://ntrs.nasa.gov/search.jsp?R=19670008526 2018-05-09T17:30:18+00:00Z I n t r o d u c t i o n I n t h e p r e s e n t paper, we p r e s e n t t h e r e s u l t s of a numerical s t u d y of phase s h i f t s which a r i s e i n c o l l i s i o n s between molecules which i n t e r a c t according t o a Lennard-Jones (12,6) p o t e n t i a l . W e c o n s i d e r , i n p a r t i c u l a r , t h e behavior of t h e phase s h i f t s under c o n d i t i o n s such t h a t t h r e e c l a s s i c a l t u r n i n g p o i n t s e x i s t . Although t h e q u a n t i t a t i v e r e s u l t s apply t o t h e p r e s e n t model p o t e n t i a l , most of t h e q u a l i t a t i v e f e a t u r e s a r e c h a r a c t e r i s t i c of any r e a l i s t i c i n t e r m o l e c u l a r p o t e n t i a 1.The L. -J. (12,6) p o t e n t i a l f u n c t i o n v m a y be w r i t t e n where c p Y ( v * ) = 4['" -1 2 -p -6 3 i s t h e depth of t h e p o t e n t i a l minimum, Ui s t h e s e p a r a t i o n a t which t h e p o t e n t i a l i s zero, and s e p a r a t i o n . The phase s h i f t s , *lp , are defined by t h e asymptotic form o f t...
Bound state contributions to transport coefficients J. Chem. Phys. 95, 2702 (1991; 10.1063/1.460922Kinetic theory of moderately dense gasesadditional computations of the triple collision contributions to the transport coefficients for the rigid sphere modelThe quantum-mechanical BBGKY equations are solved in terms of the phase-space transformation functions. Introducing the molecular-chaos assumption, the pair distribution function is obtained. In the binary-collision approximation, this leads to a generalized Boltzmann equation, in which the collisionaltransfer contributions are included. This equation is solved by the method of Chapman and Enskog, and expressions for the transport coefficients are obtained, which are given entirely in terms of the solutions of the two-particle Schrodinger equation. The classical limit (h->O) of the expressions are also derived. The heat flux is found to contain contributions which are proportional to the gradient in density. The significance of these terms, which are of the order h" is discussed.
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