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The influence of a magnetic field on the transport properties of polyatomic gases; A comparison of theory with experiments
Volume24A. number 13
THE
PHYSICS LETTERS
19 June 1967
INFLUENCE OF A MAGNETIC FIELD ON THE TRANSPORT PROPERTIES OF POLYATOMIC GASES; A COMPARISON OF THEORY WITH EXPERIMENTS J. KORVING National Magnet Laboratory, M.I.T., Cambridge, Mass H. F. P. K N A A P The Institute for Advanced Study, Princeton, New Jersey R. G. GORDON Chemistry Department, Harvard University, Cambridge, Mass
and J. J. M. BEEN KKER Kamerlingh Onnes Laboratorium, Leiden, The Netherlands Received 15 May 1967
Effects which arise in the transport properties of polyatomic gases under the influence of a magnetic field. can be explained by a theory which takes into account inelastic collissions. The inelastic collission c r o s s - s e c tions, which can be calculated from the field dependence of the viscosity, are in excellent agreement with those obtained from NMR data.
It is now well known that in a gas of polyatomic molecules under the influence of a magnetic field the following effects can occur: 1) a change in the thermal conductivity and viscosity coefficients (A~ and A~7) [I]; 2) a transverse transport of energy m o m e n t u m (ktr and 7}tr) [2]. These effects are functions of the ratio of the magnetic field and the pressure (H/p). The first group saturates for high (H/p) at a value A;~sat and A~?sat, r e a c h i n g half this value at (H/p)!. The s e c o n d group goes to z e r o for high (H/p), 2passing through a m a x i m u m X~a x and ~?tmraxat the value (H/P)max" F u r t h e r m o r e a c r o s s - e f f e c t l e a d i n g to a t o r q u e in the p r e s e n c e of a t e m p e r a t u r e g r a d i e n t h a s been o b s e r v e d [3]. In the c a s e that the g a s h a s no i n v e r sion s y m m e t r y in the a b s e n c e of m a g n e t i c f i e l d (like a g a s of o p t i c a l l y a c t i v e m o l e c u l e s ) , c r o s s effects between t h e r m a l conductivity and v i s c o s i ty m a y o c c u r [4]. We have p e r f o r m e d e x t e n s i v e e x p e r i m e n t s on A~, AT}, ~tr a n d ~ t r in 0 2 , NO, N2, CO, CO2, H2, D2, HD, CH4, CD 4 and C F 4 [1,2]. E x p l i c i t c a l c u l a t i o n s have been c a r r i e d out along the l i n e s of the C h a p m a n - E n s k o g t h e o r y by Kagan and M a k s i mov [5] f o r A;~ in p a r a m a g n e t i c g a s e s (the s o -
c a l l e d Senftleben e f f e c t ) a n d by Knaap and B e e n a k k e r [6] for the g e n e r a l c a s e of d i a m a g n e t i c m o l e c u l e s (Ak, A~7, ~tr, 77tr). T h e s e t r e a t m e n t s a r e b a s e d on an e l a s t i c c o l l i s i o n m o d e l . The t h e o r y g i v e s a good p h e n o m e n o l o g i c a l d e s c r i p t i o n of the o b s e r v e d e f f e c t s a s well a s the c o r r e c t d e p e n d e n ce on HIp f o r the t h e r m a l conductivity. However, t h e o r e t i c a l p r e d i c t i o n s f o r the v i s c o s i t y a p p e a r to be p o o r . In a c r i t i c a l a n a l y s i s of this situation we found that the a s s u m p t i o n that only e l a s t i c c o l l i s i o n s a r e of i m p o r t a n c e i s u n s a t i s f a c t o r y . The f i e l d effect is d i r e c t l y r e l a t e d to the f a c t that, if in the f i e l d f r e e c a s e a g r a d i e n t in a m a c r o s c o p i c quantity in a gas of p o l y a t o m i c m o l e c u l e s o c c u r s , the d i s t r i b u t i o n function b e c o m e s a n i s o t r o p i c in both v e l o c i t y , U , and a n g u l a r m o m e n t u m , M . One can e x p r e s s this a n i s o t r o p y in t e r m s of i r r e d u c i b l e t e n s o r s m a d e up of U and M. The work of D a h l e r et at. [7] shows that the l e a d i n g t e r m c o n talning M in the c a s e of t h e r m a l conductivity i s [U] [ M ~ and f o r the v i s c o s i t y [MA4J. A t e r m of type [U] [MM] i s , b e c a u s e of the d e p e n d e n c e on U, s e n s i t i v e to e l a s t i c c o l l i s i o n s , but [MM] i s only influenced by i n e l a s t i c c o l l i s i o n s that change M. The influence of the m a g n e t i c f i e l d is brought '755
Volume24A. number 13
PHYSICS LETTERS
about by the p a r t i a l d e s t r u c t i o n of t h e s e a n i s o t r o p i e s by m a g n e t i c a l l y induced r e a r r a n g e m e n t s of M, which a r e c a u s e d by the L a r m o r p r e c e s s i o n . Thus in the c a s e of the v i s c o s i t y the u s e of an e l a s t i c c o l l i s i o n m o d e l s u p p r e s s e s the c o n t r i b u tion of the leading [MM] t e r m , r e t a i n i n g only an a n i s o t r o p y t e r m of h i g h e r o r d e r : [UU] [MM]. This explains why the e l a s t i c model, which g i v e s a f a i r l y good d e s c r i p t i o n of the heat conductivity e x p e r i m e n t s , is quite inadequate f o r the v i s c o s i t y results. We found that a m o d e l allowing a l s o f o r i n e l a s tic c o l l i s i o n s (A mj ¢ 0 and A j ¢ 0) can explain m o s t of the o b s e r v e d d i s c r e p a n c i e s . Using a m e thod s i m i l a r to the v e r y elegant t r e a t m e n t for z~)t i n t r o d u c e d by M c C o u r t and Snider [8] f o r an a r b i t r a r y potential with s m a l l n o n - s p h e r i c i t y , we found f o r the contribution of [MM] to the v i s c o s i ty [9].
AT r/
- -~
40 2
;
~
1 + 402
± r/
7/tr -
~ -
O 1+O2
- -~
02 1+
02
of HIP that a r e roughly a f a c t o r 10 h i g h er . The s a m e will be t r u e f o r the o t h e r H 2 - i s o t o p e s . b) Eq. (I) g i v es I A~?,,/7/I - IAT/±/T/ i >~ 0 in a g r e e m e n t with e x p e r i m e n t a l r e s u l t s for i n t e r m e diate v a l u e s of HIp [12]. At h i g h e r v a l u e s of H/p, w h e r e [UU] [MM] b e c o m e s i m p o r t a n t one ex p ect s a change of sign of this d i f f e r e n c e [cf. 13]. c) Although f o r the heat conductivity the a l l o w ance for i n e l a s t i c c o l l i s i o n s does not change the f i e l d dependence, it does influence the absolute magnitude of the effect, si n ce now both the heat t r a n s p o r t through t r a n s l a t i o n a l and r o t a t i o n a l d e g r e e s of f r e e d o m a r e affected, [ s e e 8]. T h i s c o r r e s p o n d s to the o b s e r v a t i o n that f o r m o s t m o l e cu l es (02, N2, CH4, . . . ) I Z~,/;t {sat > t ~7//~? I sat as opposed to e l a s t i c t h e o r y . The situation f o r HD, w h e r e i n e l a s t i c c o l l i s i o n s a r e much l e s s f r e quent, is different. Only the r o t a t i o n a l t r a n s p o r t is e f f e c t i v e l y influenced and one finds e x p e r i m e n tally: I ~ / ; t | s a t < I AT/fiT{sat [14].
(1)
(2)
w h e r e 0 =KH/p and ~ and K a r e as yet unknown c onst an t s depending on the potential m o d e l . O can a l s o be wr i t t en as O = 00LT, w h e r e coL is the L a r m o r f r e q u e n c y and 1- the t i m e c h a r a c t e r t s i n g the c o l l i s s i o n p r o c e s s . The c o n c l u s i o n s that can be drawn f r o m this a p p r o a c h a r e : a) The t i m e s c a l e i n v o l v e d in the v i s c o s i t y is now the one r e l a t e d to the r e o r i e n t a t i o n of M. T h i s t i m e , which is known independently f r o m NMR data, can be a p p r e c i a b l y h i g h e r than the t i m e between two s u c c e s s i v e (elastic) c o l l i s i o n s . T h i s explains why (H/p). and (H/P)max f o r the v i s c o s i t y a r e a p p r e c i a b l y s m a l l e r than the c o r r e s p o n d i n g v a l u e s f o r the t h e r m a l conductivity . T h i s is e s p e c i a l l y s t r i k i n g in the c a s e of the H 2i o s t o p e s w h e r e the e a r l i e r d i s c r e p a n c y of a f a c t o r 10 can be fully explained. C a l c u l a t i o n s b a s e d on this m o d e l using i n e l a s t i c s c a t t e r i n g c r o s s s e c t i o n s obtained f r o m d e u t e r o n NMR work [10] a g r e e within 5%. F o r t t 2 - i s o t o p e s the t i m e s c a l e c o r r e s p o n d i n g to [M/Ill and to [UU] [/14/14] d i f f e r s by a f a c t o r 10 so that the two c o n t r i b u t i o n s will no l o n g e r o v e r l a p . Indeed one finds f o r the e x i s t i n g data on ttD, that do not extend to high v a l u e s of H/p, that the [M,M] t e r m alone g i v e s v e r y a c c u r a t e l y the r a t i o s of (HiP)max and (H/p)~ and of 0/tr/~ )max and (AT//7/)sat. We e x p e c t a s e c o n d contribution of 7/tr/7/ and A7//~7 at v a l u e s
756
19 June 1967
References 1. J.Korving, Thesis, Leiden (1967}; J. Korving, H. Hulsman, G. Scoles. H . F . P . Knaap and J. J.M. Beenakker, Physica 33 (1967), to be published; J. Korving, W.I. Honeywell, T.K. Bose and J. J. M. Beenakker, Physica 33 (1967), to be published. 2. J. Korving, H.Hulsman, H.F.P.Knaap and J . J . M . Beenakker, Phys. Letters 21 (1966) 5; L. L. Gorelik. V.G. Nikolajewski and V. V. Sinitsyn, JETP Letters 4 (1966) 307; L. J. F. Hermans, P.H. Fortuin, H. F. P. Knaap and J. J. M. Beenakker. Phys. Letters, to be published. 3. G.G. Scott, H.W. Turner and R. M. Williamson, Bull. Amer. Phys. Soc. 12 (1967) 94. 4. We acknowledge a discussion with Dr. H.C.Andersen of Harvard University. 5. Y.Kagan and L.Maksimov, Soviet Phys. JETP 14 (1962) 604. 6. H. F. P. Knaap and J. J. M. Beenakker. Physica 33 (1967) 643. 7. D.W. Condiff, Wei-Kao Lu and J.S.Dahler, J. Chem. Phys, 42 (1965} 3445. 8. F.R. McCourt, Thesis, University of British Columbia, Vancouver (1966}. 9. H . F . P . Knaap and R.G.Gordon, J. Chem. Phys. (1967) to be published. 10. W. Hardy, Thesis, University of British Columbia, Vancouver (1964). 11. L.L. Gorelik, J. N. Redkoborody and V. V. Sinitsyn, Soviet Phys. JETP 48 (1965} 751. 12. J. Korving, H. F. P. Knaap and J. J. M. Beenakker, Physica 33 (1967} to be published. 13. J.K.Kikoin, K.J. Balashov, S.D. Lasarev and R.E. Neushtadt, Phys. Letters 24A (1967) 165. 14. T.K. Bose, J. Kokx, J. Korving and J. J. M. Beenakker, Physica (1967} to be published.
THE
PHYSICS LETTERS
19 June 1967
INFLUENCE OF A MAGNETIC FIELD ON THE TRANSPORT PROPERTIES OF POLYATOMIC GASES; A COMPARISON OF THEORY WITH EXPERIMENTS J. KORVING National Magnet Laboratory, M.I.T., Cambridge, Mass H. F. P. K N A A P The Institute for Advanced Study, Princeton, New Jersey R. G. GORDON Chemistry Department, Harvard University, Cambridge, Mass
and J. J. M. BEEN KKER Kamerlingh Onnes Laboratorium, Leiden, The Netherlands Received 15 May 1967
Effects which arise in the transport properties of polyatomic gases under the influence of a magnetic field. can be explained by a theory which takes into account inelastic collissions. The inelastic collission c r o s s - s e c tions, which can be calculated from the field dependence of the viscosity, are in excellent agreement with those obtained from NMR data.
It is now well known that in a gas of polyatomic molecules under the influence of a magnetic field the following effects can occur: 1) a change in the thermal conductivity and viscosity coefficients (A~ and A~7) [I]; 2) a transverse transport of energy m o m e n t u m (ktr and 7}tr) [2]. These effects are functions of the ratio of the magnetic field and the pressure (H/p). The first group saturates for high (H/p) at a value A;~sat and A~?sat, r e a c h i n g half this value at (H/p)!. The s e c o n d group goes to z e r o for high (H/p), 2passing through a m a x i m u m X~a x and ~?tmraxat the value (H/P)max" F u r t h e r m o r e a c r o s s - e f f e c t l e a d i n g to a t o r q u e in the p r e s e n c e of a t e m p e r a t u r e g r a d i e n t h a s been o b s e r v e d [3]. In the c a s e that the g a s h a s no i n v e r sion s y m m e t r y in the a b s e n c e of m a g n e t i c f i e l d (like a g a s of o p t i c a l l y a c t i v e m o l e c u l e s ) , c r o s s effects between t h e r m a l conductivity and v i s c o s i ty m a y o c c u r [4]. We have p e r f o r m e d e x t e n s i v e e x p e r i m e n t s on A~, AT}, ~tr a n d ~ t r in 0 2 , NO, N2, CO, CO2, H2, D2, HD, CH4, CD 4 and C F 4 [1,2]. E x p l i c i t c a l c u l a t i o n s have been c a r r i e d out along the l i n e s of the C h a p m a n - E n s k o g t h e o r y by Kagan and M a k s i mov [5] f o r A;~ in p a r a m a g n e t i c g a s e s (the s o -
c a l l e d Senftleben e f f e c t ) a n d by Knaap and B e e n a k k e r [6] for the g e n e r a l c a s e of d i a m a g n e t i c m o l e c u l e s (Ak, A~7, ~tr, 77tr). T h e s e t r e a t m e n t s a r e b a s e d on an e l a s t i c c o l l i s i o n m o d e l . The t h e o r y g i v e s a good p h e n o m e n o l o g i c a l d e s c r i p t i o n of the o b s e r v e d e f f e c t s a s well a s the c o r r e c t d e p e n d e n ce on HIp f o r the t h e r m a l conductivity. However, t h e o r e t i c a l p r e d i c t i o n s f o r the v i s c o s i t y a p p e a r to be p o o r . In a c r i t i c a l a n a l y s i s of this situation we found that the a s s u m p t i o n that only e l a s t i c c o l l i s i o n s a r e of i m p o r t a n c e i s u n s a t i s f a c t o r y . The f i e l d effect is d i r e c t l y r e l a t e d to the f a c t that, if in the f i e l d f r e e c a s e a g r a d i e n t in a m a c r o s c o p i c quantity in a gas of p o l y a t o m i c m o l e c u l e s o c c u r s , the d i s t r i b u t i o n function b e c o m e s a n i s o t r o p i c in both v e l o c i t y , U , and a n g u l a r m o m e n t u m , M . One can e x p r e s s this a n i s o t r o p y in t e r m s of i r r e d u c i b l e t e n s o r s m a d e up of U and M. The work of D a h l e r et at. [7] shows that the l e a d i n g t e r m c o n talning M in the c a s e of t h e r m a l conductivity i s [U] [ M ~ and f o r the v i s c o s i t y [MA4J. A t e r m of type [U] [MM] i s , b e c a u s e of the d e p e n d e n c e on U, s e n s i t i v e to e l a s t i c c o l l i s i o n s , but [MM] i s only influenced by i n e l a s t i c c o l l i s i o n s that change M. The influence of the m a g n e t i c f i e l d is brought '755
Volume24A. number 13
PHYSICS LETTERS
about by the p a r t i a l d e s t r u c t i o n of t h e s e a n i s o t r o p i e s by m a g n e t i c a l l y induced r e a r r a n g e m e n t s of M, which a r e c a u s e d by the L a r m o r p r e c e s s i o n . Thus in the c a s e of the v i s c o s i t y the u s e of an e l a s t i c c o l l i s i o n m o d e l s u p p r e s s e s the c o n t r i b u tion of the leading [MM] t e r m , r e t a i n i n g only an a n i s o t r o p y t e r m of h i g h e r o r d e r : [UU] [MM]. This explains why the e l a s t i c model, which g i v e s a f a i r l y good d e s c r i p t i o n of the heat conductivity e x p e r i m e n t s , is quite inadequate f o r the v i s c o s i t y results. We found that a m o d e l allowing a l s o f o r i n e l a s tic c o l l i s i o n s (A mj ¢ 0 and A j ¢ 0) can explain m o s t of the o b s e r v e d d i s c r e p a n c i e s . Using a m e thod s i m i l a r to the v e r y elegant t r e a t m e n t for z~)t i n t r o d u c e d by M c C o u r t and Snider [8] f o r an a r b i t r a r y potential with s m a l l n o n - s p h e r i c i t y , we found f o r the contribution of [MM] to the v i s c o s i ty [9].
AT r/
- -~
40 2
;
~
1 + 402
± r/
7/tr -
~ -
O 1+O2
- -~
02 1+
02
of HIP that a r e roughly a f a c t o r 10 h i g h er . The s a m e will be t r u e f o r the o t h e r H 2 - i s o t o p e s . b) Eq. (I) g i v es I A~?,,/7/I - IAT/±/T/ i >~ 0 in a g r e e m e n t with e x p e r i m e n t a l r e s u l t s for i n t e r m e diate v a l u e s of HIp [12]. At h i g h e r v a l u e s of H/p, w h e r e [UU] [MM] b e c o m e s i m p o r t a n t one ex p ect s a change of sign of this d i f f e r e n c e [cf. 13]. c) Although f o r the heat conductivity the a l l o w ance for i n e l a s t i c c o l l i s i o n s does not change the f i e l d dependence, it does influence the absolute magnitude of the effect, si n ce now both the heat t r a n s p o r t through t r a n s l a t i o n a l and r o t a t i o n a l d e g r e e s of f r e e d o m a r e affected, [ s e e 8]. T h i s c o r r e s p o n d s to the o b s e r v a t i o n that f o r m o s t m o l e cu l es (02, N2, CH4, . . . ) I Z~,/;t {sat > t ~7//~? I sat as opposed to e l a s t i c t h e o r y . The situation f o r HD, w h e r e i n e l a s t i c c o l l i s i o n s a r e much l e s s f r e quent, is different. Only the r o t a t i o n a l t r a n s p o r t is e f f e c t i v e l y influenced and one finds e x p e r i m e n tally: I ~ / ; t | s a t < I AT/fiT{sat [14].
(1)
(2)
w h e r e 0 =KH/p and ~ and K a r e as yet unknown c onst an t s depending on the potential m o d e l . O can a l s o be wr i t t en as O = 00LT, w h e r e coL is the L a r m o r f r e q u e n c y and 1- the t i m e c h a r a c t e r t s i n g the c o l l i s s i o n p r o c e s s . The c o n c l u s i o n s that can be drawn f r o m this a p p r o a c h a r e : a) The t i m e s c a l e i n v o l v e d in the v i s c o s i t y is now the one r e l a t e d to the r e o r i e n t a t i o n of M. T h i s t i m e , which is known independently f r o m NMR data, can be a p p r e c i a b l y h i g h e r than the t i m e between two s u c c e s s i v e (elastic) c o l l i s i o n s . T h i s explains why (H/p). and (H/P)max f o r the v i s c o s i t y a r e a p p r e c i a b l y s m a l l e r than the c o r r e s p o n d i n g v a l u e s f o r the t h e r m a l conductivity . T h i s is e s p e c i a l l y s t r i k i n g in the c a s e of the H 2i o s t o p e s w h e r e the e a r l i e r d i s c r e p a n c y of a f a c t o r 10 can be fully explained. C a l c u l a t i o n s b a s e d on this m o d e l using i n e l a s t i c s c a t t e r i n g c r o s s s e c t i o n s obtained f r o m d e u t e r o n NMR work [10] a g r e e within 5%. F o r t t 2 - i s o t o p e s the t i m e s c a l e c o r r e s p o n d i n g to [M/Ill and to [UU] [/14/14] d i f f e r s by a f a c t o r 10 so that the two c o n t r i b u t i o n s will no l o n g e r o v e r l a p . Indeed one finds f o r the e x i s t i n g data on ttD, that do not extend to high v a l u e s of H/p, that the [M,M] t e r m alone g i v e s v e r y a c c u r a t e l y the r a t i o s of (HiP)max and (H/p)~ and of 0/tr/~ )max and (AT//7/)sat. We e x p e c t a s e c o n d contribution of 7/tr/7/ and A7//~7 at v a l u e s
756
19 June 1967
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