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Effect of a superflow on a vortex lattice
Volume 31A, number 5
PHYSICS LETTERS
EFFECT
OF
A SUPERFLOW
9March1970
ON A V O R T E X
LATTICE
M. Le RAY, J. B A T A I L L E , M. FRANQOIS and D. L ' H U I L L I E R L a b o r a t o i r e de M4canique des F l u i d e s *, Facult@ d ' O r s a y - 91 - F r a n c e Received 21 January 1970
This paper provides experimental evidence that the vortex lines due to rotation very probably exhibit the theoretically predicted triangular structure which persists in the presence of a superflow.
As m e n t i o n e d in ref. 1 e x p e r i m e n t s of s i m u l tan e ous t h e r m o m e c h a n i c a l l y d r i v e n s u p e r f l o w p a r a l l e l to OX and u n i f o r m r o t a t i o n about OZ w e r e c a r r i e d out in a channel s u r r o u n d e d with a r o t a tion shield, but o t h e r w i s e i d e n t i c a l with those p r e v i o u s l y used [1,2]. M e a s u r e m e n t s of the s e cond sound e x c e s s attenuation ( a - ao) w e r e m a d e in both Y and Z d i r e c t i o n s as a function of s u p e r fluid m e a n v e l o c i t y Vs, f o r a given a n g u l a r v e l o city w, and a l s o at a g i v e n Vs f o r v a r i a b l e o~. * Partly sponsored by D.R.M.E.
4r~-~ [~=- =.).j]3
T = 1.385 °K
3
i I""
i
J'J
I
J i
The c o m p a r a t i v e l y c l e a n e r c h a r a c t e r of such e x p e r i m e n t s (superflow i n st ead of counterflow, p r e s e n c e of a r o t a t i n g shield) made it p o s s i b l e to d e t e r m i n e p r e c i s e l y the d i f f e r e n t s t e p s of the int e r a c t i o n between the su p er f l o w and the v o r t e x l a t t i c e due to r o t at i o n , in t e r m s of the non d i m e n sional c i r c u l a t i o n K* = V s d / ( h / m ) , w h e r e d stands f o r the i n t e r v o r t e x d i s t a n c e of the l a t t i c e , which has been a s s u m e d t r i a n g u l a r , a c c o r d i n g to the t h e o r e t i c a l p r e d i c t i o n s of Kiknadze et al. [3] at T ~ T~ and Tkachenko [4] at T = 0, f o r p u r e r o tation (Vs = 0), following t h e o r e t i c a l and e x p e r i m e n t a l work of A b r i k o s o v [5], K l e i n e r et al. [6], M a t r i c o n [7] and C r i b i e r et al. [8] in type II sup e r c o n d u c t o r s . We r e c a l l that i n t e r v o r t e x d i s t ance in the c a s e of a s q u a r e l a t t i c e should be 6.94% l o w e r than in the c a s e of a t r i a n g u l a r l a t tice. The e x p e r i m e n t a l p r e c i s i o n on K* is e s t i m ated to be b e t t e r than 2%. A n u m b e r of points of s p e c i a l i n t e r e s t (inflexions, slope c h a n g e s , i n t e r s e c t i o n s , junction with the "additivity c u r v e " , e x t r e m a of the " i s o t r o p y d e g r e e " ) of the c u r v e s shown in figs. 1 and 2,
i i
.JJ--L/i
i I
/!i
fsotropy degree
I
T =1.385 "K
0.75
Fig. I. [(Ot-Oeo)*]I/2= [(Or-Oto)/(OZ-Oto)R]V~, (ot-ok))R being the attenuation due to rotation only, in the d i r e c tion Y, versus K* = V s d / ( h / ~ ). With w = 4.29 rad~sec and variable Ys: • Superflow only, [(ot- c~o)~]v~, + Superflow and rotation second sound in Y dir~ctinn, [(ot-~)~.]V2; 0 Superflow mpd rotation, second sound in Z ~hr'ection [ ( ~ - ~ o ) ~ ] ~ ; • Rotation only, second sound, in Y dirr~tion_ . . . . : [{(OL"~o)R + (Or- ~.o)S}/(~ -~o)R] , ~addltivity (calculated). With Vs = 0.250 c m / s e c and variable o~: x Superflow and rotation, second sound in Z direction.
~
i
i
i-~+
0.5
025
,..,ll •
Ji
Ji
,
11
132-V3 } '/'3 2 ~4
.
.
I
3~V3
2V~
4
V_.~./
l
I
}
3V3
Fig. 2. (Or- ~o)Z/(Ot- C~o)Y versus K*. 249
Volume 31A. number 5
PHYSICS LETTERS
c o r r e s p o n d v e r y c l o s e l y to a set of r e m a r k a b l e v a l u e s of K*. Of c o u r s e , it is i m p o s s i b l e to be a b s o l u t e l y p o s i t i v e about such a c o m p l e t e c o i n c i dence, at l e a s t f o r s o m e of the l a r g e s t n u m b e r s of the set. H o w e v e r the r a t i o of the a b s c i s s a e of points of s i m i l a r n a t u r e on c u r v e s Y and Z , r e s p e c t i v e l y , is equal to ~-, with a good p r e c i s i o n (Cf. for i n s t a n c e , the slope changes at 2 / ( 3 - o n c u r v e Yand 2 on c u r v e Z). On the other hand, f o r many of the r e m a r k able v a l u e s of the set, the " i s o t r o p y d e g r e e " (o~- Oto)Z/(Ot- Cto)Y d i f f e r s by l e s s than 2% f r o m v e r y s i m p l e r a t i o s e a s i l y r e l a t e d with 1, 2, ~'3, 3. In view of t h e s e s t r i k i n g f e a t u r e s , it is suggested that d i s t a n c e s like d, d/~3, d'¢~ play a p r o m i n e n t r o l e , a s s o c i a t e d with s o m e kind of t r i a n g u l a r s t r u c t u r e of the v o r t e x l i n e s , c o n s i s t ent with the above m en t i o n e d t h e o r e t i c a l p r e d i c tions. Obviously, such a s t r u c t u r e should not be identified with the f a m i l i a r t r i a n g u l a r a r r a y of r e c t i l i n e a r v o r t i c e s c o r r e s p o n d i n g to Vs = 0. It m u s t be modified in s o m e way by the s u p e r f l o w which i n v o l v e s the d e f o r m a t i o n of the v o r t e x l i n e s and p e r h a p s the f o r m a t i o n of v o r t e x r i n g s regularly distributed. It will be notided that the attenuations in d i r e c t i o n s Yand Z due to s u p e r f l o w and r o t a t i o n s t a r t be in g additive f o r K* = 2 ~ - a n d K* = 6 r e s p e c t i v e l y (for the l a t t e r value K*, i s o t r o p y is r e a c h e d ) w h e r e a s , in ref. 1 a p p r o x i m a t e a d d i t i v i ty has b e e n o b s e r v e d even f o r low v a l u e s of
I Vs - Vn[ at 1.47°K. It is believed that in our first experiment [i], the lack of a shield preventing the creation of quasi isotropic turbulence at
250
9March 1970
the e n t r a n c e of the r o t a t i n g channel i s responsibl~ f o r this d i f f e r e n c e . It is worth mentioning that the attenuation, f o r su p er f l o w only, (fig. l) c l e a r ly exhibits a change of slope quite analogous to that o b s e r v e d by K r a m e r s [9]. It should be e m p h a s i z e d that s i m i l a r p e c u l i a r i t i e s of quantization have been o b s e r v e d at d i f f e r ent t e m p e r a t u r e s and angular v e l o c i t i e s and e v e n in counterflow (between 1.35°K and 2.08°K). D et a i l e d r e s u l t s and c o m m e n t s will be published in a n e a r future. We wish to thank Dr. Y. Simon f o r valuable discussions.
Refer~ces 1. M. Le Ray, J. Bataille and M. Franqois, Phys. Letters 29A (1969) 699. 2. M. Le Ray and J. Batallle, Phys. Letters 26A (1968) 283 and Errata 27A (1968) 68. 3. L. V. Kiknadze, Yu. G. Mamaladze and O. D. Cheisvllt, Zh. Eksp. i Teor. Fiz. 48 (1965) 1520. 4. V.K. Tkachenko, Zh. Eksp. i Teor. Fiz. 49 (1965) 1875. 5. A. A. Abrikosov, Zh. Eksp. i Teor. Fiz. 32 (1957) 1442. 6. W.H. Kleiner, L. M. Roth and S. H. Autler, Phys. Rev 133 (1964) A1226. 7. J. Matricon, Low temperature physics, LT9 (Plenum Press, 1964), Part A, p. 544. 8. D. Cribier, B.Jacrot, L. Madhav Rao and B. Farnoux Progress in low temperature physics (North Holland, 1967) p. 161. 9. H. C. Kramers, Symposium of St. Andrews, 1965 in: Superfluid helium (Academic Press, 1966) p. 199.
PHYSICS LETTERS
EFFECT
OF
A SUPERFLOW
9March1970
ON A V O R T E X
LATTICE
M. Le RAY, J. B A T A I L L E , M. FRANQOIS and D. L ' H U I L L I E R L a b o r a t o i r e de M4canique des F l u i d e s *, Facult@ d ' O r s a y - 91 - F r a n c e Received 21 January 1970
This paper provides experimental evidence that the vortex lines due to rotation very probably exhibit the theoretically predicted triangular structure which persists in the presence of a superflow.
As m e n t i o n e d in ref. 1 e x p e r i m e n t s of s i m u l tan e ous t h e r m o m e c h a n i c a l l y d r i v e n s u p e r f l o w p a r a l l e l to OX and u n i f o r m r o t a t i o n about OZ w e r e c a r r i e d out in a channel s u r r o u n d e d with a r o t a tion shield, but o t h e r w i s e i d e n t i c a l with those p r e v i o u s l y used [1,2]. M e a s u r e m e n t s of the s e cond sound e x c e s s attenuation ( a - ao) w e r e m a d e in both Y and Z d i r e c t i o n s as a function of s u p e r fluid m e a n v e l o c i t y Vs, f o r a given a n g u l a r v e l o city w, and a l s o at a g i v e n Vs f o r v a r i a b l e o~. * Partly sponsored by D.R.M.E.
4r~-~ [~=- =.).j]3
T = 1.385 °K
3
i I""
i
J'J
I
J i
The c o m p a r a t i v e l y c l e a n e r c h a r a c t e r of such e x p e r i m e n t s (superflow i n st ead of counterflow, p r e s e n c e of a r o t a t i n g shield) made it p o s s i b l e to d e t e r m i n e p r e c i s e l y the d i f f e r e n t s t e p s of the int e r a c t i o n between the su p er f l o w and the v o r t e x l a t t i c e due to r o t at i o n , in t e r m s of the non d i m e n sional c i r c u l a t i o n K* = V s d / ( h / m ) , w h e r e d stands f o r the i n t e r v o r t e x d i s t a n c e of the l a t t i c e , which has been a s s u m e d t r i a n g u l a r , a c c o r d i n g to the t h e o r e t i c a l p r e d i c t i o n s of Kiknadze et al. [3] at T ~ T~ and Tkachenko [4] at T = 0, f o r p u r e r o tation (Vs = 0), following t h e o r e t i c a l and e x p e r i m e n t a l work of A b r i k o s o v [5], K l e i n e r et al. [6], M a t r i c o n [7] and C r i b i e r et al. [8] in type II sup e r c o n d u c t o r s . We r e c a l l that i n t e r v o r t e x d i s t ance in the c a s e of a s q u a r e l a t t i c e should be 6.94% l o w e r than in the c a s e of a t r i a n g u l a r l a t tice. The e x p e r i m e n t a l p r e c i s i o n on K* is e s t i m ated to be b e t t e r than 2%. A n u m b e r of points of s p e c i a l i n t e r e s t (inflexions, slope c h a n g e s , i n t e r s e c t i o n s , junction with the "additivity c u r v e " , e x t r e m a of the " i s o t r o p y d e g r e e " ) of the c u r v e s shown in figs. 1 and 2,
i i
.JJ--L/i
i I
/!i
fsotropy degree
I
T =1.385 "K
0.75
Fig. I. [(Ot-Oeo)*]I/2= [(Or-Oto)/(OZ-Oto)R]V~, (ot-ok))R being the attenuation due to rotation only, in the d i r e c tion Y, versus K* = V s d / ( h / ~ ). With w = 4.29 rad~sec and variable Ys: • Superflow only, [(ot- c~o)~]v~, + Superflow and rotation second sound in Y dir~ctinn, [(ot-~)~.]V2; 0 Superflow mpd rotation, second sound in Z ~hr'ection [ ( ~ - ~ o ) ~ ] ~ ; • Rotation only, second sound, in Y dirr~tion_ . . . . : [{(OL"~o)R + (Or- ~.o)S}/(~ -~o)R] , ~addltivity (calculated). With Vs = 0.250 c m / s e c and variable o~: x Superflow and rotation, second sound in Z direction.
~
i
i
i-~+
0.5
025
,..,ll •
Ji
Ji
,
11
132-V3 } '/'3 2 ~4
.
.
I
3~V3
2V~
4
V_.~./
l
I
}
3V3
Fig. 2. (Or- ~o)Z/(Ot- C~o)Y versus K*. 249
Volume 31A. number 5
PHYSICS LETTERS
c o r r e s p o n d v e r y c l o s e l y to a set of r e m a r k a b l e v a l u e s of K*. Of c o u r s e , it is i m p o s s i b l e to be a b s o l u t e l y p o s i t i v e about such a c o m p l e t e c o i n c i dence, at l e a s t f o r s o m e of the l a r g e s t n u m b e r s of the set. H o w e v e r the r a t i o of the a b s c i s s a e of points of s i m i l a r n a t u r e on c u r v e s Y and Z , r e s p e c t i v e l y , is equal to ~-, with a good p r e c i s i o n (Cf. for i n s t a n c e , the slope changes at 2 / ( 3 - o n c u r v e Yand 2 on c u r v e Z). On the other hand, f o r many of the r e m a r k able v a l u e s of the set, the " i s o t r o p y d e g r e e " (o~- Oto)Z/(Ot- Cto)Y d i f f e r s by l e s s than 2% f r o m v e r y s i m p l e r a t i o s e a s i l y r e l a t e d with 1, 2, ~'3, 3. In view of t h e s e s t r i k i n g f e a t u r e s , it is suggested that d i s t a n c e s like d, d/~3, d'¢~ play a p r o m i n e n t r o l e , a s s o c i a t e d with s o m e kind of t r i a n g u l a r s t r u c t u r e of the v o r t e x l i n e s , c o n s i s t ent with the above m en t i o n e d t h e o r e t i c a l p r e d i c tions. Obviously, such a s t r u c t u r e should not be identified with the f a m i l i a r t r i a n g u l a r a r r a y of r e c t i l i n e a r v o r t i c e s c o r r e s p o n d i n g to Vs = 0. It m u s t be modified in s o m e way by the s u p e r f l o w which i n v o l v e s the d e f o r m a t i o n of the v o r t e x l i n e s and p e r h a p s the f o r m a t i o n of v o r t e x r i n g s regularly distributed. It will be notided that the attenuations in d i r e c t i o n s Yand Z due to s u p e r f l o w and r o t a t i o n s t a r t be in g additive f o r K* = 2 ~ - a n d K* = 6 r e s p e c t i v e l y (for the l a t t e r value K*, i s o t r o p y is r e a c h e d ) w h e r e a s , in ref. 1 a p p r o x i m a t e a d d i t i v i ty has b e e n o b s e r v e d even f o r low v a l u e s of
I Vs - Vn[ at 1.47°K. It is believed that in our first experiment [i], the lack of a shield preventing the creation of quasi isotropic turbulence at
250
9March 1970
the e n t r a n c e of the r o t a t i n g channel i s responsibl~ f o r this d i f f e r e n c e . It is worth mentioning that the attenuation, f o r su p er f l o w only, (fig. l) c l e a r ly exhibits a change of slope quite analogous to that o b s e r v e d by K r a m e r s [9]. It should be e m p h a s i z e d that s i m i l a r p e c u l i a r i t i e s of quantization have been o b s e r v e d at d i f f e r ent t e m p e r a t u r e s and angular v e l o c i t i e s and e v e n in counterflow (between 1.35°K and 2.08°K). D et a i l e d r e s u l t s and c o m m e n t s will be published in a n e a r future. We wish to thank Dr. Y. Simon f o r valuable discussions.
Refer~ces 1. M. Le Ray, J. Bataille and M. Franqois, Phys. Letters 29A (1969) 699. 2. M. Le Ray and J. Batallle, Phys. Letters 26A (1968) 283 and Errata 27A (1968) 68. 3. L. V. Kiknadze, Yu. G. Mamaladze and O. D. Cheisvllt, Zh. Eksp. i Teor. Fiz. 48 (1965) 1520. 4. V.K. Tkachenko, Zh. Eksp. i Teor. Fiz. 49 (1965) 1875. 5. A. A. Abrikosov, Zh. Eksp. i Teor. Fiz. 32 (1957) 1442. 6. W.H. Kleiner, L. M. Roth and S. H. Autler, Phys. Rev 133 (1964) A1226. 7. J. Matricon, Low temperature physics, LT9 (Plenum Press, 1964), Part A, p. 544. 8. D. Cribier, B.Jacrot, L. Madhav Rao and B. Farnoux Progress in low temperature physics (North Holland, 1967) p. 161. 9. H. C. Kramers, Symposium of St. Andrews, 1965 in: Superfluid helium (Academic Press, 1966) p. 199.