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Independence of the density of rotating liquid helium on the angular velocity
Volume 18, number 1
PHYSICS LETTERS
The e v a l u a t i o n of e q s . (4) and (5) by the e x p r e s s i o n s (2) and (3) gives explicit f o r m u l a e f o r d e t e r m i n i n g the r e s p o n s e function in the p r e s e n c e of a filter:
Dfl(p)
=
_I
,,
F2(1 +~fl2)a
+
4fl(fl- 2)p2a
+
~(I +~") L -
4~2p4a + 2 ( ~ - ~ 2 _
I)pE + 4g(l _ ~)13p3~ +
- ~0~o5~ + ~¢~o3E3],
(6)
w h e r e a = a r c c o s p and ~ = ~ - ~ 2 . If we l e t ~ --, 0, the r e s p o n s e function i s e a s i l y s e e n to tend to the f o r m Do(O) =
2 ¥(a
INDEPENDENCE
- ~),
(7)
1 August 1965
which i s the known r e s u l t f o r c o n s t a n t r e l e a s e . F i g . 1 i s showing the f r e q u e n c y r e s p o n s e of above m e n t i o n e d c a s e s of r e l e a s e . F r o m t h i s c u r v e s i t m a y e a s i l y shown t h a t p r a c t i c a l l y in the whole r e g i o n u s i n g t h i s a s s u m p t e d f i l t e r the r e s p o n s e function h a s b e t t e r v a l u e s a s the o p t i c a l s y s t e m with c o n s t a n t r e l e a s e . I a m i n d e b t e d to P r o f e s s o r H a v e l k a f o r m a n y helpful d i s c u s s i o n s .
References 1. Ma~'eoh~ and F r ~ o n , D t ~ o n structure des ~m, 1960). 2. H.H.Hcpkins, The frequemcy r u p c ~ e of a defocusseal optical system Proc.Soc. 231 (1955) p. 91.
OF THE DENSITY oF ROTATING ON THE ANGULAR VELOCITY
LIQUID
HELIUM
E. L. AND/K)NIKASHVILI a n d J. S. TSAKADZE l ~ t t t ~ e of Plzysics, Academy of Sciences of the Goo~zm SSR, Tbtltst, USSR Received 7 July 1965
The p r o b l e m of the d e n s i t y change in r o t a t i n g h e l i u m H, in c o m p a r i s o n with that when i t i s at r e s t , i s of g r e a t i n t e r e s t b e c a u s e of the p r e s e n c e of O n s a g e r - F e y n m a n ' s q u a u t i z e d v o r t i c e s in a quantum liquid [1]. In p a r t i c u l a r , one often e n c o u n t e r s an a s s u m p tion, which i s s t a t e d in t a l k s and in p u b l i c a t i o n s , that the e x i s t e n c e of v o r t e x l i n e s s e e m s to m a k e r o t a t i n g h e l i u m II m o r e " l o o s e " . We have m a d e an e x p e r i m e n t to find the angul a r v e l o c i t y d e p e n d e n c e of the d e n s i t y of h e l i u m H. A s e n s i t i v e p y c n o m e t e r w a s b r o u g h t into r o t a t i o n . A c o p p e r c y l i n d e r of the v o l u m e equal to 52.57 c m 3 with a g l a s s c a p i l l a r y (60 m m long and with the i n n e r d i a m e t e r of 1.75 mm) a t t a c h e d to i t w a s u s e d a s a p y c n o m e t e r (fig. 1). The top of the c a p i l l a r y w a s blown into a s m a l l s p h e r e and i t s s e c ond end w a s glued into a v a l v e , t h i s end r e g u l a t e d the amount of liquid h e l i u m in the p y c n o m e t e r . D u r i n g the m e a s u r e m e n t s the v a l v e w a s c l o s e d and the p y c n o m e t e r , i l l u m i n a t e d b y a l a m p of daylight, w a s e x a m i n e d t h r o u g h a c a t h e t o m e t e r . The a c c u r a c y of a s i n g l e r e a d i n g then r e a c h e d ~- 0.005 ram, u n d e r t h e s e conditions the s p r e a d
26
Y Fig. 1.
Volume 18, number 1
PHYSICS LETTERS
of the e x p e r i m e n t a l d a t a did not e x c e e d 0.2 m m . This f a c t gave the p o s s i b i l i t y to m e a s u r e the r e l a t i v e change of the d e n s i t y with the a c c u r a c y A p / p = * 0.0009%. The p y c n o m e t e r w a s put into a v e s s e l m a d e of o r g a n i c g l a s s and i m m e r s e d into a b a t h with liquid helium. This v e s s e l was brought into m o t i o n in an o r d i n a r y way when the d r i v e through a gland was c o a x i a l to the p y c n o m eter. The d e v i c e c o n s t r u c t e d by us had the s e n s i t i v i t y allowing to r e g i s t e r c h a n g e s in the s i x t h sign of the d e n s i t y . T h i s s e n s i t i v i t y i s so high that bY m e a n s of s u c h a p y c n o m e t e r it w a s i m p o s s i b l e to m e a s u r e the d e p e n d e n c e of d e n s i t y on t e m p e r a t u r e d i r e c t l y , a s the l e v e l of the liquid w a s out of the c a p i l l a r y at t h e s m a l l e s t change of t e m p e rature. The r e s u l t s of the study of the dependence of h e l i u m II d e n s i t y on the a n g u l a r v e l o c i t y of r o t a tion o b t a i n e d at a c o n s t a n t t e m p e r a t u r e a r e given in fig. 2. At the t r a n s i t i o n f r o m one t e m p e r a t u r e to a n o t h e r the l e v e l of the liquid in the c a p i l l a r y was e s t a b l i s h e d a g a i n by m e a n s of the valve. It i s s e e n f r o m fig. 2 that d u r i n g t w i s t i n g hel i u m II b e c o m e s the d e n s e r the h i g h e r a r e the angular velocities. The e x p l a n a t i o n of the o b s e r v e d phenomenon b y an i n c r e a s e of the c e n t r i f u g a l p r e s s u r e equal to 0.5 P0Jo2r2 (w o i s the a n g u l a r v e l o c i t y and r i s the d i s t a n c e to the a x i s of r o t a t i o n ) would d e m a n d the value of the c o e f f i c i e n t of c o m p r e s s i b i l i t y P"l(ap/ap) T ten t i m e s a s l a r g e a s the value obt a i n e d by e x t r a p o l a t i o n of the a v a i l a b l e d a t a to s m a l l p r e s s u r e s . (The c e n t r i f u g a l p r e s s u r e und e r the conditions of o u r e x p e r i m e n t i s s m a l l and d o e s not e x c e e d 2 x 10-6 arm.) B e s i d e s it should be noted that if the o b s e r v e d p a c k i n g of liquid h e l i u m would be a s s o c i a t e d with i t s c o m p r e s s i o n due to c e n t r i f u g a l p r e s s u r e then, due to the known p r o p e r t i e s of the c o m p r e s s i b i l i t y
1 August 1.9~$
_ t.~K
o
s
•
s~
2b
~
sO
,~o s e c - '
Fig. 2. coefficient, the effect should be i n c r e a s e d with the r i s e of t e m p e r a t u r e and the d e r i v a t i v e ~p/aoJ o should be d e c r e a s e d with the i n c r e a s e of the a n g~]~r v e l o c i t y . In r e a l i t y the r i s e of t e m p e r a t u r e d e c r e a s e s the o b s e r v e d packing mad the d e r i v a t i v e ap/aoJ o i n c r e a s e s with the i n c r e a s e of Wo" Ii h e l i u m I i s b r o u g h t into r o t a t i o n with the s a m e a n g u l a r ,~eiocities one d o e s not m a n a g e to o b s e r v e a change in i t s density. Thus one h a s to a s s u m e that t h e r e e x i s t s s o m e s p e c i f i c m e c h a n i s m of h e l i u m II packing which i s not a s s o c i a t e d with the p r e s e n c e of c e n t r i f u g a l p r e s s u r e , but i s a p p a r e n t l y c o n n e c t e d with the e x i s t e n c e of quantized v o r t i c e s of O n s a g e r - F e y n man' s type. The a u t h o r s a r e g r a t e f u l to Dr. Yu. G. M a m a l a d z e for the d i s c u s s i o n of the r e s u l t s .
Reference 1. R.P. Feynman, Progress in low temperature physics, e d . C . J . G o r t e r (North-Holland Publ.Co., Amsterdam, 1955) v o l . I , c h . H .
27
PHYSICS LETTERS
The e v a l u a t i o n of e q s . (4) and (5) by the e x p r e s s i o n s (2) and (3) gives explicit f o r m u l a e f o r d e t e r m i n i n g the r e s p o n s e function in the p r e s e n c e of a filter:
Dfl(p)
=
_I
,,
F2(1 +~fl2)a
+
4fl(fl- 2)p2a
+
~(I +~") L -
4~2p4a + 2 ( ~ - ~ 2 _
I)pE + 4g(l _ ~)13p3~ +
- ~0~o5~ + ~¢~o3E3],
(6)
w h e r e a = a r c c o s p and ~ = ~ - ~ 2 . If we l e t ~ --, 0, the r e s p o n s e function i s e a s i l y s e e n to tend to the f o r m Do(O) =
2 ¥(a
INDEPENDENCE
- ~),
(7)
1 August 1965
which i s the known r e s u l t f o r c o n s t a n t r e l e a s e . F i g . 1 i s showing the f r e q u e n c y r e s p o n s e of above m e n t i o n e d c a s e s of r e l e a s e . F r o m t h i s c u r v e s i t m a y e a s i l y shown t h a t p r a c t i c a l l y in the whole r e g i o n u s i n g t h i s a s s u m p t e d f i l t e r the r e s p o n s e function h a s b e t t e r v a l u e s a s the o p t i c a l s y s t e m with c o n s t a n t r e l e a s e . I a m i n d e b t e d to P r o f e s s o r H a v e l k a f o r m a n y helpful d i s c u s s i o n s .
References 1. Ma~'eoh~ and F r ~ o n , D t ~ o n structure des ~m, 1960). 2. H.H.Hcpkins, The frequemcy r u p c ~ e of a defocusseal optical system Proc.Soc. 231 (1955) p. 91.
OF THE DENSITY oF ROTATING ON THE ANGULAR VELOCITY
LIQUID
HELIUM
E. L. AND/K)NIKASHVILI a n d J. S. TSAKADZE l ~ t t t ~ e of Plzysics, Academy of Sciences of the Goo~zm SSR, Tbtltst, USSR Received 7 July 1965
The p r o b l e m of the d e n s i t y change in r o t a t i n g h e l i u m H, in c o m p a r i s o n with that when i t i s at r e s t , i s of g r e a t i n t e r e s t b e c a u s e of the p r e s e n c e of O n s a g e r - F e y n m a n ' s q u a u t i z e d v o r t i c e s in a quantum liquid [1]. In p a r t i c u l a r , one often e n c o u n t e r s an a s s u m p tion, which i s s t a t e d in t a l k s and in p u b l i c a t i o n s , that the e x i s t e n c e of v o r t e x l i n e s s e e m s to m a k e r o t a t i n g h e l i u m II m o r e " l o o s e " . We have m a d e an e x p e r i m e n t to find the angul a r v e l o c i t y d e p e n d e n c e of the d e n s i t y of h e l i u m H. A s e n s i t i v e p y c n o m e t e r w a s b r o u g h t into r o t a t i o n . A c o p p e r c y l i n d e r of the v o l u m e equal to 52.57 c m 3 with a g l a s s c a p i l l a r y (60 m m long and with the i n n e r d i a m e t e r of 1.75 mm) a t t a c h e d to i t w a s u s e d a s a p y c n o m e t e r (fig. 1). The top of the c a p i l l a r y w a s blown into a s m a l l s p h e r e and i t s s e c ond end w a s glued into a v a l v e , t h i s end r e g u l a t e d the amount of liquid h e l i u m in the p y c n o m e t e r . D u r i n g the m e a s u r e m e n t s the v a l v e w a s c l o s e d and the p y c n o m e t e r , i l l u m i n a t e d b y a l a m p of daylight, w a s e x a m i n e d t h r o u g h a c a t h e t o m e t e r . The a c c u r a c y of a s i n g l e r e a d i n g then r e a c h e d ~- 0.005 ram, u n d e r t h e s e conditions the s p r e a d
26
Y Fig. 1.
Volume 18, number 1
PHYSICS LETTERS
of the e x p e r i m e n t a l d a t a did not e x c e e d 0.2 m m . This f a c t gave the p o s s i b i l i t y to m e a s u r e the r e l a t i v e change of the d e n s i t y with the a c c u r a c y A p / p = * 0.0009%. The p y c n o m e t e r w a s put into a v e s s e l m a d e of o r g a n i c g l a s s and i m m e r s e d into a b a t h with liquid helium. This v e s s e l was brought into m o t i o n in an o r d i n a r y way when the d r i v e through a gland was c o a x i a l to the p y c n o m eter. The d e v i c e c o n s t r u c t e d by us had the s e n s i t i v i t y allowing to r e g i s t e r c h a n g e s in the s i x t h sign of the d e n s i t y . T h i s s e n s i t i v i t y i s so high that bY m e a n s of s u c h a p y c n o m e t e r it w a s i m p o s s i b l e to m e a s u r e the d e p e n d e n c e of d e n s i t y on t e m p e r a t u r e d i r e c t l y , a s the l e v e l of the liquid w a s out of the c a p i l l a r y at t h e s m a l l e s t change of t e m p e rature. The r e s u l t s of the study of the dependence of h e l i u m II d e n s i t y on the a n g u l a r v e l o c i t y of r o t a tion o b t a i n e d at a c o n s t a n t t e m p e r a t u r e a r e given in fig. 2. At the t r a n s i t i o n f r o m one t e m p e r a t u r e to a n o t h e r the l e v e l of the liquid in the c a p i l l a r y was e s t a b l i s h e d a g a i n by m e a n s of the valve. It i s s e e n f r o m fig. 2 that d u r i n g t w i s t i n g hel i u m II b e c o m e s the d e n s e r the h i g h e r a r e the angular velocities. The e x p l a n a t i o n of the o b s e r v e d phenomenon b y an i n c r e a s e of the c e n t r i f u g a l p r e s s u r e equal to 0.5 P0Jo2r2 (w o i s the a n g u l a r v e l o c i t y and r i s the d i s t a n c e to the a x i s of r o t a t i o n ) would d e m a n d the value of the c o e f f i c i e n t of c o m p r e s s i b i l i t y P"l(ap/ap) T ten t i m e s a s l a r g e a s the value obt a i n e d by e x t r a p o l a t i o n of the a v a i l a b l e d a t a to s m a l l p r e s s u r e s . (The c e n t r i f u g a l p r e s s u r e und e r the conditions of o u r e x p e r i m e n t i s s m a l l and d o e s not e x c e e d 2 x 10-6 arm.) B e s i d e s it should be noted that if the o b s e r v e d p a c k i n g of liquid h e l i u m would be a s s o c i a t e d with i t s c o m p r e s s i o n due to c e n t r i f u g a l p r e s s u r e then, due to the known p r o p e r t i e s of the c o m p r e s s i b i l i t y
1 August 1.9~$
_ t.~K
o
s
•
s~
2b
~
sO
,~o s e c - '
Fig. 2. coefficient, the effect should be i n c r e a s e d with the r i s e of t e m p e r a t u r e and the d e r i v a t i v e ~p/aoJ o should be d e c r e a s e d with the i n c r e a s e of the a n g~]~r v e l o c i t y . In r e a l i t y the r i s e of t e m p e r a t u r e d e c r e a s e s the o b s e r v e d packing mad the d e r i v a t i v e ap/aoJ o i n c r e a s e s with the i n c r e a s e of Wo" Ii h e l i u m I i s b r o u g h t into r o t a t i o n with the s a m e a n g u l a r ,~eiocities one d o e s not m a n a g e to o b s e r v e a change in i t s density. Thus one h a s to a s s u m e that t h e r e e x i s t s s o m e s p e c i f i c m e c h a n i s m of h e l i u m II packing which i s not a s s o c i a t e d with the p r e s e n c e of c e n t r i f u g a l p r e s s u r e , but i s a p p a r e n t l y c o n n e c t e d with the e x i s t e n c e of quantized v o r t i c e s of O n s a g e r - F e y n man' s type. The a u t h o r s a r e g r a t e f u l to Dr. Yu. G. M a m a l a d z e for the d i s c u s s i o n of the r e s u l t s .
Reference 1. R.P. Feynman, Progress in low temperature physics, e d . C . J . G o r t e r (North-Holland Publ.Co., Amsterdam, 1955) v o l . I , c h . H .
27