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λ-point shift in turbulently flowing 4He
Volume 26A. n u m b e r 8
PHYSICS
a s s u m e d c o n s t a n c y of t h e o r d e r p a r a m e t e r ( a l though plausible in the radial direction since d << ~(T) * m a y not b e v a l i d i n t h e a z i m u t h a l d i r e c t i o n b e c a u s e of t h i c k n e s s v a r i a t i o n s . S u c h v a r i a t i o n s in I,~f2 w o u l d p r o b a b l y b e g r e a t e s t a t t h e lower temperatures and lead to increased phase change around the cylinder resulting in a relativ e l y h i g h e r f i e l d p e r i o d at h i g h e r t e m p e r a t u r e s , as observed. Secondly the assumed time indep e n d e n c e of t h e o r d e r p a r a m e t e r m a y n o t b e j u s t i f i e d i n t h e t h i n n e s t r e g i o n s n e a r Tc b e c a u s e of fluctuations. Presumably such fluctuations could also affect phase change around the cylinder. • A t e m p e r a t u r e dependence of the filed periodicity in r e c t a n g u l a r loops was previopslv o b s e r v e d by one of us [5] but in that case d~60O0 ./k and a v a r i a t i o n in I~t 2 p e r p e n d i c u l a r to the film plane s e e m e d very plausible.
X-POINT
SHIFT
IN
LETTERS
11 March 1968
W e c o n c l u d e t h a t , a l t h o u g h t h e c a u s e of t h i s temperature variation needs further investigat i o n , i t s e x i s t e n c e s h o u l d b e c o n s i d e r e d in p r e c i s i o n m e a s u r e m e n t s of hc/2e b a s e d o n t h e m a g n e t i c f i e l d p e r i o d i c i t y in t h i n f i l m s u p e r c o n d u c t ing cylinders. 1. W.A. Little and R. D. P a r k s , Phys. Rev. L e t t e r s 9 (1962) 9: R. D. P a r k s and W. A. Little, Phys. Re:v. 133 (1964) A97. 2. L. M e y e r s and W. A. Little, Phys. Rev. J~etters 13 (1964) 325. 3. M. Tinkham, Phys. Rev. 129 (1963) 2413. 4. D.H. Douglass J r . . Phys. Rev. 132 (1963) 513. 5. R. Meservey, Proc. 9th Intern. Conf. on Low temp. physics, Columbus (1964) (Plenum P r e s s , P a r t A) p.455.
TURBULENTLY
FLOWING
4He
K. D. E R B E N a n d F. P O B E L L
Physik Dept. der Technischen Hochschule Miinchen, Germany Received 12 F e b r u a r y 1968
Shifts of the k - t e m p e r a t u r e of 4He up to A T k = 1.4 × 1 0 - 3 ° K due to turbulent flow caused by a heat c u r r e n t in a wide tube have been observed.
Recently there has been considerable interest i n d e t e c t i n g a s h i f t of t h e X - p o i n t due to t h e c r e a t i o n of q u a n t i z e d v o r t i c e s in r o t a t i n g l i q u i d 4He [1,2]. T h e k - p o i n t w a s f o u n d t o b e i n d e p e n d e n t of r o t a t i o n w i t h i n 10 - 5 OK u p t o a n a n g u l a r v e l o c i t y of co = 110 s e c -1 [1,2]. T h e p h e n o m e n o l o g i c a l t h e o r y [3,4] p r e d i c t s t h a t t h e s e a n g u l a r v e l o c i t i e s a r e t o o s m a l l to p r o d u c e m e a s u r a b l e s h i f t s of Tk [5,6]. H o w e v e r , a h i g h e r a v e r a g e v e l o c i t y of t h e s u p e r f l u i d c o m p o n e n t can b e a t t a i n e d , p a r t i c u larly just below the k-point, by a heat current. H e r e w e r e p o r t o n a s h i f t of t h e k - p o i n t u p to
o
A T x = 1.4 × 10 -3 O K c a u s e d by a heat current in
z o
helium contained in a very wide tube. The order of m a g n i t u d e of t h i s s h i f t i s i n a g r e e m e n t w i t h t h e o r e t i c a l e x p e c t a t i o n s [5,6]. Fig. 1 shows the bucket which is immersed into the helium bath. Helium was introduced into c h a m b e r A (1.40 c m c m i. d . , 9.6 c m l o n g ) t h r o u g h valve V until the meniscus could be seen in the g l a s s c a p i l l a r y C. D u e t o t h e v a c u u m b e t w e e n chambers A and B, thermal contact between the 368
2
~---
~_ A
C
/.-
R
J
e T ~ e e •. •f
LD
Q
ee i
20 ]
HEATING
30 I
RATE
40 I
[mW
50 J
60 I
cm -2 ]
Fig. 1. Shift of the k-point t e m p e r a t u r e of 4He as a function of the heat input at the bottom of c h a m b e r A. Insert: Experimental set-up.
Volume 26A, number 8
PHYSICS LETTER S
h e l i u m in A and the c o l d e r h e l i u m bath was p o s sible only at the top of c o n t a i n e r A. Heating the bottom of c h a m b e r A by the h e a t e r H r e s u l t e d in a counterflow of the s u p e r f l u i d and n o r m a l f l u i d c o m p o n e n t s of h e l i u m in A. S t a r t i n g below the h - p o i n t the liquid i n s i d e A was slowly heated to a t e m p e r a t u r e above the h - p o i n t by a c o n s t a n t heat input into the h e a t e r H. The c a r b o n r e s i s t o r R (~ W A l l e n - B r a d l e y ) was u s e d as a t h e r m o m e t e r at a p o w e r level of 2 × 10-7W and the v o l tage drop a c r o s s it as a function of t i m e was r e g i s t e r e d on a c h a r t r e c o r d e r . T h i s t h e r m o m e t e r was placed 7 m m above the b o t t o m of A. The k - p o i n t was identified by a s h o r t stop in the h e l i u m w a r m i n g r a t e followed by a r a p i d o v e r heating of R due to the poor t h e r m a l conductivity of h e l i u m in the n o r m a l state. The m e a s u r e d shift AT k of the t r a n s i t i o n t e m p e r a t u r e as a function of the heat c u r r e n t d e n s i t y supplied to h e a t e r H i s shown in fig. 1. T h e r e i s no d e t e c t a b l e shift at h e a t i n g p o w e r s below 2 m W / c m 2. But above about 4 m W / c m 2 the shift of T~ i s p r o p o r t i o n a l to Q with a slope of 2.2 x 1 0 - 2 0 K / ( W / c m 2 ) . A c c o r d i n g to the p h e n o m e n o l o g i c a l t h e o r y of G i n z b u r g and P i t a e v s k i i [3,4], which is a s s u m e d to be valid n e a r T k , the r e l a t i v e v e l o c i t y ( v s - v n ) b e t w e e n the s u p e r f l u i d and n o r m a l p h a s e of the liquid l e a d s to a d e c r e a s e d s u p e r f l u i d d e n s i t y and t h e r e b y to a d e p r e s s i o n of the h-point. In p a r t i c u l a r , even at t e m p e r a t u r e s below the u s u a l h - p o i n t s u p e r f l u i d i t y should only e x i s t up to a c h a r a c t e r i s t i c v e l o c i t y (v s - Vn)m. A c c o r d i n g to M a m a l d z e [5,6] the d e p r e s s i o n of the h - p o i n t and this c h a r a c t e r i s t i c v e l o c i t y and r e l a t e d by
11 March 1968 Table 1
(~ (mW/cm2) ATk, exper. (10 -3 OK) (vs - Vn)m (cm/sec) (vs - Vn)° (cm/sec)
20 0.52 37.4 4.5
40 0.97
60 1.42
56.8 5.7
73.0 7.0
r a n g e they a r e a p p r o x i m a t e l y t e n t i m e s s m a l l e r than the v e l o c i t i e s n e c e s s a r y to explain the obs e r v e d shifts by m e a n s of the equation for ATk. The actual local v e l o c i t i e s in the t u r b u l e n t flow of the liquid a r e unknown. The d i f f e r e n c e between (Vs - V n ) o and (Vs - vn)m is p r o b a b l y due to the t u r b u l e n t c h a r a c t e r of the flow and the c o r r e s ponding high v o r t e x density, the s t r o n g d e p e n dence of Ps on velocity and t e m p e r a t u r e n e a r Tk, and the state of the p r e s e n t t h e o r y , which may not be valid at high t u r b u l e n c e . T h i s e x p e r i m e n t shows the p o s s i b i l i t y of c r e a t i n g h i g h e r r e l a t i v e v e l o c i t i e s and h i g h e r v o r t e x d e n s i t y in liquid h e l i u m n e a r T k by heat flow e x p e r i m e n t s than by rotation. It should t h e r e f o r e be of p a r t i c u l a r i n t e r e s t to r e p e a t some of the e x p e r i m e n t s done on r o t a t i n g h e l i u m with methods s i m i l a r to the one d e s c r i b e d here. We a r e much indebted to K. D r a n s f e l d for support of this work and for many helpful s u g g e s t i o n s . In addition we thank K. Kehr for c l a r i f y i n g d i s c u s s i o n s , W. Veith for advice c o n c e r n i n g the method to m e a s u r e Tk, and the B a v a r i a n A c a d e m y of S c i e n c e s for the h o s p i t a l i t y extended to us.
ATk = 2.28 X 10 -6 (vs - Vn)m ~ (OK) . F o r comparison with our results the shift of the h-point at a p a r t i c u l a r power level (~ and the velocity (vs - Vn)m derived from it by the above equation are listed in table 1. We can estimate a lower limit of the mean velocities along A from the equation for laminar
counterflow in a c l o s e d s y s t e m u s i n g only the known t e m p e r a t u r e dependence of Ps [7,8], n e g l e c t i n g its velocity dependence: (Vs - Vn)o = - Q/Ps ST ( c m / s e c ) . T h e s e v a l u e s a r e also given in table 1. O v e r the whole v e l o c i t y
1. F. Pobell, W. Schoepe and W. Veith. Phys. Letters 25A (1967) 209. 2. G.Ahlers, Phys.Rev. 164 (1968) 259. 3. V.L. Ginzburg and L. P. Pitaevskii, Soy. Phys. JETP 7 (1958) 858. 4, L.P. Pitaevskii, Soy.Phys. JETP 8 (1959) 282. 5, L.V. Kiknadze, Yu. G. Mamaladze and O, D. Cheishvili, Sov.Phys.JETP 21 (1965) 1018. 6. Yu.G,Mamaladze, Sov.Phys.JETP 25 (1967) 479. 7. J.R.Clow and J.D.Reppy, Phys.Rev. Letters 16 (1966) 887.
8. J.A. Tyson and D. H. Douglass J r . . Phys.Rev. Letters 17 (1966) 472.
* $ * * *
369
PHYSICS
a s s u m e d c o n s t a n c y of t h e o r d e r p a r a m e t e r ( a l though plausible in the radial direction since d << ~(T) * m a y not b e v a l i d i n t h e a z i m u t h a l d i r e c t i o n b e c a u s e of t h i c k n e s s v a r i a t i o n s . S u c h v a r i a t i o n s in I,~f2 w o u l d p r o b a b l y b e g r e a t e s t a t t h e lower temperatures and lead to increased phase change around the cylinder resulting in a relativ e l y h i g h e r f i e l d p e r i o d at h i g h e r t e m p e r a t u r e s , as observed. Secondly the assumed time indep e n d e n c e of t h e o r d e r p a r a m e t e r m a y n o t b e j u s t i f i e d i n t h e t h i n n e s t r e g i o n s n e a r Tc b e c a u s e of fluctuations. Presumably such fluctuations could also affect phase change around the cylinder. • A t e m p e r a t u r e dependence of the filed periodicity in r e c t a n g u l a r loops was previopslv o b s e r v e d by one of us [5] but in that case d~60O0 ./k and a v a r i a t i o n in I~t 2 p e r p e n d i c u l a r to the film plane s e e m e d very plausible.
X-POINT
SHIFT
IN
LETTERS
11 March 1968
W e c o n c l u d e t h a t , a l t h o u g h t h e c a u s e of t h i s temperature variation needs further investigat i o n , i t s e x i s t e n c e s h o u l d b e c o n s i d e r e d in p r e c i s i o n m e a s u r e m e n t s of hc/2e b a s e d o n t h e m a g n e t i c f i e l d p e r i o d i c i t y in t h i n f i l m s u p e r c o n d u c t ing cylinders. 1. W.A. Little and R. D. P a r k s , Phys. Rev. L e t t e r s 9 (1962) 9: R. D. P a r k s and W. A. Little, Phys. Re:v. 133 (1964) A97. 2. L. M e y e r s and W. A. Little, Phys. Rev. J~etters 13 (1964) 325. 3. M. Tinkham, Phys. Rev. 129 (1963) 2413. 4. D.H. Douglass J r . . Phys. Rev. 132 (1963) 513. 5. R. Meservey, Proc. 9th Intern. Conf. on Low temp. physics, Columbus (1964) (Plenum P r e s s , P a r t A) p.455.
TURBULENTLY
FLOWING
4He
K. D. E R B E N a n d F. P O B E L L
Physik Dept. der Technischen Hochschule Miinchen, Germany Received 12 F e b r u a r y 1968
Shifts of the k - t e m p e r a t u r e of 4He up to A T k = 1.4 × 1 0 - 3 ° K due to turbulent flow caused by a heat c u r r e n t in a wide tube have been observed.
Recently there has been considerable interest i n d e t e c t i n g a s h i f t of t h e X - p o i n t due to t h e c r e a t i o n of q u a n t i z e d v o r t i c e s in r o t a t i n g l i q u i d 4He [1,2]. T h e k - p o i n t w a s f o u n d t o b e i n d e p e n d e n t of r o t a t i o n w i t h i n 10 - 5 OK u p t o a n a n g u l a r v e l o c i t y of co = 110 s e c -1 [1,2]. T h e p h e n o m e n o l o g i c a l t h e o r y [3,4] p r e d i c t s t h a t t h e s e a n g u l a r v e l o c i t i e s a r e t o o s m a l l to p r o d u c e m e a s u r a b l e s h i f t s of Tk [5,6]. H o w e v e r , a h i g h e r a v e r a g e v e l o c i t y of t h e s u p e r f l u i d c o m p o n e n t can b e a t t a i n e d , p a r t i c u larly just below the k-point, by a heat current. H e r e w e r e p o r t o n a s h i f t of t h e k - p o i n t u p to
o
A T x = 1.4 × 10 -3 O K c a u s e d by a heat current in
z o
helium contained in a very wide tube. The order of m a g n i t u d e of t h i s s h i f t i s i n a g r e e m e n t w i t h t h e o r e t i c a l e x p e c t a t i o n s [5,6]. Fig. 1 shows the bucket which is immersed into the helium bath. Helium was introduced into c h a m b e r A (1.40 c m c m i. d . , 9.6 c m l o n g ) t h r o u g h valve V until the meniscus could be seen in the g l a s s c a p i l l a r y C. D u e t o t h e v a c u u m b e t w e e n chambers A and B, thermal contact between the 368
2
~---
~_ A
C
/.-
R
J
e T ~ e e •. •f
LD
Q
ee i
20 ]
HEATING
30 I
RATE
40 I
[mW
50 J
60 I
cm -2 ]
Fig. 1. Shift of the k-point t e m p e r a t u r e of 4He as a function of the heat input at the bottom of c h a m b e r A. Insert: Experimental set-up.
Volume 26A, number 8
PHYSICS LETTER S
h e l i u m in A and the c o l d e r h e l i u m bath was p o s sible only at the top of c o n t a i n e r A. Heating the bottom of c h a m b e r A by the h e a t e r H r e s u l t e d in a counterflow of the s u p e r f l u i d and n o r m a l f l u i d c o m p o n e n t s of h e l i u m in A. S t a r t i n g below the h - p o i n t the liquid i n s i d e A was slowly heated to a t e m p e r a t u r e above the h - p o i n t by a c o n s t a n t heat input into the h e a t e r H. The c a r b o n r e s i s t o r R (~ W A l l e n - B r a d l e y ) was u s e d as a t h e r m o m e t e r at a p o w e r level of 2 × 10-7W and the v o l tage drop a c r o s s it as a function of t i m e was r e g i s t e r e d on a c h a r t r e c o r d e r . T h i s t h e r m o m e t e r was placed 7 m m above the b o t t o m of A. The k - p o i n t was identified by a s h o r t stop in the h e l i u m w a r m i n g r a t e followed by a r a p i d o v e r heating of R due to the poor t h e r m a l conductivity of h e l i u m in the n o r m a l state. The m e a s u r e d shift AT k of the t r a n s i t i o n t e m p e r a t u r e as a function of the heat c u r r e n t d e n s i t y supplied to h e a t e r H i s shown in fig. 1. T h e r e i s no d e t e c t a b l e shift at h e a t i n g p o w e r s below 2 m W / c m 2. But above about 4 m W / c m 2 the shift of T~ i s p r o p o r t i o n a l to Q with a slope of 2.2 x 1 0 - 2 0 K / ( W / c m 2 ) . A c c o r d i n g to the p h e n o m e n o l o g i c a l t h e o r y of G i n z b u r g and P i t a e v s k i i [3,4], which is a s s u m e d to be valid n e a r T k , the r e l a t i v e v e l o c i t y ( v s - v n ) b e t w e e n the s u p e r f l u i d and n o r m a l p h a s e of the liquid l e a d s to a d e c r e a s e d s u p e r f l u i d d e n s i t y and t h e r e b y to a d e p r e s s i o n of the h-point. In p a r t i c u l a r , even at t e m p e r a t u r e s below the u s u a l h - p o i n t s u p e r f l u i d i t y should only e x i s t up to a c h a r a c t e r i s t i c v e l o c i t y (v s - Vn)m. A c c o r d i n g to M a m a l d z e [5,6] the d e p r e s s i o n of the h - p o i n t and this c h a r a c t e r i s t i c v e l o c i t y and r e l a t e d by
11 March 1968 Table 1
(~ (mW/cm2) ATk, exper. (10 -3 OK) (vs - Vn)m (cm/sec) (vs - Vn)° (cm/sec)
20 0.52 37.4 4.5
40 0.97
60 1.42
56.8 5.7
73.0 7.0
r a n g e they a r e a p p r o x i m a t e l y t e n t i m e s s m a l l e r than the v e l o c i t i e s n e c e s s a r y to explain the obs e r v e d shifts by m e a n s of the equation for ATk. The actual local v e l o c i t i e s in the t u r b u l e n t flow of the liquid a r e unknown. The d i f f e r e n c e between (Vs - V n ) o and (Vs - vn)m is p r o b a b l y due to the t u r b u l e n t c h a r a c t e r of the flow and the c o r r e s ponding high v o r t e x density, the s t r o n g d e p e n dence of Ps on velocity and t e m p e r a t u r e n e a r Tk, and the state of the p r e s e n t t h e o r y , which may not be valid at high t u r b u l e n c e . T h i s e x p e r i m e n t shows the p o s s i b i l i t y of c r e a t i n g h i g h e r r e l a t i v e v e l o c i t i e s and h i g h e r v o r t e x d e n s i t y in liquid h e l i u m n e a r T k by heat flow e x p e r i m e n t s than by rotation. It should t h e r e f o r e be of p a r t i c u l a r i n t e r e s t to r e p e a t some of the e x p e r i m e n t s done on r o t a t i n g h e l i u m with methods s i m i l a r to the one d e s c r i b e d here. We a r e much indebted to K. D r a n s f e l d for support of this work and for many helpful s u g g e s t i o n s . In addition we thank K. Kehr for c l a r i f y i n g d i s c u s s i o n s , W. Veith for advice c o n c e r n i n g the method to m e a s u r e Tk, and the B a v a r i a n A c a d e m y of S c i e n c e s for the h o s p i t a l i t y extended to us.
ATk = 2.28 X 10 -6 (vs - Vn)m ~ (OK) . F o r comparison with our results the shift of the h-point at a p a r t i c u l a r power level (~ and the velocity (vs - Vn)m derived from it by the above equation are listed in table 1. We can estimate a lower limit of the mean velocities along A from the equation for laminar
counterflow in a c l o s e d s y s t e m u s i n g only the known t e m p e r a t u r e dependence of Ps [7,8], n e g l e c t i n g its velocity dependence: (Vs - Vn)o = - Q/Ps ST ( c m / s e c ) . T h e s e v a l u e s a r e also given in table 1. O v e r the whole v e l o c i t y
1. F. Pobell, W. Schoepe and W. Veith. Phys. Letters 25A (1967) 209. 2. G.Ahlers, Phys.Rev. 164 (1968) 259. 3. V.L. Ginzburg and L. P. Pitaevskii, Soy. Phys. JETP 7 (1958) 858. 4, L.P. Pitaevskii, Soy.Phys. JETP 8 (1959) 282. 5, L.V. Kiknadze, Yu. G. Mamaladze and O, D. Cheishvili, Sov.Phys.JETP 21 (1965) 1018. 6. Yu.G,Mamaladze, Sov.Phys.JETP 25 (1967) 479. 7. J.R.Clow and J.D.Reppy, Phys.Rev. Letters 16 (1966) 887.
8. J.A. Tyson and D. H. Douglass J r . . Phys.Rev. Letters 17 (1966) 472.
* $ * * *
369