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The shift of the lambda point by a heat current Robinson, Brad J. 1976

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THE SHIFT OF THE LAMBDA POINT BY A HEAT CURRENT by BRAD J. ROBINSON B.Sc, University of Toronto, 1972 B.Ed., University of Toronto, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1976 © Brad J. Robinson, 1976 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th i s thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thesis fo r f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date h^fajfa IWt ABSTRACT The velocity of second sound, with and without a heat flux present, has been measured using a time of f l i g h t technique. The velocity changes resulting from the presence of the heat flux are interpreted as changes in the temperature of the He II - He I transition. The null results in-dicate that the depression of the transition temperature, 6T . . , by a heat A flux Q is 6TX 4 1.3 (± 1.6) x lO" 4 Q K for Q in W/cm2 and Q 4 3.4 * 10~2 W/cm2. i i i TABLE OF CONTENTS Chapter Page I INTRODUCTION AND THEORY I n t r o d u c t i o n 1 R e l a t i v e V e l o c i t y and S u p e r f l u i d Density 1 The S h i f t i n T^- T h e o r e t i c a l C o n s i d e r a t i o n s 4 The S h i f t i n T - Experimental C o n s i d e r a t i o n s 7 A Experiment 8 I I APPARATUS Cryogenic Apparatus 10 Primary Thermometer 12 D e t a i l s o f Second Sound Apparatus 12 E l e c t r o n i c s 16 I I I EXPERIMENTAL PROCEDURES Thermometry 20 Second Sound Measurements 25 IV DATA REDUCTION AND ERROR ANALYSIS Time of F l i g h t Data 30 AT Data 31 Heat Flu x per U n i t Area 32 V ANALYSIS AND RESULTS A n a l y s i s 33 Re s u l t s 35 VI DISCUSSION AND CONCLUSIONS D i s c u s s i o n 37 Conclusions 38 REFERENCES 41 i v LIST OF FIGURES Figure Page 1 Cryogenic Apparatus 11 2 Second Sound C e l l (Exploded View) 13 3 Bolometer or Second Sound Generator 15 4 Pulse Sequence 15 5 E l e c t r o n i c s - Block Diagram 17 6 A Warming Curve Showing R^ P l a t e a u 22 7 Thermometer Aging 24 8 Detected Second Sound Pulses 27 9 The S h i f t of T\ by a Heat F l u x 36 V ACKNOWLEDGEMENTS I t i s a pleasure to express my g r a t i t u d e to Dr. M.J. Crooks f o r h i s e f f o r t s i n the s u p e r v i s i o n of t h i s p r o j e c t . I would a l s o l i k e to thank J . Choi and C P . Barry f o r t h e i r work wi t h the computer. CHAPTER I INTRODUCTION AND THEORY (1) Introduction At present the theoretical understanding of the microscopic nature of the superfluid transition occurring in ^He is incomplete and subject 1 2 to controversy ' . In an attempt to describe the properties of super-fluidity, phenomenological approaches have been developed which employ the concepts of a normal fluid component (with density p ) and a super-3 fluid component (with density p g) as in the theory of Landau . In par-ticular there has been experimental and theoretical work to determine factors which influence the temperature, T^, at which the transition occurs, that is, the temperature at which the superfluid component van-ishes. Factors which alter the superfluid density and also the value of T^  include the presence of boundaries and surfaces, impurities such o 6 as dHe, pressure (as produced by a gravitational field , for example), and a relative velocity between the normal and superfluid components such as that produced by rotation or a heat flux. The last of these is the present subject of interest. (2) Relative Velocity and Superfluid Density The following discussion will illustrate why a relative velocity, -> w .--v -. v , between the normal and superfluid components may alter the 7 superfluid density p g and normal density pn- This treatment is quant-itatively applicable only for small values of w in the low temperature limit (T < 1 K) . We consider the normal component as a gas of elementary excitations, phonons and rotons, while the superfluid is a 'background' for the gas. 2 The energy of an e x c i t a t i o n , e, i s r e l a t e d t o the momentum, p, of the e x c i t a t i o n v i a a spectrum e ( p ) . For helium at r e s t the d i s t r i b u t i o n 7 f u n c t i o n f o r the ( i d e a l ) gas o f phonons i s given by the Planck f u n c t i o n njCp) = { exp(e/kT) - 1 } - 1 while f o r the higher energy rotons the Boltzmann f u n c t i o n i s used: n 2 ( p ) = exp(-e/kT). Now consider the s i t u a t i o n i n which the normal and s u p e r f l u i d components have v e l o c i t i e s and v s r e s p e c t i v e l y , as measured w i t h r e s p e c t to a la b o r a t o r y frame. The momentum of an e x c i t a t i o n i s i n v a r i a n t under a G a l i l e a n t r a n s f o r m a t i o n from one frame t o another. However, the energy as measured i n the s u p e r f l u i d r e s t frame i s r e l a t e d to the energy E of the e x c i t a t i o n i n the l a b o r a t o r y frame by E(p) = e(p) + P*v • The gas of e x c i t a t i o n s d r i f t s w i t h v e l o c i t y and t h e r e f o r e E(p) i s expressed i n terms of the energy E'(p) o f an e x c i t a t i o n i n the normal f l u i d r e s t frame by E(p) = E'(p) + p*v . Thus, u s i n g E'(p) = -e (p) - p-w i n the d i s t r i b u t i o n f u n c t i o n f o r phonons, we have VP) = { e x p f ^ E ^ ) - 1 }"1 w i t h a s i m i l a r m o d i f i c a t i o n i n the r o t o n d i s t r i b u t i o n f u n c t i o n . With these d i s t r i b u t i o n f u n c t i o n s we can determine what i s meant by the s u p e r f l u i d d e n s i t y , the normal d e n s i t y , and how the r e l a t i v e v e l -o c i t y a l t e r s them. In the l a b o r a t o r y frame the momentum per u n i t volume, l, i s composed o f two terms: i = p v + p v where p and p are con-•" r J n n s s s n s t r a i n e d by p g + p n = p = d e n s i t y of the l i q u i d . The momentum i n the frame moving w i t h the s u p e r f l u i d i s i = i - p v = p w . (1) Jo J s n This momentum can be w r i t t e n u s i n g the t o t a l d i s t r i b u t i o n f u n c t i o n n of a l l elementary e x c i t a t i o n s as J 0 = / P n t(e-p*w) h~ 3d 3p (2) where h i s Planck's constant. Comparing (1) and (2) g i v e s the d e f i n i t i o n of p as n p^w = / p n (e-p'w) h 3 d 3 p Evaluating the phonon c o n t r i b u t i o n to p^ by u s i n g the phonon d i s p e r s i o n r e l a t i o n e = cp (c i s the v e l o c i t y of sound) i n the a p p r o p r i a t e d i s t r i -b u t i o n , n^, gives P n p h - { 1 - Cw/O* T3. The roton c o n t r i b u t i o n p can a l s o be c a l c u l a t e d and leads to a more nr complicated e x p r e s s i o n . The f i n a l e x p r e s s i o n f o r p ^ i s found by adding the c o n t r i b u t i o n s so t h a t p = p , + p . Thus, the normal d e n s i t y i s n nph nr ' ' a f u n c t i o n of the r e l a t i v e v e l o c i t y and. t h i s i n t u r n i m p l i e s a s u p e r f l u i d d e n s i t y dependence on w through the equation p g + p n = p . By s i m i l a r c a l c u l a t i o n s i t can be shown th a t a l l thermodynamic v a r i a b l e s (entropy, i n t e r n a l energy, s p e c i f i c heat e t c . ) are f u n c t i o n s of w. In a d d i t i o n to the s t a t i s t i c a l m i c r o s c o p i c approach o u t l i n e d above pu r e l y thermodynamic arguments'' i l l u s t r a t e the general r o l e of w = v n - v s through the i n t r o d u c t i o n of P n / p and %w2 as an a d d i t i o n a l p a i r of con-jugate v a r i a b l e s upon which a l l other q u a n t i t i e s depend. For example, these v a r i a b l e s appear i n the thermodynamic i d e n t i t y s a t i s f i e d by the chemical p o t e n t i a l u: dy = -odT + i d p - ( p n / p ) d ( % w 2 ) where a i s the entropy per u n i t mass and the other independent v a r i a b l e s are temperature T and pressure p. A Maxwell's r e l a t i o n which f o l l o w s from t h i s equation i s 4 8p 2 9 /• / ^ 3Giw2) = P F p ( p n / p ) • From this expression the dependence of p on w can be determined to order p'(p,T,w) = p(p,T) + hp2P | p ( P n / p ) • (3) The Shift in T^-Theoretical Considerations The changes in the superfluid and normal density as produced by a relative velocity w are small. In order to detect a shift in the trans-ition temperature (as defined by p g -> 0 or pn/p -* 1) i t is necessary to produce large values of w and to perform the experiment in the c r i t i c a l region (T^- T << T^) where shifts in would have their most pronounced effect. As a result of these factors the theoretical treatment of the problem of a X-point shift in bulk helium involves (a) consideration of superfluid vortex motion and (b) the use of the theory of second order phase transitions. In this theory thermodynamic potentials are expanded in terms of an order parameter which is related to the superfluid density. The theory of the generation and growth of vortex lines in He II ^ 8 i s , at present, not complete. However, i t is proposed that for s u f f i c -iently high values of the normal and superfluid velocities the flow can become turbulent and the liquid supports an isotropic tangled mass of 5 9 vortex lines. To study the nature of vortices near T Mamaladze ' A considers a non-turbulent situation involving the equilibrium rotation of He II as a means of generating relative velocities. In this case his theory shows that the He II is threaded by a regular two dimensional 5 s u p e r f l u i d v o r t e x l a t t i c e of t r i a n g u l a r symmetry i n which the vor t e x l i n e s are p a r a l l e l t o the a x i s of r o t a t i o n . He a l s o shows that v o r t e x motion i s thermodynamically p r e f e r r e d (minimized f r e e energy) over a motion i n which the normal component r o t a t e s w h i l e the s u p e r f l u i d r e -mains s t a t i o n a r y . In comparing these two p o s s i b l e motions he shows t h a t , as a r e s u l t o f vo r t e x motion, the s u p e r f l u i d d e n s i t y a t t a i n s a v a l u e , on the average,closer to th a t of a l i q u i d at r e s t . N e v e r t h e l e s s , there s t i l l e x i s t r e l a t i v e v e l o c i t i e s between the s u p e r f l u i d and normal comp-onents which r e s u l t i n a depression o f the X-temperature ( f o r example, at the centre of a vor t e x v s 0 0 a n Q j the s u p e r f l u i d d e n s i t y vanishes l o c a l l y ) . Using e x p e r i m e n t a l l y determined parameters i n h i s equations, Mamaladze c a l c u l a t e s the A-point s h i f t i n an i n f i n i t e f l u i d r o t a t i n g at angular v e l o c i t y co (sec *) as AT X = -5.4 x l o - 9 t o 3 / l t K. Thus, at exp e r i m e n t a l l y a t t a i n a b l e r o t a t i o n r a t e s , the depression of T A would be impossible to detect w i t h present thermometry techniques. As an a l t e r n a t i v e t o r o t a t i n g a can of helium, a r e l a t i v e or counterflow v e l o c i t y w can be produced by a heat i n p u t . I f there i s no mass flow then P s v g + P n v n = 0 . Since entropy i s c a r r i e d only by the normal f l u i d , a heat f l u x per u n i t a r e a , C;, r e s u l t s i n a counterflow v e l o c i t y given by 3 = pSTv = p STw n s where S i s the entropy per u n i t mass. In the event that w i s l a r g e enough (greater than some c r i t i c a l v e l o c i t y ) the uniform counterflow w i l l break down t o a t u r b u l e n t flow i n v o l v i n g t a ngled v o r t i c e s . Then, v^ and v^ represent averages over the v o r t e x r e g i o n . Bhagat and Lasken"^ have performed experiments i n which they used such a method t o generate r e l -a t i v e v e l o c i t i e s . To account f o r t h e i r observed s h i f t s i n the t r a n s -i t i o n temperature, they propose a theory which i n v o l v e s v o r t i c e s . They consider a channel ( a x i s i n 2 d i r e c t i o n ) i n which there i s a uniform s u p e r f l u i d v e l o c i t y v . They expand the f r e e energy d i f f e r e n c e between He II and He I i n terms o f an order parameter > f ( x , y , z , t ) as f o l l o w s : F I I - F l = AF = / ( - a ^ l 2 + J^M4 + |vy|2) dV . The values of a and 6 are determined by e q u i l i b r i u m c o n d i t i o n s and are evaluated u s i n g experimental data. The order parameter Y = xe 1^ i s r e l a t e d t o the s u p e r f l u i d d e n s i t y and v e l o c i t y by p = m M 2 and v = ^-^<j) . s 1 1 s 2imi They then f o r c e the order parameter t o d e s c r i b e a v o r t e x p a r a l l e l to the channel a x i s ; t h a t i s , they w r i t e V = xCO expi<j>(r,6) where e - 0 > a ) XOO = { 1 - — — - > f o r r > a Q j X ( r ) = 0 r = a 0 and p = mx 2 r >> a . s A°° o TO. £ i - J i • * * ,K sine K cos6 , The s u p e r f l u i d v e l o c i t y components are ( — - — , — - — , v g ) where K IS the vortex c i r c u l a t i o n . £ i s a c o r r e l a t i o n l e n g t h which i s t y p i c a l l y on the same order of s i z e as the v o r t e x core r a d i u s a . Bhagat and Lasken s u b s t i t u t e f o r the terms i n AF and evaluate the i n t e g r a l . The subsequent r e s u l t s need not be quoted here. I t i s s u f f i c i e n t to p o i n t out t h a t they modify the expr e s s i o n f o r AF w i t h one v o r t e x by g e n e r a l i z i n g the r e s u l t to N s i m i l a r v o r t i c e s . This f e a t u r e , the number of v o r t i c e s , r e -mains an a d j u s t a b l e parameter i n t h e i r theory. I t i s used to a r r i v e at-a consistency between t h e i r experimental values of the X-point s h i f t and a t h e o r e t i c a l c r i t i c a l heat f l u x which f o l l o w s from the f r e e energy d i f f e r e n c e c a l c u l a t i o n o u t l i n e d above. To account f o r t h e i r observed s h i f t s i n the X-point they r e q u i r e ^ 4 * 10 9 v o r t e x l i n e s per square centimeter f o r a s h i f t of 6T ^ -1 x 10~ 3 K. A Mikeska*''' considers the e f f e c t of a counterflow w produced by a heat f l u x by s o l v i n g the l i n e a r i z e d equations of two f l u i d hydrodynamic to a l l orders i n w. V o r t i c e s do not enter i n t o h i s treatment but he does p r e d i c t a s h i f t i n the valu e of T^. However, i t i s d i f f i c u l t t o apply h i s t h e o r e t i c a l statements concerning the depression of T t o the experimental s i t u a t i o n i n t h i s work. (4) The S h i f t i n T^- Experimental C o n s i d e r a t i o n s Bhagat and Lasken*^ express t h e i r experimental r e s u l t s as T x ( 0 ) - T X(QJ = 0.059 Q (3) where T^  (0) i s the e q u i l i b r i u m t r a n s i t i o n temperature and T (Q) i s the A A s h i f t e d t r a n s i t i o n temperature i n the presence of a heat f l u x Q i n W/cm f o r 10" 3 W/cm2 < Q < 1 0 _ 1 W/cm2. 12 Bhagat and Davis employed a technique d i f f e r e n t than Bhagat and Lasken and a r r i v e d at e s s e n t i a l l y ( w i t h i n 50%) the same r e s u l t . 13 / - o Erben and P o b e l l observed changes i n T x of t y p i c a l l y -1 x 10 i K f o r a Q of 40 mW/cm2. 14 15 Others ' have report e d n u l l r e s u l t s t h a t are i n c o n f l i c t w i t h 14 the X-point s h i f t s mentioned above and one of them, A h l e r s , proposes an a l t e r n a t i v e i n t e r p r e t a t i o n o f the experiments. In view of the experimental and t h e o r e t i c a l s i t u a t i o n d e s c r i b e d above, i t i s of i n t e r e s t t o perform measurements of T i n the presence of a heat f l o w u s i n g a technique d i f f e r e n t from those employed i n the 8 experiments above, and av o i d i n g some of the d i f f i c u l t i e s i n i n t e r p r e t -a t i o n inherent i n those previous techniques. (5) Experiment The experiment t h a t has been performed used the v e l o c i t y o f second sound as the probe of the value of.T^(Q). L i n e a r i z e d hydrodynamics p r e d i c t s the second sound v e l o c i t y to be, i n s t a t i o n a r y helium, r TS 2 p s u2 = 1 F- P ~ * ' P n T h e . v a l i d i t y o f t h i s e xpression has been e x p e r i m e n t a l l y v e r i f i e d * ^ to w i t h i n a few microdegrees of T (0). In the c r i t i c a l r e g i o n of temper-atures T where AT = T^- T < 60 * 10~ 3 K we c o n s i d e r the v e l o c i t y of second sound as a f u n c t i o n of AT o n l y ; that i s , we c o n s i d e r = u^(AT) w i t h U £ 0 as AT 0 and the f u n c t i o n remains unchanged by a heat f l u x Q. Thus, by measuring the v e l o c i t y of second sound f o r a given value of T w i t h and without a heat f l o w p r e s e n t , we may be a b l e t o a t -t r i b u t e any changes i n v e l o c i t y to a change i n T . I f the temperature dependence of second sound v e l o c i t y remains unchanged w i t h a heat f l u x p resent, we conclude t h a t T.. remains unchanged. A In t h i s experiment the v e l o c i t y was determined by measuring the time of f l i g h t o f a second sound p u l s e . The p u l s e propagated across the width of a long channel which was c l o s e d at one end. By p l a c i n g a heater at the c l o s e d end of the channel a r e l a t i v e v e l o c i t y without mass flow c o u l d be developed along the l e n g t h o f the channel and, i n p a r t i c u l a r , at the r e g i o n o f second sound propagation. In t h i s f a s h i o n the mutual o b s t r u c t i o n of the heat f l u x arid second sound by the assoc-9 i a t e d experimental devices was avoided. A l s o , t h i s geometry removed 11 17 entrainment ' e f f e c t s which would change the v e l o c i t y of second sound and p o s s i b l y obscure a s h i f t i n T.. 10 CHAPTER I I APPARATUS (1) Cryogenic Apparatus As i l l u s t a t e d i n F i g u r e 1, the helium i n the c r y o s t a t c o n s i s t e d of two baths. The 2.5 l i t r e outer bath served as a helium reserve and con-t r i b u t e d to general temperature s t a b i l i t y w h i l e the in n e r bath of 0.6 l i t r e s contained the second sound c e l l . The temperature of the outer bath could be r e g u l a t e d t o b e t t e r than 10~k K over a h a l f hour i n t e r v a l 18 by u s i n g a Walker pressure r e g u l a t o r . The inner bath was i s o l a t e d from the outer bath by a vacuum j a c k e t formed by the w a l l s o f the s t a i n l e s s s t e e l outer can and brass inner can. The inner can was suspended by two t h i n w a l l e d s t a i n l e s s s t e e l tubes con-n e c t i n g the brass f l a n g e s which formed the tops of the two cans. One tube (0.5" diameter) served as a pump l i n e t o the inner b a t h , w h i l e the other tube (0.313" diameter) served as a f i l l l i n e through which helium was admitted to the i n n e r bath by means of a low temperature needle v a l v e . There was a l s o a porous s t a i n l e s s s t e e l f i l t e r over the v a l v e entrance t o prevent s o l i d a i r from e n t e r i n g the in n e r bath and v a l v e seat. The major thermal l i n k between the two baths was through the helium f i l m on the i n s i d e s urface of the two tubes. A l s o , the two baths could be con-nected by means o f a press u r e e q u a l i z i n g v a l v e at room temperature. E l e c t r i c a l leads were brought down through the vacuum j a c k e t pump l i n e . They were f ed through a short segment of copper tube and f i x e d i n plac e w i t h a high thermal c o n d u c t i v i t y epoxy. This arrangement provided the leads w i t h a thermal l i n k to the o u t s i d e bath. The leads were brought to the i n s i d e bath through a f i t t i n g c o n s i s t i n g of 2" s e c t i o n s VACUUM JACKET PUMP LINE TO PUMP LINE INNER CAN 11 / / / / / / / / / / / / / / o W W O o B O o O o o STEM \ LOWTEMR F I L T E R - ' V A L V E HEAT SINK LEADS FILL LINE INDIUM "0" RING SEAL OUTER BATH INNER BATH SECOND SOUND CELL (Schematic cut-away) OUTER CAN VACUUM JACKET INNER CAN FIG. I CRYOGENIC APPARATUS of copper wire (AWG #20) which were fed through holes d r i l l e d i n a brass plug and subsequently f i x e d i n p l a c e w i t h an epoxy. The epoxy formed the vacuum s e a l and provided e l e c t r i c a l i n s u l a t i o n . The second sound c e l l (Figure 2) sat at the bottom of the i n n e r bath. I t was machined out of three p i e c e s o f nylon which when b o l t e d together formed a square channel 1.6 cm across and 10 cm long w i t h 0.25" t h i c k w a l l s and a 0.125" t h i c k bottom p l a t e . The bolometer and second sound generator were mounted on opposite s i d e s o f the v e r t i c a l channel. A l s o , at the same height as the bolometer and second sound generator, but at the s i d e of the channel, was l o c a t e d a carbon r e s i s t o r which served as a thermometer. There were heaters a t each end of the channel. The heater at the c l o s e d end produced the s u p e r f l u i d - normal f l u i d counter-flo w . When t h i s counterflow heater was o f f the heater at the open end of the channel (the dummy heater) was turned on to simulate the hea t i n g c o n d i t i o n s t h a t e x i s t e d when a counterflow was present i n the channel. In a d d i t i o n , b o l t e d onto the o u t s i d e o f the second sound c e l l , there was a t h i r d heater t h a t was p a r t o f a feedback c i r c u i t used t o c o n t r o l the temperature of the i n n e r bath. (2) Primary Thermometer I n i t i a l temperature measurements f o r the c a l i b r a t i o n of the carbon 19 r e s i s t o r thermometer were performed w i t h a b u t y l p h t h a l a t e manometer The vapour pressures t h a t r e s u l t e d from the measurements w i t h the man-4 ometer were converted t o temperatures by means of the " 1958 He Scale 20 of Temperatures " (3) D e t a i l s o f the Second Sound C e l l Apparatus Figure 2 presents an exploded view of the c e l l and a s s o c i a t e d ap-paratus . OPEN END DUMMY A = THERMOMETER B = BOLOMETER C = SECOND SOUND GENERATOR HIDDEN FROM VIEW OPPOSITE B FIG. 2 SECOND SOUND CELL (EXPLODED VIEW) 14 (a) Heaters The dummy and counterflow heaters were wound t o the same value ( w i t h i n 0.3%) of 764 ft from v a r n i s h i n s u l a t e d r e s i s t a n c e wire (72 Q per f o o t ) . The counterflow heater was wound and p o s i t i o n e d i n such a f a s h i o n as t o cover most o f the cross s e c t i o n a l area of the c l o s e d end of the channel thereby producing a uniform heat i n p u t . The dummy heater at the channel mouth occupied a small f r a c t i o n (^  10%) o f the cross s e c t i o n a l area of the channel and t h e r e f o r e presented l i t t l e o b s t r u c t i o n to the counterflow. The temperature c o n t r o l heater was wound from s i m i l a r wire and had a value of 9.85 KQ. The power input d u r i n g o p e r a t i o n was t y p i c a l l y 1.5 x 10" 3 W. (b) Bolometer and Second Sound Generator The bolometer (see F i g u r e 3) was made from commercial p r i n t e d c i r -c u i t board. Using p h o t o f a b r i c a t i o n techniques a 0.014 cm wide s l i t was etched through the .008- cm t h i c k copper s u r f a c e o f a l a r g e p i e c e of board. From t h i s l a r g e p i e c e was cut the d e s i r e d 1.3 cm square s e c t i o n w i t h the s l i t c e n t r a l l y l o c a t e d and p a r a l l e l to the top and bottom edges. Leads were soldered t o the two copper surfaces as i l l u s t r a t e d and the bolometer was completed by spraying the sur f a c e w i t h a suspension o f 16 m i l l i - m i c r o n diameter carbon p a r t i c l e s i n xylene. The xylene evaporated l e a v i n g a la y e r of carbon. Thus, the bolometer e f f e c t i v e l y c o n s i s t e d o f a carbon f i l m r e s i s t o r w i t h l a t e r a l dimensions 0.00.3 cm by 1.3 cm and length 0.014 cm. At l i q u i d helium temperatures the r e s i s t a n c e of the bolometer de-pended on the v a l u e of the constant c u r r e n t b i a s that was a p p l i e d . In-crea s i n g the b i a s c u r r e n t r e s u l t e d i n a decrease i n the r e s i s t a n c e FIBERGLASS LEAD 15 COPPER SLIT CONTAINING CARBON LEAD FIG. 3 BOLOMETER OR SECOND SOUND GENERATOR PULSES AVAILABLE FOR X -Y RECORDER r t SIGNAL SOURCE i k BOX-CAR # | , , N T R ? r r F R R SECOND TRIGGER SOUND t PULSE —>i REPETITION INTERVAL-SIGNAL SOURCE FIG. 4 PULSE SEQUENCE ( ^- * -3 x 105 fl/yA at 1.25 x i o " 6 A) i n d i c a t i n g perhaps a hea t i n g e f -f e c t . For a b i a s c u r r e n t of 1.25 * 10 6 A, the value used i n the exper-iment, the r e s i s t a n c e o f the bolometer at the X-point was ^ 6 x i o 5 fl. 1 dR The thermal s e n s i t i v i t y , ( ) , was found to be i n s e n s i t i v e t o the R d 1 value of the b i a s c u r r e n t and was -^ ( ^  ) ^ -1 K 1 at the X-point. The second sound generator was s i m i l a r t o the bolometer i n s i z e and c o n s t r u c t i o n and d i s p l a y e d s i m i l a r c h a r a c t e r i s t i c s . The generator and bolometer were mounted on opposite s i d e s of the channel w i t h the carbon f i l m s l i t s at the same height ( r e l a t i v e t o the bottom of the channel) and p e r p e n d i c u l a r to the long a x i s of the channel. This alignment en-sured that the detected second sound propagated p e r p e n d i c u l a r t o the counterflow and hence d i d not experience entrainment. The o v e r a l l s e n s i t i v i t y of the second sound d e t e c t i o n system was u s u a l l y 6 x 10 8 K per c h a r t cm. The detected second sound p u l s e rep-resented temperature excursions which were t y p i c a l l y ^ 10 6 K. Second sound noise ( r e a l and apparent) was ^ 2 x 10 7 K p-p. (c) Thermometer The thermometer was an A l l e n - B r a d l e y 0.1 W 150 fl carbon r e s i s t o r . I t s lambda p o i n t r e s i s t a n c e , R^, was ^ 11900 fl w i t h a s e n s i t i v i t y R~ ^  aT "'x ^ ^ With a P o w e r d i s s i p a t i o n i n the thermometer of 10 9 W the s e n s i t i v i t y of the thermometry system was ^ 20 x 10 6 K per chart i n c h . Thermal n o i s e ( r e a l and apparent) was o r d i n a r i l y about 6 x 10 6 K peak t o peak when o p e r a t i n g w i t h a bandwidth of 0.3 Hz. (4) E l e c t r o n i c s Figure 5 i s a block diagram of the e l e c t r o n i c s , (a) Pulse Inputs and Timing The v a r i a b l e r e p e t i t i o n i n t e r v a l of the p u l s e r t r i g g e r determined PULSER TRIGGER X-Y RECORDER SIGNAL SOURCE BOXCAR TRIGGER FEEDBACK POT.* BOXCAR INTEGRATOR CALIBRATION I CALIBRATION 2 COUNTER POWER SUPPLY (OFFSET) PREAMP x 105 BOLOMETER BIAS HELIUM A C AAAA/—l SI U D WW-i B CHART-RECORDER AMMETER HEATER POWER SUPPLY LOCK-IN AMP' I PREAMP * 100 TRANSFORMER BRIDGE A = SECOND SOUND GENERATOR B = BOLOMETER C + D = HEATERS-COUNTERFLOW AND DUMMY E - THERMOMETER F = TEMPERATURE CONTROL HEATER FIG. 5 ELECTRONICS - BLOCK DIAGRAM 4^ 18 the r e p e t i t i o n i n t e r v a l of the p u l s e s produced by each of the other p u l -sers (Tektronix 160 S e r i e s ) . . The s i g n a l source p u l s e r produced pulses having a d j u s t a b l e amplitude and d u r a t i o n . The other three p u l s e r s (box-car t r i g g e r , c a l i b r a t i o n 1, c a l i b r a t i o n 2) produced p u l s e s t h a t had i n -dependently v a r i a b l e durations and amplitudes as w e l l as a d j u s t a b l e de-l a y s (0 t o 100% o f the r e p e t i t i o n i n t e r v a l ) r e l a t i v e t o the s i g n a l source. The r e p e t i t i o n i n t e r v a l as w e l l as the time i n t e r v a l between any p a i r of p u l s e s could be measured on the counter. The sequence of pulses i s i l l u s t r a t e d i n F i g u r e 4. The s i g n a l source pulse ( t y p i c a l l y 15 x 10 6 sec d u r a t i o n ) was a p p l i e d t o the sec-ond sound generator where J o u l e h e a t i n g produced a second sound p u l s e . At an appropriate time t ^ a f t e r the s i g n a l source p u l s e the boxcar i n t e -g r a t o r ( P rinceton A p p l i e d Research model CW-1) was turned on by the t r i g -ger so t h a t the detected second sound p u l s e , delayed by the time of f l i g h t T , would occur i n the a c t i v e or 'on' p e r i o d of the i n t e g r a t o r . The output of the i n t e g r a t o r was d i s p l a y e d on an X-Y r e c o r d e r . Due t o an u n c e r t a i n t y inherent i n the time base o f the i n t e g r a t o r i t was neces-sary t o i n s e r t two c a l i b r a t i o n p u l s e s . These p u l s e s , separated by a measured time i n t e r v a l , were i n t e g r a t e d and d i s p l a y e d on the X-Y recorder w i t h the s i g n a l t r a c e and thereby c a l i b r a t e d the time base, (b) S i g n a l D e t e c t i o n The bolometer was b i a s e d w i t h a constant D.C. current (1.25 x 10 6 A ) . Temperature excursions a s s o c i a t e d w i t h the second sound pulses produced r e s i s t a n c e changes i n the bolometer which appeared as v o l t a g e changes across the bolometer. These v o l t a g e changes were a m p l i f i e d ( P r i n c e t o n A p p l i e d Research model 114 w i t h model 185 p r e a m p l i f i e r ) , i n t e g r a t e d and d i s p l a y e d on an X-Y r e c o r d e r as d e s c r i b e d above. 19 (c) Counterflow and Dummy Heaters The heaters ( r e s i s t o r s ) i n the channel were d r i v e n by a v a r i a b l e D.C. power supply. Measurement of the input current and heater r e s i s -tance y i e l d e d the power input t o the channel; (d) Temperature Measurement and C o n t r o l The reference s i g n a l from the l o c k - i n a m p l i f i e r ( P r i n c e t o n A p p l i e d Research model 122) was used t o d r i v e (at ^ 145 Hz.) a Wheatstone b r i d g e . Two arms of the bri d g e c o n s i s t e d of f i x e d matched ( w i t h i n 1%) r e s i s t o r s , a t h i r d arm c o n s i s t e d of the r e s i s t a n c e thermometer w h i l e the f o u r t h arm was a p r e c i s i o n decade r e s i s t a n c e . A v a r i a b l e capacitance across the f o u r t h arm of the bri d g e was used t o n u l l out that component of the s i g -n a l which was i n quadrature w i t h the r e f e r e n c e . The unbalanced s i g n a l across the b r i d g e was brought through an i s o l a t i o n t r a n sformer, a m p l i f i e d . and then fed to the l o c k - i n a m p l i f i e r . The output of the l o c k - i n am-p l i f i e r was d i s p l a y e d on a s t r i p chart r e c o r d e r . From the c a l i b r a t i o n of.the chart recorder s c a l e ( i . e . the ohms per chart i n c h conversion f a c t o r ) and the thermometer r e s i s t a n c e versus temperature c a l i b r a t i o n , the temperature of the inner bath c o u l d be determined. In a d d i t i o n , the output of the l o c k - i n a m p l i f i e r was combined w i t h a D.C. o f f s e t s i g n a l and fed t o a r e s i s t o r i n the inner bath. This feedback arrangement en-hanced temperature s t a b i l i t y ( d r i f t s < 20 x 10 5 K per hour) and per-m i t t e d some a d j u s t a b i l i t y (y ± 30 * 10~ 6 K) of the temperature. 20 CHAPTER I I I EXPERIMENTAL PROCEDURES (1) Thermometry Thermometer c a l i b r a t i o n p o i n t s f o r temperatures l e s s than the lambda temperature were obtained by connecting the in n e r and outer baths v i a the low temperature v a l v e and pressure e q u a l i z i n g v a l v e . A pressure c o r r e s -ponding t o a s u i t a b l e temperature was set i n the r e f e r e n c e volume of the Walker r e g u l a t o r and the system then allowed t o come to e q u i l i b r i u m . (The temperature c o n t r o l l i n g loop was not used i n c a l i b r a t i o n runs; equ-i l i b r a t i o n time was *v< 15 minutes.) The vapour pressure o f the outer bath was measured u s i n g the primary thermometer and the corresponding r e s i s t a n c e was measured w i t h the b r i d g e . In t h i s f a s h i o n s e v e r a l values-of the r e s i s t a n c e R and temperature T were obtained f o r temperatures \ from 2.0 K to w i t h i n a few m i l l i d e g r e e s of the X-point. The r e s i s t a n c e o f the thermometer at T^ (that i s T^ (0) , the t r a n -s i t i o n temperature without a heat f l u x ) was obtained as an a d d i t i o n a l c a l i b r a t i o n p o i n t as f o l l o w s . The e n t i r e system was brought to thermal e q u i l i b r i u m at a temperature o f about 10 3 K below T The two baths were then i s o l a t e d (connecting v a l v e s closed) and the pressure on the outside bath set f o r a temperature o f a few mK above T . The inner bath was held below T^ by pumping w h i l e the outer bath warmed t o the new temp-erature. A f t e r having reached the new temperature the pumping on the inner bath was terminated, the pressure e q u a l i z i n g v a l v e was opened (to prevent pressure b u i l d u p and subsequent departures from the sa t u r a t e d vapour pressure l i n e ) and the i n n e r bath allowed t o warm through the lambda p o i n t towards the outer bath temperature. The r e s i s t a n c e R was continuously d i s p l a y e d on a chart r e c o r d e r . This warming curve ( i . e . R(T) where T = T ( t ) ) had a r e g i o n of zero s l o p e , or p l a t e a u , which was i d e n t i f i e d as the thermometer r e s i s t a n c e at the lambda temperature (see Figure 6 ) . The p l a t e a u i n the warming curve has been used by others to a c c u r a t e l y determine the v a l u e R^. The p l a t e a u occurs as a r e s u l t of the s p e c i f i c heat anomaly a t T and a change i n the means of heat t r a n s f e r A from a second sound mode i n He I I t o a convection and conduction process" i n He I. For T <T^ and small heat i n p u t s the s u p e r f l u i d bath remains v i r t u a l l y i n e q u i l i b r i u m as i t approaches T^. At T^ the formation of the He I f l u i d 'blanket' at the heat source e f f e c t i v e l y i s o l a t e s the remain-der of the bath from the heat source. Consequently the temperature of the i n t e r n a l r e g i o n o f the bath becomes constant w i t h the very l a r g e s p e c i f i c heat c o n t r i b u t i n g t o the temperature s t a b i l i t y . Continued heat input overcomes the divergence i n the s p e c i f i c heat, the thermometer be-comes engulfed w i t h He I and the p l a t e a u breaks as warming proceeds to the outer bath temperature. The purpose o f the thermometer c a l i b r a t i o n was t o enable measurement of AT = T . . - T. Besides being the more convenient parameter, AT could A be measured more a c c u r a t e l y than the a b s o l u t e temperature s i n c e systematic e r r o r s disappear on t a k i n g the d i f f e r e n c e T^- T. Thus, the data were f i t t e d by computer t o the e x p r e s s i o n l o g R = A + B/T i n order t o d e t e r -mine the v a l u e of B (B - 3 ) . Rearrangement of t h i s expression and e l -i m i n a t i o n o f A y i e l d s T x l o g (R/R x) A T = l o g (R/R x) + B/Tx ( 4 ) where T^= T^ (0)= 2.1720 K. T h i s was used to determine AT from the meas-22 T ( increasing) R (decreasing) II93I& ^ TIME FIG. 6 A WARMING CURVE SHOWING Rk PLATEAU (Bandwidth of ~ I Hz) ured values R, R^, B. Several c a l i b r a t i o n runs were performed t o determine the e f f e c t s of thermal c y c l i n g between T and room temperature. The valu e of B was A found to be constant to w i t h i n ± 0.1%. I t was observed t h a t the low temperature r e s i s t a n c e of the t h e r -mometer was a f u n c t i o n of time. T h i s d r i f t or aging e f f e c t was most e a s i l y observed by measuring R^(t) (by the p l a t e a u method), however i t was determined t h a t both R.. and R d r i f t at the same r a t e . F i gure 7 il-A l u s t r a t e s the decaying time dependence of R ^ ( t ) . The valu e of t , the time at which aging begins or r e c y c l e s , was d i f f i c u l t to determine. 21 Tyson suggests t h a t the aging begins w i t h the a p p l i c a t i o n of power t o the b r i d g e . While t h i s appeared to be a f a c t o r i n t h i s experiment, the dominant c o n t r i b u t i o n t o the r e c y c l i n g o f the aging process seemed to be the pressure i n c r e a s e 4 cm Hg to 1 atm.) accompanying a prolonged time i n t e r v a l i n which the system was l e f t at 4.2 K or room temperature. This systematic e r r o r due t o aging was overcome by keeping the c r y o s t a t at. a temperature T = T^ f o r s e v e r a l days u n t i l the thermometric d r i f t had decayed, to l e s s than 0.2 ohms per hour. Then, during second sound runs, frequent checks of R (t) were made and the a p p r o p r i a t e values of A determined by i n t e r p o l a t i o n . As a l r e a d y mentioned, the power input t o the thermometer was h e l d at about 10 9 W f o r the c a l i b r a t i o n and second sound runs. Checks were made t o determine whether or not t h i s f i n i t e power input was i n t r o d u c i n g any systematic e r r o r t o temperature measurement. R^ and R at AT ^ 10 mK were measured as f u n c t i o n s of power. E x t r a p o l a t i o n to zero power i n d i c -ated that the s h i f t due to the f i n i t e power o f 10 9 W was approximately 0.2 ft. Since t h i s s h i f t was the same f o r both R and R , the e f f e c t was 24 'Rx(f)Xi FIG. 7 THERMOMETER AGING i n s i g n i f i c a n t i n the c a l c u l a t i o n of AT and no c o r r e c t i o n s were made. (2) Second Sound Measurements (a) Time of F l i g h t Data Second sound time o f f l i g h t data w i t h and without a counterflow present were c o l l e c t e d i n the f o l l o w i n g manner. F i r s t , s e v e r a l warming curves were performed to determine R^ and the slope of R.. (t) . Then, w i t h the o u t s i d e and i n s i d e baths i s o l a t e d , A the o u t s i d e bath was placed at a temperature c o n s i d e r a b l y lower (AT ^  10's of mK u s u a l l y ) than the d e s i r e d temperature f o r the inner .bath. As the inner bath temperature dropped, the power to the counterflow or dummy heater was turned on. The heat leak to the outer bath and adjustments t o the pumping speed on the inner bath brought i t s temperature to ap-proximately the d e s i r e d v a l u e . At t h i s p o i n t p ulses were a p p l i e d to the second sound generator and the feedback loop o f the thermometry system was turned on. F i n a l adjustments to the temperature were made u s i n g the feedback c o n t r o l s and, w i t h the temperature s t a b l e , the s e n s i t i v i t y of the thermometry system was c a l i b r a t e d i n ohms per chart i n c h by making known changes i n the decade r e s i s t a n c e arm o f the b r i d g e and r e c o r d i n g the subsequent output. The second sound p u l s e s were observed on an os-c i l l o s c o p e to determine the approximate time o f f l i g h t o r , when they were not observable due to a t t e n u a t i o n e f f e c t s c l o s e to T , the time of f l i g h t A was estimated u s i n g the known values o f v e l o c i t y without counterflow. This i n f o r m a t i o n was used t o set the t i m i n g c i r c u i t r y of the p u l s e r s and boxcar i n t e g r a t o r thereby ensuring that the s i g n a l would a r r i v e i n the scan window o f the i n t e g r a t o r . Two scans were performed. In one case there was power input to the counterflow heater w h i l e i n the other scan the power input was to the dummy heater and t h e r e f o r e no counter-f l o w was present. The time i n t e r v a l between scans was made as short as p o s s i b l e (eg. 10 minutes) so that temperature d r i f t s , ( t ) aging, and other p o s s i b l e systematic changes (such as those a s s o c i a t e d w i t h a drop-ping bath l e v e l ) were minimal. While the scan was on the l e a d i n g edge of the p u l s e t r a c e the time was noted ( f o r R^(t) i n t e r p o l a t i o n ) as was the value of R(T). F i n a l l y the c a l i b r a t i o n p ulses were pla c e d on the t r a c e . F o l l o w i n g the completion of one such c h a r t the temperature feedback was removed and a sma l l e r v a l u e of AT achieved by r a i s i n g the o u t s i d e bath temperature and/or decreasing the pumping speed and/or i n c r e a s i n g the power input t o the counterflow heater. The temperature was then s t a -b i l i z e d w i t h the feedback and the data c o l l e c t i o n procedure repeated. The value of R was determined every two or three hours. Values of AT A ranged from 90 uK to 30 mK w h i l e the heat f l u x i n the channel v a r i e d from 0.4 mW/cm2 to 160 mW/cm2. Figu r e 8 i l l u s t r a t e s the e s s e n t i a l f e a t u r e s o f a t y p i c a l t r a c e . The arrow and bars a s s o c i a t e d w i t h each p u l s e i n d i c a t e the a r r i v a l time and estimated e r r o r . Pulse #3 was i n the presence o f counterflow w h i l e p u l s e s #1 and #2 occurred without counterflow. Pulse #1 was put on f o r i l l u s t r a t i v e purposes as the s h i f t i n a r r i v a l time between #1 and #2 r e -presents an a c t u a l temperature change o f about 16 * 10 6 K (#1 i s c o l d e r ) . A l s o note t h a t #2 occurred at a temperature approximately 8 x 10 6 K co l d e r than #3. For these pulses AT ^  1 x 10~ 3 K. The c a l i b r a t i o n p u l s e s are a l s o i l l u s t r a t e d . F i g u r e 8 a l s o c o n t a i n s two a d d i t i o n a l f e a t u r e s r e l a t e d t o the p u l s e s t r u c t u r e . The r e c e i v e d pulses are c o n s i d e r a b l y wider than the 15 ysec input p u l s e s . T h i s e f f e c t i s a consequence of the generator-bolometer FIG. 8 DETECTED SECOND SOUND PULSES. geometry which r e s u l t s i n a delayed c o n t r i b u t i o n to the s i g n a l from the propagation of second sound along paths which are angled w i t h r espect to the sh o r t e s t d i r e c t r o u t e across the channel. This p u l s e broadening i s more pronounced f o r sm a l l e r second sound v e l o c i t i e s which occur as AT 4- 0 . A l s o , t h ere i s a c o l d r e g i o n which f o l l o w s the pu l s e or hot 22 23 reg i o n . This e f f e c t has been s t u d i e d by others ' (b) Amplitude Dependence Checks Several attempts were made w i t h and without counterflow to determine g whether the second sound amplitude dependence o f the v e l o c i t y was of any s i g n i f i c a n c e i n t h i s experiment. No amplitude e f f e c t on the time of f l i g h t was observed and i t was concluded, t h a t f o r the range of temper-atures and amplitudes encountered i n t h i s experiment, t h i s f a c t o r need not be considered i n the a n a l y s i s of the data. "(c) Alignment Checks The d e s i r e d alignment o f the bolometer and generator was such t h a t the second sound should propagate p e r p e n d i c u l a r to the heat f l u x . I f 17 there was some misalignment then the r e s u l t i n g entrainment o f the sec-ond sound could i n t r o d u c e e r r o r . T h i s p o s s i b l e entrainment e f f e c t was checked by r e v e r s i n g the bolometer and generator r o l e s . T h u s , i f the misalignment was such t h a t the entrainment i n c r e a s e d the second sound v e l o c i t y then r e v e r s i n g the d i r e c t i o n o f propagation would decrease the second sound v e l o c i t y by the same amount. Therefore a time o f f l i g h t d i f f e r e n c e between the two cases would i n d i c a t e a misalignment. Tests were made at s e v e r a l temperatures and heat f l u x e s i n the r e g i o n of exp-erimental i n t e r e s t w i t h the r e s u l t that no entrainment was observed. Using t h e o r e t i c a l v a l u e s f o r the entrainment c o e f f i c i e n t and assuming a uniform normal f l u i d v e l o c i t y d i s t r i b u t i o n across the channel, i t was estimated that a misalignment of ^ 4° could have been detected. CHAPTER IV DATA REDUCTION AND ERROR ANALYSIS 30 I n t e r p r e t a t i o n of the f i n a l r e s u l t s depends fundamentally on the probable e r r o r i n the experiment. Thus, the methods used f o r the e r r o r estimates w i l l be o u t l i n e d i n some d e t a i l . (1) Time o f F l i g h t Data Reference t o Figur e s 4 and 8 i n d i c a t e s t h a t the time of f l i g h t x was a v a i l a b l e as the sum of two time i n t e r v a l s . One i n t e r v a l was the time, measured by the counter, from the s i g n a l source t o the f i r s t c a l -i b r a t i o n puisne. The other i n t e r v a l was the time from the f i r s t c a l i b -r a t i o n p u l s e t o the a r r i v a l time p o i n t determined from the e x t r a p o l a t i o n of the l e a d i n g edge of the second sound p u l s e t o the l o c a l b a s e l i n e . The e r r o r i n the e s t i m a t i o n o f the a r r i v a l time o f the pulse (see Fi g u r e 8 f o r e r r o r bars) was the o n l y s i g n i f i c a n t e r r o r i n the q u a n t i t y T . Due to the v a r i a t i o n i n pul s e shapes and b a s e l i n e s introduced by a t t e n u a t i o n e f f e c t s and thermal n o i s e depending on the value of AT and the heat f l o w , there was no r i g o r o u s method f o r determining the probable e r r o r i n x. Consequently, the l i m i t s to t h i s u n c e r t a i n t y were determined in"each case by q u a l i t a t i v e c o n s i d e r a t i o n s as to whether the pul s e had d e f i n i t e l y a r r i v e d or d e f i n i t e l y not a r r i v e d . The probable e r r o r i n the time o f f l i g h t was t y p i c a l l y ± 0.2%. For t h i s work the more u s e f u l time i n t e r v a l was the d i f f e r e n c e Sx between the times of f l i g h t o f the second sound w i t h and without counter-f l o w present. The absolute e r r o r i n 6x was c a l c u l a t e d by adding i n quadrature the absolute e r r o r s i n the two times used to form the d i f -f e rence. (2) AT Data In the c a l c u l a t i o n of AT = T (0) - T (see equation (4) i n Chapter A 3) there were the f o l l o w i n g sources of probable e r r o r : (a) B was determined from s e v e r a l c a l i b r a t i o n runs as the slope of a logR vs 1/T r e l a t i o n . I t s v a l u e and probable e r r o r e(B) were B ± e(B) = 3.084 ± 0.004. (b) Measurement of R and R^ introduced u n c e r t a i n t y i n the value of the r a t i o x = R/R^ o c c u r r i n g i n equation (4). The probable e r r o r i n R and R x (denoted by ± e(R) and ± e(R^) ) w a s c a l c u l a t e d approximately as the zero to peak noise l e v e l i n the temperature t r a c e s . e(R) and e(R^) c o n t r i b u t e d to an e r r o r i n x c a l c u l a t e d as E ( x ) = ± {'{ e(RD/R x> 2 +' { e ( R x ) / R x } 2 ) H . . The subsequent probable e r r o r e(AT) i n AT r e s u l t i n g from e(x) was taken as £(AT) = 0.67 e(x) which f o l l o w s from a f i r s t order v a r i a t i o n of equ-a t i o n (4) . Combining the independent f r a c t i o n a l e r r o r c o n t r i b u t i o n s i n quad-r a t u r e , the f i n a l absolute probable e r r o r e(AT) i n AT was e(AT) = ± { {e(AT)/AT} 2 + {e(B)/B} 2 }^ AT. T y p i c a l e r r o r s i n R and R x were ± 0.05 ft and ± 0.1 ft r e s p e c t i v e l y w i t h e r r o r s i n AT ranging from ± 40 uK t o ± 5 uK. As w i t h the times of f l i g h t , i t was u s u a l l y o n ly necessary to know the change i n r e l a t i v e temperature AT r e s u l t i n g from an a c t u a l change i n the absolute temperature o c c u r r i n g between p u l s e s propagating w i t h and without counterflow. This d i f f e r e n c e i s denoted by 6(AT)^,. The u n c e r t a i n t y a s s o c i a t e d w i t h these r e l a t i v e temperature changes was smaller than that o u t l i n e d above. In the c a l c u l a t i o n o f the probable e r r o r i n 6 ( A T ) T , the u n c e r t a i n t y e(B) was ignored and the p r e c i s i o n i n R.. was improved s i n c e changes i n R, (t) were p r i m a r i l y r e q u i r e d . The un-A A c e r t a i n t y i n R remained unchanged as e (R). Denoting r e l a t i v e values w i t h a t i l d a , the r e l a t i v e probable e r r o r i n AT became e(AT) = 0.67 e(x) where g(x) was c a l c u l a t e d as e(x) above but u s i n g e(R) and the improved £(R X). In c a l c u l a t i n g the probable e r r o r i n the d i f f e r e n c e 6(kT)^, the r e l a t i v e probable e r r o r s £(AT) a s s o c i a t e d w i t h each term i n the d i f f e r -ence were added i n quadrature t o g i v e e(6(AT)^,)". The r e l a t i v e probable e r r o r i n R^ was t y p i c a l l y ± 0.02 U w i t h r e l a t i v e e r r o r s e(AT) ranging from 6 uK t o 2 uK. (3) Heat F l u x per U n i t Area The power input t o the c o u n t e r f l o w heater was determined from the measured values o f c u r r e n t and heater r e s i s t a n c e . In determining the heat f l u x per u n i t area the major source o f e r r o r was due to the un-c e r t a i n t y i n the c r o s s s e c t i o n a l area o f the channel at the bolometer and generator. T h i s area was 2.03 cm 2 w i t h an u n c e r t a i n t y of ± 3%. Thus,.the probable e r r o r i n the heat f l u x per u n i t area was t y p i c a l l y between ± 3 and 4 per cent and was l a r g e l y systematic due t o the uncer-t a i n t y i n area. CHAPTER V ANALYSIS AND RESULTS (1) A n a l y s i s The conversion of the v e l o c i t y u^, or more s p e c i f i c a l l y the time of f l i g h t data, t o i n f o r m a t i o n r e g a r d i n g the s h i f t 6T^ i n the t r a n s i t i o n temperature r e q u i r e d t h a t the values of the q u a n t i t y du2(AT)/dAT be known f o r the values of AT o c c u r r i n g i n the experiment. T h i s q u a n t i t y was c a l c u l a t e d by e v a l u a t i n g the hydrodynamic exp r e s s i o n r TS 2 p s x h p ^n f o r a number of values of AT. The d e s i r e d d ^ f A ^ / d A T was then approx-imated, by t a k i n g d i f f e r e n c e s 6^(AT)/<5AT. The d i f f e r e n c e between suc-c e s s i v e values o f AT was small enough so t h a t no s i g n i f i c a n t e r r o r was introduced by t h i s numerical method. The values of the terms contained i n the hydrodynamic expression f o r were a v a i l a b l e from experimental - - 21 work.. For 6 x 10 5 K < AT < 5 x 10 2 K, measurements o f p g g i v e p n / p s = 0.699 ( A T ) 2 / 3 . (5) 24 A h l e r s : give s an e m p i r i c a l r e l a t i o n f o r the s p e c i f i c heat c^ which, a f t e r a change of v a r i a b l e s and u n i t s , reduces t o = c ^ = 4.866 - 1.274 AT - 9.23 InAT + 1.668 ATlnAT J/gmK. (6) This expression i s good to b e t t e r than 1% f o r AT < 10 1 K. The entropy S can be determined from S(T) = SCT^ + , (c s/T) dT where c o i s given by equation (6) and SCT^ i s obtained from H i l l and 25 26 Lounasmaa : S(2.10 K) = 1.24 J/gmK. A f t e r an approximate e v a l u a t i o n of the i n t e g r a l we have S(AT) = 1.54 - 2.83(AT) + 1.73(AT) 2 + 0.587(AT)InAT - 0.249(AT) 2lnAT S u b s t i t u t i o n of expressions ( 5 ) , ( 6 ) , (7) i n t o the hydrodynamic equation f o r u» and e v a l u a t i o n y i e l d e d the d e s i r e d numerical values u (AT) and For each time of f l i g h t datum T i n the absence of counterflow the value of was determined from - SL/T , £ being the d i s t a n c e of f l i g h t as measured (to ±1%) at room temperature and c o r r e c t e d f o r the thermal 27 c o n t r a c t i o n of nylon by 1.39%. The experimental v e l o c i t i e s were con-s i s t e n t l y l a r g e r than the hydrodynamic v e l o c i t i e s by f r a c t i o n a l values ranging from 0.3% t o 4.4% w i t h an average o f 1.4%. This suggests t h a t there was a systematic e r r o r i n £ however no attempt was made t o norm-a l i z e the data t o theory by a l t e r i n g £ s i n c e the t h e o r e t i c a l values of U£ are probably only accurate to about 2%. As w i l l be seen, a sys-tematic e r r o r i n £ of 'v* 1% i s of no s i g n i f i c a n c e i n e s t i m a t i n g 6T^. The change i n the time o f f l i g h t , 6x, t h a t occurred between the heat f l u x on and heat f l u x o f f s i t u a t i o n s was i n t e r p r e t e d as a change SAT i n the r e l a t i v e temperature. D i f f e r e n t i a t i o n o f the expression u_ = £/x w i t h respect to AT r e l a t e s the q u a n t i t i e s 6T,6A T as f o l l o w s : J/gmK. (7) du 2(AT)/dAT. (8) The change 6AT may be a t t r i b u t e d to (a) an a c t u a l change i n the temp-erature T which i s denoted, as above, by 6(AT) and (b) a change i n the t r a n s i t i o n temperature which i s denoted by 6T^. Thus, 6AT can be w r i t -ten as the sum of two c o n t r i b u t i o n s : 6(AT) = 6 ( A T ) T + 6T X . (9) Comparing expressions (8) and ( 9 ) , we have f o r the X-point s h i f t 6T X = - 6 ( A T ) T - ~2 ( d u 2 / d A T ) _ 1 6x . (10) The q u a n t i t y 6(AT)^ was a v a i l a b l e from the temperature t r a c e s and T , 6T from the pulse c h a r t s . The probable e r r o r i n 6T was obtained by adding i n quadrature the terms e( 6(AT)^, ) and ~2 (.du^/d&T)1 e ( 6 r ) . The quan-t i t i e s e( 6(AT)^, ) and e(St) are d e f i n e d i n Chapter IV. For a l l data e(6x)/6x >> 10 2 and t h e r e f o r e a 1% e r r o r i n I i s unimportant. (2) R e s u l t s The m a j o r i t y of the r e s u l t s (some p o i n t s were omitted f o r c l a r i t y ) , as c a l c u l a t e d from equation (10), are presented i n Figure 9. A l e a s t squares f i t to the data of the form 6T = AQ was performed. Each data A p o i n t was weighted w i t h the square of the r e c i p r o c a l of the probable error, of that p o i n t . The v a l u e of A i s A = - 1.3 (± 1.6) x 10" 4 (11) f o r Q i n W/cm2 and 6T X i n K. The estimated e r r o r on the parameter A 28 i s c a l l e d the ' s t a t i s t i c a l e r r o r o f estimate' and i t represents ± one standard d e v i a t i o n ( i . e . 68% confidence l i m i t ) . STX (/iK) 120* 90 60 30 0 - 3 0 - 6 0 - 9 0 -120 ll 1 ( »| / i f f Lio M • > $20 30" ! 1 40 36 FIG. 9 THE SHIFT OF Tv BY A HEAT FLUX 37 CHAPTER VI DISCUSSION AND CONCLUSIONS (1) D i s c u s s i o n The i n t e r p r e t a t i o n of the r e s u l t s of the experiment i s based on the assumption t h a t , f o r 90 uK < AT < 30 mK (the range o f AT i n the e x p e r i -ment), the v e l o c i t y o f second sound remains the same f u n c t i o n u 2(AT) f o r AT = T A ( 0 ) - T or AT = T^(Q) - T. I f t h i s assumption i s v a l i d then the v e l o c i t y of second sound i s a good probe of the value of T^ (Q) . The v a l i d i t y o f t h i s assumption i s subject t o c o n s i d e r a t i o n of the 29 t h e o r e t i c a l work of Mehl . He proposes an i n t e r a c t i o n between second sound and v o r t i c e s which b a s i c a l l y i n v o l v e s r o t o n s c a t t e r i n g from v o r t e x cores. T h i s i n t e r a c t i o n r e s u l t s i n a decrease Au 2 i n the second sound' v e l o c i t y t h a t i s p r o p o r t i o n a l t o w z (w = v - v g and v^, v g are averaged over a r e g i o n c o n t a i n i n g many v o r t i c e s ) . Thus, he proposes A u 2/u 2 = - kw 2 (12) where k depends, i n p a r t , on the r e c i p r o c a l of the frequency of the second sound and the r a t i o of two f o r c e 'constants'. These f o r c e con-st a n t s are temperature dependent and the t h e o r e t i c a l l y c a l c u l a t e d r a t i o i n the c r i t i c a l r e g i o n near T.. depends on the p a r t i c u l a r v o r t e x model A used. B a s i c a l l y the d i s t i n c t i o n between the two models considered by Mehl i s t h a t i n one case the r a t i o o f the f o r c e 'constants' approaches a small v a l u e (0.1).while i n the other model the r a t i o q u i c k l y vanishes as T approaches T. . The data and r e s u l t s of t h i s experiment cannot be used t o s e l e c t one of the two v o r t e x t h e o r i e s as being c o r r e c t . The reason f o r t h i s i s e s s e n t i a l l y a r e s u l t of the frequency dependence i n the term k above. The f a c t t h a t ( 15 ysec) p u l s e s have been used i n t h i s experiment r e s u l t s i n a l o s s of d e t a i l e d frequency i n f o r m a t i o n . A l s o , the r e l a t i v e l y l a r g e value of the main frequency component ( f ^  3 x 101* Hz) a s s o c i a t e d w i t h the p u l s e e f f e c t i v e l y decreases the s e n s i t -i v i t y to the changes i n u^ as proposed i n equation (12) due to the de-pendence o f k on f 1 . Thus, the change i n v e l o c i t y Au 2, as p r e d i c t e d by Mehl, i s small f o r the h i g h frequency pulses used i n t h i s experiment and i t may be van-i s h i n g l y small depending on the v o r t e x model used i n the c a l c u l a t i o n . However, such v e l o c i t y changes, i f they are d e t e c t a b l e , would be f a l s e l y i n t e r p r e t e d i n t h i s experiment as a depression i n the lambda temperature. In t h i s event the t r u e depression of the lambda p o i n t would be l e s s than t h a t measured by the experiment. (2) Conclusions Keeping the above d i s c u s s i o n i n mind and r e c a l l i n g the r e s u l t s of the l e a s t squares f i t as given i n equation (11) of Chapter V, i t i s con-cluded that the depression o f the t r a n s i t i o n temperature, 6T i n K, by a heat current Q i n W/cm2 i s 6T X 4 1.3 (± 1.6) x 10~k Q (13) f o r Q ^  3.4 x 10 2 W/cm2. (There were o n l y two data p o i n t s w i t h l a r g e r v a l u e s of Q and there were l a r g e probable e r r o r s a s s o c i a t e d w i t h these p o i n t s . T h i s i s the reason f o r the r e s t r i c t i o n on Q). Since t h i s experiment was performed w i t h a channel width and heat flows approximately the same as those i n the works of Bhagat, Lasken and D a v i s ^ ' t h e i r r e s u l t s as expressed i n equation (3) of Chapter I are i n c o n f l i c t w i t h the r e s u l t g i ven by equation (13). S i m i l a r remarks 13 apply to the work of Erben and P o b e l l . However, equation (13) i s an order of magnitude smaller i n the upper l i m i t and c o n s i s t e n t w i t h the r e s u l t s reported by A h l e r s 1 4 : 6T 4 3 x i o " 3 Q f o r 0.2 4 Q 4 2 W/cm2. A l s o , equation (13) i s c o n s i s t e n t w i t h and a f a c t o r of 6 sma l l e r than the work of Le i d e r e r and P o b e l l 1 5 : 6T^ 4 1.6 x 10~ 3 Q f o r Q = 1.55 * 10" 2 W/cm2. There are important d i s t i n c t i o n s between the technique used i n t h i s work and the techniques used i n the experiments r e f e r r e d to above. 12 With exception o f the work by Bhagat and Davis , each o f the above ex-periments r e l i e d p r i m a r i l y on the measurement of temperature a t one or more places along a channel i n which there was a heat f l u x . In p a r t i c -14 u l a r , A h l e r s measured the heat f l u x r e q u i r e d t o ma i n t a i n v a r i o u s temp-erature d i f f e r e n c e s between the ends o f a c a p i l l a r y f i l l e d w i t h helium; Bhagat and Lasken*^ analyzed the warming curves o f s e v e r a l thermometers 13 placed along a channel c a r r y i n g a heat f l u x ; Erben and P o b e l l made use of the p l a t e a u of the warming curve of a thermometer i n a channel c o n t a i n i n g heat f l o w ; L e i d e r e r and Pobell**' measured the warming r a t e s of thermometers a t each end of a narrow s l i t through which there was a heat flow . Thus, i n these experiments the i d e n t i f i c a t i o n of T^(Q) was made on the b a s i s o f temperature measurements o f an environment i n or near a temperature g r a d i e n t . As A h l e r s s u g g e s t s * 4 ' * ^ , i n the i n t e r p r e t -a t i o n of such measurements one may have t o cons i d e r the e f f e c t of a 31 mutual f r i c t i o n which diverges as T -> T . In t h i s experiment the A i d e n t i f i c a t i o n of T.. (Q) i s based on the time o f f l i g h t of a second A sound p u l s e . As the propagation o f the pul s e i s p e r p e n d i c u l a r to the d i r e c t i o n o f heat f l o w , temperature g r a d i e n t s a s s o c i a t e d w i t h mutual f r i c t i o n are e f f e c t i v e l y removed by the geometry. 40 12 Bhagat and Davis measured the v e l o c i t y o f second sound propag-a t i n g p a r a l l e l t o a heat f l u x by u s i n g a resonance technique. They observed changes i n the v e l o c i t y due to the presence of a heat f l u x and i n t e r p r e t e d those changes i n terms of a s h i f t e d T^. However, i t i s f e l t that the method of i n t e r p r e t a t i o n of v e l o c i t y changes as s h i f t s i n T^ i n the present experiment i s more accurate and i n v o l v e s fewer assumptions, (eg. They d e f i n e T^fQJ by the s o l u t i o n , f o r T, of the equation u 2(Q,T) = 0. To solve t h i s equation they have e x t r a p o l a t e d t h e i r data f a r beyond measured values.) In a d d i t i o n , i t i s f e l t t h a t t h i s experiment i s geo-m e t r i c a l l y simpler i n t h a t i t removes entrainment and mutual f r i c t i o n e f f e c t s i n a f a s h i o n which i s sub j e c t t o l e s s i n t e r p r e t a t i o n than the experiment of Bhagat and D a v i s . 41 REFERENCES 1 W.A.B. Evans, Proceedings of the 14th I n t e r n a t i o n a l Conference on  Low Temperature P h y s i c s , Ed. M. Kru s i u s and M. V u o r i o , (North- H o l -land P u b l i s h i n g Co., Amsterdam, 1975) V o l . 1, p. 127. 2 G.S. Grest and A.K. Rajagopal, Proceedings of the 14th I n t e r n a t i o n a l  Conference on Low Temperature P h y s i c s , Ed. M. Krusius and M. V u o r i o , (North-Holland P u b l i s h i n g Co., Amsterdam, 1975) V o l . 1, p. 139. 3 L.D. Landau, J . Phys., Moscow 5_, 71 (1941). (Reprinted i n 7 below) 4 V.L. Ginzburg and L.P. P i t a e v s k i i , S o v i e t Physics - JETP, 34_, 858 (1958). 5 Yu. G. Mamaladze, S o v i e t P h y s i c s - JETP, 2_5, 479 (1967). 6 G. A h l e r s , Phys. Rev., 171, 275 (1968). 7 I.M. Kh a l a t n i k o v , I n t r o d u c t i o n to the Theory of S u p e r f l u i d i t y (W.A. Benjamin, Inc., New York, 1965). 8 J . W i l k s , The P r o p e r t i e s o f L i q u i d and S o l i d Helium (Clarendon Press, Oxford, 1967) . 9 L.V. Kiknadze, Yu. G. Mamaladze, O.D. C h e i s h v i l i , S o v i e t Physics -JETP, 21_, 1018 (1965). An i n t e r e s t i n g review of the hydrodynamics of r o t a t i n g He I I i s : E.L. A n d r o n i k a s h v i l i and Yu. G. 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Low Temp. Phys., 22_, 293 (1976). 23 R.W. Guernsey et a l . , C ryogenics, 7j 110 (1967). 24 G. A h l e r s , The Physics o f L i q u i d and S o l i d Helium, Part I , Ed. K.H. Benneman and J.B. Ket t e r s o n (John Wiley and Sons, New York, 1976) p. 106. 25 R.W; H i l l and O.V. Lounasmaa, P h i l . Mag., 2_, Ser. 8, 145 (1957). 26 D.L. Johnson, Ph.D. t h e s i s , U.B.C. (1969). The approximation i n -troduces an e r r o r of 0.1%. 27 . Thermal Expansion o f T e c h n i c a l S o l i d s a t Low Temperatures, N a t i o n a l Bureau o f Standards, Monograph 29 (1961). 28 For a b r i e f o u t l i n e of the s t a t i s t i c s i n v o l v e d r e f e r to the manual U.B.C. Curve. A more d e t a i l e d treatment i s i n M.E. Rose, Phys. Rev., 91, 610 (1953). 29 J.B. Mehl, Phys. Rev. A, 10, 601 (1974). 30 G. A h l e r s , Phys. Rev. L e t t e r s , 22, 54 (1968). 31 C.J. Gorter and J.H. M e l l i n k , Physica,15, 285 (1949). Gorter and M e l l i n k o r i g i n a l l y proposed a mutual f r i c t i o n f o r c e to e x p l a i n ob-s e r v a t i o n s that Q « (grad T) . Extensions of t h i s have l e d to the i d e a of a divergent mutual f r i c t i o n constant as i n reference 

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