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The velocity of second sound near the Lambda point Johnson, David Lawrence 1969

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1  S\oo  c  THE  VELOCITY OP SECOND SOUND NEAR THE LAMBDA POINT by  DAVID LAWHENCE JOHNSON B.Sc,  The U n i v e r s i t y of B r i t i s h Columbia, 1963  M.So., The U n i v e r s i t y o f B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics  We accept t h i s t h e s i s as conforming t o the required  THE  standard  UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1969  In p r e s e n t i n g an  this  thesis  advanced degree at  the  Library  I further for  shall  the  his  of  this  agree that  written  of  be  for extensive  g r a n t e d by  for financial  gain  Physics  15,  the  It i s understood  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  May  British  available for  permission.  Department of  Date  University  permission  representatives. thesis  f u l f i l m e n t of  make i t f r e e l y  s c h o l a r l y p u r p o s e s may  by  in p a r t i a l  1969  Columbia  shall  requirements  Columbia,  Head o f my  be  I agree  r e f e r e n c e and copying of  that  not  the  that  Study.  this  thesis  Department  copying or  for  or  publication  allowed without  my  11  DEDICATED  to my wife ELIZABETH  whose contribution to education was twofold and without whose help t h i s work would have been impossible.  iii  ABSTRACT D i r e c t measurements have been made of the v e l o c i t y of second sound i n l i q u i d helium over t h e temperature range T - T A  from 1.3  x 10"  2  K t o 5 i 10"  6  K.  Using  p r e v i o u s l y determined r e l a t i o n s h i p s f o r the s p e c i f i c heat, superfluld density,  and thermal c o n d u c t i v i t y near the  lambda p o i n t , c o n s i s t e n c y  has been demonstrated between  the measurements, v e l o c i t i e s p r e d i c t e d  by s u p e r f l u l d  hydrodynamics, and c e r t a i n s c a l i n g law p r e d i c t i o n s .  If  TABLE OP CONTENTS Chapter I  Introduction and Theory Introduction Experiment The Two Pluld Model C r i t i c a l Point Arguments  II  Apparatus Cryogenic Apparatus Electronics Second Sound Cavity  III  Experimental Procedure Second Sound T r i a l s Phase S h i f t and Dispersion T r i a l  IV  Data Reduction Extraction of Data from the Charts Analysis of Probable Errors  V  Results and Analysis Results Comparison with the Hydrodynamic Theory Comparison with the Scaling Prediction  VI  Conclusions and Conclusions Discussion  Discussion  Glossary of Symbols References and Footnotes Appendix 1  LIST OP TABLES  Table I  P h y s i c a l Q u a n t i t i e s near  T  c  vi  LIST OF FIGUBES Figure  Page  1  ( k, I" )  2  Cryogenic Apparatus  20  3  B l o c k Diagram o f E l e c t r o n i o s  22  4  Bolometer E l e c t r o d e P a t t e r n  25  5  T y p i c a l Recorder T r a c e  29  6  Experimental R e s u l t s  39  7  Deviation Plot  42  1  Plane  15  ACKNOWLEDGEMENTS Dr. M.J.Crooks gave many hours of h i s time f o r thought-provoking discussions and h e l p f u l supervision of my work, f o r which I thank him h e a r t i l y . I am indebted also to Mr. G.Brooks and Mr. R.Weisbach f o r t h e i r t e c h n i c a l assistance, to Mr. C.R.Brown f o r discussions about wave propagation, and to Senor J.BeJar f o r assistance with the computer programming. Dr. G.Ahlers very kindly gave h i s thermal condu c t i v i t y data and provided some insights into the problem of c r i t i c a l phenomena.  1  CHAPTER I . INTRODUCTION AND THEORY  (1) INTRODUCTION In the two decades preceding t h i s one, a slowly Increasing c o l l e c t i o n of experimental data began to indicate remarkable s i m i l a r i t i e s i n physical behaviour near the c r i t i c a l points of some otherwise very d i s s i m i l a r systems.  I t was r e a l i z e d that t r a n s i t i o n s such ass  - the superconducting-normal metal t r a n s i t i o n , - the liquid-vapour c r i t i c a l t r a n s i t i o n of a pure f l u i d , - the order-disorder t r a n s i t i o n of some m e t a l l i c binary alloys, - the misolble-immisoible c r i t l o a l point of some binary l i q u i d mixtures, - the ferromagnetic Curie point and the antiferromagnetic Neel point, and '  - the l i q u i d helium lambda point, a l l behaved i n a q u a l i t a t i v e l y s i m i l a r way.  This r e a l i z a t i o n ,  coupled with a r a p i d l y improving experimental technology, led to a recent stimulation of interest In the t h e o r e t i c a l and experimental i n v e s t i g a t i o n of c r i t i c a l phenomena. C l a s s i c a l c r i t i c a l point theories such as the Landau theory of the second-order phase t r a n s i t i o n , the van der Waals» equation f o r a l i q u i d , and the Weiss molecular f i e l d theory f o r a ferromagnet, were found both theore t i c a l l y and experimentally* to be unsatisfactory very  2  close to the c r i t i c a l temperature • It has been known f o r many years that the temperature dependence of a physical quantity near a c r i t i c a l point may often be described by ( T - f ) f i where T i s the absolute c  temperature, T  0  i s the c r i t i c a l temperature, and e i s  a constant " c r i t i c a l e x p o n e n t P a r t of the recent t h e o r e t i c a l work has been the development of "scaling laws" which attempt to r e l a t e to each other the c r i t i c a l exponents or temperature dependences of d i f f e r e n t physical quantities.  One such theory; applied to l i q u i d helium,  predicts a r e l a t i o n s h i p between the v e l o c i t y of second sound below the lambda point and the thermal d i f f u s i v i t y of the l i q u i d above the lambda point. C l a s s i c a l hydrodynamic arguments, based on the two-fluid model f o r l i q u i d helium, lead to an expression for the velooity of second sound l n terms of the s p e c i f i c heat of the l i q u i d , and the density of i t s superfluld component*  Experimental measurements of t h i s s p e c i f i c  heat and superfluld density had been obtained over a temperature range much c l o s e r to the lambda point than the  range covered by the e x i s t i n g d i r e c t measurements  of the second sound v e l o c i t y . The above considerations a l l Indicated i t would be i n t e r e s t i n g to measure the magnitude and temperature dependence of the v e l o c i t y of second sound as close to the lambda point as possible.  3  (2)  EXPERIMENT  The experiment we performed b a s i c a l l y  consisted  of the measurement of second sound resonances, or thermal s t a n d i n g waves, l n a p a r a l l e l p l a t e r e s o n a t o r ( c a l l e d the c a v i t y ) Immersed l n and c o n t a i n i n g helium I I . The use of resonant a m p l i f i c a t i o n t o d e t e c t and measure second sound i s w e l l known.  In g e n e r a l ,  the c o n d i t i o n s f o r resonance l n a c a v i t y depend on the c a v i t y geometry,  the frequency of the waves gener-  a t e d i n the c a v i t y , and t h e i r v e l o c i t y i n the medium c o n t a i n e d w i t h i n the c a v i t y .  The u s u a l technique f o r  second sound measurements has been t o f i x the  tempera-  t u r e (and t h e r e f o r e the second sound v e l o c i t y ) and v a r y the frequency t o s e a r c h f o r the d i f f e r e n t modes.  resonance  In t h i s experiment, a b a s i c a l l y d i f f e r e n t  proach was used i n t h a t the frequency was the temperature was  ap-  f i x e d and  not.  The temperature i n t h i s experiment was allowed to d r i f t s l o w l y i n time.  As a consequence,  the second  sound v e l o c i t y i n the l i q u i d w i t h i n the c a v i t y was slow f u n c t i o n o f time.  Resonances  a  were t h e r e f o r e ob-  served w i t h the frequency and r e s o n a t o r geometry  fixed,  s e p a r a t e d i n time due t o the time dependence o f the second sound v e l o c i t y l n the l i q u i d h e l i u m .  The exper-  imental d i f f i c u l t i e s a s s o c i a t e d with the p r e c i s e c o n t r o l o f t h e l i q u i d temperature were thus e l i m i n a t e d , o r more a c c u r a t e l y , were reduced t o t h e problems o f p r e e l s e measurement o f temperature.  I t was f e l t t h a t w i t h t h i s  t e c h n i q u e o b s e r v a t i o n s o f t h e second sound v e l o c i t y c o u l d be made much c l o s e r t o t h e lambda p o i n t than 1-1 had been done i n p r e v i o u s experiments. J  (3) THE TWO  FLUID MODEL  Many o f t h e p r o p e r t i e s o f l i q u i d helium below the lambda p o i n t may be d e s c r i b e d i n terms o f a model which assumes two i n t e r p e n e t r a t i n g f l u i d s — ( mass M , n  d e n s i t y f>  n  the normal f l u i d  ) and the s u p e r f l u l d  ( mass  M, s  d e n s l t y y O g ) , which may move Independently o f each o t h e r , and which t o g e t h e r c o n s t i t u t e helium I I , ( mass M = ^ n ^ g  (a)  n The Hydrodynamlc E q u a t i o n s .  A system o f hydro-  dynamic equations f o r helium I I may above m o d e l , a n d  be deduced from the  t h r e e assumptions}  (i)  t h a t the s u p e r f l u l d f r a c t i o n has zero entropy;  (ii)  t h a t below some c r i t i c a l v e l o c i t y the motion  of the s u p e r f l u l d i s l r r o t a t i o n a l ; (iii)  and  t h a t a l l the c o n s e r v a t i o n laws a r e v a l i d .  Assumption ( l i ) i s w r i t t e n curl v  s  «= 0 ,  where v . i s t h e s u p e r f l u l d v e l o c i t y .  We d e f i n e j , the  5 momentum/unit votume o f t h e f l u i d . and normal f l u i d v e l o c i t i e s •  For small  and v , J can be expanded Q  i n powers o f t h e v e l o c i t i e s and t o a f i r s t (neglecting  terms i n v  superfluld  approximation  )  2  ~f _ Ar + Aj . As s y n n J  The  c o n s e r v a t i o n o f mass equation i s  .  ^/>/^t + d i v ~ J " * 0 The  first  ....(2)  equation o f motion, u s i n g t h e c o n s e r v a t i o n  o f momentum law, i s ^/frt  +^ /^x _ « lk  1  0 ,  ...(3)  l.k - 1,2.3, where t h e summation i s c a r r i e d out over the twice subscripts,  repeated  and where Xj_ a r e t h e C a r t e s i a n c o o r d i n a t e s .  T / ^ j . i s t h e momentum f l u x t e n s o r which f o r s m a l l v e l o c i t i e s (neglecting  viscous effects)  i s 7£j_ « p£(l,l-)  +  y°  v n  n  i  v n k  +y^s^slVgi-, where p I s t h e p r e s s u r e , and £(i,-r) i s the Kronecker d e l t a  function.  Again assuming no d l s s i p a t i v e processes we mayw r i t e t h e c o n s e r v a t i o n o f entropy  equation  ^(/>S)/d t + d i v "F = 0 , where F i s t h e entropy u n i t mass o f f l u i d .  f l u x v e c t o r and S i s t h e entropy/  As entropy  normal f l u i d , F = y ° v s  & n  nd  d(/>S)/at + d i v The helium  i s c a r r i e d only by the  (/>Sv ) n  =0.  ...(*0  i n t e r n a l energy o f an i n c r e m e n t a l mass o f l i q u i d  i s g i v e n by  6 dU - TdS - pdV + GdM  ...(5)  where V i s the s p e c i f i c volume and G i s the Gibbs free energy per unit mass. now be increased  Let the mass of t h i s b i t of helium  a t constant volume (hence dV = 0 ) by  the addition of p a r t i c l e s which carry no entropy (hence dS «B 0), that i s by the addition of superfluld.  Then  Eq.(5) becomes dU *= GdM and we see that the p o t e n t i a l energy per unit mass of the superfluld must be G, and that the acceleration of the superfluld must be dv /dt * ^ v / d t + ( v * g r a d ) v = -gradG. s  8  8  g  ...(6)  The t o t a l Gibbs energy NG of the f l u i d i s the sum of the Gibbs energy MG (p,T) of the stationary 0  fluid,  and the k i n e t i c energy of the r e l a t i v e motion of the normal and superfluld  parts. MG - MG  + (Pn/ZMjj) ,  0  where P  n  « ^n^n'^s^  i  *  s  n  ...(7)  momentum of the normal f l u i d  e  with respect to the s u p e r f l u l d .  Recalling that the masses  and densities of the f l u i d f r a c t i o n s are related by Mn-Cn/*)!*  c  (/n//n /s » +  )M  w  e  n  o  w  d i f f e r e n t i a t e Eq.(7)  with respect to M, obtaining *MG/dM « G  - (/^n/Z/^M ) - G 2  Q  Substituting P n ^ ^ C v ^ - T g ) ^ we have  and s u b s t i t u t i n g t h i s into d v / o t + (v «grad)Vg 8  s  Eq.(6)  gives  = -grad[G - (/JQ/2y)) ( ) 0  2  ] .  ...(8)  The term  (Vg'gpadJVg may be r e w r i t t e n ( v • grad )"r - £ g r a d v a  8  2 8  - j ^ x(curlVg Q.  . . . (9)  S u b s t i t u t i o n o f Eq.(9) and E q . ( l ) i n t o Eq.(8) g i v e s grad{G  +  0  +  (Vg2/2) - (/n/2/>) ( v - T s ) } = 0. .(10) 2  n  (2), (3), (4), and (10) c o n s t i t u t e t h e system  Equations  o f hydrodynamic equations f o r helium I I . (b)  L i n e a r i z a t i o n o f t h e Hydrodynamic Equations,  In  d e r i v i n g t h e v e l o c i t i e s o f low amplitude sound; we may assume t h a t v ^ and "v^ a r e s m a l l and thaty°,  S;- p and T  e x h i b i t o n l y s m a l l f l u c t u a t i o n s from t h e i r e q u i l i b r i u m v a l u e s JQ*  S  O» o p  a  n  d T  simplified.  o»  1 1 1 6  -Dove equations may then be  N e g l e c t i n g terms In "v  i  2  n  Eqs.(3) and (10),  and taxingyOS out o f t h e d i v e r g e n c e i n Eq.(4) we o b t a i n t h e f o l l o w i n g system o f l i n e a r i z e d hydrodynamic equations f o r helium I I :  (b/>/bt) + d i v f - 0;  ...(2)  ( d j / d t ) + grad p * 0;  (dy>S)/dt)  +/>Sdiv-v  ( d v / a t ) + grad G g  (c)  Q  n  ...(11) =0;  - 0.  C a l c u l a t i o n o f Sound V e l o c i t i e s .  ...(12) ...(13)  We now make use  o f Eqs.(2) t o (13) t o c a l c u l a t e the v e l o c i t i e s o f wave propagation.  D i f f e r e n t i a t i n g Eq.(2) w i t h r e s p e c t t o  time and s u b s t i t u t i n g E q . ( l l ) we o b t a i n ^^)/91  2  = d i v grad p .  ...(HO  o  8  The i d e n t i t y as/at  (s^nayat)  -  s (l£>)0(/*S)/Zt)  on substitution of Eqs.(2) and (12) gives dS/dt « (S ^^c9)div(v -"? ). y  8  The Gibbs free energy Is G dG  Q  ...(15)  n  « U + pV - TS whence  m -SdT + Vdp * -SdT + (l^o)dp.  Q  Rearranging t h i s and taking the gradient we obtain grad p =yOSgradT +^agradG . 0  Substituting grad p from E q . ( l l ) and gradG from E q . ( 1 3 ) 0  gives y ^ t v ^ - V g J / d t ) +/>SgradT « 0.  ...(16)  D i f f e r e n t i a t i n g E q . ( 1 5 ) with respect to time and substituting Eq.(16) we f i n d  ^/B//^  s2  <^ S/dt « 2  2  d  *  v  T  «  ...(17)  Equations (14) and (17) govern the propagation of waves i n helium I I . Our previous assumption of small variations i n the thermodynamic observables may be written e x p l i c i t l y  S°=/o  +  T as T  /V  ( t )  »  s  =  s  V*** P P  +  e  0  + p 0  v  ( t a ) n d  + T ( t ) wherey^, S , p^, and T are independent  Q  v  of time.  o  Q  Under these conditions we may further write P T  v  v  =  Ov/^/>) fy.  +  s  =  />  a  +  r  (^p/^Sj S , and 0  (^ ^ > T  s  s v  v  -  Equations (14) and (17) are then rewritten c> / /£t 2  v  2  «= Op/^/)) V /> Z  S  T  + Qp/Zs^S^  and  ^%/dt = (/> ^° )S {OT/^) '7 >0 + O T / a s j ^ S ^ . 2 where V i s the Laplacian operator. 2  2  s  n  2  s  v  ...(18) (19)  9 We now wish t o know i f i t I s p o s s i b l e f o r d i s t u r b a n c e s i n the thermodynamic v a r i a b l e s , o f frequency co t o p r o pagate as wares i n t h e l i q u i d w i t h v e l o c i t y u . To determine t h i s we now seek simultaneous s o l u t i o n s o f Eqs. (18) and (19) o f the form f and S « S  Q  =  + S»exp[i<_>(t-x/u)] .  +/)• erp[lco(t-x/u)]  S u b s t i t u t i o n o f these  i n t o the above equations l e a d s t o (dT/d^gteS/fcTjayO* - { ( u / u ) - l } s « = 0 ,  ...(20)  2  2  -[(u/u ) -l)r 0» 2  1  >  + <dp/dS^ (^-/_p) S» = 0 ,  ...(21)  s  where  u | m Op/3/>) , and 3  The c o n d i t i o n f o r t h e simultaneous s o l u b i l i t y o f Eqs. (20) and (21), t h a t t h e determinant o f the c o e f f i c i e n t s be z e r o , g i v e s {(u/u ) -l |{(u/u ) -l} 2  l  1  QT/d/>) Os/m ^p/dS) (^/dp)  =  2  2  -  s  ( C  P- v C  /Q  /)  s  p •  ) / C  where C and C_ a r e t h e constant p r e s s u r e and constant p  volume s p e c i f i c heats o f t h e l i q u i d .  Using the f a c t  Cp -BS. C^, we now s e t (C -C_.)/Cp « 0.  that  With t h i s approximation  p  we then o b t a i n t h e two s o l u t i o n s  If u « U j ,  u -u  x  - [Qp/<^) l* ,  ...(22)  u «u  2  - [TS /> // C :i*.  ... ( 2 3 )  s  2  s  n  p  Eq.(21) shows t h a t S»-»0, and that t o the  f i r s t order In which we a r e working the entropy f l u c t u a t i o n s vanish.  T h i s mode Is a t r a v e l l i n g wave o f d e n s i t y  f l u c t u a t i o n s under a d i a b a t i c c o n d i t i o n s , I.e. o r d i n a r y  10 o r f i r s t sound.  I f u • Ugt Eq.(20) shows t h a t the d e n s i t y  fluctuations vanish.  T h i s i s a t r a v e l l i n g wave of  entropy f l u c t u a t i o n s , and t h e r e f o r e temperature f l u c t u a t i o n s , at  oonstant d e n s i t y .  T h i s i s t h e mode c a l l e d second  sound. (4) (a)  CRITICAL POINT ARGUMENTS  General Discussion.  critical  Recent i n v e s t i g a t i o n s ? o f  p o i n t phenomena have l e d t o t h e r e a l i z a t i o n  t h a t not o n l y do c e r t a i n p h y s i c a l p r o p e r t i e s e x h i b i t r a t h e r simple behaviours i n the v i c i n i t y o f a  critical  p o i n t , but t h a t t h e i r behaviours a r e v e r y s i m i l a r near the c r i t i c a l transitions.  points of apparently  v e r y d i f f e r e n t phase  A r i s i n g from such d i s c o v e r i e s i s the i d e a  t h a t each phase t r a n s i t i o n i s d e s c r i b a b l e i n terms o f an o r d e r parameter p .  ( P a r t i c u l a r examples a r e the  m a g n e t i z a t i o n o f a ferromagnetic  m a t e r i a l , the condensate  wave f u n c t i o n o f a s u p e r f l u l d , and t h e c o n c e n t r a t i o n i n a b i n a r y l i q u i d system). of t h e o r d e r i n g present critical  T h i s parameter i s a measure  i n t h e system, and near the  p o i n t i t may undergo l a r g e f l u c t u a t i o n s with  change l n t h e f r e e energy o f t h e system.  small  The s p e c i a l  behaviours o f the o t h e r p h y s i c a l q u a n t i t i e s i n t h e critical  r e g i o n a r e thought t o be r e l a t e d t o t h i s l a r g e  susceptibility to fluctuations. To d e s c r i b e these f l u c t u a t i o n s , a c o r r e l a t i o n f u n c t i o n o f the g e n e r a l  form  11 C?(r) - < [ ? ( r \ ) r - r  t  -  t  <J(?i)>3[;(r) 2  - <?(r )>]> . 2  2  i s used, r e l a t i n g the deviations of the order parameter at point r ^ from i t s expectation deviations a t point r ^ .  value there to the  In general, t h i s c o r r e l a t i o n  function i s a monotonically decreasing function of 1% As such, c r i t e r i a may be set up to define a p a r t i c u l a r value of |"r| c a l l e d the c o r r e l a t i o n length, or range £ of the c o r r e l a t i o n function.  This c o r r e l a t i o n length ^ i s  generally a function of temperature, and divergent  near  the c r i t i c a l point. As noted above, many quantities depend on £ as e — where e i s some exponent ( c a l l e d the o r l t i o a l exponent) which i s f i x e d i f the sign of 6 i s f i x e d , and € » ( T - T ) / T . 0  c  Table I l i s t s some of the physical quantities pertinent to t h i s experiment, t h e i r expected behaviours, and the conventional notation f o r t h e i r c r i t i c a l exponents. Recently l t has been suggested®"  12  that the c r i t i c a l  indices are not independent of each other.  Through  the use of p l a u s i b i l i t y arguments and assumptions about the functional forms of thermodynamic observables i n the c r i t i c a l region, c e r t a i n relationships c a l l e d s c a l i n g laws have been proposed between various of the c r i t i c a l exponents.  Some of these relationships are subject to  experimental v e r i f i c a t i o n .  I t should be noted that  none of the s c a l i n g laws predict the magnitude of any c r i t i c a l exponent.  What they do predict i s the relationship  12  TABLE I  Physical Quantities near T  Physical  Critical  Expected  Quantity  Exponent  Behaviour  e  <*P> - o,  Order parameter  •  < P > ~ ilel^.^o  P  Correlation function  V  C-lr^l-^2-7 for a d dimensional  d(r)  —>  oo  fi.  >  —-.  r = r -r x  system  2  Correlation length.  V  J ~ € ,e >o  or range of C.  V'  f ~ l € r v ' , e<o  f Superfluid  >*•  density  yOg-lef^.  e<o  13 of one c r i t i c a l exponent to another, usually l n the form of a sum  involving two  or more c r i t i c a l exponents*  Correlation functions  may  be defined also f o r  operators or physical quantities a other than the order parameter*  Halperin and Hohenberg  arguments may  11  point out that s c a l i n g  then be subdivided into s t a t i c or dynamic  s c a l i n g depending on whether one assumes time independence or dependence of the c o r r e l a t i o n function, ( ^ ( r )  or  C ^ ( r , t ) ) , and dynamic s c a l i n g into r e s t r i c t e d or extended dynamic s c a l i n g depending on whether or not only  the  order parameter c o r r e l a t i o n function i s expeeted to obey the dynamic s c a l i n g laws. (b)  A S p e c i f i c Prediction concerning Seeond Sound.  In  a recent p a p e r , Halperin and Hohenberg propose a 11  dynamic s c a l i n g hypothesis which leads to a s p e c i f i c p r e d i c t i o n that may  be tested l n t h i s experiment.  They  note that a dynamic c o r r e l a t i o n function C*(r,t) f o r some operator a, may and may  be Fourier transformed to c|(k,6>)  i n general be written i n the form  C*(k,u) « Z T r - c J ^ - f ^ / ^ C k ) ) * ^ * ^ ) ] - , 1  where +00  f  loo  f (x)dx  «= 1 ,  * _  and where the c h a r a c t e r i s t i c frequency u> (k) by the  i s determined  constraint +1 /  '-1  f(x)dx = I  .  A mental p i c t u r e of the s c a l i n g hypothesis  may  Ik  be obtained a s f o l l o w s . (k,jJ~*) plane*  In P i g . 1 we r e p r e s e n t t h e  (where k i s t h e wave number and |  the i n v e r s e c o r r e l a t i o n l e n g t h ) .  is  Three r e g i o n s may be  d e f i n e d i n t h i s planet - Region I .  k £ « l , T<T .  The macroscopic  C  critical  r e g i o n , o r hydrodynamic c r i t i c a l r e g i o n , below t h e c r i t i c a l temperature.  In t h i s r e g i o n phenomena occur  o v e r d i s t a n c e s r l a r g e compared w i t h  Hydrodynamic  arguments a r e expected t o be v a l i d i n t h i s and r e g i o n I I I . - Region I I .  k£»l.  The m l c r o s c o p i o  critical  region  i n which phenomena o c c u r over d i s t a n c e s r s m a l l compared w i t h | . - Region I I I .  k | « l , T>T  e#  r e g i o n above t h e c r i t i c a l r  The macroscopic  temperature.  The c r i t i c a l p o i n t i s t h e l i n e  = 0.  The dynamic s c a l i n g h y p o t h e s i s assumption t h a t c£  critical  has as i t s b a s i c  ( r e s t r i c t e d s c a l i n g ) , o r Cf (extended  s c a l i n g ) v a r y smoothly i n t h e ( k , ^ " ) p l a n e except a t 1  the o r i g i n , and t h a t e i t h e r o f these c o r r e l a t i o n f u n c t i o n s i s e s s e n t i a l l y s p e c i f i e d by i t s l i m i t i n g behaviour i n regions  I , I I , and I I I .  forms o f t h e f u n c t i o n  That i s t o say, I f t h e asymptotic  In regions  I ( o r I I I ) and I I a r e  s e p a r a t e l y e x t r a p o l a t e d t o t h e l i n e L j ( o r L^) i n F i g . 1, d e f i n e d by k£«l, T<T ( o r T>T ) the two r e s u l t i n g expressions J O c must agree t o w i t h i n a f a c t o r o f o r d e r u n i t y . The asymptotic forms C* , C * - C * 1  1 1  1 1 1  a r e d e f i n e d as  15  FIGURE 1 .  The ( k, p  1  ) Plane  16 the l i m i t i n g forms of Cj(k) under the conditions: C*  c  1  : J f i x e d , k-* 0, T<T ; C  a I I I .^  f l x e d >  w  0 f  T > T o  .  The s c a l i n g hypothesis then takes the form of the matching conditions C* ^) 1  =oCC  aII  (k),  C * ( k ) -oc'c* ^), m  11  for k = f for k  _ 1  -p . 1  whence C  a I I I  ( k ) * AC^^k),  f o r k ^ p  ...(24)  1  where oc,oc», A are constants of order unity.  Halperin  and Hohenberg also propose that s i m i l a r expressions may be assumed f o r the c h a r a c t e r i s t i c frequencies o u>  aII  (k), c j  a I  (k),  ( k ) i n the three regions.  a I I I  We now consider a p a r t i c u l a r c r i t i c a l t r a n s i t i o n , the lambda t r a n s i t i o n of l i q u i d helium. temperature T T  A  Q  The  critical  f o r t h i s t r a n s i t i o n i s the lambda temperature  = 2.172 K, and C «= (T-TjJ/T*. The s p e c i f i c p r e d i c t i o n we w i l l test concerning  * l i q u i d helium evolves from using as operator a the heat operator q ( r , t ) = E ( r , t ) + (<E^>;/)(r".t) )/</>> where E(r,t) i s the energy density of the l i q u i d . Halperin and Hohenberg  11  assume that the heat operator  c o r r e l a t i o n function c|(k,<o) s h a l l he dominated i n region I ( £ < 0 ) by second sound, and i n region III ( £ > 0 ) by thermal d i f f u s i o n , and that the asymptotic forms of the  17 c h a r a c t e r i s t i c frequencies i n these two hydrodynamic regions may  then be derived and are given by  ^ (t)  ...(25)  , (€>0),  ...(26)  2  coqHI(k) = D k  2  t  where D  (6<0)r  «= u k .  T  = fc^0C i s the thermal d i f f u s i v i t y of the l i q u i d , p  t  and A i s the thermal conductivity.  The extended dynamic  soaling p r e d i c t i o n i s then, using Eqs.  (24), (25),  (26), the matching condition f o r k «=j*~  and  (see l i n e s  1  k£ « 1 i n Pig. 1) that D J" (€>0) - Au^" (fe<0). 2  ...(27)  1  t  The two temperature dependent c o r r e l a t i o n lengths J may  be written (see Table I) | ( 0  ^*^f—-e-|"" '* where  £  >  Q  = ^Jt"  are constants.  u  and f (-6)  Equation  _  (27) i s  then rewritten u (-6) * {K[e )//>^))'l^l(^)" y e \-e\^\ i  Now  x  ...(28)  zv  :  2  making use of a s t a t i c s c a l i n g l a w  1 2  and the s t a t i c s c a l i n g a s s e r t i o n 3 thatyO 1  a  to J* ~  1  i s proportional  we see that  /.~l-€|£~f- ~ l-e| ' . l  and  v  therefore  f = y* = y . Experimental measurements ^"* ^ have shown 1  £  «= 0.666 * .006  = 2/3.  1  Thus we obtain f o r the exponent  of € i n Eq.(28)  2V - y» = 2/3  18 and Eq.(28) becomes U (-€) = 2  (^(6)^C (6)).[A^(^)-l]-l 2/3  T h i s r e s u l t was  p  t  #  f i r s t d e r i v e d by A h l e r s .  19  CHAPTER I I . APPARATUS (1)  CRYOGENIC APPARATUS  The cryogenic apparatus; shown l n Pig* 2, consisted of a large outer bath f o r environmental s t a b i l i t y , and a small inner bath containing the seoond sound cavity. The 3 l i t e r outer bath could be temperature regulated to a short term accuracy of about 10 diaphragm r e g u l a t o r *  18  K using a Walker  The 0*33 l i t e r inner bath was  separated from the outer bath on the sides and bottom by a vacuum jacket, and on the top by a 3/8" s t a i n l e s s steel plate.  This moderate thermal l i n k gave the inner  bath a thermal response time with respect to the outer bath of about one hour*  The inner bath was f i l l e d from  the outer bath through a porous s t a i n l e s s s t e e l f i l t e r (to prevent s o l i d nitrogen p a r t i c l e s from entering) and a small s t a i n l e s s s t e e l needle valve*  The inner bath  could be pumped v i a a pumping l i n e (containing also the e l e c t r i c a l leads) terminating i n a s t a i n l e s s s t e e l plug bored with a 1 mm diameter hole.  E l e c t r i c a l leads  were brought through the s t e e l plug by s e a l i n g varnish insulated copper wires (AWG #37) into short lengths of s t a i n l e s s s t e e l c a p i l l a r y (1/16" outer diameter, 0.007" inner diameter) using A r a l d i t e ^ epoxy. 1  The wires were  thus e l e c t r i c a l l y insulated from the s t e e l but the holes i n the c a p i l l a r i e s were sealed. These cased  20  VALVE  STEM  PUMPING  LINE  FILTER s  s \ \  INNER  BATH  VACUUM SPACE  PIGUBE 2. erator,  Cryogenic Apparatus.  OUTER BATH  A = second sound  B = r e s i s t a n c e thermometer,  D = Perspex s p a c e r s .  gen-  C = bolometer,  21  wires were sealed with epoxj into 1/16" the plug.  holes through  Film flow heat transfer between the two baths  occurred only through the 1 mm hole; and was therefore small.  The second sound cavity, near the bottom of the  Inner bath, consisted of a 2.5 cm square bolometer separated by 2 mm Perspex spacers from a 2.5 cm square second sound generator (heater).  The cavity was oriented  h o r i z o n t a l l y ( i . e . second sound propagated v e r t i c a l l y ) to reduce the g r a v i t a t i o n a l l y induoed temperature range of the lambda t r a n s i t i o n was open on two sides.  2 0  to 2.5 x 10"?K.  The cavity  The thermometer, i n the center  of the c a v i t y / was a carbon r e s i s t o r . (2)  ELECTRONICS  A block diagram of the electronics i s shown i n F i g . 3.  An o s c i l l a t o r supplied  a s i g n a l of frequency  f/2 to the generator, at which the Joule heating produced a second sound plane wave of frequency f ( t y p i c a l l y 200Hz to 5KHz) i n the l i q u i d .  The same f/2 signal,  supplied to a frequency doubler, emerged as a signal necessarily coherent with the second sound, and was used as the reference s i g n a l f o r coherent amplifier A (Fig. 3).  x  Thermally induced resistance changes i n  the bolometer," biased with a constant DC current (ranging from 10 to 30/tA)/appeared  as voltage changes a-  cross the bolometer load r e s i s t o r which were a m p l i f i e d  2 2  22  P H A S E AND TEMPERATURE C H A R T RECORDERS  SIGNAL OSCILLATOR  I  I  FREQUENCY DOUBLER  COHERENT AMPLIFIER A  COHERENT AMPLIFIER B  FREQUENCY COUNTER  PRE-AMP  PRE-AMP  I BOLOMETER  I RESISTANCE  B I A S . LOAD  H E IE  GENERATOR  BOLOMETER  THERMOMETER  FIGURE 3.  B l o c k Diagram o f E l e c t r o n i c s .  I  BRIDGE  23  and fed to coherent a m p l i f i e r A.  The output from co-  herent a m p l i f i e r A was fed to one channel of a two channel chart recorder. 3 2  was  This "second sound trace"  proportional to the produot of the second sound  amplitude i n the cavity and the cosine of the phase of the received second sound with respect to the reference signal. Coherent a m p l i f i e r ^ B generated i t s own reference 2  s i g n a l which was also fed to a Wheatstone bridge, one arm  of which was the resistance thermometer.  The bridge  unbalance s i g n a l was amplified -* and fed to coherent 2  a m p l i f i e r B. the output of which went to the other channel of the chart recorder. was  This "temperature trace"  proportional to the resistance difference between  the thermometer and the preset value of a p r e c i s i o n resistance decade. (3) (a)  SECOND SOUND CAVITY  Bolometer.  Some desirable properties of a bolo-  meter f o r second sound detection are: (1) (ii)  high thermal s e n s i t i v i t y (1/H)(dB/dT); low e l e c t r i c a l resistance B to minimize the  problems of impedance matching to transmission  cables  leading out of the cryostat; (ill)  small heat capacity to enable i t to respond  to very rapid temperature fluctuations; and (iv)  large a c t i v e area.  24 The carbon f i l m bolometers constructed by Cannon and Chester ^ exhibit properties (1) and ( i l l ) .  Sig-  2  n i f i c a n t improvement over t h e i r design was achieved using phot©fabrication techniques rather than t h i n f i l m evaporation techniques to prepare the bolometer  elect-  rodes • The electrode pattern P i g . > was photographically reduoed onto Kodak Ortho Type I I I f i l m .  The scale i n  Pig. 4 represents the f i n a l s i z e of the bolometer. two inch square of 1/16"  A  thiok f i b e r g l a s s was bonded  with A r a l d l t e ^ epoxy to a 0.001" thick layer of brass. 1  A f t e r cleaning with organio solvents i n an ultrasonic cleaner, the brass was spray ooated with a t h i n (less than 10~* inch) l a y e r of Kodak KPH p h o t o r e s i s t . ? 2  The  photographio negative was contact printed onto the ooated brass using u l t r a v i o l e t l i g h t , and the r e s u l t i n g latent image developed.  A f t e r development/ the photoresist  has the property that a l l parts which were exposed to the l i g h t become Insoluble l n most a c i d s / while the unexposed portions dissolve i n the developer.  The  plate was then etohed to remove a l l the brass which was unprotected by developed photoresist. ing photoresist was  The  remain-  then removed, leaving the pattern  of P i g . 4 i n 0.001" brass bonded to the fiberglass substrate.  25  26 The bolometer was completed by spraying the prepared electrodes with a suspension of nominally 16 milli-mioron carbon p a r t i c l e s  2 8  i n xylene.  The xylene  evaporated leaving an extremely t h i n layer of semiconducting carbon covering the electrodes.  Reference to  F i g . 4 w i l l show that the bolometer therefore consisted of 64 t h i n f i l m carbon r e s i s t o r s wired i n p a r a l l e l , each approximately 2*5  om wide and 0.02  cm long.  Con-  s i d e r i n g the f i l m to be a homogeneous slab of bulk graphite; and using Ohm's law and the measured room temperature resistances* a thickness of 0.1 i s indicated f o r the f i l m .  milli-microns  As the carbon granules had  diameters of the order of 16 milll-mlcrons/ one can conolude that the f i l m was microscopically inhomogeneous and roughly the thickness of one carbon granule.  This  bolometer design achieved the desired properties of low thermal capacity without excessive e l e c t r i c a l r e sistance. The bolometer resistance and s e n s i t i v i t y while operating at the lambda point were ~70KXL and  ~3.4  K"  respectively; giving the second sound detection system a maximum s e n s i t i v i t y of 3 x 10* K rms/chart inch. 8  The maximum temperature wave amplitude observed i n the experiment was 10"*?K rms.  Second sound noise (real and  1  27 apparent) was less than 3 x 10"^K n s ,  The detection  system bandwidth was t y p i c a l l y 0•25 He• (b)  Thermometer-  An Allen-Bradley ^ 1/10 watt carbon 2  r e s i s t o r ; nominally 33 ohms a t room temperature? was used as the thermometer.  I t s resistance and s e n s i t i v i t y  at the lambda point were  ~ 120011 and (1/B)(dB/dT)  A  ~ 1.2 K*"^ giving the thermometry system a maximum sensi t i v i t y of ~10"°" K/chart inch.  Low frequency thermal  noise (real and apparent) was about 10~^K peak to peak. The thermal response time of the thermometer was measured i n l i q u i d helium and found to be 22 msec at 4.2K and 5 mseo a t 2•OK. (o)  Second Sound Generator.  The second sound gener-  ator was a pleoe of commercial carbon resistance board^ nominally 25 ohms/square at room temperature. I t s resistance a t l i q u i d helium temperatures was about 50 ohms/square.  0  28  (1)  SECOND SOUND TRIALS  Data were acquired l n the following manner. (Reference w i l l be made to P i g . 5, a t y p i c a l section of the output from the two channel chart recorder). (1)  With the inner-outer bath valve closed and  the outer bath s t a b i l i z e d a t a temperature a few m l l l l degrees below T , the inner bath temperature was held A  steady a t AT ~ 1 0 "  2  t o 5 x I O - ^ K , where  AT « T -T. A  The frequeney and coherent a m p l i f i e r pass-band were set  t o optimize seoond sound detection, and the sens-  itivity  (ohms/chart d i v i s i o n ) of the thermometry system  was measured by making d i s c r e t e changes l n the zero s e t t i n g of the Wheatstone bridge and recording i t s output. (ii)  With both recording systems and the second  sound generator and bolometer operating; pumping of the inner bath was terminated o r reduced allowing the temperature to climb slowly up to and through the lambda point, (see P i g . 5 l i n e s #2 and #3), the Inner bath dT/dt being t y p i c a l l y of the order of 1 0 " K/sec. 7  The  second sound v e l o c i t y and the temperature were thus functions of time.  When v e l o c i t i e s occurred such that;  29  FIGURE 5.  T y p i c a l Recorder Trace  30 f o r the p a r t i c u l a r frequency i n use; the cavity was resonant, peaks were recorded on the second sound trace, (trace #2 In P i g . 5 ) . (Ill)  The Inner bath temperature was Immediately  brought s l i g h t l y below the lambda point, the seoond sound generator and bolometer were turned o f f , and the temperature was again allowed to r i s e through the lambda temperature (see traces #1 and #4 i n P i g . 5) to establish Rpt, the thermometer resistance at the lambda point. Step ( i l l ) was necessary because the accurate i d e n t i f i c a t i o n of the lambda point i s dependent on knowing the t o t a l power d i s s i p a t i o n (P) In the c a v i t y . ^ 1  The combined power input to the cavity from the second sound generator; the bolometer, and the thermometer -varied between 25 and 300 microwatts. power was 0.1 to 1.0 microwatts.  The thermometer  The generator power  was always twice the bolometer power so that the DC power inputs (as opposed to the power input at the second sound frequency) of the two were equal, and so that the maximum second sound output signal f o r a given t o t a l power input was obtained.  With only the thermometer  on, the warming curve (the "temperature trace") showed a zero slope region, or B i n P i g . 5).  x  plateau (traces #1 and #4  With the thermometer, generator, and b o l -  ometer on, the warming curve broke (point A Pig. 5 )  31 at a temperature AT*(P) s l i g h t l y below the true lambda temperature and took up a much higher slope dfi/dt. AT'(P) was measured and found to be approximately proportional to the t o t a l power input P to the cavity. I t was assumed that the plateau temperature was the true, or zero power; lambda temperature.  I f , however, the  power dependent s h i f t i n the apparent lambda temperature was s t i l l i n e f f e c t during measurements made with the thermometer only, i t was i = £ T « ( 1 0 " 16,20 In previous work,  6  watt) = 3 x 10" K. 8  a time dependent s h i f t of the  lambda point resistance R  A  has been observed.  The e f -  f e c t i s apparent i n the non-zero slope of the R  A  plat-  eaux and of the dashed l i n e l n F i g . 5* The thermometer was c a l i b r a t e d by opening f u l l y the valve between the two baths and measuring thermometer resistance and outer bath vapour pressure f o r a number of temperatures s l i g h t l y below T . A  The data were f i t t e d  to the expression l o g R = A + B/T which was taken to be exact over the c r i t i c a l region. Data were extracted from the recorder traces by noting the thermometer resistance R(t) a t the time of occurrence of a peak i n the second sound trace corresponding to resonance mode n.  The experimental second  sound v e l o c i t y was then u = 2fd/n, 2 e  ...(30)  32 (where the subscript e denotes the experimental value f o r the v e l o c i t y , and d i s the cavity spacing, i . e . the thickness of the Perspex spacers), and the corresponding temperature difference AT was a function of R (t)-B(t). A  (2)  PHASE SHIFT AND DISPERSION TRIAL  One run was performed at a fixed AT ~  5 x  10**  to determine (a) whether the generator and bolometer were introducing any detectable phase s h i f t s i n the second sound s i g n a l , (b) whether the cavity had any observable resonance modes other than the a x i a l modes, and (o) whether any dispersion was observable. was made from 20 Hz to 5 KHz  A search  (covering the range of  frequencies used i n the experiment) f o r the second sound resonances. , Resonances corresponding to the f i f t e e n a x i a l modes up to t h i s frequency were observed;* — no others.  and  The extrapolated zero frequency phase s h i f t  was 0° + 15°t  and no departure from l i n e a r i t y i n the  frequency-phase s h i f t curve was observed.  We concluded  that (a) at frequencies up to 5 KHz neither the generator nor the bolometer were near t h e i r upper frequency response l i m i t s , and (b) no dispersion was detectable.  33 CHAPTER IV. (I)  DATA REDUCTION  EXTRACTION OP DATA PROM THB CHARTS.  Line #2 i n P i g . 5» the seoond sound trace, shows the peaks which ooourred as the cavity resonated, contains the v e l o c i t y information.  Lines #1,  #4 contain the temperature information.  and  #3,  and  Data were  extracted from the chart as follows. (i)  The zero-power lambda l i n e  R\ ( t ) ; the dashed  l i n e i n P i g . 5, was drawn between the R  A  plateaux of  two successive traces of the zero power warming curve (#1 and #4 l n P i g . (II)  5).  The r e l a t i v e peak numbers n', n*+  were assigned. resonance.  1,  The integer n i s the mode number of the  I t i s the number of h a l f wavelengths of  seoond sound i n the c a v i t y . (iii)  The distances ^ i ,  & t+ i* n  •  (see Pig. 5)  representing B ( t ) - R(t) at the times of occurrence A  the peaks n', n'+ 1,  were measured, and  of  converted  to A T t , A T t + i * using the previously measured sensn  n  i t i v i t y (ohms/chart d i v i s i o n ) of the thermometry system and the previously established thermometer c a l i b r a t i o n R(T). (Iv)  The distance of point A (Pig. 5) below the  zero-power lambda l i n e was measured and oonverted  to  AT«(P), P being the t o t a l power used during the recording  3* of the trace being examined.3 (v) was  1  The absolute peak number n of the f i r s t peak  established. To do t h i s , the second sound v e l o c i t y Equation (30)  had to be known roughly.  i  f l  rewritten  ...(3D  n •. 2 f d / u ( A T ) . 2  Taking the measured A T , n  f o r the f i r s t peak on the chart, inserted Into Eq.  a rough value of U2(AT ») was a  g i v i n g n m r e a l number.  (31)  This r e a l number, say 3.04*7*  would be within a few percent of some integer. absolute peak number n was  The  then taken to be that i n t e g e r .  It should be pointed out that the rough value of U 2 ( - T i ) used to e s t a b l i s h n must be erroneous by n  at  l e a s t Z$% before an error oould be inourred i n es-  t a b l i s h i n g n by t h i s method.  Not only must n be an  integer? but i t s p a r i t y i s known from the d i r e c t i o n of deviation of the peak (see P i g . 5 ) of  the chart.  from the center l i n e  Thus, i f the correct n were to be i n -  c o r r e c t l y i d e n t i f i e d as rf, the error i n the value of ugC&Tgt) used to e s t a b l i s h T/L would have had to be a t l e a s t * 2/(  n*2  )•  In t h i s experiment the highest value of  n f o r the f i r s t peak on a chart was n = 6 giving 2$% as the minimum error i n U2(AT ,) necessary to produce n  an incorreot i d e n t i f i c a t i o n of n.  As the experimental  r e s u l t s w i l l show a maximum deviation of the measured v e l o c i t y from theory of about 10#, we conclude that no erroneous i d e n t i f i c a t i o n s of n were made.  35 (vi)  The absolute mode numbers of a l l peaks on  the chart having been established v i a (11) and (•) the corresponding experimental v e l o c i t i e s u using E q . ( 3 0 ) .  2 e  were calculated  Por each peak recorded on the charts  therefore? a data point (u # AT) has been established 2e  and the raw data have been extracted from the charts. In the course of the experiment 6 3 charts were measured, producing a t o t a l of 276 data points. (2)  ANALYSIS OP PROBABLE ERRORS  (i)  Errors l n u  2 e  . Error l n the absolute v e l o c i t y  can a r i s e from the measurements of f and d (Eq. (30)  ).  As noted above/ the error i n establishing n i s presumed to be zero.  The cavity spacing d was determined by  measuring the thickness of the Perspex spaoers several times with a micrometer and looking a t the mean and standard error of the measurements.  To obtain the value  of d a t helium temperatures; the t o t a l thermal contraction from room temperature to helium temperature of 1.13/S^  2  was subtracted giving the low temperature value of d = 1.913 x 10~ M * 0,6%, 3  The frequency f was measured to * 1Hz using a d i g i t a l counter. \00%/t  The uncertainty i n f was therefore  which was less than 0,5%, The t o t a l probable uncertainty i n the absolute  value of the measured v e l o c i t y was therefore taken to  36  be of the order of !%• (11)  Errors l a AT. The experimental value of AT  f o r any point i may  be written.  ATi « (B )(A ) CfOH)] , 1  where  1  i s the thermometry system s e n s i t i v i t y (ohms/  chart d i v i s i o n ) f o r the trace containing point 1, A^ i s the measurement of the temperature trace (chart d i v i s i o n s ) , and f ( A ) i s the resistance thermometer c a l i b 1  r a t i o n f i g u r e i n ( K/ohm ). The r e p r o d u c i b i l i t y of the measurements of B^ depended on the (variable) s e n s i t i v i t y of the system. The probable error i n B  1#  based on repeated measurements  at a f i x e d temperature, ranged from n e g l i g i b l e to about 1%, *  We therefore take the probable error i n B^ to be \%.  The resistance thermometer c a l i b r a t i o n figure f (A^) was very nearly a constant  equal to (dT/dR)^,  the inverse slope of the thermometer resistance vs. temperature c a l i b r a t i o n curve at the lambda point. As such, i t s accuracy was  determined by the accuracy  of the c a l i b r a t i o n curve, and i t s probable error Is estimated as ± 1%, A^ was  obtained by i d e n t i f y i n g two points on the  chart and measuring the distance between them. probable error involved f o r each point was  The  arbitrarily  37 taken to be *0.03 ehart d i v i s i o n s (one inch d i v i s i o n s ) and the t o t a l uncertainty l n A^ was therefore «*0.06 ohart d i v i s i o n s giving a t o t a l f r a c t i o n a l error i n A * (  chart d i v i s i o n s ) %,  i n the experiment  The values of  4  of measured  ranged from 0*3 to 9*0 chart d i v i s i o n s  g i v i n g f r a c t i o n a l errors of from 20% to 0*7% with an average of about 2%, The average probable error i n the absolute values of the experimental data i s therefore taken to be 1% i n the v e l o c i t y u ^ AT.  D  e  and 3% i n the temperature  difference  38 CHAPTER V. (1)  RESULTS AND ANALYSIS.  RESULTS  The experimental r e s u l t s a r e shown I n P i g . 6. points  (u » AT), 2 e  The  (The s u b s c r i p t e d e n o t i n g t h e experimental  v a l u e o f t h e v e l o c i t y ) , cover t h e range AT «= 4.75 x 10"^ K to  1.25 x 1 0 " K• 2  The temperature  d i f f e r e n c e s AT and  the measured v e l o c i t i e s u„ a r e l i s t e d i n t h e f i r s t two ze columns o f Appendix I i n o r d e r o f i n c r e a s i n g AT. The d a t a o v e r l a p t h e p r e v i o u s measurements o f Peshkov  1  ( A T > 1 0 * 3 K ) , o f Pearce, L i p a , and Buckingham (PLB) 2  ( A T > 2 x 10"** K ), and o f Tyson and D o u g l a s s 9 x 10"^ K ) •  I n t h e ranges  3  ( AT >  o f o v e r l a p , o u r data a r e i n  e x c e l l e n t agreement w i t h those o f Peshkov and o f PLB, i n both cases w e l l w i t h i n the experimental u n c e r t a i n t i e s . In  t h e range A T «= 10*"**Kto 1 0 " K t h e Tyson and Douglass 2  measurements g i v e a u to  which i s 5# t o 7% low compared  2  o t h e r experimental d a t a , and t o t h e p r e d i c t i o n s o f  hydrodynamics. (2)  COMPARISON WITH THE HYDRODYNAMIC THEORY.  The e x p r e s s i o n f o r t h e v e l o c i t y of second U  was  2h  = (T S  ^Wp ^  sound,  •'•  (32)  d e r i v e d above u s i n g hydrodynamic arguments.  We have  here used t h e s u b s c r i p t h to denote t h i s hydrodynamic velocity. to  put Eq. (32)  macroscopic of  Two p i e c e s o f experimental work enable one i n an a n a l y t i c form useable i n the  critical  r e g i o n below T^.  the s u p e r f l u l d d e n s i t y y O  Q  14-16 Measurementa ^ x  l e a d t o the e m p i r i c a l r e l a t i o n  39  FIGURE 6.  Experimental R e s u l t s .  40 /*//O  a  ...(33)  « 0.699AT"^- 1 ,  where j> = 0.666 * X%, based on experimental data AT 2s 6 x 10"^ K . (BPK)  Buckingham, Palrbank, and K e l l e r s  measurements  by t h e e m p i r i c a l C  with 1  o f t h e s p e c i f i c heat a r e d e s c r i b e d  33  relation  = 4.55 - 3.00 l o g | A T | - 5.202 ( J / g - K ) ,  p  1 Q  where £ • 0 f o r AT > 0 , S « l experimental data with The  ...(34)  f o r AT < 0, based on  I ATI > 10"*^ K .  entropy S o f t h e l i q u i d may be w r i t t e n S(T)  = S(T )  +/  X  (C /T)dT. p  \ Making a change o f v a r i a b l e t o AT, and w r i t i n g Eq; (34) as C ( A T ) ss A + B i nAT ( A T > 0 ) we o b t a i n p  S(AT)  « S(T-)  AT* AT., + A? d(AT) + B / i n AT d(AT). & (T -AT) AT (T -AT) A  where AT^ • T ^ - T^. Eq*  Expanding t h e denominators o f  (35) I n powers o f A T / T  order  (AT/T )  ...(35)  A  A  and n e g l e c t i n g terms o f  obtain  A  ( T -AT r  1  A  3?  ...(36)  ( 1 / T ) ( 1 + (AT/T ) ) , A  A  w i t h a maximum f r a c t i o n a l e r r o r o f 0,1% i f T^ = 2.10 K• S u b s t i t u t i n g Eq. (36) Into Eq. (35), performing the I n t e g r a t i o n s , and c o l l e c t i n g terms we o b t a i n  finally  ...(37)  S(AT) m S ( T ) X  Aln(T -AT ) + B ^ A T j / T ^ d n ^ - l ) + i ( A T / T ) ( l n A T - i ) l 2  A  1  A l n ( T - A T ) - B((AT A  We s e t ^  1  /T )(lnAT A  x  1  -1) + £(AT / T ) ( l n d T - % ) \ . 2  A  = 2.10 K , and o b t a i n S f T j ) from t h e data o f  34 H i l l and Lounasmaa^ u s i n g t h e i r  value  +  41  S( 2.10 K ) • 1.24 ( J / g - K ) .  ...(38)  Equation (38) and Eq. (37) with AT « 0 give the entropy a t the lambda point S(T ) * 1.55 ( J / g - K ) . X  Substitution of Eqs. (33). (34). (37)# and (38) Into Eq. (32) gives an a n a l y t i c form f o r the hydrodynamic v e l o c i t y of second sound dependent f o r i t s accuracy on the v a l i d i t y of the hydrodynamic theory and the extrapolation of Eq. (33) from AT • 6 x 1 0 ' K to 1 0 ~ 5  6  K.  To compare the experimental data with the hydrodynamic theory, a computer was used t o evaluate the above mentioned a n a l y t i c form of u  using the AT from each of the  2 n  experimental points ( u. . A T ) , and to calculate the ze ratio Rj^AT) £ u ( A T ) / u ( A T ) , 2e  f o r each AT.  2h  A d e v i a t i o n p l o t of logfR^) against AT  i s shown i n P i g . 7. and the corresponding values of u ( A T ) are tabulated i n the t h i r d column of Appendix I. 2n  and i t s 95^ confidence l i m i t s are  The mean value of  *1 " ^ 2 e 2 h \ v * u  / u  1  ,  0  0  ° *  # 0 0 l f  •••(39)  *  In obtaining the above r e s u l t we have extrapolated Eq. (33).  As a check on the self-consistency of t h i s  extrapolation, a weighted l e a s t squares f i t ^  of the  data t o the form of Eq. (32) was performed to predict the values of Q and £ l n the expression u  - ( T S / C ) * » ( QAT-^ - 1 ) " * . 2  2 e  p  ...(40)  42  CM  O  10  O  o  p -  —  .X. ~ „•  o  o •  ^o•o «  l  O  CP  <f-  o °  o  °  o° u  in o d  o d  O  i  004  ti  FIGURE ?.  8  O  m o d 901  I  D e v i a t i o n p l o t comparing the d i r e c t  and the c a l c u l a t i o n s  from thermal c o n d u c t i v i t y  to the hydrodynamic v e l o c i t y . open c i r c l e s = Rg = 2 s ^ 2 h * u  U  Dots = Rj, = u  2 e  measurements, and  /u  2 h  scaling »  43 We obtained Q * O.67 * .02 and j? * +.671 * .004 ( 95% confidence l i m i t s ) i n agreement with the more precise values given above i n Eq. ( 3 3 ) . (3)  COMPARISON WITH THE SCALING PREDICTION.  Recent experimental measurements by A h l e r s ? * ^ of 1  3  the thermal conductivity of helium I near the lambda point, combined with our seoond sound v e l o c i t y measurements i n helium I I , make p o s s i b l e a cheek of the extended dynamic s c a l i n g p r e d i c t i o n derived above. Equation ( 2 9 ) may be rewritten K(AT-) *  u (AT)/>C (AT-){A52(^)-l]| T-r , 2/3  2  p  where AT- indicates T -T i s negative. A  u ( A T ) i n Eq. (41)  Substituting f o r  the hydrodynamic v e l o c i t y  2  ,..(4l)  A  u (AT), 2h  Ahlers performed ? a l e a s t squares f i t of the r e s u l t i n g 1  equation to h i s thermal conductivity measurements, obtaining f o r the constant Aj ^)" 2  1  « ( 0.87 * .06 ) x l O  - 8  om.  ...(42)  In performing t h i s f i t Ahlers used f o r C (AT-) and p  Cp(AT),  ( contained l n u  ) , his own measurements of p » " ^ C  2 n  As the AT dpendences of Ahlers s p e c i f i c heat and of the BPK s p e c i f i c heat are s l i g h t l y d i f f e r e n t , h i s data are not d i r e c t l y comparable with the hydrodynamic v e l o c i t i e s we have calculated.  We have therefore repeated the  calculations using the BPK s p e c i f i c heat and Ahlers' o r i g i n a l thermal conductivity data ( f t ( A T - ) , A T - ) ,  3 8  44  obtaining f o r the constant A  Jo U " (  )  1  "  * *°  ( 0 , 8 6  6) x 1 0  "  8e m  "  •••( *3) l  i n agreement with Eq. ( 4 2 ) . Rewriting Eq. (41) we f i n d u ( A T ) = (fc (AT-)^>C (AT-) ) • 2s  p  C f 2<5i>" 3 - ! AT-| 2 / 3 . . . . A  1  1  (^)  where the subscript s Indicates the v e l o c i t y i s a s c a l i n g prediction.  For eaoh of the thermal conductivity points  ( K ( A T - ) , AT-), Eq. (44) was evaluated (with substitution of Eqs. (34) and (43) ) and the r a t i o R (cxT) s u ( A T ) / u ( A T ) 2  2s  2h  was c a l c u l a t e d . The r e s u l t s are shown i n F i g . 7. For the 66 points the mean value of H  2  and i t s 95% confidence  l i m i t s are Rg =  <  u  / u 2 s  z h  )  - 1.005 * .004 .  ...(45)  45 CHAPTER VI. (1)  CONCLUSIONS AND DISCUSSION.  CONCLUSIONS  The hydrodynamic and the s c a l i n g law predictions of the v e l o c i t y of second sound were tested by comparison with the experimentally determined values. value of the r a t i o of u  2 e  to u  g h  The average  over the temperature  range of the experiment i s given by S i l n Eq. ( 3 9 ) . The equality of H i and unity therefore demonstrates the  agreement between the measured v e l o c i t i e s and those  predloted by two f l u i d hydrodynamics.  Further calcu-  l a t i o n s showed, and F i g . 7 indicates, that within the measured temperature range there were no s i g n i f i c a n t departures of R^ from u n i t y .  In other words, R± was  observed to be independent of temperature.  The hydro-  dynamic theory therefore appears to properly predict both the magnitude and the temperature dependence of the  second sound v e l o c i t y i n the c r i t i c a l region. A comparison of R^ with Rg implies a comparison  of U 2  e  and U2».with ugh and hence of u  2 e  with U2s»  The equality of Rj and R2 demonstrates the agreement between the measured v e l o c i t i e s and those calculated from dynamic s c a l i n g arguements. to be Independent of temperature.  R2 was also shown As u  2 s  contains a  46  constant term (evaluated i n Eq*  (43)  ) determined by  f i t t i n g to the thermal conductivity data, i t i s t h i s temperature independence of Bg which constitutes the a f f i r m a t i v e t e s t of the s e a l i n g law p r e d i c t i o n . On the basis of these r e s u l t s (Eq. (39)  and  (45)  we conclude that i n the c r i t i c a l region f o r A T £ 5 x the hydrodynamic p r e d i c t i o n Eq. (23) law p r e d i c t i o n Eq. (29)  ) 10"^K  and the s c a l i n g  of the v e l o c i t y of low amplitude,  low frequency second sound have been v e r i f i e d . The v e r i f i c a t i o n of the hydrodynamic prediction, and the p r e d i c t i o n of the values of Q and j> i n Eq.  (40)  both imply that the extrapolation pf the superfluid density r e l a t i o n Eq. (33)  to 5 x 10~^K  was  valid.  The experiment also proved that when properly  ap-  p l i e d , the method of d r i f t i n g temperatures i s well suited for making measurements of physical phenomena near the c r i t i c a l point. (2)  DISCUSSION  In the derivation of Eq. (23)  i t was assumed ex-  p l i c i t l y that the v e l o c i t i e s v l and s  are small, and n  i m p l i c i t l y that inhomogeneitles ( i n the absence of seoond sound) i n the thermodynamic properties of the f l u i d occur only over distances small compared with the second sound wave length ( i . e . that k £ « l ) .  In other words.  47 the  derivation was f o r low amplitude ( v and w g  are  proportional to the seoond sound amplitude), low f r e quency second sound. In deriving the s c a l i n g law prediction, the c r i t i c a l frequency co (k) was related (Eq. (25) ) to second sound q  by using the assumption that second sound i s the dominant mode of heat transport i n Region I (Pig. 1), defined by kf « 1 . Both the above conditions (low v e l o c i t i e s , and k | « l ) were met i n the experiment, the highest observed seoond sound amplitude being <~ 10"?K RMS, and the largest value of kj" being ~ 0 . 0 2 . A departure from either the low amplitude, or the low frequency condition would presumably be accompanied by a departure of the observed v e l o c i t y from the hydrodynamic value.  V e l o c i t i e s of large amplitude second  sound (analogous to shock waves i n ordinary sound) d i f f e r e n t from the hydrodynamic v e l o c i t y have been observed.  39,  "*°  In the c r i t i c a l region very near T^ the c o r r e l a t i o n function range j* i s diverging r a p i d l y with decreasing AT.  I t would therefore seem interesting, and not im-  p r a c t i c a l , to make use of higher frequency second sound to enter Region I I (Pig. 1) near the c r i t i c a l point  48 i n which kl* i s o f the o r d e r o f , o r g r e a t e r than u n i t y , and i n which two f l u i d hydrodynamics c o u l d no he presumed  valid.  longer  49  GLOSSARY OP SYMBOLS  A  Constant o f o r d e r u n i t y .  a  An o p e r a t o r o r p h y s i c a l q u a n t i t y o t h e r than p .  B^  Thermometry system s e n s i t i v i t y chart d i v i s i o n .  (ohms p e r  C  Constant p r e s s u r e s p e c i f i c h e a t . Constant volume s p e c i f i c heat.  c* c* c* " 1  c* d D  t  Correlation function f o r a. F o u r i e r t r a n s f o r m o f c£. Asymptotic forms o f cf and I I I ( F i g . 1 ) .  i n regions  C o r r e l a t i o n f u n c t i o n f o r p. Resonant c a v i t y s p a c i n g . Thermal  diffusivity.  E  Energy d e n s i t y o f t h e l i q u i d .  e  A critical  t  Entropy f l u x v e c t o r .  t  Second sound frequency.  G  Gibbs f r e e energy p e r u n i t mass.  T  Momentum p e r u n i t volume.  k  Wave number.  M  Mass o f l i q u i d helium sample.  M  n  M  s  exponent.  Mass o f normal f l u i d  component.  Mass o f s u p e r f l u i d component.  50 n  Resonance mode number. (Equal t o t h e number o f h a l f wavelengths o f second sound i n the cavity).  n'  R e l a t i v e mode number. Only d i f f e r e n c e s between t h e n a r e c o r r e c t . 1  P  P  T o t a l power d i s s i p a t i o n i n t h e c a v i t y .  n  Normal f l u i d - s u p e r f l u i d r e l a t i v e momentum.  p *  Pressure.  p  Order parameter.  Q  Coefficient of AT  q  Heat o p e r a t o r .  R(t)  Thermometer r e s i s t a n c e a t time t .  R^  T  i n e x p r e s s i o n .  E l e o t r i c a l resistance  o f d e v i c e a t T^.  S  Entropy p e r u n i t mass.  T  A b s o l u t e temperature. A c r i t i c a l temperature.  c  T^  The lambda temperature 2.172 K.  t  Time.  U  Internal  energy p e r u n i t mass.  u^  V e l o c i t y o f f i r s t sound i n l i q u i d helium.  u  V e l o c i t y o f second sound i n l i q u i d h e l i u m .  2  2e  u  2  2h u  u  2 l * l a t e d from hydrodynamics. u c a l c u l a t e d from thermal c o n d u c t i v i t y and s c a l i n g t h e o r y .  u  u  2 a  V  d e t e r r a  c a  *  experimentally.  n e d  c t  2  S p e c i f i c volume ( equal t o  v  n  Normal f l u i d  v  s  Superfluid  velocity.  velocity.  ).  51 S t a t i s t i c a l weight of a data point. Cartesian coordinates ( k =1,2,3 )•  Constants of the order of unity. C r i t i c a l exponent of p. A measured distance on a recorder chart. A i s subsequently converted to a temperature d i f f e r e n c e . T  A  - T .  T\ - (the temperature of the sharp d i s continuity i n the slope of the warming curve) The Kroneoker delta function. ( T - T  c  )/T . c  C r i t i c a l exponent ofytfg. C r i t i c a l exponent of C$. Thermal conductivity. C r i t i c a l exponents of Correlation length, or range of C?. Momentum f l u x tensor. Density of l i q u i d helium sample. Density of normal f l u i d component. Density of superfluld component. 2=0 i f T < T Angular  V  2*1 i f T>T . X  frequency.  Characteristic frequency f o r c|. Div grad .  52  REFERENCES AND FOOTNOTES 1.  V.P.Peshkov, Soviet Phys. - JETP, U ,  580 (I960).  2.  C.J.Pearce, J.A.Llpa, and M.J.Buckingham* Phys. Rev. Letters, 20, 1471 (1968).  3.  J.A.Tyson and D.H.Douglass, Phys. Rev. Letters, 21, 1308 (1968).  4.  D.L.Johnson and M.J.Crooks, Phys. Letters, 27A, 688 (1968).  5.  A good review of the two f l u i d model i s contained l n : J.Wllks, The Properties of Liquid and Solid Helium, (Clarendon Press, Oxford, 1967)* Chapter 3.  6.  L.D.Landau, J . Phys., Moscow 71 (1941). Reprinted l n : I.M.Khalatnikov, Introduction to the Theory of Superfluidity (V.A.Benjamin, Inc. New York, 1965).  7.  A good review paper i s : L.P.Kadanoff, W.Gotze, D.Hamblen, R.Hecht, E.A.S.Lewis, V.V.Palclauskas, M.Rayl, and J.Swift, Rev. Mod. Phys., 22, 395 (1967).  8.  M.E.Fisher, J . Math. Phys.,  9.  B.Wldom, J . Chem. Phys., 43, 3892 (1965). and 42, 3898 (1965).  944 (1964).  10.  L.P.Kadanoff, Physics. 2, 263 (1966).  11.  B.I.Halperin and P.C.Hohenberg, Phys. Rev., 177. 952 (1969).  12.  R.A.Ferrell, N.Menyhard, H.Schmidt, F.Schwabl and P.Szepfalusy, (a) Phys. Rev. Letters, 18, 891 (1967). (b) Phys. Letters, 24A, 493 (1967). and (c) Ann. Phys. (N.I.) , 47., 3&7 (1968).  13.  See Reference 12(c), Equation (5.4).  14.  J.R.Clow and J.D.Reppy, Phys.Rev. Letters, 16, 887 (1966).  15.  J.A.Tyson and D.H.Douglass, Phys. Rev. Letters, 12, 472 (1966); 12. 622(E) (1966).  16.  J.A.Tyson, Phys. Rev., 166. 166 (1968).  53  17.  G.Ahlers, Phys. Rev. L e t t e r s , 21, 1159 (1968).  18.  E.J.Walker, Rev. S c i . I n s t r . , 20, 834 (1959).  19.  CIBA (A.R.L.) L t d . , Duxford, Cambridge, England.  20.  G.Ahlers, Phys. Rev., 121, 275 (1968).  21.  P r i n c e t o n A p p l i e d Research Corp. Model 122. T y p i c a l s e n s i t i v i t y 0.1 t o 1.0 mV rms f u l l s c a l e . T y p i c a l i n t e g r a t i o n time 1 t o 3 seconds.  22.  P r i n c e t o n A p p l i e d Research Corp. Model 112 preamplifier. F i x e d g a i n o f 40db.  23.  Hewlett-Packard / Moseley D i v i s i o n Model 7100B. T y p i c a l gain 1 to 5 V f u l l scale.  24.  T e l t r o n l c s Inc. Model CA-2. T y p i c a l s e n s i t i v i t y 5ju.V rms f u l l s c a l e . T y p i c a l i n t e g r a t i o n time 1 second.  25.  Tektronix o r 60db.  26.  W.C.Cannon and M.Chester, Bev. S c i . I n s t r . , 38,  27.  An I n t r o d u c t i o n t o P h o t o f a b r l c a t l o n u s i n g Kodak P h o t o s e n s i t i v e R e s i s t s , Kodak P u b l i c a t i o n No. P-79. Eastman Kodak Co., Rochester, N.Y.  28.  No. 999 B l a c k .  29.  Allen-Bradley  30.  IRC Inc., Boone D i v i s i o n , Boone, North C a r o l i n a .  31*  Some data were a c q u i r e d before t h e n e c c e s s l t y t o measure zero-power warming curves was r e a l i z e d . On these e a r l y c h a r t s , t h e p o i n t A was assumed t o be the lambda p o i n t . As P was known, these data were subsequently c o r r e c t e d by t h e a d d i t i o n t o t h e i r AT v a l u e s o f AT'(P) obtained from e x t r a p o l a t i o n and i n t e r p o l a t i o n o f t h e subsequent measurements of AT»(P) as a f u c t i o n o f P.  32.  Thermal Expansion o f T e c h n i c a l S o l i d s a t Low Tempe r a t u r e s , N a t i o n a l Bureau o f Standards Monograph 29, [United S t a t e s Department o f Commerce, Government P r i n t i n g O f f i c e , Washington, D . C , 1961). Note that Perspex and P l e x i g l a s s a r e both manufacturers trade names f o r polymethylmethacrylate.  Inc. Model BM 122. F i x e d gains o f 40db  318 (1967).  Columbian Carbon Co. of New York, N.Y. Co., Milwaukee, Wisconsin.  54  33*  C . P . K e l l e r s , T h e s i s , Duke U n i v e r s i t y ( i 9 6 0 ) unpubl i s h e d 1 M.J.Buckingham and W.M.Pairbank, I n Progress In Low Temperature P h y s i c s . ed. C.J.Gorter (NorthH o l l a n d P u b l i s h i n g Co.,Amsterdam, I 9 6 D , V o l . I l l , C h . 3 .  34.  R.W.Hill  145  35.  and O.V.Lounasmaa, P h i l . Mag., 2, S e r . 8,  (1957).  The " o o r r e c t " weight W g i v e n t o a p o i n t should be p r o p o r t i o n a l t o the inverse o f the variance of t h a t p o i n t . The b e s t a v a i l a b l e estimate o f t h e v a r i a n c e i s t h e square o f t h e p r o b a b l e e r r o r . The p o i n t s were t h e r e f o r e a s s i g n e d weights W Q p r o p o r t i o n a l t o ( 2 + 6 / A )" » t h e number i n b r a c k e t s b e i n g t h e estimated p r o b a b l e f r a c t i o n a l e r r o r {%) i n t h e measurement o f A T . N  n  n  36.  G.Ahlers, Phys. L e t t e r s , 28A, 507 ( 1 9 6 9 ) .  37.  G.Ahlers, B u l l . Am. Phys. S o c , 1^, 506 ( 1 9 6 8 ) , and t o be p u b l i s h e d .  38.  G.Ahlers, p r i v a t e communication.  39.  A . J . D e s s l e r and W.M.Fairbank, Phys. Rev., 1 0 4 , 6  4 0 .  (1956).  D.V.Osbourne, P r o c . Phys. S o c ,  A 6 4 , 1 1 4 ( 1 9 5 1 ) .  55 APPENDIX I EXPERIMENTAL AND THEORETICAL SECOND SOUND VELOCITIES  TEMPERATURE DIFFERENCE (K) 4.75E-06 5.27E-06 5.33E-06 5.66E-06 5.66E-06 5.75E-06 5.98E-06 6.10E-06 6.18E-06 6.59E-06 6.63E-06 6.90E-06 7.31E-06 7.61E-06 7.75E-06 7.90E-06 7.94E-06 8.08E-06 8.36E-06 8.44E-06 8.44E-06 8.74E-06 8.74E-06 8.87E-06 8.93E-06 9.03E-06 9.42E-06 9.43E-06 9.68E-06 9.88E-06 1.03E-05 1.05E-05 1.07E-05 1.09E-05 1.10E-05 1 .12E-05 1.13E-05 1.I5E-05 1.17E-05 1.23E-05  . EXPERIMENTAL ,, HYDRODYNAMIC VELOCITY VELOCITY (M/SEC) (M/SEC) 0.335 0.353 0.380 0.389 0.353 0.389 0.342 0.383 0.389 0.383 0.392 0.383 0.392 0.389 0.383 0.444 0.383 0.389 0.425 0.415 0.418 0.418 0.418 0.383 0.410 0.399 0.399 0.410 0.383 0.446 0.441 0.444 0.453 0.446 0.425 0.465 0.446 0.441 0.425 0.465  0.321 0.334 0.335 0.342 0.342 0.344 0.349 0.352 0.354 0.362 0.363 0.368 0.376 0.382 0.384 0.387 0.388 0.390 0.395 0.396 0.396 0.401 0.401 0.404 0.405 0.406 0.413 0.413 0.417 0.420 0.426 0.429 0.432 0.435 0.437 0.440 0.441 0.444 0.447 0.455  56  MPERATURE FFERENCE (K) 1.24E-05 1.26E-05 1.29E-05 1 .31E-05 1.42E-05 1.45E-05 1.46E-05 1.46E-05 1.47E-05 1.48E-05 1.49E-05 1.50E-05 1.51E-05 1.52E-05 1.52E-05 1.52E-05 1.52E-05 1.54E-05 1.57E-05 1.58E-05 1.61E-05 1.61E-05 1.62E-05 1.63E-05 1.66E-05 1.68E-05 1.73E-05 1.76E-05 1.78E-05 1.78E-05 1.80E-05 1.80E-05 1.81E-05 1.83E-05 1 .89E-05 1.93E-05 1.95E-05 2.00E-05 2.02E-05 2.04E-05 2.05E-05 2.09E-05 2.12E-05 2.14E-05 2.26E-05  . EXPERIMENTAL ,, HYDRODYNAMIC VELOCITY VELOCITY (M/SEC) (M/SEC) 0.478 0.465 0.511 0.478 0.478 0.478 0.478 0.478 0.479 0.504 0.480 0.486 0.480 0.478 0.479 0.486 0.519 0.480 0.478 0.523 0.517 0.478 0.478 0.498 0.486 0.504 0.523 0.523 0.540 0.519 0.536 0.536 0.511 0.536 0.511 0.510 0.547 0.547 0.574 0.553 0.540 0.547 0.547 0.540 0.540  0.456 0.459 0.463 0.466 0.480 0.483 0.485 0.485 0.486 0.487 0.488 0.489 0.491 0.492 0.492 0.492 0.492 0.494 0.498 0.499 0.502 0.502 0.503 0.505 0.508 0.510 0.516 0.519 0.521 0.521 0.523 0.523 0.524 0.527 0.533 0.537 0.539 0.544 0.546 0.548 0.549 0.553 0.556 0.558 0.569  57  TEMPERATURE DIFFERENCE (K) 2.29E-05 2.30E-05 2.34E-05 2.5IE-05 2.53E-05 2.55E-05 2.55E-05 2.57E-05 2.58E-05 2.63E-05 2.63E-05 2.68E-05 2.72E-05 2.75E-05 2.76E-05 2.79E-05 2.86E-05 2.87E-05 2.90E-05 2.97E-05 2.98E-05 2.99E-05 3.01E-05 3.01E-05 3.02E-05 3.03E-05 3.03E-05 3.05E-05 3.09E-05 3.09E-05 3.10E-05 3.11E-05 3.14E-05 3.15E-05 3.16E-05 3.18E-05 3.20E-05 3.30E-05 3.33E-05 3.42E-05 3.44E-05 3.54E-05 3.55E-05 3.57E-05 3.59E-05  . EXPERIMENTAL ,, HYDRODYNAMIC VELOCITY VELOCITY (M/SEC) (M/SEC) 0.574 0.574 0.597 0.574 0.597 0.574 0.589 0.574 0.574 0.618 0.589 0.623 0.597 0.574 0.622 0.618 0.598 0.622 0.618 0.638 0.638 0.639 0.638 0.598 0.599 0.638 0.598 0.599 0.638 0.639 0.648 0.638 0.618 0.639 0.638 0.638 0.648 0.639 0.648 0.670 0.639 0.656 0.646 0.702 0.670  0.572 0.573 0.577 0.592 0.593 0.595 0.595 0.597 0.598 0.602 0.602 0.606 0.609 0.612 0.613 0.615 0.621 0.622 0.624 0.630 0.630 0.631 0.633 0.633 0.633 0.634 0.634 0.636 0.639 0.639 0.640 0.640 0.643 0.643 0.644 0.646 0.647 0.655 0.657 0.663 0.665 0.672 0.673 0.674 0.675  58  MPERATURE FFERENCE (K) 3.81E-05 3.84E-05 3.86E-05 3.86E-05 3.89E-05 3.93E-05 3.97E-05 4.07E-05 4.13E-05 4.16E-05 4.17E-05 4.20E-05 4.24E-05 4.31E-05 4.33E-05 4.36E-05 4.36E-05 4.36E-05 4.37E-05 4.48E-05 4.63E-05 4.67E-05 4.68E-05 4.78E-05 4.93E-05 4.96E-05 4.96E-05 4.96E-05 5.00E-05 5.03E-05 5.21E-05 5.23E-05 5.30E-05 5.52E-05 5.57E-05 5.67E-05 6.08E-05 6.13E-05 6.13E-05 6.24E-05 6.27E-05 6.43E-05 6.45E-05 6.61E-05 6.67E-05  . EXPERIMENTAL ,• HYDRODYNAMIC VELOCITY VELOCITY (M/SEC) (M/SEC) 0.670 0.638 0.718 0.697 0.718 0.697 0.721 0.717 0.697 0.721 0.718 0.712 0.706 0.706 0.656 0.765 0.765 0.718 0.721 0.717 0.767 0.721 0.765 0.765 0.738 0-.765 0.765 0.765 0.765 0.765 0.765 0.767 0.778 0.765 0.765 0.778 0.820 0.765 0.797 0.797 0.820 0.797 0.830 0.865 0.836  0.690 0.692 0.694 0.694 0.696 0.698 0.701 0.707 0.711 0.713 0.714 0.716 0.718 0.723 0.724 0.726 0.726 0.726 0.726 0.733 0.742 0.744 0.745 0.751 0.760 0.761 0.761 0.761 0.764 0.765 0.775 0.776 0.780 0.792 0.795 0.800 0.821 0.824 0.824 0.829 0.830 0.838 0.839 0.847 0.850  59  TEMPERATURE DIFFERENCE (K) 6.78E-05 6.87E-05 6.89E-05 6.95E-05 6.97E-05 6.97E-05 7.06E-05 7.10E-05 7.25E-05 7.34E-05 7.35E-05 7.38E-05 7.54E-05 7.62E-05 7.63E-05 8.04E-05 8.53E-05 8.54E-05 8.59E-05 9.09E-05 9.10E-05 9.10E-05 9.15E-05 9.16E-05 9.17E-05 9.21E-05 9.39E-05 9.39E-05 9.44E-05 9.56E-05 9.75E-05 9.84E-05 9.88E-05 1.02E-04 1.03E-04 1.07E-04 1.22E-04 1.27E-04 1.28E-04 1.28E-04 1.31E-04 1.33E-04 1.33E-04 1.36E-04 1.41E-04  . EXPERIMENTAL ,, HYDRODYNAMIC VELOCITY VELOCITY (M/SEC) (M/SEC) 0.836 0.850 0.852 0.842 0.862 0.836 0.865 0.865 0.850 0.862 0.852 0.861 0.865 0.893 0.852 0.893 0.957 0.957 0.957 0.918 0.957 0.957 0.957 0.957 0.918 0.957 0.959 0.957 0.959 0.957 0.957 0.957 0.957 0.985 0.957 0.985 1.030 1.081 1.081 1.052 1.081 1.090 1.090 1.081 1.150  0.855 0.859 0.860 0.863 0.864 0.864 0.868 0.870 0.877 0.881 0.881 0.882 0.889 0.893 0.893 0.911 0.931 0.932 0.934 0.954 0.954 0.954 0.956 0.956 0.957 0.958 0.965 0.965 0.967 0.972 0.979 0.982 0.984 0.995 0.999 1 .013 1 .064 1 .080 1.084 1.084 1.093 1 .099 1 .099 1 .108 1.124  60  TEMPERATURE DIFFERENCE (K) 1.45E-04 1.51E-04 1.58E-04 1.59E-04 1.79E-04 1.83E-04 1.99E-04 1 .99E-04 1.99E-04 2.06E-04 2.11E-04 2.22E-04 2.62E-04 2.82E-04 3.06E-04 3.11E-04 3.98E-04 4.35E-04 4.43E-04 5.27E-04 5.57E-04 5.88E-04 6.01E-04 6.05E-04 6.11E-04 6.62E-04 6.64E-04 7.25E-04 8.70E-04 9.15E-04 9.22E-04 9.74E-04 1.03E-03 1.08E-03 1.20E-03 1.23E-03 1.39E-03 1.67E-03 1.75E-03 2.01E-03 2.03E-03 2.17E-03 2.65E-03 2.65E-03 2.95E-03  • EXPERIMENTAL .. HYDRODYNAMIC VELOCITY VELOCITY (M/SEC) (M/SEC) 1.150 1.149 1.1 46 1.149 1.200 1.200 1.290 1.280 1.280 1.279 1.279 1.379 1.379 1.480 1.530 1.530 1 . 720 1.724 1.724 1.910 1.910 1.913 1.913 1.913 1.960 2.060 2.070 2.150 2.240 2.300 2.300 2.350 2.540 2.460 2.620 2.580 2.740 2.870 2.960 2.960 3.140 3.290 3.440 3.440 3.550  1.135 1.153 1.173 1.175 1.229 1.239 1.279 1.279 1.279 1.295 1.307 1.332 1.418 1.458 1.503 1 .512 1 .660 1.717 1.729 1.846 I .885 1.924 1.941 1.945 1.953 2.014 2.017 2.085 2.235 2.278 2.285 2.333 2.382 2.426 2.525 2.549 2.671 2.866 2.918 3.077 3.089 3.174 3.426 3.426 3.570  61  TEMPERATURE DIFFERENCE (K) 3.18E-03 3.74E-03 3.93E-03 4.62E-03 5.32E-03 5.53E-03 7.35E-03 7.90E-03 9.99E-03 1.15E-02 1.25E-02  EXPERIMENTAL VELOCITY (M/SEC) 3.550 3.920 4.110 4.310 4.440 4.440 5.480 5.230 5.740 5.920 5.920  HYDRODYNAMIC VELOCITY (M/SEC) 3.674 3.914 3.989 4.244 4.482 4.549 5.079 5.222 5.715 6.042 6.233  

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