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A rotation experiment with liquid helium II Crooks, Michael John Chamberlain 1957

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A ROTATION EXPERIMENT WITH LIQUID HELIUM II by . MICHAEL JOHN CHAMBER LAIN CROOKS B.A., Reed College, 1953 A THESIS,SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS THE UNIVERSITY OF BRITISH COLUMBIA August, 1957 ABSTRACT The thesis contains a brief introduction to,the problem of He II, including mention of the salient experimental work, of the last fifty years. The treatment of He II in rotating systems is then discussed in more detail and a review of the previous.rotation experiments is presented. In a rotating mass of He II, it has been shown by. Osborne that the parabolic rise of the liquid surface should be proportional to the amount of normal fluid present and that a radial temperature gradient should be.set up. For reasonable angular velocities (on the order of 3 rev/sec.) and for beakers of about 2 cm. radius, the temperature difference between the axis.and the periphery would be on the order of IO'^OK . It can be shown that this temperature would lead to an effective distillation of liquid from the warmer to the cooler regions which it was thought might be the mechanism leading to the known result that the whole mass.of liquid partakes.of the rotary motion. An experiment was designed whereby a volume of helium might be rotated but the possibility of such a distillation process was elimi-nated. Measurements of the surface shape during rotation showed that even in this,case the whole fluid rotated at all angular velocities and temperatures used. Finally, further experiments which might shed further light on this problem are proposed. In presenting t h i s thesis i n p a r t i a l fulfilment of the requirements fo r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representative. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of P h y s i c s  The University of B r i t i s h Columbia, Vancouver tt, Canada. Date s,^ /qr y- ACKNOWLE DGEMENTS I should like first to express my sincere thanks to Dr. J. B. Brown who first suggested the problem with which this thesis is concerned, and whose.constant assistance and advice made the completion of the work possible. I should also like to thank all the other members;of the Low Temperature Group for their help, in particular Mr. H. Zerbst who is responsible for the production of liquid helium and who helped greatly in the construction of apparatus. Special thanks is also due.to Mr. J. Lees for his construction of all the glassware used, and.to the machine shop staff, headed by Mr. A. Fraser, who were,always ready with advice and help in the construction of the,apparatus. iv TABLE OF CONTENTS Page CHAPTER 1 - LIQUID HELIUM II 1 CHAPTER 2 - HE II IN ROTATING SYSTEMS .10 CHAPTER 3 - APPARATUS 21 CHAPTER 4 - EXPERIMENTAL PROCEDURE 28 CHAPTER 5 - RESULTS 33 CHAPTER 6 - DISCUSSION 37 APPENDIX 39 BIBLIOGRAPHY 41 LIST OF ILLUSTRATIONS FIGURE 1 - Overall photograph, of apparatus 21 FIGURE 2 - Photograph of early containers 22 FIGURE 3 - Photograph of container and holder .22 FIGURE 4 - Diagram, of.container and holder (x-section) 23 FIGURE 5 - Block diagram.of rotating system 23 FIGURE 6 - Electrical arrangements (a) d.c. motor 24 (b) motor-alternator (c) counting devices FIGURE 7 - Vacuum system 26 FIGURE 8 - Graph of results. 35 CHAPTER I LIQUID HELIUM .11 In order to explain fully the nature of the problem dealt with in this thesis it is.desirable to present to the reader a background of the development of low temperature researches.over the past fifty years, and to describe some of the unique properties of liquid helium. .Below 2.18°K. liquid helium undergoes a radical transformation into a so-called fourth state of matter. At these.temperatures the.behaviour.of theTiquid appears to be,dominated by quantum effects and the appearance of this new state is one of the major factors that have led.to the wide interest in low temperature physics over the past three decades. Helium, the last of the atmospheric gases to be discovered, was also the last to be liquefied. It was not until July 10, 1908 that H. Kammerlingh Onnes (1908) working at the University, of Leiden succeeded in producing liquid helium, and simultaneously reached the lowest tempera-ture then attained by man, an e sbLma ted. 1.4°K. The liquefaction of purified helium gas was achieved by compression, cooling and utilization of the Joule-Thomson effect. The.final cooling was a regenerative process, that is the cold gas flowing, out of the. collecting; volume was used to cool the incoming gas. By pumping as .rapidly as possible on the liquid the boiling point, which is 4.2°K at atmbspheric pressure, was reduced to 2. a temperature limited by. the speed of the pumping system and the rate.of heat influx to the. system. The. same day evidence.was. sought for the appearance of solid helium. It has.since become evident (London, 1954, pp. 3 - 6) that helium is unlikely.to solidify at atmospheric pressure, even as the temperature approaches absolute zero. During the.next thirty years a variety of experiments were conducted, largely at Leiden and Oxford Universities, to.determine the mechanical and thermal propertie s. of helium above 1°K. The variation of density.with temperature, studied by Kammerlingh Onnes (1911) and later by Kammerlingh Onnes and Boks (1924) showed a surprising maximum at 2.2°K. Later Wolfke and Keesom (1927) found evidence.for a.discon-tinuity of the .dielectric.constant of the liquid at the same temperature, and in the face;of the. accumulating evidence.suggested that liquid helium undergoes.a phase.transition at this temperature. They referred to the phase stable at the higher temperatures as Helium I, and to that stable at the lower temperatures as Helium II. They felt that, in their.own words, "at pressure .of about 38 mm. something very peculiar takes place." Again, in measurements of the specific heat of liquid helium made by Keesom and Keesom (1932) a most remarkable discontinuity was found at 2.19°K. and the resemblance of the shape.of the curve to the Greek character ^ led Ehrenfest to suggest that this transition temperature be.referred to as the; "lambda point". Measurements.of the thermal conductivity made by.the.same.authors (Keesom and Keesom, 1936) 3. and independently by.Rollin (1935) found that helium slightly, below the ^ -point has a conductivity better than one hundred times that of copper at room temperature. The.disagreement between two methods.of measuring . viscosity, in He II was .crucial in leading to the phenomenological "two fluid" theory. Measurements made in TorontoXWilhelm, Miserier and Clarke, 1935) using an oscillating cylinder led to a value;of 7[ = 33 ytt-poise.for helium just below the ^ -point. However, measurements made by.Kapitza (1938) using flow between optical flats, and Allen and Misener (1938) using flow through long capillaries led to values.lower by a factor of 10^. In fact it was.found that in the very narrow channels ;(10"4 to 10"5 cm.), the flow was apparently non-viscous and independent of pressure head. By 1939 the large body, of experimental work had established that there .occurred in He.II at least four phenomena which are without parallel in other forms of m atter. (1) Film Flow. The. first evidence for the movement of liquidover the surface of solid bodies in contact with He II was noted by Kammerlingh Onnes (1922). However, it was not until fourteen years later that Rollin and Simon (1939) definitely established the existence of these films, which make.it impossible to maintain independent liquid levels.in He II if the two regions are.connected by a solid.surface. .4. (2) Fountain Effect. It was.observed by Allen and Jones (1938) that if a beam of light is directed on a cylinder of fine pow^r open at the bottom to a bath of He II, a jet of liquid several centimeters in height is emitted from,the.top.of the cylinder. (3) Anomalous Heat Conduction. As mentioned above, the: thermal conductivity of bulk. He II is so large as to suggest that the mechanism involved is of a type quite different from that in other states of matter. (4) Viscosity Measurements . Measurements ;of the. coefficient of viscosity made using the method of damped torsional vibrations are in direct disagreement with those made using methods of Poiseuille flow. It seemsxlear then, that the problem of.liquid helium involves phenomena totally unlike those encountered in classical physics, and will require some radically new concepts in its solution. The.first major theoretical contribution was the.suggestion by London (1938) that the specific heat anomaly of liquid helium might be ,analagous to the,condensation of an ideal Bose-Einstein gas, which had been predicted at low temperatures for gases with the.type of statistics of the He^ atom. This suggestion was.adopted by Tisza (1938a,b,c) as the basis for his phenomenological "two fluid" theory, which he used to,construct a model which would account for the occurrence of many of the unusual properties of He. II. 5. In essence, this, theory considers He l i a s consisting of an interpenetrating mixture of two fluids - the normal fluid of density gn and.the. superfluid of density . The density of normal fluid decreases with the.temperature from the ^-point where 9^ .= l , to 0°K. where = 0. .At any temperature below the ^ -point the total, density of the fluid ^ , is given by 9 * ?"+ft Experimentally it appears that g has an almost constant value of 0.145 gm./cc. in this region. These fluids are considered, at least when moving slowly, to be non-interacting,and to have completely independent velocities. Thus J, the density flow of mass, where v s is the superfluid velocity and y n is the normal fluid velocity, is given by J • f s V s + f n V n , The superfluidcomponent is said to consist of.atoms in their ground state and bence the flow of superfluid will transport no entropy; S s = 0 Also, since, the superfluid is. entirely in a. single quantum state .momentum transfer cannot occur, and the.viscosity of the fluid must be zero. n s - o This theory explains satisfactorily most of the phenomena mentioned above. The extremely high thermal conductivity in bulk.He II can be considered to be essentially a convection process. The two fluids 6. flow in opposite ^directions transporting heat very rapidly, but with no,net flow of mass. The "fountain effect" or mermomechanical effect.which occurs when two regions at different temperatures are_connected only by extremely narrow channels, ("superleaks") can be.explained as follows.. In the warmer region the density of isuperfluid is,considerably less than in the cooler one, and the superfluid, as predicted.by Le^Chatelier's principle, flows in through the superleak to reduce the ratio to that of the liquid as.a whole. However, the normal fluid, because.of its viscosity, cannot flow in the^opposite direction and me, initial tempera-ture gradient leads.to a resulting pressure gradient. The disparity of the two types of viscosity measurements .is clearly explained in terms of the type of apparatus used. In the measurements made using oscillating cylinders, it is assumed that.only the normal component exerts a drag.on the.disc, and hence only the viscosity of the normal fluid is measured. In measurements made by flow methods the. channels are. so narrow that the normal, component is held stationary by its viscosity and only the superfluid takes part in the flow. The nature of the normal and superfluid, components is still a matter of controversy. London's theory (London, 1954, pp.40-64) involving the condensation of a Bose-Einstein gas is capable.of explaining the presence;of the A -point, and the behaviour in general of He_H in the 7. region relatively- close to this temperature. The superfluid is considered to consist of the "condensed" atoms which are in their ground state-and ordered.in momentum space. The normal fluid consists of, the atoms which are in excited states and which behave in all respects like those of an ordinary fluid. London calculated the temperature-at which this degeneracy should occur, and found it to be 3.13°K., which is fairly, close to the ^ observed ^ -point of 2.189K. However, it is questionable that liquid helium can be adequately described by a model relating to the behaviour of an ideal gas, and.the shape of the specific heat curve;of a Bose-Einstein gas is only, very roughly related, to that found experi-mentally for liquid helium. Some three years later, Landau (1941) suggested a .different basic concept for the two fluids in He H. In this approach there are no,"normal" and "superfluid" atoms, but only, at any tempera-ture .above absolute zero, the liquid in its ground state and excitations within the body of the-fLuid. These.excitations are of two kinds which Landau named."phonons" and "rotons". The first of these-are quantized units.of sound-energy, identical with the.Debye-lattice waves of-a solid, and the latter are.described as."quantized vortices". The.nature of these vortices has been further elaborated by Feynman (Gorter 1955, Chap.II). Superfluid flow is.then restricted to cases where the velocities involved are insufficient to produce excitations. By judicious.choice.of the parameters involved in the energy spectrum of this model Landau was able.to produce a.specific heat curve which agrees, especially, at very low temperatures, with that found experimentally. However, this latter theory makes no prediction of the occurrence.of the X -point discontinuity. In one.of his original papers on the two fluid theory. Tisza (1938b) predicted the occurrence of thermal waves, or second sound .(the .latter name is due to Landau) in He II. These waves were, first observed by.Peshkov (1946). The velocity/of second sound.could be predicted from both the London and Landau theories.and the general shape of the.curve against temperature is the same for both from the ^ -point down to about 1°K. Below 1° the curves diverge. London's ^ original theory predicted a second sound velocity approaching zero at 0°K.,. and Landau's treatment predicts a constant value U2 = uj / p f (where u^ and U2 are the velocities of normal and second sound) as the temperature approaches absolute zero. Measurements made down to.0.IOK. by Atkins and Osborne (1950) appear to vindicate the Landau prediction. It should.also be noted, however, that as experiments have preceded with the rare.isotope.He3 that it has been indicated (Daunt and Heer, 1950) that here no.superfluidity.can occur above 0.25°K. These results would indicate that if Landau's.theory is to be believed that the.energies involved must be.surprisingly.different from those involved in the He^ transition. However, on the basis of London's theory one would not expect any super-fluid effect since He^.obeys. Fermi-Dirac statistics. It has been suggested by Temperley (1952) that since London's predictions are borne out in the region near the /% -point and Landau's at very low temperatures, that the true picture can only be obtained by a fusion of the.two. However, at the present time, this has not been achieved and the basic theoretical concepts :of He II are still very much in question. CHAPTER 2 HE II IN ROTATING SYSTEMS .During the past ten yearsiconsiderable.interest has developed in the problem.of liquid helium in rotating and oscillating systems. Many experiments, of this.type have been performed and most of these have shown results.that require modification of the two fluid theory as.it was first proposed. For abnormal fluid having no viscosity the equations of motion are well known. For linear motion F = a Dv * Dt where F is the net force.on a unit volume and ^ and v are the fluid density and velocity. For a uniformly rotating system in a gravitational field, this takes on the form where F is the force oh. a unit volume. of fluid which is rotating with an angular velocity a;, at a distance r from the axis of rotation; g is the gravitational constant, and z the height above some arbitrary level. It can easily be. shown that the free surface of a. volume of fluid rotating about a vertical axis takes on a parabolic cross-section, the steepness ;of which is proportional to the square;of the angular velocity, The height of the surface above the lowest (axial) point at a distance r from theaxis is given by Z = r2_a>2 2g This equation is independent of the density of the liquid. In He II the general equations of motion must be modified to.agree with the two fluid theory, and.allow for thermo-mechanical effects. They become, in linear approximation, ignoring irreversible effects, e s Pvs = - grad p - gs S grad T*f Dt * $ n Dyn a " ^  P + <?s s T+f where f is the external force per unit volume. (See Daunt and Smith, 1945, p. 219). In a further treatment of the. equations-of motion of the superfluid component (London 1945, pp. 128 - 29) it becomes.apparent that the superfluid behaves as an ideal fluidand hence must have curl v s - 0 i .e. , all superfluid flow must be potential flow. Since, in addition to the above condition, the superfluid component is said to be completely non-viscous, one would expect that if a cylinder containing He II is. rotated on its central axisOnly the normal component would.rotate with it, and one expects a somewhat smaller curvature, of the surface, dependent upon the fraction of normal fluid present. Osborne (1950) shows.the surface shape would be governed b y „ 2 Z = $Z r 2 f 2g Experiments reporting observations of the surface, of a rotating volume have been carried out by a number of investigators (Osborne, 1950. Andronikashvili, 1952. Andronikashvili and Kaverlin, 1955. Donnelly, Chester, Walmsley and Lane, 1956) 12. and all have.reported that for all velocities,observed He II behaves like a .completely normal fluid. The se experiments .covered a wide range of angular and peripheral velocities: 0.5 to 15 rev. /sec. and 4 to 70 cm./sec. Over this.entire.range it is.evident that both the superfluid and normal components are moving together. It is interesting to compare the results of the above experiments with those carried but somewhat earlier on the problem of He II in oscillating systems. Here it was-found (Andronikashvili 1946, 1948) that when a system of closely spaced discs suspended in He II was.set oscillating that only the normal component of the fluid moved with them, and indeed ..these measurements provided the first direct measurement of as it varied with temperature. Hollis Hallett (1950) extended these measurements by, varying the.amplitudes, of oscillation. He found that at amplitudes of >0.1 radian the damping increased with amplitude, though for all.smaller values in He.II, and all values in He I, the damping was independent of amplitude. The peripheral velocity above which this.effect occurred was.0.1 cm/sec. The. first explanation attempted for the occurrence of this non-linear behaviour was made in terms of turbulence. However, Hollis.Hallett (1952)has shown that the Reynolds' numbers for which these effects.first occur are too low for the appearance of turbulence (compared to the onset of turbulence in He.I), and that these Reynolds' numbers are not independent of temperature as one would expect. Inxarder to explain a variety of non-linear effects in He II such as critical velocity in film flow,. and the irregular dependence of heat flow pn temperature difference at high rates of flow, a variety of modifications of the first order equations ;of motion as given above have been suggested. A review of the various. forms of these equations has been given by Daunt and Smith (1954, pp. 218 - 21). The .most successful of these, and the only one.to have any real experimental verification, was that prpposed by Gorter and Mellink (1949). They suggested that a mutual friction term proportional to .the cube of the velocity difference :of the two fluids be. included in each equation with the appropriate, sign. Explicitly the force per unit volume was written F = £ n $s A ( v s - v n)3 where A (evaluated from the heat flow experiments) is a constant with a value.of *» 50 cm. sec./gm. Hollis Hallett (1952) pointed out that these -forces wouldJpnly account for approximately. 10% of the additional energy dissipation in his.oscillating disc.experiments, and that if they are present there must also be other forces,, as yet not understood. In order to provide a further check on these mutual friction types;of forces mother rotational experiments were undertaken. Such forces must decrease to zero as a steady state of rotation is attained. Hollis Hallett (1953) measured the. coefficient of viscosity of He .II usinga rotating viscometer and found that above .a certain critical velocity the torque, on the inner.cylinder was not proportional to the angular velocity of the outer rotating cylinder. This experiment was later repeated (Heikkila and Hollis Hallett, 1955) and the variation of the.critical velocity with temperature was found. Hall (1955) has.also reported measurements :of the frictional forces occurring during the deceleration of the superfluid .component after a steady state ;of rotation had been achieved. His results show that.in addition to Gorter-Mellink forces, and possible turbulent effects, there is an additional force proportional to the absolute value.of the angular velocity. Experiments have also been carried out using the attenuation of second sound to measure.the additional non-linear processes occurring in rotating He.II. Hall and Vinen (1955 a, b) find that when the liquid.is rotating additional attenuation occurs which is proportional to .the angular velocity. This is in direct agreement with Hall's results quoted above, usingmechanical measurements. Hall and Vinen suggest a dissipative force of form F = B f af» jmJlVs - V n ) S Using a somewhat different geometry, another group (Wheeler, Blakewood and Lane, 1955,,, Blakewood, Walmsley and Lane, 1956),has done a similar series of experiments, finding that the additional attenuation is .related exponentially, to the angular velocity. It has been suggested by London (1954, pp. 151 - 55) that there, is another mode,of rotation of superfluid helium which might agree with the observations of Osborne and others on the shape of the surface of rotating He II, and yet still preserve potential flow (curl v s = JO). This model would picture the rotating superfluid broken up into a.series of concentric-cylinders within each of which the velocity varies as 1/r, there being discontinuities.in v s at the boundaries of each cylinder. London finds, that for very small velocities (tv< 4^.^^ where m is the mass of the helium atom and a the radius of the container) the superfluid component of the liquid would not move with the container. For higher angular velocities the motion would break.into curl free.motion in concentric cylinders, the width of the individual cylinders becoming less as «J.increases. Landau and Lifschitz.(1955) have further discussed this model and made allowance for a surface.energy occurring at the boundaries between cylinders. Bhagat and.Pathria (1957) discuss two particular experiments .in terms of this model and show that if pne.assumes .mat turbulence Occurs when the rings become.narrower than a critical width, these two experiments are in complete agreement. Feynman (Gorter, 1955. Chap. II) has recently suggested a third po ssible model. By attempting to find the lowest pos sible. energy state for a volume of rotating He II, he shows that another type of irrotational flow would occur if the liquid should form within itself a number of quantized vortex; "lines", the density of these vortices being proportional to the angular velocity of the rotating system. The smallest of these vortices he.identifies with the Landau rotons. All these, theoretical views show that for a steady state rotation of Liquid He II the whole body of fluid is moving. However, as has. been mentioned above, for low amplitude oscillations it is known experimentally that only the.normal fluid is in motion. An experiment was therefore designed by Esel'son, Lazarev, Sinel'nikov and Shvets (1957) to measure, the.relaxation time between one state.and the other. This has shown that for initial velocities of 0.5 rev./sec. relaxation times, if they occur, must be less than 2 sec. and thus cannot account for the failure of the superfluid to move with low amplitude.oscillations the, ease o f and/|oscillations of infinite.amplitude (steady.rotation) remains unresolved at the present time. The present experiment was proposed to test a possible further explanation of why rotating liquid helium with a free surface fails tO;obey the two fluid.equations of motion. Osborne.(1950) showed that in a rotating vessel of He II there should be a radial, temperature gradient of the form T(r) - T(0), = J_ t*J 2 r 2  2 S;ri where Wis the angular velocity of me container, r the radius, and S n the entropy of the.normal component of the fluid. Thus for a container . of radius 2 cm. and rotating at 2 rev. /sec., arid at a temperature of about 1. 5 U K. (S^ = 0.1 cal. /gm. deg.) a temperature .difference.of ~ 10"^°K. will exist between the center and the periphery of the liquid. This difference.can be,considered to be set up by the .centrifugal force acting on the norm al component as it rotates and tending to move it away from the axis. The superfluid is not considered to be.rotating and hence experiences ho force. The result is that the ratio t*j^ increases with radius, causing a temperature gradient. It was thought that the existence :of this temperature gradient wouldlead to a difference of rate of evaporation across the liquid which is effectively a distillation of liquid from the warmer to the cooler regions. The tendency of this process to annul the temperature difference would be overcome by the energy of rotation which is constantly being fed.in. However, on evaporating, a normal.atom will take up energy from other . normal atoms in the liquid and presumably one or more of these may drop into the superfluid state and maintain its rotation, though now it will not interact with the normal rotating fluid. On recondensation nearer the axis, the atom will give up energy to the fluid and presumably convert one or more superfluid atoms into the normal state where .they experience, rotation, and begin to migrate, away from the axis. In this manner it is possible to suppose that, the entire mass qf fluid would eventually be rotating, though both the superfluid and normal components are still present. Of course the superfluid would still have.to experience curl-free flow, perhaps in concentric-cylinders, as proposed by London, or in a system containing Feynman's vortex lines. From kinetic theory we know that "0, the total number of molecular collisions/on either side.of a liquid surface at equilibrium with I-unit area. the vapour above it, is given by -0, = n ? .-= n 4 2 and the variation of the number of collisions with temperature is given by where we have.calculated n and -jrp from the ideal gas.relation n = mc 2 kT If we use a more exact relation for the pressure p, such as that given by Keesom (1942, p. 191) i Q T O O 7-978 0.13628.4.3634 l ogPcm - 3-729 Y~ ' T 2 ~ + ~W~ we have dPcm = Pern T 7.978- . 0.27256 _ 13.090 1 ~dT~ .43429 L T 2 T4 J evaluating this for helium, at a temperature of 1.5°K. (p z 0.360 cm. Hg) we find JP_ ~ l .15 x 10 4 dynes/em2 - deg. dT and finally = 8.9 x IO 2 2 collisions/deg. cm. 2 If we now consider the total energy transported by each atom in this .distillation process we find that, taking the heat of vapourization of helium as approximately 6 cal/g. the heat of vapourization per atom is.approximately 1.7 x lO"^' erg. This i s the approximate.amount of energy taken up by each atom as it evaporates and given up as.it condenses back into the fluid. In adjusting the Bose-Einstein condensation theory to the experimental thermal data for helium II London (1954, pp. 53 - 56) made use of an energy spectrum with an energy gap /& between the ground state and the first excited state. The value of this gap which gives the best fit to the specific heat curve is A / k */4Tj ~ 8.8° The temperature corresponding to the heat of vapourization per atom is given by 1.7 x.10-15- _ 1 2 o k .and it is.seen that these temperatures are.of the same magnitude. Therefore if we assume that every atom colliding with the. surface passes into the other phase.(even in a normal fluid the evidence on this point is rather contradictory) (Partington 1949, pp.930 - 32) and that on the average each normal atom escaping near the periphery carries off enough energy to cause one other normal atom to drop into the ground state, and on recon-densing gives,up enough energy to lift one superfluid atom out of the ground state, the given temperature difference will be.sufficient to change 2 x (AT) atoms /sec -cm2. If, as.a rough approximation, we divide the surface of the liquid into two equal regions of uniform temperature differing by A T = 10"^°K. we find that the total number of .atoms converted per second is f - 2 ( ^ ) ( A T ) ^ 1 0 2 0 where A is the area of the whole surface (--v 12.cm.2). It would appear then that this.effect might have some influence in setting the superfluid into rotation and in an experiment designed to eliminate this effect it would be of interest to note if the parabolic rise of the surface is proportional to f n/^ as.Osborne predicted. CHAPTER 3 APPARATUS The apparatus used in the present set of experiments was developed in order to allow the.rotation of a small quantity of liquid helium at temperatures ranging from the normal boiling point of the fluid (4.2° K) to about 1°K., and at a reasonably wide.range of angular velocities. The shape of the container was so.chosen that the possibility of a distil-lation process occurring between the outermost portions :of the rotating fluid and the axial region would be eliminated. By making the container completely of transparent material, any difference in levels at the outside and.centre of the fluid was observable. A number of different types_of container were tried before .the final design was achieved (see Figure 2). The glasscontainers were all slightly eccentric, and the joins.in the first Lucite container did not withstand the strains introduced in cooling to low temperatures The design which was finally successful is shown in Figure.3. The container was constructed from three sections Of Lucite, machined and then polished using #1000 grit on a cotton buffer. Seals were made by dissolving.the surface .of the plastic with ethylene dicbjtoride, and leaving the join under pressure overnight. The overall dimensions of the container were 1 1/2" by 2 7/8", and the radius of the inner face :of the outer wall was. 11/16". The central columnwas of 1/4" tubing with a 1/8" bore and the container was filled through a tiny hole drilled in the central column just below the level of the top of the outside ring. Ia He II the.rate of filling was quite rapid: the whole container filled in less than one minute. To give added rigidity to the central column a. collar of bakelite was fitted to sit on the topiqf the outer ring. This collar had holes drilled in it to allow helium to flow easily past it. At the top of the central column and fixed to it by a pin, a brass insert was fitted: this was attached to a flexible coupling which linked it to the driving,shaft. The container was.mounted in a brass holder about 10 cm. high aaa" had a diameter just slightly smaller than the inner diameter of the Dewar. This holder could move vertically over a range of about 26 cm. slidingon three brass;rods. This apparatus is shown in cross-section in Figure 4. These.rods.were spring loaded.to the inner wall of the Dewar andremained in a given position while the holder was moved over their length. Movement was provided, by a germ an silver tube which passed through an O-ring seaLand could be raised or lowered, from above ..the cryostat. The bearing arrangement for the container, offered some difficulty. At first a small ball-bearing collar was used to support the top of the central column, but this proved unreliable, since it was easily fouled by any small amount of solid air contamination that might be present. Also, since it had to be.run without lubricant, it was found rod c o n t r o l l i n g v e r t i c a l motion holder l u c i t e container A, 0 pc; .drive shaft brass sheet polyethylene c o l l a r f i l l i n g hole ways for v e r t i c a l motion FIGURE 4. Rotating Container and Holder variable speed d „ C o motor 5*1 reduction Rear Selsyn motors h e l i c a l gears f l e x i b l e d rive telescoping drive f l e x i b l e drive FIGURE 5 . Block Diagram of rotating Rotating System container that unless the apparatus was dismantled and tfioroughly. dried immediately following an experimental run, rusting soon made the bearing useless. The bearing that finally proved satisfactory was merely a small piece of 1/64" brass sheet soldered to a small collar which was press-fitted into the cup which had previously held3the ball-bearing. A hole was then drilled in the centre.of the brass sheet, and.it was .machined to fit, with a very small clearance, around the brass insert above the Lucite. The bearing at the base.of the.container merely cpnsistedof a pointed tip machined on the bottom of the.container which fitted into a small V-shaped hole in the centre of the holder. The container was prevented from coming out of alignment when being raised or lowered by the presence of a small polyethylene collar over the brass insert just below the upper bearing. Since the coefficient of linear expansion of Lucite is :considerably larger than that of brass, it was found necessary to force this collar very tightly up against the bearing when assembling the apparatus at room temperature, in order that it would not be too loose when the apparatus was .cooled to liquid helium temperatures. A schematic diagram of the driving system is shown in Figure 5. The drive shaft consisted:of two pieces^of 4 and. 10 mm. german silver tubing telescoping into each other, to allow for vertical movement, and keye d together to transmit the rotary motion. It was connected through a pair, of helical gears, mounted above the cap on the head plate, and a second flexible drive, to a Selsyn motor also mounted on the head plate. The use of this Selsyn made it pos sible to remove the main variable speed d. c. motor SP3T switch 350 Ik potentio-meter rotor f i e l d A. motor i e l d 40 W. lamp alter-" nator^ f i e l d B. 190 <; 95 V . T 350 contactor 2.2 K A - A A A A A / v -0.25 /*fd. scaler FIGURE 6. E l e c t r i c a l Arrangements (A) d.c. motor (B) motor-alternator (C) counting devices from the cryostat and.reduce the amount of vibration. Also it was possible to mount, the. receiving Selsyn inside the vacuum system and thus eliminate the need for a moving vacuum joint. The Selsyns .were, run-well under their rated power to remove any danger of overheating. Rotation was provided by a d.c. motor (Figure 6a) whose speed was;made variable by tapping the.rotor voltage from a 350 ohm slide-wire potentiometer connected across the 110 V. line. The output of this motor geared down by a ratio.of 5:1 was used'to drive the transmitting Selsyn. Originally the Selsyns were supplied at 25V. 60 cycles. Though they were designed for 400 cycle.operation, the simplicity'of this arrangement made it very desirable. However, it was found that the torque output from the following Selsyn was not sufficient to rotate the.container reliably, and .coukLonly be increased by stepping up the voltage, which led to serious overheating. A 500 cycle, 220 V. d.c. motor-alternator (Figure 6b) was finally obtained, and it was found that when this was joperated at 110 V. and with an external 40 W. lamp placed in series with the alternator winding the output of 95 V. at 350 cycles.was sufficient to provide.the necessary torque by the Selsyn. A power resistor of 190 ohms was used to provide a load when the Selsyns were shut off. When using the earlier rotating containers, which had a closed stand pipe at the outer edge instead of the complete annular ring, it was necessary to light the system stroboscopically in order to make observations. For this purpose a General Radio Strobotac was .used, synchronization being provided by a contactor connected to the shaft of the external Selsyn (see . Figure 6c). The rotations were counted over a known time interval by feeding pulses from the Strobotac through a smoothing network and into a scaler. For the.rotatingcontainer which finally proved successful, it was not necessary to use stroboscopic lighting, and a 6 V. projection lamp and focussing system was used. A copper sulphate xe l l was provided to filter out the red end. of the spectrum. The stroboscope arrangement was .also retained, however, because of the ease of counting rotations by this.means. Measurement of the liquid levels was ;carried out with a Gaertner cathetometer having, a vertical range of 102.cm. and an accuracy of 0.002 cm. The telescope of the cathetometer was provided with an additional objective lens which produced a focal, length of approximately 25 cm. and afieldxrf vision of 1.3.cm. diameter. The high magnification gave readings, accurate to the. limit aUowed.bythe stability of the helium-surface. Since it was necessary to.change the focus of the telescope to take.readings, of the two.liquid surfaces the telescope had;to be carefully levelled. The method.of carrying out this adjustment is-describedin the Appendix. The -arrangement of the,Dewars. was the. standard one used in this laboratory. An inner Dewar of inner diameter 6.5 cm. and volume <"<2.6 litres was attached to the cap by a vacuum-tight rubber and glycerine seal. An outer Dewar to hold liquid nitrogen was also used. The b^ ody of the latter had, over the main part of its length, inner diameter of ^8.5 cm. However, the top 12 cm. had an inner diameter about 1 cm. larger, the high speed pump gate valve needle valve return l i n e ^ He Dewa 5 l i t r e reservoir siphon jacket 4 -vacuum space«*_4) i n He Dewar 4] d i f f u s i o n pump discharge tube l i q u i d a i r trap out rotary pump o i l manometer L FIGURE f . Vacuum Syatem extra space, acting.as a reservoir of liquid nitrogen. A loose cap was fitted over the outer Dewar to reduce to a minimum the solid carbon dioxideand water vapour impurities which collect in the bath, and which can seriously impede the visibility. Both Dewars had unsUvered viewing strips from 1.5.to 2.0 cm. wide down their whole-length. Provision was also made for oooling the cap of the inner Dewar, but in practice this was rarely done. The vacuum system is shown in Figure 7. The pumps used .were a conventional mercury diffusion pump backed by a Welch "Duo-Seal" single stage mechanical pump. A 5 litre reservoir was also provided between the diffusion and mechanical pumps so that if necessary, the mechanical pump,could be turned off for periods up to half an hour to reduce vibration. For pumping on the helium bath to obtain temperatures below the normal boiling point, a.Kinney high speed -mechanical pump was used. This pump had a capacity of 1200 litres/min. and was.connected to the cryostat by a 4" pumping line. Pressures were controlled at the cryostat by a large gate valve in the main pumping line, .and.a small needle valve in parallel with it. Both valves were specially treated.to make them vacuum tight. Pressures were measured-using two manometers. One contained mercury, the other Apiezon Boil having, a. density of 0.8641 gm./cc so that 1 cm. of oil was the equivalent of 0.06377 cm. of.mercury. Liquid helium for the. low temperature group is produced by a Collins.Helium Cryostat capable of liquefying4.5 litres/hour after an initial precooling period of 3 hours. A full description of this apparatus is.available in the.literature (Collins, 1947). The.helium is,then stored in a 25 litre.double Dewar transport vessel where, it is surrounded by a liquid nitrogen bath. The rate of evaporation of helium from this vessel is approximately 0.2 litres /day. Liquid nit rogen for the laboratory, is provided.by a Joy Nitrogen Generator (Type.NXL-45) which produces .20 litres.of liquid per hour after an initial cooling period of 3 hours, and which has a storage .capacity, of 450 litres. All helium used in the-experimental apparatus in the laboratory is collected and.returned.to a gas .holder. It is compressed, cleaned and stored in cylinders until the next liquefaction. The overall collection cycle including liquefaction is .about 80% efficient. CHAPTER 4 EXPERIMENTAL PROCEDURE The.necessary preparations for an experiment were fortunately not lengthy and.could beicompleted in a few hours.time. The Dewars.were first thoroughly cleaned with soap and water, rinsed several times with distilled water, then with, acetone and then dried. Meanwhile the container was mounted within the.holder, making sure that the bearing-of the holder pressed very tightly on the polyethylene .collar. The complete apparatus was then assembled on the rotating shaft and the holder fastened to the.rodIcontrolling vertical movement. The inner Dewar was put in place, care being taken that the viewing slits were not obstructed by the apparatus. The rubber and glycerine seal was wired to the top of the Dewar, and the outer Dewar and its cap.put in place. Finally the brass plate of the Dewar support was fixed and the bottom of the support bolted to a strut extending from thexryostat frame, to give greater rigidity. The siphon for transferring liquid helium into the apparatus was inserted through the.cap, and its.Outer end sealed with a small piece of rubber tubing. It was then possible to flush all the . air from the inner Dewar and siphon, and replace it by clean helium gas. Generally the Dewar was flushed three.times and finally left with one. atmosphere of helium gas in it. The pumping and fillingof the Dewar was.always carried out very slowly to prevent a large pressure difference occurring across the walls.of the Lucite container. The vacuum space.of the inner Dewar was pumped with the .fore pump until the discharge, tube showed a pressure of about 0.5 mm. Hg., i.e., a molecular mean free path of the order of the width of the vacuum space. To precool the system the outer Dewar was filled with liquid nitrogen and the system allowed to stand for at least three hours to.ensure that all the apparatus was at liquid nitrogen temperature . During this time it was necessary periodically, to add more helium gas to the inner Dewar to maintain atmospheric pressure. After precooLtng was judged.to be.complete the vacuum jacket of the siiphon and the.vacuum space of the inner Dewar were pumped hard with the diffusion pump, and the apparatus was,ready to receive liquid helium. The transfer of liquid helium was accomplished by moving the 25 litre storage vessel to the cryostat, inserting the.siphon into the liquid and exerting a slight overpressure (3,to 7 cm. Hg) of helium gas on the surface of the liquid. It is most important that air be kept out of the siphon while inserting.it into the storage vessel, sjnce.evena small amount of solid air could block.the siphon and prevent the passage of the liquid. During the transfer the.return line was.kept open to allow evap-orating helium to.collect in the gasholder. It was.found that about 0.5 litres of liquid.were evaporated in cooling th e apparatus from nitrogen temperatures (77°K.) to helium temperatures (4.2°K.): after this temperature had been reached liquid helium collected at its.normal boiling point. Usually from 1.5 to 2.0 litres of liquid were transferred, since after pumping this,left the maximum quantity of liquid which it was possible to use. To pump down to temperatures below the normal boiling point the high speed mechanical pump was used. The.lowest pressure -which could be attained with the present arrangement was .15.6 mm. of oil which corres-ponds to a temperature of 1.26° K. (T55 E scale). The container was ;dipped below the. surf ace. of the liquid helium and-allowed to fill to within about 1 cm. of the top of the annular ring. It was then raised until only its bottom.region was immersed in the bath. The. cathetometer, which had meanwhile been levelled-and focussed, was used to check for any differences :of level while the container was at rest. Such a difference occurred only.once, and this.set of readings, after being, corrected for this spurious difference in. height, was in close agreement with previous.data. The procedure.used in taldngreadings was as follows. The container, was started rotating, then the counter and stop watch were started simultaneously. Readings were taken of the.liquid levels .in the outside ring.and in the central column, then each of these.readings was .repeated. The manometers were read to.determine the vapour pressure. Finally the watch and .counter were stopped and me number of-rotations and time.interval noted. This procedure usually took.between two and three minutes, and the evaporation rates were low enough that during the time of the readings the loss of liquid from the container was negligible. The rheostat controlling the d.c. motor speed was then r'ejset and the procedure.repeated at a lower velocity. About five, different angular velocities were usually taken at any particular temperature, and in the coursejof an experiment three.or four different temperatures might be used. The major difficulties.to be overcome in obtaining;an accurate set of readings were the. elimination of vibration and maintenance of a.steady rotation. In the glass containers which were first used it was impossible to obtain accurate readings because.of slight eccentricities,in the.rotating container. It was also found that for angular velocities below about 4.5 radians/sec. (0.75 rev. /sec.) the motion became irregular due to the tendency of the Selsyn rotor to take up.discrete positions, and since the differences.in height involved were very small, it was riot possible to get accurate readings. Finally, it was attempted to obtain a comparative set of data for He I. However, it proved impossible to get such readings with the present apparatus. When the container was dipped below the bath level helium would fill only the bottom region and central, column, but would not rise into the outer ring, even when left for more than one hour. If the pressure was then reduced until the A -point was reached, the container filled with He II, but when the pressure.was.allowed to rise again, the gas above the.liquid in the outer ring expanded, forcing the outer liquid level down, and raising that in the central colum. Again it was found that the levels would not equalize, even after a period of almost two hours. This.attempt was then abandoned since it was :quite certain that the.liquid levels.in rotating He.I would correspond to those;of any normal fluid. CHAPTER 5 RESULTS The results of measurements of liquid level differences in the central column and the outside, ring of the container at a variety . of different angular velocities and temperatures are given in Table I, and shown graphically in Figure 8. These.values represent the most reliable re suits obtained and were taken during two separate experimental runs about two weeks apart. In all cases the_irregular "bouncing" of the liquid helium surface was at a.minimum and an average-error of 0.006cm. is probably valid for all the figures given with the: exception of the readings taken at 1.92°K. In the latter case the given values have been corrected to take.account of the.spuriousTevel difference which was found when the container was.stopped at the.conclusion of the readings. Unfortunately the value for this correction was :of limited accuracy and the third significant figure of the quoted values for Z are in considerable doubt. The source .of this spurious A Z, is not known, and it occurred only in this one .instance. The. channel connecting the two measured regions was wide enough ( > 1/8") that any thermomech-anical effect can be ruled out, and it seems unlikely that any surface tension effects would produce the difference when they, were, smaller by an order of magnitude.at all lower temperatures; there is.no abnormal TABLE I. Trial (rev/sec) Z (cm) Z t A Z (cm) V..P.. /mm Hg) ,T A - 1 - 2 - 3 .-4 - 5 .- 6 3.20 2.66 2.12 1.56 1.10 0.704 0.562 0.395 0.245 0.150 0.095 0.038 0.577 0.410 0.260 0.165 0.110 0.053 0.931 1.26°K ~ 4 % B - 1 3.13 -2 .2.66 -3 2.13 -4 1.57 -5 1.03 -6 0.649 C - 1 3.17 -2 2.36 -3 1.57 - 4 0.821 0.505 0.520 3.40 0.393 0.408 0.252 0.267 0.145 0.160 0.070 0.085 0.025 0.040 0.520 0.535 5.78, 0.300 -0.315 0.150 0.165 0.038 0.053 1.49°K -^10% 1.60°K ~ 1 5 % D - 1 3.16 0.55(5) 0.56(9) 18.4 1.92°K - 2 2.59 0.37(0) 0.38(4) - 3 2.08 0.24(2) 0.25(6) 4 1.56 0.15(0) 0.16(4) '54% Data for sets A, Band D were, recorded June 25, 1957. Data of set C was recorded July 9, 1957. During the reading pf set C there was a slow drop in temperature ;of 0.01 °,K. shown by a change in vapour pressure.of 0.22 mm. Hg over ^30 minutes. The values given for Z and Z + A Z in set D are corrected to allow for a spurious height difference of •¥ 0.11(0) cm. which was found to occur with the . container at rest. FIGURE 8 . Graph of R©aults + T - 1 . 2 6 0 K. n T = 1.49° K. T « 1.60° K. T = 1.92° K. increase in the surface.tension itself in this region. Small surface tension corrections ( AZ) have been added to.all readings taken before they were plotted. These.were calculated as follows. In the central column the rise due to surface tension h^ is given by h x = 2S_ where S is the coefficient of surface-tension and _r is the inner radius o f the tube. The rise Jin the outer ring is calculated using the formula for rise between two parallel plates h 2 = 2S byg where b is the separation of the two walls. In the container used b = 2r = 1/8" and consequently hi = 2h2» Thus the correction to be added to the observed reading is given by Z = hv- h2.z h 2 = 2S For the values.of surface tension given by Keesom (1942. p. 263) the correction at T ° 1.92°K. was.0.014cm. and at the lower temperatures the correction was 0.015 cm. Vapour pressures quoted were.all read on the.oil mano-meter and converted to mm. of Hg. Temperature s given are. in accord-ance with the T55E scale (Clement, Logan and Gaffney, 1955. See "Note added in proof"). Values.of § n/g are .taken from .second sound and entropy data as.shown by London (1954. p. 109). In. Figure 8 the. difference in liquid level is graphed against the frequency, of rotation. (T) = ^ /^TJ1)' The solid curve shows the expected difference for a normal fluid, i.e. where $ n/^ .= 1. It is immediately evident mat me experimental values.unanimously show almost exactly classical behaviour, and.fail.completely to exhibit the much reduced rise which would occur ifonly. the normal fluid were rotating. In the latter case, as shown by Osborne, the rise would be proportional to 9?-/^ and for example.the data taken at 1.49° K. should show a rise of only one-tenth that of a classical fluid. CHAPTER 6 DISCUSSION From the results plotted in Figure 8 it is clear that under the condition of the present experiment He II is behaving as a. classical liquid, and both components partake of the.rotary motion. These.results are in complete-agreement with previous.experiments:of this type performed by Osborne (1950) and.other workers.since. The proposed distillation mechanism can have, at most, a very small effect and one which is masked by some.other coupling of the two fluids. It would, seem, as pointed out by Hollis Hallett, that the Gorter-Mellink forces do,not satisfactorily explain these.steady state experiments, as;any term.in (v s - v n ) must reduce to zero. It is poisible that a forceof the type proposed by Hall and Vinen (1955b), proportional to the absolute angular velocity may exist between the two.fluids, though the .expression proposed by them also contained as a factor the difference in velocity of the two components. However, the present experiments will give us no information of the variation in magnitude of this force, other than showing that it is present over the whole, rangex>f angular velocities tested. Further, experiments :of the type carriedout by Andrqnikashvili and.Kaverlin (1955) and by. Donnelly, etal (1956) where.the surface shape was.studied as a function of time.at the beginning.of the rotary motion might conceivably, lead to further information about the kind of forces .involved, in the present experiment .no.large scale, unusual surface effectswvere noted as the rotation commenced, but the surface elements were, difficult to see with the naked eye, and no systematic ^ observations -of this.kind were attempted. One or two further experiments.are suggested by the present results. First it would be interesting, if a steady rotation could be maintained, to carry, out the present measurements at lower angular velocities in order to find whether there is. a critical velocity below which the.superfluid.does not rotate. However, in view of Hollis Hallett's estimateof a critical velocity of 0.1 cm./sec. it is doubtful that the . liquid rise would be observable at the angular velocities of interest. Secondly, it would be of considerable interest.to.attempt to actually measure the temperature difference between the center and the outside :of the.liquid. Since this .difference is. on the order of 10 _4oK. there would be many practical difficulties involved.in its measurement in a rotating system, but if the temperature was found Osborne's prediction would be vindicated. Finally, it would be of great interest to. suspend small hydrogen -deuterium particles in the liquid helium, at the.lowest temper-ature, as done by Chopra and Brown (1957). For it is possible, that direct evidence of the mode of rotational.flow of the superfluid.could be obtained. It might then be possible to decide between the stratification model of London and Landau and Lifshitz-and the vortex model of Feynman as to which presents the true picture of rotating.superfluid. APPENDIX I Since it was necessary to .change.the focus of the catheto-meter telescope between the two readingsjof the liquid level, great care had to be taken that the telescope itself was accurately levelled. The L. following method was used to achieve this adjustment. (1) The cathetometer was set up at the midpoint between two scales .Sand S'^  (~ 12-m. apart) and the column made vertical by adjustment of the feet of the cathetometer. (2) The.telescope was sighted on scale Sand then rotated through 180° and sighted on scale S*. These.readings were noted, and since both scales are the same distance from the .telescope, the readings were known to be in a horizontal line, even if the telescope made some angle & with the horizontal. (3) The cathetometer was. then moved close ( < 1 an.) to scale Sand again adjusted so thexolumn was vertical. The telescope was.moved vertically until the crosshairs were.on the same.reading.S^as previously. (4) The telescope was turned through 180° and .the levelling screw on the tele scope, adjusted, until the correct readingon scale. S* was aligned with the crosshair. The telescope.was then refocussed on scale.S and again moved vertically until the. reading ;on S was correct. These adjustments to scales Sand S^  were repeated until each reading;appearedon the appropriate, scale without further adjustment. The telescope was then accurately horizontal. (5) The adjustment screws ;on the spirit level attached to the telescope were used.to make the level itself horizontal. It was now guaranteed that when the spirit level is horizontal the telescope is also. After this adjustment was made all trouble with spurious height differences with the liquid at rest was removed. BIBLIOGRAPHY Allen, J . F . & H. Jones (1938). Nature 141, 243. Allen, J. F. & A. D. Misener (1938). Nature 14^, 75 Andronikashvili, E . L . (1946). Jour. Phys. U.S.S..R. I0..20I. Phys. Abstracts 50, #321 (1947). Andronikashvili, E. L. (1948). Jour. Expt. Theor. Phys. 18_, 424. Phys. Abstracts 5l_, #3945 (1949). Andronikashvili, E . L . (19.52). Jour. Expt. Theor. Phys. 22, 62. Phys. Abstracts. 56, #325.6 (19 52). Andronikashvili, E . L . & I. P. Kaverlin (t955). Soviet Phys. J. E. T. P.J_, .174. Atkins, K. R. & D. V. Osborne (1950). Phil. Mag. 41, 1078. Bhagat, S. M. & R. K. Pathria (1957). Phys. Rev. 106, 3. Blakewood, C. H., R. H.. Walmsley, & C. T. Lane (1956). Phys. Rev. 104, 1495. Chopra, K. L. & J. B. Brown (1957). Phys. Rev. To be published. Clement, J. R., J.K. Logan &.J. Gaffney (1955). Phys. Rev. 100, 743. Collins, S.C. (1947). Rev. Sci. Inst. 18_, 157. Daunt, J. G. & C . V . Heer (1950). Phys. Rev. 79, 46. Daunt, J. G. & R. S. Smith (1954). Rev. Mod..Phys. 26, 172 Donnelly, R. J., G..V. Chester, R. H..Walmsley & C. T. Lane (1956). Phys. Rev. 102, 3. Esel'son, B. N., B. G. Lazarev, K.D.Sinel'nikov & A. D. Shvets (1957). Soviet Phys. J. E. T. P. 4, 774 . Gorter, C. J. (1955). Progress in low temperature physics, vol. I. Amsterdam: North-Holland. Gorter, C.J. & J. H. Mellink (1949). Physica 15, 285. Hall, H. E . (1955). Conf. Phys. Basses Temp. Paris. Abstract #21. Hall, H. E. & W. F. Vinen (1955a). Conf. Phys. Basses Temp. Paris. Abstract #22. Hall, H..E. & W.F.,Vinen (1955b). Phil. Mag. 46 , 546. Heikkila, W. J. & A. C. Hollis HaUett (1955). Can. Jour. Phys. 33, 420. HoUis Hallett, A..C. (1950). Proc. Phys. Soc. A63, 1367. HoUis HaUett, A. C. (1952). Proc. Roy. Soc. A2J0, 404. Hollis Hallett, A. C. (1953). Proc.Camb. Phil. Soc. 49_, 717. Kammerlingh Onnes, H. (1908). Commun. Leiden #108. Cited by Keesom (1942) p. 152. Kammerlingh Onnes, H. (1911). Commun. Leiden #119. Cited by Keesom (1942) p. 206. Kammerlingh Onnes, H. (1922). Commun. Leiden.#159. Kammerlingh Onnes, H. |c.J. D..A. Bbks (1924). Commun. Leiden #170b. Kapitza, P. (1938). Nature 141_, 74. Keesom, W. H. (1942). Helium. Amsterdam: Elsevier. Keesom, W.H. & A. P. Keesom (1932). Commun Leiden #221d. Keesom.W. H. & A. P. Keesom (1936). Physica 3, 359. Landau, L. D. (1941). Phys. Rev. 60, 356. Landau, L ..D. & E . M. Lifshitz (1955). Dokl.Akad. Nauk. S.S.S.R. 100, 669. London, F. (1938). Phys. Rev. 54, 947. London, F. (1954)-. Superfluids, vol. I. New York; Wiley. Osborne, D. V. (1950). Proc. Phys. Soc. 63A, 909. Partington, J.R. (1949). Advanced treatise on physical chemistry, vol. I. London: Longmans. Peshkov, V. (1946). Physical Society. Report of Internal. Conf. Low Temp. vol. H, 19. Rollin, B. V. (1935). Thesis. Oxford. Cited in Kurti, Rollin & Simon (1936). Physica 2, 269. Rollin, B. V. & F. Simon (1939). Physica 6, 219. Temperley, H . N. V. (1952). Proc.. Phys.. Soc. A65, 490. Tisza, L. (1938a). Nature 141, 913. Tisza, L. (1938b), Compt. Rend. 207, 1035. Tisza, L. (1938c). Compt. Rend. 207, 1186. Wheeler, R. G., C..H. Blakewood & C. T.Lane (1955). Phys. Rev. 99, 1667. Wilhelm, J.O., A. D. Misener, & A.. R.Clarke (1935). Proc. Roy. Soc. A151, 342. Wolfke, M. & W.H. Keesom (1927). Commun. Leiden #190a. 

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