SOME APPLICATIONS OF SECOND SOUND IN LIQUID HELIUM r A. G. M. VAN DER BOOG SOME APPLICATIONS OF SECOND SOUND IN LIQUID HELIUM SOME APPLICATIONS OF SECOND SOUND IN LIQUID HELIUM PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE RIJKSUNIVERSITEIT TE LEIDEN, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. A. A. H. KASSENAAR. HOOGLERAAR IN DE FACULTEIT DER GENEESKUNDE, VOLGENS BESLUIT VAN HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP WOENSDAG 12 SEPTEMBER 1979 TE KLOKKE 15.15 UUR DOOR ALOYSIUS GERARDUS MARIA VAN DER BOOG GEBOREN TE LEIDEN IN 1948 KRIPS REPRO MEPPEL 1979 PROMOTOR: DR. H.C. KRAMERS. This investigation is part of the research program of the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM), which is financial- ly supported by the "Nederlandse Organisatie voor Zuiver-Wetenschap- pelijk Onderzoek11 (ZWO). \/SXr\ KWMV> ouofers CONTENTS. GENERAL INTRODUCTION. CHAPTER I. AC JOSEPHSON EFFECT IN % II. 11 1. Introduction 11 2. Order parameter and phase slippage by quantized vortex motion 12 3. The Richards-Anderson experiment 14 CHAPTER II. THE PARALLEL-FLOW APPARATUS. 17 1. Introduction 17 2. Principle of the apparatus 17 3. Experimental set-up 21 r 3.1. The apparatus 21 3.2. Registration of a helium level 22 3.3. Detection and generation of second sound 24 3.4. Constrictions 26 CHAPTER III. ATTEMPTS TO OBSERVE QUANTIZED VORTEX CREATION. 29 1. Introduction 29 2. Experiment on a single orifice 30 3. Experiment on a large number of parallel channels 33 3.1. Introduction 33 3.2. Experiment with CFI 34 3.3. Experiment with CFII 34 3.3.1. Example of a run with CFII 34 3.3.2. Velocity in the constriction 36 3.3.3. Second-sound attenuation 38 4. Second-sound generation due to vortex creation 41 5. Conclusion 42 CHAPTER IV. EXPERIMENTAL RESULTS ON THE VELOCITY OF SECOND SOUND AND THE VISCOSITY IN DILUTE 3HE-4HE MIXTURES. 44 1. Introduction 44 2. Theory of the resonator 45 2.1. Introduction 45 4 46 2.2. Sound in bulk He I 2.3. Second sound in a tube 47 2.4. Helmholtz resonator 50 2.4.1. Resonator and energy balance 50 2.4.2. Energy currents 51 2.4.3. Eigenfrequencies 56 2.4.4. Additional energy currents and dissipation 57 2.4.5. Response of the resonator to an external heat input 62 2.5. Dilute 3He-4He mixtures 63 3. Experimental set-up 64 3.1. Resonator 64 3.2. Second-sound detection and generation 66 3.3. Thermometry 68 4. Calibration measurements 68 5. Measurements in dilute He- He mixtures 74 5.1. Measurements and fitting procedure 74 5.2. Results on the second-sound velocity 76 5.3. Results on the viscosity 79 6. Concluding remarks 83 Chapter V. CALCULATION OF THE 3HE QUASIPARTICLE EFFECTIVE MASS FROM EXPERIMENTAL RESULTS ON THE SECOND-SOUND VELOCITY. 85 1. Introduction 85 2. Theory 86 2.1. He quasi particles 86 2.2. Low-concentration fixed-pressure expansion 87 j§£3. Effective masses mQ and m 88 '' 2.4. Second-sound velocity 89 2.5. Contributions of the He quasiparticle gas 90 2.6. Contributions of the potential energy density e* 92 2.7. Contributions of phonons and rotons 93 3. Results on m and e 95 4. Conclusion 97 SAMENVATTING 101 CURRICULUM VITAE 103 GENERAL INTRODUCTION. In the Hell region, that is, at temperatures below its "lambda temperature" (T = 2.18K), liquid He can be described in terms of a two-fluid model. Accor- ding to this model, Hell can be regarded as consisting of two components: a non-viscous superfluid component with density p ,and a normal viscous component with density p . The total density p equals the sum of p and p , and is approx- imately independent of temperature. The densities of the two components are temperature-dependent. At the lambda temperature, p = 0 and p = p, at zero temperature, p = 0 and p = p. In addition to a pressure wave called first sound, a second longitudinal hydrodynamic mode exists in Hell. This mode is a temperature or entropy wave and is called second sound. In a second-sound wave, the normal and superfluid com- ponents of the liquid are oscillating in such a way that the total density of the liquid remains approximately constant. This hydrodynamic mode also exists 3 4 3 in superfluid He- He mixtures; in this case, the He impurities take part in the motion of the normal component. In this thesis, two different investigations will be presented. In both investigations a second-sound resonance technique was employed. The first part of this thesis deals with the creation of quantized vor- tices at a constriction between two parts of an apparatus filled with Hell. A number of experiments have been carried out, which were attempts (a) to observe the average rate at which vortices are created at the constriction, and (b) to observe vortex synchronisation (ac Josephson effect) by making use of second sound to modulate the flow in the constriction. The constriction was either a single orifice or a large number of parallel channels. In chapter I, some aspects of the ac Josephson effect in Hell will be re- viewed. In this chapter, a short discussion will be given of the Richards- Anderson experiment. The results obtained in this last experiment were at one time interpreted as being an observation of the ac Josephson effect in Hell. In chapter II, the apparatus used in the present investigation will be des- cribed. In chapter III, a review will be given of the various experimental at- tempts. The conclusion must be that the sought-for effects were not observed. This is in agreement with the now generally accepted negative result of the Richards-Anderson experiment. The second part of this thesis deals with the properties of dilute He- He 3 mixtures in the temperature region 100 mK < T < 650 mK and at He molar concen- trations in the region O.173xlO"2 < X < l.OOxlO"2. A dilute 3He-4He mixture can be described in terms of models based on the assumption that the excita- tions of this system behave like a weakly interacting gas. These excitations 3 4 are both the He quasiparticles and, as in pure He, phonons and rotons. In chapter IV, an experiment will be presented in which a second-sound resonance technique was used to obtain values for both the velocity of second- 3 4 sound Ujrand the viscosity n in dilute He- He mixtures. The resonator was of the Helmholtz-type, with two low-frequency resonance modes. Because of the low resonance frequencies, the bulk dissipation could be neglected compared to the transverse viscous dissipation. Consequently, the attenuation of the resonator was determined mainly by the viscosity of the liquid. A detailed analysis of the resonator will be presented. The analysis provided a theoretical expres- sion for the shape of the resonance curves which could be fitted to the expe- rimental data for a derivation of values for UJT and n. Results on these quan- tities will be compared with results from other authors. In chapter V, the results on the second-sound velocity will be used to determine values for various parameters in the models describing the He qua- si particle energy spectrum and the effective interaction between the He quasi- particles. In particular, a value for the He quasiparticle effective mass m0 will be derived. To this end, a theory, based on these models and succesfully used at temperatures below the fermi temperature, was modified to be used in 3 the classical regime for the He quasiparticles. 10 CHAPTER I. AC JOSEPHSON EFFECT IN 4HE II. 1. INTRODUCTION. In 1962, Josephson [1] predicted the transmission of supercurrents by means of quantum-mechanical tunnelling through thin insulating barriers sepa- rating two superconductors. Since 1962, a large number of experiments were carried out which were a clear experimental verification of the existence of these quantum interference effects, or "Josephson effects", in superconduc- tors. In these experiments, the transmission of supercurrents was demonstrated not only through thin insulating barriers (oxide barriers, e.g. Shapiro [2]), but also through other "weak links", such as point contacts [3] and narrow constrictions (e.g. [4]). The quantum interference effects, or Josephson effects, are generally di- vided into two types. First, a dc tunnel current can flow through a weak link in the absence of a voltage difference across this junction. This effect is called the dc Josephson effect. Secondly, when a constant voltage difference V is established across the weak link, an oscillating supercurrent flows through this junction. The frequency v of the current is given by v=(2e/h)V where e is the charge of an electron and h is Planck's constant. This effect is called the ac Josephson effect. By electromagnetically irradiating the junc- tion at a frequency v1, current steps at constant voltage can be observed in the dc current-voltage characteristic of the junction when V=(h/2e)(n/m)v', with n and m integers. The analogy between superconductivity and superfluidity leads to the pre- diction that, under the right experimental conditions, one should also be able to observe the Josephson effects in superfluid helium. In 1965, Richards and Anderson[5 Ireported an experiment the result of which they interpreted to be 4 an observation of the ac Josephson effect in He II. Improved results were published by Richards [6] in 1970; the experiment was repeated by Khorana and Chandrasekhar [7] in 1967 and Khorana [8] in 1969, who obtained the same re- 11 suits.