Helium-4 normal-superfluid transition

Helium-4 Normal-Superfluid Transition Liquid helium has some unique and interesting properties, including a transition into a phase described as a superfluid. Unlike most materials where the isotopic nature of the atoms has little influence on the phase behavior, 4He and 3He have a very different phase behavior at low temperatures, and so we will consider them separately Figure 13.11 shows the phase diagram for 4He at low temperatures. The normal liquid phase of 4He is called liquid I. Line ab is the vapor pressure line along which (gas + liquid I) equilibrium is maintained, and the (liquid + gas) phase transition is first order. Point a is the critical point of 4He at T= 5.20 K and p — 0.229 MPa. At this point, the (liquid + gas) transition has become continuous. Line be represents the transition between normal liquid (liquid I) and a superfluid phase referred to as liquid II. Along this line the transition... 

Chapters 13 and 14 use thermodynamics to describe and predict phase equilibria. Chapter 13 limits the discussion to pure substances. Distinctions are made between first-order and continuous phase transitions, and examples are given of different types of continuous transitions, including the (liquid + gas) critical phase transition, order-disorder transitions involving position disorder, rotational disorder, and magnetic effects the helium normal-superfluid transition and conductor-superconductor transitions. Modem theories of phase transitions are described that show the parallel properties of the different types of continuous transitions, and demonstrate how these properties can be described with a general set of critical exponents. This discussion is an attempt to present to chemists the exciting advances made in the area of theories of phase transitions that is often relegated to physics tests. 

Obviously 0 =0 corresponds to the SmA phase. This transition is analogous to the normal-superfluid transition in liquid helium and the critical behaviour is described by the XT model. Fmther details can be found elsewhere [18. 19 and 20]. 

Helium is unusual in that it is the only known substance for which there is no triple point (i.e., no combination of pressure and temperature at which all three phases can co-exist). This is because the interatomic forces, which normally participate in the formation of solids, are so weak that they are of the same order as the zero-point energy. At 2.2 K helium undergoes a transition from liquid helium I to liquid helium II, the latter being a true liquid but exhibiting superconductivity and an immeasurably low viscosity [superfluidity). 

The SmA liquid crystalline phase results from the development of a one-dimensional density wave in the orientationally ordered nematic phase. The smectic wave vector q is parallel to the nematic director (along the z-axis) and the SmA order parameter i/r= i/r e is introduced by P( ) = Po[1+R6V ]- Thus the order parameter has a magnitude and a phase. This led de Gennes to point out the analogy with superfluid helium and the normal-superconductor transition in metals [7, 59]. This would than place the N-SmA transition in the three-dimensional XY universality class. However, there are two important sources of deviations from isotropic 3D-XY behavior. The first one is crossover from second-order to first-order behavior via a tricritical point due to coupling between the smectic order parameter y/ and the nematic order parameter Q. The second source of deviation from isotropic 3D-XY behavior arises from the coupling between director fluctuations and the smectic order parameter, which is intrinsically anisotropic [60-62]. 

Another type of transition is the second-order phase transition in which the first derivative of the chemical potential is continuous while the second derivative is not. This means that enthalpy, volume, and entropy vary continuously with temperature through a second-order phase transition temperature. This behavior is qualitatively different from that of a first-order phase transition, as illustrated in Figure 4.5. Whereas first-order phase transitions occur at a definite temperature for a given pressure, and with separation of the phases, second-order transitions do not exhibit a separation of phases and occur over a range of temperatures. The transition from superfluid helium to normal liquid helium and the transition from being a superconducting metal to being an ordinary conductor are examples of second-order transitions. 

Helium Purification and Liquefaction. HeHum, which is the lowest-boiling gas, has only 1 degree K difference between its normal boiling point (4.2 K) and its critical temperature (5.2 K), and has no classical triple point (26,27). It exhibits a phase transition at its lambda line (miming from 2.18 K at 5.03 kPa (0.73 psia) to 1.76 K at 3.01 MPa (437 psia)) below which it exhibits superfluid properties (27). 

Liquid helium-4 can exist in two different liquid phases liquid helium I, the normal liquid, and liquid helium II, the superfluid, since under certain conditions the latter fluid ac4s as if it had no viscosity. The phase transition between the two hquid phases is identified as the lambda line and where this transition intersects the vapor-pressure curve is designated as the lambda point. Thus, there is no triple point for this fluia as for other fluids. In fact, sohd helium can only exist under a pressure of 2.5 MPa or more. 

Helium is an interesting example of the application of the Third Law. At low temperatures, normal liquid helium converts to a superfluid with zero viscosity. This superfluid persists to 0 Kelvin without solidifying. Figure 4.12 shows how the entropy of He changes with temperature. The conversion from normal to superfluid occurs at what is known as the A transition temperature. Figure 4.12 indicates that at 0 Kelvin, superfluid He with zero viscosity has zero entropy, a condition that is hard to imagine.v... 

In this analysis the transition is defined as a step change in the heat capacity of the sample as a function of temperature. By far the most important transition that is generally considered to be second order is the glass transition, Tg. However, for completeness, other examples of second-order transitions include Curie point transitions where a ferromagnetic material becomes paramagnetic, the transition from an electrical superconductor to a normal conductor, and the transition in helium from being a normal liquid to being a superfluid at 2.2 K. 

The liquefied helium is subdivided into two states He I and He II with a sharp transition point of 2.18 K at 5.04 kPa, the so-called A,-point. He I behaves like a normal liquid, whereas He II exhibits interesting properties of a superfluid or quantum fluid. During expansion of liquid He I below this pressure, the previously even surface forms a sharp meniscus at the wall of the container since at the 7.-point the viscosity decreases by the factor 10 and the thermal conductivity rises by the same factor. The thermal conductivity of He II is about 200 times higher than that of copper at 20 °C. Close to the absolute zero point, the viscosity turns zero and He II becomes an inviscid superfluid. He II flows over obstacles, which lie higher than the surface of the liquid, to reach the lowest level. If two containers of different temperatures are filled with He 11 and connected to each other by a capillary or another He Il-film, He II flows from the cold container into the warmer one. 

In the smectic-A phase the molecules are oriented on average in a direction perpendicular to the layers, whereas in the smectic-C phase the director is tilted with respect to the layer normal. The tilted smectic-C phase has been compared to superfluid helium, and the smectic A-smectic C transition is predicted to exhibit critical behavior similar to that of helium. 

The understanding of continuous phase transitions and critical phenomena has been one of the important breakthrough in condensed matter physics in the early seventies. The concepts of scaling behavior and universality introduced by Kadanoff and Wi-dom and the calculation of non-gaussian exponents by Wilson and Fisher are undeniably brilliant successes of statistical physics in the study of low temperature phase transitions (normal to superconductor, normal to superfluid helium) and liquid-gas critical points. 

A modulus 6=0 corresponds to the SmA state. A Landau-Ginzburg functional similar to Eq. (20) with 5n = 0, and therefore to the superfluid-normal helium problem, can be constructed to describe the SmA-SmC transition. 

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