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8 Low-Temperature Physics DEFINITION OF SUBFIELD There are two central themes that define the field of low-temperature physics. One is the study of the collective behavior in the motion of quantum-mechanical fluids, such as the electrons in superconductors and liquid helium. The other is the development of technology to go to lower temperatures. Any experiment attempted at the limit of this technology is usually classified as low-temperature physics. QUANTUM FLUIDS The term quantum fluid is used as the generic expression to cover any material in which the particles of interest do not solidify when in the ground state. Quantum fluids remain as a liquid or gas at T = 0 K because the particles are sufficiently light that the ground-state kinetic energy is larger than the interparticle potential energy that normally causes crystallization. Interest in such materials is frequently centered on how they satisfy the third law of thermodynamics. The entropy, which is a measure of the disorder in a system, must go to zero as the temperature goes to zero. Other substances achieve the condition of perfect order through spatial arrangement, usually in a lattice structure. In quantum fluids the 164

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LOW-TEMPERATURE PHYSICS 165 order takes place in the motion of the particles. Superfluidity, the ability to flow without resistance, is the manifestation of such order. The particles are coordinated to move in a coherent way. A single wave function can serve to describe the behavior of an entire mole of atoms, nuclei, or electrons, if they are in a fluid state at T = 0 K. Superconductivity, discovered in 1911, was the first example found of such order. It describes the ability of electrical currents to flow without resistance in certain conductors. The superfluidity of liquid 4He was discovered in 1937, 30 years after it was first liquified, and manifests itself in such properties as the ability to flow through capillaries of such small diameters that ordinary fluids cannot pass through them. Basic research in both superconductivity and superfluid 4He remains active. Rapid progress in the understanding of superconductivity followed the development of the microscopic theory by Bardeen, Cooper, and Schrieffer in 1957. The BCS theory attributes the phe- nomenon to an attractive interaction between pairs of electrons in a spin singlet (antiparallel spins) state caused by the lattice vibrations of the solid. The theory was so successful that superconductivity has now become an important tool for studies of the electron interactions in many different types of metals. In the case of 4He, a microscopic theory of the interactions that lead to superfluidity has still not appeared nearly 50 years after the phenomenon first came under study. Modern research into the fundamental questions related to superflu- idity has frequently been directed toward changing the nature of the transition by producing it in thin films on planar surfaces and a variety of packed powder geometries. In the past decade there has also been much work in which superfluidity and superconductivity have been applied to entirely new types of questions. Thus, liquid 4He is now widely used as a model substance for fundamental investigations in other research areas. It is a simple material with few defects and no impurities. For example, it is being used in tests of recent ideas about the onset of disorder in fluid flow. In the case of superconductivity, commercially significant tech- nologies that depend on it are likely to become commonplace in the near future. Most hospitals will have superconducting magnets for use in nuclear magnetic resonance (NMR) tomography, and devices based on the Josephson effect will be widely used. In a significant development, new areas of research into quantum fluids have recently emerged, to complement the knowledge that we have gained about motional order in superconductors and superfluid 4He. Liquid 3He has been found to undergo a phase transition similar to that occurring in superconductors, and an active search is under way

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166 A DECADE OF CONDENSED-MATTER PHYSICS for similar phenomena in other low-temperature fluids, such as spin- aligned atomic hydrogen. Superfluid 3He A high point of the research of the last decade was the discovery of the superfluidity of 3He at temperatures below 3 mK. This is the only new superfluid to be discovered in nearly 50 years. The possible existence of a paired state of atoms in liquid 3He was predicted as early as the late l950s. By 1970, the usual estimate of the transition temperature T`. for a state like that in superconductors was in the range between 10-6 and 10-9 K, well below that possible for the cryogenic technology of the day. Thus, it came as a rather dramatic surprise in 1971 when the superfluid transition was accidentally found in a series of experiments in which cryogenic methods for cooling solid 3He were being developed. The reason that the transition occurs at such an unexpectedly high temperature is that the nature of the pairing interaction in 3He is quite different from that found in superconductors known to date. In 3He the bound pairs of atoms form in a triplet magnetic state (parallel spins) with total spin S = 1 (so that Sz = -1, 0, + 1), whereas the pairs in superconductors and in 4He are in the singlet state (antiparallel spins) with total spin 5 = 0. In the former case the orbital part of the pair wave function is required to be antisymmetric in the interchange of the coordinates of the two atoms; in the latter it must be symmetric. The members of a pair having parallel spins must be in states of odd angular momentum, and in the case of 3He they are in a state with L = 1. As a consequence of this the order parameter of 3He is not a simple scalar function of position, as it is in 4He, but is a much more complex entity that can undergo a variety of different distortions. In liquid 3He the magnetic interaction between pair constituents is almost ten times larger than that between the paired electrons in a superconductor. Unlike the cases of superfluid 4He and superconductivity, where only one phase is found at low temperatures, liquid 3He has three distinctly different low-temperature phases, called the A, Al, and B phases. The Al phase exists only in the presence of a magnetic field. The boundaries between these phases as functions of temperature, pressure, and magnetic field are shown in Figure 8.1. Even at the low temperatures indicated, 3He remains a fluid until a pressure greater than about 34 bar is applied. The three phases have quite different experimental properties. Regardless of which superfluid phase is entered from the normal fluid,

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FOW-TEMPERATURE PHYSICS 167 o 30 o - a, 2^ ~ V In ~ 1 0 CL o / ~cJ~4~/ onto / * 2 / , B Liquid A ~ L i qu id ` Melting Plane / B Liquid o ~ J , PCP ~ , Norma l ~J-zero Field Plane Temperature (mK) FIGURE 8.1 The phase diagram of liquid 3He. however, a second-order phase transition like that in superconductors is observed. However, the transitions between the B phase and the A phase is a first-order transition with a minute latent heat, and the transition between the A and Al phases is also a first-order transition. The interpretation of the phase diagram is that the three superfluid phases correspond to the separate onset of the pairing transition for each of the three spin polarizations of pairs in the triplet state (see Table 8.11. The B phase contains all three polarizations: Sz = + 1, O. and -1. The A phase contains Sz = + 1 and -1; and the Al phase contains only Sz = + 1. An important consequence of the odd angular momentum of the orbital wave function is that superfluid 3He has an anisotropic spatial character. The fluid in all the superfluid phases has an intrinsic bending energy that favors the persistence of a particular orientation of the pair wave functions over rather long ranges in the liquid. This property of superfluid 3He has been given the name texture. At surfaces the wave function must be oriented with the orbital axis normal to the surface (just as the stable angular momentum configura- tion of a top is the one with the spin perpendicular to the surface). It costs energy for the wave function to change direction in space. However, the textures can be readily bent through a variety of means such as magnetic fields and fluid flow. The characteristic bending length of the texture in the B phase has been found experimentally to

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168 o Ct .= .~ Ct so o by Cat o an rid ._ ._ o Cal Cal Cal ._ ma: V) 4_ o Cal . ._ so o o Cal o Cal V: o L, ~ o o ~ s- ao Cal ~ ,~ o CD Ce s 50 t ~ o ~Q z C~ o C) o Ct ~ ~ 3 O {O A _ _ Cq: Ct _ O ~ ~ ~ o ._ ._ .:: CL O V, ~ C~ O ~ ~3 S~ ~ ~, S~ O ._ Ct 11 a' ~ ._ 11 C,~ C~ Ct O ~ z C ~o O C4 - _ - O ~L) _ . - E-, (,, ~ I ~ ~ ao ,,= ~ ce S~ . ~ ~ Q ', 3 c, ~ o - o - + 1 - o - + 1 C4- o ._ C~ Ct CL p, V) . ~ ~ C~ ~ ~ ;^ ~ 3 o.= ~ C,0 o ' o ._ L~ C~ 11 C ~ o ~ ~ ._ .= ~ - o .- _ CL = U' ~ Ct .= Ct C) o au ~ -~ s~ ._ ~ - - , o o ._ - ._ 3 o oc o, Ce ._ ._ ~ o U, _ ~ U, C~ o ~ o . _ _ oo Ct o ~ .= G . o a,

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169 o ~ 0 0, ~ ~O ~ 3 ~ ~ 3 ct 3 o ;> ~ _ v: Z L ~V) cn , , I o ~_ ~ C ~ ~ ~ C ~ Ct ~o -o, ~, C ', D o E o o c G C E 3 - o u G ::] ~Z - c O ~ oc ' ~ C~ O .- C O >) ~ 0(,, 0 au ~ ,_ C = - _ ~ ~ O ~ ._ C ~s~ ,S ~ ~ O O - ~ O ~O - CQ V) Z Z Z O O ~ O ~ ~ ~ O O ~ ~ O - O ~' - O 3 ~ O 3 ~.- ~ ' _ E3 C ~-' ~c ~ ~Z Z ~ ~ ~t_ E ~c~ ~ o ' 2 ~ ~o ~ 3 ~ o ~ o C C, t- ~ ~7 ~ _ _ ~_ 3 ~3 ~ ~ ~ S_ Ct O Ct ~ O 0 3 3 _ 0 3 3 ~ o 04 ~ c: o oo ~ ~O z z o 1 z _ _ - - O ,O + 1 + 1 1Z o 5 C~ C) O C O Z ~ ~> ~l C~ l ~ .= O - O o ~_ ~ 0) (,, CL ~ ~o 3 C: _ V) C ~Z

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170 A DECADE OF CONDENSED-MATTER PHYSICS be inversely proportional to the field strength and is of the order of millimeters in a field of 20 gauss. A striking difference between superfluid 4He and superfluid 3He was found in studies of the fluid flow. In both substances the fluid can flow through spaces too small for penetration by the normal-state liquid. In liquid 4He, there are no preferred orientations for such a flow. How- ever, when the flow of superfluid 3He in the A phase was studied in magnetic fields, it was found that the flow was much more rapid along the direction of the field than in the plane perpendicular to the field. The magnetic field aligned the fluid texture to produce an anisotropy in one of its most fundamental properties, the fluid flow. When the same flow experiments were repeated in the B phase, no dependence of the flow on the direction of the magnetic field was observed. NUCLEAR MAGNETIC RESONANCE IN SUPERFLUID SHE The macroscopic quantum nature of 3He has profound effects on its NMR. In liquid 3He, the nuclear magnetic dipoles exhibit a coherent response to external perturbations and are coupled to the orbital character of the macroscopic state. Because of this the frequency of the NMR absorption is shifted as if it were in internal fields of the order of 100 gauss. Both the A and B phases have a longitudinal resonance not found in NMR in any other physical system except solid 3He. A step change in the magnetic field produces a ringing in the amplitude of the magneti- zation parallel to the field. The ringing frequency depends on the temperature ratio T/T`.. Similarly, a radio-frequency field polarized along the steady external field can produce a detectable resonant energy absorption at the same ringing frequency. UNSOUND It has been possible to use high-frequency sound to perform an unusual type of spectroscopy in superfluid 3He. Sound at frequencies much larger than the atomic collision frequency of the liquid, the so-called zero sound, can be propagated with only weak damping and has been studied in all the superfluid phases of 3He and in the normal fluid. Unlike 4He, where only a single zero sound mode exists, a large number of narrow absorption modes have been observed in superfluid 3He because of the complex nature of the order parameter. The excitations associated with the distortion of the order parameter have been studied most extensively in the B phase because its isotropic

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~ O W- TEMPERA TURK PH YSI CS 1 7 1 order parameter makes the experiments easier to interpret. They correspond to the resonant excitation of J 74 0 states from the J = 0 ground state. Careful examination of the modes revealed that they were analogues of phenomena first studied for single particles in the early days of atomic physics, the Stark effect and the Zeeman effect. In the case of 3He there is a remarkable difference. These line-splitting phenomena occur because of distortions that affect the wave function that simultaneously describes the behavior of all the atoms in the container at once. OTHER SOUND MODES Most of the sound modes that have long been studied in superfluid 4He have also been fruitfully examined in superfluid 3He. Normal, or first, sound, the usual long-wavelength mode of most fluids, has been used to determine the viscosity parameters of the fluid. Second sound, the famous entropy wave in 4He, is also a spin wave in 3He. The 5~ = + 1 polarization of the spin triplet in the Al phase was determined through comparing a mechanically induced second-sound pulse with the magnetization change measured in an NMR coil. Fourth sound, a compression wave in a superleak, was used to make the first convinc- ing demonstration of superfluid flow in 3He. DEFECTS Sudden discontinuities in the pair-wave functions are called defects. In any real container there must be at least one defect because it is impossible to meet the boundary condition that the wave function be perpendicular to the surface at all points in even the simplest three-di- mensional geometry, the sphere. Modern topological methods have been used to classify the types of defects that might appear in the texture fabric. Sharp walls in which the direction of the orbital axis is reversed by 180 degrees are solitons. Solitons can be created in either the spin or the spatial portion of the superfluid wave function through the appli- cation of intense radio-frequency pulses at the NMR resonance. The existence and specific nature of such defects have been investigated through examination of, and the changes in, the resonance modes in the NMR spectrum. These types of defects persist in the liquid almost indefinitely after their creation if neither the fluid temperature nor the magnetic field is changed. They can be erased by removing the magnetic field or by flow of the fluid.

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172 A DECADE OF CONDENSED-MATTER PHYSICS Vortices, which are the only defect known in superfluid 4He, have been studied in both the A and B phases in an apparatus capable of producing a steady rotation of the fluid. The presence of the vortices was demonstrated, and estimates of their density made, again through NMR methods. Recently it has been found that a spontaneous magne- tization appears in the vortex cores of fluid in the B phase. This result has strong implications for our understanding of the magnetic field about neutron stars. Modern models of the dense centers of such stars suggest that there also the fermions, neutrons in this case, become paired in a triplet state similar to the B phase of superfluid 3He. SUPERFLUID FLOW AND HYDRODYNAMICS In addition to the rotation experiments discussed above, a variety of flow and hydrodynamic measurements have been performed. Fluid flow has been studied in cylindrical, spherical, and parallel plate geometries. The motion of a wire moving through the fluid has also been carefully analyzed. The critical velocities limiting the superfluid flow in both the A and B phases appear to be much smaller than originally expected. The viscosity measured in the fluid at low temper- atures is also much smaller than expected. These deviations from theory are probably related to both the long mean free path of the normal fluid excitations and the peculiar kinetic behavior of these excitations at boundaries. One useful product of the flow studies has been the determination of the superfluid mass fraction in 3He. The results are important for understanding other hydrodynamic experiments. The superfluid mass fraction can be found from determinations of the amount of fluid that is not changed through viscous contact with chamber surfaces. Novel Quantum Fluids There is a variety of gases and fluids that can be cooled to low enough temperatures for quantum-mechanical effects to become im- portant in understanding their physical behavior. Examples that have been studied for quite some time include the bulk behavior of liquid 3He, liquid 4He, liquid mixtures of the two helium isotopes, and the conduction electrons in metals. The temperature at which quantum statistics becomes important is that at which the thermal de Broglie wavelength becomes comparable to the interparticle spacing. In the case of fermions, 3He, and electrons, this temperature is the Fermi temperature, and in the case of bosons, for example 4He, the temper

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LOW-TEMPERATURE PHYSICS 173 ature is roughly the Bose condensation temperature. Statistical me- chanics predicts a macroscopic occupation of the ground state for Bose particles cooled below the Bose condensation temperature. In liquid 4He this temperature is calculated to be 3 K, and it seems likely that the superfluid transition of bulk 4He at 2.2 K is related to this result. In the latter part of the 1970s attention was turned to these fluids under new, extreme physical conditions. The Fermi systems are being investigated under conditions of large polarization where only one spin population is present. The Bose systems are being investigated in unusual geometries and under conditions of increasing dilution of the distance between the particles. The newest materials in this class that have come under investigation are the gases of atomic hydrogen and deuterium, which can be stabilized against the formation of H2 and D' through polarization of the atomic electrons in large magnetic fields at low temperatures. MIXTURES OF SHE IN SHE At low temperatures 3He is soluble in 4He in concentrations up to 6 percent for fluid with no external pressure, and up to 10 percent for fluid under pressures greater than 10 bar. The gas fermions in superfluid 4He have weak interactions. The 4He acts as an ether medium supporting a dilute gas that behaves almost like an ideal Fermi gas. The properties of such mixtures in low magnetic fields were studied extensively in the late 1960s and early 1970s to measure the small deviations from ideal gas behavior and to develop the dilution refrigerator (this is discussed later in this chapter in the section on Low-Temperature Technology). SPIN-POLARIZED HYDROGEN AND DEUTERIUM Atomic hydrogen and deuterium can be stabilized against the forma- tion of molecules if their electronic spins are polarized. The spin-triplet potential has no bound states. Hence, if all the atoms could be forced to interact only through the triplet potential large densities of the material could be collected, and because of the weak interaction between the atoms they would remain as a gas to absolute zero temperature. The hydrogen atom has an even number of fundamental particles, one electron and one proton. It thus is a boson and should display a statistical condensation like that in 4He. Deuterium is a fermion, and it should have properties similar to the weakly interacting Fermi gases discussed in other sections of this report. The search for

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174 A DECADE OF CONDENSED-MA TTER PH YSI CS Bose condensation in atomic hydrogen has been a topic that has attracted a great deal of interest in the early 1980s. It offers an entirely new example of a superfluid that can be compared with 4He. Hydrogen would be an especially important example because the interparticle coupling is so weak that virtually all of the substance properties can be calculated from simple principles. In addition, hydrogen has a nuclear magnetic moment, whereas 4He does not. A large class of collective spin phenomena, similar to those in superfluid 3He, are expected to be found. Research in this area is in its infancy. The experimental problems are formidable, but significant progress has been made. Atomic hydrogen densities of 10~8 atoms per cm3 have been achieved at a temperature of 0.5 K. The Bose condensation temperature for such a density is ~17 mK. At a temperature of 100 mK a density of 10'9 atoms is required (T`. varies as the 2/3 power of the density) for Bose condensation. It is not obvious that the problem can be solved by improvements in the cooling techniques. The hydrogen in the gas phase is in pressure equilibrium with atoms bound to the container surface. The dominant processes for atomic recombination take place on the surface. The surface recombi- nation rate increases rapidly with the density of surface atoms. The lowest binding energy of hydrogen bound on any surface is that on helium films. However, even using surfaces preplated with liquid helium, gas densities of 10~7 atoms per cm3 would be expected to saturate the surface completely at temperatures of a few millikelvins. On the other hand, the time constants for the decay of the collected hydrogen can be quite long several hours. Several research groups have succeeded in performing transient experiments in which the hydrogen density is increased to ~3 x 10'8/cm3 by compressing a bubble of the gas in liquid helium. LIQUID 4HE IN UNUSUAL GEOMETRIES During the past decade, the superfluid transition in liquid 4He has been extensively studied in thin films on a variety of substrates. Films deposited on smooth and flat surfaces provided the first model for measuring the way in which the order of a two-dimensional system is disrupted by thermally activated defects. In helium films, the order in the low-temperature phase appears in the fluid momentum. The inertial response of the mobile atoms in the film is correlated. As the temper- ature is raised, the order is disrupted by thermally activated vortex pairs. When the vortex density becomes sufficiently large, the long- range order in the fluid motion becomes completely disrupted. The film

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LO W- TEMPkRA 7 URE PH Y5 arcs 179 progress toward a very-high-performance, Josephson-effect-based computer, potentially the highest-speed, most-compact, and most- power-efficient approach known. The above abbreviated list is necessarily only representative of the much broader totality of superconductivity research. Even with that restriction a comprehensive account of the major developments in each of these areas cannot be presented here, and so what follows are brief descriptions and highlights selected from the above areas. Nonequilibrium Superconductivity The study of nonequilibrium superconductivity is the study of the response of superconductors to externally applied perturbations. Be- cause superconductors have much longer relaxation times than normal metals, they can be driven out of equilibrium more readily than normal metals. For example, the electronic and ionic (phonon) effective temperatures can be decoupled in superconductors. This observation, together with the seminal discovery that the chemical potentials of the excitations and condensate in a superconductor could be made signif- icantly different by the injection of charged particles, and could readily be measured, opened up a fertile and exciting area of many-body, condensed-matter research. The rich variety of studies in this field has included the microwave radiation enhancement of the superconducting energy gap, the explanation of excitation-to-condensate conversion processes, the elucidation of unusual behavior in superconducting microstructure and contacts, and the as-yet-unexplained anomalously large thermoelectric generation of flux in a superconducting bimetallic loop. More recent studies focus on regimes far from equilibrium such as the transition to the normal state triggered by supercritical currents. Although much progress in this area has been made, many research opportunities remain, particularly in the area of extreme nonequili- brium conditions. Novel Superconducting Materials Searches for superconductivity in a variety of novel or exotic materials have been richly rewarded in the past decade. Some quasi- one-dimensional systems, such as TaSe~, NbSe~, and polymerized sulfur nitride, and quasi-two-dimensional conductors, such as interca- lated dichalcogenides and artificially layered structures, have been found to superconduct. Superconductivity has been found in organic systems, the first of which was (TMTSF)'PF`,.

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180 A DECADE OF CONDENSED-MATTER PHYSICS New metallic compounds have been found to have unusual superconducting properties. For example the Chevrel compounds have high transition temperatures and record critical fields from 50 to 70 T. Superconductivity has been observed in low-electron-density com- pounds such as BaPb~_xBxO3. Finally, heavy-fermion superconductors with enormous electronic specific heats have been found. The first example was CeSi2Cu2, and its discovery was followed by the discov- ery of three more heavy-fermion superconductors, UBe~3, CeCu6, and UPt3. These systems are particularly exciting because they may be the first examples of superconductors in which the electrons are paired with parallel spins. Renewed interest is developing in amorphous and metastable sys- tems such as the record transition temperature A15 materials. Still more unusual phenomena await diligent materials synthesists. Magnetic Superconductors In 1958 it was shown that as little as 1 percent of magnetic impurities can destroy superconductivity in the host metal. Beginning in the mid-1970s, the interplay between superconductivity and magnetism has been re-explored. Unexpected results were found in a family of ternary rare-earth compounds typified by ErRh4B4. In these com- pounds the superconductivity is associated with the transition-metal electrons that are confined in clusters and are therefore relatively isolated from the magnetic rare-earth ions. This sets up a competition between superconductivity and magnetism that reveals itself in unusual behavior. For example, ErRh4B4 becomes a superconductor near 9 K. As it is cooled further, the rare-earth ions begin to order magnetically until the superconductivity is destroyed near the Curie temperature just below 1 K. Recent small-angle neutron-scattering experiments suggest that some of these compounds exhibit a new phase of matter in which superconductivity coexists with magnetic order in periodic structures with a wavelength of about 200 A with superconductivity . . survlvlng. High-Transition-Temperature, High-Magnetic-Field Materials Efforts to increase the high-magnetic-field performance of supercon- ducting devices have gone in two directions. One is the improvement of existing materials, and the other is the search for new materials. The past decade has seen striking progress in the technical use of existing supermagnet materials. Part of the progress is due to improved '

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LOW-TEMPERATURE PHYSICS 181 understanding of flux jumping and related thermal effects. This has led to the development of multifilamentary cables optimized for maximum performance in specific applications. Progress has also been made through painstaking improvement in production processes. Even so, studies of optimized short samples show that much improvement in commercial materials can still be made. The quest for new technologically tractable materials has proved more difficult. Although Nb3Sn with its superior superconducting properties has been fully stabilized, the more ductile Nb-Ti alloys, currently produced in the thousands of tons, remain the workhorse of high-field superconductors (Figure 8.31. Nevertheless, as noted earlier, significant progress in the understanding of the conditions for the occurrence of superconductivity have been made. Inevitably, this must contribute to the development of truly superior metals. FIGURE 8.3 High-performance multifilament superconducting conductor used in the Mirror Fusion Test Facility superconducting magnets at Lawrence Livermore National Laboratory. Nb-Ti alloy superconducting filaments (dark regions seen end on in center square) are embedded in normal metal matrix. (Courtesy of Oxford Airco.)

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1 8 2 A DECADE OF CONDENSED-MA TTER PH YSl CS The Josephson Elects The Josephson effects are among the most beautiful and novel manifestations of the macroscopic quantum nature of superconductiv- ity. Their study and their technological application have continued to be exciting. Central to rapid progress in this area during the past decade has been the highly successful development of techniques for fabricating micrometer- and submicrometer-size junction structures of high quality, uniformity, and reliability. This has made possible the application of nearly ideal Josephson structures to a number of exciting scientific and technological endeavors. One scientific focus has been the investigation of dynamic Josephson effects in thin-film microbridges. Another has been the study of the quantum limits of noise. The most sensitive magnetic-field detectors, the so-called superconducting quantum interference devices (SQUIDs), have been fabricated with an energy resolution only slightly greater than the uncertainty principle limit h (Figure 8.41. In addition to being direct objects of study, these sensors are the essential compo- nents in a wide variety of other low-temperature quantum-noise studies. Josephson devices are being used to address a variety of fundamen- tal questions in condensed-matter physics. The investigation of mac- roscopic quantum tunneling is a current, active example. The purpose of this work is to test efforts aimed at including dissipation (friction) in quantum-mechanical descriptions of macroscopic physical systems. This work is just getting under way and appears to have a bright future. The use of large, two-dimensional arrays of Josephson junctions to examine concepts of two-dimensional phase transitions is but another example of how superconductivity serves as a novel means of testing general theoretical hypotheses. The study of chaotic behavior in Josephson junctions is another recent example. QUANTUM CRYSTALS Quantum crystals are solids in which the atoms have a large- amplitude zero-point motion. The most important quantum crystals are helium, hydrogen, and deuterium, the same elements found in quantum fluids. In the case of hydrogen and deuterium, the atoms are in a molecular form, and interest in these crystals centers mostly around the fact that they are the simplest of the molecular crystals. The molecules solidify in several angular momentum states, and the low- temperature properties including the crystalline structure depend on

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FOW-TEMPERATURE PHYSICS 183 FIGURE 8.4 Ultralow-noise Josephson analog superconducting quantum interference device (SQUID) with spiral input coil. Devices such as this are utilized for extremely high-sensitivity magnetic-field measurements. (Courtesy of IBM Corporation.) the fraction of the molecules that have decayed to the lowest angular momentum state. The molecules in solid hydrogen and deuterium interact by means of a quadrupolar interaction. Crystalline fields in the solid couple with the interatomic quadrupolar field of the molecule. The quantum motion is much larger in helium than in solid hydrogen because of the weaker binding potential. A major consequence of the quantum motion in solid 3He appears in its magnetic properties. The overlap between the wave functions of neighboring atoms leads to an

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184 A DECADE OF CONDENSED-MATTER PHYSICS atomic exchange. The exchange energy is of the order of 1 mK for crystals with the largest molar volume, at the melting pressure. The size of the exchange energy decreases rapidly with decreasing molar volume V of the solid, varying approximately as Vie. The large exchange energy in solid 3He produces magnetic order in the crystal at 1 mK. The transition occurs at the highest temperature of any nuclear magnetic transition. In copper, for example, the nuclear ordering transition occurs at 60 nK. The details of the solid 3He magnetic transition are still under intensive investigation. There are quite a number of surprising features to the transition. It is first order in nature, and the entropy drops discontinuously by a factor of 0.44 R in 2 at Tc The solid expands in volume by 1 x 104 at the transition, and the magnetic susceptibility drops by more than a factor of 2 at Tc. The microscopic nature of the sublattice orientation was determined through an elegant set of nuclear resonance measurements in which the antiferromagnetic spin-wave modes were studied. The crystal has a body-centered cubic lattice, and the sublattice structure is one in which the direction of the spins on successive planes alternates every two planes. The sublattice planes are parallel to the cubic faces of the crystal. The up-down-down structure has been abbreviated u2d2. This low-field phase is unstable in magnetic fields greater than 0.45 T. In fields greater than 0.45 T another magnetic phase appears. Little is known about the microscopic nature of the high-field phase. It is likely to be similar to the spin-Hopped phase often found in antiferromagnets. The most successful theory of the magnetic properties of solid 3He suggests that the transition occurs as the result of the competition between three and four particle rings of exchange in the bee crystal. The odd number of interchanges favors a ferromagnetic order, and the even number favors an antiferromagnetic order. Using plausible values for the exchange rates, a two-parameter theory seems to give reason- ably accurate descriptions of the observed phenomena. There have been recent investigations into the nature of the interface between liquid and solid helium, to determine whether the surface has facets at low temperatures. In all other crystals there is a roughening transition at which the low-temperature state, with flat faces related to the crystal structure, is disrupted by thermally activated defects so that the facets disappear. It had been speculated that such a transition would not exist in solid helium because the zero-point motion would keep the surface rough even at zero temperature. However, it was found that facets do, in fact, appear in crystals of solid 4He. Experi- ments have not yet been performed on solid 3He, which has a larger zero-point motion.

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LO W- TEMPERA TURK PH YSl CS 1 85 LOW-TEMPERATURE TECHNOLOGY One of the most important traditions of this research field has been that of extending the experimental working regime of physical mea- surements to lower temperatures. The historical progress of the field is illustrated in Figure 8.5, where the minimum equilibrium temperature achieved after each new technological advance is plotted versus the year of the advance. The minimum temperature has decreased by a factor of 10 approximately every 15 years, since the first liquefaction of air a little over 100 years ago. Despite the apparently continuous nature of the progress suggested by the graph, the new advances have always come after long periods of consolidation of the most recently devel- oped methods. As is the case with many other technologies, the cryogenic advances have been transferred to related areas of scientific research. This has typically happened 10 years after the workers in the low-temperature physics community have consolidated the experimen- tal method. The use of apparatus that requires working temperatures of 4 K is common today. The present frontier of the field is in the region of 10 ~K. The cooling method in modern apparatuses working at the lowest loll _ 10 By ~ I 0 ILL 1 10-5 . . . . . . . . . . 900 1 9so 2000 YEAR FIGURE 8.5 Graph representing progress in cryogenic technology. Each point in this semilogarithmic graph represents a major advance in technology.

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186 A DECADE OF CONDENSED-MA TTER PHYSICS temperatures relies on two technologies that have been refined in the 1970s, the dilution refrigerator and superconducting magnets. The dilution refrigerator is an apparatus that takes advantage of the phase separation in mixtures of 3He in 4He. 3He is driven from a 3He-rich phase into the dilute phase by a concentration gradient in the dilute phase. The thermodynamic process is similar to the evaporation of atoms from a liquid to a gaseous phase. There is a large entropy increase when the 3He atom passes from the 3He-rich phase into the dilute phase. After the 3He atom passes through a dilute column of liquid mixture, it is evaporated and collected by pumps at room temperature. The 3He gas is then recycled, and after heat exchange with the dilute column returns to the 3He-rich phase at low tempera- tures. The process can operate continuously, and temperatures as low as 2 mK have been maintained with this cycle. Equipment of this type for experiments down to 5 mK can now be purchased from several different commercial manufacturers. The combination of a constant low-temperature thermal sink at 5 mK and modern superconducting magnets capable of producing fields of is T make it possible to achieve large values of the ratio BIT, the significant variable for achieving large polarizations in any paramagnetic system. To obtain even lower temperatures, magnetic cooling cycles are employed. The dilution refrigerator is used to precool a paramagnetic refrigerant placed in a large magnetic field. The refrigerant is thermally isolated after it has been polarized. The next stage of cooling is achieved by reducing the magnetic field. Nuclear moments are used for the refrigerant material. A significant simplification of the magnetic cooling technology was accomplished by the introduction of a new magnetic material, Prim. In this material, there is a hyperfine interac- tion between the electrons and the nucleus of the Pr atoms. The hyperfine field acts as an amplifier for the external applied field. The local field at the site of the nuclei is enhanced by more than a factor of 20. Complete polarization of the Pr nuclei can be achieved with modest dilution refrigerator, operating down to 10 mK, and an 8-T magnet constructed with NbTi wire, the least expensive magnet wire. Unfortunately, the hyperfine interaction that assists in producing the large polarization also limits the minimum temperature of the material. The minimum temperature is 0.4 mK, a limit imposed by magnetic order in the metal. The lowest temperatures have been obtained by using two cascaded stages of nuclear magnetic cooling. Typically PrNis has been used as the first magnetic stage, to remove the heat of magnetization from copper. After the copper has been demagnetized, stable lattice tem

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LOW-TEMPERATURE PHYSICS 187 peratures as low as 20 OK have been measured. The spin temperature of the copper goes much lower. A nuclear ordering transition has been measured in the copper nuclei at a temperature of 60 nK. It is not clear what limits this technology. Only a few materials have been tested. Metals with good electrical conductivity must be used to have reasonable equilibrium times in the lowest temperature stages. The ultimate lattice temperature is governed by the balance between the metal conductivity and the heat leak from external sources. In the most successful apparatuses, it has been estimated that the heat leak from cosmic radiation might be playing a significant role in determining the minimum temperature. RESEARCH OPPORTUNITIES IN LOW-TEMPERATURE PHYSICS Superfluid 3He has been studied for one decade, and a great number of interesting questions remain to be answered. The analogues of some of the most important superconductivity experiments have not yet been repeated. For example, the Josephson effect should be observable in 3He. Persistent currents have never been created in superfluid 3He, and the question of the quantization of fluid circulation in the various phases has not been tested. Most of the texture phenomena suggested by theory remain to be investigated. Size effects and the limits on the superfluidity imposed by the dimensionality of the fluid container have not been studied. The Fermi temperature of pure liquid 3He is roughly 1 K. It is not practical to produce a highly polarized specimen through the brute- force application of a large magnetic field (at any temperature), because the 1300-T field required is several orders of magnitude too large for present technology. Instead, several groups are developing transient methods for polarizing liquid 3He. Two methods look promising. In one case, the 3He starts off in the solid phase at 34 bar, where it is polarized by cooling at low temperatures in a large magnetic field. It is then converted to the liquid phase by expansion of the chamber volume. In the second method, the 3He is polarized by optical pumping of the atoms in the gaseous phase at room temperature. The gas is then cooled to the condensation temperature to produce a polarized liquid. In both cases, the lifetime of the polarized state is limited by spin- relaxation processes at surfaces or interfaces. Under some circum- stances these times can be long enough that interesting measurements are possible. When there is only one spin population present, the fluid properties are likely to be different from those of unpolarized liquid

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188 A DECADE OF CONDENSED-MATTER PHYSICS 3He because of changes in the interactions between particles. When the nuclear spins are parallel, the Pauli principle requires them to have a larger separation than atoms when the spins are antiparallel. Thus, the density of the fluid will change slightly. The average interaction between the 3He atoms should be different from those in unpolarized liquid 3He so that quantities like the heat capacity and the magnetic susceptibility are expected to change. From a theoretical standpoint, the properties of the polarized liquid are expected to be easier to calculate because the interatomic potential is simpler. There are two areas of especially interesting research with mixtures of 3He in 4He that are likely to be significant in the coming decade. One is the search for a pairing transition between 3He atoms as the fluid is cooled to lower temperatures. The second is the study of the transport properties of the fluid in large magnetic fields. There are no reliable estimates for the pairing transition temperature in the dilute mixtures; nor is it known whether the pairing will be a triplet state like that in pure liquid 3He or a singlet state similar to that in superconductors. In pure 3He and in superconductors Tc is ~ 10-3 Tf, where Tf is the Fermi temperature. With each advance in cryogenic technology the dilute mixtures have been re-examined to see whether there are sudden changes in the magnetic susceptibility or heat capacity that would mark the onset of the pairing transition. By 1983 the experiments had been extended down to a little over 200 OK for solutions with a Fermi temperature of 100 mK. No pairing transition has yet been observed. The dilute mixtures are technically easier to polarize than pure 3He because the Fermi temperature can be adjusted to be small enough to match the polarization energy available with practical magnets. For a mixture with a 3He fraction of 10-4, the Fermi temperature is 5 mK. When such a solution is cooled to a few millikelvins the nuclear moments can be almost completely polarized in a magnetic field of 8 T. a field that is relatively easy to obtain. To date, there have been few investigations of this system, but the fluid should have some remark- able properties. The scattering cross section for polarized 3He atoms in solution is much weaker than that of the unpolarized atoms. Thus, the mean free path between particle collisions is expected to be more than a factor of 10 longer than that in the unpolarized liquid. The most interesting quantities to measure are the transport coefficients: the thermal conductivity should be larger than that of good metals; the viscosity should approach that of liquids about to form a glass; and the diffusion coefficient should be larger than that measured in any other

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FOW-TEMPERATURE PHYSICS 189 liquid. Many fruitful studies are likely to come from this system in the near future. Even if the Bose condensation of spin-polarized hydrogen is not achieved, there are likely to be many useful by-products of the research that should have an impact on technology. Studies of wall relaxation phenomena have already led to a specific design for a better frequency standard. Other technically significant developments are likely to be an improvement of the techniques for storing excited atomic populations for work with lasers and the demonstration of a particularly clean system for studies of chemical reaction kinetics in hydrogen. The future of superconductivity in the next 10 years seems to be readily apparent from its past 10 years. Even without the unpredictable discovery of a material with a much higher Tc, of an alternative pairing mechanism other than the electron-phonon interaction, or of new phenomena with the impact of the Josephson effects, the field is likely to continue to prosper along the lines of the recent past. It is reasonable to predict that new and unusual superconducting materials will con- tinue to be discovered and avidly studied. Future improvements in the theory of dynamic phenomena in superconductors seem likely so are improvements in device performance and in high field materials. There also seems as yet to be no end to the use of superconductivity as the test vehicle of basic theoretical advances in condensed-matter science. A question of current interest in studies of molecular solids is that of how order is achieved in the orientation of J = 1 molecules. There are speculations that a glassy state of order exists in which there is short-range orientational order but no long-range order. Nuclear reso- nance experiments have been performed that support both sides of the argument. This is an issue that is likely to be resolved in the near future.