Louis Goldstein

On the theory of liquid and solid He3

Abstract The object of the present paper is to develop quantitatively several features of the theory of the nuclear spin system in liquid and solid He3, elaborated previously. The effect of the application of external pressure on the spin system will thus be described in a more rigorous way than heretofore. The attempt to correlate the various experimentally determined thermal properties of the compressed liquid from the standpoint of the theory of the spin system leads one to recognize the need of new and precise nuclear paramagnetic susceptibility measurements in order to verify various predictions of the theory. One of these concerns the strictly asymptotic temperature approach of the nuclear paramagnetic susceptibility of the liquid, or the solid, toward its limiting ideal Langevin-Brillouin limit. This feature of the susceptibility appears to be necessary for an explanation of the locus of the vanishing expansion coefficients located, at already moderate pressures, at relatively high temperatures, T ≳ 1.0°K. On assuming the absence of nuclear magnetic anomalies, the spin system of the solid is expected to render this phase also anomalous in its thermal properties, up to relatively high temperatures and pressures, as suggested by recent findings of two independent groups of experimental investigators in this Laboratory. Within certain relatively mild limitations, it is shown that solid He3 may exhibit two possible types of thermal excitations of its normal degrees of freedom. One of these, denoted as case (a), corresponds to excitations which appear to be particle like, as those of the liquid all over its phase diagram. They impose at low temperatures negative latent heats in the solid-liquid phase change, through the dominance of the entropy decrease of the nuclear spin system in this transformation. In case (b), phonon types of thermal excitations appear to be allowed, provided the solid-liquid phase separation line has a positive though small temperature slope at low temperatures. The importance of the indicated empirical decision between these two types of thermal behaviors for the theory of the dense phases of He3 hardly needs to be emphasized.

ON THE THEORY OF LIQUIDS

Abstract The present paper concerns the exact analytic forms of correlation functions of liquids and some of their macroscopic properties which these correlations determine. Since these functions are expressible in terms of the scattering structure factors of liquids, the macroscopic properties they define will be described in terms of the properties of the structure factors. One is then led, with some limitations, to a rigorous result on the behavior of pair potential energies of isolated stationary atoms at close separations. Part of the paper is devoted to a discussion of some of the limitations of the experimentally derived structure factors as well as the liquid correlation functions to which they give rise. It will be shown, on the basis of liquid helium data, that improvements in the empirical correlation functions as well as the various macroscopic liquid properties they define, require the knowledge of the structure factors over a far wider range of the momentum changes on scattering than obtained heretofore. The final topic concerns a detailed study of the asymptotic, large atomic separation, behavior of the correlation functions. The temperature limitations of a direct method yielding these asymptotic forms will be obtained. A new method of derivation of these forms, involving the scattering structure factors, will be described. This same method also leads to the asymptotic form of the structure factors themselves, at large values of the momentum changes on scattering. These latter asymptotic structure factors retain their validity over the whole temperature interval of existence of liquids.

Low-temperature melting properties of strongly magnetized solid helium-3

Various equilibrium thermal properties of strongly magnetized solid3He at melting are discussed in terms of the asymptotic, isotropic, nearest-neighbor pair-exchange interaction model at temperatures above 3 mK. Experimental verification of the predictions of the model to a limited degree of approximation suggests the possibility of using solid3He, at or near melting, for the production of fractional or small fractional millikelvin temperatures. This requires its adiabatic demagnetization from currently accessible several millikelvin initial temperatures and large magnetic field strengths to final field strengths above but close to its critical magnetic field strength where its paramagnetism still prevails.

Mean-field theory of some magnetized antiferromagnets

Two-sublattice general spin-s exchange antiferromagnets, assumed to have a stable configuration in applied uniform external magnetic fields parallel to their spontaneous spin-ordering direction in the absence of a field, are treated in the molecular- or mean-field approximation. The magnetic phase-boundary line of this antiferromagnetic configuration is shown to have simple closed parametric expressions. In the present approximate formalism the peculiar nonmonotonic phase-boundary line becomes of monotonic variation in the classical limit ofs → ∞. The possibility of strongly anisotropic two-sublattice parallel-field-stable antiferromagnets to function as efficient cooling systems is explored. This arises from their magnetic entropy increase on isothermal magnetization as a consequence of the large asymmetry developed through the respective cooperation and competition of the exchange-coupled spins with the external applied field in determining the effective fields acting on the spins of the two sublattices. The competing or negative sublattice has anomalous behavior in that its component molar spin entropy exhibits its maximum limit of1/2 R ln(2s+1) at low temperatures compared to the zero-field transition temperature, as long as the applied magnetic field strength is a substantial fraction of the critical field strength, above which the system remains paramagnetic down to the absolute zero. The transformation into the antiferromagnetic state is then prevented. The enormous magnetic or spin entropy of the system, persisting down to very low temperatures, according to the mean-field formalism, makes it possible to reach quite low temperatures on adiabatic magnetization from easily accessible initial states in the presence of an external magnetic field. Within the limitations of the model and of its treatment experimental investigations into the preparation of antiferromagnets with the required anisotropy may be of particular interest in connection with their possible use for the production of low temperatures.

Theory of a magnetic cooling system at very low temperatures

The totally anisotropic magnetic cooling compound, hydrated cerium magnesium nitrate, CMN, is treated at very low entropy values with the formalism of molecular-field theory. The magnetic phase-boundary line separating the ordered antiferromagnetic type region from the disordered paramagnetic region is well accounted for by the theory when compared with recent data of the Berkeley Low-Temperature Experimental Group. The ordered region refers to the canted antiferromagnetic or transverse magnetic field configuration of the Ce ions in CMN single crystals. One of the main experimental results refers to the temperature-scale functions obtained indirectly through largely arbitrary functions fitted to enthalpy-entropy data, treating simultaneously and indiscriminately both the ordered antiferromagnetic and disordered paramagnetic regions, with the data in the latter outweighing those in the former. The theoretical enthalpy-entropy function in the ordered range is in fair agreement with the corresponding data in absence of magnetic field with these data exceeding the calculated enthalpies. This may refer to the possible inclusion in the data of excitations in excess of the magnetic excitations alone accounted for by the theory. The entropy derivatives of the empirically fitted multi-parameter enthalpy-entropy functions define the indirectly derived temperature scales of strictly numerical significance at the very low temperatures. The molecular-field-theoretical entropy-temperature function, while approximate, is conceptually well grounded and may be preferable to, or at least competitive with, the indirectly derived Berkeley CMN temperature scales. There is fair agreement between them at the larger entropies approaching the zero magnetic-field critical transition entropy.

Model formalism of paramagnetic solid helium-3

The statistical-thermodynamic formalism of a collection of localized spin-1/2 atoms whose spin Hamiltonian refers to the isotropic antiferromagnetic Heisenberg exchange-interaction scheme is applied here to account for a set of important equilibrium-thermodynamic measurements on paramagnetic solid3He performed some time ago by University of Florida investigators. The measured properties were the temperature-dependent modulations of the pressure, which were proved earlier to arise overwhelmingly from the nuclear spin system. The present formalism of the pressure modulations or of the spin pressures, along specified isochores of the solid, includes the density- or molar-volume-dependent microscopic exchange energy parameter and its derivative. In this paper we have derived directly hitherto unavailable exact values of these parameters from spin pressure data on magnetized solid3He, as well as indirectly through the intermediary of spin pressures in the absence of a magnetic field. The directly derived exact parameters result from a single characteristic equilibrium-thermodynamic state of the solid in the presence of a constant and uniform magnetic field of adequate strength. The indirectly derived but exact parameters, of possibly somewhat lower accuracy, require the knowledge of a single directly derived exchange-energy parameter together with a set of spin pressures of asymptotic high-temperature equilibrium states in the absence of a magnetic field. The indirectly derived microscopic parameters along two of the three experimentally explored magnetized solid isochores yielded calculated spin pressures in fair and acceptable agreement, respectively, with their measured values. The directly derived exact parameters used in calculating the spin pressures along the third experimentally investigated isochore, in the absence of a magnetic field and at three different field strengths, led to complete agreement with the data. These results lend support to the tentative proposition advanced in early work that over a range of temperatures and molar volumes of paramagnetic solid3He, the statistical-thermodynamic formalism based on the antiferromagnetic exchange-interaction scheme may give an acceptable account of the spin pressures as well as of other thermal properties of this quantum solid.

Theory of magnetized ising antiferromagnets at low temperatures

The exact low-temperature series-expansion formalism of spin-1/2 anisotropic Ising two-sublattice antiferromagnets will be discussed in their parallel magnetic field configuration. In terms of reduced temperature and reduced field strength, defined through the molecular-field spin-ordering transition temperature and the associated critical magnetic field strength of the system, at asymptotically low temperatures, the various thermal properties are expressed through a few exponential terms. At large values of a unique dependent variable of the theory and in leading order the exact formalism is identical with that of molecular-field theory of the above systems. The entropy at constant field strength, approaching critical strength, remains very large down to quite low temperatures. As a consequence, in eventual real systems of low transition temperature which behave predominantly as Ising antiferromagnets or which closely simulate the latter, adiabatic magnetization up to the neighborhood of the critical field strength should produce substantial cooling, provided certain possible competing nuclear warming effects remain reasonably small. The heat capacity at constant applied field strength, approaching critical strength, develops a low-temperature maximum in addition to its transition anomaly along the antiferromagnetic-paramagnetic phase boundary line. This exact result, found earlier in, and attributed to, the approximate molecular-field theory, might be useful for experimentally identifying the presence or absence of dominant Ising-like interactions in certain eventual real anisotropic anti- ferromagnets.

Thermal behavior of ordered liquid helium-3

Heat capacity and phase-boundary line data on very low-temperature liquid 3He lead one to predict the thermally anomalous higher temperature liquid to become of normal thermal behavior over a limited pressure and temperature range of its ordered B phase. Well below the phase boundary temperatures, this ordered phase may become thermally anomalous again. If confirmed experimentally, this alternation in thermal behavior may point toward the respective dominance of two groups of thermal excitations over the indicated temperature ranges, a situation realized in liquid 4He II.

On the theory of the liquid He-4 phase transformation

Abstract The previously obtained mean kinetic energy and kinetic heat capacity of saturated liquid He4 around its phase transformation temperature will be used here in deriving some qualitative properties of the momentum space distribution function of the liquid atoms. The method will be first illustrated through a description of the local momentum space anomalies of ideal Bose-Einstein fluids which are responsible for their macroscopic first-order phase change. The derived liquid He4 momentum space distribution functions will then be applied to predict, within the approximate formalism of the gas kinetic momentum transport mechanism, the qualitative temperature dependence of the viscosity coefficient around the transition temperature. Finally, modifications of various thermal properties of liquid He4 around its transition temperature will be considered on the basis of a third-order type of phase change of liquid He4, suggested by its recently observed very large, possibly infinite, heat capacity peak.

Thermal properties of paramagnetic solid helium 3

It was shown in recent work that over a limited molar volume range and at asymptotically high temperatures the thermal modulations of the pressure along isochores of paramagnetic solid3He could be accounted for through the formalism of the Heisenberg model of an antiferromagnetically interacting localized spin-1/2 system. The internal consistency of this formalism requires the characteristic exchange-interaction parameter of the model derived from pressure modulation data to be identical with that appearing in the other thermal properties of this quantum solid. In a restricted temperature region where the spin excitations are the dominant thermal excitations of the solid, heat capacity data yield exchange-interaction parameters in fair agreement with those derived from pressures along isochores of larger molar volume. At higher temperatures, within well-defined limitations, thermal excitations involve both spin and phonon excitations. Here, because of the opposite temperature variations of the spin and phonon heat capacity components, the ensuing heat capacity minimum determines exactly the exchange-energy parameter and the relevant limiting Debye temperature as a function of the measured temperature location and value of the heat capacity extremum along the experimentally explored isochore. The exchange-energy parameters so derived display larger deviations from their predicted pressure-based values than those resulting from the lower temperature but still asymptotic spin-only heat capacities. At the present time, ambiguities in the experimental determinations of the characteristic Weiss temperatures of the asymptotic paramagnetic susceptibilities prevent one from deriving exchange-energy parameters with them. The present work leads to the prediction, within the limitations of the model formalism, of thermal properties of magnetized solid3He. Experimental investigations of these properties offer new approaches for probing the validity of the model formalism applied to paramagnetic solid3He.