Nuclear Susceptibility Research Articles

Electrodynamic Properties of a Gas of Spin-½ Particles in a Uniform Magnetic Field

A rather complete discussion of the electrodynamic behavior of a gas of spin-\textonehalf{} particles in a uniform external magnetic field is presented. The method presented here is based on the general principles of elementary nonrelativistic quantum theory and takes account of the Doppler-broadening effects arising from the random thermal motion of the particles and the Compton recoil effects arising from the emission (spontaneous plus the induced or the stimulated emission) and absorption of photons by the spin-\textonehalf{} particles. In the first half of the paper a method is presented for calculating the spin permeability tensor which determines the propagation of transverse electromagnetic radiation through a gas of spin-\textonehalf{} particles in a uniform external magnetic field. It is shown that the spin permeability tensor thus obtained yields the familiar results of solid-state physics, such as the nuclear and electronic-spin radiofrequency susceptibility and the spin paramagnetism. In the second half of this paper the subject of spin-radiative equilibrium of a gas of spin-\textonehalf{} particles in a uniform external magnetic field is discussed in some detail. Simple closed-form expressions for the steady-state photon number density and the spin-radiative relaxation time of the system under study are thus derived. It is shown that the steady-state photon number density is the familiar distribution function corresponding to Bose---Einstein statistics for "complete thermodynamic equilibrium." The conditions under which the system of spin-\textonehalf{} particles in a uniform magnetic field can become unstable for transverse electromagnetic radiations are also discussed.

Nuclear spin-lattice relaxation time in copper below 10 −2° K

THE nuclear spin-lattice relaxation in metals is governed primarily by the interaction of the nuclear spins with the conduction electrons. The corresponding relaxation time has been calculated by Korringa, ~ who gave an expression for r which is inversely proportional to the conduction electron temperature and to the square of the Knight shift. The theory has been modified subsequently by Pines, 2 Redfield, 3 and Hebel and Slichter. 4 Experimental v, ork on copper at low temperatures has been done by Anderson and Redfieid s in the liquid helium temperature range and by Kurti 6 and others in the adiabatic demagnetization temperature range down to 1-2 x 10 -2° K. The first authors used resonance techniques for the determination of the relaxation time, whereas the method of the Oxford group was based on a nuclear cooling of the copper spin system. Also in the investigations reported here a nuclear cooling method was used. The nuclear spin system was precooled by rotational demagnetization of CMN (cerium magnesium nitrate) which is highly anisotropic and which has only small interactions, so that temperatures down to a few millidegrees can easily be attained. The nuclear stage consisted of a bundle of copper wires in thermal contact with the salt. Copper was used as a nuclear sample because of its large nuclear susceptibility. Both salt and copper were located in the pole gap of an electromagnet, which could be rotated for precooling the salt and removed after the nuclear demagnetization. The spin-lattice relaxation time was determined by recording the decay of the nuclear susceptibility of the copper after the final demagnetization. The susceptibility was measured by means of an electronic mutual inductance bridge, the output of which was detected and recorded. A ballistic method was used for the determination of the susceptibility of the salt. The corresponding temperature was calibrated in the usual way by comparison with the vapour pressure of the liquid helium between 1 ° K and 4.2 ° K. The nuclear spin temperature was not calibrated; it was calculated using the theoretical value of the nuclear spin susceptibility and the sensitivity of the mutual inductance measurement. The salt tempera-

Measurement of the Spin Dependence of theHe3(n, p)TReaction and of the Nuclear Susceptibility of AdsorbedHe3

The spin dependence of the ${\mathrm{He}}^{3}(n, p)\mathrm{T}$ reaction was investigated by studying the transmission of polarized thermal neutrons through a polarized ${\mathrm{He}}^{3}$ target which consisted of ${\mathrm{He}}^{3}$ adsorbed on zeolite. It was found that the interaction cross section was essentially all associated with an $I\ensuremath{-}\frac{1}{2}=0$ channel, i.e., $\frac{{\ensuremath{\sigma}}_{I\ensuremath{-}\frac{1}{2}}}{{\ensuremath{\sigma}}_{0}}=1.010\ifmmode\pm\else\textpm\fi{}0.032$. Once the spin dependence of the cross section was established, polarized neutrons provided a convenient method of studying the nuclear spin susceptibility of ${\mathrm{He}}^{3}$. Susceptibility measurements were made on our adsorbed-phase target over the temperature range 0.06-1.0\ifmmode^\circ\else\textdegree\fi{}K. It was found that the nuclear susceptibility began to fall below Curie's law at about 0.2\ifmmode^\circ\else\textdegree\fi{}K, indicating the onset of some type of antiparallel ordering in the nuclear spin system.

Nuclear Paramagnetism of Helium-3—Helium-4 Solutions above 0.9°K

The temperature dependence of the nuclear susceptibility of pure ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures of ${\mathrm{He}}^{3}$ concentrations 0.81, 0.70, 0.38, and 0.14 has been determined with an accuracy of \ifmmode\pm\else\textpm\fi{}1% for $0.92<T<2.9$\ifmmode^\circ\else\textdegree\fi{}K. This work is an extension of that of Fairbank, Ard, and Walters, and shows that within experimental error the temperature dependence for the solutions is the same as that of the pure liquid. The maximum deviation of the susceptibilities from an inverse temperature dependence is approximately 5%. These results indicate that the susceptibility is insensitive to the average separation of the ${\mathrm{He}}^{3}$ atoms, contrary to the behavior that might be expected from a Fermi system in which exchange is important. A perturbation calculation, however, suggests that such behavior can be understood as a near cancellation of terms arising from the exclusion principle and the exchange interaction.

The nuclear susceptibility of 3He in dilute solution in liquid 4He

The nuclear susceptibility of 3He in liquid 4He has been measured between 0.35°K and 1.3°K for concentrations up to 15%. The results are very close to the ideal Fermi-Dirac values calculated with m ∗ = 2.5m 3 , the effective 3He 3He interaction being too small to detect for concentrations below 10%.

Nuclear Susceptibility of Solid Helium-3

Abstract : Measurements of the nuclear susceptibility of solid He3 in the alpha phase between 0.07 and 1.7K are presented. The purity of this helium was better than 99.95%. All the results could be represented by a Curie-Weiss law chi = C/(T + 0) indicating an antiferromagnetic exchange. While theta is smaller than about 0.01K at molar volumes larger than 22 cu cm, it appears to increase systematically when the density increases further. The magnetic exchange between the nuclei, as deduced from our data, is fairly consistent at molar volumes above 22 cu cm with that obtained from specific heat and relaxation measurements. The higher density data, however, must be regarded as very tentative, in view of their disagreement with other data and theoretical expectations and are not presented in detail. Measurements of the longitudinal relaxation time were also carried out. In particular, for V = 19.5 cu cm/mole, T sub 2 was found to increase approximately as exp (2/T) below 1K, in contradiction with relaxation data of Reich, who found T sub 1 to be constant in this range. New susceptibility data are also presented for a solution of 99.0% He3 1% He4 and a tentative explanation is given to account for the difference with the results on the 'pure' He3. A method for measuring the pressure inside the solid He3 cavity is described. On the basis of these measurements, it is argued that the phase boundary in the V-T plane is unlikely to bend upwards at temperatures below 0.8K, as deduced from Reich's relaxation measurements. (Author)

Nuclear Susceptibility of Liquid Helium-3 under Pressure

New measurements of the nuclear susceptibility of liquid helium-3 between 0.045 and 1.3\ifmmode^\circ\else\textdegree\fi{}K and up to 30.7 atm will be described. At the lowest temperatures, the susceptibility for all pressures has almost reached a constant value. The effective Fermi temperature ${{T}_{F}}^{**}$ from these measurements is found to vary from 0.43\ifmmode^\circ\else\textdegree\fi{}K at 0.5 atm to about 0.26\ifmmode^\circ\else\textdegree\fi{}K at 30.7 atm. From these measurements and recent specific-heat data of several authors, the spin-dependent exchange constant is calculated and is found to decrease with increasing pressure and density. The spin specific heat calculated from susceptibility according to Goldstein's theory is compared to available data. The susceptibility results are compared to recent data of Anderson, Reese, and Wheatley.

On the dense phases of He 3 at very low temperatures

A class of physical properties of the dense phases of He 3 are dominated at low temperatures by their nuclear spin systems. The latter determine the lower limits of these properties, and yield thereby approximately their behavior and limiting values at very low temperatures and at the absolute zero. The theory of the spin system allows one to estimate the lowest temperatures at which the various thermal properties of this class have to be determined experimentally to approach with a preset deviation their limiting values in the ground state of these dense phases. These determinations, with a common relative value of this difference, of the nuclear paramagnetic susceptibility, heat capacity, expansion coefficient, and their temperature rates of variation require decreasingly lower temperatures, with the latter finite quantities imposing temperatures one order of magnitude lower than the susceptibility. These studies suggest that in order to obtain satisfactory, experimentally inaccessible, extrapolated very low temperature behaviors and finite limits of the above class of properties, careful prior analysis of their finite temperature values will have to be performed in terms of the partial properties of the spin systems as well as those of the degrees of freedom other than spin of the dense phases of He 3.

Self-Diffusion Coefficient and Nuclear Susceptibility of LiquidHe3

The self-diffusion of the magnetization was measured at pressures of 5 to 9 cm Hg in pure liquid He/sup 3/ and in liquid He/sup 3/ containing 1.2% He/ sup 4/ at 0.032 to 0.15 deg K. The nuclear susceptibility relative to its value at the He/sup 4/ bath temperature was measured at similar pressures in pure He/ sup 3/ from 0.031 to 1.3 deg K, and in He/sup 3/ containing 1.2% He/sup 4/ at 0.032 to 1.3 deg K. No evidence was found for a transition of the liquid into any kind of cooperative state. Graphical representations of the results are included. Results indicate that the addition of 1.2% He/sup 4/ does not change the values for either case within the limits of experimental error. (B.O.G.)

On the theory of liquid and solid He 3

The object of the present paper is to develop quantitatively several features of the theory of the nuclear spin system in liquid and solid He 3, 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 He 3 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 He 3 hardly needs to be emphasized.

Direct observation of static nuclear susceptibilities at room temperature

Direct observation of static nuclear susceptibilities at room temperature

Electron Spin Paramagnetism of Lithium and Sodium

The paramagnetic susceptibility ${{\ensuremath{\chi}}_{p}}^{e}$ of conduction electron spins is isolated experimentally from the total magnetic susceptibility in metallic lithium and sodium by studying the intensity of the conduction-electron spin resonances. The absolute intensity of absorption is calibrated by comparison with the nuclear resonance of the metal nuclei in the same sample and at the same frequency, the two resonances being observed merely by changing the static magnetic field. In this manner ${{\ensuremath{\chi}}_{p}}^{e}$ is measured in terms of the nuclear static susceptibility, ${{\ensuremath{\chi}}_{p}}^{n}$, which in turn can be calculated accurately from the Langevin-Debye formula. A narrow band modulation technique gives improved signal to noise over our earlier work. The values of ${{\ensuremath{\chi}}_{p}}^{e}$ are (2.08\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ cgs volume units for lithium at 300\ifmmode^\circ\else\textdegree\fi{}K and (0.95\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ cgs volume units for sodium at 79\ifmmode^\circ\else\textdegree\fi{}K, in rather good agreement with the theory of Pines and Bohm, but in substantial disagreement with the simple Pauli model, or the results of Sampson and Seitz. Experimental precision does not permit conclusions to be drawn about the diamagnetism of conduction electrons.

Nuclear Susceptibility of Liquid and SolidHe3

An interpretation of the observed deviation from Curie's law of the nuclear paramagnetic susceptibility of liquid helium ${\mathrm{He}}^{3}$, based on a "cell" type of model of the liquid, is presented. The theory is able to account qualitatively, but not precisely, for the deviation. The reasons for its imprecision are discussed. The bearing of the point of view of the theory on the paramagnetism of the solid is considered: it is concluded that there should be no deviation from Curie's law at the lower temperatures, but that a small deviation near the rising part of the melting curve is quite possible.

On the Theory of LiquidHe3

The difficulties raised by the elementary interpretation of the recently observed temperature variation of the nuclear paramagnetic susceptibility of liquid ${\mathrm{He}}^{3}$ as an ideal antisymmetric fluid type of degeneration phenomenon are discussed. The susceptibility data are then shown to lead to a direct and rigorous evaluation of the entropy of spin disorder and the associated heat capacity. The observed enormous total entropy of liquid ${\mathrm{He}}^{3}$ at low temperatures is thus explained to be overwhelmingly entropy of spin disorder. The spin heat capacity is about half of the observed total heat capacity at the lowest temperature of observations. It should tend to become the dominant part of the total liquid heat capacity below this temperature. The maximum of the spin heat capacity at low enough temperatures may become observable indirectly through the total heat capacity of the liquid.