The spin-lattice relaxation time, $T_1$, has been determined in single crystals of nickel ferrite, by using the ferromagnetic resonance effect at a frequency of 9000 Mc/sec. At high levels of microwave power, the saturation effect is observed both in the resonance absorption and in the component of magnetization along the static field. $T_1$ is computed from these data.

The experimental method is described, and the results presented. Some of the results have not been encountered before in magnetic resonance, and are not predicted by available theories. Because of this, the relaxation time cannot be calculated from the classical theory of the magnetic resonance. By equating the energy absorbed to the energy transferred to the lattice, we obtain a value $T_1\sim 3\times {10}^{-5\mathrm{}}$ sec. This is longer than the decay time observed directly, which is less than a few tenths of a microsecond. The results are compared with available theories of the magnetic resonance and spin-lattice interaction.

This paper surveys current theoretical work on spin-spin and spin-lattice relaxation in ferromagnetic substances with reference to ferromagnetic resonance. It is shown that it is possible to give a reasonable explanation of the principal features of the experimental results. The calculations, carried out for nickel in particular, make use of a macroscopic spin-lattice interaction determined by the observed magnetostriction and a pseudo-dipolar spin-spin interaction determined by the observed anisotropy. It is not sufficient to use magnetic dipole coupling alone for either of the two relaxation processes. The zero-point fluctuations in the pseudo-dipolar field play a central role in the development of the theory.

The microwave resonance absorption in gadolinium metal has been measured at 9000 and 24 000 Mc/sec. The $g$ value in the paramagnetic region is found to be 1.95±0.03. In the ferromagnetic region extreme line breadth prevents accurate determination of the $g$ value. Calculations neglecting effects of line breadth give a ferromagnetic $g$ value of 1.94.

This paper is intended to supplement the review articles published by Stoner in 1948 and 1951, firstly by considering in greater detail the quantum-mechanical and statistical-mechanical foundations of the collective electron theory and secondly by considering briefly a wider range of relevant experimental results.

In Sec. 1 previous theoretical work is recalled. In Sec. 2 the difficulties of a rigorous quantum mechanical derivation of the internal energy of a ferromagnetic metal at absolute zero are outlined. In order to determine at least the form of the expressions, a calculation based on the tight binding approximation is described for a crystal containing $N$ singly charged ions, which are fairly widely separated, and $N$ electrons. The forms of the Coulomb and exchange contributions to the energy are discussed in the two instances of maximum and minimum multiplicity. The need for a correlation correction is stressed, and the effects of this correction are discussed with special reference to the state of affairs at infinite ionic separation. The fundamental difficulties involved in calculating the energy as function of magnetization are considered below; it is shown that they are probably less serious for tightly bound than for free electrons, so that the approximation of neglecting them in the first instance is not too unreasonable. The dependence of the exchange energy on the relative magnetization $\zeta$ is then of the form $\Sigma n^{}\frac{A_n{\zeta }^{2n}}{2n}$, and the relative orders of magnitude of $A_1$ and $A_2$ are considered. A previous calculation by Slater is critically reviewed.

In an application of statistical mechanics the difficulty arises that only if $\zeta =1\mathrm{} or 0$ will the zero-order wave function be in general a single Slater determinant. A calculation by Lidiard on this problem, that of spin degeneracy, is referred to. The relevance of the dependence of the interaction energies on wave vector is stressed. If these energy contributions are constant, then the free energy expression is that derived by Stoner, based on Fermi-Dirac statistics, even if spin degeneracy is taken into account. Several related difficulties are exemplified by a discussion of the properties of the free-electron gas, for which the exchange energy varies rapidly with wave vector ${\mathbf{k}}_i$, ${\mathbf{k}}_j$ when $|{\mathbf{k}}_i-{\mathbf{k}}_j|\to 0$. For tightly bound electrons the dependence on wave vector is much less rapid, and it is suggested that spin degeneracy effects are here less serious. A simple calculation of the low temperature electronic heat of tightly bound electrons, including exchange and correlation, is referred to in exemplification.

In Sec. 3 several experimental results are briefly discussed, including the following: (1) The variation with composition of the Curie temperature and saturation magnetization, and the variation with temperature of the susceptibility of nickel alloys; (2) the variation with temperature of the magnetization, electronic heat, and magnetocaloric effect of cubic cobalt, nickel, and nickel alloys below the Curie point; (3) the high temperature electronic heat of nickel and palladium; (4) the electronic properties of chromium and some other transition metals; (5) the electronic properties of palladium and its alloys.

By an analysis of a magnetization curve is meant the determination of the character and extent of the various elementary changes occurring over any part of the curve. Even if the quantitative characteristics of the various elementary processes are known, an analysis cannot in general be made simply on the basis of the information provided by a given magnetization curve, owing to the varied and complicated way in which the different processes are superposed with an ordinary polycrystalline specimen. Further information is obtainable from measurements such as those of Bates on the thermal changes accompanying adiabatic magnetization, but little progress can be made in the analysis of curves in the hysteresis range unless the magnetic and thermal effects due to the reversible and irreversible changes separately can be unambiguously determined.

A broad survey of the general problem is given with particular reference to recent theoretical and experimental work at Leeds. The theoretical approach and the methods of analysis are along the same lines as in the treatment of the magnetocaloric effect by Stoner and Rhodes (1949). The experimental work reviewed includes studies of the reversible susceptibility, the Barkhausen effect, and the reversible change of magnetization with temperature at constant field (Tebble, with Corner, Newhouse, Skidmore, Wood, and others, 1948 onwards).

Through this work, it is now possible to determine not only the relative contributions from reversible and irreversible processes to the change of magnetization over any range but also the thermal changes associated with each. The part of the reversible change due to change in intrinsic magnetization can be determined, and in many cases analysis will show whether the remainder of the reversible change is mainly due to rotation of the magnetization against the crystalline and strain anisotropy forces, or to other processes such as boundary movements. Much detailed information about irreversible processes is obtained from determinations of the number-size relations of Barkhausen discontinuities, which in general correspond not to spontaneous movements of primary domain walls as wholes, but to smaller scale sectional irregularities in the motion. The thermal effect associated with these irreversible processes may be a heating or a cooling.

Partial analyses are presented for annealed nickel, hard-drawn iron, and other materials. These show clearly the potentiality of the general method for obtaining detailed quantitative information about the elementary processes contributing to the change of magnetization along a magnetization curve.

Two rival methods of handling molecular problems, the Heitler-London and the molecular orbital methods, have been used for many years, and they have their counterparts in magnetism in the Heisenberg and the energy-band methods. The first part of this paper shows that the Heitler-London and Heisenberg methods, as usually applied, are not appropriate for problems containing more than a few electrons. The reason is the lack of orthogonality of the one-electron orbitals of atomic type. This involves the presence of overlap integrals which, if included in the calculation, make the method essentially divergent and unsuitable for numerical calculation if the product of the number of electrons in the system, times an overlap integral between nearest neighbors, is large compared to unity, which in practice confines the application to problems of a few electrons. Most applications of the method, including all applications of the Heisenberg method to ferromagnetism and antiferromagnetism, disregard many integrals which are not as a matter of fact negligible, and for this reason cannot be regarded as valid.

Van Vleck many years ago tried to overcome this difficulty, but his results are only partially satisfactory on account of lack of generality. The straightforward way to overcome it is to use really orthogonal one-electron orbitals, either molecular orbitals or Wannier functions, and the remainder of the paper considers how the familiar facts of ferromagnetism are to be explained on this basis. If we use a single determinantal wave function composed of energy-band orbitals, we have the familiar energy-band or collective electron theory of ferromagnetism. This simple method may well be fairly accurate at small internuclear distances; it predicts correctly that ferromagnetism must be impossible when the atoms are close enough together to broaden the energy bands greatly. For large internuclear distances, however, it leads to quite wrong limiting behavior, since it gives a wave function involving a considerable contribution of ionic states. To eliminate these ionic states, at large internuclear distances, we must make linear combinations of different determinantal functions, corresponding to different assignments of electrons to orbitals. The results of such a calculation are sketched.

To treat magnetic problems properly, we must make such calculations for different total spins, calculate the energy of the states of different magnetizations as a function of internuclear distance, and see which states lie lower, the magnetic or nonmagnetic. At infinite separation, the energy will be independent of total magnetization, since in this limit the spins of different atoms can be oriented in any arbitrary way without affecting the energy. At very small distances, the elementary band theory shows that the nonmagnetic state must lie below the magnetized state, so that ferromagnetism is impossible. At intermediate distances, the elementary theory can show only that ferromagnetism is possible, not that it is necessary; to find whether it actually exists, we should really have to carry out a calculation of the type described. If we find ferromagnetism, we shall necessarily find that the energy difference between nonferromagnetic and ferromagnetic states was zero at infinite internuclear distance, increased as the internuclear distance decreased, went through a maximum, and then went to zero and changed sign. This behavior is often postulated for the Heisenberg exchange integral, although we do not feel that this integral has the direct theoretical meaning often ascribed to it.

These problems are closely related to the correlation energy, which is discussed. We do not feel that the familiar Wigner-Seitz calculation of correlation energy is very accurate; the correlation energy is in fact a function of magnetization, decreasing numerically with decreasing internuclear distance, and its interpretation cannot be given properly except by considering its intimate connection with magnetization. The net result of the present discussion is that to give a proper account of ferromagnetism, we are forced to use orthogonal orbitals and, therefore, must make close connection with the band theory. We must, however, carry our theory further than has usually been done, though the prediction of the elementary band theory, i.e., that ferromagnetism is impossible for broad bands and can exist only for narrow bands and electronic wave functions which overlap only slightly, is verified by the present more accurate approach.

As an illustration of the general method of calculating magnetic energies described in this paper, we may mention the recent calculation of Meckler, not yet published, on the oxygen molecule. This has been carried out by setting up many determinantal wave functions formed from orthogonal atomic orbitals made out of the original nonorthogonal orbitals. The secular equations involving these determinantal functions were solved by use of a digital computer. The resulting energy levels and wave functions reduce to the proper type to represent the ${}_{}{}^3{}_{}{}^{}P$ ground state of the oxygen atom at infinite internuclear distance. Meckler's energy for the triplet ground state is very accurate; that for the next singlet excited state is quite good. The energy separation between singlet and triplet shows the behavior expected for the energy difference between a ferromagnetic and nonferromagnetic state, being zero at infinite distance, then increasing, going to a maximum, then decreasing again. As far as the writer is aware, this is the first time when such a behavior has been found as a result of straightforward calculation from quantum mechanics.

THIS symposium consisted of a round-table discussion on exchange forces in ferromagnetics, and began with the presentation of invited papers by C. Zener, J. C. Slater, E. P. Wohlfarth (read by E. C. Stoner), and J. H. Van Vleck. A lengthy but lively discussion period, partly with the audience participating, followed the invited papers. It is unfortunately not possible to reproduce here the discussions, but R. Smoluchowski has been kind enough to prepare a brief summary and reconciliation of the invited papers: his discussion follows here after the papers.

The magnet is an air core and water-cooled massive Helmholtz type with rectangular cross section. It consists of a field coil 10 cm in mean radius which can produce 35 000 oersteds at a power of 1700 kw with a uniformity of about 10 ppm at the center within a region of 10 percent of the mean radius, and a gradient coil 5.2 cm in mean radius which can produce a gradient of 100 oersteds per cm at a power of 2.5 kw with a uniformity of about 140 ppm within the same region. The fields and gradients are separately controlled.

Bromine, even in small quantities, is known to alter very radically properties of graphite such as conductivity, Hall effect, and others. In particular, the relatively large diamagnetic susceptibility of graphite is rapidly decreased by addition of bromine and the effect cannot be explained by assuming presence of bromine in either molecular or atomic form. It appears that this effect can be interpreted in terms of the Brillouin zone structure of graphite. Using Peierls' theory of diamagnetism of free electrons in a solid and Wallace's results concerning the band structure of graphite an expression is derived which relates the change in the magnetic susceptibility to the change in the number of electrons in the Brillouin zones. The theory is in good agreement with experiment if the effectiveness of bromine in accepting electrons is assumed to be that obtained from Hall effect and conductivity data. The probable influence of lattice imperfections is briefly considered.

Following a suggestion by A. Kastler, an attempt has been made to produce a high degree of nuclear orientation in ${\mathrm{Hg}}^{199}$ vapor by the absorption of circularly polarized resonance radiation. A method is described for detecting the nuclear orientation by observing the intensity of the $\pi$-component of the resonance radiation. No indication of orientation was found. The probable cause for this failure is considered to be the imprisonment and consequent depolarization of the effective component of the resonance radiation.

The molar specific heat $C_{\mathrm{I}}$ of neodymium ethyl sulfate has been measured on three samples of separated neodymium isotopes in the temperature range 0.95°K-2.15°K by the magnetic method of Benzie and Cooke at frequencies from 300 to 1200 cps. The measured $C_{\mathrm{I}}$ may be described within 1 percent by $C_{\mathrm{I}}=\frac{b}{T^2}$, where $T$ is the temperature and $b$ is the constant. All three samples were shown by x-ray diffraction measurements to have the same structure as that given for neodymium ethyl sulfate by Ketelaar. For the electron contribution to the specific heat we obtained $b_e=0.146\mathrm{}\pm 0.004\times {10}^5$, and for the hyperfine splitting contributions we obtained $b_{143}=5.08\mathrm{}\pm 0.05\times {10}^5$ and $b_{145}=1.7\mathrm{}\pm 0.2\times {10}^5$ erg degree. These values of $b_{143}$ and $b_{145}$ agree within experimental error with the corresponding values computed from the microwave measurements of Scovil. Thus, the hyperfine splitting contribution to the specific heat for ${\mathrm{Nd}}^{143}$ ${\left({\mathrm{C}}_2{\mathrm{H}}_5\mathrm{S}{\mathrm{O}}_4\right)}_3$·9${\mathrm{H}}_2$O is some 35 times larger than the electron contribution which makes this material of interest for the study of nuclear effects below 1°K.

A brief description is given of the method used by Bates and his co-workers which permits the measurement of the thermal changes which accompany the step-by-step magnetization of ferromagnetic metals. The method has now been used to examine the behavior of 0.5 and 4 percent silicon iron in the untreated and annealed states when magnetized along the virgin curve and in cycles with fields up to 380 oersteds. In this work, some twenty copper-constantan thermocouples generate currents when a magnetic field acting on the ferromagnetic specimen is changed, and these are transmitted ballistically via a toroidal Mu Metal transformer to a highly sensitive moving coil galvanometer of long period. As the thermal changes in the present work were very small, a new technique had to be developed to deal with them. The measured thermal changes are plotted as functions both of the intensity of magnetization and of the effective magnetic field. The several errors and experimental difficulties, including asymmetry in thermal readings, stray induction effects, heating due to eddy currents, and thermal and electromagnetic disturbances experienced during the measurements are briefly discussed, and the methods used to overcome them are briefly outlined.

The results are examined in the light of current domain theory and a theoretical procedure developed by Stoner and Rhodes, in which the thermal contribution resulting from change in intrinsic magnetization, often described as the magnetocaloric effect, is substracted from the recorded thermal change, and the residue is considered in detail. Another procedure recently applied to results for nickel was also examined, but it was found inapplicable to the silicon iron data. An unusual depression or dip is found in the curves of thermal change plotted as a function of the applied field when the region of low field values is reached upon reducing the field from the maximum used in describing a hysteresis cycle. This observation provides evidence in support of Kittel's theory of flux closure which Bozorth has used to explain the low remanence of such alloys which exhibit little magnetostriction and magnetic anisotropy and small internal strain. It is suggested that the strange depression may be correlated with some peculiar domain boundaries recorded by the Bitter powder technique on the (100) surface of a single crystal of silicon iron when it is demagnetized either thermally or in the above manner. These boundaries "wriggle" in an extraordinary manner which suggests that they separate regions either of flux closure in opposite directions or domains whose magnetization vectors meet head-on.