and the up and down velocities of the molecules will exceed the horizontal velocities, until after a short time involving many collisions, a redistribution, as required by the principle of equipartition, will have occurred, in which the component squared velocities are equalized and the whole mass of gas has a temperature greater than before. If Do denotes the density of the gas at the bottom of the enclosure, D the density at any height x, m the mass of one molecule, r the gas constant for one molecule, T the absolute temperature and g the acceleration of gravity, we have the relation in which wmgx is the work necessary to raise one molecule through the distance x against gravity. Now suppose each molecule to have a magnetic moment μ and imagine a vertical magnetic field applied throughout the enclosure instead of the gravitational field. The molecules will be driven to set themselves with their magnetic axes parallel to the magnetic intensity just as before the molecules were driven downward, and rotational velocities about lines normal to the field intensity will be favored, but thermal agitation will redistribute them as before until the law of equipartition is satisfied. If now denotes the angle made by the axis of any molecular magnet with the (vertical) magnetic intensity H, p the number of molecules per unit volume with their axes between 0 and 0 + do, and the number between 0 and do, we have, Po by strict analogy with the gravitational case, (9) Starting from this formula we can readily calculate the total change produced in the magnetic moment of the gas (0 before the application of the field) and thus the intensity of magnetization I. If a is written for The susceptibility is thus independent of H, and inversely proportional to T. So far as temperature is concerned it expresses the law of Curie, which holds for the paramagnetic gas oxygen over a great range of temperatures, and which holds over a great range in many other cases in which the molecular magnets are so far apart as not to act appreciably on one another. Inasmuch as r is known, and as N is known for any value of T at known pressure, we can calculate μ from the observed value of K. We thus obtain for oxygen, reckoning from 0° C. and 760 mm. pressure, Langevin's theory of paramagnetism is not an electron theory, as it has been developed without regard to the permanent electrical rotations assumed on this theory to account for the permanent magnetic moment of the elementary magnet. Nevertheless, it has rendered great services and has important relations to the electron theory. Investigation of the behavior of free electron orbits, as distinguished from the fixed orbits of Weber, in a magnetic field, have been made by Voigt" and J. J. Thomson,8 who independently, in 1902 and 1903, reached the conclusion that the existence, without damping, of such orbits in a substance would give it neither diamagnetic nor paramagnetic properties. The diamagnetic effects arising from change of velocities produced by the (10) magnetic intensity are just balanced by the paramagnetic effects due to the change of orbital orientation. With suitable dissipation 8 Phil. Mag. (6), 6, 1903, p. 673. }, (11) of energy, however, Thomson has concluded that paramagnetism may result, and Voigt that either paramagnetism or diamagnetism may result, according to circumstances. But the conceptions they have presented of the manner in which these results may be brought about do not seem probable, and have not gained wide acceptance. Voigt and, after him, Lorentz and Gans, have examined the behavior in a magnetic field of magnetic elements, or magnetons, consisting of homogeneous uniformly charged solids or symmetrical electron systems, in rotation, and have reached interesting and important conclusions. One of the most important cases is that of a magneton which may be treated as a solid of revolution, with initial angular velocity greater than eH/2m about the unique axis. In this case in accordance with classical electromagnetic theory, the rotation proceeds undamped about the unique axis, while it is damped about the other (equal) axes, and the action of the field on the magneton is as follows: When the field is applied, precession of the magneton's axis about the direction of the field begins, accompanied by nutation. The nutation is damped out by dissipation or radiation, and the precession is retarded for the same reason. Hence the direction of the axis of the magneton gradually approaches coincidence with the direction of the field, when it is in stable equilibrium. During this process the velocity of rotation diminishes slightly, the motion being affected as in the case of the electricity in Weber's molecular grooves. If there are N such magnetons in the unit of volume, and if the demagnetizing and molecular fields and the upsetting effect of collisions are negligible, all the magnetons will ultimately become oriented with their axes in the direction of the magnetic field. In this case the moment of unit volume will be manent rotation, u its angular velocity about this axis, and H the intensity of the applied field. The first and principal term is entirely independent of H. The orientation is, of course, produced by the magnetic field, but only the time taken to arrive at the steady state is affected by its magnitude. The second term is a diamagnetic term, and arises from the fact that owing to the change of flux through the magneton during the process of its orientation its velocity is decreased, just as in the case of the Weber-Langevin theory. In this case we have, except for the small diamagnetic term, which vanishes with the intensity, saturation for even the weakest fields; and we have less nearly complete saturation for stronger fields. When collisions are not absent, a magneton's axis will be repeatedly deflected in its approach toward coincidence with the direction of the field, and the intensity of magnetization will not reach saturation; but it will increase with the field strength, being greater for a given field strength, the greater the mean time between collisions and the weaker the molecular and demagnetizing fields. Increase of temperature, shortening this time between collisions, and increasing their violence, will, if the magnetons remain unchanged, thus diminish the magnetization for a given field strength. The precessional process described above is doubtless similar in a general way to the process by which in every case in paramagnetic and ferromagnetic substances the magnetons are aligned more or less completely with the magnetic field. The exceedingly interesting ring electron recently proposed by A. L. Parson and extensively applied by him and others to a wide range of chemical and physical phenomena, is a special case of Voigt's magneton, and will be discussed by one of my colleagues. Bearing in mind that, on the electron theory, the molecule or magneton must, with Voigt, be treated as a gyroscope and can not L 7 execute true rotations,10 such as Langevin assumed, except as very special cases of precession, Gans11 has recently developed a general theory of diamagnetism and paramagnetism, proceeding in accordance with the methods of statistical mechanics. He assumes as his magneton a body rigidly built of negative electrons and placed inside a uniformly and positively charged sphere whose center is coincident with the center of mass of the electrons, and whose charge is equal in magnitude to that of the magneton, so that electrical actions do not have to be considered. The energy is assumed to be entirely electromagnetic. For simplicity it is assumed that two of the principal (electromagnetic) moments of inertia are equal, but it is not assumed in general that the magneton is a body of revolution; thus the cross-section normal to the unique axis might be a square, and rotation about it subject to the effects of thermal collisions, instead of a circle, with rotation independent of such collisions. The method of statistical mechanics is then applied to the two cases to be considered: first, that in which the magneton is not a body of revolution so that the rotations about the three axes must all be treated as statistical coordinates; and second, that in which the magneton is a body of revolution so that rotation about the axis of figure is not affected by collisions and can not be treated as statistical coordinate. a In the first case it is found that the susceptibility is always negative, or the substance diamagnetic. When the three principal moments of inertia are equal, the susceptibility is independent of the temperature and of the intensity of the magnetic field, which is the case with many diamagnetic substances. When but two of the moments are equal, however, the susceptibility depends on both. the temperature and the intensity in somewhat complicated ways. Fig. 1 shows the general 10 See also F. Krueger, Ann. der Phys. (4), 50, 1916, p. 364. 11 Ann. der Phys. (4), 49, 1916, p. 149. 1.4 -1,6 FIG. 3. 20 Krogauss Fig. 4 shows the way in which, according to the theory, the susceptibility x depends. upon the absolute temperature 0, while the type of curve found in Honda's experiments is shown in Fig. 5. Little weight can be given the lower part of the theoretical curve, inasmuch as equipartition of energy and also absence of inter-molecular action were both assumed in its derivation, and it is improbable 12 Ann. der Phys. (4), 61, 1920, p. 585. curve can not be explained by the presence of iron, as the positive susceptibility of the iron would become less with temperature increase. We come now to the second case, in which the magneton has a true axis of figure and an essentially permanent angular momentum about this axis, and therefore a magnetic moment in the direction of this axis, unchangeable by collisions. On account of this permanent magnetic moment and angular momentum, paramagnetism results very much as in the theory of Voigt already presented; and on account of the slight diminution of this angular momentum in the magnetic field and on account of the rotation of the magneton about the other axes brought about or modified by the thermal agitation in accordance with the law of equipartition, diamagnetism results and is superposed upon the paramagnetism. This diamagnetism does not appear in Langevin's theory, because instead of a permanently rotating magneton he assumed a permanent magnet without angular momentum about the axis except as produced by thermal collisions. Langevin, however, assumed that Weber's diamagnetism was superposed upon the paramagnetism, and this corresponds in part to the diamagnetism of Gans's theory. Returning to the results of Gans's statistical treatment for the case of the magneton in permanent rotation about a unique axis, we find that the susceptibility is a function of both field strength and temperature. It planation, has been observed by Weber and Overbeck13 in the case of copper-zinc alloys, and by Honda in the case of indium. Weber and Overbeck, who have taken great precautions and believe their alloys free from iron, have called the phenomenon metamagnetism. The downward trend of paramagnetic susceptibility with increase of field strength is apparent in some of the curves obtained by Honda. For weak fields at low temperatures, but with H/T finite, Gans's formula approaches that of Langevin as a limit. Here the paramagnetic rotations are prominent in comparison with diamagnetic thermal rotations about the other axes. As the field intensity approaches zero with finite values of the temperature the susceptibility approaches a limit which is the sum of two terms, a paramagnetic term identical with that of Langevin and a diamagnetic term independent of the temperature like that of Weber. The theory of Gans thus covers a wide range of cases, but so far has been applied in detail to but few. By taking account of the molecular field, and by applying the quantum theory, although not in the most thorough way, he has more recently extended his theory to cover more accurately the paramagnetism exhibited by dense bodies and at low temperatures.14 In a similar way the quantum theory has been set into the theory of Langevin by Oesterhuis15 and Keesom16; and it has been thoroughly applied, for the case of rotation with one degree of freedom, by Weyssenhoff,17 and for the case of rotation with two degrees of freedom by Reiche18 and by Rotzajn,19 to the system of elementary magnets, without permanent angular momentum, assumed by Langevin. These theories are thus not electron theories, like that of Gans. They reduce to the theory of Langevin at high tempera 13 Ann. der Phys. (4), 46, 1915, p. 677. 17 Ann, der Phys. (4), 51, 1916, p. 285. tures when equipartition exists, and the rigorous theories agree well with experimental results obtained at low temperatures, where Langevin's theory completely fails. The next step should be the rigorous application of the quantum theory to the case in which the magneton has a permanent angular momentum,, with gyroscopic properties, as required by the electron theory. According to experiment hydrogen and helium are diamagnetic although according to Bohr's models their molecules have strong magnetic moments. This is apparently consistent with the theory of Gans, but inconsistent with the theory of Weber and Langevin. Honda and Okubo,20 in a part of a paper dealing more generally with the kinetic theory of magnetism, have proposed the following explanation of this diamagnetic effect. Suppose the magnetic axis to be rotating about one of the other axes in a plane parallel to the magnetic intensity. On account of the presence of the field, the velocity of rotation, which would be uniform without the field, is now variable, the motion being more rapid when the moment points in the direction of the field than when it points the other way. Hence the time mean of its directions is opposite to that of the field and the mean effect is diamagnetic. If the magnetic axis is rotating in a plane not parallel to the direction of the field, we must resolve the effect in the direction of the field. Doing this for all the elementary magnets, originally pointing uniformly in all directions, we get a resultant diamagnetic effect. This, however, is only a part of the total effect found in Langevin's theory to be paramagnetic, though it is only implicit in his treatment, unless we assume permanent rotations, independent of the temperature, about an axis normal to the magnetic axis. This assumption they have made. From what we have seen there seems to be no way to account satisfactorily for paramagnetism and ferromagnetism except on the assumption of an elementary magnet which is a permanent electrical whirl, as Ampère assumed; which has also mass, as Weber as20 Phys. Rev., 13, 1919, p. 6. |