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and the up and down velocities of the mole- where N= the number of molecules per unit cules will exceed the horizontal velocities, volume. until after a short time involving many
When a is small, as it is except for very collisions, a redistribution, as required by the intense fields and very low temperatures, this principle of equipartition, will have occurred, equation becomes, with negligible error, in which the component squared velocities are
· H, equalized and the whole mass of gas has a
(12) temperature greater than before. If D. de- which gives for the susceptibility notes the density of the gas at the bottom of
Νμ? the enclosure, D the density at any height
K = I/H
(13) , m the mass of one molecule, the gas con
The susceptibility is thus independent of H, stant for one molecule, T the absolute tem
and inversely proportional to T. So far as perature and g the acceleration of gravity,
temperature is concerned it expresses the law we have the relation
of Curie, which holds for the paramagnetic D/D.
(8) gas oxygen over a great range of temperarᎢ
tures, and which holds over a great range in in which w=mgx is the work necessary to
many other cases in which the molecular raise one molecule through the distance &
magnets are so far apart as not to act appreagainst gravity.
ciably on one another. Now suppose each molecule to have a mag
Inasmuch as r is known, and as N is known netic moment
for any value of T at known pressure, we can netic field applied throughout the enclosure
calculate u from the observed value of K. We instead of the gravitational field. The mole
thus obtain for oxygen, reckoning from 0° C. cules will be driven to set themselves with
and 760 mm. pressure, their magnetio axes parallel to the magnetic
3rTK intensity just as before the molecules were
(14) driven downward, and rotational velocities about lines normal to the field intensity will
Langevin's theory of paramagnetism is not be favored, but thermal agitation will redis
an electron theory, as it has been developed tribute them as before until the law of equi.
without regard to the permanent electrical
rotations assumed on this theory to account partition is satisfied. If now @ denotes the angle made by the axis of any molecular
for the permanent magnetic moment of the magnet with the (vertical) magnetic inten
elementary magnet. Nevertheless, it has
rendered great services and has important sity H, p the number of molecules per unit volume with their axes between 0 and 0 + dd,
relations to the electron theory. and the number between 0 and do, we have,
Investigation of the behavior of free Po by strict analogy with the gravitational case,
electron orbits, as distinguished from the
fixed orbits of Weber, in a magnetic field, mH (1 – cos ) p/po = e
(9) have been made by Voigt) and J. J. Thomson,8
who independently, in 1902 and 1903, reached calculate the total change produced in the
the conclusion that the existence, without magnetic moment of the gas (0 before the
damping, of such orbits in a substance would application of the field) and thus the inten
give it neither diamagnetic nor paramagnetic
properties. The diamagnetic effects arising sity of magnetization 1. If a is written for
from change of velocities produced by the mH T
(10) magnetic intensity are just balanced by the we get the expression
paramagnetic effects due to the change of
orbital orientation. With suitable dissipation eo te I = Nu
& Phil. Mag. (6), 6, 1903, p. 673.
of energy, however, Thomson has concluded manent rotation, u its angular velocity about that paramagnetism may result, and Voigt this axis, and I the intensity of the applied that either paramagnetism or diamagnetism field. may result, according to circumstances. But
The first and principal term is entirely the conceptions they have presented of the
independent of H. The orientation is, of manner in which these results may be brought
course, produced by the magnetic field, but about do not seem probable, and have not
only the time taken to arrive at the steady gained wide acceptance. Voigt and, after him, Lorentz and Gans,
state is affected by its magnitude. The second have examined the behavior in a magnetic
term is a diamagnetic term, and arises from field of magnetic elements, or magnetons, con
the fact that owing to the change of flux sisting of homogeneous uniformly charged through the magneton during the process of solids or symmetrical electron systems, in
its orientation its velocity is decreased, just rotation, and have reached interesting and
as in the case of the Weber-Langevin theory. important conclusions.
In this case we have, except for the small One of the most important cases is that diamagnetic term, which vanishes with the of a magneton which may be treated as a intensity, saturation for even the weakest solid of revolution, with initial angular fields; and we have less nearly complete velocity greater than eH/2m about the unique saturation for stronger fields. axis. In this case in accordance with class- When collisions are not absent, a magneton's ical electromagnetic theory, the rotation pro- axis will be repeatedly deflected in its apceeds undamped about the unique axis, while
proach toward coincidence with the direction it is damped about the other (equal) axes, of the field, and the intensity of magnetizaand the action of the field on the magneton tion will not reach saturation; but it will inis as follows: When the field is applied, pre
crease with the field strength, being greater cession of the magneton's axis about the
for a given field strength, the greater the direction of the field begins, accompanied by
mean time between collisions and the weaker nutation. The nutation is damped out by
the molecular and demagnetizing fields. Indissipation or radiation, and the precession is retarded for the same
crease of temperature, shortening this time Hence the
between collisions, and increasing their viodirection of the axis of the magneton gradually approaches coincidence with the direction
lence, will, if the magnetons remain unof the field, when it is in stable equilibrium.
changed, thus diminish the magnetization for
a given field strength. During this process the velocity of rotation diminishes slightly, the motion being affected
The precessional process described above is as in the case of the electricity in Weber's
doubtless similar in a general way to the molecular grooves.
process by which in every case in paramagIf there are N such magnetons in the unit
netic and ferromagnetic substances the magof volume, and if the demagnetizing and netons are aligned more or less completely molecular fields and the upsetting effect of col- with the magnetic field. lisions are negligible, all the magnetons will The exceedingly interesting ring electron ultimately become oriented with their axes recently proposed by A. L. Parson and extenin the direction of the magnetic field. In this sively applied by him and others to a wide case the moment of unit volume will be
range of chemical and physical phenomena, is eNC el
a special case of Voigt's magneton, and will I =
(15) 2m 2m
be discussed by one of my colleagues. when e is the charge of the magneton, C its Bearing in mind that, on the electron moment of inertia about the axis of per- theory, the molecule or magneton must, with • Gött. Nachr., 1910, p. 197.
Voigt, be treated as a gyroscope and can not
relation between the susceptibility and the intensity according to Gans's theory, while Fig. 2 shows the type of curve found experimentally by Honda in many cases. The importance of carrying the measurements down
methinproceedinga magnetism and ped a gen
execute true rotations, 10 such as Langevin assumed, except as very special cases of pre cession, Gans11 has recently developed a general theory of diamagnetism and paramag netism, 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 a statistical coordinate.
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.
shown in Fig. 7. Thus while diamagnetism may exist without paramagnetism, paramagnetism is always accompanied by diamag
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
netism, as on all other theories. In weak fields and at low temperatures the paramagnetism may prevail; but as the temperature and field strength increase it goes over into diamagnetism.
A transition from paramagnetic to diamagnetic susceptibility, whatever may be the explanation, 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 para magnetic 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 Keesom10, and it has been thoroughly applied, for the case of rotation with one degree of freedom, by Weyssen hoff, 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.
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 as
20 Phys. Rev., 13, 1919, p. 6.
Olve thehe directicotating in