Paramagnetism

PARAMAGNETISM The distinctive characteristics of paramagnetics have already been mentioned. In this section some of the quantitative experi mental data will be considered and their theoretical significance discussed. A considerable number of susceptibility measurements were made on paramagnetics before Curie's investigation was carried out (1895) ; but the results were not very accurate, and in particular no reliable data as to variation with temperature had been obtained. Curie measured the susceptibility of oxygen, one of the few paramagnetic gases, over a range of temperatures from 2o° to 45o° C, and found that the specific susceptibility was in versely proportional to the absolute temperature. The relation x = — was also found to hold approximately for paramagnetic solutions. A number of attempts were made to formulate a precise theory of paramagnetism on the basis of the electron theory, but Langevin's treatment of the problem was the first to give a satisfactory interpretation of the leading experimental obser vations.

Kinetic Theory of Paramagnetism.

Langevin considered the case of a paramagnetic gas, in which the molecules had a re sultant magnetic moment, due to lack of balance of the moments corresponding to the individual electron orbits. The diamagnetic effect due to precession will occur for such molecules, but if the molecules have a resultant permanent moment, there will be a tendency for them to aline themselves in the direction of the magnetic field. The alinement will not be complete owing to the collisions between the molecules, but the intensity of magnetiza tion in the direction of the field due to the change in orientation of the molecules may be, and, as experiment shows, generally is far greater than that in the opposite direction due to the dia magnetic effect, and the gas will then be paramagnetic. The paramagnetic effect may be considered independently of the dia magnetic. Let /u be the magnetic moment of the molecules, supposed all alike. The magnetic potential energy of a molecule in a field H is given by where 0 is the angle between the magnetic axis and the field.

In the absence of a magnetic field the molecules will be orien tated at random, and the gas as a whole will have no resultant magnetic moment ; but when a field is applied, the molecules acquire different energies according to the direction of their axes, and a uniform distribution of the directions of the axes is no longer compatible with thermal equilibrium. Just as, in a column

of gas in the earth's gravitational field, the density increases downwards, the number of molecules being greater where their potential energy is smaller, so in a magnetic field, the axes of the molecules crowd together towards the direction of the mag netic field. The higher the temperature, the more nearly uniform is the distribution. According to Boltzmann's theorem, if certain conditions are fulfilled, the number of molecules with their mag netic axes pointing in a direction 0, per unit solid angle, will be proportional to In this expression, e is the base of the Naperian logarithms, and k is the gas constant per molecule. (The kinetic energy of a mole cule at a temperature T is given by ik T.) In order that this equilibrium may be set up, it is necessary to suppose that there is equipartition of energy among the degrees of freedom of the gas molecules, in particular that a molecule possesses mean rotational ener gies about axes perpendicular to p. equal to 2 kT. A change in orientation of a molecule implies a change in its energy, and also a change in the direc tion of the angular momentum associated with the magnetic moment of the electron ro tating in an orbit ; this change cannot be brought about by the agency of the magnetic field alone, which can only produce the precessional effect ; but the change in orientation can occur if there are collisions between the molecules (or radiational pro cesses) when the energy and momentum conditions can be satis fied. The contribution of a molecule with its axis in a direction 0 to the intensity of magnetization in the field direction is ,ucos0, and, since the directional distribution of the molecules is known, the total intensity of magnetization due to a field may be calcu lated. Let be the total resultant magnetic moment, in a field H, of n molecules of moment ,u; then The Langevin function, L(a), is shown in fig. 27.