Diamagnetism

The magnetic moment of an atom (or molecule) as a whole will be equal to the resultant of the moments of the electrons, but whether there is a resultant moment or not the application of an external magnetic field will modify the motion of the electrons in such a way as to produce a diamagnetic effect. For simplicity the effect of a field applied perpendicularly to a circular orbit of radius r may be considered. Let ,u be the magnetic moment, v the velocity of the electron, then eS evr = — •12= CT = c2rr 2C A change of magnetic flux through the orbit produces an electro motive force E round it, which will accelerate the electrons, so that dH 2 rrE= — c dt where is the mean square radius of the projected orbit on a plane at right angles to the lines of force. The atomic suscep tibility, XA t is the ratio of the total resultant change of magnetic moment, obtained by summing the above expression over all the electrons, to the applied field: For an atom which is spherically symmetrical E = 3 E r2, where is the mean square distance of the electron from the nucleus, while refers to the projected orbit. For the atomic susceptibil ity of such an atom, or for the mean atomic susceptibility of atoms orientated at random with respect to the field, It is usually more convenient to deal with the gram-atomic susceptibility xA, equal to the product of the mass susceptibility X and the atomic weight A. Let Z be the number of atoms in a gram atom (Avogadro's number), then (For gram molecular susceptibility, the symbol x„, may be used.) The diamagnetic effect will occur whether the atoms (or mole cules) have a resultant magnetic moment or not, but it may be masked, if there is a resultant moment, by the paramagnetic effect, which, as will be discussed later, is usually much stronger. If there is no resultant moment the substance will be diamagnetic. The theory which has been outlined applies strictly to systems (atoms or ions) consisting of electrons rotating about a single centre of force, or to aggregates of such systems, such as a mon atomic gas. For such a gas, the diamagnetic susceptibility, since it depends only on the structure of the atoms, and not on their state of motion, should be independent of the temperature. The precise manner in which the Larmor theorem is to be applied to molecular systems in which the electrons are under the influence of more than one centre of force is by no means clear; but a rough proportionality between the area of the electronic orbits, or the region over which they are diffused, and so of the "size" of the molecule and the susceptibility may be expected. An inde pendence of temperature of diamagnetic susceptibility would, however, only be anticipated for diamagnetic substances which are constituted of quasi-independent simple or complex systems (ions, atoms, or molecules) which do not change with the tem perature.

Some Experimental Results.

The first extensive series of susceptibility measurements were made by P. Curie (Ann. de Chim. et Phys., 1895), who found that for almost all the diamagnetics investigated there was practically no change in mass suscepti bility with temperature, and that frequently it was independent of the physical state. Thus and yellow phosphorus showed no change in passing through the melting point, and the suscepti bilities of different forms of sulphur were the same. As an exam ple of exceptional behaviour, for bismuth the numerical value of the susceptibility decreased linearly with the temperature (from x = —1.23X io 6 at 20° C to x = —.87x at 273° C), changed abruptly at the melting point to a much smaller value ( —.035X and then remained constant.

The influence of chemical combination on magnetic properties was studied by P. Pascal (Ann. de Chim. et Phys. 1908-13), who made valuable measurements, particularly on organic liquids, by the U-tube method. He concluded that the molecular suscep tibility could be expressed as the sum of the atomic suscep tibilities XA, with a correcting factor X depending on the nature of the chemical linkages between the atoms: The susceptibilities of the halogens were measured directly; then from the change produced, e.g., by the substitution of a Cl atom for an H atom in an organic compound, the susceptibility constant for the H atom could be found ; the constant for the C atom could This was first shown by J. Larmor -(Phil. Mag., 1897), and the theorem is of fundamental importance in connection both with diamagnetism and the Zeeman effect (q.v.). tinder the influence of a magnetic field there is no change in the shape and size or orien tation of an electronic orbit but simply a precession.

An expression may be readily obtained for the diamagnetic susceptibility of an atom containing N electrons. For each electron the change in the corresponding magnetic moment is then be deduced from measurements on different members of the aliphatic C„H2„ series. In this way constants for the different atoms and radicles, and correcting constitutive constants, were deduced, some of which are given in the following tables :— cated in connection with Pascal's work. It is found that the greater number of ions are diamagnetic, exceptions occurring for the ions of the "transition" elements in the periodic table (such as the elements from titanium to iron) which will be discussed in the next section. Accurate quantitative data for the susceptibilites of diamagnetic ions are somewhat scarce. In the following table are given some values which have been arrived at by G. Joos (Zeits. fur Phys., 1923) for the susceptibilities of inert-gas like ions, that is ions having the same number of electrons, and the same elec tronic configuration, as atoms of the inert gases. In the first row the number of electrons in the ions is given.

The observed and calculated values usually agreed to within one or two per cent. The constants giving the contribution of a particular atom to the total diamagnetism in some cases agree fairly closely with those calculated from the directly measured susceptibilities of the elements. (Thus, for carbon values ranging from X to 6.14X have been found.) It would not be expected, however, that the contribution of the atom to the diamagnetism when in combination with other atoms (when it may gain or lose electrons, or share them) would in general be the same as the diamagnetic susceptibility character istic of the atom in a free state, or even when combined with other atoms of a similar kind. An oxygen atom in combination with different atoms has a diamagnetic effect ; but molecular oxygen is strongly paramagnetic. The actual magnitude of the diamagnetic constants is of the order indicated by the theory. The equation given leads to values for the mean radii of the elec tron orbits in H, C and N of about cm., which is in agree ment with other evidence ; but the data do not enable conclusions to be drawn as to the free atoms. A large number of elements are diamagnetic, but their consideration will be deferred until the elements generally are discussed. They present a number of peculiar features, and it is not possible to draw conclusions directly from the data as to the magnetic character of the constituent atoms themselves.