x = A cos (p1t + [beta]) + B cos (p2t + [beta]1)
y = A sin (p1t + [beta]) - B sin (p2t + [beta]1)
z = C cos (pt + [gamma])
where
a - mp1² = - He p1
a - mp2² = He p2
p² = [alpha]/m,
or approximately
He He
p1 = p + ½ ---, p2 = p - ½ ---.
m m
Thus the motion of the ion on the xy plane may be regarded as made up
of two circular motions in opposite directions described with
frequencies p1 and p2 respectively, while the motion along z has the
period p, which is the frequency for all the vibrations when H = 0.
Now suppose that the cadmium line is due to the motion of such an ion;
then if the magnetic force is along the direction of propagation, the
vibration in this direction has its period unaltered, but since the
direction of vibration is perpendicular to the wave front, it does not
give rise to light. Thus we are left with the two circular motions in
the wave front with frequencies p1 and p2 giving the circularly
polarized constituents of the doublet. Now suppose the magnetic force
is at right angles to the direction of propagation of the light; then
the vibration parallel to the magnetic force being in the wave front
produces luminous effects and gives rise to a plane-polarized ray of
undisturbed period (the middle line of the triplet), the plane of
polarization being at right angles to the magnetic force. The
components in the wave-front of the circular orbits at right angles to
the magnetic force will be rectilinear motions of frequency p1 and p2
at right angles to the magnetic force--so that they will produce
plane-polarized light, the plane of polarization being parallel to the
magnetic force; these are the outer lines of the triplet.
If Zeeman's observations are interpreted from this point of view, the
directions of rotation of the circularly-polarized light in the doublet
observed along the lines of magnetic force show that the ions which
produce the luminous vibrations are _negatively_ electrified, while the
measurement of the charge of frequency due to the magnetic field shows
that e/m is of the order 10^7. This result is of great interest, as this
is the order of the value of e/m in the negatively electrified particles
which constitute the Cathode Rays (see CONDUCTION, ELECTRIC III.
_Through Gases_). Thus we infer that the "cathode particles" are found
in bodies, even where not subject to the action of intense electrical
fields, and are in fact an ordinary constituent of the molecule. Similar
particles are found near an incandescent wire, and also near a metal
plate illuminated by ultra-violet light. The value of e/m deduced from
the Zeeman effect ranges from 10^7 to 3.4 × 10^7, the value of e/m for
the particle in the cathode rays is 1.7 × 10^7. The majority of the
determinations of e/m from the Zeeman effect give numbers larger than
this, the maximum being about twice this value.
A more extended study of the behaviour of the spectroscopic lines has
afforded examples in which the effects produced by a magnet are more
complicated than those we have described, indeed the simple cases are
much less numerous than the more complex. Thus Preston[47] and Cornu[48]
have shown that under the action of a transverse magnetic field one of
the D lines splits up into four, and the other into six lines; Preston
has given many other examples of these quartets and sextets, and has
shown that the change in the frequency, which, according to the simple
theory indicated, should be the same for all lines, actually varies
considerably from one line to another, many lines showing no appreciable
displacement. The splitting up of a single line into a quartet or sextet
indicates, from the point of view of the ion theory, that the line must
have its origin in a system consisting of more than one ion. A single
ion having only three degrees of freedom can only have three periods.
When there is no magnetic force acting on the ion these periods are
equal, but though under the action of a magnetic force they are
separated, their number cannot be increased. When therefore we get four
or more lines, the inference is that the system giving the lines must
have at least four degrees of freedom, and therefore must consist of
more than one ion. The theory of a system of ions mutually influencing
each other shows, as we should expect, that the effects are more complex
than in the case of a single ion, and that the change in the frequency
is not necessarily the same for all systems (see J. J. Thomson, _Proc.
Camb. Phil. Soc._ 13, p. 39). Preston[49] and Runge and Paschen have
proved that, in some cases at any rate, the change in the frequency of
the different lines is of such a character that they can be grouped into
series such that each line in the series has the same change in
frequency for the same magnetic force, and, moreover, that homologous
lines in the spectra of different metals belonging to the same group
have the same change in frequency.
A very remarkable case of the Zeeman effect has been discovered by H.
Becquerel and Deslandres (_Comptes rendus_, 127, p. 18). They found
lines in iron when the most deflected components are those polarized in
the plane at right angles to the magnetic force. On the simple theory
the light polarized in this way is not affected. Thus the behaviour of
the spectrum in the magnetic field promises to throw great light on the
nature of radiation, and perhaps on the constitution of the elements.
The study of these effects has been greatly facilitated by the invention
by Michelson[50] of the echelon spectroscope.
There are some interesting phenomena connected with the Zeeman effect
which are more easily observed than the effect itself. Thus Cotton[51]
found that if we have two Bunsen flames, A and B, coloured by the same
salt, the absorption of the light of one by the other is diminished if
either is placed between the poles of a magnet: this is at once
explained by the Zeeman effect, for the times of vibration of the
molecules of the flame in the magnetic field are not the same as those
of the other flame, and thus the absorption is diminished. Similar
considerations explain the phenomenon observed by Egoroff and
Georgiewsky,[52] that the light emitted from a flame in a transverse
field is partially polarized in a plane parallel to the magnetic force;
and also Righi's[53] observation that if a sodium flame is placed in a
longitudinal field between two crossed Nicols, and a ray of white light
sent through one of the Nicols, then through the flame, and then through
the second Nicol, the amount of light passing through the second Nicol
is greater when the field is on than when it is off. Voight and Wiechert
(_Wied. Ann._ 67, p. 345) detected the double refraction produced when
light travels through a substance exposed to a magnetic field at right
angles to the path of the light; this result had been predicted by
Voight from theoretical considerations. Jean Becquerel has made some
very interesting experiments on the effect of a magnetic field on the
fine absorption bands produced by xenotime, a phosphate of yttrium and
erbium, and tysonite, a fluoride of cerium, lanthanum and didymium, and
has obtained effects which he ascribes to the presence of positive
electrons. A very complete account of magneto- and electro-optics is
contained in Voight's _Magneto- and Elektro-optik_.