Experimental Proof of the Discreteness of the Directions of the Atomic Angular Momentum Vector in a Magnetic Field.

1. Stern. Ein Weg zur experimentellen Prüfung der Richtungsquantelung im Magnetfeld.
   Zeitschrift für Physik. 7, p. 249, 1921.

2. W. Gerlach u. O. Stern. Der experimentelle Nachweis des magnetischen Moments des Silberatoms.
   Zeitschrift für Physik. 8, p. 110, 1921.

3. W. Gerlach u. O. Stern. Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld.
   Zeitschrift für Physik. 9, p. 349, 1922.

4. W. Gerlach u. O. Stern. Das magnetische Moment der Silberatoms.
   Zeitschrift für Physik. 9, p. 353, 1922.

Bohr’s theory, created for the purpose of explaining the remarkable regularities of the serial spectra of the elements, leads to a number of consequences of primary importance in other areas of physics as well. In recent years there has begun a systematic, consistent study of these consequences, both from the theoretical and from the experimental point of view.

For the physicist it is extremely important to determine whether Bohr’s model of the atom corresponds to the properties of an individual atom, or whether it expresses only the average statistical properties of a large number of atoms.

In the quantum theory of magnetism and of the Zeeman effect, constructed on the basis of Bohr’s model, the angular momentum vector of each atom can form with the direction of the magnetic force of the field \(H\) only quite definite discrete angles: the component of the angular momentum in the direction of \(H\) had to be an integral multiple of the moment \(\frac{h}{2\pi}\), where \(h\) is Planck’s constant. If we are dealing with a gas whose atoms possess a total angular momentum \(\frac{h}{2\pi}\), then, according to the indicated theo-

in a magnetic field only two positions of the atoms are possible, determined by the fact that the component of the angular momentum in the direction \(H\) has the value \(+\dfrac{h}{2\pi}\) or \(-\dfrac{h}{2\pi}\). In classical theory the situation is quite different. The action of the magnetic field amounts to this: all atoms acquire an additional uniform rotation about the direction \(H\); the angle formed by the direction of the angular momentum with the direction \(H\) may then have all possible values.

O. Stern proposed the following simple experiment to test this consequence of quantum theory. Let the magnetic field of strength \(H\) be directed parallel to the \(z\)-axis; let \(m\) be the vector of the magnetic moment of the atom, and let it be connected with the angular momentum of the atom \(J\) by the relation:

\[ m=\frac{1}{2}\frac{e}{m}\,J \tag{1} \]

If the magnetic field is inhomogeneous, with gradient \(\dfrac{dH}{ds}\), then the force acting on the atom is:

\[ K=|m|\frac{dH}{ds} \]

The force \(K\) is composed of three components:

\[ K=m_x\frac{\partial H}{\partial x}+m_y\frac{\partial H}{\partial y}+m_z\frac{\partial H}{\partial z} \]

In the present case, when \(H\) is directed parallel to the \(z\)-axis, the atom performs a uniform rotation about \(H\), with \(m_z\) remaining constant, while the mean value of \(m_x\) and \(m_y\) over the time of a complete revolution is zero. Thus the mean force acting on the atom is

\[ K=m_z\frac{dH}{dz} \tag{2} \]

Let us consider the simplest case, when

\[ J=\frac{h}{2\pi} \]

Then \(m_z\), according to quantum theory, can have only the values:

\[ m_z=\pm\frac{1}{2}\cdot\frac{e}{m}\cdot\frac{h}{2\pi} \tag{3} \]

Imagine that we have an inhomogeneous field formed by the pole of an electromagnet shaped like a wedge. Near the edge of the wedge there passes a thin parallel beam of atoms. According to quantum theory, under the action of the force (2) the beam must split into two, since by formula (3) \(m_z\) can have only two values. In classical theory the beam should only broaden.

If \(\mu\) is the mass of the atom, then the acceleration experienced by it under the action of the force (2) is:

\[ g=\frac{K}{\mu}=\frac{m_z}{\mu}\cdot\frac{\partial H}{\partial z} \]

If \(t\) is the duration of the atom’s flight, and \(v\) its velocity, then the displacement \(s\) is determined as

\[ s=\frac{1}{2}gt^2=\frac{1}{2}g\frac{l^2}{v^2} =\frac{1}{2}\frac{m_z}{\mu}\cdot\frac{\partial H}{\partial z}\cdot\frac{l^2}{v^2} \]

where \(l\) is the length of the path traversed by the atom near the edge of the wedge. Denoting by \(n\) the number of molecules in a mole, \(M=n\mu\) the atomic weight, \(N=m_z n=5600\) C. G. S. (Bohr magneton), we find:

\[ s=\frac{N}{2Mv^2}\frac{\partial H}{\partial z}\cdot l^2 \]

Replacing \(v^2\) by the mean-square molecular velocity, we have: \(Mv^2=3RT\) (\(R\)—gas constant, \(T\)—absolute temperature). Hence:

\[ s=\frac{N}{6R}\cdot \frac{\partial H}{\partial z}\cdot \frac{l^2}{T} =1.12\cdot 10^{-5}\cdot \frac{\partial H}{\partial z}\cdot \frac{l^2}{T}\ \mathrm{cm}. \tag{3} \]

Let \(l=3.3\) cm; \(T=1000^\circ\), \(\dfrac{\partial H}{\partial z}=2\cdot 10^5\) gauss per centimeter, then

\[ s=2.24\cdot 10^{-2}\ \mathrm{cm}=0.224\ \mathrm{mm}. \]

O. Stern and W. Gerlach carried out this experiment. Silver served as the substance under investigation, offering considerable practical advantages. A beam of silver atoms emerges from a small chamotte furnace, heated by an electric current, through an aperture of \(1\ \mathrm{sq.\ mm}\). Farther on, at a distance of 2.5 cm, the beam passes through a circular diaphragm with an aperture of \(0.003\ \mathrm{sq.\ mm}\). At 3.3 cm beyond this diaphragm the beam of silver atoms passes through a slit 0.8 mm long and 0.03–0.04 mm wide. Both diaphragms are made in platinum foil. The slit is situated at the very wedge-shaped pole of an electromagnet. The length of the wedge edge is 3.5 cm. At the other end of the edge a glass plate is placed, on which the silver is deposited. Both diaphragms, the magnetic poles, and the glass plate are fastened to the walls of a brass box (wall thickness 1 cm). In this way the possibility of deformation of the system when the electromagnet is switched on is avoided. The space inside the box is evacuated to a pressure of approximately \(10^{-5}\) mm of mercury. The duration of the exposure, in the absence of a magnetic field, is 4.5 hours; when the field is switched on—8 hours. The reason for this difference is that in the absence of the field the silver trace should give an unsplit line, while when the field is switched on the line, according to quantum theory, should split into two. To obtain traces of equal density, approximately double exposure is required. But even an eight-hour exposure gives no noticeable deposition of silver on the glass plate, owing to the extreme fineness of the beam of atoms. A preliminary “development” of the glass plate is required, achieved by depositing fresh silver from a silver solution.

The results obtained by the authors fully confirmed the theory both qualitatively and quantitatively. It should be noted that the correct installation of the unusually thin diaphragms used by the authors is very difficult; an error of a few hundredths of a millimeter inevitably entails failure of the experiment. The excellent results obtained in several experiments are considered by the authors themselves a fortunate accident. In the accompanying drawing an approximate representation is given of the most successful “photograph.” Fig. a corresponds to the absence of the field. The somewhat irregular shape of the unsplit trace is explained by the irregularity of the slit. Fig. b was obtained when the field was switched on. The trace is distinctly split. The long projection is directed toward the very wedge of the pole of the electromagnet, where the gradient \(\dfrac{\partial H}{\partial z}\) is extremely large. The blurred shape of the split branches is explained by the fact that the silver atoms fly with different velocities. The drawing corresponds to the picture obtained when viewing the original photograph under a microscope (approximately at 20-fold magnification).

The inhomogeneity of the magnetic field near the edge of the wedge was found by direct measurement of the repulsive force experienced by a small test bismuth body, and also by measuring the resistance of a thin bismuth wire stretched parallel to the wedge edge. Table 1 gives the gradient in gauss per 1 cm.

TABLE 1.

The theoretical magnitude of the deflection was found from formula (3), with a correction also introduced for the velocity of flight of the atoms, on which we shall not dwell here.

The measurement and calculation of two photographs gave the results presented in Table 2.

TABLE 2.

The authors estimate the limits of error at 10% and regard the exceptionally good agreement of the tabulated numbers as accidental.

The experiments of Stern and Gerlach thus prove that: 1) silver atoms possess a magnetic moment; 2) the magnitude of this moment is equal to Bohr’s magneton; 3) the directions of the angular-momentum vector are discrete, as required by Bohr’s theory.

S. Vavilov.