ON THE MAGNETIC DEFLECTION OF HYDROGEN MOLECULES AND ON THE MAGNETIC MOMENT OF THE PROTON*

R. Frisch and O. Stern, Hamburg

The aim of the investigation carried out by Frisch and me was to determine the magnetic moment of the proton. Although these experiments are still far from complete, I should nevertheless like to report some results which seem to me interesting.

The mechanical moment of the proton is known with great accuracy: it is equal to the electronic one,

\[ \frac{1}{2}\,\frac{h}{2\pi}. \]

The magnetic moment of the electron is equal to

\[ 2 \cdot \frac{e}{2mc}\cdot \frac{1}{2}\,\frac{h}{2\pi} \]

(the Bohr magneton \(= 0.9 \cdot 10^{-20}\) CGS for one electron and, correspondingly, \(5600\) CGS per mole). If one assumes that the same formula is applicable to the magnetic moment of the proton, then, by virtue of the mass ratio, it would be \(1840\) times smaller (\(0.5 \cdot 10^{-23}\) CGS for one proton and, correspondingly, \(3\) CGS per mole). We shall call this quantity the nuclear magneton. At present, for direct measurement of such small moments, only the method of deflection of molecular beams in an inhomogeneous magnetic field can be used.**

Experiments with direct deflection of free protons by the Lorentz force are impossible (and, as Bohr has shown, impossible in principle). Hydrogen atoms also practically cannot be used, since their electric moment is 2000 times greater than the nuclear moment to be measured. Consequently, the simplest system with which measurements can be carried out is a molecule that has no electronic moment.

In our experiments, as in the corresponding apparatus of Gerlach and Stern, a beam of \(H_2\) molecules passed through an inhomogeneous magnetic field, and the deflection of the molecule was measured. The change in comparison with their experiments consisted only in the fact that,

* The present article is a translation of a report delivered in Leipzig at the conference on magnetism in 1933: Leipziger Vorträge, Magnetismus, herausgegeben von P. Debye, Verlag S. Hirzel 1933. Translation by D. Goberidze. Ed.

** The hyperfine structure of spectral lines makes it possible in some cases to determine the magnetic moment of the nucleus; the discrepancies that occurred in doing so have at present apparently been overcome (private communication from Fermi), but this method is at present not applicable to the proton.

that, in order to obtain a noticeable deflection with a small moment, the field was made longer and inhomogeneous. Still earlier experiments had shown that by this means it is possible to measure moments of the order of magnitude of a nuclear magneton. ¹˒² But whereas at that time only an order-of-magnitude determination was possible, experimental technique has since been so far improved as to make quantitative measurements possible. An essential improvement

Fig. 1. Diagram of the apparatus.

consists in the fact that, by the method now employed, a quantitative measurement of the intensity of the H₂ rays is possible. This method, developed simultaneously by Knauer and by me,³ consists in passing the molecular rays into a closed vessel and measuring the change in pressure produced by them with a sensitive hot-filament manometer.

The apparatus used is shown schematically in Fig. 1. The length of the field was 10 cm, the width of the channel was 1 mm, and the knife edge was 0.5 mm from the plane of the channel. The inhomogeneity thus obtained was approximately \(2 \cdot 10^{5}\) gauss/cm. It caused, for H₂ molecules having a velocity of 900 m/sec (the probable velocity at the temperature of liquid air), a deflection of 0.04 mm per nuclear magneton. The width of the beam varied in different experiments, but was of the order of several hundredths of a millimeter. The intensity of the beam obtained in this way was very small; when a slit-type receiver was used, the beam produced in the receiver a pressure of about \(2 \cdot 10^{-8}\) mm (measurement limit \(2 \cdot 10^{-9}\) mm). To obtain higher pressures we used an artificial device developed by us earlier—namely, at the receiver we used a slit which did not disturb the motion of the incoming molecules, but offered very strong resistance to the outgoing ones. In our case, owing to the use of an especially narrow and long channel (0.02 mm wide, 0.5 mm high, and 4 mm long), the pressure was raised approximately 50-fold. In doing this, however, a very long time is required for the final pressure to be established. In our case, with the manometer size adopted (about

20 cm³), this took about half an hour. We also constructed a manometer of smaller volume, the design of which is shown in Fig. 2. Its volume was only about 0.5 cm³, so that the filling time was reduced to \(30/40 = 3/4\) min. The receiving slit was movable and could be displaced relative to the beam. A sample of the curve obtained in this way is shown in Fig. 3.

At first the beams of ordinary \(\mathrm{H}_2\) at low temperatures (liquid air) were investigated. To understand these experiments, it is necessary to note the following: ordinary hydrogen consists of 25% parahydrogen and 75% orthohydrogen. In parahydrogen the two protons are antiparallel and, consequently, should not have

Fig. 3. Example of a splitting curve.

any magnetic moment depending on the nuclear spin; but one should expect that the rotation of the molecules will produce a magnetic moment. But at liquid-air temperatures, almost all (99%) para-\(\mathrm{H}_2\) molecules have a rotational quantum number equal to zero; consequently, at such temperatures para-\(\mathrm{H}_2\) should not have had a magnetic moment. We confirmed this by experiments on pure para-\(\mathrm{H}_2\). In ortho-\(\mathrm{H}_2\) the two protons are parallel and, consequently, it should have a magnetic moment from the two protons. In addition, rotation should also influence the magnetic moment, and this part of the magnetic moment does not disappear when the temperature is lowered, since the lowest rotational state of the ortho-\(\mathrm{H}_2\) molecule has quantum number 1. Since the coupling between the two moments (rotation and nuclear spin) is very small and in the fields used for deflection, of order about 20,000 gauss, is probably completely destroyed, then for ortho-\(\mathrm{H}_2\) a beam of uniform velocity should undergo splitting (Fig. 4). Each of the two moments has in the field 3 components for quantum number 1; in our figure it is assumed that the magnetic moment of rotation and the deflection \(S_R\) caused by it are considerably smaller than the other \(S_p\). With the act...

structurally applied beams with a Maxwellian distribution of velocities. Each line in our drawing (except for the middle one) corresponds to a Maxwellian curve; the measured intensity is the superposition of these curves.

In principle, it is possible from the measured distribution of intensities to determine both unknowns \(S_p\) and \(S_R\) (Fig. 4), but this would require very great accuracy of measurement. Therefore we determined one of the unknowns—the rotational moment \(S_R\)—in the following way: we investigated pure para-\(\mathrm{H}_2\), besides the temperature of liquid air, also at higher temperatures (solid \(\mathrm{CO}_2\), i.e. \(195^\circ \mathrm{K}\), and room temperature \(292^\circ \mathrm{K}\)). At the temperature of liquid air, as was to be expected, it proved to be nonmagnetic*, while at higher temperatures it exhibited a moment arising from higher rotational quantum states. We calculated, by Boltzmann’s formula, the probability of such quantum states: if we denote the rotational quantum number by \(n\), then the calculation gives that at \(T=95^\circ\), \(73\%\) of the molecules had \(n=0\) and \(27\%\) \(n=2\). At room temperature (\(T=292^\circ\)) we find \(52.5\%\) with \(n=0\), \(46.1\%\) with \(n=2\), and \(1.4\%\) with \(n=4\%\). On the assumption that all the components entering into the magnetic moment are integral multiples (see Fig. 5) of the principal moment, which is expressed by \(n=1\), we can obtain this principal moment from the measurements, and it proves to be equal to one nuclear magneton or a little less. It is assumed that this moment is equal to the rotational moment of one-quantum orthohydrogen. If now, with this value of the rotational moment, we calculate from our measurements the proton-dependent moment of ordinary ortho-\(\mathrm{H}_2\), it proves to be approximately equal to four nuclear magnetons per ortho-\(\mathrm{H}_2\) molecule. Thus the magnetic moment of the proton proves to be equal not to one, but to two nuclear magnetons. The numerical value of this quantity is not very exact; it may prove to be equal also to 3, but apparently a value of this quantity equal to unity is not compatible with the measurement.

Fig. 4. Theoretically expected splitting of a beam of ortho-\(\mathrm{H}_2\) molecules having the same velocity at low temperatures (rotational quantum number 1).

Fig. 5. Theoretically expected splitting of a beam of para-\(\mathrm{H}_2\) molecules having the same velocity for rotational quantum numbers 2 (top) and 4 (bottom).

* The insignificant admixture of magnetic molecules is probably explained by contamination (3–4%) with ortho-\(\mathrm{H}_2\) molecules.

Concerning the rotational moment, the following may also be said: at first we made measurements only with ordinary \(H_2\) and tried to substitute for the moment of rotation the theoretically calculated value. At Fermi’s suggestion, Bethe calculated the electric moment of inertia of the \(H_2\) molecule. Assuming that the \(H_2\) molecule rotates as a rigid body, he obtained (for rotational quantum number 1) a value for the moment of rotation close to 3 nuclear magnetons. Only later did we come to realize that, by the method described above, the moment of rotation can be determined directly by experiment, by direct measurements on pure para-\(H_2\).

These measurements, as indicated, gave a value not exceeding one nuclear magneton. Since this discrepancy lies far beyond both the experimental errors and the theoretical uncertainty, we again turned to Fermi, who then expressed the following view: the assumption that the \(H_2\) molecule rotates as a rigid body is unacceptable. It must be assumed that the electron shell lags behind (“slips”) during rotation. An estimate of this effect, made at Fermi’s suggestion by Wick, showed that the moment of rotation should lie between 0.35 and 0.92 nuclear magneton; we would be inclined to suppose that the true value lies closer to the upper limit.

Literature

1. Frisch R. und Stern O., Ztschr. f. Phys., 85, 4, 1933.
   1a. Estermann I. und Stern O., Ztschr. f. Phys., 85, 17, 1933.
       Wick G. C., Ztschr. f. Phys., 85, 25, 1933.
2. Knauer und Stern, Ztschr. f. Phys., 39, 780, 1926.
3. Knauer und Stern, Ztschr. f. Phys., 53, 765, 1929.