NEW DATA ON THE MAGNETIC MOMENT OF THE DEUTERON

In 1940, Kellogg, Rabi, Ramsey, and Zacharias¹, studying the radio-frequency spectra of the molecules HD and D₂, came to the conclusion that the deuteron possesses an electric quadrupole moment
\(Q = (2.73 \pm 0.005)\times 10^{-27}\ \text{cm}^2\). If this is so, then the ground state of the deuteron cannot be a \({}^{3}S_{1}\) state, characterized by a wave function with spherical symmetry.

On the other hand, the experimental information on the magnetic moment of the deuteron, together with all general theoretical considerations, definitely excludes the assumption that the ground state of the deuteron is a “pure” \({}^{3}P_{1}\) or \({}^{3}D_{1}\) state.

Thus one has to accept that the ground state of the deuteron is a superposition of \({}^{3}S_{1}\)- and \({}^{3}D_{1}\)-states (superpositions of \({}^{3}S_{1}\)- and \({}^{3}P_{1}\)-states are impossible, since the wave functions of \({}^{3}S_{1}\)- and \({}^{3}P_{1}\)-states have different parity). From this follows a conclusion of primary importance for the theory of nuclear forces: the nuclear force field is noncentral.

Naturally, many investigators have therefore sought to verify by other methods whether the ground state of the deuteron is in fact a “mixture” of \({}^{3}S_{1}\)- and \({}^{3}D_{1}\)-states. For this purpose, for example, the following fact may be used: if the ground state of the deuteron is a “pure” \({}^{3}S_{1}\)-state, its magnetic moment must be exactly equal to the difference \(\mu_p - \mu_n\), where \(\mu_p\) is the magnetic moment of the proton and \(\mu_n\) the magnetic moment of the neutron; if the ground state is a superposition of \({}^{3}S_{1}\)- and \({}^{3}D_{1}\)-states, there must be some additional magnetic moment, owing to the rotation of the proton and neutron in the nucleus.

The theoretical calculations of Rarita and Schwinger² showed that the “admixture” of the \({}^{3}D_{1}\)-state should amount to only about 4%. These authors proceeded from the assumption that the nuclear fields of the proton and neutron are, in form, analogous to dipole fields. To determine the field constants they used experimental data on the binding energy of the deuteron, its quadrupole moment, the scattering of slow neutrons by protons, and the range of nuclear forces (from experiments on the scattering of protons by protons). Starting from the values \(\mu_p = 2.785 \pm 0.2\) nuclear magnetons, \(\mu_D = 0.855 \pm 0.006\) nuclear

magneton, Rarita and Schwinger obtained for \(\mu_n\) the value \(-1.901 \pm 0.02\) nuclear magnetons, whereas simple subtraction, \(\mu_D-\mu_p\), gives \(-1.950 \pm 0.2\) nuclear magnetons.

According to the measurements of Alvarez and Bloch\(^3\), the magnetic moment of the free neutron was \(-1.935 \pm 0.02\) nuclear magnetons. As is evident from the data cited, the experimental material available at that time could not be used to check the effect considered above because of the insufficient accuracy of the measurements.

In 1945 Arnold and Roberts\(^4\) undertook precise measurements of the ratios

\[ \frac{\mu_D}{\mu_p} \quad \text{and} \quad \frac{\mu_n}{\mu_p} \]

and obtained complete agreement with the theoretical results of Rarita and Schwinger. Their experiments are a combination of the nuclear-induction method\(^5\) and the Alvarez and Bloch technique for measuring the magnetic moment of the free neutron. As is known, the determination of the magnetic moments of nuclei by means of resonance methods, which include the nuclear-induction method and the Alvarez and Bloch method, amounts to measuring the resonant values of the intensity \(H\) of a homogeneous magnetic field and of the frequency \(\omega\) of an oscillating magnetic field, satisfying the relation

\[ \omega=\frac{2\pi}{h}\,\frac{\mu H}{I}, \tag{1} \]

where \(I\) is the spin of the nucleus. Since the frequency \(\omega\) can easily be measured with an accuracy up to \(0.01\%\) by means of an ordinary heterodyne frequency meter, the principal error in determining \(\mu\) arises from the inaccuracy of measuring \(H\).

However, this error can be eliminated when determining ratios of the magnetic moments of nuclei if the measurements are made with one and the same magnetic-field setting \(H\). Arnold and Roberts made use of this. In their experiments for determining \(\dfrac{\mu_n}{\mu_p}\), a small amount of a hydrogen-containing substance (distilled water, paraffin, benzine) was placed in the interpolar gap of a magnet producing a homogeneous field \(H\), and for a certain intensity of the homogeneous field \(H\) the resonant value \(\omega_p\) satisfying relation (1) for the proton was selected. Then, with the same field setting \(H\), by means of the Alvarez and Bloch technique the resonant value of the frequency \(\omega_n\) for the neutron was selected. Then, as is easy to see,

\[ \frac{\mu_n}{\mu_p}=\frac{\omega_n}{\omega_p}. \]

In an analogous way the ratio \(\dfrac{\mu_D}{\mu_p}\) was found. Since for comparison with theory it is necessary to know only these ratios, one may take \(\mu_p\) to be known exactly and calculate, from the quantities \(\dfrac{\mu_n}{\mu_p}\) and \(\dfrac{\mu_D}{\mu_p}\), \(\mu_n\) and \(\mu_D\).

The results of Arnold and Roberts and their comparison with the theoretical value of \(\mu_n\) calculated by Rarita and Schwinger are contained in the table.

As is evident from the table, there is excellent agreement of experiment with theory, which may even seem somewhat surprising if one bears in mind the obvious imperfection of the theory.

Nevertheless, what is beyond doubt is that the results of Arnold and Roberts confirm the conclusion, discussed above, about the structure of the wave function of the ground state of the deuteron and, consequently, the noncentral character of nuclear forces.

Magnetic Moments of the Proton, Deuteron, and Neutron

I. S. Shapiro

Cited Literature

1. J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey and J. R. Zacharias, Phys. Rev. 57, 677, (1940).
2. William Rarita and Julian Schwinger, Phys. Rev. 59, 436 (1941).
3. L. W. Alvarez and F. Bloch, Phys. Rev. 57, 111 (1940).
4. Wayne R. Arnold and Arthur Robert, Phys. Rev. 70, 766, (1946).
5. F. Bloch, Phys. Rev. 70, 460 (1946).
   F. Bloch, N. W. Hansen and M. Packard, Phys. Rev. 70, 474 (1946).