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A New Definition of the Charge of an Atomic Nucleus
J. Chadwick. The charge of the Atomic Nucleus and the Law of Force. Phil. Mag. 40, p. 734 (1920).
The charge of an atomic nucleus is one of the most important constants of an element, and therefore its precise determination is a problem of the highest importance. Already from the early observations of Geiger and Marsden1 on the scattering of α-rays Rutherford concluded that this charge is equal to \(\tfrac{1}{2}Ae\), where \(A\) is the atomic weight of the element, and \(e\) the charge of the electron. Further experiments by the same Geiger and Marsden2 confirmed this conclusion. However, experimental difficulties allowed them to determine the charge of the nucleus only roughly, with an error within 20%.
Van den Broek3 put forward the hypothesis that the nuclear charge is equal to the atomic number \(Z\) of the element. This hypothesis was brilliantly used by Moseley in his classic work on the X-ray spectra of the elements4 to explain the fact he discovered: a linear dependence between the frequency of oscillation of the corresponding lines of one and the same series (for example, the \(K\)-series or \(L\)-series) and a certain whole number that changes by one in passing from one element to the next.
But the most direct method for determining the nuclear charge still remains the study of the scattering of α-rays. The chief difficulty, which caused the particularly large error in the experiments of Geiger and Marsden, comes down to the fact that the intensities of the primary and scattered beams differ very greatly from one another, and therefore one must resort to different methods for measuring it in the two cases. According to Rutherford’s idea, Chadwick carried out an arrangement that made it possible to count the number of α-particles both in the primary and in the scattered beam on one and the same screen of zinc sulfide. In his experiments the scattering foil had the form not of a small disk, as in Geiger and Marsden, but of a ring subtending a considerably larger solid angle. In the drawing \(R\) is the source of α-rays,
\(S\)—a screen of zinc sulfide. The scattering ring \(AA'\) is arranged so that \(RA = AS\). Under these conditions it can be shown that the number of scattered \(\alpha\)-particles will be
\[ \frac{Qntb^{2}}{64r^{2}} \left( \log \tg \frac{\varphi_{2}}{4} -\log \tg \frac{\varphi_{1}}{4} +\cotg \frac{\varphi_{1}}{2}\,\cosec \frac{\varphi_{1}}{2} -\cotg \frac{\varphi_{2}}{2}\,\cosec \frac{\varphi_{2}}{2} \right), \]
where
\[ \begin{aligned} Q&=\text{the number of }\alpha\text{-particles emitted by the source per unit time,}\\ n&=\text{the number of atoms in a unit volume of the sheet,}\\ t&=\text{the thickness of the sheet,}\\ b&=\frac{2E}{mu^{2}}\,Ne, \end{aligned} \]
where \(E\), \(m\), and \(u\) are respectively the charge, mass, and velocity of the \(\alpha\)-particle, and \(Ne\) is the sought charge of the nucleus.
For the angles \(\varphi_{1}\) and \(\varphi_{2}\), see the drawing.
The number of \(\alpha\)-particles falling directly on a unit area of the screen \(S\) will obviously be
\[ \frac{Q}{4\pi l^{2}}, \quad \text{where } l=RS. \]
It is not difficult to calculate that if the number of scattered \(\alpha\)-particles is of the order of 30 per minute, then the number of \(\alpha\)-particles in the primary beam will be 30,000 per minute.
In order, nevertheless, to be able to count the scintillations in both cases on one and the same screen, the author resorted to the following device. When the number of particles in the scattered beam was counted, the aperture of the ring was covered by a thick lead disk; when the number of particles in the primary beam was counted, the disk was removed and, in front of the screen, a rotating sector was set in motion in the path of the beam. In this way the observed number of particles could be diminished to any desired extent, and, knowing the aperture of the sector, it was easy from the observed number to determine also the full number of particles in the primary beam.
The experiments were carried out with platinum, silver, and copper. Their results are compared in the following table.
| Atomic number | Nuclear charge | |
|---|---|---|
| Platinum | 78 | 77.4 |
| Silver | 47 | 46.3 |
| Copper | 29 | 29.3 |
Thus, within an accuracy of 1%, the number of elementary charges of the nucleus is equal to the atomic number of the element.
Along the way, Chadwick investigated the question of the dependence of the force on the distance close to the nucleus. Namely, Darwin1 showed that if the force
varies with distance according to the formula \(1/r^p\), then the number of scattered \(\alpha\)-particles, depending on the velocity, will be, caeteris paribus,
\[ \left(\frac{1}{u^2}\right)^{\frac{2}{p-1}}, \]
where \(u\) is the velocity of the \(\alpha\)-particles in the primary beam. In order to vary this velocity in front of the source, one or another number of mica sheets was placed. As a result it turned out that the number of scattered \(\alpha\)-particles is inversely proportional to the fourth power of the velocity, i.e. \(p = 2\). It may be calculated that, in the case of platinum, the faster \(\alpha\)-particles approach the nucleus to a distance of \(7 \cdot 10^{-12}\) cm, and the slower ones to a distance of \(14 \cdot 10^{-12}\) cm. Hence it follows that Coulomb’s law is fulfilled even at distances of the order of \(10^{-11}\) cm from the nucleus.
E. Shpolsky.