Size and Structure of the Nucleus

Figure 18.5: Schematic illustration of Robert Hofstadter's results for scattering of electrons off of atomic nuclei. The solid line shows the relative probability (in log-log coordinates) of elastic scattering as a function of the momentum transfer. The dashed curve illustrates the observed probability distribution. The difference between the curves is the logarithm of the form factor, $F(q)$.

In the late 1950s and early 1960s Robert Hofstadter of Stanford University extended the Geiger-Marsden experiment to much shorter de Broglie wavelengths using high energy electrons from an accelerator rather than alpha particles as the probe. The type of results obtained by Hofstadter are shown in figure 18.5. After accounting for some effects having to do with the electron spin, these experiments should agree with the Rutherford formula if the nucleus is truly a point particle. However, the actual results show probabilities which drop off more rapidly with increasing momentum transfer $q$ than is predicted by the Rutherford model. The ratio of the actual to the Rutherford probability distributions is called the form factor, $F(q)$, for this process:

$$P_{obs} (q) = F(q) P_{Ruth} (q) \propto F(q) q^{-4} .$$ (19.3)

Taking the logarithm of this equation results in

$$\log[ P_{obs} ] = \log [ F(q) ] - 4 \log (q) + const .$$ (19.4)

These results are related to the fact that the nucleus is actually of finite size. The diffraction effects discussed in the section on the scattering of moonlight come into play here, in that little scattering takes place for scattering angles larger than roughly $\lambda /(2d)$, where $\lambda$ is the de Broglie wavelength of the probing particle and $d$ is the diameter of the target. For small scattering angle (which we now call $\theta$), it is clear from figure 18.4 that

$$\theta \approx q/p ,$$ (19.5)

where $p$ is the momentum of the incident electron and $q$ is the momentum transfer. If $q_{max}$ is the maximum momentum transfer for which there is significant scattering, then we can write

$$q_{max} / p = \theta_{max} \approx \lambda /d ,$$ (19.6)

where the factor of $2$ in the denominator on the right side has been dropped since this is an approximate analysis. However, since $\lambda = h/p$, we find that

$$q_{max} \approx \frac{h}{d} .$$ (19.7)

Thus, the momentum transfer for which the measured form factor becomes small compared to one gives us an immediate estimate of the diameter of an atomic nucleus: $d \approx h / q_{max}$. The results obtained by Hofstadter show that nuclear diameters are typically a few times $10^{-15} \mbox{ m}$.

More than just size information can be extracted from the form factor. Hofstadter's experiments also led to a great deal of information about the internal structure of atomic nuclei.