Nuclear Size and Density

Nuclear size is defined by nuclear radius; nuclear density can be calculated from nuclear size.

Learning Objective

Explain relationship between nuclear radius, nuclear density, and nuclear size.

Key Points

The first estimate of a nuclear charge radius was made by Hans Geiger and Ernest Marsden in 1909, under the direction of Ernest Rutherford, in the gold foil experiment that involved the scattering of α-particles by gold foil, as shown in Figure 1.
An empirical relation exists between the charge radius and the mass number, $A$, for heavier nuclei ($A>20$): where $r$ is an empirical constant of 1.2–1.5 fm.
The nuclear density for a typical nucleus can be approximately calculated from the size of the nucleus: $n = \frac{A}{\frac{4}{3}\pi R^3}$

Terms

α-particle
two protons and two neutrons bound together into a particle identical to a helium nucleus
atomic spectra
emission or absorption lines formed when an electron makes a transition from one energy level of an atom to another
nucleus
the massive, positively charged central part of an atom, made up of protons and neutrons

Full Text

Nuclear size is defined by nuclear radius, also called rms charge radius. It can be measured by the scattering of electrons by the nucleus and also inferred from the effects of finite nuclear size on electron energy levels as measured in atomic spectra.

The problem of defining a radius for the atomic nucleus is similar to the problem of atomic radius, in that neither atoms nor their nuclei have definite boundaries. However, the nucleus can be modelled as a sphere of positive charge for the interpretation of electron scattering experiments: because there is no definite boundary to the nucleus, the electrons "see" a range of cross-sections, for which a mean can be taken. The qualification of "rms" (for "root mean square") arises because it is the nuclear cross-section, proportional to the square of the radius, which is determining for electron scattering.

The first estimate of a nuclear charge radius was made by Hans Geiger and Ernest Marsden in 1909, under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester, UK. The famous Rutherford gold foil experiment involved the scattering of α-particles by gold foil, with some of the particles being scattered through angles of more than 90°, that is coming back to the same side of the foil as the α-source, as shown in Figure 1. Rutherford was able to put an upper limit on the radius of the gold nucleus of 34 femtometers (fm).

Figure 1

Top: Expected results: alpha particles passing through the plum pudding model of the atom undisturbed.Bottom: Observed results: a small portion of the particles were deflected, indicating a small, concentrated positive charge. Note that the image is not to scale; in reality the nucleus is vastly smaller than the electron shell.

Later studies found an empirical relation between the charge radius and the mass number, $A$, for heavier nuclei ($A>20$): $R \approx r\cdot A^{1/3}$ where $r$ is an empirical constant of 1.2–1.5 fm. This gives a charge radius for the gold nucleus ($A=197$) of about 7.5 fm.

Nuclear density is the density of the nucleus of an atom, averaging about $4 \cdot 10^{17} \text{kg/}\text{m}^3$. The nuclear density for a typical nucleus can be approximately calculated from the size of the nucleus:

$\displaystyle n = \frac{A}{\frac{4}{3}\pi R^{3}}$

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