42.1.4 Nuclear Size and Density

Unlike the atom’s electron cloud, which is quite diffuse, the nucleus has a fairly sharp boundary. Experimentally, the radius of a nucleus with mass number $A$ is found to be

R = r_{0} A^{1 / 3}

(42.2)

where $r_{0} = 1.2 fm$ . Recall that $1 fm = 1 femtometer = 10^{- 15} m$ .

As FIGURE 42.2 shows, the radius is proportional to $A^{1 / 3}$ . Consequently, the volume of the nucleus (proportional to $R^{3}$ ) is directly proportional to $A$ , the number of nucleons. A nucleus with twice as many nucleons will occupy twice as much volume. This finding has three implications:

Nucleons are incompressible. Adding more nucleons doesn’t squeeze the inner nucleons into a smaller volume.
The nucleons are tightly packed, looking much like the drawing in Figure 42.1.
Nuclear matter has a constant density.

The figure illustrates a graph of nuclear radius and volume as a function of mass number “A”.

FIGURE 42.2 Full Alternative Text

In fact, we can use Equation 42.2 to calculate the density of nuclear matter. Consider a nucleus with mass number $A$ . Its mass, within 1%, is $A$ atomic mass units. Thus

\begin{array}{l} ρ_{nuc} & \approx & \begin{matrix} \frac{A u}{\frac{4}{3} π R^{3}} = \frac{A u}{\frac{4}{3} π r_{0}^{3} A} = \frac{1 u}{\frac{4}{3} π r_{0}^{3}} = \frac{1.66 \times 10^{- 27} kg}{\frac{4}{3} π {(1.2 \times 10^{- 15} m)}^{3}} \end{matrix} \\ = & 2.3 \times 10^{17} {kg/m}^{3} \end{array}

(42.3)

The fact that $A$ cancels means that all nuclei have this density. It is a staggeringly large density, roughly $10^{14}$ times larger than the density of familiar liquids and solids. One early objection to Rutherford’s model of a nuclear atom was that matter simply couldn’t have a density this high. Although we have no direct experience with such matter, nuclear matter really is this dense.

FIGURE 42.3 shows the density profiles of three nuclei. The constant density right to the edge is analogous to that of a drop of incompressible liquid, and, indeed, one successful model of many nuclear properties is called the liquid-drop model. Notice that the range of nuclear radii, from small helium to large uranium, is not quite a factor of 4. The fact that $^{56} Fe$ is a fairly typical atom in the middle of the periodic table is the basis for our earlier assertion that the nuclear diameter is roughly $10^{- 14} m,$ or 10 fm.

The figure consists of two parts that illustrate three nuclei and their density profiles as a graph.

FIGURE 42.3 Full Alternative Text

42.1.4 Nuclear Size and Density

FIGURE 42.2 The nuclear radius and volume as a function of $A$ .

FIGURE 42.3 Density profiles of three nuclei.