10.1 Properties of Nuclei

Counts of Nucleons

The number of protons in the nucleus is given by the atomic number, Z. The number of neutrons in the nucleus is the neutron number, N. The total number of nucleons is the mass number, A. These numbers are related by

A nucleus is represented symbolically by

where X represents the chemical element, A is the mass number, and Z is the atomic number. For example, ${}_6^{12}\text{C}$ represents the carbon nucleus with six protons and six neutrons (or 12 nucleons).

A graph of the number N of neutrons versus the number Z of protons for a range of stable nuclei (nuclides) is shown in Figure 10.3. For a given value of Z, multiple values of N (blue points) are possible. For small values of Z, the number of neutrons equals the number of protons $\left(N=P\right),$ and the data fall on the red line. For large values of Z, the number of neutrons is greater than the number of protons $\left(N>P\right),$ and the data points fall above the red line. The number of neutrons is generally greater than the number of protons for $Z>15.$

Figure 10.3 This graph plots the number of neutrons N against the number of protons Z for stable atomic nuclei. Larger nuclei, have more neutrons than protons.

A chart based on this graph that provides more detailed information about each nucleus is given in Figure 10.4. This chart is called a chart of the nuclides. Each cell or tile represents a separate nucleus. The nuclei in this chart are arranged in order of ascending N (along the horizontal direction) and ascending Z (along the vertical direction).

Figure 10.4 Partial chart of the nuclides. For stable nuclei (dark blue backgrounds), cell values represent the percentage of nuclei found on Earth with the same atomic number (percent abundance). For the unstable nuclei, the number represents the half-life.

Atoms that contain nuclei with the same number of protons (Z) and different numbers of neutrons (N) are called isotopes. For example, hydrogen has three isotopes: normal hydrogen (1 proton, no neutrons), deuterium (one proton and one neutron), and tritium (one proton and two neutrons). Isotopes of a given atom share the same chemical properties, since these properties are determined by interactions between the outer electrons of the atom, and not the nucleons. For example, water that contains deuterium rather than hydrogen (“heavy water”) looks and tastes like normal water. The following table shows a list of common isotopes.

Why do neutrons outnumber protons in heavier nuclei (Figure 10.5)? The answer to this question requires an understanding of forces inside the nucleus. Two types of forces exist: (1) the long-range electrostatic (Coulomb) force that makes the positively charged protons repel one another; and (2) the short-range strong nuclear force that makes all nucleons in the nucleus attract one another. You may also have heard of a “weak” nuclear force. This force is responsible for some nuclear decays, but as the name implies, it does not play a role in stabilizing the nucleus against the strong Coulomb repulsion it experiences. We discuss strong nuclear force in more detail in the next chapter when we cover particle physics. Nuclear stability occurs when the attractive forces between nucleons compensate for the repulsive, long-range electrostatic forces between all protons in the nucleus. For heavy nuclei $\left(Z>15\right),$ excess neutrons are necessary to keep the electrostatic interactions from breaking the nucleus apart, as shown in Figure 10.3.

Figure 10.5 (a) The electrostatic force is repulsive and has long range. The arrows represent outward forces on protons (in blue) at the nuclear surface by a proton (also in blue) at the center. (b) The strong nuclear force acts between neighboring nucleons. The arrows represent attractive forces exerted by a neutron (in red) on its nearest neighbors.

Detailed examination reveals greater stability and more attractive nuclear forces when when neutrons and protons are in pairs. German-born physicist Maria Goeppert-Mayer identified these characteristics based on certain quantities of nucleons, leading to her development of nuclear shell theory. Goeppert Mayer and other researchers recognized that "closed" nuclear shells are more stable than others. The theory has been very successful in explaining nuclear energy levels, nuclear decay, and the greater stability of nuclei with closed shells. Along with Johannes Jensen and Eugene Wigner, Maria Goeppert Mayer received the Nobel Prize for this work, becoming the second woman to win the award.

Because of the existence of stable isotopes, we must take special care when quoting the mass of an element. For example, Copper (Cu) has two stable isotopes:

Given these two “versions” of Cu, what is the mass of this element? The atomic mass of an element is defined as the weighted average of the masses of its isotopes. Thus, the atomic mass of Cu is $m_{\text{Cu}}=\left(62.929595\right)\left(0.6909\right)+\left(64.927786\right)\left(0.3091\right)=63.55\text{g/mol}.$ The mass of an individual nucleus is often expressed in atomic mass units (u), where $\text{u}=1.66054\times {10}^{\mathrm{-27}}\text{kg}$. (An atomic mass unit is defined as 1/12th the mass of a ${}^{12}\text{C}$ nucleus.) In atomic mass units, the mass of a helium nucleus (A = 4) is approximately 4 u. A helium nucleus is also called an alpha (α) particle.

Nuclear Size

The simplest model of the nucleus is a densely packed sphere of nucleons. The volume V of the nucleus is therefore proportional to the number of nucleons A, expressed by

where r is the radius of a nucleus and k is a constant with units of volume. Solving for r, we have

where $r_0$ is a constant. For hydrogen ($A=1$), $r_0$ corresponds to the radius of a single proton. Scattering experiments support this general relationship for a wide range of nuclei, and they imply that neutrons have approximately the same radius as protons. The experimentally measured value for $r_0$ is approximately 1.2 femtometer (recall that $1\text{fm}={10}^{\mathrm{-15}}\text{m}$).