42-2 Some Nuclear Properties

42-2 Some Nuclear Properties#

Prompts

  • What are \(Z\), \(N\), and \(A\) for a nucleus? What are isotopes?

  • The nuclear radius is \(r \approx r_0 A^{1/3}\) with \(r_0 \approx 1.2\) fm. What does the \(A^{1/3}\) dependence tell you about nuclear density?

  • Define binding energy and mass defect. Why is the mass of a nucleus less than the sum of its free nucleon masses?

  • How do you calculate the binding energy from atomic masses? What is the conversion factor \(1\,\text{u} = 931.5\,\text{MeV}/c^2\)?

  • The binding energy per nucleon \(E_b/A\) is roughly 8 MeV for most nuclei and peaks near iron. What does that imply about fission and fusion?

Lecture Notes#

Overview#

  • A nucleus is characterized by \(Z\) (proton number), \(N\) (neutron number), and \(A = Z + N\) (mass number). Isotopes share the same \(Z\) but differ in \(N\).

  • The nuclear radius scales as \(r \approx r_0 A^{1/3}\), implying roughly constant nuclear density.

  • The binding energy \(E_b\) is the energy required to separate a nucleus into its nucleons; it equals \((\Delta m)c^2\), where \(\Delta m\) is the mass defect.

  • The binding energy per nucleon \(E_b/A\) is ~8 MeV for most nuclei and peaks near iron — explaining why fusion releases energy for light nuclei and fission for heavy ones.

The Nuclides#

A nuclide is a specific nuclear species. It is specified by:

  • \(Z\) (atomic number): number of protons.

  • \(N\) (neutron number): number of neutrons.

  • \(A\) (mass number): total number of nucleons,

(436)#\[ A = Z + N \]

Isotopes are nuclides with the same \(Z\) but different \(N\) (e.g., \(^{12}\text{C}\), \(^{13}\text{C}\), \(^{14}\text{C}\)). Notation: \(_Z^A\text{X}\) or \(^A\text{X}\) when \(Z\) is implied by the element symbol.

Nuclear Radius#

Nuclei have a mean radius given empirically by

(437)#\[ r \approx r_0 A^{1/3},\quad r_0 \approx 1.2\ \text{fm} \]

Since volume \(V \propto r^3 \propto A\), the nuclear density \(\rho \propto A/V \approx\) constant. Nuclei behave like incompressible droplets of similar density.

Atomic Mass Unit#

The atomic mass unit (u) is defined so that \(^{12}\text{C}\) has mass 12 u exactly:

(438)#\[ 1\ \text{u} = 1.66054\times 10^{-27}\ \text{kg} \]

The mass–energy equivalence gives

(439)#\[ 1\ \text{u} = 931.494\ \text{MeV}/c^2 \]

The mass excess \(\Delta = M - A\) (in u) is often tabulated; it simplifies binding-energy calculations when combined with Eq. (439).

Mass Defect and Binding Energy#

The mass defect \(\Delta m\) is the difference between the sum of the masses of the free nucleons and the nuclear mass:

(440)#\[ \Delta m = Z m_p + N m_n - M_{\text{nuc}} \]

where \(M_{\text{nuc}}\) is the nuclear mass. The nucleus has less mass than its constituents because energy was released when the nucleons bound together.

The binding energy \(E_b\) is the energy required to separate the nucleus into free nucleons:

(441)#\[ E_b = (\Delta m) c^2 \]

Using atomic masses

Nuclear masses are rarely tabulated; atomic masses are. For a neutral atom, the atomic mass includes \(Z\) electrons. When computing \(\Delta m\) from atomic masses, the electron masses cancel in the difference (for beta-stable nuclei). The conversion \(1\,\text{u} = 931.5\,\text{MeV}/c^2\) is used to get \(E_b\) in MeV.

Binding Energy per Nucleon#

The binding energy per nucleon is

(442)#\[ \frac{E_b}{A} = \frac{(\Delta m)c^2}{A} \]

  • For most stable nuclei, \(E_b/A \approx 7\)–8 MeV.

  • \(E_b/A\) peaks near iron (\(A \approx 56\)) at ~8.8 MeV.

  • Light nuclei (\(A\) small): fusion (combining) releases energy.

  • Heavy nuclei (\(A\) large): fission (splitting) releases energy.

Why the peak?

The curve reflects a balance: the strong force attracts nucleons (favors binding), while the Coulomb repulsion between protons favors separation. For heavy nuclei, Coulomb effects grow and \(E_b/A\) decreases.

Summary#

  • Nuclides: \(Z\), \(N\), \(A = Z + N\); isotopes share \(Z\).

  • Nuclear radius: \(r \approx r_0 A^{1/3}\) (\(r_0 \approx 1.2\) fm); density ~constant.

  • Mass defect \(\Delta m\) and binding energy \(E_b = (\Delta m)c^2\).

  • \(E_b/A\) peaks near iron; fusion (light) and fission (heavy) release energy.