Premed · Premed · Physics 2
Lecture 23: Nuclear Physics
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the structure of the atomic nucleus and define key nuclear terminology
- Explain the concept of nuclear binding energy and the mass defect
- Interpret the binding energy per nucleon curve and its implications for nuclear stability
- Describe the strong nuclear force and its properties
- Explain nuclear fission and fusion and their energy release mechanisms
Lecture Content
I. Nuclear Structure and Terminology
The nucleus of an atom contains protons and neutrons, collectively called nucleons. The proton carries a positive charge (+e) and has a mass of 1.6726 x 10^-27 kg (938.3 MeV/c^2). The neutron is electrically neutral and slightly heavier, with a mass of 1.6749 x 10^-27 kg (939.6 MeV/c^2).
Nuclear notation uses the form ^A_Z X, where Z is the atomic number (number of protons, which defines the element), N is the neutron number, and A = Z + N is the mass number (total number of nucleons). Isotopes are atoms of the same element (same Z) but with different numbers of neutrons. For example, carbon has isotopes ^12_6 C (6 protons, 6 neutrons), ^13_6 C (6 protons, 7 neutrons), and ^14_6 C (6 protons, 8 neutrons).
The nuclear radius follows the empirical formula r approximately equals r_0 A^(1/3), where r_0 is approximately 1.2-1.3 fm (1 fm = 10^-15 m). The nucleus is about 100,000 times smaller than the atom. Nuclear density is approximately constant at roughly 2.3 x 10^17 kg/m^3, independent of A, indicating that nucleons are packed at roughly the same density in all nuclei.
II. The Strong Nuclear Force
The strong nuclear force is responsible for holding the nucleus together, overcoming the intense electrostatic repulsion between the closely packed protons. It is attractive between all nucleon pairs (proton-proton, proton-neutron, and neutron-neutron) and is charge-independent, meaning the strong force between any pair of nucleons is approximately equal. It is very short-ranged, effective only over distances of about 1-2 fm and essentially zero beyond 3 fm. At nuclear distances, it is much stronger than the electromagnetic force. The strong force also saturates, meaning each nucleon interacts primarily with its nearest neighbors rather than with all other nucleons in the nucleus.
The balance between the strong force (attractive, short range) and the electromagnetic force (repulsive, long range) determines nuclear stability. For small nuclei, roughly equal numbers of protons and neutrons (N approximately Z) produce the most stable configurations. For larger nuclei, more neutrons than protons are needed because the additional neutrons contribute to the attractive strong force without adding to the electromagnetic repulsion.
III. Nuclear Binding Energy and Mass Defect
The mass of a nucleus is always less than the sum of the masses of its individual constituent nucleons. This difference is the mass defect: Delta m = [Z m_p + N m_n] - m_nucleus. The "missing mass" has been converted to the energy that holds the nucleus together, in accordance with Einstein's mass-energy equivalence.
The binding energy (BE) is the energy required to completely disassemble the nucleus into its separate nucleons: BE = Delta m c^2 = [Z m_p + N m_n - m_nucleus] c^2. A useful conversion factor is 1 u (atomic mass unit) = 931.5 MeV/c^2.
As an example, consider the ^4_2 He nucleus (an alpha particle). The combined mass of 2 protons and 2 neutrons is 4.0320 u, while the measured mass of the He-4 nucleus is 4.0015 u. The mass defect is 0.0305 u, giving a binding energy of 0.0305 x 931.5 = 28.4 MeV.
IV. Binding Energy Per Nucleon Curve
The binding energy per nucleon (BE/A) serves as a key indicator of nuclear stability. Plotting BE/A versus mass number A reveals a characteristic curve. It rises steeply for light nuclei, from hydrogen through helium-4. He-4 (the alpha particle) is unusually stable with BE/A approximately 7.07 MeV. The curve peaks near iron-56, which has BE/A approximately 8.79 MeV and is therefore the most tightly bound nucleus per nucleon. For heavier nuclei, the curve gradually declines, reaching about 7.6 MeV for uranium.
The shape of this curve has profound implications. Fusion of light nuclei, which moves up the curve toward the peak, releases energy. Fission of heavy nuclei, which moves from the right side of the curve toward the peak, also releases energy. Nuclei near iron-56 sit at the bottom of the nuclear energy valley and are the most stable.
<image>A graph of binding energy per nucleon (BE/A in MeV) versus mass number (A). The vertical axis ranges from 0 to about 9 MeV. Key nuclei are labeled on the curve: ^2H (deuterium, ~1.1 MeV), ^3He (~2.6 MeV), ^4He (sharp peak at ~7.07 MeV), ^12C (~7.68 MeV), ^16O (~7.98 MeV), ^56Fe (peak at ~8.79 MeV), ^235U (~7.59 MeV). The curve rises steeply for light nuclei, peaks at Fe-56, and gradually declines for heavier nuclei. Arrows indicate that fusion of light nuclei (left side) and fission of heavy nuclei (right side) both release energy by moving toward the peak.</image>
V. Nuclear Fission
Fission is the process in which a heavy nucleus splits into two or more lighter nuclei, along with neutrons and a large release of energy. In induced fission, a neutron is absorbed by a heavy nucleus, making it unstable. A typical reaction is ^235_92 U + ^1_0 n yielding ^141_56 Ba + ^92_36 Kr + 3 ^1_0 n + approximately 200 MeV. The fission products are variable, with many different fragment pairs possible.
The energy released is roughly 200 MeV per fission event, mostly as kinetic energy of the fragments. This is about a million times more energy per reaction than chemical reactions produce. The neutrons released can trigger further fissions, creating a chain reaction. The critical mass is the minimum amount of fissile material needed to sustain a chain reaction. Below critical mass, the chain reaction dies out (subcritical); above it, the reaction can grow exponentially (supercritical).
Nuclear reactors use controlled fission for electricity generation. Control rods that absorb neutrons regulate the reaction rate. A moderator (water or graphite) slows neutrons to increase the probability of inducing fission. The fuel is typically enriched uranium (^235U) or plutonium (^239Pu). Nuclear weapons, by contrast, rely on an uncontrolled supercritical chain reaction.
VI. Nuclear Fusion
Fusion is the process in which two light nuclei combine to form a heavier nucleus, releasing energy. It is the energy source of stars. In the sun, the proton-proton chain converts four hydrogen nuclei into one helium-4 nucleus, two positrons, two neutrinos, and 26.7 MeV of energy. The sun's core temperature of approximately 15 million K provides the thermal energy needed to overcome the Coulomb repulsion between protons. The sun converts roughly 4 million tons of mass to energy every second. In more massive stars, the CNO cycle dominates, using carbon as a catalyst.
Fusion releases more energy per unit mass than fission. Thermonuclear fusion on Earth remains an active area of research, requiring extreme temperatures of approximately 100 million K to confine a plasma. Approaches include magnetic confinement (tokamaks and stellarators) and inertial confinement (laser fusion). The most promising fuel is the deuterium-tritium (D-T) reaction: ^2H + ^3H yielding ^4He + n + 17.6 MeV. Fusion offers the promise of abundant fuel (deuterium from seawater) and far less radioactive waste than fission, but achieving sustained net energy gain remains a significant engineering challenge.
<image>Two panels comparing fission and fusion. Panel A (Fission): A large uranium-235 nucleus absorbs a neutron and splits into two medium-sized fragments (e.g., barium-141 and krypton-92) plus 3 neutrons and gamma rays. The released neutrons are shown triggering additional fissions in a branching chain reaction diagram. Energy released: ~200 MeV. Panel B (Fusion): Two small nuclei (deuterium and tritium) collide at high speed and fuse into a helium-4 nucleus plus a neutron. Energy released: 17.6 MeV. The binding energy curve is shown schematically between the two panels, with arrows indicating that both processes move toward the peak at iron-56.</image>
