Nuclear Physics

Liquid Drop Model

The liquid drop model, proposed by George Gamow and later developed by Niels Bohr and John Archibald Wheeler, provides a macroscopic view of the atomic nucleus. It treats the nucleus as a droplet of incompressible nuclear fluid, where the nucleons (protons and neutrons) are analogous to molecules in a liquid drop. This model is particularly useful for understanding nuclear binding energy and nuclear fission.

The core idea is that the nuclear force, which holds nucleons together, is short-range and behaves similarly to the surface tension forces in a liquid. The nucleons interact strongly with their nearest neighbors, but not with nucleons far away.

Assumptions of the Liquid Drop Model:

• The nucleus is like a drop of incompressible liquid.
• The nuclear matter is uniformly distributed within the nuclear volume.
• The nuclear force is short-ranged, acting only between adjacent nucleons.
• The nucleus has a surface tension, similar to a liquid drop.

Application to Binding Energy: Semi-Empirical Mass Formula (Weizsäcker Formula)

The liquid drop model forms the basis of the semi-empirical mass formula, also known as the Weizsäcker formula. This formula provides an approximate value for the binding energy (BE) of a nucleus with mass number A and atomic number Z. It consists of several terms, each corresponding to a physical effect:

BE(A, Z) = a_VA - a_SA^2/3 - a_CZ(Z-1)A^-1/3 - a_A(A-2Z)²A^-1 ± δ(A, Z)

Let's break down each term:

• Volume Energy (a_VA): This term represents the binding energy due to the strong nuclear force. It assumes that each nucleon interacts with a fixed number of neighboring nucleons, so the total binding energy is proportional to the total number of nucleons (A). This is analogous to the volume energy of a liquid drop. The coefficient a_V is positive, indicating a stabilizing effect.
• Surface Energy (-a_SA^2/3): Nucleons on the surface of the nucleus have fewer neighbors to interact with, thus contributing less to the binding energy. This effect is proportional to the surface area of the nucleus, which is proportional to A^2/3. The coefficient a_S is positive, making this term a reduction in binding energy.
• Coulomb Energy (-a_CZ(Z-1)A^-1/3): Protons within the nucleus repel each other due to the electrostatic (Coulomb) force. This repulsion reduces the binding energy. The term accounts for the interaction between every pair of protons. The number of proton pairs is approximately Z(Z-1)/2. The term A^-1/3 comes from the average distance between protons, which is related to the nuclear radius. The coefficient a_C is positive, contributing negatively to the binding energy.
• Asymmetry Energy (-a_A(A-2Z)²A^-1): Nuclei with a large excess of neutrons or protons are less stable. This term accounts for the energy associated with the imbalance between neutrons (N) and protons (Z). Since N = A - Z, the difference is A - 2Z. This term is related to the Pauli exclusion principle, which states that nucleons of the same type cannot occupy the same quantum state. The coefficient a_A is positive, reducing binding energy for asymmetric nuclei.
• Pairing Energy (± δ(A, Z)): This term accounts for the empirical observation that nuclei with even numbers of protons and neutrons are more stable than those with odd numbers.
  • + δ for even-even nuclei (even Z, even N)
  • 0 for odd-even or even-odd nuclei (odd Z, even N or even Z, odd N)
  • - δ for odd-odd nuclei (odd Z, odd N)
  The term δ is roughly proportional to A^-1/2.

The Weizsäcker formula accurately describes the general trend of binding energy per nucleon, showing a peak around A=56 (Iron).

Shell Model

The shell model, developed independently by Maria Goeppert Mayer and J. Hans D. Jensen, provides a quantum mechanical description of the nucleus, similar to the atomic shell model for electrons. It explains the observed "magic numbers" of nucleons (2, 8, 20, 28, 50, 82, 126) which correspond to particularly stable nuclei with high binding energies.

In this model, nucleons move independently in an average potential created by all other nucleons. This potential is often approximated by a harmonic oscillator potential or a Woods-Saxon potential. The nucleons then fill energy levels or "shells" according to the Pauli exclusion principle. When a shell is completely filled, the nucleus exhibits enhanced stability.

Key Concepts of the Shell Model:

• Independent Particle Motion: Each nucleon moves in an average potential field created by the other nucleons, rather than interacting pairwise with every other nucleon.
• Energy Levels and Shells: The potential creates discrete energy levels for the nucleons. These levels are grouped into shells.
• Pauli Exclusion Principle: Nucleons of the same type (protons or neutrons) must occupy different quantum states.
• Magic Numbers: The number of nucleons (protons or neutrons) that completely fill a major energy shell. These numbers (2, 8, 20, 28, 50, 82, 126) correspond to nuclei with exceptionally high binding energies and stability, and are also characterized by higher ionization energies (or separation energies for nucleons).

Spin-Orbit Coupling:

A crucial addition to the basic shell model is the concept of spin-orbit coupling. This interaction couples the intrinsic angular momentum (spin) of a nucleon with its orbital angular momentum. This splitting of energy levels is essential for correctly predicting the magic numbers. For a given orbital angular momentum l, spin-orbit coupling splits the energy level into two: one with total angular momentum j = l + 1/2 and another with j = l - 1/2. The energy level with higher j is typically lower in energy due to the interaction.

The sequence of energy levels, including spin-orbit splitting, correctly predicts the filling of shells and the emergence of magic numbers. For example, the 1d_5/2 and 1d_3/2 levels, when filled, account for the magic number 20. The 1g_7/2 and 1h_11/2 levels contribute to the magic number 82.

Evidence for the Shell Model:

• Magic Numbers: The most direct evidence.
• High Abundance: Nuclei with magic numbers of protons or neutrons are more abundant in nature.
• Binding Energies: Higher binding energies per nucleon for nuclei with magic numbers.
• Separation Energies: Higher energy required to remove a neutron or proton from nuclei with magic numbers.
• Nuclear Spin and Parity: The ground state spin and parity of nuclei with a single nucleon outside a closed shell are predicted correctly by the shell model.

Collective Models

While the liquid drop model is macroscopic and the shell model is microscopic (independent particles), collective models attempt to bridge the gap by considering the coordinated motion of many nucleons. These models are particularly successful in describing the properties of deformed nuclei, which are not well explained by the simple shell model.

Collective models treat the nucleus as a whole, focusing on collective excitations such as rotations and vibrations of the entire nucleus. These collective motions arise from the coherent interaction of many nucleons.

Vibrational Model:

This model treats the nucleus as a vibrating liquid drop. The vibrations can be of various modes, such as surface vibrations (changing the shape) or volume vibrations. The energy levels associated with these vibrations are quantized, leading to excited states with specific energies. This model is useful for nuclei near closed shells that are slightly deformed.

Rotational Model:

This model is applied to nuclei that are significantly deformed, typically prolate (cigar-shaped) or oblate (pancake-shaped). These deformed nuclei behave like rigid rotors. The collective rotation of the nucleus leads to a series of energy levels with spins 0⁺, 2⁺, 4⁺, etc., with energies proportional to I(I+1), where I is the spin.

The energy spectrum predicted by the simple rotational model is E(I) = (ħ² / 2I) * I(I+1), where I is the moment of inertia of the nucleus. This is analogous to the rotational spectra of molecules.

Interacting Boson Model (IBM):

A more sophisticated collective model, the Interacting Boson Model, treats pairs of nucleons (fermions) as bosons. These bosons then interact with each other. The IBM has been very successful in describing the collective properties of medium and heavy nuclei, including transitional nuclei that exhibit characteristics of both vibrational and rotational behavior.

Limitations:

Collective models often struggle to explain phenomena related to individual nucleon behavior, such as detailed spectroscopic properties or the ground state spins of nuclei with few nucleons outside closed shells.

Nuclear Fission

Nuclear fission is a nuclear reaction in which the nucleus of an atom splits into two or more smaller, lighter nuclei. This process releases a large amount of energy, neutrons, and gamma rays. Fission is typically induced by the absorption of a neutron by a heavy nucleus, such as uranium or plutonium.

The process can be understood using the liquid drop model. When a heavy nucleus absorbs a neutron, it becomes excited and oscillates. If the excitation energy is sufficient, the nucleus can deform to such an extent that the short-range nuclear force can no longer hold the two distorted parts together. The long-range Coulomb repulsion between the positively charged fragments then overcomes the remaining nuclear force, causing the nucleus to split.

The Fission Process:

1. Initiation: A neutron strikes a fissile nucleus (e.g., ²³⁵U).
2. Absorption: The nucleus absorbs the neutron, forming a highly excited compound nucleus (e.g., ²³⁶U*).
3. Deformation: The compound nucleus oscillates and deforms, elongating like a liquid drop. The Coulomb repulsion between the protons starts to dominate over the nuclear attraction.
4. Scission: The nucleus splits into two (or sometimes three) lighter fission fragments, which are themselves radioactive nuclei.
5. Neutron Emission: Several neutrons (typically 2-3) are emitted during the fission process. These are called prompt neutrons.
6. Energy Release: A significant amount of energy (around 200 MeV per fission event) is released, primarily in the form of kinetic energy of the fission fragments and neutrons, and also as gamma rays. This energy comes from the difference in binding energy between the parent nucleus and the fission fragments. The fragments are generally more tightly bound than the parent nucleus.

Fissile vs. Fissionable Nuclei:

• Fissile: Nuclei that can undergo fission with slow (thermal) neutrons. Examples: ²³³U, ²³⁵U, ²³⁹Pu.
• Fissionable: Nuclei that can undergo fission but require fast neutrons to do so. Examples: ²³²Th, ²³⁸U.

Chain Reaction:

The neutrons released during fission can go on to induce fission in other fissile nuclei, leading to a self-sustaining chain reaction.

• Controlled Chain Reaction: Used in nuclear reactors to generate power. The rate of fission is controlled by using moderators to slow down neutrons and control rods to absorb excess neutrons.
• Uncontrolled Chain Reaction: Used in nuclear weapons, where the reaction proceeds very rapidly, releasing a massive amount of energy in a short time.

The condition for a sustained chain reaction is that, on average, at least one neutron from each fission event must cause another fission event. This is described by the multiplication factor, k.

• k = 1: Critical (sustained chain reaction)
• k < 1: Subcritical (reaction dies out)
• k > 1: Supercritical (reaction grows exponentially)

Energy Release Calculation:

The energy released in fission can be estimated by comparing the mass of the initial nucleus and the neutron with the total mass of the fission fragments and the emitted neutrons.

Energy Released = [Mass(Parent Nucleus) + Mass(Neutron) - Sum of Masses(Fission Fragments) - Sum of Masses(Emitted Neutrons)] × c²

For example, the fission of ²³⁵U by a thermal neutron typically yields:

²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n + Energy (approx. 200 MeV)

This energy is equivalent to about 3.2 × 10^-11 Joules per fission event.

Applications of Fission:

• Nuclear Power Plants: Generate electricity by using the heat produced from controlled fission to boil water and drive turbines.
• Nuclear Weapons: Utilize uncontrolled fission for destructive purposes.
• Production of Isotopes: Fission products include many radioactive isotopes used in medicine, industry, and research.
• Research Reactors: Used for scientific experiments and neutron production.