Models of Nuclear Physics

The atomic nucleus is one of the most fascinating objects in the universe. Groups of protons and neutrons, themselves composed of quarks, bind together as a result of the strong nuclear force that, at distances of around 1fm = 1×10-15m, is potent enough to overcome proton-proton electromagnetic repulsion. The continuous tug of war between these two forces gives rise to the stability properties of nuclei i.e. how inclined the nucleus is to remain intact without undergoing radioactive decay.

A large proportion of nuclear physics is dedicated to formulating models of the nucleus that explain the emergent stability of different nuclei. The binding energy is defined as the amount of energy required to break the nucleus up into individual protons and neutrons. The larger the binding energy the higher the nuclear stability. 

Liquid-Drop Model

Early models of the nucleus drew comparisons between nuclear matter and drops of classical liquids. This analogy was motivated by two main observations…

  1. The mass density of the nucleus is approximately constant and the same for all nuclei.
  2. The binding energy of a nucleus is approximately proportional to its mass.

Similarly…

  1. The mass density of an incompressible liquid is constant.
  2. The heat required to evaporate a drop of liquid is approximately proportional to its mass.

Protons and neutrons are likened to the molecules within a liquid. Properties such as the surface tension of the drop are considered in the calculation of binding energies and seem to increase their accuracy. As well as making sensible predictions for stability, the liquid-drop model also offers an intuitive conceptualisation of nuclear fission.

Fission describes the process of a large nucleus, known as the parent, splitting into two smaller nuclei, known as daughters. A diagram of fission is shown in Figure 1. Modelling the parent nucleus as a spherical drop of liquid, we can consider the beginning of fission as the deformation of the sphere into an oblate spheroid. Altering the shape of the drop changes the energetics related to the surface tension, motivating the nucleus to undergo further distortion, assuming a dumbbell-like shape. Eventually, perturbations within the drop cause it to split and the process is complete.

Fermi-Gas Model

All the physics within the liquid-drop model is classical. The problem with trying to describe small objects such as nuclei classically is that quantum effects are very significant. There are consequently deviations from the liquid-drop models binding energy predictions, especially for very stable nuclei.

The Fermi-Gas model is a fundamentally quantum approach. Protons and neutrons move freely within the nucleus but are subject to the Pauli Exclusion Principle, allowing only two protons/neutrons to occupy each energy level. Nuclear particles fill up a ladder of possible energies from the bottom up, as shown in Figure 2.

Figure 2: Protons and neutrons fill up a ladder of energy levels within the Fermi-Gas model. Note that the proton levels begin above the neutrons due to the electrostatic repulsion between nearby protons.

The consideration of quantum effects results in the addition of extra terms during binding energy calculations. One of these terms arises from the nucleus’s preference to be symmetric i.e. contain an equal number of protons and neutrons. Many nuclei don’t possess this property, adding to their instability, and thus including this effect within calculations increases the achieved accuracy.

Shell Model

Asymmetry isn’t the only effect that arises from the quantum nature of the atomic nucleus. Protons and neutrons possess an orbital angular momentum due to their movement within the nucleus and an intrinsic angular momentum known as spin. Both of these are vectors and can thus be pictured heuristically as tiny arrows pointing away from each particle. The relative direction of the spin and orbital angular momentum of nearby protons and neutrons effects the energies of the respective particles, causing single energy levels to split into multiple. This effect is known as spin-orbit coupling (SOC).

Implementing SOC into binding energy calculations results in the emergence of a shell structure similar to that observed when studying electrons in atoms. A diagram of the shell model energy levels, for two different nuclei, is shown in Figure 3. Notice the gaps between shells are not equal across the spectrum. Nuclei whose most energetic proton/neutron occupies an energy level with a large gap above are more stable, as more energy is needed to promote it and cause change. Nucleus 2 would therefore be more stable than nucleus 1 – the highest neutron/proton needs more energy to be promoted to the next energy level.

Unification

All the models described above have their shortcomings. Scientists are still in search of a universal theory of nuclear interactions. The collective model incorporates ideas from each of the models discussed and also accounts for nuclear vibrations. This model makes the most accurate predictions available but is still incomplete.

The long-term goal of nuclear physics is to formulate a model based on quark-quark interactions as oppose to protons and neutrons. The theory of quarks and their interactions is known as Quantum Chromodynamics (QCD). The problem with QCD is that the strength of the nuclear strong force means we can’t use the same methods to investigate particle interactions as we do for other forces, such as electromagnetism, and have to turn to computational methods instead. Advancements in computing power over the next decade should lead scientists closer to a unified theory of nuclear physics at the quark level.

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