# 2.1.4 Nuclear Models

Created October 6, 1995

### 2.1.4.1 Nuclear Force

The nuclear force is purely quantum mechanical, something we cannot envision in the real world. It involves the exchange of virtual pions, which are allowed to exist by the Heisenberg Uncertainty Principle of Quantum Mechanics only for a very short period of time. This short time period translates into a range or distance over which the exchange particle can travel - a range of about 5 fermis.
One way to envision the nuclear force is to imagine the relatively new and highly useful product "velcro." Velcro involves one surface comprised of loops and another surface comprised of hooks. If the distance between the two surfaces is close enough for the hooks to engage the loops, a very strong adhesion is observed. However, if the distance is slightly greater than the required proximity, no force whatsoever exists between the two. The nuclear force has such a distance dependence, as can be seen in this plot:

### 2.1.4.2 Liquid Drop Model

Bohr suggested that the nucleus may behave as a suspended drop of liquid behaves, with a surface tension. This model is referred to as the "liquid drop" model. This model does not explain energy levels well, but is useful in providing an explanation of how a very heavy nucleus, such as U-235, might begin oscillating and thus elongating until it becomes so thin at the "waist" that it splits in two - it undergoes fission.

One of the great successes of the liquid drop model is its description of the binding energy per nucleon. The liquid drop model assumes that three processes compete to produce the binding energy per nucleon: the volume effect, which gives a constant BE/A as A is increased and tends to bind the nucleus; the surface effect, which is proportional to the surface area and hence R**2 or A**2/3 which tends to unbind the nucleus; and the coulomb effect, which is proportional to the number of proton pairs and tends to unbind the nucleus. These contributions can be shown to account for the binding energy curve which is seen in nature:

The liquid drop model contains assumptions which approximate the true nature of the interactions within the nucleus, although it fails to predict many of the finer details of nuclear structure.

### 2.1.4.3 Nuclear Shell Model

This model is an extension of the ideas behind the theory of atomic structure: that there are energy levels with vacancies to be filled, and that energetically stable states can be achieved as these shells are filled. Paired nucleons of a particular type (e.g., protons) are more energetically favorable than unpaired nucleons. To demonstrate this, we have the following table:

There are 281 stable nuclides in this table. Odd-odd configurations have difficulty forming, only the four *lowest possible mass* odd-odd nuclides exist (these have Z=N as well, Z=1,3,5, and 7): H-2, Li-6, B-10, and N-14. For higher masses the number of ergetics of spin pairing predominate over the forces that hold these odd-odd nuclei together.
### TOTAL ANGULAR MOMENTUM J

The shell model is successful at predicting the total nuclear angular momentum J. This vector is the sum of three contributors: 1) the intrinsic angular momentum (spin) of the protons; 2) the spin of the neutrons; and the *orbital* angular momentum from their position withing the nucleus. All even-enen nuclei have J =0. Even-odd combinations can have 1/2; 3/2; 5/2, etc., and odd-odd have been found to have J=1 or 3.
### PARITY PI

The parity of the nucleus is often given in conjunction with the total angular momentum J. Parity is a concept of great importance in the quantum mechanical world. It has to do with the wave function of the nucleus: If the wave function becomes negative when the coordinates of the nucleus are reversed (i.e, when the nucleus is reflected through the origin, or when x becomes -x, y becomes -y, and z becomes -z), then the nucleus is said to have "NEGATIVE PARITY". If the wave function remains unchanged upon reflection through the origin, it is said to have "POSITIVE PARITY".
### MAGIC NUMBERS

The nuclear shell model predicts energetically favorable nuclear configurations if either the number of neutrons or the number of protons equals 2, 8, 20, 28, 50, 82 and 126. These numbers are refered to as the "magic numbers".fff

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Douglas J. Wagenaar, Ph.D., wagenaar@nucmed.bih.harvard.edu