Models of the Atomic Nucleus

Type of physical science: Nuclear physics

Field of study: Nuclei

The precise structure of the atomic nucleus is too complex to be fully known. In order to gain an understanding of the nucleus, simplifying assumptions are used to create models which are hoped to have some connection with the real world.

Overview

The atomic nucleus occupies only about one-trillionth of the volume of the atom but contains almost all of its mass. The nucleus is held together by the strongest of the forces of nature, but this force extends only slightly past its surface. Electrons, which are responsible for the chemical properties of the atom, shield the nucleus from the outside. The result of these observations is that the atomic nucleus is not fully understood.

The nucleus contains protons and neutrons. Protons carry a positive electrical charge (opposite that of the electrons), while neutrons are electrically neutral. Thus, protons are subject to electromagnetic interactions, but neutrons are not. The strongest interaction between two nucleons (a term that refers to both protons and neutrons) is the strong nuclear force, but electromagnetic interactions are also important.

It is known that the nucleus is made up of nucleons and that these nucleons can interact through nuclear and electromagnetic forces. This leads to the question: What is the arrangement of the nucleons in the nucleus? There are two major problems encountered in answering this question. First, the nuclear force is very complicated; it is not possible to write down a simple formula for it. Rather, it is necessary to approximate the nuclear force using models that are derived from experimental data. Second, all nuclei that are heavier than deuterium (a form of hydrogen containing one proton and one neutron) contain at least three nucleons. In nature, each of these nucleons interacts with each of the others. Only problems with pairwise interactions, however, have solutions that are not prohibitively difficult.

Instead of trying to determine an exact solution to a system with some large number of nucleons, simplifying assumptions must be made about the physics that are involved. These assumptions lead to models of the structure of the atomic nucleus. The test of a model is how well it can explain and predict experimental results, including the nuclear mass, electromagnetic properties of the nucleus, and excitation energies of the nucleus.

One of the earliest models of nuclear structure was the liquid drop model, which suggests that the lowest energy state of the nucleus (the ground state in which the nucleus is generally found) can be represented as a spherical drop of fluid. This is not to say that the nucleus is really a drop of ordinary fluid, but the nucleus and the drop have two important properties in common: Neither is very compressible and both have well-defined surfaces. An initially spherical drop of liquid can vibrate in a variety of shapes. The vibrating drop is at a higher energy than its spherical ground state. Each of these vibrational modes also has a certain angular momentum. Experimentally, the nucleus is observed to undergo similar vibrations. Thus, the liquid drop model can be used to represent vibrational excitations of nuclei.

The most widely used model of the nucleus is the shell model. The underlying assumption in the shell model is that the protons and neutrons do not interact with each other, but they instead exist in an average potential "well" arising from all the nucleons. If this assumption were not made, then it would be necessary to consider interactions between every pair of nucleons in the nucleus. The resulting many-body problem would be virtually impossible to solve, even with massive computing power. When this average shell-model potential is inserted in the Schrodinger equation of quantum mechanics, the resulting solutions are a series of levels, or "shells," in which the nucleons can move. These shells are similar in principle to those involved in the electronic structure of atoms. In the case of atoms, the potential arises from the central nucleus. In the case of the nucleus, however, the nucleons themselves generate the potential: There is no core at the center. This, in conjunction with uncertainty in the actual interaction between two nucleons, results in a problem that is very computationally rich and complex. Certain assumptions are usually made as to the shape of the potential well in order to simplify the solution.

The shell model predicts that configurations with 2, 8, 20, 28, 50, 82, or 126 nucleons (either protons or neutrons) will be particularly stable. Thus, helium 4, with two protons and two neutrons, is very stable, as is oxygen 16 (eight protons and eight neutrons) and lead 208 (82 protons and 126 neutrons). This prediction of the stability of closed-shell nuclei is an important facet of the shell model.

Most nuclei are not closed-shell nuclei. If a nucleon is added to a closed-shell nucleus, then the resulting nucleus will have the angular momentum of the added nucleon. This results from the fact that the closed shell can be viewed as a noninteracting, spherical core. The same holds if a nucleon is removed from the outermost level. This missing nucleon is known as a "hole."

Instead of adding a nucleon to or removing a nucleon from a nucleus, one can instead excite the nucleus. If one nucleon is taken from the highest shell of the ground state of the nucleus (leaving a hole) and is put in a higher level, then a "particle-hole excitation" is obtained.

It is an "excitation," or excited state of the nucleus, because its energy is higher than that of the ground state. The angular momenta of the particle and the hole can combine to give definite values of the total angular momentum because states with different angular momentum will have different excitation energies. These states of given angular momentum can be seen experimentally, usually in the scattering of protons or electrons from the nucleus. The agreement between theory and experiment tends to be quite good, but the shell model does not work as well for more complicated excitations.

The shell model is at its best when describing nuclei that are not too far removed from closed-shell configurations. When the number of nucleons outside the closed shell (or the number of holes in the outermost level) becomes too large, the nucleus ceases to be spherical and becomes ellipsoidal. This violates one of the assumptions of the shell model, which breaks down under these conditions. Instead of treating the properties of the nucleus as arising from a small number of nucleons outside the shell, it must be assumed that the behavior of the nucleus depends on all the nucleons. The resulting model is known as the collective model.

In many respects, the collective model treats the nucleus like a rigid body rotating around an axis, somewhat like a spinning top. The ground state of many nuclei--those that are still roughly spherical--can be pictured as not turning, while the first excited state spins with two units of angular momentum, the second with four units, and continuing in this manner. These rotational excitations can be observed experimentally. The collective model predicts the energy and strength of these levels well.

The liquid drop, shell, and collective models do not provide a full description of the atomic nucleus, and they each have their own strengths and weaknesses. Other models have been proposed to explain regimes in which the previously discussed models do not work well. For example, interacting boson approximation uses as its building blocks groups of two or four nucleons. The entire nucleus is then built up out of these groups. One application of this model is to nuclei at the border between vibration and rotation. Some of the other models involve assuming that the nucleus contains particles more exotic than the proton and neutron. For example, one can use more massive versions of the nucleons or "strange" particles. Elementary particle physics maintains that nucleons are made up of three particles called quarks. Some approaches take two closely interacting nucleons in the nucleus and replace them with a single entity consisting of six quarks.

Applications

The shell model and the collective model are, in general, much better models of the atomic nucleus than is the liquid drop model because they explain the properties of the nucleus better. They suffer, however, from a lack of significant applications interesting to anyone other than a nuclear physicist. Most often, they are studied for their own sake in order to foster an understanding of the structure of the nucleus. The liquid drop model, on the other hand, yields applications that are less obscure and have some contact with everyday life.

As a first guess, one might assume that the mass of a nucleus is equal to the sum of the masses of the individual protons and neutrons that make up that nucleus, but this is not the case.

The mass of the nucleus is less than the sum of its constituent parts. What happens to this missing mass? The famed equation E = mc² of Albert Einstein states that mass and energy are related. In the case of the nucleus, the missing mass becomes the energy that holds the nucleus together. This energy is called the binding energy.

One result of the liquid drop model is the "semiempirical mass formula" (SEMP), which provides a means of estimating the binding energy of the nucleus. It is "semiempirical" because the basic form of the SEMP is determined by theory, but the actual values fit experimental data. The SEMP has several terms, each corresponding to some aspect of the model. The largest term in the SEMP measures the total energy of attraction between each of the nucleons and each of its neighbors. Since it involves each nucleon, the binding energy is proportional to the volume of the nucleus, and thus to the total number of nucleons. The binding energy is reduced by a term proportional to the surface area of the nucleus (and therefore to the two-thirds power of the number of nucleons). This term arises because the nucleons on the surface do not have neighbors in all directions. This is analogous to the surface tension of the liquid drop. The protons carry positive electric charges and thus repel one another. This repulsion also serves to decrease the binding energy because it tends to push the nucleus apart. There are two other terms in the SEMP. The first of these decreases the binding energy when the number of protons does not equal the number of neutrons. The second increases the binding energy when the numbers of protons and neutrons are both even and decreases it when both are odd. It arises because there is a tendency for protons to pair up with protons and neutrons to pair up with neutrons. When fitted to data for several nuclei, the semiempirical mass formula does a good job of estimating the binding energy.

Another application of the liquid drop model is to describe nuclear fission. Fission is a process in which a heavy nucleus, usually uranium or plutonium, is split into two lighter nuclei by the absorption of a neutron. (From the standpoint of nuclear forces, a proton would work just as well. The incident positively charged proton, however, would be repelled by the protons in the nucleus. The neutron, without an electric charge, does not have this problem.) This splitting is accompanied by the release of a large amount of energy. The source of this energy is the binding energy of the heavy nucleus. Early nuclear weapons (the atomic bomb) and nuclear power reactors draw their energy from fission.

In treating fission by the liquid drop model, one begins with a heavy nucleus which is already somewhat deformed into an ellipsoidal shape (heavier nuclei are not as well represented by spherical drops as are medium-weight nuclei). A neutron hits this drop and causes it to vibrate. One possible mode of vibration is that the nucleus expands and contracts along its long axis. At this stage, the nucleus can be pictured as a dumbbell. If it stretches far enough, then the "neck" connecting the two halves of the dumbbell can part, leaving two lighter daughter nuclei where there had originally been one heavier parent nucleus. The liquid drop model is a good way to look at fission because the shell and collective models do not provide a mechanism by which the nucleus can break in two.

Context

At the beginning of the twentieth century, it was known that the atom contains negatively charged electrons and positively charged protons. In 1911, Ernest Rutherford discovered the nucleus by scattering α particles (nuclei of the most common form of helium) off of metal foils. The discovery of the neutron by James Chadwick in 1932 put physicists in the position to start serious investigations into the structure of the nucleus.

The development of quantum mechanics in the mid-1920's led to the first microscopic model of the nucleus--the liquid drop model, which was the dominant model through the 1940's.

What could be termed "modern" nuclear physics began in 1955 when the shell model was proposed by Maria Goeppert Mayer and J. Hans D. Jensen in Elementary Theory of Nuclear Shell Structure. Since then, much work has gone into trying to refine the shell model, as well as the collective model. A large portion of this work has resulted from the introduction of better models of nuclear forces. The rise of large-scale computing in the late 1960's and early 1970's dramatically increased the sizes of problems for which solutions could be attempted.

Despite its importance in the development of nuclear weapons and nuclear power, the study of nuclear models is probably less in touch with the real world than are most other areas of physics. Many regard it (and nuclear physics in general) as a field for the previous generation of physicists. In large extent, this is attributable to the shift in general interest in the physics community from nuclear physics to elementary particle physics. From the standpoint of both theory and experiment, however, it is a very technically challenging field and is interesting because it does attempt to explain nature.

There will be future attempts at modeling the atomic nucleus because a definitive description of it still does not exist. The models all have their own strengths and weaknesses, predicting some phenomena well but doing a very poor job with others. The 1980's saw the introduction into nuclear structure physics of techniques used in the study of elementary particles, a field spawned from nuclear physics in the late 1950's. From the perspective of nuclear physics, the most significant result that could come from elementary particle physics would be an accurate determination of the interaction between two nucleons. There is no clear indication, however, that this will happen in the foreseeable future. It remains to be seen how soon nuclear physics and elementary particle physics may be reunited.

Principal terms

ANGULAR MOMENTUM: the inertia of a body in rotating around an axis that quantum mechanics allows to take on only certain values

GROUND STATE: the configuration in which the energy of a nucleus (or an atom) is at a minimum

NEUTRON: an electrically neutral particle, which is a component of the nucleus

NUCLEUS: the massive body at the center of an atom, analogous in some ways to the sun in the solar system

PROTON: a positively charged particle, which is a component of the nucleus

Bibliography

Elton, L. R. B. INTRODUCTORY NUCLEAR THEORY. Philadelphia: W. B. Saunders, 1966. A book suited for advanced undergraduates. Provides a nice background into the entirety of theoretical nuclear physics. Contains a good chapter on nuclear models.

Gamow, George. THIRTY YEARS THAT SHOOK PHYSICS. Garden City, N.Y.: Doubleday, 1966. An account of nuclear physics from its beginnings until after World War II that is written by a major figure in the field. Geared for the general reader.

Motx, Lloyd, and Jefferson Hane Weaver. THE STORY OF PHYSICS. New York: Plenum Press, 1989. A history of physics written for the lay reader. Noteworthy because, instead of being in chronological order, it is divided into the fields of physics. Chapter 18 deals with nuclear physics.

Rhodes, Richard. THE MAKING OF THE ATOMIC BOMB. New York: Simon & Schuster, 1986. More than an account of the Manhattan Project, this is the story of the physicists involved in the creation of the first nuclear weapons. The physics receives a narrative, not mathematical, treatment.

Weinberg, Steven. THE FIRST THREE MINUTES. New York: Basic Books, 1977. Primarily a treatment of the time shortly after the beginning of the universe. It deals more with elementary particle physics than with nuclear physics, but it does examine the origin of the elements.

The Structure of the Atomic Nucleus

Fission and Thermonuclear Weapons

Nuclear Reactions and Scattering

Nuclear Reactors: Design and Operation

The Periodic Table and the Atomic Shell Model

Radioactive Nuclear Decay and Nuclear Excited States