Radioactive Nuclear Decay And Nuclear Excited States

Type of physical science: Nuclear physics

Field of study: Nuclei

When atomic nuclei become unstable, their restabilization involves the emission of particles and/or radiation. Attempts to understand the process of radioactivity and excited-state decay have led to the development of many cornerstones of modern physics.

Overview

The dynamic properties of nuclei are of two types: radioactive decay and the decay of excited states of the nucleus, and nuclear reactions. The former occur spontaneously, while the latter must be induced. Radioactive nuclei and excited states of nuclei can be produced by nuclear reactions. Radioactive and excited-state decays occur when unstable nuclei stabilize themselves by emitting particles and/or radiation. For this reason, it is important to examine the stability of physical systems in general and of nuclei in particular.

A physical system tends toward a configuration which minimizes the system's potential energy. This tendency is a result of the law of conservation of energy: The sum of the kinetic energy and the potential energy of a system is constant. Thus, a ball balanced at the crest of a hill, when slightly disturbed, will minimize its gravitational potential energy by rolling to the bottom of the hill. Conversely, a ball resting in the trough between two hills will return to the trough when slightly disturbed since it already has the minimum gravitational potential energy that the system can possess. The ball balanced at the top of the hill is said to be in unstable equilibrium, while the ball resting in the trough is said to be in stable equilibrium.

In the same manner, the nuclei of atoms can be stable or unstable. Like any quantum-mechanical bound system, the nucleus of an atom has a lowest-energy state, called the ground state, and a number of discrete, higher-energy excited states. The excited states will be unstable, since the release of some energy from the excited nucleus will lead to a state of lower potential energy. The ground state of a particular nucleus may also be unstable. When the actual mass of a nucleus is compared with the sum of the masses of the protons and neutrons of which it is composed, a small difference in the two values is found to exist. The energy equivalent (E = mc²) of this mass difference is known as the binding energy of the nucleus.

Since higher binding energies correspond to lower potential energies, nuclei will tend toward configurations having higher total binding energies.

The average binding energy per nucleon is found by dividing the total binding energy of a particular nucleus by the number of nucleons contained in that nucleus. Naturally occurring nuclei show a rapid increase in the binding energy per nucleon as the number of nucleons increases from zero, with nuclei having small nucleon number values that are multiples of four being particular strongly bound. The binding energy per nucleon peaks at about 60 nucleons, after which it decreases only slowly as the number of nucleons approaches its largest value of 260. The binding energy is the result of several competing influences: the strong nuclear force, the Coulomb force, and some quantum-mechanical effects.

The strong nuclear force is an attractive force which acts between nucleons, tending to bind them together. The existence of stable nuclei suggests that this force must be very strong indeed, since the competing Coulomb repulsion between the positively charged protons in the nucleus would otherwise cause an assembly of protons and neutrons to fly apart violently.

Although the nuclear force is quite strong, it has a very limited range. The strong nuclear force is said to "saturate," meaning that a given nucleon interacts only with its very nearest neighbors.

This characteristic can be contrasted with Coulomb repulsion, in which any one proton is repelled by all the other protons in the nucleus. In fact, the helium nucleus (or α particle) having four nucleons is an extremely stable arrangement, while almost no attractive force is exerted on an additional fifth nucleon, and there is no stable nucleus with five nucleons.

As the number of nucleons in the nucleus increases, a point is finally reached where the Coulomb force, with its longer range and lack of saturation, dominates the short-range, saturating nuclear force. It is essentially for this reason that no stable nuclei exist with more than 84 protons or more than 210 nucleons. Furthermore, with increasing numbers of nucleons, the respulsive Coulomb interactions between the protons in the nucleus become relatively more important, and the most stable nuclei will be found to have slightly more neutrons than protons for a given number of nucleons. The ratio of neutrons to protons is 1.0 for low-mass nuclei and increases to 1.6 for the heaviest nuclei. The addition of more and more neutrons to a given nucleus would seem to increase its stability, since neutrons contribute to the attractive strong nuclear force but are not affected by the repulsive Coulomb force. This effect is limited, however, by the laws of quantum mechanics.

A quantum-mechanical treatment of the nucleus reveals the existence of nuclear energy levels, analogous to the energy levels occupied by electrons in a quantum-mechanical model of the atom as a whole. As with electrons, a quantum treatment of nucleons requires that they be identified as fermions, that is, particles possessing an intrinsic spin of one half. This requirement is important since all fermions obey the Pauli exclusion principle, which requires that no two identical fermions occupy the same energy level in a quantum system. A particular nucleus having a given number of neutrons and protons can then be visualized with the neutrons stacked into the lowest neutron energy levels of that nucleus and with the protons stacked into the lowest proton energy levels of that nucleus. Because of the additional energy required to confine protons in the nucleus against their mutual Coulomb repulsion, the proton energy levels start off higher than their neutron counterparts.

This gap between the bottom of the proton energy-level stack and the bottom of the neutron energy-level stack increases in size as the proton number increases. It is energetically most favorable for the tops of the proton and neutron stacks to be as nearly even as possible. For example, if the last neutron in a nucleus were required to go into a neutron energy level which was much higher than the highest occupied proton level, then the nucleus could increase its binding energy (decrease its potential energy) and enhance its stability by replacing the last neutron with a proton in a lower-energy proton level. Therefore, the energy gap at the bottom of the proton and neutron stacks accounts for the relatively larger number of neutrons, as found especially in nuclei with large numbers of nucleons.

Any nucleus, in its ground state or an excited state, that has too many or too few neutrons relative to protons will be in a state of unstable equilibrium and will undergo a decay of some type. This fact explains why there are only about one hundred elements found in nature, since for eighty-four protons there is no number of neutrons that is capable of making the nucleus stable. It also explains the presence of unstable isotopes neighboring stable isotopes in the lighter elements.

Applications

There are almost two dozen specific mechanisms, each accompanied by the emission of some type of particle, by which nuclei stabilize themselves. Seven of these mechanisms are fairly common: α decay, proton emission, β-negatron decay, β-positron decay, electron capture, γ decay (internal conversion), and spontaneous fission. The first six processes constitute what is usually referred to as radioactivity. Spontaneous fission is a decay mechanism which is energetically favorable for large unstable nuclei. These nuclei are able to reduce the Coulomb repulsion between protons by splitting (usually asymmetrically) into two smaller nuclei that have higher total binding energies than the original nucleus.

When considering an unstable nucleus, the laws of quantum mechanics, which must be used to obtain an accurate quantitative description of the subsequent decay, give only the probabilities of specific processes occurring. Each nucleus of the same isotope has identical decay probabilities. In a large collection of radioactive nuclei, the number that actually decay within some time interval depends only on the length of the time interval and the number of radioactive nuclei present at the beginning of the interval. There is an exponential decrease in the number of radioactive nuclei with time. Equivalently, the rate at which the radioactive decays take place, also known as the activity, decreases exponentially with time. The time required for the original number of radioactive nuclei present (or the original activity of the sample) to decrease by a factor of two is known as the half-life. After the passage of one half-life, there is a fifty-fifty chance that any specific nucleus will have decayed. Of those nuclei that have not yet decayed, each has a fifty-fifty chance of decaying within the next half-life period, and so on.

Thus it is impossible to determine when a given nucleus will decay, and the probability of that decay occurring is independent of the past history of the sample. Furthermore, since the energies typically involved in radioactive decays are so large compared to those associated with thermal agitation, the absorption of light, or chemical interaction, the decay rates are essentially independent of the environment in which the radioactive nucleus finds itself. Half-lives of radioactive substances vary from a few microseconds to more than ten billion years. Many radioactive isotopes can decay by more than one mechanism. When this is the case, each mechanism is indeed observed with its distinct decay probability and associated half-life.

In α decay, the radioactive parent nucleus expels an α particle, or helium nucleus. When the binding energy of the parent nucleus is less than the sum of the binding energies of the daughter nucleus and the α particle, the parent nucleus will be unstable against α decay. Because of the very high binding energy of the α particle, it is possible for α decay to occur for both neutron-rich and neutron-deficient nuclei, which is especially true when the daughter nucleus also has a relatively high binding energy. In fact, because of the extreme stability of the α particle, the decay process can be viewed as an α particle moving about within the parent nucleus under the influence of the forces exerted by the other nucleons. At the nuclear surface, the α particle encounters a potential-energy barrier which its energy is insufficient to overcome. The laws of quantum mechanics predict, however, that the α particle has a small, but non-zero, probability of "tunneling" through the barrier (Gamow tunneling) to appear outside the nuclear surface as an emitted α particle. The actual spectrum of α particles emitted by a given isotope is found to consist of a number of groups, the α particles of each group having a common distinct kinetic energy. These groups arise since the daughter nucleus may be left in an excited state which will subsequently decay by the emission of a γ ray or a conversion electron.

Occasionally, a neutron-deficient isotope may emit a proton, thereby increasing its neutron-to-proton ratio. This process generally occurs only for extremely neutron-deficient nuclei. As with α decay, the proton must tunnel through the potential-energy barrier surrounding the nucleus.

In β decay, which involves the emission of an electron or its antiparticle the positron, the nucleon number of the parent remains unchanged but the proton number is increased or decreased by one. Beta-negatron decay occurs for neutron-rich nuclei and serves to reduce the neutron-proton ratio, while β-positron decay occurs for neutron-deficient nuclei and causes an increase in the neutron-proton ratio. An alternative process to β-positron decay can also occur for nuclei that are neutron-deficient but lack sufficient energy to make β-positron emission possible. This process is known as electron capture. Rather than a proton emitting a positron to become a neutron, a proton can capture one of the atomic electrons from an inner electron shell (usually the K shell) to transform itself into a neutron. Quantum theory predicts a non-zero probability that one of the innermost orbital electrons can be found within the nucleus for the very brief instant needed for this process to occur.

Experimentally, it is found that the energy spectrum of the emitted β particles is continuous, extending from zero up to a specific maximum value. Since the binding energies of the parent and daughter nuclei have fixed values that are the same for each individual decay, it would seem that the conservation of energy law is violated. Additional analysis shows that the conservation of linear momentum and the conservation of angular momentum are also apparently violated. In 1930, Wolfgang Pauli proposed that there must be a third, then unknown, particle which carried away the "missing" energy. This new particle, which was detected in 1956, was the antineutrino. This particle has no electric charge, is massless or nearly massless (so that it travels at or near the speed of light), and is unaffected by the strong nuclear force. These properties account for the fact that neutrinos exhibit almost no interaction with ordinary matter. The existence of antineutrinos explains β-negatron decay. Beta-positron decay cannot be viewed as the exact reverse of the β-negatron decay, however, since this would require the improbable simultaneous encounter of a proton, an electron, and an antineutrino. This apparent paradox is resolved with the existence of the neutrino, whose antiparticle is the antineutrino and which is emitted in β-positron decay.

The theoretical description of the β-decay process was begun in 1934 by Enrico Fermi, who developed a theory of allowed β decays, that is, decays between initial and final states satisfying certain selection rules. In 1941, E. J. Konopinski and G. E. Uhlenbeck extended the theory to include so-called forbidden β decays, which occur much more slowly because they violate the selection rules. The electroweak theory, developed by Steven Weinberg, Sheldon L. Glashow, and Abdus Salam in the late 1960's, gives a more complete picture of all the processes involved in the weak nuclear force, including β decays.

Gamma decay takes place between two excited energy levels of a nucleus, or between an excited level and the ground state of the nucleus. The emitted particle in γ decay is a photon, or quantum of electromagnetic energy. Since a nucleus in its ground state cannot emit a γ ray, γ decay occurs only in conjunction with some other radioactive-decay mechanism, or more generally, in connection with any process that leaves the nucleus in an excited state. The emitted photons have very high energies, being one million to one billion times more energetic than visible-light photons. As a result of the γ decay, the parent nucleus is unchanged except for its lower final energy state. Occasionally, the excitation energy of the nucleus is transferred by the emitted photon directly to one of the atomic electrons, which then has sufficient energy to leave the atom. This process, known as internal conversion, has the outward appearance of the emission of an electron by the radioactive nucleus, except that the identity of the nucleus is unchanged. The atom is left in an ionized state in this process, accounting for the apparent charge imbalance in this reaction.

Context

The discovery of radioactivity is usually credited to the French physicist Antoine-Henri Becquerel. In 1896, Becquerel was investigating fluorescent materials, in particular some compounds that contained uranium, to see if X rays were among the emitted radiation. Becquerel found that the materials emit penetrating radiations even when they are not fluorescing. Marie Curie investigated these penetrating radiations, which were also found to be capable of ionizing air, and designated the phenomenon as radioactivity. Curie also showed that the intensity of the radiations was proportional to the amount of uranium contained in the compound, indicating that it was the uranium atom that was radioactive. She also found a second radioactive element, thorium.

Soon after the discovery of radioactivity, it became apparent that these radiations were not homogeneous. When passed through a magnetic field, the radiations showed three types of behavior: One type was very slightly deflected in one direction, a second type was strongly deflected in the opposite direction, and the third type was undeflected. Ernest Rutherford coined the terms for these three types as "α," "β," and "γ" radiation, respectively. Alpha rays were found to be stopped after penetrating only a few centimeters of air, β rays were capable of penetrating through thin metal foils, and γ rays were as fully penetrating as X rays. Becquerel speculated in 1899 that the β rays were identical to the cathode rays, or electrons, recently discovered by Joseph John Thomson.

Experiments which followed confirmed that β particles are actually high-energy electrons. The γ rays, undeflected in electromagnetic fields, were confirmed to be very high-energy electromagnetic waves, or photons. Alpha rays, with deflection opposite that of the negatively charged β particles, had to be positively charged. In addition, the small amount of deflection suggested that α rays must be much more massive than the β particles. In 1906, Rutherford measured the charge-to-mass ratio of the α particles and found it to be identical to that of a helium nucleus. He confirmed this identification in an experiment which made electrons available to emitted α particles and resulted in the creation of helium atoms.

In the period from 1900 to 1902, experiments performed by several scientists, including Rutherford and Becquerel, showed that highly purified uranium and thorium were only mildly radioactive but that the intensity of the radioactivity increased with the passage of time.

Rutherford and his collaborator, Frederick Soddy, correctly suggested that what was being seen was not a single radioactive isotope, but rather the development of a radioactive decay series in which a long-lived parent isotope produced a daughter which was itself radioactive, and so on. It was some of these daughter isotopes, produced within the radioactive decay series, that were the intensely radioactive sources. Marie and Pierre Curie used chemical techniques to follow the development of the members of the radioactive uranium decay series, and in the process, they discovered two new, highly radioactive, isotopes: polonium and radium. The decay of these two isotopes is so rapid that any quantity of them that was present when the earth formed would have disappeared long since. These isotopes are only found in nature today as a result of the much longer-lived, and less radioactive, uranium and thorium parent isotopes. There are three naturally occurring radioactive series, headed by the long-lived isotopes uranium 238, thorium 232, and uranium 235, each terminating with a stable isotope of lead. These series are known as the uranium series, the thorium series, and the actinium series, respectively, and their parent isotopes have half-lives of 4.5 billion years, 13.9 billion years, and 713.0 million years. Note that each of these half-lives either exceeds, or is of the order of, the age of the earth. These decay series are responsible for the presence of most of the naturally occurring radioactive isotopes.

Finally, in 1934, Frederic Joliot and Irene Joliot-Curie bombarded stable nuclei with α particles and observed that the resulting nuclei were radioactive. This experiment was the first example of artificial radioactivity. Since that time, similar nuclear reactions have been used to produce more than one thousand radioisotopes, including at least one radioisotope of every stable element. All these radioisotopes have relatively short half-lives, and it is for this reason that they are not naturally occurring.

Principal terms

BINDING ENERGY: the energy needed to disassemble a nucleus into its constituent parts

DAUGHTER: the nucleus that remains after radioactive decay has occurred

EQUILIBRIUM: a condition of balance, which may be stable or unstable, existing between a number of forces

HALF-LIFE: the period of time required for one-half of the original number of radioactive nuclei to decay

ISOTOPE: a particular species of a given element which differs from other isotopes by its number of neutrons and is identified by its number of nucleons

NUCLEON: a proton or neutron; the nucleon (or mass) number is the sum of the proton (or atomic) number and the neutron number

NUCLEUS: the central portion of an atom, composed of protons and neutrons, that contains the positive charge of the atom and most of the atomic mass

PARENT: the original radioactive nucleus that decays into a daughter nucleus and emitted particles

RADIOISOTOPE: an artificially produced radioactive isotope

Bibliography

Asimov, Isaac. THE HISTORY OF PHYSICS. New York: Walker, 1984. This comprehensive history of the development of physics includes an excellent section on nuclear structure, radioactivity, and nuclear chemistry in chapters 36 through 40. An easily accessible work which is nevertheless technically thorough.

Davies, P. C. W. THE FORCES OF NATURE. New York: Cambridge University Press, 1979. This relatively short book examines the path taken to the present understanding of the structure of matter. Examines the fundamental forces that operate in the universe. Of particular interest on the topic of radioactivity are the chapters on the strong and weak nuclear forces and the description of nuclear stability as a result of these forces. The treatment is nonmathematical.

Segre, Emilio. FROM X-RAYS TO QUARKS: MODERN PHYSICISTS AND THEIR DISCOVERIES. San Francisco: W. H. Freeman, 1980. This book is evenly divided between an exposition of the discoveries of modern physics and the modern physicists themselves. As the title indicates, the story is taken up just at the point in time when radioactivity was discovered. Chapters 2, 3, and 6 are of special interest to the student of radioactivity. The personal stories of the individuals involved are of great interest.

Weaver, Jefferson, ed. THE WORLD OF PHYSICS. Vol. 2. New York: Simon & Schuster, 1987. This second volume of a three-volume set contains eight articles written by the pioneers of research into radioactivity. Included are articles by Pierre and Marie Curie, Ernest Rutherford, Frederick Soddy, and Frederic Joliot and Irene Joliot-Curie. These articles are written in nontechnical language for the layperson and give interesting insights into the thinking of the people who were attempting to unravel the mystery of radioactivity.

Weinberg, Steven. THE DISCOVERY OF SUBATOMIC PARTICLES. New York: W. H. Freeman, 1990. This book by one of the architects of the electroweak theory has a goal similar to that of P. C. W. Davies' book but focuses on the discovery of the particles that make up the atom and nucleus, as opposed to the forces that act among the particles. In this sense, these two books complement each other.

Models of the Atomic Nucleus

The Structure of the Atomic Nucleus

Fission and Thermonuclear Weapons

Forces on Charges and Currents

The Periodic Table and the Atomic Shell Model

Reactor Fuels and Waste Products