CHAPTER 1
ATOMIC STRUCTURE AND RADIOACTIVE DECAY

1. Objectives
2. Introduction
3. Atomic and nuclear structure
   1. Atomic units
   2. Mass defect and binding energy
   3. Electron energy levels
   4. Nuclear stability
4. Radioactive decay
5. Types of radioactive decay
   1. Alpha decay
   2. Beta decay
   3. Gamma emission and internal conversion
6. Radioactive equilibrium
7. Natural radioactivity and decay series
8. Artificial production of radionuclides
9. Summary
10. Problems
11. References

Objectives

After studying this chapter, the reader should be able to:

• Understand the relationship between nuclear instability and radioactive decay.
• Describe the different modes of radioactive decay and the conditions in which they occur.
• Interpret decay schemes.
• State and use the fundamental equations of radioactive decay.
• Perform elementary computations for sample activities.
• Describe the principles of transient and secular equilibrium.
• Discuss the principles of the artificial production of radionuclides.

Introduction

The composition of matter has puzzled philosophers and scientists for centuries. Even today, the mystery continues as strange new particles are detected in high-energy accelerators. Various models proposed to explain the composition and mechanics of matter are useful in certain applications, but invariably fall short in others. One of the oldest models, the atomic theory of matter devised by early Greek philosophers,1 remains a useful approach to understanding many physical processes, including those important to the study of the physics of radiation therapy. The atomic model is used in this text, but it is important to remember that it is only a model, and that the true composition of matter remains an enigma.

Atomic and nuclear structure

The atom is the smallest unit of matter that possesses the physical and chemical properties characteristic of one of the 118 elements, 92 of which occur naturally and the others are produced artificially. The atom consists of a central positive core, termed the nucleus, surrounded by a cloud of electrons moving in orbits around the nucleus. The nucleus is composed of protons and neutrons, collectively termed nucleons, with a diameter on the order of 10−14 meters (m). Protons are subatomic particles with a mass of 1.6734 × 10−27 kilograms (kg) and a positive charge of +1.6 × 10−19 Coulombs. Neutrons are subatomic particles with a mass of 1.6747 × 10−27 kg and no electrical charge. The electron cloud surrounding the nucleus has a diameter of about 10−10 m.

Electrons have a mass of 9.108 × 10−31 kg and a negative charge of –1.6 × 10−19 Coulombs. In the neutral atom, the number of protons in the nucleus is balanced by an equal number of electrons in the surrounding orbits. An atom with a greater or lesser number of electrons than the number of protons is termed a negative or positive ion.

An atom is characterized by the symbolAZX, in which A is the number of nucleons in the nucleus, Z is the number of protons in the nucleus (or the number of electrons in the neutral atom), and X represents the chemical symbol for the particular element to which the atom belongs. The number of nucleons, A, is termed the mass number of the atom and Z is called the atomic number of the atom. The difference A – Z is the number of neutrons in the nucleus, termed the neutron number, N. Each element has a characteristic atomic number but can have several mass numbers depending on the number of neutrons in the nucleus. For example, the element hydrogen has the unique atomic number of 1, signifying the solitary proton that constitutes the hydrogen nucleus, but can have zero (11H), one (21H), or two (31H) neutrons. The atomic forms 1H, 2H, and 3H (the subscript 1 can be omitted because it is redundant with the chemical symbol) are said to be isotopes of hydrogen because they contain different numbers of neutrons combined with the single proton characteristic of hydrogen. Isotopes of an element have the same Z but different values of A, reflecting a different neutron number, N. Isotones have the same N but different values of Z and A. 3H, 4He, and 5Li are isotones because each nucleus contains two neutrons (N = 2). Isobars have the same A, but different values of Z and N.  3H and  3He are isobars (A = 3). Isomers are different energy states of the same atom and therefore have identical values of Z, N, and A. For example 99mTc and 99Tc are isomers because they are two distinct energy states of the same atom. The m in 99mTc signifies a metastable energy state that exists for a finite time (6 hours half-life) before changing to 99Tc. The term nuclide refers to an atomic nucleus in any form.

Atomic units

Units employed to describe dimensions in the macroscopic world, such as kilograms, Joules, meters, and Coulombs, are too large to use at the atomic level. Units more appropriate for the atomic scale include the atomic mass unit (amu) for mass, electron volt (eV) for energy, nanometer (nm) for distance, and electron charge (e) for electrical charge.

The amu is defined as 1/12 of the mass of an atom of the most common form of carbon, 12C, which has 6 protons, 6 neutrons, and 6 electrons. One amu = 1.66 × 10−27 kg. By definition, the atomic mass of an atom of 12C is 12.00000 amu. In units of amu, the masses of atomic particles are as follows:

Every atom has a characteristic atomic mass, Am. The gram-atomic mass of an isotope is an amount of the isotope in grams that is numerically equivalent to the isotope's atomic mass. For example, one gram-atomic mass of 12C is exactly 12 grams. One gram-atomic mass of any isotope contains 6.0228 × 1023 atoms, which is a constant value that is known as Avogadro's number NA. With these expressions, the following quantities can be computed:

Example 1-1

Compare the number of electrons/g for 12C to the number of electrons/g for 40Ar.

For 12C, the atomic number is 6 and the atomic mass is 12.000. Consequently, the number of electrons/g is 6.0228 × 1023 × 6/12.000 = 3.0114×1023 electrons/g.

For 40Ar, the atomic number is 18 and the atomic mass is 39.948. Consequently, the number of electrons/g is 6.0228 × 1023 × 18/39.948 = 2.714 × 1023 electrons/g.

Note that, although the atomic masses and atomic numbers of carbon and argon are widely different from one another, the electron densities are within 10% of each other. Because for most materials the mass number is approximately twice the atomic number, the electron densities will be relatively constant.

The electron volt (eV) is a unit of energy equal to the kinetic energy of a single electron accelerated through a potential difference (voltage) of 1 volt. One keV = 103 eV and 1 MeV = 106 eV. One nanometer (nm) is 10−9 meters. The electron unit of electrical charge = 1.6 × 10−19 Coulombs. One eV is equal to 1.6 × 10−19 Joules of energy.

Example 1-2

What is the kinetic energy (Ek) of an electron accelerated through a potential difference of 400,000 volts [400 kilovolts (kV)]?

Mass defect and binding energy

The neutral 12C atom contains 6 protons, 6 neutrons, and 6 electrons. The mass of the components of this atom can be computed as:

The mass of an atom of 12C, however, is 12.00000 amu by definition. That is, the sum of the masses of the components of the 12C atom exceeds the actual mass of the atom. There is a mass defect of 0.09888 amu in the 12C atom. The difference in mass must be supplied to separate the 12C atom into its constituents. The mass defect can be described in terms of energy according to Einstein's expression E = mc2, for the equivalence of mass and energy. In this expression, E is energy, m is mass, and c is the speed of light in a vacuum (3 × 108 m/sec). From the formula for mass-energy equivalence, 1 amu of mass is equivalent to 931 MeV of energy. For example, the energy equivalent to the mass of the electron is (0.00055 amu) (931 MeV/amu) = 0.511 MeV.

The energy associated with the mass defect of 12C is (0.09888 amu)(931 MeV/amu) = 92.0 MeV. The energy equivalent to the mass defect of an atom is known as the binding energy of the atom and is the energy required to separate the atom into its constituent parts. Almost all of the binding energy of an atom is associated with the nucleus and reflects the influence of the strong nuclear force that binds particles together in the nucleus. For 12C, the average binding energy per nucleon is 92.0 MeV/12 = 7.67 MeV/nucleon. When computing the average binding energy per nucleon as the...