# Atomic Physics

## Patterns in Spectra Reveal More Quantization

### Learning Objectives

By the end of this section, you will be able to:
1. State and discuss the Zeeman effect.
2. Define orbital magnetic field.
3. Define orbital angular momentum.
4. Define space quantization.
High-resolution measurements of atomic and molecular spectra show that the spectral lines are even more complex than they first appear. In this section, we will see that this complexity has yielded important new information about electrons and their orbits in atoms.
In order to explore the substructure of atoms (and knowing that magnetic fields affect moving charges), the Dutch physicist Hendrik Lorentz (1853–1930) suggested that his student Pieter Zeeman (1865–1943) study how spectra might be affected by magnetic fields. What they found became known as the Zeeman effect, which involved spectral lines being split into two or more separate emission lines by an external magnetic field, as shown in . For their discoveries, Zeeman and Lorentz shared the 1902 Nobel Prize in Physics.
Zeeman splitting is complex. Some lines split into three lines, some into five, and so on. But one general feature is that the amount the split lines are separated is proportional to the applied field strength, indicating an interaction with a moving charge. The splitting means that the quantized energy of an orbit is affected by an external magnetic field, causing the orbit to have several discrete energies instead of one. Even without an external magnetic field, very precise measurements showed that spectral lines are doublets (split into two), apparently by magnetic fields within the atom itself.
Bohr’s theory of circular orbits is useful for visualizing how an electron’s orbit is affected by a magnetic field. The circular orbit forms a current loop, which creates a magnetic field of its own,  as seen in . Note that the orbital magnetic field  and the orbital angular momentum  are along the same line. The external magnetic field and the orbital magnetic field interact; a torque is exerted to align them. A torque rotating a system through some angle does work so that there is energy associated with this interaction. Thus, orbits at different angles to the external magnetic field have different energies. What is remarkable is that the energies are quantized—the magnetic field splits the spectral lines into several discrete lines that have different energies. This means that only certain angles are allowed between the orbital angular momentum and the external field, as seen in .
We already know that the magnitude of angular momentum is quantized for electron orbits in atoms. The new insight is that the direction of the orbital angular momentum is also quantized. The fact that the orbital angular momentum can have only certain directions is called space quantization. Like many aspects of quantum mechanics, this quantization of direction is totally unexpected. On the macroscopic scale, orbital angular momentum, such as that of the moon around the earth, can have any magnitude and be in any direction.
Detailed treatment of space quantization began to explain some complexities of atomic spectra, but certain patterns seemed to be caused by something else. As mentioned, spectral lines are actually closely spaced doublets, a characteristic called fine structure, as shown in . The doublet changes when a magnetic field is applied, implying that whatever causes the doublet interacts with a magnetic field. In 1925, Sem Goudsmit and George Uhlenbeck, two Dutch physicists, successfully argued that electrons have properties analogous to a macroscopic charge spinning on its axis. Electrons, in fact, have an internal or intrinsic angular momentum called intrinsic spin .  Since electrons are charged, their intrinsic spin creates an intrinsic magnetic field , which interacts with their orbital magnetic field . Furthermore, electron, analogous to the situation for orbital angular momentum. The spin of the electron can have only one magnitude, and its direction can be at only one of two angles relative to a magnetic field, as seen in . We refer to this as spin up or spin down for the electron. Each spin direction has a different energy; hence, spectroscopic lines are split into two. Spectral doublets are now understood as being due to electron spin.
These two new insights—that the direction of angular momentum, whether orbital or spin, is quantized, and that electrons have intrinsic spin—help to explain many of the complexities of atomic and molecular spectra. In magnetic resonance imaging, it is the way that the intrinsic magnetic field of hydrogen and biological atoms interact with an external field that underlies the diagnostic fundamentals.

### Section Summary

1. The Zeeman effect—the splitting of lines when a magnetic field is applied—is caused by other quantized entities in atoms.
2. Both the magnitude and direction of orbital angular momentum are quantized.
3. The same is true for the magnitude and direction of the intrinsic spin of electrons.

### Conceptual Questions


# Atomic Physics

## Quantum Numbers and Rules

### Learning Objectives

By the end of this section, you will be able to:
1. Define quantum number.
2. Calculate angle of angular momentum vector with an axis.
3. Define spin quantum number.
Physical characteristics that are quantized—such as energy, charge, and angular momentum—are of such importance that names and symbols are given to them. The values of quantized entities are expressed in terms of quantum numbers, and the rules governing them are of the utmost importance in determining what nature is and does. This section covers some of the more important quantum numbers and rules—all of which apply in chemistry, material science, and far beyond the realm of atomic physics, where they were first discovered. Once again, we see how physics makes discoveries which enable other fields to grow.
The energy states of bound systems are quantized, because the particle wavelength can fit into the bounds of the system in only certain ways. This was elaborated for the hydrogen atom, for which the allowed energies are expressed as , where . We define
 to be the principal quantum number that labels the basic states of a system. The lowest-energy state has
, the first excited state has
, and so on. Thus the allowed values for the principal quantum number are
This is more than just a numbering scheme, since the energy of the system, such as the hydrogen atom, can be expressed as some function of , as can other characteristics (such as the orbital radii of the hydrogen atom).
The fact that the magnitude of angular momentum is quantized was first recognized by Bohr in relation to the hydrogen atom; it is now known to be true in general. With the development of quantum mechanics, it was found that the magnitude of angular momentum  can have only the values
where  is defined to be the angular momentum quantum number. The rule for  in atoms is given in the parentheses. Given , the value of  can be any integer from zero up to . For example, if , then  can be 0, 1, 2, or 3.
Note that for ,  can only be zero. This means that the ground-state angular momentum for hydrogen is actually zero, not

   as Bohr proposed. The picture of circular orbits is not valid, because there would be angular momentum for any circular orbit. A more valid picture is the cloud of probability shown for the ground state of hydrogen in . The electron actually spends time in and near the nucleus. The reason the electron does not remain in the nucleus is related to Heisenberg’s uncertainty principle—the electron’s energy would have to be much too large to be confined to the small space of the nucleus. Now the first excited state of hydrogen has , so that  can be either 0 or 1, according to the rule in  . Similarly, for ,  can be 0, 1, or 2. It is often most convenient to state the value of , a simple integer, rather than calculating the value of  from . For example, for , we see that
It is much simpler to state .
As recognized in the Zeeman effect, the direction of angular momentum is quantized. We now know this is true in all circumstances. It is found that the component of angular momentum along one direction in space, usually called the -axis, can have only certain values of . The direction in space must be related to something physical, such as the direction of the magnetic field at that location. This is an aspect of relativity. Direction has no meaning if there is nothing that varies with direction, as does magnetic force. The allowed values of  are
where  is the  and  is the angular momentum projection quantum number. The rule in parentheses for the values of  is that it can range from  to  in steps of one. For example, if , then  can have the five values –2, –1, 0, 1, and 2. Each  corresponds to a different energy in the presence of a magnetic field, so that they are related to the splitting of spectral lines into discrete parts, as discussed in the preceding section. If the -component of angular momentum can have only certain values, then the angular momentum can have only certain directions, as illustrated in .

### Intrinsic Spin Angular Momentum Is Quantized in Magnitude and Direction

There are two more quantum numbers of immediate concern. Both were first discovered for electrons in conjunction with fine structure in atomic spectra. It is now well established that electrons and other fundamental particles have intrinsic spin, roughly analogous to a planet spinning on its axis. This spin is a fundamental characteristic of particles, and only one magnitude of intrinsic spin is allowed for a given type of particle. Intrinsic angular momentum is quantized independently of orbital angular momentum. Additionally, the direction of the spin is also quantized. It has been found that the magnitude of the intrinsic (internal) spin angular momentum, , of an electron is given by
where  is defined to be the spin quantum number. This is very similar to the quantization of  given in , except that the only value allowed for  for electrons is 1/2.
The direction of intrinsic spin is quantized, just as is the direction of orbital angular momentum. The direction of spin angular momentum along one direction in space, again called the -axis, can have only the values
for electrons.  is the  and  is the spin projection quantum number. For electrons,  can only be 1/2, and  can be either +1/2 or –1/2. Spin projection  is referred to as spin up, whereas  is called spin down. These are illustrated in .
To summarize, the state of a system, such as the precise nature of an electron in an atom, is determined by its particular quantum numbers. These are expressed in the form  —see  For electrons in atoms, the principal quantum number can have the values . Once  is known, the values of the angular momentum quantum number are limited to . For a given value of , the angular momentum projection quantum number can have only the values . Electron spin is independent of  and , always having . The spin projection quantum number can have two values, .
 shows several hydrogen states corresponding to different sets of quantum numbers. Note that these clouds of probability are the locations of electrons as determined by making repeated measurements—each measurement finds the electron in a definite location, with a greater chance of finding the electron in some places rather than others. With repeated measurements, the pattern of probability shown in the figure emerges. The clouds of probability do not look like nor do they correspond to classical orbits. The uncertainty principle actually prevents us and nature from knowing how the electron gets from one place to another, and so an orbit really does not exist as such. Nature on a small scale is again much different from that on the large scale.
We will see that the quantum numbers discussed in this section are valid for a broad range of particles and other systems, such as nuclei. Some quantum numbers, such as intrinsic spin, are related to fundamental classifications of subatomic particles, and they obey laws that will give us further insight into the substructure of matter and its interactions.

### Section Summary

1. Quantum numbers are used to express the allowed values of quantized entities. The principal quantum number  labels the basic states of a system and is given by

2. The magnitude of angular momentum is given by

where  is the angular momentum quantum number. The direction of angular momentum is quantized, in that its component along an axis defined by a magnetic field, called the -axis is given by
 is the -component of the angular momentum and  is the angular momentum projection quantum number. Similarly, the electron’s intrinsic spin angular momentum  is given by
     is defined to be the spin quantum number. Finally, the direction of the electron’s spin along the -axis is given by

where  is the -component of spin angular momentum and  is the spin projection quantum number. Spin projection  is referred to as spin up, whereas  is called spin down.  summarizes the atomic quantum numbers and their allowed values.

### Conceptual Questions

### Problem Exercises


# Atomic Physics

## The Pauli Exclusion Principle

### Learning Objectives

By the end of this section, you will be able to:
1. Define the composition of an atom along with its electrons, neutrons, and protons.
2. Explain the Pauli exclusion principle and its application to the atom.
3. Specify the shell and subshell symbols and their positions.
4. Define the position of electrons in different shells of an atom.
5. State the position of each element in the periodic table according to shell filling.

### Multiple-Electron Atoms

All atoms except hydrogen are multiple-electron atoms. The physical and chemical properties of elements are directly related to the number of electrons a neutral atom has. The periodic table of the elements groups elements with similar properties into columns. This systematic organization is related to the number of electrons in a neutral atom, called the atomic number, . We shall see in this section that the exclusion principle is key to the underlying explanations, and that it applies far beyond the realm of atomic physics.
In 1925, the Austrian physicist Wolfgang Pauli (see ) proposed the following rule: No two electrons can have the same set of quantum numbers. That is, no two electrons can be in the same state. This statement is known as the Pauli exclusion principle, because it excludes electrons from being in the same state. The Pauli exclusion principle is extremely powerful and very broadly applicable. It applies to any identical particles with half-integral intrinsic spin—that is, having  Thus no two electrons can have the same set of quantum numbers.
Let us examine how the exclusion principle applies to electrons in atoms. The quantum numbers involved were defined in Quantum Numbers and Rules as , and . Since  is always  for electrons, it is redundant to list , and so we omit it and specify the state of an electron by a set of four numbers . For example, the quantum numbers  completely specify the state of an electron in an atom.
Since no two electrons can have the same set of quantum numbers, there are limits to how many of them can be in the same energy state. Note that  determines the energy state in the absence of a magnetic field. So we first choose , and then we see how many electrons can be in this energy state or energy level. Consider the  level, for example. The only value  can have is 0 (see  for a list of possible values once  is known), and thus  can only be 0. The spin projection  can be either  or , and so there can be two electrons in the  state. One has quantum numbers , and the other has .  illustrates that there can be one or two electrons having , but not three.

### Shells and Subshells

Because of the Pauli exclusion principle, only hydrogen and helium can have all of their electrons in the  state. Lithium (see the periodic table) has three electrons, and so one must be in the  level. This leads to the concept of shells and shell filling. As we progress up in the number of electrons, we go from hydrogen to helium, lithium, beryllium, boron, and so on, and we see that there are limits to the number of electrons for each value of . Higher values of the shell  correspond to higher energies, and they can allow more electrons because of the various combinations of , and  that are possible. Each value of the principal quantum number  thus corresponds to an atomic shell into which a limited number of electrons can go. Shells and the number of electrons in them determine the physical and chemical properties of atoms, since it is the outermost electrons that interact most with anything outside the atom.
The probability clouds of electrons with the lowest value of  are closest to the nucleus and, thus, more tightly bound. Thus when shells fill, they start with , progress to , and so on. Each value of  thus corresponds to a subshell.
The table given below lists symbols traditionally used to denote shells and subshells.
To denote shells and subshells, we write  with a number for  and a letter for . For example, an electron in the  state must have , and it is denoted as a  electron. Two electrons in the  state is denoted as . Another example is an electron in the  state with , written as . The case of three electrons with these quantum numbers is written . This notation, called spectroscopic notation, is generalized as shown in .
Counting the number of possible combinations of quantum numbers allowed by the exclusion principle, we can determine how many electrons it takes to fill each subshell and shell.
The number of electrons that can be in a subshell depends entirely on the value of . Once  is known, there are a fixed number of values of , each of which can have two values for  First, since  goes from  to l in steps of 1, there are  possibilities. This number is multiplied by 2, since each electron can be spin up or spin down. Thus the maximum number of electrons that can be in a subshell is .
For example, the  subshell in  has a maximum of 2 electrons in it, since  for this subshell. Similarly, the  subshell has a maximum of 6 electrons, since . For a shell, the maximum number is the sum of what can fit in the subshells. Some algebra shows that the maximum number of electrons that can be in a shell is .
For example, for the first shell , and so . We have already seen that only two electrons can be in the  shell. Similarly, for the second shell, , and so . As found in , the total number of electrons in the  shell is 8.

### Shell Filling and the Periodic Table

 shows electron configurations for the first 20 elements in the periodic table, starting with hydrogen and its single electron and ending with calcium. The Pauli exclusion principle determines the maximum number of electrons allowed in each shell and subshell. But the order in which the shells and subshells are filled is complicated because of the large numbers of interactions between electrons.
Examining the above table, you can see that as the number of electrons in an atom increases from 1 in hydrogen to 2 in helium and so on, the lowest-energy shell gets filled first—that is, the  shell fills first, and then the  shell begins to fill. Within a shell, the subshells fill starting with the lowest , or with the  subshell, then the , and so on, usually until all subshells are filled. The first exception to this occurs for potassium, where the  subshell begins to fill before any electrons go into the  subshell. The next exception is not shown in ; it occurs for rubidium, where the  subshell starts to fill before the  subshell. The reason for these exceptions is that  electrons have probability clouds that penetrate closer to the nucleus and, thus, are more tightly bound (lower in energy).
 shows the periodic table of the elements, through element 118. Of special interest are elements in the main groups, namely, those in the columns numbered 1, 2, 13, 14, 15, 16, 17, and 18.
The number of electrons in the outermost subshell determines the atom’s chemical properties, since it is these electrons that are farthest from the nucleus and thus interact most with other atoms. If the outermost subshell can accept or give up an electron easily, then the atom will be highly reactive chemically. Each group in the periodic table is characterized by its outermost electron configuration. Perhaps the most familiar is Group 18 (Group VIII), the noble gases (helium, neon, argon, etc.). These gases are all characterized by a filled outer subshell that is particularly stable. This means that they have large ionization energies and do not readily give up an electron. Furthermore, if they were to accept an extra electron, it would be in a significantly higher level and thus loosely bound. Chemical reactions often involve sharing electrons. Noble gases can be forced into unstable chemical compounds only under high pressure and temperature.
Group 17 (Group VII) contains the halogens, such as fluorine, chlorine, iodine and bromine, each of which has one less electron than a neighboring noble gas. Each halogen has 5  electrons (a   configuration), while the  subshell can hold 6 electrons. This means the halogens have one vacancy in their outermost subshell. They thus readily accept an extra electron (it becomes tightly bound, closing the shell as in noble gases) and are highly reactive chemically. The halogens are also likely to form singly negative ions, such as , fitting an extra electron into the vacancy in the outer subshell. In contrast, alkali metals, such as sodium and potassium, all have a single  electron in their outermost subshell (an

     configuration) and are members of Group 1 (Group I). These elements easily give up their extra electron and are thus highly reactive chemically. As you might expect, they also tend to form singly positive ions, such as , by losing their loosely bound outermost electron. They are metals (conductors), because the loosely bound outer electron can move freely.
Of course, other groups are also of interest. Carbon, silicon, and germanium, for example, have similar chemistries and are in Group 4 (Group IV). Carbon, in particular, is extraordinary in its ability to form many types of bonds and to be part of long chains, such as inorganic molecules. The large group of what are called transitional elements is characterized by the filling of the  subshells and crossing of energy levels. Heavier groups, such as the lanthanide series, are more complex—their shells do not fill in simple order. But the groups recognized by chemists such as Mendeleev have an explanation in the substructure of atoms.

### Section Summary

1. The state of a system is completely described by a complete set of quantum numbers. This set is written as .
2. The Pauli exclusion principle says that no two electrons can have the same set of quantum numbers; that is, no two electrons can be in the same state.
3. This exclusion limits the number of electrons in atomic shells and subshells. Each value of  corresponds to a shell, and each value of  corresponds to a subshell.
4. The maximum number of electrons that can be in a subshell is .
5. The maximum number of electrons that can be in a shell is .

### Conceptual Questions

### Problem Exercises


# Radioactivity and Nuclear Physics

## Introduction to Radioactivity and Nuclear Physics

There is an ongoing quest to find substructures of matter. At one time, it was thought that atoms would be the ultimate substructure, but just when the first direct evidence of atoms was obtained, it became clear that they have a substructure and a tiny nucleus. The nucleus itself has spectacular characteristics. For example, certain nuclei are unstable, and their decay emits radiations with energies millions of times greater than atomic energies. Some of the mysteries of nature, such as why the core of the earth remains molten and how the sun produces its energy, are explained by nuclear phenomena. The exploration of radioactivity and the nucleus revealed fundamental and previously unknown particles, forces, and conservation laws. That exploration has evolved into a search for further underlying structures, such as quarks. In this chapter, the fundamentals of nuclear radioactivity and the nucleus are explored. The following two chapters explore the more important applications of nuclear physics in the field of medicine. We will also explore the basics of what we know about quarks and other substructures smaller than nuclei.


# Radioactivity and Nuclear Physics

## Nuclear Radioactivity

### Learning Objectives

By the end of this section, you will be able to:
1. Explain nuclear radiation.
2. Explain the types of radiation—alpha emission, beta emission, and gamma emission.
3. Explain the ionization of radiation in an atom.
4. Define the range of radiation.
The discovery and study of nuclear radioactivity quickly revealed evidence of revolutionary new physics. In addition, uses for nuclear radiation also emerged quickly—for example, people such as Ernest Rutherford used it to determine the size of the nucleus and devices were painted with radon-doped paint to make them glow in the dark (see ). We therefore begin our study of nuclear physics with the discovery and basic features of nuclear radioactivity.

### Discovery of Nuclear Radioactivity

In 1896, the French physicist Antoine Henri Becquerel (1852–1908) accidentally found that a uranium-rich mineral called pitchblende emits invisible, penetrating rays that can darken a photographic plate enclosed in an opaque envelope. The rays therefore carry energy; but amazingly, the pitchblende emits them continuously without any energy input. This is an apparent violation of the law of conservation of energy, one that we now understand is due to the conversion of a small amount of mass into energy, as related in Einstein’s famous equation . It was soon evident that Becquerel’s rays originate in the nuclei of the atoms and have other unique characteristics. The emission of these rays is called nuclear radioactivity or simply radioactivity. The rays themselves are called nuclear radiation. A nucleus that spontaneously destroys part of its mass to emit radiation is said to decay (a term also used to describe the emission of radiation by atoms in excited states). A substance or object that emits nuclear radiation is said to be radioactive.
Two types of experimental evidence imply that Becquerel’s rays originate deep in the heart (or nucleus) of an atom. First, the radiation is found to be associated with certain elements, such as uranium. Radiation does not vary with chemical state—that is, uranium is radioactive whether it is in the form of an element or compound. In addition, radiation does not vary with temperature, pressure, or ionization state of the uranium atom. Since all of these factors affect electrons in an atom, the radiation cannot come from electron transitions, as atomic spectra do. The huge energy emitted during each event is the second piece of evidence that the radiation cannot be atomic. Nuclear radiation has energies of the order of  per event, which is much greater than the typical atomic energies (a few ), such as that observed in spectra and chemical reactions, and more than ten times as high as the most energetic characteristic x rays. Becquerel did not vigorously pursue his discovery for very long. In 1898, Marie Curie (1867–1934), then a graduate student married to the already well-known French physicist Pierre Curie (1859–1906), began her doctoral study of Becquerel’s rays. She and her husband soon discovered two new radioactive elements, which she named polonium (after her native land) and radium (because it radiates). These two new elements filled holes in the periodic table and, further, displayed much higher levels of radioactivity per gram of material than uranium. Over a period of four years, working under poor conditions and spending their own funds, the Curies processed more than a ton of uranium ore to isolate a gram of radium salt. Radium became highly sought after, because it was about two million times as radioactive as uranium. Curie’s radium salt glowed visibly from the radiation that took its toll on them and other unaware researchers. Shortly after completing her Ph.D., both Curies and Becquerel shared the 1903 Nobel Prize in physics for their work on radioactivity. Pierre was killed in a horse cart accident in 1906, but Marie continued her study of radioactivity for nearly 30 more years. Awarded the 1911 Nobel Prize in chemistry for her discovery of two new elements, she remains the only person to win Nobel Prizes in physics and chemistry. Marie’s radioactive fingerprints on some pages of her notebooks can still expose film, and she suffered from radiation-induced lesions. She died of leukemia likely caused by radiation, but she was active in research almost until her death in 1934. The following year, her daughter and son-in-law, Irene and Frederic Joliot-Curie, were awarded the Nobel Prize in chemistry for their discovery of artificially induced radiation, adding to a remarkable family legacy.

### Alpha, Beta, and Gamma

Research begun by people such as New Zealander Ernest Rutherford soon after the discovery of nuclear radiation indicated that different types of rays are emitted. Eventually, three types were distinguished and named alpha, beta, and gamma, because, like x-rays, their identities were initially unknown.  shows what happens if the rays are passed through a magnetic field. The s are unaffected, while the  s and  s are deflected in opposite directions, indicating the  s are positive, the  s negative, and the  s uncharged. Rutherford used both magnetic and electric fields to show that  s have a positive charge twice the magnitude of an electron, or . In the process, he found the  s charge to mass ratio to be several thousand times smaller than the electron’s. Later on, Rutherford collected  s from a radioactive source and passed an electric discharge through them, obtaining the spectrum of recently discovered helium gas. Among many important discoveries made by Rutherford and his collaborators was the proof that  radiation is the emission of a helium nucleus. Rutherford won the Nobel Prize in chemistry in 1908 for his early work. He continued to make important contributions until his death in 1934.
Other researchers had already proved that  s are negative and have the same mass and same charge-to-mass ratio as the recently discovered electron. By 1902, it was recognized that  radiation is the emission of an electron. Although  s are electrons, they do not exist in the nucleus before it decays and are not ejected atomic electrons—the electron is created in the nucleus at the instant of decay.
Since  s remain unaffected by electric and magnetic fields, it is natural to think they might be photons. Evidence for this grew, but it was not until 1914 that this was proved by Rutherford and collaborators. By scattering  radiation from a crystal and observing interference, they demonstrated that  radiation is the emission of a high-energy photon by a nucleus. In fact,  radiation comes from the de-excitation of a nucleus, just as an x ray comes from the de-excitation of an atom. The names " ray" and "x ray" identify the source of the radiation. At the same energy,  rays and x rays are otherwise identical.

### Ionization and Range

Two of the most important characteristics of , , and  rays were recognized very early. All three types of nuclear radiation produce ionization in materials, but they penetrate different distances in materials—that is, they have different ranges. Let us examine why they have these characteristics and what are some of the consequences.
Like x rays, nuclear radiation in the form of  s,  s, and  s has enough energy per event to ionize atoms and molecules in any material. The energy emitted in various nuclear decays ranges from a few  to more than , while only a few  are needed to produce ionization. The effects of x rays and nuclear radiation on biological tissues and other materials, such as solid state electronics, are directly related to the ionization they produce. All of them, for example, can damage electronics or kill cancer cells. In addition, methods for detecting x rays and nuclear radiation are based on ionization, directly or indirectly. All of them can ionize the air between the plates of a capacitor, for example, causing it to discharge. This is the basis of inexpensive personal radiation monitors, such as pictured in . Apart from , , and , there are other forms of nuclear radiation as well, and these also produce ionization with similar effects. We define ionizing radiation as any form of radiation that produces ionization whether nuclear in origin or not, since the effects and detection of the radiation are related to ionization.
The range of radiation is defined to be the distance it can travel through a material. Range is related to several factors, including the energy of the radiation, the material encountered, and the type of radiation (see ). The higher the energy, the greater the range, all other factors being the same. This makes good sense, since radiation loses its energy in materials primarily by producing ionization in them, and each ionization of an atom or a molecule requires energy that is removed from the radiation. The amount of ionization is, thus, directly proportional to the energy of the particle of radiation, as is its range.

Radiation can be absorbed or shielded by materials, such as the lead aprons dentists drape on us when taking x rays. Lead is a particularly effective shield compared with other materials, such as plastic or air. How does the range of radiation depend on material? Ionizing radiation interacts best with charged particles in a material. Since electrons have small masses, they most readily absorb the energy of the radiation in collisions. The greater the density of a material and, in particular, the greater the density of electrons within a material, the smaller the range of radiation.
Different types of radiation have different ranges when compared at the same energy and in the same material. Alphas have the shortest range, betas penetrate farther, and gammas have the greatest range. This is directly related to charge and speed of the particle or type of radiation. At a given energy, each
,
, or
 will produce the same number of ionizations in a material (each ionization requires a certain amount of energy on average). The more readily the particle produces ionization, the more quickly it will lose its energy. The effect of charge is as follows: The  has a charge of  , the  has a charge of  , and the  is uncharged. The electromagnetic force exerted by the  is thus twice as strong as that exerted by the  and it is more likely to produce ionization. Although chargeless, the  does interact weakly because it is an electromagnetic wave, but it is less likely to produce ionization in any encounter. More quantitatively, the change in momentum  given to a particle in the material is , where  is the force the
,
, or
 exerts over a time
. The smaller the charge, the smaller is  and the smaller is the momentum (and energy) lost. Since the speed of alphas is about 5% to 10% of the speed of light, classical (non-relativistic) formulas apply.
The speed at which they travel is the other major factor affecting the range of  s,  s, and  s. The faster they move, the less time they spend in the vicinity of an atom or a molecule, and the less likely they are to interact. Since  s and  s are particles with mass (helium nuclei and electrons, respectively), their energy is kinetic, given classically by . The mass of the  particle is thousands of times less than that of the  s, so that  s must travel much faster than  s to have the same energy. Since  s move faster (most at relativistic speeds), they have less time to interact than  s. Gamma rays are photons, which must travel at the speed of light. They are even less likely to interact than a , since they spend even less time near a given atom (and they have no charge). The range of  s is thus greater than the range of  s.
Alpha radiation from radioactive sources has a range much less than a millimeter of biological tissues, usually not enough to even penetrate the dead layers of our skin. On the other hand, the same  radiation can penetrate a few centimeters of air, so mere distance from a source prevents  radiation from reaching us. This makes  radiation relatively safe for our body compared to  and  radiation. Typical  radiation can penetrate a few millimeters of tissue or about a meter of air. Beta radiation is thus hazardous even when not ingested. The range of  s in lead is about a millimeter, and so it is easy to store   sources in lead radiation-proof containers. Gamma rays have a much greater range than either s or s. In fact, if a given thickness of material, like a lead brick, absorbs 90% of the s, then a second lead brick will only absorb 90% of what got through the first. Thus, s do not have a well-defined range; we can only cut down the amount that gets through. Typically, s can penetrate many meters of air, go right through our bodies, and are effectively shielded (that is, reduced in intensity to acceptable levels) by many centimeters of lead. One benefit of s is that they can be used as radioactive tracers (see ).

### Test Prep for AP Courses

### Section Summary

1. Some nuclei are radioactive—they spontaneously decay destroying some part of their mass and emitting energetic rays, a process called nuclear radioactivity.
2. Nuclear radiation, like x rays, is ionizing radiation, because energy sufficient to ionize matter is emitted in each decay.
3. The range (or distance traveled in a material) of ionizing radiation is directly related to the charge of the emitted particle and its energy, with greater-charge and lower-energy particles having the shortest ranges.
4. Radiation detectors are based directly or indirectly upon the ionization created by radiation, as are the effects of radiation on living and inert materials.

### Conceptual Questions


# Radioactivity and Nuclear Physics

## Radiation Detection and Detectors

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the working principle of a Geiger tube.
2. Define and discuss radiation detectors.
It is well known that ionizing radiation affects us but does not trigger nerve impulses. Newspapers carry stories about unsuspecting victims of radiation poisoning who fall ill with radiation sickness, such as burns and blood count changes, but who never felt the radiation directly. This makes the detection of radiation by instruments more than an important research tool. This section is a brief overview of radiation detection and some of its applications.

### Human Application

The first direct detection of radiation was Becquerel’s fogged photographic plate. Photographic film is still the most common detector of ionizing radiation, being used routinely in medical and dental x rays. Nuclear radiation is also captured on film, such as seen in . The mechanism for film exposure by ionizing radiation is similar to that by photons. A quantum of energy interacts with the emulsion and alters it chemically, thus exposing the film. The quantum  come from an -particle, -particle, or photon, provided it has more than the few eV of energy needed to induce the chemical change (as does all ionizing radiation). The process is not 100% efficient, since not all incident radiation interacts and not all interactions produce the chemical change. The amount of film darkening is related to exposure, but the darkening also depends on the type of radiation, so that absorbers and other devices must be used to obtain energy, charge, and particle-identification information.
Another very common radiation detector is the Geiger tube. The clicking and buzzing sound we hear in dramatizations and documentaries, as well as in our own physics labs, is usually an audio output of events detected by a Geiger counter. These relatively inexpensive radiation detectors are based on the simple and sturdy Geiger tube, shown schematically in (b). A conducting cylinder with a wire along its axis is filled with an insulating gas so that a voltage applied between the cylinder and wire produces almost no current. Ionizing radiation passing through the tube produces free ion pairs (each pair consisting of one positively charged particle and one negatively charged particle) that are attracted to the wire and cylinder, forming a current that is detected as a count. The word count implies that there is no information on energy, charge, or type of radiation with a simple Geiger counter. They do not detect every particle, since some radiation can pass through without producing enough ionization to be detected. However, Geiger counters are very useful in producing a prompt output that reveals the existence and relative intensity of ionizing radiation.
Another radiation detection method records light produced when radiation interacts with materials. The energy of the radiation is sufficient to excite atoms in a material that may fluoresce, such as the phosphor used by Rutherford’s group. Materials called scintillators use a more complex collaborative process to convert radiation energy into light. Scintillators may be liquid or solid, and they can be very efficient. Their light output can provide information about the energy, charge, and type of radiation. Scintillator light flashes are very brief in duration, enabling the detection of a huge number of particles in short periods of time. Scintillator detectors are used in a variety of research and diagnostic applications. Among these are the detection by satellite-mounted equipment of the radiation from distant galaxies, the analysis of radiation from a person indicating body burdens, and the detection of exotic particles in accelerator laboratories.
Light from a scintillator is converted into electrical signals by devices such as the photomultiplier tube shown schematically in . These tubes are based on the photoelectric effect, which is multiplied in stages into a cascade of electrons, hence the name photomultiplier. Light entering the photomultiplier strikes a metal plate, ejecting an electron that is attracted by a positive potential difference to the next plate, giving it enough energy to eject two or more electrons, and so on. The final output current can be made proportional to the energy of the light entering the tube, which is in turn proportional to the energy deposited in the scintillator. Very sophisticated information can be obtained with scintillators, including energy, charge, particle identification, direction of motion, and so on.
Solid-state radiation detectors convert ionization produced in a semiconductor (like those found in computer chips) directly into an electrical signal. Semiconductors can be constructed that do not conduct current in one particular direction. When a voltage is applied in that direction, current flows only when ionization is produced by radiation, similar to what happens in a Geiger tube. Further, the amount of current in a solid-state detector is closely related to the energy deposited and, since the detector is solid, it can have a high efficiency (since ionizing radiation is stopped in a shorter distance in solids fewer particles escape detection). As with scintillators, very sophisticated information can be obtained from solid-state detectors.

### Section Summary

1. Radiation detectors are based directly or indirectly upon the ionization created by radiation, as are the effects of radiation on living and inert materials.

### Conceptual Questions

### Problems & Exercises


# Radioactivity and Nuclear Physics

## Substructure of the Nucleus

### Learning Objectives

By the end of this section, you will be able to:
1. Define and discuss the nucleus in an atom.
2. Define atomic number.
3. Define and discuss isotopes.
4. Calculate the density of the nucleus.
5. Explain nuclear force.
What is inside the nucleus? Why are some nuclei stable while others decay? (See .) Why are there different types of decay (,  and )? Why are nuclear decay energies so large? Pursuing natural questions like these has led to far more fundamental discoveries than you might imagine.
We have already identified protons as the particles that carry positive charge in the nuclei. However, there are actually two types of particles in the nuclei—the proton and the neutron, referred to collectively as nucleons, the constituents of nuclei. As its name implies, the neutron is a neutral particle () that has nearly the same mass and intrinsic spin as the proton.  compares the masses of protons, neutrons, and electrons. Note how close the proton and neutron masses are, but the neutron is slightly more massive once you look past the third digit. Both nucleons are much more massive than an electron. In fact,  (as noted in Medical Applications of Nuclear Physics and .
 also gives masses in terms of mass units that are more convenient than kilograms on the atomic and nuclear scale. The first of these is the unified  (u), defined as
This unit is defined so that a neutral carbon  atom has a mass of exactly 12 u. Masses are also expressed in units of . These units are very convenient when considering the conversion of mass into energy (and vice versa), as is so prominent in nuclear processes. Using  and units of  in , we find that  cancels and  comes out conveniently in MeV. For example, if the rest mass of a proton is converted entirely into energy, then
It is useful to note that 1 u of mass converted to energy produces 931.5 MeV, or
All properties of a nucleus are determined by the number of protons and neutrons it has. A specific combination of protons and neutrons is called a nuclide and is a unique nucleus. The following notation is used to represent a particular nuclide:
where the symbols , ,  , and  are defined as follows: The number of protons in a nucleus is the atomic number , as defined in Medical Applications of Nuclear Physics. X is the symbol for the element, such as Ca for calcium. However, once  is known, the element is known; hence,  and  are redundant. For example,  is always calcium, and calcium always has .  is the number of neutrons in a nucleus. In the notation for a nuclide, the subscript  is usually omitted. The symbol  is defined as the number of nucleons or the total number of protons and neutrons,
where  is also called the mass number. This name for  is logical; the mass of an atom is nearly equal to the mass of its nucleus, since electrons have so little mass. The mass of the nucleus turns out to be nearly equal to the sum of the masses of the protons and neutrons in it, which is proportional to . In this context, it is particularly convenient to express masses in units of u. Both protons and neutrons have masses close to 1 u, and so the mass of an atom is close to  u. For example, in an oxygen nucleus with eight protons and eight neutrons, , and its mass is 16 u. As noticed, the unified atomic mass unit is defined so that a neutral carbon atom (actually a  atom) has a mass of exactly 12 . Carbon was chosen as the standard, partly because of its importance in organic chemistry (see Appendix A).
Let us look at a few examples of nuclides expressed in the  notation. The nucleus of the simplest atom, hydrogen, is a single proton, or
 (the zero for no neutrons is often omitted). To check this symbol, refer to the periodic table—you see that the atomic number
 of hydrogen is 1. Since you are given that there are no neutrons, the mass number
 is also 1. Suppose you are told that the helium nucleus or
 particle has two protons and two neutrons. You can then see that it is written
. There is a scarce form of hydrogen found in nature called deuterium; its nucleus has one proton and one neutron and, hence, twice the mass of common hydrogen. The symbol for deuterium is, thus,
 (sometimes
 is used, as for deuterated water
). An even rarer—and radioactive—form of hydrogen is called tritium, since it has a single proton and two neutrons, and it is written
. These three varieties of hydrogen have nearly identical chemistries, but the nuclei differ greatly in mass, stability, and other characteristics. Nuclei (such as those of hydrogen) having the same
 and different  s are defined to be isotopes of the same element.
There is some redundancy in the symbols , , , and  . If the element  is known, then  can be found in a periodic table and is always the same for a given element. If both  and  are known, then  can also be determined (first find ; then, ). Thus the simpler notation for nuclides is
which is sufficient and is most commonly used. For example, in this simpler notation, the three isotopes of hydrogen are  and  while the  particle is . We read this backward, saying helium-4 for , or uranium-238 for . So for , should we need to know, we can determine that  for uranium from the periodic table, and, thus, .
A variety of experiments indicate that a nucleus behaves something like a tightly packed ball of nucleons, as illustrated in . These nucleons have large kinetic energies and, thus, move rapidly in very close contact. Nucleons can be separated by a large force, such as in a collision with another nucleus, but resist strongly being pushed closer together. The most compelling evidence that nucleons are closely packed in a nucleus is that the radius of a nucleus, , is found to be given approximately by
where  and  is the mass number of the nucleus. Note that . Since many nuclei are spherical, and the volume of a sphere is , we see that  —that is, the volume of a nucleus is proportional to the number of nucleons in it. This is what would happen if you pack nucleons so closely that there is no empty space between them.
Nucleons are held together by nuclear forces and resist both being pulled apart and pushed inside one another. The volume of the nucleus is the sum of the volumes of the nucleons in it, here shown in different colors to represent protons and neutrons.

### Nuclear Forces and Stability

What forces hold a nucleus together? The nucleus is very small and its protons, being positive, exert tremendous repulsive forces on one another. (The Coulomb force increases as charges get closer, since it is proportional to , even at the tiny distances found in nuclei.) The answer is that two previously unknown forces hold the nucleus together and make it into a tightly packed ball of nucleons. These forces are called the weak and strong nuclear forces. Nuclear forces are so short ranged that they fall to zero strength when nucleons are separated by only a few fm. However, like glue, they are strongly attracted when the nucleons get close to one another. The strong nuclear force is about 100 times more attractive than the repulsive EM force, easily holding the nucleons together. Nuclear forces become extremely repulsive if the nucleons get too close, making nucleons strongly resist being pushed inside one another, something like ball bearings.
The fact that nuclear forces are very strong is responsible for the very large energies emitted in nuclear decay. During decay, the forces do work, and since work is force times the distance (), a large force can result in a large emitted energy. In fact, we know that there are two distinct nuclear forces because of the different types of nuclear decay—the strong nuclear force is responsible for  decay, while the weak nuclear force is responsible for  decay.
The many stable and unstable nuclei we have explored, and the hundreds we have not discussed, can be arranged in a table called the chart of the nuclides, a simplified version of which is shown in . Nuclides are located on a plot of  versus . Examination of a detailed chart of the nuclides reveals patterns in the characteristics of nuclei, such as stability, abundance, and types of decay, analogous to but more complex than the systematics in the periodic table of the elements.
In principle, a nucleus can have any combination of protons and neutrons, but  shows a definite pattern for those that are stable. For low-mass nuclei, there is a strong tendency for  and  to be nearly equal. This means that the nuclear force is more attractive when . More detailed examination reveals greater stability when  and  are even numbers—nuclear forces are more attractive when neutrons and protons are in pairs. For increasingly higher masses, there are progressively more neutrons than protons in stable nuclei. This is due to the ever-growing repulsion between protons. Since nuclear forces are short ranged, and the Coulomb force is long ranged, an excess of neutrons keeps the protons a little farther apart, reducing Coulomb repulsion. Decay modes of nuclides out of the region of stability consistently produce nuclides closer to the region of stability. There are more stable nuclei having certain numbers of protons and neutrons, called magic numbers. Magic numbers indicate a shell structure for the nucleus in which closed shells are more stable. Nuclear shell theory has been very successful in explaining nuclear energy levels, nuclear decay, and the greater stability of nuclei with closed shells. We have been producing ever-heavier transuranic elements since the early 1940s, and we have now produced the element with . There are theoretical predictions of an island of relative stability for nuclei with such high  s.

### Test Prep for AP Courses

### Section Summary

1. Two particles, both called nucleons, are found inside nuclei. The two types of nucleons are protons and neutrons; they are very similar, except that the proton is positively charged while the neutron is neutral. Some of their characteristics are given in  and compared with those of the electron. A mass unit convenient to atomic and nuclear processes is the unified atomic mass unit (u), defined to be

2. A nuclide is a specific combination of protons and neutrons, denoted by
       is the number of protons or atomic number, X is the symbol for the element,  is the number of neutrons, and  is the mass number or the total number of protons and neutrons,

3. Nuclides having the same  but different  are isotopes of the same element.
4. The radius of a nucleus, , is approximately

where . Nuclear volumes are proportional to . There are two nuclear forces, the weak and the strong. Systematics in nuclear stability seen on the chart of the nuclides indicate that there are shell closures in nuclei for values of  and  equal to the magic numbers, which correspond to highly stable nuclei.

### Conceptual Questions

### Problems & Exercises


# Radioactivity and Nuclear Physics

## Nuclear Decay and Conservation Laws

### Learning Objectives

By the end of this section, you will be able to:
1. Define and discuss nuclear decay.
2. State the conservation laws.
3. Explain parent and daughter nucleus.
4. Calculate the energy emitted during nuclear decay.
Nuclear decay has provided an amazing window into the realm of the very small. Nuclear decay gave the first indication of the connection between mass and energy, and it revealed the existence of two of the four basic forces in nature. In this section, we explore the major modes of nuclear decay; and, like those who first explored them, we will discover evidence of previously unknown particles and conservation laws.
Some nuclides are stable, apparently living forever. Unstable nuclides decay (that is, they are radioactive), eventually producing a stable nuclide after many decays. We call the original nuclide the parent and its decay products the daughters. Some radioactive nuclides decay in a single step to a stable nucleus. For example,  is unstable and decays directly to , which is stable. Others, such as , decay to another unstable nuclide, resulting in a decay series in which each subsequent nuclide decays until a stable nuclide is finally produced. The decay series that starts from  is of particular interest, since it produces the radioactive isotopes  and , which the Curies first discovered (see ). Radon gas is also produced ( in the series), an increasingly recognized naturally occurring hazard. Since radon is a noble gas, it emanates from materials, such as soil, containing even trace amounts of  and can be inhaled. The decay of radon and its daughters produces internal damage. The  decay series ends with , a stable isotope of lead.
Note that the daughters of  decay shown in  always have two fewer protons and two fewer neutrons than the parent. This seems reasonable, since we know that  decay is the emission of a  nucleus, which has two protons and two neutrons. The daughters of  decay have one less neutron and one more proton than their parent. Beta decay is a little more subtle, as we shall see. No  decays are shown in the figure, because they do not produce a daughter that differs from the parent.

### Alpha Decay

In alpha decay, a  nucleus simply breaks away from the parent nucleus, leaving a daughter with two fewer protons and two fewer neutrons than the parent (see ). One example of  decay is shown in  for . Another nuclide that undergoes  decay is  The decay equations for these two nuclides are
and
If you examine the periodic table of the elements, you will find that Th has , two fewer than U, which has . Similarly, in the second decay equation, we see that U has two fewer protons than Pu, which has . The general rule for  decay is best written in the format . If a certain nuclide is known to  decay (generally this information must be looked up in a table of isotopes, such as in Appendix B), its  decay equation is
where Y is the nuclide that has two fewer protons than X, such as Th having two fewer than U. So if you were told that   decays and were asked to write the complete decay equation, you would first look up which element has two fewer protons (an atomic number two lower) and find that this is uranium. Then since four nucleons have broken away from the original 239, its atomic mass would be 235.
It is instructive to examine conservation laws related to  decay. You can see from the equation   that total charge is conserved. Linear and angular momentum are conserved, too. Although conserved angular momentum is not of great consequence in this type of decay, conservation of linear momentum has interesting consequences. If the nucleus is at rest when it decays, its momentum is zero. In that case, the fragments must fly in opposite directions with equal-magnitude momenta so that total momentum remains zero. This results in the  particle carrying away most of the energy, as a bullet from a heavy rifle carries away most of the energy of the powder burned to shoot it. Total mass–energy is also conserved: the energy produced in the decay comes from conversion of a fraction of the original mass. As discussed in Atomic Physics, the general relationship is
Here,  is the nuclear reaction energy (the reaction can be nuclear decay or any other reaction), and  is the difference in mass between initial and final products. When the final products have less total mass,  is positive, and the reaction releases energy (is exothermic). When the products have greater total mass, the reaction is endothermic ( is negative) and must be induced with an energy input. For  decay to be spontaneous, the decay products must have smaller mass than the parent.

### Beta Decay

There are actually three types of beta decay. The first discovered was “ordinary” beta decay and is called  decay or electron emission. The symbol  represents an electron emitted in nuclear beta decay. Cobalt-60 is a nuclide that  decays in the following manner:
The neutrino is a particle emitted in beta decay that was unanticipated and is of fundamental importance. The neutrino was not even proposed in theory until more than 20 years after beta decay was known to involve electron emissions. Neutrinos are so difficult to detect that the first direct evidence of them was not obtained until 1953. Neutrinos are nearly massless, have no charge, and do not interact with nucleons via the strong nuclear force. Traveling approximately at the speed of light, they have little time to affect any nucleus they encounter. This is, owing to the fact that they have no charge (and they are not EM waves), they do not interact through the EM force. They do interact via the relatively weak and very short range weak nuclear force. Consequently, neutrinos escape almost any detector and penetrate almost any shielding. However, neutrinos do carry energy, angular momentum (they are fermions with half-integral spin), and linear momentum away from a beta decay. When accurate measurements of beta decay were made, it became apparent that energy, angular momentum, and linear momentum were not accounted for by the daughter nucleus and electron alone. Either a previously unsuspected particle was carrying them away, or three conservation laws were being violated. Wolfgang Pauli made a formal proposal for the existence of neutrinos in 1930. The Italian-born American physicist Enrico Fermi (1901–1954) gave neutrinos their name, meaning little neutral ones, when he developed a sophisticated theory of beta decay (see ). Part of Fermi’s theory was the identification of the weak nuclear force as being distinct from the strong nuclear force and in fact responsible for beta decay. Chinese-born physicist Chien-Shiung Wu, who had developed a number of processes critical to the Manhattan Project and related research, set out to investigate Fermi’s theory and some experiments whose failures had cast the theory in doubt. She first identified a number of flaws in her contemporaries’ methods and materials, and then designed an experimental method that would avoid the same errors. Wu verified Fermi’s theory and went on to establish the core principles of beta decay, which would become critical to further work in nuclear physics.
The neutrino also reveals a new conservation law. There are various families of particles, one of which is the electron family. We propose that the number of members of the electron family is constant in any process or any closed system. In our example of beta decay, there are no members of the electron family present before the decay, but after, there is an electron and a neutrino. So electrons are given an electron family number of . The neutrino in  decay is an electron’s antineutrino, given the symbol , where  is the Greek letter nu, and the subscript e means this neutrino is related to the electron. The bar indicates this is a particle of antimatter. (All particles have antimatter counterparts that are nearly identical except that they have the opposite charge. Antimatter is almost entirely absent on Earth, but it is found in nuclear decay and other nuclear and particle reactions as well as in outer space.) The electron’s antineutrino , being antimatter, has an electron family number of . The total is zero, before and after the decay. The new conservation law, obeyed in all circumstances, states that the total electron family number is constant. An electron cannot be created without also creating an antimatter family member. This law is analogous to the conservation of charge in a situation where total charge is originally zero, and equal amounts of positive and negative charge must be created in a reaction to keep the total zero.
If a nuclide  is known to  decay, then its  decay equation is
where Y is the nuclide having one more proton than X (see ). So if you know that a certain nuclide  decays, you can find the daughter nucleus by first looking up
 for the parent and then determining which element has atomic number
. In the example of the
 decay of
 given earlier, we see that  for Co and
 is Ni. It is as if one of the neutrons in the parent nucleus decays into a proton, electron, and neutrino. In fact, neutrons outside of nuclei do just that—they live only an average of a few minutes and
 decay in the following manner:
We see that charge is conserved in  decay, since the total charge is  before and after the decay. For example, in  decay, total charge is 27 before decay, since cobalt has
. After decay, the daughter nucleus is Ni, which has
, and there is an electron, so that the total charge is also  or 27. Angular momentum is conserved, but not obviously (you have to examine the spins and angular momenta of the final products in detail to verify this). Linear momentum is also conserved, again imparting most of the decay energy to the electron and the antineutrino, since they are of low and zero mass, respectively. Another new conservation law is obeyed here and elsewhere in nature. The total number of nucleons . In
 decay, for example, there are 60 nucleons before and after the decay. Note that total
 is also conserved in
 decay. Also note that the total number of protons changes, as does the total number of neutrons, so that total
 and total  are not conserved in  decay, as they are in  decay. Energy released in  decay can be calculated given the masses of the parent and products.
The second type of beta decay is less common than the first. It is decay. Certain nuclides decay by the emission of a positive electron. This is antielectron or positron decay (see ).

The antielectron is often represented by the symbol , but in beta decay it is written as  to indicate the antielectron was emitted in a nuclear decay. Antielectrons are the antimatter counterpart to electrons, being nearly identical, having the same mass, spin, and so on, but having a positive charge and an electron family number of . When a positron encounters an electron, there is a mutual annihilation in which all the mass of the antielectron-electron pair is converted into pure photon energy. (The reaction, , conserves electron family number as well as all other conserved quantities.) If a nuclide  is known to  decay, then its decay equation is
where Y is the nuclide having one less proton than X (to conserve charge) and  is the symbol for the electron’s neutrino, which has an electron family number of . Since an antimatter member of the electron family (the ) is created in the decay, a matter member of the family (here the ) must also be created. Given, for example, that

 decays, you can write its full decay equation by first finding that  for , so that the daughter nuclide will have
, the atomic number for neon. Thus the  decay equation for  is
In  decay, it is as if one of the protons in the parent nucleus decays into a neutron, a positron, and a neutrino. Protons do not do this outside of the nucleus, and so the decay is due to the complexities of the nuclear force. Note again that the total number of nucleons is constant in this and any other reaction. To find the energy emitted in  decay, you must again count the number of electrons in the neutral atoms, since atomic masses are used. The daughter has one less electron than the parent, and one electron mass is created in the decay. Thus, in  decay,
since we use the masses of neutral atoms.
Electron capture is the third type of beta decay. Here, a nucleus captures an inner-shell electron and undergoes a nuclear reaction that has the same effect as  decay. Electron capture is sometimes denoted by the letters EC. We know that electrons cannot reside in the nucleus, but this is a nuclear reaction that consumes the electron and occurs spontaneously only when the products have less mass than the parent plus the electron. If a nuclide  is known to undergo electron capture, then its electron capture equation is
Any nuclide that can  decay can also undergo electron capture (and often does both). The same conservation laws are obeyed for EC as for  decay. It is good practice to confirm these for yourself.
All forms of beta decay occur because the parent nuclide is unstable and lies outside the region of stability in the chart of nuclides. Those nuclides that have relatively more neutrons than those in the region of stability will  decay to produce a daughter with fewer neutrons, producing a daughter nearer the region of stability. Similarly, those nuclides having relatively more protons than those in the region of stability will  decay or undergo electron capture to produce a daughter with fewer protons, nearer the region of stability.

### Gamma Decay

Gamma decay is the simplest form of nuclear decay—it is the emission of energetic photons by nuclei left in an excited state by some earlier process. Protons and neutrons in an excited nucleus are in higher orbitals, and they fall to lower levels by photon emission (analogous to electrons in excited atoms). Nuclear excited states have lifetimes typically of only about  s, an indication of the great strength of the forces pulling the nucleons to lower states. The  decay equation is simply
where the asterisk indicates the nucleus is in an excited state. There may be one or more  s emitted, depending on how the nuclide de-excites. In radioactive decay,  emission is common and is preceded by  or  decay. For example, when   decays, it most often leaves the daughter nucleus in an excited state, written . Then the nickel nucleus quickly  decays by the emission of two penetrating  s:
These are called cobalt  rays, although they come from nickel—they are used for cancer therapy, for example. It is again constructive to verify the conservation laws for gamma decay. Finally, since  decay does not change the nuclide to another species, it is not prominently featured in charts of decay series, such as that in .
There are other types of nuclear decay, but they occur less commonly than ,

, and  decay. Spontaneous fission is the most important of the other forms of nuclear decay because of its applications in nuclear power and weapons. It is covered in the next chapter.

### Test Prep for AP Courses

### Section Summary

1. When a parent nucleus decays, it produces a daughter nucleus following rules and conservation laws. There are three major types of nuclear decay, called alpha  beta  and gamma . The   decay equation is
2. Nuclear decay releases an amount of energy  related to the mass destroyed  by
3. There are three forms of beta decay. The decay equation is

4. The  decay equation is
5. The electron capture equation is
6.  is an electron,  is an antielectron or positron,  represents an electron’s neutrino, and   is an electron’s antineutrino. In addition to all previously known conservation laws, two new ones arise— conservation of electron family number  and conservation of the total number of nucleons.  The  decay equation is  is a high-energy photon originating in a nucleus.

### Conceptual Questions

### Problems & Exercises

In the following eight problems, write the complete decay equation for the given nuclide in the complete  notation. Refer to the periodic table for values of .
In the following four problems, identify the parent nuclide and write the complete decay equation in the  notation. Refer to the periodic table for values of .


# Radioactivity and Nuclear Physics

## Half-Life and Activity

### Learning Objectives

By the end of this section, you will be able to:
1. Define half-life.
2. Define dating.
3. Calculate age of old objects by radioactive dating.
Unstable nuclei decay. However, some nuclides decay faster than others. For example, radium and polonium, discovered by the Curies, decay faster than uranium. This means they have shorter lifetimes, producing a greater rate of decay. In this section we explore half-life and activity, the quantitative terms for lifetime and rate of decay.

### Half-Life

Why use a term like half-life rather than lifetime? The answer can be found by examining , which shows how the number of radioactive nuclei in a sample decreases with time. The time in which half of the original number of nuclei decay is defined as the half-life, . Half of the remaining nuclei decay in the next half-life. Further, half of that amount decays in the following half-life. Therefore, the number of radioactive nuclei decreases from  to  in one half-life, then to  in the next, and to  in the next, and so on. If  is a large number, then many half-lives (not just two) pass before all of the nuclei decay. Nuclear decay is an example of a purely statistical process. A more precise definition of half-life is that each nucleus has a 50% chance of living for a time equal to one half-life . Thus, if  is reasonably large, half of the original nuclei decay in a time of one half-life. If an individual nucleus makes it through that time, it still has a 50% chance of surviving through another half-life. Even if it happens to make it through hundreds of half-lives, it still has a 50% chance of surviving through one more. The probability of decay is the same no matter when you start counting. This is like random coin flipping. The chance of heads is 50%, no matter what has happened before.
There is a tremendous range in the half-lives of various nuclides, from as short as  s for the most unstable, to more than  y for the least unstable, or about 46 orders of magnitude. Nuclides with the shortest half-lives are those for which the nuclear forces are least attractive, an indication of the extent to which the nuclear force can depend on the particular combination of neutrons and protons. The concept of half-life is applicable to other subatomic particles, as will be discussed in Particle Physics. It is also applicable to the decay of excited states in atoms and nuclei. The following equation gives the quantitative relationship between the original number of nuclei present at time zero () and the number () at a later time :
where  is the base of the natural logarithm, and  is the decay constant for the nuclide. The shorter the half-life, the larger is the value of , and the faster the exponential  decreases with time. The relationship between the decay constant  and the half-life  is
To see how the number of nuclei declines to half its original value in one half-life, let  in the exponential in the equation . This gives . For integral numbers of half-lives, you can just divide the original number by 2 over and over again, rather than using the exponential relationship. For example, if ten half-lives have passed, we divide  by 2 ten times. This reduces it to . For an arbitrary time, not just a multiple of the half-life, the exponential relationship must be used.
Radioactive dating is a clever use of naturally occurring radioactivity. Its most famous application is carbon-14 dating. Carbon-14 has a half-life of 5730 years and is produced in a nuclear reaction induced when solar neutrinos strike  in the atmosphere. Radioactive carbon has the same chemistry as stable carbon, and so it mixes into the ecosphere, where it is consumed and becomes part of every living organism. Carbon-14 has an abundance of 1.3 parts per trillion of normal carbon. Thus, if you know the number of carbon nuclei in an object (perhaps determined by mass and Avogadro’s number), you multiply that number by  to find the number of
 nuclei in the object. When an organism dies, carbon exchange with the environment ceases, and
 is not replenished as it decays. By comparing the abundance of
 in an artifact, such as mummy wrappings, with the normal abundance in living tissue, it is possible to determine the artifact’s age (or time since death). Carbon-14 dating can be used for biological tissues as old as 50 or 60 thousand years, but is most accurate for younger samples, since the abundance of
 nuclei in them is greater. Very old biological materials contain no
 at all. There are instances in which the date of an artifact can be determined by other means, such as historical knowledge or tree-ring counting. These cross-references have confirmed the validity of carbon-14 dating and permitted us to calibrate the technique as well. Carbon-14 dating revolutionized parts of archaeology and is of such importance that it earned the 1960 Nobel Prize in chemistry for its developer, the American chemist Willard Libby (1908–1980).
One of the most famous cases of carbon-14 dating involves the Shroud of Turin, a long piece of fabric purported to be the burial shroud of Jesus (see ). This relic was first displayed in Turin in 1354 and was denounced as a fraud at that time by a French bishop. Its remarkable negative imprint of an apparently crucified body resembles the then-accepted image of Jesus, and so the shroud was never disregarded completely and remained controversial over the centuries. Carbon-14 dating was not performed on the shroud until 1988, when the process had been refined to the point where only a small amount of material needed to be destroyed. Samples were tested at three independent laboratories, each being given four pieces of cloth, with only one unidentified piece from the shroud, to avoid prejudice. All three laboratories found samples of the shroud contain 92% of the  found in living tissues, allowing the shroud to be dated (see ).
There are other forms of radioactive dating. Rocks, for example, can sometimes be dated based on the decay of . The decay series for  ends with , so that the ratio of these nuclides in a rock is an indication of how long it has been since the rock solidified. The original composition of the rock, such as the absence of lead, must be known with some confidence. However, as with carbon-14 dating, the technique can be verified by a consistent body of knowledge. Since  has a half-life of  y, it is useful for dating only very old materials, showing, for example, that the oldest rocks on Earth solidified about  years ago.

### Activity, the Rate of Decay

What do we mean when we say a source is highly radioactive? Generally, this means the number of decays per unit time is very high. We define activity   to be the rate of decay expressed in decays per unit time. In equation form, this is
where  is the number of decays that occur in time . The SI unit for activity is one decay per second and is given the name becquerel (Bq) in honor of the discoverer of radioactivity. That is,
Activity  is often expressed in other units, such as decays per minute or decays per year. One of the most common units for activity is the curie (Ci), defined to be the activity of 1 g of , in honor of Marie Curie’s work with radium. The definition of curie is
or  decays per second. A curie is a large unit of activity, while a becquerel is a relatively small unit. . In countries like Australia and New Zealand that adhere more to SI units, most radioactive sources, such as those used in medical diagnostics or in physics laboratories, are labeled in Bq or megabecquerel (MBq).
Intuitively, you would expect the activity of a source to depend on two things: the amount of the radioactive substance present, and its half-life. The greater the number of radioactive nuclei present in the sample, the more will decay per unit of time. The shorter the half-life, the more decays per unit time, for a given number of nuclei. So activity  should be proportional to the number of radioactive nuclei, , and inversely proportional to their half-life, . In fact, your intuition is correct. It can be shown that the activity of a source is
where  is the number of radioactive nuclei present, having half-life . This relationship is useful in a variety of calculations, as the next two examples illustrate.
Human-made (or artificial) radioactivity has been produced for decades and has many uses. Some of these include medical therapy for cancer, medical imaging and diagnostics, and food preservation by irradiation. Many applications as well as the biological effects of radiation are explored in Medical Applications of Nuclear Physics, but it is clear that radiation is hazardous. A number of tragic examples of this exist, one of the most disastrous being the meltdown and fire at the Chernobyl reactor complex in the Ukraine (see ). Several radioactive isotopes were released in huge quantities, contaminating many thousands of square kilometers and directly affecting hundreds of thousands of people. The most significant releases were of , , , , , and . Estimates are that the total amount of radiation released was about 100 million curies.

### Human and Medical Applications

Activity  decreases in time, going to half its original value in one half-life, then to one-fourth its original value in the next half-life, and so on. Since , the activity decreases as the number of radioactive nuclei decreases. The equation for  as a function of time is found by combining the equations  and , yielding
where  is the activity at . This equation shows exponential decay of radioactive nuclei. For example, if a source originally has a 1.00-mCi activity, it declines to 0.500 mCi in one half-life, to 0.250 mCi in two half-lives, to 0.125 mCi in three half-lives, and so on. For times other than whole half-lives, the equation  must be used to find .

### Test Prep for AP Courses

### Section Summary

1. Half-life  is the time in which there is a 50% chance that a nucleus will decay. The number of nuclei  as a function of time is

where  is the number present at , and  is the decay constant, related to the half-life by

2. One of the applications of radioactive decay is radioactive dating, in which the age of a material is determined by the amount of radioactive decay that occurs. The rate of decay is called the activity :
3. The SI unit for  is the becquerel (Bq), defined by
4.  is also expressed in terms of curies (Ci), where

5. The activity  of a source is related to  and  by
6. Since  has an exponential behavior as in the equation , the activity also has an exponential behavior, given by
where  is the activity at .

### Conceptual Questions

### Problems & Exercises

Data from the appendices and the periodic table may be needed for these problems.


# Radioactivity and Nuclear Physics

## Binding Energy

### Learning Objectives

By the end of this section, you will be able to:
1. Define and discuss binding energy.
2. Calculate the binding energy per nucleon of a particle.
The more tightly bound a system is, the stronger the forces that hold it together and the greater the energy required to pull it apart. We can therefore learn about nuclear forces by examining how tightly bound the nuclei are. We define the binding energy (BE) of a nucleus to be the energy required to completely disassemble it into separate protons and neutrons. We can determine the BE of a nucleus from its rest mass. The two are connected through Einstein’s famous relationship . A bound system has a smaller mass than its separate constituents; the more tightly the nucleons are bound together, the smaller the mass of the nucleus.
Imagine pulling a nuclide apart as illustrated in . Work done to overcome the nuclear forces holding the nucleus together puts energy into the system. By definition, the energy input equals the binding energy BE. The pieces are at rest when separated, and so the energy put into them increases their total rest mass compared with what it was when they were glued together as a nucleus. That mass increase is thus . This difference in mass is known as mass defect. It implies that the mass of the nucleus is less than the sum of the masses of its constituent protons and neutrons. A nuclide  has  protons and  neutrons, so that the difference in mass is
Thus,
where  is the mass of the nuclide ,  is the mass of a proton, and  is the mass of a neutron. Traditionally, we deal with the masses of neutral atoms. To get atomic masses into the last equation, we first add  electrons to , which gives , the atomic mass of the nuclide. We then add  electrons to the  protons, which gives , or  times the mass of a hydrogen atom. Thus the binding energy of a nuclide  is
The atomic masses can be found in Appendix A, most conveniently expressed in unified atomic mass units u (). BE is thus calculated from known atomic masses.
What patterns and insights are gained from an examination of the binding energy of various nuclides? First, we find that BE is approximately proportional to the number of nucleons  in any nucleus. About twice as much energy is needed to pull apart a nucleus like  compared with pulling apart , for example. To help us look at other effects, we divide BE by  and consider the binding energy per nucleon, . The graph of  in  reveals some very interesting aspects of nuclei. We see that the binding energy per nucleon averages about 8 MeV, but is lower for both the lightest and heaviest nuclei. This overall trend, in which nuclei with  equal to about 60 have the greatest  and are thus the most tightly bound, is due to the combined characteristics of the attractive nuclear forces and the repulsive Coulomb force. It is especially important to note two things—the strong nuclear force is about 100 times stronger than the Coulomb force, and the nuclear forces are shorter in range compared to the Coulomb force. So, for low-mass nuclei, the nuclear attraction dominates and each added nucleon forms bonds with all others, causing progressively heavier nuclei to have progressively greater values of . This continues up to , roughly corresponding to the mass number of iron. Beyond that, new nucleons added to a nucleus will be too far from some others to feel their nuclear attraction. Added protons, however, feel the repulsion of all other protons, since the Coulomb force is longer in range. Coulomb repulsion grows for progressively heavier nuclei, but nuclear attraction remains about the same, and so  becomes smaller. This is why stable nuclei heavier than  have more neutrons than protons. Coulomb repulsion is reduced by having more neutrons to keep the protons farther apart (see ).

There are some noticeable spikes on the  graph, which represent particularly tightly bound nuclei. These spikes reveal further details of nuclear forces, such as confirming that closed-shell nuclei (those with magic numbers of protons or neutrons or both) are more tightly bound. The spikes also indicate that some nuclei with even numbers for  and , and with , are exceptionally tightly bound. This finding can be correlated with some of the cosmic abundances of the elements. The most common elements in the universe, as determined by observations of atomic spectra from outer space, are hydrogen, followed by , with much smaller amounts of  and other elements. It should be noted that the heavier elements are created in supernova explosions, while the lighter ones are produced by nuclear fusion during the normal life cycles of stars, as will be discussed in subsequent chapters. The most common elements have the most tightly bound nuclei. It is also no accident that one of the most tightly bound light nuclei is , emitted in  decay.
There is more to be learned from nuclear binding energies. The general trend in  is fundamental to energy production in stars, and to fusion and fission energy sources on Earth, for example. This is one of the applications of nuclear physics covered in Medical Applications of Nuclear Physics. The abundance of elements on Earth, in stars, and in the universe as a whole is related to the binding energy of nuclei and has implications for the continued expansion of the universe.

### Problem-Solving Strategies

### For Reaction And Binding Energies and Activity Calculations in Nuclear Physics

1.   Identify exactly what needs to be determined in the problem (identify the unknowns). This will allow you to decide whether the energy of a decay or nuclear reaction is involved, for example, or whether the problem is primarily concerned with activity (rate of decay).
2. Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
3.  For reaction and binding-energy problems, we use atomic rather than nuclear masses. Since the masses of neutral atoms are used, you must count the number of electrons involved. If these do not balance (such as in  decay), then an energy adjustment of 0.511 MeV per electron must be made. Also note that atomic masses may not be given in a problem; they can be found in tables.
4. For problems involving activity, the relationship of activity to half-life, and the number of nuclei given in the equation  Owing to the fact that number of nuclei is involved, you will also need to be familiar with moles and Avogadro’s number.
5. Perform the desired calculation; keep careful track of plus and minus signs as well as powers of 10.
6. Check the answer to see if it is reasonable: Does it make sense? Compare your results with worked examples and other information in the text. (Heeding the advice in Step 5 will also help you to be certain of your result.) You must understand the problem conceptually to be able to determine whether the numerical result is reasonable.

### Test Prep for AP Courses

### Section Summary

1. The binding energy (BE) of a nucleus is the energy needed to separate it into individual protons and neutrons. In terms of atomic masses,
where  is the mass of a hydrogen atom,  is the atomic mass of the nuclide, and  is the mass of a neutron. Patterns in the binding energy per nucleon, , reveal details of the nuclear force. The larger the , the more stable the nucleus.

### Conceptual Questions

### Problems & Exercises


# Radioactivity and Nuclear Physics

## Tunneling

### Learning Objectives

By the end of this section, you will be able to:
1. Define and discuss tunneling.
2. Define potential barrier.
3. Explain quantum tunneling.
Protons and neutrons are bound inside nuclei, that means energy must be supplied to break them away. The situation is analogous to a marble in a bowl that can roll around but lacks the energy to get over the rim. It is bound inside the bowl (see ). If the marble could get over the rim, it would gain kinetic energy by rolling down outside. However classically, if the marble does not have enough kinetic energy to get over the rim, it remains forever trapped in its well.
In a nucleus, the attractive nuclear potential is analogous to the bowl at the top of a volcano (where the “volcano” refers only to the shape). Protons and neutrons have kinetic energy, but it is about 8 MeV less than that needed to get out (see ). That is, they are bound by an average of 8 MeV per nucleon. The slope of the hill outside the bowl is analogous to the repulsive Coulomb potential for a nucleus, such as for an  particle outside a positive nucleus. In  decay, two protons and two neutrons spontaneously break away as a  unit. Yet the protons and neutrons do not have enough kinetic energy to get over the rim. So how does the  particle get out?
The answer was supplied in 1928 by the Russian physicist George Gamow (1904–1968). The  particle tunnels through a region of space it is forbidden to be in, and it comes out of the side of the nucleus. Like an electron making a transition between orbits around an atom, it travels from one point to another without ever having been in between.  indicates how this works. The wave function of a quantum mechanical particle varies smoothly, going from within an atomic nucleus (on one side of a potential energy barrier) to outside the nucleus (on the other side of the potential energy barrier). Inside the barrier, the wave function does not become zero but decreases exponentially, and we do not observe the particle inside the barrier. The probability of finding a particle is related to the square of its wave function, and so there is a small probability of finding the particle outside the barrier, which implies that the particle can tunnel through the barrier. This process is called barrier penetration or quantum mechanical tunneling. This concept was developed in theory by J. Robert Oppenheimer (who led the development of the first nuclear bombs during World War II) and was used by Gamow and others to describe  decay.
Good ideas explain
          more than one thing. In addition to qualitatively explaining how the four nucleons in an  particle can get out of the nucleus, the detailed theory also explains quantitatively the half-life of various nuclei that undergo  decay. This description is what Gamow and others devised, and it works for  decay half-lives that vary by 17 orders of magnitude. Experiments have shown that the more energetic the  decay of a particular nuclide is, the shorter is its half-life. Tunneling explains this in the following manner: For the decay to be more energetic, the nucleons must have more energy in the nucleus and should be able to ascend a little closer to the rim. The barrier is therefore not as thick for more energetic decay, and the exponential decrease of the wave function inside the barrier is not as great. Thus the probability of finding the particle outside the barrier is greater, and the half-life is shorter.
Tunneling as an effect also occurs in quantum mechanical systems other than nuclei. Electrons trapped in solids can tunnel from one object to another if the barrier between the objects is thin enough. The process is the same in principle as described for  decay. It is far more likely for a thin barrier than a thick one. Scanning tunneling electron microscopes function on this principle. The current of electrons that travels between a probe and a sample tunnels through a barrier and is very sensitive to its thickness, allowing detection of individual atoms as shown in .

### Section Summary

1. Tunneling is a quantum mechanical process of potential energy barrier penetration. The concept was first applied to explain  decay, but tunneling is found to occur in other quantum mechanical systems.

### Conceptual Questions

### Problems-Exercises


# Medical Applications of Nuclear Physics

## Introduction to Applications of Nuclear Physics

Applications of nuclear physics have become an integral part of modern life. From the bone scan that detects a cancer to the radioiodine treatment that cures another, nuclear radiation has diagnostic and therapeutic effects on medicine. From the fission power reactor to the hope of controlled fusion, nuclear energy is now commonplace and is a part of our plans for the future. Yet, the destructive potential of nuclear weapons haunts us, as does the possibility of nuclear reactor accidents. Certainly, several applications of nuclear physics escape our view, as seen in . Not only has nuclear physics revealed secrets of nature, it has an inevitable impact based on its applications, as they are intertwined with human values. Because of its potential for alleviation of suffering, and its power as an ultimate destructor of life, nuclear physics is often viewed with ambivalence. But it provides perhaps the best example that applications can be good or evil, while knowledge itself is neither.