# 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

## Connection for AP® Courses

Applications of nuclear physics have become an integral part of modern life. From a bone scan that detects a cancer to a radioiodine treatment that cures another, nuclear radiation has many diagnostic and therapeutic applications in medicine. In addition nuclear radiation is used in other useful scanning applications. The fission power reactor and the hope of controlled fusion have made nuclear energy a part of our plans for the future. That said, the destructive potential of nuclear weapons haunts us, as does the possibility of nuclear reactor accidents.
Nuclear physics revealed many secrets of nature, but full exploitation of the technology remains controversial as it is intertwined with human values. Because of its great potential for alleviation of suffering and its power as a giant-scale destroyer of life, nuclear physics is typically viewed with ambivalence. Nuclear physics is a classic example of the truism that applications of technology can be good or evil, but knowledge itself is neither.
This chapter focuses on medical applications of nuclear physics. The sections on fusion and fission address the ideas that objects and systems have properties, such as mass (Big Idea 1), and that interactions between systems can result in changes in those systems (Big Idea 4). The changes that occur as a result of interactions always satisfy conservation laws (Big Idea 5). The mass conservation (Enduring Understanding 1.C) and energy conservation (Enduring Understanding 5.B) are replaced by the law of conservation of mass-energy.
In nuclear fusion and fission reactions, so much potential energy is lost that the mass of the products of a reaction are measurably less than the mass of the reactants (Essential Knowledge 1.C.4, Essential Knowledge 4.C.4) in accordance with the equation
 . This equation explains that mass is part of the internal energy of an object or system (Essential Knowledge 5.B.11). In addition, the number of nucleons is conserved in these nuclear reactions (Enduring Understanding 5.G), and that determines which nuclear reactions are possible (Essential Knowledge 5.G.1).
Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure.
Enduring Understanding 1.C Objects and systems have properties of inertial mass and gravitational mass that are experimentally verified to be the same and that satisfy conservation principles.
Essential Knowledge 1.C.4 In certain processes, mass can be converted to energy and energy can be converted to mass according to
 , the equation derived from the theory of special relativity.
Big Idea 4 Interactions between systems can result in changes in those systems.
Enduring Understanding 4.C Interactions with other objects or systems can change the total energy of a system.
Essential Knowledge 4.C.4 Mass can be converted into energy and energy can be converted into mass.
Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws.
Enduring Understanding 5.B The energy of a system is conserved.
Essential Knowledge 5.B.11 Beyond the classical approximation, mass is actually part of the internal energy of an object or system with
 .
Enduring Understanding 5.G Nucleon number is conserved.
Essential Knowledge 5.G.1 The possible nuclear reactions are constrained by the law of conservation of nucleon number.


# Medical Applications of Nuclear Physics

## Diagnostics and Medical Imaging

### Learning Objectives

By the end of this section, you will be able to:
1. Explain the working principle behind an anger camera.
2. Describe the SPECT and PET imaging techniques.
Most medical and related applications of nuclear physics are driven, at their core, by the difference between a radioactive substance and a non-radioactive substance. One of the first such methods is the precision measurement and detection method known as radioimmunoassay (RIA). Developed by Rosalyn Sussman Yalow and Solomon Berson in the late 1950s, RIA relies on the principle of competitive binding. For the particular substance being measured, a sample containing a radioactive isotope is prepared. A known quantity of antibodies is then introduced. By measuring the amount of "unbound" antibodies after the reaction, technicians can detect and measure the precise amount of the target substance. Radioimmunoassay is essential in cancer screening, hepatitis diagnosis, narcotics investigation, and other analyses.
A host of medical imaging techniques employ nuclear radiation. What makes nuclear radiation so useful? First,  radiation can easily penetrate tissue; hence, it is a useful probe to monitor conditions inside the body. Second, nuclear radiation depends on the nuclide and not on the chemical compound it is in, so that a radioactive nuclide can be put into a compound designed for specific purposes. The compound is said to be tagged. A tagged compound used for medical purposes is called a radiopharmaceutical. Radiation detectors external to the body can determine the location and concentration of a radiopharmaceutical to yield medically useful information. For example, certain drugs are concentrated in inflamed regions of the body, and this information can aid diagnosis and treatment as seen in . Another application utilizes a radiopharmaceutical which the body sends to bone cells, particularly those that are most active, to detect cancerous tumors or healing points. Images can then be produced of such bone scans. Radioisotopes are also used to determine the functioning of body organs, such as blood flow, heart muscle activity, and iodine uptake in the thyroid gland.

### Medical Application

 lists certain medical diagnostic uses of radiopharmaceuticals, including isotopes and activities that are typically administered. Many organs can be imaged with a variety of nuclear isotopes replacing a stable element by a radioactive isotope. One common diagnostic employs iodine to image the thyroid, since iodine is concentrated in that organ. The most active thyroid cells, including cancerous cells, concentrate the most iodine and, therefore, emit the most radiation. Conversely, hypothyroidism is indicated by lack of iodine uptake. Note that there is more than one isotope that can be used for several types of scans. Another common nuclear diagnostic is the thallium scan for the cardiovascular system, particularly used to evaluate blockages in the coronary arteries and examine heart activity. The salt TlCl can be used, because it acts like NaCl and follows the blood. Gallium-67 accumulates where there is rapid cell growth, such as in tumors and sites of infection. Hence, it is useful in cancer imaging. Usually, the patient receives the injection one day and has a whole body scan 3 or 4 days later because it can take several days for the gallium to build up.
Note that  lists many diagnostic uses for , where “m” stands for a metastable state of the technetium nucleus. Perhaps 80 percent of all radiopharmaceutical procedures employ  because of its many advantages. One is that the decay of its metastable state produces a single, easily identified 0.142-MeV

 ray. Additionally, the radiation dose to the patient is limited by the short 6.0-h half-life of

. And, although its half-life is short, it is easily and continuously produced on site. The basic process for production is neutron activation of molybdenum, which quickly

  decays into

. Technetium-99m can be attached to many compounds to allow the imaging of the skeleton, heart, lungs, kidneys, etc.
 shows one of the simpler methods of imaging the concentration of nuclear activity, employing a device called an Anger camera or gamma camera. A piece of lead with holes bored through it collimates  rays emerging from the patient, allowing detectors to receive  rays from specific directions only. The computer analysis of detector signals produces an image. One of the disadvantages of this detection method is that there is no depth information (i.e., it provides a two-dimensional view of the tumor as opposed to a three-dimensional view), because radiation from any location under that detector produces a signal.
Imaging techniques much like those in x-ray computed tomography (CT) scans use nuclear activity in patients to form three-dimensional images.  shows a patient in a circular array of detectors that may be stationary or rotated, with detector output used by a computer to construct a detailed image. This technique is called single-photon-emission computed tomography(SPECT) or sometimes simply SPET. The spatial resolution of this technique is poor, about 1 cm, but the contrast (i.e. the difference in visual properties that makes an object distinguishable from other objects and the background) is good.
Images produced by
 emitters have become important in recent years. When the emitted positron (
) encounters an electron, mutual annihilation occurs, producing two  rays. These  rays have identical 0.511-MeV energies (the energy comes from the destruction of an electron or positron mass) and they move directly away from one another, allowing detectors to determine their point of origin accurately, as shown in . The system is called positron emission tomography (PET). It requires detectors on opposite sides to simultaneously (i.e., at the same time) detect photons of 0.511-MeV energy and utilizes computer imaging techniques similar to those in SPECT and CT scans. Examples of
 -emitting isotopes used in PET are

,

,

, and

, as seen in . This list includes C, N, and O, and so they have the advantage of being able to function as tags for natural body compounds. Its resolution of 0.5 cm is better than that of SPECT; the accuracy and sensitivity of PET scans make them useful for examining the brain’s anatomy and function. The brain’s use of oxygen and water can be monitored with
. PET is used extensively for diagnosing brain disorders. It can note decreased metabolism in certain regions prior to a confirmation of Alzheimer’s disease. PET can locate regions in the brain that become active when a person carries out specific activities, such as speaking, closing their eyes, and so on.

### Section Summary

1. Radiopharmaceuticals are compounds that are used for medical imaging and therapeutics.
2. The process of attaching a radioactive substance is called tagging.
3.  lists certain diagnostic uses of radiopharmaceuticals including the isotope and activity typically used in diagnostics.
4. One common imaging device is the Anger camera, which consists of a lead collimator, radiation detectors, and an analysis computer.
5. Tomography performed with -emitting radiopharmaceuticals is called SPECT and has the advantages of x-ray CT scans coupled with organ- and function-specific drugs.
6. PET is a similar technique that uses  emitters and detects the two annihilation  rays, which aid to localize the source.

### Conceptual Questions

### Problems & Exercises


# Medical Applications of Nuclear Physics

## Biological Effects of Ionizing Radiation

### Learning Objectives

By the end of this section, you will be able to:
1. Define various units of radiation.
2. Describe RBE.
We hear many seemingly contradictory things about the biological effects of ionizing radiation. It can cause cancer, burns, and hair loss, yet it is used to treat and even cure cancer. How do we understand these effects? Once again, there is an underlying simplicity in nature, even in complicated biological organisms. All the effects of ionizing radiation on biological tissue can be understood by knowing that ionizing radiation affects molecules within cells, particularly DNA molecules.
Let us take a brief look at molecules within cells and how cells operate. Cells have long, double-helical DNA molecules containing chemical codes called genetic codes that govern the function and processes undertaken by the cell. It is for unraveling the double-helical structure of DNA that James Watson, Francis Crick, and Maurice Wilkins received the Nobel Prize. Damage to DNA consists of breaks in chemical bonds or other changes in the structural features of the DNA chain, leading to changes in the genetic code. In human cells, we can have as many as a million individual instances of damage to DNA per cell per day. It is remarkable that DNA contains codes that check whether the DNA is damaged or can repair itself. It is like an auto check and repair mechanism. This repair ability of DNA is vital for maintaining the integrity of the genetic code and for the normal functioning of the entire organism. It should be constantly active and needs to respond rapidly. The rate of DNA repair depends on various factors such as the cell type and age of the cell. A cell with a damaged ability to repair DNA, which could have been induced by ionizing radiation, can do one of the following:
1. The cell can go into an irreversible state of dormancy, known as senescence.
2. The cell can initiate programmed cell death.
3. The cell can go into unregulated cell division leading to tumors and cancers.
Since ionizing radiation damages the DNA, which is critical in cell reproduction, it has its greatest effect on cells that rapidly reproduce, including most types of cancer. Thus, cancer cells are more sensitive to radiation than normal cells and can be killed by it easily. Cancer is characterized by a malfunction of cell reproduction, and can also be caused by ionizing radiation. Without contradiction, ionizing radiation can be both a cure and a cause.
To discuss quantitatively the biological effects of ionizing radiation, we need a radiation dose unit that is directly related to those effects. All effects of radiation are assumed to be directly proportional to the amount of ionization produced in the biological organism. The amount of ionization is in turn proportional to the amount of deposited energy. Therefore, we define a radiation dose unit called the rad, as  of a joule of ionizing energy deposited per kilogram of tissue, which is
For example, if a 50.0-kg person is exposed to ionizing radiation over her entire body and she absorbs 1.00 J, then her whole-body radiation dose is
If the same 1.00 J of ionizing energy were absorbed in her 2.00-kg forearm alone, then the dose to the forearm would be
and the unaffected tissue would have a zero rad dose. While calculating radiation doses, you divide the energy absorbed by the mass of affected tissue. You must specify the affected region, such as the whole body or forearm in addition to giving the numerical dose in rads. The SI unit for radiation dose is the gray (Gy), which is defined to be
However, the rad is still commonly used. Although the energy per kilogram in 1 rad is small, it has significant effects since the energy causes ionization. The energy needed for a single ionization is a few eV, or less than . Thus, 0.01 J of ionizing energy can create a huge number of ion pairs and have an effect at the cellular level.
The effects of ionizing radiation may be directly proportional to the dose in rads, but they also depend on the type of radiation and the type of tissue. That is, for a given dose in rads, the effects depend on whether the radiation is  x-ray, or some other type of ionizing radiation. In the earlier discussion of the range of ionizing radiation, it was noted that energy is deposited in a series of ionizations and not in a single interaction. Each ion pair or ionization requires a certain amount of energy, so that the number of ion pairs is directly proportional to the amount of the deposited ionizing energy. But, if the range of the radiation is small, as it is for  s, then the ionization and the damage created is more concentrated and harder for the organism to repair, as seen in . Concentrated damage is more difficult for biological organisms to repair than damage that is spread out, so short-range particles have greater biological effects. The relative biological effectiveness (RBE) or quality factor (QF) is given in  for several types of ionizing radiation—the effect of the radiation is directly proportional to the RBE. A dose unit more closely related to effects in biological tissue is called the roentgen equivalent man or rem and is defined to be the dose in rads multiplied by the relative biological effectiveness.
So, if a person had a whole-body dose of 2.00 rad of  radiation, the dose in rem would be . If the person had a whole-body dose of 2.00 rad of  radiation, then the dose in rem would be . The  s would have 20 times the effect on the person than the  s for the same deposited energy. The SI equivalent of the rem is the sievert (Sv), defined to be , so that
The RBEs given in  are approximate, but they yield certain insights. For example, the eyes are more sensitive to radiation, because the cells of the lens do not repair themselves. Neutrons cause more damage than  rays, although both are neutral and have large ranges, because neutrons often cause secondary radiation when they are captured. Note that the RBEs are 1 for higher-energy  s,  s, and x-rays, three of the most common types of radiation. For those types of radiation, the numerical values of the dose in rem and rad are identical. For example, 1 rad of  radiation is also 1 rem. For that reason, rads are still widely quoted rather than rem.  summarizes the units that are used for radiation.
A high level of activity doesn’t mean much if a person is far away from the source. The activity  of a source depends upon the quantity of material (kg) as well as the half-life. A short half-life will produce many more disintegrations per second. Recall that . Also, the activity decreases exponentially, which is seen in the equation .
The large-scale effects of radiation on humans can be divided into two categories: immediate effects and long-term effects.  gives the immediate effects of whole-body exposures received in less than one day. If the radiation exposure is spread out over more time, greater doses are needed to cause the effects listed. This is due to the body’s ability to partially repair the damage. Any dose less than 100 mSv (10 rem) is called a low dose, 0.1 Sv to 1 Sv (10 to 100 rem) is called a moderate dose, and anything greater than 1 Sv (100 rem) is called a high dose. There is no known way to determine after the fact if a person has been exposed to less than 10 mSv.
Immediate effects are explained by the effects of radiation on cells and the sensitivity of rapidly reproducing cells to radiation. The first clue that a person has been exposed to radiation is a change in blood count, which is not surprising since blood cells are the most rapidly reproducing cells in the body. At higher doses, nausea and hair loss are observed, which may be due to interference with cell reproduction. Cells in the lining of the digestive system also rapidly reproduce, and their destruction causes nausea. When the growth of hair cells slows, the hair follicles become thin and break off. High doses cause significant cell death in all systems, but the lowest doses that cause fatalities do so by weakening the immune system through the loss of white blood cells.
The two known long-term effects of radiation are cancer and genetic defects. Both are directly attributable to the interference of radiation with cell reproduction. For high doses of radiation, the risk of cancer is reasonably well known from studies of exposed groups. Hiroshima and Nagasaki survivors and a smaller number of people exposed by their occupation, such as radium dial painters, have been fully documented. Chernobyl victims will be studied for many decades, with some data already available. For example, a significant increase in childhood thyroid cancer has been observed. The risk of a radiation-induced cancer for low and moderate doses is generally assumed to be proportional to the risk known for high doses. Under this assumption, any dose of radiation, no matter how small, involves a risk to human health. This is called the linear hypothesis and it may be prudent, but it is  controversial. There is some evidence that, unlike the immediate effects of radiation, the long-term effects are cumulative and there is little self-repair. This is analogous to the risk of skin cancer from UV exposure, which is known to be cumulative.
There is a latency period for the onset of radiation-induced cancer of about 2 years for leukemia and 15 years for most other forms. The person is at risk for at least 30 years after the latency period. Omitting many details, the overall risk of a radiation-induced cancer death per year per rem of exposure is about 10 in a million, which can be written as .
If a person receives a dose of 1 rem, his risk each year of dying from radiation-induced cancer is 10 in a million and that risk continues for about 30 years. The lifetime risk is thus 300 in a million, or 0.03 percent. Since about 20 percent of all worldwide deaths are from cancer, the increase due to a 1 rem exposure is impossible to detect demographically. But 100 rem (1 Sv), which was the dose received by the average Hiroshima and Nagasaki survivor, causes a 3 percent risk, which can be observed in the presence of a 20 percent normal or natural incidence rate.
The incidence of genetic defects induced by radiation is about one-third that of cancer deaths, but is much more poorly known. The lifetime risk of a genetic defect due to a 1 rem exposure is about 100 in a million or , but the normal incidence is 60,000 in a million. Evidence of such a small increase, tragic as it is, is nearly impossible to obtain. For example, there is no evidence of increased genetic defects among the offspring of Hiroshima and Nagasaki survivors. Animal studies do not seem to correlate well with effects on humans and are not very helpful. For both cancer and genetic defects, the approach to safety has been to use the linear hypothesis, which is likely to be an overestimate of the risks of low doses. Certain researchers even claim that low doses are beneficial. Hormesis is a term used to describe generally favorable biological responses to low exposures of toxins or radiation. Such low levels may help certain repair mechanisms to develop or enable cells to adapt to the effects of the low exposures. Positive effects may occur at low doses that could be a problem at high doses.
Even the linear hypothesis estimates of the risks are relatively small, and the average person is not exposed to large amounts of radiation.  lists average annual background radiation doses from natural and artificial sources for Australia, the United States, Germany, and world-wide averages. Cosmic rays are partially shielded by the atmosphere, and the dose depends upon altitude and latitude, but the average is about 0.40 mSv/y. A good example of the variation of cosmic radiation dose with altitude comes from the airline industry. Monitored personnel show an average of 2 mSv/y. A 12-hour flight might give you an exposure of 0.02 to 0.03 mSv.
Doses from the Earth itself are mainly due to the isotopes of uranium, thorium, and potassium, and vary greatly by location. Some places have great natural concentrations of uranium and thorium, yielding doses ten times as high as the average value. Internal doses come from foods and liquids that we ingest. Fertilizers containing phosphates have potassium and uranium. So we are all a little radioactive. Carbon-14 has about 66 Bq/kg radioactivity whereas fertilizers may have more than 3000 Bq/kg radioactivity. Medical and dental diagnostic exposures are mostly from x-rays. It should be noted that x-ray doses tend to be localized and are becoming much smaller with improved techniques.  shows typical doses received during various diagnostic x-ray examinations. Note the large dose from a CT scan. While CT scans only account for less than 20 percent of the x-ray procedures done today, they account for about 50 percent of the annual dose received.
Radon is usually more pronounced underground and in buildings with low air exchange with the outside world. Almost all soil contains some  and , but radon is lower in mainly sedimentary soils and higher in granite soils. Thus, the exposure to the public can vary greatly, even within short distances. Radon can diffuse from the soil into homes, especially basements. The estimated exposure for  is controversial. Recent studies indicate there is more radon in homes than had been realized, and it is speculated that radon may be responsible for 20 percent of lung cancers, being particularly hazardous to those who also smoke. Many countries have introduced limits on allowable radon concentrations in indoor air, often requiring the measurement of radon concentrations in a house prior to its sale. Ironically, it could be argued that the higher levels of radon exposure and their geographic variability, taken with the lack of demographic evidence of any effects, means that low-level radiation is less dangerous than previously thought.

### Radiation Protection

Laws regulate radiation doses to which people can be exposed. The greatest occupational whole-body dose that is allowed depends upon the country and is about 20 to 50 mSv/y and is rarely reached by medical and nuclear power workers. Higher doses are allowed for the hands. Much lower doses are permitted for the reproductive organs and the fetuses of pregnant women. Inadvertent doses to the public are limited to  of occupational doses, except for those caused by nuclear power, which cannot legally expose the public to more than  of the occupational limit or 0.05 mSv/y (5 mrem/y). This has been exceeded in the United States only at the time of the Three Mile Island (TMI) accident in 1979. Chernobyl is another story. Extensive monitoring with a variety of radiation detectors is performed to assure radiation safety. Increased ventilation in uranium mines has lowered the dose there to about 1 mSv/y.
To physically limit radiation doses, we use shielding, increase the distance from a source, and limit the time of exposure.
 illustrates how these are used to protect both the patient and the dental technician when an x-ray is taken. Shielding absorbs radiation and can be provided by any material, including sufficient air. The greater the distance from the source, the more the radiation spreads out. The less time a person is exposed to a given source, the smaller is the dose received by the person. Doses from most medical diagnostics have decreased in recent years due to faster films that require less exposure time.

### Problem-Solving Strategy

You need to follow certain steps for dose calculations, which are

Examine the situation to determine that a person is exposed to ionizing radiation.
Identify exactly what needs to be determined in the problem (identify the unknowns). The most straightforward problems ask for a dose calculation.
Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Look for information on the type of radiation, the energy per event, the activity, and the mass of tissue affected.
For dose calculations, you need to determine the energy deposited. This may take one or more steps, depending on the given information.
Divide the deposited energy by the mass of the affected tissue. Use units of joules for energy and kilograms for mass. To calculate the dose in Gy, use the definition that 1 Gy = 1 J/kg.
To calculate the dose in mSv, determine the RBE (QF) of the radiation. Recall that .
Check the answer to see if it is reasonable: Does it make sense? The dose should be consistent with the numbers given in the text for diagnostic, occupational, and therapeutic exposures.

### Risk versus Benefit

Medical doses of radiation are also limited. Diagnostic doses are generally low and have further lowered with improved techniques and faster films. With the possible exception of routine dental x-rays, radiation is used diagnostically only when needed so that the low risk is justified by the benefit of the diagnosis. Chest x-rays give the lowest doses—about 0.1 mSv to the tissue affected, with less than 5 percent scattering into tissues that are not directly imaged. Other x-ray procedures range upward to about 10 mSv in a CT scan, and about 5 mSv (0.5 rem) per dental x-ray, again both only affecting the tissue imaged. Medical images with radiopharmaceuticals give doses ranging from 1 to 5 mSv, usually localized. One exception is the thyroid scan using . Because of its relatively long half-life, it exposes the thyroid to about 0.75 Sv. The isotope  is more difficult to produce, but its short half-life limits thyroid exposure to about 15 mSv.

### Test Prep for AP Courses

### Section Summary

1. The biological effects of ionizing radiation are due to two effects it has on cells: interference with cell reproduction, and destruction of cell function.
2. A radiation dose unit called the rad is defined in terms of the ionizing energy deposited per kilogram of tissue:

3. The SI unit for radiation dose is the gray (Gy), which is defined to be
4. To account for the effect of the type of particle creating the ionization, we use the relative biological effectiveness (RBE) or quality factor (QF) given in  and define a unit called the roentgen equivalent man (rem) as

5. Particles that have short ranges or create large ionization densities have RBEs greater than unity. The SI equivalent of the rem is the sievert (Sv), defined to be

6. Whole-body, single-exposure doses of 0.1 Sv or less are low doses while those of 0.1 to 1 Sv are moderate, and those over 1 Sv are high doses. Some immediate radiation effects are given in . Effects due to low doses are not observed, but their risk is assumed to be directly proportional to those of high doses, an assumption known as the linear hypothesis. Long-term effects are cancer deaths at the rate of and genetic defects at roughly one-third this rate. Background radiation doses and sources are given in . World-wide average radiation exposure from natural sources, including radon, is about 3 mSv, or 300 mrem. Radiation protection utilizes shielding, distance, and time to limit exposure.

### Conceptual Questions

### Problems & Exercises