Chapter 5. Radiation and Materials

The word “radiation” will be taken to embrace all particles, whether they are of material or electromagnetic origin. We include those particles produced by both atomic and nuclear processes and those resulting from electrical acceleration, noting that there is no essential difference between X-rays from atomic collisions and gamma rays from nuclear decay; protons can come from a particle accelerator, from cosmic rays, or from a nuclear reaction in a reactor. The word “materials” will refer to bulk matter, whether of mineral or biological origin, as well as to the particles of which the matter is composed, including molecules, atoms, electrons, and nuclei.

When we put radiation and materials together, a great variety of possible situations must be considered. Bombarding particles may have low or high energy; they may be charged, uncharged, or photons; and they may be heavy or light in the scale of masses. The targets may be similarly distinguished, but they may also exhibit degrees of binding that range from (a) none, as for “free” particles, to (b) weak, as for atoms in molecules and electrons in atoms, to (c) strong, as for nucleons in nuclei. In most interactions, the higher the projectile energy in comparison with the energy of binding of the structure, the greater is the effect.

Out of the broad subject we will select for review some of the reactions that are important in the nuclear energy field. Looking ahead, we will need to understand the effects produced by the particles and rays from radioactivity and other nuclear reactions. Materials affected may be in or around a nuclear reactor, as part of its construction or inserted to be irradiated. Materials may be of biological form, including the human body, or they may be inert substances used for protective shielding against radiation. We will not attempt to explain the processes rigorously but be content with qualitative descriptions based on analogy with collisions viewed on an elementary physics level.

5.1. Excitation and Ionization by Electrons

These processes occur in the familiar fluorescent light bulb, in an X-ray machine, or in matter exposed to beta particles. If an electron that enters a material has a very low energy, it will merely migrate without affecting the molecules significantly. If its energy is large, it may impart energy to atomic electrons as described by the Bohr theory (Chapter 2), causing excitation of electrons to higher energy states or producing ionization, with subsequent emission of light. When electrons of inner orbits in heavy elements are displaced, the resultant high-energy radiation is classed as X-rays. These rays, which are so useful for internal examination of the human body, are produced by accelerating electrons in a vacuum chamber to energies in the kilovolt range and allowing them to strike a heavy element target. In addition to the X-rays as a result of transitions in the electron orbits, a similar radiation called bremsstrahlung (German: braking radiation) is produced. It arises from the deflection and resulting acceleration of electrons as they encounter nuclei.

Beta particles as electrons from nuclear reactions have energies in the range 0.01 to 1 MeV and thus are capable of producing large amounts of ionization as they penetrate a substance. As a rough rule of thumb, approximately 32 eV of energy is required to produce one ion pair. The beta particles lose energy with each event and eventually are stopped. For electrons of 1 MeV energy, the range, as the typical distance of penetration, is no more than a few millimeters in liquids and solids or a few meters in air.

5.2. Heavy Charged Particle Stopping by Matter

Charged particles such as protons, alpha particles, and fission fragment ions are classified as heavy, being much more massive that the electron. For a given energy, their speed is lower than that of an electron, but their momentum is greater, and they are less readily deflected on collision. The mechanism by which they slow down in matter is mainly electrostatic interaction with the atomic electrons and with nuclei. In each case the Coulomb force, varying as 1/r² with distance of separation r, determines the result of a collision. Figure 5.1 illustrates the effect of the passage of an ion by an atom. An electron is displaced and gains energy in a large amount compared with its binding in the atom, creating an ion. Application of the collision formulas of Exercise 4.19 leads to the energy change when a heavy particle of mass m^H and energy E⁰ collides head-on with an electron of mass m^e, as approximately 4(m^e/m^H) E⁰. For example, for an alpha particle of 5 MeV, the loss by the projectile and the gain by the target are 4(0.000549/4.00) 5 = 0.00274 MeV or 2.74 keV. The electron is energetic enough to produce secondary ionization, whereas hundreds of collisions are needed to reduce the alpha particle's energy by as little as 1 MeV. As the result of primary and secondary processes, a great deal of ionization is produced by heavy ions as they move through matter.

Figure 5.1. Interaction of heavy ion with electron.

In contrast, when a heavy charged particle comes close to a nucleus, the electrostatic force causes it to move in a hyperbolic path as in Figure 5.2. The projectile is scattered through an angle that depends on the detailed nature of the collision (i.e., the initial energy and direction of motion of the incoming ion relative to the target nucleus) and the magnitudes of electric charges of the interacting particles. The charged particle loses a significant amount of energy in the process, in contrast with the slight energy loss on collision with an electron. Unless the energy of the bombarding particle is very high and it comes within the short range of the nuclear force, there is a small chance that it can enter the nucleus and cause a nuclear reaction.

Figure 5.2. Interaction of heavy ion with nucleus.

A measure of the rate of ion energy loss with distance traveled is the stopping power, symbolized by −dE/dx. It is also known as the linear energy transfer (LET). Two separate components, atomic and nuclear, add to give the total, as tabulated in the NIST Web site (see References). Theoretical formulas giving the dependence on electric charges, masses, and energy are given by Mayo (see References). A related quantity is the range, which is the maximum distance of travel of a projectile as it makes multiple collisions in matter. Integration of the reciprocal of the stopping power yields values of the range, also given by NIST. For example, in its ASTAR database, the range of 4 MeV alpha particles is listed as 3.147E-03 cm²/g, and with an air density of 0.001293 g/cm³, a distance of 2.43 cm. An alpha particle has a very small range in solid materials: a sheet of paper is sufficient to stop it and the outer layer of human skin provides protection for sensitive tissue.

5.3. Gamma Ray Interactions with Matter

We now turn to a group of three related processes involving gamma ray photons produced by nuclear reactions. These have energies as high as a few MeV. The interactions include simple scattering of the photon, ionization by it, and a special nuclear reaction known as pair production.

(a). Photon-Electron Scattering

One of the easiest processes to visualize is the interaction of a photon of energy E = hν and an electron of rest mass m⁰. Although the electrons in a target atom can be regarded as moving and bound to their nucleus, the energies involved are very small (eV) compared with those of typical gamma rays (keV or MeV). Thus the electrons may be viewed as free stationary particles. The collision may be treated by the physical principles of energy and momentum conservation. As sketched in Figure 5.3, the photon is deflected in its direction and loses energy, becoming a photon of new energy E′ = hν′. The electron gains energy and moves away with high-speed υ and total mass-energy mc², leaving the atom ionized. In this Compton effect, named after its discoverer, one finds that the greatest photon energy loss occurs when it is scattered backward (180 °) from the original direction. Then, if E is much larger than the rest energy of the electron E⁰ = m⁰c² = 0.51 MeV, it is found that the final photon energy E′ is equal to E⁰/2. On the other hand, if E is much smaller than E⁰, the fractional energy loss of the photon is 2E/E⁰ (see also Exercise 5.3). The derivation of the photon energy loss in general is complicated by the fact that the special theory of relativity must be applied. The resulting formulas are displayed in the computer program COMPTON, which is used in several Computer Exercises to find photon energy losses.

Figure 5.3. Photon electron scattering (Compton effect).

The probability of Compton scattering is expressed by a cross section, which is smaller for larger gamma energies as shown in Figure 5.4 for the element lead, a common material for shielding against X-rays or gamma rays. We can deduce that the chance of collision increases with each successive loss of energy by the photon, and eventually the photon disappears.

Figure 5.4. Gamma ray cross sections in lead, Pb. Plotted from data in National Bureau of Standards report NSRDS-NSB-29.

(b). Photoelectric Effect

This process is in competition with scattering. An incident photon of high enough energy dislodges an electron from the atom, leaving a positively charged ion. In so doing, the photon is absorbed and thus lost (see Figure 5.5). The cross section for the photoelectric effect decreases with increasing photon energy, as sketched in Figure 5.4 for the element lead.

Figure 5.5. Photoelectric effect.

The preceding two processes are usually treated separately even though both result in ionization. In the Compton effect, a photon of lower energy survives, but in the photoelectric effect, the photon is eliminated. In each case, the electron released may have enough energy to excite or ionize other atoms by the mechanism described earlier. Also, the ejection of the electron is followed by light emission or X-ray production, depending on whether an outer shell or inner shell is involved.

(c). Electron-Positron Pair Production

The third process to be considered is one in which the photon is converted into matter. This is entirely in accord with Einstein's theory of the equivalence of mass and energy. In the presence of a nucleus, as sketched in Figure 5.6, a gamma ray photon disappears and two particles appear—an electron and a positron. Because these are of equal charge but of opposite sign, there is no net charge after the reaction, just as before, the gamma ray having zero charge. The law of conservation of charge is thus met. The total new mass produced is twice the mass-energy of the electron, 2(0.51) = 1.02 MeV, which means that the reaction can occur only if the gamma ray has at least this amount of energy. The cross section for the process of pair production rises from zero as shown in Figure 5.4 for lead. The reverse process also takes place. As sketched in Figure 5.7, when an electron and a positron combine, they are annihilated as material particles, and two gamma rays of energy totaling at least 1.02 MeV are released. That there must be two photons is a consequence of the principle of momentum conservation.

Figure 5.6. Pair production.

Figure 5.7. Pair annihilation.

The reverse process, in which two high-energy photons collide to form an electron-positron pair, is believed to have been common in early times after the Big Bang.

Figure 5.4 shows that the total gamma ray cross section curve for lead (Pb), as the sum of the components for Compton effect, photoelectric effect, and pair production, exhibits a minimum at approximately 3 MeV energy. This implies that gamma rays in this vicinity are more penetrating than those of higher or lower energy. In contrast with β particles and α particles, which have a definite range, a certain fraction of incident gamma rays can pass through any thickness of material. The exponential expression e^−Σz as used to describe neutron behavior can be carried over to the attenuation of gamma rays in matter. One can use the mean free path λ = 1/Σ or, better, the half-thickness 0.693/Σ, the distance in which the intensity of a gamma ray beam is reduced by a factor of two.

Cross section data for the interaction of photons with many elements are found in the NIST Web site (see References).

5.4. Neutron Reactions

For completeness, we review again the interaction of neutrons with matter. Neutrons may be scattered by nuclei elastically or inelastically, may be captured with resulting gamma ray emission, or may cause fission. If their energy is high enough, neutrons may induce (n, p) and (n, α) reactions as well.

We are now in a position to understand the connection between neutron reactions and atomic processes. When a high-speed neutron strikes the hydrogen atom in a water molecule, a proton is ejected, resulting in chemical dissociation of the H²O. A similar effect takes place in molecules of cells in any biological tissue. The proton compared with the electron is a heavy charged particle. It passes through matter, slowing and creating ionization along its path. Thus two types of neutron radiation damage take place—primary and secondary.

After many collisions, the neutron arrives at a low enough energy that it can be readily absorbed. If it is captured by the proton in a molecule of water or some other hydrocarbon, a gamma ray is released, as discussed in Chapter 4. The resulting deuteron recoils with energy that is much smaller than that of the gamma ray but still is far greater than the energy of binding of atoms in the water molecule. Again dissociation of the compound takes place, which can be regarded as a form of radiation damage.

5.5. Summary

Radiation of special interest includes electrons, heavy charged particles, photons, and neutrons. Each of the particles tends to lose energy by interaction with the electrons and nuclei of matter, and each creates ionization in different degrees. The ranges of beta particles and alpha particles are short, but gamma rays penetrate in accord with an exponential law. Gamma rays can also produce electron-positron pairs. Neutrons of both high and low energy can create radiation damage in molecular materials.

5.6. Exercises

1. The charged particles in a highly ionized electrical discharge in hydrogen gas—protons and electrons, mass ratio m^p/m^e = 1836—have the same energies. What is the ratio of the speeds υ^p/υ^e? Of the momenta p^p/p^e?
2. A gamma ray from neutron capture has an energy of 6 MeV. What is its frequency? Its wavelength?
3. For 180 ° scattering of gamma or X-rays by electrons, the final energy of the photon is
   • What is the final photon energy for the 6 MeV gamma ray of Exercise 5.2?
   • Verify that if E >> E⁰, then E′ ≅ E⁰/2 and if E << E⁰, (E − E′)/E ≅ 2 E/E⁰.
   • Which approximation should be used for a 6 MeV gamma ray? Verify numerically.
4. An electron-positron pair is produced by a gamma ray of 2.26 MeV. What is the kinetic energy imparted to each of the charged particles?
5. Estimate the thickness of paper required to stop 2 MeV alpha particles, assuming the paper to be of density 1.29 g/cm³, about the same electronic composition as air, density 1.29 × 10⁻³ g/cm³.
6. The element lead, M = 206, has a density of 11.3 g/cm³. Find the number of atoms per cubic centimeter. If the total gamma ray cross section at 3 MeV is 14 barns, what is the macroscopic cross section Σ and the half-thickness 0.693/Σ?
7. The range of beta particles of energy greater than 0.8 MeV is given roughly by the relation Find what thickness of aluminum sheet (density 2.7 g/cm³) is enough to stop the betas from phosphorus-32 (see Table 3.1).
8. A radiation worker's hands are exposed for 5 seconds to a 3 × 10⁸ cm⁻² s⁻¹ beam of 1 MeV beta particles. Find the range in tissue of density 1.0 g/cm³ and calculate the amounts of charge in coulombs (C) and energy deposition in C/cm³ and J/g. Note that the charge on the electron is 1.60 × 10⁻¹⁹ C. For tissue, use the equation in Exercise 5.7.
9. Calculate the energy gain by an electron struck head-on by an alpha particle of energy 4 MeV. How many such collisions would it take to reduce the alpha particle energy to 1 MeV?
10. At a certain time after the Big Bang, high-speed photons collided to form electrons and positrons. Assuming energies of 0.51 MeV each, what temperature is implied?

Computer Exercises

1. The scattering at any angle of a photon colliding with a free electron is analyzed by the program COMPTON, after Arthur Holly Compton's theory. (a) Run the program and use the menus. (b) Find the maximum and minimum photon energies of 50 keV X-rays passing through a thin aluminum foil and making no more than one collision.
2. With the program COMPTON, compare the percent energy change of 10 keV and 10 MeV photons scattered at 90 °. What conclusion do these results suggest?
3. (a) Find the fractional energy loss for a 20 keV X-ray scattered from an electron at angle 180 ° and compare with 2E/E⁰. (b) Find the final energy for a 10 MeV gamma ray scattered from an electron at 180 ° and compare with E⁰/2.

5.7 References

Segrè, 1977 Emilio Segrè, Nuclei and Particles 2nd Ed. 1977 Reading, MA Benjamin-Cummings A classic book on nuclear theory and experiments for undergraduate physics students written by a Nobel Prize winner

Mayo, 1998 Robert M. Mayo, Introduction to Nuclear Concepts for Engineers 1998 American Nuclear Society La Grange Park, IL Chapter 6 is devoted to the interaction of radiation with matter

Bethe et al., 1986 Hans A. Bethe, Robert F. Bacher, M. Stanley Livingston, Basic Bethe, Seminal Articles on Nuclear Physics, 1936–1937 1986 American Institute of Physics and Springer Verlag New York347 ff- Reprints of classic literature on nuclear processes. Discussion of stopping power

National Institute of Science and Technology (NIST) National Institute of Science and Technology (NIST)

http://www.nist.gov http://www.nist.gov

For stopping powers and ranges of electrons, protons and alpha particles, use Search with keyword “stopping power.” For stopping powers and ranges of electrons, protons and alpha particles, use Search with keyword “stopping power.”

For photon cross sections for many elements, use Search with keyword “XCOM.” Data by Berger, Hubbell, et al For photon cross sections for many elements, use Search with keyword “XCOM.” Data by Berger, Hubbell, et al.

Was, 2007 S. Gary, Was, Fundamentals of Radiation Materials Science 2007 Springer New York

Shultis and Faw, 1996 J. Kenneth Shultis, E. Richard Faw, Radiation Shielding 1996 Prentice-Hall Upper Saddle River, NJ Basics and modern analysis techniques

Faw and Shultis, 1999 Richard E. Faw, J. Kenneth Shultis, Radiological Assessment: Sources and Doses 1999 American Nuclear Society La Grange Park, IL Includes fundamentals of radiation interactions