32.8

Concepts & Calculations

When considering the biological effects of ionizing radiation, the concept of the biologically equivalent dose is especially important. Its importance lies in the fact that the biologically equivalent dose incorporates both the amount of energy per unit mass that is absorbed and the effectiveness of a particular type of radiation in producing a certain biological effect. Example 9 examines this concept, as well as reviewing the notions of power (Section 6.7) and intensity (Section 16.7) of a wave.

| Concepts & Calculations Example 9 | | | The Biologically Equivalent Dose

A patient is being given a chest X-ray. The X-ray beam is turned on for 0.20 s, and its intensity is 0.40 W/m2. The area of the chest being exposed is 0.072 m2, and the radiation is absorbed by 3.6 kg of tissue. The relative biological effectiveness (RBE) of the X-rays for this tissue is 1.1. Determine the biologically equivalent dose received by the patient.
Concept Questions and Answers How is the power of the beam related to the beam intensity?

> Answer The power P is equal to the intensity I times the cross-sectional area A of the beam, as indicated by Equation 16.8 (P = IA).

How is the energy absorbed by the tissue related to the power of the beam?

> Answer The energy E is equal to the product of the power P delivered by the beam and the time t of exposure, according to Equation 6.10b (E = Pt).

What is the absorbed dose?

> Answer The absorbed dose is defined by Equation 32.2 as the energy E absorbed by the tissue divided by its mass m, or Absorbed dose = E/m. The unit of the absorbed dose is the gray (Gy).

How is the biologically equivalent dose related to the absorbed dose?

> Answer The biologically equivalent dose is equal to the product of the absorbed dose and the relative biological effectiveness (RBE), as indicated by Equation 32.4. However, the absorbed dose must be expressed in rad units, not in gray units. The relation between the two units is 1 rad = 0.01 Gy.

Solution The biologically equivalent dose is given by Equation 32.4 as

At this point we need to convert the absorbed dose from Gy units to rad units:

The biologically equivalent dose then becomes

Equation 32.2 indicates that the absorbed dose (in Gy) is equal to the energy E divided by the mass m of the tissue, so that we have

From the answers to the first and second Concept Questions, we can conclude that the absorbed energy E is related to the intensity I, the cross-sectional area A, and the time duration t of the beam via E = Pt = IAt. Substituting this result into the equation for the biologically equivalent dose gives

The next example illustrates the decay of a particle into two photons, so matter is completely converted into electromagnetic waves. The example discusses how to find the energy, frequency, and wavelength of each photon. It also provides an opportunity to review the principle of conservation of energy and the principle of conservation of linear momentum.

| Concepts & Calculations Example 10 | | |

The Decay of the Meson The meson is a particle that has a rest energy of 135.0 MeV (see Table 32-3). It lives for a very short time and then decays into two -ray photons: . Suppose that one of the -ray photons travels along the +x axis. If the is at rest when it decays, find (a) the energy (in MeV), (b) the frequency and wavelength, and (c) the momentum of each -ray photon.
Concept Questions and Answers How is the energy E of each -ray photon related to the rest energy E0 of the particle?

> Answer According to the principle of conservation of energy, the sum of the energies of the two -ray photons must equal the rest energy E0 of the particle, since it is at rest. Therefore, the energy E of each photon is E0.

How can the frequency and wavelength of a photon be determined from its energy?

> Answer A photon is a packet of electromagnetic energy, and each contains an amount of energy given by Equation 29.2 as E = hf, where h is Planck’s constant and f is the frequency. Equation 16.1 indicates that the wavelength of the photon is inversely proportional to its frequency, , where c is the speed of light.

How is the total linear momentum of the photons related to the momentum of the particle, and what is the momentum of each photon?

> Answer The principle of conservation of linear momentum states that the total linear momentum of an isolated system is constant (see Section 7.2). Since the particle is at rest, the total linear momentum before the decay is zero. Since the particle is an isolated system, momentum is conserved, and the total linear momentum of the two -ray photons must also be zero. The magnitude p of the momentum of a photon is given by Equation 29.6 as . Thus, if one photon travels along the +x axis with a momentum of , the other must travel along the −x axis with a momentum of , ensuring that the total momentum is zero.

Solution

a. | Since the sum of the energies of the two -ray photons must be equal to the rest energy E0 of the particle, the energy of each photon is

b. | The frequency of each photon is given by Equation 29.2 as its energy (in units of joules) divided by Planck’s constant. Using the fact that 1 eV = 1.60 × 10−19 J, we find that the frequency of each photon is

The wavelength of each photon can be found directly from Equation 16.1:

c. | The magnitude of each photon’s momentum is related to its wavelength by Equation 29.6:

Since the photons move along the x axis in opposite directions, their momenta are and .

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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