Radiation Dosimetry PDF

361 12 Radiation Dosimetry 12.1 Introduction Radiation dosimetry is the branch of science that attempts to quantitatively relate specific measurements made in a radiation field to physical, chemical, and/or...

361 12 Radiation Dosimetry 12.1 Introduction Radiation dosimetry is the branch of science that attempts to quantitatively relate specific measurements made in a radiation field to physical, chemical, and/or bio- logical changes that the radiation would produce in a target. Dosimetry is essential for quantifying the incidence of various biological changes as a function of the amount of radiation received (dose–effect relationships), for comparing different experiments, for monitoring the radiation exposure of individuals, and for surveil- lance of the environment. In this chapter we describe the principal concepts upon which radiation dosimetry is based and present methods for their practical utiliza- tion. When radiation interacts with a target it produces excited and ionized atoms and molecules as well as large numbers of secondary electrons. The secondary electrons can produce additional ionizations and excitations until, finally, the energies of all electrons fall below the threshold necessary for exciting the medium. As we shall see in detail in the next chapter, the initial electronic transitions, which produce chemically active species, are completed in very short times (!10–15 s) in local re- gions within the path traversed by a charged particle. These changes, which require the direct absorption of energy from the incident radiation by the target, represent the initial physical perturbations from which subsequent radiation effects evolve. It is natural therefore to consider measurements of ionization and energy absorption as the basis for radiation dosimetry. As experience and knowledge have been gained through the years, basic ideas, philosophy, and concepts behind radiation protection and dosimetry have continu- ally evolved. This process continues today. On a world scale, the recommendations of the International Commission on Radiological Protection (ICRP) have played a major role in establishing protection criteria at many facilities that deal with radia- tion. In the United States, recommendations of the National Council on Radiation protection and Measurements (NCRP) have provided similar guidance. There is close cooperation among these two bodies and also the International Commission on Radiation Units and Measurements (ICRU). As a practical matter, there is a cer- tain time delay between the publication of the recommendations and the official Atoms, Radiation, and Radiation Protection. James E. Turner Copyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40606-7 362 12 Radiation Dosimetry promulgation of statutory regulations by organizations responsible for radiation protection. As a result, at any point in time, some differences might exist among particular procedures in effect at different locations, even though they are based on publications of the ICRP or NCRP. We shall discuss the implementation of radia- tion protection criteria and exposure limits in Chapter 14. The present chapter will deal with radiation quantities and units of historical and current importance. 12.2 Quantities and Units Exposure Exposure is defined for gamma and X rays in terms of the amount of ionization they produce in air. The unit of exposure is called the roentgen (R) and was intro- duced at the Radiological Congress in Stockholm in 1928 (Chap. 1). It was origi- nally defined as that amount of gamma or X radiation that produces in air 1 esu of charge of either sign per 0.001293 g of air. (This mass of air occupies 1 cm3 at standard temperature and pressure.) The charge involved in the definition of the roentgen includes both the ions produced directly by the incident photons as well as ions produced by all secondary electrons. Since 1962, exposure has been defined by the International Commission on Radiation Units and Measurements (ICRU) as the quotient !Q/!m, where !Q is the sum of all charges of one sign produced in air when all the electrons liberated by photons in a mass !m of air are completely stopped in air. The unit roentgen is now defined as 1 R = 2.58 × 10–4 C kg–1. (12.1) The concept of exposure applies only to electromagnetic radiation; the charge and mass used in its definition, as well as in the definition of the roentgen, refer only to air. Example Show that 1 esu cm–3 in air at STP is equivalent to the definition (12.1) of 1 R of exposure. Solution Since the density of air at STP is 0.001293 g cm–3 and 1 esu = 3.34 × 10–10 C (Appen- dix B), we have 1 esu 3.34 × 10–10 C = = 2.58 × 10–4 C kg–1. (12.2) cm3 0.001293 g × 10–3 kg g–1 Absorbed Dose The concept of exposure and the definition of the roentgen provide a practical, mea- surable standard for electromagnetic radiation in air. However, additional concepts  12.2 Quantities and Units 363 are needed to apply to other kinds of radiation and to other materials, particularly tissue. The primary physical quantity used in dosimetry is the absorbed dose. It is defined as the energy absorbed per unit mass from any kind of ionizing radiation in any target. The unit of absorbed dose, J kg–1 , is called the gray (Gy). The older unit, the rad, is defined as 100 erg g–1. It follows that 1 J 107 erg erg 1 Gy ≡ = 3 = 104 = 100 rad. (12.3) kg 10 g g The absorbed dose is often referred to simply as the dose. It is treated as a point function, having a value at every position in an irradiated object. One can compute the absorbed dose in air when the exposure is 1 R. Photons produce secondary electrons in air, for which the average energy needed to make an ion pair is W = 34 eV ip–1 = 34 J C–1 (Sect. 10.1). Using a more precise W value,1) one finds 2.58 × 10–4 C 33.97 J 1R= × = 8.76 × 10–3 J kg–1. (12.4) kg C Thus, an exposure of 1 R gives a dose in air of 8.76 × 10–3 Gy (= 0.876 rad). Calculations also show that a radiation exposure of 1 R would produce a dose of 9.5 × 10–3 Gy (= 0.95 rad) in soft tissue. This unit is called the rep (“roentgen- equivalent-physical”) and was used in early radiation-protection work as a measure of the change produced in living tissue by radiation. The rep is no longer employed. Dose Equivalent It has long been recognized that the absorbed dose needed to achieve a given level of biological damage (e.g., 50% cell killing) is often different for different kinds of radiation. As discussed in the next chapter, radiation with a high linear energy transfer (LET) (Sect. 7.3) is generally more damaging to a biological system per unit dose than radiation with a low LET (for example, cf. Fig. 13.16). To allow for the different biological effectiveness of different kinds of radiation, the International Commission on Radiological Protection (ICRP), National Coun- cil on Radiation Protection and Measurements (NCRP), and ICRU (Chap. 1) intro- duced the concept of dose equivalent for radiation-protection purposes. The dose equivalent H is defined as the product of the absorbed dose D and a dimensionless quality factor Q, which depends on LET: H = QD. (12.5) In principle, other multiplicative modifying factors can be included along with Q to allow for additional considerations (e.g., dose fractionation), but these are not ordinarily used. Until the 1990 recommendations made in ICRP Publication 60, the dependence of Q on LET was defined as given in Table 12.1. Since then, the 1 See p. 29 in Attix reference, Section 12.11. 364 12 Radiation Dosimetry Table 12.1 Dependence of Quality Factor Q on LET of Radiation as Formerly Recommended by ICRP, NCRP, and ICRU LET (keV µm–1 in Water) Q 3.5 or less 1 3.5–7.0 1–2 7.0–23 2–5 23–53 5–10 53–175 10–20 Gamma rays, X rays, electrons, positrons of any LET 1 Table 12.2 Dependence of Quality Factor Q on LET as Currently Recommended by ICRP, NCRP, and ICRU LET, L (keV µm–1 in Water) Q 100 300/ L ICRP, NCRP, and ICRU have defined Q in accordance with Table 12.2. In the con- text of quality factor, LET is the unrestricted stopping power, L∞ , as discussed in Section 7.3. For incident charged particles, it is the LET of the radiation in water, expressed in keV per µm of travel. For neutrons, photons, and other uncharged radiation, LET refers to that which the secondary charged particles they generate would have in water. Like absorbed dose, dose equivalent is a point function. When dose is expressed in Gy, the (SI) unit of dose equivalent is the sievert (Sv). With the dose in rad, the older unit of dose equivalent is the rem (“roentgen-equivalent- man”). Since 1 Gy = 100 rad, 1 Sv = 100 rem. Dose equivalent has been used extensively in protection programs as the quantity in terms of which radiation limits are specified for the exposure of individuals. Dose equivalents from different types of radiation are simply additive. Example A worker receives a whole-body dose of 0.10 mGy from 2-MeV neutrons. Estimate the dose equivalent, based on Table 12.1. Solution Most of the absorbed dose is due to the elastic scattering of the neutrons by the hydrogen in tissue (cf. Table 12.6). To make a rough estimate of the quality fac- tor, we first find Q for a 1-MeV proton—the average recoil energy for 2-MeV neu- trons. From Table 5.3 we see that the stopping power for a 1-MeV proton in water is 270 MeV cm–1 = 27 keV µm–1. Under the current recommendations of the ICRP, NCRP, and ICRU, Q is defined according to Table 12.2. However, the older recom-  12.3 Measurement of Exposure 365 mendations, which include Table 12.1, are still in effect at a number of installations. We see from Table 12.1 that an estimate of Q ∼ 6 should be reasonable for the re- coil protons. The recoil O, C, and N nuclei have considerably higher LET values, but do not contribute as much to the dose as H. (LET is proportional to the square of a particle’s charge.) Without going into more detail, we take the overall quality factor, Q ∼ 12, to be twice that for the recoil protons alone. Therefore, the estimated dose equivalent is H ∼ 12 × 0.10 = 1.2 mSv. [The value Q = 10 is obtained from detailed calculations (cf. Table 12.5).] We note that Table 12.2 implies a comparable value, Q = 6.4, for the protons. Changes in the recommendations are discussed more fully in Chapter 14. By the early 1990s, the ICRP and NCRP replaced the use of LET-dependent qual- ity factors by radiation weighting factors, w, specified for radiation of a given type and energy. The quantity on the left-hand side of the replacement for Eq. (12.5), H = wD, is then called the equivalent dose. In some regulations the older terminol- ogy, dose equivalent and quality factor, is still employed. However, the latter has come to be specified by radiation type and energy, rather than LET. 12.3 Measurement of Exposure Free-Air Ionization Chamber Based on its definition, exposure can be measured operationally with the “free-air,” or “standard,” ionization chamber, sketched in Fig. 12.1. X rays emerge from the target T of an X-ray tube and enter the free-air chamber through a circular aperture of area A, defining a right circular cone TBC of rays. Parallel plates Q and Q′ in the chamber collect the ions produced in the volume of air between them with center P′. Fig. 12.1 Schematic diagram of the “free-air” or “standard” ionization chamber. 366 12 Radiation Dosimetry The exposure in the volume DEFG in roentgens would be determined directly if the total ionization produced only by those ions that originate from X-ray interac- tions in the truncated conical volume DEFG could be collected and the resulting charge divided by the mass of air in DEFG. This mass is given by M = ρA′ L, where ρ is the density of air, A′ is the cross-sectional area of the truncated cone at its mid- point P′ , and L, the thickness of the cone, is equal to the length of the collecting plates Q and Q′. Unfortunately, the plates collect all of the ions between them, not the particular set that is specified in the definition of the roentgen. Some electrons produced by X-ray interactions in DEFG escape this volume and produce ions that are not collected by the plates Q, Q′. Also, some ions from electrons originally pro- duced outside DEFG are collected. Thus, only part of the ionization of an electron such as e1 in Fig. 12.1 is collected, while ionization from an “outside” electron, such as e2 , is collected. When the distance from P to DG is sufficiently large (e.g., ∼ 10 cm for 300-keV X rays), electronic equilibrium will be realized; that is, there will be almost exact compensation between ionization lost from the volume DEFG by electrons, such as e1 , that escape and ionization gained from electrons, such as e2 , that enter. The distance from P to DG, however, should not be so large as to attenuate the beam significantly between P and P′. Under these conditions, when a charge q is collected, the exposure at P′ is given by q EP′ =. (12.6) ρA′ L In practice, one prefers to know the exposure EP at P, the location where the entrance port is placed, rather than EP′. By the inverse-square law, EP = (d′ /d)2 EP′. Since A = (d/d′ )2 A′ , Eq. (12.6) gives ! ′ "2 d q q EP = ′ =. (12.7) d ρA L ρAL Example The entrance port of a free-air ionization chamber has a diameter of 0.25 cm and the length of the collecting plates is 6 cm. Exposure to an X-ray beam produces a steady current of 2.6 × 10–10 A for 30 s. The temperature is 26◦ C and the pressure is 750 torr. Calculate the exposure rate and the exposure. Solution We can apply Eq. (12.7) to exposure rates as well as to exposure. The rate of charge col- lection is q̇ = 2.6 × 10–10 A = 2.6 × 10–10 C s–1. The density of the air under the stated conditions is ρ = (0.00129)(273/299)(750/760) = 1.16 × 10–3 g cm–3. The entrance- port area is A = π (0.125)2 = 4.91 × 10–2 cm2 and L = 6 cm. Equation (12.7) implies, for the exposure rate, q̇ 2.6 × 10–10 C s–1 ĖP = = ρAL 1.16 × 10–3 × 4.91 × 10–2 × 6 g 1R × = 2.95 R s–1. (12.8) 2.58 × 10–7 C g–1 The total exposure is 88.5 R.  12.3 Measurement of Exposure 367 Measurement of exposure with the free-air chamber requires some care and at- tention to details. For example, the collecting plates Q and Q′ in Fig. 12.1 must be recessed away from the active volume DEFG by a distance not less than the lateral range of electrons produced there. We have already mentioned minimum and maximum restrictions on the distance from P to DG. When the photon energy is increased, the minimum distance required for electronic equilibrium increases rapidly and the dimensions for a free-air chamber become excessively large for photons of high energy. For this and other reasons, the free-air ionization chamber and the roentgen are not used for photon energies above 3 MeV. The Air-Wall Chamber The free-air ionization chamber is not a practical instrument for measuring routine exposure. It is used chiefly as a primary laboratory standard. For routine use, cham- bers can be built with walls of a solid material, having photon response properties similar to those of air. Chambers of this type were discussed in Section 10.1. Such an “air-wall” pocket chamber, built as a capacitor, is shown schematically in Fig. 12.2. A central anode, insulated from the rest of the chamber, is given an initial charge from a charger-reader device to which it is attached before wearing. When exposed to photons, the secondary electrons liberated in the walls and enclosed air tend to neutralize the charge on the anode and lower the potential difference between it and the wall. The change in potential difference is directly proportional to the total ionization produced and hence to the exposure. Thus, after exposure to photons, measurement of the change in potential difference from its original value when the chamber was fully charged can be used to find the exposure. Direct- reading pocket ion chambers are available (Fig. 10.6). Example A pocket air-wall chamber has a volume of 2.5 cm3 and a capacitance of 7 pF. Initially charged at 200 V, the reader showed a potential difference of 170 V after the chamber was worn. What exposure in roentgens can be inferred? Solution The charge lost is !Q = C!V = 7 × 10–12 × (200 – 170) = 2.10 × 10–10 C. The mass of air [we assume standard temperature and pressure (STP)] is M = 0.00129 × 2.5 = 3.23 × 10–3 g. It follows that the exposure is 2.10 × 10–10 C 1R –3 × = 0.252 R. (12.9) 3.23 × 10 g 2.58 × 10–7 C g–1 Fig. 12.2 Air-wall pocket ionization chamber, having a plastic wall with approximately the same response to photons as air. 368 12 Radiation Dosimetry Literally taken, the data given in this problem indicate only that the chamber was partially discharged. Charge loss could occur for reasons other than radiation (e.g., leakage from the central wire). Two pocket ion chambers can be worn simultaneously to improve reliability. In practice, air-wall ionization chambers involve a number of compromises from an ideal instrument that measures exposure accurately. For example, if the wall is too thin, incident photons will produce insufficient ionization inside the chamber. If the wall is too thick, it will significantly attenuate the incident radiation. The optimal thickness is reached when, for a given photon field, the ionization in the chamber gas is a maximum. This value, called the equilibrium wall thickness, is equal to the range of the most energetic secondary electrons produced in the wall. In addition, a solid wall can be only approximately air equivalent. Air-wall chambers can be made with an almost energy-independent response from a few hundred keV to about 2 MeV—the energy range in which Compton scattering is the dominant photon interaction in air and low-Z wall materials. 12.4 Measurement of Absorbed Dose One of the primary goals of dosimetry is the determination of the absorbed dose in tissue exposed to radiation. The Bragg–Gray principle provides a means of relating ionization measurements in a gas to the absorbed dose in some convenient mate- rial from which a dosimeter can be fabricated. To obtain the tissue dose, either the material can be tissue equivalent or else the ratio of the absorbed dose in the ma- terial to that in tissue can be inferred from other information, such as calculations or calibration measurements. Consider a gas in a walled enclosure irradiated by photons, as illustrated in Fig. 12.3. The photons lose energy in the gas by producing secondary electrons there, and the ratio of the energy deposited and the mass of the gas is the absorbed dose in the gas. This energy is proportional to the amount of ionization in the gas Fig. 12.3 Gas in cavity enclosed by wall to illustrate Bragg–Gray principle.  12.4 Measurement of Absorbed Dose 369 when electronic equilibrium exists between the wall and the gas. Then an electron, such as e1 in Fig. 12.3, which is produced by a photon in the gas and enters the wall before losing all of its energy, is compensated by another electron, like e2 , which is produced by a photon in the wall and stops in the gas. When the walls and gas have the same atomic composition, then the energy spectra of such electrons will be the same irrespective of their origin, and a high degree of compensation can be realized. The situation is then analogous to the air-wall chamber just dis- cussed. Electronic equilibrium requires that the wall thickness be at least as great as the maximum range of secondary charged particles. However, as with the air- wall chamber, the wall thickness should not be so great that the incident radiation is appreciably attenuated. The Bragg–Gray principle states that, if a gas is enclosed by a wall of the same atomic composition and if the wall meets the thickness conditions just given, then the energy absorbed per unit mass in the gas is equal to the number of ion pairs produced there times the W value divided by the mass m of the gas. Furthermore, the absorbed dose Dg in the gas is equal to the absorbed dose Dw in the wall. Denoting the number of ions in the gas by Ng , we write Ng W Dw = Dg =. (12.10) m When the wall and gas are of different atomic composition, the absorbed dose in the wall can still be obtained from the ionization in the gas. In this case, the cavity size and gas pressure must be small, so that secondary charged particles lose only a small fraction of their energy in the gas. The absorbed dose then scales as the ratio Sw /Sg of the mass stopping powers of the wall and gas: D g Sw Ng WSw Dw = =. (12.11) Sg mSg If neutrons rather than photons are incident, then in order to satisfy the Bragg– Gray principle the wall must be at least as thick as the maximum range of any secondary charged recoil particle that the neutrons produce in it. As with the air-wall chamber for measuring exposure, condenser-type chambers that satisfy the Bragg–Gray conditions can be used to measure absorbed dose. Prior to exposure, the chamber is charged. The dose can then be inferred from the re- duced potential difference across the instrument after it is exposed to radiation. The determination of dose rate is usually made by measuring the current due to ionization in a chamber that satisfies the Bragg–Gray conditions. As the following example shows, this method is both sensitive and practical. Example A chamber satisfying the Bragg–Gray conditions contains 0.15 g of gas with a W value of 33 eV ip–1. The ratio of the mass stopping power of the wall and the gas is 1.03. What is the current when the absorbed dose rate in the wall is 10 mGy h–1 ? 370 12 Radiation Dosimetry Solution We apply Eq. (12.11) to the dose rate, with Sw /Sg = 1.03. From the given conditions, Ḋw = 10 mGy h–1 = (0.010 J kg–1 )/(3600 s) = 2.78 × 10–6 J kg–1 s–1. The rate of ion- pair production in the gas is, from Eq. (12.11), Ḋw mSg Ṅg = WSw 2.78 × 10–6 J kg–1 s–1 × 0.15 × 10–3 kg = = 7.67 × 107 ip s–1. (12.12) 33 eV ip–1 × 1.60 × 10–19 J eV–1 × 1.03 Since the electronic charge is 1.60 × 10–19 C, the current is 7.67 × 107 × 1.60 × 10–19 = 1.23 × 10–11 C s–1 = 1.23 × 10–11 A. Simple electrometer circuits can be used to mea- sure currents smaller than 10–14 A, corresponding to dose rates much less than 10 mGy h–1 in this example. Note that Eq. (12.12) implies that the current is given by Ḋw meSg I = Ṅg e =. (12.13) WSw Expressing W = 33 J C–1 and remembering that the conversion factor from eV to J is numerically equal to the magnitude of the electronic charge e, we write, in SI units, Ḋw mSg I= (12.14) WSw 2.78 × 10–6 J kg–1 s–1 × 1.5 × 10–4 kg = = 1.23 × 10–11 A. (12.15) 33 J C–1 × 1.03 The arithmetic is thus shortened somewhat by using Eq. (12.14). However, one must be careful to keep units straight. 12.5 Measurement of X- and Gamma-Ray Dose Figure 12.4 shows the cross section of a spherical chamber of graphite that encloses CO2 gas. The chamber satisfies the Bragg–Gray conditions for photons over a wide energy range, and so the dose DC in the carbon wall can be obtained from the measured ionization of the CO2 by means of Eq. (12.11). Since carbon is a major constituent of soft tissue, the wall dose approximates that in soft tissue Dt. Calcu- lations show that, for photon energies between 0.2 MeV and 5 MeV, Dt = 1.1DC to within 5%. Thus soft-tissue dose can be measured with an accuracy of 5% with the carbon chamber. Generally, the dose in low-Z wall materials will approximate that in soft tissue over a wide range of photon energies. This fact leads to the widespread use of plas- tics and a number of other low-Z materials for gamma-dosimeter walls. The ratio of the absorbed dose in many materials relative to that in soft tissue has been cal- culated. Figure 12.5 shows several important examples. For photon energies from  12.6 Neutron Dosimetry 371 Fig. 12.4 Cross section of graphite-walled CO2 chamber for measuring photon dose. Fig. 12.5 Ratio of absorbed doses in bone, air, and carbon to that in soft tissue, Dt. ∼0.1 MeV to ∼10 MeV, the ratios for all materials of low atomic number are near unity, because Compton scattering dominates. The curve for bone, in contrast to the other two, rises at low energies due to the larger cross section of photoelectric absorption in the heavier elements of bone (e.g., Ca and P). 12.6 Neutron Dosimetry An ionization device, such as that shown in Fig. 12.4, used for measuring gamma- ray dose will show a reading when exposed to neutrons. The response is due to ionization produced in the gas by the charged recoil nuclei struck by neutrons in the walls and gas. However, the amount of ionization will not be proportional 372 12 Radiation Dosimetry Table 12.3 Principal Elements in Soft Tissue of Unit Density Element Atoms cm–3 H 5.98 × 1022 O 2.45 × 1022 C 9.03 × 1021 N 1.29 × 1021 Table 12.4 Relative Response of C CO2 Chamber to Neutrons of Energy E and Photons [Eq. (12.16)] Neutron Energy, E (MeV) P(E) 0.1 0.109 0.5 0.149 1.0 0.149 2.0 0.145 3.0 0.151 4.0 0.247 5.0 0.168 10.0 0.341 20.0 0.487 to the absorbed dose in tissue unless (1) the walls and gas are tissue equivalent and (2) the Bragg–Gray principle is satisfied for neutrons. As shown in Table 12.3, soft tissue consists chiefly of hydrogen, oxygen, carbon, and nitrogen, all having different cross sections as functions of neutron energy (cf. Fig. 9.2). The carbon wall of the chamber in Fig. 12.4 would respond quite differently from tissue to a field of neutrons of mixed energies, because the three other principal elements of tissue are lacking. The C CO2 chamber in Fig. 12.4 and similar devices can be used for neutrons of a given energy if the chamber response has been calibrated experimentally as a function of neutron energy. Table 12.4 shows the relative response P(E) of the C CO2 chamber to photons or to neutrons of a given energy for a ftuence that delivers 1 rad to soft tissue. If DnC (E) is the absorbed dose in the carbon wall due to 1 tissue rad of neutrons of energy E and Dγ C is the absorbed dose in the wall due to 1 tissue rad of photons, then, approximately, DnC (E) P(E) = γ. (12.16) DC Example A C CO2 chamber exposed to 1-MeV neutrons gives the same reading as that ob- tained when gamma rays deliver an absorbed dose of 2 mGy to the carbon wall. What absorbed dose would the neutrons deliver to soft tissue?  12.6 Neutron Dosimetry 373 Solution From Table 12.4, the neutron tissue dose Dn would be given approximately by the relation P(E)Dn = 0.149 Dn = 2 mGy, (12.17) or Dn = 13.4 mGy. Tissue-equivalent gases and plastics have been developed for constructing cham- bers to measure neutron dose directly. These materials are fabricated with the ap- proximate relative atomic abundances shown in Table 12.3. In accordance with the proviso mentioned after Eq. (12.11), the wall of a tissue-equivalent neutron cham- ber must be at least as thick as the range of a proton having the maximum energy of the neutrons to be monitored. More often than not, gamma rays are present when neutrons are. In monitoring mixed gamma–neutron radiation fields one generally needs to know the separate contributions that each type of radiation makes to the absorbed dose. One needs this information in order to assign the proper quality factor to the neutron part to obtain the dose equivalent. To this end, two chambers can be exposed—one C CO2 and one tissue equivalent—and doses determined by a difference method. The response RT of the tissue-equivalent instrument provides the combined dose, RT = Dγ + Dn. The reading RC of the C CO2 chamber can be expressed as RC = Dγ + P(E)Dn , where P(E) is an appropriate average from Table 12.4 for the neutron field in question. The individual doses Dγ and Dn can be inferred from RT and RC. Example In an unknown gamma–neutron field, a tissue-equivalent ionization chamber regis- ters 0.082 mGy h–1 and a C CO2 chamber, 0.029 mGy h–1. What are the gamma and neutron dose rates? Solution The instruments’ responses can be written in terms of the dose rates as ṘT = Ḋγ + Ḋn = 0.082 (12.18) and ṘC = Ḋγ + P(E)Ḋn = 0.029. (12.19) Since we are not given any information about the neutron energy spectrum, we must assume some value of P(E) in order to go further. We choose P(E) ∼ 0.15, representa- tive of neutrons in the lower MeV to keV range in Table 12.4. Subtracting both sides of Eq. (12.19) from (12.18) gives Ḋn = (0.082 – 0.029)/(1 – 0.15) = 0.062 mGy h–1. It follows from (12.18) that Ḋγ = 0.020 mGy h–1. Very often, as the example illustrates, the neutron energy spectrum is not known and the difference method may not be accurate. 374 12 Radiation Dosimetry Fig. 12.6 Hurst fast-neutron proportional counter. Internal alpha source in wall is used to provide pulses of known size for energy calibration. (Courtesy Oak Ridge National Laboratory, operated by Martin Marietta Energy Systems, Inc., for the Department of Energy.)  12.6 Neutron Dosimetry 375 As mentioned in Section 10.7, the proportional counter provides a direct method of measuring neutron dose, and it has the advantage of excellent gamma discrim- ination. The pulse height produced by a charged recoil particle is proportional to the energy that the particle deposits in the gas. The Hurst fast-neutron proportional counter is shown in Fig. 12.6. To satisfy the Bragg–Gray principle, the polyethylene walls are made thicker than the range of a 20-MeV proton. The counter gas can be either ethylene (C2 H4 ) or cyclopropane (C3 H6 ), both having the same H/C = 2 ratio as the walls. A recoil proton or carbon nucleus from the wall or gas has high LET. Unless only a small portion of its path is in the gas it will deposit much more energy in the gas than a low-LET secondary electron produced by a gamma ray. Re- jection of the small gamma pulses can be accomplished by electronic discrimina- tion. Fast-neutron dose rates as low as 10–5 Gy h–1 can be measured in the presence of gamma fields with dose rates up to 1 Gy h–1. In very intense fields signals from multiple gamma rays can “pile up” and give pulses comparable in size to those from neutrons. The LET spectra of the recoil particles produced by neutrons (and hence neutron quality factors) depend on neutron energy. Table 12.5 gives the mean quality fac- tors (based on Table 12.1) and fluence rates for monoenergetic neutrons that give a dose equivalent of 1 mSv in a 40-h work week. The quality factors have been com- puted by averaging over the LET spectra of all charged recoil nuclei produced by the neutrons. For practical applications, using Q = 3 for neutrons of energies less than 10 keV and Q = 10 for higher energies will result in little error. Using Q = 10 for all neutrons is acceptable, but may be overly conservative. Thus, in monitoring neutrons for radiation-protection purposes, one should generally know or estimate the neutron energy spectrum or LET spectrum (i.e., the LET spectrum of the re- coil particles). Measurement of LET spectra is discussed in Section 12.8. Several methods of obtaining neutron energy spectra were described in Section 10.7. The neutron rem meter, shown in Fig. 10.44, was discussed previously. Figure 12.7 shows an experimental setup for exposing anthropomorphic phan- toms, wearing various types of dosimeters, to fission neutrons. A bare reactor was positioned above the circle, drawn on the floor, with an intervening shield placed between it and the phantoms, located 3 m away. In this Health Physics Research Reactor facility, the responses of dosimeters to neutrons with a known energy spec- trum and fluence were studied. Intermediate and fast neutrons incident on the body are subsequently moder- ated and can be backscattered at slow or epithermal energies through the surface they entered. Exposure to these neutrons can therefore be monitored by wearing a device, such as a thermoluminescent dosimeter (TLD) enriched in 6 Li, that is sensitive to slow neutrons. Such a device is called an albedo-type neutron dosime- ter. (For a medium A that contains a neutron source and an adjoining medium B that does not, the albedo is defined in reactor physics as the fraction of neutrons entering B that are reflected or scattered back into A.) 376 12 Radiation Dosimetry Table 12.5 Mean Quality Factors Q and Fluence Rates for Monoenergetic Neutrons that Give a Maximum Dose-Equivalent Rate of 1 mSv in 40 h Neutron Energy (eV) Q Fluence Rate (cm–2 s–1 ) 0.025 (thermal) 2 680 0.1 2 680 1.0 2 560 10.0 2 560 102 2 580 103 2 680 104 2.5 700 105 7.5 115 5 × 105 11 27 106 11 19 5 × 106 8 16 107 6.5 17 1.4 × 107 7.5 12 6 × 107 5.5 11 108 4 14 4 × 108 3.5 10 Source: From Protection Against Neutron Radiation, NCRP Report No. 38, National Council on Radiation Protection and Measurements, Washington, D.C. (1971). In its 1987 Report No. 91, the NCRP recommends multiplying the above values of Q by two (and reducing the above fluence rates by this factor). 12.7 Dose Measurements for Charged-Particle Beams For radiotherapy and for radiobiological experiments one needs to measure the dose or dose rate in a beam of charged particles. This is often accomplished by measuring the current from a thin-walled ionization chamber placed at different depths in a water target exposed to the beam, as illustrated in Fig. 12.8. The dose rate is proportional to the current. For monoenergetic particles of a given kind (e.g., protons) the resulting “depth–dose” curve has the reversed shape of the mass stopping-power curves in Fig. 5.6. The dose rate is a maximum in the region of the Bragg peak near the end of the particles’ range. In therapeutic applications, absorbers or adjustments in beam energy are employed so that the beam stops at the location of a tumor or other tissue to be irradiated. In this way, the dose there (as well as LET) is largest, while the intervening tissue is relatively spared. To further spare healthy tissue, a tumor can be irradiated from several directions. If the charged particles are relatively low-energy protons (!400 MeV), then es- sentially all of their energy loss is due to electronic collisions. The curve in Fig. 12.8 will then be similar in shape to that for the mass stopping power. Higher-energy  12.8 Determination of LET 377 Fig. 12.7 Anthropomorphic phantoms, wearing a variety of dosimeters in different positions, were exposed to neutrons with known fluence and energy spectra at the Health Physics Research Reactor. (Courtesy Oak Ridge National Laboratory, operated by Martin Marietta Energy Systems, Inc., for the Department of Energy.) protons undergo significant nuclear reactions, which attenuate the protons and de- posit energy by nuclear processes. The depth–dose curve is then different from the mass stopping power. Other particles, such as charged pions, have strong nuclear interactions at all energies, and depth–dose patterns can be quite different. 12.8 Determination of LET To specify dose equivalent, one needs, in addition to the absorbed dose, the LET of incident charged particles or the LET of the charged recoil particles produced by incident neutral radiation (neutrons or gamma rays). As given in Tables 12.1 and 12.2, the required quality factors are defined in terms of the LET in water, which, for radiation-protection purposes, is the same as the stopping power. Stopping-power values of water for a number of charged particles are available and used in many applications. Radiation fields more often than not occur with a spectrum of LET values. H. H. Rossi and coworkers developed methods for inferring LET spectra directly 378 12 Radiation Dosimetry Fig. 12.8 Measurement of dose or dose rate as a function of depth in water exposed to a beam of charged particles. from measurements made with a proportional counter.2) A spherically shaped counter (usually tissue equivalent) is used and a pulse-height spectrum measured in the radiation field. If energy-loss straggling is ignored and the counter gas pres- sure is low, so that a charged particle from the wall does not lose a large fraction of its energy in traversing the gas, then the pulse size is equal to the product of the LET and the chord length. The distribution of isotropic chord lengths x in a sphere of radius R is given by the simple linear expression x P(x) dx = dx. (12.20) 2R2 Thus, the probability that a given chord has a length between x and x + dx is P(x) dx, this function giving unity when integrated from x = 0 to 2R. Using analytic tech- niques, one can, in principle, unfold the LET spectrum from the measured pulse- height spectrum and the distribution P(x) of track lengths through the gas. How- ever, energy-loss straggling and other factors complicate the practical application of this method. Precise LET determination presents a difficult technical problem. Usually, practi- cal needs are satisfied by using estimates of the quality factor or radiation weighting factor based on conservative assumptions. 2 Cf. Microdosimetry, ICRU Report 36, Units and Measurements, Bethesda, MD International Commission on Radiation (1983).  12.9 Dose Calculations 379 12.9 Dose Calculations Absorbed dose, LET, and dose equivalent can frequently be obtained reliably by calculations. In this section we discuss several examples. Alpha and Low-Energy Beta Emitters Distributed in Tissue When a radionuclide is ingested or inhaled, it can become distributed in various parts of the body. It is then called an internal emitter. Usually a radionuclide en- tering the body follows certain metabolic pathways and, as a chemical element, preferentially seeks specific body organs. For example, iodine concentrates in the thyroid; radium and strontium are bone seekers. In contrast, tritium (hydrogen) and cesium tend to distribute themselves throughout the whole body. If an in- ternally deposited radionuclide emits particles that have a short range, then their energies will be absorbed in the tissue that contains them. One can then calculate the dose rate in the tissue from the activity concentration there. Such is the case when an alpha or low-energy beta emitter is embedded in tissue. If A denotes the average concentration, in Bq g–1 , of the radionuclide in the tissue and E denotes the average alpha- or beta-particle energy, in MeV per disintegration, then the rate of energy absorption per gram of tissue is AE MeV g–1 s–1. The absorbed dose rate is MeV J g Ḋ = AE × 1.60 × 10–13 × 103 gs MeV kg = 1.60 × 10–10 AE Gy s–1. (12.21) Note that this procedure gives the average dose rate in the tissue that contains the radionuclide. If the source is not uniformly distributed in the tissue, then the peak dose rate will be higher than that given by Eq. (12.21). The existence of “hot spots” for nonuniformly deposited internal emitters can complicate a meaningful organ-dose evaluation. Nonuniform deposition can occur, for example, when in- haled particulate matter becomes embedded in different regions of the lungs. Example What is the average dose rate in a 50-g sample of soft tissue that contains 1.20 × 105 Bq of 14 C? Solution The average energy of 14 C beta particles is E = 0.0495 MeV (Appendix D). (As a rule of thumb, when not given explicitly, the average beta-particle energy can be assumed to be one-third the maximum energy.) The activity density is A = (1.20 × 105 s–1 )/(50 g). It follows directly from Eq. (12.21) that Ḋ = 1.90 × 10–8 Gy s–1. 380 12 Radiation Dosimetry Example If the tissue sample in the last example has unit density and is spherical in shape and the 14 C is distributed uniformly, make a rough estimate of the fraction of the beta-particle energy that escapes from the tissue. Solution We compare the range of a beta particle having the average energy E = 0.0495 MeV with the radius of the tissue sphere. The sphere radius r is found by writing 50 = 4π r3 /3, which gives r = 2.29 cm. From Table 6.1, the range of the beta particle is R = 0.0042 cm. Thus a beta particle of average energy emitted no closer than 0.0042 cm from the surface of the tissue sphere will be absorbed in the sphere. The fraction F of the tissue volume that lies at least this close to the surface can be calculated from the difference in the volumes of spheres with radii r and r – R. Alternatively, we can differentiate the expression for the volume, V = 4π r3 /3: dV 3 dr 3 × 0.0042 F= = = = 5.50 × 10–3. (12.22) V r 2.29 If we assume that one-half of the average beta-particle energy emitted in this outer layer is absorbed in the sphere and the other half escapes, then the fraction of the emitted beta-particle energy that escapes from the sphere is F/2 = 2.8 × 10–3 , a very small amount. Charged-Particle Beams Figure 12.9 represents a uniform, parallel beam of monoenergetic charged parti- cles of a given kind (e.g., protons) normally incident on a thick tissue slab with fluence rate ϕ̇ cm–2 s–1. To calculate the dose rate at a given depth x in the slab, we consider a thin, disc-shaped volume element with thickness !x in the x direc- tion and area A normal to the beam. The rate of energy deposition in the volume element is ϕ̇A(–dE/dx) !x, where –dE/dx is the (collisional) stopping power of the beam particles as they traverse the slab at depth x. [We ignore energy straggling (Chap. 7).] The dose rate Ḋ is obtained by dividing by the mass ρA !x of the vol- ume element, where ρ is the density of the tissue: ! " ϕ̇A(–dE/dx) !x dE Ḋ = = ϕ̇ –. (12.23) ρA !x ρ dx It follows that the dose per unit fluence at any depth is equal to the mass stopping power for the particles at that depth. If, for example, the mass stopping power is 3 MeV cm2 g–1 , then the dose per unit fluence can be expressed as 3 MeV g–1. This analysis assumes that energy is deposited only by means of electronic collisions (stopping power). As discussed in connection with Fig. 12.8, if significant nuclear interactions occur, for example, as with high-energy protons, then accurate depth– dose curves cannot be calculated from Eq. (12.23). One can then resort to Monte Carlo calculations, in which the fates of individual incident and secondary particles are handled statistically on the basis of the cross sections for the various nuclear interactions that can occur.  12.9 Dose Calculations 381 Fig. 12.9 Uniform, parallel beam of charged particles normally incident on thick tissue slab. Fluence rate = ϕ̇ cm–2 s–1. Point Source of Gamma Rays We next derive a simple formula for computing the exposure rate in air from a point gamma source of activity C that emits an average photon energy E per disin- tegration. The rate of energy release in the form of gamma photons escaping from the source is CE. Neglecting attenuation in air, we can write for the energy fluence rate, or intensity, through the surface of a sphere of radius r centered about the source %̇ = CE/(4π r2 ). For monoenergetic photons, it follows from Eq. (8.61) that the absorbed dose rate in air at the distance r from the source is µen CE µen Ḋ = %̇ =. (12.24) ρ 4π r2 ρ Here, µen /ρ is the mass energy-absorption coefficient of air for the photons. In- spection of Fig. 8.12 shows that this coefficient has roughly the same value for photons with energies between about 60 keV and 2 MeV: µen /ρ ∼ = 0.027 cm2 g–1 = 0.0027 m2 kg–1. Therefore, we can apply Eq. (12.24) to any mixture of photons in this energy range, writing CE 0.0027 2.15 × 10–4 CE Ḋ = =. (12.25) r2 4π r2 With C in Bq (s–1 ), E in J, and r in m, Ḋ is in Gy s–1. This relationship can be brought into a more convenient form. Expressing the activity C in Ci and the energy E in MeV, we have 2.15 × 10–4 × C × 3.7 × 1010 × E × 1.60 × 10–13 Ḋ = r2 1.27 × 10–6 CE = Gy s–1. (12.26) r2 382 12 Radiation Dosimetry Using hours as the unit of time and changing from dose rate Ḋ to exposure rate Ẋ [Eq. (12.4)] gives 1.27 × 10–6 CE Gy s 1R Ẋ = × 3600 × r2 s h 0.0088 Gy 0.5CE = R h–1. (12.27) r2 This simple formula can be used to estimate the exposure rate from a point source that emits gamma rays. The specific gamma-ray constant, &˙ , for a nuclide is defined by writing C Ẋ = &˙. (12.28) r2 This constant, which numerically gives the exposure rate per unit activity at unit distance, is usually expressed in R m2 Ci–1 h–1. Comparison with Eq. (12.27) shows that the specific gamma-ray constant in these units is given approximately by &˙ = 0.5E, with E in MeV. Example (a) Estimate the specific gamma-ray constant for 137 Cs. (b) Estimate the exposure rate at a distance of 1.7 m from a 100-mCi point source of 137 Cs. Solution (a) The isotope emits only a 0.662-MeV gamma ray in 85% of its transformations (Appendix D). The average energy per disintegration released as gamma radiation is therefore 0.85 × 0.662 = 0.563 MeV. The estimated specific gamma-ray constant for 137 Cs is therefore &˙ = 0.5E = 0.28 R m2 Ci–1 h–1. (b) From Eq. (12.28), the exposure rate at a distance r = 1.7 m from a point source of activity C = 100 mCi = 0. 1 Ci is R m2 0.1 Ci Ẋ = 0.28 × = 9.7 × 10–3 R h–1 = 9.7 mR h–1. (12.29) Ci h (1.7 m)2 The accuracy of the approximations leading to Eq. (12.27) varies from nuclide to nuclide. The measured specific gamma-ray constant for 137 Cs, 0.32 R m2 Ci–1 h–1 , is somewhat larger than the estimate just obtained. For 60 Co, which emits two gamma photons per disintegration, with energies 1.173 MeV and 1.332 MeV, the estimated specific gamma-ray constant is 0.5(1.173 + 1.332) = 1.3 R m2 Ci–1 h–1 , in agreement with the measured value. In addition to gamma rays, other photons can be emitted from a radionuclide. 125 I, for example, decays by electron capture, giving rise to the emission of char- acteristic X rays (the major radiation) plus a relatively infrequent, soft (35-keV) gamma photon. Internal bremsstrahlung from a beta particle (β – or β + ) or cap- tured electron accelerated near the nucleus can also occur, though this contribution is often negligible. The exposure-rate constant, &˙ δ , of a radionuclide is defined like  12.9 Dose Calculations 383 the specific gamma-ray constant, but includes the exposure rate from all photons emitted with energies greater than a specified value δ. In the case of 125 I, the spe- cific gamma-ray constant is 0.0042 R m2 Ci–1 h–1 and the exposure-rate constant is 0.13 R m2 Ci–1 h–1 for photons with energies greater than about 10 keV. Neutrons As discussed in Chapter 9, fast neutrons lose energy primarily by elastic scattering while slow and thermal neutrons have a high probability of being captured. The two principal capture reactions in tissue are 1 H(n,γ)2 H and 14 N(n,p)14 C. Slow neutrons are quickly thermalized by the body. The first capture reaction releases a 2.22-MeV gamma ray, which could deposit a fraction of its energy in escaping the body. In contrast, the nitrogen-capture reaction releases an energy of 0.626 MeV, which is deposited by the proton and recoil carbon nucleus in the immediate vicinity of the capture site. The resulting dose from exposure to thermal neutrons can be calculated, as the next example illustrates. Example Calculate the dose in a 150-g sample of soft tissue exposed to a fluence of 107 thermal neutrons cm–2. Solution From Table 12.3, the density of nitrogen atoms in soft tissue is N = 1.29 × 1021 cm–3 , 14 N being over 99.6% abundant. The thermal-neutron capture cross section is σ = 1.70 × 10–24 cm2 (Section 9.7). Each capture event by nitrogen results in the depo- sition of energy E = 0.626 MeV, which will be absorbed in the unit-density sample (ρ = 1 g cm –3 ). The number of interactions per unit fluence per unit volume of the tissue is Nσ. The dose from the fluence ϕ = 107 cm–2 is therefore ϕNσ E D= ρ 107 cm–2 × 1.29 × 1021 cm–3 × 1.70 × 10–24 cm2 × 0.626 MeV = 1 g cm–3 1.6 × 10–13 J 1 × × –3 = 2.20 × 10–6 Gy. (12.30) MeV 10 kg g–1 Some additional dose would be deposited by the gamma rays produced by the 1 H(n,γ)2 H reaction, for which the cross section is 3.3 × 10–25 cm2. However, in a tissue sample as small as 150 g, the contribution of this gamma-ray dose is negligi- ble. It is not negligible in a large target, such as the whole body. The absorbed dose from fast neutrons is due almost entirely to the energy trans- ferred to the atomic nuclei in tissue by elastic scattering. As discussed in Sec- tion 9.6, a fast neutron loses an average of one-half its energy in a single colli- sion with hydrogen. For the other nuclei in soft tissue, the average energy loss is 384 12 Radiation Dosimetry approximately one-half the maximum given by Eq. (9.3). These relationships facil- itate the calculation of a “first-collision” dose from fast neutrons in soft tissue. The first-collision dose is that delivered by neutrons that make only a single collision in the target. The first-collision dose closely approximates the actual dose when the mean free path of the neutrons is large compared with the dimensions of the tar- get. A 5-MeV neutron, for example, has a macroscopic cross section in soft tissue of 0.051 cm–1 , and so its mean free path is 1/0.051 = 20 cm. Thus, in a target the size of the body, a large fraction of 5-MeV neutrons will not make multiple col- lisions, and the first-collision dose can be used as a basis for approximating the actual dose. The first-collision dose is, of course, always a lower bound to the actual dose. Moreover, fast neutrons deposit most of their energy in tissue by means of collisions with hydrogen. Therefore, calculating the first-collision dose with tissue hydrogen often provides a simple, lower-bound estimate of fast-neutron dose. Example Calculate the first-collision dose to tissue hydrogen per unit fluence of 5-MeV neu- trons. Solution The density of H atoms is N = 5.98 × 1022 cm–3 (Table 12.3) and the cross section for scattering 5-MeV neutrons is σ = 1.61 × 10–24 cm2 (Fig. 9.2). The mean energy loss per collision, Qavg = 2.5 MeV, is one-half the incident neutron energy. The dose per unit neutron fluence from collisions with hydrogen is therefore (tissue density ρ = 1 g cm–3 ) Nσ Qavg 5.98 × 1022 cm–3 × 1.61 × 10–24 cm2 × 2.5 MeV D= = ρ 1 g cm–3 1.6 × 10–13 J MeV–1 × 10–3 kg g–1 = 3.85 × 10–11 Gy cm2. (12.31) Note that the units of “Gy per (neutron cm–2 )” are Gy cm2. Similar calculations of the first-collision doses due to collisions of 5-MeV neu- trons with the O, C, and N nuclei in soft tissue give, respectively, contributions of 0.244 × 10–11 , 0.079 × 10–11 , and 0.024 × 10–11 Gy cm2 , representing in total about an additional 10%. Detailed analysis shows that hydrogen recoils contribute approxi- mately 85–95% of the first-collision soft-tissue dose for neutrons with energies be- tween 10 keV and 10 MeV. Table 12.6 shows the analysis of first-collision neutron doses. Detailed calculations of multiple neutron scattering and energy deposition in slabs and in anthropomorphic phantoms, containing soft tissue, bone, and lungs, have been carried out by Monte Carlo techniques (Section 11.13). Computer pro- grams are available, based on experimental cross-section data and theoretical algo- rithms, to transport individual neutrons through a target with the same statistical  12.9 Dose Calculations 385 Table 12.6 Analysis of First-Collision Dose for Neutrons in Soft Tissue Neutron First-Collision Dose per Unit Neutron Fluence for Collisions with Various Energy Elements (10–11 Gy cm2 ) (MeV) H O C N Total 0.01 0.091 0.002 0.001 0.000 0.094 0.02 0.172 0.004 0.001 0.001 0.178 0.03 0.244 0.005 0.002 0.001 0.252 0.05 0.369 0.008 0.003 0.001 0.381 0.07 0.472 0.012 0.004 0.001 0.489 0.10 0.603 0.017 0.006 0.002 0.628 0.20 0.914 0.034 0.012 0.003 0.963 0.30 1.14 0.052 0.016 0.003 1.21 0.50 1.47 0.122 0.023 0.004 1.62 0.70 1.73 0.089 0.029 0.005 1.85 1.0 2.06 0.390 0.036 0.007 2.49 2.0 2.78 0.156 0.047 0.012 3.00 3.0 3.26 0.205 0.045 0.018 3.53 5.0 3.88 0.244 0.079 0.024 4.23 7.0 4.22 0.485 0.094 0.032 4.83 10.0 4.48 0.595 0.157 0.046 5.28 14.0 4.62 1.10 0.259 0.077 6.06 Source: From “Measurement of Absorbed Dose of Neutrons and Mixtures of Neutrons and Gamma Rays,” National Bureau of Standards Handbook 75, Washington, D.C. (1961). distribution of events that neutrons have in nature. Such Monte Carlo calculations can be made under general conditions of target composition and geometry as well as incident neutron spectra and directions of incidence. Compilations of the re- sults for a large number of neutrons then provide dose and LET distributions as functions of position, as well as any other desired information, to within the statis- tical fluctuations of the compilations. Using a larger number of neutron histories reduces the variance in the quantities calculated, but increases computer time. Figure 12.10 shows the results of Monte Carlo calculations carried out for 5-MeV neutrons incident normally on a 30-cm soft-tissue slab, approximating the thick- ness of the body. (The geometry is identical to that shown for the charged particles in Fig. 12.9.) The curve labeled ET is the total dose, Ep is the dose due to H recoil nu- clei (protons), Eγ is the dose from gamma rays from the 1 H(n,γ)2 H slow-neutron capture reaction, and EH is the dose from the heavy (O, C, N) recoil nuclei. The total dose builds up somewhat in the first few cm of depth and then decreases as the beam becomes degraded in energy and neutrons are absorbed. The proton and heavy-recoil curves, Ep and EH , show a similar pattern. As the neutrons penetrate, they are moderated and approach thermal energies. This is reflected in the rise of the gamma-dose curve, Eγ , which has a broad maximum over the region from 386 12 Radiation Dosimetry Fig. 12.10 Depth–dose curves for a broad beam of 5-MeV neutrons incident normally on a soft-tissue slab. Ordinate gives dose per unit fluence at different depths shown by the abscissa. [From “Protection Against Neutron Radiation Up to 30 Million Electron Volts,” in National Bureau of Standards Handbook 63, p. 44, Washington, D.C. (1957).] about 6 cm to 14 cm. Note that the total dose decreases by an order of magnitude between the front and back of the slab. The result of our calculation of the first-collision dose, D = 3.85 × 10–11 Gy cm2 = 3.85 × 10–9 rad cm2 , due to proton recoils in the last example can be compared with the curve for Ep in Fig. 12.10. At the slab entrance, Ep = 4.8 × 10–9 rad cm2 is greater than D, which, as we pointed out, is a lower bound for the actual dose from proton recoils. A number of neutrons are back-scattered from within the slab to add to the first-collision dose deposited directly by the incident 5-MeV neutrons.  12.10 Other Dosimetric Concepts and Quantities 387 12.10 Other Dosimetric Concepts and Quantities Kerma A quantity related to dose for indirectly ionizing radiation (photons and neutrons) is the initial kinetic energy of all charged particles liberated by the radiation per unit mass. This quantity, which has the dimensions of absorbed dose, is called the kerma (Kinetic Energy Released per unit MAss). Kerma was discussed briefly for photons in Section 8.9 in connection with the mass energy-transfer coefficient [Eq. (8.62)]. By definition, kerma includes energy that may subsequently appear as bremsstrahlung and it also includes Auger-electron energies. The absorbed dose generally builds up behind a surface irradiated by a beam of neutral particles to a depth comparable with the range of the secondary charged particles generated (cf. Fig. 12.10). The kerma, on the other hand, decreases steadily because of the attenuation of the primary radiation with increasing depth. The first-collision “dose” calculated for neutrons in the last section is, more pre- cisely stated, the first-collision “kerma.” The two are identical as long as all of the initial kinetic energy of the recoil charged particles can be considered as being absorbed locally at the interaction site. Specifically, kerma and absorbed dose at a point in an irradiated target are equal when charged-particle equilibrium exists there and bremsstrahlung losses are negligible. It is often of interest to consider kerma or kerma rate for a specific material at a point in free space or in another medium. The specific substance itself need not actually be present. Given the photon or neutron fluence and energy spectra at that point, one can calculate the kerma for an imagined small amount of the material placed there. It is thus convenient to describe a given radiation field in terms of the kerma in some relevant, or reference, material. For example, one can specify the air kerma at a point in a water phantom or the tissue kerma in air. Additional information on kerma can be found in the references listed in Sec- tion 12.11. Microdosimetry Absorbed dose is an averaged quantity and, as such, does not specifically reflect the stochastic, or statistical, nature of energy deposition by ionizing radiation in mat- ter. Statistical aspects are especially important when one considers dose in small regions of an irradiated target, such as cell nuclei or other subcellular components. The subject of microdosimetry deals with these phenomena. Consider, for exam- ple, cell nuclei having a diameter ∼5 µm. If the whole body receives a uniform dose of 1 mGy of low-LET radiation, then 23 of the nuclei will have no ionizations at all and 13 will receive an average dose of ∼3 mGy. If, on the other hand, the whole body receives 1 mGy from fission neutrons, then 99.8% of the nuclei will receive 388 12 Radiation Dosimetry no dose and 0.2% will have a dose of ∼500 mGy.3) The difference arises from the fact that the neutron dose is deposited by recoil nuclei, which have a short range. A proton having an energy of 500 keV has a range of 8 × 10–4 cm = 8 µm in soft tissue (Table 5.3), compared with a range of 0.174 cm for a 500-keV electron (Table 6.1). Both particles deposit the same energy. The proton range is comparable to the cell-nucleus diameter; the electron travels the equivalent of ∼1740/5 = 350 nuclear diameters. Specific Energy When a particle or photon of radiation interacts in a small volume of tissue, one refers to the interaction as an energy-deposition event. The energy deposited by the incident particle and all of the secondary electrons in the volume is called the energy imparted, ϵ. Because of the statistical nature of radiation interaction, the energy imparted is a stochastic quantity. The specific energy (imparted) in a volume of mass m is defined as ϵ z=. (12.32) m It has the dimensions of absorbed dose. When the volume is irradiated, it experi- ences a number of energy-deposition events, which are characterized by the single- event distribution in the values of z that occur. The average absorbed dose in the volume from a number of events is the mean value of z. Studies of the distributions in z from different radiations in different-size small volumes of tissue are made in microdosimetry. Similarly, one can regard an ensemble of identical small volumes throughout an irradiated body and the distribution of specific energy in the volumes due to any number of events. Thus, the example cited from the BEIR-III Report in the next-to-last paragraph can be conveniently described in terms of the distribution of specific energy z in the cell nuclei. For the low-LET radiation, 23 of the nuclei have z = 0; in the other 13 , z varies widely with a mean value of ∼3 mGy. For the fission neutrons, 99.8% of the nuclei have z = 0; in the other 0.2%, z varies by many orders of magnitude with a mean value of ∼500 mGy. The stochastic specific energy plays an important role as the microdosimetric analogue of the conventional absorbed dose, which is a nonstochastic quantity. Lineal Energy The lineal energy y is defined as the ratio of the energy imparted ϵ from a single event in a small volume and the mean length x̄ of isotropic chords through the volume: ϵ y=. (12.33) x̄ 3 The Effects on Populations of Exposure to Low Report, p. 14, National Academy of Sciences, Levels of Ionizing Radiation: 1980, BEIR-III Washington, D.C. (1980).  12.11 Suggested Reading 389 Lineal energy, which is a stochastic quantity, has the same dimensions as LET, which is nonstochastic (being the mean value of the linear rate of energy loss). Lin- eal energy is the microdosimetric analogue of LET. Unlike specific energy, however, it is defined only for single events. A relationship between lineal energy and LET can be seen as follows. Consider a small volume containing a chord of length x traversed by a charged particle with LET = L. We ignore energy-loss straggling and assume that the energy lost by the particle in the volume is absorbed there. We assume further that the chord is so short that the LET is constant over its length. The energy imparted by the single traversal event is ϵ = Lx. For isotropic irradiation of the volume by particles travers- ing it with LET = L, the mean value of the imparted energy is ϵ̄ = Lx̄. Under these conditions, it follows from the definition (12.33) of the lineal energy that its mean value is the LET: ȳ = ϵ̄/x̄ = L. For any convex body, having surface area S and volume V, traversed by isotropic chords, the mean chord length is given quite generally by the Cauchy relation, x̄ = 4V/S. For a sphere of radius R, it follows that x̄ = 4R/3 (Problem 60). Proposals have been made to use lineal energy instead of LET as a basis for defining quality factors in radiation-protection work. Whereas the measurement of LET spectra is a difficult technical problem, distributions of lineal energy and its frequency- and dose-mean values can be readily measured for many radiation fields. Disadvantages of using lineal energy include the necessity of specifying a universal size for the reference volume, usually assumed to be spherical in shape. There does not appear to be a compelling reason for any particular choice, and the y distributions depend upon this specification. In addition, concepts associated with chords are probably inappropriate for application to the tortuous paths of electrons, especially at low energies. 12.11 Suggested Reading 1 Attix, F. H., Introduction to Radiolog- 4 ICRU Report 60, Fundamental Quan- ical Physics and Radiation Dosimetry, tities and Units for Ionizing Radiation, Wiley, New York (1986). [Clear and International Commission on Ra- rigorous treatments (and in much diation Units and Measurements, greater depth) of subjects in this chap- Bethesda, MD (1998). [Provides de- ter. Dosimetry fundamentals and, finitions of fundamental quantities especially, instrumentation and mea- related to radiometry, interaction co- surements are described.] efficients, dosimetry, and radioactiv- 2 Cember, H., Introduction to Health ity. Gives standardized symbols and Physics, 3rd Ed., McGraw-Hill, New units.] York (1996). 5 Martin, J. E., Physics for Radiation Pro- tection: A Handbook, 2nd Ed., Wiley, 3 ICRU Report 36, Microdosimetry, In- New York (2006). ternational Commission on Radiation Units and Measurements, Bethesda, 6 Rossi, H. H., and M. Zaider, Micro- MD (1983). dosimetry and its Applications, Springer Verlag, Berlin (1996).