# Relationship Between Radionuclide Gamma Emission and Exposure Rate

George Chabot, CHP, PhD

The quantity exposure, usually symbolized X, and commonly expressed in units of roentgens (R) or milliroentgens (mR), is a quantity that reflects the extent of ionization events taking place when air is irradiated by ionizing photons (gamma radiation and/or x rays). In particular, it is a measure of the amount of electric charge of one sign (either positive or negative) produced per unit mass of air when electrons produced by photon interactions in a small mass of air are allowed to dissipate all of their kinetic energy in air. In the international system of units exposure is measured in coulombs per kg (C/kg), and 1 R = 2.58 x 10^{-4} C/kg. The exposure rate expresses the rate of charge production per unit mass of air and is commonly expressed in roentgens per hour (R/h) or milliroentgens per hour (mR/h). Embodied in the definition of exposure is a condition referred to as charged particle equilibrium (CPE) which, stated briefly, says that when a material is irradiated with ionizing radiation under CPE conditions the types, numbers, energies, and directions of charged particles entering a small volume of the material are equal to the types, numbers, energies, and directions of charged particles leaving that same volume element. While this condition may presently seem somewhat obscure and of dubious relevance, it happens to be a most-important principle that lies at the foundation of the calculations that we frequently do, and will do in the examples that follow, that relate radiation exposure to gamma emissions from selected radionuclides. It also is a critical principle that must be invoked in the design of instrumentation intended to measure exposure rate from gamma radiation or x-ray sources. The same principle has equally important implications when doing other radiation dose calculations and measurements.

The degree of production of charged particles by gamma- and/or x-ray interactions in air depends on (1) the intensity of these radiations, which we shall express through a quantity called the fluence rate, which has commonly used c.g.s. units of number of photons per square centimeter per second, cm^{-2}s^{-1}, (2) the energies of the photons, E, commonly expressed in millions of electron-volts, MeV, (3) the average fraction of the photon energy transferred to charged particles and available to be absorbed per photon interaction per unit mass density thickness traversed, expressed through a quantity called the mass energy absorption coefficient, symbolized μ_{en}/ρ, and having dimensions of square centimeters per gram, cm^{2}g^{-1}, and (4) the average amount of energy required to produce an ionization event in air, measured through a quantity called the w-value for air, and whose value varies slightly depending on the amount of humidity in the air, and is about 33.8 electron-volts per ion pair (an ion pair is the positive ion and negative electron produced by an ionization event) at 50% relative humidity at 22°C.

In the initial evaluation below we shall assume that we are concerned with a single radionuclide that emits gamma rays or x rays of a single energy. We will later extend the discussion to multiple photon energies. We shall also assume that the radionuclide source is a physically small source that emits photons uniformly in all directions; such sources are often referred to as point isotropic sources. In practice, if photon attenuation in the source is negligible, a source may be treated as a point isotropic source, from a calculation viewpoint, if the distance from the center of the source to the exposure point is at least three times the largest source dimension. We shall make the additional assumptions that there is no attenuation of photons either within the source itself or in any intervening material between the source and the dose point of interest and that the dose point lies at a distance r (cm) in air from the source.

The fluence rate, φ, can be calculated easily if we recognize that all of the photons emitted from the source will pass uniformly through the surface of an imaginary sphere whose surface area is given by 4πr^{2}. Thus, if the source emits S photons per second,

φ = S/4πr^{2} (1)

Note that if A represents the activity of the source, in becquerels (1 Bq = 1 disintegration per second), and y represents the yield of photons (the number of photons of energy E emitted per disintegration of the radionuclide), we can write S as

S = Ay, and φ = Ay/4πr^{2}. (2)

If we multiply the fluence rate by the photon energy we obtain what is referred to as the photon energy fluence rate, symbolized ψ:

ψ = AyE/4πr^{2}. (3)

We can now obtain the energy absorption rate per unit mass, resulting from energy made available to charged particles by the photon interactions, by multiplying ψ by the mass energy absorption coefficient. Thus,

ψμ_{en}/ρ = AyEμ_{en}/ρ/4πr^{2}. (4)

The exposure rate can be obtained from the energy absorption rate per unit mass by converting MeV to eV, dividing by the w-value for air to convert to ion pairs per unit mass per unit time, multiplying by a conversion constant that represents the charge of one sign associated with one ion pair (1.602 x 10^{-19} coulombs), converting mass from grams to kilograms, converting time from seconds to hours, and finally applying the coulomb/kilogram-to-roentgen conversion factor mentioned earlier, all as follows:

. | ||||

X | = | AyE(μ_{en}/ρ)(10^{6}eV/MeV)(1/w)(1.602x10^{-19} C/ip)(1000 g/kg)(3600 s/h)(1R/2.58x10^{-4} C/kg)/4πr^{2}. |

Using a w-value of 33.8 ev/ip and combining all the numerical terms, including π, we obtain

. | ||||

X | = | 5.263 x 10^{-6} AyE(μ_{en}/ρ)/r^{2} (5) |

To use equation 5, we must maintain appropriate unitsi.e., A in Bq, E in MeV, μ_{en}/ρ in cm^{2}/g, and r in cm.

If the radionuclide of interest emits other photons with additional energies, we can use equation 5 independently for each energy and add the results at the end. We can combine this operation into a single equation by simply modifying equation 5 as follows:

. | ||||

X | = | 5.263 x 10^{-6} A∑(y_{i}E_{i}(μ_{en}/ρ)_{i})/r^{2}, (6)
5.263 x 10 ^{-6} A ^{i}(y_{i}E_{i}(μ_{en}/ρ)_{i})/r^{2}, (6) |

where the summation symbol,

∑ |

^{i} |

_{i}E

_{i}(μ

_{en}/ρ)

_{i}, should be evaluated at each energy, i, and the resulting products added together.

As an example we will calculate the expected exposure rate at a point in air 50 cm away from a 100 mCi small source of ^{131}I. We can locate required decay data, in particular the gamma energies and yields, at a National Nuclear Data website established by Brookhaven National Laboratory. If we enter the nucleus of interest, ^{131}I, and select "Gamma" from the "Radiation Type" drop down list, and select the "Ordering" sequence "I,E,Z,A,T12" and click on "Search" we see the list of photons emitted by ^{131}I. There are 24 listed photons (gamma rays and x rays) arranged in order of increasing yield. We shall restrict our consideration here to photons that have yields greater than 1%. These are listed in the table below. The first energy listed is the average of two closely spaced energies given on the Web page. The values of the mass energy absorption coefficients for the photons in air were obtained (by interpolation) from a Web page maintained by the National Institute of Standards and Technology.

Energy, MeV | Fractional yield | (μ_{en}/ρ)_{air} |
---|---|---|

0.02962 | 0.0392 | 0.160 |

0.08019 | 0.0262 | 0.0241 |

0.2843 | 0.0614 | 0.0284 |

0.3645 | 0.817 | 0.0292 |

0.6370 | 0.0717 | 0.0294 |

0.7229 | 0.0177 | 0.0291 |

Applying equation 6 after converting the activity of 100 mCi to Bq (3.7x10^{9} Bq), we obtain

. | ||||

X | = | 5.263 x 10^{-6} (3.7 x 10^{9})[(0.0392)(0.02962)(0.160)+(0.0262)(0.08019)(0.0241) + (0.0614)(0.2843)(0.0284) + (0.817)(0.3645)(0.0292) + (0.0717)(0.637)(0.0294) + (0.0177)(0.7229)(0.0291)]/(50)^{2} = 0.0868 R/h or 86.8 mR/h. |

When using equation 6 or similar expressions to estimate exposure rates from radionuclide sources, we must sometimes make judgments about whether certain photons that are part of the decay scheme will actually contribute to the exposure rate. This is especially true when low-energy photons are prevalent and the source material or encapsulation may prevent such photons from reaching the point of interest. Additionally, if the point of interest is a relatively large distance from the source, the air between the source and exposure point may cause significant attenuation of the low-energy photons.

As a final note we should point out that it is common practice, when estimating exposure rates from point isotropic sources, to use "rule-of-thumb" expressions. The most common such expression for exposure rate in R/h at one foot in air from C curies of activity of a radionuclide that emits n photons of energy E (MeV) per disintegration is

. | ||||

X | = | 6CEn. (7) |

This expression can be justified from equation 5 if we can assume a constant value for the mass energy absorption coefficient. A review of the values of (μ_{en}/ρ)_{air} as a function of energy show that from roughly 0.1 MeV to 2 MeV the value does not change by more than about 25%, being largest at about 0.5 MeV with a value of about 0.03 cm^{2}/g. If we use this value of 0.03 cm^{2}/g in equation 5, use 3.7 x 10^{10} Bq (1 Ci) for A, and 30.5 cm (one foot) for r, equation 5 reduces to the equivalent of equation 7 (when C = 1) with the constant 6.28 which is rounded down to 6. Such simplified equations may often be reasonable for quick estimations, but it is important to recognize that they are approximations; they can lead to significantly erroneous results when applied outside of their relevant energy range, especially energies appreciably lower than 0.1 MeV where the values of (μ_{en}/ρ)_{air} may be considerably larger than the assumed value of 0.03 cm^{2}/g.