Appendix IDosimetry of Alpha Particles

Publication Details

The name alpha particle was given to the energetic helium nuclei (that is, helium atoms stripped of their two electrons) emitted in radioactive decay. They are emitted with energies in the range of 4 to 9 MeV. Because they are always the same kind of particle, regardless of the nucleus from which they come, all alpha particles of a given energy have the same properties.

Properties of Alpha Particles


The tracks of alpha particles are nearly straight lines, meaning that they are scattered very little by the material through which they pass. In dosimetric calculations it is usual to assume that the tracks are straight lines.

Alpha particles do undergo scattering, however. One of the earliest developments in nuclear physics was the discovery, by Rutherford, of the alpha-particle scattering law. He found that very small deflections, a degree or so, are frequent and that larger deflections do occur but are quite rare. Consequently, along a typical track, each time an alpha particle scatters it changes direction only slightly and in a random direction, with the result that the track winds back and forth a small amount around a straight line. This multiple scattering contributes to the straggling discussed below.

Stopping Power

As alpha particles go through a material, they interact with the material's molecules, losing a little energy in each interaction. Thus, they gradually slow down. At very low energies (less than 1 eV) they acquire two electrons and become atoms of helium gas in thermal equilibrium with the material. Alpha particles of the same energy lose different energies when they go equal distances because of randomness in the number and in the kinds of interactions with the material. The differences in the energy losses are small, however. The actual rates of energy loss are close to the average rate.

The stopping power (S) of a charged particle of specified energy is its average energy loss per unit distance along its path. The mass stopping power (S/ρ) is the quotient of the stopping power by the density of the material.

The experimental and theoretical determinations of the alpha-particle stopping powers are in good agreement.1,14 Figure I-1 shows the stopping power of alpha particles in soft tissues of unit density. For comparison, the stopping power of electrons is also shown in Figure I-1.6 (Beta particles, the other common kind of charged particle emitted in radioactive decay, are electrons.) Note that for the energies shown, the stopping powers of the alpha particles are 2 to 3 orders of magnitude larger than those of the electrons. As a consequence, the damage (for example, caused by ionization, excitation, chemical alteration, and biological damage) is generally about that much higher along the alpha-particle tracks.

Figure I-1. Stopping powers of alpha particles and electrons in soft tissue of unit density.

Figure I-1

Stopping powers of alpha particles and electrons in soft tissue of unit density.

The mass stopping powers of alpha particles are slightly higher in gases than in liquids or solids.13 The stopping powers in Figure I-1 are for solid tissue and are about 5% higher than the mass stopping powers for gases.

Linear Energy Transfer

The energy lost by a charged particle produces damage of various kinds in the material with an effectiveness that depends on how densely the energy it loses is spread in the material. The stopping power is one indication of that density, but it is not a complete measure: Some of the energy lost by the particle is transferred to secondary radiations, electrons and photons, that penetrate to distances from the particle track. Several investigators have used various other quantities to characterize the results of this spreading process.3

One of the quantities used for this purpose is the linear energy transfer (LET; also referred to as L). While the stopping power gives the rate of energy loss of the particle, the LET gives the rate at which energy is laid down close to the particle track. This is done by excluding the energy carried away by photons and energetic electrons from the energy lost by the particle. Two criteria have been used to specify which electrons are to be excluded: (1) electrons with energies above some limit, and (2) electrons with ranges above some limit. In either case it is important to specify the limit; this is usually done by writing it as a subscript to the symbol L. The energy limit is the current preference, because the LET can be calculated from theoretical equations for the Stopping power by simply excluding energy losses to secondary electrons above the selected limit.

As an example, a 4-MeV alpha particle has a stopping power of 102 keV µm-1; the LET excluding losses of more than 100 eV is L 100 = 56 keV µm-1; 1,000 eV, L 1,000 = 81 keV µm-1; 10,000 eV, L 10,000 = 102 keV µm-1. Losses of 10 keV are above the maximum energy, about 2,000 eV, that a 4-MeV alpha particle can lose to an electron; therefore, the LET for that cutoff does not differ from the stopping power. It is customary to designate the LET for energy limits larger than the maximum energy transferable by L Image img00167.jpg. L Image img00167.jpg is the largest value possible for the LET, and it equals the stopping power S.


Alpha particles have a fairly sharply defined range (R), which is the average distance that they travel in coming to rest in a material. Figure I-2 shows the ranges of alpha particles in soft tissue of unit density. For comparison, the average distances traveled by electrons, neglecting energy straggling (see below), are also given. Because of their very much lower stopping powers, the electrons travel 2 or 3 orders of magnitude farther than alpha particles.

Figure I-2. Ranges of alpha particles and electrons in soft tissue of unit density.

Figure I-2

Ranges of alpha particles and electrons in soft tissue of unit density.

The stopping powers and ranges in tissue suffice for most applications to alpha-particle dosimetry. Occasionally, however, it is necessary to consider alpha-particle paths that lie partly in one material, partly in another (e.g., partly in soft tissue and partly in bone). Fortunately, the ranges of an alpha particle of a given energy in different materials are proportional to one another, independently of the energy, to within a few percent.


Random variations in the energy lost and in the change in direction in individual interactions with the molecules of the material produce distributions in the actual distances traveled by different alpha particles; this is known as range straggling. Similarly, there is a distribution in the energy remaining after traveling a given distance; this is known as energy straggling.

The probability distribution of the actual ranges is represented fairly accurately by a normal distribution (it neglects the effects of occasional large energy losses in individual collisions). Theoretical derivations of parts of the variance, σ2(R), of the normal distribution have been made, but the observed variances are always larger. Evans2 recommended a relation that he estimated to be accurate to within 10% for range straggling in air:

Image img00116.jpg

where R is the range. This relation shows that, in air, most of the actual distances traveled are within a few percent of the mean distance (the range); theoretical analyses suggest similar narrow distributions for the distances in tissue.


Exposure and Dose

A basic step in the study of the risk associated with an agent is the establishment of the relation between the degree of harm it produces and some physically measurable quantity that characterizes its prevalence. The measurable quantity is often referred to as the exposure to the agent. The exposure is seldom the concentration of the agent (or of a product of the agent) in the specific cells or tissues where the harm is thought to arise. The latter, or some closely related quantity, is often called the ''dose." Because of the difficulty in making measurements within the body, it is usually hard to determine the dose. The exposure, on the other hand, is usually the concentration outside the body m some material in which it is easier to measure. Usually it is the concentration in the material that is the main carrier of the agent into the body.

The working level used in the study of radon and its daughters is a good example of these general statements; it is treated at length in Annex 2B to Chapter 2. In brief, the working level, an exposure quantity, is a concentration of radon and its daughters in air, and is reasonably easy to measure. The corresponding dose quantity might be the number of atoms of radon and its daughters deposited at some point in the respiratory tract The relation between the concentration in air and the number of atoms deposited depends on a host of variables (for example, the structure of the respiratory tract and breathing rate; see Chapter 2). This lack of a unique correlation between the exposure and dose quantities is typical.

There is danger of confusion in discussing the exposure and dose concepts: A particular quantity named exposure was introduced early into radiation studies to characterize x-ray and gamma-ray fields; this exposure is the one measured in roentgens.4 Fortunately, a need to use this particular exposure seldom arises in studying internally deposited alpha emitters; thus, the term can usually be used here in its general sense.

Absorbed Dose

Radiation studies employ the dose concept in the quantity called absorbed dose. The determination of absorbed dose is called dosimetry. The International Commission on Radiation Units and Measurements (ICRU) defines absorbed dose to be the mean energy imparted to the irradiated medium, per unit mass, by ionizing radiation. (For definitions of absorbed dose and the other quantities used in dosimetry, see ICRU.4 )

The energy of ionizing radiation is imparted to the medium in a series of individual interactions with it. The number of interactions and the amount of energy lost in each are random variables. The word mean used in the definition of absorbed dose requires that the average of the energies imparted be used. In what follows, up to the section "Microdosimetry," it is assumed that the average has been taken; in the section "Microdosimetry" the probability distribution of the energy imparted whose mean is the one required for the absorbed dose will be dealt with.

Average Absorbed Dose

Often one can be satisfied (see the section "Nonequilibrium Doses" below) with the average absorbed dose in some volume. If the range of the alpha particle is much smaller than the dimensions of the volume, most of the alpha particles emitted within that volume are absorbed within it; that is, they impart their energy within it. Only those emitted close to the surface and headed through the bounding surface can escape and impart their energy elsewhere (or particles emitted Outside t lie volume can impart energy to it). In many circumstances, this leakage in and out is negligible, because far more particles are emitted within the volume than are emitted close enough to the surface to leak in or out. In these circumstances the average absorbed dose, <D>, in the particular tissue equals the product of the number of alpha particles emitted within it and their energy, E, divided by the mass of the tissue. Let <C> be the number of particles emitted divided by the mass of the tissue, that is, the mean number emitted per unit mass. Then:

Image img00117.jpg

Charged-particle Equilibrium

The leakage is also negligible (actually, zero) in another situation that is representative of many experimental situations. Suppose the tissue and its surroundings are of uniform composition and the number of alpha particles emitted per unit mass (C) is constant throughout the volume of interest and for some distance, greater than the range, into the material on all sides of it (see the next section). The net leakage is then zero, because there is as much leakage into the volume of interest as leakage out of it. This condition is known as charged-particle equilibrium, and the dose (D) is given by:

Image img00118.jpg

Nonequilibrium Doses

When the average dose does not suffice or when charged-particle equilibrium does not exist, the dose at a point can be calculated from the local density of alpha-particle emission (C) in all elementary volumes (dV) within the alpha-particle range of the point. The number of alpha particles emitted from a particular dV is C ρ dV, where ρ is the density of the medium (assumed constant in the neighborhood of the point). The number of these alpha particles per unit area at the point is (C ρ dV)/(4πr2), where r is the distance to the point. The number entering an elementary target volume at the point and with area dA facing dV is dA times the number per unit area. Each particle imparts an energy, denoted by e(r) dx;, to the target, where dx is the thickness of the target. The mass of the target volume is ρdAdx;. Thus, the dose to the target from the alpha particles emitted in the particular dV is:

Image img00119.jpg

The total dose is obtained by integrating over all dV within range of the point. Several factors cancel to give, for the dose:

Image img00120.jpg

Different approximations have been used for the kernel e(r). One approximation is to equate it to the stopping power: e(r) = S. This expression is approximate for two reasons. First, S gives the energy lost by the alpha particle, not the energy imparted to the medium in the target. Secondary radiations, electrons (called delta rays) and photons, leak energy into and out of the target, as discussed above. Because of the very short ranges of these secondary radiations, leakages in and out tend to compensate each other and make the approximation a good one. Second, because of the straggling described above, there is a spread in the energies of the alpha particles arriving at the target element. This straggling causes, a larger error than the leakage just mentioned. An average, <S>, of the stopping power over the straggling spectrum (σ) would be a better approximation to e(r).

A basic datum, called the Bragg curve, collected for alpha particles and other heavy ions provides another good estimate for e (r). The curve is the ionization as a function of distance in a thin, broad ionization chamber held so that the particles strike the broad face perpendicularly. The ion chamber does the averaging and leakage compensation required for e(r). Furthermore, by being broad, it allows for the multiple scattering discussed above. Use of the stopping power or an average stopping power does not allow for particles emitted in dV and headed for the target element that do not get there because they scatter away from it; it also does not allow for those not headed for it but scattered so that they hit it. The data in a Bragg curve are converted to e(r) by normalizing the curve so it equals the stopping power near the point of emission, where the effects of straggling are smallest.

If one does not require much accuracy, for example, if one is making just a trial or illustrative calculation, the variation of the stopping power with the distance traveled can be neglected; that is, one can approximate e(r) with the average stopping power E/R. Figure I-3 shows the results of a calculation done in the e(r) = E/R approximation to illustrate the doses from nonequilibrium distribution of alpha-particle emitters. In this instance, C alpha particles per unit mass of energy E were emitted from spherical regions of radius a in uniform tissue in which the range of the particles is R. The doses are shown as functions of x, the distance from the center of the sphere. All distances are normalized to the alpha-particle range R. Under charged-particle equilibrium conditions, the density C would produce a uniform dose, CE; all the doses in Figure I-3 are normalized to CE.

Figure I-3. Absorbed dose from alpha, particles of energy E and range R as a function of distance x form the centers of spherical regions of radius a that contain uniform concentrations C of the emitter [in the e(r) = E/R approximation].

Figure I-3

Absorbed dose from alpha, particles of energy E and range R as a function of distance x form the centers of spherical regions of radius a that contain uniform concentrations C of the emitter [in the e(r) = E/R approximation].

Figure I-3 illustrates the conditions required to obtain charged-particle equilibrium. For the two largest spheres, radii of 2 and 3 times the range, the dose equals the equilibrium dose CE in the central region of the sphere. There each point is surrounded by emitters out to a distance at least equal to the range of the alpha particles; emitters farther away have no influence on the dose because the particles cannot reach the point. This example illustrates the requirement discussed above that C must be uniform out to distances equal to the range beyond the edge of the region before charged-particle equilibrium can exist within it.

Inside the largest spheres, at points less than the alpha-particle range from the edge, the dose is less than CE because there are regions within the range that contain no emitters. At the edge of these spheres (that is, for x = a), the dose is roughly one-half CE (the dose would be exactly one-half CE at a plane surface in an equilibrium region if all the emitters were removed from one side of the plane; small parts of the surfaces of large spheres are shaped much like planes). Outside a sphere of any size, at distances greater them the alpha-particle range from the edge, the dose drops to sero because particles cannot penetrate that far.

The figure also suggests when the average dose <D> discussed above in the sphere is a reasonable estimate of the dose throughout the sphere. Clearly, it is reasonable if the lower doses at the inner edge of the sphere are negligible in the average. Since the mass involved is proportional to the square of the distance from the center, the sphere must be quite large relative to the range of the alpha particles. For about 10% accuracy, the radius of the sphere must be roughly 30 times the range; this usually means that spheres with radii of 1 to 2 mm are needed to give meaningful averages.

When the radius of the sphere equals the range, the dose CE is attained only at the exact center. For smaller spheres, the dose CE is never attained. [In the e(r) = E/R approximation only, the relative dose at the center is a/R.] In general, for uniform distributions in C, the absorbed dose does not exceed CE anywhere. If the sphere (or any other small volume) is very small, then at distances several times its radius:

Image img00122.jpg

where V is the volume of the sphere and, hence, CV is the number of alpha particles emitted.


The ICRU and U.S. National Bureau of Standards10 recommend the use of the International System of Units (SI). Absorbed dose is a quotient of a quantity with dimensions of energy by one with dimensions Of mass; therefore, its unit in the SI is joules per kilogram (J kg-1). In the radiological sciences, this unit is called the gray (Gy); 1 Gy = 1 J kg-1. The gray recently replaced another popular unit, the rad; 1 Gy = 100 rad.

The SI units for the quantities in Equations I-2 and I-3 are the gray, inverse kilograms, and joules. These are seldom convenient units. In particular, radiation energies are universally given in electron volts (eV) or a multiple thereof. Also, C and <C> are often given in either becquerels (Bq) times a time per unit mass or curies (Ci) times a time per unit mass. The becquerel is a unit of activity, the rate of radioactive disintegrations; 1 Bq = 1 s-1, that is, 1/s. The curie is an older unit for activity; 1 Ci = 3.7 × 1010 s-1.

To accommodate mixed systems of units, Equations I-2 and I-3 are rewritten:

Image img00123.jpg
Image img00124.jpg

where k is the same in both and is introduced solely to provide for the different units used. For example, if D or <D> is in rads, C or <C> is in microcurie hours per gram, and E is million electron volts, then k = 2.13 J kg-1/µCi h MeV g-1.

Relative Biological Effectiveness

Equal doses of different radiations do not always produce the same effect. In radiobiology, therefore, the relative biological effectiveness (RBE) was introduced to compare the effects of different radiations. The RBE of a test radiation with respect to a reference radiation, for a given effect, is defined as the ratio:

Image img00125.jpg

where D test and D reference are the doses of the two radiations that produce the same degree of the given effect. If the radiation being tested required less dose than the reference radiation, it would be said to be the more effective one, and its RBE would be greater than 1.

An RBE is a number. But the effect considered can be defined either with numbers (e.g., the number of tumors) or without numbers (e.g., degree of erythema). All that is required is a way of identifying equality of effect (or of identifying one effect as greater than or equal to another).8

RBEs are used for comparing radiations. To give a meaningful comparison, everything else that might affect the outcome should be the same during the experimental comparisons. For example, the absorbed dose distribution, the exposure time, the temperature, the atmosphere in which the cells or animals are exposed, and the growth conditions after the exposure should all be the same.

The RBE may depend, in particular, on how long observations are continued after exposure to radiation. In experiments with cells or animals, it is conventional to follow the exposed populations for their lifetime, or at least until new occurrences of the effect cease to appear. In epidemiological studies of human populations, few studies have reached this degree of completion, and caution is required in interpreting the data derived from them.

Making the absorbed dose distributions the same during the irradiations may be difficult. There is seldom difficulty in in vitro cell experiments where the absorbed dose can ordinarily be made uniform throughout the exposed population. In animal experiments, on the other hand, dose uniformity is the exception rather than the rule: Radiations incident from outside the body are subject to different attenuations; for internally deposited radionuclides, the distributions of dose reflect the distributions of the radionuclides and can be. very erratic. When one or both of the test and reference dose distributions are nonuniform, the dose to be entered in the definition of RBE is not defined. If the spread in doses is not too great, an average dose can be used, with a consequent uncertainty in the RBE.

Dose Averaging

Because of the nonuniformity of dose typically encountered with alpha particles, the following argument is often made. For low doses the yield is proportional to the dose; thus, if the yield is averaged throughout some tissue, the average yield would be proportional to the average dose (the <D> dealt with above). One could then assess RBEs with average doses, which would be a great simplification because the generally difficult determination of nonuniform doses is avoided. But this argument hides several critical assumptions. One is that the cells are equally sensitive throughout the tissue, something that is not obvious in view of the differences in oxygenation and nutrient supply throughout a typical tissue. Probably even more critical is the assumption that the cells are uniformly distributed throughout the tissue (implied by the uniform weighting of the dose in the tissue during the averaging). For example, if the critical cells were the epithelial cells lining small blood vessels and the radionuclide were one that deposited in or near these cells, the dose to them could be very much higher than the average dose in the tissue.

In spite of these criticisms and because of the practical difficulties in determining nonuniform doses, average doses have normally been used in alpha-particle dosimetry. In comparing like situations, the practice of using <D> values is a useful, practical expedient. The practice leads to difficulty when data for one radionuclide are applied to another or when data for one Species are applied to another. In these applications, the actual doses to the relevant cells should be used in determining RBEs.


If the energy imparted by radiation to the mass in a small volume (usually called a site in the microdosimetric literature) were measured repeatedly under apparently identical conditions, the values obtained would differ. These differences are not experimental errors; the errors can be made much smaller than the differences observed. The differences are inherent; they are due to the randomness in the number of charged particles that impart energy to the site and to the randomness in the energy imparted to the medium in the individual interactions between a particle and the medium. These random features are particularly important for alpha particles and other high-LET radiations where (as can be seen in autoradiographs) the particle density is often so low that many sites are struck by only a few particles and some sites are not struck at all. In ordinary dosimetry, that is, in the determination of absorbed dose, the different values of the energy imparted to the mass would be averaged; information about the extent of the randomness would thus be discarded. In microdosimetry this information is kept and exploited.

While the effect of the randomness is present, no matter what the dose or size of the site studied, the degree of variation encountered will be less the larger the dose or the larger the mass. As a consequence, although microdosimetry applies to sites of all sizes, it generally focuses on low doses and small masses-where the differences are larger. Usually the attention is on masses the size of cells or cellular components.

Specific Energy

Dosimetry deals with the absorbed dose, the mean energy per unit mass imparted to matter by radiation; microdosimetry deals with the actual energy per unit mass. The latter is given another name (specific energy) and symbol (z) to distinguish it from the former. Dose and specific energy have the same units. The definitions of close and specific energy are framed so that <z>, the average value of the specific energy over many repetitions of the irradiation, equals the absorbed dose:

Image img00126.jpg

The specific energy is the result of the energies imparted by individual charged particles. The number of events in which individual particles impart energy is random with s Poisson distribution. If the mean number for that Poisson distribution is m and if the mean specific energy for single events is denoted by <z: 1>, then Kellerer and Rossi9 (see also ICRU5 ) proved that:

Image img00127.jpg

that is, the average specific energy due to all the charged particles (the absorbed dose) equals the product of the average number of events and the average specific energy for a single event.

The mean square specific energy, <z 2>, is given9 by:

Image img00128.jpg

where <z 2 : 1> is the mean square for individual events. The mean square can be used to calculate the variance of the specific energy:

Image img00129.jpg
Image img00130.jpg
Image img00131.jpg

The variance, the standard deviation (σ), or the coefficient of variation (σ/<z>) can be used to indicate the breadth of the distribution in specific energy. According to Equation I-15, the variance is the product of the absorbed dose and a factor that is independent of dose; the factor depends only on the characteristics of single events. Table I-1 lists values of this factor and of the mean specific energy for single events for a number of radiations.

TABLE I-1. Representative Values of Microdosimetric Parameters for a 1-µm Sphere.


Representative Values of Microdosimetric Parameters for a 1-µm Sphere.

Distributions in Specific Energy

The mean values just discussed are the means and mean squares of probability densities in specific energy. Two basic kinds of densities are of interest: densities for individual events and densities for a given absorbed dose. The probability density far single events will be denoted by f(z : 1); this means that f(z : 1)dz is the probability that the specific energy due to a single event is in a range dz that includes z. The probability density for a given dose will be denoted by. f(z).

In the microdosimetry of alpha particles, each distribution of the alpha, particle emitters can produce a different probability density. Here, only two examples will be given: the charged-particle equilibrium situation considered above, and the situation in which the emitters are agglomerated into particulates from which many alpha particles are emitted. The densities for nonequilibrium distributions of emitters can also be calculated.11

Distributions for Single Events

Figure I-4 shows the single-event densities for 60Co gamma rays, a low-LET radiation, and for 239Pu alpha particles, a high-LET radiation, for charged-particle equilibrium. To cover a wide range of the abscissa, the probability density is multiplied by z and then plotted on a logarithmic scale, because equal areas anywhere under such a curve represent equal probabilities of occurrence. On this logarithmic plot the two distributions differ only slightly in shape, but there is a large distance along the abscissa between them due to the difference in the stopping powers of the particles (discussed earlier in this appendix).

Figure I-4. Probability densities in specific energy for single events, f(z : 1) for 60Co and 239Pu.

Figure I-4

Probability densities in specific energy for single events, f(z : 1) for 60Co and 239Pu.

The location of the single-event distributions on the specific energy scale also depends strongly on the size of the site considered. The energy imparted to the site by a particle increases in proportion to the diameter of the site, but the mass of the site increases in proportion to the cube of the diameter. The energy per unit mass, therefore, is inversely proportional to the square of the diameter. The following expression relates, approximately, the mean specific energy in single events, <z : 1>, the mean stopping power of the particles, <S>, and the diameter, d, of a spherical site in tissue of unit density:

Image img00133.jpg

where the units are grays, kiloelectron volts per micrometer, and micrometers, respectively. Thus, for larger sites the distribution is moved to the left in Figure I-4; for smaller sites the distribution is moved to the right. There are also changes in the details of the shapes of the distributions.

The mean number (m) of events in a site is proportional to the cross-sectional area it presents to the charged particles, that is, proportional to the square of the site size. This fact and the inverse dependence of <z : 1> on the square of the site size are the reason that the dose D = m<z : 1> is independent of the site size.

For alpha particles, the difficulties in measuring such a short-range radiation have forced investigators to use calculations to obtain approximate single-event distributions.11 The f<z : 1> for radiations with longer ranges are determined experimentally with proportional counters.5,12

Distribution for a Given Dose

While f(z : 1) is the distribution for a single event, f(z) is the distribution for a number of events. The number of events is random with a Poisson distribution. The f(z) distribution is calculated from f(z : 1) by Fourier-transform methods.7,11

Figure I-5 shows the distributions for 239Pu alpha particles for different absorbed doses. For small doses, the chance of a site being hit by more than one alpha particle is very small. The area under f(z : 1) is, by definition, unity; that under f(z) at these doses is approximately equal to the probability of the one event. Consequently, f(z) has the same shape as f(z : 1) but is smaller by a factor equal to the one-event probability.

Figure I-5. Probability densities f(z) in specific energy for different absorbed doses (i.

Figure I-5

Probability densities f(z) in specific energy for different absorbed doses (i.e., different mean numbers of events) for 239Pu alpha particles compared with the density for single events f(z : 1).

As the dose increases, the area under f(z) increases because the chance of a site being missed by alpha particles decreases. The increase is seen in two ways: the part similar in shape to f(z : 1) increases, reflecting more single hits. In addition, a small bulge begins to develop on the high-z side due to hits by two alpha particles.

At much higher doses f(z) begins to move to the right. The chance of just one hit has grown small; many now occur and give a higher total z. The area under the curve is close to unity, because the chance of any site being missed is now small. As f(z) moves farther and farther to the right (with increasing dose), the shape of the curve approaches that of a normal distribution.

Distributions for Particulate Sources

The distributions shown in Figures I-4 and I-5 are for radionuclides randomly dispersed in the tissue. But random dispersion does not always occur. Under some circumstances the molecules of a nuclide coalesce with each other (and with other molecules). These agglomerations of radionuclides are called particulates. For a particulate, many alpha particles may emerge from nearly the same point in the medium. Sites near such a particulate stand a larger chance of receiving energy than if the activity were spread more uniformly; sites far away stand a smaller chance.

Figure I-6 illustrates what the agglomeration into particulates can do.11 It shows the distributions in specific energy, f (z), for the same absorbed dose (0.75 Gy) for different average numbers of alpha particles per particulate. To get the sine absorbed dose, the number of particulates per unit volume is changed in inverse proportion to the number of alphas per particulate. When the number of alphas per particulate is small, 1 or 10 for this site size, the f (z) values do not differ much; they are actually very close to the f (z) for no agglomeration into particulates. In these circumstances, even though many alpha particles are emitted at a common point, the chance of a significant number of them going in nearly the same direction so as to affect a common site is small. For particulates that emit up to about 100 alpha particles and for this site size, f (z) differs only slightly from f (z) for nonagglomeration, that is, for a uniform distribution of molecules of the radionuclide. For about 1,000 alpha particles and higher, it is distinctly different. A site close to a particulate that emits so many alpha particles stands a good chance of receiving energy from more than one alpha particle, with the result that its f (z) is pushed to higher specific energies. But, when so many alpha particles are emitted from each particulate, there are fewer particulates for a given dose, with the result that there is an increased chance that some sites will not be close enough to any particulate to be hit by any alpha particles. This causes the area beneath f (z) to decrease. At 0.75 Gy, for 1 alpha per particulate and this site size, the chance of being missed is 0.52. For 100,000, it is 0.9978. In other words, only 0.0022 sites get hit at all; but, the sites that are hit are apt to be hit many times.

Figure I-6. Probability densities in specific energy for particulates of different sizes for the same absorbed dose (0.

Figure I-6

Probability densities in specific energy for particulates of different sizes for the same absorbed dose (0.75 Gy).


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