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OCR for page 397
APPENDIX I
Dosimeiry of Alpha Particles
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
S CATTERING
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.
397
OCR for page 398
398
STOPPING POWER
HEALTH RISKS OF RADON AND OTHER ALPHA-EMITTERS
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/p) is the quotient of the stopping power by the density of the
material.
The experunental and theoretical determinations of the alpha-
pa~ticle stopping powers are in good agreement.~4 Figure I-1 shows the
stopping power or alpha particles In soft tissues of unit density. For com-
parison, the stopping power of electrons is also shown in Figure I-1.6 (Beta
particles, the other common kind of charged particle emitted in radioac-
tive 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.
The mass stopping powers of alpha particles are slightly higher in
gases than in liquids or solids.~3 The stopping powers in Figure I-1 are for
solid tissue and are about 5% higher than the mass stopping powers for
gases.
~ d ~ ~ ~
· .
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
OCR for page 399
DOSIMETRY OF ALPHA PARTICLES
1 000-
_ 100-
A
Ct
O 10-
CL
by
CL
1-
0.1
0.01
0.1
-
-
-
399
Alpha Particles
Electrons
-
1
10
ENERGY (MeV)
FIGURE I-1 Stopping powers of alpha particles and electrons in soft tissue of unit
density.
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 ,um-~; the LET excluding losses of more than 100 eV is Logo = 56
keV ,um-~; 1,000 eV, L~,ooo = 81 keV ,um-~; 10,000 eV, L~o,ooo = 102 keV
Emu. 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
transferrable by Loo. Loo is the largest value possible for the LET, and it
equals the stopping power S.
RANGE
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
OCR for page 400
400
105—
104
3
, 1 0
At
tar
1o2 -
10—
0.01
HEALTH RISKS OF RADON AND OTlIER ALPHA-EMITTERS
-
Electrons
/
/
-
-
Alpha Particl" _
0.1
ENERGY (MeV)
FIGURE I-2 Ranges of alpha particles and electrons in soft tissue of unit density.
10
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.
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.
STRAGGLING
Random variations in the energy lost and in the change in direc-
tion 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.
OCR for page 401
DOSIM~TRY OF ALPHA PARTICLES
401
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, a2(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:
a(R) = 0.015R,
(I-1)
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 sunilar narrow distributions for the
distances in tissue.
DOSIMETRY
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 in 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
OCR for page 402
402
HEALTH RISKS OF RADON AND OTHER ALPHA-EMITTERS
particular exposure seldom arises in studying internally deposited alpha
emitters; thus, the term can usually be used here in its general sense.
ABSORBED D OSE
Radiation studies employ the dose concept in the quantity called ab-
sorbed dose. The determination of absorbed dose is called dosime try. 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,n 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 the 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:
(D>= (C)E.
CHARGED_ PARTICLE EQUILIBRIUM
(I-2)
The leakage is also negligible (actually, zero) in another situation
that is representative of many experimental situations. Suppose the tissue
OCR for page 403
DOSIMETRY OF ALPHA PAR T7CLES
403
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 glVeI1 by:
D= CE.
N ONEQUTLIBRTUM D OSES
(I-3)
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 p dV, where p 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 p dV)/~4'rr2),
where r is the distance to the point. The number entering an elementary
target volume at the point and with area dA facing d Y is dA times the
number per unit area. Each particle imparts an energy, denoted by eked,
to the target, where do is the thickness of the target. The mass of the
target volume is pdAdx. Thus, the dose to the target from the alpha
particles emitted in the particular d V is:
tCpdVdAe (r~dx/4,rr2~/ pdAdx. (I-4)
The total dose is obtained by integrating over all dV within range of the
point. Several factors cancel to give, for the dose:
D= /efr)(CdV/47rr2~.
(I-5)
Different approximations have been used for the kernel etr). One
approximation is to equate it to the stopping power: etr) = 5. 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),
OCR for page 404
404
HEALTH RISKS OF RADON AND OTHER ALPHA-EMITTERS
of the stopping power over the straggling spectrum (a) would be a better
approximation to etr).
A basic datum, called the Bragg curve, collected for alpha particles
and other heavy ions provides another good estimate for eked. 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 etr).
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(T) 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
efr) with the average stopping power E/R. Figure I-3 shows the results
of a calculation done in the etr) = 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-pa~ticle 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 illustrates the conditions required to obtain charged- par-
ticle 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,
OCR for page 405
DOSIMETRY OF ALPHA PARTICLES
405
0.8—
In
o
C)
0.6—
0.0
0.2
O-
\
-
2
\ 1
\
o
1
4
x/R
FIGURE I-3 Absorbed dose from alpha particles of energy E and range R as a
function of distance z from the centers of spherical regions of radius a that contain
uniform concentrations C of the emitter [in the etr) = E/R approximations.
at distances greater than the alpha-particle range from the edge, the dose
drops to zero 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. tin the e(~) = 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:
D = t.CV e~x)~/~4~x2),
(I-6)
where V is the volume of the sphere and, hence, CV is the number of
alpha particles emitted.
OCR for page 406
406
U NITS
HEALTH RISKS OF RADON AND OTHER ALPHA-EMITTERS
The ICRU and U.S. National Bureau of Standardsi° 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-~. In the
radiological sciences, this unit is called the gray (Gy); 1 Gy = 1 J kg-~.
The gray recently replaced another popular unit, the red; 1 Gy = 100 red.
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~i, that is, 1/s. The curie is an older unit for activity; 1 Ci =
3.7 x 10~° sol.
To accommodate mixed systems of units, Equations I-2 and I-3 are
rewritten:
(D) = k(C>E, and
D = kCE,
(I-7)
(I-8)
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 reds, C or (C) is in
microcurie hours per gram, and E is million electron volts, then k = 2.13
J kg-~/,uCi h MeV gal.
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:
RBE = DreferenCe/Dle8~,
(I-9)
where Dress and Preference 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 eRect as greater than or equal to another).8
OCR for page 407
DOSIMETRY OF ALPHA PARTICLES
407
RBEs are used for comparing radiations. To give a meaningful com-
parison, everything else that might affect the outcome should be the same
during the experimental comparisons. For example, the absorbed dose dis-
tribution, 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 irradia-
tions 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 uni-
formity is the exception rather than the rule: Radiations incident from
outside the body are subject to different attenuations; for internally de-
posited 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 deter-
mination 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 oxy-
genation 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
OCR for page 408
408
HEALTH RISKS OF RADON AND OTHER ALPHA-EMITTERS
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.
MICRO DO SIMETRY
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.
SPECIFTC 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 (a) to distinguish it from the former. Dose and specific
energy have the same units. The definitions of dose and specific energy
are framed so that (a>, the average value of the specific energy over many
repetitions of the irradiation, equals the absorbed dose:
D= (a).
(I-10)
The specific energy is the result of the energies imparted by individual
charged particles. The number of events in which individual particles
OCR for page 409
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Representative terms from entire chapter:
absorbed dose
DOSIMETRY OF ALPHA PAR TICIES
TABLE I-1 Representative Values of Microdosimetric
Parameters for a 1-pm Sphere
Radiation (Gy) (Z2 1> (G )
65-kVp x rays 0.45 1.05
250-kVp x rays 0.25 0.78
60Co gamma rays 0.08 0.34
0.43-MeV neutrons 10 15
1.8-NIeV neutrons 7.1 12.6
14-MeV neutrons 2.5 30
5.3 MeV alpha particles 13
19
409
impart energy is random with a Pomson 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:
(I-11)
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, (z2>, is given9 by:
D= (a) = mid: 1~;
(z2) = mazy 1) +m2(z 1~2,
(I-12)
where (z2 1) is the mean square for individual events. The mean square
can be used to calculate the variance of the specific energy:
tr2 = ((z_ ~z>~2>
= mazy: 1), and
= D(z2 :1~/(z :1~.
(I-13)
(I-14)
(I-15)
The variance, the standard deviation All, or the coefficient of variation
(cr/
410
1.0
-
N 0.5
N
o
NBALTU RISKS OF RADON AND OTHER ALPNA-EMITTERS
A239PU
-
, \ 1
10-2 10-1 10° 2 5 10
SPECIFIC ENERGY, Z/Gy
FIGURE I-4 Probability densities in specific energy for single events, fez: 1) for
60Co and 239Pu.
interest: densities for individual events and densities for a given absorbed
dose. The probability density for single events will be denoted by fez: 1~;
this means that fez: 1) do is the probability that the specific energy due to
a single event is in a range do that includes a. The probability density for
a given dose will be denoted by fix).
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 densi-
ties for nonequilibrium distributions of emitters can also be calculated.'
Distributions for Single Events
a. . . .
.
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).
The location of the single-event distributions on the specific energy
scale also depends strongly on the size of the site considered. The energy
DOSIMETRY OF ALPHA PARTICLES
411
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:
(z: 1) ~ 0.2~5~/~,
tI-16)
where the units are grays, kiloelectron volts per micrometer, and microm-
eters, 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 (x: 1)
on the square of the site size are the reason that the dose D = mid: 1) is
independent of the site size.
For alpha particles, the difficulties in measuring such a short-range ra-
diation have forced investigators to use calculations to obtain approximate
single-event distributions. The fez: 1) for radiations with longer ranges
are determined experimentally with proportional counters.5~2
Distributions for a Given Dose
While fez: 1) is the distribution for a single event, fix) is the distri-
bution for a number of events. The number of events is random with a
Poisson distribution. The fizz distribution is calculated from fez: 1) by
Fourier-transform methods.7~' ~
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 fix: 1) is, by
definition, unity; that under f (a) at these doses is approximately equal to
the probability of the one event. Consequently, f (z) has the same shape
as f (x: 1) but is smaller by a factor equal to the one-event probability.
As the dose increases, the area under f (a) increases because the chance
of a site being missed by alpha particles decreases. The increase ~ 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 (x) begins to move to the right. The chance
of just one hit has grown small; many now occur and give a higher total a.
The area under the curve is close to unity, because the chance of any site
being missed is now small. As f (a) moves farther and farther to the right
412
N
N
HEALTH RISKS OF RADON AND OTHER ALPHA-EMITTERS
1.5
1.0
0.5
0.0
o
O SINGLE EVENT
O 0.1 EVENT AVERAGE
0.4
· 1.0
· 4.0
· 10.0
/
i-:
~ I ~111\
lo2
10
LOG-SPECIFIC ENERGY, Z/Gy
FIGURE I-5 Probability densities f (a, in specific energy for different absorbed doses
(i.e., different mean numbers of evente) for 239Pu alpha particles compared with the
density for single events f (z: 1~.
(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-S 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 radionu-
clides 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. It shows the distributions in specific energy, fizz, for the same
absorbed dose (0.75 Gy) for different average numbers of alpha particles
per particulate. To get the same absorbed dose, the number of particulates
per unit volume is changed in inverse proportion to the number of alphas
DOSIM~TRY OF ALPHA PAR T7CLES
0.3
0.2
N
0.1
0.0
413
l _
~ \
it- ~ ___
1o~2 10-1 1 10
Particulate
Slze
· 1 DECAY
0 10
· 100
0 1,000
· 1 0,000
1 00,000
1o2 103
SPECIFIC ENERGY, z/Gy
FIGURE I-6 Probability densities in specific energy for particulates of different sizes
for the same absorbed dose (0.75 Gy).
per particulate. When the number of alphas per particulate is small, 1 or
10 for this site size, the faze values do not differ much; they are actually
very close to the faze 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, fix) differs only
slightly from f (a) 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 fizz 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 (a) 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
414
HEALTH RISKS OF RADON AND OTHER ALPHA-EMITTERS
other words, only 0.0022 sites get hit at all; but, the sites that are hit are
apt to be hit many times.
REFERENCES
1. Bicheel, H. 1972. Passage of charged particles through matter. P. 159 in American
Institute of Physics Handbook, D. E. Gray, ed. New York: McGraw-Hill.
2. E`rans, R. D. 1955. The Atomic Nucleus. New York: McGraw-Hill.
3. International Commission on Radiation Units and Measurements. 1970. Linear
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Radiation Units and Measurements.
4. International Commission on Radiation Units and Measurements. 1980. Radial
tion Quantities and Units. Report 33. Bethesda, Md.: International Commission
on Radiation Units and Measurements.
International Commission on Radiation Units and Measurements. 1983. Micro-
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Units and Measurements.
6. International Commission on Radiation Units and Measurements. 1984. Stopping
Powers for Electrons and Positrons. Report 37. Bethesda, Md.: International
Commission on Radiation Units and Measurements.
7. Kellerer, A. M. 1970. Analysis of patterns of energy deposition; a survey of
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Kellerer, A. M., and-H. H. Rossi. 1970. Summary of the quantities and functions
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Microdosimetry, H. G. Ebert, ed. Brussels: Euratom.
10. National Bureau of Standards. 1976. Guidelines for Use of the Metric System.
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11. Roesch, W. C. 1977. Microdosimetry of internal sources. Radiat. Res. 70:494-510.
12. Rossi, H. H. 1968. Microscopic energy distribution in irradiation matter. Pp.
4~92. In Radiation Dosimetry, Vol. I, 2nd ea., F. H. Attix and W. C. Roesch,
ede. New York: Academic Press.
13. Thwaites, D. I., and D. E. Watt. 1978. Similarity treatment of phase effects
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