128
THE SHORTER-TERM BIOLOGICAL HAZARDS OF A FALLOUT FIELD
MATHEMATICAL AIDS IN UNDERSTANDING OF BIOLOGICAL
HAZARDS
The problem remaining then is to evaluate
Thus the areaof a thin ring is proportionalto its
radius, but the contribution (o dose at P, is
the geometric factor:
eee
2 poyBPE
oH
Let gy?fi=2.
2uindr =
Then
wr'+ih=Z? and
PAYA
iog|
Z, or rds we °
This change in variable requires a change in
limits as follows:
When r=0, z=uh
When r=a, ' z= pfa?--h?
When ro, g= @,
* rete _
(both neglecting build-up foctor)
Including build-up factor
“/
/
/
/
as in Ref.(3)——_y“
“
2
a
% of Dose ———
ale
==
BY Yeti
OTN ¢ pe
oy
—— dz.
to
The similar case of two thin spherical shell
50
Pe im the plone
of the rings
eet
zr dex" a z dz
=Bi(—nJeFh)—Ei(—ph).
The physical meaningof the limits is that the
area of the disk in question is the difference in
areas between r=0 to ©, and r=@ to ~.
.
A specific application of this analysis is given in
Figure2.
.
A useful conclusion to be drawn from Figure 2 is
that 50 percent of the total dose at P comes
from an area of radius 8 meters. This is
rather less area than one might gucss considering that: the mean free path of the photonsin air
is ~ 100 meters.
on?
aretorrart ond lim=7,
Contribution to dose at the center of the sphere
Vary?
is « ~+A—=1,
Thus, spherical shells of equal thickness make
equal contributions to dose at the center of a
which has been evaluated by Jahnke and Emde.
Therefore:
f
Vy __4f8a[ r+ Ar)?—r5]
Va 4/8e[ (Got Ary —ry4]
Var?
f,”
eo de=—Ei(~2)
4 2
© ert
If the shell volumes are
_ or 3rAr+ Ar?
The two integrals on the right are of the form
VGH pe
sources is of interest.
VY, and V,,
meter).
Thefigure of 8 meters is more plausible if“one
considers the cage of two narrow rings of width
Ar as in Figure 3.
%
The effect of air absorption is to further depress
the importance of distant rings. This latter
depression is only partially compensated for by
scattered rays going through P,. This may
help to make the figure of 8 meters seem more
reasonable.
Fieune 2.-—- Percent of total dose as a function of r (h==1
2
fo reae= f ferde~ f. S(eddz.
y
lo
e
2
From the theory of limits
y
Solid wave calculated (Moupin}
Pointe read from EAW Fig.026
I
:
VEEP pe~tdz
J “FFE =f
% of Totals (uh)
El (- «Jah )
El
Thus, dose-wise, the relative importance of a
ring is inversely proportional to its radius.
Fiaune 3.---P, in the plane of the rings.
The ratio of the area of the two rings is:
sphere, regardless of how large r may be. This
conclusion is geometrical only, of course, and
neglects scattering and absorption. Hence in
the case of internally deposited gamma emitters,
seatter is much more important than it is with
the plane source.
In the case of a one-dimensional, or line
source, the relative contribution by any increment of line is inversely proportional to the
squareofits distance from the point of measurement.
To those schooled in the exact sciences this
sort of explanation may amount to belaboring
the obvious. It is hoped that biologists who
find such exposition illuminating will be for-
given.
and lim =7!
Ars
Tr
In April 1949 Condit, Dyson, and Lamb [2]
mede the first calculation of the ratio of beta
129
dose to gamma dose near a plane contaminated
withfission products. The approach was very
simple and-amounts to saying that if two betas
are emitted per gammaphoton, and if the
energy loss per unit path length for the beta
particle is about 75 times that for the gamma,
then the cnergy absorbed per unit volume
(cx dose) will be about 275 == 150.
Slightly modified the derivation was as
follows:
E,=Ee~™ for both betas and gammas
where #,=Energy flux at a distance z from
the source
E,= Energyflux at the source
+=Energy absorption coefficient
z== distance from source
Dose is closely related to the space rate of
energy loss:
dE,
»
Geet
Forbetas, 7 is replaced bythe energy absorp-
tion coefficient x, which is obtained from
empirical formulas
Ha
22
«Ey
where d==densityof absorber in gms/em’.
For gammas, + is replaced by o,=3.5X
10-5 em7},
Then the desired ratio of doses is:
ope 130
near the ground.
This result was not widely known until
about the time of Operation Greenhouse in
1951. In general, it was known from the early
radium and radon days that gamma dose near
a beta-gamma emitter is apt to be relatively
negligible by comparison with beta dose.
Using the same geometrical analysis outlined
in reference 1 above, reference 2 continued on