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