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X. APPENDIX: SURVIVAL TIME OF AN IRRADIATED POPULATION We consider a population of No organisms, each having mean density p gm cm"^, characteristic size a cm, and mean lethal dose for a given kind of electromagnetic or corpuscular ionizing radia- tion of D rep. The population is irradiated with an intensity of I erg cm"2 sec"1 of the given kind of radiation, which has a mass absorption coefficient in organic matter of \j-lp cm2 gun"1. We are interested in the time, t, in seconds, for the population to be de- pleted from N0 to N organisms. Let J be the energy absorbed by unit cross-section of or- ganism due to a dose of d rep. Then, since one rep corresponds to the absorption of 93 ergs gm"1, J^ 93 p a d. (A-l) On the other hand, if the energy incident on unit cross-section of the organism is EQ, then, by Beer's law, the energy transmitted through the organism is Et = EQ e-pa . (A-2) Consequently, the energy absorbed by the organism is Ea= EQ -Et= EQ [l - e-(^M<>a] . (A-3) Now if Ea ergs absorbed by 1 cm2 corresponds to a dose of d rep, Ea = J, and from equations (A-l) and (A-3), i-l Eo /d = 93 p a [ 1 - e"(ti/p)pa J erg cm-2 rep-l . (A-4) Consequently, the time, T , for one organism to accumulate D rep due to an incident flux of I erg cm"2 sec"1 is r= (D/I) (E0/d) . (A-5) Assuming an exponential survival curve for the population of organ- isms, the number surviving after time t will be N = N0 e -t/r . (A-6) 43
Solving equation (A-6) for t, substituting from equations (A-4) and (A- 5), and converting from natural to common logarithms, we ob- tain for the time in which the population will have been depleted to N organisms, t=214ap(D/I) f 1 - e-fc/'J'aJlogjo (NQ/N). (A-7) In the case that the mean lethal dose, D, is given directly in units of erg cm"2 instead of rep, as is the case for ultraviolet ir- radiation, equation (A-7) is replaced by t = 2. 3 (D/I) [ 1 - e-(n/p)pa | lOglQ (NQ/N). (A-8) Table II was constructed from equations (A-7) and (A-8); p was taken as unity throughout. For an organism opaque in the given radiation, (\i./p)pa » 1, and equations (A-7) and (A-8) reduce respectively to t = 214 a p (D/I) logic (N0/N), (A-9) and t- 2.3 (D/I) 1og10 (NQ/N) . (A- 10) For an organism which is almost transparent in the given radiation, (n/p)pa « 1, and a Taylor series expansion of the exponential re- duces equations (A-7) and (A-8) respectively to t = 214 (,/ji) (D/I) log10(N0/N) (A-ll) and t=(2.3/^a) (D/I) 1og10 (N0/N). (A"12) 44
