80 WORLDIFIDE EFFECTS OF ATOMIC WEAPONS APPENDIX II Here we note that one can use Eq. (21) to get the particle-size distribution instead of using Eq. (5). If this is done, Eq. (8) has a slightly more complicated form. The heat of vaporization to be taken should be about 50,000 cal/mole. (This would give more than 100,000 cal/mole for Al,O,, since the unit we are taking is only part of a molecule.) Thus (AH/9RT)% =~ 1 for T = 2500. Substitution of a from Eq. (25) into Eqs. (27) and (28) gives ESTIMATION OF CONSTANTS The purpose of this development is the correlation of the properties of the precipitate with known properties of the oxides (or other molecules) that condense to form the precipitate. Because of lack of time and data, essentially no start has as yet been made in this direction. In this section we shall examine a few magnitudes to show that a consistent description seems to be within the realm of possibility. Examination of particle-size data suggests a particle-size distribution for particles in the range from 2 to 10 p# to be - 1% % I(tg) = 10°exp |- (# ar. @ "| ~ Cet. ©) If we take AH/RT, = 10, T, = 2500, a = 500, the solution is a = 25. This pair of solutions gives 2/8 = 5 X 10‘, which is somewhat larger than the probable value, although about the correct magnitude. It is certain that for values of all the physical constants that would seem reasonable, the value 10% for a/8 can be obtained. One of the most uncertain quantities N(R) o exp [—(a/B)R}], with a value of the constant a/8 = 10¢cm-'. We shall estimate the magnitude to be expected for a/f from physical consideration. The magnitude of £ is easy to obtain, e.g., B= Fup | “x 10 “10° kK 2K 103= 5 & 1074. Here we have assumed that there are 10!® oxide molecules (say, AIO) per cubic centimeter at the time of condensation; their velocity is 10° cm/sec; the volume each adds to a particle is 2 X 10°74 cm’. The magnitude of « is somewhat more difficult to obtain because of the arbitrariness of the quantity /,, 1.¢., the time that is set as zero in Eq. (5). Clearly this time must be near to that at which the process is complete. The total volume of precipitate will finally he =2 Bt of this calculation is the value of a, the cooling rate in degrees per second. If instead of 500 we take 100, the corresponding value of « ts 7 and wa! y For JF, Bo This calculation suggests that the extreme diminution in the cooling rate, as the radiative cooling becomes ineffective at about 2000°K, may be important in the precipitation process, As a final check on the consistency of this last calculation, we obtain the value of x* at ¢ = f,. Using Eqs. (14) and (25), we can write w= 2o(1\( Fe). 10-% cm’. If we use Eq. (6), V(t.) = 2 108% = im e [6I(to)], (26) fo) - > GAat, (27) in which 8 == 5 X 10 ' has been used. 5 (30) Fora = 7,7,/a = 25, and RT,/AH = 1/10, x* = 35, which shows that the nucleation mechanism would still be operating. For suffi- ciently low temperatures, x* approaches unity and nucleation is no longer the slow process. With the use of Eq. (20) wefind thatif A 4 T ” AH Ito) & ' x 10 X 108 & 4 & 10°15 [at 101 exp [- 4(_ : | pa where S = 47(3v,/4r)"* = 4 & 10°1* has beenused, then 4 T, \* AH DISCUSSION The mechanism proposed here for condensation by means of self-nucleation of oxide molecules appears to be promising enough to warrant further investigation. If such a mechanism can be established as being responsible for the particle forma- tion, it will yield valuable information regarding scaling of particle sizes with bomb