80
O'BRIEN, LOWDER, AND SOLON
Because the rate of energy absorption per gram equals the rate of energy production (the equilibrium condition), 7 = #’p and therefore
wa
(4)
i e(1 + at)’ dt = a®e'"P(B +1, L/a) = y
0
where y= Uy/Be.
Because of the way that @ and 8 appear in Iq. (4) it ts difficult to perform
uumerical operations using it, and for convenience we have made use of two
upproximations, For y < 2.8 we use
(5)
p=
Infy(y — 1) + 1]
In (1 + ay)
and for y > 2.8
(\
(6)
B
y— 1
oe
In (1 + ay)
These approximations are discussed in the appendix.
The buildup function now has the form
_
| exp Intyy
=
D+
In (1 + ww}
In(1 + ay)
y < 28
a
exp in
(Tb ay) In (L + ay
y > 2 28
Fitting Eq. (4) to the experimental data (7-9) we find that a ~~ 1/y.' Substituting this in Eq. (7) we have for the buildup function
(8)
“ex
jin Ly Cy DFM ln ¢ + ‘Y}
| XP 1
b(t) = 1
fy
1
In 2
)
y—
b
exp 4 ind In (1 + )
y
y < 28
——
.
y> »28
Figures 1, 2, and 3 show comparisonof Ieq. (8) with the calculations of Goldstein
and Wilkins (4). The calculations for lead and for low-energy photons on iron
represent cases where departures from our assumptions become important.
500-kev photons on iron have energies close to the photoelectric region, and
lead is hardly a Compton scatterer even at 1 and 2 Mev. Equation (8) agrees
very poorly with the calculations of Goldstem and Wilkins for 0.255 Mev and
‘The work of M.A. Van Dilla and G. J. Hine [Nucleonics 10, No. 7,54 (1952)] furnishes
different results. For a discussion of this experiment see Ref. (4)