3802
JACOBI AND ANDRE
and
a= 1
we obtain from (2) the following homogeneous
system of differential equations for the ¢,:
(Ke,’)’ = rL*p;
(5)
The boundary conditions for? = 1 are
gi(o) = 0
K(0)¢1’(0) = —&£,
and forz > 1 they are
v,(o) = 0
a
9,(0) = o;,171(0)
After evaluating ¢,, g2, °** successively, we determined the functions n, from (3). For the evalu-
ation the exhalation rates of Rn™ and Rn™ were
standardized to H = 1 atom/cm’sec.
For the numerical treatment the given K profile was approximated by a step function. If z. =
0, 2:1, °**, 2, are the points of discontinuity of K,
and if K(z) = K;, forz; <z < 2, ,4,, the yare
linear combinations of exp(@,z) and exp(— 8,2),
where 8, = (A,/K,)’, when z is restricted to an
interval in which XK is constant. », and Ky,’ must
remain continuous at the points of discontinuity
of K.
In particular,
¢.' (2; -—-O= (K;/K;-e.'(z; + 0)
(6)
where 9,’(z; — 0) and ¢,’(z,; + 0) denote left
and right limits, respectively.
An easy computation shows that the function
values at the left and right ends of the interval
are related by the linear transformation
e(2;)
| = ul
g' (2; + 0).
gz; +1)
|
(7)
¢’(2,.1 — 0)
where
—8;"' sinh a
M, = = 8,4;
— 8; sinh 6; 6;
cosh 8,6;
and 8, = 2;,1— 2).
Using (6) and (7) we determine the values
gs(0)
and 9,’(0)
from 9s(z,)
and gi (2a). It
seems reasonable to replace the boundary condition ¢.(0) = 0 by g.(z,) = 0. For the K profiles
given in Figure 1 this is sufficient for Rn™ anc
its short-lived decay products, if z, = 3 = 10
cm, as in our case. However, the concentratio:
of the long-lived Rn™ decay productsis still not
negligible at this altitude. For this reason the
calculation was, in general, extended to the region z, = 30 km < z < o by putting K =
constant = 3 x 10‘ em?/sec for z > z. We then
have ¢’(z,)/(z,) = —8,. Starting with an arbi-
trary initial value, 9(z,) = c, and the initial
value, y’(z,) = —B,c, we computed the values
g(z,) and ¢’{z,) for j = n — 1, ***, 0, successively. The boundary condition at z was then
satisfied by multiplication with a suitable factor.
To avoid floating-point overflow—the 9, increase rapidly as z decreases—it was sufficient to
choose ¢ sufficiently small in most cases. The
computer used (Siemens model 2002) admits
values between 10° and 10” for variable point
computations. To includethe less favorable cases
we started with z, instead of with z, for which
pal
> a,6;< 100
1
e
> ia,s, > 100
1
This procedure is suggested by the exponential
behavior of the solutions (e= 10°). In fact,
after satisfying the boundary conditions at z =
0, the values at z, are negligible if p <n.
To estimate the error caused by replacing the
K profile by a step function the computation
was repeated in some cases with a larger number
of steps. The deviations were negligible.
THEORETICAL RESULTS AND COMPARISON
WITH EXPERIMENTAL Data
Rn™. Figure 2 shows the Rn™ profiles which
were calculated with the typical K profiles given
in Figure 1. They are standardized to a mean
exhalation rate of 1 atom/cm* sec. Exhalation
measurements at several places having normal
Ra™ content of the soil material indicate an
average exhalation rate of 0.2 to 1.5 atoms/cm?
sec [/srael, 1962], in rather good agreement with
the mean value of 1 atom/em?® sec which was
estimated theoretically by Jsrael [1958, 1962]
from the diffusion transport of Rn™ in the surface layer of the ground. It follows that the
calculated Rn™ profiles in Figure 2 should be
directly comparable with the results of measurements over continental areas having normal
Ra™ content.