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.