GEOPHYSICS

Yu. A. IZRAEL

ON THE DETERMINATION OF THE COEFFICIENTS OF FRACTIONATION AND BIOLOGICAL AVAILABILITY OF THE PRODUCTS OF NUCLEAR EXPLOSIONS IN RADIOACTIVE FALLOUT

(Presented by Academician E. K. Fedorov on 22 VIII 1964)

Radioactive products associated in the cloud of a nuclear explosion with particles formed in the high-temperature zone of the explosion constitute the principal part of radioactive fallout. Below we consider the distribution of radioactive products on large spherical particles (with diameter greater than 10–20 μ) formed in large quantities in surface explosions (mainly as a result of the melting of particles of entrained soil, i.e., the material of the underlying surface \((^{1,2})\)). For air bursts, in which particle formation occurs as a result of condensation of the vaporized bomb material and coagulation of the submicron particles thereby formed \((^3)\), a number of the propositions of the present work require clarification.

The capture of radioactive isotopes in the cloud by particles before and after the solidification of the latter occurs selectively \((^1)\), since nuclides in any one mass chain (isobars), as a result of radioactive decay, are successively transformed into isotopes of different elements with different boiling temperatures. This leads to fractionation of the products of nuclear explosions \((^2)\), which is characterized by the coefficient \((^4)\)

\[ f_{i-j}= \frac{n_i(t,r)_{\mathrm{e}}}{n_j(t,r)_{\mathrm{e}}} \frac{n_j(t)_{\mathrm{T}}}{n_i(t)_{\mathrm{T}}}, \tag{1} \]

where \(n_i(t,r)_{\mathrm{e}}\), \(n_j(t,r)_{\mathrm{e}}\) are the experimentally determined activities of isotopes belonging to the \(i\)-th and \(j\)-th mass chains, respectively, in a particle (or group of particles) of size (diameter) \(r\) at time \(t\) after the explosion; \(n_i(t)_{\mathrm{T}}/n_j(t)_{\mathrm{T}}\) is the theoretical ratio of the activities of these same isotopes formed in the explosion (at time \(t\)).

Isotopes of elements (or their oxides) having boiling temperatures above the solidification temperature of the substance that is the basis of the radioactive particles (for example, soil in surface explosions) enter still-liquid particles and are distributed relatively uniformly throughout their volume \((^1)\). Isotopes of elements having a lower boiling temperature condense on the surface of solid particles. As a result, different isotopes will have unequal densities of distribution of activity over particle sizes \(N_i(t,r)_{\mathrm{e}}\).

Knowledge of the value of \(f_{i-j}\) for particles of different sizes falling out at different distances from the site of the explosion \((^5)\) is of considerable interest; however, at present the literature contains no indications of methods for determining the dependence of \(f_{i-j}\) on particle size. There are only fragmentary data indicating that small particles prove to be enriched in such isotopes as \(\mathrm{Sr}^{89}\), \(\mathrm{Ba}^{140}\) \((^{1,6})\) and \(\mathrm{Sr}^{90}\) \((^7)\), which have volatile precursors. In the present work one possible way of determining this dependence is considered.

One may write:

\[ n_i(t,r)_{\mathrm{e}}/n_j(t,r)_{\mathrm{e}} = N_i(t,r)_{\mathrm{e}}/N_j(t,r)_{\mathrm{e}} = \]

\[ = \left[ N_i^{v}(t,r)_{\mathrm{e}} + N_i^{s}(t,r)_{\mathrm{e}} \right]/ \left[ N_j^{v}(t,r)_{\mathrm{e}} + N_j^{s}(t,r)_{\mathrm{e}} \right] = \]

\[ = \left[ \frac{n_i(t)_{\mathrm{T}} F_i}{n_j(t)_{\mathrm{T}} F_j} + \frac{n_i(t)_{\mathrm{T}}(1-F_i)\sum_k n_k(t)_{\mathrm{T}} F_k N^s(t,r)_{\mathrm{e}}} {n_j(t)_{\mathrm{T}}F_j\sum_k n_k(t)_{\mathrm{T}}(1-F_k)N^v(t,r)_{\mathrm{e}}} \right] : \]

\[ : \left[ 1+ \frac{(1-F_j)\sum_k n_k(t)_{\mathrm{T}}N^s(t,r)_{\mathrm{e}}} {F_j\sum_k n_k(t)_{\mathrm{T}}(1-F_k)N^v(t,r)_{\mathrm{e}}} \right], \tag{2} \]

since it is evident that

\[ N_i^v(t,r)_{\mathrm{e}} = \left[ n_i(t)_{\mathrm{T}}F_i\Big/\sum_k n_k(t)_{\mathrm{T}}F_k \right]N^v(t,r)_{\mathrm{e}}, \]

\[ N_i^s(t,r)_{\mathrm{e}} = \left[ n_i(t)_{\mathrm{T}}(1-F_i)\Big/\sum_k n_k(t)_{\mathrm{T}}(1-F_k) \right]N^s(t,r)_{\mathrm{e}}. \]

In equation (2), \(N_i^v(t,r)_{\mathrm{e}}\), \(N_i^s(t,r)_{\mathrm{e}}\), and \(N_i(t,r)_{\mathrm{e}}\) are the distribution densities of the volume, surface, and total activities of the \(i\)-th isotope (or of the total mixture of isotopes for \(N^v(t,r)_{\mathrm{e}}\), \(N^s(t,r)_{\mathrm{e}}\), and \(N(t,r)_{\mathrm{e}}\)) with respect to particle sizes (at time \(t\)), respectively; \(n_i(t)_{\mathrm{T}}\) is the activity of the \(i\)-th isotope produced in the explosion at time \(t\) (calculated theoretically); \(F_i\) is the fraction of the elements in the \(i\)-th mass chain of elements that had condensed by the moment of particle solidification \((^2)\); \(k\) is the number of mass chains whose isotopes at time \(t\) make a substantial contribution to the total activity. Choosing, as the isotope belonging to the \(j\)-th mass chain, \(Zr^{95}\), which has refractory precursors, one may set \(F_j=1\) in (2). Then from (1) and (2)

\[ f_{i-95} = F_i+ (1-F_i) \left[ \sum_k Y_k\lambda_k e^{-\lambda_k t}F_k N^s(t,r)_{\mathrm{e}} \Big/ \sum_k Y_k\lambda_k e^{-\lambda_k t}(1-F_k)N^v(t,r)_{\mathrm{e}} \right], \tag{3} \]

since

\[ \sum_k n_k(t)_{\mathrm{T}}F_k \Big/ \sum_k n_k(t)_{\mathrm{T}}(1-F_k) = \sum_k Y_k\lambda_k e^{-\lambda_k t}F_k \Big/ \sum_k Y_k\lambda_k e^{-\lambda_k t}(1-F_k), \]

where \(Y_k\) is the yield of the \(k\)-th mass chain in fission.

It follows from formula (3) that the quantity \(f_{i-95}\) is a function of particle size, explosion yield \((F_i)\) \((^2)\), type of fission \((Y_i)\) \((^8)\), type of explosion, and the character of the soil composing the underlying surface, since the dependence \(N(t,r)_{\mathrm{e}}\) differs for underground, surface, and air explosions \((^{5,9,10})\), and also differs greatly for different soils \((^9)\). Obviously, the distribution density of total activity with respect to particle sizes is

\[ N(t,r)_{\mathrm{e}}=N^v(t,r)_{\mathrm{e}}+N^s(t,r)_{\mathrm{e}}, \tag{4} \]

whence

\[ N^s(t,r)_{\mathrm{e}}/N^v(t,r)_{\mathrm{e}} = N(t,r)_{\mathrm{e}}/N^v(t,r)_{\mathrm{e}}-1. \]

The dependence \(N(t,r)_{\mathrm{e}}\) is well known for various types of explosions \((^{5,9,10,11})\). This distribution may be expressed either by a logarithmic-normal law \((^{3,5,10,11})\), or by an exponential function \((^{10})\), or in the form of the two-parameter function used in \((^{12})\) to represent the distribution density of a polydisperse impurity with respect to particle settling velocities. Obviously,

\[ \int_0^\infty N(t,r)_{\mathrm{e}}\,dr=Q(t), \]

where \(Q(t)\) is the total number of all fission fragments formed in the explosion at time \(t\).

Assuming the presence of thermodynamic equilibrium and the applicability of Raoult’s law in the process of particle formation in the fireball \((^9)\), it may be considered that the volume and surface concentrations of radioactive-

isotopes in the formed particles of different sizes (for \(r \gg 10\ \mu\)), which remained in the cloud for a time sufficient for completion of the condensation of fission-fragment elements on these particles, are constant, i.e. \(\sigma_v=\mathrm{const}\) and \(\sigma_s=\mathrm{const}\).

Hence

\[ N^s(t,r)_{\mathrm{e}}/N^v(t,r)_{\mathrm{e}}=6\sigma_s/\sigma_v r = \alpha/r,\qquad \alpha=\mathrm{const}. \tag{5} \]

In view of the fact that the quantity \(F_i\) can be approximately calculated if the yield of the explosion is known (2), \(N(t,r)_{\mathrm{e}}\) is selected for the case under consideration \((^{5,9,10,11,13})\), and \(Y_i\) is known for the given type of fission \((^8)\), the main difficulty in determining \(f_{i-j}\) is finding the value \(\alpha\). From (4) and (5)

\[ N^v(t,r)_{\mathrm{e}}=rN(t,r)_{\mathrm{e}}/(r+\alpha). \tag{6} \]

Fig. 1. Distribution density of activity \(\dfrac{1}{Q}N(r)\) by particle sizes

The quantity \(\alpha\) may be found, for example, from the equation expressing the fact that practically all isotopes of elements boiling at a temperature exceeding the solidification temperature of the particle material will enter these particles:

\[ \int_0^\infty N^v(t,r)_{\mathrm{e}}\,dr = \frac{\sum\limits_k Y_k\lambda_k e^{-\lambda_k t}F_k} {\sum\limits_k Y_k\lambda_k e^{-\lambda_k t}} \int_0^\infty N(t,r)_{\mathrm{e}}\,dr . \tag{7} \]

For illustration, let us find the dependence \(f_{i-95}(r)\) for a surface explosion with a yield of 3 Mt. The values of \(F_i\) for this case may be taken from (2) in accordance with the particle solidification time, equal to 35 sec after the explosion \((^5)\). We take the dependence \(N(t,r)_{\mathrm{e}}\) from \((^{13})\); in Fig. 1 the dependence \(\dfrac{1}{Q(t)}N(t,r)_{\mathrm{e}}\) is shown as a histogram (curve 2). We note that in works \((^{5,9,10,11,13})\) it is assumed that the distribution density \(N(t,r)_{\mathrm{e}}\) for fresh fallout (days, weeks) is practically independent of time. The dependence \(\dfrac{1}{Q(t)}N(t,r)_{\mathrm{e}}\) chosen by us (see Fig. 1) can be approximately represented by the two-parameter function

\[ \frac{1}{Q(t)}N(t,r)_{\mathrm{e}} = \frac{a^{n+1}}{\Gamma(n+1)}r^n e^{-ar}, \tag{8} \]

where \(n=2,\ a=0.06\) (Fig. 1, curve 1). From formulas (6), (7), and (8) we obtain:

\[ \frac{1}{2}\{0.06\alpha+(0.06\alpha)^3e^{0.06\alpha}[-\operatorname{Ei}(-0.06\alpha)]-(0.06\alpha)^2\} = \sum_k Y_k\lambda_k e^{-\lambda_k t}F_k \Big/ \sum_k Y_k\lambda_k e^{-\lambda_k t}. \tag{9} \]

An approximate calculation carried out by us using the data of tables \((^8)\) showed that radioactive isotopes with refractory precursors constitute, in fresh fallout (\(t\leq 20\) days), about half of all fission fragments formed in the explosion (at the same point in time), i.e. the quantity standing on the right-hand side of equation (9), for the case described, is approximately 0.5. Substituting this value into equation (9) and solving it graphically, we find \(\alpha\approx40\ \mu\). Figure 1 shows the dependences \(\dfrac{1}{Q(t)}N^v(t,r)_{\mathrm{e}}\) and \(\dfrac{1}{Q(t)}N^s(t,r)_{\mathrm{e}}\) (curves 3 and 4, respectively), obtained from equations (6) and (4) and the value \(\alpha=40\ \mu\).

Figure 2 shows the dependences of the fractionation coefficients \(f_{i-95}\) on particle size for the isotopes Sr\(^{90}\) and Ba\(^{140}\) relative to the isotope Zr\(^{95}\), determined from formula (3) for the case considered. We note that the quantity \(f_{i-95}\) defined above does not depend on time, i.e., it is valid not only for “fresh” fallout.

If the substance forming the basis of the particles (for example, SiO\(_2\)) is practically insoluble in water and weak acids \((^{1})\), knowledge of the quantity \(N_i^s(t,r)\) makes it possible to determine the soluble fraction of the \(i\)-th isotope (located on the surface of the particles), and consequently also its biological availability, since solubility in 1 N HCl is an appropriate measure of biological availability \((^{14})\). The quantity (coefficient) of biological availability \(b_i\) of the \(i\)-th isotope is defined \((^{14})\) as the quotient obtained by dividing the relative accumulation by a biological system of this isotope contained in the radioactive particles of the explosion by the relative accumulation of the isotope from solution. Taking the maximum value of \(b_i\) to coincide quantitatively with the fraction of the \(i\)-th isotope located on the surface of insoluble particles (since diffusion of isotopes from inside such particles is practically excluded), one may write

Fig. 2. Dependence of the fractionation coefficients (solid curves) and biological availability (dashed curves) on particle size for Sr\(^{90}\) (1) and Ba\(^{140}\) (2)

\[ b_i = N_i^s(t,r)_e/[N_i^s(t,r)_e+N_i^v(t,r)_e] = \]

\[ = \left\{1+ \left[ F_i \sum_k Y_k \lambda_k e^{-\lambda_k t}(1-F_k)N^v(t,r)_e \right] \left/ \left[ (1-F_i)\sum_k Y_k \lambda_k e^{-\lambda_k t}F_k N^s(t,r)_e \right]\right. \right\}^{-1}. \tag{10} \]

The limited and highly scattered material \((^{14-17})\) containing information on the coefficients of biological availability and on the leaching of various isotopes from particles by water and weak acids in most cases confirms formula (10). Thus, Sr\(^{90}\) in large particles is more soluble than the sum of all fission fragments; the solubility of the sum of fragments in such particles is small \((^{14,16,17})\), and the fraction of the soluble part in them is smaller than in small particles; solubility in particles from explosions of the very highest yields is the least \((^{14,17})\). On the surface of vitrified spheres larger than 50 \(\mu\), collected in the state of Nevada, there was no more than 2% of the activity contained in the particles \((^{16})\).

In Fig. 2, as an illustration, the dependence of \(b_i\) on particle size is shown for the isotopes Sr\(^{90}\) and Ba\(^{140}\) (calculated for the same conditions and assumptions as for the fractionation coefficients shown in the same figure).

Received
15 VIII 1964

CITED LITERATURE

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\(^{10}\) E. C. Freiling, ibid., p. 47.
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