DECREASE OF TOTAL RADIOACTIVITY IN TIME AT A LONG DISTANCE FROM NUCLEAR EXPLOSION OR NUCLEAR ACCIDENT
The total activity at long distance from a source is inversely proportional to time,
if the moment of a nuclear explosion or the moment of a nuclear accident is used as time origin.
On the other hand,
Way and Wigner have shown that the (theoretical) slope in the log-log scale has a value of
1.2 = 6/5. According to this derivation, the value 1+1/5 is a consequence of the proportionality between
the decay constant x and the fifth power of disintegration energy E
(the dependence follows from the experimental results).
The fission products were considered sort of a statistical assembly;
calculations were made of the beta-disintegrations per second and
of the total energy emitted per second at any time after fission.
They assumed all energies to be equally frequent
(f(E)=const, where f(E) is a frequency function of energy E).
Consequently, they used the following formula:
The model with the power 1.2 is too optimistic and it does not describe the data at a log distance from nuclear explosion.
The decrease of exposure rate along with time in the log-log scale in Hradec Králové (eastern Bohemia)
after the Chernobyl accident. The violet straight line has its slope exactly equal to –1, and the blue straight
line has the slope equal to –1.2. Both these straight lines go exactly through the maximum.
The green bow shows the exponential decrease from the maximum to the minimum.
It goes exactly through the maximum and through the minimum.
At long distance from a nuclear event, the total activity of the mixture of independent non-chain radionuclides is,
in general, the sum of exponential terms exp(-xi.t) with different decay constants xi and with different weights in
the entire sum (i.e. with different constants Co(xi) independent of the time, where Co(xi)
is a concentration of specific radionuclides for t = 0).
It is valid:
The total activity may be a linear - but no concave - function of time in the log-log scale.
The equation is not valid in the two following cases:
1) As regards natural radioactivity, the first decay product is in the secular equilibrium with
the other radioactive isotopes (i.e., x1 is very small) and the total activity is approximately constant with time.
2) The second exception of the relationship Eq.(3) immediately occurs, when the activity of a partial decay chain
depends on the processes in other decay chains (parameters Co(xi) are not invariables and f(x) can not be used).
For this case, Way and Wigner assumed the index a in Eq. 1 to be 6/5.
Unlike to the situation immediately after a nuclear explosion, the quantity of a certain nuclide at long distance
from the nuclear explosion site is not affected by the decay of other (non-chain) isotopes.
In fact, more than one decay product of a particular decay chain that are independent on each other
can be present in the mixture, but such a group of nuclides, i.e. primarily a parent-daughter pair, can be described
by one exponential decay term that decreases with time (e.g. 140Ba - 140La). The composition of a mixture at
long distance from the site is influenced by many external factors, such as meteorological variation,
aerosol size, type of landscape, physico-chemical quality of radionuclides.
Let factual(x;t1) be the frequency function of decay constant, x, that describes the existent composition of the
mixture at long distance at time t1 (one portion of the radionuclide with a decay constant equal to x or a
quantity Co(xi)). The parameter t1 is the period from the moment of explosion/accident to the moment of fallout
(e.g., t1 = 5.5 d for 1 May 1986 or t1=16 d for 30 May 1965; t1 is unknown for 2 December 1961).
Let factual(x; t1) be affected by many factors and one of them, radioactive disintegration,
be dominating and separated from the others, if the age of fallout is known.
For example, the half-life of 137Cs is about 11 000 days, whereas the half-life of 131I is only 8 days.
Hence, the quantity of 131I in the mixture at time t1 is more affected by radioactive decay.
The effect of radioactive decay on the mixture composition within the period t1 can be eliminated
if the true frequency function factual(x; t1) is replaced by a virtual frequency function fv(x) at time t = 0.
The virtual frequency function fv(x) can be defined as: fv(x) = factual(x; t1) exp(+t1x)
The frequency function fv(x) is virtual, because the true composition differs at t = 0).
The result, fv(x), of all factors that affect the composition of the mixture during the period t1 with
the exception of the radioactive decay. This virtual mixture composition is due to the fact that different nuclides
have different probabilities to achieve a target, and the frequency function fv(x) describes this aspect
(e.g., some inert gases could disappear and their values of f(x) could be zero).
The total activity within the period t (which is counted from the moment of the explosion) is the sum of particular
activities xfv(x)exp(-xt) of all nuclides, and the following equation can be applied:
, where N(t) is the number of radioactive atoms, aklfa>0 for t > t1,
Min and Max are the minimum and maximum decay constants in the mixture at the moment of fallout
(or at the moment when the measurement of exposure rate has started).
Under the assumption of xmin = 0 and xmax = infinity, A(t) is the Laplace transformation of the product xfv(x).
This mathematical formalism is identical with that of
the theory of congenital individual risks. The total activity of a mixture of fission products and the mortality rate of
a heterogeneous population could be described using the same mathematical formalism.
This theory may explain the linear decrease in the log-log scale and it agrees with the calculation of
the slope -alfa. Thus, the congenital individual risk in a human population corresponds with the decay constant
in the mixture of fission products.
In compliance with the theory of congenital individual risks, the slope depends on the distribution type of
congenital individual risks and on the distribution types of decay constants x, respectively.
The theoretical value a does not depend on values of the particular distribution parameters
(if x is normally or log-normally distributed in the wide range of these parameters).
The slope is equal to -2 if the frequency function f(x) is that of normal distribution with some variety or
if the frequency function f(x) is invariable (due to uniform distribution). The slope is equal to -1,
if f(x) is a frequency function of log-normal distribution with notable geometric deviation or
if the frequency function can be approximated by f(x)=const./x.
The slope of the integral converges to -2, if x is normally distributed.
In general, it is valid:
and, consequently, the activity for one value of the slope, is additive.
The activity within a particular period, t, is also an additive quantity (activities from two sources can be added)
and the convergence condition of the slope to -1 could be more generalized. The set of sources of the total activity
may be divided into partial subsets. The satisfactory condition is the fact that the decay constants are approximately
log-normally distributed with a notable geometric deviation in the partial sources or that the formula
f(x)"const./x is valid (however not necessarily fulfilled for all sources together). The territory after fallout can
possibly be divided into more sections. If the decay constants, x, are log-normally distributed with a great geometric
deviation in the partial sections, then the slope -a is equal to -1 in all sections. Because Eq. 1 is also additive
(for one value of the parameter a), the slope -a on the whole territory could be -1. The activity sources can be
divided not only into the territories, but into any quantities with the exception of time. The mixture could be
divided into subsets in line with the chemical types of the radionuclides
(At the same time, the range of x has to be wide in these subsets).
The total activity at long distance from a source is inversely proportional to time,
if the moment of a nuclear explosion or the moment of a nuclear accident is used as time origin.
The model with the power 1.2 is too optimistic and it does not describe the data.
The changes of the amount of a specific nuclide do not depend upon the other nuclides and their decay constants for
the fallout at a long distance and, consequently, the function f(x; t=0) could be used.
Such function is independent on the time and the mixture activity is the sum of exponential terms.
It can be assumed that a great number of "external" factors is manifested.
If the decay constants are approximately log-normally distributed (its envelop could be approached by
the frequency function of the log-normal distribution or by the function f(x)"const./x,
then the theoretical slope -a is equal to -1.
The estimate of cumulative exposure rate is simple if the relationship between exposure rate and
time contains only two parameters (the linear relationship in the log-log scale).
If the parameter alfa is known ahead, the cumulative exposure rate can be estimated based on a single measurement.
This can be applied if the time of accident or explosion is known and if the composition of fission products mixture
depends only on a radioactive decay for t>t1. These conditions were fulfilled both for the majority of all measurements
in former Czechoslovakia after the Chernobyl accident and it was automatically fulfilled for the fallout sample taken
in Bratislava in 1965. The slope -a is equal to -1 in both cases. This evidence could be explained as a consequence of
a log-normal distribution of decay constants in the partial sources or as a consequence of the formula f(x)=const/x.
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