Radioactive decay occurs at an exponential rate, meaning that it can be described in terms of a half life. After one half live, half of the original radioactive isotope material in the system under consideration decays. Another half life and half of the remaining material decays, and so on. This is for unforced decay. Forced decay is when the isotopic material is packed densely enough that a decay in one unstable atom sends out a particle that hits another atom and causes it to decay. If it is packed too densely there is a run away reaction and one of those unpopular mushroom clouds or meltdowns. Normal concentrations of radioactive material on earth are well below the levels where forced decay occurs so we can use mathematics of exponential decay to describe the process.
So, to begin with an explanation of the chemistry behind radionuclide decay, the precise mathematical law that this process obeys, and how that law is derived, both empirically and theoretically.
The decay law was first derived empirically, courtesy of work by scientists such as Henri Becquerel, Marie Curie, Ernest Rutherford and others. However, the underlying physics had to wait until the advent of detailed and rigorous quantum theories before it could be elucidated, and is based upon the fact that the nuclei of radionuclides are in an excited state with respect to the sum total of the quantum energy states of the constituent particles.
In order for the system to move to a lower energy state, and settle upon a stable set of quantum numbers, various transformations need to take place, and these transformations result in the nucleus undergoing specific and well-defined structural changes, involving the emission of one or more particles.
Empirical Determination Of The Decay Law
Initially, the determination of the decay law was performed empirically, by observing the decay of various radionuclides in the laboratory, taking measurements of the number of decay events, and plotting these graphically, with time along the x-axis, and counts along the y-axis. Upon performing this task, the data for many radionuclides is seen to lie upon a curve, and determination of the nature of that curve requires a little mathematical understanding.
To determine the nature of a curve, various transformations can be performed upon the data. The result of each of these transformations is as follows:
*note this site will not allow scientific notation* _ denotes lower subscript, ^ higher subscript.
[1] Plot log_e(y) against x - if the result is a straight line, then the relationship is of the form:
log_e(y) = kx + C (where C is some constant, in particular, the y-intercept of the straight line)
which can be rewritten:
log_e(y) - log_e(C_0) = kx (where C = log_e(C_0))
which rearranges to:
y = C_0e^kx
where C_0 is derived from the y-intercept of the straight line produced by the transformed plotting, and k is the gradient of the transformed line.
[2] Plot y against log_e(x) - if the result is a straight line, then the relationship is of the form:
y = k log_e(x) + C, where k is the gradient of the line, and C is the y-intercept of the straight line.
[3] Plot log_e(y) against log_e(x) - if the result is a straight line, then the relationship is of the form:
log_e(y) = k log_e(x) + C (where C is the y-intercept of the straight line thus produced)
This rearranges to:
log_e(y) - log_e(C_0) = k log_e(x) (where C = log_e(C_0))
Which in turn rearranges to:
log_e(y/C_0) = log_e(x^k)
Which finally gives us the relationship:
y = C_0x^k, where k is the gradient of the straight line produced by the transformed data, and C_0 is derived from the y-intercept of the straight line produced by the transformed data.
The above procedures allow us to determine the nature of the mathematical relationships that govern large bodies of real world data, when those bodies of real world data yield curves as raw plots of y against x. By applying the relevant transformations to radionuclide decay data, it was found that transformation [1] transformed the data into a straight line plot and consequently, this informed the scientists examining the data that the decay law was of the form:
N = C_0e^kt
where C_0 and k were constants to be determined from the plot, and which were regarded as being dependent upon the particular radionuclide in question.
So then, if we are start with a known amount of radionuclide, and observe it decaying, then each decay event we detect with a Geiger counter represents one nucleus undergoing the requisite decay transformation. Since the process is random, over a long period of time, decaying nuclei will emit α or β particles in all directions with equal frequency, so we don't need to surround the material with Geiger counters in order to obtain measurements allowing a good first approximation to the decay rate. Obviously if we're engaged in precise work, we do set up our experiments to do this, especially with long-lived nuclei, because the decay events for long-lived nuclei are infrequent, and we need to be able to capture as many of them as possible in order to determine the decay rate with precision. Let's assume that we're dealing with a relatively short-lived radionuclide which produces a steady stream of decay events at a reasonably fast rate, in which case we can simply point a single Geiger counter at it, and work out what proportion of these events we are actually capturing, because that proportion will be the ratio of the solid angle subtended by your Geiger counter, divided by the solid angle of an entire sphere (this latter value being 4π). When we have computed this ratio, let's call it R, which will necessarily be a number less than 1 unless we have surrounded your sample with a spherical shell of Geiger counters, we then start collecting count data, say once per second, and plotting that data. In a modern setup we'd use a computer to collect this mass of data, in order to have as large a body of data as possible to work with. Before working with the raw data, we transform it by taking each of the data points and dividing it by R to obtain the true count.
Once the data has been collected, transformed and plotted, the end result should be a nice curve. At this point, we're interested in knowing what sort of curve we have, and there are two ways we can determine this. One way is to take the transformed data set, comprising count values c1_, c_2, c_3, ... , c_n, where n is the number of data points collected, compute the following values:
r_1 = c-_2 - c_1
r_2 = c_3 - c_2
r_3 = c_4 - c_3
...
r_n-1 = c_n - c_n-1
Then plot a graph with r_k on the vertical axis, and c_k on the horizontal axis. This should give a reasonable approximation to a straight line, and the slope of that straight line, obtained via regression analysis, will give the first approximation to the decay constant k. At this point, we know we are dealing with a relationship of the form dN/dt = -kN, and you can then apply the integral calculus to that equation. Technically, what we are doing here is approximating the derivative by computing first differences.
However, as a double check, we can also perform a logarithmic regression on the data, plotting log_e(c_k) against time, which should also reveal a straight line, and again, the slope of that line will give you the value of k, which should be in good agreement with the value obtained earlier using the more laborious plot of r_k against c_k. In other words, applying the transformation [1] above to the data set, and extracting an exponential relationship from the data. Since we now know that the data is of the form:
log_eN = -kt
we can then derive the exponential form and check that it tallies with the integral calculus result.
Once we have that function coupling the decay rate to time, we can then work backwards, and feed in the values of the known starting mass and the experimentally obtained decay constant k, and see if the function obtained reproduces the transformed data points. If the result agrees with observation to a very good fit, we're home and dry.
This is, essentially, how the process was done when the decay law was first derived, lots of data points were collected from observation of real radionuclide decay, and the above processes applied to that data, to derive the exponential decay law. When this was done for multiple radionuclides, it was found that they all obeyed the same basic law, namely:
N = N_0e^-kt
where N_0 is your initial amount of radionuclide, N is the amount remaining after time t, and k is the decay constant for the specific radionuclide.
Theoretical Derivation Of The Decay Law And Comparison With The Above Empirical Result
Using the calculation of first differences in the empirical determination above, that the rate of change of material with time, plotted against the material remaining, is constant, this immediately leads us to conclude that the decay law is governed by a differential equation. An appropriate differential equation is therefore:
dN/dt = -kN
which states that the amount of material undergoing decay is a linear function of the amount of material present, the minus sign indicates that the process results in a reduction of material remaining. Rearranging this differential equation, we have:
dN/N = -k dt
Integrating this, we have:
∫dN/N = - ∫ k dt
Our limits of integration are, for the left hand integral, the initial amount at t=0, which we call N_0, and the amount remaining after time t, which we call N_t. Our limits of integration for the right hand integral are t=0 and t=t_p, the present time.
So, we end up with:
log_eN -log_eN0 = -kt_p
By an elementary theorem of logarithms, this becomes:
log_e(N/N_0) = -kt_p
Therefore, exponentiating both sides, we have:
N/N_0 = e^-kt
or, the final form:
N = N_0e^-kt
The half-life of a radionuclide is defined as the amount of time required for half the initial amount of material to decay, and is called T_½. Therefore, feeding this into the equation for the decay law,
½N_0 = N_0e^-kt
Cancelling N_0 on both sides, we have:
½ = e^-kt
log_e½ = -kt
By an elementary theorem of logarithms, we have:
log_e2 = kt
Therefore T_½ = log_e2/k
Alternatively, if the half-life is known, but the decay constant k is unknown, then k can be computed by rearranging the above to give:
k = log_e2/T_½
Which allows us to move seamlessly from one system of constants, half-lives to another, decay constants and back again.
If the initial amount of substance N_0 is known (e.g., we have a fresh sample of radionuclide prepared from a nuclear reactor), and we observe the decay over a time period t, then measure the amount of substance remaining, we can determine the decay constant empirically as follows:
N = N_0e^-kt
N/N_0 = e^-kt
log_e(N/N_0) = -kt
Therefore:
(1/t) log_e(N_0/N) = k
On the left hand side, the initial amount N_0, the remaining amount N and the elapsed time t are all known, therefore k can be computed using the empirically observed data.
Once again, this agrees with the empirical data from which the law was derived in the earlier exposition above, and consequently, we can be confident that we have alighted upon a correct result.
Once we have the decay law in place, it simply remains for appropriate values of k to be determined, which will be unique to each radionuclide. This work has been performed by scientists, and as a result of decades of intense labour in this vein in physics laboratories around the world, vast bodies of radionuclide data are now available.
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