While the underlying process of radioactive decay is subatomic, historically and in most practical cases it is encountered in bulk materials with very large numbers of atoms. This section discusses models that connect events at the atomic level to observations in aggregate.
Terminology The
decay rate, or
activity, of a radioactive substance is characterized by the following time-independent parameters: • The
half-life, , is the time taken for the activity of a given amount of a
radioactive substance to decay to half of its initial value. • The
decay constant, "
lambda", the reciprocal of the mean lifetime (in ), sometimes referred to as simply
decay rate. • The
mean lifetime, "
tau", the average lifetime (1/
e life) of a radioactive particle before decay. Although these are constants, they are associated with the
statistical behavior of populations of atoms. In consequence, predictions using these constants are less accurate for minuscule samples of atoms. In principle a half-life, a third-life, or even a (1/√2)-life, could be used in exactly the same way as half-life; but the mean life and half-life have been adopted as standard times associated with exponential decay. Those parameters can be related to the following time-dependent parameters: •
Total activity (or just
activity), , is the number of decays per unit time of a radioactive sample. •
Number of particles, , in the sample. •
Specific activity, , is the number of decays per unit time per amount of substance of the sample at time set to zero (). "Amount of substance" can be the mass, volume or moles of the initial sample. These are related as follows: :\begin{align} t_{1/2} &= \frac{\ln(2)}{\lambda} = \tau \ln(2) \\[2pt] A &= - \frac{\mathrm{d}N}{\mathrm{d}t} = \lambda N = \frac{\ln(2)}{t_{1/2}}N \\[2pt] S_A a_0 &= - \frac{\mathrm{d}N}{\mathrm{d}t}\bigg|_{t=0} = \lambda N_0 \end{align} where
N0 is the initial amount of active substance — substance that has the same percentage of unstable particles as when the substance was formed.
Assumptions The mathematics of radioactive decay depend on a key assumption that a nucleus of a radionuclide has no "memory" or way of translating its history into its present behavior. A nucleus does not "age" with the passage of time. Thus, the probability of its breaking down does not increase with time but stays constant, no matter how long the nucleus has existed. This constant probability may differ greatly between one type of nucleus and another, leading to the many different observed decay rates. However, whatever the probability is, it does not change over time. This is in marked contrast to complex objects that do show aging, such as automobiles and humans. These aging systems do have a chance of breakdown per unit of time that increases from the moment they begin their existence. Aggregate processes, like the radioactive decay of a lump of atoms, for which the single-event probability of realization is very small but in which the number of time-slices is so large that there is nevertheless a reasonable rate of events, are modelled by the
Poisson distribution, which is discrete. Radioactive decay and
nuclear particle reactions are two examples of such aggregate processes. The mathematics of
Poisson processes reduce to the law of
exponential decay, which describes the statistical behaviour of a large number of nuclei, rather than one individual nucleus. In the following formalism, the number of nuclei or the nuclei population
N, is of course a discrete variable (a
natural number)—but for any physical sample
N is so large that it can be treated as a continuous variable.
Differential calculus is used to model the behaviour of nuclear decay.
One-decay process Consider the case of a nuclide that decays into another by some process (emission of other particles, like
electron neutrinos and
electrons e− as in
beta decay, are irrelevant in what follows). The decay of an unstable nucleus is entirely random in time so it is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant in time. Therefore, given a sample of a particular radioisotope, the number of decay events expected to occur in a small interval of time is proportional to the number of atoms present , that is : - \frac{\mathrm{d}N}{\mathrm{d}t} \propto N Particular radionuclides decay at different rates, so each has its own decay constant . The expected decay is proportional to an increment of time, : The negative sign indicates that decreases as time increases, as the decay events follow one after another. The solution to this first-order
differential equation is the
function: :N(t) = N_0\,e^{-{\lambda}t} where is the value of at time = 0, with the decay constant expressed as The sum of these two terms gives the law for a decay chain for two nuclides: :\frac{\mathrm{d}N_B}{\mathrm{d}t} = -\lambda_B N_B + \lambda_A N_A. The rate of change of , that is , is related to the changes in the amounts of and , can increase as is produced from and decrease as produces . Re-writing using the previous results: The subscripts simply refer to the respective nuclides, i.e. is the number of nuclides of type ; is the initial number of nuclides of type ; is the decay constant for – and similarly for nuclide . Solving this equation for gives: : N_B = \frac{N_{A0}\lambda_A}{\lambda_B - \lambda_A} \left( e^{-\lambda_A t} - e^{-\lambda_B t}\right) . In the case where is a stable nuclide ( = 0), this equation reduces to the previous solution: : \lim_{\lambda_B\rightarrow 0} \left[ \frac{N_{A0}\lambda_A}{\lambda_B - \lambda_A} \left( e^{-\lambda_A t} - e^{-\lambda_B t} \right) \right] = \frac{N_{A0}\lambda_A}{0 - \lambda_A} \left( e^{-\lambda_A t} - 1 \right) = N_{A0} \left( 1- e^{-\lambda_A t} \right), as shown above for one decay. The solution can be found by the
integration factor method, where the integrating factor is . This case is perhaps the most useful since it can derive both the one-decay equation (above) and the equation for multi-decay chains (below) more directly.
Chain of any number of decays For the general case of any number of consecutive decays in a decay chain, i.e. , where is the number of decays and is a dummy index (), each nuclide population can be found in terms of the previous population. In this case , , ..., . Using the above result in a recursive form: : \frac{\mathrm{d}N_j}{\mathrm{d}t} = - \lambda_j N_j + \lambda_{j-1} N_{(j-1)0} e^{-\lambda_{j-1} t}. The general solution to the recursive problem is given by '''Bateman's equations''': {{Equation box 1 N_D &= \frac{N_1(0)}{\lambda_D} \sum_{i=1}^D \lambda_i c_i e^{-\lambda_i t} \\[3pt] c_i &= \prod_{j=1, i\neq j}^D \frac{\lambda_j}{\lambda_j - \lambda_i} \end{align} }}
Multiple products In all of the above examples, the initial nuclide decays into just one product. Consider the case of one initial nuclide that can decay into either of two products, that is '
and ' in parallel. For example, in a sample of
potassium-40, 89.3% of the nuclei decay to
calcium-40 and 10.7% to
argon-40. We have for all time : : N = N_A + N_B + N_C which is constant, since the total number of nuclides remains constant. Differentiating with respect to time: : \begin{align} \frac{\mathrm{d}N_A}{\mathrm{d}t} & = - \left(\frac{\mathrm{d}N_B}{\mathrm{d}t} + \frac{\mathrm{d}N_C}{\mathrm{d}t} \right) \\ - \lambda N_A & = - N_A \left ( \lambda_B + \lambda_C \right ) \\ \end{align} defining the
total decay constant in terms of the sum of
partial decay constants and : : \lambda = \lambda_B + \lambda_C . Solving this equation for : : N_A = N_{A0} e^{-\lambda t} . where is the initial number of nuclide A. When measuring the production of one nuclide, one can only observe the total decay constant . The decay constants and determine the probability for the decay to result in products or as follows: : N_B = \frac{\lambda_B}{\lambda} N_{A0} \left ( 1 - e^{-\lambda t} \right ), : N_C = \frac{\lambda_C}{\lambda} N_{A0} \left ( 1 - e^{-\lambda t} \right ). because the fraction of nuclei decay into while the fraction of nuclei decay into .
Corollaries of laws The above equations can also be written using quantities related to the number of nuclide particles in a sample; • The activity: . • The
amount of substance: . • The
mass: . where = is the
Avogadro constant, is the
molar mass of the substance in kg/mol, and the amount of the substance is in
moles.
Decay timing: definitions and relations Time constant and mean-life For the one-decay solution '''': :N = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}, \,\! the equation indicates that the decay constant has units of , and can thus also be represented as 1/, where is a characteristic time of the process called the
time constant. In a radioactive decay process, this time constant is also the
mean lifetime for decaying atoms. Each atom "lives" for a finite amount of time before it decays, and it may be shown that this mean lifetime is the
arithmetic mean of all the atoms' lifetimes, and that it is , which again is related to the decay constant as follows: :\tau = \frac{1}{\lambda}. This form is also true for two-decay processes simultaneously '''', inserting the equivalent values of decay constants (as given above) : \lambda = \lambda_B + \lambda_C \, into the decay solution leads to: :\frac{1}{\tau} = \lambda = \lambda_B + \lambda_C = \frac{1}{\tau_B} + \frac{1}{\tau_C}\,
Half-life A more commonly used parameter is the half-life . Given a sample of a particular radionuclide, the half-life is the time taken for half the radionuclide's atoms to decay. For the case of one-decay nuclear reactions: :N = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/\tau}, \,\! the half-life is related to the decay constant as follows: set and = to obtain :t_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2. This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer.
Half-lives of known radionuclides vary by almost 54 orders of magnitude, from more than years ( sec) for the very nearly stable nuclide
128Te, to seconds for the highly unstable nuclide
5H. For example,
chemical bonds can affect the rate of electron capture to a small degree (in general, less than 1%) depending on the proximity of electrons to the nucleus. In 7Be, a difference of 0.9% has been observed between half-lives in metallic and insulating environments. This relatively large effect is because beryllium is a small atom whose valence electrons are in 2s
atomic orbitals, which are subject to electron capture in 7Be because (like all s atomic orbitals in all atoms) they naturally penetrate into the nucleus. In 1992, Jung et al. of the Darmstadt Heavy-Ion Research group observed an accelerated β− decay of 163Dy66+. Although neutral 163Dy is a stable isotope, the fully ionized 163Dy66+ undergoes β− decay
into the K and L shells to 163Ho66+ with a half-life of 47 days.
Rhenium-187 is another spectacular example. 187Re normally undergoes beta decay to 187Os with a half-life of 41.6 billion years, but studies using fully ionised 187
Re atoms (bare nuclei) have found that this can decrease to only 32.9 years. This is attributed to "
bound-state β− decay" of the fully ionised atom – the electron is emitted into the "K-shell" (1s atomic orbital), which cannot occur for neutral atoms in which all low-lying bound states are occupied. A number of experiments have found that decay rates of other modes of artificial and naturally occurring radioisotopes are, to a high degree of precision, unaffected by external conditions such as temperature, pressure, the chemical environment, and electric, magnetic, or gravitational fields. Comparison of laboratory experiments over the last century, studies of the Oklo
natural nuclear reactor (which exemplified the effects of thermal neutrons on nuclear decay), and astrophysical observations of the luminosity decays of distant supernovae (which occurred far away so the light has taken a great deal of time to reach us), for example, strongly indicate that unperturbed decay rates have been constant (at least to within the limitations of small experimental errors) as a function of time as well. Recent results suggest the possibility that decay rates might have a weak dependence on environmental factors. It has been suggested that measurements of decay rates of
silicon-32,
manganese-54, and
radium-226 exhibit small seasonal variations (of the order of 0.1%). However, such measurements are highly susceptible to systematic errors, and a subsequent paper has found no evidence for such correlations in seven other isotopes (22Na, 44Ti, 108Ag, 121Sn, 133Ba, 241Am, 238Pu), and sets upper limits on the size of any such effects. The decay of
radon-222 was once reported to exhibit large 4% peak-to-peak seasonal variations (see plot), which were proposed to be related to either
solar flare activity or the distance from the Sun, but detailed analysis of the experiment's design flaws, along with comparisons to other, much more stringent and systematically controlled, experiments refute this claim.
GSI anomaly An unexpected series of experimental results for the rate of decay of heavy
highly charged radioactive
ions circulating in a
storage ring has provoked theoretical activity in an effort to find a convincing explanation. The rates of
weak decay of two radioactive species with half-lives of about 40 s and 200 s are found to have a significant
oscillatory modulation, with a period of about 7 s. The observed phenomenon is known as the
GSI anomaly, as the storage ring is a facility at the
GSI Helmholtz Centre for Heavy Ion Research in
Darmstadt,
Germany. As the decay process produces an
electron neutrino, some of the proposed explanations for the observed rate oscillation invoke neutrino properties. Initial ideas related to
flavour oscillation met with skepticism. A more recent proposal involves mass differences between neutrino mass
eigenstates. ==Nuclear processes==