Instead of counting the number of nuclei ( ),tN it is easier to count the rate at which decays occur in the radioactive sample, and this is called the activity, defined as ( )0 tdN N e Ndt λλ λ−= = t (47.5) and follows the same exponential decay law (47.3). The SI unit for activity is the Becquerel (Bq), equal to one decay per second, and the common unit is the Curie (Ci), defined as It is also convenient to introduce the half-life as the time interval in which the activity is reduced to one-half .Bq107.3Ci1 10×= 2/1t 2ln2ln2/1 τλ ==t (47.6) The probability of radioactive decay through the direct transition of the nucleus from an initial state to a final state Ψi Ψ f , in time t, is given by (47.7) P t a tif f( ) | ( )|= 2 where the wave function amplitudes a are obtained from Eq.(37.11), which can be approximated by tf ( ) == /)()()()( tifiif ifetVtadt tda i εε −= (47.8) Since the probability of finding our decaying system in the initial state Ψi must decrease with time according to the radioactive decay law (47.3), which gives 2 2| ( ) | | | ti it e λ−Ψ = Ψ (47.9) we have to choose / 2( ) tia t e λ−= (47.10) rather than as considered in Eq.(37.13). Substituting this result into Eq.(47.8), and assuming a constant perturbation which is switched on at time t = 0, yields ai = 1, i εελ −+−= (47.11) Integrating, we get an oscillatory behaviour of the wave function amplitude, given by ( ) ( ) /2 V e a t i ε ε λ − +⎡ ⎤−⎣ ⎦= − + = == = (47.12) Figure 47.1. Energy spectrum of a decaying state of width Γ At the end of the nuclear transition we should consider the amplitude corresponding to ,∞→t which reads )( λεε = = i V a if fi f +−=∞ (47.13) and this yields the transition probability (47.7) in the form )( || )( Γ+−=∞ if fi if V P εε = (47.14) Hence, the probability to observe the nucleus with energy fε in the vicinity of iε follows a Lorentzian distribution, illustrated in Figure 47.1. By analogy with the atomic transitions, Γ = = 〈 〉= =λ / t gives the natural linewidth of the emitted radiation, which is the same as the width of the decaying nuclear state. Nuclear states which are populated in ordinary decays typically have lifetimes greater than corresponding to sec,10 12− ( ) τ/1066.0eV 15−×=Γ (47.15) which is of the order of Since this width is small compared with the energy spacing of nuclear levels, which is of the order of the nuclear decaying states may be regarded to be discrete quasi-stationary states.
The decay law, $N={N}_{0}{e}^{\ensuremath{-}\ensuremath{\lambda}t}$, is based on the assumption that the probability
w of an atom disintegrating in unit time is constant. The question arises whether this assumption is necessary. It might happen that atoms of different ages have different decay probabilities; and
w might depend on age in such a way that exponential decay curves would result. A theorem of Wolfke, however, can be interpreted in such a manner as to show that
w cannot depend on the age of the atom. This result is generalized. Consider a series whose members
A,
B, etc., have decay probabilities
wA,
wB, etc. If all members decay exponentially, if all
w′s are smooth and continuous functions of their variables and if each
w is, at most, a function of the age of the atom, the ages of its ancestors and the ages of its descendants; then the
w′s are constant; and so in this more general case, the assumption of constant decay probability which underlies the wave mechanical theory of disintegration is necessary as well as sufficient.
The framework to describe natural phenomena at their basics being quantum mechanics, there exist a large number of common global phenomena occurring in different branches of natural sciences. One such global phenomenon is spontaneous quantum decay. However, its long time behavior is experimentally poorly known. Here we show, that by combining two genuine quantum mechanical results, it is possible to infer on this large time behavior, directly from data. Specifically, we find evidence for nonexponential behavior of alpha decay of
8Be at large times from experiments.
Schrödinger's original Cat explicitly refers to both “one of the atoms decays” and “the first atomic decay.” This paper examines the first decay occurrence of a finite radioactive sample. Let t₁ denote the time of the first atomic decay. Existing quantities such as half-life, decay constant, mean lifetime, and activity describe radioactive decay collectively, whereas t₁ identifies the first actual decay occurrence of a particular sample. Probability theory contains well-developed treatments of first occurrences, including waiting times, first-arrival times, first-passage times, hitting times, and related stochastic processes. Such treatments require information about the underlying process. This raises two distinct questions for radioactive decay: what information determines the probability distribution of t₁, and what information, if any, determines the actual t₁ of a particular finite sample of a given radioactive isotope before the first decay occurs?