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Physical Review

The Exponential Law of Radioactive Disintegration

Arthur E. Ruark

Published Oct 15, 1933• 3 citations
Publisher

Abstract

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 ${w}_{A}$, ${w}_{B}$, etc. If all members decay exponentially, if all $w'\mathrm{s}$ 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 $w'\mathrm{s}$ 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.

DOI
10.1103/physrev.44.654

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The decay law, N=N0e\ensuremath−\ensuremathλtN={N}_{0}{e}^{\ensuremath{-}\ensuremath{\lambda}t}N=N0​e\ensuremath−\ensuremathλt, is based on the assumption that the probability www 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 www 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 www cannot depend on the age of the atom. This result is generalized. Consider a series whose members AAA, BBB, etc., have decay probabilities wA{w}_{A}wA​, wB{w}_{B}wB​, etc. If all members decay exponentially, if all w′sw'\mathrm{s}w′s are smooth and continuous functions of their variables and if each www 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′sw'\mathrm{s}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.
Application of quantum mechanics to a simple model of the nucleus gives the phenomenon of radioactive disintegration. The statistical nature of the quantum mechanics gives directly disintegration as a chance phenomenon without any special hypothesis. \textsection{}1 contains a presentation of those features of quantum mechanics which are here used and gives a simple calculation of the disintegration constant. \textsection{}2 discusses the qualitative application of the model to the nucleus. \textsection{}3 presents quantitative calculations amounting to a theoretical interpretation of the Geiger-Nuttall relation between the rate of disintegration and the energy of the emitted $\ensuremath{\alpha}$-particle. In getting this relation one arrives at the rather remarkable conclusion that the law of force between emitted $\ensuremath{\alpha}$-particle and the rest of the nucleus is substantially the same in all the atoms even where the decay rates stand in the ratio 1022{10}^{22}1022. \textsection{}4 calls attention to the natural way in which the paradoxical results of Rutherford and Chadwick on the scattering of fast $\ensuremath{\alpha}$-particles by uranium receive explanation with the model here used. \textsection{}5 discusses certain limitations inherent in the methods employed.
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.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
The quantum-mechanical theory of the decay of unstable states is revisited. We show that the decay is non-exponential both in the short-time and long-time limits using a more physical definition of the decay rate than the one usually used. We report results of numerical studies based on Winter's model that may elucidate qualitative features of exponential and non-exponential decay more generally. The main exponential stage is related to the formation of a radiating state that maintains the shape of its wave function with exponentially diminishing normalization. We discuss situations where the radioactive decay displays several exponents. The transient stages between different regimes are typically accompanied by interference of various contributions and resulting oscillations in the decay curve. The decay curve can be fully oscillatory in a two-flavor generalization of Winter's model with some values of the parameters. We consider the implications of that result for models of the oscillations reported by GSI.
Results of theoretical studies of the quantum unstable systems caused that there are rather widespread belief that a universal feature of the quantum decay process is the presence of three time regimes of the decay process: the early time (initial) leading to the Quantum Zeno (or Anti Zeno) Effects, “exponential” (or “canonical”) described by the decay law of the exponential form, and late time characterized by the decay law having inverse-power law form. Based on the fundamental principles of the quantum theory we give the proof that there is no time interval in which the survival probability (decay law) could be a decreasing function of time of the purely exponential form but even at the “exponential” regime the decay curve is oscillatory modulated with a smaller or a large amplitude of oscillations depending on parameters of the model considered.