Seeking Non-Exponential Decay Other Times Are Important": -R£ = H/E0 Where E0 Is the Energy Released in the P

298 --------------------------------------N8NSAN0VIBNS--------------~N~A~TU~R=E~v~o=L~.3~3s~n~sE~ff~EM~BE~R~I9~~ Radioactivity and very large times. The intrinsic mem ory time 'l'w, and the exponential decay time -r have already been identified. Two Seeking non-exponential decay other times are important": -r£ = h/E0 where E0 is the energy released in the P. T. Greenland decay and h is Planck's constant, and rL - 3rlog.,(E0r/li). One always has ONE of the most famous laws of physics <'P,Iexp( -iHt,) I'P.(t,)> , where 'P. is the rw < <r. For decays with very small energy is the exponential-decay law of radio state of the decay products at timet,. The release, however, rE increases and rL activity': the decay of any radioactive second term is the 'memory' term that is can decrease into regions where obser isotope is characterized solely by its time additional in the quantum-mechanical vation is possible. It can be shown that constant -r or equivalently by its half-life treatment', modifying the classical effect exponential decay sets in only after the t,n = -r log,2. During a time t, N0exp-th represented solely by the first term.) Thus maximum of -rE and <wand lasts until -rD nuclei will decay, where N0 is the number quantum mechanics implies that the so for decays near threshold (E0 --+ 0) 0 of radioactive atoms at the beginning of system has a memory, so that it is possible there is no exponential decay region' • the interval. This law has now been to determine the absolute age of a sample But to see deviations from exponential checked by Norman et al.' for extremely by examining its decay law, which is not decay, the decay products must be allow short periods, down to 0.01 per cent of a possible for pure exponential decay. ed to recombine. This is not a problem at half-life, and long periods, up to 45 half In the new experiments, Norman eta/.' short times; indeed a straightforward esti lives. In neither case was any deviation searched for deviations from exponential mate of the time required for the decay from exponential decay found. Why products to separate by more than a de should so much effort be expended testing Broglie wavelength (the characteristic dis a law with such a venerable history? tance for quantum systems) gives -r£, but Because it is incompatible with the equally for long times the situation is different. 3 venerable theory of quantum mechanics • Any fluctuations, including those associ Classically, the exponential law holds ated with observation in quantum mech because each radioactive atom in an en anics, will destroy the recombination semble decays independently and with effect and therefore produce exponential a probability that is independent of time. decay. Furthermore, the decay constant As decay proceeds, there are fewer will be -r, the undisturbed value, and inde remaining radioactive atoms and the pendent of the characteristic timescale of sample's activity falls. Exponential decay the fluctuations if these occur pre is seen in atomic and molecular decay dominantly in the exponential region of 9 processes as well as in radioactivity. the decay • This is likely to prevent the In quantum mechanics, the decay law of observation of post-exponential decay, a prepared system should be derived from particularly in nuclear systems, where -rL is first principles. The system is described by large. For 56Mn, -rL = 3 weeks, and the a state vector 'P which includes informa Rutherford exponen- decay products must be able to recombine tion on how many atoms have aJ:eady tial after this time; -r£ =10-" s so the pre-ex decayed. A 'hamiltonian' operator H con decay for short and long times in the radio ponential decay will also not be observed. trols the dynamics of the system, which activity of 60Co and 56Mn. 60Co, which has a Thus it seems unlikely that nuclear evolves according to the •Schrodinger half-life of 5.271 yr, was made from 59Co decays will show deviations from the equation as 'P(t) = exp( -iHt)'P,, where by neutron irradiation in a pulsed reactor. exponential-decay law which they made 'P, is the state at timet = 0. This defines a precise starting time for the famous. The most promising process for The decay law is obtained by evaluating experiment, and Norman et a/. examined observing non-exponential decay is the the survival probability, that is, the com the decay law for the first 10-4 half-lives. near-threshold photodetachment of elec ponent of the initial state remaining in Similarly, the decay of 56Mn, with a half trons from negative ions using highly wavefunction 'P(t) at time t (denoted life of 2.5785 h, was examined over 45 stabilized lasers". In this case, there is a P(t) = IA(t)l ' = I<'P,I'P(t)>l'). C~rtain half-lives, by which time the initial activity chance of extending -r£, the length of the 14 physical restrictions must be put on Hand had fallen by a factor of about 3x 10 • pre-exponential region, and of shrinking 'P, (ref. 4). Briefly, there must be a Neither isotope revealed any deviation -rD the time of onset of post-exponential lower limit to the possible energy of the from exponential decay, and nor has any decay. The correct combination of cir decayed nuclei, and for decay to take other test'. This seems to be a paradox. cumstances does not seem to arise natur place, the spectrum of energies must The resolution of this paradox lies in the ally to produce deviations from exponen contain a continuum. This is sufficient to form that 'P, must have if we are to speak tial decay, which is why it is such an show that the decay rate of any state must of a decaying state. Essentially the decay accurately fulfilled law, even out to 45 ultimately become slower than exponen time -r of 'P, must be long in comparison half-lives. 0 tial at very large times. The mean energy with the characteristic evolution times 'l'w I. Rutherford, E. Sber. Akad. Wil's. Wien. Abt. 2A 120. 300 (1911) . of the initial state 'P, must be associated with the interactions which 2. Norman, E.B. eta/. Phys. Rev. Lett. 60,2246--2249 (1988) . finite, which is sufficient to prove that the (nearly) bind the decay products of 'P,. Or 3. Fonda, L. , Ghirardi, G .C. & Rimini, A. Rep. Prog. Phys. initial decay rate of any state must be equivalently, 'P, must be constructed from 41 ,587-631 (1978). 4. Khalfin , L.A. / £ TP6, 1053-1063(1958). slower than exponential for small times. a wavepacket which is much narrower in 5. Williams, O.N. Commun. Math. Phys. 21 ,314--333(1971). In fact the decay of an isolated quantum energy than the range spanned by the 6. Nikolaev . N.N. Soviet Phys. Usp. II, 522 (1968) . 7. Rice, O.K. Phys.Rev. 35, 1538--1550; 1551-1558 (1930). state can never be exponential. This is interaction responsible for the decay'. 8. Chiu, C.B., Sudarshan, E.C.G. & Misa, B. Phys. Rev. because the products of the decay at an Under these circumstances the memory 016,520-529 (1977). earlier time can recombine to form the 9. Greenland, P.T. & Lane, A.M. Phys. Lett. All7, 181 (1986). time is of the order of rw, very much less 10. Rz~ewski , K. eta/. J. Phys. BIS, 1.661-1.667 (1982). initial state later. (This can be seen by than the decay time -r, and the memory II. Haan, S.L. & Cooper, J. J. Phys. 817,3481-3492 (1984). subdividing the interval t into two inter term then becomes negligible. P. T. Greenland is in the Theoretical Physics vals t, and t2• The probability amplitude Significant deviations from exponential Division, Harwell Laboratory, Didcot OXll A(t) must satisfy A(t,+t,) = A(t,)A(t,) + decay occur actually only for very small8 ORA, UK. .

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