Non-Exponential Decay of Nuclei – A Search for a
Fundamental Quantum Effect in Nuclei.
W.W. SCATES, D.P. WELLS, J.F. HARMON and R.J. SPAULDING Idaho State University, Idaho Accelerator
Center, Campus Box 8263, Pocatello, ID 83209
Abstract: Modern tests of grand unification theories have spent considerable experimental effort in pursuit of rare
decays. A common feature of these experiments is that they involve extremely rare decay processes and probe regions of
the systems’ decay curves which are very short in time compared to their expected mean lifetimes. A potential
complication to interpretation of such experiments is the approximate nature of the exponential decay law for quasistationary states. Using the decay of the isomeric nuclear state 207mPb in the short time limit, we search for predicted
deviations from the exponential decay law. These experiments address the short-time electromagnetic decays of nuclei
with half-lives on the order of a few seconds, and explore the as-yet untapped electromagnetic sector for short-time
(tmin/t½ ≈ 10-5) violations of the exponential decay law. Isomeric states are photo-populated via (γ, n) reactions with
bremsstrahlung beams.
discrete states, (2) that the final group of states is truly
continuous, (3) that the spectrum is flat and unbounded
(the so-called Weisskopf-Wigner approximation) and
(4), that the continuum couples only to the initial state
or, equivalently, that the products of the decay are
stable.
It should be noted that several authors [6, 7]
have pointed out that an unstable state, when monitored
for its existence at sufficiently small time intervals,
should live longer than one that is not periodically
observed. Others contend the opposite. That is, the
monitored state decays faster than the unperturbed
systems [10]. This effect is sometimes called the
quantum Zeno effect (or Anti-Zeno effect for the latter)
and is intimately related to the deviation from the
exponential decay law at small times in that the
measurement of the existence of the particle amounts to
a collapse of its wave-function and a return to the
beginning of its decay curve.
The existence of a deviation in the decay curve
at short times would be especially important for
quantum computing. Non-exponential decay (NED) for
excited states and its relevance to quantum computing
has been theoretically discussed [11], but has not been
observed.
Thus in addition to testing a basic prediction of
quantum mechanics and fundamental conservation
laws, and shedding light on tests of grand unification
theories, the observation of non-exponential decay
offers the possibility of altering the decay rate (and
production rate) of radio-nuclides, and has important
Introduction
Modern tests of grand unification theories
have spent considerable experimental effort in pursuit
of proton decay and neutrinoless double beta decay.
These experiments are designed to test theories of the
fundamental forces and new grand unification schemes.
A common feature of these experiments is that they
probe regions of the systems’ decay curves which are
very short compared to their expected mean lifetimes.
A potential complication to interpretation of such
experiments is the approximate nature of the
exponential decay law for quasi-stationary states, which
has been acknowledged by numerous authors [1,2,3,4].
Merzbacher has noted that, “the exponential
decay law can be derived only as an approximate, and
not a rigorous, result of quantum mechanics and ... it
holds only if the decay process is essentially
independent of the manner in which the decaying state
was formed and of the particular details of the incident
wave packet.” [3]
Kalfin [5] showed that quantum mechanics
predicts that the exponential decay law cannot hold in
the short and long time regime and that the decay rate
must approach zero as t →0. Here short and long mean
with respect to the half life of the system when
measured from the time of preparation of the unstable
state. Deviations from the exponential law can be
traced to a number of approximations made in its
derivation. These include (1) the assertion that the
initial level is well separated in energy from other
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
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consequences for quantum computing. Another
application may be the production of new medical or
industrial radio-nuclides through intermediate states
that would ordinarily be too short-lived to serve as a
production intermediary.
Most historical theoretical and experimental
research on NED has focused upon the long-time limit.
No deviation from the exponential decay “law” in this
regime has ever been observed.
Experiments on the short-time limit for NED
in nuclei were conducted by Norman et al. [8, 9].
These experiments studied the weak decays of 56Mn
down to 0.3 half lives and 60Co down to around 10-4
half lives and found no deviations. In the experiments
on 56Mn and 60Co, the isotopes were produced in a
reactor over a very short time interval, and their
subsequent beta decay curves were observed. More
recently, Norman et al. performed a weak-decay
experiment on 40K [9], where the decay rate of a newly
prepared sample was compared to that of naturally
occurring 40K and found to be the same to an accuracy
of ±11%. This experiment pushed the observational
time-scale tmin/t½ for β decay down to the 10-10 level. It
may be that deviations from exponential decay are too
small in nuclear and/or particle systems to be detected
with current experimental techniques. However, the
absence of experimental limits in a variety of nuclear
and particle systems leaves open the question of the
importance of this effect in interpreting rare decays.
Currently, we have extended the weak-decay
studies conducted by Norman [8, 9] et al. to the
electromagnetic sector using nuclear isomeric decays at
time scales of tmin/t½ = 10-2, and will soon decrease this
limit to tmin/t½ = 10-5.
above neutron threshold (nominally 8 MeV). The
energy spread of the bremsstrahlung almost completely
spanned the predominantly (γ, n) giant dipole resonance
of the nuclei. The peak cross section for the giant
dipole resonance of 208Pb is 1,200 mb.
Decay gammas were detected with a highefficiency germanium detector. The detector was
initially shielded with 10 cm of lead and a 7.5 cm
diameter collimator. We began counting tens of µs (t ≈
10-4 t1/2) after the flash, continuing until the subsequent
pulse at t ≈ 0.25 t1/2. Energy information was recorded
for all events, and temporal information was recorded
for the 1.064-MeV 207mPb line separately using a singlechannel analyzer in conjunction with a multi-channel
scaler.
The pulsed time yields were repeated until
adequate statistics were obtained.
Pulses were
separated by approximately 0.25 half-lives. The
residual activity from previous pulses was a source of
complication, which we address below. The 5-cmthick lead target consisted of natural isotopic
abundance, having roughly 99% purity.
The yield of isomeric nuclei Ytot recorded over
the duration of an experiment was:
Ytot = εγ(E) Ιγ N TR / т
(1)
where εγ(E) is the absolute efficiency of the detector
(including solid angle and other geometric effects), Ιγ is
the relative intensity (branching ratio) of the isomeric
transition, N is the number of isomers produced in a
pulse, TR is the duration of the experiment and т is the
time between pulses. Assuming exponential decay, the
yield (Y∆t) for a given time decade of width « t½ from
activation over the duration of the experiment is then:
Experimental Methods
Y∆t = λ ∆t Ytot
We have investigated the short-time
electromagnetic decays of the isomeric nucleus 207mPb
which has a half-life of 0.805 seconds. This isomeric
nucleus was formed by the 208Pb (γ, n) reaction. The
subsequent purely electromagnetic isomeric transitions
(IT) were recorded as a function of energy and also as a
function of time after irradiation. (γ, n) reactions were
produced with high-intensity, pulsed bremsstrahlung
beam. The bremsstrahlung beam (maximum energy =
20 MeV) was produced with a pulsed electron beam
from the IAC LINAC. Burst widths ranged from 2-30
ns, with 5-200 nC per pulse. These beams were
incident upon tungsten targets of either 0.1mm or 1mm
thickness. This produced a large photon flux per pulse
(3)
where λ is the decay constant of the reaction product
and ∆t is the time width of the decade in question.
However, the target is not allowed to completely decay
before the subsequent bremsstrahlung burst arrives.
As a result, Y∆t does not represent the total recorded
yield in that time division, but the yield in that time
division from the last pulse only. The total yield in the
time division (Ytot∆t), therefore, is Y∆t plus the yield
from all residual activity stemming from all pulses
previous to the most recent. The total number of counts
recorded from the isomeric states in question, in a time
division is therefore:
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Ytot∆t = Y∆t / (1 – e-λт)
(4)
approximately: Y∆t = 0.5 λ ∆t Ytot. Thus the expected
experimental signature of NED in this simple model is a
yield of approximately half of that expected from
exponential decay.
Our present experimental apparatus involving
the HPGe detector and accompanying electronics is
capable of probing for NED effects down to t/t½ ≈ 10-5
in 207mPb, as the resolving time of this system is roughly
tens of microseconds. Note that because we compared
different parts of the decay curve in these experiments,
we are not sensitive to the usual systematic errors
associated with absolute photo-nuclear cross sections,
photon flux, target thickness, etc. The predominant
uncertainties in these experiments are counting
uncertainties.
The background rates from natural background
are small when compared to the expected rates from
these experiments. Other backgrounds stem from
neutron activation of the detector’s environment, from
neutron capture in the detector, and from photoninduced activation of materials other than the target
isotope.
Pileup and dead-time due to the lower-energy
portion of these beam-related background rates can
complicate data analysis by “mis-routing” useful
events, adding background counts to the useful part of
the spectrum, and “losing” useful counts from the
spectrum. These difficulties have not been addressed at
this time.
Recall that Y∆t is the portion of the total that arises from
the most recent pulse. This portion of the decay curve
is most likely to exhibit NED. The remainder of the
yield in the time division is produced by nuclei
relatively late in their decay process, and therefore is
presumably exponential.
Our signal, therefore, is defined as Y∆t, while
Ytot∆t - Y∆t constitutes a temporally-dependent
background. An example of the relative number of
“good” counts (Y∆t) with respect to the total counts
(Ytot∆t) appears in Figure 1. In this figure, the counts
below the dashed line are generated by the decays from
the previous pulses (which, as suggested above, are
assumed exponential). The counts above the dashed
line are counts generated by decays from the most
recent pulse, and are depicted here with a greatly
exaggerated non-exponential effect for demonstrative
purposes.
Results
207m
Pb has two major gamma rays associated with its
decay. The gamma ray energies are 0.570 MeV and
1.064 MeV. The spectrum collected by our high purity
germanium detector is displayed in Figure 2 below. The
net areas of the 0.570-MeV and 1.064-MeV peaks are
11,408 ± 121 and 15,903 ± 140, respectively. The data
in Figure 2 and 3 were collected over a three-and-a-half
hour period at an LINAC repetition rate of 5 Hz. The
half-life of 207mPb is ~ 0.8 sec, therefore т ≈ ¼ t½. The
total yield of the 1.064-MeV peak, Ytot∆t, in this
configuration equated to 0.25 counts per pulse. The
temporal history of the counts in the 1.064-MeV
gamma line is plotted in Figure 3.
Figure 1 – Counts vs. Time for NED measurement with four pulses
per half-life.
To convincingly demonstrate a deviation from
exponential in a particular time-decade, the total
counting error of the time decade, ∆Ytot∆t, must be
statistically significantly smaller than the counts
generated by the decays from the most recent pulse,
namely Y∆t.
For short-lifetime (t½ ~ seconds) experiments
such as our initial studies to date, the expected yield in
a given decade of the decay (∆t « t½), if the decay is
exponential is: Y∆t = λ ∆t Ytot. On the other hand, if
the decay follows the simple linear model from one end
of the time-decade to the other, the yield will be
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correct the energy and efficiency changes during the
recovery of the detector due to the pulse. Once this
enhancement of the experimental setup is integrated, a
better understanding of the decay curve near the pulse
will be possible.
We have searched for the much discussed, but
never observed effect of NED. Preliminary analysis of
our data does not show statistically significant NED
effects at the level of t/t½ = 10-2 for the isomeric
transition of the 207mPb nucleus.
Acknowledgements
*Supported in part by the U.S. Department of
Energy under DOE Idaho Field Office Contract 684045-95.
Figure 2 – HPGe spectrum, showing the 0.570-MeV and 1.064-MeV
lines produced from the isomeric transition of 207mPb.
The time axis of the decay curve in Figure 3 is
with respect to the most recent pulse. Preliminary
analysis of these data does not suggest non-exponential
behavior at t/t½ = 10-2; however further statistical
analysis of our data, down to t/t½ =10-5 is required
before strong conclusions can be drawn.
References
1. Jacob, R., R.G. Sachs, Phys. Rev., 121, no. 1,
(1961) 350.
2. Goldberger, M.L., K.M. Watson, Phys. Rev., 136,
no. 5B, (1964) 1472.
3. Merzbacher, E., Quantum Mechanics, second
edition, 1970.
4. Peres, A., Ann. of Phys., 129 (1980) 33.
5. Khalfin, L.A., JETP Lett., 8, (1968) 65.
6. Ekstein, H., A.J.F. Siegert, Ann. of Phys. 68
(1971)509.
7. Sudbery, A., Ann. of Phys. 157 (1984) 512.
8. Norman, E., S.B. Gazes, S.G. Crane, D.A Bennett,
Phys. Rev. Lett. 60, no. 22, (1988) 2246.
9. Norman, E., et al. Phys. Lett. B, 357 (1995) 521.
10. Kofman A.G., and G. Kurizki, Nature 405, 546
(2000).
11. Flambaum V.V., and F.M. Izrailev, Phys. Rev. E,
64, 026124 (2001).
Figure 3 – Time information recorded for the 1.064-MeV Pb line. The
solid curve represents the least-squares exponential decay fit of the
data. t=0 corresponds to the LINAC pulse.
Future Research and Conclusion
To compensate for the pileup and dead time
due to the lower-energy portion of the beam-related
background, several “references lines” shall be
employed in future experiments. One reference line
will stem from a source with long half-life which is
placed in the geometry. Another reference line will
stem from a short-lived isomer from photon activation
with a half-life of ~100 µs. The gamma rays recorded
in the reference lines by the detector will be used to
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