ABSTRACT
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.
