Equations of Radioactive Decay and Growth
EXPONENTIAL DECAY
Half Life. You have seen (Meloni) that a given radioactive species decays according to an
exponential law: N  N0e  t or A  A0e  t , where N and A represent the number of
atoms and the measured activity, respectively, at time t, and N0 and A0 the
corresponding quantities when t = 0, and λ is the characteristic decay constant for the
species. The half life t1/2 is the time interval required for N or A to fall from any
particular value to one half that value. The half life is conveniently determined from a
plot of log A versus t when the necessary data are available, and is related to the
decay constant:
t1/2 
ln2


0.693

Average Life. We may determine the average life expectancy of the atoms of a
radioactive species. This average life is found from the sum of the times of existence
of all the atoms divided by the initial number. If we consider N to be a very large
number, we may approximate this sum by an equivalent integral, finding for the average
life 





1
1
1
 t  1 t 
tdN 
tNdt    te tdt   
e  


NO 0
NO 0
 
0 
0
We see that the average life is greater than the half life by the factor 1/0.693; the
difference arises because of the weight given in the averaging process to the fraction
of atoms that by chance survive for a long time.
It may be seen that during the time 1/ λ an activity will be reduced to just 1 / e of its
initial value.
Mixtures of Independently Decaying Activities. If two radioactive species, denoted
by subscripts 1 and 2, are mixed together, the observed total activity is the sum of
the two separate activities: A = A1 + A2 = c1 λ1N1+ c2 λ2N2. The detection coefficients
c1 and c2 are by no means necessarily the same and often are very different in
magnitude. In general, A1  A2  A3  …… An for mixtures of n species.
For a mixture of several independent activities the result of plotting log A versus t is
always a curve concave upward (convex toward the origin). This curvature results
because the shorter-lived components become relatively less significant as time
passes. In fact, after sufficient time the longest-lived activity will entirely
predominate, and its half life may be read from this late portion of the decay curve.
Now, if this last portion, which is a straight line, is extrapolated back to t = 0 and the
extrapolated line subtracted from the original curve, the residual curve represents
the decay of all components except the longest-lived. This curve may be treated again
in the same way, and in principle any complex decay curve may be analyzed into its
components. In actual practice experimental uncertainties in the observed data may be
expected to make it difficult to handle systems of more than three components, and
even two-component curves may not be satisfactorily resolved if the two half lives
differ by less than about a factor of two. The curve shown in figure 1 is for two
components with half lives differing by a factor of 10.
Time (h)
Figure 1- Analysis of composite decay curve: (a) composite decay curve; (b) longerlived component (t1/2 = 8.0 h); (c) shorter-lived component (t1/2 = 0.8 h).
The resolution of a decay curve consisting of two components of known but not very
different half lives is greatly facilitated by the following approach. The total activity
at time t is
A  A10 e 1t  A20 e 2t
By multiplying both sides by e 1t we obtain
A  A10  A20 e  1 2 t
Since A1 and A2 are known and A has been measured as a function of t, we can construct
a plot of Ae 1t versus e 1 2 t ; this will be a straight line with intercept A10 and slope A20 .
Least-squares analysis is a more objective method for the resolution of complex decay
curves than the graphical analysis described. Computer programs for this analysis have
been developed (J. B. Cumming, "CLSQ, The Brookhaven Decay-Curve Analysis Program,"
in Application of Computers to Nuclear and Radiochemistry (G. D. O'Kelly, Ed.), NASNRC, Washington, 1963, p. 25.) that give values of A° and its standard deviation for each
of the components. Some of the programs can also be used to search for the "best
values" of the decay constants.
Calculate the weight in grams w of 1 mCi of 14C from its half-life of 5720 years.

ln2
0,693

 3,83x 1012 s 1
t1 / 2 5720x 365x 24x 3600
dN
 N
dt
1mCi  3,70x 107 dps

A
3.7 x 107
N  
 2,24x 10 4 g
12
 3,83x 10
Growth of radioactive products
General Equation. We considered briefly a special case in which a radioactive daughter
substance was formed in the decay of the parent. Let us take up the general case for
the decay of a radioactive species, denoted by subscript 1, to produce another
radioactive species, denoted by subscript 2.
The behavior of N1 is just as has been derived; that is,

and
dN1
 1N1
dt
N1  N10 e 1t
where we use the symbol N10 to represent the value of N1 at t = 0.
Now the second species is formed at the rate at which the first decays, 1N1 , and itself
decays at the rate 2N2 . Thus
dN2
 1N1  2N2
dt
dN2
 1N10 e 1t  2N2
dt
dN2
 2N2  1N10e 1t
dt
By multiplying both sides by e 2t :
e2t
dN2  t 
 2N2e2t  1N10e 2 1 t
dt
what to be rewritten:
d
N2e2t   1N10e 2 1 t

dt
Integrating:
1
N10 e 2 1 t  C
2  1
1
N2 
N10 e 1t  Ce 2t
2  1
N2e 2t 
for t=0, N2 = N20 :
C
N2 
1
N10  N20
2  1
1
N10  e 1t  e 2t   N20 e 2t (2)
 2  1
dN2
 1N1  2N2  0 this linear differential equation of the first order
dt
may be obtained by standard methods and gives
The solution of
N2 
1
N10  e 1t  e 2t   N20 e 2t
 2  1
where N20 is the value of N2 at t = 0. Notice that the first group of terms shows the
growth of daughter from the parent and the decay of these daughter atoms; the last
term gives the contribution at any time from the daughter atoms present initially.
Transient Equilibrium.
In applying (2) to considerations of radioactive (parent and daughter) pairs, we can
distinguish two general cases, depending on which of the two substances has the
longer half life.
If the parent is longer-lived than the daughter (λ1<λ2), a state of so-called radioactive
equilibrium is reached; that is, after a certain time the ratio of the numbers of atoms
and, consequently, the ratio of the disintegration rates of parent and daughter
become constant.
This can be readily seen from (2); after t becomes sufficiently large, e 2t is negligible
compared with e 1t , and N20 e 2t also becomes negligible; then
N2 
and, since N1  N10 e 1t
1
N10 e 1t
 2  1
The relation of the two measured activities is
found from A1  c11N1 , A2  c22N2 to be
A1 c1  2  1 
(4)

A2
c22
In the special case of equal detection coefficients (c 1 =c2) the ratio of the two
A

activities, 1  1  1 , may have any value between 0 and 1, depending on the ratio of
A2
2
λ1 to λ2 that is, in equilibrium the daughter activity will be greater than the parent
activity by the factor λ2/( λ2 – λ1).
In equilibrium both activities decay with the parent's half life.
As a consequence of the condition of transient equilibrium (λ2>λ1), the sum of the parent
and daughter disintegration rates in an initially pure parent fraction goes through a
maximum before transient equilibrium is achieved.
This situation is illustrated in figure 2.
Figure 2 - Transient equilibrium: (a) total activity of an initially pure parent fraction; ( b ) activity
due to parent (t1/2 = 8.0 h); (c) decay of freshly isolated daughter fraction (t1/2 = 0.80 h); (d)
daughter activity growing in freshly purified parent fraction; ( e ) total daughter activity in
parent-plus-daughter fractions
The more general condition for the total measured activity (A1+A2) of an initially pure
parent fraction to exhibit a maximum is found to be c2/c1 > λ1/λ2. This condition holds
   1   c2  1
regardless of the relative magnitudes of λ1, and λ2. The 2
condition
2
c1 2
will give a maximum in the total measured activity that occurs at a negative time.
Secular Equilibrium. A limiting case of radioactive equilibrium in which 1
2 and in
which the parent activity does not decrease measurably during many daughter half lives
is known as secular equilibrium.
Derive the equation as a useful approximation of (3):
N1

 1 or 1N1  2N2
N2 2
In the same way (4) reduces to
A1 c1

A2 c2
and the measured activities are equal if c1 =c2.
Figure 2 presents an example of transient equilibrium with 1  2 (actually with λ1/λ2 =
0.1); the curves represent variations with time of the parent activity and the activity of
a freshly isolated daughter fraction, the growth of daughter activity in a freshly
purified parent fraction, and other relations; in preparing the figure we have taken
c1=c2.
Figure 3 is a similar plot for secular equilibrium; it is apparent that as λ1, becomes
smaller compared to λ2 the curves for transient equilibrium shift to approach more
and more closely the limiting case shown in figure 3.
Figure 3 - Secular equilibrium: (a) total activity of an initially pure parent fraction; (b)
activity due to parent (t1/2   ); this is also the total daughter activity in parent-plus-
daughter fractions; (c) decay of freshly isolated daughter fraction (t1/2  0.8h ); (d)
daughter activity growing in freshly purified parent fraction.
The Case of No Equilibrium. If the parent is shorter-lived than the daughter ( λ1>λ2),
it is evident that no equilibrium is attained at any time. If the parent is made initially
free of the daughter, then as the parent decays the amount of daughter will rise,
pass through a maximum, and eventually decay with the characteristic half life of the
daughter. This is illustrated in figure 4; for this plot we have taken λ1/λ2= 10, and c1=c2.
In the figure the final exponential decay of the daughter is extrapolated back to t=0.
Figure 4 - The case of no equilibrium: (a) total activity; (b) activity due to parent
(t1/2  0.8h ); (c) extrapolation of final decay curve to time zero; (d) daughter activity
in initially pure parent.
This method of analysis is useful if 1
2 , for then this intercept measures the
activity c22N the N1 atoms give rise to N2 atoms so early that N10 may be set equal to
0
1
0
the extrapolated value of N2 at t = 0. The ratio of the initial activity c11N10 to this
extrapolated activity gives the ratios of the half lives if the relation between c1 and c2
is known:
 
 t1 
c11N10
c1 1  2 2
 x

c22N10 c2 2  
 t1 
 2 1
If λ2 is not negligible compared to λ1, it can be shown that the ratio λ1/λ2 in this equation
 1  2  and the expression involving the half lives changed
should be replaced by
2
accordingly.
Both the transient-equilibrium and the no-equilibrium cases are sometimes analyzed in
terms of the time tm for the daughter to reach its maximum activity when growing in
a freshly separated parent fraction.
This time we find from the general equation (2) by differentiating,
dN2
12
1  2

N10 e 1t 
N0 e 2t
dt
  2  1 
  2  1  1
and setting
dN2
 0 when t = tm:
dt
1

2
ln 2
 e 2 1 tm or tm 
 2  1 1
1
At this time the daughter decay rate 2N2 is just equal to the rate of formation 1N1 ,
[this is obvious from (1)]; in figures 2 and 4, in which we assumed c1=c2, we have the
parent activity A1 intersecting the daughter growth curve d at the time tm. (The time
tm is infinite for secular equilibrium.)
Many Successive Decays. If we consider a chain of three or more radioactive
products, it is clear that the equations already derived for N1 and N2 as functions of
time are valid, and N3 may be found by solving the new differential equation:
dN2
This is entirely analogous to the equation for dN3   N   N (5)
, but the
2 2
3 3
dt
dt
solution calls for more labor, since N2 is a much more complicated function than Nì. The
next solution for N4 is still more tedious. H. Bateman (H. Bateman. "Solution of a
System of Differential Equations Occurring in the Theory of Radio-active
Transformations," Proc. Cambridge Phil. Soc. IS, 423 (1910) has given the solution for a
chain of n members with the special assumption that at t = 0 the parent substance
alone is present, that is, that N20  N30  ........  Nn0  0 . This solution is
Ni  C1e 1t  C2e 2t  ....  Cn e n t
where
C1 
12 ...n 1
N10






.....



 2 1  3 1   n 1 
C2 
12 ...n 1
N10
 1  2   3  2  .....  n  2 
............
Cn 
12 ...n 1
N10






.....



 1 n  2 n   n1 n 
If we do require a solution to the more general case with N20 , N30 ,........, N30  0 , we may
construct it by adding to the Bateman solution for Nn, in an n-membered chain a
Bateman solution for Nn in an (n-1)-membered chain with substance 2 as the parent,
and, therefore, N2  N20 at t = 0, and a Bateman solution for Nn in an (n-2)-membered
chain, and so on.
Branching Decay. The case of branching decay when a nuclide can decay by more than
one mode is illustrated by
B
B 
A
C 
C
The two partial decay constants B and C must be considered when the general
relations in either branch are studied because, for example, the substance B is
formed at the rate
dNB
 BNB
dt
dNA
   B  C  NA
dt
The nuclide A has only one half life
0.693
t1 
t
2
but A is consumed at the rate
where At = AB + AC + • • •. By definition the half life is related to the total rate of
disappearance of a substance, regardless of the mechanism by which it disappears.
If the Bateman solution is to be applied to a decay chain containing branching decays,
the  ’s in the numerators of the equations defining C1, C2, and so on, should be
replaced by the partial decay constants; that is, Ai in the numerators should be
replaced by i* , where i* is the decay constant for the transformation of the i th
chain member to the (i+1)th member. If a decay chain branches, and subsequently the
two branches are rejoined as in the natural radioactive series, the two branches are
treated by this method as separate chains; the production of a common member
beyond the branch point is the sum of the numbers of atoms formed by the two paths.
EQUATIONS OF TRANSFORMATION DURING NUCLEAR REACTIONS
Stable Targets. When a target is irradiated by particles that induce nuclear
reactions, a steady state can be reached in which radioactive products disintegrate
at just the rate at which they are formed; the situation is analogous to that of
secular equilibrium. If the irradiation is terminated before the steady state is
achieved, then the disintegration rate of a particular active nuclide is less than its
rate of formation R. The differential equation that governs the number of product
atoms N present at time t during the irradiation is
dN
 R  N
dt
the solution to which is
R
N
1  e t
For very large irradiation times ( T > > 1 /  ) the disintegration rate  N approaches the
saturation value R. The factor ( 1  e t ) is often called the saturation factor. If the
disintegration rate of a particular radioactive product at the end of a steady
bombardment of known duration is divided by this saturation factor, the rate at which
the product was formed during the bombardment is obtained.
Occasionally a product is formed during irradiation both directly by nuclear reaction
and by the decay of an active parent that is produced by another reaction [e.g., the
product of a (p,pn) reaction, if unstable, may decay by positron emission or EC into the
product of the (p, 2p) reaction on the same target]. Under these circumstances the
number of atoms of the product of interest present at a time t, after the end of a
bombardment of duration tb has three sources:
1. Those formed directly in nuclear reactions.
2. Those formed by the decay of the parent during bombardment.
3. Those formed by the decay of the parent during the interval tb (which may, for
example, be the time between the end of bombardment and the chemical
separation of daughter from parent).
If R1 and R2 are the rates of the nuclear reactions that directly form the parent and
daughter products, respectively, then the number of daughter atoms (characterized by
subscript 2) arising from each of the three sources is
N2' 
R2
1  e 2tB  e 2tS
2
R

R1
N2''   1 1  e 2tB  
e 1tS  e 2ts   e 2tS

1   2
 2

N 
'''
2
R2 1  e 2tB  e 1tS  e 2tS 
1   2
Experimentally it is, of course, only the totality of the daughter atoms
N
'
2

 N2''  N2''' that is observed, but from a knowledge of the times tB and ts the
decay constants 1 , and  2 , and of the rate of formation R1 of the parent (which can be
determined in a separate experiment), it is possible to calculate R2.
Radioactive Targets in a High-Flux Reactor. When nuclear reactions are induced in a
radioactive nuclide, the rate of disappearance of the substance is no longer governed
by the law of radioactive transformation alone but by a modified law that takes into
account the disappearance by transmutation reactions also. Under most practical
bombardment conditions the rate of transformation of radioactive species by nuclear
reactions is negligible compared to the rate of radioactive decay. However, in the case
of long-lived nuclides, and with the large neutron fluxes available in nuclear reactors,
transformations by both mechanisms sometimes have to be considered. We state the
modified transformation equations for the case of a neutron flux; they are equally
applicable for any other bombarding particle. The treatment given here follows that
developed by W. Rubinson .
Consider N atoms of a single radioactive species of decay constant  (in reciprocal
seconds) and total neutron reaction cross section σ (in square centimeters) in a
constant neutron flux nv (neutrons cm-2s-1). The rate of radioactive transformation is
 N, the rate of transformation by neutron reactions is nvσN, and the total rate of
disappearance is
dN

    nv  N  N (10)
dt
where  may be considered as a modified decay constant. Equation 10 has the same
form as the standard differential equation of radioactive decay and is integrated to
give
N  N0e t (11)
If we consider a parent-daughter pair, the parent disappears by both transmutation
dN
and decay:  1   1  nv1  N1  1N1 ; but the daughter grows by decay of the mother
dt
dN2
only and disappears by both processes:
 1N1  2N2 , or, in more general notation,
dt
dNi1
 iNi  i1Ni1
dt
Actually we may want to consider chains in which the transformation from one member
to the next may occur by nuclear reaction as well as by radioactive decay. Then A,
must be replaced by a modified decay constant i*  i*  nvi* , where the asterisks
serve as a reminder that, if either the decay or reaction of the parent does not always
lead to the next chain member, then i* must be the partial decay constant and
i* must be the partial reaction cross section leading from the i th member to the
(i+1)th member of the chain. With this notation the general solution is written, as in
the Bateman equations, for N20  N30  ........  Nn0  0 :
Ni  C1e 1t  C2e 2t  ....  Cn e n t (12)
where
C1 
C2 
1 2 ...n 1
N10
 2  1   3  1  .....  n  1 
1 2 ...n 1
N10 and so on.
 1  2   3  2  .....  n  2 
As an illustration, we compu
reactions when 1 g 197Au is e
of reactions is
The numerical values to be substituted are
t = 1.08xl05 s,
nv=1.x1014 cm-2s-1
197 = 9.9x1023 cm2
198 = 2.5x1020 cm 2
N
0
197
1x6.02x1023

 3.05x1021
197
*
197
 197  nv197  9.9x10 9 s 1
We use (12) for three-mem
Using these values, we get

e 0.00107
e 0.594
N199  7.85x107 


6
6
5.5x10 6 x2.95x10 6 2.55
 5.5x10 x2.55x10
N199  7.55x107  7.12x1010  3.40x1010  1.01x1011   3.2x1017
and
The disintegration rate of
199
Au at the end of the irradiation is 199N199  0.82x1012 s1 . For
comparison we compute the disintegration rate of
two-membered chain]:
198
Au in the sample [again from (12) for a
 e 197 t
e 198t 
0.999  0.522
7
198N198  198nv197N 

 7.36x1012 s1
  9.06x10 x
6
5.5x10
 198  198 197  198 
Thus about 10 percent of the radioactive disintegrations in the sample occur in 199Au.
0
197