22.101 Applied Nuclear Physics (Fall 2006)

Lecture 1 3 (10/30/06) Radioactive-Series Decay

References :

R. D. Evan s , The Atomic Nucleus (M cGraw-Hill, New York, 1955).

W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967).

We begin with an experim e ntal observa tion that in radioactiv e decay th at the probability of a decay during a sm all tim e interv al  t, which we will den o te as P(  t), is proportional to  t. Given this as a fact one can write

P (  t )    t (12.1)

where  is th e proportionality consta n t which we will call the d ecay cons ta nt. Notice that this exp r ession is m eaningf ul on ly when   t < 1, a condition which defines what we m ean by a sm all tim e inte rval. In other words ,  t < 1/  , which will turn out to be the m ean lif e tim e of the radio i sotop e .

Suppose we are in terested in the su rv ival p r obability S(t), th e probability that the radioisotope does not decay duri ng a n arbitra r y tim e interval t. To calcula te S(t) using (12.1) we can take th e tim e interval t and di vide it into m a ny sm all, equal segm ents, each one of m a gnitude  t. For a given t the num b er of such segm ents will be t /  t = n. To survive the e n tir e tim e interva l t, we need to surv ive the f i rst segm ent (  t) 1 , then the n e xt segm ent (  t) 2 , …, all the way up to the nth segm ent (  t) n . Thus we can write

S ( t )    1  P ((  t ) i ) 

i  1

=  1   ( t / n )  n  e   t

(12.2)

where the arrow indicated the lim it of n   ,  t  0 . Unlike (12.1), (12.2) is valid for any t. W h en  t is sufficien tly sm all co mpared to u n ity, it redu ces to (12.1 ) as expected. Stated another way, (12.2) is extension of 1 – P(t) for arbitrary t. One should also notice a close sim ilarity between (12.2) and the pr obability that a particle will go a distance x without collision, e   x , where  is the m acroscopic co llision cros s section (recall Lec1).

The role of the decay co nstant  in th e probability of no decay in a tim e t is the sam e as

the m acroscopic cros s section  in the probability of no collis ion in a dis t a n ce x. The exponential attenu ation in tim e or space is qu ite a general result, which o n e encounters frequently. There is another way to derive it. Suppose the radioisotope has not decayed up to a tim e inte rval of t 1 , f o r it to su r v ive the n e x t sm all segm ent  t the pro b ability is just 1 - P (  t) = 1 -   t. Then we have

S ( t 1   t )  S ( t 1 )  1    t ]  (12.3)

which we can rearrange to read

S ( t   t )  S ( t )    S ( t ) (12.4)

 t

Taking the lim it of sm all  t, w e get

dS ( t )    (12.5)

dt

which we can readily integrate to give (12.2) , since the initial condi tion in this case is S(t=0) = 1.

The decay o f a single radioisotop e is de scrib e d b y S(t) which depends on a single physical constant  . Instead of  one can speak of two equi valent quantities, the half lif e t 1/ 2 and the m ean life . They are defined as

S ( t 1 / 2 )  1 / 2  t 1 / 2   n 2 /   0 . 693 /  (12.6)



 dt ' t ' S ( t ' )

and   0  (12.7)



 dt ' S ( t ' )

0

Fig. 12.1 shows the relationship be tween these quantities an d S(t).

Fig. 12.1 . The half lif e a nd m ean lif e of a surviva l probab ility S(t).

Radioactivity is m easured in term s of th e rate of radioactive d ecay. The q u antity

 N(t), where N is the nu mber of radi oisotope ato m s at tim e t, is ca lled activity . A standard unit of radioactivity has been the cur i e , 1 Ci = 3.7 x 10 10 disintegrations/sec, w h ich is rou ghly the a c ti vity of 1 gram of Ra 226 . Now it is replaced by th e becquerel (Bq), 1 Bq = 2.7 x 10 -1 1 Ci. An old unit which is not often used is the ru therford (10 6 disintegrations/sec).

Radioisotope Production by Bombardment

There are tw o general ways of producing radioisotopes, activat ion by particle or radiation bo mbardm ent such as in a nuclear re actor or an accelerator, and the decay o f a radioactive series. Both m e t hods can be discussed in term s of a differential equation that governs the num b er of radioisotope s at tim e t, N(t). This is a first-order linear differential equation with constant coefficients, to wh ich the solution can be read ily o b tained.

Although there are different situa tions to which one can apply this equation, the analysis

is fundam e ntally quite straigh t forward. W e will treat the ac tivation p r oblem first. Le t Q(t), the rate of production of the radioisotope, have the form shown in the sketch below.

This m eans the production takes place at a constant Q o f o r a tim e interval ( 0 , T), af ter which production ceases. During production, t < T, the equation governing N(t) is

dN ( t )  Q dt o

  N ( t ) (12.8)

Because we have an external sou r ce term , th e equation is seen to be inhom ogeneous. The solution to ( 12.8) w ith the initia l co nditi on that there is no radioisotope prior to production, N(t = 0) = 0, is

N ( t )  Q o  1  e   t  , t < T (12.9)



For t > T, the governing equation is (12.8) without the source term . The solution is

N ( t )  Q o ( 1  e   T ) e   ( t  T ) (12.10)



A sketch of the solutions (12.9) and (12.10) is shown in Fig. 12.2. One sees a build up of N(t) during production wh ich approaches the as ym ptotic value of Q o /  , and after production is stopped N(t) undergoes an exponential decay, so that if  T >>1,

N ( t )  Q o e   ( t  T ) (12.11)



Fig. 12.2 . Tim e variation of num b er of radioiso tope atom s produced at a constant rate Q o f o r a tim e inte rval of T af ter w h ic h the system is lef t to de cay.

Radioisotope Production in Series Decay

Radioisotop e s also are p r oduced as the produc t(s) of a series of sequential decays.

Consider th e case of a three-m e m b er chain,

 1  2

N 1  N 2  N 3 (stab l e)

where  1 and  2 are th e decay constan t s of the parent (N 1 ) and the daughter (N 2 ) respectively. The gove rning equations are

dN 1 ( t )    N

( t ) (12.12)

dt 1 1

dN 2 ( t )   N

( t )   N

( t ) (12.13)

dt 1 1 2 2

dN 3 ( t )   N dt 2 2

( t ) (12.14)

For the initial conditions we assum e there are N 10 nuclid es of species 1 and no nuclides of species 2 an d 3. The solutions to (12.12) – (12.14) then becom e

N ( t )  N e   1 t (12.15)

N 2 ( t )  N 10 

 1  e   1 t  e   2 t  (12.16)

2   1

   1  e   1 t 1  e   2 t 

N ( t )  N 1 2   

(12.17)

 2   1   1  2 

Eqs.(12.15) through (12.17) are known as the Batem a n equations. One can use them to analyze situ ations when the decay co nstants  1 and  2 take on different relative values. W e consider two such scenarios, the case where the paren t is short-liv ed,  1 >>  2 , and the opposite case where the pa rent is long- lived,  2 >>  1 .

One should notice from (12.12) – (12.14) that the sum of these three differential equation s is zero. This m eans that N 1 (t) + N 2 (t) + N 3 (t) = co nstant for an y t. W e also know from our initia l co nditions tha t this cons tan t m u st be N 10 . One can use this inf o rm ation to f i nd N 3 (t) given N 1 (t) and N 2 (t), o r use this as a check that the solutions given by (12.15) – (12.17) are indeed correct.

Series Deca y with Short-Lived Parent

In this case one expects the parent to decay quick ly and the d a ughter to b u ild up quickly. Th e daughter then decays more sl owly which m eans that the grand daughter w ill build up slow ly, eve n tually appr oaching th e initial num ber of the par e nt. Fig. 12. 3 show s sche m a tically th e behavio r o f the three is otopes. The initial v a lue s of N 2 (t) an d N 3 (t) can be readily deduced from an ex am ination of (12.16) and (12.17).

Fig. 12.3. Tim e variatio n of a three-m e mber decay chain for the cas e  1 >>  2 .

Series Deca y with Long-Lived Parent

W h en  1 <<  2 , we expect th e paren t to d ecay slowly so the daugh ter and g r an d daughter will build up slowly. Since the da ughter decays qu ickly the lon g -tim e behavior of the daughter follows that of the parent. Fig. 12.4 shows the general behavior

Fig. 12.4. Tim e variatio n of a three-m e mber chain w ith a lon g -liv ed pare nt. (adm ittedly the N 2 behavior is not sketched accura te ly). In this c a se w e f i nd

N 2 ( t )  N

 1 e   t (12.18)

10 

2

o r  2 N 2 ( t )   1 N 1 ( t ) (12.19)

The condition of approxim a tely equal activities is called secular equilibrium . G e neralizin g this to an a r bitrary chain, we can say for the series

N 1  N 2  N 3  . . .

i f  2   1 ,  3   1 , …

then  1 N 1   2 N 2   3 N 3  ... (12.20)

This condition can be used to estim ate the half life of a very long-lived radioisotope. An exam ple is U 238 whose half life is so long that it is difficult to determ ine by directly m easuring its decay. However, it is known that U 238  Th 234  …  Ra 226  …, and in uranium m i neral the ratio of N(U 238 )/ N(Ra 226 ) = 2.8 x 10 6 has been m easured, with t 1/2 (Ra 226 ) = 1620 yr. Using these data we can write

N ( U 238 )  N ( Ra 226 )

238 6 9

t 1 / 2

( U 238 ) t

1 / 2

( Ra 226 )

or t 1 / 2 ( U

) = 2.8 x 10

x 1620 = 4.5 x 10

yr.

In so doing we assum e t h at all the in term edia te d ecay cons tants are larger than that o f U 238 . It turns out that this is indeed true, and that the above estim ate is a good result. For an extensive treatm ent of ra dioactiv e series decay, the st ud ent should consult Evans.