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Radioactivity – decay rates and half life. presentation for April 30, 2008 by Dr. Brian Davies, WIU Physics Dept. Probability of radioactive decay.

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Radioactivity – decay rates and half life

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## Radioactivity – decay rates and half life

presentation for April 30, 2008 by

Dr. Brian Davies, WIU Physics Dept.

• Radioactive decay obeys an exponential decay law because the probability of decay does not depend on time: a certain fraction of nuclei in a sample (all of the same type) will decay in any given interval of time.

• The rate law is: DN = - l N Dt where

N is the number of nuclei in the sample

l is the probability that each nucleus will decay

in one interval of time (for example, 1 s)

Dt is the interval of time (same unit of time, 1s)

DN is the change in the number of nuclei in Dt

• The rate law DN = - l N Dt can also be written:

• DN/N = - lDt As an example,

• Suppose that the probability that each nucleus will decay in 1 s is l = 1x10-6 s-1 In other words, one in a million nuclei will decay each second.

• To find the fraction that decay in one minute, we multiply by Dt = 60 s to get:

• DN/N = - lDt = - (1x10-6 s-1) x (60s) = - 6x10-5

• Equivalently: l = 6x10-5 min-1 and Dt = 1 min

• Now write the rate law DN = - l N Dt as:

• DN/Dt = - l.N ( -DN/Dt is the rate of decay)

• Stated in words, the number of nuclei that decay per unit time is equal to l.N , the decay constant times the number of nuclei present at the beginning of a (relatively short) Dt.

• Example: if l = 1x10-6 s-1 and N = 5x109. then

• DN/Dt = - l.N = -1x10-6 s-1. 5x109 = - 5x103 s-1

• If each of these decays cause radiation, we would have an activity of 5000 Bq. (decays per second)

### Decrease of parent population

• DN = - l N Dt represents a decrease in the population of the parent nuclei in the sample, so the population is a function of time N(t).

• DN/ Dt = - l.N can be written as a derivative:

• dN/dt = - l.N and this can be solved to find:

• N(t) = No exp(-lt) = No e -lt

• Recall that e0 = 1; we see that No is the population at time t = 0 and so the population decreases exponentially with increasing time t.

exp(x)

+

exp(0) = 1

exp(x)<1 if x<0

x

### Graph of the exponential exp(-lt)

+

exp(-lt)

exp(0) = 1

exp(-0.693) = 0.5 = ½

+

+

exp(-1) = 1/e = 0.37

lt

### Half-life of the exponential exp(-lt)

+

exp(-lt)

The exponential decays to ½

when the argument is -0.693

exp(-0.693) = 0.5 = ½

+

lt½

+

lt

### Half-life of the exponential exp(-lt)

• Because the exponential decays to ½ when the argument is -0.693, we can find the time it takes for half the nuclei to decay by setting

• exp(-lt½) = exp(-0.693) = 0.5 = ½

• The quantity t½ is called the half-life and is related to the decay constant by:

• lt½ = 0.693 or t½ = 0.693/l

• In our previous example, l = 1x10-6 s-1

• The half-life t½ = 0.693/l = 6.93x105 s = 8 d

### Half-life of number of radioactive nuclei

• Because the exponential decays to ½ after an interval equal to the half-life this means that the population of radioactive parents is reduced to ½ after one half-life:

• N(t½) = No .exp(-lt½) = ½ No

• In our example, the half-life is t½ = 8 d, so half the nuclei decay during this 8 d interval.

• In each subsequent interval equal to t½ , half of the remaining nuclei will decay, and so on.

### Half-life of activity of radioactive nuclei

• Because the activity (the rate of decay) is proportional to the population of radioactive parent nuclei:

• DN/Dt = - l . N(t) but N(t) = No e -lt

• the activity has the same dependence on time as the population N(t) (an exponential decrease):

• DN/Dt = (DN/ Dt)o . e -lt

• In our example, the half-life is t½ = 8 d, so the activity is reduced by ½ during this 8 d interval.

### Multiple half-lives of radioactive decay

• The population N(t) decays exponentially, and so does the activity DN/Dt .

• After n half-lives t½, the population N(t) = N(n.t½) is reduced to No/(2n).

• In our example, the half-life is t½ = 8 d, so the activity is reduced by ½ during this 8 d interval, and the population is also reduced by ½.

• After 10 half-lives, the population and activity are reduced to 1/(210) = 1/1024 = 0.001 times (approximately) their starting values.

• After 20 half-lives, there is about 10-6 times No.

### Plotting radioactive decay (semi-log graphs)

• DN/Dt = (DN/ Dt)o . e -lt or N(t) = No . e -lt

• can be plotted on semi-log paper in the same way as the exponential decrease of intensity due to absorbers in X-ray physics.

• ln(N) = ln( No . e -lt ) = ln(No) + ln(e -lt)

• ln(N) = -lt + ln(No)

• which has the form of a straight line if y = ln(N) x = t and m = - l

• y = m . x + b with b = ln(No)

### Semi-log graph of the exponential exp(-x)

exp(0) = 1

+

exp(-0.693) = 0.5 = ½

exp(-x)

+

+

exp(-1) = 1/e = 0.37

x

### Plotting radioactive decay (semi-log graphs)

• Starting with N(t) = No . e -lt , we want to plot this on semi-log paper based on the common logarithm log10.