Radioactive decay

1. The law of the decay

Let N₀ the population of radioactive nuclei at the beginning; that is at the time t = t₀.

At a time t = t, the short time after t = t₀, the population N₀ decreases and becomes N(t).

The difference dN(t) = N₀ - N(t) occurs at the time t = t, after the time t₀; that is during the period dt = t - t₀.

Let's remark that dN is negative because N₀ is greater than N(t).

The ratio: - dN(t)/dt is called the Activity of the substance; that is the population decay rate. It is the rate at which the process occurs. In other words, it is the rate at which the desintegration occurs. In The SI, the activity is measured in becquerel (Bq); that is the number of desintegration (or emission)per second.

The ratio: - dN(t)/N is called the fractional reduction of the population. It means also the percentage of population reduction. It is proportional to the time dt. Let's write:

- dN(t)/N = λ dt (1)

λ is called the decay constant or the desintegration constant.

Solving for N(t), the integration gives:
ln[N(t)] = - λt + Constant
Or:
N(t) = K exp (- λt)
Where the constant K = exp (Constant)
Solving for K, we have:
At t = t₀, N(t) = N₀ the equation (1) can be written then:
N₀ = K exp (- λt₀)
Hence:
K = N₀exp (λt₀)
N(t) = K exp (- λt) K = N₀exp (λt₀) (2)

In the case of t₀ = 0, K = N₀.
N(t) = N₀ exp (- λt) N₀ = N(0) (3)

2. Other expression of the activity

Deriving the relationship (2), we obtain:
A = - dN(t)/dt = K λ exp (- λt) = λ N(t) = A₀exp (- λt) With A₀ = K λ
A(t) = A₀ exp (- λt) (4)

3. The half-time of a sample

The half-time T_1/2 is the time at which the population falls off by a half (by a factor 1/2).
Let's write:
N(T_1/2) = N(0)/2 = N₀/2
N₀exp (- λT_1/2) = N₀/2
Hence:
1/2 = exp (- λT_1/2); and And:
λT_1/2 = ln 2
That is:
T_1/2= (ln 2)/λ (3)

4. The rate of the acticity

From the relationship (1), the rate of the acticity is expressed by:
R = - dN(t)/N dt = λ
Or from the relationship (4), it takes the expression: R = - dA(t)/A = λ
λ is also the rate at which the population decays regarding the initial population.
Rate of activity = λ