Theory of Radioactive Disintegration

Radioactive disintegration is a spontaneous type of disintegration of atomic nuclei. The process was firstly examined in 1896 by the French scientist Henri Becquerel. Marie Curie and her husband Pierre Curie are a couple, and they also contributed to the clear understanding of radioactive disintegration. Their discovery of a couple of the latest radioactive elements, radium and polonium, forced scientists to change their views about the shape and size of atoms.

Radioactive disintegration takes place when the atom tries to reach a stable and better nuclear configuration. The radioactive disintegration process can take place with the help of 3 methods; such as a nucleus can alter its one neutron to a proton via simultaneous emission of electrons (beta disintegration), via spontaneous fission (splitting) into two fragments, by releasing helium nucleus (alpha decay). These events often include the release of higher energy photons known as gamma rays. 

Radioactive Disintegration

Radioactive disintegration, known by various names such as nuclear decay, radioactivity, nuclear disintegration, radioactive decay, and also radioactive disintegration, is the most suitable example of a first-order reaction. A first-order reaction is a technique in which an unsteady atomic nucleus loses energy via radiation. Any substance which contains risky nuclei is considered radioactive.

Three of the most common types of radioactive disintegration are:

• Alpha radioactive disintegration (α-disintegration)

• Beta radioactive disintegration(β-disintegration)

• Gamma radioactive disintegration(γ-disintegration)

All of these radioactive disintegrations include the release of single or multiple particles. Beta disintegration takes place via weak forces, while on the other hand, the other two disintegrations are held via electromagnetic or strong forces.

• Alpha Radioactive Disintegration (α-decay) occurs when the nucleus ejects an alpha particle (helium nucleus).

• Beta Radioactive Disintegration (β-decay) occurs when positrons (particles with the same mass as electrons, but with a charge exactly opposite to that of an electron) or electrons are released, which is known as beta radioactive disintegration.

• Gamma Radioactive Disintegration (γ-decay) occurs when the release of high-energy photons takes place(hundreds of keV or more).

Law of Radioactive Disintegration

When the radioactive material goes through an α, β, or γ-disintegration, then various nuclei undergo the disintegration, per unit time, which is known as proportional to the sum total of nuclei within the material sample.

If N = total no. of nuclei within the sample and ΔN = number of nuclei that disintegrate in time Δt, then,

ΔN/ Δt ∝ N

Or,             ΔN/ Δt = λN … (1)

Where the change in the number of nuclei in the sample is, dN = – ΔN in time Δt.

Now, λ = radioactive decay constant or disintegration constant.

Hence, the rate of change of N (in the limit Δt → 0) is,

dN/dt = – λN

Or,                dN/N = – λ dt

Now, integration of both the sides of the above equation, we get,

NN0 ∫ dN/N = λ tt0∫ dt … (2)

Or,   ln N – ln N0 = – λ (t – t0) … (3)

Where N0 is the number of radioactive nuclei in the sample at some arbitrary time t0 and N is the number of radioactive nuclei at any subsequent time t.

Now, set t0 = 0 and rearrange the above-mentioned equation (3) to find,

ln (N/N0) = – λt

Or,           N(t) = N0e-λt … (4)

Equation (4) is the Law of Radioactive Disintegration.

The Decay Rate

In radioactive disintegration or the law of radioactive disintegration calculations, the more interesting is the decay rate.

Where R ( = – dN/dt) than in N itself. Decay rate provides us the per unit time decay of nuclei. Even if the number of nuclei in the sample is unknown, via simply calculating the emissions of α, β, or γ particles in 10 – 20 seconds, then the decay rate will be calculated.

Let’s assume that the time interval is dt and get a decay count ΔN (= –dN). Then the decay rate is defined as,

R = – dN/dt

Differentiating equation 4 on both sides, then

R = λ N0e-λt

Or,             R = R0 e-λt … (5)

Where R0 is known as the radioactive decay rate in the time t = 0, and R is considered as the rate at any subsequent time t. Equation (5) is another form of the Law of Radioactive Disintegration or Decay. Now it can be rewritten equation (1) as follows,

R = λN … (6)

where the radioactive nuclei and R that have not undergone decay should be judged at the same time.

Half-Life and Mean Life

The complete decay rate of the sample is commonly known as the process of the sample. The SI unit for measuring the process is ‘becquerel’ and is elaborated as,

1 becquerel = 1 decay per second = 1 Bq

A curie is a historic unit which is in use now also,

1 curie = 1 Ci = 3.7 × 1010 Bq (decays per second)

Two methods to calculate the total time. Radionuclide scan last,

• Half-life T1/2 – Half-life is the time in which N and R are reduced to initial values or half values.

• Mean life τ – Meantime is the time in which both N and R get reduced to e-1, which is the initial value also.

Calculating Half-Life

To find the connection between the disintegration constant λ and T1/2. For this, let’s put the following values to the equation (5),

R = (1/2)R0 and t = T1/2

So, we find    T1/2 = (ln2)/ λ

Or,             T1/2 = 0.693/ λ … (7)

Calculating Mean life

Next, find the association between the disintegration constant λ and mean life τ.

For this, let’s assume the equation (5),

The number of nuclei that disintegrate during the time interval:

‘t’ to ‘t + Δt’ is: R(t)Δt = (λN0e-λt Δt)

All of them have lived for ‘t’ time.

So, the total life of all the nuclei is,

tλN0e-λt Δt

Hence, to get the mean life, we will merge this derivation over whole times from 0 to ∞  and divide it by the total no. of nuclei at t = 0 (which is N0).

τ = (λN0 0∞∫ te-λt dt)/N0

= λ0∞∫ te-λt dt

On solving this integral, we get

τ = 1/λ

Therefore, we can summarize the findings as follows:

T1/2 = (ln2)/λ = τ ln 2 … (8)

Conclusion

Radioactive disintegration takes place when the atom tries to reach a stable and better nuclear configuration, and radioactive disintegration is further divided into alpha, beta, and gamma radioactive disintegration. Furthermore, it also concluded the law of radioactive disintegration.