Cause of Radioactivity and How to Measure it? JEE Important Topic

What causes radioactivity? Nucleus is made up of protons and neutrons. It is said to be stabilised when the number of protons and neutrons are in equal number in the nuclei. If there is excess of protons or neutrons present in the nuclei, it causes instability inside the nucleus, which in turn tries to stabilise by emitting these excess protons or neutrons. These can be disintegrated or emitted by electromagnetic radiation or by some particles. Radioactivity is due to the spontaneous decay of the nucleus accompanied by the emission of alpha particles, beta particles or gamma rays. The nucleus of a radioactive element causes the low binding energy. As the nucleus disintegrates its elements for stabilisation, it loses its energy too. 

Types of Radioactivity

Radioactivity takes place in terms of radiation. Initially, Rutherford and his co workers found these radiations and named them as alpha particles, beta particles and gamma rays. These are the decay modes of the radioactive nuclei. Each one is summarised below. 

1. Alpha  Decay

In this process, the parent nuclei which has either excess number of protons or neutrons disintegrates to the daughter nuclei which is the stable nuclei and the alpha particle. Mass number of the parent nuclei decreases by four units and atomic number or proton number decreases by two units. The reaction is given below:

${ }_{Z}^{A} X \rightarrow{ }_{Z-2}^{A-4} X+\alpha \text { particle }$

Later, it was found that the alpha particle is nothing but the 4He nuclei or simply the helium. Take the example of uranium which is radioactive in nature that becomes thorium i.e. the daughter nuclei through alpha decay. The reaction is as follows:

${ }_{92}^{238} U \rightarrow{ }_{90}^{234} \mathrm{Th}+{ }_{2}^{4} \mathrm{He}$

2. Beta Decay

In this process, the parent nuclei disintegrates to produce daughter nuclei and a beta particle. There are three types of beta decay that take place: electron emission, positron emission or electron capture. 

3. Electron Emission or $\beta^-$ Decay:

In this, the parent nucleus disintegrates to produce daughter nuclei, an electron and a new element known as antineutrino ($\bar v$). Here, the mass number remains constant and the atomic number increases by one unit. The reaction is as follows:

${ }_{6}^{14} C \rightarrow{ }_{7}^{14} N+e^{-}+\bar{v}$

4. Positron Emission or $\beta^+$ Decay:

The parent nucleus disintegrates to produce daughter nuclei, a positron (mass is same as electron, just the charge is positive) and a new element neutrino (v). Mass number of the daughter nuclei remains the same, but the atomic number decreases by one unit. Example is as follows:

${ }_{11}^{22} N a \rightarrow{ }_{10}^{22} N e+e^{+}+v$

5. Electron Capture:

In this process, the parent nuclei captures one of its orbital electrons with the emission of a neutrino. Mass number of the daughter nucleus remains the same and the atomic number decreases by one unit. Reaction is as follows: 

${ }_{25}^{54} M n+e^{-} \rightarrow{ }_{24}^{54} C r+v$

6. Gamma Decay

Alpha and beta decay usually leave the daughter nuclei in an excited state. When the daughter nucleus goes from excited state to the ground state, it loses its energy. That energy is in the form of electromagnetic radiations known as gamma decay. The atomic mass and mass number is constant in this decay. Reaction is as follows: 

${ }_{56}^{137} B a^* \rightarrow{ }_{56}^{137} B a+\gamma$

How to Measure Radioactivity?

Radioactivity is measured by various devices. Geiger Counters is the most common approach to determine the radioactivity of radioactive elements. It counts the number of particles present during that time. Mathematically, it is given as the rate of disintegration of radioactive elements in proportion to the number of nuclei present at that time. It can be written as: 

$\dfrac{d N}{d t} \propto N$ 

Where N is the number of particles present at time t. 

Removing the proportionality sign we get:

$\dfrac{d N}{d t}=-\lambda N$

Where $\lambda$ is the proportionality constant known as decay constant that depends on the material. Negative sign is  there because particles keep decreasing with respect to time.

By integrating above equation we get:

$\begin{align} &\int \dfrac{d N}{N}=-\lambda \int d t \\ &\Rightarrow \ln (N)=-\lambda t+C \end{align}$

Where C is the constant of integration that can be found out by applying some boundary conditions. 

At t=0, all the particles are present so the radioactive nucleus at initial is N0. Using this we get:

$\Rightarrow \ln \left(N_{0}\right)=C$

If we put this in the above equation, we get:

$\begin{align} &\ln (N)=-\lambda t+\ln \left(N_{0}\right) \\ &\Rightarrow \ln \left(\dfrac{N}{N_{0}}\right)=-\lambda t \\ &\Rightarrow N=N_{0} e^{-\lambda t} \end{align}$

This equation is known as radioactive decay law in which N0 is the number of radioactive nuclei in the sample at t=0.

Half Life of Radiation

Half life is the time required for the number of radioactive nuclides to reduce to half by disintegration. If N0 is the initial radioactive nuclei present, then in one half life this number will reduce to $\dfrac{N_0}{2}$. i.e. Exactly half of its original. Now consider the radioactive decay law:

$N=N_{0} e^{-\lambda t}$

At half life: $N=\dfrac{N_0}{2}$ and $t=t_{\dfrac{1}{2}}$

$\begin{align} &\Rightarrow \dfrac{N_{0}}{2}=N_{0} e^{-\lambda t_{\dfrac{1}{2}}} \\ &\Rightarrow \dfrac{1}{2}=e^{-\lambda t_{\dfrac{1}{2}}} \\ &\Rightarrow t_{\dfrac{1}{2}}=\dfrac{\ln 2}{\lambda} \end{align}$

This is the equation for calculating the half life of any given radioactive element.

Decay Rate and its Units

The decay rate is defined as a rate at which the number of disintegration occurs per unit time. Mathematically it is given as:

$\begin{align} &R=-\dfrac{d N}{d t}=\lambda N \\ &\Rightarrow R=\lambda N_{0} e^{-\lambda t} \end{align}$

Substituting $\lambda N_0=R_0$, we get:

$\Rightarrow R=R_{0} e^{-\lambda t}$

This is an alternative form of the law of radioactive decay. Here $R_{0}$ is the decay rate at time t=0 and R is the decay rate at any subsequent time t. 

The total decay rate R of a sample of one or more nuclides is known as activity. The SI unit of activity is the becquerel. 

1 becquerel = 1 Bq = 1 decay per second

Another unit is curie;

$1 \text { curie }=1 \mathrm{Ci}=3.7 \times 10^{10} \mathrm{~Bq}$

Mean Life

This is another parameter to find the life of the radioactive element. In this, both N and R have been reduced to $\dfrac{1}{e}$ of its original element. Mathematically, this can be given as:

$R=\dfrac{1}{e}R_0$ and the time $t=\tau$

From radioactive decay rate:

$\begin{align} &\Rightarrow \dfrac{R_{0}}{e}=R_{0} e^{-\lambda \tau} \\ &\Rightarrow \tau=\dfrac{1}{\lambda} \end{align}$

This is the expression for mean life. It depends only on the decay constant.

Conclusion

Radioactive is the total probabilistic phenomenon. No one actually knows whether the decay will happen or not in the next second. Radioactivity is only due to the instablization of nuclei. Excess protons or neutrons lead to the instability of nuclei. There are mainly three types of radioactivity decay, i.e., alpha, beta and gamma decay. With the help of Geiger Counters and radioactive decay law, one can measure the radioactivity of the given radioactive sample.