Subject: Physics

The spontaneous breaking of a nucleus is known as radioactive disintegration. Rutherford and Soddy made experimental study of the radioactive decay of various radioactive materials and gave the following laws:

Radioactivity decay is a random and spontaneous process. It is not influenced by external conditions such as temperature, pressure, electric field etc. Each decay is an independent event occurs by a chance to take first.

In any radioactive decay, either a α-particle or β-particle is emitted by the atom. Emission of both types is impossible at a time. Moreover, an atom does not emit more than one α-particle or more than one β-particle at a time.

When a nuclide emits α-alpha particle, its mass number is reduced by four and atomic number by two.

\begin{align*} _ZX^A \rightarrow _{Z-2}Y^{A-4} + _2He^4 \\ \text {Example}: _{92} U^{238} \rightarrow _{90} Th^{234} + _2He^4 \end{align*}

When a nuclide emits a β-particle, its mass number remains unchanged but atomic number increases by 1.

\begin{align*} _ZX^A \rightarrow _{Z+1}Y^{A} + _{-1}e^0 \\ \text {Example}: _{6} C^{14} \rightarrow _{7} N^{14} + _2He^4 + _{-1}e^0 \end{align*}

When a nuclide emits a gamma ray, neither the mass number nor the atomic number changes.

$$ _ZX^A \rightarrow _{Z}Y^{A} +\gamma $$

The gamma radiation is emitted by the excited nucleus. These laws are called the displacement laws.

The rate of disintegration of a radioactive substance is directly proportional to the number of atoms present at that instant. This is called decay law.

Let N_{0} be the number of atoms present in the radioactive sample at t =0 and N be the number of atoms left after time t. then, the rate of disintegration, dN/dt is proportional to N. so,

\begin{align*} \frac {dN}{dt} &\propto N \\ \text {or,} \: \frac {dN}{dt} &= -\lambda N \\ \end{align*}

Where λ is a constant of proportionality called as disintegration constant or decay constant. The – sign indicates that N decreases as time increases. The number of disintegration per second, dN/dt is called the activity of radioactive sample.

The above equation can be written as

\begin{align*} \frac {dN}{N} &= -\lambda dt \\ \text {integrating on both sides, we get} \\ \int _{N_0}^N \frac {dN}{N} &= -\lambda \int _0^t dt\\ \text {or,} \: [\log _e N]_{N_0}^N &= -\lambda [t]_0^t \\ \text {or,} \: \log _e N - \log_e N_0 &= -\lambda t \\ \text {or,} \log _e \frac {N}{N_0} &= -\lambda t \\ \text {or,} N &= N_0 e^{-\lambda t} \\ \end{align*}

This equation is known as decay constant. It shows that number of active nuclei in a radioactive sample decreases exponentially with time as shown in the figure.

From the law of radioactive disintegration, we have

\begin{align*} \frac {dN}{dt} &= -\lambda N \\ \text {or,} \: \lambda &= \frac {-\frac {dN}{dt}}{N} \\ \end{align*} Hence the decay constant is defined as the ratio of rate of decay per unit atom present. \begin{align*} \\ \text {If we put t} = \frac {1}{\lambda } \text {in decay equation }\\ N &= N_0 e^{-\lambda t} we get, \\ N &= N_0 e^{-1} = \frac {N_0} {e} \\ &= \frac {N_0}{2.718} = 0.37 N_0 = 37\%\: of \:N_0 \\ \end{align*}

Since decay constant may also be defined as the reciprocal of time during which the number of radioactive atoms of a radioactive substance falls to 37% of its original value.

The time is taken by a radioactive substance to disintegrate half of its atoms is called the half-life of that substance. It is denoted by T_{1/2} or simply T. Its value is different for a different substance.

Relation between half life and decay constant:

Let N_{0} be the initial number of atoms in a radioactive substance of decay constant λ . Then after time T, the number of atoms left behind N_{0}/2. So,

\begin{align*} t = T\: \text {and} \: N = \frac {N_0}{2} \\ \text {Substituting these values in the equation,} \: N =N_0 e^{-\lambda t}, \\\text {we get} \\ \frac {N_0}{2} &= N_0 e^{-\lambda T} \\ \text {or,} \: \frac 12 &= e^{-\lambda T} \\ e^{\lambda T} &= 2 \\ \text {or,} \: \lambda T&= \log _e 2 = 0.693 \\ T &= \frac {0.693}{\lambda } \\ \end{align*}

This is the relation between the half-life and decay constant. Thus the half life of the radioactive substance is inversely proportional to its decay constant.

**Average Life or Mean Life**

The average life or mean life of a radioactive substance is equal to the sum of total life of the atoms divided by the total number of atoms of element.

\begin{align*} \text {Mean life} &= \frac {\text {sum of life of all the atoms}}{\text {total number of atoms}} \\ \end{align*} It can be shown that the mean life of a radioactive substance is equal to the reciprocal of the decay constant. \begin{align*} T_{mean} &= \frac {1}{\lambda } \\ \text {But} \lambda = \frac {0.693}{T} \text {where T is the half life of the substance.} \\ \therefore T_{mean} &= \frac {T}{0.693} = 1.443 T \\ \end{align*}

Thus the mean life of a radioactive substance is longer than its half life.

**Activity of Radioactive Substance**

The rate of decay of a radioactive substance is called the activity (R) of the substance.

\begin{align*} R &= \frac {dN}{dt} = -\lambda N\\ \text {or,} \: |R| &= \lambda N = \frac {0.693} {T} N \\ \text {So,} R \propto N. \text {If} \: R_0 \text {is the activity of a substance at time}\: t = 0, \text {then} \\ R_0 &= \lambda N_0 \\ \text {or,} \frac {R}{R_0} &= \frac {N}{N_0} = \frac {N_0 e^{-\lambda t} }{N_0} = e^{-\lambda t} \\ \text {or,} \: R &= R_0 e^{-\lambda t} \\ \end{align*}

**Number of atoms left behind after n half lives**

Let N_{0} be the total number of atoms of a radioactivity substance present at time t = 0. Then, the number of atoms present after one half life, \( T = N = \frac {N_0}{2} \)

After two half life, the number present \( = \frac 12 \frac {N_0}{2} = \left ( \frac 12 \right ) ^2 N_0 \)

After n-half life time, the number present \( = \left ( \frac 12 \right ) ^n N_0 \)

In general after time t = nT, the number of atoms left is given by \( \left ( \frac 12 \right ) ^{1/T} N_0 \)

Also, after time t = nT, the activity of a radioactive substance is \( \left ( \frac 12 \right ) ^{1/T} R_0 \)

Further, if R be the activity after n half life times, and R_{0} be the initial activity, then

\begin{align*} R &= \left ( \frac 12 \right ) ^{1/T} R_0 \\ \text {or,} \: R_0 &= 2^n R \\ \end{align*}

**Units of Radioactivity**

The activity of a radioactive substance is measured in terms of disintegration per second. Following are units of radioactivity.

- Curie (Ci)

It is defined as the activity of a radioactive substance which gives 3.7×10^{10}integration per second. It is equal to the activity of 1 g of pure radium.

$$ 1 \text {Ci} = 3.7 \times 10^{10} \text {disintegration}/\text {second} $$ - Rutherford (rd)

It is defined as the activity of radioactive substance which gives rise to 10^{6}disintegration per second.

$$ 1 \text{rd} = 10^6 \text {disintegration}/\text {second} $$ - Becquerel (Bq)

It is SI-unit of radioactivity.

\begin{align*} 1\text {Bq} = 1 \text {disintegration}/\text {second} \\ \text {Ci} = 3.7 \times 10^{10} \text {disintegration}/\text {second} = 3.7 \times 10^{10} \text {Bq} = 3.7 \times 10^4 \text {rd} \\ \end{align*}

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. *Principle of Physics*. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. *A text Book of Physics*. Kathmandu: Surya Publication, 2003.

The spontaneous breaking of a nucleus is known as radioactive disintegration.

The rate of disintegration of a radioactive substance is directly proportional to the number of atoms present at that instant. This is called decay law.

The time is taken by a radioactive substance to disintegrate half of its atoms is called the half-life of that substance.

The average life or mean life of a radioactive substance is equal to the sum of total life of the atoms divided by the total number of atoms of element.

The activity of a radioactive substance is measured in terms of disintegration per second.

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