Of more than 3000 known nuclides only 266 are stable. Many unstable nuclei can decay spontaneously to a nucleus of lower mass but different combinations of nucleons. This process of spontaneous emission
of radiation is called Radioactivity. Here, the term radiation refers to particles as well as electromagnetic radiation.

Three types of radiations are emitted by radioactive substances: the simplest is gamma rays, which are photons energy state (without change in N or Z); alpha $(\alpha)$ in which emitted particles are $_{2}^{4}He$ nuclei; beta $(\beta)$ , in which the emitted particles are electrons or positrons. A positron is a particle like electron in all respects except that the positron has charge (+e). The symbol $e^-$ is used to designate an electron and $e^+$ designates a positron.

Some Important Features of Radioactivity
1. Radioactive decay is a statistical process; we can not precisely predict the timing of a particular radioactivity of a particular nucleus. The nucleus can disintegrate immediately or it may take infinite time. We can predict the probability of the number of nuclei is being disintegrated at an instant.
2. Radioactivity is independent of all the external conditions; we can not induce radioactivity by applying strong electrical field, magnetic field, high temperature, and high pressure, etc.
3. The energy liberated during radioactive decay comes within individual nuclei.
4. When a nucleus undergoes alpha or beta decay, its atomic number changes and it transforms into a new element; thus elements can be transformed from one to another.
5. The rate, at which a particular decay process occurs in a radioactive sample, is proportional to the number of radioactive nuclei present (i.e., those nuclei that have not decayed). If N is the number of radioactive nuclei present at some instant, the rate of change of N is

$\dfrac{dN}{dt} = -\lambda N$ …(1)

where $\lambda$ is called decay constant, the minus sign indicates that $\dfrac{dN}{dt}$ is negative, i.e., N is decreasing with time. We can also express equation (1) in another form :

$\dfrac{dN}{N} = -\lambda dt$ …(2)

$\dfrac{dN}{N}$ expresses the fraction of nuclei decayed in time dt, or the probability that out or N nuclei dN number of nuclei will decay in time dt.

We can integrate Equation (2) to get

$\int_{N_{\circ}}^{N} \dfrac{dN}{N} = -\lambda \int_{0}^{t} dt$

$\ln \left ( \dfrac{N}{N_{\circ}} \right ) = -\lambda t \ \mathrm{or} \ N = N_\circ e^{-\lambda t}$ …(3)

Where the constant $N_{\circ}$ represents the number of nuclei or radioactive nuclei at t = 0. Equation (3) shows that the number of nuclei in a sample decays exponentially with time.
The number of decays per unit time or decay rate is called activity. If N is the number of nuclei present in the sample at a certain time, its activity R is given by differentiating Equation (3) with respect to time :

$|R| = \left | \dfrac{dN}{dt} \right | = N_\circ \lambda e^{-\lambda t} = R_\circ e^{-\lambda t}$

where $R_\circ = N_\circ \lambda$ is the decay rate at t = 0 and $R = N \lambda;$ the SI unit of activity is named after Becquerel

1 becquerrel = 1Bq = 1 decays/s

Traditionally, curie $C_i$ has been used as unit of activity.

$1 \ \mathrm{curie} = 1 C_i = 3.70 \times 10^{10}$

The other unit of radioactivityis rutherford (Rd) and is defined as,

$1 \ \mathrm{rutherford} = 10^6 \mathrm{distintegrations/s}$

Important point to remember is that both R and N decreases exponentially with time. The plot of N versus t shown in the figure. illustrates the exponential decay law.

6. Half life : Half life of a radioactive substance is the time that it takes half of a given number of radioactive nuclei to decay.

Setting $N = N_\circ / 2$ and $t = T_{1/2}$ in Equation (3), we get

$\dfrac{N_\circ}{2} = N_\circ e^{-\lambda T_{1/2}}$

Writing the above equation in the form

$e^{\lambda T_{1/2}} = 2 \ \mathrm{or} \ T_{1/2} = \dfrac{\ln 2}{\lambda} = \dfrac{0.693}{\lambda}$ …(4)

In numerical problems, instead of specifying decay constant, usually half life of a nucleus is specified. The equation (4) is a convenient expression for relating half life to the decay constant.

Note that after one half life, $N_\circ/2$ radioactive nuclei remain; after two half lives $N_\circ/4$ radioactive nuclei are remain, after one half life, $N_\circ/8$ are left; and so on. In general, the number of nuclei remaining after n half lives is $N_\circ/2^n$ . The decay of a radioactive nucleus is a statistical process. If we take a radioactive sample of 1 mg with a half life of 1 h, about 50% of the 1 mg sample will decay in 1 h; during the second hour, probability of decay is still 50% for each remaining nucleus.

The total probability that a given nucleus did not decay in 2h is $0.5 \times 0.5 = 0.25$ or 25%. The probability of decay is 75%, which is a fraction of the original nucleus expected to be disintegrated in 2h.

7. The mean life time : mean life (average life) $\tau$ is defined as the average time the nucleus survive before it decays. Mean life can be determined by calculating total life time of all the nuclei initially present $N_\circ$ and dividing it by total number of nuclei. Let the number of nuclei decaying in time interval t and t + dt be dN, and the life time of these nuclei be t. Then the total life time or these nuclei is t dN.

The total life time of all the nuclei = $\int_{N = N_{\circ}}^{N} t dN$

From Equation (3), $dN = N_\circ \lambda e^{-\lambda t} dt,$

Now recognizing the fact that N = 0 for $t \rightarrow \infty$ and $N = N_\circ$ for $t \rightarrow 0,$ total life time of all the nuclei is $= \int_{N = N_\circ}^{N} t dN = \int_{0}^{\infty} t N_\circ \lambda e^{-\lambda t} dt = N_\circ \lambda \int_{0}^{\infty} te^{-\lambda t}dt = \dfrac{N_0}{\lambda}$ (we can carry out integration by parts to get the result).

Thus,

$\tau = \dfrac{1}{\lambda} = \dfrac{T_{1/2}}{0.693}$

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