Radioactivity is the phenomenon exhibited by the nuclei of an atom as a result of nuclear instability. It is a process by which the nucleus of an unstable atom loses energy by emitting radiation. Radioactivity was discovered by Henry Becquerel completely by accident. He wrapped a sample of a Uranium compound in a black paper and put it in a drawer that contained photographic plates. On examining these plates later, he found that they had already been exposed. This phenomenon was termed as **Radioactive Decay**. The element or isotope which emits radiation and undergoes the process of radioactivity is called **Radioactive Element**.

The radioisotope of an element has unstable nuclei and thus do not have sufficient binding energy to hold all the particles of an atom. To stabilize, these isotopes constantly decay. In this entire process of radioactive decay, they release a lot of energy in the form of radiations and often transform into a new element.

This process of transformation of an isotope into an element with a stable nucleus is known as **Transmutation**. Transmutation can occur naturally or can be done artificially.

**Types of Radioactive Decay:**

There are four kinds of radioactive decays namely: Alpha decay, Beta decay, and Gamma decay.

**1. Alpha decay:**

The process of alpha decay involves the emission of a nucleus from an alpha particle.The alpha particle is usually a nucleus of helium which is very stable. It is a group of two neutrons and two protons. The example of alpha decay involves the uranium-238 as shown below-

\(_{92}^{238}\textrm{U}\rightarrow _{90}^{234}\textrm{Th}+_{2}^{4}\textrm{He}\) |

This process of transformation of one element into another element is known as transmutation.

**2. Beta Decay:**

An electron is often a beta particle, but it can also be a positron, which is a positively-charged particle. If the electrons are involved in the reaction, then the number of neutrons decreases in the nucleus one by one. Also, the proton number increases one by one. The example for the beta decay process is as shown below:

\(_{90}^{234}\textrm{Th}\rightarrow _{91}^{234}\textrm{Pa}+_{-1}^{0}\textrm{e}\) |

3. Gamma Decay:

3. Gamma Decay:

The electrons orbiting around the nucleus have energy levels, and that each time an electron moves from a high energy level to a low energy level, it emits a photon. The same thing happens in the nucleus: when it rearranges into a lower energy state, it shoots out a high-energy photon known as a gamma ray.

**Radioactive Decay Law:**

In a radioactive material, it is found that the radioactive decays per unit time are directly proportional to the total number of nuclei of radioactive compounds in the sample. Through this, we can mathematically quantify the rate of radioactive decay.

If the number of nuclei in a sample is N and the number of radioactive decays per unit time Δt is ΔN then,

\(\frac{\Delta N}{\Delta t}\propto N\)

or \(\frac{\Delta N}{\Delta t}=\lambda N\),

Where, λ is the constant of proportionality called the radioactive decay constant or disintegration constant. Also, the number of radioactive decays ΔN is reducing the total number present in the sample. Convention tells us that this should be termed negative.

\(\frac{dN}{dt}=-\lambda N\)

Rearranging this,

\(\frac{dN}{N}=-\lambda dt\)

Integration of both sides then results in,

\(\int_{N_{0}}^{N}\frac{dN}{N}=-\lambda \int_{t_{0}}^{t}dt\)

\(ln\; N-ln\; N_{0}=-\lambda (t-t_{0})\)

Here, N0 represents the original number of nuclei in the sample at a time t0, i.e. t=0. Applying that in the equation results in;

\(ln\frac{N}{N_{0}}=-\lambda t\)

This further leads to,

\(N(t)=N_{0}e-\lambda t\)

One thing we must address here is that radioactive decay is exponential. If one sees the time it takes to burn incense sticks (agarbatti), they notice that it will all burn out at approximately the same time. Here the decrease is represented by the term half-life. We will discuss it further down in the article.

**Rate of Decay:**

Getting back to the expression, to see radioactive decays clearly, we should focus not on the number but on the rates. Rate here is the change per time. Calculating the rate of decay,

\(R =-\frac{dN}{dt}\)

Substituting N(t) in the equation and differentiating it,

\(N(t)=N_{0}e-\lambda t\)

Differentiation result is’

\(R =-\frac{dN}{dt}=\lambda N_{0}e^{-\lambda t}\\ \\ R=R_{0}e^{-\lambda t}\)

R0 here represents the Radioactive decay rate at time, t=0.

Substituting the original equation back here,

\(\frac{\Delta N}{\Delta t}=\lambda N\)

We get,

\(R=\lambda N\)

The total decay rate R of a radioactive sample is called the activity of that sample. SI unit of the activity is Becquerel, in the honour of Radioactivity’s discoverer, Henry Becquerel.

1 becquerel = 1 Bq = 1 decay per second

Another unit is the curie.

1 curie = 1 Ci = \(3.7\times 10^{10}Bq\)

**What is half-life?**

In radioactivity, the life of a sample is measured using two different measurements.

Half-Life, T\(\frac{1}{2}\): Half-life is the time period in which both the number of nuclei, N and the rate of decay, R have been reduced to a half of the original value. If we start with 100 nuclei, after one half-life the number left will be 50. Second, 25. Third, 12.5. Fourth, 6.25 and so on.

- \(R = R_{0} e^{-\lambda t}\)

At the first half-life, \(R = \frac{1}{2}\times R_{0}\) and t = T\(\frac{1}{2}\). Substituting and solving for T\(\frac{1}{2}\).

\(T_{\frac{1}{2}}=\frac{ln\; 2}{\lambda }\\ \\ =\frac{0.693}{\lambda }\)

Average or Mean life, tav: Average life refers to the average number of radioactive decays in a given span of time. Assuming that the time period is Δt then the rate R(t) is;

- \(R(t)\Delta t = (\lambda N_{0}e^{-\lambda t}\Delta t)\)

Though some nuclei decay in a short while, some live much longer. In order to take these decays into consideration, we integrate these from zero to infinity.

- \(\tau =\frac{\lambda N_{0}\int_{0}^{\infty }te^{-\lambda t}dt}{N_{0}}=\lambda \int_{0}^{\infty }te^{-\lambda t}dt\)
- From this we get,
- \(\tau =\frac{1}{\lambda }\)
- Which can then be summarized as;
- \(T_{\frac{1}{2}}=\frac{ln\; 2}{\lambda }=\tau \; ln\; 2\)

Half-life doesn’t mean that if there are 15 nuclei, then after one half-life there will 7.5 atoms left. Half-life just tells us the probability of the atoms decaying. The probability of a radioactive atom decaying within its half-life is 50%. But since the graph is exponential, it never really reaches zero. It approaches zero asymptotically. It just reduces to small number of atoms without ever becoming zero.

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