Radioactivity and Decay Laws

Radioactivity

Radioactivity is the spontaneous emission of radiation from unstable atomic nuclei. Certain isotopes of elements have nuclei that are not stable. These unstable nuclei undergo spontaneous transformations, emitting particles and/or electromagnetic radiation to attain a more stable configuration. This process is called radioactive decay. The substances that exhibit radioactivity are called radioactive substances.

Radioactivity was discovered by Henri Becquerel in 1896 while studying phosphorescence in uranium salts. Later, Pierre and Marie Curie did pioneering work in isolating new radioactive elements like Polonium and Radium and characterised the nature of the emitted radiations.

The radiations emitted by radioactive substances were found to be primarily of three types, which were named Alpha ($\alpha$), Beta ($\beta$), and Gamma ($\gamma$) rays based on their penetrating power and deflection in electric and magnetic fields.

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Law Of Radioactive Decay ($ N(t) = N_0 e^{-\lambda t} $)

Radioactive decay is a random process at the level of individual nuclei. We cannot predict when a particular nucleus will decay. However, for a large sample of radioactive nuclei, the decay process follows a definite statistical law.

The Law of Radioactive Decay states that:

"The rate of disintegration of a radioactive substance at any instant is directly proportional to the number of radioactive atoms present in the sample at that instant."

Let $N$ be the number of radioactive nuclei present in a sample at time $t$. The rate of disintegration is $-\frac{dN}{dt}$ (the negative sign indicates that $N$ is decreasing). According to the law:

$ -\frac{dN}{dt} \propto N $

$ -\frac{dN}{dt} = \lambda N $

Where $\lambda$ is a constant called the decay constant or disintegration constant. The decay constant is characteristic of the particular radioactive isotope and represents the probability of decay per unit time for a single nucleus. A larger $\lambda$ means a faster decay rate.

The differential equation can be written as $\frac{dN}{N} = -\lambda dt$. Integrating both sides:

$ \int_{N_0}^{N(t)} \frac{dN}{N} = \int_0^t -\lambda dt $

Where $N_0$ is the number of radioactive nuclei at time $t=0$, and $N(t)$ is the number of radioactive nuclei remaining at time $t$.

$ [\ln N]_{N_0}^{N(t)} = -\lambda [t]_0^t $

$ \ln N(t) - \ln N_0 = -\lambda t $

$ \ln \left(\frac{N(t)}{N_0}\right) = -\lambda t $

Taking the exponential of both sides:

$ \frac{N(t)}{N_0} = e^{-\lambda t} $

$ N(t) = N_0 e^{-\lambda t} $

This is the Radioactive Decay Law. It describes how the number of radioactive nuclei in a sample decreases exponentially with time. The number of remaining nuclei never truly reaches zero in finite time according to this model, although the rate of decay becomes very small.

The rate of decay ($dN/dt$) is also called the activity ($R$) of the sample (discussed in Section I3).

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Alpha Decay ($\alpha$-Decay)

In alpha decay, an unstable nucleus emits an alpha particle ($\alpha$). An alpha particle is a helium nucleus ($_2^4He$), consisting of 2 protons and 2 neutrons.

When a parent nucleus $_Z^A X$ undergoes alpha decay, it transforms into a daughter nucleus $_ {Z-2}^{A-4} Y$ and emits an alpha particle.

$ _Z^A X \longrightarrow _{Z-2}^{A-4} Y + _2^4He $

Properties of Alpha Particles:

• They are positively charged ($+2e$).
• They are relatively heavy (mass $\approx 4u$).
• They have a high ionising power (can ionise atoms in the medium they pass through).
• They have low penetrating power (can be stopped by a sheet of paper or a few centimetres of air).
• They are deflected by electric and magnetic fields.

Alpha decay typically occurs in heavy, unstable nuclei. The energy released in alpha decay ($Q$-value) appears as kinetic energy of the daughter nucleus and the alpha particle. The daughter nucleus is usually in an excited state and may subsequently emit gamma rays.

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Beta Decay ($\beta$-Decay)

In beta decay, a neutron in the nucleus is converted into a proton, or a proton is converted into a neutron. There are three main types of beta decay: $\beta^-$, $\beta^+$, and electron capture.

1. Beta-minus decay ($\beta^-$): A neutron is converted into a proton, an electron (called a beta particle, $\beta^-$), and an electron antineutrino ($\bar{\nu}_e$).

$ n \longrightarrow p + e^- + \bar{\nu}_e $

The parent nucleus $_Z^A X$ transforms into a daughter nucleus $_{Z+1}^A Y$ with the same mass number but atomic number increased by 1.

$ _Z^A X \longrightarrow _{Z+1}^A Y + e^- + \bar{\nu}_e $

$\beta^-$-decay occurs in neutron-rich nuclei. The electron and antineutrino are created at the moment of decay; they do not exist in the nucleus beforehand.

2. Beta-plus decay ($\beta^+$): A proton is converted into a neutron, a positron (a positively charged electron, $\beta^+$), and an electron neutrino ($\nu_e$).

$ p \longrightarrow n + e^+ + \nu_e $

The parent nucleus $_Z^A X$ transforms into a daughter nucleus $_{Z-1}^A Y$ with the same mass number but atomic number decreased by 1.

$ _Z^A X \longrightarrow _{Z-1}^A Y + e^+ + \nu_e $

$\beta^+$-decay occurs in proton-rich nuclei. The positron and neutrino are created at the moment of decay.

3. Electron Capture: The nucleus captures an inner atomic electron. This electron combines with a proton in the nucleus to form a neutron and an electron neutrino.

$ p + e^- \longrightarrow n + \nu_e $

The parent nucleus $_Z^A X$ transforms into a daughter nucleus $_{Z-1}^A Y$. This is an alternative to $\beta^+$-decay for proton-rich nuclei.

Properties of Beta Particles ($\beta^-$ and $\beta^+$):

• They are charged ($e^-$ is negative, $e^+$ is positive).
• They are much lighter than alpha particles (same mass as electrons).
• They have lower ionising power than alpha particles.
• They have higher penetrating power than alpha particles (can penetrate several millimetres of aluminium).
• They are deflected by electric and magnetic fields (in opposite directions for $\beta^-$ and $\beta^+$).

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Gamma Decay ($\gamma$-Decay)

In gamma decay, a nucleus in an excited energy state releases energy in the form of a gamma ray photon ($\gamma$). The nucleus does not change its composition (number of protons or neutrons); it simply transitions from a higher energy state to a lower energy state.

$ _Z^A X^* \longrightarrow _Z^A X + \gamma $ (where $X^*$ denotes an excited nucleus)

Gamma decay often follows alpha or beta decay, as the daughter nucleus is frequently produced in an excited state.

Properties of Gamma Rays:

• They are electromagnetic radiation (photons), not particles with mass.
• They are electrically neutral.
• They have very low ionising power.
• They have very high penetrating power (can penetrate thick sheets of lead or concrete).
• They are not deflected by electric or magnetic fields.
• They travel at the speed of light.
• Their energy is quantised (photons of specific energy).

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Answer:

Given:

Parent nucleus is Uranium-238, $_{92}^{238}U$. Here $Z=92$, $A=238$.

Alpha decay involves the emission of an alpha particle, $_2^4He$.

Let the daughter nucleus be $_ {Z'}^{A'} Y$. The nuclear equation for alpha decay is $_Z^A X \longrightarrow _{Z-2}^{A-4} Y + _2^4He$.

Applying this to Uranium-238:

Atomic number of daughter nucleus, $Z' = Z - 2 = 92 - 2 = 90$.

Mass number of daughter nucleus, $A' = A - 4 = 238 - 4 = 234$.

The element with atomic number 90 is Thorium (Th).

So, the daughter nucleus is Thorium-234, $_{90}^{234}Th$.

The nuclear equation is:

$ _{92}^{238}U \longrightarrow _{90}^{234}Th + _2^4He $

The daughter nucleus is Thorium-234 ($_{90}^{234}Th$).