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Modern Physics

Alpha Radiation and Alpha Decay

Alpha radiation consists of a stream of energetic helium-4 nuclei (^4_2He) emitted spontaneously from heavy, unstable atomic nuclei. The process reduces the mass number by 4 and the atomic number by 2, governed by quantum mechanical tunneling through a Coulomb potential barrier.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Characteristics of alpha particles, conservation laws in radioactive decay, charge and mass changes, ionization and penetration properties, and Geiger-Nuttall law.

PGT extension

Quantum mechanical theory of alpha decay, barrier penetration probability, derivation of decay constant via Gamow's factor, and kinetic energy distribution in fine structure.

Prerequisites

Binding energy per nucleon and mass defect
Quantum mechanical potential barrier and tunneling
Conservation of energy and linear momentum

Nature and Properties of Alpha Particles

An alpha particle is identical to a doubly ionized helium atom (^4_2He^{2+}), containing two protons and two neutrons.
They possess high ionizing power due to their heavy mass and double positive charge, but low penetrating power (stopped by a thin sheet of paper or a few centimeters of air).
Alpha particles are deflected by both electric and magnetic fields, confirming their positive charge.

Kinetic Energy and Q-Value of Decay

In a two-body decay from a stationary parent nucleus (mass M) to a daughter nucleus (mass M_d) and an alpha particle (mass M_alpha), the Q-value is given by $Q = (M - M_d - M_alpha)c^2$.
Due to momentum conservation, the kinetic energy of the alpha particle is slightly less than the total Q-value, sharing it with the recoiling daughter nucleus.
Alpha spectra are discrete, indicating that both parent and daughter nuclei occupy definite nuclear energy states.

Gamow's Theory of Alpha Decay

Alpha decay is treated as a pre-formed alpha particle trapped inside a nuclear potential well, bounded by a Coulomb potential barrier.
Classically, the alpha particle lacks sufficient energy to surmount the Coulomb barrier of height $V_c = \frac{1}{4piepsilon_0} \frac{2(Z-2)e^2}{R}$.
Quantum mechanical tunneling allows the particle to penetrate the barrier with a finite transmission probability T.

Geiger-Nuttall Law

An empirical relationship exists between the decay constant lambda of an alpha emitter and the range R of the emitted alpha particles in air: \log lambd$a = A \log R + B$.
A linear relationship also exists between \log lambda and the inverse square root of the alpha particle's kinetic energy.
This law is naturally explained by Gamow's barrier penetration theory.

Derivations

Kinetic Energy of Alpha Particle from Q-Value

Conditions: Stationary parent nucleus undergoing two-body alpha decay

Apply conservation of linear momentum: p_\alpha + $p_d = 0 \implies p_\alpha = -p_d$.
Express kinetic energy in terms of momentum: $E_\alpha = \frac{p_\alpha^2}{2M_\alpha}$ and $E_d = \frac{p_d^2}{2M_d}$.
Relate the energies using mass ratio: $E_d = E_\alpha \left(\frac{M_\alpha}{M_d}\right)$.
Substitute into the total Q-value equation: $Q = E_\alpha + E_d = E_\alpha \left(1 + \frac{M_\alpha}{M_d}\right) \approx E_\alpha \left(1 + \frac{4}{A-4}\right)$.

Result: $E_\alpha = Q \left(\frac{M_d}{M_d + M_\alpha}\right) \approx Q \left(\frac{A-4}{A}\right)$

Barrier Penetration Probability (Gamow Factor)

Conditions: Square-well nuclear potential with a long-range Coulomb barrier

Write the WKB approximation for the transmission coefficient T through a potential barrier V(r) from classical turning point r_a to r_b.
T \approx \exp \left( -2 \int_{r_a}^{r_b} \sqrt{\frac{2m}{\hbar^2} (V(r) - E)} dr \right).
Substitute the Coulomb potential V(r) = \frac{1}{4piepsilon_0} \frac{2(Z-2)e^2}{r} and evaluate the integral.
Express the result in terms of the fine-structure constant and velocity of the emitted alpha particle.

Result: $T = \exp \left[ -2 \left( \frac{2\pi Z_1 Z_2 e^2}{4piepsilon_0 \hbar v} \right) f(\frac{E}{V_c}) \right]$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Alpha Decay Transformation	$$^{A}_{Z}X arrow ^{A-4}_{Z-2}Y + ^{4}_{2}\alpha + Q$$	X = Parent nucleus, Y = Daughter nucleus, Q = Energy released Atomic mass units (u) or MeV	Applicable for heavy nuclei with Z > 82
Q-Value Equation	$$Q = [m(^{A}_{Z}X) - m(^{A-4}_{Z-2}Y) - m(^{4}_{2}\alpha)] c^2$$	m = Rest masses of nuclei MeV	Masses in atomic mass units
Alpha Kinetic Energy	$$E_\alpha = Q \left( \frac{A-4}{A} \right)$$	A = Mass number of parent nucleus, Q = Total decay energy MeV	Parent nucleus initially at rest
Geiger-Nuttall Law	$$\log \lambda = A \log R + B$$	lambda = Decay constant, R = Range of alpha particle in air, A and B = Constants s^{-1} for lambda, cm for R	Valid for a given radioactive series
Coulomb Barrier Height	$$V_c = \frac{1}{4piepsilon_0} \frac{2(Z-2)e^2}{R_0 A^{1/3}}$$	Z = Atomic number, R_0 \approx 1.2 \times 10^{-15} \text{ m} Joules or MeV	Spherical nucleus approximation
Decay Constant Expression	$$\lambda = f \cdot T$$	f = Attempt frequency of alpha particle hitting the barrier, T = Transmission coefficient s^{-1}	Quantum mechanical tunneling model
Velocity of Ejected Alpha Particle	$$v_\alpha = \sqrt{\frac{2 E_\alpha}{m_\alpha}}$$	$E_\alpha = Kinetic energy$, $m_\alpha = Mass of \alpha particle$ m/s	Non-relativistic approximation valid for alpha particles
Daughter Recoil Energy	$$E_d = E_\alpha \left( \frac{4}{A-4} \right)$$	$E_d = Recoil energy of daughter$, A = Mass number MeV	Momentum conservation

Graphs and experiments

Nuclear Potential Well and Coulomb Barrier

Draw: Potential energy V(r) versus radial distance r, showing the deep attractive nuclear well, the potential barrier peak, and the energy level E of the alpha particle.

Interpret: Illustrates that the alpha particle is bound inside by nuclear forces but escapes via quantum mechanical tunneling through the Coulomb barrier without having classical energy to go over the top.

Exam use: Frequently asked in descriptive questions regarding why heavy nuclei emit alpha particles instead of individual nucleons.

Geiger-Nuttall Plot

Draw: A straight-line graph plotting \log \lambda on the y-axis against \log R on the x-axis for different radioactive series.

Interpret: Shows that alpha emitters with higher energy (longer range R) have a much shorter half-life (larger decay constant lambda).

Exam use: Tested via conceptual interpretation of slopes and intercepts for uranium or thorium series.

Wilson Cloud Chamber Track of Alpha Particles

Draw: Straight, dense, uniform tracks of droplets ending abruptly with a slight scatter at the end.

Interpret: Demonstrates the heavy mass, constant specific ionization, and discrete range of monoenergetic alpha particles in a gas medium.

Exam use: Identification of radiation types based on track characteristics.

Solved numericals

TGT core

Find the kinetic energy of the alpha particle emitted in the decay of ^{226}_{88}Ra into ^{222}_{86}Rn, given that the total Q-value of the reaction is 4.87 MeV.

Identify the mass number of the parent nucleus: A = 226.
Use the formula relating alpha kinetic energy to Q-value: $E_\alpha = Q \left( \frac{A-4}{A} \right)$.
Substitute the values: $E_\alpha = 4.87 \times \left( \frac{226 - 4}{226} \right) = 4.87 \times \left( \frac{222}{226} \right)$.
Calculate the result: $E_\alpha = 4.87 \times 0.9823 = 4.78 \text{ MeV}$.

Answer: 4.78 MeV

PGT extension

Calculate the height of the Coulomb barrier for an alpha particle approaching a daughter nucleus of Uranium (Z = 90, A = 234). Assume $R_0 = 1.2 fm$.

Identify atomic number of daughter nucleus $Z_d = 90$, charge of alpha $Z_\alpha = 2$.
Calculate nuclear radius: $R = R_0 A^{1/3} = 1.2 \times 10^{-15} \times (234)^{1/3}$.
Evaluate (234)^{1/3} \approx 6.162, so $R = 1.2 \times 10^{-15} \times 6.162 = 7.394 \times 10^{-15} \text{ m}$.
Use Coulomb barrier formula: $V_c = \frac{1}{4piepsilon_0} \frac{Z_\alpha Z_d e^2}{R}$.
Substitute constants: \frac{1}{4piepsilon_0} $e^2 = 1.44 \times 10^{-15} \text{ MeV m}$.
$V_c = \frac{1.44 \times 10^{-15} \times 2 \times 90}{7.394 \times 10^{-15}} = \frac{259.2}{7.394} = 35.06 \text{ MeV}$.

Answer: 35.06 MeV

PGT extension

The decay constant for an alpha emitter is 1.2 \times 10^{-11} \text{ s}^{-1}. If the constant B in the Geiger-Nuttall relation \log $\lambda = A \log R + B$ is 45.2 and A is -32.5, find the range R of the alpha particle in air.

Take the common logarithm (base 10) of lambda: \log_{10}(1.2 \times 10^{-11}) = \log_{10}(1.2) - $11 = 0.0792 - 11 = -10.9208$.
Substitute into Geiger-Nuttall equation: -$10.9208 = A \log R + B$.
Substitute $A = -32.5$ and $B = 45.2: -10.9208 = -32.5 \log R + 45.2$.
Rearrange for \log R: 32.5 \log $R = 45.2 + 10.9208 = 56.1208$.
Calculate \log R: \log R = \frac{56.1208}{32.5} = 1.7268.
Find R by taking antilog: $R = 10^{1.7268} = 53.31 \text{ cm}$.

Answer: 53.31 cm

Assumptions and limitations

Assumes the alpha particle exists as a pre-formed cluster inside the heavy nucleus prior to emission.
The potential barrier is treated as a static one-dimensional Coulomb barrier, ignoring dynamic shape deformations of the nucleus during tunneling.

Common misconceptions

Common error: Alpha particles have enough classical kinetic energy to overcome the repulsive electrostatic barrier of the nucleus.

Correct: Alpha particles possess kinetic energy (4-9 MeV) much lower than the Coulomb barrier height (approx 30-35 MeV).

They escape exclusively via quantum mechanical tunneling, permitted by wave-particle duality.

Common error: The emitted alpha particle takes all the energy released in the decay (Q-value).

Correct: The alpha particle shares the Q-value with the recoiling daughter nucleus.

Linear momentum must be conserved, forcing the heavy daughter nucleus to recoil in the opposite direction with a small fraction of kinetic energy.

Quick revision

Alpha emission reduces mass number A by 4 and atomic number Z by 2.
Kinetic energy of an alpha particle is always less than the Q-value due to daughter nucleus recoil.
Gamow's theory explains alpha decay successfully using quantum mechanical barrier penetration.
Geiger-Nuttall law connects decay constant directly with the range of emitted alpha particles.

Must remember

An alpha particle is a doubly ionized helium nucleus ($^4_2He^{2+}$) consisting of two protons and two neutrons.
In alpha decay, the parent nucleus transforms as: $^{A}_{Z}X arrow ^{A-4}_{Z-2}Y + ^{4}_{2}\alpha + Q$.
The kinetic energy of the emitted alpha particle is given by $E_\alpha = Q \left( \frac{A-4}{A} \right)$ due to recoil momentum conservation.
Classically, alpha particles lack sufficient energy to surmount the Coulomb barrier ($V_c \approx 30-35$ MeV); they escape via quantum mechanical tunneling.
The Geiger-Nuttall law relates the decay constant $\lambda$ to the range $R$ of alpha particles in air as $\log \lambda = A \log R + B$.
Alpha spectra are discrete, indicating that parent and daughter nuclei occupy definite nuclear energy states.

Glossary

Alpha ParticleA helium nucleus ($^4_2He^{2+}$) emitted during the radioactive decay of heavy nuclides.

Q-ValueThe total energy released in a nuclear decay process, equal to $\Delta m \cdot c^2$.

Coulomb BarrierThe electrostatic potential energy barrier that an alpha particle must overcome or tunnel through to escape the nucleus.

Gamow FactorThe exponential factor governing the quantum mechanical transmission probability of an alpha particle through a potential barrier.

Geiger-Nuttall LawAn empirical logarithmic relation between the radioactive decay constant of an alpha emitter and the range of its emitted alpha particles.

Properties of Nuclear Radiations

Property	Alpha Radiation ($\alpha$)	Beta Radiation ($\beta$)	Gamma Radiation ($\gamma$)
Nature	Helium nuclei ($^4_2He^{2+}$)	Fast-moving electrons or positrons	High-frequency electromagnetic photons
Ionizing Power	Very high	Moderate	Low
Penetrating Power	Low (stopped by paper)	Moderate (stopped by aluminum foil)	High (stopped by thick lead)
Deflection in Fields	Deflected by both electric and magnetic fields	Deflected by both electric and magnetic fields	Not deflected by fields

Practice Quiz

Choose an option to reveal the explanation.

1. Which of the following best describes the fundamental constitution of an $\alpha$ particle?

Explanation: An $\alpha$ particle is identical to a doubly ionized helium atom ($^4_2He^{2+}$), containing two protons and two neutrons.

2. In the $\alpha$ decay process represented by $^{A}_{Z}X arrow ^{A-4}_{Z-2}Y + ^{4}_{2}\alpha + Q$, what is the change in the atomic number $Z$ of the parent nucleus?

Explanation: During $\alpha$ decay, two protons and two neutrons are emitted, so the atomic number decreases by $2$ and the mass number decreases by $4$.

3. Why do $\alpha$ particles possess high ionizing power compared to $\beta$ and $\gamma$ rays?

Explanation: $\alpha$ particles have a heavy mass and a double positive charge ($+2e$), causing intense electrostatic interactions with orbital electrons along their path.

4. What is the primary reason why $\alpha$ particles have low penetrating power?

Explanation: Because of their high ionizing power, $\alpha$ particles lose their kinetic energy very rapidly in matter, limiting their range and penetration.

5. If a stationary parent nucleus of mass number $A$ undergoes $\alpha$ decay, what fraction of the total Q-value is approximately carried away as kinetic energy by the $\alpha$ particle?

Explanation: From momentum conservation, the kinetic energy of the $\alpha$ particle is given by $E_\alpha = Q \left( \frac{A-4}{A} \right)$.

6. What does the discrete nature of $\alpha$ particle spectra imply about nuclear energy levels?

Explanation: Discrete $\alpha$ particle energies indicate that both parent and daughter nuclei possess definite, quantized nuclear energy states.

7. Classically, why is $\alpha$ decay considered paradoxical in terms of potential energy barriers?

Explanation: Classically, an $\alpha$ particle with $4-9$ MeV cannot surmount a Coulomb barrier of $30-35$ MeV, yet $\alpha$ decay occurs abundantly.

8. Which quantum mechanical phenomenon explains how $\alpha$ particles escape the nuclear potential well?

Explanation: Gamow's theory treats $\alpha$ decay as quantum mechanical tunneling, where a particle with energy less than the barrier height has a finite transmission probability.

9. The Geiger-Nuttall empirical law establishes a linear relationship between $\log \lambda$ and:

Explanation: The Geiger-Nuttall law is expressed as $\log \lambda = A \log R + B$, linking the decay constant $\lambda$ logarithmically to the range $R$ in air.

10. In the Geiger-Nuttall relation $\log \lambda = A \log R + B$, what do parameters $A$ and $B$ represent?

Explanation: Constants $A$ and $B$ vary depending on the particular radioactive series (e.g., Uranium, Thorium, Actinium series).

11. What happens to the recoil energy of the daughter nucleus as the mass number $A$ of the parent nucleus increases?

Explanation: Daughter recoil energy is given by $E_d = E_\alpha \left( \frac{4}{A-4} \right)$. As $A$ increases, the fraction decreases, reducing the daughter recoil energy.

12. Which of the following nuclei is primarily expected to undergo $\alpha$ decay?

Explanation: $\alpha$ decay is predominantly observed in heavy nuclei with atomic numbers $Z > 82$ where Coulomb repulsion is exceptionally high.

13. In Gamow's theory, the decay constant $\lambda$ is expressed as the product of:

Explanation: The decay constant is given by $\lambda = f \cdot T$, where $f$ is the attempt frequency of the $\alpha$ particle hitting the barrier and $T$ is the transmission coefficient.

14. Are $\alpha$ particles deflected by electric and magnetic fields?

Explanation: $\alpha$ particles possess a double positive charge ($+2e$), making them susceptible to deflection by both electric and magnetic fields.

15. What is the typical kinetic energy range of $\alpha$ particles emitted by natural radioactive sources?

Explanation: Natural $\alpha$ particles typically possess kinetic energies in the range of $4$ to $9$ MeV.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A parent nucleus $X$ with mass number $A = 220$ undergoes $\alpha$ decay to form daughter nucleus $Y$. If the total Q-value of the decay is $5.0$ MeV, what is the approximate kinetic energy of the emitted $\alpha$ particle?

Explanation: Using $E_\alpha = Q \left( \frac{A-4}{A} \right) = 5.0 \times \left( \frac{220-4}{220} \right) = 5.0 \times \left( \frac{216}{220} \right) = 5.0 \times 0.9818 \approx 4.91$ MeV.

2. Which of the following experimental facts provided direct historical confirmation that $\alpha$ particles are doubly ionized helium atoms?

Explanation: Rutherford and colleagues measured the charge-to-mass ratio ($e/m$) of $\alpha$ particles using crossed electric and magnetic fields, proving they were $^4_2He^{2+}$ ions.

3. According to Gamow's theory of $\alpha$ decay, the transmission probability $T$ through the Coulomb barrier depends primarily on:

Explanation: In quantum mechanical tunneling, the transmission probability $T$ depends exponentially on the height and width of the potential barrier encountered by the particle.

4. If the range of an $\alpha$ particle in air is doubled, how does the decay constant $\lambda$ roughly change according to the Geiger-Nuttall law ($\log \lambda = A \log R + B$)?

Explanation: Taking antilog of $\log \lambda = A \log R + B$, we get $\lambda = e^B R^A$. If $R$ is doubled, $\lambda$ scales by $2^A$.

5. In a two-body $\alpha$ decay, if the parent nucleus is stationary, the momentum of the emitted $\alpha$ particle ($p_\alpha$) and the momentum of the recoiling daughter nucleus ($p_d$) satisfy:

Explanation: By conservation of linear momentum, since the initial momentum is zero, $p_\alpha + p_d = 0$, which means $p_\alpha = -p_d$.

6. What is the approximate height of the Coulomb barrier for a heavy nucleus with $Z = 90$ and mass number $A = 232$?

Explanation: Evaluation of the Coulomb barrier formula $V_c = \frac{1}{4\pi\epsilon_0} \frac{2(Z-2)e^2}{R_0 A^{1/3}}$ for heavy actinide nuclei yields approximately $30 - 35$ MeV.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Which conservation law is fundamentally violated if an $\alpha$ decay occurs without a recoiling daughter nucleus?

Explanation: Linear momentum must be conserved. If the $\alpha$ particle moves in one direction, the daughter nucleus must recoil in the opposite direction.

2. Calculate the recoil energy $E_d$ of a daughter nucleus if the $\alpha$ particle kinetic energy is $4.8$ MeV and the parent mass number is $A = 216$.

Explanation: Using $E_d = E_\alpha \left( \frac{4}{A-4} \right) = 4.8 \times \left( \frac{4}{216-4} \right) = 4.8 \times \left( \frac{4}{212} \right) = 4.8 \times 0.01887 \approx 0.09$ MeV.

3. Why is $\alpha$ decay rare in nuclei with mass numbers below $150$?

Explanation: For lighter nuclei, the binding energy per nucleon does not favor the emission of a tightly bound $^4_2He$ cluster, resulting in negative or insufficient Q-values.

4. What is the primary physical implication of a large decay constant $\lambda$ in the Geiger-Nuttall law?

Explanation: A larger decay constant $\lambda$ means a shorter half-life ($\tau = \ln 2 / \lambda$), which corresponds to higher energy $\alpha$ particles and a longer range in air.

5. In the context of Gamow's barrier penetration, what does a higher transmission coefficient $T$ signify?

Explanation: A higher transmission coefficient $T$ means the $\alpha$ particle escapes more readily, leading to a higher decay rate ($\lambda$) and shorter half-life.

6. If an $\alpha$ particle and a proton have the same kinetic energy, which of the two has a larger de Broglie wavelength?

Explanation: de Broglie wavelength is $\lambda = \frac{h}{\sqrt{2 m E}}$. Since the proton has a much smaller mass than the $\alpha$ particle ($m_p \approx \frac{1}{4} m_\alpha$), its wavelength is larger.

7. The disintegration energy (Q-value) for $\alpha$ decay can be evaluated directly from:

Explanation: The Q-value is fundamentally derived from the mass difference between the parent nucleus and the total mass of the decay products multiplied by $c^2$.

8. Which feature of $\alpha$ decay spectra provides evidence for excited states in daughter nuclei?

Explanation: When daughter nuclei are left in excited states, $\alpha$ particles are emitted with correspondingly lower kinetic energies, producing distinct groups in the spectrum.

9. What is the approximate nuclear radius scaling factor $R_0$ used in calculating the Coulomb barrier height?

Explanation: The nuclear radius empirical constant is $R_0 \approx 1.2 \times 10^{-15}$ m ($1.2$ fm).

10. Which of the following statements about $\alpha$ particles is INCORRECT?

Explanation: $\alpha$ particles have mass and travel at speeds much less than the speed of light (typically $\sim 0.05c$). Thus, stating they travel at the speed of light is incorrect.