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Modern Physics
Beta Radiation in Nuclear Physics
Beta radiation consists of high-speed electrons (beta-minus) or positrons (beta-plus) emitted from atomic nuclei during radioactive decay. This process involves the weak nuclear force, which transforms a neutron into a proton or vice versa, thereby changing the atomic number by one while keeping the mass number constant.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic types of beta decay (beta-minus and beta-plus), conservation laws in beta decay, properties of beta particles, and neutrino hypothesis.
PGT extension
Fermi's theory of beta decay, continuous beta spectrum analysis, Kurie plots, parity non-conservation in weak interactions, and orbital electron capture.
Prerequisites
Radioactive decay law and half-life
Conservation of energy, momentum, and angular momentum
Pauli's exclusion principle and neutrino concept
Types of Beta Decay
Beta-minus decay occurs when an unstable neutron-rich nucleus emits an electron and an electron antineutrino, converting a neutron into a proton: n → p + e&sup- + ν̄e.
Beta-plus decay occurs in proton-rich nuclei where a proton converts into a neutron, emitting a positron and an electron neutrino: p → n + e&sup+ + νe.
Electron capture is a competing process where an inner-shell electron (typically K-shell) is captured by the nucleus, converting a proton into a neutron and emitting a neutrino: p + e&sup- → n + νe.
Continuous Beta Spectrum and the Neutrino
Unlike alpha decay, which exhibits discrete energy lines, beta decay yields a continuous energy spectrum of emitted electrons ranging from zero up to a maximum kinetic energy (Emax).
To preserve the conservation of energy, momentum, and angular momentum (spin 1/2), Wolfgang Pauli postulated the existence of a neutral, nearly massless spin-1/2 particle called the neutrino.
The available decay energy Q is shared continuously between the beta particle, the recoiling daughter nucleus, and the neutrino (or antineutrino).
Energetics of Beta Decay
For beta-minus decay to occur spontaneously, the mass of the parent neutral atom must exceed the mass of the daughter neutral atom: M(Z, A) > M(Z+1, A).
For beta-plus decay, the mass of the parent neutral atom must exceed the mass of the daughter atom by at least two electron masses: M(Z, A) > M(Z-1, A) + 2mec².
For electron capture, the mass condition is simply M(Z, A) > M(Z-1, A) (neglecting binding energy of the captured electron).
Fermi's Theory of Beta Decay
Enrico Fermi formulated a quantitative theory of beta decay modeled after quantum electrodynamics, treating the beta particle and neutrino as being created at the moment of emission.
The transition rate (decay probability per unit time) is governed by Fermi's Golden Rule, proportional to the square of the matrix element of the weak interaction and the density of final states (phase space factor).
The shape of the allowed beta spectrum is explicitly derived by integrating over the available momenta of the electron and neutrino.
Derivations
Kinematic Q-Value for Beta-Minus Decay
Conditions: Rest frame of the parent nucleus, utilizing atomic masses to account for orbital electrons.
Write the nuclear mass conservation equation for parent mass Mnuc(Z, A) decaying to daughter nuclear mass Mnuc(Z+1, A) plus electron mass me and neutrino mass mν ≈ 0.
$Q = [M_{nuc}(Z$, A) - Mnuc(Z+1, A) - me] c².
Add Z electron masses me to both parent and daughter nuclear masses to convert them into neutral atomic masses M(Z, A) and M(Z+1, A).
Q = [M(Z, A) - Z me - (M(Z+1, A) - (Z+1) me) - me] c².
Simplify the electron mass terms: Z me - (Z + 1) me + me = 0.
Result: Q = [M(Z, A) - M(Z+1, A)] c²
Kinematic Q-Value for Beta-Plus Decay
Conditions: Rest frame of the parent nucleus, accounting for positron emission and atomic electron balance.
Write the nuclear mass conservation equation for positron decay: Mnuc(Z, A) → Mnuc(Z-1, A) + me + mν.
$Q = [M_{nuc}(Z$, A) - Mnuc(Z-1, A) - me] c².
Convert nuclear masses to neutral atomic masses by adding Z electron masses to the parent and (Z-1) electron masses to the daughter.
Mnuc(Z, A) = M(Z, A) - Z me and Mnuc(Z-1, A) = M(Z-1, A) - (Z-1) me.
Substitute these into the Q-value equation: Q = [M(Z, A) - Z me - (M(Z-1, A) - (Z-1) me) - me] c².
Combine the electron mass terms: -Z me + (Z-1) me - me = -2 me.
Result: Q = [M(Z, A) - M(Z-1, A) - 2me] c²
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Beta-Minus Q-Value
$$Q = (M(Z, A) - M(Z+1, A)) c^2$$
M(Z,A) = atomic mass of parent, M(Z+1,A) = atomic mass of daughter, c = speed of light Joules (J) or MeV
M(Z,A) > M(Z+1,A)
Beta-Plus Q-Value
$$Q = (M(Z, A) - M(Z-1, A) - 2m_e) c^2$$
$m_e = rest mass of electron$ Joules (J) or MeV
M(Z,A) > M(Z-1,A) + 2m_e
Electron Capture Q-Value
$$Q_{EC} = (M(Z, A) - M(Z-1, A)) c^2 - B_n$$
$B_n = binding energy of the captured electron$ Joules (J) or MeV
M(Z,A) > M(Z-1,A)
Radioactive Decay Law
$$N(t) = N_0 e^{-\lambda t}$$
$N_0 = initial nuclei$, \lambda = decay constant, t = time Dimensionless (number)
Constant decay probability per unit time
Kurie Plot Linearization Function
$$K(E) = \sqrt{\frac{N(E)}{F(Z, E) p E}} \propto (Q - E)$$
N(E) = beta count rate, F(Z,E) = Fermi function, p = momentum, E = kinetic energy Arbitrary units
$M_{if} = matrix element$, \rho($E_f) = density of final states$ s^{-1}
Time-dependent perturbation theory
Beta Particle Range-Energy Relation
$$R = c_1 E_{max}^n$$
R = range in absorber, c_1 and n = empirical constants g cm^{-2}
Non-relativistic to moderately relativistic energies
Maximum Kinetic Energy Conservation
$$E_{max} \approx Q$$
$E_{max} = maximum kinetic energy of \beta particle$ Joules (J) or MeV
Assuming negligible recoil energy of the heavy daughter nucleus
Graphs and experiments
Beta Decay Energy Spectrum
Draw: A plot of number of emitted beta particles N(E) versus kinetic energy E, showing a continuous curve rising from zero, peaking around 1/3 to 1/2 of Emax, and dropping sharply to zero at $E = E_{max}$.
Interpret: Demonstrates that beta particles are emitted with a continuous distribution of energies, proving the existence of a third unseen particle (the neutrino) sharing the decay energy.
Exam use: Frequently tested to contrast with discrete alpha and gamma spectra.
Kurie Plot
Draw: A straight line plot of [N(E)/F(Z,E)pE]1/2 against kinetic energy E, intersecting the energy axis at E = Q.
Interpret: Linearizes the beta spectrum data, confirming Fermi's theory and allowing an accurate determination of the maximum endpoint energy Q (and upper limit of neutrino mass).
Exam use: Tested in PGT exams regarding graphical verification of Fermi theory and endpoint energy determination.
Absorption of Beta Radiation by Matter
Draw: A semi-logarithmic plot of transmitted beta intensity versus absorber thickness (e.g., aluminum foil), showing an initial near-exponential decrease followed by a tail due to Bremsstrahlung X-rays.
Interpret: Beta particles have a well-defined maximum range in matter due to continuous ionization energy losses, unlike gamma rays which exhibit pure exponential attenuation.
Exam use: Distinguishing the interaction mechanisms of alpha, beta, and gamma radiations.
Solved numericals
TGT core
In a beta-minus decay, if the mass of the parent atom 24Na is 23.990963 u and the mass of the daughter atom 24Mg is 23.985042 u, calculate the maximum kinetic energy of the emitted beta particle in MeV. (1 $u = 931.5 MeV/c^{2})$
Identify the Q-value formula for beta-minus decay: Q = [M(Z, A) - M(Z+1, A)] c².
Substitute the given atomic masses: $Q = (23.990963 u - 23.985042 u) c^{2}$.
Calculate the mass difference: ΔM = 0.005921 u.
Convert mass difference to energy: $Q = 0.005921 \times 931.5 MeV = 5.515 MeV$.
Answer: 5.515 MeV
PGT extension
Determine the threshold atomic mass difference for a nucleus to undergo spontaneous positron (beta-plus) decay, and calculate the minimum mass excess required for 11C (atomic mass 11.011434 u) to decay to 11B (atomic mass 11.009305 u). Given me = 0.0005486 u.
Write the Q-value condition for beta-plus decay: Q = [M(Z, A) - M(Z-1, A) - 2m_e] c² > 0.
Condition for spontaneous decay: M(Z, A) > M(Z-1, A) + 2m_e.
Substitute the atomic masses for 11C and 11B: M(C-11) = 11.011434 u, M(B-11) = 11.009305 u.
Calculate 2me = 2 × 0.0005486 u = 0.0010972 u.
Calculate M(B-11) + 2$m_e = 11.009305 u + 0.0010972 u = 11.0104022 u$.
Compare parent mass: 11.011434 u > 11.0104022 u, hence decay is energetically allowed.
A radioactive sample undergoes beta-minus decay with a half-life of 14.3 days. Calculate the decay constant λ and the time required for 75% of the initial nuclei to decay.
Take natural logarithm on both sides: ln(0.25) = -λ t ⇒ -$1.3863 = -0.04846 t$.
Solve for t: $t = 1.3863 / 0.04846 = 28.6 days (which$ is exactly 2 half-lives).
Answer: t = 28.6 days
Assumptions and limitations
Assumption: The neutrino mass is treated as zero or extremely small in standard non-relativistic Q-value calculations. Limitation: Precise measurements of beta spectra near the endpoint require non-zero neutrino mass corrections.
Assumption: Orbital electron binding energies are neglected in crude Q-value estimates. Limitation: Binding energy becomes critical when calculating accurate threshold energies for electron capture and low-energy beta transitions.
Common misconceptions
Common error: Beta particles are orbital electrons emitted from the atom.
Correct: Beta particles are created within and emitted from the atomic nucleus via the conversion of a neutron into a proton (or vice versa).
Orbital electrons do not have sufficient energy nor do they originate from nuclear transitions; the weak force governs beta decay.
Common error: Beta decay has a sharp, discrete energy spectrum similar to alpha decay.
Correct: Beta decay produces a continuous energy spectrum.
The available decay energy is shared dynamically between the beta particle, the recoiling nucleus, and an undetected neutrino.
Common error: A free neutron decays into a proton without emitting any other particle.
Correct: A free neutron decays into a proton, an electron, and an electron antineutrino.
Lepton number, angular momentum, and energy conservation require the emission of an antineutrino.
Quick revision
Beta-minus decay converts a neutron into a proton: n → p + e&sup- + ν̄e.
Beta-plus decay converts a proton into a neutron: p → n + e&sup+ + νe.
Electron capture involves a nucleus absorbing an inner-shell electron: p + e&sup- → n + νe.
The continuous beta energy spectrum led Pauli to postulate the neutrino.
Beta-minus Q-value condition: M(Z, A) > M(Z+1, A).
Beta-plus Q-value condition: M(Z, A) > M(Z-1, A) + 2me.
Kurie plots linearize the beta spectrum to find the endpoint energy Q.
Fermi's theory successfully explains the transition probabilities of allowed beta decays.
Must remember
Beta-minus decay involves the conversion of a neutron into a proton: $n arrow p + e^- + \bar{\nu}_e$.
Beta-plus decay requires the parent neutral atom mass to exceed the daughter atom mass by at least $2m_e c^2$.
Electron capture is a competing process where an inner-shell electron is captured by the nucleus: $p + e^- arrow n + \nu_e$.
Unlike alpha decay, beta decay exhibits a continuous energy spectrum due to the sharing of decay energy $Q$ with a neutrino or antineutrino.
Wolfgang Pauli postulated the neutrino to conserve energy, momentum, and angular momentum (spin $1/2$) in beta decay.
Fermi's Golden Rule governs the transition rate of beta decay, combining the weak interaction matrix element and density of final states.
Glossary
Beta-minus DecayA radioactive decay process in which an unstable neutron-rich nucleus emits an electron and an antineutrino.
Beta-plus DecayA radioactive decay process in which a proton-rich nucleus emits a positron and a neutrino.
Electron CaptureA process where a nucleus absorbs an inner-shell orbital electron, converting a proton into a neutron and emitting a neutrino.
Kurie PlotA linearization plot of $K(E) = \sqrt{\frac{N(E)}{F(Z, E) p E}} \propto (Q - E)$ used to analyze beta spectra and determine endpoint energy.
Fermi's Golden RuleExpression giving the transition rate per unit time: $\lambda = \frac{2\pi}{\hbar} |M_{if}|^2 \rho(E_f)$.
Comparison of Beta Decay Modes
Parameter
Beta-Minus Decay
Beta-Plus Decay
Electron Capture
Emitted Particles
$e^-$ and $\bar{\nu}_e$
$e^+$ and $\nu_e$
$\nu_e$ (neutrino only)
Nuclear Transition
$n arrow p + e^- + \bar{\nu}_e$
$p arrow n + e^+ + \nu_e$
$p + e^- arrow n + \nu_e$
Energetic Condition
$M(Z,A) > M(Z+1,A)$
$M(Z,A) > M(Z-1,A) + 2m_e c^2$
$M(Z,A) > M(Z-1,A)$
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following particles is emitted alongside a positron during $\beta$-plus decay?
Explanation: In $\beta$-plus decay, a proton converts into a neutron, emitting a positron ($e^+$) and an electron neutrino ($\nu_e$) to conserve lepton number.
2. What is the mass condition required for spontaneous $\beta$-plus decay of a neutral parent atom $M(Z, A)$ into a daughter atom $M(Z-1, A)$?
Explanation: For $\beta$-plus decay, accounting for the rest mass of the created positron and the loss of one orbital electron from the daughter atom, the parent atomic mass must exceed the daughter atomic mass by at least $2m_e c^2$.
3. Why does $\beta$ decay exhibit a continuous energy spectrum of emitted electrons?
Explanation: Unlike $\alpha$ decay (two-body kinematics yielding discrete energies), $\beta$ decay is a three-body decay process where the $Q$-value is shared continuously.
4. Who postulated the existence of the neutrino to resolve the apparent violation of energy and momentum conservation in $\beta$ decay?
Explanation: Wolfgang Pauli postulated the neutrino in 1930 as a neutral, nearly massless spin-$1/2$ particle carrying away the missing energy and momentum.
5. In electron capture, which atomic shell electron is most commonly captured by the nucleus?
Explanation: Inner-shell electrons, specifically K-shell electrons, have the highest probability density near the nucleus and are most commonly captured.
6. What is the nuclear reaction representing $\beta$-minus decay?
Explanation: $\beta$-minus decay occurs when an unstable neutron converts into a proton, emitting an electron ($e^-$) and an electron antineutrino ($\bar{\nu}_e$).
7. What is the mass condition for electron capture to occur spontaneously (neglecting binding energy)?
Explanation: In electron capture, an orbital electron combines with a proton to form a neutron and a neutrino. The mass condition is simply $M(Z, A) > M(Z-1, A)$.
8. What does a Kurie plot linearize in the analysis of $\beta$ spectra?
Explanation: The Kurie plot graphs $K(E) = \sqrt{\frac{N(E)}{F(Z, E) p E}}$ versus kinetic energy $E$, yielding a straight line whose intercept gives the maximum decay energy $Q$.
9. Which framework forms the basis of Fermi's theory of $\beta$ decay?
Explanation: Fermi formulated his $\beta$ decay theory using time-dependent perturbation theory, treating the creation of the electron-neutrino pair through weak interactions.
10. What is the approximate relation between the maximum kinetic energy $E_{max}$ of the $\beta$ particle and the total decay energy $Q$ (assuming negligible daughter nucleus recoil)?
Explanation: Because the heavy daughter nucleus takes away a negligible amount of recoil energy, the maximum kinetic energy of the $\beta$ particle approximately equals the total decay energy $Q$.
11. Which conservation law necessitates the emission of an antineutrino in $\beta$-minus decay?
Explanation: Lepton number (since an electron lepton number +1 is created, an antilepton of lepton number -1 must be created), angular momentum (spin 1/2), and energy conservation require the antineutrino.
12. What is the expression for the Q-value of $\beta$-minus decay in terms of neutral atomic masses?
Explanation: For $\beta$-minus decay, $Q = (M(Z, A) - M(Z+1, A)) c^2$, as the electron mass cancels when using neutral atomic masses.
13. What is the typical spin of a neutrino or antineutrino as postulated and later verified?
Explanation: Neutrinos are fermions with spin $1/2$, which is crucial for accounting for the half-integral spin changes in $\beta$ decay transitions.
14. In Fermi's Golden Rule expression $\lambda = \frac{2\pi}{\hbar} |M_{if}|^2 \rho(E_f)$, what does $\rho(E_f)$ represent?
Explanation: $\rho(E_f)$ represents the density of final states available to the emitted $\beta$ particle and neutrino at energy $E_f$.
15. Which misconception about $\beta$ particles is scientifically incorrect?
Explanation: Orbital electrons are not emitted in $\beta$ decay; $\beta$ particles are created via the weak interaction conversion of nucleons inside the nucleus.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. In a $\beta$-minus decay process, a neutron transforms into a proton. Which of the following correctly lists all the particles produced in this elementary decay?
Explanation: The fundamental neutron $\beta$ decay is $n arrow p + e^- + \bar{\nu}_e$, producing a proton, an electron, and an electron antineutrino.
2. The continuous nature of the $\beta$ decay energy spectrum was a major puzzle before the introduction of which hypothesis?
Explanation: Pauli proposed the neutrino to explain how energy could appear continuous while obeying conservation laws in a two-body apparent decay.
3. What is the minimum mass excess condition required for $\beta$-plus ($e^+$) decay of a parent neutral atom $M(Z,A)$ into a daughter atom $M(Z-1,A)$?
Explanation: Because a positron is created and the daughter atom has one less electron, the mass of the neutral parent atom must exceed that of the daughter atom by at least $2m_e c^2$.
4. Which of the following processes competes directly with $\beta$-plus decay in proton-rich nuclei?
Explanation: Electron capture is an alternative decay mode for proton-rich nuclei where an inner shell electron is captured, yielding the same nuclear daughter product.
5. A Kurie plot is used in $\beta$ decay studies to determine which of the following quantities?
Explanation: Extrapolating the linear fit of a Kurie plot to the energy axis yields the maximum kinetic energy (endpoint energy) of the $\beta$ spectrum.
6. According to Fermi's theory of $\beta$ decay, the transition rate probability is proportional to which of the following terms?
Explanation: Fermi's Golden Rule dictates that the decay rate $\lambda$ is proportional to $|M_{if}|^2 \rho(E_f)$ (matrix element squared multiplied by density of final states).
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. In $\beta$-minus decay, if the rest mass of the parent neutral atom is $M(Z,A)$ and the daughter atom is $M(Z+1,A)$, what is the $Q$-value expression?
Explanation: The $Q$-value for $\beta$-minus decay is given by $Q = (M(Z, A) - M(Z+1, A)) c^2$ when using neutral atomic masses.
2. Which particle is emitted during electron capture alongside the conversion of a proton into a neutron?
Explanation: Electron capture follows $p + e^- arrow n + \nu_e$, emitting an electron neutrino.
3. Why is the continuous $\beta$ spectrum historically significant?
Explanation: The apparent loss of energy in the continuous $\beta$ spectrum prompted Pauli to propose the neutrino.
4. What is the condition on atomic masses for electron capture to be energetically allowed?
Explanation: Electron capture requires $M(Z, A) > M(Z-1, A)$ (neglecting atomic electron binding energy differences).
5. In the Kurie plot function $K(E) = \sqrt{\frac{N(E)}{F(Z, E) p E}} \propto (Q - E)$, what does $F(Z,E)$ represent?
Explanation: $F(Z, E)$ is the Fermi function that corrects the $\beta$ spectrum for the Coulomb attraction or repulsion experienced by the electron/positron in the daughter nucleus electric field.
6. Which of the following interactions is responsible for $\beta$ decay in atomic nuclei?
Explanation: $\beta$ decay is mediated by the weak nuclear force, which allows flavor-changing quark or nucleon transformations.
7. If a nucleus undergoes $\beta$-plus decay, what happens to its atomic number $Z$ and mass number $A$?
Explanation: In $\beta$-plus decay, a proton converts into a neutron, so the atomic number $Z$ decreases by 1 while the mass number $A$ remains constant.
8. What is the primary reason why $\beta$ particles have a variable range in matter compared to $\alpha$ particles of a given energy?
Explanation: Because $\beta$ particles have a continuous energy spectrum and undergo severe multiple scattering due to their small mass, their range is distributed rather than sharply defined like $\alpha$ particles.
9. Which conservation law is NOT tested or involved in standard leptonic $\beta$ decay processes?
Explanation: Leptons do not carry color charge; quantum chromodynamics (color charge conservation) applies to quarks and gluons, not leptons or electroweak $\beta$ decay.
10. What is the expected value of the Kurie plot $K(E)$ when the kinetic energy $E$ of the $\beta$ particle equals the maximum decay energy $Q$?
Explanation: Since $K(E) \propto (Q - E)$, when $E = Q$, the value of $K(E)$ drops to zero, providing the intercept to determine the endpoint energy.