GATE Physics Nuclear and Particle Physics Syllabus

How does quantum mechanics explain alpha decay despite the Coulomb barrier?

By quantum tunneling: the alpha particle penetrates the Coulomb barrier (Gamow theory). The transmission probability depends exponentially on the Gamow factor, explaining the steep energy dependence of half-lives.

Write the three types of beta decay processes.

$\beta^{-}$: $\ce{n -> p + e^{-} + \bar{\nu}_{e}}$; $\beta^{+}$: $\ce{p -> n + e^{+} + \nu_{e}}$; electron capture: $\ce{p + e^{-} -> n + \nu_{e}}$. All conserve charge and lepton number.

Why is the beta-decay electron energy spectrum continuous, and what did this imply historically?

Because the decay energy is shared among three bodies (daughter nucleus, electron, and antineutrino). The continuous spectrum led Pauli to postulate the neutrino to conserve energy, momentum, and angular momentum.

Which fundamental interaction governs beta decay, and what conserved quantity may it violate?

The weak interaction. Beta decay violates parity (the Wu experiment showed parity non-conservation in $\ce{^{60}Co}$ decay) and does not conserve flavor in the quark sector.

What is the $Q$-value condition for $\beta^{-}$ decay in terms of atomic masses?

$Q_{\beta^{-}} = [M(Z,A) - M(Z+1,A)]c^{2} > 0$, using atomic masses (electron masses cancel automatically for $\beta^{-}$). Decay occurs only when $Q > 0$.

What are electromagnetic transitions (gamma decay) in nuclei?

A nucleus in an excited state de-excites to a lower state by emitting a photon (gamma ray): $\ce{^{A}_{Z}X^{*} -> ^{A}_{Z}X + \gamma}$. The photon energy equals the difference between nuclear energy levels; $A$ and $Z$ are unchanged.

State the selection rules for the multipolarity of a nuclear electromagnetic transition.

For a transition between states $J_i$ and $J_f$, the photon multipole order $L$ satisfies $|J_i - J_f| \leq L \leq J_i + J_f$ (with $L \geq 1$). Parity change determines electric vs magnetic: $E_L$ has $\Delta\pi = (-1)^{L}$, $M_L$ has $\Delta\pi = (-1)^{L+1}$.

What is internal conversion as an alternative to gamma emission?

The nuclear de-excitation energy is transferred directly to an atomic electron, ejecting it instead of emitting a photon. It competes with gamma decay, especially for high-$Z$ nuclei and low transition energies; a $0^{+} \to 0^{+}$ transition can only proceed this way.

What is the formula relating nuclear magnetic moment to angular momentum?

$\vec{\mu} = g\,\mu_{N}\dfrac{\vec{J}}{\hbar}$, where $\mu_{N} = \dfrac{e\hbar}{2m_{p}}$ is the nuclear magneton and $g$ is the nuclear g-factor. The nuclear magneton is about $1836$ times smaller than the Bohr magneton.

What does a nonzero nuclear electric quadrupole moment indicate about nuclear shape?

It indicates a non-spherical charge distribution: a positive quadrupole moment means a prolate (cigar-shaped) nucleus, negative means oblate (disk-shaped), and zero means spherical.

State the Rutherford scattering differential cross-section formula.

$$\frac{d\sigma}{d\Omega} = \left(\frac{Z_{1}Z_{2}e^{2}}{16\pi\varepsilon_{0}E}\right)^{2}\frac{1}{\sin^{4}(\theta/2)}$$ where $E$ is the projectile kinetic energy and $\theta$ the scattering angle.

What did the Rutherford scattering experiment (Geiger-Marsden) establish about the atom?

The large-angle back-scattering of alpha particles showed that atomic positive charge and mass are concentrated in a tiny dense nucleus, refuting Thomson's plum-pudding model and establishing the nuclear model of the atom.

How does the Rutherford cross-section depend on scattering angle and on incident energy?

It scales as $\dfrac{1}{\sin^{4}(\theta/2)}$ (strongly peaked at small angles) and as $\dfrac{1}{E^{2}}$ (more deflection at lower energy). Deviations at large angles/high energy reveal the finite nuclear size and strong-force effects.

List the additive quantum numbers conserved in all interactions (strong, EM, weak).

Electric charge $Q$, baryon number $B$, lepton number $L$ (and individual lepton flavors except in oscillations), and energy-momentum and angular momentum. Color charge is also conserved.

Which symmetries/quantum numbers are conserved by the strong and electromagnetic interactions but violated by the weak interaction?

Parity ($P$), charge conjugation ($C$), strangeness, charm, and other quark flavors, and isospin (strong only). The weak interaction violates $P$, $C$, flavor, and even $CP$ (in some processes).

State the Gell-Mann-Nishijima formula relating charge to other quantum numbers.

$$Q = I_{3} + \frac{Y}{2} = I_{3} + \frac{B + S + C + B' + T}{2}$$ where $I_3$ is the third isospin component and $Y$ the hypercharge (sum of baryon number and flavor quantum numbers).

What distinguishes the three classes of elementary particles: leptons, hadrons, and gauge bosons?

Leptons ($e, \mu, \tau$ and neutrinos) feel weak and (if charged) EM forces but not the strong force. Hadrons (baryons and mesons) are made of quarks and feel the strong force. Gauge bosons ($\gamma, W^{\pm}, Z^{0}, g$) mediate the forces.

What is a baryon, and what is its quark content and baryon number?

A baryon is a hadron made of three quarks ($qqq$), e.g. the proton ($uud$) and neutron ($udd$). Each baryon has baryon number $B = +1$ (antibaryons $B = -1$) and half-integer spin, so baryons are fermions.

What is a meson, and what is its quark content and statistics?

A meson is a hadron made of a quark-antiquark pair ($q\bar{q}$), e.g. the pion ($\pi^{+} = u\bar{d}$). Mesons have baryon number $B = 0$ and integer spin, so they are bosons.

Name the six quark flavors and their electric charges.

Up ($+\tfrac{2}{3}$), charm ($+\tfrac{2}{3}$), top ($+\tfrac{2}{3}$); down ($-\tfrac{1}{3}$), strange ($-\tfrac{1}{3}$), bottom ($-\tfrac{1}{3}$). They come in three generations and carry color charge.

What are the properties of the photon as a gauge boson?

The photon is the gauge boson of electromagnetism: massless, electrically neutral, spin-1, travels at $c$, and has infinite range. It couples to electric charge and is its own antiparticle.

Which particle did Yukawa predict to mediate the nuclear force, and what is it?

The pion (pi meson), $\pi^{+}, \pi^{0}, \pi^{-}$, with mass $\approx 140\ \text{MeV}/c^{2}$. Yukawa related the meson mass to the force range via $R \approx \hbar/(m_{\pi}c) \approx 1.4\ \text{fm}$.

What is the relationship between the range of a force and the mass of its mediating particle (Yukawa)?

$R \approx \dfrac{\hbar}{m c}$. A massless mediator (photon) gives infinite range; a massive mediator gives short range. This is why the weak force ($W, Z$ massive) is short-ranged and EM ($\gamma$ massless) is long-ranged.

What are the four fundamental interactions in order of relative strength?

Strong (strength $\sim 1$), electromagnetic ($\sim 10^{-2}$), weak ($\sim 10^{-6}$), and gravitational ($\sim 10^{-39}$). Their mediators are gluons, photons, $W^{\pm}/Z^{0}$ bosons, and (hypothetically) gravitons.

What is the working principle of a cyclotron?

Charged particles spiral in a constant magnetic field while an alternating voltage across the dees accelerates them each half-revolution. The cyclotron frequency $f = \dfrac{qB}{2\pi m}$ is constant for non-relativistic particles, keeping the accelerating field in resonance.

Why must a synchrotron vary its magnetic field, unlike a cyclotron?

Because at relativistic speeds the particle mass increases ($m = \gamma m_{0}$), so to keep particles on a fixed-radius orbit the magnetic field and RF frequency must increase synchronously as the energy rises.

How does a scintillation detector detect radiation?

Ionizing radiation excites atoms in a scintillator material, which emit flashes of light upon de-excitation. A photomultiplier tube converts these light flashes into an amplified electrical pulse proportional to the deposited energy.

How does a Geiger-Muller counter operate, and what is its main limitation?

Incoming radiation ionizes the gas; the high voltage causes a Townsend avalanche producing a large pulse for each event. It detects individual particles but gives no energy information (all pulses are the same size) and has appreciable dead time.

Compare the proton and neutron: charge, mass, and stability.

Proton: charge $+e$, mass $\approx 938.3\ \text{MeV}/c^{2}$, stable (or lifetime $> 10^{34}$ yr). Neutron: charge $0$, mass $\approx 939.6\ \text{MeV}/c^{2}$ (slightly heavier), free neutron is unstable with mean lifetime $\approx 880\ \text{s}$, decaying by $\beta^{-}$.