Nuclei - Class 12

A hypothetical spherical celestial body is composed of $A$ identical particles, each of mass $m_0$. The density of the body varies with the radial distance $r$ from its center as $\rho(r) = k r$, where $k$ is a universal constant for this type of matter. A reference body made of this exact same matter containing $A_0$ particles has a known radius $R_0$. What is the magnitude of the gravitational self-energy of the body containing $A$ particles? (Assume $G$ is the universal gravitational constant).

A fully stripped heavy ion of atomic number $Z$ and mass number $A$ moves with a constant highly relativistic velocity $v$ (Lorentz factor $\gamma = (1 - v^2/c^2)^{-1/2} \gg 1$) along the $x$-axis. It passes by a stationary proton (charge $e$, mass $m_p$) located at the origin. The impact parameter is such that the surface of the ion just grazes the proton at the point of closest approach. The rest-frame radius of the ion is given by $R = R_0 A^{1/3}$. Assuming the proton's displacement during the interaction is negligible, what is the magnitude of the total momentum transferred to the proton?

A rigid, heavy deformed nucleus of mass $M$ is initially at rest. Its moment of inertia about an axis passing through its center of mass and perpendicular to the scattering plane is $I_0$. The surface of the nucleus in the scattering plane is given by the polar equation $r(\theta) = R_0(1 + \beta \cos 2\theta)$, where $\beta \ll 1$ is a dimensionless deformation parameter, and $\theta$ is the polar angle. An alpha particle of mass $m$ is fired along the radial line $\theta = \pi/4$ towards the center of the nucleus with an initial speed $v_0$. It strikes the frictionless surface of the nucleus. Assuming the collision is perfectly elastic, what is the angular velocity $\omega$ acquired by the nucleus after the collision? (Assume $I_0 \gg m R_0^2$ such that the rotational recoil energy of the nucleus is negligible, and neglect translational recoil).

While standard nuclear models often assume density invariance (uniform density $\rho \approx \text{constant}$), a more refined classical model of a heavy nucleus accounts for a diffuse surface. Consider a spherical nucleus of total mass $M$ and radius $R$ whose mass density varies radially as $\rho(r) = \rho_0 \left(1 - \frac{r^2}{R^2}\right)$ for $r \le R$, where $\rho_0$ is the central density. What is the magnitude of the gravitational self-energy of this nucleus? (Assume $G$ is the universal gravitational constant).

Consider a hypothetical heavy nucleus of mass number $A$ and atomic number $Z$, modeled classically as a rigid solid sphere of uniform mass density and uniform charge density. The nucleus rotates about its diametric axis with a constant angular velocity $\omega$. Given that the nuclear radius follows the empirical relation $R = R_0 A^{1/3}$ (where $R_0$ is a constant) and the mass of a single nucleon is $m_p$, which of the following expressions correctly represents the magnitude of the magnetic dipole moment $\mu$ of this rotating nucleus?

In the semi-empirical Liquid Drop Model, atomic nuclei are treated as incompressible spherical droplets. Consider two nuclei, Aluminum ($^{27}\text{Al}$) and Tellurium ($^{125}\text{Te}$), completely submerged in a hypothetical, non-viscous 'pion fluid' of uniform density $\rho_\pi = 1.0 \times 10^{17} \text{ kg/m}^3$. Both nuclei are released from rest and subjected to a uniform strong-field acceleration $g_s = 10^{20} \text{ m/s}^2$. Assume the nuclear density $\rho_0$ is invariant for all nuclei and equals $2.3 \times 10^{17} \text{ kg/m}^3$. Let $a_{\text{Al}}$ and $a_{\text{Te}}$ be their respective initial accelerations, and $F_{\text{Al}}$ and $F_{\text{Te}}$ be the magnitudes of the buoyant forces exerted on them by the pion fluid. Which of the following represents the correct ratios $a_{\text{Al}} : a_{\text{Te}}$ and $F_{\text{Al}} : F_{\text{Te}}$?

A heavy nucleus of atomic number $Z$ and radius $R$ is modeled as a non-conducting solid sphere with a uniform volume charge density $\rho = \frac{3Ze}{4\pi R^3}$, where $e$ is the elementary charge. Suppose a spherical cavity of radius $R/2$ is formed within this nucleus. The cavity touches the outer surface of the nucleus, and its center is located at a distance of $R/2$ from the center of the original nucleus. What is the absolute electrostatic potential at the center of the original nucleus? (Assume the potential at infinity is zero, and $\epsilon_0$ is the vacuum permittivity).

A hypothetical superheavy nucleus of initial mass number $A_0$ undergoes a continuous decay process where it loses nucleons. The decay rate is governed by $\frac{dA}{dt} = -\lambda A$, where the decay parameter $\lambda$ is not a constant. Instead, $\lambda$ depends on the instantaneous surface area $S$ and volume $V$ of the nucleus as $\lambda = c \frac{S}{V}$, where $c$ is a positive constant with dimensions of $[L T^{-1}]$. Assuming the strong nuclear force maintains a strictly invariant nuclear density $\rho_n$ (nucleons per unit volume) throughout the decay process, what is the time $T$ required for the nucleus to completely decay ($A \to 0$)?

A heavy nucleus of mass $M$ initially at rest in the laboratory frame undergoes an isotropic shattering process. It breaks into a continuous distribution of fragments, converting a total mass defect $\Delta m$ into the kinetic energy of the fragments. The mass of the fragments $dm$ having a velocity magnitude between $v$ and $v+dv$ is given by the distribution $dm = k v^2 (v_0 - v) dv$ for $0 \le v \le v_0$, where $v_0$ is the maximum velocity of any fragment and $k$ is a positive constant. Assuming $\Delta m \ll M$ and that all fragment velocities are non-relativistic, what is the maximum velocity $v_0$ of the fragments? (Here $c$ is the speed of light in vacuum).

A sealed capsule of volume $V$ contains $N$ nuclei of a specific isotope. The total mass of the empty capsule plus the sum of the rest masses of all individual constituent nucleons of these $N$ nuclei is $M_0$. The capsule is fully submerged in an incompressible liquid of density $\rho$ inside a tank. The tank undergoes a constant horizontal acceleration $a = \sqrt{3}g$, where $g$ is the acceleration due to gravity. An external agent moves the capsule quasi-statically through a distance $L$ exactly along the direction of the effective gravitational field (deeper into the liquid). If the work done by the external agent is $W$, which of the following expressions correctly represents the total binding energy $E_B$ of the $N$ nuclei? (Let $c$ be the speed of light in vacuum, and assume $\rho V > M_0$).

A uniform conducting rod of mass $M$ (kg) and length $L$ (m) is pivoted at one end and rotates in a horizontal plane. Its free end slides on a frictionless, perfectly conducting circular ring of radius $L$. A uniform vertical magnetic field $B$ (T) is applied perpendicular to the plane of rotation. A stationary resistor of resistance $R$ ($\Omega$) connects the pivot and the ring. The rod is composed of a radioactive material with a decay constant $\lambda$ (s$^{-1}$). The energy released from the radioactive decay is continuously and entirely converted into mechanical power that drives the rotation of the rod. The initial total available decay energy if all nuclei were to decay is $E_0$ (J). Assuming the mass $M$ of the rod remains practically constant, and the rod starts from rest at $t=0$, at what time $t_m$ (s) will the angular velocity of the rod reach its maximum value?

A radioactive sample initially contains $N_0$ nuclei of isotope $X$. It undergoes a successive decay chain $X \xrightarrow{\lambda_1} Y \xrightarrow{\lambda_2} Z$, where $Z$ is stable. The total mass defect for the complete transformation of one nucleus of $X$ to $Z$ is $\Delta m$.

To model the population of $Y$, a physicist constructs a source-free series L-C-R circuit. The charge $q(t)$ on the capacitor is scaled such that $q(t) = \alpha N_Y(t)$, where $\alpha$ is a constant with units of Coulombs. At $t=0$, the capacitor is uncharged. The circuit parameters $R$, $L$, and $C$ are tuned such that the differential equation governing $q(t)$ perfectly matches the second-order differential equation governing $N_Y(t)$ for all $t \ge 0$.

If the total electrical energy dissipated as heat in the resistor over $t \in [0, \infty)$ is exactly equal to the total rest mass energy released by the complete decay of the initial sample, which of the following expressions gives the required resistance $R$? (Here $c$ is the speed of light in vacuum).

A parent nucleus ${}^{A}_{Z}X$ at rest undergoes alpha decay to form a daughter nucleus ${}^{A-4}_{Z-2}Y$ and an alpha particle ${}^{4}_{2}\text{He}$. The total mass-energy released in the decay is $Q$. The nuclear radius is given by $R = R_0 A^{1/3}$, where $A$ is the mass number and $R_0$ is a constant. Let $M_Y$ and $M_\alpha$ be the exact masses of the daughter nucleus and the alpha particle, respectively. Assuming the daughter nucleus and alpha particle are spherical and just touching at the exact moment of creation, what is the kinetic energy of the alpha particle at this specific instant? (Let $k = 1/(4\pi\epsilon_0)$ and $e$ be the elementary charge. Assume no gamma emission occurs.)

A stationary unstable particle of rest mass $M = 5m$ decays into two identical particles, each of rest mass $m$. One of these ejected particles is collimated and directed towards a double-slit apparatus with a slit separation $d$. An interference pattern is observed on a screen placed at a distance $D$ ($D \gg d$) from the slits. What is the fringe width $\beta$ of the resulting interference pattern? (Assume $h$ is Planck's constant and $c$ is the speed of light in vacuum).

A fixed ideal electric dipole of dipole moment $\vec{p} = p \hat{i}$ is located at the origin of a coordinate system. A stationary radioactive nucleus is located at point $A$ with polar coordinates $(r, \pi)$ in the $xy$-plane, where the polar angle $\theta$ is measured from the positive $x$-axis. The nucleus undergoes a reaction, emitting a charged fragment of mass $m$ and quantized charge $q = +ne$ (where $e$ is the elementary charge and $n$ is a positive integer). The heavy residual nucleus remains fixed at $A$ and is uncharged. The $Q$-value of the reaction is entirely converted into the initial kinetic energy of the emitted fragment. The fragment is constrained to move along a fixed, frictionless, non-conducting circular track of radius $r$ centered at the origin. It just reaches point $B$ with polar coordinates $(r, \pi/3)$ in the $xy$-plane, where its kinetic energy momentarily becomes zero. Assuming the only force doing work on the fragment is the electrostatic force from the dipole, which of the following expressions gives the correct value of the integer $n$?

An elevator of height $h$ is accelerating upwards with a constant acceleration $a$ in a region of uniform downward gravitational field $g$. A massive nucleus, initially at rest on the floor of the elevator, undergoes a nuclear transition. It releases an internal energy $Q$ by emitting a single gamma-ray photon vertically upwards. Assume the mass of the nucleus is sufficiently large such that its recoil kinetic energy is negligible. The photon is subsequently absorbed by a perfectly absorbing detector fixed to the ceiling of the elevator. What is the magnitude of the impulse imparted to the detector by the photon, as measured in the reference frame of the elevator? (Let $c$ be the speed of light in vacuum, and assume $(g+a)h \ll c^2$).

A stationary nucleus in free space undergoes an annihilation event, releasing a total energy $Q$ instantaneously as an isotropic burst of gamma radiation. A perfectly absorbing, rigid, flat circular disk of mass $M$ and radius $R$ is positioned such that its center is at a distance $d$ from the nucleus. The central axis of the disk passes through the nucleus. Assuming the disk is initially at rest and free to move, what is the recoil speed $v$ of the disk immediately after the radiation is completely absorbed? (Assume $v \ll c$, where $c$ is the speed of light in vacuum, and ignore any gravitational effects).

A radioactive sample of instantaneous mass $m$ is attached to a light horizontal spring of spring constant $k$. The system oscillates on a smooth horizontal surface in a viscous medium that exerts a drag force $F_d = -cv$, where $v$ is the instantaneous velocity of the sample and $c$ is a positive constant. The sample undergoes radioactive decay with a decay constant $\lambda$, such that its mass decreases according to $\frac{dm}{dt} = -\lambda m$. The decay products are continuously ejected such that they are left completely stationary with respect to the horizontal surface. If $E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$ represents the instantaneous mechanical energy of the oscillating sample, which of the following expressions gives the rate of change of this mechanical energy, $\frac{dE}{dt}$?

A non-conducting solid sphere of radius $R$ is centered at the origin. It contains a radioactive isotope such that its volume charge density is given by $\rho(r, t) = \rho_0 \left(\frac{r}{R}\right) e^{-\lambda t}$, where $r$ is the distance from the center, $t$ is the time, and $\lambda$ is the decay constant. A point charge $q$ is moved from the center of the sphere ($r=0$) starting at $t=0$ to its surface ($r=R$) along a radial line at a constant speed $v = \lambda R$. What is the work done by the electric field on the point charge $q$ during this process?

A radioactive source containing $N_0$ nuclei with a decay constant $\lambda$ is placed at the origin. It undergoes $\beta^-$ decay, emitting electrons of charge $-e$. Through a constant external magnetic confinement mechanism, all emitted electrons are immediately captured into a single circular orbit of radius $R$ centered at the origin. The electrons continue to orbit indefinitely with a constant speed $v$. A small stationary conducting loop of radius $r$ ($r \ll R$) is placed at the origin, coplanar with the electron orbit. Assuming the external confinement field does not change with time, which of the following expressions represents the magnitude of the induced non-conservative electric field at the rim of the small loop as a function of time $t$?

A sample contains $N_0$ identical hypothetical heavy nuclei, each initially having a mass number $A_0$. These nuclei undergo a continuous decay process by emitting neutrons. The binding energy per nucleon for these nuclei is given by the functional approximation $B(A) = \alpha - \beta A$, where $\alpha$ and $\beta$ are positive constants, and $A$ is the instantaneous mass number. The energy released per unit decrease in mass number is defined as the instantaneous Q-value, $Q(A)$. The decay process is statistical but can be modeled continuously such that the rate of decrease of mass number for each nucleus is directly proportional to its instantaneous Q-value, given by $\frac{dA}{dt} = -k Q(A)$ (where $k$ is a positive constant). Assuming $A_0 > \frac{\alpha}{2\beta}$, which of the following expressions represents the total power $P(t)$ released by the sample as a function of time $t$?

A specialized radioisotope thermoelectric generator (RTG) is powered by a successive radioactive decay chain $A \xrightarrow{\lambda_1} B \xrightarrow{\lambda_2} C$ (stable). At time $t=0$, the RTG contains $N_0$ nuclei of $A$ and zero nuclei of $B$ and $C$. The decay constants are $\lambda_1 = \lambda$ and $\lambda_2 = 2\lambda$.

The RTG is modeled as a time-varying ideal voltage source $E(t)$ in series with a time-varying internal resistance $r(t)$. The electromotive force is directly proportional to the instantaneous number of $B$ nuclei: $E(t) = E_0 \frac{N_B(t)}{N_0}$. The internal resistance is directly proportional to the instantaneous number of $A$ nuclei: $r(t) = R_0 \frac{N_A(t)}{N_0}$.

An external load resistance $R$ is connected to the RTG. An onboard controller continuously adjusts $R$ to extract the maximum possible electrical power at every instant $t$. What is the absolute maximum power $P_{max}$ extracted by the load during the entire operational lifetime of the RTG? (Assume $E_0$ and $R_0$ are positive constants with standard SI units).

A toroidal coil of rectangular cross-section has an inner radius $a$, outer radius $b$, height $h$, and a total of $N$ turns. A thin, straight capillary tube passes exactly along the axis of symmetry of the toroid. The tube contains a sample of a radioactive isotope that undergoes alpha decay, emitting $\alpha$ particles (each with charge $+2e$). At time $t=0$, the number of radioactive nuclei is $N_0$, and the decay constant of the isotope is $\lambda$. An external mechanism ensures that all emitted $\alpha$ particles are immediately swept along the tube in the same direction, forming a continuous narrow beam. Assume the transit time of the particles through the toroid is negligible, so the beam acts as a long straight wire carrying a spatially uniform, time-varying current. What is the magnitude of the maximum induced electromotive force (EMF) in the toroidal coil? (Assume all quantities are in SI units and $\mu_0$ is the permeability of free space)

A beam of unstable particles, each of mass $m$ (in $\text{kg}$) and initial velocity $v_0$ (in $\text{m s}^{-1}$), enters a gaseous medium at $t=0$. The particles undergo radioactive decay with a decay constant $\lambda$ (in $\text{s}^{-1}$). The medium has a uniform number density $n$ (in $\text{m}^{-3}$) and the interaction cross-section between the particles and the medium molecules is $\sigma$ (in $\text{m}^2$). As the particles traverse the medium, they experience a viscous drag force $F = -bv$, where $v$ is the instantaneous velocity and $b$ (in $\text{kg s}^{-1}$) is a positive constant. Given that the decay constant is exactly equal to the drag coefficient per unit mass ($\lambda = b/m$), what is the total probability $P$ that a particle will undergo a collision with a medium molecule before it decays? Let the dimensionless parameter $\beta = \frac{n \sigma m v_0}{b}$.

A parallel plate capacitor of vacuum capacitance $C_0$ is filled with a radioactive gas. The gas acts as a dielectric whose relative permittivity varies with time as $K(t) = 1 + \beta A(t)$, where $A(t)$ is the instantaneous radioactive activity of the gas and $\beta$ is a positive constant. The initial activity of the gas at $t=0$ is $A_0$ and its decay constant is $\lambda$. The capacitor is connected across an ideal DC voltage source $V_0$. The charging current flows to the capacitor through a very long straight wire. A square conducting loop of side length $a$ is placed in the same plane as the long straight wire, with its nearest edge parallel to the wire at a distance $a$. What is the magnitude of the induced EMF $\varepsilon$ in the square loop as a function of time $t$?

In a geological rock sample, the ratio of the number of stable daughter nuclei ($N_d$) to the number of radioactive parent nuclei ($N_p$) is measured to be $R = \frac{N_d}{N_p} = 15.0 \pm 0.48$. It is assumed that the sample initially contained no daughter nuclei and that all daughter nuclei present are solely due to the decay of the parent nuclei. If the decay constant $\lambda$ of the parent nuclei is known exactly, what is the percentage error in the estimated age $t$ of the rock sample?

In a phenomenological model of a heavy nucleus of radius $R$, the effective potential energy of a single nucleon at a distance $r$ from the center ($0 < r \le R$) is given by: $U(r) = -U_0 \left[ 1 - \left(\frac{r}{R}\right)^2 \right] + \frac{k}{r^2}$ where $U_0$ and $k$ are positive constants. The first term represents the attractive mean field of the saturated nuclear matter, and the second term represents a hard-core repulsion. A nucleon is initially at its stable equilibrium position $r_0$ inside the nucleus. What is the minimum external work required to move this nucleon slowly to the nuclear surface ($r = R$)? Assume the nucleus remains otherwise unperturbed.

In a theoretical model of nuclear decay, a stationary heavy nucleus emits a nucleon. The total energy released due to the mass defect is $Q = 15.00 \text{ MeV}$. However, before emission, the nucleon must be localized within a surface layer of thickness $\Delta x = 1.25 \text{ fm}$ due to the short-range saturation of the strong nuclear force. The kinetic energy required for this localization is provided entirely by the decay energy $Q$. Assuming the daughter nucleus is infinitely massive and using the minimum uncertainty $\Delta x \Delta p = \hbar/2$, calculate the final kinetic energy of the emitted nucleon far from the nucleus. (Given: $\hbar c = 200 \text{ MeV fm}$, rest mass energy of the nucleon $mc^2 = 1000 \text{ MeV}$).

A stationary parent nucleus undergoes beta decay into a daughter nucleus of mass $M$, an electron of mass $m$, and a massless antineutrino. The total energy released in the decay (Q-value) is $Q$. In a specific decay event, the electron is emitted with a kinetic energy $K$ (where $0 < K < Q$). Assume that the daughter nucleus and the electron are non-relativistic ($Q \ll mc^2$), and that $M$ is sufficiently large such that the recoil kinetic energy of the daughter nucleus is negligible in the energy conservation equation. Which of the following expressions gives the minimum possible recoil kinetic energy of the daughter nucleus? (Let $c$ be the speed of light in vacuum).

In a certain $\beta^{-}$ decay process, a parent nucleus at rest emits an electron and an antineutrino. Assume the daughter nucleus is infinitely heavy (negligible recoil) and the antineutrino is massless. The maximum kinetic energy of the emitted electron is observed to be $K_{\text{max}} = 4 m_0 c^2$, where $m_0$ is the rest mass of the electron and $c$ is the speed of light. According to Fermi's theory of beta decay, the transition rate $d\lambda$ into an electron momentum interval $dp$ is proportional to the available phase space volume of the electron-antineutrino pair. What is the ratio of the emission probability per unit momentum interval, $\frac{d\lambda}{dp}$, for an electron emitted with total relativistic energy $E_1 = 3 m_0 c^2$ to that for an electron emitted with total relativistic energy $E_2 = 2 m_0 c^2$?

A system of mass $6m$ is initially at rest in the laboratory frame. It explodes into three fragments $A$, $B$, and $C$ of masses $m$, $2m$, and $3m$ respectively. The total kinetic energy released in the explosion is $Q$. If fragment $C$ is observed to have a kinetic energy of $\frac{Q}{6}$, what is the maximum possible kinetic energy of fragment $A$ in the laboratory frame?

In a theoretical statistical model of beta decay, the probability density of emitting a non-relativistic beta particle of mass $m$ with kinetic energy $E$ is given by $\rho_c(E) = A E \left(1 - \frac{E}{Q}\right)^n$, where $Q$ is the endpoint kinetic energy, $A$ is a normalization constant, and $n$ is a very large integer ($n \gg 1$) representing the phase space degrees of freedom.

To account for the wave nature of the beta particle, a quantum suppression factor $S(E)$ is introduced. $S(E)$ is defined as the ratio of the nuclear volume $V_0$ to the spherical volume $V_\lambda$ associated with the electron's de Broglie wavelength $\lambda_e$ (i.e., $V_\lambda = \frac{4}{3}\pi \lambda_e^3$).

The modified quantum probability density is $\rho_q(E) = B \rho_c(E) S(E)$, where $B$ is another normalization constant. Let $E_c$ and $E_q$ be the most probable kinetic energies of the beta particle in the classical and quantum models, respectively. What is the limiting value of the ratio $\frac{E_q}{E_c}$ for $n \to \infty$?

A hypothetical super-heavy nucleus of mass $M = 2 \sqrt{\frac{hc}{G}}$ undergoes fission into two identical fragments. The mass defect $\Delta m$ of this process is given by $\Delta m = \frac{3}{5} \frac{G M^2}{R c^2}$, where the nucleus radius is $R = \frac{h}{2\pi M c}$. Here $h$ is the Planck constant, $c$ is the speed of light, and $G$ is the universal gravitational constant.

A new system of units is proposed where the fundamental base units are:

1. Action: $A_0 = h$
2. Power: $P_0 = \frac{c^5}{2G}$
3. Force: $F_0 = \frac{c^4}{G}$

What is the numerical value of the energy released in the fission process in this new system of units?

A stationary nucleus of mass $2m$ undergoes nuclear fission into two identical fragments, each of mass $m$. The total mass defect of the reaction corresponds to an energy equivalent $Q$. During the fission process, a gamma-ray photon of energy $E_\gamma$ is emitted. Assume the momentum carried by the gamma-ray photon is negligible. One of the fission fragments enters a medium where it experiences a velocity-dependent drag force of magnitude $F_d = kv^2$ (where $k$ is a positive constant and $v$ is the instantaneous velocity) in addition to a constant kinetic friction force of magnitude $f_k = \mu mg$, where $\mu$ is the coefficient of kinetic friction and $g$ is the acceleration due to gravity. Which of the following expressions gives the total distance $x$ traveled by this fragment before coming to rest?

A stationary nucleus undergoes nuclear fission, releasing a total kinetic energy $Q$ and splitting into two fragments of masses $m$ and $2m$. The lighter fragment, carrying a charge $q$, is directed towards a massive, stationary target nucleus of charge $Q_t$. Assume the target nucleus is infinitely massive and fixed in space. If the impact parameter of the lighter fragment is $b = \frac{3\sqrt{2} q Q_t}{8 \pi \epsilon_0 Q}$, what is the distance of closest approach between the lighter fragment and the target nucleus?

A projectile nucleus of mass number $A_1 = 8$ and atomic number $Z_1 = 4$ is fired from infinity with an initial kinetic energy $K$ towards a stationary, free-to-move target nucleus of mass number $A_2 = 27$ and atomic number $Z_2 = 13$. The strong nuclear force is assumed to have a negligible range beyond the nuclear surface. Assuming the nuclear radius follows the relation $R = R_0 A^{1/3}$ (where $R_0$ is a constant) and the masses of the nuclei are exactly proportional to their mass numbers, what is the minimum value of $K$ required for the nuclei to just touch each other, thereby allowing the strong nuclear force to initiate a reaction? (Let $k = \frac{1}{4\pi\epsilon_0}$ and $e$ be the elementary charge).

A spacecraft of mass $M$ (in $\text{kg}$) is traveling in deep space along the positive $x$-direction with a velocity $v_0$ (in $\text{m/s}$). It is propelled by a rocket engine that ejects exhaust gas at a constant mass rate $\mu = -\frac{dM}{dt}$ (in $\text{kg/s}$) with a constant relative speed $u$ (in $\text{m/s}$) in the negative $x$-direction. At $t=0$, the spacecraft encounters a localized 'fusion barrier' that exerts an instantaneous repulsive impulse of magnitude $J$ (in $\text{N}\cdot\text{s}$) in the negative $x$-direction. Assuming the mass $M$ remains constant during the infinitesimally short duration of the impulse, what is the instantaneous change in the rate of change of the spacecraft's kinetic energy, $\Delta \left( \frac{dK}{dt} \right) = \left. \frac{dK}{dt} \right|_{t=0^+} - \left. \frac{dK}{dt} \right|_{t=0^-}$?

A beam of particles, each of mass $m$ and kinetic energy $E$, is incident from the left ($x < 0$) on a one-dimensional potential step defined by $V(x) = 0$ for $x < 0$ and $V(x) = 2E$ for $x \ge 0$. The incident wavefunction is $\psi_{\text{inc}}(x) = A e^{ikx}$ and the reflected wavefunction is $\psi_{\text{ref}}(x) = B e^{-ikx}$. The reflection amplitude ratio is $r = B/A = e^{i\delta}$, where $\delta \in (-\pi, \pi]$ is the phase shift upon reflection. In the classically forbidden region ($x \ge 0$), the probability density decays exponentially. Let $\Delta x$ be the penetration depth, defined as the distance over which the probability density decreases by a factor of $1/e^2$. Using the minimum uncertainty relation $\Delta x \Delta p = \hbar/2$, determine the phase shift $\delta$ and the minimum momentum uncertainty $\Delta p$ in the region $x \ge 0$.

Consider a hypothetical heavy nucleus of mass number $A$ and atomic number $Z$. Instead of a uniform charge distribution, assume the proton charge density within the nucleus varies radially as $\rho(r) = k r^2$ for $r \le R$, and $\rho(r) = 0$ for $r > R$, where $k$ is a positive constant and $r$ is the radial distance from the center. The nuclear radius scales with mass number as $R = R_0 A^{1/3}$, where $R_0$ is a constant. Calculate the electrostatic self-energy $U$ of this nucleus. (Let $e$ be the elementary charge and $\epsilon_0$ be the vacuum permittivity).

According to the Liquid Drop Model, a heavy nucleus of mass number $A$ and atomic number $Z$ can be modeled as a spherical drop of incompressible nuclear fluid of radius $R = R_0 A^{1/3}$. The strong nuclear force creates an effective surface tension $\gamma$ and an internal pressure $P_{\text{int}}$ to maintain mechanical equilibrium. The protons are assumed to be uniformly distributed throughout the nuclear volume, creating an electrostatic Coulomb energy $E_c = \frac{3}{5} \frac{(Ze)^2}{4\pi \epsilon_0 R}$. Which of the following expressions gives the correct internal pressure $P_{\text{int}}$ required to maintain mechanical equilibrium of the nucleus?

Consider a muonic atom where an electron is replaced by a muon of mass $m_\mu$ and charge $-e$. The nucleus has mass $M_N$, atomic number $Z$, and is modeled as a uniformly charged sphere of radius $R$. For a sufficiently heavy nucleus, the muon's lowest energy orbits lie entirely inside the nuclear volume. Assuming the Bohr quantization rule for angular momentum holds, which of the following expressions gives the total energy $E_n$ of the muon in its $n^{\text{th}}$ orbit?

(Assume the orbit is entirely inside the nucleus, $\hbar = \frac{h}{2\pi}$, and $\epsilon_0$ is the vacuum permittivity.)

In a realistic model of a heavy nucleus of mass number $A$, the nucleon number density $\rho(r)$ is not perfectly uniform. It is modeled as being constant at $\rho_0$ from the center up to a radial distance $R - t$, and then decreasing linearly to zero at the geometric surface radius $R$. Here, $t$ is a constant 'skin thickness' ($t \ll R$) arising from the finite range of the strong nuclear force, and is independent of $A$. Let $R_0$ be the standard nuclear radius parameter defined by the relation $\rho_0 = \left(\frac{4}{3}\pi R_0^3\right)^{-1}$. Which of the following expressions best represents the geometric surface radius $R$ of the nucleus as a function of $A$, to the first order in the small parameter $(t/R)$?

In a hypothetical model of dense nuclear matter, a nucleon of rest mass $m$ is confined within a nucleus of mass number $A$. The nuclear radius follows the standard empirical relation $R = R_0 A^{1/3}$, where $R_0$ is a constant. Due to strong short-range correlations, the nucleon is localized to a small sub-region such that its position uncertainty is $\Delta x = \frac{R}{A}$. The corresponding momentum uncertainty is given by the Heisenberg-like relation $\Delta p \approx \frac{\hbar}{\Delta x}$. Let $K$ be the minimum kinetic energy of the nucleon associated with this confinement. Which of the following correctly describes the scaling of $K$ with $A$ in the limits of small $A$ (where the nucleon is non-relativistic) and very large $A$ (where the nucleon becomes ultra-relativistic)?

In a theoretical liquid-drop model, the binding energy per nucleon $B$ of a nucleus with mass number $A$ depends on a dimensionless deformation parameter $x$ as $B(x) = B_0 - \alpha x^2 + \beta x^3$ where $B_0$, $\alpha$, and $\beta$ are positive constants in $\text{MeV}$. The state $x=0$ corresponds to the stable spherical equilibrium of the nucleus. To induce nuclear fission, the nucleus must be deformed beyond its restoring stability limit. What is the minimum total kinetic energy $K_{\text{min}}$ that must be imparted to the deformation mode of the nucleus at $x=0$ to cause it to undergo fission?

An intrinsic semiconductor sample with an intrinsic carrier concentration $n_i = 1.0 \times 10^{10} \text{ cm}^{-3}$ is illuminated by a short laser pulse. This generates an initial excess electron-hole pair concentration of $\Delta n_0 = \Delta p_0 = 1.0 \times 10^{16} \text{ cm}^{-3}$. The recombination rate of electrons and holes is given by $R = rnp$, where $n$ and $p$ are the total electron and hole concentrations respectively, and the recombination rate constant is $r = 2.0 \times 10^{-10} \text{ cm}^3 \text{ s}^{-1}$. The thermal generation rate is constant at $G_{th} = r n_i^2$. If the laser is turned off at $t=0$, what is the time $t_{1/2}$ required for the excess carrier concentration to drop to half of its initial value? (Assume uniform generation and no drift or diffusion currents).

A block of mass $m$ is attached to a spring of force constant $k$ and undergoes lightly damped oscillations on a horizontal frictionless surface. The damping force is given by $F_d = -b(t)v$, where $v$ is the velocity of the block. The damping coefficient $b(t)$ is linked to the activity of a radioactive sample such that $b(t) = b_0 e^{-\lambda t}$, where $\lambda$ is the decay constant of the sample. Assume that the damping is very light ($b_0 \ll \sqrt{mk}$) and the radioactive decay is very slow compared to the oscillation frequency ($\lambda \ll \sqrt{k/m}$). If the initial amplitude of oscillation at $t=0$ is $A_0$, what is the final amplitude of the block as $t \to \infty$?

To determine the decay constant $\lambda$ of a radioactive isotope, an experiment is conducted using a detector with $100\%$ efficiency. At $t = 0$, the detector records $C_0$ counts in a small time interval $\Delta t$. At a later time $t = t_1$ (where $t_1 \gg \Delta t$), the detector records $C_1$ counts in an identical time interval $\Delta t$. Assume $\Delta t \ll 1/\lambda$ such that the activity remains constant during each measurement interval. If the statistical fluctuations in the recorded counts are the only source of uncertainty, which of the following expressions gives the relative statistical uncertainty $\frac{\sigma_\lambda}{\lambda}$ in the calculated decay constant?

A stationary, initially neutral radioactive sample is located at the origin of a vacuum chamber. At $t=0$, it contains $N_0$ radioactive nuclei and begins to decay with a decay constant $\lambda$. Each decay event produces a heavy daughter nucleus (which remains at rest at the origin) and emits a light particle of charge $+q$ that travels radially outward at a constant speed $v$. Consider a fixed spherical Gaussian surface of radius $R$ centered at the origin. What is the magnitude of the displacement current $I_d$ through this Gaussian surface at a time $t > \frac{R}{v}$?

A spherical black body of surface area $A$ contains a radioactive isotope that generates heat at a time-dependent rate $P(t) = P_0 e^{-\lambda t}$, where $P_0$ is the initial power and $\lambda$ is the decay constant. The sphere is located in deep space where the ambient temperature is $0\text{ K}$. An ideal Carnot engine operates between the sphere (acting as the hot reservoir) and a cold reservoir maintained at a constant temperature $T_c$.

The engine is designed to continuously extract a constant fraction $\alpha$ ($0 < \alpha < 1$) of the heat generated by the isotope, while the remaining $(1-\alpha)$ fraction of the generated heat is radiated away into space by the sphere. Assume the heat capacity of the sphere is negligible, so it is always in instantaneous thermal equilibrium. The engine operates until the temperature of the sphere drops to $T_c$.

Given that the initial temperature of the sphere is $T_0 = \left( \frac{(1-\alpha)P_0}{\sigma A} \right)^{1/4}$ (where $\sigma$ is the Stefan-Boltzmann constant), what is the total work $W$ done by the Carnot engine during its entire period of operation?

A mechanical oscillator consists of a block of mass $m$ (in kg) attached to a spring of force constant $k$ (in N/m). The block is rigidly connected to a rectangular conducting loop of width $L$ (in m) and total resistance $R$ (in $\Omega$). The loop moves in a uniform magnetic field $B$ (in T) perpendicular to its plane, such that only one side of length $L$ remains in the field. A radioactive sample with initial number of nuclei $N_0$ and decay constant $\lambda$ (in s$^{-1}$) is placed on the block. An external mechanism exerts a time-varying driving force on the block proportional to the instantaneous activity of the sample: $F_{ext}(t) = \alpha A(t)$, where $A(t) = -\frac{dN}{dt}$ is the activity and $\alpha$ (in N$\cdot$s) is a positive coupling constant. Assuming the system's displacement from equilibrium can be modeled with a particular solution of the form $x_p(t) = X_0 e^{-\lambda t}$, what is the amplitude coefficient $X_0$ (in m)?