Quiz — Chapter 8: Collective Motion: Vibrations, Rotations, and Nuclear Deformation

DataField.Dev

Quiz — Chapter 8: Collective Motion: Vibrations, Rotations, and Nuclear Deformation

Instructions: Select the best answer for each question. Unless otherwise stated, assume even-even nuclei.

Question 1

A nucleus has a first excited $2^+$ state at 800 keV, interpreted as a one-phonon quadrupole vibration. What energies and $J^\pi$ values does the harmonic vibrator model predict for the two-phonon states?

(A) $0^+$, $2^+$, $4^+$ at 1200 keV
(B) $0^+$, $1^+$, $2^+$, $3^+$, $4^+$ at 1600 keV
(C) $0^+$, $2^+$, $4^+$ at 1600 keV
(D) $2^+$, $4^+$ at 1600 keV

Question 2

The $J = 1$ and $J = 3$ states are absent from the two-phonon quadrupole multiplet because:

(A) They are forbidden by parity conservation
(B) Coupling two $J = 2$ angular momenta cannot yield $J = 1$ or $J = 3$
(C) Bose symmetry requires symmetric coupling, which excludes odd-$J$ states for two identical quadrupole phonons
(D) The Pauli exclusion principle forbids these states

Question 3

The parity of a one-phonon octupole ($\lambda = 3$) vibration is:

(A) Positive, because all vibrational excitations have positive parity
(B) Negative, because $(-1)^\lambda = (-1)^3 = -1$
(C) Positive, because the phonon consists of pairs of nucleons
(D) Undefined, because parity is not a good quantum number for collective states

Question 4

For a perfectly rigid axially symmetric rotor, the ratio $E(4^+)/E(2^+)$ equals:

(A) 2.00
(B) 2.50
(C) 3.00
(D) 3.33

Question 5

A well-deformed even-even nucleus has $E(2^+) = 75$ keV. The value of $\hbar^2/2\mathcal{J}$ is:

(A) 75 keV
(B) 37.5 keV
(C) 25 keV
(D) 12.5 keV

Question 6

In the ground-state rotational band ($K = 0$) of an axially symmetric even-even nucleus, which set of spin-parity values is allowed?

(A) $0^+, 1^+, 2^+, 3^+, 4^+, \ldots$
(B) $0^+, 2^+, 4^+, 6^+, 8^+, \ldots$
(C) $0^-, 2^-, 4^-, 6^-, 8^-, \ldots$
(D) $0^+, 2^-, 4^+, 6^-, 8^+, \ldots$ (alternating parity)

Question 7

Measured nuclear moments of inertia fall between the rigid-body and irrotational-flow values. The primary reason $\mathcal{J}_{\text{exp}}$ is less than $\mathcal{J}_{\text{rigid}}$ is:

(A) The nucleus is not actually deformed
(B) Relativistic corrections reduce the effective mass
(C) Pairing correlations suppress contributions to the moment of inertia
(D) The nuclear surface is not sharp

Question 8

Backbending in rotational nuclei refers to:

(A) A sudden decrease in excitation energy at high spin
(B) A sudden increase in the moment of inertia at a critical angular momentum, producing an S-shaped curve in a $2\mathcal{J}/\hbar^2$ vs. $(\hbar\omega)^2$ plot
(C) The reversal of the rotational direction at high spin
(D) The transition from prolate to oblate shape during rotation

Question 9

The physical mechanism responsible for backbending is:

(A) Fission of the nucleus at high spin
(B) The breaking and alignment of a pair of high-$j$ nucleons by the Coriolis force
(C) A shape transition from quadrupole to octupole deformation
(D) The emission of nucleons from the rotating nucleus

Question 10

Superdeformed bands, first discovered in $^{152}$Dy, correspond to nuclear shapes with an approximate axis ratio of:

(A) 1.2:1 (slightly deformed)
(B) 1.5:1 (moderately deformed)
(C) 2:1 (highly elongated)
(D) 3:1 (hyperdeformed)

Question 11

In the Interacting Boson Model (IBA), the collective degrees of freedom are represented by:

(A) Quadrupole phonons only
(B) $s$ bosons ($J = 0$) and $d$ bosons ($J = 2$), representing correlated valence nucleon pairs
(C) Individual valence protons and neutrons
(D) Alpha particles and deuterons

Question 12

The three dynamical symmetry limits of the IBA are:

(A) U(5) — vibrator, SU(3) — rotor, O(6) — $\gamma$-unstable
(B) U(5) — rotor, SU(3) — vibrator, O(6) — rigid sphere
(C) SU(2) — vibrator, SU(3) — rotor, SU(5) — $\gamma$-unstable
(D) O(3) — vibrator, O(5) — rotor, O(6) — deformed

Question 13

The energy ratio $R_{4/2} = E(4^+)/E(2^+) = 2.5$ is characteristic of which IBA symmetry limit?

(A) U(5)
(B) SU(3)
(C) O(6)
(D) No symmetry limit gives this ratio

Question 14

In the Casten triangle, the transition from U(5) to SU(3) represents:

(A) A second-order quantum phase transition from vibrational to rotational structure
(B) A first-order quantum phase transition from spherical vibrational to deformed rotational structure
(C) A smooth crossover with no phase transition
(D) A triple point where all three phases coexist

Question 15

The deformation parameter $\beta_2$ relates to the nuclear shape. A nucleus with $\beta_2 = +0.3$ is:

(A) Spherical
(B) Oblate (disk-shaped)
(C) Prolate (cigar-shaped)
(D) Octupole deformed (pear-shaped)

Question 16

Which of the following nuclei would you expect to exhibit the most well-developed rotational spectrum?

(A) $^{208}$Pb ($Z = 82$, $N = 126$) — doubly magic
(B) $^{116}$Sn ($Z = 50$, $N = 66$) — semi-magic
(C) $^{168}$Er ($Z = 68$, $N = 100$) — mid-shell rare earth
(D) $^{40}$Ca ($Z = 20$, $N = 20$) — doubly magic

Question 17

The beta band ($K^\pi = 0^+$) of a deformed nucleus corresponds to:

(A) Oscillations in the deformation magnitude $\beta$ while maintaining axial symmetry
(B) Oscillations in the triaxiality parameter $\gamma$
(C) Rotation about the symmetry axis
(D) Single-particle excitations within the deformed potential

Question 18

The critical-point symmetry X(5) describes:

(A) A stable spherical vibrator
(B) The exact solution at the first-order phase transition between spherical and deformed shapes
(C) A rigidly deformed rotor
(D) The triple point of the Casten triangle

Answer Key

(C) — Two identical quadrupole phonons (bosons) couple symmetrically to $J = 0, 2, 4$ at energy $2 \times 800 = 1600$ keV.
(C) — Identical bosons must have symmetric wave functions; coupling two $J = 2$ bosons antisymmetrically gives $J = 1, 3$, which are forbidden.
(B) — The parity of a multipole vibration is $(-1)^\lambda$; for $\lambda = 3$, parity is negative.
(D) — $E(4^+)/E(2^+) = 4(5)/[2(3)] = 20/6 = 3.33$.
(D) — From $E(2^+) = [\hbar^2/2\mathcal{J}] \times 2(3) = 6\hbar^2/2\mathcal{J}$, so $\hbar^2/2\mathcal{J} = 75/6 = 12.5$ keV.
(B) — The $\mathcal{R}_2$ symmetry of an axially symmetric even-even nucleus with reflection symmetry restricts the ground-state band to even spins with positive parity.
(C) — Pairing creates a gap that suppresses the low-energy particle-hole excitations contributing to the cranking moment of inertia.
(B) — Backbending is defined by the characteristic S-shaped curve in the $2\mathcal{J}/\hbar^2$ vs. $(\hbar\omega)^2$ plot, reflecting a sudden jump in the moment of inertia.
(B) — The Coriolis force in the rotating frame breaks a pair of high-$j$ nucleons, which align their angular momenta along the rotation axis (Coriolis anti-pairing).
(C) — Superdeformation corresponds to $\beta_2 \approx 0.6$, an axis ratio of approximately 2:1.
(B) — The IBA represents collective states using $s$ ($J = 0$) and $d$ ($J = 2$) bosons, which are images of correlated valence nucleon pairs.
(A) — The three limits are U(5) (vibrator), SU(3) (rotor), and O(6) ($\gamma$-unstable).
(C) — The O(6) $\gamma$-unstable limit gives $R_{4/2} = 2.5$, intermediate between the vibrational (2.0) and rotational (3.33) values.
(B) — The U(5) to SU(3) transition is first-order, characterized by a discontinuous change in equilibrium deformation and shape coexistence near the critical point.
(C) — Positive $\beta_2$ indicates prolate (elongated along the symmetry axis) deformation.
(C) — $^{168}$Er is in the middle of the rare-earth shell, far from any magic number, and exhibits strong deformation and rotational behavior. Doubly magic nuclei are spherical.
(A) — The beta band involves oscillations of $\beta$ (the magnitude of deformation) about its equilibrium value, preserving axial symmetry ($\gamma = 0$).
(B) — X(5) is an exactly solvable model at the first-order critical point between the spherical [U(5)] and axially deformed [SU(3)] phases, first applied to $^{152}$Sm.