Hindered $\Delta K=1$ Dipole Strength in octupole bands in $N=90$ $^{154}$Gd from Lifetime Measurements with $\gamma-\gamma$ fast timing technique

Using the $\gamma$-$\gamma$ fast-timing technique with the VENTURE array at VECC, Kolkata, researchers measured the lifetimes of low-lying negative-parity states in $^{154}$Gd to determine that their $B(E1)$ transition strengths are strongly hindered, providing evidence for weak $\Delta K=1$ dipole strength in octupole bands.

The Gist

Imagine the nucleus of an atom not as a solid marble, but as a squishy, dancing drop of liquid. Sometimes, this drop wobbles in a simple, round way (like a sphere). Other times, it stretches into a football shape. But in some special cases, like the one studied in this paper, the nucleus does something even stranger: it wobbles in a way that makes it look like a pear. It has a "top" and a "bottom" that are different, breaking its mirror symmetry. This is called octupole correlation.

The scientists in this paper were investigating a specific atom, Gadolinium-154 (specifically the version with 90 neutrons), to see how this "pear-shaped" wobbling behaves.

Here is the story of what they found, broken down into simple concepts:

1. The Mystery of the "Hidden" Dance Moves

Inside this nucleus, there are different groups of energy levels, which we can think of as different "dance troupes."

• Troupe A (The Strong Dancers): This group moves in a way that is very easy to see and measure. They are like a loud, clear drumbeat. In physics terms, these are transitions where a specific number called K stays the same ($\Delta K = 0$).
• Troupe B (The Shy Dancers): This group is supposed to be similar, but they move in a way that is very hard to detect. They are like a whisper in a noisy room. These are transitions where the number K changes by 1 ($\Delta K = 1$).

For a long time, scientists knew Troupe A existed and was loud. They suspected Troupe B existed too, but they weren't sure how "shy" (or "hindered") they really were. They needed to measure exactly how long these "shy" states lasted before they decayed (stopped dancing) to figure out their strength.

2. The Experiment: Catching the Whisper

To measure these fleeting moments, the team at the Variable Energy Cyclotron Centre in India used a high-tech stopwatch called the $\gamma-\gamma$ fast timing technique.

• The Setup: They created the Gadolinium-154 atoms by smashing protons into a target. These atoms were excited and then settled down, emitting gamma rays (packets of light).
• The Stopwatch: They used special detectors (like high-speed cameras) to measure the tiny fraction of a second between the emission of two gamma rays.
• The Challenge: The "shy" states they were looking for (specifically at energy levels of 1398 keV and 1414 keV) lived for only about 35 to 46 picoseconds. That is 35 to 46 trillionths of a second. It's like trying to time a blink of an eye, but the eye is a billion times faster.

3. The Discovery: The "Shy" Dancers are Extremely Quiet

Once they measured the time, they could calculate the "strength" of the transition (how much energy was released).

• The Result: They found that the "shy" dancers (the $\Delta K = 1$ transitions) were extremely weak. Their strength was thousands of times weaker than the "loud" dancers ($\Delta K = 0$).
• The Analogy: Imagine Troupe A is a rock band playing a guitar solo at full volume. Troupe B is a single person trying to hum a tune in the same room. The paper confirms that in Gadolinium-154, the "hum" is so quiet it's almost non-existent.

This is a big deal because it proves that in this specific type of atom, the rules of the "dance" (quantum mechanics) strictly forbid the "shy" moves from happening easily. The nucleus resists changing its internal "K" number.

4. Why the Order is Backwards

The paper also discusses a confusing history about which states belong to which troupe.

• Usually, scientists expect the "loud" troupe to have the lowest energy (start the dance first).
• However, in Gadolinium-154, the "shy" troupe actually has a state that is slightly lower in energy than the "loud" one.
• The authors confirmed that the state at 1414 keV and the one at 1398 keV belong to the "shy" troupe ($K=1$), while the state at 1241 keV belongs to the "loud" troupe ($K=0$). This ordering is a bit unusual and changes as you add more neutrons to the atom, but this experiment helped pin down exactly where they stand in Gadolinium-154.

5. The Theoretical Explanation

The scientists used a computer model (based on something called the Interacting Boson Model) to simulate the nucleus.

• The Model: They tried to predict how the nucleus should behave. The model correctly predicted the energy levels (where the dancers stand), but it overestimated the strength of the "shy" dancers.
• The Fix: To make the model match the real data, they had to assume two things:
  1. The "shy" dancers are naturally very weak (intrinsic hindrance).
  2. The two troupes don't mix much. They stay in their own lanes. If they mixed too much, the "shy" dancers would become louder. The fact that they are so quiet means the nucleus is very good at keeping these two groups separate.

Summary

In simple terms, this paper is a precise measurement of how a specific atomic nucleus wobbles. The scientists found that while some wobbles are loud and obvious, others are incredibly faint and suppressed. They proved that in Gadolinium-154, the nucleus is very strict about its internal rules, preventing certain types of "shy" movements from gaining any strength. This helps physicists understand the fundamental rules that govern how atomic nuclei are shaped and how they move.

Technical Summary

Technical Summary: Hindered $\Delta K = 1$ Dipole Strength in Octupole Bands in $N=90$ $^{154}\text{Gd}$

Problem and Motivation
Octupole correlations in atomic nuclei are characterized by reflection-asymmetric surface degrees of freedom, often manifesting as low-lying negative-parity rotational bands. In deformed nuclei, these excitations split into bands defined by the projection quantum number $K$. While enhanced electric dipole ($E1$) transitions are a hallmark of octupole collectivity, the strength of $\Delta K = 1$ transitions is theoretically expected to be hindered relative to $\Delta K = 0$ transitions due to the scarcity of available configurations near the Fermi surface.

While octupole collectivity is well-established in actinides and the Ba–Sm region, experimental constraints on heavier rare-earth nuclei, specifically Gadolinium (Gd) isotopes, remain limited. In $^{154}\text{Gd}$ ($N=90$), the assignment of low-lying negative-parity states to specific $K$ bands ($K^\pi = 0^-$ vs. $K^\pi = 1^-$) has been historically contentious. Recent spectroscopic studies suggested a reassignment of the lowest odd-spin states to the $K^\pi = 1^-$ band, contrasting with earlier interpretations. However, a definitive assessment of these band assignments requires quantitative data on absolute $B(E1)$ transition strengths, particularly for the lowest even-spin $2^-$ state, for which no direct lifetime measurements existed in the $N=90$ region.

Methodology
The authors performed the first lifetime measurements of the low-lying negative-parity $2^-$ state at 1398 keV and the $1^-$ state at 1414 keV in $^{154}\text{Gd}$.

• Production: The states were populated via the $\beta$-decay of $^{154}\text{Tb}$, produced through the $^{154}\text{Gd}(p,n)^{154}\text{Tb}$ reaction using a 12 MeV proton beam from the K130 cyclotron at VECC, Kolkata. An enriched $^{154}\text{Gd}$ target was used to minimize background from competing reactions.
• Detection: The experiment utilized the VENTURE array, comprising eight fast CeBr$_3$ scintillator detectors for timing and two Compton-suppressed Clover HPGe detectors for high-resolution energy spectroscopy.
• Analysis: Lifetimes were extracted using the $\gamma$-$\gamma$ fast-timing technique with the Generalized Centroid Difference (GCD) method. This involved measuring centroid shifts between delayed and anti-delayed time distributions of specific $\gamma$-$\gamma$ cascades. Corrections for Compton background contributions and the Prompt Response Distribution (PRD) of the setup were applied to determine the true centroid differences and subsequent lifetimes.

Key Results

1. Measured Lifetimes:

   • The lifetime of the $2^-$ state (1398 keV) was determined to be 46(5) ps.
   • The lifetime of the $1^-$ state (1414 keV) was determined to be 35(5) ps.
   • These measurements provided the first experimental data for the lowest $2^-$ state in $^{154}\text{Gd}$ and the first reported lifetime for the $1^-$ state at 1414 keV.

2. Transition Strengths ($B(E1)$):

   • Using the measured lifetimes and known branching ratios, absolute $B(E1)$ values were deduced.
   • The $B(E1; 2^-_1 \to 2^+_1)$ strength was found to be $2.4 \times 10^{-6}$ W.u.
   • The $B(E1; 1^-_1 \to 0^+_1)$ and $B(E1; 1^-_1 \to 2^+_1)$ strengths were calculated as $5.1 \times 10^{-7}$ W.u. and $2.4 \times 10^{-6}$ W.u., respectively.
   • These values are orders of magnitude smaller than the $B(E1)$ strength of the $1^-$ state at 1241 keV ($4.1 \times 10^{-2}$ W.u.), which is associated with the $K^\pi = 0^-$ band.

3. Systematics and Comparison:

   • The extracted $B(E1)$ values for the 1398 keV and 1414 keV states fall into a "weak-strength" region when compared to neighboring Gd isotopes and $\Delta K = 0$ transitions.
   • This strong hindrance supports the assignment of these states to the $K^\pi = 1^-$ band, while the $K^\pi = 0^-$ band contains the yrast odd-spin negative-parity levels (e.g., the 1241 keV state).

Theoretical Interpretation
The experimental results were analyzed using two theoretical frameworks:

• sdf-IBM Calculations: Gogny-HFB-based Interacting Boson Model (IBM) calculations with $s$, $d$, and $f$ bosons were performed. While the model successfully reproduced the excitation energy trends (including the unusual $E(2^-_1) < E(1^-_1)$ ordering in $^{154}\text{Gd}$), it significantly overestimated the $B(E1)$ strengths for the $K^\pi = 1^-$ states.
• Band-Mixing Analysis: A band-mixing calculation was performed to quantify the hindrance. The analysis considered mixing between the $K^\pi = 0^-$ and $K^\pi = 1^-$ bands driven by Coriolis coupling. The results indicated:
  • A very small mixing amplitude ($\epsilon \approx 0.0028$) between the two bands.
  • A strongly hindered intrinsic $\Delta K = 1$ dipole matrix element ($\zeta \approx -0.0095$) relative to the $\Delta K = 0$ element.
  • The observed suppression of $E1$ strength is attributed to this weak intrinsic dipole strength combined with limited configuration mixing.

Significance
The paper establishes that the $\Delta K = 1$ dipole strength in $^{154}\text{Gd}$ is strongly hindered compared to $\Delta K = 0$ transitions, providing experimental evidence for weak dipole strength in the $K^\pi = 1^-$ octupole band at $N=90$. The measured $B(E1)$ values serve as a critical anchor point for the systematics of low-lying dipole strengths in the Gd isotopic chain. The findings suggest that strong octupole-related dipole collectivity in this mass region is concentrated in the $K^\pi = 0^-$ branch, while the $K^\pi = 1^-$ sequence carries significantly weaker strength. The work highlights the necessity of precise lifetime measurements to resolve band assignments and understand the microscopic origins of dipole excitations in deformed nuclei.