Upbend Resonance (UBR) in Atomic Nuclei

• Upbend Resonance (UBR) is a low-energy enhancement of the gamma-ray strength function in nuclei, emerging from thermally induced p–p and h–h excitations.
• Experimental observations in samarium isotopes reveal an exponential increase in dipole strength, exceeding the GDR tail by up to 10³ at Eγ ≲ 2–3 MeV.
• UBR challenges the Brink–Axel hypothesis and significantly impacts (n,γ) reaction rates in r-process nucleosynthesis, altering astrophysical models.

The upbend resonance (UBR) constitutes a pronounced low-energy enhancement of the radiative strength function (RSF) for $\gamma$-decay in atomic nuclei. Specifically, it manifests as an exponential increase in dipole strength for $\gamma$-ray energies $E_\gamma \lesssim 2$–$3$ MeV, exceeding the extrapolated tail of the giant dipole resonance (GDR) by factors of up to $10^3$ in neutron-rich nuclei. The physical origin of the UBR has recently been elucidated as arising from non-collective particle–particle (p–p) and hole–hole (h–h) excitations, becoming significant only at finite nuclear temperatures. The UBR impacts statistical model calculations, with direct consequences for $(n,\gamma)$ reaction rates in r-process nucleosynthesis (Simon et al., 2016, Naqvi et al., 2019, Phuc et al., 3 Nov 2025).

1. Radiative Strength Function and Definition of UBR

The radiative $\gamma$-strength function $f(E_\gamma)$ quantifies the average reduced probability of $\gamma$ emission (or absorption) per MeV and serves as a fundamental input to statistical Hauser–Feshbach reaction models. For a multipole order $L$ and character $X$ (electric or magnetic), the strength function is given by:

$$f_{XL}(E_\gamma) = \frac{\langle\Gamma_{XL}(E_\gamma)\rangle}{D\, E_\gamma^{2L+1}}$$

where $\langle\Gamma_{XL}(E_\gamma)\rangle$ is the average partial radiative width, $D$ is the average level spacing, and $E_\gamma^{2L+1}$ reflects the Weisskopf estimate. For $L=1$ (dipole), and in Oslo-method extractions:

$$f(E_\gamma) = \frac{1}{2\pi} \frac{{\cal T}(E_\gamma)}{E_\gamma^3}$$

The upbend resonance refers to a marked enhancement of $f(E_\gamma)$ for $E_\gamma \lesssim 2$–$3$ MeV, parameterized empirically by an exponential form:

$f_{\rm upbend}(E_\gamma) = C\, e^{-\eta E_\gamma}$

where $C$ and $\eta$ are fitted parameters, with $C \sim 2 \times 10^{-6}\, \mathrm{MeV}^{-3}$ and $\eta \sim 5\, \mathrm{MeV}^{-1}$ for mid-shell rare-earth nuclei such as ${}^{151,153}\mathrm{Sm}$ (Simon et al., 2016, Naqvi et al., 2019).

2. Experimental Observations and Methodologies

The UBR has been systematically observed in samarium isotopes through $(p,d)$ reactions on enriched targets (${}^{152,154}\mathrm{Sm}$ and ${}^{148,150}\mathrm{Sm}$), employing segmented $\Delta E$–$E$ silicon telescopes for particle identification and HPGe clover detectors with BGO Compton shields for $\gamma$-spectroscopy. Notable features of the setups include:

• Extension of reliable $\gamma$-strength measurements down to $E_\gamma \approx 500$ keV.
• Photopeak efficiencies of $\sim4.8\%$ at 100 keV and energy resolutions $2.6$ keV at $122$ keV, $3.5$ keV at $963$ keV.

Data processing involves:

• Sorting particle–$\gamma$ coincidences into excitation versus $\gamma$ energy matrices.
• Unfolding detector response via GEANT4 simulations.
• Oslo-method iterative subtraction to extract primary $\gamma$-ray spectra $P(E_x, E_\gamma)$.

The resultant $f(E_\gamma)$ exhibits a low-energy enhancement by roughly an order of magnitude compared to the GDR extrapolation for $E_\gamma \lesssim 2$ MeV. Representative $f(E_\gamma)$ values extracted for ${}^{153}\mathrm{Sm}$ are:

$E_\gamma$ [MeV]	$f(E_\gamma)$ [$\mathrm{MeV}^{-3}$]
0.6	$1.2 \times 10^{-6}$
1.0	$6.5 \times 10^{-7}$
1.5	$3.8 \times 10^{-7}$
2.0	$2.8 \times 10^{-7}$
3.0	$1.5 \times 10^{-7}$ (SR onset)

3. Microscopic Interpretation and Thermodynamic Origin

The EP+PDM (exact thermal pairing plus phonon damping model) provides a microscopic foundation for the UBR as a thermally induced dipole excitation (Phuc et al., 3 Nov 2025). Under EP+PDM, the RSF for each resonance (GDR, UBR) assumes:

$$f^{R}(E_\gamma,T) = \frac{\sigma_R\, \gamma_q^R(E_\gamma,T)\, S^R(E_\gamma,T)}{3\pi^2 \hbar^2 c^2 E_\gamma}$$

with the strength function $S^R$ given by:

$$S^R(E_\gamma,T) = \frac{1}{\pi} \frac{\gamma_q^R(E_\gamma,T)}{(E_\gamma-E_R)^2 + [\gamma_q^R(E_\gamma,T)]^2}$$

The total (temperature-dependent) width $\gamma_q^R$ for the UBR splits into quantal (particle–hole) and thermal (particle–particle, hole–hole) contributions. Critically, thermal p–p and h–h excitations only emerge at finite temperature, with coupling matrix elements to the UBR phonon approximately three times stronger than for the GDR:

$$F_{ph}^{\rm UBR} \simeq 3\, F_{ph}^{\rm GDR}, \quad F_{ss'}^{\rm UBR} \simeq 3\, F_{ss'}^{\rm GDR}$$

This threefold enhancement accounts for the prominence of the UBR at low $E_\gamma$ across a broad mass range ($44 \leq A \leq 153$).

4. Systematics, Isotopic Dependence, and Coexistence with Scissors Resonance

Analysis across the samarium chain reveals systematic behavior:

• In near-spherical ${}^{147,149}\mathrm{Sm}$: Upbend visible below $\sim2$ MeV; no distinct scissors resonance (SR) at $E_\gamma \sim3$ MeV.
• In well-deformed mid-shell ${}^{151,153}\mathrm{Sm}$: Coexistence of upbend and pronounced SR ($E_\gamma \sim3$ MeV, $B_{\rm SR} \sim 7.8\,\mu_N^2$).
• The total low-energy $M1$ strength (0–5 MeV) remains nearly constant ($\sim8.3\,\mu_N^2$) from $A=147$ to $153$, matching shell-model estimates (Naqvi et al., 2019).

The extracted upbend parameters for lighter Sm isotopes are:

Nucleus	$C$ ($\times 10^{-7}\,\mathrm{MeV}^{-3}$)	$\eta$ ($\mathrm{MeV}^{-1}$)
${}^{147}$Sm	$10 \pm 5$	$3.2 \pm 1.0$
${}^{149}$Sm	$20 \pm 10$	$5.0 \pm 1.0$

The UBR fraction of the total low-energy RSF, $R(A)$, declines steeply with mass, described globally by:

$$R(A) \approx 243.18\,e^{-A/11.45} + 1.17\,\ln A - 5.05$$

This empirical relation holds over $40 \leq A \leq 160$ ($\mathcal{R}^2=0.95$) (Phuc et al., 3 Nov 2025).

5. Theoretical Implications and Validity of the Brink–Axel Hypothesis

UBR's thermodynamic nature has profound consequences. In EP+PDM:

• UBR is absent at $T=0$ (ground-state absorption) and emerges only at finite temperature, invalidating the conventional Brink–Axel hypothesis in the low-energy region.
• The total RSF at low $E_\gamma$ is a strongly temperature-dependent function, $$f_{\rm tot}(E_\gamma, T) = f^{\rm GDR}(E_\gamma, T) + f^{\rm UBR}(E_\gamma, T)$$.

This temperature dependence manifests in decay (hot compound nucleus) but not in ground-state photoabsorption, implying that absorption and decay RSFs diverge at low energies—directly falsifying Brink–Axel at $E_\gamma \lesssim 2$ MeV.

6. Astrophysical Consequences and Reaction Rate Sensitivity

The (n,γ) cross sections for r-process nucleosynthesis are highly sensitive to the low-energy $\gamma$-ray strength. Hauser–Feshbach calculations using TALYS or equivalent reaction codes, including measured $f_{\rm upbend}(E_\gamma)$ and SR, result in:

• Enhancement of Maxwellian-averaged (n,γ) rates for neutron-rich Sm isotopes beyond $N=126$ by factors of $10^{2}$–$10^{3}$ at $T \approx 0.15$ GK (“cold” r-process), and by factors of several at $T=1.0$ GK (Simon et al., 2016).
• Substantial alteration of r-process abundance peaks and waiting-point lifetimes (Phuc et al., 3 Nov 2025).

A plausible implication is that inclusion of UBR in statistical model calculations is essential for reliable nucleosynthesis modeling, reducing rate uncertainties via a microscopic thermodynamically consistent description.

7. Nuclear Structure Perspectives and Future Directions

Multiple microscopic mechanisms have been proposed for UBR:

• Thermal coupling in the continuum enhancing low-energy $E1$ transitions.
• Strong $M1$ transitions among high-$\ell$ orbitals with shears-like character.
• Coherent recoupling of quasi-particles in the same shell.

Current angular-distribution and shell-model data favor a dominant $M1$ (magnetic dipole) origin, though admixture with other multipoles cannot be excluded. The systematic observation of UBR in near-spherical and well-deformed systems supports its universality as a non-collective, thermally induced mode.

A plausible implication is that the UBR challenges existing nuclear structure paradigms of the quasi-continuum and residual interactions. Future experimental studies—such as those involving polarized photon beams, high-resolution $\gamma$-spectroscopy, and fully microscopic modeling—are expected to further clarify the origin, evolution, and application of the UBR in nuclear physics and astrophysics.