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The upbend resonance (UBR) is a pronounced enhancement of dipole γ-ray strength observed at low γ-ray energies ($E_\gamma \lesssim 2$–$3$ MeV) in the radiative strength function (RSF) of atomic nuclei. It emerges in a broad range of nuclei—including both near-spherical and deformed systems—and has a direct impact on key processes such as neutron-capture cross sections and r-process nucleosynthesis. Empirically, the UBR rises above the extrapolated tail of higher-lying giant dipole (GDR) and spin-flip resonances and has been robustly characterized in experimental and theoretical investigations. The underlying microscopic origin is attributed to thermally induced, non-collective two-quasiparticle (particle-particle and hole-hole) excitations with a coupling strength significantly exceeding that of the GDR.

1. Definition and Characterization of the UBR

The γ-ray strength function $f(E_\gamma)$ quantifies the mean reduced probability of γ-ray emission or absorption per unit energy. For a multipole of order $L$ and character $X$ (electric or magnetic), it is defined as

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

where $\langle \Gamma_{XL}(E_\gamma) \rangle$ is the mean partial radiative width for transitions of type $XL$, $D$ is the average level spacing, and the exponent arises from the Weisskopf single-particle estimate.

In the Oslo-method extraction, which underpins much of the contemporary experimental work, the dipole ($L=1$) strength is

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

with $\mathcal{T}(E_\gamma)$ the γ-ray transmission coefficient.

The hallmark of the UBR is an exponential low-energy upturn: $$f_{\text{upbend}}(E_\gamma) = C \, e^{-\eta E_\gamma}$$ where empirical parameters $C$ and $\eta$ capture the magnitude and rate of decline. Typical parameter values in $^{151,153}$Sm are $C = 2.0 \times 10^{-6}\;\mathrm{MeV}^{-3}$ and $\eta = 5.0~\mathrm{MeV}^{-1}$, with close analogs in neighboring isotopes (Simon et al., 2016, Naqvi et al., 2019).

2. Experimental Probes and Systematics

Experimental investigation of the UBR employs advanced segmented detection arrays for both ejected light ions (particle identification) and coincident γ spectroscopy, such as:

• Highly segmented $\Delta E$–$E$ silicon telescopes for ion identification, achieving $\sim$130 keV FWHM energy resolution.
• HPGe clover detectors with BGO Compton suppression, attaining photopeak efficiency $\sim$4.8% at 100 keV and energy resolution of 2.6 keV (122 keV) and 3.5 keV (963 keV).
• Extension of reliable γ energy reach down to $E_\gamma \sim$ 0.5 MeV, well below prior NaI-based setups ($\sim$1 MeV limit).

The Oslo method is used to extract the primary γ-ray spectra, followed by a simultaneous fit to the level density and γ-strength, with normalization to known discrete levels at low energy and neutron-resonance data at the neutron separation energy. This technique has enabled the observation of UBR in both light ($^{147,149}$Sm) and mid-shell deformed ($^{151,153}$Sm) rare-earth isotopes.

A representative table of extracted $f(E_\gamma)$ values in $^{153}$Sm (Simon et al., 2016):

$E_\gamma$ (MeV)	$f(E_\gamma)$ (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 (SR onset)	$1.5 \times 10^{-7}$

3. Microscopic Origin and Theoretical Interpretation

The microscopic origin of the UBR is rooted in non-collective two-quasiparticle (particle-particle and hole-hole) excitations. The exact thermal-pairing-plus-phonon-damping (EP+PDM) framework provides a unified, temperature-dependent model for both the GDR and UBR (Phuc et al., 3 Nov 2025). The RSF for each resonance $R$ is presented as: $$f^R(E_\gamma,T) = \frac{1}{3\pi^2\hbar^2c^2} \frac{\sigma_R\,\gamma_q^R(E_\gamma, T) \, S^R(E_\gamma, T)}{E_\gamma}$$ with $S^R(E_\gamma, T)$ a Breit–Wigner strength function and $\gamma_q^R(E_\gamma, T)$ the total damping width.

For the UBR, $\gamma_q^{\mathrm{UBR}}$ acquires a finite value only for $T > 0$, as the relevant p–p and h–h poles are thermally activated: $$\gamma_q^{\rm UBR}(E_\gamma, T) \propto [F^{\rm UBR}_{ss'}]^2 \sum_{s>s'} [u_su_{s'}-v_sv_{s'}]^2 [n_s-n_{s'}]\delta(E_\gamma - E_s + E_{s'})$$ where $(u_k, v_k)$ are Bogolyubov amplitudes and $n_k$ are thermal occupations.

Numerically, the UBR-phonon couples to non-collective two-quasiparticle states with matrix elements approximately three times stronger than the GDR-phonon. This enhanced coupling directly explains the observed strength of the upbend at low $E_\gamma$ across a wide mass region.

Shell-model calculations in the jj56pn model space (relative to $^{132}$Sn core), with allowance for up to $2p$–$2h$ excitations, accurately reproduce the constant-temperature slope of $\rho(E)$ and the shape of $f_{M1}(E_\gamma)$ between 1–2.5 MeV, including the evolution of the upbend and the scissors resonance with deformation (Naqvi et al., 2019).

4. Systematics, Parameterization, and Mass Dependence

The UBR is well described by a purely exponential form without a distinct centroid or width, unlike the Lorentzian profile of collective resonances. Parameters extracted from $^{147,149}$Sm and $^{151,153}$Sm indicate the following systematics:

Nucleus	$C$ $(\times 10^{-7}\,\mathrm{MeV}^{-3})$	$\eta$ (MeV$^{-1}$)
$^{147}$Sm	$10 \pm 5$	$3.2 \pm 1.0$
$^{149}$Sm	$20 \pm 10$	$5.0 \pm 1.0$
$^{151,153}$Sm	$20$	$5.0$

The integrated low-energy $M1$ strength to $E_\gamma=5$ MeV is nearly constant, e.g., $B(M1)_{\rm tot} \approx 8.3\,\mu_N^2$ across $A=147$–$153$.

A global mass dependence of the integrated UBR fraction $R(A)$ is established within the EP+PDM framework (Phuc et al., 3 Nov 2025): $$R(A) \approx 243.18\,e^{-A/11.45} + 1.17\,\ln A - 5.05 \qquad (\mathcal{R}^2=0.95)$$ This reveals a steep drop—from $5$–$6\%$ in light systems to a few tenths of a percent by $A \approx 150$—in the fractional UBR strength with increasing mass.

Empirical observations across the samarium chain indicate that while lighter isotopes (near-spherical) exhibit an upbend below $\sim2$ MeV with no appreciable scissors resonance, the well-deformed mid-shell isotopes display both phenomena, with the scissors mode peaking at $E_\gamma \approx 3$ MeV.

5. Coexistence with the Scissors Resonance and Angular Momentum Systematics

The simultaneous observation of both the UBR and the scissors resonance (SR) in $^{151,153}$Sm represents a critical structural distinction (Simon et al., 2016). The UBR (predominantly dipole $M1$) is centered at $E_\gamma \lesssim 2$ MeV, while the SR (also $M1$) peaks near $3.0$ MeV with fitted Lorentzian parameters:

• $\omega_{\rm SR} = 3.0(3)$ MeV,
• $\Gamma_{\rm SR} = 1.1(3)$ MeV,
• $\sigma_{\rm SR} = 0.6(2)$ mb,
• summed strength $B_{\rm SR} = 7.8(3.4)\,\mu_N^2$.

Shell-model calculations find the upbend and SR to be distinctly separated in energy, mapping to different physical mechanisms: the SR as a collective oscillation of protons and neutrons in deformed potentials, and the UBR to strong $M1$ transitions stemming from quasiparticle reorientation or thermal continuum effects. The total low-energy $M1$ strength summed over upbend and SR remains approximately constant across relevant isotopes.

6. Thermodynamic Nature and Violation of the Brink–Axel Hypothesis

Within the EP+PDM approach, the UBR is strictly a thermal mode: $f^{\mathrm{UBR}}(E_\gamma) = 0$ at $T = 0$ and becomes nonzero only as finite temperature populates non-collective quasiparticle states (Phuc et al., 3 Nov 2025). This introduces clear $T$-dependence in low-energy RSFs, violating the Brink–Axel hypothesis, which asserts that the RSF depends solely on $E_\gamma$ and is independent of temperature or initial state. Experimentally, the upbend is absent in ground-state photoabsorption but arises in the hot compound state following, e.g., neutron capture or inelastic reactions.

A plausible implication is that predictions of (n,γ) cross sections and nucleosynthesis rates based solely on ground-state strength functions systematically underestimate neutron-rich reaction rates unless the UBR is explicitly included.

7. Astrophysical Impact and Model Uncertainties

The UBR exerts a substantial influence on astrophysical (n,γ) cross sections, particularly in neutron-rich nuclei near the r-process path. Hauser–Feshbach calculations incorporating measured upbend parameters indicate up to $10^2$–$10^3$-fold enhancements in Maxwellian-averaged rates at “cold” r-process temperatures ($T \approx 0.15$ GK) and factors of a few at $T = 1.0$ GK (Simon et al., 2016).

Network calculations demonstrate that the presence and magnitude of the UBR modulate final abundance distributions and the position of r-process peaks. By providing theoretically constrained, parameter-free UBR descriptions anchored in discrete-level schemes, the EP+PDM significantly reduces uncertainties in reaction-rate evaluation, directly benefitting nucleosynthesis modeling (Phuc et al., 3 Nov 2025).

8. Outlook and Open Questions

Current analyses confirm a strong $M1$ character for the UBR, though possible $E1$ admixtures are not excluded. The mechanism’s persistence in near-spherical and deformed systems argues for a nearly universal, thermally induced, non-collective origin. Nonetheless, detailed microscopic theory—including the relative roles of shell structure, pairing correlations, and deformation—remains under development. The relationship between the upbend and other low-energy modes, as well as its evolution with isospin and excitation energy, are active areas of inquiry.

Further experimental progress—leveraging polarized photon beams, improved γ-ray detectors, and systematic isotopic surveys—will be central to elucidating the detailed nature of the UBR. The astrophysically critical role of the UBR in r-process synthesis continues to motivate precision measurements and theory developments.