---
title: Wasserstein Barycenters in Optimal Transport
url: https://www.emergentmind.com/topics/wasserstein-barycenters
type: topic
---

# Wasserstein Barycenters in Optimal Transport

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 [1602.05824, 1905.13461].

## 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 [1602.05824]:

| $E_\gamma$ (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 [2511.01696]. 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 [1905.13461].

## 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 [2511.01696]:
\[
R(A) \approx 243.18\,e^{-A/11.45} + 1.17\,\ln A - 5.05 \qquad (\mathcal{R}^2=0.95)
\]
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 [1602.05824]. 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 [2511.01696]. 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 [1602.05824].

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 [2511.01696].

## 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.

Source: https://www.emergentmind.com/topics/wasserstein-barycenters