Charge-Exchange Oslo Method

• Charge-Exchange Oslo Method is an adaptation of the conventional Oslo technique that uses charge-exchange reactions to extract nuclear level densities and gamma-ray strength functions.
• It employs coincidence measurements and iterative subtraction of unfolded gamma-ray spectra to isolate primary gamma rays from an excitation-energy versus gamma-ray energy matrix.
• The method enables simultaneous determination of neutron-capture cross sections and Gamow–Teller strengths, with results validated by comparison to direct measurements.

The charge-exchange Oslo method is an adaptation of the Oslo technique to charge-exchange reactions at intermediate energies, designed to extract nuclear level densities and $\gamma$-ray strength functions from a matrix of excitation energy versus emitted $\gamma$-ray energy, and then use those quantities to estimate neutron-capture cross sections. In the demonstrated implementation, the $^{93}$Nb($t$,$^{3}$He) reaction at 115 MeV/u was used to populate $^{93}$Zr, reconstruct an $E_x$–$E_\gamma$ matrix, extract the level density and $\gamma$-ray strength function, and estimate the $^{92}$Zr($\gamma$0,$\gamma$1)$\gamma$2Zr cross section; good agreement with direct measurements was found, and the result was presented as enabling experiments in which ($\gamma$3,$\gamma$4) cross sections and Gamow–Teller strengths can be measured simultaneously (Pathirana et al., 11 Sep 2025).

1. Definition and methodological scope

The conventional Oslo method has traditionally used reactions such as ($\gamma$5,$\gamma$6), ($\gamma$7He,$\gamma$8), or $\gamma$9 to extract nuclear level densities (NLD) and $^{93}$0-ray strength functions ($^{93}$1SF). The charge-exchange Oslo method extends this framework to charge-exchange reactions, specifically reactions such as ($^{93}$2,$^{93}$3He), while preserving the characteristic Oslo analysis of a two-dimensional matrix whose axes are excitation energy and $^{93}$4-ray energy (Pathirana et al., 11 Sep 2025).

In the demonstrated case, the method was applied to the $^{93}$5Nb($^{93}$6,$^{93}$7He$^{93}$8) reaction in order to study $^{93}$9Zr and indirectly provide the $t$0Zr($t$1)$t$2Zr cross section. The stated motivation was astrophysical: the method makes it possible to combine observables usually associated with weak-interaction studies in charge-exchange experiments with observables needed for neutron-capture modeling. Since charge-exchange reactions at intermediate energies have long been used for extracting Gamow–Teller strengths, the successful demonstration of the method establishes a route to simultaneous measurements of Gamow–Teller strengths and ($t$3,$t$4) cross sections (Pathirana et al., 11 Sep 2025).

A plausible implication is that the method is not merely a transfer of analysis software from one reaction class to another. It depends on whether the charge-exchange reaction populates a statistical ensemble suitable for Oslo-type factorization, and on whether the spin population induced by the reaction can be incorporated consistently in the NLD and $t$5SF normalization.

2. Charge-exchange reaction framework and spin representation

The broader charge-exchange literature emphasizes that the representation of the underlying nucleon–nucleon process is not innocuous. In the study of the quasi-elastic process $t$6 at $t$7, two mathematically distinct descriptions of the same physical interaction were compared: the charge-exchange representation $t$8 and the elastic representation $t$9 with the neutron scattered to $^{3}$0. The transition from one representation to the other is provided by the Majorana operator, with the relation

$^{3}$1

and, in the spin sector,

$^{3}$2

The same analysis showed that amplitude decompositions into Flip and Non-Flip parts depend on the chosen representation (Shindin et al., 2017).

Within impulse approximation, the reaction $^{3}$3 leads, in the charge-exchange picture, to the Dean formula. At $^{3}$4, only the Flip part survives:

$^{3}$5

In the alternative elastic representation, the corresponding expression at $^{3}$6 is

$^{3}$7

The paper explicitly noted that the elastic representation contains contributions from both Flip and Non-Flip amplitudes, whereas the Dean formula yields a selective dependence on the Flip part (Shindin et al., 2017).

For charge-exchange Oslo analyses, this background is relevant because the extracted nuclear information is mediated by reaction selectivity in spin and isospin channels. The paper on $^{3}$8 stated that extraction of structure information from quasi-elastic reactions depends critically on the proper representation of the underlying $^{3}$9 amplitude and on correct transformation using the Majorana operator. This suggests that, in charge-exchange Oslo applications, reaction selectivity and population mechanisms cannot be treated as purely kinematic preliminaries; they condition the statistical ensemble on which the Oslo factorization is imposed.

3. Experimental realization in $^{93}$0Nb($^{93}$1,$^{93}$2He$^{93}$3)

The first explicit demonstration used the $^{93}$4Nb($^{93}$5,$^{93}$6He) reaction at 115 MeV/u. Outgoing $^{93}$7He ions were detected and momentum-analyzed with the S800 Spectrograph, which allowed reconstruction of the excitation energy in $^{93}$8Zr through missing-mass calculations. Coincident $^{93}$9 rays emitted by the excited $E_x$0Zr nuclei were measured with the GRETINA $E_x$1-ray detector. Event by event, this produced pairs $E_x$2 from which a two-dimensional histogram was constructed (Pathirana et al., 11 Sep 2025).

The raw $E_x$3–$E_x$4 matrix covered the excitation region up to the neutron separation energy and somewhat beyond. In this experiment, neutron emission starts above $E_x$5 MeV, but statistical $E_x$6 decays persist up to approximately $E_x$7 MeV because the onset of neutron emission is delayed by angular momentum barriers. This feature was used in choosing the excitation-energy region appropriate for the Oslo analysis (Pathirana et al., 11 Sep 2025).

The experimental configuration was therefore tailored to the two observables required by the method. The spectrograph provided excitation-energy reconstruction, while the segmented high-purity germanium array provided the coincident $E_x$8-ray spectrum needed for the Oslo matrix. A plausible implication is that the method is intrinsically coincidence-based: its basic observable is not a singles spectrum but an experimentally reconstructed distribution in two variables.

4. Oslo analysis adapted to charge-exchange data

After acquisition of the raw matrix, the first step was unfolding of the detector response. The measured $E_x$9-ray spectrum is distorted by detector resolution and efficiency, so the analysis used a detector response matrix derived from GEANT4 simulations and calibrated against source data. The unfolded matrix was then used as the input for extraction of primary, or first-generation, $E_\gamma$0 rays (Pathirana et al., 11 Sep 2025).

Primary $E_\gamma$1 rays were obtained with an iterative subtraction technique under two stated assumptions: the $E_\gamma$2-decay pattern of a state does not depend on whether it was directly populated or reached via cascade, and the nucleus is fully equilibrated after the charge-exchange reaction so that the subsequent decay is statistical. The resulting primary matrix $E_\gamma$3 satisfies the standard Oslo factorization

$E_\gamma$4

where $E_\gamma$5 is the nuclear level density of the final state and $E_\gamma$6 is the $E_\gamma$7-ray transmission coefficient (Pathirana et al., 11 Sep 2025).

To ensure the validity of the factorization, explicit energy cuts were imposed. Low $E_\gamma$8-ray energies below $E_\gamma$9 MeV were excluded because discrete levels and non-statistical effects dominate there. Excitation energies below $\gamma$0 MeV were excluded to avoid strong influence from low-lying states, and energies above $\gamma$1 MeV were excluded because neutron emission begins to dominate and non-$\gamma$2Zr nuclei can be formed (Pathirana et al., 11 Sep 2025).

The simultaneous extraction of $\gamma$3 and $\gamma$4 was performed by $\gamma$5 minimization. As in the standard Oslo method, only the product of these functions is uniquely defined. The analysis therefore confronted the standard ambiguity

$\gamma$6

$\gamma$7

so the absolute normalizations and slope parameter had to be fixed externally (Pathirana et al., 11 Sep 2025).

Step	Quantity or operation	Function
Unfolding	Detector response matrix from GEANT4	Recover true $\gamma$8-ray distribution
First-generation extraction	Iterative subtraction technique	Isolate primary $\gamma$9 rays
Factorization fit	$^{92}$0	Determine NLD and transmission coefficient
External normalization	Parameters $^{92}$1, $^{92}$2, $^{92}$3	Remove Oslo scaling ambiguity

The charge-exchange adaptation is therefore not a change in the core mathematical factorization, but in the reaction channel from which the matrix is derived and in the constraints required to make the statistical interpretation credible.

5. Normalization of NLD and $^{92}$4SF and inference of capture cross sections

The level density normalization combined low-energy spectroscopy with neutron-resonance information. At low energies, the extracted NLD was matched to the known number of discrete levels for $^{92}$5Zr from ENSDF up to $^{92}$6 MeV. At the neutron separation energy, given as $^{92}$7 MeV, the normalization used the value derived from the neutron resonance spacing $^{92}$8. The conversion from $^{92}$9 to total level density employed the Ericson spin-distribution model

$\gamma$00

and the Constant Temperature model was used to interpolate or extrapolate $\gamma$01 up to $\gamma$02 (Pathirana et al., 11 Sep 2025).

The $\gamma$03SF normalization was fixed by requiring reproduction of the average total radiative width at $\gamma$04, $\gamma$05, as obtained from neutron resonance data. The transmission coefficient was then converted to the $\gamma$06-ray strength function through

$\gamma$07

This provided the experimentally constrained $\gamma$08SF needed for reaction modeling (Pathirana et al., 11 Sep 2025).

The extracted and normalized NLD and $\gamma$09SF for $\gamma$10Zr were then used as input to the Hauser–Feshbach reaction code TALYS to calculate the $\gamma$11Zr($\gamma$12,$\gamma$13)$\gamma$14Zr cross section. For this purpose, the total $\gamma$15SF was decomposed into $\gamma$16 and $\gamma$17 components using HFB+QRPA and D1M+QRPA calculations, together with an empirical low-energy upbend:

$\gamma$18

The contribution from the upbend was reported to be important for achieving consistency with direct neutron-capture measurements; without it, the neutron-capture cross section is significantly below the direct measurements (Pathirana et al., 11 Sep 2025).

The sequence from matrix construction to Hauser–Feshbach modeling is central to the method’s scientific role. It converts a charge-exchange coincidence measurement into quantities directly usable in compound-nucleus reaction calculations, rather than treating charge-exchange data only as spectroscopy of discrete transitions.

6. Reaction selectivity, limitations, and scientific significance

A defining issue for the charge-exchange Oslo method is spin population. In charge-exchange reactions near $\gamma$19 at intermediate energies, population is mainly to higher-spin states, with $\gamma$20; for the $\gamma$21Nb case, the analysis considered population in $\gamma$22Zr up to $\gamma$23. The normalization therefore used a spin range chosen to reflect the states populated and deexcited in this reaction. Two analyses were performed: a main analysis with $\gamma$24–$\gamma$25 and a systematic alternative with $\gamma$26–$\gamma$27 (Pathirana et al., 11 Sep 2025).

The method also depends on the statistical validity of the decay. The analysis explicitly stated that one must check whether the nuclear system is fully equilibrated after charge-exchange excitation and before decay. This is not a merely formal requirement: the first-generation extraction and the factorization of the primary matrix both assume statistical decay patterns. Likewise, the empirical upper limit on $\gamma$28 was chosen because neutron emission begins to dominate and because nuclei other than $\gamma$29Zr may be formed at higher excitation (Pathirana et al., 11 Sep 2025).

The broader charge-exchange literature provides a cautionary example regarding model dependence. In the quasi-elastic $\gamma$30 problem, the measured ratio

$\gamma$31

was reported as approximately $\gamma$32 over neutron energies $\gamma$33–$\gamma$34 GeV, whereas the theoretical prediction from the elastic representation under impulse approximation was $\gamma$35, underestimating by about $\gamma$36. The paper listed possible reasons: breakdown of impulse approximation at large Fermi-momenta, neglect of the deuteron $\gamma$37-state, meson-exchange currents, and possible intermediate resonance excitations (Shindin et al., 2017).

For the charge-exchange Oslo method, this suggests a general methodological caution. Even when the Oslo extraction itself is internally consistent, the reaction mechanism and the spin–isospin selectivity of the entrance channel remain consequential. The reported success for $\gamma$38Zr($\gamma$39,$\gamma$40)$\gamma$41Zr demonstrates that, when those constraints are controlled, charge-exchange reactions can provide NLD and $\gamma$42SF information of sufficient quality for neutron-capture calculations, while also preserving access to Gamow–Teller strengths for astrophysical studies (Pathirana et al., 11 Sep 2025).