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Microscopic Folding Analysis of

Neutron Elastic Scattering on ⁹⁰Zr

Elastic and inelastic angular distributions using folding-derived Vᵥ in TALYS 2.2

İsmail Hakkı SARPÜN¹²³, Ferhan AKDENİZ¹

¹ Akdeniz University, Science Faculty, Physics Department

² Akdeniz University, Faculty of Medicine, Radiation Oncology

³ Akdeniz University, Nuclear Sciences Research and Application Centre

4th International Workshop on Nuclear Theory (INTW 2026)

27 September – 3 October 2026 • Antalya, Türkiye

⁹⁰Zr

N = 50

Z = 40

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Motivation and objective

From microscopic ground-state densities to measurable angular distributions

1

Microscopic

densities

→

2

Single-folding

potential

→

3

TALYS 2.2

with modified Vᵥ

→

4

Angular differential

cross sections

Central question: How much do the angular distributions change when the phenomenological real volume depth is replaced by Vᵥ values obtained from microscopic density-based single-folding calculations?

SHF-WS

HFB-S

HFB-G

Three density descriptions are propagated through exactly the same TALYS framework.

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Microscopic ground-state density models

Three descriptions of the ⁹⁰Zr proton and neutron distributions

SHF-WS

Hartree–Fock–Skyrme

Microscopic density obtained within the Skyrme mean-field approach and represented with a Woods–Saxon-type radial description.

HFB-S

Hartree–Fock–Bogoliubov + Skyrme

Pairing correlations are included in the HFB framework using a Skyrme effective interaction.

HFB-G

Hartree–Fock–Bogoliubov + Gogny

Finite-range Gogny interaction provides an alternative microscopic density description including pairing.

These densities are not used to refit the entire optical potential. Their role in this work is to generate alternative single-folding values of the real volume depth Vᵥ.

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Proton and neutron density distributions of ⁹⁰Zr

The three models are globally similar, but small radial differences remain

Protons

Neutrons

Small density differences can therefore produce measurable angular-distribution changes even when the bulk proton and neutron profiles look almost identical.

Interior differences�→ shell occupancy & pairing modify the central density

Surface fall-off (≈5–7 fm)�→ rms radius / diffuseness; controls the folded-potential surface

N = 50 shell closure�→ reduced neutron pairing; interior density is more tightly constrained

Tail separation (>5 fm)�→ changes peripheral overlap and the scattering phase

Reaction sensitivity is surface dominated: ρ(r) differences → folded V(r) → phase shifts → diffraction minima and amplitudes.

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Single-folding potential and TALYS 2.2 implementation

Only the real volume potential depth Vᵥ is changed

Single-folding concept

V_F (R) = ∫ ρ(r) · v_NN(|R − r|) d³r

ρ(r): microscopic nucleon density

v_NN: effective nucleon–nucleon interaction

V_F: folded real interaction potential

TALYS 2.2 implementation

Default TALYS → Koning–Delaroche OMP

Microscopic runs → replace only Vᵥ

rᵥ, a, imaginary terms, spin–orbit and all remaining OMP parameters are kept at their TALYS default values.

Interpretation rule: differences between SHF-WS, HFB-S and HFB-G curves isolate the effect of the folding-derived Vᵥ values within an otherwise unchanged TALYS optical-model framework.

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Folding-derived real volume depths Vᵥ

Comparison with the TALYS default Koning–Delaroche value

Eₙ (MeV)

HFB-S

HFB-G

SHF-WS

Default

(Koning–Delaroche)

8 MeV

HFB-G differs from default by only ≈ 0.19 MeV.

11 MeV

HFB-S differs from default by only ≈ 0.18 MeV.

24 MeV

All folding Vᵥ values are 2.78–3.92 MeV deeper than default.

The microscopic–default separation is energy dependent; it is smallest near 8–11 MeV and largest at 24 MeV.

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Elastic angular distributions

Default = Koning–Delaroche; microscopic curves differ only through Vᵥ

8 MeV

11 MV

24 MeV

8 MeV: curves nearly overlap → the elastic data have weak power to discriminate the three density models.

11 MeV: minima and maxima occur at the correct angles → the phase pattern and potential geometry are reproduced.

24 MeV: model separation grows around extrema → elastic scattering becomes more sensitive to the folding-derived Vᵥ.

First minimum position�→ effective radius + surface gradient of the real potential

Peak / minimum spacing�→ partial-wave interference; phase shifts encode potential geometry

Larger spread at 24 MeV�→ deeper Vᵥ changes phases most near constructive/destructive interference

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Inelastic angular distributions: Eₓ = 2.186 MeV

Energy evolution of the same excited-state channel

8 MeV

11 MeV

24 MeV

The angular shape is partly captured, but the measured strength is systematically lower at several angles. Because the three Vᵥ curves remain close, the residual mismatch points to transition-density and coupling physics beyond the ground-state real depth.

2₁⁺ at 2.186 MeV�→ dominant quadrupole excitation; ΔL≈2

Node / phase location�→ shape of the quadrupole transition form factor

Absolute magnitude�→ transition strength / deformation + channel coupling, not Vᵥ alone

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Inelastic angular distributions: Eₓ = 2.748 MeV

A second excited-state channel shows the same controlled Vᵥ sensitivity

8 MeV

11 MeV

24 MeV

The calculated curves reproduce part of the oscillatory structure, but not the full normalization. Since changing the folding-derived Vᵥ produces a much smaller effect than the experiment–theory gap, the missing sensitivity is more naturally associated with octupole transition strength and channel coupling.

3₁⁻ at ≈2.748 MeV�→ octupole surface vibration; ΔL≈3

Oscillation / node sequence�→ multipolarity fingerprint of the octupole transition form factor

Theory–data gap > model spread�→ transition density / coupled-channel dynamics dominate over Vᵥ variation

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

Structural signatures from elastic and inelastic scattering

What can be read directly from the angular distributions of ⁹⁰Zr?

ELASTIC SCATTERING

INELASTIC SCATTERING

Minima / maxima

surface geometry and real optical potential

Oscillation spacing

interference of contributing partial waves

Energy evolution

redistribution of ℓ contributions

Model separation

sensitivity to folding-derived Vᵥ

2₁⁺, 2.186 MeV

quadrupole collectivity; ΔL≈2

3₁⁻, 2.748 MeV

octupole collectivity; ΔL≈3

Node / phase

multipolarity and transition form factor

Normalization

transition strength and channel coupling

Density profile → folded potential → elastic diffraction | transition density → inelastic multipole pattern

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Integrated interpretation

What is constrained by changing only Vᵥ — and what is not?

Elastic channel

• Direct sensitivity to the real optical potential

• Diffraction minima/maxima respond to Vᵥ

• Very good agreement at 8 and 11 MeV

• Larger model separation at 24 MeV, where folding Vᵥ is substantially deeper than default

Inelastic channels

• Depend on more than the ground-state real potential

• Transition form factors and coupling dynamics matter

• The spread among SHF/HFB curves is often smaller than the experiment–theory difference

• Therefore Vᵥ modification alone is not expected to reproduce all inelastic strengths

Controlled conclusion

The calculations isolate the reaction sensitivity to the folding-derived real depth Vᵥ rather than refitting the full optical model.

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Conclusions

Main messages of the ⁹⁰Zr analysis

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Microscopic densities

SHF-WS, HFB-S and HFB-G provide closely related but non-identical proton and neutron radial distributions.

2

Folding-derived Vᵥ

These density differences generate energy-dependent real volume depths through the single-folding procedure.

3

Controlled TALYS test

Only Vᵥ is replaced in TALYS 2.2; all remaining optical-model parameters stay at their default Koning–Delaroche values.

4

Elastic sensitivity

The elastic angular distributions are reproduced well at 8–11 MeV and become more sensitive to the microscopic Vᵥ choice at 24 MeV.

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Inelastic limitation

For the studied excited states, changing Vᵥ alone does not account for the full experimental strength, indicating sensitivity to transition/coupling dynamics beyond the ground-state real depth.

Microscopic density information can be propagated into reaction calculations in a transparent, parameter-controlled way.

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr

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Thank you

Questions and discussion

Key references / data sources

• Koning–Delaroche global optical model (TALYS default)

• TALYS 2.2 nuclear reaction code

• Experimental angular distributions from the data sets shown in the figures

• Microscopic SHF-WS, HFB-S and HFB-G ground-state densities used in this work

⁹⁰Zr

isarpun@gmail.com

INTW 2026 • Antalya, Türkiye

Microscopic folding analysis of ⁹⁰Zr