Double-Differential Cross Sections (DDCS)

• Double-differential cross sections are probability distributions for particle emission with specified energy and angle, essential for understanding nuclear reaction dynamics.
• The STLN framework decomposes DDCS into pre-equilibrium and compound nucleus components while rigorously conserving energy, angular momentum, and parity.
• Refined DDCS calculations, validated against experimental data, enhance nuclear data libraries for reactor physics, astrophysics, and material irradiation applications.

Double-differential cross sections (DDCS) are central quantitative observables in nuclear, particle, and atomic physics. Defined as distributions that characterize the probability for an outgoing particle to be emitted with specified energy and angle (and, in some contexts, for two correlated variables such as invariant mass and rapidity), the DDCS encapsulate the dynamical mechanisms of underlying interactions, conservation laws, and nuclear structure effects. Advances in the precise measurement and theoretical description of DDCS—especially for complex nuclear targets and outgoing composite particles—have been crucial for both fundamental insights and practical applications, ranging from material irradiation damage assessment to reactor neutron transport and nuclear astrophysics.

1. Fundamental Definition and Physical Significance

For a reaction $a + A \rightarrow b + B + \ldots$, the double-differential cross section

$\frac{d^2\sigma}{d\Omega\, d\varepsilon}$

quantifies the likelihood for emission of a particle $b$ with energy $\varepsilon$ into solid angle $\Omega$, measured in the appropriate frame (lab or center-of-mass). In many-body nuclear reactions, the DDCS is sensitive not only to the direct interaction but also to detailed properties of the compound and residual nuclei, including discrete and continuum energy level structure, angular momentum couplings, and cluster formations.

These distributions are essential for elucidating reaction mechanisms—distinguishing, for example, between direct, pre-equilibrium, and compound processes. The DDCS reveal features such as spectral peaks associated with transitions to specific nuclear levels, angular anisotropies due to conservation laws, and the relative roles of sequential versus simultaneous emission mechanisms. Their accurate calculation, especially for outgoing clusters (e.g., deuterons, tritons, α-particles), remains a theoretical challenge due to the need to respect all conservation laws in multistep emission scenarios and the complexity of nuclear level structures, as highlighted for p- and sd-shell nuclei (Cao et al., 19 Sep 2025).

2. Statistical Theory for Light Nuclear Reactions (STLN) and DDCS Formulation

The Statistical Theory for Light Nuclear reactions (STLN) provides a microscopic framework for the calculation of DDCS in nucleon-induced reactions on light nuclei. The formalism decomposes the overall DDCS into pre-equilibrium (exciton) and compound nucleus (equilibrium) emission components, with explicit conservation of energy, angular momentum, and parity. The DDCS for an outgoing particle $b$ in the center-of-mass frame is given by: $$\frac{d^2\sigma}{d\Omega^c\, d\varepsilon_b^c} = \sum_{n=3}^{n_{\text{max}}} \frac{d\sigma(n)}{d\varepsilon_b^c} A(n, \varepsilon_b^c, \Omega^c) + \frac{1}{4\pi} \frac{d\sigma}{d\varepsilon_b^c}$$ where $n$ indexes exciton configurations, and $A(n, \varepsilon_b^c, \Omega^c)$ introduces angular dependence via normalized expansions in Legendre polynomials $P_l(\cos\theta^c)$.

The energy spectrum for each phase space component includes discrete transitions to residual nuclear levels,

$$\frac{d\sigma(n)}{d\varepsilon_b^c} = \sum_{j\pi} \sigma_a^{j\pi} P^{j\pi}(n) \frac{W_b^{j\pi}(n, E^*, \varepsilon_b^c)}{W_T^{j\pi}(n, E^*)}$$

with terms enforcing conservation of angular momentum $j$, parity $\pi$, and including absorption cross sections, occupation probabilities, and emission rates $W_b^{j\pi}$ to discrete levels (Cao et al., 19 Sep 2025).

For emission to a specific residual level $k_1$, the emission rate reads: $$W_{b, k_1}^{j\pi}(n, E^*, \varepsilon_b^c) = \frac{1}{2\pi\, \hbar\, \omega^{j\pi}(n,E^*)} \sum_{S=|j_{k_1} - s_b|}^{j_{k_1} + s_b} \sum_{l=|j-S|}^{j+S} T_l(\varepsilon_b^c, k_1) g_l(\pi,\pi_{k_1}) F_{b[1,m]}(\varepsilon_b^c) Q_{b[1,m]}(n)$$ where $T_l$ is the optical model transmission coefficient, $g_l$ ensures parity and angular momentum conservation, and $F_{b[1,m]}$ is the pre-formation probability of clusters.

3. Treatment of Complex Particle Emission: Pick-Up Mechanism

The emission of composite charged particles (deuterons, tritons, α-particles) is governed by the cluster or “pick-up” mechanism, which describes the likelihood that these clusters are pre-formed within the compound nucleus prior to emission. The improved STLN framework applies an enhanced Iwamoto–Harada model, introducing an empirical parameterization for the pre-formation probability: $$F_{b[1,m]}(\varepsilon_b^c) = (a_1 + a_2 \varepsilon_f) + (b_1 + b_2 \varepsilon_f) \varepsilon + (c_1 + c_2 \varepsilon_f) \varepsilon^2$$ where $\varepsilon = \varepsilon_b^c + B_b$ ($B_b$ is the binding energy of particle $b$), and coefficients $a_i, b_i, c_i$ encode empirical fits (Cao et al., 19 Sep 2025). This mechanism is critical for reproducing not only the gross features, but also the fine discrete spectral peaks observed experimentally—these originate from direct transitions to discrete residual levels.

The combination of this refined cluster pick-up probability, exact conservation of quantum numbers, and detailed treatment of the residual nuclear level scheme leads to a self-consistent and realistic prediction of the DDCS spectra across all outgoing particle types.

4. Energy Levels, Angular Momentum, and Parity: Conservation Effects on DDCS

The complexity of sd-shell (and heavier) nuclei manifests in more intricate level schemes compared to 1p-shell systems, leading to higher-level density and a richer set of possible transition pathways. The STLN explicitly incorporates transitions to all energetically accessible discrete levels, enforcing that every emission step—both primary and secondary—satisfies conservation of energy, angular momentum, and parity:

• The sums over possible couplings in the emission rate (e.g., $S$, $l$) are determined from angular momentum algebra given the spins of the residual nucleus and the emitted particle.
• Parity conservation is enforced through factors $g_l(\pi,\pi_{k_1})$ to exclude disallowed transitions.
• The inclusion of these constraints naturally gives rise to the observed structure (discrete peaks, angular distributions) in the DDCS.

Especially for $n + \mathrm{^{19}F}$, the paper demonstrates that such detailed treatment is necessary for the accurate reproduction of the experimental DDCS spectra, particularly at outgoing particle energies and angles where transitions to discrete residual levels dominate. This feature is generally not captured by database evaluations (ENDF/B, JENDL, etc.) that assume isotropic angular distributions and smooth over discrete structure in the DDCS.

5. Comparison with Experiment and Major Data Libraries

Direct comparison of the improved STLN-based DDCS predictions with experimental measurements at neutron energies near 14.2 MeV demonstrates high fidelity in both the shape and magnitude of the angular and energy distributions for all light charged particle types ($p, d, t, \alpha$) (Cao et al., 19 Sep 2025). Features such as distinct peaks associated with specific residual nucleus levels (e.g., $^{19}$O, $^{18}$O, $^{17}$O, $^{16}$N) are well reproduced. In contrast, major evaluated data libraries, which often employ statistical models assuming angular isotropy and/or neglect discrete level structure in their DDCS evaluations, fail to account for these observed substructures—particularly at high outgoing energy where discrete emissions dominate.

This improved agreement is not restricted to a narrow energy range or a specific angle but extends over the full kinematically allowed regime, establishing the essential role of a detailed, conservation-law respecting, and microstate-resolving approach.

6. LUNF Code and ENDF-6 Data Generation for Practical Applications

The theoretical advances are realized in the LUNF code for $n + ^{19}\mathrm{F}$ reactions, which systematically generates ENDF-6 formatted DDCS datasets for all relevant emission channels and outgoing light composite particles. The code leverages the full improved STLN formalism, including the pick-up mechanism and the explicit energy level structure, integrating over initial neutron energies up to 20 MeV.

The resulting ENDF-6 files with high-fidelity DDCS are directly applicable in reactor physics, nuclear engineering simulations, and astrophysical reaction network calculations, where accurate predictions of secondary reaction products and energy/angle distributions are critical for neutron transport, material irradiation effects, and nuclear heating analysis.

7. Implications and Broader Impact

The refined prediction of DDCS for the $n + ^{19}\mathrm{F}$ system establishes a benchmark for future theoretical and evaluation efforts in light and medium-mass nuclei, especially as experimental capabilities continue to probe finer details of angular and energy correlations in emission spectra. The approach sets a standard for the incorporation of nuclear structure details, level-resolved transitions, and cluster emission mechanisms. It further highlights the limitations of currently adopted models in major nuclear data libraries and demonstrates the necessity of advancing microscopic theoretical treatments for accurate, application-ready nuclear data.

The methodology and findings have direct applications in the improvement of nuclear databases, nuclear technology development, and in the understanding of nuclear processes fundamental to astrophysics and basic science.

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Table: Key Formal Expressions Used in DDCS Calculation for n + $^{19}$F (Cao et al., 19 Sep 2025)

Formula Number	Expression/Description	Physical Content
(1)	$$\frac{d^2\sigma}{d\Omega^c\, d\varepsilon_b^c} = \sum_{n=3}^{n_{\max}} ...$$	General STLN DDCS for outgoing particle $b$ with energy/angular terms
(2)	$$\frac{d\sigma(n)}{d\varepsilon_b^c} = \sum_{j\pi}\sigma_a^{j\pi}...$$	Pre-equilibrium energy spectrum with conservation constraints
(3)	$W_{b, k_1}^{j\pi}(n, E^*, \varepsilon_b^c) = ...$	Emission rate to specific residual level, all conservations enforced
(4)	$A(n, \varepsilon_b^c, \Omega^c) = ...$	Angular distribution (sum of Legendre polynomials times coefficients)
(5)	$F_{b[1,m]}(\varepsilon_b^c) = ...$	Cluster pre-formation probability (pick-up mechanism)

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This comprehensive integration of nuclear structure, cluster formation dynamics, and exact quantum-number conservation establishes the current state-of-the-art for DDCS calculations and sets a new benchmark for data generation, evaluation, and the interpretation of experimental observables in nuclear reaction studies (Cao et al., 19 Sep 2025).