Weighted Proton-to-Neutron Yield Ratio Analysis

• Weighted proton-to-neutron yield ratios are observables that combine proton and neutron yields with tailored weights to cancel trivial factors such as volume and temperature, thereby revealing specific isospin-sensitive physics.
• They employ methodologies like coalescence models, density normalizations, and detector-response corrections to probe neutron density fluctuations, effective temperatures, and symmetry energy dynamics.
• Applications span heavy-ion collisions, laser-driven fusion, and thick-target nuclear reactions, highlighting challenges like model dependency and systematic corrections in experimental analysis.

A weighted proton-to-neutron yield ratio is not a single universal observable but a family of ratio constructions in which proton-associated and neutron-associated yields are combined with additional weights—most commonly coalescence factors, density normalizations, efficiency and solid-angle corrections, or phase-space weights—to suppress trivial dependencies and isolate specific isospin-sensitive physics. In heavy-ion coalescence studies, the observable that most directly plays this role is $R \equiv N_tN_p/N_d^2$, which connects proton, deuteron, and triton yields to neutron density fluctuations and neutron–proton correlations (Zhang et al., 2022). In laser-driven fusion, the density-weighted ratio $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$ is used to infer an effective plasma temperature (Zhu et al., 13 Aug 2025). In low-energy reaction-yield studies, the central quantity can instead be the corrected neutron-to-proton thick-target yield ratio $Y_n/Y_p$, valued because common factors in beam current, target thickness, and irradiation time cancel in the ratio (Li et al., 2019).

1. Scope and principal definitions

Across the cited literature, the phrase is used for several non-identical observables. The common feature is that the ratio is not a bare $N_p/N_n$: it is deliberately weighted to encode coalescence, reaction geometry, detector response, density, or fragment composition. This makes the ratio useful in settings as different as relativistic heavy-ion collisions, intermediate-energy transport studies, thick-target resonance measurements, ionization detectors, and laser-driven fusion plasmas (Zhang et al., 2022, Zhu et al., 13 Aug 2025, Li et al., 2019, Yang et al., 2021, Ma et al., 2011).

Context	Observable	Weighting principle
Heavy-ion coalescence	$N_tN_p/N_d^2$	Cancels volume and temperature; retains neutron fluctuations and $np$ correlations
Laser-driven fusion	$(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$	Density normalization of proton and neutron fusion yields
Thick-target reactions	$Y_n/Y_p$ or inverse	Efficiency, solid angle, and energy-loss weighting
Intermediate-energy heavy ions	$R(n/p)$, $R_{ci}(n/p)$	Energy and coalescence-invariant weighting
Isobaric fragmentation	$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$0	Fixed-$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$1 substitution of one proton by one neutron

A persistent source of confusion is the assumption that all such ratios probe the same physics. They do not. In some cases the observable is a coalescence proxy for neutron density fluctuations; in others it is a transport observable for the symmetry energy and neutron–proton effective mass splitting; in others it is a detector-response ratio or a temperature diagnostic. The weighting determines what information survives the cancellations.

2. Coalescence-based weighted ratios in heavy-ion collisions

In the coalescence framework of light-nucleus production, the ratio

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$2

is the central weighted proton-to-neutron yield ratio. Here $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$3, $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$4, and $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$5 are the proton, deuteron, and triton yields. The basic reason this combination is singled out is that deuterons probe $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$6 coalescence whereas tritons probe $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$7 coalescence, so the triton contribution weights neutron content more strongly than proton content (Zhang et al., 2022).

The underlying fluctuation variables are the relative neutron density fluctuation

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$8

and the neutron–proton correlation

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$9

In the AMPT treatment, $Y_n/Y_p$0, $Y_n/Y_p$1, and $Y_n/Y_p$2 is volume independent. The coalescence yields are written as

$Y_n/Y_p$3

$Y_n/Y_p$4

with the three-nucleon correlation $Y_n/Y_p$5 neglected (Zhang et al., 2022).

Combining these expressions eliminates the explicit temperature and volume dependence: $Y_n/Y_p$6 The AMPT study further shows that $Y_n/Y_p$7 increases monotonically with $Y_n/Y_p$8, while for positive $Y_n/Y_p$9 and $N_p/N_n$0 it decreases with increasing $N_p/N_n$1 correlation. In the special case $N_p/N_n$2,

$N_p/N_n$3

so $N_p/N_n$4 becomes a direct proxy for the relative neutron density fluctuation (Zhang et al., 2022).

The Angantyr study rewrites the same logic with $N_p/N_n$5 for the relative neutron density fluctuation and $N_p/N_n$6 for the neutron–proton density correlation coefficient: $N_p/N_n$7 leading to

$N_p/N_n$8

That study emphasizes that $N_p/N_n$9 should be treated as a weighted measure of both neutron fluctuations and proton–neutron correlations rather than as a pure neutron-fluctuation observable (Zuman et al., 2023).

In this coalescence setting, the designation “weighted proton-to-neutron yield ratio” is literal. The ratio includes $N_tN_p/N_d^2$0 explicitly, includes $N_tN_p/N_d^2$1 and $N_tN_p/N_d^2$2 yields as $N_tN_p/N_d^2$3 and $N_tN_p/N_d^2$4 coalescence products, and is designed to cancel trivial volume and temperature factors while isolating neutron-side fluctuation physics (Zhang et al., 2022).

3. Baselines, systematics, and QCD phase interpretation

The phenomenological value of $N_tN_p/N_d^2$5 depends on what constitutes the non-critical baseline. In AMPT, the ratio decreases with rapidity coverage, increases with collision centrality, and increases slightly and monotonically with collision energy in Au+Au collisions from $N_tN_p/N_d^2$6 to 200 GeV; it does not exhibit any non-monotonic behavior in collision energy dependence (Zhang et al., 2022). AMPT also shows that the impact of $N_tN_p/N_d^2$7 is centrality dependent: in central 200 GeV collisions $N_tN_p/N_d^2$8 nearly vanishes, whereas in peripheral 39 GeV collisions it is sizable, so neglecting $N_tN_p/N_d^2$9 in peripheral collisions would underestimate $np$0 (Zhang et al., 2022).

Angantyr provides a different hadronic baseline. In that model, the light-nuclei yield ratio remains unchanged even as the rapidity coverage and collision centrality increase, and it experiences only a slight increase with increasing collision energy. The same study finds that the effect of color reconnection is entirely dependent on the presence of multiple-parton interactions; color reconnection has no impact on the yield ratio if MPI is off (Zuman et al., 2023). A plausible implication is that baseline systematics are themselves model dependent even before any critical contribution is invoked.

The first-order chiral phase-transition study makes the contrast explicit. In that transport calculation, the trajectory with $np$1 enters the spinodal region, the net-baryon scaled density moment $np$2 grows to about 2, and the predicted light-nucleus ratio reaches

$np$3

compared with

$np$4

for the crossover case and $np$5 for a uniform-density baseline (Sun et al., 2020). In that framework the enhancement is tied to spinodal amplification of density inhomogeneities.

The experimental motivation for treating the ratio as a phase-structure probe comes from the contrast between these monotonic baselines and the reported non-monotonic energy dependence. Both the AMPT and Angantyr baselines fail to reproduce the pronounced peak around $np$6–30 GeV seen in STAR and NA49 data, while the first-order-transition calculation interprets an enhanced $np$7 as a consequence of amplified baryon density fluctuations (Zhang et al., 2022, Zuman et al., 2023, Sun et al., 2020). The central controversy is therefore not whether the ratio is fluctuation sensitive, but which component of the fluctuation spectrum—non-critical transport, coalescence dynamics, or critical/phase-transition physics—dominates the observed energy dependence.

4. Direct neutron-to-proton yield ratios in nuclear reactions and intermediate-energy transport

Outside the light-nucleus coalescence context, weighted proton-to-neutron yield ratios appear in more direct forms. In thick-target studies of $np$8 and $np$9, the neutron and proton yields are

$(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$0

so the central observable is the thick-target neutron-to-proton yield ratio

$(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$1

Because beam current, target thickness, and irradiation time are common, the ratio is effectively an energy-weighted ratio of cross sections as the deuteron slows in the target. After correcting neutron counts by detector efficiency and both channels by their solid angles, the ratio suppresses major systematics and highlights resonance structure. In this system the ratio was used to identify resonances at 1.4, 1.7, and 2.5 MeV in $(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$2 and at 1.6 and 2.7 MeV in $(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$3 (Li et al., 2019).

In intermediate-energy heavy-ion transport, the basic observables are the free-nucleon ratio

$(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$4

and the coalescence-invariant ratio

$(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$5

where $(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$6 and $(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$7 are built by summing cluster yields with neutron and proton multiplicity weights up to $(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$8, $(N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$9. In the ImQMD-L analysis of $Y_n/Y_p$0 at 200 MeV per nucleon, the low-kinetic-energy region of $Y_n/Y_p$1 is proposed as a probe of the slope of the symmetry energy $Y_n/Y_p$2, while the inclination of $Y_n/Y_p$3 with respect to $Y_n/Y_p$4 is proposed as a probe of the neutron–proton effective mass splitting. With fixed effective mass splitting, the high-kinetic-energy part can also be used to learn the symmetry energy at suprasaturation density (Yang et al., 2021).

The same general observable generates a long-standing transport-theory tension. In IBUU11 with an improved isospin- and momentum-dependent interaction, the neutron–proton effective mass splitting $Y_n/Y_p$5 has a relatively stronger effect than the density dependence of the symmetry energy $Y_n/Y_p$6, and the assumption $Y_n/Y_p$7 leads to a higher neutron/proton ratio. Yet calculations using $Y_n/Y_p$8 and $Y_n/Y_p$9 within their current uncertainty ranges remain too low compared with the NSCL/MSU double neutron/proton ratio data, prompting the suggestion that additional mechanisms are required (Kong et al., 2015).

A more algebraic weighting appears in projectile-fragmentation isobaric yield ratios. In the modified Fisher model,

$R(n/p)$0

where $R(n/p)$1 and fixed $R(n/p)$2 means that moving from $R(n/p)$3 to $R(n/p)$4 corresponds to replacing one proton with one neutron. The logarithm of this ratio depends on $R(n/p)$5, $R(n/p)$6, $R(n/p)$7, and $R(n/p)$8, so it acts as a weighted proton-to-neutron substitution observable rather than a free-nucleon yield ratio (Ma et al., 2011).

5. Fragment substitutions, two-dimensional yields, and odd–even structure

When yields are resolved in fragment proton and neutron number, weighting can be built directly into the fragment distribution. In the generalized Brownian shape-motion approach to fission, the central object is the two-dimensional fragment yield $R(n/p)$9, generated by random walks on a macroscopic–microscopic potential-energy surface that depends on elongation, neck, fragment deformations, and the proton and neutron numbers in each fragment. The extension introduces fragment-specific odd–even staggering in both variables and also allows correlated transfer of nucleon pairs in one step, in addition to sequential transfer (Moller et al., 2015).

This framework does not define a single proton-to-neutron yield ratio, but it makes such ratios natural to construct. Marginal yields,

$R_{ci}(n/p)$0

permit weighted quantities such as global averages of $R_{ci}(n/p)$1 and $R_{ci}(n/p)$2, heavy-fragment-restricted averages, or ratios over selected domains in $R_{ci}(n/p)$3 and $R_{ci}(n/p)$4. The significance of this construction is that the ratio then inherits explicit pairing and shell effects. Because the model raises the potential of odd-$R_{ci}(n/p)$5 and odd-$R_{ci}(n/p)$6 fragment splits and allows pair-transfer steps that can bypass odd configurations, the resulting $R_{ci}(n/p)$7 should display odd–even staggering in both variables (Moller et al., 2015).

A related but simpler substitution logic underlies isobaric yield ratios in projectile fragmentation. At fixed mass number $R_{ci}(n/p)$8, the ratio $R_{ci}(n/p)$9 is exactly the relative yield of a fragment in which one proton has been replaced by one neutron. This is already a weighted proton-to-neutron yield ratio in a strict combinatorial sense: the weighting is supplied by the symmetry, Coulomb, pairing, and chemical-potential terms of the modified Fisher model (Ma et al., 2011).

These fragment-based constructions make clear that “proton-to-neutron yield ratio” need not refer to free nucleons at all. In fragmentation and fission, the ratio can be embedded in the way yields reorganize across integer changes in $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$00 and $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$01, and the dominant weights can be symmetry energy, shell structure, and pairing rather than kinematic acceptance or density normalization.

6. Detector-response and fusion-plasma diagnostics

In detector physics, the phrase becomes more operational. The tetramethylsilane time-projection-chamber study does not define an explicit proton-to-neutron yield ratio, but it provides the ingredients needed to construct one conceptually. The measured quantity is the ionization charge yield from neutron-induced proton recoils, and the paper explicitly defines the proton-to-electron quenching factor

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$02

A possible weighted proton-to-neutron yield ratio is then the average ionization yield per incident neutron,

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$03

or the counting ratio

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$04

where $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$05 is the neutron detection efficiency in the proton-recoil channel. The study reports proton recoil quenching factors from $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$06 at 3.9 kV/cm to $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$07 at 5.8 kV/cm, showing that proton ionization yield is strongly quenched relative to electron recoils (Wu et al., 2021).

The laser-driven deuterium–helium-3 plasma study uses an explicit density-weighted proton-to-neutron yield ratio as its central diagnostic: $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$08 Here the proton-producing channel is $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$09, while neutrons come from $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$10. In a thermal model,

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$11

Using PIC plus stochastic fusion, that study finds $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$12, corresponding to an effective temperature $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$13 keV, whereas a Maxwell–Boltzmann fit to the low-energy part of the final deuteron spectrum gives $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$14 keV (Zhu et al., 13 Aug 2025).

The discrepancy is physically important. Because the deuteron numbers in deuterium clusters follow a log-normal distribution, the low-energy part of the final deuteron spectrum can be fitted by a Maxwell–Boltzmann distribution, while there are more deuterons in the intermediate- and high-energy region compared to a thermal distribution, and those deuterons dominate the weighted yield ratio. The same study shows that local density fluctuation, intrinsic to the log-normal cluster-size distribution, further enhances hot deuteron–deuteron collisions and significantly affects the weighted yield ratio (Zhu et al., 13 Aug 2025). A common misconception is therefore that a yield-ratio thermometer necessarily measures the same temperature as a spectral fit; in this case it does not.

7. Structure-sensitive and geometric analogues

Several related observables extend the logic of weighted proton-to-neutron ratios into nuclear structure and collision geometry. In quasielastic electron scattering on $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$15 mirror nuclei, the ratio of $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$16 to $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$17 cross sections is used as a proxy for the proton-to-neutron momentum-distribution ratio in $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$18: $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$19 A weighted version divides the yields by luminosity, kinematic factors, and the off-shell electron–proton cross section. In the proposal discussed, the ratio is expected to fall from about 3 at low momentum to about 1 in the 300–500 MeV/$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$20 region, reflecting the transition from mean-field dominance to $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$21-SRC dominance (Hen et al., 2014).

For proton-rich nuclei, the ratio method compares breakup to summed quasi-elastic angular distributions,

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$22

to reduce reaction-mechanism dependence and isolate projectile structure. For one-proton halos such as $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$23 and $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$24, this remains structurally informative, but it works less well than for neutron halos because the Coulomb interaction between the valence proton and the target is non-negligible (Yun et al., 2018). This is a ratio of proton-rich to reaction-normalized yields rather than a direct proton-to-neutron ratio, but it follows the same weighting philosophy.

At much higher energy, proton–lead collisions admit a geometric analogue. In the neutron-skin study of $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$25 production, the event generator tracks the average numbers of hard interactions on protons and neutrons in a centrality bin,

$$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$26

and these weighted proton and neutron interaction counts enter the centrality-dependent $$R_{p/n}^{\mathrm{(w)}} \equiv (N_p/N_n)(\rho_D/\rho_{^3\mathrm{He}})$$27 yields. The study qualitatively confirms the neutron-skin expectation that peripheral collisions preferentially sample neutrons, but with a realistic centrality trigger the effect is a factor of two smaller than the original estimate (Alvioli et al., 2018).

These analogues clarify a broader point. A weighted proton-to-neutron yield ratio need not be a literal quotient of free proton and neutron counts. It can be a coalescence ratio, a fragment-substitution ratio, a response-normalized yield, a detector-efficiency ratio, a momentum-distribution proxy, or an effective proton-versus-neutron interaction count. What defines the class is the deliberate use of weights to remove one set of trivial dependencies and expose another, more specific layer of isospin-sensitive physics.