---
title: Proton vs Deuteron Hops (PH)
url: https://www.emergentmind.com/topics/proton-or-deuteron-hops-ph
type: topic
---

# Proton vs Deuteron Hops (PH)

Searching arXiv for the specified paper and closely related knockout-reaction work.
arXiv search query: 2512.17236 one-proton knockout deuteron proton DWIA-BU
Proton or Deuteron Hops (PH) denotes the comparison between proton-induced and deuteron-induced one-proton knockout reactions, particularly the question of when a deuteron projectile yields a larger inclusive one-proton knockout cross section than a proton projectile. In the formulation analyzed in "Systematic analysis of proton- and deuteron-induced one-proton knockout reactions" [2512.17236], the relevant processes are proton-induced $A(p,2p)B$ and deuteron-induced $A(d,xp)B$, with the latter including both $A(d,dp)B$ and breakup-assisted $A(d,d^{*}p)B$. The central result is that the deuteron-to-proton knockout ratio is set by a competition between a larger elementary deuteron–proton strength and stronger deuteron absorption in the nucleus: the former tends to raise the ratio toward $\approx 3.5$, whereas the latter reduces it to $\approx 1.4$–$1.5$, consistent with experiment.

## 1. Reaction channels, observables, and empirical motivation

The PH problem is formulated for one-proton knockout in inverse kinematics at about $240\ \mathrm{MeV}/u$, with energy averaging over $150$–$240\ \mathrm{MeV}/u$ to emulate energy loss in the target. The systems studied are neutron-rich nuclei in the K–Ca–Sc–Ti–V region, specifically 12 nuclei from $^{51}\mathrm{K}$ to $^{62}\mathrm{V}$. The experimental motivation is the RIKEN SEASTAR finding that the ratio of deuteron-induced to proton-induced one-proton knockout cross sections is $\approx 1.5$ at $\sim 240\ \mathrm{MeV}/u$ [2512.17236].

For proton projectiles, the process is the quasi-free scattering reaction
$$
A(p,2p)B,
$$
which removes a bound proton from $A$, leaving the residue $B$ and two outgoing protons. For deuteron projectiles, the bound-channel reaction is
$$
A(d,dp)B,
$$
and the breakup-assisted contribution is
$$
A(d,d^{*}p)B,
$$
which is described at the elementary level as $A(d,pn)B$ with the outgoing proton detected. The paper collectively denotes these deuteron-induced channels as
$$
A(d,xp)B.
$$

The principal observables are the exclusive triple-differential cross sections
$$
\frac{d^3\sigma}{dE_1^L d\Omega_1^L d\Omega_2^L},
$$
and the inclusive knockout cross section $\sigma$ obtained by integrating the triple-differential cross section over detected proton energies and angles. Laboratory-frame variables carry the superscript $L$, while internal variables are defined in the center-of-mass frame. Missing momentum and Jacobians are treated explicitly.

The empirical issue is not merely that deuteron-induced knockout is larger than proton-induced knockout, but that the enhancement is only moderate. A naive reading of the elementary collision strength might suggest a much larger ratio. The analysis resolves this by showing that the inclusive knockout ratio is governed by both the elementary $dp$ versus $pp$ strength and the different attenuation of deuteron and proton distorted waves in the nuclear medium.

## 2. DWIA formulation for proton- and deuteron-induced knockout

The basic reaction theory is the standard distorted-wave impulse approximation (DWIA) for $A(p,2p)B$ and $A(d,dp)B$ [2512.17236]. In factorized form, the transition amplitude is
$$
T_c = \langle \chi^{(-)}_{1, K1}(R1) \chi^{(-)}_{2, K2}(R2) \mid t_{cp}(s) \mid \phi_p(R2) \chi^{(+)}_{0, K0}(R0) \rangle,
$$
where $c=p$ or $d$ denotes the incident particle, $\chi$ are distorted waves for incoming and outgoing particles, $\phi_p$ is the bound-state wave function of the struck proton, and $t_{cp}$ is the effective $c$–$p$ interaction.

Using the asymptotic momentum approximation (AMA) for short-range propagation,
$$
\chi_i(R + \Delta R) \approx \chi_i(R)e^{iK_i \cdot \Delta R},
$$
the amplitude becomes
$$
T_c = \tilde{t}_{cp}(\kappa', \kappa)\int dR\, \chi^{(-)*}_{1,K1}(R)\chi^{(-)*}_{2,K2}(R)\chi^{(+)}_{0,K0}(R)e^{-iK_0\cdot R/A}\phi_p(R),
$$
with
$$
\tilde{t}_{cp}(\kappa', \kappa)=\int ds\, e^{-i\kappa'\cdot s} t_{cp}(s)e^{i\kappa\cdot s}.
$$

The on-the-energy-shell approximation relates the squared $t$-matrix to the elementary elastic differential cross section:
$$
\left( \frac{\mu_{cp}}{2\pi\hbar^2} \right)^2 |\tilde{t}_{cp}(\kappa', \kappa)|^2 \approx \frac{d\sigma_{cp}}{d\Omega_{cp}}(\theta^{cp}, E^c).
$$

The triple-differential cross section is then written as
$$
\frac{d^3\sigma_c^{\mathrm{DWIA}}}{dE_1^L d\Omega_1^L d\Omega_2^L}
= \left( \frac{(2\pi)^4}{\hbar v} \right) S_p F_{\mathrm{kin}}^L
\times \frac{E_1E_2E_B}{E_1^L E_2^L E_B^L}
\times \frac{1}{2l+1}
\times \left( \frac{2\pi\hbar^2}{\mu_{cp}} \right)^2
\left( \frac{d\sigma_{cp}}{d\Omega_{cp}} \right)(\theta^{cp}, E^c)
\times |\bar{T}_c|^2,
$$
where
$$
\bar{T}_c = \int dR\, \chi^{(-)*}_{1,K1}(R)\chi^{(-)*}_{2,K2}(R)\chi^{(+)}_{0,K0}(R)\phi_p(R),
$$
and the Jacobian factor is
$$
F_{\mathrm{kin}}^L
= \frac{E_1^L K_1^L E_2^L K_2^L}{(\hbar c)^2}
\left[
1 + \frac{E_2^L}{E_B^L}
- \frac{E_2^L}{E_B^L}\frac{q^L\cdot K_2^L}{(K_2^L)^2}
\right]^{-1},
$$
with
$$
q^L = K_0^L + K_A^L - K_1^L.
$$

The inclusive knockout cross section is
$$
\sigma_c^{\mathrm{DWIA}}
= \int dE_1^L d\Omega_1^L d\Omega_2^L\,
\frac{d^3\sigma_c^{\mathrm{DWIA}}}{dE_1^L d\Omega_1^L d\Omega_2^L}.
$$

This formulation makes the structure–reaction factorization explicit. The spectroscopic factor $S_p$ carries the overlap strength, the elementary two-body process appears through $d\sigma_{cp}/d\Omega_{cp}$, the attenuation and distortion enter through $|\bar{T}_c|^2$, and kinematic phase-space effects enter through $F_{\mathrm{kin}}^L$ and the relativistic energy factors.

## 3. Breakup-inclusive extension: DWIA-BU

The defining methodological step for PH is the extension from ordinary DWIA to DWIA-BU for deuteron-induced knockout [2512.17236]. In this approach, deuteron breakup is incorporated at the elementary level by replacing the elastic $dp$ elementary cross section with the sum of elastic and breakup contributions:
$$
\frac{d\sigma_{dp}}{d\Omega_{dp}}(\theta^{dp}, E^d)
\rightarrow
\frac{d\sigma_{dp}}{d\Omega_{dp}}(\theta^{dp}, E^d)
+
\frac{\sigma_{dp}^{\mathrm{BU}}(E^d)}{4\pi}.
$$

The total deuteron-induced knockout cross section is therefore
$$
\sigma_d^{\mathrm{DWIA-BU}}
=
\sigma_d^{\mathrm{DWIA}}
+
\sigma_{dd^{*}p}^{\mathrm{DWIA}},
$$
where the first term corresponds to $A(d,dp)B$ and the second to $A(d,d^{*}p)B$.

The paper introduces three ratios:
$$
\mathcal{R}^{\mathrm{DWIA-BU}}
\equiv
\frac{\sigma_d^{\mathrm{DWIA-BU}}}{\sigma_p^{\mathrm{DWIA}}},
\qquad
\mathcal{R}^{\mathrm{DWIA}}
\equiv
\frac{\sigma_d^{\mathrm{DWIA}}}{\sigma_p^{\mathrm{DWIA}}},
\qquad
\mathcal{R}^{\mathrm{PWIA-BU}}
\equiv
\frac{\sigma_d^{\mathrm{PWIA-BU}}}{\sigma_p^{\mathrm{PWIA}}}.
$$

A central finding is that breakup is essential. The ratio without breakup, $\mathcal{R}^{\mathrm{DWIA}}$, is only $\approx 0.2$, which significantly underestimates the measured value. By contrast, including breakup produces $\mathcal{R}^{\mathrm{DWIA-BU}} \approx 1.4$, in reasonable accord with the experimental $\approx 1.5$.

The isotropic approximation used for the breakup term was checked against angular-dependent $dp\to p+n$ breakup input from a previous study. For the inclusive $\sigma$, the ratio of the angular-dependent treatment to the isotropic one is $\approx 0.93$ at $250\ \mathrm{MeV}/u$, implying only a few-percent effect. This supports the use of the isotropic breakup approximation for inclusive one-proton knockout, although it does not imply that the same approximation would be sufficient for fully differential observables.

## 4. Ratio factorization and the mechanism setting the PH enhancement

The paper’s most important conceptual result is an approximate factorization of the deuteron-to-proton knockout ratio into an elementary-cross-section factor and an attenuation factor [2512.17236]. Starting from a discrete-bin representation with isotropic elementary processes,
$$
\bar{\sigma}_c \approx \sum_i \bar{\sigma}_i^c,
\qquad
\bar{\sigma}_i^c
=
\left[
\frac{\sigma_{cp}^{\mathrm{tot}}(E_i^c)}{4\pi}
\right]
w_i^c,
$$
one groups configurations into bins of elementary energy $\epsilon_j$:
$$
\bar{\sigma}_c
\approx
\sum_j
\left[
\frac{\sigma_{cp}^{\mathrm{tot}}(\epsilon_j)}{4\pi}
\right]
W_j^c,
\qquad
W_j^c
=
\sum_{i\,|\,E_i^c\in[\epsilon_j-\Delta\epsilon,\epsilon_j+\Delta\epsilon]} w_i^c.
$$

In the plane-wave limit, the weights are nearly identical for proton- and deuteron-induced reactions,
$$
W_j^p \approx W_j^d,
$$
so that
$$
\mathcal{R}^{\mathrm{PWIA-BU}}
\approx
\frac{\sigma_{dp}^{\mathrm{tot}}}{\sigma_{pp}^{\mathrm{tot}}}
\approx 3.5.
$$
This ratio is weakly dependent on energy in the range considered and is essentially independent of nucleus.

Once distorted waves are restored, the ratio acquires an attenuation factor:
$$
\mathcal{R}
\equiv
\frac{\sigma_d}{\sigma_p}
\approx
\left[
\frac{\sigma_{dp}^{\mathrm{tot}}}{\sigma_{pp}^{\mathrm{tot}}}
\right]
\times
\left[
\frac{\sum_j W_j^d}{\sum_j W_j^p}
\right].
$$
In the energy region where the weights are concentrated,
$$
\frac{W_j^d}{W_j^p} \approx 0.4,
$$
because the deuteron is more strongly absorbed than the proton. The resulting estimate is
$$
\mathcal{R}^{\mathrm{DWIA-BU}}
\approx
3.5 \times 0.4
\approx 1.4,
$$
which matches the measured $\sim 1.5$.

This resolves a common misconception: the larger deuteron-induced yield is not determined solely by the deuteron’s larger elementary interaction space. The inclusive ratio is instead set by two competing effects. First, the deuteron has both $p$–$p$ and $n$–$p$ interactions available in the underlying quasi-free collision, and breakup channels add to the inclusive yield. Second, the deuteron distorted wave is more strongly attenuated by the optical potential, making the reaction more surface-dominated.

The paper also provides an optical-theorem-based derivation of the elementary ratio. Under the simplifying assumptions $t_{np}\approx t_{pp}$, a $\delta$-like deuteron wave function, and neglected Coulomb effects,
$$
f_{pp}(0)
=
-
\left(
\frac{\mu_{pp}}{2\pi\hbar^2}
\right)
\int dR\, t_{pp}(R),
$$
$$
f_{dp}(0)
=
-
\left(
\frac{\mu_{dp}}{2\pi\hbar^2}
\right)
2\int dR\, t_{pp}(R),
$$
and with
$$
\mathrm{Im}\,f(0)=\frac{C_{\mathrm{id}}k}{4\pi}\sigma^{\mathrm{tot}},
$$
one obtains
$$
\frac{\sigma_{dp}^{\mathrm{tot}}}{\sigma_{pp}^{\mathrm{tot}}}\approx 4
$$
for equal energies per nucleon. Empirically, however, the fitted value is $\approx 3.5$ across $5$–$800\ \mathrm{MeV}$, and this empirical value is what controls the PWIA-BU ratio.

## 5. Systematics, orbital dependence, and robustness of the ratio

The numerical systematics of PH can be summarized as follows [2512.17236].

| Quantity | Value | Interpretation |
|---|---:|---|
| Experimental deuteron-to-proton ratio | $\approx 1.5$ | Measured at $\approx 240\ \mathrm{MeV}/u$ |
| $\mathcal{R}^{\mathrm{DWIA-BU}}$ | $\approx 1.4$ | Agrees reasonably with experiment |
| $\mathcal{R}^{\mathrm{DWIA}}$ | $\approx 0.2$ | Breakup omitted; severe underestimate |
| $\mathcal{R}^{\mathrm{PWIA-BU}}$ | $\approx 3.5$ | Elementary-strength limit |
| $W^d/W^p$ in DWIA-BU | $\approx 0.4$ | Attenuation reduction from deuteron absorption |

Averaging over $150$–$240\ \mathrm{MeV}/u$ changes the ratio negligibly compared with a calculation at $240\ \mathrm{MeV}/u$. This is attributed to the weak variation of both $\sigma_{dp}^{\mathrm{tot}}/\sigma_{pp}^{\mathrm{tot}}$ and the distortion balance over that interval.

The dependence on the removed proton’s orbital angular momentum $l$ is a notable structural effect. For $^{58}\mathrm{Ti}$, both $\sigma_p$ and $\sigma_d$ increase with $l$, but $\sigma_d$ increases faster, so $\mathcal{R}^{\mathrm{DWIA-BU}}$ also increases with $l$. The stated mechanism is geometric: higher-$l$ proton orbits are more peripheral, their radial wave functions peak further out, and the reduction in attenuation benefits the more strongly absorbed deuteron disproportionately. A plausible implication is that PH becomes progressively more favorable for peripheral proton removal.

The ratio is also robust against spectroscopic-factor uncertainties because $S_p$ appears in both numerator and denominator and therefore cancels in $\mathcal{R}$ by construction. This makes the ratio a cleaner observable than the absolute knockout cross section.

The reaction-model ingredients used to generate these results are standard but specific. Proton distorted waves employ the EDAD1 Dirac optical potential. Deuteron distorted waves are generated by folding proton and neutron optical potentials with the deuteron density. Nonlocality corrections are included through Darwin factors for protons and Perey factors for deuterons. The study does not provide a quantitative uncertainty band from varying optical potentials, but its central qualitative conclusion rests on the consistent appearance of $W^d/W^p \approx 0.4$ with these choices.

## 6. Experimental use, modeling requirements, and open issues

For applications, the paper gives explicit practical guidance on choosing proton versus deuteron projectiles in one-proton knockout [2512.17236]. Around $200$–$250\ \mathrm{MeV}/u$, switching from a proton to a deuteron is expected to increase the inclusive one-proton knockout cross section by a factor $\approx 1.4$–$1.5$ for the neutron-rich mid-mass nuclei studied. Equivalently, deuteron projectiles provide roughly $40$–$50\%$ higher inclusive yields, and the gain increases somewhat for higher-$l$ proton removal.

This advantage comes with constraints. Breakup channels are intrinsic to deuteron-induced reactions and enhance the inclusive yield, but they also generate more complex final states and potential backgrounds in exclusive analyses. Because deuterons are more strongly absorbed, the reaction becomes more surface-dominated. For deeply bound, low-$l$ proton orbits, the relative advantage of the deuteron can therefore be reduced.

The study’s limitations are explicit. Breakup is treated inclusively through the isotropic term $\sigma_{dp}^{\mathrm{BU}}/(4\pi)$; although this is accurate to a few percent for inclusive $\sigma$ at $250\ \mathrm{MeV}/u$, fully differential observables would require explicit angular dependence, with CDCC or IAV/UT inclusive-breakup formalisms given as examples. The calculations assume the residue remains in the ground state and that a single dominant proton orbit describes each nucleus; full configuration mixing and decay feeding are not included. Two-proton knockout, for which a very large $d/p$ ratio has been reported experimentally, lies outside the present scope. The on-shell and AMA approximations remain standard DWIA ingredients, but off-shell effects and nonlocalities beyond Darwin and Perey factors are not explored.

Within these bounds, the PH picture is internally consistent. The larger deuteron-induced one-proton knockout cross section is not anomalous and does not require a novel reaction mechanism. It follows from a quantitatively separable balance: an elementary deuteron advantage of $\approx 3.5$ and an attenuation penalty of $\approx 0.4$. The resulting product, $\approx 1.4$, explains why deuteron-induced one-proton knockout is more efficient than proton-induced knockout while remaining far below the naive elementary limit.

Source: https://www.emergentmind.com/topics/proton-or-deuteron-hops-ph