⚛️ nuclear theory

Coulomb and nuclear polarization of the nucleon density in the dinuclear configuration of the $^{40}$ Ca + $^{208}$ Pb system

This study investigates the Coulomb and nuclear polarization of nucleon density in the $^{40}$ Ca + $^{208}$ Pb system using a deformed Woods-Saxon mean field, revealing that the repulsive Coulomb tidal field dominates the induced oblate deformation while the attractive nuclear field partially counteracts it at small separations, with polarization energy primarily driven by dipole rather than quadrupole effects.

Original authors: Bakhodir Kayumov

Published 2026-07-31

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CC BY 4.0

Original authors: Bakhodir Kayumov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic Dance of Atomic Giants

Imagine the universe as a grand ballroom where the smallest building blocks of matter, called atomic nuclei, are constantly waltzing, bumping, and sometimes merging together. This is the world of nuclear physics, a field dedicated to understanding how these tiny, super-dense spheres of protons and neutrons behave when they get close to one another. Two of the most important forces in this dance are the "Coulomb force," which acts like a powerful magnet pushing two positively charged objects apart, and the "nuclear force," a sticky glue that only works when the objects are practically touching, pulling them together.

Scientists often study what happens when two heavy nuclei, like a giant lead ball and a smaller calcium ball, approach each other to see if they will fuse into a new, super-heavy element. To predict the outcome, they usually pretend the nuclei are like rigid, unchanging billiard balls that keep their shape no matter how close they get. But in reality, nuclei are more like soft, squishy water balloons. When a giant balloon gets near another, the electric push and the sticky pull should theoretically squish and stretch it, changing its shape before they even touch. The big question is: does this squishing actually matter enough to change the rules of the dance, or is the "rigid ball" idea good enough for now?

The Squishy Ball Experiment

In this study, a researcher named Bakhodir Kayumov decided to test the "rigid ball" assumption by treating the nuclei as the squishy water balloons they really are. He set up a computer simulation of a specific dance partner: a heavy lead nucleus (208Pb) and a lighter calcium nucleus (40Ca). He wanted to see exactly how the presence of one partner changes the shape and density of the other as they approach, without them actually touching or merging yet.

To do this, he didn't just look at the nuclei from the outside; he zoomed in on the individual particles inside. He calculated how the "cloud" of protons and neutrons inside the lead nucleus shifts when the calcium nucleus gets close, and vice versa. He treated the interaction as a two-part force: the electric push (Coulomb) and the nuclear glue (nuclear force). He ran the numbers for distances ranging from 20 femtometers down to 15.5 femtometers (a femtometer is a quadrillionth of a meter, so imagine the distance between two grains of sand if they were scaled up to the size of the solar system).

The Findings: Squishy, Not Stiff

The results revealed that the nuclei are indeed squishy, but the way they squish is a bit surprising. As the two nuclei approached, the electric repulsion acted like a strong, invisible hand pushing the protons away from the other nucleus. This caused the nuclei to flatten out, like a pancake being pressed from the top and bottom, rather than stretching out like a rubber band. This "pancake" shape is called an oblate deformation.

Interestingly, the "nuclear glue" tried to do the opposite. It wanted to pull the nuclei into an elongated shape, like a rugby ball. However, because the nuclear glue only works when the surfaces are very close, it was too weak to win the fight at the distances studied. The electric push was the boss, dominating the interaction and forcing the nuclei to flatten.

The study found that this flattening creates a tiny change in the nucleus's shape, but it's a very small change—about one part in a thousand. For the heavy lead nucleus, the change is almost invisible. For the lighter calcium nucleus, the effect is about twice as strong relative to its size, but still quite small. The energy released by this squishing (called polarization energy) is also tiny, amounting to less than 1 MeV (a unit of energy used in nuclear physics) at the closest distances tested.

When the Rules Break Down

One of the most important things the paper discovered is exactly when this "one-nucleus" way of thinking stops working. The simulation showed that as long as the nuclei are separated by more than 15.3 femtometers, the "rigid ball" idea is a safe shortcut. The squishing is so small that ignoring it doesn't mess up the predictions for how the nuclei will move.

However, if they get closer than 15.3 femtometers, the math breaks down. At this point, the electric and nuclear forces become so strong that the individual nuclei start to lose their identity. The particles inside begin to spill over into the other nucleus, and the two start to merge into a single, giant blob. The simulation couldn't handle this "merging" phase because it was designed to look at one nucleus in the field of another, not two nuclei becoming one. This means the "rigid ball" shortcut is valid for the early part of the dance, but scientists need a different, more complex set of rules to describe the moment they actually touch.

The Verdict

So, what does this mean for the big picture? The paper concludes that for the specific case of two round, spherical nuclei like lead and calcium, the "rigid ball" assumption used in many nuclear models is actually quite good. The tiny squishing that happens doesn't change the outcome of the dance enough to worry about, at least until the nuclei get very close to touching.

The study confirms that the electric repulsion is the main character in this story, flattening the nuclei like a pancake, while the nuclear glue plays a supporting role that only kicks in at the very last second. While the effect is real and measurable in the simulation, it is small enough that scientists can continue to use their simpler models for many calculations, saving them the headache of doing the super-complex math required to track every single squishy particle. However, the researchers warn that if they were studying nuclei that were already squashed or stretched to begin with, or if they got even closer to the point of fusion, this "squishiness" might become a much bigger deal, and the simple models would need an upgrade.

Technical Summary: Coulomb and Nuclear Polarization in the $^{40}\text{Ca} + ^{208}\text{Pb}$ Dinuclear System

Problem Statement
In the theoretical description of the entrance channel for heavy-ion reactions, particularly within the Dinuclear System (DNS) approach, the densities of the colliding nuclei are typically treated as "frozen" in their ground-state configurations. While this approximation simplifies the construction of the nucleus-nucleus interaction potential and driving potential, it neglects the microscopic polarization of one nucleus induced by the static external field of the other. Previous fully dynamical methods, such as Time-Dependent Hartree-Fock (TDHF), incorporate this polarization but yield the potential of the combined system rather than isolating the density response of a single partner. Furthermore, linear-response studies have often omitted the nuclear part of the external field constructed from the same effective interaction used in DNS calculations. This work addresses the gap by explicitly calculating the induced density change, $\delta\rho(r; R)$ , of one nucleus in the static field of a partner, separating Coulomb and nuclear contributions, and quantifying the validity limits of the one-centre description.

Methodology
The study focuses on the $^{40}\text{Ca} + ^{208}\text{Pb}$ system, chosen because both partners are doubly magic and spherical in their ground states, ensuring that any observed deformation is purely induced. The calculation is performed within the framework of a deformed Woods–Saxon mean field.

Hamiltonian and Basis: The single-particle problem is solved by diagonalizing a Hamiltonian $\hat{h} = \hat{h}_0 + U_{\text{ext}}(r; R)$ in a cylindrical harmonic oscillator basis. The unperturbed Hamiltonian $\hat{h}_0$ includes the axially symmetric Woods–Saxon potential, spin-orbit term, and Coulomb potential of the isolated nucleus.
External Field ( $U_{\text{ext}}$ ): The external field of the partner nucleus is composed of two parts:
- Coulomb: Calculated assuming a uniformly charged sphere.
- Nuclear: Obtained via double folding of the Migdal density-dependent effective nucleon-nucleon interaction. Crucially, this uses the exact same interaction form and parameters as those employed in the DNS approach to construct the nucleus-nucleus potential, ensuring consistency.
Treatment Schemes: Two methods are compared to assess the importance of higher-order terms:
- Non-perturbative (Self-Consistent): The full Hamiltonian including $U_{\text{ext}}$ is diagonalized to all orders.
- Linear Response (First-Order): The density change is calculated using first-order perturbation theory on the unperturbed eigenstates.
Validity Criterion: A quantitative criterion is established for the range of separations $R$ where the one-centre description (treating one nucleus in the fixed field of the other) remains physically meaningful. This requires that the external field is not strong enough to bind a nucleon in the partner's potential well, which would cause the single-particle states to delocalize.

Key Results
Calculations were performed for centre-to-centre distances $R = 15.5$ –$20$ fm.

Nature of Polarization: The induced density change is localized in the surface region of the nuclei. The polarization is dominated by the repulsive Coulomb tidal field, which compresses the density along the internuclear axis, resulting in a negative (oblate) induced quadrupole moment ( $\Delta Q_{20}$ ). The attractive nuclear field acts in the opposite sense (promoting elongation) but is negligible at larger separations due to the zero-range nature of the Migdal force.
Distance Dependence:
- The Coulomb contribution to $\Delta Q_{20}$ follows the expected $R^{-3}$ tidal law.
- The nuclear contribution grows exponentially as the surfaces approach, becoming significant only at small separations. At $R=15.5$ fm, the nuclear contribution reaches about one-third of the Coulomb value, partially cancelling the oblate deformation.
- The total induced quadrupole moment exhibits a maximum in magnitude around $R \approx 17$ fm for $^{208}\text{Pb}$ and $R \approx 16\text{--}17$ fm for $^{40}\text{Ca}$ before decreasing as the nuclear attraction begins to counteract the Coulomb compression.
Deformation Parameters: The induced quadrupole deformation parameters ( $\Delta \beta_2$ ) are small, of the order of $10^{-3}$ . Although the absolute induced moment of $^{208}\text{Pb}$ is larger, the lighter partner ( $^{40}\text{Ca}$ ) is more strongly polarized in relative terms ( $\Delta \beta_2$ is roughly twice that of $^{208}\text{Pb}$ ) due to its smaller size and the stronger field it experiences.
Polarization Energy: The polarization energy is a dipole quantity, with the quadrupole deformation contributing only a few percent. The energy is negative (attractive), reaching approximately $-0.61$ MeV for $^{208}\text{Pb}$ and $-0.89$ MeV for $^{40}\text{Ca}$ at $R=15.5$ fm.
Linearity: The response of the heavy nucleus ( $^{208}\text{Pb}$ ) is essentially linear (within 10%) down to $R=17$ fm. The response of the lighter nucleus ( $^{40}\text{Ca}$ ) shows significant saturation (deviating from linearity by ~45%) even at large separations due to the stronger field of the heavy partner.

Validity Limits
The one-centre description breaks down when the barrier between the two single-particle wells disappears, allowing nucleons to transfer to the partner. For the $^{40}\text{Ca} + ^{208}\text{Pb}$ system with the folded nuclear field, this occurs at $R \lesssim 15.3$ fm. Below this distance, the calculation becomes dependent on the basis size (variational collapse), and the concept of a single nucleus's density loses meaning. The touching configuration ( $R \approx 12.1$ fm) is inaccessible to this one-centre approach.

Significance and Conclusions
The paper establishes that for two spherical partners at separations where the one-centre description is valid ( $R > 15.3$ fm), the neglect of density polarization in the equations of relative motion within the DNS approach is justified. The induced quadrupole deformations are smaller than the uncertainties associated with tabulated static deformation parameters, and the resulting change in the radial force is only a few percent.

However, the study highlights that the dipole component of the response, which dominates the polarization energy, is absent from standard DNS models. While the magnitude of this effect at the Coulomb barrier cannot be quantified by this one-centre method (as it lies outside the valid range), the calculated scale at $R \approx 15.5$ fm (a few hundred keV) suggests that polarization acts as an additional attraction, steepening the potential and lowering the barrier. The authors conclude that while the frozen-density approximation is adequate for spherical nuclei at these separations, future work must address statically deformed nuclei and separations closer to the barrier, where induced deformations may become comparable to static uncertainties and require explicit inclusion in the equations of motion.