Theoretical estimates for the synthesis of $Z=119$ superheavy nuclei with Ca, Ti, V, and Cr projectiles: effects of reaction $Q$ values and mass-model dependence

This paper theoretically estimates the evaporation-residue cross sections for synthesizing superheavy nuclei with $Z=119$ using Ca, Ti, V, and Cr projectiles via a hybrid fusion framework, revealing that reaction $Q$ values relative to Coulomb barriers and significant nuclear mass-model uncertainties are the primary factors governing the survival probability and resulting cross sections.

The Gist

Imagine you are a master chef trying to bake the most difficult cake in the universe: a "Superheavy Element" cake. You want to create a new ingredient for the periodic table, specifically Element 119, which has never been seen before.

This paper is essentially a theoretical recipe book written by a team of physicists. They are trying to figure out the best way to bake this cake, how likely it is to succeed, and why their predictions might be wrong.

Here is the breakdown of their work, using simple analogies:

1. The Goal: Baking the "Unbakeable" Cake

For decades, scientists have been adding new ingredients to the periodic table. They usually do this by smashing two smaller atomic "doughs" together to make a bigger one.

• The Old Way: They used to use a very special, heavy "dough" called Calcium-48. It worked great for making elements up to 118.
• The Problem: To make Element 119, they need to use even heavier doughs (like Titanium, Vanadium, or Chromium). But the "target" doughs they need to smash these into are rare, radioactive, and fall apart (decay) very quickly. It's like trying to bake a cake using a target that melts before you can even turn on the oven.

2. The Three Stages of the "Baking" Process

The authors explain that making these elements isn't just about hitting them together; it's a three-step dance that is incredibly hard to get right.

• Step 1: The Catch (Capture)
  Imagine two magnets flying toward each other. They need to get close enough to stick. If they fly too fast, they bounce off. If they fly too slow, they might not stick. The scientists use a complex math model to figure out the perfect speed to get the two atoms to "kiss" and stick together.
• Step 2: The Merge (Fusion vs. Quasi-Fission)
  This is the tricky part. Once they stick, they don't always merge into one big ball. Often, they get stuck in a "limbo" state where they touch but then immediately rip apart again. The paper calls this Quasi-Fission. It's like two dancers trying to hold hands but immediately spinning away from each other instead of dancing together. The scientists use a "Langevin approach" (a fancy way of simulating random movements) to predict how often they actually merge into a single, stable ball.
• Step 3: The Cool Down (Survival)
  Even if they merge, the new "Superheavy Cake" is super hot and unstable. It's vibrating with energy. To survive, it needs to cool down by shooting out tiny particles (neutrons). If it gets too hot, it will explode (fission) before it cools down. The goal is to calculate the Survival Probability: How likely is the cake to survive the heat and become a solid element?

3. The Four Recipes They Tested

The team tested four different combinations of "projectile" (the thing thrown) and "target" (the thing hit) to see which one makes the best Element 119:

1. Calcium + Einsteinium (The classic, but the target is very rare).
2. Titanium + Berkelium
3. Vanadium + Curium
4. Chromium + Americium

The Surprise Result:
You might think the heavier the projectile, the harder it is to smash them together (like throwing a bowling ball vs. a tennis ball). So, they expected the Calcium reaction to be the easiest.

• The Twist: The Vanadium + Curium reaction actually had the worst results, even though it wasn't the heaviest.
• Why? It's about the Energy Balance. Think of it like a bank account. The reaction releases a certain amount of energy (the "Q value"). For Vanadium, the "bank account" was too low, meaning the resulting cake was too hot (too much excitement energy). Because it was so hot, it exploded before it could cool down. The heavier Chromium reaction, surprisingly, resulted in a slightly cooler cake that survived better.

4. The "Crystal Ball" Problem (Mass Models)

This is the most important part of the paper. To predict if the cake will survive, the scientists need to know the "weight" and "structure" of the atoms. But since Element 119 doesn't exist yet, they have to guess these properties using different mathematical models (called Mass Models).

The authors tested four different "Crystal Balls" (FRDM2012, FRDM1995, WS4, KTUY05) to predict the future.

• The Shock: Depending on which Crystal Ball they used, the predicted success rate changed by orders of magnitude.

• The Analogy: Imagine you are betting on a horse race.

  • Model A says: "There is a 1 in 1,000 chance this horse wins."
  • Model B says: "There is a 1 in 10,000,000 chance."
  • Model C says: "There is a 1 in 100 chance."

  The paper shows that for Element 119, different models give wildly different answers. One model suggests the Calcium reaction could be 10 times more successful than another model predicts. This is because the models disagree on tiny details like how tightly the neutrons are holding on (binding energy) and how strong the "glue" is inside the nucleus (shell correction).

5. The Conclusion: What Does This Mean?

The paper concludes that we cannot just guess the best way to make Element 119.

1. It's not just about the ingredients: You can't just pick the heaviest projectile. You have to balance the energy so the new atom doesn't get too hot and explode.
2. Our maps are blurry: Because we don't know the exact properties of these super-heavy atoms yet, our predictions are very uncertain. The "Survival Probability" is the biggest unknown.
3. The Path Forward: Before scientists spend millions of dollars and years of time building a new experiment, they need to realize that their theoretical "recipe" might be off by a factor of 10 or 100. They need to keep testing different models and be ready to adjust their plans when they finally get real data.

In short: The scientists are saying, "We have a good idea of how to bake Element 119, but our oven thermometer is broken. We think the Vanadium recipe might be too hot, and the Calcium recipe might be the best, but we aren't 100% sure because our math models disagree on how the ingredients behave."

Technical Summary

1. Problem Statement

The synthesis of superheavy elements (SHE) with atomic numbers $Z \ge 119$ represents a major frontier in nuclear physics, aiming to extend the periodic table and explore the predicted "Island of Stability." However, the production of these nuclei faces significant challenges:

• Limitations of $^{48}$Ca: The successful synthesis of elements up to $Z=118$ relied heavily on $^{48}$Ca beams. However, for $Z \ge 119$, suitable target nuclei (e.g., $^{254}$Es) have extremely short half-lives and limited availability, making $^{48}$Ca reactions impractical.
• Need for Heavier Projectiles: Alternative reactions using heavier projectiles ($^{50}$Ti, $^{51}$V, $^{54}$Cr) with more stable actinide targets are being pursued. However, these reactions involve higher Coulomb barriers and increased competition from quasi-fission, leading to lower compound-nucleus formation probabilities.
• Theoretical Uncertainties: Predicting the evaporation-residue cross sections ($\sigma_{ER}$) for these unmeasured systems is fraught with uncertainty. Current models suffer from significant discrepancies due to uncertainties in nuclear mass models, which dictate critical inputs like neutron binding energies ($B_n$) and fission barrier heights ($B_f$).

2. Methodology

The authors employ a hybrid theoretical framework that divides the fusion reaction into three distinct stages, calculating the total $\sigma_{ER}$ as the product of probabilities for each stage:
$$\sigma_{ER} = \pi \sum_{\ell=0}^{\infty} (2\ell+1) T_{\ell}(E_{c.m.}, \ell) P_{CN}(E^*, \ell) W(E^*, \ell)$$

• Stage 1: Capture (Entrance Channel):
  • Method: Coupled-channels method implemented in the CCFULL code.
  • Physics: Accounts for the rotational excitations of deformed target nuclei using a sudden approximation. Calculates the capture probability ($T_\ell$) considering the orientation angle of the deformed target.
• Stage 2: Fusion vs. Quasi-fission (Dynamics):
  • Method: Multi-dimensional Langevin approach.
  • Physics: Describes the competition between compound-nucleus formation and quasi-fission. The nuclear shape evolution is modeled using a two-center parametrization ($z_0, \delta, \alpha$). Trajectories are solved on a potential-energy surface defined by liquid-drop energy, shell corrections (with temperature damping), and centrifugal terms.
  • Output: Determines the compound-nucleus formation probability ($P_{CN}$).
• Stage 3: De-excitation (Survival):
  • Method: Statistical model.
  • Physics: Calculates the survival probability ($W$) of the compound nucleus against fission during neutron evaporation. This depends on the ratio of neutron evaporation width ($\Gamma_n$) to fission width ($\Gamma_f$).
  • Key Inputs: Neutron binding energy ($B_n$) and fission barrier height ($B_f$, derived from shell-correction energy $V_{shell}$).

Reactions Studied:
The study evaluates four specific reactions targeting $Z=119$:

1. $^{48}\text{Ca} + ^{254}\text{Es}$
2. $^{50}\text{Ti} + ^{249}\text{Bk}$
3. $^{51}\text{V} + ^{248}\text{Cm}$
4. $^{54}\text{Cr} + ^{243}\text{Am}$

Mass Models Tested:
To assess uncertainty, the authors compare results using nuclear properties from four different mass tables: FRDM2012 (standard), FRDM1995, WS4 (Weizsäcker-Skyrme), and KTUY05 (Koura-Tachibana-Unno-Yamada).

3. Key Contributions

• Systematic Comparison of Projectiles: Provides the first comprehensive theoretical comparison of $^{48}$Ca, $^{50}$Ti, $^{51}$V, and $^{54}$Cr projectiles for $Z=119$ synthesis within a unified dynamical framework.
• Identification of Q-Value vs. Coulomb Barrier Interplay: Demonstrates that the reaction $Q$ value relative to the Coulomb barrier height is a critical, often overlooked factor. A smaller magnitude $Q$ value leads to higher excitation energy ($E^*$) at the barrier, which drastically reduces survival probability.
• Quantification of Mass Model Dependence: Rigorously quantifies how different mass models affect the predicted cross sections, revealing that uncertainties in nuclear properties can alter $\sigma_{ER}$ by orders of magnitude.
• Mechanism of Survival Probability Enhancement: Explains the physical mechanisms (neutron binding energy and shell-correction evolution) that cause survival probabilities to vary between different mass models.

4. Key Results

A. Cross Section Rankings and the "Inversion" Effect

Using the standard FRDM2012 model, the maximum calculated $\sigma_{ER}$ (summed over all $xn$ channels) are:

• $^{48}\text{Ca} + ^{254}\text{Es}$: 233 fb (Highest)
• $^{50}\text{Ti} + ^{249}\text{Bk}$: 206 fb
• $^{54}\text{Cr} + ^{243}\text{Am}$: 38 fb
• $^{51}\text{V} + ^{248}\text{Cm}$: 33 fb (Lowest)

Critical Finding: The ranking is not determined solely by the projectile-target charge product ($Z_p Z_t$). Although $^{54}\text{Cr} + ^{243}\text{Am}$ has a higher charge product ($2280$) than $^{51}\text{V} + ^{248}\text{Cm}$ ($2208$), the Cr reaction yields a slightly higher cross section.

• Reason: The $^{51}\text{V} + ^{248}\text{Cm}$ reaction has a relatively small $Q$-value magnitude compared to its Coulomb barrier. This results in a higher excitation energy ($E^*$) for the compound nucleus at the optimal beam energy. Higher $E^*$ significantly suppresses the survival probability ($W$) due to increased fission competition.

B. Mass Model Dependence

The choice of nuclear mass model introduces massive uncertainties, particularly in the survival probability ($W$):

• FRDM1995 vs. FRDM2012: For the $^{48}\text{Ca} + ^{254}\text{Es}$ reaction, FRDM1995 predicts a maximum $\sigma_{ER}$ of 2067 fb, nearly 10 times larger than the 233 fb predicted by FRDM2012.
  • Cause: FRDM1995 predicts lower neutron binding energies ($B_n$) and a significant increase in shell-correction energy ($V_{shell}$) as neutrons are evaporated. This cooperative effect suppresses fission at high excitation energies, maintaining a higher $W$.
• WS4 vs. KTUY05:
  • WS4: Predicts $W$ values 2–3 orders of magnitude smaller than FRDM2012 due to significantly lower shell-correction energies (lower fission barriers).
  • KTUY05: Predicts $W$ values ~1 order of magnitude larger than FRDM2012 due to higher shell-correction energies and larger neutron binding energies.

5. Significance and Conclusions

• Experimental Guidance: The study suggests that while $^{48}\text{Ca} + ^{254}\text{Es}$ remains the most promising candidate (assuming FRDM2012 properties), the $^{50}\text{Ti} + ^{249}\text{Bk}$ reaction is a strong alternative. The $^{51}\text{V} + ^{248}\text{Cm}$ reaction is less favorable due to high excitation energy penalties.
• Theoretical Uncertainty: The results highlight that theoretical predictions for SHE synthesis are intrinsically uncertain. The choice of mass model can change the predicted optimal synthesis conditions and cross sections by orders of magnitude.
• Future Directions: The authors conclude that reliable planning for future experiments requires not just accurate descriptions of capture and fusion, but a rigorous assessment of nuclear mass model uncertainties, specifically regarding neutron binding energies and shell corrections. A stage-by-stage analysis of the full reaction process is essential to disentangle these effects.

In summary, this paper provides a robust dynamical framework for predicting $Z=119$ synthesis, revealing that the interplay between reaction $Q$ values and nuclear mass model uncertainties is the dominant factor governing the feasibility of creating new superheavy elements.