Reply to "Threefold error in the reported zero-field cooled magnetic moment of single crystal $La_2SmNi_2O_7$ (arXiv: 2602.23240)"

This paper refutes Korolev and Talantsev's critique of the superconducting phase fraction calculations in Li et al. (2026) by demonstrating that their underestimation of the fraction stems from incorrectly treating the demagnetization field as a constant rather than accounting for its dependence on the measured magnetic moment, while also confirming the absence of paramagnetic Meissner effects and the homogeneity of the single-crystal sample.

The Gist

Imagine you are a detective trying to figure out how much of a mysterious, super-cooled material is actually doing the "super" thing (superconducting) versus how much is just ordinary material.

In a recent scientific paper, a team of researchers claimed they found a new material that becomes a superconductor (conducts electricity with zero resistance) at a surprisingly high temperature. They calculated that about 62% of their sample was superconducting.

However, two other scientists (Korolev and Talantsev) looked at the same data and said, "Wait a minute! You made three big mistakes. If we fix them, the real number is only about 23%."

This new paper is the original team's reply, essentially saying, "No, we didn't make mistakes. Here is why your math is wrong and our original result of 62% is correct."

Here is a simple breakdown of their three main arguments using everyday analogies:

1. The "Backpack" vs. The "Ghost" (Field-Cooled Data)

The Accusation: The critics said, "You used data taken while cooling the sample in a magnetic field (Field-Cooled). But sometimes, superconductors get confused and act like magnets instead of anti-magnets (the Paramagnetic Meissner Effect). This makes your data unreliable."

The Reply: The authors say, "That confusion (the 'ghost') only happens in specific, rare materials. Our material is well-behaved. The tiny weird blip at the bottom of our graph? That wasn't a ghost; it was just dust on the camera lens (background noise). Because our sample is clean and behaves normally, we are allowed to use that data. It's like saying you can't use a thermometer because sometimes it breaks; ours didn't break."

2. The "Crowded Elevator" (The Demagnetization Effect)

The Accusation: The critics did a simple math calculation: Total Magnetism / Ideal Magnetism = Percentage. They got 23%.

The Reply: The authors say, "Your math is too simple. You treated the magnetic field inside the sample as if it were a constant, empty hallway. But in reality, the sample is like a crowded elevator.

When you have a superconductor, it pushes magnetic fields away. The more superconductor you have, the more it pushes back, changing the internal environment.

• The Critics' Mistake: They assumed the 'crowd' (the magnetic field) stayed the same size regardless of how many people (superconducting parts) were in the elevator.
• The Reality: As more people get in, the space shrinks and the pressure changes. You have to recalculate the math to account for this 'crowding' effect (called the demagnetization effect).

When you do the math correctly, accounting for how the sample pushes back on itself, the number jumps from 23% back up to 62%. The critics' method was like trying to count people in a room by ignoring that the walls are moving inward."

3. The "Patchwork Quilt" vs. The "Solid Block" (Sample Quality)

The Accusation: The critics suggested, "Maybe your sample isn't one solid piece. Maybe it's a patchwork quilt with tiny, random islands of superconductivity. If that's true, your math for a solid block doesn't apply."

The Reply: The authors say, "We checked our sample with a dozen different high-tech microscopes and tests. It is a perfect, solid block of crystal, not a patchwork quilt.

• The electricity flows smoothly through it.
• The atoms are lined up perfectly.
• The transition to superconductivity happens all at once, sharply.

If it were a patchwork quilt, the transition would be messy and slow. Since it's sharp and clean, it proves the whole sample is uniform. Therefore, our method for calculating the percentage is valid."

The Bottom Line

The authors conclude that the critics' math was based on a misunderstanding of how magnetic fields behave inside a material (the "elevator" analogy) and a false assumption that the sample was messy (the "quilt" analogy).

Once you fix the math to account for the magnetic "crowding" and confirm the sample is a solid block, the original claim stands: About 62% of their material is indeed superconducting.

Technical Summary

Based on the provided text, here is a detailed technical summary of the paper "Reply to 'Threefold error in the reported zero-field cooled magnetic moment of single crystal La2SmNi2O7'".

1. Problem Statement

This paper is a formal rebuttal to a critique (arXiv: 2602.23240) by Aleksandr V. Korolev and Evgeny F. Talantsev regarding the authors' previous publication in Nature (Li et al., 2026). The critique challenged the validity of the reported superconducting phase fraction ($f$) in pressurized single-crystal La$_2$SmNi$_2$O$_7$. The critics raised three specific objections:

1. Data Selection: They argued that Field-Cooled (FC) data cannot be used for calculating the superconducting phase fraction due to the potential presence of the Paramagnetic Meissner Effect (PME).
2. Quantitative Discrepancy: They calculated a phase fraction of 22.8% using the authors' Zero-Field-Cooled (ZFC) data, which is nearly three times lower than the ~62.1% reported in the original Nature paper.
3. Sample Homogeneity: They questioned the applicability of the authors' method, suggesting the sample might consist of multiple discrete superconducting regions rather than a homogeneous bulk, implying the calculated $f$ is physically unrealistic.

2. Methodology and Response Strategy

The authors address each critique point-by-point using theoretical magnetic principles, experimental verification, and established community standards:

• Addressing PME (FC vs. ZFC): The authors re-examined their low-temperature data. They confirmed that the weak upturn observed in the low-temperature tail is an experimental background artifact, not a physical Paramagnetic Meissner Effect. Since no positive magnetization (the defining feature of PME) was observed, they argue that FC data is valid for phase fraction calculations, consistent with usage in other superconductivity studies.
• Correcting the Demagnetization Calculation: The core of the rebuttal focuses on the mathematical treatment of the demagnetization effect.
  • The critics assumed a constant internal field ($H_{int}$) independent of the superconducting volume fraction ($f$), leading to a linear relationship: $f = -\chi_{meas}(1-N)$.
  • The authors demonstrate that this assumption is physically inconsistent. They apply the standard self-consistent relation between measured susceptibility ($\chi_{meas}$) and intrinsic susceptibility ($\chi_{int}$), accounting for the demagnetization factor ($N$):
    $$\chi_{meas} = \frac{\chi_{int}}{1 + \chi_{int}N}$$
  • By solving for $f$ (where $\chi_{int} \approx -f$), they derive the correct non-linear relationship:
    $$f = \frac{-\chi_{meas}}{1 - N\chi_{meas}}$$
  • They show that the critics' method underestimates $f$ by a factor of approximately 1/3 (specifically $(1-N\chi_{meas})(1-N)$) because they ignored the dependence of the internal field on the shielding fraction.
• Verifying Sample Homogeneity: The authors cite multiple characterization techniques (Energy Dispersive Spectroscopy, Single-crystal X-ray Diffraction, Nuclear Quadrupole Resonance, and Scanning Transmission Electron Microscopy) to prove the sample is a homogeneous high-quality bulk single crystal. They further point to sharp superconducting transitions in both resistivity and magnetic susceptibility, as well as uniform electrical transport, to refute the claim of discrete, fragmented superconducting regions.

3. Key Contributions

• Mathematical Correction: The paper provides a rigorous derivation showing that the critics' calculation of the superconducting phase fraction was flawed due to an incorrect assumption of a constant demagnetization field. It reaffirms the standard self-consistent method used in the superconductivity community for extracting $f$ from susceptibility data.
• Validation of Data Integrity: It confirms that the experimental data (specifically the FC curves) are free from the Paramagnetic Meissner Effect, validating the use of FC data for quantitative analysis in this specific material system.
• Defense of Material Quality: It consolidates evidence from various structural and transport characterization techniques to establish the sample as a homogeneous bulk single crystal, invalidating the "fragmented region" hypothesis.

4. Results

• Quantitative Resolution: When the correct demagnetization formula is applied, the discrepancy between the critics' calculation (22.8%) and the authors' original result (~62.1%) is fully explained. The factor of ~3 difference is mathematically accounted for by the term $(1-N\chi_{meas})(1-N)$, where $N=0.849$ and $\chi_{meas}=-1.313$.
• Consistency: The authors demonstrate that their shielding fraction estimations are consistent (~60%) across multiple independent measurements (ZFC, FC, and different field orientations), reinforcing the reliability of their original findings.
• Conclusion: The superconducting phase fraction calculations reported in the original Nature paper remain valid and are not invalidated by the critique.

5. Significance

This reply is significant for the field of high-pressure superconductivity, particularly regarding the recently discovered high-temperature superconductivity in nickelates.

• Scientific Rigor: It underscores the critical importance of correctly handling demagnetization effects when calculating superconducting volume fractions. Misapplying linear approximations in highly diamagnetic samples can lead to severe underestimation of the superconducting phase.
• Reinforcement of Discovery: By successfully defending the methodology and data interpretation, the authors reinforce the credibility of their claim regarding bulk superconductivity up to 96 K in La$_2$SmNi$_2$O$_7$.
• Community Standard: The paper serves as a reminder of the standard self-consistent relations required for magnetic susceptibility analysis in superconductors, ensuring future studies in this emerging field utilize correct physical models.