On the estimating the superconducting volume fraction from the internal magnetic susceptibility

This paper challenges the widely used postulate that the superconducting volume fraction equals the amplitude of internal magnetic susceptibility, arguing that this relationship is incorrect and citing a specific counterexample in pressurized $Pr_4Ni_3O_{10}$ to call for a reevaluation of the methodology across the entire field of superconductivity.

The Gist

The Big Picture: A Dispute Over a "Superconducting Score"

Imagine a group of scientists (let's call them Team Zhang) who discovered a new material that acts like a superconductor (a material that conducts electricity with zero resistance) when squeezed under immense pressure. They measured how much this material "fights back" against a magnetic field and claimed that 85% of the material is a perfect superconductor.

They used a specific math formula to get this number. They said: "We measured the magnetic pushback, plugged it into our calculator, and the result tells us exactly how much of the sample is superconducting."

Team Korolev and Talantsev (the authors of this paper) are saying: "Wait a minute. That calculator is broken."

They argue that the formula Team Zhang used is flawed. It's like using a thermometer to measure the weight of a rock; it might give you a number, but that number doesn't actually tell you what you think it does.

---

The Analogy: The "Empty Room" vs. The "Full Room"

To understand why the authors think the math is wrong, let's use an analogy of a room full of people.

The Scenario

Imagine you have a large, empty room (the sample). You want to know how many people are actually inside.

• Team Zhang's Method: They shine a flashlight through the room and measure how much light is blocked. If 85% of the light is blocked, they conclude, "Aha! 85% of the room is full of people!"
• The Flaw: What if the room is actually 90% empty, but the few people inside are wearing giant, light-blocking suits and are standing right in front of the flashlight? The light is blocked just as much as if the room were full, but the volume of people is tiny.

The Paper's Argument

The authors (Korolev and Talantsev) say that the formula Team Zhang uses assumes the "superconducting people" are spread out evenly throughout the whole room.

But, they argue, what if the superconducting part is actually just a tiny, thin slice (a lamella) hiding inside the sample, while the rest is just normal, non-superconducting junk?

1. The Real Situation: The sample might be 90% "junk" and only 10% "superconductor."
2. The Geometry Trick: Because the superconducting part is shaped like a thin disk, it interacts with the magnetic field in a very specific way. It creates a "shadow" (magnetic shielding) that looks huge, even though the actual amount of superconducting material is small.
3. The Result: The formula sees the huge "shadow" and says, "Wow, that's 85% superconducting!" But in reality, it's only 10%.

The "Counter-Example" Experiment

To prove their point, the authors built a fake scenario (which they call "Sample A"):

• They imagined a sample that is 90% non-superconducting and only 10% superconducting.
• They arranged the superconducting part as a tiny, thin disk inside the larger chunk.
• They ran the numbers through the exact same formula that Team Zhang used.

The Shocking Result:
Even though the sample was only 10% superconducting, the formula spat out a result of 82% (or 0.82).

This proves that the formula cannot distinguish between:

1. A sample that is truly 82% superconducting.
2. A sample that is only 10% superconducting but shaped in a way that tricks the formula.

Why Does This Matter?

The authors are not just nitpicking about one nickel crystal. They are saying that this specific math trick (called the "internal susceptibility postulate") is used everywhere in the field of superconductivity.

If this formula is wrong, then:

• Many previous claims about "bulk" (whole-volume) superconductivity might be exaggerated.
• Scientists might think they have found a material that is 80% superconducting when it's actually only a tiny speck.
• The "demagnetization factor" (a correction for the shape of the sample) changes depending on where the superconducting part is located, making the standard calculation unreliable.

The Conclusion

The authors are asking the scientific community to stop and rethink how they calculate superconducting volume.

They are essentially saying: "You can't just measure the magnetic pushback and assume it equals the volume of the superconductor. The shape and location of the superconducting parts can trick the math. We need a better way to measure this, or we might be celebrating discoveries that aren't actually as big as we think."

In short: The paper is a warning that a popular ruler used to measure superconductors might be giving false readings, and we need to find a new ruler before we trust the measurements.

Technical Summary

1. Problem Statement

The paper addresses a critical methodological flaw in the estimation of the superconducting volume fraction ($f$) in bulk materials, specifically targeting recent high-profile claims regarding bulk superconductivity in pressurized Ruddlesden-Popper nickelates (e.g., $\text{Pr}_4\text{Ni}_3\text{O}_{10}$).

• The Controversy: Recent studies (Zhang et al., and others) reported high superconducting volume fractions ($f \approx 85\%$) in single crystals based on Zero-Field Cooled (ZFC) magnetic susceptibility data.
• The Flawed Postulate: These studies rely on the assumption that the superconducting volume fraction is equal to the magnitude of the internal magnetic susceptibility ($|\chi_{\text{internal}}|$). The methodology uses the equation:
  $$f = |\chi_{\text{internal}}|$$
  where $\chi_{\text{internal}}$ is derived from experimental susceptibility ($\chi_{\text{experimental}}$) and the demagnetization factor ($N$) via:
  $$\chi_{\text{internal}} = \frac{\chi_{\text{experimental}}}{1 - N \cdot \chi_{\text{experimental}}}$$
• The Authors' Critique: Korolev and Talantsev argue that this postulate is fundamentally incorrect. They posit that a sample can exhibit a high calculated $|\chi_{\text{internal}}|$ (suggesting high $f$) while the actual physical superconducting volume fraction is very low ($f < 0.10$), provided the non-superconducting material is distributed in a specific geometric configuration that alters the sample's effective demagnetization factor.

2. Methodology

The authors employ a rigorous theoretical and computational approach to demonstrate the inconsistency of the standard methodology:

1. Re-evaluation of Existing Data: They re-analyze the experimental data from Zhang et al. for a $\text{Pr}_4\text{Ni}_3\text{O}_{10}$ single crystal (Sample S3) at 40.2 GPa. They calculate the experimental susceptibility ($\chi_{\text{experimental}}$) using the measured magnetic moment ($m_{\text{ZFC}}$), applied field ($H$), and sample volume ($V$).
2. Demagnetization Factor Calculation: They calculate the demagnetization factor ($N$) for the specific disk geometry of the sample using precise analytical formulas (Equations 8 and 9 in the text) rather than the approximations used in the original reports.
3. Counterexample Construction (The "Sample A" Model):
   • The authors construct a hypothetical model ("Sample A") with the exact same physical dimensions and exact same measured magnetic moment as the original $\text{Pr}_4\text{Ni}_3\text{O}_{10}$ sample.
   • However, in this model, the superconducting phase is not a uniform bulk but a thin, disk-shaped lamella embedded within a non-superconducting matrix.
   • This configuration results in a true superconducting volume fraction of only $f \approx 9.8\%$.
4. Comparative Calculation: They calculate the $\chi_{\text{internal}}$ and the resulting $f$ for this "Sample A" using the same equations (1 and 2) employed by the original authors.

3. Key Results

The calculations yield a paradoxical result that invalidates the standard postulate:

• Standard Analysis (Zhang et al.):
  • Measured moment: $-2.77 \times 10^{-9} \text{ Am}^2$.
  • Calculated $\chi_{\text{internal}} \approx -0.82$.
  • Reported $f$: $\approx 82\%$ (interpreted as bulk superconductivity).
• Counterexample Analysis (Sample A):
  • True Physical State: The sample consists of a superconducting lamella occupying only 9.8% of the total volume ($f_{\text{true}} = 0.098$).
  • Geometric Effect: The specific geometry of the superconducting lamella changes the effective demagnetization factor of the superconducting part to $N \approx 0.96$.
  • Calculated Output: Despite the low volume fraction, the calculation using the standard formula (Eq. 1 & 2) yields:
    • $\chi_{\text{internal}} \approx -0.82$.
    • Calculated $f$: $\approx 82\%$.
• The Contradiction: The standard methodology yields $f = 0.82$ for a sample where the actual superconducting volume is only $f = 0.098$.
  $$f_{\text{true}} (0.098) \ll |\chi_{\text{internal}}| (0.82)$$

4. Key Contributions

• Refutation of the $f = |\chi_{\text{internal}}|$ Postulate: The paper provides a definitive mathematical counterexample proving that internal susceptibility amplitude is not a direct measure of superconducting volume fraction when the superconducting phase is non-uniform or geometrically distinct from the bulk sample shape.
• Demonstration of Geometric Ambiguity: The authors show that the demagnetization factor ($N$) is sensitive to the distribution of the superconducting phase. If the superconducting region is a thin lamella, $N$ approaches 1, which mathematically inflates the calculated $\chi_{\text{internal}}$ regardless of the actual volume fraction.
• Correction of Previous Claims: The authors address and refute the "Replies" given by the original authors (Zhang et al. and others), who claimed their calculations were based on susceptibility rather than magnetic moment ratios. The authors demonstrate that even using susceptibility values, the methodology remains flawed.

5. Significance

• Broad Impact on Superconductivity Research: The authors argue that this issue is not limited to nickelates but applies to the entire field of superconductivity where ZFC/FC data are used to claim "bulk" superconductivity.
• Re-evaluation of "Bulk" Claims: The findings suggest that many recent claims of high-volume-fraction superconductivity in complex materials (including high-pressure hydrides and nickelates) may be overestimated. A high $|\chi_{\text{internal}}|$ value does not guarantee that the material is a bulk superconductor; it could be a small fraction of superconducting material arranged in a specific geometry.
• Call for Methodological Change: The paper calls for the scientific community to reconsider the validity of using Equation 2 ($f = |\chi_{\text{internal}}|$) as a standard metric. It implies that without independent verification of the spatial distribution of the superconducting phase (e.g., via microscopy or other probes), claims of bulk superconductivity based solely on magnetic susceptibility are scientifically unfounded.

In conclusion, Korolev and Talantsev demonstrate that the standard method for calculating superconducting volume fraction from magnetic susceptibility is mathematically insufficient to distinguish between a bulk superconductor and a sample with a small, geometrically favorable superconducting fraction, thereby casting doubt on recent high-profile claims of bulk superconductivity in pressurized nickelates.