Research spotlight: Grzegorz Szklarz

For a physicist trying to hold a temperature steady near 4 kelvin, the weakest link is not always the sensor. It is often the equation chosen to translate sensor resistance into temperature. Research from the Włodzimierz Trzebiatowski Institute of Low Temperature and Structure Research in Wrocław, Poland, shows that high-degree polynomials, a familiar default in cryogenic calibration, fit rhodium-iron (RhFe) and platinum-cobalt (PtCo) resistance thermometers poorly, while a less fashionable class of equations, rational functions, tracks their behavior more faithfully below 25 K.

Seventeen sensors were put through the pipeline, covering the three families in the study: platinum resistance devices, PtCo alloy capsules and RhFe alloy sensors manufactured with two different wire lots. The findings, published in Measurement, point toward a standardized calibration route that could eventually stretch a single function from the boiling point of helium at 4.2 K to the triple point of water at 273.16 K.

Why should a non-scientist care tomorrow morning? Quantum processors, superconducting single-photon detectors and the liquid-hydrogen lines feeding clean transport all operate in this cold regime, and their performance depends on knowing temperature to within thousandths of a kelvin. An error of 0.1 K at 4 K distorts internal energy by about 2.5 percent, an effect analogous to a 3 K error at room temperature. Yet laboratories have lacked an agreed, automated way to judge whether a calibration curve is trustworthy. The new framework targets that gap.

Why RhFe and PtCo sensors matter below 25 K

Standard platinum resistance thermometers are the workhorses of precision thermometry, prized for stability and a nearly linear response. Their sensitivity fades as temperatures drop, though. In practice they become unreliable below the triple point of neon at 24.5561 K, and high measurement currents generate troublesome self-heating. The International Temperature Scale of 1990, or ITS-90, sets the lower limit of standard platinum resistance thermometers at the triple point of hydrogen, 13.8033 K.

Rhodium-iron sensors, developed in the 1970s and typically containing 0.5 atomic percent iron, offer the opposite behavior: their sensitivity improves as temperature decreases. They are difficult to manufacture reproducibly because material homogeneity is hard to control. Platinum-cobalt sensors have emerged as a promising alternative. Their advantage is reproducibility. Their cost is that they need higher measurement currents to reach a signal-to-noise ratio comparable to RhFe devices.

Calibration has long been the bottleneck. Traditional RhFe calibration could require more than 30 points, and choosing a polynomial that hugs the data often reads imaginary curvature as real sensor physics. The new work replaces that ad hoc process with a systematic pipeline.

Inside the Python calibration framework

Researchers build the framework in Python at the Institute's Temperature Standard Laboratory. It ingests raw resistance data, detects statistical outliers, compares candidate models and scores them with information criteria. The deeper change is diagnostic: instead of relying on one fit statistic, the framework asks whether a model is physically coherent. That matters because sensor data arrive with noise, statistical outliers, calibration drift after thermal cycling and hysteresis, and the act of removing an outlier assumes you already have a reasonable model. Model choice and outlier removal form a loop.

Outlier detection uses z-scores, interquartile range thresholds, studentized residuals based on hat-matrix leverage and repeated cross-validation checks. Those methods identify points that distort the fitted model, after which the optimization repeats. Because the process is iterative, the model and the outlier set are refined together rather than in a one-pass script.

Two statistical tools sit at the center. The Akaike information criterion, AIC, and Bayesian information criterion, BIC, penalize models that use extra parameters merely to chase noise. The Durbin-Watson test looks for autocorrelation in the residuals, the prediction errors left over after fitting. If a calibration curve leaves residuals with structure, the test exposes it. The framework also compares polynomial forms, logarithmic forms, rational functions and piecewise variants, then cross-validates the winner against independent heating and cooling cycles.

Photo by Fulvio Ciccolo on Unsplash

Why rational functions beat high-degree polynomials

A useful analogy: fitting a curve through a set of calibration points is like routing a flexible ruler through pins. A high-degree polynomial is a long ruler with many loose segments. It can pass through every pin and still shake badly between them. That oscillation is called Runge's phenomenon. A rational function, which expresses temperature as one polynomial divided by another, uses the denominator to adapt to sharp changes in sensor response without the same fragility.

The measurements confirmed the idea. Across the 17 sensors, the study found that rational functions described RhFe and PtCo characteristics with higher numerical stability, lower residuals and a stronger match to the physical changes in sensitivity. They avoid the oscillations that can make a polynomial look accurate in a plot while being wrong in the gaps.

The full analysis in Measurement describes the validation. A version of the tool, including documentation and example datasets, is archived as PrecisionThermometryFramework on Zenodo, and the paper lists a separate archive under DOI 10.5281/zenodo.18921803.

What the experiments actually used

The sensors were measured in a dual-chamber vacuum cryostat with two thermally coupled, independently controlled stages. Both stages sit on a Sumitomo SHI CH204-S two-stage Gifford-McMahon cryocooler driven by an HC-4E compressor. Before cooling, each sensor's resistance was recorded at the triple point of water as a reference, R(TPW). Subsequent readings were converted to a dimensionless ratio, W(T) = R(T) / R(TPW), so sensors with different nominal resistances could be compared on the same scale.

The normalized step is common in thermometry and helps expose drift. The framework then fits W(T) across the range below 25 K. Although the paper reports on 17 devices, the authors are explicit that the contribution is the method, not the ranking of one manufacturer's sensor against another.

Szklarz and his co-authors put the stakes in the paper's discussion: modern quantum technologies require moving "beyond simplistic calibration routines toward evidence-based, automated procedures." That phrase captures the intended audience, a national metrology institute that must defend its numbers under audit or a cryogenics lab that cannot afford a hidden calibration error.

One independent metrologist not involved in the DireK-T project offered a cautionary view: "Rational functions are strong interpolators, but their denominator can produce wild behavior outside the calibration range. The real test is whether this standardized approach reproduces across independent laboratories and datasets, not just the sensors reported here." That caution matches the paper's own framing: it presents a validation on the DireK-T dataset without claiming a final universal equation has already been proven.

The European metrology backdrop

The work sits inside a larger series of European projects. It was conducted as part of the European Partnership in Metrology project 22IEM02, "Dissemination of the redefined kelvin," or DireK-T. Since the kelvin was redefined in 2019 by fixing the value of the Boltzmann constant, national laboratories have worked to make thermodynamic temperature practical between 4 K and 300 K. A target uncertainty near 0.3 mK below 25 K forces rigorous calibration, and better transfer sensors plus transparent analytical tools are part of that system. The DireK-T project summary describes the dissemination goals in plain terms.

RhFe and PtCo sensors often serve as transfer standards, moving temperature traceability between laboratories. If each institute uses a different polynomial with hidden artefacts, comparison results become hard to untangle. A shared open-source framework provides a common statistical vocabulary. Project records also note that rational functions proved effective when data from different cryogenic systems were merged into a single calibration range, which supports the idea that the approach can travel beyond one setup.

Photo by Jonathan Cosens Photography on Unsplash

Five-year horizon

If rational-function calibration catches on, the next decade could see a single, shared model form spanning from liquid-helium temperatures to the triple point of water. That would shrink the calibration workload for RhFe and PtCo sensors and reduce the chance that two laboratories disagree because their equations wiggle differently. The framework's standardized diagnostics also create a natural foundation for machine-learning-driven model selection, a direction the authors identify explicitly.

The software is already public on Zenodo, which matters more than the journal version for practical uptake. A lab can clone it, run its own data, and test heating against cooling datasets before publishing. Better mathematical self-discipline at 4 K may quietly decide how sound tomorrow's quantum and hydrogen technologies turn out to be.