What Johnson noise thermometry actually is.

Temperature is reflected in the thermal motion within a material. In a resistor, that motion causes charge carriers to move randomly, producing a tiny, constantly changing voltage across it. This electrical noise is extremely small, but does increase proportionally as the resistor gets hotter.

Johnson measured it at Bell Labs in 1928 and Nyquist explained it the same year, with an equation that has one property we care about enormously: there are no material constants in it. It makes no difference whether the resistor is platinum, tungsten, or a corroded lump of something that used to be a sensor. Temperature, resistance, bandwidth and the Boltzmann constant are the only terms, which is what makes the whole thing useful to us.

Every conventional industrial thermometer measures a secondary effect of temperature and then uses its calibration data to determine temperature. A platinum resistance thermometer measures resistance and turns it into a temperature using a curve established when the sensor was calibrated; a thermocouple measures a voltage and does much the same. Both depend on the sensor still behaving the way it behaved on the day somebody certified it.

Sensors do not always stay the same; platinum can become contaminated, thermocouple alloys can change under neutron irradiation, and strain, oxidation, work-hardening and grain growth can all alter a sensor’s response. When that happens, the instrument sees nothing wrong. It continues to report a temperature, but the relationship between the sensor and temperature has changed, so the reading may no longer be correct. This is calibration drift, and in places such as reactor cores or sealed waste stores, retrieving the probe for recalibration may be impractical or impossible.

It measures the Johnson noise generated by the resistor and uses Nyquist’s equation to determine thermodynamic temperature.

There is a catch, and it is the first thing any metrologist says to us when we explain this. Resistance is in the equation. If you have to know the sensor’s resistance then we are back where we started, because the sensor’s resistance is the thing that changes. In fact, we rely on a platinum thermometer’s resistance changing when using a standard PRT.

So we measure it. The resistance and the noise can be calculated from the same measured data. In the work we published in 2019 that measured data is updated every 6.5536 seconds. Contamination, oxidation and transmutation are slow processes, and nothing meaningful happens to a sensor in six and a half seconds. The resistance we put into the calculation is therefore always the resistance the sensor has at that moment, not the one it had when somebody last certified it.

Which leaves the sensor free to change. It can corrode, oxidise or transmute into something with an entirely different resistance without that change, by itself, invalidating the temperature result. It is the property the whole technique exists to exploit, and the part people find hardest to believe until they watch it happen.