# Johnson noise thermometry

You may not have noticed but the world uses two different temperature scales: not the Fahrenheit and Celsius scale, but the thermodynamic and the International Practical Temperature scales.

Ideally, the temperature scale should be thermodynamic i.e. based on equations of state such as the ideal gas law:

which relates the pressure and volume of *N* moles of an ideal gas to temperature. Similar equations of state exist enabling us to measure temperature in terms of the speed of sound, the radiance of a blackbody, or the dielectric constant of a gas. However, despite decades of research into such thermometers, none have been found to offer sufficient accuracy, convenience, and low cost to satisfy our needs for a temperature standard. For this reason the SI has adopted a recipe called the International Practical Temperature scale of 1990, which is based on a series of fixed points with defined temperatures, and thermometers and equations for interpolating or extrapolating from these points (see http://www.bipm.org/en/publications/ for more information on ITS-90 and the *mise en pratique* for the kelvin).

MSL has been exploring the possibility of using a Johnson noise thermometer to measure thermodynamic temperature. Johnson noise is an electronic effect caused by the random movement of electrons in a conductor. It was first measured by Johnson and explained by Nyquist in 1929, although Einstein had predicted the effect in his 1905 on Brownian motion. As far as is known, the theory underpinning Johnson noise thermometry is exact. This is in contrast with other thermodynamic methods where the theory is imperfect because there is, for example, no such thing as an ideal gas or an ideal blackbody. Other thermodynamic thermometers tend to require months of secondary measurements just to characterise imperfections in the apparatus. Johnson noise thermometers, because they are purely electronic, do not suffer so much from this problem. They are also very much more affordable than most other thermodynamic thermometers.

Johnson noise thermometers infer temperature from measurements of the mean square thermal noise voltage generated by a resistor:

where *T* is the temperature, *k* is Boltzmann’s constant, *R* is the resistance of the sensing resistor, and Δ*f* is the bandwidth over which the noise is measured.

One disadvantage of noise thermometry is that the typical sensor signal is less than 1 μV rms and spread over bandwidths of many tens of kilohertz. To measure such small voltages requires a couple of tricks. Firstly, a cross correlator is used to eliminate the effects of amplifier noise. Two amplifiers are connected to the sensor, each of which amplifies the sensor signal, *V _{T}*, plus its own noise

*V*

_{n}. The product of the outputs of the two amplifiers gives

Note that the first term on the right hand side is always positive while the other noise products fluctuate about zero. If the output of the correlator is averaged over a long time, the result is mean square noise voltage, as required.

**Caption**: Block diagram of the MSL noise thermometer. The analogue signal at the output of the filters is digitised and the multiplier and integrator are implemented in software.

Secondly, to eliminate the uncertainty and drifts associated with the gain of the thermometer, the correlator is switched periodically (every 10 seconds or so) between two sensors; one maintained at the triple point of water, the other at the unknown temperature. The ratio of the two noise-power measurements of the two sensors is therefore independent of any gain drifts so that the unknown temperature is given by

where *T*_{0} is the temperature of the triple point of water. An unknown temperature can be inferred from measurements of the ratio of the noise powers and the ratio of the resistance of the two sensors.

Another disadvantage of noise thermometry is that the noise is random. This means that the uncertainty in the measurement falls as the square root of the number of samples of the noise. Very approximately, to measure a temperature near 300 K with an accuracy of 3 mK requires in excess of 10^{10} measurements. Additional inefficiencies arise from the two correlator channels, the amplifier noise, and the finite bandwidth of the thermometer, so the relative uncertainty in a measured temperature is

where ζ is the measurement time and Δ*f* is the bandwidth. Typically, in excess of 1000 GB of data must be gathered over periods of days, sometimes weeks, and averaged to obtain temperature measurements with the required uncertainty. Obviously, noise thermometry is not suited to the measurement of rapidly changing temperatures! But it is useful for the measurement of the temperatures of the fixed points, and in a few industrial applications where the required uncertainty is much less, and measurement time is a few seconds.

At the time we developed an interest in noise thermometry the basic theory was well established, the correlators were used to eliminate the biasing effects of preamplifier noise, lead resistances and non-linearity, and digital correlators based on analogue-to-digital converters (ADC) and digital multiplication had just been introduced to overcome the imperfections of analogue multipliers. However, there were many sources of error and uncertainty that were not well understood, and these were an impediment to the acceptance of noise thermometry as a high-accuracy thermodynamic technique. Effects we have considered include quantisation and differential non-linearity in the ADCs, aliasing effects on the uncertainty, preamplifier topology, preamplifier noise currents, noise-current-noise-voltage correlation, preamplifier input impedance, transmission-line effects, clipping, the detection and measurement of electromagnetic interference, and non-linearity. The long-term aim is to measure the thermodynamic temperature of some of the ITS-90 fixed points with an uncertainty of about 3 mK. Research at MSL is currently focussed on measurement of the non-linearity effects and to measure the indium freezing point.

In 2001 and again in 2007, Rod White was asked to assist the National Institute of Standards and Technology (NIST, USA) with their noise thermometry project. Rather than use a second sensor at the triple point of water, the NIST thermometer uses a pseudo-random noise generator based on a quantised ac-voltage source. The noise source exploits the Josephson effect to generate voltages with quantum accuracy. This means that their noise thermometer will measure the noise voltage i.e. temperature, directly in terms of the charge of the electron, Planck's constant, and a suitable time standard. The use of the pseudo-random noise signal raises a number of additional concerns relating to the averaging process and non-linearities. The aim of the NIST project is to measure Boltzmann’s constant with an accuracy of about 0.0005%.

For further information contact **Rod White**

### Key Publications

D R White and C P Pickup, “Systematic Errors in Digital Cross Correlators Due to Quantization and Differential Non-linearity”, IEEE Trans. Instrum. Meas. **IM-36**, 47-53, 1987.

D R White, “The noise bandwidth of sampled data systems”, IEEE Trans. Instrum. Meas. **IM-38**, 1036-1043, 1989.

R G Vaughan, N L Scott and D R White, “The Theory of Bandpass Sampling”, IEEE Trans. Signal Proc. **SP-39**, 1973-1984, 1991.

D R White, “Calibration of a Digital Cross Correlator for Johnson Noise Thermometry”, Metrologia, **29**, 23-35, 1992.

D R White, R Galleano, A Actis, H Brixy, M De Groot, J Dubbeldam, A L Reesink, F Edler, H Sakurai, R L Shepard and J C Gallop, “ The Status of Johnson Noise Thermometry”, Metrologia, **33**, 325-335, 1996

D R White and S J Bonsey “Preamplifier Limitations on the Accuracy of Noise Thermometers”. Proc. TEMPMEKO ’96, (Ed P. Marcarino, Levrotta & Bello, Torino) pp147-153, 1997

R Willink and D R White “The Detection of Corruption in Gaussian Processes with Application to Noise Thermometry”, Metrologia, **35**, 787-798, 1998

D R White and E Zimmermann, “Preamplifier Limitations on the Accuracy of Johnson Noise Thermometers”, Metrologia, **37**, 11-23, 2000

D R White, R S Mason, and P Saunders, Progress Towards A Determination Of The Indium Freezing Point By Johnson Noise Thermometry, Proc. TEMPMEKO 2001, (Ed Fellmuth, Seidel and Scholz, VDE Verlag, Berlin) pp129-134, 2002

D R White and R S Mason, An Emi Test for Johnson Noise Thermometry, Proc. TEMPMEKO 2004, (Ed Zvizdic, Bermanec, Stasic, and Veliki, Faculty of mechanical Engineering and Naval Architecture, Zagreb) pp485-490, 2005

D R White and S P Benz, “Constraints on a Synthetic Noise Source for Johnson Noise Thermometry”, Metrologia, **45**, 93-101, 2008

D.R. White, S.P. Benz, J. R. Labenski, S.W. Nam, J.F. Qu1, H. Rogalla, W.L. Tew, Measurement Time and Statistics for a Noise Thermometer with a Synthetic-noise Reference Metrologia, **45**, 395-405, 2008

Benz, S., White, D.R., Qu, J.F., Rogalla, H., Tew, W., Electronic Measurement of the Boltzmann Constant with a Quantum-Voltage-Calibrated Johnson-Noise Thermometer, Comptes rendus de l'Academie des Sciences, In press

Jifeng Qu, S. P. Benz, H. Rogalla, and D. R. White, “Reduced Nonlinearities and Improved Temperature Measurements for the NIST Johnson Noise Thermometer” Metrologia .