## Abstract

Using exceptionally accurate measurements of the speed of sound in argon, we have made estimates of the difference between thermodynamic temperature, *T*, and the temperature estimated using the International Temperature Scale of 1990, *T*_{90}, in the range 118 K to 303 K. Thermodynamic temperature was estimated using the technique of relative primary acoustic thermometry in the NPL-Cranfield combined microwave and acoustic resonator. Our values of (*T*−*T*_{90}) agree well with most recent estimates, but because we have taken data at closely spaced temperature intervals, the data reveal previously unseen detail. Most strikingly, we see undulations in (*T*−*T*_{90}) below 273.16 K, and the discontinuity in the slope of (*T*−*T*_{90}) at 273.16 K appears to have the opposite sign to that previously reported.

## 1. Introduction

### (a) Background

The International Temperature Scale of 1990 (ITS-90) [1,2] is an approximation to thermodynamic temperature, and provides a practical scale that is the reference for dissemination of the kelvin, the unit of temperature in the International System of Units (SI). By following the procedures of ITS-90, users worldwide are able to produce approximations, called *T*_{90}, to thermodynamic temperature, *T*, based on the best primary thermometry results published prior to 1990.

However, advances in primary thermometry since 1990—most notably developments in acoustic gas thermometry (AGT) [3]—have revealed differences between *T*_{90} and *T* which approach one part in 10^{4} [4]. We have used the apparatus and techniques developed in our determination of the Boltzmann constant [5] to estimate (*T*−*T*_{90}) at 19 temperatures across the range 118 K to 303 K. The quality of the acoustic data is exceptional, with excess half-widths typically less than one part in 10^{6} of the resonance frequency, and mode-to-mode agreement at a similar level. Our results substantially agree with previous AGT results, but the small temperature intervals between our measurements and the low measurement uncertainty reveal more detail in (*T*−*T*_{90}).

### (b) The structure of the ITS-90

From the triple point of equilibrium hydrogen (13.8033 K) to the freezing point of silver (1234.93 K), the ITS-90 is defined by means of a platinum resistance thermometer (PRT) calibrated at specified sets of defining fixed points, and using specified reference and deviation functions for interpolation at intervening temperatures [1,2]. The fixed points are a set of 14 well-defined states of highly pure elements, each of which has been assigned a value of *T* from the best data available in 1990. Within the ITS-90 PRT range, several subranges are defined, each of which is associated with a subset of fixed points.

Every determination of *T*_{90} is made in terms of *W*(*T*_{90}), the ratio of resistance of a PRT at *T*_{90} to that at the triple point of water (TPW): *W*(*T*_{90})=*R*(*T*_{90})/*R*_{TPW}. The ITS-90 defines a reference function *W*_{r}(*T*_{90}), derived from measurements using a number of high-quality PRTs, and the functional form of subrange deviation functions, which approximate the difference between a given PRT and the reference function. To ensure that the thermometric platinum is of sufficient purity that *W*(*T*_{90}) is close to *W*_{r}(*T*_{90}), the ITS-90 specifies a minimum value of *W*(*T*_{90}) at the gallium melting point, and a maximum value at the mercury triple point. Thermometers which fulfil these criteria are known as standard PRTs (SPRTs). Calibrating an SPRT involves measuring *W*(*T*_{90}) at the fixed points specified for the subrange of interest, and calculating the coefficients of the deviation function for that particular subrange.

### (c) Measuring *T* using acoustic gas thermometry

At present, the most accurate technique for measuring thermodynamic temperature between approximately 7 K and 550 K is AGT [3]. The technique exploits the simple relationship between *T* and the limiting low-density speed of sound, *u*_{0}, in a monatomic gas: *u*_{0}^{2}=*γ*_{0}*k*_{B}*T*/*m*, where *k*_{B} is the Boltzmann constant, *m* is the average molecular mass of the gas, and *γ*_{0} is the limiting low-pressure value of the adiabatic index, which is equal to exactly in a monatomic gas.

In the current version of the SI, *T* is defined by fixing the temperature of the triple point of water, *T*_{TPW}=273.16 K. Most applications of AGT make use of this by measuring ratios of *u*_{0}^{2} at *T*_{TPW} and an unknown temperature, a technique known as relative primary AGT [3]. The unknown temperature *T* is determined from the simple expression
1.1where *m*_{T} and *m*_{TPW} are the average molecular masses of the gases used at *T* and *T*_{TPW}, respectively. The mass terms will cancel if the same gas is used for both temperatures, and if the composition of this gas remains stable with temperature and time.

In high accuracy AGT, the speed of sound *u*(*T*,*p*) is determined from the acoustic resonance frequencies of a gas-filled cavity. The shape of this cavity is either cylindrical, spherical or—as in this work—quasi-spherical [3]. The resonance frequencies of a subset of acoustic modes are measured at several gas pressures and a single temperature (an isotherm), and a model describing the pressure dependence of *u*(*T*,*p*) is fitted to the data to find *u*_{0,T}.

The remainder of this article is structured as follows: §2 describes the apparatus used in this work; §3 explains how we estimated the ITS-90 temperature of the argon gas; §4 describes how measurements of the acoustic and microwave resonance frequencies were used to estimate *u*_{0,T}; finally, §5 reports the values (*T*−*T*_{90}) and compares them with recent AGT results. In all sections, uncertainties are expressed as standard uncertainties.

## 2. Experimental apparatus

### (a) Overview

The apparatus used in this work was substantially the same as that used in our estimate of the Boltzmann constant [5]. It is based around a quasi-spherical combined acoustic and microwave resonator with an internal volume of 1 l. The resonator is made from copper, and has an exceptionally accurate and smooth inner surface prepared by diamond turning at the Cranfield Unit for Precision Engineering, Cranfield University, UK. For the Boltzmann constant measurements, the resonator and its enclosing pressure vessel were thermostatted in a stirred liquid bath, and a constant flow of ultra-pure argon was maintained through the apparatus. Small transducers (GRAS-type 40DP) were used to measure the radial acoustic resonance frequencies by sweeping the excitation frequency and measuring the response with a lock-in amplifier. Microwave resonance measurements were performed simultaneously using a network analyser (Agilent PNA-L) and two coaxial antennas in the resonator wall.

To enable operation at lower temperatures, both the apparatus and its method of operation were modified in several steps. We use the reference M0 to refer to the original configuration, and label the modified configurations M1–M4.

### (b) Modifications

#### (i) M0 (273 K to 303 K)

For the measurements between 273 K and 303 K, we used the Boltzmann constant apparatus but increased the temperature of the liquid bath which surrounded the pressure vessel. Improvements were made to the control and data acquisition software that enabled automated operation. During measurements, fluctuations in the resonator temperature were typically less than 0.1 mK (1 *mK*=10^{−3} K), and the argon pressure was stable to approximately 1 Pa. The argon flow rate was set to 3.7×10^{−6} mol s^{−1} (5 sccm) to minimize contamination through outgassing from the pipework and resonator interior. Previous measurements have shown that this flow rate has a negligible effect on the acoustic resonance frequencies in our apparatus [5].

#### (ii) M1 (207 K to 273 K)

This configuration was a substantial change from M0, which enabled reliable operation at temperatures down to 207 K.

The first modification was to change the microphone, which required 200 V polarization, to a pre-polarized version (GRAS-type 40DD). This eliminated an intermittent fault which we attributed to electrical arcing in the microphone. The new microphone required a different pre-amplifier (GRAS 26CB) which dissipated approximately 100 mW, around 50 times more than the previous pre-amplifier, and caused an unacceptably large, vertical temperature gradient across the resonator. To reduce the extra dissipation, we inserted a 10 cm length of semi-rigid stainless-steel co-axial cable between the pre-amplifier and the microphone, and embedded the pre-amplifier in a copper block that was thermally anchored to the top of the isothermal shield by a 12 mm diameter copper rod. This modification reduced the temperature gradient to an undetectable level, but reduced the signal-to-noise ratio by approximately 50%.

The ITS-90 thermometry on the resonator was also substantially improved. Previously, we had noted that the thermometers placed in the resonator support (position N1 in figure 1) were in relatively poor thermal contact with the argon in the resonator. To improve this situation, we attached a copper collar to the top of the resonator that could accommodate two SPRTs (positions N2 and N3). All the SPRTs were removed from the apparatus and re-calibrated, and three SPRTs were added to the ensemble as described in §3c.

Modifications were also made to enable low-temperature operation. A wider and deeper liquid bath was installed, and the neck of the pressure vessel was lengthened by 10 cm to allow for deeper immersion. The bath was filled with ethanol, into which a coil of copper tubing was immersed for cooling. A Huber Unistat 360 w thermoregulator pumped temperature-controlled ethanol through the coil, and a motor-driven propeller vigorously stirred the bath. The cooling was effective down to 207 K, as limited by the cooling power of the thermoregulator.

A small cartridge heater was mounted on the top plate of the inner isothermal shield for fine temperature control. Previously, this function was performed by a foil heater on the inner shield; the new configuration had a faster response and improved short-term stability. During an isotherm, the heater power was typically approximately 10 mW.

#### (iii) M2 (163 K to 191 K)

This configuration, used between 163 K and 191 K, was an extension of configuration M1 to lower temperatures. This was achieved by replacing the pumped ethanol coolant with liquid nitrogen, delivered slowly from a 160 l pressure-controlled dewar. A 50 W cartridge heater and Pt100 temperature sensor were used in a proportional-integral loop for temperature control of the bath.

Although the configuration was still functioning at 163 K, the ethanol was close to freezing and its increased viscosity made the stirring ineffective. A rudimentary check of the bath uniformity revealed differences in excess of 1 K, resulting in temperature gradients of approximately 1 mK across the resonator as explained in §3d. Consequently, the data obtained at 163 K using configuration M2 are not included in our (*T*−*T*_{90}) estimates, but are included in the electronic supplementary material. The isotherm at 163 K was repeated in configuration M3 (see below), and we observed no evidence of the temperature gradients found in configuration M2. We also used configuration M3 to make spot-checks at 178 K, and found good agreement with the results from configuration M2.

#### (iv) M3 (118 K to 163 K)

The third modification was designed to enable operation down to our lowest temperature, 118 K. At these temperatures, a stirred liquid bath cannot be used, so the pressure vessel was cooled by two copper plates in which liquid nitrogen was evaporated. The ethanol was drained from the bath and replaced with loosely packed cryogenic superinsulation film. A large disc made from 10 mm thick aluminium was mounted on top of the pressure vessel to provide uniform cooling. Two copper plates with internal fluid channels were attached to the disc, through which liquid nitrogen was flowed at a rate of 1–2 l per hour from a 160 l pressure-controlled dewar. The cold nitrogen exhaust gas was circulated around the outside of the pressure vessel.

To improve the temperature uniformity within the pressure vessel, a second copper isothermal shield was placed around the existing shield inside the pressure vessel, as shown in figure 1. The shield had a 10 mm thick aluminium top-plate bolted to the underside of the pressure vessel lid, and a small cartridge heater and PRT for temperature control.

The cooling configuration performed very well—fluctuations in the resonator temperature were less than 0.1 mK—but was inconvenient to use, as the dewar needed refilling every few days. In principle, this configuration could be used below 100 K, but we decided to limit our measurements to 118 K as AGT in argon becomes increasingly inaccurate as *T* approaches the argon boiling point [6].

#### (v) M4 (273 K)

Finally, a measurement at *T*_{TPW} was made using a configuration identical to M3, except that the liquid nitrogen was replaced with chilled ethanol, pumped and temperature-stabilized by the Huber thermoregulator.

## 3. Measuring the ITS-90 temperature of the argon gas

### (a) Overview

For each acoustic measurement, we assigned an ITS-90 temperature to the argon gas in the resonator from measurements of the resistance of several SPRTs attached to the resonator. Figure 1 shows the position of the SPRTs on the resonator, and the differences in their respective *T*_{90} from the mean on each isotherm. In this section, we describe the calibration of these thermometers and estimate the uncertainty in our values of *T*_{90}.

For all measurements, the resistance of the SPRTs was compared against a calibrated Wilkins type 100 ohm standard resistor using an ASL F18 bridge. The 100 ohm standard was submerged in a stirred oil bath and temperature-stabilized at 20.00 °C. The bridge linearity was assessed using an Isotech RBC100A resistance bridge calibrator, and for ratios between 0.1 and 0.3 relevant to this work the ratio errors ranged from −2.2×10^{−8} to +5.6×10^{−8}, with an average ratio error of +1.5×10^{−8}, equivalent to a resistance error of 1.5 *μΩ*. The full results of the linearity check are included in the electronic supplementary material.

Our ensemble of SPRTs was changed between configuration M0 and subsequent configurations. In the following two sections, we describe the ensembles and their calibrations in detail.

### (b) Measurements 273 K to 303 K (SPRT ensemble 1)

The measurements between 273 K and 303 K were performed during the period April to July 2012 using configuration M0. The resonator was equipped with five Tinsley-type 5187L SPRTs, these being the same thermometers that we employed in our Boltzmann constant experiment [5].

The SPRTs were calibrated in November 2009 at the triple point of water and the melting point of gallium (302.9146 K). The TPW cells and gallium melting point apparatus have both been verified by comparison with National Physical Laboratory (NPL) standards. During measurements, each SPRT was placed in a light, coiled wire spring to position it centrally within, and a few millimetres above, the base of the thermowell. The TPW and gallium cell thermowells were partially filled with liquid paraffin to improve thermal conduction. Several measurements at each fixed point were made to assess the stability and self-heating of the SPRTs; typical reproducibility was less than 0.1 mK.

In configuration M0 (*T*≥273 K), the temperatures indicated by SPRT B436 (position N1 in figure 1) were systematically lower than all the other SPRTs. This difference is ascribed to a vertical heat flux through the resonator, originating from the microphone pre-amplifier on the underside of the resonator. Previous thermal modelling [5] demonstrated that the gas temperature is close to the temperature of the equatorial SPRTs, and for this reason our estimates of *T*_{90} in configuration M0 are based on the average of SPRTs B409 and B442 (positions E1 and E2 in figure 1).

### (c) Measurements 118 K to 273 K (SPRT ensemble 2)

In the period November 2012–September 2013, the apparatus was modified for low temperature operation as described in §2b. The SPRTs were removed from the resonator and re-calibrated in TPW cells, and all were found to have changed by less than 0.1 mK.

At this point, we had no means of calibrating our SPRTs for use below the triple point of mercury (234.3156 K), as the argon triple point (83.8058 K) cells currently maintained at NPL are unsuitable for capsule SPRTs. The situation was resolved by adding two Leeds and Northrup (L&N) type 8614 thermometers (serial numbers 1676928 and 1832689) to our SPRT ensemble. These thermometers were built in the early 1970s and were part of the SPRT intercomparison by Ward & Compton [7]. Since then, they have proved to be exceptionally stable. The most recent argon TP calibration of these thermometers was made in 1999, using an NPL-built sealed argon TP cell [8]. The *W*_{Ar} values from this calibration were less than 0.1 mK different from the Ward and Compton calibration. In addition, another Tinsley-type 5187L SPRT was added (serial number 221476), which also has a long calibration history and is the same SPRT used by Ewing and Trusler in their (*T*−*T*_{90}) measurements [9].

The updated SPRT ensemble was calibrated at the TPW, mercury and gallium points. Following this, the SPRTs were placed in a copper comparison block and immersed in liquid nitrogen (LN2) at 77.3 K and in liquid argon (LAr) at 87.3 K. The temperature of the block was determined from the two L&N SPRTs using the Ward and Compton calibration from the triple point of oxygen (54.3584 K) to TPW. The calibration measurements in LAr were used in the calculation of the Ar–TPW subrange coefficients of the Tinsley SPRTs. A detailed account of the comparison procedure is given in [10]. The *R*_{TPW} values of SPRTs B385 and B409, which had previously been stable over a period of several years, drifted by the equivalent of 0.3 mK during the mercury TP and LN2 calibrations, and they were consequently removed from the ensemble.

The comparison apparatus was also used to crosscheck the calibrations of the L&N SPRTs with three other SPRTs from the Ward and Compton ensemble. The comparisons were performed in LN2 at 77 K and a mixture of LN2 and LAr at 84 K. The largest difference in the average SPRT temperatures at 83 K was only 0.13 mK [10], giving us confidence in the long-term stability of the L&N SPRTs.

For configurations M1–M4, our estimate of *T*_{90} was taken as the mean of the two L&N SPRTs, owing to their direct traceability to the argon TP, low self-heating effect and superior stability. Although the *T*_{90} values from the other SPRTs do not feature in our final result, they provide valuable information on temperature uniformity and calibration drift.

### (d) Uncertainty in our estimates of *T*_{90}

Our estimated uncertainty in *T*_{90} was guided by document CCT/08-19/rev [11], and consists of five components which are discussed in the following sections, and shown as rows 1–5 in table 1.

#### (i) Fixed-point calibration uncertainties and propagation

The standard uncertainties in the fixed-point calibrations are estimated to be *u*(Ar)=0.13 mK, *u*(Hg)=0.16 mK, *u*(TPW)=0.08 mK and *u*(Ga)=0.12 mK. The dominant sources of uncertainty in the temperature of the fixed-point cells are due to the purity of the fixed-point material, the residual gas pressure and uncertainty in the hydrostatic pressure correction. Other significant sources of uncertainty are due to perturbing heat exchanges in the cells (estimated by immersion tests of the SPRTs) and extrapolation of the SPRT self-heating to zero measuring current. The *u*(Ar) estimate includes a 0.09 mK uncertainty for drift in the L&N Ar TP calibrations, as estimated from the comparison in LN2. Full uncertainty budgets for the fixed-point calibrations are included in the electronic supplementary material.

The propagation of fixed-point calibration uncertainties to intermediate temperatures can be described by a sum of interpolation functions that depend on the calibration subrange [11]. The propagated uncertainties for the Ar–TPW and TPW–Ga subranges relevant to this work are plotted as curve 2 in figure 2*a*, and listed as row 1 in table 1.

#### (ii) Thermometer drift and uncertainty in resistance measurement

With the apparatus assembled, it is not possible to measure *R*_{TPW}, so we include an estimate of the stability of *R*_{TPW} between calibrations in our *T*_{90} uncertainty. Based on the observed changes in *R*_{TPW} between calibrations, we assign an uncertainty in this component of 10 *μΩ* for all the SPRTs. This is equivalent to 0.1 mK at *T*_{TPW} (row 2 of table 1). At the end of our measurements, we considered performing a final check of *R*_{TPW}. However, the excellent agreement of the SPRTs on the four isotherms at *T*_{TPW} (table 2) led us to conclude that removing the SPRTs from the apparatus would probably cause a greater shift in *R*_{TPW} than the drift since the last calibration.

Uncertainty in the resistance measurements results from the nonlinearity of the bridge, uncertainty in the standard resistor value, and the extrapolation to zero measuring current. We estimate this to be 5 *μΩ*, equivalent to 0.05 mK at all temperatures as shown in row 3 of table 1.

#### (iii) Uncertainty owing to non-uniqueness of *T*_{90}

One undesirable feature of the ITS-90 in the PRT range (13.8 K to 1235 K) is that the temperature assigned by the scale to a single thermodynamic state is not unique. This ‘non-uniqueness’ arises in part from variations in the *W*(*T*_{90}) functions of the thermometric platinum used in SPRTs. A group of SPRTs calibrated over the same subrange will agree on *T*_{90} at the fixed-point temperatures, but will disagree away from these temperatures owing to the inability of their deviation functions to accurately describe *W*(*T*_{90})−*W*_{r}(*T*_{90}) everywhere in the subrange. This is known as type 3 non-uniqueness [12]. We must also consider type 1 non-uniqueness [13], also called subrange inconsistency, which arises from the overlap of the subranges. In the overlapping ranges where two or more deviation functions are defined, all functions are equally valid but assign different values of *T*_{90} to a single value of *W*.

The consequence of non-uniqueness is that if our experiment was repeated identically with different SPRTs a different set of (*T*−*T*_{90}) values would be obtained, irrespective of experimental uncertainties. The ITS-90 gives equal precedence to all SPRTs, so we are obliged to include a component of uncertainty in (*T*−*T*_{90}) owing to the intrinsic ambiguity of *T*_{90}. One case of particular relevance to this work concerns the non-uniqueness of *T*_{90} close to the gallium melting temperature, because this fixed point is not included in all subranges. Consequently, our overall uncertainty in *T*_{90} at 303 K is 0.33 mK, whereas if we considered only subranges including calibration at *T*_{Ga} the non-uniqueness at 303 K would be effectively zero, and our overall uncertainty in *T*_{90} would be 0.16 mK.

White *et al.* [11] summarize the results of several experimental studies which were performed with the aim of quantifying type 1 and type 3 non-uniqueness. The concomitant uncertainty in *T*_{90} is plotted as curve 3 in figure 2, and listed as row 4 in table 1.

#### (iv) Uncertainty owing to non-isothermal conditions

In the absence of temperature gradients, we expect the gas within the resonator to equilibrate to the temperature of the walls of the resonator. Our measurements indicate that temperature gradients had an insignificant effect on the inference of *T*_{90} over the entire temperature range. In support of this assertion, we report two experimental tests.

In the first test, we deliberately induced temperature gradients across the resonator in configuration M3 by increasing the temperature differential between the inner and outer heat shields (figure 2*b*). We found that, for heater powers below 200 mW, no changes in the acoustic temperature with respect to the average SPRT temperature were observed, even when temperature differences across the resonator approached 1 mK. During each isotherm, heater powers never exceeded 100 mW and were typically less than 10 mW. Although this experiment was carried out at 178 K in configuration M3, we expect similar results would apply for other temperatures and configurations.

The second test concerns the two isotherms measured at 163 K. As mentioned earlier, the isotherm in configuration M2 suffered from large temperature gradients in the liquid bath, which led to SPRT temperatures with a typical range of 1 mK. (Note that these data are not plotted on figure 1 as the isotherm was not included in our (*T*−*T*_{90}) values.) Conversely, the measurements in configuration M3 showed excellent agreement between SPRTs, and the residuals from fitting the acoustic model (§4c(iv)) were notably smaller. Despite these deficiencies, the resulting (*T*−*T*_{90}) from configuration M2 was only 0.15 mK different from that from configuration M3. We thus conclude that the temperatures inferred from the SPRTs fairly estimate the temperature of the gas within the resonator. However, we include an uncertainty of 0.05 mK at all temperatures (row 5, table 1) to allow for the largest temperature difference that could have been present, but not reliably detected.

## 4. Measuring the thermodynamic temperature of the argon gas

### (a) Overview

Our estimates of *T*, determined from equation (1.1), require estimates of the limiting low-pressure speed of sound on isotherms, *u*_{0,T}. These estimates were obtained from measurements of the acoustic resonance frequencies, , of several (0,*n*) radial modes at a number of gas pressures. To compute *u*_{0,T} from requires an estimate of the volume of the resonator, *V* , which can also be expressed as an equivalent radius: *a*=[3*V*/(4*π*)]^{1/3}. As in our previous work [5], this volume measurement was accomplished by measuring microwave resonance triplets at the same time as the acoustic measurements. The microwave measurements and analysis are described in §4b. The acoustic measurements and analysis are described in §4c.

### (b) Resonator volume from microwave resonances

#### (i) Microwave measurements

Our estimates of the equivalent radius at temperature *T*, *a*_{T}, were obtained from the frequencies of the TM1*n* (*n*=1,2,…,5) electromagnetic modes, which are threefold degenerate in a spherical cavity. An Agilent PNA-L network analyser was used to measure the microwave transmission through the resonator (the *S*_{21} parameter) over a sweep of 200 discrete frequencies spanning each TM1*n* triplet. The triaxial shape of the resonator separates the triplet components, labelled TM1*nm* (*m*=x,y,z), into three resolvable peaks. Each sweep was fitted with a summation of three resonance functions with background terms (equation (15) in [14]), resulting in three fitted centre frequencies , and half-widths . At each pressure on an isotherm, approximately 100 sweeps of the TM11–TM15 triplets were acquired. After averaging, the microwave dataset for each isotherm consisted of (3×5×number of pressures) sets of .

#### (ii) Microwave model and corrections

In a perfectly spherical cavity, *a*_{T} and *f*^{m}_{nm} are related through
4.1where *c*_{0} is the speed of light in vacuum, *n*(*p*,*T*) is the refractive index of the gas and are the mode eigenvalues. The term represents the sum of several small frequency perturbations. In this work, we applied corrections for two types of perturbation: those owing to the surface resistivity of the resonator walls, which reduces and increases *g*^{m}_{nm} by equal amounts [15]; and those owing to the coaxial antennas used to excite and detect the resonances [16]. At each temperature, the wall resistivity was calculated as the sum of the resistivity of pure copper [17] and a residual resistivity of 5.8×10^{−10} *Ω*m, estimated from previous measurements with the resonator [18].

Small corrections were applied to the pressure data for the aerostatic head effect, which arises from the variations in gas density between the resonator and the Bourdon tube pressure sensor, and the viscous pressure drop along the 62 mm outlet tube leading from the resonator into the pressure vessel. The corresponding (molar) gas densities, *ρ*, were calculated using a third-order virial expansion,
4.2where *p* is the gas pressure, *R* is the molar gas constant [19] and the virial coefficients *B*, *C* and *D* were taken from the *ab initio* calculations of Jäger *et al.* [20]. The corrected pressures and corresponding densities were also used in the analysis of the acoustic data (§4c).

Following Moldover *et al.* [3], the argon refractive index was then calculated by solving the truncated Lorentz–Lorenz equation,
4.3with values of *A*_{ϵ} and *A*_{μ} taken from table 1 of [3], and *b*_{ϵ}(*T*) fitted to the data in table 3 of Rizzo *et al.* [21].

Our model for the microwave data is based around equation (4.1) with several refinements. First, the resonator is not spherical; it is a triaxial ellipsoid with minor axis lengths *a*_{x}=0.062 m, *a*_{y}=*a*_{x}(1+*ϵ*_{y}) and *a*_{z}=*a*_{x}(1+*ϵ*_{z}), where *ϵ*_{y}≈0.5×10^{−3} and *ϵ*_{z}≈1×10^{−3}. The resulting shift in the TM1*nm* eigenvalues has been calculated by Mehl [22] using second-order perturbation theory:
4.4where *P*_{nm} is a polynomial of order 2, and *O*(*ϵ*^{3}) indicates terms of order *ϵ*^{3}. This perturbation was included in our model, with a single pair of values (*ϵ*_{y},*ϵ*_{z}) fitted for each isotherm.

Second, the compressibility of the resonator is accounted for by making *a*_{T} a linear function of pressure
4.5where *κ*_{T} is the effective isothermal compressibility. A single value of *κ*_{T} was fitted for each isotherm.

Third, an extra parameter, *δ*, was included to describe the dispersion of the radii inferred from the TM1*n* modes that we observed in our previous work (figure 11 of [5]). The eigenvalue of each TM1*n* mode was allowed to shift according to
4.6The *n*-dependent term in square brackets was not chosen arbitrarily; it appears in the surface resistivity, waveguide and dielectric layer correction formulae [18]. Typical values of *δ* were 4×10^{−6}, resulting in Δ(*ξ*^{2})/*ξ*^{2} of 4×10^{−7} for the TM11 modes, 1×10^{−7} for the TM12 modes, and approximately 10^{−8} for the higher TM1*n* modes.

The model was fitted by nonlinear least-squares regression to the data pairs. Each point was given equal weighting as the statistical uncertainty was practically independent of mode or pressure—see §4b(iv). The residuals were defined by
4.7where
4.8and
4.9For every isotherm, the residuals of the TM11z mode were much larger than all the other modes, which we attribute to the particular sensitivity of this mode to imperfections in the join between the two halves of the resonator [18]. Including or excluding this mode from the fitting process has a negligible effect on *a*_{0,T}, but we chose to exclude it because it introduces a subtle bias into the *ϵ*_{z} parameter.

#### (iii) Microwave results

For every isotherm, the model was found to be an excellent fit to the data. A typical set of residuals is shown in figure 3*a*. The residuals are small, 10^{−7} being equivalent to around 3 nm in *a*_{0,T}, but exhibit systematic variations with pressure. This variation is due to drift in the Bourdon tube pressure sensor, a characteristic of this type of sensor that was corrected for by re-zeroing at each pressure. From the zero readings, we estimate an average drift of 10 Pa over a measurement set, which is consistent with the variation in the residuals.

Figure 3*b* plots the excess-half widths, defined as the difference between the measured and those calculated from the resistivity correction: Δ*g*=*g*_{meas}−*g*_{calc}. These are also small—typically, (Δ*g*/*f*)<0.2×10^{−7}—although they are significantly larger for the TM1*n*z modes, which provides further justification for omitting them from the fitting. The (Δ*g*/*f*) values for the TM11x and TM11y modes are also larger than those for the *n*>1 modes, but this is simply because (*g*/*f*) becomes smaller with increasing *n*. If we had instead plotted (Δ*g*/*g*_{calc}), the TM11 data would not appear anomalous. The slight discontinuity at 273.16 K is where the microphone was replaced in modification M1. A small change in the equivalent radius and shape parameters was also observed after this modification.

Figure 4 plots the fitted parameters for each isotherm. The values of *a*_{0,T} are plotted as a ratio against the first TPW isotherm, minus the relative expansion of pure copper [23]. The differences are significant, emphasizing the necessity of the measurements, and smooth. The shift of 0.8×10^{−6}*a*_{0,T} (equivalent to 48 nm) between the measurements above and below 273 K occurred when the microphone was replaced in modification M1. The corresponding shift in the values of *ϵ*_{y} and *ϵ*_{z} suggests that the replacement microphone did not protrude as far into the cavity as the original microphone. The temperature dependence of *ϵ*_{z} indicates that the copper hemispheres that constitute the resonator have a greater thermal expansivity than the stainless-steel bolts that join them, causing an axial compression as *T* increases. No such trend is visible in *ϵ*_{y}, as would be expected from the symmetry of the resonator.

Likewise, the fitted values of *κ*_{T} are not obviously temperature-dependent; for pure copper *κ*_{T,Cu} increases by only 3% from 118 K to 303 K [24], a change which is much smaller than the scatter in the *κ*_{T} values. However, all the values are around 40% larger than *κ*_{T,Cu}(295 K)=7.5×10^{−12} Pa^{−1} as calculated by May *et al.* [14]. We investigated several possible causes of this difference. The simplest explanation would be to assume that the discrepancy arose from a linear error in our estimates of the gas pressure within the resonator. At *T*_{TPW} this would amount to approximately 37 Pa per 100 kPa, a factor of 10 larger than the calibration uncertainty in the pressure indicator. If such an error were present, the fitted *κ*_{T} would be implausibly temperature-dependent—*κ*_{T} at 273 K would be approximately twice as large as *κ*_{T} at 118 K—so it is unlikely that this was the cause of the difference. Similarly, we could find no combination of plausible errors in the gas flow and aerostatic head pressure corrections, or the calculation of refractive index, that would make *κ*_{T} and *κ*_{T,Cu} consistent at all temperatures. In short, we found no plausible explanation for the discrepancy between *κ*_{T} and *κ*_{T,Cu}, but we emphasize that none of the investigated errors had a significant impact on our inference of *a*_{0,T}.

The increase in the values of *δ* at lower temperatures indicates a slight worsening in mode-to-mode agreement. This is possibly due to progressive distortion of the resonator shape as it contracts in size. The mode-to-mode agreement is nonetheless excellent across the entire temperature range.

#### (iv) Microwave uncertainties

Because our values of *T* are calculated from ratios of *u*_{0,T}, many systematic errors in our estimates of *a*_{0,T} will cancel to some extent. For example, uncertainties in the waveguide correction and thickness of the surface oxide layer, which are both assumed to be temperature-independent, will have a negligible impact on the uncertainty in *T*. This contrasts with our measurement of the Boltzmann constant [5], where these were the dominant uncertainties in our microwave measurements. Our microwave uncertainty analysis has four components, which are listed in rows 6–9 of table 1. Uncertainties such as the drift in the rubidium frequency reference (less than 10^{−9}) and statistical uncertainty in the mean (less than 10^{−8}) are omitted owing to their insignificant contribution to the overall uncertainty.

Uncertainty component 6, the surface resistivity correction, can either be calculated from copper resistivity data or inferred from the measured . Its value was taken as the difference in the (*T*−*T*_{90}) values obtained from these two approaches.

Uncertainty component 7 arises from the mode dependence of *a*_{0,T}^{2}. This was estimated from the values of parameter *δ*, which characterizes the systematic dispersion of the TM1*n* modes on each isotherm (figure 4). The implicit assumption in equation (4.9) is that the higher *n* modes are ‘correct’, and the lower *n* modes, especially the TM11, are perturbed by some unknown cause. The effect of reversing this assumption—i.e. assuming that the TM11 mode is ‘correct’—can be seen by multiplying all the *a*_{0,T}^{2} values by (1+0.181*δ*). This factor is equation (4.6) with *n*=1. Uncertainty component 7 was taken as the resulting difference in (*T*−*T*_{90}).

Component 8 arises from the short-term drift in the pressure sensor, which has already been mentioned as the source of the structure in the residuals in figure 3. The corresponding uncertainty in *a*_{0,T} was estimated by a Monte Carlo simulation, in which a normally distributed pressure offset with a standard deviation of 10 Pa was added to the data. A new offset was drawn at every point in the data where the pressure sensor was re-zeroed. The modified data were refitted, the process repeated approximately 10^{4} times, and *u*(*a*_{0,T}^{2}) estimated from the standard deviation of *a*_{0,T}^{2}.

Component 9 arises from the overall uncertainty in the pressure measurement. It was estimated by adding a 10 Pa pressure offset to all readings, and then a 0.01%-of-reading pressure error. The effect of the linear error was negligible, and component 9 shows the small effect of the 10 Pa offset.

### (c) Speed of sound from acoustic resonances

#### (i) Acoustic measurements

The acoustic measurements were performed using two -inch microphones (GRAS models 40DP and 40DD), mounted flush with the inner wall of the cavity. One microphone was used as a variable-frequency acoustic source, whereas the other measured the magnitude and phase of the acoustic pressure field using a lock-in amplifier. For each (0,*n*) radial mode (*n*=2,3,…,9), the lock-in signal was measured over a sweep of 30 excitation frequencies spanning a range of eight times the resonance half-width. Each sweep was fitted to a resonance function with constant and linear background terms (equation (4.1) in [25]), yielding values for the centre frequency, , and half-width, . At all but the lowest pressures, the fractional uncertainties in the fitted were less than 10^{−6}. The (0,7) and (0,9) modes required an additional quadratic background term at low pressures, owing to the broadening and consequent overlap of neighbouring modes. At each pressure on an isotherm at least 15 (and typically approx. 50) sweeps of each mode were acquired, and the average values of *p*, *T*_{90} and were fitted to our acoustic model (§4c(iv)).

#### (ii) Gas purity

AGT experiments are especially susceptible to error caused by contamination of the gas, usually by the outgassing of water vapour. A continual flow of gas through the resonator mitigates this problem, along with careful design and preparation of the pipe work [26]. The problem of contamination is also reduced at low temperatures owing to the exponential temperature dependence of the vapour pressure of water. We therefore did not anticipate that water contamination would be a significant issue in our experiment. This assumption was confirmed by performing tests at 178 K and 273 K, in which the argon flow rate was temporarily reduced to 0.7×10^{−6} mol *s*^{−1} (1 sccm); the shift in the (0,3) acoustic mode frequency was undetectable at the level of one part in 10^{7}.

The system of gas purifiers used on our apparatus effectively removes all impurities from the source gases except noble gases. The noble gas impurity levels in BIP argon are small [5], and any such impurities will be accounted for by our relative molecular mass measurements (§4c(iii)).

#### (iii) Relative molecular masses of the argon gases

The measurement of each isotherm consumed a significant amount of argon gas, so four different argon sources were used in this work. It is known that the average molecular mass of commercial argon can vary by several parts in 10^{6}, and that this variation is due to differences in the argon isotope ratios [27]. The relative molecular masses of our four argon sources were estimated by measuring the shift in the (0,3) acoustic mode frequency as the gas source was changed over. These measurements were typically performed at *T*_{TPW} and a pressure of 150 kPa. For each gas, more than 100 mode sweeps were measured, and the average values of were normalized to a reference temperature using the average *T*_{90} value from the SPRTs. We estimate the uncertainty in the mass ratio measurements to be 0.07×10^{−6}, which is predominantly owing to the achievable reproducibility of the normalization temperature, estimated to be 0.02 mK. The equivalent uncertainty in (*T*−*T*_{90}) is listed in row 10 of table 1.

The relative masses of argon sources I, III and IV were identical within the uncertainty of our measurements. Argon source II, which was used for the isotherms between 243 K and 263 K, and one of the isotherms at 233 K, was found to be (1.40±0.07) parts in 10^{6} lighter. This difference was factored into our calculations as described in §4c(vi).

#### (iv) Acoustic model and corrections

On each isotherm, we seek the limiting low-density speed of sound *u*_{0,T}, in order that the thermodynamic temperature can be established from equation (1.1). To achieve this, we used linear least-squares regression to fit the data from each isotherm to a model that linked *u*_{0,T} to the measured quantities, and *p*. The model begins with the simple relationship between the pressure-dependent speed-of-sound on an isotherm, *u*_{T}(*p*) and :
4.10where are the acoustic mode eigenvalues and *a*_{T} is the equivalent radius of the resonator at temperature *T*. The term represents the sum of several small frequency perturbations, for which corrections were applied. The most important of these is the thermal boundary layer [28,29], which reduces the mode frequencies by approximately , and increases by a near-equal amount. In applying this correction, we used the argon thermal conductivity values recently calculated by Mehl, as reported in [3]. Smaller corrections, of the order of approximately , were applied for the dissipation of energy in the bulk of the gas [25], the presence of gas inlet and outlet ducts [30], the mechanical compliance of the transducers [25], the second-order acoustic eigenvalue perturbation owing to the ellipsoidal shape [31] and the temperature-jump effect [32]. For the latter, we used a value of *h*=0.777 for the thermal accommodation coefficient at all temperatures, which was derived from low-pressure measurements at *T*_{TPW} in our previous work [5]. We investigate the effects of a temperature-dependent *h* in §4c(vii). Another small correction was applied for the fact that the ‘isotherms’ are not truly isothermal owing to small variations in the average *T*_{90} at each pressure. Each on an isotherm was corrected to a common *T*_{90} using an expression derived from equation (4.11). The uncertainty in the acoustic corrections is discussed in §4c(vii).

The model approximates the pressure dependence of *u*_{T} as a third-order expansion in *ρ*(*p*,*T*):
4.11where *u*_{0}^{2}=*γ*_{0}*k*_{B}*T*/*m* is the limiting low-density speed-of-sound squared. The coefficients *K*, *L* and *M* are the second, third and fourth acoustic virial coefficients, which are connected via exact relations to the density virial coefficients *B*, *C* and *D* [33]. The gas densities *ρ*(*p*,*T*) were calculated using equation (4.2).

Combining equations (4.10) and (4.11) leads to a model which relates *u*_{0,T}^{2} to the measured quantities . However, this simple model does not give an adequate description of the actual pressure dependence of . First, the calculated values of *K*, *L* and *M* have uncertainties which are comparable to the uncertainties in our data, so observable differences would be expected. More importantly, the coupling between the gas resonances and the mechanical resonances of the resonator is expected to shift by an amount proportional to gas density. Calculations based on the (0,*n*) radial acoustic modes coupling exclusively to the ‘breathing’ mode of a spherically symmetric resonator [34,35] predict frequency shifts approximately given by
4.12where *C*_{sh} is a constant and *f*_{sh} is the breathing-mode frequency, both of which depend on the geometry and material properties of the resonator. At first, we corrected the dataset using values of *C*_{sh} and *f*_{sh} calculated from the properties of copper and approximate geometry of the resonator, but found poor agreement between data and model. This is probably due to the non-spherical exterior of our resonator. Attempts to correct the entire dataset using best-fit values of *C*_{sh} and *f*_{sh} were more successful, but the discrepancies between data and model were still too large in places. Finally, we added a set of terms, *A*_{1,n}*ρ*, to the model, one for each (0,*n*) mode. A different set of *A*_{1,n} parameters were fitted to each isotherm. The fitted *A*_{1,n} parameters are discussed in §4c(v).

The residuals were further reduced by adding another term, *A*_{2}*ρ*^{2}, to the model. The inclusion of the parameter *A*_{2}, which takes a single value on each isotherm, is justified by the comparable uncertainties in the *ab initio* virial calculations and our data.

The averaged and corrected data pairs for each isotherm were fitted to the model using linear least-squares to minimize the quantity
4.13where *W*_{i} are the data weights, and the residuals *r*_{i} are defined by
4.14where
4.15and
4.16The data were weighted according to
4.17where *σ*_{i} is the sample standard deviation of each *y*_{i}, and *N*_{i} the number of averaged measurements. The value of 0.01 accounts for the estimated 0.03 mK uncertainty in the correction of each point to a reference value of *T*_{90}. This uncertainty is an estimate of the short-term drift in the resistance measurements and uncertainty in the self-heating correction, which is re-measured at each pressure.

The estimation of *a*_{T}(*p*) in equation (4.15) using microwave measurements was described in §4b. Note also that equation (4.15) contains *T*, which is of course unknown. The model could be fitted iteratively, but it is much easier (and entirely satisfactory) to substitute *T*_{90} in place of *T*.

#### (v) Acoustic results

The quality of our acoustic data can be assessed by examination of the excess half-widths, and the residuals from the least-squares fits. The values of these quantities for isotherms at 118 K, *T*_{TPW} and 303 K are plotted in figure 5. These isotherms were chosen as a representative sample of the data; residuals from every isotherm are available in the electronic supplementary material.

We first consider the excess half-widths, which are defined as the difference between the measured and those calculated from the thermal boundary layer and other corrections: Δ*g*=*g*_{meas}−*g*_{calc}. Low excess half-widths indicate that the physics within the resonator is well understood. The normalized excess half-widths, Δ*g*/*f*, plotted in figure 5*b*, increase approximately linearly with gas density, and extrapolate to near zero in the limit of zero density. The slope of Δ*g*/*f* versus *ρ* is related to the gas–shell coupling, and is discussed below. The excellent agreement between theory and measurements in the low-pressure limit gives us confidence in the acoustic corrections which, in the case of the thermal boundary layer, can exceed .

We now discuss the residuals, defined by equation (4.14) and plotted as fractional residuals (*r*_{i}/*y*_{i}) in figure 5*a*. Generally, these are very small—less than one part in 10^{6}—indicating excellent agreement between model and experiment. However, every isotherm contained several residuals which lay outside of the typical spread. In AGT experiments, ‘anomalous’ residuals are often indicative of an underlying perturbation, in which case the corresponding data point should be excluded from the least-squares fit to avoid biasing the estimate of *u*_{0,T}^{2}. The dominant perturbation is the gas–shell coupling, which has the greatest effect on modes with frequencies close to the ‘breathing’ frequency of the resonator. However, when considering variations in the resonant frequencies with magnitudes below one part in 10^{6} there are many small, unpredictable perturbations which can affect the acoustic frequencies, such as spurious mechanical resonances, acoustic resonances in the pressure vessel and overlap of neighbouring modes. The boundary layer corrections and acoustic eigenvalue calculations also have some degree of uncertainty, which will cause systematic trends in the residuals.

We considered both the residuals and excess half-widths when deciding whether to exclude data from an isotherm. For each isotherm, modes with frequencies in the approximate range 11–17 kHz exhibited large slopes of Δ*g*/*f* and residuals with a systematic trend. These modes were excluded from the fit, as they were clearly perturbed by the shell resonance. The (0,7) and (0,9) modes often had large residuals at low densities owing to interference from neighbouring modes (see §4c(i)), and these points were also excluded from the fits. The (0,4) mode was excluded from several isotherms owing to anomalously large residuals and excess half-widths. We do not know the origin of this perturbation.

The boundary between what could be considered a ‘normal’ or ‘anomalous’ residual is not well defined; to some extent, all the data were affected by underlying, unspecified perturbations. We assessed the impact on our estimates of *u*_{0,T}^{2} by including or excluding ‘borderline’ modes in our isotherm fits; the maximum equivalent shift in (*T*−*T*_{90}) on any isotherm was 0.09 mK, and typically much less than this. These differences are consistent with the ‘mode dependence’ uncertainty estimated in §4c(vii).

If the acoustic corrections described in §4c(iv) fully accounted for the perturbations, we would expect the *A*_{1,n} parameters and excess half-widths Δ*g*/*f* to be zero. Figure 6 shows that these two terms are not zero, but that they are strongly linked in a way that is reasonably described by a shell interaction. Figure 6*a* plots the *A*_{1,n} parameter values, scaled in such a way that they reveal the underlying shell perturbation predicted by equation (4.12). The parameters *C*_{sh} and *f*_{sh} were adjusted to fit the data, yielding a ‘breathing’ frequency of *f*_{sh}=14 kHz. This corresponds to the centre of a peak in the slope of Δ*g*/*f* versus *ρ*, plotted in figure 6*b*. Interestingly, the calculations of Greenspan reported in [34] and Mehl [35] predict insignificant mode broadening from the shell effect, although it has since been speculated that frictional losses in the joint between hemispheres may explain the observed effects.

One detail in figure 6*a* is additionally worthy of note. At the lowest temperatures, the scaled *A*_{1,n} values deviate from the shell model. This behaviour is primarily visible in the (0,2) to (0,6) modes, because, at low temperatures, the higher-order modes were more strongly affected by the shell. We ascribe these deviations to the increasing magnitude, and correspondingly large uncertainty, in the virial coefficient *K*(*T*) (equation (4.11)).

#### (vi) Calculation of *T* from ratios of *u*_{0,T}^{2}

The thermodynamic temperature estimates *T* were calculated from the fitted *u*_{0,T}^{2} using equation (1.1). The value of *u*_{0,TPW}^{2} was taken as the mean of the four isotherms measured at *T*_{TPW}, the results of which are shown in table 3. These measurements were made at the start, end and approximate middle of the measurement campaign, using argon sources I, III and IV, and configurations M0, M1 and M4. The four values of *u*_{0,TPW}^{2} are in remarkable agreement—to an extent which is probably fortuitous—and give an indication of the excellent reproducibility of our measurements.

Following the results of the mass measurements (§4c(iii)), the factor *m*_{T}/*m*_{TPW} in equation (1.1) was set to (1−1.40×10^{−6}) for the argon source II data, and unity otherwise. A full results table showing the values of *u*_{0,T}^{2} for each isotherm is provided in the electronic supplementary material.

In addition to the four TPW isotherms, repeat isotherms were also measured at 163 K and 233 K. As mentioned previously (§3d(iv)), the 163 K measurement was repeated using configuration M3 because of the poor thermal uniformity in configuration M2. Despite these shortcomings, the estimates of (*T*−*T*_{90}) varied by only 0.15 mK. The repeat isotherm at 233 K was made owing to its proximity to the mercury triple point. The two values of (*T*−*T*_{90}) differ by 0.18 mK, which again demonstrates the repeatability of the apparatus and also confirms the mass correction.

#### (vii) Acoustic uncertainties

We now discuss the sources of uncertainty in our estimates of *u*_{0,T}^{2}, and how these propagate to our estimates of (*T*−*T*_{90}). The values assigned to these uncertainties are presented in rows 11–17 of table 1. Each estimate of *T* is derived from the ratio (*u*_{0,T}/*u*_{0,TPW})^{2}, and so the uncertainty in *u*_{0,TPW}^{2} affects all measurements. In estimating these ratios and their uncertainties, we used the average value of the four TPW isotherms.

First, we consider the uncertainty in the values of *u*_{0,T}^{2} that results from fitting the acoustic model. The data weighting factors *W*_{i} (equation (4.17)) were based on estimates of the statistical uncertainty in each point on an isotherm. The statistical uncertainties in *u*_{0,T}^{2} were estimated from the parameter covariance matrices in the least-squares fits. These uncertainties are comparatively small, owing to the quality and quantity of the acoustic data, and did not vary much from isotherm to isotherm (rows 11 and 13, table 1). However, close examination of the residuals reveals small, systematic trends, similar to those in our Boltzmann experiment [5], which we discussed [36] in response to criticism by Macnaughton [37]. We emphasize that these trends are weak, and only evident because of the low ‘noise’ in our data. We estimated the possible impact of these trends on *u*_{0,T}^{2} in three ways.

In our first estimate, we attempted to specifically evaluate the effect of mode-to-mode variability. To do this, we randomly re-sampled the data from each isotherm according to mode, refitted the data, and repeated the process approximately 10^{3} times. This procedure, sometimes called the Monte Carlo bootstrap technique [38], allowed us to assess the variability of the fitted *u*_{0,T}^{2} arising from the choice of modes included in the analysis. The resulting distributions of *u*_{0,T}^{2} were approximately normal, and the standard deviation of the *u*_{0,T}^{2} values on each isotherm was taken as an estimate of the uncertainty associated with mode choice.

Our second estimate followed our Boltzmann constant work [5], where the statistical uncertainty in each on each isotherm was inflated by a pressure-independent factor, such that approximately 95% of the residuals fell within ±2 standard uncertainties of the data. The inflation factors for the isotherms ranged from 1.0 to 2.0, compared with a factor of 2.2 in our Boltzmann constant work [5]. The resulting uncertainties were on average 30% smaller than the quadrature sum of the Monte Carlo estimate and the un-inflated statistical uncertainty. This difference arises, because the Monte Carlo procedure partially double-counts the statistical uncertainty.

We also derived a third estimate following Pitre *et al.* [6], who fitted each mode individually, resulting in several estimates of *u*_{0,T}^{2} on each isotherm. These estimates were averaged, and the standard deviation taken as the uncertainty in *u*_{0,T}^{2}. Applying their method to our data resulted in no systematic shift from our estimates of (*T*−*T*_{90}), with a maximum shift of 0.06 mK, and uncertainties which were typically 25% larger than our combined statistical and Monte Carlo uncertainty. The disadvantage of this method is that it is not clear how the *u*_{0,T}^{2} values should be weighted in the average.

All three uncertainty estimates produced broadly consistent values. Our preferred estimate is that derived using the first technique, as it partially separates the statistical and systematic uncertainty components; our estimated values are shown in rows 12 and 14 of table 1.

We evaluated the effect of a systematic pressure error by adding a pressure offset of 10 Pa to the data and refitting. The effect is most significant at low temperatures, where the magnitude of ∂*u*/∂*p* is greatest. The resulting uncertainties in (*T*−*T*_{90}) are shown in row 16 of table 1.

We evaluated the effect of a 0.1% shift in the thermal conductivity of argon (row 17, table 1). Although the thermal boundary layer is the largest correction to the data, and strongly mode-dependent, the resulting uncertainty estimate is small. This assessment is confirmed by the excellent mode-to-mode agreement, which would not be possible if there was a significant error in the thermal conductivity.

Finally, we consider the uncertainty arising from our estimate of the thermal accommodation coefficient *h*. This coefficient characterizes a modification to the thermal boundary layer arising from the non-zero mean free path of the gas molecules, and has the effect of increasing the mode frequencies by an amount inversely proportional to pressure [32]. Significantly, *h* varies with the material and condition of the resonator surface, and so must be estimated for each resonator.

In this work, we used a value of *h*=0.777±0.013 estimated from our previous work carried out at *T*_{TPW} [5]. The low uncertainty arose from time-consuming measurements at low pressure, which were impractical to perform on every isotherm in this work. Because *h* is determined by the interaction between the gas and the surface of the resonator, there exists the possibility that *h* may be changing with time, owing to changes in the condition of the resonator wall. Additionally, *h* may also be temperature-dependent.

If *h* were changing significantly with time, then we would expect our four *T*_{TPW} isotherms (table 3) taken over a period of 3 years to show inconsistent values of *u*_{0}^{2}. The sensitivity of the fitted values *u*_{0}^{2} to small changes in the assumed value of *h*=0.777 is plotted in figure 7*c*. Using this sensitivity, we can interpret the data of table 3 as evidence that *h* has not changed by more than 0.03. Additionally, we can monitor the condition of the surface oxide layer from the *δ* parameter in equation (4.9). At *T*_{TPW}, the values of *δ* have not changed over the 4 years since the resonator was first used. This stability is not surprising, as the resonator surface has only been exposed to ultra-pure argon over that period. We thus conclude that there is no evidence for a time dependence of *h* in our resonator.

Investigating possible changes in *h* with temperature is considerably more difficult. To carry out our analysis, we added a term *A*_{−1}*ρ*^{−1} to the right-hand side of equation (4.16), which is equivalent to allowing *h* to vary independently on each isotherm. We re-fitted the data, and the resulting *A*_{−1} parameters are shown in figure 7*a*. Also plotted as smooth curves are the values of *A*_{−1} that would result from selected values of *h*. Values corresponding to *h*>1 are not physically meaningful, and arise from the difficulties in estimating this small term from noisy data. We consider that the data provide no evidence for a temperature dependence in *h*, but the large uncertainty in the data means that a temperature dependence cannot be ruled out.

We estimate our uncertainty in (*T*−*T*_{90}) owing to uncertainty in *h* as the quadrature combination of two terms: the effect of shifting *h* by 0.013 across all isotherms; and the effect of a linear temperature dependence in *h* with a range of 0.1 over the extremes of our temperature range (figure 7*b*). The shift in *h* is correlated across all isotherms, and results in only a small uncertainty in (*T*−*T*_{90}) (approx. 0.01 *mK*). The total uncertainty (row 15 in table 1) thus arises mainly from the possibility of a temperature dependence in *h*. Our simple model of this hypothetical temperature dependence causes the uncertainty to increase to almost 0.1 mK at the extreme ends of our temperature range. A more complicated model of the temperature dependence of *h* would require either a theoretical model for *h*(*T*) or an experimental study at low pressure, similar to that which we previously performed at *T*_{TPW} [5]. However, after modification M1, the reduced signal-to-noise ratio did not permit such a study.

Although the *u*_{0}^{2} values obtained with the additional *A*_{−1} parameter make no assumptions about *h*, the increased variability in the other fitted parameters is clearly unphysical. For example, considering our four isotherms at *T*_{TPW}, the inclusion of *A*_{−1} increases the range of *u*_{0}^{2} values by a factor of 20, from 0.008 (table 2) to 0.181 *m*^{2} s^{−2}. For this reason, we consider that our best estimates of *u*_{0}^{2}, and hence (*T*−*T*_{90}), are obtained using a value of *h*=0.777 on every isotherm.

## 5. Results of this work

### (a) The differences (*T*−*T*_{90})

Table 4 and figure 8 show our estimates for (*T*−*T*_{90}) at 19 temperatures across the range 118 K to 303 K, together with their estimated standard uncertainties. The value at 233 K is the mean of the two isotherms at this temperature. The general trend of the results agrees well with other estimates of (*T*−*T*_{90}) made by AGT also shown in figure 8. With the exception of one marginal case, all the (*T*−*T*_{90}) values are consistent within two combined standard uncertainties.

Before discussing the detailed features of the dataset and the significance of the small differences between experiments, it is important to note that there are significant differences between the AGTs used to derive the data in figure 8. The material of the resonators varied from copper, to aluminium, to steel; some resonators were bolted together and others welded; the volume varied from 0.5 to 3 l; and the gas used was variously flowed or static, helium or argon. Given this diverse experimental base, this level of agreement is already a significant improvement on the situation described by Fischer *et al.* [4].

### (b) Behaviour near *T*_{TPW}

Our five data points above *T*_{TPW} show a clear linear trend that agrees closely with estimates of (*T*−*T*_{90}) at the gallium point by Moldover *et al.* [41] and Benedetto *et al.* [40], and to a lesser extent Ewing & Trusler [9]. Our estimated slope d(*T*−*T*_{90})/*d*(*T*_{90}) just above *T*_{TPW} is 0.14±0.02 *mK* K^{−1}, slightly larger than the estimate by Fischer *et al.* [4] of 0.10 mK *K*^{−1}.

Below *T*_{TPW}, the situation is more complicated. In previous work, the next lowest temperatures measured below *T*_{TPW} were in the region of the mercury triple point, *T*_{90}(Hg)=234.3156 K. With the exception of Ewing & Trusler [9], our data in this range agree well with other measurements. However, our data in the region of *T*_{90}(Hg) suggest a much-reduced slope in (*T*−*T*_{90}) compared with the CCT-WG4 recommended curve [4]. Additionally, the points just below TPW fall significantly below the CCT-WG4 curve. It is likely that this ‘wiggle’ in the form of (*T*−*T*_{90}) is real, because the uncertainties in both the acoustic data and *T*_{90} are exceptionally low in this region (table 4). As a consequence, the slope *d*(*T*−*T*_{90})/d(*T*_{90}) immediately below *T*_{TPW} is not well represented by assuming (*T*−*T*_{90}) varies linearly between *T*_{Hg} and *T*_{TPW}, but the precise functional form is unknown. Based on a variety of polynomial fits to the data below *T*_{TPW}, we estimate d(*T*−*T*_{90})/d(*T*_{90}) to be 0.16±0.01 mK K^{−1}, considerably larger than the figure of 0.07 mK K^{−1} estimated by Fischer *et al.* [4]. Our estimate for the slope discontinuity in (*T*−*T*_{90}) at *T*_{TPW} is (0.14−0.16) mK K^{−1}=−0.02 *mK* K^{−1}, which has the opposite sign to the estimate of +0.03 mK K^{−1} from Fischer *et al.* [4]. However, our experimental uncertainty (±0.02 mK K^{−1}) does not allow us to conclude that the discontinuity is significantly different from zero. Rusby [42] analysed the ITS-90 deviation coefficients of more than 40 SPRTs above and below *T*_{TPW}, and found slope discontinuities that varied considerably, but had a negative bias.

### (c) Behaviour below *T*(Hg)

Below the triple point of mercury, the data show increasingly negative values of (*T*−*T*_{90}) with a minimum at approximately 150 K. The trend of our data indicates values of (*T*−*T*_{90}) which are between 1 and 2 mK smaller than those estimated by Pitre *et al.* [6] and Ewing & Trusler [9]. Although these discrepancies are small, the uncertainties are also small and we consider that these differences may well be significant, but we do not have any convincing explanation for them.

The differences could arise from estimates of *T*_{90} or *T*. The thermometer used by Ewing and Trusler was also used in this work and the thermometer used by Pitre *et al.* was extensively calibrated at the National Institute of Standards and Technology (NIST), so it seems unlikely that a simple error in *T*_{90} could be the cause. Both Ewing & Trusler and Pitre *et al.* estimated the effect of temperature gradients, but each used only a single SPRT, whereas we used six thermometers and were thus able to definitively evaluate temperature gradients at all temperatures. Acoustically, our mode-to-mode agreement and excess half-widths are exceptionally small, whereas Ewing & Trusler and Pitre *et al.* both have excess half-widths of the order of 10 times larger. Ewing and Trusler used a static gas charge of argon, whereas Pitre *et al.* [6] flowed helium gas through their resonator to address issues around outgassing. Finally, and very significantly, at 150 K, Pitre *et al.* measured with both helium and argon and achieved excellent agreement, which rules out many systematic biases in their estimates of *T*. In short, we do not know the origin of the discrepancy between these works.

## 6. Future work

After the conclusion of research focused on the estimation of the Boltzmann constant for the redefinition of the kelvin [43] in 2018, it is possible that the techniques developed will be used for precision estimates of (*T*−*T*_{90}). Dielectric constant gas thermometry, Doppler broadening thermometry or Johnson noise thermometry may all contribute data in this temperature range with significantly small uncertainty and hence lead to more robust estimates of the differences (*T*−*T*_{90}) based on multiple physical principles [44].

One obvious test of the consistency of our data above and below *T*_{TPW} would be to examine the smoothness of *W*(*T*) for our SPRTs without reference to the ITS-90 reference function, *W*_{r}(*T*_{90}). However, for the highest accuracy, this work requires that the data above *T*_{TPW} be repeated with the same SPRT ensemble as was used below *T*_{TPW}. This work is underway, along with an extension to higher temperatures, the results from both of which will be presented in future publications.

## Data accessibility

Supplementary data are available to accompany this article; these data comprise: chronological summary of isotherms; SPRT calibration uncertainty tables; table of SPRT *W* versus *T*; full (*T*−*T*_{90}) uncertainty table for all isotherms; plots of acoustic and microwave residuals and excess half-widths for all isotherms; resistance bridge nonlinearity results; and raw microwave, acoustic and thermometric data.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported in part by the Engineering and Flow programme of the UK Department for Business, Innovation and Skills, and by the EMRP project Implementing the New Kelvin. The EMRP is jointly supported by the EMRP participating countries within EURAMET and the European Union.

## Acknowledgements

The authors thank Rafael Van Den Bossche for help with data preparation, and Jonathan Pearce for assessing the resistance bridge nonlinearity, and a careful reading of the manuscript. We are also grateful to the reviewers, whose constructive comments have led to significant improvements in the manuscript.

## Footnotes

One contribution of 16 to a Theo Murphy meeting isssue ‘Towards implementing the new kelvin’.

- Accepted November 16, 2015.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.