1 Introduction

In May 2019, new definition of some base units of the International System of Units (SI) has been effective by fixing the numerical values of fundamental constants [1]. For thermodynamic temperature, the unit kelvin has come to be defined based on the Boltzmann constant. Acoustic gas thermometry (AGT) is one of the methods for measuring thermodynamic temperature, which utilizes the dependence of the speed of sound on thermodynamic temperature in a dilute gas. Moldover et al. [2] first reported in detail the development of a spherical acoustic resonator for the speed of sound measurement in argon, and determined the Boltzmann constant based on measurements at the triple point of water. Since in principle, based on the fixed Boltzmann constant, the AGT can accurately measure the thermodynamic temperature, it was then also applied for thermodynamic temperature measurements [3,4,5,6,7,8,9,10]. Recent reports consistently evidenced systematic differences between thermodynamic temperature T and the temperature T90 based on the International Temperature Scale of 1990 (ITS-90) [11]. Since the International Temperature Scale was defined in 1990 as the best approximation to thermodynamic temperature at that time, the evaluation of the difference (T − T90) by using state-of-the-art technique becomes important to improve practical temperature measurements based on the ITS-90 [12].

Taking into account the importance of thermodynamic temperature measurement, we built a new AGT system at the National Metrology Institute of Japan (NMIJ). This new system is a continuation of our work on developing a prototype acoustic gas thermometry (AGT) system, as already reported elsewhere [13]. The new AGT system introduces a 1-L quasi-spherical resonator (QSR) made of oxygen-free copper (OFC) fabricated using a diamond-turn finishing of its internal surface. A new mounting configuration of the resonator, intended to reduce the deformation caused by the suspension of the cavity under its own weight, was evaluated by relative measurements of the thermodynamic temperature based on the ratio of speed of sound at several temperatures.

In the following sections, we report about the design and the construction of our new AGT system including the improvement of the performance, which was achieved compared to our earlier system. We also consider two different QSR assemblies, and the corresponding measured thermodynamic temperatures as well as their measurement uncertainty are evaluated. The comparison of the results with the previous existing determinations of (T − T90) at the overlapping temperature range are also evaluated.

2 Measurement Apparatus

The AGT system developed in this work consists of a QSR, a gas-handling system, an acoustic measurement system, and a microwave measurement system. The set-up of the AGT system is shown schematically in Fig. 1. Each part is described in the following subsections.

Fig. 1
figure 1

Measurement system set-up

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2.1 Resonator

The resonator developed in this work was an assembly of two half quasi-spheres, each made of oxygen-free copper by employing a diamond-turn tool. The inner shape of the QSR was designed to be an ellipsoid as expressed by the following equation, where a is the nominal radius, and \({\epsilon_1}\) and \({\epsilon_2}\) are the deformation parameters.

$$\frac{{x^{2} }}{{a^{2} \left( {1 + \epsilon_{1} } \right)^{2} }} + \frac{{y^{2} }}{{a^{2} }} + \frac{{z^{2} }}{{a^{2} \left( {1 + \epsilon_{2} } \right)^{2} }} = 1.$$
(1)

The QSR is illustrated in Fig. 2. The nominal values of a, \({\epsilon_1}\), and \({\epsilon_2}\) are 62 mm, 0.0005, and 0.001, respectively, and the wall thickness of the QSR is 15 mm. These specifications were the same as in the design by Underwood et al. [14]. The equatorial part of the QSR was made like a belt shape, where stainless steel bolts for aligning and fixing the hemispheres, and capsule-type standard platinum resistance thermometers (cSPRTs) were placed. Six ports through the wall of the QSR were made to accommodate OFC plugs used to hold two microphones, two antennas, and two tubes allowing gas to flow through. The plug for transmitter microphones was set at angle 39.2° from the plug for receiver microphones set on the south end. This setting was following the design proposed by Mehl et al. [15] in order to have the best possible separation of (0,2) acoustic radial mode from the neighboring (3,1) acoustic non-radial mode. Plugs for gas-flowing tubes were set on the upper (or north) hemisphere at angular position 45° from the north end. One port for an antenna was placed in the upper hemisphere at the angle position 45° from north end, and one port for another antenna in the lower hemisphere at the angle position 45° from the south end. In the north end, two holes were drilled as housing of cSPRTs. The same holes were also made in the south end.

Fig. 2
figure 2

Quasi-spherical resonator

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The QSR was enclosed in a stainless steel pressure vessel by fixing it to the cap of the vessel with three rods. Figure 3 illustrates two alternative fixing configurations that were evaluated in this work. In the first configuration, shown in Fig. 3a, an OFC circular plate was fixed to the lower surface of the belt-shaped equator, and three stainless steel long bolts (6 mm OD) connected the plate to the pressure vessel cap. In this configuration, as shown in Fig. 3a, the QSR was in the ‘hung at the equator’ position (called hereafter the ‘hung’ position). Small heaters (Sakaguchi E.H VOC, MS-3, 10 × 10 mm, max 40 W) for controlling the temperature of the QSR were attached on the surface of the circular plate and the fixing rods. In the second configuration, shown in Fig. 3b, an OFC made star-shaped plate was fixed on the flat surface at the south end of the QSR, and three OFC rods (15 mm OD) connected the plate to the pressure vessel cap. In this configuration, the QSR was in the ‘sit on plate’ position (called hereafter the ‘sit’ position). A small heater (Sakaguchi E.H VOC, MS-M5, 5  × 5 mm, max 15 W) was attached on each rod.

Fig. 3
figure 3

Fixing configuration of the resonator (up) and fixing plate (down): (a) ‘Hung’ configuration, (b) ‘Sit’ configuration

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2.2 Gas System

A gas-handling system was built to maintain argon gas flowing through the QSR, as an attempt to minimize the effect of impurities from outgassing. Two argon gas samples of 99.9999 % purity (Japan Fine Product, G1) were used. All tubes used in the gas-handling system were made of electro-polished stainless steel. The inlet and outlet tubes connected directly to the gas port of the QSR were 1/16-in. nickel tube with length of 1.1 m. A filter for reducing oxygen content (Nikka Seiko, GC-RX) and a filter for reducing water content (Nikka Seiko, DC-A4) were installed in the upstream side of the gas-handling system. Two mass flow controllers (MFC, Fujikin, FCST1005MZFC) were used to set a constant desired flow rate. Flow rates for both mass flow controllers fixed at 2 sccm for the ‘hung’ position and 5 sccm for the ‘sit’ position were used in this work. One was installed in the upstream side (MFC1), while the other in the downstream side (MFC2). A pressure gauge (Paroscientific, Model 745, max 1.4 MPa) was installed in the upstream side (identified as P1) and the same model pressure gauge was in the downstream side (identified as P2). Before being installed in the handling system, the pressure gauge P2 was calibrated in the range up to 700 kPa by the Pressure Standard Laboratory at NMIJ/AIST, while the pressure gauge P1 was calibrated by comparison with the pressure gauge P2. A third pressure gauge (Paroscientific, Model 2100A, max 700 kPa, identified as P3) was introduced and also calibrated by comparison with the pressure gauge P2. The pressure gauge P3 was once installed at the entrance of the gas line into the liquid bath (62 cm above the gas inlet into the QSR) in order to measure the gas pressure at this point under the flow rate of 2 sccm and 5 sccm. The relation between the pressure P3 the pressure P1 is shown in Fig. 4, implying that the effect of the flow on the indicated pressure is small. The pressure gauge P3 was then removed from the upstream gas line and directly connected to the pressure vessel (62 cm above the gas outlet from the QSR). The installation of the pressure gauge P3 here was intended to measure pressure inside the pressure vessel. The pressure head and the pressure drop induced by the gas flow, which was calculated based on the Hagen–Poiseuille law, were added as corrections to the pressure measured by pressure gauges P1 and P3, to have an average value of pressure inside the QSR.

Fig. 4
figure 4

Pressure difference under gas flow

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2.3 Temperature Measurement System

The temperature of the QSR was measured by employing four cSPRTs. As shown in Fig. 2, one cSPRT was placed at the north end, one at the south end, and two around the equator. The cSPRT at the south end was Fluke 5686-B (S/N HS225), and the rest cSPRTs were Chino R800-0 (S/N RS913-2, RS143-02, and RS143-03). These cSPRTs were calibrated before use at the triple point of water and the triple point of gallium (302.9166 K). Three of them (HS225, RS913-2, and RS143-03) were additionally calibrated at the triple point of mercury (234.3156 K) and the triple point of argon (83.8058 K). Two of them (HS225 and RS913-2) have been used for a long time since our earlier work [13], the stability of which was evaluated using the maintenance system in NMIJ/AIST [16] to estimate a maximum long-term change within ± 0.15 mK. The measurement of the resistance of the cSPRTs in this work was conducted by employing a resistance measuring device (Fluke, 1595A Super-Thermometer) coupled with a temperature-controlled 100-ohm standard resistor (Tinsley, 5685A). The resistance measurements for cSPRTs were compensated for the self-heating effect. For maintaining the temperature of the QSR, the temperature at the south end was used as the reference point.

2.4 Microwave Measurement System

Speed of sound measurements in the QSR require an accurate value of the radius of the QSR. To obtain this value at various temperatures and pressures, the microwave resonance method [7] was adopted in this work. The microwave measurement was conducted by employing two loop antennas, each of which was fixed to the related plug using an epoxy adhesive (stycast) that filled the gap between tip part of the antenna and the plug, as described in earlier report [13]. These antennas were connected to a vector network analyzer (Agilent Technology, ENA5071C) by coaxial cables through hermetic connectors. The vector network analyzer was synchronized to a rubidium clock (Stanford Research Systems, FS725), which served as frequency reference.

2.5 Acoustic Measurement System

Speed of sound measurements in argon within the QSR were performed by employing two free-field externally polarized microphones (G. R. A. S., 40BF), whose diameter was 1/4 in.: one for exciting and one for detecting the acoustic field. A pre-amplifier (G. R. A. S., 26AC) was directly connected to the receiver microphone. A signal generator (Agilent Technology, 33220A) was employed for generating sinusoidal signals. The signal generator referred to the same rubidium clock as the vector network analyzer so that the ratio between the acoustic and microwave frequency measurement could be accurate. An actuator amplifier (G. R. A. S., 14AA) amplified and provided this signal to the source microphone. Then the transmitted signal was detected by the receiver microphone and amplified by the conditioning amplifier (Brüel and Kjær, 2690 OS1). A lock-in amplifier (Stanford Research Systems, SR830) was used to measure the signal detected by the receiver microphone.

2.6 Measurement Set-up and Procedures

As illustrated in Fig. 1, the QSR was enclosed in the pressure vessel. The pressure vessel was immersed in a liquid bath, which maintains a temperature slightly lower than that to measure. As aforementioned in Sect. 2.1, small heaters were employed to adjust the temperature of the QSR to the target temperature. A liquid-circulating chiller (Julabo, FP50) was used to adjust the temperature of the liquid bath. Four cylindrical rods hold the pressure vessel at the cap plate of the liquid bath. The pressure vessel was surrounded by an aluminum basket with a punched wall, so that the effect from liquid flow was minimized. A connection tube was used as a feedthrough of the cables. A vacuum pump was employed to pump out the argon gas for realizing a continuous flow.

In this work, thermodynamic temperature was determined by implementing the method of relative acoustic gas thermometry, which is based on the evaluation of the ratio between the speed of sound at the temperature T of interest and the speed of sound at TPW [3, 4]. We adopted 273.16 K as the value of the TPW and 0.1 mK as its standard uncertainty on the basis of the Appendix 2 of SI Brochure—9th edition [17]. The measurements at the TPW were conducted at first, preceding those at other temperatures. For one isotherm, microwave resonance frequencies (based on the S-parameter S21) were first measured under vacuum, then series of the microwave and the acoustic resonance frequency were measured at pressures from 700 kPa down to 50 kPa, and terminated by measurement of the microwave resonance frequencies under vacuum. At each pressure, a series of acoustic and microwave resonance frequency measurements were repeated three times. In one series, the acoustic resonance frequencies for radial modes (0,n) with n = 2 to 8 were measured, and the microwave resonance frequency for transverse magnetic modes TM1l and transverse electric modes TE1k with l = 1 to 4 and k = 1 to 3.

At first, the abovementioned procedures were done with the QSR in the ‘hung’ configuration at the TPW and the melting point of gallium. The argon gas used in the first trial was identified as Sample 1 in this work. The QSR configuration was then made to the ‘sit’ configuration described in Sect. 2.1. The change in the QSR configuration was performed very carefully so that changes in the QSR alignment, plugs, and tubes position were minimized. We needed, however, to remove temporarily the receiver microphone at the south-end when installing the star-shaped plate (see Sect. 2.1). The microphone was reinstalled to the same port with the original condition. The same measurement procedures were then executed with the QSR in the ‘sit’ configuration at the TPW, 283.15 K, 293.15 K, and the melting point of gallium. The argon gas used in this ‘sit’ configuration was different from that at the ‘hung’ configuration, and was identified as Sample 2.

3 Measurement Results

3.1 Microwave Measurement

The measured microwave frequencies at each TM or TE mode were fitted to the resonance function reported in [18] introducing linear and quadrature backgrounds to have three resonance frequencies and their half-widths for each mode. These resonance frequencies were then corrected for the effects from the microwave penetration length [18] and from the waveguide [14]. The second-order shape correction proposed by Mehl [19] was applied to derive the deformation parameters, \({\epsilon_1}\) and \({\epsilon_2}\) in Eq. 1, as well as the nominal radius of the QSR, a. The values of the refractive index of argon used for these derivations were calculated using the method in [18] (Eq. 15 truncated after the second term). From values a obtained at various temperatures and pressures, the nominal radius of the QSR can be expressed by the following relation [20].

$$a\left( {T,P} \right) = a_{0} \left[ {1 - \frac{{K_{P} }}{3}P + \left( {T - 273.16} \right)K_{T} } \right],$$
(2)

where KP and KT in Eq. 2were empirically determined as the slopes of the dependence of a on pressure P and temperature, respectively. These may refer to isothermal compressibility and thermal expansion coefficient of the QSR, respectively. The values of KP in Eq. 2 at the TPW were 1.05 × 10−11 Pa−1, while KT was 1.66 ×  10−5 K−1. a0 in Eq. 2 represents the nominal radius of the QSR at T = 273.16 K and zero pressure, which was the average of the extrapolated values at P = 0 of mode TM12 to TM14 and TE11 to TE13, as shown in Fig. 5 by a dotted line for ‘hung’ configuration and a dashed line for ‘sit’ configuration. The discrepancy in a0 between the two configurations was 1.2 ppm. In both configurations, inconsistency of mode TM11 was estimated to be 2.6 ppm. The deformation parameters \({\epsilon_1}\) and \({\epsilon_2}\) obtained from the microwave measurements at various modes are summarized in Fig. 6, where a slight discrepancy was observed between the ‘hung’ and ‘sit’ condition. We found that \({\epsilon_1}\) and \({\epsilon_2}\) were significantly smaller than designed. It is likely that this leads to an imperfect lift of the threefold degeneracy of TM1l and TE1k modes, whose triplet peaks appeared in a narrow range of frequency, f, as shown in Fig. 7, where the vertical axis is S21 parameter.

Fig. 5
figure 5

Effective radius of the resonator for various modes

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Fig. 6
figure 6

Deformation parameters of resonator for various modes: (a) \({\epsilon_1}\), (b) \({\epsilon_1}\)

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Fig. 7
figure 7

Microwave resonance at mode TM11: (a) measured data and correlation, (b) residuals of the fit

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3.2 Acoustic Measurement

In the ‘hung’ position, the resonance frequencies of several acoustic modes were measured at pressures from 700 kPa down to 50 kPa at the TPW, and from 500 kPa down to 60 kPa at the melting point of gallium. In the ‘sit’ configuration, on the other hand, the measurements at all isotherms were performed in the pressure range from 500 kPa down to 60 kPa. The measured acoustic frequencies at each radial mode were fitted to the resonance function reported in [18] by introducing linear background to derive the resonance frequency and its half-width for each mode. The derived resonance frequencies were corrected for the effects from the thermal boundary layer and the acoustic transducer perturbation following the method compiled in [18]. The thermal accommodation coefficient, h, was assumed as h = 1 for the thermal boundary correction here [6]. They were also corrected for the duct perturbation coming from the inlet and outlet gas tubes, by employing the model proposed by Gillis et al. [15, 21].

Contribution of the thermal boundary layer and bulk dissipation to the resonance half-width were calculated following [18], and the calculation for the contribution of the inlet and outlet gas tubes to the resonance half-width was following [15]. The difference between the frequency half-width and the above-calculated contributions, which is called the excess half-width hereafter, is depicted in Fig. 8. It is obvious from Fig. 8 that at frequencies around 14 kHz, the excess half-width, especially at the TPW, became quite large. This frequency was near the acoustic mode (0,6). This large excess half-width is related to the existence of resonance frequency of the QSR itself, which is often called the breathing frequency of the QSR, fbr. This frequency should be determined and compensated for. Moldover et al. [3] derived a method to determine the breathing frequency using only the properties of the material of the QSR. When this method was applied to our QSR, the fbr was 15.2 kHz, which was not near the mode (0,6), which was around 13.6 kHz. Pitre et al. [22] proposed a method for determining fbr by optimizing the acoustic frequency that produces second acoustic virial coefficient independent on the observed radial modes, and in this work, this strategy was adopted. The speed of sound, w, was derived from the resonance frequency, by introducing a and KP determined from the microwave resonance frequencies measured at the respective isotherms for mode TM12 to TM14 and TE11 to TE13. The speed of sound measured at the TPW for modes (0,2) to (0,8), with exception of the mode (0,6), was fitted to the following equations, A0 to A2 are the fitting parameters, and A3 is 1.45 × 10−9 m2·s−2·kPa−3 for the TPW as reported in [3, 6].

$$w^{2} - {\text{A}}_{3} P^{3} = {\text{A}}_{0} + {\text{A}}_{1} P + {\text{A}}_{2} P^{2}$$
(3)
$$w = \frac{2\pi a}{{Z_{0,n} }}\left( {f_{0,n} - \Delta f} \right).$$
(4)
Fig. 8
figure 8

Excess half-width of acoustic resonance for various modes, inset for whole scale

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Δf is the sum of corrections to the unperturbed resonance frequency f0,n, and Z0,n is the eigenvalue of acoustic mode (0,n). The following correction for the resonance frequency due to the shell effect is adopted [22], where ρwall and wwall are the density of and the speed of sound in the resonator material (copper), and t is the thickness of the wall of the resonator.

$$\frac{{\Delta f_{\text{sh}} }}{{f_{0,n} }} = \frac{5a}{{6t\rho_{\text{wall}} w_{\text{wall}}^{2} }}\frac{1}{{1 - \left( {\frac{{f_{0,n} }}{{f_{\text{br}} }}} \right)^{2} }}.$$
(5)

Adopting the method proposed in [22] gives the value of fbr to be 13.5 kHz. This value is shown in Fig. 8 as a vertical broken line.

The speed of sound obtained at the TPW, 283.15 K, 293.15 K, and the melting point of gallium, after the correction of resonance frequency described above, was fitted to Eq. 3 to obtain the value of A0, which corresponds to the square of the speed of sound at the limit condition of P = 0, w20. For fitting the data at the TPW, the value of A3 reported in [3, 6] was introduced. For the other temperatures, value of A3 was obtained from quadratic fitting of those reported in [23] to obtain the values of 1.09 × 10−9, 7.9 × 10−10, and 5.4 × 10−10 m2·s−2·kPa−3 for 283.15 K, 293.15 K, and 302.9146 K, respectively. Figure 9 summarizes the obtained data, w, from Eq. 3, wcorr, for selected modes and configurations at aforementioned temperatures. Most of the data agreed with Eq. 3 within ± 2 ppm, except those at lower pressures below 150 kPa, which showed a large scatter. At a dilute gas condition, electrical noise occurred more frequently than at a dense gas condition, and the determination of the resonance frequency became erratic. For the mode (0,2) at the TPW, as shown in Fig. 9a, c, even at higher pressures the scatter was larger than ± 2 ppm. This may be caused by the effect from the neighboring non-radial mode (3,1) (see [3]), which was not completely suppressed by the present system.

Fig. 9
figure 9figure 9

Residual speed of sound: (a) at 273.16 K under ‘Hung’ configuration, (b) at 302.91 K under ‘Hung’ configuration, (c) at 273.16 K under ‘Sit’ configuration, (d) at 283.15 K under ‘Sit’ configuration, (e) at 293.15 K under ‘Sit’ configuration, (f) at 302.91 K under ‘Sit’ configuration; small-sized symbols for data excluded in fitting

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3.3 Discussion

From the speed of sound extrapolated to the limit P = 0 described above, the thermodynamic temperature, T, was determined by employing the following relationship.

$$T = T_{\text{ref}} \frac{{\mathop {\lim }\limits_{P \to 0} w^{2} \left( {P,T} \right)}}{{\mathop {\lim }\limits_{P \to 0} w^{2} \left( {P,T_{\text{ref}} } \right)}} = T_{\text{ref}} \frac{{w_{0}^{2} \left( T \right)}}{{w_{0}^{2} \left( {T_{\text{ref}} } \right)}}.$$
(4)

As Tref, we adopted the TPW, that Tref = 273.16 K, the value based on the Appendix 2 of SI Brochure—9th edition [17], and its uncertainty was 0.1 mK, corresponding to a relative uncertainty 3.7 × 10−7 [17, 24]. Figure 10 represents temperatures calculated using Eq. 4 for four isotherms based on the speed of sound obtained by each mode. The numerical value of T for each T90 is summarized in Table 1. T in Table 1 is the average value of those of related acoustic modes. Also listed in Table 1 are the difference between T and T90, (T − T90) and its uncertainty, u(T − T90).

Fig. 10
figure 10

Thermodynamic temperatures at various modes: (a) at 283.15 K, (b) at 293.15 K, (c) at 302.91 K

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Table 1 Results of (T − T90)
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In Fig. 10c, the results obtained at the gallium melting point under the ‘hung’ configuration are compared with those under the ‘sit’ configuration. The T values obtained under the ‘hung’ configuration for various acoustic modes scattered within similar temperature range to those under the ‘sit’ configuration. This implies that under the present system, the difference between the ‘hung’ and ‘sit’ configurations was not significant. The results of the isotherm 293.15 K, as shown in Fig. 10b, seemed not to depend on the acoustic mode.

The uncertainty of (T − T90), namely u(T − T90), is mainly due to the contributions of temperature, pressure, microwave, and acoustic measurements. We estimated u(T − T90) based on the categories and components as listed in Table 2. The uncertainty of the pressure measurements affects both the microwave and acoustic measurements, so it is included in both the microwave and acoustic categories.

Table 2 Estimated uncertainty of (T − T90)
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The uncertainty related to the determination of T90 is the combination of several contributions from the cSPRTs and the imperfect temperature uniformity and stability of the cavity within the QSR. The uncertainty of the cSPRTs includes the calibration uncertainty at the TPW and at the gallium melting point, the long-term stability, which was estimated from the change in the resistance at the TPW, and the non-uniqueness. The non-uniqueness was estimated following the report by White and Strouse [25] and White et al. [26]. The uncertainty of the temperature realization within the QSR is the combination of the stability of temperature realized and the temperature inhomogeneity across the QSR. Figure 11 represents the temperature measured by four cSPRTs, described in Sect. 2.3, under the ‘hung’ position and the ‘sit’ position. As shown in Fig. 11, the effect of heating from the pre-amplifier, attached at the south end microphone, in the ‘hung’ configuration was not satisfactorily compensated, resulting in a temperature inhomogeneity across the QSR within ± 0.6 mK. On the other hand, in the ‘sit’ configuration, the effect of heating from the pre-amplifier was satisfactorily compensated and the temperature of the QSR was uniform within ± 0.3 mK.

Fig. 11
figure 11

Temperature stability and uniformity across the resonator at the TPW: (a) ‘Hung’ configuration, (b) ‘Sit’ configuration

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The uncertainty related to the microwave measurements includes the mode inconsistency, the observed deviation of the thermal expansion, and compression of the radius from the calculations of the simplified model in Eq. 2 and the uncertainty of pressure measurement propagated to the microwave resonance frequency. The mode inconsistency was estimated from the discrepancy of a of the modes TM12 to TM14 and TE12 to TE13, represented in Fig. 5, from the average of a. This component may relate to effects that have not been completely compensated, namely a possible oxide layer on the inner surface of the QSR, slits, grooves, and holes formed due to misalignment of microphones, antennas, and hemispheres. Concerning the deviations of the effective radius from Eq. 2, these effects were included in the uncertainty for the acoustic measurements, so it is not listed in Table 2. The pressure effect was estimated based on the observed variation of the resonance frequency caused by the variation of the argon gas flow.

The uncertainty of our speed of sound determinations was combined from the acoustic frequency measurements at both TPW and T, including the mode discrepancy and the fitting uncertainty of the virial expansion for extrapolating to the limit zero pressure. Residuals from these fits are shown in Fig. 9, while the contribution from the discrepancies among different modes was calculated based on the speed of sound extrapolated to the limit zero pressure, w20 in Eq. 4, for various acoustic modes. As in the case of microwave measurements, the effect of pressure measurements on the uncertainty of the acoustic measurements was estimated based on the difference in resonance frequency with and without argon gas flow.

In Sect. 3.2, it has been already stated that h = 1 was used for the thermal accommodation parameter in this work. Using the assessment method of the coefficient h proposed in [22], we found h in the present system would be in the range between 0.7 and 1. Therefore, we consider the uncertainty due to h selection by comparing the case of h = 1 and h = 0.7.

The values of (T − T90) listed in Table 1 exist in an overlapping temperature range including those reported by Moldover et al. [4], Ewing and Trusler [5], Benedetto et al. [6], Underwood et al. [9] and Gavioso et al. [10]. Figure 12 is a plot of these reported data in a range covering our results. The error bars show the standard uncertainty of each data. To make a better view, the error bar for the uncertainty in Table 1 is arrowed. It is seen from Fig. 12 that T at the gallium melting point agrees within the estimated uncertainty with the data reported in [4, 5, 6, 9, 10]. T corresponding to T90 = 283.15 K and T90 = 293.15 K are both larger than those by Moldover et al. [3] and by Underwood et al. [9]. These differences are within the estimated uncertainty.

Fig. 12
figure 12

Existing (T − T90) and the present work

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4 Conclusions

A new acoustic gas thermometry (AGT) system was built based on a diamond-turn QSR, where two different configurations for fixing the QSR were evaluated based on the measurement of the speed of sound in argon at the triple point of water and the melting point of gallium under the pressure range from 700 kPa down to 50 kPa. It was found from the thermodynamic temperatures obtained from these speed of sound measurements at the melting point of gallium that different QSR fixing configurations exhibit no significant difference to the value of the thermodynamic temperature determined. The speed of sound measurements at the isotherm of 283.15 K, 293.15 K, and 302.9146 K was also conducted and compared with the currently reported values that exist in overlapping temperature range with the present work. No significant difference was found between the present results and the reported values. They were in agreement within the estimated uncertainty.