Jiangfeng Hu

Haiyang Zhang

Yaonan Song

Changzhao Pan

Bo Gao
email: bgao@mail.ipc.ac.cn

Wenjing Liu

Dongxu Han

Zhen Zhang

Ercang Luo

Laurent Pitre

Gao Bo

Han Dongxu

Luo Ercang

Pitre Laurent

Investigation of High-Stability Temperature Control in Primary Gas Thermometry

Keywords: temperature control, temperature stability, temperature resolution, rhodium-iron resistance thermometer, primary gas thermometry

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

On May 20th, 2019, World Metrology Day, the Bureau International des Poids et Mesures announced a major revision to the International System of Units (SI) in which the definitions of four of the base units were redefined implicitly by fixing the values of four constants of nature [START_REF] Phillips | The end of artefacts[END_REF]. With the help of fundamental constants, one can establish and realize units at the highest level of precision in multiple ways at any place or time and on any scale, without losing accuracy [START_REF] Gibney | New definitions of scientific units are on the horizon[END_REF][START_REF] Klitzing | Metrology in[END_REF]. The practical realizations of the base unit, the kelvin, symbol K, are indicated in the "Mise en pratique for the definition of the kelvin in the SI", in which the absolute and relative Measuring time/h primary refractive-index gas thermometry have been included [START_REF]Mise en pratique for the definition of the kelvin in the SI[END_REF][START_REF] Rourke | Refractive-index gas thermometry[END_REF].

Nomenclature

To realize accurate thermodynamic temperature measurements worldwide, temperature stability is a key aspect of state-of-the-art primary gas thermometry [START_REF] Gao | Measurement of thermodynamic temperature between 5 K and 24.5 K with single-pressure refractive-index gas thermometry[END_REF][START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF][START_REF] Pitre | Determination of the Boltzmann constant using a quasi-spherical acoustic resonator[END_REF][START_REF] Gaiser | Primary Dielectric-Constant Gas Thermometry in the Range from 2.4K to 26K at PTB[END_REF][START_REF] Pitre | New measurement of the Boltzmann constant k by acoustic thermometry of helium-4 gas[END_REF][START_REF] Rourke | NRC Microwave Refractive Index Gas Thermometry Implementation Between 24.5 K and 84[END_REF][START_REF] Gavioso | Determination of the thermodynamic temperature between 236 K and 430 K from speed of sound measurements in helium[END_REF][START_REF] Osadchii | Stabilization of the Temperature of Resonator of an Acoustic Gas Thermometer[END_REF][START_REF] Berry | NPL-75: A Low Temperature Gas Thermometry Scale from 2.6 K to 27.1[END_REF][START_REF] Gaiser | Primary dielectric-constant gas thermometry in the range from 2.4 K to 26 K at PTB[END_REF], in which there are some strong coupling effects between different physical quantities especially at low temperatures. For example, with the help of better temperature stability, the measurement of pressure and capacitance changes was improved in the new dielectric constant gas thermometry at Physikalisch-Technische Bundesanstalt in Germany [START_REF] Gaiser | Primary dielectric-constant gas thermometry in the range from 2.4 K to 26 K at PTB[END_REF]. It is also the case for single-pressure refractive-index gas thermometry (SPRIGT), where the temperature stabilities not only are included in the total standard uncertainty of the determined thermodynamic temperature [START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF] but also have a direct influence on microwave measurements [START_REF] Zhang | Resonance frequency measurement with accuracy and stability at the 10-12 level in a copper microwave cavity below 26 K by experimental optimization[END_REF] as well as pressure control [START_REF] Gao | Realization of ppm level pressure stability for primary thermometry using a primary piston gauge[END_REF]. Besides, high-stability temperature control is also necessary for facilitating high-accuracy international data comparison [START_REF] Rusby | KEY COMPARISON: Final Report on CCT-K1: Realizations of the ITS90, 0.65 K to 24.5561 K, using rhodium iron resistance thermometers[END_REF][START_REF] Gaiser | EURAMET key comparison no. EURAMET. T-K1: realisations of the ITS-90 from 2.6 K to 24.5561 K, using rhodium-iron resistance thermometers[END_REF] and for temperature calibration [START_REF] Nakano | Comparison System for the Calibration of Capsule-Type Standard Platinum Resistance Thermometers at NMIJ/AIST[END_REF]. In the above primary gas thermometry, data comparison and calibration systems, rhodium-iron resistance thermometers (RIRT) were used to control and measure the temperatures under the International Temperature Scale of 1990 (ITS-90). Since RIRTs are the practical thermometers on which realizations of the ITS-90 [START_REF]The international temperature scale of 1990[END_REF] are most accurately maintained, disseminated and compared [START_REF] Rusby | KEY COMPARISON: Final Report on CCT-K1: Realizations of the ITS90, 0.65 K to 24.5561 K, using rhodium iron resistance thermometers[END_REF]. Moreover, the stable temperature measurements of T 90 by RIRTs are also used to estimate the values of TT 90 , the discrepancy between thermodynamic temperature and that of the ITS-90 in primary gas thermometry.

To provide high-stability work conditions for high-accuracy temperature measurements and international data comparisons in the future, temperature control using RIRTs was further studied in the present work. The rest of the paper is structured as follows. In section 2, we describe the apparatus pertinent to the discussion of the cryostat, namely the temperature measurement and control system. A detailed description of the microwave measurement, pressure control and measurement lie beyond the scope of the present paper, they can be respectively found in our previous papers [START_REF] Zhang | Resonance frequency measurement with accuracy and stability at the 10-12 level in a copper microwave cavity below 26 K by experimental optimization[END_REF][START_REF] Gao | Realization of ppm level pressure stability for primary thermometry using a primary piston gauge[END_REF]. In section 3, a relationship between the temperature stability limit and temperature resolution was present. In section 4, the experimental results of the temperature control and measurement are described in detail in the range from 5 K to 24.5 K. In section 5, a conclusion and perspectives were present.

Apparatus and methods

Cryostat for SPRIGT

To achieve a highly stable temperature environment at low temperatures for SPRIGT measurements, a cryogen-free cryostat, refrigerated by an available Sumitomo model RP-082B2 GM type pulse tube cooler (GM-PTC), was used. The cryostat has been described in detail in our previous work [START_REF] Gao | Realization of an ultra-high precision temperature control in a cryogen-free cryostat[END_REF][START_REF] Chen | Thermal response characteristics of a SPRIGT primary thermometry system[END_REF]. It is only briefly described here.

Fig. 1 shows the simplified schematic diagram of the cryostat for SPRIGT. The declared cooling powers of the GM-PTC are about 1 W at 4.2 K and 45 W at 45 K. A multi-layer radiation shield was added, and a thermal-resistance method and a gas-type heat switch were used. The multi-layer radiation shield and thermal-resistance method were used to reduce the thermal noise from surroundings as shown in our previous work [START_REF] Gao | Realization of an ultra-high precision temperature control in a cryogen-free cryostat[END_REF]. Heat switch on mode with helium-4 gas was employed all the time in the present work, with which the resonator and the pressure vessel can be easily cooled down to about 4.3 K [START_REF] Chen | Thermal response characteristics of a SPRIGT primary thermometry system[END_REF], although it is less good for the temperature control. To reduce the perturbation from the surroundings still further, a laboratory with good room temperature regulation was constructed. Besides, a novel active control temperature oscillation method was developed [START_REF] Pan | Active suppression of temperature oscillation from a pulse-tube cryocooler in a cryogen-free cryostat: Part 1. Simulation modeling from thermal response characteristics[END_REF][START_REF] Pan | Active suppression of temperature oscillation from a pulse-tube cryocooler in a cryogen-free cryostat: Part 2. Experimental realization[END_REF].

Temperature control method

To realize long-term stability temperature control at low temperatures, proportion-integration-differentiation (PID) control on the 2 nd flange and resonator were simultaneously implemented with Cernox and RIRT sensors respectively in the present work. As for the temperature control on flange, it has been described in detail in our previous work [START_REF] Gao | Realization of an ultra-high precision temperature control in a cryogen-free cryostat[END_REF][START_REF] Chen | Thermal response characteristics of a SPRIGT primary thermometry system[END_REF]. Here below only that for the resonator is briefly presented.

Table 1 lists the specified instruments and sensors for resonator temperature control and measurement. The resonator temperature was controlled by one RIRT (Tinsley, SN 226245), calibrated by National Physical Laboratory (NPL) UK and denoted NPL2, using a manual AC resistance bridge (ASL F18) and a Tinsley 5685A standard 10 Ω resistors (SN 1631808). While the resonator temperature stability was monitored by another NPL-calibrated RIRT (Tinsley, SN 226242, denoted NPL1) using a ASL F900 AC resistance bridge combining a Tinsley 5685A standard 10 Ω resistor (SN 1580409). The amplified voltage signal of NPL2, read by 8½ Keithley 2002, is available to be used as the feedback for the PID control. The key implementation steps of the PID control are: (1) set the desired ratio for the controlled thermometer NPL2 with F18 bridge operated in manual balance mode [START_REF]F18 Operator's Handbook[END_REF]; (2) read the feedback signal, the voltage V of the F18 bridge; (3) heat the pressure vessel by Heater 3 according to the feedback; (4) adjust the PID parameters until the resonator temperature measured by NPL1 is stable. 3. Temperature stability limit and temperature resolution

Limit of temperature control and measurement

In the present work, two assumptions were made as shown in Fig. 2:

(1) The number of temperatures larger than the average value T avg , N(T n + ), is equal to that of temperatures smaller than T avg , N(T n -), which is usual the case for long-term temperature control and measurement.

    n n N T N T N     (1) avg avg
,

n n T T T T     (2) 1 1 avg 2 N N n n n n T T T N         (3)
where N is half of the total measured data point number. By PID control, the measured temperatures are preferred to approach the temperature set point T set point and usually the measured average value T avg is very close to or even equal to T set point , i.e., set avg point

T T  (4)
(2) The difference between two continuous temperature measurements should be not less than the sensor's temperature resolution  T . 1 .

1, 2, ..., 2 1

x x T T T x N       (5)
where x means the data point of the x-th measurement. The stability of the measured temperature, defined by the standard deviation of the measured temperature corresponding to an integration time of τ, can be calculated as follows:

      2 2 avg avg 1 + 2 1 N n n n T T T T T N          (6)
For n positive numbers a i , we have the following inequalities (see page 163 in [START_REF] Angel | Problem-solving strategies (Problem books in Mathematics)[END_REF]):

2 2 2 1 2 1 2 ... ... . 1, 2, 3,... n n a a a a a a n n n         (7)
where the equality sign is valid only if a 1 =a 2 =…=a n . Combined Eqs. ( 2), ( 6) and ( 7), we can get:

  2 2 1 2 T T T N N        ( 8
)
where the equality sign is valid only if:

1 avg 2 avg avg avg 1 avg 2 avg n n T T T T T T T T T T T T                      (9)
i.e.,

+ 1 2 1 2 + avg 2 n n T T T T T T T T T T T                     (10)
In this case, Eq. ( 10) is also equivalent to say that there are only two measured points T + and T -. From Eq. ( 5), we know

+ T T T     (11)
Combined Eqs. ( 8) and ( 11), we can get the following equation for long-term PID temperature control and measurement:

2 T T    (12)
where the equality sign is valid only if Eq. ( 10) is satisfied, T + -T -= T , and N→+∞.

From Eq. ( 12), we can know that with PID control the smallest stability of the measured temperature (i.e., the theoretical temperature stability limit  T,L ), defined by the standard deviation of temperature corresponding to an integration time of τ, amounts to half the resolution of the temperature sensor  T .

 

,L min 2

T T T      ( 13
)
where the temperature sensor resolution  T is described by Eq. ( 22).

Indicator for evaluating stability

Fig. 3 shows the measurement sequence with an integration time or sampling time of τ and the method for generated integration time τ g for a period of measuring time t. To evaluate the controlled and measured stability of physical quantity X, the following indicators were employed in the present work.

The stability of the measured physical quantity X, corresponding to an integration time of τ for a measuring time t, σ X (), is defined by the following equation:

      2 1 1 1 , 1 N N n n X n X n N t N                  (14)         0 0 2 0 0 0 1 1 0 1 , , , 1 N N X n n nt nt N u t N                      ( 15
)
where N and N 0 are the numbers of total measured data point for an integration time of τ and t 0 with t=Nτ=N 0 t 0 .

Temperature resolution model

The temperature resolution  T is the smallest detectable increment of measurement that can be determined by the measurement system and the chosen sensor. It is a combination of sensor sensitivity and instrument resolution.

According to the handbooks of F900 [START_REF]Operator's Handbook[END_REF] (page 38) and F18 [START_REF]F18 Operator's Handbook[END_REF] (page 34), the typical voltage resolution of the two bridges for 1 ohm matching impedance with integration time τ (in seconds) for a single measurement, i.e. the specific voltage resolution  v (in nV• -1 ), is:

v c    ( 16
)
where voltage noise coefficient

c = 0.3 in nV•Hz -1/2 • -1 .
Usually, the integration time τ (in second) is reciprocal of bandwidth (in Hz) of the AC bridges, and can be chosen by setting different bandwidths. For convenience, integration time is used throughout the present work. While the output impedance R out (in ) for the measured sensor is equal to (page 18 of F18 handbook and page 23 of F900 handbook):

  2 out s 1 t 2 2 2 R R R n R R     ( 17
)
where R s and R 1 are values of the standard resistor and its potential lead resistance, while R t and R 2 are values of the thermometer resistance and its potential lead resistance with a unit in ohm, n is the transformer ratio at balance for the bridges with n=R t /R s . Note: when measuring low resistances especially for cryogenic applications, the potential lead resistances are significant and cannot be neglected in the calculation of the sensor impedance, while the current lead resistance has no effect on the output impedance of the sensor [START_REF]F18 Operator's Handbook[END_REF][START_REF]Operator's Handbook[END_REF].

From Eqs. ( 16) and ( 17), the voltage resolution  V (in nV) of the measured thermometer can be estimated by:

out = V v R    (
= V R I   ( 19
)
Using the resistance-temperature characteristic of a thermometer, the temperature resolution  T (in K) can be calculated as follows:

R T S    (20)
where S (in •K -1 ) is the sensitivity of the thermometer: t = dR S dT [START_REF]The international temperature scale of 1990[END_REF] Combining Eqs. ( 16)-( 20), we can determine the temperature resolution  T (in K) of a thermometer by:

  2 s 1 t 2 s,pl t,pl s t = 2 2 = 1 1 T c R R n R R IS R R c n R R I                                   (22) s,pl 1 t,pl 2 t = , 2 , 2 S R R R R R    (23)
where  (in K -1 ) denotes the temperature coefficient of the thermometer. R s,pl and R t,pl are values of total potential lead resistances of standard resistors and RIRT thermometers with a unit in ohm.

One can see that the temperature resolution mainly has three types of influence factors: (1) instrument parameters (voltage noise, excitation current, integration time or bandwidth); (2) sensor characteristic (resistance, sensitivity) and potential lead resistances; (3) standard resistor characteristic and potential lead resistances. However, the temperature resolution mainly depends on the adjustable parameters of the AC resistance bridge for a fixed temperature once chosen the matched sensors, AC resistance bridges and standard resistors, which is the case in this work and it will be described in section 4.1.

Uncertainty estimation for the temperature limit

Note: unless otherwise stated, all uncertainties in the present work are standard uncertainties with coverage factor k=1, corresponding to a 68% confidence interval.

The standard uncertainty of the temperature resolution, i.e., u( T ) (in K), can be deduced from Eq. ( 22) based on the error propagation formula:

  2 2 2 R,1 R,2 R,3 T T u u u u      (24)         2 2 2 2 s s , p l R,1 out 2 2 R R u c u u I u n u n c I R                                  ( 25
)
where u(c), u(), u(I) and u(n) respectively are standard uncertainties of voltage noise coefficient c, integration time τ, excitation current I, transformer ratio n.

    2 2 2 R,2 s s,pl out + n u u R u R R   ( 26
)
where u(R s ) and u(R s,pl ) respectively are standard uncertainties of the standard resistor resistance R s and its total potential lead resistance R s,pl .

    2 2 t,pl t R,3 out out u R R u u T R R                                ( 27
)
where u(R t,pl ) and u(T) respectively are standard uncertainties of total potential lead resistance R t,pl and the measured temperature T.  (in K -1 ) denotes the ratio of the second derivative to the first derivative of resistance with respect to temperature:

1 dS S dT    (28)
The standard uncertainty of temperature stability limit, u( T, L ) in K, can be determined by Eq. ( 13) based on the error propagation formula:

    ,L ,L T T T T u u      ( 29
)
where the right side of the equation has been given in Eq. (24).

AC bridge parameters

The typical value of the voltage noise coefficient c is equal to 0.3, however, the standard uncertainty was not declared in the handbooks of the AC bridges [START_REF]F18 Operator's Handbook[END_REF][START_REF]Operator's Handbook[END_REF]. According to the rounding rule, the standard uncertainty u(c) was estimated as 0.05. In this work the default integration time =33.6 s with the standard uncertainty u()=0.23 s. The declared error for transformer ratio n is 0.1 parts per million (ppm). The declared accuracy of the excitation current I is about ±0.1 %. A uniform distribution was applied and thus the standard uncertainty of excitation current, u(I), is determined as:

  0.001 3 u I I  (30)

Rhodium-iron resistance thermometers

The two RIRT sensors, NPL1 and NPL2, were originally calibrated from 2.6 K to 27.1 K by NPL [START_REF] Berry | NPL-75: A Low Temperature Gas Thermometry Scale from 2.6 K to 27.1[END_REF][START_REF] Rusby | KEY COMPARISON: Final Report on CCT-K1: Realizations of the ITS90, 0.65 K to 24.5561 K, using rhodium iron resistance thermometers[END_REF]. The declared calibration standard uncertainties of T 90 lie below 0.50 mK for 2.6 K<T 90 <27.1 K. The standard uncertainty u(T 90 ) (in K) for NPL1 and NPL2 can be described as a function of T 90 by a linear fitting with Goodness of Fit R 2 =0.96:

    5 90 90 10 21 u T T     (31)
For simplicity, hereafter all the T 90 in the present work are denoted as T.

During the system cooling down, the resistances of the potential and current leads of the two RIRTs were manually measured at 4.66 K and 82.3 K using a multimeter (SATA 03015 mode) as shown in Fig. 4. Once the resistances, R + and R -, were measured, the total potential lead resistance R pl can be simply determined by supposing that R pl + =R cl + and R pl  =R cl  , which are nearly the truth because the four lead wires of each thermometer are with the same material and length.

t,pl + = 2 R R R   (32)
where one uncertainty component for R t,pl , u 1 (R t,pl ), was introduced, which is half of the difference between R + and R  .

  1 t,pl = 2 R R u R    (33)
The other uncertainty component is from the accuracy of the multimeter with a value of 1%. A uniform distribution hypothesis was supposed, thus the standard u 2 (R t,pl ) was determined:

  2 t,pl pl 0.01 3 u R R  (34)
Then the total standard uncertainty of R t,pl , u(R t,pl ), was calculated as follows by supposing that the two components are independent.

      2 2 t,pl 1 t,pl 2 t,pl + u R u R u R  (35)
Based on Eq.s (32)-( 35), R pl and u(R pl ) for 4.66 K and 82.3 K were determined and the values are listed in Table 2. From 82.3 K to 4.66 K, the total potential lead resistances have respectively changed only 6% and 9% for a 77.64 K temperature difference. Thus, to estimate the R t,pl and u(R t,pl ) from 5 K to 24.5 K, a linear function on R pl was supposed as R t,pl =aT+b (36) where the two constants a (in Ω•K -1 ) and b (in Ω) need to be determined.

In order to achieve uncertainty estimation for the interpolated R pl between 4.66 K and 82.3 K, a Monte-Carlo simulation was performed on Eq. (36). The input quantities T and R pl are varied by using random uniform distributed numbers, where the variations were carried out at the level of the estimated uncertainty of each specific quantity. Table 2 lists the two parameters and their standard uncertainties, with which R pl and u(R pl ) for temperatures from 5 K to 24.5 K can be estimated by Eq. ( 36) combining with the error propagation formula. a During the SPRIGT system being cooled down, the temperature of a Cernox sensor (CX-1050-CU-HT-1.4L, SN X118726) T 10 were recorded all the time, while those for NPL1 and NPL2 were only read below 26 K

Standard resistors

The standard resistors for AC bridges were placed in an oil bath (Aikom Instruments MR 5100-L) at 20°C with temperature stability better than 1 mK. The standard uncertainties for the standard resistors and its total potential lead resistances are summarized in Table 3.

Uncertainty estimation for the temperature limit

Fig. 5 shows the temperature stability limit band for the two RIRTs (I=1 mA and =33.6 s). The values of  T, L for NPL2 is similar to but slightly lower than that for NPL1 over the whole range from 4.2 K to 26 K, i.e., higher temperature resolution, mainly because the total potential lead resistance R pl of NPL2 is lower than that of NPL1. To be convenient, polynomial fitting for the temperature stability limit  T, L for NPL1 and NPL2

respectively with respect to temperature were carried out with an appropriate order. In the present work, the polynomial order, 4, was determined by judging the fitted standard deviation. The fitted coefficients are given in Eq. s (37) and ( 38) respectively for the NPL1 and NPL2 with I m =I c =1.0 mA and =33.6 s, the maximum absolute residuals of which are less than 0.005 μK. In this case, the theoretical temperature stability limit of NPL2 is 0.43(1) times of the temperature resolution of NPL1, determined by Monte-Carlo simulation. The standard uncertainty of  T,L , u( T, L ), was estimated using Eq. ( 29). The biggest uncertainty component is the voltage noise coefficient of the AC bridges, and the second one is the lead resistances of the RIRTs. While the relative uncertainties of the temperature stability limit, u R ( T, L )=u( T, L )/ T, L , are nearly constant for the two RIRTs NPL1 and NPL2 with values of 0.1672(2) and 0.1675( 5) respectively. One should know that these temperature stability limit will have different values when the parameters in Eq. ( 22) were changed.

Effect of excitation current

To investigate how the excitation current influences the temperature stability, three excitation currents, namely 0.5 mA, 1.0 mA and 2 mA, were used in the temperature control and measurement with resonator at 24.5 K under vacuum. Since the self-heating of the controlled RIRT will increase with the current, to keep the same temperature, the resistance ratio set point for the AC temperature control should also be increased. The experimental results are plotted in Figs. 6 and7 with =33.6 s and t=21 h. The standard uncertainties of the predicted  T, L , u( T, L ) in K, were estimated using the Eq. ( 29). While that for the experimentally measured standard uncertainties were estimated using Eq. ( 15).

Figs. 6(a) and 7(a) show the temperature stability response of the sensor NPL1, measured with a constant excitation current I m =1.0 mA, for different excitation currents I c of the control thermometer NPL2. We can see that the measured temperature stability of NPL1 for I c =1.0 mA and I c = 2 mA is close to the measuring limit of RIRTs and AC bridges. When I c was increased from 0.5 mA to 2 mA, the temperature resolution of NPL2 was improved by a factor of 2.8, which in turn improves the measured temperature stability of NPL1 by 33% from 10.1 K to 6.8 K. However, when I c was increased from 1.0 mA to 2 mA, the measured temperature stability of NPL1 was only improved by 8% from 7.4 K to 6.8 K, mainly due to the increase of the

I m =1
.0 mA and I m = 2 mA, while NPL2 with excitation current I c = 1.0 mA. When I m was increased from 1.0 mA to 2 mA, the measured temperature stability of NPL1 conversely increases from 7.0 K to 7.8 K, mainly because of its increasing self-heating effect. The two stabilities are consistent with each other within their combined uncertainties. The measuring stability 7.8 K of NPL1 lies outside the limit of NPL2 (5.0±0.83) K with a deviation of about 3.4 times of u( T, L ). It shows that an excitation current of 1.0 mA is enough for NPL1 to detect the temperature control stability. Thereafter, the excitation current of NPL1 was determined to I m =1.0 mA and used throughout the rest of the present work.

The above results show that a larger excitation current of the controlled sensor is helpful to realize higher temperature stability due to the improved temperature resolution. While a larger excitation current of the monitoring sensor RIRT_NPL1 cannot improve the temperature control limit. It also implies that the measured sensor should have a matched resolution with the controlled one to well reflect the temperature control stability, as is the case in the present work.

Effect of Integration time

With the aforementioned excitation currents 1.0 mA, long-term temperature control and measurement of the resonator for 180 h (7.5 days) was carried out under vacuum at 5 K and 24.5 K with =33.6 s. The measured temperatures are plotted in Fig. 8(a) as a function of the measuring time with temperature stabilities of NPL1 of 2.2 K for 5 K and 7.4 K for 24.5 K. Fig. 8(b) shows a plot of the generalized variance [START_REF] Allan | Statistics of atomic frequency standards[END_REF] for resonator temperature as a function of integration time for the measurements at 5 K and 24.5 K. We can see that the experimental temperature stabilities are not exactly on the white noise line, which means there is some small noise in the cryostat. For T=5 K, the measured temperature stability decreases with the increase of the integration time nearly in the whole range. While for T=24.5 K the measured temperature stability begins to be stable until τ g ≈310 4 s (i.e., 9 h), which means it is close to the measuring limit of the instrument. Over the whole integration time range, NPL1 temperature stability for 5 K is at least three times better than that for 24.5 K with the same integration time, mainly because the relatively higher temperature resolution of NPL2 at 5 K (3.6 times better than that at 24.5 K) improves the temperature control stability. Besides, the measured temperature stabilities with different integration times are equivalent. Therefore, when provided one measured temperature stability, the integration times of the measuring sensor should be given, otherwise, it will be meaningless and cannot be comparable.

Uncertainty estimation and comparison

The influence of the measuring time t on the value of measured temperature stability was estimated for =33.6 s based on the above long-term stability measurement results. Fig. 9(a) plots the temperature stability as a function of the measuring time t increasing by a step of t 0 =. The standard deviations of the measured temperature stabilities were calculated using Eq. ( 15) and used as the standard uncertainties of the measured temperature stability, u( T ) in K, with values of 0.44 K and 0.45 K for 5 K and 24.5 K respectively.

In our previous work [START_REF] Chen | Thermal response characteristics of a SPRIGT primary thermometry system[END_REF], temperature stability of about 19 μK was achieved between 5 K and 25 K for These temperature control stabilities have been improved by (70±7)% and (44±8)% for 5 K and 24.5 K respectively as in Fig. 9(b). Because RIRT is more competent in high-stability temperature control at low temperatures, even the Cernox sensor has relative higher resolution at this temperature region. The temperature resolution is the key factor for temperature control, however, the stability of the controlled sensor is another factor in practice. In this aspect, RIRT is ten times stable more than the used Cernox at this concerned temperature region, which might account for the above observed phenomenon.

Application in primay gas thermometry

With the above parameters, i.e., excitation currents of 1.0 mA and =33.6 s. The temperature of NPL1 was measured at temperatures from 5 K to 24.5 K and pressures of helium-4 gas in the resonator up to 120 kPa. The measurements were firstly implemented in vacuum, then repeated under four pressures namely (30, 60, 90, and 120) kPa. The experimental temperature stability results and the theoretical temperature stability limit of NPL1 are compared in Fig. 10. The standard uncertainties of the theoretical  T, L , u( T, L ) in K, were given in Eqs. (37) and (38), while the measured standard uncertainties u( T ) were determined from Eq. ( 15). We can see that:

(1) The measured temperature stabilities under pressures are better than those of measured in vacuum for the same temperature. There are two main reasons. Comparison between experimental temperature stabilities and theoretical temperature stability limit band (k=1) of RIRTs NPL2 and NPL1 with resonator under different pressures. NPL1 and NPL2 are respectively used for measuring and controlling the temperature of the microwave resonator

One is that the enhanced thermal effect (greater conductivity) of the working gas helium-4, filled inside the pressure vessel and the microwave resonator, improves the thermal response characteristic of the controlled and measured RIRT sensors, which improves the temperature control and measured stability. One evidence was that the self-heating of the thermometer NPL1 under pressures was reduced with respect to those in vacuum, as clearly shown in Fig. 17 in our previous work [START_REF] Gao | Measurement of thermodynamic temperature between 5 K and 24.5 K with single-pressure refractive-index gas thermometry[END_REF]. The other reason is that temperature measurements of NPL1 under pressures are carried out after those implemented in vacuum, the temperature cycle effect improves the temperature control stability as shown in Fig. 10. Some points have two temperature stabilities, the lower one was measured after the normal measuring sequence mentioned above, i.e., in vacuum then under pressures (30, 60, 90, and 120) kPa.

(2) The measured temperature stabilities at lower temperatures are better than those of measured in higher temperatures mainly due to the temperature resolution, as observed in section 4.1.1. Besides, lower temperature means greater thermal diffusivity for the resonator, making from electrolytic tough pitch (ETP) copper (ISO norm), which leads to rapid thermal response characteristic and thus improves the temperature control stability.

(3) The experimental temperature stability limit of NPL1,  T , is 0.48(1) times of the temperature resolution of NPL1, determined by Monte-Carlo simulation. These temperature stabilities are close to the limit that the two RIRTs and the two AC bridges can control and measure.

In the future, the temperature control stability can be further improved by using low-noise instruments, highly sensitive and stable temperature sensors, and low-resistance leads.

Conclusion and perspectives

Relationship between the temperature control stability limit and temperature resolution is developed, which indicates the temperature control stability can be further improved by increasing its resolution, i.e., using low-noise instruments, high sensitivity and stable sensors, and low-resistance leads. The present results show that the temperature control stability is mainly determined by the temperature resolution of the controlled sensor and instrument rather than that of the measuring sensor and instrument. However, to well reflect the temperature control stability, the controlled and measuring sensors should have a matched temperature resolution for the same integration time. Besides, temperature control with helium-4 gas filled in the resonator can improve the measured temperature stabilities due to its greater conductivity, and higher temperature resolution helps lower temperatures to realize better temperature control stability. The measured stability lies close to the limit that the RIRT sensors and the AC bridges can control and measure. By using highly stable and positive temperature coefficient RIRT sensors, the temperature stability can be maintained to better than 8 K with an integration time of 33.6 s over 180 h, even longer if needed. These temperature stabilities were improved by (70±7) % and (44±8) % respectively at 5 K and 24.5 K comparing with our previous work [START_REF] Chen | Thermal response characteristics of a SPRIGT primary thermometry system[END_REF].

Fig. 1

 1 Fig. 1 Simplified schematic diagram of the cryostat for SPRIGT

Fig. 2

 2 Fig. 2 Schematic diagram for the theoretical temperature stability limit  T,L

Fig. 3

 3 Fig. 3 Simplified schematic diagram for measuring time t, integration time  and generated integration time  g , with the integer variate h changing from 1 up to half of N

  [START_REF] Rusby | KEY COMPARISON: Final Report on CCT-K1: Realizations of the ITS90, 0.65 K to 24.5561 K, using rhodium iron resistance thermometers[END_REF] For an excitation current I (in milliamps) of the AC resistance bridge, we can simply estimate the equivalent resistance resolution  R (in ) by:

Fig. 4

 4 Fig. 4 Lead resistances for the two RIRTs

Fig. 5

 5 Fig. 5 Theoretical temperature stability limit band of RIRT sensors NPL1 and NPL2 for the temperature region from 4.2 K to 26 K ,L 5 4 4 3 3 2 1.5653 10 +4.02 10 +4.12 10 +0.19916 +0.

Fig. 6 Fig. 7

 67 Fig. 6 Comparing results between theoretical limit and experimental temperature stability of NPL1 under vacuum with =33.6 s and t=21 h at 24.5 K: (a) response to different NPL2 excitation current with constant I m =1.0 mA; (b) response to different NPL1 excitation current with constant I c =1.0 mA

Fig. 8

 8 Fig. 8 180-hour temperature stability measurement of NPL1 at 5 K and 24.5 K under vacuum: (a) resonator temperature evolution as a function of measuring time with =33.6 s; (b) measured temperature stability of NPL1 versus its integration time

Fig. 9

 9 Fig. 9 Measured temperature stability with =33.6 s: (a) as a function of the measuring time t; (b) temperature stability comparison between our previous work [23] and this work by respectively controlling Cernox and RIRTs for t=24 h =0.8 s and t=24 h by controlling a Cernox sensor (CX-1050-CU-HT-1.4L, SN X118726). The stabilities are equivalent to (10.7±1.9) K for 5 K and (11.7±1.5) K for 25 K with =33.6 s and t=24 h, recalculated via methods in sections 3.2. From Fig. 9(a), resonator temperature stabilities of about (3.2±0.45) K and (6.5±0.45) K was respectively realized for 5 K and 24.5 K with =33.6 s in the first 24-hour measurement.

Fig. 10

 10 Fig. 10Comparison between experimental temperature stabilities and theoretical temperature stability limit band (k=1) of RIRTs NPL2 and NPL1 with resonator under different pressures. NPL1 and NPL2 are respectively used for measuring and controlling the temperature of the microwave resonator

Table 1

 1 Information for temperature control and measurement on the resonator

	Temperature control		Temperature measurement
	Instrument	RIRT	Instrument	RIRT
	F18 AC bridge and 8½ Keithley 2002	NPL2	F900 AC bridge	NPL1

Table 2

 2 Total potential lead resistance R t,pl and standard uncertainty u(R t,pl ) of the two RIRTs

	T 10	a	NPL1		NPL2
	T/K R 4.66 u(T)/K 0.005 21.2	0.81	15.35	0.17
	82.3	0.2	22.55	0.66	16.9	0.14
	Linear fitting		R t,pl = aT + b
	a/Ω•K -1	0.017±0.008	0.020±0.002
	b/Ω	21.1±0.5	15.3±0.1

t,pl /Ω u(R t,pl )/Ω R t,pl /Ω u(R t,pl )/Ω

Table 3

 3 Information for Tinsley 5685A standard resistors and their potential lead resistances

	Temperature control	Temperature measurement
	10 Ω (SN 1631808)	10 Ω (SN 1580409)
	R s /Ω	u(R s )/Ω	R s /Ω	u(R s )/Ω
	9.9999682	0.000005	9.9999818	0.000005
	R s,pl /Ω	u(R s,pl )/Ω	R s,pl /Ω	u(R s,pl )/Ω
	0.9	0.01	0.9	0.01

Acknowledgment

This work was supported by the National Key Research and Development Program of China (Grant No. 2016YFE0204200), the National Natural Science Foundation of China (Grant No. 51627809 and 52006231), the International Partnership Program of the Chinese Academy of Sciences (Grant No. 1A1111KYSB20160017), the Scientific Instrument Developing Project of the Chinese Academy of Sciences (Grant No. ZDKYYQ20210001), the European Metrology Research Program (EMRP) Joint Research Project 18SIB02 "Real K", the CAS Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry (Grant No. CRYOQN202110). This work was also supported by the project "15SIB02 InK 2" which has received funding from the EMPIR programme co-financed by the Participating States and from the European Union's Horizon 2020 research and innovation programme. Changzhao Pan is grateful for the funding provided by a Horizon 2020 Marie Skłodowska Curie Individual Fellowship 2018 (No. 834024).

The authors are deeply grateful to our colleague Ying Ma for expert technical assistance. We gratefully acknowledge Richard Rusby from NPL for calibrating three rhodium-iron resistance thermometers, sharing his long experience and giving constant advice on temperature measurements.


Haiyang Zhang

Bo Gao
email: bgao@mail.ipc.ac.cn.

Wenjing Liu

Changzhao Pan

Dongxu Han

Mark Plimmer

Ercang Luo

Laurent Pitre

Resonance frequency measurement with accuracy and stability at the 10 -12 level in a copper microwave cavity below 26 K by experimental optimization

Keywords: Stability, Microwave resonance frequency, Resonator, Allan deviation, Primary thermometry, Thermodynamic temperature

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Single-pressure refractive index gas thermometry (SPRIGT), developed jointly by Technical Institute of Physics and Chemistry of Chinese Academy of Sciences (TIPC-CAS) in China and Laboratoire national de métrologie et d'essais-Conservatoire national des arts et métiers (LNE-Cnam) in France [START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF], is a type of thermometry methods based on gas polarizability. As a relative primary thermometry technique [START_REF] Rourke | Refractive-index gas thermometry[END_REF],

SPRIGT measurements are conducted on single isobars rather than isotherms, which alleviates the need for accurate absolute pressure measurement and increases the measurement speed ten-fold. With state-of-the-art ab initio calculations of helium-4 properties [START_REF] Bruch | Diamagnetism of helium[END_REF][START_REF] Rizzo | The effect of intermolecular interactions on the electric properties of helium and argon. III. Quantum statistical calculations of the dielectric second virial coefficients[END_REF][START_REF] Garberoglio | Improved first-principles calculation of the third virial coefficient of helium[END_REF][START_REF] Cencek | Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potential and thermophysical properties of helium[END_REF][START_REF] Shaul | Path-integral Mayer-sampling calculations of the quantum Boltzmann contribution to virial coefficients of helium-4[END_REF][START_REF] Puchalski | Theoretical determination of the polarizability dispersion and the refractive index of helium[END_REF], a competitive uncertainty of 0.25 mK 1 is expected for the measurement of thermodynamic temperature below the neon triple point at 24.5561 K, which is of important strategic significance for the development of both cutting-edge scientific researches and large scientific facilities, such as the Spallation Neutron Source in China and Large Hadron Collider in Europe [START_REF] Balle | Calibration of Cryogenic Thermometers for the LHC[END_REF].

Microwave measurements are widely used in the field of gas metrology [START_REF] Pitre | Acoustic thermometry: new results from 273 K to 77 K and progress towards 4[END_REF][START_REF] Pitre | Measurement of the Boltzmann constant k B using a quasi-spherical acoustic resonator[END_REF][START_REF] Pitre | Determination of the Boltzmann constant using a quasi-spherical acoustic resonator Phil[END_REF][START_REF] Moldover | Acoustic gas thermometry[END_REF][START_REF] Moldover | Correlations among acoustic measurements of the Boltzmann constant[END_REF][START_REF] Pitre | New measurement of the Boltzmann constant k by acoustic thermometry of helium-4 gas[END_REF][START_REF] Gavioso | Determination of the thermodynamic temperature between 236 K and 430 K from speed of sound measurements in helium[END_REF].

SPRIGT allows the thermodynamic temperature to be determined from the ratio of gas refractive indices at a single pressure, one measured at the unknown temperature and the other at the reference temperature [START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF][START_REF] Zhang | A high-stability quasi-spherical resonator in SPRIGT for microwave frequency measurements at low temperatures[END_REF], while the refractive index can be measured from the microwave resonance frequencies in a quasi-spherical highconductivity resonator. The reference temperature can be the fixed point of neon or a known thermodynamic temperature measured using another absolute primary thermometry method, such as acoustic gas thermometry (AGT). In refractive index gas thermometry (RIGT) [START_REF] Rourke | Refractive-index gas thermometry[END_REF][START_REF] Rourke | Progress Toward Development of Low-Temperature Microwave Refractive Index Gas Thermometry at[END_REF], microwave measurements are used to determine the gas refractive index n(p,T). From a knowledge of n(p,T) and the pressure p, the thermodynamic temperature T is deduced. RIGT has been successfully implemented at 1 On May 20th, 2019, the Bureau International des Poids et Mesures announced a major revision to the SI, in which the base units, the kelvin, symbol K, was redefined by fixing the value of Boltzmann constant. The practical realizations of the kelvin by primary thermometry are indicated in the "Mise en pratique for the definition of the kelvin in the SI", in which low-uncertainty primary thermometry is required to promote the realizations of the new kelvin and the spread of high-accuracy, low-temperature metrology. (see https://www.bipm.org/utils/en/pdf/simep/SI-App2-kelvin.pdf).

NRC by Rourke to measure the temperatures 24.5561 K (neon triple point), 54.3584 K (oxygen triple point) and 83.8058 K (argon triple point) which are three ITS-90 defining fixed points [START_REF] Rourke | NRC Microwave Refractive Index Gas Thermometry Implementation Between[END_REF]. Recent progress has also been achieved in the same laboratory on the measurement of the thermodynamic temperature of the triple point of xenon T = 161.40596 K by extrapolating the experimentally-determined compressibility at the triple point of water [START_REF] Rourke | Thermodynamic temperature of the triple point of xenon measured by refractive index gas thermometry[END_REF]. In a complementary application of RIGT, where the pressure is deduced from the refractive index and temperature measurements, a quantum standard for absolute pressure measurements is currently being developed jointly by LNE in France and INRiM in Italy in the range from 0.2 kPa to 20 kPa by a superconducting microwave cavity [START_REF] Gambette | Towards a Quantum Standard for Absolute Pressure Measurements in the Range 200 Pa to 20 kPa Based on a Superconducting Microwave Cavity[END_REF]. In acoustic gas thermometry [START_REF] Pitre | Acoustic thermometry: new results from 273 K to 77 K and progress towards 4[END_REF][START_REF] Moldover | Acoustic gas thermometry[END_REF][START_REF] Moldover | Correlations among acoustic measurements of the Boltzmann constant[END_REF][START_REF] Pitre | New measurement of the Boltzmann constant k by acoustic thermometry of helium-4 gas[END_REF][START_REF] Gavioso | Determination of the thermodynamic temperature between 236 K and 430 K from speed of sound measurements in helium[END_REF], microwave measurements are used to determine the dimensions or volume of the resonator so that the thermodynamic temperature can be deduced from acoustic measurements. Besides, microwave measurements have also been used in other precise measurements, such as density and critical phenomena of helium [START_REF] Yeh | Precise measurements of the density and critical phenomena near the phase transitions in helium using high-Q niobium microwave cavities Czech[END_REF], frequency [START_REF] Yeh | Superconducting-cavitystabilized oscillators (SCSO) for precise frequency measurements[END_REF] and quantum-gas and gravitational physics [START_REF] Corcovilos | Mugti-frequency Superconducting Cavity Stabilized Oscillators (SCSO) for Quantum-Gas Measurements and Gravitational[END_REF]. These were successfully implemented by using high-quality factor niobium microwave cavities. The aforementioned gas metrology methods (SPRIGT, RIGT, AGT and absolute pressure in a quantum standard) and precise measurements would benefit from any improvement in microwave measurements.

The best current primary standards can produce the SI second with a relative standard uncertainty approaching 10 -16 , while the relative uncertainty of secondary frequency standards based on optical clocks is at the level of 10 -18 [25]. A convenient way to perform experiments is to use a servo loop to lock the oscillator frequency to the maximum of a resonance. The frequency stability then depends on the shape and sharpness of the resonance, the signal-to-noise ratio, and the type of noise that predominates at the time scale under consideration [START_REF] Allan | Statistics of atomic frequency standards[END_REF][START_REF] Audoin | The measurement of time: time, frequency and the atomic clock[END_REF][START_REF] Drever | Laser phase and frequency stabilization using an optical resonator[END_REF]. One of the most commonly used techniques for servo-locking is the sideband method due to Pound and its optical equivalent [START_REF] Drever | Laser phase and frequency stabilization using an optical resonator[END_REF]. By contrast, in SPRIGT, it is essential to perform scans over the resonance lines to check for any change in the width and shape (see Section 3.1). These parameters are used both to measure electrical conductivity and monitor systematic effects. A scan takes typically two minutes. To implement SPRIGT, a system has been built at TIPC-CAS in China, which includes three subsystems, namely a temperature control system, a pressure control system and a microwave system. The temperature and pressure control systems are designed to provide a very stable working environment for the microwave resonator and connecting cables. High-stability temperature control (0.2 mK), pressure control (4 ppm, 1 ppm ≡ 10 -6 ) and microwave resonance frequency (2 ppb, 1 ppb ≡ 10 -9 ) are required for thermodynamic temperature measurements with an uncertainty of 0.25 mK. A simple description of the three subsystems is presented in Section 2.

In this paper, the structure of the quasi-spherical resonator is described and the experimental optimization of microwave measurement is presented. The rest of the paper is structured as follows. In Section 2, we describe the experimental apparatus involving the main instrumentations for microwave, temperature and pressure measurements. Detailed descriptions of the temperature [START_REF] Gao | Realization of an ultra-high precision temperature control in a cryogen-free cryostat[END_REF][START_REF] Chen | Thermal response characteristics of a SPRIGT primary thermometry system[END_REF] and pressure measurement [START_REF] Han | Ultra-stable pressure is realized for Chinese single pressure refractive index gas thermometry in the range 30-90 kPa[END_REF] can be found elsewhere and lie beyond the scope of the present paper.

In Section 3, we present how to realize high-stability and low-uncertainty of microwave resonance frequency measurement by optimizing the microwave background, the microwave emission power, mechanical stability, temperature control method, time reference, and microwave signal.

Experimental set-up

The apparatus of SPRIGT is shown in Fig. 1a, including microwave, temperature control and pressure control subsystems. For the microwave system, the core element is a copper (Cu-EPT) microwave cavity, i.e., a quasi-spherical resonator (QSR), shown in Fig. 1b. The high-quality factor 2 QSR was built from two hemispheres whose inner surfaces were machined by precision diamond turning. The dimensions are the same as those of the resonator used at LNE-Cnam for a determination of the Boltzmann constant [START_REF] Pitre | Measurement of the Boltzmann constant k B using a quasi-spherical acoustic resonator[END_REF]. The inner shape is designed to be a triaxial ellipsoid defined by 2 Quality factors (Q = f/2g) for the 4 microwave modes at 5 K ~26 K are: Q(TM11)≈140 000, Q(TE11)≈210 000, Q(TM12)≈170 000, Q(TE13)≈290 000. The QSR has been closed successfully at room temperature using a microwave method by monitoring the change of relative excess half-width [START_REF] Liu | Microwave Method for Closure of Quasi-spherical Resonator Conference on Precision Electromagnetic Measurements[END_REF]. The resonator was designed to be used under vacuum and then with helium gas. For this reason, to avoid contamination, Argon (purity of 99.999 %, supplied by Beijing AP BAIF Gases Industry Co. Ltd.) was flowed through it at a continuous rate of 80 SCCM 3 until the resonator was coupled in the pressure vessel of the cryostat. After putting the resonator into the pressure vessel of the cryostat, a preliminary test on the performance of the quasi-spherical resonator under vacuum was implemented without using any optimization methods involved in the present work. Frequency stability and uncertainty of 0.02 ppb and 0.03 ppb were realized by using 0 dBm microwave emission power with an integration time of 3 h [START_REF] Zhang | A high-stability quasi-spherical resonator in SPRIGT for microwave frequency measurements at low temperatures[END_REF], equivalently 0.063 ppb and 0.095 ppb for -10 dBm microwave emission power [START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF].

For the temperature control system, a cryogen-free cryostat was developed for SPRIGT in our previous work [START_REF] Gao | Realization of an ultra-high precision temperature control in a cryogen-free cryostat[END_REF], using a two-stage GM type pulse tube cryocooler as the cooling resource. Based on multi-layer radiation shields combined with the thermal-resistance method, gas type heat switch and PID control method, the pressure vessel temperature stabilities of 0.021 mK ~ 0.050 mK were realized with an integration time of 0.8 s in the temperature range from 5 K to 25 K. Based on further investigation on thermal characterization of the cryostat, the pressure vessel temperature stabilities were controlled at 0.019 mK in the same range [START_REF] Chen | Thermal response characteristics of a SPRIGT primary thermometry system[END_REF]. In this work, the temperature of the resonator was measured by a rhodium iron thermometer (Tinsley, SN226242) coupled to an AC resistance bridge (ASL F900). The 10 Ω standard reference resistance (Tinsley 5685A, SN1580409) was placed in an oil bath (Aikom Instruments MR 5100-L) with temperature stability better than 1 mK.

For the pressure control system, the detailed content is not emphasized in this work

since it has been presented in our previous work [START_REF] Han | Ultra-stable pressure is realized for Chinese single pressure refractive index gas thermometry in the range 30-90 kPa[END_REF]. The uncertainty contribution of pressure to measurements of thermodynamic temperature has also been reported in our

3 SCCM = standard cubic centimeters per minute, corresponding to 1.6667×10 -8 m 3 •s -1 in the International System of Units (SI). We define the volumetric flow 1 sccm as the flow of 1 cubic centimeter per minute of argon at the pressure 101.325 kPa and temperature 20℃.

previous work [START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF] and Fig. 2 in the review paper of RIGT [START_REF] Rourke | Refractive-index gas thermometry[END_REF]. To fill helium-4 gas (purity of 99.9999 %, supplied by Air Liquide) into the resonator and keep a single pressure, a gas compensation loop was built at room temperature to compensate for the leak of the piston gauge. The gas pressure was measured by an absolute-pressure piston gauge (Fluke PG 7601) and conducted by maintaining the piston at a constant height using a laser interferometer (KEYENCE LK-G80). Pressure stabilities of 0.0032 Pa at 30 kPa and 0.002 Pa at 90 kPa were realized with an integration time of 1.4 s at room temperature, corresponding to relative stabilities of 0.1 ppm and 0.02 ppm, respectively [START_REF] Han | Ultra-stable pressure is realized for Chinese single pressure refractive index gas thermometry in the range 30-90 kPa[END_REF].

Experimental optimization and discussions

To realize high-stability and low-uncertainty microwave resonance frequency measurements at low temperatures, several experimental optimizations were implemented by using Allan analysis of variance in the present work. Detailed results are listed as below.

Microwave background polynomial

In this work, the microwave resonance frequencies fn and half-widths gn were determined from non-linear least-squares fitting (LM method [START_REF] Moré | The Levenberg-Marquardt algorithm: implementation and theory[END_REF]) of the measured complex scattering parameters S21 as a function of frequency [START_REF] May | Quasi-spherical cavity resonators for metrology based on the relative dielectric permittivity of gases[END_REF]. Here we rewrite equation 15 in the literature to the following form

21 * 22 0 () () J j n j nj nn f S f f f f ig       A B ( 2
)
where the fitting parameters were the complex constants An, Bj and the three complex resonance frequencies (fn + ign), one for each component of the triplet. In equation 2, J is the background polynomial order, f is the source frequency and f* is an arbitrary constant, here we made the same choice as the literature [START_REF] May | Quasi-spherical cavity resonators for metrology based on the relative dielectric permittivity of gases[END_REF], i.e., f*= fx, to avoid numerical problems in the fitting program.

Fig. 2 shows the fitting results of TM11 microwave mode for different background polynomial orders at 5 K and 24.5 K under vacuum. With the polynomial order increasing, the average fitted triplet frequency f + g and the uncertainty u(f + g)

have a little change, as shown in Fig. 2a and Fig. 2c; the maximum relative changes on f + g are about 2.0×10 -10 and 1.9×10 -10 for 5 K and 24.5 K, respectively. These changes are less than half of the relative standard uncertainty of f + g, indicating that increasing the microwave background polynomial order cannot obviously reduce their influence on the average fitted triplet frequency and the uncertainty in our system.

Considering other microwave modes and those measured at room temperatures, a 2 ndorder background polynomial is competent in this work. Table 1 lists the fitted frequencies and half-widths of the triplet for 2 nd -order background polynomial. Fig. 2b and Fig. 2d compare the results of the measured and calculated components of the scattering parameters S21, where good agreement was found. Besides, the background noise from cables and feed-throughs also influences the microwave measurement. To reduce the noise as much as possible, one can use some low-loss cables and feed-throughs, covering the working frequency range. For further improvement, the optimal combination of cables and feed-throughs, even the working microwave modes can be selected by measuring and analyzing the S21 parameters.

Microwave emission power

In this work, the microwave signal was generated by a two-port N5241A vector network analyzer. The signal was transmitted from port 1 of the vector network analyzer, passing through the cables, feed-throughs, and resonator, later received by port 2. Since the relative loss of the cables and feed-throughs are nearly constant at stable operating conditions, the microwave emission power has a direct influence on the measurement inside the resonator. If the power is too small, the microwave signal may be drowned in the background noise and thus leads to a relatively larger uncertainty of the fitted frequency or even no triplet. On the contrary, the microwave signal will heat the resonator too much, this will increase the temperature control instability and in turn reduce the frequency stability. Thus, it is necessary to optimize the microwave emission power.

Fig. 3 shows the optimized results of the power at 5.0 K under vacuum based on the TM11, TE11, TM12 and TE13 microwave modes, which were employed in our preliminary measurements [START_REF] Zhang | A high-stability quasi-spherical resonator in SPRIGT for microwave frequency measurements at low temperatures[END_REF]. A bigger power is easier to increase the temperature instability, as shown in Fig. 3a and Fig. 3b, where the temperature stabilities are about 0.042 mK and 0.024 mK for microwave emission power PMW = -7 dBm and PMW = -10 dBm, respectively. Fig. 3c plots the temperature Allan standard deviation against the integration time for these two powers. It can be seen that there is an oscillation of the temperature Allan standard deviation for PMW = -7 dBm, which is coupling with the measurements of the 4 microwave modes, measured in the TM11-TE11-TM12-TE13 loop. While for PMW = -10 dBm, no oscillation was observed on the temperature Allan standard deviation, which means the microwave power has negligible disturbance effect on temperature stability. Thus the optimal microwave emission power was found to be Popt = -10 dBm. In the present work, we use one single microwave emission power Popt during all the low-temperature microwave frequency measurements no matter whether the resonator is under vacuum or filled with high-purity helium-4 gas.

Mechanical stability, temperature control method, time reference, and signal intensity

In this section, mechanical stability, temperature control method, time reference and signal intensity were also studied in this work under vacuum unless otherwise specified.

Mechanical stability

When cooling the resonator from room temperature down to low temperatures (5 K to 25 K) for each independent run in this work, it is important to keep the resonator in good mechanical stability before doing the microwave measurements. A simple way to release the mechanical stress is to do temperature cycling in the objective temperature range, for example, cooling and warming between 25 K and 5 K in this work. Because releasing mechanical stress needs some time. Besides, it is also good for temperature and frequency measurements. At the same set point for temperature control, after the temperature cycling, it is easier to realize highly stable temperature measurements, which has been observed in our system. The temperature stability, in turn, will improve the frequency measurement as shown in Section 3.3.2 below. When the temperature of the resonator reached the objective temperatures (5 K and 24.5 K), temperature regulations together with microwave measurements were implemented for several hours. If the frequencies of each microwave mode had no obvious change anymore (changing within microwave measurement uncertainty) at the objective temperatures, then it can be considered that the mechanical stress of the resonator is completely released.

A cooling-warming temperature cycling for 55 hours was carried out in this work.

The result is shown in Fig. 4a, where the cooling process and warming process are

Temperature control method

Due to thermal expansion effect of the resonator, its dimensions will change if the temperature is not stable. The microwave resonance frequency is proportional to the reciprocal of the radius of the resonator, thus the temperature stability has a direct influence on microwave measurements. Fig. 4b shows the influence on microwave resonance frequency stability of TM11 mode for two temperature control methods at T = 15.0 K and p = 30 kPa. Method 1 uses an 8½ digit multimeter (Keithley 2002) with Cernox sensor as in our previous work [START_REF] Chen | Thermal response characteristics of a SPRIGT primary thermometry system[END_REF], while method 2 uses an AC resistance bridge (ASL F18) with a calibrated rhodium-iron sensor as in CCT-K1 [START_REF] Rusby | Key comparision: Final Report on CCT-K1: Realizations of the ITS90, 0.65 K to 24.5561 K, using rhodium iron resistance thermometers[END_REF]. The temperature stabilities of the two methods are 38 K and 8 K with an integration time of 33.6 s, respectively. The frequency stability of method 2 is better than that of method 1 by about an order of magnitude with an integration time τ = 3 h. Good temperature stability can improve the microwave resonance frequency stability. Thus, in our later measurements, method 2 was adopted all the time to maintain a good frequency stability.

Time reference

Usually, the vector network analyzer is connected to an external time reference, such as rubidium atomic clock or GPS clock, to realize high-accuracy and high-stability frequency measurements. In this work, rubidium atomic clock (FS725) and GPS clock (FS740 locked to GPS) were used. While the long-term stability of GPS is excellent, its short-term stability is rather poor in comparison to modern oscillators [36]. So in this work, two different oscillators, an OCXO and a rubidium atomic frequency standard, were configured for FS740.

Fig. 4c shows the typically measured stability of microwave resonance frequency of TM11 mode at 24.5 K in two days with the three kinds of time references, namely, FS725 (rubidium atomic clock), FS740 locked to GPS with rubidium atomic clock and with OCXO clock, with GPS configured indoor antenna. The frequency stability of FS740 locked to GPS with rubidium atomic clock configured indoor antenna and that of the FS725 are comparatively equivalent, both stabilities are better than that of OCXO clock. A resonance frequency stability of 0.02 ppb was realized using FS740 locked to GPS with rubidium atomic clock configured indoor antenna and FS725 with an integration time τ = 3 h in the two-days microwave measurements. The SPRIGT measurement at different temperatures and pressures may take one or two months, the time reference stability of the reference should be tested at this level to check if any other effect may perturb the measurement.

Furthermore, long-term microwave measurements at 5 K in seven days were carried out by FS740 locked to GPS with rubidium atomic clock configured indoor antenna. The stability result is plotted in Fig. 4d. A good resonance frequency stability of 0.04 ppb was achieved with τ = 3 h.

Signal intensity

Amplifier is usually used to enhance the microwave signal intensity as shown in literature [START_REF] Yang | Investigating the adequacy of a lowcost vector network analyser for microwave measurements in quasispherical resonators[END_REF], where the amplitude of |S21| was amplified by nearly ten times with the help of a set of two 10 dBm amplifiers (Model ZX60-14012L manufactured by Mini-Circuits, bandwidth from 0.3 MHz to 14 GHz) connected in series at room temperature.

The uncertainty was reduced by a factor of 5 -12 even with a low-cost vector network analyzer.

In this work, with the same method and two amplifiers like the literature [START_REF] Yang | Investigating the adequacy of a lowcost vector network analyser for microwave measurements in quasispherical resonators[END_REF],

measurements were carried out at 5 K under vacuum. Fig. 4e shows the comparison results of the real and imaginary components of |S21| with and without the two amplifiers.

With amplifiers, the amplitudes were amplified by about 10 times and in turn, the fitted relative standard uncertainty of resonance frequency was reduced by a factor of 10 (from 0.3 ppb to 0.031 ppb). This is because the amplifiers increase the resolution of the peaks of the triple. A similar situation appears in TE11, TM12, and TE13 microwave modes, with the fitted relative standard uncertainty of resonance frequency reduced by a factor of 5, 5, and 10, respectively. It is a simple and effective way to solve signal problems, such as high-loss of microwave cable and feed-through, even low-cost vector network analyzer. Besides, low-temperature microwave amplifiers are expected to have a better performance than room-temperature amplifiers, and this will be one solution of ultra-high accuracy microwave measurements in the future work.

Overall performance

In Section 3.3.3, the microwave measurements were carried out by using the indoor antenna, which was placed outside a window of the laboratory building, facing north against another building. Actually, the FS740 was supplied with two types of antenna, indoor antenna and outdoor antenna. According to the user's manual of the FS740 time reference, best results can be realized if the antenna has a clear unobstructed view of the sky. It is highly recommended to put the outdoor antenna on the roof of the building, within which the FS740 is located. Because it improves the quality and reliability of the GPS signals as more satellites will typically be visible and with a higher signal-tonoise ratio. This can improve the long-term stability of the FS740 time reference by a factor of three. This configuration was used for the following test described in this Section.

Combining the above optimized results, namely, second-order background polynomial, optimized power Popt = -10 dBm, good mechanical stability, temperature control using an AC resistance bridge and rhodium-iron resistance sensor, the time reference FS740 locked to GPS with rubidium atomic clock but configured outdoor antenna and two room-temperature amplifiers connected in series, preliminary stability measurements on TM11 mode were implemented at 5 K under vacuum. times better than those in our previous work [START_REF] Zhang | A high-stability quasi-spherical resonator in SPRIGT for microwave frequency measurements at low temperatures[END_REF], 0.064 ppb and 0.095 ppb with the same integration time and Popt = -10 dBm [START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF].

In SPRIGT, the thermodynamic temperature uncertainty has many influence factors, frequency is one of the major factors as shown in our previous work [START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF] and the RIGT review paper [START_REF] Rourke | Refractive-index gas thermometry[END_REF]. The improvement in frequency measurement accuracy can reduce its influence on thermodynamic temperature measurement by a factor of 20, which is good for realizing the high-accuracy measurement of thermodynamic temperature by SPRIGT. The above optimization methods have the potential to be used in gas metrology and other research fields, where high-stability and low-uncertainty microwave measurements are necessary.

Results with pressures

After many experimental optimizations under vacuum, preliminary microwave measurements were performed from 5 K to 26 K with the resonator filled by high-purity helium-4 at the following conditions: 1) fitting S21 with 2 nd order background polynomial; 2) setting microwave emission power P = Popt = -10 dBm; 3) doing temperature cycle to make the resonator with a good mechanical stability; 4) controlling temperature by AC resistance bridge and rhodium iron sensor; 5) using time reference FS725; and 6) without the two amplifiers. The measurements were implemented on isobars and the results are described in the Fig. 5.

Fig. 5a shows the total relative standard uncertainty of <f+g> of TM11 mode at temperatures from 5 K to 26 K and pressures up to 120 kPa with τ = 80 s 4 . Most uncertainties are located within 0.9 ppb, except the p-T region with pressures near 120 kPa and temperatures near 5 K. The main reason is that at lower temperatures, the pressure stability inside the resonator becomes a little worse than that at higher temperatures, which disturbed the frequency stability by the change of helium-4 density inside the resonator. Fig. 5b plots a one-day microwave measurement of TM11 mode at 24.5 K with different pressures. The frequency stabilities are about (0.03, 0.02, 0.04, 0.05) ppb for (30, 60, 90, 120) kPa with τ = 3 h, respectively, and the temperature stabilities are about (6.9, 8.3, 7.9, 6.6) K with τ = 33.6 s. Due to the pressure stability, 60 kPa has the best frequency stability in the one-day stability measurement. These are still better than our previous work (0.063 ppb under vacuum provided with Popt = -10 dBm) even the pressure instability increases the microwave frequency measurement 4 It is usually used a constant integration time for all the microwave modes. Figure 5(a) only shows the uncertainty of TM11 microwave mode for different pressures and temperatures. However, for TE13 mode, the uncertainties are very near 2 ppb. Besides, the integration time was also chosen to be in the same "time" as the other instruments, such as the resistance bridge. The integration time of 20 000 s is used to check and characterize the stability of the measurements.

instability. The stability and uncertainty still have room for improvement, provided later using the time reference, FS740 locked to GPS with rubidium atomic clock configured outdoor antenna, and microwave amplifiers.

Thermodynamic temperatures calculated from different modes were determined for the isobar p = 60 kPa, based on the SPRIGT principle [START_REF] Gao | Feasibility of primary thermometry using refractive index measurements at a single pressure[END_REF]. Fig. 5c and Fig. 5d presents the differences between mode thermodynamic temperatures Tmode and the average values Tavg, (Tmode -Tavg), at temperatures from 5 K to 26 K, where microwave measurements with τ = 80 s were used. Good mode consistency was observed on (Tmode -Tavg) within ±80 K, and a small relative standard uncertainty component for the thermodynamic temperatures, from mode consistency, was realized with u(Tmode)

values less than 60 K 5 .

Conclusion

The object of this study was the optimization of microwave frequency measurements at low temperatures. The measurements in question concern the resonance frequencies of a high quality factor, tri-axial ellipsoidal copper resonator either under vacuum or else filled with high-purity helium-4 gas (at pressures from 30 kPa to 120 kPa). The ratio of frequencies with and without gas yields the refractive index from which temperature can be determined with high accuracy once the effects of pressure are taken into account. This is called refractive index gas thermometry (RIGT). The application here concerns a novel variant of the technique exploiting single isobars such that pressure-dependent effects almost cancelsingle-pressure RIGT or

SPRIGT. Microwave resonances are detected using pairs of antennas linked to a vector network analyzer.

The study dealt with both experimental parameters and data analysis. Frequency was characterized using the two-sample (Allan) variance widely used in time and frequency metrology. To minimize self-heating yet with an acceptable signal-to-noise ratio, a microwave power of -10 dBm was found to be optimal. A slight improvement of the quality of the fitted lineshape compared with our previous work [START_REF] Zhang | A high-stability quasi-spherical resonator in SPRIGT for microwave frequency measurements at low temperatures[END_REF] was obtained using a quadratic function for the background rather than a linear one. The use of temperature cycling between 5 K and 25 K turned out to improve the reproducibility of frequency measurements. Fine control of the resonator temperature is achieved using a calibrated rhodium-iron resistance thermometer read using an AC resistance bridge instead of an 8½ digit multimeter. The local frequency reference was a GPS referenced commercial rubidium clock. It was essential to use a roof-based outdoor antenna for the GPS system. Frequency resolution was also improved by the introduction of two microwave pre-amplifiers in series, albeit close to the network analyzer located at room temperature. Increased performance is expected with the installation of cryogenically cooled pre-amplifiers located closer to the microwave resonator.

In this way, for an integration time of three hours, microwave resonance frequency measurements with a stability at the 10 -12 level were demonstrated, which represents an almost 20 fold enhancement compared with those of our previous work [START_REF] Zhang | A high-stability quasi-spherical resonator in SPRIGT for microwave frequency measurements at low temperatures[END_REF]. The present observations could help in other high-stability, low-uncertainty frequency measurements in copper microwave cavities, not only for improving thermodynamic temperature measurement in SPRIGT, but also for other precise measurements, where high-stability and low-uncertainty microwave measurements are necessary.

The final uncertainty achieved using SPRIGT in our laboratory is equal to or better than 0.17 mK in the range 5 K to 25 K. An extensive description of the uncertainty budget will be presented in a future publication. As far as frequency measurements are concerned (i.e., the subject of the present manuscript), had they not been possible at the 10 -12 level, the subtle influence of microwave cables and feed-thoughs would perhaps not have been detected as early as it was. The effect of thermal cycling is not one of uncertainty but rather of potential mechanical instability.

Fig. 1 .

 1 Fig.1. The structure schematic diagram of SPRIGT. (a) Simplified system diagram. (b) 3D drawing of the quasi-spherical resonator.

Fig. 2 .

 2 Fig.2. Fitting results of TM11 microwave mode for different background polynomial orders under vacuum: (a) f + g against microwave background polynomial order J at 5 K. (b) Real and imaginary components of the parameter S21 at 5 K, measured and fitted using 2 nd -order background polynomial. (c) f + g against microwave background polynomial order J at 24.5 K. (d) Real and imaginary components of the parameter S21 at 24.5 K, measured and fitted values using 2 nd -order background polynomial.

Fig. 3 .

 3 Fig.3. Microwave emission power optimization at 5.0 K under vacuum. (a) Temperature evolution with PMW = -7 dBm. (b) Temperature evolution with PMW = -10 dBm. (c) Temperature Allan standard deviation σT as a function of integration time for PMW = -7 dBm and PMW = -10 dBm.

  Fig.4. Optimized results of mechanical stability, temperature control, time reference and signal on TM11 microwave mode under vacuum unless otherwise specified. (a) Evolution of resonator temperature and average amplitude of |Ap| in equation 2 during temperature cycle. (b) Frequency Allan standard deviation with two temperature control methods at T = 15.0 K and p = 30 kPa. (c) Two-days frequency stability at 24.5 K with different time references. (d) Seven-days frequency stability at 5.0 K for FS740 time reference locked to GPS with Rubidium atomic clock configured indoor antenna. (e) Real and imaginary components of |S21| of the triplet at 5.0 K with and without the two microwave amplifiers. (f) Frequency Allan standard deviation at 5.0 K using FS740 locked to GPS with Rubidium atomic clock configured outdoor antenna and amplifiers.

  Fig. 4f plots the Allan standard deviation of the resonance frequency. With the same time reference, FS740 locked to GPS with rubidium atomic clock, the stability for outdoor antenna with amplifiers is about an order of magnitude better than that for only indoor antenna as shown in Fig. 4d and Fig. 4f. High-stability of 3.7 parts per trillion (ppt, 1 ppt ≡ 10 -12 ≡ 10 -3 ppb) and low-uncertainty of 5.2 ppt were realized with an integration time of 3 h. The frequency stability and uncertainty in the present work are nearly 20

Fig. 5 .

 5 Fig.5. Microwave measurements from 5 K to 26 K at pressures up to 120 kPa. (a) Total relative standard uncertainty of f + g for TM11 mode with τ = 80 s. (b) One-day frequency stability measurements at 24.5 K for (30, 60, 90, 120) kPa. (c) Thermodynamic temperature differences

Table 1 .

 1 Fitted values of the triplet with 2 nd -order background polynomial.

	T / K	f1 / MHz	g1 / MHz	f2 / MHz	g2 / MHz	f3 / MHz	g3 / MHz
	5	2617.0688336(11) 0.0093815(11) 2617.6300911(15) 0.0093725(15) 2618.2397203(16) 0.0098880(16)
	24.5	2617.0614026(12) 0.0094027(12) 2617.6226012(16) 0.0093916(16) 2618.2321434(17) 0.0099124(17)

Acknowledgments

This work is supported financially by the National Key R&D Program of China (Grant No. 2016YFE0204200), the National Natural Science Foundation of China (Grant No. 51627809), the International Partnership Program of the Chinese Academy of Sciences (Grant No. 1A1111KYSB20160017), and the EMRP project Real-K (No. 18SIB02).

Author Changzhao Pan was supported by the funding provided by the Marie Skłodowska-Curie Individual Fellowships-2018 (834024). The authors gratefully acknowledge Richard Rusby from National Physical Laboratory UK for sharing his long experience and constant advice on temperature measurements. We are deeply grateful to Wei Wu from City University of Hong Kong China for his kind reading and helpful suggestions on the manuscript. We would also like to thank Kun Liang from the National Institute of Metrology China and Chongxia Zhong from Beijing Institute of Metrology China for their helpful discussion on time reference.