Linestrength ratio spectroscopy as a new primary thermometer for redefined Kelvin dissemination

Experimental methods for primary thermometry, after Kelvin unit redefinition on May 2019, become based on a known value of the Boltzmann constant rather than by measuring temperature with respect to a reference point. In this frame, we propose linestrength ratio thermometry (LRT) as a candidate method for primary thermometry in the 9–700 K temperature range. Temperature accuracies at the ppm level are prospected for LRT applied to optical transitions of the CO molecule in the range 80–700 K and of a rare-earth-doped crystal in the 9–100 K one. Future implementations of this technique can contribute to measure the calibration-discrepancies in the ITS-90 metrological scale of thermodynamic temperature which can have a measurable impact in applications ranging from fundamental-physics to meteorology and climatology.

Up to now, the International System (SI) of units has defined the Kelvin in terms of the temperature of the triplet point of 'pure' water, which was assigned the exact value of T TPW =273.16 K. It was the absolute reference value for each accurate thermodynamic temperature measurement, traceable trough the primary thermometers (PTs). In these thermometers, there is a basic relation between the observable and T that can be written down explicitly without having to introduce unknown, temperature-dependent quantities. The PTs are impractical for most part of the applications that need accurate T measurements due to their size, measurement speed and cost. Instead, secondary thermometers (ST's), as the platinum resistance temperature detector, are the practical-instrumentation used to measure precisely thermodynamic temperature. But such a thermometers must be calibrated against a defined fixed temperature points to ensure worldwide accuracy and primary traceability. Indeed, the International Temperature Scale of 1990 (ITS-90 [15]) and the Provisional Low Temperature Scale of 2000 (PTLS-2000 [16]) were established by the Consultative Committee for Thermometry of the International Committee for Weights and Measures (CIPM) as a practical reference scales for dissemination of the Kelvin unit. These scales have a number of fixed temperature values defined by the reproducible freezing or melting temperature of pure substances measured against T TPW by means of accepted PTs.
On May 2019, the CIPM has anticipated the most radical revision of the definition of the SI base units in terms of exact fundamental constants, and in particular the Kelvin unit in terms of the assigned exact value of the Boltzmann constant, k [17,18]. The mise en pratique for the Kelvin redefinition (MeP-K-19) establishes, as first, that the accurate methods used to determine k [19][20][21][22][23], become PTs [24] because in their basic equations T always appears as thermal energy kT. Recently, we proposed linestrength ratio thermometry (LRT) [25] to determine Boltzmann constant with accuracy comparable or even better than other primary techniques. Consequently, we propose the LRT method as a potential candidate as a PT, prospecting its accuracy and precision performances to measure some reference temperatures of the ITS-90, and by extending the application towards the low T range where other optical techniques are not available. Secondly, by MeP-K-19, the worldwide dissemination of Kelvin will be provided equally by those PTs and through the defined scales ITS-90 and PLTS-2000. In this way the practice of thermometry will remain largely unchanged with a smoothed impact in the calibration of ST's and the T-measurement-based applications. The first immediate implication on the scales of the new definition is that the T TPW would lose its status of reference temperature of the primary thermometry, becoming just another fixed T point. The uncertainty of such point would be the order of 10 −4 K transferred by the k uncertainty at the moment of the redefinition. Because T TPW still remains the reference point for the ITS-90, this uncertainty will be propagated to the other fixed points of the scale with a larger impact in the extreme T's. The new PTs are called to calibrate these uncertainties as first issue.
Equally important for a proper Kelvin dissemination is to calibrate systematic discrepancies, T-T 90 between temperature points of the ITS-90 (T 90 ) and the measured T by using the new defined PTs [26,27]. Although such systematic errors seem to be small if compared with precision of ST's, they can produce measurable effects in applications that need precise temperature determinations. For example measured shift of 1.5×10 −4 of T-T 90 for temperatures in the range of meteorological relevance (−50°C to 90°C), even if smaller than the Tvariations of the ST's in the short time due to other environment factors, might be just detectable in homogenization procedures using annual or decadal means [27]. More important, the metrological relevance of these discrepancies prospects a possible future revision of the ITS-90, where current scale errors will certainly be removed if they takes place, and the temperatures estimated according the new ITS will be likely to be much closer to true (i.e. thermodynamic) temperature T than at the present.
In this framework, new accurate thermometry methods as LRT are welcome. To this aim, we report the prospected accuracy performances of LRT to measure absolute T's in the range between 9 and 700 K. The temperature uncertainties of some fixed points of the ITS-90 between triple point of argon (83.8058 K) and freezing point of zinc (692.677 K) were systematically evaluated by applying LRT to ro-vibrational transitions of the CO molecule. In addition, the applicability of LRT to measure T's in the low range of ITS-90 (9-100 K) by laser spectroscopy in a rare-earth-doped crystal is proposed. Similar accuracy performances are also prospected in this temperature range 2. LRT method for primary thermometry LRT method is based on the precise measurement of the linestrength ratio, R of two transitions a and b of the targeted specie at two different temperatures, T and T r : , linestrengths of the a, b transitions, respectively [25]. Laser absorption spectroscopy experiments in highly controlled temperature environment can be used to measure these ratios with high precision (few ppm). Other techniques [28,29] based on the same principle but with different experimental implementation have been used to measure high temperatures with lower accuracy.
In LRT, a determination of the temperature T with respect to the other T r is obtained by: , k Boltzmann constant and ν a the frequency corresponding to the energy of the lower level of the a transition, and by considering b as a ground state transition (ν b =0) for simplicity. This equation was deduced by considering the relationship between the linestrengths of a transition at two different temperatures [30]. Hence, LRT can be considered to fulfill the MeP-K-19 criteria of relative primary thermometry: equation (1) is the simple equation to measure thermodynamic temperatures with respect to a reference one, in a wide temperature range with a low uncertainty δT: The experimental implementation of LRT uses two cells, highly stabilized at the T, T r temperatures to perform simultaneous absorption spectroscopic detection of the a, b transitions with high resolution and signal-to-noise ratio (S/N). In this way, a precise measurement of F is achieved by numerical integration of the detected signals. Moreover, common systematic experimental effects present in individual spectroscopic measurements of linestrengths are removed by using the ratio approach, and hence improving the final uncertainty, dF. Nevertheless, it actually turns to be the limiting factor of the final error of T in LRT even if the functional dependence of T on F, according with equation (1), shows a large slope and mitigates the F-error propagation: an uncertainty δF on F measurement corresponds to a small uncertainty δT on T. This is the strength of LRT as thermometric method which can be able to provide precise T measurements even if measuring a challenge parameter as the linestrength ratios of optical transitions.
The key criteria for LRT applicability as PT is to individuate the best combination of the transitions which minimize the total error given by equation (2). In our previous paper [25], we did this evaluation with the aim of an accurate determination of k by LRT of the CO molecule at two known temperatures: the T r =T TPW and the triplet point temperature of Argon, T=T TPAr . If LRT is used as PT, the individuated CO transitions in that case could be appropriated to measure T TPA against T TPW (or the contrary), but not for other temperature points of the ITS-90 scale. In view of a real PT applicability of LRT, now, we do a more esaustive analysis both in terms possible transitions and targeted species to cover all T 90 points in the range 9-700 K.

LRT on CO molecule: primary thermometry in the 80-700 K range
To illustrate the performances of LRT as PT in this range, we have selected which is probably the most promising molecule: carbon monoxide. The reasons of that are the following.
-Low freezing point (68.13 K) enabling wide range of applicability for LRT thermometry.
-As biatomic molecule, it shows a simple ro-vibrational spectrum in the mid-infrared (MIR) spectral region.
As a consequence it is feasible to find target transitions with negligible interference effects by off-resonance absorptions of CO.
-Ro-vibrational transitions in the MIR with large dipole moment, allowing strong absorption at low pressure.
In addition, the current availability of narrow linewidth laser sources [31][32][33] and high performance detection technology [34] in the MIR allow to perform high-resolution spectroscopy of CO transitions with very high S/N, as required to get low uncertainty on F.

Selection of the LRT transitions
In the following we describe the procedure to individuate the best transition of CO (a transition) to perform LRT primary thermometry in defined points of ITS-90 scale between 83.8058 and 692.677 K by considering T r =T TPW as reference temperature. The analysis is made in terms of minimum error δT evaluated by propagating the prospected errors in dT r , dF and dν a for each considered CO transition. The R(0) of the (2-0) band of CO at 4263.837 197cm −1 (having ν b =0) is the b transition, common to all temperatures. Our algorithm downloads a list of linestrengths for the CO molecule in 1000-10 000 cm −1 wavenumber range [35] and the corresponding energy (divided by h) of lower state, ν a [36,37] for each analyzed temperature, and calculates δT by using equation (2). In this calculation, we consider an experimental uncertainty on T r of dT r =0.3mK, easily achievable by using temperature-controlled commercial cells. We assume to achieve a S/N of the CO spectra to allow a R measurement uncertainty at the best reported for absorption spectroscopy [23], i.e. dR/R=dR r /R r =2.8 ppm or dF/F=4 ppm.
For ν a , dν a is evaluated taken into account the error due to pressure shift for each transition in the pressure range of a practicable LRT experiment (around 10 Pascal) [37]. As shown in the following, this contribution should be considered negligible to the total T-error.
Let first analyzed the behavior of T error with the spectral region by considering CO transitions from different vibrational bands. In figure 1, the relative error δT/T for T TPAr =83.8058 K calculated for the CO rovibrational transitions of the fundamental, first and second overtone bands is shown, as well as the partial contributions δT r /T, δF/T and δν a /T. In all cases, the final error is dominated by the linestrength ratio uncertainty as expected. We note that similar relative temperature errors were found for transitions in the fundamental and overtone bands.This is fundamental in view of a practical implementation of LRT, because of the major availability of the laser, optics and optics and electro-optics technology at the wavelengths of overtone bands. Between them, the (2-0) band transitions are preferred to those of the (3-0) band due to the strong intensity of the former by considering the same final T-error level. For that, the final best a transition candidates were chosen from the first overtone (2-0). The results are shown in figure 2 in two three dimensional graphs with the calculated δT/T in the z-axis as function of the measured thermodynamic temperature T (x-axis) and the wavenumber of the transition (y-axis). All the three CO vibrational bands (fundamental and 1st and 2nd overtones) haven been considered. The best transition for LRT determination of the T 90 ʼs at the ppm level (signed as ST in the figure 2) was selected taking into account the criteria of minimum δT for transitions of the first overtone band, excluding those with linestrength lower than 10 −22 cm mol −1 . For the ST transitions, high S/N spectra can be allowed at low CO gas pressure in a technically better spectral region.
Considering these ST as the a transition candidates for LRT primary thermometry at the ITS-90 fixed temperatures between 80 and 700 K, we report in detail in figure 3 and table 1, the prospected relative uncertainty δT/T by considering the T TPW as reference temperature. Data about linestrengths and wavenumbers of the a and b transitions for each temperature are included in table 1.
In figure 3, in addition, the main temperature error contributions due to dT r and dF are shown. The Fcontribution is the dominant source of error and it is almost constant in all considered temperature range. Instead, the T r -contribution begin to be consistent for T above 300 K, and it is almost negligible at low temperatures. Nevertheless, the total error is always lower than 2.5 ppm and becomes lower than 1 ppm for T under 300 K. Figure 1. Relative error δT/T (black points) in ppm on temperature measurement of T TPAr by LRT with T TPW as reference temperature and by using different rotational transitions of the fundamental (0<J<16) (left graph), first overtone (0<J<15) (center graph) and second overtone (0<J<10) (right graph) vibrational bands. Green, red and blue points for each graph are the partial contribution to this error due to the uncertainties of the reference temperature, frequency of the lower level of the transition and F ratio measurement, respectively (δT r , δν a and δF contribution in equation (2)). 3.2. Uncertainty analysis on spectroscopic measurement of F ratios All the above analysis about uncertainties at ppm level on LRT primary thermometry are supported if the considered experimental errors on ν a , T r and F are achieved. The ν a frequencies of the ST transitions are known with a relative accuracy of about 0.1 ppm (see figure 1) for the experimental conditions of the CO samples in LRT, and hence we consider a negligible contribution of dν a /ν a on T's at the ppm level. For T r , temperature stability at the 0.3mK level can be achieved with commercial control systems. Only for above 300 K temperatures where the dT r becomes a significative contribution to the δT, a further technological effort in the T-stability should be realized if the dF/F contribution will be improved better than 4 ppm. Finally for F, we evaluate that a few ppm F-measurement can be achieved by numerical integration of the absorption area for the spectra with S/N of thousands [25].
As a preliminary experimental support of this estimation on error in F, a spectroscopic measurement in the 4.5 μm wavelength region was performed. Because we have not yet available a laser source resonant with the CO transitions in this spectral region, the test was realized on the (0 2 01-0 0 00) R(16)e transition of N 2 O molecule at 2209.0854 cm −1 , which has similar spectroscopic characteristics to those the CO transitions at these wavelengths. Linear absorption spectroscopy of this transition was performed by interaction of a quantum cascade laser (QCL) at 4.5 μm with the N 2 O gas contained in a cell of 18 cm of length at 1.1 mbar of pressure. Eleven consecutive transmission spectra were recorded by scanning the QCL frequency in an interval of about 500 MHz around N 2 O resonance at acquisition times of 50 ms per spectrum. The results are reported in figure 4 in terms of absorbance. Then, the integrated absorption (IA) area for each spectrum has been calculated by using numerical Romberg integration method and the area ratio R with respect to the mean value were calculated, as well as the standard deviation. As a result a relative uncertainty on the absorbance area of about 1.1% was measured, only two times larger than we have prospected for spectra with a noise level ò=0.05 and a QCL linewidth of about 500 kHz [25]. We ascribe this discrepancy to temperature, pressure and absorption path fluctuations whose do not particularly taken care in this simple experimental set-up, differently of a real LRT setup which provides simultaneous measurement of both transitions in a cell at a highly controlled temperature. In these conditions, error contributions due to absorption path and gas pressure fluctuations should be canceled and highly minimized those due to temperature fluctuations. Hence, we are confident to achieve the prospected F uncertainties when the experiment will be performed in the conditions of the LRT method. Nevertheless, F measurement can be affected by systematic effects on the measured area due to the limited spectral range detected for each spectrum or due to the presence of off-resonance molecular absorbances different of the . Total relative error on temperature measurement using LRT (black points), contribution due to T r uncertainty (blue points) and contribution due to the F uncertainty (red points). The errors were calculated at ITS-90 fixed points using the ST transitions. Table 1. Prospected total relative temperature error of a LRT-measurement of ITS-90 fixed points in the range 83.8058-692.677 K with respect to the reference temperature T TPW =273.1600 (3) K. dF/ F=4 ppm was considered in this calculation. targeted ones. In the following, we evaluate the contributions of these systematics in view of a practical realization of LRT in CO. A benefit of LRT spectroscopy is the measurement of the absorbance of the different spectra without requiring a sophisticated line shape model, but just by numerical integration. In order to guarantee the few ppm F-accuracy in LRT methods, we evaluate that a scan bandwidth of about 2 GHz is sufficient to numerically capture the entire absorption area with a systematic error below ppm. The CO transition line-shapes in the thermodynamic conditions of the sample gas relevant for LRT experiments are due to Doppler broadening and pressure broadening.
The line shape I D (ν) due to Doppler broadening effect is given by: The pressure broadening generates a Lorentzian contribution to the line-shape: where γ is pressure broadening coefficient [35]. For our analysis, we consider this profile at pressure P of about 10 Pascal, sufficient to have an adequate S/N of the absorption spectra of the ST and b transitions. If we consider each spectral profile separately to calculate S a and S b at T and T r , a 2 GHz scan bandwidth is not sufficient to measure their area with the required precision because of the truncated-wings contribution. The error in this case is: and hence very far from the requested ppm. But the method is based on the ratio between to linestrength ratios, F. If the CO pressure P for the measurement of R was be exactly equal to the P for R r , the error on F should be 0 even with a limited 2 GHz scan. If the P for measurement of R was be different to the P for R r , the error on F should be 0 by using an infinite integration bandwidth. Assuming an error of 1% on the pressure gauge, the pressure broadening for R will be different of the pressure broadening of R r and the error on F in the case of 2 GHz scan bandwidth is: Finally, let us consider the uncertainty consequences on the F measurement when numerical integration of the spectra instead of lineshape fit is used to measure it. This consideration is not only important in terms that LRT is a line-shape model independent, avoiding, in this way, to accurately consider all velocity-changing collision effects, normally present in spectroscopy of gas species. Even in terms of how the uncertainty of the numerical integrated area is lower than fitted one. To evaluate it, we have simulated a Voigt profile of an absorption transition with a Doppler width contribution, ω much larger than the collisional one and with an area A. This is the usual situation in optical spectroscopy-based thermometric techniques: LRT and Doppler Broadening Thermometry, DBT [38,39]. A random white noise of the 1% was added to the spectral profile to simulate a real experimental spectrum of the transition. Then, we have determined the relevant spectral parameters for thermometry by using a fit procedure to measure ω F and A F , and a numerical integration to measure A I . The fit and numerical integration is repeated a significant number of times to get a normal statistical distribution of the measured parameters. As a result, the standard deviations of the fitted area (s A F ) and Doppler linewidth contribution (s w F ) are one order of magnitude worst than the numerical integrated area (s A I ).
Consequently, even if with an adequate line-shape model, better accurate results could be expected for LRT if compared with DBT due F is measured by the numerical integration and ω by the fit procedure.
4. LRT on Er:YLF crystal: primary thermometry in the 9-100 K range The use of the carbon monoxide for LRT is limited in the range 80-700 K because of technical issues. Below 80 K, high resolution CO absorption spectroscopy is limited by the gas sample vapor pressure at these T's. Above 700 K instead, the not negligible absorption of interferences from off-resonance neighbor transitions limits the LRTapplication. A possible system useful to extend LRT range towards few Kelvin is a rare-earth-doped crystal. In such materials, the shielding of f electrons by closed shells makes possible optical transitions between these quasi-unperturbed f levels [40] with very narrow achieved linewidths as, for example, the transitions of Erbium doped YLiF 4 ( + Er : YLF 3 ) in the near-IR region [41,42]. For this reason, we propose this crystal as a target specie to perform LRT in the low temperature range of ITS-90.
In YLiF 4 , erbium ions substitute yttrium ions in one crystallographic site of S4 point symmetry. Since for T=30 and 9 K. Consequently, as done above for the CO, the T and its total relative uncertainty from equations (1), (2) and the partial uncertainties δT r /T determination and δF/T were estimated (as for CO, δν a /T is negligible if compared with the others). The results are shown in the left part of figure 5. The same values and criteria as in CO are used for the dF, dT r and dν a in this calculation. As a result, the total T uncertainty is few ppm proving the validity of LRT as primary thermometry technique also in few Kelvin range. This would be, to the best of our knowledge, the first optical primary thermometry technique working in this low temperature range. Indeed, in figure 5 is also shown the range of applicability of the optical spectroscopy based PTs: DBT and LRT. DBT uses Doppler broadening due to the T of gas samples, and, in principle, a low temperature applicability up to 80K could be prospected if CO is used 5 .LRT is also down limited to this T if CO is used, but its range of applicability can be extended up to 9 K if transitions in rare-earth doped materials as Erbium doped YLiF 4 crystal are measured.
Comparison with the most accurate primary thermometry, acoustic gas thermometry (AGT), in the range of 80-300 K is also shown in figure 5. Similar accuracies to the AGT ones are prospected for T's around water triplet point and even better for low temperatures.

Discussion
On the frame of the Kelvin unit dissemination prospected by MeP-K-19 by using the ITS-90 scale corrected by the new PTs, we are advancing the LRT method as candidate for a no-contact PT. The thermometric capabilities of LRT envisage accuracies of thermodynamic temperature at the ppm level when applied to transitions of the CO molecule and of the Erbium doped YLiF 4 crystal in a temperature range between 80 and 700 K and between 9 and 100 K, respectively. This accuracy makes LRT comparable even with the most precise PTs as those based in AGT. Even more, if compared with the other optical-based thermometry, the DBT [38], an accuracy improvement of a factor of 15 for T;300 K is prospected. Discrepancies under mK level of T-T 90 in 9-700 K range can be tested by this accurate LRT thermometry, and in particular, the T range of meteorological interest [27]. In addition, LRT technique would be, to the best of our knowledge, the first optical based primary thermometry working at few kelvin with ppm accuracies. For the above discussion, crucial for LRT success as a PT is to achieve experimental accuracy at the few ppm level in the spectral measurements of F in a LRT experiment. The experimental feasibility of LRT is enabled by technological development of laser sources in Figure 5. Relative T uncertainty (black points) and their partial contributions due to δT r (green points) and δR for LRT applied to hyperfine transitions of Erbium doped YLiF 4 crystal around 800 nm at T of 30 and 9 K with respect to the 100 K reference temperature. Uncertainty performances of AGT, DBT and LRT in the 80-300 K range are shown. Freezing points of CO, CO 2 and H 2 O are also indicated. 5 A constant relative uncertainty of about 9ppm was considered in all the range of applicability corresponding to that reported for CO 2 at 300 K [38]. Different values could be achieved with different sample gases at other T's.