A New Precision Spectroscopy Based Method for Boltzmann Constant

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A New Precision Spectroscopy Based Method for Boltzmann Constant Determination and Primary Thermometry Luigi Santamaria Amato, Mario Siciliani de Cumis, Daniele Dequal, Giuseppe Bianco, and Pablo Cancio Pastor J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b05523 • Publication Date (Web): 26 Jun 2018 Downloaded from http://pubs.acs.org on June 27, 2018

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The Journal of Physical Chemistry

A New Precision Spectroscopy Based Method For Boltzmann Constant Determination And Primary Thermometry †

Luigi Santamaria Amato,

∗,†

Mario Siciliani de Cumis,

Bianco,

†Agenzia

†

†

Daniele Dequal,

and Pablo Cancio Pastor

Giuseppe

∗,‡

Spaziale Italiana, Centro di Geodesia Spaziale, Matera 75100, Italy

‡CNR-INO,

Istituto Nazionale di Ottica, Largo E. Fermi 6, 50125 Firenze, Italy

E-mail: [email protected]; [email protected]

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We propose a new independent thermometry method, line strength ratio thermometry (LRT), based on optical spectroscopy measurement of linestrength intensity ratio R between pairs of molecular transitions. Due to strong dependence of R on kT , a given measurement uncertainty δR on R reects in a small uncertainty on kT determination. By assuming experimental uncertainties on R and T as those reported in literature, a

k determination at the 5 ppm level is foreseen which is better than the most precise k determination by using Doppler broadening thermometry (DBT). In the frame of new denition of the SI Kelvin unit, based on k as xed constant once k constant will be exactly established, LRT is proposed as high resolution non contact thermometry technique for absolute temperature measurements of gas samples at the ppm level.

Introduction Measuring the temperature is fundamental in the every-day life: the importance of this observable spans from health to meteorology, from industry to environment, from metrology to science. However temperature denition and consequently temperature measurement is quite dierent with respect to the others SI units: temperature is an intensive quantity, so it is not really directly measurable as other real physical-chemical observables. Actually, in order to dene the Kelvin, a physical property related to the concept of thermodynamic temperature is projected to an order relationship between the thermodynamic systems with respect to the direction in which the heat would ow if they were contacted. For other observable, only the choice of a unit of measurement is arbitrary, instead, in the case of temperature, also the measurement scale is arbitrary. For a real observable the conversion between units must be linear whereas, for temperature, the only requirement to preserve the order relationship is that the scale is monotonic and it explains the nonlinear relation between, for example, Celsius and Fahrenheit scales. The rst temperature unit, relative temperature scale, was based on phase transitions and developed before thermodynamic theory so it was fully empirical

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and needs at least two reference transition temperatures to be dened (two degrees of freedom). The absolute temperature scales are completely dierent: they are based on the concept of absolute temperature obtained by thermodynamic theory. The temperature unit in SI is the Kelvin and it is dened as 1/273.16 of the triple point of water (one degree freedom); the zero is xed by thermodynamic principles. The statistical mechanics showed the absolute temperature is proportional to thermal motion of the gas particles. Consequently, the temperature can be converted in energy unit and the conversion factor is the Boltzmann constant which the recommended value by the Committee on Data for Science and Technology

(CODATA) 1 is k = 1.38064852(79) ∗ 10−23 JK −1 as

given in 2014. Nowadays the Kelvin is dened using the triple point of water, which is of impractical use at thermodynamic temperatures very far from this point. Concerning the standard use of this unit, the Consultative Committee for Thermometry (CCT) of the International Committee for Weights and Measures (CIPM) published The International temperature Scale of 1990 (ITS-90) to map the thermodynamic temperature as closely as possible throughout its range through many dierent thermometers. 2 In next future, before the end of 2018, the CIPM foresee a new denition of SI units based on exact fundamental physics constants with undoubted advantages in units conversion factors and reduction of uncertainties 3 . 4 In addiction, this new approach in redenition of SI decouples denitions from measurement procedures. In this way many dierent measurements methods can be used and compared. Concerning the temperature denition, the dening constant will be the Boltzmann constant whose current value is known with an uncertainty of 0.57 ppm. 1 To provide k without error, as required by the new approach to dene SI units, and to maintain continuity with the actual temperature denition, k should be measured with best accuracy as many as possible independent methods. With the new denition, the thermodynamic temperature can be measured directly at any point in the scale. In this context, after Kelvin redenition, the present techniques for k determination become automatically primary thermometer methods. Using the k value candidate for Kelvin redenition, some discrepancy appears in the comparison of points measured using ITS-90. So it is crucial developing of independent

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primary thermometers methods. Nowadays, the most accurate technique to measure k is through speed of sound measurement in an acoustic resonator lled with noble gas. 5,6 Using acoustic gas thermometry the present relative uncertainty on k determination is 0.71 ppm. 7 A second method to determine k is through Clausius-Mossotti equation by measuring the pressure dependence of helium electric susceptibility; such technique has recently produced an uncertainty of 1.9 ppm. 8 Recently, an independent approach unrelated to the thermal properties of gases, called Johnson noise thermometry (JNT) has achieved an uncertainty of 2.7 ppm. An optical method to obtain an accurate k determination is the Doppler broadening thermometry (DBT), which is based on an accurate spectroscopic measurement of Doppler width of a transition of an atomic or molecular gas at temperature T. 9,10 At low gas pressures, the main source of line-shape broadening is the Doppler eect which links the thermal energy of the gas to an optical frequency, according to the following equation:

ν0 ∆νD = c

s

2 log 2

kT M

(1)

where ∆νD is the Doppler half-width at half maximum (HWHM), ν0 is the transition frequency, c is the light velocity in vacuum, T is the thermodynamic temperature of the gas and M is the molecule or atom mass. By measuring ∆νD and inverting Eq. 1, it is possible to obtain kT if v0 and M are known. Then, if the measurement is performed at dened temperature, as for example in a triple point cell, the Boltzmann constant can be obtained. Actually, the best DBT measurement of k has been carried out on

H2 O molecule with an uncertainty of 24 ppm. 11 Recently, DBT on CO2 transitions has been demonstrated as a primary thermometer at the 14 ppm level. 12 In this work we propose a new spectroscopic method: Linestrength Ratio thermometry (LRT). Using this approach, it is possible to extract kT through optical measurement completely alternative to DBT. Hence, LRT is proposed as another independent method to the nal proposed k value or, alternatively, as a primary no-contact thermometer.

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Methods Unlike DBT that exploits the link between the thermal energy and optical frequency, LRT method exploits the link between the occupation probability of rotational levels and temperature in a way similar to techniques for ame temperature measuring through laser induced uorescence. 13,14 It is based on the precise measurement of the integrated absorbance of particular molecular ro-vibrational lines. In Fig. 1 we show how kT enters in the measured spectral quantities in DBT and LRT; in DBT (Fig. 1, left side), the width of the Doppler-limited Gaussian contribution to the lineshape takes into account the kT energy of the molecules, whereas in LRT (Fig. 1, right side), the

kT enters in the occupation probability by measuring the line intensity. This value can be precisely measured by using simple integration techniques of the detected absorption, free of systematics errors present in DBT due to the needed accurate lineshape model in that method. Indeed, in DBT the line shape model used to t experimental spectra of the transition should be suciently rened to extract the ∆νD with high accuracy, free of systematic contributions due to other broadening mechanism of the spectra. LRT, instead, does not need sophisticated line shape model to measure line intensity, minimizing the occurrence of such systematic errors. LRT is based on accurate measurement of linestrength Sa , Sb of two ro-vibrational transitions of the tested molecule at two dierent temperatures T and Tr . According to Rothman et al., 15 the transition linestrength Sa at temperature T is:

−hν0 −hνa Q(Tr ) exp kT 1 − exp kT , Sa (k, T ) = Sa (k, Tr ) Q(T ) exp −hνa 1 − exp −hν0 kTr

(2)

kTr

where Tr is a reference temperature, h the Planck constant, νa the frequency corresponding to the energy of lower level transition, ν0 the transition frequency and Q(T ) is the partition function. At optical frequencies, the term exp

−hν0 kTr

is completely

negligible (∼ 10−11 ) compared to unity. In this way, by measuring the ratio between the linestrength intensity at two dierent temperatures is possible to obtain the Boltzmann constant. Unfortunately the partition function is not known with a sucient accuracy

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for k determination. But dening R(k, T ) as the ratio between the linestrength intensity

Sb (k, T ) and Sa (k, T ) for the b and a transitions at temperature T, and the quantity F (k, T, Tr ) ≡

R(k,T ) R(k,Tr ) ,

through the measurement of F (k, T, Tr ), T and Tr it is possible

to obtain a measurement of k being F (k, T, Tr ) independent of partition function:

1 b 1 exp −hν h (νa − νb ) 1 R(k, T ) 1 k ( T − Tr ) = exp F (k, T, Tr ) = = − 1 a 1 R(k, Tr ) k T Tr ) exp −hν ( − k T Tr

(3)

Inverting eq. 3 we obtain the Boltzmann constant as function of T , Tr , R(k, T ) and

R(k, Tr ), (from now on to simplify the notation R(k, T ) ≡ R and R(k, Tr ) ≡ Rr )

k(T, Tr , R, Rr , νa − νb ) =

h( T1 −

1 Tr ) (νa

− νb )

ln (R) − ln (Rr )

=

h( T1 −

1 Tr ) (νa

ln (F )

− νb )

(4)

It is important to note that the functional form of F as function of k shows a large slope so an uncertainty δF on F determination corresponds to a small uncertainty δk on k . To estimate δk in a quantitative way, we dierentiate k(T, Tr , R, Rr , νa − νb ) respect its variables obtaining the corresponding error contribution:

δT =

∂k dT ∂T

(5)

δTr =

∂k dTr ∂Tr

(6)

δR =

∂k dR ∂R

(7)

∂k dRr ∂Rr

(8)

δRr =

δ (νa − νb ) =

∂k d (νa − νb ) ∂ (νa − νb )

6


(9)

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and the total error:

δk =

q

(δT )2 + (δTr )2 + (δR)2 + (δRr )2 + (δ (νa − νb ))2

(10)

To get an estimation of this error, we applied previous formulas to mid-infrared transitions of carbon monoxide molecule. From the experimental point of view, CO is a good target molecule for LRT determination of k because very intense ro-vibrational transitions around 2.3 µm can be found. Moreover, being a bi-atomic molecule, CO provides very isolated lines avoiding interference eects from absorptions of isotopologues and even other molecular species. 16 The ro-vibrational transitions R(17) (a line) and R(0) (b line) of the (2-0) band at 4317.159007 cm−1 and 4263.837197 cm−1 were considered for the present estimation, respectively. In the spectral determination of R factors for these transitions, the estimated contribution due to other possible interfering absorptions is under ppm level. By accurate absorption spectroscopy measurement on lines a and b at two different temperatures (T and Tr ), the linestrengths Sa (k, T ), Sa (k, Tr ), Sb (k, T ) and

Sb (k, Tr ) are measured. Triple point temperatures of Argon, T = 83.8058 K , and water, Tr = 273.16 K guarantee the needed experimental temperature control to achieve the challenge k accuracy at the ppm level. The energies of the lower rotational state of the transition are 587.7209 cm−1 for a line and νb = 0 cm−1 for b line. The critical point of the experiment is the measurement of the ratios R(k, T ) and R(k, Tr ) with high accuracy. The error correlated to the measurement of these ratios can be very small; indeed, while the measurement of the absolute linestrength can show an error of 1000 ppm in the best case, being the uncertainty on transmission spectrum very low, 10 the measurement of the ratios between two linestrengths can be orders of magnitude lower because of uncertainty correlation between measurement of Sa (k, T ) and Sb (k, T ). It is useful to remind that integrated absorbance is equal to the product of the linestrength, the gas density n (which in turn is proportional the gas sample pressure p), and the absorption path

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(which in turn is proportional to cell length L). The two principal error components on linestrength measurement are caused by the error on pressure gauge and error on cell length that generates a measured linestrength inversely proportional to the measured cell length times measured cell pressure. When the ratio between measured Sa (k, T ) and Sb (k, T ) is calculated, the error due to pressure and length on Sa (k, T ) simplify with error on Sb (k, T ) and only systematic errors like those due to instrumental interference fringes remains. It is important to note that in the pressure range necessary for the experiment (below Torr), the expected energy level shift due to particles interaction is about 200 kHz. 17 Because in eq. 4 the frequency involved in the formula is νa −

νb = 587.7209 cm−1 (corresponding to 17.6 THz) 18 , the relative error is less than 11 ppb, and hence negligible in the k determination from LRT. By assuming the error on transmission spectra as obtained by 10 the corresponding error on linestrength ratio is negligible in comparison to cell temperature error. We assume an error

dR(k,T ) R(k,T )

and

dR(k,Tr ) R(k,Tr )

of order of 2.8 ppm if we consider to achieve

the state of the art uncertainty reported in transmission spectroscopy. 10 Regarding the temperature achievable in a laboratory, commercial triple point cells get an error on T and Tr of about dT = dTr ' 0.3 mK as reported, for example, by Moretti et al. 11 Substituting the above assumed numerical values for δR, δRr , δT and δTr , the calculated value of the ratios R(T ) and R(Tr ) by using HITRAN database linestrengths 19 and temperatures T , Tr in Eqs.5-10, we will prospect a relative uncertainty on Boltzmann constant of δk ' 5ppm. The error budget for each single contribution is reported in Table 1. Table 1: Estimated error budget of LRT for CO ro-vibrational transitions R(17) and R(0) of the (2-0) band at temperatures of triple points of argon and water

Error contribution (ppm) δT 5.1 δTr 0.49 δR 0.40 δRr 0.40 δ (νa − νb ) 0.011 Total Error 5.2 8


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The estimated δk from LRT is about 5 times better than the best DBT determination. 11 Moreover, δk as the best DBT (24 ppm) could be achieved by LRT from R measurements at 100 ppm relative uncertainty level, that is easily obtainable using conventional laser spectroscopy. In addition, LRT can be applied to a larger number of linestrength ratios from dierent couple of transitions of the targeted molecule in order to increase the accuracy in the k determination at the ppm level, close to the uncertainty provided by acoustic thermometry. 7 Alternatively, if k is considered as an exact constant, LRT on CO could provide a measurement of the absolute temperature of the triplet point of Ar with an accuracy of 0.5 ppm, limited by the 0.3 mK accuracy of the triplet point of water temperature. Again, it is a T accuracy 28 times better than the T record of DBT in CO2 . 12

Figure 1: Representation of DBT and LRT thermometry techniques. (DBT) the kT is related to Gaussian width component of transition lineshape, a variation of kT value inuences the Doppler broadening by acting on Maxwell-Boltzmann velocity distribution. (LRT) the kT is related to linestrength intensity of transition, a variation of kT value inuences the occupation probability by acting on Boltzmann distribution. The nal accuracy in linestrength ratio measurement will be limited by the signalto-noise ratio, S/N, of the acquired absorption spectra and the sampling frequency resolution used in its recording. Both, higher is the spectra S/N and narrow is the sampling frequency interval, more precise will be measured the integrated absorbance. Spectroscopic techniques will limit the nal S/N whereas laser linewidth (LFWHM)

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will provide frequency resolution. For the targeted CO transitions, being the Doppler FWHM about 300 MHz, a laser frequency scan of 2 GHz is sucient to collect the entire molecular transition prole. In principle there are no particular experimental constraints for the scanning range. In advance, hypothesizing a fast measurement time, we evaluated the consistency of this range for numerical acquisition at the ppm level. For Doppler contribution is suciently a scanning bandwidth of about 7 times the FWHM of CO lines. For the Lorentzian contribution, we have to consider the eect on the areas due to dierent broadening coecient of the two transitions at a given Temperature. However, in the computation of F , this error becomes negligible because the trunked Lorentzian contribution simplies. This frequency interval will be sampled with a number of the steps N equal to 2 GHz/LFWHM. In order to evaluate the laser linewidth requirements for a feasible LTR experiment in CO, we calculated the area under a simulated Voigt prole as expected for dierent S/N and LFWHM. The result is depicted in Fig. 2. As a result shown in the left graph of this gure, with a 1 kHz laser linewidth and a noise level of 1%, a 40 ppm relative error on the area is obtained, corresponding to 12 ppm on k which is a factor of two better than the best DBT uncertainty. Nowadays, near-infrared and mid-infrared laser sources with linewidths at the few kHz level (and less) are available by using dierent stabilization techniques as those based on ultrastable cavities 20,21 or whispering gallery mode microresonators. 22,23 In addition, optical frequency comb (OFC) disciplined laser sources 24,25 or directly OFC's as spectroscopic sources 2628 provide sub Hz control level of the absolute frequency. Recently QCL-combs 29 are integrating metrological tools in the Mid-Ir Region. Optical spectroscopy techniques with this kind of laser sources have provided detection of strong transitions of simple molecular gases with high S/N, 30 and hence the experimental feasibility of LRT. Because of the prospected accuracy and the possibility of its experimental realization, LRT is proposed as a new spectroscopic method for a dierent primary realization of the new Kelvin. In addition, LRT provides a new spectroscopy-basedprimary Thermometry, alternative to DBT, with prospected implications in control of

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Figure 2: The gure shows the behavior of the relative error on the absorbance area measurement due to the detection noise and frequency resolution. Several simulated Voigt spectra of one CO transition aected by dierent amount of noise and recorded at dierent sampling number N=2GHz/LFWHM are shown on right in the gure (the noise increases from down to up of the vertical axis and record sampling from right to left on horizontal axis) In the left side of the gure the relative error of the measured absorbance area (vertical axis) as a function of the laser linewidth (horizontal axis) is shown for three dierent noise levels ( = 1%, 10%, 30%). We note that a white random noise level was used in all cases. cold molecules 31,32 . In principle LRT is a less demanding with respect to DBT method. The justication of that is the dierent nature of the limiting factor of the uncertainty for these two approaches. In DBT, the main source of uncertainty is given by the complex line-shape model needed to remove molecular collision contributions to the total linewidth, whereas the measurement of the integrated absorbance, as needed in LRT, is line-shape model independent. On the contrary, spectroscopic experiment in LRT is more complicate because the two transitions at each temperature must be recorded simultaneously, whereas for DBT only one spectroscopic recording of the transition at one temperature must be provided.

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Conclusions We proposed a new method for Boltzmann constant evaluation with uncertainty at least comparable with well established technique as DBT. In view of Boltzmann constant xing and Kelvin redenition, the LRT approach enables a complementary method with DBT 12 and acoustic gas approach 6 for accurate temperature measurements: using linestrengths accuracy of Truong et al 10 and the stability value of commercial cell of Water we evaluate an uncertainty of Temperature of 0.5 ppm comparable with acoustic gas Thermometry and better of present accuracy of DBT. 12

Acknowledgement This work was partially supported by Progetti di Ricerca di Interesse Nazionale (PRIN), Project No. 20152MRAKH of the Italian Ministry of University and Research.

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for next-generation environmental sensor. Appl. Phys. B: Lasers Opt. 2011, 102, 255269. (31) Santamaria, L.; Di Sarno, V.; Ricciardi, I.; De Rosa, M.; Mosca, S.; Santambrogio, G.; Maddaloni, P.; De Natale, P. Low-temperature spectroscopy of the 12 C2 H2 (ν1 + ν3 ) band in a Helium buer gas. Astrophys. J. 2015, 50, 801. (32) Santamaria, L.; Di Sarno, V.; De Natale, P.; De Rosa, M.; Inguscio, M.; Mosca, S.; Ricciardi, I.; Calonico, D.; Levi, F.; Maddaloni, P. Comb-assisted cavity ring-down spectroscopy of a buer-gas-cooled molecular beam. Phys. Chem. Chem. Phys.

2016, 18, 1671516720.

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Relative Error

0,1 0,01

  

Noise

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


B

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1E-3 1E-4 1E-5

0,1

1

10

100

1000 10000

Laser FWHM (kHz)

Laser FWHM ACS Paragon Plus Environment

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A New Precision Spectroscopy Based Method for Boltzmann Constant

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