1. Columbus, 18 June 2013 International Symposium on Molecular Spectroscopy 68 th Meeting - June 17-21, 2013, Ohio State University Determination of the Boltzmann constant by means of Doppler-Broadening Thermometry on water at 1.39  m. L. Moretti, A. Castrillo, E. Fasci, M.D. De Vizia, G. Galzerano, P. Laporta, A. Merlone, L. Gianfrani Department of Mathematics and Physics Second University of Naples Caserta (ITALY) Email: antonio.castrillo@unina2.it antonio.castrillo@unina2.it

2. Columbus, 18 June 2013 1. Outline of the presentation Introduction: Motivations; Doppler Broadening Thermometry (DBT). Experimental setup: MASTER laser (ML) and offset-frequency locking of the SLAVE laser (SL) to the ML; temperature stabilized cell. Data analysis and results: line-shape modeling; k B determination. Conclusions and perspectives.

3. Columbus, 18 June 2013 2. The “new SI” and the so-called “explicit- constant formulation” Each of the SI unit should be defined by specifying explicitly an exact value for a well-recognized fundamental constant. General Conference of Weights and Measures http://www.bipm.org/en/si/new_si/

4. Columbus, 18 June 2013 3. The “new kelvin” linked to k B Present format for the definition of the kelvin: The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. It follows that the thermodynamic temperature of the triple point of water is exactly 273.16 kelvin, T tpw = 273.16 K. Recommended new format with new definition: The kelvin, K, is the unit of thermodynamic temperature; its magnitude is set by fixing the numerical value of the Boltzmann constant (k B ) to be equal to exactly 1.38064XX×10 -23 J/K when it is expressed in the unit s -2 m 2 kgK -1, which is equal to JK -1. Thus we have the exact relation k B = 1.38064XX×10 -23 J/K. The effect of this definition is that the kelvin equals the change of thermodynamic temperature that results in a change of thermal energy k B T by 1.38064XX×10 -23 J/K.

5. Columbus, 18 June 2013 k B = 1.3806488 (13) 10 -23 J/K  k B / k B = 9.1 × 10 -7 In 2010, CODATA recommended this value: Based on Acoustic Gas Thermometry (AGT) at NPL London, LNE-CNAM Paris, and NIST Gaithesburg Speed of sound in a real gas: u 2 = u 0 2 (1 + a p + b p 2 +…) when p → 0 P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod. Phys. 84(4), 1527-1605 (2012) 4. The current value of k B

6. Columbus, 18 June 2013 5. Upcoming method: DBT Advantages: general method; direct approach to determine k B T; it does not require absolute intensity determinations. Main requirements: high spectral fidelity; refined line shape model. Laser Intensity stabilization Frequency stabilization and control Detector Beer-Lambert’s law: Isothermal cell Proposed by Ch. J. Bordé, Metrologia 39, 435 (2002). First experimental demonstrations: Daussy et al. Physical Review Letters 98, 250801 (2007) on NH 3 Casa et al. Physical Review Letters 100, 200801 (2008) on CO 2 u r (k B )≈10 -4

7. Columbus, 18 June 2013 6. The experimental set-up A few details: 1)Master ECDL: absolutely referenced to a H 2 O transition observed in a high- finesse cavity; linewidth of ~30 kHz. 2)Slave ECDL: offset- frequency locked to the Master; intensity stabilized at a level of 10 -4 ; frequency tuning range of 3 GHz (or larger). 3)Isothermal cell with a temperature stability and uniformity at a level below 1 mK. 4)Spectroscopic detection sensitivity of 10 -4 in 1 Hz. Castrillo, et al. Optics Express 18, 21851 (2010)

8. Columbus, 18 June 2013 Performances: long-term stability of 0.075 mK; temperature uniformity better than 0.4 mK; overall temperature uncertainty of the order of 0.3 mK. 7. The isothermal cell and its temperature stability A. Merlone et al., Int. J. Thermophysics 31, 1360 (2010). Glass Capsule SPRT F 18 Primary Thermometry Bridge

9. Columbus, 18 June 2013 8. The master laser: absolute frequency stability Galzerano, et al. Optics Letters 34, 3107 (2009) Galzerano, et al. Appl. Phys. B 102, 725-729 (2011)

10. Columbus, 18 June 2013 9. The slave laser (frequency locked to the master) Castrillo, et al. Optics Express 18, 21851 (2010) Gate time = 1 s rms ~ 1 kHz White-type frequency noise

11. Columbus, 18 June 2013 10. Example spectrum: linearity of the frequency scale Main questions: Is it possible to extrapolate the Doppler width? How accurate could this retrieval be? De Vizia M.D., et al. Phys. Rev. A 83, 052506 (2011) De Vizia M.D., et al. Phys. Rev. A 85, 062512 (2012) H 2 18 O

12. Columbus, 18 June 2013 The proposed new line profile (pcSDKS) includes: 1)Doppler broadening; 2)Collisions which perturb the molecular internal state; 3)Collisions which change the velocity of the absorbing molecule (Dicke narrowing); 4)Speed-dependence of collisional parameters; 5)A partial correlation between velocity and internal-state changing collisions. 11. Key information Comparisons with experimental spectra show that: 1)semiclassical pcSDHC profile is the most appropriate to describe our physical situation; 2)correlation parameter  ≈0.2.

13. Columbus, 18 June 2013 In addition to speed-dependent and hard-collisions effects on the shape, it takes into account the correlation between collisions that produce a dephasing of the molecular dipole (internal-state) and those producing a change of the absorber velocity. A. S. Pine, JQRST 62, 397-423 (1999). 12a. The pcSDHC model

14. Columbus, 18 June 2013 13. pcSDHC model: example of line fitting L. Moretti et al., Physical Review Letters (submitted on April 2013). free parameters

15. Columbus, 18 June 2013 Number of spectra: 718; q=5.017,  =0.2; Pressure range: 200-500 Pa 14. Experimental results k B = 1.380631 (22) 10 -23 J/K L. Moretti et al., Physical Review Letters (submitted on April 2013).

16. Columbus, 18 June 2013 Error sourceType AType B Experimental reproducibility16 x 10 -6 Frequency scale uncertainty 2.6 x 10 -6 Line-center frequency 0.278 x 10 -6 Laser emission width and FM broadening ~10 x 10 -6 Detector non-linearity Negligible Spurious etalon effects Negligible Cell’s temperature1.1 x 10 -6 Hyperfine structure effectsNegligible Line shape model14.9 x 10 -6 Total uncertainty = 24 x 10 -6 15. Uncertainty budget on k B L. Moretti et al., Physical Review Letters (submitted on April 2013).

17. Columbus, 18 June 2013 16. Conclusions Precise control, stabilization and synchronization of the laser frequency, as well as the removal of any amplitude variations in the background signals are essential for accurate k B determinations; A very refined and reliable line shape model is of great importance for a successful DBT experiment; Speed-dependent, Dicke narrowing effects and correlations between collisions must be taken into account; The combined uncertainty in the spectroscopic determination of k B is 24 ppm.

18. Columbus, 18 June 2013 Global fitting of sets of spectra as a function of the gas pressure should help in reducing uncertainty associated to line shape modeling and correlations among fitting parameters; 17. Perspectives To increase the SNR a long path- length technique is being implemented (also lower gas pressures). NICE-OHMS for the MASTER laser (reducing FM laser broadening).

19. Columbus, 18 June 2013 Thank you for your attention!