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FFK-2017 Advances in doppler broadening thermometry for the spectroscopic determination of the Boltzmann constant Livio Gianfrani Department of Mathematics and Physics Università degli studi della Campania Warsaw, May 18, 2017
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FFK-2017 Advances in doppler broadening thermometry for the spectroscopic determination of the Boltzmann constant Livio Gianfrani Department of Mathematics and Physics Università degli studi della Campania Warsaw, May 18, 2017
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Outline Introduction and motivations DBT: Basic principles
The 3rd generation experiment on C2H2 The line shape problem Concluding remarks I will start with a brief introduction to the field; then, I will move to the basic principles and state of the art of DBT
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Motivations EMPIR Program
To provide a spectroscopic determination of the Boltzmann constant; To make DBT competitive with AGT and DCGT; To contribute to the development of a coherent set of primary thermometry methods for the aims of the practical realization of the new kelvin. InK #2 – Implementing the new Kelvin Coordinator: Graham Machin, NPL EMPIR Program
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The roadmap towards the new SI
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D.R. White and J. Fischer, Metrologia 52 (2015) S213–S216.
The history of kB DBT values This slide resumes the history of kB. The relative uncertainty in the value of Boltzmann’s constant versus time expressed as parts per million (ppm). Also shown is the relative reproducibility of the practical temperature scales, in the vicinity of 100 °C, over the same period. First measurements: M. Planck, A. Einstein, and Jean Perrin. D.R. White and J. Fischer, Metrologia 52 (2015) S213–S216.
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The Doppler effect 1842 At the age of 38, Doppler gave a lecture to the Royal Bohemian Society of Sciences and subsequently published his most notable work, "Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels" (On the coloured light of the binary stars and some other stars of the heavens). 1842 Usually to be suppressed in order to perform precision spectroscopy. For the aims of primary gas thermometry, it is a gift of nature!
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The basic idea of DBT The method consists in measuring very precisely the Doppler width of an absorption profile corresponding to a single atomic or molecular spectral line in a gas at thermodynamic equilibrium. Advantages: general method applicable to any gas, at any temperature, in any spectral region; direct approach to determine kBT; it does not require absolute intensity determinations; it is not sensitive to the purity of the gas. Requirements: high spectral resolution; high detection sensitivity and experimental reproducibility; high spectral fidelity: high linearity of the detection electronics; precise and absolute control of the laser frequency; refined methods of line shape modelling and fitting. Requirements: high spectral resolution; high detection sensitivity and experimental reproducibility; high spectral fidelity: high linearity of the detection electronics; precise and absolute control of the laser frequency; refined methods of line shape modelling and fitting. Proposed by Ch.J. Bordé in Metrologia 39, 440 (2002)
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Sketch of a DBT spectrometer
Laser Intensity stabilization Frequency stabilization and control Thermostat Detector Isothermal cell Schematics of a DBT experiment.…… Repeated measurements of the profile in coincidence with a well isolated line are performed, possibly at different gas pressures; this has to be done with extremely high fidelity; then, a refined ….. Is required to retrieve….. Primary method of gas thermometry
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Our values of kB CO2, 2008: (1.38058 ± 0.00022) ×10-23 J/K
H2O, 2013: ( ± ) ×10-23 J/K G. Casa, A. Castrillo, G. Galzerano, R. Wehr, A. Merlone, D. Di Serafino, P. Laporta, and L. Gianfrani Phys. Rev. Lett. 100, (2008) L. Moretti, A. Castrillo, E. Fasci, M.D. De Vizia, G. Casa, G. Galzerano, A. Merlone, P. Laporta, and L. Gianfrani, Phys. Rev. Lett. 111, (2013) APS Synopsis:
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Our new target: C2H2 Legenda:
Centrosymmetric LINEAR molecule (D∞h group) Non-polar molecule Weak but measurable absorption in the telecom region Five fundamental modes of vibration Negligible hyperfine structure Legenda: In green In red
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Previous DBT works on C2H2
Group Vibrational band and wavelength Isotopologue Precision (ppm) Year National Metrology Institute of Japan (Onae and coworkers) n1+n3 band at 1.54 mm 13C2H2 ~1000 2009 University of Lethbridge Canada (Predoi-Cross and coworkers) 12C2H2 85 2014 Hefei National Laboratory for Physical Sciences at Microscale, China (Hu and coworkers) n1+3n3 band at 787 nm 6 2015 There have been already some attempts to implement DBT using acetylene as thermometric substance!
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Overall uncertainty: ~90 ppm
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The 3rd generation experiment
l = 1.4 mm
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The reference oscillator
OFCS Zerodur cavity Zerodur: glass-ceramic material with an extremely low CTE, of the order of 10^-6 1/K fbeat = nECDL – (nfrep + fCEO)
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The relative frequency stability
Flicker phase noise 1/tau White frequency noise 1/radq(tau)
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The isothermal cell Type Spherical Herriott Optical path length
Variable between 8 and 12 m Length of the cell’s body 256 mm Diameter 91 mm Number of passes 42 (for ≈10 m of path length) Entrance angle ≈ 5° Material Stainless steel Front of the cell.
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Temperature stability
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Refined line shape model
The lineshape problem n Refined line shape model Fitting code Doppler width
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Collisional lineshape effects
The frequency response of a system interacting with radiation and a surrounding medium is given by: The most general dipole correlation function should include: Thermal motion State-perturbing collisions Collisions that change the velocity of the absorbing molecules (Dicke narrowing effect) Dependence of the collision parameters on molecular velocities Correlation between collisions
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C2H2 spectra and Voigt fits
This slide demonstrates the failure of the Voigt model
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The failure of the Voigt model
<T> = (4) K Set point: K Negative shift due to the occurrence of line-narrowing effects!
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A realistic lineshape model
Thermal motion Dephasing collisions pcSDHCP Hard collision HTP Dicke effect Speed dependence
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The quadratic speed dependence
Berman: J. Quant. Spect. Rad. Transf. 12, (1972) Pickett: J. Chem. Phys. 73, 6090 (1980) G (va): Quadratic approximation Collisional theory G (vr): Berman-Pickett model Maxwell distribution of va Empirical collisional interaction potential of the form V(R) R‑q Power-law dependence on the relative speed of the absorber/perturber system G(vr) (vr)m, with m = (q‑3) / (q‑1) Robert & Bonamy theory (calculations for CO in O2); conditional probability, distribution of relative speeds. We are neglecting the influence of those molecules moving at the highest velocities, for which the quadratic approximation departs from the actual behavior of the relaxation rate as a function of the molecular speed. M.D. De Vizia, A. Castrillo, E. Fasci, L. Moretti,, F. Rohart, and L. Gianfrani, Phys. Rev. A 85, (2012).
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h = (30)
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Multi-spectrum fitting approach
Simultaneous analysis of a number of absorption spectra across a given pressure range, by means of a non-linear least-squares method, under the MATLAB environment. Parameters’ vector for the pcSDHCP model with quadratic speed-dependence: PG = [A1, P0,1, P1,1, AM, P0,M, P1,M, n0j; G0, g2, DnD, and b] shared h set at the value of 0.124 The shape of an individual spectrum (j = 1...M) is determined by the following parameters: pj = [Aj, P0,j, P1,j; n0j, G0, g2, DnD, and b] Global CHI-SQUARE: Advantages of the multi-spectrum approach: It reduces significantly statistical correlation issues among free parameters; It provides strong elements for the selection of the most appropriate lineshape model. P. Amodio, L. Moretti, A. Castrillo, and L. Gianfrani, J. Chemical Physics 140, (2014) P. Amodio, M.D. De Vizia, L. Moretti, and L. Gianfrani, Phys. Rev. A 92, (2015)
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First test on acetylene
Isolated line: P(14) – 2n3+n51 band Model: pcSDHC Number of spectra: 80; Pressure range: Torr Precision: 110 ppm
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From an isolated line to a doublet
2% of difference in self-broadening coefficient R(15) n2+n3+n5 0.135 cm-1/atm P(17) 2n2+n4+n5 0.132 cm-1/atm dEi,i’ and dEf,f’ 80 cm-1 │no,1-no,2 │≈ 1980 MHz >>Dncoll= 5-45 MHz
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Example of fit rms value = 0.25 mV
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Absolute frequency and pressure shift measurements
d1 = (4) MHz/Torr d2 = (7) MHz/Torr n0,1 = (3) kHz n0,2 = (3) kHz R(15) P(17)
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Extensive measurements and spectral analysis in progress!
Current status <T> = (6) K Set point = (3) K Total number of spectra ≈ 1000 Extensive measurements and spectral analysis in progress!
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Uncertainty budget of the 2013 determination
Component Type A Type B Statistical uncertainty 15.7 × 10-6 Frequency scale < 2 × 10-6 Line-center frequency 0.278 × 10-6 Line emission width and FM broadening 10 × 10-6 Optical saturation effects Negligible Detector nonlinearity AM modulation effects Relativistic effects Finite detection bandwidth < 10-9 Cell’s temperature 3.7 × 10-8 1.1 × 10-6 Hyperfine structure effects (Ortho transitions) < 10-6 Line shape model 14.9 × 10-6 Combined relative uncertainty = 24 × 10-6 A. Castrillo, L. Moretti, E. Fasci, M.D. De Vizia, G. Casa and L. Gianfrani, J. Molecular Spectroscopy 300, (2014).
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Remarks and future prespectives
The 3rd generation is in progress; The new DBT spectrometer sounds very powerful; The multi-spectrum fitting approach helps a lot; Expected global uncertainty smaller than 10 ppm; DBT will be soon tested at other fixed points; Keep working on the development of DBT as a very effective link between temperature and frequency; The same apparatus can be used for precision determinations of spectroscopic parameters and for measuring absolute frequencies at 1.4 mm.
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Molecules & Precision Measurements Research Group
Permanent staff: Antonio Castrillo, Luigi Moretti, and Livio Gianfrani Post-doctorate fellows: Eugenio Fasci, Maria Domenica De Vizia PhD student: Tanya Odintsova Main collaborations:
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Extra slides
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Possible non-linearity
Our typical photocurrent range: 20 →50 mA Nonlinearity smaller than 2 ppm Type B uncertainty on the Doppler width of: 0.2 ppm for DI/I ≈ 60% 1 ppm for DI/I ≈ 80%
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Atomic or molecular sample?
Alkali metal and alkali-earth metal atoms: Advantages Disadvantages Very low-pressures (10-5 – 10-4 Pa) Hyperfine structure No perturbation from collisions Pertubation from magnetic fields Optical pumping effects Different species for different temperatures Natural broadening Molecules: Advantages Disadvantages No perturbation from magnetic fields (diamagnetic molecules) Collisional line-shape perturbations Negligible non-linear effects Hyperfine structure (depending on the molecule) One species for a broad T-range Negligible natural broadening (vibration-rotation transitions) The main difference between a low-pressure atomic vapor system, like Rb, and molecular experiments is that atomic motion is effusive, so collisions are extremely rare. An advantage of moving to this effusive regime is the avoidance of pressure-induced systematic changes, such as collisional line-shape perturbations. Very weak probes to avoid optical pumping! Zeeman effects in ordinary molecules (singlet sigma states); a magnetic moment is associated to nuclear rotation and it is a factor of 1000 smaller than that of electrons.
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Atoms vs temperatures L = 10 cm
Operating along the metal’s sublimation curve means that the equilibrium vapor density increases exponentially with temperature, and thus the absorption depth is much more sensitive to temperature for atomic systems than in the molecular experiments. The range of temperatures over which we can use Rb as a thermometric substance is comparatively smaller than for molecules. Gar-Wing Truong et al., PHYSICAL REVIEW A 83, (2011)
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Strontium vapor pressure
At T= K P ≈ Pa At T= K P ≈ Pa ≈ 2 atoms/m3 Hg: at 20°C its vapor pressure is 0.16 Pa; at 100°C the vapor pressure is 36 Pa.
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What about relativistic effects?
H2O In a recent work…..relativistic formulation of the Voigt Profile. Omega_0 = frequency in the moving frame Omega = frequency in the laboratory frame Formula valid for an emitted photon V ≈ 500 m/s → b ≈ → g -1 ≈ 10-12
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Primary methods of gas thermometry
Physical law Best implementation Global uncertainty (ppm) Acoustic gas thermometry (AGT) NPL - Teddington UK 0.7 Dielectric constant gas thermometry (DCGT) PTB - Berlin Germany 4.0 Johnson noise thermometry (JNT) NIST – Boulder US 3.5 Doppler broadening thermometry (DBT) UniNA2 – Caserta Italy 24 What you need for measuring kb is a primary gas thermometer!! The most consolidated and accurate method, so far, is.... Gamma is the low density limit of the heat capacity ratio; Simultaneous measurements of acoustic and microwave resonances in a quasi-spherical resonator This method has already benefited from 40 years of technical developments. JNT implementation based upon ac-Josephson voltage synthesizers Newest method !!! Livio Gianfrni
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Isolated lines Low pressure regime
The sum of separated line profiles is possible. It works in the low pressure regime!! At low pressures, lines can be considered as isolated each other. The additive rule holds!
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Line mixing Requirements: Rosenkranz model (1975): n0,2 n0,1
The resulting spectrum can not be considered as a superposition of two isolated lines due the population exchanges. Requirements: di,i’ and df,f’ ≤ kBT │no,1-no,2 │≈ Dncoll If the two lines are close enough and the levels are collisionally coupled. Two different channels are possible to bring a molecule from the state i to the state f: a direct optical transfer and an indirect path involving inelastic collisions. Yi first order line mixing coefficients Wij depending on collision dynamics, intermolecular potential, velocity distribution Rosenkranz model (1975): Wij are the off-diagonal elements of the rotational relaxation matrix related to inelastic collisions
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