1. 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

2. 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

3. 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

4. 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

5. The roadmap towards the new SI

6. 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.

7. 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!

8. 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)

9. 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

10. 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:

11. 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

12. 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!

13. Overall uncertainty:
~90 ppm

14. The 3rd generation experiment
l = 1.4 mm

15. 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)

19. The relative frequency stability
Flicker phase noise 1/tau
White frequency noise 1/radq(tau)

20. 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.

21. Temperature stability

22. Refined line shape model
The lineshape problem n
Refined line shape model
Fitting
code
Doppler
width

23. 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

24. C2H2 spectra and Voigt fits
This slide demonstrates the failure of the Voigt model

25. The failure of the Voigt model
<T> = (4) K
Set point: K
Negative shift due to the occurrence of line-narrowing effects!

26. A realistic lineshape model
Thermal
motion
Dephasing
collisions
pcSDHCP
Hard
collision
HTP
Dicke
effect
Speed
dependence

27. 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).

28. h = (30)

29. 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)

30. First test on acetylene
Isolated line:
P(14) – 2n3+n51 band
Model: pcSDHC
Number of spectra: 80; Pressure range: Torr
Precision: 110 ppm

31. 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

32. Example of fit
rms value = 0.25 mV

33. 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)

34. 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!

35. 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).

36. 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.

37. 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:

38. Extra slides

39. 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%

40. 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.

41. 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)

42. 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.

43. 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

44. 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

45. 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!

46. 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