Spectral line-shape model to replace the Voigtprofile in spectroscopic databases

D. Lisak1, N.H. Ngo2, H. Tran2, J.-M. Hartmann2

1 Institute of Physics, Faculty of Physics, Astronomy and Informatics, NicolausCopernicus University, Grudziadzka 5, 87-100 Torun, Poland,

2 Laboratoire Interuniversitaire des Systèmes Atmosphériques, UMR CNRS 7583,Université Paris Est Créteil (UPEC), Université Paris Diderot (UPD), Institut Pierre-Simon Laplace, 94010 Créteil Cedex, France

Outline

• Semi-classical line shape models

• Recommended profile and its advantages

• Verification of models on calculated

and experimental spectra

• Conclusions

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VP

BBP

H2O 10687.36 cm-1 + SF6 26.7 kPa

13.3 kPa

53.3 kPa

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SDVP

SDBBP

SDNGP

Semi-classical line shape models

Free motion of absorber, pressure (collisional) and Doppler broadening statistically independent – Voigt profile (VP)

Dependence of collisional width and shift on absorber velocity – Speed-dependent Voigt profile (SDVP) [1]

)(xBW )(xBS

/)()(AmW xvxB , /)()(

AmS xvxB , ← speed dependence of collisional width and shift

where AmA vvx / ; vA is the absorber speed and

Amv is the most probable absorber speed.

Quadratic speed-dependence model [2]:

)2/3(1)( 2 xaxB WW and )2/3(1)( 2 xaxB SS 2 more parameters: aW, aS

or equivalent notation:

)2/3( 2

20 x and )2/3( 2

20 x

Hypergeometric speed-dependence model (absorber-perturber interaction qrrV ~)( )

2

23

22

3)22/()3(,,1),( xMxB

q

qqq

where α is perturber to absorber mass ratio, M – confluent hypergeometric function

[1] P.R. Berman, JQSRT 12, 1331 (1972)[2] F. Rohart et al., J. Chem. Phys. 101, 6475 (1994)

Semi-classical line shape models

Velocity-changing collisions (Dicke narrowing or collisional narrowing) [3]

VC – frequency of velocity-changing collisions 1 more parameter VC

Can be calculated from mass diffiusion coeffeicent, but it does not agree with values obtained from fits of simple models

soft collision model – Galatry profile (GP) [4]

diffusion motion of abasorber, should work better with light perturbers

hard collision model – Nelkin-Ghatak profile (NGP) [5] (or Rautian profile) vA after collision does not depend on vA before collision

soft and hard collisions together – Rautian-Sobelman profile (RSP) [6]

ε - vc-collision hardness parameter or S, H 1 more parameter

More complex models of vc:

Keilson-Storer model (KS) [7]

billiard-ball model – (BB) [8]

Correlations between velocity-changing and dephasing (state changing) collisions [6]

η - correlation parameter 1 more parameter η

)( iVCVC

[3] R. H. Dicke, Phys. Rev. 89, 472 (1953)[4] L. Galatry, Phys. Rev. 122, 1218 (1961)[5] M. Nelkin and A. Ghatak, Phys. Rev. 135, A4 (1964)

[6] S. G. Rautian and I. I. Sobelman, Sov. Phys. Usp. 9, 701 (1967)[7] Keilson, Storer, Quart J Appl Math 1952;10:243[8] R. Ciuryło et al., Phys. Rev. A 65, 012502 (2002).

Simplifications of profiles(by setting some parameters to zero)

Profile Parameters Parameters of CSDRSP set to 0

pCqSDRSP , , 2, 2, S, H,

pCqSDNGP , , 2, 2, H, S

pCqSDGP , , 2, 2, S, H

qSDNGP , , 2, 2, H S,

qSDGP , , 2, 2, S H,

qSDVP , , 2, 2, S S,H,

NGP , , H S, , 2, 2

GP , , S H, , 2, 2

VP , S, H, , 2, 2

q - means quadratic speed-dependenceSimilar list can be done with hypergeometric SD

– collisional width – collisional shift2 – speed-dependence of 2 – speed-dependence of S – freq. of soft v-c collisionsH – freq. of hard v-c collisions – corellation coeff.

pCqSDNGP was proposedas a new standard profile for databases

It was tested on line shapes:H2O lines perturbed by N2 (Keilson-Storer

model for velocity changes, semiclassicalspeed-depedence calculations) [1]

O2 (requantized Classical Molecular Dynamics Simulations - rCMDS) [2]

CO2 (rCMDS) [3]

H2 (velocity-changing collisions from CMDSand experimental speed-dependencesof and ) [4]

[1] N. H. Ngo, et al., J. Chem. Phys. 137, 064302 (2012)[2] J.-M. Hartmann et al. Phys.Rev. A 87, 013403 (2013)[3] J.-M. Hartmann et al. Phys.Rev. A 87, 032510 (2013)[4] H. Tran et al. JQSRT 134, 104 (2013)

pCqSDNGP can be calculated fast –only a few times slower than VP [4]

• physically based and robust to represent the experimental lineshapes of many different molecules to ~ 0.1%

• line-by-line parameters with linear dependences on the gas pressure• compatible with previously used profiles• fast computation, comparable to Voigt

Partially corellated quadratic speed-dependent Nelkin-Ghatak profile pCqSDNGP *

* A. S. Pine, JQSRT 62, 397 (1999)P. Joubert et al., JQSRT 61, 519 (1999)Ciuryło et al. JQSRT 68, 257 (2001)

Takes into account:velocity-changing collisions in the hard-collision approximationspeed-dependence of collisional broadening and shifting in the quadratic approximationcorrelations between velocity- and phase/state changing collisions

Parameters:w0 – unperturbed line center0 – collisional width0 – colisional shift2 – (or aW) speed-dependence of collisional width2 – (or aS) speed-dependence of collisional shiftvvc – frequency of velocity-changing collisionsη – correlation parameter

First order line mixing** can be easily incorporated by adding a dispersive term of the profile

** Rosenkranz P. IEEE Trans Antennas Propag 23 498 (1975)

Quadratic speed-dependence parameters

For multi-component gas (e.g. air):

Speed-dependent collisional width

where 𝑥 = 𝑣/𝑣𝑚𝐴

Speed-dependence parameter can be calculated from temperature dependence of Γ [1]

K components Speed-dependence parameter of mixture

For quadratic speed-dependence the line-shape parameters for gas mixture can be calculatedfrom parameters for individual components.So, parameters e.g. for air are still compatible with these for O2, N2, Ar

[1] D. Lisak, A. Cygan, P. Wcisło, R. Ciuryło, submtted to JQSRT (2014)

– perturber to absorber mass ratio

In the other notation:

a = 2 /0

Voigt profile vs pCqSDNGP

H2O, ν3, 404←303 line, p = 0.009 - 1.039 atm

VP: Rmax = 2.5%

pCqSDNGP: Rmax = 0.05%

VP: Rmax = 2%

pCqSDNGP: Rmax = 0.4%

VP: Rmax = 3%

pCqSDNGP: Rmax = 0.5%

VP: Rmax = 60%

pCqSDNGP: Rmax = 3%

H2 Q(1) line of 1−0 Raman band, p = 0.29 – 18.45 atm

O2 P9P9 line, p = 0.1 - 1 atm

CO2 R16 line, p = 0.01 – 0.4 atm

Improvementby at leastone order of magnitude

Computation time of pCqSDNGP

pCqSDNGP can be written in terms of two Voigt (or complex probability) functions [1]Fast algorithms for Voigt already exist, e.g. J. Humlicek JQSRT 1979;21:309–13.

Maximum relative error < 10-4 canbe obtained with computationtime < 5 x VP time (with slightlymodified Humlicek algorithm) [2]

Fortran code available in [2]

[1] Ngo NH, Lisak D, Tran H, Hartmann J-M. JQSRT 2013;129:89–100 & 2014;134:105[2] Tran H, Ngo NH, Hartmann J-M. JQSRT 2013;134:104 & 2014;134:104

H2O 3837.869 cm-1 3

404 – 303 broadened by N2

Data simulated with Keilson-Storer(KS) model for velocity changesand semi- classical calculations for speed-dependent line widths and shifts.

N.H. Ngo, D. Lisak, H. Tran, J.-M. Hartmann, JQSRT 129 (2013) 89–100

Multi-spectrum fit of 14 pressures

pressure

Quality of the fit in a large pressure range

0.2109 0.2109 0.2109

0.2052 0.2057

0.2023

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H2O 3837.869 cm-1 3 404 – 303 broadened by N2

2.6%4.1%

Collisional broadening and Dicke-narrowing parameters from multi-spectrum fit of 14 pressures

=0.55

Fitted line-shape parameters vs profile

H2O 3837.869 cm-1 3 404 – 303 broadened by N2

Relative error of line area from multi-spectrum fit atdifferent max. pressures

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pressure (bar)

pCqSDNGP

qSDNGP

qSDVP

NGP

GP

VP

Fitted line intensity vs profile & range of pressures

Experimental data from FS-CRDS at NIST: D. Lisak, J. T. Hodges, R. Ciurylo, PRA 73, 012507 (2006).

H2O 10687.362 cm-1 23 + 3

404 – 303 broadened by N2

Speed-dependent profilesgive good fit quality

Tests on experimental results

Experimental data from FS-CRDS at NIST: D. Lisak, J. T. Hodges, R. Ciurylo, PRA 73, 012507 (2006).

H2O 10687.362 cm-1 23 + 3

404 – 303 broadened by N2

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profile

0.21370.2151 0.2157

0.20620.2081

0.19530.19

0.195

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g/p

profile

3.5%

8.6%

Collisional broadening and Dicke-narrowing parameters from multi-spectrum fit of 9 pressures

Fitted line-shape parameters vs profile

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qSDNGP

qSDVP

NGP

GP

VP

Line area ratio to the pCqSDNGP from multi-spectrum fit at different pressures

Examples of good fit quality with recommended profile

H2O line @ 7222.298050 cm-1

M. D. De Vizia, A. Castrillo, E. Fasci, L. Moretti, F. Rohart, and L. Gianfrani, Phys. Rev. A85, 062512 (2012)

Examples of good fit quality with recommended profile

Oxygen (16O2) B-band R7Q8 line at 933 Pa

signal-to-noise ratio = 220000

A. Cygan, D. Lisak, S. Wójtewicz, J. Domysławska, J. T. Hodges, R. S. Trawiński, R. Ciuryło, Phys. Rev. A 85, 022508 (2012)

P. Wcisło, A. Cygan, D. Lisak, R. Ciuryło, Phys. Rev. A 88, 012517 (2013)

Line asymmetry caused by the speed-dependence of collisional shift

[1] Tennyson J, Bernath PF, Campargue A, Csaszar AG, Daumont L, Gamache RR, Hodges JT, Lisak D, Naumenko OV, Rothman LS, Tran H, Zobov NF, Buldyreva J, Boone CD, De Vizia MD, Gianfrani L, Hartmann J-M, McPheat R, Weidmann D, Murray J, Ngo NH, Polyansky OL. Recommended isolated-line profile for representing high-resolution spectroscopic transitions. Pure Appl Chem 2014 accepted.

IUPAC Task Group recommendation [1]

pCqSDHCP (pCqSDNGP) should be adopted as the appropriate line profile for high-resolution spectroscopy moving beyond the Voigt profile

For simplicity it was proposed to call this profile with fast computationalgorithm the Hartmann-Tran profile (HTP)

Conclusions

• pCqSDNGP (HTP) is a powerful model and easy to calculate (lowcomputation time)

• easy to implement into databases (4 extra parameters, comparing to Voigt profile)

• can be reduced to simpler profiles by setting some parameters to 0• very well describes H2O, O2, CO2, CO lines – fit residuals below 0.1% (+

random noise)

• More flexible and more physically justified profiles exist, but there is no fast algorithm developed for pCSDRSP

• high quality data (high SNR, wide pressure range) are needed to verifyexperimentally advantages of more complex models(SD+VC+correlations) and to provide line parameters to new databases

• Experimental parameters of new profiles should be obtained from multi-spectrum fits (big range of pressures) because of numerical correlationsbetween parameters

• Temperature dependence of new parameters needs further studies