Lecture 6: Spectral Lineshapes - Princeton University

1 Lecture 6: Spectral LineshapesA typical lineshape of line of quantitative examples 0 N ( 0)/2 ( 0) Beer s Law21. Background introductionRecall:LIo( )I( )Collimated light @ Gas kThggAnkThBnchS/exp18/exp1scm12211212111 12 sscmcm,11121 Sk LkIIIIT exp//00intensity or power @ Spectral intensity @ absorption coefficient @ , cm-1 Line strength, line k d 1, ddkkline The lineshape function A common form of S Another common form Alternate forms of , , S1231. Background introduction S12 sscmcm,11121 Sk 11cm,s, cc s,cm, c cSS/scm,cm,1112212 atm,scm,atm,/cm,atm/cm,1212212212iicPSPS S kThggAPnci/exp1atm,8122112 Partial pressure of absorber Notes: iinnkTkTnnatm,1iPn cm,atm,/atmcm,cm,2121 iPSk iiPP atm,atm, Boltzmann fractionMole fraction 321*1212cmmolec,molec/cmcm,atm,atmcm,iin SPSHITRAN database lists S* (cm/molec), usually at Tref= 296K_ How are S12and measured?

2 41. Background introduction High-resolution absorption experiments 0 T 0/lnII 0 k Area=S12 0 FWHM line dkk Area=1 Shape determined by main broadening mechanismLorentzian+ GaussianGaussianInhomogeneous (affects certain class of molecule)LorentzianHomogeneous (affects all molecules equally) Brief overview52. Types of line broadening1. Natural broadening Result of finite radiative lifetime2. Collisional/pressure broadening Finite lifetime in quantum state owing to collisions3. Doppler broadening Thermal motion4. Voigt profile Convolution of 1-3 Natural line broadening62. Types of line broadening1. Heisenberg uncertainty principle:2. In general 2/htEuu h 0u (upper level)l(lower level)uulA 1 Decay rate raduuuuthhEutuE /2/ of occupation of in timey uncertaint the, ofenergy in y uncertaintradu 2/1 lifetime limited luluN 11210 for ground state (natural broadening) Natural line broadening72.

3 Types of line broadening3. Typical values Electronic transitions: Vib-rot transitions These are typically much smaller than Dand C4. Lineshape function Lorentzian follows from fourier transform141178cm105/cm, ~s10~ cNNNu 110112cm105cm,,s16~s10~ NNu 2202/2/1 NNN Note: a)b) 2/2/12000max NN 0 N ( 0)/2 ( 0) Natural line broadening82. Types of line broadeningLineshape derivation from damped oscillator model (Ref. Demtr der) 00,0/ ,002020 xxxmkxxx 2/122004/sin2/cos2/exp tttxtx ttxtx00cos2/exp Small damping ( << 0)Amplitude of x(t) decrease frequency of emitted radiation is no longer monochromatictxx0tx0xexp( t/2)F 0|A( )|2 0|A( )|2 dtiAtxexp2210 2/12/18exp21000 iixdttitxA 0*/,IILAAI 22002/2/1 L = Damping ratio n in units of s-1 Collision broadening92. Types of line broadening1. Also lifetime limited time set by collision time optical collision diameter of BA B Effective areaOptical cross-section2 ABA 2 BAAB v ABABABAkTcnZ 8A all with B single a of scollision/ # 2#/ccBABAAB mmmm AABABAAABABABkTXPkTnZ 8822 For a mixture, atm, ,62PP Collision broadening102.

4 Types of line broadening1. Also lifetime limited time set by collision time interval2. Lineshape function Lorentzian3. Crude approximation AABABAAABABABkTXPkTnZ 8822 Since AABABAB lowercolluppercollC-AkTXPZ /atms,262,, ,1121s, ccCC/atm/s,2atm/cm,2/s,cm,1111 Notes: AAACXP 2atm,s,12 A= colli. halfwidth, , FWHM per atm. pressure 2202/2/1 CCcoll nTT/30022300 cm-1/atm for hard sphere Collision broadening112. Types of line broadeningExample: Pressure broadening of COR(9) line of CO s 2ndovertone, 50ppm in Air, 300K, population: 77% N2, 20% O2, 2% H2O (85% humidity) 380ppm CO2 Species, AMole Fraction, XA2 CO-A(300K) cm-1 OCOOCOCOCOCOCOCONOHOHNCONCXXXXXP AAACXP 2atm,cm,1with 2 Ain cm-1/atm Collision broadening122. Types of line broadeningSome collisional broadening coefficients 2 [cm-1/atm] in Ar and N2at 300 KSpeciesWavelength [nm] collisional broadening coefficients 2 [cm-1/atm] in Ar and N2at 2000 KSpecies00 Wavelength [nm] Doppler broadening132.

5 Types of line broadening1. Moving molecules see different frequency (Doppler shift)2. Gaussian velocity distribution function (leads to Gaussian ) cuactapp/1 //ucuactactapp molec. velocity along beam path 202ln2exp2ln2 DD 0 g/mole of emitter/absorber kTmUkTmUfxx2exp222/1 Aside:Maxwellian velocity distribution Stark broadening Important in charged gases, , plasmas. Coulomb forces perturb energy levels Types of instrument broadening Instruments have insufficient resolution Powerful lasers can perturb populations away from equilibrium (saturation effect) Transit-time broadening Another type of lifetime-limited broadening is transit-time broadening142. Types of line broadeningReference: Demtr der time Transitfor apparent broadening of an abs. lineDLaser beamGasV Examples152. Types of line broadening1stExample:T = 300K, M = 30g/mole, P = 1atmCDCNDD ~s10~ ~ Electronic transition( =600nm, =5x1014s-1) Vib-rot transition( =6 m, =5x1013s-1)CDCD ~ ~ ~2ndExample:T = 2700K, M = 30g/mole, P = ~ ~ CD Electronic transition( =600nm, =5x1014s-1) Vib-rot transition( =6 m, =5x1013s-1) ~ ~ CD ~T1/2~T-1/2 IR=10 vis Conclusions Doppler broadening most significant at: Collision broadening most significant at: Many conditions require consideration of botheffects162.

6 Types of line broadeningLow P, high T, small High P, low T, large Together Voigt profile!3. Voigt types of broadening Collision broadening Doppler mechanisms Dw /2ln20 00, DDVkkwaV a=0(pure Doppler)a=1a= function Collision broadening review Lorentzian form lifetime limited Typical value of 2 A~ (or ) A type of Homogenous broadening , , same for all molecules of absorbing species183. Voigt Dominant types of broadening 2202/2/1 CCC mixture of atm,2s,1 PXAAAC mole fraction of Acoll. width/atm for A as coll. partner,T/1 ,2261 if ABis constant Doppler broadening review Gaussian form Typical value This is a type of Inhomogenous broadening , , depends on specific velocity class of molecule193. Voigt Dominant types of ,0702/121 FWHMg/mole of absorber/emitter 202ln2exp2ln20 DD ,nm600 MD Comparison of Dand C(for same (FWHM)) Some exceptions/improved models Collision narrowing (low-pressure phenomenon) Galatry profiles, others, with additional parameters Stark broadening Plasma phenomenon203.

7 Voigt profiles Dominant types of broadening Both have same area (unity) Peak heightsfor CCcollDDDopp DC Gaussian: higher near peak Lorentzian: higher in wingsReady to combine Doppler & collision broadening; done via Voigt profile Physical Voigt profileThe physical argument employed in establishing the Voigt profile is that the effects of Doppler & collision broadening are decoupled. Thus we argue that every point on a collision-broadened lineshape is further broadened by Doppler : dDDCCV22022ln2exp2ln22/2/1 duuuCDCDV * waVywadyyaDDV,exp2ln22220 the Voigt function (V 1) out integrated /2ln2/2ln2/2ln/2ln0 DDDCDNCywa Voigt profile Notes:1., so waVDV,0 waVkk,0 Spec. abs. coeff., the line-center spec. abs. coeff. for Doppler broadening 0 Dk aaaaaVDVerfcexperfcexp0,2002 Recall: Sk Dw /2ln20 00, DDVkkwaV a=0(pure Doppler)a=1a= :a=2: aaaa waVywadyyaDDV,exp2ln22220 the Voigt function (V 1) Voigt Voigt profile Voigt profile Given: T, M, 0, P, , or 2 Desire: ( ) : Dand D( 0) : w, enter table (for a) and for 0(and hence ) for that : ( ) vs 0 DCa /2ln 00// DDkk Procedure Refinements Galatry profiles (collision narrowing) Berman profiles (speed-dependent broadening) Voigt profile Given: T, M, 0, P, , or 2 Desire: ( ) : Dand D( 0) : w, enter table (for a) and for 0(and hence ) for that : ( ) vs 0 DCa /2ln 00// DDkk Pressure shift of absorption lines Doppler shift263.

8 Voigt Line shifting mechanisms Interaction between two collision partners can have a perturbing effect on the intermolecular potential of the molecule differences in the energy level spacings pressure shift AAASXP MAATTTT 00 cm-1/atm Notes:1. While 2 >0, can be + or 2. , average values for IR H2O spectra: = , M= ( 0)u = |v|cos v cu/0 0k Abs. line for static sample = shift in frequency required to excite this transition! Species concentration and pressure Temperature FWHM of lineshape gives T in Doppler-limited applications Two-line technique with non-negligible pressure broadening274. Uses of quantitative lineshape measurements Integrated absorbance area LPXSdAjii Line strength of the transitionPressureSpecies mole fractionPathlengthLXSAPPLSAX jiiiij 0"2"120102111exp,,,,TTEEkhcTSTSTSTSR 0"2"10102"2"1lnlnTEEkhcTSTSREEkhcT T sensitivity:Large E" for higher sensitivity;Absorbance: < < Tradeoff between acceptable absorbance and T sensitivity.

9 100/%12"2"1 TEEkhcKdTdRR Examples284. Uses of quantitative lineshape measurements 1stExample: Spectrally resolved absorption of sodium (Na) in a heated cell = 589nm, T = 1600K, P = 1atm1)Find2)Find3)Find Pi 0000/ln/1 IILk PPC 117cm16978cm10589 12 D DCa Interpolate Voigt table , , VwaV DD VD Solve for Piusing 0 SkPi What is PNa?Could also have solved for T from lineshape data Examples294. Uses of quantitative lineshape measurements 2ndExample: Atomic H velocityLIF (Laser Induced Fluorescence) in an arcjet thruster is used to measure the Doppler shift of atomic hydrogen at shift: = corresponding velocity component is cuUse line position to infer velocityIncident laser light, variable uLIF detectorSupersonic arcjet exhaust CW laser strategies for multi-parameter measurements of high-speed flows containing NO305. Working examples - 1 Schematic for NO LIF experiments CW laser strategies for multi-parameter measurements of high-speed flows containing NO315.

10 Working examples - 1 TDL mass flux sensor Full-scale aero-engine inlet 325. Working examples - 2 TDL mass flux sensor Sensor tests in Pratt and Whitney engine inlet 335. Working examples - 2 Bellmouth installed on inlet of commercial engine (Airbus 318) Sensor hardware remotely operated in control room TDL beams mounted in engine bellmouth TDL mass flux sensor P & W mass flux versus TDL sensor measurements 345. Working examples - 2 TDL data agrees well ( in V and in ) w/ test stand instrumentation Flow model employed to account for non-uniformities Success in non-uniform flow suggest other potential applicationsNext: Electronic Spectra of Diatomics Term Symbols, Molecular Models Rigid Rotor, Symmetric Top Hund s Cases Quantitative Absorptio