Lecture 5: Quantitative Emission/Absorption

Transcription

1 Lecture 5: Quanttatve Emsson/bsorpton. Eqn. of radatve transfer / Beer s Law o (ν) Gas (ν). Ensten theory of radaton 3. pectral absorpton coeffcent Collmated ν L 4. Radatve lfetme 5. Lne strenths Lht transmsson throuh a slab of as Beer s Law () = o () (- L)

2 . Eqn. of radatve transfer / Beer s Law Enery balance ν hn sample of emttn/absorbn as ν +d ν Collmated ν dx absorpton reflecton scattern transmsson spectral absorptvty, or absorbance dx d no unts spectral transmssvty pectral absorpton coeffcent (the d / dx - fracton of ncdent lht ν over cm frequency rane ν ν+dν whch s absorbed per unt lenth dx W/cm spectral ntensty n or cm nterate over ν W/cm d lso apples ν for total W/cm Hz

3 . Eqn. of radatve transfer / Beer s Law Enery balance ν hn sample of emttn/absorbn as ν +d ν Collmated ν dx Consder emsson from the as slab em pectral no unts emssvty em no unts Blacbody spectral radancy Krchhoff s Law emssvty equals absorptvty emsson absorpton d emsson absorpton Dfferental form of the eqn. of radatve transfer d dx 3

4 . Eqn. of radatve transfer / Beer s Law Enery balance Dfferental form of the eqn. of radatve transfer d dx nterate over L nterated form of the eqn. of radatve transfer L L Optcal depth o (ν) Gas (ν) Collmated ν L Consder two nterestn cases: Emsson, bsorpton 4

5 . Eqn. of radatve transfer / Beer s Law Case : Emsson erment (no external radaton source) Emsson types: pectral radancy: pectral emssvty: L L L, L nterate over ν L Ld L L L L nle/multple lne nle/multple bands Contnuum Optcal depth: Optcally thc: Optcally thn: L L 4 L, L, Note: d d L 4 d -4 tefan-boltzmann constant er cm s K L L L, ε L 5

6 . Eqn. of radatve transfer / Beer s Law Case : bsorpton erment L L L L = v (- v ) = absorbance Beer s Law / Beer-Lambert Law lternate form: L Observatons:. he same equaton would apply to the transmsson of a pulse of laser exctaton, wth enery E [/cm /cm - ],.e., =E /E. he fundamental parameter controlln absorpton over lenth L s the spectral absorpton coeffcent,. How s related to fundamental molecular parameters? 6

7 . Ensten theory of radaton mplfed theory (Mlne heory) tate ranston probablty/s of process per atom n state or pontaneous Emsson nduced bsorpton nduced Emsson B ρ(ν) B ρ(ν) E Enery E h tate otal transton rate [molec/s] N N B ρ(ν) N B ρ(ν) Ensten coeffcents of radaton B ρ(ν) B ρ(ν) he probablty/s that a molecule n state osed to radaton of spectral densty ρ(ν) [/(cm 3 Hz)] wll absorb a quantum hν and pass to state. he Ensten B-coeffcent thus carres unts of cm 3 Hz( s). he probablty/s that a molecule n state osed to radaton of spectral densty ρ(ν) wll emt a quantum hν and pass to state. he probablty/s of spontaneous transfer from state to wth release of photon of enery hν (wthout reard to the presence of ρ(ν)). 7

8 . Ensten theory of radaton mplfed theory (Mlne heory) Equlbrum Detaled balance N N B N B eq rad molec/s entern state N B h N B eq Planc s blacbody dstrbuton B rad. equl. 3 8h 3 B c statstcal equl h / c B eq / h / molec/s leavn state Note: for collmated lht eq 3 - eq np h /cm s - n h cw/cm s B B / B 3 8h / c h / Radatve lfetme Where s the ln to? p h / 3 / c 8

9 . Ensten theory of radaton Fnd for a structureless absorpton lne of wdth δν Recall Beer s Law: L d dx ν δν Gas ν δν+(d ν )δν bsorbed power Optcally thn lmt P abs ncdent power over fracton absorbed W/cm L W/cm s - s - dx P dx Now, let s fnd fracton absorbed usn Ensten coeffcents abs dx Pabs fracton absorbed dx 9

10 . Ensten theory of radaton Fnd ν for a structureless absorpton lne of wdth δν ν δν Enery balance d nduced emsson spontaneous emsson nduced absorpton nduced emsson = nduced absorpton = for collmated lht ndx B h molec/cm ndx B h molec/cm n state n state prob/s of emsson prob/s of emsson d n B n B d dx enery per photon enery per photon h cm n B h / c h n B c h c dx n B Gas dx Recall: ν δν+(d ν )δν nce ν s a functon of δν, we conclude depends on lnewdths + hence shape; next, repeat wth realstc lneshape / c

11 Where are we headed next? mproved Ensten heory, Radatve Lfetme, Lne trenth 3. pectral absorpton coeffcent wth proper lneshape 4. Radatve lfetme 5. Lne strenths emperature dependence = absorbance X HO =. L = 5 cm Band strenth Wavenumber [cm - ] Water vapor absorpton spectrum smulated from HRN

12 . pectral absorpton coeffcent Eqn. of radatve transfer Recall: o Gas (ν) (ν) Collmated ν L L L ndependent of lneshape! d, cm dx For structureless absorpton lne of wdth δν (Hz), we found h n B h cm / c Note n, B, and /δ Next: use realstc lneshape to replace /δ

13 . pectral absorpton coeffcent Repeat dervaton of ν usn an mproved lneshape model tructureless absorpton lne of wdth δν Replace wth realstc lneshape typcal absorpton lne wth typcal structure 3

14 3. pectral absorpton coeffcent ν Recall Beer s Law: Defne: Note: d, max p,max d lne d L L Normalzed lneshape functon verae wdth cmor s, d lne nverse frequency ln Relevant transton probabltes have the same spectral dependence (shape) as and () nd we can antcpate that /v wll be replaced by n v equaton 4

15 3. pectral absorpton coeffcent Modfed model tate ranston probablty/s/molec (n level or ) for rane ν to ν +dν pontaneous Emsson nduced bsorpton nduced Emsson φ(ν)dν B φ(ν)dνρ(ν) B φ(ν)dνρ(ν) E Enery E h tate Ensten coeffcents of radaton φ(ν)dν B φ(ν)dνρ(ν) he probablty/s of a molecule underon spontaneous emsson, n the rane ν ν+dν. [Note that the nteral of ths quantty over the rane of allowed s just [s - ],.e., d.] he probablty/s of a molecule underon a transton from, n the rane ν ν+dν. B φ(ν)dνρ(ν) Recall: he probablty/s of a molecule underon a transton from, n the rane ν ν+dν. / c 5

16 3. pectral absorpton coeffcent Enery balance ν dν Gas ν dν+(d ν )dν d d [W/cm n ν to ν+dν] dx emsson n d absorpton n #/cc n dx molec/cm B d / c h ndx B d c h / prob/s molec for d d dx h n c d enery/photon B nb h nb h / c nterated absorpton / Lne strenth d cm s - - lne h c n B h / 6

17 3. pectral absorpton coeffcent Lne strenth alternate forms Lne strenth does not depend on lneshape, but s a functon of n,, B Oscllator strenth n 8 e n mec f f where h / cm s h / cm s.65cm Hzn f h actual /, p n 7 cm / P, n =n=.7x 9 cm -3, ( hν /)<<, actual h /, classcal e, classcal n mec e, mec f f.65cm Hz f 7

18 3. pectral absorpton coeffcent mportant observatons. From the ornal defnton of ν and we have. When h / cm hν/= 4 hν/=x 4 hν/=4x 4 K h / Hz as s common for electronc state transtons e n mec.65cm Hz n 8 f f n f / f.5 cm Radatve lfetme of the transton / 8

19 3. pectral absorpton coeffcent Example: Resonance ranston Resonance transton one that couples the round state to the frst excted state lower (L) upper (U) Electronc transton of a sodum atom Na3 / 3 P/ 5, 589nm 5.89 cm Conventons: atoms: (L-U) molecules:(u L), arrow denotes absorpton or emsson f j : denotes ntal state, j denotes fnal 9 f 589nm.5 cm 5.4 s 9 8 Measured: 6. s.6 s f.35 tron atomc transton: snle electron Much smaller for molecular transtons: ~

20 3. pectral absorpton coeffcent Oscllator strenth ranstons f λ [nm] 3 / 3 P / / 3 P 3/ P Oscllator strenths of selected sodum transtons Molecule CO OH v' v Electronc ranston Band center [cm - ] x x x -6 Σ Π 36.x -3 CN Π Σ 97.x - bsorpton oscllator strenths of selected vbratonal and vbronc bands of a few molecules f

21 4. Radatve lfetme u l dn dt Radatve and non-radatve lfetmes u Rate equaton for radatve decay n Upper level u u u l l spontaneous emsson only Lower level l Radatve lfetme (zero-pressure lfetme) Rate equaton for non-radatve decay dnu nu nrnu dt nr nr Rate parameter [s - ] Non-radatve decay tme, depends on the transton consdered and on the surroundn molecules r n u t n t ul l u ul l ntal number densty multaneous presence of radatve and non-radatve transtons dnu nu nu nu, r nr Lfetme of level u dt r nr

22 5. Lne strenths lternate forms Lne strenths cm cm cm cm ccm/s s d cm / ccm /s cm / atm cm / Patm cm / cm cm / cd s Number densty of absorbn speces n state n c h / P atm 8 * cm / moleccm nmolec/ cc cm / atm deal as law * cm / moleccm 35dynes/ cm atm cm / atm er/k * 7.34 cm =96K HRN unt P * atm cm atm

23 5. Lne strenths lternate forms Beer s Law,, L n L PL PL n = number densty of the absorbn speces [molecules/cm 3 ] σ ν = absorpton cross-secton [cm /molec] = lne strenth [cm - atm - ] or [cm - sec - /atm] β ν = frequency-dependent absorpton coeffcent [cm - /atm] P = partal pressure of speces [atm] φ ν = frequency-dependent lneshape functon [cm] or [s] v = v L = absorbance Common to use atmosphere and wavenumber unts n R cm /atm d cm s cp atm 8.8 c 8 n P 3 P n atm f h / h / / P absorpton coeffcent per atmosphere of pressure 3

24 emperature dependence 4 5. Lne strenths,, hc hc hce Q Q Lne strenth n unts of [cm - atm - ] Lne strenth n unts of [cm - /(molecule cm - ],, * * hc hc hce Q Q * *

25 Band strenth 5 5. Lne strenths Example: Heteronuclear Datomc Band trenth ' ' ' v v R P band band lnes 6 6 / 8.3 / 8.3 n n c P n n c R / n n c h n n c R R / ' /.3 8 ' 6 ' atm, P n P Based on normalzed Hönl-London factor

26 5. Lne strenths Band strenth band lnes band Example: Heteronuclear Datomc Band trenth 6.3 c 8 Band strenth of CO: 3. 8 CO 73K 8cm /atm 5cm 36s s.8s Compare wth prevous example of τ Na 6ns R transtons have much lower values of and loner radatve lfetme than UV/Vsble transtons due to ther smaller chanes n dpole moment 6

27 Next: pectral Lneshapes Doppler, Natural, Collsonal and tar Broadenn Vot Profles