7. Introduction to rotational spectroscopy

Transcription

1 7 Introdction to rotationa spectroscopy See Bernath here For Incredibe detai, go to Herzberg, Spectra of Diatomic Moeces (in a the ibraries) Diatomic moeces r m 1 m r 1 r COM center of mass (or inertia): m 1 r 1 = m r I: moment of inertia = mr mr m r rm rm r ; r 1 1 m1m m1m i i 11 rmm 1 rmm 1 mm 1 I r r, where is the redced mass (for rotation, ( m1m) ( m1m) m1m in this case) Note the common case of hydrogenic redced mass: For HX moeces (X=F, C, Br, I, O, ) moeces, the redced mass for rotation is sma, simiar to the mass of the hydrogen atom In genera, for axis i of a three-dimensiona object, from the axis and dm i the differentia in mass I rdm, where r is the distance i i i The moment of inertia tensor has diagona eements, eg, I xx mi( yi zi ) and offdiagona eements, eg, I I mx y The principa axes are the choice of i axes xy yx i i i i that diagonaize the moment of inertia tensor for an object: I I I 0 The rotationa energy L I, or, nits of xy xz yz E I /, cassicay, and the rotationa angar momentm L E However, qantm mechanics says that L mst be qantized in I h, where h is Panck s constant: nh ( nh / ) L ; E I The formation of the qantm mechanica Hamitonian operator (tota energy operator for a conservative system) for angar momentm probems (deveoped for atomic spectra, bt jst as appicabe to rotationa angar momentm) and its sotions are presented in Bernath, Section 5

2 Jˆ ( 1) The Hamitonian expression for rotation abot one axis is J J E I I h E BJ( J 1), where B B is the rotationa constant Noninear I 8 I poyatomic moeces have or rotationa constants, depending on the moece s symmetry (more ater) Expressing B in cm -1 : and B (H 5 C) 1059 B (HF) 094 B (OH) 1887 B (O ) 1446 not hydrogenic The spectroscopic conseqences of having sma, hydrogenic, moments of inertia, with corresponding arge rotationa constants, are that sch moeces (or the corresponding rotationa degrees of freedom in poyatomic moeces having one or more hydrogenic redced masses abot rotationa axes) 1 have smaer rotationa partition fnctions, and ths distribte the rotationa spectra into fewer, stronger transitions; have Botzmann popation distribtions that extend to higher energies This corresponds to the far infrared or sbmiimeter (or terahertz) region, where intensities are stronger becase of the σ factor in the backbody expression These are NOT hydrides In hydrides, the hydrogen atom has a forma negative charge (eg, NaH, LiH, LiAH 4 ) ˆ J E has the set of sotions (cos ) im P, JM e which are spherica harmonics, I where J = 0, 1,, and M J There are J + 1 states of same energy for each J The degeneracy (in the absence of m and E ) = J + 1: J g J 5 1 M = -1, 0, M = 0 This shows why we need to modify or statement from Einstein Bnm Bmn to gb n nm gb m mn The microscopic probabiities are the same bt the smmed qantities are not Detaied baance reqires the above reationship, eg, 1 B01 B10 Degeneracy in genera is given by F + 1, where F is the tota angar momentm of the state (incding rotation, orbita, spin, eectronic, ncear) Reminders: Backbody radiancy Rd e hc d hc / kt 1

3 8 hc The radiation density ( ) hc / kt e 1 erg cm- /cm -1 = erg cm - hc Since ( ) F( ) (F = fx), F( ) ( )/ h in cm - s -1 c Intensities (withot qantm mechanics) Detais on intensities are given in severa paces in Bernath (see Preface to the nd Edition) Aso, I am very fond of the hard-nosed derivations in Penner (especiay Chapter 7) r + Dipoe moment: x exdx x, y, z (say a sm over charges) The nit of is the Debye (D) 1 es-å One Debye: COM 1 1Å +1 (1 Debye) = 10-6 erg cm The intensity of a transition is proportiona to : 1 d 0 (direction cosine matrix eement) For ftre reference, (1 Bohr magneton) = erg cm 0 (HC) = 11 D 0 (OH) = 17 D 0 (O ) = 0, bt B = Bohr magnetons ( means amost exacty, with tiny corrections, incding a reativistic correction) For or simpe diatomic moeces, J 1 J 1 JJ 1 0 ; JJ 1 0 J 1 J The sa seection res for rotationa transitions of poar diatomics are J 1 (from the symmetry of ab) Then, the rotationa ines occr at energies of E = B, 4B, 6B

4 1B 6B 1 B J = 0 E = 0 Transition dipoe moment: Rmn mnd It is independent of degeneracy as defined: Rmn J J 1 (J 1) J J 1 (J ) Then, qantm mechanics gives: Amn Rmn / gm mn, where m is the pper state h h Bmn R /, mn gm mn Bnm R / mn gn nm The standard definition of intensity, S Treating the indced absorption and emission (for now): N N ( f B f B ) ( I, ) in cm - s -1, or c 0 0 N N f (1 e ) ( I, ) 8 c N N0f (1 e ) F( I0, ) N 8 c f (1 e ) S NF 0 The above version is in absorption form The entirey eqivaent emission form can be derived from spontaneos emission sing the Einstein A coefficient N N o f A 8 c After normaizing to backbody sterradiancy, the rest is S ( e 1) f ce ce 8 ( e e ) In genera: S R Q

5 f is the fraction of moeces in the ower state and f in the pper state S is in cm (my preference), or in cm -1 /(moeces cm - ), which is more standard sage This is becase mtipying S by the comn density of moeces, in moeces cm -, gives the ine s eqivaent width (W) in cm -1 (ignoring the Beer-Lambert aw, for the moment), W (cm -1 ) = S (cm) n (cm - ), where n is the comn density area in cm -1 Introdction to ine shapes Given a ineshape ( ) in cm, S( ) the cross section ( ) in cm ( ) n (cm - ) ( ) S (cm) (cm) n (cm - ) (jst as before!) Then, we get the acta eqivaent width after appying the Beer-Lambert aw and ( ) integrating over the whoe ine W (1 e ) d, for absorption What is it for emission?