5. Molecular rotation 5.1 Moments of inertia
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1 5. Moleulr rottion 5. Moments of inerti The key moleulr prmeter needed to desribe moleulr rottions is the moment of inerti, I, of the moleule defined s I m i r i i Definition of moment of inerti. In this moleule there re three identil toms tthed to the B tom nd three different but mutully identil toms tthed to the C tom. In this exmple, the enter of mss lies on n xis pssing through the B nd C tom, nd the perpendiulr distnes re mesured from the xis. 5. Rottionl properties The rottionl properties of ny moleule n be expressed in terms of the moment of inerti bout three perpendiulr xes set in the moleule. For liner moleules the moment of inerti round the internuler xis is zero. (why??) An symmetri rotor hs three different moments of inerti; ll three rottion xes oinide t the entre of mss of the moleule.
2 5.3 Rigid rotors We shll suppose initilly tht moleules re rigid rotors (i.e., do not distort under the stress of rottion). There re four types then: hve one moment of inerti equl to zero (CO, HCl, HCCH) (ditomis: I=mR ) hve three equl moments of inerti (CH 4, SF 6 ) hve two equl moments of inerti (NH 3, CH 3 Cl) hve three different moments of inerti (H O, CH 3 OH) 5.4 Rottionl energy levels The rottionl energy levels of rigid rotor my be obtined by solving the pproprite SE. However, there is short ut to the ext expressions: The lssil energy of body rotting bout n xis is E I A body free to rotte bout three xes hs n energy E I b I b Angulr momentum bout the xis is I I, therefore E I b I b I
3 5.5 Spheril rotors When ll three moments of inerti re equl to some vlue, I, (CH 4, SF 6 ), the lssil expression for energy is E b I I where is the mgnitude of the ngulr momentum. The quntum expression is obtined by ( ) =0,,, therefore E ( ) I =0,,, 5.6 The rottionl onstnt B Definition of the rottionl onstnt B B 4I E ( ) hb =0,,, rottionl term ( wvenumber): F( ) B ( ) The rottionl energy levels of liner or spheril rotor. Note tht the energy seprtion between neighboring levels inreses s inreses. 3
4 5.7 Degeneries The ngulr momentum of the moleule hs omponent on n externl, lbortory-fixed xis, M whih is quntized: M 0,,..., The signifine of the quntum number M. () When M is lose to its mximum vlue,, most of the moleulr rottion is round the lbortory z-xis. (b) n intermedite vlue of M. () When M =0 the moleule hs no ngulr momentum bout the z-xis. 5.8 Centrifugl distortion The effet of rottion on moleule. The entrifugl fore rising from rottion distorts the moleule, opening out bond ngles nd strething bonds slightly. The effet is to inrese the moment of inerti of the moleules nd hene to derese its rottionl onstnt. The effet is tken into ount empirilly: F ( ) B ( ) D ( D : entrifugl distortion onstnt ) 4
5 5.9 Rottionl trnsitions Typil vlues of B for smll moleules: 0. 0 m -. Therefore, rottionl trnsitions lie in the mirowve region of the spetrum! A moleule must hve permnent eletri dipole moment (it must be polr) for the observtion of pure rottionl spetrum. Ditomi homonuler nd symmetri liner moleules s well s spheril rotors re normlly intive Rottionl seletion rules For liner moleule the trnsition moment vnishes unless: M 0, When photon is bsorbed by moleule, the ngulr momentum of the ombined system is onserved. If the moleule is rotting in the sme sense s the spin of the inoming photon, then inreses by (bsorption). 5
6 5.0 The pperne of rottionl spetr When seletion rules re pplied for rigid symmetri or liner rotor llowed wvenumbers for + bsorptions re: ( ) B( ) =0,,, The rottionl energy levels of liner rotor, the trnsitions llowed by the seletion rule D=±, nd typil pure rottionl bsorption spetrum. Mesurement of n s gives B nd hene bond lengths in se of ditomi moleules. 5. The intensities of spetrl lines The Boltzmnn distribution implies tht the popultion deys exponentilly with : N Ng e E kt nd if the degenery g of level is + (liner rotor): N ( ) e hb ( ) kt For typil moleule (OCS, B=0. m - ) t room temperture, mx is. 6
7 5. Rottionl Rmn spetr An eletri field pplied to moleule results in its distortion, nd the distorted moleule quires ontribution to its dipole moment (even if it is nonpolr initilly). The polrizbility my be different when the field is pplied () prllel or (b) perpendiulr to the moleulr xis (or, in generl, in different diretions reltive to the moleule); if tht is so, then the moleule hs n nisotropi polrizbility. 5.. Rottionl Rmn spetr-seletion rules The distortions indued in moleule by n pplied eletri field returns to its initil vlue fter rottion of only 80 o (tht is, twie revolution). This is the origin of the D=± seletion rule in rottionl Rmn spetrosopy. 7
8 5.. Rottionl Rmn spetr-seletion rules All liner moleules nd ditomis (whether homonuler or heteronuler) hve nisotropi polrizbilities, nd so re rottionlly Rmn tive. Note tht spheril rotors (e.g. CH 4, SF 6 ) re rottionlly Rmn intive s well s mirowve intive! Rottionl Rmn seletion rules re: Liner rotors: 0, 5.. Rottionl Rmn spetr energy levels The rottionl energy levels of liner rotor nd the trnsitions llowed by the D=± Rmn seletion rules. The form of typil rottionl Rmn spetrum is lso shown. The Ryleigh line is muh stronger thn depited in the figure, it is shown s weker line to improve visuliztion of the Rmn lines. 8
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