= m ; where m =j, j-1,,-j; (2j+1) possible values.

Transcription

1 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese A Eiee's Guide to uatized Aula Mometum he maitude of a aula mometum vecto j with quatum umbe j is ive by: h j j( j + 1) j( j + 1) ; j ca take itee o half-itee values oly π Obital aula mometum vectos ae esticted to itee quatum umbes but spi aula mometa may have eithe itee o half-itee quatum umbes A aula mometum state is ot completely specified by the sile quatum umbe j he pojectio of the vecto j o ay space-fixed axis (taditioally oe uses the z-axis) is also quatized he maitude j z is thus a additioal quatum umbe j z m ; whee m j, j-1,,-j; (j+1) possible values If space is isotopic (o electic o maetic fields) the aula mometum states with the same j, but diffeet j z have the same eey, ie j is (j+1)-fold deeeate Whe two aula mometa couple with each othe the esultat depeds o the oietatio of the vectos Sice specifyi the quatum umbe j, does ot specify the diectio, oe ca et seveal diffeet esults whe aula mometa of maitudes j 1 ad j ae added he discussio below uses spi as a example of a aula mometum, but the ules ae the same fo ay type of aula mometum coupli, ie spi-spi, spi-obit, obit-obit, I use s fo spi quatum umbe desiatio, which is a memoic fo spi his is the covetioal symbol fo idividual electo spi, uclea spi quatum umbes ae usually desiated by I If thee ae two spi chaacteized by spi quatum umbes s 1 ad s, the the quatum ules fo addi aula mometum say that the esultat vecto has a spi quatum umbe that obeys the followi sum ule: s s 1 + s s ca have ay oe of the followi values: (s 1 +s ), (s 1 +s -1), (s 1 +s -), s 1 -s Fo example, if s 1 1, s, the s,, o 1 Similaly, if s 1 1, s 1, the the esultat s 1 o 0 I eeal, the eeies of the states with diffeet esultat aula mometum ae diffeet, althouh diffeeces may be small Whe aula mometa with quatum umbes s 1 ad s couple, the thee ae (s 1 +1)(s +1) possible combiatio, coespodi to the diffeet possible oietatio of the oiial two vectos (ie diffeet values of s z ) ad emembei that oe is doi a vecto additio If you compae the (s+1) fold deeeacy of the aula mometa eeated by the sum ule ive above you fid that it matches Specifically, fo the example above whee s 1 1, s, we have (s 1 +1), (s +1) 5, so thee should be 515 possible vecto combiatio If you use the esults of the sum ule fo the possible esults you et: 1

2 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese s (s+1)-fold deeeacy otal 15 Similaly, if s 1 s 1 the (s+1) (coespodi to sz ± 1 spi "up" o "dow") thee should be 4 combiatio Aai: s (s+1)-fold deeeacy otal 4 Fially, if the paticles ae idetical, ad the aula mometum quatum umbes ae the same, the the vecto combiatio ca also be examied fo thei symmety with espect to exchae of these idetical paticles If s 1 s s the thee ae: (s+1)(s+1) combiatio that ae symmetic, ad s(s+1) combiatio that ae ati-symmetic with espect to exchae of idetical paticles Retui to the s 1 s 1 example, thee ae ( 1 +1) ( 1 +1) symmetic combiatio, ad (you do the math) 1 ati-symmetic combiatio Icidetally, do't be fooled by the coicidece of ad 1 with the values i the table above he (s 1 +s ) 0 combiatio is symmetic; two othe symmetic states ad oe atisymmetic state ae obtaied fom the thee (s 1 +s ) 1 states Oe fial example, you wok out the details: s 1 s 1 he esultat (s 1 +s ), 1, o 0 hee ae 9 idividual spi quatum states possible, distibuted 5,, ad 1 betwee (s 1 +s ), 1, ad 0 espectively If these ae idetical boso, e 14 N uclei, the we must distiuish symmety of the combiatio hee ae 6 symmetic ad ati-symmetic spi quatum states that ca be fomed he symmetic ad ati-symmetic spi combiatio ae associated with symmetic ad atisymmetic otatioal, vibatioal, ad electoic state desciptio so that the oveall desciptio is symmetic (fo idetical boso) o ati-symmetic (fo idetical femio)

3 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese Rotatioal states ad uclea spi Heteouclea molecules Fo heteouclea molecules, e CO, HD, D, 16 O 17 O, etc the effects of uclea spi ca be safely elected hey ae oly eeded if oe eeds to eumeate the total umbe of micostates pecisely Fo a eeic molecule AB with uclea spi I a ad I b espectively, the uclea spi deeeacy (I a +1)(I b +1) appeas as a cotat multiplie fo the deeeacy of each otatioal (actually aula mometum) state (it is tempeatue- ad -idepedet) hus the otatioal patitio fuctio ca be computed as follows: ot (I a + 1)(I b + 1)( k k he spi deeeacy tems ca be factoed out of the sum because they ae the same fo all states Hee ε() is the eey of otatioal state with quatum umbe If oe uses a iid oto appoximatio fo the otatioal eey, ε() k (+1), the: 0,1,,, ( + 1) i the usual limit >> Whe calculati populatio factio i a paticula otatioal state descibed by quatum umbe, we have k k k k k hus the factio ca be computed coectly electi the spi deeeacy completely ad computi the otatioal patitio fuctio fom ot k ( he otatioal patitio fuctio so computed is too small by the cotat facto Howeve, sice themodyamic popeties deped o ot o eo is iduced by electi it It is tue that thee is a eo i absolute etopy associated with eumeati micostates because Ω will be udecouted by a facto ie Howeve: Ω tue Ω electi spi tue [ Ω ] S k Ω k + electi spi

4 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese Because Ω electi spi is so lae (typically ~10 0 ) the facto which is ivaiably ~1 is eliible uless oe is deali with systems at tempeatues close to absolute zeo I ay case chaes i etopy ae coectly calculated because is a cotat Homouclea molecules Fo homouclea molecules, e H, D, O ( 16 O 16 O), etc the effects of uclea spi must be coideed whe computi both the otatioal patitio fuctio ad the factioal populatio he oveall wave fuctio descibi the system must be symmetic with espect to exchae of idetical boso ad ati-symmetic with espect to exchae of idetical femio Rotatioal motio of the uclea famewok exchaes the uclei i space ad hece spi-symmety effects affect the otatioal states Fo the simplest case of a liea diatomic molecule a otatio thouh 180 exchaes the uclea positio 1 he wave fuctio descibi the uclea otatioal motio must have the coect symmety fo femio o boso (o both, e acetylee which is H-C C-H) Fo a otatio thouh 180 the otatioal wavefuctio ae ati-symmetic (ie chae si, ) fo, ad symmetic (ie do ot chae si, +) fo Now the oveall wavefuctio must be symmetic fo boso o ati-symmetic fo femio with espect to the exchae of the uclei (ad oly the uclei) Howeve, whe the uclei otate they cay the electo with them o bi the electo back to whee they stated fom, leavi the uclei whee they wee caied by the otatio (say about the z-axis) oe eeds two successive opeatio: (i) eflect electo i xy-plae [(x,y,z) (x,y,-z)] ; (ii) ivet all electo thouh the oii [(x,y,z) (-x,-y,-z)] Explicitly, if the electo coodiates wee (x e,y e,z e ) iitially, the the fist otatio thouh 180 about the z-axis ives (x e,y e,z e ) (-x e,-y e,z e ) Now (-x e,-y e,z e ) (i) (-x e,-y e,-z e ) (ii) (x e,y e,z e ), ad the electo is back whee it stated he electo wave fuctio desiatio tell us how the wave fuctio behaves ude opeatio (i) ad (ii) Usually the electoic oud state has zeo esultat electoic obital aula mometum (deoted as a Σ state) I this case the state is desiated Σ + o Σ - depedi o whethe eflectio opeatio (i) leaves the wave fuctio si uchaed o chaed espectively If the chae distibutio has a cete of symmety (as it must fo homouclea molecules) the a subscipt o u is added to idicate the esult of the ivesio opeatio (ii), fo symmetic, ad u fo ati-symmetic 1 he same eeal piciple applies to moe complicated molecules, e i NH at equilibium the thee H atoms fom a equilateal tiale i a plae with the N atom above o below Hece a otatio thouh 10 o 40 exchaes H atom positio We will ot coide these moe complicated cases, but estict ouselves to homouclea diatomics 4

5 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese Nuclea spi, I I fixed Rotatio, vaies Electoic eflectio Electoic ivesio Symmety diffeetiato Otho (symmetic) Paa (ati-symmetic) Eve Odd + u Deeeacy Effect o wave cotibutio fuctio (I+1)(I+1) + I(I+1) +1) + +1) a + a + Example 1: 1 + Goud state (X) of N : Σ, uclei ae boso with I 1 Because uclei ae boso the oveall wavefuctio must be symmetic (+) with espect to exchae Electoic Nuclea spi Rotatio Oveall Nuclea spi deeeacy (+) (+)(+) otho (+) Eve (+) (+) (I+1)(I+1) 6 ( ) + paa ( ) Odd ( ) (+) I(I+1) ( ) Example : + Excited (A) state of N : Σ u, uclei ae boso with I1 Because uclei ae boso the oveall wavefuctio must be symmetic (+) with espect to exchae Electoic Nuclea spi Rotatio Oveall Nuclea spi deeeacy (+) ( )( ) otho (+) Odd ( ) (+) (I+1)(I+1) 6 ( ) + u paa ( ) Eve (+) (+) I(I+1) ( ) Note: I the A excited state of N it is the states that have eate deeeacy 5

6 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese Example : 1 Goud (X) state of O : Σ, uclei ae boso with I0 Because uclei ae boso the oveall wavefuctio must be symmetic (+) with espect to exchae Electoic Nuclea spi Rotatio Oveall Nuclea spi deeeacy ( ) (+)( ) otho (+) Odd ( ) (+) (I+1)(I+1) 1 ( ) paa ( ) Eve (+) (+) I(I+1) 0 ( ) Example 4: Goud (X) state of H : 1 + Σ, uclei ae femio with I 1 / Because uclei ae femio the oveall wavefuctio must be ati-symmetic ( ) with espect to exchae Electoic Nuclea spi Rotatio Oveall Nuclea spi deeeacy (+) (+)(+) otho (+) Odd ( ) ( ) (I+1)(I+1) ( ) + paa ( ) Eve (+) ( ) I(I+1) 1 ( ) Patitio fuctio calculatio We assume that oly the oud electoic state is siificatly populated, so oly sums ove otatioal states i the oud electoic state eed to be coideed Hece if excited states have diffeet statistics tha the oud state, e the A-state of N, the eo i the calculatio of patitio fuctio is iiificat Whe computi the otatioal patitio fuctio oe has to sum ove ad sepaately because they ae multiplied by diffeet uclea spi deeeacies ot all spi k 0,,4, k + 1,,5, k If oe uses a iid oto appoximatio fo the otatioal eey, ε() k (+1), ad i the usual limit >>, the: 0,,4, ( + 1) 1,,5, ( + 1) Hece, i this limit 6

7 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese ot ( + ) (( I + 1)( I + 1) + I (I + 1) ) (I + 1) he factioal populatio i ay otatioal state is the ive by: / ( + ) ( + 1) / + ) ( + 1) / I + 1 I he umeical value of the factio o aisi fom uclea spi ( + ) I + 1 I + 1 statistics depeds o the type of the ucleus (femio o boso), the uclea spi quatum umbe, ad the electoic state symmeties as discussed above Example 1b: N (see Example 1 above) ( + 1) ( 1) + ( 1) 6 ( 1) + e + e ; total + ) 9 ( + 1) ( 1) + ( + 1) ( + 1) e e total + ) 9 Hece the evelopes of the otatioal distibutio fuctio fo - ad - ae i the atio 6: ie, :1 he fiue below shows a otatioal Rama spectum of N measued i ou laboatoy he elative iteities of idividual lies withi a bach show the :1 iteity alteatio pedicted by the theoy (he sto elastic liht scattei at Δν0 is suppessed with a atomic filte) 7

8 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese Rotatioal Rama Spectum of 14 N (I N 1) Rama sial (abitay uits) Stokes bach Ati-Stokes bach Fequecy shift of Rama scatteed liht, Δν / cm -1 Example b: O (see Example above) ( + 1) ( + 1) 0 0; total + ) 1 1 total + ) 1 ( + 1) ( + 1) I this case the - states ae missi etiely because 0 8

9 ASE 8R6 Molecula Gas Dyamics ME 814 Molecula Pocesses 00 by Philip L Vahese Example 4b: H (see Example 4 above) total total + + ) ) ( + 1) ( + 1) ( + 1) ( + 1) ; Hece the evelopes of the otatioal distibutio fuctio fo ad - ae i the atio 1: States with diffeet uclea spi symmety (otho- ad paa-) ae ot easily ite-coveted, ie they do ot each themodyamic equilibium with each othe, but athe behave as diffeet chemical species Hece if homouclea diatomic ases ae cooled they maitai the atio of otho- to paa- that pevails at hih (>> ) tempeatue his is most obvious fo H which has a lae (~85 K) ad codees at 0 K at atmospheic pessue Most othe ases have much smalle (~ K typically) ad ae ot commoly used i the vapo phase at tempeatues whee ~ he uclea spi deeeacy must be teated coistetly whe computi equilibium cotats Suppose, fo example, that we ae computi K p fo the eactio: N + O NO, with the commo isotopes 14 N (I1) ad 16 O(I0) If the thee-fold uclea spi deeeacy of the NO is elected, oe must oly use a facto of 1 i the otatioal patitio fuctio of N NO NO ad O Covesely, if oe uses 1 whe calculati ot, the the NO NO coespodi otatioal patitio fuctio of N ad O ae (I + 1) 9 (I + 1) N O ot N N N ot O O O Whe calculati the equilibium cotat the spi factos cacel ad we et the same esult as befoe: NO ( ) ot N ot O ot NO 9 N O NO N O 9