q-Deformed Vibron model for diatomic molecules

PHYSICAL REVIEW AVOLUME 50, NUMBER 2 q-de o med ib on model o dia omic molecnles AUGUST 1994 R. N. Al a ez, 'Dennis Bona sos, and Yu. F.Smi no ' 'Ins i u e o Nuclea Physics, Moscow S a e Uni e si y, 117234 Moscow, Russia Ins i u e o Nuclea Physics, Na ional Cen e o Scien i c Resea ch Demok i os, GR 153-10Aghia Pa aske i, A iki, G eece (Recei ed 29 No embe 1993) Ade o med e sion o he ib on model o dia omic molecules is cons uc ed. Bo h he O(4) and U(3) dynamical symme ies o he model a e ew i en, using he concep o complemen a y subalgeb as, in amo e con enien o m, which is subsequen ly de o med. The p esen model uni ies he so a in- dependen success ul quan um-algeb aic app oaches o o a ional and o ib a ional spec a o dia omic molecules. In addi ion, he me hod can be used o he cons uc ion o de o med e sions o he U(5) and O(6) limi s o he in e ac ing boson model o nuclea s uc u e. PACS numbe (s): 33.10.Cs, 31.15.+q, 02.20.S I. INTRODUCTION The ma hema ical s uc u e o quan um algeb as (quan um g oups) [1— 4] has ecen ly been a ac ing much a en ion. They a e de o med e sions o he usual Lie algeb as, o which hey educe when he de o ma ion pa ame e qis se equal o 1. In pa allel, applica ions o quan um algeb as in physics ha e begun o de elop in pa icula in cases in which Lie algeb as a e known o de- sc ibe app oxima ely he symme ies o aphysical sys- em. The quan um algeb a su (2) has been success ully used o desc ibing o a ional spec a o dia omic mole- cules [5— 7], de o med nuclei [8— 10], and supe de o med nuclei [11]. Vib a ional spec a o dia omic molecules ha e been desc ibed in e ms o de o med oscilla o s [12— 16], as well as in e ms o an SU (1,1) symme y [17,18]. Po en ials gi ing spec a equi alen o hose o he de o med oscilla o s jus men ioned ha e been con- s uc ed [19,20] and ound o be de o med e sions o he modi ied Poschl- Telle po en ial o , equi alen ly, he Mo se po en ial. On he o he hand, he ib on model [21— 23], ha ing an o e all U(4} symme y, is known o p o ide auni ied desc ip ion o molecula o a ions and ib a ions h ough he use o algeb aic echniques, in away simila o he desc ip ion o collec i e nuclei in e ms o he in e ac ing boson model (IBM) [24]. The O(4} limi ing symme y o he ib on model has been ound o be app op ia e o di- a omic molecules, while he U(3) limi ing symme y has been used o he desc ip ion o clus e ing e ec s in nu- clei, as well as o he quasimolecula desc ip ion o hea y-ion esonances (see [25] o lis s o e e ences). The ques ion is he e o e c ea ed i ade o med e sion o he ib on model can accommoda e in auni ied ame- wo k he imp o ed desc ip ions o o a ional and ib a- 'On lea e o absence om he Ins i u e o Nuclea Physics, Moscow S a e Uni e si y, 117234Moscow, Russia. ional molecula spec a ob ained so a in e ms o sepa a e quan um algeb as. The p oblem o cons uc ing he de o med e sion o he ib on model (o o he IBM) is no asimple one since he cons uc ion o he educ ion chains o Ue(4) and U(6) has no been achie ed ye . I su ices o be men ioned ha he educ ion o n SU (3) o SO (3) has been ca ied ou only o ully symme ic i educible ep esen a ions (i eps} o SU (3) [26]. How- e e , a ew e o s owa ds cons uc ing de o med e - sions o he ib on model [27,28] and he IBM [29,30] al- eady exis . In his pape ade o med e sion o bo h he O(4) and U(3) dynamical symme ies o he U(4) ib on model will be cons uc ed, aking ad an age o he echniques o complemen a y algeb as, in oduced by Quesne and co- wo ke s [31— 33],which bypass he di icul ies in he con- s uc ion o educ ion chains o quan um algeb as. In ad- di ion o uni ying he exis ing independen quan um- algeb aic desc ip ions o o a ional and o ib a ional molecula spec a, he p esen app oach allows, in asim- ple way, o he in oduc ion o c oss e ms desc ibing he coupling be ween hese wo exci a ion mechanisms. Ab ie accoun o he ib on model o dia omic mole- cules will be gi en in Sec. II, while in Sec. III he model will be o mula ed in ano he way, using he echniques o complemen a y algeb as. In Sec. IV he q-de o med e sion o he complemen a y analogs o bo h he O(4) and U(3) dynamical symme ies o he ib on model will be gi en. Sec ion Vwill con ain discussion o he p esen esul s and plans o u he wo k. II. THK VIBRON MODEL FOR DIATOMIC MOLKCULES In his sec ion he b ie es possible accoun o he ib- on model [21— 23] is gi en in i s usual o m. In he ib- on model he o a ions and ib a ions o adia omic mol- ecule a e desc ibed in e ms o ou bosons: ascala bo- son o posi i e pa i y and angula momen um 1=0, deno ed by s+, and he h ee componen s o a ec o bo- 1050-2947y94g5Og)g1O88(8)A%06. 0O 50 Qc 1994 The Ame ican Physical Socie y 50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1089 [T 'T '] '= g(k, uk~ u3~k3 u3)T„'T„', Q)Q2 (2.1) one obse es ha he 16 possible bilinea quan i ies [bi+eh&. ]gene a e he algeb a U(4), which is, he e o e, he o e all symme y o he ib on model. The e a e wo chains o subalgeb as o u(4) con aining he angula -momen um algeb a so(3) as asubalgeb a. These a e I: u(4) Do(4) Dso(3) Dso(2), II: u(4) Ou(3) Dso(3) Dso(2) . (2.2) (2.3) In he case o chain I he basis has he o m ~N ELM ), whe e he a ious quan um numbe s a e de ined as ol- lows. (i) Nis he o al numbe o bosons. I cha ac e izes he i eps o U(4), which a e ully symme ic, since we a e dealing wi h asys em o bosons. (ii) co is he senio i y quan um numbe , cha ac e izing he i eps o O(4) and ob aining he alues co=N, N— 2, ...,1o 0. (iii) Lis he angula momen um quan um numbe , la- beling he i eps o SO(3) and aking he alues L=a),co — 1, .. . ,1,0. (i ) Mdeno es he zcomponen o he angula momen- um, labeling he i eps o SO(2) and ha ing he alues — L+M+L. When he Hamil onian is cha ac e ized by he dynami- cal symme y o chain I, i can be w i en in e ms o he Casimi ope a o s o he algeb as appea ing in his chain: H =ep+ eC~ (u(4) )+63C3(U(4)) +ACz(o(4) )+BC'(so(3)), (2.4) whe e Nand Na e ela ed o he i s - and second-o de Casimi ope a o s o u(4). The eigen alues o he Hamil- onian in he basis gi en abo e a e hen E(N, co,L)=ep+e)N+e3N(N+3) +Ace(co+2)+BL (L +1}.(2.5) Usually he ib a ional quan um numbe N— co 2(2.6) is in oduced, and he ene gy eigen alues a e ew i en as E(N, u, L)=op+ e',N+e~N 4A (N +2)(u +—, — '} +4A (u+ , '}+BL(L+1), — (2.7) whe e E'(), E'),Ep a e ela ed o E'0 E'~ 6'p A. I should be no- iced ha he ou h and i h e ms on he igh -hand son o nega i e pa i y and 1=1, deno ed by p„+, p=0,+1. The co esponding annihila ion ope a o s ans o ming as sphe ical enso s a e s=sand P„=(— 1)' "p „.Deno ing hese bosons by b&+„, I=0,1 and — 1&p&I, and bI „=(— 1)' "b~ „,and de6ning he enso p oduc o wo ope a o s T„and T„as 12 +aC3(u(3))+pC, (so(3)) .(2.8) The eigen alues o he Hamil onian in he basis gi en abo e a e E(N, n&,L)=Fp+F.,N+ezN(N+3) +en +an (n +2)+PL(L+1) . III. ALTERNATIVE FORMULATION OF THE VIBRON MODEL (2.9) An al e na i e o mula ion o he ib on model can be achie ed in e ms o complemen a y algeb as. The no- ion o complemen a y algeb as was in oduced by Moshinsky, Quesne, and co-wo ke s [31— 33]. I is espe- cially ui ul in he case o mul idimensional ha monic oscilla o s o many-pa icle sys ems o ew kinds o bo- sons. In he p esen case o ou kinds o bosons (s+,p„+, @=0,+1) he hos algeb a is sp(S,E). Two chains o subalgeb as a e sp(8, E)Du(4) Do(4) Dso(3) Dso(2), sp(S, E)Dsp(2, R)Du(1) . (3.1) (3.2) The quan um numbe s N, p3, L,M, labeling he i eps o he subalgeb as o he i s chain, ha e been desc ibed in Sec. II. sp(2,R) is isomo phic o su(1,1). The i eps o su(1,1) and u(1) a e labeled by he quan um numbe s j and m, espec i ely. Two subalgeb as A& and Az o a la ge algeb a Aa e complemen a y wi hin ade ini e i - ep o Ai he e is aone- o-one co espondence be ween all he i eps o A& and o Az con ained in his i ep o A [31]. In he example gi en abo e, he only i eps o he hos algeb a sp(S,E} ha can be ealized in aFock boson space a e he e en i ep [0], including he ec o s ~NcoLM )wi h Ne en, and he odd i ep [1],including he ec o s wi h Nodd. I can hen be p o ed ha o(4} and sp(2,E) [and hus also o(4) and su(1,1)] a e comple- men a y. The same holds o u(4) and u(1). Fo con enience le us deno e he ou kinds o bosons in oduced in Sec. II by b, =1,2,3,4, co esponding o p+&,p„po,s, espec i ely. To each kind o bosons co esponds an algeb a sp"(2,E},gene a ed by K+ =,'bP„, K' =, 'b b„,—ECp= —, '(N, +——, '), (3.3) side co espond o he spec um o he Mo se po en ial [34]. In he case o chain II he basis is ~Nn LM ),whe e N is again he o al numbe o bosons, while he o he quan- um numbe s a e de ined as ollows. (i) nis he numbe o pbosons labeling he i eps o U(3) and ob aining he alues n=0,1, ...,N (ii) Lis labeling he i eps o SO(3}, ob aining he alues L=n,n— 2, ...,1o 0. (iii) Mis labeling he i eps o SO(2), wi h alues — L+M+L. When he Hamil onian is cha ac e ized by he dynami- cal symme y o chain II, i can be w i en in e ms o he Casimi ope a o s o he algeb as appea ing in i : H„=ep+ e,C,(u(4) )+ezCz(u(4) )+eC,(u(3)) R. N. ALVAREZ, DENNIS BONATSOS, AND YU. F.SMIRNOV 50 whe e N„=bP .These gene a o s sa is y he commu a- ion ela ions [K(),K+ ]=+K~, [K+,K" ]= 2— K() .(3.4) The sp(2,E}=su( 1, 1}algeb a, men ioned abo e, is eal- ized in he space o ou kinds o bosons. The e o e we a e going o use o i he symbol sp" '(2, E)=su" '(l, l). This algeb a is gene a ed by =—, 'gbP„, K=—, 'gb„b„, K=—, '(N+2), ~NcoLM) he quan um numbe s Nco by he quan um numbe s jm o he complemen a y subalgeb as. Fu he - mo e, in he Hamil onian o Eq. (2.4) one is en i led o e- place he second-o de Casimi ope a o o o(4) by he second-o de Casimi ope a o o su" '(l, l) and he i s - and second-o de Casimi ope a o s o u(4) [N and N(N +3)]by he i s - and second-o de Casimi ope a- o s o u(1) (Ko and Ko ). In he case o chain II, he u(3) subalgeb a o u(4) in- ol es only he pbosons. The hos algeb a is hen sp(6,E),ha ing he wo chains o subalgeb as (3.5) whe e N=g„b„b, The. se gene a o s sa is y he com- mu a ion ela ions sp(6, E)Du(3) Dso(3) Dso(2), sp(6, E)Dsu" '(l, l)Du(1), (3.15) (3.16) [Kp,K~]=+K~, [K+,K]=— 2K() . The Casimi ope a o is Cz(sp" '(2,E))=— K+K +K()(K()— 1), (3.6) (3.7) 1n— 4 J—N+ 22(3.8) wi h eigen alue j(j+1).I is known ha when o(n) and su(1, 1) a e complemen a y, he quan um numbe s co and j cha ac e izing hei i eps a e connec ed by [35] whe e he supe sc ip (123) means ha only he bosons b„bz, b3a e in ol ed in he o ma ion o su" '(1,1}=sp" '(2, E). Fu he de ails on hese chains a e gi en below [see Eqs. (3.26)— (3.28}]. The building up o he bases ela ed o he wo limi ing symme ies o he ib on model can hen be achie ed as ollows. To each kind o boson b„, an sp"(2,E) algeb a co esponds, as al eady men ioned, gene a ed by he ope a o s gi en in Eq. (3.3). The s a es wi h N„e en co - espond o he i ep D,while he s a es wi h N„odd co espond o he i ep D'.Thus o each boson s a e co N— U J=2= 4(3.9) In he p esen case su" '(l, l) is complemen a y o o(4), so ha 1 lN)NzN, N4) =QN)!Nz!N3!N4! X(bi) '(bz) '(bz) '(b4) '0) (3.17) Cz(A) )=c)cz(Az)+cz . In he case o o(n) and su(1,1} his ela ion is (3.10) I is also known ha he Casimi ope a o s o wo alge- b as complemen a y o each o he a e connec ed by a simple, usually linea , ela ion o he ype one can co espond ase o ou noncompac "angula momen a" j„=— —, 'o — —, ', =1,2,3,4, he alue o each angula momen um jdepending on he pa i y o he co esponding boson numbe N. We can now p oceed o he ec o coupling o he i s wo angula momen a j, and jz, using he SU(1,1) Clebsch-Go dan coe icien s [36,37] Cz(su(1, 1))=—, 'Cz(o(n ))+n(n — 4) (3.11) lj)jzj)zm)z&= g&J(m)jzmzlj(zmiz}sU(), )) which in he p esen case o o(4} educes o C,(su"""(1,1))=-, 'C,(o(4)).(3.12) 1n m= —X+— 22(3.13) which in he p esen special case o u(4) and u(1) educes o m=—, '(N+2) .(3.14) The chain o Eq. (3.1) al eady s udied is o in e es in he case o he chain Io he ib on model. I implies ha in s udying chain I, one can eplace in he basis The u(1) subalgeb a o su" '( l, l) is gene a ed by he ope a o Eo alone, he eigen alues o which we label by m. In he gene al case o he complemen a y algeb as u( n)and u(l), he quan um numbe s Nand mcha ac e - izing hei i eps a e connec ed by [35] m&m& X~j)m))~jzmz) .(3.18) wi h p=0, +1. The hos algeb a o his space o wo kinds o bosons is sp(4, E}.The ollowing wo chains o subalgeb as exis : sp(4, E)&u(2) Dso(2), sp(4, E)Dsu' )( l, l)DU(1), (3.20} (3.21) whe e he i eps o u(2) a e labeled by he o al numbe o bosons N, z=N, +Nz, while he i eps o so(2) a e labeled by M=N, Nz. SO(2) is — complemen a y o su" '( 1,1)=sp" '(2, E), he i eps o which a e labeled by j,z=—, '(M — 1), (3.22} This means ha he in e media e su" '(l, l}algeb a has been in oduced, gene a ed by (3.19) 50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1091 m(z =— '(N(z+1), (3.23) acco ding o Eq. (3.13). The nex s ep along his line is o couple j12 wi h j3. In his case h ee kinds o bosons a e in ol ed, so ha he hos algeb a is sp(6,R). The ele an chains o his case ha e been gi en in Eqs. (3.15) and (3.16), so(3) being com- plemen a y o su" '(1,1},which is gene a ed by g123 g12 +~3 (3.24} wi h p=0, +1. The esul ing eigen ec o s a e IJljz{4lz V3'J123m 123 & {j(2mlzj3m3IJ123m123 &SU(1, () m12m3 xIj(jz.j(zm &z )Ij3m 3)(3.25) The Casimi ope a o s o so(3} and su" '(l, l) a e con- nec ed by acco ding o Eq. (3.8), while u(2) is complemen a y o u(1), he i eps o which a e labeled by The hos algeb a in his case is sp(8,R), he ele an chains ha ing been gi en in Eqs. (3.31) and (3.32). The basis ec o s, deno ed by Ij,jz(j,z)j3(j,z3)j4. .jm ), o by Ij(zj3(j(z3 )j4.jm )in he case in which he sho ened e - sion o Eq. (3.30) is used o he ec o s wi h angula momen um j12, co espond o he i eps o he sp' (2,R)=su' (l, l) algeb a, gene a ed by g1234 g123 +g4(3.32) I wi h @=0,+1. He e Ka e he gene a o s o he su (1,1) algeb a, associa ed wi h he s-bosons. The o al noncompac angula momen um j, cha ac e izing he i - eps o su" 4'(l, l), is connec ed o he senio i y quan- um numbe (0, cha ac e izing he i eps o o(4), by Eq. (3.8), while he Casimi ope a o s o hese wo comple- men a y algeb as a e connec ed by Eq. (3.12). Gi en he abo e, i is clea ha o he chain Io he ib on model, ins ead o he basis INcoLM), he basis Ij,zjz{j,z3)j4.jm )can be used. Fu he mo e, in he case o chain II, ins ead o he basis INn~LM), he basis Ij,zj3:j,z,m, z, )Ij4m4 )can be used. I is clea ha he connec ion be ween he wo new bases o he dynamical symme ies o he ib on model is Cz(so(3) )=4Cz(su" "(1,1))+— ', ,{3.26) Ij(zjz(j]z3V4:Jm & acco ding o Eq. (3.11), while he quan um numbe s la- beling hei i eps, Land j123, espec i ely, a e connec ed by m&23 m4 &j(z3m(z3j4m4ljm &sU((, (i j(z3=I'(L (3.27) xIj(zj3 j(z3m]z3 &Ij4m4 &(3.33) acco ding o Eq. (3.8). The eigen alues o Cz(su" '(l, l)) in he abo e-men ioned basis a e gi en by j,z3(j,z3+1). Fu he mo e, in he chains o Eqs. (3.15) and (3.16), u(3) and u(1) a e complemen a y, he quan um numbe s n~ and m,23 labeling, espec i ely, hei i eps being connec ed by The Hamil onian o chain I, gi en in Eq. (2.4), can be ew i en using he complemen a i y ela ions desc ibed in Eqs. (3.1), (3.2), (3.15), and (3.16},as H) =ep+e(C((u(l))+ezCz(u(1)) +4ACz(su" '(1,1))+4BCz(su" '(1,1)).(3.34) m,z3=—, '(n +— ', ), (3 28) The eigen alues o his Hamil onian a e Ij„m„&= (b$) '(bz) 'I0&, QN (!Nz! wi h (3.30) J(z 2(N( Nz — 1), m„=—, '(N, +N, +1), (3.31) [which a e in ag eemen wi h Eqs. (3.22) and (3.23)] a e eigen ec o s o he Casimi ope a o Cz(su"z'(1, 1)),wi h eigen »ues j»(j»+1). In his way one can a oid he couphng o j1 and j2 o j». The coupling o j» and j3 canno be a oided howe e . The esul ing ec o s in his case we deno e by Ij,zj3: j,z3m (z3 ). In he las s ep he coupling o j123 o j4 is pe o med. acco ding o Eq. (3.13). The coupling o he wo angula momen a j, and jz, pe o med abo e, can be a oided by no icing ha he su" '(l, l) can be gene a ed by E'+ =b,bz, E' =b(bz, Ep =—, '(N, +Nz+1) .(3.29) The ec o s E(m,jj(z3)=Ep+eIm +ezm '+4Aj(j+1) +4Bj123{j123+1).(3.35) Hn &O+e1m +edam +e'm123+a'm 123 +4P'Cz{su" '(1,1)), (3.36) whe e on he igh -hand side he second and hi d e m co espond o he i s - and second-o de Casimi ope a- o o he u(1) algeb a o he chain o Eq. (3.2), while he ou h and i h e ms co espond o he i s - and second-o de Casimi ope a o s o he u(1) algeb a ap- pea ing in he chain o Eq. (3.16). The eigen alues o his Using Eqs. (3.14), (3.9), and (3.27), which connec he quan um numbe s m, j,j,z3 o he p e ious ones (N, co,L), i is easily e i ied ha Eq. (3.35) is an al e na i e way o w i ing Eq. (2.5}. Simila ly he Hamil onian o chain II, gi en in Eq. (2.8), can be ew i en, aking in o accoun he com- plemen a i y ela ions gi en in Eqs. (3.1},(3.2), (3.15), and (3.16), as 1092 R. N. ALVAREZ, DENNIS BONATSOS, AND YU. F.SMIRNOV 50 Hamil onian a e E(m, m, 23 J)23)= eP+ eIm+e2m +e m)23+a'm(23 one can p o e ha su" '(1,1) is gene a ed by [38— 40] K''=a"a K" '=a aK" '=— '(N, +N2+1), +4&'J123(j123+1).(3.37) Using Eqs. (3.14), (3.28), and (3.27), which connec m, m,23,j,23 o N, n~, L, i is easily e i ied ha Eq. (3.37) is an al e na i e way o w i ing Eq. (2.9). In his sec ion we ha e he e o e ew i en he bases and he Hamil onians co esponding o he wo dynami- cal symme ies o he ib on model in e ms o comple- men a y subalgeb as. This o mula ion is use ul because i can be qde o med in a e y simple way. IV. qDEFORMATION OF THE VIBRON MODEL he ele an commu a ion ela ions being [K(12) K(12) )+K(12) 0~+j—+ [K(12) K(12) ]= (2K()2) ] The ec o s ~j»m»), =(a)) '(a2) '~0), V'[N(]q)[N2)q( (4.10) (4.11) The co esponding Hamil onians we e hen w i en in e ms o he Casimi ope a o s o he new educ ion chains. An e iden possibili y o qde o ming hese Hamil onians is o subs i u e he su(1, 1) algeb as o Eq. (4.1) by hei q-de o med coun e pa s su (1,1) [38— 40], su" '(1,1)Dsu" '(1,1)Dsu" '(1, 1)ZU (1) .(4.2) In his sec ion we shall explain how his can be achie ed, a e gi ing ab ie accoun o he necessa y ma hema ical de ails. qnumbe s a e de6ned as (4.3) Fo q eal (q =e'wi h eal), hey can be w i en as sinh x sinh~ (4.4) while in he case o qbeing aphase (q =e"wi h eal), hey ob ain he o m sin x qsin7- (4.5) In he limi ing q~ 1( ~0), qnumbe s educe o usual numbe s. q-de o med oscilla o s [41,42] a e in oduced h ough he ela ions aalu — q— 'a 'a =q, [N, a]=a, [N,a]=— a, (4.6) whe e aand aa e he q-de o med boson c ea ion and annihila ion ope a o s and N he ele an numbe ope a- o . Using Eq. (4.6) one can easily show ha aa=[N]~, aa =[N+1]~ .(4.7) q-de o med algeb as can be exp essed in e ms o q- de o med bosons. In oducing a, as he q-de o med ana- logs o b; (i =1,2, 3,4), wi h he p ope ies [a ,a„"]=[a, a„]=[a„, a„]=0, ip, (4.8) In he p eceding sec ion he subalgeb a chains o he ib on model we e educed o equi alen chains o com- plemen a y subalgeb as su" '(1 1)~su" '(1 1)~su" '(1 1)~u(1) .(4.1) wi h j,2, m)2 s ill gi en by Eq. (3.31), a e eigen ec o s o he de o med Casimi ope a o C(su" '(1 1))= K" 'K — ''+[K" '] [K" '— 1] (4.12) (4.13) whe e Nis he numbe o bosons. These gene a o s sa is y he commu a ion ela ions [K(),K+ ]=+K+ [K+ K]= [2K()]2.(4.14) We a e no going o use he su'(l, l) and su (1,1) alge- b as explici ly in couplings, since o he su" '( l, l) alge- b a we al eady ha e he o m gi en in Eq. (4.9), which a oids he di ec coupling. In o de o be able o couple su (1,1) and su (1,1) o su" '(l, l), i is use ul o ha e he same de o ma ion pa ame e in all o hese algeb as, i.e.,i is use ul o ha e he same de o ma ion pa ame e in he commu a ion ela ions o Eqs. (4.10) and (4.14). In o de o achie e ha , we eplace in Eqs. (4.13) and (4.14) qby q. As a esul , o =3,4, Eq. (4.6) is mean om now on wi h q eplaced by &q. Then one also has a~„=[N,]&—, a,,a„"=[N,+I]&- .(4.15) Equa ion (4.12), gi ing he Casimi ope a o , is he e o e alid in his case wi h he usual qnumbe s. The su" '(1,1) algeb a is gene a ed by he ope a o s g3 g(12) +(123) +(12) o+~3o +q+ ~(123) ~(12)+~3 00 (4.16) 112+ i.e.,i is as anda d cop oduc o he i eps Dand D'o he su (1,1) algeb a. The e o e he basis ec- o s, in analogy o Eq. (3.25), a e o he o m wi h eigen alues [j)2]~[j)2+1]». Fo he su'(1, 1) algeb as one has he boson ealiza ion [38] K' =aa, K" =aa, K() ='(N +-'),— 11 50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1093 Ij)2j3:j)23m)23 &q while he ec o s analogous o Eq. (3.33) a e Ij)2J3(J123 }J4:Jm & &j)23m)23j4m4IJm &sU, )1,1) m123 m4 xIj)2j3.j)23m]23 &ql j4m4&, .(4.19) These ec o s a e he qanalogs o he eigen ec o s o he dynamical symme y Io he ib on model. Simila ly he ec o s IJ,2J3.j)23m)23 &, Ij4m4 &, (4.20) =g&j)2m)2J3m3IJ)23m)23&SU (),)) m12m 3 XIj)2m )2 &,Ij3m3 &, ,(4.17) &j,m,j,m, Ijm &s„„„a e Clebsch-Go dan coe icien s o he enso p oduc o wo su (1,1) i eps. Explici analy ical o mulas o hese coe icien s, as well as o he ele an su (2) coe icien s, can be ound in [43— 47]. The su"234'(1, 1)algeb a is gene a ed by he ope a o s ~4 g(123) ~(1234) ~(123) o+~4 o(4.18) ha e been used. Using Eq. (2.6), Eq. (4.23) can be w i en as E(N, , L)=6p+E') [N +2]~—+e2[N +2]~— +A — — — 1—— 22 +B'[L]~ [L— +1]~ —, (4.26) which educes o Eq. (2.7) in he limi q~l, up o a ede ini ion o E'0 In he case o he dynamical symme y II he Hamil- onian can be w i en as H)) =~P+~)[m]q+~2[m]q'+~[m)23)q +a[m)23]q+PC2(suq) '( l, l)), (4.27) which is he qanalog o Eq. (3.36) (wi h he p imes o he coe icien s d opped). The eigen alues o his Hamil oni- an a e E(m, m)23 L)=Ep+E')[m]q+e2[m]q+E'[m)23]q +a[m)23], +P[j)23]q[j)23+1]q .(4.28) In he limi q~1,Eq. (3.37) is ob ained. Assuming ha m, m,23,j,23 a e connec ed o N, n~, L h ough Eqs. (3.14), (3.28), and (3.27), Eq. (4.28) can be ew i en in away esembling i s classical coun e pa , Eq. (2.9), as a e he qanalogs o he eigen ec o s o he dynamical symme y II o he ib on model. The e o e Eq. (4.19) connec s he eigen ec o s o he wo dynamical sym- me ies, as Eq. (3.33) does in he classical case. In he case o dynamical symme y I he Hamil onian eads E(N, n&,L)=Ep+ E,[N +2]&- +ez[N +2]&— +e'[n +— ', ]g- +a'[n~+ —' ,]g-, +P'[L]~[L +l]g- .(4.29} H) =6p+6)[m]q +'e2[m ]q +4AC2(suq '( 1,1)) +4BC2(su"2'(1,1)), (4.21) The esul s ob ained in his sec ion call o he ollow- ing commen s. (i) Ro a ional- ib a ional spec a o dia omic molecules a e desc ibed empi ically by he Dunha n expansion [48] which is he qanalog o Eq. (3.34) (wi h he p imes o he coe icien s d opped). The eigen alues o his Hamil oni- an a e E(u,L)=g Yk(u+ —, ')'[L(L+1)]", ik (4.30) E(m,j,j)23)=op+@)[m],+e2[m] +4A [j] [j+1] +[»»)q[»»+ )q (422} In he limi q~ 1, Eq. (3.35) is ob ained. Assuming ha m, j,j123 a e s ill connec ed o quan um numbe s N, co,L h ough Eqs. (3.14}, (3.9), and (3.27}, Eq. (4.22) can be ew i en in away esembling i s classical coun e pa , Eq. (2.5), as E(N, a),L)=ep+eI [N+2]~—+e2[N+2]~— +A'[co)~— [p)+2]~—+B'[L]~ [L +1]~ —.— (4.23) In p oducing Eq. (4.23), iden i ies such as H) =a)+DC2(su" '(1 1))C2(su" '(1 1)) (4.31) Then Eq. (4.26}is modi ied as whe e is he ib a ional quan um numbe , L he angu- la momen um, and Y;k he Dunham coeScien s, i ed o expe imen . I is clea ha he Dunham expansion con- ains powe s o ( +—, '), powe s o L(L +1), as well as c oss e ms. Equa ions (4.23) and (4.29) con ain no c oss e ms. This is due o he ac ha in he Hamil onians o Eqs. (4.21) and (4.27), only e ms up o quad a ic in he gene a o s a e included, as in he case o he classical ib- on model. C oss e ms can be aken in o accoun in he dynamical symme y I, o example, by modi ying Eq. (4.21) as ollows: q — [](1/2+ — 1/2) — 1(4.24) E'(N, , L)=E(N, u, L)+D' —— N — 1—— q2q 2)q [L +— ']q =[L]q[L +1]q [—, ']q [— ', ]q (4.25) X[L]~ [L +1]~ —— (4.32) 1094 R..N. ALVAREZ, DENNIS BONATSOS, AND YU. F.SMIRNOV 50 (ii) Ro a ional spec a in bo h dynamical symme ies [Eqs. (4.26) and (4.29)] a e desc ibed by he e m [L]& [L—+I]&—. This is known o be he Casimi ope a o o su&— (2). The su (2) model has been ex en- q si ely used o he desc ip ion o o a ional spec a o di- a omic molecules [5— 7] and de o med [8— 10] and supe - de o med [11]nuclei. I has been ound [9] ha his e m is equi alen o an expansion in e ms o powe s o L(L +1), [L]q[L+1] =(jp( )L(L+1) j i( '){L(L+1)j +~ j2( )jL(L+1)]3 l jjp«)]' — — ', j( )jL (L +1)] +,', j~—( )jL (L +1)] —.), (4.33) whe e j„( )a e he sphe ical Bessel unc ions o he i s kind and q=e". This expansion is simila o he one con ained in he Dunham expansion. In he case o su (2), howe e , all he expansion coe icien s a e ela ed o powe s o ~, hus esul ing in economy o pa ame e s. No ice ha he dec easing o he coe icien s o inc easing powe s o L(L +1),as well as he al e na ing signs o he e ms, ac s ha a e known empi ically o hold, occu in Eq. (4.33) au oma ically, since is known [5— 11] o ob- ain small posi i e alues. Fu he mo e, i has been p o ed [9] ha he su~(2) model is equi alen o he a i- I able momen o ine ia (VMI) model, which desc ibes o- a ional s e ching e ec s. The qpa ame e has been ound [9] o co espond o he so ness pa ame e o he VMI model. The implica ions o he su (2) model on he elec omagne ic ansi ion p obabili ies connec ing he o a ional le els o nuclei ha e been conside ed [10]. (iii) The ou h e m in Eq. (4.26) co esponds o he Casimi ope a o o su (1,1), al eady used [17] o he desc ip ion o ib a ional spec a o dia omic molecules. I has been p o ed [17] ha his e m, o q=e",can be expanded as N U2u— 1——=j—, '(cos( ) — cosj (N+2)]) — sinj (N+2)](u+ —, ') sin( ) +Hcosj (N+2) j( +—, ') +— 2 sinj (N+2)}(u+ —, ') — —, ' cosj (N+2)]( + —, ') +(4.34) We ema k ha ase ies o powe s o ( +—, ')is ob ained, simila o he one con ained in he Dunham expansion. In he p esen case, howe e , he expansion coeScien s a e all ela ed o (and N, which in he ib on model is a cons an o agi en molecule), hus esul ing in economy in pa ame e s. (i ) The anha monici y cons an (i.e., he a io Y2p /Yip )in he classical case [Eq. (2.7)] is ixed o — 1/(N+2). In he de o med case o Eq. (4.26), howe - e , i is equal o — / an[ (N+2)], as i is easily seen om he expansion o Eq. (4.34). The ex a eedom gained his way has been ound [17] o imp o e he i s o ib a ional molecula spec a. ( ) Since Nis ixed o agi en molecule ( ela ed o he maximum numbe o bound s a es below he dissocia ion limi ), he i s h ee e ms in Eqs. (4.26) and (4.29) ha e no inhuence on he spec um. ( i) In Eq. (4.26) i is clea ha he de o ma ion pa am- e e o he ib a ional pa o he spec um is ~, while o he o a ional pa i is /2. The e o e a ela ion is implied be ween he o a ional s e ching and he anha - monici y co ec ions. Ca e ul empi ical i s a e needed in o de o decide i his is a es ic ion o an ad an age o he p esen model. The e is no ap io i eason, howe e , ha hese wo physically di e en mechanisms be de- sc ibed by he same pa ame e . Amo e gene al e sion o he model, allowing o hese wo de o ma ion pa ame- e s o be independen o each o he , migh gi e be e e- sul s. V. DISCUSSION In his pape ade o med e sion o he O(4) and U(3) dynamical symme ies o he ib on model o dia omic molecules has been cons uc ed. This has been achie ed by i s ew i ing, h ough he use o he concep o com- plemen a y subalgeb as, he model in amo e con enien o m, which is subsequen ly de o med. The p esen ap- p oach uni ies in o acommon amewo k he so a sepa a e algeb aic app oaches o o a ional and o ib a- ional spec a o dia omic molecules. Fo he O(4) limi o he p esen model, i ings o ex- pe imen al da a o dia omic molecules a e equi ed. I s U(3) limi can be used o he desc ip ion o clus e ing phenomena in nuclei [49], as well as o he quasimolecu- la desc ip ion o hea y-ion esonances [50]. The p esen wo k can be ex ended o he s udy o ia omic molecules [23]. The me hod o complemen a y subalgeb as can also be used in cons uc ing [30] he de o med e sions o he U(5) and O(6) dynamical symme ies o he in e ac ing boson model [24] o nuclea s uc u e. 50 q-DEFORMED VIBRON MODEL FOR DIATOMIC MOLECULES 1095 [1]P. P. Kulish and N. Yu. Reshe ikhin, Zap. Nauchn. Se- min. LOMI 101,101 (1981). [2]E.K. Sklyanin, Func . Anal. Appl. 16, 262 (1982). [3]V. G. D in eld, in P oceedi ngs o he In e na ional Cong ess o Ma hema icians, edi ed by A. M. Gleason (Ame ican Ma hema ical Socie y, P o idence, RI, 1987), p. 798. [4]M. Jimbo, Le . Ma h. Phys. 11,247 (1986). [5]D. Bona sos, P. P. Rayche , R. P. Rousse , and Yu. F. Smi no , Chem. Phys. Le . 175, 300 (1990). [6]Z. Chang and H. Yan, Phys. Le . A154, 254 (1991). [7]J. G. Es e e, C. Tejel, and B.E.Villa oya, J.Chem. Phys. 96, 5614 (1992). [8]P. P, Rayche , R. P. Rousse , and Yu. F. Smi no , J. Phys. G16,L137 (1990). [9]D. Bona sos, E.N. A gy es, S.B.D enska, P. P. Rayche , R. P. Rousse , and Yu. F.Smi no , Phys. Le . B251, 477 (1990). [10]D. Bona sos, A. Faessle , P. P. Rayche , R. P. Rousse , and Yu. F.Smi no , J.Phys. A25, 3275 (1992). [11]D. Bona sos, S.B.D enska, P. P. Rayche , R. P. Rousse , and Yu. F.Smi no , J.Phys. G17,L67 (1991). [12]Z. Chang, H. Y. Guo, and H. Yan, Phys. Le . A156, 192 (1991). [13]Z. Chang and H. Yan, Phys. Re . A44, 7405 (1991). [14]Z. Chang, H. Y.Guo, and H. Yan, Commun. Theo . Phys. 17, 183 (1992). [15]D. Bona sos and C. Daskaloyannis, Phys. Re . A46, 75 (1992). [16]D. Bona sos and C. Daskaloyannis, Chem. Phys. Le . 203, 150(1993)~ [17]D. Bona sos, E.N. A gy es, and P. P. Rayche , J.Phys. A 24, L403 (1991). [18]D. Bona sos, P. P. Rayche , and A. Faessle , Chem. Phys. Le . 178,221 (1991). [19]D. Bona sos, C. Daskaloyannis, and K. Kokko as, Phys. Re . A45, R6153 (1992). [20]D. Bona sos, C. Daskaloyannis, and K. Kokko as, J. Ma h. Phys. 33, 2958 (1992). [21]F.Iachello, Chem. Phys. Le . 78, 581 (1981). [22] F. Iachello and R. D. Le ine, J. Chem. Phys. 77, 3046 (1982). [23]O. S. an Roosmalen, F. Iachello, R. D. Le ine, and A. E. L.Diepe ink, J.Chem. Phys. 79, 2515 (1983). [24] F. Iachello and A. A ima, The In e ac ing Boson Model (Camb idge Uni e si y P ess, Camb idge, 1987). [25] D. Bona sos, In e ac ing Boson Models o Nuclea S uc u e (Cla endon, Ox o d, 1988). [26] J.Van de Jeug , J.Phys. A25, L213 (1992). [27]Y. F. Cao and H. L. Lin, Chem. Phys. Le . 207, 317 (1993). [28]F.Pan, J.Phys. B26, IA7 (1993). [29]D. Bona sos, A. Faessle , P. P. Rayche , R. P. Rousse , and Yu. F.Smi no , J.Phys. A25, L267 (1992). [30]F. Pan, In e na ional Cen e o Theo e ical Physics T ies e Repo , 1992(unpublished). [31]M. Moshinsky and C. Quesne, J. Ma h. Phys. 11, 1631 (1970). [32]G. Cou eu , J. Deenen, and C. Quesne, J. Ma h. Phys. 24, 779 (1983). [33]C. Quesne, J.Phys. A18, 2675 (1985). [34]P. M. Mo se, Phys. Re . 34, 57 (1929). [35]F.Pan and Y.F.Cao, J.Ma h. Phys. 29, 2384 (1988). [36]H. Ui, Ann. Phys. (N.Y.)49, 69 (1968). [37]W. J. Holman III and L. C. Biedenha n, J .,Ann. Phys. (N.Y.)39, 1(1966). [38]P. P. Kulish and E. V. Damaskinsky, J.Phys. A23, IA15 (1990). [39]H. Ui and N. Aizawa, Mod. Phys. Le . A5, 237 (1990). [40]N. Aizawa, J.Phys. A26, 1115(1993). [41]L. C.Biedenha n, J.Phys. A22, L873 (1989). [42] A. J.Mac a lane, J.Phys. A22, 4581 (1989). [43]T.Maekawa, J.Ma h. Phys. 32, 2598 (1991). [44] Yu. F. Smi no , V. N. Tols oy, and Yu. I. Kha i ono , Yad. Fiz. 53, 959 (1991) [So . J. Nucl. Phys. 53, 593 (1991)]. [45] Yu. F. Smi no , V. N. Tols oy, and Yu. I. Kha i ono , Yad. Fiz. 53, 1746 (1991) [So . J. Nucl. Phys. 53, 1068 (1991)]. [46] N. Aizawa, J.Ma h. Phys. 344, 1937(1993). [47] R. N. Al a ez, Diploma hesis, Moscow S a e Uni e si y, 1992. [48]J.L.Dunham, Phys. Re . 41, 721 (1932). [49]J.Cseh, J.Phys. G19,L63 (1993). [50]Z. Chang and H. Yan, Academia Sinica Repo No. ASITP-91-25, 1991(unpublished).