Classical Chaos on Double Nonlinear Resonances in Diatomic Molecules

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1 Junal f Mden Physics, 05, 6, Published Online Mach 05 in SciRes. Classical Chas n Duble Nnlinea Resnances in Diatmic Mlecules G. V. López, A. P. Mecad Depatament de Física, Univesidad de Guadalajaa, Guadalajaa, Mexic gulpez@udgsev.cenca.udg.mx, en-gel-8903@htmail.cm Received 5 Febuay 05; accepted 5 Mach 05; published 7 Mach 05 Cpyight 05 by auths and Scientific Reseach Publishing Inc. This wk is licensed unde the Ceative Cmmns Attibutin Intenatinal License (CC BY). Abstact Classical chatic behavi in diatmic mlecules is studied when chas is diven by a ciculaly plaized esnant electic field and expanding up t futh de f appximatin the Mse s ptential and angula mmentum f the system. On this duble esnant system, we find a weak and a stng statinay ( citical) pints whee the chatic chaacteistics ae diffeent with espect t the initial cnditins f the system. Chatic behavi aund the weak citical pint appeas at much weake intensity n the electic field than the electic field needed f the chatic behavi aund the stng citical pint. This classical chatic behavi is detemined thugh Lyapunv expnent, sepaatin f tw neaby tajecties, and Fuie tansfmatin f the time evlutin f the system. The theshld f the amplitude f the electic field f appeaing the chatic behavi nea each citical pint is diffeent and is fund f seveal mlecules. Keywds Classical Chas, Duble Resnace, Nnlinea Dynamics, Diatmic Mlecules. Intductin Beside the clea imptance f the study f diatmic mlecules [] and [], ne f the actual inteests in classical chatic behavi f diatmic mlecules, due t duble nnlinea esnances, is the cnnectin with its assciated quantum dynamics [3]-[5]. F a quantum system assciated t nn chatic classical ne, it is mstly believed that classical dynamical behavi must ccu f lage quantum numbes high value f the actin vaiable [6] [7]. Hweve, f the quantum cunte pat f a chatic classical system the situatin can be vey diffeent [8] [9], whee the assciated actin quantum numbe when chas has his manifestatin n classical system is small [0] []. In this case the quantum manifestatin f chas is athe a subtle matte. These studies have been dne s fa using the cdinates f angle-actin in the Hamiltnian fmalism f diatmic mlecule sys- Hw t cite this pape: López, G.V. and Mecad, A.P. (05) Classical Chas n Duble Nnlinea Resnances in Diatmic Mlecules. Junal f Mden Physics, 6,

2 G. V. López, A. P. Mecad tem [], whee a smewhat atificial nnlinea actin tem is intduced n the system [0], keeping the angula mmentum at ze appximatin. Hweve, the nnlinea tems can be als intduced natually by taking highe tems n the appximatin n the ptential enegy f lage amplitude f scillatins f the system, and by ding the same type f appximatin with the angula mmentum f the systems. On the the hand, when nnlinea esnances appea n a classical system, chatic behavi f the system is deteminated by Chiikv s citeia f velapping esnances [3] [4]. Hweve, this citein is nt cnvenient f u study since ne f the esnances is weak (small stability egin in phase space) and the the is vey stng (lage stability egin in phase space). T detemine the chatic behavi n the system we use Lyapunv expnent, sepaatin f tw neaby tajecties, and Fuie tansfmatin f the time evlutin f the system. In this study we shw that it is pssible t bseve the types f chatic behavi whee chas can depend n cnditins aund the citical pints (initial cnditins chatic behavi), and we pceed in the fllwing way: we establish the evlutin equatins f a diatmic mlecule within a cicula esnant electic field f lage amplitude scillatins, making up t futh de f appximatin n the ptential inteactin between atms and the angula mmentum f the system. We slve numeically the esulting Hamiltnian equatins and calculate the Lyapunv, distance between tw neaby tajecties, and Fuie tansfmatin t detemine whethe nt the tajecty is chatic nt [5]. F ne selected diatmic mlecule, we chse initial cnditins nea the weak and the stng citical pints and incease the magnitude f the electic field until the chatic behavi appeas n each case (expeimentally, this chatic behavi can be measued by electn diffactin technique [6]). Finally, the same study is dne in the diatmic mlecules.. Equatin f Mtin The study f diatmic mlecule is a typical tw bdies pblem with adial fce as shwn in Figue, whee m and m ae the masses f the tw atms, and ae thei psitin, = is the elative cdinate, and R = ( m+ m ) ( m+ m) is the cente f mass cdinate. It is well knwn that with these last tw cdinates, the equatins f mtin ae educed fm 6-D t 3-D pblem, and the equatins ae witten as U + = 0, =, () ( m m ) ˆ R µ whee µ = mm ( m + m ) is the educed mass f the system, and U U( ) = is the ptential due t the cental fce between de mlecules. Due t the symmety unde tatin f the system, the elative mtin is educed t -D pblem and its Figue. Tw bdies cental fce case. 497

3 G. V. López, A. P. Mecad equatin is given in spheical cdinates by whee the effective ptential is being l the angula mment f the system with µ = Veff ( ), () l l ϕ Veff ( ) = + U ( ) l = lθ +, (3) µ sin θ l = µ θ l = µ sin θϕ. (4) θ ϕ The cnstant f mtin (enegy) assciated t this system is and its Lagangian is Theefe, its Hamiltnian is l K(, ) = µ + + U ( ), (5) µ l ( ), (6) µ L = µ U L = µ ( + θ + ϕ sin θ) U( ). (7) P l H = + + U ( ) with P = µ. (8) µ µ 3. Appximatin n Ptential and Angula Mment Let aund this pint. Defining the vaiable as and be the minimum f the effective ptential U( ), and let us expand the functins ( ) = (with < ), it fllws that iv ( ) ( ) ( ) U and U U U 3 4 U( ) = U( ) + U ( ) , (9)! 3! 4! ( + ) ( + ) k ( ) ( k + ) k = = =, (0) k + k = Since ne has that U ( ) = 0, let us define ω (the natual fequency f scillatin f the mlecule) fm the elatin µω = U ( ). S, u Hamiltnian a futh de is whee ne has that k k iv ( ) ( k + ) ( ) ( ) P 4 l U U 3 4 H = + µω + +, k µ + + µ k = 3! 4! () P = P. The ptential assciated t the mlecula inteactin ( ) ( ) U U is just the Mse s ptential [7], a ( ) = D( e ), () whee D,, and a ae paamete detemined f each mlecule. The paamete D epesents the dissciatin enegy f the mlecule and the deep f the ptential, is the minimum f this ptential and the equilibium distance f the atms, and a is elated with the width f the ptential, and it fllws that 498

4 G. V. López, A. P. Mecad 4 3 ( iv ( ) ( ) ( ) ( ) ) 7a U = 0, U = 0, U = a D, U = a D, U ( ) = D. (3) Then, the abve Hamiltnian can be witten f the fm whee and H and W ae defined as Let us ecall that ( ) ( ) ( ) H, P, l = H, P, l + W, l, (4) H (, P, l) P = + + (5) l µω µ µ W( l) l ad 7 ad , = µ l can be witten in tems f the genealized linea mmenta as Then, Hamiltn s equatins f mtin ae P ϕ l = Pθ +, P, P sin. θ = µ θ ϕ = µ ϕ θ (7) sin θ P =, (8) µ l 6 0 P = µω + 3 ad ad, (9) 3 µ 3 4 Pθ θ = (0) µ 3 4 cs P P θ ϕ θ =, () sin θµ 3 4 Pϕ ϕ = () µ sin θ P ϕ = 0 (3) Fm the last equatin ne has that P ϕ = cnstant, and the ttal angula mmentum l is anthe cnstant f mtin. Thus, by chsing the mtin at θ = π, the dynamical system is educed t the fllwing tw dimensinal autnmus system P (6) =, µ (4) l 6 0 P = µω + 3 ad ad (5) 3 µ The set f citical pints f this system, ( j ) {, 0, 0 } Ω c = j P R = P =, is given by l 6 j j 0 j Ω c = ( j,0) µω j + 3aDj ad j + + = µ that is, the citical pints ae lcated ve the -axis, and they ae detemined by the eal ts f a thid de plynmial, which means that ne will have ne thee eal ts, depending n the values f the cefficients. (6) 499

5 G. V. López, A. P. Mecad As it is knwn [8] [9], the natue f this citical pints is detemined by the tace and deteminate f the Jacbian matix, J j 0 P µ = = P P 3 4 l 3 4 µω + 6aDj 7aD j ( 6 4 j + 60 j ) 0 P µ j Since ne has that t ( J j ) = 0, this implies that the citical pints ae cente pints (if det ( J j ) > 0 ) hypeblic pints(if det ( J j ) < 0 ). F example, f the BeO mlecule (Beilium Oxide), the paamete ae (in MKS units) and its chaacteistic fequency is a = = m, D J, = µ = The set citical pints is c {(,0 ),(,0 ),( 3,0) } and ne has m, kg, Hz. j (7) ω = (8) Ω =, whee, j =,, 3 ae given by = m, = m, = m, (9) ( ) ( ) ( ) 3 Theefe, ( ) and ( ) epesent centes, and ( ) j det J > 0, det J < 0, det J > 0. (30),0,0 3,0 epesents an hypeblic pint. Sme tajecties n the phase space (, P ) can be seen in Figue f this mlecule, btained numeically by slving (4) and (5). These tajecties epesent the egula behavi f the system (Lyapunv expnent is negative, tw neaby tajecties emain always neaby, Fuie tansfmatin f any f these tajecties has nly peaks). The values j, j =,, 3 f sme diatmic mlecules ae shwn n the table f appendix A, whee the Mse s paametes assciated t each mlecule wee taken fm [7]. 4. Adding Electic Field and Nn Autnmus Dynamical System Diatmic mlecules with a dipla mment p can inteact with an extenal electic field. The diple electic Figue. Tajecties n the phase space f the mlecule BeO. 500

6 G. V. López, A. P. Mecad mment is just the chage times the distance between atms, p = q, whee in spheical cdinates = ( sinθcs ϕ,sinθsin ϕ,csθ). If the electic field E is chsen f the fm E = ( E cs ωte, sin ωt,0), the enegy f inteactin is U = p E, and the Hamiltnian f inteactin is then ( ) sinθcs( ϕ ω ) H = q + E t (3) int Let d = q sinθcs ϕ ωt, and let us absb the cnstant tem de n the definitin f the Hamiltnian. Thus, ne cnside the Hamiltnian f inteactin as d be the aveage f dipla mment ve the angles and time, ( ) In this way, using (4), (5) and (6), the full Hamiltnian is and the equatins f mtin ae nw ( ) H = qe sinθcs ϕ ωt. (3) int ( ) ( ) ( ) int ( θ ) H, P, l, t = H, P, l + W, l + H,, t, (33) P =, µ (34) ( ) 3 P = qesinθ cs ϕ ω t µω + 3a D 7 l 6 0, ad µ (35) 3 4 Pθ θ = (36) µ ( ) P = qe csθcs ϕ ωt θ 3 4 csθ Pϕ , sin θµ (37) 3 4 Pϕ ϕ = (38) µ sin θ ( ) P ϕ = qesinθsin ϕ ωt. (39) By chsing the study f mtin at θ = π as befe, ne btains that P θ = cnstant, and system is educed t a fu dimensinal nn autnmus system P =, µ (40) ( ) 3 P = qe cs ϕ ω t µω + 3a D 7 l 6 0, ad µ (4) 3 4 Pϕ ϕ = (4) µ ( ) P ϕ = qesin ϕ ωt. (43) These equatin ae slved numeically t find the dynamical behavi f the system. What we ae inteested in is n the theshld f the intensity f the electic field E f the system t becme chatic. T d this, we use the Pincaé stbscpic map [9], Lyapunv paamete [9], distance between tw neaby tajecties, and the Fuie tansfmatin t see the the pwe spectum [9]. If Lyapunv expnent f the tajecty is psitive, 50

7 G. V. López, A. P. Mecad if the distance between tw neaby tajecties gws, and if the Fuie tansfmatin f the tajecty has a cntinuus cmpnent, ne can be sue that the tajecty is chatic. 5. Numeical Results Let us cnside the diatmic mlecule BeO and the initial cnditins nea the weak citical pint, with 5 5 between 5 0 m t m, and with P = 0, θ = π, P θ = 0.0 kg m sec, ϕ = 0.0, P ϕ = 0.0 kg m sec, and ω = ω ( π) ( ) = a D µ. Taking 0 diffeent initial cnditins n the mentined ange f values nea pint, we make the analysis f each tajecty with the tl mentined n the 3 last sectin as a functin f the magnitude f the electic field E. F electic fields such that qe < 0 N the tajecties ae quasi-peidic, the tajecties n the phase space (, P ) ae clsed ellipses-like cuves, the Lyapunv is nt psitive, tajecties which ae initially infinitesimally sepaated emain infinitesimally sepaated, and the Fuie tansfmatin shw nly peaks (as it was mentined befe whee the egula mtin was shwn n the phase space). These elements shw that the behavi f the system is egula at these magnitude f electic field. 3 F an intensity f the electic field such that qe = 0 N, Figue 3 shws the Lyapunv as a functin f time, which becmes psitive. Figue 4 shws the Pincaé map (stbscpic map), which becmes diffused. Figue 5 shws the sepaatin (as a functin f time) between tw neaby tajecties, with sudden vey big values, and Figue 6 shws the discete Fuie tansfmatin f ne f the ten tajecties, shwing a cntinus Figue 3. Lyapunv expnent behavi. Figue 4. Pinceé map. 50

8 G. V. López, A. P. Mecad Figue 5. Distance between tw tajecties. Figue 6. Discete Fuie tansfmatin f ( t). cmpnent. As we can see clealy fm these figues, this tajecty is chatic and the system behaves as chatic system (the same was dne f the the nine tajecties). 5 5 When initial cnditins ( ) ae chsen utside the ange ( 5 0 m, m) and with P = 0, clse t the citical pint with 3, the behavi f the tajecties is egula at this intensity f the electic field. 3 The tansitin egin (just f qe 0 N ) is nt pesented in this study. This analysis was dne f each mlecule listed n Appendix A, finding the theshld f the system t becme chatic with initial cnditins clse t its assciated (, P 0 = ) citical value. F the abve initial cnditins and f seveal highe values f the electic field, we checked the chatic behavi f the the tajecties. Nw, chsing the initial cnditins clse t the citical value (, P 0 3 = ), the behavi f the diatmic 3 9 mlecule is egula f intensities f the electic field such that qe <.5 0 N. Ten tajecties wee chsen with in the ange m, m. The phase space has ellipse like figue, the Lyapunv expnent is nn psitive, tw tajecties, initially infinitesimally sepaated, emain infinitesimally sepaated, and the Fuie tansfmatin pesents just peaks. This mean that up t this amplitude f electic field, the tajecties with these initial cnditins pesents a egula behavi (figues ae nt shwn f these statements, but Figue befe can be taken as a efeence f this egula behavi). 9 F an intensity field such that qe =.5 0 N, Figue 7 shws the Lyapunv expnent as a functin f time, Figue 8 shws the distance between tw neaby tajecties as a functin f time, Figue 9 shws the stbscpic map, and Figue 0 shws the discete Fuie tansfmatin f ne f the tajecties. 503

9 G. V. López, A. P. Mecad Figue 7. Lyapunv s expnent as a functin f time. Figue 8. Distance between tw tajecties. Figue 9. Stbscpic map. 504

10 G. V. López, A. P. Mecad Figue 0. Discete Fuie tansfmatin f a ( t). These figues shw that the tajecties ae chatic with this aptitude f electic field. In fact we checked that the same happen independently f the initial cnditins chsen and f highe values f the magnitude f electic 9 field. The tansitin egin (just f qe 0 N ) is nt pesented in this study. The same analysis was dne f each mlecule listed n Appendix A t find its theshld electic field f appeaing the chatic behavi. The table n Appendix B shws the theshld values f the electic field f the appeaing f chatic behavi f the tajecties aund the weak citical pint (, P 0 = ) and the stng citical pint (, P 0 ) 3 =. Of 3 cuse, if a tajecty is chatic aund the citical pint (, P 0 3 = ), it will be chatic with espect the ci- 3, P 0 =. tical pint ( ) 6. Cnclusin and Cmments We pesented the study f the classical chatic behavi f a diatmic mlecule diven by a ciculaly plaized esnant electic field. The duble esnance system appeas fm expanding up t futh de f appximatin the Mse s ptential and angula mmentum. Chatic behavi f tajecties aund the weak citical pint appeas at much weake electic field stength than the stength f the electic field needed t appea the chatic behavi f tajecties aund the stng citical pints. This esult pints ut the pssible chatic behavi f duble nnlinea esnant systems depending n its initial cnditin. The exact tansitin egin t chatic behavi will be pesented in the aticles. The gap (weak-stng) n the theshlds f the electic field stength t ccu the chatic behavi may be imptant f the study f diatmic mlecules in diffeent envinments and f quantum dynamical studies. Refeences [] Butn, M. (987) Ast. Sc., 8, 69. [] Chevalie, R. (999) The Astphysical Junal, 5, [3] Shuyak, É.V. (976) Sv. Phys. JEPT, 44, 070. [4] Pasn, R.P. (987) The Junal f Chemical Physics, 88, [5] Dadi, P.S. and Gay, K. (98) The Junal f Chemical Physics, 77, [6] Messiah, A. (964) Quantum Mechanics I. Nth Hlland, Jhn Wiley & Sns, Inc., New Yk, Lndn, 9. [7] Lmbadi, M., Labastie, P., Bdas, M.C. and Bye, M. (988) The Junal f Chemical Physics, 89, [8] Beman, G.P. and Klvsky, A.R. (989) Sv. Phys. JEPT, 68, 898. [9] Beman, G.P. and Klvsky, A.R. (99) Sviet Physics Uspekhi, 35, [0] Beman, G.P., Bulgakv, E.N. and Hlm, D.D. (995) Physical Review A, 5,

11 G. V. López, A. P. Mecad [] López, G.V. and Zanud, J.G.T. (0) Junal f Mden Physics,, [] Reichl, L.E. (004) The Tansitin t Chas. Spinge-Velag, Belin. [3] Lichtenbeg, A.J. and Libeman, M.A. (983) Regula and Stchastic Mtin. Spinge-Velag, Belin. [4] Chiikv, B.V. (979) Physics Repts, 5, [5] Dazin, P.G. (99) Nnlinea Systems. Cambidge Univesity Pess, Cambidge. [6] Geshitv, A.G., Spiidnv, V.P. and Butayev, B.S. (978) Chemical Physics Lettes, 55, [7] Mse, P.M. (99) Physical Review, 34, [8] Pek, L. (996) Diffeential Equatins and Dynamical Systems. nd Editin, Spinge, Belin. [9] Stgatz, S.H. (994) Nnlinea Dynamics and Chas. Peseus Bks, New Yk City. 506

12 G. V. López, A. P. Mecad Appendix A N. Mlecule (m) (m) 3 (m) BeO BeO 3 BO 4 BO 5 BO 6 AlO 7 AlO 8 C 9 C 0 CN CN CN 3 CO 4 CO 5 CO 6 CO 7 CO 8 CO CO CO CO F F H H H H H H H H H H H I I N

13 G. V. López, A. P. Mecad Cntinued 38 N 39 N 40 N 4 N NO 45 NO 46 NO 47 NO 48 O 49 O 50 O SiN 56 SiN N N O O O O Appendix B N. Mlecule (,0 ) N (,0) 3 BeO BeO 3 BO 4 BO 5 BO 6 AlO 7 AlO 8 C 9 C 0 CN CN N CN 3 CO 4 CO 5 CO 6 CO 7 CO

14 G. V. López, A. P. Mecad Cntinued 8 CO 9 CO + 0 CO + CO + F 3 F 4 H 5 H 6 H 7 H 8 H 9 H 30 H 3 H 3 H 33 H I 36 I 37 N 38 N 39 N 40 N 4 N NO 45 NO 46 NO 47 NO 48 O 49 O 50 O SiN 56 SiN H + 4 N + 3 N O + 3 O O + 3 O