Absorption and Recurrence Spectra of Li Rydberg Atom in Perpendicular Electric and Magnetic Fields

Transcription

1 Commun. Theor. Phys. (Bejng, Chna) 46 (2006) pp c Internatona Academc Pubshers Vo. 46, No. 3, September 15, 2006 Absorpton and Recurrence Spectra o L Rydberg Atom n Perpendcuar Eectrc and Magnetc Feds WANG De-Hua, 1,2, LIN Sheng-Lu, 3 WANG Me-Shan, 1 and YANG Chuan-Lu 1 1 Coege o Physcs and Eectronc Engneerng, Ludong Unversty, Yanta , Chna 2 Schoo o Physcs and Mcroeectroncs, Shandong Unversty, Jnan , Chna 3 Department o Physcs, Shandong Norma Unversty, Jnan , Chna (Receved Juy 18, 2005; Revsed October 17, 2005) Abstract We deveop the sem-cosed orbt theory rom two degrees o reedom to three non-separabe degrees o reedom and put orward a new mode potenta or the L Rydberg atom, whch reduces the study o the system to an eectve one-partce probem. Usng ths mode potenta and the cosed orbt theory or three degrees o reedom, we cacuate the recurrence spectra o L Rydberg atom n perpendcuar eectrc and magnetc eds. The cosed orbts n the correspondng cassca system have aso been obtaned. The Fourer transormed spectra o L atom have aowed drect comparson between the resonance peas and the scaed acton vaues o cosed orbts, whereas the nonhydrogenc resonance can be expaned n terms o the new orbts created by the core scatterng. Our resut s n good agreement wth the quantum spectra, whch suggests that our cacuaton s correct. PACS numbers: r, , Sq Key words: cosed orbt theory, mode potenta, recurrence spectra, core scatterng 1 Introducton Hgh Rydberg states o atoms n strong externa eds have payed an mportant roe n atomc physcs. These atoms represent quantum mechanca systems whose cassca counterparts show chaotc behavor. M.L. Du and J.B. Deos s semcassca cosed orbt theory ntroduced a new way o oong at the photoabsorpton spectra o Hamtonans havng chaotc cassca dynamcs. [1] For atoms n pure magnetc ed and n parae eectrc and magnetc eds, [1 6] the system st possesses a cyndrca symmetry about the drecton o the magnetc ed, thus the anguar moton perpendcuar to the magnetc ed can be separated, whch reduces the probem to a two-dmensona one. However, n perpendcuar eectrc and magnetc eds, the cyndrca symmetry s broen, and the Hamtonan s nonseparabe n three degrees o reedom, thereore the theoretca treatment s more compcated. Rao and Tayor [7] put orward the rst quantum mechanca cacuaton or non-hydrogenc atoms n scaed crossed eds and deduced the occurrence o scatterng n such systems between cosed orbts cacuated or the correspondng hydrogen system. Moreover, they obtaned a the out-o pane cosed orbts manested n the correspondng quantum recurrence spectrum. Recenty, a number o expermenta and theoretca evdence on spectra and wave-pacet dynamcs suggests that core eects have mportant dynamca eects not seen n hydrogen. In nonhydrogenc atoms, the core nduces pure quantum eects, especay addtona spectra moduatons, whch cannot be anayzed reaby n terms o cassca orbts and ther stabty parameters. Through quantum deect theory, Dando et a. extended the cosed orbt theory rom hydrogen atom to nonhydrogenc atoms and put orward or the rst tme the scatterng eect o a hydrogenc cosed orbts by a non-hydrogenc core. [8] However, as wth Du and Deos, they ony treated two-dmensona systems. In ths paper, we deveop the cosed orbt theory rom two-dmensona to three-dmensona one and treat the same scatterng eects exsted n the three-dmensona systems. Furthermore, a new mode potenta or the L Rydberg atom s used, whch ncude not ony the eectrostatc core potenta produced by a nuceus o charge z = 3 but aso the exchange potenta between the excted eectron and the other two 1s eectrons. Thereore, t s much more accurate n contrast to the hydrogenc potenta n the prevous wor. [8] We nd that the ntroducton o the core mode potenta eads to an extreme ncrease o the number o cosed orbts as compared to hydrogen atom. The cacuaton o the recurrence spectra by usng cosed orbt theory requres the numerca search or cosed orbts startng and endng exacty at the nuceus. Ths resuts n numerca dcutes because o the Couomb snguarty o the potenta at the orgn. In order to remove ths nd o snguarty, we adopted the Kustaanhem and Stee transormaton, [9] whch transorms the system rom a three-dmensona one to a our-dmensona one. Recurrence spectra are the Fourer transormaton o The project supported by Natona Natura Scence Foundaton o Chna under Grant No and the Doctora Research Foundaton o Ludong Unversty under Grant No To whom correspondence shoud be addressed, E-ma: jnwdh@sohu.com

2 546 WANG De-Hua, LIN Sheng-Lu, WANG Me-Shan, and YANG Chuan-Lu Vo. 46 the photoabsorpton spectra. It can be shown that each cosed cassca orbt o the eectron generates a pea n the recurrence spectra at the acton o the orbt. Consequenty, the recurrence spectra provde a quantum pcture o the cassca behavor. Studes o recurrence spectra have ead to observatons o the creaton o new orbts through burcatons, [3,10] symmetry breang n crossed eds, [11] etc. In the past, many researchers have cacuated the recurrence spectra o L Rydberg atom n strong externa ed, [8,12,13] but they ony cacuated the spectra n pure eectrc ed or magnetc ed or n parae eectrc and magnetc eds. As or the spectra n perpendcuar eectrc and magnetc eds, due to the compcaton o the theoretca treatment, none has gven the computaton. In the present wor, by usng cosed orbt theory or three degrees o reedom and a new mode potenta, we cacuate the spectra o L Rydberg atom n perpendcuar eectrc and magnetc eds. Our resuts are n good agreement wth the quantum spectra, whch suggest that our resuts are correct. Our paper s organzed as oows. In Sec. 2, we gve a bre descrpton o the cassca dynamcs o the system and put orward a new mode potenta or the onc core. The cosed orbts o the correspondng system have aso been ound. In Sec. 3, we extend the cosed orbt theory rom two degrees o reedom to three nonseparabe degrees o reedom. Secton 4 gves the cacuaton expressons or the Fourer transormed recurrence spectra o L atom. In Sec. 5, we cacuate the absorpton and recurrence spectra o L atom and compare t wth those o H atom and the quantum spectra. Secton 6 s the concusons o ths paper. 2 Cassca Dynamcs 2.1 Mode Potenta For the L Rydberg atom, where ony one eectron s hghy excted to the onzaton threshod whe the other two eectrons reman n the 1s ground state. In ths stuaton, the nucear charge s screened by the nner eectrons. Hence, n the prevous study, some authors consder that the potenta actng on the excted eectron s ony an attractve Couomb potenta moded by a short-ranged core potenta, [6,14] but negect the nuence o the exchange nteracton potenta between ths excted eectron and other eectrons. In act, the cosed orbts are senstve to the exchange nteracton. Thereore, the potenta o the L Rydberg atom actng on the excted eectron shoud be the eectrostatc core potenta produced by a nuceus o charge z = 3 and the exchange potenta between the excted eectron and the other two 1s eectrons. It may be ormuated as V (r) = V n (r) + V m (r), (1) where V n (r) s the nucear-attractve potenta to the excted eectron, V n = z/r, and V m (r) s a mode potenta. The potenta V (r) shoud satsy the correct asymptotc condtons V (r) (z 2)/r and V (r) z/r. r r 0 In the Hartree Foc (HF) approach, the wave uncton or a vaence eectron n the presence o a 1s 2 core s the souton o the Schrödnger equaton: wth [H HF ε]ϕ n (r) = 0 (2) H HF = z r + V core(r) + V ex (r), (3) where V core (r) s the eectrostatc core potenta arses rom the 1s 2 core and V ex (r) s the exchange potenta between the excted eectron and the other two 1s eectrons, V core (r ) = ϕ 1s (r j ) r 1 j ϕ 1s(r j ) rj, (3a) V ex (r ) = ϕ 1s(r ) ϕ n (r ) ϕ 1s(r j ) r 1 j ϕ n(r j ) rj. (3b) The subscrpt r j ndcates that the ntegraton must be done over ths coordnate. Thereore, the mode potenta can be wrtten as V m (r) = V core (r) + V ex (r). (4) Wth respect to L Rydberg atom, two eectrons e n the 1s ground state whe the vaence eectron s excted to the 3s state. We can use hydrogenc unctons to descrbe the 1s eectron and the 3s excted eectron. Then, the eectrostatc core potenta taes the we-nown orm, [15] V core (r) = 2 r 2 r (1 + 3r) e 6r. V ex s the exchange potenta between the excted eectron and the other eectrons, or the 1s 2 3s state, t taes the orm 81 ( V ex = 2r 2 r 3 + 4r r r + 9 ) e 6r. (5) 16 It must be noted that the exchange potenta has snguartes at r = r ± wth r ± = (3/2)(3 ± 7). However, snce ths potenta s a short-ranged potenta, we can use t n the core regon (r 0), where the exchange wth the core eectrons s more mportant. The core potenta has the usua Couomb snguarty at the orgn, and ater Kustaanhemo Stee transormaton, [9] Couomb snguarty vanshes. By tang the mode potenta Eq. (1) (ncudng the exchange potenta) nto the anaytca equaton o the quantum deect, [6] µ = 2 π m R [ R V (r) r 0 ( + 1/2)2 2r 2 dr R r 0 1 r ( + ] 1/2)2 2r 2 dr,

3 No. 3 Absorpton and Recurrence Spectra o L Rydberg Atom n Perpendcuar Eectrc and Magnetc Feds 547 where r 0 = (1/2)[ + (1/2)] 2, we cacuate the quantum deects o the L atom and compare our resut wth those gven n Re. [6]. It shows that our resut s coser to the quantum mechanca resut, see Tabe 1. Tabe 1 Quantum mechanca and semcassca quantum deects µ o L atom n Re. [6]. µ s our resut. µ (quantum) µ (semc.) < 10 4 < 10 4 < Hamtonan and the Scaed Varabes The exact Hamtonan or an N-eectron atom n externa eds contans a nteractons o the eectrons, the nuceus and the externa eds. Snce we are deang wth the exctaton o Rydberg states, where ony one eectron s hghy excted, whe the onc core remans n the ground state. We can smpy the probem by consderng the Hamtonan o a snge eectron movng n an attractve Couomb potenta moded by a mode potenta combned wth the perpendcuar externa eds. Assumng the magnetc ed pontng n the z-axs and the eectrc ed aong the x-drecton, then the Hamtonan o the hghy excted eectron o hydrogenc atom s descrbed n atomc unts as [14] H = 1 2 P 2 + γ 2 z γ2 ρ 2 z r + F x, (6) where γ = B/ (B s magnetc ed strength), F s eectrc ed strength. The cassca moton o Hamtonan (Eq. (6)) exhbts an mportant scang property. I we transorm varabes accordng to r = rγ 2/3, P = P γ 1/3, ε = Eγ 2/3, = F γ 4/3, t = tγ, then the cassca moton s governed by the scaed Hamtonan, H = 1 2 P z ρ2 + x, (7) z r where z = x p y ỹ p x. I we omt and rewrte H as ε, we get H = 1 2 P z ρ2 z + x = ε. (8) r From Eq. (8), we nd that the scaed Hamtonan does not depend on the energy E and ed strengths B and F separatey, but ony on the scaed energy ε and the scaed eectrc ed, thus reduces a parameter B. Wth the addtona mode potenta, the scang property s no onger exact. But snce the range o the mode potenta s short compared to the extenson o the excted state. Thereore, apart rom a sma regon around the nuceus, the µ Hamtonan does st exhbt the above scang propertes. We can st cacuate the cassca dynamcs o L atom ncudng the mode potenta. 2.3 Cosed Orbt Search In order to sove numercay the moton equatons and remove the snguarty generated by Hamtonan (8), t s convenent to mae a reguarzng transormaton. For the case consdered here, we mpement the Kustaanhemo Stee transormaton [9] as prevousy done by Rao and Tayor, [7] whch transorms the system rom a threedmensona to a our-dmensona one. Remova o the Couomb snguarty maes the numerca cacuaton very stabe and an orbt aunched rom the nuceus can evove qute a ong tme wthout oss o numerca accuracy. Due to the ncreased dmensonaty o the phase space nvoved, the number o the cosed orbts s arge and a search or the cosed orbts s by no means trva. The dcuty es n the act that or a system o three degrees o reedom, one has to scan two ndependent parameters (θ, ϕ) to ocate the cosed orbts, whe n the pure magnetc ed ony one parameter θ needs to be scanned. [7] We use the th-order Cash Karp Runge Kutta method to ntegrate the moton equatons. In prevous wors, [6,14] the core potenta s to be swtched-o at the begnnng and end o the orbts. In contrast, n our cacuaton, we aways tae nto account the mode potenta. Thereore, ths treatment s more compete and more accurate. By ntegratng the Hamtonan moton equatons, we nd that the cosed orbts have the mnmum scaed actons correspondng to the semcassca spectra n the ower resouton,.e. the scaed actons smaer than 10 at ε = 0.45, = For each orbt, we evauate the cassca acton, the cassca amptude as we as the Masov ndex. Some o the panar cosed orbts are gven n Fg. 3. For the panar orbts yng n the pane perpendcuar to the magnetc ed, due to the z-symmetry, ths pane s nvarant under the underyng cassca dynamcs. Thus, the nta drecton o an orbt can be speced by means o the azmutha ange ony. As soon as a sma crossed eectrc ed s present, the one-parameter amy o the damagnetc Keper orbt s destroyed and spt nto two soated cosed orbts. These orbts start n opposte drectons wth respect to the eectrc ed, so that ther azmutha startng anges der by π. An addtona compcaton to obtan the non-panar cosed orbts arses because the poar startng ange θ s no onger bound to the xed vaue π/2, so that the two orbts w n genera have derent vaues o θ. 3 Cosed-Orbt Theory Extended to Three Degrees o Freedom 3.1 Basc Theory and Physca Pcture Accordng to the basc deas presented by Du and Deos, [1] the exact quantum expresson or the oscator

4 548 WANG De-Hua, LIN Sheng-Lu, WANG Me-Shan, and YANG Chuan-Lu Vo. 46 strength densty or dpoe transtons o an nta state ψ to na states at energy E can be expressed as (E) = 2 π (E E ) Im d 3 r (Dψ ) ( r )ψ( r ), (9) ψ out (r, θ, ϕ) = = m where D s the dpoe operator, ψ s the nta wave uncton, and ψ( r) s a souton o the nhomogeneous Schrödnger equaton, (E H)ψ( r ) = (Dψ )( r ). (10) The physca pcture can be descrbed as oows. The eectron s ntay n a state ocated cose to the nuceus, where the externa eds are neggbe compared wth the Couomb orce and the mode potenta. Ater the atom absorbs a photon, an outgong wave ψ out s produced. At sucent arge dstances rom the nuceus, the quantum wave ψ( r ) can be approxmated semcasscay. In the outer regon, where the externa eds must no onger be negected and the wave uncton s now propagated semcasscay aong cassca trajectores. These trajectores are drected raday at rst, and then ee the nuence o the addtona eectrc and magnetc eds. They evove accordng to the Hamtonan moton equatons, where the u potenta s apped. Fnay, some o the trajectores are turned bac to the nuceus by the externa eds, where they orm a returnng wave ψ ret. Thereore, the u wave uncton s a superposton o the outgong and the returnng waves: ψ( r ) = ψ out ( r ) + ψ ret ( r ). (11) The ntererence between the ncomng and outgong waves eads to oscatons n the absorpton spectra. 3.2 Outgong Wave Cose to the nuceus, the nuence o the externa eds can be negected. The outgong wave can be seen as the same as no externa eds were present, [6] Y,m(θ, ϕ )g E=0 (r, r )Y,m (θ, ϕ)dψ ( r )d r, (12) where g E=0 (r, r ) s the rada Green s uncton, g E=0 (r, r ) = 2R0,reg (r < )R 0,out (r > ) r 2 w[r 0,reg (r ), R 0,out (13) (r )] wth r < = mn{r, r }, r > = max{r, r } and w beng the Wronsan determnant. The unctons R 0,reg and R 0,out are the reguar and outgong soutons o the rada part o the Schrödnger equaton at E = 0 wth the asymptotc orms [16] ( R 0,reg 2 ) 1/2 1 [ ( (r) = π cos 8r + 1 ) π π ] 8r 8r δ, (14) ( R 0,out 2 ) 1/2 1 { [ ( (r) = π exp 8r + 1 ) π π ]} 8r 8r δ, (15) where δ s the phase sht caused by the mode potenta. Fnay, the outgong wave can be wrtten as ψ out (r, θ, ϕ) = π where B m = d r Dψ (r )R 0,reg (r )Y,m (θ, ϕ ). 3.3 Semcassca Propagaton = m R 0,out (r)y,m (θ, ϕ)b m, (16) Outsde the regon, where the externa eds become mportant, the outgong wave ψ out can be propagated semcasscay. At a gven pont (r, θ, ϕ), two derent nds o trajectores contrbute to the semcassca wave uncton: trajectores comng drecty rom the nta surace wthout ever eavng the vcnty o the nuceus and trajectores that have traveed away rom the nuceus and have been turned bac by the externa eds. The semcasscay propagated wave then reads ψ sem ( r ) = J (0, θ, ϕ )/J (t, θ, ϕ ) e [s (π/2)µ ] ψ out ( r ). (17) The sum ncudes a cassca trajectores wth energy E started raday on the nta sphere, J(t, θ, ϕ) s the Jacoban determnant, t s the tme when the trajectory reaches (r, θ, ϕ), S s the acton and µ s the Masov ndex. Due to the nuence o the externa eds, some o the trajectores are turned bac. The returnng wave ψ ret contans, on the one hand, the ncomng trajectores whch pass through a gven pont (r, θ, ϕ) that comes rom the regon outsde the vcnty o the nuceus, on the other hand, the outgong trajectores that trave around the onc core and pass through (r, θ, ϕ) when eavng the core agan. The ncomng contrbutons to ψ ret can approxmatey be cacuated rom cosed orbts startng and endng exacty at the nuceus. Let S 0 and µ 0 be the acton and Masov ndex o the cosed orbt. Then µ = µ 0 1, (18)

5 No. 3 Absorpton and Recurrence Spectra o L Rydberg Atom n Perpendcuar Eectrc and Magnetc Feds 549 S = S 0 8r 2 r(1 + cos γ ). (19) Here r s the radus o the nta sphere, and γ s the ange between (r, θ, ϕ) and the drecton (θ, ϕ ) rom whch the cosed orbt returns to the nuceus. Usng the expressons or the Jacoban determnant as gven n Re. [14], we obtan the semcassca approxmaton or ψ ret, ( ) 2 1/4 r 3/4 ) + ψ scat,out (r). (20) ψ ret sem(r) = e (S 0 (π/2)µ 0 ) exp[ 2 r(1 + cos γ )] e 8r ψ out (r M [r(1 + cos γ 0 )] 1/4 The sum now contans a cosed orbts startng at and returnng exacty to the nuceus, M 0 s reated to the stabty matrx o the cosed orbt. [14] The term ψ scat,out (r) represents the contrbutons rom trajectores passng through (r, θ, ϕ) when gong out agan. Expanded equaton (20) n terms o spherca harmoncs, we nay obtan ψsem(r) ret = ( 32π 2 e (S 0 (π/2)µ 0 ) ( 1) e δ B M m Y m (θ, ϕ ) 0,m Here, θ and ϕ 3.4 Returnng Waves ( 1) Y m(θ, ϕ )Y m(θ, ϕ) 1 ( 2 8r π 8r,m ( [ ( exp 8r + 1 ) π π ]) 2 4 are the startng anges o the -th cosed orbt. + ψ scat,out (r) ) 1/2 ). (21) Equaton (21) s vad ony sucenty ar away rom the nuceus. In the vcnty o the nuceus, the returnng waves can be wrtten as ψ ret (r) = A m R 0,reg (r)y m (θ, ϕ). (22),m Usng the asymptotc orm o R 0,reg (r), we get ψ ret (r) = 1 ( 2 8r π 8r ) 1/2,m ( ( A m Y m (θ, ϕ) cos 8r + 1 ) π π ) δ. (23) In order to determne the expanson coecents A m, we spt the cosne nto ncomng and outgong parts, and compare the ncomng parts o Eqs. (23) and (21), we obtan the na resut or the returnng wave, ψ ret sem(r) =,m 64π 2 e (S ( 1) Y m(θ 0 (π/2)µ 0 ) M 0,m ( 1) e δ B m Y m (θ, ϕ ), ϕ )Y m(θ, ϕ) e δ R 0,reg (r). (24) 4 Fourer Transormed Recurrence Spectra By substtutng the above unctons nto Eq. (9), we can get the oscator strength densty, (E) = 2 π (E E ) Im d 3 r (Dψ ) ( r )ψ out ( r) 2 π (E E ) Im d 3 r (Dψ ) ( r )ψ ret ( r ) = 0 (E) + osc (E). (25) The rst term s a sowy varyng bacground o the spectra, whch can be obtaned wthout the externa eds. The second part, whch s composed o contrbutons rom the returnng cosed orbts, eads to oscatons n the absorpton spectra. Insertng ψ ret nto the above equaton, we get osc (E) = 64π e (S 0 (π/2)µ 0 ) M 0,m ( 1) e δ B m Y m (θ, ϕ ),m ( 1) Y m(θ, ϕ )B m e δ. (26) For the Rydberg L atom, the externa-she eectron s excted to the 3s state, thus the nta state wave uncton s ψ (r, θ, ϕ) = R 30 (r)y 00 (θ, ϕ). I we use the x-poarzed ght, then the nonvanshng coecents o B m are B 11 and B 1 1 wth the reaton B 11 = B 1 1. In ths case, the oscatng part o the absorpton spectra s osc (E) = B 11 64π sn θ sn θ cos ϕ cos ϕ ( sn M 0 S 0 π 2 µ ) 0 + 2δ 1. (27)

6 550 WANG De-Hua, LIN Sheng-Lu, WANG Me-Shan, and YANG Chuan-Lu Vo. 46 where Ater the scaed transormaton S z = γ 1/3, M 0 = γ 1/3 S 0 and 0 = γ 2/3 M 0, we obtan osc (z) = 1 ( A sn S 0 z z π ) 2 µ 0 + 2δ 1, (28) A = B 11 64π sn θ sn θ cos ϕ cos ϕ. M 0 The Fourer transormed recurrence spectra can be cacuated n the nterva [z 1, z 2 ]. Wth z = (z 1 + z 2 )/2, z = z 2 z 1, and approxmate 1/z by 1/ z, we get the recurrence spectra, F ( S) = 1 z sn[ z( S 0 S)] [ ( A exp S 0 z π )] ( S 0 S) 2 µ 0 + 2δ 1 e z S. (29) For nte ength z, F ( S) s a compex number. In our cacuaton, we tae the square vaue o F ( S). 5 Resuts and Dscussons Fg. 1 The Photo-absorpton spectra o L atom n perpendcuar eectrc and magnetc eds wth B = 5.96 T and F = 10 V/cm (wthout the smooth part). Fg. 2 The recurrence spectra o atoms n perpendcuar eectrc and magnetc eds at ε = 0.45, = 0.02, z = γ 1/3 n the range o (a) Semcassca cacuaton or the L atom; (b) Semcassca cacuaton or the H atom. Fgure 1 dspays the photo-absorpton spectra o L atom n perpendcuar eectrc and magnetc eds wth B = 5.96 T and F = 10 V/cm. Each oscaton corresponds to a cosed orbt. That means every cosed orbt causes a snusoda oscaton n the absorpton spectra varyng wth energy. In order to show the reaton between the absorpton spectra and the cosed orbt, we mae a Fourer transormaton o the absorpton spectra at constant scaed energy and eectrc ed. Fgure 2 gves the recurrence spectra o atoms n perpendcuar eectrc and magnetc ed at ε = 0.45, = 0.02, z = γ 1/3 n the range o Fgure 2(a) s the recurrence spectra o L atom, and gure 2(b) s the recurrence spectra o H atom. Each pea s assocated wth a cosed orbt. Some o the panar cosed orbts are shown n Fg. 3. From Fg. 2, we can see that or smaer S, the spectra structure o L and H atoms s anaogues. As S ncreases, some new peas appear n the spectra o L atom (as ndcated by arrows). Ths can be nterpreted as oows. In order to cacuate the recurrence spectra o L atom, we have taen nto account the eect o the core potenta and the exchange potenta. The man eect o the core potenta and the exchange potenta es n the creaton o a huge number o new cosed orbts. These new orbts appear to be composed o two or more hydrogenc orbts orgnate rom the scatterng o hydrogenc orbts by the onc core. [7] Ths can be ceary seen rom the cosed orbts o Fg. 3. For exampe, gure 3(b) can be seen as the hydrogenc orbt (a) scattered by the core once and gure 3(c) s scattered by a second tme, whereas gure 3() s scattered by many tmes beore t returns to the orgn. Fgure 4 dspays the semcassca recurrence spectra o Rydberg L atom n perpendcuar externa eds at ε = 0.6, = 0.06, γ 1/3 n the range o In order to compare the resut between the semcassca and the quantum spectra, we cacuate the recurrence spectra o L Rydberg atom n perpendcuar eectrc and magnetc ed by usng quantum mechanca method as gven n Re. [7]. In our cacuaton, we chose the bass set members wth abes rangng over n = 1 to 150, = 0 to 149 and m = 100 to 100 yedng a bass o sze

7 No. 3 Absorpton and Recurrence Spectra o L Rydberg Atom n Perpendcuar Eectrc and Magnetc Feds 551 Fgure 4(a) s the semcassca cacuaton resut, and gure 4(b) s the quantum cacuaton resut. It s apparent rom the gure that the quantum recurrence spectrum retans the pronounced pea structures as the semcassca spectra. Even or the argest acton consdered here, the quantum and semcassca recurrence spectra agree we quanttatvey or some peas. However, the pea heghts n the two gures dsagree a tte. Some peas o the quantum spectra are smaer or even competey absent n the semcassca spectra. They can be attrbuted to the mssng orbts. On the contrary, some semcassca peas are hgher than the quantum peas. These hgh peas can be traced bac to the eects o the burcatng orbts n the semcassca cacuaton. Ths phenomenon s rather common, even or the Star spectra o the hydrogen atom n an eectrc ed, where there are ony uph and downh orbts and no other cosed orbts arsng rom the core scatterng. As or ths probem, we w dscuss t by usng the unorm approxmaton method deveoped by T. Bartsch et a. [17] n the uture wor. Fnay, n Fg. 5 we cacuate the Star recurrence spectra o L atom n eectrc ed (wthout externa magnetc ed) at ε = 1.6 and compare t wth the expermenta one gven n Re. [13]. The resuts are agreed we wth each other, or exampe, the peas n the recurrence spectra o our resut and the expermenta one occur neary at the same poston, whch suggest that our cacuaton method s correct. Fg. 3 Some cosed orbts o Rydberg L atom n perpendcuar eectrc and magnetc ed at ε = 0.45, = These orbts are symmetrc wth respect to x axs and are drawn n the x-y pane (the horzonta axs s x and vertca axs s y). Fg. 4 The recurrence spectra o Rydberg L atom n perpendcuar eectrc and magnetc ed at ε = 0.6, = (a) Semcassca cacuaton resut; (b) Quantum cacuaton resut.

8 552 WANG De-Hua, LIN Sheng-Lu, WANG Me-Shan, and YANG Chuan-Lu Vo. 46 Fg. 5 The star recurrence spectra o L atom at ε = 1.6. The resut s n good agreement wth the expermenta one gven n Re. [13]. 6 Concuson A new mode potenta or the L Rydberg atom s put orward, whch ncudes not ony the Couomb nteracton potenta and the eectrostatc core potenta, but aso the exchange potenta between the excted eectron and other two 1s eectrons. By usng ths mode potenta and the cosed orbt theory or three non-separabe degrees o reedom, we cacuated the recurrence spectra o L Rydberg atom n perpendcuar eectrc and magnetc eds. For gettng the cosed orbts that startng and endng exacty at the nuceus, the non-couombc core potenta and exchange potenta has been consdered n cassca trajectory cacuatons and some core-scattered nonhydrogenc cosed orbts have been dscovered. The new peas n the recurrence spectra o L atom have been consdered as eects caused by the core scatterng o the returnng waves at the onc core, rom whch we gan a deep understandng o the hghy excted Rydberg atoms n perpendcuar eectrc and magnetc eds. Snce these atoms represent a system nonseparabe n three degrees o reedom whch behaves casscay chaotc. The good genera agreement between the expermenta recurrence spectra and the spectra n our paper or the L atom n eectrc ed demonstrates the correctness o our method. Presenty no experments on the recurrence spectra o L atom n perpendcuar externa eds have been carred out. We hope that our cacuatons may gude uture measurements. Reerences [1] M.L. Du and J.B. Deos, Phys. Rev. A 38 (1988) 1986; 38 (1988) [2] D.H. Wang, S.L. Dng, and S.L. Ln, J. Phys. B 36 (2003) [3] A.D. Peters, C. Jae, and J.B. Deos, Phys. Rev. Lett. 73 (1994) [4] J. Man, G. Webusch, K. Wege, et a., Phys. Rev. A 49 (1994) 847. [5] P.A. Dando, T.S. Montero, D. Deande, and K.T. Tayor, Phys. Rev. Lett. 74 (1995) [6] B. Hupper, J. Man, and G. Wunner, Phys. Rev. A 53 (1996) 744. [7] J.G. Rao and K.T. Tayor, J. Phys. B 35 (2002) L1; 35 (2002) 2627; 34 (2001) L391. [8] P.A. Dando, T.S. Montero, D. Deande, and K.T. Tayor, Phys. Rev. A 54 (1996) 127. [9] P. Kustaanemo and E. Stee, J. Ang. Math. 218 (1965) 204. [10] M. Courtney, N. Spemeyer, H. Jao, e a., Phys. Rev. Lett. 74 (1991) [11] C. Neumann, R. Ubert, S. Freund, et a., Phys. Rev. Lett. 78 (1997) [12] Ln Sheng-Lu, Zhang Qu-Ju, et a., Chn. Phys. Lett. 19 (2002) 29. [13] M. Courtney, et a., Phys. Rev. A 51 (1995) [14] K. Webert, J. Man, and G. Wunner, Ann. Phys. 268 (1998) 172. [15] A. Hbbert, Adv. At. Mo. Phys. 18 (1982) 309. [16] J. Gao and J.B. Deos, Phys. Rev. A 46 (1992) [17] T. Bartsch, J. Man, and G. Wunner, Phys. Rev. A 66 (2002)