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Journal of Modern Physics, 2010, 1, 399-404 doi:10.4236/jmp.2010.16057 Published Online December 2010 (http://www.SciRP.org/journal/jmp) Copyright © 2010 SciRes. JMP Calculation of the Effective G-Factor for the 222 12 3212 ns SnpPns S Transitions in Hydrogen-Like Atoms and Its Application to the Atomic Cesium Ziya Saglam1, S. Burcin Bayram2, Mesude Saglam3 1Department of Physics, Faculty of Science, Aksaray University, Aksaray, Turkey 2Department of Physics, Miami University Ohio, USA 3Department of Physics, Ankara University, Ankara, Turkey E-mail: saglam@science.ankara.edu.tr Received August 14, 2010; revised October 25, 2010; accepted October 27, 2010 Abstract We have calculated the effective g-factor * g for the 22 2 12 3212 ns Snp Pns S transitions in hydrogen-like atoms and applied it to atomic cesium. We have identified that not only the g* factor in this case is an integer number g* = 1, but also the existence of possible entangled states related to the above tran- sitions. Furthermore we have used the above result to calculate the transition energies which are in complete agreement (within the 1% margin error). Such results can give access to the production of new laser lights from atomic cesium. Keywords: Photonic Transitions, Hydrogen-Like (Hydrogenic) Atoms, Landé G-Factor, Quantum Entanglement 1. Introduction The study and accurate determination of the excited-state properties of atomic and molecular systems, such as the fine and hyperfine coupling constants and oscillator strengths play an important role in testing high-precision atomic theories and quantum mechanics. In particular the investigation of hydrogen-like atoms and its applications in modern technology go beyond the hydrogen atom itself. Amongst hydrogenic atoms, ce- sium with its low ionization potential and simplicity of its outer shell structure has attracted a lot of attention. In particular cesium has been involved in a number of sig- nificant studies within the field of laser cooling as well as in the development of atomic clocks. Furthermore investigations of parity violation [1] in cesium have been able to yield high precision results. Recently it was also shown that cesium atoms are ideal candidates for optical computers, since cesium vapor is optically highly nonlinear, as well as possessing sensitiv- tion a new series of scientific experiments on the proper- ties and behavior of cesium atoms have been used to prove fundamental connections between chaos theory [3] and quantum entanglement. The aim of the present st ity much greater than most semiconductors [2]. In addi- udy is to investigate the ef- fective g-factor ( g ) for the 3 12 P transitions in hy- dro-gen-like atoms and apply it to ling a one-electron hydrogen-like at 22 2 12 2 nsSnpn sS atomic cesium. We identify that the effective g factor has an integer value of g*=1 as well as the existence of possible entangled states related to the above transitions. We show that the al- lowed transitions occur vertically from one crossing point to another with respect to g* variable. Moreover calculations of their corresponding energy values reveal complete agreement with our previous results [4] (within the 1% margin of error), giving access to new laser lights from atomic cesium. We start by mode om in the presence of a magnetic field ˆ BBz . We assume the sources of the magnetic field [5e the ,6] to b proton’s magnetic moment, p and theag- electron’s m Z. SAGLAM ET AL. 400 netic moment e (or j ) which has two components, namely, the orbital part, l and the spinning part s . We note that thCoub potential in both cases re- quires planar orbits for thlectron, therefore the dir- tion of e lom e e ec p is taken to be in z-direction. Moreover it was recently shown by Saglam et al. [5,6] that theantized magnetic flux through quthe electronic orbits of the Dirac hydrogen atom corresponding to the quantum state ,, j nlm is given by 0 ,, jj nlmn lm where 72 hc is the flu 04.1410 Gc x quantum. the aboationship is i m nd e ve rel Since ependent of , p e , it can gen-like atoms. Note that be easily gneralized to ehydro in such a cases p would be replaced bagnetic moment of the positively charged ion core. The above calculated flux relts in very high magnetic field values (such as 78 10 10GG) inside a hydrogen-like atom. Therefore when such an atom is placed in an external magnetic f its effect would be to orient y the m su B ield ˆ Bz p in the z-direction and e in the opposite direction. Nevertheless the agnetic field inside the atom will be of the same orr with the initial value 78 10 10GG because of the fact that 7 10BG . Such high magnetic field values have not been achieved ex- so far. The total magnetic field inside the atom will be called the local magnetic f total m still de , B perime ield . The investigation of th ef ntally e fective g-factor * g requires to study the Zeeman effect in detail, therefore theigenstates must be distin- guished in the Ze sense including the quantum number e eman j ls mmm [7-8]. Using a non relativistic Hamiltonian [9] we proceed to investigate the Zeeman fine enere shown that for the above mag- netic field values the spin-orbit coupling and the quad- rupole moment energy can be neglected. Considering diamagnetic and paramagnetic effects the energy eigen- values become gies. It can b * ,, ,C Enlm BBmgm j 2Bls n corresping to the ond,, j nlm eigenstate. Note th magnetic feld is replaced by the local magnetic f factor is at the ieldiB B and the Lande-g replaced by the effective Lande-g factor, g which is treated here as a varying ameter. It should be noted that the same expression is also applicable in the case of an atom subject to a laser beam, where due to the photon’s magnetic moment [11] and hence the large intrinsic magnetic field comparable with the above large values [12], the same diamagnetic and paramagnetic effects are present inside the atom. To proceed further, by using the above energy relation we define a new dimensionless function par * 2 ,,,, l ljc m 24 g C fm gEnlmBn which allows us to investigate the effective-g-factor va ues l- * g more easily. For and 0sl(1)pl levels where we have 0, 1 l m, the plotting of fm, lg as a function of g that c of these lines correspond to either 1g, shows rossings s p r 2g crossiongs , which corresponds to pp crossings. As the subject of the present w t tigate the effectctor * ork iso inves- ive g-fa g for the 22 2 12 3212 nsSnpPn sS transitions in hydro- gen-like atoms, our graphic arly shows clethat for the above transitions we have: g ese cr 1. At these crossing points, two states with opposite spin have the same en- ergy value. We believe that thossing points corre- spond also to the entangled states [12]. After finding the effective g-factor value which is equal to one 1g , we set 1g in the energy expressions ,, ,Enlm B j . Substituting the value of 2 Be mc and 12 s m , we find: ** 22 l m C En n ,, 4 c c j m B . Here the constant is the characteristic of each hy calculated from the ionization ene ab tiv- ist C drogen-like atom and rgies. Applying the ove results to atomic cesium and calculating the cor- responding Zeeman fine energies at the s-p crossings, reveals complete agreement with experiment [4,13] (within the 1% margin of error). The above treatment can give access to new laser lights from atomic cesium. The outline of the present study is as follows: In Sec- tion 2 we calculate the energy levels of the non-rela ic hydrogen-like atom in the presence of an effec- tive(local) magnetic field, * B . From the above energy relation we calculate the effective g-factor for the 22 12 3212 nsSnpPn s transitions. In Sec- tion 3 we apply the present energy results to atomic ce- oncluding remarks. 2. Calculation of the Energy Levels of 2 S sium. In Section 4 we present the c LeHamiltonian [9] for a ne-electron hydrogen-like atom in the presence of an Hydrogen-Like Atoms in the Presence of a Uniform Magnetic Field t us consider the non-relativistic o external magnetic field, ˆ BBz : 2 2 11 . 24 Ze 0 22 2 8 B H prLS m r e LgSB Br mc (1) Copyright © 2010 SciRes. JMP Z. SAGLAM ET AL.401 The first term corresponds to the kinetic energy op- erator, the second term is the Coulomb potential, the third term is the spin-orbit coupling, the fo dipole moment energy and the last term is the quadrupole moment energy. As was discussed in [9] for fields up to rth term is the 4T the spin-orbit coupling and the quadratic terms can be neglected so the Hamiltonian is reduced to: 10 2 2 0 0 11 24 B reduced B Ze H pLgSBHH mr (2) where 0 H is the Hamiltonian for a free electron and B B H LgSB is the perturbing Hamiltonian Using Dirac notation, we denote the eigenstates of 0 H by ,,l nlm thus writing: 02 ,, ,, l l ,, ln C H nlm Enlm nlm n (3) where 2 C En is the energy of the free hydrog nen- like atom (here the constant C is determined t ionization energy). If we denote the eigenstates of the Hamiltonian, by hrough the reduced H ,, j nl m where mj = ml + ms, the energy eigenvalues ,, , j nZeeman Enlm BEE will contain the Zeeman correction which is first order in B, i.e.: 0 ,, ,, ,, reduced jBj ,, , jj H nlmHHnlm EBnlm (4) Here the field dependent part nlm . B B H L BgS corresponds to the Zeeman energy, Z eeman E i.e.: ,, ,, ,, B jZeeman j Bl sj H nlm Enlm (5) and (5quation (4) gives the energy eigenvalue, Bm gm nlm Substitution of the Equations (3)) in E ,, ,,, jj Enlm BEnm B rection: including the Zeeman cor- 2 ,, j Bl s C EnmBBmgm n (6) where 2 B ohr magneton and e mc is the B g is the Lande-g factor which is equal to 2 for a free electron an atom or ion in a free space the Lande-g fac given by: . For tor [14] is 111JSS LL (7) 121 J gJJ agnetic field [12], we will have diamagnetic and paramagnetic effects [10] inside the atom. Therefore the magnetic field inside the atom must be the local magnetic field When an atom is subject to a laser beam because of the photon’s magnetic moment [11] and hence the large intrinsic m replaced by B [15]. The Lande-g factor is also replaced by the effective Lande-g factor, g which is treated as a varying parameter [16,17]. Therefore the magnetic field B and the Lande-g factor in Equation (6) must be replaced by the effective values, B and g respectively, such that: 2 ,, j Bl s C EnmBBmgm n (8) Substituting the value of 2 Be mc and 1 2 s m in Equation (8), we find: * 2 ,, 24 lc c j mg C EnmBn * (9) Where * ceB mc is the cyequenc- clotron fry corre sponding to B . Let us proceed by defining a dimensionles using the above energy expression given in E such as: s function quation (9) * *l m 2 ,,, 24 j cl g C fm gEnmBn (10) which does not have an explicit n dependence: Plotting , l f mg for 0, 1m as a function of l g leawingure 1): The h ds to the follo crossings of t g results (Fi ese , 1) l f mg lines co spond to rre- 1g or 2g . 2) Atts, with opposite these crints correspoangled st F these crossing pointwo states spin have the same energy value. We believe that ossing pond to entates [12]. 3) or the crossings (entanglements) of the [ S an 2 ns 12 d 2 32 np P] as well as [ 2 12 ns S and 2 12 np P] states, ther is f effective g- factoound to be equal to one: 1g . 4) The crossing for the pp states occur at 2g . In the present study since we concentrate on the S n trans at th for all energy calculations. Here we note the allowed phically f point tonother one. 22 2 12 3212 nspPn sS sitione crossing points, we substitute 1g in Equation (9) otonic transitions occur vertrom one crossing a To find the energy eigenvalues, we start with 0sl . orientations we have two We note that depending on the spin 2 12 ns S states. In Dirac nn, these states are otatio Copyright © 2010 SciRes. JMP Z. SAGLAM ET AL. 402 Figure 1. ,g 24 l l m g fm as a function of g for . Identifying the effective g values (g*).The ,01 l m s p crossings occur at 1g; the pp crossings occur at g 2 . ,0,0,n,0,0,1 2n and ,0,0,,0,0, 2n . ation (9) the corresponding energy eigenvalues ossing points are given by: 1n From Equ at the cr * 2 ,0,0,124 c C En n (11) and * ,0,0,12C En 24 c n (12) respectively. Equations (11) and (12) we see that the state From 12 , 0,n0,,0,0,n has the lowest energy so it is identified as the ground state. Next we consider the 1pl up states states, w states in total. The spin hich are six 12 ,1,1, , n12 ,1,0,n and 12 ,1,1,n states are denoted by 2 32 e spin down states np P while th ,1,1,12n , ,1,0,12nand ,1,1,12n are written as 2 12 np P. Using Equation (9) the corresponding energy eigenvalues are: * 2 4 * * 2 3 ,1,1,1 224 cc c CC En nn ) (13 ** * 22 ,1,1,1 2244 C En nn cc c C (14) * c C 2 ,1,0,1 24 En n (15) * 2 ,1,0,124 c C En n (16) ** 22 ,1,1,1 224 * cc c CC En nn 4 (17) ** 22 3 ,1,1,1 224 * cc c CC En nn 4 (18) 3. Application to Atomic Cesium The ionization energy for cesium is , while the smallest amount of energy th the ground state to the neares herefore in Equation (11) setting th 3.89 eV at allows a transition t excited state is e lowest en from 1.38 eV . ergy to T 3.89 eV we can write: 2 6, 0, 0,124 6 c EV (19) From Equation (12) and Equation (17) the nearest excited state energy is given by 3.89 Ce 24 6 c C . Since the smallest amount of energy that allows a transition from the ground is then the energ level will be 1.38 eV ,y of this 23.89 1.38 4 6 (20) 2.51 ceV C From Equations (19) and (20), we find that the un- knowns: 115 CeV and * 1.38 ceV . Thus for the 2 cesium atom Equation (9) is written as: 2 1115 , ,,, ,,1.380.69 2 jl l E nlmBE nlmBmg n ange of es the values ( lot the energies bers: (21) where the rl tak-1,0,1) for the m present transitions. In Figure 2 we p given in Equation (21) for quantum num6n and 10n . Plotting the energies corresponding to Equation (21) against g for 6,7,8, 9,10n gives crossings at 1g as ex which are denoted pected0 main crossing points . We observe 1 by the letters: A , ' A B, ' B, C, Copyright © 2010 SciRes. JMP Z. SAGLAM ET AL.403 Figure 2. The plots of ,, l E nm g with respect to g for the 6,s p and ,s10 p stat occur at g respect to es for at it iscrossand transitios take plly with omic cesium. As n seen the ace vertical ings 1 g . and ' C, D, ' D F , ' F . Their energies and related n in from states are give Table 1. To excite electron A to ' A ,which orre- sponds to a c 22 32 66 12 s S ener pP transition, the re- quiredgy is '2.513.891.38 AA eV . To excite electron from ' EE A to F , which corresponds to 22 12 632 10pP sS tsiran 2.51 e giv 44 erim ti en e oe required en- rgy is When electron rela nts involving cesium clusions n, th 2.06 by: Th e' xes bac toe ground state the energy of the corresponding laser will 0.45 FA EE eV . k ent with th b exp the 0.453.89 3. FA E eV .ese values are in good agreem lasers [4]. 4. Con We have calculated the effective g-factor E g for the 22 2 ns Snp P transitions in hydro- gen-like atoms and applied it to atomic cesium. We have found that the value of 12 32 12 n sS g is exactly equal to 1 for the bove mena p tioned photonan resent results to atomic cesium gives complete agree- nd the raleted states at the crossing ,10. ic trsitions. Application of the Table 1. Energies a oints for n = 6,7,8,9p Energies Mixture of states 3.89 A EeV 6,0,0, and 61,1, '2.51 A EeV 6, 0, 0, and 6,1, 1, 3.04 B EeV 7, 0, 0, and 7 , 1, 1, '1.66 B EeV 7,0,0, and 7,1,1, 2.49 C Ee V8, 0, 0, and 8,1,1, '1.11 C Ee V8, 0, 0, and 8,1,1, 2.09 D Ee V9,0,0, and 9 , 1, 1, '0.71 D Ee V9, 0,0, and 9,1, 1, 0.45 F Ee V10,0,0, and 10,1,1, '0.93 F Ee V10,0,0, and 10,1, 1, ment (within the margin of error) with the previous experimental results. We have also s the extence of possible entangled states related to the above transi- tions. Application the above tre ac- cess to the production of new laser from atomic cesium. . References ipkin, “Cesium Atoms for Optical Computers,” Sci- 146, No. 14, 1994, pp. 214-215. ating Ions Get Entangled,” Nature, 2009. Internet Available: http://www.nature.com/news/2009/ rnal of Russian Laser 377-382. 1% hown is s ofatment can give lights 5 [1] J. Guena, D. Chauvat, P. Jacquier, M. Lintz, M. D. Plimmer and M. A. Bouchiat, “Sensitive Pulsed Pump-probe Atomic Polarimetry for Parity-violation Measurements in Caesium,” Journal of Quantum and Semiclassical Optics, Vol. 10, No. 6, 1998, pp. 733-752. [2] R. L ence News, Vol. 3] Z. Merali, “Vibr[ 090603/full/news.2009.540.html [4] S. B. Bayram, S. Kin, M. J. Welsh and J. D. Hinkle, “Collisional Depolarization of Zeeman Coherences in the 133Cs 6p 2P3/2 Level: Double-resonance Two-photon Polarization Spectroscopy,” Physical Review A (Atomic, Molecular, and Optical Physics), Vol. 73, No. 4, 2006, pp. 42713-1-6. [5] M. Saglam, Z. Saglam, B. Boyacioglu and K. K. Wan, “Quantized Magnetic Flux Through the Excited State Orbits of Hydrogen Atom,” Jou Research, Vol. 28, No. 3, 2007, pp. 267-271. [6] M. Saglam, B. Boyacıoglu and Z. Saglam, “Spin-Flip Investigation of 1s-2p and 2p-3d Transitions of Dirac Hydrogen Atom in Terms of the Flux-quantization Argument,” Journal of Russian Laser Research, Vol. 28, No. 4, 2007, pp. [7] M. Saglam, B. Boyacioglu and Z. Saglam, “Spin Depen- Copyright © 2010 SciRes. JMP Z. SAGLAM ET AL. Copyright © 2010 SciRes. JMP 404 09. Hill, New York, 1985. ,” Springer-Verlag, Berlin, 1985. s,” 8th 33-142. ca E, dent Selection Rules for Dipole Transitions in Hydrogen Atom,” The Journal of Old and New Concepts in Physics, Vol. 3, 2006, pp. 181-189. [8] Z. Saglam and M. Saglam, “Quantized Magnetic Flux through the Electronic Orbits of Dirac Hydrogen Atom and its Relation with the Spin Dependent Selection Rules and Photons Intrinsic Flux,” Journal of Physics: Con- ference Series, Vol. 194, 20 [9] G. Drake, “Handbook of Atomic, Molecular and Optical Physics,” Springer, New York, 2006. [10] E. Purcell, “Electricity and Magnetism,” Berkley Physics Course, 2nd Edition, McGraw Vol. [11] G. Sahin and M. Saglam, “Calculation of the Magnetic Moment of the Photon,” Journal of Physics: Conference Series, Vol. 194, 2009. [12] M. Saglam and G. Sahin, “Photon in the Frame of the Current Loop Model,” International Journal of Modern Physics B, Vol. 23, No. 24, 2009, pp. 4977-4985. [13] A. Radzig and B. M. Simirnow, “Reference Data on At- oms, Molecules and Ions [14] M. Born, “Atomic Physics,” 7th Edition, Hafner, New York, 1962. [15] C. Kittel, “An Introduction to Solid State Physic Edition, John Wiley & Sons Ltd., Chichester, 2004. [16] M. Saglam and B. Boyacioglu, “The Absence of the Decimal g-Factor,” Physica Status Solidi B, Vol. 230, No. 1, 2002, pp. 1 [17] M. Saglam, “Flux Quantization Associated with Electron Spin for Correlated Electron System in QHE,” Physi 17, 2003, pp. 345-346. [18] I. N. Levine, “Quantum Chemistry”, 5th Edition, Pren- tice-Hall, New Jersey, 2000. |