Calculation of the Isotope Shifts on 5S 1/2 4D 3/2,5/2 Transitions of 87,88 Sr +

Transcription

1 Commun. Theor. Phys. (Beijing, China) 37 (22) pp 76 7 c International Academic Publishers Vol. 37, No. 6, June 5, 22 Calculation of the Isotope Shifts on 5S /2 4D 3/2,5/2 Transitions of 87,88 Sr + LI Yong,,2 WU Li-Jin, ZHU Xi-Wen, and GAO Ke-Lin Wuhan Institute of Physics and Mathematics, the Chinese Academy of Sciences, Wuhan 437, China 2 Institute of Theoretical Physics, the Chinese Academy of Sciences, Beijing 8, China (Received July 9, 2; Revised November, 2) Abstract A simple method is applied to calculating the isotope shifts (ISs) on 5S /2 4D 3/2,5/2 transitions of 87,88 Sr +. First we have calculated the ISs of lower transitions on a series of alkali-like systems such as B 2+, Ca + and Ba +, which are in agreement with other works. Then the ISs on 5S /2 4D 3/2,5/2 transitions of 87,88 Sr +, which are useful to study the Sr + optical frequency standard, are evaluated. PACS numbers: 3.3.Gs Key words: isotope shift, specific mass shift, normal mass shift, field shift Introduction Studies of isotope shift and hyperfine structure for atoms have a long history, but are still useful in giving some information about the charge distribution and other atomic properties. They have been used to separate the isotopes and measure high-resolution spectra. They are also particularly useful in the frequency standard applications. There are many species of atoms, which could be used as a basis for an atomic frequency standard. One of the most powerful techniques in approximating a nearly ideal reference atomic system has been investigated in the ion trap. Optical frequency standards based on transitions in laser cooled trapped ions offer significant advantages over more established optical frequency standards. It has been proposed that certain ion systems may yield accuracy eventually approaching the level of 8. [] Optical frequency standards based on odd-isotope ions are particularly appealing for their half-integral nuclear spin and have m F = m F = Zeeman transitions, where there is not the first-order Zeeman shift and greatly suppressed broadening from residual ac magnetic fields. A number of candidate ions, such as 99 Hg +, [2] 43 Ca +[3] and 7 Yb +, [4] have been investigated worldwide. Recently at the U.K. National Physical Laboratory (NPL) an optical frequency standard based on odd-isotope strontium ion has been developed. [5,6] The studies of the new optical frequency standards make the theoretical investigation of the isotope shift and hyperfine structure significant. The hyperfine constants of 87 Sr + have been evaluated by L. Wu et al. [7] This work mainly describes the calculation of the isotope shifts on 5S /2 4D 3/2, 5/2 transitions of 87,88 Sr +, which are useful for NPL group to study the new odd-isotope Sr + optical frequency standard. Researchers have studied the ISs of many elements on both the alkali-like systems and the alkaline earth metal systems. The calculation of the IS includes three parts: normal mass shift (NMS), specific mass shift (SMS) and field shift (FS). The NMS can be easily evaluated from the experimental energies. The calculation of the FS is not difficult by taking advantage of the changes in meansquare nuclear change radii δ r 2 from optical isotope shifts. So our interest in this work focuses on the SMS effect. Bauche [8] has performed Hartree Fock (HF) calculations of SMS in the transition between the two lowest lying states for a number of atoms. To go beyond the HF approximation many ways are open. More accurate calculation of SMS has been performed for helium by Pekeris and coworkers [9] in their extensive work on two-electron systems. However, their method is not easily applicable to larger systems. Keller and Labarthe [] have used the multi-configuration Hartree-Fock (MCHF) method to include the second-order correlation effects. Martensson [] has used the many-body perturbation theory (MBPT) and treated the pair correlation by numerical solution of the pair equation. Here we use the MCHF method, which has been simplified according to the specific properties of the alkali-like system. First, the ISs of a series of alkalilike elements have been calculated and compared with the experimental and other theoretical data in order to be assured that the modified method is reasonable. Then, the ISs on 5S /2 4D 3/2,5/2 transitions of 87,88 Sr + are calculated. 2 Theory of the Isotope Shift In the HF approach, the nucleus can be treated as an infinitely heavy point charge. In fact, the nucleus is built from protons and neutrons and has both a finite mass The project supported by the Joint Projects Between the U.K. National Physical Laboratory and Wuhan Institute of Physics and Mathematics, the Chinese Academy of Sciences

2 No. 6 Calculation of the Isotope Shifts on 5S /2 4D 3/2,5/2 Transitions of 87,88 Sr + 77 and an extended charge distribution. These properties of the nucleus affect the energy level structure of an atomic system. The shift of electronic energies between different isotopes of one element is due to the finite mass and the volume of the nucleus. The field shift (also called volume shift) is caused by the penetration of electrons into the nucleus. The NMS and SMS effects are caused by the finite mass of nucleus and the mass of electrons. 2. Theory of the NMS and SMS on an Alkali- Like System In the HF approximation, the nucleus is considered as an infinitely heavy point charge. So in atomic units the Hamiltonian is given as [2,3] H = i p 2 i 2m + i Z r i + i<j r ij, () where m is the mass of the electron, Z is the nuclear charge of the atom or ion, r i is the distance of the electron i from the nucleus and r ij is the distance between the electron i and the electron j, and with E and Ψ being, respectively, the eigenvalue and eigenfunction. But in the centre of mass system the nucleus has a momentum p N = i p i, where p i is the momentum of the individual electron i, that can still be expressed in terms of the gradient operator with respect to its coordinate relative to the nucleus. So the Hamiltonian must be [2,3] H M = i p 2 i 2µ + M p i p j + i<j i Z r i + i<j r ij, (2) where M is the mass of the nucleus, and µ = mm/(m+m) is the reduced electron mass. The first term includes a correction to the electron mass in which the mass m is replaced by µ. This change in mass leads to an energy correction [2] E nms M = m M E. (3) This is the NMS effect. The inclusion of the second term leads to an additional energy correction called SMS, [2] E sms M = Ψ M i<j p i p j Ψ. (4) One can see from above that the SMS operator is a two-body operator. We will perform ab initio calculations of the SMS in this work. In the MCHF approach [3] the wavefunction Ψ for a state labelled by γls, where γ represents the configuration and any other quantum numbers required to completely specify the state, is approximated by an expansion over configuration state functions (CSFs) with the same LS term, Ψ(γLS) = i c i Φ(γ i LS). (5) Since the CSFs including high excited states have only a few contributions, we do not consider those CSFs. But as regards the alkali-like system we should consider them here, it has only one open shell and has only one electron in the open shell. So it has only one term in the expansion. The momentum operator p = i has odd parity and can only combine orbitals of opposite parity in a matrix element. The rank of p is and it has matrix elements only between orbitals whose angular momenta differ by one. [] This implies that it has only exchange contributions. The SMS operator is a scalar and it has contributions arising from the core alone in addition to contributions involving the valence electron, o. So the expectation value of SMS is given by E sms M = M ) oa p p 2 ao + ab p p 2 ba = M Ssms, (6) (core a where S sms is called the SMS factor, a, b are core electrons. First we consider the term oa p p 2 ao. The coupling of angular-momentum is taken into account, the state o is thought of as a shortening of n o (l o s o )j o m o j, so we have a<b oa p p 2 ao = n o (l o s o )j o m o j, n a (l a s a )j a m a j 2 n a (l a s a )j a m a j, n o (l o s o )j o m o j, (7) where = r C (), r is the radial part and C () is the angular part. Making use of the Wigner Eckart theorem and vector-coupling properties, [4] then l o l a oa p p 2 ao = oa p p 2 ao = ( ) +jo+3ja 6(2j a + ) /2 /2 a n al aj am a n j al aj a j o j a l a l o /2 /2 l o C () l a l a C () l o f(a, o), (8) j a j o

3 78 LI Yong, WU Li-Jin, ZHU Xi-Wen, and GAO Ke-Lin Vol. 37 where Similarly f(a, o) = = r 2 R o (r) r R a (r)dr r 2 R a (r) r R o (r)dr [ d r 2 R o (r) dr l o(l o + ) 2 l a (l a + ) 2r [ d r 2 R a (r) dr l a(l a + ) 2 l o (l o + ) 2r ] R a (r)dr ] R o (r)dr. (9) ba p p 2 ab = ba p p 2 ab = ( ) +jb+3ja 6(2j a + )(2j b + ) 2 2 a<b n al aj am a j,n bl b j b m b n j al aj a n b l b j b l b l a l a l b /2 /2 /2 /2 l b C () l a l a C () l b f(a, b). () j b j a j a j b In the formulae (8) and (), the 9-j coefficient and the reduced matrix of C () are easy to be calculated by means of the angular momentum theory. [4] In the calculation of f(a, o) and f(a, b), the radial function R(r) is obtained by making use of the nonrelativistic approximate result of the program GRASP, [5] which is a relativistic program. 2.2 Theory of the FS on an Alkali-Like System Due to the finite size of the nucleus, the potential deviates from the Coulomb potential of a point charge Z. Since s electrons have a finite probability within the nuclear volume, the potential deviation will lead to an energy shift. The expression for this energy correction of a transition is [2] E fs M = 2πZ 3 Ψ() 2 r 2 M, () where Ψ() 2 is the change in electron density at the nucleus between the lower ( Ψ () 2 ) and upper states ( Ψ 2 () 2 ) in the transition, so it is expressed by Ψ() 2 = Ψ 2 () 2 Ψ () 2, (2) where Ψ() 2 is expressed by Ψ() 2 N N = Ψ δ( r i ) Ψ = Ψ 4π r 2 i δ(r i ) Ψ = i= i= N i= 4π R i (r)r 2 δ(r)r i (r)r 2 dr, (3) and R(r) is the radial function. Here r 2 M is the so-called mean-square radius of nucleus (the mass is M). It is difficult to calculate the r 2 M since often the charge distribution is not known. In this work, we use the mean-square charge radii from the optical isotope shifts. [6] 3 Calculation of the Isotope Shift 3. Calculation of the SMS According to Eqs (6) and (8) (), the SMS of a single level state on an alkali-like system can be computed. Then for an isotope pair M and M, the SMS of a state is EM,M sms = Esms M Esms M = ( M M ) S sms. (4) After having calculated the SMSs of the lower state and the upper state respectively, the SMS of a transition can be obtained. The SMSs of np states of 6,7 Li have been calculated and compared with the theoretical calculations of Martensson [] and other experimental data. The results agree each other, as shown in Table. Table The SMS results for the np states of 6,7 Li (MHz). Martensson et al. did not point out that the np state is np /2 or np 3/2. In this work, the results of np /2 and np 3/2 are almost the same. Our following results belong to np /2. State This work Other theor. work Experiment 2P [] 368 [7] 3P 4 34 [] 298 [8] 4P [] 5P [] 6P []

4 No. 6 Calculation of the Isotope Shifts on 5S /2 4D 3/2,5/2 Transitions of 87,88 Sr + 79 In Ref. [] Martensson has used the many-body perturbation theory and treated the higher-order effect. Virtually Mantensson also calculated the first-order SMS results according to his method in Ref. [], and the firstorder results are more adjacent to the experimental results than that of the higher-order effect. So do our results. Then it is seen that our results are reasonable. We also calculated the SMSs on a lot of alkali-like systems which are used to obtain the transition ISs and the results are listed in Table Calculation of the FS For an isotope pair M and M, the field shift of a transition is [2] E fs M,M = Efs M Efs M = 2πZ 3 Ψ() 2 rm 2 2πZ 3 Ψ() 2 rm 2 = K fs δ r 2, (5) where δ r 2 = rm 2 r2 M is the change in mean-square nuclear charge radii from optical isotope shift, and Kfs = (2πZ/3) Ψ() 2 is the FS factor. In our calculation of K fs, the radial function R(r) is obtained as done in the calculation of SMS part as given above. As regards the light element [9] and high excited states of heavy element, [2] the FS effect is much smaller than the SMS effect. But for the low excited states of the heavy atoms the FS effect may be much larger than the SMS effect. The FSs of both the light and heavy elements have been calculated and compared with other works in Table 2. Table 2 The FS results of some alkali-like and alkiline earth elements (MHz). Isotope pair Transition FS of this work Other theor. results Exp. results 4,43 Ca + 4S /2 4P / [2] 4,43 Ca + 4S /2 3D 3/ [2] 4,43 Ca + 3D 3/2 4P /2.6 [2] 39,4 K 4S /2 4P / [22] 36,38 Ba + 6S /2 6P 3/ [23] From Table 2, both the light elements such as Ca +[2] and K, [22] and the heavy elements such as Ba +[23] have good agreement with other results. So we can ensure that the FS in this work should be reasonable. The FSs on 5S /2 4D 3/2,5/2 transitions of 87,88 Sr + are very near (both about 7.6 MHz) because the two fine structure states have similar wavefunctions. They are too little indeed compared with the cases of 86,88 Sr + (about 65 MHz) because the δ r 2 of 87,88 Sr + is much smaller than that of 86,88 Sr Calculation of Transition Isotope Shift So far the NMS, SMS, and FS effects have been calculated. We calculated the total ISs of a series of alkali-like systems and compared them with other theoretical and experimental works, as shown in Table 3. Table 3 The isotope shifts of some alkali-like elements (MHz). Isotope pair Transition NMS SMS FS IS Other works (exp.), B 2+ 2S /2 2P / [24], B 2+ 2S /2 2P 3/ [24] 4,43 Ca + 4S /2 3D 3/ [2] 86,88 Sr + 5S /2 4D 3/ ,88 Sr + 5S /2 4D 3/ ,88 Sr + 5S /2 4D 5/ ,38 Ba + 6S /2 5D 5/ [25] From the first and second rows of Table 3, we can see that the ISs of the ns /2 np /2 and ns /2 np 3/2

5 7 LI Yong, WU Li-Jin, ZHU Xi-Wen, and GAO Ke-Lin Vol. 37 almost have the same results. The other experimental results of other cases are similar. The calculated IS results on 5S /2 4D 3/2,5/2 transitions of 87,88 Sr + almost have the same results too. This is reasonable since they have the same isotope pair, similar wavefunctions and very close energies. So the three parts of isotope shifts almost have the same results. 4 Conclusion We use the above simplified method to calculate the SMSs and FSs of many alkali-like systems. And the results agree with other works, especially on the transitions ns (n )D. The ISs of 5S /2 4D 3/2,5/2 transitions of 87,88 Sr + are 27.7 MHz and 26.3 MHz, respectively. The δ r 2 will effect our FS results much, because it may give large deviations. If the more accurate data of δ r 2 can been used, the more precise results of IS will be obtained. The SMS part may be of some uncertainty, because our method only includes a few configurations. If we consider the coupling configurations of not only valance electrons but also some core electrons (for example the most outside core shell), then we will obtain many configurations. In a later work we hope to perform that method and obtain the more accurate IS results. Acknowledgments We thank H.A. Klein, G.P. Barwood, and G. Huang who encourage us to evaluate the isotope shift of 87 Sr + ion. References [] H.G. Dehmelt, IEEE Trans. Instrum. Meas. 3 (982) 83. [2] D.J. Berkeland, et al., Phys. Rev. Lett. 8 (998) 289. [3] F. Plumelle, M. Desaintfusien, and M. Houssin, IEEE Trans Instrum. Meas. 42 (993) 462. [4] D. Engelke and C. Tamn, Europhys. Lett. 33 (996) 347. [5] G.P. Barwood, K. Gao, P. Gill, G. Huang, and H.A. Klein, IEEE Trans. Instrum. Meas. April (2). [6] M.G. Boshier, G.P. Barwood, G. Huang, and H.A. Klein, Appl. Phys. B7 (2) 5. [7] Li-jin Wu, Yong Li, and Ke-lin Gao, Calculation of Hyperfine Constants a and b by Using Finite Basis Sets Constructed with B-splines, Phys. Rev. A (to be published). [8] J. Bauche, J. Physique 35 (974) 9. [9] C.L. Pekeris, Phys. Rev. 27 (962) 59. [] J.-C. Keller, J. Phys. B: At. Mol. Phys. 6 (973) 77. [] A.-M. Martensson and S. Salomoson, J. Phys. B5 (982) 25. [2] C.F. Fisher, T. Brage, and P. Jonsson, Computational Atomic Structure, Institute of Physics Publishing, London (997) p. 5. [3] P. Jonsson, C.F. Fisher, and M.R. Godefroid, J. Phys. B32 (999) 233. [4] I. Lindgren and J. Morrison, Atomic Many-Body Theory, Springer Series in Chemical Physics (982) p. 3. [5] K.G. Dyall, I.P. Grant, C.T. Johnson, F.A. Parpia, and E.P. Plummer, Comp. Phys. Commun. 55 (989) 425. [6] K. Heilig and A. Steudel, Atomic Data and Nuclear Data Tables 4 (974) 63. [7] R.J. Mariella, Appl. Phys. Lett. 35 (979) 58. [8] R.H. Hughes, Phys. Rev. 99 (955) 837. [9] C.J. Lorenzen and K. Niemax, Opt. Commun. 43() (982) 26. [2] K. Niemax and L.R. Pendrill, J. Phys. B: At. Mol. Phys. 3 (98) L46. [2] F. Kurth and T. Gudjons, Z. Phys. D34 (995) 227. [22] A.-M. Martensson-Pendrill, L. Pendrill, S. Salomonsson, A. Ynnerman, and H. Warston, J. Phys. B: Mol. Opt. Phys. 23 (99) 749. [23] A.-M. Martensson-Pendrill and A. Ynnerman, J. Phys. B: Mol. Opt. Phys. 25 (992) L55. [24] U. Litzen and R. Kling, J. Phys. B: Mol. Opt. Phys. 3 (998) L933. [25] X. Zhao, N. Yu, H. Dehmelt, and W. Nagourney, Phys. Rev. A5 (995) 4483.