Level magnetizabilities of the alkaline metal atoms

Chemical Physics 282 (2002) 289–304 www.elsevier.com/locate/chemphys

Level magnetizabilities of the alkaline metal atoms P. Otto, M. Gamperling, M. Hofacker, T. Meyer, V. Pagliari, A. Stifter, M. Krauss, W. H€ uttner * Abteilung Chemische Physik, Universit€at Ulm, D-89069 Ulm, Germany Received 2 April 2002

Abstract The Zeeman effects of Doppler-free two-photon transitions as induced between the ground and several low-quantum electronic states of Li, Na, K, Rb, and Cs have been recorded in fields up to 4.5 T. Adequate selection rules were chosen to suppress the dominating orbital and spin first-order frequency field splittings, and atomic magnetizabilities were then extracted from the higher-order effects. The values can be accounted for quantitatively using quantum defect singleelectron hydrogen wave functions. Fine structure and hyperfine structure coefficients have been obtained from the zerofield spectra. Ó 2002 Elsevier Science B.V. All rights reserved.

1. Introduction Any ordinary matter, when placed in a magnetic field, experiences repelling forces. This has recently been utilized for levitation experiments on small macroscopic objects by Berry and Geim [1], in the inhomogeneous-field region of a magnet at fields as large as 13 T. These forces act on microscopic magnetic moments built up during electronic induction processes, and are directed antiparallel to the inducing field as required by Lenz’ Law [2]. On the molecular or atomic scale, these induced moments, m, are expressed in terms of a secondrank tensor, n, via

* Corresponding author. Tel.: +49-731-502-2831; fax: +49731-502-2839. E-mail address: [email protected] (W. H€ uttner).

m ¼ n  B;

ð1Þ

where B is the magnetic field. n is called the magnetic susceptibility or the magnetizability. The electric analogue is the (static) polarizability. Van Vleck [3] has shown that the elements of n bear fundamental information on electronic structure. A wealth of data has been collected for molecules during the last 30 years, mostly from microwave Zeeman spectroscopy and other high-resolution spectroscopic methods in external fields [4], but this is almost entirely restricted to molecular electronic ground states. In the case of atoms, experimental knowledge of n is rather scarce, especially for excited electronic states. The available pertinent literature has been summarized in a previous publication [5]. Difficulties arise from the fact that most atomic states show orbital and spin magnetism, and therefore strong first-order Zeeman effects which tend to mask the second-order contribution,

0301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 7 1 8 - 8

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P. Otto et al. / Chemical Physics 282 (2002) 289–304

1^ ^ ^ ð2Þ H^Z ¼  B  n  B; 2

ð2Þ

caused by the induced moments in Eq. (1). This is true especially for low excitation when the contrið2Þ bution of H^Z is very small. In this respect it is interesting that atomic polarizabilities have already been determined spectroscopically some thirty years ago [6], obviously because atomic Stark effects are not affected by permanent-electric-dipole contributions. Recently it has been shown that first-order Zeeman effects can be suppressed at high fields using the Doppler-free two-photon absorption Zeeman technique with suitable selection rules, and the magnetizabilities of the sodium atom in the 5S and 5D states have been determined [5]. It is the purpose of the present contribution to repeat this investigation with improved equipment and to extend it to higher states and to the whole series of the stable alkaline atoms. The magnetizabilities of potassium were of special interest because this atom represents the first case in the periodic table where a new shell is beginning to be occupied by the outermost electron before completion of the inner ones. In what follows we will briefly give the theory required for analysing the Zeeman data, inform about some experimental details, and then present and discuss the results.

It is advantageous to average these terms over ^ be the the atomic-state functions of interest. Let L orbital angular momentum operator and jgLML i the standard wave function diagonalizing L^2 and L^z we define   * + X  e2  2 gLML  nI ¼  r gLML ð4Þ  i i 6m as the isotropic magnetizability of the state (g represents all quantum numbers other than angular momentum). The anisotropic part can be expressed in terms of the corresponding spherical ð2Þ zero-component, T0 ðLij Þ, in angular momentum space. Using the Wigner–Eckart and tensor replacement theorems, the result is   gLkT ð2Þ ðrij ÞkgL ð2Þ ð2Þ  : T0 ðrij Þ ¼ T0 ðLij Þ  ð5Þ LkT ð2Þ ðLij ÞkL The reduced matrix element in the denominator is readily available [7,8], while that in the nominator can be expressed in terms of a suitably defined state dependent parameter [9], nA ¼

e2 12mð2L  1ÞL   !+ * X    2 2 ; ð6Þ

gLL r ð3 cos #i  1ÞgLL  i i 

where #i is the polar angle of electron i in the spherical coordinate system. The final result, 2. Theoretical background Choosing B ¼ ð0; 0; BÞ along the laboratory z-axis, Eq. (2) reads [3] 1 e2 X 2 ð2Þ H^Z ¼  n^zz B2 ¼ ðxi þ yi2 ÞB2 2 8m i ¼

e2 X 2 2 e2 X 1 2 ð3z  ri2 ÞB2 ; rB  12m i i 8m i 3 i

ð3Þ

where ri ¼ ðxi ; yi ; zi Þ is the Cartesian atomic space vector centered at the nucleus and pointing to the ith electron, and e and m are the elementary charge and electronic mass, respectively. The second line of this equation represents the diamagnetic operator as the sum of a scalar and an anisotropic term which is proportional to the zero-component, ð2Þ T0 ðrij Þ, of a second-rank spherical tensor [7,8].

1 1 ð2Þ0 H^Z ¼  nI B2  nA ð3L^2z  L^2 ÞB2 ; 2 2

ð7Þ

is in agreement with a0 similar derivation by Miller ð2Þ and Freund [9]. H^Z is an effective Hamiltonian valid for the state jgLi. The complete Hamiltonian to be used reads ^ þ Ahfs J^  ^I  l ge L^z B ^ S H^ ¼ H^0 þ Afs L B L ð2Þ0

 lB gSe S^z B  lN gI I^z B þ H^Z ;

ð8Þ

where the first three terms represent the zero-field contributions [10]. lB and lN are the Bohr and nuclear magnetons, respectively, gLe , gSe , gI are the electronic orbital, electronic spin, and nuclear spin g-factors, respectively, and L^z , S^z , I^z are the z components of the electronic orbital, electronic spin, and nuclear angular momentum operators,

P. Otto et al. / Chemical Physics 282 (2002) 289–304

^ , and Afs and Ahfs are the ^ þS respectively. J^ ¼ L fine structure (fs) and hyperfine structure (hfs) coupling constants, respectively. Since we are to apply strong fields, and the alkaline atoms show weak fs and hfs couplings, the decoupled basis is appropriate. The diagonal terms then read D E gLML SMS IMI jH^ jgLML SMS IMI 0 ¼ EnL þ Afs ML MS þ Ahfs ðMS þ ML ÞMI

 lB gLe ML B  lB gSe MS B  lN gI MI B  12ðnI þ nA ð3ML2  LðL þ 1ÞÞB2 Þ;

ð9Þ

0 where EnL is the zero-field atomic energy apart from hf and hfs terms. For some transitions the Paschen–Back limit could not be achieved, so off^ diagonal elements of the form hLML SMS jL 0 0 0 0 ^ S jLML SMS i, ML ¼ ML 1, MS ¼ MS  1 had to be considered in a perturbation treatment. This will later be explained in detail.

3. Experiment Doppler-free two-photon absorption spectroscopy was used throughout. The basic set-up is shown in Fig. 1. Since absolute transition fre-

291

quencies are not of importance, the field-induced shifts of the Zeeman components are measured relative to the zero-field frequency of a suitably chosen hyperfine component of the electronic transition under investigation. Frequency interpolation between this zero-field signal, observed in the reference cell, and the Zeeman signals, detected in fields up to 4.5 T in a superconducting magnet, was carried out with the aid of the fringe pattern of a Fabry–Perot interferometer. Different from previously in [5] the two absorption cells were mounted in series in order to utilize nearly the complete available laser power in each cell. This works in favour of a better signal to noise ratio and also helps to cancel the small perturbing shifts caused by the dynamic Stark effect. The interferometer etalon was replaced by a longer one, in which the two mirrors were mounted on the ends of a quartz rod. Temperature stabilization was provided but proved to be unnecessary because of the low coefficient of expansion of quartz. The etalon was calibrated by using the frequency difference between the partly resolved hyperfine components J ¼ 5=2 F ¼ 4 and J ¼ 5=2 F ¼ 3 of the 11 D5=2 6 S two-photon transition of 133 Cs as a standard [11] where F is the total angular momentum quantum number including

Fig. 1. The two-photon spectrometer used in this work. Three channels are recorded and displayed: the signals from the Zeeman and reference cells and the fringe pattern of the Farby–Perot interferometer which serves to interpolate the frequencies between the zerofield and the Zeeman components. The laser light is coupled back and forth through the two cells in series.

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nuclear spin. The frequency difference between two fringe maxima (free spectral range, FSR) turned out to be (61:0479 0:0088) MHz where the error corresponds to three standard deviations. All other laser optical instrumentation employed was that used earlier [5]. The cross-section of the field and probe regions of the split-coil magnet is shown in the lower part of Fig. 1. The absorption cell in use is placed along the 19 mm bore, and is oriented perpendicularly to the field in the large bore which has a diameter of 85 mm. The fluorescence radiation is collected by a lens and imaged to the entrance slit of a monochromator, in the direction of the field, and therefore perpendicular to the exciting laser beam. Some problems arose when the probe had to be heated to temperatures much higher than 300 K, because the liquid He coolant is kept away from the probe volume by a wall thickness of merely a few millimeters. In order to achieve sufficient number densities in the absorption cell, maximum temperatures of 500 °C had to be applied for lithium, 300 °C for sodium, 250 °C for potassium, and 170 °C for rubidium and cesium. Two types of cells were in use. The low-temperature quartz assembly, shown in Fig. 2, is approximately 0.5 m long. Its heated central part, the probe region, has a somewhat enlarged cross-section to allow for the condensed material to flow back to the bottom of the reser-

voir. It is surrounded by a doubly walled glass cylinder which is filled with glycerine and contains a heating wire. Three openings were provided for the installation of a Pt 100 temperature sensor, for feeding in and out the heating wire, and for filling in the glycerine. The fluorescence could readily be detected through this transparent heating device. The glycerine was heated to no more than 250 °C where it was sufficient to provide cooling by circulating air. Temperatures higher than 1000 °C have been accomplished with the cell-assembly shown in Fig. 3. Its central part, the evaporation crucible, is sketched in Fig. 3(b). It consists of a small steel vessel which is surrounded by a bifilar winding of heating wire held in place and electrically insulated by ceramic glue. Thermal insulation is accomplished by several layers of ceramic material which are wrapped by a graphite foil. This foil prevents the direct contact of the chemically aggressive metal vapours and the insulation material. As shown in Fig. 3(a), the crucible is mounted to two metal slabs which provide the connection to the heating power supply. They are held in place by a flange which fits to its counterpart in the body of the cell. The dimensions are such that, when the crucible is mounted vacuum tightly, the laser beam will cross right on top of it. The laser light is led in and out by two glass tubes, also shown in Fig. 3(a), which can be mounted by ground glass joints on

Fig. 2. The low-temperature cell, used up to 250 °C. The two-layered glass cylinder (above) embraces the sample reservoir. The heating coil, embedded in glycerine, leaves space for the fluorescence to be detected outside. The length of the sample reservoir is 40 mm, the distance between the Brewster windows is 540 mm.

P. Otto et al. / Chemical Physics 282 (2002) 289–304

293

Fig. 3. Water-cooled high temperature brass cell, used up to 1000 °C. (a) The sample is evaporated in the crucible shown in (b). It is held in place by two copper bars which also serve to lead in and out the heating current via a flange connection. The laser light is directed in and out by two quartz pipes, and adjusted to cross the crucible right on top of it. (b) Side view of the crucible. (c) Top view of the mounted cell in the cryo-magnet.

both sides of the body. Connection to the vacuum system is provided by an opening in one of the tubes. The fluorescence can be collected through a big window which tops the cell body on the side opposite the flange system. The body is made from

brass, and the central cylindric wall is hollow in order to allow water cooling. A top view of the completely mounted system is given in Fig. 3(c). The cells must be loaded in a glove box in order to prevent any contact of the alkaline metals with

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oxygen. The vacuum line is steadily in operation during measurement in order to maintain a cell pressure lower than 1 Pa. Under these conditions the low-temperature cell can be used over several days without refilling and recleaning, while appropriate measurement conditions can be maintained only for a few hours in the hightemperature set-up. In order to prevent quenching of the cryomagnet, maximum field settings were limited to B ¼ 4:5 T. An average of 16 single frequency sweeps was normally taken for each field setting where the fringe pattern of the etalon, the zerofield spectrum, and the Zeeman spectrum were recorded simultaneously. One sample spectrum (single sweep) of the 11 S 6 S two-photon transition of cesium is shown in Fig. 4. The dynamic Stark effect (light shift) may cause frequency shifts of two-photon transitions in strong laser fields [12]. These show, however, the same direction under the present conditions, irrespective of the magnetic quantum number. They cancel therefore partly when the zero-field and field-on spectra are obtained with the same beam intensity as is the case here. As a result of some quantitative measurements, we estimate the effec-

tive light shifts to be less than 1 MHz which can be disregarded. Line widths achieved (fwhm) were between 5 and 20 MHz. Time-of-flight broadening was observed when small transition matrix elements enforced the application of narrow laser light foci. The accuracy of the final results, the magnetizability differences between the excited electronic states and the ground state, is limited by the uncertainty in the determination of the frequency shifts and by the uncertainty of the field settings. The field is an unequivocal function of the current when the coil dimensions do not change. We have calibrated the magnet repeatedly over three years, and did not observe any systematic deviations. We found a threefold relative standard deviation of 0.2%. This value is added to the individual uncertainty arising from the frequency statistics for each transition, see Table 1 below. The field calibrations were carried out with the aid of a Rawson–Lush type 940 MI rotating coil gaussmeter which in turn has been checked from time to time against the 2.1142 T proton resonance in an NMR spectrometer.

4. Results The selection rules for two-photon transitions have been discussed by Cagnac et al. [13]. The general requirement that the initial and final states must have the same parity was met in all our measurements. For weak coupling between L, S, and I like in the alkali atoms, and in strong magnetic fields DMS ¼ DMI ¼ 0

ð10aÞ

holds as well as DML ¼ q1 þ q2 ; Fig. 4. Magic-doublet Zeeman spectroscopy of the 11 S 6S transition of the cesium atom. The upper trace shows the Fabry–Perot fringe pattern used to determine the frequency shift Dmdia of the magic-doublet-Zeeman component, see text. I ¼ 7=2, the zero-field hyperfine components are F ¼ 8 (left) and F ¼ 7, DF ¼ 0: The DMS ¼ DMI ¼ 0 Zeeman transitions are designated by the products of their projection quantum numbers. The small splitting of the magic doublet was unexpected and appeared only with cesium.

ð10bÞ

where qi ¼ 1, 0, 1 describes the polarization status of the two incoming beams ðr , p, rþ in the above order). 4.1. 2 S

2

S Transitions

DL ¼ 0 transitions are strongly allowed. Here we have L ¼ 0 for both states, thus DML ¼ 0

P. Otto et al. / Chemical Physics 282 (2002) 289–304

295

Table 1 The magnetizability differences of the alkaline atoms determined in this work, defined in Eq. (11) 2a ne–g zz in units of MHz/T

Transition

Experiment

Theory Hartree–Fock

n

b

HCCM

c

HF86

MCHF

Li

3D 4D 4S

2S 2S 2S

2.9985 3.9983 3.5982

1.40(24) 6.70(40) 8.28(34)

1.440 6.790 8.106

1.434 6.792 8.168

1.430 6.776 8.086

Na

3D 4D 5D 6D 5S 6S 7S

3S 3S 3S 3S 3S 3S 3S

2.9897 3.9877 4.9868 5.9863 3.6474 4.6492 5.6501

1.322(84) 6.57(18) 18.66(36) 40.8(16) 8.54(22) 22.96(38) 51.00(76)

1.380 6.672 18.55 40.82 8.540 23.02 50.52

1.362 6.680 18.64 41.02 8.784 23.54 51.46

1.362 6.636 18.50 40.80 8.508 22.98 50.42

K

5D 4 S
6D 4 S
7D 4 S
8D 4S 9D 4S 10 D 4S 11 D 4S 7S 4S 8S 4S 9S 4S 10 S 4S 11 S 4S 12 S 4S 13 S 4S 14 S 4S

4.7695 5.7544 6.7459 7.7403 8.7365 9.7337 10.731 4.8138 5.8158 6.8117 7.8177 8.8181 9.8185 10.819 11.821

14.56(40) 33.44(34) 66.34(68) 118.08(84) 195.098(88) 303.2(13) 451.6(22) 26.24(44) 56.84(50) 107.6(13) 184.8(11) 298.8(20) 462.2(32) 679.4(54) 972.4(72)d

15.08 34.36 67.40 119.5 196.8 306.2 455.7 26.36 56.58 107.1 185.4 300.3 461.6 680.4 968.6




7D 5S 8D 5 S
9D 5 S
10 D 5 S
8S 5S 9S 5S 10 S 5S 11 S 5S

5.6728 6.6668 7.6630 8.6604 4.8609 5.8633 6.8647 7.8654

30.2(28) 63.1(12) 114.9(16) 189.5(13) 27.20(50) 58.84(72) 110.9(18) 189.9(11)




7.5295 8.5293 9.5291 10.529 6.9453 7.9463 8.9469 9.9472

103.8(58) 175.8(40) 279.2(30) 418.6(28) 115.36(90) 197.9(16) 316.4(34) 488.0(26)

Rb

Cs

10 D 11 D 12 D 13 D 11 S 12 S 13 S 14 S

6S 6 S
6 S
6 S
6S 6S 6S 6S

16.45 37.16 72.26 127.2 208.2

–



–



–

27.64 58.88 110.8 191.0

15.84

– –

26.65 – – –



–



–



–



–

32.27 64.10 114.6 189.8 27.38 58.42 110.1 190.0

35.10 68.96 122.2 201.0 29.22 61.70 115.4 197.9

–

106.4 178.2 280.7 421.6 115.3 197.8 318.1 486.2

112.3 292.6

–



–



–

123.1 209.5 507.5

–



–

– – – – – – –

–

– –

The experimental values are given in the fourth column, their errors in parentheses in units of the least significant figure are three standard deviations plus 0.2% from the field uncertainty. Predictions from HCCM calculated with the n parameters in column three are listed in column five. Ab initio results obtained in this work with public routines [19,20] are given in the last two columns. a The Values must be multiplied by 106 h in order to obtain them in units of J=T2 (h ¼ 6:626075 1034 Js is Planck’s constant [16]). b Dðn; ‘Þ ¼ n  n is the quantum defect, see text. c Hydrogen-constant-core model, Eqs. (14), (15). d Harper and Levenson give (930 100) MHz=T2 [17]. h The deperturbation procedure as explained in the Appendix A had to be applied for this transition.  No convergence achieved. – Not calculated.

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P. Otto et al. / Chemical Physics 282 (2002) 289–304

which can be realized with linearly polarized radiation, q1 þ q2 ¼ 0. It is convenient to concentrate on transitions effected between states of maximum angular momentum, Fmax ¼ S þ I, and maximum projections MFmax ¼ ðS þ IÞ. As MS þ MI is a good quantum number in L ¼ 0 subspace, off-diagonal matrix elements cannot exist for the two states jgFmax MFmax i and jgFmax  MFmax i, and Eq. (9) yields their energies. Furthermore, setting MS ¼ S and MI ¼ I in this equation, all linear field terms cancel in a transition jg0 Fmax MFmax i jgFmax MFmax i, L ¼ 0, irrespective of the remaining quantum numbers g and g0 , and the same is true for MF ¼ MF max . The two transitions coincide and form the so-called magic doublet. This result follows, of course, also from the well-known Breit–Rabi formula [14]. The magic-doublet concept is easily extended to molecular problems where it can be used, for example, to measure diamagnetic properties of paramagnetic species [15]. The magic-doublet Zeeman shift, Dmdia , of the two-photon transition 11 S 6 S of cesium, has been indicated in Fig. 4, at a field strength of B ¼ 4 T. 133 Cs has a nuclear spin of I ¼ 7=2 which leads to the two hyperfine states F ¼ 4 and F ¼ 3. The two transitions 4 4 and 3 3 which take place in the L ¼ 0, DL ¼ 0 pattern are seen in the zero-field (middle) trace. There are seven more Zeeman doublets besides the magic one which can be distinguished by the products MI MS . They would pairwise merge into single lines at still higher fields as has been shown for sodium in Fig. 2 of ref. [5]. According to the last line of Eq. (9), the shifts Dmdia of a magic doublet, as a function of the field strength, should lay on a straight line when plotted over B2 . This has indeed been found for all S S magic-doublet transitions of all alkali atoms. An example is shown in Fig. 5, for a transition of rubidium. The parameter nIe–g , the difference between the isotropic magnetizabilities of the excited state and of the ground state, is directly obtained from the slope of this straight line. In Fig. 4 it is seen that the magic doublet is split into two barely separated components which is probably caused by a perturbation of a near-by L state. None of the other magic doublets recorded in this work has shown this feature.

Fig. 5. Plot of the Zeeman shift of the magic doublet as a function of B2 for the transition 8 S 5 S of rubidium. The slope is Dmdia =B2 ¼ ð13:60 0:25Þ MHz=T2 .

4.2. 2 D

2

S Transitions

These DL ¼ 2 two-photon transitions are also strong and readily observed. They do not show magic doublets, but the first-order Zeeman effects can as well be suppressed in the Paschen–Back regime, on grounds of the selection rules Eq. (10a) and DML ¼ 0. The latter can again be realized with linearly polarized light, see Eq. (10b). Setting ML ¼ 0 in Eq. (9) as excitation starts from an S state, it is predicted that a multiplet of 2I þ 1 lines will be observed, with a constant frequency separation of 12h1 DAhfs (DAhfs is the difference of the hyperfine coupling constants of the two states under consideration). Each of these lines is a doublet which starts to separate with decreasing field strength. The 2I þ 1 multiplet members, according to the last line of Eq. (9), undergo a common frequency shift which amounts to e e 2 12h1 ðne–g I  6nA ÞB where nA is the anisotropic magnetizability parameter of the D state (ne–g was I defined earlier in the text). This behaviour has been observed previously in the high-field Zeeman effect of the 42 D–32 S transition of atomic sodium [5]. It has also been seen in all transitions investigated for lithium and sodium in the present work, as well as for the n2 D states of potassium with n P 8. Thus, in these cases the magnetizability differences could easily be obtained from the slopes of the Dmdia (B2 ) plots.

P. Otto et al. / Chemical Physics 282 (2002) 289–304

In the transitions of potassium involving n2 D states n < 8, and in all transitions investigated for rubidium and cesium the Paschen–Back limit had not been reached in fields up to B ¼ 4:5 T. Here it was necessary to explicitly take into account the spin–orbit decoupling effects of L and S in the upper states. It is shown in Appendix A that this problem can be solved exactly by the diagonalization of two 2 2 matrices. The zero-field and Zeeman spectra of the DML ¼ DMS ¼ DMI ¼ 0, 62 D 42 S transition of 39 K, I ¼ 3=2, are shown in Fig. 6. The zero-field selection rule is DF 6 2. So all four transitions between the upper and the lower doublet are allowed (a–d in the figure). The vertical dotted line marks the hypothetic frequency position which would appear if no fine and hyperfine couplings were present. h1 Afs ð6 DÞ ¼ ð3184:14 0:16Þ MHz is determined from the distance of lines a and c as well as of b and d, and h1 Ahfs ð6 DÞ ¼ ð230:8598601 0:0000003Þ MHz [11] by the frequency differences of a and b as well as of c and d. The constant Afs is needed for the deperturbation treatment. The weak extra lines belong to the 41 K isotope with the natural abundance of 6.9%, its nuclear spin is also I ¼ 3=2. The two MS ¼ 1=2 and MS ¼ 1=2 quartets are well separated at low fields. The arithmetic mean of the frequencies of the two far left lines, Dm, is used for the determination of the magnetizability difference, see the Appendix A. Their MI quantum numbers are here 3/2 for the left, and )3/2 for the right quartet, and designate the magic-doublet Zeeman states of the ground state. Dm first decreases as a consequence of the L–S uncoupling, and then, markedly above 2.5 T, the diamagnetic interaction causes it to increase with increasing field strength. In the Paschen–Back limit, the two groups of lines eventually overlap completely and form the quartet of doublets usually observed in the weak-coupling cases. Without diamagnetism the far left doublet would take on the position of the vertical line in Fig. 6. The fine structure constant of the 5 D state of 39 K, h1 Afs ð5 DÞ ¼ ð6037:2 1:2Þ MHz, is almost twice as large as that of 6 D and leads, therefore, to more pronounced decoupling effects. Fig. 7 shows the shift of the mean

297

Fig. 6. The two-photon transition 6 D 4 S of 39 K at different field strengths (the weaker signals stem from the 41 K isotope). The perpendicular dotted line gives the hypothetical center frequency expected for vanishing fine and hyperfine constants. The development of the groups of lines with increasing field is discussed in the text.

frequency Dm as a function of B2 , relative to the hypothetical transition frequency for vanishing fs and hfs. The low-field negative shift is much more pronounced than in Fig. 6. The positive slope at high fields is due to the diamagnetic term in the last line of Eq. (9). The solid curve is the result of the first iteration step of the deperturbation procedure described in Appendix A, and agrees satisfactorily with the measured entries. 4.3. Tabulation of the results The diamagnetic frequency shift, Dmdia , according to Eq. (3) can be written

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structure constants of the D states, important input parameters for this procedure, can be obtained from Table 3 below. For the sake of completeness, all Afs values measured in this work have been listed, and can be compared with those from the literature. The accuracy of these has been greatly improved in some cases. Since not all hyperfine coupling constants are available in the literature, we have also listed the Ahfs values measured in this work. They can be found in Table 4 below. 4.4. The hydrogen-constant-core model (HCCM) Fig. 7. Result of the perturbation treatment of the 5 D 4S Zeeman transition of 39 K. The mean frequency Dm of the MI ¼ 3=2 components (compare Fig. 6) is plotted as a function of B2 . The L–S coupling is stronger than in the transition of Fig. 6, and therefore the frequency drop at low fields more pronounced than there. The slope of the high-field tail determines the magnetizability difference.

Dmdia ¼ 

1 e 1 ðnzz  ngzz ÞB2 ¼  ne–g B2 ; 2h 2h zz

ð11Þ

where ne–g zz

e2 ¼ 4m

*

X

+e ðx2i þ yi2 Þ

* 

i

X

+g ! ðx2i þ yi2 Þ

i

ð12Þ is the difference of the expectation values of the magnetizability operator in the excited state (e) and in the ground state (g). We have collected these differences in the fourth column of Table 1, for all transitions measured for the alkali atoms. They can be written e–g ne–g for the S zz ¼ nI

S transitions;

ð13aÞ

and e–g e 2 ne–g zz ¼ nI þ nA ð3ML  LðL þ 1ÞÞ

for the D

S transitions;

ð13bÞ

where ML ¼ 0 and L ¼ 2, see Eq. (7). The values of ne–g are listed in units of zz MHz=T2 . They can be converted to J=T2 by applying the factor 106 h where h is Planck’s constant [16]. The transitions which needed a deperturbation treatment are marked by an open square. The fine

The values ne–g in Table 1 vary from )1.40 zz MHz/T2 for the 3 D 2 S transition of lithium to )972.4 MHz/T2 for the 14 S 4 S transition of potassium. So the dynamic range is quite large. This might not surprise at a first glance since the number of electrons increases considerably within the series of the alkali atoms. However, it should be remembered that these are not absolute magnetizabilities as the ground-state value is always subtracted. If we assume, in a crude approach, that the contribution of the core electrons remains constant in Eq. (3) on excitation, we can consider the entries in the table as those of the valence electrons alone. Fortunately, this can easily be checked because the magnetizabilities of the atomic states of hydrogen, nn‘ zz , can be predicted exactly. Writing jn‘m‘ i for the (hydrogen) oneelectron functions, we obtain n‘ n‘ 2 nn‘ zz ¼ nI þ nA ð3m‘  ‘ð‘ þ 1ÞÞ

¼

 ‘ð‘ þ 1Þ þ m2‘  1 e2  : n‘jr2 jn‘ 3 ð2‘  1Þð2‘ þ 3Þ 6m

ð14Þ

We have used h‘‘j3 cos2 #  1j‘‘i ¼ 2‘=ð2‘ þ 3Þ [8], Eqs. (7), (6), and (4), and the fact that r2 does not act on the spherical harmonics. This expression has previously been given by Harper and Levenson [17]. The well-known expression for hr2 i is [3,10] 

 n2 n‘jr2 jn‘ ¼ 2 ð15n2  9‘2  9‘ þ 3Þa20 ; 6Z

ð15Þ

where a0 is Bohr’s radius and Ze the nuclear charge. We use here Z ¼ 1 which gives the net charge of the nucleus-core system of the alkalies, but we replace n by n ¼ n  Dðn; ‘Þ where D(n; ‘)

P. Otto et al. / Chemical Physics 282 (2002) 289–304

is the quantum defect introduced to adjust the main quantum number in such a way that the Rydberg formula, En‘ ¼ Z 2 R1 =n2 , fits the experimental term values exactly (R1 is the Rydberg constant [10]). The values of n used are listed in Table 1 for the excited states, and in Table 2 for the ground states. They have been taken from Gallagher [18] or, if not listed there, determined directly from the zero-field energies. In cases where accurate zero-field energies were not found in the literature they have been determined experimentally relative to known iodine transitions, the latter in Doppler-limited resolution. The iodine frequencies were obtained from the tabulations of Gerstenkorn and Luc [30,31]. The magnetizability differences as predicted by the hydrogen-constant-core model (HCCM), Eq. (14), can be found in the fifth column of Table 1. In order to facilitate comparison with experiment, we have displayed the deviations of the measured values jne–g zz j from the predicted ones in Fig. 8. The HCCM values of the ground-state valence elec-

Table 2 Theoretical magnetizabilities of the alkaline atoms for the ground state Valence electron

Atom

ngzz

ng in units of MHz/T2 a

in units of MHz/T2 a n Li Na K Rb Cs

2s 2 S 3s 2 S 4s 2 S 5s 2 S 6s 2 S

b

1.5885 1.6271 1.7705 1.8048 1.8692

c

HCCM

HF86

Ab initio [21,33]

0.340 0.374 0.518 0.558 0.640

0.352 0.410 0.626 0.718 0.892

0. 0. 0. 1. 1.

364d 515d 948d 27e 86e

Values for the valence electrons are given in column four (HCCM, with the n parameters in column three) and five (computed with HF86 [19]). Atomic ab initio values from the literature are listed in the last column. a See footnotes Table 1. b See footnotes Table 1. c See footnotes Table 1. d Calculations using core polarization potentials. The author’s [21] estimated uncertainty range is better than 2%. e Relativistic Hartree–Fock calculations [33]. These values are probably high by 5–10 % as can be estimated in comparison to the results from [21].

299

Table 3 The fine structure constants Afs measured in this work, and comparision with literature values 1 fs A h

in units of MHz

State

This worka

Literature values

Li

3D 4D

433.4(12) 182.4(7)

444(12) 180(24)

[23] [23]

Na

3D 4D 5D 6D

)597.17(29) )411.24(74) )248.2(9) )155.8(11)

)609(4) )411.196(12) )247(5) )154(2)

[24] [25] [26] [24]

K

5D 6D 7D 8D 9D 10 D 11 D

)6037.2(13) )3184.14(61) )1860.39(50) )1176.30(47) )790.33(49) )556.87(41) )406.65(70)

)6032(12) )3142(12) )1895(12) )1160(18) )786(11) )550(6) )412(10)

[27] [27] [27] [28] [28] [28] [28]

Rb

7D 8D 9D 10 D

18 078(12) 12 105.3(77) 8376.0(53) 5988(12)

18 071(12) 12 148(12) 8358(12) 5760(240)

[29] [29] [29] [22]

Cs

10 D 11 D 12 D 13 D

55 783(222) 38 494(35) 27 642(25) 20 515(18)

56 120(240) 38 970(240) 27 940(240) 20 390(240)

[22] [22] [22] [22]

a The errors in parentheses, in units of the least significant figure, are three standard deviations or determined by the uncertainty of the etalon FSR, whatever is larger.

trons have been listed in Table 2. They can be used to estimate the excited-state parameters, nezz , for the valence electron. Ground-state ab initio values for the complete atom (core plus valence electron) from the literature have been listed in the last column. They must be added to the measured ne–g zz values in Table 1 to obtain the atomic excited-state magnetizabilities, nezz . Fig. 8 shows a surprisingly good agreement for the results of the S S transitions for all atoms. The results of the D S transitions for potassium suggest that the model overestimates the magnitudes of the magnetizabilities of the 2 D states, but its predicting power must still be considered very good. This trend seems to hold also for the other atoms, though there the experimental uncertainties do not allow a clear statement. We had previously discussed [5] the replacement of ‘ by ‘ ¼ ‘  Dðn; ‘Þ

300

P. Otto et al. / Chemical Physics 282 (2002) 289–304

Table 4 The hyperfine structure constants Ahfs measured in this work, and comparision with values from the literature State

7

Li

1 fs A h

in units of MHz

This worka

Reference [18]

4S 3 D3=2 3 D3=2 6S 7S

35.32(72) 1.14(49) 0.31(13) 38.0(15) 19.2(25)

– – – 39(3) –

39

K

7S 8S 9S 10 S 11 S 12 S 13 S 14 S

10.41(93) 6.2(12) 4.0(11) 2.6(12) 2.1(9) 1.5(12) 1.9(14) 1.0(16)

10.79(5) 5.99(8) – 2.41(5) – – – –

41

K

7S 8S 9S 10 S 11 S

6.5(10) 3.5(13) 2.6(13) 2.0(24) 2.0(11)

– – – – –

83

Rb

8S 9S 10 S 11 S 7 D3=2 7 D5=2

47.1(20) 33.5(15) 22.2(16) 17.1(19) 1.40(10) Not resolved

45.2(20) – – – – )0.55(10)

87

Rb

8S 9S 10 S 11 S 7 D3=2 7 D5=2 8 D3=2 8 D5=2 9 D3=2

159.3(30) 91.6(47) 56.1(23) 37.2(35) 4.69(23) )1.85(80) 2.97(15) )1.00(13) 2.01(17)

159.2(15) 90.8(8) 56.27(12) 37.4(3) 4.53(3) )2.0(3)

2:840ð15Þ )1.20(15) 1.90(1)

11 S 12 S 13 S 14 S 10 D3=2 10 D5=2 11 D3=2 11 D5=2

39.4(17) 26.4(16) 18.6(18) 13.9(15) 1.503(91) Not resolved 1.11(11) Not resolved

39.4(2) 26.31(10) 18.40(11) 13.41(12) 1.51(2) )0.35(10)

1:055ð15Þ

0:24ð6Þ

133

a

Cs

See footnote Table 3.

in Eq. (15), because this would keep the number of nodes in the wave functions constant. This, however, leads to a worse overall agreement with experiment, so we do not report the details.

4.5. Hartree–Fock calculations We have used the programmes HF86 [19] and selectively MCHF [20] for non-relativistic computations of the measured magnetizability differences. The results are given numerically in the last two columns of Table 1, and again graphically in Fig. 8. The agreement with experiment is satisfactory for lithium and sodium, but not for the heavier atoms, notably their higher levels. In some cases convergence could not be achieved which is indicated in the Table. Normally MCHF gave, expectedly, better agreement. We conclude that ab initio computations of atomic properties for systems of more than 20 electrons, for the non-expert, is still a difficult task.

5. Discussion and conclusion Magnetizability differences between excited electronic 2 S and 2 D states and the 2 S ground state have been measured for the series of the six stable alkali atoms. Doppler-free two-photon transitions have been utilized, where the number of investigated transitions was limited by the accessible frequency range of the dye laser in use. The high fields of a cryomagnet had to be applied in order to discriminate the generally weak diamagnetic second-order Zeeman effect, Eq. (7). It was possible to approach the Paschen–Back limit, and to utilize selection rules defined in the decoupled basis. The disturbingly strong spin and orbital magnetisms could be either suppressed completely in magic-doublet transitions or taken into account satisfactorily with the aid of perturbation theory. The results (Table 1) can be understood by assuming that the magnetizability contributions of the inner-shell electrons cancel in a Zeeman transition so that the measured effect can be attributed to the state magnetizabilities of the valence electron alone. The corresponding one-electron expression, Eq. (14), easily utilized using hydrogen wave functions, leads to an excellent agreement with the experimental 2 S–2 S magnetizability differences while a small systematic deviation is found for the lower 2 D–2 S quantum transtitions, for all alkaline atoms. The quantitative success of

P. Otto et al. / Chemical Physics 282 (2002) 289–304

301

Fig. 8. A graphical representation of Table 1: The magnitudes of the magnetizability differences determined in this work in comparison with those predicted from the hydrogen-constant-core model (HCCM). The ordinates give the percentage deviations, the HCCM model values are represented by the 0% horizontal lines. All transitions started from an S ground state, they can be identified by the upper-state nS and nD designations given below the error bars. The ab initio HF and MCHF results, when available, are also indicated.

302

P. Otto et al. / Chemical Physics 282 (2002) 289–304

the model is rather surprising, especially for the S–S transitions which involve penetrating orbits. Replacing n by n in Eq. (15) obviously mimics the use of the correct energy eigenfunction. This differs from an ab initio calculation where much effort is required to iterate a wave function to just minimize the energy. In the case of the D states it might be speculated that polarization effects cause a contraction of the wave function, so that the model would overestimate the quantity jne–g zz j as it is observed for the lower states. The results for potassium, the first atom in the periodic table where the successive filling-up of shells is interrupted, do not show any irregularities. The individual excited-state magnetizabilities for the valence electron can be obtained by adding the ground-state hydrogen-constant-core values from Table 2 to the experimental differences in Table 1. Some ab initio atomic ground-state magnetizabilities found in the literature [21,33] have also been collected in Table 2. They can be used, in a similar way, to estimate the total level magnetizabilities of the excited states. However, comparing the values for the valence electrons and the atoms, Table 2 shows that the contributions of the core electrons are surprisingly small. It turns out that the core contributions amount to less than the experimental errors for most of the ne–g zz values in Table 1. The overall consistency of the data in Fig. 8 suggests that the experimental procedure adopted for measuring atomic susceptibilities is correct, so the measurements could be extended to other atoms of interest. First experimental results for some alkaline earth metals have recently been obtained [32]. These atoms are two-electron systems and show strong spin–orbit interactions in their triplet states. Their second-order Zeeman effects require, therefore, different kinds of analysis from those used in the present work. Transitions into excited singlet states, on the other hand, exhibit simple Zeeman patterns, and the susceptibilities can easily be extracted. The singlet S states, especially, are purely diamagnetic. Atoms in such states could be trapped, like the macroscopic objects referred to above [1], in inhomogeneous fields.

In conclusion, accurate structural electronic P parameters, in the form of combinations of h i ri2 i P and h i ð3z2i  ri2 Þi, can be obtained for atoms in excited electronic states using methods discussed in this paper. These novel quantities should prove to be valuable for testing wave functions, especially of heavy atoms.

Acknowledgements The support of the Deutsche Forschungsgemeinschaft and of the Fonds der Chemischen Industrie are gratefully acknowledged. One of us (P.O.) thanks Professor Froese Fischer for advice in using the HF86 and MCHF programmes. Thanks are due to Dr. U. Seifert for critically reading the manuscript.

Appendix A Off-diagonal terms of the Hamiltonian Eq. (8) have to be taken into account when Afs is large enough to prevent L–S uncoupling in fields B < 2 T. This was the case with the D states marked by an open square in Table 1. Since hyperfine couplings in these states are negligibly weak, the energy matrix to be considered reads D E LSML0 MS0 jH^ jLSML MS D E ¼ Afs ML0 MS0 jL^x S^x þ L^y S^y þ L^z S^z jML MS þ ðlB gLe ML B  lB gSe MS B þ Edia ÞdML0 ML MS0 MS ; ðA:1Þ where Edia represents the last term in Eq. (9). We 0 have omitted Enl because this term is not involved in the perturbation procedure. The x and y components of the angular momenta produce off-diagonal (ðML0  ML Þ ¼ 1, ðMS0  MS Þ ¼ 1), the z components diagonal contributions. As our experiment specializes to DML ¼ 0, ML ¼ 0 transitions it is sufficient to consider the four states j0; 1=2i and j 1; 1=2i. Furthermore, since ML þ MS is a good quantum number, (A.1) above decomposes into two 2 2 sub matrices. Setting

P. Otto et al. / Chemical Physics 282 (2002) 289–304

up the two matrices by aid of the well-known angular momentum algebra [7,8], diagonalizing, and selecting the two desired solutions jML MS i ¼ j0; 1=2i on grounds of their high-field behaviour, we arrive at 1 1 1 D D EM ¼  Afs  lB gLe B þ ðEdia ðML ¼ 0Þ S ¼ 1=2 4 2 2 D ðML ¼ 1ÞÞ þ Edia " 1 1 1

 Afs  ‘ lB gLe B  lB gSe B 4 2 2 2 1 D D

ðEdia ðML ¼ 0Þ  Edia ðML ¼ 1ÞÞ 2 #1=2 3 fs 2 þ ðA Þ : ðA:2Þ 2 The two expressions show vanishing orbital and diverging spin-up and spin-down Zeeman effects with increasing field which is the expected behaviour. The corresponding terms for the groundstate magic doublets, MF max ¼ ðI þ 1=2Þ, are  1 S e S l EM ¼  g g ðA:3Þ þ Il N I B þ Edia : S ¼ 1=2 2 B S The mean frequency of the two DMS ¼ 0 transitions connecting Eqs. (A.2) and (A.3), Dm, is given by i 1h D S D S hDm ¼ ðE1=2 Þ þ ðE1=2  E1=2 Þ  E1=2 2 1 D D S ¼ ðE1=2 þ E1=2 Þ  Edia ; ðA:4Þ 2 and it is seen that the linear Zeeman contributions for the ground state cancel completely. There is also some simplification in the contributions from Eq. (A.2). We have determined ne–g zz by fitting it to the experimental mean frequencies Dm (for example, the two marked lines in Fig. 6 and their field dependence). This was done in two iteration steps D where the start values for Edia ðML ¼ 1Þ, D S Edia ðML ¼ 0Þ, and Edia have been calculated with the aid of Eqs. (14) and (15) and the constant-core model. As this model is so successful it is not surprising that normally the experimental field function DmðBÞ, Eq. (A.4), is already well reproduced with these parameters. This is demonstrated in Fig.

303

7. It is barely seen here that the high-field values of Dm are actually somewhat overestimated showing a maximum difference of 5 MHz at B ¼ 4 T. Plotting the deviations as a function of B2 gives a slightly sloped straight line. The experimental result of ne–g zz is then obtained by adding its slope to D S 2h1 ðEdia ðM ¼ 0Þ  Edia Þ. In case the deviations did not obey a linear function of B2 a further iteration step would have to be carried out. This, however, was not necessary for the items in Table 1.

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