fulltext

Quadratic Stark effect of the 6snf

1,3

F

3

and

3

F

4

Rydberg series of BaI

M. A. ZAKIEWISS(*)

Physics Department, Faculty of Science, Cairo University - Giza, Egypt (ricevuto il 24 Giugno 1997; approvato il 14 Luglio 1997)

Summary. — The scalar and tensor polarizabilities of the barium odd-parity 6snf

1_F

3and3F3and3F4Rydberg sequences with 11 GnG30, 12 GnG40 and 12 GnG 40, respectively, have been theoretically studied. The theoretical values of the polarizabilities have been calculated in the framework of the multichannel quantum defect theory. The contributions to the polarizabilities of all known even-parity states of the 6snd and 6sng configurations together with their perturbers connected via the electric-dipole operator with the 6 snf1_F

3and3F3and3F4levels are considered .The results are compared with the available experimental values, with few exceptions good agreement is found.

PACS 32.10.Dk – Electric and magnetic moments, polarizabilities. PACS 32.60 – Zeeman and Stark effects.

1. – Introduction

Over the last years, the odd-parity Rydberg series of barium have been subjected to several spectroscopic studies. The 6 snp levels have been studied by Garten and Tomkins in a classical absorption experiment [1]. In addition to the level energies of the Rydberg P-states up to n 475, valuable information regarding the autoionizing states with orbital angular momentum l 41 was obtained. Armstrong et al. [2] expanded the

study of bound l 41 states to levels with 3_P

1 and 3P2 character in a three-step

excitation scheme employing pulsed tunable dye laser. They also performed a

multichannel quantum defect theory (MQDT) analysis of their results. The 6 snp 1

P1

series was found to be more strongly perturbed than the 6 snp 3_P

1 series. The

perturbers belong to the 5 dnp and 5 dnf configurations. The energy levels of these autoionizing configurations have been determined by Abu-Taleb et al. [3, 4] using two-step laser excitation.

There exist accurate energy values of the 6snf1

F3and 3F2, 3, 4 levels with 10GnG45. These energy levels were measured by Post et al. [5] using a high-resolution laser

spectroscopy technique. Beside a strong singlet-tripling mixing between the 1F3 and

3_F

3 levels, it has been found that these levels are strongly perturbed with the J 43

levels of the 5 d8 p configuration. Also the 6 snf 3_F

2 and the 6 snf 3F4 levels are

perturbed by the J 42 and J44 levels of the 5d8p and 5d4f configurations [6]. The

(*) E-mail: mzewissHfrcu.eun.eg

Stark effect of these odd-parity sequences has been measured by Zaki Ewiss [7], using high-resolution UV-laser radiation. Besides testing the perturbation of these levels, these measurements aimed at deriving information on the contributed unknown (at that time) even-parity 6 sng1_G

4and3G3 , 4 , 5 Rydberg series. The lack of information of these even-parity states hampered the theoretical fitting of the polarizabilities.

Although the results revealed that the MQDT wave functions of the odd-parity 6 snf1_F

3 and3F2 , 3 , 4levels derived by Post et al. [6] are sufficiently accurate, the data available on the contributed even-parity levels 6 snd configuration required some corrections.

Nowadays, there exist accurate level energies and hyperfine splittings of the even-parity 6 sng1

G4 and3G3 , 4 , 5Rydberg series, together with their MQDT analyses, see Vassen et al. [8]. Recently, Zaki Ewiss and Al-Ahdali [9] have studied the natural

radiative lifetimes of the 6 snf 3F2 Rydberg sequence. The results showed deviations

from the hydrogenic scaling law.

In the present work, the quadratic Stark effect of the 6 snf 1F3 and 3F3 and 3F4

Rydberg levels of Ba I, with 11 GnG30, 12 GnG40 and 12 GnG40, respectively, has been theoretically studied. The calculation of the polarizabilities based on the MQDT wave functions of these levels and all known contributed even-parity levels of the 6 snd and 6 sng configurations connected with them via the electric dipole operator, normalized to their experimental energy values.

2. – Theoretical consideration

2.1. The quadratic Stark effect. – The Hamiltonian describing the interaction of an atom with a uniform electric field (directed along the z-axis) with strength Ez is given by [9]

HE_{4 2P}zEz,

(1)

where Pz is the z-component of the electric-dipole operator P

–

(P–₄

!

i

eri with ri the

position vector of the i-th electron and e the electron charge). The Schrödinger equation may be written as

(H01 HE) Nbb 4WbNbb

(2)

with Wb the energy and Nbb the zeroth-order electronic state of the system. The

zero-field unperturbed case (HE

4 0 ) is assumed to be solved. The unperturbed level energies are given by the Rydberg formula [7]

W04 Wn , l0 4 I 2

R (n 2ml)2

, (3)

I is the ionization limit, R the Rydberg constant, n is the principal quantum number

and ml is the quantum defect of the level with orbital angular momentum l. ml

deter-mines the deviation from a hydrogenic level energy. In barium I 442 034.92 cm21 _and

R 4109 736.882 cm21_.

The diagonal matrix elements of HE_{for electronic states of definite parity are zero,}

thus the stark operator contributes, in general, only in second (and higher) order to the energy of an atomic state. The effective operator describing the second-order Stark effect can be expressed as [9]

HS₄

!

b HE Nbb abNHE W02 Wb . (4)

The summation extends over all zeroth-order electronic states Nbb with energy

WbcW0, where W0is the zeroth-order energy of the degenerate set of states spanning

the model space in which HS_{operates. When the model space consists of the magnetic}

substates NbJMb of one zero-field state NbJb (J is the total angular momentum and b

denotes all other quantum numbers), the matrix elements of HS_{can be written as}

abJMNHSNb 9 J 9 M 9 b 4

!

b 8 J 8 M 8 abJMNPzNb 8 J 8 M 8 b ab 8 J 8 M 8 NPzNb 9 J 9 M 9 b Ez2 W02 Wb 8 J 8 . (5)

The matrix elements (3) are evaluated using the graphical representation of angular-momentum calculus as presented by Lindgren and Marrison [10], therefore

abJMNHS Nb 9 J 9 M 9 b 4

!

b 8 J 8 3 (6) 3a bJVPz V_{b 8 J 8b a bJ 8 VPz}V_{bJb Ez}2 W02 Wb 8 J 8 , with the sum over M 8 implicit in the composite diagram. Separating the diagram in eq. (6) into the four open lines according to the theorems of Jucys, Levinson and Vanagas results in 4 (7) 4

!

k 40 Q ( 2 k 11) .

By adding the phase factor, the first diagram on the right-hand side of eq. (7) can be transformed into the representation of a 6 j-symbol and the second diagram can be separated into two 3 j-symbols:

4

!

k( 2 k 11) (21)

2 J

.

Transforming the right-hand side into conventional notation and rearranging the phase factors finally leads to

4 (8) 4

!

k 40( 2 k 11)(21) 2 J 1J 92M 9._/ ´ k J 8 1 J 1 J 9 ˆ ¨ ˜

u

J 9 2M 9 k 0 J M

vu

1 0 k 0 1 0

v

.

The second 3 j-symbol is non-zero only for k 40 and k42. The terms with k40 and k 42 have the azimuthal dependence of a scalar and a second-order spherical tensor, respectively, as shown by the first 3 j-symbol in eq. (8). The separation of the k 40 and k 42 terms after insertion of eq. (8) into eq. (4) yields the definition of the scalar stark effect H0Sand the tensorial stark effect H2S, respectively. Therefore

a bJMNHS

Nb 9 J 9 M 9 b 4 a bJMNH0S1 H2SNb 9 J 9 M 9 b , (9)

for J 4J 9 one gets

a bJMNHSNb 9 J 9 M 9 b 4 dMM 9

g

abJ ( 0 ) 1 abJ ( 2 ) 3 M 2 2 J(J 1 1 ) J( 2 J 21)

h

g

2 1 2Ez 2

h

, (10)

where the scalar polarizability a( 0 )_{bJ is defined by}

a( 0 )_bJ 4 22 3( 2 J 11) b 8 J 8

!

(21) J 82J abJVPzVb 8 J 8b ab8 J 8 VPzVbJb W02 Wb 8 J 8 (11)

and the tensor polarizability a( 2 )_{bJ is defined by}

a( 2 )_bJ 42(21) 2J11_{[ 20 J( 2 J21)/3]}1 /2 [ ( 2 J13)(2J12)(2J11) ]1 /2

!

J 8 . / ´ 2 J 8 J 1 J 1 ˆ ¨ ˜ abJVPzVb 8 J 8b ab8 J 8 VPzVbJb W02 Wb 8 J 8 . (12)

The summation in eqs. (11) and (12) extends over all electronic states (including continuum states) with b 8 J 8c bJ. Thus the center of gravity of state NbJb is shifted and the magnetic substates NbJMb are split in energy. The shift and splitting are determined by a0bJ, and a2bJ, respectively; both are proportional to Ez2.

The evaluation of the polarizabilities requires knowledge of the level energies and wave functions of all states involved. Once the wave functions specified in a basis of MQDT channels are known (see below), the dipole matrix elements presented in eqs. (11) and (12) can be reduced to single electron radial integrals of the type [9]

R_{n 8 l 8}nl 4



0 Q Rnl(r) Rn 8 l 8r3dr , (13)

where Rnl(r) is the radial part of the wave function of the nl electron.

2.2. MQDT analysis. – It is now common to analyze data on Rydberg series with the

multichannel quantum defect theory (MQDT). This theory, which was developed by Seaton in the Sixties [11] and reformulated by Fano and coworkers in the Seventies [12, 13], gives level energies, wave functions and oscillator strength distributions of mutually interacting Rydberg series using two sets of physical meaningful parameters. These parameters are extracted, independently for each total angular momentum J and parity, with the help of energy data in an (always) limited energy range and may in principle be extrapolated outside this range. They describe the short-range interaction between the Rydberg electron and the ion core

characterized by the close coupling channel a. The eigen quantum defects ma of these

close coupling channels form one set of parameters. The other set is the Uia matrix which describes the transformation of the jj-coupling basis i of the collision channels (core + electron dissociated system) to the a basis and thus is a measure of the change of angular-momentum coupling when the electron approaches the core.

Following ref. [14], an LS-coupled intermediate basis a– is used to build the Uia matrix as Uia4

!

a –Uia –_Vaa–_, (14)

where Uia– is the jj-LS transformation matrix. This choice is based on the assumption

that the close coupling channels are close to LS coupled. All deviations are then

included in the matrix Vaa–, whose matrix elements are expressed in terms of

generalized Eulerian angles uij; uij couples the pure LS-coupling channel j and is a

measure of configuration interaction when the channels i and j have a different core. When both electrons of the i and j channel are in the same orbits, it is a measure of the departure of LS-coupling within a configuration.

3. – Experiment

A high-resolution laser spectroscopy technique has been used for the Stark effect

measurements of the odd-parity 6 snf 1_F

3 and 3F2 , 3 , 4 Rydberg sequences of barium. The experimental set-up has been described in detail elsewhere [7]. Briefly, the transitions to the selected levels are induced in a collimated beam of metastable atoms. The atomic beam is produced by heating a sample of neutral barium in a tantalum oven with a small orifice. The oven is heated to 800 K by a tungsten filament in front of the orifice. The metastable 6 s5 d levels are populated by small discharge maintained

Fig. 1. – An example of an excitation spectrum 6 s 5 d3_D

3-6 s16 f 3F4, a) in the absence of an electric field, b) in the presence of electric field of strength 66.3 V/cm.

TABLEI. – Theoretical and experimental values of the scalar (a0) and tensor (a2) polarizabilities of the 6snf3_F 3levels of BaI. n Energy ( cm21₎ a0(MHzO(VOcm)2) a2(MHzO(VOcm)2)

this work experiment (a_{) this work experiment (}a₎

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 38 40 41247.658 41327.376 41463.372 41537.100 41597.454 41651.300 41690.521 41725.853 41756.125 41782.178 41804.769 41824.455 41841.733 41856.962 41870.462 41882.484 41893.234 41902.890 41911.589 41919.454 41926.589 41933.086 41939.017 41944.442 41949.418 41958.219 41965.719 0.047 0.157 0.234 0.343 0.503 2.05 1.34 2.12 2.60 3.46 4.87 7.03 10.03 7.15 (b₎ 10.0 14.1 18.8 24.4 31.1 40.0 50.6 62.6 77.5 96.3 118.0 144.4 212 308 0.045 (2) 0.155 (6) 0.224 (9) 0.347 (14) 0.502 (20) 2.02 (8) 1.32 (5) 1.83 (7) 2.58 (10) 3.62 (14) 4.95 (20) 7.06 (28) 6.59 (26) 10.3 (4) 14.4 (6) 19.0 (8) 24.4 (1.0) 31.3 (1.2) 39.4 (1.6) 50.9 (20) 63.8 (2.6) 78.5 (3.6) 98.7 (4.0) 121.9 (4.5) 141.3 (5.7) 200 (8) 290 (12) 20.017 20.060 20.093 20.139 20.200 20.728 20.489 20.413 20.962 21.27 21.82 22.67 1.13 4.02 (b₎ 23.23 24.89 26.60 28.63 211.0 214.0 217.8 222.2 227.4 234.4 242.2 252.8 273.1 2108.9 20.015 (1) 20.061 (1) 20.094 (4) 20.133 (5) 20.193 (8) 20.698 (28) 20.470 (19) 20.653 (27) 20.935 (37) 21.32 (5) 21.85 (7) 22.71 (11) 4.01 (16) 23.40 (14) 24.96 (20) 26.90 (27) 28.63 (35) 211.1 (5) 214.1 (6) 218.1 (7) 222.8 (9) 228.2 (1.1) 234.8 (1.4) 242.7 (1.7) 251.7 (2.1) 273.3 (2.9) 2107.8 (4.1) (a) The experimental values of the polarizabilities obtained from ref. [7].

(b) Data obtained with a modified MQDT wave function and slight variation of the energy level values of the 6 snd1_D 2,3

series and their perturber (see text).

between the filament and the oven. The atomic beam is intersected perpendicularly by the beam of an intracavity actively frequency-stabilized CW ring dye laser (Spectra Physics380D), producing UV-radiation in the wavelengths region 292–330 nm. Both

temperature-tuned ADA (Ammonium Dihydrogen Arsenate) and angle-tuned LiIo3

(lithium iodate) non-linear crystals have been used. The ultraviolet light excites the

barium atoms e.g. from the 6 s 5 d 3D2 and 3D3 levels to the selected 6 snf

1_F

3and3F3 , 4Rydberg levels, respectively. In this experiment, the spectral width of the laser is less than 2 MHz, which is of the same order as the residual Doppler width. The excitation region is centered between two small capacitor plates which sustain the applied dc-electric field. The excited Rydberg atoms are field ionized 2 cm down stream in a small capacitor with one mesh electrode and the detached electrons are detected by the electron multiplier. The signal is calibrated using an interferometer locked to a

TABLE II. – Theoretical and experimental values of the scalar (a0) and tensor (a2) polarizabilities of the 6snf1_F 3levels of BaI. n Energy ( cm21₎ a0(MHzO(VOcm)2) a2(MHzO(VOcm)2)

experiment (a_{) this work experiment (}a_{) this work}

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 41120.299 41259.053 41364.251 41456.048 41529.068 41568.238 41646.618 41684.727 41717.218 41769.403 41791.435 41811.445 41829.480 41845.646 41860.106 41873.043 41884.650 41895.070 41904.460 41912.952 20.873 (33) 0.118 (5) 0.071 (4) 0.101 (4) 0.113 (5) 0.062 (3) 0.550 0.300 (12) 0.023 (1) 24.79 (19) 5.04 (20) 4.90 (20) 229.1 (1.2) 244.51 (1.8) 28.78 (36) 213.6 (5) 220.5 (8) 237.0 (1.5) 249.7 (2.0) 275.0 (30) 20.857 0.120 0.075 0.1 0.113 0.060 0.066 (b₎ 0.476 0.296 20.003 0.012 (b₎ 25.20 24.73 (b) 4.78 5.01 (b₎ 3.47 4.87 (b₎ 228.0 229.4 (b) 246.8 244.4 (b) 28.73 214.2 222.4 234.4 252.5 277.6 0.330 (13) 20.048 (2) 20.024 (1) 20.028 (1) 20.032 (1) 0.003 20.216 (9) 20.076 (3) 0.146 (6) 4.54 (18) 25.17 (21) 25.82 (23) 27.3 (1.1) 16.5 (7) 2.47 (10) 4.15 (17) 6.95 (28) 12.6 (5) 19.0 (8) 27.7 (1.1) 0.323 20.047 20.027 20.028 20.031 0.008 0.004 (b₎ 20.208 20.070 0.169 0.149 (b₎ 4.96 4.49 (b₎ 24.91 25.14 (b) 24.35 25.75 (b) 26.1 27.4 (b₎ 19.3 16.9 (b₎ 2.64 4.45 7.35 11.7 18.7 28.2 (a) The experimental values of the polarizabilities obtained from ref. [7].

(b) Data obtained with a modified MQDT wave function and slight variation of the energy level values of the 6 snd1_D 2,3

series and their perturber (see text).

stabilized He-Ne laser. Figure 1(a) shows an example of an excitation spectrum obtained for the 6 s 5 d 3_D

3-6 s16 f 3F4 in the absence of the electric field. Figure 1(b)

shows the Stark spectrum of the 6 s16 f3_F

4 level in the presence of an electric field of strength 66.3 V/cm. The peaks have been identified by the isotope mass number and

absolute value of the excited-state magnetic quantum number NMJN (with electric

field). Small unmarked peaks are hyperfine components of the odd isotopes 135 , 137Ba .

Shift and splittings of the peak of the most abundant even isotope 138_{Ba (nuclear spin}

I 40) as a function of the applied voltage have been measured for the 6 snf3_F

3,1F3and 3

F4 Rydberg levels. The polarizabilities of these levels were deduced from the

extrapolation of the low-field limit. The experimental values of the scalar a0 and the

tensor a2 polarizabilities of these levels are given in tables I, II and III,

TABLE III. – Theoretical and experimental values of the scalar (a0) and tensor (a2) polarizabilities of the 6snf3_F 4levels of BaI. n Energy ( cm21₎ a0(MHzO(VOcm)2) a2(MHzO(VOcm)2)

this work experiment (a_{) this work experiment (}a₎

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 38 40 41365.383 41458.069 41532.449 41592.791 41641.819 41680.542 41733.062 41760.009 41784.447 41806.266 41825.533 41842.538 41857.599 41870.976 41882.903 41893.588 41903.186 41911.842 41919.678 41926.787 41933.259 41939.170 41944.580 41949.541 41958.316 41965.808 0.083 0.254 0.228 0.235 0.248 0.092 20.297 — 19.4 12.4 12.9 13.2 18.0 22.5 28.4 35.4 48.3 55.5 68.2 82.8 103.7 124.1 148.6 180.3 250.6 356.4 0.083 (3) 0.254 (10) 0.223 (9) 0.234 (9) 0.249 (5) 0.093 (4) 20.331 (13) — 19.1 (8) 12.3 (5) 12.9 (5) 13.8 (6) 18.0 (7) 22.4 (9) 28.3 (1.1) 35.3 (1.4) 48.3 (1.9) 55.4 (2.2) 68.3 (2.7) 83.0 (3.3) 101.4 (4.5) 123.8 (4.9) 149.6 (6.0) 181.4 (7.3) 254.5 (10.2) 365.5 (14.5) 20.037 20.182 20.12 20.098 20.091 0.024 20.222 — 27.66 25.31 25.55 28.06 28.42 210.4 213.1 216.2 222.4 225.5 231.3 237.9 247.7 256.5 268.4 283.1 2116.4 2167.5 20.038 (2) 20.180 (7) 20.098 (4) 20.096 (4) 20.092 (4) 0.020 (1) 20.200 (8) — 27.44 (30) 25.29 (2) 25.57 (22) 28.18 (33) 28.33 (33) 210.3 (4) 213.0 (5) 216.3 (7) 222.2 (9) 225.7 (1.0) 231.4 (1.3) 238.4 (1.5) 247.1 (1.9) 257.3 (2.3) 269.9 (2.8) 283.4 (3.4) 2119.0 (4.8) 2172.0 (6.0) (a) The experimental values of the polarizabilities obtained from ref. [17].

4. – Results

The evaluation of a0and a2of the 6 snf 3F3, 1F3 and 3F4 Rydberg levels requires knowledge of energies and wave functions of these odd-parity J 43 and J44 levels themselves as well as of the even-parity states coupled in the field to these levels. Involved are 6 sng1_G

4,3G3 , 4and 6 snd3D2 , 3and1D2for the 6 snf3F3,1F3levels and 6 sng 1

G4,3G3 , 4 , 5and 6 snd 3D3levels for the 6 snf3F4levels. Also, contributions of perturber states of the 6 snd and 6 sng configurations with triplet Rydberg character have to be considered.

There exist, accurate level energies (0.01 cm21_{) for odd-parity J 43 and 4 series in}

the interval n 410–45 and an MQDT analysis has been reported by Post et al. [6]. The 6 snf 1

F3 series is found to be perturbed near n 411 by the doubly excited 5 d 8p 3F3

level, near n 416 by the 5d 8p 3_D

3 level and near n 420 by the 5d 8p 1F3 level.

Figure 2 shows the Lu-Fano plot of the 6 snf1 , 3F3Rydberg levels. It is clear that the

perturbation of the 5 d 8 p 1_F

Fig. 2. – Lu-Famo plot of the 6 snf J 43 series of Ba I as reported in ref. [5].

Rydberg levels. The MQDT wave functions of these 6 snf J 43 series are given in terms of singlet-tripling coefficients and admixtures of perturber character for most values of n.

The 6 snf3

F4levels are perturbed by the odd-parity 5 d 8 p 3F4doubly excited state. This is shown from the Lu-Fano plot of fig. 3. In the calculation, the wave functions of the 6 snf J 43 and J44 series have been generated using the MQDT parameters given in ref. [6]. These wave functions are normalized to the experimental level energies.

Vassen et al. [15] have reported accurate level energies of the even-parity 6 sng1_G

4, 3_G

3 , 4 , 5 ( 6 GnG45) series with accuracy 0.01 cm21. They performed a J 44 and J45 MQDT analysis using five and two channels, respectively. The 6 sng J 44 levels are

perturbed around n 410 by the 5dnd3G4level and around n 424 by the 5dnd3F4level.

Also the 5 d 7 d1_G

4autoionizing resonance located 7 cm21 above the 6 s ionization limit

affects the 6 sng J 44 levels with high n. These perturbations of the series are reflected

in the behavior of the singlet-tripling mixing. The 6 sng3_G

5Rydberg series is found to

be perturbed by the 5 d 7 d3_G 5level.

Experimental level energies of the 6 snd 3

D3series with accuracy A0.1 cm21 have

been reported by Camus et al. [ 16 ]. Aymar and Camus [ 17 ] have reported an MQDT

model on the even-parity J 43 states. They fitted the heavily perturbed 6 snd 3D3

series and the 6 sng 3_G

3 series (as far as data were available at that time) with the

introduction angle between the 5 dnd3_G

3and the 6 sng3G3channels. Here, we used the

MQDT information on the wave functions of the even-parity J 43 states as given in ref. [ 16 ]. From the hyperfine structure and Landé-factor measurement, Aymar [14] has reported the nine-channel MQDT analysis of the 6 snd1 , 3D2levels. Beside a strong

singlet-triplet mixing character in these series, she found that the 1_D

2 levels are

perturbed near n 412, 14 and 26.

The specified wave functions on the basis of MQDT channels of the 6 snf 3_F

Fig. 3. – Lu-Famo plot of the 6 snf J 44 series of BaI as reported in ref. [5].

and 3F4 levels with 12 GnG40, 11 GnG30 and 13 GnG40, respectively, and the

contributed even-parity levels of the 6 snd and 6 sng configurations, together with their

perturbers are used in the calculation of the scalar (a0) and the tensor (a2)

Fig. 4. – Experimental (m) and theoretical (!) values of a_j/(n * )7as a function of n * for barium 3_F

Fig. 5. – Experimental (m) and theoretical (!) values of a2/(n *)7as a function of n * for barium3F3states.

Fig. 6. – Experimental (m) and theoretical (!) values of a_j/(n * )7as a function of n * for barium 1_F

Fig. 7. – Experimental (m) and theoretical (!) values of a2/(n * )7as a function of n * for barium1F3 states.

Fig. 8. – Experimental (m) and theoretical (!) values of a_j/(n * )7as a function of n * for barium 3_F

Fig. 9. – Experimental (m) and theoretical (!) values of a2/(n * )7as a function of n * for barium3F4 states.

polarizabilities of these levels. The integral (13) has been calculated numerically in Coulomb approximation [18], assuming generalized hydrogenic radial functions for the highly excited state, normalized to the experimental level energies.

The theoretical values of a0and a2of the 6 snf3F3,1F3and3F4levels with 12 GnG 40, 11 GnG30 and 12 GnG40 are presented in tables I, II, III, respectively. These values are compared with the corresponding experimental data reported in ref. [ 7 ].

Since, in a hydrogen model, a0 and a2 are expected to scale as n *7, therefore, for

convenience, the theoretical and experimental values of a0On *7and a2On *7are plotted as a function of n * in figs. 4, 5 for the3F3levels, figs. 6, 7 for the1F3levels and figs. 8, 9, respectively.

5. – Discussion

5.1. The polarizabilities of the 6 snf3F3levels. – The experimental values of a0and

a2of the 6 snf 3F3levels showed deviations from the hydrogenic scaling law at n 417

and 24. These deviations have been attributed to the localized perturbation in some of

the Rydberg series with the 5 d 8 p3

D3 and 5 d 8 p3F3levels as discussed above. From table I, the calculated a0and a2for the 6 snf3F3levels are in good agreement with the corresponding experimental values, except that deviations at n 419 and 24 are found.

At n 419 the deviation is 16% for a0and 37% for a2far from the experimental values.

For the 6 s19 f 3_F

3 level located at 41 725.85(1) cm21, it has been found that the

contributions of the 6 snd 3_D

analysis for the even-parity J 43 states as presented by Aymar and Camus [17] gives

equal fraction of the 6 snd3D3and the 6 sng 3G3Rydberg character into the perturber

5 d 7 d 3_F

3. Variation in the wave functions and/or level energies involved does not

produce consistent values for the 6 s 19 f3_F

3level. For this reason, more accurate data

on the 6 snd 3

D3 series are needed to improve the MQDT J 43 analysis. This

improvement may remove much of these discrepancies. For the 6 s 24 f3_F

3level located

at 41 841.73 cm21_{, the contribution of the 6 snd} 1 , 3

D2 series is dominant. These

even-parity J 42 levels are perturbed between n426 and 27 by the presence of the 5 d7 d 1_D

2 doubly excited state. According to the MQDT analysis of Aymar [ 14 ], the

perturber energy is located at 41 841.55 cm21 _{with 6.2% admixture of the 6 snd} 3

D2

character and 56 % of the 6 snd1_D

2 character into its wave function. Considering the

systematic deviation in level energies (0.14 cm21_{) of the 6 snd}1 , 3

D2series as discussed in ref. [ 14 ], consistent results for a0and a2of the 6 s 24 f3F3level can only be obtained

by changing the level energy to 41 841.49 cm21_{for the perturber, with 12.5% admixture}

of the 6 snd 3_D

2 character and 46 % of the 6 snd 1D2 character into its wave function.

These changes are well within the uncertainty of the MQDT analysis.

5.2. The polarizabilities of the 6 snf 1_F

3 levels. – From figs. 5, 6, the experimental

values a0 and a2 of the 6 snf 1F3 levels showed strong deviations from the hydrogenic

model. These are mainly due to the heavy perturbation with the 5 d 8 p 1_F

3level. This perturbation extends over a large value of n as discussed above. For this reason, difficulties were found in measuring the polarizabilities of the 6 snf1

F3levels for n F31. In this case, the excited levels are very sensitive to the applied dc-electric field. The recorded spectrum showed the appearance of the Stark manifold, indicating the break-down of the orbital angular momentum quantum number l , hence the breakbreak-down of the

parity conservation law. The Stark manifolds of the 6 s 40 f1_F

3, 6 s 40 g 1G4and 6 s 40 h 1_H

5levels have been studied in more details by Lahaye et al. [ 19 ].

From table II, the calculated a0 and a2 for the 1F3 levels are in good agreement

with the corresponding experimental values, except that deviations at n 416, 19, 20, 21,

22, 23 and 24 are found. The major contribution to a0 and a2 of the 6 s16 f 1F3 level

located at 41 586.24 cm21_{comes from the 6 s18 d}1

D2level located at 41 567.18 cm21. Due

to the large energy separation the systematic variation of 0.14 cm21_{has no influence on}

the calculation. Only a small decrease in the singlet character (99 % according to the MQDT analysis of Aymar [ 14 ]) to 90 % and a corresponding increase in the triplet character in the 6 s18 d1_D

2level are sufficient to improve the discrepancy between the

theoretical and experimental values of the polarizabilities of the 6 s16 f1F3level. It is believed that the deviation from the experimental values of the polarizabilities of the 6 s 19 f1_F

3level is due to the misinterpreted 6 s 21 d1D2level which is located at

41 707.30 cm21_{. Again, a change of 0.14 cm}21 _{of this energy level value has no}

significant influence. However, a small increase in the singlet character of the 6 s 21 d 1

D2level leads to the modified result given in table II.

The large discrepancies between the calculated and experimental values of the polarizabilities of the 6 snf 1_F

3 levels for n 420–24 arose from the uncertainty and

deficiencies of the energy level values and the wave functions of the 6 snd3_D

3and1 , 3D2 levels. The 6 s 20 f 1_F

3 level located at 41 769.40 cm21 is influenced by the 6 s 23 d 1D2 level located at 41 768.24 cm21_{instead of 41 768.35 cm}21_{, which results in the corrected} values given in table II. This is also the case for the 6 s 21 f1_F

3level at 41 791.44 cm21, where the polarizabilities are determined by the positive contribution of the 6 s 24 d1D2 level at 41 792.62 cm21_{[ 14 ] and 6 s 24 d} 3_D

contribution of the 6 s 24 d 3_D

2 level at 41 790.84 cm21[ 14 ]. With small energy

corrections for the three levels, we obtained good agreement with the experimental values of a0 and a2 of the 6 s 21 f 1F3 level by lowering the 6 s 24 d 1D2 and 3D2 level

energy by 0.18 cm21 _{to 41 792.44 cm}21 _{and 41 790.66 cm}21_{, respectively, and increasing}

the energy of the 6 s 24 d3

D3level by 0.05 cm21to 41 791.77 cm21. The results are given

in table II. These correction, indicate that the energy values of the 6 snd3_D

3series as reported by Camus et al. [ 16 ] are probably incorrect. These changes do not influence the results for the3_F

3levels, as they have much larger energy difference with the 6 snd

series.

The values of a0 and a2 of the 6 s22 f 1F3 level located at 41 811.45 cm21 are

influenced by the position of the 6 s 25 d 3_D

2 level at 41 811.82 cm21 instead of

41 811.94 cm21_{[ 14 ]. It is noticeable that the energy of the 6 s 23 f}1

F3 level located at 41 829.48 cm21_{[ 5 ] nearly coincides with the 6 s 26 d}1_D

2level at 41 829.49 cm21[ 14 ]. The sign of the polarizabilities of the 6 s 23 f1F3level indicates that the 6 s 26 d1D2level must have a slightly lower energy than this level. The results for a0and a2of the 6 s 23 f1F3

level have been obtained by positioning the 6 s26 d1_D

2 level at 41 829.35 cm21. Finally, the values of a0and a2of the 6 s 24 f1F3level at 41 845.65 cm21[ 5 ] are dominated by the

negative contribution of the 5 d 7 d 1_D

2 perturber of the 6 snd 1 , 3D2 series at

41 841.49 cm21 _{and the positive contributions of the 6 s 27 d} 3

D2 and 1D2 level at

41 848.26 cm21 _{and 41 852.06 cm}21_{, respectively. The corrected values of a}

0and a2can

only be obtained by lowering the energies of the 6 s 27 d 1 , 3_D

2 levels to be at

41 848.10 cm21_{for the 6 s 27 d}3

D2and at 41 851.90 cm21for the 6 s 27 d1D2level. Also the singlet character of the 6 s 27 d3_D

2level has to be increased from 9 % according to the

MQDT analysis made in ref. [ 14 ] to 46 % with a corresponding decrease in the triplet

character to 48 % . The singlet character of the 27 d1_D

2 level has been enhanced from

78 % to 85 % .

5.3. The polarizabilities of the 6 snf 3F4levels. – Again, figs. 7, 8, the experimental values a0and a2of the 6 snf3F4levels showed deviations from the hydrogenic model at

n 413–24. These deviations are referred to the extended perturbation of these

Rydberg series by the 5 d 8 p 3_F

4level. This perturber state is located between n 418

and 19 (fig. 2). Post et al. [5] have found that the 6 s 20 f 3_F

4 level has zero quantum

defect. This level showed strong deviation from the quadratic dependence of the Stark effect as discussed in ref. [7]. In this case, the higher L-levels are excited in the presence of the applied dc-electric field and rapidly evolve into the linear Stark regime. For this reason, the evaluations of a0 and a2 of the 6 s 20 f 3F4 level are excluded. A remeasure of the Stark manifolds for n 420, using laser excitation with the theoretical analysis approach of Lahaye et al. [19], could explain this phenomenon.

At n 414, the polarizabilities are affected by the 5 d 8p 3_D

3perturber of the 6 snd

3

D3levels [16], by the 5 d8 p 3F4 perturber of the 6 sng 1 , 3G4 Rydberg series at n 424 and by the 5 d 7 d3_G

5perturber of the 6 sng3G5series at n 415 and 16 [16].

For n F25, the plotted quantities of a0 and a2 are constant, indicating the

verification of the hydrogenic model in this region.

In table III, the theoretical values a0 and a2 of the 6 snf 3F4 levels are presented and compared with the corresponding experimental values. Good agreement is obtained except small deviations at n 416 and 19. These discrepancies are attributed to the inaccuracy in the level energy and/or deficiencies in the MQDT wave functions of the 6 snd3_D

energies or in the mixing MQDT coefficients of the 5 d 7 d 3_D

3 perturber state could

remove these discrepancies.

6. – Conclusion

In conclusion, the calculated values of the polarizabilities of the 6 snf 1_F

3, 3F3 and 3

F4 Rydberg series served as a sensitive probe of the perturbation of these levels

themselves as well as of the even-parity levels of the 6 snd and 6 sng configurations. The results showed some discrepancies in comparison with the corresponding experimental values of a0and a2, especially at n 416 and 19–24 for the 1F3levels, at n 419, 24 for the 3_F

3 levels and at n 416, 19 for the 3F4 levels. These discrepancies have been

attributed to two factors. First of all, deficiencies and inaccuracies in the MQDT

analysis of the even-parity J 43 states. Second, inaccuracy in the 6snd1_D

2 and 3D2 , 3 energy level values, especially in the region close to the perturber states.

* * *

The author wishes to thank the anonymous referee for his helpful remarks.

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