Optics and Spectroscopy, Vol. 99, No. 4, 2005, pp. 536–539. Translated from Optika i Spektroskopiya, Vol. 99, No. 4, 2005, pp. 558–561. Original Russian Text Copyright © 2005 by Semenov, Kapel’kina, Tsygankova, Tsygankov.

ATOMIC SPECTROSCOPY

Semiempirical Calculation of the Atomic Characteristics of the 2p5 3d, 4d, and 2p5 ns (n = 3–10) Configurations of Neon R. I. Semenov, E. L. Kapel’kina, G. A. Tsygankova, and M. A. Tsygankov Institute of Physics, St. Petersburg State University, Peterhof, 198504 Russia e-mail: [email protected] Received November 11, 2004

Abstract—Semiempirical calculation of the fine-structure parameters, the intermediate-coupling wave functions, and the gyromagnetic ratios is carried out for a number of configurations of the neon atom in the singleconfiguration approximation. Comparative analysis of the results obtained is performed. © 2005 Pleiades Publishing, Inc.

INTRODUCTION

TECHNIQUE OF CALCULATION

It is known that neglect of interconfiguration and weak magnetic (spin–other-orbit and spin–spin) interactions significantly reduces the accuracy of semiempirical calculations. To increase the accuracy of such calculations, a number of empirical parameters were used: α [1], 3Scorr [2], Fi and Gi [3], and some others. As many researchers believe (see, for example, [4]), these parameters partially take into account the superposition of configurations and weak interactions (for example, contact interactions) [5].

The technique of semiempirical calculation of the fine-structure parameters of the npn'd configurations was described in detail in [10]. The energy-operator matrix was also given in [10]. In contrast to [10], we did not use the effective electrostatic parameters F1 and G2 or the experimental gyromagnetic ratios in the system of equations for the fine structure parameters. Independent comparison with the g factors was performed.

In our semiempirical calculations, beginning with [6], along with the conventional (electrostatic and spin– own-orbit) interactions, we also take into account the spin–other-orbit, spin–spin, and orbit–orbit interactions in the energy-operator matrix. These interactions are described by the same radial integrals of Marvin spin interactions Mk and Nk [7] in the single-configuration approximation. For the majority of p5p [8] and p5f [9] configurations of inert gases considered by us previously, this approach turned out to be valid since the obtained residuals ∆E (the differences between the experimental and calculated energies) did not exceed the experimental error, and comparison of the calculated and experimental gyromagnetic ratios showed satisfactory agreement.

The experimental data for the calculation were the absolute energies of the fine-structure levels taken from [11]. These values were measured with errors of 10–4 and 10–3 cm–1 for the 2p5 3d and 2p5 4d configurations, respectively. Refined experimental energies for the 2p5 3d and 2p5 4d configurations were obtained later in [12]. We performed a calculation using both sets of experimental energies and obtained almost the same results.

The above-said is also valid for the 2p5 3d and 4d configurations (which do not overlap in energy with the 2p5 4s and 5s configurations, respectively) considered in this study and also the four-level 2p5 ns configurations.

For the four-level 2p5 ns configuration, the system of equations included three linear equations for the matrix trace, a quadratic equation for the pair product of the roots of the second-order secular equation (the Vieta theorem), and two quadratic equations for the unitary transformation of the matrix of rank 2 to the diagonal form. The unknowns for this system were five finestructure parameters and one coupling coefficient α11. This system of equations was also solved by the Newton iterative method. The zero-order approximations for the fine-structure parameters were obtained by the Slater sum rule.

For the 2p5 3d and 2p5 4d configurations, the experimental gyromagnetic ratios of all levels are known. Hence, we restricted our analysis to these configurations.

For the 2p5 ns configurations, the energy intervals from [11] were used as empirical data. The energy of the lowest level of the 3P2 configuration was assumed to be zero.

0030-400X/05/9904-0536$26.00 © 2005 Pleiades Publishing, Inc.

SEMIEMPIRICAL CALCULATION OF THE ATOMIC CHARACTERISTICS

537

Table 1. Expansion coefficients of the intermediate-coupling wave functions in the LS coupling and the gyromagnetic ratios of the 2p5 3d configuration of the neon atom Level (in Paschen notation)

gcalcd

LS component 3P 2

1D

3F 2

3D 2

d3

0.7997

0.3882

–0.0073

d 1''

–0.0114

0.6445

s 1''''

–0.1062

s 1''

0.5908

gexp [14]

this study

[13]

–0.4580

1.356

1.357

1.356

–0.5460

0.5351

0.948

0.947

0.948

0.5057

0.8246

0.2300

0.787

0.786

0.781

–0.4220

0.1476

0.6716

1.243

1.243

1.242

2

3D 3

1F 3

3F 3

d4 d'

0.7507

–0.0075

0.6606

1.036

1.036

1.034

1

0.3644

0.8388

–0.4045

1.249

1.248

1.249

s 1'''

–0.5511

0.5444

0.6324

1.132

1.133

1.125

1P 1

3D 1

3P 1

d5 d2

0.8978 0.2780

–0.4264 0.7430

–0.1104 –0.6088

1.398 0.853

1.397 0.854

1.397 0.860

s 1'

0.3416

0.5159

0.7856

0.749

0.749

0.752

Table 2. Expansion coefficients of the intermediate-coupling wave functions in the LS coupling and the gyromagnetic ratios of the 2p5 4d configuration of the neon atom Level (in Paschen notation)

gcalcd

LS component 3P 2

1D

3F 2

3D 2

d3

0.7484

0.4421

–0.0168

d 1''

–0.0266

0.6563

s 1''''

–0.1930

s 1''

0.6340

gexp [14]

this study

[13]

–0.4941

1.321

1.325

1.322

–0.5006

0.5639

0.970

0.966

0.990

0.5084

0.8283

0.1345

0.792

0.767

0.783

–0.3395

0.2510

0.6479

1.250

1.275

1.230

2

1F 3

3D 3

3F 3

d4 d'

0.7527

–0.0061

0.6583

1.036

1.036

1.040

0.3339

0.8653

–0.3738

1.262

1.261

1.248

s 1'''

–0.5673

0.5012

0.6534

1.120

1.119

1.116

1P 1

3D 1

1

3P 1

d5 d2 s 1'

0.8977 0.2271

–0.4159 0.7195

–0.1453 –0.6563

1.393 0.810

1.393 0.809

1.391 0.812

0.3775

0.5562

0.7404

0.797

0.798

0.797

RESULTS AND DISCUSSION The fine-structure parameters for the 2p5 3d, 4d, and ns configurations were obtained by solving the systems of nonlinear equations. The corresponding intermediate-coupling wave functions found in the singleconfiguration approximation are listed in Tables 1–3.

2p5

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These tables also contain the calculated gyromagnetic ratios, which are compared with both experiment and the calculated data of [13]. The results of [13] were obtained taking into account the superposition of the configurations 2p5 nd + 2p5 (n + 1)s. It can be seen from Tables 1 and 2 that the results of our calculation of the gyromagnetic ratios are in good agreement with both

SEMENOV et al.

538

Table 3. Coefficients of expansion of the wave functions in LS components and the gyromagnetic ratios of the 2p5 ns configurations of Ne(I) (n = 3–10) α11 (3P1)

α12 (1P1)

g1

2p53s

0.9641

0.2657

1.466

1.464

1.035

1.034

4s

0.7472

0.6646

1.280

1.276*

1.221

1.223*

5s

0.6453

0.7639

1.209

1.207

1.292

1.295

6s

0.6100

0.7924

1.186

1.184

1.315

1.313

7s

0.5956

0.8033

1.178

–

1.323

1.315

8s

0.5894

0.8079

1.174

–

1.327

–

9s

0.5845

0.8114

1.171

–

1.330

–

10s

0.5823

0.8129

1.170

–

1.331

–

Configuration

exp

exp

g2

g1

g2

Notes: The row of coefficients corresponds to the lower level with J = 1. The corresponding gyromagnetic ratio is g1 . For the upper level with J = 1, the values of the coefficients are obtained from the orthogonality condition (the gyromagnetic ratio g2). The experimental values of the g factors and the values denoted by asterisks were taken from [14] and [15], respectively. The calculated values of the g factors are g(3P1) = 1.50115, g(1P1) = 1, g([3/2, 1/2]1) = 1.16705, and g([1/2, 1/2]1) = 1.33411.

experiment and the data of [13]. We neglected the superposition of the above-mentioned configurations since they are completely isolated [11, 14]. However, all possible magnetic interactions in the energy-operator matrix were taken into account, which made it possible to obtain almost zero residuals between calculated and experimental energies (within experimental error). Table 1 shows that the gyromagnetic ratios calculated by us are in good agreement with the corresponding experimental data for the 2p5 3d configuration. For the 2p5 4d configuration, there is a discrepancy (about 0.02) between the calculated and experimental gyromagnetic ratios for some levels (Table 2). A possible reason for this discrepancy is the nonlinear dependence of the energies of the Zeeman sublevels on the magnetic field (see, for example, [16]), which was disregarded in measuring the g factors [14]. It is generally assumed in such measurements that this dependence is linear (this holds true for weak magnetic fields), and, therefore, some average value of the g factor is taken, which differs from the actual one. This explanation is also confirmed by the fact that, for the lowest configuration of the neon atom, 2p5 3d, for which the nonlinearity is smaller, the calculated gyromagnetic ratios are within the experimental error. The calculated gyromagnetic ratios of the 2p5 ns configurations (Table 3) are also in good agreement with the existing experimental data. Moreover, it can be seen from Table 3 that, with an increase in the principal quantum number n of the s electron, the wave-function components α11 and α12 change gradually, approaching the jj coupling (see note to Table 3, where the gyromagnetic ratios in the LS and jj couplings are indicated). To conclude, we should note the following. The results reported here were obtained with the physically justified fine-structure parameters at zero residuals

between calculation and experiment. The latter circumstance is very important in calculation of Zeeman sublevels, especially the crossing and anticrossing fields of magnetic components. As our numerical experiments for the neon atom showed, residuals as small as several tenths of an inverse centimeter lead to errors in determining the crossing fields of up to 10 kOe. A comparison of our results with the data in the literature (including the experimental results) shows good agreement, which indicates sufficient reliability of the single-configuration wave functions listed in Tables 1–3. With an increase in the quantum number n of the s electron, the 2p5 ns configurations show a deviation from the LS coupling and transition to the jj coupling. REFERENCES 1. S. Feneuille, M. Klapisch, E. Koenig, and S. Liberman, Physica A 48, 571 (1970). 2. J. E. Hansen, J. Phys. B 6 (9), 1751 (1973). 3. A. V. Loginov and P. F. Gruzdev, Opt. Spektrosk. 45 (5), 839 (1978) [Opt. Spectrosc. 45 (5), 725 (1978)]. 4. B. Judd and B. Wyborn, Theory of Complex Atomic Spectra (Mir, Moscow, 1973) [in Russian]. 5. G. P. Anisimova, E. L. Kapel’kina, R. I. Semenov, and V. I. Tuchkin, Opt. Spektrosk. 81 (4), 543 (1996) [Opt. Spectrosc. 81 (4), 493 (1996)]. 6. G. P. Anisimova and R. I. Semenov, Opt. Spektrosk. 48 (4), 625 (1980) [Opt. Spectrosc. 48 (4), 345 (1980)]. 7. A. P. Yutsis and A. Yu. Savukinas, Mathematical Foundations of the Atomic Theory (Mintis, Vilnius, 1973) [in Russian]. 8. G. P. Anisimova, E. L. Kapel’kina, R. I. Semenov, and M. Choffo, Opt. Spektrosk. 88 (3), 366 (2000) [Opt. Spectrosc. 88 (3), 321 (2000)]. OPTICS AND SPECTROSCOPY

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SEMIEMPIRICAL CALCULATION OF THE ATOMIC CHARACTERISTICS 9. G. P. Anisimova, E. L. Kapel’kina, and R. I. Semenov, Opt. Spektrosk. 89 (6), 885 (2000) [Opt. Spectrosc. 89 (6), 811 (2000)]. 10. G. P. Anisimova, R. I. Semenov, and V. I. Tuchkin, Opt. Spektrosk. 79 (3), 443 (1995) [Opt. Spectrosc. 79 (3), 409 (1995)]. 11. V. Kaufman and L. Minnhagen, J. Opt. Soc. Am. 62 (1), 92 (1972). 12. E. S. Chang, W. G. Schoenfeld, E. Biemont, et al., Phys. Scr. 49, 26 (1994).

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13. S. Liberman, Physica A 69, 598 (1973). 14. C. E. Moore, Atomic Energy Levels (NBS, Washington, 1949), Vol. 1. 15. V. D. Galkin and R. I. Semenov, Opt. Spektrosk. 29 (6), 1021 (1970). 16. E. L. Kapel’kina and R. I. Semenov, Opt. Spektrosk. 87 (5), 773 (1999) [Opt. Spectrosc. 87 (5), 703 (1999)].

Translated by Yu. Sin’kov