Jj coupling treatment to the nonequivalent fnfn’ electrons for jj terms and j levels determination Текст научной статьи по специальности «Математика»

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CHEMICAL PROBLEMS 2020 no. 2 (18) ISSN 2221-8688

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jj COUPLING TREATMENT TO THE NONEQUIVALENT fnfn' ELECTRONS FOR jj

TERMS AND J LEVELS DETERMINATION

P. L. Meena

Department of Chemistry, University of Rajasthan, Jaipur, India, 302004 E-mail: parmeshwar1978@gmail.com

Received 08.04.2020 Accepted 10.06.2020

Abstract: Various coupling schemes are applied to smaller to heavy atoms to describe atomic structure and investigate atomic spectra. jj-coupling is important for heavy atoms in which spin-orbit interactions are dominant over the electrostatic interactions. Here jj-terms and J levels are determined for the nonequivalent

ff (n & n' = 1-2) electrons from the analysis of all possible microstates. The determined jj terms are

designated of the form — }&^notation.

Keywords: nonequivalent electron, ff configuration, jj terms, jj-coupling DOI: 10.32737/2221-8688-2020-2-257-272

Introduction

To obtain the energy levels from multielectron atomic systems, various types of interactions such as electron- electron interaction, spin-orbit interaction etc. are taken into consideration in the Hamiltonian. To describe these energy levels, they are designated by labels which called term symbols. The derivation of the electronic states arising from any given electronic configuration with equivalent or nonequivalent electron is by now a conventional topic in a chemistry course [1-2].

To determine the energy levels two coupling schemes LS-coupling and jj-coupling are the most widely used in atomic theory and spectroscopy. However, they are not suitable for investigation of some elements, so some other coupling schemes also used which are LK and JK-coupling schemes. The total number of levels with given J is the same for all coupling schemes. LS and jj-couplings are homogenous couplings since two electrons are coupled in pairs completely in a symmetric mode, whereas LK and JK-coupling are inhomogeneous, they are characterized by successive coupling of the momenta [3].

The LS coupling is most likely realized

in smaller atoms, whereas the jj-coupling in the case of a heavy system or in atoms which are multiply ionized in which the effect of the spinorbit coupling is no more negligible in relation to electrostatic effects [4-7].

The jj-coupling assumes that the orbital angular momenta, l of each electron is coupled to its spin-momneta, s to give its own angular momenta, j and all j coupled to give total angular momentum J, j values define a term and J values to the levels [8-9]. Symbolically the jj-coupling scheme is written as given by equation (1).

(U)=; (i)

For an equivalent electronic system, the electron-electron interaction is almost larger than the spin-orbit interaction. Thus, for the nonequivalent electronic system the jj-coupling is of the greatest interest [10].

Here the possible jj terms arising from the nonequivalent electrons of ff' (n & n' = 12) configurations occupying to the nonequivalent subshells are obtained from the analysis of all possible microstates and J levels to each jj term are determined.

Methodology

Different method have been developed nonequivalent electrons [8, 11, 13, 14, 18-21]. for determination of the jj terms and J levels, for To derive the jj terms and J levels for the the equivalent electrons [8, 11-18] and nonequivalent ff electrons, first of all, the total

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CHEMICAL PROBLEMS 2020 no. 2 (18)

number of microstates for nonequivalent //^electrons are calculated by possible permutations. The calculated total microstates for nonequivalent / f, / / & / / and / / electrons are 196, 1274 and 8281. After this the possible jj terms are determined by using equation (2) [13].

K*+1}X (*' + II] (2) where n and n' are represent the total number of electrons in subshells.

The determined possible jj terms are designated as [(/jy^.....Js)j notation [8,

9], where the ji, j2, j3, jN are the total angular momenta of electrons. The possible number of jj terms for the nonequivalent /i /i, /i /2 & /2 /i and // electrons are 4, 6 and 9. The j values for /-orbital are 7/2 and 5/2, therefore these jj terms in ...../»)J notation are given in

Table 1.

Table 1. jj terms for f/^'configurations

//'configurations

jj terms

J1 J1 configuration

/7 n/T SVS ArS ft

la'aAa'aAa'aAa'a;

J1 J2 configuration

/7 ? 7\/7 ? 5\f7 5 5U5 ? 7\/5 ? SWS S Ö'2rZ/*5rZr2/tïrZr2/\2' 2' 2f 2' 212' 2/

/ J1 configuration

(L1H(L1 i}(l £ (11ËVË t Tift Ë -h

J2J2 configuration

/7 7 7 7\/7 7 7 5\ /7 7 S 5\/7 S ? 7W? 5 ? S\

lï'ï'ï'ïAz'ï'ï'zi lï'ï'ï'âAï'ï'ï' ïi V.ï'ï'ï'îi

/7 & & &WE & 7 7WË S ? &W& S S \2'2'2'2}\2'2'2'2)\2'2'2'2)\ï2'2'l)

Further, total number of microstates associated with a particular jj term counted by using equation (3) [13].

(2i )!(2i + 2)!

i! (2i - i)! (n - i)! (2i + 2 + i-n)!

x

(2 i')!(2 ii + 2)!

i!(2i -i)!(n -i)!(2i + 2 + i-n )!

(3)

where n and n' represent the number of electrons in partially filled subshells, i and i represent values of orbital angular quantum numbers and the value of i is constrained such that, [»¡> t & O]for subshell less than half full,

[2f ^ £ i Pi for subshell half full and |Si > i > a - Si- 2] for subshell more than half. The microstates for each jj terms for the nonequivalent f f, f f & f f and f f electrons are represented in Table 2.

Table 2. Microstates for each jj term for f/' configuration

ff configurations jj terms with microstates

J1 J1 configuration jj terms

(H) U! (irD (Ü'a) 196

64 48 48 36

J1 J configuration jj terms

Ç 7 7\ ?! ? S\ f? S Ss b'î'îi & 7 \2'2'V /S r S\ fi S 1274

224 384 120 168 288 90

J2 J1 configuration jjj terms

Once the jj terms and total microstates associated with jj terms have been determined, a microstate table for each jj term is constructed and Mj values for each microstate is determines and all microstates for given jj term are arranged in another microstate table with Mj (J,

J-1,......., -J) from which J levels for each jj

term are determined by elimination of

microstates associated with particular Mj starting from maximum Mj and finally move to lowest possible Mj. The microstate table (Table 3) all jj terms to the nonequivalent ff electrons is constructed by arranging two electrons with different j values is and Mj values for all microstates are calculated and same is done for other nonequivalent ff electrons.

Table 3. Microstates for different jj terms offf configuration

jj term {7/2, 7/2}

ji j2 Mj ji j2 Mj ji j2 Mj ji j2 Mj ji j2 Mj

7/2 7/2 7 5/2 7/2 6 3/2 7/2 5 1/2 7/2 4 -1/2 7/2 3

7/2 5/2 6 5/2 5/2 5 3/2 5/2 4 1/2 5/2 3 -1/2 5/2 2

7/2 3/2 5 5/2 3/2 4 3/2 3/2 3 1/2 3/2 2 -1/2 3/2 1

7/2 1/2 4 5/2 1/2 3 3/2 1/2 2 1/2 1/2 1 -1/2 1/2 0

7/2 -1/2 3 5/2 -1/2 2 3/2 -1/2 1 1/2 -1/2 0 -1/2 -1/2 -1

7/2 -3/2 2 5/2 -3/2 1 3/2 -3/2 0 1/2 -3/2 -1 -1/2 -3/2 -2

7/2 -5/2 1 5/2 -5/2 0 3/2 -5/2 -1 1/2 -5/2 -2 -1/2 -5/2 -3

7/2 -7/2 0 5/2 -7/2 -1 3/2 -7/2 -2 1/2 -7/2 -3 -1/2 -7/2 -4

-3/2 7/2 2 -3/2 -1/2 -2 -5/2 7/2 1 -5/2 -1/2 -3 -7/2 7/2 0

-3/2 5/2 1 -3/2 -3/2 -3 -5/2 5/2 0 -5/2 -3/2 -4 -7/2 5/2 -1

-3/2 3/2 0 -3/2 -5/2 -4 -5/2 3/2 -1 -5/2 -5/2 -5 -7/2 3/2 -2

-3/2 1/2 -1 -3/2 -7/2 -5 -5/2 1/2 -2 -5/2 -7/2 -6 -7/2 1/2 -3

-7/2 -1/2 -4 -7/2 -3/2 -5 -7/2 -5/2 -6 -7/2 -7/2 -7

jj term {7/2, 5/2}

ji j2 Mj ji j2 Mj ji J2 Mj ji J2 Mj ji J2 Mj

7/2 5/2 6 5/2 5/2 5 3/2 5/2 4 1/2 5/2 3 -1/2 5/2 2

7/2 3/2 5 5/2 3/2 4 3/2 3/2 3 1/2 3/2 2 -1/2 3/2 1

7/2 1/2 4 5/2 1/2 3 3/2 1/2 2 1/2 1/2 1 -1/2 1/2 0

7/2 -1/2 3 5/2 -1/2 2 3/2 -1/2 1 1/2 -1/2 0 -1/2 -1/2 -1

7/2 -3/2 2 5/2 -3/2 1 3/2 -3/2 0 1/2 -3/2 -1 -1/2 -3/2 -2

7/2 -5/2 1 5/2 -5/2 0 3/2 -5/2 -1 1/2 -5/2 -2 -1/2 -5/2 -3

-3/2 5/2 1 -3/2 -3/2 -3 -5/2 1/2 -2 -7/2 5/2 -1 -7/2 -3/2 -5

-3/2 3/2 0 -3/2 -5/2 -4 -5/2 -1/2 -3 -7/2 3/2 -2 -7/2 -5/2 -6

-3/2 1/2 -1 -5/2 5/2 0 -5/2 -3/2 -4 -7/2 1/2 -3

-3/2 -1/2 -2 -5/2 3/2 -1 -5/2 -5/2 -5 -7/2 -1/2 -4

jj term (5/2, 7/8

j1 j2 Mj ji j2 Mj ji j2 Mj ji j2 Mj ji J2 Mj

5/2 7/2 6 3/2 7/2 5 1/2 7/2 4 -1/2 7/2 3 -3/2 7/2 2

5/2 5/2 5 3/2 5/2 4 1/2 5/2 3 -1/2 5/2 2 -3/2 5/2 1

5/2 3/2 4 3/2 3/2 3 1/2 3/2 2 -1/2 3/2 1 -3/2 3/2 0

5/2 1/2 3 3/2 1/2 2 1/2 1/2 1 -1/2 1/2 0 -3/2 1/2 -1

5/2 -1/2 2 3/2 -1/2 1 1/2 -1/2 0 -1/2 -1/2 -1 -3/2 -1/2 -2

5/2 -3/2 1 3/2 -3/2 0 1/2 -3/2 -1 -1/2 -3/2 -2 -3/2 -3/2 -3

5/2 -5/2 0 3/2 -5/2 -1 1/2 -5/2 -2 -1/2 -5/2 -3 -3/2 -5/2 -4

5/2 -7/2 -1 3/2 -7/2 -2 1/2 -7/2 -3 -1/2 -7/2 -4 -3/2 -7/2 -5

-5/2 7/2 1 -5/2 3/2 -1 -5/2 -1/2 -3 -5/2 -5/2 -5

-5/2 5/2 0 -5/2 1/2 -2 -5/2 -3/2 -4 -5/2 -7/2 -6

jj term OS/2, 5/2)

ji j2 Mj ji j2 Mj ji J2 Mj ji J2 Mj ji J2 Mj

5/2 5/2 5 3/2 5/2 4 1/2 5/2 3 -1/2 5/2 2 -3/2 5/2 1

5/2 3/2 4 3/2 3/2 3 1/2 3/2 2 -1/2 3/2 1 -3/2 3/2 0

5/2 1/2 3 3/2 1/2 2 1/2 1/2 1 -1/2 1/2 0 -3/2 1/2 -1

5/2 -1/2 2 3/2 -1/2 1 1/2 -1/2 0 -1/2 -1/2 -1 -3/2 -1/2 -2

5/2 -3/2 1 3/2 -3/2 0 1/2 -3/2 -1 -1/2 -3/2 -2 -3/2 -3/2 -3

5/2 -5/2 0 3/2 -5/2 -1 1/2 -5/2 -2 -1/2 -5/2 -3 -3/2 -5/2 -4

-5/2 5/2 0 -5/2 1/2 -2 -5/2 -3/2 -4

-5/2 3/2 -1 -5/2 -1/2 -3 -5/2 -5/2 -5

Determination of J levels for jj terms

J level for jj term are obtained by elimination of microstates from microstates associated with a particular J level starting from the maximum Mj value and followed for next J levels and finally move to lowest possible Mj value.

J levels determination and microstates elimination for nonequivalent//* electrons

For (t'^)jj term of the nonequivalent

ff electrons, 15 microstates (Mj =7 to Mj = -7) associated with maximum Mj =7 are eliminated

from Table 4 which result into J=7 level and the next maximum Mj =6, which yield another J=6 level by elimination of 13 microstates (Mj =6 to Mj = -6), and follow the same procedure the continuous elimination of 11, 9,7, 5, 3 and 1 microstates associated with Mj = 5, 4, 3, 2, 1 and 0, result into J=5, 4, 3, 2, 1 and 0 levels for this term. Using the same procedure for other jj terms for the nonequivalent ff electrons and for the nonequivalent f f, f f and f f electrons the J levels are determined. J levels determination for all jj terms for the nonequivalent f f electrons are represented in Table 4 to Table 6.

Mj Microstates Microstates left after the sequential elimination of J levels

7 1 0

6 2 1 0

5 3 2 1 0

4 4 3 2 1 0

3 5 4 3 2 1 0

2 6 5 4 3 2 1 0

1 7 6 5 4 3 2 1 0

0 8 7 6 5 4 3 2 1 0

-1 7 6 5 4 3 2 1 0

-2 6 5 4 3 2 1 0

-3 5 4 3 2 1 0

-4 4 3 2 1 0

Table 4. J levels for (7/^7/2) jj term for// configuration

-5 3 2 1 0

-6 2 1 0

-7 1 0

Microstates 64 49 36 25 16 9 4 1 0

J levels 7 6 5 4 3 2 1 0

Table 5. J levels for (7/3,3/2) and (5/3,7/2) jj terms for ff configuration

Mj Microstates Microstates left after the sequential elimination of J levels

6 1 0

5 2 1 0

4 3 2 1 0

3 4 3 2 1 0

2 5 4 3 2 1 0

1 6 5 4 3 2 1 0

0 6 5 4 3 2 1 0

-1 6 5 4 3 2 1 0

-2 5 4 3 2 1 0

-3 4 3 2 1 0

-4 3 2 1 0

-5 2 1 0

-6 1 0

Microstates 48 35 24 15 8 3 0

J levels 6 5 4 3 2 1

Table 6. J levels for (5^2,-5/2) jj term for ff configuration

Mj Microstates Microstates left after the sequential elimination of J levels

5 1 0

4 2 1 0

3 3 2 1 0

2 4 3 2 1 0

1 5 4 3 2 1 0

0 6 5 4 3 2 1 0

-1 5 4 3 2 1 0

-2 4 3 2 1 0

-3 3 2 1 0

-4 2 1 0

-5 1 0

Microstates 36 25 16 9 4 1 0

J levels 5 4 3 2 1 0

J levels determination and microstates J levels determination for all jj terms for the

elimination for nonequivalent ff2 and f2f nonequivalent ff and f f electrons are electrons represented in Table 7 to Table 12.

Table 7. J levels for (7/2, 7/2, 7/2 ) jj term for//and// configurations

Mj Micro states Microstates left after the sequential removal of different J levels

19/2 1 0

17/2 2 1 0

15/2 4 3 2 0

13/2 6 5 4 2 0

11/2 9 8 7 5 3 0

9/2 12 11 10 8 6 3 0

7/2 16 15 14 12 10 7 4 0

5/2 19 18 17 15 13 10 7 3 0

3/2 21 20 19 17 15 12 9 5 2 0

1/2 22 21 20 18 16 13 10 6 3 1 0

-1/2 22 21 20 18 16 13 10 6 3 1 0

-3/2 21 20 19 17 15 12 9 5 2 0

-5/2 19 18 17 15 13 -10 7 3 0

-7/2 16 15 14 12 10 7 4 0

-9/2 12 11 10 8 6 3 0

-11/2 9 8 7 5 3 0

-13/2 6 5 4 2 0

-15/2 4 3 2 0

-17/2 2 1 0

-19/2 1 0

Micro states 224 204 186 154 126 90 60 28 10 2 0

J levels 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 1/2

(2) (2) (3) (3) (4) (3) (2)

Table 8. J levels for (5/2. 5/1, 5/2 ]jj term for//and// configurations

Mj Microstates Microstates left after the sequential removal of different J levels

13/2 1 0

11/2 2 1 0

9/2 4 3 2 0

7/2 6 5 4 2 0

5/2 9 8 7 5 3 0

3/2 11 10 9 7 5 2 0

1/2 12 11 10 8 6 3 1 0

-1/2 12 11 10 8 6 3 1 0

-3/2 11 10 9 7 5 2 0

-5/2 9 8 7 5 3 0

-7/2 6 5 4 2 0

-9/2 4 3 2 0

-11/2 2 1 0

-13/2 1 0

Microstates 90 76 64 44 28 10 2 0

J levels 13/2 11/2 9/2(2) 7/2(2) 5/2(3) 3/2(2) 1/2

Table 9 J levels for (7/2- 7/2- 5/3 ) jj term for// configuration and <7/2- S/2, 5/2 ) jj term

for f f configurations

Mj Microstate s Microstates left after the sequential removal of different J levels

19/2 1 0

17/2 3 2 0

15/2 6 4 2 0

13/2 10 8 6 4 0

11/2 15 13 11 9 5 0

9/2 21 19 17 15 11 6 0

7/2 27 25 23 21 17 12 6 0

5/2 33 31 29 27 23 18 12 6 0

3/2 37 35 33 31 27 22 16 10 4 0

1/2 39 37 35 33 29 24 20 14 8 4 0

-1/2 39 37 35 33 29 24 20 14 8 4 0

-3/2 37 35 33 31 27 22 16 10 4 0

-5/2 33 31 29 27 23 18 12 6 0

-7/2 27 25 23 21 17 12 6 0

-9/2 21 19 17 15 11 6 0

-11/2 15 13 11 9 5 0

-13/2 10 8 6 4 0

-15/2 6 4 2 0

-17/2 3 2 0

-19/2 1 0

Microstates 384 364 312 280 224 152 108 60 24 8 0

J levels 19/2 17/ 2 (2) 15/ 2 (2) 13/ 2 (4) 11/ 2 (5) 9/2 (6) 7/2 (6) 5/2 (6) 3/2 (4) 1/2 (4)

Table 10. J levels for (7/2r 3/2, 5/2 ) jj term for//configuration and £5/2,n 5/1, 7/2 ) jj term

for f f configuration

Mj Microstates Microstates left after the sequential removal of dil fferent J levels

15/2 1 0

13/2 2 1 0

11/2 4 3 2 0

9/2 6 5 4 2 0

7/2 9 8 7 5 3 0

5/2 11 10 9 7 5 2 0

3/2 13 12 11 9 7 4 2 0

1/2 14 13 12 10 8 5 3 1 0

-1/2 14 13 12 10 8 5 3 1 0

-3/2 13 12 11 9 7 4 2 0

-5/2 11 10 9 7 5 2 0

-7/2 9 8 7 5 3 0

-9/2 6 5 4 2 0

-11/2 4 3 2 0

-13/2 2 1 0

-15/2 1 0

Microstates 120 104 90 66 46 22 10 2 0

J levels 15/2 13/2 11/2(2) 9/2(2) 7/2(3) 5/2(2) 3/2(2) 1/2

Table 11. J levels for £5/2, 7/2r 7/2 ) jj term for//configuration and (7/2r 7/2r 5/2) jj

term for f2 / configuration

Mj Microstates Microstates left after the sequential removal of different J levels

17/2 1 0

15/2 2 1 0

13/2 4 3 2 0

11/2 6 5 4 2 0

9/2 9 8 7 5 3 0

7/2 12 11 10 8 6 3 0

5/2 15 14 13 11 9 6 3 0

3/2 17 16 15 13 11 8 5 2 0

1/2 18 17 16 14 12 9 6 3 1 0

-1/2 18 17 16 14 12 9 6 3 1 0

-3/2 17 16 15 13 11 8 5 2 0

-5/2 15 14 13 11 9 6 3 0

-7/2 12 11 10 8 6 3 0

-9/2 9 8 7 5 3 0

-11/2 6 5 4 2 0

-13/2 4 3 2 0

-15/2 2 1 0

-17/2 1 0

Microstates 168 150 134 106 81 52 28 10 2 0

J levels 17/2 15/2 13/2(2) 11/2(2) 9/2(3) 7/2(3) 5/2(3) 3/2(2) 1/2

Table 12. J levels for ï/3? 5/2 !)jj term for//configuration and [7/5* &/2J jj

term for f2 / configuration

Mj Microst ates Microstates left after sequential removal of different J levels

17/2 1 0

15/2 3 2 0

13/2 6 5 3 0

11/2 10 9 7 4 0

9/2 16 15 13 10 6 0

7/2 21 20 18 15 11 5 0

5/2 26 25 23 20 16 10 5 0

3/2 29 28 26 23 19 13 8 3 0

1/2 32 31 29 26 22 16 11 6 3 0

-1/2 32 31 29 26 22 16 11 6 3 0

-3/2 29 28 26 23 19 13 8 3 0

-5/2 26 25 23 20 16 10 5 0

-7/2 21 20 18 15 11 5 0

-9/2 16 15 13 10 6 0

-11/2 10 9 7 4 0

-13/2 6 5 3 0

-15/2 3 2 0

-17/2 1 0

Micros tates 288 270 238 196 148 88 48 18 6 0

J levels 17/2 15/2(2 ) 13/2(3 ) 11/2(4 ) 9/2(6 ) 7/2(5 ) 5/2(5 ) 3/2(3 ) 1/2(3 )

J levels determination and microstates J levels determination for all jj terms for the elimination for the nonequivalent f2f nonequivalent f2 f2 electrons are represented in electrons Table 13 to Table 18.

Table 13. J levels for (7/2r 7/2, 7/2,7/2-) jj term for//configuration

Mj Micro states Microstates left after the sequential removal of different J levels

12 1 0

11 2 1 0

10 5 4 3 0

9 8 7 6 3 0

8 14 13 12 9 6 0

7 20 19 18 15 12 6 0

6 30 29 28 25 22 16 10 0

5 38 37 36 33 30 24 18 8 0

4 49 48 47 44 41 35 29 19 11 0

3 56 55 54 51 48 42 36 26 18 7 0

2 65 64 63 60 57 51 45 35 27 16 9 0

1 68 67 66 63 60 54 48 38 30 19 12 3 0

0 72 71 70 67 64 58 52 42 34 23 16 7 4 0

-1 68 67 66 63 60 54 48 38 30 19 12 3 0

-2 65 64 63 60 57 51 45 35 27 16 9 0

-3 56 55 54 51 48 42 36 26 18 7 0

-4 49 48 47 44 41 35 29 19 11 0

-5 38 37 36 33 30 24 18 8 0

6 30 29 28 25 22 16 10 0

-7 20 19 18 15 12 6 0

-8 14 13 12 9 6 0

-9 8 7 6 3 0

-10 5 4 3 0

-11 2 1 0

-12 1 0

Micro states 784 759 736 673 616 514 424 294 206 107 58 13 4 0

J 12 11 10 9 8 7 6 5 4 3 2 1 0

levels (3) (3) (6) (6) (10) (8) (11) (7) (9) (3) (4)

Table 14 J levels for (7/Zr 7/Zr //Jand $/Zr 't/'i ,'//2,Jjj terms for// ___configuration_

Mj Micro states Microstates left after sequential removal of different J levels

12 1 0

11 3 2 0

10 7 6 4 0

9 13 12 10 6 0

8 22 21 19 15 9 0

7 34 33 31 27 21 12 0

6 49 48 46 42 36 27 15 0

5 66 65 63 59 53 44 32 17 0

4 83 82 80 76 70 61 49 34 17 0

3 99 98 96 92 86 77 65 50 33 16 0

2 112 111 109 105 99 90 78 63 46 29 13 0

1 121 120 118 114 108 99 87 72 55 38 22 9 0

0 124 123 121 117 111 102 90 75 58 41 25 12 3 0

-1 121 120 118 114 108 99 87 72 55 38 22 9 0

-2 112 111 109 105 99 90 78 63 46 29 13 0

-3 99 98 96 92 86 77 65 50 33 16 0

-4 83 82 80 76 70 61 49 34 17 0

-5 66 65 63 59 53 44 32 17 0

-6 49 48 46 42 36 27 15 0

-7 34 33 31 27 21 12 0

-8 22 21 19 15 9 0

-9 13 12 10 6 0

-10 7 6 4 0

-11 3 2 0

-12 1 0

Micro states 1344 1319 1273 1189 1075 922 742 547 360 207 95 30 3 0

J levels 12 11 (2) 10 (4) 9 (6) 8 (9) 7 (12) 6 (15) 5 (17) 4 (17) 3 (16) 2 (13) 1 (9) 0 (3)

Table 15. J levels for <5/2, Sf2f 5f2rSfl'r} jj term for//configuration

Mj Microstates Microstates left after the sequential removal of different J levels

8 1 0

7 2 1 0

6 5 4 3 0

5 8 7 6 3 0

4 14 13 12 9 6 0

3 18 17 16 13 10 4 0

2 24 23 22 19 16 10 6 0

1 26 25 24 21 18 12 8 2 0

0 29 28 27 24 21 15 11 5 3 0

-1 26 25 24 21 18 12 8 2 0

-2 24 23 22 19 16 10 6 0

-3 18 17 16 13 10 4 0

-4 14 13 12 9 6 0

-5 8 7 6 3 0

-6 5 4 3 0

-7 2 1 0

-8 1 0

Microstates 225 208 193 154 121 67 39 9 3 0

J levels 8 7 6(3) 5(3) 4(6) 3(4) 2(6) 1(2) 0(3)

Table 16 J levels for (7/2, 7/2- 5/S ,5/2,) and (5/2- 5/2- 7/S ,7/2-) jj terms for ff

configuration

Mj Microstates Microstates left after t ie sequential removal of different J leve s

10 1 0

9 2 1 0

8 5 4 3 0

7 8 7 6 3 0

6 14 13 12 9 6 0

5 19 18 17 14 11 5 0

4 27 26 25 22 19 13 8 0

3 32 31 30 27 24 18 13 5 0

2 39 38 37 34 31 25 20 12 7 0

1 41 40 39 36 33 27 22 14 9 2 0

0 44 43 42 39 36 30 25 17 12 5 3 0

-1 41 40 39 36 33 27 22 14 9 2 0

-2 39 38 37 34 31 25 20 12 7 0

-3 32 31 30 27 24 18 13 5 0

-4 27 26 25 22 19 13 8 0

-5 19 18 17 14 11 5 0

-6 14 13 12 9 6 0

-7 8 7 6 3 0

-8 5 4 3 0

-9 2 1 0

-10 1 0

Microstates 420 399 380 329 284 206 151 79 44 9 3 0

J levels 10 9 8(3) 7(3) 6(6) 5(5) 4(8) 3(5) 2(7) 1(2) 0(3)

Table 17. J levels for (7/2- 5/2, 5/5,5/2-) and (5/2- 7/S jj terms for ff ___configuration_

Mj Microstates Microstates left after the sequential removal o: " different J levels

10 1 0

9 3 2 0

8 7 6 4 0

7 13 12 10 6 0

6 22 21 19 15 9 0

5 33 32 30 26 20 11 0

4 45 44 42 38 32 23 12 0

3 57 56 54 50 44 35 24 12 0

2 67 66 64 60 54 45 34 22 10 0

1 74 73 71 67 61 52 41 29 17 7 0

0 76 75 73 69 63 54 43 31 19 9 2

-1 74 73 71 67 61 52 41 29 17 7 0

-2 67 66 64 60 54 45 34 22 10 0

-3 57 56 54 50 44 35 24 12 0

-4 45 44 42 38 32 23 12 0

-5 33 32 30 26 20 11 0

-6 22 21 19 15 9 0

-7 13 12 10 6 0

-8 7 6 4 0

-9 3 2 0

-10 1 0

Microstates 720 699 661 593 503 386 265 157 73 23 2

J levels 10 9(2) 8(4) 7(6) 6(9) 5(11) 4(12) 3(12) 2(10) 1(7) 0(2)

Table 18. J levels for (7Д 5/3, 7/2 jj term for f f configuration

Mj Micro Microstates left after the sequential removal of different J levels

states

12 1 0

11 4 3 0

10 10 9 6 0

9 20 19 16 10 0

8 35 34 31 25 15 0

7 56 55 52 46 36 21 0

6 82 81 78 72 62 47 26 0

5 112 111 108 102 92 77 56 30 0

4 143 142 139 133 123 108 87 61 31 0

3 172 171 168 162 152 137 116 90 60 29 0

2 196 195 192 186 176 161 140 114 84 53 24 0

1 212 211 208 202 192 177 156 130 100 69 40 16 0

0 218 217 214 208 198 183 162 136 106 75 46 22 6 0

-1 212 211 208 202 192 177 156 130 100 69 40 16 0

-2 196 195 192 186 176 161 140 114 84 53 24 0

-3 172 171 168 162 152 137 116 90 60 29 0

-4 143 142 139 133 123 108 87 61 31 0

-5 112 111 108 102 92 77 56 30 0

-6 82 81 78 72 62 47 26 0

-7 56 55 52 46 36 21 0

-8 35 34 31 25 15 0

-9 20 19 16 10 0

-10 10 9 6 0

-11 4 3 0

-12 1 0

Micro states 2304 2279 2210 2084 1894 1639 1324 986 656 377 174 54 6 0

J levels 12 11 10 9 8 7 6 5 4 3 2 1 0

(3) (6) (10) (15) (21) (26) (30) (31) (29) (24) (6) (6)

Results and Discussion

J levels determined for all the jj terms for the for ff (f * |) for //and// and (7, r)

nonequivalent// electrons {n = 1-2, n' = 1-2) Uaa/ U 2 2 1/

are represented in Table 19. The ground state jj ff electrons. The jj terms and J levels for/

, ,, , /b 5"i /electrons are shown in Figure 1. terms determined for these electrons are \Tt~)

(7/2, 7/2) —- *t|l

f/1

---------

(7/2, 5/2) (5/2, 7/2)

\ (5/2, 5/2)

•air--::—

J= 7

J= =6

J= =5

J= =4

J= =3

J= 2

. J = 1

J= =0

J =6

J= =5

J= 4

J= 3

J= 2

J= 1

J= 5

J= 4

J= 3

J= 2

J= 1

_ J= 0

Configuration

jj terms

J levels

Figure 1. Schematic representation of jj terms and J levels for the nonequivalent J1 J1 electrons Table 19. jj terms and J levels for different //'configurations

ff configuration

(r I) ' [(I* f)& (f'!) s^3'2'1] [(I' f) s^^^iiQ

f / f configuration

(H' D5ß(Z)fZß(2)Af2\

(M' S 11/2(5), 3/2(4). 1/2(4)]

(2, 2> j)15A 13/2, 11 5/2(2), 3/2(2), 1/2

[(I Ï' I)1?Al 15 A 9/2(3?, 7/2(3), 5/2(3), 3/2(2), I/2]

(IÎ 2) !E/2(ï)rl3/2 (3), 11/2(4), 9/2(Ê), 7/2(5), E/2(5),3/2(3),1 /2 (3)]

(j, ¿I 2) 13A11 & sftW' V2®1 r±

ff configuration

(I'l 1?/2, ^^W^^ffi11^® E/2(3), 3/2(2)., 1/2]

[(II' S ^WM 13/2(4), 11/2(E), Ë/2{£Ti., 3/2(4):1/S(4)]

(I ID15 A 13A 11 W2), ^

(I' f ' I)1E A (ft 9/2(3), 7/2(3), 5/2(3), 3/2(2), 1/2

(2' 2)171/2' 13/2№ 11 /2(4), ?/ï(5), 5/2(5), 3/2(3), 1/2(3)]

(111)11/2, 1/21

ff configuration

Î \> D ^W3)'**3)^7^10?' 5(^4(11),3(7),2(S),1(3),0(*)]

Kï' Î I D S (? I' ilS^ÎÏ^Œi

[(| 1I)'l^ll<3),10(^,î(ie),6CL5),?(ïl), 6(26),! 5(30), ^31)3(23), 2ÎÏ4).1(1S),Q<6?]

[f(|: f, I f) & (f, f, i D) 1W), 5 (11), 4(12), 3(12 ), 2 (10), 1 (7), 0(2)]

(2' 2' 2' 2) 5(3),4(6),3(4),2(£>,1(2),0(3)]

Conclusion

Using a simple method the jj terms and J levels helpful in analysis of atomic structure and are determined for the nonequivalent electrons spectral properties of f f configurations and for different f /'configurations (f /, f f2& f f enrich knowledge about J levels and jj terms in and ff ). These jj terms and J levels will be theoretical inorganic chemistry.

Acknowledgement

Author would like to thank Dr. K. S. Meena for useful suggestions.

References

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ОПРЕДЕЛЕНИЕ jj ТЕРМОВ И J УРОВНЕЙ НЕЭКВИВАЛЕНТНЫХ ЭЛЕКТРОНОВ

П.Л. Меена

Индия, г.Джайпур, университет Раджастан e-mail: parmeshwar1978@gmail.com

jj-Термы н J уровни определены для неэквивалентных ff (п & п' = 1-2) электронов из анализа всех возможных микросостояний. Установленные jj термы обозначены в виде [O^fa^i1,11/я• Ключевые слова: неэквивалентные электроны, ff конфигурации, jj-термы

jj TERMLdRINiN VO J SO ViYYOL ORINiN TOYiN EDiLMOSi UQUN QEYRi EKViVALENT ELEKTRONLARA CUTLd§Md i^LdNiLMdSiNiN TOTBiQi

P. L. Meena

Rajastan Universiteti, Jaypur, Hindistan e-mail: parmeshwar1978@gmail.com

Atom qurulu§unu tasvir etmak vd atom spektrlarini oyranmak ugun yungul vd agir atomlara muxtalif cutla§ma sxemlari tatbiq olunur. jj -cutla§ma, spin-orbit qar§iliqli tasirin elektrostatik qar§iliqli tasir uzarinda ustunluk ta§kil edan agir atomlar ugun vacibdir. Burada, butun mumkun olan mikrovaziyyatlarin analizindan jj termlari va J saviyyalari mumkun olan qeyri-ekvivalent

ff (n & n' 1-2) elektronlar iigim mi\dyydn edilmisdir. Tsyyin olunmus jj termlar [ Cik/i./s.____-iff)/] §dkHndd isar,i edilmisdir.

Agar sozlw. jj -cutla§ma, qeyri-ekvivalent, atom spektrlari