Capability of the Free-Ion Eigenstates for Crystal-Field Splitting

Mulak, Jacek; Mulak, Maciej

doi:10.4236/jmp.2011.211170

Cited by 14 publications

(57 citation statements)

References 22 publications

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“…The data needed for the calculation coming from the

^{3}

_{4}

and

^{3}

_{5}

states are given below in this order. The total splitting of the state

Δ E

(cm

^{- 1} false)

: 703 and 451;

σ^{2}

((cm

^{- 1})^{2} false)

: 56 844 and 35 967;

A_{4}

- 0.7693

and

- 0.6833

;

A_{6}

: 0.6555 and 0.4451 (). Substituting the above data into Eq.…”

Section: Usage Of the Multipolar Additivity Of The Squares Of The Cf mentioning

confidence: 99%

“…where J is a good quantum number of the resultant angular momentum of the state

false| J 〉

E_{i}

is the i‐th CF sublevel energy, and

3 \overset{E}{true} false(J false)

is the energy baricenter (centroid) of the split

false| J 〉

state. Simultaneously, one can use the complementary algebraic expression for the

σ^{2} false(J false)

consisting in additivity of the squares of the partial multipolar components of the considered CF splitting second moments

σ^{2} false(J false) = \sum_{k} σ_{k}^{2} false(J false) = \frac{1}{2 J + 1} \sum_{k} A_{k}^{2} false(J false) S_{k}^{2},

where

k = 2, 4

, and 6, and

A_{k} (J) = 〈 J false| false| C^{false(k false)} false| false| J 〉

is a dimensionless scalar that depends only on the angular distribution of the

false| J 〉

state electron density and reflects its

2^{k}

‐pole‐type asphericity , the term

S_{k}^{2} = \frac{1}{2 k + 1} \sum_{q} false| B_{kq} |^{2}

stands for the square of the conventional CF strength of the respective

2^{k}

‐pole . The

S_{k}

represents modulus of

2^{k}

‐pole of the central ion surroundings, whereas

A_{k} false(J false)

represents modulus of the same type multipole (k) of the central ion electron state.…”

Section: Introductionmentioning

confidence: 99%

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A direct algebraic parametrization of the high-symmetry crystal-field Hamiltonians

Mulak

2015

Phys. Status Solidi B

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An algebraic, calculational, non‐fitting method of parametrization of the cubic and hexagonal crystal‐field (CF) Hamiltonians false(HCFfalse) with only two independent CF parameters B40 and B60 is proposed. It is based on the linear relation σ2false(Jfalse)=∑kAk2false(Jfalse)Sk2 between σ2false(Jfalse) the square of the second moment of the electron state false|J〉 CF splitting and the squares of the multipolar CF strengths Sk2=12k+1∑qfalse|Bkq|2, where AkJ=〈Jfalse|false|Cfalse(kfalse)false|false|J〉. For a pair of false|Ji〉 and false|Jj〉 electron states an algebraic solution for S42 and S62, and hence for the B40 and B60 CF parameters can be gained. The necessary conditions for physical correctness of the solutions false(Sk2≥0false) impose limitations on the observed σ2false(Jifalse)/σ2false(Jjfalse) ratios depending on the respective ratios Ak2false(Jifalse)/Ak2false(Jjfalse). These restrictions can be used to verify postulated CF splitting diagrams. In the high‐symmetry crystal fields, the electron states with J=2 or 5/2 undergo splitting only by the k=4 HCF multipole, and from their CF splitting second moments the B40 parameter can be directly obtained. For J=2, false|B40false|=6310Δfalse(Jfalse)false|A4false(Jfalse)false|, whereas for J=5/2, false|B40false|=7Δfalse(Jfalse)false|A4false(Jfalse)false|, where Δ is the energy gap between the two Stark's levels of the false|J〉 state. The proposed method is demonstrated and discussed for several systems of RE3+ ions in some high‐symmetry crystal matrices.

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“…The data needed for the calculation coming from the

^{3}

_{4}

and

^{3}

_{5}

states are given below in this order. The total splitting of the state

Δ E

(cm

^{- 1} false)

: 703 and 451;

σ^{2}

((cm

^{- 1})^{2} false)

: 56 844 and 35 967;

A_{4}

- 0.7693

and

- 0.6833

;

A_{6}

: 0.6555 and 0.4451 (). Substituting the above data into Eq.…”

Section: Usage Of the Multipolar Additivity Of The Squares Of The Cf mentioning

confidence: 99%

“…where J is a good quantum number of the resultant angular momentum of the state

false| J 〉

E_{i}

is the i‐th CF sublevel energy, and

3 \overset{E}{true} false(J false)

is the energy baricenter (centroid) of the split

false| J 〉

state. Simultaneously, one can use the complementary algebraic expression for the

σ^{2} false(J false)

consisting in additivity of the squares of the partial multipolar components of the considered CF splitting second moments

σ^{2} false(J false) = \sum_{k} σ_{k}^{2} false(J false) = \frac{1}{2 J + 1} \sum_{k} A_{k}^{2} false(J false) S_{k}^{2},

where

k = 2, 4

, and 6, and

A_{k} (J) = 〈 J false| false| C^{false(k false)} false| false| J 〉

is a dimensionless scalar that depends only on the angular distribution of the

false| J 〉

state electron density and reflects its

2^{k}

‐pole‐type asphericity , the term

S_{k}^{2} = \frac{1}{2 k + 1} \sum_{q} false| B_{kq} |^{2}

stands for the square of the conventional CF strength of the respective

2^{k}

‐pole . The

S_{k}

represents modulus of

2^{k}

‐pole of the central ion surroundings, whereas

A_{k} false(J false)

represents modulus of the same type multipole (k) of the central ion electron state.…”

Section: Introductionmentioning

confidence: 99%

A direct algebraic parametrization of the high-symmetry crystal-field Hamiltonians

Mulak

2015

Phys. Status Solidi B

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“…The obtained CEF parameters for B 60 = +5.9 cm −1 and B 66 = +500 cm −1 . They enable an easy calculation of the multipolar CEF strengths S 2 k [26]. With known from the atomic physics the ion aspherities of the rank k of the multiplet 4 I 9/2 , given in Refs.…”

Section: Table Imentioning

confidence: 99%

“…Such good agreement, according to Refs. [26,27], is strong argument for the localization of 5f electrons.…”

Section: Table Imentioning

confidence: 99%

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Three Localized f Electrons in UPd₂Al₃and in UGe₂Intermetallics

Radwański

Nalecz

Ropka³

2016

Acta Phys. Pol. A

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We have calculated the strength of the excitations between the crystal-field states which are in agreement with inelastic-neutron-scattering results. This agreement confirms the existence in the heavy-fermion superconductor UPd2Al3 the crystal-field electronic structure being the finger-print of the U 3+ ions with three localized f electrons forming strongly-correlated atomic-like quantum system 5f 3 . The ionic integrity and the low-energy crystal-field electronic structure is preserved in this metallic system in the meV scale as has been postulated in the Quantum Atomistic Solid State theory (QUASST). We provide preliminary results with the U 3+ ion in UGe2 showing the ground-state eigenfunction which reproduces the ordered magnetic-moment value of 1.48 µB. This moment is composed from the dominant orbital contribution (2.6 µB) and the opposite spin moment (1.12 µB).

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Verification of the fitted crystal‐field Hamiltonian parametrization by the crystal‐field splitting second moment

Mulak

2012

Physica Status Solidi (b)

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A physically valid parametrization of the crystal-field (CF) Hamiltonian H CF has to reproduce not only the energy of Stark sublevels but also the multipolar structure of the central ion surroundings. The square of the second moment s 2 J j i ð Þ of J j i electronic state CF splitting must be a superposition of squares of the second moments of the partial splittings produced separately by the individual multipoles:is the k-rank CF strength. The relationship between s 2 J j i ð Þ known from experiment and S 2 k obtained from fitting procedure allows us to verify whether the parametrization is physically valid. It is demonstrated by several examples of CF splittings of electronic states of trivalent lanthanide ions in different crystal matrices. The approach is confined only to the states with good quantum numbers J. Nevertheless, by finding such states in any analyzed spectrum, the method can provide the actual multipole characteristics of H CF , and plays the role of a preparametrization. In general, the stepwise fitting procedure can never lead to parametrizations with a correct multipolar structure due to the limitation of the initial eigenfunctions. In consequence, the pertinent crystalfield parameters (CFPs) are burdened with a methodical inherent fault. The loss of physical meaning of fitted CFPs, its source and consequences, constitute the main theme of the paper.

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Capability of the Free-Ion Eigenstates for Crystal-Field Splitting

Cited by 14 publications

References 22 publications

A direct algebraic parametrization of the high-symmetry crystal-field Hamiltonians

A direct algebraic parametrization of the high-symmetry crystal-field Hamiltonians

Three Localized f Electrons in UPd₂Al₃and in UGe₂Intermetallics

Verification of the fitted crystal‐field Hamiltonian parametrization by the crystal‐field splitting second moment

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Capability of the Free-Ion Eigenstates for Crystal-Field Splitting

Cited by 14 publications

References 22 publications

A direct algebraic parametrization of the high-symmetry crystal-field Hamiltonians

A direct algebraic parametrization of the high-symmetry crystal-field Hamiltonians

Three Localized f Electrons in UPd2Al3and in UGe2Intermetallics

Verification of the fitted crystal‐field Hamiltonian parametrization by the crystal‐field splitting second moment

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Product

Resources

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Three Localized f Electrons in UPd₂Al₃and in UGe₂Intermetallics