Magnetic Functions Beyond the Spin-Hamiltonian by Roman Boča (auth.), D. M. P. Mingos (eds.)

By Roman Boča (auth.), D. M. P. Mingos (eds.)

Using the spin-Hamiltonian formalism the magnetic parameters are brought during the elements of the Lambda-tensor regarding merely the matrix components of the angular momentum operator. The power degrees for various spins are generated and the modeling of the magnetization, the magnetic susceptibility and the warmth skill is completed. Theoretical formulae worthwhile in appearing the power point calculations for a multi-term procedure are ready with the aid of the irreducible tensor operator strategy. The objective of the programming lies within the undeniable fact that the total correct matrix parts (electron repulsion, crystal box, spin-orbit interplay, orbital-Zeeman, and spin-Zeeman operators) are evaluated within the foundation set of free-atom phrases. The modeling of the zero-field splitting is finished at 3 degrees of class. The spin-Hamiltonian formalism bargains easy formulae for the magnetic parameters via comparing the matrix components of the angular momentum operator within the foundation set of the crystal-field phrases. The magnetic capabilities for dn complexes are modeled for quite a lot of the crystal-field strengths.

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ML q ML MS ,MS S,S 38 R. Boˇca Since the first-rank (orbital) unit tensor yields a simple expression ln vLS U 1 ln v L S (107) 1/2 =[L(L + 1)(2L + 1)] –1/2 [l(l + 1)(2l + 1)] δv,v δL,L , then ln vLSML MS L1q ln v L S ML MS (108) = [L(L + 1)(2L + 1)]1/2 (– 1)L–ML · L 1 L δ δ δ . δ – ML q ML MS ,MS v,v L,L S,S Finally, we get the complete scalar product in the form +1 –1 n (– 1)q B1–q κq L1q ln v L S ML MS µB l vLSML MS (109) q=–1 –1 = µB ln vLSML MS – B1+1 κ–1 L1–1 + B10 κ0 L10 – B1–1 κ+1 L1+1 ln v L S ML MS = δv,v δL,L δS,S δMS ,MS [L(L + 1)(2L + 1)]1/2 (– 1)L–ML µB · – B1+1 κ–1 – B1–1 κ+1 L 1 L L 1 L + B10 κ0 – ML – 1 ML – ML 0 ML L 1 L – ML + 1 ML .

0 ⎜ T2g (c) ⎟ . Lˆ y . i ⎜ ⎜. ⎟ ⎜ ⎟ ⎜. ⎜T1g (a)⎟ . 0 – 3/ 8 3/ 8 ⎟ ⎜ ⎟ ⎝T (b)⎠ ⎝. . . 0 0 ⎠ 1g T1g (c) . . ⎞ 0 ⎛ . ⎞ ⎛ 0 2 0 0 0 0 0 A2g ⎜. 0 0 0 0 √0 0 ⎟ ⎜T2g (a)⎟ ⎜ ⎟ ⎟ ⎜ ⎜ . 1/2 0 0 15/2 ⎟ 0 ⎜T2g (b)⎟ ⎜ ⎟ √ ⎟ ⎜ ⎟ ⎜ T2g (c) ⎟ . Lˆ z . . ⎜ 15/2 . . – 1/2 0 0 – ⎜ ⎟ ⎟ ⎜ ⎜ T (a) ⎜ 1g ⎟ . 0 0 0 ⎟ ⎜. . ⎟ ⎝T (b)⎠ ⎝. . . 3/2 0 ⎠ 1g T1g (c) . . . – 3/2 ⎛ D4h symmetry, |Γ (γ , a) > Reduction of F-term Oh symmetry, |Γ (γ , a) > Reduction of F-term Table 8 Matrix elements of the angular momentum operator a 48 R.

In this way the energies of the CFTs are obtained. The associated (unitary) eigenvector matrix U contains all the symmetry-adaptation coefficients that transform the atomic-term kets to the CF-term kets: G : (ln αvLSMS )Γγ a i = Uij R3 : (ln α)vLSML MS j . (47) j Γγ a The symmetry-adaptation coefficients SLML ≡ LML |LΓγ a through the orbital-part transformation |Γγ a ≡ |(vLSMS )Γγ a = |vLSML MS · LML |LΓγ a introduced (48) ML could be obtained, for example, by a projection technique |Γγ a = d(Γ ) d(G) ∗ Γγ R Γγ R |LML , (49) R leading to a system of equations [61, 62] Γγ a a Γγ a SLM · SLML = L d(Γ ) d(G) ∗ Γγ R Γγ · LML R LML .

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