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Spin-Hamiltonian Analysis for a Pair of Spins Featuring Exchange and Spin-Spin Interactions

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This document presents a comprehensive analysis of the Spin-Hamiltonian for a pair of spins, emphasizing the terms related to spin interactions, such as the Zeeman effect, exchange coupling, and anisotropic interactions. The mathematical framework outlines the effects of magnetic fields on the energy states of spin pairs, exploring different configurations and interactions. The significance of antiferromagnetic and ferromagnetic coupling, along with the pertinence of Goodenough-Kanamori rules in magnetic materials, is discussed, providing insights into magneto-structural correlations.

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Spin-Hamiltonian Analysis for a Pair of Spins Featuring Exchange and Spin-Spin Interactions

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  1. Spin Hamiltonian for a Pair H= BB.g1.S1+ S1.D1.S1+ j S1.A1j.Ij+.. + BB.g2.S2+ S2.D2.S2+ j S2.A2j.Ij+.. +S1.J12.S2 S1.J12.S2 = J12S1.S2+ S1.D12.S2+ d12.S1xS2 isotropic anisotropic antisymmetric Spin-spin interaction

  2. Decomposizione di J12

  3. Spin totale

  4. S=0 Sa=1/2 E=Jex S=1 S= 5 S= 4 S= 3 S= 2 S= 1 S= 0 Energie degli stati S E(S)=(J/2)[S(S+1)-S1(S1+1)-S2(S2+1)]

  5. Two spin 1/2

  6. Suscettività =

  7. S=0 Sa=1/2 E=Jex S=1 Sa=1/2 e Sb=1/2 Sb=1/2 E(S=1)=1/2Jex(1*2-1/2*3/2-1/2*3/2)=-¼ Jex E(S=0)= 1/2Jex(-1/2*3/2-1/2*3/2)=3/4Jex E=Jex the energy of the four states are E(1,-1)=-gBH E(1,0)=0 E(1,1)=+gBH E(0,0)=Jex

  8. Sa=1/2 e Sb=1/2 S=0 Sa=1/2 E=Jex Sb=1/2 E(S=1)=1/2Jex(1*2-1/2*3/2-1/2*3/2)=-¼ Jex E(S=0)= 1/2Jex(-1/2*3/2-1/2*3/2)=3/4Jex E=Jex 2Ng2B2 =------------------ kT[3+exp(Jex/kT)] eq. Bleaney - Bowers S=1

  9. S=1 T= S=0 Magnetic field

  10. S=1 T J/kB S=0 Magnetic field

  11. Cu2(CH3COO)4.2H2O

  12. Il modello di Anderson A-C-B →A+-C-B- Lo scambio cinetico favorisce il singoletto Lo scambio potenziale il tripletto

  13. Regole di Goodenough-Kanamori • Se gli orbitali magnetici si sovrappongono l’accoppiamento è antiferromagnetico • Se gli orbitali magnetici sono ortogonali ed hanno ragionevoli zone di sovrapposizione lo scambio è ferromagnetico • Se un orbitale magnetico sovrappone con un orbitale vuoto l’accoppiamento è ferromagnetico

  14. Interazione di scambio Orbitali magnetici (quelli che hanno l’elettrone spaiato) con sovrapposizione diversa da zero: accoppiamento antiferromagnetico

  15. Interazione di scambio (2) Orbitali magnetici ortogonali: interazione ferromagnetica (regola di Hund)

  16. Interazione di superscambio

  17. Interazione di superscambio (2)

  18. Interazione di superscambio (3) La frazione di elettrone trasferita nell’orbitale z2 polarizza gli spin degli altri elettroni spaiati, tenendoli paralleli a sé: accoppiamento ferromagnetico

  19. Alcuni Esempi: Dimeri di Rame(II) > 96°  < 96° R.D.Willett, D.Gatteschi,O.Kahn, Magneto-Structural Correlations in Exchange Coupled Systems, NATO ASI C140,Reidel, 1985

  20. Rame(II)-Vanadile(IV) Indipendente dall’angolo J> 100 cm-1

  21. Un po’ di MO - Hay-Thibeault-Hoffman + J’ è l’integrale di scambio, k sono integrali coulombiani

  22. Il modello di Kahn J=j-ks2 J integrale di scambio s integrale di sovrapposizione

  23. Prussian Blue Type Compounds CnAp[B(CN)6]q.xH2O C monovalent cation A is N coordinated B is C coordinated The sign of the coupling can be easily understood considering the magnetic orbitals: if they are orthogonal the coupling is ferromagnetic, otherwise antiferromagnetic

  24. Ion Dependence of TC

  25. Doppio Scambio L’elettrone passa dal Mn(III) al Mn(IV) mantenendo lo spin parallelo a quello degli altri elettroni: accoppiamento ferromagnetico

  26. Doppio Scambio Mn3+ Mn4+

  27. Un sistema a valenza mista di nichel Formalmente Ni(II)-Ni(I) Stato fondamentale S= 3/2. Nessuna evidenza di S= 1/2

  28. Interazione spin-spin anisotropa

  29. Operatori di shift

  30. Altre relazioni importanti

  31. Zero field splitting S=1

  32. Zero field splitting S=1

  33. Ancora operatori di shift

  34. Origin of the Spin-spin interaction • Through space (magnetic dipolar) • Through bonds (exchange)

  35. Magnetic Dipolar J12dip= (B2/r3) [g1.g2- 3(g1.r)(g2.r)/r2]

  36. Dipolar matrix in B2/r3 units gxxge 0 0 0 gyyge(1-3sin2) -3sin cos  gyyge 0 -3sin cos  gzzgegzzge(1-3cos2)

  37. Decomposition of the interaction matrix J= (1/3)(Jxx+Jyy+Jzz) dxx=(Jyz-Jzy)/2 Dij=(Jij+Jji)/2

  38. Dipolar interaction calculated r=2.5 Å r=3.5 Å r=4.5 Å J 18 7 3 D -3519 -1283 -603 E 28 11 5 dx -83 -30 -14 The values are given in 10-4 cm-1. gxx=gyy=2.2; gzz=2.0. The principal direction of D is parallel to the Mn-Cu direction

  39. Origin of the Exchange Contributions J<g1g2Hexg1g2> D <n1g2Hexn1g2>2/2 D(g/g)2J d <n1g2Hexg1g2>/ d(g/g)J

  40. Spin-orbit coupling

  41. SH Parameters for Pairs In the strong exchange limit, J>>D,d the total spin S=S1+S2 is a good quantum number: gS= c1g1+ c2g2 AS= c1A1+ c2A2 DS= d1D1+ d2D2+ d12D12 c1=(1+c)/2; c2= (1-c)/2; d1= (c++c-)/2; d2= (c+-c-)/2; d12= (1-c+)/2

  42. Zeeman

  43. Coupling coefficients

  44. Some numerical coefficients S1 S2 S c1 c2 d1 d2 d12 1/2 1/2 1 1/2 1/2 0 0 1/2 1 1 1 1/2 1/2 -1/2 -1/2 1 1 1 2 1/2 1/2 1/6 1/6 1/3 3/2 3/2 1 1/2 1/2 -6/5 -6/5 17/10 3/2 3/2 2 1/2 1/2 0 0 1/2 3/2 3/2 3 1/2 1/2 1/5 1/5 3/10

  45. More coefficients S1 S2 S c1 c2 d1 d2 d12 2 2 1 1/2 1/2 -21/10 -21/10 13/5 2 2 2 1/2 1/2 -3/14 -3/14 5/7 2 2 3 1/2 1/2 1/10 1/10 2/5 2 2 4 1/2 1/2 3/14 3/14 2/7 5/2 5/2 1 1/2 1/2 -16/5 -16/5 37/10 5/2 5/2 2 1/2 1/2 -10/21 -10/21 41/42 5/2 5/2 3 1/2 1/2 -1/45 -1/45 47/90 5/2 5/2 4 1/2 1/2 1/7 1/7 5/14 5/2 5/2 5 1/2 1/2 2/9 2/9 5/18

  46. And More S1 S2 S c1 c2 d1 d2 d12 1/2 1 1/2 -1/3 4/3 0 0 0 1/2 1 3/2 1/3 2/3 0 1/3 1/3 1/2 3/2 1 -1/4 5/4 0 3/2 -1/4 1/2 3/2 2 1/4 3/4 0 1/2 1/4 1/2 2 3/2 -1/5 6/5 0 7/5 -1/5 1/2 2 5/2 1/5 4/5 0 3/5 1/5 1/2 5/2 2 -1/6 7/6 0 8/6 -1/6 1/2 5/2 3 1/6 5/6 0 4/6 1/6

  47. A test ground pair AF coupling J> 500 cm-1

  48. Single Xtal spectra of Mn(II) doped

  49. Spin Hamiltonian Parameters gi= -1/6 g1i + 7/6 g2i

  50. D tensor

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