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Spin Hamiltonian for a Pair. H =  B B.g 1 .S 1 + S 1 .D 1 .S 1 +  j S 1 .A 1j .I j +. +  B B.g 2 .S 2 + S 2 .D 2 .S 2 +  j S 2 .A 2j .I j +. +S 1 .J 12 .S 2. S 1 .J 12 .S 2 = J 12 S 1 .S 2 + S 1 .D 12 .S 2 + d 12 .S 1 xS 2. isotropic. anisotropic. antisymmetric.

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Spin hamiltonian for a pair
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


Sh parameters for pairs
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



Some numerical coefficients
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


More coefficients
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


And more
And More

S1 S2 S c1 c2 d1 d2 d12

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

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


Origin of the spin spin interaction
Origin of the Spin-spin interaction

  • Through space (magnetic dipolar)

  • Through bonds (exchange)


Magnetic dipolar
Magnetic Dipolar

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


Dipolar matrix in b 2 r 3 units
Dipolar matrix in B2/r3 units

gxxge 0 0

0 gyyge(1-3sin2) -3sin cos  gyyge

0 -3sin cos  gzzgegzzge(1-3cos2)


Decomposition of the interaction matrix
Decomposition of the interaction matrix

J= (1/3)(Jxx+Jyy+Jzz)

dxx=(Jyz-Jzy)/2

Dij=(Jij+Jji)/2


Dipolar interaction calculated
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


Origin of the exchange contributions
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


Spin hamiltonian for a pair

A

B

Potential exchange- the case of non-degenerate terms

1. One half-filled orbital per ion:

The effective Hamiltonian of the exchange interaction: one orbital per center:

s-s molecule:

Exchange integral (potentialenergy):

2.Non-degenerate terms: Many-electron exchange parameter (all bi-orbital interactions, half-filled orbitals):

The effective Hamiltonian of the exchange interaction: many orbitals per center:

Heisenberg-Dirac- Van Vleck model (HDVV model)


Spin hamiltonian for a pair

Ground

Kinetic exchange-illustration for the simplest case of a dimer-one orbital/one electron per center

P.W. Anderson,

mechanism of the

kinetic exchange:

Charge transfer

A*B, A*B

AB

Antiferromagnetic effect(J<0), singlet-triplet gap: |J |


Spin hamiltonian for a pair

-parameter of the isotropicexchange, incorporates contributions of all mechanisms:

Lande’s rule for the intervals:

Heisenberg-Dirac-Van Vleck (HDVV) model

Full spin S numerates the energy levels (“good” quantum number):

Further generalization: summation over all pairwise interactions ij in many-electron ions with full spinsSi and Sj

Zeeman interaction (orbital part disappears in HDVV model):

This result holds for any direction of the magnetic field H

HDVV- isotropic model


Spin hamiltonian for a pair

Orbital configurations: degenerate ions

Orbital configurations: non-degenerate ions

Orbital doublets

Orbital triplets

Orbital triplets

HDVV systems

Non-Heisenberg systems

When is the HDVV model applicable ?


Spin hamiltonian for a pair

HDVV modelisotropic interactions >>anisotropic interactions

Heisenberg-Dirac-Van Vleck (HDVV) model

The main condition of applicability-orbitally non-degenerate well isolated ground term in crystal field

Under this condition the orbital angular momentum is strongly reduced andtheanisotropic terms arerelatively small (second and higher order corrections):

Antisymmetric exchange:

Local anisotropy:

High order isotropic terms: biquadratic exchange,

symmetric part of the anisotropic exchange tensor,etc


Il modello di anderson
Il modello di Anderson

A-C-B →A+-C-B-

Lo scambio cinetico favorisce il singoletto

Lo scambio potenziale il tripletto


Regole di goodenough kanamori
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


Interazione di scambio
Interazione di scambio

Orbitali magnetici (quelli che hanno l’elettrone spaiato) con sovrapposizione diversa da zero: accoppiamento antiferromagnetico


Interazione di scambio 2
Interazione di scambio (2)

Orbitali magnetici ortogonali: interazione ferromagnetica (regola di Hund)




Interazione di superscambio 3
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


Alcuni esempi dimeri di rame ii
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


Rame ii vanadile iv
Rame(II)-Vanadile(IV)

Indipendente dall’angolo

J> 100 cm-1


Un po di mo
Un po’ di MO

-

Hay-Thibeault-Hoffman

+

J’ è l’integrale di scambio, k sono integrali coulombiani


Il modello di kahn
Il modello di Kahn

J=j-ks2

J integrale di scambio

s integrale di sovrapposizione


A test ground pair
A test ground pair

AF coupling

J> 500 cm-1






Mixed valence manganese dimers
Mixed Valence Manganese Dimers

Manganese(III), d4, S=2

Manganese(IV), d3, S= 3/2

Antiferromagnetic coupling, S= 1/2


Epr spectra of mn iii mn iv
EPR Spectra of MnIII-MnIV

The measurement of the g anisotropy possible at high frequency allows different fits of the hyperfine at low frequency

9 GHz

95 GHz

285 GHz


G anisotropies in mn iii mn iv
g Anisotropies in MnIII-MnIV

giso gx gy gzDg

bisimMe 1.9927 2.0022 1.9963 1.9796 0.0026

bispicenMe 1.9968 2.0055 1.9970 1.9878 0.0177

bisimH2 1.9920 2.0020 1.9935 1.9806 0.0214

bipy 1.9917 2.0005 1.9942 1.9850 0.0200

phen 1.9922 2.0002 1.9950 1.9814 0.0188

Un et al J Phys Chem B 1998, 102 10391


Coefficients for clusters
Coefficients for Clusters

In the assumption of dominant isotropic exchange the coefficients for the spin hamiiltonian in an S multiplet can be obtained using recurrence formulae

The coefficients depend on the intermediate spins


A trinuclear cluster
A trinuclear cluster

c1(S1S2S12S3S)=c1(S12S3S)c1(S1S2S12)

c2(S1S2S12S3S)=c1(S12S3S)c2(S1S2S12)

c3(S1S2S12S3S)=c2(S12S3S)

d1(S1S2S12S3S)=d1(S12S3S)d1(S1S2S12)

d2(S1S2S12S3S)=d1(S12S3S)d2(S1S2S12)

d3(S1S2S12S3S)=d1(S12S3S)

d12(S1S2S12S3S)=d1(S12S3S)d12(S1S2S12)

d13(S1S2S12S3S)=d12(S12S3S)c1(S1S2S12)

d23(S1S2S12S3S)=d12(S12S3S)c2(S1S2S12)


Resonance fields for s states
Resonance fields for S states

H(MM+1)=(ge/g)[H0+(2M+1)/D’/2];

D’=(3cos2-1)D/(geB)


Hf epr provides the sign of d
HF-EPR Provides the Sign of D

Negative D:±S lie lowest

Easy axis type anisotropy

At low T only the -S-S+1 transition is observed


An example cu6
An Example: Cu6

Ground S= 3 state




Single molecule magnets
Single-Molecule Magnets

  • In molecular clusters with large spin S and Ising type anisotropy the magnetization relaxes slowly at low temperature

  • Intermediate behavior between classic and quantum magnets

  • HF-EPR is unique tool for determining the axial and transverse magnetic anisotropy


The first single molecule magnet mn12 acetate

MS=-10

Easy axis

of magnetization

MS= 10

The first single molecule magnet: Mn12-acetate

top view

S4||z

Prepared by a comproportionation reaction:

T. Lis Acta Cryst.1980, B36, 2042.

Mn(AcO)2•4H2O + KMnO4 in 60% v/v AcOH/H2O

[Mn12O12(OAc)16(H2O)4]·2AcOH·4H2O

lateral view

z

Manganese(IV) (s = 3/2, 3d3,)

Manganese(III) (s =2, 3d4)

Oxygen

Carbon

Ground state

S = 8*2 - 4*3/2 = 10

Msaturation = 2.S = 20B



Which are the conditions for tunneling
Which are the conditions for tunneling?

  • The two wave functions must overlap

  • A transverse field must couple the two wavefunctions

  • The coupling splits the two states: tunnel splitting

  • The larger the tunnel splitting the higher the tunnelling probability


Spin hamiltonian for a pair

Zero Field EPR of Mn12Ac

9  8

10  9

8  7


Local probes
Local Probes

  • Electron spin → EPR

  • Nuclear spin → NMR, NQR

  • Muon spin → μSR

  • Neutron spin → PND, INS

Endogenous

Exogenous



55 nmr of mn12 at low t
55NMR of Mn12 at low T

Zero field

Goto et al; Furukawa et al.


Parallel field dependence of 55 nmr in mn12
Parallel field dependence of 55NMR in Mn12

1.5 K



A tetragonal mn12 tbuac
A tetragonal Mn12: tBuAc

Wernsdorfer et al. PRL 2006, 96 057208; Hill et al. Polyhedron 2005 24 2284


The strategy

EPR Spectra

MSH

GSH S= 10

The strategy



Single xtal spectra of mn12tbuac
Single Xtal spectra of Mn12tBuAc

H along c

345 GHz

30 K

The uneven spacings of the lines show the effect of fourth- and higher-order terms

S= 9


Single xtal spectra in the ab plane
Single Xtal spectra in the ab plane

ab plane

115 GHz

5 K

The resonance oscillations indicate high order tetragonal terms


Angular dependence of the resonance fields
Angular dependence of the resonance fields

Parameters in cm-1

B20=-0.15

B40=-2.2x10-5

B44=+1.9x10-5

B60=-1.0x10-8

B64=-1.16x10-7


Physical origin of the parameters
Physical origin of the parameters

  • Projection of individual spin high order parameters

  • Spin admixture

  • A comparison with a Multi Spin Hamiltonian is needed

But the Hilbert space is 100,000,000x100,000,000



Spin hamiltonian for a pair

Axial Field

Transverse Field



Local anisotropy axes vs energy levels
Local anisotropy axes vs. Energy levels

Tetragonal axis

Local Jahn-Teller distortion

Local Jahn-Teller distortions determine transverse tetragonal anisotropy