Spin hamiltonian for a pair
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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

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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 = J12S1.S2+ S1.D12.S2+ d12.S1xS2




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

Coupling coefficients

Coupling coefficients

Some numerical coefficients

Some numerical coefficients








More coefficients

More coefficients


2211/21/2-21/10-21/10 13/5




5/25/211/21/2-16/5-16/5 37/10

5/25/221/21/2-10/21-10/21 41/42

5/25/231/21/2-1/45-1/45 47/90



And more

And More










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


0gyyge(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)



Dipolar interaction calculated

Dipolar interaction calculated

r=2.5 År=3.5 År=4.5 Å





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


D <n1g2Hexn1g2>2/2


d <n1g2Hexg1g2>/


Spin hamiltonian for a pair



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


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


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

Interazione di superscambio

Interazione di superscambio 2

Interazione di superscambio (2)

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


Indipendente dall’angolo

J> 100 cm-1

Un po di mo

Un po’ di MO




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

Il modello di kahn

Il modello di Kahn


J integrale di scambio

s integrale di sovrapposizione

A test ground pair

A test ground pair

AF coupling

J> 500 cm-1

Single xtal spectra of mn ii doped

Single Xtal spectra of Mn(II) doped

Spin hamiltonian parameters

Spin Hamiltonian Parameters

D tensor

D tensor

G tensor

g Tensor

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 gygzDg

bisimMe1.9927 2.0022 1.99631.9796 0.0026

bispicenMe1.9968 2.0055 1.9970 1.9878 0.0177

bisimH21.9920 2.0020 1.9935 1.9806 0.0214

bipy1.9917 2.0005 1.9942 1.9850 0.0200

phen1.9922 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










Resonance fields for s states

Resonance fields for S states



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

Cu6 x band spectra

Cu6: X-band Spectra

Cu6 245 ghz spectra

Cu6: 245 GHz Spectra

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


Easy axis

of magnetization

MS= 10

The first single molecule magnet: Mn12-acetate

top view


Prepared by a comproportionation reaction:

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

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


lateral view


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

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



Ground state

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

Msaturation = 2.S = 20B

Spin hamiltonian for a pair

Very High Field EPR Spectra of Mn12acetate


525 GHz

T= 30 K

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



Nmr and tunnelling

NMR and tunnelling

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

Transverse field dependence of 55 mn nmr of mn12

Transverse field dependence of 55Mn NMR of Mn12

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


GSH S= 10

The strategy

The sh for giant spin

The SH for giant spin

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






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

A tractable model system

A tractable model system


Spin hamiltonian for a pair

Axial Field

Transverse Field

Calculated angular dependence of the transverse resonances

Calculated angular dependence of the transverse resonances

-10 -9

-9 -8

-8 -7

-7 -6

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

Angular dependence of the transverse resonances in the molecule

Angular dependence of the transverse resonances in the molecule

Spin hamiltonian for a pair

Barra et al. JACS 2007 in press

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