Spin Hamiltonian for a Pair
H= B B.g1.S1+ S1.D1.S1+ j S1.A1j.Ij+..
+ B B.g2.S2+ S2.D2.S2+ j S2.A2j.Ij+..
+S1.J12.S2
S1.J12.S2 = J12 S1.S2+ S1.D12.S2+ d12.S1xS2
isotropic
anisotropic
Spin-spin interaction
antisymmetric
Decomposizione di J12
J S A
1
1
S  ( J   J  ); A  ( J   J  )
2
2
J  J 1  (S  J 1)  A
1
J  Tr (J )
3
Spin totale
S  S1  S 2
S 2  S12  S 22  2S1S 2
1 2
S1 .S 2  (S  S12  S 22 )
2
J
W ( S )  S ( S  1)  S1 ( S1  1)  S 2 ( S 2  1)
2
Energie degli stati S
E(S)=(J/2)[S(S+1)-S1(S1+1)-S2(S2+1)]
S= 5
S=0
Sa=1/2
E=Jex
S= 4
S= 3
S=1
S= 2
S= 1
S= 0
Two spin 1/2
SM
11  
00 
1
   
10 
2
1
   
2


1  1  
Suscettività
2 2  S(S  1)(2S  1) exp(  E(S) / kT)
N Bg S
=
 S (2S  1) exp(  E(S) / kT)
kT
Sa=1/2 e Sb=1/2
S=0
Sa=1/2
E=Jex
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
S=1
Sb=1/2
Sa=1/2 e Sb=1/2
S=0
Sa=1/2
E=Jex
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
Sb=1/2
S=1
T=
S=0
Magnetic field
S=1
T J/kB
S=0
Magnetic field
Cu2(CH3COO)4.2H2O
Il modello di Anderson
A-C-B →A+-C-B-
b122
J
 J12
U
2
e
*
*
J12    A1(1) B 2  A1(2) B 2 (1)d 1d 2
r12
Lo scambio cinetico favorisce il singoletto
Lo scambio potenziale il tripletto
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
Orbitali magnetici (quelli che hanno l’elettrone spaiato) con
sovrapposizione diversa da zero: accoppiamento antiferromagnetico
Interazione di scambio (2)
Orbitali magnetici ortogonali: interazione ferromagnetica (regola di
Hund)
Interazione di superscambio
Interazione di superscambio
(2)
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)

> 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
-
Hay-Thibeault-Hoffman
(     )
J  2 j'
k0  k
2
J’ è l’integrale di scambio, k sono integrali
coulombiani
+
Il modello di Kahn
J=j-ks2
J integrale di scambio
s integrale di sovrapposizione
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
Ion Dependence of TC
Doppio Scambio
Mn3+
Mn4+
L’elettrone passa dal Mn(III) al Mn(IV)
mantenendo lo spin parallelo a quello degli
altri elettroni: accoppiamento
ferromagnetico
Doppio Scambio
Mn3+
Mn4+
Un sistema a valenza mista di
nichel
Formalmente Ni(II)-Ni(I)
Br
N
N
N
N
N
Ni
Ni
N
N
N
Br
Stato fondamentale S= 3/2. Nessuna evidenza di S= 1/2
Interazione spin-spin
anisotropa
Operatori di shift
S  S x  i S y
S   S x  iS y
1
i
S x  S   S  
S y   S   S  
2
2
1 2
1 2
2
2
2
S x  S   S  S   S  S   S  ; S y   S   S  S   S  S   S 2
4
4
S  S , M  ( S  M )( S  M  1) S , M  1



S  S , M  ( S  M )( S  M  1) S , M  1

Altre relazioni importanti
D XX
DYY
DZZ
1
  DE
3
1
  DE
3
2
 D
3
Zero field splitting S=1
[11>
<11]
Dzz+(Dxx+Dyy)/2
D/3
[1-1>
(Dxx-Dyy)/2
E
(Dxx+Dyy)
-2D/3
<10]
<1-1]
[10>
(Dxx-Dyy)/2
E
Dzz+(Dxx+Dyy)/2
D/3
Zero field splitting S=1
[>
< ]
D12zz/4
D12/6
(D12xx-D12yy)/4
E12/2
(-D12zz +D12xx+D12yy)
-D12/3
{<β]+<β
]}/2
< ββ]
{[β>+[β>}/2 [ββ>
(D12xx-D12yy)/4
E12/2
D12zz/4
D12/6
Ancora operatori di shift

1
Six S jx  Si  S j   Si  S j   Si  S j   Si  S j 
4


1
Siy S jy   Si  S j   Si  S j   Si  S j   Si  S j 
4

Origin of the Spin-spin
interaction
• Through space (magnetic dipolar)
• Through bonds (exchange)
Magnetic Dipolar
J12dip= (B2/r3) [g1.g2- 3(g1.r)(g2.r)/r2]
y
Mn
x

Cu
z
Dipolar matrix in B2/r3 units
gxxge
0
0
0
gyyge(1-3sin2)
-3sin cos  gzzge
0
-3sin cos  gyyge
gzzge(1-3cos2)
Decomposition of the
interaction matrix
J= (1/3)(Jxx+Jyy+Jzz)
dxx=(Jyz-Jzy)/2
Dij=(Jij+Jji)/2
Dipolar interaction calculated
r=2.5 Å
r=3.5 Å
r=4.5 Å
7
3
J
18
D
-3519
E
28
11
5
dx
-83
-30
-14
-1283
-603
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
J<g1g2Hexg1g2>
D <n1g2Hexn1g2>2/2
D(g/g)2J
d <n1g2Hexg1g2>/
d(g/g)J
Spin-orbit coupling
 J ( x  y , xy )

Dex  
/
32

2
2
2
2
2
2
 2 J ( x  y , xz ) y  2 J ( x  y , yz ) x 
2
2
2
z
Dex __(g / g e )
2
SH Parameters for Pairs
In the strong exchange limit, J>>D,d the total spin
S=S1+S2 is a good quantum number:
gS= c1 g1+ c2 g2
AS= c1 A1+ c2 A2
DS= d1 D1+ d2 D2+ d12 D12
c1=(1+c)/2;
c2= (1-c)/2;
d1= (c++c-)/2;d2= (c+-c-)/2;
d12= (1-c+)/2
Zeeman
11 S z 11  1
1 2 s1z 1 2
1
c1  c2 
2
1

2
1 2 s2 z 1 2
1

2
Coupling coefficients

S1 (S1  1)  S2 (S2  1)
c
S(S  1)
3S1 (S1  1)  S2 (S2  1)2  S(S  1)3S(S  1)  3  2S1 (S1  1)  2S2 (S2  1)
c 
(2S  3)(2S  1)S(S  1)
4S(S  1)S1 (S1  1)  S2 (S2  1)  3S1 (S1  1)  S2 (S2  1)
c 
(2S  3)(2S  1)S(S  1)
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
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
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
A test ground pair
AF coupling
J> 500 cm-1
Single Xtal spectra of Mn(II)
doped
Spin Hamiltonian Parameters
gi= -1/6 g1i + 7/6 g2i
D tensor
g Tensor
Origin of the Spin-spin
interaction
• Through space (magnetic dipolar)
• Through bonds (exchange)
Magnetic Dipolar
J12dip= (B2/r3) [g1.g2- 3(g1.r)(g2.r)/r2]
y
Mn
x

Cu
z
Dipolar matrix in B2/r3 units
gxxge
0
0
0
gyyge(1-3sin2)
-3sin cos  gzzge
0
-3sin cos  gyyge
gzzge(1-3cos2)
Decomposition of the
interaction matrix
J= (1/3)(Jxx+Jyy+Jzz)
dx=(Jyz-Jzy)/2
Dij=(Jij+Jji)/2
Dipolar interaction calculated
r=2.5 Å
r=3.5 Å
r=4.5 Å
7
3
J
18
D
-3519
E
28
11
5
dx
-83
-30
-14
-1283
-603
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
Coefficients for Clusters
In the assumption of dominant isotropic exchange the
coefficients for the spin hamiltonian in an S multiplet
can be obtained using recurrence formulae
The coefficients depend on the intermediate spins
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)=d2(S12S3S)
d12(S1S2S12S3S)=d1(S12S3S)d12(S1S2S12)
d13(S1S2S12S3S)=d12(S12S3S)c1(S1S2S12)
d23(S1S2S12S3S)=d12(S12S3S)c2(S1S2S12)
Problemi di Calcolo
Il numero degli stati da calcolare sale rapidamente con il numero
di centri. Infatti per N centri con spin S gli stati da calcolare
sono
(2S+1)N
Stati per 8 spin 5/2
S
n
S
n
S
n
20
17
1
84
19
16
7
210
18
15
28
462
14
11
8
916
4,333
11,200
13
10
7
1,660
6,328
13,600
12
9
6
2,779
8,680
15,520
5
2
16,576
11,900
4
1
16,429
7,700
3
0
14,875
2,666
Zfs parameters for three iron(III)
rings
LiFe6
NaFe6
Fe10
S=1 1.16(1)
4.32(3)
2.24(2)
S=2 0.30(1)
--------
0.599(3)
S=3 -------
--------
0.291(1)
S=4 -------
--------
0.180(1)
S=5 -------
--------
0.123(1)
Dipolar Contribution to zfs of
six membered rings
Spin states:
|S1 S3 S13 S5 S135 S2 S4 S24 S6 S246 SM>
5/2 5/2 5 5/2 15/2
5/25/2 5 5/2
15/2
Dipolar Sum
D is axial with D> 0
Coefficients for Fe6 Rings
S
ci
di
di,i+1
di,i+2
di,i+3
1
.16667
-2.4
2.856 -3.00 2.856
2
.16667
-0.54
0.690 -0.68 0.690
3
.16667
-0.235
0.330 -0.29 0.330
[NaFe6(OCH3)12(pmdbm)6]ClO4
Dipolar Interactions
Inelastic Neutron Scattering
2K
D1= 4.57(2) cm-1
15.31(1) cm-1
S=1
S=0
dipolar interactions
(1.16 cm-1)
[NaGa6-xFex(OCH3)12(pmdbm)6]ClO4
x = 0.1 (Fe/Ga = 1.7%)
[NaGa5Fe] = 9.3%
[NaGa6] = 90.3%
Bulk Susceptibility
H   B S  g  B  DFe [S 2z  SS  1 / 3]  E Fe [S 2x  S 2y ]
—— DFe = 0
0.3 T (
1.0 T (
2.0 T (
3.0 T (
4.0 T (
)
)
)
)
)
B = 0.0255
T
—— DFe = 0.45 cm-1
- - - - DFe = -0.42
cm-1
High-Frequency EPR (240
GHz)
H   B S  g  B  DFe [S 2z  SS  1 / 3]  E Fe [S 2x  S 2y ]
15 K
DFe = 0.43(1) cm-1
EFe = 0.066(3) cm-1
g = 2.003