Magnetism, Magnetic Properties,
Magnetochemistry
1
2
Lecture 1: Basic notions from atomic and molecular magnetism (2 h)
Atom in magnetic field
Diamagnetism and paramagnetism
Spin-orbit interaction
Zeeman effect
Van Vleck theory of magnetic susceptibility
Curie law
Magnetization of atoms and molecules
3
Lecture 2: Exchange interaction in magnetic molecules (2 h)
Exchange interaction between electrons
Spin operators
Heisenberg-Dirac-Van Vleck Hamiltonian
Origin of paramagnetism in molecules
Contributions to the exchange interaction in magnetic molecules
Examples: exchange-coupled transition metal complexes
4
Lecture 3: Magnetic anisotropy in molecules (2 h)
Origin of magnetic anisotropy
Ligand field splitting of orbital shells
Quenching of orbital and spin momenta in molecules
Magnetic anisotropy in mononuclear complexes:
zero-field splitting
g tensor
susceptibility tensor
Example: Ni(II) complexes
Magnetism
All matter is electronic
Positive/negative charges - bound by Coulombic forces
Result of electric field E between charges, electric dipole
Electric and magnetic fields = the electromagnetic interaction
(Oersted, Maxwell)
Electric field = electric +/ charges, electric dipole
Magnetic field ??No source?? No magnetic charges, N-S
No magnetic monopole
Magnetic field = motion of electric charges
(electric current, atomic motions)
Magnetic dipole – magnetic moment  = i  A [A m2]
5
Electromagnetic Fields
6
Magnetism
Magnetic field = motion of electric charges
• Macro = electric current
• Micro = spin + orbital momentum
Ampère 1822
Magnetic dipole – magnetic (dipole) moment  [A m2]
Ai 
7
Poisson model
Ampere model
Magnetism
Microscopic explanation of source of magnetism
= Fundamental quantum magnets
Unpaired electrons = spins (Bohr 1913)
Atomic building blocks (protons, neutrons and electrons = fermions)
possess an intrinsic magnetic moment
Gerlach and Stern 1921
Relativistic quantum theory (P. Dirac 1928)  SPIN
(quantum property ~ rotation of charged particles)
Spin (½ for all fermions) gives rise to a magnetic moment
8
Atomic Motions of Electric Charges
The origins for the magnetic moment of a free atom
Motions of Electric Charges:
1) The spins of the electrons S. Unpaired spins give a
paramagnetic contribution. Paired spins give a diamagnetic
contribution.
2) The orbital angular momentum L of the electrons about the
nucleus, degenerate orbitals, paramagnetic contribution.
The change in the orbital moment induced by an applied
magnetic field, a diamagnetic contribution.
3) The nuclear spin I – 1000 times smaller than S, L
nuclear magnetic moment  =  I
 = gyromagnetic ratio 9
Magnetic Moment of a Free Electron
the Bohr magneton = the smallest quantity of a magnetic moment
μB = eh/(4πme) = 9.2742  1024 J/T (= A m2)
(μB = eh/(4πmec) = 9.2742  1021 erg/Gauss)
S = ½, the spin quantum number
g = 2.0023192778 the Lande constant of a free electron
for a free electron (S = ½)
μeff = 2 3/4  μB = 1.73 μB
    B
e
eff SSg
m
eh
SSg 

 1
4
1 
10
A Free Electron in a Magnetic Field
11
An electron with spin S = ½ can have two orientations in a
magnetic field
mS = +½ or mS = −½
In a magnetic field
- degeneracy of the two states is removed
HE  0
In SI units
0 = permeability of free space
= 4π 107 [N A2 = H m1]
BE  
Magnetic energy
A Free Electron in a Magnetic Field
12
An electron with spin S = ½
The state of lowest energy = the moment aligned with the magnetic field
mS = −½
The state of highest energy = aligned against the magnetic field
mS = +½
The energy of each orientation E = μ H
For an electron μ = ms g B,
B = the Bohr magneton
g = the spectroscopic g-factor of the free electron 2.0023192778 (≈ 2.00).
E = g B H
E = ½ g B H
E =  ½ g B H
mS = +½
mS = −½
At r.t.
kT = 205 cm-1
H = 1 T
E = 28 GHz
= 1 cm-1
Origin of Magnetism and Interactions
13
S L
I H
Magnetism and Interactions
14
Magnetic field – splitting + mixing of energy levels
Zeeman-Effect: splitting of levels in an applied magnetic field
the simplest case with S = ½ :
splitting of the levels with mS = + ½ and mS =  ½
BE  
Magnetism and Interactions
Zero Field Splitting (ZFS): The interactions of electrons with
each other in a given system (fine interaction), lifting of the
degeneracy of spin states for systems with S > 1/2 in the absence
of an applied magnetic field, a weak interaction of the spins
mediated by the spin–orbit coupling. ZFS appears as a small
energy gap of a few cm−1 between the lowest energy levels.
15
S = 3/2
Zero Field Splitting in dn Ions
16
Magnetism and Interactions
Hyperfine Interactions: The interactions of the nuclear spin I
and the electron spin S (only s-electrons).
17
Spin-Orbit Coupling: The interaction of the orbital L and spin
S part of a given system, more important with increasing
atomic mass.  = L × S
Magnetism and Interactions
Ligand Field: States with different orbital momentum differ in
their spatial orientation, very sensitive to the presence of
charges in the nearby environment.
In coordination chemistry these effects and the resulting
splitting of levels is described by the ligand field.
18
19
Interactions of Spin Centers
Isotropic interaction
The parallel alignment of spins favored = ferromagnetic
The antiparallel alignment = antiferromagnetic
Non-isotropic interactions (like dipole–dipole interactions)
Antisymmetric exchange
Excluded by an inversion center
without orbital contributions (pure spin magnetism) the last two
terms are omitted
20
Lenz's Law
21
(~1834)
When a substance is placed within a magnetic field, H, the
field within the substance, B, differs from H by the induced
field, M, which is proportional to the intensity of
magnetization, M.
B = μ0(H + M)
Magnetization does not exists outside of
the material.
Magnetic Variables SI
Magnetic field strength (intensity) H [A m1]
fields resulting from electric current
Magnetization (polarization) M [A m1]
Vector sum of magnetic moments () per unit volume /V
spin and orbital motion of electrons [A m2/m3 =A m1]
Additional magnetic field induced internally by H, opposing or supporting H
Magnetic induction (flux density) B [T, Tesla = Wb m2 = J A1m2]
a field within a body placed in H resulting from electric current and spin and
orbital motions (Earth’s magnetic field = 50 microtesla)
Field equation
(infinite system)
μ0 = 4π 107 [N A2 = H m1 = kg m A2s2] permeability of free space
)(0 MHB  
)0(0  HB In vacuum:
22
Magnetic Variable Mess (cgs)
Magnetic field strength (intensity) H (Oe, Oersted)
fields resulting from electric current (1 Oe = 79.58 A/m)
Magnetization (polarization) M (emu/cm3)
magnetic moment per unit volume
spin and orbital motion of electrons 1 emu/g = 1 Am2/kg
Magnetic induction B (G, Gauss) 1T = 104 G
a field resulting from electric current and spin and orbital motions
Field equation
μ0 = 1 permeability of free space, dimensionless
)4(0 MHB  
See:
Magnetochemistry in SI Units, Terence I. Quickenden and Robert C. Marshall,
Journal of Chemical Education, 49, 2, 1972, 114-116 23
Important Variables, Units, and
Relations
24
= Wb m2
103/4
4
Magnetic Susceptibility 
(volume) magnetic susceptibility  of a sample [dimensionless]
 = how effectively an applied magnetic field H induces magnetization M in
a sample, how susceptible (receptive) are dipoles to reorientation
measurable, extrinsic property of a material, positive or negative
M = magnetization [A m1]
H = the macroscopic magnetic field strength (intensity) [A m1]
H
M

HM  


H
M



25
M is a vector, H is a vector, therefore χ
is a second rank tensor.
If the sample is magnetically isotropic,
χ is a scalar.
If the magnetic field is weak enough and T not too low,
χ is independent of H and
Mass and Molar Magnetic Susceptibility
molar magnetic susceptibility M of a sample
(intrinsic property)
mass magnetic susceptibility m of a sample







g
cm
m
3










mol
emu
mol
cm
MmM
3

Typical molar susceptibilities
Paramagnetic ~ +0.01 μB
Diamagnetic ~ 1×10-6 μB
Ferromagnetic ~ +0.01 - 10 μB
Superconducting ~ Strongly negative, repels fields completely (Meisner effect)
  )1()1( 00
H
M
H
B
 = density
26
Relative Permeability μr
Magnetic field H generated by a current is enhanced in
materials with permeability μ to create larger fields B
H
B
 HB  
r
H
HM
H
B


 00
0
)1(
)(



μ < 1 diamagnetic μ > 1 paramagnetic
μ0 = 4π 107 [N A2 = H m1 = kg m A2s2] permeability of free space
HHHHMHB   )1()()( 000
)1(0   27
0

 r
  1r
Magnetic Susceptibility
M is the algebraic sum of contributions associated with
different phenomena, measurable:
M =  M
D +  M
P + M
Pauli
M D = diamagnetic susceptibility due to closed-shell (core) electrons.
Always present in materials. Can be calculated from atom/group additive
increments (Pascal’s constants) or the Curie plot. Temperature and field
independent. Small negative values.
M P = paramagnetic susceptibility due to unpaired electrons, increases
upon decreasing temperature. Large positive values.
M Pauli = Pauli, in metals and other conductors - due to mixing excited
states that are not thermally populated into the ground (singlet) state temperature
independent.
28
Dimagnetic Susceptibility
M
D is the sum of contributions from atoms and bonds:
M
D = D atom + bond
D atom = atom diamagnetic susceptibility increments (Pascal’s constants)
bond = bond diamagnetic susceptibility increments (Pascal’s constants)
Diamagnetic Corrections and Pascal’s Constants
Gordon A. Bain and John F. Berry: Journal of Chemical Education Vol. 85,
No. 4, 2008, 532-536
29
For a paramagnetic substance, e.g. Cr(acac)3 it is difficult to measure its
diamagnetism directly.
Synthesize Co(acac)3, Co3+: d6 low spin.
Use the dia value of Co(acac)3 as that of Cr(acac)3.
Pascal’s Constants
30
Magnetic Susceptibility
31
T
N
k
M
BA
B
eff 


2
1
2
0
3







M
P = paramagnetic susceptibility relates to number of
unpaired electrons
  1
3
22
 SS
k
gN
T
B
BAP
M


Caclculation of μeff (microscopic quantity) from χ (macroscopic quantity)
2
0
3
BA
BM
eff
N
Tk


 
Magnetic Properties
32
Magnetic Properties
Magnetic behavior of a substance = magnetic polarization in a mg field H0
33
H0
M M
Magnetic Properties
Magnetic behavior of a substance = magnetic polarization in a mg field H0
34
Magnetism of the Elements
35
Diamagnetism and Paramagnetism
)(0 MHB  
Diamagnetic Ions
a small magnetic moment
associated with electrons
traveling in a closed loop
around the nucleus.
Paramagnetic Ions
The moment of an atom with unpaired
electrons is given by the spin, S, orbital
angular momentum, L and total
momentum, J, quantum numbers.
Inhomogeneous mg field
36
(Langevine) Diamagnetism
Lenz’s Law – when magnetic field acts on a conducting loop, it generates a
current that counteracts the change in the field
Electrons in closed shells (paired) cause a material to be repelled by H
Weakly repulsive interaction with the field H
All the substances are diamagnetic
 < 0 = an applied field induces
 a small moment opposite to the field
 = 105 to 106
Superconductors  = 1 perfect
diamagnets
HM  
37
Diamagnetism
Large and heavy atoms have large diamagnetic susceptibilities
2
2
2
r
mc
NZe

Substance Molar Magnetic Susceptibility M
38
(Curie) Paramagnetism
Paramagnetism arises from the interaction of H with the magnetic field of
the unpaired electron due to the spin and orbital angular momentum.
Randomly oriented, rapidly reorienting magnetic moments
no permanent spontaneous magnetic moment
M = 0 at H = 0
Spins are non-interacting, non-cooperative, independent, dilute system
Weakly attractive interaction with the field
 ˃ 0 = an applied field induces a small moment in the same
direction as the field
 = 103 to 105
39
(Curie) Paramagnetism for S = ½
ms = 1/2
ms = -1/2
ms = 1/2
H = 0 H 0
E=gBH.0
E
S = 1/2 at T
H
Energy diagram of ONE S = 1/2 spin in an external magnetic field
E = g B H about 1 cm-1 at 1 T (10 000 G)
B = Bohr magneton (= 9.27 1024 J/T)
g = the Lande constant (= 2.0023192778) 40

Parallel to H

Opposite to H
Magnetic moment  = g B S
The interaction energy of magnetic moment with the applied magnetic field
E =  H = g B S H = ms g B H
Zeeman effect
(Curie) Paramagnetism for S = ½
41
MANY SPINS
Relative populations P of ½ and –½ states
For H = 25 kG = 2.5 T E ~ 2.3 cm1
At 300 K kT ~ 200 cm-1
Boltzmann distribution
The populations of ms = 1/2 and –1/2 states are almost equal with only a
very slight excess in the ms = –1/2 state.
Even under very large applied field H, the net magnetic moment is very
small.
1
2/1
2/1




Tk
E
B
e
P
P
(Curie) Paramagnetism for S = ½
42
To obtain magnetization M (or M), need to consider all the energy states that
are populated
E =  H = g B S H = ms g B H
The magnetic moment, n (the direction // H) of an electron in a quantum state n
Bs
n
n gm
H
E
 



Consider:
- The magnetic moment of each energy state
- The population of each energy state
M = NA  n Pn
Pn = probability in state n
Nn = population of state n
NT = population of all the states
 = ms g B
E = ms g B H
Tk
E
Tk
E
Tot
n
n
B
n
B
n
e
e
N
N
P




(Curie) Paramagnetism for S = ½
43
=
N[g/2 -g/2
g/2kT -g/2kT
e
+
e ]
[eg/2kT
e-g/2kT
]
M=
N ms
n e
-En/kT
ms
e
-En/kT
= [ 1 + g/2kT 1( g/2kT)
]g/2kT+1 + )g/2kT(1
Ng
2
e±x
~ 1 x±
For x << 1
g B H << kT when H ~ 5 kG
= Ng2
4kT H
Tk
gN
M
B
BA
M
4
22


(Curie) Paramagnetism for S = ½
44
m
1/T
slope = C
Curie Law:
T
C
Tk
gN
H
M
B
BA
M 
4
22


T
C
M 
M
(Curie) Paramagnetism for general S
45
En = ms g B H ms = S, S + 1, …. , S  1, S
M=
N ms=-S
ms
S
(-msg)e-msgH/kT
-msgH/kT
e
= Ng2H
3kT
S(S+1)
)1(
3
22
 SS
Tk
gN
H
M
B
BA
M


m
1/T
slope = Curie const.
S=1/2
S=1
S=3/2
For S = 1
For S = 3/2
For S = 1/2 Tk
gN
B
BA
M
4
22

 
Tk
gN
B
BA
M
3
2 22

 
Tk
gN
B
BA
M
4
5 22

 
non-interacting, non-cooperative,
independent, dilute ions
M
(Curie) Paramagnetism
46
)1(
3
22
 SS
Tk
gN
H
M
B
BA
M


)1()1(  nnSSgeff
2
0
3
BA
BM
eff
N
Tk


 
(in BM, Bohr Magnetons)
n = number of unpaired electrons
g = 2
Curie Law
 vs. T plot
1/ = T/C plot - a straight line of gradient C-1
and intercept zero
T = C - a straight line parallel to the x-axis at
a constant value of T
showing the temperature independence of the
magnetic moment.
47
Plot of eff vs Temperature
48
temperature
eff
(BM)
3.87
S=3/2 eff = 2[S(S+1)]1/2
eff = 2.828(T)1/2
2
0
3
BA
BM
eff
N
Tk


 
Spin Equilibrium and Spin Crossover
49
temperature
eff
(BM)
3.87
1.7
S=3/2
S=1/2
temperature
eff
(BM)
3.87
1.7
S=3/2
S=1/2
Spin Crossover
50
See C4010 Bonding – Bond stretch isomers
Curie Plot
at high temperature if is small
T
T
C
 exp
T
T
C


 

exp
T = dia + Pauli = temperature independent contributions
Plot exp vs 1/T
slope = C; intercept = T
51
1/1.8 K = 0.5561/273 K = 0.00366
M
Curie Plot
at high temperature if is small
T
T
C
 exp
T
T
C


 

exp
T = dia + Pauli = temperature independent contributions
Plot 1/exp vs T
slope = 1/C; intercept = /C
52
1/M
Temperature, K
Curie-Weiss Law
53
m
T1
Deviations from paramagnetic behavior
The system is not magnetically dilute (pure paramagnetic) or at low
temperatures
The neighboring magnetic moments may align parallel or antiparallel
(still considered as paramagnetic, not ferromagnetic or antiferromagnetic)
Θ = the Weiss constant
(the x-intercept)
Θ = 0 paramagnetic
spins independent of each
other
Θ is positive, spins align
parallell
Θ is negative, spins align
antiparallell


T
C
M
1/M
Curie-Weiss Paramagnetism
Plots obeying the Curie-Weiss law with a positive Weiss constant
 = intermolecular interactions among the moments
 > 0 - ferromagnetic interactions = spins align parallell
(NOT ferromagnetism)
54
Curie-Weiss Paramagnetism
Plots obeying the Curie-Weiss law with a negative Weiss constant
 = intermolecular interactions among the moments
 < 0 - antiferromagnetic interactions
= spins align antiparallell
 (NOT antiferromagnetism)
55
Temperature-independent
Paramagnetism
56
Induced in mg. field
Saturation of Magnetization
57
The Curie-Wiess law does not hold where the system is
approaching saturation at high H – M is not proportional to H
)1(
3
22
 SS
Tk
gN
H
M
B
BA
M


Approximation for g B H << kT not valid
e±x
~ 1 x±
M/
mol-1
H/kT
S=1/2
S=1
S=3/2
S=2
1
2
3
4
1 20
follow Curie -Weiss law
SgNM BAsat 
Saturation of Magnetization
58
The Curie-Wiess law does not hold where the system is
approaching saturation at high H – M is not proportional to H
)1(
3
22
 SS
Tk
gN
H
M
B
BA
M


Approximation for g B H << kT not valid
e±x
~ 1 x±
Fe3O4 films
Saturation of Magnetization
59
The Curie-Wiess law does not hold where the system is
approaching saturation at high H – M is not proportional to H
)1(
3
22
 SS
Tk
gN
H
M
B
BA
M


Approximation for g B H << kT not valid
e±x
~ 1 x±
Curves I, II, and III refer to ions
of chromium potassium alum,
iron ammonium alum, and
gadolinium sulfate octahydrate
for which g = 2 and j = 3/2, 5/2,
and 7/2, respectively.
Magnetism in Transition Metal
Complexes
60
Many transition metal salts and complexes are paramagnetic due to partially
filled d-orbitals.
The experimentally measured magnetic moment (μ) can provide important
information about the compounds :
• Number of unpaired electrons present
• Distinction between HS and LS octahedral complexes
• Spectral behavior
• Structure of the complexes (tetrahedral vs octahedral)
Paramagnetism in Metal Complexes
61
Orbital motion of the electron generates
ORBITAL MAGNETIC MOMENT (μl)
Spin motion of the electron generates
SPIN MAGNETIC MOMENT (μs)
l = orbital angular momentum
s = spin angular momentum
For multi-electron systems
L = l1 + l2 + l3 + …………….
S = s1 + s2 + s3 + ……………
Paramagnetism in Transition Metal
Complexes
62
The magnetic properties arise mainly from the d-orbitals.
The energy levels of d-orbitals are perturbed by ligands – ligand field,
spin-orbit coupling is less important, the orbital angular momentum is often
“quenched” by special electronic configuration, especially when the
symmetry is low, the rotation of electrons about the nucleus is restricted
which leads to L = 0
    B
e
s SS
m
eh
SSg 

 14
4
1 
Spin-Only Formula
  Bs nn  2
μs = 1.73, 2.83, 3.88, 4.90, 5.92, 6.93 BM for n = 1 to 6, respectively
Mn2+, Fe3+, Gd3+
Ground states of ions with partially filled
d-shells (l = 2)
63
Orbital Angular Momentum
Contribution
64
There must be an unfilled / half-filled orbital similar in energy to that of the
orbital occupied by the unpaired electrons. If this is so, the electrons can
make use of the available orbitals to circulate or move around the center of
the complexes and hence generate L and μL
Conditions for orbital angular momentum contribution:
1. The orbitals should be degenerate (t2g or eg)
2. The orbitals should be similar in shape and size, so that they are
transferable into one another by rotation about the same axis (e.g. dxy
is related to dx2-y2 by a rotation of 45 about the z-axis.)
3. Orbitals must not contain electrons of identical spin
x
y
dx2-y2
x
y
dxy
Orbital Contribution
in Octahedral Complexes
65
dx2-y2 + dz2
Spin-Orbit Coupling
66
E dxy
dx2-y2
dx2-y2 and dxy orbitals have
different energies in a certain
electron configuration, electrons
cannot go back and forth
between them
E
dxydx2-y2
Electrons have to change
directions of spins to
circulate
E
dxydx2-y2
Little contribution from orbital angular momentum
Orbitals are filled
E
dxydx2-y2
E
dxydx2-y2
Spin-orbit couplings are significant
Magic Pentagon
67
dxydx2-y2
dz2
dxz dyz
66
8
2
22
22
ml
0
1
2
g = 2.0023 +
E1-E2
n
2.0023: g-value for free ion
+ sign for <1/2 filled subshell
 sign for >1/2 filled subshell
n: number of magic pentagon
: free ion spin-orbit
coupling constant
Spin-orbit coupling influences g-value
orbital sets that may give spin-orbit coupling
no spin-orbit coupling contribution for dz2/dx2-y2 and dz2/dxy
Orbital Contribution
in Octahedral Complexes
68
Orbital Contribution
in Octahedral Complexes
69
Orbital Contribution
in Tetrahedral Complexes
70
Orbital Contribution
in Low-symmetry Ligand Field
71
If the symmetry is lowered, degeneracy will be destroyed and the
orbital contribution will be quenched.
Oh D4h
D4h: all are quenched except d1 and d3
eff = g[S(S+1)]1/2 (spin-only) is valid
[Fe6(HAIPA)12(OH)6]
72
300 K, spin-only (g = 2), six independent (N = 6) ions:
HS Fe(III) S = 5/2 T = 26.25 cm3 K mol1
LS Fe(III) S = 1/2 T = 2.25 cm3 K mol1
11.6 cm3 K mol1
NSST )1(
2
1

[Fe6(HAIPA)12(OH)6]
73
Magnetic Properties of Lanthanides
74
4f electrons are too far inside 4fn 5s2 5p6
as compared to the d electrons in transition metals
Thus 4f normally unaffected by surrounding ligands
The magnetic moments of Ln3+ ions are generally well-described
from the coupling of spin and orbital angular momenta to give J
vector
Russell-Saunders Coupling (J = L + S)
• spin-orbit coupling constants are large (ca. 1000 cm-1)
• ligand field effects are very small (ca. 100 cm-1)
– only ground J-state is populated
– spin-orbit coupling >> ligand field splitting
• magnetism is essentially independent of coordination environment
Magnetic Properties of Lanthanides
75
Magnetic Properties of Lanthanides
76
Magnetic moment of a J-state is expressed by the Landé formula:
  BJJ JJg  1 J = L+ S, L+ S  1,……L  S
     
 12
111
1



JJ
LLSSJJ
gJ
For the calculation of g value, use
minimum value of J for the configurations up to half-filled;
i.e. J = L − S for f0 - f7 configurations
maximum value of J for configurations more than half-filled;
i.e. J = L + S for f8 - f14 configurations
For f0, f7, and f14, L = 0, hence μJ becomes μS
g-value for free ions
g = ?
For singlet
For spin-only
Magnetic Properties of Lanthanides Ln3+
77
eff
eff of Nd3+ (4f3)
78
+3 +2 +1 0 -1 -2 -3ml
Lmax = 3 + 2 + 1 = 6
Smax = 3  1/2 = 3/2 M = 2S + 1 = 2  3/2 + 1 = 4
Ground state J = L  S = 6  3/2 = 9/2
Ground state term symbol: 4I9/2
g = 1+
2x(9/2)(9/2+1)
3/2(3/2+1)-6(6+1)+(9/2)(9/2+1)
= 0.727
eff = g[J(J+1)]1/2 = 0.727[(9/2)(9/2 + 1)] = 3.62 BM
MLJ
Term symbol of electronic state
Magnetic Properties of Pr3+(4f2)
79
Pr3+ [Xe]4f2
Find Ground State from Hund's Rules
Maximum Multiplicity S = 1/2 + 1/2 = 1 M = 2S + 1 = 3
Maximum Orbital Angular Momentum L = 3 + 2 = 5
Total Angular Momentum J = (L + S), (L + S) - 1, …L - S = 6 , 5, 4
f2 = less than half-filled sub-shell - choose minimum J  J = 4
g = (3/2) + [1(1+1)-5(5+1)/2(4)(4+1)] = 0.8
μJ = 3.577 B.M. Experiment = 3.4 - 3.6 B.M.
Magnetic Properties of Lanthanides Ln3+
80
Experimental _____Landé Formula -•-•-Spin-Only Formula - - Landé
formula fits well with observed magnetic moments for all but Sm(III) and
Eu(III) ions. Moments of these ions are altered from the Landé expression by
temperature-dependent population of low lying excited J-state(s)
eff at 300 K (B.M.)
Van Vleck Equation
81
To obtain magnetization M (or M), need to consider all the energy states that
are populated
E =  H = g B S H = ms g B H
The magnetic moment, n (the direction // H) of an electron in a quantum state n
Bs
n
n gm
H
E
 



Consider:
- The magnetic moment of each energy state
- The population of each energy state
M = NA  n Pn
Pn = probability in state n
Nn = population of state n
NT = population of all the states
 = ms g B
E = ms g B H
Tk
E
Tk
E
Tot
n
n
B
n
B
n
e
e
N
N
P




Van Vleck Equation
82
Brillouin Function
83
N1 = Population of the lower level
N2 = Population of the upper level
Many-electron systems 2J + 1 states, small energy gaps between states,
populates according to Boltzman distribution
Brillouin Functions for different S
84
Spin Hamiltonian in Cooperative Systems
j
ij
i SSJH

 .2
The coupling between pairs of individual spins, S, on atom i and atom j
J = the magnitude of the coupling
85
J  0 J  0
Magnetism in Solids
Cooperative Magnetism
Diamagnetism and paramagnetism are characteristic of compounds with
individual atoms which do not interact magnetically (e.g. classical complex
compounds)
Ferromagnetism, antiferromagnetism and other types of cooperative
magnetism originate from an intense magnetical interaction between
electron spins of many atoms.
86
Magnetic Ordering
Ferromagnets - all interactions ferromagnetic, a large overall magnetization
Ferrimagnets - the alignment is antiferromagnetic, but due to different
magnitudes of the spins, a net magnetic moment is observed
Antiferromagnets - both spins are of same magnitude and are arranged antiparallel
Weak ferromagnets – spins are not aligned anti/parallel but canted
Spin glasses – spins are correlated but not long-range ordered
Metamagnets
87
0
Magnetic Ordering
88
Critical temperature – under Tcrit the magnetic coupling energy between
spins is bigger than thermal energy resulting in spin ordering
TC = Curie temperature
TN = Neel temperature
Para-, Ferro-, Antiferromagnetic
89
Curie Temperature
90
Curie Temperature of Ni
Tc = 358.28 C
1832, Pouillet
nickel, iron and cobalt
Observed a limit for the
temperature of magnetism
1895 Curie
a transition from
ferromagnetic to
paramagnetic
A second order transition
Lambda shape of the Cp
versus T
a maximum
= the Curie point
not associated with an
enthalpy change
Magnetic Ordering
91
Magnetic Ordering
92
Para-, Ferro-, Antiferromagnetic
Ordering
93
Ferromagnetism
J positive with spins parallel below Tc
a spontaneous permanent M (in absence of H)
Tc = Curie Temperature, above Tc = paramagnet
94
Ferromagnetism
95
Ferromagnetism
96
Antiferromagnetism
J negative with spins antiparallel below TN
no spontaneous M, no permanent M
critical temperature: TN (Neel Temperature), above TN = paramagnet
97
Neutron Diffraction
Single crystal may be anisotropic
Magnetic and structural unit cell may be different
The magnetic structure of a crystalline sample can be determined
with „thermal neutrons“ (neutrons with a wavelength in the order of
magnitude of interatomic distances): de Broglie equation: λ = h/mnvn
(requires neutron radiation of a nuclear reactor)
98
Neutron Diffraction
99
Ferrimagnetism
J negative with spins of unequal magnitude antiparallel below critical T
requires two chemically distinct species with different moments coupled
antiferomagnetically:
no M; critical T = TC (Curie Temperature)
bulk behavior very similar to ferromagnetism, Magnetite is a ferrimagnet
T
FiM
Paramagnetic
behaviour


100
Ferromagnetism
Ferromagnetic elements: Fe, Co, Ni, Gd (below 16 C), Dy
Moments throughout a material tend to align parallel
This can lead to a spontaneous permanent M (in absence of H)
In a macroscopic (bulk) system, it is energetically favorable for spins to
segregate into regions called domains in order to minimize the
magnetostatic energy E = H  M
Domains need not be aligned with each other
may or may not have spontaneous M
Magnetization inside domains is aligned
along the easy axis and is saturated
101
Magnetic Anisotropy
102
Magnetic anisotropy = the dependence of the magnetic properties on the
direction of the applied field with respect to the crystal lattice, result of spinorbit
coupling
Depending on the orientation of the field with respect to the crystal lattice a
lower or higher magnetic field is needed to reach the saturation magnetization
Easy axis = the direction inside a crystal, along which small applied magnetic
field is sufficient to reach the saturation magnetization
Hard axis = the direction inside a crystal, along which large applied magnetic
field is needed to reach the saturation magnetization
Magnetic Anisotropy
103
bcc Fe - the highest density of atoms in the <111> direction = the hard axis, the atom
density is lowest in <100> directions = the easy axis.
Magnetization curves show that the saturation magnetization in <100> direction requires
significantly lower field than in the <111> direction.
fcc Ni - the <111> is lowest packed direction = the easy axis. <100> is the hard axis.
hcp Co the <0001> is the lowest packed direction (perpendicular to the close-packed
plane) = the easy axis. The <1000> is the close-packed direction and it corresponds to
the hard axis. Hcp structure of Co makes it the one of the most anisotropic materials.
Magnetic Anisotropy
104
bcc Fe - the highest density of atoms in the <111> direction = the hard axis, the atom
density is lowest in <100> directions = the easy axis.
Magnetization curves show that the saturation magnetization in <100> direction requires
significantly lower field than in the <111> direction.
fcc Ni - the <111> is lowest packed direction = the easy axis. <100> is the hard axis.
hcp Co - the <0001> is the lowest packed direction (perpendicular to the close-packed
plane) = the easy axis. The <1000> is the close-packed direction and it corresponds to
the hard axis
hcp structure makes Co one of the most anisotropic materials
Magnetostatic Energy
A single domain behaves as a block magnet and a demagnetising field is present
around the domain
Demagnetising field has a magnetostatic energy that depends on the shape
It is the field that allows work to be done by the magnetised sample (e.g.
lifting another ferromagnetic material)
Minimise the total magnetic energy - the magnetostatic energy must be
minimised - decreasing the external demagnetising field by dividing the
material into domains
Adding extra domains increases the exchange energy
The total energy is decreased as the magnetostatic energy is the dominant
effect. The magnetostatic energy can be reduced to zero by a domain structure
that leaves no external demagnetising field
105
Magnetic Domains
The external field (magnetostatic) energy is decreased by dividing into domains
The internal energy is increased because the spins are not parallel
When H external is applied, saturation magnetization can be achieved
through the domain wall motion, which is energetically inexpensive, rather
than through magnetization rotation, which carries large anisotropy energy
penalty
Application of H causes aligned domains to grow at the expense of misaligned
Alignment persists when H is removed
106
Domain Walls
107
A domain wall is a transition region between the different magnetic domains of
uniform magnetization that develops when a magnetic material forms domains to
minimize the magnetostatic energy
Wall energy is the energy required to maintain the wall
When domains form, the magnetostatic energy decreases, and the wall energy and the
magnetocrystalline anisotropy energy increase
Domain Walls
108
the magnetic anisotropy energy is: E = K sin2 , where  is the angle
between the magnetic dipole and the easy axis
Large exchange integral yields wider walls
High anisotropy yields thinner walls
The domain wall width is determined by the balance between the exchange
energy and the magnetic anisotropy:
the total exchange energy E is a sum of the penalties between each pair of
spins
Domain Walls
109
180° walls = adjacent domains have opposite vectors of magnetization
90° walls = adjacent domains have perpendicular vectors of magnetization
Depends on crystallografic structure of ferromagnet (number of easy axes)
One easy axis = 180° DW (hexagonal Co)
Three easy axes = both180° and 90° DW (bcc-Fe, 100)
Four easy axes = 180°, 109°, and 71° DW (fcc-Ni, 111)
Domain Wall Motion
At low Hext = bowing/relaxation of DWs, after removing Hext DWs return back
Volume of domains favorably oriented wrt H increases, M increases
At high Hext = irreversible movements of DW
a) Continues without increasing Hext
b) DW interacts with an obstacle (pinning)
110
Magnetic Hysteresis Loop
Important parameters
Saturation magnetization, Msat
Remanent magnetization, Mr
Remanence: Magnetization of sample
after H is removed
Coercivity, Hc
Coercive field: Field required to flip M
(from +M to M)
111
Magnetic Hysteresis Loop
Saturation magnetization, Msat
Remanent magnetization, Mr
Remanence: Magnetization of sample after H is removed
Coercivity, Hc
Coercive field: Field required to flip M (from +M to M)
112
Magnetic Hysteresis Loop
"Hard" magnetic material = high Coercivity
"Soft" magnetic material = low Coercivity
Electromagnets
• High Mr and Low HC
Electromagnetic Relays
• High Msat, Low MR, and Low HC
Magnetic Recording Materials
• High Mr and High HC
Permanent Magnets
• High Mr and High HC
113
Magnetic Hysteresis Loop
114
Single-Molecule Magnets (SMM)
Macroscopic magnet
= magnetic domains (3D regions with aligned spins) + domain walls
Hysteresis in M vs H plots because altering the magnetisation requires the
breaking of domain walls with an associated energy-cost
Magnetisation can be retained for a long time after removal of the field
because the domains persist
115
Single-Molecule Magnets (SMM)
Single-molecule magnet
= individual molecules, magnetically isolated and non-interacting, no
domain walls
Hysteresis in M vs H plots at very low temperatures
Magnetisation is retained for relatively long periods of time at very low
temperatures after removal of the field because there is an energy barrier U
to spin reversal (1.44 K = 1 cm-1 = 1.986 10-23 J)
The larger the energy barrier to spin reversal (U) the longer magnetisation
can be retained and the higher the temperature this can be observed at
116
U
Single-Molecule Magnets (SMM)
117
The First SMM: Mn12
Orange atoms are Mn(III) with S = 2, green are Mn(IV) with S = 3/2
Some discrete molecules can behave at low temperature as tiny magnets
[Mn12O16(CH3COO)16(H2O)4].4H2O.2CH3COOH
118
S = 8  2  4  3/2 = 10
Antiferromagnetic coupling
Mn12 Spin Ladder
119
U = anisotropy energy barrier
 D   S2 for integer spins
 D  (S2  1/4) for non-integer spins
D = the axial zero-field splitting (ZFS) parameter
S = the spin ground state of the molecule
MS
Single-Molecule Magnets (SMM)
120
The anisotropy of the magnetisation = the result of zero-field splitting (ZFS)
A metal complex with a total spin S, 2S + 1 possible spin states, each sublevel with a
spin quantum number MS (the summation of the individual spin quantum numbers (ms)
of the unpaired electrons;
MS from S to −S
MS = S ‘spin up’
MS = −S ‘spin down’
In the absence of ZFS,
all of the MS sublevels are degenerate
ZFS lifts degeneracy, doublets ±MS
For D negative: MS = ±S are lower in energy than the intermediary sublevels MS
with S > MS > −S
At low temperatures, the magnetisation remains trapped in one of the two MS= ±S
because of the energy required to transition through high-energy intermediary states and
over the barrier U (its size is related to both D and S) to the other well
Anisotropy Barrier in SMMs
121
(a) effect of a negative zero-field splitting parameter D on a S = 10 system
(b) magnetization of the sample by an external magnetic field (Zeeman effect)
(c) frozen magnetized sample showing a slow relaxation of the magnetization over
the anisotropy barrier after turning off the external magnetic field
(d) quantum tunneling of the magnetization through the anisotropy barrier for
magnetic fields leading to interacting MS substates at the same energy
Anisotropy Barrier in SMMs
122
Single-Molecule Magnets (SMM)
123
Magnetic anisotropy = a molecule can be more easily magnetized along one direction
than along another = the different orientations of the magnetic moment have different
energies
Easy axis = an energetically most favorable anisotropic axis in which to orient the
magnetisation
Hard plane = a plane perpendicular to the easy axis, the least favorable orientation for
the magnetisation
The greater the preference for the easy axis over other orientations the longer the
magnetisation retained in that direction
M-H Hysteresis
124
Hysteresis: the change in the magnetisation as the field is cycled from +H to −H
and back to +H, at a range of (very low) temperatures
If the magnetisation is retained despite the field being removed (M  0 at H = 0),
the complex has an energy barrier to magnetisation reversal within the temperature
and scan rate window of the measurement
Hysteresis in Mn12
125
Superparamagnets
126
Superparamagnets
Tunable magnetic properties:
• Saturation magnetization (Ms)
• Coercivity (Hc)
• Blocking temperature (TB)
• Neel and Brownian relaxation
time of nanoparticles (tN & tB)
Parameters and variables:
• Shape
• Size
• Composition
• Architecture
127
Superparamagnets
Small particles of ferromagnetic and ferrimagnetic materials
Nanoparticles below a critical size (depends on material)
a single magnetic domain
• Nanoparticles (NP) with a size distribution
• Molecular particles which also display hysteresis – effectively behaving as
a Single Molecule Magnet (SMM)
Above blocking temperature random spin flipping = no net magnetization
A zero average magnetic moment in the absence of an external field
All the spins simultaneously flip by thermal fluctuation. Each NP behaves as
a paramagnetic spin with a giant magnetic moment
Magnetic moment strongly increases (as compared to paramagnetic materials)
under application of an external field in the direction of the field 128
Superparamagnets
Above blocking temperature random spin flipping = no net magnetization
The thermal fluctuations within the nanoparticles are comparable to or greater
than the energy barrier for moment reversal = random spin flipping
When the number of the constitutional atoms is small enough, all the
constitutional spins simultaneously flip by thermal fluctuation. Each
NP then behaves as a paramagnetic spin with a giant magnetic
moment
 = −g J B
g = the g factor
B = the Bohr magneton
J = the angular momentum quantum number, which is on the order of the
number of the constitutional atoms of the NP
129
Superparamagnets
55 and 12 nm sized iron oxide nanoparticles
130
Blocking Temperature
131
Above the blocking temperature (TB), ferromagnetic and ferrimagnetic
nanoparticles exhibit superparamagnetic behavior = rapid random
magnetization reversals = a zero time-average magnetic moment
The value of TB, associated with the energy barrier, depends on the characteristic
measuring time (100 to 10−8 s)
The magnetic behavior arises from the relative difference between the
measuring time and the relaxation time
The measuring time > the relaxation time = the superparagmagnetic regime
The measuring time < the relaxation time = a “blocked” (ferromagnetic) regime
Blocking Temperature
Blocking Temperature
V = particle volume
Particles with volume smaller than Vc will be at T < TB superparamagnetic
B
B
k
KV
T
25

K
Tk
V B
C
25

132
Blocking Temperature
133
Blocking temperature by Moessbauer spectroscopy
TB corresponds to the “merging point” of the zero-field cooled (ZFC) and field-cooled
(FC) magnetization curves
ZFC and FC Magnetization Curves
134
ZFC - a sample is first cooled to low temperature (2–10 K) in the absence of an
external field (zero-field), at this point, a small external field is applied, and the
temperature is gradually increased while measuring the sample magnetization as a
function of temperature
the magnetic particles are cooled below their blocking temperature in zero applied
filed, the direction of particle's moment will be frozen along the easy direction of
magnetization which will be at random orientation,
FC - the sample is cooled in the presence of an external field (~50 Oe) and the same
external field is applied as the temperature is increased
the point where the two curves merge is the irreversibility temperature, Tirr,
the maximum on the ZFC curve is the blocking temperature, TB
the system of particles is cooled below their blocking temperature in an applied field,
the particles direction of magnetization will be frozen in a direction other than the easy
axis
Blocking Temperature
Zero-field cooling curves and TEM images of Co nanoparticles
135
Size Dependent Mass Magnetization
iron oxide Fe3O4 nanoparticles hysteresis loops
mass magnetization values at 1.5 T
136
Compositional Modification of Magnetism of
Nanoparticles
Fe3O4 (inverse spinel) nanoparticles - ferrimagnetic spin structure
Fe2+ and Fe3+ occupying Oh sites align parallel to the external magnetic field
Fe3+ in the Td sites of fcc-packed oxygen lattices align antiparallel to the field
Fe3+ = d5 high spin state = 5 unpaired electrons (upe)
Fe2+ = d6 high spin state = 4 upe
the total magnetic moment per unit (Fe3+)Td (Fe2+ Fe3+)Oh O4 = 4.9 μB
Incorporation of a magnetic dopant M2+ (Mn 5 upe, Co 3 upe, Ni 2 upe)
replace Oh Fe2+ = change in the net magnetization
137
Compositional Modification of Magnetism of
Nanoparticles
138
Superferromagnetic, SFM
139
Spin Frustration
140
Magnetocaloric Effect
141