Magnetochemistry
1
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]
2
Magnetism
Magnetic field = motion of electric charges
• Macro = electric current
• Micro = spin + orbital momentum
1822 Ampère
Magnetic dipole – magnetic (dipole) moment  [A m2]
Ai 
3
Poisson model
Ampere model
Electromagnetic Fields
4
Magnetism
Microscopic explanation of source of magnetism
= Fundamental quantum magnets
1913 Bohr - Unpaired electrons
1921 Gerlach and Stern experiment (Ag)
1925 Uhlenbeck and Goudsmit - postulated the existence of a new intrinsic
property of particles that behaved like an angular momentum
Pauli later termed it spin
Unlike mass and charge, there is no classical analog to spin!
1928 Dirac - Relativistic quantum theory  spins
Quantum property, two-state quantum mechanical system
(~ rotation of charged particles?)
Spin arises in a correct relativistic formulation of the quantum theory
the relativistic generalization of the Schrödinger equation = the Dirac equation
5
Magnetism
Atomic building blocks (protons, neutrons, and electrons = fermions)
possess an intrinsic magnetic moment
Spin (½ for all fermions) gives rise to a magnetic moment
6
According to the Uhlenbeck and Goudsmit proposal, the spin of a particle
should behave like an angular momentum and, therefore, should have an
associated magnetic moment
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
7
Spin
angular
momentum
Orbital
angular
momentum
Atomic Motions of Electric Charges
The origins for the magnetic moment of unparied electrons
1) The spin of the electron S - Spin motion
2) The orbital angular momentum L - Orbital motion
8
Quantum number S
For more than 1 e
S = n × s , s = 1/2 and n is number of unparied electrons
Number of magnetic levels – spin multiplicity MS = 2S + 1
SPIN MAGNETIC MOMENT (μs)
ORBITAL MAGNETIC MOMENT (μl)
Magnetic Moment of a Free Electron
The Bohr magneton μB = 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 (g  2)
For a free electron (S = ½)
μeff = 2 3/4  μB = 1.73 μB
    B
e
eff SSg
m
eh
SSg 

 1
4
1 
9
A Free Electron in a Magnetic Field
10
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
= Zeeman-Effect
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
11
An electron with spin S = ½
The state of lowest energy = the moment aligned along 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
Magnetism and Interactions
12
S L
I H
Magnetism and Interactions
Zero Field Splitting (ZFS): The interactions of electrons with each other,
lifting of the degeneracy of spin states for systems with S > ½ in the absence
of an applied magnetic field, interaction of the spins mediated by the spinorbit
coupling and dipole-dipole interactions
ZFS = as a small energy gap of a few cm−1 between the lowest energy levels
D = the axial zero-field splitting (ZFS) parameter
13
S = 3/2
D < 0
Zero Field Splitting in dn Ions
14
Magnetism and Interactions
Hyperfine Interactions: The interactions of the nuclear spin I and
the electron spin S (only s-electrons) – NMR spectroscopy
15
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
Ligand Field: States with different orbital momentum L 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
Magnetism and Interactions
16
1 eV = 8065.73 cm1 = 1.60210 1019 J
Magnetization
17
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 exist outside of the
material
μ0 = permeability of vacuum
Hd = demagnetizing or stray field
(ferromagnets)
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:
18
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] (1 T = 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 19
Important Variables, Units, and
Relations
20
= Wb m2
103/4
4
Magnetic Field
21
950 MHz NMR
B = 22.3 T
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



22
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 H is weak enough and T not too
low, χ is independent of H and thus:
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
23
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 < μ0 diamagnetic μ > μ0 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   24
0

 r
  1r
HM  
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
25
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
26
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
27
Magnetic Susceptibility
28
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


 
eff = 0.7977  m T  m in cm3 mol1
eff = 2.828  m T  m in emu Oe1 mol1
Magnetic Susceptibility
29
Magnetic Properties
Magnetic behavior of a substance = magnetic polarization in a mg field H0
30
H0
M M
Diamagnetic materials
- a slight negative
magnetic susceptibility
- repel the magnetic
lines of force
Paramagnetic materials
- a positive magnetic
susceptibility
- attract the magnetic
lines of force
Ferromagnetic materials
a stronger attractive
effect
Magnetic Properties
Magnetic behavior of a substance = orientation of magnetic moments
in/outside a magnetic field H0
31
Atomic/ionic properties Cooperative (bulk) properties
Magnetically dilute, noninteracting Magnetically concentrated, interacting
Magnetism of the Elements
32
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
33
(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  
34
Diamagnetism
Large and heavy atoms have large diamagnetic susceptibilities
2
2
2
r
mc
NZe

35
(Curie) Paramagnetism
Paramagnetism arises from the interaction of H with the magnetic field of
the unpaired electron due to the spin (S) and orbital angular (L)
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
36
(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 = ½ spin in an external magnetic field H
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) 37

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
38
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
Bext
Eupper
Elower
(Curie) Paramagnetism for S = ½
39
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
NTot = 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
40
=
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 Law for S = ½
41
m
1/T
slope = C 1895 Curie Law:
T
C
Tk
gN
H
M
B
BA
M 
4
22


T
C
M 
M
Pierre Curie
(1859 – 1906)
NP in physics 1903
(Curie) Paramagnetism for general S
42
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


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 (spins)
m
1/T
slope = Curie const.
S=1/2
S=1
S=3/2
M
(Curie) Paramagnetism
43
)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
44
T
C
M 
Plot of eff vs Temperature
45
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
The individual spins in the solid
material switch cooperatively, rather
than independently of each other
temperature
eff
(BM)
3.87
S=3/2 eff = 2[S(S+1)]1/2
eff = 2.828(T)1/2
Spin Equilibrium Spin Crossover
)1(  SSgeff
The individual spins in the
solid material switch
independently
Curie-Weiss Law
46
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!)
T  Θ continental convention
T + Θ anglo-american convention
Θ = 0 paramagnetic
spins independent of each other
Θ is positive, spins align parallell
Θ is negative, spins align
antiparallell


T
C
M
m
T1
1/M
C
T
CM


11

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)
47
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)
48
Saturation of Magnetization
49
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
50
Fe3O4 films
Curves I, II, and III refer to chromium
potassium alum, iron ammonium alum,
and gadolinium sulfate octahydrate
g = 2
Gd3+
Cr3+
Fe3+
3/2
5/2
7/2
SgNM BAsat 
Lande g-factor
51
     
 12
111
1



JJ
LLSSJJ
gJ
J = the total electronic angular
momentum
J = L + S
L = the orbital angular momentum
S = the spin angular momentum
a) For a single s electron:
L= 0, S = ½, J = L + S = 0 + ½ = ½
g = 2
b) For a single p electron:
L = 1, S = ½, J = L + S = 1 + ½ = 3/2
g = 4/3
c) For a single d electron:
L = 2, S = ½, J = L + S = 2 + ½ = 5/2
g = 6/5
A dimensionless proportionality constant of the total magnetic moment μ of
a particle and the total angular momentum J
μB = eh/(4πme)
Jg B
JJ


 
Magnetism in Transition Metal
Complexes
52
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
• Oxidation state
• Distinction between HS and LS octahedral complexes
• Spectral behavior
• Structure of the complexes (tetrahedral vs octahedral vs tetragonal)
Magnetism in Transition Metal
Complexes
53
eff in Bohr magnetons, MT in cm3 mol1 K
Paramagnetism in Transition Metal
Complexes
54
μl+s = μl + μs
Orbital motion of the electron generates
ORBITAL MAGNETIC MOMENT (μl)
Spin motion of the electron generates
SPIN MAGNETIC MOMENT (μs)
L-S coupling (Russel- Saunders - assumes strong interaction
between total orbital and total spin angular momenta)
l = orbital angular momentum
s = spin angular momentum
For multi-electron systems
L = l1 + l2 + l3 + … total orbital angular momentum
S = s1 + s2 + s3 + … total spin angular momentum
Paramagnetism in Transition Metal
Complexes
55
The magnetic properties arise mainly from the d-orbitals
For the first Transition Metal series, spin-orbital interaction is
small
Orbital angular momentum and spin angular momentum act
independently – no spin-orbit coupling
Free ions:
Paramagnetism in Transition Metal
Complexes
56
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+
S = ½ n
Ground States of Free Ions with
Partially Filled d-shells (l = 2)
57
J = L+ S, L+ S  1,……L  S
See Hunds Rules
2S+1LJ
Orbital Contribution
in Octahedral Complexes
58
eg
t2g
Orbital Angular Momentum
Contribution
59
There must be an unfilled / half-filled orbital similar in energy to that of the
orbital occupied by the unpaired electrons
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
Spin-Orbit Coupling
60
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
61
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
62
dx2-y2 + dz2
dxz to dyz dxy to dyz dxz to dxy
degenerate
rotation
spin
Orbital Contribution
in Octahedral Complexes
63
Orbital Contribution
in Octahedral Complexes
64
OAM = orbital angular
momentum contribution
SO = Spin-Only Formula
S+L = Spin-Orbit Coupling
 14  SSSO
   114  LLSSLS
Orbital Contribution
in Tetrahedral Complexes
65
If μobs > μso
Contribution from
excited states to the
magnetic moment
Co(II) d7
ground: e4 t2
3 no OAM
excited: e3 t2
4 yes OAM
Orbital Contribution
in Low-symmetry Ligand Field
66
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
Magnetic Properties of Lanthanides
67
4f electrons are too far inside 4fn 5s2 5p6
as compared to the d electrons in transition metals
Thus 4f are 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)
• spin-orbit coupling >> ligand field splitting
• only ground J-state is populated
• magnetism is essentially independent of coordination
environment (ligand field)
Valence Shell of Lanthanides
68
Xe [Kr] 4d10 5s2 5p6 E(4f)  E(6s)
Cs [Xe] 6s1 4f0 5d0
Ba [Xe] 6s2 4f0 5d0
La [Xe] 5d1 6s2 4f0 transition metal
Ce [Xe] 4f1 5d1 6s2 E(4f)  E(6s), E(5d)
Pr [Xe] 4f3 5d0 6s2
Nd [Xe] 4f4 5d0 6s2
Pm [Xe] 4f5 5d0 6s2
Sm [Xe] 4f6 5s2 5p6 5d0 6s2
Eu [Xe] 4f7 5s2 5p6 5d0 6s2
Gd [Xe] 4f7 5s2 5p6 5d1 6s2 4f half-filled
Tb [Xe] 4f9 5s2 5p6 5d0 6s2
Dy [Xe] 4f10 5s2 5p6 5d0 6s2
Ho [Xe] 4f11 5s2 5p6 5d0 6s2
Er [Xe] 4f12 5s2 5p6 5d0 6s2
Tm [Xe] 4f13 5s2 5p6 5d0 6s2
Yb [Xe] 4f14 5s2 5p6 5d0 6s2
Lu [Xe] 4f14 5s2 5p6 5d1 6s2 4f completely filled
Magnetic Properties of Lanthanides
69
Gd [Xe] 4f7 5s2 5p6 5d1 6s2
Magnetic Properties of Lanthanides
70
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+
71
eff
Russell-Saunders Coupling (J = L + S), only ground J-state is populated
MLJ
Magnetic Properties of Nd3+ (4f3)
72
+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)
73
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 = L  S  J = 4
g = (3/2) + [1(1+1)-5(5+1)/2(4)(4+1)] = 0.8
μJ = 3.577 BM Experiment = 3.4 - 3.6 BM
Magnetic Properties of Lanthanides Ln3+
74
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 temperaturedependent
population of low lying excited J-state(s)
effat300K(BM)
Magnetic order is due to the coupling between discrete microscopic
magnetic moments
Two exchange-coupled unpaired electrons
The Heisenberg-Dirac-Van Vleck Hamiltonian
- an empirical operator that models interaction
(coupling) of unpaired electrons
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 exchange coupling constant
75
J  0
parallel
(ferromagnetic)
alignment
J  0
antiparallel
(antiferromagnetic)
alignment
Magnetism in Solids
Cooperative Magnetism
Diamagnetism and paramagnetism are characteristic of compounds with
individual atoms which do not interact magnetically (e.g., classical metal
complexes)
Ferromagnetism, antiferromagnetism and other types of cooperative
magnetism originate from an intense magnetic interaction between electron
spins of many atoms in bulk materials
76
Magnetic Ordering
77
0
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 – random orientation of frozen spin orientations, spins are correlated but
not long-range ordered, spin coupling mediated through the conduction electrons
Metamagnets - a field-induced magnetic transition from a state of low to high
magnetization
Superparamagnets - ferromagnets with particle size too small to sustain the
multidomain structure, the particle behaves as one large paramagnetic ion
Magnetic Ordering
78
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
Curie Temperature
79
Curie Temperature of Ni
Tc = 358.28 C
1832 Pouillet
Ni, Fe, Co
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
80
cooperativeindividual
Magnetic Ordering
81
Exchange Interactions
82
In order for a material to be magnetically ordered, the spins on one atom
must couple with the spins on neighboring atoms
Direct Exchange - a direct overlap between the localized orbitals of
electrons on adjacent magnetic ion sites
Superexchange - an indirect exchange interaction between the localized
electrons on magnetic ions separated by the nonmagnetic ion or ligand
Double Exchange - interactions between localized spin and delocalized
spins
Direct Exchange Interaction
83
Bethe-Slater curve - J as a function of interatomic distance and radius of
partially filled d-shell of an atom
The interatomic distance
- Small - the electrons spent most of the time in between the atoms and
give rise to the antiferromagnetic order (Pauli's exclusion principle)
- Large - the electrons spent most of the time away from each other and
give rise to the ferromagnetic order
j
ij
i SSJH

 .2
J  0
J  0
Superexchange Interaction
84
The most common mechanism for the magnetic coupling (particularly in
insulators) - the spin information is transferred through covalent bonds
with the intervening ligand (oxygen, halogen)
No movement of electrons from one magnetic site to other magnetic site
because the oxidation states of magnetic ions are same or differ by two
antiferromagnetic coupling
ferromagnetic coupling
Double Exchange Interaction
85
Movement (hopping) of electrons from one magnetic ion to 2p orbital of
oxygen from which one electron simultaneously hops the other magnetic
ion because oxidation states of two nearest neighbor ions differs by one
This hopping occurs with preservation of the spin sign - ferromagnetic
The assumption - intraatomic exchange interactions between localized
spin and delocalized (hopping) spins are very strong which aligns the
spins of delocalized electrons always parallel to the localized ion spin
ferromagnetic coupling
Antiferromagnetism
1936
J negative with spins antiparallel below TN
No spontaneous M, no permanent M
Critical temperature: TN (Neel Temperature)
Above TN = paramagnet
86
paramagnetic
Louis Néel
1904 – 2000
1970 NP in Physics
Antiferromagnetism
87
MnO – alternating planes of the O a Mn atoms (111)
Magnetic moments of Mn atoms are in each plane organized in
antiparallel manner
MnO = NaCl structure type
Antiferromagnetism
88
Material TN (K)
NiO 525
Cr 308
Cr2O3 307
CoO 291
MnS 160
MnO 116
FeF2 79
NiCl2 50
CoCl2 25
CoI2 12
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
Neutron radiation of a nuclear reactor
89
Neutron Diffraction
90
Ferrimagnetism
1948 Néel
J negative with spins of unequal magnitude antiparallel below critical T
Requires two chemically distinct species with different moments coupled
antiferromagnetically
Spontaneous M, Critical T = TC (Curie Temperature)
Bulk behavior very similar to ferromagnetism
Magnetite Fe3O4 is a ferrimagnet
T
FiM
Paramagnetic
behaviour


91
Ferrimagnetism
92
Magnetite Fe3O4 = Spinel
Magnetite Fe3O4
93
An inverse spinel (Fe3+ )8[Fe2+ Fe3+]16O32
A = (Fe3+) on the Td sites (5μB)
B = [Fe2+ / Fe3+] a 1:1 mixture on the Oh sites
[Fe2+] 3d6 (4μB)
Fe3+ 3d5 (5μB)
[Fe2+] to [Fe3+] ferrimagnetic double exchange alignment
[Fe3+] to (Fe3+) antiferrimagnetic super-exchange
interactions through the O2− anions
These Fe3+ moments cancel each other, leaving a net
moment of 4μB per formula unit from the Fe2+ ions
Ferrimagnetic
Curie Temperature = 858 K
Ferromagnetism
94
Ferromagnetism is a quantum mechanical effect arising from Coulomb
(electric) repulsion – electrons with parallel spins tend to avoid each other
spatially due to the Pauli's exclusion principle
This gives rise to the exchange interactions – energy is lowered
The exchange interactions split the electronic density of states (DOS) with
spin up and spin down states in a magnetic metal
At Fermi level the DOS are spin polarized which gives rise to the
ferromagnetism in a material
Ferromagnetism
J positive with spins parallel below Tc
A spontaneous permanent M (in absence of H)
Tc = Curie Temperature
Above Tc = paramagnet
95
Material Tc, K
Fe 1063
Co 1404
Ni 631
Gd 293
Dy 88
EuO 77
GdCl3 2.2
SmCo5 1015
Nd2Fe14B 670
Without an external magnetic field
the atomic moments are oriented
parallel in large areas (Weiss
domains)
Ferromagnetism
Ferromagnetic elements: Fe, Co, Ni, Gd, 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
96
Magnetic Anisotropy
97
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
98
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
Magnetostatic Energy
A single domain behaves as a block magnet
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
99
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
100
Domain Walls
101
A domain wall (DW) 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
102
The magnetic anisotropy energy: E = K sin2
 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
103
180° domains walls = adjacent domains have opposite vectors of
magnetization
90° domains 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 = both 180° 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)
104
Magnetic Force Microscopy
105
Magnetized tip scans across a
magnetic sample
Interaction between the tip and
sample are detected and used to
reconstruct the magnetic structure
of the sample surface
Magnetic Hysteresis Loop
Important parameters
Saturation magnetization, Ms
Remanent magnetization, Mr
Remanence: Magnetization of
sample after H is removed
Coercivity, Hc
Coercive field: Field required to flip
M (from +M to M)
106
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
107
Magnetic Hysteresis Loop
108
Magnetic Hysteresis Loop
109
Dictionary of Used Terms
110
Angular momentum – moment hybnosti
Magnetic lines of force – magnetické siločáry
Easy axis – snadná osa
Hard axis - obtížná osa