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General concepts
• There are three principal origins for the magnetic moment of a free atom:
• The spins of the electrons. Unpaired spins give a paramagnetic contribution.
• The orbital angular momentum of the electrons about the nucleus also contributing to paramagnetism.
• The change in the orbital moment induced by an applied magnetic field giving rise to a diamagnetic contribution.
• The molar magnetic susceptibility of a sample can be stated as:
= M/H
M is the molar magnetic moment
H is the macroscopic magnetic field intensity
• In general is the algebraic sum of two contributions associated with different phenomena:
= D + P
D is diamagnetic susceptibility
P is paramagnetic susceptibility
Curie paramagnetism
Energy diagram of an S=1/2 spin in an external magnetic field along the z-axis
E = gBH, which for g = 2 corresponds to about 1 cm-1 at 10000G
Brillouin Function
T)H/kgμ2
1exp(-T)H/kgμ
2
1exp(
T)H/kgμ2
1exp(
BBBB
BB
2
1P
T)H/kgμ21
exp(-T)H/kgμ21
exp(
T)H/kgμ21
exp(-
BBBB
BB
2
1P
=
=
Brillouin Function
• Substituting for P we obtain the Brillouin function
T)H/kgμ21
exp(-T)H/kgμ21
exp(
T)H/kgμ21
exp(- - T)H/kgμ21
exp(μ
BBBB
BBBB
B
NgM
2
1
Tkg BBB 2tanhNg2
1 M μ
Curie Law
where C = Ng2B2/(4kB) is the Curie constant
Since the magnetic susceptibility is defined as = M/H
the Curie Law results:
T
C
T
CHTkNgTkg BBBBB 42Ng
2
1 M 22μμ
vs. T plot 1/ = T/C gives a straight line of gradient C-1 and intercept zero T = C gives a straight line parallel to the X-axis at a constant value of T showing the temperature independence of the magnetic moment.
Ferromagnetism
J positive with spins parallel below Tc
T
F e r r o m a g n e t i c
b e h a v i o u r ( F M )
P a r a m a g n e t i c
b e h a v i o u r ( P M )
χ
C u r i e P o i n t
Antiferromagnetism
• J negative with spins antiparallel below TN
T
A n t i f e r r o m a g n e t i c
b e h a v i o u r A F M
P a r a m a g n e t i c
b e h a v i o u r ( P M )
T N
χ
Ferrimagnetism
• J negative with spins of unequal magnitude antiparallel below critical T
T
FiM
Paramagnetic
behaviour
Spin Hamiltonian in Cooperative Systems
jij
i SSJH
.2
This describes the coupling between pairs of individual spins, S, on atom i and atom j with J being the magnitude of the coupling
Magnetisation
Knowing how M depends on B through the Brillouin
function and assuming that B = 0 we can plot the two
sides of the equation as functions of M/T
SUPERPARAMAGNETS
• These are particles which are so small that they define a single magnetic domain.
• Usually nanoparticles with a size distribution
• It is possible to have molecular particles which also display hysteresis – effectively behaving as a Single Molecule Magnet (SMM)