Magnetism in Chemistry

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

M = N nPn = N (½P½ + -½P-½)

n= -sgB, Pn= Nn/N with Nn

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

Brillouin Functions for different S

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.

Curie-Weiss paramagnetism

jij

iB S.SJ2S.HgH

BB

B

k

zJ

k

NgCwith

T

HCM

24

22

is the Weiss constant

Curie-Weiss paramagnetism

Plots obeying the Curie-Weiss law with a negative Weiss constant

Curie-Weiss paramagnetism

Plots obeying the Curie-Weiss law with a positive Weiss constant

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

Temperature dependence of M

Ferromagnets

Ferromagnets

Ferromagnets

Ferromagnets

Domains

Domains

Hysteresis

Spin Frustration

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)

Mn12

Orange atoms are Mn(III) with S = 2, green are Mn(IV) with S = 3/2

Mn12

Mn12 Spin Ladder

Hysteresis in Mn12