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 = g  B H, which for g = 2 corresponds to about 1 cm -1 at 10000G

Brillouin Function M = N   n Pn = N (  ½ P ½ +  -½ P -½ )  n = -  s g  B, P n = N n /N with  N n

Brillouin Function = =

Substituting for P we obtain the Brillouin function

Brillouin Functions for different S

Curie Law where C = Ng 2  B 2 /(4kB) is the Curie constant Since the magnetic susceptibility is defined as  = M/H the Curie Law results:

 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  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 T c

Antiferromagnetism J negative with spins antiparallel below T N

Ferrimagnetism J negative with spins of unequal magnitude antiparallel below critical T

Spin Hamiltonian in Cooperative Systems 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

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