Magnetics- Ferromagnetism Flashcards
How are magnetic moments aligned in ferromagnetic materials?
Their neighbouring atoms have aligned magnetic moments even under zero applied magnetic field
What gives rise to atomic magnetic moments?
Unpaired electrons in the electronic configuration
What is Pauli exclusion principle?
Interacting electrons must have a different set of quantum numbers.
Alternatively, the overall wave function of a system must be anti-symmetric
What are the 4 quantum numbers, n, l, ml, s for electrons?
n is principle
l is angular momentum
ml is magnetic
s is spin
What is true about spatial symmetry for 2 electrons?
If they are spatially symmetric, there is region of constructive interference so spins must be anti-symmetric (opposite).
If they are spatially anti-symmetric, there is region of destructive interference so spins must be symmetric (aligned).
Spatially symmetric total spin and spin angular momentum
Total spin is S=0
Spin angular momentum ls=2S+1=1
This is singlet state
Spatially anti-symmetric total spin and spin angular momentum
S=1/2+1/2=1
ls=2S+1=3
This is triplet state
Why do symmetric and anti-symmetric configurations have different overall energies?
Electrons are electrically charged so have an electrostatic interaction. Different spatial arrangement of electrons mean a difference in electrostatic energy. Therefore the different configurations have different overall energies. This is called the exchange energy.
What does sign of exchange energy depend on?
Meaning whether S=0 or S=1 has a lower energy.
Depends on separation of atoms
Closely spaced atoms have electrons concentrated between them requiring symmetric wave functions and opposite spins so S=0.
Widely spaced atoms have electrons separated from region between them requiring anti-symmetric wavefunctions and aligned spins S=1
Bethe-Slater curve
Shows dependence of exchange energy with interatomic spacing.
y axis Jex (exchange constant) and x axis interatomic separation.
Line curves up from below x axis to peak above it then curves back down exponentially. Above is ferromagnetic alignment (aligned atomic moments). Below is anti-ferromagnetic alignment (oppositely aligned atomic moments).
Formula for exchange energy
Eex =-2JexS1•S2
S1 and S2 are spin vectors dot producted together
Jex is positive for ferromagnetic materials
What happens if exchange constant Jex is negative?
Anti-ferromagnetic: equal opposite adjacent moments mean no net magnetisation at zero field
Ferrimagnetic: unequal opposite adjacent moments mean overall net magnetisation but weaker than ferromagnets.
What does exchange energy do?
It acts to stabilise a particular atomic arrangement of atomic moments
Comparing susceptibilities of ferromagnets, ferrimagnets, antiferromagnets, paramagnets and diamagnets on M vs H graph
All through origin. Ferromagnet highest gradient as χ»0, then ferrimagnet with lower χ. Then antiferromagnet with χ of 10 to 100. The. Paramagnet with χ>0 still. Then diamagnet with χ<0 and shallowest gradient magnitude.
Common crystal structures of iron, nickel and cobalt
Iron BCC
Nickel FCC
Cobalt HCP
What is spin-orbit coupling?
Atomic magnetic moments interact with charge of atomic nuclei. Means there are preferred crystallographic orientations for magnetisation. Principle known as magnetocrystalline anisotropy.
Easy and hard axes for BCC, FCC, HCP
BCC: easy is 100 (along an edge), hard is 111 (through diagonal)
FCC: easy is 111, hard is 100
HCP: easy is 1000 (up through centre), hard is 1010 (from centre of bottom face out equally between 2 atoms on same face)
What does easy and hard axis mean?
Easy is direction magnetisation prefers. Hard is least preferred as is energetically most costly
M vs H graph for easy axis and hard axis
Easy rapidly gets to saturation at low applied fields and stays there. Takes hard longer and stronger applied field to get to saturation
Formulae for directional cosines of vector M in cubic anisotropy
See page 23 lecture 3
Formula for magnetocrystalline anisotropy energy density, Ea/V for cubic anisotropy
See page 24 lecture 3
What magnetocrystalline anisotropy energies are in preferred direction?
Low values of energy
Formula for magnetocrystalline anisotropy energy density, Ea/V for uniaxial anisotropy
See page 26 lecture 3
How to find energy to rotate from different crystallographic directions
Difference in energy density of one - that of other gives energy density change
