Download Band theory

Band theory
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The pauli principle: each energy level in an atom can contain
a maximum of two electrons and they must have opposite
spin.
The 2P subshell is actually composed of 3 sub-subshells of
almost the same energy, each capable of holding two
electrons.
3d and 4s level have nearly the same energy and they shift
their relative positive positions almost from atom to atom.
The transition elements, those in which an incomplete 3d
shell is being filled are the ones of most interest to us
because they include 3 ferromagnetic metals.
When atoms are brought close together to from a solid, the
positions of the energy level are profoundly modified.
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 In the molecule orbital theory of diatomic molecules, an atomic
orbital from atom 1 overlaps with an atomic orbital on atom 2,
resulting in the formation of two molecular orbitals that are
delocalized over both atoms.
 “Bonding” has lower energy than that of the atomic orbitals. The
other is “Antibonding” and is of higher energy.
 For each atomic orbital that is put into the system, one molecule
orbital is created. As the number of molecule orbitals increases, the
average energy gap between adjacent molecular orbitals must
decrease.
 The gap between bonding and antibonding orbitals also may
decrease until the situation is reached in which there is essentially
a continuum of energy levels.
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When two atoms approach so closely that their electron clouds begin
to overlap.
In the transition elements, the outermost electrons are the 3d and 4s;
these electron clouds are the first to overlap as the atoms are brought
together, and the corresponding level are the first to split.
When the interatomic distance d→do, the 3d levels are spread into a
band extending from B → C, and 4s levels are spread into a much
wider band from A → D.
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(because the 4s electrons are farther from the nucleus)
However, the inner core electrons (1s and 2s) are too far apart to
have much effect on one another, and the corresponding energy
levels show a negligible amount of splitting.
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N(E) is not constant but a function of the energy E.
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density of state
The product of the density N(E) and any given energy range gives
the number of levels in that range; thus N(E)dE is the number of
levels between E and E+dE.
Since the 3d and 4s bands overlap in energy.
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the corresponding density curve as shown as following.
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The density of 3d levels far greater than that of 4s levels, because
there are five 3d levels per atom, with a capacity of 10 electrons,
whereas there is only one 4s level with a capacity of 2 electrons.
Filled energy levels can’t contribute a magnetic moment, because the
two electrons in each level have opposite spin and thus cancel each
other out.
Suppose that 10 atoms are brought together to form a “crystal”.
Then the single level in the free atom will split into 10 levels, and
the lower 5 will each contain 2 electrons.
If one electron reverses its spin, as in (b), then a spin imbalance of 2
is created, and the magnetic moment, μH=2/10 μB/atom.
The force creating this spin imbalance in a ferromagnetic is just the
exchange force.
To create a spin imbalance requires that one or more electrons be
raised to higher energy levels; evidently these levels must not be too
widely spaced or the exchange force will not be strong enough to
effect a transfer.
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The ferromagnetism of Fe, Co, and Ni is due to spin imbalance in
the 3d band.
4s electrons are assumed to make no contribution. The density of
levels in the 4s band is low, which means that the levels themselves
are widely spaced
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The maximum imbalance in 3d (the saturation magnetization), is
achieved when one half-band is full of 5 electrons.
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Suppose we let
n = no. of (3d+4s) electron per atom
x = no. of 4s electron per atom
n - x = no. of 3d electron per atom
At saturation, five 3d electrons have spin up and (n - x - 5) have spin
down → μH= [5 - (n - x - 5)]μB
= [10 - (n - x)] μB
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the magnetic moment
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atom
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This eq. shows that the max spin imbalance is equal to the no. of
unfilled electron states in the 3d band.
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For Ni, n = 10 and the experimental value of μH= 0.6μB
insert
μH= [10 - (n - x)]μB, we found x = 0.6
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To assume that the no. of 4s electrons is constant at 0.6 for elements
near Ni, we have μH= (10.6 - n)μB.
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The magnetic moments per atom predicted by this eq. agree well
with Fe, Co, Ni, and that the predicted negative moment for Cu has
no physical meaning, since 3d band of Cu is full.
In Fe, we have assumed 5.00 electrons 
 net 2.60 
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2.40 electrons 
Since the observed spin imbalance in Fe is about 20% less than this
predicted value, and in Mn actually zero, it appears that the
exchange force can’t keep one half-band full of electrons if the other
half-band is less than about half full.
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◎ Magnetic ceramics
Each electron in an atom contributes a quantized amount of
(1) orbital angular momentum
magnetic moment form its 

(2)
spin
angular
momentum
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For the transition metal ions used in most ferrites, the
contribution from the orbital angular momentum is negligible,
and the magnetic moment of an ion is determined by the no. of
unpaired electron spins.
Each unpaired electron contributes a moment of one Bohr
magneton
B  9.27 1021erg / gauss
(μ0)
In the transition metal series where 3d-shells are partially filled,
the moment is determined by the net no. of unpaired spins.
◎ The resulting magnetic moments for the transition metal series.
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P orbitals
d orbitals
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σ bonding
• Head-to-head overlap
Bonding
Antibonding
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π bonding
• Sideway overlap
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Energies of bonding and antibonding MOs
Energy as a function of distance for the bonding and
antibonding orbitals of the H2 molecule
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• Molecular Orbital Theory predicts that O2 has unpaired electrons,
so it will be paramagnetic.
Bond order = ½ (no. of bonding e- −no. of anti-bonding e-)
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• Molecular Orbital Theory predicts that O2 has paired electrons, so
it will be diamagnetic.
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Ferrimagnetism refers to the condition where the moments of
ions on type of site are partially offset by antiparallel interaction
with ions of another site, but there remains a net magnetization.
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Where a metal oxide containing magnetic ions is ferromagnetic,
antiferromagnetic, or ferrimagnetic depends on (1) the
magnitude of individual moments, (2) the type and no. of sites
that are occupied, (3) the nature of the interaction between sites.
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The exchange interaction between any two cations is mediated
by the intervening oxygen ions, and is known as a superexchange
interaction.
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The superexchange interaction involves the temporary transfer of
an electron from one of the oxygen ions dumbbell-shaped 2P
orbitals to one of the adjacent cations, leaving behind an
unpaired 2P electron interacting with the opposing cation.
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For cations with more than half-filled d levels, this interaction
generally results in antiparallel spin between the cations.
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⊙Molecular field theory
We expect that exchange forces between the metal ions in a
ferrimagnetic will act through the O ions by means of the indirect
exchange (super exchange) mechanism, just as in antiferromagnetic.
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The exchange interaction in antiferromagnetic ionic solid takes place
by the mechanism of indirect exchange (super exchange).
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The positive metal ions, which carry the magnetic moment are too
far apart for direct exchange forces. Instead, they act indirectly
through the neighboring anions (negative ions).
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For example, two Mn2+ ions being brought up an O2- ion from a
large distance as in (a). The moment on these two ions are at first
only randomly related and O2- has no net moment.
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However, the outer electrons of the O2- ion constitute two
superimposed orbits, one with net spin up, the other with net spin
down.
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When a Mn2+ ion with an up spin is brought close to O2- ion the upspin part will be displaced as in (b), because parallel spins repel one
another.
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If another Mn2+ ion is brought up from the right, it is forced to have
a down-spin when it comes close to the up-spin side of the
“unbalanced” O2- ion.
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The strength of the antiparallel coupling between metal ions M
depends on the band angle AOB and is generally greatest when this
angle is 180O (M-O-M colinear).
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However, molecular field theory for a ferrimagnetic is inherently
more complicated than for an antiferromagnetic.
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The a-a interaction in a ferrimagnetic will differ from the b-b
interaction, even though the ions involved are identical.
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The basic reason is that an ion on an A site has a different no. and
arrangement of neighbors than the same ion on a B site.
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The interaction is stronger for more closely separated cations and
for metal-oxygen-metal angles closer to 180 o , in spinel structure
the a-b interaction > b-b interaction > a-a interaction.
Ex. Transition metal monoxides, MnO, FeO, CoO, NiO, in each
the cation has at least a half-filled d-level, there are no a-site
cations, we expect an antiparallel b-b interaction to domain.
These oxides are anti ferromagnetic, ordered cations within a
(111) plane have parallel spins adjacent (111) planes have
antiparallel spins.
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