We have seen so far how the exchange interaction
between localized electrons spins leads to
ferromagnetism and other form of magnetic
ordering as a result of the Dirac Isenberg
exchange interaction.
But of course, this is somewhat not our going
to limited interest, because we know that
the well-known ferromagnets do not have localized
spins.
Namely the ferromagnetic metals, such as iron,
cobalt and nickel are metals with delocalized,
not localized electrons. So in order to account
for the ferromagnetic ordering in these metals,
it is necessary to consider exchange interaction
between itinerant or conduction electrons
in a d band metals. We can use the same concept
that we consider for the exchange interaction
between a pair of electrons and build in the
idea of the electron wave function for itinerant
electron.
So we can write the effective wave function
for such a pair of electrons i and j as one
by root two V e to the power i k i r i e to
the power i k j dot r j minus e to the power
i k i dot r j e to the power i k j dot r i.
This is the positions at r i and r j, the
electrons at r i and r j and wave vector k
and k i and k j. Now this is the exchange
because of the in disguise visibility of the
electrons this is the exchanged term and the
two electron wave function should be anti-symmetry
with respect to the spatial part in order
to give raise to a ferromagnetic or spin parallel
wave function.
So rewriting this, we can write this as into
one minus… Therefore, psi i j square, mod
psi i j square dr i dr j will be from this,
we can easily see that this will have the
form 
taking r i minus r i j as the separation r
between the two electrons. We can now write
the probability for two spin being parallel,
two electrons spin being parallel, and being
separated as just this. Now this should be
multiplied by d r here, and this will giving
you n up.
Where n up is the number of electrons spins.
In this spin up sign band which is equal to
n by 2, so that using this we can write this
exchange charge density 
as by multiplying the electronic charge and…
We take this average, this dash, this dash
over this form – bracketed term is just
the average, now we average over the Fermi
sphere, so that we have rho exchange over
r as.
So that would be the form of this exchange
charge density. Now to this of course, we
must add the charge density e n by 2 due to
anti parallel plus the charge density, due
to anti parallel spin.
Doing this, we finally arrive at the effective
charge density as…
So that would be the form of the effective
charge.
And this is shown in the form of a plot of
this effective charge normalized by e n versus
as a function of k F r and that shows so called
the exchange hole. This means that the presence
of the exchange interaction leads to a situation
where the effective charge density is reduced
because of this exchange correlation.
So this leads to a renormalization of the
electron energies, which is the stating points
of the Hahree -Fock approximation.
We will not go into details of this, but use
this idea to discuss the so called band model
of ferromagnetism. This was first proposed
by Stoner and Wohlfarth, effectively that
means that the energy of the electrons in
the spin up band and in the spin down band
can be written as basic original energy minus
I into n up divided by N, and this is minus
I down by N. Where N is of course, n up plus
n down that is the total number of electrons,
and I is the Stoner parameter, which describes
the energy reduction due to electron correlation.
We define a parameter R which is n up minus
n down by N. So this is the difference between
the number of electrons with up and down spins
and therefore, this should be proportional
to the magnetization. In order to put this
electron energy, which we have written in
a slightly more transparent form, we redefine
the zero of energy with respect to by subtracting.
I times n up plus n down by 2 N, so subtract,
this from the energy. And redefine the energy,
so we denote this by e tilt 
minus I R by 2. This can be easily verified
and similarly e down turns out to be given
by…
Where e tilt of k equals e of k minus this
quantity. So starting with these, these are
the renormalized electrons sub band energies
and our aim is to calculate the magnetization,
which is proportional to parameter R. So we
can write R as 1 by N sigma f up k minus f
down k the summation over, all k values, where
f k is the Fermi Dirac distribution function
which we have discussed already. So substituting
for this, R turns out to be one by N sigma
over K writing the actual form of the Fermi
Dirac distribution function and substituting
for e up and e down values the energies. So
this will be e minus e tilt minus e F minus
I R by 2 by k B T, because there is a negative
sign there plus 1 minus 1 by…
So we simplify this by noting that we have
a function f of X minus delta x here, and
the function with f of X plus delta x. So
this is given because of the exponential,
we can write this as 
plus higher order term involving delta x cube
into f x by 3 factorial etcetera. So using
this, and applying it to this, we get the
parameter R as 1 by N sigma K d f k by d e
k times I R, neglecting the other terms, which
is necessarily positive, the times f dash.
So this is we know the Fermi Dirac distribution
function has a negative sign here, whereas
the next term involving the third order derivative
is positive.
So if we want a positive magnetization, a
non-zero magnetization, which means that R
should be positive. We arrive at the Stoner
criterion for ferromagnetism, we can see readily
from this, this criterion to be…
This will have a maximum value at t equal
to 0, and it will have a particularly simple
form at absolute 0. So you will evaluate this
at absolute 0 over the summation can be written
as… And we know this is going to give you
a delta function, therefore, we can simplify
this as… So that would become, where D of
e f is the electron density of states of the
Fermi energy.
So from this, we get Stoner criterion as…
And we can redefine V by 2 N D of E F as some
D tilt of E F, in which case we get particularly
compact form for this Stoner criterion for
ferromagnet. Now this has been calculated,
the electronic density of states at the Fermi
level for the various metal have been calculated
and using these values, the product I times
D tilt of E F can be calculated and that is
shown in the picture.
And it can be seen that the Stoner criterion
is fulfilled only for iron, cobalt and nickel,
so that is a very remarkable result predicting
ferromagnetism according to the simple Stoner
criterion in D band metals namely iron, cobalt
and nickel, which are well-known to be metallic
ferromagnets.
So that is how this simple Stoner model accounts
for ferromagnetism in these metals.
Now the next question is what happens in an
external magnetic field. It is quite simple
and straight forward, so this Stoner parameter
the R becomes… Where this is the two mu
B B is the Zeeman splitting in the presence
of the applied magnetic field. So instead
of I R, it becomes I R plus two mu B B and
the magnetization is nothing but N by V times
R. So that can be written straight a way in
the form, so the magnetization therefore we
get the magnetization is given in this form
in the presence of an applied field. Therefore,
we can define the susceptibility as the ratio
between M and B which is given by, so it has
the form chi 0 by 1 minus I times D tilt e
F, so this is known as an enhancement factor,
this is called a Stoner enhancement of the
magnetic susceptibility. So this is the Stoner
enhancement factor, which increases the magnetic
susceptibility.
Next, we would like to calculate the spontaneous
magnetization and its temperature dependence.
You know to calculate this, we assume a delta
function behavior for the electron density
of states, D electron density of states at
the Fermi energy. In order to keep the calculation
simple and with that assumption, we get the
parameter R as 
we have same as before exponential mu B B
naught minus I R by 2 plus 1 minus 1 by exponential
mu B B naught plus I R by 2 plus 1. We set
to bring it to a simpler form, we make the
following substitution, we set T c, a parameter
T c as I and mu B effective by mu B into 4
k B.
We also take R tilt as mu B effective by mu
B times R, so that in terms of this, the R
tilt becomes simply 1 by exponential two R
T c by T plus 1 minus… This shows the correct
behavior, this tends to equals to 1 for T
equal to 0 and equal to 0 for T equal to T
c. So T c defined in this way is the Curie
temperature of this ferromagnet. In addition
for T very small compared to T c, it is well
below the Curie temperature, this R tilt is
given by 1 minus 2 e to the power minus 2
T c by T. And the neighborhood of T c, this
is given by root three times 1 minus T by
T c.
So this gives a number of things to compare
with experiment, the figure shows the factor
parameter R tilt as the function of T by T
c.
So this is the near the critical temperature,
the expected behavior according to this model
is root 3 times 1 minus T by T c to the power
half, giving rise to a so called a critical
exponent of half. But what is experimentally
observed is one-thirds has can be seen from
the next figure.
So the critical exponent magnetization goes
as one minus T by T c to the power one-thirds
in the neighborhood of the critical temperature,
so there is a strong deviation in the critical
behavior at the ferromagnetic Curie temperature.
Also the low temperature behavior, there is
considerable deviation of the experimental
results data points from the expected theoretical
curve.
So these are due to the shortcomings of the
Stoner model, especially that gives Stoner
model does not take proper account of the
excited states. Because in addition to spin
flips, a cumbering the excitation from band
to another, other element excitation with
a smaller quantum of energy or possible and
they can also cause spin flip.
This is not taken into account in the Stoner
model.
Now for T above the Curie temperature, T greater
than T c, we can expand the exponentials and
write R as mu B by 2 K B T into B naught plus
T c by T into R, so that is leads to a susceptibility
which goes as C by T minus T c. And this we
can readily recognizes as the Curie Weiss
behavior. So in short, we have described in
terms of the simple Stoner model, how one
can account for ferromagnetism in a d-band
metal, such as iron, cobalt and nickel and
how this leads to features, which predict
the correct Curie Weiss behavior and also
leads to a Stoner enhancement of the susceptibility.
The temperature dependence of the spontaneous
magnetization of course you have the correct
overall behavior for the order parameter namely
the magnetization, but the critical behavior
as well as the low temperature behavior are
not very well described by Stoner model, because
of the improper treatment of the excited states.
So with we have some idea of how band of model
of ferromagnetism can be used to describe
magnetic ordering in metals. With this we
conclude our discussion of magnetism.
