### Ferromagnetism (F.Lüttner)

```Ferromagnetism
Characteristics experiments and its
„origin“
(by Florian Lüttner)
Contents
• General expressions for describing magnetism and ferromagnetism
• Different kinds of magnetism
- short overview
• What is ferromagnetism?
- the phenomenon and ist characteristics
• Ferromagnetism ferrimagnetism and antiferromagnetism
• Observation of magnetic structures
• „Origin of ferromagnetism“
- Singlet and tripplet states for a two electron system
- Spin Hamiltonian
- Heisenbergmodel
- Meanfield approximation
- curie-temperature
General expressions
Magnetization:

=

• Vectorfield that expresses the density of permanent or induced magnetic
moments

=
=
=
=   ℎ
=
=
Magnetic susceptibility:
Dimensionless proportionality constant between the degree of
magnetization and the applied magnetic field

χ =

0
χ =

=
=
0 =
Magnetic moment:
Determines the angular momentum (or torque) a magnet will experience
in an external applied magnetic field
=×
=
=
=
is the vector that relates the aligning torque on an object from an
externally applied magnetic field to the field vector itself.
With an unknown sample or object you can measure the torque by an
applied known external magnetic field and get the magnetic moment.
Gyromagnetic ratio:
Ratio of its magnetic moment to its angular momentum (classical body)
The gyromagnetic ratio can be written as

=
2
Therefore the magnetic moment is
=
=
=   ℎ
= ℎ  ℎ
For an isolated electron (quantummechanical)
An electron has a spin  no classical rotation (quantummechanical phenomenon)
so

≠
2
But with a correction factor or electron g-factor
−
=

2
=
= Bohr magneton

ℏ
= 2 1 +
=
1
137

+⋯
2
−
For this  is
= 2.0023193043617
And therefore  is calculated as
= 1.760859708 ∗ 1011

∗
Electron spin:
Follows the same mathematical laws as quantized angular momenta, but
• No classical rotation  quantummechanical phenomenon
• Spin as an intrinsic form of angular momentum for elementary particles,
hadrons and nuclei with a definite magnitude and a direction (up and down)
• Spin quantum number only takes half-integer values (0, ½, 1, 3/2, 2, …)
• Only spindirection can be changed in spin up or spin down but not ist value
• The calculation of the magnetic moment of an electron needs an electron g
factor different from 1 (like in classical cases)

=
2
(n is any non − negative integer)
Spin angular momentum
=
ℎ
( + 1)
2
Spin angular momentum
=
So S also can be calculated with  =
ℎ
( + 1)
2

2
ℎ
=
( + 2)
4
For this the electron (fermion) spin quantum number and the angular momentum can be
calculated with
=1
for this
and in this case
1
=
2
ℎ
=
3
4
Short overview
• Paramagnetism
-χ<0
- magnetization is different from 0 as long as an external applied magnetic field
exists
- atoms, molecules and lattice defects with an odd number of electrons
(the total spin of a system must not be zero)
- free atoms and ions with a partly filled inner shell
- metals
- few compounds with an even number of electrons (molecular oxygen)
• Diamagnetism
-χ>0
- magnetization is different from 0 as long as an external applied magnetic field
exists
- the inner magnetization is in the opposite direction to the applied external field
(want to vanish the field)
- Bismut
- Carbon
- superconducter
(crowds the magnetic field lines of the applied field out of the superconducter)
• Ferromagnetism
- spontanious magnetic moment (saturation moment)
- electron spins and magnetic moment have to be arranged in a regular
manner
What is ferromagnetism?
Ferromagnetic orders
• Spontanious magnetization  saturation magnetization
• Vanishes not  temperature under the Curie-temperature ( <  )
•  >  the order of electronspins vanishes after the external magnetic field
is turned off (Paramagnetism)
• From internal interaction of the magnetic moments of the spins (exchange
field)
Characteristics
(hysteresis)
Materials with different forms of
magnetism
Ferromagnetism, ferrimagnetism and
antiferromagnetism
• Use the language appropriate of solids (magnetic ions localized at lattice points)
ferromagnetism
• Here we have in general non vanishing vector moments of the magnetic ions below
a temperature  (ordered spin orientation)
• Same direction of magnetic moments (spins)  add up to a net magnetization
density
 ferromagnetism
• Microscopic magnetic ordering is revealed by the existance of a macroscopic bulk
magnetization density
• Not all magnetic moments have to be the same
Antiferromagnetism
• Magnetic ordering of the individual local moments vavanishes to zero
• No spontanious magnetization
• Microscopic magnetic ordering can not be revealed by a macroscopic bulk
magnetization density
• Local moments orientated like two interpenetrating sublattices with the same
structure
ferrimagnetism
• Here we have even a non vanishing vector moments of the magnetic ions
• Not the same direction of magnetic moments (spins) at all
• Exchange coupling between nearest neighbours may favor antiparallel orientation
• Neighbour has not the same value of magnetic moment
• Leaving a net magnetic moment for the solid as a whole
• Some more complex arrangements are possible
• Describtion not with the above three types
• Specifiing the ordering in terms of spin density
At any point  along any direction  the density  () is described by
1
= ↑  − ↓ ()
2
↑   ↓   ℎ   ℎ    ℎ   (  )
therefore
() ≠ 0 for ferrimagnets and ferromagnets and
= 0 for antiferromagnets
Observation of magnetic structure
• Can be observed by the scattering pattern of neutron scattering
• Neutrons have a magnetic moment  couples to the elctronic spin
• also nonmagnetic Bragg reflection of neutrons by ionic nuclei
 additional peaks in the elastic neutron scattering cross section
• Over  the magnetic scattering peaks are vanished
• Crosssection for neutron electron scattering of the same order of magnitude as
for neutron nuclei interaction
• Determination of the distribution direction and the order of the magnetic
moments
• Detemine the magnetic structure of antiferromagnets
Origin of ferromagnetism
• Magnetic effects in a material by the pauli priciple
(even with no spin dependent terms in the Hamiltonian)
• consider Two-electron system (with spin independet Hamiltonian)
• Ψ stationary state is the product of a purely orbital stationary state
• Ψ(1 , 2 ) satisfies orbital Schrödinger equation
Ψ = −
ℏ
1 2 + 2 2 Ψ +  1 , 2 Ψ
2
Ψ = Ψ
• Four spin states with both electrons in levels of definite
|↑↑ , |↑↓ , |↓↑ , |↓↓
• Choose linearcombinations of these states to get definite values of the total spin
(and its component  )
Singlet state
Triplet states
1
Combination of two spin − 2 particles can only carry a total spin of 1 or 0
(occupy triplet or singlet state)
Any combination (like |↑↑ ) can be calculated by
1 , 1 2 , 2 = |2 , 2 ⊗ |2 , 2
1
1
Span a 4-dimensional room (1/2 = 2 for spin − 2 particles)
Calculated with Clebsch-Gordon coefficients

|,  =
2
11
1 , 1 2 , 2
2
1 +2 =
Substituting in the four basis states
1
1
,
+
2
2
1
1
,
+
2
2
1
1
,
−
2
2
1
1
,
−
2
2
1
1
,
+
2
2
1
1
,
−
2
2
1
1
,
+
2
2
1
1
,
−
2
2
for ↑↑
for ↑↓
for ↓↑
for ↓↓
Therefore you get three states with total spin angular momentum 1 and one
state with total spin angular momentum 0
• Total wave function change sign under simultanious ninterchange of the spin
(1 , 2 ) and orbital parts (1 , 2 )
• Ψ is the product of its spin and orbital parts
• Therefore if Ψ do not change sign under interchange of 1  2
 solution is symmetric
 must describe states with  = 0
• Ψ change sign under interchange of 1  2
 antisymmetric solution
 must describe states with  = 1
• If  and  are the lowest eigenvalues of the spin-independent orbital
schrödinger equation associated to the symmetric and antisymmetric solution of
it then the groundstate will only have spin 0 or spin 1
<  →  = 0
>  →  = 1
• This one can require only by an examination of the spin-independent Schrödinger
equation
• For two electron problem the ground-state wave function have to be symmetric
• Only for two electron system
• Have to estimate  −  that can be generalized to the anologous problem of a
N-atom solid
Spin Hamiltonian
• A way to express dependence of a two-electron configuration spin on the singlettriplet energy splitting
• Important for the analyzationof energies of the spin configuration of real insulating
solids
• protons with great distances  two independent hydrogen atoms described by the
ground state
(fourfold degenerated with two possible orientations of each electron spin)
• Bring the protons closer to each other (molecule)  splitting of the fourfold
degeneracy (due to atominteractions)
• This interaction is small compared with other excitation energies of a two-electron
system
• Simplifying by ignoring higher states over fourfold degenerated
• Molecule as a simple four-state system
• Represent general state of the molecule by linear combination of the forur lowest
states
• Convenient to have an operator
Spin Hamiltonian
• Same eigenvalues as in the original Hamiltonian within the four state manifold
• Eigenfunctions give the spin of each state
• Each individual spin operator satisfies

Therefore the total spin satisfies
2
1 1
=
+1
2 2
3
2 =
4
2 = 1 + 2 2
3
2
= + 21 ∙ 2
2
2 has the eigenvalues ( + 1) and with the equation above for 2 we get the eigenvalues
3
1
of the operator 1 ∙ 2 as − 4 for the singlet state and 4 for each triplet state
Therefore   is
1

ℋ
= ( + 3 ) − ( −  )1 ∙ 2
4
1
Redefining the zero of energy and omit the constant 4 ( + 3 ) which is common to all
four states we get
ℋ  = −1 ∙ 2
Heisenberg model
• ℋ  is the scalar product of the spin operators 1  2  parallel spins
for  > 0 (leads to ferromagnets) and antiparallel spins for  < 0 (leads to
ferrimagnets and antiferromagnets)
• Coupling in the spin Hamiltonian do not depend on the spatial direction with
respect to 1 − 2 but on the ralative orientation of the two spins
• ℋ  is Isotropic but we need unisotropic terms to describe ferromagnetism
• Include terms that braeak rotational symmetrie in spin space
• Spin Hamiltonian only for insulating materials (Hubbard model for metals)
• Therefore N widely seperated ions with a small overlap of the electron
wavefunction
• Only interaction between nearest neighbours
• Ground state will be 2 + 1  −  gegenerated
• Spin Hamiltonian describes the splitting of this ground state when ions are
some closer together that the splittings are small compared with any other
excitation energies
• The eigenvalues of the    gives the split levels
• For many cases of interest ℋ  is in the form of the two-spin case summed
over all pairs of ions
ℋ  =
∙
,
•  is the exchange coupling constant (exchange energy)
• Exchange interaction between localized spins only from coulomb repulsion
and from pauli principle
• For angular momentums depending on orbital as well as on spin parts the
Hamiltonian depends on the absolute spin oriantations as well as on the
relative ones
Mean field approximation
• Early analysis of the ferromagnetic effects by P. Weiss with the mean field
theory
• This theory fails to predict spinwaves at low and high temperatures but a good
approximation at
Suppose we focus our attention on a particular Bravais lattice site R in the
Heisenberg Hamiltonian
ℋ  =
∙
,
• With   − ´ =  ´ −  > 0 (ferromagnet)
• Represents for each ion the total angular momentum with spin and orbital part
usual to take these fictitious spins to be parallel to the magnetic moment of the
ion
1
ℋ=−
∙  ´   − ´ −
()
2 ´

• And isolating from ℋ  those terms containing
− ´  ´ +
∆ℋ = −  ∙
≠´
• Has the form of an energy of a spin in an effective external field

1
=+

− ´  ´
´
• But this is an operator depending on a complicated way on the detailed spin configuration
of all the other spin at different sites from R
Mean field approximation
• Replace  with its termal equilibrium mean value
(replace all spin values by its termal equilibrium)
• In ferromagnets all spins have the same mean value
• In terms of the total magnetization density it is

() =

Replace each spin in
=  +
1

− ´  ´
´
by its mean value
() =

We arrive at the effective field
=  + λ

with λ =

and 0 =
()
Every magnetic atom experiences not only the magnetization of its nearest neighbour but
an average magnetization of all the other magnetic atoms
With a spontanious magnetization at it is usual in ferromagnetics with no applied field we
can assume  as
= λ
Curie temperature
1
• It is known that  ∝
• Since  is given by
= λ
• without an applied field the magnetization is given by
= 0
λ

• With 0 the magnetization density with no applied field
• Seperate in a pair of equations like
= 0 x

=
λ
• Put both equations as graphs in a plot

• Important are the intersections of 0 x and λ

• If and only if the slope of the straight line λ is less than that one of 0 x at the
origin, 0 ´ (0) at a nonzero value of x
• 0 ´ (0) can be calculated in terms of zero field susceptibility χ0 , calculated in the
absence of interactions, for
0
χ0 =

=0
0 ´ (0)
χ0 =

• From the analysis of a set of identical ions of angular momentum J one can get
curies law
( )2 ( + 1)
χ=

3

with
≫
• Compare curies law with χ0 =
0 ´ (0)

(can read off the value of 0 ´ (0) )
• Critical temperature is given by
=

3
2
+1 λ
+1
=
0
3
• Not exactly
• More precise with the Isingmodel
• Simplification of the Heisenbergmodel only with spins on one axes (z-axes)
and only spins with the value  = ±1
•  ≠ 0 only for neighboring ions
1
ℋ=−
∙  −

2
,
=1
Sources
Books
• Ashcroft/Mermin, Solid State Physics Ch. 31, 32, 33
• Charles Kittel, Introductions to Solid State Physics (eight edition) Ch.12, 13
•
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http://de.wikipedia.org/wiki/Ferromagnetismus
http://en.wikipedia.org/wiki/Gyromagnetic_ratio
http://de.wikipedia.org/wiki/Diamagnetismus
http://de.wikipedia.org/wiki/Paramagnetismus
http://en.wikipedia.org/wiki/Magnetic_susceptibility
http://en.wikipedia.org/wiki/Magnetization
http://de.wikipedia.org/wiki/Elektronenspin