40
S. J. Blundell
where β = 1/k B T , k B is the Boltzmann constant, T is the temperature and i =
1, . . . , N . Here E({r i , p i }) is the energy associated with the N charged particles
having positions r 1 , r 2 , . . . , r N , and momenta p 1 , p 2 , . . . , p N . The integral is, therefore, over a 6N -dimensional phase space (3N position coordinates, 3N momentum
coordinates). The effect of a magnetic field is to shift the momentum of each particle
by an amount q A. We must, therefore, replace p i by p i − q A. The limits of the
momentum integrals go from −∞ to +∞ so this shift can be absorbed by shifting
the origin of the momentum integrations. Hence the partition function is not a function of magnetic field, and so neither is the free energy F = −k B T log Z . Thus the
magnetic moment m = −(∂ F/∂ B) T must be zero in a classical system.
Thus, we need quantum mechanics to make further progress. In this chapter, I
will not provide an exhaustive review of magnetism (fuller treatments can be found
elsewhere, e.g. [2–5]) but focus on a few key issues and some selected examples.
To begin our discussion, it is helpful to note the energy scales inherent in magnetic
problems. First, there is the kinetic energy which is on the eV scale. This typically
takes a value like
2
π
2
/(2m L
2 ), where L is a length scale and this expression is
the familiar one for particle in a box. This is an energy cost and arises because it
takes energy to put an electron in a small box. Second comes the potential energy
which is also on a similar scale and takes a form such as e
2
/(4ππ 0 L). This will be
a negative energy if considering the attraction between an electron and a nucleus
(and becomes larger and more negative as L decreases) and positive if considering
electron–electron repulsion. Atoms are the size they are because of a compromise
between kinetic energy wanting the atom to be infinite size and the potential energy
wanting the atom to be zero size. Because one energy goes as L
−2 and the other as
−L
−1 a compromise can be reached (and this is essentially the derivation of the Bohr
radius). Both the kinetic and potential energies are large and are typically k B T .
Next, we have to add the spin–orbit interaction which is typically much smaller,
usually in the meV, and the magnetocrystalline anisotropy, which in cubic materials
is in the µeV. These effects will turn out to be very important in magnetic materials,
but they are small perturbations to the main interactions and will mainly come into
play only once the magnetic order is established by the dominant interactions.
2.2 Exchange
The exchange interaction arises from the kinetic and potential energy in bonds
between atoms. To see how this comes about, we begin by recalling simple results
for the molecular orbitals in H 2 [see Fig. 2.1a]. We label the two hydrogen atoms A
and B and write the wave function |ψ as a linear combination of atomic orbitals
|ψ A and |ψ B so that
|ψ = c A |ψ A + c B |ψ B .
(2.2)
S. J. Blundell
where β = 1/k B T , k B is the Boltzmann constant, T is the temperature and i =
1, . . . , N . Here E({r i , p i }) is the energy associated with the N charged particles
having positions r 1 , r 2 , . . . , r N , and momenta p 1 , p 2 , . . . , p N . The integral is, therefore, over a 6N -dimensional phase space (3N position coordinates, 3N momentum
coordinates). The effect of a magnetic field is to shift the momentum of each particle
by an amount q A. We must, therefore, replace p i by p i − q A. The limits of the
momentum integrals go from −∞ to +∞ so this shift can be absorbed by shifting
the origin of the momentum integrations. Hence the partition function is not a function of magnetic field, and so neither is the free energy F = −k B T log Z . Thus the
magnetic moment m = −(∂ F/∂ B) T must be zero in a classical system.
Thus, we need quantum mechanics to make further progress. In this chapter, I
will not provide an exhaustive review of magnetism (fuller treatments can be found
elsewhere, e.g. [2–5]) but focus on a few key issues and some selected examples.
To begin our discussion, it is helpful to note the energy scales inherent in magnetic
problems. First, there is the kinetic energy which is on the eV scale. This typically
takes a value like
2
π
2
/(2m L
2 ), where L is a length scale and this expression is
the familiar one for particle in a box. This is an energy cost and arises because it
takes energy to put an electron in a small box. Second comes the potential energy
which is also on a similar scale and takes a form such as e
2
/(4ππ 0 L). This will be
a negative energy if considering the attraction between an electron and a nucleus
(and becomes larger and more negative as L decreases) and positive if considering
electron–electron repulsion. Atoms are the size they are because of a compromise
between kinetic energy wanting the atom to be infinite size and the potential energy
wanting the atom to be zero size. Because one energy goes as L
−2 and the other as
−L
−1 a compromise can be reached (and this is essentially the derivation of the Bohr
radius). Both the kinetic and potential energies are large and are typically k B T .
Next, we have to add the spin–orbit interaction which is typically much smaller,
usually in the meV, and the magnetocrystalline anisotropy, which in cubic materials
is in the µeV. These effects will turn out to be very important in magnetic materials,
but they are small perturbations to the main interactions and will mainly come into
play only once the magnetic order is established by the dominant interactions.
2.2 Exchange
The exchange interaction arises from the kinetic and potential energy in bonds
between atoms. To see how this comes about, we begin by recalling simple results
for the molecular orbitals in H 2 [see Fig. 2.1a]. We label the two hydrogen atoms A
and B and write the wave function |ψ as a linear combination of atomic orbitals
|ψ A and |ψ B so that
|ψ = c A |ψ A + c B |ψ B .
(2.2)