THE MINISTRY OF EDUCATION AND SCIENCE

OF THE RUSSIAN FEDERATION

NATIONAL RESEARCH NUCLEAR UNIVERSITY MEPhI

(Moscow Engineering Physics Institute)

M.M. Maslov, K.P. Katin, L.A. Openov

INTRODUCTION TO PHYSICS OF SECOND-ORDER

MAGNETIC PHASE TRANSITIONS

This textbook is recommended by UMO “Nuclear Physics and Technologies” as a textbook for students of higher educational institutions of Russia

Moscow 2015

3

4

Prephace

A phase transition is the transformation of a thermodynamic system from one phase (or state) of matter to another one. During a phase transition certain properties change as a result of the change of some external condition, such as temperature, pressure, or others. Examples of phase transitions include: the transitions between the solid, liquid, and gaseous phases of a single component; the transition between the ferromagnetic and paramagnetic phases of magnetic materials at the Curie point; changes in the crystallographic structure such as between ferrite and austenite of iron; order-disorder transitions such as in alpha- titanium aluminides; the emergence of superconductivity in certain metals and ceramics when cooled below a critical temperature; quantum condensation of bosonic fluids (Bose–Einstein condensation); the breaking of symmetries in the laws of physics during the early history of the universe as its temperature cooled, etc.

Phase transitions occur when the thermodynamic free energy of a system is non-analytic for some choice of thermodynamic variables. This condition generally stems from the interactions of a large number of particles in a system, and does not appear in systems that are too small. At the phase transition point (for instance, boiling point) the two phases of a substance, liquid and vapor, have identical free energies and therefore are equally likely to exist. Below the boiling point, the liquid is the more stable state of the two, whereas above the gaseous form is preferred.

In the modern classification scheme, phase transitions are divided into two broad categories. First-order phase transitions are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume. During this process, the temperature of the system will stay constant as heat is added: the system is in a “mixed-phase regime” in which some parts of the system have completed the transition and others have not. Familiar examples are the melting of ice or the boiling of water. Second-order phase transitions are also called continuous phase transitions. They are characterized by a divergent susceptibility, an infinite correlation length, and a power-law decay of correlations near criticality. Examples of sec-

5

ond-order phase transitions are the ferromagnetic transition, superconducting transition, and the superfluid transition.

In this textbook, we outline the general features of second-order phase transitions by the specific examples of ferromagnetic and antiferromagnetic transitions in a solid state system of local magnetic moments. Author contributions: L.A. Openov wrote Preface, §§ 1–10, §§ 24–27; K.P. Katin and M.M. Maslov wrote §§ 11–23, §§ 28–30; M.M. Maslov coordinated all aspects of the work on this textbook.

1. Atomic magnetic moment

The net magnetic moment of every atom is determined by magnetic moments of constituting nucleus and electrons. Let us first make a rough estimate of the magnetic moment µ of a single atomic electron. Starting

with a classical equation

where e, r , and v are, respectively, the electron charge, radius-vector, and velocity, and c is the speed of light, we have

where a and p are characteristic values of, respectively, electron-nucleus separation and electron momentum, m is the electron mass, is the

Planck constant divided by 2π, and we made use of the Heisenberg uncertainty relation p ~ ħ/a.

More rigorous expression for µ follows from quantum mechanics. Since the classical relation between the magnetic moment of an electron

erator l as

6

where mB = |2emc| ≈ 9.974 10-21 erg/G is the so called Bohr magneton.

The operator of the magnetic moment projection on an arbitrary axis z is

For example, if an electron is in the state with an orbital quantum num-

ber l, then the quantum-mechanical eigenvalues of the operator lz (magnetic quantum numbers) are –l, –l + 1, …, l – 1, l. and hence possible magnetic moment projections µz are –µBl, –µB(l – 1), …, µB(l – 1),

µBl. In particular, µz = 0 for an electron occupying the symmetric s- orbital (l = 0).

Except an orbital angular momentum, every electron has an “intrinsic” mechanical angular momentum ħS, where S = 1/2 is the electron spin. As is shown in quantum electrodynamics, this results in the spin

magnetic moment µS whose operator is µS = −2µB S , so that the operator of the total electron magnetic moment becomes

and µz ≠ 0 even if l = 0.

The magnetic moments µi of all atomic electrons add up to give a net resulting magnetic moment of the atom µat , the corresponding operator being

angular momentum and total electron spin respectively (the contribution to µat from nucleus is negligible because of its large mass, see (1.2)). It

can be shown that L + 2S = g (L + S ), where g ~ 1 is the dimensionless Lande factor. For the sake of simplicity let us take L = 0 and S = 1/2. In

7

this case the operator of the atomic magnetic moment is (we omit the subscript “at”)

where µ0 = gµB/2, and hence the z-component of the atomic magnetic

moment takes only two possible values, either +µ0 or –µ0. Such a moment is local in the sense that it is associated with a particular atom, being localized at that atom. In what follows, we shall consider macroscopic solid state systems of local magnetic moments.

2. Physical quantities characterizing the magnetic properties of matter

Any atom in the solid may carry a local magnetic moment µn , where n = 1, …, N is the number of a given atom and N 1 is the number of

atoms in the sample. The total magnetic moment is a vector sum ∑N µn

n=1

and depends on N. It is convenient to consider a specific magnetic moment called magnetization:

n=1

where V is the sample volume. In general magnetization can also result from electric currents flowing inside the sample.

As a rule (but not necessarily) magnetization arises upon application

of an external magnetic field of strength H and is linear in H at small

H:

where (α,β)= (x, y, z) and χαβ is the tensor of magnetic susceptibility.

8

for any direction in space. In the case that M is not linear in H, the differential magnetic susceptibility is used:

Magnetic induction B (magnetic field in the sample) is the sum of external and induced fields:

is the magnetic permeability.

3. Classification of materials for their magnetic properties

All materials (not only solids) may be categorized according to their magnetization M in an external magnetic field H :

1) nonmagnetic materials – M = 0 , i.e., B = H , χ = 0, µ = 1 (in an-

tiferromagnets, M = 0 as well, while the differential magnetic susceptibility is nonzero);

this case, the magnetic susceptibility defined by Eq. (2.3) is meaningless, while the differential magnetic susceptibility (2.4) makes sense. Ferromagnetism is observed only below the material-dependent Curie

temperature θ.

9

4. Isolated local magnetic moment in an external magnetic field

We consider an isolated local magnetic moment µ subjected to an

external magnetic field of strength H . In the case that this moment is associated with an atom in the solid, its kinetic energy is negligibly small, so that its total energy E is just an interaction energy U:

where µ is the magnetic moment operator. Taking a frame of reference with z-axis oriented along H , so that H = Hez , we have

corresponding to orientation of the magnetic moment along the z axis, i.e., along the applied field (“up”) and in the opposite direction (“down”) respectively. Note that the electron magnetic moment is antiparallel to the vector of its spin because of the negative electron charge.

where µ0 = gµB/2, see §1.

Since the Hamiltonian, Eq. (4.3), is linear in , the states (4.4.) are

µz

its eigenstates as well:

10