Marder M.P. Condensed Matter Physics

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Critical Phenomena

753

/

drdr

(n(r)n(r

)) j - (N)

(24.119)

{

/•

Ï Follows steps similar to those

\+n

dr{g(r)-\)\.

around Eq.

(3.51),

and use (24.120)

V6W

'j Eq. (5.45) for g.

Because near the critical point

one has

~ drg(r). (24.122)

Changing variables to s = r/£ and using Eq. (24.121), one obtains

KT^^C

Jds^

(24.123)

~£

-^|,p(2-r,)_

(24.124)

However, the compressibility must diverge as

|?|~

(2-77)1/=

(24.125)

the Fisher-Essam relation. Finally, there is an apparently improbable argument

that because G = S/kgTV has dimensions of inverse length to the third power, but

the only length in the system that should be important near the critical point is the

correlation length £, one must have that

r-!^~ki

126

)

=>2-a = 3v, (24.127)

the Josephson or hyperscahng

relation.

All of these scaling relations are obeyed

within a few percent by the experimental values listed in Table 24.3.

Scaling Form for Magnetization. Some of the most detailed experimental results

are expressed using an alternate form of the scaling hypothesis, one that relates the

magnetic field and magnetization through

m&h

iwiß) ■

(24

128)

The function h can be measured by choosing temperature T and field H, measuring

M, and then constructing the ratio |//|/|M|

and plotting it versus the variable

W*-

(24

129)

When the external magnetic field H vanishes, h(x) must vanish. Because \M\ ~

for H = 0, the conclusion is that in vanishing field x = t/\M\

—XQ

is a constant,

and h{—

XQ)

= 0.

754 Chapter 24. Classical Theories of Magnetism and Ordering

0 -

0 1 2

gio(

Ao+l)

Figure 24.14. Log-log plot of scaling function h = I^I/IMI"

versus x = t/lM}

^, using

ö = 4.32, ß

—

0.364, and

= 0.596 for CrBr

. The exponents and the critical temperature

= 32.841 K are determined together as part of the process of trying to ensure that data

taken at different temperatures lie on top of a single scaling curve. [Source: Vicentini-

Missoni(1972), p. 68.]

If the scaling hypothesis is correct, then measurements of |//|/|M|

versus x +

should fall on a single line. As shown in Figure 24.14, the data do collapse in

this way, and furthermore the function h takes the form of a power law over a large

range of temperature near the critical point.

Problems

Magnetic dipole moment: Consider a small loop of wire in the x-y plane

with area A, and current J flowing through it. Show that the vector potential

A far from the loop is given by

mx r

(24.130)

where r is a vector from the middle of the loop of wire to an observation point.

Magnetic dipole energies: Verify Eqs. (24.36) and (24.37):

(a) Consider a distribution of current j(r) that is localized and is in steady state,

which means that V

■

j = 0. By considering

dr r

■

j = 0,

(24.131)

Problems

755

show that

J dr(r

)=0.

(24.132)

(b) Using Eq. (24.132), show that for any vector C,

J dr]r-C=

-J dr[rxj\xC. (24.133)

Thermodynamic potential: Show that Eq. (24.27) describes the thermody-

namic potential appropriate for a system where external currents

t(r)

rather

than magnetic induction B(r) are specified. Write B as the curl of a vector

potential. Show that for fixed temperature, any changes in S can be written

as an integral over changes in jexti^), and find the thermodynamic variable

conjugate to

ext

Mean field theory: Consider a two-dimensional Ising model on a square

lattice. Suppose that

£ = - E

- E

(

134

)

(RR') R

and that J is negative.

(a) Let the mean value of spins on alternating sites be denoted by cr-p and a

Find the free energy corresponding to this Hamiltonian within mean field the-

ory.

(b) Find the self-consistent equations that determine the mean spins

a-\

and <7j

within mean field theory.

<T|

= ay. Find the high-temperature form

of the magnetic susceptibility, reproducing an equation in this chapter. Relate .

the unknown quantity in that equation to J.

One-dimensional Ising model: The goal of this problem is to analyze the

statistical mechanics of the one-dimensional Ising model.

(a) Let /(<7i, (72) = exp[/3/<7i<72]- Show that

f(cr

) = cosh ßj + u\u

sinh ßJ. (24.135)

(b) The partition function for an Ising chain with N spins is

TV—1

n f^

'+I)-

136

)

CTl...crjve{-l,l} '=1

Inserting

Eq.

(24.135), show that only one term survives the sum over o\. . .a^.

756

Chapter 24. Classical Theories of Magnetism and Ordering

ruling out any phase transition.

(d) What is the thermal average of o\ as a function of temperature?

6. Superlattices: Consider a 3:1 mixture of atoms A and B in the structure

depicted in Figure 5.4.

Construct a mean field theory analogous to Eq. (24.72) for atoms in this ge-

ometry.

(a) Find two self-consistent equations for o

and a

. One of the equations

should be

= ta.nh(ßn

H + e

ß[4a

}).

(24.137)

(b) Write down the simple linear relation between a

and a

which follows from

requiring atoms A and B to be in 3:1 ratio.

obtain a single self-consistent equation for a

(d) Draw a picture illustrating the solution of this equation. Carefully trace out

the solutions as a function of temperature, looking closely at the onset of

order. According to mean field theory, is the transition first or second order?

Properties of scaling function h: Consider the scaling function h defined in

Eq. (24.128).

(a) Send T

—>

. What is the behavior of h(x) in the vicinity of x = 0? Does it

have some type of singularity, or not?

(b) While holding t fixed away from zero, send M toward zero. How must h(x)

behave in this limit?

(d) h(x) vanishes as a power of x +

for

close to but greater than zero.

Find the power.

8. Preisach model: The Preisach model views a magnet as a large collection

of noninteracting hysteretic domains. Each domain can be described by the

hysteretic function

HH,,H

(H). The way /x works is this: If /i = 0, it remains

so until H exceeds

//T.

Then // jumps to [i

. It remains at [i

until H drops

below H\, at which point it returns to 0. If N is the probability density of

domains with lower and upper fields H\ and

H2,

then the total response of a

ferromagnet is given by

/•oo

/ dH

^{H

)liH

{H) (24.138)

-oo

JHt

Argue that any magnet described by Eq. (24.138) exhibits return-point mem-

ory, as shown in Figure 24.9. That is, suppose field H is changed as follows:

References 757

• H is initially made large and negative.

• H increases monotonically to some value

Ho,

where the magnetization is Mo.

• H next decreases below Ho.

• Finally, it increases back up to Ho- What needs to be argued is that the

magnetization has returned to

MQ.

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758 Chapter 24. Classical Theories of Magnetism and Ordering

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25.

Magnetism of

Ions

and Electrons

25.1 Introduction

As late as the first quarter of the twentieth century, the magnetism of solids re-

mained a great mystery. The assumption that magnetic solids were assemblies

of vast numbers of smaller magnets was an hypothesis that could successfully be

studied from mechanical and statistical points of view. However, the origin of the

microscopic magnets was not explained.

The greatest difficulty was presented by a theorem of Bohr (1911) and van

Leeuwen (1921), which stated that if

one

considers a collection of classical charged

particles interacting with each other and with external potentials, their partition

function must be completely insensitive to the presence of any applied magnetic

field. The simple proof of this statement is the subject of Problem 2. Therefore, no

solid in thermal equilibrium should have any magnetic properties. Indeed "when

one attempts to apply classical statistics to electronic motions within the atom, the

less said, the better" [van Vleck (1932), p. 104].

If one thinks of a collection of electrons in a box, moving with various veloci-

ties,

this result at first seems strange, because upon introduction of a magnetic field

the electrons will begin to circle in orbits and thus generate magnetic moments.

However, the magnetism of electrons turning in circles in the middle of a sample

is exactly compensated by electrons at the boundaries, as illustrated in Problem

Only with the appearance of quantum mechanics could one explain how solids

could have magnetic properties at all.

In part, the answer given by quantum mechanics is the simple assertion that

electrons possess a magnetic moment of strength /xg, together with the directive

that one must not question its origin in classical terms. The magnetism of solids

does not end, however, with the primitive magnetic moments of the elementary

particles, but continues to emerge in new ways from assemblies of particles. The

unfilled inner shells of the transition metals and rare earth metals have a strong

paramagnetic response. They sometimes have permanent magnetic moments, and

even filled atomic shells are weakly diamagnetic. An ensemble of free electrons in

a box is diamagnetic.

The existence of magnetism is due to quantum-mechanical restrictions on the

orbits of charged particles that prevent the perfect classical cancellation from oc-

curring. Many details of how the magnetic response works out are best viewed as

attempts by tightly packed electrons to minimize Coulomb repulsion. Two elec-

trons circling a nucleus reduce their Coulomb interaction by adopting an antisym-

metric wave function that vanishes whenever they come near each other. The Pauli

759

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

760

Chapter

25.

Magnetism of Ions and Electrons

principle demands that the overall wave function be antisymmetric, so the spin

wave function must be symmetric. That is, the electrons lower their energy by

adopting the same spin state and developing a local magnetic moment. This ten-

dency to develop ferromagnetism fights against other energies that tend the oppo-

site way and that are of the same general size. The helium atom does not have a

ground state with total spin S = 1, but instead has S = 0. The reason is that the

lowest-energy state for a single electron is a nondegenerate n = 0, / = 0 state. To

build a spatially antisymmetric wave function, there is no choice but to have at least

one electron in an excited state, n = 1, / = 1, and this excited state costs enough in

energy to outweigh the benefits of antisymmetry. In fact, it can be proved in gen-

eral that the ground state of any two-electron system has total spin 5 = 0, which

shows that filling atomic shells in the proper order is the most important first step

in placing electrons in their ground state.

No general statements will be true in all cases because magnetism results fre-

quently from competition between delicately balanced competing effects. How-

ever, it may be helpful to indicate some of the energies that are likely to be in-

volved.

Shell energies 1-10 eV: To good approximation, electrons fill up each atom with

ascending values of index n and angular momentum quantum number /, as de-

scribed in the periodic table, and rarely employ states with quantum number n

until all states with number n

— 1

have been filled. The closed inner shells of

the atoms have no net magnetic moment and are weakly diamagnetic. Excep-

tions to this rule in the transition metals and rare earths can produce metals

with spontaneous magnetic moments.

Hund's rules 1-10 eV: Within an incompletely filled atomic shell, electrons try to

minimize Coulomb repulsion by choosing total spin and angular momentum

in accord with Hund's rules, to be described in more detail below. For unfilled

/ shells of the rare earth solids, the energy to be gained from Hund's rules

overrides any interaction with neighboring atoms.

Band energies 1-10 eV: In Fe, Ni, and Co, delocalized electrons acquire a net

magnetic moment, essentially also as a means of reducing Coulomb repulsion.

Crystal field splitting 0.1-1 eV: The presence of neighboring atoms in a crystal

breaks the spherical symmetry of the individual atomic nucleus. For iron

compounds, this effect is often large enough that Hund's rules no longer apply,

and the total spatial angular momentum L equals 0.

Interatomic exchange 10~

-0.1 eV: Just as the combination of Coulomb repul-

sion and Fermi statistics produces magnetic correlations within the atom, it

also creates them between neighboring atoms. Whether the effect produces

ferromagnetic or antiferromagnetic correlations is extremely difficult to cal-

culate or predict, because it typically results from the near cancellation of

terms whose energy might be ~ 1.0 eV.

Indirect exchange, superexchange 10

—0.1

eV: Magnetic atoms can induce mag-

Atomic Magnetism

761

netic moments in their surroundings, and the induced moments can interact

with other magnetic atoms, producing a net indirect interaction. The sur-

rounding might include a sea of otherwise nonmagnetic conduction electrons

or neighboring nonmagnetic atoms.

Interaction with magnetic fields 10~

-10

eV: The energy of a single magnetic

moment in a 1-T magnetic field is only around 10

-4

eV. The interaction en-

ergy of magnetic moments with the magnetic fields produced by other mo-

ments therefore appears to be small. However, the interaction is so long

ranged that the number of relevant interactions is enormous, and it effectively

grows without bound as samples grow in size. For this reason, magnetic sam-

ples spontaneously develop elaborate domain structures in which magnetic

correlations on large scales are quite different from the magnetic correlations

on short scales. It is also for this reason that magnetic fields of a strength

achievable in the laboratory are able to flip the moments in permanent mag-

nets.

25.2 Atomic Magnetism

Isolated atoms have magnetic properties, and the goal of this section is to lay out the

combination of semiempirical wisdom and quantum mechanics that permits them

to be calculated. Speaking of isolated atoms may raise thoughts of dilute gases,

but magnetic experiments on dilute vapors of rare elements are difficult and rarely

performed, partly because magnetic measurements so often rely upon measuring

torques or forces created when magnetic fields are applied to large solid samples.

What is referred to theoretically as an isolated atom is more likely a rare magnetic

atom dissolved in a largely nonmagnetic host, in which case the possibility that the

magnetic atom is influenced by the host has to be considered. Nevertheless, the

discussion begins with the assumption that any desired atom can be isolated and its

magnetic properties can be measured.

The interaction of isolated atoms with external magnetic fields occurs in two

ways.

First, there is the interaction of the orbital magnetic momentum with the

magnetic induction. This contribution is found by including the vector potential

A in the momentum operator for the electron motion. Second, one must include

the interaction of B with the electron spin. One has therefore a Hamiltonian of the

form

The factor of 2 sitting in front of ßa is a con-

sequence of Dirac's relativistic theory of the

electron. The positive sign of the term results

from the negative charge of the electron, the

convention that e be positive, and the interac-

tion energy (24.37).

(25.1)

where /xg is the Bohr magneton defined in Eq. (24.39).

The sum is taken over all electrons in the atom. Two additional comments are

in order. The first is that all of the magnetic interactions involve the inverse of

+ -A(Ri)

2fi

BSf,

762 Chapter

25.

Magnetism of Ions and Electrons

the electron mass. This fact is significant, because it demonstrates why nuclear

contributions to magnetism can be ignored in this context. These would involve

the inverse of the nuclear mass instead and are therefore negligible by comparison.

In order to obtain definite results from the Hamiltonian, assume that

A(R) = --RxBz, (25.2)

which describes a uniform induction B, taken to point in the z direction. Define the

angular momentum operator

hL^^RjXPj

(25.3)

p 7 _p

D y.

3 _ 3 D

R and P can be treated here as commuting ,-^r A\

-/1

2 J 2

operators, because the cross product always W-'-^V

multiplies different components together.

^^=

]

^{L

2s)-B+^-

B^^j

+ yl)- (25-5)

L and S denote the total orbital and spin angular momenta of the atom, and have in-

teger and half-integer eigenvalues respectively. If the states |/) index the electronic

orbital states in the absence of the magnetic interaction, then keeping all terms up

to quadratic order in the induction B, perturbation theory to second order gives

V+l

(25.6)

The first term on the right hand side of Eq. (25.6) is linear, and it dominates mag-

netic response unless the relevant matrix element vanishes. The reason is that the

natural dimensionless parameter in terms of which to discuss the strength of the

magnetic interactions is the ratio of a typical magnetic to a typical atomic energy.

The typical magnetic energy is

= 5.79

•

[5/Tesla] eV. (25.7)

On the other hand, a typical separation between atomic energy levels is on the order

eV, so the magnetic energies are small by comparison.

25.2.1 Hund's Rules

In order to determine whether the first term of

Eq.

(25.6) vanishes or not, it is nec-

essary to find the properties of the atom in its ground state. The ground states are

described for all but the very heaviest atoms by Hund's rules, which originated in

spectroscopic observation. They are easy to describe, but could only rigorously

be justified by full solution of the quantum-mechanical problem of many electrons

circling the nucleus. In order to express the rules, it must first be observed that