Dionne G.F. Magnetic Oxides

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Chapter 5

Anisotropy and Magnetoelastic Properties

In this chapter, we discuss the local origins of the two measurable macroscopic

effects that occur from interactions between the ionic magnetic moments and the

lattices in which they reside: magnetocrystalline anisotropy and magnetostriction.

In the preceding chapters, the focus has been on the molecular origin of the mag-

netic moments in crystal lattices. For the 3d

transition group in particular, the

disposition of spin alignments as determined by covalent-induced superexchange

and the randomizing effect of temperature has been reviewed. The spin system is

also inﬂuenced by geometrical shape of the specimen in which it resides (described

in Chap.1) and the symmetry of the lattice itself and its elastic properties, each of

which contribute to the anisotropy that inﬂuences the magnetization process and

other magnetic properties. In addition, large anisotropic magnetic effects can result

from asymmetry of the local crystal ﬁelds and their interactions with magnetoelastic

cations. In this sense, magnetoelasticity refers to the coupling between the magnetic

moment of the cation and local crystal ﬁeld of the anion coordination. All of these

mechanisms, however, involve interaction between the spins and the elastic proper-

ties of the lattice, which can be collective, as in the case of dipole–dipoleinteractions

in ﬁxed array of lattice sites, or individual through orbital angular momentum cou-

pling to the crystal ﬁeld. The conventional macroscopic phenomenological model

is presented later in this chapter, but it is the molecular origins of these properties

where our initial attention will be focused.

Following the context established by the preceding chapters, we begin by exam-

ining the local origins of the local anisotropy. In particular, self-induced anisotropy

in the form of crystal-ﬁeld distortions derived from spin–orbit coupling and the

Jahn–Teller effect will be emphasized. The underlying physics is reviewed ﬁrst

through the properties of individual ions. With the single-ion concepts in hand, we

then examine the ions in an exchange-coupled ferromagnet (or ferrimagnet) to de-

termine how the macroscopic anisotropy and magnetostriction effects inﬂuence the

collective magnetization statically, and then dynamically in Chap. 6.

G.F. Dionne, Magnetic Oxides, DOI 10.1007/978-1-4419-0054-8 5,

 Springer Science+Business Media, LLC 2009

201

202 5 Anisotropy and Magnetoelastic Properties

5.1 Quantum Paramagnetism of Single Ions

In the previous chapter, the concept of magnetic anisotropy was introduced some-

what incidentally in order to explain the variation in magnetic moment of the

rare-earth ion Ho

in iron garnet through the effects of crystal ﬁelds on the to-

tal angular momentum. Although crystal-ﬁelds inﬂuence the magnetic properties

of rare-earth ions, they are critically important in the 3d

transition iron group se-

ries. Because the coupling to the electrostatic ﬁelds of the crystalline environment

is with the orbital angular momentum, the reason why the iron group ions are so

sensitive to their surroundings is the unshielded 3d shell. Not only do the d elec-

trons sense the electric ﬁelds of the ligands, but they also participate in the chemical

bonding by sharing their orbital states with the oxygen anion 2p electrons. Much

of the underlying theory is reviewed in Chap. 2, but the actual relation between

the electron–lattice effects and the properties of the oxides (and other families of

transition-metal compounds) was not discussed. The role of spin–orbit coupling as

the intermediary between spin and lattice and the interaction between orbit and spin

angular momentum with a magnetic ﬁeld must now be examined.

5.1.1 Theory of Anisotropic g Factors

The classical approach to paramagnetism from a collection of isolated magnetic

moments m was treated in Sect. 1.2.1. From this model Curie’s law of  / 1=T

was derived with versions that employed the Langevin or semiclassical Brillouin

functions. In either case, the applicable variable is m  H =kT (or gm

HJ=kT for

the Brillouin function). In later developments spearheaded by Van Vleck, the theory

was reﬁned to take into account an angular momentum energy level structure above

the ground state [1]. Because these excited levels have differing values of m,a

Boltzmann probability distribution can be used to determine a weighted average

susceptibility shown schematically in Fig. 5.1 for a ﬁve-level orbital group (with

spin implied) split in a crystal ﬁeld. The general expression for the paramagnetic

susceptibility of a molecule with quantum numbers L, S,andJ can be written as

 D

LCS

J DjLS j

.2J C 1/  .LSJ / exp ŒJ .J C 1/ =2kT

LCS

J DjLSj

.2J C 1/ exp ŒJ .J C 1/ =2kT

; (5.1)

where .LSJ/ is the susceptibility contribution from the energy level designated

2SC1

. Details of the derivation of (5.1), including an expression for .LSJ/,

may be found in Grifﬁth’s book [2].

The most successful application of this theory was carried out by Van Vleck for

the 4f

lanthanide series. The results are shown in Fig.5.2, where the effective

5.1 Quantum Paramagnetism of Single Ions 203

Fig. 5.1 Schematic diagram of Boltzmann population distribution function showing the relative

occupancies of the ﬁve d orbital states controlled by the exp

E=kT

= g[J(J+1)]

1/2

experiment

(Bohr magnetons)

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Fig. 5.2 Rare-earth ion magnetic moments as a function of the number of 4f

electrons. The

lanthanide series is used as the x-axis coordinate, which extends from 0 to 14. The experimental

data and calculated ﬁt were reported by Van Vleck and Frank [1]. Note that the lack of agreement

with the simple theory for Eu

and Sm

indicates that the more complex model of (5.1)is

needed

moments are graphed in terms of Bohr magnetons n



3kT =N



1=2

where N

is Avogadro’s number. These data may be found in tabular form in

most textbooks on magnetism. Despite intense efforts to extract information from

204 5 Anisotropy and Magnetoelastic Properties

susceptibility measurement data, results of analysis based on this theory have gen-

erally fallen short of expectations. The difﬁculty in sorting out the individual

contributions from the ladder of energy levels with varying separations, com-

pounded by the uncertainty in the values of the Land´e g factors that are implicit

within the .LSJ/ of individual states led researchers to adopt more sophisticated

approaches in both measurement and theory. Because this necessary spectral infor-

mation is so sensitive to the effects of the crystal ﬁeld, more powerful tools were

developed to probe the lowest energy states and directly measure g factors by the

Zeeman effect in many of the transition metal ions. In these situations, the analysis

applied to electron paramagnetic resonance (EPR) spectroscopy proved to be fer-

tile ground for theorists with skill in the application of group theory and quantum

mechanical perturbation theory.

If we now return to the general perturbation Hamiltonian for an ion in a crys-

talline environment introduced as (4.38) for the lanthanide series, H

D H

C H

, we can examine the effects of the remaining interactions that in-

ﬂuence the magnetic moment of an isolated ion. For the individual ions of the 3d

transition series H

does not apply, and the relation is changed to

D H

C H

D H

C L  S C m

.L C g

S /  H ; (5.2)

D H

C L  S C m

;

where g

D 2 is the g factor for an electron and g

is the tensor that results

from the orbital quenching and its attendant anisotropy effects from the crystal

ﬁeld symmetry. Since the effects of H

on the orbital states have already been

dealt with in Chap. 2 for D and F terms of the iron group, we can now continue

with an examination of the next perturbations for this series: spin–orbit coupling

L  S and the combined effect of the orbital and spin moments in a magnetic ﬁeld

.L C 2S /  H . Smaller energy terms involving spin–spin and electron–nuclear

spin interactions that produce hyperﬁne spectral structure will be omitted from this

discussion.

For the 3d

series, we recall from Chap. 2 that an undistorted octahedral crys-

tal ﬁeld will stabilize either a triplet (from a free-ion D-state d

and d

, from

an F -state d

and d

), a doublet (from a D-state d

and d

), or a singlet (from

an F -state d

and d

). However, in cases where the ground state is expected to

be degenerate based on the host lattice symmetry, local ligand distortions usually

occur spontaneously to lower the symmetry. In the context of magnetocrystalline

anisotropy, it is the extent to which the crystal ﬁeld lifts the orbital degeneracy that

determines the magnetoelastic properties. To this end, we will review the various

electronic structures that exist in the transition metal ions residing in ligand ﬁelds of

initially cubic symmetry.

Analysis of any of these electronic conﬁgurations is solvable by conventional

degenerate perturbation theory with the use of high-speed digital computation tech-

nology once the appropriate Hamiltonian matrices are set up. To demonstrate the

5.1 Quantum Paramagnetism of Single Ions 205

origin of single-ion anisotropy as reﬂected in the variation of the g factor with mag-

netic ﬁeld direction, a detailed examination of the simplest case will be presented to

sensitize the reader to the important physics issues involved in these critical spin–

lattice interactions.

5.1.2 Conventional Perturbation Solutions

For the general case of a single d electron in an octahedral crystal ﬁeld with an or-

thorhombic distortion dominated by a z-axis contraction,

the energy level structure

isshowninFig.5.3. For cases where the free-ion term is a

2SC1

, the appearance

of the crystal-ﬁeld energy levels in the ground-state occupancy diagram conforms

to the actual energy states of a multiple d -electron ion (as opposed to the

2SC1

cases to be reviewed in subsequent sections). The ground term is indicated by

3=2

and the eigenvectors of the crystal-ﬁeld states are listed as j0i; j1i; j2i; j3i,andj4i,

where from (2.12)

D d

D Y



 r



;

D d

y

C d

2

/ D



C Y

2





 y



;

Fig. 5.3 3d

energy-level model in an orthorhombic crystal ﬁeld, showing spin stabilization in a

Zeeman splitting of the ground-state Kramers doublet

In other situations, the distortion can be an expansion, which is manifested by the orbital functions

with a z-dependence lying below the in-plane x-ory-dependent states. Note also that a trigonal

[111]-axis expansion is compatible with a [100]-axis contraction.

206 5 Anisotropy and Magnetoelastic Properties

D d

D i

C d

1

/ D i



C Y

1



3yz; (5.3)

D d

D

 d

1

/ D



 Y

1



3xz;

D d

Di

 d

2

/ Di



 Y

2



3xy:

The solutions of these perturbation effects will be carried out in two steps: ﬁrst,

the eigenstates of H

term will be determined and a new ground state will be

found; then the Zeeman term will be applied to determine the anisotropy of the

magnetic moment which will appear in the inequality of the elements of the g

tensor. The problem will be analyzed in terms of the Ti

-substituted hydrated

alum salt



; Ti



.SO

2

 12H

O from three approaches, begin-

ning with the most general [3]. First, degenerate theory will be applied whereby

a full perturbation matrix is established for the elements hkjH

C L  S jni,

where hkj and jni are eigenfunctions from the list in (5.3) with the spin states in-

cluded. Since each orbital state has a twofold spin degeneracy .m

D˙1=2/ this

exercise involves a total of ten wavefunctions, requiring the diagonalization of a

10  10 matrix. For the spin–orbit contributions a convenient form of the operator

is L  S D L

C .1=2/ .L



C L



Details of the solution are given in Appendix 5A, where the secular equation

is shown to be separable into two identical functions that yield degenerate energy

states, each representing a twofold spin degeneracy (known as Kramers doublets,

which occur in ions with odd numbers of d electrons). The resulting ground state

eigenfunctions from (5.72)are

D a

C b

1; 

C c

2; 

C d

; (5.4)



D a

0; 

C b

 c

 d

3; 

;

where the highest energy j4i level has been dropped to simplify the calculation with

only a small loss in accuracy, are then used for the 2 2 ground state Zeeman matrix

according to hkjm

.L C 2S /  H jni.

To compute the Zeeman splittings for a particular direction of H , we deter-

mine the matrix elements by applying the appropriate operators, recalling that

D .1=2/ .L

C L



/ and L

D – .i=2/ .L

 L



/ and likewise for the spin

operators. In each case the Zeeman matrix is diagonalized and the expressions for

the energies of the split spin doublet are subtracted to give E

D g

, from

In this lattice, the ligands of the Ti

ions are an octahedron of H

O molecules that mimic

2

ions.

For this exercise we consider only the lowest Kramers doublet (Zeeman state) of the three orig-

inating from the T

of the cubic ﬁeld. Implicit in this choice is that the ground state is a singlet

that results from either an axial ﬁeld component of the appropriate sign or a Jahn–Teller effect.

5.1 Quantum Paramagnetism of Single Ions 207

which expressions for g

, g

,andg

in terms of the a, b, c,andd coefﬁcients can be

deduced, as shown in Appendix 5A. To solve for the g factors, values are assigned

to the crystal ﬁeld splittings ı

, ı

,and;avaluefor must also be assigned,

usually the free-ion value. Conversely, if g factors are found from measurement,

paramagnetic resonance at microwave frequencies in many cases, the reverse pro-

cedure could be followed to determine the energy splittings. This method was used

to compute energy-level splittings from measured g factors for a series of Ti

substituted alum salts starting with Rb

.SO

2

 12H

O, and the results

are included in Table 5.1. Among these compounds are two types of crystal ﬁeld:

for Rb, K, Tl, and Na alum, orthorhombic with complete splitting of all ﬁve d or-

bitals that produces three g factors [3–6], and for the Cs alum-based compounds, a

trigonal ﬁeld directed along the lattice <111> axes produces a ground state singlet

and two excited doublets as depicted in Figs. 2.13 and 5.3 (for ˛>60

)[7], [8]. In

each case the degenerate method outlined in Appendix 5A was employed. In the Cs

alum cases, accuracy of ﬁts between theory and experiments were limited by the ap-

proximations to the theory because the values of the lower energy splittings was on

the order of 200 cm

1

so that the assumption that =ı < 1 could not be justiﬁed, i.e.,

off-diagonal elements are as large as the diagonal ones. Among the orthorhombic

alums, variations in the g factors and their interpreted lower symmetry orbital split-

tings were discussed by Dionne and MacKinnon [6] in terms of the possible effect

on the crystal ﬁelds by differing locations of the .SO

2

radicals. The ionic radius

of the large monovalent cation relative to that of the Al

host ion was correlated

Table 5.1 Anisotropic g factors and crystal ﬁeld splittings of Ti





in octahedral sites.

(

D154 cm

1

)

host

Radius

(

Theory g



1





1







1



Refs.

Rb alum 1.48 SpinHam 1.895 1.715 1.767 1,070 1,310 11,500 [4]

nondeg. ”””1,0501,32017,000[4]

degen.”””1,0501,32020,300

[4]

Tl Alum 1.40 degen. 1.938 1.790 1.834 1,462 1,843 20,300 [5]

K Alum 1.33 degen. 1.975 1.828 1.897 1,780 2,950 20,300 [6]

Na Alum 0.95 degen. 2:00 1:86 1:86 >2;000 >2;000 20,300 [6]

host Radius

(

Theory g



1







1



Refs.

CsTi Alum 1.69 Degenerate 1.24 0.93 200 20,300 [8]

” 1.19 0.70 ” ” [8]

” 1.17 0.23 ” ” [8]

CsAlum 1.69 Degenerate 1.25 1.14 300 20,300 [7]

– – 1.067 <0:1 5.19

The value  D 20;300 cm

1

was measured by optical absorption in an aqueous solution [10],

where the ligands are octahedra of water molecules. This situation is close to that of oxygen ligands

because of the relative location of the O in the H

O molecule. To obtain the measured value of

D 1:895 by the (corrected) degenerate method,  D 22;000 cm

1

208 5 Anisotropy and Magnetoelastic Properties

with the magnitude of the apparent orthorhombic departures of the crystal ﬁeld from

its normal alum trigonal symmetry that is seen only in the Cs alum case. One result

that supported this contention was the observation that Cs(TiAl) alum crystals could

be grown from aqueous solution over the full range of solid solution, while the rest

of the family with the apparent orthorhombic distortions would accept only small

concentrations of Ti as substitutions for Al.

If we assume that lower symmetry exists from local site distortions to produce

a singlet ground state as depicted in Fig. 5.3, we can also treat the problem as a

conventional nondegenerate case whereby the matrix elements of interest become

the off-diagonal ones hkjL  S jni for use in the standard relation



k¤n

L  S

 E

; (5.5)

where jni refers to j0; C1=2i and j0; 1=2i and jki is any of the other eigenfunc-

tions in (5.2) with the jC1=2i and j1=2i spin functions added. Applying the

matrix elements from Appendix 5A, we obtain for the ground states of (5.5)

i .=2ı

1; 

 .=2ı

2; 

C i .=/

; (5.6)



0; 

i .=2ı

C .=2ı

i .=/

3; 

To calculate the Zeeman energy splitting of the Kramers doublet je

˙i we follow

the same procedure for the degenerate solution above to produce matrix elements

for the z direction



1 

4







2ı















;

DH

; (5.7)

D H

D 0:

By solving the secular equation and relating E D 2H

D g

,andthen

repeating the procedure for the x and y directions, the diagonal elements of the g

tensor can be expressed as

D 2 

8















2



;

D 2 

2









2





2



; (5.8)

D 2 

2









2





2



5.1 Quantum Paramagnetism of Single Ions 209

As listed in Table 5.1 for Rb alum, splitting energies for the measured g

D 1:895,

D 1:715,andg

D 1:767 are ı

D 1;050, ı

D 1;320,and D 17;000 cm

1

[3,4].

5.1.3 The Spin Hamiltonian for 3d

Ions

To complete this instructional example, it is important that we introduce albeit

brieﬂy the “spin” Hamiltonian that was developed by Pryce and Abragam as an

approximation to the nondegenerate model outlined in Sect. 5.1.2 [9]. This method

was used to analyze the paramagnetic resonance spectrum of multiple-d -electron

ions with an orbital singlet ground state, in particular the

F -state Cr

ion that

was the key to the early ruby maser and laser demonstrations. If the excited orbital

states of the crystal ﬁeld splittings are high enough to be ignored in following calcu-

lations, the approach differs from those of the two conventional methods in that the

perturbation matrix elements are computed for a combination of spin–orbit and Zee-

man interactions, hkjH

jniDhkjLS Cm

.L C 2S / H jni, with the result that

the spin S is treated as the only angular momentum quantum number and the con-

tribution of L becomes absorbed in the g factors, consistent with (5.8) but with less

precision. Since the je

i states are singlets that carry no orbital angular momentum,

i.e., S-state behavior, the only ﬁrst-order nonzero elements will be he

j2m

SHje

and the second-order correction will simplify to

.1/

D

n¤0

L  S C m

H  L

 E

: (5.9)

After collecting all ﬁrst- and second-order terms that are linear in m

H ,(5.9) can

be reduced to

D 2m



 



; (5.10)

where



n¤0

 E

: (5.11)

From (5.9) we see that the matrix elements used are only those linked to the ground

state, thereby substantially simplifying the labor involved. With this approximation

only the terms to ﬁrst order in  appear, and

D 2 

8



;

D 2 

2

; (5.12)

210 5 Anisotropy and Magnetoelastic Properties

D 2 

2

For Rb alum measured g factors are ﬁtted to splitting energies of ı

D 1;070,

D 1;310,and D 11;500 cm

1

, as listed in Table 5.1. Note that only the larger

cubic ﬁeld component  deviates signiﬁcantly from the measured reference value

of 20;300 cm

1

[10].

For ions with more complicated spin multiplets, i.e., S>1=2,suchasCr

and

, higher-order terms in  become important for spectral analysis and a more

general form of (5.10) can be expressed as [9]

.2/

D 2m



 



 



; (5.13)

leading to the general form of the spin Hamiltonian expressed to second order in  as

D m

g  H  S C DS

C E



 S





C S



; (5.14)

where the anisotropic g is now expressed in tensor format. The zero-ﬁeld splitting

parameters D and E (both functions of 

) can be determined from experiment and

can be used to deduce information about the electronic structure of the ions in their

crystal ﬁeld. Included in this version of the spin Hamiltonian is the fourth-order

term characterized by the a parameter which becomes signiﬁcant in S-state ions

(Fe

,Mn

)and4f

(Gd

,Eu

) for which the ﬁrst-order orbital angular

momentum contributions are absent. The subject of S-state ions will be reviewed

further in Sect. 5.2.5.

Equations (5.8)and(5.12) expose the role of spin–orbit coupling in determining

the magnitude and anisotropy of the magnetic moments in these transition-metal

ions. The ratios of the constant  to the various orbital level splittings can dictate

not only the degree of anisotropy of the magnetic moment as the magnetic ﬁeld is

applied to different directions, but also whether the moment increases or decreases

as in this case of Ti

ion. According to Table 5.2,the constants increase in mag-

nitude and reverse sign in the upper half of the 3d

series, thereby alerting us to the

fact that g factors greater than 2 as well as increased anisotropy is to be expected

from the ions with larger d -electron populations. Through the rest of the series, the

Aufbau principle is employed to show how the peculiarities of the orbital ground

states of the various ions are related to their magnetoelastic properties.

5.1.4 The Crystal-Field Hamiltonian for 4f

Ions

For the lanthanide series, analysis of isolated ion magnetic properties in solids fol-

lows the same perturbations approach as that for the iron group. The hierarchy of

perturbations is totally different, however. In the 4f

ions spin–orbit coupling ex-

ceeds that of the 3d

group by at least a factor of 3. Moreover, crystal ﬁeld effects