Dionne G.F. Magnetic Oxides

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5.2 Anisotropy of Single Ions 221

Fig. 5.10 Electronic structure of d

and d

conﬁgurations: one-electron and combined electron

models

S D 3=2 that provides three Kramers doublets. Its EPR spectrum has been studied

in paramagnetic salts [34]andalsoinAl

(ruby)[35] where it not only served

as the vehicle for the invention of the maser [36] by utilization of its comparatively

large trigonal D parameter that required a millimeter-wave frequency, but also pro-

duced the earliest laser utilizing the 10Dq



17;000 cm

1



red optical transition.

To analyze these singlet ground states, the expanded spin Hamiltonian of (5.14)is

employed, with emphasis on the second-order D and E terms. In octahedral sites,

neither Cr

nor its d

counterpart Ni

are expected to be active in a spin–lattice

sense because of the l

D 0 singlet ground state.

However, despite the absence

of J–T or S–O stabilization possibilities in the octahedral sites that they almost al-

ways occupy, the Ni

ions have exhibited an isotropic g  2:25 in MgO [37]. The

absence of anisotropy is consistent with the ﬁrst-order cubic crystal ﬁeld, and the

origin of the apparent larger spin–lattice contribution can be found in a considera-

tion of the relative magnitudes of  and Dq in the ﬁrst-order approximation given by

(5.12), g D 2 8=10Dq. If we apply  D335 cm

1

from Table 5.2 and assume

In nickel spinel ferrite, a small fraction of Ni

enters the tetrahedral sublattice, where a triplet

similartothatoftheCo

case is stabilized.

222 5 Anisotropy and Magnetoelastic Properties

Dq  1;000 cm

1

for divalent cations taken from optical absorption measurements

of Ni

in octahedral ﬁelds such as Ni.H

[38–41], we arrive at g values

2:26 without even taking into account the inﬂuence of covalent bonding on reduc-

ing orbital contributions. For Cr

with  D 87 cm

1

and Dq  1;700 cm

1

for

trivalent cations, this calculation yields g  1:96 in good agreement with the mea-

sured value. However, the EPR spectrum of this ion features large angular variations

because of the D and E terms.

5.2.5 3d

S-State Singlet

There remains to be discussed the d

conﬁguration exempliﬁed most commonly

by Fe

and Mn

, which is the least sensitive to its ligand environment while

providing the strongest superexchange interaction of all the transition-metal ions.

To explain this apparent paradox, we must begin by pointing out that although ﬁve

d electrons in the high-spin state produce an S D 5=2 total spin, the addition of

orbital angular momentum leaves an L D 0 state and a

5=2

free-ion ground term.

As a consequence, orbit–lattice effects are nonexistent in the pure ground state, but

the †b

=2U exchange term involves all ﬁve half-ﬁlled orbital states to produce a

large JS S energy. Small contributions to the anisotropy are introduced through the

excited free-ion terms that are discussed in Sect. 2.3 as part of the exposition of the

“strong-ﬁeld” approach to crystal ﬁeld theory. In the L D 0 case, any orbital angular

momentum inﬂuence must be derived from coupling between the ground

S term

and the excited terms from the Russell–Saunders coupling listed in Table 5.4.

As indicted by the sketches in Fig. 5.11,

P ,

F , and others are sequen-

tially laddered above the ground

A singlet, each with its own set of degeneracies.

The states that are sensitive to the cubic crystal ﬁeld parameter are shown with

limiting-case electron occupancies. These energy levels were computed by Tanabe

and Sugano using the Slater integrals and A, B,andC Racah parameters also de-

ﬁned in Sect. 2.3. The procedure employed is outlined in Appendix 5C.

In a cubic ﬁeld, the ﬁrst excited state with orbital angular momentum becomes the

from the

G term for both octahedral and tetrahedral sites. Orbital inﬂuence felt

in the ground state through spin–orbit coupling that links the two separate terms in-

stead of states within a J manifold, can be analyzed by the theoretical tools outlined

by Grifﬁth [2], with the result that a hybrid ground state '

is formed according to

Table 5.4 d

orbital energy terms subject to splitting in a cubic crystal ﬁeld

Russell–Saunders (free-ion) term Cubic crystal-ﬁeld states

5.2 Anisotropy of Single Ions 223

Fig. 5.11 Variation of d

energy terms as a function of cubic ﬁeld strength 10Dq, showing

proposed electron conﬁgurations



C "

; (5.17)

where " 

L  S

1g;n

 E

: (5.18)

where n corresponds to any of the three orbital states within the T

triplet (anal-

ogous to a P state with l

D 1; 0; 1). The approximations in (5.18)and(5.18)

represent only part of a more comprehensive computation of " for the spin–orbit in-

teraction with

5=2

excited state that was reported by Watanabe [42] who produced

the result for the g-factor reduction as

g D



1  "



C "

; (5.19)

where g

and g

are the g factors for the

5=2

and

5=2

states, and " D

2 .E

 E

/, with .E

 E

/ representing the denominator of (5.18)[43].

The

5=2

excited state is featured here as the logical mixing state because it has

224 5 Anisotropy and Magnetoelastic Properties

Fig. 5.12 Electronic structure of d

conﬁgurations: one-electron and combined electron models,

plus the more complete diagram showing the relation of the A

ground state and the

G term

the same J.D5=2/ and would therefore blend readily with

5=2

under a L  S

perturbation. As demonstrated by Watanabe, a complete treatment would require

a summation over the full ladder of spin states, rather than the single excited state

approximation given by (5.19) shown here to illustrate how the excited terms can

affect the orbital angular momentum of an S state in a crystal ﬁeld. The inﬂuence

of the cubic crystal ﬁeld can be seen in Fig. 5.12 where the inverse dependence of

E on Dq is illustrated. As a consequence, the degree of mixing of the excited state

into the ground singlet would be greater for an octahedral site than a tetrahedral site

because Dq

tet

D.4=9/Dq

oct:

This conclusion is supported by observations of the

behavior of magnetic anisotropy in the dilution of ferrimagnetic spinels and garnets

discussed in Sect. 4.3.

Paramagnetic anisotropy of single d

ions, therefore, is a higher order effect

than for the other members of the series. g-factors are, in effect, equal to the pure

spin value of 2. However, paramagnetic resonance measurements reveal anisotropic

effects in the angular dependence of the microwave spectra that are attributed to the

a parameter. These data have been interpreted by the spin Hamiltonian of the

5=2

state in a cubic crystal ﬁeld with lower symmetry components given by

5.2 Anisotropy of Single Ions 225

D m

g  H  S C



C S



S.SC 1/



C 3S  1









S.SC 1/



C E



 S



(5.20)

35S

 30S .S C 1/ S

C25S

 6S .S C 1/C3S

.S C1/

;

where the x, y,andz axes of the spin system are chosen here to coincide with those

of the crystal-ﬁeld coordinates.

In most cases, the cubic parameter a can be treated as negligible. D and E,

however, have inﬂuence where signiﬁcant lower symmetry ﬁelds are present. Such

a situation occurs when Fe

ions occupy the trigonal bipyramid sites (Fig. 2.7)in

magnetoplumbites, e.g., M-type Ba hexaferrite, which exhibit very large collective

anisotropies. In general, the a term of (5.20) causes the

S state to split into a

Kramers doublet, m

D˙5=2 and a quartet with m

D˙3=2 and ˙1=2 [44]. In

an axial ﬁeld the spin state group divides further into three Kramers doublets, m

˙5=2, ˙3=2,and˙1=2 [32, 33]. These Zeeman states are depicted in Fig. 5.13.

Experimental results and their analysis have been reported by a number of authors,

and a review of this work can be found in Bleaney and Stevens [45]andLow[13].

More relevant to the present discussion perhaps is the work of Folen on Fe

lithium spinel ferrite with Al

and Ga

dilutions [46].

Fig. 5.13 Higher-order energy level stabilizations of the

5=2

ground term of Fe





in cubic

and cubic+ axial crystal ﬁelds (based on discussions in Section 19 of Low [13]

226 5 Anisotropy and Magnetoelastic Properties

Throughout the ions of the transition series, competition between crystal ﬁelds

and spin–orbit coupling for control of the orbital angular momentum dictates the

crystal magnetocrystalline anisotropy and magnetostriction, which in turn deter-

mines the relative ease (or difﬁculty) in magnetizing a material and the degree to

which the magnetized state is retained once the magnetic ﬁeld is removed, i.e., the

hysteresis properties. A strong spin–orbit coupling will also provide a ready path

for electromagnetic energy to ﬂow into the lattice from the spin system when the

magnetic material becomes a microwave frequency transmission medium. We can

now begin to examine these phenomena in the context of these discussions.

5.2.6 4f

Ion Anisotropy

To illustrate the difference between the crystal-ﬁeld effects on the exposed 3d elec-

trons of the iron-group ions and the shielded 4f electrons of the lanthanide group,

we consider the energy-level diagrams of the respective three-electron cases in

Fig. 5.14. These examples represent idealized situations for typically Cr





and Nd





. Note that the exchange ﬁeld is indicated as acting after spin–

orbit coupling and possibly the crystal ﬁeld as well in the 4f

case because of the

reduced covalency, i.e., smaller b exchange integrals. An example of crystal-ﬁeld

splittings of the lowest multiplets for rare earths in chlorides is given in reference

[15]. Table 5.5 lists the prominent crystal ﬁeld parameters for operator equivalent

use in cubic, C

,andC

symmetries. It should also be commented that rare-

earth ions condense into sites of lower symmetry in many compounds, although

in magnetic oxides they are found in cubic coordinations such as in the garnets and

perovskite families.

The absence of quenching of the orbital angular momentum leaves spin–orbit

coupling as the dominant interaction in the 4f

group. For this reason, only a small

crystal ﬁeld stabilization energy is sufﬁcient to create large anisotropy of the g

factors in single ions. The strong coupling between the spin and the lattice that

results will be examined in Chap. 6 in connection with high-frequency properties.

Microwave spectroscopy has provided data on the g-factor anisotropy for the vari-

ous ions of the rare-earth series and some of these are listed in Table 5.6 as compiled

by Elliott and Stevens [47]. For more details of individual ion behavior through the

series, the reader is directed to reviews by Orton [48], Low [13], and Bowers and

Owen [49].

Among the 4f

group we have seen in Chap. 4 how the heavy rare earths

(n D 8–13 with J DjL C Sj) participate in the extraordinary magnetization

properties of the iron garnets. These ions also strongly inﬂuence the anisotropy and

magnetostriction behavior when exchange ﬁelds are signiﬁcant. The light rare earths

(n D 1–6, with J DjL–S j) do not offer strong magnetism because of their lower J

values, but certain paramagnetic and optical properties have proven to be important

in laser and magneto-optical applications.

5.2 Anisotropy of Single Ions 227

Fig. 5.14 Ground-term energy-levels: (a) the 3d

conﬁguration, typical of the Cr

F-state ion

in point-charge cubic crystal ﬁeld with an axial distortion characterized by the parameter D. The

Zeeman splitting energy h D gm

H for S D 1 can occur at three values of H (not indicated).

In this case g  2 and is reduced only slightly by the multiplet interaction because the ground term

after the crystal ﬁeld splitting is an orbital singlet (

A with L D 0); and (b) the 4f

conﬁguration,

typical of the light rare-earth ion Nd

I-state ion. For this series, J DjL  S j in the lower

half (light rare earths) and jL CSj in the upper half (heavy rare earths). In the manifold of energy

levels induced by a magnetic ﬁeld, the signs of J

are reversed between the light and heavy rare

earth ions

Table 5.5 Crystal ﬁeld symmetries and operator equivalent terms

common to 4f

ions in solids

Compound Site symmetry V

terms

CaF

,ThO

Cubic O

; O

Double Nitrate C

; O

Ethyl Sulfate; Chloride C

; O



6



Compiled by Taylor and Darby [14]

228 5 Anisotropy and Magnetoelastic Properties

Table 5.6 Rare-earth ion anisotropy parameter data for ethyl sulfate hosts

4f Ion n g

i A

i Ground

state

–cm

1

––10

4

1

4

1

4

1

4

1

1 640 0.995 2.185 15 40 –92 1,150 One/Two

3.725 0.20 Kramers

doublets

2 800 1.525 50 100 48 660 Split

doublet

3 900 3.535 2.072 15 35 60 640 Kramers

doublet

4 (1,070)

5 1,200 0.596 0.604 0 30 54 590 Kramers

doublet

6 1,410

7 1,540 1.991 1.991

1.991 1.991

8 1,770 17.72 <0:3 37 20 20 220 Split

doublet

9 1,860

10 2,000 15.36 0 18 22 170

11 2,350 1.47 8.85 0 –40 –30 330 Kramers

doublet

12 2,660

13 2,940

Data compiled by Elliott and Stevens [47]

The single-ion anisotropy of the 3d

and 4f

ions seen in paramagnetic

systems can also have a strong inﬂuence on spontaneous magnetism to produce

magnetocrystalline anisotropy that is responsible for the directional magnetic capa-

bilities that are essential to many applications.

5.3 Magnetocrystalline Anisotropy and Magnetostriction

The foregoing discussions have focused on magnetization established sponta-

neously through the medium of covalent bonding by the mechanism called magnetic

exchange. As the molecular-orbital models showed, the exchange creates an en-

ergy stabilization of ionic spins in parallel or antiparallel ordering that we initially

assumed to be insensitive to the symmetry of the crystal lattice. When a ferro- or fer-

rimagnetically ordered system is placed in an external magnetic ﬁeld, the anisotropic

tendencies created by the interaction of magnetic ions (collectively or as individuals)

with the lattice are acquired by the resultant magnetic moment to create preferred

directions of magnetization called “easy” axes. The corresponding magnetically

“hard” directions alternate with the easy directions in any crystallographic plane of

5.3 Magnetocrystalline Anisotropy and Magnetostriction 229

the lattice. Magnetocrystalline anisotropy energy is the magnetostatic work neces-

sary to rotate the magnetic moment from an easy to hard direction, and the ﬁeld

required is the anisotropy ﬁeld. Magnetostriction is the corresponding reaction by

the lattice in the form of an elastic strain that occurs along any axis, but is usually

characterized by the deformation along the magnetic ﬁeld direction. Although both

of these effects are manifestations of a magnetoelastic interaction, the relation be-

tween them can vary widely, depending on the origin of the crystalline source of the

anisotropy and the strength of the spin–orbit coupling of the magnetic ion.

5.3.1 Phenomenological Anisotropy Theory

The beginnings of magnetocrystalline anisotropy theory were phenomenological,

resulting from interpretation of measurements and based on symmetry arguments.

This early history will not be reviewed here. However, the fundamental relations

and parameter deﬁnitions are essential elements of the dialogue. For background

and derivations the reader is directed to accounts found in standard magnetism texts

by Chikazumi [50], Smit and Wijn [51], Bozorth [52], and Morrish [53].

Referring to the coordinate-axe/M-vector sketches of Fig. 5.15a for the case of

uniaxial anisotropy, we can express the magnetocrystalline anisotropy energy as a

series expansion according to

D K

sin

 C K

sin

 C; (5.21)

where K

and K

are the ﬁrst- and second-order uniaxial anisotropy constants, and

 is the polar angle made by M with the z-axis. Here the situation is straightforward

Fig. 5.15 Vector diagrams of the magnetocrystalline anisotropy (a) the uniaxial case, and (b) the

cubic case, with rotation conﬁned to a f110g plane, i.e.,  D 45

230 5 Anisotropy and Magnetoelastic Properties

because the analysis is two-dimensional, involving only the polar angle  between

M and the axis of symmetry. The cubic symmetry case is more complex because of

the anisotropy in the azimuthal plane. In polar coordinates (depicted for rotation of

M in a

110

plane in Fig.5.15b), in-plane variation of E

is characterized by the

azimuthal angle , according to

D K



sin

 sin

' cos

 C sin

 cos





sin 4 cos

 sin

' cos

'; (5.22)

where K

and K

are the respective ﬁrst- and second-order cubic anisotropy

constants.

A static method for determining the anisotropy constants in a particular plane

is to measure the mechanical torque imposed on the specimen when it is rotated

from an equilibrium position where M is initially aligned with H along an easy

axis. After rotation of the specimen easy axis through angle ı away from H , M

assumes an equilibrium angle  with respect to the easy axis. The new equilibrium

is established when the anisotropy torque balances the torque from the magneto-

static energy E

DHM cos .ı–/. The offsetting torques as function of  are

determined by differentiation, according to



./ D

./

@

DK

sin 2 4K

sin

 cos ; (5.23)

and



./ D

./

@

DHM sin .ı  /: (5.24)

If the specimen is magnetically saturated,  D ı and measurement of 

as a

function of  will provide the means for measurement of 

as a function of K

and K

For the cubic case



./ K



2 sin  cos

  sin

 cos 



 K



cos 4 cos

 

sin

 sin 2



(

5.25)

In these models, H sets the azimuthal plane . D 0/ and M and H share the

same value of .

If the specimen is constrained from rotating, the effect of magnetization rotation

away from an easy direction as the magnetic ﬁeld is increased and can be observed

by studying the change in the M component along the direction of H . The equi-

librium positions of the M vector at each value of H is given by the value of 

for which the torque 

is balanced by 

as obtained from (5.23, 5.24,and5.25),

according to

@.

C 

@

D2K

cos 2  K



4 sin

 C 12 sin

 cos





(5.26)

H

M cos .ı  / D 0;