3

Chapter 1

Magnetism of the Rare-Earth Ions in Crystals

4

1. Magnetism of the Rare-Earth ions in Crystals

In this chapter we present the basic facts regarding the physics of magnetic phenomena of rare-earth ions in crystals. We discuss the theory of quantized angular momentum and the theory of the crystal field (CF) splitting of the electronic energy levels of the rare-earth ions in these crystals as required to give a better understanding for the material discussed in detail in later chapters in this book. Additional data and analyses can be found in corresponding references, which expand on topics that are discussed in each of the chapters in this book.

1.1. Electronic Structure and Energy Spectra of the “Free” Rare-Earth Ions

In rare earth (RE) compounds, the lanthanide ions (from Ce to Yb) are usually found in the trivalent state RE3+. The ground electronic configuration of the RE ions may be written as [Xe]4f n, where [Xe] = 1s22s22p63s23p63d104s24p64d10

5s2p6 is the closed-shell configuration of the noble gas xenon, and n is the number of electrons in the unfilled 4f n shell, ranging from n = 1 for Ce3+ to n = 13 for Yb3+. The characteristic magnetic moment for each RE ion leads to an interac-tion between that ion and an applied external magnetic field H, producing interesting magnetic and magnetooptical fea-tures in the RE compounds.

Over the years, methods of numerical analysis have been developed to calculate the energy states of “free” RE ions (that is, for ions that are not in a ligand or crystal-field environment). These methods allow evaluation of the multiplic-ities of states and the energy-level positions of excited electronic configurations relative to the ground state. Energy intervals for 4f n, 4f n-15d, 4f n-16s, and other excited electronic configurations for free RE ions are presented in Refs. [1,2]. Also presented are energy level schemes for the energy levels of the 4f n configuration for all trivalent RE ions. From these results it follows that the excited RE ion configurations, such as 4f n-15d, 4f n-16s, are separated from the 4f n ground state by an energy interval typically on the order of 105 cm–1.

The repulsion between the equivalent 4f electrons within the shell, usually called the correlation Coulomb interaction of the RE ion, splits the states into terms characterized by the orbital (L) and spin (S) momenta. A term with fixed val-ues of L and S has (2L + 1)(2S + 1) degenerate states distinguished by the mL and mS projections of the orbital and spin momenta. The wave functions of these degenerate states are given by |LSmLmS>, where –L mL L, –S mS S and the index distinguishes between terms with the same L and S. Neighboring terms are separated from each other by an energy interval on the order of 104 cm–1 [3-5].

To determine the ground term of the 4f n electronic configuration, one usually applies Hund’s rules and the Pauli ex-clusion principle [4,6]. Hund’s first two rules assert:

1) For a given electronic configuration the term (i.e., a quantum state with fixed L and S) with maximum multiplicity (i.e., with maximum S) has the lowest energy.

2) For a given multiplicity (i.e., S = Smax), the term with the largest value of L has the lowest energy. For instance, let us apply Hund’s rules for the determination of the ground term of the rare-earth Tb3+ ion that has

eight electrons in the unfilled 4f 8 electronic configuration. In this regard, we can construct Table 1.1 for the orbital (ml) and spin (ms) momentum projections of the eight f electrons. Hund’s first rule indicates that the first seven electrons will fill states of the same spin momentum (ms); Hund’s second rule indicates that the eighth electron will fill one of the re-maining (opposite-spin) states having the largest angular momentum (ml). This is shown in Table 1.1. In spectroscopic language, the ion Tb3+ has a ground term of 7F, where the conventional notation is (2S + 1)L with (2S + 1) being the multiplicity.

The quantum degeneracy of the RE ion terms is removed by the spin-orbit interaction WSO that has a value on the or-der of 103 cm–1. The effective Hamiltonian of the spin-orbit interaction that describes the splitting of the term with fixed values of L and S has the form,

LSH L S

, (1.1)

Table 1.1. Arrangement of the orbital (ml) and spin (ms) momentum projections in the 4f 8 electronic configuration of Tb3+. The max-imum number of f electrons in the shell is N = 14.

ml ms

–3 –2 –1 0 +1 +2 +3

5

where L

and S

are the operators of the orbital and spin momenta, respectively, and is the spin-orbit coupling con-stant defined by the well-known Goudsmit formula [5]. From a qualitative point of view, the spin-orbit interaction WSO corresponds to the magnetic interaction between the spin magnetic momentum and the magnetic field caused by the motion of the 4f electron around the nucleus. In the one-electron approximation, WSO can be written as [7],

2 2

2 2 32SO

e l sW

m c r

, (1.2)

where is the Plank constant, m and e are the mass and charge of the electron, c is the speed of light, r is the radius of electron orbit, and l and s are the orbital and spin moments of the electron, respectively.

The spin-orbit interaction splits the (2S + 1)L-terms into multiplets characterized by the total angular momentum J (with |L – S| J L + S) whose wavefunctions are spherical functions expressed in terms of |J,MJ> [8]. Each multiplet is many-fold degenerate in terms of its angular momentum projection MJ; this degeneracy can be removed by an external action (relative to the RE ion), such as crystalline electric or magnetic fields. In the case of an applied external magnetic field, Н, a complete lifting of the degeneracy takes place, with the (2S + 1)LJ multiplet split into (2J + 1) equidistant sub-levels. The energy interval between sublevels is defined by the magnetic field intensity and by the value of the g-factor. We write the Hamiltonian (1.1) in the form,

2 2 21

2LSH J L S

. (1.3)

For the diagonal matrix elements that define the multiplet energy E we obtain,

11 1 1

2E J J J L L S S . (1.4)

From this expression, the Lande’ “rule of intervals” is determined as,

11 1 1

2E J E J J J J J J , (1.5)

which gives the difference in energy between neighboring multiplets having the same L and S [3]. The nomenclature of the multiplets in a term depends on the sign of the spin-orbit coupling constant . For the

“heavy” RE ions (from Tb3+ to Yb3+) with the 4f shell more than half filled < 0, so the lowest-energy multiplet will have the largest possible value of J for a given L and S, that is, J0 = L0 + S0. For the “light” RE ions (from Ce3+ to Gd3+) with less than a half-filled 4f shell, > 0. In this case, the lowest-energy multiplet will have the smallest possible value of J for a given L and S, that is, J0 = |L0 S0|. This is known as Hund’s third rule, which asserts that:

3) For the less-than-half filled shell, the state with the smallest allowed value of J is the lowest energy state; for the more-than-half filled shell, the state with the largest allowed value of J is the lowest energy state.

A general scheme for the energy spectrum for the 4f n configuration of the free rare earth ion based on these energy terms is presented in Figure 1.1. The ground term for each trivalent RE ion and the ground and first excited multiplets associated with each of these terms in the 4f n configuration are presented in Table 1.2.

cm–1

cm–1

Figure 1.1. Splitting scheme for the energy levels of the free rare-earth ions [3].

6

Table 1.2. Ground and first excited energy levels of the free RE ions.

RE3+ Ground electronic configuration Ground term Ground multiplet First excited multiplet E1 – E0 (cm–1)

Ce 4f 1 2F 2F5/2 2F7/2 2200

Pr 4f 2 3H 3H4 3H5 2200

Nd 4f 3 4I 4I9/2 4I11/2 1800

Pm 4f 4 5I 5I4 5I5 1600

Sm 4f 5 6H 6H5/2 6H7/2 1000

Eu 4f 6 7F 7F0 7F1 350

Gd 4f 7 8S 8S7/2 – – Tb 4f 8 7F 7F6

7F5 2300 Dy 4f 9 6H 6H15/2

6H13/2 3400 Ho 4f 10 5I 5I8

5I7 5000 Er 4f 11 4I 4I15/2

4I13/2 6400 Tm 4f 12 3H 3H6

3H5 8200 Yb 4f 13 2F 2F7/2

2F5/2 10100

The classification of free RE ion states is based on the Russell-Saunders approximation (also called normal- or LS-

coupling), which requires that the energy separation between the terms be much greater than the value of the term-splitting into multiplets by the spin-orbit interaction. For the ground term of the 4f n configuration, this approximation is generally valid. However, significant deviation from LS-coupling is observed for excited RE ion states [3,4]. Neverthe-less, LS-coupling is still a sufficiently good approximation to calculate the energy spectra and the classification of states for both the ground 4f n configuration and the lower states of the first excited 4f n-15d and 4f n-16s configurations of the free RE ions. As shown by the results from direct calculations (see, for example, Refs. [9,10]), the energy value of the d electron interaction with the f electrons of the 4f n-1 “core” of the 4f n-15d configuration is on the order of 104 cm–1, while the value of the spin-orbit interaction for the d electron is on the order of 103 cm–1. The lower states of the 4f n-15d (or 4f n-16s) configuration of the free RE ion can be described in the LS-coupling approximation as vector sums of the quantum numbers L and S that characterize the ground state of the 4f n-1 “core” with the quantum numbers l and s of the “va-lence” 5d (or 6s) electron [4,5,7].

1.2. Paramagnetism of the “Free” RE Ions

Let us consider the interaction between the “free” RE ion (as defined in Sec. 1.1) with an external magnetic field H. An interaction Hamiltonian of the ion with an external magnetic field (denoted the Zeeman Hamiltonian) is usually written as,

2Z B BH L S H J S H

. (1.6)

If the field H is directed along the z-axis of the coordinate system, the Hamiltonian can be written in the following form,

Z B Z ZH J S H

. (1.7)

Based on the state wave functions, |LSJMJ>, that are distinct for each state, we can write the matrix elements for this Hamiltonian as [3],

J Z J J B JLSJM H LSJM g M H

, (1.8)

where

1 1 1

12 1J

J J S S L Lg

J J

is the Lande’ factor for the multiplet of the RE ion,

and

1/22 21 1J Z J J B JLS J M H LSJM g H J M

(1.9)

where

1/2

2

2 1 1

4 1 2 1 2 3J

J L S L J S J L S J L Sg

J J J

.

Let us consider now the behavior of an ensemble of free RE ions in an external magnetic field. In this case, the mag-netic field tends to orient the magnetic moments M

of the ions, whereas the thermal motion tends to disorient them. As

M

is spatially quantized, the energy of the interaction between the magnetic moment M

of the RE ion with the

7

magnetic field must also be quantized. This additional interaction energy WH is given by the expression,

H J J BW g M H . (1.10)

As the projection MJ of the angular momentum J

takes on (2J + 1) values, with –J MJ J, WH splits the (initially degenerate) energy level (multiplet) of the ion into sublevels located above and below the unperturbed energy level. These sublevels are equally spaced in the energy spectrum, with the energy separation between the magnetic (or Zeeman) sublevels equal to H J Bg H .

The magnetization of N ion ensembles, each having similar equidistant structure in the energy spectrum, is isotropic and can be represented as [3,11],

0 JI I B x , (1.11)

where JB x is the Brillouin function [11],

2 1 2 1 1 1coth coth

2 2 2 2J

J JB x x x

J J J J

, (1.12)

and J Bg JHx

kT

. In eqn. (1.11), I0 = J BNg J is the saturation magnetization of N ion ensembles at T = 0 K (for ex-

ample, for the RE ion Gd3+, I0 = 7B). Keeping only the first terms in the series expansions of the coth(x) terms in eqn. (1.12) allows us to simplify eqn.

(1.11) for high T and a low H (i.e., x << 1) as,

2 21

3J BNJ J g

I HkT

, (1.13)

where k is the Boltzmann constant. I corresponds to the expression for the paramagnetic susceptibility , which can be written as,

C

T , (1.14)

where C is the Curie constant1 2 2 21

3 3J B JNJ J g NM

Ck k

, and MJ is the atomic magnetic momentum. Expression

(1.14) is called the Curie law, and this expression is used in this text for several compounds in which the RE ions can be considered as free ions. For the most part, rare-earth compounds obey the Curie-Weiss law [3,11,12],

p

C

T

, (1.15)

where p is the Weiss constant, often called the paramagnetic or Curie temperature [10], which takes into account both magnetic and electric interactions between magnetic ions in paramagnets.

In addition to the orientation-dependent paramagnetism just described, we also encounter Van Vleck paramagnetism, caused by the mixing of wave functions of closely spaced electronic states of certain RE ions by an external magnetic field H. This situation is especially common in compounds containing the RE ions Eu3+ and Sm3+ [3,12].

1.3. Energy Spectra and Wavefunctions of Kramers and Non-Kramers RE Ions in the Paramagnetic Garnets and Orthoaluminates

The crystalline environment plays an important role in the formation of the electronic properties of the rare earth (RE) ions in crystals. In this section, we will consider in particular the properties of the paramagnetic garnets (gallates and

1If the number of ions N equals Avogadro's number NA, then we can write [11]:

2 221 1

13 8

A J BJ

N J J gC g J J

k

and the magnetic susceptibility

m (calculated on the basis of one mole) is: 3 211

8m Jcm g J JT

. We can also write the specific magnetic susceptibility (calculated on the basis

of one gram) as: (cm3/gram) = m

M

, where M is the molecular weight. In addition, the magnetic susceptibility V calculated based on a unit of

volume (cm–3) is: mV M

, where is the density (in gram/cm3). In this way we see that V is a dimensionless quantity! Therefore, the

magnetization MV that is calculated on the basis of a unit of volume (cm-3) has identical dimensionality to the external magnetic field H. But in this case, the magnetization MV can be expressed in units of Gauss (gs), whereas the dimensionality of the external magnetic field is in units of Oersted (Oe).

8

aluminates) and orthoaluminates. For RE ions having an odd number of 4f electrons (e.g., Sm3+(4f 5) and Dy3+(4f 9)), the low crystallographic site symmetry of the RE3+ ion in the garnets (D2) and orthoaluminates (Cs) splits each multiplet of the 4f n electronic configuration into (J + 1/2) discrete Kramers’ doublets. By contrast, for RE ions having an even number of 4f electrons (e.g., Tb3+(4f 8),and Ho3+(4f 10)), the crystal field completely removes the degeneracy of the en-ergy (Stark) sublevels, resulting in (2J + 1) singlets for each multiplet.

Due to the “shielding” of the 4f valence electrons from the environment by the filled 5s and 5p shells (see §1.1), the crystal field (CF) interaction is substantially smaller than the spin-orbit coupling (WCF << WSO). This allows the CF to be considered as a weak perturbation acting on a free atomic or ionic energy level, with the multiplet structure (due to the interaction between the orbital and spin momenta of the 4f electron system) remaining nearly constant. However, the be- havior of the RE ions will be quite different in the presence of the crystal field, depending on whether there is an even or an odd number of electrons in the unfilled (4f n) electronic shell. The differences in the observed spectra are explained by the Kramers theorem [3,13]. That is, for ions with an odd number of electrons in the 4f electronic shell, resulting in a half-integral total spin, the orbital degeneracy is removed completely by the low-symmetry CF because the crystal field directly influences the orbital motion of the electron. The spin degeneracy is reduced by the “pairing” of electrons with oppositely oriented spins, i.e., the RE ion goes from a high-spin state to a low-spin state. However, there remains one extra “unpaired” electron with spin in a degenerate state. Such degeneracy cannot be removed by either crystalline or orbital interaction, but can be removed by an external magnetic field (or an exchange field). By contrast, the spin degen- eracy can be removed completely for ions with an even number of electrons, as all of the electrons may be paired up. Therefore, in a low-symmetry CF the energy levels of the Kramers ions are split into doubly degenerate states (Kramers doublets), while the energy levels of non-Kramers ions are split into non-degenerate or singlet states.

Thus, the symmetry of the crystal field (CF) acting on the RE ion will define both the energy and symmetry charac-teristics of the electronic states. As examples, we will consider RE ions in sites with low symmetry in orthoaluminates (Cs) and garnets (D2), and discuss briefly the effect of these low-symmetry environments on the form of the wave func- tions and the character of the RE ion energy spectra.

RE ions in a CF of Cs symmetry For orthoaluminates, orthochromites, and orthoferrites, the RE ions lie in sites sur-rounded by a distorted perovskite structure having Cs point-group symmetry. The orthoaluminates, usually written as RAlO3, where R3+ is the trivalent rare-earth ion, have a crystalline structure that is described by the spatial group D16

2h - Pbnm [14,15]). An elementary cell consists of 4 formula units of RAlO3, with the 4 RE ions located in two crystal-lographically non-equivalent Cs sites. The similar site characteristics of the RE ions in the rare-earth orthoaluminate structure, their small radius and the crystallographic distortion from the ideal perovskite structure, lead to an environ- ment of nearest-neighbor oxygen atoms surrounding the RE ions in such a way that the only element of symmetry is a reflection h in the “ab” plane perpendicular to the monoclinic crystalline c-axis. Consequently, the Cs group possesses only two one-dimensional irreducible representations, А and В, (also known as Γ1 and Γ2, respectively [22]) satisfying the following group theory multiplication rules: А В = В А = В; А А = В В = А. The Cartesian components of an arbitrary polar vector A

are transformed with the following irreducible representations (see also Ref. [3]): Az → В;

Ay, Ax → А. A scheme for the two crystallographic non-equivalent positions of the RE ion in the RAlO3 structure is given in Figure 1.2; these positions are distinguished by the orientation of the low-symmetry crystalline environment (Сs symmetry). Consequently, anisotropy axes appear in the structure. As a result, the crystal field is responsible for the orientation of the magnetic moments in RAlO3. The magnetic moments of the RE ions (e.g., Tb3+ or Dy3+) in RAlO3 are set in the “ab” plane of the crystal at corresponding angles 0 to the “a” or “b” axes (i.e., along the so called “Ising” axes, or the magnetic anisotropy axes)2.

For non-Kramers ions in a crystal field of Сs symmetry, each multiplet is split into (2J + 1) singlets. The wavefunc-tions for each singlet transform according to the irreducible representation (irrep) A or B, with the function |A> being invariant and the function |B> changing its sign upon reflection h in the symmetry plane. Each Stark singlet can be characterized by the irrep corresponding to the transformation properties of its wavefunction. In some cases, a quasi-doublet structure observed in the spectra is formed by two closely spaced singlet levels (the value of the energy “gap” generally being between 1 and 3 cm–1 [3,14]).

The optical and magnetic properties of TbAlO3 have recently been examined [15]. The authors used crystal-field modeling techniques to assign all 58 experimentally determined Stark levels within the 7FJ (J = 6, 5, … 1, 0) and 5D4 multiplet manifolds, with a fitting standard deviation of 4.5 cm–1 (3.8 cm–1 rms error). As a further test, the theoretical Stark levels and calculated wavefunctions were used to determine the temperature dependence of the magnetic suscepti-bility of the TbAlO3 crystal. Good agreement was obtained between the calculated susceptibility and the previously re- 2The forces responsible for the ordering of the RE ions in the RAlO3 structure and defining the type of the magnetic (or spin) configuration of the RE ions at low temperatures are relatively small (the Néel temperature ТN being on the order of several K) and have an exchange (dipole) character [3].

9

a

c

b

Figure 1.2. Schematic diagram of the two crystallographic non-equivalent positions of the RE3+ ion in the RAlO3 structure, which differ by both the orientation of the crystalline environment and the orientation of the magnetic anisotropy axes. ported temperature-dependent magnetic data, confirming the predicted ordering of the states within the 7F6 multiplet.

Calculated eigenvectors (wavefunctions) for the 13 Stark components of the ground multiplet manifold 7F6 of non-Kramers Tb3+ ion in the CF symmetry Cs are given in Table 1.3 (see also Ref. [16]). The left-hand set of wavefunctions in this table are calculated in c-axis notation (i.e. in the crystallographic coordinate system) using the crystal-field parameters from Table IV of Ref. [16]. The calculated eigenvectors (wavefunctions) for the non-degenerate Stark singlets are real. That is, they satisfy the reality condition, | * |i i , and transform according to the irreps A and B of the Cs group, taking into account the transformation properties of the non-spin |J,MJ> states:

| |i h iA A and | |i h iB B . The wavefunctions given in the right-hand side of Table 1.3 were calculated in the Ising coordinate system of Tb3+ in the CF of Cs symmetry (see §1.5). Ising notation wavefunctions have been used for the TbAlO3 magnetic susceptibility calculations presented in §1.5.

Now let us consider the RE ions having an odd number of 4f electrons. The Kramers-ion energy spectrum consists of (J + 1/2) Kramers doublets, each doublet described by the Kramers-conjugate wavefunctions,

1/2

0

| | 2J

pp

C M J p

1/2

*

0

| | 2J

pp

C M p J

. (1.16)

The coefficients Cp are defined by the values of the CF parameters (each doublet has its own set of Cp coefficients). As an example of a Kramers-ion system, we consider the recent examination of Er3+ ions in Er3+:YAlO3 [17]. The au-

thors modeled 134 experimentally observed Stark levels, split out from the 30 multiplets with energies below 44000 cm−1, using a parametrized Hamiltonian defined to operate within the 4f 11 electronic configuration for Er3+ ions doped into Cs sites of Er3+:YAlO3. For convenience of discussion, the total Hamiltonian can be partitioned as,

A CFH H H

(1.17)

where AH

is the “atomic” Hamiltonian defined to include all spherically symmetric interactions. The free-ion (or “atomic”) Hamiltonian is characterized by a set of three electron repulsion parameters (F2, F4, F6), the spin-orbit cou-pling constant 4 f , the Trees configuration interaction parameters , , , the three-body configuration parameters (T 2, T 3, T 4, T 6, T 7, T 8) and parameters that describe magnetic interactions (M 0, M 2, M 4, P2, P4, P6). Parameter Eave takes into account the kinetic energy of the electrons and their interactions with the nucleus. Treated as a parameter, it shifts the barycenter of the whole 4f n configuration. As a result, one can write AH

as,

2 7 41k i k jA ave k i f so k j

k i k j

H E F f L L G G G R T t A P p M m

, (1.18)

where k = 2,4,6; i = 2,3,4,6,7,8; j = 0,2,4; fk and Aso represent the angular part of the electrostatic and spin-orbital interact-tions, respectively; L is the total orbital momentum; G(G2) and G(R7) are the Casimir operators for the Lie groups G2 and R7; the ti are the three-particle operators; and pk and mj represent the operators for the magnetic interactions.

The CFH

denotes the non-spherically-symmetrical part of the one-electron crystal field. The crystal-field Hamilto-nian may be expressed (in Wybourne notation [4]) as,

10

,

kkCF q q

k q

H B C

(1.19)

where k = 2,4,6 and |q| k is restricted by the Cs site symmetry to be even. Radially dependent parts of the one-electron crystal-field interactions are contained in the k

qB parameters, and the kqC are many-electron spherical tensor operators

acting within the 4f n electronic configuration. For lanthanide ions occupying Cs symmetry sites, group-theoretical con-siderations allow for 15 crystal field parameters: three pure-real 0

kB (k = 2,4,6) parameters plus six complex k kq qB iS

parameters (k = 2,4,6), with q = 2,4,6 and |q| k. However, the x-axis of the crystal-field quantization may be arbitrarily chosen within the crystallographic xy-plane, allowing one of the 15 crystal-field parameters to be arbitrarily fixed, leav-ing 14 independent crystal-field parameters [18,19]. Standard convention selects the imaginary part of the rank-2 pa-rameter, 2

2S , to be set equal to zero.

Table 1.3. Wavefunctions in the |J, MJ basis of the 7F6 ground multiplet Stark sublevels in TbAlO3, from Ref. [16].

Energy (сm–1) CS Irrep Wavefunctions in c-axis notation Real wavefunctions in Ising system notation

4.3 A

1 = e30.8i [0.7567|6,0> + 0.4262(e9.5i |6,+2 + e9.5i |6,2) + 0.0890(e55.9i |6,+4 + e55.9i |6,4)]

1 = + 0.6724(|6,+6 + |6,6) + 0.1412(|6,+4 + |6,4)

4.5 B 2 = e85.1i [0.6525(e86.5i |6,+1 e86.5i |6,1)

+ 0.2197(e67.8i |6,+3 e67.8i |6,3)] 2 = + 0.6734(|6,+6 |6,6)

+ 0.1378(|6,+4 |6,4)

168 B 3 = e18.2i [0.6327(e6.4i |6,+1 e6.4i |6,1)

+ 0.2557(e2.7i |6,+3 e2.7i |6,3) + 0.0803(e87.9i |6,+5 e87.9i |6,5)]

3 = 0.5330(|6,+5 + |6,5) 0.3347(|6,+3 + |6,3) 0.2608(|6,+1 + |6,1)

185 A 4 = e20.1i [ 0.6471(e89.7i |6,+2 + e89.7i |6,2)

0.1909(e72.4i |6,+4 + e72.4i |6,4) + 0.1540|6,0>]

4 = 0.6241(|6,+5 |6,5) 0.2371(|6,+3 |6,3) 0.1426(|6,+2 + |6,2)

211 A 5 = e34.8i [ 0.5797|6,0>

+ 0.5333(e3.6i |6,+2 + e3.6i |6,2) + 0.1360(e51.5i |6,+4 + e51.5i |6,4)]

5 = + 0.4639(|6,+2 + |6,2) + 0.3315(|6,+4 + |6,4) + 0.4659 |6,0 0.1540(|6,+5 |6,5) 0.1006(|6,+6 + |6,6)

253 B 6 = e22.6i [ 0.6357(e22.6i |6,+3 e22.6i |6,3)

+ 0.2442(e2.2i |6,+1 e2.2i |6,1) 0.1041(e5.5i |6,+5 e5.5i |6,5)]

6 = 0.4753(|6,+1 + |6,1) + 0.4072(|6,+5 + |6,5) 0.2592(|6,+3 + |6,3) 0.1001(|6,+2 |6,2)

296 B 7 = e59.3i [0.6265(e70.0i |6,+3 e70.0i |6,3)

0.2134(e69.6i |6,+1 e69.6i |6,1) + 0.1860(e89.7i |6,+5 e89.7i |6,5)]

7 = + 0.6156(|6,+4 |6,4) + 0.2494(|6,+2 |6,2) 0.1225(|6,+1 + |6,1) 0.1264(|6,+6 |6,6)

349 A 8 = e48.3i [ 0.6712(e13.1i |6,+4 + e13.1i |6,4)

0.1334(e36.1i |6,+6 + e36.1i |6,6) + 0.0771(e13.0i |6,+2 + e13.0i |6,2)]

8 = + 0.5584(|6,+4 + |6,4) 0.4451 |6,0 0.1909(|6,+2 + |6,2) 0.1099(|6,+6 + |6,6)

366 A

9 = e42.3i [0.6285(e77.8i |6,+4 + e77.8i |6,4) 0.1902(e73.5i |6,+2 + e73.5i |6,2) 0.1866(e56.4i |6,+6 + e56.4i |6,6) + 0.1278|6,0>]

9 = + 0.5816(|6,+3 |6,3) 0.2275(|6,+5 |6,5) + 0.2208(|6,+1 |6,1) + 0.1233(|6,+4 + |6,4) 0.1787 |6,0

433 B 10 = e39.0i [ 0.6790(e7.2i |6,+5 e7.2i |6,5)

+ 0.0991(e1.0i |6,+3 e1.0i |6,3)]

10 = 0.4506(|6,+3 + |6,3) + 0.3940(|6,+1 + |6,1) 0.2533(|6,+2 |6,2) + 0.1834(|6,+4 |6,4) + 0.1348(|6,+5 + |6,5)

443 B 11 = e50.8i [0.6584(e82.4i |6,+5 e82.4i |6,5)

+ 0.1604(e82.7i |6,+3 e82.7i |6,3) 0.1231(e37.4i |6,+1 e37.4i |6,1)]

11 = + 0.5743(|6,+2 |6,2) 0.2990(|6,+3 + |6,3) 0.1973(|6,+4 |6,4) + 0.1051(|6,+1 + |6,1)

514 A 12 = e13.9i [0.6725(e32.5i |6,+6 + e32.5i |6,6)

0.1238(e15.4i |6,+4 + e15.4i |6,4) + 0.0801(e12.4i |6,+2 + e12.4i |6,2)]

12 = 0.4952 |6,0 + 0.4923(|6,+1 |6,1) 0.2624(|6,+3 |6,3) + 0.1905(|6,+2 + |6,2)

519 A 13 = e80.4i [ 0.6602(e57.5i |6,+6 + e57.5i |6,-6)

0.1556(e76.0i |6,+4 + e76.0i |6,-4) + 0.1180(e74.0i |6,+2 + e74.0i |6,-2)]

13 = + 0.5042 |6,0 + 0.4108(|6,+1 |6,1) 0.4030(|6,+2 + |6,2) + 0.1190(|6,+4 + |6,4)

11

Crystal-field energy-level parameters may be determined using a Monte-Carlo method originally developed for inten-sity parametrizations [20]. The parameter values are then optimized using standard least-squares fitting between calculated and experimental energy levels. Calculated energy levels are presented in Ref. [17] for all energy levels up to 55000 cm−1.

Wavefunctions generated from the crystal-field splitting modeling calculations were used to predict the temperature-dependent and orientation-dependent magnetic molar susceptibility (m) of the ground-state multiplet 4I15/2, as described in Ref. [17]. Excellent agreement was obtained between the calculated and the experimental susceptibility data (see also §1.4). The agreement also serves as an independent check of the crystal-field splitting analysis for Er3+ substituted into Y3+ sites of Cs symmetry in the orthoaluminate structure. Wavefunctions for the lowest Stark components of the ground multiplet 4I15/2 of Er3+ in the orthoaluminate structure are given in Table 1.4 (excerpted from Table III of Ref. [17]).

RE ions in a crystal field of D2 symmetry The RE paramagnetic garnets having a general formula of R3M5O12 or R3+:Y3M5O12 (where R3+ is the trivalent rare earth that replaces some or all of the yttrium ions in the crystal, and M is the metal ion Al3+ or Ga3+) have a cubic structure described by the space group symmetry 10Oh . An elementary cell of the crystal contains eight formula units, i.e., 24 R3+/Y3+ ions, 40 M3+ ions and 96 O2– ions. In the elementary cell of R3M5O12, the rare earth ions, having a large ionic radius, occupy the c-sites, while the Al3+ and Ga3+ ions with a smaller radius occupy the a- and d-sites [3,21]. It is usually the case that the RE ions, when substituted for Y3+, are located ran-domly in the six non-equivalent c-sites differing by orientation of their local axes of symmetry (the symmetry axes of the crystalline environment at those sites characterized by the D2 point group). The symmetry axes of all six c-sites are formed by rotating the crystalline system of coordinates (x, y, z) at angles /2 around the crystal axes [100], [010], [001], respectively, as shown in Figure 1.3.

For rare-earth garnets and isomorphous compounds, where the non-Kramers RE ions occupy the dodecahedral posi-tions described by the D2 point symmetry, the symmetry group contains four elements: the identity transformation Е, and the rotation by 180° around the three mutually perpendicular x-, y- and z-axes. The D2 group has four one-dimensional irreducible representations (irreps) A, B1, B2, and B3, which is equivalently designated in the literature as Гi, where the identification is given: A = Γ1, B1 = Γ2, B2 = Γ3, and B3 = Γ4 [3,22,23]. These irreps satisfy the requirement of group theory multiplication given in Table 1.5. Because the majority of the rare-earth garnet literature uses the Гi nota- tion, this is the notation we will use in what follows.

Table 1.4. Wavefunctions in the |J,MJ> basis for the lowest two Kramers’ doublets of the 4I15/2 multiplet in Er3+:YAlO3 [17].

Energy (cm–1) Expt. Calc.

Irrep Wavefunction

0 −2 Γ3/2 Ψ1 = +0.5661 e+88.44i |15/2,15/2> +0.4517 e−51.60i |15/2,7/2> +0.3625 e−53.66i |15/2,3/2>

+0.3400 e−53.64i |15/2, −5/2> +0.3106 e+76.52i |15/2,11/2> +0.2382 e−60.14i |15/2,−1/2> −0.1891 e−11.78i |15/2, −13/2> −0.1014 e+12.76i |15/2,−9/2>

0 −2 Γ1/2 Ψ2 = +0.5661 e−88.44i |15/2,−15/2> +0.4517 e+51.60i |15/2,−7/2> +0.3625 e+53.66i |15/2,−3/2>

+0.3400 e+53.64i |15/2,5/2> +0.3106 e−76.52i |15/2,−11/2> +0.2382 e+60.14i |15/2,1/2> −0.1891 e+11.78i |15/2,13/2> −0.1014 e−12.76i |15/2,9/2>

49 52 Γ3/2 Ψ3 = e+6.44i [−0.6001 e−29.78i |15/2,−5/2> −0.4381 e−68.84i |15/2,15/2>

+0.3850 e−35.88i |15/2,7/2> +0.3804 e+9.33i |15/2,−13/2> −0.3282 e−37.88i |15/2,3/2> +0.1261 e+13.73i |15/2,−9/2>]

49 52 Γ1/2 Ψ4 = e+6.44i [+0.6001 e+29.78i |15/2,−5/2> +0.4381 e+68.84i |15/2,15/2>

−0.3850 e+35.88i |15/2,7/2> −0.3804 e−9.33i |15/2,−13/2> +0.3282 e+37.88i |15/2,3/2> −0.1261 e−13.73i |15/2,−9/2>]

x

y

z z

y

z z

y

x

= 90°

= 90° = 0°

[001]

Figure 1.3. The sequence of rotations of the Euler’s angles: = 90°, = 90°, = 0° necessary for the calculations of the Wigner D-functions for the coordinate transformation between wavefunctions listed in columns 3 and 5 of Table 1.7.

12

Table 1.5. Multiplication table for D2 symmetry irreps Γi. (Alternative A, Bi notation is given in parentheses.)

D2 Γ1 (A) Γ2 (B1) Γ3 (B2) Γ4 (B3) Γ1 (A) Г1 Г2 Г3 Г4 Γ2 (B1) Г2 Г1 Г4 Г3 Γ3 (B2) Г3 Г4 Г1 Г2 Γ4 (B3) Г4 Г3 Г2 Г1

The D2 symmetry crystal field splits non-Kramers ion multiplets into (2J + 1) singlets. The wavefunction for each

Stark sublevel singlet transforms according to one of the four one-dimensional irreps Γi of the D2 point group that it is characterized by, as presented in Table 1.6.

The electronic energy level structure of R3+:YAG (or R3+:YGG) is analyzed by means of a model Hamiltonian de-fined to operate within the 4f n electronic configuration of RE ions, as described by eqns. (1.17 - 1.19) above. The crys-tal-field parameters k

qB are constrained by the expression, 1k qk k

q qB B

, which, for D2 site symmetry, yields 9 in-dependent pure-real crystal-field parameters: 2 2 4 4 4 6 6 6

0 2 0 2 4 0 2 4, , , , , , ,B B B B B B B B , and 66B .

The crystal-field parameterization of the RE garnet systems is more complicated than the parametrization of other single-crystal systems, because there are six crystallographically-equivalent, but magnetically inequivalent D2 sites per unit cell [19, 24-26]. Additionally, because there exist three mutually orthogonal but inequivalent C2 symmetry axes in D2 symmetry, different orientations of the crystal-field axes will yield differing crystal-field parameter sets with identi- cal calculated energy levels [25,27]. That is, for the D2 symmetry system examined here, the crystal-field z-axis may be chosen parallel to any one of three orthogonal C2 symmetry axes. Additionally, for each of these three z-axis orienta- tions, there exist two orientations of the x- and y-axes along the two remaining C2 symmetry axes, resulting in six alter- native sets of the nine crystal-field parameters [28-30]. Although the CF parameter sets have very different parameter values, expressions have been given to allow transformations between parameter sets [29]. These six CF parameter sets, corresponding to the six possible permutations of the orthogonal crystallographic a, b, and c axes, also result in the six possible permutations of the D2 symmetry irreps Г2, Г3 and Г4, with Г1 remaining invariant.3 This has resulted in some potential confusion in the literature, as different authors have used differing CF orientations when reporting their results. For example, in the local coordinate system for Tb3+:YAG, Bayerer et al. [31], chose x-parallel to the [001] axis, per- pendicular to the yz-plane, while Gruber et al. [32] chose z-parallel to the [001] axis, perpendicular to the xy-plane. Thus, the Bayerer et al. [31] symmetry label set {Г1,Г3,Г4,Г2} corresponds to the Gruber et al. [32] set {Г1,Г2,Г3,Г4}. Once this notational difference is accounted for, the calculated crystal-field splitting and symmetry labels of Bayerer et al. [31] and Gruber et al. [32] for Tb3+:YAG closely correspond, even though they superficially appear to be very dif-ferent. Detailed symmetry identification of the experimental crystal-field sublevels of the multiplet manifolds of the non-Kramers RE ions Tb3+ [32], Ho3+ [33], Tm3+ [34], Pr3+ and Eu3+ [35,27] in the garnet structure were performed on the basis of an algorithm using the ED and MD selection rules for D2 symmetry.

In these systems, there arise cases where two closely spaced non-degenerate singlet levels may not be directly resolv- able in the optical experiments, as the energy differences are about 1 to 3 cm–1 [3]. These unresolved pairs of levels are called “quasidoublets.” Traditional spectroscopy faces some difficulties in interpreting the optical spectra containing these quasidoublets, since the number of the components observed in the spectra proves to be less than the theoretically predicted number of Stark sublevels. However, interesting magnetooptical effects arise from these quasidoublets, as the values of the g-tensor components of the quasidoublets depend on the degree of mixing of these wavefunctions by the external magnetic field H [27,31]. Likewise, the g-tensor components determine the Zeeman contribution to the magne- tooptical and magneto-resonance properties of the non-Kramers RE ion states. Indeed, according to the data obtained

Table 1.6. Wavefunction symmetries corresponding to each of the D2 irreps.

Label MJ Wavefunction

Γ1 (A) even , 1 ,J M

J M J M

Γ2 (B1) even 1, 1 ,

J MJ M J M

Γ3 (B2) odd , 1 ,J MJ M J M

Γ4 (B3) odd 1, 1 ,

J MJ M J M

3This is clear in A, Bi notation, where the six axes orientations result in the six possible permutations of the three Bi (i = 1, 2, 3) symmetry irreps, with Aremaining invariant.

13

from optical, magnetic, and magnetooptical studies, as well as the results obtained from numerical calculations carried out within the framework of the CF theory [3,28-30], these quasidoublets are quite typical in the spectra of non-Kramers ions such as Tb3+ and Ho3+ in YAG and YGG crystals [32,33,36]. For example, according to the numerically calculated energy spectra and wavefunctions of the Stark sublevels of Tb3+ ions in YAG, presented in Table 1.7, both the ground and first excited states of the 7F6 and 5D4 multiplets are actually quasidoublet states whose initial splittings 0 (in the absence of an external magnetic field) are very small. To carry out the CF calculations for these sub-levels, the authors of Refs. [32,37] used a local coordinate system for Tb3+ in the garnet structure in which the z-axis is parallel to the [001] crystal axis and perpendicular to the xy-plane [32]. But in what follows, for the evaluation of the results obtained from the magnetooptical measurements of Ref. [37], the wavefunctions were transformed into the coordinates used by Guillot et al. [38] for Tb3+ in Y3Ga5O12 (YGG) and Bayerer et al. [31] for Tb3+ in Y3Al5O12 (YAG), where x-was chosen parallel to the [001] axis and perpendicular to the yz-plane. The coordinate transformations of the wave-functions were carried out using the Wigner D-functions Ref. [39] (for J = 6 [40]) with the sequence of rotations for the Euler's angles: = 90°, = 90°, = 0°, illustrated in Figure 1.3. If prime labels ( nA ) are used to identify the symmetry labels used by Guillot et al. [38] and Bayerer et al. [31], the corresponding labels used by Gruber et al. [32] are related as follows: Г1 1A , Г2 3A , Г3 4A and Г4 2A . In this case the wavefunction | JM > specified regarding the z-axis of an initial local coordinate system (x, y, z) can be represented by wavefunctions | JM > specified regarding the new quantization z'-axis (directed along [010] axis of crystal) of the transformed (rotated) local coordinate system (x, y, z) as,

JJM MM JM

M

D , (1.20)

, ,J Ji M i MMM MMD e d e

, (1.21)

where , ,JD are the Wigner D-functions and functions JMMd are calculated and tabulated for the value of

J = 6 in Ref. [40]. As a result, the wavefunctions of the first excited quasidoublet of the 7F6 multiplet of Tb3+: YAG can be represented in the transformed local coordinate system as given in column 5 of Table 1.7, Таble 1.7. Wavefunctions of some Stark sublevels of the 7F6 and 5D4 multiplets of Tb3+:YAG given in two alternative orientation schemes [37].

Wavefunctions in the |J, MJ basis Local axis-z || axis 001 of crystal Local axis-x || axis 001 of crystal

Stark level energy (сm–1)

Irrep Wavefunction Irrep Wavefunction

0.0 Г1

1 = 0.562 (|6,+6 + |6,6) 0.387 (|6,+4 + |6,4) 0.172 (|6,+2 + |6,2)

1Г

1 = 0.915 |6,0 + 0.233 (|6,+2+|6,2) 0.142 (|6,+4 + |6,4)

5.0 Г3 2 = 0.577 (|6,+6 |6,6)

+ 0.382 (|6,+4 |6,4) + 0.141 (|6,+2 |6,2)

4Г 2 = 0.701 (|6,+1 +|6,1)

212 Г1 5 = 0.630 |6,0

+ 0,536 (|6,+2 + |6,2) 0,106 (|6,+4 + |6,4)

1Г

5 = 0.672 (|6,+6 +|6,6) + 0.129 (|6,+4 + |6,4) + 0.083 (|6,+2 + |6,2)

214 Г2 6 = 0.638 (|6,+1 |6,1)

0.292 (|6,+3 |6,3) + 0.083 (|6,+5 |6,5)

3Г

6 = 0.696 (|6,+6 |6,6) 0.076 (|6,+4 |6,4) 0.038 (|6,+2 |6,2)

20514 Г1 50 = 0.532 (|4,+4 + |4,-4)

0.366 (|4,+2 + |4,2) 0.407 |4,0

1Г

50 = 0.095 (|4,+4 + |4,4) + 0.696 (|4,+2 + |4,2) + 0.114 |4,0

20516 Г3 51 = 0.493 (|4,+2 |4,2)

0.512 (|4,+4 |4,4) 4Г

51 = 0.643 (|4,+3 + |4,3) 0.305 (|4,+1 + |4,1)

20585 Г4 54 = 0.449 (|4,+1 + |4,1)

+ 0.546 (|4,+3 + |4,3) 2Г 54 = 0.7065 (|4,+3 |4,3)

20587 Г1 55 = 0.381 (|4,+4 + |4,4)

+ 0.101 (|4,+2 + |4,2) + 0.830 |4,0

1Г

55 = 0.415 (|4,+4 + |4,4) 0.126 (|4,+2 + |4,2) +0.797 |4,0

20594 Г3 56 = 0.508 (|4,+2 |4,2)

+ 0.490 (|4,+4 |4,4) 4Г

56 = 0.637 (|4,+1 + |4,1) 0.302 (|4,+3 + |4,3)

20638 Г2 57 = 0.692 (|4,+1 |4,1)

0.145 (|4,+3 |4,3) 3Г

57 = 0.698 (|4,+4 |4,4) 0.105 (|4,+2 |4,2)

14

5 1| | [0.672(| 6, 6 | 6, 6 ) 0.129(| 6, 4 | 6, 4 )

0.083(| 6, 2 | 6, 2 ) 0.146 | 6,0 ]

Г

(1.22)

6 3| | [0.696(| 6, 6 | 6, 6 ) 0.076(| 6, 4 | 6, 4 )

0.038(| 6, 2 | 6, 2 )]

Г

(1.23)

The wavefunctions of other Stark sublevels of the 7F6 and 5D4 multiplet manifolds are given in columns 2 and 3 of Table 1.7 in the local coordinate system used by Gruber et al. [32], with transformed wavefunctions listed in columns 4 and 5. Stark sublevel energy values are given in column 1 of the table. The set of wavefunctions listed in columns 4 and 5 are used in the magnetooptical analyses given in Chapters III and IV of this book.

1.4. Influence of the Symmetry of the Crystal Field on the Magnetic Susceptibility of RE Ions in Crystals

The crystal field (CF) produced by the environment surrounding the RE ions can have a great effect in changing the character of the Zeeman splitting of the energy levels of the rare-earth (RE) ion multiplets in a crystal as compared with the Zeeman splitting observed in the “free” RE ion. To determine the magnetic properties of the RE ion in a crystal, it is necessary to find their dependence on the value and orientation of an external magnetic field H relative to the crystal in the form of the matrix CF ZH H

eigenvalues, where ZH

is the Zeeman Hamiltonian (eqn. 1.6, §1.2). This problem

was first solved by Penny and Schlap [41] with a calculation of the magnetic susceptibility of the RE compound Pr2(SO4)3x-H2O. The matrix CF ZH H

has been constructed in Ref. [41] for the wavefunctions for the 3H4 ground

multiplet of the Pr3+ ion. In addition, Penny and Schlap [41] showed that the crystal field produces significant anisotropy of the magnetic sus-

ceptibility of RE compounds at low temperatures; there is a qualitatively different character in its temperature de- pendence for different crystal axes. Indeed, according to the experimental data from Ref. [3], the behavior of the mag- netic susceptibility becomes more complicated with decreasing of temperature T. In this case it becomes necessary to use the general expression for ( = x, y, z) obtained by Van-Vleck [12], that associates the magnetic susceptibility with the spectrum of the RE ion in a crystal field.

Let us consider the behavior of T for RE ions with J = 1 in a crystal field (CF) having the simplest form,4 2

CF zH D J

. (1.24)

Here the energy levels and the wavefunctions of the RE ion in the CF of (1.24) have the following form,

E0 = 0, | 0 > = |J,MJ > = |1, 0> ,

1E = D, | 1 > = |J,MJ > = |1, 1> . (1.25)

By calculating the components of the magnetic susceptibility using the Van-Vleck formula, we obtain [3],

2 2

2 2

exp 2

exp

n m nn m n

J Bn

n

E kT n J n kT n J m E E

g NE kT

(1.26)

where are the components of the magnetic susceptibility of the RE ion in the crystal field, En are the energy levels of the ion in the CF and gJ is the Lande’ g-factor. From (1.26), we can obtain the following expressions [3],

2 2

||

2

2 expJ B

zz

g N

kT D kT

, (1.27)

2 22 exp 1

2 expJ B

xy

g N D kT

D D kT

. (1.28)

The temperature dependence of the magnetic susceptibility components || and are given in Figure 1.4, which ill-ustrates the strong increase in the magnetic susceptibility anisotropy with a decrease in temperature. Note that magnetic susceptibility is isotropic at high temperatures (T >> |D|) [3], 4It is possible to show that at D < 0, we will have magnetic anisotropy of a type along the “easy” axis [3].

15

T/D

zz

/D/

/1>

/0>

Figure 1.4. Temperature dependence for the RE paramagnet magnetic susceptibility | and at D < 0.

2 22

3J Bg N

kT

. (1.29)

The susceptibility along the “easy” axis (z-axis) increases proportionally with 1/T, and at T << |D| it corresponds to the magnetic susceptibility of a two-level system with “effective” spin S = 1/2 for the decrease of T (known as an “Ising” case). At the same time, the susceptibility along the z-direction, perpendicular to the “easy” axis, tends to a constant limit with a decrease of T, and becomes equal to [3],

2 2J Bg N

D

. (1.30)

Thus, we see that the magnetic properties and their behavior with changing temperature essentially depend on the CF splitting structure of the energy spectrum of the magneto-active ion in the crystal field.

For example, the calculation of the anisotropic magnetic susceptibility of the RE orthoaluminates with Kramers ions (Dy3+, Er3+ as examples) is made by using the CF Hamiltonian [42],

,

Re Imk k k k k kCF q q q q q q

k q

H B C C i B C C

, (1.31)

where the complete specification of the CF requires the fitting of fifteen kqB crystal field parameters, and the () signs

correspond to two crystallographically-inequivalent positions of the RE ion in the unit cell. The matrix elements of the kqC operators are in the usual way expressed in terms of the 3j- and 6j-symbols [6,7,39,43], and the spectroscopic coef-

ficients [44] in the intermediate coupled basis defined in Ref. [45] for the 4f 9 electronic configuration of Dy3+. The components g of the g-tensor of the ground Kramers doublet and other parameters required for the calculation

of the magnetic susceptibility according to expression (1.26) can be found in the usual way [3,42]5,

2 | |zz J zg g m J m

, (1.32)

2 Re | |xx J xg g m J m

, (1.33)

2 Im | |yy J yg g m J m

. (1.34)

Here | m is the Kramers-conjugate wavefunction of the Kramers doublet. RE ions in a CF of Cs symmetry To give an example, the wavefunctions of the ground Kramers doublet of Er3+ ion

in the CF symmetry Cs (Table 1.4, 1.3) can be written as, 5Note that the g-tensor is diagonal with the execution of the following condition: Im Re 0x ym J m m J m

16

88.4 51.6 53.7 53.6

76.5 60.1 11.8 12.8

0.57 |15 / 2,15 / 2 0.45 |15 / 2,7 / 2 0.36 |15 / 2,3 / 2 0.34 |15 / 2, 5 / 2

| 0.31 |15 / 2,11/ 2 0.24 |15 / 2, 1/ 2 0.19 |15 / 2, 13 / 2 0.10 |15 / 2, 9 / 2

| 0.57

i i i i

i i i i

i

e e e e

m e e e e

m e

88.4 51.6 53.7 53.6

76.5 60.1 11.8 12.8

|15 / 2, 15 / 2 0.45 |15 / 2, 7 / 2 0.36 |15 / 2, 3 / 2 0.34 |15 / 2,5 / 2

0.31 |15 / 2, 11/ 2 0.24 |15 / 2,1/ 2 0.19 |15 / 2,13 / 2 0.10 |15 / 2,9 / 2

i i i

i i i i

e e e

e e e e

(1.35) where all angles (in the exponents) are given in degrees. Here we can see that the wavefunctions of the Kramers dou-blets are described by the doubly-degenerate states corresponding to the “effective” spin S = 1/2 and satisfy the follow-ing condition for Kramers conjugation,

| | ,JMM

m C J M , (1.36)

| ( 1) | ,J MJM

M

m C J M , (1.37)

where CJM are the coefficients of the wavefunction on the |J,M> basis states). These wavefunctions can be treated as spherical spinors ,L

JM of rank 1/2 [6,39]. Spherical spinors ,LJM are

eigenfunctions of the operators of full angular, orbital and spin (for spin S = 1/2) moments and can be expressed through spherical functions YLM (,) and spin functions 1/2 z for particles with a spin of 1/2, according to the formula [39],

,1/2 1/2,z z

L JMJM Lm Lm

m

C Y , (1.38)

where ,1/2

1

2z

JMLm zC Lm JM

are Clebsch-Gordan coefficients [6,7,39] and the spin functions 1/2 z are real and

represent the 1 2 matrices of type a

b

[39]. Spherical tensors ,LJM for complex conjugation are transformed by

using the formula [39],

, 1 ,J L ML L

JM y J Mi , (1.39)

where the 2 2 matrix yi can be written as: 0 1

1 0yi

.

Now let us apply expression (1.38) to the wavefunctions (1.35) of the ground state (Kramers doublet) of the 4I15/2

multiplet of Er3+ in Cs symmetry, which can be represented in a 1 2 matrix. We then obtain the complex conjugate wavefunction corresponding to the sublevel of the ground Kramers doublet which will then be,

88.4 51.61 1

53.7 53.6 76.5

60.1 11.8 12.8

| (| ) | 0.57 15 / 2,15 / 2 | 0.45 15 / 2,7 / 2 |

0.36 15 / 2,3 / 2 | 0.34 15 / 2, 5 / 2 | 0.31 15 / 2,11/ 2 |

0.24 15 / 2, 1/ 2 | 0.19 15 / 2, 13 / 2 | 0.10 15 / 2, 9 / 2

i iJM

M

i i i

i i i

C JM e e

e e e

e e e

|

(1.40)

The numerical calculation obtained by using eqn. (1.32) for the g-tensor z-component for the ground state (Kramers doublet) of the 4I15/2 multiplet for Er3+ in the CF symmetry of Cs is gzz = 10.68.

To give another example, we can use the above mentioned considerations regarding the nature of the magnetic prop-erties of the Kramers RE ions in a low-symmetry CF to show that the anisotropy of the temperature dependence of the paramagnetic susceptibility of DyAlO3 can be described by the following expression [42],

1 1 CFi i i , (1.41)

where,

2 28

/2 2

1

| | | ( ) | | | | ( ) |2n i ix iy iz i ix iy izE kTCF

i J Bn m nCF m n

n J n n n J m mNT g e

Z kT E E

. (1.42)

In this expression, the Lande’ g-factor is gJ = 4/3, N is the number of Dy3+ ions per gram, ZCF is the statistical sum, |n> and | n~ > are the wavefunctions of the Kramers doublets, ik is the Kronecker delta, the (x, y, z) directions coincide with the (a, b, c) crystallographic axes, and i are the paramagnetic Curie temperatures describing the magnetic-dipole Dy3+ Dy3+ interaction. Experimental data from Ref. [42] for the magnetic susceptibility of DyAlO3 confirm the theo-retical expressions given in (1.41-1.42).

17

Figure 1.5 shows the temperature dependence of the inverse magnetic susceptibility –1 for crystallographic directions [010] (b-axis) and [001] (c-axis) of the orthorhombic Er3+:YAlO3 (Er0.5Y0.5AlO3) crystal as determined in Ref. [17], with the results of magnetic measurements reported in Ref. [46] included for comparison. The measured values closely follow the Curie-Weiss Law over the entire temperature range (from 20K to 300K) measured above the magnetic phase-transition (Ne’el temperature), which has been reported to be 0.6K along the c-axis [47]. The maximum magnetic suscep- tibility of Er3+:YAlO3 at low temperatures (T < 100 K) is observed along the c-axis while the susceptibility c is approxi-mately twice as large as the magnetic susceptibility b along the b-axis. Despite the decrease in the susceptibility values with increasing temperature, the anisotropic character is preserved in the high temperature region, as shown in Figure 1.5.

It appears to be the interaction between the Er3+ ion in Cs symmetry and the RE sublattice that leads to the strong ani-sotropy of the magnetic moment observed in the orthoaluminate structure, especially at low temperatures. Indeed, the additional contribution to the magnetization of the Er3+:YAlO3 crystal that arises from the magnetic moments of the RE sublattice of the Er3+ ions associated with the Van-Vleck mechanism is due to the “mixing” of excited states of the 4I15/2 multiplet with the ground-state Kramers-doublet when an external field H is applied [3,17]. As a result, a substantial con- tribution is made to the anisotropy of the Er3+: YAlO3 magnetization, while the average magnetic moment of Er3+ is con- nected with the difference in the population of the sublevels of the ground Kramers-doublet and defines the value of the Er3+:YAlO3 magnetization. The Van-Vleck correction to the magnetic moment of the RE sublattice in Er3+:YAlO3 is due to a mixing between the wavefunctions of the excited 4I15/2 Kramers-doublets with energies of 54, 166, 214, 267 and 386 cm–1 and the wavefunctions of the 4I15/2 ground-state Kramers-doublet. The corresponding levels, their energies and the wavefunctions involved, listed in part in Table 1.4, are presented in Table III of Ref. [17] (see also inset of Figure 1.5).

In the high temperature region (T 300 K), the behavior of the magnetic properties of Er3+:YAlO3 can be explained by significant contributions from excited states located at energies of 166, 214 and 267 cm–1 in the 4I15/2 manifold, which become thermally populated as the temperature is increased. These states are “mixed” (and split) by an external magnetic field H directed along the crystalline c-axis. Furthermore, it is necessary to take into consideration that the RE ions in the orthoaluminate structure are located on two magnetically-nonequivalent sites of monoclinic point symmetry Cs [17], but which are optically equivalent in the absence of an external magnetic field. As a result, we choose the z-axis of the local coordinate system of the RE ion Er3+ to be parallel to the c-axis of the orthorhombic crystal. At the same time, the local x- and y-axes will lie in the ab-plane oriented at an angle to the crystalline a-axis (the ± signs indi-cate the two crystallographically-nonequivalent sites differing by the orientation of the local axes).

The eigenvectors from the crystal-field calculations given in Table III of Ref. [17] can be used to calculate the molar magnetic susceptibility ( )m

c along the c-axis of Er3+:YAlO3. The corresponding expression, given in Ref. [42], can be written (in the local coordinate system of Er3+) as,

Figure 1.5. The inverse molar magnetic susceptibility 1c Er3+:YAlO3 in cgs units (mole/cm3) as a function of the absolute tempera-

ture (T in K) [17]: (1) experimental data for the Er0.5Y0.5AlO3 crystal measured along the c-axis, (2) experimental data measured along the b-axis, (3) results of the numerical calculations for the c-axis, and (4) and (5) are data obtained from Ref. [46] for the c- and b-axes, respectively. The inset presents the schematic of Van-Vleck “mixing” between six Kramers doublets of the 4I15/2 ground multiplet. Note that the Van-Vleck “mixing” between the Kramers-doublet states at 0.0 and 54 cm−1 is forbidden by selection rules.

18

2 28( ) ( ) 2 2

010

| | | | | | | |exp( ) 2m m nA z z

c z Bn m n m n

EN n J n n J mg

Z kT kT E E

. (1.43)

Using the wavefunctions and energy levels given in Ref. [17], we can evaluate eqn. (1.43) as,

( ) 1 2 3 1 4 5 6

1 3 1 4 3 4 4 5

28.41 23.41 23.82 28.41 25.3 19.56 43

8

0.00092( ) 0.0459( ) 0.4744( ) 0.0266( )

mc T

. (1.44)

In these equations, g0 = 6/5 is the Lande’ g-factor of the 4I15/2 manifold, NA is Avogadro’s number,

Z0 = 8

1

exp nn

E kT

, En and Em are the energies of the “mixing” states given by the wavefunctions |n> and |m>. In

eqn. (1.44), which is valid over the temperature range between 20 K and 300 K, 1 is the Boltzmann population of the Stark sublevels of the ground-state Kramers doublet and 2-6 are the Boltzmann populations of the Kramers-doublets at energies of 54, 166, 214, 267, and 386 cm–1, respectively. The temperature dependence of the inverse molar magnetic susceptibility calculated from Eq. (1.44) is plotted in Figure 1.5. The inset in the figure shows a schematic diagram of the Van-Vleck mixing between states representing the six lowest-energy degenerate Kramers doublets of the 4I15/2 mul- tiplet. There is reasonable agreement between the experimental and calculated values of the molar magnetic susceptibil- ity of Er3+ in Er3+:YAlO3 investigated in the temperature range 20 to 300K.

At the same time, the components of the magnetic moment of the RE sublattice in Er3+:YAlO3. along the b- and a-axes are determined by the x- and y-components of the magnetic moment of the ion in the local coordinate system and as well by the value of the angle as,

cos sin

sin cosx a b

y a b

M M M

M M M

, (1.45)

where , ,x y x yM H , and the corresponding expressions for the molar magnetic susceptibility ( )mx and ( )m

y can be written (in the local coordinate system of Er3+) as [42],

2 28, ,( ) 2 2

, 010

| | | | | | | |exp 2x y x ym nA

x y Bn m n m n

n J n n J mENg

Z kT kT E E

, (1.46)

where |n> and | n are the Kramers-conjugate wavefunctions of the degenerate Stark sublevels of the Kramers doublet and |m> and | m are the Kramers conjugate wavefunctions of the mixing states of the Kramers doublets (see also the right column of Table III in Ref. [17]). However, the numerical calculation of the temperature dependence of the ( )m

X and ( )m

Y susceptibilities in Er3+:YAlO3 is complicated (in Cs symmetry) by the situation that each Kramers doublet of the RE ion is characterized by its own local coordinate system and also has a different value of the angle (see Refs. [3,17]). In this case, we can only determine the value of angle for the ground Kramers doublet at the temperature of 20K when the populations of the excited Kramers doublets of the 4I15/2 multiplet can be neglected. Indeed, using the measured values of the molar magnetic susceptibilities ( )m

a = 0.310 and ( )mb = 0.276 cm3K/mole at T = 20K from Ref. [46], along with

the values of the molar susceptibilities ( )mx = 0.384 and ( )m

y = 0.036cm3K/mole calculated from eqn. (1.46), we can find the value of angle = 33.5 degrees for the ground Kramers doublet of the 4I15/2 manifold in Er3+:YAlO3, as illus- trated in Figure 1.6. This angle for the ground state of the Er3+ ion in the orthoaluminate structure is in satisfactory agree- ment with the analogous data found for the Er3+ ion in the orthoferrite structure reported by Wood et al. [48], providing an independent confirmation regarding the results of the crystal-field calculations we used to interpret the spectroscopic data in Ref. [17].

RE ions in a crystal field of D2 symmetry As far as the paramagnetic rare-earth garnets are concerned, within the microscopic theory that considers the real RE ion spectrum in a crystal field of D2 symmetry and its perturbation under an external magnetic field Н, the rM momentum components of the Kramers RE ion in one of the six non-equivalent positions r (in the local coordinate system) can be given in the following form [3]

(2)1 1tanh 2

2 2

r rr rn n

n na nrn n

M H gkTH

, (1.47)

1

0 0exp expn n mm

E kT E kT

19

2 4(1) (3) ....r r rn na a na a

a a

g H g H

where, for the nth Kramers doublet, 0nE is the energy, n is the thermal (Boltzmann) population, and r

n is the Zeeman splitting of the sublevels. The rH are the magnetic field components relative to the local axes of the coordinate system, and ( )i

nag are the coefficients that depend on the RE ion wave functions in the crystal field that characterize the correc-tions to the energy of the ith order perturbation theory.

The first term in (1.47) is associated with the Zeeman splitting of the Kramers doublet levels, and the second term is associated with the Van-Vleck contribution to the RE ion magnetic field due to the shifting of the line center of the doublets in the magnetic field. The isotherms of the experimental and theoretically-calculated field dependence (1.47) of the magnetic momentum M of Dy3Al5O12 are presented in Figure 1.7 as a typical representative of this type of RE garnet [49].

As indicated earlier (in §1.3), the ground states of the non-Kramers RE ions Tb3+ and Ho3+ have pairs of closely spaced singlet levels (quasidoublet) that are well-distinguished from the higher energy states of the ground multiplets. In general, the quasidoublet behavior in an external magnetic field is similar to that of the Kramers doublet, so the mag-netic features of the garnets doped with the non-Kramers RE ions Tb3+ and Ho3+ are in many respects analogous to the magnetic features of the garnets containing the Kramers RE ions. Within the low temperature range (T < 10 K) the RE ion magnetization is anisotropic and becomes saturated in a strong magnetic field (H 30 kOe) [50-52].

a

b

c

y

z

x

y

z

Figure 1.6. The local coordinate systems corresponding to the two magnetically-nonequivalent sites of Er3+ ions in the orthoalumi-nate structure.

Figure 1.7. Isotherms of experimentally measured and theoretically calculated dependence of Dy3Al5O12 magnetization.

20

At higher temperatures the magnetic susceptibility of this type of RE garnets decreases with the rise in temperature approximately according to the Curie law [51].

Suppose that the eigenfunctions |A> and |B> of the Stark sublevels which form the ground-state quasidoublet are transformed according to the irreducible representations ГA and ГB of the D2 symmetry group, and have energies

01,2 0 2E , where 0 is the energy distance between the quasidoublet levels in zero magnetic field. Then for the

components of the magnetic moment of the non-Kramers RE ions Tb3+ and Ho3+ in garnets the formula (1.47) still ap-plies, but with the splitting r of the quasidoublet in the magnetic field written as [3],

2

2 2

0r r r

g gH q H

, (1.48)

where 2/ | |g J B gA g J B

is the magnetic moment of the quasidoublet, which is transformed according to the ire-ducible representation Гg of the D2 group, and g A BA A A . The parameters g, qa, ga for Tb3Al5O12 and Ho3Al5O12 are given in Ref. [51]. As an example, in Figure 1.8 the experimental field dependence of the magnetization of the RE garnet Tb3Al5O12 is compared with calculated values using the formulas (1.47) and (1.48) [51].

1.5. Ising RE Ions and Behavior of Their Magnetic Features

Significant interest in the study of the magnetic properties of the rare-earth (RE) compounds that have an orthoalu-minate or garnet structure has been generated in these crystals to a large extent by the features of the anisotropic mag-netization of the Kramers (or non-Kramers) RE ions in the presence of an external magnetic field H in these crystals (i.e., their “Ising” behavior [3,14]). This definition underscores a certain analogy between the behavior of these RE ions in an ex- ternal magnetic field and the spin moments in the famous Ising model [3], since the Ising ion can be magnet-ized only in the direction of an axis of magnetic anisotropy (the “Ising” axis). A specific feature of the Ising ion is that the magnetic field applied in any direction perpendicular to the Ising axis does not magnetize a given RE ion (to a first magnetic field approximation) [3,14].

Ising RE ions in the orthaluminate structure. A special characteristic of the energy spectrum of the Stark splitting of a Kramers RE ion such as Dy3+ or Sm3+ in a low-symmetry crystal field (Cs site symmetry) leads to the appearance of the Kramers doublets having a strongly anisotropic g-factor, i.e., they become Ising states [3,14]. It is important to note the significant role played by the uniaxial component of the electrostatic potential (i.e., the m = 0 terms of the CF Ham- iltonian, eqn. (1.31), §1.4)6 in the crystalline environment of the RE ion in an orthoaluminate structure. For example, for the Ising RE ion system DyAlO3, the magnetic anisotropy (Ising) axis lies in the symmetry plane of the crystal field. Therefore, the thermodynamic characteristics of the rhombic DyAlO3 crystal are characterized by a strong anisotropy in

Figure 1.8. The field dependence of the magnetization of RE garnet Tb3Al5O12, measured at T = 4.2 K. Solid lines are the results from the calculations based on eqns. (1.47) and (1.48). Symbols are the experimentally measured field dependence: 1 - H || [100]; 2 - H || [110]; 3 - H || [111]. 6It is possible to show that the Kramers doublet having Ising character has all diagonal matrix elements as zero except for the matrix element of the zJ

operator, with the z-component of the g-tensor 2 , | | ,zz J zg g J M J J M

0 (see also Ref. [3]). At the same time, all non-diagonal matrix elements are equal to zero.

21

the ab-plane at low temperatures T. The anisotropic character of the inverse magnetic susceptibility temperature de-pendence of the RE orthoaluminate DyAlO3 measured for the crystallographic directions [010] (b-axis) and [001] (c-axis) of the orthorhombic crystal is shown in Figure 1.9 [54], as well as the results of magnetic measurements from Ref. [55] for comparison. It is easy to see that the temperature dependence of b and c are in good agreement (within the error of measurements) with the data [55] over the temperature range under study and for temperatures below 100 K, where the maximum values of the magnetic susceptibility of DyAlO3 are observed in the ab-plane of the crystal. More- over, the susceptibility along the c-axis (c) is essentially lower and tends to a constant limit with decrease in tempera- ture. This observation indicates its Van-Vleck origin, according to Refs. [54,55].

As reported by Ref. [54], the behavior of the magnetic susceptibility of DyAlO3 (particularly within the range of low T) is associated with the fact that the ground state of the Dy3+ ion in a crystal field having Cs symmetry is a Kramers doublet described by the wavefunctions (0)

1,2| = |15/2, ±15/2>, that is, a nearly pure |J,MJ> state. Its axis of quantize- tion is in the ab-plane at an angle of aa= ±33.5˚ with respect to the b-axis (representing two crystallographic non- equivalent positions of the RE ion in the orthoaluminate structure). This means that the Dy3+ ion can be considered as an Ising RE ion with the direction of magnetization coinciding with the axis of quantization or the z'-axis of the local coordinate system corresponding to one of the non-equivalent positions of the RE Dy3+ ion in the orthoaluminate struc- ture. In this case, the susceptibility b corresponds to the susceptibility of a two-level system (1.27) with an “effective” spin S = 1/2 and a magnetic moment close to a maximum possible value of (0) 9.2g B (where B is the Bohr magne- ton) [3,54,55]. At the same time, the magnetic susceptibility along the direction perpendicular to the “easy” (Ising) axis of magnetization is connected with a mixing of the first excited state doublet with the ground state, under an external magnetic field H. Within the range of low T, the susceptibility can be written as [3,54],

(1) 2

2 20

1,2 1 0

| 15 / 2, 15 / 2 | | |2 z i

c Bi

JN g

E E

, (1.49)

where (1)| i are the wavefunctions of the first excited state doublet of the ground multiplet 6H15/2 of the Dy3+ ion, g0 = 4/3 is the Lande’ factor of the ground multiplet of the Dy3+ ion, N is the number of the RE ions per mole, and (E1 – E0) = 54 cm–1 is the energy of the first excited state doublet [56,57].

The wavefunction of the first excited state (admixed with the ground state) can be approximated by the pure |J,MJ> state |15/2, ±13/2>. As a result, it is not difficult to obtain from eqn. (1.49) the calculated value of c = 0.128 cm3/mole that is in good agreement with the data of c measured in Ref. [55] at low temperature (and extrapolated to T = 0). Here we use the equality, 15 / 2, 15 / 2 | |15 / 2, 13 / 2 15 2xJ

. Thus, the analysis of the behavior of the temperature

dependence of the transverse magnetic susceptibility c (for low T) demonstrates that the wave functions of the ground (I) and first excited (II) Kramers doublets of the 6H15/2 multiplet of the Dy3+ ion in the orthoaluminate structure can be represented with sufficient accuracy as “pure” (“Ising”) states |15/2, ±15/2> and |15/2, ±13/2>, respectively, in the local coordinate system of the ion. The local axes orientation is presented in the inset of Figure 1.9.

Figure 1.9. Temperature dependence of the inverse magnetic susceptibility 1/ measured along the [010] and [001] axes of the ortho-

rhombic crystal DyAlO3. Experimental values of 1c (1, 3) are from Ref. [55] and experimental values of 1

b (2, 4) are from Ref.

[54], respectively. The inset shows the local axes (x, y, z) of the coordinate system corresponding to one of the two crystallographic non-equivalent positions of the Dy3+ ion in the DyAlO3 structure. Letters (a, b, c) are the axes of orthorhombic DyAlO3 crystal.

22

On the other hand, for a non-Kramers ion, if the ground state is a quasidoublet which is separated by a sufficiently large energy interval (i.e. >> kT) from the excited states of the ground multiplet, the RE subsystem magnetization can be described by the following expression [3] (see also eqn. (1.47), §1.4),

(0) (0)(0)

0

1 1tanh

2 2r r

r r

MH kT

, (1.50)

where 0 is the Boltzmann population of ground quasidoublet sublevels, (0)r is the Zeeman splitting of the quasidoublet

sublevels (see the discussion given below), Hr is the component of the external magnetic field relative to the axes of the local coordinate system corresponding to the RE ion non-equivalent position (c-site) in the crystal, and r is the index of the c-site.

It is important to note that the mechanism of Van-Vleck “mixing” of states by an external magnetic field is necessary when taking into account a review of the magnetism of non-Kramers RE ions (with an even number of electrons in un- filled 4f n-subshell). The existence of the quasidoublet states is a characteristic feature of the optical spectrum of the non-Kramers ions in the orthoaluminate as well as in the garnet structure. They are formed by two closely spaced (in energy) Stark singlets separated by an energy interval of about 1 сm-1 that is not directly resolved in optical experiments. The Zeeman splitting of quasidoublet states in the presence of an external magnetic field (Н) is determined by the fol-low- ing expression [3],

(0) 2 2 20 ||r B rg H , (1.51)

where 0 is the zero-field splitting; g|| gk = 2g0 <A| kJ

|B> is the k-component of the g-tensor of the quasi-doublet and |A>, |B> are the wavefunctions of the quasidoublet sublevels. A magnetic field H applied in any direction perpendicular to the k-axis (for example, z-axis) does not split the quasidoublet states. Calculation of the Van-Vleck mixing of the quasidoublet sublevels by an external field H by using formula (1.47) shows that the magnetization of the RE sublattice in this case can be represented as (H|| 001 axis),

2 2||(0) cos tanh

2 2B

H

g HM N

kT

, (1.52)

where (0)HM is the projection of the magnetization on the direction of an external magnetic field H, is the angle be-

tween the direction of the field H and the z-axis, and N is the number of RE ions in the unit of volume (cm3). In other words, we see that the behavior of the magnetization of the Non-Kramers ions in the magnetic field, where temperatures kT , coincides with the behavior of the temperature dependence of the magnetization of the Kramers ions, and cor-responds to the magnetization of a two-level system with effective spin S = 1/2.

Moreover, action by the time-reversal operator T

(equivalent to a change in sign of the external magnetic field H → H) on the quasidoublet wavefunctions of the non-Kramers ions leave them invariant, while the wavefunctions for the Kramers doublet turn into each other (i.e. T

|A>= |A>, T

|B>= |B>, while T

|+1/2>= |1/2>, T

|1/2>= |+1/2>)7. This

feature arises due to the different transformation properties of the eigenfunctions |A>, |B> of the quasidoublet and the Kramers doublet |1/2>, relative to the change of time sign. The predicted difference in the mechanism of magnetiza- tion between the quasidoublet and the Kramers doublet states leads to an essential change in the Zeeman patterns of the 4f → 4f transitions between the Zeeman sublevels in an external magnetic field of the opposite sign with an invariant sign regarding circular polarization (see also §3.7 Chapter III).

Let us now find an orientation of the magnetic anisotropy axis (the Ising axis) for the positions with Cs symmetry. If the quasidoublet of the non-Kramers ion is formed by two Stark singlets with wavefunctions having the same irreps (both A or both B) of the Cs point group, then the Ising axis will coincide with the z-axes of an initial crystallographic coordinate system, which is defined by the CF Hamiltonian (1.19)8. Indeed, as provided by the selection rules for the matrix elements of angular momentum other than zero, only matrix elements | |zA J A

and | |zB J B

are

non-zero, i.e., g = (0, 0, gz), since the wavefunctions |A> and |B> of the quasi-doublet are determined by the linear com- binations of |J,MJ> states having only even (or odd) values of the MJ projections.

By contrast, if the quasidoublet of the non-Kramers ion is formed by the Stark singlets with wavefunctions that trans-form using different irreps (A and B) of the Cs group, then | | 0zA J B

, but | | 0xA J B

and | | 0yA J B

.

Therefore g = (gx, gy, 0), and the Ising axis lies in the xy-plane. In this case the orientation of the magnetic anisotropy axis relative to the axes of the initial crystallographic coordinate system, which is defined by the CF Hamiltonian, is 7In this case, we suppose that between the wavefunctions of the Kramers doublet and the spin wavefunctions there is a direct “physical” isomorphism (see also Ref. [3]). 8Note that the axes of the coordinate system under consideration do not coincide with the axes of orthorhombic crystal (a, b, c).

23

determined by the values of the matrix elements | |xA J B

and | |yA J B

[3]. For example, using the wavefunc- tions (A and B) of the ground quasidoublet of Tb3+ in the CF symmetry Cs that are given in in the right-hand column of Table 1.3, we find,

02 | | 16.428yg A J B

(1.53)

02 | | 1.083xg A J B

. (1.54)

Therefore, 2 2|| 16.46x yg g g and = 1tan y

x

g

g 3.8°. Thus, the Ising magnetic moment (0) 8.23g B of the

non-Kramers Tb3+ ion is found in the ab-plane of the rhombic crystal TbAlO3 and the angle between the magnetic moment (0)

g and the x-axis of the initial crystallographic coordinate system (x, y, z) is equal to approximately 4.0°, as illustrated in Figure 1.10.

Earlier experimental studies [3,14] have shown that Tb3+ in the RE orthoaluminate TbAlO3 can be treated as an Ising ion with its Ising axis taken as the magnetic anisotropy axis lying in the ab-plane of the orthorhombic crystal at an angle of 0 = 36° to the a-axis of crystal. The signs belong to the two crystallographically-nonequivalent sites differing in the orientation of the local axes. As a result it is not hard to choose the Ising axis as the z-axis of the local coordinate system of the Tb3+ ion (located at one of the non-equivalent sites) so that the y-axis will be parallel to the c-axis of the orthorhombic crystal (the Ising local coordinate system) [3].

We can now determine the wavefunctions for the Stark sublevels of the ground quasidoublet of the Tb3+ ion in the 7F6 multiplet, with the quantization axis coinciding with the Ising axis9. For this purpose we use a coordinate transformation of the eigenvectors (from column three of Table 1.3) determined in the initial crystallographic coordinate system (x, y, z) by means of the Wigner D-functions [39,40]. Appropriate rotation by Euler angles , , orients the z-axis of the initial coordinate system parallel to the Ising axis in the direction of the ground quasidoublet magnetic moment, (0)

g . For the convenience of following the calculations of the Wigner D-functions in Figure 1.11, the sequence of rotations on the Euler's angles: = 90°, = 90°, = 0° are given in an inverse order. The eigenvector | JM > specified regarding the z-axis of an initial crystallographic coordinate system can be represented by eigenvectors | JM > of a “pseudo-Ising” coordinate system (with the z-axis, or quantization axis, lying in ab-plane of crystal) as,

(6)JM MM JM

M

D , (1.55)

(6) 90 (6)90 , 90 , 0 90i MMM MMD e d

, (1.56)

where 6 , ,MMD are the Wigner D-functions. Functions (6) 90MMd are calculated and tabulated for the value of

J = 6 in Ref. [40].

a

b

c z'

x

y

)0(g

Figure 1.10. (a, b, c) are the axes of the orthorhombic crystal and (x, y, z) are the axes of the initial crystallographic coordinate sys-

tem (in which the CF Hamiltonian is defined). (0)g is the magnetic moment of the ground quasi-doublet of Tb3+ in the orthoalumi-

nate structure.

9It is important to note that the wavefunctions determined for the Ising local coordinate system (column four of Table 1.3) are more convenient for the calculation of the TbAlO3 magnetic susceptibility in comparison with the set of wavefunctions found for the initial crystallographic coordinate system (in which the CF Hamiltonian is defined), given in column three of Table 1.3.

24

a

b

c

X2

Y2

Z2

a

b

c

Y

X

Z

a

b

c

Z1

Y1

X1

a

b

c

Y ′

Z ′

X ′

α = 90˚β = 90˚

γ = 0˚β = 90˚

Figure 1.11. The sequence of rotations on the Euler’s angles: = 90°, = 90°, = 0° necessary for the calculations of Wigner D-functions.

The wavefunctions of the ground quasi-doublet represented as the low sublevel A can be written (in the crystallo-

graphic coordinate system) as (see column IV Table 1.3): 30.8 9.5 9.5

55.9 55.9

| |1 [0.7567 | 6,0 0.4262( | 6, 2 | 6, 2 )

0.0890( | 6, 4 | 6, 4 )]

i i i

i i

A e e e

e e

(1.57)

(where all angles are given in degrees). At the same time, using eqns. (1.56) and (1.57) we can present the | , JJ M states in eqn. (1.55) in the following form,

90 (6),0

1| 6,0 | 6, [ 20 | 6,0 2 105(| 6, 2 | 6, 2 )

64

6 14(| 6, 4 | 6, 4 ) 2 231(| 6, 6 | 6, 6 )}

iMM

M

e d M

(1.58)

90 (6), 2

1| 6, 2 | 6, [2 105 | 6,0

64

2 10(| 6, 1 | 6, 1 ) 17(| 6, 2 | 6, 2 )

18 (| 6, 3 / 6, 3 ) 30(| 6, 4 | 6, 4 )

2 165(| 6, 5 | 6, 5 ) 3 55(| 6, 6 | 6, 6 )]

iMM

M

e d M

i

i

i

(1.59)

90 (6), 2

1| 6, 2 / 6, [2 105 | 6,0

64

2 10(| 6, 1 | 6, 1 ) 17(| 6, 2 | 6, 2 )

18 (| 6, 3 | 6, 3 ) 30(| 6, 4 | 6, 4 )

2 165(| 6, 5 | 6, 5 ) 3 55(| 6, 6 | 6, 6 )]

iMM

M

e d M

i

i

i

(1.60)

Substituting (1.58) - (1.60) into eqn. (1.57), we find that in the “pseudo-Ising” coordinate system the ground quasidou-

25

blet for the lower sublevel A wavefunction has a simple form,

| |1 [0.6641(| 6, 6 | 6, 6 ) 0.1530(| 6, 4 | 6, 4 )]A . (1.61)

Similarly, the wavefunctions of the ground quasidoublet sublevel B can be represented (in the “pseudo-Ising” coordi-nate system) as,

85.1 86.5 86.5

67.8 67.8

| | 2 [0.6525( | 6, 1 | 6, 1 )

0.2197( | 6, 3 / 6, 3 )]

[0.6670(| 6, 6 | 6, 6 ) 0.1492(| 6, 4 | 6, 4 )].

i i i

i i

B e e e

e e

(1.62)

Finally, the ground-state wavefunctions of Tb3+ for the Ising local coordinate system can be established using the co-ordinate transformation of the wavefunctions (1.61) and (1.62) by means of an additional rotation of the “pseudo-Ising” coordinate system about the y-axis (or c-axis) by an angle of +4.0° in the ab-plane of crystal. In this case the z-axis of the “pseudo-Ising” coordinate system becomes parallel to the direction of the Ising magnetic moment (0)

g , that is, a new z- axis of the rotated local coordinate system can be identified with an axis of crystallographic magnetic anisotropy (the “Ising” axis). As a result of the similar coordinate transformation of calculated wavefunctions and inclusion of this magnetic information, we can now unambiguously remove the rotational ambiguity arising in the crystal-field splitting calculations of low-symmetry systems.

Wavefunctions for the 13 Stark components of the ground multiplet 7F6 calculated in Ref. [16] using the same orienta-tion of the local axes (in Ising orientation) as we just described are given in the far right column of Table 1.3. It is impor- tant to note that the wavefunctions of the ground quasidoublet, formed by these two nearly degenerate Stark singlets that have different irreps A and B of the Cs group, can also be approximated very well by a linear combination of pure |J,±MJ> states of the type |6, ±6> [14,16]. At the same time, the wavefunctions of the first excited states at 163 cm–1 and 199 cm–1 of the ground multiplet 7F6 also have different symmetries A and B [16]. They are “admixed” by the external magnetic field H with the wavefunctions of the ground quasidoublet. The wavefunctions for the excited states are represented by linear combinations of |6,±5> states [16]. As a result, the Van-Vleck correction to the resulting magnetic moment of Tb3+ in TbAlO3 structure, can be ascertained by examining the transverse magnetic moment along the y-axis (or x-axis) of the local coordinate system. Figure 1.12 shows the temperature dependence of the inverse magnetic susceptibility of the TbAlO3 crystal measured along the crystallographic direction [110]. The experimental data (in black squares) show that the temperature dependence of the inverse magnetic susceptibility [110]

–1 is non-linear for high temperature T. Further-more, the inverse magnetic susceptibility of TbAlO3 shown in Refs. [14,16,59] has a noticeable anisotropy in the ab-plane of the rhombic crystal over the same temperature range. Similar behavior of the magnetic properties of TbAlO3 can be explained by the significant contribution to the magnetic moment of the RE sublattice that results from the mixing of the wavefunctions of states of excited Stark levels within the ground 7F6 multiplet of Tb3+ ion. This mixing is especially important for the interpretation of the experimental data above 100 K. Indeed, with an increase in temperature up to 300 K, the behavior of the magnetic properties of TbAlO3 becomes more complicated. In this region, the temperature depend- ence of [110] can be explained by a significant contribution from the Van-Vleck mixing of thermally populated excited states represented by Stark levels found between 160 and 365 cm–1. The inset in Figure 1.12 shows a diagram of the nine lowest-energy Stark levels of 7F6 multiplet involved in Van-Vleck “mixing” [58].

The eigenvectors representing these states are listed in Table 1.3 and were used to calculate the molar susceptibility ( )[110]

m for TbAlO3 in the ab-plane of the rhombic crystal as,

22

( ) ( ) ( ) ( ) ( ) 2 2[110] 0

| | | || | | |1 12 2

2 2m m m m m xz

a b V V B A n kn m i km n i k

i J kn J mg N

E E E E

(1.63)

( ) ( ) 2 ( )0cosm m m

a V V (1.64)

( ) ( ) 2 ( )0sinm m m

b V V , (1.65)

where NA is Avogadro’s number; En and Em are the energies of the “mixing” states; |n>, |m> and |i>, |k> are the real wave functions of the non-degenerate Stark sublevels; n is the Boltzmann population; ( )m is the contribution to a magnetic susceptibility due to differences in the population of the Stark levels shown in the inset (longitudinal suscepti-bility); and ( )m

V V is the Van-Vleck contribution to a magnetic susceptibility (having an isotropic nature in comparison with the magnetic susceptibilities ( )

,m

a b along the a- and b-axes of rhombic crystal) [58]. Using the wave functions and the Stark levels from the far right column of Table 1.3, expressions for the longitudinal ( )m and transverse ( )m

V V magnetic susceptibility to ( )[110]

m for the temperature range between 10 and 300 K may be calculated [56] to obtain the following expressions,

26

0 50 100 150 200 250 3000

1

2

3

4

5

6 Aρ7ρ

6

ρ5

ρ4

ρ3

ρ2

ρ1

ρ0

A

AA

B

B

B

BA

7F6

(257)

(295)

(343)(365)

(207)

(201)

(163)

(0)In

vers

e m

ag

netic

sus

cept

ibili

ty (

cm3 /g

ram

m),

10

3

alo

ng

[11

0]-a

xis

T(K)

Figure 1.12. The TbAlO3 inverse magnetic susceptibility 1

[110] in CGS units (gm/cm3) as a function of the absolute temperature (T

in K): squares present experimental data from [58], circles give 1

[110] calculated using expressions (1.63) - (1.67). The inset presents

the schematic diagram of Van-Vleck mixing (red arrows) between nine Stark singlets of the 7F6 ground multiplet.

( ) 20 0 1 2 2 4 3 5

4 7 5 6

3 70.224 32.42 10.068 11.361

16 52 83.5 127

9.371 16.724

158 69

m

P

gT

(1.66)

( ) 20 0 1 0 2 1 6 2 5

3 7.728 9.345 1.318 14.95

16 234 286 207 138m

V V g

, (1.67)

where 0 is the Boltzmann population of the Stark levels of the ground state quasi-doublet, and 1-7 are the Boltzmann populations of Stark levels at energies, 163, 199, 207, 257, 295, 343, and 365 (cm–1), respectively; p = 5 K is the para- magnetic Curie temperature [3,14]; and g0 = 1.5 is the Lande’ factor of the 7F6 ground multiplet.

In determining values for ( )[110]

m , it is important to take into account that the Stark singlets are non-spin states, so that the Van-Vleck’s magnetic moment can be oriented only along the direction of an external magnetic field. The inverse magnetic susceptibility as a function of temperature, based on the calculated susceptibility obtained from Eqs. (1.63) - (1.67) is plotted in Figure 1.12. Here ( )

[110] [110] /m M , where M is the molecular weight of TbAlO3. Good agreement is found between the experimental and calculated values of the magnetic susceptibility [110] investigated in the tem- perature range between 80 K and 300 K. We conclude from the observed agreement that the group-theoretical irrep. labels and Stark level energies have been properly assigned for the nine lowest-energy Stark levels of the ground 7F6 multiplet for the non-Kramers Tb3+ ion in the orthoaluminate structure [58].

Ising RE ions in the garnet structure The Ising model has also been used successfully to describe the behavior of the magnetic properties of the rare-earth (RE) ions Dy3+, Tb3+, and Ho3+ in the D2 crystal field symmetry site of garnet (gallate or aluminate) systems. According to the Ising model, the magnetic moment of the Kramers (or non-Kramers) RE ion in the crystal is directed along a given symmetry axis of the garnet crystal [3]. For example, a typical Ising mag- net is dysprosium aluminum garnet Dy3Al5O12 (or diluted dysprosium-yttrium aluminum garnet DyxY3-xAl5O12), in which the ground state of the Dy3+ ion in the crystal field of symmetry D2 is a Kramers doublet with a highly anisot- ropic g-factor, whose z-component is gz g|| = 18.4 (gx, gy = g = 0.7) [60]10. It has been shown experimentally from 10The g-factor anisotropy is a consequence of the distribution of the highly anisotropic orbital 4f electronic distribution. For the different axes of the crystal there are various components of the orbital moment L

of the RE ion having the same value of the spin moment S

on which the CF does not

operate. That is, in the formula for the Zeeman energy H BE g mH the value of g is a tensor. If the magnetic properties of the crystal have a symmetry axis, it is possible to introduce values of g|| and g, relative to the symmetry axis, with the direction of the field Н along (or perpendicular to) this symmetry axis. In 4f magnets this distinction can be significant. These 4f ions that have this extremely anisotropic (uniaxial) behavior of the g-factor are called Ising ions. This behavior is similar to that of spin momenta in the original Ising model [3], in which moments MJ are established along one selected direction.

27

magnetic and magnetooptical data that the ground doublet is well described by the pure wavefunctions |15/2, ±15/2> (in a given coordinate system) with a quantization axis coinciding with the Ising axis, i.e., the “easy” magnetization axis of the Dy3+ ion in the structure of the Y3Al5O12 garnet (YAG) [60].

The Ising behavior of the magnetic moments of the Stark sublevels of the RE ions in the crystal field having D2 symmetry is what produces some of the unique magnetic properties of these RE garnets. Indeed, for Kramers RE ions, the Ising behavior of the g-tensor in the garnet structure may arise only in the case of a low-symmetry crystal-field envi-ronment and only for certain wavefunctions of Kramers doublets, whose expansions include nearly pure |J, MJ> states. According to the results of the crystal-field analysis of dysprosium-yttrium aluminum garnet (Dy3+:YAG) carried out by Grünberg et al. [60] the wavefunctions 1,2| of the ground-state Kramers doublet of the 6H15/2 multiplet of Dy3+can be written as,

1,2| 0.846 |15 / 2, 15 / 2 0.497 |15 / 2, 11/ 2 0.100 |15 / 2, 7 / 2

0.149 |15 / 2, 3 / 2 0.000 |15 / 2, 1/ 2 0.020 |15 / 2, 5 / 2

0.041|15 / 2, 9 / 2 0.019 |15 / 2, 13 / 2

(1.68)

The ground state 1| is predominantly |15 / 2, 15 / 2 , which explains the large magnitude of the gz-factor (18, as compared with 20 for a “pure” |15 / 2, 15 / 2 state) and its pronounced anisotropy [60].

The magnetic behavior of Tb3+ ions in (gallate and aluminate) garnets is very similar to that of Dy3+, despite their electronic structure differences (Tb3+ has an even number of 4f electrons and Dy3+ has an odd number of 4f-electrons). This similarity arises from the fact that the Tb3+ ground state in the garnets consists of two nearly degenerate Stark singlets which form a quasidoublet state (Δ0 less than 3 cm–1). Furthermore, the wavefunctions of the quasidoublet ground state can be approximated by linear combinations of the pure |J,±MJ> states, |6,±6> [61]. Therefore, according to Griffith’s Theorem [62], the Tb3+ ion in the garnet structure is a true Ising ion. Since the Tb3+ and Dy3+ ions in the garnets are characterized by similar magnetic behavior of their ground states (Tb3+ an Ising quasidoublet and Dy3+ a quasi-Ising Kramers doublet), we find similar magnetic property behavior at low temperatures [51].

For a non-Kramers quasidoublet, the orientation of the Ising axis depends on the site symmetry of the RE ion and the symmetry of the wavefunctions forming the quasidoublet. In the case of D2 symmetry, the “Ising” axis coincides with one of the three two-fold rotation axes [3]. To find the precise orientation of the Ising axis, it is necessary to use a group-theoretical approach. Table 1.5 shows that the point group, D2, consists of four one-dimensional irreducible rep-resentations Γi (i = 1, 2, 3, 4). The operators xJ

, yJ

, zJ

transform as irreps Γ4, Γ2, and Γ3, respectively. If the quasidou-blet is formed by two singlets with wavefunctions |Гi> and |Гj>, the matrix element for the operator will differ from zero if the direct product of the irreducible representations satisfies the following condition: Гi Гk Гj = Γ1 (where Γ1 ≡ A is the unit representation of the D2 group). Alternatively, it may be stated that the irrep Гk is contained in the decomposi-tion of the direct product Гi Гj.

If the wavefunctions of quasidoublet |Гi> and |Гj> have different irreps, there will be one non-zero | |i k jГ J Г

matrix element. For example, for the wavefunctions |Γ2> and |Γ3>, the matrix element 2 3| |xГ J Г

is non-zero be-

cause the direct product Γ2 Γ3 = Γ4 and xJ

is transformed as the Γ4 irrep. In this case g = (gx, 0, 0) and the Ising axis coincides with coordinate x-axis. The “Ising” axis orientation for the other irrep combinations of the quasidoublet may be found using the irrep multiplication table for D2 symmetry given in Table 1.5. If the product Гi Гj = Γ4, the x-axis is the “Ising” axis. Similarly, if the product Гi Гj = Γ2 (or Γ3) the y-axis (or z-axis) is the Ising axis. And if the product Гi Гj = Γ1 ≡ A, (i.e, the irreps of the two states are the same) then it follows that the quasidoublet does not split in an external magnetic field, as all matrix elements | |i k iГ J Г

are identically zero.

It is important to note that the anisotropy of the RE ion g-tensor leads to an appearance of anisotropy in the Zeeman splitting in the RE garnets at low temperatures. From the investigations of the absorption spectra using linearly-polarized light in a magnetic field up to 20 kOe, Kolmakova et al. [63] found the components of the g-tensor and the magnitude of the crystal-field splitting of the ground state (quasidoublet) of the Tb3+ ion in Tb3Al5O12 and Tb3Ga5O12. Their results show that the Zeeman effect is strongly anisotropic, indicating that Tb3+ in the garnets exhibits “Ising” behavior.

The CF parameters determined by least-squares fitting in Ref. [63] are equal to D = (1.8 ± 0.4) cm–1 and gz = (14.7 ± 0.4) in Tb3Ga5O12, and D = (2.1 ± 0.4) cm–1 and gz = (16.3 ± 0.4) in Tb3A15O15. Within the limits of the experimental error, these values are identical to data obtained from magnetic measurements [51,61,64]. The splitting of the quasidou-blet by the x- and y-components of the magnetic field results in broadening of the absorption lines when H || [100] and [011]. The observed broadening falls within the measurement error and does not exceed 0.5 cm–1, which gives for the upper limit of the effective transverse components of the g-tensor: gx, gy < 0.5.

The calculations carried out by the authors of Ref. [63] have shown that the magnetic field mixes higher-energy states

28

of the ground multiplet into the ground-state quasidoublet by a very small amount. The splittings of the quasidoublet in a field of 20 kOe oriented along the local x- and y-axes (i.e., perpendicular to the Ising z-axis), caused by the admixing of the excited states, are equal to 0.6 cm–1 for Tb3Ga5O12 and 0.4 cm–1 for Tb3Al5O12. However the mixing of wavefunc- tions in the multiplets 7F6 and 5D4 in a magnetic field should affect more strongly the intensity of the absorption lines. Analysis of the mixing of the wavefunctions of an isolated quasidoublet has shown that the order of magnitude of the rate of enhancement of the optical transition from the lower level of the quasidoublet is close to that observed experi- mentally [63].

Thus the observed anisotropy of the Zeeman effect, as well as the influence of the mixing of the wavefunctions of an isolated quasidoublet on the intensity of the absorption lines confirm that the Tb3+ ion in paramagnetic garnets behaves like an “Ising” ion.

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