Chapter 1. Magnetism of the Rare-Earth Ions in Crystals

Chapter Magnetism of the Rare-Earth Ions in Crystals 3

. 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... 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 RE 3+. The ground electronic configuration of the RE ions may be written as [Xe]4f n, where [Xe] = s s p 6 3s 2 3p 6 3d 0 4s 2 4p 6 4d 0 5s 2 p 6 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 = for Ce 3+ to n = 3 for Yb 3+. The characteristic magnetic moment for each RE ion leads to an interaction between that ion and an applied external magnetic field H, producing interesting magnetic and magnetooptical features 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 multiplicities of states and the energy-level positions of excited electronic configurations relative to the ground state. Energy intervals for 4f n, 4f n- 5d, 4f n- 6s, and other excited electronic configurations for free RE ions are presented in Refs. [,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- 5d, 4f n- 6s, are separated from the 4f n ground state by an energy interval typically on the order of 0 5 cm. 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 values of L and S has (2L + )(2S + ) degenerate states distinguished by the m L and m S projections of the orbital and spin momenta. The wave functions of these degenerate states are given by LSm L m S >, where L m L L, S m S 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 0 4 cm [3-5]. To determine the ground term of the 4f n electronic configuration, one usually applies Hund s rules and the Pauli exclusion principle [4,6]. Hund s first two rules assert: ) 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 = S max ), 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 Tb 3+ ion that has eight electrons in the unfilled 4f 8 electronic configuration. In this regard, we can construct Table. for the orbital (m l ) and spin (m s ) 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 (m s ); Hund s second rule indicates that the eighth electron will fill one of the remaining (opposite-spin) states having the largest angular momentum (m l ). This is shown in Table.. In spectroscopic language, the ion Tb 3+ has a ground term of 7 F, where the conventional notation is (2S + ) L with (2S + ) being the multiplicity. The quantum degeneracy of the RE ion terms is removed by the spin-orbit interaction W SO that has a value on the order of 0 3 cm. 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, H L S, (.) LS Table.. Arrangement of the orbital (m l ) and spin (m s ) momentum projections in the 4f 8 electronic configuration of Tb 3+. The maximum number of f electrons in the shell is N = 4. m s m l 3 2 0 + +2 +3 4

where L and S are the operators of the orbital and spin momenta, respectively, and is the spin-orbit coupling constant defined by the well-known Goudsmit formula [5]. From a qualitative point of view, the spin-orbit interaction W SO 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, W SO can be written as [7], e l s WSO, (.2) 3 2mc r 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 + ) 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,M J > [8]. Each multiplet is many-fold degenerate in terms of its angular momentum projection M J ; 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 + ) L J multiplet split into (2J + ) equidistant sublevels. The energy interval between sublevels is defined by the magnetic field intensity and by the value of the g-factor. We write the Hamiltonian (.) in the form, 2 H LS J L S 2. (.3) For the diagonal matrix elements that define the multiplet energy E we obtain, EJ JJ LLSS 2. (.4) From this expression, the Lande rule of intervals is determined as, EJEJ JJ JJ J 2, (.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 Tb 3+ to Yb 3+ ) 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, J 0 = L 0 + S 0. For the light RE ions (from Ce 3+ to Gd 3+ ) 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, J 0 = L 0 S 0. 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.. 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.2. cm cm Figure.. Splitting scheme for the energy levels of the free rare-earth ions [3]. 5

Table.2. Ground and first excited energy levels of the free RE ions. RE 3+ Ground electronic configuration Ground term Ground multiplet First excited multiplet E E 0 (cm ) Ce 4f 2 F 2 F 5/ F 7/200 Pr 4f 2 3 H 3 H 4 3 H 5 2200 Nd 4f 3 4 I 4 I 9/2 4 I /2 800 Pm 4f 4 5 I 5 I 4 5 I 5 600 Sm 4f 5 6 H 6 H 5/2 6 H 7/2 000 Eu 4f 6 7 F 7 F 0 7 F 350 Gd 4f 7 8 S 8 S 7/2 Tb 4f 8 7 F 7 F 6 7 F 5 2300 Dy 4f 9 6 H 6 H 5/2 6 H 3/2 3400 Ho 4f 0 5 I 5 I 8 5 I 7 5000 Er 4f 4 I 4 I 5/2 4 I 3/2 6400 Tm 4f 2 3 H 3 H 6 3 H 5 8200 Yb 4f 3 2 F 2 F 5/2 000 The classification of free RE ion states is based on the Russell-Saunders approximation (also called normal- or LScoupling), which requires that the energy separation between the terms be much greater than the value of the termsplitting 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]. Nevertheless, 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- 5d and 4f n- 6s configurations of the free RE ions. As shown by the results from direct calculations (see, for example, Refs. [9,0]), the energy value of the d electron interaction with the f electrons of the 4f n- core of the 4f n- 5d configuration is on the order of 0 4 cm, while the value of the spin-orbit interaction for the d electron is on the order of 0 3 cm. The lower states of the 4f n- 5d (or 4f n- 6s) 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- core with the quantum numbers l and s of the valence 5d (or 6s) electron [4,5,7]..2. Paramagnetism of the Free RE Ions Let us consider the interaction between the free RE ion (as defined in Sec..) 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, H Z BL2S H B J S H. (.6) If the field H is directed along the z-axis of the coordinate system, the Hamiltonian can be written in the following form, H Z B J Z S Z H. (.7) Based on the state wave functions, LSJM J >, that are distinct for each state, we can write the matrix elements for this Hamiltonian as [3], LSJM H LSJM g M H, (.8) where and g J 2JJ J J S S L L 2 F 7/2 J Z J J B J is the Lande factor for the multiplet of the RE ion, LS J M J H Z LSJM J g JBH J M J J LS 2LJ SJ LS J LS 2 4J 2J 2J 3 where g J. Let us consider now the behavior of an ensemble of free RE ions in an external magnetic field. In this case, the magnetic 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 /2 /2 (.9) 6

magnetic field must also be quantized. This additional interaction energy W H is given by the expression, W g M H. (.0) H J J B As the projection M J of the angular momentum J takes on (2J + ) values, with J M J J, W H 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 gjbh. The magnetization of N ion ensembles, each having similar equidistant structure in the energy spectrum, is isotropic and can be represented as [3,], I IB 0 J x, (.) where BJ x is the Brillouin function [], 2J 2J BJ x coth x coth x 2J 2J 2J 2J, (.2) g JBJH and x. In eqn. (.), I 0 = Ng JBJ is the saturation magnetization of N ion ensembles at T = 0 K (for example, for the RE ion Gd 3+, I 0 = 7 B ). kt Keeping only the first terms in the series expansions of the coth(x) terms in eqn. (.2) allows us to simplify eqn. (.) for high T and a low H (i.e., x << ) as, NJ J g J I B H, (.3) 3kT where k is the Boltzmann constant. I corresponds to the expression for the paramagnetic susceptibility, which can be written as, C, (.4) T 2 where C is the Curie constant NJ J gjb NM J C, and M J is the atomic magnetic momentum. Expression 3k 3k (.4) 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,,2], C, (.5) T p where p is the Weiss constant, often called the paramagnetic or Curie temperature [0], 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 Eu 3+ and Sm 3+ [3,2]..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 N AJ J gjb 2 If the number of ions N equals Avogadro's number N A, then we can write []: C gj JJ and the magnetic susceptibility 3k 8 3 2 m (calculated on the basis of one mole) is: cm g J J of one gram) as: (cm 3 /gram) = m M 7 m J. We can also write the specific magnetic susceptibility (calculated on the basis 8T, where M is the molecular weight. In addition, the magnetic susceptibility V calculated based on a unit of volume (cm 3 m ) is: V, where is the density (in gram/cm 3 ). In this way we see that V is a dimensionless quantity! Therefore, the M magnetization M V 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 M V can be expressed in units of Gauss (gs), whereas the dimensionality of the external magnetic field is in units of Oersted (Oe).

aluminates) and orthoaluminates. For RE ions having an odd number of 4f electrons (e.g., Sm 3+ (4f 5 ) and Dy 3+ (4f 9 )), the low crystallographic site symmetry of the RE 3+ ion in the garnets (D 2 ) and orthoaluminates (C s ) splits each multiplet of the 4f n electronic configuration into (J + /2) discrete Kramers doublets. By contrast, for RE ions having an even number of 4f electrons (e.g., Tb 3+ (4f 8 ),and Ho 3+ (4f 0 )), the crystal field completely removes the degeneracy of the energy (Stark) sublevels, resulting in (2J + ) singlets for each multiplet. Due to the shielding of the 4f valence electrons from the environment by the filled 5s and 5p shells (see.), the crystal field (CF) interaction is substantially smaller than the spin-orbit coupling (W CF << W SO ). 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 behavior 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 [3,3]. 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 degeneracy 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 characteristics of the electronic states. As examples, we will consider RE ions in sites with low symmetry in orthoaluminates (C s ) and garnets (D 2 ), and discuss briefly the effect of these low-symmetry environments on the form of the wave functions and the character of the RE ion energy spectra. RE ions in a CF of C s symmetry For orthoaluminates, orthochromites, and orthoferrites, the RE ions lie in sites surrounded by a distorted perovskite structure having C s point-group symmetry. The orthoaluminates, usually written as RAlO 3, where R 3+ is the trivalent rare-earth ion, have a crystalline structure that is described by the spatial group D 6 2h - P bnm [4,5]). An elementary cell consists of 4 formula units of RAlO 3, with the 4 RE ions located in two crystallographically non-equivalent C s 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 environment 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 C s group possesses only two one-dimensional irreducible representations, А and В, (also known as Γ and Γ 2, 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]): A z В; A y, A x А. A scheme for the two crystallographic non-equivalent positions of the RE ion in the RAlO 3 structure is given in Figure.2; these positions are distinguished by the orientation of the low-symmetry crystalline environment (С s symmetry). Consequently, anisotropy axes appear in the structure. As a result, the crystal field is responsible for the orientation of the magnetic moments in RAlO 3. The magnetic moments of the RE ions (e.g., Tb 3+ or Dy 3+ ) in RAlO 3 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 С s symmetry, each multiplet is split into (2J + ) singlets. The wavefunctions 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 quasidoublet structure observed in the spectra is formed by two closely spaced singlet levels (the value of the energy gap generally being between and 3 cm [3,4]). The optical and magnetic properties of TbAlO 3 have recently been examined [5]. The authors used crystal-field modeling techniques to assign all 58 experimentally determined Stark levels within the 7 F J (J = 6, 5,, 0) and 5 D 4 multiplet manifolds, with a fitting standard deviation of 4.5 cm (3.8 cm rms error). As a further test, the theoretical Stark levels and calculated wavefunctions were used to determine the temperature dependence of the magnetic susceptibility of the TbAlO 3 crystal. Good agreement was obtained between the calculated susceptibility and the previously re- 2 The forces responsible for the ordering of the RE ions in the RAlO 3 structure and defining the type of the magnetic (or spin) configuration of the RE ions at low temperatures are relatively small (the Néel temperature Т N being on the order of several K) and have an exchange (dipole) character [3]. 8

c b a Figure.2. Schematic diagram of the two crystallographic non-equivalent positions of the RE 3+ ion in the RAlO 3 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 7 F 6 multiplet. Calculated eigenvectors (wavefunctions) for the 3 Stark components of the ground multiplet manifold 7 F 6 of non- Kramers Tb 3+ ion in the CF symmetry C s are given in Table.3 (see also Ref. [6]). 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. [6]. 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 C s group, taking into account the transformation properties of the non-spin J,M J > states: A A and B B i h i i h i. The wavefunctions given in the right-hand side of Table.3 were calculated in the Ising coordinate system of Tb 3+ in the CF of C s symmetry (see.5). Ising notation wavefunctions have been used for the TbAlO 3 magnetic susceptibility calculations presented in.5. Now let us consider the RE ions having an odd number of 4f electrons. The Kramers-ion energy spectrum consists of (J + /2) Kramers doublets, each doublet described by the Kramers-conjugate wavefunctions, J /2 C M J 2p p0 p J /2 * p 2 p0 C M p J. (.6) The coefficients C p are defined by the values of the CF parameters (each doublet has its own set of C p coefficients). As an example of a Kramers-ion system, we consider the recent examination of Er 3+ ions in Er 3+ :YAlO 3 [7]. The authors modeled 34 experimentally observed Stark levels, split out from the 30 multiplets with energies below 44000 cm, using a parametrized Hamiltonian defined to operate within the 4f electronic configuration for Er 3+ ions doped into C s sites of Er 3+ :YAlO 3. For convenience of discussion, the total Hamiltonian can be partitioned as, H H A H CF (.7) where H A 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 (F 2, F 4, F 6 ), the spin-orbit coupling 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, P 2, P 4, P 6 ). Parameter E ave 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 H A as, H E F k f L L G G G R T i t A P k p M j m, (.8) 2 7 4 A ave k i f so k j k i k j where k = 2,4,6; i = 2,3,4,6,7,8; j = 0,2,4; f k and A so represent the angular part of the electrostatic and spin-orbital interacttions, respectively; L is the total orbital momentum; G(G 2 ) and G(R 7 ) are the Casimir operators for the Lie groups G 2 and R 7 ; the t i are the three-particle operators; and p k and m j represent the operators for the magnetic interactions. The H CF denotes the non-spherically-symmetrical part of the one-electron crystal field. The crystal-field Hamiltonian may be expressed (in Wybourne notation [4]) as, 9

H k k CF BC q q kq, where k = 2,4,6 and q k is restricted by the C s site symmetry to be even. Radially dependent parts of the one-electron k k crystal-field interactions are contained in the B q parameters, and the C q are many-electron spherical tensor operators acting within the 4f n electronic configuration. For lanthanide ions occupying C s symmetry sites, group-theoretical considerations allow for 5 crystal field parameters: three pure-real B k k k 0 (k = 2,4,6) parameters plus six complex Bq isq 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 5 crystal-field parameters to be arbitrarily fixed, leaving 4 independent crystal-field parameters [8,9]. Standard convention selects the imaginary part of the rank-2 parameter, S, to be set equal to zero. 2 2 Table.3. Wavefunctions in the J, M J basis of the 7 F 6 ground multiplet Stark sublevels in TbAlO 3, from Ref. [6]. Energy (сm ) C S Irrep Wavefunctions in c-axis notation Real wavefunctions in Ising system notation 4.3 A = e 30.8i [0.7567 6,0> + 0.4262(e 9.5i 6,+2 + e 9.5i 6,2) = + 0.6724( 6,+6 + 6,6) + 0.0890(e 55.9i 6,+4 + e 55.9i + 0.42( 6,+4 + 6,4) 6,4)] 4.5 B 2 = e 85.i [0.6525(e 86.5i 6,+ e 86.5i 6,) 2 = + 0.6734( 6,+6 6,6) + 0.297(e 67.8i 6,+3 e 67.8i 6,3)] + 0.378( 6,+4 6,4) 68 B 85 A 2 A 253 B 296 B 349 A 366 A 433 B 443 B 54 A 59 A 3 = e 8.2i [0.6327(e 6.4i 6,+ e 6.4i 6,) + 0.2557(e 2.7i 6,+3 e 2.7i 6,3) + 0.0803(e 87.9i 6,+5 e 87.9i 6,5)] 4 = e 20.i [ 0.647(e 89.7i 6,+2 + e 89.7i 6,2) 0.909(e 72.4i 6,+4 + e 72.4i 6,4) + 0.540 6,0>] 5 = e 34.8i [ 0.5797 6,0> + 0.5333(e 3.6i 6,+2 + e 3.6i 6,2) + 0.360(e 5.5i 6,+4 + e 5.5i 6,4)] 6 = e 22.6i [ 0.6357(e 22.6i 6,+3 e 22.6i 6,3) + 0.2442(e 2.2i 6,+ e 2.2i 6,) 0.04(e 5.5i 6,+5 e 5.5i 6,5)] 7 = e 59.3i [0.6265(e 70.0i 6,+3 e 70.0i 6,3) 0.234(e 69.6i 6,+ e 69.6i 6,) + 0.860(e 89.7i 6,+5 e 89.7i 6,5)] 8 = e 48.3i [ 0.672(e 3.i 6,+4 + e 3.i 6,4) 0.334(e 36.i 6,+6 + e 36.i 6,6) + 0.077(e 3.0i 6,+2 + e 3.0i 6,2)] 9 = e 42.3i [0.6285(e 77.8i 6,+4 + e 77.8i 6,4) 0.902(e 73.5i 6,+2 + e 73.5i 6,2) 0.866(e 56.4i 6,+6 + e 56.4i 6,6) + 0.278 6,0>] 0 = e 39.0i [ 0.6790(e 7.2i 6,+5 e 7.2i 6,5) + 0.099(e.0i 6,+3 e.0i 6,3)] = e 50.8i [0.6584(e 82.4i 6,+5 e 82.4i 6,5) + 0.604(e 82.7i 6,+3 e 82.7i 6,3) 0.23(e 37.4i 6,+ e 37.4i 6,)] 2 = e 3.9i [0.6725(e 32.5i 6,+6 + e 32.5i 6,6) 0.238(e 5.4i 6,+4 + e 5.4i 6,4) + 0.080(e 2.4i 6,+2 + e 2.4i 6,2)] 3 = e 80.4i [ 0.6602(e 57.5i 6,+6 + e 57.5i 6,-6) 0.556(e 76.0i 6,+4 + e 76.0i 6,-4) + 0.80(e 74.0i 6,+2 + e 74.0i 6,-2)] 3 = 0.5330( 6,+5 + 6,5) 0.3347( 6,+3 + 6,3) 0.2608( 6,+ + 6,) 4 = 0.624( 6,+5 6,5) 0.237( 6,+3 6,3) 0.426( 6,+2 + 6,2) 5 = + 0.4639( 6,+2 + 6,2) + 0.335( 6,+4 + 6,4) + 0.4659 6,0 0.540( 6,+5 6,5) 0.006( 6,+6 + 6,6) 6 = 0.4753( 6,+ + 6,) + 0.4072( 6,+5 + 6,5) 0.2592( 6,+3 + 6,3) 0.00( 6,+2 6,2) 7 = + 0.656( 6,+4 6,4) + 0.2494( 6,+2 6,2) 0.225( 6,+ + 6,) 0.264( 6,+6 6,6) 8 = + 0.5584( 6,+4 + 6,4) 0.445 6,0 0.909( 6,+2 + 6,2) 0.099( 6,+6 + 6,6) 9 = + 0.586( 6,+3 6,3) 0.2275( 6,+5 6,5) + 0.2208( 6,+ 6,) + 0.233( 6,+4 + 6,4) 0.787 6,0 0 = 0.4506( 6,+3 + 6,3) + 0.3940( 6,+ + 6,) 0.2533( 6,+2 6,2) + 0.834( 6,+4 6,4) + 0.348( 6,+5 + 6,5) = + 0.5743( 6,+2 6,2) 0.2990( 6,+3 + 6,3) 0.973( 6,+4 6,4) + 0.05( 6,+ + 6,) 2 = 0.4952 6,0 + 0.4923( 6,+ 6,) 0.2624( 6,+3 6,3) + 0.905( 6,+2 + 6,2) 3 = + 0.5042 6,0 + 0.408( 6,+ 6,) 0.4030( 6,+2 + 6,2) + 0.90( 6,+4 + 6,4) (.9) 0

Crystal-field energy-level parameters may be determined using a Monte-Carlo method originally developed for intensity 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. [7] for all energy levels up to 55000 cm. Wavefunctions generated from the crystal-field splitting modeling calculations were used to predict the temperaturedependent and orientation-dependent magnetic molar susceptibility (m) of the ground-state multiplet 4 I 5/2, as described in Ref. [7]. Excellent agreement was obtained between the calculated and the experimental susceptibility data (see also.4). The agreement also serves as an independent check of the crystal-field splitting analysis for Er 3+ substituted into Y 3+ sites of C s symmetry in the orthoaluminate structure. Wavefunctions for the lowest Stark components of the ground multiplet 4 I 5/2 of Er 3+ in the orthoaluminate structure are given in Table.4 (excerpted from Table III of Ref. [7]). RE ions in a crystal field of D 2 symmetry The RE paramagnetic garnets having a general formula of R 3 M 5 O 2 or R 3+ :Y 3 M 5 O 2 (where R 3+ is the trivalent rare earth that replaces some or all of the yttrium ions in the crystal, and M is the metal ion Al 3+ or Ga 3+ 0 ) have a cubic structure described by the space group symmetry O h. An elementary cell of the crystal contains eight formula units, i.e., 24 R 3+ /Y 3+ ions, 40 M 3+ ions and 96 O 2 ions. In the elementary cell of R 3 M 5 O 2, the rare earth ions, having a large ionic radius, occupy the c-sites, while the Al 3+ and Ga 3+ ions with a smaller radius occupy the a- and d-sites [3,2]. It is usually the case that the RE ions, when substituted for Y 3+, are located randomly 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 D 2 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 [00], [00], [00], respectively, as shown in Figure.3. For rare-earth garnets and isomorphous compounds, where the non-kramers RE ions occupy the dodecahedral positions described by the D 2 point symmetry, the symmetry group contains four elements: the identity transformation Е, and the rotation by 80 around the three mutually perpendicular x-, y- and z-axes. The D 2 group has four onedimensional irreducible representations (irreps) A, B, B 2, and B 3, which is equivalently designated in the literature as Г i, where the identification is given: A = Γ, B = Γ 2, B 2 = Γ 3, and B 3 = Γ 4 [3,22,23]. These irreps satisfy the requirement of group theory multiplication given in Table.5. Because the majority of the rare-earth garnet literature uses the Г i notation, this is the notation we will use in what follows. Table.4. Wavefunctions in the J,M J > basis for the lowest two Kramers doublets of the 4 I 5/2 multiplet in Er 3+ :YAlO 3 [7]. Energy (cm ) Expt. Calc. Irrep 0 2 Γ 3/2 0 2 Γ /2 49 52 Γ 3/2 49 52 Γ /2 Wavefunction Ψ = +0.566 e +88.44i 5/2,5/2> +0.457 e 5.60i 5/2,7/2> +0.3625 e 53.66i 5/2,3/2> +0.3400 e 53.64i 5/2, 5/2> +0.306 e +76.52i 5/2,/2> +0.2382 e 60.4i 5/2, /2> 0.89 e.78i 5/2, 3/2> 0.04 e +2.76i 5/2, 9/2> Ψ 2 = +0.566 e 88.44i 5/2, 5/2> +0.457 e +5.60i 5/2, 7/2> +0.3625 e +53.66i 5/2, 3/2> +0.3400 e +53.64i 5/2,5/2> +0.306 e 76.52i 5/2, /2> +0.2382 e +60.4i 5/2,/2> 0.89 e +.78i 5/2,3/2> 0.04 e 2.76i 5/2,9/2> Ψ 3 = e +6.44i [ 0.600 e 29.78i 5/2, 5/2> 0.438 e 68.84i 5/2,5/2> +0.3850 e 35.88i 5/2,7/2> +0.3804 e +9.33i 5/2, 3/2> 0.3282 e 37.88i 5/2,3/2> +0.26 e +3.73i 5/2, 9/2>] Ψ 4 = e +6.44i [+0.600 e +29.78i 5/2, 5/2> +0.438 e +68.84i 5/2,5/2> 0.3850 e +35.88i 5/2,7/2> 0.3804 e 9.33i 5/2, 3/2> +0.3282 e +37.88i 5/2,3/2> 0.26 e 3.73i 5/2, 9/2>] [00] z z = 90 = 90 = 0 x y z z x y y Figure.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.7.

Table.5. Multiplication table for D 2 symmetry irreps Γ i. (Alternative A, B i notation is given in parentheses.) D 2 Γ (A) Γ 2 (B ) Γ 3 (B 2 ) Γ 4 (B 3 ) Γ (A) Г Г 2 Г 3 Г 4 Γ 2 (B ) Г 2 Г Г 4 Г 3 Γ 3 (B 2 ) Г 3 Г 4 Г Г 2 Γ 4 (B 3 ) Г 4 Г 3 Г 2 Г k q k q Bq 4 4 4 6 6 6 0, 2, 0, 2, 4, 0, 2, 4 The D 2 symmetry crystal field splits non-kramers ion multiplets into (2J + ) singlets. The wavefunction for each Stark sublevel singlet transforms according to one of the four one-dimensional irreps Γ i of the D 2 point group that it is characterized by, as presented in Table.6. The electronic energy level structure of R 3+ :YAG (or R 3+ :YGG) is analyzed by means of a model Hamiltonian defined to operate within the 4f n electronic configuration of RE ions, as described by eqns. (.7 -.9) above. The crys- k tal-field parameters q, which, for D 2 site symmetry, yields 9 independent pure-real crystal-field parameters: B B B B B B B B, and B 6 6. 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 D 2 sites per unit cell [9, 24-26]. Additionally, because there exist three mutually orthogonal but inequivalent C 2 symmetry axes in D 2 symmetry, different orientations of the crystal-field axes will yield differing crystal-field parameter sets with identical calculated energy levels [25,27]. That is, for the D 2 symmetry system examined here, the crystal-field z-axis may be chosen parallel to any one of three orthogonal C 2 symmetry axes. Additionally, for each of these three z-axis orientations, there exist two orientations of the x- and y-axes along the two remaining C 2 symmetry axes, resulting in six alternative 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 D 2 symmetry irreps Г 2, Г 3 and Г 4, with Г 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 Tb 3+ :YAG, Bayerer et al. [3], chose x-parallel to the [00] axis, perpendicular to the yz-plane, while Gruber et al. [32] chose z-parallel to the [00] axis, perpendicular to the xy-plane. Thus, the Bayerer et al. [3] symmetry label set {Г,Г 3,Г 4,Г 2 } corresponds to the Gruber et al. [32] set {Г,Г 2,Г 3,Г 4 }. Once this notational difference is accounted for, the calculated crystal-field splitting and symmetry labels of Bayerer et al. [3] and Gruber et al. [32] for Tb 3+ :YAG closely correspond, even though they superficially appear to be very different. Detailed symmetry identification of the experimental crystal-field sublevels of the multiplet manifolds of the non-kramers RE ions Tb 3+ [32], Ho 3+ [33], Tm 3+ [34], Pr 3+ and Eu 3+ [35,27] in the garnet structure were performed on the basis of an algorithm using the ED and MD selection rules for D 2 symmetry. In these systems, there arise cases where two closely spaced non-degenerate singlet levels may not be directly resolvable in the optical experiments, as the energy differences are about to 3 cm [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 [27,3]. Likewise, the g-tensor components determine the Zeeman contribution to the magnetooptical and magneto-resonance properties of the non-kramers RE ion states. Indeed, according to the data obtained k B are constrained by the expression, B Table.6. Wavefunction symmetries corresponding to each of the D 2 irreps. Label M J Wavefunction Γ (A) even JM J, M J, M J Γ 2 (B ) even Γ 3 (B 2 ) odd J Γ 4 (B 3 ) odd M J, M J, M JM J, M J, M M J, M J, M 3 This is clear in A, B i notation, where the six axes orientations result in the six possible permutations of the three B i (i =, 2, 3) symmetry irreps, with A remaining invariant. 2

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 Tb 3+ and Ho 3+ in YAG and YGG crystals [32,33,36]. For example, according to the numerically calculated energy spectra and wavefunctions of the Stark sublevels of Tb 3+ ions in YAG, presented in Table.7, both the ground and first excited states of the 7 F 6 and 5 D 4 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 Tb 3+ in the garnet structure in which the z-axis is parallel to the [00] 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 Tb 3+ in Y 3 Ga 5 O 2 (YGG) and Bayerer et al. [3] for Tb 3+ in Y 3 Al 5 O 2 (YAG), where xwas chosen parallel to the [00] axis and perpendicular to the yz-plane. The coordinate transformations of the wavefunctions 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.3. If prime labels ( A n ) are used to identify the symmetry labels used by Guillot et al. [38] and Bayerer et al. [3], the corresponding labels used by Gruber et al. [32] are related as follows: Г A, Г2 A 3, Г3 A 4 and Г 4 A 2. 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 [00] axis of crystal) of the transformed (rotated) local coordinate system (x, y, z) as, J MM J JM DMM JM M im J,, MM im, (.2) J dmm are calculated and tabulated for the value of, (.20) D e d e J where D,, are the Wigner D-functions and functions J = 6 in Ref. [40]. As a result, the wavefunctions of the first excited quasidoublet of the 7 F 6 multiplet of Tb 3+ : YAG can be represented in the transformed local coordinate system as given in column 5 of Table.7, Таble.7. Wavefunctions of some Stark sublevels of the 7 F 6 and 5 D 4 multiplets of Tb 3+ :YAG given in two alternative orientation schemes [37]. Stark level energy (сm ) Wavefunctions in the J, M J basis Local axis-z axis 00 of crystal Local axis-x axis 00 of crystal Irrep Wavefunction Irrep Wavefunction = 0.95 6,0 Г + 0.233 ( 6,+2+ 6,2) 0.42 ( 6,+4 + 6,4) 0.0 Г 0.387 ( 6,+4 + 6,4) = 0.562 ( 6,+6 + 6,6) 0.72 ( 6,+2 + 6,2) 5.0 Г 3 22 Г 24 Г 2 2054 Г 2 = 0.577 ( 6,+6 6,6) + 0.382 ( 6,+4 6,4) + 0.4 ( 6,+2 6,2) 5 = 0.630 6,0 + 0,536 ( 6,+2 + 6,2) 0,06 ( 6,+4 + 6,4) 6 = 0.638 ( 6,+ 6,) 0.292 ( 6,+3 6,3) + 0.083 ( 6,+5 6,5) 50 = 0.532 ( 4,+4 + 4,-4) 0.366 ( 4,+2 + 4,2) 0.407 4,0 2056 Г 5 = 0.493 ( 4,+2 4,2) 3 Г 0.52 ( 4,+4 4,4) 4 20585 Г 54 = 0.449 ( 4,+ + 4,) 4 Г + 0.546 ( 4,+3 + 4,3) 2 20587 Г 20594 Г 3 20638 Г 2 55 = 0.38 ( 4,+4 + 4,4) + 0.0 ( 4,+2 + 4,2) + 0.830 4,0 Г 4 Г Г 3 Г Г 56 = 0.508 ( 4,+2 4,2) Г + 0.490 ( 4,+4 4,4) 4 57 = 0.692 ( 4,+ 4,) Г 0.45 ( 4,+3 4,3) 3 2 = 0.70 ( 6,+ + 6,) 5 = 0.672 ( 6,+6 + 6,6) + 0.29 ( 6,+4 + 6,4) + 0.083 ( 6,+2 + 6,2) 6 = 0.696 ( 6,+6 6,6) 0.076 ( 6,+4 6,4) 0.038 ( 6,+2 6,2) 50 = 0.095 ( 4,+4 + 4,4) + 0.696 ( 4,+2 + 4,2) + 0.4 4,0 5 = 0.643 ( 4,+3 + 4,3) 0.305 ( 4,+ + 4,) 54 = 0.7065 ( 4,+3 4,3) 55 = 0.45 ( 4,+4 + 4,4) 0.26 ( 4,+2 + 4,2) +0.797 4,0 56 = 0.637 ( 4,+ + 4,) 0.302 ( 4,+3 + 4,3) 57 = 0.698 ( 4,+4 4,4) 0.05 ( 4,+2 4,2) 3

5 Г [0.672( 6, 6 6, 6 ) 0.29( 6, 4 6, 4 ) 0.083( 6, 2 6, 2 ) 0.46 6,0 ] 6 Г 3 [0.696( 6, 6 6, 6 ) 0.076( 6, 4 6, 4 ) (.23) 0.038( 6, 2 6, 2 )] The wavefunctions of other Stark sublevels of the 7 F 6 and 5 D 4 multiplet manifolds are given in columns 2 and 3 of Table.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 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..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 HCF H Z eigenvalues, where H Z is the Zeeman Hamiltonian (eqn..6,.2). This problem was first solved by Penny and Schlap [4] with a calculation of the magnetic susceptibility of the RE compound Pr 2 (SO 4 ) 3 x-h 2 O. The matrix HCF H Z has been constructed in Ref. [4] for the wavefunctions for the 3 H 4 ground multiplet of the Pr 3+ ion. In addition, Penny and Schlap [4] showed that the crystal field produces significant anisotropy of the magnetic susceptibility of RE compounds at low temperatures; there is a qualitatively different character in its temperature dependence for different crystal axes. Indeed, according to the experimental data from Ref. [3], the behavior of the magnetic 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 [2], 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 = in a crystal field (CF) having the simplest form, 4 2 H D J. (.24) CF Here the energy levels and the wavefunctions of the RE ion in the CF of (.24) have the following form, E 0 = 0, 0 > = J,M J > =, 0>, E = D, > = J,M J > =, >. (.25) By calculating the components of the magnetic susceptibility using the Van-Vleck formula, we obtain [3], expen kt n J n kt 2 n J m Em En g N J B n n where are the components of the magnetic susceptibility of the RE ion in the crystal field, E n are the energy levels of the ion in the CF and g J is the Lande g-factor. From (.26), we can obtain the following expressions [3], 2gJBN zz, (.27) kt 2 expd kt xy z exp E n mn kt 2gJBN exp D kt D 2 exp D kt (.22) (.26). (.28) The temperature dependence of the magnetic susceptibility components and are given in Figure.4, which illustrates 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], 4 It is possible to show that at D < 0, we will have magnetic anisotropy of a type along the easy axis [3]. 4

zz /D/ /0> /> T/D Figure.4. Temperature dependence for the RE paramagnet magnetic susceptibility and at D < 0. 2gJBN. (.29) 3kT The susceptibility along the easy axis (z-axis) increases proportionally with /T, and at T << D it corresponds to the magnetic susceptibility of a two-level system with effective spin S = /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], gjbn. (.30) D 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 (Dy 3+, Er 3+ as examples) is made by using the CF Hamiltonian [42], k k k k k k H Re B C C iim B C C, (.3) CF q q q q q q kq, k where the complete specification of the CF requires the fitting of fifteen Bq 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 k C q operators are in the usual way expressed in terms of the 3j- and 6j-symbols [6,7,39,43], and the spectroscopic coefficients [44] in the intermediate coupled basis defined in Ref. [45] for the 4f 9 electronic configuration of Dy 3+. 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 (.26) can be found in the usual way [3,42] 5, gzz 2 gj m Jz m, (.32) gxx 2gJ Re m Jx m, (.33) g 2g Im m J m. (.34) yy J y Here m is the Kramers-conjugate wavefunction of the Kramers doublet. RE ions in a CF of C s symmetry To give an example, the wavefunctions of the ground Kramers doublet of Er 3+ ion in the CF symmetry C s (Table.4,.3) can be written as, 5 Note that the g-tensor is diagonal with the execution of the following condition: Im mj x m Re mjy m 0 5

i88.4 i5.6 i53.7 i53.6 0.57 e 5 / 2,5 / 2 0.45 e 5 / 2,7 / 2 0.36 e 5 / 2,3 / 2 0.34 e 5 / 2, 5 / 2 i76.5 i60. i.8 i2.8 m 0.3 e 5 / 2,/ 2 0.24 e 5 / 2, / 2 0.9 e 5 / 2, 3 / 2 0.0 e 5 / 2, 9 / 2 i m 88.4 5.6 53.7 53.6 i i i 0.57e 5 / 2, 5 / 2 0.45 e 5 / 2, 7 / 2 0.36 e 5 / 2, 3 / 2 0.34 e 5 / 2,5 / 2 i76.5 i60. i.8 i2.8 0.3 e 5 / 2, / 2 0.24 e 5 / 2,/ 2 0.9 e 5 / 2,3 / 2 0.0 e 5 / 2,9 / 2 (.35) where all angles (in the exponents) are given in degrees. Here we can see that the wavefunctions of the Kramers doublets are described by the doubly-degenerate states corresponding to the effective spin S = /2 and satisfy the following condition for Kramers conjugation, m CJM J, M, (.36) M JM M JM m C ( ) J, M, (.37) where C JM are the coefficients of the wavefunction on the J,M> basis states). L L These wavefunctions can be treated as spherical spinors JM, of rank /2 [6,39]. Spherical spinors JM, are eigenfunctions of the operators of full angular, orbital and spin (for spin S = /2) moments and can be expressed through spherical functions Y LM (,) and spin functions /2 z for particles with a spin of /2, according to the formula [39], L JM, C Y, (.38),/ 2 / 2 JM Lm z Lm m JM where CLm,/ 2 Lm z z JM 2 are Clebsch-Gordan coefficients [6,7,39] and the spin functions /2 z are real and a L represent the 2 matrices of type [39]. Spherical tensors JM, for complex conjugation are transformed by b using the formula [39], L, J LM L JM i yj M,, (.39) where the matrix i 0 y can be written as: i y. 0 Now let us apply expression (.38) to the wavefunctions (.35) of the ground state (Kramers doublet) of the 4 I 5/2 multiplet of Er 3+ in C s symmetry, which can be represented in a 2 matrix. We then obtain the complex conjugate wavefunction corresponding to the sublevel of the ground Kramers doublet which will then be, i88.4 i5.6 ( ) C JM 0.57e 5/2,5/2 0.45e 5/2,7/2 JM M i53.7 0.36e 5 / 2,3 / 2 i53.6 0.34e 5 / 2, 5 / 2 i76.5 0.3e 5 / 2,/ 2 i60. i.8 i2.8 0.24e 5 / 2, / 2 0.9e 5 / 2, 3 / 2 0.0e 5 / 2, 9 / 2 The numerical calculation obtained by using eqn. (.32) for the g-tensor z-component for the ground state (Kramers doublet) of the 4 I 5/2 multiplet for Er 3+ in the CF symmetry of C s is g zz = 0.68. To give another example, we can use the above mentioned considerations regarding the nature of the magnetic properties of the Kramers RE ions in a low-symmetry CF to show that the anisotropy of the temperature dependence of the paramagnetic susceptibility of DyAlO 3 can be described by the following expression [42], where, CF i J B z (.40) CF, (.4) i i i 8 N n J ( ) ( ) n / i n ix E kt iy niz n Ji m ix iy m iz T g e 2. (.42) ZCF n kt mn Em En In this expression, the Lande g-factor is g J = 4/3, N is the number of Dy 3+ ions per gram, Z CF 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 Dy 3+ Dy 3+ interaction. Experimental data from Ref. [42] for the magnetic susceptibility of DyAlO 3 confirm the theoretical expressions given in (.4-.42). 6

Figure.5 shows the temperature dependence of the inverse magnetic susceptibility for crystallographic directions [00] (b-axis) and [00] (c-axis) of the orthorhombic Er 3+ :YAlO 3 (Er 0.5 Y 0.5 AlO 3 ) crystal as determined in Ref. [7], 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 phasetransition (Ne el temperature), which has been reported to be 0.6K along the c-axis [47]. The maximum magnetic susceptibility of Er 3+ :YAlO 3 at low temperatures (T < 00 K) is observed along the c-axis while the susceptibility c is approximately 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.5. It appears to be the interaction between the Er 3+ ion in C s symmetry and the RE sublattice that leads to the strong anisotropy of the magnetic moment observed in the orthoaluminate structure, especially at low temperatures. Indeed, the additional contribution to the magnetization of the Er 3+ :YAlO 3 crystal that arises from the magnetic moments of the RE sublattice of the Er 3+ ions associated with the Van-Vleck mechanism is due to the mixing of excited states of the 4 I 5/2 multiplet with the ground-state Kramers-doublet when an external field H is applied [3,7]. As a result, a substantial contribution is made to the anisotropy of the Er 3+ : YAlO 3 magnetization, while the average magnetic moment of Er 3+ is connected with the difference in the population of the sublevels of the ground Kramers-doublet and defines the value of the Er 3+ :YAlO 3 magnetization. The Van-Vleck correction to the magnetic moment of the RE sublattice in Er 3+ :YAlO 3 is due to a mixing between the wavefunctions of the excited 4 I 5/2 Kramers-doublets with energies of 54, 66, 24, 267 and 386 cm and the wavefunctions of the 4 I 5/2 ground-state Kramers-doublet. The corresponding levels, their energies and the wavefunctions involved, listed in part in Table.4, are presented in Table III of Ref. [7] (see also inset of Figure.5). In the high temperature region (T 300 K), the behavior of the magnetic properties of Er 3+ :YAlO 3 can be explained by significant contributions from excited states located at energies of 66, 24 and 267 cm in the 4 I 5/2 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 C s [7], 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 Er 3+ 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 indicate 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. [7] can be used to calculate the molar ( ) magnetic susceptibility m c along the c-axis of Er 3+ :YAlO 3. The corresponding expression, given in Ref. [42], can be written (in the local coordinate system of Er 3+ ) as, Figure.5. The inverse molar magnetic susceptibility c Er 3+ :YAlO 3 in cgs units (mole/cm 3 ) as a function of the absolute temperature (T in K) [7]: () experimental data for the Er 0.5 Y 0.5 AlO 3 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 4 I 5/2 ground multiplet. Note that the Van-Vleck mixing between the Kramers-doublet states at 0.0 and 54 cm is forbidden by selection rules. 7