Room-temperature magnetic anisotropy of lanthanide complexes: A model study for various coordination polyhedra

JOURNAL OF CHEMICAL PHYSICS

VOLUME 116, NUMBER 11

15 MARCH 2002

Room-temperature magnetic anisotropy of lanthanide complexes: A model study for various coordination polyhedra Vladimir S. Mironov Institute of Crystallography, Russian Academy of Sciences, Leninskii prosp. 59, 117333 Moscow, Russia

Yury G. Galyametdinov Physical-Technical Institute, Kazan Branch RAS, Sibirsky Tract 10/7, 420029 Kazan, Russia

Arnout Ceulemans, Christiane Go¨rller-Walrand, and Koen Binnemansa) Katholieke Universiteit Leuven, Department of Chemistry, Celestijnenlaan 200F, B-3001 Leuven, Belgium

共Received 11 October 2001; accepted 20 December 2001兲 The dependence of the room-temperature magnetic anisotropy ⌬␹ of lanthanide complexes on the type of the coordination polyhedron and on the nature of the lanthanide ion is quantitatively analyzed in terms of a model approach based on numerical calculations. The aim of this study is to establish general regularities in the variation of the sign and magnitude of the magnetic anisotropy of lanthanide complexes at room-temperature and to estimate its maximal value. Except for some special cases, the variation of the sign of the magnetic anisotropy over the series of isostructural lanthanide complexes is found to obey a general sign rule, according to which Ce共III兲, Pr共III兲, Nd共III兲, Sm共III兲, Tb共III兲, Dy共III兲, and Ho共III兲 complexes have one sign of ⌬␹ and Eu共III兲, Er共III兲, Tm共III兲, and Yb共III兲 complexes have the opposite sign. Depending on the specific coordination polyhedron, a maximal magnetic anisotropy is observed for Tb共III兲, Dy共III兲, or Tm共III兲 complexes, and its absolute value can reach 50 000⫻10⫺6 cm3 mol⫺1 or more. Results of the present study can be helpful for the analysis of the orientational behavior of lanthanide-containing liquid crystals and lanthanide-doped bilayered micelles in an external magnetic field. The use of the Bleaney theory in the quantitative analysis of the magnetic anisotropy of lanthanide compounds is shown to have limitations because of a large ratio between the crystal-field splitting energy of the ground multiplet of the lanthanide ion and the thermal energy at room-temperature. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1450543兴 I. INTRODUCTION

bilities associated with the principal magnetic axes x, y, and z 共here, we assume that ␹ 1 ⭐ ␹ 2 ⭐ ␹ 3 兲. The maximum magnetic anisotropy is characterized by the difference ␹ 3 ⫺ ␹ 1 . In the following, the magnetic anisotropy can be expressed as the difference ⌬ ␹ ⫽ ␹ z ⫺( ␹ x ⫹ ␹ y )/2, where ␹ x , ␹ y , and ␹ z are the magnetic susceptibilities along the x, y, and z quantization axes which are normally related to some symmetry axes. Strong magnetic anisotropy is a general property of lanthanide compounds 关except Gd共III兲 compounds兴 which always takes place provided the local symmetry of the ligand surrounding of Ln共III兲 ions is low enough.1 It originates mainly from the crystal-field 共CF兲 splitting of the ground multiplet of the Ln共III兲 ion into individual CF energy levels which have different thermal populations. Most studies on magnetic properties of lanthanide compounds were done at low temperatures when only the ground electronic state and a few low-lying CF levels of the 4 f N configurations of lanthanide ions are thermally populated. The low-temperature magnetic anisotropy of lanthanide compounds is often so high that the magnetic susceptibilities ␹ 1 , ␹ 2 , and ␹ 3 in Eq. 共2兲 can differ from each other by several orders of magnitude.9 At higher temperatures, more CF levels are thermally populated and the magnetic anisotropy decreases. However, even at room-temperature and above, lanthanide compounds exhibit a pronounced magnetic anisotropy so that

Magnetic properties of nonmetallic lanthanide compounds have been studied extensively for several decades.1– 8 The interest in this subject has especially increased since the discovery of high-temperature superconducting cuprates and related compounds, many of which contain lanthanide ions. Normally, lanthanide compounds are characterized by a strong magnetic anisotropy ⌬␹, which is the difference between the minimum and maximum magnetic susceptibility along different crystallographic directions. More quantitatively, the anisotropic magnetic susceptibility is described by a second rank-tensor ␹ M⫽ ␹H,

共1兲

where M is the molar magnetization and H is the external magnetic field. The tensor ␹ is represented by a 3⫻3 matrix ␹ ␣␤ , which can always be transformed to a diagonal matrix by an appropriate choice of the coordination frame x, y, z

冉

␹1

0

0

0

␹2

0

0

0

␹3

冊

,

共2兲

where ␹ 1 , ␹ 2 , and ␹ 3 are the principal magnetic susceptia兲

Electronic mail: [email protected]

0021-9606/2002/116(11)/4673/13/$19.00

4673

© 2002 American Institute of Physics

4674

Mironov et al.

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

the ␹ 1 , ␹ 2 , and ␹ 3 principal susceptibilities often differ from each other by a factor up to ten.2– 8 In recent years, lanthanide complexes with a high roomtemperature magnetic anisotropy have attracted interest due to their orientational behavior in the external magnetic field. This property is potentially important for practical uses, such as designing magnetically active liquid crystals.10–15 The magnetic anisotropy of lanthanide ions doped in bilayered micelles or protein molecules is of interest for biological applications.16 –20 Thus, the magnetic anisotropy of lanthanide ions incorporated in the calcium protein calbindin D9k was recently studied.20 Lanthanide-containing liquid crystals are particularly interesting materials since, in contrast to conventional organic diamagnetic liquid crystals, they can be easily oriented in a rather weak magnetic field.10–14 The director n of the liquid crystal is oriented either parallel or perpendicular to the external magnetic field, depending on the sign of the magnetic anisotropy 共for liquid crystals in the mesophase, the sign is given by the difference between magnetic susceptibilities ␹ 储 – ␹⬜ parallel and perpendicular to the director n兲. A high magnetic anisotropy is required to obtain a low threshold of the magnetic field causing the magnetic alignment. As shown in our previous work,15 the sign and magnitude of the magnetic anisotropy of lanthanide-containing liquid crystals is determined by an interplay of several factors, of which the maximum anisotropy ␹ 3 – ␹ 1 of the molecular tensor of magnetic susceptibility is of primary importance. Being closely related to the geometry of the coordination polyhedron of the central metal ion, the ⌬␹ value depends strongly on the nature of the Ln共III兲 ion and the strength and symmetry of the CF potential created by the nearest ligand surrounding. There are only a few studies in the literature devoted to the theoretical description of the magnetic anisotropy of lanthanide complexes at room-temperature or at higher temperatures.2,21–26 The magnetic anisotropy of lanthanide compounds is often analyzed in terms of the Bleaney theory22 or its more recent modifications, which are based on the high-temperature expansion of the magnetic susceptibility. This approximation may be limitedly applicable to real lanthanide compounds due to the fact that the crystal-field splitting energy of the ground multiplet of a lanthanide ion is often larger than the thermal energy. This matter is discussed in the next section. The aim of this paper is to study systematically the room-temperature magnetic anisotropy of lanthanide complexes using a numerical model approach developed in our previous work on this topic.15,26 We study the variation of the magnetic anisotropy on the nature of the Ln共III兲 ion and the shape of the coordination polyhedron for the entire lanthanide series. Particularly, we focus on the dependence of ⌬␹ on the degree of the distortion of high-symmetry polyhedra 共such as an octahedron or cube兲 to low-symmetry polyhedra. Although the cubic parent structures may show anisotropic EPR spectra,27 their magnetic susceptibility is isotropic, implying that ⌬␹ will be zero in the absence of distortions. For these purposes, we study the variation of ⌬␹ upon distortions of the octahedron to an elongated and compressed trigonal prism and antiprism and distortion of the

cube to a prism, antiprism, and dodecahedron. Other eightand nine-coordinated polyhedra typical of lanthanide complexes are also examined. Apart from establishing general regularities in the variation of the magnetic susceptibility of lanthanide complexes, this study is also aimed at the search for the specific combination of the lanthanide ion and coordination polyhedron that provides maximum magnetic anisotropy at room-temperature. The latter is particularly important in designing lanthanide-containing metallomesogenes with a very high magnetic anisotropy required for their easy alignment in a weak magnetic field.10–14 An additional aim of this paper is to examine in some details the validity of the Bleaney approach in the description of the roomtemperature magnetic anisotropy for real lanthanide complexes in which the ⌬E CF splitting energy is normally larger than the thermal energy k BT. We revise the general conclusions of the Bleaney theory on the basis of the results of our numerical calculations of the anisotropic magnetic susceptibility for model lanthanide complexes with various types of the coordination polyhedra. II. THE BLEANEY THEORY

The magnetic anisotropy of lanthanide compounds at room-temperature or at higher temperatures is often analyzed in terms of the Bleaney theory,22 which is based on the hightemperature expansion of the magnetic susceptibility in a power series in the inverse temperature. In this approximation, magnetic anisotropy of a lanthanide complex arises from the term in T ⫺2

␹ x ⫺ ␹ 0 ⫽N

␮ B2 具 r 2 典 共 A 02 ⫺A 22 兲共 1⫹ p 兲 ␰ , 30共 k BT 兲 2

共3兲

␹ y ⫺ ␹ 0 ⫽N

␮ B2 具 r 2 典 共 A 02 ⫹A 22 兲共 1⫹ p 兲 ␰ , 30共 k BT 兲 2

共4兲

␮ B2 具 r 2 典 2A 02 共 1⫹ p 兲 ␰ , 30共 k BT 兲 2

共5兲

␹ z ⫺ ␹ 0 ⫽⫺N

where ␹ x , ␹ y , and ␹ z , are the magnetic susceptibilities along x, y, and z quantization axes. These relations are obtained from Eqs. 共18兲 and 共30兲 of Ref. 22. Here, 具 r 2 典 A 02 and 具 r 2 典 A 22 are rank-two (k⫽2) crystal-field parameters 共CFPs兲 共which are related to the conventional B kq parameters by B 20 ⫽2 具 r 2 典 A 02 and B 22 ⫽(2/3) 1/2具 r 2 典 A 22 兲,28 and ␹ 0 is the magnetic susceptibility of a free lanthanide ion

␹ 0⫽

Ng 2J ␮ B2 3k BT

J 共 J⫹1 兲 ,

共6兲

where N is the Avogadro number, g J is the g factor, ␮ B is the Bohr magneton, k B is the Boltzmann constant, and J is the total angular momentum of the ground multiplet of the lanthanide ion. The quantity ␰ in Eqs. 共3兲–共5兲 is given by

␰ ⫽g 2J 具 J 储 ␣ 储 J 典 J 共 J⫹1 兲共 2J⫺1 兲共 2J⫹3 兲 ,

共7兲

in which 具 J 储 ␣ 储 J 典 is a numerical coefficient. In Table I of Ref. 22 the ␰ value was tabulated for all Ln共III兲 ions, ⫺11.8 共Ce兲, ⫺20.7 共Pr兲, ⫺8.08 共Nd兲, ⫹0.943 共Sm兲, ⫺157.5 共Tb兲, ⫺181 共Dy兲, ⫺71.2 共Ho兲, ⫹58.8 共Er兲, ⫹95.3 共Tm兲, and

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

Magnetic anisotropy of lanthanide complexes

⫹39.2 共Yb兲. The factor (1⫹p) in Eqs. 共3兲–共5兲 reflects the thermal population of excited multiplets of the lanthanide ion, where p⫽a(k BT/⌬W)⫹b(k BT/⌬W) 2 ⫹¯, in which a and b numerical coefficients are presented in Table II of Ref. 22 for all lanthanide ions, and ⌬W is the energy gap between the ground and the first excited multiplet of the lanthanide ion. The thermal population correction p is small for all lanthanide ions 共it ranges from ⫺0.003 to 0.075兲 except for Sm共III兲 and Eu共III兲.22 According to Eqs. 共3兲–共5兲, the magnetic anisotropy ⌬ ␹ ⫽ ␹ z ⫺( ␹ x ⫹ ␹ y )/2 can be written as

␮ B2 ⌬ ␹ ⫽⫺N B 2 共 1⫹ p 兲 ␰ , 20共 k B T 兲 2 0

共8兲

where B 20 ⫽2 具 r 2 典 A 02 is the conventional B kq CF parameter 共see below兲. As can be seen from Eqs. 共3兲–共5兲 and 共7兲, 共8兲 the Bleaney theory predicts some important regularities in the behavior of the high-temperature magnetic anisotropy in a series of isostructural lanthanide complexes with the same CF potential 共a兲

共b兲

共c兲

Depending on the sign of ␰, all lanthanide ions can be classified into two groups, i.e., Ce共III兲, Pr共III兲 Nd共III兲, Tb共III兲, Dy共III兲, and Ho共III兲 ions form the first group ( ␰ ⬍0), and Eu共III兲, Er共III兲, Tm共III兲, and Yb共III兲 ions form the second group ( ␰ ⬎0). Two lanthanide ions not belonging to the same group always have the opposite sign of ⌬␹. In fact, despite a small positive ␰ value, Sm共III兲 ion belongs to the first group because of the thermal population of its low-lying excited multiplets; for the same reason, Eu共III兲 ion belongs to the second group although its 7 F 0 ground multiplet (J ⫽0) cannot be split by the crystal field. According to the variation of the absolute value of ␰ over the lanthanide series, the maximum absolute magnetic anisotropy ⌬␹ should always be observed for Dy共III兲 complexes regardless of the specific coordination polyhedron. Other lanthanide ions having a high magnetic anisotropy are Tb共III兲 and Tm共III兲. Equations 共3兲–共5兲 and 共8兲 show that the magnetic anisotropy is proportional to the rank-two (k⫽2) CFPs only, while rank four (k⫽4) and rank six (k⫽6) CFPs have no influence on ⌬␹. In particular, this implies that the magnetic anisotropy should vanish when all the rank two CFPs are zero.

These conclusions are expected to be valid only when rank-two CFPs are large and/or the ratio between the CF splitting energy ⌬E CF and thermal energy k BT is small enough. The basic assumption of the Bleaney theory is that the overall CF splittings ⌬E CF of the ground multiplet of a lanthanide ion do not exceed k BT in energy. This implies that all CF levels of the ground multiplet have a comparable thermal population.22 On closer inspection, however, this assumption is rather far from reality since in many lanthanide compounds the ⌬E CF splitting energy can easily reach a value of 400 cm⫺1 or more 共see Refs. 29, 30兲, so that we have typically ⌬E CF /k BT⭓2 (k BT⫽200 cm⫺1 at roomtemperature兲. In particular, a strong magnetic anisotropy can appear at ⌬E CF /k BT⬎1 even if B 2q ⫽0, as evidenced from

4675

numerical calculations.15 The use of the T ⫺2 term only therefore cannot be enough for an accurate quantitative description of the magnetic anisotropy. There have been several attempts to revise the Bleaney theory by including the higher-order CFPs and the next terms in the inverse temperature.23,25 Golding and Pyykko¨ included rank-four and rank-six CFPs in the magnetic susceptibility calculations for the specific case of a D 3h or C 3 v symmetry of the CF potential and they found from numerical calculations for LnCl3 compounds that corrections to the Bleaney theory are within 20%.23 More recently, McGarvey extended the Bleaney theory to the T ⫺3 term, which involves both rank-four and rank-six CFPs.25 The relative magnitude of this term was found to be less than 10% for such compounds as LnCl3 or rare-earth ethylsulfates. Unfortunately, these compounds seem to be unfavorable from the point of view of evaluating the true degree of the correctness of the Bleaney approach at room-temperature, because of a small CF splitting energy. For a large ⌬E CF /k BT ratio, not only the Bleaney theory, but even the high-temperature expansion of the magnetic susceptibility in a power series in the inverse temperature may be inadequate. In this case, even the inclusion of higher terms T ⫺n does not guarantee a good accuracy in the calculation of the magnetic anisotropy, because of a poor convergence of the power series. This question is still open and requires a detailed study. Below, we will show that the difference between the magnetic anisotropy obtained from the Bleaney theory and from rigorous numerical calculations is often far beyond 20%, in contrast to the results obtained in Refs. 23 and 25.

III. COMPUTATIONAL DETAILS

In the general case, the ␹ ␣␤ components of the tensor of the magnetic susceptibility ␹ in Eq. 共1兲 of a lanthanide complex can be calculated using the Gerloch–McMeeking formula31

␹ ␣␤ ⫽

N 兺 i exp共 ⫺E i /kT 兲

⫺

兺j

再

兺i 兺j

具 i 兩 ␮ ␣ 兩 j 典具 j 兩 ␮ ␤ 兩 i 典 kT

具 i 兩 ␮ ␣ 兩 j 典具 j 兩 ␮ ␤ 兩 i 典 ⫹ 具 i 兩 ␮ ␤ 兩 j 典具 j 兩 ␮ ␣ 兩 i 典

⫻exp共 ⫺E i /kT 兲 ,

E i ⫺E j

冎 共9兲

where ␣, ␤ ⫽x, y or z, N is the Avogadro number, E i is the energy of the CF state 兩 i 典 , k is the Boltzmann constant, T is the absolute temperature 共in K兲, and ␮ ␣ 共␣⫽x, y, z兲 are the components of the operator of the total magnetic momentum ␮

␮⫽⫺ ␮ B共 L⫹2S兲 ,

共10兲

in which L and S are the total orbital momentum and spin operators, respectively, and ␮ B is the Bohr magneton. In Eq. 共9兲, the first sum in 兵¯其 runs over CF states 兩 j 典 for which E i ⫽E j ; in the second sum the index j runs over 兩 j 典 states, for which E i ⫽E j . In contrast to the commonly used Van Vleck formula,32 which is adapted for calculations of diagonal matrix elements of the tensor ␹ only, formula 共9兲 allows

4676

Mironov et al.

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

calculations of nondiagonal ␹ ␣␤ matrix elements and thus is more favorable for analysis of the magnetic susceptibility in low-symmetry systems. The very similar Urland formula33 differs from 共9兲 by the relation 具 i 兩 ␮ ␣ 兩 j 典 ⫽ 具 i 兩 ␮ ␣ 兩 i 典 ␦ i j assumed for the first term. This condition for the wave functions 兩 i 典 makes it less convenient in the practical use, especially for low-symmetry complexes. The energies E i and wave functions 兩 i 典 of CF states of a lanthanide ion in a coordination complex are obtained from the diagonalization of the model Hamiltonian H⫽H0 ⫹HCF ,

共11兲

where H0 is the free-ion Hamiltonian H 0⫽

兺

k⫽2,4,6

f kF k⫹ ␨ 4 f

⫹␥G共 R7兲,

兺i l i s i ⫹ ␣ L 共 L⫹1 兲 ⫹ ␤ G 共 G 2 兲 共12兲

which involves the electrostatic repulsion between equivalent 4 f electrons and the spin–orbit interaction; for configurations of two or more 4 f electrons, the ␣, ␤, and ␥ are the parameters related to the two-body correction associated with the angular momentum L and the Casimir operator G for the groups G 2 and R 7 , respectively.28,34 –36 Electron repulsion parameters F k , spin–orbit coupling constants ␨ 4 f , and ␣, ␤, and ␥ parameters were taken from Ref. 36. Higher free-ion correction terms 共such as Judd three-body operators兲 determining fine details of the energy spectra of lanthanide ions in solids28,35 are of minor importance in calculations of the magnetic susceptibility due to the fact that ␹ ␣␤ values 共9兲 are mainly determined by the CF splitting of the ground multiplet and are insensitive to small variations in the energy position of excited multiplets of the lanthanide ion. Therefore, they can safely be omitted in model calculations of the high-temperature magnetic susceptibility of lanthanide complexes. HCF is the crystal-field Hamiltonian HCF⫽

B kq C kq , 兺 kq

共13兲

where B kq are crystal-field parameters and C kq are spherical tensor operators. Details of parametric CF calculations for Ln共III兲 ions were described elsewhere.28,35 Since in this paper we are mainly interested in establishing general trends in the variation of the high-temperature magnetic anisotropy as a function of the nature of the Ln共III兲 ion and the shape of the coordination polyhedron rather than in quantitative calculations for specific lanthanide compounds, we used a model approach based on a number of simplifications 共1兲 In the series of model Ln共III兲 complexes with the same coordination polyhedron, the same sets of CFPs are used. 共2兲 CFPs are calculated in terms of the point charge electrostatic model 共PCEM兲.28,35 Since simple PCEM calculations considerably overestimate rank CFPs, we use an improved PCEM approach involving shielding factors ( ␴ 2 , ␴ 4 , ␴ 6 ) to obtain more realistic B kq parameters. For all Ln共III兲 ions we used the 具 r 2 典 ⫽1.40, 具 r 4 典 ⫽4.20,

具 r 6 典 ⫽39.0 radial parameters and ␴ 2 ⫽0.8, ␴ 4 ⫽0.1, ␴ 6

⫽0.05 shielding factors. No expansion factors ␶ 共which describe the expansion of the 4 f wave functions in the solid state when compared to the free ion兲 are applied. Although PCEM calculations are not very accurate for real lanthanide compounds, they provide in many cases a reasonable approximate set of B kq parameters, which is then used as a starting point in the fitting CF calculations. Alternative approaches for estimating CFPs, such as the superposition model,37,38 angular overlap model 共AOM兲,39 or simple orbital model 共SOM兲,40,41 involve adjustable empirical parameters and thus are more relevant to the CF analysis for specific lanthanide complexes rather than to model systems used to establish general trends in the variation of ⌬␹. 共3兲 Since in homoligand coordination polyhedra of Ln共III兲 ions the influence of angular distortions on the magnetic anisotropy is often more important than that of radial distortions, in all model coordination polyhedra under study all the ligands are taken to be equidistant from the central lanthanide ion. The metal–ligand distance 共2.3 Å兲 and the electric charge on ligands (q⫽⫺1.0 e) are chosen so as to have a moderate CF splitting strength typical of real lanthanide compounds. Quantitatively, the strength of the CF splitting is monitored by the Changparameter S 共represented by a weighed sum of squares of B kq CFPs兲,42 which is about 500 cm⫺1 or less for all the model polyhedra under study. The calculations of the magnetic susceptibility are performed with the CF wave functions 兩 i 典 of the lowest CF energy levels belonging to the ground and several excited multiplets. The number of multiplets involved in calculations of the tensor ␹ is chosen so as to cover the thermal population effect well above the room-temperature 关this number is 2 for Ce共III兲 and Yb共III兲, 7 for Sm共III兲 and Eu共III兲 ions having many low-lying multiplets, and 3 for other Ln共III兲 ions兴. For each of the model coordination complexes under study, the ␹ ␣␤ components of the tensor ␹ in Eq. 共1兲 are calculated at 298 K using a computer program described in Ref. 15. The directions of the principal magnetic axes and the ␹ 1 , ␹ 2 , and ␹ 3 principal components of the tensor ␹ are obtained by diagonalization of the 3⫻3 matrix ␹ ␣␤ determined by Eq. 共9兲 共for coordination polyhedra with a high enough symmetry, these axes coincide with symmetry axes兲. The quantization axis is chosen to be z. To bring the magnetic anisotropy ⌬␹ into correspondence to that of lanthanide-containing liquid crystals 共in which it is given by ⌬ ␹ ⫽ ␹ 储 ⫺ ␹⬜ 兲, we use the relation ⌬ ␹ ⫽ ␹ z ⫺( ␹ x ⫹ ␹ y )/2 rather than ⌬ ␹ ⫽ ␹ 3 ⫺ ␹ 1 . In contrast to the ␹ 3 ⫺ ␹ 1 quantity, which is always positive, the magnetic anisotropy defined in such a way may be both positive and negative, as observed in lanthanide-containing liquid crystals.11,12 The sign of ⌬␹ is also important in analyzing the paramagnetic shifts in NMR spectra of lanthanide ions.16 –20 Throughout the paper, ⌬␹ is expressed in units of 10⫺6 cm3 mol⫺1 .

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

FIG. 1. Distortions of the octahedron to 共a兲 elongated and compressed trigonal antiprism, 共b兲 trigonal prism 共via intermediate figures of D 3 symmetry兲; and 共c兲 elongated and compressed trigonal prism.

IV. RESULTS AND DISCUSSION

In this section, we present results of comparative numerical calculations of the magnetic anisotropy ⌬␹ for Ce共III兲, Pr共III兲, Nd共III兲, Sm共III兲, Eu共III兲, Tb共III兲, Dy共III兲, Ho共III兲, Er共III兲, Tm共III兲, and Yb共III兲 ions in various model coordination polyhedra with the coordination number 共CN兲 ranging from six to nine 共the case of the icosahedron with CN⫽12 is analyzed too兲. We establish regularities in the variation of the room-temperature magnetic susceptibility over the lanthanide series and analyze quantitatively the limitations of the Bleaney theory.22 To this end, for selected coordination polyhedra we compare in detail magnetic anisotropy obtained from numerical calculations with full and truncated 共to the rank two B kq ’s only兲 sets of CFPs with that obtained from the Bleaney approach with the use of Eqs. 共3兲–共5兲 and 共8兲. The results of calculations are compared in tables; the variation of the magnetic anisotropy is presented in graphical form. A. Distortion of the octahedron to an antiprism

We started with calculations of the magnetic anisotropy with six-coordinated lanthanide ions. Although coordination polyhedra with CN⫽6 are not common in lanthanide compounds, the regular or distorted octahedron can be found in lanthanide complexes with bulk ligands and in some highsymmetry crystals, such as elpasolites.43– 45 In the regular octahedron, the magnetic anisotropy vanishes for all Ln共III兲 ions. The magnetic anisotropy increases with increasing degree of distortion of the octahedron. We illustrate this effect for three modes of distortion of the octahedron, as shown in Figs. 1共a兲–1共c兲. Compression and elongation of the octahedron along the three-fold rotation axis C 3 results in a trigonal antiprism with D 3d symmetry 关Fig. 1共a兲兴. In this distortion mode, the polar angle ␪ varies 45° 共elongated trigonal antiprism兲 to 65° 共compressed trigonal antiprism兲. The angle ␪ cubic ⫽arccos(1/31/2)⫽54.736° 共the so-called cubic angle兲 corresponds to the regular octahedron, in which the vertical quantization axis z coincides with the C 3 rotation axis. Results of numerical calculations of the magnetic anisotropy for Ln共III兲 ions are presented in Table I. As one can expect, the magnetic anisotropy reverses sign at the cubic angle ␪ cubic for all lanthanide ions simultaneously. It is interesting to note that ⌬␹ is proportional to the degree of distortion of the octahedron only for small devia-

Magnetic anisotropy of lanthanide complexes

4677

tions of the polar angle ␪ from ␪ cubic 共within a few degrees兲, while for strongly distorted antiprisms ⌬␹ is not symmetric with respect to ␪ ⫺ ␪ cubic . In Table I, the magnetic anisotropy obtained from numerical calculations is compared with ⌬␹ values calculated using Eq. 共8兲 resulting from the Bleaney theory 共series B兲 and the variation of the B 20 CF parameter on the ␪ angle. This comparison shows that, although ⌬␹ is roughly proportional to B 20 , the ratio between the magnetic anisotropies for various Ln共III兲 ions deviates considerably from that expected from the variation of the quantity ␰ in Eq. 共7兲 over the lanthanide series. Indeed, Eqs. 共3兲–共5兲 and 共7兲, 共8兲 imply that the ratios between the magnetic anisotropies ⌬ ␹ (Tb)/⌬ ␹ (Dy)⫽0.87 and ⌬ ␹ (Tm)/⌬ ␹ (Dy)⫽⫺0.53 are independent of the B 20 parameter and the polar angle ␪. In fact, numerical calculations show that these ratios are noticeably larger and that they are rather sensitive to ␪, since they vary from 1.6 to 1.0 for Tb/Dy and from ⫺0.65 to ⫺1.28 for Tm/Dy. In particular, the magnetic anisotropy of Tb共III兲 is larger than that of Dy共III兲 for both elongated and compressed trigonal antiprism. Moreover, Table I shows that maximum magnetic anisotropy can be observed even for Tm共III兲 ion 共at ␪ ⫽65°兲, despite the fact that the ␰ value of the Tm共III兲 ion is considerably smaller than that of Dy共III兲 or Tb共III兲 ions. To explore further limitations of the Bleaney approach in the quantitative analysis of the room-temperature magnetic anisotropy of lanthanide complexes, we calculated the magnetic anisotropy of the compressed and elongated trigonal antiprism with a truncated set of CFPs involving the ranktwo B kq parameters only 共Table I, series C兲. As can be seen from the comparison of data of Table I, results of approximate calculations of the magnetic anisotropy 共series B and C兲 differ noticeably both from each other and from the results of exact numerical calculations of ⌬␹ with the use of the full set of CFPs 共series A兲. This indicates that taking into account only rank-two B kq parameters does not provide good accuracy in quantitative calculations of the roomtemperature magnetic anisotropy, neither in terms of the high-temperature expansion approach 共8兲 nor with the use of numerical calculations. In fact, results of numerical calculations involving only the rank-two CFPs are close to those of the Bleaney approach only for small ␪ ⫺ ␪ cubic differences, when the B 20 parameter and the resulting CF splitting are small 关for instance, ⌬E CF⬍50 cm⫺1 for all Ln共III兲 ions at ␪ ⫽54°兴. On the other hand, these results show that for the D 3d antiprism the sign of ⌬␹ obeys strictly the general sign rule 共a兲, being positive for Ce共III兲, Pr共III兲, Nd共III兲, Sm共III兲, Tb共III兲, Dy共III兲, and Ho共III兲 ions and negative for Eu共III兲, Er共III兲, Tm共III兲, and Yb共III兲 ions for an elongated trigonal antiprism 共for which ␪ ⬍ ␪ cubic and B 20 ⬎0兲. For the compressed antiprism, ⌬␹ reverses its sign for all Ln共III兲 ions. B. Distortion from the octahedron to a trigonal prism

Rotation of the upper triangular face of the octahedron with respect to its base face results in a trigonal prism passing via intermediate polyhedra of D 3 point symmetry 关Fig. 1共b兲兴. The rotation angle increases from ␸ ⫽0° 共octahedron兲

4678

Mironov et al.

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

TABLE I. Room-temperature 共298 K兲 magnetic anisotropy ⌬␹ 共in 10⫺6 cm3 mol⫺1 兲 and B 20 CF parameter of lanthanide ions in a compressed and elongated D 3d trigonal antiprism. Magnetic anisotropy obtained from 共A兲 numerical calculations using the full set of B kq CFPs, 共B兲 from Eq. 共8兲 of the Bleaney theory, and 共C兲 from numerical calculations using the B 20 parameter only. Polar angle ␪, degrees Ln

45

47

49

51

53

54

␪ cubic

56

57

59

61

63

65

A

Ce Pr Nd Sm Eu Tb Dy Ho Er Tm Yb

3 317 3 948 2 340 481 ⫺1 546 60 318 37 756 23 705 ⫺12 571 ⫺24 642 ⫺4 621

2 738 2 955 1 866 426 ⫺1 288 50 463 28 311 18 971 ⫺9 979 ⫺22 434 ⫺3 673

2 053 2 092 1 379 350 ⫺1 004 38 262 20 068 14 086 ⫺7 415 ⫺19 027 ⫺2 723

1 313 1 323 890 249 ⫺6 89 24 584 12 679 9 126 ⫺4 858 ⫺13 992 ⫺1 788

582 6 06 407 124 ⫺3 38 10 852 5 787 4 187 ⫺2 277 ⫺7 142 ⫺852

238 256 170 53 ⫺147 4426 2439 1758 ⫺969 ⫺3127 ⫺369

0 0 0 0 0 0 0 0 0 0 0

⫺378 ⫺440 ⫺286 ⫺95 267 ⫺6898 ⫺4164 ⫺2952 1679 5544 676

⫺646 ⫺790 ⫺506 ⫺170 491 ⫺11 672 ⫺7 438 ⫺5 212 3 016 9 909 1 260

⫺1 102 ⫺1 488 ⫺926 ⫺315 969 ⫺19 428 ⫺13 916 ⫺9 499 5 685 18 009 2 601

⫺1 458 ⫺2 172 ⫺1 318 ⫺443 1 478 ⫺25 071 ⫺20 213 ⫺13 449 8 299 24 701 4 198

⫺1 731 ⫺2 819 ⫺1 682 ⫺549 2 004 ⫺29 055 ⫺26 174 ⫺17 060 10 795 29 788 6 006

⫺1 937 ⫺3 405 ⫺2 022 ⫺630 2 528 ⫺31 821 ⫺31 632 ⫺20 366 13 121 33 441 7 902

B

Ce Pr Nd Sm Eua Tb Dy Ho Er Tm Yb

3 814 6 708 2 547 439 ¯ 52 146 61 032 24 137 ⫺19 969 ⫺32 334 ⫺13 320

3 015 5 302 2 014 347 ¯ 41 224 48 249 19 081 ⫺15 786 ⫺25 562 ⫺10 530

2 220 3 904 1 482 255 ¯ 30 347 35 519 14 047 ⫺11 621 ⫺18 817 ⫺7 752

1 435 2 523 958 165 ¯ 19 612 22 954 9 078 ⫺7 510 ⫺12 161 ⫺5 009

659 1 160 440 76 ¯ 9 016 10 553 4 173 ⫺3 453 ⫺5 590 ⫺2 303

279 490 186 32 ¯ 3811 4460 1764 ⫺1459 ⫺2363 ⫺973

0 0 0 0 ¯ 0 0 0 0 0 0

⫺469 ⫺825 ⫺313 ⫺54 ¯ ⫺6413 ⫺7506 ⫺2968 2456 3977 1638

⫺840 ⫺1 477 ⫺561 ⫺97 ¯ ⫺11 479 ⫺13 435 ⫺5 313 4 396 7 118 2 932

⫺1 557 ⫺2 738 ⫺1 040 ⫺179 ¯ ⫺21 285 ⫺24 912 ⫺9 852 8 151 13 198 5 437

⫺2 247 ⫺3 951 ⫺1 501 ⫺258 ¯ ⫺30 719 ⫺35 954 ⫺14 219 11 764 19 048 7 846

⫺2 910 ⫺5 117 ⫺1 943 ⫺335 ¯ ⫺39 781 ⫺46 560 ⫺18 413 15 234 24 667 10 161

⫺3 539 ⫺6 223 ⫺2 363 ⫺407 ¯ ⫺48 378 ⫺56 623 ⫺22 393 18 526 29 998 12 357

C

Ce Pr Nd Sm Eu Tb Dy Ho Er Tm Yb

3 520 6 746 2 589 345 ⫺1 869 52 937 62 610 23 523 ⫺18 206 ⫺21 199 ⫺7 938 1 122

2 982 5 525 2 037 296 ⫺1 583 43 436 50 752 18 449 ⫺14 883 ⫺18 521 ⫺7 072 887

2 302 4 138 1 489 236 ⫺1 246 32 529 37 637 13 445 ⫺11 310 ⫺15 050 ⫺5 868 653

1 512 2 660 952 165 ⫺857 20 852 23 989 8 571 ⫺7 518 ⫺10 661 ⫺4 242 422

684 1 191 432 81 ⫺417 9 285 10 674 3 882 ⫺3 549 ⫺5 320 ⫺2 153 194

282 491 180 35 ⫺180 3817 4395 1622 ⫺1513 ⫺2322 ⫺946 82

0 0 0 0 0 0 0 0 0 0 0 0

⫺449 ⫺790 ⫺302 ⫺63 319 ⫺6079 ⫺7042 ⫺2704 2623 4171 1709 ⫺138

⫺768 ⫺1 360 ⫺532 ⫺116 579 ⫺10 417 ⫺12 117 ⫺4 764 4 708 7 575 3 103 ⫺247

⫺1 298 ⫺2 347 ⫺969 ⫺223 1 104 ⫺17 810 ⫺20 919 ⫺8 660 8 873 14 420 5 864 ⫺458

⫺1 694 ⫺3 134 ⫺1 372 ⫺329 1 622 ⫺23 606 ⫺28 012 ⫺12 248 12 977 20 956 8 399 ⫺661

⫺1 978 ⫺3 748 ⫺1 739 ⫺429 2 115 ⫺28 056 ⫺33 620 ⫺15 520 16 961 26 850 10 559 ⫺856

⫺2 178 ⫺4 217 ⫺2 072 ⫺522 2 570 ⫺31 443 ⫺38 009 ⫺18 479 20 769 31 920 12 301 ⫺1 041

B 20 , cm⫺1 a

Magnetic anisotropy of Eu共III兲 ion cannot be calculated in terms of Eq. 共8兲 because the ground 7F0 multiplet (J⫽0) is not split by the CF effect.

to ␸ ⫽60° 共trigonal prism兲. To avoid a zero B 20 parameter resulting in a low magnetic anisotropy, the polar angle ␪ was set not to ␪ cub , but to 50°. Results of calculations are shown in Fig. 2 for selected lanthanide ions. In accordance with the fact that the B 20 parameter remains unaltered throughout the rotation (B 20 ⫽⫺561 cm⫺1 ), the magnetic anisotropy is really rather insensitive to the rotation for most lanthanide ions, although not strictly constant as predicted by Eq. 共8兲. The most noticeable relative variation in ⌬␹ is observed for Yb共III兲 ion and, to a lesser extent, for Ce共III兲 ion: ⌬␹ decreases from 5185 to 3368 ⫻10⫺6 cm3 mol⫺1 for Yb共III兲 and from ⫺1723 to ⫺1292 ⫻10⫺6 cm3 mol⫺1 for Ce共III兲 in going from the D 3d antiprism to the D 3h prism. This effect is therefore beyond the Bleaney theory. C. Compressed and elongated trigonal prism

A trigonal prism 共D 3h symmetry兲 is obtained from the octahedron by rotation of the top plane by the angle of 60° 关Fig. 1共b兲兴. We study the variation of the magnetic anisotropy

on the degree of distortion of the trigonal D 3h prism when the polar angle varies from 45° 共elongated prism兲 to 65° 共compressed prism兲 关Fig. 1共c兲兴. Results of calculations of the magnetic susceptibility are presented in Fig. 3. Since the magnetic anisotropy of light Ln共III兲 ions is too small to be seen in the ⌬␹ scale of Fig. 3, only data for heavy lanthanide ions are shown. The general behavior of the magnetic susceptibility as a function of ␪ is reminiscent of that in the above case of the D 3d trigonal antiprism except that ⌬␹ is no longer zero at ␪ cubic for all Ln共III兲 ions simultaneously, but it changes sign at different ␪ for different lanthanide ions. At ␪ ⫽ ␪ cubic, the sign of ⌬␹ 关33 共Ce兲, 485 共Pr兲, ⫺115 共Nd兲, ⫺28 共Sm兲, 12 共Eu兲, 2188 共Tb兲, 4500 共Dy兲, ⫺1391 共Ho兲, ⫺ 1704 共Er兲, 3503 共Tm兲, and 1307 共Yb兲 共in 10⫺6 cm3 mol⫺1 兲兴 differs considerably from that predicted from the general sign rule 共a兲. This indicates once more a discrepancy of the Bleaney approach. D. Distortion from the cube to a square prism

Next, we study the behavior of the magnetic anisotropy in the group of coordination polyhedra with CN⫽8 resulting

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

Magnetic anisotropy of lanthanide complexes

4679

FIG. 4. Distortions of the 共a兲 cube to 共b兲 elongated and compressed square prism, 共c兲 and 共d兲 square antiprism, 共e兲 dodecahedron; and 共f兲 capped square antiprism.

FIG. 2. Variation of the magnetic anisotropy ⌬␹ of lanthanide ions upon the rotation of the top plane of the octahedron 关Fig. 1b兴. The octahedron is somewhat elongated 共␪ ⫽50°, see the text兲. The rotation angle ␸ increases from 0° 共octahedron兲 to 60° 共trigonal prism兲. Hereafter, the variations of ⌬␹ for light lanthanide ions 关i.e., Ce共III兲, Pr共III兲, Nd共III兲, Sm共III兲, and Eu共III兲兴 ions are not shown in figures since their magnetic anisotropy is too low to be seen well in the present scale.

from the distortion of the regular cube according to several schemes shown in Fig. 4. As is the case of the octahedron, all lanthanide ions are magnetically isotropic in a cubic ligand environment. When the cube 关Fig. 4共a兲兴 is compressed or elongated along the C 4 symmetry z axis, a square prism of D 4h sym-

FIG. 3. Variation of the magnetic anisotropy of lanthanide ions in the trigonal prism. The polar angle ␪ of the three top ligands increases from 45° 共elongated prism兲 to 65° 共compressed prism兲.

metry is formed 关Fig. 4共b兲兴 and some magnetic anisotropy ⌬␹ arises which increases with increasing degree of distortion of the cube. We performed calculations for a number of square prisms in which the ␪ angle increases from 45° 共elongated prism兲 to 65° 共compressed prism兲 passing via the cubic angle ␪ cubic⫽54.736° corresponding to a regular cube. The dependence of the magnetic anisotropy on the ␪ angle is presented in Table II, series A. The variation of the magnetic anisotropy for the D 4h square prism is very similar to that of the D 3d trigonal antiprism considered above in the point 共a兲. As is the case of the D 3d antiprism, for all lanthanide ions the magnetic anisotropy changes sign at the cubic angle ␪ cubic . However, due to the fact that at the same ␪ angle the B 20 parameter is larger by a factor of 4/3, the ⌬␹ value of the square prism is larger than that of the trigonal antiprism. An interesting competition between the largest magnetic anisotropy of Tb共III兲, Dy共III兲, and Tm共III兲 ions is seen. The magnetic anisotropy of Tb共III兲 ion is considerably larger than that of other lanthanide ions for elongated ( ␪ ⬍ ␪ cub) and slightly compressed square prism; for moderately and strongly compressed prism 共at ␪ ⬎57°兲 the absolute value of ⌬␹ is comparable for Tb共III兲, Dy共III兲, and Tm共III兲 ions. Table II also presents the dependence of the B 20 parameter on the polar angle ␪ and ⌬␹ values obtained from both the Bleaney approach using Eq. 共8兲 共series B兲 and from numerical calculations with the rank-two CFPs only 共series C兲. Again, strong deviations from the rules 共b兲 and 共c兲 can be seen. The ⌬␹共Tb兲/⌬␹共Dy兲 ratio ranges from 2.02 to 0.80 and the ⌬ ␹ (Tm)/⌬ ␹ (Dy) ratio from ⫺1.07 to ⫺0.57 关vs 0.87 and ⫺0.57, respectively, as obtained from the ␰ (Tb)/ ␰ (Dy) and ␰ (Tm)/ ␰ (Dy) ratios兴. Table II shows that the magnetic anisotropy resulting from approximate calculations with the use of the rank-two CFPs only 共series B and C兲 differs considerably from the results of precise calculations using the full set of CFPs 共series A兲. In this context, it is interesting to compare the CF splitting patterns of Tb共III兲 and Dy共III兲 ions in the square prism at ␪ ⫽45°. There is a large energy gap of 370 cm-1 between the ground CF state and first excited CF state of the Tb共III兲 ion, so that at room-temperature the thermal population of ex-

4680

Mironov et al.

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

TABLE II. Room-temperature 共298 K兲 magnetic anisotropy ⌬␹ 共in 10⫺6 cm3 mol⫺1 兲 and B 20 CF parameter of lanthanide ions in a compressed and elongated D 4h square prism. Magnetic anisotropy obtained from 共A兲 numerical calculations using the full set of B kq CFPs, 共B兲 from Eq. 共8兲 of the Bleaney theory, and 共C兲 from numerical calculations using the B 20 parameter only. Polar angle ␪ degrees Ln

45

47

49

51

53

54 Cubic

56

57

59

61

63

65

A

Ce Pr Nd Sm Eu Tb Dy Ho Er Tm Yb

4 363 3 989 3 341 2 347 1 090 447 4 959 3 757 2 709 1 759 830 356 2 597 2 027 1 462 918 409 356 556 506 430 318 164 72 ⫺1 640 ⫺1 353 ⫺1 047 ⫺717 ⫺352 ⫺154 76 813 69 311 57 060 39 272 17 885 7273 48 889 38 007 28 230 18 817 9 051 3903 24 876 18 978 13 393 8 267 3 644 1507 ⫺22 180 ⫺18 427 ⫺14 208 ⫺9 566 ⫺4 561 ⫺1952 ⫺27 834 ⫺25 951 ⫺22 676 ⫺17 263 ⫺9 127 ⫺4060 ⫺4 484 ⫺3 369 ⫺2 309 ⫺1 381 ⫺598 ⫺248

0 ⫺679 ⫺1 119 ⫺1 742 ⫺2 097 ⫺2 282 ⫺2 371 0 ⫺627 ⫺1 131 ⫺2 132 ⫺3 053 ⫺3 823 ⫺4 409 0 ⫺282 ⫺498 ⫺916 ⫺1 331 ⫺1 755 ⫺2 196 0 ⫺129 ⫺232 ⫺424 ⫺583 ⫺702 ⫺783 0 282 522 1 048 1 629 2 246 2 872 0 ⫺10 910 ⫺17 904 ⫺27 785 ⫺33 496 ⫺36 648 ⫺38 366 0 ⫺6 885 ⫺12 394 ⫺23 089 ⫺32 549 ⫺40 123 ⫺45 695 0 ⫺2 484 ⫺4 375 ⫺8 053 ⫺11 737 ⫺15 586 ⫺19 719 0 3 391 6 077 11 339 16 261 20 638 24 322 0 7 370 13 265 24 209 32 984 39 236 43 274 0 430 796 1 710 3 082 5 120 7 753

B

Ce Pr Nd Sm Eua Tb Dy Ho Er Tm Yb

5 085 4 021 2 961 1 914 880 371 8 943 7 072 5 207 3 366 1 548 652 3 396 2 686 1 977 1 278 588 247 585 463 341 220 101 43 ¯ ¯ ¯ ¯ ¯ ¯ 69 523 54 977 40 478 26 164 12 036 5066 81 372 64 347 47 376 30 623 14 088 5929 32 180 25 447 18 736 12 111 5 571 2345 ⫺26 624 ⫺21 053 ⫺15 501 ⫺10 019 ⫺4 609 ⫺1940 ⫺43 110 ⫺34 090 ⫺25 099 ⫺16 224 ⫺7 464 ⫺3141 ⫺17 758 ⫺14 043 ⫺10 339 ⫺6 683 ⫺3 074 ⫺1294

0 ⫺629 ⫺1 118 ⫺2 077 ⫺2 998 ⫺3 882 ⫺4 722 0 ⫺1 106 ⫺1 967 ⫺3 652 ⫺5 272 ⫺6 827 ⫺8 303 0 ⫺420 ⫺747 ⫺1 387 ⫺2 002 ⫺2 593 ⫺3 153 0 ⫺72 ⫺129 ⫺239 ⫺345 ⫺446 ⫺543 ¯ ¯ ¯ ¯ ¯ ¯ ¯ 0 ⫺8 597 ⫺15 290 ⫺28 395 ⫺40 989 ⫺53 072 ⫺64 551 0 ⫺10 063 ⫺17 895 ⫺33 234 ⫺47 974 ⫺62 117 ⫺75 552 0 ⫺3 980 ⫺7 077 ⫺13 143 ⫺18 973 ⫺24 565 ⫺29 879 0 3 292 5 855 10 874 15 697 20 324 24 719 0 5 331 9 481 17 607 25 416 32 909 40 027 0 2 196 3 905 7 253 10 470 13 556 16 488

Ce Pr Nd Sm Eu Tb Dy Ho Er Tm Yb

4 101 3 637 2 940 2 003 919 378 8 270 7 030 5 436 3 567 1 604 659 8 270 2 732 2 000 1 278 578 241 407 356 292 210 106 47 ⫺2 235 ⫺1 935 ⫺1 562 ⫺1 101 ⫺548 ⫺239 64 654 55 131 42 739 28 017 12 526 5123 78 003 65 420 49 900 32 326 14 396 5895 31 598 24 842 18 109 11 524 5 201 2168 ⫺22 947 ⫺19 024 ⫺14 650 ⫺9 854 ⫺4 698 ⫺2012 ⫺24 211 ⫺21 783 ⫺18 313 ⫺13 450 ⫺6 940 ⫺3071 ⫺8 828 ⫺8 118 ⫺7 003 ⫺5 288 ⫺2 797 ⫺1249 1 496 1 183 871 563 259 109

C

B 20 , cm⫺1 a

0 0 0 0 0 0 0 0 0 0 0 0

⫺589 ⫺989 ⫺1 605 ⫺2 010 ⫺2 263 ⫺2 418 ⫺1 039 ⫺1 765 ⫺2 953 ⫺3 819 ⫺4 432 ⫺4 858 ⫺401 ⫺705 ⫺1 273 ⫺1 787 ⫺2 245 ⫺2 648 ⫺86 ⫺157 ⫺302 ⫺443 ⫺569 ⫺679 429 781 1 492 2 179 2 809 3 361 ⫺7 981 ⫺13 471 ⫺22 277 ⫺28 574 ⫺32 989 ⫺36 092 ⫺9 261 ⫺15 727 ⫺26 367 ⫺34 283 ⫺40 055 ⫺44 251 ⫺3 591 ⫺6 304 ⫺11 369 ⫺15 941 ⫺20 020 ⫺23 621 3 511 6 319 11 948 17 490 22 833 27 880 5 615 10 227 19 352 27 590 34 411 39 728 2 301 4 182 7 788 10 821 13 113 14 740 ⫺185 ⫺329 ⫺611 ⫺882 ⫺1142 ⫺1389

Magnetic anisotropy of Eu共III兲 ion cannot be calculated in terms of Eq. 共8兲 because the ground 7F0 multiplet (J⫽0) is not split by the CF effect.

cited CF states is very small in comparison to that of the ground CF level. By contrast, in the Dy共III兲 ion several CF levels lie below 200 cm⫺1 and thus have comparable thermal populations.

the prism to an antiprism 共from 2323 to 6165 ⫻10⫺6 cm3 mol⫺1 兲. This effect cannot be described by Eq. 共8兲, which predicts no dependence of ⌬␹ on the ␸ angle, because the B 20 parameter is not changed throughout this rotation.

E. Distortion from the cube to a square antiprism

Rotation of the top square face of the cube or square prism with respect to its base plane over an angle ␸ ⫽45° results in a square antiprism with D 4d symmetry; rotation over an angle 0⬍ ␸ ⬍45° results in an intermediate polyhedron with D 4 symmetry 关Fig. 4共c兲兴. To make the model more realistic, we start rotation not for the regular cube, but for a compressed square prism with the polar angle ␪ ⫽60°, which almost coincides with the angle of the closest ligand stacking in the square antiprism 共59.3°兲. Results of calculations are given for heavy lanthanide ions in Fig. 5, which show that the rotation of the top plane does not influence much the magnetic anisotropy of most lanthanide ions except Yb共III兲, for which ⌬␹ increases by a factor of about 3 in going from

F. Compressed and elongated square antiprism

Square antiprisms with D 4d symmetry can be found in molecular eight-coordinated lanthanide complexes. We study the variation of the magnetic anisotropy for a series of elongated and compressed antiprisms in which the polar angle ␪ increases from 45° to 65° 关Fig. 4共d兲兴. The dependence of the magnetic anisotropy on the ␪ angle is shown in Fig. 6 for heavy lanthanide ions. Interestingly, Tb共III兲 ion has maximal magnetic anisotropy for the elongated antiprism ( ␪ ⬍ ␪ cub), Tm共III兲 has maximal magnetic anisotropy for the compressed antiprism ( ␪ ⬎ ␪ cub), while ⌬␹ of Dy共III兲 ion is always intermediate between that of Tb共III兲 and Tm共III兲 ions. Such a strong de-

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

FIG. 5. Variation of the magnetic anisotropy of Ln共III兲 ions upon the rotation of the top plane of a compressed square prism ( ␪ ⫽60°). The angle ␸ increases from 0° 共compressed tetragonal prism兲 to 45° 共compressed antiprism兲.

viation from the conclusion 共b兲 of the Bleaney theory can be accounted for by differences in the CF splitting patterns of Tb共III兲, Dy共III兲, and Tm共III兲 ions in the square antiprism at various ␪. For instance, at ␪ ⫽65° the CF splitting pattern of the ground 3H6 multiplet of Tm共III兲 ion is given by 0 ( 兩 3 H6 ,⫾6 典 ), 498 ( 兩 3 H6 ,⫾5 典 ), 603 ( 兩 3 H6 ,⫾4 典 ), 655

Magnetic anisotropy of lanthanide complexes

4681

( 兩 3 H6 ,⫾3 典 ), 740 ( 兩 3 H6 ,⫾2 典 ), 821 ( 兩 3 H6 ,⫾1 典 ), and 852 cm⫺1 ( 兩 3 H6 ,⫾0 典 ), while the CF splitting of the ground 7 F6 multiplet of Tb共III兲 is described by 0 ( 兩 7 F6 , 0典 ), 21 ( 兩 7 F6 ,⫾1 典 ), 86 ( 兩 7 F6 ,⫾2 典 ), 205 ( 兩 7 F6 ,⫾3 典 ), 388 ( 兩 7 F6 , ⫾4 典 ), 617 ( 兩 7 F6 ,⫾5 典 ), and 727 cm⫺1 ( 兩 7 F 6 ,⫾6 典 ). A very large energy gap between the highly anisotropic ground 兩 3 H6 ,⫾6 典 doublet of Tm共III兲 ion (g 储 ⬇14,g⬜ ⫽0) and the first excited 兩 3 H6 ,⫾5 典 doublet lying at 498 cm⫺1 is therefore the reason for the enhanced magnetic anisotropy of Tm共III兲 ion. Indeed, the thermal population factor of the ground 兩 3 H 6 ,⫾6 典 doublet is close to unity, while that of excited CF levels of Tm共III兲 is at least one order of magnitude smaller. Therefore, the high-temperature expansion of the magnetic susceptibility in the inverse temperature cannot be applied to this system since it requires comparable thermal populations of all CF levels of the ground multiplet of the lanthanide ion. Note that a magnetic anisotropy of about 80 000 ⫻10⫺6 cm3 mol⫺1 found for Tb共III兲 ion in the elongated square antiprism with ␪ ⫽45° seems to be close to the upper limit of magnetic anisotropy that can be achieved for lanthanide complexes at room-temperature. In fact, the CF splitting pattern of Tb共III兲 ion in this coordination polyhedron is most favorable for obtaining maximum magnetic anisotropy since the ground CF state is the well isolated 共by a gap of 378 cm-1 兲 pure 兩 7 F6 ,⫾6 典 doublet, whose extremely anisotropic g tensor has the largest leading component (g 储 ⫽17.9,g⬜ ⫽0) of all lanthanide ions. Accordingly, the magnetic susceptibility ␹ z along the z axis differs from that in the perpendicular direction ( ␹ x , ␹ y ) by about one order of magnitude, ␹ z ⫽90 024⫻10⫺6 cm3 mol⫺1 and ␹ x ⫽ ␹ y ⫽12 296 ⫻10⫺6 cm3 mol⫺1 . The magnetic anisotropy calculated for the capped square antiprism 共CN⫽9兲 at ␪ ⫽ ␪ cubic 关Fig. 4共f兲兴 is 3006 共Ce兲, 3492 共Pr兲, 1081 共Nd兲, 311 共Sm兲, -1119 共Eu兲, 48 422 共Tb兲, 34 958 共Dy兲, 9882 共Ho兲, -17 728 共Er兲, -17 351 共Tm兲, and 2778⫻10⫺6 cm3 mol⫺1 共Yb兲. G. Distortion from the cube to a dodecahedron

FIG. 6. Dependence of the magnetic anisotropy of lanthanide ions on the polar angle ␪ in the elongated and compressed square antiprism 关see Fig. 4共d兲兴.

A rhombic dodecahedron occurs as a coordination polyhedron in a number of lanthanide compounds, such as phosphates 共LnPO4 兲 and vanadates 共LnVO4 兲 with the zircon-type structure 共ZrSiO4 兲. The dodecahedron is obtained from the cube by compression of one of the two interpenetrating tetrahedra and elongation of the other one 关Fig. 4共e兲兴. In the idealized dodecahedron, the eight ligands are equidistant from the central metal ion and form the polar angles ␪ A ⫽36.9° and ␪ B ⫽69.5°. 46 Throughout the distortion of cube to the dodecahedron, we keep the ratio ( ␪ A ⫺ ␪ cub)/( ␪ B ⫺ ␪ cub) constant and equal to that in the idealized dodecahedron. Results of calculations of the magnetic anisotropy are shown in Fig. 7 for heavy lanthanide ions. The absolute value of ⌬␹ is considerably lower than that in the other coordination polyhedra with CN⫽6 and 8 considered previously, and the sign of ⌬␹ obeys the normal regularity 共a兲. In the idealized dodecahedron, the magnetic susceptibility was found to be 1261 共Ce兲, 1279 共Pr兲, 752 共Nd兲, 211 共Sm兲, ⫺646 共Eu兲, 17 864 共Tb兲, 10 375 共Dy兲, 6829 共Ho兲,

4682

Mironov et al.

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

FIG. 8. Tricapped trigonal prism 共a兲 and related coordination polyhedra resulted from 共b兲 rotations of the top and base planes of the prism around the threefold axis in opposite directions and 共c兲 moving three capping ligands from the equatorial plane. Figures 共d兲 and 共f兲 represent the capped and bicapped trigonal prism.

FIG. 7. Variation of the magnetic anisotropy of lanthanide ions upon the distortion of the cube to a dodecahedron. The degree of the distortion is given in percent 共0% corresponds to the regular cube and 100% to the dodecahedron; see the text for details兲.

⫺6189 共Er兲, ⫺14 355 共Tm兲, and ⫺3355⫻10 ⫺6 cm3 /mol⫺1 共Yb兲. The magnetic anisotropy is rather far from being proportional to ␰ 共7兲; in particular, the largest 兩 ⌬ ␹ 兩 follows the order Tb共III兲⬎Tm共III兲⬎Dy共III兲 rather than Dy共III兲 ⬎Tb共III兲⬎Tm共III兲 expected from the conclusion 共b兲 of the Bleaney theory. H. Tricapped trigonal prism and related coordination polyhedra

Of coordination polyhedra with CN⫽9, the tricapped trigonal prism 共regular or distorted兲 can be frequently found in lanthanide compounds. LnCl3 or Ln共III兲 ethylsulfates are typical examples. For the regular tricapped trigonal prism of

D 3h symmetry 关Fig. 8共a兲兴, the optimum value of the polar angle ␪ is close to 45° as estimated from the hard-sphere model or the most favorable polyhedron approach.28 On the other hand, PCEM calculations predict that the B 20 parameter is zero at ␪ ⫽45°. 47 One can therefore expect that this type of coordination is characterized by a small B 20 parameter and low magnetic anisotropy for all lanthanide ions. To explore the behavior of the magnetic anisotropy, we performed numerical calculations for the regular tricapped trigonal prism with D 3h symmetry and for coordination polyhedra resulting from its distortion to lower symmetries. We considered two distortion schemes for the tricapped trigonal prism: rotations of the base and top of the tricapped prism in opposite directions 关Fig. 8共b兲兴 at fixed ␪ ⫽45° and umbrella motion of the three capping ligands from the equatorial plane 关Fig 8共c兲兴. In the latter case, the ␪ angle for the three top and three lower ligands of the prism is set to 47° and 133°, respectively, while for the three capping ligands the deviation from the equatorial plane is described by the angle ␪ eq ⫽90°⫺ ␪ , which is varied from 0 to 30°.

TABLE III. Room-temperature 共298 K兲 magnetic anisotropy ⌬␹ 共in 10⫺6 cm3 mol⫺1 兲 of lanthanide ions in a tricapped trigonal prism in which top and base planes rotates in opposite directions 关Fig. 8共b兲兴. The rotation angle ␸ varies from 0 关D 3h tricapped trigonal prism, Fig. 8共a兲兴 to 30° 关Fig. 8共b兲兴. The polar angle ␪ of ligands in the top and base planes is set to 45° and 135°, respectively. Note that B 20 ⫽0 throughout the rotation.

Ln

0

5

Ce Pr Nd Sm Eu Tb Dy Ho Er Tm Yb

33 ⫺62 143 4 5 248 ⫺1164 991 626 ⫺1067 ⫺24

24 ⫺89 171 9 ⫺1 ⫺14 ⫺1481 1451 1336 ⫺1452 ⫺90

Rotation angle ␸, degrees 10 15 1 ⫺162 245 20 ⫺15 ⫺699 ⫺2339 2670 3193 ⫺2441 ⫺267

⫺23 ⫺262 343 35 ⫺35 ⫺1566 ⫺3486 4247 5548 ⫺3661 ⫺498

20

25

30

⫺44 ⫺361 437 50 ⫺54 ⫺2363 ⫺4607 5725 7702 ⫺4749 ⫺716

⫺57 ⫺433 503 60 ⫺68 ⫺2905 ⫺5409 6745 9157 ⫺5472 ⫺867

⫺61 ⫺459 526 63 ⫺73 ⫺3096 ⫺5700 7105 9665 ⫺5723 ⫺920

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

Magnetic anisotropy of lanthanide complexes

4683

⫻10⫺6 cm3 mol⫺1 共Yb兲 共monocapped trigonal prism兲 and 1178 共Ce兲, 1094 共Pr兲, 733 共Nd兲, 261 共Sm兲, ⫺755 共Eu兲, 17 729 共Tb兲, 10 006 共Dy兲, 6780 共Ho兲, ⫺7441 共Er兲, ⫺16012 共Tm兲, and ⫺2834⫻10⫺6 cm3 mol⫺1 共Yb兲 共bicapped trigonal prism兲. The comparison with the ⌬␹ values for the regular tricapped trigonal prism 共Table III, the ␸ ⫽0 column兲 shows that the magnetic anisotropy increases when the equatorial ligands are progressively removed from the equatorial plane. This can be rationalized in terms of an increasing asymmetry in the ligand surrounding. I. Icosahedron

FIG. 9. Dependence of the magnetic susceptibility of lanthanide ions in the tricapped trigonal prism on the ␪ eq angle describing the degree of deviation of the three capping ligands from the equatorial plane 共␪ eq⫽0 corresponds to the regular tricapped trigonal prism of D 3h symmetry兲 关see Fig. 8共c兲兴.

Results of calculations for the distortion scheme shown in Fig. 8共b兲 are presented in Table III. These data indicate that the magnetic anisotropy is indeed very low for the regular tricapped prism with ␪ ⫽45°. As is the case for the above coordination polyhedra with zero or small rank-two CFPs, one can observe that ⌬␹ develops in a very unusual way with increasing the rotation angle ␸ 共note that B 20 ⫽0 throughout the rotation at fixed ␪ ⫽45°兲. Indeed, maximal magnetic anisotropy is observed not for Dy共III兲 or Tb共III兲 ions, as could be expected from the variation of the ␰ parameter 共7兲, but for Er共III兲 and Ho共III兲 ions. The sign of the magnetic anisotropy is also quite different from the normal regularity 共a兲; moreover, for Ce共III兲 and Eu共III兲 ions ⌬␹ changes the sign during the rotation. This is once more evidence of the fact that the behavior of the magnetic anisotropy at small rank-two CFPs cannot be described in terms of Eq. 共8兲. The variation of the magnetic susceptibility for the distortion scheme given by Fig. 8共c兲 is shown in Fig. 9. Although for all lanthanide ions ⌬␹ correlates with the B 20 parameter 共which increases from ⫺235 at ␪ eq⫽0° to ⫹607 cm⫺1 at ␪ eq⫽30°兲, this dependence is rather complicated. In particular, while B 20 reverses sign at around ␪ eq ⫽15°, the sign reversal of ⌬␹ takes place at different ␪ eq angles for different lanthanide ions ranging from ␪ eq⫽14° 共Pr and Dy兲 to ␪ eq⫽22° 共Yb兲. This again demonstrates the inadequacy of the Bleaney approach for small rank-two CFPs. We also calculated the magnetic anisotropy for the monocapped and bicapped trigonal prism at ␪ ⫽45° 关Figs. 8共d兲 and 8共e兲兴, 2647 共Ce兲, 2612 共Pr兲, 1392 共Nd兲, 390 共Sm兲, ⫺1251 共Eu兲, 42474 共Tb兲, 24 770 共Dy兲, 13 188 共Ho兲, ⫺14 229 共Er兲, ⫺22189 共Tm兲, and ⫺4336

In the regular icosahedron (CN⫽12), all CFPs are zero except B 65 and B 60 , for which the B 65 /B 60 ⫽(7/11) 1/2 ratio holds.23 In the regular icosahedron 共I h point group兲, multiplets with J⬍3 cannot be split by the CF effect. Both numerical calculation and the Bleaney approach confirm that the magnetic anisotropy is strictly zero for all lanthanide ions in the icosahedron. The regular icosahedral coordination is not common for real lanthanide compounds, but distorted icosahedra were found in rare-earth double nitrates R2 M3 (NO3 ) 12•24H2 O 共R⫽Ce– Eu, M⫽Mg, Zn兲.48 The magnetic anisotropy of lanthanide ions calculated with the use of CFPs obtained for Eu2 Zn3 (NO3 ) 12•24H2 O compound48 is ⫺355 共Ce兲, ⫺391 共Pr兲, ⫺491 共Nd兲, ⫺100 共Sm兲, 473 共Eu兲, ⫺4783 共Tb兲, ⫺4381 共Dy兲, ⫺7147 共Ho兲, ⫺3626 共Er兲, 4769 共Tm兲, and 2161 ⫻10⫺6 cm3 mol⫺1 共Yb兲 for C 3 v point group, and ⫺410 共Ce兲, ⫺527 共Pr兲, ⫺502 共Nd兲, ⫺94 共Sm兲, 429 共Eu兲, ⫺5911 共Tb兲, ⫺5588 共Dy兲, ⫺5799 共Ho兲, 166 共Er兲, 5053 共Tm兲, and 1959⫻10⫺6 cm3 mol⫺1 共Yb兲 for the C 3 symmetry. A very low magnetic anisotropy of all lanthanide ions indicates that the coordination polyhedron is really close to the regular icosahedron. Note that the magnitude and sign of ⌬␹ of Er共III兲 ion is sensitive to the point-group symmetry, in which CFPs are determined. Our systematic calculations for various coordination polyhedra considered above in points 共a兲–共i兲 have allowed to establish some regularities in the behavior of the roomtemperature magnetic anisotropy of lanthanide ions. First of all, these results show that the magnetic anisotropy of lanthanide complexes at room-temperature is generally very high. Magnetic anisotropies of about 50 000 ⫻10⫺6 cm3 mol⫺1 can be obtained in many coordination polyhedra, which is more than two orders of magnitude larger than that for transition metal complexes or organic compounds.10–15 The upper limit of the room-temperature anisotropy is estimated to be about 80 000 ⫻10⫺6 cm3 mol⫺1 关Tb共III兲 ion in the elongated square antiprism兴. The magnetic anisotropy of the light lanthanide ions 共Ce–Eu兲 is considerably lower 共typically, by one order of magnitude兲 than that of heavy lanthanide ions 共Tb–Yb兲. Model numerical calculations show that in most coordination polyhedra maximum magnetic anisotropy can be found only for three lanthanide ions, Tb共III兲, Dy共III兲, and Tm共III兲. Metallomesogenic complexes containing these ions are therefore most promising for designing lanthanide-containing liquid crystals with a very high magnetic anisotropy which is re-

4684

Mironov et al.

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002

quired for their easy alignment in the magnetic field. The largest magnetic anisotropy of real lanthanide-containing liquid crystals obtained to the moment is about 17 000 ⫻10⫺6 cm3 mol⫺1 , as found in Dy共III兲 and Tb共III兲 metallomesogenic complexes.11,12 This value is still considerably lower than that found from our calculations. Magnetic behavior of Tm共III兲 metallomesogenic complexes is less studied than that of Tb共III兲 or Dy共III兲 complexes.10–14 Since the sign of the magnetic anisotropy of Tm共III兲 complexes is opposite that of Tb共III兲 and Dy共III兲 complexes, thuliumcontaining metallomesogens having a high magnetic anisotropy can be used instead of related terbium or dysprosium complexes when another orientational behavior in the magnetic field is required. Our comparative model analysis of the magnetic anisotropy evidences a pronounced discrepancy between the results of the Bleaney theory and those obtained from more accurate numerical calculations. Indeed, results of numerical calculations for coordination polyhedra 共a兲–共i兲 indicate that considerable discrepancies in calculations of the roomtemperature magnetic anisotropy of lanthanide complexes in terms of the Bleaney approach seem to the rule rather than the exception. The conclusion 共b兲 of the Bleaney theory that maximum magnetic anisotropy should always obey the regularity Dy共III兲⬎Tb共III兲⬎Tm共III兲⬎¯ is very often not true, since in many cases maximal magnetic anisotropy can be found for Tb共III兲 or even for Tm共III兲. The conclusion 共c兲 that the magnetic anisotropy is proportional to the rank two CFPs and is not influenced by rank-four and rank-six CFPs is also shown to be very approximate. In particular, magnetic anisotropy is strictly zero for all lanthanide ions only in the regular cube and octahedron, while in other polyhedra with zero or small rank-two CFPs magnetic anisotropy may be rather noticeable. The most reliable result of the Bleaney theory seems be the sign rule 共a兲, according to which Ce共III兲, Pr共III兲, Nd共III兲, Sm共III兲, Tb共III兲, Dy共III兲, and Ho共III兲 ions always have one sign and Eu共III兲, Er共III兲, Tm共III兲, and Yb共III兲 ions have the opposite sign. It is found to be valid in most coordination polyhedra except those in which rank-two CFPs are small or zero. In these cases, the sign of ⌬␹ may be irregular. We can therefore conclude that the Bleaney approach is not well suited for quantitative calculations of the roomtemperature magnetic anisotropy of lanthanide complexes, since its use can give rise to an error of multiple ten percents or even more. The general reason is that the underlying hightemperature expansion of the magnetic susceptibility in the inverse temperature limited to T ⫺2 terms is not a good approximation since the necessary condition ⌬E CF /k B T⬍1 共implying that all CF levels of the ground multiplet of the lanthanide ion should have comparable thermal populations兲 does not often hold in real lanthanide complexes. In particular, this should be borne in mind in the interpretation of paramagnetic shifts in NMR spectra of lanthanide complexes at room-temperature, in which the Bleaney approach is often used.16 –20

V. CONCLUSIONS

In this study, the room-temperature magnetic anisotropy of lanthanide ions in various coordination polyhedra is analyzed in detail using a model approach based on numerical calculations of the tensor of the magnetic susceptibility. Principal regularities in the variation of the magnetic anisotropy over the entire lanthanide series are established. The roomtemperature magnetic anisotropy of lanthanide complexes is generally very high and can often reach a value of 50 000 ⫻10⫺6 cm3 mol⫺1 or more. The upper limit of the room temperature magnetic anisotropy is estimated to be as high as 80 000⫻10⫺6 cm3 mol⫺1 . Magnetic anisotropy of light lanthanide ions is always considerably lower than that of heavy lanthanide ions. In the series of isostructural complexes, maximum magnetic anisotropy can be observed for three lanthanide ions only, Tb共III兲, Dy共III兲, and Tm共III兲, which are therefore the most attractive for obtaining lanthanidecontaining liquid crystals with a high magnetic anisotropy. The comparison with numerical calculations indicates that the Bleaney theory is of limited applicability for quantitative calculations of the room-temperature magnetic anisotropy of lanthanide complexes, since in many cases its use results in considerable errors. ACKNOWLEDGMENTS

K.B. is a Postdoctoral Fellow of the Fund for Scientific Research Flanders 共Belgium兲. Financial support by the K.U. Leuven 共No. GOA 98/03兲 and by the F.W.O.-Flanders 共No. G.0234.99兲 is gratefully acknowledged. V.M. wishes to acknowledge partial financial support of the Russian Foundation for Basic Research, Grant No. 01-03-32210. S. Hu¨fner, Optical Spectra of Transparent Rare Earth Compounds 共Academic, New York, 1978兲, p. 147. 2 M. Gerloch and D. J. Mackey, J. Chem. Soc. Dalton Trans. 1972, 415, and references therein. 3 C. Cascales, G. Lozano, C. Zaldo, and P. Porcher, Chem. Phys. 257, 29 共2000兲. 4 J. Ho¨lsa¨, R. J. Lamminma¨ki, M. Lastusaari, P. Porcher, and R. SaezPuche, J. Alloys Compd. 303, 498 共2000兲. 5 C. Cascales, P. Porcher, and R. Saez-Puche, J. Alloys Compd. 250, 391 共1997兲. 6 J. Ho¨lsa¨, E. Kestila, P. Ylha, R. Saez-Puche, P. Deren, W. Strek, and P. Porcher, J. Phys.: Condens. Matter 8, 1575 共1996兲. 7 C. Cascales, P. Porcher, and R. Saez-Puche, J. Phys.: Condens. Matter 8, 6413 共1996兲. 8 H. De Leebeeck and C. Go¨rller-Walrand, J. Alloys Compd. 225, 75 共1995兲. 9 L. Holmes, R. Sherwood, and L. G. van Uitert, J. Appl. Phys. 39, 1373 共1968兲. 10 Yu. G. Galyametdinov, M. A. Athanassopoulou, K. Griesar, O. Kharitonova, E. A. Soto Bustamante, L. Tinchurina, I. Ovchinnikov, and W. Haase, Chem. Mater. 8, 922 共1996兲. 11 K. Binnemans, Yu. G. Galyametdinov, R. Van Deun et al., J. Am. Chem. Soc. 122, 4335 共2000兲. 12 Yu. G. Galyametdinov, W. Haase, L. Malykhina, A. Prosvirin, I. Bikchantaev, A. Rakmatullin, and K. Binnemans, Chem.-Eur. J. 7, 99 共2001兲. 13 A. N. Turanov, I. V. Ovchinnikov, Yu. G. Galyametdinov, G. I. Ivanova, and V. A. Goncharov, Russ. Chem. Bull. 48, 690 共1999兲. 14 I. Bikchantaev, Yu. G. Galyametdinov, O. Kharitonova, I. V. Ovchinnikov, D. W. Bruce, D. A. Dunmur, D. Guillon, and B. Heinrich, Liq. Cryst. 20, 489 共1996兲. 15 V. S. Mironov, Yu. G. Galyametdinov, A. Ceulemans, and K. Binnemans, J. Chem. Phys. 113, 10293 共2000兲. 1

J. Chem. Phys., Vol. 116, No. 11, 15 March 2002 16

R. S. Prosser, S. A. Hunt, J. A. DiNatale, and R. R. Vold, J. Am. Chem. Soc. 118, 269 共1996兲. 17 R. S. Prosser, H. Bryant, R. G. Bryant, and R. R. Vold, J. Magn. Reson. 141, 256 共1999兲. 18 R. S. Prosser, J. S. Hwang, and R. R. Vold, Biophys. J. 74, 2405 共1998兲. 19 R. S. Prosser and I. V. Shiyanovskaya, Concepts Magn. Reson. 13, 19 共2001兲. 20 I. Bertini, M. B. L. Janik, Y. M. Lee, C. Luchinat, and A. Rosato, J. Am. Chem. Soc. 123 4181 共2001兲. 21 W. De W. Horrocks, Jr. and J. P. Sipe, Science 177, 994 共1972兲. 22 B. Bleaney, J. Magn. Reson. 8, 91 共1972兲. 23 R. M. Golding and P. Pyykko, Mol. Phys. 26, 1389 共1973兲. 24 W. De W. Horrocks, Jr., J. Magn. Reson. 26, 333 共1977兲. 25 B. R. McGarvey, J. Magn. Reson. 33, 445 共1979兲. 26 V. S. Mironov, Yu. G. Galyametdinov, A. Ceulemans, C. Go¨rller-Walrand, and K. Binnemans, Chem. Phys. Lett. 345, 132 共2001兲. 27 A. Ceulemans, G. Mys, and S. Walcerz, New J. Chem. 17, 131 共1993兲. 28 C. Go¨rller-Walrand and K. Binnemans, ‘‘Rationalization of Crystal Field Parametrization,’’ in Handbook on the Physics and Chemistry of Rare Earths, edited by K. A. Gschneidner, Jr. and L. Eyring 共North-Holland, Amsterdam, 1996兲, Vol. 23, Chap. 155, p. 121. 29 C. A. Morrison and R. P. Leavitt, ‘‘Spectroscopic Properties of Trivalent Lanthanides in Transparent Host Crystals,’’ in Handbook on the Physics and Chemistry of Rare Earths, edited by K. A. Gschneidner, Jr. and L. Eyring 共North-Holland, Amsterdam, 1982兲, Vol. 5, Chap. 46, p. 461. 30 A. A. Kaminskii, Crystalline Lasers 共CRC Press, New York, 1996兲. 31 M. Gerloch and R. F. McMeeking, J. Chem. Soc. Dalton Trans. 1975, 2443.

Magnetic anisotropy of lanthanide complexes

4685

J. H. VanVleck, The Theory of Electric and Magnetic Susceptibilities 共Oxford University Press, Oxford, 1932兲. 33 W. Urland, Chem. Phys. Lett. 46, 457 共1976兲. 34 B. G. Wybourne, Spectroscopic Properties of Rare Earths 共Interscience, New York, 1965兲. 35 D. Garcia and M. Faucher, ‘‘Crystal Field in Non-metallic Rare Earth Compounds,’’ in Handbook on the Physics and Chemistry of Rare Earths, edited by K. A. Gschneidner, Jr. and L. Eyring 共North-Holland, Amsterdam, 1995兲, Vol. 21, Chap. 144, p. 263. 36 W. T. Carnall, P. R. Fields, and K. Rajnak, J. Chem. Phys. 49, 4412 共1968兲. 37 D. J. Newman, Aust. J. Phys. 31, 79 共1978兲. 38 Y. Y. Yeung and M. F. Reid, J. Less-Common Met. 148, 213 共1988兲. 39 W. Urland and R. Kremer, Inorg. Chem. 23, 1550 共1984兲. 40 O. L. Malta, S. J. L. Ribeiro, M. Faucher, and P. Porcher, J. Phys. Chem. Solids 52, 587 共1991兲. 41 P. Porcher, M. Couto dos Santos, and O. Malta, Phys. Chem. Chem. Phys. 1, 397 共1999兲. 42 N. C. Chang, J. B. Gruber, R. P. Leavitt, and C. A. Morrison, J. Chem. Phys. 76, 3877 共1982兲. 43 M. V. Hoehm and D. G. Karraker, J. Chem. Phys. 60, 393 共1974兲. 44 G. Meyer and A. Scho¨nemund, Mater. Res. Bull. 15, 89 共1980兲. 45 H. Maghrawy, W. Strek, J. Legendziewicz, J. Hanuza, and E. Lukowiak, Pol. J. Chem. 67, 1959 共1993兲. 46 M. C. Favas and D. L. Kepert, Proc. Inorg. Chem. 26, 325 共1980兲. 47 K. Binnemans and C. Go¨rller-Walrand, Chem. Phys. Lett. 245, 75 共1995兲. 48 C. Go¨rler-Walrand, I. Hendrickx, L. Fluyt, M. P. Gos, J. D’Olieslager, and G. Blasse, J. Chem. Phys. 96, 5650 共1992兲. 32