Influence of cubic crystal field on the magnetic susceptibility of defect-perovskite RTa3O9 (R=rare earth)

PCS 1859

Journal of Physics and Chemistry of Solids 61 (2000) 45–65 www.elsevier.nl/locate/jpcs

Influence of cubic crystal field on the magnetic susceptibility of defect-perovskite RTa3O9 (R ˆ rare earth) S. Ebisu*, H. Morita, S. Nagata Department of Materials Science and Engineering, Muroran Institute of Technology, 27-1 Mizumoto-cho, Muroran, Hokkaido 050-8585, Japan Received 24 May 1999; accepted 4 June 1999

Abstract Perovskite-related compounds RTa3O9 (R ˆ rare earth elements except Sc, Pm, Yb, Lu) have been prepared and magnetic susceptibility measurements have been carried out. The characteristic behavior in the temperature dependence of the susceptibility for each sample has been analyzed in terms of the crystalline electric field of cubic symmetry yielded from the nearest neighboring 12 O 22 and the next-nearest neighboring eight Ta 51 around R 31 ions. These analytical results demonstrate clearly that the contribution of the next-nearest neighbors to the susceptibility is important and fairly large because of the high valence state of Ta 51. The ground states of R 31 ions in the compounds are determined as follows in the Bethe’s notation: Ce 31, G8; Pr 31, G3; Nd 31, G6; Tb 31, G5; Dy 31, G8; Ho 31, G5; Er 31, G8; and Tm 31, G5. The precise analysis for RTa3O9 based on the crystal field theory has systematically provided the energy splitting scheme, the f-electron wave functions and significant crystalline field parameters containing both fourth and sixth order terms. The splitting into G6, G7 and G8 from the degenerated ground state 8S7/2 of Gd 31 is negligibly small and not distinguishable above 2 K. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: A. Inorganic compounds; A. Oxides; D. Magnetic properties; D. Crystal fields

1. Introduction Rare earth tantalate RTa3O9 possesses a novel crystal structure, related to the perovskite structure, as shown in Fig. 1 [1]. Here R stands for rare earth elements except Sc and Pm [1–5]. A frame of a unit cell for this structure, called defect-perovskite, is formed by stacking of two unit cells of cubic perovskite, then the unit cell is completed by taking off all the R atoms from the interface and 1=3 of the R atoms from the basal planes. Magnetic ions in these compounds are only R 31 ions, whereas Y 31, La 31 and Lu 31 are nonmagnetic because of no electrons in the 4f orbitals. All the R 31 ions, randomly distributed in the basal planes, are surrounded by 12 oxygen ligands. The 12-fold coordination exists rarely in compounds, while the four-, six- and eightfold coordinations appear in general. RTa3O9 are novel compounds which play an important role to study the magnetic properties of R 31 ions in a 12-fold cubic coordination in the perovskite structure. RTa3O9 are insulators with* Corresponding author. Fax: 1 81-143-46-5601. E-mail address: [email protected] (S. Ebisu)

out conduction electrons and the f-electrons in RTa3O9 are localized. We have successfully prepared the RTa3O9 system by solid state reaction method. Some exceptions are scandium, promethium, ytterbium and lutetium tantalates. The scandium tantalate had not crystallized into the defect-perovskite structure according to Ref. [1]. Information on the synthesis including the radioactive element Pm is not available on this defect-perovskite. The preparations of singlephase samples of YbTa3O9 and LuTa3O9 [5] have been unsuccessful in our laboratory. YbTa3O9 was obtained as the main phase, however impure phases of YbTaO4 and Ta2O5 were always detected. LuTa3O9 was not obtained by the solid state reaction method below 1873 K. The crystal systems and the lattice parameters of these series compounds RTa3O9 are tabulated and compared with the results obtained by other workers. The dc paramagnetic susceptibility of RTa3O9 has been measured over the temperature range 2–300 K. The temperature dependence of the susceptibility indicates a successive coherent change, depending on the rare earth element. In this paper, we will present the detailed analysis

0022-3697/00/$ - see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00227-9

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CeO2 (99.99% pure) and Tb4O7 (99.9% pure) than Ce2O3 and Tb2O3 were used for preparing CeTa3O9 and TbTa3O9. The reaction formulae are as follows: R2 O3 1 3Ta2 O5 ! 2RTa3 O9 ;

Fig. 1. Crystal structure of RTa3O9. The crystal structure belongs to one of tetragonal, orthorhombic or monoclinic system. The b-axis should be regarded as a-axis for the tetragonal system, and the angle g is equal to a right angle for the tetragonal and the orthorhombic systems.

of the paramagnetic susceptibility on the basis of the crystalline electric field (CEF) theory. R 31 ion in RTa3O9 has 12 O 22 ions as the nearest neighboring (n.n.) ions and eight Ta 51 ions as the next-nearest neighboring (n.n.n.) ions as shown in Fig. 2. The significant influence from the n.n.n. ions is emphasized on the assumption of predominant cubic CEF. Since Ta ion has the high valence state of Ta 51, the influence of the CEF is strongly modified by the n.n.n. Ta 51, which will be demonstrated and discussed below.

…1†

2CeO2 1 3Ta2 O5 ! 2CeTa3 O9 1

1 O "; 2 2

…2†

Tb4 O7 1 6Ta2 O5 ! 4TbTa3 O9 1

1 O "; 2 2

…3†

here R in formula (1) excludes Ce and Tb. The powder was mixed thoroughly in an agate mortar and the mixture was heated on an alumina boat in air to 1773–1873 K for 20– 30 h. High quality specimens were prepared by repeating this process 3–4 times. The crystal structures and the lattice parameters were determined by powder X-ray diffraction measurements using CuKa-radiation. The dc magnetic susceptibility x , which refers to magnetization M divided by a constant magnetic field H …x ˆ M=H†; was measured in the temperature range over 2–300 or 340 K using an rf-SQUID magnetometer (Quantum Design). The magnetic field of 100 Oe or 10 kOe was applied after cooling the sample to 2 K and measurements were performed on warming. The data of x are corrected by subtracting the diamagnetic contribution, x core, due to the orbitals of each ions. The values tabulated in Ref. [6] are used for Ta 51 and O 22; they are 21:4 × 1025 and 21:2 × 1025 emu …mol-f:u:†21 ; respectively. The x core of 23:8 × 1025 emu …mol-f:u:†21 for La 31 estimated by Pauling [7] is used for all R 31. The deviations in the values of x core for R 31 from the value for La 31 must be small. The estimated values of x core for all the RTa3O9 are commonly 21:88 × 1024 emu …mol-f:u:†21 :

2. Experimental methods 3. Experimental results Polycrystalline samples of RTa3O9 were synthesized with a solid state reaction technique. R2O3 and Ta2O5 (Furuuchi Chemicals Corp.; La2O3, Sm2O3, Eu2O3, Er2O3 and Ta2O5 are 99.99% pure and the others are 99.9% pure) were used as starting materials in most cases. More stable compounds

Fig. 2. Simple idealized cubic sub-cell for RTa3O9.

3.1. Sample preparation The X-ray-diffraction (XRD) patterns for the powder samples of RTa3O9 are shown in Figs. 3–5. Fig. 3 shows XRD patterns for RTa3O9 …R ˆ La; Gd, Er and Tm). All the peaks have been indexed to the tetragonal unit cells. The right side of double peaks observed at a higher angle than 408 is due to CuKa2-radiation, which is realized by the fact that its intensity is about half of the left side peak. The lattice parameters are listed in Table 1. The parameter c is slightly larger than 2a. If we take the pseudocubic sub-cell obtained by separating the unit cell by (002) plane, the degree of distortion of the sub-cell from cubic symmetry along the c-axis is measured by the value of c=2a. They are made into list in the column c=…a 1 b† in Table 1, where these values are close to unity. Although a single-phase sample of YbTa3O9 was not synthesized, the main peaks of YbTa3O9 show the tetragonal symmetry and their lattice parameters are included in Table 1.

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

47

Fig. 3. X-ray diffraction pattern of RTa3O9 …R ˆ La; Gd, Er and Tm). All the peaks are indexed to the tetragonal unit cells.

The compounds of RTa3O9 …R ˆ Ce; Pr, Nd, Sm and Eu) show slight distortion to the orthorhombic symmetry from the tetragonal one. The XRD patterns for these compounds are shown in Fig. 4. All the peaks have been indexed to the orthorhombic unit cells. The index hkl 1 means that split of the (hkl) peak into the (hkl) and the (khl) peaks is not observed. The splits are observed restrictively in the higher angular region, for an example, the peaks of (302), (310), (312) and (313) above 2u ˆ 778 split in the XRD pattern for CeTa3O9. The values of b=a, as a measure of the distortion in the ab-plane, are slightly larger than unity. The largest value is 1.004 for PrTa3O9 and the other values are less than 1.001 (see Table 1). The distortion of the pseudocubic sub-cell along the c-axis is also small. Only for CeTa3O9 the c value is 0.8% larger than the value of a 1 b; while for the others the c values are 0.4–0.5% smaller than the values of a 1 b: The XRD patterns for RTa3O9 …R ˆ Tb; Dy, Ho and Y) are shown in Fig. 5. All the peaks have been indexed to the monoclinic unit cells. The index hkl 1 means that split of

the (hkl) peak into the (hkl) and the (khl) peaks is not observed. In the measured region no such split has been observed, hence the lattice parameter a is equal to b. Therefore, strictly speaking, the Bravais lattice of these compounds belongs to base-centered orthorhombic system. However, we regard these systems as monoclinic for good correspondence with crystal structure shown in Fig. 1. The lattice parameters a 0 and b 0 for orthorhombic system are related to a and g for monoclinic symmetry as a 0 ˆ  p p  a 2…1 1 cos g† and b 0 ˆ a 2…1 2 cos g†; and the c parameter is same between both the symmetries. They are also listed in Table 1. The Miller indices h 0 , k 0 and l 0 for orthorhombic system are h 0 ˆ uh 1 ku; k 0 ˆ uh 2 ku and l 0 ˆ l: The distortion parameters b=a in the orthorhombic systems are less than 1.019, and c=…a 1 b† in the monoclinic systems are less than 1.014. The distortion of the pseudocubic sub-cell is, as we have seen, small in all compounds. The decrease in the volume of the unit cell from La to Yb compounds matches well with the lanthanide contraction.

Fig. 4. X-ray diffraction pattern of RTa3O9 …R ˆ Ce; Pr, Nd, Sm and Eu). All the peaks are indexed to the orthorhombic unit cells. The index hkl 1 means that split of the (hkl) peak into the (hkl) and the (khl) peaks is not observed. Such splits are observed in higher angular region.

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Fig. 5. X-ray diffraction pattern of RTa3O9 …R ˆ Tb; Dy, Ho and Y). All the peaks are indexed to the monoclinic unit cells. In the measured region no split of the (hkl 1 ) peaks into the (hkl) and (khl) peaks is observed. Hence, strictly speaking, the Bravais lattice of these compounds belongs to base-centered orthorhombic system.

3.2. Paramagnetic susceptibility The temperature dependence of dc magnetic susceptibility x and x21 for RTa3O9 …R ˆ La; Ce, Pr, Nd, Sm and Eu) is shown in Fig. 6(a) and (b). Here, the magnetic field H ˆ 100 Oe was applied. The field H of 10 kOe was applied to LaTa3O9. The x21 for LaTa3O9 is omitted in Fig. 6(b). The x of LaTa3O9 has a positive and a nearly constant value between 2 and 3 × 1024 emu mol21 : The deviation from the Curie–Weiss law is clearly seen in the x21 in Ce and Nd compounds, which originates from the CEF effect. These x21 of these compounds at low temperature decrease acceleratingly with decreasing temperature. These results are

consistent with remaining of the Kramers doublet in Ce 31 and Nd 31 ions having an odd number of 4f electrons. The x of PrTa3O9, in which Pr 31 has two 4f electrons, is strongly suppressed by the CEF effect. The abrupt increase below 10 K is presumably due to the Curie term from the impurity phase. The net susceptibility after subtraction of this term shows a tendency toward saturation, which indicates the ground state of PrTa3O9 to be non-magnetic having a singlet state. In the Sm compound, the temperature dependence of susceptibility is small in the higher temperature region above 50 K and is very large in the lower temperature region. The temperature dependence of susceptibility for

Table 1 Lattice parameters of RTa3O9. The notation of T, O and M in the column of “crystal system” means tetragonal, orthorhombic and monoclinic system, respectively. The values in the column c=…a 1 b† for the tetragonal systems are calculated by c/2a. All the values of b/a and c=…a 1 b† are close to unity. The volume of the unit cell, V, decreases from La to Yb, associated with lanthanide contraction Compound

Crystal system

˚) a (A

˚) b (A

˚) c (A

g (8)

b=a

c=…a 1 b†

˚ 3) V (A

YTa3O9 YTa3O9 a LaTa3O9 CeTa3O9 PrTa3O9 NdTa3O9 SmTa3O9 EuTa3O9 GdTa3O9 TbTa3O9 TbTa3O9 a DyTa3O9 DyTa3O9 a HoTa3O9 HoTa3O9 a ErTa3O9 TmTa3O9 YbTa3O9

M O T O O O O O T M O M O M O T T T

3.831 5.366 3.916 3.907 3.911 3.908 3.898 3.886 3.868 3.849 5.409 3.839 5.387 3.832 5.368 3.836 3.829 3.820

3.831 5.470 – 3.910 3.927 3.913 3.902 3.892 – 3.849 5.476 3.839 5.472 3.832 5.469 – – –

7.765 7.765 7.906 7.876 7.801 7.784 7.758 7.744 7.787 7.790 7.790 7.779 7.779 7.767 7.767 7.745 7.739 7.733

91.09 – – – – – – – – 90.71 – 90.90 – 91.07 – – – –

1.000 1.019 1.000 1.001 1.004 1.001 1.001 1.001 1.000 1.000 1.012 1.000 1.016 1.000 1.019 1.000 1.000 1.000

1.013 – 1.010 1.008 0.995 0.995 0.995 0.996 1.007 1.012 – 1.013 – 1.014 – 1.009 1.011 1.012

114.0 227.9 121.2 120.3 119.8 119.0 118.0 117.1 116.5 115.4 230.7 114.6 229.3 114.0 228.0 114.0 113.4 112.8

a

The values for base-centered orthorhombic system are also listed for monoclinic systems (see text).

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

49

Fig. 6. Temperature dependence of dc magnetic susceptibility for RTa3O9 …R ˆ La; Ce, Pr, Nd, Sm and Eu) shown as (a) x vs. T plot and (b) x 21 vs. T plot. The magnetic field of H ˆ 100 Oe was applied except for LaTa3O9 …H ˆ 10 kOe†:

EuTa3O9 is extremely weak. The behavior in the Sm and Eu compounds originates from the Van Vleck paramagnetism superposed by the CEF effect. Fig. 7(a) shows the temperature dependence of inverse susceptibility x21 for the system of Gd, Tb, Dy, Ho, Er and Tm, and the enlargement of the temperature region below 50 K is shown in Fig. 7(b). All the compounds except Gd show the deviation from the Curie–Weiss behavior in the x21 vs. T plot, more clearly seen in the lower temperature region.

4. Discussion 4.1. Crystal system The comparison of the crystal system with the other researcher’s results is given. The compounds RTa3O9 are classified into three crystal systems in the present work. These results are listed in Table 2. Other workers also classified these compounds into two or three systems, however the results show rather poor agreement each other. LaTa3O9

is the only one compound which all the researchers have specified unanimously as the tetragonal symmetry. Sirotinkin et al. [5] used optical zone fusion method for the preparation of RTa3O9, while the other workers have used the method of solid-state reaction or fusion. The results by Iyer et al. [1] are obtained from single crystals except ErTa3O9, whereas the results by the other workers are obtained from polycrystalline samples. It should be pointed out that the temperatures of the solid-state reaction made by other workers are below 1773 K, whereas those carried out in the present study are between 1773 and 1873 K. The difficulties for obtaining single-phase sample by solidstate reaction below 1773 K were reported in Refs. [2,3,5]. The results by Keller et al. [2] and by Iyer et al. [1] agree well, and do not include the monoclinic system. Rooksby et al. [3] have reported three monoclinic systems. In their monoclinic system the lattice parameters a and b have slightly different values, so those crystals do not belong to the base-centered orthorhombic system. Since the lattice parameters are unknown in the monoclinic system reported by Romashov et al. [4], we cannot realize whether their data agree with our monoclinic system or not.

Fig. 7. Temperature dependence of the inverse susceptibility x 21 for RTa3O9 …R ˆ Gd; Tb, Dy, Ho, Er and Tm). The figure (b) is the enlargement of the temperature region below 50 K. The solid curves in the figure (b) are guides to the eye.

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S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

Table 2 Comparison of crystal systems of RTa3O9 with the other worker’s results. The notation of T, O and M means tetragonal, orthorhombic and monoclinic system, respectively. Here M p means exact monoclinic system, in which a is not equal to b. For M #, we do not know the details because the lattice parameters were not reported. O † stands for the base-centered orthorhombic system, hence these systems correspond to our monoclinic system Compound

Present work

Ref. [3]

Ref. [2]

Ref. [1]

Ref. [4]

Ref. [5]

YTa3O9 LaTa3O9 CeTa3O9 PrTa3O9 NdTa3O9 SmTa3O9 EuTa3O9 GdTa3O9 TbTa3O9 DyTa3O9 HoTa3O9 ErTa3O9 TmTa3O9 YbTa3O9 LuTa3O9

M(O) T O O O O O T M(O) M(O) M(O) T T T

Mp T O O T T

T T T O O O O T T T T T

T T T O O O

M# T

T T

T T T O O M# M# M# M# M# M#

O O O O O O† O† O† O† T T T

O Mp

Mp

The orthorhombic system in the latest report by Sirotinkin et al. [5] stands for base-centered orthorhombic system, hence it corresponds to our monoclinic system. Our results agree mainly with the results by Sirotinkin et al. except GdTa3O9, ErTa3O9 and YTa3O9. The compound CeTa3O9 is not included in their work. They have pointed out that the XRD pattern of GdTa3O9 is similar to one of the tetragonal group. There may be two reasons for disagreement among the researchers. One is due to an accuracy of X-ray analysis. Since the distortion from tetragonal system is weak in these compounds, the lower resolution of 2u in XRD measurements may lead missing the small split of peak. It is difficult to distinguish the peak by Ka1-radiation from one by the Ka2-radiation. The other reason is due to quality or conditions in the sample preparation. For example, we have observed the monoclinic phase in the earlier sample of ErTa3O9. By repeating the process of grinding and heating, which is the conventional technique to obtain high purity sample in the solid-state reaction method, the monoclinic phase diminished and the tetragonal phase grew. Finally, after processing four times, only the tetragonal phase remained. Sirotinkin et al. have pointed out the existence of superstructural peaks in the XRD pattern for ErTa3O9 before annealing at 1433 K for 2–4 days [5]. The fact that such peaks disappear after the annealing at 1433 K indicates the preparation temperature is one of the decisive factors for changing the crystal system. For the other monoclinic system in our work, any change of the crystal system was not observed even after repeating the process 3–4 times. The crystal system sensitive to the preparation conditions may be due to the formation of random vacancy in (001) plane.

T T T T

4.2. Simplified ideal cubic coordination We assume a simple idealized cubic sub-cell as shown in Fig. 2 in order to analyze the magnetic properties of RTa3O9. The R 31 ion is coordinated by 12 O 22 ligands and by eight Ta 51 ions in cubic symmetry. The nearest neighbors are 12 O 22 and next-nearest neighbors are eight Ta 51. The 12 O 22 form a 14-faced polyhedron and the eight Ta 51 form a cube around R 31. The refinement of the accurate crystal structure parameters for RTa3O9 is a difficult problem because of the random distribution of R atoms in (001) plane. Iyer et al. [1] have reported them for LaTa3O9 which is only one datum about the crystal structure parameters for RTa3O9 system. They have shown these parameters for LaTa3O9 assuming the space group P4/mmm, using single crystal data. The R atom is in one-fold position a in P4/mmm, two Ta atoms are in two-fold position h with positional parameter z of 0.26040. Here, the letter a or h stands for special point in P4/mmm symmetry using Wyckoff notation in crystallography [8], each letter of i, c, d, p or q described below also means it. There are three kinds of sites for O atoms. Four O atoms are located in four-fold position i with z parameter of 0.22617. Therefore, two TaO6 octahedra in a unit cell are distorted and the central Ta atoms deviate toward the center of the unit cell. The space group P4/mmm requires the other two O sites are in one-fold position c (0.5, 0.5, 0) and d (0.5, 0.5, 0.5), however their result shows those are in eight-fold position p and q. The positional parameters x, y, z are 0.505, 0.510, 0 and 0.505, 0.510, 0.5. They have explained for this inconsistency that each O atom in (001) or (002) plane deviates from the position c or d in the disordered manner. Our understanding for their explanation is that the directions

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

of the deviation of these O atoms in one lattice are different from those in another lattice, according to the difference of the distribution of R atoms around each lattice. Since there are such difficulties in the refinement of the accurate crystal structure, the simplified ideal cubic coordination is assumed in the present CEF analysis mentioned below.

trigonometric functions, which are VCEF …r† ˆ

VCEF …r† ˆ

X …2Zi e2 † ; i ur 2 Ri u

…4†

where e stands for the absolute value of charge of an electron …e . 0† and Zie means the charge of the i-th neighboring ion. It should be noticed that the integer Zi has a sign in our definition. If the neighboring ion is an anion then Zi has a negative value and if it is a cation then Zi is positive. Eq. (4) is expanded in terms of spherical harmonics Ylm as VCEF …r† ˆ

∞ X l X

Al;m rl Ylm …u; w†:

…5†

lˆ0 mˆ2 l

Here, Al,m is a constant which depends on the electron configuration and the charges of the neighboring ions around the central ion, and it is given by Al;m

4p X …2Zi e2 † mp ˆ Yl …Qi ; Fi †; 2l 1 1 i Ril11

…6†

where Ylmp means the complex conjugate of Ylm : Eqs. (5) and (6) are expressed by associated Legendre functions Pm l and

2 X ∞ X l X

Cl;m;k rl Pm l …cos u†ftri;k …mw†;

…7†

kˆ1 lˆ0 mˆ0

Cl;m;k ˆ

4.3. Quantitative analysis for paramagnetic susceptibility 4.3.1. CEF Hamiltonian We investigate the CEF effect on the magnetic susceptibility of RTa3O9. The level splitting of only the ground state multiplet of R 31 ion for the total angular momentum J is considered. Since it is not sufficient for Sm and Eu compounds which have the first excited multiplet not much higher than the ground state multiplet, the detailed analyses for these systems are not shown in this paper. An extensive study of SmTa3O9 and EuTa3O9 is in progress now and will be found elsewhere. The CEF effects on the rare earth elements are usually treated by regarding the CEF is smaller than the spin–orbit coupling but larger than the spin–spin interactions. We assume the validity of Russell–Saunders coupling between the total value of angular momentum L and the total value of spin S. We assign the notation J the coupled total angular momentum and M the zcomponent of J. We treat the CEF effect by taking the point charge model for simplicity in this paper. The definition of notations related to the CEF is different sometimes among literature, then we specify our notations used below to avoid confusion. We consider a magnetic ion located at an origin of a polar coordinate. The position vector of 4f electron around the central ion is taken as r…r; u; w†; and the position vector of i-th neighboring ion is taken as Ri …Ri ; Qi ; Fi †: The Coulomb potential energy VCEF on a 4f electron is expressed as

51

X …2Zi e2 † …l 2 m†! m …2 2 dm;0 † P …cos Qi †ftri;k …mFi †: …l 1 m†! l Rl11 i i

…8† Here, the trigonometric function ftri;1 …x† means cos x and ftri;2 …x† is sin x, respectively, and dm;0 is Kronecker’s d . In most cases, we can select the coordinate axes by which Cl;m;1 or Cl;m;2 vanishes. In the present system, by selecting the directions of the axes x, y, z to be along the edges of the cube shown in Fig. 2, Cl;m;2 disappears. Therefore, Cl;m;k is rewritten by Cl;m , ftri …x† by cos x and the summation related k is ignored hereafter. The CEF Hamiltonian HCEF is obtained by summing VCEF over the number of the valence electrons of the central ion as X HCEF ˆ VCEF …rj †: …9† j

Eq. (9) is rewritten by using Eq. (7) as HCEF …r† ˆ

∞ X l X X ‰ Cl;m rl Pm l …cos u†cos mwŠ lˆ0 mˆ0

;

∞ X l X

j

…10†

Vlm :

lˆ0 mˆ0

The number of terms needed is much reduced by the symmetry and the finite value of the orbital angular momentum of the central ion [9]. If l is odd number then Cl,m equals 0. The V00 affects as the energy level shift of the lowest multiplet for J, however it does not affect the level splitting in the J multiplet. Considering the 4f electrons system, the Vlm …l . 6† vanishes in the calculation process. Moreover, the coordination of the neighboring ions reduces the number of the terms of Vlm : Thus, in the cubic symmetry, the number of the terms needed is only four (see for example [10]) as HCEF …r† ˆ V40 1 V44 1 V60 1 V64 :

…11†

Without the CEF and the magnetic field, the energy levels of the wave functions uJ; Ml; here M is J, J 2 1; …, 2 J, are 2J 1 1 times degenerate in the J multiplet. The wave function uJ; Ml is given by a linear combination of single-electron wave functions by using the Clebsh–Gordan coefficients [11]. The level splitting due to the CEF is led by diagonalizing the Hamiltonian matrix: HCEF ˆ 3 … kJ; JuHCEF uJ; J 2 1l kJ; JuHCEF uJ; 2Jl kJ; JuHCEF uJ; Jl 7 6 6 kJ; J 2 1uH uJ; Jl kJ; J 2 1uH uJ; J 2 1l … kJ; J 2 1uH uJ; 2Jl 7 7 6 CEF CEF CEF 7 6 7: 6 . .. .. 7 6 . 7 6 . . ] . 5 4 kJ; 2JuHCEF uJ; J 2 1l … kJ; 2JuHCEF uJ; 2Jl kJ; 2JuHCEF uJ; Jl 2

…12†

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Table 3 CEF parameters A04 and A06 based on the point-charge model in various types of coordination. It should be noticed that e is positive, Z is positive for coordination of cations or Z is negative for coordination of anions

Number

X

…11z2j 2 rj2 †…x4j 2 6x2j y2j 1 y4j †

j

ˆ

g 6 kr l{…J 41 1 J42 †‰11J 2z 2 J…J 1 1† 2 38Š 1 ‰11J 2z 4 2 J…J 1 1† 2 38Š…J 41 1 J 42 †}

Type of coordination Polyhedron

V64 /

A04

A06 =A04

A06

Tetrahedron 4

1

7 Ze2 36 R5

2

1 Ze2 18 R7

2

2 1 7 R2

Octahedron

6

2

7 Ze2 16 R5

2

3 Ze2 64 R7

1

3 1 28 R2

Cube

8

1

7 Ze2 18 R5

2

1 Ze2 9 R7

2

2 1 7 R2

14-faced

12

1

7 Ze2 32 R5

1

39 Ze2 256 R7

1

39 1 56 R2

Instead of carrying out these complicated calculations, the convenient and useful method of Stevens operator equivalents [12] is applied, which is applicable only to the ground state multiplet for J. In this method, the summation of the polynominal equations in Vlm over the number of the valence electrons is rewritten by the operator equivalents consisting of J x ; J y and Jz. The Jx and Jy are transformed into the operators J ^ ˆ J x ^ iJ y for convenience. The list of the equivalent operators has been given by Hutchings [13]. The operator equivalents needed are as follows: X V40 / …35z4j 2 30rj2 z2j 1 3rj4 † j

; gkr6 lO46 :

…13d†

Here, b and g are numerical factors called as Stevens coefficients and listed in Ref. [12] (g for Pm 31 has been corrected in Ref. [14]). The matrix elements are obtained from J and M as follows: kJ; MuJz uJ; M 0 l ˆ M 0 dM;M 0 ;

…14a†

kJ; MuJ^ uJ; M 0 l ˆ ‰…J 7 M 0 †…J ^ M 0 1 1†Š1=2 dM;M 0 ^1 : …14b† The CEF parameters Am l and Bl used are defined as XX m l HCEF …r† ˆ Al kr lul Om l l

m

ˆ bkr4 l…A04 O04 1 A44 O44 † 1 gkr6 l…A06 O06 1 A46 O46 † ˆ B4 …O04 1 5O44 † 1 B6 …O06 2 21O46 †:

Here, ul is the generalized Stevens coefficient and Bl is defined as Bl ˆ A0l krl lul ;

…16†

and the relations satisfied in the cubic symmetry as A44 ˆ 5A04 ;

ˆ bkr4 l‰35J 4z 2 30J…J 1 1†J2z 1 25J 2z 2 6J…J 1 1†

…17a†

A46 ˆ 221A06

1 3J 2 …J 1 1†2 Š ; bkr4 lO04 ;

…13a†

…15†

…17b†

are used. The explicit expression of

Am l

is

X …2Zi e † …l 2 m†! m …2 2 dm;0 † P …cos Qi †cos mFi ; l11 …l 1 m†! l R i i 2

V44

/

X

…x4j

2

6x2j y2j

1

m Am l ˆ Nl

y4j †

…18†

j

ˆ

b 4 4 kr l‰J 1 1 J 42 Š 2

; bkr4 lO44 ; V60 /

X

Nlm

…13b†

…231z6j 2 315rj2 z4j 1 105rj4 z2j 2 5rj6 †

j

ˆ gkr6 l{231J 6z 2 ‰315J…J 1 1† 2 735ŠJ 4z 1 ‰105J 2 …J 1 1†2 2 525J…J 1 1† 1 294ŠJ 2z 2 5J 3 …J 1 1†3 1 40J 2 …J 1 1†2 2 60J…J 1 1†} ; gkr6 lO06 ;

…13c†

Pm l ;

where is a numerical coefficient appearing in i.e. N40 ˆ 1=8; N44 ˆ 105; N60 ˆ 1=16 and N64 ˆ 945=2: One should pay attention to the difference in CEF parameters between the Am l and the Al;m given in Eq. (6). Lea, Leask and Wolf (LLW) [10] have introduced the tractable parameters x, W and F…l†; and the parameters are related to B4 and B6 in the cubic CEF as B4 F…4† ˆ Wx;

…19a†

B6 F…6† ˆ W…1 2 uxu†:

…19b†

Here, F(4) and F(6) are positive numerical factors and W is a scaling factor of the width of the level splitting. The x parameter characterizes the ratio between B4 and B6, and by permitting x to take the value in the range of 21 # x #

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

53

Fig. 8. Energy level splitting by cubic CEF as a function of LLW parameter x, which shows modified diagrams given by LLW [10]. The energy level measured from the ground state is normalized by the whole splitting width D . It should be noticed that the diagrams (a), (b) and (e) are reversed upside down to the LLW diagrams, because the LLW parameter W is negative in these three cases.

11; all the possible ratios between the fourth and sixth order terms are covered. By using the LLW parameters, the Hamiltonian is described as " 0 # O 1 5O44 O0 2 21O46 HCEF …r† ˆ W x 4 1 …1 2 uxu† 6 : …20† F…4† F…6† The energy level splitting due to the CEF and the resultant wave functions are able to be calculated by diagonalizing the Hamiltonian matrix shown in Eq. (12).

If the exact values of A04 kr4 l and A06 kr6 l are known, then the Hamiltonian in Eq. (15) should be utilized to solve the CEF effect. If this is the case, the LLW Hamiltonian in Eq. (20) is unnecessary. Although the estimation of the expectation values of r 4 and r 6 is not easy, the values are given in Refs. [15,16]. The CEF parameters A04 and A06 are obtained from Eq. (18) based on the point-charge model, however these values are not so accurate. The reason is the wave functions of the neighboring ions have finite extension, so the valence electrons play a role in the shielding of the CEF. Therefore we

54 Table 4 Regions and boundaries for LLW parameter x, introduced for the fitting to magnetic susceptibility. The boundaries for x denote the location at which two or more Gn degenerate accidentally or at which the energy level of Gn has a maximum or a minimum (a) J ˆ 7=2 (for Yb 31) Number of the regions is 4 Region ←I! Boundary 21.000 20.583 (27/12) Situation G7 ; G8

(c) J ˆ 9=2 (for Nd 31) Number of the regions is 4 Region ←I! Boundary 21.000 20.57377 Situation G…1† 8 max

← III ! 0.200 (1/5) G6 ; G7

← II !

20.375 (23/8) G4 ; G5

← II !

20.80777 G…1† 5 max

(d) J ˆ 6 (continued) Region VII ! Boundary Situation

0.22951 G…2† 5 min

← VIII !

20.500 (21/2) G7 ; G…3† 8

(e) J ˆ 15=2 (continued) Region VI ! Boundary Situation

0.167 (1/6) G6 ; G…2† 8

← VII ! 0.07177 G…3† 8 min

(f) J ˆ 8 (continued) Region VII ! Boundary Situation

20.286 (22/7) …2† G…2† 3 ; G4

(f) J ˆ 8 (continued) Region XIII Boundary Situation

0.63343 G…2† 5 min

← II !

← VIII !

← IV !

← III !

20.545 (26/11) G2 ; G…2† 5 G3 ; G…1† 5

20.08064 G…1† 3 max

XIV !

20.46774 G…2† 8 min

← IV !

← III !

20.509 (228/55) G2 ; G4

← IV !

20.459 (217/37) G6 ; G 7 G…3† 8 max

← IX !

←X! 0.182 (2/11) G1 ; G…2† 5

←V!

20.375 (23/8) G6 ; G…3† 8

← VI

← XI !

20.500 (21/2) …2† G…2† 4 ; G5 …1† G…1† 4 ; G5

← XI !

1.000

← VI !

20.458 (238/83) G3…1† min G4…1† min G1 ; G…1† 5

← XII ! 0.375 (3/8) …2† G…1† 4 ; G5

← XVII ! 0.76238 G…1† 4 max

← VII

1.000

0.833 (5/6) G…2† 8 min G6 ; G…1† 8

0.26480 G…1† 4 max ← XVI !

0.72177 …2† G…1† 4 ; G5

←V!

20.354 (274/209) G1 ; G…2† 5

← XII !

←X!

20.545 (26/11) …2† G…2† 3 ; G5 …1† G…1† 3 ; G5

← VI !

0.824 (14/17) G2 ; G3 ; G…2† 5

0.750 (3/4) G7 ; G…2† 8

← IV !

20.455 (25/11) G1 ; G2 G4 ; G…2† 5

← XI !

← IX !

20.57262 G…1† 4 max

←V!

0.677 (21/31) G4 ; G…1† 5

0.583 (7/12) G7 ; G…3† 8 G…2† 8 max

← XV ! 0.667 (2/3) …1† G…1† 3 ; G5

1.000

←X!

← VIII !

20.59788 G…1† 3 max

1.000

← IV !

0.585 (38/65) G4 min G1 ; G…1† 5 ← III !

←V! 0.857 (6/7) G1 ; G3 ; G4

0.833 (5/6) G6 ; G…2† 8

← IX !

(e) J ˆ 15=2 (for Dy 31, Er 31) Number of the regions is 11 Region ←I! ← II ! Boundary 21.000 20.65668 …3† Situation G8 min

1.000

0.526 (10/19) G1 ; G5 ← III !

(d) J ˆ 6 (for Tb 31, Tm 31) Number of the regions is 12 Region ←I! ← II ! Boundary 21.000 20.83338 Situation G…2† 5 min

(f) J ˆ 8 (for Ho 31) Number of the regions is 19 Region ←I! Boundary 21.000 20.83395 Situation G…2† 3 min …1† G1 ; G…1† 3 ; G4

← III !

0.375 (3/8) G6 ; G…1† 8

0.211 (4/19) G1 ; G3 ; G4

← IV ! 0.750 (3/4) G6 ; G8

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

(b) J ˆ 4 (for Pr 31, Pm 31) Number of the regions is 5 Region ←I! Boundary 21.000 20.737 (214/19) Situation G3 ; G5

← II !

0.792 (19/24) …2† G…1† 4 ; G5

← VII

← XIII 0.48285 G4…2† min

← XVIII ! 0.857 (6/7) …2† G1 ; G…2† 3 ; G4

← XIX ! 1.000

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

A0l krl l

have to treat the as fitting parameters, and actual fitting has been made by using LLW parameters x and W. 4.3.2. Essential significance of the next-nearest neighbors The estimation of the CEF parameters A04 and A06 based on the point-charge model is useful for analysis of the magnetic susceptibility. Firstly, we consider only the n.n. ions around the central ion R 31. The n.n. ions are 12 O 22 ions as shown in Fig. 2, and they form the 14-faced polyhedron surrounding the R 31 ion. The cubic symmetry of the CEF is expressed as a point group Oh. The CEF parameters A04 and A06 are calculated from Eq. (18), and they are tabulated in Table 3 with parameters in other types of coordination. It is stressed that the value of 39=56R2 for the ratio of the sixth order term to the fourth order term for the 14-faced type is much larger than the values for others. Now, we will consider the effect of the n.n.n. The eight Ta 51 ions surround the R 31 ions as the n.n.n., and its symmetry is also cubic. In order to take account of the effect of the n.n.n., we have to sum the Am l for the n.n. and those for n.n.n. Here, we take a as the length of edge for the cube 31 22 shown in Fig. 2. is pThe distance R1 between R and O 31 given by R1 ˆp2a=2; and the distance R2 between R and Ta 51 is R2 ˆ 3a=2: The Z number with a sign is 2 2 for O 22 and 1 5 for Ta 51. Consequently the CEF parameters are given by A04 ˆ 1

7 …22e2 † 7 5e2 e2 p  p 1 ˆ 11:52 ; 32 … 2a=2†5 18 … 3a=2†5 a5

…21†

A06 ˆ 1

39 …22e2 † 1 5e2 e2 p p 2 ˆ 24:97 7 : 7 7 256 … 2a=2† 9 … 3a=2† a

…22†

It should be noticed that the sign of each term of fourth order terms is different, so the effect of the fourth order term is weakened by taking account of the n.n.n. Moreover the sign of A04 changes because of the large positive valence state of Ta 51 as n.n.n. The large value 1 5 of Z is a novel case, it originates from the characteristic feature in the defectperovskite structure. While, the signs of both the six order terms are negative, so the effect of the sixth order term is strengthened. Hence, the ratio A06 =A04 is too large as 23:28=a2 . Even qualitatively, the CEF is strongly modified owing to the existence of the n.n.n. It is hard to judge the actual sign of A04 because of the shield effect mentioned above, while it must be correct that A06 has a negative value. From the negative sign of A06 ; it is concluded that the signs of LLW parameter W are plus for Nd 31, Tb 31, Ho 31 and Tm 31, and they are minus for Pr 31, Pm 31, Dy 31, Er 31 and Yb 31. 4.3.3. Energy level splitting by CEF The level splitting due to the cubic CEF for the ground J multiplet of R 31 ions as a function of LLW parameter x have been shown by Lea et al. [10]. In the consideration of the magnetic susceptibility, the evaluation of the whole splitting

55

width in the energy level is essentially important. Let us propose new modified diagrams of LLW diagrams in Fig. 8. In our diagrams, the energy level measured from the ground state is normalized by the whole splitting width D . The diagrams for J ˆ 7=2; 4, 9=2, 6, 15=2 and 8, which correspond to the values J of the ground state multiplet of R 31, are shown. The diagram for J ˆ 5=2 corresponding to the case of Ce 31 and Sm 31 is not shown, because the sixth order terms vanish inherently in these systems, i.e. the value of x is restricted to 1 1 or 2 1. Here we reverse the LLW diagram upside down for the systems having the negative W, i.e. for Pr 31, Pm 31, Dy 31, Er 31 and Yb 31 systems. The diagonalization of the matrix was performed with our personal computer at intervals in the step of 0.005 for x. Each J-manifold obtained by decomposition of J is expressed as the irreducible representation Gn using the Bethe’s notation [17]. The multiplicity of each state is as follows: 1 for G1, G2; 2 for G3, G6, G7; 3 for G4, G5; and 4 for G8. The states of G1 –G5 occur when J is a integer, while G6 – G8 occur when J is a half-integer. The same Gn are distinguished by the superscript (i), and the i is numbered from the lower state in our diagram. We divide the diagram into some regions in order to avoid to be trapped in the incorrect region of x, in the process of scanning x for fitting the magnetic susceptibility data. We take the boundaries for x at which two or more Gn accidentally degenerate or at which the energy level of Gn is a maximum or a minimum. The regions and the boundaries obtained numerically are tabulated in Table 4. Each value of x at the boundary was calculated precisely to the order of 10 215, and the significant figures might be reliable down to about the order of 10 212. If recurring decimals arise, the values are rounded off the fractions to three decimal places, and they are shown with fractional expressions in parentheses. The calculated values of the magnetic susceptibility are carefully fitted to the experimental data by scanning x in all of the regions.

4.3.4. Magnetic susceptibility If weak magnetic field relative to thermal energy is applied along the z-axis, the magnetic susceptibility in which the first and the second order Zeeman terms are included by the perturbation method, is given by the Van Vleck formula [18,19], Ng2J m2B xz ˆ Z

"

vn 1 X X E ukCn;k uJ z uCn;k 0 lu2 exp 2 n kB T kB T n;k k 0 ˆ1

vn 0 X X X ukCn;k uJ z uCn 0 ;k 0 lu2 E 12 exp 2 n 0 E 2 E k n n BT n;k n 0 ±n k 0 ˆ1

!

!#

: …23†

Here, N is Avogadro’s number, gJ the Lande´ g-factor, m B the Bohr magneton, kB the Boltzmann constant and Z is a

56

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

partition function given by   X E Zˆ vn exp 2 n : kB T n

…24†

The wave function of n-th state which has energy of En ; is represented by uCn;k l; where the subscript k is attached to distinguish the wave functions among the n-th state having the multiplicity vn : The calculation of kCn;k uJ z uCn;k 0 l in the first term in Eq. (23) is made among the states in the same energy level, while kCn;k uJz uCn 0 ;k 0 l in the second term is made between the states in the different energy levels. The first term in Eq. (23) is the Curie-like components and the second term is the Van Vleck contributions. Careful calculations are required near the point x at which the accidental degeneracy occurs. When En equals En 0 …n ± n 0 †; the related matrix element ukCn;k uJ z uCn 0 ;k 0 lu should be calculated in the first term in Eq. (23), because the energy levels of n- and n 0 -th states are same. While, it is analytically confirmed that the methods in which this calculation of ukCn;k uJ z uCn 0 ;k 0 lu is made in the second term, gives the same result. However, in the numerical calculation with a computer, when these energy levels come close each other sufficiently, this calculation should be made in the first term in order to avoid the leading the incorrect result owing to the divergence of the second term. We take account of the exchange interactions between R 31 ions by a molecular field theory. The observed magnetic susceptibility x is expressed as 1 1 ˆ 2 l; …25† x xCEF where x CEF is the magnetic susceptibility perturbed by the CEF given by Eq. (23). l is a certain constant value detected as a shift in the ordinate of x 21 vs. T plot and the l denotes the molecular field constant which comes mainly from the exchange interactions. If l is positive then the exchange interactions are ferromagnetic and negative l means antiferromagnetic interactions. Next, we show the details of the analyses of the magnetic susceptibility of RTa3O9. The fitting to the experimental results is performed by using three parameters x, D and l . Here D is a whole width of the splitting of the lowest J multiplet, and has a relation with LLW parameter W as

D ˆ uW…Emax 2 Emin †u;

…26†

where Emax and Emin are the maximum and the minimum eigenvalues obtained from the determinant of the CEF Hamiltonian. It is very hard to obtain the optimized fit by varying three parameters simultaneously. Since l is the vertical shift of the inverse susceptibility in the x 21 vs. T plot, firstly we fit the slope of the x 21 data to the slope of the theoretical x CEF. We adopt acute angle f measured from the horizontal T-axis as

f ˆ tan21

dx21 : dT

…27†

If we use the original data including experimental errors, the

errors are enhanced in f . We introduce a new function expanded in polynominal which reproduces experimental data of the susceptibility. This function and its derivative are given by

x21 ˆ

n X

ai T i ;

…28†

iˆ0 n X dx21 iai T i21 : ˆ dT iˆ1

…29†

Here, taking the value of n as 7 or 8 is sufficient in the present case. On the contrary, theoretical values for the slope are calculated from the equation    dx21 1 1 X En CEF ˆ v E exp 2 n n dT kB T Ng2J m2B kB T 2 X n X   1 E 1 v exp 2 n X n n kB T ×

X X vn  n;k k 0 ˆ1

12

  vn 0 X X X En fnk;nk 0 2 2 En fnk;n 0 k 0 ; kB T n;k n 0 ±n k 0 ˆ1 …30†

with Xˆ

vn 0 vn X X X 1 X X fnk;nk 0 1 2 fnk;n 0 k 0 ; kB T n;k k 0 ˆ1 n;k n 0 ±n k 0 ˆ1

  E fnk;nk 0 ˆ ukCn;k uJ z uCn;k 0 lu2 exp 2 n ; kB T fnk;n 0 k 0 ˆ

  kCn;k uJ z uCn 0 ;k 0 l2 E exp 2 n : En 0 2 En kB T

…31†

…32a†

…32b†

The following steps of procedure were taken. (1) One value of x is chosen for an initial step in a certain region of x given in Table 4. (2) The determinant of the CEF Hamiltonian is diagonalized. Subsequently, the energy levels and the wave functions are obtained for each state. (3) One value of D associated with the whole width of the splitting is taken as a initial value. (4) f ˆ tan21 …dx21 CEF =dT† is calculated and compared with f of the experimental data. (5) D is temporarily optimized by the method of least squares and the sum of the squared errors is written into the computer memory. This value is useful to be compared with that for another x. (6) The optimization of x is made by comparison of the sum of the squared errors obtained in the Step (5) for each x. This process is repeated until leading to the sufficiently small variation of the sum of the square errors. (7) After optimization of x and D , next, l is optimized by the method of least squares with comparison of x 21 between the experimental and the calculated values.

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

Thus, a set of the optimized values of x, D and l is obtained for each region of x. Consequently, the most reliable parameters set is obtained on the following strict criteria: 1. Evaluation of the all the parameters of x, D and l should be accompanied with sufficiently low magnitude of the sum of the squared errors. 2. The calculated curve for x 21 should reproduce the experimental data well, in particular, at the lower temperature range where the CEF affect strongly the magnetic susceptibility. In addition, when the definitive value of x is assigned to be on the just boundary shown in Table 4, one can recognize that optimum x is not found inside of the region. In the case, actually, the result does not satisfy the criteria mentioned above. 4.4. Analyses of magnetic susceptibility for RTa3O9 4.4.1. General remarks Generally in the cubic symmetry, not only for the onefold states G1, G2 but also for the two-fold state G3, the value of ukCn;k uJ z uCn;k 0 lu is absolutely zero. The wave functions of G1, G2 are generally written by the equations with symmetrical coefficients as G1 : a1 u 1 8l 1 a2 u 1 4l 1 a3 u0l 1 a2 u 2 4l 1 a1 u 2 8l; …33† G2 : b1 u 1 6l 1 b2 u 1 2l 1 b2 u 2 2l 1 b1 u 2 6l;

…34†

where the number of terms in the linear combination varies, depending on J. It is easily understood that the diagonal matrix element is zero. The wave functions of two-fold state G3 have the same type as G1 and G2, hence the diagonal and nondiagonal matrix elements of G3 are always zero. While the value of ukCn;k uJ z uCn;k 0 lu is not necessarily zero for the other states Gn …n $ 4†: Thus, the Curie-like component in the x CEF formula in Eq. (23) is zero for G1, G2 and G3 while it is not zero for the other Gn …n $ 4†: When we consider the x CEF at sufficiently lower temperature near 0 K, the ground state dominates in the Curie-like components. Hence, if G1, G2 or G3 is ground state then the magnetic susceptibility tends to saturate near T ˆ 0 K because only the Van Vleck contributions, which are independent of temperature, are effective. 4.4.2. LaTa3O9 The magnetic susceptibility x for LaTa3O9 is almost independent of temperature and has positive values between 2 and 3 × 1024 emu mol21 as shown in Fig. 6(a). The value is as small as the diamagnetic susceptibility due to the core orbitals of each ion. The abrupt increase below 10 K suggests the existence of the Curie contribution. The susceptibility was separated into the Curie term and the temperature independent term as x ‰emu mol21 Š ˆ …1:53 ×

23

57

24

10 †=T 1 …1:98 × 10 †: From the Curie term, i.e. the first term, the effective magnetic moment is estimated to be 0.11m B for a formula unit of LaTa3O9. This small Curie term might be due to the existence of the localized magnetic moment at impurity sites and/or at other kinds of lattice imperfections. Then, the increase of x at low temperatures is not intrinsic. For the temperature independent term, there are some origins, for examples, inaccuracy in the corrections of diamagnetic susceptibility or the background of the experimental data. The temperature independent term might appear in the other systems of RTa3O9, however such small value of 2 × 1024 emu mol21 does not influence the following analyses because of the much larger magnitude in the magnetic susceptibility. 4.4.3. CeTa3O9 The lowest J multiplet of Ce 31 obtained from the Hund rule is 2 F5=2 : The first excited multiplet 2 F7=2 is about 3200 K higher than the lowest multiplet [20], and the perturbation from this higher state does not influence the analyses. It is led by using group theory that the six times degenerated multiplet of J ˆ 5=2 splits into two irreducible representations G7 and G8 of the Bethe’s notation [17], under the cubic symmetry operation in the cubic CEF. Hereafter, we describe such reduction by the cubic CEF as Jˆ

5 ! G7 1 G8 : 2

…35†

In the case of J ˆ 5=2; all of the matrix elements for O06 and O46 are zero, thus the CEF effect of the sixth order terms vanishes. Therefore, only the sign of CEF parameter A04 is a factor to decide which state is in the ground level. If A04 is positive then the sign of B4 F…4† ˆ A04 kr4 lbF…4† ˆ Wx is plus, and negative A04 leads to negative Wx, because b for 2 F5=2 is positive [12]. We can determine the sign of W arbitrarily because the sixth order terms vanish. If we fix the sign of W in plus then the value of x is 1 1 in the case of A04 . 0; or the value of x is 2 1 in the case of A04 , 0: The fitting was carried out in these two cases, and we obtained the result shown in Fig. 9. It corresponds to the case of x ˆ 21; i.e. A04 , 0: The calculated curve …x ˆ 21† reproduces the data well, while the curve in the reverse case …x ˆ 11† does not. This result is consistent with the point-charge model taking account of only the n.n. In this case the effect of the n.n.n. is not sufficiently strong to change the sign of A04 : The ground state is four-fold G8, and the split width D=kB between G8 and G7 is 446 K. The molecular field constant l is 25.4 mol emu 21. The wave functions and the energy levels are given as follows: G7 :

0:4082u ^ 5=2l 2 0:9129u 7 3=2l;

EG7 ˆ 1297 K;

G8 :

0:9129u ^ 5=2l 1 0:4082u 7 3=2l;

EG8 ˆ 2149 K;

1:0000u ^ 1=2l; where constant value of J ˆ 5=2 in uJ; Ml is omitted, the

58

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energy levels are measured from the level of 2 F5=2 without CEF splitting. 4.4.4. PrTa3O9 The lowest J multiplet of Pr 31 is 3 H4 : We do not consider the first excited multiplet 3 H5 ; which is about 3100 K higher than 3 H4 [20,21]. The lowest J multiplet 3 H4 for Pr 31 splits into four states by the cubic CEF as J ˆ 4 ! G1 1 G3 1 G4 1 G5 :

…36†

The ground state is one of G1, G3, G4 or G5 as shown in Fig. 8(b), where the G4 ground state appears at just one point of x. The magnetic susceptibility below 10 K shows the abrupt increase in Fig. 6(a). This increase is presumably the Curie contribution from small amount of impurities. In order to subtract this Curie term, we have to consider qualitatively about the ground state before the detailed analysis. If G1 or G3 is ground state then the magnetic susceptibility tends to saturate near T ˆ 0 K; while the G4 and G5 ground states lead to the divergent magnetic susceptibility near T ˆ 0 K (see Section 4.4.1). Our data of the magnetic susceptibility in Fig. 6(a) shows obviously a pronounced tendency toward the saturation, which supports the ground state to be G1 or G3. Fig. 10 shows the xT vs. T plot at lower temperatures. The linearity is clearly observed, here the solid line is a fit to the data below 8 K. The Curie term and the saturated value of x are evaluated from this line as 2:85 × 1023 =T and 1:06 × 1022 emu mol21 ; respectively. We carried out the fitting of the magnetic susceptibility in all the regions shown in Table 4, using the data after the subtraction of the Curie term evaluated above. Now, for a concrete example, we show the CEF Hamiltonian matrix used for the calculation as

Fig. 9. Optimized fit to the x 21 data of CeTa3O9.

equals that of the left side boundary of the region, and the opposite arrow ! shows x of the right side boundary of the region, and the arrow toward the bottom # indicates the optimized x is inside of the region. The value of x only for region III is found to be inside of the region, while each value for the other regions is on one side boundary of the region. We show the calculated curve in Fig. 11, here the fitting parameters are x ˆ 10:482; D=kB ˆ 275 K and l ˆ 252 mol emu21 : The calculated curve reproduces the data fairly well, while all the curves using x in the other regions do not. We cannot explain the positive sign of the obtained x if we take account of only the n.n. This fact verifies the drastic change in the sign of A04 by the effect of the n.n.n. The value of D=kB seems too small, while the value of l seems too large in comparison with the values of the other

…37† where the numerical factors of F…4† ˆ 60 and F…6† ˆ 1260 for Pr 31 [10] was used (see Eq. (20)). The LLW parameters x and W associated with D were optimized by varying those step by step. We use following simplified expression which indicates the location of the optimized x in each region, as ← I; II !; III #; ← IV; V ! : Here, the arrow toward the left ← means the optimized x

RTa3O9. It is just pointed out the value of l is close to them observed in the intermetallic compounds, for examples; l ˆ 126:6 mol emu21 …11:60 × 1024 T2 J21 † for PrAl2, l ˆ 143:3 mol emu21 …12:61 × 1024 T2 J21 † for NdAl2, l ˆ 255:0 mol emu21 for CeCu2Si2, the former two values are cited from Ref. [22] and the latter from Ref. [23]. The ground state of Pr 31 in PrTa3O9 is two-fold G3, and the states G4, G5, G1 follow in this order. The wave functions and the

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

Fig. 11. Optimized fit to the x 21 data of PrTa3O9.

Fig. 10. x T vs. T plot for PrTa3O9 below 20 K. The solid line is a fitting of the data below 8 K.

energy levels are as follows: G1 :

0:4564u 1 4l 1 0:7638u0l 1 0:4564u 2 4l; EG1 ˆ 1122 K;

G3 :

0:5401u 1 4l 2 0:6455u0l 1 0:5401u 2 4l; EG3 ˆ 2153 K; 0:7071u 1 2l 1 0:7071u 2 2l;

G4 :

0:7071u 1 4l 1 0:0000u0l 2 0:7071u 2 4l; EG4 ˆ 238 K; 0:3536u ^ 3l 1 0:9354u 7 1l;

G5 :

0:7071u 1 2l 2 0:7071u 2 2l;

EG5 ˆ 1100 K;

0:9354u ^ 3l 2 0:3536u 7 1l:

We show the explicit expression of x CEF for PrTa3O9 without the contribution from l for an example, which is given by

xCEF ˆ

Ng2J m2B kB









1



59

The values of the optimized x are found to be the locations in each region as I #; ← II; III #; ← IV: Two x values are obtained as candidates to lead the true energy splitting. The value of x ˆ 10:3751 close to the boundary at x ˆ 10:3750 is obtained in the region III, however the calculated curve deviates largely from the experimental data. We take the value of x ˆ 20:760 in the region I. The calculated curve is shown in Fig. 12, and the fitting parameters are x ˆ 20:760; D=kB ˆ 1057 K and l ˆ 20:58 mol emu21 : The negative sign of x shows that the n.n.n. play an important role of changing the sign of A04 ; 



 ×



   1 38:49 0:5000exp T T

153:0 38:49 99:87 121:8 1 3exp 1 3exp 2 1 exp 2 T T T T          99:87 d 153:0 38:49 99:87 112:50exp 2 12 2 0:06382exp 2 0:1794exp 2 0:4246exp T T T T 274:8   121:8 20:1814exp 2 ; T 2exp

where d stands for the difference between the Emax and Emin shown in Eq. (26), and the value of d is 62.99 for x ˆ 0:4822: 4.4.5. NdTa3O9 We treat only the lowest J multiplet 4 I9=2 for Nd 31, disregarding the 2700 K higher state 4 I11=2 [20,21]. The lowest J multiplet 4 I9=2 splits into three states as 9 J ˆ ! G6 1 2G8 : 2

…39†

We have four regions for x to find the optimum conditions.

…38†

as the same to the case of PrTa3O9. The ground state of Nd 31 …2† in NdTa3O9 is two-fold G6, and the states G…1† 8 ; G8 follow in this order. The wave functions and the energy levels are as follows: G6 : G…1† 8

:

0:6124u ^ 9=2l 1 0:7638u ^ 1=2l 1 0:2041u 7 7=2l;

EG6 ˆ 2673 K;

0:7640u ^ 9=2l 2 0:5053u ^ 1=2l 2 0:4012u 7 7=2l;

EG…1† ˆ 247 K; 8

0:3142u ^ 5=2l 1 0:9494u 7 3=2l; G…2† 8 :

0:2033u ^ 9=2l 2 0:4016u ^ 1=2l 1 0:8930u 7 7=2l; 0:9494u ^ 5=2l 2 0:3142u 7 3=2l:

EG…2† ˆ 1384 K; 8

60

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

Weiss law gives the effective magnetic moment of meff ˆ 7:82mB ; which is close to the calculated value 7.94m B for the free Gd 31 ion, and the Weiss constant of Q ˆ 21:6 K: The result indicates the eight-fold lowest J multiplet 8 S7=2 is not split by CEF above 2 K. This fact originates from the isotropic wave function of 8 S7=2 in Gd 31 ion having half-filled 4f orbital.

Fig. 12. Optimized fit to the x 21 data of NdTa3O9.

4.4.6. SmTa3O9 The magnetic susceptibility of SmTa3O9 shows a large variation in lower temperature range, and a small variation in higher temperature range as shown in Fig. 6. In the higher temperature range than 300 K, the slight increase of the susceptibility was also observed. This characteristic behavior is qualitatively similar to the theoretical calculations by Frank [24] based on the Van Vleck paramagnetism. The lowest J multiplet for Sm 31 is 6 H5=2 ; and the first excited state is 6 H7=2 : The interval between these states is so small as about 1400–1500 K [20,25] that we cannot neglect the mixing effect. Hence, the Stevens method of the operator equivalents is not applicable in this case any longer. The detailed analysis for the magnetic susceptibility of SmTa3O9 will be presented elsewhere.

4.4.9. Common notes in systems from TbTa3O9 to TmTa3O9 In the almost ions from Tb 31 to Yb 31 which have more than half electrons in 4f orbital, the interval between the ground multiplet and the first excited multiplet is larger than that of the ions having less than half electrons in 4f orbital [20,21,25]. One exception is the case of TbTa3O9. The interval between the lowest 7 F6 and the first excited 7 F5 is about 2900 K [20,25], which is as small as the value for Pr 31. We do not consider the perturbation from 7 F5 for the CEF analysis in TbTa3O9. From Dy 31 to Yb 31, the perturbation effect from the higher state is negligible. The magnetic susceptibilities from TbTa3O9 to TmTa3O9 reflect the CEF effect strongly in lower temperature range, and weakly in higher temperature range. In other words, the slope of the susceptibility at lower temperatures is characteristic. We use the data only below 150 K to fit to the susceptibility calculation in order to avoid the incorrect values of the fitting parameters. The validity of the analysis is confirmed by a comparison between the calculations and the data in all the temperature range measured. 4.4.10. TbTa3O9 The lowest J multiplet 7 F6 for Tb 31 splits into six states as J ˆ 6 ! G1 1 G2 1 G3 1 G4 1 2G5 :

…40†

The optimum x varies in each region as 4.4.7. EuTa3O9 The temperature dependence of magnetic susceptibility for EuTa3O9 is very weak and the susceptibility holds an almost constant value of about 7 × 1023 emu mol21 below 100 K, as shown in Fig. 6. This characteristic feature, being similar to the theoretical calculation by Frank [24], is one of the typical Van Vleck paramagnetism. The lowest J multiplet for Eu 31 is singlet 7 F0 ; so the expectation value of Jz is zero. Nevertheless, the magnetic moment is induced from the perturbed ground state by higher states. The interval between the 7 F0 and the first excited multiplet 7 F1 is too small as about 550 K [26]. Well below 550 K, nearly all the population is in the ground state and the susceptibility is independent of the temperature. We do not deal with the CEF effects on the higher states in this paper, and will present elsewhere with the analysis for SmTa3O9. 4.4.8. GdTa3O9 The magnetic susceptibility of GdTa3O9 is almost linear to the temperature in the measured temperature range of 2– 300 K as shown in Fig. 7. The fit of the data to the Curie–

← I; ← II; III #; IV !; V #; ← VI; ← VII; ← VIII; IX #; X #;

XI #; XII ! : Five x values are candidates to find the best x value. The curves calculated by using the optimized values obtained in the regions IX, X and XI deviate largely from the data below 20 K. We obtained the optimized values x ˆ 20:566; D=kB ˆ 232 K and l ˆ 10:031 mol emu21 in the region III, and x ˆ 20:497; D=kB ˆ 265 K and l ˆ 10:16 mol emu21 in the region V. The ground state is two-fold G3 for III, and three-fold G5 for V. Both of the calculated curves for these regions reproduce the data above 50 K well. We select x ˆ 20:497 in the region V because of the better fitting to the data below 50 K. The results of the fit is shown in Fig. 13. It is only one case among RTa3O9 that the molecular field constant l is positive. However its small value of 0.16 might be within the errors. The negative sign of x indicates that the n.n.n. do not change the sign of A04 in the present case. The ground state of Tb 31 in TbTa3O9 is three-fold G…1† 5 ; and it is followed by the

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

G3, G…2† 5 ;

states G4, G2, G1 in this order. The wave functions and the energy levels are as follows:

61

4.4.12. HoTa3O9 The lowest J multiplet 5 I8 for Ho 31 splits into seven

G1 :

0:6614u 1 4l 2 0:3536u0l 1 0:6614u 2 4l;

EG1 ˆ 1120 K;

G2 :

0:3953u 1 6l 2 0:5863u 1 2l 2 0:5863u 2 2l 1 0:3953u 2 6l;

EG2 ˆ 194 K;

G3 :

0:2500u 1 4l 1 0:9354u0l 1 0:2500u 2 4l;

EG3 ˆ 2137 K;

0:5863u 1 6l 1 0:3953u 1 2l 1 0:3953u 2 2l 1 0:5863u 2 6l; G4 :

0:7071u 1 4l 1 0:0000u0l 2 0:7071u 2 4l;

EG4 ˆ 188 K;

0:5863u ^ 5l 2 0:4330u ^ 1l 1 0:6847u 7 3l; G…1† 5

:

0:5648u 1 6l 1 0:4254u 1 2l 2 0:4254u 2 2l 2 0:5648u 2 6l;

EG…1† ˆ 2145 K; 5

0:5695u ^ 5l 1 0:8214u ^ 1l 1 0:0318u 7 3l; G…2† 5

:

0:4254u 1 6l 2 0:5648u 1 2l 1 0:5648u 2 2l 2 0:4254u 2 6l;

EG…2† ˆ 178 K; 5

0:5762u ^ 5l 2 0:3712u ^ 1l 2 0:7282u 7 3l:

4.4.11. DyTa3O9 The lowest J multiplet 6 H15=2 for Dy 31 splits into five states as Jˆ

15 ! G6 1 G7 1 3G8 : 2

…41†

The location of the optimum x varies in each region as I #; ← II; ← III; IV !; V #; VI !; VII !; VIII !; IX #; ← X;

states as J ˆ 8 ! G1 1 2G3 1 2G4 1 2G5 :

…42†

The location of the optimized x varies as ← I; ← II; ← III; IV !; ← V; VI #; VII #; ← VIII; ← IX; ← X;

← XI; ← XII; XIII #; ← XIV; XV #; ← XVI; ← XVII; ← XVIII; ← XIX:

← XI: There are three candidates for the x value to be selected. Two negative x values in the region I and V and the associated values of D and l do not reproduce the data at all. The set of the values of x ˆ 10:700; D=kB ˆ 721 K and l ˆ 20:082 mol emu21 in the region IX reproduces the data well as shown in Fig. 14. The positive sign of x indicates the reverse of the sign of A04 is caused by the n.n.n. The ground state of Dy 31 in DyTa3O9 is four-fold G…1† 8 ; and the …3† states G6, G…2† 8 ; G7, G8 follow in this order. The wave functions and the energy levels are as follows:

There is not such a large difference among the curves calculated by using the optimized values in the regions VI, VII, XIII and XV above 50 K. However, the curve obtained in the region XV shows most similar behavior to the experimental result below 50 K. We show this fit in Fig. 15, here the fitting parameters are as follows; x ˆ 10:707; D=kB ˆ 843 K and l ˆ 20:51 mol emu21 : The positive sign of x indicates the reverse of the sign of A04 does not occur. The ground state of Ho 31 in HoTa3O9 is three-fold G…1† 5 ; and two…2† fold G…1† 3 is in the slightly higher level, and the states G5 ; …1† …2† …2† G4 ; G1, G4 ; G3 follow in this order. The wave functions

G6 :

0:5818u ^ 15=2l 1 0:3307u ^ 7=2l 1 0:7181u 7 1=2l 1 0:1909u 7 9=2l;

G7 :

0:6333u ^ 13=2l 1 0:5818u ^ 5=2l 2 0:4507u 7 3=2l 2 0:2394u 7 11=2l; EG7 ˆ 1204 K;

G…1† 8 : 0:8092u ^ 15=2l 2 0:3270u ^ 7=2l 2 0:4743u 7 1=2l 2 0:1156u 7 9=2l;

EG6 ˆ 2353 K; EG…1† ˆ 2405 K; 8

0:0284u ^ 13=2l 2 0:6445u ^ 5=2l 2 0:7620u 7 3=2l 2 0:0568u 7 11=2l; G…2† 8 : 0:0733u ^ 15=2l 1 0:7463u ^ 7=2l 2 0:2391u 7 1=2l 2 0:6169u 7 9=2l;

EG…2† ˆ 1164 K; 8

0:7726u ^ 13=2l 2 0:4614u ^ 5=2l 1 0:4075u 7 3=2l 1 0:1553u 7 11=2l; G…3† 8 : 0:0367u ^ 15=2l 1 0:4762u ^ 7=2l 2 0:4497u 7 1=2l 1 0:7547u 7 9=2l; 0:0347u ^ 13=2l 1 0:1822u ^ 5=2l 2 0:2241u 7 3=2l 1 0:9567u 7 11=2l:

EG…3† ˆ 1315 K; 8

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S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

4.4.13. ErTa3O9 The lowest J multiplet for Er 31 is 4 I15=2 ; which is reduced into five representations as Eq. (41). The location of the optimized x varies as I !; II #; ← III; IV !; ← V; VI #; ← VII; ← VIII; IX !; X #; XI ! :

Fig. 13. Optimized fit to the x 21 data of TbTa3O9.

and the energy levels are as follows: G1 : G…1† 3

The calculated curves obtained in the region II and X deviate largely from the data, especially at the higher temperatures. Whereas the curve obtained in the region VI shows good agreement with the data as shown in Fig. 16. The optimized values are x ˆ 10:056; D=kB ˆ 646 K and l ˆ 20:23 mol emu21 : The positive sign of x indicates that the reverse of the sign of A04 does not occur. However, the small value of x ˆ 10:056 prove the n.n.n. play a role to weaken the strength of A04 effectively. Consequently the sixth order term of the CEF is dominant in ErTa3O9. The ground state of

0:4114u 1 8l 1 0:2700u 1 4l 1 0:7181u0l 1 0:2700u 2 4l 1 0:4114u 2 8l; EG1 ˆ 1414 K; :

0:0537u 1 8l 1 0:6036u 1 4l 2 0:5154u0l 1 0:6036u 2 4l 1 0:0537u 2 8l; EG…1† ˆ 2347 K; 3

0:7068u 1 6l 2 0:0200u 1 2l 2 0:0200u 2 2l 1 0:7068u 2 6l; G…2† 3 :

0:5726u 1 8l 2 0:2506u 1 4l 2 0:4677u0l 2 0:2506u 2 4l 1 0:5726u 2 8l; EG…2† ˆ 1493 K; 3

0:0200u 1 6l 1 0:7068u 1 2l 1 0:7068u 2 2l 1 0:0200u 2 6l; G…1† 4 :

0:0841u 1 8l 1 0:7021u 1 4l 1 0:0000u0l 2 0:7021u 2 4l 2 0:0841u 2 8l; EG…1† ˆ 2139 K; 4

0:6656u ^ 7l 2 0:5395u ^ 3l 1 0:3716u 7 1l 1 0:3576u 7 5l; G…2† 4 :

0:7021u 1 8l 2 0:0841u 1 4l 1 0:0000u0l 1 0:0841u 2 4l 2 0:7021u 2 8l; EG…2† ˆ 1470 K; 4

0:0482u ^ 7l 2 0:5846u ^ 3l 2 0:7971u 7 1l 2 0:1434u 7 5l; G…1† 5 :

0:6955u 1 6l 2 0:1274u 1 2l 1 0:1274u 2 2l 2 0:6955u 2 6l;

EG…1† ˆ 2350 K; 5

0:1266u ^ 7l 1 0:3950u ^ 3l 2 0:4266u 7 1l 1 0:8037u 7 5l; G…2† 5 :

0:1274u 1 6l 1 0:6955u 1 2l 2 0:6955u 2 2l 2 0:1274u 2 6l;

EG…2† ˆ 2216 K; 5

0:7339u ^ 7l 1 0:4595u ^ 3l 2 0:2110u 7 1l 2 0:4535u 7 5l:

Fig. 14. Optimized fit to the x 21 data of DyTa3O9.

Fig. 15. Optimized fit to the x 21 data of HoTa3O9.

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

G8…1† ;

31

Er in ErTa3O9 is four-fold and it is followed by the …3† states G…2† 8 ; G6, G8 ; G7 in this order. The wave functions and the energy levels are as follows:

5. Summary A systematic investigation of the paramagnetic suscep-

G6 :

0:5818u ^ 15=2l 1 0:3307u ^ 7=2l 1 0:7181u 7 1=2l 1 0:1909u 7 9=2l;

G7 :

0:6333u ^ 13=2l 1 0:5818u ^ 5=2l 2 0:4507u 7 3=2l 2 0:2394u 7 11=2l; EG7 ˆ 1342 K;

G…1† 8

:

63

0:6628u ^ 15=2l 2 0:7183u ^ 7=2l 2 0:2108u 7 1=2l 1 0:0169u 7 9=2l;

EG6 ˆ 125 K; EG…1† ˆ 2304 K; 8

0:3777u ^ 13=2l 2 0:7284u ^ 5=2l 2 0:5271u 7 3=2l 1 0:2212u 7 11=2l; G…2† 8

:

0:0980u ^ 15=2l 1 0:0164u ^ 7=2l 1 0:1736u 7 1=2l 2 0:9798u 7 9=2l;

EG…2† ˆ 255 K; 8

0:4190u ^ 13=2l 2 0:3480u ^ 5=2l 1 0:4979u 7 3=2l 2 0:6749u 7 11=2l; G…3† 8 :

0:4610u ^ 15=2l 1 0:6119u ^ 7=2l 2 0:6402u 7 1=2l 2 0:0571u 7 9=2l;

EG…3† ˆ 1176 K; 8

0:5298u ^ 13=2l 1 0:0990u ^ 5=2l 1 0:5207u 7 3=2l 1 0:6621u 7 11=2l: 4.4.14. TmTa3O9 The lowest J multiplet for Tm 31 is 3 H6 ; and reduced into six representations as Eq. (40). The location of the optimized x varies as ← I; II !; III !; IV #; V !; VI #; ← VII; ← VIII; IX #; X #; ← XI; XII ! : The calculated curves obtained in the regions IX and X deviate largely from the data below 30 K. Although both of the curves obtained in the regions IV and VI fit well to the data above 20 K, the former curve reproduces the behavior of the x 21 data below 20 K better than the latter curve. The fit to the data for region IV is shown in Fig. 17. The fitting parameters are x ˆ 20:545; D=kB ˆ 413 K and l ˆ 21:0 mol emu21 : Here, more precise value of x is 20.54499, which is not on the boundary having the value of x ˆ 26=11 ˆ 20:54545 between III and IV but extremely close to it. The negative sign of x indicates the reverse of the sign of A04 does not arise. The ground state of Tm 31 in TmTa3O9 is three-fold G5…1† ; and two-fold G3 is in the slightly higher level, and the states G…2† 5 ; G2, G4, G1 follow in this order. The wave functions and the energy levels are as follows:

tibility for RTa3O9 has been performed. On the assumption that R 31 is subjected to a CEF predominantly cubic, fair agreement between experiment and calculations is obtained over the whole temperature range. It is stressed that the n.n.n. of high valence Ta 51 play significant role to CEF effect on paramagnetic susceptibility of defect-perovskite compounds RTa3O9. The analyzed results are summarized in Table 5. The ground state of each compound is clarified as shown in Table 5. The energy level scheme is shown in Fig. 18. The following points should be noted: 1. The reverse of the sign of A04 is caused by the n.n.n. in three compounds PrTa3O9, NdTa3O9 and DyTa3O9. We have demonstrated clearely the powerful change of CEF due to the presence of the n.n.n. with high valence Ta 51. 2. Among three compounds mentioned above, the x values in NdTa3O9 and DyTa3O9 are fairly larger than the values in PrTa3O9. The large value of x shows that the fourth order CEF terms from the n.n.n. strongly affect. 3. The value of x close to zero is obtained for ErTa3O9, hence the sixth order CEF terms are dominant in this compound. It is an interesting result that the strength

G1 :

0:6614u 1 4l 2 0:3536u0l 1 0:6614u 2 4l;

EG1 ˆ 1196 K;

G2 :

0:3953u 1 6l 2 0:5863u 1 2l 2 0:5863u 2 2l 1 0:3953u 2 6l;

EG2 ˆ 1113:8 K;

G3 :

0:2500u 1 4l 1 0:9354u0l 1 0:2500u 2 4l;

EG3 ˆ 2216:7 K;

0:5863u 1 6l 1 0:3953u 1 2l 1 0:3953u 2 2l 1 0:5863u 2 6l; G4 :

0:7071u 1 4l 1 0:0000u0l 2 0:7071u 2 4l;

EG4 ˆ 1144 K;

0:5863u ^ 5l 2 0:4330u ^ 1l 1 0:6847u 7 3l; G…1† 5

:

0:5861u 1 6l 1 0:3956u 1 2l 2 0:3956u 2 2l 2 0:5861u 2 6l;

EG…1† ˆ 2216:8 K; 5

0:5389u ^ 5l 1 0:8395u ^ 1l 1 0:0695u 7 3l; G…2† 5 :

0:3956u 1 6l 2 0:5861u 1 2l 1 0:5861u 2 2l 2 0:3956u 2 6l; 0:6049u ^ 5l 2 0:3282u ^ 1l 2 0:7255u 7 3l:

EG…2† ˆ 1113:6 K; 5

64

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

Fig. 17. Optimized fit to the x 21 data of TmTa3O9.

Fig. 16. Optimized fit to the x 21 data of ErTa3O9.

Fig. 18. Energy splitting components by a cubic CEF for ground state J multiplet for R 31 in RTa3O9. The multiplicity of each state is as follows: 1 for G1, G2; 2 for G3, G6, G7; 3 for G4, G5; 4 for G8.

Table 5 Summarized results of the CEF analyses for RTa3O9. All the values including the unit of energy are shown in temperature unit (K) Compound

CeTa3O9

PrTa3O9

NdTa3O9

TbTa3O9

DyTa3O9

HoTa3O9

ErTa3O9

TmTa3O9

Ground state x W (K) D (K) l (mol emu 21) B4 (10 23 K) B6 (10 25 K) 10 2B6/B4 ˚) A04 a5 (10 5 K A ˚) A06 a7 (10 5 K A A06 2 a A04

G8 21.000 174.3 446 25.4 21240

G3 10.482 24.36 275 252 235.1 2179 15.12 11.68 210.3

G6 20.760 114.9 1057 20.58 2189 1142 20.752 125.8 215.7

G5 20.497 11.69 265 10.16 214.0 111.3 20.803 27.65 285.6

G8 10.700 21.82 721 20.082 221.3 23.94 10.186 125.9 236.3

G5 10.707 11.60 843 20.51 118.9 13.39 10.179 244.4 227.8

G8 10.056 21.16 646 20.23 21.08 27.87 17.30 22.07 244.9

G5 20.545 12.58 413 21.0 223.5 115.5 20.663 213.1 236.1

2 6.11

20.607

111.2

21.40

10.626

121.7

12.76

25.83

S. Ebisu et al. / Journal of Physics and Chemistry of Solids 61 (2000) 45–65

4.

5.

6.

7.

8.

of the fourth order CEF terms are effectively weakened by the effect due to the n.n.n. The ratio of B6 to B4 is also tabulated in Table 5. When we compare the values of 10 2B6 =B4 among RTa3O9 except Pr, Nd and Dy compounds, the outstandingly large value of 17.30 for ErTa3O9 is found. While, the value of 10.179 in HoTa3O9 is rather small. The influence from the n.n.n. is weak in HoTa3O9. The molecular field constant l in PrTa3O9 is fairly large. Only the l in TbTa3O9 has a positive value. Since its value is relatively small, it might be regarded within errors. B4 and B6 are selected for the set of the CEF parameters tabulated. The values of B6 for almost R 31 in RTa3O9 are 2–6 times larger than those in the intermetallic compounds RAl2 reported by Purwins et al. [22]. One exception is TbTa3O9, the value of B6 is 37 times larger than the value for TbAl2. On the other hand, the ratios of the B4 values to those for RAl2 are largely scattered from 0.7 for Pr 31 to 33 for Dy 31. The point charge model requires constant values for A04 a5 and A06 a7 ; where a is a length of the edge of the cube shown in Fig. 2. Actually Purwins et al. have reported the almost constant values for RAl2 [22]. In the present case, the value of A04 a5 changes from compound to compound, while A06 a7 gives relatively close values. Here A04 and A06 are calculated by using the values of kr4 l and kr6 l reported by Freeman et al. [15,16]. The value of a is used on the assumption of simplified cube having the same volume as actually distorted sub-cell. The value of …A06 =A04 †a2 is estimated to be 23.28 from the simple point charge model. All the values obtained are also fairly large, reflecting the characteristic defectperovskite structure of the present compounds.

Acknowledgements The authors would like to thank Messrs. Satoshi Sasaki, Takashi Komatsu, Takeshi Sogabe, Masanori Hayashi and Takuya Ueno for their valuable help.

65

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