CRYSTAL FIELD EFFECTS IN RARE EARTH IONS

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CRYSTAL FIELD EFFECTS IN RARE EARTH IONS

I. Nowik, I. Felner, R. Yanovsky

To cite this version:

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CHEMICAL STRUCTURE AND

BONDING,

**CRYSTAL FIBLD EFFECTS IN RARE EARTH IONS (*)**

I. NOWIK, I. FELNER and R. YANOVSKY

The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel

RBsumB. - L'ktude par spectromktrie Mossbauer d'isotopes de terre rare places dans des environnements cubiques peut donner des informations dktaillkes sur le champ cristallin. L'6tude de 170Yb3f dans TmPd3, YPd3, YbBe13, TmBel3, TrnAuNi4 et YbAuNi4 montre que 1'6tat fondamental de Yb3+ est

r7

dans les deux premiers composks et r 6 dans les trois derniers. La struc-

ture paramagnetique bien rtsolue des spectres Mossbauer dans TmPd3, TmBels, TmAuNi4 et surtout dans YbAuNi4 sont les premiers

A

avoir etk observks dans des systkmes paramagnktiques concentres, c'est la preuve que Tm3+ dans les composks de Tm a un &tat fondamental qui est un singulet bien defini. L'ktude de 166Er dans LaA12, YPd3 et YCu aurait da conduire A des spectres de type

r8

purs, pour lesquels le rapport A s

<

r6

>

/ A4

<

r4

>

aurait pu &re r6duit. Les spectres ne correspondent pas B une structure pure et peuvent &re expliques par une distorsion de la sym&- trie cubique qui skpare 1'Btat r8en deux doublets de Kramer. Le spectre Mossbauer de la transition k122keVdu ls2SmdansSmBel3conduitBA4

<

r4

>

< 2cm-1.

Abstract. - Mossbauer studies of rare earth isotopes located in cubic environments may yield quite detailed information on the crystalline fields. Studies of 170Yb3+ in TmPd3, YPd3, YbBel3, TmBe13, TmAuNi4 and YbAuNi4 show that the Yb3+ ground state is r 7 in the first two compounds

and Ts in the last three. The well resolved cubic paramagnetic hyperfine structure Mossbauer spectra in TmPd3, TmBel3, TmAuNi4 and in particular in YbAuNi4 are the first to be observed in concentrated paramagnetic metallic systems. They prove that Tm3+ in the Tm compounds has a well isolated singlet ground state. Studies of 166Er in LaA12, YPd3 and YCu were expected to yield pure r8spectra from which the ratio A6

<

r6

>

/

A4

<

r4

>

could be deduced. The spectra do not correspond to pure r8spectra and are explained in terms of local distortions from cubicity which split the Ts state into two Kramers' doublets. The Mossbauer spectra of the 122 keV transition of I5ZSm in SmBel3 (first 152Sm metallic system to show an observable Mossbauer effect) yield A 4

<

r4

>

< 2cm-1.

Introduction.

-

The role of crystalline fields In determining the magnetic behavior of rare earth ions has been a subject of many experimental and theoretical studies and is discussed in detail in several books [I]. The importance of crystalline field effects on the hyperfine structure observable in a Mossbauer study has been realized quite early [2] and in fact almost each Mossbauer spectrum of a rare earth isotope has in it some information on the crystalline fields acting on the ion. Pure quadrupole spectra of an S state ion (Gd3

+,

Eu2+) yield directly the second order crystalline field parameter A: [3]. Quadrupole spectra of E U ~ + (J = 0 ground state, J = 1 at 480 K) also yield almost directly A: [4]. In the non S state rare earth ions the quadrupole interaction is composed of a direct, temperature independent, lattice contribution and a temperature dependent 4f contribution determined by the energies and wave functions of the various Stark levels. Thus, if the quadrupole interaction is studied as a function of temperature, several crystalline field parameters can be deduced [5]. Mossbauer spectra

which exhibit both quadrupole and magnetic hyperfine splittings yield more detailed information on the crystalline fields 161. The measurement of the temperature dependence of both the magnetic and quadrupole hyperfine interactions may yield accurate values for the crystal field parameters, in particular if only few parameters have to be taken into account, as e. g. in cubic systems [ 6 ] . In cubic systems the observed Mossbauer spectrum is determined by two crystal field parameters (A, and A,) only. The spectrum a t low temperatures depends only on the sign of A, and on the ratio A,/A,. These determine which

I

Ti

>

state is lowest 171. A full discussion of the hyperfine structure Hamiltonian for the various cubic Ti states is given by Abragam and Bleaney [I]. In particular it is convenient to study rare earth ions in cubic environments using 2* + Of Mossbauer transitions. In this case the spec-

trum, for a

r,

or

r6

ground state, is composed of only two lines corresponding to F =

3

and F =

3

produced by the interaction A(I.S) where I = 2 and S =

t.

The measurement of the line positions and intensities determines whether it is a

r6

or T, state (the splitting is (*) This research was supported by a grant from the United independent of the ratio &/A,). Such studies were States-Israel Binational Science Foundation @SF), Jerusalem, reported for yb170 in AU [81, RbzNaYbF, [91 and

Israel. Cs,NaYbCl, [lo] and ~r~~~ in Au 1111, Ag 1121

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C6-432 I. NOWIK, I. FELNER AND R. YANOVSKY

and Th [13]. In the metallic systems studied, the rare earth ion was introduced as a dilute probe. Here the first observations of a resolved cubic magnetic hyperfine structure Mossbauer spectrum in concentrated paramagnetic metallic systems is reported. These include the spectrum of Yb170 in TmBe,,, TmAuNi, and YbAuNi, with a

r6

ground state and the spectrum in TmPd, with a

T,

ground state. Spectra of 166Er3+ in LaAl,, YPd, and YCu were not similar to the theoretical

r,

expected spectra and were interpreted as due to local distortions from perfect cubicity. Finally, a detailed study of the hyperfine structure of Sm15' in SmBe,, at various temperatures is reported and A,

<

r 4

>

< 2 cm-l and the exchange field in satu- ration ,uB H,,,,/k = 26 K were derived.

1. Paramagnetic hyperfine structure of Yb17' in

cubic metallic environment.

-

Several Mossbauer studies of 17'Yb3+ in paramagnetic insulators [9-101 or of dilute 170Y b3+ in diamagnetic metals [8] reveal a well resolved paramagnetic cubic hyperfine structure at 4.2 K. These spectra are composed of two absorption lines at positions A and

-

3

A, where A ( S . 0 is the magnetic hyperfine interaction between the nuclear spin I = 2 and the effective spin S = for cubic

r6

and

r,

doublets. Raising the temperature leads to an increase in the spin relaxation rates and the spectra are smeared out, until they converge to single narrow absorption lines. In all these cases the spin spin interactions, which lead to the destruction of the resolved hyperfine spectrum, are avoided either by the relatively large Yb-Yb distances (in the insulators Rb,NaYbF6 and Cs,NaYbCl,) or by extreme dilution of the paramagnetic species in metallic systems

(Yb resulting from the decay of in 200 ppm Tm

in Au). Now well resolved cubic paramagnetic hyperfine structure in an Yb concentrated metallic paramagnetic system (YbAuNi,) and in Tm concentrated metallic systems (yb170 in TmPd,, TmBe,

,,

TmAuNi,) have also been observed.

YbAuNi, AND Yb : TmAuNi,.

-

Following

studies of the structure of the system RAuNi, [14] (R = rare earth), which revealed that the rare earth ion

is located in a cubic symmetry site, YbAuNi, and TmAuNi, were prepared by melting stoichiometric amounts of the metals in an induction furnace under a dry argon atmosphere.

The structure was checked by X-ray analysis and susceptibility and YbI7O Mossbauer studies were performed. The susceptibility studies show that both TmAuNi, and YbAuNi, are paramagnetic down to 4.1 K. The Mossbauer spectrum of YbAuNi, at 4.1K obtained with a TmAl, source is shown in figure 1. The

-2.0 -1.0 0.0 1.0 2.0 3.0

VELOCITY fcrnls)

Rci. 1. - Mossbauer spectrum a t 4.1 K of Ybi70 in YbAuNi4. Solid curve is a theoretical least square fit spectrum.

shape of the spectrum indicates that the ground state of Yb3' in YbAuNi, is

r6.

A detailed computer analysis yields the various parameters given in table I. The question arises why in a concentrated metallic paramagnetic system, such as YbAuNi,, the spin spin interactions enhanced through conduction electrons, are not effective enough to lead to exchange narrowing of the Mossbauer spectrum. The crystal structure of YbAuNi, is similar to that of the Laves phase compound [I51 YbNi,. In YbNi, the magnetic ordering point is 5.7 K. Since in YbAuNi, one Yb ion is exchang- ed by a diamagnetic ion (Au), it is not surprising that the ordering temperature is much below 4.1 K. The low Curie point indicates a weak J,, coupling which also leads to low spin-conduction electron relaxation rates. In YbNi, it was found 1151 that A,

<

r4

>

-

44 cm-I leading to a T6 ground state for the Yb3+ ion. It

Hyperfine constants, crystalline Jields and relaxation rates in some cubic systems

Compound

-

YbAuNi, Yb : TmAuNi, Yb : TmPd, Yb : YPd, Yb : TmBe,, YbBe,, SmBe

,,

Hyperfine constant Ground state 7 A (mmls)

-

r 6

-

10.3(4) r6

-

10.3 r7 13.2 r 7 13.2(5) l-6

-

8.5 Yb2+ (?)

-

F7 or

r8

gp, Hetr (4.1 K) = 9.98(5) Relaxation rate (W) logs-' at 4.1 K

-

0.52(2)

Cubic crystalline fileld

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CRYSTAL FIELD EFFECTS IN RARE EARTH IONS C6-4 33

seems that the sign- of A,

C

r4

>

in YbAuNi, is the same as that in YbNi,. If also in TmAuNi,

the ground state of Tm3+ in TmAuNi, would be the singlet TI. This would lead to long spin relaxation for Yb3+ in TmAuNi, as is indeed observed. The spectrum of the 84.3 keV transition of Yb170 emitted from a TmAuNi, source in a Yb metal absorber is very similar to the absorption spectrum obtained in YbAuNi,. The spin relaxation rates of Yb3+ in TmAuNi, and YbAuNi, seem therefore quite similar. One prominent difference between the TmAuNi, and YbAuNi, spectra is a relatively strong axial component present in the TmAuNi, spectrum and absent in the YbAuNi4 spectrum. This may be a direct result of the local distortion created by after decay effects or due to neutron irradiation damage. A repeated experiment with an annealed source will clarify this point.

Ybl'O : TmPd,.

-

Previous studies of ybl'O in YbPd, [16] have shown that the ground state of Yb3+ in YbPd, is

r7

and the first excited

r8

state is at 29 cm-I above the ground state. The spin relaxation rate at

4.2 K of Yb3+ in YbPd, was measured to be

--

1.25 x 10'' s-l. Considering the crystalline field parameters A,

<

r4

>

= - 12 cm-I and A ,

<

r6

>

-0

obtained for Yb3+ in YbPd,, one expects that Tm3+

in TmPd, will have the non-magnetic doublet

r3

(or the

r,

singlet) as ground state, with the first excited

r,

triplet state close to the ground state (few K). Under these conditions, relatively long spin spin relaxation times are expected for Yb3+ in TmPd, at low temperatures.

TmPd, was prepared by the standard methods, analyzed by X-ray to have the CuAu, structure, and then neutron irradiated to produce radioactive Tm170 which decays to Yb170. The absorber was an Yb metal disc enriched to 30

%

in yb170. The spectrum obtained is shown in figure 2. Since all hyperfine parameters are

L L I I I I I I l J

-4.0 -2.0 0.0 2.0 4.0 VELOCITY ( c m l s ]

FIG. 2. - Mossbauer spectrum at 4.1 K of Yb170 in YPd3 (a) and in TmPd3 .(b)..Solid line-represents a theoreticaLleast.square

fit spectrum.

known from the YbPd, spectra, the only adjustable parameter needed to fit the experimental spectrum is the spin relaxation rate, which is obtained as 1.3 x 10' s-I. The difference in the relaxation rates between Yb : TmPd, and YbPd, may be explained as resulting from the Yb-Yb spin spin interactions. The spin relaxation rate measured in TmPd, is mainly due to interactions with conduction electrons.

yb170 : YPd3.

-

Since we have observed low spin relaxation rates in TmPd, at 4.1 K, we expected even lower rates in YPd,. A radioactive source of Tm170 in 200 ppm Tm in YPd3 was investigated at 4.1 K

with the Yb170 metallic absorber. The spectrum is also shown in figure 2. One observes in this case, in addition to the resolved T, spectrum, also an axial spectrum seen as the additional lines in the figure. It seems that even at 200 ppm Tm in YPd, local distortions from cubicity are very dominant, probably due to the fact that the rare earths and Y metal were only 99.9

%

pure. The solid curve in figure 2 is a least square fit of a smeared axial spectrum of relative intensity 90

%

and a

r,

spectrum with A = 13.5 mm/s.

yb170 : TmBe,,. - Susceptibility studies of

YbBe,, [I71 have shown that Yb in this compound is probably in the mixed valence state. TmBe,, is paramagnetic down to 4.1 K. We have studied ybl'O :

TmBe13 produced after neutron irradiation of TmBe,, and the ~m~~~ -, yb170 decay. The TmBe,, source was put in a lucite oven and the Yb170 metal absorber was kept at 4.1 K. Spectra were obtained at various temperatures from 4.1 to 77 K. Some of the spectra are shown in figure 3. The spectra resemble pure T , spectra,

-2 -1 0 1 2

VELOCITY ( c m l s )

FIG. 3.

-

Mossbauer spectrum at 4.1 K of Yb170 in T m B e l ~ and YbBels. Solid curves are theoretical least square fit spectra.

though theoretical

r6

spectra cannot be fitted in a

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C6-434 I. NOWIK, I. FELNER AND R. YANOVSKY

mental spectra are obtained, however, this may be just due to the fact that more free parameters were allowed in the theoretical reconstruction. The analysis of the various spectra using the theory given by Gonzalez

et al. 181 for the T, relaxation spectra (Fig. 3) yields the

spin relaxation rate. If plotted against temperature (4.1

< T <

39 K) it follows the formula

W ( T ) = w s s + p(r5)

+

W,r,(ex~(AllT)

+

1)

.

The first term is due to spin spin relaxation. Since Tm3+ in TmBe,, has a singlet T2 ground state [17], only the population of the first excited state (P(T,)) leads to spin spin relaxation. The second term is due to the Orbach spin lattice relaxation process. The spin- conduction electron relaxation term was neglected as it is very small in comparison with the other terms. We obtain for W,, the value 2.8(5)

x

lo9 s-I, for W,,, the value 2.3(4) x lo9 s-I and for A the value 50(10) K. The value of A corresponds probably to the energy position of the

r,

state above the T, ground state. The fact that

r,

is the ground state proves that unlike for Tm3

'

in TmBe13 [17], A,

<

r 4

>

is positive for Yb3+ in TmBe,,. Since

we conclude, according to the LLW diagrams that A4

<

r 4

>

for Yb3+ : TmBe13 may extend from 15 K for A,

<

r 6

>

= 8 to 40 K for A,

<

r 6

>

= 3 K. Yb170 : YbBe13.

-

The spectrum of Yb170 in YbBe,, is composed of a single absorption line of 4.1 mm/s width. Such a spectrum can be obtained either with a divalent Yb ion (4fl4) or with a trivalent Yb ion (4fi3) with extremely fast spin spin relaxation rates. Because of the fact that the Yb-Yb distances are quite large and the J,, integral is quite low (as demons- strated by the low Curie points and by the absence of magnetic order in TmBe,, and YbBe,,) the first possibility is much more likely. Considering the fact that previous studies have found [17] that Yb in YbBe,, is in the mixed valence state, it is not surprising that at 4.1 K Yb in YbBe,, is divalent (or of mixed valency) and is dominantly trivalent in TmBe,,. The experimental value of A(T6) =

-

8.5 mm/s (Table I) is substantially lower than that for a pure Yb3+ T, state (- 10.6 mm/s).

2. Hyperfine structure of ions in the cubic

I

'

,

ground state. - The observation of the magnetic hyperfine structure in Mossbauer spectra of rare earth ions in the T , and T7 states yield limited information on crystalline fields because the doublet wave functions are independent of the absolute values of A, and A,. The observation of the hyperfine structure of the cubic quartet (T,) state would yield much more information. In this case the hyperfine structure Hamiltonian depends also on the quadrupole interaction parameter and on the parameter tt x >> as defined by LLW 171. The T, hyperfine structure Mossbauer spectrum (consisting of seven absorption lines : four doublets and three quartets)

yields, therefore, in addition to the hyperfine interaction parameters, also the ratio A,

<

r 4 >/A6

<

r 6

>

of the crystalline field. Unfortunately, it is extremely difficult to observe a pure

r ,

spectrum, since small non-cubic distortions will split, in first order perturba- tion, the

r,

state into two Kramers' doublets. A splitting of the

r,

state by lo-' K is enough to distort completely the expected T, spectrum. This situation is demonstrated by three examples : Er16, in YPd3, in YCu and LaAI,.

According to the LLW [7] diagrams and the crystalline field parameters in YPd, [18], LaA1, [19] and YCu [20], Er3+ in these systems should have a T 131 ground state. These three systems were studied by introducing 200-1 000 ppm of Ho in the respective compounds. 1

%

Ho was melted with Y or La metal in an induction furnace. These samples were neutron irradiated and then a nice grain was remelted together with additional Y or La and the other components of the compound and served as the source for the 80.8 keV gamma ray transition of Erl 6 6 . Homogeneous

compounds were formed which showed no clustering effects. Clustering phenomena are observed in the Mossbauer spectra by the appearance of an exchange narrowed central subspectrum [ l l , 121. However, due to the fact that the rare earth metals and the Y were of purity 99.9

%

only, noncubic local distortions could not be eliminated. An absorber of 80 mg/cm2 of ErAI, at 15 K was used. This absorber gives a single narrow Mossbauer absorption line. The spectra observed are shown in figure 4. They do not resemble any T6, T,

I I I I I I I I

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 VELOCITY (cm/s)

FIG. 4. - Mossbauer spectra of Erl66 : YCu, Er166 : Y P d 3 and ER166 : LaAlz at 4.1 K.

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CRYSTAL FIELD EFFECTS I N RARE EARTH IONS C6-435

that these spectra do not resemble the expected

r,

spectra is that noncubic distortions split the T, state into two Kramers' doublets. Even very small splittings (lo-' K) will give completely different spectra. More- over, the spectra depend on the direction of the noncubic distortion since the T, state is anisotropic. In order to demonstrate these arguments quantitatively we have calculated the Er3+ T8 [3] spectrum for x = 0.6, and hyperfine constants, A = 4.1 mm/s,

B = 0.007 mm/s as a function of the splitting 6

resulting from a noncubic axial distortion along the [001], [ l l l ] and [I101 directions (Fig. 5). We observe in figure 5 that the pure T, seven line spectrum is split

1 1 -i -7.5 -5.0 -2.5 0.0 2 . 5 5.0 7 . 5 VELOCITY (cm/s\

FIG. 5. - Mossbauer line positions of Er166 for Er3+ in the

Ts 131 state with x = 0.6 and AFI = 4.1 mm/s, B = 0.07 mmls, as a function of an axial distortion which splits the Ts state into two doublets separated in energy by an amount 6.

into 20 lines for 6 comparable to the hyperfine splitting (1 K = 320 mm/s, velocity units of Er166). When 6 increases above 0.5 K two five line patterns corresponding to the T, two Kramers' doublets are formed. The line positions are independent of 6 but their relative intensities are given by the Boltzmann factor e-dkT. As can be seen from figure 5, 6

-

lo-' K is indeed enough to distort the

r ,

spectrum. Even more distress- ing is the fact that depending on the direction of the distortion the spectrum changes drastically. In a given sample the noncubic distortions may have a random distribution [I] and any effort to fit the experimental spectra is a hopeless task. Nevertheless, we observe in figure 4 that the Er : LaAl, spectrum resembles very much a relaxation of two doublets (T, [3], for x

-

0.6), and the extension and general features of the YPd, and YCu spectra resemble T, spectra with 6 splittings of the order of the magnitude of 0.1-0.2 K.

We conclude that as a rule, because of small local

noncubic distortions, the observation of a pure

r,

Mossbauer spectrum is almost impossible. Only highly pure, good quality crystals, in which the investigated ions are all equivalent (achieved either by extreme dilution or by 100 %'concentration), a pure T, spectrum could perhaps be observed. In sources distortions may be caused by after decay effects.

3. Crystalline and exchange Fields in SmBe13.

-

Mossbauer studies of the 122 keV transition of SmlS2 in SmBe,, were performed. The source was

in (Euo,,Gdo ,,),03, which gives a broad single emis- ion line [211 at 4.1 K. The observation of the Mossbauer spectrum in SmBe,, is the first observation of this high energy transition in a metallic system. Due to the high Debye temperature [17] relatively large effects were observed and the observed hyperfine structure could be followed to above TN = 8.8 K. Some of the spectra are shown in figure 6 . Since both, the crystalline fields and

0 . 8 ,

,

;

I

, ,

-3.0 -2.0 -1.0 0.0 1.0 5.0 3.0 VELOCITY ( c m / s )

FIG. 6. - Mossbauer spectra of Srnls2 in SmBel3 at 4.12, 5.7

and 12 K.

exchange interactions, are low [I 71 in comparison to the splitting between the H,/, state and the Sm3+ H51,

ground state (1 100 cm-I) we analyze the results within the H,,, levels. Assuming the molecular field approximation we can express the temperature dependence of the effective magnetic hyperfine field (Heff(T)) and of the electric field gradient (q(T)), in terms of the parameters A,

<

r 4

>

and Hm = lo(T) where o is the Sm3' reduced magnetization (identical to the reduced hyperfine field within the above approximations). Since we know the ordering temperature

T,

= 8.8 K the parameters A,

<

r 4

>

and

A

are not independent. If o(T) = P ( A ~

<

r 4

>,

H,, T), then

l / l = d l d ~ , F(A,

<

r 4

>,

0, T,)

.

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C6-436 I. NOWIK, I. FELNER AND R. YANOVSKY

values of the parameter A,

<

r4

>

and then calculated

q(T). H, was along [ill] for A , < 0 and along [001] for A, > 0 to guarantee the minimum of the free energy.

The experimental values obtained for gp, He,, (4.1 K) and eq (4.1 K) Q were 980(5) Mc/s and 705(20) Mc/s, respectively and the values for T = 5.7 K were 800(30) Mc/s and 420(130) Mc/s respectively. These values were compared to those expected theoretically for various A,

<

r 4

>

values. The theoretical cal- culation was based on the Hartree Fock free ion

( A , = 0) values for gp,, He,, (0 K ) = 1 020(50) Mc/s and eq (0 K) Q = 805(160) Mc/s where g = 0.416(2) and Q =

-

1.22(20) barn. We conclude that agree- ment between the theoretical and experimental values can be obtained only for positive A,

<

r 4

>

<

2 cm-l. This very small value for A ,

<

r 4

>

is consistent with the general trend of A ,

<

r 4

>,

it is positive

(-- 15 cm-I) in PrBe,, and negative (- 3 cm-l) in TmBe,, [17].

We are grateful to E. R. Bauminger and S. Ofer for helpful discussions.

References [I] Most prominent is ABRAGAM, A. and BLEANEY, B., Elec-

tron Paramagnetic Resonance of Transition Ions (Clarendon Press) 1970.

[2] EICHER, H., 2. Phys. 169 (1962) 178.

[3] In this case eqQ = - 4 A! Q(l - y,) where y, is the Sternheimer anti shielding factor.

[4] ARMON, H., BAUMINGER, E. R., DIAMANT, A., NOWIK, I. and OFER, S., Phys. Lett. A 44 (1973) 279 ;

BAUMINGER, E. R., DIAMANT, A., FELNER, I., NOWIK, I. and OFER, S., Phys. Lett. A 50 (1974) 321.

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Chem. Solids 23 (1962) 1381.

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Colloq. 35 C 6 (1974) 429.

[Ill SHENOY, G. K., STORH, J. and KALVIUS, G. M., Solid State Commun. 13 (1973) 909.

[12] STOHR, J. and SHENOY, G. K., Solid State Commun. 14

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[13] STOHR, J., WAGNER, W. and SHENOY, G. K., Phys. Lett.

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[14] DWIGHT, A. E., J. Less, Common Metals 43 (1975) 117.

1151 N o w r ~ , I. and DUNLAP, B. D., J. Phys. Chem. SoIids 34 (1973) 465.

[16] NOWIK, I., DUNLAP, B. D. and KALVIUS, G. M., Phys. Rev. 6B (1972) 1048.

[17] BUCHER, E., MAITA, J. P., HULL, G. W., FULTON, R. C.

and COOPEN, A. S., Phys. Rev. B 11 (1975) 440.

[la] DAVIDOV, D., BUCHER, E., RUPP, L. W. Jr., LONGI-

NOTTI, L. D. and RETTORI, C., Phys. Rev. B 9 (1974) 2879.

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