Crystalline electric field effects in CeCd11 and PrCd11

Journal of Magnetism and Magnetic Materials 109 (1992) 316-322 North-Holland

Crystalline electric field effects in CeCd

and PrCd

S.K. Malik Tata b~stitute of Fundamental Research, Bombay 400 005, b~dia

J. Tang ~ and K.A. Gschneidner Jr. Ames LaboratoO, *~ and Departments of Physics and of Materials" Science and Eng#~eering, Iowa State Unieersity, Ames, IA 50011, USA Received 12 August 1991; in revised form 1 October 1991

A simultaneous analysis of the low temperature heat capacity, electrical resistivity and magnetic susceptibitlity of CeCd~ has been carried out on the basis of crystalline electric fields (CEF) acting on the Ce 3+ ion. The ground multiplet level (2F5/.,) of the Ce 3+ ion splits into three doublets under the influence of a CEF of tetragonal symmetry. Detailed analysis yields an overall splitting of -- 70 K with the intermediate doublet lying at an energy separation of = 16 K from the ground doublet. Two sets of CEF parameters are obtained from this analysis waich equally fit all the experimental observations in the low temperature range investigated. Experimental results on the b .... t capacity and magnetic susceptibility of PrCdtt are also presented and analyzed in the same manner using CEF paramet .s derived from those of CeCd~.

|. Introduction In a previous paper by Tang and Gschneidner [1], the results of low temperature heat capacity, electrical resistivity and magnetic susceptibility measurements on CeCd~ have been reported. They found that CeCd~ is paramagnetic down to 1.5 K, obeys the Curie-Weiss law (p~ff = 2.57,t~ and O r , = - 5 . 5 K), and exhibits a Schottky anomaly at an unusually low temperature (-- 7 K) due to a low lying first excited crystal field level at 17.5 K. In order to understand better the crystalline electric field effects and to quantify the crystalline electric field Hamiltonian, a simultaneous analysis of the heat capacity, electrical resistivity and magnetic susceptibility has been

I Present address: Department of Physics, University of New Orleans, Lakefront, New (, '.ans, LA 70148, USA. * Operated lbr the US Department of Energy by Iowa State University under contract no. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences.

made. In this paper we present the details of such an analysis. Further, heat capacity and magnetic susceptibility of isostructural PrCd~ has also been measured and the results are reported here. The CEF parameters obtained for C e C d ~ are scaled for PrCd~ and used in the analysis. The experimental details of the sample preparation, heat capacity and magnetic susceptibility measurements on PrCdt~ are the same as those given earlier for CeCd t~ [1]-

2. Crystal field analysis 2.1. Hamiltonian

The compound CeCd~I crystallizes in the cubic BaHg~ type structure (space group Pm3m). There are 4 inequivalent Cd sites but a unique cerium site in the unit cell. Each cerium atom is surrounded by 12 nearest neighbor Cd atoms and 8 next nearest neighbor Cd atoms. The site symmetry at the cerium ion is tetragonal. The Ce 3+ ion

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S.K. Malik et at / Crystalline electric field effects in CeCd t~ and PrCd tt

in this compound is subject to a crystalline electric field of tetragonal symmetry given by the Hamiltonian '~CEF

0 0 + B 440 ~4 , = B 200 20 + B40~

317

tibility arising due to non-4f terms. In the simplest molecular field approximation, the exchange constant is related to the magnetic ordering temperature T M by

(1)

1

where B,m are the coefficients giving the strength of different C E F terms and O,~" are the operators given in terms of angular momenta. The ground multiplet level of the Ce 3+ ion (2F5/z) is split into three doublets under the influence of the above crystalline electric field. The energy splitting and the resulting wave functions manifest themselves in determining the physical properties of interest, such as the heat capacity, magnetic susceptibility, electrical resistivity, etc., as discussed below. 2.2. Susceptibility

Ixl

-- X C E F ( T M )

where XCEv(TM) is the CEF only susceptibility at the temperature T M. 2.3. Heat capacity From the energy eigenvalues obtained by diagonalizing eq. (1), the heat capacity may be calculated using the expression

CcEv = ~

Z,~. tkar l

In the presence of a crystalline electric field and the applied magnetic field H, the total Hamiltonian is given by a ~ = a~C~CEF q- g~txBJ" H .

(2)

The Hamiltonian given by eq. (2) is diagonalized within the ground multiplet level of Ce 3+ ion to obtain the energy eigenvalues and the wave functions. Calculations are performed with the field H applied parallel and perpendicuiar to the c-axis which we take to be the quantization direction. The susceptibilities in the two directions are designated as XII and g ±, respectively. The polycrystalline averaged susceptibility is given by I

2

XCEF = 3XII+ .~X _L•

(3)

The exchange interaction between the cerium ions is included by writing the susceptibility as [2] -XM

~CEF •

1 --

"'XCEF

,

ex.[+]

k+WeXO(---kB T

'

(7)

where E i is the energy of the crystal field split state l i) relative to the ground level, R is the universal gas constant and Z is the partition function i.e. Z = ,~. exp

(Ei) kaT

"

Note that eq. (7) represents only the 4f contribution to the heat capacity. It is assumed that the non-4f contribution to the heat capacity in CeCd ~ (i.e. the electronic and the lattice contribution) is nearly the same as that of nonmagnetic isostructural LaCd11. A small linear term, y0 T, was also added while fitting the data [1]. Thus the heat capacity due to Ce ions in CeCdl~ and including the linear term is written as

(4~ "

where A is the molecular field coefficient, which, for simplicity, is assumed to be isotropic. The total susceptibility of the compound is given by

x=xM+xo,

(6)

,

(5)

where Xo is the temperature independent suscep-

CcE v + yoT = Ccccd,, - CL,,Cd,'

(8)

2.4. Resisticity The spin disorder resistivity arises due to the exchange scattering of conduction electrons by the localized magnetic moments (the cerium ion

318

S.K. Malik et al. / Crystalline electric fieM effects in CeCd t t and PrCd t t

moments in the present case). The exchan~,e interaction between the conduction electron spins (s) and the cerium spin (S) is given by the Hamiitonian (9)

~,,,= - 2h,.S . s ,

where J,~f is the exchange constant. In the paramagnetic state, this interaction gives rise to a contribution to the resistivity which can be evaluated in the first Born approximation. In the absence of crystalline electric fields this term is given by [3] 3arNm j,,~( _ 1) P0 = 2h e:E~ g.i 1)2J( J + ,

(10)

where N is the number of scattering centers, E F is the Fermi energy and other symbols have their usual meanings. It is to be noted that the resistivity given by eq. (10) is temperature independent. In the presence of the crystalline electric fields, the spin-disorder resistivity becomes temperature dependent and is given by the following equation, [31, 2 PCEF Poj ( j + 1) =

×

~_,

(m~,i'ls'JIm.,.i)

2

pifii,,

m~.m'.,,i,i'

(11) where P0 is the spin-disorder resistivity in the absence of CEF, m~ and m~ are the spin quantum numbers of the conduction electron in the initial and the final states. The probability p~ of a rare-earth ion being in a certain crystal-field state l i) of energy E~ is given by exp( - E i / k T ) P~ = ~, e x p ( - E ~ / k T ) ' J

(12)

and the factor f.,..,., stands for 2 .I),, = I + cxp(-E~,,/kT)

(13)

The tt~tal resistivity is given by the sum of three terms: Ptot,,I = PCEF -¢- Pin, p "¢- Pphonon"

(14)

Here PCEF is the spin-disorder resistivity as discussed above, Pphonon is due to the scattering of conduction electrons by the phonons and P~mp is the temperature independent residual resistivity. The phonon contribution may be estimated by measuring the resistivity of a nonmagnetic analogue. In the low temperature range in which the resistivity of CeCdi~ has been analyzed, the phonon contribution is very small and thus has been neglected. 2.5. Analysis

A simultaneous analysis of the complete susceptibility data (4.2-300 K), complete heat capacity data (4.2-70 K) and low temperature resistivity data (4.2-18 K) has been carried out. The parameters involved in the fit are the crystalline electric field strength parameters (B~°, B4° and B44), molecular field constant A, temperature independent susceptibility X0, linear specific heat coefficient ~'0, exchange constant J.~t, or rather P0, the spin disorder resistivity in the absence of CEF effects, and the residual resistivity Pimp" Since the absolute values of heat capacity, magnetic susceptibility and resistivity were measured over different temperature ranges, a least-squares analysis in terms of minimizing the sum of square deviation would tend to give large weight to the quantities which have large magnitude. Therefore, to avoid this problem during fitting, the heat capacity, susceptibility and resistivity data were normalized to their values at one temperature, in this case the highest temperature (see above). This brings the data points for each measurement to the order of unity, at least in the region of the respective maximum temperature. However, in the temperature range of the fit, the heat capacity shows much larger variation than the susceptibility and the resistivity. Therefore, effectively a larger weight is given to the heat capacity than to the other two quantities. This is justifiable because of the accuracy of the heat capacity data involved and the fact that resistivity and susceptibility do not show any special features, and as such may yield a large range of crystal field parameters which can fit these values. The heat capacity data, on the other hand, show a pro-

319

S.K. Malik et al. / Crystalline electric fieM effects in C'eCd ~t and PrCd ~t 4

360

'

I

'

I

'

I

'

I

'

I ~ 1 ~

320 .

I

x

t

-'.

X

x.

X

2 •~

260

x

XX

"

2

g

X

X X

o

~

240

~-~ ;~

X

E

200

~ o

160

E

120

T~

~

x

80

40

,

1

I0

,

I

20

,

I

,

30

Temperature

I

,

40

I

50

,

I

,

60

0

I

40

80

70

120

160

200

240

Temperoture(K)

(K)

Fig. 2. Magnetic susceptibility as a function of temperature between 4.2-240 K. The solid line is based on the crystal field model, using set I of B~ parameters (see text).

Fig. 1. Differential heat capacity (Ccecd,,- CLued,,) VS. temperature. The solid line is the best fitted curve (CcE F + TOT), based on CEF calculations using set I of B," parameters (see text).

n o u n c e d p e a k at a certain t e m p e r a t u r e . This requires a particular crystal field scheme of levels and, therefore, constrains the crystal field p a r a m eters which can fit the data. A n extensive search of C E F p a r a m e t e r s was m a d e to fit all the data. P a r a m e t e r s which fit s o m e of the e x p e r i m e n t a l results are not included. For instance, some set of crystal field p a r a m e t e r s yielded a good fit to the heat capacity a n d susceptibility d a t a but not to the resistivity data. Such crystal field p a r a m e t e r s are not included. T h e fitting was tried by starting from various different starting parameters. All these trials eventually led to two sets of crystal field p a r a m e t e r s which explain all the experimental results. However, the uniqueness of the p a r a m e ters cannot be ascertained. Figs. 1, 2 and 3, respectively, show the experimental and calculated variation of h e a t capacity, m a g n e t i c suscep-

3.0

'

I

'

I

'

I

'

I

CeCd ~l 2.5-

-

.* ~¢x

,~.

E

5

_

0.5

, O

I 4

,

I 8

,

Temperature

!

12

I

I 16

~ 20

(K)

Fig. 3. Electrical resistivity vs. temperature. The solid line is the fitted curve calculated from CEF splitting, using set 1 of B~ parameters (see text).

Table 1 C"ystal field and other parameters

s2 Set I Set II

X0

Y0

P0

Pimp

[K]

[KI

[KI

[mol/emul

[emu/mol]

[J/mol K2I

[1~ cml

[~ cml

- 2.523 1.694

0.1948 9.856× 10 -4

- 0.02806 - 1.318

- 0.1569 -0.465

- 0 . 6 3 6 × 10 - 3 - 0 . 6 1 7 × 10 - 3

10.646× 10 -3 10.74× 10 -3

3 . 7 2 × 10 -6 4 . 0 2 3 × 10 -6

0.0734× 10 -6 0

S.K. Malik et aL / Crystalline electric field effects in CeCd n and erCd l!

320

,

,

!

10

-

'

I

'

I

'

I

'

I

'

I

'

i

'

I

'

I

'

I

- (a) +

+ Cprcd~t ]

I

0 CLoCd~t

#

_

0 CprCd .-CLocdtt 0

+

_

E ÷

÷

_

0

~+#-+++ ~+

_

~0~0~~.++++ +~~++

.m

+

~]

~ I~I

.

-

_

(J 0

_

0

ooo

0 _

0 O~

"I-

I

, ., ~ ' n - m - 1 - 1 1 1 " r T ~

I

2

,

3

4

I

o@ od. ,. ,

5

I

,

6

I

,

I

7

,

I

8

9

10

T e m p e r a t u r e (K) 50

'

I

'

I

'

I

'

(b) +

+ Cprcd~t I

+.I

0 CLacd~t 13 Cprcd~i-CLoCd.

40

m

0

E

0 30

+

0

+

0

+ 0 + 0 + 0 +++~; O0

20

0 c~ 0 10 0

3E

~ ........

i

(i(ii[I|)-llq~il~[~ OO ~ 0

~

, 5

!

,

10

! 15

O

Oo@~pi !

2O

Temperature (K) Fig. 4. Heat Capacity of PrCd~ and l,,aCdll as a function of temperature: (a) in the temperature range of 1.2-10 K, (b) in the temperature range of 1.2-20K.

tibility and electrical resistivity with temperature for one set of crystal field parameters. These parameters along with the energy eigenvalues and wave functions are given in tables 1 and 2. In either case, the overall splitting is about

73.6 K with an intermediate doublet 16.5 K above the grouzad state. These value.~ are slightly lower (but in agreernent) with the values deduced ear, e r [1] from just the heat capacity results, namely, 17.5 and 80 K, respectively. Furthermore, the

32 i

S.K. Malik et aL / Crystalline electric fieM effects in CeCd n and PrCd tt

constants Xo and 7o are approximately 10% larger than th6~d reported earlier [1] (i.e. X0 = - 0 . 5 7 × 10 - 3 e m u / m o l and ~o = 9.0 m J / m o l K 2, respectively), which were obtained by analyzing the magnetic susceptibility and the heat capacity independently. We believe these new values to be more reliable since they were obtained by fitting the CEF contributions to all these measurements simultaneously. The two sets of parameters given above differ mainly in the magnitude of B4° and the sign of B~.

9

*

'

>,

s

i

"g

4

i~

g ~

Tr

~ ~~ ~~ ~

'~

E [KI

Ig

3e

~O

73.62 16.55 13

60

?e

Fig. 5.

D i f f e r e n t i a l hem capacity ( C p r c d t l - C~Cd~ ~) as a f ~ . c tion of temperature. The calculated c u ~ e s for the two sets of C E F paEameIeEs a~e shown in the figure labeled as I (set I)

and II (set II).

(r") values (assuming that the shielding factors are nearly the same for C e 3+ a n d p r 3 + ) . It should be noted that the sixth order crystal field terms, although allowed by the symmetry, have no effect on the J = 5 / 2 state of the C e 3+ ion and thus cannot be determined from the analysis of the data on CeCd11. Once these are not determined for CeCd~l, they cannot be projected for P r 3+ Therefore, the sixth order crystal field terms have not been included in the analysis of PrCdt~. This should not be a serious problem because, in most cases, when present, the second order terms dom-

150

oJ ~) ~ .

T~ ;K

I_+ ~-> 0.998961 + ~-~> - 0.045541 -T-~> 0.9c,896 [ T- ~ ) + 0.04554 i __+~ >

50

• O

Set II 73.642 16.67 0

5e

4e

Temperolure ( K )

-~

Set I

/~ /

O

..-. :3

1/,

*~~ " ,' ~ -, "'~.....: ~

._,. ....

1

2.6. Scaling of CEF parameters for PrCd n

Table 2 Energy eigenvalues

,

, ~,., ~:

~ ~

Jif O

Heat capacity of LaCd ~t and PrCd ~t has been measured in the temperature range of 1.2-60 K. Fig. 4 shows the results of these measurements at low temperatures: (a) in the 1.2-10 K and (b) in the 1.2-20 K temperature ranges, the two heat capacities are nearly parallel above 10 K. The difference between the heat capacities of PrCdl~ and LaCd~ gives the contribution due to Pr 3÷ ions and is shown in fig. 5. The inverse molar susceptibility of PrCdt~ is plotted in fig. 6 as a function of temperature and shows nearly Curie-Weiss behavior with an effective magnetic moment close to that of the P r 3÷ free ion value. In order to decide between the two sets of CEF parameters obtained from the analysis of experimental results on CeCdll we have scaled these crystal field parameters to that for the P r 3+ ion with appropriate Stevens' coefficients and

CPrCd~ ~- CLaCd.

0.799751 + -~)-0.60031-T- -~>

I+_-~> 0.799751 -T- ~-~>+ 0.6003 [ +_ :~>

I 50

'

I leo

'

I 150

'

I

ZOO

'

I ~58

'

308

Temperature (K) Fig. 6. Inverse magnetic susceptibility of PrCd~l vs. temperature. The calculated curves for the two sets of crystal field parameters are almost identical.

322

S.K. Malik et al. / C~ystallinc electric field effects in CeCd tt and PrCd tt

inate. The results of the calculations of the heat capacity of the Pr 3+ ion are shown in fig. 5, appropriately marked. Both sets predict a Schottky peak in the heat capacity of PrCd~ at about the same temperature, namely around 15 K (see fig. 5). However, the peak height and the behavior below 10 K is different in the two cases and experimental results are closer to the values predicted by set II. The large scatter in data is due to the small difference between the two heat capacities at higher temperatures. Both sets of CEF parameters, projected in this manner for Pr 3÷, do not give rise to a sizeable deviation of the susceptibility of Pr 3+ ions in PrCdt~ from the free ion value in agreement with the exgerimental results shown in fig. 6 and also those given in ref. [4].

3. Point charge estimate of CEF parameters Finally, we have used a point charge model to calculate the sign and magnitude of the CEF parameters. Appropriate lattice sums were calculated over a sphere of 60 ~ radius with a Ce ion at the origin. The charges assumed are 3+ at the rare-earth site and 2+ at the Cd site. The contributions to the CEF parameters, in particular, to B~° is strongly shielded by the non-4f electrons. The shielding factor for the trivalent rare earth ions for various crystal field terms are not well known. Theoretical calculations give values in the range of 0.5-0.7 for the shielding factor of B~ terms for different rare earth ions. However, comparison with experiments would require this shielding factors to range from 0.48 for Lu 3+ to "about approximately 0.9 for Pr 3÷ [5]. Assuming that the shielding factors are nearly the same for Ce 3+ and Pr 3+ and taking an average value of 0.8 for the shielding of the B~ term and no shielding for the B4° term, the following values of CEF parameters are obtained for Ce 3÷ in CeCd~:

B~= - 1 1 K, B4° = 0.023 K,

B 2 = -0.804 K. The sign of B2° and the magnitude of B4° favors the first set of CEF contrary to the analysis of various experimental results presented above. Moreover, the magnitude of B~ is rather high. It should be cautioned that such point charge calculations are not always reliable and may even lead to a wrong sign of CEF parameters. If the magnitude of B~ is to be relied upon, the present calculations would suggest an even larger shielding factor for the B2° term than assumed. In conclusion, a detailed analysis of the susceptibility, resistivity and heat capacity of CeCd~! has been carried out on the basis of CEF effects. Two sets of CEF parameters are obtained which explain the experimental results. When scaled for the Pr 3+ ion, these CEF parameters also predict a Schottky anomaly in the heat capacity of PrCd i~. Inelastic neutron scattering experiments may help to resolve the sign of B2° and narrow down the sets of crystal field parameters.

References [1] J. Tang and K.A. Gschneidn~.r Jr., J. Magn. Magn. Mater. 75 (1988) 355. [2] B.D. Dunlap, L.N. Hall, F. Behroozi, G.W. Crabtree and D.G. Niarchos, Phys. Rev. B 29 (1984) 6244. H. Zhou, S.E. Lambert, M.B. Maple, S.K. Malik and B.D. Dunlap, Phys. Rev. B 36 (1987) 594. H.A. Kierstead, B.D. Dunlap, S.K. Malik, A.M. Umarji and G.K. Shenoy, Phys. Rev. B 32 (1985) 135. [3] V.U.S. Rao and W.E. Wallace, Phys. Rev. B 2 (1970) 4613. [4] W.E. Wallace, Prog. Solid State Chem. 16 (1985) 127. [5] J. Blok and D.A. Shirley, Phys. Rev. 143 (1966) 278. C.J. Lenander and E.Y. Wong, J. Chem. Phys. 38 (1963) 2750. D.K. Ray, Proc. Phys. Soc. (London) 82 (1963) 47.