Crystalline fields and exchange interactions in YbNi2

1. /‘II?.<. Chrn~. So&k

1973. Vol. 34. pp. 465-47 I.

Pergnmon Prccs.

Printed in Great Rritain

CRYSTALLINE FIELDS INTERACTIONS

AND EXCHANGE IN YbNi,*

I. NOWIKt and B. D. DUNLAP Argonne

National

Laboratory,

(Receioed

Argonne,

30 May

III. 60439,

U.S.A.

1972)

Abstract- Mossbauer studies of ““Yb in YbNiL at temperatures of I .4-15°K and in external fields up to 44 kOe reveal that YbNi, is magnetically ordered below 5.7 2 0.4 K. Analysis of the experimental results within molecular field and crystal field theory approximations yield the cubic crystalline field constants A.,(P) = 44 f 2 cm-‘, IA, 1 < 2 cm-’ and the Yb-Yb first nearest neighbor exchange constant y = 16°K. These results are compared with a previous calculation of Bleaney and with crystal field constants in the rare-earth-Al, systems. Effects of crystal field mixing on details of the hyperflne spectra and on the systematics of the Curie temperature for the rare-earth-Ni? compounds are considered. 1. JNTRODUCTION THE RARE-EARTH cubic Laves phase com-

pounds have been the subject of many experimental and theoretical studies in recent years [ 11. Among these investigations have been magnetic measurements of the RN& (R = rare earth) compounds by Wallace and co-workers [21. The results of that study have been discussed by Bleaney[3], who attempted to estimate the magnitude of crystal field interactions and the resulting effect on the magnetic properties. While this approach qualitatively explains the experimental results, no direct measurement of the crystal field energies has been performed. In particular, the ytterbium compound, which has the configuration Yb:‘+ : 4f13, and so is very simply treated within crystal field theory, has not even been studied experimentally. In a recent Mossbauer effect study [4] of 17”Yb in YbPdIl, we have shown how data obtained as a function of temperature and external magnetic field can be analyzed to provide crystal field constants, provided the external field is sufficiently large to admix the crystal field levels. Here we report a similar study for YbNi,. The results *Based on work performed under the auspices of the U.S. Atomic Energy Commission. ton leave from the Hebrew University, Jerusalem, Israel.

thus simultaneously provide an addition to the magnetic properties study of Wallace et al., and a check on Bleaney’s general discussion of those results. 2. EXPERIMENTAL

Mossbauer spectra have been obtained for the 0+ + 2+, 84.3 keV resonance of i70Yb as a function of temperature and external magnetic field. The source material was 170Trn in a TmAI, matrix. The absorber was unenriched YbNi, powder having a total material thickness of 300 mglcm’. For this 0+ + 2+ transition, magnetic hyperfine splitting will ordinarily produce a hyperfine spectrum consisting of five equally spaced lines having equal intensity. If, however, both source and absorber are placed in the external magnetic field (H,,,) with gamma ray propagation along the field direction, then both the emission spectrum and the absorption spectrum will consist of polarized doublets, provided the magnetic anisotropy is sufficiently low to align the electronic moments along H,,,. The resulting resonance spectrum then consists of two lines. In the present case, the Yb ion in the source is diamagnetic (Y b 2+; 4f’3 while that in the absorber is magnetic (Yb 3+; 4f’3. Under these circumstances, the separation of the two lines is given by 2g,,&f,rr where g, is the nuclear g

466

1. NOWIK

and B. D. DUNLAP

factor for the 2+ state and Iferr is the hyperfine field produced by alignment of the electronic moment in the absorber. The direct splitting due to H,,, does not appear explicitly[4,5]. .Spectra obtained at various temperatures with Hext = 0 are shown in Fig. 1. At the lowest temperatures, a clear magnetic splitting is observed, showing that the material is magnetically ordered. The spectra obtained, however, are not the typical form expected for this case (i.e., five equally intense lines). Instead, a considerable amount of line broadening is observed, with unequal line widths for the different transitions. Such effects may occur as a result of spin relaxation processes with a relaxation time comparable to the nuclear Lannor precession time [61, or as a result of a distribution of hyperfine interactions, as will be discussed below. Above approximately 6 K, the spectra are all identical, indicating that these temperatures are greater than the magnetic ordering temperature. The broad spectrum which is observed instead of the usual single line is interpreted again as being

c

I1 -30

I -20

I

_

I,

I

-10

0

VELOCITY

r-.*I=-

5.6K

8

6.5K

due to long spin relaxation times. From the temperature dependence of the overall linewidth, we estimate a magnetic ordering temperature of 5 97-+ 0.4 K. Spectra taken at 4.2 K for various H,,, are shown in Fig. 2. In addition to the doublet structure mentioned above, one also can see that the AZ, = 0, +2 lines do not vanish because of a finite magnetic anisotropy in this material. There is also substantial line broadening of the various transitions, which becomes more pronounced as Hext increases. The spectra were analyzed assuming a single magnetic hyperfine field I?,,( and electric field gradient &, but with different line widths for the three types of transition AI, = 0, + 1, ?2. The hyperfme interaction parameters so obtained are given in Table 1. Note added in proof: A second sample kindly provided by K. H. J. Buschow, Phillips Research Laboratories, gave spectra identical with those shown here. 1

1

t

I

I

IO

I

I

20

I

I

30

(mmkcl

Fig. 1. Variation of Massbauerspectraof ““Yb in YbNir with temperature.

-25

-20 -15 -10 -5 VELOCITY

I

,

0

5

,

,

,

IO 15 20

,

25

(mm/sac)

Fig. 2. Variation of Miissbauer spectra of ““Yb in YbNi, with external field at 4.2 K.

CRYSTALLINE

FIELDS

AND

EXCHANGE

Table 1. Experimental values of the hyper-ne coupling constants for 170Yb in YbNi, at various values of tern erature and external field H ext T(K) I .4 1.4 1.4 4.2 4.2 4.2 4.2 4.2 4.2 [a] Hrs

(kOe) 0 15 44 0 2 8 15 30 44

IN

H,dkOe)

374k48 504k68 864&61 258k20 258241 340234 496k48 633k54 687k34

467

crystalline field is given by

A?= vc-g.,PJ%

(1)

where

1741244 1929-c50 2261242 1196?16 1296k20 154Ok26 1721 -t34 2013k35 2158k24

HedHda’ 0.41*0.01 0.46kO.01 0.54kO.01 0.282kOJJO6 0.306+-O+lO6 0.364-cO.006 0~407~OGl8 0.476kO.008 0~511a0~008

is the free ion hyperfine

field.

+~G(rs)(JllyllJ)(Oso-21.0s4)

AND

ANALYSIS

In a cubic crystalline field, the ground state of Yb3’(?F7,,) is split into two doublets (I, and I,) and a quartet (I,). For any one of these states, the electronic wavefunctions may be written explicitly and are independent of the relative magnitudes of the pertinent crystal field parameters [7]. Thus, the hyperfine spectrum expected from any isolated level may be calculated exactly. We have previously described in detail the type of spectra one expects to obtain from each level, and have discussed means of calculating the variation of the hyperfine parameters due to admixtures of the crystal field levels by an external field [4]. In the present case, the situation is more complicated because both the exchange field in the ordered state and Hext will mix the electronic states. Since one does not know a priori the relative magnitudes of the molecular field interactions, external magnetic field interactions and crystal field interactions, a full solution of the total Hamiltonian must be attempted. We have analyzed the experimental results within the framework of simple molecular field and crystalline field approximations, and will now outline the procedure followed. The Hamiltonian for an ion in a molecular field, external magnetic field, and cubic

(2)

is the crystal field potential in the notation of Lea, Leask and Wolf[7], and H = Hn, + He,,

(3) where H, and H,,, are the molecular field and external field, respectively. Within the molecular field approximation, we take H, = ha(H,

3. THEORY

YbNi,

V, = A4(r4) (JIIPIIJ >(02+5-044)

ezqeO (MHz)

= 4230 kOe

INTERACTIONS

T)

(4)

where o is the reduced magnetization. Diagonalization of 2 yields the eigenvalues and thus the partition function of the system, from which desired quantities may be calculated. With the local coordinate system chosen such that the z axis is in the direction of ff,,i, the magnetic moment in the direction of H,,, will be given by the thermal average of the expectation value of g,pJ,. If the magnetic anisotropy is small, then (T will lie along H,,, and we will have u(H, Z-1 = ((J,>)/J

(5)

or u(H,

T) = Trace [J,exp

(-R/kT)]/

Trace [J exp (-%f/kT)].

(6)

This result and u(H, T) = W--H,,,)/A,

(7)

obtained from equations (3) and (4) for this case, provide two coupled equations containing the three parameters Ad, A, and X which may be solved simultaneously to obtain u(H,,~. T). This procedure is identical to that which one follows in simple molecular field discussions of magnetism where the crystal field is not considered and, in that case, equation (6) becomes a Brillouin function, B.,(H, T).

I. NOWIK

468

and

If we assume that the effective hyperhne field Herr is proportional to the magnetization Herr = HFdHeett 7-J (8) where HP, = 4230 kOe is the free ion value, H,, can be calculated as a function of H,,, and T. Comparison of the calculated dependence with the experimental results then can in principal provide the parameters Ad, A6 and A. The major difficulty in this procedure is that the present data is obtained from a powder sample, and equation (6) and (7) should be averaged over all angles. For calculations involving only the Is and Ir doublets, which are isotropic, this presents no problem. However, the IS quartet state is anisotropic and, since H,,, points in various directions relative to the local coordinate axes in a powder sample, the averaging process may become quite complex. We will partially account for the average by the same procedure we have previously followed [4]. We first ignore the finite magnetic anisotropy, and thus assume that u lies along He,,. We then consider those cases which can be easily calculated, namely those in which H,,, lies along the three symmetry axes [OOl]. [ 1101 and [ 1111. Choosing coordinate systems in which the z axis is along Hex,, the crystal field potential can be easily expressed[8] and

B. D.

DUNLAP

the three quantities u(OOl), ~(110) and (+( 111) calculated, where ~(001) denotes the value of u when H,,, is along [OOl], etc. We then approximate the complete average by a(H,T)

= [12(+(110)+8(+(111) +6(r(OO1)]/26

(9)

where each quantity is weighted by the number of times the pertinent axis appears on the unit sphere. Equations (6) and (7) are then solved with cr(H, T) replaced by a(H, T). Solution of the two equations has been performed graphically. Assuming values of A, and &, equation (6) is plotted for various temperatures as a function of the variable H. Similarly, assuming a value of A, equation (7) is plotted for different H,,, as a function of H. One then attempts by trial and error to find parameters which reproduce the observed values of (+ = Heff/HFI. For the same parameters, the dependence on Hext at both temperatures must be reproduced, and the temperature for which equation (6) has the same initial slope as equation (7) must give the observed magnetic ordering temperature, according to the usual molecular field critieria. Agreement is obtained for all data with A,(r4) =44&2cm-I, IA,(P)] G 2cm-’ and A = 118 +2 kOe, as shown in Fig. 3.

0.6 -

X=IIBkOe

Fig. 3. Determination of the crystal field and molecular field parameters. The light lines are calculated from equation (7) for the indicated values of H,,,. The dashed are calculated from equation (6) for the indicated temperatures. The solid rectangles circles give the data points at 1.4 and 4.2 K, respectively.

solid lines and

CRYSTALLINE

FIELDS

AND

EXCHANGE

4. DISCUSSION

The crystal field parameters obtained give a T6 ground state for Yb3+ in YbNif, with TH and r7 lying at 78 and 208 K, respectively. Bleaney [3] has previously obtained a rough estimate of the crystal field interactions for the RNi, series which is about twice as large as the present results. That result was primarily due to an analysis of the susceptibility of TmNi, which included only the ground state and first excited state of the Tm3+ ion. Bleaney’s conclusion based on this assumption was that the first excited state in TmNi, was at 37 K and the second excited state at about 70 K. The present results, giving crystal field interactions only half as large, indicate that the second excited state in TmNi., should lie at approximately the position that Bleaney has assigned the first excited state. Thus the neglect of the second excited state in the susceptibility analysis is questionable. Nonetheless, the qualitative conclusions reached by Bleaney are seen to be correct in that the measured A,(r4) is positive and comparable with his estimate, and Ac(r6) = 0. These results are also in rough agreement with the point-charge model, as described by Bleaney. Such a calculation gives Ad(r4) = 20 cm-’ and AJ(r4)/A6(r6) = 50. If we assume the exchange interaction is significant only between an Yb ion and its four nearest Yb neighbors, and of the form ZeXt = -2 ,P”_si.Sj,then A as defined in equation (5) is given by

INTERACTIONS

IN

YbNi,

469

ture is to cause a distribution of hyperfine fields, related to the anisotropy of the TRstate. In Table 2, values for H,rr and qerr (calculated in a way [41 analagous to that described above for H,r,) are given for H,,, = 44 kOe, T = 4.2 K, and for H,,, lying along the three principal Tuble 2. Dependence of calculated hyperfine coupling constants on orientation of H,,, for H,,, = 44 kOe and T = 4.2 K (A4(r4) = 43.5 cm-‘, A6 = 0 and

A= 118kOe) Direction of He,,

Heulff~r

Qenh 0.099

[0011

0.497 0.480 0.549

Average as in equation (9)

0.504

0.127

KY;

0.017 0.329

axes. One may first note that the values of Helf do not vary substantially with orientation, lending some confidence to the approximate averaging procedure of equation (9). However, the values of qeff vary by a factor of thirty. This causes the dff to be questionable, and hence a detailed comparison between calculated and experimental qeff is impossible. The large variation in qeff also provides a plausible explanation for the discrepancies between calculated and experimental spectra such as are seen in Fig. 2. The spectrum for Hext = 44 kOe and T = 4.2 K is reproduced in (10) Fig. 4, along with bar diagrams showing the position of resonance lines for the parameters With the value of A obtained above, we then given in Table 2. The lines in the upper part of find the nearest neighbor Yb exchange interthe figure show the range of positions allowed action to be 2 = 16 K, somewhat larger than for the various resonance lines, and it is clear the values estimated by Bleaney [3]. that the observed line broadenings can be With the parameters obtained here, it can obtained from this effect alone. In addition, be seen that a noticeable amount of the TX the observed dff extrapolate to dff = 0 for state will be mixed into the ground state at H = 0, indicating that the quadrupole intertemperatures well below the Curie temperaactions arise solely from admixturb of the Te ture, or in moderate external fields. In our level. measurement, the major effect of this admixThe line broadening phenomena observed

470

1. NOWIK

and B. D. DUNLAP

earth compounds, and use the (fl) values of Freeman and Watson [ 111, we than may estimate a value A4(fl) = 22 K appropriate to Yb3+ in LaAl,. This is just half of the present result for YbNi,. One can qualitatively understand this difference on the basis of a point charge model. For the cubic Laves phase structure, Bleaney has shown [3] AIz=-2 -2 I

I -30

I

88 -20

I

-I I -I ! -1-2 0 ” I I -10

I 0

T I I

[HO]

AqC(--&-1.22,)

2 I

[Ill1

I I I

I 0

I

: bll I IO

I

I, 20

J 30

VELOCITY hm/sed

Fig. 4. Variation of hyperhne spectra with direction of H ext. due to admixture of the Is state. Bar diagrams are given for calculated line positions with H,,, in each of the three major symmetry directions. The range of possible positions for each resonance line is indicated at the top of the figure.

in the H,,, = 0 spectra may arise in a similar way if the magnetization does not point in a unique direction throughout the material. In fact, the calculated spread in hyperfme constants for different orientations of H, in the absence of H,,, is quite adequate to explain these spectra. Another possible interpretation, however, is that the magnetization has a well defined direction and the line broadening is caused by spin relaxation phenomena. It is clear that relaxation effects are visible above the transition (see Fig. 1). Such phenomena are generally expected to vanish in magnetic saturation; however, this has recently been observed to not be so in some intermetallic actinide systems [9]. At present it is difficult to give a thorough discussion of these spectra. It is of interest to compare the crystal field parameters obtained here with those in similar systems. White et al., have measured [lo] the susceptibility of Ce dilute in LaAl, and interpreted the results as showing a low-lying doublet (I,) with a quartet (I,) at approximately 200 K. From this one obtains [7] a value of A4(r4)Ce3+ = 87 K. If we assume A4 approximately constant across a series of rare-

where a, is the unit cell dimension, while Z, and Z, are the charges on first and second neighbors, respectively. From crystallographic data[ 11 one finds that a,[51 for RN& is only half of that for RA12, which is already sufficient to explain the observed results. From the magnetic data, Bleaney argues [3] that Z, = 0 for RN&. The comparison of the values of A, for the two systems within the confines of the point charge model indicates that Z, is also rather small in RAl,; however, this approach is probably too simple to allow a strong conclusion and at present a more rigorous treatment is not available. We have attempted to explain the systematics of the Curie points of the RNi2 systems, including the present result for YbNi*, using the theory developed by Levy. This approach, which successfully described the transition temperature for RAl?, RIr, [ 121 and ROs, [ 131 does not work here. The reason for this failure is probably due again to the fact that the crystal field interactions, which were not included in Levy’s calculation, are larger here than in the other systems. REFERENCES 1. For a review, see TAYLOR K. N. R.. Adv. Phvs._ 20. 551(1971). 2. FARRELL J. and WALLACE W. E., Inorg. Chem. 5. 105 (1966); SKRABEK E. A. and WALLACE W. E., J. appl. Phys. 34.1356 (1963). 3. BLEANEY B., Proc. R. Sot. Land. 276.28 (1963). 4. NOWIK I., DUNLAP B. D. and KALVIUS G. M., Phys. Rev., B6. 1048 (1972). 5. KORNER H. J., WAGNER F. E. and DUNLAP B. D., Phys. Rev. Left. 27, 1593 (197 1). 6. WICKMAN H. H., in Miissbauer Efecr Merhodology (Edited by 1. J. Gruverman), Vol. 2, p. 38. Plenum Press, New York (1966).

CRYSTALLINE

FIELDS

AND

EXCHANGE

7. LEA K. R., LEASK M. J. M. and WOLF W. P., J. Phys. Chem. Solids 23, 138 1 (1962). 8. HUTCHINGS M. T., Solid Srate Phys. 16, 277 ( 1966). 9. DUNLAP B. D., NOWIK I. and LAM D. J., to be Dublished. 10. WHITE J. A., WILLIAMS H. J., WERN~CK J. H.

INTERACTIONS

IN

471

YbNi,

and SHERWOOD R. C., Phys. (1963). I I. FREEMAN A. J. and WATSON 127,2058 (1962). 12. LEVY P. M., J. appl. Phys. 41,902 13. LEVY P. M..SolidSrote Commun.

Rev.

131,

1039

R. E., Phys.

Rev.

(I 970). 7. 1813 (1969).