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KONDO SIDE-BANDS IN CERIUM INTERMETALLIC COMPOUNDS
H. van Daal, F. Maranzana, K. Buschow
To cite this version:
H. van Daal, F. Maranzana, K. Buschow. KONDO SIDE-BANDS IN CERIUM INTER- METALLIC COMPOUNDS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-424-C1-431.
�10.1051/jphyscol:19711150�. �jpa-00213970�
JOURNAL DE PHYSIQUE Colloque C 1, supplement au n° 2-3, Tome 32, Fevrier-Mars 1971, page C 1 - 424
KONDO SIDE-BANDS
IN CERIUM INTERMETALLIC COMPOUNDS
H. J. van DAAL, F. E. MARANZANA and K. H. J. BUSCHOW
Philips Research Laboratories N. V. Philips' Gloeilampenfabrieken, Eindhoven, The Netherlands Résumé. —La résistivité électrique O) des composés hexagonaux Cei-aLa^AU, Cei-*Y*A13 et Ce1_/rh3!Al3 (0 < x <, 1) a été mesurée entre 1.3 et 300 °K. Dans les différents composés le comportement de p{T) montre soit un seul maximum large soit deux maxima dont celui situé à la température la plus basse peut être très aigu. La présence de ces maxima est attribuée à un type spécial de la diffusion d'échange de Kondo qui se manifeste en présence du dédou- blement, dû au champ cristallin, du multiplet fondamental de Ce3+. Ce modèle de « Kondo side-bands » permet d'expli- quer la variation systématique du comportement de p(T) si l'on remplace le Ce dans CeAl3 par La, Y ou Th. Il est montré qu'il existe une forte corrélation entre la variation de p(T) et ces substitutions qui causent un changement du champ cristallin et dans le cas de Th aussi un changement de la densité d'états.
Abstract. — The electrical resistivity of compounds of the hexagonal systems Cei_Xa^AI3, Cei-sYzAh and Cei-ZThzAb (0 =S x <i 1) has been measured in the temperature range 1.3 to 300 °K. In the different compounds the resistivity as a function of temperature shows either one broad maximum or two maxima of which the one at the lower temperature can be very sharp. The presence of these maxima will be shown to be due to the occurrence of a special type of Kondo exchange scattering in the presence of crystal-field splitting of the Ce3+ ground multiplet: the « Kondo side- band » scattering mechanism. On the basis of the Kondo side-bands model, the systematic variation of resistivity beha- viour upon replacement in CeAl3 of Ce by La, Y or Th will be shown to be correlated with the changes in the crystal-field parameters and in the case of Th also with a variation of the electronic density of states.
I. Introduction. — Evidence for the occurrence of a Kondo effect in the electrical resistivity (p) of some cerium intermetallic compounds has been pre- sented recently. The relevant compounds are CeAl2
[1, 2] and C e3A lu [3]. It has furthermore been sugges- ted that the Kondo effect is active in CeAl3 [4]. The behaviour of p as a function of temperature in the above mentioned compounds is presented in figure 1.
AO
O
p(pn.cm)
- !
i i i i
\ c e A l3
/ " C e A l ^
- " ^ A l , ,
I 1 1 1 1 1 1 1 i i j i i i i i r l
FIG. 1. —The presence of the Kondo effect in the resistivity of the orthorhombic compound Ce3Al«, the cubic compound
CeAU and the hexagonal compound CeAl3.
Magnetic measurements have shown that below a certain temperature magnetic ordering takes place in C e A l2( < 4°K) and Ce3Aln (<, 6°K), whereas no such effect could be observed in CeAl3 down to the lowest temperatures considered. The gradual
rise of p with decreasing temperature below about 15 °K in CeAl2 and 20 °K in CesAIu (see Fig. 1), followed by a sharp maximum in the vicinity of the respective ordering temperatures, is considered to be due to the presence of the Kondo effect in the para- magnetic range. In CeAl3, with decreasing tempera- ture, a gradual rise of p, starting already not far below room temperature, continues down to about 35 °K.
There p passes through a maximum, despite the absence of magnetic ordering effects, and then falls rapidly.
The interpretation of p behaviour in CeAl3 in terms of a special type of Kondo effect, which takes a giant shape in this compound, is the main purpose of this paper.
Traditionally, the Kondo effect is coupled with magnetically dilute alloys such as Cu, Ag or Au as a solvent with very small amounts (<; 0.1 at %) of transition metals such as Cr, Mn or Fe as a solute.
Kondo [5] has shown that the s-d exchange interaction between the spins of the conduction electrons and the localized magnetic moments of the transition elements leads in second-order perturbation theory, under certain conditions, to an increase of magnetic resistivity with decreasing temperature. These condi- tions are : the freedom of the localized moments to flip, i. e. to change by one magnetic quantum number, and an effective s-d exchange integral with a negative sign. The expression for the resistivity in the presence of Kondo exchange scattering reads :
PK = ps[l+2N(EF)rin(£) ; (1) it is valid only as long as the second-order contribu-
tion to pK does not exceed the first-order term ps. The temperature marking this limit is commonly referred to as the Kondo temperature :
TK = TF exp { - (2 N(EF) | r I)"1 } . (2) In the above expressions, ps denotes (first-order) ther- mal spin-disorder resistivity in the paramagnetic
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711150
KONDO SIDE-BANDS IN CERIUM INTERMETALLIC COMPOUNDS C 1-425 range, N(EF) the electronic density-of-states at the
Fermi level, r the effective S-d exchange integral and TF the Fermi temperature.
It is still a matter of controversy whether a Kondo effect can exist in magnetically non-dilute systems such as the Ce compounds considered in this paper.
Nagaoka [6, 71 concludes to the appearance below TK of a quasi-bound spinless state between conduction and localized electron spins. At a certain temperature below TK there exists a critical concentration of the localized moments above which the formation of this state cannot become fullv realised. However, a conclusive indication of such an effect on the occur- rence of a Kondo anomaly at temperatures T >, TK does not seem to follow from these considerations.
In this respect it is interesting to remark that recent experiments performed by Hauser et al. [g] on the superconductive proximity effect between Pb and CeAI, indicate that above the Nkel temperature and at least up to 7.2OK, CeAI, behaves as a spinless system.
Correlation between the localized magnetic moments hampers the freedom of these moments to flip and can eventually quench the Kondo anomaly. Harrison and KIein [g] have suggested that the disappearance of the Kondo anomaly in systems such as Au(Fe) at Fe concentrations above 1 at % is due to short- range order effects. Abrikosov [l01 has shown that long-range ordering in a magnetic alloy eventually leads to extermination of the Kondo effect. However, in rare-earth systems, due to the highly localized character of the 4 f electrons, direct magnetic coupling between the localized moments is negligibly small.
Because, in addition, indirect coupling can be rather weak, these systems can be conceived of showing a Kondo anomaly in the non-dilute case, too. Maran- zana [l11 has shown that, if magnetic ordering is present in the Ce compounds, the Kondo effect is quenched in the ordered region but may become mani- fest again above the ordering temperature.
The compound CeAl,, considered in this paper, meets the requirements for the occurrence of a Kondo effect :
1. It follows from magnetic measurements that, at least down to 100 OK, Ce is in the trivalent state and thus carries a localized moment.
2. The absence of magnetic-ordering effects in the whole temperature range considered guarantees the freedom of the Ce3+ magnetic moment to flip.
3. It follows from data for the Knight shift of the AIz7 nuclear magnetic resonance that the effective S-f exchange integral has a high negative value
(r
-
- 3 eV [12, 131).Attempts to explain the observed p behaviour in CeA1, by means of traditional second-order pertur- bational Kondo theory failed. The reason for this failure is thought to be connected with the crystal- field splitting of the Ce3 + ground multiplet. Recently, Maranzana [l41 proposed a novel model for the Kondo exchange scattering in the presence of crystalline electric fields : the Kondo side-bands model D. It will be shown that p behaviour observed in CeAI, and related pseudo-binary compounds Ce, -,LaxA13, Ce, -,Y,Al, and Cel -,Th,Al, (see section II) 'supports
the validity of this model. In section 111, an outline is given of the nature of the crystal-field spIitting in CeA1, and the changes introduced in the latter by partial replacement of Ce by La, Y or Th. The substi- tution of Th affects in addition the electronic density- of-states at the Fermi level. The main features of the Kondo side-bands model are given in section IV.
On the basis of this model an interpretation of p data is presented in section V, taking into account variations of the crystal-field parameters and of the electronic density-of-states. Finally, a summary of this inter- pretation can be found in section V.
11. Experimental. - The binary compound CeAI, as well as the sets of pseudo-binary compounds Ce,-,La,Al,, Ce,-,YxA13 and Ce,-,Th,Al, have the hexagonal Ni3Sn structure [15]. The lattice cons- tants of the binary compounds CeAI,, LaAl,, cc-YAl, and ThA1, are given in table I. The lattice constants
Compound Lattice constants (A) ~ / a
a C
- -
CeAl, 6.545 4.609 0.704
LaA1, 6.662 4.609 0.692
yAl3 (*) 6.276 4.582 0.73 1
ThA1, 6.499 4.626 0.712
(*) BAILEY (D. M.), Acta Cryst., 1967, 23, 729.
of the pseudo-binary compounds can be obtained from a linear interpolation between the data of the relevant binary compounds.
The p data obtained on the three sets of compounds in the temperature range of 1.3 OK to 300 OK are presented in figures 2-4. For each set of compounds
a ~ C { I J ~ & ) C ~ I - X ~ X A ~ ~
t i l l
FIG. 2. - The variation of resistivity behaviour in CeA13 upon substitution of La for Ce. All compounds are isomorphous (hexagonal NisSn structure). The curves have been shifted on
the p scale so that they match at 300°K.
C 1-426 H. J. VAN DAAL, F. E. MARANZANA AND K. H. J. BUSCHOW
elements La, Yor Th leads to significant changes in p behaviour. The salient feature of the experimental curves is the presence of one or two peaks. The position of these peaks on the temperature scale depends on the nature and on the molar fraction of the substituent.
The composing elements of the compounds are trivalent with the exception of Th which is tetravalent.
Therefore, substitution of Th for Ce can lead to changes in the electronic density of states. Indica- tions of an appreciable variation of N(EF) induced in CeA1, by substitution of Th for Ce have been obtained from susceptibility (X) measurements carried out on the sets of isostructural compounds La, -,ThxA13 and Y,-,ThJ13 (see Fig. 5). In these compounds X
rPK1
FIG. 3. - The variation of resistivity behaviour in CeAl3 upon replacement of Ce by Y. See also caption to figure 2. YA13 has
the rhombohedral BaPb3 structure.
*X
FIG. 5. - Variation of the magnetic susceptibility of compounds of the systems Y I - ~ T ~ , A I ~ and La1--~Th~Al3 as a function of composition. The variation of the susceptibility of compounds of the system Lal-xYzA13 as a function of composition remains limited within the range determined by the values indicated for La& and YA13. All data were taken at roomtemperature.
is mainly determined by Pauli paramagnetism of the conduction electrons and diamagnetism of conduc- tion and core electrons. The X data suggest that, with increasing Th concentration, N(EF) rises at first or is a constant but then ( X > 0.3) decreases appreciably.
111. Crystal field. - In the compound CeA1, the hexagonal crystal field splits the ground state multiplet of the Ce3+ ion (J = 3) into three Kramers doublets.
The relative positions of these doublets will be esti- mated with the aid of a point-charge model. The crystal-
- 1 [*K1 field Hamiltonian is given by :
FIG. 4. - The variation of resistivity behaviour in CeAI3 upon
substitution of Th for Ce. See also caption to figure 2. 32, = B ; O ~ + B ~ o ~ , (3) the - T curves have been shifted on the p scale where 0; and 09 denote second and fourth order so that they match at 300 OK. It is seen that partial equivalent operators and B: and B: the relevant replacement in CeN3 of Ce by the non-magnetic crystal-field intensities. When considering only the
KONDO SIDE-BANDS IN CERIUM INTERMETALLIC COMPOUNDS C 1 - 427 effect of the 12 nearest AI neighbours of the Ce3+ ion
one finds, using the data tabulated in ref. [l 61 : B: = e2 F(c/a) ZA1 < r2 > &/a3
and (4)
B: = e2 Ff(c/a) ZA1 < r4 > p/a3 . ( 5 ) It is important to note that the ratio B:/@ depends on the ratio c/a through the constants F(c/a) and Ff(c/a), but is independent of the charge of the A1 ions ( I e I ZAl). In the case of CeAI,, with c/a = 0.704, F(c/a) E - 3 and Ff(c/a) 151164 (these values and the corresponding value of @/B: are at variance with those calculated by Mader and Swift [17]). With the data listed for the expectation values < r 2 >
and < r4 > (r denotes the modulus of the position vector of the 4 f electron) and for the constants a and p [16, 181 one arrives at B;/B~ E 110. This value of B~/B: corresponds to a relative position of the doublets as shown on the left hand side of figure 6, denoted by A.
FIG. 6 . -Relative position of the three Kramers doublets of the split ground multiplet of the Ce3+ ion in a hexagonal crystal-field as a function of the ratio of the crystalline-field
intensities B: and B:.
It will subsequently be shown that the behaviour of p as a function of T depends markedly on the rela- tive positions of the three doublets (Secs IV and V).
Therefore, we have considered the possible relative positions of the doublets corresponding to different values of B;/B: (see Fig. 6). In the calculations, for a given c/a ratio, viz. the ratio appropriate to CeAl,, the value of B;/B~ changes if one takes into account also the effect of neighbours of a Ce3+ ion other than the nearest AI neighbours. In that case the value of
&/B: proves to depend on the charge! of the A1 and Ce ions. In the following, it has been assumed that ZA1 = - l and Z,, = + 3 119, 201. When taking into account either the effect of nearest A1 and next- nearest Ce neighbours or of all A1 and Ce neighbours lying within spheres of 10 A or 20 a, one arrives at B;/B: - 1 300, - 490 or - 440, respectively.
In view of the insensitivity of the relative positions of the doublets with respect to BX/B~ in the range
of the AI and Ce ions which can differ appreciably from those used above and which vary with the dis- tance to the central Ce3+ ion. This means that in reality in CeAl, the relative level scheme can be any of the situations represented in figure 5.
An important pararheter in the subsequent inter- pretation ofp data (Sec. V) is the c/a ratio of the pseudo- binary compounds obtained by partial replacement of Ce in CeAI,. On the one hand substitution of La leads to a decrease, whereas on the other substitution of Y or Th leads to an increase of the c/a ratio (see Table I). Calculations on the variation of BS/B: with
c/a in the case where all neighbours within a sphere of 20 A contribute to the crystal field, show that La substitution in CeAl, produces in figure 5 a shift on the B;/B: scale to the left, while a shift in the opposite direction would be the result of Y or Th substitution.
It seems likely that in the presence of screening effects the parameter as a function of cla behaves qualitatively in the same manner.
IV. Kondo side-bands. - An essential feature of Kondo exchange scattering [5] is the occurrence of spin flip of the localized spin moment in the inter- mediate state of the scattering process : the z-compo- nent of the localized spin moment in the intermediate state differs by one quantum from the corresponding initial and/or final state. In a normal Kondo system the localized spin, e. g. the doublet I _f 4 >, can
flip in the paramagnetic range without energy expen- diture. If the S-f exchange integral has a negative sign, the transition probability l/z, at T = 0 for scattering of the conduction electrons with energy Ek shows a positive divergence proportional to - In I Ek - E, I.
In other words, normal Kondo exchange scatter~ng leads to a sharp resonance peak in the transition pro- bability at T = 0 as a function of E, centred around E, = E,. This peak broadens at T # 0 and with increasing temperature eventually disappears due to the diffuseness of the Fermi level.
In the present case the situation is more compli- cated. Spin-flip scattering involves either the doublet ( + + > solely or each of the pairs of doublets
< f + 1 , l + + > and < f $ 1 , ( & S > . In the absence of a crystalline electric field and of a magnetic field each of these scattering processes is elastic and leads to a resonance of zk for E, = E,. In the presence of a hexagonal crystal field, scattering processes invol- ving the pairs of doublets become inelastic. Maran- zana [l41 has shown that for each pair of doublets Kondo exchange-scattering leads to a sharp reso- nance peak at T = 0 in the transition probability l/zk at an energy shifted away from EF of an amount equal to the energy separation of the pair. These resonance peaks are designated as c< Kondo side- bands >>. If the splitting of the Ce3+ manifold is such that the doublet I + 3 > is the highest level, the rela- xation time at T = 0 does not even show the normal B~"/B: 5 - 100 (see Fig. 6), it can be said that the Kondo resonance at Ek = EF because, according to extra effect of neighbours other than the nearest A1 the Boltzmann distribution, the doublet is unoccupied neighbours leads to the level scheme as indicated on at very low temperatures.
the right hand side of figure 6, denoted by E. In the As a consequence of the resonance peaks in the above considerations no account has been taken of relaxation time at energies E, # EF one expects intui- screening effects. These effects lead to effective charges tively the existence of two peaks in the p - T curve
C 1 - 428 H. J. VAN DAAL, F. E. MARANZANA A N D K. H. J. BUSCHOW at temperatures equivalent to the distances of the
energy levels in the two pairs of doublets. Numerical calculations of p, based on the usual formula for the conductivity
have been made for different positions of the doublets.
In eq. (6), ,f, is the Fermi-Dirac distribution function.
The approximations used in the evaluation of eq. (6) have been indicated in ref. [14]. The results of these calculations are presented in section V. It can be seen there that the peaks in the p - T curve are usually shifted away, on the temperature scale, from the intuitively expected values. This shift is mainly brought about by the derivative of the Fermi-Dirac distribu- tion function (see Eq. 6) and by the temperature dependence of the relaxation time through the Boltz- mann distribution over the doublets. The higher the temperature at which they occur, the broader are the peaks. This is due to the Fermi level becoming diffuse at higher temperatures.
V. Interpretation. - The experimental p data pre- sented in sec. I1 (Fig. 2-4) will now be interpreted on the basis of the Kondo side-bands model (see Sec. IV), employing the results of the calculations made on the crystal-field splitting of the Ce3+ ground multiplet (see Sec. 111). In the calculations it has been assumed that the total splitting is a constant, equivalent to 50 OK.
The first-order contribution to p as a function of T, valid in the absence of a Kondo effect, has been pre- sented in figures 7 and 8. The calcuIations have been
Plarbitrary units 1
K l l
RG. 7. - The calculated temperature dependence of the resis- tivity due to first-order S-f exchange scattering for different relative positions of the three doublets I & 3 >, I $ > and I -+ + >, successively indicated (in OK) for each situation by the figures in parentheses. The results correspond to the crystal
field splitting range ABC indicated in figure 6.
FIG. 8. - See caption to figure 7. The results of this figure correspond to the range CDE of figure 6.
made for different relative positions of the three dou- blets corresponding t o values of B:/B: (see Fig. 6 ) in the ranges ABC (Fig. 7) and CDE (Fig. 8). In these figures, just as in the following figures 9 and 10, only the relative differences in temperature behaviour of p
FIG. 9. -The combined influence of first-order and second- order s-f exchange scattering on the resistivity, calculated for the different relative positions of the three doublets as indicated.
See also caption to figure 7.
KONDO SIDE-BANDS IN CEREUM INTERMETALLIC COMPOUNDS C 1 - 429
for the various crystal-field splittings have relevance.
It is seen that in general p increases continuously with temperature except in a small region beyond D.
In that region a minimum occurs in the p - T curve without the presence of a Kondo effect. An expla- nation of this anomaly can be found when one consi- ders the influence on p behaviour of the competing elastic and inelastic first-order scattering processes.
In the region beyond D the ( _f 4 > doublet is the
FIG. 10. - See caption to figure 9. The results of this figure correspond to the range CDE of figure 6.
lowest state, separated by a small energy distance from the I + 3 > doublet, while the I + 3 > doublet
is at a relatively high position (e. g. the situation (0, 1, 50) in figure 8). With increasing temperature, starting at T = 0, the initial increase of p is due to the occurrence of inelastic processes, involving tran- sitions with spin flip between the doublets I & 4 >
and I + 3 >, accompanied by spin flip of the conduc- tion electrons. At the same time, however, population of the ( _f 3 > doublet leads to a decrease in the effectiof elastic scattering processes on p. This is because for each doublet, elastic scattering is proportional to the square of the expectation value of the z-compo- nent of the magnetic moment. The competing effect of elastic and inelastic scattering leads to a maximum in the p - T curve. In the partially opposite case (viz. (1, 0, 50), see Fig. S), a maximum in the p - T curve is absent because at low temperatures elastic as well as inelastic scattering processes lead to an increase of p with T. In all cases of figure 8, at higher temperatures, the increase of p with T is due to the dominant influence of inelastic scattering processes, involving the pair of doublets < t- -3 I , I + 3 >.
No maximum at low temperatures results in the first-
order p - T curve in the region beyond B, where scattering is determined by the I -L % > doublet
as the lowest state and the I t_ + > doublet at a small energy distance (see Fig. 7). This is because in this situation the decrease of p with T due to elastic processes is small compared to the increase due to inelastic spin-flip processes.
Results of calculations of the combined first and second-order contribution to p as a function of T, corresponding to the various relative positions of the doublets, are given in figure 9 (region ABC) and figure 10 (region CDE). The calculations have been performed at the limit of the validity of perturbation theory. At this limit first and second-order contri- butions to the relaxation time for scattering of the conduction electrons are of the same order of magni- tude. It is seen in figures 9 and 10 that above about 50 OK, p behaviour in all situations is almost the same, showing the increase with decreasing T expected in normal Kondo systems. This is because at tempera- tures sufficiently larger than the total splitting, the Ce3+ ground multiplet may be regarded as being degenerate. At very low temperatures, p increases markedly with T except in region AB and at point D, where p decreases sharply upon a decrease of T. In the latter cases the normal Kondo effect is present also at very low temperatures because the lowest state consists either of the doublet 1 f + > (region AB) or of the coinciding doublets I f 3 > and 1 f >
(point D). The effect of Kondo side-bands becomes manifest in the presence of either one broad maximum if B ~ / B : has values in the vicinity of A, C or E, or in the presence of two maxima for values of B ~ / B : near B or D. In the cases where two maxima are present, the one occurring at the lower temperature is seen to be the sharpest.
When comparing experimental and theoretical results, a considerable systematic resemblance can be seen between on the one hand the sets of experi- mental curves presented in figures 2-4 and on the other the sets oi' theoretical curves shown in figures 9 and 10, if it is assumed that in CeAl, the crystal-field splitting of the Ce3+ ground state corresponds to the region around the point C in figure 6.
It has already been pointed out that in the set of compounds Ce,-,La,A13, with increasing X, the c/a ratio decreases, leading to a shift to the left on the B;/& scale of figure 6 (See sec. 111). The successive theoretical curves in figure 9, when going from C to B, show a remarkable resemblance to the set of expe- rimental curves obtained on Ce, -,La,A13 with increa- sing x (Fig. 2). This suggests that the crystal-field splitting in CeA1, changes indeed following an increa- sing replacement of Ce by La from a situation near to point C to a situation near to point B.
Partial replacement of Ce by Y leads to an increase of the c/a ratio and to a corresponding shift to the right on the &/B: scale of figure 6 . In the successive theoretical curves of figure 10, when going from C to D, first one broad maximum remains at about the same temperature. Only when approaching D does the presence of a second maximum at lower temperatures become clearly noticeable. In a limited region beyond the point D this low-temperature maximum becomes
C 1-430 H. J. VAN DAAL, F. E. MARANZANA AND K. H. J. BUSCHOW enhanced. The set of experimental curves for
Ce,,,Y,Al, with increasing x shows essentially the same trend of temperature dependence as sketched above (Fig. 3). With x increasing from 0.00 to 0.20, the temperature dependence of p does not change appreciably, p behaviour being characterized by one broad maximum. The presence of a second low- temperature maximum becomes visible for X > 0.33.
It seems plausible then that substitution in CeA1, of Y for Ce leads to a change in the relative positions of the doublets from a situation near to point C to a situation slightly beyond D.
Confirmation of the above explanation of the change in p behaviour following dilution of CeAI, with YA1, can be obtained from the data for the set of Ce, -,Th,Al, although complications arise from compounds, the additional effect of Th substitution on the electronic density-of-states. Replacement in CeAI, of Ce by Th, just as by Y, leads to an increase of the c / a ratio, albeit smaller than in the case of Y substitution. When disregarding variations in the densi- ty-of-states, one may thus expect in the set of experi- mental curves for Ce,-,Th,Al,, with increasing X, a trend of the temperature dependence similar to that of the theoretical set when going from C to or beyond D (see Fig. 10). The general trend in both sets of curves is indeed the same : with increasing X, likewise when going from C to D, there appears a second low-temperature maximum. In a further comparison, allowance should be made for the varia- tion of N(EF). It has been suggested (see Sec. 11) that in Gel-,ThxA13 with X increasing from 0.00 to 0.30, N(EF) either goes through a maximum or is a constant, but then for X > 0.3 decreases appreciably. One might thus expect that Kondo side-band effects become enhanced in the region 0 < x < 0.3 but weakened for x increasing above 0.3. It is interesting to note that the experimental curve for the case of 67 % Th shows very much the same temperature dependence as the theoretical first-order curves for situations immediately beyond point D, e. g. The curve marked (0, 2, 50) in figure 8. This similarity might suggest that such a Th concentration brings indeed the crystal-field splitting into the region beyond D and moreover that, due to the corresponding lowering of N(EF), Kondo side- band effects have totally disappeared. However, when considering the order of magnitude of first and second- order effects, it appears that the above mentioned similarity can only be accidental.
Refer [l] DAAL (H. J. van) and BUSCHOW (K. H. J.), Proc.
Colloque International du CNRS, no 180, Les 616ments des Terres Rares, Paris-Grenoble, mai 1969, tome 11, 551-557, 1970.
[21 BUSCHOW (K. H. J.) and DAAL (H. J. van), Phys.
Rev. Letters, 1969, 23, 408.
131 DAAL (H. J. van) and BUSCHOW (K. H. J.), Phys.
Letters, 1970, 31 A, 103.
[4] BUSCHOW (K. H. J.) and DAAL (H. J. van), Solid State Comm., 1970, 8, 363.
[51 KONDO (J.), Progr. of Theor. Phys., Kyoto, 1964, 4 . 194.
[6] N A G ~ O K A W.), Phys. Rev., 1965, 138, A 11 12.
VI. Summary. - It is proposed in this paper that in CeA1, the hexagonal crystal field splits the Ce3+ ground multiplet in such a way that the doublet
I _+ 2 > is the lowest state and either I + 3 > or
I + 4 > the highest one. The total splitting is pro- posed to be of the order of 50 OK. The relative position of the levels is such that Kondo side-band effects lead in the resistivity to two resonant peaks, merging together to one broad peak, the two peaks corres- ponding to the pairs of doublets < + Q 1, ( + 4 >
and < f 3 I, I .f 4 >. The normal Kondo effect due to the ( $. 3 > state is quenched at temperatures which are low in comparison to the total splitting because there this state is practically unoccupied.
Partial replacement of Ce in CeAl, by La, due t o a decrease of the average c / a ratio, leads to a re- arrangement of doublet levels such that the I _+ + >
doublet ultimately approaches closely the ground state 1 + 9 >, while the position of the upper
I f 5 > doublet remains essentially unaltered. This leads to two well-resolved Kondo side-band peaks in the p - T curve, a sharp peak a t low temperatures corresponding to the pair of doublets < + + 1,
1 + 5 > at a small energy distance, and another broad one a t higher temperatures, determined by the pair of doublets < + + 1, ( + 5 > at a larger energy distance.
Partial replacement of Ce by Y or Th, due to an increase of the average c / a ratio, ultimately leads either to the situation where the I + 3 > doublet is the lowest state with the 1 _f + > doublet at a small energy distance (the case of Y) or to the reverse of this situation (the case of Th), while in all cases. The l]+ 4 >
doublet is the highest state. This leads, just as in the case of La substitution, to the occurrence of two reso- nant peaks in the p - T curve. The low-temperature peak corresponds to the pair of doublets < + $ I,
( + 3 > at short energy distance, the high-temperature one to the pair of doublets < + 9 I, ( f + > at a
larger energy distance. In the set of compounds Ce,-,Th,Al,, with increasing X, Kondo side-band effects are enhanced in the lower range of X values but weakened in the higher range due to the corresponding variation of the electronic density of-states at the Fermi level.
AcknowIedgements. - Thanks are due to P. B.
van Aken and to Dott. P. Bianchessi for their assis- tance in the performance of the measurements and the numerical calculations.
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