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Electronic specific heat of heavy fermion systems
Solid State Communications, Vol. 85, No. 3, pp. 239-242, 1993. Printed in Great Britain.
0038--1098/93 $6.00 + .00 Pergamon Press Ltd
ELECTRONIC SPECIFIC HEAT OF HEAVY FERMION SYSTEMS S. Panwar and I. Singh Department of Physics, University of Roorkee, Roorkee - 247 667, India
(Received 14 May 1992 by G. Bastard) We study the electronic specific heat of the heavy-fermions and mixedvalence systems using the periodic Anderson model. A variational method has been used in these calculations. The electronic specific heat increases rapidly in the low temperature region, shows a maxima and then decreases. This maxima may well be associated to the formation of a narrow quasiparticle band near Fermi level and to the narrow gap in the energy spectrum evident from the density of states curves. 1. INTRODUCTION
Recently, we used a variational method to study the ground state and thermodynamic properties of the periodic Anderson model (PAM) [12, 13]. In this paper we use this variational method to study electronic specific heat of heavy fermions within the PAM. The details of the variational method may be found in Panwar and Singh [12]. In Section 2 we give basic formulation for specific heat. In Section 3 we discuss our results.
THE EXPERIMENTAL results of electronic specific heat C(T) and electrical resistivity p(T) of many heavy fermion systems, e.g. Ce compounds CePd3, CeA13 etc. at low temperatures show anomalous behaviour (just like anomalous behaviour of magnetic susceptibility at low T~ in that C ( T ) / T = ~ + A T 2 and p(T) ~_ p(O) + B T ~ with A and B positive (see, e.g. [1-4]). The low temperature electronic specific heat C(T) of systems like CeA13, CeCu6, etc. show an 2. BASIC FORMULATION enormous enhancement of the specific heat coefficient 7(T) (= C ( T ) / T ) , suggesting a very heavy effective We consider orbitally nondegenerate periodic mass of Fermi liquid state. The temperature- Anderson model which is described by the dependence of 7(T) of these compounds show a Hamiltonian peak 7max at a (relative low) temperature, p(T) + increases with increasing temperature in the very low ktr ia temperature region (known as Fermi liquid region or U coherence region), reaching a maximum and then -- E Vk(c~bk~ + h.c.) + - ~ E n j a fn j _fa , (1) decreasing slowly like In T (known as Kondo region k~r Ja or independent-impurity region) [5-7]. These systems, thus, show a sort of cross-over from low-T where symbols have their usual meanings. In the weak interaction case, where Coulomb coherent Fermi-liquid behaviour to the high-T dense Kondo behaviour. In the recent past it has been interaction U is small, all the three configurationsf°, suggested that the low-temperature coherent Fermi f 1 and f 2 are energetically possible. However, in the liquid phase may very well be represented by the high interaction case (which we are considering here), periodic Anderson model (PAM) where one con- U is very large and the probability o f f 2 configuration siders the coherent hybridization between conduc- is very small. The variational wavefunction which tion-states and the f-states on all N sites [8-13]. The projects f 2 configuration out, may be written as appearance of a possible pseudo gap near Fermi (Panwar and Singh [12]) energy due to the coherent hybridization has been f + (2) attributed to the cause of anomalous low-tempera- [~b) = H [1 + Ak~(1 -- n _,~)bk,, Ck,,llr ), ks ture behaviour of these systems [14]. However, it is not clear, precisely how the heavy fermion systems, where [F ) = IIk(k~Ck~[O + ) is the Fermi sea of conwhich at high temperatures behave like a collection of duction electrons and Aka the variational parindependent magnetic ions, start behaving like ameters. It can be seen that the resultant states are coherent Fermi liquid at very low temperatures. in the form of two quasi-particle bands; the lower ( - ) 239
240
ELECTRONIC SPECIFIC HEAT OF HEAVY F E R M I O N SYSTEMS
and upper (+) branches of quasiparticle spectra are given by
E:,~=½[(ek+~fPf)+x/(ek-~fPf)2+4V2p:].
Vol. 85, No. 3
density of perturbed f-states. Then
NC(Eka) dEk~r = NCa(~k)dek nCkcr
(3) or
The Ak'S are given by A~:~-
1 2Vkefi
[(ek
-
efPf) q: x/(ek - efPf)2 + 4V2kp~]
NS(Ek
) -
(dEkddek)
=
N~(ek) 1 (dEka/dek) 1 + AkaP/2 2" (7)
(4)
Similarly
where
Nf(Eka) =
Pf = (1 - nf~).
(5)
At T = 0 K, number of conduction electrons and f-electrons are given by (taking total number of electrons such that Fermi level lies in the lower band)
Akc~Pf NC(ek) 2 3 2 (dEkddek) 1 + Ak~P f2"
(8)
Here the factor dEk~/de k for the lower as well as upper quasi-particle bands may be evaluated from equation (3),
1
n~,~ = 1 + (A~-,~)2P~
nf'~ - 1 + (Ak,,)2P~"
(6)
dE~o _
±(E~ -
EfPf)
--
iPi) 2 +
(9)
4w2uElU2 " , k'I
2.1. Density of states Let )~(ek) denote the density of the unperturbed conduction band with spin tr, Nt(Ek~) the total density of quasiparticle states, N~(Eko) the density of perturbed conduction states and Nf(Ek~) the
2.2. Electronic specific heat At finite temperature, the number of conduction electrons, number off-electrons and the ground state 16
16
(a)
(b)
1"-
12
IC
1C
Nt(E)
Nf(E) 8
8
6
0
-0.7
-
0.5
- °'~ F
- o.~
E (¢v)
o.1
o.3
0
-o,5
-o.3
-o.~
EF
o.1
0-3
o5
E (ev)
Fig. 1. Variation off-state density of states Nf(E) with quasiparticle energy E for flat conduction band case with V = 0.25. (a) ~y = -0.4, (b) ~f = 0.0.
Vol. 85, No. 3
E L E C T R O N I C SPECIFIC HEAT OF HEAVY F E R M I O N SYSTEMS
120
(ct)
241
(a)
50
8C
C
T
C
.-f-
t.0
4C I 400 I /400
I 800
T2
I 1200
I 1600
I 800
I 2000
32ooj
I 1200 T2
I 1600
240C I
(b)
I 2000
(b)
C 160(
2400 C
80( 1600 0
I 100
0
I 200
800
I 100
| 200
I 300
l 400
I 500
T
Fig. 2. (a) Linear coefficient of specific heat 7(= C(T)/T) as a function of temperature for tight binding conduction band case. V = 0.25. Curve I is for ef = -0.4, Curve II for cf = -0.2 and Curve III for Cf = 0.0. (b) Specific heat Cv for tight binding conduction band case. V = 0 . 2 5 . Curve I for Cf = -0.4, Curve II for cf = -0.2 and Curve III for ~f= 0.0.
I 300 T
i /-,00
I 500
Fig. 3. (a) Linear coefficient of specific heat 7(= C(T)/T) as a function of temperature for tight binding conduction band with ~f = -0.2• Curve I for V = 0.25 and Curve II for V = 0.5. (b) Specific heat Cv for tight binding conduction band ef = -0.2.Curve I for V = 0.25 and Curve II for V = 0•5. bands• It is given by
Cv -
0(g)
0 .~
OT - O-T
/co
_ [(E/o - #)fk~ + (E/+ - #)f~](141
Equation (14) may be written as energy are given by
Cv = E
f ~ " ~ (E['o - #) + f ~
(E~o- #)
ka
1 + (A~-o)2P~ + 1 + (Ako)
no = ~
PfJ
[ l + ( A Z a ) 2 p j + r - - ~(Ako) + P?J, (E) = E[(E~
- #)fk~ + (E+o - #)f~].
. Of~ + "E + "oft+ ] +(EL-#)-'~ L ko--#)--~J"
(121
The first two terms in the summation give the temperature dependence of quasiparticle bands, which gives a very small contribution to specific heat in the non-magnetic case. We have neglected these two terms. The dominating terms are the 3rd and 4th terms which give temperature dependence of Fermi function. Hence C~ takes the final form
(13)
Cv= y ~ ( E ~ - # ) ~ +
(11)
ka
Here Fermi functions f ~ and f ~ are given by fg~=
1 exp[3(E~:o-#)]+ 1
•
(15)
# is the chemical potential and/3 = 1/kBT. The electronic specific heat Cv is obtained by differentiating energy ( E ) with respect to temperature T. The total specific heat gets the contribution from both the lower as well as the upper quasiparticle
ko
Off~+ y~(E~+-#l-~-f.
(16)
ko
3. RESULTS AND DISCUSSIONS In these calculations we have considered two forms of conduction states (i) the flat conduction
242
ELECTRONIC SPECIFIC HEAT OF HEAVY FERMION SYSTEMS
band and (ii) the tight binding band. Both these bands have bandwidth 2D = 2.0eV. With their centres at 0.0 eV. Total number of electrons per site (n c + n f ) has been taken to be 1.5. Figures l(a) and (b) show f-state density of states (for the flat conduction band states) as a function of energy E. Figure 1(a) corresponds to ef = -0.4 while Fig. 1 (b) to ef = 0.0 eV. It is clear from the figures that the fspectral density is concentrated around E = EF and consists of two sharp peaks separated by a narrow gap. Figures 2(a) and (b) show the variation of linear coefficient of specific heat "Y[= C(T)/T] and specific heat Cv for different positions of f-level el, i.e. ¢f = -0.4, -0.2, 0.0 eV for tight binding conduction band. In Figs 3(a) and (b) we show variation of"/and specific heat Cv for different values of hybridization V with fixed ef = -0.2 eV. In 7 vs T 2 curve at low temperatures, 7 increases as temperature increases, and has a peak around 400K 2 beyond that 7 decreases. This maxima in 7 is the characteristics of the many heavy fermion systems. Also the specific heat Cr increases linearly with temperature with a broad peak around 50 K, beyond which it decreases. As f-level ef shifts downward with respect to Fermi energy eF, Cv and 7 increase more sharply in the lowT regime as a result of which the peaks in Cv and 7 become more sharper. When V is increased, the peak broadens and disappears in some cases. The variational approach along with coherent hybridization scheme adopted in our formualtion, explains the maxima in 7 or Cv at low temperature as the f-level contributes a sharp peak in the electronic density of states in the vicinity of Fermi level. Now as temperature increases from T = 0, the quasiparticle states are excited to higher energy states available in the lower band at low temperatures. On further increasing the temperature, the quasiparticles are excited to upper quasiparticle band which is separated by a narrow gap A _ VE/D. Due to this, Cv has a peak at low temperature. The narrow gap is due to the large correlation in f-states. Hence, the maxima in 7 or Cv is associated with the formation of a narrow band near eF and a narrow gap. As f-level goes down, the gap becomes more narrower and we have a more sharp peak at low temperature. While when V is increased, the gap becomes broader. The effect is the shifting of the peak towards high temperature regime. The systematic study of Bredl et al. [14] shows that for the periodic system of Ce ions in CeCu2Si 2 and CeAl3 the 7 increases as temperature increases in
Vol. 85, No. 3
the low-temperature region, shows a peak and then decreases with increasing temperature. In their study the maxima occurs around 0.SK. Since in our calculations we have not taken realistic bands (and considered only simple flat and tight binding bands) we get maxima in 7 around 20 K. Bredl et al. [14] in their study found that if the periodicity of Ce ions is lost by substituting (up to 20 at.%) Y or La in place of Ce, the peak in 7(T) disappears and the alloy shows 3' vs T behaviour characteristic of spin-l/2 Kondo impurity system, i.e. the monotonic decrease of 7 with increasing temperature. Our results of 7 vs T 2 match qualitatively with the experimental results of Trainer et al. [15] for the compound NpSn 3. Our specific heat results correspond qualitatively to the experimental results of Mattens [16] for the specific heat difference of the compounds YbCuA1 and LuCuA1.
Acknowledgements - - One of the authors (S.P.) is thankful to the Council of Scientific and Industrial Research (India) for financial support. I.S. is thankful to the Department of Science and Technology (India) for financial support. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
G.R. Stewart, Rev. Mod. Phys. 56, 755 (1984). T. Kasuya & T. Saso (eds), Theory of Heavy Fermions and Valence Fluctuations. Springer (1985). P. Fulde, J. Keller & G. Zwichnage, Solid State Physics (Edited by H. Ehrenreich & D. Turnbull), 41, 1. Academic Press (1988). U. Rauchschwalbe, Physica 147B, 1 (1987). T. Koyama & M. Tachiki, Phys. Rev. B36, 437 (1987). K.F. Quader, K.S. Bedell & Q.E. Brown, Phys. Rev. B36, 156 (1987). H. Kaga, H. Kubo & T. Fujiwara, Phys. Rev. B37, 341 (1988). S. Doniach, Physica 91B, 231 (1977). H.J. Leder & B. Muhlschlegel, Z. Phys. B29, 341 (1978). K. Yamada & K. Yoshida, Prog. Theor. Phys. 76, 621 (1986). K. Yamada, K. Okada & K. Yoshida, Prog. Theor. Phys. 77, 1097, 1297 (1987). S. Panwar & I. Singh, Solid State Commun. 72, 711 (1989). S. Panwar & I. Singh, Phys. Status Solidi (b) 168, 583 (1991). C.D. Bredl, S. Horn, F. Steglich, B. Luthi & R.M. Martin, Phys. Rev. Lett. 52, 1982 (1984). R.J. Trainor, Brodsky, B.D. Dunlap & G.K. Shenoy, Phys. Rev. Lett. 37, 1511 (1976). W.C.M. Mattens, Ph.D. Thesis. Univ. of Amsterdam (1980) (unpublished).
0038--1098/93 $6.00 + .00 Pergamon Press Ltd
ELECTRONIC SPECIFIC HEAT OF HEAVY FERMION SYSTEMS S. Panwar and I. Singh Department of Physics, University of Roorkee, Roorkee - 247 667, India
(Received 14 May 1992 by G. Bastard) We study the electronic specific heat of the heavy-fermions and mixedvalence systems using the periodic Anderson model. A variational method has been used in these calculations. The electronic specific heat increases rapidly in the low temperature region, shows a maxima and then decreases. This maxima may well be associated to the formation of a narrow quasiparticle band near Fermi level and to the narrow gap in the energy spectrum evident from the density of states curves. 1. INTRODUCTION
Recently, we used a variational method to study the ground state and thermodynamic properties of the periodic Anderson model (PAM) [12, 13]. In this paper we use this variational method to study electronic specific heat of heavy fermions within the PAM. The details of the variational method may be found in Panwar and Singh [12]. In Section 2 we give basic formulation for specific heat. In Section 3 we discuss our results.
THE EXPERIMENTAL results of electronic specific heat C(T) and electrical resistivity p(T) of many heavy fermion systems, e.g. Ce compounds CePd3, CeA13 etc. at low temperatures show anomalous behaviour (just like anomalous behaviour of magnetic susceptibility at low T~ in that C ( T ) / T = ~ + A T 2 and p(T) ~_ p(O) + B T ~ with A and B positive (see, e.g. [1-4]). The low temperature electronic specific heat C(T) of systems like CeA13, CeCu6, etc. show an 2. BASIC FORMULATION enormous enhancement of the specific heat coefficient 7(T) (= C ( T ) / T ) , suggesting a very heavy effective We consider orbitally nondegenerate periodic mass of Fermi liquid state. The temperature- Anderson model which is described by the dependence of 7(T) of these compounds show a Hamiltonian peak 7max at a (relative low) temperature, p(T) + increases with increasing temperature in the very low ktr ia temperature region (known as Fermi liquid region or U coherence region), reaching a maximum and then -- E Vk(c~bk~ + h.c.) + - ~ E n j a fn j _fa , (1) decreasing slowly like In T (known as Kondo region k~r Ja or independent-impurity region) [5-7]. These systems, thus, show a sort of cross-over from low-T where symbols have their usual meanings. In the weak interaction case, where Coulomb coherent Fermi-liquid behaviour to the high-T dense Kondo behaviour. In the recent past it has been interaction U is small, all the three configurationsf°, suggested that the low-temperature coherent Fermi f 1 and f 2 are energetically possible. However, in the liquid phase may very well be represented by the high interaction case (which we are considering here), periodic Anderson model (PAM) where one con- U is very large and the probability o f f 2 configuration siders the coherent hybridization between conduc- is very small. The variational wavefunction which tion-states and the f-states on all N sites [8-13]. The projects f 2 configuration out, may be written as appearance of a possible pseudo gap near Fermi (Panwar and Singh [12]) energy due to the coherent hybridization has been f + (2) attributed to the cause of anomalous low-tempera- [~b) = H [1 + Ak~(1 -- n _,~)bk,, Ck,,llr ), ks ture behaviour of these systems [14]. However, it is not clear, precisely how the heavy fermion systems, where [F ) = IIk(k~Ck~[O + ) is the Fermi sea of conwhich at high temperatures behave like a collection of duction electrons and Aka the variational parindependent magnetic ions, start behaving like ameters. It can be seen that the resultant states are coherent Fermi liquid at very low temperatures. in the form of two quasi-particle bands; the lower ( - ) 239
240
ELECTRONIC SPECIFIC HEAT OF HEAVY F E R M I O N SYSTEMS
and upper (+) branches of quasiparticle spectra are given by
E:,~=½[(ek+~fPf)+x/(ek-~fPf)2+4V2p:].
Vol. 85, No. 3
density of perturbed f-states. Then
NC(Eka) dEk~r = NCa(~k)dek nCkcr
(3) or
The Ak'S are given by A~:~-
1 2Vkefi
[(ek
-
efPf) q: x/(ek - efPf)2 + 4V2kp~]
NS(Ek
) -
(dEkddek)
=
N~(ek) 1 (dEka/dek) 1 + AkaP/2 2" (7)
(4)
Similarly
where
Nf(Eka) =
Pf = (1 - nf~).
(5)
At T = 0 K, number of conduction electrons and f-electrons are given by (taking total number of electrons such that Fermi level lies in the lower band)
Akc~Pf NC(ek) 2 3 2 (dEkddek) 1 + Ak~P f2"
(8)
Here the factor dEk~/de k for the lower as well as upper quasi-particle bands may be evaluated from equation (3),
1
n~,~ = 1 + (A~-,~)2P~
nf'~ - 1 + (Ak,,)2P~"
(6)
dE~o _
±(E~ -
EfPf)
--
iPi) 2 +
(9)
4w2uElU2 " , k'I
2.1. Density of states Let )~(ek) denote the density of the unperturbed conduction band with spin tr, Nt(Ek~) the total density of quasiparticle states, N~(Eko) the density of perturbed conduction states and Nf(Ek~) the
2.2. Electronic specific heat At finite temperature, the number of conduction electrons, number off-electrons and the ground state 16
16
(a)
(b)
1"-
12
IC
1C
Nt(E)
Nf(E) 8
8
6
0
-0.7
-
0.5
- °'~ F
- o.~
E (¢v)
o.1
o.3
0
-o,5
-o.3
-o.~
EF
o.1
0-3
o5
E (ev)
Fig. 1. Variation off-state density of states Nf(E) with quasiparticle energy E for flat conduction band case with V = 0.25. (a) ~y = -0.4, (b) ~f = 0.0.
Vol. 85, No. 3
E L E C T R O N I C SPECIFIC HEAT OF HEAVY F E R M I O N SYSTEMS
120
(ct)
241
(a)
50
8C
C
T
C
.-f-
t.0
4C I 400 I /400
I 800
T2
I 1200
I 1600
I 800
I 2000
32ooj
I 1200 T2
I 1600
240C I
(b)
I 2000
(b)
C 160(
2400 C
80( 1600 0
I 100
0
I 200
800
I 100
| 200
I 300
l 400
I 500
T
Fig. 2. (a) Linear coefficient of specific heat 7(= C(T)/T) as a function of temperature for tight binding conduction band case. V = 0.25. Curve I is for ef = -0.4, Curve II for cf = -0.2 and Curve III for Cf = 0.0. (b) Specific heat Cv for tight binding conduction band case. V = 0 . 2 5 . Curve I for Cf = -0.4, Curve II for cf = -0.2 and Curve III for ~f= 0.0.
I 300 T
i /-,00
I 500
Fig. 3. (a) Linear coefficient of specific heat 7(= C(T)/T) as a function of temperature for tight binding conduction band with ~f = -0.2• Curve I for V = 0.25 and Curve II for V = 0.5. (b) Specific heat Cv for tight binding conduction band ef = -0.2.Curve I for V = 0.25 and Curve II for V = 0•5. bands• It is given by
Cv -
0(g)
0 .~
OT - O-T
/co
_ [(E/o - #)fk~ + (E/+ - #)f~](141
Equation (14) may be written as energy are given by
Cv = E
f ~ " ~ (E['o - #) + f ~
(E~o- #)
ka
1 + (A~-o)2P~ + 1 + (Ako)
no = ~
PfJ
[ l + ( A Z a ) 2 p j + r - - ~(Ako) + P?J, (E) = E[(E~
- #)fk~ + (E+o - #)f~].
. Of~ + "E + "oft+ ] +(EL-#)-'~ L ko--#)--~J"
(121
The first two terms in the summation give the temperature dependence of quasiparticle bands, which gives a very small contribution to specific heat in the non-magnetic case. We have neglected these two terms. The dominating terms are the 3rd and 4th terms which give temperature dependence of Fermi function. Hence C~ takes the final form
(13)
Cv= y ~ ( E ~ - # ) ~ +
(11)
ka
Here Fermi functions f ~ and f ~ are given by fg~=
1 exp[3(E~:o-#)]+ 1
•
(15)
# is the chemical potential and/3 = 1/kBT. The electronic specific heat Cv is obtained by differentiating energy ( E ) with respect to temperature T. The total specific heat gets the contribution from both the lower as well as the upper quasiparticle
ko
Off~+ y~(E~+-#l-~-f.
(16)
ko
3. RESULTS AND DISCUSSIONS In these calculations we have considered two forms of conduction states (i) the flat conduction
242
ELECTRONIC SPECIFIC HEAT OF HEAVY FERMION SYSTEMS
band and (ii) the tight binding band. Both these bands have bandwidth 2D = 2.0eV. With their centres at 0.0 eV. Total number of electrons per site (n c + n f ) has been taken to be 1.5. Figures l(a) and (b) show f-state density of states (for the flat conduction band states) as a function of energy E. Figure 1(a) corresponds to ef = -0.4 while Fig. 1 (b) to ef = 0.0 eV. It is clear from the figures that the fspectral density is concentrated around E = EF and consists of two sharp peaks separated by a narrow gap. Figures 2(a) and (b) show the variation of linear coefficient of specific heat "Y[= C(T)/T] and specific heat Cv for different positions of f-level el, i.e. ¢f = -0.4, -0.2, 0.0 eV for tight binding conduction band. In Figs 3(a) and (b) we show variation of"/and specific heat Cv for different values of hybridization V with fixed ef = -0.2 eV. In 7 vs T 2 curve at low temperatures, 7 increases as temperature increases, and has a peak around 400K 2 beyond that 7 decreases. This maxima in 7 is the characteristics of the many heavy fermion systems. Also the specific heat Cr increases linearly with temperature with a broad peak around 50 K, beyond which it decreases. As f-level ef shifts downward with respect to Fermi energy eF, Cv and 7 increase more sharply in the lowT regime as a result of which the peaks in Cv and 7 become more sharper. When V is increased, the peak broadens and disappears in some cases. The variational approach along with coherent hybridization scheme adopted in our formualtion, explains the maxima in 7 or Cv at low temperature as the f-level contributes a sharp peak in the electronic density of states in the vicinity of Fermi level. Now as temperature increases from T = 0, the quasiparticle states are excited to higher energy states available in the lower band at low temperatures. On further increasing the temperature, the quasiparticles are excited to upper quasiparticle band which is separated by a narrow gap A _ VE/D. Due to this, Cv has a peak at low temperature. The narrow gap is due to the large correlation in f-states. Hence, the maxima in 7 or Cv is associated with the formation of a narrow band near eF and a narrow gap. As f-level goes down, the gap becomes more narrower and we have a more sharp peak at low temperature. While when V is increased, the gap becomes broader. The effect is the shifting of the peak towards high temperature regime. The systematic study of Bredl et al. [14] shows that for the periodic system of Ce ions in CeCu2Si 2 and CeAl3 the 7 increases as temperature increases in
Vol. 85, No. 3
the low-temperature region, shows a peak and then decreases with increasing temperature. In their study the maxima occurs around 0.SK. Since in our calculations we have not taken realistic bands (and considered only simple flat and tight binding bands) we get maxima in 7 around 20 K. Bredl et al. [14] in their study found that if the periodicity of Ce ions is lost by substituting (up to 20 at.%) Y or La in place of Ce, the peak in 7(T) disappears and the alloy shows 3' vs T behaviour characteristic of spin-l/2 Kondo impurity system, i.e. the monotonic decrease of 7 with increasing temperature. Our results of 7 vs T 2 match qualitatively with the experimental results of Trainer et al. [15] for the compound NpSn 3. Our specific heat results correspond qualitatively to the experimental results of Mattens [16] for the specific heat difference of the compounds YbCuA1 and LuCuA1.
Acknowledgements - - One of the authors (S.P.) is thankful to the Council of Scientific and Industrial Research (India) for financial support. I.S. is thankful to the Department of Science and Technology (India) for financial support. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
G.R. Stewart, Rev. Mod. Phys. 56, 755 (1984). T. Kasuya & T. Saso (eds), Theory of Heavy Fermions and Valence Fluctuations. Springer (1985). P. Fulde, J. Keller & G. Zwichnage, Solid State Physics (Edited by H. Ehrenreich & D. Turnbull), 41, 1. Academic Press (1988). U. Rauchschwalbe, Physica 147B, 1 (1987). T. Koyama & M. Tachiki, Phys. Rev. B36, 437 (1987). K.F. Quader, K.S. Bedell & Q.E. Brown, Phys. Rev. B36, 156 (1987). H. Kaga, H. Kubo & T. Fujiwara, Phys. Rev. B37, 341 (1988). S. Doniach, Physica 91B, 231 (1977). H.J. Leder & B. Muhlschlegel, Z. Phys. B29, 341 (1978). K. Yamada & K. Yoshida, Prog. Theor. Phys. 76, 621 (1986). K. Yamada, K. Okada & K. Yoshida, Prog. Theor. Phys. 77, 1097, 1297 (1987). S. Panwar & I. Singh, Solid State Commun. 72, 711 (1989). S. Panwar & I. Singh, Phys. Status Solidi (b) 168, 583 (1991). C.D. Bredl, S. Horn, F. Steglich, B. Luthi & R.M. Martin, Phys. Rev. Lett. 52, 1982 (1984). R.J. Trainor, Brodsky, B.D. Dunlap & G.K. Shenoy, Phys. Rev. Lett. 37, 1511 (1976). W.C.M. Mattens, Ph.D. Thesis. Univ. of Amsterdam (1980) (unpublished).