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Mixed-valent systems as Fermi liquids
Solid State Communications, Vol. 41, No. 11, pp. 853-855. 1982. Printed in Great Britain.
0038-1098/82/I 10853-03 $02.00/0 Pergamon Press Ltd.
MIXED-VALENT SYSTEMS AS FERMI LIQUIDS P. F. de Ch~tel Natuurkunding Laboratorium, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands (Received 11 November 1981 by A.R. Miedema) The perturbation treatment of the asymmetric Anderson model in the U--, oo limit is shown to lead to a description of mixed-valent compounds in terms of Landau's Fermi liquid theory. The quasi-particle interaction is independent of spin, and hence the ratio between the low-temperature susceptibility and specific heat is the same as in a Fermi gas. THE LOW-TEMPERATURE BEHAVIOUR of mixedvalent compounds can be briefly characterized by stating [1 ] that they act like Fermi liquids. The two outstanding features this statement is based on are: the linear temperature dependence of the specific heat and the constancy of the susceptibility far below the temperature where the latter ceases to follow the Curie-Weiss law. Both of these features are well-known consequences of Fermi statistics; for independent (noninteracting) electrons x(O)T/C = 3 ~ / n Z k 2,
(1)
where #B is the Bohr magneton and kB is Boltzmann's constant. For interacting electrons, Landau's Fermi liquid theory [2[ predicts the same qualitative behaviour, with the low-temperature susceptibility enhanced by a factor D = (1 + Zo/4) -1, and the ratio x(O)T/C correspondingly modified. Here, Z0 is the Landau interaction parameter that can be associated with exchange; for short-range (intra-atomic) interaction we have Zo = -- 4Up(ee), and D becomes the familiar exchangeenhancement factor (1 -- Up). For mixed-valent compounds, Lustfeld and Bringer [3] have derived a different relationship,
x(O)T/C = g~,uM(J + 1)/~r~k~,
(2)
which reflects the (2J + 1)-fold degeneracy of the f-electron shell. This result was found by summing the most divergent terms in an expansion of the thermodynamic potential in a perturbation series in the hybridization interaction [4]. The hybridization is tuppooed to take place between the degenerate confilluratk)n chafac. terized by J and f j , and a singiet confi~ratiom. Equation (2) is valid in the strongly mixed-valet state, where the dmJcription of a mixed-valeat eamllm~md ia terms of the asymmetric Anderson model, which repnmeats a aia#e impurity, is adequate. A R h o u ~ similar to equation (1), equation (2) does not arise as a 853
direct consequence of Fermi statistics, and hence the absence of an enhancement factor did not pose a question in the context of [3] and [4]. Recently, Newns and Hewson [5] gave a "local Fermi liquid theory" for mixed-valent systems. Their model could be better characterized as a Fermi gas theory, because it represents a system of independent electrons exposed to a localized potential which gives a sharp, (2J + 1)-fold degenerate resonance in the vicinity of the Fermi energy. The potential is supposed to represent a mixed-valent impurity, and the degenerate resonance peak in the density of states originates from the (2J + 1)-fold degenerate state of a free rare-earth ion, in a fashion similar to Anderson's model [6] of the Friedel resonance [7]. The ratio found by Newns and Hewson between the susceptibility and specific heat associated with the resonance peak obeys equation (2), which is perhaps not surprising in a Fermi gas theory which takes due account of the angular momentum and strong spin-orbit coupling in the resonant state. While it is not at all clear why there should not be an exchange enhancement of the susceptibility associated with rare-earth ions, for which U is expected to be large, equation (2) has gained some credit through its successes. Experimentally, this relationship has been found valid within 10% for a number of intermetaUic compounds involving Ce and Yb in the mixed-valent state [3, 5]. The purpose of the present paper is to show that applying to excited states the perturbation treatment Rarnakrithnan [8] developed for the ground state of the asymmetric dellmerate Amtermn model results in a Fermi liquid theory of the low-energy excitations in which equatkm (2) holds. Atmrt from demonstrating the pomibtlity of a vaniahdq exchaa~ enhancement in tim U-* -. limit, the Fermi ~ description, whida ames in a natural way from an esaentialiy configurationbated tl)l~roach, ~ves an i m ~ t into the physical orisin of tim s~'pdsmg result.
854
MIXED-VALENT SYSTEMS AS FERMI LIQUIDS
A mixed-valent impurity can be described in the configuration-based representation of the asymmetric Anderson model,
do(j) = Co(j)°+
M
(3)
+ Z Y. (vs~,noX~,oCno + h.c.). M n,o
Here the mixed-valent impurity is characterized by two configurations, the singlet 1(3)at energy eo and a (2J + 1)-fold degenerate configuration, involving one electron more, at energy ca. The states within the latter manifold are denoted by IJM),M = - - J . . . . . + J. The projection operators X/j = Ii)(/'1 act in the space of the 2 J + 2 ionic states. The matrix element VaTn.ko is the mixing amplitude between the state IJM; Cn), in which the impurity is in state IJM) and the conduction electrons in some N-electron state I~U), and the state [0; c~of)N), in which the impurity is in the singlet state 10) and the conduction electrons in the (N + 1)-electron state le~o~N)~If the degenerate configuration is eliminated, for instance by setting ea = 0% or dropping the last two terms in equation (3), the Hamiltonian becomes that of a Fermi gas, with particle energies % and appropriate creation and annihilation operators c~o and cno. When the conduction-electron system is in its ground state I~0), the shift of the energies eo and ea due to the hybridization can be calculated in first-order Brillouin-Wigner perturbation theory [4, 8, 9]: 3-, (I VaM,no1 2 ( 1 - f ~ ) Eg--e~--% dk,
(4a)
f IVa-M'n°12fff o o
(4b)
Eg--e~
=~M o d
Ey-e3
= E a o
E.~ - - C o + %
dk,
where f~' = 0(eF - - % ) is the Fermi-Dirac function at T = 0, 0 being the Heavyside step function. Due to symmetry, the summation over M in equation (4a) results in a factor 2 J +1 and the sum in equation (4b) is independent of M. The mixed-valent behaviour briefly described in the introduction is found when Eg < Eft, that is, the hybridized singlet is the lowest perturbed state. The same perturbation treatment can be applied to the excited states 10; ~N+I) and IJM; CN), where the unperturbed conduction-electron states I¢i)are characterized by the distribution of occupied states f~o. The energy of a perturbed excited state with the impurity in the singlet configuration is E~ where
•+ V
V fl V~M,kol 2(1- s %i)
dk,
(6)
k(f~o - f k °) dk. O" ¢
H = E ~nc~o~no + ,oXo, o + ~ Y x ~ , ~ k,o
Vol. 41, No. 11'
(5)
As the states modified by hybridization can be brought into a one-to-one correspondence with the unperturbed 10; ¢i), the states of an independent-electron gas, and furthermore their energy E~ = E~[~a] is given in equation (5) as a functional of the distribution function fka, the criteria for the applicability of Landau's Fermi liquid theory are satisfied, and a description of the lowenergy excitations in terms of quasiparticles can be given. The quasi-particle energies appropriate to the ground state can be obtained [2] by writing equation (5) in the form
=
+X
nosno dk,
(7)
where 6f~o = f~o --fff. Taking due account of the dependence of both the numerator and the denominator of the integrand in equation (5) on f~o, one obtains ekÜ
8Eo --
6fko Y Iv ~ = en-
no =/[Eg - e.,o -
e
n]
M
1 + X X [I V,,,,,n,o,I2 M .,J
1
(8)
o
[Eg - e~ - en']
Due to the spherical symmetry, the sum in the numerator is independent of the spin index o, and so is the quasiparticle energy, ~n,o - ~n-
(9)
An equation similar to equation (8) holds for the quasiparticle energies appropriate to excited states, but the integral becomes nontrivial if the upper indices are not kept at zero. In what follows, only the ground-state distribution function will be considered, and all upper indices will be omitted. With the constant density of states p assumes by Ramasrishnan [8], the quasiparticle energies become
~n = en +
1
Eo--ej
P Eo -- e j -- %
,
(10)
where ( 2 J + 1)A (Xaa) = ~M (Xawt,JM) = ( 2 J + 1)A--Eo + ej
(11)
is the "valence", i.e. the occupation probability of the J ¢ 0 configuration in the ground state. The incremental density of states, due to hybridization, can be obtained as ~p(en ) =
- p
d(en -- %) _ den
Eo -- ea [Eo -- e # - - % ]2"
Vol. 41, No. 11
MIXED-VALENT SYSTEMS AS FERMI LIQUIDS
The singularity at ek = Eo -- ea will be attenuated i f f ~ is replaced by the distribution function appropriate to T :/: 0. At any rate, the Fermi liquid description breaks down as soon as kBTbecomes comparable to E j --Eo, because of the non-negligible thermal occupancy of the degenerate configuration. However, in the T--> 0 limit, and for energies leh I "< E j -- Eo, equation (12) is valid. Consequently, the additional low-temperature specific heat, due to hybridization, is given by
~2
if2
Chybr. = "~k~TAp(O) = -~k~T
1
ej -- E 0
~ko
(
1
1
× ~1 -+-I2 E o - - e j - - e k
)
Eo--ej--e~' '
(14)
where
1~ = Z Z° f Iv~'~°l=(1 - ~ ) M
leo
dk.
-- ea -- ek ]
In view of equation (9), the interaction function (14) is evidently independent of the spin indices o and a'. With the constant density of states assumption, for two quasiparticles at the Fermi energy (ek = ek' = 0) equation (14) gives
f(k, o; k'o') = ((Xjj)/p) 2 2 + (Xjj) ej_Eo
~2Eo[ ~H2 [H=o
_
(13)
_ (~ko - ek)(~k'o' -- ek')
6 fk'o'
expression X(0) = -- a2Eo/aH 2 directly. To find the contribution due to the vicinity of the J ¢ 0 configuration, only the field dependence of the levels within this configuration needs to be considered. Substituting eaM = ej -- #BgjMH for ej in equation (4a), the desired second derivative is easily determined:
J(J+ 1) p~g} 213 3 1 +/2 "
(16)
Assuming again a constant density of states, we find [10]
.
Like in the "local Fermi liquid theory" [5], the incremental density of states (12) is built up of 2 J + 1 contributions, each associated with a particular M value. However, the quasiparticle energy spectrum defined by equation (11) can be used to derive the susceptibility along the lines followed by Newns and Hewson only if there is no spin-dependent interaction between quasiparticles. To show that this is the case, we evaluate the quasiparticle interaction function [2]:
f(k, o; k'o') -
855
(15) '
that is, a repulsive interaction between quasiparticles. As the origin of the interaction is that two conductionelectron states cannot simultaneously be hybridized with the localized state, the sign of the interaction is in accordance with the expectation on physical grounds. In the present U -->oo limit not only the Pauli principle prevents simultaneous hybridization with two states; double occupancy is impossible, irrespective of the relative orientation of the spins. This explains the lack of spin dependence in the resulting interaction. In the absence of a spin-dependent interaction, deriving the susceptibility from the quasiparticle spectrum should thus lead to the "local Fermi liquid" result (2). An alternative way of deriving the susceptibility at T = 0 was followed by Ramakrishnan [8], who used the
Xhybr.(0) --
J(J + 1) z 2 1 Id~gsl(Xaj) 3 ea -- Eo'
(17)
The absence of exchange enhancement anticipated on the basis of the spin-independent interaction function (14) can be verified by determining the x(O)T/C ratio from equations (13) and (17): [x(O)T/f]hybr"
= Y ( S q-
1)u~g~/zrZk~,
(18)
which agrees with the "local Fermi liquid" result (2). In conclusion, a Fermi liquid description has been shown to follow from the perturbative treatment of the asymmetric Anderson model. The quasi-particle-energy correction and the interaction due to hybridization with the higher-lying configuration have been determined and found to be independent of spin. Accordingly, no exchange-enhancement factor appears in the ratio of the hybridization contributions to the susceptibility and the coefficient of the linear specific heat at low temperatures. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
9. 10.
C.M. Varma,Rev. Mod. Phys. 48,219 (1976). P. Nozieres, The Theory of Interacting Fermi Systems. Benjamin, New York (1963). H. Lustfeld & A. Bringer, Solid State Commun. 28, 119 (1978). A. Bringer & H. Lustfeld, Z. Physik B28, 213 (1977). D.N. Newns & A.C. Hewson, 3".Phys. F10, 2429 (1980). P.W. Anderson,Phys. Rev. 124, 41 (1961). J. Friedel, Can. J. Phys. 34, 1190 (1956). T.V. Ramakrishnan, Valence Fluctuations in Solids (Edited by L.M. Falicov, W. Hanke & M.B. Maple), p. 13. North-Holland, Amsterdam (1981). C.M. Varma & Y. Yafet, Phys. Rev. B13,2950 (1976). The discrepancy between this result and equation (10) of [8] is presumably due to a typographical error in [8]. In a recent reprint, "Theory of a Mixed Valent Impurity" by T.V. Ramakrishnan and K. Sur (October 1981), the low-temperature susceptibility is given as in the present work.
0038-1098/82/I 10853-03 $02.00/0 Pergamon Press Ltd.
MIXED-VALENT SYSTEMS AS FERMI LIQUIDS P. F. de Ch~tel Natuurkunding Laboratorium, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands (Received 11 November 1981 by A.R. Miedema) The perturbation treatment of the asymmetric Anderson model in the U--, oo limit is shown to lead to a description of mixed-valent compounds in terms of Landau's Fermi liquid theory. The quasi-particle interaction is independent of spin, and hence the ratio between the low-temperature susceptibility and specific heat is the same as in a Fermi gas. THE LOW-TEMPERATURE BEHAVIOUR of mixedvalent compounds can be briefly characterized by stating [1 ] that they act like Fermi liquids. The two outstanding features this statement is based on are: the linear temperature dependence of the specific heat and the constancy of the susceptibility far below the temperature where the latter ceases to follow the Curie-Weiss law. Both of these features are well-known consequences of Fermi statistics; for independent (noninteracting) electrons x(O)T/C = 3 ~ / n Z k 2,
(1)
where #B is the Bohr magneton and kB is Boltzmann's constant. For interacting electrons, Landau's Fermi liquid theory [2[ predicts the same qualitative behaviour, with the low-temperature susceptibility enhanced by a factor D = (1 + Zo/4) -1, and the ratio x(O)T/C correspondingly modified. Here, Z0 is the Landau interaction parameter that can be associated with exchange; for short-range (intra-atomic) interaction we have Zo = -- 4Up(ee), and D becomes the familiar exchangeenhancement factor (1 -- Up). For mixed-valent compounds, Lustfeld and Bringer [3] have derived a different relationship,
x(O)T/C = g~,uM(J + 1)/~r~k~,
(2)
which reflects the (2J + 1)-fold degeneracy of the f-electron shell. This result was found by summing the most divergent terms in an expansion of the thermodynamic potential in a perturbation series in the hybridization interaction [4]. The hybridization is tuppooed to take place between the degenerate confilluratk)n chafac. terized by J and f j , and a singiet confi~ratiom. Equation (2) is valid in the strongly mixed-valet state, where the dmJcription of a mixed-valeat eamllm~md ia terms of the asymmetric Anderson model, which repnmeats a aia#e impurity, is adequate. A R h o u ~ similar to equation (1), equation (2) does not arise as a 853
direct consequence of Fermi statistics, and hence the absence of an enhancement factor did not pose a question in the context of [3] and [4]. Recently, Newns and Hewson [5] gave a "local Fermi liquid theory" for mixed-valent systems. Their model could be better characterized as a Fermi gas theory, because it represents a system of independent electrons exposed to a localized potential which gives a sharp, (2J + 1)-fold degenerate resonance in the vicinity of the Fermi energy. The potential is supposed to represent a mixed-valent impurity, and the degenerate resonance peak in the density of states originates from the (2J + 1)-fold degenerate state of a free rare-earth ion, in a fashion similar to Anderson's model [6] of the Friedel resonance [7]. The ratio found by Newns and Hewson between the susceptibility and specific heat associated with the resonance peak obeys equation (2), which is perhaps not surprising in a Fermi gas theory which takes due account of the angular momentum and strong spin-orbit coupling in the resonant state. While it is not at all clear why there should not be an exchange enhancement of the susceptibility associated with rare-earth ions, for which U is expected to be large, equation (2) has gained some credit through its successes. Experimentally, this relationship has been found valid within 10% for a number of intermetaUic compounds involving Ce and Yb in the mixed-valent state [3, 5]. The purpose of the present paper is to show that applying to excited states the perturbation treatment Rarnakrithnan [8] developed for the ground state of the asymmetric dellmerate Amtermn model results in a Fermi liquid theory of the low-energy excitations in which equatkm (2) holds. Atmrt from demonstrating the pomibtlity of a vaniahdq exchaa~ enhancement in tim U-* -. limit, the Fermi ~ description, whida ames in a natural way from an esaentialiy configurationbated tl)l~roach, ~ves an i m ~ t into the physical orisin of tim s~'pdsmg result.
854
MIXED-VALENT SYSTEMS AS FERMI LIQUIDS
A mixed-valent impurity can be described in the configuration-based representation of the asymmetric Anderson model,
do(j) = Co(j)°+
M
(3)
+ Z Y. (vs~,noX~,oCno + h.c.). M n,o
Here the mixed-valent impurity is characterized by two configurations, the singlet 1(3)at energy eo and a (2J + 1)-fold degenerate configuration, involving one electron more, at energy ca. The states within the latter manifold are denoted by IJM),M = - - J . . . . . + J. The projection operators X/j = Ii)(/'1 act in the space of the 2 J + 2 ionic states. The matrix element VaTn.ko is the mixing amplitude between the state IJM; Cn), in which the impurity is in state IJM) and the conduction electrons in some N-electron state I~U), and the state [0; c~of)N), in which the impurity is in the singlet state 10) and the conduction electrons in the (N + 1)-electron state le~o~N)~If the degenerate configuration is eliminated, for instance by setting ea = 0% or dropping the last two terms in equation (3), the Hamiltonian becomes that of a Fermi gas, with particle energies % and appropriate creation and annihilation operators c~o and cno. When the conduction-electron system is in its ground state I~0), the shift of the energies eo and ea due to the hybridization can be calculated in first-order Brillouin-Wigner perturbation theory [4, 8, 9]: 3-, (I VaM,no1 2 ( 1 - f ~ ) Eg--e~--% dk,
(4a)
f IVa-M'n°12fff o o
(4b)
Eg--e~
=~M o d
Ey-e3
= E a o
E.~ - - C o + %
dk,
where f~' = 0(eF - - % ) is the Fermi-Dirac function at T = 0, 0 being the Heavyside step function. Due to symmetry, the summation over M in equation (4a) results in a factor 2 J +1 and the sum in equation (4b) is independent of M. The mixed-valent behaviour briefly described in the introduction is found when Eg < Eft, that is, the hybridized singlet is the lowest perturbed state. The same perturbation treatment can be applied to the excited states 10; ~N+I) and IJM; CN), where the unperturbed conduction-electron states I¢i)are characterized by the distribution of occupied states f~o. The energy of a perturbed excited state with the impurity in the singlet configuration is E~ where
•+ V
V fl V~M,kol 2(1- s %i)
dk,
(6)
k(f~o - f k °) dk. O" ¢
H = E ~nc~o~no + ,oXo, o + ~ Y x ~ , ~ k,o
Vol. 41, No. 11'
(5)
As the states modified by hybridization can be brought into a one-to-one correspondence with the unperturbed 10; ¢i), the states of an independent-electron gas, and furthermore their energy E~ = E~[~a] is given in equation (5) as a functional of the distribution function fka, the criteria for the applicability of Landau's Fermi liquid theory are satisfied, and a description of the lowenergy excitations in terms of quasiparticles can be given. The quasi-particle energies appropriate to the ground state can be obtained [2] by writing equation (5) in the form
=
+X
nosno dk,
(7)
where 6f~o = f~o --fff. Taking due account of the dependence of both the numerator and the denominator of the integrand in equation (5) on f~o, one obtains ekÜ
8Eo --
6fko Y Iv ~ = en-
no =/[Eg - e.,o -
e
n]
M
1 + X X [I V,,,,,n,o,I2 M .,J
1
(8)
o
[Eg - e~ - en']
Due to the spherical symmetry, the sum in the numerator is independent of the spin index o, and so is the quasiparticle energy, ~n,o - ~n-
(9)
An equation similar to equation (8) holds for the quasiparticle energies appropriate to excited states, but the integral becomes nontrivial if the upper indices are not kept at zero. In what follows, only the ground-state distribution function will be considered, and all upper indices will be omitted. With the constant density of states p assumes by Ramasrishnan [8], the quasiparticle energies become
~n = en +
1
Eo--ej
P Eo -- e j -- %
,
(10)
where ( 2 J + 1)A (Xaa) = ~M (Xawt,JM) = ( 2 J + 1)A--Eo + ej
(11)
is the "valence", i.e. the occupation probability of the J ¢ 0 configuration in the ground state. The incremental density of states, due to hybridization, can be obtained as ~p(en ) =
- p
d(en -- %) _ den
Eo -- ea [Eo -- e # - - % ]2"
Vol. 41, No. 11
MIXED-VALENT SYSTEMS AS FERMI LIQUIDS
The singularity at ek = Eo -- ea will be attenuated i f f ~ is replaced by the distribution function appropriate to T :/: 0. At any rate, the Fermi liquid description breaks down as soon as kBTbecomes comparable to E j --Eo, because of the non-negligible thermal occupancy of the degenerate configuration. However, in the T--> 0 limit, and for energies leh I "< E j -- Eo, equation (12) is valid. Consequently, the additional low-temperature specific heat, due to hybridization, is given by
~2
if2
Chybr. = "~k~TAp(O) = -~k~T
1
ej -- E 0
~ko
(
1
1
× ~1 -+-I2 E o - - e j - - e k
)
Eo--ej--e~' '
(14)
where
1~ = Z Z° f Iv~'~°l=(1 - ~ ) M
leo
dk.
-- ea -- ek ]
In view of equation (9), the interaction function (14) is evidently independent of the spin indices o and a'. With the constant density of states assumption, for two quasiparticles at the Fermi energy (ek = ek' = 0) equation (14) gives
f(k, o; k'o') = ((Xjj)/p) 2 2 + (Xjj) ej_Eo
~2Eo[ ~H2 [H=o
_
(13)
_ (~ko - ek)(~k'o' -- ek')
6 fk'o'
expression X(0) = -- a2Eo/aH 2 directly. To find the contribution due to the vicinity of the J ¢ 0 configuration, only the field dependence of the levels within this configuration needs to be considered. Substituting eaM = ej -- #BgjMH for ej in equation (4a), the desired second derivative is easily determined:
J(J+ 1) p~g} 213 3 1 +/2 "
(16)
Assuming again a constant density of states, we find [10]
.
Like in the "local Fermi liquid theory" [5], the incremental density of states (12) is built up of 2 J + 1 contributions, each associated with a particular M value. However, the quasiparticle energy spectrum defined by equation (11) can be used to derive the susceptibility along the lines followed by Newns and Hewson only if there is no spin-dependent interaction between quasiparticles. To show that this is the case, we evaluate the quasiparticle interaction function [2]:
f(k, o; k'o') -
855
(15) '
that is, a repulsive interaction between quasiparticles. As the origin of the interaction is that two conductionelectron states cannot simultaneously be hybridized with the localized state, the sign of the interaction is in accordance with the expectation on physical grounds. In the present U -->oo limit not only the Pauli principle prevents simultaneous hybridization with two states; double occupancy is impossible, irrespective of the relative orientation of the spins. This explains the lack of spin dependence in the resulting interaction. In the absence of a spin-dependent interaction, deriving the susceptibility from the quasiparticle spectrum should thus lead to the "local Fermi liquid" result (2). An alternative way of deriving the susceptibility at T = 0 was followed by Ramakrishnan [8], who used the
Xhybr.(0) --
J(J + 1) z 2 1 Id~gsl(Xaj) 3 ea -- Eo'
(17)
The absence of exchange enhancement anticipated on the basis of the spin-independent interaction function (14) can be verified by determining the x(O)T/C ratio from equations (13) and (17): [x(O)T/f]hybr"
= Y ( S q-
1)u~g~/zrZk~,
(18)
which agrees with the "local Fermi liquid" result (2). In conclusion, a Fermi liquid description has been shown to follow from the perturbative treatment of the asymmetric Anderson model. The quasi-particle-energy correction and the interaction due to hybridization with the higher-lying configuration have been determined and found to be independent of spin. Accordingly, no exchange-enhancement factor appears in the ratio of the hybridization contributions to the susceptibility and the coefficient of the linear specific heat at low temperatures. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
9. 10.
C.M. Varma,Rev. Mod. Phys. 48,219 (1976). P. Nozieres, The Theory of Interacting Fermi Systems. Benjamin, New York (1963). H. Lustfeld & A. Bringer, Solid State Commun. 28, 119 (1978). A. Bringer & H. Lustfeld, Z. Physik B28, 213 (1977). D.N. Newns & A.C. Hewson, 3".Phys. F10, 2429 (1980). P.W. Anderson,Phys. Rev. 124, 41 (1961). J. Friedel, Can. J. Phys. 34, 1190 (1956). T.V. Ramakrishnan, Valence Fluctuations in Solids (Edited by L.M. Falicov, W. Hanke & M.B. Maple), p. 13. North-Holland, Amsterdam (1981). C.M. Varma & Y. Yafet, Phys. Rev. B13,2950 (1976). The discrepancy between this result and equation (10) of [8] is presumably due to a typographical error in [8]. In a recent reprint, "Theory of a Mixed Valent Impurity" by T.V. Ramakrishnan and K. Sur (October 1981), the low-temperature susceptibility is given as in the present work.