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Effect of strong potential scattering on the local moment relaxation in anisotropic superconductors
Volume 146, number 5
PHYSICS LETTERSA
28 May 1990
EFFECT OF STRONG POTENTIAL SCATTERING ON THE LOCAL MOMENT RELAXATION IN ANISOTROPIC SUPERCONDUCTORS Hocine BAHLOULI TheoreticalPhysics Institute, University of Minnesota, Minneapolis, MN 55455, USA and Department ofPhysics, K.F. U.P.M., Dhahran 31261, SaudiArabia Received 21 February 1990; accepted for publication 21 March 1990 Communicated by A.A. Maradudin
The relaxation rate for the local moment of an impurity in an anisotropic superconductor is calculated, taking into account the effect of strong scalar potential scattering-off the same impurity. In applying our results to heavy fermion superconductors, the phase shift associated with this potential scattering turns out to be close to it/2. As a result a strong enhancement of the relaxation rate is obtained for states with lines of nodes in their energy gap, and a reduction for states with point nodes. Such signatures might be helpful in identifying the correct microscopic structure of the energy gap in these systems.
Nuclear spin relaxation and local moment relaxation have always been valuable tools in probing the microscopic structure of the superconducting state. Both are local probes and in this respect are much different from other transport properties such as thermal conductivity, ultrasound attenuation, etc. In the past, to interpret the experimental results on local moment relaxation one usually used the theory of nuclear spin relaxation in superconductors [11 and substituted the coupling constantby the exchange interaction between the magnetic impurity and conduction electrons. Despite the similarity between these two phenomena, we want to emphasize that in the magnetic impurity case one is dealing not only with the local spin of the impurity but also with the ordinary scattering potential due to the same impurity. In fact the impurity is chemically different from the host and this will be responsible for an additional potential scattering for the conduction electron. A long time ago this problem has been studied by Kondo [2], who concluded that, in the normal state, the will be renormalized 2öN exchange with oN isinteraction the phase shift associated with by cOs the s-wave scalar potential of the impurity.ofConsequently, since in the past most calculations relaxation phenomena in superconductors have assumed that the phase shift is small, the similarity between
nuclear spin relaxation and local moment relaxation remains valid to a high degree of accuracy [31. In heavy fermion systems (HFS) things are quite different, even in the normal state one observes a strong enhancement (~10~)of the nuclear spin relaxation rate [4], while local moment relaxation is enhanced [5,6] by just a factor of four. Due to the large enhancement ofthe effective mass in these systems, one concludes that a corresponding strong reduction of the exchange coupling constant is expected in these systems so as to make the local moment relaxation rate have its experimental value. Thus most probably we are close to the resonant scattering limit where 5N ,t/2. The reason being that the cancellation effect seen in ESR experiments roughly requires (m* / m) cos2ö~ 0(1), where m* and m stand for the effective and bare masses respectively. At this point I should stress the fact that we are talking about the s-wave potential scattering phase shift associated with the impurity substituting the U atom in the case of UBe 13 (or Ce in the case of CeCu2Si2) in the heavy electrontomaterial, not the f-site itself. Numerical application UBe and UPt3 0N ranges between 0.9613 x it/2 and showsx ,t/2 thatin these systems. Under such circumstan0.98 ces the usual similarity between nuclear spin relaxation and local moment relaxation is lost dtie to the 265
Volume 146, numberS
28 May 1990
PHYSICS LETTERS A
strong potential scattering off the same impurity, It is the purpose of this paper to give an explicit calculation ofthe local moment relaxation rate in the superconducting state of an anisotropic superconductor taking into account the strong potential scattering off the same impurity. Using time dependent perturbation theory in the four-dimensional space as defined by Maki [7], we find the following expression for the relaxation rate, R~(T)=CJ2Re J dwtr[a~G>(w)aG<(w)],
at the Fermi surface for a single spin. Dyson equation we obtain
By
solving the
G>(w)=M(G~(w))G~>(w)M(G~(w)) ,
(5)
where M( G~(w) ) = [1 a = a, r.
—
0N
it
3
G~( w)
]
-‘
tan
(6)
In the weak coupling approximation G~(w) and G(~(w) are defined by G 0(w)
(1) where C is a numerical constant, J is the exchange coupling constant between the local moment spin and conduction a±=t3a1 ±it0a2with t, and a, being electron the Paulispin, matrices operating in the partide—hole space and in the ordinary spin space respectively. R~(T) is the local moment relaxation rate in the superconducting state which isjust the inverse of the local moment relaxation time. The symbols tr and Re stand for the trace and the real part, respectively, and the quantity G> (w) is defined in terms of the full Green function G(p, w) by G> (w) = itN( 0) ~ G> (p, to)
,
(2)
which in its turn is defined through the Dyson equation G(p,w)=G0(p w) + V>~G~(p’,w)G(p, to)
w .
,
47~ ~/( w + it) 2_ ~‘
= —i
J
dQ~ ~
forjA~l
12
w 2 (w+i)
for
I4~>w~ (7)
and G~(w)= —f(w)N~(w) G 0> (w) [1 .—f(w)]N~(w) (8) with f( w) being the usual Fermi function, N~ (w) is the real part of G~(w).The quantities G~(w)and G>(w) are obtained from eq. (7) by just changing the sign of the real part, and from eq. (5) by replacing > by < respectively. It is easy to see from eq. (6) that in the weak scatteringlimit (ON~O)we get back our usual result which is analogous to the one for the nuclear spin relaxation rate. We will be interested here in seeing what happens in the other extreme limit where the phase shift is close to ic/2, because as we have explained before, this limit seems to be of practical interest in HFS. In this 2),case we obtam the following expression (ON~7t/ R~(T) T dw~ / ~af\ IG~(w)~4’ N~(w) (9) R~(T~) where R~( T~)stands for the normal state relaxation .
,
(3)
1’
with V= trr3 (v= const) being the scatteringpotential due to the impurity in the s-wave approximation. G 0(p, to) is the single particle Green function in the absence of impurities andbyis defined (in the weak coupling approximation) (to + ~ G 0 (p, to) = —I 12 ~ — — (4 _____________
—
4, is a unitary matrix in spin space that characterizes the superconducting state, ~,, is the normal state quasiparticle energy reckoned from the Fermi energy, and N( 0) is the density of quasiparticle states 266
1dQ~
=j—
7 2~j~J
rate evaluated at the transition temperature, T~A similar expression holds in the weak scattering limit (O~0) but without IG~(w)Iin the denominator of eq. (9). The superconducting states we shall consider in this paper are the polar and axial (also called Anderson—Brinkman—Morel or ABM) p-wave states with triplet pairing and the axial d-wave singlet pair-
Volume 146, number 5
PHYSICS LETTERS A
ing state which is consistent with hexagonal and cubic symmetries. In the case of triplet states the gap matrix is given by [81 A •A’ ‘10 —.
p
28 May 1990
nuclear spin relaxation one [3]. In contrast, in the resonant scattering limit (ON~7t/2), fig. 2 shows a drastic change in the behavior of the local moment relaxation rate for the previous superconducting
—ia 2~
~
‘S
/
where
states. A net enhancement is obtained in the case of order parameters having lines of nodes in the energy
A’ ~ —A( p1‘d
(11) ‘S
gapthe (d-wave, polarrate p-wave states), for while a decrease of relaxation is obtained states having
4(p)=4(T)cosO and for the axial state by
(12)
only point nodes in their energy gap (actually this conclusion holds even for isotropic superconducting states). It should be stressed here that our previous results will not be affected in any major way if we
A (p5) = A ( T) sin 0 e~.
(13)
—
with zlQ3) given for the polar state by
d is a fixed unit vector in spin space satisfying the unitarity condition dxd*=O. In the case of the dwave state we have [91 A =ia A(j3) (14) 2
p
‘
where
0.
(15)
A(j~)=2A(T)sinOcosOe’ In eqs. (l0)—(15), A(T) is the maximum value of the energy gap on the Fermi surface, 0 and 0 give the direction of p in polar coordinates. The numerical computations of these relaxation rates is shown in fig. 1 in the case of weak scattering (ON 0) for the previous classes of anisotropic order parameters. The weak scattering behavior, as expected, resembles the
conclude non-magnetic impurity scattering self-consistently in our calculations [10]. Unlike other transport properties which are dominated by nonmagnetic impurity scattering in HFS, local moment and nuclear spin relaxations are only slightly affected at very low temperatures where impurity renormalization effects become important [111 due to the development of a virtual bound state near the nodes ofthe gap. Up to the present time local moment relaxation measurements have been performed only in Er~U 1~Ru2Si2in the superconducting state [121 where the magnetic impurity substitutes the U atom. Now, it is believed that if the impurity substitutes the U atom, then its potential scattering phase shift should be close to it/2 because it represents a missing local f-electron (Kondo hole) in the lattice. On
1.25
15~
__
T/T~ Fig. 1. Plots of the ratio of the local moment relaxation in the superconducting state to the one in the normal state evaluated at T~for the axial (p-wave), polar (p-wave) and d-wave states. The phase shift oN is equal to zero.
T1F~ Fig. 2. Plots of the ratio of the local moment relaxation in the superconducting state to the one in the normal state evaluated at T~for theaxial 0N is(p-wave), equal to i~/2. polar (p-wave) and d-wave states. The phase shift
267
Volume 146, number 5
PHYSICS LETTERS A
the other hand if the impurity substitutes the Be atom, then one expects to be in the weak scattering limit (ON 0). The experimental data on the local moment relaxation of Er3~in the superconducting state of Er~U 1~Ru2Si2(x= 0.005) show a steady increase in the relaxation rate below T~.Such an increase is not to be expected if the phase shift was very small, as argued before under such circumstances the behavior would be similar to the nuclear spin relaxation rate which, in the case of HFS, shows a decrease [4] below T~.From fig. 2 we see that both polar p-wave and d-wave states do show an increase in the relaxation rate below T~which is in qualitative accord with the available experimental results. But the lack of availability of experimental data in the whole range oftemperatures below T~ will prevent us from doing any quantitative comparison with our work. We believe that the realization of such experiments should constitute a big test for this theoretical work and might shed more light on the detailed microscopic structure of the superconducting state in heavy fermion superconductors. Meanwhile we stress the fact that our calculation is valid only for non-bottlenecked systems [13] (i.e. electrons are strongly coupled to the lattice to achieve thermal equilibrium with it, and only weakly coupled to the magnetic impurity, so as to make our perturbative treatment legitimate). The electron spin resonance experiments [5,6] seem to indicate that doped UBe13 and UPt3 are indeed non-bottlenecked systems, thus our analysis should be valid in these compounds. To conclude, we have calculated the effect of the impurity potential scattering on the local moment relaxation rate of the same impurity in an anisotropic superconductor. Our results are then applied to heavy fermion superconductors. In the resonant scattering limit, which we believe should be the case in heavy fermion systems, we have obtained a notable enhancement of the relaxation rate for states having lines of nodes in their energy gap and a net decrease for states having point nodes in their energy gap. In almost all previous experiments in heavy fermion systems it proved very difficult to distinguish
268
28 May 1990
between states having points and lines of nodes in their energy gap because of their similar contributions to transport and relaxation phenomena [11]. Due to the sharp distinctive results of the local moment relaxation rates for different classes of anisotropic superconductors (see fig. 2), we think that the performance of such experiments should help in selecting the correct nodal structure of the energy gap in heavy fermion systems. Now whether or not the above considerations in fact explain the experimental data on local moment relaxation in heavy fermion superconductors, they constitute in any case an interesting exhibition of how strong potential scattering might affect the local moment relaxation rate. We wish to acknowledge useful and stimulating conversations with Professor Oriol Valls and correspondence with Dr. Daniel Cox on the subject matter of this paper. This work was supported by the Theoretical Physics Institute at the University of Minnesota and the Physics Department at KFUPM University.
References [1] D.E. Maclaughlin, Solid State Phys. 31(1976)1. [2].!. Kondo, Phys. Rev. 169 (1968) 437. [31 L.R. Tagirov and K.F. Trutnev, J. Phys. F 17 (1987) 695. [4]D.E. Maclaughlin, Cheng Tien, W.G. Clark, M.D. Lan, Z. Fisk, J.L. Smith and H.R. Ott, Phys. Rev. Lett. 53(1984) 1833. [5] F.G. Gandra, M.J. Pontes, S. Schultz and S.B. Oseroff, Solid State Commun. 64 (1987) 859. [6] F. Gandra, S. Schultz, S.B. Oseroff, Z. Fisk and J.L. Smith, Phys. Rev. Lett. 55(1985) 2719.
[71K. Maki, in: Superconductivity, ed. R.D. Parks [8]A.J.Leggett,Rev. Mod. Phys. 47 (1975) 331.
(1969) p.
[9] P.W. Anderson and P. Morel, Phys. Rev. 123 (1961) 1911. [101 H. Bahlouli, Ph.D. thesis, University of Illinois, unpublished (1988). Hirshfeld, P. Wolfle and D. Einzel, Phys. Rev. B 37 [12] S. Taleb, W.G. Clark, P. Armstrong, C. Rossel and M.B. Maple, Solid State Commun. 68 (1988) 231. [13] K. Baberschke, Z. Phys. B 24 (1976) 53.
[11] P.J.
PHYSICS LETTERSA
28 May 1990
EFFECT OF STRONG POTENTIAL SCATTERING ON THE LOCAL MOMENT RELAXATION IN ANISOTROPIC SUPERCONDUCTORS Hocine BAHLOULI TheoreticalPhysics Institute, University of Minnesota, Minneapolis, MN 55455, USA and Department ofPhysics, K.F. U.P.M., Dhahran 31261, SaudiArabia Received 21 February 1990; accepted for publication 21 March 1990 Communicated by A.A. Maradudin
The relaxation rate for the local moment of an impurity in an anisotropic superconductor is calculated, taking into account the effect of strong scalar potential scattering-off the same impurity. In applying our results to heavy fermion superconductors, the phase shift associated with this potential scattering turns out to be close to it/2. As a result a strong enhancement of the relaxation rate is obtained for states with lines of nodes in their energy gap, and a reduction for states with point nodes. Such signatures might be helpful in identifying the correct microscopic structure of the energy gap in these systems.
Nuclear spin relaxation and local moment relaxation have always been valuable tools in probing the microscopic structure of the superconducting state. Both are local probes and in this respect are much different from other transport properties such as thermal conductivity, ultrasound attenuation, etc. In the past, to interpret the experimental results on local moment relaxation one usually used the theory of nuclear spin relaxation in superconductors [11 and substituted the coupling constantby the exchange interaction between the magnetic impurity and conduction electrons. Despite the similarity between these two phenomena, we want to emphasize that in the magnetic impurity case one is dealing not only with the local spin of the impurity but also with the ordinary scattering potential due to the same impurity. In fact the impurity is chemically different from the host and this will be responsible for an additional potential scattering for the conduction electron. A long time ago this problem has been studied by Kondo [2], who concluded that, in the normal state, the will be renormalized 2öN exchange with oN isinteraction the phase shift associated with by cOs the s-wave scalar potential of the impurity.ofConsequently, since in the past most calculations relaxation phenomena in superconductors have assumed that the phase shift is small, the similarity between
nuclear spin relaxation and local moment relaxation remains valid to a high degree of accuracy [31. In heavy fermion systems (HFS) things are quite different, even in the normal state one observes a strong enhancement (~10~)of the nuclear spin relaxation rate [4], while local moment relaxation is enhanced [5,6] by just a factor of four. Due to the large enhancement ofthe effective mass in these systems, one concludes that a corresponding strong reduction of the exchange coupling constant is expected in these systems so as to make the local moment relaxation rate have its experimental value. Thus most probably we are close to the resonant scattering limit where 5N ,t/2. The reason being that the cancellation effect seen in ESR experiments roughly requires (m* / m) cos2ö~ 0(1), where m* and m stand for the effective and bare masses respectively. At this point I should stress the fact that we are talking about the s-wave potential scattering phase shift associated with the impurity substituting the U atom in the case of UBe 13 (or Ce in the case of CeCu2Si2) in the heavy electrontomaterial, not the f-site itself. Numerical application UBe and UPt3 0N ranges between 0.9613 x it/2 and showsx ,t/2 thatin these systems. Under such circumstan0.98 ces the usual similarity between nuclear spin relaxation and local moment relaxation is lost dtie to the 265
Volume 146, numberS
28 May 1990
PHYSICS LETTERS A
strong potential scattering off the same impurity, It is the purpose of this paper to give an explicit calculation ofthe local moment relaxation rate in the superconducting state of an anisotropic superconductor taking into account the strong potential scattering off the same impurity. Using time dependent perturbation theory in the four-dimensional space as defined by Maki [7], we find the following expression for the relaxation rate, R~(T)=CJ2Re J dwtr[a~G>(w)aG<(w)],
at the Fermi surface for a single spin. Dyson equation we obtain
By
solving the
G>(w)=M(G~(w))G~>(w)M(G~(w)) ,
(5)
where M( G~(w) ) = [1 a = a, r.
—
0N
it
3
G~( w)
]
-‘
tan
(6)
In the weak coupling approximation G~(w) and G(~(w) are defined by G 0(w)
(1) where C is a numerical constant, J is the exchange coupling constant between the local moment spin and conduction a±=t3a1 ±it0a2with t, and a, being electron the Paulispin, matrices operating in the partide—hole space and in the ordinary spin space respectively. R~(T) is the local moment relaxation rate in the superconducting state which isjust the inverse of the local moment relaxation time. The symbols tr and Re stand for the trace and the real part, respectively, and the quantity G> (w) is defined in terms of the full Green function G(p, w) by G> (w) = itN( 0) ~ G> (p, to)
,
(2)
which in its turn is defined through the Dyson equation G(p,w)=G0(p w) + V>~G~(p’,w)G(p, to)
w .
,
47~ ~/( w + it) 2_ ~‘
= —i
J
dQ~ ~
forjA~l
12
w 2 (w+i)
for
I4~>w~ (7)
and G~(w)= —f(w)N~(w) G 0> (w) [1 .—f(w)]N~(w) (8) with f( w) being the usual Fermi function, N~ (w) is the real part of G~(w).The quantities G~(w)and G>(w) are obtained from eq. (7) by just changing the sign of the real part, and from eq. (5) by replacing > by < respectively. It is easy to see from eq. (6) that in the weak scatteringlimit (ON~O)we get back our usual result which is analogous to the one for the nuclear spin relaxation rate. We will be interested here in seeing what happens in the other extreme limit where the phase shift is close to ic/2, because as we have explained before, this limit seems to be of practical interest in HFS. In this 2),case we obtam the following expression (ON~7t/ R~(T) T dw~ / ~af\ IG~(w)~4’ N~(w) (9) R~(T~) where R~( T~)stands for the normal state relaxation .
,
(3)
1’
with V= trr3 (v= const) being the scatteringpotential due to the impurity in the s-wave approximation. G 0(p, to) is the single particle Green function in the absence of impurities andbyis defined (in the weak coupling approximation) (to + ~ G 0 (p, to) = —I 12 ~ — — (4 _____________
—
4, is a unitary matrix in spin space that characterizes the superconducting state, ~,, is the normal state quasiparticle energy reckoned from the Fermi energy, and N( 0) is the density of quasiparticle states 266
1dQ~
=j—
7 2~j~J
rate evaluated at the transition temperature, T~A similar expression holds in the weak scattering limit (O~0) but without IG~(w)Iin the denominator of eq. (9). The superconducting states we shall consider in this paper are the polar and axial (also called Anderson—Brinkman—Morel or ABM) p-wave states with triplet pairing and the axial d-wave singlet pair-
Volume 146, number 5
PHYSICS LETTERS A
ing state which is consistent with hexagonal and cubic symmetries. In the case of triplet states the gap matrix is given by [81 A •A’ ‘10 —.
p
28 May 1990
nuclear spin relaxation one [3]. In contrast, in the resonant scattering limit (ON~7t/2), fig. 2 shows a drastic change in the behavior of the local moment relaxation rate for the previous superconducting
—ia 2~
~
‘S
/
where
states. A net enhancement is obtained in the case of order parameters having lines of nodes in the energy
A’ ~ —A( p1‘d
(11) ‘S
gapthe (d-wave, polarrate p-wave states), for while a decrease of relaxation is obtained states having
4(p)=4(T)cosO and for the axial state by
(12)
only point nodes in their energy gap (actually this conclusion holds even for isotropic superconducting states). It should be stressed here that our previous results will not be affected in any major way if we
A (p5) = A ( T) sin 0 e~.
(13)
—
with zlQ3) given for the polar state by
d is a fixed unit vector in spin space satisfying the unitarity condition dxd*=O. In the case of the dwave state we have [91 A =ia A(j3) (14) 2
p
‘
where
0.
(15)
A(j~)=2A(T)sinOcosOe’ In eqs. (l0)—(15), A(T) is the maximum value of the energy gap on the Fermi surface, 0 and 0 give the direction of p in polar coordinates. The numerical computations of these relaxation rates is shown in fig. 1 in the case of weak scattering (ON 0) for the previous classes of anisotropic order parameters. The weak scattering behavior, as expected, resembles the
conclude non-magnetic impurity scattering self-consistently in our calculations [10]. Unlike other transport properties which are dominated by nonmagnetic impurity scattering in HFS, local moment and nuclear spin relaxations are only slightly affected at very low temperatures where impurity renormalization effects become important [111 due to the development of a virtual bound state near the nodes ofthe gap. Up to the present time local moment relaxation measurements have been performed only in Er~U 1~Ru2Si2in the superconducting state [121 where the magnetic impurity substitutes the U atom. Now, it is believed that if the impurity substitutes the U atom, then its potential scattering phase shift should be close to it/2 because it represents a missing local f-electron (Kondo hole) in the lattice. On
1.25
15~
__
T/T~ Fig. 1. Plots of the ratio of the local moment relaxation in the superconducting state to the one in the normal state evaluated at T~for the axial (p-wave), polar (p-wave) and d-wave states. The phase shift oN is equal to zero.
T1F~ Fig. 2. Plots of the ratio of the local moment relaxation in the superconducting state to the one in the normal state evaluated at T~for theaxial 0N is(p-wave), equal to i~/2. polar (p-wave) and d-wave states. The phase shift
267
Volume 146, number 5
PHYSICS LETTERS A
the other hand if the impurity substitutes the Be atom, then one expects to be in the weak scattering limit (ON 0). The experimental data on the local moment relaxation of Er3~in the superconducting state of Er~U 1~Ru2Si2(x= 0.005) show a steady increase in the relaxation rate below T~.Such an increase is not to be expected if the phase shift was very small, as argued before under such circumstances the behavior would be similar to the nuclear spin relaxation rate which, in the case of HFS, shows a decrease [4] below T~.From fig. 2 we see that both polar p-wave and d-wave states do show an increase in the relaxation rate below T~which is in qualitative accord with the available experimental results. But the lack of availability of experimental data in the whole range oftemperatures below T~ will prevent us from doing any quantitative comparison with our work. We believe that the realization of such experiments should constitute a big test for this theoretical work and might shed more light on the detailed microscopic structure of the superconducting state in heavy fermion superconductors. Meanwhile we stress the fact that our calculation is valid only for non-bottlenecked systems [13] (i.e. electrons are strongly coupled to the lattice to achieve thermal equilibrium with it, and only weakly coupled to the magnetic impurity, so as to make our perturbative treatment legitimate). The electron spin resonance experiments [5,6] seem to indicate that doped UBe13 and UPt3 are indeed non-bottlenecked systems, thus our analysis should be valid in these compounds. To conclude, we have calculated the effect of the impurity potential scattering on the local moment relaxation rate of the same impurity in an anisotropic superconductor. Our results are then applied to heavy fermion superconductors. In the resonant scattering limit, which we believe should be the case in heavy fermion systems, we have obtained a notable enhancement of the relaxation rate for states having lines of nodes in their energy gap and a net decrease for states having point nodes in their energy gap. In almost all previous experiments in heavy fermion systems it proved very difficult to distinguish
268
28 May 1990
between states having points and lines of nodes in their energy gap because of their similar contributions to transport and relaxation phenomena [11]. Due to the sharp distinctive results of the local moment relaxation rates for different classes of anisotropic superconductors (see fig. 2), we think that the performance of such experiments should help in selecting the correct nodal structure of the energy gap in heavy fermion systems. Now whether or not the above considerations in fact explain the experimental data on local moment relaxation in heavy fermion superconductors, they constitute in any case an interesting exhibition of how strong potential scattering might affect the local moment relaxation rate. We wish to acknowledge useful and stimulating conversations with Professor Oriol Valls and correspondence with Dr. Daniel Cox on the subject matter of this paper. This work was supported by the Theoretical Physics Institute at the University of Minnesota and the Physics Department at KFUPM University.
References [1] D.E. Maclaughlin, Solid State Phys. 31(1976)1. [2].!. Kondo, Phys. Rev. 169 (1968) 437. [31 L.R. Tagirov and K.F. Trutnev, J. Phys. F 17 (1987) 695. [4]D.E. Maclaughlin, Cheng Tien, W.G. Clark, M.D. Lan, Z. Fisk, J.L. Smith and H.R. Ott, Phys. Rev. Lett. 53(1984) 1833. [5] F.G. Gandra, M.J. Pontes, S. Schultz and S.B. Oseroff, Solid State Commun. 64 (1987) 859. [6] F. Gandra, S. Schultz, S.B. Oseroff, Z. Fisk and J.L. Smith, Phys. Rev. Lett. 55(1985) 2719.
[71K. Maki, in: Superconductivity, ed. R.D. Parks [8]A.J.Leggett,Rev. Mod. Phys. 47 (1975) 331.
(1969) p.
[9] P.W. Anderson and P. Morel, Phys. Rev. 123 (1961) 1911. [101 H. Bahlouli, Ph.D. thesis, University of Illinois, unpublished (1988). Hirshfeld, P. Wolfle and D. Einzel, Phys. Rev. B 37 [12] S. Taleb, W.G. Clark, P. Armstrong, C. Rossel and M.B. Maple, Solid State Commun. 68 (1988) 231. [13] K. Baberschke, Z. Phys. B 24 (1976) 53.
[11] P.J.