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The impurity resistivity and interactions in supersaturated Al(Mn) and Al(Cr) alloys
Materials Science and Engineering, 23 ( 1 9 7 6 ) 271 - 274 © Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in t h e N e t h e r l a n d s
271
The Impurity Resistivity and Interactions in Supersaturated Al(Mn) and AI(Cr) Alloys*
A. HAMZIC, E. BABIC and B. L E O N T I ~ Institute of Physics of the University, Zagreb (Yugoslavia)
1. I N T R O D U C T I O N
Although it has long been recognized that in alloys of sufficiently high concentration the bulk properties may be dominated by the effects of interactions between the solute atoms, it is only in the last few years that the role of interactions has been more fully appreciated in supposedly "dilute" alloys. In this paper we give new and extended results on the residual resistivity for two of the most interesting systems (A1CMn) and A1CCr)) which modify some of the previously established concepts. The presence of interactions in supersaturated AI(Mn) alloys is also confirmed by electrical and magnetoresistivity measurements.
2. E X P E R I M E N T A L M E T H O D S
The constituent metals used were A1 (99.997%), Mn (99,9%) and Cr (99.9%), all obtained from Johnson-Matthey Metals Ltd., London. Master alloys were prepared by are melting on a water cooled copper hearth in an atmosphere of pure argon. The samples were obtained by an ultra rapid quenching technique [ 1 ] ; their average thickness was about 20 tam. The alloy concentrations were determined to within a few percent accuracy by electron microprobe analysis. The electrical resistivity and magnetoresistivity measurements were made using the standard four point probe potentiometrie technique with an accuracy of 5 parts in l 0 s. Residual resistivity values were obtained from the resistance by measuring the geometrical shape * P a p e r p r e s e n t e d at t h e S e c o n d I n t e r n a t i o n a l C o n f e r e n c e o n Rapidly Quenched Metals, held at t h e Massachusetts Institute of Technology, Cambridge, Mass., N o v e m b e r 17 - 19, 1 9 7 5 .
factor of each sample; the overall constant error in the absolute resistivity values is estimated to be less than + 5%. The transversal magnetoresistance was measured in a superconducting coil, with a m a x i m u m field of 35 kG and 0.1% inhomogeneity.
3. R E S I D U A L R E S I S T I V I T Y
For our alloys the variation in resistivity below 4.2K is either not detectable or shows a very w e a k - T 2 dependence. Therefore there is no significant difference between the value at 4.2K' and the residual resistivity p(0). Previous measurements [2] on Al(Mn) and AI(Cr) alloys with concentrations higher than 2 and 3 at.%, respectively, showed an increase in the slope of p (0) which was faster than linear. Attempts were made to explain this increase either by the presence of interactions or by possible changes in the crystallographic structure of the alloys. The new measurements show that the anomalous increase in the residual resistivity is not an inherent property of these alloys, but that it is caused by the difference between the nominal composition (determined by weight analysis) and the actual solute concentration (determined by electron microprobe analysis). Figure 1 shows that there is a smooth continuation of the plot through previously "critical" concentrations for Mn and Cr and indicates that the linear concentration dependence of p (0) can be extended up to quite high concentrations for all alloys in which Mn and Cr remain in solid solution. (This was proved by X-ray investigations which showed no precipitates in the samples.) The p (O)/c value for AI(Mn) is somewhat higher than the one determined earlier [ 3], and t h e new value is 8.2 + 0.3 tal2cm/at.%, while the
272
~ktCr
20
•
e/
!
0
1
2
3
4
5
6
Sc
T~
V Cr Mn Fe C'o N'i Cu
c O~to/o]
Fig. 1. A plot of the impurity resistivity as a function of impurity concentration; • Al(Mn), • Al(Mn) (precipitated), o Al(Cr).
corresponding value for Al(Cr) alloys is 8.4 + 0.3 p~2cm/At.%. However, for some AI(Mn)samples much higher values have also been obtained, but X-ray analysis showed the presence of All1 Mn4 [4] in these cases. The occurrence of this phase (its low temperature modification) is unusual since the equilibrium phase at room temperature is A16Mn. In Fig. 2 values for p(O)/c are plotted against the atomic number of the impurity. Two features of these data are that p(O)/c for these alloys is about one order of magnitude greater than in aluminium alloys with normal impurities and that there is a single peak around Cr and Mn. These facts are understood on the basis of Friedel's model [5] for the resonant scattering of conduction electrons by 3d impurities, where the residual resistivity is given by
~p(0)
- P u s i n 2 ~?l •
c
Here ~l is the resonant phase shift (l =2) and Pu is the unitarity limit. It is shown in Fig. 2 that there is quite good qualitative agreement between the measured and calculated values, although the latter are smaller by a factor of about 1.5 (owing to band structure effects). The inclusion of nonresonant phase shifts
Fig. 2. Impurity resistivity normalized to unit impurity concentration v s . the atomic number of the impurity; o 4.2K, • 500K. Solid line is the theoretical curve based on Friedel's model.
(7/0, Wz) can probably account for the asymmetry in the experimental curve (i.e., the fact that the values for Ti, V and Cr are larger than those for Ni, Co and Fe). The p(O)/c value for Sc was recently obtained [6]. The impurity resistivity of Al(Mn) and Al(Cr) at 500K clearly shows a decrease and tends to give a double peak distribution. While the existence of a single broad maximum at 4.2K confirms the "nonmagnetic" nature 0f aluminium-based 3d metal alloys, the two maxima at higher temperatures which are centered around V and Fe point to a magnetic state for Cr and Mn, and can also be explained within the framework of Friedel's model.
4. I N T E R A C T I O N S
The concentration dependence of p(0) shown in Fig. ! provides a good criterion for solid solubility, but it is not very sensitive to interactions between the impurities. Some of the investigated Al(Mn) and AI(Cr) samples have had concentrations much higher than the critical ones; as defined by the empirical relation c < Ox/TF; 0x is the Curie-Weiss temperature and TF the Fermi temperature.
273
".
C r.l
6%
55.104 2
O.
54.90
S .10~
".
•
0 . 2 0 5 5 9 ~ 54,7C
°
I
0.5
g
T(K)
55.026
38.0~
•
37.95 ~ - , , ,
..
e~
50 c 3
\, \ \* .
38.018
\ t,,
O
30.20'. •
4%
6 37.987 5 30.218
30.00 •
0
\
4
6%
o
o. •
5
\
500
3%
1000
T2IK 2)
6 30.200
Fig. 3. The i m p u r i t y resistivity vs. T 2 for three Al(Mn) alloys. The inset shows the low t e m p e r a t u r e resistance of an A I - 6 a t . % Mn alloy.
16.104 16.100
It is possible to separate the various interactions into two categories. First, there are the long range "ordering interactions", which take place between the solute transition metal atoms whose magnetic m o m e n t has survived, and which manifest themselves through the spin polarization of the conduction electrons. Interactions of the second type (called "coupling interactions") cause some impurities to behave differently from the way truly isolated solute atoms behave; in this way an alloy can be thought of as containing different kinds of scattering centers (with a corresponding different 0 ). The presence of "coupling interactions" was observed in the investigations of the TEP [7] and magnetic susceptibility [8] of AI(Mn) alloys. Recent measurements of the magnetic susceptibility at low temperatures (T < 20K) showed that for Mn concentrations between 0.5 and 2 at.% the Curie-Weiss term was dominant, with 0 ~ 3K; the coefficient of this term was proportional to c 3. Although the saturation of the magnetization could not be attained, the simple "paramagnetic m o d e l " (i.e. identical contributions of each Mn atom in a triplet of this type to the effective m o m e n t , with Pet~ ~ 1.5 PB) was in good agreement with the experimental results. The data indicate that in concentrated Al(Mn) alloys short range interactions lead to the formation of Mn triplets which have a more pronounced magnetic behaviour than that of isolated Mn atoms. The conduction electron scattering by these triplets also affects the tow temperature electrical
~
4 . 5 ~1 ,.-,--
0
1.9% _.------~0.,~, ~ H (.kG l
Fig. 4. The transverse magnetoresistivity of AI(Mn) alloys vs. H2; 0 4.2K, • 1.6K. The inset shows the slope S o f the negative c o m p o n e n t of the magnetoresistivity (S = d p / d ( H 2 ) ) as a f u n c t i o n of c 3.
resistivity. Figure 3 shows the resistivity results for three typical samples v e r s u s T 2 in the temperature region of from 1.6 to 40K. For the temperature interval between 20 and 40K there is good agreement of these results with the empirical relation
=p(o) and the characteristic temperature of this part is consistent with the 0 value [9] of 530 -+ 30 K previously found for AI(Mn). For temperatures below 20K there is an increase in the resistivity (due to the presence of triplets) and it masks the single i m p u r i t y - - T 2 term. In order to determine more precisely the characteristic temperature of the triplets (because magnetic susceptibility could not give simultaneously the values for TKtri p and Pert), it was necessary to measure the resistivity at the lowest possible temperatures. Our data for an A1-6 at.% Mn sample in the temperature region of from 80 mK to 1.1K are shown in the inset of Fig. 3. Clearly, there is again a - - T 2 behaviour, but this time it is due to the triplets. It was possible to --T 2
274
determine TKtri p independently as 3.0 -+ 0.5K from this measurement. The effect of the magnetic field (for g u B H / k B T ~ 2) on the scattering of the conduction electrons manifests itself in the freezing out of the spin flip scattering, leading to negative magnetoresistivity. In this regime, the leading term in the magnetoresistivity is given [10] by the square of the magnetization (M2). This result is obtained theoretically and experimentally for systems having TK ~ Texp; it can be extended [11] to temperatures close to TK b y replacing the Brillouin function for M by the experimental magnetization. Theoretically, magnetoresistivity has also been treated in the second Born approximation [11, 12] ; as well as recently [13] for Texp ~ TK, but only near OK, where the same M 2 dependence holds. Our experiments were made at two temperatures (1.6 and 4.2K) and the results are shown in Fig. 4. Except for the lowest concentration, we obtained a negative transverse magnetoresistivity which was proportional to H 2 up to the highest obtainable fields. The change was also more significant at the lower temperature. The H 2 dependence for the A1-6 at.% Mn sample could have only been established for H > 13 kG. The A1-0.55 at.% Mn sample showed no detectable change in the magnetoresistivity with temperature and fitted well in a Kohler diagram. This is understandable, because the magnetic field used could not affect the isolated Mn impurities which have a high characteristic temperature. The observed negative magnetoresistivity manifests itself as a deviation from Kohler's rule, but the variation of P~np with the field could be separated, and the slope of this negative component is shown versus c 3 on the inset of Fig. 4. There is a good fit only for concentrations of up to 4 at.% Mn. These magnetoresistivity results directly confirm the existence of magnetic triplets,
because they show that an increase in Mn concentration shifts the sign of the low temperature magnetoresistivity from positive to negative, which is typical for a solid solution containing localised magnetic moments. To summarize we can say that in Al(Mn) alloys with concentrations between 0.55 and 4 at.% the main contribution to the temperature and magnetic field dependent resistivity at the lowest temperatures is made by magnetic Mn triplets with 0 = 3K. At higher concentrations (c/> 4 at.%) clusters with more than three Mn atoms should also be taken into account.
ACKNOWLEDGEMENTS
We would like to thank Mr. M. O~ko for the X-ray analysis of the samples and Dr. J. R. Cooper for useful suggestions.
REFERENCES 1 E. Babi~, E. Girt, R. Krsnik and B. Leonti~, J. Phys. E, 3 (1970) 1014. 2 E. Babi~, E. Girt, R. Krsnik, B. Leonti~, Z. Vu6i~ and I. Zori~, Solid State Commun., 10 (1972) 691. 3 G. Boato, M. Bugo and C. Rizzuto, Nuovo Cimento, 45 (1966) 226. 4 T. G6decke and W. K6ster, Z. Metallk., 62 (1971) 727. 5 J. Friedel, Nuovo Cimento, Suppl. 7 (1958) 287. 6 M. Ocko, E. Babi~, R. Krsnik, E. Girt and B. Leonti4, to be published. 7 J. R. Cooper, Z. Vu6i~ and E. Bahia, J. Phys. F, 4 (1974) 1489. 8 J. R. Cooper and M. Miljak, t o be published. 9 A. D. Caplin and C. Rizzuto, Phys. Rev. Letters, 21 (1968) 746. 10 K. Yoshida, Phys. Rev., 107 (1957) 396. 11 M.-T. B~al-Monod and R. A. Weiner, Phys. Rev., 170 (1968) 552. 12 S. Nagai and J. Kondo, J. Phys. Soe. Japan, 38 (1975) 129. 13 P. Nozi~res, J. Low Temp. Phys., 17 (1974) 31.
271
The Impurity Resistivity and Interactions in Supersaturated Al(Mn) and AI(Cr) Alloys*
A. HAMZIC, E. BABIC and B. L E O N T I ~ Institute of Physics of the University, Zagreb (Yugoslavia)
1. I N T R O D U C T I O N
Although it has long been recognized that in alloys of sufficiently high concentration the bulk properties may be dominated by the effects of interactions between the solute atoms, it is only in the last few years that the role of interactions has been more fully appreciated in supposedly "dilute" alloys. In this paper we give new and extended results on the residual resistivity for two of the most interesting systems (A1CMn) and A1CCr)) which modify some of the previously established concepts. The presence of interactions in supersaturated AI(Mn) alloys is also confirmed by electrical and magnetoresistivity measurements.
2. E X P E R I M E N T A L M E T H O D S
The constituent metals used were A1 (99.997%), Mn (99,9%) and Cr (99.9%), all obtained from Johnson-Matthey Metals Ltd., London. Master alloys were prepared by are melting on a water cooled copper hearth in an atmosphere of pure argon. The samples were obtained by an ultra rapid quenching technique [ 1 ] ; their average thickness was about 20 tam. The alloy concentrations were determined to within a few percent accuracy by electron microprobe analysis. The electrical resistivity and magnetoresistivity measurements were made using the standard four point probe potentiometrie technique with an accuracy of 5 parts in l 0 s. Residual resistivity values were obtained from the resistance by measuring the geometrical shape * P a p e r p r e s e n t e d at t h e S e c o n d I n t e r n a t i o n a l C o n f e r e n c e o n Rapidly Quenched Metals, held at t h e Massachusetts Institute of Technology, Cambridge, Mass., N o v e m b e r 17 - 19, 1 9 7 5 .
factor of each sample; the overall constant error in the absolute resistivity values is estimated to be less than + 5%. The transversal magnetoresistance was measured in a superconducting coil, with a m a x i m u m field of 35 kG and 0.1% inhomogeneity.
3. R E S I D U A L R E S I S T I V I T Y
For our alloys the variation in resistivity below 4.2K is either not detectable or shows a very w e a k - T 2 dependence. Therefore there is no significant difference between the value at 4.2K' and the residual resistivity p(0). Previous measurements [2] on Al(Mn) and AI(Cr) alloys with concentrations higher than 2 and 3 at.%, respectively, showed an increase in the slope of p (0) which was faster than linear. Attempts were made to explain this increase either by the presence of interactions or by possible changes in the crystallographic structure of the alloys. The new measurements show that the anomalous increase in the residual resistivity is not an inherent property of these alloys, but that it is caused by the difference between the nominal composition (determined by weight analysis) and the actual solute concentration (determined by electron microprobe analysis). Figure 1 shows that there is a smooth continuation of the plot through previously "critical" concentrations for Mn and Cr and indicates that the linear concentration dependence of p (0) can be extended up to quite high concentrations for all alloys in which Mn and Cr remain in solid solution. (This was proved by X-ray investigations which showed no precipitates in the samples.) The p (O)/c value for AI(Mn) is somewhat higher than the one determined earlier [ 3], and t h e new value is 8.2 + 0.3 tal2cm/at.%, while the
272
~ktCr
20
•
e/
!
0
1
2
3
4
5
6
Sc
T~
V Cr Mn Fe C'o N'i Cu
c O~to/o]
Fig. 1. A plot of the impurity resistivity as a function of impurity concentration; • Al(Mn), • Al(Mn) (precipitated), o Al(Cr).
corresponding value for Al(Cr) alloys is 8.4 + 0.3 p~2cm/At.%. However, for some AI(Mn)samples much higher values have also been obtained, but X-ray analysis showed the presence of All1 Mn4 [4] in these cases. The occurrence of this phase (its low temperature modification) is unusual since the equilibrium phase at room temperature is A16Mn. In Fig. 2 values for p(O)/c are plotted against the atomic number of the impurity. Two features of these data are that p(O)/c for these alloys is about one order of magnitude greater than in aluminium alloys with normal impurities and that there is a single peak around Cr and Mn. These facts are understood on the basis of Friedel's model [5] for the resonant scattering of conduction electrons by 3d impurities, where the residual resistivity is given by
~p(0)
- P u s i n 2 ~?l •
c
Here ~l is the resonant phase shift (l =2) and Pu is the unitarity limit. It is shown in Fig. 2 that there is quite good qualitative agreement between the measured and calculated values, although the latter are smaller by a factor of about 1.5 (owing to band structure effects). The inclusion of nonresonant phase shifts
Fig. 2. Impurity resistivity normalized to unit impurity concentration v s . the atomic number of the impurity; o 4.2K, • 500K. Solid line is the theoretical curve based on Friedel's model.
(7/0, Wz) can probably account for the asymmetry in the experimental curve (i.e., the fact that the values for Ti, V and Cr are larger than those for Ni, Co and Fe). The p(O)/c value for Sc was recently obtained [6]. The impurity resistivity of Al(Mn) and Al(Cr) at 500K clearly shows a decrease and tends to give a double peak distribution. While the existence of a single broad maximum at 4.2K confirms the "nonmagnetic" nature 0f aluminium-based 3d metal alloys, the two maxima at higher temperatures which are centered around V and Fe point to a magnetic state for Cr and Mn, and can also be explained within the framework of Friedel's model.
4. I N T E R A C T I O N S
The concentration dependence of p(0) shown in Fig. ! provides a good criterion for solid solubility, but it is not very sensitive to interactions between the impurities. Some of the investigated Al(Mn) and AI(Cr) samples have had concentrations much higher than the critical ones; as defined by the empirical relation c < Ox/TF; 0x is the Curie-Weiss temperature and TF the Fermi temperature.
273
".
C r.l
6%
55.104 2
O.
54.90
S .10~
".
•
0 . 2 0 5 5 9 ~ 54,7C
°
I
0.5
g
T(K)
55.026
38.0~
•
37.95 ~ - , , ,
..
e~
50 c 3
\, \ \* .
38.018
\ t,,
O
30.20'. •
4%
6 37.987 5 30.218
30.00 •
0
\
4
6%
o
o. •
5
\
500
3%
1000
T2IK 2)
6 30.200
Fig. 3. The i m p u r i t y resistivity vs. T 2 for three Al(Mn) alloys. The inset shows the low t e m p e r a t u r e resistance of an A I - 6 a t . % Mn alloy.
16.104 16.100
It is possible to separate the various interactions into two categories. First, there are the long range "ordering interactions", which take place between the solute transition metal atoms whose magnetic m o m e n t has survived, and which manifest themselves through the spin polarization of the conduction electrons. Interactions of the second type (called "coupling interactions") cause some impurities to behave differently from the way truly isolated solute atoms behave; in this way an alloy can be thought of as containing different kinds of scattering centers (with a corresponding different 0 ). The presence of "coupling interactions" was observed in the investigations of the TEP [7] and magnetic susceptibility [8] of AI(Mn) alloys. Recent measurements of the magnetic susceptibility at low temperatures (T < 20K) showed that for Mn concentrations between 0.5 and 2 at.% the Curie-Weiss term was dominant, with 0 ~ 3K; the coefficient of this term was proportional to c 3. Although the saturation of the magnetization could not be attained, the simple "paramagnetic m o d e l " (i.e. identical contributions of each Mn atom in a triplet of this type to the effective m o m e n t , with Pet~ ~ 1.5 PB) was in good agreement with the experimental results. The data indicate that in concentrated Al(Mn) alloys short range interactions lead to the formation of Mn triplets which have a more pronounced magnetic behaviour than that of isolated Mn atoms. The conduction electron scattering by these triplets also affects the tow temperature electrical
~
4 . 5 ~1 ,.-,--
0
1.9% _.------~0.,~, ~ H (.kG l
Fig. 4. The transverse magnetoresistivity of AI(Mn) alloys vs. H2; 0 4.2K, • 1.6K. The inset shows the slope S o f the negative c o m p o n e n t of the magnetoresistivity (S = d p / d ( H 2 ) ) as a f u n c t i o n of c 3.
resistivity. Figure 3 shows the resistivity results for three typical samples v e r s u s T 2 in the temperature region of from 1.6 to 40K. For the temperature interval between 20 and 40K there is good agreement of these results with the empirical relation
=p(o) and the characteristic temperature of this part is consistent with the 0 value [9] of 530 -+ 30 K previously found for AI(Mn). For temperatures below 20K there is an increase in the resistivity (due to the presence of triplets) and it masks the single i m p u r i t y - - T 2 term. In order to determine more precisely the characteristic temperature of the triplets (because magnetic susceptibility could not give simultaneously the values for TKtri p and Pert), it was necessary to measure the resistivity at the lowest possible temperatures. Our data for an A1-6 at.% Mn sample in the temperature region of from 80 mK to 1.1K are shown in the inset of Fig. 3. Clearly, there is again a - - T 2 behaviour, but this time it is due to the triplets. It was possible to --T 2
274
determine TKtri p independently as 3.0 -+ 0.5K from this measurement. The effect of the magnetic field (for g u B H / k B T ~ 2) on the scattering of the conduction electrons manifests itself in the freezing out of the spin flip scattering, leading to negative magnetoresistivity. In this regime, the leading term in the magnetoresistivity is given [10] by the square of the magnetization (M2). This result is obtained theoretically and experimentally for systems having TK ~ Texp; it can be extended [11] to temperatures close to TK b y replacing the Brillouin function for M by the experimental magnetization. Theoretically, magnetoresistivity has also been treated in the second Born approximation [11, 12] ; as well as recently [13] for Texp ~ TK, but only near OK, where the same M 2 dependence holds. Our experiments were made at two temperatures (1.6 and 4.2K) and the results are shown in Fig. 4. Except for the lowest concentration, we obtained a negative transverse magnetoresistivity which was proportional to H 2 up to the highest obtainable fields. The change was also more significant at the lower temperature. The H 2 dependence for the A1-6 at.% Mn sample could have only been established for H > 13 kG. The A1-0.55 at.% Mn sample showed no detectable change in the magnetoresistivity with temperature and fitted well in a Kohler diagram. This is understandable, because the magnetic field used could not affect the isolated Mn impurities which have a high characteristic temperature. The observed negative magnetoresistivity manifests itself as a deviation from Kohler's rule, but the variation of P~np with the field could be separated, and the slope of this negative component is shown versus c 3 on the inset of Fig. 4. There is a good fit only for concentrations of up to 4 at.% Mn. These magnetoresistivity results directly confirm the existence of magnetic triplets,
because they show that an increase in Mn concentration shifts the sign of the low temperature magnetoresistivity from positive to negative, which is typical for a solid solution containing localised magnetic moments. To summarize we can say that in Al(Mn) alloys with concentrations between 0.55 and 4 at.% the main contribution to the temperature and magnetic field dependent resistivity at the lowest temperatures is made by magnetic Mn triplets with 0 = 3K. At higher concentrations (c/> 4 at.%) clusters with more than three Mn atoms should also be taken into account.
ACKNOWLEDGEMENTS
We would like to thank Mr. M. O~ko for the X-ray analysis of the samples and Dr. J. R. Cooper for useful suggestions.
REFERENCES 1 E. Babi~, E. Girt, R. Krsnik and B. Leonti~, J. Phys. E, 3 (1970) 1014. 2 E. Babi~, E. Girt, R. Krsnik, B. Leonti~, Z. Vu6i~ and I. Zori~, Solid State Commun., 10 (1972) 691. 3 G. Boato, M. Bugo and C. Rizzuto, Nuovo Cimento, 45 (1966) 226. 4 T. G6decke and W. K6ster, Z. Metallk., 62 (1971) 727. 5 J. Friedel, Nuovo Cimento, Suppl. 7 (1958) 287. 6 M. Ocko, E. Babi~, R. Krsnik, E. Girt and B. Leonti4, to be published. 7 J. R. Cooper, Z. Vu6i~ and E. Bahia, J. Phys. F, 4 (1974) 1489. 8 J. R. Cooper and M. Miljak, t o be published. 9 A. D. Caplin and C. Rizzuto, Phys. Rev. Letters, 21 (1968) 746. 10 K. Yoshida, Phys. Rev., 107 (1957) 396. 11 M.-T. B~al-Monod and R. A. Weiner, Phys. Rev., 170 (1968) 552. 12 S. Nagai and J. Kondo, J. Phys. Soe. Japan, 38 (1975) 129. 13 P. Nozi~res, J. Low Temp. Phys., 17 (1974) 31.