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The superconducting specific heat jump of UBe13 and a narrow peak in the electronic density-of-states
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
Solid State Communications, Vol. 55, No. 8, pp. 775-778, 1985. Printed in Great Britain.
THE SUPERCONDUCTING SPECIFIC HEAT JUMP OF UBe~3 AND A NARROW PEAK IN THE ELECTRONIC DENSITY-OF-STATES W. Stephan and J.P. Carbotte Department of Physics, McMaster University, Hamilton, Ontario LSS 4M1, Canada
(Received 11 April 1985 by M.F. Collins) The superconducting specific heat jump for the s - f hybridization model of UBels of Overhauser and Appel [2] is calculated. The narrow peak in the density-of-states is found to enhance the size of the jump relative to the BCS value of 1.43, but in weak coupling the increase is insufficient to agree with the measured value for UBe~a. The Kondo lattice model of Razafimandimby, Fulde and Keller [9] is considered as a possible modification, but in weak coupling a simple estimate shows that this cannot increase the specific heat jump significantly. It is speculated, however, that a consistent picture may still be possible with conventional strong coupling corrections. THE HEAVY FERMION superconductor UBel3 has been the subject of much recent research [1]. Although there is currently no consensus as to the appropriate theoretical description for this and other heavy fermion alloys, a recent proposal of Overhauser and Appel [2] seems to merit further investigation. The basic assumption of this s - f hybridization model is that the crystal field F7 doublet, which derives from the free uranium atom 5fs/2 levels, will lie at the Fermi energy and will hybridize with the Be(2s)-U(7s) conduction band. This results in a very narrow Lorentzian peak in the conductionelectron density-of-states centered at the Fermi energy. Using a model density-of-states with three parameters fit to the experimental heat capacity and magnetic susceptibility Overhauser and Appel [2] are able to explain quantitatively many of the anomalous properties of UBela using a simple single quasi-particle picture. The suggestion is also made that the superconductivity which appears below 0.9 K is of the conventional singlet paired BCS type. The deviation of the temperature dependence of the low temperature specific heat from the exponential BCS prediction (C, cc e - n I T for T < A) is attributed to anisotropy of the gap, which arises in this model from the anisotropy of the hybridization matrix element. The purpose of this letter is to consider the magnitude of the jump in the specific heat at Tc in the context of this model. Recall that BCS theory predicts that AC/Cn(Tc)= 1.43, where AC = Cs--CN and Cs and CN are the heat capacities of the superconducting and normal states respectively. On the other hand the latest experimental evidence indicates that for UBet3 one has AC/Cn(Tc) ~ 2.5 [3]. A specific heat jump of this size would be reasonable for a conventional strong
coupling semiconductor, but such an explanation goes against the spirit of the model of Overhauser and Appel [2], where many-body effects are assumed to be small. However, another factor which may not be so generally known is that a peak in the density-of.states near the Fermi energy also enhances AC[Cn(Te) compared to the BCS value. Previous results for the free energy of a superconductor with a Lorentzian peak in the densityof-states due to Carbotte and Abd E1-Rahman [4] and Zarate and Carbotte [5] in weak coupling and to Mitrovid and Carbotte [6] and Schachinger and Carbotte [7] in strong coupling are not directly applicable here because the width of the peak needed to agree with the normal state heat capacity of UBe13 is 6K, which is comparable to the critical temperature T e = 0 . 9 K . The previous calculations [4-7] have assumed that the scale of variation of the density-of-states is on the order of the Debye energy cot), which would be approximately 620 K for UBe13. We therefore present here the results for a numerical evaluation of the heat capacity of a BCS superconductor with a narrow peak in the density-of-states centred at the Fermi energy. Following Overhauser and Appel [2] the model density-of-states is N(co) = No
1 -
-
+
0.004 2
:l I:
'
(I)
where 8 = 6 K and W= 150K are the widths of the Lorentzian and S-band respectively which are needed to agree with normal state properties. The calculation by Skalski, Betbeder-Matibet and Weiss [8] of the difference in free energy between
775
776
Vol. 5S, No. a
THE SUPERCONDUCTING SPECIFIC HEAT JUMP OF UBela
normal and superconducting states for systems with paramagnetic impurities can easily be generalized to the case of a pure system with energy dependent densityof-states. Note that our units are such that k s = l~ = 1.
I
dco2N(¢o) Etanh
~2.40 1.79 ~1.74
/
i
--¢otanh
0
3 1.0
A2 V'
0.(
Ss = 4
d~oU(co) l n 0 + e - 0 ~ + ~
,
0
sN = 4 ;d~oN(co)
(1 + e - ~ ° ) + T - g - ~
0
(2)
F N - s = E N - s -- T(SN -- Ss).
Here E N_ s is the normal state energy minus the superconducting state energy, SN(S) is the entropy of the normal (superconducting) state and F N - s is the difference in free energy. One also has E = ~ and = 1/T, where T is absolute temperature. N ( ~ ) is the normal state density-of-states as given in (1), N o V is the BCS coupling constant and A is the order parameter, which is of course the solution of a generalized BCS equation:.~D
tanh ( - ~ )
1 -- V J d ~ N ( ~ o ) - o
(3)
E
The major step involved in the derivation of equations (2) and (3) is the confirmation that with the normal state density.of-states N(co) as given in equation (1) one has the superconducting density-of.states given
by Ns(co) _ =0
I¢ol
N ( ~ - J - A2)
for
I¢ol > A
for
I~l < A. (4)
This is easily seen by direct substitution of the model density-of-states, equation (1), into the equation for the superconducting density-of.states in terms of the Green's function. It is now a simple matter to solve equation (3) numerically by iteration for the order parameter A, and then to evaluate equation (2) to get F N_ s ( T ) and the entropies for a series of temperatures. The heat capacity may then be found by interpolating the entropy as a function of temperature and then evaluating the derivative of the interpolating function, using Cs = T(aSs/aT ). The free energy is also found to allow evaluation of the thermodynamic critical field deviation
I
~.CTO
~ .0 EN-S =
I
~G
/,
• .........
,LO
•
4
~
~
_
f3K
I
7..0 T/T¢
I
5.0
4,0
Fig. 1. The specific heat as a function of temperature for various widths 8 of the Lorentzian peak in the densityof-states. The experimental points are from [10]. The normalization of the curves for 6 = 5 and 7 K was chosen to fit the points near the peak at T/Tc ~- 3. The curve for 8 = 3 K obviously cannot fit the data for the specific heat of the normal state, so it was normalized to agree with the normal state specific heat at To.
function D(t) = He(t)/Hc(O)-- (1 -- t2), where H2e(t)/8n = FN_ s(t) and t = T/Te. The results for the heat capacity may be seen in Fig. 1, which shows the specific heat as a function of temperature for various values of 8, the width of the Lorentzian peak in the density-of-states. Note that because these results are very insensitive to W as long as I¢ ~ 8, the width of the S-band has been fixed at the value chosen by Overhauser and Appel [2], I¢ = 15OK. Also, all curves are for the same critical temperature Tc = 0.9 K, so the coupling constant No V varies with changes in 8. The width 8 = 6 K as chosen by Overhauser and Appel [2] causes a peak in the normal state heat capacity near 3 K, which is in agreement with experiment. Note however that when the normalization is chosen to fit the data near 3 K one fmds that C~(Tc) is significantly larger than the experimental results. This, when combined with a superconducting specific heat at T c which is significantly smaller than the value found by extrapolating from the data between T/Tc = 0.8 and 0.9 to T = To, results in a specific heat jump of 1.79 compared to an estimated ideal value of 2.5 from the experimental data. The agreement here cannot really be improved by varying 8. For example, although the curve for ~i = 7 K does a better job of fitting the data for the normal state near Tc (at the expense of slightly worse agreement at higher temperatures) the jump in this case is only 1.74, and the superconducting specific heat near Tc is obviously too small. Going in the other direction, for 5 = 3 K one does find AC/CN(Tc)= 2.4, roughly in agreement with experiment, but for this width the peak in the normal state specific heat is down at T/Tc "" 1.6
Vo'l. 55, No. 8
THE SUPERCONDUCTING SPECIFIC HEAT JUMP OF UBels
Table I. Values of 2A/T c and the deviation function at t = 0.7 (D(t = 0.7)) for the sameparameters as used for the curves in Fig. 1: Tc = 0.9K, wD = 620K and W= 150K
5(K)
NoV
2AlTo
D(t =0.7)
3 6 7
0.685 0.481 0.449
3.70 3.62 3.60
- 0.041 -- 0.036 -- 0.036
or T " 1.4 K, much too low to agree with experimental results. One should also note that a peak in the density.ofstates does not significantly alter the low temperature limit of the specific heat; one still has C8 ~ e - a / r for T < A. Although the ratio 2A/T, does increase as decreases (see Table 1), one must still invoke anisotropy to explain the observed deviation from exponential behaviour for T < Te. Since anisotropy is known to reduce the size of the specific heat jump even in the case of a peak in the density-of-states [5], our results might well be considered as best case for comparison of this model with experiment. The inclusion of anisotropy can only increase the discrepancy between the widths required to agree with the location of the peak in the normal state specific heat and the magnitude of the jump. It has now been shown that when the s - f hybridization model of Overhauser and Appel [2] is combined with conventional BCS theory one does find that the specific heat jump is significantly larger than the BCS value of 1.43. However, the increase is not large enough to account for the experimental value of approximately 2.5 for UBe13. At this point one might speculate about additional mechanisms which could possibly be added without drastically altering the model which might account for the present discrepancy. For example, Razafimandimby, Fulde and Keller [9] have recently proposed a Kondo lattice model of the superconductivity of CeCu2Si2, a material which is similar to UBe~3 in many respects. They find an effective pairing interaction which is strongly energy dependent, and which includes pair breaking effects for quasiparticles with energies more than the Kondo temperature TK away from the Fermi energy. When this is included phenomenologically in our model by using a square well model for the pairing interaction with a cutoff at TK, with TK as an adjustable parameter, one quickly realizes that such a mechanism cannot provide a significant further enhancement of the specific heat jump. If 5 and Te are kept constant as TK is varied one f'mds that for TK ~" 8 the jump stays roughly constant near the value for Tx = 600, where this modified model
777
becomes identical with the original one. For Tg "--/5 o n e finds that AC/CN(T¢) is enhanced by a few percent and then actually decreases as Tg is reduced below 6. One should also note that this additional energy dependence in the pairing interaction does not affect the existence of a gap in the excitation spectrum, so that one must still invoke anisotropy to explain the non-exponential behaviour of the low temperature superconducting specific heat. Speculating further, one might continue in this direction by supposing that the large specific heat jump of UBela is due to both the peak in the density-ofstates and strong coupling corrections. In this case the low energy cutoff in the pairing interaction at Tg (rather than at WD) could explain how the low critical temperature of Tc = 0.9 K for UBet3 may be consistent with large strong coupling effects. The problem in this case, of course, is that the results may be strongly model dependent, and an exhaustive search of the various possibilities would be a rather large task without more experimental information as a guide. One further point worth mentioning is that the thermodynamic critical field deviation function D(t) in the model of Overhauser and Appel [2] decreases as the width ~ decreases, as indicated in Table 1. Since anisotropy is also well-known to decrease D(t), one would predict that if UBe13 could be described in a weak coupling model then D(t) would lie below the BCS curve. On the other hand, if strong coupling effects are important no clear prediction is possible because of the competition between the strong coupling corrections and the anisotropy and structure in the density-of-states, which tend to increase and decrease D(t) respectively. In conclusion, one can say that while the s - f hybridization model of Overhauser and Appel [2] does correctly predict an enhancement of the specific heat jump over the BCS value, within the weak coupling approximation the increase is too small to agree with the measured value for UBe13. The most obvious additional correction which might be included within the conf'mes of this model is the addition of strong coupling effects. However, this could not be reasonably undertaken without further experimental information, for example the critical magnetic field deviation function.
REFERENCES 1. 2. 3.
For a review, see G.R. Stewart, Rev. Mod. Phys. 56,755 (1984). A.W. Overhauser & J. Appel, Phys. Rev. B31,193 (1985). H.R. Ott, H. Rudiger, T.M. Rice, K. Ueda, Z. Fisk & J.L. Smith, Phys. Roy. Lett. 52, 1915 (1984).
778 4. 5. 6. 7.
THE SUPERCONDUCTING SPECIFIC HEAT JUMP OF UBela J.P. Carbotte & A. Abd EI.Rahman, Can. J. Phys. 60, 1029 (1982). H.G. Zarate & J.P. Carbotte, Can. J. Phys. 61, 825 (1983). B. Mitrovid & J.P. Carbotte, Can. J. Phys. 61, 872 (1983). E. Schachinger & J.P. Carbotte, J. Phys. F13, 2615 (1983).
8. 9. 10.
Vol. 55, N~. 8
S. Skalski, O. Betbeder-Matibet & P.R. Weiss, Phys. Rev. 136, A1500 (1964). H. Razafimandimby, P. Fulde & J. Keller, Z. Phys. B54, 111 (1984). H.R. Ott, H. Rudiger, Z. Fisk & J.L. Smith, in Proceedings of the NA TO A dvanced Study Institute on Moment Formation in Solids, Vancouver Island, (1983), p. 305, (Edited by W.J.L. Buyers), Plenum, New York, (1984).
Solid State Communications, Vol. 55, No. 8, pp. 775-778, 1985. Printed in Great Britain.
THE SUPERCONDUCTING SPECIFIC HEAT JUMP OF UBe~3 AND A NARROW PEAK IN THE ELECTRONIC DENSITY-OF-STATES W. Stephan and J.P. Carbotte Department of Physics, McMaster University, Hamilton, Ontario LSS 4M1, Canada
(Received 11 April 1985 by M.F. Collins) The superconducting specific heat jump for the s - f hybridization model of UBels of Overhauser and Appel [2] is calculated. The narrow peak in the density-of-states is found to enhance the size of the jump relative to the BCS value of 1.43, but in weak coupling the increase is insufficient to agree with the measured value for UBe~a. The Kondo lattice model of Razafimandimby, Fulde and Keller [9] is considered as a possible modification, but in weak coupling a simple estimate shows that this cannot increase the specific heat jump significantly. It is speculated, however, that a consistent picture may still be possible with conventional strong coupling corrections. THE HEAVY FERMION superconductor UBel3 has been the subject of much recent research [1]. Although there is currently no consensus as to the appropriate theoretical description for this and other heavy fermion alloys, a recent proposal of Overhauser and Appel [2] seems to merit further investigation. The basic assumption of this s - f hybridization model is that the crystal field F7 doublet, which derives from the free uranium atom 5fs/2 levels, will lie at the Fermi energy and will hybridize with the Be(2s)-U(7s) conduction band. This results in a very narrow Lorentzian peak in the conductionelectron density-of-states centered at the Fermi energy. Using a model density-of-states with three parameters fit to the experimental heat capacity and magnetic susceptibility Overhauser and Appel [2] are able to explain quantitatively many of the anomalous properties of UBela using a simple single quasi-particle picture. The suggestion is also made that the superconductivity which appears below 0.9 K is of the conventional singlet paired BCS type. The deviation of the temperature dependence of the low temperature specific heat from the exponential BCS prediction (C, cc e - n I T for T < A) is attributed to anisotropy of the gap, which arises in this model from the anisotropy of the hybridization matrix element. The purpose of this letter is to consider the magnitude of the jump in the specific heat at Tc in the context of this model. Recall that BCS theory predicts that AC/Cn(Tc)= 1.43, where AC = Cs--CN and Cs and CN are the heat capacities of the superconducting and normal states respectively. On the other hand the latest experimental evidence indicates that for UBet3 one has AC/Cn(Tc) ~ 2.5 [3]. A specific heat jump of this size would be reasonable for a conventional strong
coupling semiconductor, but such an explanation goes against the spirit of the model of Overhauser and Appel [2], where many-body effects are assumed to be small. However, another factor which may not be so generally known is that a peak in the density-of.states near the Fermi energy also enhances AC[Cn(Te) compared to the BCS value. Previous results for the free energy of a superconductor with a Lorentzian peak in the densityof-states due to Carbotte and Abd E1-Rahman [4] and Zarate and Carbotte [5] in weak coupling and to Mitrovid and Carbotte [6] and Schachinger and Carbotte [7] in strong coupling are not directly applicable here because the width of the peak needed to agree with the normal state heat capacity of UBe13 is 6K, which is comparable to the critical temperature T e = 0 . 9 K . The previous calculations [4-7] have assumed that the scale of variation of the density-of-states is on the order of the Debye energy cot), which would be approximately 620 K for UBe13. We therefore present here the results for a numerical evaluation of the heat capacity of a BCS superconductor with a narrow peak in the density-of-states centred at the Fermi energy. Following Overhauser and Appel [2] the model density-of-states is N(co) = No
1 -
-
+
0.004 2
:l I:
'
(I)
where 8 = 6 K and W= 150K are the widths of the Lorentzian and S-band respectively which are needed to agree with normal state properties. The calculation by Skalski, Betbeder-Matibet and Weiss [8] of the difference in free energy between
775
776
Vol. 5S, No. a
THE SUPERCONDUCTING SPECIFIC HEAT JUMP OF UBela
normal and superconducting states for systems with paramagnetic impurities can easily be generalized to the case of a pure system with energy dependent densityof-states. Note that our units are such that k s = l~ = 1.
I
dco2N(¢o) Etanh
~2.40 1.79 ~1.74
/
i
--¢otanh
0
3 1.0
A2 V'
0.(
Ss = 4
d~oU(co) l n 0 + e - 0 ~ + ~
,
0
sN = 4 ;d~oN(co)
(1 + e - ~ ° ) + T - g - ~
0
(2)
F N - s = E N - s -- T(SN -- Ss).
Here E N_ s is the normal state energy minus the superconducting state energy, SN(S) is the entropy of the normal (superconducting) state and F N - s is the difference in free energy. One also has E = ~ and = 1/T, where T is absolute temperature. N ( ~ ) is the normal state density-of-states as given in (1), N o V is the BCS coupling constant and A is the order parameter, which is of course the solution of a generalized BCS equation:.~D
tanh ( - ~ )
1 -- V J d ~ N ( ~ o ) - o
(3)
E
The major step involved in the derivation of equations (2) and (3) is the confirmation that with the normal state density.of-states N(co) as given in equation (1) one has the superconducting density-of.states given
by Ns(co) _ =0
I¢ol
N ( ~ - J - A2)
for
I¢ol > A
for
I~l < A. (4)
This is easily seen by direct substitution of the model density-of-states, equation (1), into the equation for the superconducting density-of.states in terms of the Green's function. It is now a simple matter to solve equation (3) numerically by iteration for the order parameter A, and then to evaluate equation (2) to get F N_ s ( T ) and the entropies for a series of temperatures. The heat capacity may then be found by interpolating the entropy as a function of temperature and then evaluating the derivative of the interpolating function, using Cs = T(aSs/aT ). The free energy is also found to allow evaluation of the thermodynamic critical field deviation
I
~.CTO
~ .0 EN-S =
I
~G
/,
• .........
,LO
•
4
~
~
_
f3K
I
7..0 T/T¢
I
5.0
4,0
Fig. 1. The specific heat as a function of temperature for various widths 8 of the Lorentzian peak in the densityof-states. The experimental points are from [10]. The normalization of the curves for 6 = 5 and 7 K was chosen to fit the points near the peak at T/Tc ~- 3. The curve for 8 = 3 K obviously cannot fit the data for the specific heat of the normal state, so it was normalized to agree with the normal state specific heat at To.
function D(t) = He(t)/Hc(O)-- (1 -- t2), where H2e(t)/8n = FN_ s(t) and t = T/Te. The results for the heat capacity may be seen in Fig. 1, which shows the specific heat as a function of temperature for various values of 8, the width of the Lorentzian peak in the density-of-states. Note that because these results are very insensitive to W as long as I¢ ~ 8, the width of the S-band has been fixed at the value chosen by Overhauser and Appel [2], I¢ = 15OK. Also, all curves are for the same critical temperature Tc = 0.9 K, so the coupling constant No V varies with changes in 8. The width 8 = 6 K as chosen by Overhauser and Appel [2] causes a peak in the normal state heat capacity near 3 K, which is in agreement with experiment. Note however that when the normalization is chosen to fit the data near 3 K one fmds that C~(Tc) is significantly larger than the experimental results. This, when combined with a superconducting specific heat at T c which is significantly smaller than the value found by extrapolating from the data between T/Tc = 0.8 and 0.9 to T = To, results in a specific heat jump of 1.79 compared to an estimated ideal value of 2.5 from the experimental data. The agreement here cannot really be improved by varying 8. For example, although the curve for ~i = 7 K does a better job of fitting the data for the normal state near Tc (at the expense of slightly worse agreement at higher temperatures) the jump in this case is only 1.74, and the superconducting specific heat near Tc is obviously too small. Going in the other direction, for 5 = 3 K one does find AC/CN(Tc)= 2.4, roughly in agreement with experiment, but for this width the peak in the normal state specific heat is down at T/Tc "" 1.6
Vo'l. 55, No. 8
THE SUPERCONDUCTING SPECIFIC HEAT JUMP OF UBels
Table I. Values of 2A/T c and the deviation function at t = 0.7 (D(t = 0.7)) for the sameparameters as used for the curves in Fig. 1: Tc = 0.9K, wD = 620K and W= 150K
5(K)
NoV
2AlTo
D(t =0.7)
3 6 7
0.685 0.481 0.449
3.70 3.62 3.60
- 0.041 -- 0.036 -- 0.036
or T " 1.4 K, much too low to agree with experimental results. One should also note that a peak in the density.ofstates does not significantly alter the low temperature limit of the specific heat; one still has C8 ~ e - a / r for T < A. Although the ratio 2A/T, does increase as decreases (see Table 1), one must still invoke anisotropy to explain the observed deviation from exponential behaviour for T < Te. Since anisotropy is known to reduce the size of the specific heat jump even in the case of a peak in the density-of-states [5], our results might well be considered as best case for comparison of this model with experiment. The inclusion of anisotropy can only increase the discrepancy between the widths required to agree with the location of the peak in the normal state specific heat and the magnitude of the jump. It has now been shown that when the s - f hybridization model of Overhauser and Appel [2] is combined with conventional BCS theory one does find that the specific heat jump is significantly larger than the BCS value of 1.43. However, the increase is not large enough to account for the experimental value of approximately 2.5 for UBe13. At this point one might speculate about additional mechanisms which could possibly be added without drastically altering the model which might account for the present discrepancy. For example, Razafimandimby, Fulde and Keller [9] have recently proposed a Kondo lattice model of the superconductivity of CeCu2Si2, a material which is similar to UBe~3 in many respects. They find an effective pairing interaction which is strongly energy dependent, and which includes pair breaking effects for quasiparticles with energies more than the Kondo temperature TK away from the Fermi energy. When this is included phenomenologically in our model by using a square well model for the pairing interaction with a cutoff at TK, with TK as an adjustable parameter, one quickly realizes that such a mechanism cannot provide a significant further enhancement of the specific heat jump. If 5 and Te are kept constant as TK is varied one f'mds that for TK ~" 8 the jump stays roughly constant near the value for Tx = 600, where this modified model
777
becomes identical with the original one. For Tg "--/5 o n e finds that AC/CN(T¢) is enhanced by a few percent and then actually decreases as Tg is reduced below 6. One should also note that this additional energy dependence in the pairing interaction does not affect the existence of a gap in the excitation spectrum, so that one must still invoke anisotropy to explain the non-exponential behaviour of the low temperature superconducting specific heat. Speculating further, one might continue in this direction by supposing that the large specific heat jump of UBela is due to both the peak in the density-ofstates and strong coupling corrections. In this case the low energy cutoff in the pairing interaction at Tg (rather than at WD) could explain how the low critical temperature of Tc = 0.9 K for UBet3 may be consistent with large strong coupling effects. The problem in this case, of course, is that the results may be strongly model dependent, and an exhaustive search of the various possibilities would be a rather large task without more experimental information as a guide. One further point worth mentioning is that the thermodynamic critical field deviation function D(t) in the model of Overhauser and Appel [2] decreases as the width ~ decreases, as indicated in Table 1. Since anisotropy is also well-known to decrease D(t), one would predict that if UBe13 could be described in a weak coupling model then D(t) would lie below the BCS curve. On the other hand, if strong coupling effects are important no clear prediction is possible because of the competition between the strong coupling corrections and the anisotropy and structure in the density-of-states, which tend to increase and decrease D(t) respectively. In conclusion, one can say that while the s - f hybridization model of Overhauser and Appel [2] does correctly predict an enhancement of the specific heat jump over the BCS value, within the weak coupling approximation the increase is too small to agree with the measured value for UBe13. The most obvious additional correction which might be included within the conf'mes of this model is the addition of strong coupling effects. However, this could not be reasonably undertaken without further experimental information, for example the critical magnetic field deviation function.
REFERENCES 1. 2. 3.
For a review, see G.R. Stewart, Rev. Mod. Phys. 56,755 (1984). A.W. Overhauser & J. Appel, Phys. Rev. B31,193 (1985). H.R. Ott, H. Rudiger, T.M. Rice, K. Ueda, Z. Fisk & J.L. Smith, Phys. Roy. Lett. 52, 1915 (1984).
778 4. 5. 6. 7.
THE SUPERCONDUCTING SPECIFIC HEAT JUMP OF UBela J.P. Carbotte & A. Abd EI.Rahman, Can. J. Phys. 60, 1029 (1982). H.G. Zarate & J.P. Carbotte, Can. J. Phys. 61, 825 (1983). B. Mitrovid & J.P. Carbotte, Can. J. Phys. 61, 872 (1983). E. Schachinger & J.P. Carbotte, J. Phys. F13, 2615 (1983).
8. 9. 10.
Vol. 55, N~. 8
S. Skalski, O. Betbeder-Matibet & P.R. Weiss, Phys. Rev. 136, A1500 (1964). H. Razafimandimby, P. Fulde & J. Keller, Z. Phys. B54, 111 (1984). H.R. Ott, H. Rudiger, Z. Fisk & J.L. Smith, in Proceedings of the NA TO A dvanced Study Institute on Moment Formation in Solids, Vancouver Island, (1983), p. 305, (Edited by W.J.L. Buyers), Plenum, New York, (1984).