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Thermal conductivity of holmium at low temperatures
Volume
30A. number
52 = (1.3 f 0.3) x 10
1
PHYSICS
-18 ( ‘;c?
j-1*3*o’2
LETTERS
8 September
1969
eq. (1) must be replaced by an expression such as r=AK3, A being independent of T-T,. More sophisticated experiments now under way, seem to confirm this last assumption.
cm2. (3)
C. If K[ ., 1, the values of D and 52 deduced from eq. (1) are significantly different from those given by eqs. (2) and (3). In particular the pointby-point computation of t2 from an experiment performed at T - Tc = 4 x 10m3 OK using the value of D deduced by extrapolation from eq. (2), shows that the values of 4 2 depend 011 K. as soon as {K > 1; only the values obtained when 5 K S 1 are in a good agreement with those given by eq. (3). We think that these facts can be explained only if we admit that eq. (1) is no Iongev valid when (Kz 1. Probably the case
References 1. W. Botch and M. Fixman.
196.
J. Chem. Phys. 42 (1965)
2. M. E. Fisher. J. Math. Phys. 5 (1964) 944. 3. J. Swift, Phys. Rev. 173 (1968) 257. 4. B. I. Halperin and P. C. Hohenberg. Phgs. Rev. Letters 19 (1967) 700; B. Chu and F. J. Schoenes. Phys. Rev. Letters 21 (1968) 6; P. Berge. P. Calmettes and B. Volochine. Phys. Letters 27A (1968) 637; H. L. Swinney and H. Z. Cummins, Phys. Rev. 171 (1968) 152.
the transition between “hydrodynamic region” and “critical region”. In this last case. the “dynamic scaling law” [4] theory assumes that
*****
THERMAL
CONDUCTIVITY
OF
HOLMIUM
AT
LOW
TEMPERATURES
R. RATNALINGAM and J. B. SOUSA * Clarendon Laboratory. Oxford. UK Received
26 June 1969
The thermal conductivity of a single crystal and a polycrystal of holmium. measured between 0.8 and 4.2’K. can be separated into a linear (2’) and a small quadratic (2’2) contribution. The Lorentz numbers. as estimated from the linear term by electrical resistivity measurements. are in reasonable agreement with the theoretical value.
The thermal conductivity of heavy rare earths has been previously investigated in the liquid 4He range of temperatures [l-3], but the results reported were unexpected, the thermal conductivity largely exceeding the value derived from the electrical conductivity via the Wiedemann-Franz law. For example, Lorentz numbers greater than the theoretical value by a factor of 5 have been reported, and this would imply an extra contribution to the thermal conductivity exceeding the electronic part by a similar factor. More recently another anomaly has been reported [4] for holmium at very low temperatures (below N 1.3oK) where the thermal conductivity suddenly begins to rise and passes through a maximum around 1°K. Since holmium, at these temperatures, has a very high nuclear specific heat, it has been suggested that this might provide an extra contribution to the thermal conductivity, via a RudermanKittel-Kasuya-Yosida type interaction between the nuclei, 4f electrons and conduction electrons. 8
In view of these anomalies, we have carried out a detailed investigation of the thermal conductivity in the range of temperature 0.3OK-4.2OK on most of the rare earth metals. In all cases the Lorentz numbers do not show the discrepancies mentioned above, the values obtained differing at most by a few percent from the theoretical value of 2.45 x 10s8 W 51 cm-l. A full report of these results is being prepared for publication. In this note we are only concerned with holmium, in view of the anomaly of nuclear origin previously reported for this metal [4]. We have measured the temperature dependence of the thermal conductivity (K) of a polycrystal and a single crystal of holmium using the experimental technique previously described [5]. The resistivity ratios of these samples (table 1) are * On leave of absence from the Department of Physics and Centro Estudos Fisica Nuclear e Electronica, University of Porte. Portugal, granted by the Institute Alta Cultura. and with a Calouste Gulbenkian Scholarship.
Volume 30A. number 1
PHYSICS
LETTERS
8 September 1969
Table 1 Sample
P3OO/P4.2
o!
p4.2 (PQcrn)
108 x L
(mW,‘OK2 cm)
P (mW;‘OK3cm)
(aW cm-l)
Ho I
7.23
9.73
2.85
0.087
2.76
Ho II
9.95
9.33
2.55
0.053
2.38
comparable to the value quoted by Rae [4] for his holmium sample (16.6). Fig. 1 shows a plot of K versus T for our specimens. Both curves can be accurately described by an expression of the form
K=(Y T+pT2 which is the same as that found on our previously reported measurements on gadolinium [6] (for numerical values of Q and /3 see table 1). Assuming the linear term due to electrons with mean
P.
free path limited by impurity scattering we find, using electrical resistivity measurements, values for the Lorentz number (L) reasonably close to the theoretical value (see table 1). The origin of the small T2 contribution is not clear, although we can say that it is also present in other rare earth metals investigated, even on those which do not have any appreciable nuclear contribution to the specific heat at the temperatures considered. In conclusion, we can say that the thermal conductivity of the holmium samples investigated is essentially due to electrons with mean free path 2 limited by impurity scattering (1 - low4 cm) and there is no evidence of any anomaly down to 0.8oK, where the nuclear contribution to the specific heat is already very large. This fact, combined with a rather low thermal conductivity, causes a long relaxation time (- 2 hours at 0.8OK) and this makes it extremely difficult to extend the measurements to lower temperatures. The authors wish to express their gratitude to Dr. K. Mendelssohn, for his encouragement and interest in this work and to Dr. D. Bagguley for the loan of the holmium single crystal. It is also a pleasure to acknowledge the financial support provided for this work by the Calouste Gulbenkian Foundation (J.S.) and a Rhodes Scholarship (R. R ).
polycryrtal
References 1. S. Arajs and R. V. Colvin. J. Appl. Phys. 35 (1964)
1043. 2. N. G. Aliev and N. V. Volkenshtein, Soviet Phys. JETP 49 (1965) 17. 3. D. W. Boyds and S. Legvold. Phys. Rev. 174 (1968) 377. 4. K. V. Rao, Phys. Rev. Letters 22 (1969) 943. 5. J. B. Sousa, Cryogenics 8 (1968) 105. 6. R. Ratnalingam and ‘J. B. Sousa, Phys. Letters. to be published.
Fig. 1.
*****
9
30A. number
52 = (1.3 f 0.3) x 10
1
PHYSICS
-18 ( ‘;c?
j-1*3*o’2
LETTERS
8 September
1969
eq. (1) must be replaced by an expression such as r=AK3, A being independent of T-T,. More sophisticated experiments now under way, seem to confirm this last assumption.
cm2. (3)
C. If K[ ., 1, the values of D and 52 deduced from eq. (1) are significantly different from those given by eqs. (2) and (3). In particular the pointby-point computation of t2 from an experiment performed at T - Tc = 4 x 10m3 OK using the value of D deduced by extrapolation from eq. (2), shows that the values of 4 2 depend 011 K. as soon as {K > 1; only the values obtained when 5 K S 1 are in a good agreement with those given by eq. (3). We think that these facts can be explained only if we admit that eq. (1) is no Iongev valid when (Kz 1. Probably the case
References 1. W. Botch and M. Fixman.
196.
J. Chem. Phys. 42 (1965)
2. M. E. Fisher. J. Math. Phys. 5 (1964) 944. 3. J. Swift, Phys. Rev. 173 (1968) 257. 4. B. I. Halperin and P. C. Hohenberg. Phgs. Rev. Letters 19 (1967) 700; B. Chu and F. J. Schoenes. Phys. Rev. Letters 21 (1968) 6; P. Berge. P. Calmettes and B. Volochine. Phys. Letters 27A (1968) 637; H. L. Swinney and H. Z. Cummins, Phys. Rev. 171 (1968) 152.
the transition between “hydrodynamic region” and “critical region”. In this last case. the “dynamic scaling law” [4] theory assumes that
*****
THERMAL
CONDUCTIVITY
OF
HOLMIUM
AT
LOW
TEMPERATURES
R. RATNALINGAM and J. B. SOUSA * Clarendon Laboratory. Oxford. UK Received
26 June 1969
The thermal conductivity of a single crystal and a polycrystal of holmium. measured between 0.8 and 4.2’K. can be separated into a linear (2’) and a small quadratic (2’2) contribution. The Lorentz numbers. as estimated from the linear term by electrical resistivity measurements. are in reasonable agreement with the theoretical value.
The thermal conductivity of heavy rare earths has been previously investigated in the liquid 4He range of temperatures [l-3], but the results reported were unexpected, the thermal conductivity largely exceeding the value derived from the electrical conductivity via the Wiedemann-Franz law. For example, Lorentz numbers greater than the theoretical value by a factor of 5 have been reported, and this would imply an extra contribution to the thermal conductivity exceeding the electronic part by a similar factor. More recently another anomaly has been reported [4] for holmium at very low temperatures (below N 1.3oK) where the thermal conductivity suddenly begins to rise and passes through a maximum around 1°K. Since holmium, at these temperatures, has a very high nuclear specific heat, it has been suggested that this might provide an extra contribution to the thermal conductivity, via a RudermanKittel-Kasuya-Yosida type interaction between the nuclei, 4f electrons and conduction electrons. 8
In view of these anomalies, we have carried out a detailed investigation of the thermal conductivity in the range of temperature 0.3OK-4.2OK on most of the rare earth metals. In all cases the Lorentz numbers do not show the discrepancies mentioned above, the values obtained differing at most by a few percent from the theoretical value of 2.45 x 10s8 W 51 cm-l. A full report of these results is being prepared for publication. In this note we are only concerned with holmium, in view of the anomaly of nuclear origin previously reported for this metal [4]. We have measured the temperature dependence of the thermal conductivity (K) of a polycrystal and a single crystal of holmium using the experimental technique previously described [5]. The resistivity ratios of these samples (table 1) are * On leave of absence from the Department of Physics and Centro Estudos Fisica Nuclear e Electronica, University of Porte. Portugal, granted by the Institute Alta Cultura. and with a Calouste Gulbenkian Scholarship.
Volume 30A. number 1
PHYSICS
LETTERS
8 September 1969
Table 1 Sample
P3OO/P4.2
o!
p4.2 (PQcrn)
108 x L
(mW,‘OK2 cm)
P (mW;‘OK3cm)
(aW cm-l)
Ho I
7.23
9.73
2.85
0.087
2.76
Ho II
9.95
9.33
2.55
0.053
2.38
comparable to the value quoted by Rae [4] for his holmium sample (16.6). Fig. 1 shows a plot of K versus T for our specimens. Both curves can be accurately described by an expression of the form
K=(Y T+pT2 which is the same as that found on our previously reported measurements on gadolinium [6] (for numerical values of Q and /3 see table 1). Assuming the linear term due to electrons with mean
P.
free path limited by impurity scattering we find, using electrical resistivity measurements, values for the Lorentz number (L) reasonably close to the theoretical value (see table 1). The origin of the small T2 contribution is not clear, although we can say that it is also present in other rare earth metals investigated, even on those which do not have any appreciable nuclear contribution to the specific heat at the temperatures considered. In conclusion, we can say that the thermal conductivity of the holmium samples investigated is essentially due to electrons with mean free path 2 limited by impurity scattering (1 - low4 cm) and there is no evidence of any anomaly down to 0.8oK, where the nuclear contribution to the specific heat is already very large. This fact, combined with a rather low thermal conductivity, causes a long relaxation time (- 2 hours at 0.8OK) and this makes it extremely difficult to extend the measurements to lower temperatures. The authors wish to express their gratitude to Dr. K. Mendelssohn, for his encouragement and interest in this work and to Dr. D. Bagguley for the loan of the holmium single crystal. It is also a pleasure to acknowledge the financial support provided for this work by the Calouste Gulbenkian Foundation (J.S.) and a Rhodes Scholarship (R. R ).
polycryrtal
References 1. S. Arajs and R. V. Colvin. J. Appl. Phys. 35 (1964)
1043. 2. N. G. Aliev and N. V. Volkenshtein, Soviet Phys. JETP 49 (1965) 17. 3. D. W. Boyds and S. Legvold. Phys. Rev. 174 (1968) 377. 4. K. V. Rao, Phys. Rev. Letters 22 (1969) 943. 5. J. B. Sousa, Cryogenics 8 (1968) 105. 6. R. Ratnalingam and ‘J. B. Sousa, Phys. Letters. to be published.
Fig. 1.
*****
9