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Thermal and electrical resistivities of rhenium from 2°K to 20°K
J. Phys. Chem. Solids
Pergamon Press 1967. Vol. 28, pp. 2581-2587.
Printed in Great Britain.
THERMAL AND ELECTRICAL RESISTIVITIES OF RHENIUM FROM 2°K TO 20°K* J. T. SCHRIBMPF U.S. Naval Research Laboratory, Washington, D.C. 20390 (Received 13 March 1967 ; in rev&d form5 June 1967)
Ahsttuot-Measurement8 of the thermal and electrical reeistivities of highly pure monocrystala of rhenium have been made at temperature8 between 2’K and 2O’K. The ideal re8i8tivitie8can be described by pt CCp (in the electrical case) and by ru, a~ T” (in the thermal cme) only if m and n are allowed to vary with temperature. The value8 of nt and n fall smoothly from 54 and 3.3 at 20°K and become 2 and 1, nspectively, at the lowe8t temperaturea. The anisotropy at all temperature8 is found to be approximately p&8 N 1.3 and r.ucL/wc~~ N 14, where the parallel orientation is along the c axie of the hexagon&close-packed lattice of rhenium. Below about llaK the ideal re&tivities obey a Wiedemann-Frana law with a Lorena number of roughly O-5 x 10m8 Va/dega.
INTRODUCTION
IN THEIRexperimental study of the thermal and electrical resistivities of the transition elements at low temperatures, WHITE and WOODS(~) found that in rhenium the ideal electrical resistivity (PJ was proportional to P1, while the ideal thermal resistivity (w,) varied as Ts3, where T is the absolute temperature. However, these results were obtained at temperatures in the neighborhood of one-tenth the Debye temperature (using White and Woods’ value of 280°K for 0,) whereas it is highly desirable to observe p1 and wf at the lowest temperatures possible. In the present work, a combination of very high specimen purity with a quite sensitive measuring technique has made it possible to deduce the temperature dependence of wi and pi down to roughly one-fiftieth of the Debye temperature. We assume throughout this report that thermal conduction via the lattice is negligible, for, although they are not definitive in this respect, the data do not indicate the presence of an appreciable lattice thermal conductivity. *Thi8 work was 8~pp01ted in part by the Material8 Science8 Division of the Advanced Research Project8 Agency. 2581
EXPERIMENTAL The stainless steel cryostat, shown schematically in Fig. 1, usee the conventional axial-flow technique. The long tube extending from the top of the cryo8tat through the helium bath is terminated in a section of 8tainless Steel and a copper block. Controlling the block temperature control8 the temperature of the sunple, with which it make8 thermal contact. Except for the heat leak along the tube, the block-sample combination ir isolated by the vacuum (1 x 10-8 torr) and the liquid helium and nitrogen temperature radiation shiel&. The u8e of a common vacuum for the sample area and the cryo8tat make8 it po88ible to have the tail 8ection demountable with only one (room-temperature) vacuum 8eal. The wall8 and bottom of the helium chamber are made of copper M)that the bottom of the helium bath can be rued a8 a reference temperature. Wire8 for heating and making measurements are brought through StupakofT 8eab in the cryo8tat wall and thermally anchored to the nitrogen and helium-temperature 8hiel&, a8 well a8 to the copper block, before they make contact with the 8pecimen. Thi8 scheme ha8 the advantage that e.m.f.‘r of thermal origin in the lead wire8 are more constant than those in the uBu(11technique, where wire8 are brought through the bath and changing liquid level8 cause the thermal e.m.f.‘a to drift. The 200&m heaters, wound of con8nunan wire, are powdered by a simple voltage divider and battery network. A control heater, not shown, is contained in the copper block. With this cryo8tat, temperature control i8 achieved in two ways. Fii, pIacing dry helium gae in the tube at a pre~un of about 3 lb/in.g (gage) cau8e8 a rapid accumulation of liquid helium in the bottom of the tube.
2582
J. T.
FLANGE
(TO
SCHRIEMPF
He WELL 1
TO PUMPS -
BOTTOM
OF He WELL
ADED
JOINTS
RE FEED-THRDUGHS FLANGE
JOINT-
HERMOMETERS
FIG. 1. Thermal and electrical conductivity cryostat. Then, by means of a conventional pump and manostat syaem, temperatures between 2 and 4~2°K are establiahed. Second, for temperatures from 4.2 to 2O”K, the of about tubei8seakdwithdryheliumgaaatapremure 0~15torr,andan~~~~pmountofdirectcumntis supplied to the heater in the copper block. The &ability above 4.2”K haa been enhanced by placing a copper bar in the tube. Thk bar (not shown in Fig. 1) rests on top of the copper block and extends to a point just above the bottom of the helium well. The introduction of the bar has cawed no appreciabk change in the behavior of the cryostat below 4~2°K. Temperature differences were measured with a differential thermowupk (gold-O~02at.% iron vs. ~ilvet--O*37at.~~ gold) which was calibrated in a separate et with a germanium resktsnce thermometer. Thi8 germanium thermometer was calibrated at the National Bureau of Standards in accordance with the 1965 Provkional Temperature Scale.(a) The therm+ couple junctknn made contact to the specimen via copper ckmps which ako served a8 potential contact8 for the electrical resktivity determinations. Thk ensured that thespecimengeametrp was identical for both electrical and thermal meaaurementa, although the electrical leads were removed during the course of the thermal conductivity aperiment in order to reduce spurious heat flow. Voltages were measured with a potentiometer (Honeywell 2768) and a Semitive Research (Guildline Inatruments, Ltd.) Type 9460 photocell galvanometer amplifier. The potentiometer was mod&d by the manufacturer to have a resolution of 1 x 10-O V, and the
system was reliable to a precision of about 2 x 10 -9 V. The NBS-calibrated germanium resistance thermometer (manufactured by CryoCal, Inc.) was used in a conventional four-wire potentiometric reaktance-memuring circuit. The potentiometer and null detector were duplicates of the thermocoupk measuring equipment, except the potentiometer was unm&ed and had a resolution of 1-x 10 -8 v. The thermocouple wires (Sigmund Cohn Corp.) were 0.003 in. in dia.. and iunctions were made bv weldinn with an oxygen-&et&e flame. The Ag-Au-wire was used as received; the Au-Fe wire (Sigmund Cohn bar No. 1) was annealed in WUCWJ (about 1 x 1Om6ton) for 12 hr at SOO‘C. In their pioneering work of applying Au-Fe thermocouples to low temperature thermal conductivity meaeuremenm, BERMANand co-worken+*’ employed two corrective techniquen: (1) a superconducting shorting switch was used in the thermocouple circuit in order to allow corrections to be made for spuriouse.m.f.‘aof thermal origin in the lead wires, and (2) the well known “two-heater” technique wa.s used to correct not only temperature di&rencee between the thermocouple junctions and the specimen, but also thermal e.m.f.‘s which occur in the leads from the junctions to the ehorting switch. In the present apparatus both techniques are employed. (The superconducting shorting switches, mounted on the bottom of the copper helium well, are not shown in Fig. 1.) The present apparatus employs an NBS-calibrated germanium resistance thermometer in pkce of the gas thermometer used by Bxmr~~, et al.(s**’ for calibration of the thermocouple. The use of the
THERMAL
AND
ELECTRICAL
RESISTIVITIES
germanium thermometer enables accurate calibration to be made fairly easily and avoids the somewhat timeconsuming gas thermometer calibration technique. Calibration was accomplished by varying the temperature of the copper block with one thermocouple junction fastened to the bottom of the helium well and the other junction fastened to the copper clamp (on the sample) which contained the germanium thermometer. The reduction of the calibration data was carried out by the use of the Naval Research Laboratory CDC 3800 computing facility. Interpolation of the NBS resistancetemperature values was accomplished by fitting the data, in In R-ln T form, to a 10th degree power series in ln T.(6) The voltage-temperature data for the thermocouple was fitted to an 8th degree power series in T, and the thermoelectric power was obtained by taking the first temperature derivative of the power series.(s) All of the machine calculations were carried out in double precision, i.e. with about 25 signilicant figures in all of the arithmetical operations. Apart from errors in establishing the shape factor of the specimens, which will be discussed below, the precision of the measurements is estimated to be f 1 per cent. Specimens 2 and 3 were prepared in this h~boratory by electron-beam zone-refining (in a vacuum of about 10ea torr) 99.9 percent pure rhenium rods obtained from Chase Brass & Copper Co. Specimen 60 was prepared by Dr. R. Soden of Bell Telephone Laboratories by zone-relining compacted powder specimens.(“) All the specimens were single crystals, the details of which are presented in Table 1. Here the orientation angle is the angle between the rod axis and the c axis of the hexagonal lattice, and ps, (wT)s, and Ls are, respectively, the values upon extrapolation to 0°K of the electrical resistivity, the product of thermal resistivity and temperature, and the Lorenx number (L = p/(wZ’)). The effective cross-sectional areas (AH) shown in the table have been determined by making optical comparator measurements at a large number (N) of equally spaced diameters, computing the areas (A,), and finding
1 Iv A,-l. A Off -1 = _ N c
OF
RHENIUM
FROM
2°K
TO
20°K
2583
The largest errors in determining the specimen geometry am due to the finite siae of the clamps; these errors cause uncertainties in the effective lengths of the specimens which are of the order of 1 per cent for specimens 2 and 60, and about 5 per cent for specimen 3. RESULTS All
the
approximately ambient
were
repeatable
1 per cent both upon
cycling
to to the
(24°C) and upon *changing
temperature
the current of two,
measurements
flow through
in the electrical
the specimen case,
and factors
by factors of about
In Fig. 2 and Fig. 3 are shown logarithmic plots of the ideal thermal and electrical resistivities vs. temperature. In order to deduce values for w,, the following expression was used :
eight for the thermal
W( =
current.
wT-(wT), T
(1)
’
where w is the thermal resistivity, i.e. w = K-l, and (wT),/T is the residual thermal resistivity. The p1 values were obtained by simply subtracting from the measured resistivities the residual resistivity, p,-,. The dashed lines in Figs. 2 and 3 indicate the data of WHITE and WOODS(~)on an annealed polycrystalline sample which had a value of pzas/po of about 1360. The two sets of data are in reasonable agreement in the region of overlap. In both the electrical and thermal ideal resistivities it is apparent that a simple power law will not describe the temperature dependence over any appreciable range. However, if the temperature-dependent indices n and m are used, with p1 cc T* and w( oc Tn, the indices fall smoothly
Table 1. Physical details of rhenium monocrystuis
Specimen number
Effective cross-sectional area (cm’)
Effective length (cm)
10s po* (ohm-cm)
2 3 60 (Soden)
0.07586 0~07518 0.03618
8.36 2.22 8.65
7.54 6.45 3.66
2.49 2.51 5.04
* These are the values obtained upon extrapolation of the data to 0°K. t ~ssv is the electrical resistivity measured at 297°K. $ This is the angle between the rod axis and the c axis.
0.307 2.260 0.147
2.46 2.48 2.49
77”
2584
J. T.
SCHRIEMPF
THERMAL
AND I
ELECTRICAL I
I
RESISTIVITIES I
I
OF RHENIUM I
I
FROM
2°K TO 20°K
I
I
2585
T*[PK)*]
FIG. 4. Ideal electrical resistivity (solid circles) and product of temperature and ideal thermal rcsistivity (open circles) as functions of the square of the temperature for rhenium specimen Re 60. The dashed and solid lines are simply straight lines drawn through the low temperature values where the data are apparently following p, H !P and w, NT.
with decreasing temperature from the White and Woods’ values of m = 5.1 and n = 3.3 at about 20°K and become, at the lowest temperatures, m = 2andn = 1. No significant differences in this temperature dependence are evident among the three specimens. Figure 4 shows the low temperature values of pr and w,T for specimen Re 60 plotted as a function of T”. (The other two specimens exhibit the same behavior, although the data have somewhat more scatter.) At temperatures below roughly 5°K it is apparent that the data in Fig. 4 can be very well represented by pt - Ta and w, N T. The effects of anisotropy in both the electrical and thermal resistivities of hexagonal-closepacked rhenium are apparent in Figs. 2 and 3 where the values for specimen 2 are, at a given temperature, generally higher than the results for specimen 3. A careful inspection of the data indicates that, at least at temperatures above lO”K,
this effect is larger than the uncertainties in the measurements. Since specimens 2 and 3 are of nearly equal purity,* but the axis of specimen 2 is 77” from the c-axis whereas that of specimen 3 is 37” from c, an anisotropy effect is indicated. The data for these two specimens yield (perpendicularto-parallel) anisotropy ratios of about 1.1 for the ideal thermal resistivity and 1.3 for not only the ideal, but also the residual and room temperature electrical resistivities. Note that the resistivities of specimen 60, which is oriented in the basal plane, do not lie higher than those of specimen 2. Thus it is apparent that Matthiessen’s rule is not valid and the ideal resistivities are dependent on purity. * By purity we mean both freedom from impurities and absence of lattice defects. Furthermore, we assume that the ratio of electrical resistivities between room temperature and absolute zero provides a meaningful comparison of the purities of different specimens of the same metal.
J.
2586
T.
SCHRIEMPF
DISCUSSION In
Fig. 5 are plotted the values of both the total [L = p/(wT)] and theideal [L, = pl/(wiZ’)] Lorenz numbers. The values of L upon extrapolation to absolute zero are indicated in Table 1 and, at worst, agree within 2 per cent with the well-known theoretical value of 2443 x 10S8 V2/deg2 which obtains when the lattice conductivity is negligible and the electrons are scattered only by defects and impurities. Furthermore, the rapid decay of L with
at the lowest temperatures not only does p, obey a T2 temperature dependence, but also wUtbecomes linear in T, in agreement with the predictions of a model of electron-electron scatuering between different branches of the Fermi surface.(g-ll) Although a quadratic temperature dependence of the ideal electrical resistivity has been seen in some of the non-magnetic transition metals,(12’ an accompanying linear temperature dependence for the ideal thermal resistivity has not been previously observed. Furthermore, the ideal Lorenz values shown in Fig. 5 clearly are independent of temperature below about 11°K. Thus the data indicate that the ideal thermal and electrical resistivities of rhenium follow a Wiedemann-Franz law with a Lorenz number of about 0.5 x lo- 8 V2/deg2 at temperatures from 11°K down to roughly 2”K, the lowest temperature obtainable in these experiments. It is hoped that careful observations of the resistivities at low temperatures of other highly pure transition metals, currently underway, will provide a broader base for the evaluation of the importance of electron-electron scattering in these metals. Since the preparation
of this manuscript, C. HERRING has suggested that electron-electron collisions in transition metals would yield, at low temperatures, a universal, temperatureindependent LC of 1.08 X 10 -8Vs/dega. The only condition imposed on Herring’s c&&ion is that the Fermi surface be sufficiently elaborate, and Umklapp processes be important enough, so that “the collision operator of the linearized quasiparticle transport equation will pretty well wipe out any preference for a particular direction of velocity that may be possessed by the distribution function on which it acta” (Zoc. cit.). Herring’s result is in good agreement with data by G. K. WHITE and R. J. TAINSH (ibid, p. 16.5) who observed, in a single nickel specimen, a constant L, of l-0 X 10 -s VP/dega at temperatures below about 20°K. The present observation, in all three rhenium specimens, of a temperature-independent L, which has a magnitude considerably smaller than 244 x 10~aV’/dega is in accord with Herring’s electron-electron collision calculation, although the actual value of L, observed, * 0.5 X10-s Va/dega, indicates that more detailed calculations may, indeed, be required.
[Phyr.Rat. L&t. 19, 167 (1%7)]
OO
I
I
4
6
I 12
I 16
I 20
J
T (DEG Kl
FIG. 5. Temperature dependence of the Lorenz numbers. The “total” curves were derived from the data in the usual way, while the “ideal” curves were obtained by using the ideal components of the reaistivities,
increasing temperature is a strong indication of the absence of a lattice thermal conductivity at nonzero temperatures, although it does not, of course, insure that thermal conduction via the lattice is completely negligible. However, in keeping with of thermal the usual practiceC8) in interpretation conductivity in pure metals, we assume herein that all of the thermal conduction is due to the electrons. In the absence of theories which account for both Umklapp processes and the details of the Fermi surface, any interpretation of the conductivities in the transition metals must be regarded as tentative. It is interesting to note, however, that
Ackrwwledgenre&s-The author is indebted to R. WILLIAMS for the electron-beam zone-refining of the NRL specimens and to C. VOLLI for the X-ray-orientations of the single crystals. He is particularly grateful to Dr. R. SODBN of Bell Telephone Laboratories for preparing and lending the specimen 60. He also wishes
THERMAL
AND ELECTRICAL
BESISTIVITIES
to thank Dr. A. C. EHRLICH for many helpful diacusaiona, Dr. P. LINDBNWLDfor advice in the early experimental stagea of the work, and especially Dr. A. I. SCHINDLBR for constant support and enccuragement throughout the course of this study. REFERENCES 1. WHITE G. K. and WOODSS. B., Can. J. Phys. 35, ___ _.__-. 656(1Y57). 2. PL.UMB H. and CATALAND G., Metrologia 2, 127 (1966). 3. BBRMANR. and H~NTLBY D. J., Cryoge&s 3, 70 (1963).
OF RHENIUM
FROM 2°K TO 20°K
2587
4. BERMANIX., BROCKJ. C. F. and HUNTLBYD. J., Crycge&s 4,233 (1964). 5. SCHRIEMPF J. T., Cryogen&s 6, 362 (1966). 6. SCHRIEMPF J. T., Cryogenics 6,301 (1966). 7. SODJSN R. R., BRENNBR T G. F. and BUBHL~RE., J. electrochem. Sot. 112,77 (1965). 8. :MPNDHLSSOHN K. and RO&NE&C H. M., Solid State Physics, Vol. 12, p. 240, Academic Press, New York (1961). 9. APPEL J., Ph$s. R&J. 125,181s (1962). 10. APPBLJ., Phil. Mug. 8,107l (1963). 11. COLQUITT L.. JR.,J. afd. Phvs. 36.2454 (1965). 12. WH& G. K.-and W00~s S. B., -Phil. itans: R. Sot. A251,273 (1959).
Pergamon Press 1967. Vol. 28, pp. 2581-2587.
Printed in Great Britain.
THERMAL AND ELECTRICAL RESISTIVITIES OF RHENIUM FROM 2°K TO 20°K* J. T. SCHRIBMPF U.S. Naval Research Laboratory, Washington, D.C. 20390 (Received 13 March 1967 ; in rev&d form5 June 1967)
Ahsttuot-Measurement8 of the thermal and electrical reeistivities of highly pure monocrystala of rhenium have been made at temperature8 between 2’K and 2O’K. The ideal re8i8tivitie8can be described by pt CCp (in the electrical case) and by ru, a~ T” (in the thermal cme) only if m and n are allowed to vary with temperature. The value8 of nt and n fall smoothly from 54 and 3.3 at 20°K and become 2 and 1, nspectively, at the lowe8t temperaturea. The anisotropy at all temperature8 is found to be approximately p&8 N 1.3 and r.ucL/wc~~ N 14, where the parallel orientation is along the c axie of the hexagon&close-packed lattice of rhenium. Below about llaK the ideal re&tivities obey a Wiedemann-Frana law with a Lorena number of roughly O-5 x 10m8 Va/dega.
INTRODUCTION
IN THEIRexperimental study of the thermal and electrical resistivities of the transition elements at low temperatures, WHITE and WOODS(~) found that in rhenium the ideal electrical resistivity (PJ was proportional to P1, while the ideal thermal resistivity (w,) varied as Ts3, where T is the absolute temperature. However, these results were obtained at temperatures in the neighborhood of one-tenth the Debye temperature (using White and Woods’ value of 280°K for 0,) whereas it is highly desirable to observe p1 and wf at the lowest temperatures possible. In the present work, a combination of very high specimen purity with a quite sensitive measuring technique has made it possible to deduce the temperature dependence of wi and pi down to roughly one-fiftieth of the Debye temperature. We assume throughout this report that thermal conduction via the lattice is negligible, for, although they are not definitive in this respect, the data do not indicate the presence of an appreciable lattice thermal conductivity. *Thi8 work was 8~pp01ted in part by the Material8 Science8 Division of the Advanced Research Project8 Agency. 2581
EXPERIMENTAL The stainless steel cryostat, shown schematically in Fig. 1, usee the conventional axial-flow technique. The long tube extending from the top of the cryo8tat through the helium bath is terminated in a section of 8tainless Steel and a copper block. Controlling the block temperature control8 the temperature of the sunple, with which it make8 thermal contact. Except for the heat leak along the tube, the block-sample combination ir isolated by the vacuum (1 x 10-8 torr) and the liquid helium and nitrogen temperature radiation shiel&. The u8e of a common vacuum for the sample area and the cryo8tat make8 it po88ible to have the tail 8ection demountable with only one (room-temperature) vacuum 8eal. The wall8 and bottom of the helium chamber are made of copper M)that the bottom of the helium bath can be rued a8 a reference temperature. Wire8 for heating and making measurements are brought through StupakofT 8eab in the cryo8tat wall and thermally anchored to the nitrogen and helium-temperature 8hiel&, a8 well a8 to the copper block, before they make contact with the 8pecimen. Thi8 scheme ha8 the advantage that e.m.f.‘r of thermal origin in the lead wire8 are more constant than those in the uBu(11technique, where wire8 are brought through the bath and changing liquid level8 cause the thermal e.m.f.‘a to drift. The 200&m heaters, wound of con8nunan wire, are powdered by a simple voltage divider and battery network. A control heater, not shown, is contained in the copper block. With this cryo8tat, temperature control i8 achieved in two ways. Fii, pIacing dry helium gae in the tube at a pre~un of about 3 lb/in.g (gage) cau8e8 a rapid accumulation of liquid helium in the bottom of the tube.
2582
J. T.
FLANGE
(TO
SCHRIEMPF
He WELL 1
TO PUMPS -
BOTTOM
OF He WELL
ADED
JOINTS
RE FEED-THRDUGHS FLANGE
JOINT-
HERMOMETERS
FIG. 1. Thermal and electrical conductivity cryostat. Then, by means of a conventional pump and manostat syaem, temperatures between 2 and 4~2°K are establiahed. Second, for temperatures from 4.2 to 2O”K, the of about tubei8seakdwithdryheliumgaaatapremure 0~15torr,andan~~~~pmountofdirectcumntis supplied to the heater in the copper block. The &ability above 4.2”K haa been enhanced by placing a copper bar in the tube. Thk bar (not shown in Fig. 1) rests on top of the copper block and extends to a point just above the bottom of the helium well. The introduction of the bar has cawed no appreciabk change in the behavior of the cryostat below 4~2°K. Temperature differences were measured with a differential thermowupk (gold-O~02at.% iron vs. ~ilvet--O*37at.~~ gold) which was calibrated in a separate et with a germanium resktsnce thermometer. Thi8 germanium thermometer was calibrated at the National Bureau of Standards in accordance with the 1965 Provkional Temperature Scale.(a) The therm+ couple junctknn made contact to the specimen via copper ckmps which ako served a8 potential contact8 for the electrical resktivity determinations. Thk ensured that thespecimengeametrp was identical for both electrical and thermal meaaurementa, although the electrical leads were removed during the course of the thermal conductivity aperiment in order to reduce spurious heat flow. Voltages were measured with a potentiometer (Honeywell 2768) and a Semitive Research (Guildline Inatruments, Ltd.) Type 9460 photocell galvanometer amplifier. The potentiometer was mod&d by the manufacturer to have a resolution of 1 x 10-O V, and the
system was reliable to a precision of about 2 x 10 -9 V. The NBS-calibrated germanium resistance thermometer (manufactured by CryoCal, Inc.) was used in a conventional four-wire potentiometric reaktance-memuring circuit. The potentiometer and null detector were duplicates of the thermocoupk measuring equipment, except the potentiometer was unm&ed and had a resolution of 1-x 10 -8 v. The thermocouple wires (Sigmund Cohn Corp.) were 0.003 in. in dia.. and iunctions were made bv weldinn with an oxygen-&et&e flame. The Ag-Au-wire was used as received; the Au-Fe wire (Sigmund Cohn bar No. 1) was annealed in WUCWJ (about 1 x 1Om6ton) for 12 hr at SOO‘C. In their pioneering work of applying Au-Fe thermocouples to low temperature thermal conductivity meaeuremenm, BERMANand co-worken+*’ employed two corrective techniquen: (1) a superconducting shorting switch was used in the thermocouple circuit in order to allow corrections to be made for spuriouse.m.f.‘aof thermal origin in the lead wires, and (2) the well known “two-heater” technique wa.s used to correct not only temperature di&rencee between the thermocouple junctions and the specimen, but also thermal e.m.f.‘s which occur in the leads from the junctions to the ehorting switch. In the present apparatus both techniques are employed. (The superconducting shorting switches, mounted on the bottom of the copper helium well, are not shown in Fig. 1.) The present apparatus employs an NBS-calibrated germanium resistance thermometer in pkce of the gas thermometer used by Bxmr~~, et al.(s**’ for calibration of the thermocouple. The use of the
THERMAL
AND
ELECTRICAL
RESISTIVITIES
germanium thermometer enables accurate calibration to be made fairly easily and avoids the somewhat timeconsuming gas thermometer calibration technique. Calibration was accomplished by varying the temperature of the copper block with one thermocouple junction fastened to the bottom of the helium well and the other junction fastened to the copper clamp (on the sample) which contained the germanium thermometer. The reduction of the calibration data was carried out by the use of the Naval Research Laboratory CDC 3800 computing facility. Interpolation of the NBS resistancetemperature values was accomplished by fitting the data, in In R-ln T form, to a 10th degree power series in ln T.(6) The voltage-temperature data for the thermocouple was fitted to an 8th degree power series in T, and the thermoelectric power was obtained by taking the first temperature derivative of the power series.(s) All of the machine calculations were carried out in double precision, i.e. with about 25 signilicant figures in all of the arithmetical operations. Apart from errors in establishing the shape factor of the specimens, which will be discussed below, the precision of the measurements is estimated to be f 1 per cent. Specimens 2 and 3 were prepared in this h~boratory by electron-beam zone-refining (in a vacuum of about 10ea torr) 99.9 percent pure rhenium rods obtained from Chase Brass & Copper Co. Specimen 60 was prepared by Dr. R. Soden of Bell Telephone Laboratories by zone-relining compacted powder specimens.(“) All the specimens were single crystals, the details of which are presented in Table 1. Here the orientation angle is the angle between the rod axis and the c axis of the hexagonal lattice, and ps, (wT)s, and Ls are, respectively, the values upon extrapolation to 0°K of the electrical resistivity, the product of thermal resistivity and temperature, and the Lorenx number (L = p/(wZ’)). The effective cross-sectional areas (AH) shown in the table have been determined by making optical comparator measurements at a large number (N) of equally spaced diameters, computing the areas (A,), and finding
1 Iv A,-l. A Off -1 = _ N c
OF
RHENIUM
FROM
2°K
TO
20°K
2583
The largest errors in determining the specimen geometry am due to the finite siae of the clamps; these errors cause uncertainties in the effective lengths of the specimens which are of the order of 1 per cent for specimens 2 and 60, and about 5 per cent for specimen 3. RESULTS All
the
approximately ambient
were
repeatable
1 per cent both upon
cycling
to to the
(24°C) and upon *changing
temperature
the current of two,
measurements
flow through
in the electrical
the specimen case,
and factors
by factors of about
In Fig. 2 and Fig. 3 are shown logarithmic plots of the ideal thermal and electrical resistivities vs. temperature. In order to deduce values for w,, the following expression was used :
eight for the thermal
W( =
current.
wT-(wT), T
(1)
’
where w is the thermal resistivity, i.e. w = K-l, and (wT),/T is the residual thermal resistivity. The p1 values were obtained by simply subtracting from the measured resistivities the residual resistivity, p,-,. The dashed lines in Figs. 2 and 3 indicate the data of WHITE and WOODS(~)on an annealed polycrystalline sample which had a value of pzas/po of about 1360. The two sets of data are in reasonable agreement in the region of overlap. In both the electrical and thermal ideal resistivities it is apparent that a simple power law will not describe the temperature dependence over any appreciable range. However, if the temperature-dependent indices n and m are used, with p1 cc T* and w( oc Tn, the indices fall smoothly
Table 1. Physical details of rhenium monocrystuis
Specimen number
Effective cross-sectional area (cm’)
Effective length (cm)
10s po* (ohm-cm)
2 3 60 (Soden)
0.07586 0~07518 0.03618
8.36 2.22 8.65
7.54 6.45 3.66
2.49 2.51 5.04
* These are the values obtained upon extrapolation of the data to 0°K. t ~ssv is the electrical resistivity measured at 297°K. $ This is the angle between the rod axis and the c axis.
0.307 2.260 0.147
2.46 2.48 2.49
77”
2584
J. T.
SCHRIEMPF
THERMAL
AND I
ELECTRICAL I
I
RESISTIVITIES I
I
OF RHENIUM I
I
FROM
2°K TO 20°K
I
I
2585
T*[PK)*]
FIG. 4. Ideal electrical resistivity (solid circles) and product of temperature and ideal thermal rcsistivity (open circles) as functions of the square of the temperature for rhenium specimen Re 60. The dashed and solid lines are simply straight lines drawn through the low temperature values where the data are apparently following p, H !P and w, NT.
with decreasing temperature from the White and Woods’ values of m = 5.1 and n = 3.3 at about 20°K and become, at the lowest temperatures, m = 2andn = 1. No significant differences in this temperature dependence are evident among the three specimens. Figure 4 shows the low temperature values of pr and w,T for specimen Re 60 plotted as a function of T”. (The other two specimens exhibit the same behavior, although the data have somewhat more scatter.) At temperatures below roughly 5°K it is apparent that the data in Fig. 4 can be very well represented by pt - Ta and w, N T. The effects of anisotropy in both the electrical and thermal resistivities of hexagonal-closepacked rhenium are apparent in Figs. 2 and 3 where the values for specimen 2 are, at a given temperature, generally higher than the results for specimen 3. A careful inspection of the data indicates that, at least at temperatures above lO”K,
this effect is larger than the uncertainties in the measurements. Since specimens 2 and 3 are of nearly equal purity,* but the axis of specimen 2 is 77” from the c-axis whereas that of specimen 3 is 37” from c, an anisotropy effect is indicated. The data for these two specimens yield (perpendicularto-parallel) anisotropy ratios of about 1.1 for the ideal thermal resistivity and 1.3 for not only the ideal, but also the residual and room temperature electrical resistivities. Note that the resistivities of specimen 60, which is oriented in the basal plane, do not lie higher than those of specimen 2. Thus it is apparent that Matthiessen’s rule is not valid and the ideal resistivities are dependent on purity. * By purity we mean both freedom from impurities and absence of lattice defects. Furthermore, we assume that the ratio of electrical resistivities between room temperature and absolute zero provides a meaningful comparison of the purities of different specimens of the same metal.
J.
2586
T.
SCHRIEMPF
DISCUSSION In
Fig. 5 are plotted the values of both the total [L = p/(wT)] and theideal [L, = pl/(wiZ’)] Lorenz numbers. The values of L upon extrapolation to absolute zero are indicated in Table 1 and, at worst, agree within 2 per cent with the well-known theoretical value of 2443 x 10S8 V2/deg2 which obtains when the lattice conductivity is negligible and the electrons are scattered only by defects and impurities. Furthermore, the rapid decay of L with
at the lowest temperatures not only does p, obey a T2 temperature dependence, but also wUtbecomes linear in T, in agreement with the predictions of a model of electron-electron scatuering between different branches of the Fermi surface.(g-ll) Although a quadratic temperature dependence of the ideal electrical resistivity has been seen in some of the non-magnetic transition metals,(12’ an accompanying linear temperature dependence for the ideal thermal resistivity has not been previously observed. Furthermore, the ideal Lorenz values shown in Fig. 5 clearly are independent of temperature below about 11°K. Thus the data indicate that the ideal thermal and electrical resistivities of rhenium follow a Wiedemann-Franz law with a Lorenz number of about 0.5 x lo- 8 V2/deg2 at temperatures from 11°K down to roughly 2”K, the lowest temperature obtainable in these experiments. It is hoped that careful observations of the resistivities at low temperatures of other highly pure transition metals, currently underway, will provide a broader base for the evaluation of the importance of electron-electron scattering in these metals. Since the preparation
of this manuscript, C. HERRING has suggested that electron-electron collisions in transition metals would yield, at low temperatures, a universal, temperatureindependent LC of 1.08 X 10 -8Vs/dega. The only condition imposed on Herring’s c&&ion is that the Fermi surface be sufficiently elaborate, and Umklapp processes be important enough, so that “the collision operator of the linearized quasiparticle transport equation will pretty well wipe out any preference for a particular direction of velocity that may be possessed by the distribution function on which it acta” (Zoc. cit.). Herring’s result is in good agreement with data by G. K. WHITE and R. J. TAINSH (ibid, p. 16.5) who observed, in a single nickel specimen, a constant L, of l-0 X 10 -s VP/dega at temperatures below about 20°K. The present observation, in all three rhenium specimens, of a temperature-independent L, which has a magnitude considerably smaller than 244 x 10~aV’/dega is in accord with Herring’s electron-electron collision calculation, although the actual value of L, observed, * 0.5 X10-s Va/dega, indicates that more detailed calculations may, indeed, be required.
[Phyr.Rat. L&t. 19, 167 (1%7)]
OO
I
I
4
6
I 12
I 16
I 20
J
T (DEG Kl
FIG. 5. Temperature dependence of the Lorenz numbers. The “total” curves were derived from the data in the usual way, while the “ideal” curves were obtained by using the ideal components of the reaistivities,
increasing temperature is a strong indication of the absence of a lattice thermal conductivity at nonzero temperatures, although it does not, of course, insure that thermal conduction via the lattice is completely negligible. However, in keeping with of thermal the usual practiceC8) in interpretation conductivity in pure metals, we assume herein that all of the thermal conduction is due to the electrons. In the absence of theories which account for both Umklapp processes and the details of the Fermi surface, any interpretation of the conductivities in the transition metals must be regarded as tentative. It is interesting to note, however, that
Ackrwwledgenre&s-The author is indebted to R. WILLIAMS for the electron-beam zone-refining of the NRL specimens and to C. VOLLI for the X-ray-orientations of the single crystals. He is particularly grateful to Dr. R. SODBN of Bell Telephone Laboratories for preparing and lending the specimen 60. He also wishes
THERMAL
AND ELECTRICAL
BESISTIVITIES
to thank Dr. A. C. EHRLICH for many helpful diacusaiona, Dr. P. LINDBNWLDfor advice in the early experimental stagea of the work, and especially Dr. A. I. SCHINDLBR for constant support and enccuragement throughout the course of this study. REFERENCES 1. WHITE G. K. and WOODSS. B., Can. J. Phys. 35, ___ _.__-. 656(1Y57). 2. PL.UMB H. and CATALAND G., Metrologia 2, 127 (1966). 3. BBRMANR. and H~NTLBY D. J., Cryoge&s 3, 70 (1963).
OF RHENIUM
FROM 2°K TO 20°K
2587
4. BERMANIX., BROCKJ. C. F. and HUNTLBYD. J., Crycge&s 4,233 (1964). 5. SCHRIEMPF J. T., Cryogen&s 6, 362 (1966). 6. SCHRIEMPF J. T., Cryogenics 6,301 (1966). 7. SODJSN R. R., BRENNBR T G. F. and BUBHL~RE., J. electrochem. Sot. 112,77 (1965). 8. :MPNDHLSSOHN K. and RO&NE&C H. M., Solid State Physics, Vol. 12, p. 240, Academic Press, New York (1961). 9. APPEL J., Ph$s. R&J. 125,181s (1962). 10. APPBLJ., Phil. Mug. 8,107l (1963). 11. COLQUITT L.. JR.,J. afd. Phvs. 36.2454 (1965). 12. WH& G. K.-and W00~s S. B., -Phil. itans: R. Sot. A251,273 (1959).