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Low temperature thermal magnetoresistance of potassium
Solid State Communications,
Vol. 13, pp. 927—930, 1973.
Pergamon Press.
Printed in Great Britain
LOW TEMPERATURE THERMAL MAGNETORESISTANCE OF POTASSIUM* R.S. Newrock and B.W. Maxfield Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14850, U.S.A. (Received 2 April 1973 by E. Burstein)
Unexpected magnet field behavior is observed in the low temperature thermal magnetoresistance of potassium. The thermal resistance does not saturate nor does it mirror the electrical magnetoresistivity. At high fields, a field dependent Wiedemann—Franz ratio is observed. IN THIS letter we report the experimental observation of unexpected behavior in the thermal magnetoresistance of potassium. Our observations are not explained by the semiclassical theory of metals and are not consistent with the well-known experimental observation of a linear electrical magnetoresistivity in potassium.
perpendicular to the field direction, the theory of LAK predicts that the electrical and thermal magnetoresistivities will increase as the square of the applied field. A nearly saturating electrical magnetoresistance is observed in some uncompensated metals having a closed Fermi surface, while quadratic behavior is common in uncompensated metals having Fermi surfaces that permit open orbits. The theory of LAK explains the principle features of magnetoresistance anisotropy of many single crystals.
Employing semiclassical methods, Lifshitz, Azbel’ and Kaganov1 (LAK) have shown that the magnetic field dependence of the electrical and thermal magnetoresistivities of a metal should, in the high-field limit, depend only on the topology of the Fermi surface; in particular, the field dependence of these resistivities should be independent of the type of electron scattering mechanisms in the metal. For an uncompensated metal having a closed Fermi surface, this theory predicts that the transverse electrical and thermal magnetoresistivities will saturate (become field independent) in the high-field limit (war ~ 1). Furthermore, the high-field Wiedemann—Franz ratio should be independent of field (though it may depend on temperature). Potassium, having a spherical Fermi surface,2 is expected to exhibit saturating behavior with a field independent Wiedemann—Franz ratio at high fields. For an uncompensated metal having a Fermi surface that permits open electron orbits *
Using a wide variety of measurement techniques, a number of investigators3 have found the high-field electrical magnetoresistivity of potassium to contain a term linear in the magnetic field; there is no sign of saturation in fields as large as 100 kG. This is inconsistent with the theory of LAK; for single crystals it is difficult to see how any combination of open and closed orbits can give a linear contribution over a wide field range. The semiclassical theory cannot be made to yield such linear behavior unless some rather drastic alterations are made. Several theories have been proposed to explain this phenomenon. First, there is the proposition of some change in the Fermi surface of potassium that allows the possibility of open orbits.4 Secondly, there are a number of theoretical arguments which propose a delay in the onset of the high-field regime by introducing field-dependent terms into the scattering
This work was supported by the U.S. Atomic Energy Commission under contract number AT(l11)3 150, Technical Report No. COO-3 150.12, and by the National Science Foundation under grant No. GH-33637 through the technical facilities of the Materials Science Center at Cornell University, Report No. 1935.
.
integral or by varying the scattering rates over selected portions of the Fermi surface.5 Finally, there are those theories which attempt ot explain the linear magnetoresistivity in terms of characteristics related to 927
928
LOW TEMPERATURE THERMAL MAGNETORESISTANCE OF POTASSIUM
0 U
6009 7033
A
8077
To further investigate these difficulties, we have studied the thermal magnetoresistivity of potassium to determine if it is consistent with LAK theory and, if not, to determine to what extent small angle
K 22
electron—phonon scattering effects the magnetoresistivity. In thisthat letter present experimental which show thewe field dependence of the results trans-
A A
5
A A
A
~
RRR = 3920
A
A
A
A
C
I 0 Do C
H
0
*
. 0
K14 C
A
~06+
~ 05
•
03LA
•
A
A
A
*
*
A *
~
ratio; the Lorenz number is field dependent up to the electricalfields highest magnetoresistivity used in this work. by the Wiedemann—Franz
~RR
•2551 4.218 503 2995——
—
5 527 0
I
2
I
4
6
8
10
12
verse thermal magnetoresistivity is in disagreement with the LAK theory. Furthermore, our measurements are not related to either the measured or theoretical
7
* A
02k-
Vol. 13, No, 7
6 0~6 14 16
8
H(kG)
FIG. I. The thermal magnetoresistance of potassium times the temperature as a function of the applied magnetic field.
defects and impurities in the material.6 The theories in the first two groups predict a Hall coefficient (RH) that either differs from the free electron value or is field dependent. Chimenti and Maxfield7 have measured RH in unconstrained single crystal specimens of potassium and found that, to within 1 per cent, it is independent of magnetic field between 20 and 100 kG. In addition, those theories in the second category only delay saturation. When comparing these theories to the experimental observations, it often proves difficult to obtain a set of physically reasonable parameters that will give a linear magnetoresistivity over a sufficiently wide field range. A variety of impurity and defect related contributions to the magnetoresistance are possible but for metals all predict effects that are much smaller than the observed values.3 To date no satisfactory explanation of the linear term has been produced. In direct contrast to the magnetic field behavior, the zero-magnetic-field electrical and thermal resistivities can be explained quantitatively by semiclassical theories.8’9
Details of the apparatus, the specimen making and6~1° Figure 1 shows mounting procedures WT, the have product been described ofthe thermal elsewhere. magnetoresistance W and the absolute temperature T, as a function of the applied magnetic field H. The percentage change in WTwith field is very large; approximately a 600 per cent effect for sample K-14 at 2.5 K and 18 kG. This large change is both ternperature and purity dependent; the percentage change decreases either as the purity decreases or as the temperature increases. For example, for a RRR of 1000, the change in WT at 2.5 K is only about 100 per cent at 18 kG. These large changes should be contrasted with the much smaller changes observed in the electrical case. Taub et al.3 found that the electrical magnetoresistivity increased less than 100 per cent at 100 kG. Furthermore, in the electrical case, the magnitude of the change decreases as the purity increases and the linear term has a small ternperature dependence. At higher temperatures, the thermal magnetoresistance is almost linear in field but as the temperature decreases the field dependence changes until it becomes nearly quadratic at the lowest temperature. This change in field dependence may be a ‘reduced-field’ effect, that is, at the higher temperatures the highfield regime for the thermal resistivity may not be achieved. For specimen K-14 at 2.5 K, we estimate that w~rreaches a value between 10 and 20. The linear electrical magnetoresistivity has been observed under these conditions so it appears that these measurements reached sufficiently high fields. The electrical magnetoresistivity of some of our specimens has been measured using helicon techniques and found to be linear over the field range in question.
Vol. 13, No. 7
LOW TEMPERATURE THERMAL MAGNETORESISTANCE OF POTASSIUM 8-
• 2.542 *4207 3.985 o
—
4.991
*6009 *7.033 .8.077
7. ~WT
WT 6
929
-
54-
0
3.
_~__j___i__:— ~
c5
10
20
30
40
50
I 60
H/pHo
70
I 80
I 90
I 100
lID
I 20
I 30
(IO~ kG/fl)
FIG. 2. The relative change in the thermal resistivity times temperature versus the reduced field as defined in text. This change from linear to nearly quadratic behavior was noted for all of our specimens. The general shape of the curves is the same for all purities; the thermal magnetoresistance is, in all cases, positive, and the exponent of the field dependence increases as the temperature decreases. In contrast to this, the lowfield electrical magnetoresistivity exhibits a wide range of behavior as the field is increased.
The high-field electrical magnetoresistivity is a linear function of the applied field and, at least at low temperatures, the thermal magnetoresistance is almost quadratic in field. Thus the high-field Lorenz number decreases as the field increases. This is in disagreement with the LAK theory which predicts that in the high-field limit the Wiedemann—Franz ratio (the Lorenz number) will be independent of field.
Figure 1 also shows a rather striking behavior that is evident at most temperatures but is most clearly defined at the lowest temperatures. The field dependence of WT is greatest at the lowest temperatures and thus a low temperature WT vs H curve will eventually cross a higher temperature one. This is illustrated by the curves in Fig. 1 and was observed to occur in all our specimens at the lowest temperatures. The higher temperature curves also show signs for such crossing at fields higher than those available to us in this work. This is difficult to understand for a metal with a closed Fermi surface.
It is probable, but not altogether certain, that the high-field limit was attained in these thermal measurements. One normally finds that due to effects of small-angle scattering, the relaxation time for thermal processes is shorter than that for electrical processes.’1 For specimen K-22, w~ris approximately 20 at the highest field (18 kG). The ci~rappropriate for thermal processes could be smaller. However, we are certainly in the regime of linear electrical magne. toresistivity.
Figure 2 shows for sample K-22 a reduced field or Koh.ler plot of the relative change in the thermal resistance L~(WT~/WT, where the magnetic fIeld has been scaled by the low-temperature electrical resistivity.1°Results at all temperatures scale in this manner for values of oi~rup to about 7. However, at higher values of w~,r(higher fields and lower temperatures) the thermal analog of Kohier’s rule does not hold. A similar plot where the field is scaled by WT does not exhibit scaling for any range of field or temperature.
In a recent paper, Babiskin and Siebenmann6 have attributed the linear term in the electrical magnetoresistivity of potassium to a variety of macroscopic and microscopic scattering centers. There are some theoretical difficulties associated with such an explanation. In determining the field dependence of the magnetoresistivity the semiclassical theory does not differentiate between the various types of scattering processes that are possible in a metal; different microscopic scattering mechanisms will only delay the onset of the high-field region (by decreasing r), assuming that these scattering processes are not
930
LOW TEMPERATURE THERMAL MAGNETORESISTANCE OF POTASSIUM
field dependent. ln any case, without rather artificial assumptions, it is difficult to obtain a linear approach to saturation over a wide field range. Our thermal measurements are not consistent with the dominance of such scattering,mechanisms. Impurities and other microscopic defects should be elastic scattering centres and thus the Wiedemann—Franz law should be valid. This, of course, would yield a linear field dependence for WT. However, it is at the lowest temperatures where the Wiedemann—Franz law is most likely to hold, that we observe the greatest field dependence in the Lorenz ratio. Voids, inclusions and other macroscopic defects (macroscopic implies a size greater than an electron mean free path) charge the boundary condictions on the electric and thermal currents in the metaL Since the constitutive equations, J = AVT and = cth are of similar form (both are a current driven by an applied force) and similar boundary con-
-~.
.~.
Vol. 13, No. 7
ditions apply to both, any changes in the boundary conditions will affect each in the same manner. Thus while it is possible to imagine macroscopic mechanisms creating a linear electrical magnetoresistivity they should also create a linear thermal magnetoresistivity; our measurements show this not to be the case. Thus the mystery of the effect of magnetic fields on transport phenomena in the alkali metals is greater than ever. The thermal resistivity does not saturate nor does it mirror the electrical magnetoresistivity. To further investigate this problem we are currently in the process of extending these measurements to higher fields on the order of 100 kG. Acknowledgements We wish to thank R. Bowers for his help in preparing the manuscript and D.K. Wagner for many helpful discussions during the course of this work. —
1.
REFERENCES LIFSHITZ I.M., YA ASBEL’ M. and KAGANOV, M.I.,Zh. Eksp. Theor. Fiz. 30, 220 (1955),Soviet Phys. JETP3, 143 (1956).
2.
SHOENBERG D. and STILES P.J.,Proc. R. Soc. A 281,62(1964).
3.
5. 6.
See TAUB H., SCHMIDT R.L. MAXFIELD B.W. and BOWERS R.,Phys. Rev. B4, 1134(1971) and references therein. See for example: OVERHAUSER A.W.,Phys. Rev. 167, 691 (1968); Phys. Rev. Lett. 27,938 (1971). O’KEEFE P.M. and GODDARD W.A., III, Phys. Rev. Lett. 23, 300 (1969). See for example: YOUNG R.A.,Phys. Rev. 175, 813 (1968). BABISKIN i.and SIEBENMANN P.G.,Phys. Rev. Lett. 27, 1361 (1971).
7. 8.
CHIMENTI D.E. and MAXFIELD B.W., to be published, Phys. Rev. B7, 1283 (1973). EKJN J.W. and MAXFIELD B.W.,Phys. Rev. B4, 4215 (1971).
9.
NEWROCK R.S. and MAXFIELD B.W., to be published, Phys. Rev. B7, 3501 (1973).
10.
The RRR was determined by extrapolation of the zero-magnetic field thermal resistivity to zero temperature and then assuming the Wiedemann—Franz law to be valid; this is not a precise procedure but is sufficient for our purposes. The values of the zero-field electrical resistivity were calculated from reference 8. It is shown there that Matthiessen’s rule holds for potassium with an RRR greater than 4000. We estimate that the electrical resistivity deduced in this manner is accurate to about 5 per cent.
11.
It is not clear that a relaxation time can be defined for the small angle scattering processes that contribute to the thermal resistivity; certainly it is not possible to define mathematically a relaxation time using the Boltzmann equation. We use the concept of a thermal relaxation time only for pedagogical purposes.
4.
Im Warmemagnetwiderstand des Kalis wurde bei niedrigen Temperaturen unerwartetes Verhalten der Magnetfeldabhängigkeit beobachtet. Der Wãrmewiderstand sättigt nicht ab, und verhält sich nicht wie der elektrische Magnetwiderstand. In hohen Magnetfeldern wurde em feldabhängiges Wiedemann—Franz Verhaltnis beobachtet.
Vol. 13, pp. 927—930, 1973.
Pergamon Press.
Printed in Great Britain
LOW TEMPERATURE THERMAL MAGNETORESISTANCE OF POTASSIUM* R.S. Newrock and B.W. Maxfield Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14850, U.S.A. (Received 2 April 1973 by E. Burstein)
Unexpected magnet field behavior is observed in the low temperature thermal magnetoresistance of potassium. The thermal resistance does not saturate nor does it mirror the electrical magnetoresistivity. At high fields, a field dependent Wiedemann—Franz ratio is observed. IN THIS letter we report the experimental observation of unexpected behavior in the thermal magnetoresistance of potassium. Our observations are not explained by the semiclassical theory of metals and are not consistent with the well-known experimental observation of a linear electrical magnetoresistivity in potassium.
perpendicular to the field direction, the theory of LAK predicts that the electrical and thermal magnetoresistivities will increase as the square of the applied field. A nearly saturating electrical magnetoresistance is observed in some uncompensated metals having a closed Fermi surface, while quadratic behavior is common in uncompensated metals having Fermi surfaces that permit open orbits. The theory of LAK explains the principle features of magnetoresistance anisotropy of many single crystals.
Employing semiclassical methods, Lifshitz, Azbel’ and Kaganov1 (LAK) have shown that the magnetic field dependence of the electrical and thermal magnetoresistivities of a metal should, in the high-field limit, depend only on the topology of the Fermi surface; in particular, the field dependence of these resistivities should be independent of the type of electron scattering mechanisms in the metal. For an uncompensated metal having a closed Fermi surface, this theory predicts that the transverse electrical and thermal magnetoresistivities will saturate (become field independent) in the high-field limit (war ~ 1). Furthermore, the high-field Wiedemann—Franz ratio should be independent of field (though it may depend on temperature). Potassium, having a spherical Fermi surface,2 is expected to exhibit saturating behavior with a field independent Wiedemann—Franz ratio at high fields. For an uncompensated metal having a Fermi surface that permits open electron orbits *
Using a wide variety of measurement techniques, a number of investigators3 have found the high-field electrical magnetoresistivity of potassium to contain a term linear in the magnetic field; there is no sign of saturation in fields as large as 100 kG. This is inconsistent with the theory of LAK; for single crystals it is difficult to see how any combination of open and closed orbits can give a linear contribution over a wide field range. The semiclassical theory cannot be made to yield such linear behavior unless some rather drastic alterations are made. Several theories have been proposed to explain this phenomenon. First, there is the proposition of some change in the Fermi surface of potassium that allows the possibility of open orbits.4 Secondly, there are a number of theoretical arguments which propose a delay in the onset of the high-field regime by introducing field-dependent terms into the scattering
This work was supported by the U.S. Atomic Energy Commission under contract number AT(l11)3 150, Technical Report No. COO-3 150.12, and by the National Science Foundation under grant No. GH-33637 through the technical facilities of the Materials Science Center at Cornell University, Report No. 1935.
.
integral or by varying the scattering rates over selected portions of the Fermi surface.5 Finally, there are those theories which attempt ot explain the linear magnetoresistivity in terms of characteristics related to 927
928
LOW TEMPERATURE THERMAL MAGNETORESISTANCE OF POTASSIUM
0 U
6009 7033
A
8077
To further investigate these difficulties, we have studied the thermal magnetoresistivity of potassium to determine if it is consistent with LAK theory and, if not, to determine to what extent small angle
K 22
electron—phonon scattering effects the magnetoresistivity. In thisthat letter present experimental which show thewe field dependence of the results trans-
A A
5
A A
A
~
RRR = 3920
A
A
A
A
C
I 0 Do C
H
0
*
. 0
K14 C
A
~06+
~ 05
•
03LA
•
A
A
A
*
*
A *
~
ratio; the Lorenz number is field dependent up to the electricalfields highest magnetoresistivity used in this work. by the Wiedemann—Franz
~RR
•2551 4.218 503 2995——
—
5 527 0
I
2
I
4
6
8
10
12
verse thermal magnetoresistivity is in disagreement with the LAK theory. Furthermore, our measurements are not related to either the measured or theoretical
7
* A
02k-
Vol. 13, No, 7
6 0~6 14 16
8
H(kG)
FIG. I. The thermal magnetoresistance of potassium times the temperature as a function of the applied magnetic field.
defects and impurities in the material.6 The theories in the first two groups predict a Hall coefficient (RH) that either differs from the free electron value or is field dependent. Chimenti and Maxfield7 have measured RH in unconstrained single crystal specimens of potassium and found that, to within 1 per cent, it is independent of magnetic field between 20 and 100 kG. In addition, those theories in the second category only delay saturation. When comparing these theories to the experimental observations, it often proves difficult to obtain a set of physically reasonable parameters that will give a linear magnetoresistivity over a sufficiently wide field range. A variety of impurity and defect related contributions to the magnetoresistance are possible but for metals all predict effects that are much smaller than the observed values.3 To date no satisfactory explanation of the linear term has been produced. In direct contrast to the magnetic field behavior, the zero-magnetic-field electrical and thermal resistivities can be explained quantitatively by semiclassical theories.8’9
Details of the apparatus, the specimen making and6~1° Figure 1 shows mounting procedures WT, the have product been described ofthe thermal elsewhere. magnetoresistance W and the absolute temperature T, as a function of the applied magnetic field H. The percentage change in WTwith field is very large; approximately a 600 per cent effect for sample K-14 at 2.5 K and 18 kG. This large change is both ternperature and purity dependent; the percentage change decreases either as the purity decreases or as the temperature increases. For example, for a RRR of 1000, the change in WT at 2.5 K is only about 100 per cent at 18 kG. These large changes should be contrasted with the much smaller changes observed in the electrical case. Taub et al.3 found that the electrical magnetoresistivity increased less than 100 per cent at 100 kG. Furthermore, in the electrical case, the magnitude of the change decreases as the purity increases and the linear term has a small ternperature dependence. At higher temperatures, the thermal magnetoresistance is almost linear in field but as the temperature decreases the field dependence changes until it becomes nearly quadratic at the lowest temperature. This change in field dependence may be a ‘reduced-field’ effect, that is, at the higher temperatures the highfield regime for the thermal resistivity may not be achieved. For specimen K-14 at 2.5 K, we estimate that w~rreaches a value between 10 and 20. The linear electrical magnetoresistivity has been observed under these conditions so it appears that these measurements reached sufficiently high fields. The electrical magnetoresistivity of some of our specimens has been measured using helicon techniques and found to be linear over the field range in question.
Vol. 13, No. 7
LOW TEMPERATURE THERMAL MAGNETORESISTANCE OF POTASSIUM 8-
• 2.542 *4207 3.985 o
—
4.991
*6009 *7.033 .8.077
7. ~WT
WT 6
929
-
54-
0
3.
_~__j___i__:— ~
c5
10
20
30
40
50
I 60
H/pHo
70
I 80
I 90
I 100
lID
I 20
I 30
(IO~ kG/fl)
FIG. 2. The relative change in the thermal resistivity times temperature versus the reduced field as defined in text. This change from linear to nearly quadratic behavior was noted for all of our specimens. The general shape of the curves is the same for all purities; the thermal magnetoresistance is, in all cases, positive, and the exponent of the field dependence increases as the temperature decreases. In contrast to this, the lowfield electrical magnetoresistivity exhibits a wide range of behavior as the field is increased.
The high-field electrical magnetoresistivity is a linear function of the applied field and, at least at low temperatures, the thermal magnetoresistance is almost quadratic in field. Thus the high-field Lorenz number decreases as the field increases. This is in disagreement with the LAK theory which predicts that in the high-field limit the Wiedemann—Franz ratio (the Lorenz number) will be independent of field.
Figure 1 also shows a rather striking behavior that is evident at most temperatures but is most clearly defined at the lowest temperatures. The field dependence of WT is greatest at the lowest temperatures and thus a low temperature WT vs H curve will eventually cross a higher temperature one. This is illustrated by the curves in Fig. 1 and was observed to occur in all our specimens at the lowest temperatures. The higher temperature curves also show signs for such crossing at fields higher than those available to us in this work. This is difficult to understand for a metal with a closed Fermi surface.
It is probable, but not altogether certain, that the high-field limit was attained in these thermal measurements. One normally finds that due to effects of small-angle scattering, the relaxation time for thermal processes is shorter than that for electrical processes.’1 For specimen K-22, w~ris approximately 20 at the highest field (18 kG). The ci~rappropriate for thermal processes could be smaller. However, we are certainly in the regime of linear electrical magne. toresistivity.
Figure 2 shows for sample K-22 a reduced field or Koh.ler plot of the relative change in the thermal resistance L~(WT~/WT, where the magnetic fIeld has been scaled by the low-temperature electrical resistivity.1°Results at all temperatures scale in this manner for values of oi~rup to about 7. However, at higher values of w~,r(higher fields and lower temperatures) the thermal analog of Kohier’s rule does not hold. A similar plot where the field is scaled by WT does not exhibit scaling for any range of field or temperature.
In a recent paper, Babiskin and Siebenmann6 have attributed the linear term in the electrical magnetoresistivity of potassium to a variety of macroscopic and microscopic scattering centers. There are some theoretical difficulties associated with such an explanation. In determining the field dependence of the magnetoresistivity the semiclassical theory does not differentiate between the various types of scattering processes that are possible in a metal; different microscopic scattering mechanisms will only delay the onset of the high-field region (by decreasing r), assuming that these scattering processes are not
930
LOW TEMPERATURE THERMAL MAGNETORESISTANCE OF POTASSIUM
field dependent. ln any case, without rather artificial assumptions, it is difficult to obtain a linear approach to saturation over a wide field range. Our thermal measurements are not consistent with the dominance of such scattering,mechanisms. Impurities and other microscopic defects should be elastic scattering centres and thus the Wiedemann—Franz law should be valid. This, of course, would yield a linear field dependence for WT. However, it is at the lowest temperatures where the Wiedemann—Franz law is most likely to hold, that we observe the greatest field dependence in the Lorenz ratio. Voids, inclusions and other macroscopic defects (macroscopic implies a size greater than an electron mean free path) charge the boundary condictions on the electric and thermal currents in the metaL Since the constitutive equations, J = AVT and = cth are of similar form (both are a current driven by an applied force) and similar boundary con-
-~.
.~.
Vol. 13, No. 7
ditions apply to both, any changes in the boundary conditions will affect each in the same manner. Thus while it is possible to imagine macroscopic mechanisms creating a linear electrical magnetoresistivity they should also create a linear thermal magnetoresistivity; our measurements show this not to be the case. Thus the mystery of the effect of magnetic fields on transport phenomena in the alkali metals is greater than ever. The thermal resistivity does not saturate nor does it mirror the electrical magnetoresistivity. To further investigate this problem we are currently in the process of extending these measurements to higher fields on the order of 100 kG. Acknowledgements We wish to thank R. Bowers for his help in preparing the manuscript and D.K. Wagner for many helpful discussions during the course of this work. —
1.
REFERENCES LIFSHITZ I.M., YA ASBEL’ M. and KAGANOV, M.I.,Zh. Eksp. Theor. Fiz. 30, 220 (1955),Soviet Phys. JETP3, 143 (1956).
2.
SHOENBERG D. and STILES P.J.,Proc. R. Soc. A 281,62(1964).
3.
5. 6.
See TAUB H., SCHMIDT R.L. MAXFIELD B.W. and BOWERS R.,Phys. Rev. B4, 1134(1971) and references therein. See for example: OVERHAUSER A.W.,Phys. Rev. 167, 691 (1968); Phys. Rev. Lett. 27,938 (1971). O’KEEFE P.M. and GODDARD W.A., III, Phys. Rev. Lett. 23, 300 (1969). See for example: YOUNG R.A.,Phys. Rev. 175, 813 (1968). BABISKIN i.and SIEBENMANN P.G.,Phys. Rev. Lett. 27, 1361 (1971).
7. 8.
CHIMENTI D.E. and MAXFIELD B.W., to be published, Phys. Rev. B7, 1283 (1973). EKJN J.W. and MAXFIELD B.W.,Phys. Rev. B4, 4215 (1971).
9.
NEWROCK R.S. and MAXFIELD B.W., to be published, Phys. Rev. B7, 3501 (1973).
10.
The RRR was determined by extrapolation of the zero-magnetic field thermal resistivity to zero temperature and then assuming the Wiedemann—Franz law to be valid; this is not a precise procedure but is sufficient for our purposes. The values of the zero-field electrical resistivity were calculated from reference 8. It is shown there that Matthiessen’s rule holds for potassium with an RRR greater than 4000. We estimate that the electrical resistivity deduced in this manner is accurate to about 5 per cent.
11.
It is not clear that a relaxation time can be defined for the small angle scattering processes that contribute to the thermal resistivity; certainly it is not possible to define mathematically a relaxation time using the Boltzmann equation. We use the concept of a thermal relaxation time only for pedagogical purposes.
4.
Im Warmemagnetwiderstand des Kalis wurde bei niedrigen Temperaturen unerwartetes Verhalten der Magnetfeldabhängigkeit beobachtet. Der Wãrmewiderstand sättigt nicht ab, und verhält sich nicht wie der elektrische Magnetwiderstand. In hohen Magnetfeldern wurde em feldabhängiges Wiedemann—Franz Verhaltnis beobachtet.