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Scattering of conduction electrons by paramagnons in Ni3Al
Volume
28A. number
7
SCATTERING
PHYSICS
OF
CONDUCTION
J. H. J. FLUITMAN, Natuurkundig
LETTERS
13 January 1969
ELECTRONS
B. R. DE VRIES,
Laboratorium,
University
Received
BY
PARAMAGNONS
IN
Ni3Al
R. BOOM and C. J. SCHINKEL of Amsterdam,
7 December
The Netherlands
1968
We report low temperature electrical resistivity data of the Ni3Al phase. The results are described terms of scattering of conduction electrons by d-band spin density fluctuations (paramagnons).
Electrical resistivity measurements (four terminals method), in the temperature region between 1.2 and 20°K, have been performed on a number of Ni3Al alloys with 74, 74.3, 74.5, 75, 75.5 and 76 at. % Ni respectively. The first three alloys are paramagnetic with increasing susceptibilities, while the last three are ferromagnetic at these low temperatures [l]. The temperature dependence of the resistivity p, is unusually strong and very similar for all alloys investigated. The temperature dependent part of p is between 10 and 20°K rou hly proportional to Tl.6 and around 3oK to T1. %. This together with the magnetic properties, invites us to a description in terms of scattering of conduction electrons by paramagnons. Schindler and Rice [2] derived for such a mechanism in a rather simplified model P(T)
= ~(0)
+ cY($f
I43
- (3J5(4
506
plotted in fig. 1. Anyhow, the minimum in 0 and the maximum in (Y as functions of nickel content (fig. 1) at 74.5% Ni (the nickel composition which is nearest to the ferromagnetic transition) is in agreement with theory. At present we have no explanation for the high values of o/e2 at the nickel rich side of the phase (fig. l), which indicate some difference in the scattering mechanisms at either side of stoichiometry. Preliminary results of magnetoresistance measurements point in the same direction. As we observed that the specimens were coarse grained (grain sizes are in the order of magnitude of 0.1 cm) and sometimes easy to fracture we might have had troubles with increased
* (1)
Here p(0) is the residual resistivity, 6 is a characteristic temperature, LYis a constant dependent on the strength of the d-electron interactions, and 52 and J5 are Bloch-Grdneisen integrals. Values for the parameters cy and 0 are obtained by fitting the experimental data to equation (1). Hoever, it appeared to be impossible to get a good fit in the whole temperature region; (Y and t? are fitted to the data at the lowest temperatures, i.e. around 3OK. Of course one can formally allow cy and 0 to be temperature dependent, that is, one can determine them by getting the data over various small temperature intervals and in that way obtain cr and 9 as functions of temperature. Unfortunately cy and 0 vary considerably (a factor two for Ni74.5A125.5), going from 3 to 6oK, the former decreasing and the latter increasing, which means that at T = 0 a and 0 will have even higher and lower values, respectively, than those
in
a
c
-40
P(293) (VQCW t
5030‘8
. .
0.
.
.
.
.
.
-30 -20
.
.
6(%) I -10 I
I b
I
I
I d
P(O)
1 .o .
WQcm)
.
t
.
12- . a-
.
4.-
(nRcml°K2)
l
.
u l*(m) 82
.
-9 t
-6
.
l
Fig. la) room temperature resistivity: sistivity: c) characteristic tern erature J2(m) (i.e. the coefficient of T P at OoK, from the best fit at 3’K) as functions of
b) residual re0 : d) ((Y/e’) x with (Y and B the nickel content
Volume
28A. number
PHYSICS
7
13 January 1969
LETTERS
“form factors” by cracks for instance leading to too high valueslof p and consequently of o. This assumption wag at least not supported by careful resistivity measurements at different parts of the specimens which showed only small scatter. Comparing our results on the paramagnetic samples with those of Schindler and Rice on Pd-Ni alloys [2] we note that in both systems the variation of (u/e2 with nickel content is considerably less rapid than the square of the susceptibility and that in both systems similar deviations from theory occur, regarding the temperature dependence. On the other hand the present values of a,/0 2 giving the coefficient of the T 2 term in the
resistivity are an order magnitude larger than those for the Pd-Ni alloys. In fact they are so large that electron-phonon scattering terms could be neglected. We thank Drs. F. R. de Boer and Drs. J. Biesterbos for their assistance in preparing the specimens and Prof. Dr. A. R. Miedema for stimulating these experiments. References 1. F. R. de Boer, C. J. Schinkel, J. Biesterbos and S. Proost, J. Appl. Phys., to be published. 2. A. I. Schindler and M. J. Rice, Phys. Rev. 164 (1967) 759.
*****
SECOND
SOUND VELOCITY IN He II, THERMAL CONDUCTIVITY AND SCALING LAWS NEAR THE SUPERFLUID TRANSITION
IN He I,
G. AHLERS Bell Telephone Laboratom’es,
Murray Hill, New Jersey,
Incorporated,
Received
6 December
USA
1968
It is shown that the measured thermal conductivity of He I and scaling arguments yield a second sound velocity for He II in agreement with that calculated from hydrodynamics for TX - T 2 10m70K. This is contradictory to the recent direct measurements by Johnson and Crooks.
The hydrodynamic expression for the second sound velocity u2 in He II
u;=TS2~,,‘(pnCp)
(1)
,
where S is the entropy, pn and ps the normal and superfluid density, and Cp the specific heat at constant pressure, was verified recently by direct measurement for TX - T 2 10-40K [l-3]. However, Johnson and Crooks [3] reported significant deviations from eq. (1) at TX - T = 5 x lo-5OK. The purpose of this note is to emphasize that such a deviation is either in contradiction with recent thermal conductivity measurements in He I [4] or requires a very precise cancellation between the breakdown of eq. (1) and scaling arguments over three decades in TX - T. In terms of the thermal conductivity K of He I, u2 is given by [4-61 u2(- E) =- K(E) [&(
lI
,
where e = (T - T~)/TA 2 0, p is the density, and A5:(5o)-l is a length independent of E . Using measured values of K [4] and Cp [?‘I, u2 may be calculated from eq. (2)x and is shown in fig. 1. The solid line corresponds to eq. (1). The agreement is seen to be excellent for lo-7oK C
(2)
= 0.865 x 10-8 cm can be determined from eq. (1) well above TX. Eqs. (1) and (2) then agree for all TX - T between 10-7 and lo-3oK. 507
28A. number
7
SCATTERING
PHYSICS
OF
CONDUCTION
J. H. J. FLUITMAN, Natuurkundig
LETTERS
13 January 1969
ELECTRONS
B. R. DE VRIES,
Laboratorium,
University
Received
BY
PARAMAGNONS
IN
Ni3Al
R. BOOM and C. J. SCHINKEL of Amsterdam,
7 December
The Netherlands
1968
We report low temperature electrical resistivity data of the Ni3Al phase. The results are described terms of scattering of conduction electrons by d-band spin density fluctuations (paramagnons).
Electrical resistivity measurements (four terminals method), in the temperature region between 1.2 and 20°K, have been performed on a number of Ni3Al alloys with 74, 74.3, 74.5, 75, 75.5 and 76 at. % Ni respectively. The first three alloys are paramagnetic with increasing susceptibilities, while the last three are ferromagnetic at these low temperatures [l]. The temperature dependence of the resistivity p, is unusually strong and very similar for all alloys investigated. The temperature dependent part of p is between 10 and 20°K rou hly proportional to Tl.6 and around 3oK to T1. %. This together with the magnetic properties, invites us to a description in terms of scattering of conduction electrons by paramagnons. Schindler and Rice [2] derived for such a mechanism in a rather simplified model P(T)
= ~(0)
+ cY($f
I43
- (3J5(4
506
plotted in fig. 1. Anyhow, the minimum in 0 and the maximum in (Y as functions of nickel content (fig. 1) at 74.5% Ni (the nickel composition which is nearest to the ferromagnetic transition) is in agreement with theory. At present we have no explanation for the high values of o/e2 at the nickel rich side of the phase (fig. l), which indicate some difference in the scattering mechanisms at either side of stoichiometry. Preliminary results of magnetoresistance measurements point in the same direction. As we observed that the specimens were coarse grained (grain sizes are in the order of magnitude of 0.1 cm) and sometimes easy to fracture we might have had troubles with increased
* (1)
Here p(0) is the residual resistivity, 6 is a characteristic temperature, LYis a constant dependent on the strength of the d-electron interactions, and 52 and J5 are Bloch-Grdneisen integrals. Values for the parameters cy and 0 are obtained by fitting the experimental data to equation (1). Hoever, it appeared to be impossible to get a good fit in the whole temperature region; (Y and t? are fitted to the data at the lowest temperatures, i.e. around 3OK. Of course one can formally allow cy and 0 to be temperature dependent, that is, one can determine them by getting the data over various small temperature intervals and in that way obtain cr and 9 as functions of temperature. Unfortunately cy and 0 vary considerably (a factor two for Ni74.5A125.5), going from 3 to 6oK, the former decreasing and the latter increasing, which means that at T = 0 a and 0 will have even higher and lower values, respectively, than those
in
a
c
-40
P(293) (VQCW t
5030‘8
. .
0.
.
.
.
.
.
-30 -20
.
.
6(%) I -10 I
I b
I
I
I d
P(O)
1 .o .
WQcm)
.
t
.
12- . a-
.
4.-
(nRcml°K2)
l
.
u l*(m) 82
.
-9 t
-6
.
l
Fig. la) room temperature resistivity: sistivity: c) characteristic tern erature J2(m) (i.e. the coefficient of T P at OoK, from the best fit at 3’K) as functions of
b) residual re0 : d) ((Y/e’) x with (Y and B the nickel content
Volume
28A. number
PHYSICS
7
13 January 1969
LETTERS
“form factors” by cracks for instance leading to too high valueslof p and consequently of o. This assumption wag at least not supported by careful resistivity measurements at different parts of the specimens which showed only small scatter. Comparing our results on the paramagnetic samples with those of Schindler and Rice on Pd-Ni alloys [2] we note that in both systems the variation of (u/e2 with nickel content is considerably less rapid than the square of the susceptibility and that in both systems similar deviations from theory occur, regarding the temperature dependence. On the other hand the present values of a,/0 2 giving the coefficient of the T 2 term in the
resistivity are an order magnitude larger than those for the Pd-Ni alloys. In fact they are so large that electron-phonon scattering terms could be neglected. We thank Drs. F. R. de Boer and Drs. J. Biesterbos for their assistance in preparing the specimens and Prof. Dr. A. R. Miedema for stimulating these experiments. References 1. F. R. de Boer, C. J. Schinkel, J. Biesterbos and S. Proost, J. Appl. Phys., to be published. 2. A. I. Schindler and M. J. Rice, Phys. Rev. 164 (1967) 759.
*****
SECOND
SOUND VELOCITY IN He II, THERMAL CONDUCTIVITY AND SCALING LAWS NEAR THE SUPERFLUID TRANSITION
IN He I,
G. AHLERS Bell Telephone Laboratom’es,
Murray Hill, New Jersey,
Incorporated,
Received
6 December
USA
1968
It is shown that the measured thermal conductivity of He I and scaling arguments yield a second sound velocity for He II in agreement with that calculated from hydrodynamics for TX - T 2 10m70K. This is contradictory to the recent direct measurements by Johnson and Crooks.
The hydrodynamic expression for the second sound velocity u2 in He II
u;=TS2~,,‘(pnCp)
(1)
,
where S is the entropy, pn and ps the normal and superfluid density, and Cp the specific heat at constant pressure, was verified recently by direct measurement for TX - T 2 10-40K [l-3]. However, Johnson and Crooks [3] reported significant deviations from eq. (1) at TX - T = 5 x lo-5OK. The purpose of this note is to emphasize that such a deviation is either in contradiction with recent thermal conductivity measurements in He I [4] or requires a very precise cancellation between the breakdown of eq. (1) and scaling arguments over three decades in TX - T. In terms of the thermal conductivity K of He I, u2 is given by [4-61 u2(- E) =- K(E) [&(
lI
,
where e = (T - T~)/TA 2 0, p is the density, and A5:(5o)-l is a length independent of E . Using measured values of K [4] and Cp [?‘I, u2 may be calculated from eq. (2)x and is shown in fig. 1. The solid line corresponds to eq. (1). The agreement is seen to be excellent for lo-7oK C
(2)
= 0.865 x 10-8 cm can be determined from eq. (1) well above TX. Eqs. (1) and (2) then agree for all TX - T between 10-7 and lo-3oK. 507