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Matthiessen's rule in some dilute nickel alloys
PHYSICS
Volume 24A, number 8
MATTHIESSEN’S
RULE
10 April 1967
LETTERS
IN SOME
DILUTE
NICKEL
ALLOYS
T. FARRELL and D. GREIG Physics
Department,
University
of Leeds,
Leeds,
England
Received 1 March 1967 Deviations from Matthiessen’s rule have been observed in both the electrical and electronic thermal resistivities of some very dilute Ni-Pd and Ni-Cu alloys. The larger deviations in the thermal resistivity case are accounted for by a higher anisotropy of the electron-phonon relaxation time over the Fermi surface.
We have observed deviations from both Matthiessen’s rule and the thermal equivalent of the rule in a number of very dilute Ni-Pd and Ni-Cu alloys, and wish to report on the correlation between them. The eight specimens all contained less than 1 atomic percent solute, so that effects of alloying such as changes in the Fermi surface and the phonon spectrum can be neglected. We therefore propose that for these alloys the deviations are due to differing anisotropies of relaxation times for phonon and impurity scattering over the Fermi surfaces. In nickel, most of the electron current arises from the s-electrons of the saturated spin direction [l-3], and the Fermi surface for these electrons is similar in form to the Fermi surfaces of noble metals [4]. As in these metals a simple approximation is to consider that conduction is due to two groups or bands of electrons from the belly and neck region of the s-Fermi surface [5,6]. For a 2-band conductor the deviations in the electrical resistivity , Ap = pmeas - (pi +p”), are given by [6]
Ap =
05
.o -
.5 -
*o-
I.5-
(a - a2PiPo p(1 +&pi+
(Y(1+@pO
’
where pi is the ideal resistivity obtained from the purest sample available, .p” the impurig resistivity, and where (Y= pi/pa and 6 = pi/p,, the ratios of the ideal and impurity resistivities in the two bands. At high temperatures, pi = pPh, the phonon resistivity, and is much greater than p”, so that Ap/p” tends to the value y = (cr-aP/p(l +(Y)2, which for the!ealloys is - 0.3. The measured thermal conductivity, KmeaS, of a metal is the sum of an electronic component, K~, and a lattice component, K~. “Matthiessen’s
0, TEMPERATURE
DEG.
K.
Fig. 1. Variations of (Wmeas -Iy~u~es) (solid curve) and (Wo -W&,) (points) with temperature. o, Ni + 0.1% Pd; 0, Ni + 0.3% Pd; A, Ni + 1.0% Pd. rule” for thermal resistivity is We = l/~~ = = Wpb+ Wo, where Wph and W” respectively represent the ideal and impurity components of We.
Deviations can again be represented by the addition of a further term AW which will have the same form as eq. (1). We assume that in the NiPd 401
PHYSICS
Volume 24A, number 8
LETTERS
alloys Kg is negligible and Kmeas = 1 We, so that meas - Wpt$“) = (W” - Wzure) + Aw. There is (w
For the belly region of Fermi surface, small angle scattering processes hardly contribute to the electrical resistivity but can change electrons from ‘hot’ to ‘cold’. They are therefore much more effective in reducing th thermal than the p’ >>T&, . In the electrical current, and so 7pb neck regions, on the other hand, the same phonon wave vector can scatter an electron through a much lar r angle so that there is less differIt follws that 7ph/7 ph ’’ ence in C n and 7&. Wn Wb ph P > ?pn $,h , and so CYWb CY~in accordance with our experimental observation that yw > yp.
clear evidence from fig. 1 for the existence of these deviations. In NiCu, phonon scattering due to mass difference effects is much smaller than in NiPd. On the other hand, the difference in valency between the two elements leads to an increase in WO so that the ratio K~/K~ is greater than in NiPd and ~~ can no longer be neglected. When a direct separation of K and Ke is made in NiCu without taking account of 1 w, the lattice component is found to be negative over the range of temperatures in which AW should be greatest. Sensible values of K can only be obtained when Ke is corrected to 1.I?elude Aw. However, the value of yw required to make this correction is -0.6, which is about 2Yp. The electrical and thermal conductivities originating from any area AY of the Fermi surface both depend on the product Arvrrr, where vy is the velocity of electrons at A,, and 7y is their relaxation time [7]. The anisotropy factors in eq. (1) are therefore Of the form AnvnTn/AbvbTb, where the subscripts n and b refer to ‘neck’ and ‘belly’ respectively. Since the ratio A,v,/Abvb (= C) is the same for p and W,, the four aniso&-oby factors we are concerned with are, (YP = = c$/r;;, ‘yw = C$;/T&, @ = CT&, /;b ,
We are indebted to Professor J. S. Dugdale for leading numerous discussions on the subject of anisotropic relaxation times, and we whould like to thank SRC for financial assistance and for a maintenance grant to T. F.
References 1. N.F.Mott, Proc.Roy.Soc.Al53 (1936) 699; Al56 (1936) 368, 2. A. I.Schindler, R. J.Smith and E. I. Salkovitz, J. Phys. Chem.Solids 1 (1956) 39. 3. A. Hasegawa, S. Wakoh and J. Yamashita, J. Phys. Sot. Japan 20 (1965) 1865. 4. L. Hodges, H. Ehrenreich and N. D. Lang, Phys.Rev. 152 (1966) 505. 5. J.M: Zim& Phys.Rev. 121 (1961) 1320. 6. Z.S. Basinski and J.S. Dugdale, Phgs.Rev. , to be published. 7. J. M. Ziman, Principles of theory of solids (Cambridge University Press, 1964) p. 196.
and pw = C’r”wnj.T&b. The validity of the Wiedemann-Franz law at low temperatures implies that the relaxation times for impurity scattering are the same in p and We, so that pp = pw. However, this will not be so for low temperature phonon scattering. **
PHONON
EFFECTS
IN
THE
10 April 1967
* * *
MOTION
OF
POSITRONS
IN
METALS
H. J. MIKESKA Physik Department,
Technische
Hochschule,
Miinchen,
Deutschland
Received 10 March 1967
The positron-phonon interaction is shown to lead to an important tail in the positron momentum distribution. The tail can account for at least part of the large so-called effective mass of positrons in sodium.
Recent experiments on the annihilation of positrons in sodium [l] have shown a thermal smearing of the Fermi-cutoff which leads to an effective 402
mass rn: = (1.9 f 0.4)m of the positron (bare mass m) when fitted by a Gaussian momentum distribution for the positron. Calculations of the positron
Volume 24A, number 8
MATTHIESSEN’S
RULE
10 April 1967
LETTERS
IN SOME
DILUTE
NICKEL
ALLOYS
T. FARRELL and D. GREIG Physics
Department,
University
of Leeds,
Leeds,
England
Received 1 March 1967 Deviations from Matthiessen’s rule have been observed in both the electrical and electronic thermal resistivities of some very dilute Ni-Pd and Ni-Cu alloys. The larger deviations in the thermal resistivity case are accounted for by a higher anisotropy of the electron-phonon relaxation time over the Fermi surface.
We have observed deviations from both Matthiessen’s rule and the thermal equivalent of the rule in a number of very dilute Ni-Pd and Ni-Cu alloys, and wish to report on the correlation between them. The eight specimens all contained less than 1 atomic percent solute, so that effects of alloying such as changes in the Fermi surface and the phonon spectrum can be neglected. We therefore propose that for these alloys the deviations are due to differing anisotropies of relaxation times for phonon and impurity scattering over the Fermi surfaces. In nickel, most of the electron current arises from the s-electrons of the saturated spin direction [l-3], and the Fermi surface for these electrons is similar in form to the Fermi surfaces of noble metals [4]. As in these metals a simple approximation is to consider that conduction is due to two groups or bands of electrons from the belly and neck region of the s-Fermi surface [5,6]. For a 2-band conductor the deviations in the electrical resistivity , Ap = pmeas - (pi +p”), are given by [6]
Ap =
05
.o -
.5 -
*o-
I.5-
(a - a2PiPo p(1 +&pi+
(Y(1+@pO
’
where pi is the ideal resistivity obtained from the purest sample available, .p” the impurig resistivity, and where (Y= pi/pa and 6 = pi/p,, the ratios of the ideal and impurity resistivities in the two bands. At high temperatures, pi = pPh, the phonon resistivity, and is much greater than p”, so that Ap/p” tends to the value y = (cr-aP/p(l +(Y)2, which for the!ealloys is - 0.3. The measured thermal conductivity, KmeaS, of a metal is the sum of an electronic component, K~, and a lattice component, K~. “Matthiessen’s
0, TEMPERATURE
DEG.
K.
Fig. 1. Variations of (Wmeas -Iy~u~es) (solid curve) and (Wo -W&,) (points) with temperature. o, Ni + 0.1% Pd; 0, Ni + 0.3% Pd; A, Ni + 1.0% Pd. rule” for thermal resistivity is We = l/~~ = = Wpb+ Wo, where Wph and W” respectively represent the ideal and impurity components of We.
Deviations can again be represented by the addition of a further term AW which will have the same form as eq. (1). We assume that in the NiPd 401
PHYSICS
Volume 24A, number 8
LETTERS
alloys Kg is negligible and Kmeas = 1 We, so that meas - Wpt$“) = (W” - Wzure) + Aw. There is (w
For the belly region of Fermi surface, small angle scattering processes hardly contribute to the electrical resistivity but can change electrons from ‘hot’ to ‘cold’. They are therefore much more effective in reducing th thermal than the p’ >>T&, . In the electrical current, and so 7pb neck regions, on the other hand, the same phonon wave vector can scatter an electron through a much lar r angle so that there is less differIt follws that 7ph/7 ph ’’ ence in C n and 7&. Wn Wb ph P > ?pn $,h , and so CYWb CY~in accordance with our experimental observation that yw > yp.
clear evidence from fig. 1 for the existence of these deviations. In NiCu, phonon scattering due to mass difference effects is much smaller than in NiPd. On the other hand, the difference in valency between the two elements leads to an increase in WO so that the ratio K~/K~ is greater than in NiPd and ~~ can no longer be neglected. When a direct separation of K and Ke is made in NiCu without taking account of 1 w, the lattice component is found to be negative over the range of temperatures in which AW should be greatest. Sensible values of K can only be obtained when Ke is corrected to 1.I?elude Aw. However, the value of yw required to make this correction is -0.6, which is about 2Yp. The electrical and thermal conductivities originating from any area AY of the Fermi surface both depend on the product Arvrrr, where vy is the velocity of electrons at A,, and 7y is their relaxation time [7]. The anisotropy factors in eq. (1) are therefore Of the form AnvnTn/AbvbTb, where the subscripts n and b refer to ‘neck’ and ‘belly’ respectively. Since the ratio A,v,/Abvb (= C) is the same for p and W,, the four aniso&-oby factors we are concerned with are, (YP = = c$/r;;, ‘yw = C$;/T&, @ = CT&, /;b ,
We are indebted to Professor J. S. Dugdale for leading numerous discussions on the subject of anisotropic relaxation times, and we whould like to thank SRC for financial assistance and for a maintenance grant to T. F.
References 1. N.F.Mott, Proc.Roy.Soc.Al53 (1936) 699; Al56 (1936) 368, 2. A. I.Schindler, R. J.Smith and E. I. Salkovitz, J. Phys. Chem.Solids 1 (1956) 39. 3. A. Hasegawa, S. Wakoh and J. Yamashita, J. Phys. Sot. Japan 20 (1965) 1865. 4. L. Hodges, H. Ehrenreich and N. D. Lang, Phys.Rev. 152 (1966) 505. 5. J.M: Zim& Phys.Rev. 121 (1961) 1320. 6. Z.S. Basinski and J.S. Dugdale, Phgs.Rev. , to be published. 7. J. M. Ziman, Principles of theory of solids (Cambridge University Press, 1964) p. 196.
and pw = C’r”wnj.T&b. The validity of the Wiedemann-Franz law at low temperatures implies that the relaxation times for impurity scattering are the same in p and We, so that pp = pw. However, this will not be so for low temperature phonon scattering. **
PHONON
EFFECTS
IN
THE
10 April 1967
* * *
MOTION
OF
POSITRONS
IN
METALS
H. J. MIKESKA Physik Department,
Technische
Hochschule,
Miinchen,
Deutschland
Received 10 March 1967
The positron-phonon interaction is shown to lead to an important tail in the positron momentum distribution. The tail can account for at least part of the large so-called effective mass of positrons in sodium.
Recent experiments on the annihilation of positrons in sodium [l] have shown a thermal smearing of the Fermi-cutoff which leads to an effective 402
mass rn: = (1.9 f 0.4)m of the positron (bare mass m) when fitted by a Gaussian momentum distribution for the positron. Calculations of the positron