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Residual resistivity of dilute binary alloys: Pseudopotential approach
Physica 71 (1974) 237-240
RESIDUAL
0 North-Holland Publishing Co.
RESISTIVITY
OF DILUTE
PSEUDOPOTENTIAL A. SWAROOP
BINARY ALLOYS:
APPROACH
and L. M. TIWARI
Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110029, India
Received 7 June 1973
Synopsis The residual resistivities of dilute binary alloys of Mg, Al, Pb and In have been studied using the model pseudopotential of Gurskii and Krasko. The theoretical values of residual resistivities are found closer to the experimental ones than the values obtained by Harrison from the pointion model potential. It has been concluded that the Gurskii and Krasko model potentials are more reliable and may be used to evaluate other properties of the solid and liquid states of metals.
1. Introduction. Residual resistivity is the enhancement in the resistivity of a pure metal when a small amount of impurity is dissolved in it. Mottl) explained this phenomenon for the tist time on the basis of a model in which (i) ions are considered as point charges surrounded by a screened Coulomb potential, and (ii) the conduction electrons are supposed to be scattered by a very weak potential, i.e., the difference between the potentials around an impurity and the solvent ion. His values of the residual resistivities were two to three times higher than the experimental values. Friede12) improved the situation by using the exact partialwave approach to calculate the scattering, but his approach suffers from a lacuna, i.e., it does not account for two impurities of same valency but belonging to different periods in the periodic table. Blatt3) studied the same, using a quite artificial form of potential, i.e., a square well, satisfying the sum rule and relating the phase shifts of partial waves to the charge on the ion. It was found later that the resistivities obtained by this method were insensitive to the shape of the potential chosen. A modern approach to the problem of scattering in alloys is due to Faber and Ziman4) via the use of pseudopotentials. These pseudopotentials are known in abundance and found to be very successful in interpreting various properties of metals. Recently Gurskii and Kraskos) (GK) have proposed a model potential which has explained satisfactorily the phonon dispersio@) binding energy, crystal 237
A. SWAROOP
238
AND
L. M. TIWARI
stability7) superconducting transition temperature and pressure dependence*) for various metals. In the light of these studies it was thought worthwhile to apply this model potential to the case of dilute binary alloys. In the present paper, we therefore have calculated the residual resistivities of dilute alloys of Mg, Al, Pb and In using the same model potential. 2. Theory. According to Harrisong) the resistivity due to an atomic per cent of atoms of type (2) in a host type (1) can be written as - WC,, IOI’ x3 dx, and
(1)
x = q/kF.
C is the characteristic of the host lattice in a dilute alloy. G
+
4
w,2,
-
WC,,
IQ = - ,
(2)
where
(k + 41 Wdk> = w
t1 + (qrc)‘l”
(3)
’
where a and rc are the parameters. The screening has been treated using a Hartree dielectric function c(q), modified according to ShamlO) to include the exchange and correlation effects. It is given by 4xze2 c(q) = 1 + -
Qoq2 x(q) = $
l-
q2 x(q) 9 2 (q2 + kZ: + kg) >
(0.5 + 4kikSqq2 In /$Qi), F
F
F-
9
where k, is Thomas-Fermi screening parameter. In the present paper we have assumed that in an alloy, the form factor of the host atom remains unchanged because of the introduction of dilute impurities but the form factor of the impurity atom does change. The modified form factor of the impurity atom can be written as
+
41 WC,, Ik> = 4xz’2e2 r
0 z
(2~’ - 1) (qrr)2 - 1 [l + m212
1
4d ’
RESIDUAL
RESISTIVITY OF DILUTE BINARY ALLOYS
239
where z’ is the valence; a’ and ri are the GK potential parameters for the impurity. e(q) and Q0 are appropriate to the host metal of which the values are quite reasonable for the case of dilute binary alloys. 3. Results. For the present theoretical investigation we have chosen eight dilute binary alloys such as Li, Al and In in Mg; Mg and Zn in Al; Na and In in Pb; and Pb in In. Our choice of dilute alloys has been restricted because of the availability of GK model potential parameters for their constituent metals. The model potential parameters a and r, for these above constituent metals have been taken from Gurskii and Krasko5). The evaluation of the resistivity has been made by integrating eq. (1). The theoretical values of the residual resistivities for eight dilute alloys have been collected in table I along with the other theoretical and experimental data for comparison. TABLEI The residual rekstivities (in @J cm/at. %) of dilute binary alloys Dilute alloy
Theor.
Exp.9)
Harrisong)
Mg*-Li Mg+-Al Mg*-In Al*-Mg Al*-Zll Pb*-Na Pb+-In In*-Pb
0.46 2.08 1.86 0.22 0.14 1.38 1.37 0.47
0.75 2.10 2.50 0.33 0.22 2.90 1.20 0.60
0.27 1.60 2.40 0.78 0.10 0.77 0.50 0.56
* Solvent metal.
4. Discussion. Table I displays that the residual resistivities of most of the dilute binary alloys except Mg in In, and In in Pb derived from the GK model potential are in better agreement with the experiment than the theoretical results from Harrison’s point - ion model potential. HaiTisong) could attain only qualitative agreement with the experiment because of the crudeness of the form factors used. Since the use of the GK model potential shows an improvement over the previous studies it can be concluded that these model-potential form factors are quite reliable and may be used to evaluate other properties of the liquid and solid states of metals and alloys. Acknowledgements. One of the authors (A.S) would like to thank C.S.I.R. India for the financial assistance. All the computations have been carried out on an ICL 1909 computer.
240
A. SWAROOP AND L. M. TIWAIU REFERENCES
1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
Mott, N.F., Proc. Cambridge Phil. Sot. 32 (1936) 281. Friedel, J., Advances in Phys. 3 (1954) 446. Blatt, F.J., Phys. Rev. 108 (1958) 285. Faber, T.E. and Ziman, J. M., Phil. Mag. 11 (1965) 153. Gurskii, Z.A. and Krasko, G.L., Soviet Physics-JETP Letters 9 (1969) 363; Soviet PhysicsSolid State 11 (1969) 2447; Soviet Physics-Dokl. 16 (1971) 298. Bajpai, R.P., Phys. Letters A42 (1972) 163. Prasad, B. and Srivastava, R.S., Phys. Letters A38 (1972) 521. Swaroop, A. and Tiwari, L. M., Indo-Soviet Conference on Solid State Materials (1972). Harrison, W.A., Pseudopotentials in the Theory of Metals, Benjamin (New York, 1966) p. 150. Sham, L.J., Proc. Roy. Sot. A283 (1965) 33.
RESIDUAL
0 North-Holland Publishing Co.
RESISTIVITY
OF DILUTE
PSEUDOPOTENTIAL A. SWAROOP
BINARY ALLOYS:
APPROACH
and L. M. TIWARI
Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110029, India
Received 7 June 1973
Synopsis The residual resistivities of dilute binary alloys of Mg, Al, Pb and In have been studied using the model pseudopotential of Gurskii and Krasko. The theoretical values of residual resistivities are found closer to the experimental ones than the values obtained by Harrison from the pointion model potential. It has been concluded that the Gurskii and Krasko model potentials are more reliable and may be used to evaluate other properties of the solid and liquid states of metals.
1. Introduction. Residual resistivity is the enhancement in the resistivity of a pure metal when a small amount of impurity is dissolved in it. Mottl) explained this phenomenon for the tist time on the basis of a model in which (i) ions are considered as point charges surrounded by a screened Coulomb potential, and (ii) the conduction electrons are supposed to be scattered by a very weak potential, i.e., the difference between the potentials around an impurity and the solvent ion. His values of the residual resistivities were two to three times higher than the experimental values. Friede12) improved the situation by using the exact partialwave approach to calculate the scattering, but his approach suffers from a lacuna, i.e., it does not account for two impurities of same valency but belonging to different periods in the periodic table. Blatt3) studied the same, using a quite artificial form of potential, i.e., a square well, satisfying the sum rule and relating the phase shifts of partial waves to the charge on the ion. It was found later that the resistivities obtained by this method were insensitive to the shape of the potential chosen. A modern approach to the problem of scattering in alloys is due to Faber and Ziman4) via the use of pseudopotentials. These pseudopotentials are known in abundance and found to be very successful in interpreting various properties of metals. Recently Gurskii and Kraskos) (GK) have proposed a model potential which has explained satisfactorily the phonon dispersio@) binding energy, crystal 237
A. SWAROOP
238
AND
L. M. TIWARI
stability7) superconducting transition temperature and pressure dependence*) for various metals. In the light of these studies it was thought worthwhile to apply this model potential to the case of dilute binary alloys. In the present paper, we therefore have calculated the residual resistivities of dilute alloys of Mg, Al, Pb and In using the same model potential. 2. Theory. According to Harrisong) the resistivity due to an atomic per cent of atoms of type (2) in a host type (1) can be written as - WC,, IOI’ x3 dx, and
(1)
x = q/kF.
C is the characteristic of the host lattice in a dilute alloy. G
+
4
w,2,
-
WC,,
IQ =
(2)
where
(k + 41 Wdk> = w
t1 + (qrc)‘l”
(3)
’
where a and rc are the parameters. The screening has been treated using a Hartree dielectric function c(q), modified according to ShamlO) to include the exchange and correlation effects. It is given by 4xze2 c(q) = 1 + -
Qoq2 x(q) = $
l-
q2 x(q) 9 2 (q2 + kZ: + kg) >
(0.5 + 4kikSqq2 In /$Qi), F
F
F-
9
where k, is Thomas-Fermi screening parameter. In the present paper we have assumed that in an alloy, the form factor of the host atom remains unchanged because of the introduction of dilute impurities but the form factor of the impurity atom does change. The modified form factor of the impurity atom can be written as
+
41 WC,, Ik> = 4xz’2e2 r
0 z
(2~’ - 1) (qrr)2 - 1 [l + m212
1
4d ’
RESIDUAL
RESISTIVITY OF DILUTE BINARY ALLOYS
239
where z’ is the valence; a’ and ri are the GK potential parameters for the impurity. e(q) and Q0 are appropriate to the host metal of which the values are quite reasonable for the case of dilute binary alloys. 3. Results. For the present theoretical investigation we have chosen eight dilute binary alloys such as Li, Al and In in Mg; Mg and Zn in Al; Na and In in Pb; and Pb in In. Our choice of dilute alloys has been restricted because of the availability of GK model potential parameters for their constituent metals. The model potential parameters a and r, for these above constituent metals have been taken from Gurskii and Krasko5). The evaluation of the resistivity has been made by integrating eq. (1). The theoretical values of the residual resistivities for eight dilute alloys have been collected in table I along with the other theoretical and experimental data for comparison. TABLEI The residual rekstivities (in @J cm/at. %) of dilute binary alloys Dilute alloy
Theor.
Exp.9)
Harrisong)
Mg*-Li Mg+-Al Mg*-In Al*-Mg Al*-Zll Pb*-Na Pb+-In In*-Pb
0.46 2.08 1.86 0.22 0.14 1.38 1.37 0.47
0.75 2.10 2.50 0.33 0.22 2.90 1.20 0.60
0.27 1.60 2.40 0.78 0.10 0.77 0.50 0.56
* Solvent metal.
4. Discussion. Table I displays that the residual resistivities of most of the dilute binary alloys except Mg in In, and In in Pb derived from the GK model potential are in better agreement with the experiment than the theoretical results from Harrison’s point - ion model potential. HaiTisong) could attain only qualitative agreement with the experiment because of the crudeness of the form factors used. Since the use of the GK model potential shows an improvement over the previous studies it can be concluded that these model-potential form factors are quite reliable and may be used to evaluate other properties of the liquid and solid states of metals and alloys. Acknowledgements. One of the authors (A.S) would like to thank C.S.I.R. India for the financial assistance. All the computations have been carried out on an ICL 1909 computer.
240
A. SWAROOP AND L. M. TIWAIU REFERENCES
1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
Mott, N.F., Proc. Cambridge Phil. Sot. 32 (1936) 281. Friedel, J., Advances in Phys. 3 (1954) 446. Blatt, F.J., Phys. Rev. 108 (1958) 285. Faber, T.E. and Ziman, J. M., Phil. Mag. 11 (1965) 153. Gurskii, Z.A. and Krasko, G.L., Soviet Physics-JETP Letters 9 (1969) 363; Soviet PhysicsSolid State 11 (1969) 2447; Soviet Physics-Dokl. 16 (1971) 298. Bajpai, R.P., Phys. Letters A42 (1972) 163. Prasad, B. and Srivastava, R.S., Phys. Letters A38 (1972) 521. Swaroop, A. and Tiwari, L. M., Indo-Soviet Conference on Solid State Materials (1972). Harrison, W.A., Pseudopotentials in the Theory of Metals, Benjamin (New York, 1966) p. 150. Sham, L.J., Proc. Roy. Sot. A283 (1965) 33.