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Deviation from Matthiessen rule in Mg -Cd and Cd -Mg
Volume 45A, number 1
PHYSICS LE~FERS
27 August 1973
DEVIATION FROM MAT~HIESSENRULE IN ~g-Cd AND Cd-Mg G. TOUSSAINT and P. PECHEUR Institut National Polytechnique de Nancy, Laboratoire associéau C.N.R.S. no 155, Parc de Saurupt, Nancy (54), France Received 9 July 1973 The interference tenn between electron-phonon and electron impurity scattering has been evaluated for ~gCd and CdMg and found to be negative, in agreement with experiment. It is shown that measurements on single crystal should give valuable information. It has been shown by Kagan and Zherov [1] that the interference term between electron-phonon and electron impurity scattering should dominate the deviation from Matthiessen rule for T> ~ODand gives a contribution ~‘ linear with T. Bhatia and Gupta [2] taking into account U-processes concluded that the sign oft’ should reverse in complementary alloys (A in B and B in A). This simple rule seems to be well obeyed by many dilute alloys, with the ~,
6
/
/
/ ~/ /WMS(WCd_WMQ)
4
exception of MgCd and CdMg where both ~ decrease in their linear range [3]. To investigate this point in more detail we have calculated ~‘ for the two symmetry directions of a single hexagonal crystal of A containing B impurities. The calculation proceeds as in ref. [4]. ~‘ is given by formula (2) of [4], where one replaces I h’~q)I2by 2W~(q)[W~(q)—W~(q)] Due to U-processes, the neighborhood of q = G (G corresponding to the first reciprocal lattice nodes) is heavily weighted in the integral. Fig. 1 shows that for Cd and Mg these two points are in a region where the sign of 2W~(q) X [W~(q)—W~(q)]changes. This means that the sign of ~‘ is strongly dependent of the form factors W~q).We performed the numerical calculation using ~‘
~‘
.
\
T(K) 100
2
/
\
300 MQCd ±CdMQ
\\
\N~
\4olio]M9 [0002JMg ~
200
~
/
\\~\• ~/KF
/ °~
\
-2
[ooo2]~/~__’
/ [orio]cd
1
4
I //
4
\
/
\\
N
\
—
-~
\
“N N
\
“N
“N
‘N
N//MgCd
WCd(WM 9_WCd)
8
—6
10
//CdMQ
-8
Fig. 1. W~(q)[W~(q)— W~(q)Jfor ~gCd and CdMg. The positions of the first lattice nodes are shown in abcissa.
Fig. 2. ~ and ~ for ~gCd and CdMg. (cp~is the impurity resistivity at T = 0). 75
Volume 45A, number 1
PHYSICS LETTERS
27 August 1973
a Born von Karman model for lattice dynamics [5,6] and the Animalu—Heine form factors [7]. The
References
impurity form factor has been corrected in the simple way of [8]: this is not very important, the atomic volumes of Mg and Cd being quite close. Fig. 2 shows our results for ~‘, both negative. The comparison with experiment is only indicative, due to the neglect of other processes contributing to Matthiesen rule (for instance, the one advocated by Kagan and Zhernov in [9] which is important at low impurity concentration and could then well change the sign of ~ [10]. However there is a possible check of the importance of the interference term. We have found that ~‘ ~ differs in the two symmetry direction by an order of magnitude (fig. 2). This is explained by the fact that lattice nodes contribute differently in the two directions and are differently weighted by 2W~(q)[W~(q)—W~(q)](fig. 1). Measurements on single crystals should then give valuable information.
[1] Y. Kagan and A.P. Zhernov, Zh. Eksp. iTeor. Fiz. 50 (1966) 1107. [2] A.B. Bhatia and O.P. Gupta, Phys. Lett. 29A (1969) 358. [3] Seth and Woods, Phys. Rev. Chem. B2 (1970) 2961. [4] R.S. P. Pecheur andS.B. G. Toussaint, J. Phys. Solids 33 (1972) 2281. [5] P.K. Iyengar, G. Venkataraman, P.R. Vijayaraghavan and A.P. Roy, Inelastic scattering of neutrons in solids and liquids (I.A.E.A. Vienna, 1965) Vol. 1, p. 153. [6] G. Toussaint (1972) 165. and G. Champier, Phys. Stat. SoL (b) 54 [7] W.A. Harrison, Pseudopotential in the theory of metals (Benjamin, New York, 1966). [8] Y. Fukai, Phys. Rev. 186 (1969) 3, 697. [9] Yu. Kagan and A.P. Zhernov, Zh. Eksp. i Teor. Fiz. 60 1832. and P. Pecheur, to be published. [10] (1971) G. Taoussaint
,
76
PHYSICS LE~FERS
27 August 1973
DEVIATION FROM MAT~HIESSENRULE IN ~g-Cd AND Cd-Mg G. TOUSSAINT and P. PECHEUR Institut National Polytechnique de Nancy, Laboratoire associéau C.N.R.S. no 155, Parc de Saurupt, Nancy (54), France Received 9 July 1973 The interference tenn between electron-phonon and electron impurity scattering has been evaluated for ~gCd and CdMg and found to be negative, in agreement with experiment. It is shown that measurements on single crystal should give valuable information. It has been shown by Kagan and Zherov [1] that the interference term between electron-phonon and electron impurity scattering should dominate the deviation from Matthiessen rule for T> ~ODand gives a contribution ~‘ linear with T. Bhatia and Gupta [2] taking into account U-processes concluded that the sign oft’ should reverse in complementary alloys (A in B and B in A). This simple rule seems to be well obeyed by many dilute alloys, with the ~,
6
/
/
/ ~/ /WMS(WCd_WMQ)
4
exception of MgCd and CdMg where both ~ decrease in their linear range [3]. To investigate this point in more detail we have calculated ~‘ for the two symmetry directions of a single hexagonal crystal of A containing B impurities. The calculation proceeds as in ref. [4]. ~‘ is given by formula (2) of [4], where one replaces I h’~q)I2by 2W~(q)[W~(q)—W~(q)] Due to U-processes, the neighborhood of q = G (G corresponding to the first reciprocal lattice nodes) is heavily weighted in the integral. Fig. 1 shows that for Cd and Mg these two points are in a region where the sign of 2W~(q) X [W~(q)—W~(q)]changes. This means that the sign of ~‘ is strongly dependent of the form factors W~q).We performed the numerical calculation using ~‘
~‘
.
\
T(K) 100
2
/
\
300 MQCd ±CdMQ
\\
\N~
\4olio]M9 [0002JMg ~
200
~
/
\\~\• ~/KF
/ °~
\
-2
[ooo2]~/~__’
/ [orio]cd
1
4
I //
4
\
/
\\
N
\
—
-~
\
“N N
\
“N
“N
‘N
N//MgCd
WCd(WM 9_WCd)
8
—6
10
//CdMQ
-8
Fig. 1. W~(q)[W~(q)— W~(q)Jfor ~gCd and CdMg. The positions of the first lattice nodes are shown in abcissa.
Fig. 2. ~ and ~ for ~gCd and CdMg. (cp~is the impurity resistivity at T = 0). 75
Volume 45A, number 1
PHYSICS LETTERS
27 August 1973
a Born von Karman model for lattice dynamics [5,6] and the Animalu—Heine form factors [7]. The
References
impurity form factor has been corrected in the simple way of [8]: this is not very important, the atomic volumes of Mg and Cd being quite close. Fig. 2 shows our results for ~‘, both negative. The comparison with experiment is only indicative, due to the neglect of other processes contributing to Matthiesen rule (for instance, the one advocated by Kagan and Zhernov in [9] which is important at low impurity concentration and could then well change the sign of ~ [10]. However there is a possible check of the importance of the interference term. We have found that ~‘ ~ differs in the two symmetry direction by an order of magnitude (fig. 2). This is explained by the fact that lattice nodes contribute differently in the two directions and are differently weighted by 2W~(q)[W~(q)—W~(q)](fig. 1). Measurements on single crystals should then give valuable information.
[1] Y. Kagan and A.P. Zhernov, Zh. Eksp. iTeor. Fiz. 50 (1966) 1107. [2] A.B. Bhatia and O.P. Gupta, Phys. Lett. 29A (1969) 358. [3] Seth and Woods, Phys. Rev. Chem. B2 (1970) 2961. [4] R.S. P. Pecheur andS.B. G. Toussaint, J. Phys. Solids 33 (1972) 2281. [5] P.K. Iyengar, G. Venkataraman, P.R. Vijayaraghavan and A.P. Roy, Inelastic scattering of neutrons in solids and liquids (I.A.E.A. Vienna, 1965) Vol. 1, p. 153. [6] G. Toussaint (1972) 165. and G. Champier, Phys. Stat. SoL (b) 54 [7] W.A. Harrison, Pseudopotential in the theory of metals (Benjamin, New York, 1966). [8] Y. Fukai, Phys. Rev. 186 (1969) 3, 697. [9] Yu. Kagan and A.P. Zhernov, Zh. Eksp. i Teor. Fiz. 60 1832. and P. Pecheur, to be published. [10] (1971) G. Taoussaint
,
76