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Temperature dependence of the cross section for electron scattering by dislocations in metals
V. F. Gantmakher and G. I. Kulesko
Solid State Institute, USSR Academy of Sciences (Submitted July 12, 1974) Zh. Eksp. Teor. Fiz. 67, 2335-2340 (December 1974) The temperature dependence of the additional electrical resistance that arises when dislocations are introduced into a metal is examined for single-crystal and polycrystalline specimens of copper, molybdenum, and zinc. In all cases a stepped curve is obtained, the position of the step on the temperature scale being independent of the degree and mode of deformation. The possibility is discussed that a temperature dependence of this type may result from a temperature dependence of the cross section for scattering of electrons by the dislocation themselves, either on account of inelastic scattering via quasilocal vibrational modes, or on account of changes in the elastic scattering cross section because of filling of electron levels localized at the dislocations.


DISCUSSIO N Thus, in all th e cases that we investigated th e r(T ) curv e has a step, and fo r zinc , it probably has two . We fee l that ther e mus t be some specific reason fo r a dependenc e of this sort. The r(T ) curv e is usuall y discusse d in terms of deviations fro m Matthiessen's rule , i.e., it is attribute d to nonadditivity of the scattering fro m scatterers of differen t types . Such nonadditivity could arise, fo r example, fro m difference s in th e angular dependences of the differential scattering cross sections or fro m difference s in the dependences of the total scattering cross sections on the position of the electron
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Various mechanisms may be proposed fo r the temperature variation of the scattering. First, if the dislocations have quasilocal vibrational modes of characteristic frequenc y w , a new inelastic scattering channelscattering with absorption or emission of a quasilocal phonon-may open up at elevated temperatures . From the position of the step on the r(T ) curve it follows that w mus t be of the order of 70°, i.e., jus t a few times lower than th e Debye frequency . Such vibrational frequencies have not been observed fo r dislocations, but , generally speaking, we can suggest reasonable physical models fo r suc h modes . For example, there may be a branch w(k )
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of the optical type corresponding to vibrations of dislocations near one minimu m of the Peierls surfac e with a characteristic frequenc y of the order of a few degrees1'. In addition, one can imagine vibrational modes in th e cores of dislocations near points at which the dislocations are pinned whose frequencies are determined by parameters associated, not with th e defect , but with the dislocation itself. This possibility, however , was apparently not realized in our experiments : otherwise, altering the impurit y concentration and y-ra y irradiation of the specimens woul d have affecte d r since they affec t the concentration of pinning points. Regardless of the specific model , the change in the nonequilibriu m addition Af to the electron distribution functio n f due to collisions with phonons is proportional to the quantity (see, e.g., Cel )

FIG. 5. Process of fitting Eq. (5) to the experimental points of curve 1, Fig. 1 (copper single crystal). The correspondin g values of a/r m a x are given at the curves.

1/T mus t be a straight line. This condition allows us to determin e a region of possible values of a (a 2 r max ) > and the position of the line fo r each valu e of a gives th e corresponding values of e and /3. The procedur e is illustrated in Fig. 5, usin g one of the copper specimens (fo r 2 r(T)=a(i + $e""]-\ (5)

CONCLUSION It mus t be said that at present there is no satisfactory theory of the contribution to electrical resistance fro m scattering of electrons by dislocations. We still do not understan d why dislocations introduce d by bending have very large scattering cross sections181. In fact , there is no explanation for the P ''V. Ya. Kravchenko called ou r attentio n to this possibility. He mad e preliminary estimates, which however, gave to values several times smaller than the experimenta l values.

where j3 is the spin degeneracy of the level and a is a proportionality constant. Of cours e Eq. (5) is also applicable when the level lies below the Fermi level and the dislocations become positively charged when the temperatur e is raised. Although Eq. (5) contains thre e parameters (a, j3, and e) , the possibilities fo r fittin g it to the experimental curve are quite limited. In fact, if the r(T) curve can be represented by Eq . (5) , the plot of ln(a/ r -1) against
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Z . S. Basinski, J. S. Dugdale , and A. Howie, Philos. Mag. 8, 1989 (1963). 2 Ch . V . Kopetskii, G . I . Kulesko , and A . M . lonov , Izv . Akad. Nau k SSSR, Metally 2, 130 (1972). 3 Ch . V. Kopetskii , G. I. Kulesko , and L. S. Kokhanchik , Fiz. Met. Metalloved. 35, 624 (1973). 4T. W . Barbee Jr. , R . A . Huggins , and W . A . Little , Philos. Mag. 14, 255 (1966). 5J. Livingstone, Direct Observation of Imperfection s in Crystals (J. B. Newkirk and J. H. Wernick , editors) , 1962. V. F. Gantmakher , Rep . Prog. Phys . 37, 317 (1974). 7R. A. Brown , Phys . Rev . 156, 889 (1967). "V. F. Gantmakher , V. A. Gasparov, G. I. Kulesko, and V. N. Matveev , Zh . Eksp . Teor. Fiz. 63, 1752 (1972) [Sov. Phys.-JETP 36, 925 (1973)]. Translated by E. Brunner .
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