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Applicability of the T-matrix unitary condition for the electrical resistivity of liquid metals
Physica 67 (1973) 564-566 0 North-Holland Publishing Co.
APPLICABILITY
OF THE T-MATRIX
FOR THE ELECTRICAL
RESISTIVITY
A. J. GREENFIELD
UNITARY
CONDITION
OF LIQUID
METALS
and N. WISER
Department of Physics, Bar-Ilan University, Ramat-Gan, Israel
Received 21 February 1973
Synopsis It is shown that the T-matrix will not satisfy unitarity if one makes the usual assumption of spherical symmetry in the total scattering potential of an electron by an assembly of ions. As a consequence, we show that the importance of multiple scattering remains an open question in the calculation of the electrical resistivity of liquid metals.
1. Introduction. It has recently been suggested by Preist, Barnes, and Simpsonl) that the calculations of Young, Meyer, and Kilby’) for the electrical resistivity of liquid alkali metals must be in error. Preist et al. contend that the procedure of Young et al. corresponds to using a T matrix which seriously violates unitarity. Preist et al. conclude that the source of the violation is the failure of the calculations of Young et al. to include multiple-scattering terms. In this paper, we wish to point out that the unitarity condition set forth by Preist et al. is meaningful only for spherically symmetric potentials. The total scattering potential seen by an electron in a metal is in fact not spherically symmetric. From a comparison with the solid, we shall demonstrate that it is a poor approximation to assume that the total scattering potential in the liquid is spherically symmetric. Thus, no conclusion can be drawn about the importance of multiple scattering of electrons in liquid metals from the application of the unitarity criterion as formulated by Preist et al. 2. Discussion. Young et al. have calculated the phase shifts l;lrfor the scattering of an electron from a single-screened ion, which is of course spherically symmetric. We shall designate by t(q) the corresponding T matrix where g is the momentum transfer upon scattering. In order to deduce T(q), the T matrix corresponding to electron scattering by an assembly of ions, Preist et al. suggest the relation T(q) + T(q) = t(q)
(1)
b(dl*,
where u(q) is the structure factor for the assembly of ions. This is equivalent to the approximation that for calculating the scattering of the electrons, one may replace 564
T-MATRIX
UNITARITY
CONDITION
FOR RESISTIVITY
565
the true positions of the assembly of ions by their ensemble average. Let us designate this procedure as the sphericalized-potential (SP) approximation. The SP approximation treats the ions as occupying ensemble-averaged positions, which gives rise to a spherically symmetric potential. Therefore, within the framework of the SP approximation, one may make the usual Legendre-polynomial expansion for T(q) T(q) = k;lC
I
(21 + 1) TIPI (1 - q2/2k;),
(2)
where P,(p) is the Legendre polynomial and Tl is the Zth partial-wave amplitude of the scattering by the assembly of ions. Preist et al. checked for unitarity the Tl they derived in the SP approximation from the phase shifts of Young et al. Unitarity requires that Im Tl = ITJ”.
(3)
Preist et al. found a serious violation of (3) for all the alkalis as shown in the second and fourth qolumns of table I. They conclude on this basis that multiple-scattering terms are important. We disagree with this conclusion, attributing the violation of unitarity instead to the inadequacy of the SP approximation. TABLEI Comparison of 1Tllz with Im TI for both the solid and liquid phases of Na and K in the vicinity of the melting point. The data for the liquid phase are taken from Preist, Barnes, and Simpson’). For the solid phase, we calculated the values using the structure factor and form factor of Kaveh and Wiser3).
lT012 Im To IT112 Im Tl lT212
Im T2
Na-Liquid
Na-Solid
K-Liquid
K-Solid
0.026 0.110 0.005 0.002 0.000 - 0.005
0.013 0.077 0.001 -0.010 0.000 0.002
0.051 0.133 0.003 -0.006 0.000 0.000
0.015 0.085 0.001 -0.013 0.000 0.002
One may demonstrate this inadequacy by making the same SP approximation, eq. (I), for the solid phase of a metal at high temperatures, deriving u(q) from the phonon frequencies3). The resulting Ti display the same violation of unitarity as found for the liquid phase, as shown in the third and fifth columns of table I. As is well known4s5), in the solid phase the small excursions of the ions from their equilibrium position justify the neglect of multiple-scattering terms. Indeed,
566
A. J. GREENFIELD
AND N. WISER
recent electrical-resistivity calculations3) for solid Na and K which ignore multiplescattering corrections give excellent agreement with experiment over a very wide temperature range. Since one cannot ascribe the violation of unitarity for the solid to neglect of multiple scattering; the fault in the solid must lie with the SP approximation. Although macroscopically a liquid metal is isotropic whereas a solid metal is not, electron scattering is a microscopic process. This is particularly evident for calculations of the electrical resistivity for which scattering with momentum transfers q N 2kF dominate the resistivity integra16). Such a q value corresponds to a wavelength comparable to the interatomic separation and thus the electron samples the detailed microscopic structure. The short-range order in the liquid insures that there exists microscopic anisotropy in the liquid to the same extent as in the solid. Therefore, in view of the fact that the SP approximation fails for the solid, there is no justification for accepting it for the liquid. It is in place to point out that one should not view the above discussion as supporting the view that multiple scattering may be neglected for the liquid. In fact, this important question remains unresolved. 3. Conclusions. We show in the above discussion that it is unwarranted to make the sphericalized-potential approximation to describe the scattering of an electron by an assembly of ions. This approximation assumes that one may replace the true (non-spherically symmetric) positions of the ions by their (spherically symmetric) ensemble-averaged positions. It follows, therefore, that the recently proposed test for unitarity of the calculated T-matrix may not be meaningfully applied. Thus, it remains an open question whether multiple scattering is important in the calculation of the electrical resistivity of liquid metals.
REFERENCES 1) 2) 3) 4) 5) 6)
Preist, T.W., Barnes, G. J. and Simpson,D. C., Phys. Letters 31A (1970) 114. Young, W. H., Meyer,A. and Kilby, C. E., Phys. Rev. 160 (1967) 482. Kaveh, M. and Wiser, N., Phys. Rev. B 6 (1972) 3648. Baym, G., Phys: Rev. 135 (1964) A1691. Prange,R. E. and Kadanoff, L. P., Phys. Rev. 134 (1964) A566. Wiser, N., Phys. Rev. 143 (1966) 393.
APPLICABILITY
OF THE T-MATRIX
FOR THE ELECTRICAL
RESISTIVITY
A. J. GREENFIELD
UNITARY
CONDITION
OF LIQUID
METALS
and N. WISER
Department of Physics, Bar-Ilan University, Ramat-Gan, Israel
Received 21 February 1973
Synopsis It is shown that the T-matrix will not satisfy unitarity if one makes the usual assumption of spherical symmetry in the total scattering potential of an electron by an assembly of ions. As a consequence, we show that the importance of multiple scattering remains an open question in the calculation of the electrical resistivity of liquid metals.
1. Introduction. It has recently been suggested by Preist, Barnes, and Simpsonl) that the calculations of Young, Meyer, and Kilby’) for the electrical resistivity of liquid alkali metals must be in error. Preist et al. contend that the procedure of Young et al. corresponds to using a T matrix which seriously violates unitarity. Preist et al. conclude that the source of the violation is the failure of the calculations of Young et al. to include multiple-scattering terms. In this paper, we wish to point out that the unitarity condition set forth by Preist et al. is meaningful only for spherically symmetric potentials. The total scattering potential seen by an electron in a metal is in fact not spherically symmetric. From a comparison with the solid, we shall demonstrate that it is a poor approximation to assume that the total scattering potential in the liquid is spherically symmetric. Thus, no conclusion can be drawn about the importance of multiple scattering of electrons in liquid metals from the application of the unitarity criterion as formulated by Preist et al. 2. Discussion. Young et al. have calculated the phase shifts l;lrfor the scattering of an electron from a single-screened ion, which is of course spherically symmetric. We shall designate by t(q) the corresponding T matrix where g is the momentum transfer upon scattering. In order to deduce T(q), the T matrix corresponding to electron scattering by an assembly of ions, Preist et al. suggest the relation T(q) + T(q) = t(q)
(1)
b(dl*,
where u(q) is the structure factor for the assembly of ions. This is equivalent to the approximation that for calculating the scattering of the electrons, one may replace 564
T-MATRIX
UNITARITY
CONDITION
FOR RESISTIVITY
565
the true positions of the assembly of ions by their ensemble average. Let us designate this procedure as the sphericalized-potential (SP) approximation. The SP approximation treats the ions as occupying ensemble-averaged positions, which gives rise to a spherically symmetric potential. Therefore, within the framework of the SP approximation, one may make the usual Legendre-polynomial expansion for T(q) T(q) = k;lC
I
(21 + 1) TIPI (1 - q2/2k;),
(2)
where P,(p) is the Legendre polynomial and Tl is the Zth partial-wave amplitude of the scattering by the assembly of ions. Preist et al. checked for unitarity the Tl they derived in the SP approximation from the phase shifts of Young et al. Unitarity requires that Im Tl = ITJ”.
(3)
Preist et al. found a serious violation of (3) for all the alkalis as shown in the second and fourth qolumns of table I. They conclude on this basis that multiple-scattering terms are important. We disagree with this conclusion, attributing the violation of unitarity instead to the inadequacy of the SP approximation. TABLEI Comparison of 1Tllz with Im TI for both the solid and liquid phases of Na and K in the vicinity of the melting point. The data for the liquid phase are taken from Preist, Barnes, and Simpson’). For the solid phase, we calculated the values using the structure factor and form factor of Kaveh and Wiser3).
lT012 Im To IT112 Im Tl lT212
Im T2
Na-Liquid
Na-Solid
K-Liquid
K-Solid
0.026 0.110 0.005 0.002 0.000 - 0.005
0.013 0.077 0.001 -0.010 0.000 0.002
0.051 0.133 0.003 -0.006 0.000 0.000
0.015 0.085 0.001 -0.013 0.000 0.002
One may demonstrate this inadequacy by making the same SP approximation, eq. (I), for the solid phase of a metal at high temperatures, deriving u(q) from the phonon frequencies3). The resulting Ti display the same violation of unitarity as found for the liquid phase, as shown in the third and fifth columns of table I. As is well known4s5), in the solid phase the small excursions of the ions from their equilibrium position justify the neglect of multiple-scattering terms. Indeed,
566
A. J. GREENFIELD
AND N. WISER
recent electrical-resistivity calculations3) for solid Na and K which ignore multiplescattering corrections give excellent agreement with experiment over a very wide temperature range. Since one cannot ascribe the violation of unitarity for the solid to neglect of multiple scattering; the fault in the solid must lie with the SP approximation. Although macroscopically a liquid metal is isotropic whereas a solid metal is not, electron scattering is a microscopic process. This is particularly evident for calculations of the electrical resistivity for which scattering with momentum transfers q N 2kF dominate the resistivity integra16). Such a q value corresponds to a wavelength comparable to the interatomic separation and thus the electron samples the detailed microscopic structure. The short-range order in the liquid insures that there exists microscopic anisotropy in the liquid to the same extent as in the solid. Therefore, in view of the fact that the SP approximation fails for the solid, there is no justification for accepting it for the liquid. It is in place to point out that one should not view the above discussion as supporting the view that multiple scattering may be neglected for the liquid. In fact, this important question remains unresolved. 3. Conclusions. We show in the above discussion that it is unwarranted to make the sphericalized-potential approximation to describe the scattering of an electron by an assembly of ions. This approximation assumes that one may replace the true (non-spherically symmetric) positions of the ions by their (spherically symmetric) ensemble-averaged positions. It follows, therefore, that the recently proposed test for unitarity of the calculated T-matrix may not be meaningfully applied. Thus, it remains an open question whether multiple scattering is important in the calculation of the electrical resistivity of liquid metals.
REFERENCES 1) 2) 3) 4) 5) 6)
Preist, T.W., Barnes, G. J. and Simpson,D. C., Phys. Letters 31A (1970) 114. Young, W. H., Meyer,A. and Kilby, C. E., Phys. Rev. 160 (1967) 482. Kaveh, M. and Wiser, N., Phys. Rev. B 6 (1972) 3648. Baym, G., Phys: Rev. 135 (1964) A1691. Prange,R. E. and Kadanoff, L. P., Phys. Rev. 134 (1964) A566. Wiser, N., Phys. Rev. 143 (1966) 393.