Physica I07B (1981) 135-136 North-Holland PublishingCompany
ELECTRICAL RESISTIVITY OF ALKALI METALS AT LOW TEMPERATURES
0. N. Awasthi
Physics Department, Regional College of Education (National Council of Educational Research & Training) University of Mysore Campus, Mysore - 570006, India
The contribution of electron-electron Umklapp scattering processes in the electrical resistivity of sodium, potassium, rubidium, and cesimn at low temperatures has been evaluated using a simplified spherical Fermi surface model with an isotropic transition probability. Our values of the electrical resistivity so obtained compare fairly well with the recent experimental results for sodium, potassium, and rubidium. Our results also compare fairly well with the other available data in the literature.
the electrons in the concerned metal. The details of our calculations are listed below.
At liquid helium temperatures the electrical resistivity of metals is governed mainly by e-e and e-phonon interactions. Recent measurements [1,2] of the electrical resistivity of sodium and potassium in the temperature range 0.5 - 4.2 K have indicated that these metals have a T 2term. The electrical resistivity of the alkali metals at low temperatures can be represented by an expression of the form: 0(T) = 0(0) + A T 2 + B T exp(-O/T)
The T2-term which is dominant below 2 K occurs due to e-e Umklapp scattering and the Texp(-8/T) term (with 8 = 19.9±0.2 K) which dominates above 2 K is due to ~-phonon scattering under phonon drag conditions. The remaining part, 0(0), which is known as the residual resistivity, is caused by the e-dislocation and e-impurity scattering processes.
It is well known that the scattering potential for the electrons in a metal is of the screened Coulomb type and can be expressed as:
V(r) = ( l / r )
where k i s the s c r e e n i n g p a r a m e t e r . The t r a n s i t i o n c p r o b a b i l i t y f o r e-e s c a t t e r i n g due to the above p o t e n t i a l can be w r i t t e n as: T(kl,k2 ÷ k3,~)
1 < ~ 3 , ~ [ V ( r ) l~l,~2>l 2
) . (3)
The 6 - f u n c t i o n i n (3) t a k e s c a r e of the energy In the isotropic limit the normal e-e scattering does not contribute to the electrical resistivity because in such collisions charge and the momentum are conserved. Umklapp processes, however, impart momentum to the lattice as a whole and they can thus contribute to the electrical resistivity. The magnEtude of this contribution is, however, greatly reduced due to the operation of the Pauli's exclusion principle. To obtain the T2-term for the alkali metals due to t h e e - e Umklapp scattering processes, we employ an approach which differs from that of Lawrence and Wilklns  in certain aspects. Firstly, for solving the linearized Boltzmann equation, they use an energy dependent trial function for the deviation of the Fermi function from its equilibrium v a l u e . Secondly, the e f f e c t of Umklapp scattering enters in their rasistlvlty expression in the form of A-function which they evaluate using two-plane wave states, ignoring the core-orthogonallzation components of the wave functions. We, on the other hand, introduce the effect of the lattice on the free electrons in terms of an overlap integral which we have evaluated using the Bloch functions for 03784363/81/0000-0000/$02~0 ~Nonh-HonandPub~JngCo~any
conservation. Using Born approximation, the matrix elements of V(r) between the initial and final states can be written as:
, k 2 ( r 2 ) d 3 r 1 d3r_2
Here ~k'S are the Bloch functions for the electrons -- in the lattlce and they obey the following relation:
Sk(D = ~xp(~.~) .k(~)
with Uk(_r+~) = uk(r_); ~ = lattice vector. On introducing (5) for the Bloch functions into Eq. (4) and simplifying it, we find that the matrix elements of V(r) contain the square of an overlap integral G given by:
Table i. (6)
where ~ = ~i + ~2 - ~3 - ~4; K = reciprocal lattice vector, and ~ is the atomic volume. The integral (6) is to be evaluated over a WignerSeitz cell. Proceeding in a similar manner as discussed by Ziman  and by Rosler , we obtain the following expression for 0ee due to e-e Umklapp scattering processes: Pee(T) = 6.146xi0 -38 (Z/rs)(I/eF3VF)G2T2~ cm (for g = 2 ~ )
where Z is the coordination number of the reciprocal lattice, r = ro/a h (where a h is the Bohr s
radius and r is the radius of a sphere that contains one o electron), gF is the Fermi energy, v F is the Fermi velocity, and G is the overlap integral for the incident and scattered electron wave functions inside the Wigner-Seitz cell of the lattice. 2.1
Evaluation of the Overlap Integral.
In order to evaluate the overlap integral (G) we use the Wigner-Seitz method  which gives us the wave-functlon for the electron at the lowest state of the metal, i.e., for k = 0. Denoting t h i s wave function by ~ (r), a fair approximation o -to the wave function for higher states will be within any one atomic sphere, ~k(r) = exp(ik.r) ~ (r) provided that k -- Ties within the f ~ s t o B-Z, but not too near its boundaries. Using this approximation we can write Eq.
G = ~i I
_ I2 exp (i~'r) d3~ l~°(r)
Na K Rb Cs
0.089 0.074 0.080 0.059
0.i0 0.24 0.43 0.38
0.015 0.17 ---
0.04 0.20 0.50 1.40
Expt. data 0.19 0.ii 0.54 --
MacDonald and Geldart . The experimental values of A for K vary from 0.075 to 0.27 p~ ¢m K -2 under various physical conditions. The cause of this variation of A in different samples of K has recently been discussed by Kaveh and Wiser  in terms of anisotropic e-e scattering processes. For K, the condition of isotropic e-e scattering can be met when pi/0d > i; where Pi' Pd are the contributions to 0(0) due to e-impurity and e-dislocation scattering processes, respectively. Under this condition we find that for K, A = 0.ii p~ cm K -2 is a fairly good representative value. Even though this value is almost 50% of our estimate, we believe that the agreement is fairly good considering the various approximations involved in the many body problem. Na, on the other hand, does not show very large sample dependence for A. In this case also our values are closer to the experimental data as compared to the other theoretical estimates, i.e., Refs. 3 and 13. Our values of A compare fairly well in Rb with experimental data as well as with the data of MacDonald and Geldart. Experimental data for Cs ere not available for comparison. It is concluded, therefore, that our approach renders fairly good results for Oee in alkali metals at low temperatures.
A(p ~ cm K -2) Lawrence and Mac. Wilkins & G.
RESULTS AND DISCUSSION
We have evaluated G numerically with the help of Eq. (8). Wigner and Seitz's values  of ~ (r) for sodium, and Callaway's values [7,8,9] o -of ~ (r) for potassium, rubidium, and cesium have ° been used in our calculations. We ignore the band-structure effects in these metals as the Fermi surface for these metals deviates from the free electron spheres by less than 0.2% [i0, Ii]. We have listed in Table i, below, the values of G and A for the alkali metals studied. For the sake of comparison the experimental values drawn from Refs. i, 2, and 12, as well as the theoretical data from from Refs. 3 and 13, are also included in Table I. The experimental data of Ref. 12 are extrapolated for the sake of comparison and should not be taken very seriously. From Table I it is obvious that our values of A for K compare fairly well with the values reported by Lawrence and Wilkins , as well as by
REFERENCES: [i]         [i0] [ii]   
Levy, B., Sinvanl, M., et al., Phys. Rev. Lett. 43 (1979) 1822. Rowlands, J.A., et al., Phys. Rev. Lett. 40 (1978) 1201. Lawrence, W.E., et al., Phys. Rev. 7B (1973) 2317. Ziman, J.M., Electrons and Phonons (Oxford Univ. Press, 1960). Rosler, M., Ann. Phys. 16 (1965)'70. Wigner, E., et al., Phys. Rev. 43 (1933) 804. Callaway, J., Phys. Rev. 119 (1960) 1012. Ibid., 112 (1958) 3349. Ibid., 112 (1958) 1061. Lee, M.J.G., Proc. Roy. Soc. (L) A295 (1966) 440. Lee, M.J.G., et al., Proc. Roy. Soc. (L) A2314 (1968) 319. Cook, J.G., Can. J. Phys. 57 (1979) 1216. MacDonald, A.H., et al., J.P.F.: Metal Phys. i0 (1980) 677. Kaveh, M., et al., J. Phys. F.: Metal Phys. i0 (1980) 137.