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The validity of the rigidmuffin-tin approximation is discussed on the basis of measur rements of the response of the Fermi surface of palladium to homogeneous volume conserving strains. The experimental strain derivatives are found to be consistently 30 % smaller than theoretical values obtained from a KKR band structure calculations. Implications of this result are discussed for the electron-phonon A and the superconducting Tc. According to the McMillan theory /1,2/ the superconducting transition temperature T of ametal depends on the electron-phonon enhancement factor A through the following relation Tc = (1.2)-lw exp {-l .04(1+~)/~-~~(1+0.62A)~ (1) 1% where uX is a constant (assumed to be 0.13) andw log is an average over the phonon frequencies. The parameter A can be expressed as a product of a purely electronic factor rl and an essentially phononic factor so that : A = ~ / M (2) Here M is the mass of the ions ; depends only weakly on the electronic properties and is determined by the phonon spectrum of the metal under consideration. The central problem is the calculation of r~ which is proportional /l/ to the gradient of the potential V. Most of the recent theoreticalwork on electron-phonon interaction in transition metals makes use of the rigid-muffin-tin (RMT) approximation of Gaspari and Gyorffy 131. In this approximation one assumes that the self-consistent muffinmetal we write the KKR secular equation as + ~ ( k , E, e.) = 0 J (3) where are scattering phase shifts of the muffintin potential and e is the j-th component of the j strain tensor. The strain dependence of the area A OS an extremal cross-section of the FS is obtained by solving the implicit equation (3) with respect + to k at an energy equal to EF. As shown by Shaw, Ketterson and Windmiller /4/ and by Griessen, Stanley and Lee /S/, the strain derivative D = j dlnA/de. can be calculated from an integral along 3 the orbit containing the term Dx/De where Dx/De i' i For a general strain 5, the derivative dlnA/dc depends explicitly on the strain dependence of EF and the scattering phase shifts. For volume conserving strains y it is however easy to show that in a cubic crystal : 6 I tin potential around an ion moves rigidly with the dlnA/dy = C ain~/ae de. ldy j=1 ion when a phonon is propagating through thelatkice. The validity of this approximation is difficult to where de./dy is entirely determined by the strain J test by means of superconducting T data, because of State of the Eq' shows that C the uncertainties in w and pX. In this pathe response of the FS to a shear depends on the log' per we present a direct method to test the validity pertubation of the electronic band structure by the of the mT approximation based on measurements of lattice deformation at constant values of 6 $ and EF. the strain response.of the Fermi surface (FS) of Therefore the shear response can be calculated from transition metals to homogeneous volume conserving a using zero-pressure for E~ and strains (shears). 6 & and the validity of the RMT-approximation can be In order to show which role the mT approxitested by comparing measured dlnhldy withcalculated mation plays in the shear response of the FS of a derivatives. Experimental values for Dj were obtaiArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786486 ned by simultaneously measuring the amplitude of The experimental numbers are found to be 30 % smalQ, quantum oscillations in the magnetization (M) and ler than the theoritical values from the relativis'L the sound velocity (vs). As shown by Testardi and tic KKR calculation. This is also true for the te~ondon/7/ the strain dependence of the FS can be tragonal shear derivative of the belly orbit rl l00 l calculated from : determined from the uniaxial stress measurements of Q, {D.D.} = V 1 J s ceff/T ' (6) where Ceff is the effective elastic constant corresponding to the ultrasonic configuration, F is the de Haas-van Alphen frequency and (D.D .) is a combi1 J nation of derivatives D j' The measurements were done on palladium single crystals of 4N purity in magnetic fields up to 6.5 tesla. The oscillations in M were detected by means of a fluxgate probe coupled to a superconducting flux transformer /8,9/ and the oscillations in v were measured with a continuous wave set-up. From the D. measured in several ultrasonic configuJ rations we determined the response to tetragonal shear yz(e,=e2=-$e3) and angular shear y of the XY hole-ellipsoids at point X and L of the Brillouin zone, listed in table I. Joss and Van der Mark 161. This information may be used to estimate the errror introduced in q by the use of the RMT approximation as in the deformation potential approximation /l01 rl is proportional to ( ~ ~ 1 ~ 6 ) ', whereas dlnA/dy is /5/ proportional to DXIDE. As a consequence of this quadraticdependence we expect the RMT approximation to give values approximately 1.7 too high for 11. This tendency has also been found by Butler /11/ in Y,Zr,Tc and Rh, while Gyorffy 1121 pointed out that the experimental value of 9 from measurements of T in niobium is about 1.6 times lower than the calculated value of Butler /Ill. It is interesting to point out that the Tc of palladium obtained with a "scaled" value of -1 8 11 = ~ ~ 1 1 . 7 is essentially zero (Tcs10 mK)while the value calculated by Butler 1111 with the RMT leads to 5.5 mK. TABLE I Experimental and theoretical shearderivatives of the area of extremal cross-sections of the FS of palladiud. yz is a tetra onal shear in the 1001 1 direction and Yxy is equal to the decrease of the angle between 71001 and l010l axes of the real space lattice induced by the shear. The parameters entering the KKR bandstructure calculations are : EF=0.4816 a.u., 60=-0.09887 rad, 6 =-0.00205 rad, 1,112 '1,312 =-0.04170 rad, 6 =-0.23918 rad, 6 = -0.29095 rad 2,312 2,512 orbit direction F( tesla) center of C ~ X P e r a)Obtained from uniaxial stress data 161 References /l/ McMillan, W.L., Phys. Rev. 167 (1968)331 / 2 / Allen, P.B. and Dynes, R.C., Phys. Rev. B12 (1975) 905 /3/ Gaspari, G.D. and Gyorffy, B.L., Phys. Rev. Lett. 28 (1972) 801 - / 4 / Shaw, J.C., Ketterson, J.B. and Windmiller, L.R., Phys. Rev. (1972) 3894 / 5 / Griessen, R., Lee, M.J.G. and Stanley, D.J., Phys Rev. B16 (1977) 4385 ,161 Joss, W. and Van Der Mark, W., Proc. Int. Conf. on Physics of Transition Metals, Toronto 1977 171 Testardi, L.R. and Condo, J.H. Phys. Rev. (1970) 3928 181 De Wilde, J., Ph.D. Thesis (1978),Vrije Universi- teit, Amsterdam I91 De Wilde, J. and Meredith, D.J., J. Phys. (1976) 62 /10/ Ziman, J.M. "Electrons and Phonons" (Clarendon Press Oxford) 1962, p. 186 /11/ Butler, W.H., Phys. Rev. B15 (1977) 5267 1121 Gyorffy, B.L. "Superconductivity in dand f- metals" (Plenum Press, Ed. D.H. Douglass) 1972, p.29 |
