ASPHERICAL COULOMB SCATTERING OF CONDUCTION ELECTRONS IN PRB6

The electrical resistivity of PrB6 contains a term arising from the temperature dependent crystal field level populations by 4f electrons which our measurements indicate is equally made up of aspherical Coulomb and exchange scattering of conduction electrons. The Hund’s rule 4f ground state of a rare earth ion is generally split ill a solid into a 13 number of crystal field levels. The level popu-~ lations change with temperature, and in a met-k al the transport properties reflect this temperature dependence through the interaction be-and tween 4f and conduction electrons.

hange, the direct Coulomb interaction and 1J Z interband mixing. This last interaction is where kB is Boltzmann's constant and Pex is most important for the case of a single 4f elec-a constant which depends on the 4f-conduction tron or hole and will not be considered here, electron interaction strength. The effect of exchange on the temperature de- The direct Coulomb interaction is evalupendence of transport properties has, in the ated by making a multipole expansion of the past, usually been thought to dominate that of Coulomb interaction energy of the 4f and conthe direct Coulomb interaction.
However, duction electrons The L = 0 term is included Fulde and coworkers1' 2 point out that in favor-in the lattice potential for a crystallographiable cases the two should be of comparable cally ordered compound. The lowest order magnitude, and that some properties of metals term which concerns us is the quadrupole (particularly the superconducting behavior) term. This gives aspherical Coulomb scatterwill be affected by the direct Coulomb interac-ing, first considered by Elliot.~Following tion in a way very different from that of ex-Fulde et al. ,2 we have for the interaction change. We find that the temperature depend-Hamiltonian: ence of the electrical resistivity due to scattering from the 4f crystal field levels of Pr3+ in +2 PrB 6 is sufficiently different for the two types H = > Q212(k', '~',k, v) of interaction for a separation of their contri-A k',k, s,v',~M= butions to be made, and that the two contributions to the resistivity in PrB6 are equal in M magnitude.
)< y2 (J) a~, , a ksv ksH irst 3 and Anderson et al. 4 have cornputed the temperature dependent contribution to the electrical resistivity due to the exchange The y~are operator equivalents (given in interaction In the latter's notation, this con-ref. 2) for L = 2. The operator akSV destroys tribution can be written p = Pex tr (pQex), a conduction electron of momentumk and spin where the trace is taken over the 2J+ 1 crystal s in band v . 1 2(k\ / , k~) is an integral which field states~' whose energies are E1 The we will assume to be a constant, an approximamatrices P and Qex are: tion also made in deriving the spin exchange interaction Hex = -2~(g-l) 3 s. Q2 is pro-ceptibility. using this impurity concentration portional to the quadrupole moment of the and the level scheme indicated, and the agreefully aligned 4f shell. ment with the data is within the experimental Hirst 3 points out that there are no cross error. The fit is not improved for the case terms between the direct and exchange inter-x = +0. 95 in ref. 6, and the resistance fit (.see actions in the electrical resistivity. The con-below) is much worse. tribution to the electrical resistivity from HA We use this level scheme to separate the calculated by second-order time-dependent aspherical Coulomb and exchange contributions perturbation theory is, therefore, similar to to the electrical resistivity; our measurements that due to exchange and of the form on single crystal specimens between 2 K and = PA~1~P~~' where has been given is a constant which depends on the interaction 1'I~1 123K strength. We expect the above formula to apply r 4 -*--s o long as each scattering event due to either r3~8K (2) spin or orbital disorder can be treated as isolated. The absolute resistivity (see below) indicates this to be the case for PrB6 .~100  The phase transition is appar-were determined by a fitting (solid line) ently second order. 8 Since F4 cannot be the to the theoretical expression assuming ground state in a cubic field, these data sug-fourth order crystal fields only. gest a F5 ground state.
Previous magnetic susceptibility meas-295 K are shown in Fig.   2(a). We assume that urements on Pr impurities in YB 6 suggested a the resistance of PrB6 is the sum of a residual F1 singlet ground state.
We now believe term p0. a lattice term which is given by these data may not be reliable; subsequent the known lattice term of LaB6 , and the aspherwork has shown that sample preparation by arc ical Coulomb and exchange scattering terms. melting causes segregation of mixed hexabor-We determine p0 by extrapolating the low temides. 1 We have re-investigated the magnetic perature PrB6 data against T 2 this gives a susceptibility of dilute Pr in this structure by good hue. We require the magnitude and slope measurements on a sample of small single of the room temperature resistance to equal crystals of nominal La 0 977 Pr0 023B6 grown'' the theoretical expression. Our fit, then, defrom molten Al, and the resultsare shown in pends on a geometrical factor for the PrB6 Fig. 1. The low temperature data indicate that sample and the ratio PA'Pex . We use this the ground state is F5 . The susceptibility of geometrical factor to give the absolute resissimilarly grown LaB6 crystals has been sub-tivity of PrB6 , since we cannot measure this tracted from the data in Fig. 1. We fitted value directly with enough accuracy for our these data to the magnetic susceptibility ex-purposes. Figure 2(b) shows the PrB6 data pression for Pr using fourth order fields only, with PL + p0 subtracted: the solid line gives an assumption which works well for NdB6 J2, 13 the theoretical expression for aspherical Con-This is the case x = 1 in ref. 6. This fit in-lomb plus exchange scattering for (PA'PeX) = 1 volves the impurity concentration and an ener-QA and Qex are computed using the wave funcgy scale factor. The concentration of Pr in tions of ref. 6. For convenience we have chosthe sample was found to correspond to en the normalization~Q~=~~= La0 9819Pr0 0181B6, indicating that some 14~j,j 3 fractionation occurred during crystallization.
(2J+1) J(J+l). The inset to Fig. 2(b) shows The solid line in Fig. 1 is the calculated sus-the sensitivity of the fit to the choice of in NdB6, using the ratio of the appropriate deGennes factors, is8.74~jQ-crn.
Thus, in NdB6 we predict that the aspherical term in the resistivity is only 6% of the exchange term. This explains why an earlier analysis of the resistivity of NdB6 which neglected aspherical Coulomb scattering was successful. 8 The measured exchange term in NdB 6 is 6.85~iQcm, indicating an exchange coupling constant ..

X 33
Coulomb scattering in PrB6 -The factor which sensitive to the mixture of exchange and aspher-

1(K)
The importance of our result is that we now have a quantitative determination of the aspherical Coulomb scattering contribution for Fig. 2 (a) Absolute resistivities of PrB6 and one case. That the magnitude is so large LaB6. PrB6 absolute value was determined means that aspherical Coulomb scattering can-as described in text. Residual terms are not be neglected in the analysis of resistance not subtracted here. Short range magnetic data, and, more importantly, that other effects order effects are evident between the Neel due to aspherical Coulomb scattering may be