Determination through symmetry arguments of the various contributions to the self polarisation field at rare earth nuclei in cubic compounds

NMR of the rare earth nuclei in cubic RE-X, compounds has shown that the self polarisation field, Hsp, gives an important contribution to the total hyperfine field at the REnuclei. Both orbital and spin polarisations of the conduction bands arise through anisotropic exchange interaction between the ordered magnetic moments of the rare earth and the conduction electrons. These polarisations are further reflected through orbital, magnetic dipole dipole and contact or core polarisation terms in H,,. Due to the complexity of the problem, group symmetry arguments are used to determine the independant parameters of the problem for cubic compounds. In a recent publication [I], we studied the nuclear magnetic resonance of RE nuclei in RE Al, compounds. The hyperfine field at the nucleus of a RE ion in this ordered system is usually assumed composed of three terms HN = H4f + Hsp + Hnn (1) where H,, represents the field created by the 4f electrons and is proportional to the magnetic moment of the 4f shell ; H,, is of the order of several Mega Oe. Hnn is the transferred hyperfine field at the nucleus under study arising from polarisation of the conduction electrons by other magnetic ions ; it is usually small, of the order of 10 kOe. We focus our attention here on H,,, the self polarisation field arising from polarisation by 4f electron-conduction electron exchange which reflects itself through hyperfine interaction of these conduction electrons with the RE nucleus. It was found that H,, varies from 200 kG for GdAl, to 1 000 kG for ErAl,. The first attempts to understand the experimental results have shown [l] that H,, could not be interpreted through spin polarisation only and that one must include orbital polarisation effects. The orbital polarisation of the 5d band at the RE site is considerable. A phenomenological and crude approach including these effects led us to try to fit the experiments with a self polarisation field given by [I], [2] H,, = a ( S , ) + b < L , ) (2) where ( S, ) and ( L, ) are the average values of the total spin and angular momentum of the RE ion. These two terms result from the spin polarisation of the 5d and 6s conduction electrons and from the orbital polarisation of 5d electrons respectively. Although great improvement is obtained by including the ( L, ) term, the agreement is not perfect and some difficulties arise with the light RE (PrAl,, NdAl,). A more sophisticated model taking into account correctly the microscopic form of the exchange interactions between the 4f and the conduction electrons, a calculation of the spin and orbital polarisation of the conduction bands and a detailed analysis of the hyperfine interactions between the conduction electrons and the RE nucleus is thus necessary. Quite generally, H,, may be written [3], [4] as where in the right side of (3), the first term represents the magnetic orbit31 and dipole-dipole interaction between the 5d conduction electrons and the RE nucleus, the second term arises from the core polarisation due to the same 5d electrons and the third term comes from the contact interaction of the 6s conduction electrons. In (3) only the z component of the conduction electron operators is involved because the self-polarisation hyperfine interaction may be consiArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979522 DETERMINATION THROUGH SYMMETRY ARGUMENTS O F THE VARIOUS CONTRIBUTIONS C5-57 dered as a 1st order perturbation compared with the direct hyperfine coupling due to the 4f electrons and represented by H4, in (1). A, B, C are constants which are not supposed to vary with the RE ion in a series like RE Al,. In order to evaluate the average value of ( 1, ), ( (s x C2); ), and ( s, ), for the d band, we use a linear response theory following the model of Huang Liu, Ling and Orbach [5] used in the case of dilute RE ions. Here, instead of having an external magnetic field, the polarisation of the d band comes from anisotropic exchange interactions with the 4f electrons which are themselves sensitive to the cubic crystalline electric field and to the molecular field responsible for the magnetic ordering of the RE ions. The 4f-5d exchange interaction averaged over the ground state of the single RE ion Hamiltonian (crystal field + molecular field) may be written [5], [6] as where the cpi (i = 1, ..., 5) are the t,, or e, orbital 5d states considered [5] as band states and mi refer to the spin state of the d electron (in the particular case of (5) we have necessarily i = j). S(cpi, pj) is a generalized susceptibility [5]. For d electrons in cubic symmetry we have only three independent terms S(t, t), S(t, e) and S(e, e). These susceptibilities depend on the bandwidth and on the splitting between the t,, and the e, states. They will be considered as independent of the RE in a given series. As ( s $ transforms like r 4 , only the combinations of ( Y :,,(L) S ) transforming themselves according to the same representation are involved. Furthermore only invariant combinations of Y,,(1) must occur. Then when the magnetization direction of the RE ion is along a fourfold axis of the cube, we obtain : where a, are exchange parameters [5], b, and c, are Using the same kind of arguments, the average values coupling coefficients ; L, S refer to the 4f shell and of ( I , ),, ( s, ), and ( (s x CZ)h ), have been derived I, s to the 5d conduction electrons. As an example, both when the magnetisation direction of the RE is a term like ( s, ), is obtained from (4) by along a fourfold (PrAl,, NdAl,, DyA1,) or a threefold (ErAl,, TbA1,) direction of the local cube. < ~ , ) d = x x ((Pimi IszI(Pjmj) x It is then necessary to project the operators Y:,(L) S, i,j m, ,mj on to the Y,,(J) operators. From (6) we obtain for ( (Pj mj I X<4f)5d 1 (Pi mi ) S((Pi, (Pj) (5)