THE SUSCEPTIBILITY OF MIXED VALENT Tm AND U IONS

Using a variational technique we calculate the susceptibility of an ion fluctuating between two magnetic valence states. We show the importance of including the first excited state. Our results can be expressed in a renormalised form in the Kondo limits and are seen to correspond to exact results for Ce. Materials with mixed valent properties fall into where HI is a small perturbation given by two distinct categories. Compounds containing Ce and Yb are examples of the first category where an ion flueH2 = hgf p s mfA fm + hgc pa me:, ckm (2) tuates between a non-magnetic state, f O, and a magm km netic state, fl. For a single impurity there exist both exact results [I] and a number of approximation techwith fm and ckm denoting localised and conduction niques based on taking an expansion in / 12, 3~ electrons respectively. The mixing term is given by where N, the degeneracy of the f1 state is large. The ground state is seen to be a singlet characterised by a Hmi. = V (12 mm') (1 ml ckrmt + h-c.) . (3) small binding energy defining an energy scale TK. k'mm' The second category, in which both valence states have magnetic moments, presents a more difficult problem and has no exact solution. Examples are Tm (fl f 2 ) , and U (f2 f3) . The expansion parameter is now 1 / R where R, the ratio of the degeneracies of the f n and fntl states depends on the coupling scheme used. R is no longer large though in analogy with Ce this may not be crucial [4]. Using a variational technique Varma et al. [5] (referred to as I) and others [6] have shown that for a single ion the ground state is again a singlet which now has a small energy scale for all values of f occupation, n f . This implies an enhanced susceptibility and specific heat even in a mixed valent regime which would provide an explanation of the experimental results that certain U systems with nf 2.5 have heavy fermion behaviour and TmSe (nf 1.5) is magnetic at T = 0. It has, however, been suggested that the existence of a small matrix element between the singlet and triplet states found in I s u p presses the enhancement. Working to zeroth order the method yields a marked asymmetry in the energy scale in the two Kondo limits [8], which would suggest that if we consider nf = m + q where m is an integer and q -+ 0 the form of the result depends on the sign of q. This asymmetry does not occur in a functional integral calculation. In I an approximate calculation to all orders is performed but only for a single set of parameters which as we shall show leads to a mistaken conclusion. In this paper we extend the methods used in I to calculate the magnetic susceptibility of a Tm ion as a function of nf to various degrees of approximation. We have considered both minimal orbital degeneracy (R = 3/4) and j j coupling in the limit of zero j j coupling (R = 7/2) . The notation for the latter is simplest so we show only that here. The Hamiltonian we use is (more details are given in I) H = Hband + Hf + Hmix + HI (1) The bare energies of the f1 and f states are El and E2 respectively. We azsume that V and the density of states, po, are constant. The lowest order singlet wave-function is where 1@0) denotes the full Fermi sea. This leads to the following equation for am where Ek = ek +Jm with h = h (gf gc) ~ B . W S is the binding energy, I? = Z R V ~ po and 6 = E2 El. The equations (5) can be solved numericgy for all nf and analytically in the two Kondo l i i t s . The susceptibility, X, can then be calculated from the field dependence of ws; x = a2ws / ah2.nf can be calculated as a function of 6; nf= 1 6ws / aS. The solutions in the f1 and f Kondo limits are respectively. w~ is the energy of the doublet state and 2 has been defined to remove the spin dependent terms. The results are clarly very different. In figure 1 we plot ws WD and 2 I as a function of ni for minimal orbital degeneracy. We see that they are closely correlated. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19888323 C8 714 JOURNAL DE PHYSIQUE Fig. 1. A plot of the energy scale (full curve) and 1/2 (broken curve) against nf for D = 20 I?. The upper curve shows the oth order results and the lower curve the results on including the state with one electron hole pair. We now consider adding a term containing one electron-hole pair to (4) . The term we add is