Bleaney factors

In a seminal contribution, Bleaney demonstrated that when the crystal-field splitting of the ground multiplet is smaller or comparable to kT, a situation often met with lanthanide complexes, the anisotropic part of the axial paramagnetic susceptibility tensor originates from second-order effects and can be simply estimated by the product of magnetic constants Cj, characteristics of the electronic configuration of each lanthanide (i.e., Bleaney factor), multiplied by the second-rank crystal-field parameter Bq (Eq. (34), Bleaney, 1972). 

Cj Bleaney s factor of lanthanide / (scaled to —100 Tr(z) trace of the magnetic susceptibility tensor... 

Spin expectation values3 (Sz) and Bleaney s factors 3 C for /f(III) free ions at 300 K, and rhombic proportionality factors0 cr lombio... 

The first numerical terms C = P2 1 + p) j / (l20(kT)2) are often referred to as Bleaney s factors and their relative values (scaled to Coy = -100) have been tabulated at 300 K for any 4f configurations including excited states contributions (table 3, Bleaney et al. (1972)). Finally, the introduction of the geometrical factors defined by eqs. (30), (31) together with C into eq. (41) gives the classical eq. (42) for the pseudo-contact shifts according to Bleaney s approach (Forsberg, 1996)... 

Magnetic anisotropies xlz (l/3)Tr/ for R = Ce-Yb except Pm, Gd (0.002 < AFj < 0.06, table 9) have been computed with eq. (58) and using five contact contributions Sfj (i = H9, H11-H14) and the geometrical G factors obtained from the crystal structures of (HHH)-[/ Co(L5)3]6+ (R = La, Lu). A qualitative good agreement (AF = 0.23) is obtained between the experimental magnetic anisotropies (scaled to -100 for Dy(III) and corrected for the variation of the crystal-field parameter near the middle of the series (vide supra), table 9) and Bleaney s factors (table 3). Further non-linear least-squares refinements of the molecular... 

A clear shift of Na and Cs NMR lines, proportional to the effective y-factor of the Ho + ion, has been observed in Cs2NaHoCl6 by Bleaney et al. (1981b). After subtracting contributions of the Lorentz and demagnetizing fields, there remains a small paramagnetic shift Ay ( 0.006 for Na and even less for Cs), which for Na and Cs atoms, placed in the cubic positions of elpasolite, can occur only due to the transferred hyperfine interaction. ENDOR experiments directly testify to the presence of the above interaction in elpasolites with lanthanide ions (Fish and Stapleton 1978). 

Abragam and Bleaney (1983) discussed the possibility of obtaining the ordered nuclear states in TmP04 and LiTmp4 crystals, similar to vanadates. The expected ordering temperature in these compounds is less than 10p.K due to the lower values of the factor and nuclear spin. To obtain such low temperatures it is necessary to apply methods used for conventional nuclear magnets. 

A, comparison between CEF parameters obtained using tensor operators and the commonly used Stevens formalism can be made by a numerical conversion (see table 1). We follow the common convention of writing the tensor parameters as and inverting the indices for the Stevens parameters, B ". Table 1 also contains the conversation factors for a comparison with parameters used by a group at the ETH Zurich. Further details about conversions between various formalisms are available elsewhere (Bleaney and Stevens 1953, Hutchings 1964, Kassman 1970), We stress that care is necessary when comparing parameters obtained by different groups and different techniques. 

Obviously, the use of eq. (38.6), which considers only the first term in the expansion of the Boltzmann factors, with g-tensor components weighted by unequal populations of the corresponding levels is a contradiction in terms and as could be anticipated leads to erroneous results. It is thus clear that Bleaney s theory accounts satisfactorily for the dipolar shifts in lanthanide complexes, whereas the criticisms of this theory, Horrocks et al. (1973), seem to be not well founded. We wish to emphasize that the generalized treatment of Golding and Pyykko (1973) is of course more rigorous. 

Values of (r" ) are based primarily on theoretical calculations of the 4f wave functions, for example, those of Freeman and Watson (1962) and of Judd (1963). In table 18.5 are listed the values of (/ N /), (r ), and Hfs for the tripositive rare-earth ions. Strictly speaking, correction factors differing from unity by a few percent should be applied to these (/ M /) values [Bleaney (1972)] to take account of intermediate coupling effects which arise from the admixture into the ground state (L, S, J) wave function of states of different L, S, but the same /, by the spin-orbit interaction. A table of these values is included in Bleaney (1972). 

The g-tensor is obtained by evaluating the matrix elements of the magnetic hamiltonian (18.105) using the wave functions of the lowest levels that result from solution of the eigenvalue problem for %cef (18.109). Illustrative calculations of the g-factors for lanthanide ions in various cases are given, for example, by Al tshuler and Kozyrev (1974) and by Abragam and Bleaney (1970), among many others. 

The ESR spectra do not themselves yield estimates of the CEF parameters. However, when taken in conjunction with other data the wave-functions of the lowest levels can be accurately checked by the g-factor values which they yield. In the case of Pr in lanthanum ethyl sulfate, for example. Baker and Bleaney (1958) originally estimated the CEF parameters on the basis of the magnetic susceptibility data then available. Subsequently, these parameters have been determined from high resolution optical data for all of the lanthanide ions in the ethylsulfates. A complete listing of these is given, for example, in Abragam and Bleaney (1970). 

A Fermi hyperfine constant for a nucleus i bJ crystal-field parameters of rank k spherical tensor operators of rank k Cj Bleaney s factor of lanthanide j (scaled to -100... 

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