Orbital Lattice Research Articles

Orbit–Lattice Interaction Hamiltonian for Iron‐Group Ions in Non‐Cubic Sites

AbstractThe orbit–lattice Hamiltonian for 3d ions in tetragonal and trigonal XY4 and XY6 complexes with D2d, C3v, D4h, and D3d symmetry is derived. A point‐ion model is adopted and the Hamiltonian is expressed in terms of operator equivalents in a form that exhibits clearly the symmetry of each component. The results are applied to a calculation of the phonon‐induced transitions in the ground state of V3+ in Al2O3. The ratio of the strengths of the ΔMs′ = 2 and the ΔMs′ = 1 transitions can depend strongly on whether the correct symmetry of the magnetic ion is accounted for – in particular it is shown that the commonly invoked 'cubic approximation' for the vibrational modes can lead to substantial errors.

Paramagnetic Spin Relaxation Due to Spin—Spin Interaction Assisted by Spin—Phonon Interaction or Vice Versa (II)

AbstractFor paramagnetic ions the electronic spin—spin and spin—lattice relaxations taking place independently are well known. In the first part of this paper a new type of “mixed spin relaxation process” has been considered in which dipolar spin—spin and spin—phonon (or orbit—lattice) interactions are jointly needed to effectively produce a relaxation transition, and the rate‐expression for this process has been obtained on phenomenological considerations. In this part of the paper the “mixed spin relaxation rates” for substitutional paramagnetic Fe2+(5D) ions in zinc carbonates are explicitely calculated and then compared with experimental data obtained from the analysis of the Mössbauer magnetic hyperfine spectra of this system at liquid helium temperatures. It appears that at low temperatures this process also provides a competing mode for the exchange of energy between electronic spins and phonons.

Tests of orbit–lattice interaction models using spin–lattice relaxation data of Er3+, Dy3+, and Yb3+ in Cs2NaYCl6

Measurements of the electron spin–lattice relaxation rate are reported for Er3+, Dy3+, and Yb3+ in the cubic host Cs2NaYCl6. Expressions for the anisotropic direct relaxation rates of the respective Γ8 and Γ6 ground states are given in terms of the eleven coupling parameters of the dynamic orbit–lattice interaction for octahedrally coordinated rare earths. Experimental direct process relaxation data are used to test the predictions of three models: (1) the electrostatic model of Buisson and Borg, (2) a nearest neighbor, effective point charge model, and (3) the nearest neighbor superposition model of Newman and co-workers. None of these proves satisfactory in providing a self-consistent explanation of the data from all three ions.

Interpretation of strain shifts of lanthanide ion energy levels

Second-order orbit lattice parameters are deduced from strain shifts of Pr3+, Er3+, Sm2+ ions substituted in fluorite lattices to be typically 103 cm-1. The radial power dependence of the second-order crystal field was found using the superposition model to be 35 for CaF2:Pr3+, 70 for SrF2:Pr3+. This is compared with the value of 3 expected from a point charge model. Strain-induced angular and radial changes of the ligand field surrounding the impurity ion are determined from the energy level shifts and from bulk elastic compliances.

ENDOR and relaxations in systems with I=S=1/2:169Tm2+and171Yb3+in CaF2

It is shown that equal intensities of the two ENDOR transitions for 169Tm2+ and 171Yb3+ in CaF2 indicate that the relaxation rates Tx-1( mod 1/2,-1/2( to or from mod -1/2,-1/2)) and Tx'-1( mod 1/2, 1/2( to or from mod -1/2,-1/2)) are approximately equal. The relative simplicity of the wavefunctions for the 4f13 configuration in a cubic crystal field, and of the Hamiltonian for the orbit-lattice interaction, enables one to make an explicit calculation of the various relaxation rates in terms of the orbit lattice parameters V( Gamma i,l). Matrix elements of the hyperfine interaction between ground and excited states of the J=7/2 manifold are found to be important in leading to calculated values for Tx-1 which are comparable to those for Tx'-1. Similar magnitudes for Tx-1 and Tx'-1 are shown to be expected for systems with large components of unquenched orbital angular momentum, in contrast to systems with little orbital angular momentum where Tx'-1 approximately=0.

The Tight-Binding and the Nearly-Free-Electron Approach to Lattice Electrons in External Magnetic Fields

The energy structure for lattice electrons in an external magnetic field is treated by extension of either the tight-binding or the nearly-free-electron method. The secular problems resulting from both methods are identical, if the coupling integrals and the real lattice characterizing the tight-binding method are interchanged with the Fourier integrals and the orbit lattice characterizing the nearly-free-electron method. A Bloch-type transformation reducing these essentially two-dimensional secular problems to one dimension is performed, following the procedure introduced in a preceding paper. For rational fluxes $\frac{2\ensuremath{\pi}N}{M}$ per lattice cell, an even finite secular determinant is left, yielding a splitting of each zero-field band into $M$ subbands. Total agreement between the level structures resulting from the two approaches is found. The relation of these structures to the de Haas-van Alphen periods in the case of magnetic breakdown is discussed.

Crystal Field in Rare-Earth Trichlorides. IV. Parameter Variations

Parameter variations are determined for changes in the ligand coordination angles and distances in Pr3+:LaCl3. Parameters for Sm3+ and Er3+ substituted in LaCl3 are also calculated. These results are used in conjunction with the superposition approximation to discuss the variation of parameters down the rare-earth series in the trichlorides and to predict values of the dynamic orbit–lattice parameters in PrCl3.

Evaluation of Certain Sums Arising in Chemical Physics

In this paper we show that the Poisson transformation of finite sums of the form ∑ i=1NF (cos 2πj/N) cos 2πjn/Ngenerally leads to a quickly convergent sum for problems of physical interest. Examples related to molecular orbital theory and lattice dynamics are discussed.