ISSN 0030-400X, Optics and Spectroscopy, 2014, Vol. 116, No. 6, pp. 897–905. © Pleiades Publishing, Ltd., 2014. Original Russian Text © I.V. Romanova, B.Z. Malkin, M.S. Tagirov, 2014, published in Optika i Spektroskopiya, 2014, Vol. 116, No. 6, pp. 973–981. 897 INTRODUCTION Over the last three decades, double lithium–rare- earth fluorides LiLnF 4 (Ln denotes the rare-earth ion) have been studied intensely due to a variety of their magnetic properties [1–5]. Simulation of magnetic- field effects in LiLnF 4 single crystals is an important problem for the development of the theory of magne- toelastic effects in magnetically concentrated crystals that contain rare-earth ions. LiLnF 4 crystals have a tetragonal structure of scheelite CaWO 4 ; their space-symmetry group is (I4 1 /a); the unit cell contains two magnetically equiv- alent Ln 3+ ions at sites with the S 4 point-symmetry group [6]. LiTmF 4 is a dielectric van Vleck paramagnet [7], a distinctive feature of which is a strong magne- toelastic interaction, which is evidenced by a giant magnetostriction, which was revealed in [8], and char- acteristic extrema in temperature dependences of elas- tic constants [9, 10]. The authors of [11, 12], using optical-spectroscopy methods, have observed a sub- stantially nonlinear and anisotropic dependence of the splitting of the first excited doublet of Tm 3+ ions on the external magnetic field that was oriented in the base plane of the LiTmF 4 crystal, which is caused by the magnetoelastic interaction. Later, the authors of [13, 14] measured the dependences of the magnetization of LiTmF 4 single crystals on the magnitude and direction of the magnetic field at different temperatures. Results of calculations of piezospectroscopic effects [15], magnetostriction [8, 10], and nonlinear Zeeman effect [11, 12], which were done in terms of the model of a linear electron–deformation interaction, agree well with measurement data; however, the values of coupling parameters of thulium ions with lattice deformations that were used in these works differ con- C 4 h 6 siderably from the values of corresponding parameters that were found from the analysis of the data of piezospectroscopic investigations of thulium and hol- mium impurity ions in an isostructural crystal LiYF 4 [16, 17]. It should be noted that the previous analysis of the experimental data was performed taking into account only the magnetic-dipole–dipole interaction between rare-earth ions, which, in the case of the van Vleck paramagnet, plays an insignificant role at low temperatures. The revealed strong interaction of Tm 3+ ions with static lattice deformations is indicative of the possibility of an efficient multipole interaction between thulium ions that is caused by exchange of phonons. The multipole interaction leads to the renormalization of parameters of an effective one-par- ticle Hamiltonian and, as was noted in [15], can be the reason for the difference between parameters of the electron–phonon interaction in magnetically concen- trated and diluted systems. The objective of this work is to calculate the parameters of the Hamiltonian of the interaction between thulium ions using previously found characteristics of the electron–phonon interac- tion and the lattice dynamics of LiLnF 4 crystals [18– 20] and to construct a self-consistent model for the calculation of magnetic and magnetoelastic charac- teristics of the LiTmF 4 crystal taking into account multipole interactions. THEORETICAL SUBSTANTIATION OF THE MODEL The Hamiltonian of the magnetic subsystem, which consists of thulium ions interacting with the crystal lattice with a free surface, will be written as XV INTERNATIONAL FEOFILOV SYMPOSIUM Multipole Interactions in a LiTmF 4 Single Crystal I. V. Romanova, B. Z. Malkin, and M. S. Tagirov Kazan (Volga Region) Federal University, Kazan, 420008 Tatarstan, Russia e-mail: Irina.Choustova@kpfu.ru Received November 18, 2013 Abstract—We have considered magnetic and magnetoelastic characteristics of a van Vleck paramagnet LiTmF 4 taking into account the interaction between thulium ions via the phonon field. We have calculated parameters of the multipole interaction that is caused by the interaction of Tm 3+ ions with dynamic lattice deformations of the B g symmetry. We have presented a self-consistent description of previously published results of measurements of temperature dependences of elastic constants and the nonlinear Zeeman effect in the optical spectrum of the LiTmF 4 single crystal, as well as dependences of the magnetostriction on temper- ature, magnitude, and direction of the external magnetic field. DOI: 10.1134/S0030400X14060198