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A B C Z. Naturforsch. 68a, 442 – 446 (2013)doi:10.5560/ZNA.2013-0019Theoretical Investigations of the Knight Shifts and the Hyperfine Structure Constants for the Tetragonal Cu2+ Sites in the Bismuth- and Thallium-Based High-Tc Superconductors1 School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, P. R. China2 International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, P. R. ChinaReceived December 3, 2012 / revised February 4, 2013 / published online May 1, 2013Reprint requests to: M.-Q. K.; E-mail: mqkuang@yeah.netThe Knight shifts and hyperfine structure constants for the tetragonal Cu2+ sites in bismuth- and thallium-based high-Tc superconductors (Bi1.6Pb0.4Sr2Ca2Cu3O10, TlSr2CaCu2O7–y, and Tl2Ba2CuOy) are theoretically investigated from the high-order perturbation formulas of these parameters for a 3d9 ion under tetragonally elongated octahedra in a unified way. The calculation results show good agreement with the observed values. The significant anisotropies of the Knight shifts are attributed to the local tetragonal elongation distortions of the five-(or six-)coordinated Cu2+ sites in these systems. The present studies would be beneficial to establish a complete physical scheme for unified understandings of electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR) spectral behaviours of Cu2+ (or other similar 3d9 ions) in the high-Tc superconductors.Key words: Knight Shifts; Hyperfine Structure Constants; 63Cu2+; Bi1.6Pb0.4Sr2Ca2Cu3O10; TlSr2CaCu2O7–y; Tl2Ba2CuOy.