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Volume 59, Issue 4, February 2011, Pages 1742-1760 Acta Materialia Electronic structure, mechanical properties and thermal conductivity of Ln2Zr2O7 (Ln = La, Pr, Nd, Sm, Eu and Gd) pyrochlore J. Fenga, b, , , B. Xiaoc, C.L. Wana, Z.X. Qua, Z.C. Huanga, J.C. Chenb, R. Zhoub, W. Pana, |
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Electronicstructure, mechanicalproperties and thermalconductivity of Ln2Zr2O7 (Ln = La, Pr, Nd, Sm, Eu and Gd) pyrochloreJ. Fenga, b, , , B. Xiaoc, C.L. Wana, Z.X. Qua, Z.C. Huanga, J.C. Chenb, R. Zhoub, W. Pana, , Purchase a State Key Laboratory of New Ceramics and Fine Processing, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, PR China b Key Laboratory of Advanced Materials of Precious-Nonferrous Metals, Education Ministry of China, Kunming University of Science and Technology, Kunming 650093, PR China c Department of Physics, School of Science and Engineering, Tulane University, New Orleans, LA 70118, USA Received 15 April 2010; revised 14 September 2010; Accepted 17 November 2010. Available online 18 December 2010. Abstract The electronicstructure, chemical bonding and mechanicalproperties of Ln2Zr2O7 (Ln = La, Pr, Nd, Sm, Eu and Gd) pyrochlore are investigated by local-density approximation of spin polarized scheme + U calculations (U is the Hubbard energy) and further verified by the experimental results. Ln2Zr2O7 compounds are wide band gap insulators, and this is consistent with the experiment results. The calculated spin polarized density of states of them indicates that they are in a ferromagnetic state and the magnetic moment is mainly attributed to the 4f shell of the Ln atoms. For the chemical bonds in the Ln2Zr2O7 crystals, the O–Zr bond is stronger than the O–Ln bond. The hardness, elastic constants, bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio of Ln2Zr2O7 compounds are investigated and the theoretical values are in good agreement with the experiments. The mechanical anisotropic properties are discussed using Zener’s indexes and universal elastic anisotropic index (AU). The sound velocities along [1 0 0], [1 1 0] and [1 1 1] directions are calculated for each Ln2Zr2O7 crystal. The thermalconductivities of Ln2Zr2O7 compounds are evaluated and the obtained thermalconductivity of Ln2Zr2O7 is lower than that of yttria-stabilized zirconia, indicating that they could be good low thermally conductive materials at high temperature. Keywords: Ceramics; Coating; Electronicstructure; Elastic behaviors; First-principles electron theory Article Outline 1. Introduction 2. Methods and details 3. Results and discussion 3.1. Cell constants and stability 3.2. Electronic structures 3.3. Mulliken populations and intrinsic hardness 3.4. Elastic constants and sound velocities 3.5. Thermal conductivity 4. Conclusions Acknowledgements References -------------------------------------------------------------------------------- Fig. 1. Crystal structure of Ln2Zr2O7 pyrochlore, Ln = La, Pr, Nd, Sm, Eu and Gd. 16d for Ln cations, 16c for Zr cations, 48f for O and 8b for O‘, while 8a sites (1/8, 1/8, 1/8) are unoccupied. Oxygen ions in the 8b (3/8, 3/8, 3/8) sites are stable, and are tetrahedrally coordinated by Ln cations. Another oxygen ions at 48f (x, 1/8, 1/8) positions are displaced toward the neighboring empty 8a sites, and which are bonded with two Ln and two Zr cations. View Within Article -------------------------------------------------------------------------------- Fig. 2. Band structures of Ln2Zr2O7 pyrochlore, Ln = La, Pr, Nd, Sm, Eu and Gd. The calculated band structures of Ln2Zr2O7 compounds within LSDA + U are all charge-transfer type insulators, which is in agreement with experiments. For Pr, Nd and Gd, U = 6.0 eV. For La, Eu, and Sm, U = 0, 7.0 and 8.0 eV, respectively. The dashed line is the Fermi energy. View Within Article -------------------------------------------------------------------------------- Fig. 3. The occupation numbers of up spin channel for 4f orbital of Eu2Zr2O7 as a function of U. The small fluctuations of the occupation number in the empty 4f down spin orbital are observed, but the values are negligible and omitted. View Within Article -------------------------------------------------------------------------------- Fig. 4. The band gap of Eu2Zr2O7 is calculated as a function of Hubbard U. For the U values between 6 and 10 eV, one can get the converged results. The minimum U value is required to open the band gap. For U < 5.0 eV, no band gap can be observed and the electronic structure of Eu2Zr2O7 is analogous to ordinary LDA calculation. View Within Article -------------------------------------------------------------------------------- Fig. 5. The calculated spin polarized density of states (SPDOS) of Eu iron in Eu2Zr2O7 using several different U values; only the 4f shell is shown in the figure. The dotted line is the Fermi level. View Within Article -------------------------------------------------------------------------------- Fig. 6. The calculated density of states (DOS) of Ln2Zr2O7 pyrochlore (Ln = La, Pr, Nd, Sm, Eu and Gd) within LDA and LSDA + U, For Pr, Nd and Gd, U = 6.0 eV. For La, Eu, and Sm, U = 0, 7.0 and 8.0 eV, respectively. Including U correction for electronic calculations basically splits the 4f band at the Fermi level and shifts the occupied part to a lower energy range. The dashed line is the Fermi energy. View Within Article -------------------------------------------------------------------------------- Fig. 7. Spin polarized partial density of states (SPDOS) of Ln2Zr2O7 (Ln = La, Pr, Nd, Sm, Eu and Gd) pyrochlore are calculated by LSDA + U. For Pr, Nd and Gd, U = 6.0 eV. For La, Eu, and Sm, U = 0, 7.0 and 8.0 eV, respectively. For Ln2Zr2O7, we also calculated total energies for other possible spin states, such as a low spin state and a high spin state. We found the high spin state has the minimum energy and we selected the high spin state as the true ground electronic configuration. The SPDOS of La2Zr2O7 is symmetric for spin up and down channels, and is consistent with the paramagnetic nature of it. LSDA + U calculations predicted a strong ferromagnetic state for Ln2Zr2O7 compounds except La2Zr2O7. This is in good agreement with previous investigations on rare-earth pyrochlore (Ln2Ti2O7, etc.) , , and . The dashed line is the Fermi energy. View Within Article -------------------------------------------------------------------------------- Fig. 8. The U dependence of mechanical properties of Eu2Zr2O7. B and G are bulk modulus and shear modulus, respectively. U is only used to correct the on-site Coulomb interactions for highly localized 4f orbital, and which is not directly related to stress–strain evaluations. So mechanical properties are nearly independent of U values and the results by LDA and LSDA + U are similar to each other for Ln2Zr2O7 compounds. View Within Article -------------------------------------------------------------------------------- Fig. 9. Bulk modulus (B) and Young’s modulus (E) of Ln2Zr2O7 pyrochlore (Ln = La, Pr, Nd, Sm, Eu and Gd) is in terms of increasing Z (atomic number). The values by calculation are in good agreement with the experiments; only the experimental results are sensitive to the samples used for measurement and LSDA is used which overestimates the cell constants. Solid symbols are theoretical values and the hollow symbols refer to experimental results. View Within Article -------------------------------------------------------------------------------- Fig. 10. Comparisons of bulk modulus (B), Young’s modulus (E), C11 and C12 of Ln2Zr2O7 pyrochlore (Ln = La, Pr, Nd, Sm, Eu and Gd). The calculated values are a little bigger than experiments. One possible reason is that the measured samples have defects; the B and G values are underestimated by experiments. The solid symbols represent the calculated values and the hollow symbols are values of experiments. View Within Article -------------------------------------------------------------------------------- Fig. 11. Anisotropic shear modulus in different crystal directions of Ln2Zr2O7 pyrochlore (Ln = La, Pr, Nd, Sm, Eu and Gd). GVRH is calculated by VRH approximation and G’ is the average of shear modulus along (1 0 0) and (1 1 0) crystal directions. View Within Article -------------------------------------------------------------------------------- Table 1. Lattice parameters, cohesive energy and formation enthalpy of Ln2Zr2O7 pyrochlore (Ln = La, Pr, Nd, Sm, Eu and Gd). View Within Article -------------------------------------------------------------------------------- Table 2. The ground electronic configurations and magnetic moments of Ln3+ ions. L, S and J are the total orbital angular momentum, total spin momentum and total angular momentum, respectively; g is the Lander factor, u is the calculated spin magnetic moment of Ln3+; us is the pure spin magnetic moment in theory and uLS is the total magnetic moment calculated by including spin–orbital coupling. 2s+1LJ is the optical notation for electronic configuration of Ln3+. The units for all magnetic moments are μB. View Within Article -------------------------------------------------------------------------------- Table 3. Population analysis and hardness of Ln2Zr2O7 pyrochlore (Ln = La, Pr, Nd, Sm, Eu and Gd). View Within Article -------------------------------------------------------------------------------- Table 4. Elastic coefficients (GPa), bulk modulus (GPa), shear modulus (GPa), Yong’s modulus (GPa), Poisson’s ratio (σ), anisotropic properties (AZ and AE), longitudinal (vl), transverse (vt) and average (vm) sound wave velocity (m s−1), Debye temperature (ΘD, K) of Ln2Zr2O7 pyrochlores (Ln = La, Pr, Nd, Sm, Eu and Gd). View Within Article -------------------------------------------------------------------------------- Table 5. The anisotropic sound velocities of Ln2Zr2O7 pyrochlore (Ln = La, Pr, Nd, Sm, Eu and Gd), and the unit of velocity (v) is km s−1. View Within Article -------------------------------------------------------------------------------- Table 6. The thermal conductivity (κ) of Ln2Zr2O7 pyrochlore (Ln = La, Pr, Nd, Sm, Eu and Gd). View Within Article Corresponding authors. Address: State Key Laboratory of New Ceramics and Fine Processing, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, PR China (J. Feng). Tel./fax: +86 01062772858. |
3楼2011-11-22 22:06:37
