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Optical floating zone (OFZ) technique is one of the most used methods for growth of single crystals. Although OFZ has been extensively investigated for past decades, the temperature distribution during growth has not been understood due to the complicated process during growth. It makes the optimization of growth parameters very difficult. In this work, we presented a simplified model to calculate the temperature field in floating zone by FEM. The optical properties of grown crystal and ceramic perform rod, lamp power, filament dimensions, ceramic rod diameter, floating zone dimensions, and ambient gas cooling rate were considered in the model. The calculated results show good agreement with the experiments. The developed model opens up a new way to optimize the growth parameters for a variety of single crystals. Introduction Man-made single crystal materials have widely been used in modern science and technology and the growth of crystals is an important way to obtain large size and high quality crystals. The growth techniques of single crystal materials include: solution growth method [1] , melt growth method [2, 3] , gas phase growth method [4] , solid phase growth method [5] etc. The optical floating zone method [6] is one of the most studied methods due to the low costs, narrow molten zone and fast growth for oxide-based materials with high melting points [7-11] . However, the optimal parameters were only obtained by experiments due to the lack of understanding of growing process, which causes high-energy consumptions and time costs [12] . The rapid development of advanced numerical simulation based on modern computer technology provides a new tool for understanding of single crystal growing process and optimizing process parameters [13] . Numerical simulation is based on the establishment of mathematical model in order to obtain the results difficult to capture from experiment. The process parameters and material properties can be tested in the model and the corresponding effects are hence observed. Then the simulated results can be applied to optimize the experimental procedures and process parameters in practice. In 1966, McCord developed the first commercial simulation software. It is the beginning of modern numerical simulation technology [14] . Arizumi and Kobayashi in the 70s of last century opened on the crystal growth process of numerical simulation [15] . Nowadays, the commonly used in crystal numerical simulation methods include Monte Carlo method, boundary unit method, finite volume method, finite difference method, and finite element method. The most used ones are finite element method and the finite difference method [15-18] . In this work, a FE model was developed to simulate the process of single crystal growth by optical floating zone technique. The optical properties of grown crystal and ceramic perform rod, lamp power, lamp filament dimensions, ceramic preform rod diameter, floating zone dimensions, and ambient gas cooling rate were considered in the model. The temperature fields during growth were simulated. The experiment with the same process parameters was also performed in order to validate 2nd Annual International Conference on Advanced Material Engineering (AME 2016) © 2016. The authors Published by Atlantis Press 571 the simulated results. According the numerical simulation, the growth parameters for TiO2 crystals were optimized, by which the high-quality TiO2 single crystal was grown. Theoretical Model and Experimental Procedure The growth system used in this work an optical floating zone furnace with four ellipsoidal mirrors (FZ-T-10000-H-VI-VP) with four halogen lamps of maximum power of 1.5kW or 1kW, as Figure 1(a). The spectrum of the halogen lamp was measured by a Spectrometer (Spectra PR715), as shown in Figure 1(b). It can be seen that the main power intensity is located in the range of 500~2700nm and the central wavelength is around 1100nm. (a) (b) Fig.1 (a) The Optical Floating Zone Furnace and (b) Normalized Spectral Distribution of Halogen Lamp Used in This Work In order to obtain the material absorption with respect to the lamp light within the main power distribution (500-2700nm), the reflectance and transmittance were measured by the spectrometer (UV-3600-230VCE). The top panel of Figure 2 demonstrates typical reflectance distribution of TiO2 perform rod, floating zone and crystal in the range of 500-2700nm. The transmittance is related to the thickness of the sample, the absorption coefficient (α) was therefore used to describe the absorption in different bands of the lamp light. The bottom panel of Figure 2 illustrates the absorption coefficients of the materials with respect to lamp wavelength. Fig.2 Reflectance and Absorption Coefficient of TiO2 Preform Rod, Floating Zone and Crystal

Numerical Simulation of Temperature Field in Single-crystal Growth by Optical Floating Zone Technique