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In this study, polarized Raman mapping technique was applied to determine the orientations of c-axis of each grain in translucent polycrystalline alumina (PCA) samples which were well-sintered in H2 atmosphere. The averaged refractive index difference Δnavg between neighboring alumina grain particles was then experimentally estimated from the mapping data. It was shown that the translucent alumina had Δnavg of 0.002 and 0.0015 for horizontal and side planes, respectively, smaller than the value for random orientation in the Apetz et al model (ΔnApetz = 0.0053). With the experimental value, then light transmission spectra was simulated in the scope of the well-known Rayleigh-Gans-Debye (RGD) theory.
©2012 Optical Society of America
1. Introduction
On the understanding of light transmission properties of translucent polycrystalline alumina (PCA) [1–3], an averaged refractive index difference Δnavg is one of the most important parameters among averaged grain size, pore-size/distribution etc. It is because the crystal structure of alumina belongs to a corundum-type hexagonal system, and exhibits birefringence with optical anisotropy [4]. Necessarily, PCA has refractive index differences on grain boundaries, which causes a decrease of in-line transmission. In the present study, polarized Raman mapping technique was applied to determine the orientation of c-axis for each grain of alumina and to estimate an averaged value of refractive index differences, Δnavg. In this letter, we promptly report our recent experimental results on the averaged refractive index difference and subsequent numerical simulation of in-line transmission spectra with the obtained Δnavg.
2. Experimentals and theory
Seven planar samples of translucent alumina with a different averaged grain size D in diameter (20~48 μm), as listed in Table 1 , were obtained by sintering high purity alumina powder in H2 atmosphere as a products of NGK (HICERAM) [5]. The materials have relatively large grain size and are rather transparent over 60% in transmission at wavelengths longer than 600 nm. Grain boundaries for the samples investigated were visualized by a thermal etching at 1450 °C for 30 min. for the observation under an optical microscope. For Samples 1~4, a horizontal plane was analyzed, while for Samples 5-1~5-3, a side plane was analyzed. The analyzed planes were well polished to optical flat before thermally etched. The thicknesses of the planar samples were all ~0.5 mm. All of the samples were in a same series of PCA samples synthesized with the same starting chemicals and the same preparation conditions except the grain-growth time.
Table 1. Averaged Grain Size D (= 2G) and Δnavg
Polarized Raman scattering spectra were collected in total 400 points of X-axis * Y-axis = 20*20 points in 2-5 μm pitch in a confocal-type micro-Raman spectrophotometer (NRS-2000, JASCO). A polarized Ar+ laser at 514.5 nm was used as an excitation light. The spot size of the laser beam at focal point was ~2 μm. For Raman mapping, the intensity at 645 cm−1, assigned to a A1g vibrational mode of alumina was mapped, as shown in Fig. 1 . It has been shown that the mapping of the relative intensity of the A1g mode correlates well with grain orientation (texture) [6] but small deviations are obtained because of the grooving effect due to surface reconstruction upon the thermal etching and to reflection off of grain-boundaries intersecting the sample surface, or the depth profiles of grains in a few μm from the top surface which was detected by the confocal micro-Raman method. Thus, the central positions of grains were analyzed for the estimation of refractive index.
Fig. 1 A1g (645cm−1) Raman mappings of translucent alumina, (a) Sample 1, (b) Sample 4. White curves show grain boundaries.
At first, such Raman maps were taken at various rotation angles of the sample about the z-axis (Fig. 1) with 30 o increments. It is known that the A1g intensity at 645 cm−1 depends on the rotation angle, ϕexp, and the relative orientation of the c-axis of the grain (θ, ϕ0) with respect to the laboratory frame, as shown in Eq. (1) [7,8].
where θ is the angle between the excitation laser beam from + Z direction and the c-axis of the crystal, and ϕ0 is the angle between the polarization vector of the incident laser beam and the projection of the c-axis on the X-Y plane analyzed. The c-axis orientation was calculated by fitting the intensity change as a function of the rotation angle in Eq. (1). The refractive index of the crystal with optical anisotropy (no = 1.768, ne = 1.760) [4] was calculated using optical index ellipsoid [9]. The refractive indexes of each grain for incident lights from X direction were obtained. More exactly to mention, because they can be polarized in two directions (Y and Z polarization; See Fig. 1), two kinds of refractive indexes were obtained. The respective incident light experienced Δn at each boundary. We considered as many optical paths as possible and successfully estimated averaged refractive index difference, Δnavg.Factors for in-line transmission of polycrystalline alumina, which was estimated by a UV-VIS-NIR spectrometer (V-570, JASCO), are surface reflectance, light scattering due to grain boundaries, pores, and light absorption by impurities. To know the origin of transluscency of our PCAs, we have here focused on the grain boundaries scattering because our PCA samples have extremely low impurities, low pore density and very smooth surface. With using Δnavg estimated experimentally, the influence of grain boundaries in the light transmission was analyzed based on Rayleigh-Gans-Debye (RGD) theory [10,11]. Light scattering coefficient (γgb) due to grain boundary is given in the product of the density (N) and scattering cross-section (Csca).
where Csca,gb can be calculated by applying Mie theory [10] but when a phase shift of the light due to the scattering body is very small, the approximated solution known as RGD scattering can be used.G is the grain size in radius (G = D/2) and λ is the wavelength of incident light. Here it is assumed that grains can be approximated to be spheres. Because two anisotropic refractive indexes of alumina are no = 1.768 and ne = 1.760, Δn should not exceed 0.008. In case of random orientation, it is commonly used to be 0.0053 [11]. N is defined by N = Vf/(4πG3/3), where Vf is an effective volume fraction of alumina scatters. The volume fraction of scatters (Vf) is defined to be a relative volume of a scatter surrounded by a transparent part without other scatters, and is deeply related to the density of scatters but not exactly same as the density of alumina grains. According to Apetz et al. [11], Vf is set to 1/2 in spite of no physical basis [12]. We believe that it should be regarded as a fitting parameter for simulation. Thus, from Eqs. (2) and (3), light scattering coefficient (γgb) due to grain boundaries is given by [13]In fitting procedures, the influence of pores was approximated with 100 nm pore size and Csca, pore = aλ-b, where a and b are also fitting parameters. Since these parameters were less dominant especially on the transmission profiles ranging from 250 to 800 nm, we here focus ourselves to the influence of grain boundaries on in-line transmission. Moreover, in the numerical simulations and subsequent fitting analysis we have taken into account the dispersion of refractive index, which was measured by using a spectroscopic ellipsometory and estimated by Sellmeier’s equation. For PCAs, the averaged value of no and ne was experimentally obtained and used for these simulations.3. Results and discussion
Averaged refractive index differences Δnavg for Sample 1~4, 5-1, 5-2 and 5-3, estimated from our Raman mapping measurement are summarized in Table 1. Δnavg of Samples 1~4 are approximately 0.002 and no significant influence of the averaged grain size is obtained. On the other hand, Δnavg of Samples 5-1~5-3 are found to be ~0.0015, having no clear size-dependence, either. As a result, Δnavg on the side-planes of planar translucent alumina is lower than that of the horizontal plane.
In-line transmission spectra simulated using Eq. (4) are shown in Figs. 2 and 3 . In Fig. 2, VfG is fixed to be 0.5 μm, and Δn is changed from 0.001 to 0.005. The circles show representative experimental transmission data of Sample 4 at given wavelengths. It is seen that transmission spectra of Δn > 0.003 are below the experimental data and cannot reflect the experiment result in simulation. Thus Δn < 0.003 is proper for this sample. In Fig. 3 Δn is fixed to be 0.0015 and VfG is varied from 0.5 μm to 12.5 μm. In the same way, it is seen that spectra of VfG > 2.5 μm is below the experimental data and then VfG < 2.5 μm is proper for the sample (D (= 2G) = 48 μm), that is Vf < 0.10. From these numerical considerations, it is suggested that small Δnavg values of 0.002 and 0.0015, experimentally obtained in this study, are valid and a volume fraction Vf is greatly different from the assumption by Apetz et al.(Vf = 1/2) [11]. To sum up, small Δn (< 0.0053 for random orientation) and small Vf (< 1/2 for Apetz’s assumption) are indispensable [12,13] for understanding of in-line transmission properties of our translucent PCA (HICERAM). Small Δn means preferable c-axis orientations of neighboring alumina grains, which greatly contribute to the translucency (See Figs. 4 and 5 ).
Least-square fitted in-line transmission of Samples 1 (D = 20 μm) and 4 (D = 48 μm) are shown in Figs. 4 and 5, respectively. Red mark( + ) and triangles show the respective experimental data. In the fitting procedures, Δn is fixed to 0.0015 (for the side plane). As for light scattering by residual pores Mie theory was applied. Surface reflection and dispersion of refractive index were also considered in this fitting procedure as well as RGD theory (the details will be given elsewhere. Here, the main results given from the fitting analysis is discussed). As seen in Figs. 4 and 5, the fitting results are in pretty good agreement with experimental curves. And the fitting parameter Vf was obtained to be 0.067 and 0.038, respectively, which are much smaller than the value (Vf = 1/2) by Apetz et al. [11] The obtained decrease in Vf is related with an increase in grain growth, which means the samples were well sintered and had high density. Such a small Vf can be explained by Cubic Model (See Fig. 6 ). When a sample is fully dense, highly pure, and grain size distribution is sharp, the volume fraction Vf should be equal to about 0.037 (as an ideal value).
Fig. 6 Cubic model for translucent polycrystalline alumina (PCA) (minimization of volume fraction Vf of scatter for an ideal PCA structure).
4. Conclusion
For understanding of transmission properties of translucent polycrystalline alumina (PCA) samples which were well-sintered in H2 atmosphere, the averaged refractive index difference Δnavg between neighboring alumina grain was experimentally estimated by a polarized Raman mapping technique. The obtained Δnavg was 0.002 and 0.0015 for horizontal and side plane, respectively.
With the value (0.0015) for side plane, in-line transmission spectra were simulated in the scope of RGD theory, The volume fraction of alumina scatter Vf was obtained to be 0.067 and 0.038, for samples with averaged grain size D (= 2G) = 20 and 48 μm, respectively, which were both much smaller than the value (Vf = 1/2) by Apetz et al. It was found that Vf was decreased with an increase in grain growth, indicating that the samples were well sintered, and had high density (low pore density) and sharp grain-size distribution.
References and links
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