CRYSTALLOGRAPHICALLY ANISOTROPIC SHAPE OF FORSTERITE: NEW PROBE FOR EVALUATING DUST FORMATION HISTORY FROM INFRARED SPECTROSCOPY

Infrared spectroscopy has shown that silicates are commonly present in protoplanetary disks, dust shells around asymptotic giant branch (AGB) stars, and planetary nebulae. Most circumstellar silicates are present as amorphous showing broad infrared features at ∼9.8 and ∼18 μm. Crystalline silicates also exist around some oxygen-rich AGB stars and in protoplanetary disks. Sharp infrared features of crystalline silicates are attributed to stretching or bending of atomic bonds within ordered arrangement of atoms, and peak wavelengths and their relative strengths change with several factors such as chemical composition (Koike et al. 2003, 2006), temperature (Bowey et al. 2001), and size (Min et al. 2005).

The importance of grain shape has also been pointed out to interpret peak wavelengths in infrared spectra of circumstellar crystalline silicates from both laboratory measurements (e.g., Tamanai et al. 2006; Koike et al. 2010) and numerical calculations (Bohren & Huffman 1983; Jäger et al. 1998; Min et al. 2003, 2005, 2008; Mutschke et al. 2009). It has been theoretically shown that grains having different shape distributions, such as uniform sphere, continuous distribution of ellipsoids (CDEs), and Gaussian random spheres, have different infrared features. However, most of these studies have not paid much attention to the relationship between a grain shape and the crystallographic orientation. For instance, grains elongated along different crystallographic axes have been assumed to exist with equal probabilities.

This is not always the case for crystalline phases because a crystal often has a specific shape related to a specific crystallographic orientation that reflects the anisotropic crystal structure and/or growth conditions as well known in the field of mineralogy. Enstatite whiskers in interplanetary dust particles elongating to the crystallographic a-axis (Bradley et al. 1983; Ishii et al. 2008) are one of the examples that reflect an anisotropic formation process.

Current understanding of anisotropy in formation processes of silicates under circumstellar conditions is limited due to the lack of experimental studies. The exceptions are evaporation of forsterite (Mg₂SiO₄), which is one of the most abundant crystalline silicates observed spectroscopically in circumstellar environments (Yamada et al. 2006; Takigawa et al. 2009; Ozawa et al. 2012) and heterogeneous nucleation and growth of forsterite on metallic substrates (Tsuchiyama 1998).

Takigawa et al. (2009) showed experimentally that forsterite evaporates anisotropically in hydrogen gas as in vacuum (Ozawa et al. 1996; Yamada et al. 2006) and that the anisotropy in evaporation rates depends on temperature and hydrogen gas pressure. They suggested that crystalline forsterite can have a specific shape with a specific relationship to a crystallographic orientation, which is called as "crystallographically anisotropic shape," depending on its evaporation condition. They showed that the change of the crystallographically anisotropic shape of forsterite by evaporation causes observable changes of infrared features.

Tsuchiyama (1998) showed that forsterite condenses on metallic substrates in hydrogen gas to form whiskers elongating along the crystallographic c-axis although condensation conditions in the experiments were not well known quantitatively and further experimental investigations on anisotropy are needed. Fabian et al. (2001) found that forsterite elongating along the c-axis shows a different infrared spectrum from those of sphere and CDEs.

These studies suggest that the crystallographically anisotropic shape of forsterite may record formation conditions of forsterite dust in circumstellar environments quantitatively and that the crystallographically anisotropic shape of forsterite can be obtained spectroscopically. It is therefore important to investigate the infrared spectra of forsterite with various crystallographically anisotropic shapes. However, systematic and exhaustive discussion has not yet been made except for the shapes formed by anisotropic evaporation (Takigawa et al. 2009). In this study, we calculate mass absorption coefficients (MACs) of ellipsoidal forsterite with various aspect ratios along different crystallographic orientations and fully investigate the relationship between crystallographically anisotropic shapes of forsterite and their spectral features. We introduce our calculation method of MACs in Section 2. In Section 3, we show the infrared cross sections along different crystallographic axes and their relationship to the grain shape. In Section 4, we present the MACs for various crystallographically anisotropic shapes and discuss the possibility of observation of forsterite with different crystallographically anisotropic shapes. The effects of grain temperature, chemical composition, grain size, and edges on the grain surface on infrared features are discussed in Section 5. We relate the crystallographically anisotropic shape to the formation condition of circumstellar forsterite in Section 6, and summarize and conclude in Section 7.

In order to calculate the MACs of forsterite grains with various crystallographically anisotropic shapes based on Bohren & Huffman (1983), we consider ellipsoids with a single aspect ratio, of which each semi-axis is parallel to one of the principal axes of the dielectric tensor (eigenvectors). We assume that particles are sufficiently smaller than the wavelength of the light (the Rayleigh limit), i.e., sub-μm in size for the infrared light. The polarizability of an ellipsoid in a uniform static electric field parallel to each semi-axis (x_j; j = 1, 2, 3) is represented by

$$\begin{equation} \;\,{\kern 1pt} \;\alpha _j (\omega) = \frac{{4\pi r_1 r_2 r_3 }}{3}\frac{{\varepsilon _j (\omega) - 1}}{{1 + L_j ({\varepsilon _j (\omega) - 1} )}}, \end{equation} \tag{ 1 }$$

where ε_j is the complex dielectric function of the ellipsoid along the x_j-axis, r₁, r₂, and r₃ are the lengths of semi-axes of the ellipsoid, and L_j is a geometrical factor generally represented by

$$\begin{eqnarray} L_j &=& \frac{{r_1 r_2 r_3 }}{2}\int_0^\infty {\frac{{dq}}{{\big({r_j^2 + q} \big)f\left(q \right)}}} \nonumber\\ f(q) &=& \left\{ {\left({q + r_1^2 } \right)\left({q + r_2^2 } \right)\left({q + r_3^2 } \right)} \right\}^{1/2} . \end{eqnarray} \tag{ 2 }$$

The geometrical factor satisfies 0 < L_j < 1 and L₁ + L₂ + L₃ = 1 (Bohren & Huffman 1983), and a unique set of L₁, L₂, and L₃ is determined for an aspect ratio of r₁:r₂:r₃. For example, the geometrical factors for a sphere are given by (L₁, L₂, L₃) = (1/3, 1/3, 1/3). If the semi-axis along the x_j-axis (r_j) is much shorter than those along the other axes, L_j approaches to unity, and if r_j is sufficiently large relative to other axes, L_j becomes zero. Thus, the geometrical factors for a disk extremely flattened to the x₁-, x₂-, or x₃-axis are (L₁, L₂, L₃) = (1, 0, 0), (0, 1, 0), or (0, 0, 1), and those for a needle exceedingly elongated along the x₁-, x₂-, or x₃-axis are (L₁, L₂, L₃) = (0, 1/2, 1/2), (1/2, 0, 1/2), or (1/2, 1/2, 0).

Because it is reasonable to assume that dust grains are randomly oriented against the static electric fields, the average absorption cross section 〈C^abs〉 is related to α_j by

$$\begin{eqnarray} \langle {C^{abs} } \rangle &=& \frac{1}{3}C_1^{abs} + \frac{1}{3}C_2^{abs} + \frac{1}{3}C_3^{abs} \nonumber \\ & = & 2\pi \omega {\mathop{\rm Im}\nolimits} \left({\frac{1}{3}\alpha _1 + \frac{1}{3}\alpha _2 + \frac{1}{3}\alpha _3 } \right). \end{eqnarray} \tag{ 3 }$$

The MAC (κ^abs) for ellipsoids with a single crystallographically anisotropic shape is given by 〈C^abs〉 per unit mass:

$$\begin{eqnarray} \kappa ^{abs} (\omega) &=& \frac{{\langle {C^{abs} } \rangle }}{{\rho V}}\nonumber \\ &=& \frac{{2\pi \omega }}{{3\rho }} \cdot {\mathop{\rm Im}\nolimits} \sum\limits_{j = 1,\,2,\,3} {\frac{{\varepsilon _j \left(\omega \right) - 1}}{{1 + L_j \cdot ({\varepsilon _j (\omega ) - 1} )}}}, \end{eqnarray} \tag{ 4 }$$

where ρ and V are the density and volume of the ellipsoid. For the dielectric function of ε_j(ω) = ε'_j(ω) + iε''_j(ω), κ^abs is expressed as

$$\begin{equation} \kappa ^{abs} (\omega) = \frac{{2\pi \omega }}{{3\rho }}\sum\limits_{j = 1,\,2,\,3} {\frac{{{\varepsilon}_j ^{\prime \prime} \left(\omega \right)}}{{({1 + L_j \cdot ({{\varepsilon}_{j}^{\prime} (\omega) - 1} )} )^2 + ({L_j {\varepsilon}_{j}^{\prime \prime} (\omega )} )^2 }}}. \end{equation} \tag{ 5 }$$

κ^abs has peaks at the wavelengths, where the geometrical factor and the dielectric function satisfy the following relation:

$$\begin{equation} ({1 + L_j \cdot ({{\varepsilon}^{\prime}_{j} (\omega) - 1} )} )^2 + ({L_{j}} {\varepsilon}^{\prime\prime}_{j} (\omega))^2 \sim 0, \end{equation} \tag{ 6 }$$

which are called the surface mode resonances.

In the case of isotropic materials (e.g., amorphous silicates and crystals belonging to the cubic system), the dielectric functions along the principal axes are identical (ε₁ = ε₂ = ε₃), and κ^abs of an isotropic material is thus affected only by its shape, i.e., L_j (Equation (5)). The dielectric functions of anisotropic crystals are not identical to each other; ε₁ = ε₂ ≠ ε₃ for crystals belonging to the tetragonal, hexagonal, and trigonal systems and ε₁ ≠ ε₂ ≠ ε₃ for orthorhombic, monoclinic, and triclinic crystals. Therefore, a spectrum of an anisotropic crystal depends both on the shape and its relationship to the orientations of principal axes of the dielectric tensor, i.e., the crystallographically anisotropic shape controls spectra of anisotropic crystals.

Forsterite, which belongs to the orthorhombic crystal system, has 11, 12, and 8 infrared optical resonances in the wavelength range of 9–70 μm along the three principle axes corresponding to the crystallographic a-, b-, and c-axes, respectively. These resonances are attributed to the stretching and bending of Si–O bonds, the rotation of SiO₄ tetrahedra, and the translation of Mg atoms in the crystal structure of forsterite (Iishi 1978). Thus, an infrared spectrum of ellipsoidal forsterite grains with a single crystallographically anisotropic shape can be expressed as superposition of 31 surface-mode resonances.

We calculated α_a,b,c for 0 ⩽ L_a, L_b, L_c ⩽ 1 (Equation (1)) and κ_abs (Equation (4)) of ellipsoidal forsterite particles, varying the aspect ratios of ${{r_b } \mathord{\left/ {\vphantom {{r_b } {r_a }}} \right. \kern-\nulldelimiterspace} {r_a }}$ and ${{r_c } \mathord{\left/ {\vphantom {{r_c } {r_a }}} \right. \kern-\nulldelimiterspace} {r_a }}$ from 100 to 0.01, as a function of wavelength. The optical constants at 100 K, which were experimentally determined by Suto et al. (2006), were used to obtain the dielectric functions of forsterite.

We also calculated κ_abs for the continuous distributions of ellipsoids for comparison (CDE and CDE2). The spectrum of the CDE gives the ensemble of κ_abs for all possible crystallographically anisotropic shapes with equal probabilities (Bohren & Huffman 1983), and that of the CDE2 also gives κ_abs as the ensemble of ellipsoids but with higher presence probabilities of near-spherical particles (CDE2; Fabian et al. 2001). The κ_abs of the CDE2 has been known to reproduce the laboratory-measured infrared spectra of well-grinded forsterite powder (Koike et al. 2006).

We also calculated κ_abs of forsterite parallelepipeds with various aspect ratios in the range of 9–12 μm to compare with those of ellipsoids because crystalline grains possibly have crystal surfaces and edges, which may affect the surface modes. The calculation of κ_abs for the forsterite parallelepipeds was made by the discrete dipole approximation (DDA) method using the FORTRAN program DDSCAT developed by Draine (1988) and Draine & Flatau (1994).

We show the absorption cross section per unit mass along the crystallographic a-axis (C^abs_a/3ρV) in the wavelength range of 8–30 μm for L_a = 0–1 in Figure 1 as an example. Superposition of C^abs_a/3ρV with C^abs_b/3ρV and C^abs_c/3ρV under the condition of L_a + L_b + L_c = 1 represents a MAC. For larger L, the peaks shift to shorter wavelengths and the peaks tend to become weaker irrespective of the crystallographic orientation as explained in the following.

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The dielectric function along one of the principal axes (x_j) of the dielectric tensor, i.e., an eigenvalue of the tensor, is represented by the following equation with the assumption of multiple harmonic oscillation as lattice vibrations (Lorentz model):

$$\begin{eqnarray} \varepsilon_{j} (\omega) &=& \varepsilon ^{\prime}_j (\omega) \,{+}\, i\varepsilon ^{\prime\prime}_j (\omega) \nonumber\\ &=& \varepsilon _{\infty j} \,{+}\, \sum\limits_{k = 1}^{N_j } {\frac{{\omega _p \left({k,j} \right)^2 }}{{\omega _T \left({k,j} \right)^2 \,{-}\, \omega ^2 \,{-}\, i\gamma (k,j)\omega }}} \nonumber \\ \varepsilon ^{\prime}_j (\omega) &=& \left({\varepsilon _{\infty j} + \sum\limits_{k = 1}^{N_j } {\frac{{\omega _p ({k,j} )^2 ({\omega _T ({k,j} )^2 - \omega ^2 } )}}{{({\omega _T ({k,j} )^2 - \omega ^2 } )^2 + \gamma (k,j)^2 \omega ^2 }}} } \right)\nonumber \\ \varepsilon ^{\prime\prime}_j (\omega) &=& \sum\limits_{k = 1}^{N_j } {\frac{{\omega _p \left({k,j} \right)^2 \gamma (k,j)\omega }}{{({\omega _T ({k,j} )^2 - \omega ^2 } )^2 + \gamma (k,j)^2 \omega ^2 }}} , \end{eqnarray} \tag{ 7 }$$

where ε_∞j is the dielectric function representing the contribution from higher wavenumbers, ω_T and ω_p represent the resonant frequencies of the oscillator and the plasma frequency, respectively, γ is the damping factor, and N_j is the number of optical modes along the x_j-axis.

The ε''(ω), the imaginary part of ε_j(ω), is a bell-shaped curve with a peak at ω_T (the transverse optical frequency). The second term of ε'(ω), the real part of ε_j(ω), has a shape similar to the derivative curve of ε''(ω) at ω ∼ ω_T but it approaches to (ω_p/ω_T)² and 0 at the limits of ω →0 and ∞, respectively. In the case of L_j = 0, C^abs_i has a peak at ω_T, where ε''(ω) is maximum. In the case of L_j = 1, a peak appear at a wavelength that minimizes ε'(ω)² + ε''(ω)². ε''(ω)² can be close to zero either at frequencies larger or smaller than ω_T, but ε'(ω)² minimizes only at a frequency larger than ω_T. The frequency where C^abs_i with L_j = 1 has a peak is called as a longitudinal optical frequency (ω_L). Thus, the peak wavelength for L = 1 is always shorter than that for L = 0. In the case of 0 < L < 1, ε'(ω) should be close to 1−1/L in order to satisfy Equation (6), which requires ε'(ω) < 0 and is achieved only at a frequency between ω_T and ω_L.

Therefore, as L_j increases from 0 to 1, all the peaks for optical resonances belonging to the x_j-axis shift to shorter wavelengths, and the intensities of the peaks become weaker in general because ε''(ω) becomes smaller as ω approaches to ω_L. We also note that the wavelength range of the peak shift for L_j between 0 and 1 differs from one resonance to another (Figure 1), depending on oscillator parameters.

The MACs should vary for different crystallographically anisotropic shapes because the MACs are given as the superposition of C^abs_a/3ρV, C^abs_b/3ρV, and C^abs_c/3ρV, which change with L_a, L_b, and L_c, respectively. Because the geometrical factors should satisfy the relation of L_a + L_b + L_c = 1, the change of the crystallographically anisotropic shape makes the peaks attributed to a crystallographic axis shift to one direction and those related to at least one of the other crystallographic axes shift to the opposite direction. This variety of spectra for different crystallographically anisotropic shapes is more complex than those caused by the changes of size, temperature, and chemical compositions, where all the peaks shift unidirectionally to either longer or shorter wavelengths.

In order to discuss the relationship between the MACs and the crystallographically anisotropic shapes of forsterite, we first classify the crystallographically anisotropic shapes into six types depending on the geometrical factor (Figure 2); the a-, b-, and c-needles represent shapes with (L_a, L_b, L_c) = (<1/3, >1/3, >1/3), (>1/3, <1/3, >1/3), and (>1/3, >1/3, <1/3) that are ellipsoids elongated along the crystallographic a-, b-, and c-axes, respectively, and the a-, b-, and c-disks represent shapes with (L_a, L_b, L_c) = (>1/3, <1/3, <1/3), (<1/3, >1/3, <1/3), and (<1/3, <1/3, >1/3) that are ellipsoids flattened to the crystallographic a-, b-, and c-axes, respectively. The optical constants along the a-, b-, c-axes at 100 K (Suto et al. 2006) were used in the calculations.

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The characteristic features of the absorption cross sections for different crystallographically anisotropic shapes, i.e., different sets of geometrical factors, are summarized in Table 1 for the five band complexes (the 10, 18, 25, 33, and 69 μm complexes), and the distinguishability of the MACs for different crystallographically anisotropic shapes is described in the following. We also show the MACs for crystallographically anisotropic spheroidal forsterite with various aspect ratios in Figures 3–7. Spectra for the spherical forsterite and the CDE and CDE2 models are also shown for comparison.

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4.1. The 10 μm Complex

4.2. The 18 μm Complex

There are two resonances along the a-axis in this complex. C^abs_a has one peak at 15.4, 16, and 16.5 μm and another peak at 18.2, 18.8, and 19.8 μm for L_a = 1, 1/3, and 0, respectively. Shapes with L_a > 1/3 (the a-disk, b-needle, and c-needle) show two peaks between 15.4 and 16 μm and between 18.2 and 18.8 μm, while those with L_a < 1/3 (the a-needle, b-disk, and c-disk) have peaks in the ranges of 16–16.5 and 18.8–19.8 μm (Figure 4(a); Table 1).

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C^abs_b shows a peak at 17.4, 18.2, and 18.9 μm for L_b = 1, 1/3, and 0, respectively, in this complex. Another peak appears distinctly at 19.7 μm only when L_b is close to 0. The MACs for ellipsoids with L_b > 1/3 (the b-disk, a-needle, and c-needle) show one peak between 17.4 and 18.2 μm, while those with L_b < 1/3 (the b-needle, a-disk, and c-disk) show a peak between 18.2 and 18.8 μm and another peak at ∼19.8 μm in the case of L_b ∼ 0 (Figure 4b; Table 2).

The c-axis has one optical resonance in this complex. A peak of C^abs_c appears at 17.3, 18.3, and 19.8 μm for L_a = 1, 1/3, and 0, respectively. The MACs for ellipsoids with L_c > 1/3 (c-disk, a-needle, and b-needle) show a peak between 17.3 and 18.3 μm, while shapes with L_c < 1/3 (c-needle, a-disk, and b-disk) show at least one peak between 18.3 and 19.8 μm (Figure 4(b); Table 2).

The diagnostic feature to deduce the crystallographically anisotropic shape in the 18 μm complex is the optical resonant mode between 15.4 and 16.5 μm, which is the only resonance in that wavelength range and is attributed to that along the a-axis. The degree of contribution from the resonance along the a-axis (L_a > 1/3 or L_a < 1/3) can be estimated from this resonance (Figure 4(b); Table 2).

For the shapes with L_a > 1/3 (a-disk, b-, and c-needle), b- and c-needles show a peak between 17.3 and 18.3 μm but a-disk has no peak in that wavelength range. The peak of b-needle (L_b < 1/3 and L_c > 1/3) between 17.3 and 18.3 μm is twice as strong as the peak between 18.3 and 18.9 μm, but that of c-needle (L_b > 1/3 and L_c < 1/3) is comparable to that of the peak between 18.3 and 18.9 μm. This is because the peak between 17.3 and 18.3 μm is due to the optical resonances along the b-axis for L_b > 1/3 and along the c-axis for L_c > 1/3 (Figure 4(a)) and the latter is stronger than the former.

The difference in spectral features between a-needle, b-disk, and c-disk (L_a < 1/3) is the relative strengths of peaks. The peak of b-disk (L_b > 1/3 and L_c < 1/3) between 18.3 and 19.8 μm is strongest in this complex and twice as strong as that the second largest peak between 16 and 16.5 μm. This is because the resonances of C^abs_a and C^abs_c for L_a and L_c < 1/3 (Figure 4(a)) make a single strong peak between 18.3 and 19.8 μm. On the other hand, the peaks between 18.3 and 19.8 μm and between 16 and 16.5 μm are comparable for a-needle and c-disk because L_c of these shapes are larger than 1/3. The a-needle (L_b > 1/3) and c-disk (L_b <1/3) may possibly be distinguished by the strength of the peak between 17 and 18.3 μm, which is strongest for a-needle.

The CDE show peaks at 16.4, 18.8, and 19.6 μm. The peaks at 16.4 and 19.6 μm are those for C^abs_a with L_a ∼ 0.05, and the 18.8 μm peak is that for C^abs_c with L_c ∼ 0.2 (Figure 4(a)). Because peaks of C^abs_b and C^abs_c appear at similar wavelengths and the peaks of C^abs_b has weaker than that of C^abs_c (Figure 4(a)), the spectral peak positions of CDE can be similar to those for the ellipsoidal b-disk with (L_a, L_b, L_c) ∼ (0.05, 0.75, 0.20) corresponding to the aspect ratio of r_a:r_b:r_c ∼ 1:0.11:0.4. The spectrum for the CDE2 shows peaks similar to that of the b-disk with (L_a, L_b, L_c) ∼ (0.29, 0.46, 0.25) (r_a:r_b:r_c ∼ 0.9:0.6:1). As mentioned above, the spectrum with predominance of a specific shape should show sharper peaks compared to the CDE and the CDE2, and we thus conclude that the dominant crystallographically anisotropic shape of forsterite can be distinguished from the infrared features in this complex.

The 25 μm complex. Nine optical resonances are present in the wavelength range of 20–30 μm (Figure 5(a)). However, in the wavelength range of 20–26 μm, the strong peaks of C^abs_a and C^abs_c appear at similar wavelengths and the peaks of C^abs_b are not strong. These resonances are thus not useful to distinguish the crystallographically anisotropic shape. The only characteristic feature in the 25 μm complex is the peak for C^abs_b between 26.5 and 28.4 μm (Figure 5(b); Table 2), which appears at 26.5, 26.7, and 28.4 μm for L_b = 1, 1/3, and 0, respectively. Because the peak appears clearly in the MAC only for L_b < 1/3 (b-needle, a-disk, and c-disk), the contribution of the b-axis dimension to the grain shape could be determined based on the presence of a peak at 26.7–28.4 μm (Figure 5(b); Table 2). The extreme shape with L_a ∼ 1 might also be distinguished by the presence of a peak at ∼26 μm although the predominance of such an extreme shape seems to be unlikely.

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The 33 μm complex. Five optical resonances are present in this complex. No diagnostic features are present to distinguish all the six types of shapes in this complex because there are no strong resonances along the a-axis and two resonances along the b- and c-axes exist in a similar wavelength range (34.2–34.4 μm).

The 69 μm complex. The 69 μm feature is a single peak reflecting an optical mode along the b-axis (Figures 7(a) and (b); Table 2). A clear 69 μm peak appears only for the shape with L_b < 1/3 (a-disk, b-needle, and c-disk), and the peak position shifts toward longer wavelengths with increasing the degree of contribution from the b-axis, i.e., decreasing L_b (Figure 7(b)). The peak shifts from 68.8 to 69.1 μm with decreasing L_b. The peak position of the 69 μm feature has been known as an indicator of temperature of circumstellar forsterite because peaks at far-infrared wavelengths are largely affected by temperature, but the present results show that the crystallographically anisotropic shape also affects the peak position.

The infrared features of forsterite reflect the crystallographically anisotropic shape in the wide range of wavelengths from 9 to 70 μm. However, peak features are also dependent of dust temperature, size, chemical composition, and grain edges. In this section, we discuss the effects of these factors on infrared features and their distinguishabilities from the features caused by crystallographically anisotropic shapes.

The MACs of forsterite calculated with the optical constants at 100 K (Suto et al. 2006) are shown in Figures 3–7, but peak positions shift to longer wavelengths at higher temperatures (Koike et al. 2006). When we use the optical constants measured at 50–295 K (Suto et al. 2006), the shifts of peak positions due to temperature change are larger for peaks at longer wavelengths; peak wavelengths of the CDE model shift by <0.1 μm in the 10 and 18 μm complex, ∼0.3 μm in the 25 and 33 μm complex, and ∼0.7 μm in the 69 μm complex.

Grain size affects both peak positions and intensities. When the size changes from 0.1 to 1.0 μm, peak wavelengths of 10.1 and 11.3 μm shift to 10.2 and 11.5 μm, respectively, but peaks at longer wavelengths are not largely affected (Min et al. 2003).

Peak intensities also change with the grain size, and they decrease with increasing size of grains, e.g., the intensity of the 11.3 μm peak of 4 μm sized forsterite grains is ∼1/5 of that for 0.1 μm sized grains.

The Mg/(Mg+Fe) ratio has influences on the positions and intensities of peaks particularly at longer wavelengths. Peak wavelengths of irregularly shaped olivine measured in laboratory tend to shift toward longer sides with decreasing the Mg/(Mg+Fe) from 1 (forsterite) to 0 (fayalite; Koike et al. 2003). Peaks of forsterite grains at 10.1, 11.2, and 11.9 μm appear at 10.6, 11.4, and 12.1 μm, respectively, for fayalite (Fe₂SiO₄) in the 10 μm complex. If the addition of Fe to forsterite is limited (Mg/(Mg+Fe) > 0.9), it has little effects on peak positions in the 10 μm complex and peaks in the 18 μm complex shifts to longer sides by <1 μm. For the peaks at longer wavelengths (25, 33, and 69 μm complexes), a small content of iron in the lattice structure shifts peak positions significantly to longer sides without broadening the peaks (Koike et al. 2006; Suto et al. 2006; de Vries et al. 2011). Combination of multiple peak positions in the 33 and 69 μm complexes can be used to evaluate the grain shape effect separately from the composition and temperature effects (Figures 5 and 6). However, the features in the 33 and 69 μm complexes are required to be emitted from the same dust component, and they may not be as useful as the features at shorter wavelengths to deduce the grain shape.

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The presence of the crystal surfaces and edges results in the increase of surface modes. For instance, cubic-shaped optically isotropic grains have six strong surface modes for one optical resonance (Fuchs 1975), while ellipsoidal-shaped grains have three surface modes for the resonance. Figure 8 shows the MACs of forsterite parallelepipeds with various aspect ratios calculated with DDA (Draine & Flatau 1994, 2010) using the optical constant at 100 K by Suto et al. (2006). Cubic forsterite shows peaks at 9.7, 10.6, and 10.8 μm, which also appear for spherical forsterite. Another strong peak appears at 11.0 μm, which is inconsistent with the spectrum of spherical forsterite. However, when the aspect ratios of parallelepipeds increase, the 11 μm feature becomes smaller and the peak features of ellipsoids and parallelepipeds look similar to each other. The euhedral forsterite has crystal faces more than six, so that the edge effect should be less than that shown in Figure 8 although the number of weak (and unobservable) surface mode resonances may increase. Therefore, crystallographically anisotropic shapes are distinguishable for grains with edges as well and that the ellipsoidal approximation is reasonable to evaluate the crystallographically anisotropic shapes of forsterite dust.

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Temperature, size, chemical composition, and crystal edges have clear effects on infrared features. However, the crystallographically anisotropic shapes of forsterite can be deduced separately from these effects in the 10 and 18 μm complexes because: (1) the peak positions are insensitive to temperature in these complexes and thus the grain shapes can be deduced from a spectrum even if circumstellar forsterite has some temperature distribution; (2) the peaks shifts due to the compositional effect are only in one direction, and are limited as long as the iron content is limited as observed in circumstellar forsterite (Mg/(Mg+Fe) > 0.9); (3) the grain size effect, which is larger in these complexes than that for longer wavelengths, is insignificant as long as the grains are smaller than 1 μm even if the grain size is beyond the Rayleigh limit and it also shifts peaks in one direction; (4) grain edges have little effects on shifts of peaks except for almost spherical shapes.

The compositional effect is not negligibly small in the 25 μm complex, but features in this complex can also be used to evaluate the predominance of the b-axis-dominant shapes (b-needle, a-disk, and c-disk) because the peaks shift unidirectionally with similar degrees by the compositional effect and the presence of a single peak at >27 μm is the criteria for judgment. Because there is only one peak in the 69 μm complex, it is not simple to separate the effect of crystallographically anisotropic shapes from other effects, particularly temperature and compositional effects.

In summary, infrared features of forsterite with crystallographically anisotropic shapes can be separated from the effects of temperature, size, composition, and grain edges in a wide range of wavelengths.

The crystallographically anisotropic shape of forsterite is not arbitrary in a mineralogical point of view because it reflects physical and chemical conditions of dust formation and subsequent evolution. The processes potentially responsible for the change of the crystallographic anisotropic shape of forsterite are condensation of solid from vapor, evaporation of solid, and crystallization of amorphous, of which anisotropy is likely to depend on temperature, pressure, and gas composition. It has been well known that forsterite evaporates anisotropically to change the crystallographically anisotropic shape and anisotropy in evaporation depends on temperature and hydrogen gas pressure (Ozawa et al. 1996; Yamada et al. 2006; Takigawa et al. 2009). The changes of the crystallographically anisotropic shape of forsterite (r_b/r_a and r_c/r_a) by evaporation from a sphere under various conditions (Takigawa et al. 2009) are shown in Figure 9. The typical crystallographically anisotropic shapes of forsterite formed by evaporation from a sphere are b-needle or c-disk at >1900 K, c-disk at ∼1800 K, c-disk or a-needle at ∼1600 K, and b-disk at <1420 K (Figure 9). Because details of anisotropic growth rates of forsterite have not yet been precisely determined, we assume the needle-like forsterite elongated to the c-axis as condensed forsterite based on the condensation experiments of forsterite on metallic substrates in hydrogen gas of 1 Pa at >1300 K (Tsuchiyama 1998).

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We discussed the infrared features of forsterite grains with a single shape, which show narrow peaks as seen in Figures 3–8. However, some degree of variations in aspect ratios of grains is expected through different degrees of anisotropic evaporation or condensation in circumstellar environments, which result in broader peaks for the ensemble of the grains. In order to discuss the infrared features for crystallographically anisotropic shapes as a potential probe to estimate the circumstellar dust-forming conditions, we consider the infrared features of different ensembles of grain shapes (LDE: limited distribution of ellipsoids) formed by evaporation at >1600 K (LDE1), evaporation at <1600 K (LDE2), and condensation (LDE3; Figure 9). We assume that the shapes within the range of aspect ratios for each ensemble appear with equal probability to calculate the infrared spectrum. Note that if all the shapes in Figure 9 appear with equal probability, the MAC for that ensemble should be that for the CDE.

Figure 10 compares the MACs of each ensemble and the CDE at 100 K. As most of grains in the LDE1 belong to c-disk, their MAC shows peaks at 9.7–10.2, 10.2–10.6, 10.8–11.5, and 11.9 μm. The MAC of the LDE2 shows peaks at 9.7–10.2, 10.2–10.6, and 10.6–10.8 μm. It shows no clear peak at 10.8–11.5 μm although grains in the LDE2 belong mostly to b-disk. The peak at 10.8–11.5 μm is attributed to C_c for L_c < 0.25 (see Section 4; Figure 3(a)); but the shapes with L_c < 0.25 occupy a small fraction of the LDE2. Therefore, the contribution from the shapes with L_c <0.25 appears only as a tail of the peak at 10.6–10.8 μm (Figure 9), which is the same as in the case of nearly spherical spheroidal b-disk (Table 2). The MAC of the LDE3, which covers the entire region of c-needle ((L_a, L_b, L_c) = (>1/3, >1/3, <1/3); Figure 9), shows peaks at 9.2–9.7, 10.6–10.8, 10.8–11.5, and 11.9 μm and no peak at 9.7–10.2 and 10.2–10.6 μm. The infrared features of the three ensembles are basically consistent with the classification described in Section 4 and Table 2, and can thus be distinguished from each other.

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The above discussion indicates that the dust-forming conditions can be evaluated from infrared spectra even for grains having a variation of crystallographically anisotropic shape. Another factor we should consider to link the infrared features to the dust-forming conditions is that most of infrared observations do not resolve stars and their dusty envelopes are observed as point sources. Especially, the dust-forming region or the region where dust particles are thermally altered is spatially limited, so that it cannot be resolved with large telescopes such as the VLTI or Subaru. We thus need to evaluate how the mixing of different ensembles of grains with different ranges of shape distributions affects the infrared spectra. Here we show the composite spectra of a grain ensemble with the dominance of a specific crystallographically anisotropic shape (LDE1, LDE2, or LDE3 in Figure 9) and the CDE with different mixing ratios (Figure 11). It is clearly seen that the composite spectrum is different from the spectrum of CDE if the fraction of each LDE is higher than a few tens of a percent.

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We therefore conclude that the dust-forming or dust-evolving conditions of forsterite (condensation and evaporation under different temperature conditions) can be evaluated from infrared observations as long as a certain fraction of forsterite particles with a dominant crystallographically anisotropic shape are present (more than a few tens of a percent) among the entire forsterite grains. We finally apply the present result to the observed spectrum of a protoplanetary disk HD100546 (Malfait et al. 1998; Juhász et al. 2010). Juhász et al. (2010) compared the spectra of forsterite derived from the Spitzer Infrared Spectrograph spectrum of HD100546, which is the protoplanetary disk showing strong features of crystalline forsterite in the 10 μm band, with the calculated spectrum of DHS forsterite (the distribution of hollow spheres; Min et al. 2003) and the absorption spectrum of forsterite measured in laboratory with the aerosol technique (Tamanai et al. 2006). However, the observed 16.1 μm peak could not be reproduced by the DHS model, which shows similar band profiles to the CDE model with some exceptions. The laboratory spectrum, which is well produced by the CDE2 model, shows a peak at ∼16 μm, but could not reproduce peak positions observed at 10.1 and 11.3 μm.

We found that the composite spectrum of CDE and the LDE1 with the mixing ratio of 3:1 reproduces the peak wavelengths at 10.1, 11.3, and 16.1 μm better than the DHS and the laboratory measurement (Figure 12). We note that the wavelength resolution of the composite spectrum in Figure 12 was reduced to be ∼0.1 μm to compare with the observed spectrum by Spitzer/IRC, of which spectral resolution was ∼0.1 μm in the 10 μm band. Moreover, the band shapes of the peaks at 11.3 and 16.1 μm in the composite spectrum reproduce the observation better than the DHS, the laboratory spectrum, and the CDE alone. The modeled peaks at 10.1 and 11.3 μm are, however, stronger than the observed peaks, which is also the cases for the DHS, the laboratory spectrum, and the CDE. This might be due to the overestimation of amorphous silicate and too much subtraction of its spectrum, which shows a broad peak centered at ∼10 μm, from the original. We here note that the grain size effect is not significant as long as the typical size of larger grains is ∼1–2 μm (Bouwman et al. 2001, 2003; van Boekel et al. 2005).

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We therefore propose that a certain fraction of forsterite dust in HD100546 experienced high-temperature evaporation processes most likely in the inner region of the disk. Rapid transient heating events such as formation of chondrules, the major constituents of primitive meteorites, could be responsible for the high-temperature evaporation of forsterite in the disk. The partially evaporated dust particles may have mixed with grains that did not experience such high-temperature events due to radial mixing of materials in the disk (e.g., Dullemond et al. 2006; Ciesla 2007).

Further experimental works for condensation are clearly needed in order to evaluate the dust-forming conditions quantitatively, but we conclude that the crystallographically anisotropic shape of forsterite deduced from infrared features leads us to understand thermal events that crystalline forsterite experienced in circumstellar environments.

The crystallographically anisotropic shape is a specific crystal shape with a specific relationship to crystallographic orientations, which changes with dust formation and/or subsequent evolution conditions reflecting the anisotropic nature of crystals. An emission spectrum from dust grains depends on their crystallographically anisotropic shapes because different crystallographically anisotropic shapes yield different wavelengths of surface mode resonances and their different degrees of activation. The crystallographically anisotropic shape can thus be used to estimate circumstellar dust-forming conditions quantitatively. In order to relate the crystallographically anisotropic shape with the mid-infrared features, we calculated MACs for spheroidal forsterite particles, the most abundant crystalline silicate in space, with various aspect ratios. We found that the crystallographically anisotropic shape of forsterite can be estimated from infrared features.

• 1.  
  The 10 μm complex. Discrete and/or relatively strong optical resonances along each crystallographic axis exist in the 10 μm complex, and the peak wavelengths and intensities of the surface mode resonances show clear changes with the crystallographically anisotropic shape. The temperature of dust particles also changes the peak wavelengths, but the wavelength shift occurs only in one direction with a limited range in these complexes. The composition (substitution of Fe for Mg) has little effect on peak positions if the iron content is limited as observed in circumstellar forsterite (Mg/(Mg + Fe) > 0.9). The effect of grain size larger than the Rayleigh limit is insignificant as long as the grains are ∼1 μm in size, which is a typical size of larger grains observed in protoplanetary disks, and it also shifts peaks in one direction. We also showed that grain edges have little effect on shifts of peaks except for almost spherical shapes. Therefore, the infrared features in the 10 μm complex can be used as an indicator of the crystallographically anisotropic shape of forsterite.
• 2.  
  The 18 μm complex. The features in the 18 μm complex can also be used to evaluate the crystallographically anisotropic shape of forsterite because characteristic variations of peak features are expected for various crystallographically anisotropic shapes. Note that the shape determination based on the features in the 18 μm complex may be more complicated than that based on the 10 μm features because not only the peak positions but the relative peak intensities should also be used for the shape determination.
• 3.  
  The 25 μm complex. The only diagnostic feature in the 25 μm complex is the contribution of the crystallographic b-axis although the compositional effect is not negligible; the shapes with the longer dimension along the b-axis (b-needle, a-disk, and c-disk) have a peak at >27 μm and can be distinguished from those with the shorter dimension along the b-axis (b-disk, a-needle, and c-needle). There are also many resonances in the range of 20–26 μm, but their interactions for different crystallographically anisotropic shapes are too complicated to be used as the shape indicator.
• 4.  
  The 33 μm complex. There are limited numbers of resonances in the 33 μm complex, where resonances along the a-axis are very weak and strong resonances along the b- and c-axes appear at similar wavelengths. The MAC in the 33 μm complex thus does not have diagnostic features to determine the crystallographically anisotropic shape.
• 5.  
  The 69 μm complex. For the 69 μm feature that is attributed to the resonance along the b-axis, the peak shift by the crystallographically anisotropic shape is comparable to that caused by varying dust temperature of 100 K.
• 6.  
  The comparison of infrared features of grain ensembles with shape distributions expected for various degrees of anisotropic evaporation at different temperatures and of condensation shows that the infrared features of the crystallographically anisotropic shape are distinguishable even when forsterite dust has a certain range of shape variations. Moreover, it is shown that the presence of such a grain ensemble can be detected even if the ensemble is mixed with the CDE spectrum as long as the fraction of the ensemble is higher than a few tens of a percent.

These findings strongly suggest that the dust-forming conditions (evaporation and/or condensation) can be evaluated from the infrared spectra of forsterite. The application of the present results to the infrared spectrum of forsterite in a protoplanetary disk HD100546 shows that the composite spectrum of the CDE (75%) and the ensemble of grains experienced high-temperature evaporation (>1600 K) (25%) well reproduces the observed spectrum. The presence of forsterite dust experienced high-temperature evaporation in the protoplanetary disk may indicate the radial outward transport of high-temperature inner-disk materials in the disk.

The crystallographically anisotropic shape of forsterite will be a new powerful probe for astromineralogy to link the infrared observations to dust formation conditions around stars via anisotropy in dust formation processes. Finally, we emphasize that the crystallographically anisotropic shape of other crystalline dust species can also be evaluated from the infrared spectra. Combining the crystallographically anisotropic shapes of various minerals and their anisotropic properties of formation processes will enable us to evaluate dust formation histories from infrared spectra in more detail. Anisotropic nature of minerals provides us in-depth scope for astromineralogy.

We are grateful to the anonymous reviewer for a careful reading of the manuscript and helpful comments and to Eric D. Feigelson for editorial handling. We thank Hiroko Nagahara for her valuable comments. We also thank Hisato Sogawa for his code for calculation of infrared spectra. The DDSCAT numerical code was kindly provided by Bruce T. Draine. This work was supported by Grant-in-Aid for JSPS Fellows (09J02084) (A.T.) and Grant-in-Aids for Young Scientists (A) (20684025) (S.T.).