Evolution of dust grain size distribution by shattering in the interstellar medium: Robustness and uncertainty
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB 2013
Received: 16 November 2012
Accepted: 9 March 2013
Published: 24 October 2013
Shattering of dust grains in the interstellar medium is a viable mechanism of small grain production in galaxies. We examine the robustness or uncertainty in the theoretical predictions of shattering. We identify P1 (the critical pressure above which the deformation destroys the original lattice structures) as the most important quantity in determining the timescale of small grain production, and confirm that the same P1/t (t is the duration of shattering) gives the same grain size distribution [n(a), where a is the grain radius] after shattering within a factor of 3. The uncertainty in the fraction of shocked material that is eventually ejected as fragments causes uncertainties in n(a) by a factor of 1.3 and 1.6 for silicate and carbonaceous dust, respectively. The size distribution of shattered fragments have minor effects as long as αf ≲ 3.5 (the size distribution of shattered fragments ), since the slope of grain size distribution n(a) continuously changes by shattering and becomes consistent with n(a) ∝ a−3.5>. The grain velocities as a function of grain radius can have an imprint in the grain size distribution especially for carbonaceous dust. We also show that the formulation of shattering can be simplified without losing sufficient precision.
Key wordsCosmic dust galaxy evolution grain size distribution interstellar medium
The evolution of dust in galaxies is important to the understanding of galaxy evolution, since dust grains govern some fundamental physical processes in the interstellar medium (ISM). First, they dominate the absorption and scattering of the stellar light, affecting the radiative transfer in the ISM. Second, the dust surface is the main site for the formation of molecular hydrogen. The former process is governed by the extinction curve (absorption and scattering coefficient as a function of wavelength; Hoyle and Wickramasinghe, 1969; Draine, 2003) and the latter by the total surface area of dust grains (Yamasawa et al., 2011). Since the extinction curve and the total grain surface area both depend strongly on the grain size distribution, clarifying the regulating mechanism of grain size distribution is of particular importance in understanding those important roles of dust.
Mathis et al. (1977, hereafter MRN) show that a mixture of silicate and graphite with a grain size distribution [number density of grains per grain radius, denoted as n(a) in this paper] proportional to a−3.5 (a grain size distribution with a power index of –3.5 is called MRN grain size distribution), where a is the grain radius (a ~ 0.001–0.25 μm), reproduces the Milky Way extinction curve. Pei (1992) shows that the extinction curves in the Magellanic Clouds are also explained by the MRN grain size distribution with different abundance ratios between silicate and graphite. Kim et al. (1994) and Weingartner and Draine (2001) have applied a more detailed fit to the Milky Way extinction curve in order to obtain the grain size distribution. Although their grain size distributions deviate from the MRN size distribution, the overall trend from small to large grain sizes roughly follows a power law with an index near to −3.5. In any models that fit the Milky Way extinction curve, the existence of a large number of small (a ≲ 0.01 μm) grains is required. The existence of such small grains is further supported by the mid-infrared excess of the spectral energy distribution in the Milky Way (e.g., Desert et al., 1990; Draine and Li, 2001).
Dust grains formed by stellar sources, mainly supernovae (SNe) and asymptotic giant branch (AGB) stars, are not sufficient to explain the total dust mass in the Milky Way ISM, because the timescale of dust enrichment by these sources is much longer than that of dust destruction by SN shocks (e.g., McKee, 1989; Draine, 1995). Moreover, both SNe and AGB stars are suggested to supply large (a ≳ 0.1 μm) grains into the ISM and cannot be the dominant source of small grains (Nozawa et al., 2007; Höfner, 2008; Mattsson and Hofner, 2011; Norris et al., 2012). Thus, some interstellar processes (or more precisely, non-stellar processes) are necessary to explain the large abundance of small grains.
There are some possible processes that efficiently modify the grain size distribution. Hellyer (1970) shows that the collisional fragmentation of dust grains finally leads to a power-law grain size distribution similar to the MRN size distribution (see also Bishop and Searle, 1983; Tanaka et al., 1996). Hirashita and Yan (2009) show that such a fragmentation and disruption process (or shattering) can be driven efficiently by turbulence in the diffuse ISM. Dust grains are also processed by other mechanisms. Various authors show that the increase of the total dust mass in the Milky Way ISM is mainly governed by grain growth through the accretion of gas phase metals onto the grains (we call elements composing dust grains “metals”) (e.g., Dwek, 1998; Zhukovska et al., 2008; Inoue, 2011; Asano et al., 2013a). Grain growth mainly works in the dense ISM, such as molecular clouds. In the dense ISM, coagulation also occurs, making the grain sizes larger (e.g., Hirashita and Yan, 2009; Ormel et al., 2009). In the diffuse ISM phase, interstellar shocks associated with supernova (SN) remnants destroy dust grains, especially small ones, by sputtering (e.g., McKee, 1989; Nozawa et al., 2006). Shattering also occurs in SN shocks (Jones et al., 1996). All these processes above other than shattering do not contribute efficiently to the increase of small (a ≲ 0.01 μm) grain abundance. Indeed, Asano et al. (2013b) show that, unless we consider shattering, the grain size distribution is biased to large (a ~ 0.1 μm) sizes. Therefore, shattering is important in reprocessing large grains into small grains.
Based on the formulation developed by Hirashita and Yan (2009), Hirashita et al. (2010, hereafter H10) consider small grain production by shattering in the context of galaxy evolution. H10 assume that Type II SNe (SNe II) are the source of the first dust grains, which are biased to large sizes (≳ 0.1 μm) because small grains are destroyed in the shocked region in the SNe II (Nozawa et al., 2007, hereafter N07). Starting with the size distribution of SNe II dust grains in N07, H10 solve the shattering equation to calculate the evolution of grain size distribution. They show that the velocity dispersions acquired by the dust grains in a warm ionized medium (WIM) is large enough for shattering to produce a large abundance of small grains on a short (≲ 10 Myr) timescale. As mentioned above, even if we consider other sources of dust such as the dust formation in the wind of AGB stars and the accretion of metals onto grains in the interstellar clouds, shattering is generally necessary to produce small grains (Asano et al., 2013b).
Considering that shattering is a unique mechanism that produces small grains efficiently in the ISM, it is important to clarify how robust or uncertain the theoretical calculations of shattering are. Since there are some basic physical parameters regulating shattering, it is crucial to clarify how they affect the grain size distribution or to which parameter shattering is sensitive. Although any models should commonly contain those basic parameters, there could be some uncertainties in the formulation itself, or there could be unnecessary complexity which cannot be constrained from observations anyway. Thus, in this paper, we examine the robustness of shattering calculations for the interstellar dust by changing some major parameters in a model and comparing the results between two different frameworks.
This paper is organized as follows. In Section 2, we explain the shattering models and pick out some major parameters. In Section 3, we examine the variation of the grain size distribution due to changes of the major parameters. In Section 4, we discuss our results and implications for the evolution of grain size distribution. In Section 5, we give our conclusions.
2. Overview of the Models for Shattering
As mentioned in Introduction, we use the framework in H10 to test the robustness of a shattering model. H10 consider the evolution of grain size distribution through shattering by solving a shattering equation, which treats the grain-grain collision rate and the redistribution of shattered fragments in the grain size distribution. They assume that grains collide with each other under the velocity dispersions induced by dynamical coupling with interstellar turbulence. The production of fragments in grain-grain collisions is the key process that we focus on below.
2.1 Initial grain size distribution
The normalization of the grain size distribution is determined as follows. We consider a solar metallicity environment as a representative case. (For other metallicities, the timescale of shattering just scales with the inverse of metallicity.) We parameterize the dust abundance by the oxygen abundance. The oxygen mass produced by a SN II of 20 M⊙ progenitor is mO = 1.58 M⊙ according to Umeda and Nomoto (2002), whose data were adopted by N07. The solar oxygen abundance is assumed to be ZO,⊙ = 5.6 × 10−3 by mass ratio (Lodders, 2003). Thus, for the solar oxygen abundance, the corresponding dust-to-gas ratio is given byD0 = ZO,⊙md/mO, where md is the dust mass ejected from a single SN II (0.14 M⊙; N07; H10). Here we assume that both oxygen and dust are supplied from SNe II. The grain size distribution is normalized so that the total grain mass density integrated for all the size range, ρdust, is equal to1.4 n HmHD0, where mH is the mass of hydrogen atom, and the factor 1.4 is the correction for the species other than hydrogen.
As mentioned in Introduction, additional contribution from AGB stars does not change the following results significantly as long as AGB stars also supply large (a ≳ 0.1 μm) grains (see also Asano et al., 2013b). In other words, the use of N07’s results for the initial condition is aimed at investigating the important role of shattering in the production of small grains from large grains.
We calculate the shattering processes in the same way as in H10 by solving the shattering equation. The collision frequency between grains with various sizes is determined by grain-size-dependent velocity dispersions. The shattering equation used in H10 is based on Hirashita and Yan (2009) (originally taken from Jones et al., 1994, 1996). Shattering is assumed to take place if the relative velocity between a pair of grains is larger than the shattering threshold velocities, vshathat (2.7 and 1.2 km s−1 for silicate and graphite, respectively; Jones et al., 1996). The results are not sensitive to vshat as long as the grain velocities driven by turbulence is much larger than the threshold, but are rather sensitive to the hardness of the grain materials as shown later.
We consider a range amin = 3 × 10−8 cm (3 Å) and amax = 3 × 10−4 cm (3 μm) for the grain radii. The grains passing through the boundary of the smallest radius are removed from the calculation. Most grain models adopt a minimum grain radius of a few × 10−8 cm (Weingartner and Draine, 2001; Guillet et al., 2009) although applying bulk material properties to such small grains could cause a large uncertainty. However, even if we adopt amin = 10−7 cm, the results does not change, except that the grain size distribution is truncated at 10−7 cm. This is because shattering occurs in a top-down manner in the grain sizes and the production of grains with a ~ a few × 10−8 cm by shattering is never enough for these small grains to have a large contribution to the total shattering rate.
Summary of experimental properties.
ρgr (g cm−3)
c0 (km s−1)
vshat (km s−1)
P1a (dyn cm−2)
3 × 1011
4 × 1010
Since the shattering equation is general enough, the major uncertainties can be produced by the treatment of shattering fragments. Thus, we explain how to treat shattering fragments below.
The relative velocity in the collision is estimated based on the grain velocity dispersion as a function of grain radius a. We used the calculation by Yan et al. (2004), who consider the grain acceleration by hydrodrag and gyroresonance in magnetohydrodynamic turbulence, and calculate the grain velocities achieved in various phases of ISM. Among the ISM phases, we focus on the warm ionized medium (WIM) to investigate the possibility of efficient shattering in actively star-forming environments. We adopt nH = 1 cm−3 for the hydrogen number density of the WIM. The resulting grain size distribution is not sensitive to nH (H10). Indeed, if nH is large, the grain–grain collision rate (i.e., the shattering efficiency) rises, while dust grains are more destroyed in SNe (i.e., the dust-to-gas ratio is lower). Because of these compensating effects, the resulting grain size distribution is not sensitive to the gas density. We adopt gas temperature T = 8000 K, electron number density ne = nH, Alfvén speed VA = 20 km s−1 and injection scale of the turbulence L = 100 pc. For grains with a ≳ 0.1 μm, where most of the grain mass is contained in our cases, the grain velocity is governed by gyroresonance. Since there is still an uncertainty in the typical grain radius (ac) above which gyroresonance effectively works, we also address the variation of results by ac (Subsection 2.4.4).
The grain velocities given above are velocity dispersions. In order to estimate the relative velocity (v12) between the grains with m1 and m2, whose velocity dispersions are v1 and v2, respectively, and we average the size distributions of fragments for four cases v12 = v1 + v2, ǀv1 − v2ǀ, v1, and v2 to take the directional variety into account. See H10 for more details.
2.3 Duration of shattering
Since we consider shattering in the WIM, it is reasonable to assume that the shattering duration (denoted as t) is determined by a timescale on which the ionization is maintained. A typical lifetime of ionizing stars is ≲ 10 Myr (Bressan et al., 1993; Inoue et al., 2000). We basically adopt t = 10 Myr in this paper, but we also consider longer durations for a longer starburst duration or shattering over multiple star-burst episodes.
P1 (dyn cm−2)
3 × 1011
4 × 1010
3 × 1011
4 × 1010
3 × 1011
4 × 1010
1 × 1011
1.3 × 1010
9 × 1011
1.2 × 1011
3 × 1011
4 × 1010
3 × 1011
4 × 1010
3 × 1011
4 × 1010
2.4.1 Ejected fraction of shocked material
How much fraction of the shocked material is finally ejected as fragments is uncertain. This factor is denoted as f M in Eq. (2). According to Jones et al. (1996), f m is between 0.3 and 0.6, while H10 adopted 0.4. Thus, we adopt 0.4 as a fiducial value (Model A) and also examine 0.3 (Model B) and 0.6 (Model C).
2.4.2 Critical pressure
Considering the uncertainty in the actual materials of astronomical grains, we vary P1 by an order of magnitude as Models D and E in Table 2. The standard value adopted from Jones et al. (1996) is P1 = 3 × 1011 and 4 × 1010 dyn cm−2 for silicate and graphite, respectively (Table 1; Model A in Table 2). We adopt three times smaller and larger values for P1 in Models D and E, respectively. We call grains with low/high P1 soft/hard grains.
Now we relate to P1. The dominant channel for the small grain production is cratering of large grains by small grains. If we assume that m1 ≫ m2 (and as long as is not much larger than unity), we obtain . On the other hand, if we approximate Eq. (2) as Mej/m2 ≃ A(ρgrv2/P1), A ~ 1 for the range of quantities in this paper. Therefore, if the two models are equivalent, .
Note that we have adopted the approximation, Mej ~ ρgrv2/P1. This is a good approximation for 4sϕ1 < 1 ; see Subsection 2.2). However, if ϕ1 ≪ 1, Mej/M2 ~ (ρgrv2/P1)/ϕ1 for Eq. (2), so that gives a better approximation for ϕ1 ≪ 1. In other words, if ϕ1 is significantly smaller than 1 (i.e., for soft dust), Eq. (9) overestimates by a factor of ϕ1. Thus, if we calculate shattering by adopting Eq. (9), grains are shattered less compared with Jones et al.’s case for soft dust. In this paper, because it is easy to interpret Eq. (9), we simply adopt it for KT10’s formulation.
2.4.3 Size distribution of shattered fragments
The power index of the size distribution of shattered fragments, αf, may be reflected in the grain size distribution after shattering. The canonical value adopted in H10 is αf = 3.3, which is based on Jones et al. (1996). Here we also examine the cases of αf = 2.3 and 4.3 (Models F and G, respectively), considering a wide range obtained in experiments (Takasawa et al., 2011).
2.4.4 Grain velocities
3.1 Ejected fraction of shocked material
We examine the effect of f M on the grain size distribution. In Fig. 2, we show the grain size distributions at t = 10 Myr for Models A, B, and C. In this paper, the grain size distributions are presented by multiplying a4 to show the mass distribution in each logarithmic bin of the grain radius. We observe that the production of small grains is the most efficient for the largest f M . This is simply because a larger fraction of the shocked material is ejected as fragments. However, the abundance of the smallest grains is not simply proportional to f M , since the catastrophic disruption does not depend on f M . The maximum ratio of n (a) between f M = 0.3 and 0.6 is 1.3 for silicate and 1.6 for carbonaceous dust.
3.2 Critical pressure
To show the dependence on the treatment of total cratering volume, we also show the results calculated by the simple formulation of KT10 (see Subsection 2.4.2 for the formulation) in Fig. 3. We observe that KT10’s formulation is in fair agreement with our formulation based on Jones et al. (1996) for the fiducial (Model A) and hard dust (Model E), but the discrepancy is relatively large for the soft dust (Model D) because Eq. (9) is not a good approximation for the soft dust (see Subsection 2.4.2 for detailed discussions). The maximum ratio between the two formulations in Model D is 1.8 for silicate and 2.2 for carbonaceous dust. For carbonaceous dust, the maximum difference occurs around the complicated feature around a ~ 10−5 cm, but if we focus on the small grain production at a < 10−6 cm, the ratio is 1.4. Thus, the uncertainty caused by the different formulations is just comparable to that caused by the difference in f M . If we adopt the standard values for P1 (i.e., Model A), it is concluded that the small grain production by shattering is described by the simple formulation by KT10 as precisely as more complicated framework of Jones et al. (1996).
3.3 Size distribution of shattered fragments
There is a qualitative difference in the evolution of grain size distribution between αf = 2.3 and 4.3. For αf = 2.3, the slope of the grain size distribution at a ≲ 10−6 cm changes, while for αf = 4.3, it evolves with a constant slope at small sizes. In the former case, the slope of the grain size distribution seems to approach n(a) ∝ a−3.5. The variation of slope occurs in a top-down manner; for example, for carbonaceous dust, the grain size distribution is nearly n(a) ∝ a−3.5 between a = 10−6 and 10−5 cm at 40 Myr, while it approaches n(a) ∝ a−3.5 at smaller sizes later at 80 Myr. This top-down behavior is because of the nature of shattering, which disrupts large grains into a lot of small pieces. This trend is also seen for αf = 3.3. Generally, if αf ≲ 3.5, the slope of grain size distribution tends to approach the MRN value (−3.5). On the contrary, if αf ≳ 4 (see also Subsection 4.2), the resulting grain size distribution is determined by the grain size distribution of shattered fragments. Indeed, if αf > 4, the smallest particles, which are too small to disrupt large grains around the peak (i.e., a ≳ 10−5 cm), have the dominant contribution to the total fragment mass. Moreover, we remove the grains reaching the smallest-size bin (a = 3 × 10−8 cm), and these grains do not affect the subsequent evolution of grain size distribution. Thus, for αf > 4, the fragments just keep their grain size distribution and eventually lost in reaching the smallest grain size.
We note that continuous shattering for such a long time as 80 Myr is not applicable to the real WIM, since the lifetime of the WIM is probably shorter than 80 Myr. Yet, the above calculations for the long shattering durations show that, if grains are repeatedly shattered with αf ≲ 3.5, the final grain size distribution can approach a power law with an index of ~ −3.5. Therefore, shattering is a promising mechanism to produce an MRN-like grain size distribution. This issue is further discussed in Subsection 4.2.
3.4 Grain velocities
In Fig. 6, we also show the results of Model A as a reference. Since the grain velocities at a < 10−5 cm drops below 20 km s−1 for Model A (Fig. 6), Model H with ac = 10−5 cm produce a more similar grain size distribution to Model A than Model H with ac = 10−6 cm. Thus, the effect of ac (i.e., the minimum size of grain size accelerated to the maximum velocity) is more important than the detailed functional form of v(a) at a < ac.
4.2 Sensitivity to the parameters
The power index of fragment size distribution, αf, is simply reflected in the resulting grain size distribution for αf = 4.3. However, if αf is smaller than 3.5 (e.g., the case with αf = 2.3), the slope of grain size distribution at a ≲ 10−5 cm continuously “steepens” and approaches to the slope consistent with MRN [n(a) ∝ a−3.5] (Fig. 4). According to the analytical studies of shattering by Hellyer (1970), the power index of grain size distribution after shattering has a steady state value of ≃ −3.5 if the size distribution of shattered fragments is assumed to be a power law with an index smaller than 4 (i.e., αf < 4) (see also Tanaka et al., 1996), although we should note that the relative velocity between grains is a complicated function of grain sizes in our case. For αf > 4, since the mass occupied by the smallest fragments is dominating, the size distribution after shattering is just dominated by the smallest fragments, keeping a power law index of −αf at small sizes.
In general, larger grains have larger velocity dispersions if they are accelerated by turbulence, primarily because they are coupled with larger-scale turbulent motion which has larger velocity dispersions. Thus, even if the initial grain size distribution is biased to large grains with a ≳ 10−5 cm, these grains are efficiently shattered to produce a large abundance of small grains. If these shattered grains have also larger velocities than shattering threshold, they collide with each other, accelerating the production of small grains. Thus, as shown in Fig. 6, the effect of ac (or the critical grain radius above which the grains have larger velocities than the shattering threshold) can be seen in the grain size distribution, causing a deviation from a monotonic power-law-like functional form. The imprint of ac is more clear in carbonaceous dust than in silicate dust, since the former species is softer. Interestingly, Weingartner and Draine (2001) have shown that grain size distributions derived from the Milky Way extinction curves are complicated for carbonaceous dust, which may be explained by imprints of ac.
4.3 Implication for the evolution of grain size distribution in galaxies
Inoue (2011) shows theoretically that the overall dust mass in metal-enriched systems such as the Milky Way is regulated by the balance between dust growth in molecular clouds and dust destruction by sputtering in SN shocks. While both these two mechanisms deplete small grains (Hirashita and Nozawa, 2013), shattering is a unique mechanism that reproduces the small grains. Indeed, Hirashita and Nozawa (2013) show that, unless we consider shattering, we underproduce the small grain abundance in the Milky Way (see also Asano et al., 2013b). As shown in Subsection 3.3, shattering not only produces small grains but also has an inherent mechanism of steepening the grain size distribution; that is, even if the grain size distribution of shattered fragments has αf smaller than 3.5, the resulting grain size distribution has a steeper dependence, approaching a slope consistent with the one derived by MRN [i.e., n(a) ∝ a−3.5]. Thus, the robust prediction is that, if shattering is the dominant mechanism of small grain production, the grain size distribution approaches an MRN-like grain size distribution as long as αf ≲ 3.5. And if shattering is the dominant mechanism of small grain production, αf > 4 is rejected, because the grain size distribution in the Milky Way ISM has a power index of −3.5 (MRN).
In this paper, we have not included dust production by AGB stars. Even if the dust supply from AGB stars has a significant contribution, the results in this paper are not altered as long as the dust grains formed by AGB stars are biased to large (≳ 0.1 μm) sizes. Rather, the importance of shattering is emphasized because the additional dust production by AGB stars enhances the dust abundance, making grain-grain collision more frequent. Production of large grains from AGB stars is indicated observationally (Groenewegen, 1997; Gauger et al., 1998; Norris et al., 2012) and theoretically (Hüfner, 2008; Mattsson and Hüfner, 2011). We also neglected grain growth in molecular clouds, which is suggested to be a major dust formation mechanism even at high redshift (Mattsson, 2011; Valiante et al., 2011). Grain growth, however, cannot be a supplying mechanism of small grains.
For the shattering duration, since it is degenerate with the critical pressure of grains (hardness of grains), it is difficult to constrain it directly from the models. If grains experience shattering for several tens of Myr, the grain size distribution possibly approaches a power law whose power index is consistent with the MRN, if αf ≲ 3.5 (Fig. 5). Since the timescale of ISM phase exchange is comparable or shorter than that required for the grain size distribution to approach the MRN size distribution (O’Donnell and Mathis, 1997), shattering may occur intermittently. Even in this case, it is expected that grains are relaxed into an MRN-like simple power law if the total shattering duration reaches several tens of Myr. Such a duration for shattering is probable since it is much shorter than the grain lifetime (~ a few × 108 Myr; Jones et al., 1996).
Shattering is a viable mechanism of small grain production in the ISM both in nearby galaxies and in high-redshift galaxies. We have examined if grain size distributions predicted by shattering models are robust against the change of various parameters and formulations. Because of the uncertainty in f M (the fraction of the shocked material that is eventually ejected as fragments), the predicted grain size distribution after shattering is uncertain by a factor of 1.3 (1.6) for silicate (carbonaceous dust). We have identified P1> (the critical pressure above which the original lattice structure is destroyed) as the most important quantity in determining the timescale of small grain production, and confirmed that the same P1/t (t is the duration of shattering) gives roughly the same grain size distributions after shattering within a factor of 3.
A simpler and more intuitive formulation by KT10 is in good agreement with our model based on Jones et al. (1996) within a factor of 1.8 (1.4) for silicate (carbonaceous dust) if we focus on the small grain production at a ≲ 10−6 cm. This is as small as the uncertainty caused by f M . Thus, as long as we have an uncertainty in f M , it is sufficient to adopt the simpler formulation by KT10. The size distribution of shattered fragments have minor effects as long as αf ≲ 3.5, since the grain size distribution is continuously steepened by shattering and become consistent with the MRN grain size distribution regardless of the value of αf (≲ 3.5).
The effect of the grain velocities as a function of grain radius can be seen more clearly in carbonaceous grains than in silicate, since the former species is shattered more easily. Thus, it is predicted that carbonaceous species has more complicated grain size distribution than silicate; in other words, the grain size distribution of carbonaceous dust can show some imprints of the grain velocity as a function of grain radius.
We are grateful to anonymous referees for helpful comments. HH has been supported through NSC grant 99-2112-M-001-006-MY3. HK gratefully acknowledges the support from Grants-in-Aid from MEXT (23103005).
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