Intergalactic dust and its photoelectric heating
© The Society of Geomagnetism and Earth, Planetary and Space Sciences, The Seismological Society of Japan 2010
Received: 31 July 2008
Accepted: 29 October 2008
Published: 7 February 2015
We have examined dust photoelectric heating in the intergalactic medium (IGM). The heating rate in a typical radiation field of the IGM is represented by Γpe=1.2x10-34 erg s-1 cm-3 (D/10-4)(nH/10-5 cm-3)43(JL/10-21 erg s-1 cm-2 HZ-1 sr-1)2/3(T104 K)-1/6), where D is the dust-to-gas mass ratio, nH is the hydrogen number density, JL is the mean intensity at the hydrogen Lyman limit of the background radiation, and T is the gas temperature, if we assume the new X-ray photoelectric yield model by Weingartner et al. (2006) and the dust size distribution in the Milky Way by Mathis et al. (1977). This heating rate dominates the HI and HeII photoionization heating rates when the hydrogen number density is less than ∼10-6 cm-3 if D = 10-4 which is 1% of that in the Milky Way, although the heating rate is a factor of 2–4 smaller than that with the old yield model by Weingartner and Draine (2001). The grain size distribution is very important. If only large (≥0.1 μm) grains exist in the IGM, the heating rate is reduced by a factor of ≏5. Since dust heating is more efficient in a lower density medium relative to the photoionization heating, it may cause an inverted temperature-density relation in the low-density IGM, as suggested by Bolton et al. (2008). Finally, we have found that dust heating is not very important in the mean IGM before the cosmic reionization.
Dust grains are formed at the end of the stellar life, in the stellar wind of asymptotic giant branch stars (e.g., Ferrarotti and Gail, 2006), in the stellar ejecta of supernovae (e.g., Nozawa et al., 2003; Rho et al., 2008), among others. Some of the grains grow in molecular clouds (e.g., Draine, 1990), others are destroyed by the interstellar shock (e.g., Williams et al., 2006), and some of them may go out from the parent galaxy and reach the intergalactic medium (IGM) (e.g., Aguirre et al., 2001a, 2001b).
The IGM is the medium between galaxies, and it occupies almost the whole volume of the Universe. The mean density of the IGM is as low as 10-7-10-4 cm-3. As found by Gunn and Peterson (1965), the IGM is highly ionized after the cosmic reionization epoch (the redshift z ≏ 6–10; Loeb and Barkana, 2001; Fan et al., 2006). Thus, its temperature is ∼104 K. The IGM is filled with the ionizing ultra-violet (UV) and X-ray background radiation which is produced by QSOs and galaxies (e.g., Haardt and Madau, 1996).
A significant amount of metals is found in the IGM (e.g., Aguirre et al., 2001c). Multiple supernova explosions (SN) caused by an active star-formation in galaxies can eject the metal elements to the IGM. However, Ferrara et al. (2000) showed that the metal enrichment of the IGM by SN explosions is limited to relatively small regions around starforming galaxies, and an additional physical mechanism is required to explain the observed global enrichment of metals in the IGM. Dust grains expelled from galaxies by the radiation pressure due to stellar light and by the galactic wind due to multiple SNe may contribute to the metal enrichment in the IGM (e.g., Aguirre et al., 2001a, 2001b). Bianchi and Ferrara (2005) showed that relatively large (>0.1 μm) dust grains are not completely destroyed and reach a significant distance (a few × 100 kpc) although the amount of this intergalactic dust is too small to make a detectable extinction.
Infrared (IR) emission from dust grains in the IGM surrounding edge-on galaxies has been already detected (e.g., Alton et al., 1999; Bendo et al., 2006). Moreover, IR emission from dust in the IGM accumulated from the distant Universe may affect the cosmic far-IR background and the cosmic microwave background (Aguirre and Haiman, 2000). Emission signature from dust even at the epoch of the cosmic reionization may be detectable with a future satellite observing the cosmic microwave background (Elfgren et al., 2007).
Xilouris et al. (2006) found a significant reddening of galaxies behind a giant cloud detected by HI 21 cm emission in the M81 group (e.g., Yun et al., 1994). Their measurements imply that the dust-to-gas ratio in the M81 group IGM is a factor of 5 larger than that in the Milky Way. Such a large amount of dust in the IGM may be ejected from M82 by its intense starburst activity (Alton et al., 1999).
Dust in the IGM affects results from the precision cosmology. Indeed, high redshift SNe Ia are dimmed by dust in the IGM, conseqently, the observational estimate of the distance to them and cosmological parameters become ambiguous (Goobar et al., 2002). Furthermore, future investigations of the ‘equation of state’ of the Dark energy will be affected by the extinction of the intergalactic dust even if its amount is too small to affect the conclusion of the presence of the Dark energy (Corasaniti, 2006; Zhang and Corasaniti, 2007).
Dust in the IGM also affects the thermal history of the IGM. In the intracluster medium, dust grains work as a coolant because they emit energy obtained from gas particles collisionally as the thermal IR radiation (Montier and Giard, 2004). Such an emission from some nearby galaxy clusters can be detectable with the current and future satellites for the IR observations (Yamada and Kitayama, 2005). Dust grains in the IGM also work as a heating source via the photoelectric effect (Nath et al., 1999). Inoue and Kamaya (2003, 2004) proposed the possibility of obtaining an upper limit of the amount of the intergalactic dust based on the thermal history of the IGM with the dust photoelectric heating.
In this paper, we revisit the effect of the dust photoelectric heating in the IGM. Recently, Weingartner et al. (2006) revised the model of the photoelectric yield of dust grains. They included a few new physical processes; the photon and electron transfer in a grain, the photoelectron emission from the inner shells of the constituent atoms of grains, the secondary electron emission, and the Auger electron emission. These new features reduce the photoelectric yield for moderate energy photons of ∼100 eV but enhance the yield for high-energy photons >1 keV. In particular, we explore the effect of this new yield model on the photoelectric heating by the intergalactic dust in this paper.
The rest of this paper consists of four sections; in Section 2, we describe the model of the photoelectric effect. In Section 3, we compare heating rates of the photoelectric effect with those of the photoionization in the IGM. In Section 4, we discuss the implications of the results of Section 3. Final section is devoted to our conclusions.
2. Dust Photoelectric Effect
2.1 Grain charging processes
2.1.1 Collisional charging rate
In fact, the “image potential” works to enhance the collisional cross section (Draine and Sutin, 1987). Although the effect becomes the most important for grains with an around neutral charge, it quickly declines for highly charged grains. Indeed, for the charge ratio of Zd/Zi < -3, which is satisfied in our case, as found below, the increment factor for the cross section by the effect of the “image potential” is less than 1.5 (Draine and Sutin, 1987). Therefore, we neglect the “image potential” in this paper.
2.1.2 Photoelectric charging rate
We have to note that there is still a large uncertainty of photoelectric yield models because of our insufficient understandings of the nature of the small-size particle effect as well as the lack of experiments. Abbas et al. (2006) reported measurements of the yield of individual grains of silica, olivine, and graphite with 0.09–5 μm radii for 8–10 eV photons. Their measurements indeed show larger yields than those of the bulk materials. However, the measurements do not agree with the yield enhancement factors adopted in the WD01 andW+06 models accounting for the small-size particle effect qualitatively as well as quantitatively. Clearly, we need more experiments and theoretical investigations of the photoelectric yield in future.
2.1.3 Equilibrium charge
We need to specify the radiation filed incident on grains in the IGM: the cosmic background radiation. We assume a simple description of the radiation. The intensity of the radiation at the Lyman limit is estimated from observations of the proximity effect and the Lyman a forest opacity (e.g., Scott et al., 2000). A typical value of the intensity at the Lyman limit is JL = 1 × 10-21 erg s-1 cm-2 Hz-1 sr-1 (e.g., Scott et al., 2000). We simply assume a power-law as the spectral shape: Jυ ∝ υ-p. A typical value of p is unity (e.g., Haardt and Madau, 1996). With such a radiation field, the grains in the IGM are positively charged.
A typical charging time-scale is very short. For example, the collisional charging rate of the electron is Re ∼ 5.6 × 10-6 s-1 for ne = 10-5 cm-3, T = 104 K, a = 0.1 μm, and Zd = 1700, which is the equilibrium charge of graphite or silicate grains for these parameters and Jυ = 10-21(υ/υL)-1 erg s-1 cm-2 Hz-1 sr-1. Thus, the typical charging time-scale is t ∼ 1/Re ∼ 6 × 10-3 year. Therefore, the grain charge can be in equilibrium. We set dZd/dt = 0 in Eq. (1) and obtain the equilibrium charge of the IGM grains.
2.2 Heating rates
2.2.1 Heating rate per a grain
2.2.2 Total photoelectric heating rate
2.2.3 Photoionization heating rates
3.1 Comparison between the two yield models
We compare the grain charge and heating rates with the WD01 and W+06 models quantitatively in the IGM environment. Weingartner et al. (2006) showed the grain charges in the QSO environments in a similar situation and similar radiation field as those reported in this paper. However, they did not show the heating rates in the environment.
3.1.1 Electric potential
In Fig. 2, we compare the electric potentials of the W+06 model (solid lines) with those of the WD01 model (dashed lines). We show two cases of the spectrum of the radiation field; one has a hard spectrum as a background radiation dominated by QSOs, which is the case with the spectral index p = 1, and the other has a soft spectrum with p = 5 for a comparison. Other assumed quantities are appropriate for the IGM at the redshift z ∼ 3 and are shown in the panels. The radiation fields assumed here correspond to the ionization parameter U = nion/nH, which is the number density ratio of ionizing photons and hydrogen nucleus, of 6.3 for p = 1 and of 1.3 for p = 5. Weingartner et al. (2006) showed electric potentials in their figures 6 and 7, with U = 0.1–100. We find that our calculations are quantitatively well-matched with theirs.
We find in Fig. 2 that for the hard spectrum case, the grain electric potentials with the W+06 yield model are much smaller than those with the WD01 model, especially for larger grain sizes. On the other hand, for the soft spectrum case, the difference is very small, less than 4%. This is because the main difference between the W+06 yield and the WD01 yield is found in the primary photoelectron The electrostatic stress on a grain yield at ∼100 eV due to the photon/electron transfer in a grain as shown in Fig. 1. In the soft spectrum case, since there are not many photons around the energy, we do not find a significant difference between the two yield models. For smaller grain sizes, the yield reduction by the photon/ electron transfer is small as found in figures 4 and 5 of Weingartner et al. (2006). Thus, we do not find a significant difference in the electric potentials for smaller grain sizes in Fig. 2 either.
The electrostatic stress on a grain may cause the grain destruction by the Coulomb explosion (e.g., Draine and Salpeter, 1979). The critical electric potential is Vmax = 1063 V (Sd/1010 dyn cm-2)½(a/0.1 μm), where Sd is the tensile strength of grains, which is very uncertain. Perfect crystal structure may have Sd ∼ 1011 dyn cm-2 (Draine and Salpeter, 1979), but imperfections would reduce the strength as Sd ∼ 1010 dyn cm-2 (Fruchter et al., 2001). Following Weingartner et al. (2006), we show two cases of the critical potential with Sd ∼ 1010 and 1011 dyn cm-2 in Fig. 2 as the dotted lines. The critical potential by the ion field emission is similar to the case with Sd ∼ 1011 dyn cm-2 (Draine and Salpeter, 1979). We find that grains smaller than 20–30 Å in the hard radiation field may be destroyed by the Coulomb explosion. As such, there may be no very small grains in the IGM.
3.1.2 Photoelectric heating rate
Figure 3 shows the photoelectric heating rate per grain in a typical z ∼ 3 IGM environment with hard radiation; graphite grains are shown by solid lines and silicate grains are shown by the dashed lines. In panel (a), we show the absolute value of the heating rate for the W+06 yield model. As expected in Eq. (9), the heating rate is nicely proportional to a2, square of the size. However, the slope becomes gradually steep for a small (<100 Å ) grain size.
In panel (b), we compare the two heating rates with the W+06 model and the WD01 model. We find that the heating rate with the W+06 model is much smaller than that with the WD01 model: a factor of 10 smaller for the largest grain size. This is because (1) reduction of the grain electric potential and (2) reduction of the mean energy of the photoelectron in the W+06 model. As found in Eq. (9), the heating rate per grain is proportional to the product of the potential and the mean photoelectron energy. As seen in Fig. 2, smaller potential is expected with the W+06 model because of the reduction of the yield at ∼100 eV. The yield reduction also causes the reduction of the mean energy of the photoelectron, as expected in Eq. (8). Therefore, we have up to about a factor of 10 reduction of the heating rate with the W+06 model.
For a comparison, we also show the HI and HeII photoionization heating rates in Fig. 4. We have assumed the ionization equilibrium for these. When we assume the dust-to-gas ratio in the IGM is 1% of that in the Milky Way (i.e., D = 10-4), the dust photoelectric heating dominates the HI and HeII photoionization heatings if the hydrogen number density is less than 10-6-10-5 cm-3, which corresponds to the redshift z ∼ 1–2. We note that the dust heating is the most important mechanism in the IGM at z = 0 even with the W+06 yield model if the IGM has dust with 1% dust-to-gas ratio of the Milky Way and with the MRN size distribution.
3.2 Effect of the grain size distribution
The size distribution of the intergalactic dust grains should be important for the photoelectric heating rate via the typical size defined by Eq. (12). However, it is quite uncertain. Thus, we examine several possibilities of the size distribution in this section. Table 1 is a summary of the size distribution considered here.
The grain size distribution in the Milky Way has been approximated to be a power-law since Mathis et al. (1977) suggested n(a) σ a -q with q = 3.5. This MRN distribution is a reference case and is already adopted in Fig. 4. During the grain transport from galaxies to the IGM, there may be size-filtering mechanisms. For example, Ferrara et al. (1991) showed that sputtering in the hot gas filling the galactic halo efficiently destroys grains smaller than ∼0.1 μm. Bianchi and Ferrara (2005) also showed that only grains larger than ∼0.1 μm reach a significant distance (a few × 100 kpc) from the parent galaxies by calculating the grain ejection by the radiation pressure and the grain destruction by the sputtering simultaneously. Here, we consider a simple size distribution of the MRN with ≥0.1 μm grains as the BF05 model.
In the early Universe, the dominant source of dust grains is different from that in the current Milky Way. Although asymptotic giant branch stars are considered to be the main dust source in the Milky Way (e.g., Dwek, 1998), there is not enough time for stars to evolve to the phase in the early Universe at the redshift z > 6. However, plenty of dust is found in QSOs at z > 6 (Bertoldi et al., 2003). SNe is the candidate of the dust source in the early Universe (e.g., Nozawa et al., 2003), and the observed extinction curve of dust associated with the QSO is compatible with those expected from the grains produced by SNe (Maiolino et al., 2004; Hirashita et al., 2005, 2008). Thus, we consider the size distribution expected from the SNe dust production model by Nozawa et al. (2003) as the N03 model. In addition, we adopt the size distribution expected by Nozawa et al. (2007), who explored the effect of the dust destruction by the reverse shock in the SN remnant, as the N07 model.
Finally, we adopt a hypothetical size distribution consisting of only small grains as a comparison case; the MRN distribution with the maximum size of 250 Å as the SG (small grain) model.
3.3 A simple formula of the dust photoelectric heating rate
The deviation of the heating rates from the formula for T = 105 K and nH > 2 × 10-5 cm-3 is due to the relative significance of the cooling by the electron capture (see Eq. (6)). Indeed, we find that the mean energy of photoelectrons from small (<0.01 μm) graphite grains is smaller than the mean kinetic energy of the 105 K gas in the case of nH = 10-4 cm-3, consequently, the net heating rate per such a graphite grain is negative. Thus, we have a reduction of the total heating rate for T = 105 K and nH > 2 × 10-5 cm-3 as found in Fig. 6(b) although the heating rate by silicate grains is still positive.
The validity of the formula presented in Eq. (16) is ensured for nH = 10-7-10-4 cm-3, JL = 10-23-10-20 erg s-1 cm-2 Hz-1 sr-1, and T = 103-105 K within a uncertainty of 30%, except for nH > 2 × 10-5 cm-3 with JL = 10-23 erg s-1 cm-2 Hz-1 sr-1 or T = 105 K. Note that there may be a much larger uncertainty in the photoelectric yield model. If one likes another size distribution rather than the standard MRN, for example the BF05 model discussed in Section 3.3, the heating rate might be scaled by a factor found in Fig. 5 or the ratio of the typical sizes in Table 1.
4.1 Amount of the intergalactic dust
Inoue and Kamaya (2003, 2004) discussed the effect of the photoelectric heating by the intergalactic dust on the thermal history of the IGM, obtaining an upper limit of the intergalactic dust amount. However, we have already seen that the W+06 yield model results in a factor of 2–4 reduction of the photoelectric heating rate relative to the WD01 model which was adopted in Inoue and Kamaya (2003, 2004). We can conclude that the upper limits obtained from the IGM thermal history are raised by a few factor. Even in this case, the final limit obtained by Inoue and Kamaya (2004), which is that the intergalactic dust mass should be less than 10% of the metal mass produced in galaxies, is not affected because the limit was obtained mainly from the reddening measurements of SNe Ia at z = 0.5, especially for ∼0.1 μm size grains.
4.2 Can grains cause an inverted temperature-density relation in the IGM? Bolton et al. (2008) recently suggest an inverted temperature-density relation in the low density IGM at z = 2–3. The temperature in the low density IGM was previously thought to be proportional to the density positively (e.g., Hui and Gnedin, 1997). However, Bolton et al. (2008) examined carefully the probability distribution function (PDF) of the flux in QSOs’ spectra through the Lyman a forest in the IGM and found that the observed PDF is explained better by the negatively proportional temperaturedensity relation; i.e., a lower density IGM is hotter. This needs a more efficient heating source for lower density IGM. Bolton et al. (2008) suggested a radiation transfer effect (e.g., Abel and Haehnelt, 1999) for the mechanism.
The intergalactic dust may contribute to the heating in the low-density IGM. As shown in Figs. 4 and 5, the importance of the dust photoelectric heating increases in lower density medium, which is plausible for the inverted temperaturedensity relation. For example, we expect a factor of ∼2 larger heating rate by dust than HeII photoionization heating in a medium with 1/10 of the mean density at z = 2 for the MRN size distribution and 1% dust-to-gas ratio of the Milky Way. Thus, the dust photoelectric heating may cause the inverted temperature-density relation observed in the low-density IGM at z = 2–3. This point should be examined further by implementing the dust heating in a cosmological hydrodynamics simulation. For this, the formula presented in Eq. (16) will be useful.
4.3 Photoelectric effect before the cosmic reionization
Finally, we examine if the dust photoelectric heating is efficient in the IGM before the cosmic reionization. Because of the prominent Gunn-Peterson trough in QSOs’ spectra (e.g., Fan et al., 2006), the cosmic reionization epoch should be at z > 6. Here, we consider the IGM at z ∼ 10.
In the nonionizing UV radiation-only case, there is no efficient heating mechanism for the whole of the Universe although primordial objects can heat up their surrounding gas locally. Thus, the temperature of the gas far away the sources is kept to be that of the cosmic background radiation at the epoch: ∼30 K. The electron fraction xe, i.e., the number density of electron relative to that of hydrogen nucleus, is ∼10-4 in this low temperature IGM (Galli and Palla, 1998). The nonionizing UV photons still cause the photoelectric effect of grains. In the assumed setting, we have found that grains are positively charged and the dust photoelectric heating rate becomes Γpe ≏ 7 × 10-36 erg s-1 cm-3 for the MRN size distribution with a graphite and silicate mixture (50% each in mass). We compare this heating rate with the gas thermal energy density: Ugas = (3/2)nHkBT. The time-scale doubling the gas temperature with the photoelectric heating is given by tpe = Ugas/Γpe ≏ 9 × 109 years. Since the age of the Universe at z = 10 is about 5×108 years, we conclude that the dust photoelectric heating is not very efficient in this case although it may be the strongest heating mechanism for the IGM.
In the case with the additional X-ray background radiation, the IGM is partially ionized by the X-ray and the temperature becomes ∼104 K (e.g., Venkatesan et al., 2001). If we assume the ionization equilibrium and optically thin for the X-ray, the electron fraction becomes xe ≏ 0.3 for the current setting of the X-ray background. In this medium, the grains are positively charged and the dust photoelectric heating rate becomes δpe ≏ 2 × 10-33 erg s-1 cm-3. We have assumed the MRN size distribution with a graphite and silicate mixture (50% each in mass) again. However, the HI photoionization heating is much more efficient as Γ pi,X HI ≏2x1030 erg s-1 cm-3. Therefore, we again conclude that the dust photoelectric heating is negligible in the early Universe filled with an X-ray background radiation.
We have updated our calculations made in Inoue and Kamaya (2003, 2004) of the dust photoelectric heating in the IGM with the new model of the dust photoelectric yield by Weingartner et al. (2006). This new yield model takes into account the effect of the photon and electron transfer in a grain, the photoelectric emission from inner shells of grain constituent atoms, the Auger electron emission, and the secondary electron emission. A comparison with the previous yield model by Weingartner and Draine (2001) shows that the new yield is smaller than the old one for ∼100 eV photons. This reduction of the yield is due to the photon/electron transfer effect and reduces the electric potential on grains and the heating rate significantly. For example, if we integrate over the grain size with the standard MRN distribution, the dust photoelectric heating rate with the new yield model is a factor of 2–4 smaller than that with the old yield model. The photoelectric heating rate is more important in lower density medium. If the dust-to-gas ratio in the IGM is 1% of that in the Milky Way and the size distribution is the standard MRN model, the dust heating rate dominates the HI and HeII photoionization heating rates when the gas number density is less than ∼10-6 cm-3, even with the new yield model.
We have examined the effect of the size distribution function on the heating rate because the heating rate is inversely proportional to the typical grain size as found in Eq. (11). Bianchi and Ferrara (2005) suggested that the size of the intergalactic dust is larger than ∼0.1 μm because smaller grains are destroyed by sputtering in the hot gas halo during the transport of grains from the parent galaxy to the IGM. In this case, the heating rate is reduced by a factor of ∼5 relative to that with the standard MRN size distribution. The size distributions expected by the dust formation model in supernova ejecta are also examined. The heating rate with the size distribution of the grains just produced in the ejecta is very similar to that with the MRN distribution. In contrast, the heating rate with the size distribution of the grains processed by the reverse shock in the supernova remnant is a factor of ∼3 smaller than that with the MRN model. The shock-processed grains have a larger size than the pristine ones because smaller grains are destroyed. On the other hand, if we put only small grains in the IGM, the heating rate increases significantly. Therefore, we conclude that the size distribution of grains in the IGM is an essential parameter for determining the dust heating efficiency. Even in the worst case considered here, the dust heating is expected to be the dominant heating mechanism in the IGM at z = 0 if the dust-to-gas ratio in the IGM is 1% of that in the Milky Way.
Since the dust photoelectric heating rate with the new yield model is reduced by a factor of 2–4 relative to that with the old yield model, the upper limit on the amount of the intergalactic dust obtained by Inoue and Kamaya (2003, 2004) may be affected. Indeed, the limit based on the thermal history of the IGM should be raised by a factor of a few. However, their final upper limit is mainly obtained from the reddening measurements of z = 0.5 supernovae Ia. Therefore, their conclusion would not be affected very much.
Bolton et al. (2008) suggested an inverted temperature-density relation in the lower density IGM at z = 2–3 based on recent observations of the Lyman α forest in QSOs’ spectra. To explain this interesting phenomenon, we need a heating mechanism more efficient in a lower density medium. The dust photoelectric heating has such a property. Indeed, the dust heating rate even with the new yield model is a factor of 2 larger than the HeII photoionization heating rate in a medium with a density of 1/10 of the mean in the Universe at z = 2 if the dust-to-gas ratio is 1% of that in the Milky Way. Thus, the possibility of the dust heating is worth examining more in detail. For this aim, the simple formula describing the dust photoelectric heating in the IGM presented in Eq. (16) will be very useful.
Finally, we have discussed the effect of the dust photoelectric heating in the early Universe. Prior to cosmic reionization, the ionizing background radiation is not established, but there may be nonionizing UV background and X-ray background radiations. In the low temperature IGM only with a nonionizing UV background radiation, the dust photoelectric heating is not very efficient although it may be the strongest heating mechanism in the medium. In the partially ionized IGM with an X-ray background radiation, the HI photoionization heating rate is three orders of magnitude larger than the dust heating rate if the dust-to-gas ratio is 1% of that in the Milky Way. Therefore, we conclude that the dust photoelectric heating in the early Universe is not very important at least in the mean density environment.
We appreciate comments from the reviewers, B. T. Draine and M. M. Abbas, which improved the quality of this paper very much. We are grateful to the conveners of the session “Cosmic Dust” in the 5th annual meeting of the Asia-Oceania Geosciences Society for organizing the interesting workshop. AKI is also grateful to all members of the Department of Physics, Nagoya University, especially the Ω Laboratory led by Tsutomu T. Takeuchi, for their hospitality during this work. AKI is supported by KAKENHI (the Grant-in-Aid for Young Scientists B: 19740108) by The Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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