# The influence of atomic alignment on absorption and emission spectroscopy

Heshou Zhang,1,2 Huirong Yan,1,2 Philipp Richter,2,3
1Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen,
Germany
2Institut f$\ddot{u}$r Physik und Astronomie, Universit$\ddot{a}$t Potsdam,
Haus 28, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany
3Leibniz-Institut f$\ddot{u}$r Astrophysik Potsdam (AIP), An der Sternwarte
16, D-14482 Potsdam, Germany E-mail<EMAIL_ADDRESS>
###### Abstract

Spectroscopic observations play essential roles in astrophysics. They are
crucial for determining physical parameters in the universe, providing
information about the chemistry of various astronomical environments. The
proper execution of the spectroscopic analysis requires accounting for all the
physical effects that are compatible to the signal-to-noise ratio. We find in
this paper the influence on spectroscopy from the atomic/ground state
alignment owing to anisotropic radiation and modulated by interstellar
magnetic field, has significant impact on the study of interstellar gas. In
different observational scenarios, we comprehensively demonstrate how atomic
alignment influences the spectral analysis and provide the expressions for
correcting the effect. The variations are even more pronounced for multiplets
and line ratios. We show the variation of the deduced physical parameters
caused by the atomic alignment effect, including alpha-to-iron ratio ([X/Fe])
and ionisation fraction. Synthetic observations are performed to illustrate
the visibility of such effect with current facilities. A study of PDRs in
$\rho$ Ophiuchi cloud is presented to demonstrate how to account for atomic
alignment in practice. Our work has shown that due to its potential impact,
atomic alignment has to be included in an accurate spectroscopic analysis of
the interstellar gas with current observational capability.

###### keywords:

ISM: magnetic fields – turbulence – (galaxies:) quasars: absorption lines –
(ISM:) HII regions – submillimetre: ISM – ultraviolet: ISM

††pubyear: 2018††pagerange: The influence of atomic alignment on absorption
and emission spectroscopy–A

##  1 Introduction

Spectroscopy plays a crucial role in studying the universe. Analysing atomic
and ionic spectral line intensity111Throughout the paper, we address atoms and
ions with the term ”atoms” for the sake of simplicity. is one of the most
important part of spectroscopic study. Absorption and emission atomic lines
have numerous applications in astronomy. They are the direct measurements of
column density of lots of elements (e.g., Fox et al. 2013; Richter et al.
2013; D’Onghia & Fox 2016), and thus help the analysis of chemical evolution
and composition of astronomical structures. Physical parameters derived from
the spectral line ratio (e.g., ionisation rate, temperature) facilitate us to
understand the astrophysical environments better (see, e.g., Draine 1978;
Hogerheijde et al. 1995; Salgado et al. 2016). Progress has been made in
modelling the physical processes, such as cloud extinction in
Photodissociation regions (PDRs) and outflows of Active Galactic Nuclei (AGN),
with atomic spectra (e.g., Kaufman et al. 1999; Coil et al. 2015; Youngblood
et al. 2016).

The rapid development of spectroscopic observations requires an accurate
measurement of the spectral lines that is compatible to the higher signal-to-
noise (S/N) ratio. Therefore, all relevant physical effects that are capable
to produce the spectral fluctuation larger than the noise amplitude have to be
counted for the high precision observations. We will demonstrate in this paper
the fluctuation of atomic/ionic lines from Interstellar Medium (ISM) induced
by ground state alignment (GSA) effect can have significant impact on the
physical parameter derivation. One may naively argue that the physical effect
will be averaged out given the possible line-of-sight dispersion of the
magnetic field line in the medium. Synthetic observations are performed
accounting for the averaging and the results show that such argument does not
hold. We shall demonstrate that the influence of GSA cannot be neglected for
an accurate spectral line intensity analysis with current observational
capability and provide analytical expressions for the correction.

This paper is organised as follows: The general physics for the modulation of
spectral lines due to GSA is briefly discussed in §2. We then demonstrate in
§3 the influence of radiative alignment on the spectrum, where the magnetic
field does not exist/vary. In §4, we further discuss the impact of magnetic
realignment, where the radiative geometry is fixed and the fluctuation of the
spectral lines originates from the change of magnetic field. In addition, the
influences of GSA on various astrophysical properties derived from spectral
line ratios, including alpha-to-iron ratio ([X/Fe]) and ionisation fraction,
are exhibited in §5. In order to illustrate how our modelling is applied to
real observation, we apply in §6 our analytical tool to the processing of raw
data from PDR in $\rho$ Ophiuchi cloud, showing how to remove the potential
impact that GSA produces. Discussions are provided in §7 with the
applicability of our modelling and conclusions are in §8.

##  2 Atomic transitions under the influence of GSA

Figure 1: (a) Theoretical frame $xyz-$coordinate system with the magnetic
direction being $z-$axis and line of sight on $xz-$plane. The calculation is
done in this frame. The direction of the anisotropic radiation is
$\Omega_{r}\equiv(\theta_{r},\phi_{r})$, and that of the line of sight is
$\Omega\equiv(\theta,\pi)$. (b) Observational frame
$x^{\prime}y^{\prime}z^{\prime}-$coordinate system with the line of sight
being $z^{\prime}-$axis, in which our results are presented.
$x^{\prime}y^{\prime}-$plane is the plane of sky. $\theta_{0}$ is the angle
between the direction of the incident radiation and the line of sight. The
change of magnetic direction alters $\theta$ and $\theta_{r}$.

Substantial part of ISM are photon-excitation dominant areas, including PDRs,
H ii Region and reflection nebulae, etc. The spectroscopy from those areas
will be influenced by GSA effect. The alignment here is in terms of the
projection of the angular momenta of the atoms and it arises from the
anisotropy in the radiation field. GSA happens when the optical/UV pumping
transfers angular momentum from the radiation field to the atoms/ions. The
atomic angular momentum is along the incident radiation direction. Using point
source as an example, the angular momentum of the photons that pump the atoms
from the ground state to upper states can only be $\pm 1$, whereas the
isotropic spontaneous emission from upper states allows the angular momentum
transfer $\pm 1,0$. Such difference leads to differential occupation among the
magnetic sublevels on the ground state (Landolfi & Landi Degl’Innocenti, 1986;
Yan & Lazarian, 2006; Zhang et al., 2015).

With the existence of magnetic field, which is ubiquitous in the universe,
occupations among different sublevels are mixed according to the magnetic
field direction due to the magnetic precession. The alignment is dependent on
the comparison between magnetic precession rate ($2\pi\nu_{L}$) and photon
excitation rate ($B\bar{J}^{0}_{0}$). For interstellar magnetic field, it is
in GSA (saturated) regime with the magnetic precession prevailing over the
photon excitation from the ground state ($2\pi\nu_{L}\gg
B_{lu}\bar{J}^{0}_{0}$), but still too weak to influence the excited level. It
has been demonstrated that for a wide range of magnetic field in ISM and IGM
($10^{-15}G\la B\la mG$), $A\gg 2\pi\nu_{L}\gg B_{lu}\bar{J}^{0}_{0}$ and thus
GSA applies (Yan & Lazarian, 2007, 2008; Shangguan & Yan, 2013; Zhang & Yan,
2018). We show in §7 that GSA is indeed the dominant regime by providing the
division boundary between the GSA (saturated) regime and the ground-level
Hanle regime, where magnetic field strength also matters.

We present below the expressions for the modulation of absorption and emission
coefficients 222Readers may directly go to §3 if they are only interested in
observational consequences.. The atomic density of different angular momentum
is depicted by the density matrix tensor, e.g., for the lower level $J_{l}$ it
is denoted as $\rho^{k}_{q}(J_{l})$333For example, the irreducible tensor for
$J/F=1$ is $\rho_{0}^{2}=[\rho(1,1)-2\rho(1,0)+\rho(1,-1)]$ (see, e.g., Fano
1957; D’Yakonov & Perel’ 1965; Bommier & Sahal-Brechot 1978).. Only
unpolarized incoming light is considered and therefore $k=0,2$ here. The ratio
between the density tensor and the total density of the level is defined as
the alignment parameter $\sigma^{k}_{q}\equiv\rho^{k}_{q}/\rho^{0}_{0}$. In
the regime of GSA, $q$ is always $0$ for the levels on the ground state. It is
most convenient to calculate the GSA in the theoretical frame $xyz-$system
(Fig.1a), where the magnetic direction is $z-$axis. The direction of the line
of sight is denoted as $\Omega$ and $\mathcal{J}^{K}_{Q}(i,\Omega)$ is the
irreducible geometric tensor for the observed light. Due to GSA, the
absorption coefficients of the atomic transitions are expressed by (Landi
Degl’Innocenti 1984, see also Yan & Lazarian 2006):

$\eta_{i}(\nu,\Omega)=\frac{h\nu_{0}}{4\pi}B_{lu}n(J_{l})\Psi(\nu-\nu_{0})\sum_{\begin{subarray}{c}K\end{subarray}}(-1)^{K}\omega^{K}_{J_{l}J_{u}}\sigma^{K}_{0}(J_{l})\mathcal{J}^{K}_{0}(i,\Omega),$
(1)

where $K=0,2$, $\omega^{0}_{J_{1}J_{2}}\equiv 1$,
$\omega^{2}_{J_{1}J_{2}}\equiv\\{1,1,2;J_{1},J_{1},J_{2}\\}/\\{1,1,0;J_{1},J_{1},J_{2}\\}$,
in which the matrix with braces "$\\{\\}$" represent the $6-j$ symbol. The
quantity $B_{lu}$ is the Einstein coefficient for absorption444The data of
Einstein coefficients used in the paper are taken from the Atomic Line List
(http://www.pa.uky.edu/~peter/atomic/) and the NIST Atomic Spectra Database..
The total atomic population $n(J_{l})$ on the lower level $J_{l}$ is defined
as $n\sqrt{[J_{l}]}\rho^{0}_{0}(J_{l})$, where $[j]=2j+1$. $\Psi(\nu-\nu_{0})$
is the line profile. As shown in Eq. (1), the radiative pumping and the
magnetic direction will modulate the absorption coefficients $\eta_{i}$ and
thus the absorption spectrum varies.

For resonance emission lines, the excited states are influenced by the
differential occupation on the ground state through radiative excitation (see,
e.g., Yan & Lazarian 2008). In diffuse ISM and IGM, the magnetic field is weak
and the decay rate of atoms from the levels on the excited states is much
higher than the magnetic precession rate ($A\gg 2\pi\nu_{L}$). Thus, $q$ can
be non-zero for the density tensor of the atoms on the excited states. the
emission coefficients $\epsilon_{i}(i=0\sim 3)$ from the upper level $J_{u}$
are (see Yan & Lazarian, 2007):

$\epsilon_{i}(\nu,\Omega)=\frac{h\nu_{0}}{4\pi}A_{ul}n(J_{u})\Psi(\nu-\nu_{0})\sum_{\begin{subarray}{c}KQ\end{subarray}}\omega^{K}_{J_{u}J_{l}}\sigma^{K}_{Q}(J_{u})\mathcal{J}^{K}_{Q}(i,\Omega),$
(2)

where $K=0,2;Q=0,\pm 1,\pm 2$. The quantity $A_{ul}$ is the Einstein
coefficient for emission. Similar to the case of absorption, the emissivity
$\epsilon_{0}$ is also influenced by the anisotropic radiative pumping and
magnetic field, as demonstrated in Eq. (2). In the following, we will evaluate
quantitatively the influence of GSA on absorption and emission spectral lines
in the observational frame (see Fig.1b).

##  3 Influence of radiative alignment on the spectrum intensity

Spectral line intensity are modulated due to the GSA induced by the
anisotropic radiation. We first consider the unmagnetized case, and study only
the dependence modulation on the scattering angle $\theta_{0}$, the angle
between line of sight and the incident radiation direction. It is worth noting
that there is no alignment (i.e., the alignment parameter $\sigma^{2}_{0}=0$)
at $\theta_{0}=54.7^{\circ}$ or $180^{\circ}-54.7^{\circ}$ (Van Vleck angle,
Van Vleck 1925; House 1974), the corresponding intensity $I_{VV}$ is used as
the “standard” for comparison in this section.

The intensity fluctuations due to radiative alignment for several selected
spectral lines are plotted in Fig. 2. We find that the intensity fluctuation
ratio due to radiative alignment can be fitted with a simple analytical
expression with $5\sigma$ confidence:

$r^{pump}(\theta_{0})\equiv I^{pump}/I_{VV}=p_{1}+p_{2}\cos 2\theta_{0},$ (3)

where $p_{1},p_{2}$ vary with spectral lines. The fitting for Fe ii$\lambda
2600\mbox{\AA}$ is presented as an example in Fig. 2. The comprehensive
fitting parameters for various transitions are presented in Table 1.

Figure 2: (a) The modulation of several selected atomic lines owing to radiative alignment; (b) The fitting of the analytical expression for Fe ii$\lambda 2600\mbox{\AA}$ emission line.  Table 1: FITTING PARAMETERS FOR RADIATIVE ALIGNMENT Species | $\lambda$ | Absorption | Emission
---|---|---|---
| $(\mbox{\AA})$ | $p^{ab}_{1}$ | $p^{ab}_{2}$ | $p^{em}_{1}$ | $p^{em}_{2}$
C i | $1193.03$ | 1 | 0 | 1.0307 | 0.0898
C ii | $1334.53$ | 0 | 0 | 1.0451 | 0.1320
O i | $1025.76$ | 0.9984 | -0.0047 | 1.0194 | 0.0566
S i | $1474.57$ | 1.0376 | 0.1099 | 1.0032 | 0.0093
$1474.38$ | 0.9625 | -0.1097 | 0.9776 | -0.0655
$1473.99$ | 1.0107 | 0.0314 | 1.0350 | 0.1024
S ii | $1250.58$ | 1.0210 | 0.0613 | 1 | 0
$1253.81$ | 0.9832 | -0.0490 | 1.0510 | 0.1494
$1259.52$ | 1.0042 | 0.0123 | 1.0378 | 0.1104
Si ii | $1190.42$ | 1 | 0 | 1.0440 | 0.1286
S iii | $1012.50$ | 1 | 0 | 1.0265 | 0.0774
S iv | $1062.66$ | 1 | 0 | 1.0623 | 0.1823
Fe ii | $1142.36$ | 1.0097 | 0.0283 | 0.9816 | -0.0537
$1143.22$ | 0.9859 | -0.0411 | 0.9817 | -0.0535
$1144.93$ | 1.0053 | 0.0154 | 1.0230 | 0.0673
Species | $\lambda$ |  | Emission |
| $({\mu}m)$ |  | $p^{em}_{1}$ | $p^{em}_{2}$ |
[C i] | $610$ |  | 1.0332 | 0.0940 |
[C ii] | $157.7$ |  | 0.9795 | -0.0599 |
[O i] | $63.2$ |  | 1.0046 | 0.0135 |
[Si ii] | $34.8$ |  | 1.0206 | 0.0602 |
[S i] | $25.2$ |  | 1.0030 | 0.0089 |
[S iii] | $33.5$ |  | 1.0325 | 0.0951 |
[S iv] | $10.5$ |  | 1.0061 | 0.0177 |
[Fe ii] | $26.0$ |  | 1.0049 | 0.0143 |

|
---
|
---|---
|

(a) Case 1

(b) Case 2

(c) Case 1 with magnetic field

(d) Case 2 with magnetic field

Figure 3: The influence of radiative alignment. Magnetic field is not
accounted for. (a) The geometric illustration, the synthetic ISM is
illuminated by the O-type star from different positions, the picture plane
projection is on the right; (b,c) variations of the line ratio of two
components in the S i triplets $\lambda\sim 1474\mbox{\AA}$
($1473.99,1474.38\mbox{\AA}$) for case 1,2, respectively; (d,e) same as (b,c),
but with constant magnetic field along $x-$axis included.

The influence is more significant for multiplets and line ratios since it
varies among different lines. We perform synthetic observation on a cloud that
is illuminated by a star located at two different positions (case 1 and case
2, whose picture plane projections are marked in Fig. 3). The resulting
fluctuations of the line ratio of two components in the S i triplets
$\lambda\sim 1474\mbox{\AA}$ are shown in Fig. 3(a) and Fig. 3(b),
respectively. Moreover, the possible fluctuations will be further enhanced if
magnetic field exists. Fig. 3(c) and Fig. 3(d) are the results of synthetic
observations for case 1 and 2 with the same magnetic field. The fluctuation of
the line ratio ranges from overestimation of $\gtrsim 30\%$ to underestimation
of $\gtrsim 10\%$. By comparing Fig. 3(c) and Fig.3(d), we further conclude
that the influence of GSA on the same spectral line also varies according to
radiation geometry even with the same underlying magnetic field.

We define $R_{th}$ to depict the possible intensity fluctuation range
introduced by GSA:

$R_{th}\equiv\frac{max\\{(I_{VV}+\Delta I)\\}}{min\\{(I_{VV}+\Delta I)\\}},$
(4)

The comprehensive results for the $R_{th}$ are shown in Table 2. Owing to GSA,
the same column density could produce an intensity variation by a factor of
$\gtrsim 1.5$, which is far beyond the noise amplitude of current high S/N
telescope.

Table 2: THE RANGE OF FLUCTUATION ARISING FROM GSA Species | $\lambda$ | Absorption | Emission
---|---|---|---
| $(\mbox{\AA})$ | $R^{ab}_{th}$ | $R^{em}_{th}$
C i | $1193.03$ | 1 | 1.3004
C ii | $1334.53$ | 1 | 1.4275
O i | $1025.76$ | 1.0682 | 1.1834
S i | $1474.57$ | 1.4121 | 1.0677
$1474.38$ | 1.3264 | 1.2442
$1473.99$ | 1.2526 | 1.6439
S ii | $1250.58$ | 1.1278 | 1
$1253.81$ | 1.1049 | 1.3949
$1259.52$ | 1.0247 | 1.3234
Si ii | $1190.42$ | 1 | 1.4261
S iii | $1012.50$ | 1 | 1.4726
S iv | $1062.66$ | 1 | 1.5979
Fe ii | $1142.36$ | 1.1560 | 1.1216
$1143.22$ | 1.1054 | 1.3509
$1144.93$ | 1.0361 | 1.1904
Species | $\lambda$ | Emission
| $({\mu}m)$ | $R^{em}_{th}(\%)$
[C i] | $610$ | 1.3632
[C ii] | $157.7$ | 1.1307
[O i] | $63.2$ | 1.1217
[Si ii] | $34.8$ | 1.1338
[S i] | $25.2$ | 1.1464
[S iii] | $33.5$ | 1.3712
[S iv] | $10.5$ | 1.0358
[Fe ii] | $26.0$ | 1.0344

##  4 Influence of magnetic realignment on spectral lines

In this section we investigate the fluctuation of spectral lines from the
magnetic realignment. The variation ratio of the line intensity
$r_{mag}=I_{ob}/I_{0}$ is used to characterize the influence of magnetic
field, where $I_{ob}$ is the actual line intensity observed and $I_{0}$
corresponds to the supposed line intensity without the magnetic realignment
(or $\theta_{r}=0$).

(a) $r^{ab}(1250\mbox{\AA})$ variation, $\theta_{0}=90^{\circ}$

(b) $r^{em}(1143\mbox{\AA})$ variation, $\theta_{0}=90^{\circ}$

(c) $r^{em}(610{\mu}m)$ variation, $\theta_{0}=90^{\circ}$

Figure 4: The variation ratio of the line intensity under the influence of
magnetic fields for (a) S ii$\lambda 1250\mbox{\AA}$ absorption, (b) Fe
ii$\lambda 1143\mbox{\AA}$ emission, and (c) C i $\lambda 610{\mu}m$ emission,
respectively. The line of sight is vertical to the direction of the incident
radiation ($\theta_{0}=90^{\circ}$), which corresponds to a face-on disk.

###  4.1 UV/optical spectra

Figure 5: Schematics for the typical environment where atomic lines are
altered by magnetic fields. Presented is a DLA region on the spiral arm of a
galaxy.

As an example, we show in Fig.5 a typical scenario, in which the effect of
magnetic realignment leads to the measurable variation in the observed line
intensity. Consider a typical late-type galaxy with an extended neutral gas
disk and an interstellar magnetic field. Star-formation is expected to take
place in distinct regions in the disk (i.e., in spiral arms). Therefore, the
interstellar radiation field is likely to be anisotropic at a given point in
the disk. In the spectrum of the background point source (e.g., a Quasi-
Stellar Object, hereafter QSO), the gas disk will leave its imprint through
many absorption lines from neutral and ionised species, where most of the
lines are located in the (rest frame) UV. Such disk absorbers are known to
contribute to the population of the so-called DLA absorbers that are
frequently observed at low and high redshift in QSO spectra. The DLA absorbers
adopted to illustrate the influence of magetic fields on the spectroscopy are
dominated by cold neutral hydrogen (see, e.g., Rao & Turnshek, 2000). As long
as the angle between the ambient magnetic field and the direction of the
incident radiation at the place where the sightline pierces the disk is not
Van Vleck angle, the GSA will cause changes in the central absorption depths
of unsaturated absorption lines. The variation of the line intensity of S
ii$\lambda 1250.58\mbox{\AA}$ with respect to different magnetic field
direction is presented in Fig.4(a) given the line of sight perpendicular to
the direction of the incident radiation ($\theta_{0}=90^{\circ}$).

As shown in Fig.4(a), the spectra observed change with the direction of
magnetic field. In addition, the error of the observation for S ii$\lambda
1250.58\mbox{\AA}$ due to photon noise is $2.5\%$ in current observation (see
Prochaska et al. 2007), which is smaller than the variations due to GSA in
most of the areas- top at $7\%$\- in Fig.4(a). The maximum enhancement and
reduction caused by magnetic realignment for different absorption lines are
presented in Table 3, in which the absorptions are from all the levels on the
ground state that have much longer life time than magnetic precession
period555These lines are detected at the places such as galactic halos
(Richter et al., 2013; Fox et al., 2014b) and circumstellar medium in
GRB(Fynbo et al., 2006).. The fluctuation of intensity
$\Delta\rm{I}/\rm{I_{0}}=r^{ab}_{mag}-1$. The influence of magnetic fields are
different among the lines. For some of the lines, the variation is more than
$20\%$. Thus, The measurable changes of absorption spectrum can be introduced
due to magnetic realignment. The influence of GSA on the derived alpha-to-iron
ratio of the DLA absorbers will be discussed in §5 with real observation
analysis.

Table 3: VARIATIONS OF UV/OPTICAL ABSORPTION LINES DUE TO GSA

Absorption from the ground level of the ground state
---
Species | $\lambda$ | $(\Delta\rm{I}/\rm{I_{0}})_{min}$ | $\theta_{0min}$ | $(\Delta\rm{I}/\rm{I_{0}})_{max}$ | $\theta_{0max}$
| $(\mbox{\AA})$ | $\%$ |  | $\%$ |
O i | $1025.76$ | $-6.37$ | $90^{\circ}$ | $+0$ | $0^{\circ}$
S i | $1474.57$ | $-27.29$ | $0^{\circ}$ | $+1.81$ | $54.7^{\circ}$
$1474.38$ | $-22.82$ | $90^{\circ}$ | $+2.62$ | $30^{\circ}$
$1473.99$ | $-19.99$ | $0^{\circ}$ | $+0.08$ | $54.7^{\circ}$
S ii | $1250.58$ | $-7.56$ | $0^{\circ}$ | $+6.33$ | $90^{\circ}$
$1253.81$ | $-4.71$ | $90^{\circ}$ | $+7.00$ | $0^{\circ}$
$1259.52$ | $-1.61$ | $0^{\circ}$ | $+1.22$ | $90^{\circ}$
Ti ii | $3072.97$ | $-1.64$ | $0^{\circ}$ | $+1.30$ | $90^{\circ}$
$3241.98$ | $-1.03$ | $90^{\circ}$ | $+1.35$ | $0^{\circ}$
$3229.19$ | $-0.33$ | $0^{\circ}$ | $+0.26$ | $90^{\circ}$
Fe ii | $1142.36$ | $-1.69$ | $0^{\circ}$ | $+4.42$ | $90^{\circ}$
$1143.22$ | $-2.86$ | $90^{\circ}$ | $+7.94$ | $0^{\circ}$
$1144.93$ | $-0.46$ | $35.3^{\circ}$ | $+3.21$ | $90^{\circ}$
Absorption from metastable levels of the ground state
C i∗ | $1260.93$ | $-24.20$ | $0^{\circ}$ | $+1.54$ | $54.7^{\circ}$
$1261.00$ | $-17.45$ | $90^{\circ}$ | $+0.46$ | $35.3^{\circ}$
$1261.12$ | $-15.76$ | $0^{\circ}$ | $+0$ | $0^{\circ}$
C ii∗ | $1037.02$ | $-9.02$ | $90^{\circ}$ | $+3.68$ | $0^{\circ}$
$1335.66$ | $-10.34$ | $0^{\circ}$ | $+1.29$ | $54.7^{\circ}$
$1335.71$ | $-5.39$ | $90^{\circ}$ | $+0.10$ | $54.7^{\circ}$
O i∗ | $1027.43$ | $-9.57$ | $90^{\circ}$ | $+0$ | $0^{\circ}$
Si ii∗ | $1264.74$ | $-5.59$ | $90^{\circ}$ | $+5.32$ | $0^{\circ}$
$1265.00$ | $-3.00$ | $0^{\circ}$ | $+0.27$ | $54.7^{\circ}$
S i∗ | $1303.11$ | $-21.05$ | $0^{\circ}$ | $+1.72$ | $54.7^{\circ}$
$1302.86$ | $-14.43$ | $90^{\circ}$ | $+0.50$ | $35.3^{\circ}$
$1302.34$ | $-12.74$ | $0^{\circ}$ | $+0$ | $0^{\circ}$
S iii∗ | $1015.50$ | $-24.67$ | $0^{\circ}$ | $+1.53$ | $54.7^{\circ}$
$1015.57$ | $-17.88$ | $90^{\circ}$ | $+0.46$ | $54.7^{\circ}$
$1015.78$ | $-16.19$ | $0^{\circ}$ | $+0$ | $0^{\circ}$
S iv∗ | $1072.97$ | $-1.20$ | $90^{\circ}$ | $+2.20$ | $0^{\circ}$
$1073.52$ | $-0.20$ | $90^{\circ}$ | $+0.56$ | $90^{\circ}$
C i∗∗ | $1193.65$ | $-24.65$ | $0^{\circ}$ | $+1.06$ | $54.7^{\circ}$
$1193.39$ | $-20.93$ | $90^{\circ}$ | $+1.51$ | $35.3^{\circ}$
$1193.24$ | $-18.74$ | $0^{\circ}$ | $+0$ | $0^{\circ}$
S iii∗∗ | $1202.12$ | $-24.20$ | $0^{\circ}$ | $+0.95$ | $54.7^{\circ}$
$1201.73$ | $-20.62$ | $90^{\circ}$ | $+1.37$ | $35.3^{\circ}$
$1200.97$ | $-18.53$ | $0^{\circ}$ | $+0$ | $0^{\circ}$

  * •

Note: The comparison is made by first considering a specific $\theta_{0}$ and
changing the magnetic direction in the full space to find the maximum
enhancement and reduction for the chosen $\theta_{0}$. Then we scan
$\theta_{0}$ for all the possible angles and compare the maximum variation for
different $\theta_{0}$. $(\Delta\rm{I}/\rm{I_{0}})_{max}$ means the maximum
enhancement of the observed intensity due to magnetic realignment and
$(\Delta\rm{I}/\rm{I_{0}})_{min}$ means the reduction. The corresponding
geometry are denoted as $\theta_{0max}$ and $\theta_{0min}$, respectively.

Simulations are performed in Fig. 6 to compare the spectrum profiles of the S
ii triplets for a synthetic absorption spectrum of a DLA with and without the
magnetic alignment included. Fig. 6 was designed up to illustrate the effect
in a realistic instrumental set up, with a given typical pixel size and S/N
for an optical spectrum taken by an 8m-class telescope. The graphical use of
steps instead of curves, which is common in absorption spectroscopy, is
intended to visualize the pixel-by-pixel noise variations in the data. The
three transitions of singly-ionised Sulfur (S ii; upper ionisation potential
23.3 eV) at $\lambda\lambda 1250.58,1253.81,1259.52$ represent important
tracers for neutral and weakly ionised gas in the local interstellar and
intergalactic medium and in distant galaxies (e.g., Kisielius et al. 2014;
Welsh & Lallement 2012; Richter et al. 2001; Fox et al. 2014a). Being an
$\alpha$ element, singly-ionised Sulfur only has a weak depletion into dust
grains (e.g., Savage & Sembach 1996). Thus the interstellar Sulfur abundance
is often used as a proxy for the $\alpha$-abundance in the gas. In addition,
Sulfur has a relatively low cosmic abundance (Asplund et al., 2009) and under
typical interstellar conditions (in particular in low-metallicity
environments) these lines are not saturated. The three lines are observed in
the same wavelength region with identical S/N. The important parameters for
simulations are presented in the caption, such as the assumption of S ii
column density, etc. The synthetic spectra were generated using the FITLYMAN
routine (Fontana & Ballester, 1995) implemented in the ESO-MIDAS software
package. Atomic data were taken from Morton (2003). To show the effect
clearly, we zoom in the spectrum to the radial velocity range
$[-10km{\cdot}s^{-1},10km{\cdot}s^{-1}]$. The enhancement and reduction of the
spectral line profile due to the magnetic realignment change among the
triplets. Given the fact that meanwhile optical QSO spectra reach up to a new
standard of S/N of a few hundred (e.g., D’Odorico et al. 2016), the predicted
effect is already VISIBLE, if the component structure of the DLA allows a
detailed investigation.

Figure 6: Synthetic spectrum (in velocity scale) of a single-component S ii
$\lambda\lambda 1250.58,1253.81,1259.52$ absorption system with a S ii column
density of $N$(S ii$)=4\times 10^{14}$ cm-2, a Doppler parameter of $b=5$ km
s-1, a spectral resolution of $R=45,000$, and a S/N of 100 per resolution
element for the source-in-disk scenario. Such a S ii column density would be
expected for a DLA with log $N$(H i$)=20.48$ and a sulfur abundance of $0.1$
solar assuming solar reference abundances of Asplund et al. (2009). The
geometric condition is $\theta_{0}=90^{\circ}$. (a) The blue solid lines show
the spectrum without the magnetic realignment, the red solid lines show the
spectrum when $\theta=90^{\circ}$, $\phi=90^{\circ}$. (b) The red-/blue-shaded
areas indicate the excess/deficiency of the absorption due to the magnetic
realignment.

The maximum and minimum variations of intensity for different emission lines
due to the magnetic realignment are presented in Table 4. The fluctuation of
intensity $\Delta\rm{I}/\rm{I_{0}}=r^{em}_{mag}-1$. The variations are
different among the lines, e.g., for $\theta_{0}=90^{o}$, S ii $\lambda
1259.52\mbox{\AA}$ is enhanced up to $21\%$; S i $\lambda 1473.99\mbox{\AA}$
is reduced more than $25\%$ whereas the line S ii $\lambda 1253.81\mbox{\AA}$
is enhanced up to $38\%$. Fe ii emission spectra in UV and optical band are
important for modelling Planetary Nebulae (PNe) (Dinerstein et al., 2006) and
cloud near AGNs (Wills et al., 1985). The variation of Fe ii$\lambda 1144.93$
emission line due to magnetic realignment are presented in Fig.4(b) as an
example. In this geometric condition, the magnetic realignment enhances more
than $10\%$ of the spectrum when the magnetic field is perpendicular to both
the line of sight and the direction of the incident radiation
($\theta=90^{\circ},\phi=90^{\circ}$).

As demonstrated in Tables 3,4, the modulation due to magnetic realignment is
more significant among all the measured multiplets for both absorption and
emission spectra.

Table 4: VARIATIONS OF UV/OPTICAL EMISSION LINES FROM GSA

Species | $\lambda$ | $(\Delta\rm{I}/\rm{I_{0}})_{min}$ | $\theta_{0min}$ | $(\Delta\rm{I}/\rm{I_{0}})_{max}$ | $\theta_{0max}$
---|---|---|---|---|---
| $(\mbox{\AA})$ | $\%$ |  | $\%$ |
C i | $1193.03$ | $-20.85$ | $0^{\circ}$ | $+21.00$ | $90^{\circ}$
C ii | $1334.53$ | $-29.17$ | $0^{\circ}$ | $+28.90$ | $90^{\circ}$
O i | $1025.76$ | $-14.30$ | $0^{\circ}$ | $+10.44$ | $90^{\circ}$
Si ii | $1190.42$ | $-27.20$ | $0^{\circ}$ | $+31.49$ | $90^{\circ}$
S i | $1474.57$ | $-6.32$ | $0^{\circ}$ | $+0.50$ | $54.7^{\circ}$
$1474.38$ | $-19.62$ | $90^{\circ}$ | $+7.49$ | $0^{\circ}$
$1473.99$ | $-38.67$ | $0^{\circ}$ | $+3.89$ | $54.7^{\circ}$
S ii | $1253.81$ | $-20.51$ | $0^{\circ}$ | $+38.79$ | $90^{\circ}$
$1259.52$ | $-23.04$ | $0^{\circ}$ | $+21.27$ | $90^{\circ}$
S iii | $1012.50$ | $-32.02$ | $0^{\circ}$ | $+6.01$ | $110^{\circ}$
S iv | $1062.66$ | $-35.44$ | $0^{\circ}$ | $+44.09$ | $90^{\circ}$
Fe ii | $1142.36$ | $-2.90$ | $65^{\circ}$ | $+12.10$ | $0^{\circ}$
$1143.22$ | $-9.37$ | $0^{\circ}$ | $+22.44$ | $0^{\circ}$
$1144.93$ | $-14.38$ | $0^{\circ}$ | $+13.64$ | $90^{\circ}$

  * •

Note: Same as Table 3, but for resonance emission lines in the UV/optical
band.

###  4.2 Submillimeter fine-structure lines

Submillimeter fine-structure spectra, which arise from magnetic dipole
transitions within the ground state, have a broad applicability in
astrophysics, such as, determining chemical properties of star-forming
galaxies(Kobulnicky et al., 1999), predicting the star burst size (Díaz-Santos
et al., 2013), etc. However, previous spectral analysis does not consider the
anisotropic radiation and the magnetic realignment. In diffuse ISM and IGM,
the influence of GSA on the submillimeter fine-structure absorption lines is
exactly the same as that on the UV/optical resonance absorption
lines.666Thanks to such equivalence, the fine-structure absorption lines are
not discussed to avoid repetition. Readers may refer to the results in §4.1.
The atoms on all the levels in the ground state are magnetically aligned
(i.e., only those matrix tensors $\rho^{k}_{q}$ with even k and q = 0 exist)
since the life time of these atoms on all the levels of the ground state are
much longer than the magnetic precession period in diffuse ISM and IGM (see
Yan & Lazarian 2006). For example, the influence of magnetic realignment on C
i $\lambda 610{\mu}m$ in a face-on disk is presented in Fig.4(c). The
configuration of the ground state of C i has 3 levels $3P_{0;1;2}$. C i
$\lambda 610{\mu}m$ line represents the transition between the levels within
the ground state $3P_{1}$ and $3P_{0}$. Table 5 presents the influence of
magnetic fields for a list of submillimeter emission lines.

Table 5: VARIATIONS OF SUBMILLIMETER LINES DUE TO GSA

Species | $\lambda$ | $(\Delta\rm{I}/\rm{I_{0}})_{min}$ | $\theta_{0min}$ | $(\Delta\rm{I}/\rm{I_{0}})_{max}$ | $\theta_{0max}$
---|---|---|---|---|---
| $(\mu m)$ | $\%$ |  | $\%$ |
[C i] | $610$ | $-24.2$ | $0^{\circ}$ | $+1.54$ | $54.7^{\circ}$
[C ii] | $157.7$ | $-9.02$ | $90^{\circ}$ | $+3.68$ | $0^{\circ}$
[O i] | $63.2$ | $-10.77$ | $0^{\circ}$ | $+0$ | $0^{\circ}$
[Si ii] | $34.8$ | $-8.76$ | $0^{\circ}$ | $+4.90$ | $90^{\circ}$
[S i] | $25.2$ | $-12.73$ | $0^{\circ}$ | $+0$ | $0^{\circ}$
[S iii] | $33.5$ | $-24.67$ | $0^{\circ}$ | $+1.53$ | $54.7^{\circ}$
[S iv] | $10.5$ | $-2.04$ | $0^{\circ}$ | $+2.00$ | $90^{\circ}$
[Fe ii] | $26.0$ | $-0.39$ | $35.3^{\circ}$ | $+3.11$ | $90^{\circ}$

  * •

Note: Same as Table 3, but for submillimeter fine-structure lines.

###  4.3 Observations from the medium with turbulent magnetic fields

In order to address the issue of how much modulation can be induced with the
line-of-sight dispersion of magnetic field, we perform synthetic observation
on ISM with turbulent magnetic field from numerical simulation. A three-
Dimensional (3D) super-Alfvenic ($M_{a}=1.43$) MHD datacube ($512\times
512\times 16$), which corresponds to a $1pc(x)\times 1pc(y)\times 0.2pc(z)$
diffuse layer of a reflection nebula, is generated by the MHD-simulation with
the PENCIL-code777See https://code.google.com/archive/p/pencil-code/ for
details.. Note that the simulation is dimensionless and not sensitive to the
choice of corresponding physical size. The line-of-sight dispersion is
evaluated by the chosen total Alfvenic-Mach Number $M_{a}$. A massive O-type
star radiates UV-photons to illuminate the medium. Synthetic observations are
performed on the medium for two emission lines in the S i triplets
$\lambda\sim 1474\mbox{\AA}$, which share the same S/N ratio. The intensity
variation is integrated along the line of sight and the influence of GSA on
these two lines are shown in Fig. 7. Magnetic alignment induces more than
$10\%$ variation and for a sufficient amount of areas more than $20\%$ for
both lines. In addition, the variations due to the GSA for the same
environment (density, magnetic field, and radiation field ) differ
substantially for the two spectral lines. Since the injection scale of
interstellar turbulence $\sim 100pc$ is much larger than the adopted value
here (see Armstrong et al. 1995; Chepurnov & Lazarian 2010), the line-of-sight
dispersion of the magnetic field in real observations can only be less than
that in the synthetic data (Fig. 7), and thus the intensity variation due to
GSA can be more significant in real circumstances.

(a) S i$\lambda 1473.99\mbox{\AA}$ emission

(b) S i$\lambda 1474.38\mbox{\AA}$ emission

Figure 7: Synthetic observations to super-Alfvenic ISM ($M_{a}=1.43$)
generated by PENCIL MHD code for S i emission lines (a)$\lambda
1473.99\mbox{\AA}$ and (b)$\lambda 1474.38\mbox{\AA}$, respectively. The ISM
corresponds to an $1pc\times 1pc$ area on the picture plane and the optical
depth is $0.2pc$. The radiation source, a massive O-type star, resides to the
south of the medium $0.1pc$ away. The color scale represents the intensity
variation ratio with and without GSA. The percentage of enhancement and
depletion are noted with color on the contour.

##  5 Influence on physical parameters derived from line ratios

Many important physical parameters in astronomy are derived from spectral line
ratio. For example, the ratio of different transitions of C ii ($\lambda
1036.34$ and $\lambda 1334.53$) is used to estimate the electron density in
different astrophysical environments (see Lehner et al. 2004; Zech et al.
2008; Richter et al. 2013 for details). As illustrated in §3,4, the influence
of the GSA varies for different spectral lines. Therefore, the variation of
the resulting spectral line ratio is expected to be more significant, e.g.,
with one line reduced and the other enhanced. According to Eq. (1), the ratio
of the intensity between two different absorption lines is given by

$\begin{split}\mathcal{R}^{ab}_{\lambda_{1},\lambda_{2}}(\theta_{0},\theta_{r},\theta)&=r^{ab}(\lambda_{1},\theta_{0},\theta_{r},\theta)/r^{ab}(\lambda_{2},\theta_{0},\theta_{r},\theta),\\\
&\propto\frac{\rho^{0}_{0}(J^{\prime}_{l})\sqrt{2}+\omega^{2}_{J^{\prime}_{l}J^{\prime}_{u}}\rho^{2}_{0}(J^{\prime}_{l})\left(1-1.5\sin^{2}\theta\right)}{\rho^{0}_{0}(J_{l})\sqrt{2}+\omega^{2}_{J_{l}J_{u}}\rho^{2}_{0}(J_{l})\left(1-1.5\sin^{2}\theta\right)}.\end{split}$
(5)

And the ratio of the intensities of two emission lines is obtained from Eq.
(2):

$\begin{split}\mathcal{R}^{em}_{\lambda_{1},\lambda_{2}}(\theta_{0},\theta_{r},\theta)&=r^{em}(\lambda_{1},\theta_{0},\theta_{r},\theta)/r^{em}(\lambda_{2},\theta_{0},\theta_{r},\theta),\\\
&\propto\frac{\rho^{0}_{0}(J^{\prime}_{u})+\sum_{\begin{subarray}{c}q\end{subarray}}\omega^{2}_{J^{\prime}_{u}J^{\prime}_{l}}\rho^{2}_{q}(J^{\prime}_{u})\mathcal{J}^{2}_{-q}}{\rho^{0}_{0}(J_{u})+\sum_{\begin{subarray}{c}q\end{subarray}}\omega^{2}_{J_{u}J_{l}}\rho^{2}_{q}(J_{u})\mathcal{J}^{2}_{-q}};\\\
q&=0,\pm 1,\pm 2.\end{split}$ (6)

A few examples will be presented in the following subsections.

###  5.1 Nucleosynthetic studies in DLAs

(a) $\mathcal{R}^{ab}_{1250,2599}$ variation, $\theta_{0}=90^{\circ}$

(b) $\mathcal{R}^{ab}_{1302,1608}$ variation, $\theta_{0}=90^{\circ}$

Figure 8: Influence of GSA on [$\alpha$/Fe] ratio with
(a)$\mathcal{R}^{ab}_{1250,2599}$ (S ii $\lambda 1250$/Fe ii $\lambda 2599$)
and (b) $\mathcal{R}^{ab}_{1302.17,1608.45}$ (O i $\lambda 1302.17$/ Fe ii
$\lambda 1608.45$) absorption lines, respectively.

The alpha-to-iron ratio is widely used in spectral analysis. It reflects the
nucleosynthetic processes in SFRs (e.g., Prochaska et al. 2001). The $\alpha$
elements include O, Si, S, Ti, etc. Iron refers to Fe peak elements such as
Cr, Mn, Co, Fe, etc. $\alpha$ elements in the medium of the galaxy with low
metallicity ([Fe/H]$\lesssim$-1.0) are produced exclusively by Type-II
supernovae (SNe II), but the [$\alpha$/Fe] ratio suffers a drop when the
delayed contribution of Fe from Type-Ia supernovae (SNe Ia) is effective
(McWilliam, 1997; Cooke et al., 2011). The alpha-to-iron ratio [X/Fe] index (X
represents the chemical elements for $\alpha$ elements) is defined by the
ratio of abundance observed from the medium in comparison with that from the
sun:

$[\rm{X}/\rm{Fe}]\equiv
log[\rm{N(X)}/\rm{N(Fe)}]-log[\rm{N(X)}/\rm{N(Fe)}]_{\odot}$ (7)

The abundance of the elements is assumed to be equal to the column density of
the element in the dominant ionisation state, e.g., Fe ii for the iron
abundance and S ii for the $\alpha-$element abundance since in DLAs these
elements are mostly singly-ionised. Thus, the ratio N(S)/N(Fe) is inferred
from the line ratio S ii/Fe ii. Nevertheless, the inferred N(S)/N(Fe) is
influenced by GSA, as shown in Eq. (5). The variation of the ratio
$\mathcal{R}^{ab}_{\rm{SII,FeII}}$ with respect to different direction of
magnetic fields is presented in Fig.8(a) for the geometric condition
$\theta_{0}=90^{\circ}$. Furthermore, taking into account of all the possible
geometric conditions ($\theta_{0}$), the maximum and minimum variations for
the inferred N(S)/N(Fe) due to GSA is [$-14\%,+10\%$], i.e., [-0.07,+0.04] for
[S/Fe].

In comparison, another $\alpha$ element, Oxygen, is presented as an example.
Most oxygen in DLAs are neutral and therefore the ratio N(O)/N(Fe) is inferred
from O i/Fe ii. The influence of GSA on the ratio
$\mathcal{R}^{ab}_{\rm{OI,FeII}}$ is presented in Fig. 8(b) when
$\theta_{0}=90^{\circ}$. Comparing Fig.8(a) and Fig.8(b), the inferred
N(S)/N(Fe) is enhanced by $10\%$ whereas the inferred N(O)/N(Fe) reduced by
$9\%$ when magnetic field is perpendicular to both the line of sight and the
direction of the incidental radiation, i.e., [S/Fe] varied by $+0.04$ while
[O/Fe] varied by $-0.06$.

Figure 9: Possible influence of GSA on the inferred [S/Fe] from the absorption
of DLAs on the QSO spectra. The original data (red points), error bars and the
redshift of the corresponding DLA ($z_{abs}$) are obtained from Table 1 in
Noterdaeme et al. (2008). Green and blue points are the maximum enhancement
and reduction of the inferred [S/Fe] when the influence of the magnetic fields
is taken into account. The error bars that result from photon noise are
independent of the variation due to GSA, and thus are applicable to the upper
and lower variation thresholds.

Furthermore, Fig.9 illustrates the influence of GSA on the [S/Fe] modelled in
DLA absorbers on QSO spectra observed by VLT/UVES (see Table 1 in Noterdaeme
et al. 2008 with X$=$S). By applying the maximum variation of [S/Fe] obtained
in this section to the observational data, the upper and lower variation
thresholds due to GSA are obtained. As shown in Fig. 9, the variations due to
GSA are more significant than the error produced by photon noise. In addition,
the depletion of iron in the dust is calculated by [S/Fe] (Noterdaeme et al.,
2008). As demonstrated in Fig. 9, some of the variations due to GSA will even
result in the changing sign of [S/Fe].

###  5.2 Ionisation studies in diffuse gas

The line ratio of the same element from different ionisation states is often
used to determine the ionisation fraction (see, e.g., Richter et al. 2016).
For instance, (S iii)/(S ii) emission line ratio is employed to determine the
ionisation fraction in extragalactic H ii region (see, e.g., Vilchez et al.
1988), because higher ionised sulphur is insignificant in most of these H ii
regions (Mathis, 1982, 1985). The line ratio S iii $\lambda 1012.49$/S ii
$\lambda 1250.58$ is adopted to represent (S iii)/(S ii). Fig.10(a)
demonstrates the influence of GSA with respect to different magnetic
directions for $\theta_{0}=90^{\circ}$. The influence of magnetic fields on
inferred ionisation fraction varies with different geometric conditions. By
taking into account all the possible geometric conditions ($\theta_{0}$) in
Fig.10(b), the range of variation under the influence of GSA is
[$-27\%,+12\%$], which means a variation of [-0.14,+0.05] in log index.

(a) $\mathcal{R}^{em}_{1012,1250}$ variation, $\theta_{0}=90^{\circ}$

(b) $\mathcal{R}^{em}_{1012,1250}$ max $\And$ min variations

Figure 10: Variation of the ionisation fraction due to GSA: line ratio of S
iii $\lambda 1012.49\mbox{\AA}$ and S ii $\lambda 1250.58\mbox{\AA}$
emissions. (a) The variation of line ratio with respect to the direction of
the magnetic field in the case of line of sight vertical to the direction of
the incident radiation ($\theta_{0}=90^{\circ}$); (b) The maximum and minimum
variations as a function of $\theta_{0}$.

##  6 Removing the GSA effect from raw data

In this section, we use the observational data of spectral lines from PDRs in
$\rho$ Ophiuchi cloud as an example to demonstrate how to account for GSA
effect. The cloud, $150pc$ from the earth, is illuminated by the UV/optical
radiation from the B iiiV star HD147889, which resides approximately
$d_{Source}=0.75pc$ behind the cloud (Liseau et al., 1999). Atomic and single-
ionized Carbon traces the neutral gas of the PDRs in the cloud whereas
molecular lines are also detected (e.g., ${\rm C^{18}O}$ Kamegai et al. 2003).
Two stripes in the cloud are used here as examples to perform the analysis.
Dashed lines plotted in Fig. 11 are adopted from the observational analysis in
Kamegai et al. (2003). We first consider only the radiative alignment. We
adopt the fitting curves for [C i] and [C ii] lines from Table 1 in §3:

$\begin{split}r^{pump}_{[C\,{\sc i}]}(\theta_{0})&=1.0332+0.094\cos
2\theta_{0},\\\ r^{pump}_{[C\,{\sc ii}]}(\theta_{0})&=0.9795-0.0599\cos
2\theta_{0},\\\
\theta_{0}=-\arctan\frac{r}{d_{Source}}&=-\arctan\left[5.82\times
10^{-2}(r_{\theta_{0}}/arcmin)\right],\\\ \end{split}$ (8)

where $r_{\theta_{0}}$ is the picture plane angular distance from the source
to the analyzed medium in $arcmin$. The intensity ($I_{VV}$) after correcting
the effect from radiative alignment is thus obtained from the observational
intensity ($I_{ob}$) by:

$I_{VV}=I_{ob}/r^{pump}(\theta_{0}).$ (9)

The results are shown with solid lines in Fig. 11. The Van Vleck angle
flipping is seen at $\sim 24\arcmin$. We further take into account the
magnetic realignment. The picture plane magnetic field in $\rho$ Ophiuchi
cloud is obtained through star polarization in Kwon et al. (2015). The unknown
component of magnetic field along the line of sight leads to some
uncertainties in the correction, as marked by the error bars in Fig. 11. The
potential neutral Carbon peak can be shifted from the original one in raw data
(dashed lines). The intensity of atomic and single-ionized Carbon can be
significantly modified by GSA. The real column density distribution of neutral
Carbon may be synchronized with or largely deviated from that of ${\rm
C^{18}O}$. Therefore, more accurate magnetic field analysis with compatible
resolution to the spectral analysis has to be performed in such area for a
proper study of the interstellar gas.

(a) Stripe 1 comparison

(b) Stripe 2 comparison

Figure 11: Modifications of different atomic line intensity due to GSA, the
influence of anisotropic pumping and magnetic realignment. Line colors
represent:  Blue for [C i]$\lambda 610{\mu}m$,  Orange for [C ii]$\lambda
157{\mu}m$, and  Yellow for ${\rm C^{18}O(1-0)}$. The raw observational data
are in dashed lines from Kamegai et al. (2003). The solid lines are the
modifications purely due to anisotropic radiative pumping. The error bars that
depict the possible range of the original column density of the corresponding
element are produced accounting for the magnetic realignment. Potential peaks
for neutral Carbon for both stripes are noted with green dotted lines. The
intensity peaks for atomic and single-ionized Carbon lines are marked on the
corresponding $y-$axis.

##  7 Discussion

As illustrated in this paper, the variation of spectral line intensity induced
by GSA varies among different spectral lines. Thus, such influence could be
precisely analysed if multi-spectral lines for the same element is achievable.
In addition, the alignment on the ground state is transferred to the levels of
atoms on the excited states through absorption process, as illustrated in §3
and 4. The intensity of the ultraviolet-pumped fluorescence lines, which are
derived from successive decays to different levels of atoms and applied to the
modelling of reflection nebulae (Sellgren, 1984, 1986), are dependent on the
initial upper levels, and thus is influenced by GSA through the scattering
process. The influence of collision is neglected in this paper, which applies
to most diffuse ISM and IGM. Collisions reduce the alignment efficiency (see
Hawkins 1955). The collision effect can become important in the case of higher
density medium where the collision rate $\tau_{c}^{-1}$ (either inelastic
collision rate or Van der Waals collision rate) dominates over optical pumping
rate $B_{lu}\bar{J}^{0}_{0}$ (see Yan & Lazarian 2006 for details). The focus
of our paper is on the GSA effect that is a saturated state. When the magnetic
precession rate is comparable to the optical pumping from the ground state
($2\pi\nu_{L}\sim B_{lu}\bar{J}^{0}_{0}$), the ground-level Hanle effect is
applicable (see Landolfi & Landi Degl’Innocenti 1986), which means the
magnetic field influence on the spectrum is not limited to the change of
direction but also the magnetic field strength. As demonstrated in Yan &
Lazarian (2008), the effect becomes saturated when $2\pi\nu_{L}\rightarrow
10B_{lu}\bar{J}^{0}_{0}$. Therefore, we set $r_{Hanle}$where
$2\pi\nu_{L}\simeq 10B_{lu}\bar{J}^{0}_{0}$ as the boundary between the
ground-level Hanle regime and the GSA regime (see Fig. 12):

$r_{Hanle}=r_{\ast}\sqrt{\frac{A_{ul}[J_{u}]}{1.76(B/{\mu}G)(\exp(h\nu/(k_{B}T))-1)[J_{l}]}}$
(10)

We calculate the boundary $r_{Hanle}$ for C ii$\lambda 1037\mbox{\AA}$ in the
presence of stars with different effective temperature $T_{eff}$ and radius
$r_{\ast}$ for the magnetic field with the strength range from ${\mu}G$ to
$mG$ in Fig. 12(a). The $r_{Hanle}$ increases as the magnetic field becomes
weaker and as the effective temperature $T_{eff}$ and radius $r_{\ast}$
increase. Nevertheless, even in the most optimistic scenario with
$T_{eff}=4\times 10^{4}K,r_{\ast}=10r_{\odot},B=1{\mu}G$ (though rarely
applies), the boundary $r_{Hanle}\simeq 180Au=8.5\times 10^{-4}pc$, which is a
thousand times smaller than the normal H ii Region which is in $pc$ scale. All
the analysis with GSA and their observational implication in this paper is
applicable to most of the ISM, except when performing very high resolution
spectral analysis on regions very close to the very bright O-type star.

Indeed, many spectral lines we measured reside in multiplets. It is worth
noting that the influence of GSA becomes more significant for multiplets and
line ratios. Furthermore, our work reveals that impact of magnetic fields on
the analysis of interstellar environment cannot be neglected. The study is
complimentary to the interstellar magnetic field diagnoses.

(a) Ground level Hanle effect Radius

Figure 12: (a)Geometric illustration of different regimes with the
corresponding scale; The circle denotes the boundary between ground-level
Hanle regime and the GSA regime in this paper; (b) The boundary scale of
ground-level Hanle effect with different physical environments: star radius
$r_{\ast}$, effective temperature of the pumping source $T_{eff}$, and the
magnetic field strength ($x-$axis). $r_{\odot}$ is the solar radius.

##  8 Conclusions

We have demonstrated the influence of GSA effect on the spectroscopy. We
emphasize that GSA is a general physical process that induces visible
systematic variations to both absorption and emission spectral lines observed
from diffuse medium, e.g., DLAs, H ii Regions, PDRs, SFRs, Herbig Ae/Be disks,
etc. Comprehensive results are provided to demonstrate the influence of the
GSA effect -including radiative alignment and magnetic realignment- on
resonance UV/optical absorption/emission lines and submillimeter fine-
structure lines observed from different astrophysical environments. Synthetic
observations are performed to present such influence on spectral line profile
and to investigate the influence of GSA in turbulent magnetic fields.
Variations of the physical parameters inferred from line ratios due to GSA are
studied. We illustrate how to remove the GSA effect from raw data. Our main
conclusions are:

  * •

Measurable modulations are induced on the absorption and emission lines
observed from diffuse ISM and IGM due to GSA.

  * •

The influence of GSA on the spectral line intensity is not diminished due to
line-of-sight dispersion of turbulent magnetic fields such as in ISM and IGM.

  * •

The enhancement and reduction of the same spectrum line change in accordance
with the direction of the magnetic field and the radiation geometry.

  * •

The variation of the line intensity changes from line to line. As a result,
the influences on the multiplets and line ratios are even more distinct,
affecting the inferred physical parameters.

  * •

The analytical model set up in this paper can be used for correcting the GSA
effect from interstellar spectroscopy. Without such correction, the physical
environment inferred can not achieve the accuracy that current instruments
promise.

  * •

GSA should be considered in the future spectral analysis.

## Acknowledgements

We are grateful to Gesa Bertrang, Reinaldo Santos de Lima, Ruoyu Liu, Michael
Vorster for the helpful discussions. We thank the referee for valuable
comments and suggestions.

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##  Appendix A BASIC FORMULAE ON ATOMIC ALIGNMENT

In this Appendix, we illustrate the basic equations on atomic alignment. The
calculations in this Appendix are all performed in the theoretical frame
$xyz-$system in Fig.1(a).

Anisotropic radiation excites atoms through photo-excitation and consequently
results in spontaneous emissions. Occupations of the atoms on different levels
of the ground state will alter when there exists anisotropic radiation field.
Magnetic realignment will redistribute the angular momenta of the atoms due to
fast magnetic precession, which only happens on the ground state in general
ISM and IGM (see Yan & Lazarian 2012, 2015 for details). The equations to
describe the evolution on upper and lower levels are (see Landolfi & Landi
Degl’Innocenti, 1986; Landi Degl’Innocenti & Landolfi, 2004):

$\begin{split}&\dot{\rho}_{q}^{k}(J_{u})+2\pi
i\nu_{L}g_{u}q\rho_{q}^{k}(J_{u})=-\sum_{\begin{subarray}{c}J_{l}\end{subarray}}A(J_{u}\rightarrow
J_{l})\rho_{q}^{k}(J_{u})+\sum_{\begin{subarray}{c}J_{l}k^{\prime}\end{subarray}}[J_{l}]\\\
&\times\left[\delta_{kk^{\prime}}p_{k^{\prime}}(J_{u},J_{l})B_{lu}\bar{J}^{0}_{0}+\sum_{\begin{subarray}{c}Qq^{\prime}\end{subarray}}r_{kk^{\prime}}(J_{u},J_{l},Q,q^{\prime})B_{lu}\bar{J}^{2}_{Q}\right]\rho^{k^{\prime}}_{-q^{\prime}}(J_{l}),\end{split}$
(11) $\begin{split}\dot{\rho}_{q}^{k}(J_{l})&+2\pi
i\nu_{L}g_{l}q\rho_{q}^{k}(J_{l})=\sum_{\begin{subarray}{c}J_{u}\end{subarray}}p_{k}(J_{u},J_{l})[J_{u}]A(J_{u}\rightarrow
J_{l})\rho_{q}^{k}(J_{u})\\\
&-\sum_{\begin{subarray}{c}J_{u}k^{\prime}\end{subarray}}\left[\delta_{kk^{\prime}}B_{lu}\bar{J}^{0}_{0}+\sum_{\begin{subarray}{c}Qq^{\prime}\end{subarray}}s_{kk^{\prime}}(J_{u},J_{l},Q,q^{\prime})B_{lu}\bar{J}^{2}_{Q}\right]\rho^{k^{\prime}}_{-q^{\prime}}(J_{l}),\end{split}$
(12)

in which

$\begin{split}&p_{k}(J_{u},J_{l})=(-1)^{J_{u}+J_{l}+1}\left\\{\begin{array}[]{ccc}J_{l}&J_{l}&k\\\
J_{u}&J_{u}&1\end{array}\right\\},\\\
&p_{0}(J_{u},J_{l})=\frac{1}{\sqrt{[J_{u},J_{l}]}},\\\
&r_{kk^{\prime}}(J_{u},J_{l},Q,q)=(3[k,k^{\prime},2])^{\frac{1}{2}}\left\\{\begin{array}[]{ccc}1&J_{u}&J_{l}\\\
1&J_{u}&J_{l}\\\
2&k&k^{\prime}\end{array}\right\\}\left(\begin{array}[]{ccc}k&k^{\prime}&K\\\
q&q^{\prime}&Q\end{array}\right),\\\
&s_{kk^{\prime}}(J_{u},J_{l},Q,q)=(-1)^{J_{l}-J_{u}+1}[J_{l}](3[k,k^{\prime},K])^{\frac{1}{2}}\\\
&\times\left(\begin{array}[]{ccc}k&k^{\prime}&2\\\
q&q^{\prime}&Q\end{array}\right)\left\\{\begin{array}[]{ccc}1&1&2\\\
J_{l}&J_{l}&J_{u}\end{array}\right\\}\left\\{\begin{array}[]{ccc}k&k^{\prime}&2\\\
J_{l}&J_{l}&J_{l}\end{array}\right\\}.\end{split}$ (13)

The quantities $J_{u}$ and $J_{l}$ are the total angular momentum quantum
numbers for the upper and lower levels, respectively. The quantities
$\rho_{q}^{k}$ and $\bar{J}_{Q}^{K}$ are irreducible density matrices for the
atoms and the incident radiation, respectively. $6-j$ and $9-j$ symbols are
represented by the matrices with $"\\{$ $\\}"$, whereas $3-j$ symbols are
indicated by the matrices with $"()"$ (see Zare & Harter 1989 for details).
The second terms on the left side of Eq. (11) and Eq. (12) stand for the
magnetic realignment. The two terms on the right side represent spontaneous
emissions and the excitations from lower levels. Note that the symmetric
processes of spontaneous emission and magnetic realignment conserve $k$ and
$q$. Therefore, the steady state occupations of atoms on the ground state are
obtained by setting the left side of Eq. (11) and Eq. (12) zero888This is the
correct version of Eq. (8) in Yan & Lazarian (2008), in which there is a typo.
The term $\Gamma$ in Eq. (8) of Yan & Lazarian (2008) denotes $B_{lu}$ as a
common factor. But that will be an error for multi-level atoms since the
Einstein coefficients for transitions between different upper and lower levels
are different.:

$\begin{split}&2\pi
i\rho_{q}^{k}(J_{l})qg_{l}\nu_{L}-\sum_{\begin{subarray}{c}J_{u}k^{\prime}\end{subarray}}\bigg{\\{}p_{k}(J_{u},J_{l})\frac{[J_{u}]}{\sum_{J^{\prime\prime}_{l}}A^{\prime\prime}/A+i\Gamma^{\prime}q}\\\
&\times\sum_{\begin{subarray}{c}J^{\prime}_{l}\end{subarray}}B_{lu}[J^{\prime}_{l}]\left[\delta_{kk^{\prime}}p_{k^{\prime}}(J_{u},J^{\prime}_{l})\bar{J}^{0}_{0}+\sum_{\begin{subarray}{c}Qq^{\prime}\end{subarray}}r_{kk^{\prime}}(J_{u},J^{\prime}_{l},Q,q^{\prime})\bar{J}^{2}_{Q}\right]\\\
&-\delta_{J_{l}J^{\prime}_{l}}\left[\delta_{kk^{\prime}}B_{lu}\bar{J}^{0}_{0}+\sum_{\begin{subarray}{c}Qq^{\prime}\end{subarray}}B_{lu}s_{kk^{\prime}}(J_{u},J_{l},Q,q^{\prime})\bar{J}^{2}_{Q}\right]\bigg{\\}}\rho^{k^{\prime}}_{-q^{\prime}}(J^{\prime}_{l})=0\end{split}$
(14)

where $\Gamma^{\prime}$ equals $2\pi\nu_{L}g_{u}/A$. Magnetic realignment on
the levels in excited states can be neglected because the spontaneous emission
rate from the excited states is much higher than magnetic precession rate in
diffuse ISM and IGM. As a result, $\Gamma^{\prime}\simeq 0$. On the other
hand, the magnetic precession rate is much higher than the photon excitation
rate of the atoms on the ground state in the diffuse media of ISM and IGM
($\nu_{L}\gg B_{lu}\bar{J}^{0}_{0}$). Thus, Eq. (14) is making sense only when
$q=0$ so that the first term on the left equals 0. By solving the above
equations, the atomic density tensors on different levels
($\rho^{k}_{q}(J_{l}),\rho^{k}_{q}(J_{u})$) under the influence of atomic
alignment are obtained.


# Sc20C60: A Volleyballene

Jing Wang1,2 Hong-Man Ma1 Ying Liu1,3<EMAIL_ADDRESS>1Department of Physics
and Hebei Advanced Thin Film Laboratory, Hebei Normal University,
Shijiazhuang 050024, Hebei, China.
2State Key Laboratory for Superlattices and Microstructures, Institute of
Semiconductors,
Chinese Academy of Sciences, Beijing 100083, China.
3National Key Laboratory for Materials Simulation and Design, Beijing 100083,
China

###### Abstract

An exceptionally stable hollow cage containing 20 scandiums and 60 carbons was
identified. This Sc20C60 molecular cluster has a $T_{h}$ point group symmetry
and a volleyball-like shape, that we refer to below as ”Volleyballene”.
Electronic structure analysis shows that the formation of delocalized $\pi$
bonds between Sc atoms and neighboring five-membered carbon rings is crucial
for stabilizing the cage structure. A relatively large HOMO-LUMO gap was
found. The results of vibrational frequency analysis and molecular dynamics
simulations also demonstrate that this Volleyballene molecule is exceptionally
stable.

###### keywords:

Metallo-Carbohedrene, First-principles density functional theory, Stability
and electronic property

Since the experimental observation of C601, many very interesting structures
have been proposed, such as $M_{8}$C12 ($M$=Ti, V, Zr, Hf, et al)2, 3, Au204,
Au325, Au426, $M$@Sin($M$=Transition Metals; $n$=14,15,16)7, Eu@Si208,
Eu2Si309, B8010, and B${}_{40}^{-/0}$11. In the present work, an exceptionally
stable hollow cage, Sc20C60 Volleyballene, has been identified by first-
principles density functional theory studies.

Our first-principles calculations were performed within the framework of spin-
polarized density functional theory (DFT). The exchange-correlation
interaction was treated within the generalized gradient approximation (GGA)
using the Perdue-Burke-Ernzerhof (PBE) exchange-correlation functionals12. A
double-numerical polarized (DNP) basis set13 was chosen to carry out the
electronic structure calculation with unrestricted symmetry. For the
transition metal atoms, relativistic effects in the core were included by
using the DFT semi-core pseudopotentials (DSPP) 14.

Figure 1: The configurations of the Volleyballene Sc20C60 viewed in ($a$) ball
and stick model (Large ball: Sc atom; small ball: C atom) and ($b$) CPK style.
Part ($c$) shows a volleyball.

Figure 1 shows the configuration of the Volleyballene Sc20C60. It is comprised
of 20 scandium atoms and 60 carbons with a volleyball-like shape. This new,
lowest energy Sc20C60 molecule had $T_{h}$ point group symmetry and robust
stability. As may be seen from Fig. 1, it may be viewed as consisting of six
Sc8C10 subunits joined together in a crisscross pattern. In this structure,
there are 12 pentagonal rings made of carbon atoms (C-pentagons) and 6
octagonal rings of scandium atoms (Sc-octagons). Every group of two
C-pentagons is surrounded by one Sc-octagon to give a Sc8C10 subunit.

The 20 Sc atoms link to form 12 suture lines, and the average Sc-Sc length is
3.222 Å. Depending on the coordination of the Sc atoms, they may be divided
into two types, Sc${}^{\textrm{I}}$ and Sc${}^{\textrm{II}}$ (see Fig. 2b).
For the C-pentagons, there are three C-C double bonds and two C-C single
bonds. Of the three C-C double bonds, one has a length of 1.466 Å and the
other two both have lengths of 1.434 Å. The two C-C single bonds are both
1.446 Å long. Along with a 1.463 Å C-C bond connecting the two C-pentagons,
the average C-C bond length is found to be 1.446 Å. The average Sc-C bond
length is 2.248 Å.

The following aspects of the stability of the Volleyballene Sc20C60 were
investigated: the bonding character, the binding energy, the vibrational
frequencies and the molecular dynamics, the latter through ensemble
simulations.

Figure 2: ($a$) PDOS for the Volleyballene Sc20C60 , ($b$,$c$) the deformation
electron density viewed from the top of Sc8C10 subunit and the suture line
($c$), and ($d$) HOMO, ($e$) LUMO orbitals for Volleyballene Sc20C60 . The
zero of energy for the PDOS curve ($a$) is taken to be the Fermi energy, and
the length of each vertical line represents the corresponding relative
amplitude for each electronic orbital. The isosurface for the deformation
electron density (parts $b$ and $c$) and the two orbitals (parts $d$ and $e$)
are taken to be 0.030 and 0.005 $e/$Å3, respectively.

Firstly, the bonding character of Volleyballene Sc20C60 was investigated by
analyzing its deformation electron density, as shown in Figs. 2$b$ and 2$c$.
For the C atoms, there are obvious characteristics of $sp^{2}$-like
hybridization, and each C atom has three $\sigma$ bonds. For the C atoms that
neighbor pairs of Sc atoms, two of the three orbital lobes point towards the
two neighboring C atoms, and the third lobe points towards the center point of
the line joining the two Sc atoms but with a slight inclination toward
Sc${}^{\textrm{I}}$. For the other cases, the C atoms have three neighboring C
atoms and form $\sigma$ bonds between all four C atoms. An outer delocalized
electron is then left over for each carbon atom. The Sc atom has three outer
electrons, one lying in the 3$d$ state and the other two lying in the 4$s$
state. In each Sc8C10 subunit, a delocalized $\pi$ bond forms between the two
Sc${}^{\textrm{I}}$ atoms and one C atom located at the waist position. This
is crucial for stabilizing the Sc8C10 subunit. For the Sc${}^{\textrm{II}}$
atoms, there is a chiral $\pi$ bond three petals pointing towards the three
neighboring C-C bonds, which strengthens the link between the Sc8C10 subunits.

In order to further validate the stability of the Volleyballene Sc20C60, we
constructed five other molecular structures with different combinations of 12
C-pentagons and 20 Sc atoms. Calculations were carried out within the same
framework as described above. After energy minimization, it was found that the
optimized structures all showed relatively large deformations and did not
retain their original topologies. However, closer observation of these
configurations indicated that although the overall configurations were not
viable, one or more Sc8C10 subunit usually appeared, which further suggests
that the Sc8C10 subunit is very stable. Figure 3 shows three of the cage-like
geometries examined before and after optimizations. Beneath each isomer is
listed the relative binding energy ($\Delta E_{b}$) with respect to the
Volleyballene. Clearly, from the energy point of view, the Volleyballene
(shown in Fig. 1) was the lowest energy structure.

Figure 3: Three kinds of typical initial and optimized configurations of
Sc20C60 clusters. Beneath each isomer is listed the relative binding energy
($\Delta E_{b}$) with respect to Volleyballene Sc20C60 . Key: large ball, Sc
atom; small ball, C atom.

Next, a vibrational frequency analysis was carried out for the lowest energy
Volleyballene, Sc20C60. There were no imaginary frequencies and the two
highest intensity frequencies were found to be 468.9 and 472.3 $cm^{-1}$.
Figure 4 gives the calculated Raman spectrum with a 300 K temperature and
488.0 nm incident light, in order to simulate a realistic Raman spectrum that
can be compared to experimental results. The calculated results show that the
Volleyballene Sc20C60 is kinetically the most stable isomer.

Figure 4: The simulate Raman spectrum for the Volleyballene Sc20C60 with a 300
K temperature and 488.0 nm incident light. The Lorentzian smearing is set to
be 20.00 cm-1. The labels present the frequencies corresponding to the peaks
of the intensities.

In addition, ab initio molecular dynamics simulations with NVE and NVT
canonical ensembles were carried out to test the thermodynamic stability of
the Sc20C60 Volleyballene. For the NVE ensemble, the total simulation time was
set to be 2.0 $ps$ with a time step of 1.0 $fs$ at initial temperatures of
1000, 1400, 2000, and 2400 $K$. The results of the NVE simulations showed that
the structure of Volleyballene was not disrupted over the course of a 2.0 $ps$
dynamical simulation at an initial temperature of 2400 K, equal to a
$\sim$1200 K effective temperature. For the NVT ensemble, the Gaussian
thermostat15 was chosen, and the total run time was set to be 1.0 $ps$ with a
time step of 1.0 $fs$ at temperatures of $T$ = 800 and 1000 $K$. The NVT
dynamical simulations indicated that the Volleyballene Sc20C60 also retained
its original topological structure up to a temperature of 1000 $K$. These
results indicate that the Sc20C60 Volleyballene has good thermodynamic
stability.

Considering further the question of stability, it is natural to explore the
electronic structure. To this end, we calculated the partial density of states
(PDOS) and frontier molecular orbitals, including the highest occupied
molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of
Volleyballene as shown in Figures 2$a$, 2$d$, and 2$e$. From the contours of
the HOMO and LUMO orbitals, it may be seen that the HOMO orbitals are mostly
localized on the C atoms and Sc${}^{\textrm{I}}$ atoms. There is also obvious
$p$-$d$ hybridization. As for the case of the LUMO, the energy level features
show that the LUMO orbital is doubly degenerate. On the Sc atom, there are
obvious $d$-orbital characteristics, with $d_{z^{2}}$-like orbitals for
Sc${}^{\textrm{II}}$ atoms and other $d$-like orbitals for
Sc${}^{\textrm{I}}$. The LUMO orbital hybridization is predominantly $sp$-$d$
hybridization. Close examination of the PDOS (see Fig. 2a) further confirms
the hybridization characteristics of the HOMO and LUMO orbitals. All these
results demonstrate that the hybridization between Sc $d$ orbitals and C $p$
orbitals is essential for stabilizing the cage structure. For the
Volleyballene Sc20C60, a relatively large HOMO-LUMO gap of 1.471 eV was
observed, mainly due to the energy of the d atomic orbitals being much lower
than that of the $p$ orbitals. The Volleyballene Sc20C60 should therefore be
an exotic buckyballene variant with exceedingly high chemical stability.

The authors thank the science writers’ reports16 on this work and Dr. N. E.
Davison for his help with the language. This work was supported by the
National Natural Science Foundation of China (Grant Nos. 11274089 and
11304076), the Natural Science Foundation of Hebei Province (Grant Nos.
A2012205066 and A2012205069), and the Science Foundation of Hebei Education
Award for Distinguished Young Scholars (Grant No. YQ2013008). We also
acknowledge partially financial support from the 973 Project in China under
Grant No. 2011CB606401.

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