Journal of Nuclear and Radiochemical Sciences, Vol. 7, No.1, pp. R7-R12, 2006

Nuclear Halo and Its Related Reactions
Zhang Huanqiao*
China Institute of Atomic Energy, 102413 Beijing, China
Received: November 8, 2005
The experimental results of the proton halo nuclei 27P, as well as probably 29S, and the neutron halo in excited statesof the nuclei 12B and 13C are presented. The probability of the valence nucleon(s) being out of the binding potential
have been extracted systematically from the relevant experimental data, and a new condition for nuclear halooccurrence is proposed. Based on our systematical study, we put forward our scaling laws of nuclear halo in termsof the analytical expressions of the expectation value for the operator r2 in a finite square-well potential. In order
to investigate the breakup effects of the weakly bound projectiles on fusion reactions and barrier distributions, thecomplete fusion cross sections of 6Li + 208Pb and the elastic and quasi-elastic excitation functions of 6,7Li, 9Be +208
Pb were measured and compared with the predictions of the coupled-channels model. The dynamical effects on
fusion of neutron-rich nuclei at the energies around the Coulomb barrier are discussed in terms of the comparisonof the measured fusion data of 48Ca + 90, 96Zr with the calculations of the improved quantum molecular dynamics
model and coupled-channels theory.

2.4

1. Introduction*Corresponding author. E-mail: huan@iris.ciae.ac.cn. Fax: 008610-69357787

N =13
2.2

2.0

σR (b)The neutron and proton density distributions in nucleus arevery important quantities for understanding various nuclear
properties such as nuclear potential, single-particle orbits, andmagic numbers. So far, the experimental and theoretical studies of the nuclear density distributions have resulted in a lot ofbasic and meaningful information on nuclear structures. Oneof the most notable results is the discovery of two-neutron halo
nucleus 11 Li by Tanihata et al.1 at Lawrence Berkeley
Laboratory in 1985. Nuclear halo is a threshold phenomenon.
As the binding energy becomes small, the wave function ofvalence particle extends more and more outward if the centrifugal barrier is small. Eventually, this leads to the wave function penetrating substantially beyond the range of nuclear force
as the binding energy approaches zero, i.e. occurrence of
nuclear halo. Nucleus 11Li can be viewed as a core plus twovalence-neutrons. The density distributions of these two neutrons spread out to distances far from the binding potential. Itsroot mean square (rms) radius can be comparable with the sizeof 208Pb nucleus.
In this paper, we will present our main experimental resultsof the proton halo nuclei 27 P, as well as 29S, and the neutron
halo in excited states of the nuclei 12B and 13C. We also systematically have extracted the probability of the valence
nucleon(s) being out of the binding potential from the relevant
experimental data and proposed a much relaxed condition for
nuclear halo occurrence. Based on our systematical study, we
put forward our scaling laws of nuclear halo in terms of theanalytical expressions of the expectation value for the operator
r2 in a finite square-well potential. One of the characters of
halo nuclei is weak bounding of the valence nucleon(s).
Nuclear reactions induced by halo and weakly bound nuclei are
a topic of current interest. In order to investigate the breakup
effects of the weakly bound projectiles on fusion reactions andbarrier distributions, the complete fusion cross sections of 6Li+ 208Pb and the elastic and quasi-elastic excitation functions of
6,7
Li + 208Pb were measured and compared with the predictionsof the coupled-channels model. Finally, we will discuss the
dynamical effects on fusion of neutron-rich nuclei at the ener-

1.8

N =12

3.2

Data
Glauber

2.4

1.6

11

12

13

14

15

Z
Figure 1. The Z dependence of the reaction cross sections for theisotones with N=12 and N=13. The solid squares with error bar represent the experimental data. The open circles illustrate the predictionof the modified Glauber model in the OL approach. The symbols areconnected by lines for each isotonic number to guide the eye.

gies around the Coulomb barrier in terms of the comparison of
the measured fusion data of 48Ca + 90,96Zr 2, 3 with the calculations of the improved quantum molecular dynamics (ImQMD)
model.4
2. Proton HaloThe existence of proton halos in 26 P and 27S was first predicted by Ren et al. 5 By further investigation, Brown and
Hansen6 pointed out that 26,27P and 27,29S are good candidates
for halo nuclei. The properties of proton-rich nuclei in the 2s1d
shell were also investigated in the framework of relativistic
mean-field model by Chen et al.7 Their results show that there© 2006 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 06/24/2006

R8

Zhang

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006
0.30

2.6

27

P

0.25

2.4

28P

<rC > =2.997 fm
2 1/2<rh > =4.016 fm

0.20

N=12
N=13

0.15

d

σR (b)

2.2

2 1/2

29

0.10

2.0

S

0.05
27

Si <r2>1/2=2.854 fm

1.8

0.00

1.6
0

20

40

60

80

100

-0.05
10

11

12

Figure 2. The measured reaction cross sections vs energy for the27
Si, 28 P+ 28 Si reactions. The predictions of the modified Glaubermodel of the OL approach (open circles) are compared with theexperimental data.
3.6

σR (b)

2.8

16

17

σ R (exp) – σ R (cal),σ R (cal)(1)

here σ R (exp) represents the measured total reaction cross sections and σ R (cal) is the theoretical values with the modified
Glauble theory of the OL and FB approaches. Figure 4 displays the Z dependence of d for N=12 and 13 isotones at 40
MeV/u. We find from the d values that 27P and 29S have a proton halo and a moderate proton halo structure, respectively.

OL a=0.9 fm
FB a=0.9 fm
OL a=1.1 fm
FB a=1.1 fm

P

15

Figure 4. Z dependence of the different factor d for N=12 and 13
isotones. The symbols are connected by lines to guide the eye.

d=

27

14

Z

E Lab (MeV/u)

3.2

13

3. Nuclear Halo of Excited State

2.4

26

2.0

Si
OL a = 0.9 fm

1.6
0

20

40

60

80

100

E Lab (MeV/u)
Figure 3. The measured reaction cross sections as a function ofenergy for the 26Si, 27P + 28Si reactions. The predictions of the modified Glauber model of OL and FB approaches (open symbols) arecompared with the experimental data. The numbers in the figure represent the diffuseness parameter of the Woods-Saxon potential for thevalence proton.

are one-proton halos in 26,27,28P and two-proton halos in 27,28,29S.
In order to search proton halo, the reaction cross sections of
27,28
P, 29S and the corresponding isotones on Si target8,9 were
measured at intermediate energies. As shown in Figure 1, themeasured reaction cross sections of the N=12 and 13 isotones
show an abrupt increase at Z=15. The experimental results forthe isotones with Z ≤ 14 as well as 28P can be well described by
the modified Glauber theory of the optical limit (OL) approach.
The enhancement of the reaction cross sections for 28P could be
explained in the modified Glauber theory with an enlarged
core as illustrated in Figure 2. The <rc2>1/2 and <r h2>1/2 are therms radii of the core and valence proton in 28P, respectively.
The difference between the rms radii of the core and barenuclei 27Si is about 0.15 fm. It is seen from Figure 3 that theoretical analysis with the modified Glauber theory of the OL
and few-body(FB) approaches10–13 underpredictes the experimental data of 27P. The difference between the rms radii of thecore and bare nuclei 26 Si is about 0.28 fm. Our theoreticalanalysis shows that an enlarged core together with proton halo
are probably the mechanism responsible for the enhancementof the cross sections for the reaction of 27 P + 28Si. Ozawa etal.14 proposed to take a difference factor as a possible measureof halo appearance. The difference factor d is defined as

Except the nuclei near or at drip-lines, halo may appear in
the excited states of stable nuclei. By means of the asymptotic
normalization coefficients (ANC’s) extracted from transfer
reactions of 11B(d, p)12 B, 12 C(d, p)13C, and H(6 He, n) 6 Li, we
have verified that the second (J π = 2 –) and third (J π = 1–) excitedstates in 12B and the first (J π = 1/ 2 +) excited state in 13C are theneutron halo states,15 while the second excited state (3.56 MeV,J π = 0 +) in 6Li is a proton-neutron halo state.16
3.1. Root-Mean-Square Radius. The rms radius of valencenucleon(s) in the orbit (nlj) can be separated into the contributions from the interior and the asymptotic regions,15,17∞

RN<r 2 >1/2 = [ ∫0 r 4 φ 2nlj (r)dr + (C BANlj )2 ∫ RN r 2 W 2– η,l+1/2 (2kr)dr ]1/2 .

(2)Here φ nlj (r) is the single-particle radial wave-function in the(nlj) bound state, C BANlj is the nuclear ANC, and W – η,l+1/2 (2kr)
the Whittaker function. The first term in the equation is somehow parameter dependant. The second term is model parameter independent, and gives more than 90%11 contribution to thevalue of rms radius. Thus the uncertainty introduced by theparameters used is small. Hence, <r 2 >1/2 evaluated in this way
is a nearly parameter independent quantity which can be served
as an ideal experimental observable for the examination of
nuclear halo. The rms radii for the valence particle extracted
from ANC values are listed in Table 1 for the excited states in
12
B and 13C. In the table, rc and Rsq = 5 (rc2 + 4) in the unit of3
fm are the radii of the core nucleus and the square-well potential, respectively.18
3.2. Probability. The probability15, 19 for valence particle
being out of the binding potential can be evaluated by,∞ 2 2

P(R,r0 ,a) =∫R r φ nlj (r)dr,∞ 2 2∫0 r φ nlj (r)dr(3)

where the r0 and a are the radius and diffuseness parameters of
the potential, respectively. The binding potential radius R is

Nuclear Halo and Its Related Reactions

R9

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006

TABLE 1: Values of <r 2 >1/2 and <r 2 >/rc, <r 2 >/Rsq2 and the probability for valence nucleon being out of the binding potential
for nucleus 12B and 13C
Nucleus

Jπ

Ex
(MeV)

rc
(fm)

<r 2>1/2(fm)<r 2>1/2 /rc<r 2>/R2sq

Probability
(%)

12

B

1+
21-

0
1.674
2.621

2.50

3.16 ± 0.10
4.01 ± 0.61
5.64 ± 0.90

1.26 ± 0.05
1.60 ± 0.29
2.26 ± 0.42

0.58 ± 0.15
1.10 ± 0.31
1.86 ± 0.60

19.9
53.6
66.8

13

C

1/21/2 +

0
3.089

2.42

3.39 ± 0.31
5.04 ± 0.75

1.413 ± 0.15
2.08 ± 0.37

0.70 ± 0.21
1.55 ± 0.47

14.3
50.3

10

-3

100

(a)

90

10

Be

80

-5

χ
10

11

C

2

10

19

-4

probability (%)

10

0.50
0.55
0.60
0.65
0.70

-6

-7

12

B(1-)

70

14

C(0 -)

13

12

-

C

B(2 )

60

14

C(1-)

50
40
30
0.0

0.2

0.4

0.6

0.8

1.0

1.2
2

2 µ Bp R / h

10

C(1/2 +)

15

-8

1.4

1.6

1.8

2.0

2

100

44

46

48

50

52

54

56

58

60

62

64

probability (%)

90
19Figure 5. Dependence of the χ on the probability (P) for valence
nucleon being out of the binding potential for the 2s excited state in
13
C. Symbols are connected by a line for each a 0 value to guide theeye.

Ctaken as an equivalent square-well potential radius Rsq. Hencethe single-particle wave functions are not determined uniquely.
We search the potential parameters in such a way 19 that thequantity,40fm

χ = Σ ( [ S ] φ (ri)–C2prr = RN(sp) 1/2
lj
nlj

B
Aplj

W– η,l+1/2 (2kri) 2
),
ri

(b)

11

Be

80

12

B(1 -)

Therefore, the range of nuclear halo occurrence is much
expended than the previous one,16 i.e., Bp A2/3 ≤ 2 MeV.20 HereBp is the binding energy of the valence nucleon and A is themass number of the nucleus.
3.3. Scaling Law. Hamamoto et al.21 have obtained theexpressions of the expectation value for the operator r 2 in aC(0 -)

13

C(1/2+ )14

B(2 -)

C

60

-

C(1 )

12

15

50

40
0

(5)

14

70

2

4

(4)

becomes minimum. Where S lj ( s p ) is the single particle
spectroscopic factor. First, the single-particle wave function is
calculated with Woods-Saxon potential. The depth of thepotential is adjusted to reproduce the binding energy. The
values of a are chosen in the range of 0.50–0.70 fm. For each
fixed a, r0 is varied in a small step till the minimum in χp2 isreached. In the calculation of χp2 , the summation runs fromri =6.0 to 40 fm in steps of 0.1 fm. We find that the values ofχp2 does not change as long as R N ≥ 6 fm. Figure 5 shows atypical example for the extraction of the probability for valence
nucleon(s) being out of the binding potential. With thisprocedure, available data regarding the nuclear halo candidates
are systematically analyzed and a number of halo nuclei are
confirmed. The extracted probabilities for 12B and 13C are also
listed in Table 1. Figure 6 shows the probability for valence
nucleon being out of the binding potential as a function of
2 µ Bp R 2 / h 2 (a) and Bp A2 / 3 (b). Based on these results andaccording to the probability of finding valence particle outside
the bounding potential larger than 50%, we have got a
condition for nuclear halo occurrence,19
Bp A2/3 ≤ 10 MeV.

probability (%)

2

6

8

Bp A

2/3

10

12

14

(MeV)

Figure 6. Probability for valence nucleon being out of the binding
potential as a function of 2 µ Bp R 2 / h 2 (a) and Bp A2/3 (b). The solid
points represent the s-wave halos in the ground state and in the
excited states with spin and parity values in brackets, respectively.
The dash-double-dotted line is the calculation value by means of thesingle-particle model with square-well potential. The other lines
show the predictions of the single-particle models with Woods-Saxon
potentials.19 The arrows in the panel (b) illustrate the up-limits ofBp A2/3 value set by Jensen et al.20 and by us.

finite square-well potential. The terms with ξ 04 in denominatorin their expressions are of negligible magnitude as compared tothe other terms for the case of χ2 < 2 which we are interested
in. After omitting them, we get the following scaling laws forthe dimensionless quantity <r 2>/R 2,

(

)(

) (

[(

)( (1 +3χ ) + χ + 3 + 25χ ) + (12+ξχ ) ] + ( 2 + 23χξ + χ )χ2< r2 >
1
1
1
1 1 + 2χ=1 − 2 1+ + 2 + χ +
2
R
χ +1
ξ0
χ 2χ
3 2ξ 02
χ2
1< r2 >
1− 2= 2
R2
χ + 3χ + 3
ξ0) at l = 0, (6)

2

2

2

2
0

2
0

at l = 1,

(7)

where
χ =R

ξ0= R

2µ Bp

,

(8)

2µ U0
.
h2

(9)

h

2

R10 J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006
1

10

10

2

10

1

10

0

40

2

10

Zhang
3

2

10

σ (mb)< r > /R

40

0neutron:2s ground state
2s excited state

10

48

90

Ca+ Zr exp.
90
Ca+ Zr ImQMD

48

90

Ca+ Zr exp.
90
Ca+ Zr ImQMD

-1

10

2

10

1

10

0

proton:2s excited state
2s
1p

40
40

-1

10

0.0

0.4

0.6

0.8

1.0

1.2
2

1.4

1.6

1.8

2.0

2.2

σfus (mb)

10

10

10

3

2

1

exp
- 208
+
6
3 ,5 in Pb, 3 in Li- 208
3 ,5 in Pb
no coupling

0

-1

24

96

Ca+ Zr exp.
96
Ca+ Zr ImQMD

26

28

30

32

34

10
0.85

0.90

0.95

1.00

1.05

1.10

1.15

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

E cm / Vb

2

Figure 7. The experimental data of <r2>/R 2 vs 2 µ Bp R 2 /h 2 for thevalence nucleon in 2s state. The solid and dashed lines show the scaling laws of eqs 6 and 7.

10

48

-1

0.2

2 µ Bp R /h

10

48

96

Ca+ Zr exp.
96
Ca+ Zr ImQMD

36

38

40

42

Ec.m.(MeV)

Figure 8. Total fusion cross sections for 6 Li + 208 Pb. The solid
squares are the results of our experiment. The solid, dashed and dashdotted lines are the results of the coupled-channels code CCFULL
calculations with coupling the 3-, 5- excitation states in 208Pb and the
3 + excitation state in 6Li and with coupling the 3-, 5- excitation statesin 208Pb, as well as without coupling, respectively.

In the above equations, U0 is the depth and R is the range of thefinite square-well potential and µ and Bp the reduced mass andbinding energy, respectively. Figure 7 displays the scaling laws
for the dimensionless quantity <r 2>/R2 of the halo nuclei withcomparison of experimental data. It can been seen from thefigure that the scaling law can account for the available
experimental data of halo candidates, even though it is derived
in a finite square-well potential.
4. Reactions Induced by Weekly Bound Nuclei
We have measured the complete fusion cross sections of 6Li+ 208 Pb, and found that the fusion cross sections above the
Coulomb barrier are suppressed due to the breakup effects ofweekly bound projectile 6 Li. 22 The experimental results
together with the theoretical calculations are shown in Figure
8. The lines in the figure are the coupled-channels code
CCFULL calculations.23-25 The deformation parameters usedin the calculations (assuming single phonon excitations) are
β 3 = 0.161 25; and β 5 = 0.056 27 for 3 - (2.615 MeV) and 5- (3.198
MeV) states in 208 Pb, respectively, and β 2 = 1.51 28 for the3+ (2.186 MeV) in 6Li. The coupled-channels model (CCFULL)
does not consider the influence of breakup effect on fusion.
We have also measured the elastic and quasi-elastic excitation functions of 6,7Li, 9Be + 208Pb at backward angles. Fromthese excitation functions, barrier distributions29 are extracted

Figure 9. The fusion cross sections for 40,48Ca + 90,96Zr. The sold circles denote the experimental data. The open triangles denote the calculation results with the ImQMD model, respectively.

and compared with the predictions of the coupled-channels
model. It is found that the barrier distributions are broaden andshift to lower energies which may be a signature of the breakup
effects of the weakly bound projectiles. Gomes et al.30 study
the behavior of the fusion, breakup reaction, and elastic scattering of different projectiles on 64Zn at near and above barrierenergies. They found that the elastic (noncapture) breakup
cross section is important at energies close to and above the
Coulomb barrier and increases the reaction cross sections. In
addition, they showed that the breakup process at near andbelow barrier energies is responsible for the vanishing of theusual threshold anomaly of the optical potential and gives rise
to a new type of anomaly. Recently, Newton et al. 31 systematically analyzed the high precision fusion cross sections for
Zp Z t < 1000 with the coupled channels models and found thatfits to the experimental fusion excitation functions need a valueof a of the Woods-Saxon potential from 0.75 to 1.5 fm, which
is much larger than the commonly accepted value of ≈ 0.65 fm.
They think the large value of a may indicate the necessary to
go beyond the potential model, and incorporate dynamical
effects as the two nuclei move towards fusion, even for lightsystems with Zp Zt < 1000.
5. Fusion Reactions with Neutron-Rich Nuclei
The dynamical effects on fusion of neutron-rich nuclei at theenergies around the Coulomb barrier attract much attention inrecent years. We have cooperated with Stefanini group from
LNL/INFN to measure the fusion cross sections of 48 Ca +9 0,9 6Zr at the XTU-16 Tandem accelerator facility of theLaboratori Nazionali di Legnaro, Italy. Scarlassara et al. 3 havereported the preliminary experimental results of the reactions
48
Ca + 90,96Zr in the International Conference FUSION03 at
Matsushima, Miyagi, Japan. The comparison of experimental
40,48
Ca + 90,96Zr fusion data 2,3 shows that fusion of 40Ca + 96Zr ismuch enhanced at near- and sub-barrier energies due to thepositive Q-values of the transfer channels. The much larger
enhancement for the 40 Ca + 96Zr as compared to other threesystems clearly indicates that neutron transfer with the positive
Q-value should play a significant role in sub-barrier fusion. In
order to extract the information on the complex dynamical
effect, especially the effect of excess neutrons on fusion reactions, we have calculated the fusion excitation functions with
the improved quantum molecular dynamics model 4,32 for thesystems of 40,48 Ca + 90,96 Zr around the Coulomb barrier andcompared with the experimental data of these four systems in
Figure 9. It is shown by this comparison that neutron transfer
starts far before the system reaches the top of the barrier andNuclear Halo and Its Related Reactions
40

90

J. Nucl. Radiochem. Sci., Vol. 7, No. 1, 2006 R1190

Ca+ Zr90
48
Ca+ Zr96
40
Ca+ Zr96
48
Ca+ Zr80

b=0 fmpotential (MeV)70

5MeV above the barrier(Bass)
90

80

90

40

Ca+ Zr48

90

Ca+ Zr96
40
Ca+ Zr96
48
Ca+ Zr70

b=0 fm3MeV below the barrier(Bass)

60
5.0

7.5

10.0

12.5

15.0

17.5

distance (fm)Figure 10. The dynamic barrier as a function of the distance between
the centers of mass of projectile and target for head on reactions of
40,48
Ca + 90,96Zr at the incident energy of 5 MeV above and 3MeV
below the Bass barrier, respectively.

the flow of neutron between the reaction partners results in theneck formation. Consequently, the fusion barriers are strongly
affected by the neutron transfer. Our calculations indicate thatthe neck grows faster and the distance between two partners is
shorter for 40 Ca + 96 Zr than other three systems, resulting astronger reduction of the dynamic barrier as shown in Figure
10. We, therefore, can attribute from this calculation that the
reduction of the barrier encountered by 40Ca + 96Zr is due to thestrong dynamical effect of the neutron transfer, which leads tothe enhancement of the sub-barrier fusion cross sections for40
Ca + 96Zr.
In very recent, Stefanini et al.33 calculated the fusion excitation functions and barrier distributions for the 48Ca + 90,96Zr
systems by means of coupled-channels theory with CCFULL
code. The sub-barrier fusion of 48Ca + 90Zr is reproduced bycoupled-channels calculations including the lowest quadrupole
and octupole vibrations of 90 Zr, and using a Woods-Saxon
potential with standard diffuseness parameter a = 0.68 fm.
However, the fusion cross sections are overestimated above thebarrier. The low-energy slope of the excitation function for48
Ca + 96Zr is steeper. This implies a larger diffuseness parameter a=0.85 fm. Fusion cross sections are well fit in the wholeenergy region, and the effect of the strong octupole vibration in96
Zr is predominant. The extracted fusion barrier distributions
are reasonably well reproduced by calculations for both systems. A comparison with previous data for 40 Ca + 90,96Zr is
made in an attempt to clarify the role of transfer couplings in
sub-barrier fusion.
5. Summary
The reaction cross sections of 27,28P, 29S and the corresponding isotones on Si target were measured at intermediate energies. The theoretical analysis with the modified Glauber
theory of the optical limit and few-body approaches underpre-

dicts the experimental data of 27 P. Our theoretical analysis
shows that an enlarged core together with proton halo are probably the mechanism responsible for the enhancement of thecross sections for the reaction of 27P + 28Si. We find from thedifferent factor values that 29S may have a moderate proton halo
structure. By means of the asymptotic normalization coefficients extracted from transfer reactions of 11B(d, p)12 B and
12C(d, p)13C, we have verified that the second (J π = 2 – ) and third(J π = 1 – ) excited states in 12 B and the first (J π = 1/2 + ) excitedstate in 13C are the neutron halo states. The probability of thevalence nucleon(s) being out of the binding potential have been
extracted systematically from the relevant experimental data
and a much relaxed condition for nuclear halo occurrence proposed. Based on our systematical study, we put forward our
scaling laws of nuclear halo in terms of the analytical expressions of the expectation value for the operator r 2 in a finite
square-well potential. In order to investigate the breakup
effects of the weakly bound projectiles on fusion reactions andbarrier distributions, the complete fusion cross sections of 6Li+ 208Pb and the elastic and quasi-elastic excitation functions of
6,7
Li,9Be + 208Pb were measured at the back angles and compared with the predictions of the coupled-channels model. It isfound that the fusion cross sections above the Coulomb barrier
are suppressed due to the breakup effects of weakly bound projectile 6Li and the barrier distributions are somehow broaden
and shift to lower energies. The dynamical effects on fusion ofneutron-rich nuclei at the energies around the Coulomb barrier
are discussed in terms of the comparison of the measured
fusion data of 48 Ca + 9 0,96 Zr with the calculations of the
ImQMD model. Our study shows that the dynamical effects
triggered by the neutron transfer play important role in fusion
reactions. We also introduce the very recent results calculatedwith coupled-channels theory.
Acknowledgments. This work was supported by the
National Natural Science Foundation of China under Grant No.
10275092, 10275095, 10235030.
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