ISSN: 0256-307X 中国物理快报 Chinese Physics Letters

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CHINESE PHYSICAL SOCIETY CHIN. PHYS. LETT. Vol. 27, No. 12 (2010) 121201

Analytical Solution for the 푆푈(2) Hedgehog Skyrmion and Static Properties of Nucleons * JIA Duo-Jie(贾多杰)1**, WANG Xiao-Wei(王晓维), LIU Feng(刘锋) Institute of Theoretical Physics, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070

(Received 5 May 2010) An analytical solution for symmetric Skyrmion is proposed for the 푆푈(2) Skyrme model, which takes the form of the hybrid form of a kink-like solution, given by the instanton method. The static properties of nucleons is then computed within the framework of collective quantization of the Skyrme model, in a good agreement with that given by the exact numeric solution. The comparisons with the previous results as well as the experimental values are also presented.

PACS: 12. 38. −t, 11. 15. Tk, 12. 38. Aw DOI: 10.1088/0256-307X/27/12/121201

[1] [15] SK 2 The Skyrme model is an effective field theory of ity for (1) implies 퐸 ≥ 6휋 (푓휋/푒)|퐵|, where 2 ∫︀ 3 푖푗푘 mesons and baryons in which baryons arise as topo- 퐵 ≡ (1/24휋 ) 푑 푥휀 푇 푟(퐿푖퐿푗퐿푘) is the topologi- logical soliton solutions, known as Skyrmions. The cal charge, known as the baryon number. Using the model is based on the pre-QCD nonlinear 휎 model of hedgehog ansatz, 푈(푥) = cos(퐹 ) − 푖(ˆ푥 · 휎) sin(퐹 ) (휎 the pion meson and was usually regarded to be consis- are the three Pauli matrices) with 퐹 ≡ 퐹 (푟) depend- tent with the low-energy limit of large-푁 QCD.[2] For ing merely on the radial coordinate 푟, the static energy this reason, among others, it has been extensively re- for (1) becomes visited in recent years (see Refs. [3–9], and Refs. [10,11] (︁푓 )︁ ∫︁ [︁ for review). Owing to the high nonlinearity, the solu- 퐸SK = 2휋 휋 푑푥 푥2퐹 2 + 2 sin2(퐹 )(1 + 퐹 2) tion to the Skyrme model was mainly studied through 푒 푥 푥 the numerical approach. It is worthwhile, however, sin4(퐹 )]︁ [12−14] + , (2) to seek the analytic solutions of Skyrmions due 푥2 to its various applications in baryon phenomenology. One of the noticeable analytic methods for studying with 푥 = 푒푓휋푟 a dimensionless variable and 퐹푥 ≡ the Skyrmion solutions is the instanton approach pro- 푑퐹 (푥)/푑푥. The equation of motion of (2) is posed by Atiyah and Manton,[8] which approximates (︂ sin2 퐹 )︂ 2 critical points of the Skyrme energy functional. 1 + 2 퐹 + 퐹 In this Letter, we address the static solution of 푥2 푥푥 푥 푥 hedgehog Skyrmion in the 푆푈(2) Skyrme model with- sin(2퐹 ) (︂ sin2 퐹 )︂ out pion mass term and propose an analytical solution + 퐹 2 − 1 − = 0, (3) 푥2 푥 푥2 for the hedgehog Skyrmion by writing it as the hybrid form of a kink-like solution and the analytic solution where the boundary condition 퐹 (0) = 휋, 퐹 (∞) = 0 [8] obtained by the instanton method. Two lowest-order will be imposed so that it corresponds to the physi- Padé approximations were used and the correspond- cal vacuum for (1): 푈 = ±1. Equation (3) is usually ing solutions for Skyrmion profile are given explicitly solved numerically due to its high nonlinearity.[3−6] by using the downhill simplex method. The Skyrmion A kink-like analytic solution was given by[5] mass and static properties of nucleon as well as delta were computed and compared to the previous results. 퐹1(푥) = 4 arctan[exp(−푥)], (4) The 푆푈(2) Skyrme action[1] without pion mass SK 2 term is given by with 퐸 = 1.24035(6휋 푓휋/푒), while an alternative Skyrmion profile, proposed based on the instanton ∫︁ [︂ 푓 2 1 ]︂ 풮SK = 푑3푥 − 휋 푇 푟(퐿 )2 + 푇 푟([퐿 , 퐿 ]2) , method, takes the form[8] 4 휇 32푒2 휇 휈 (1) [︁ (︁ 휆 )︁−1/2]︁ † 퐹 (푥) = 휋 1 − 1 + , (5) where 퐿휇 = 푈 휕휇푈, 푈(푥, 푡) ∈ 푆푈(2) is the 2 푥2 nonlinear realization of the chiral field describing 2 the 휎 field and 휋 mesons with the unitary con- with the corresponding energy 1.2432(6휋 푓휋/푒) for the † strain 푈 푈 = 1, 2푓휋 the pion decay constant, numeric factor 휆 = 2.109. The singularity at 푟 = 0 and 푒 a dimensionless constant characterizing non- is gauge dependent and can be gauged away without linear coupling. The Cauchy–Schwartz inequal- affecting the value for the Skyrme field.

*Supported in part by the National Natural Science Foundation of China under Grant Nos 10547009 and 10965005, and the Scientific Research Foundation (SRF) for the Returned Overseas Chinese Scholars (ROCS), State Education Ministry. **Email: [email protected]; [email protected] ○c 2010 Chinese Physical Society and IOP Publishing Ltd

121201-1 CHIN. PHYS. LETT. Vol. 27, No. 12 (2010) 121201

Table 1. The Skyrmion energy from different theoretical calculations. Work [3][13][6][14][12] Hyb(2/2) Hyb(4/4) Num. SK 퐸 36.5 36.47 36.484 36.474 36.47 36.4638 36.4638 36.4616 (2푓휋 /푒) 푀 2 1.233 1.2317 1.2322 1.23186 1.2317 1.23152 1.23146 1.23145 (6휋 푓휋 /푒) To find the more accurate analytic solution, we profile 퐹 (푥). For small 푥 ≪ 1 the solution 퐹 (푥) to first improve the solution (4) into 4 arctan[exp(−푐푥)] Eq. (3) is given by with 푐 a numeric factor and then take 휆 to be a 푥- 퐹 (푥) = 휋 + 퐴푥 + 퐵푥3 + 퐶푥5 + ··· dependent function: 휆 → 휆(푥). Hence, we propose a 3 Skyrmion profile function in the hybrid form mixing = 휋 − 2.007528푥 + 0.358987푥 (4) and (5), − 0.146499푥5 + ··· , (10) (also see Ref. [12], where the variable 푥 used is twice of 퐹푤(푥) = 4푤 arctan[exp(−푐푥)] 푥 in this study) while the analytic solution (6), when [︁ (︁ 휆(푥))︁−1/2]︁ + 휋(1 − 푤) 1 − 1 + , (6) (8) and (9) is used, behaves like 푥2 3 5 퐹푤(푥) = 휋−2.028368푥+0.518855푥 −0.641539푥 +··· . with 푤 ∈ [0, 1] being a positive weight factor. In prin- (11) ciple, one can find the governing equation for the un- One can see that Eq. (11) agrees well with Eq. (10) up known 휆(푥) by substituting Eq. (6) into Eq. (3) and to 푥6. For large 푥 → ∞ the series solution for 퐹 (푥) obtain a series solution of 휆(푥) by solving the gov- can be obtained by solving Eq. (3) with 푥 replaced by erning equation. Here, however, we choose the Padé 1/푦 and using the series expansion for small 푦. After approximation to parameterize 휆(푥) re-changing 푦 to 푥, one finds {︁ 2 2.1596 0.222 116.0 1 + 푎푥 + ··· 퐹 (푥) = 2 1 − 4 − 6 휆(푥) = 휆0 , (7) 푥 푥 푥 1 + 푏푥2 + ··· 0.113 2.71 }︁ + 8 + 10 + ··· . (12) since it has an equal potential as series in approximat- 푥 푥 ing a continuous function. Note that we have already On the other hand, the solution Hyb(2/2), at large 푥, written 휆(푥) as a function of 푥2 instead of 푥 since so has the asymptotic form is 퐹2(푥) in Eq. (5). The simplest nontrivial case of the 2.0348{︁ 0.91576 5.8524 9.6819 퐹 (푥) = 1 + − + above Padé approximation is the [2/2] approximant 푤 푥2 푥2 푥4 푥6 3.1677 51.943 }︁ 1 + 푎푥2 + − + ··· , (13) 휆(푥) = 휆 . (8) 푥8 푥10 0 1 + 푏푥2 which agrees globally with Eq. (12) except for a small The minimization of the energy (2) with the trial bit difference. The detailed differences between (12) function (6) with respect to the variational parame- and (13) at large 푥 can be due to the fact that the variationally-obtained solution (6) approximates the ters (푎, 푏, 푐, 푤, 휆0) was carried out numerically for the [2/2] Padé approximant (8) using the downhill simplex Skyrmion profile globally and may produce small er- method (the Neilder-Mead algorithm). The result for rors in local region, for instance, 퐹푤(50) = 8.142 × −4 −4 the optimized parameters is given by 10 while 퐹 (50) = 8.638 × 10 .

푎 = 0.330218, 푏 = 1.331975, 푐 = 2.094056, 3.5

푤 = 0.286566, 휆0 = 7.323877, (9)

3.0

Hyb(2/2)

SK 2 Hyb(4/4) with 퐸 = 1.23152(6휋 푓휋/푒). The solution (6), with 2.5

The solution (4)

휆(푥) given by Eq. (7) and the parameters (9), is re- The solution (5)

2.0 ferred as the solution Hyb(2/2) in brief in this study Num.

Ponciano and is plotted in Fig.1, compared to the solutions

Yamashita et al (4) and (5), and the numerical solution (Num.) to 1.5

Eq. (3). We also include the analytic solutions given 1.0 in Ref. [14] and the solution in the form of the purely Chiral angle Padé approximant[13] for comparison. A quite well 0.5 agreement of our solution with the numerical solution 0.0

can be seen from this plot. We note that the inequal- 0 2 4 6 8 10 SK 2 ity 퐸 /(6휋 푓휋/푒) ≥ |퐵| = 1 is fulfilled for all of cited Radial distance results of the energy (see Table1). To check how well the asymptotic behavior of (6) Fig. 1. Profiles of the chiral angle 퐹 (푥) for a hedgehog Skyrmion solution to Eq. (3). The curves with different is we apply the asymptotic expansion analysis on the styles refer to the solutions given by different approaches. 121201-2 CHIN. PHYS. LETT. Vol. 27, No. 12 (2010) 121201 The disagreement can be improved by employing Here we use the solution Hyb(2/2) and Hyb(4/4) to the Padé approximant of higher order than Eq. (8), for compute the static properties of nucleons and nucleon- example, the [4/4] approximant isobar (∆) within the framework of the bosonic quan-

2 4 tization of the Skyrme model. 1 + 푎푥 + 푎4푥 Following Adkin et al.,[3] one can choose 푆푈(2)- 휆(푥) = 휆0 2 4 . (14) 1 + 푏푥 + 푏4푥 variable 퐴(푡) as the collective variables, and substitute † The minimization of Eq. (2) using Eq. (14), as carried 푈 = 퐴(푡)푈0(푥)퐴(푡) into Eq. (1). In the adiabatic out for the [2/2] approximant, yields the numerically limit, one has optimal parameters ∫︁ [︂ † ]︂ SK SK 휕퐴 휕퐴 푆 = 푆0 + 퐼0Λ 푑푡푇 푟 , (16) 푎 = 0.2598, 푏 = 0.5446, 푐 = 1.9932, 푎4 = 0.0538, 휕푡 휕푡

푏4 = 0.1226, 푤 = 0.1839, 휆0 = 3.9439. (15) SK with 푆0 the action for the static hedgehog configu- 3 The solution (6) with 휆(푥) specified by Eqs. (14) and ration, 퐼0 = 휋/(3푒 푓휋), and (15) will be referred as the Hyb(4/4) in this study and ∫︁ ∞ is also plotted in Fig. 1. Figure 2 shows the profiles 2 2 2 2 2 Λ = 8 푥 푑푥 sin 퐹 [1 + 퐹푥 + sin 퐹/푥 ], (17) of 퐹 (푥) at large 푥 for Hyb(2/2) as well as Hyb(4/4), 0 and the numeric solution. The asymptotic expansion which is independent of 푓휋 and 푒. The Hamiltonian of the solution Hyb(4/4) shows that for small 푥 ≪ 1, associated to Eq. (16), when quantized via the quan- the profile becomes tization procedure in terms of collective coordinates, 3 5 yields an eigenvalue ⟨퐻⟩ = 푀 +퐽(퐽 +1)/(2퐼 Λ), with 퐹푤(푥) = 휋 − 2.0243푥 + 0.4654푥 − 0.4270푥 + ··· . 0 푀 = 퐸SK being the soliton energy of the Skyrmion. while for 푥 → ∞ This yields the masses of the nucleon and ∆-isobar, 2.220{︁ 0.9150 9.5908 65.358 퐹 (푥) = 1 − − − 3 15 푤 2 2 4 6 푀 = 푀 + , 푀 = 푀 + . (18) 푥 푥 푥 푥 푁 8퐼 Λ Δ 8퐼 Λ 264.78 821.88 }︁ 0 0 + − + ··· . 푥8 푥10 By adjusting 푓휋 and 푒 to fit the nucleon and delta masses in Eq. (18), one can fix the model parame- Here a better value 퐹 (50) = 8.877×10−4 is obtained 푤 ters 푓 and 푒 using the calculated 푀 and Λ through for the latter asymptotic profile in contrast with the 휋 Eqs. (2) and (17). solution Hyb(2/2). The computed Skyrmion energies The isoscalor rms radius and isoscalor magnetic (2), measured in units of 2푓 /푒, are listed in Table 휋 rms radius are given by 1, including the corresponding results obtained by the numeric solution and obtained in the relevant refer- {︂ ∫︁ ∞ }︂1/2 2 1/2 2 2 2 ences as indicated. 푒푓휋⟨푟 ⟩퐼=0 = − 푥 sin 퐹 퐹푥 , 휋 0 1/2 {︃∫︀ ∞ 4 2 }︃ 2.4 2 1/2 0 푥 sin 퐹 퐹푥푑푥 푒푓휋⟨푟 ⟩푀,퐼=0 = ∞ . (19) ) ∫︀ 2 2 Hyb(2/2) 0 푥 sin 퐹 퐹푥푑푥 -8

2.0 Hyb(4/4) Num. Combining with the masses of nucleon and the delta, ) (10

1.6 one can evaluate the magnetic moments for proton ( and neutron via the following formula

1.2 퐼=0 퐼=1 휇푝,푛 = 휇푝,푛 + 휇푝,푛 2

0.8 ⟨푟 ⟩ = 퐼=0 푀 (푀 − 푀 ) 9 푁 Δ 푁 Chiral Chiral angle 0.4 푀 ± 푁 , (20)

10000 12000 14000 16000 18000 20000 2(푀Δ − 푀푁 ) Radial distance where plus and minus correspond to proton and neu- Fig. 2. The hedgehog Skyrmion profiles at large radial tron, respectively. The calculated results for these distance 푥. The comparison between the approximated quantities using two solution schemes (Hyb.(2/2) and solutions in this work and the numerical one is given. Hyb.(4/4)) are shown explicitly in Table2, compared In solving Eq. (3) numerically, we employ the non- to the experimental values as well as that computed linear shoot algorithm for the boundary values at by the numeric solution for 퐹 (푥). The corresponding 푥 = 0.001 and 푥 = 1000 based on the asymptotic results from other predictions are also shown in this formulas (10) and (12) of the chiral angle 퐹 (푥). table. Here in Table 2, we use the experimental val- The static properties of nucleons can be extracted ues 푀푁 = 938.9 MeV, 푀Δ = 1232 MeV for fixing 푒 by semi-classically quantizing the spinning modes of and 푓휋 through Eq. (18), in contrast with the input [3] Skyrme Lagrangian using the collective variables. 푀푁 = 938 MeV, 푀Δ = 1232 MeV used by Refs. [3,14]. 121201-3 CHIN. PHYS. LETT. Vol. 27, No. 12 (2010) 121201

Table 2. Display of the nucleon properties from the theoretical calculation and the previous results as well as the experimental data. Quantities Ref. [3] Ref. [14] Hyb.(2/2) Hyb.(4/4) Num. Expt. 푀/(2푓휋/푒) 36.5 36.5 36.4638 36.4638 36.4616 − 2푓휋 (MeV) 129 130 128.730 129.453 129.260 186 푒 5.45 5.48 5.4229 5.4527 5.4446 − Λ 50.9 52.2 50.1467 51.2830 50.9782 − 2 1/2 ⟨푟 ⟩퐼=0(fm) 0.59 0.586 0.5985 0.5920 0.5938 0.72 2 1/2 ⟨푟 ⟩푀,퐼=0(fm) 0.92 0.920 0.9258 0.9208 0.9222 0.81 휇푝 1.87 1.8825 1.8764 1.8781 2.79 휇푛 −1.31 −1.3209 −1.3269 −1.3253 −1.91 |휇푝/휇푛| 1.43 1.4252 1.4141 1.4171 1.46 푔퐴 0.61 0.6332 0.5992 0.6081 1.23 푔휋푁푁 8.9 9.2364 8.6921 8.8343 13.5

To check the solution further, we also list, in Table the chiral angle 퐹 (푥) at infinity. We expect that our 2, the axial coupling constant and the 휋푁푁-coupling, solution can be useful in the dynamics study of the which are given by Skyrmion evolution and interactions. D. J thanks C. Liu and ChuengRyong Ji for dis- 휋 푀푁 cussions. 푔퐴 = − 2 퐺, 푔휋푁푁 = 푔퐴, (21) 3푒 푓휋 where the numeric factor 퐺 is References ∫︁ ∞ 2[︁ sin 2퐹 sin 2퐹 2 [1] Skyrme T H R 1962 Nucl. Phys. 31 556 퐺 = 4 푑푥푥 퐹푥 + + (퐹푥) 0 푥 푥 [2] Witten E 1983 Nucl. Phys. B 223 422 [3] Adkins G S, Nappi C R and Witten E 1983 Nucl. Phys. B 2 sin2 퐹 sin2 퐹 ]︁ + 퐹 + sin 2퐹 . (22) 228 552 푥2 푥 푥3 [4] Jackson A, Jackson A D and Pasquier V 1985 Nucl. Phys. A 432 567 In summary, we have shown that the hybrid form [5] Sutcliffe P M 1992 Phys. Lett. B 292 104 of a kink-like solution and the one given by the in- [6] Battye R and Sutcliffe P M 1997 Phys. Rev. Lett. 79 363 [7] Atiyah M F and Manton N S 1993 Commun. Math. Phys. stanton method are suitable to approximate the exact 152 391 solution for the hedgehog Skyrmion, when combined [8] Atiyah M F and Manton N S 1989 Phys. Lett. B 222 438 with the Padé approximation. The resulting analytic [9] Battye R and Sutcliffe P M 2006 Phys. Rev. C 73 055205 solution (6) has two remarkable features: (1) it is sim- [10] Zahed I and Brown G 1986 Phys. Rep. 142 1 [11] Manton N S and Sutcliffe P M 2004 Topological Solitons ple in the sense that it is globally given in whole re- (Cambridge: Cambridge University) gion; (2) it well approaches the asymptotic behavior [12] Ananias J, Galain R M and Ferreira E 1991 J. Math. Phys. of the exact solution. We note that the further gen- 32 1949 eralization of (6), made by approximating 푐 in (6) via [13] Ponciano J A, Epele L N, Fanchiotti H and García Canal C A 2001 Phys. Rev. C 64 045205 Padé approximation, does not exhibit remarkable im- [14] Yamashita J and Hirayama M 2006 Phys. Lett. B 642 160 provement, particularly in the asymptotic behavior of [15] Faddeev L D 1976 Lett. Math. Phys. 1 289

121201-4 Chinese Physics Letters Volume 27 Number 12 2010

GENERAL 120201 Special Lie–Mei Symmetry and Conserved Quantities of Appell Equations Expressed by Appell Function XIE Yin-Li, JIA Li-Qun 120202 Effects of Degree on Cooperation in Evolutionary Prisoner’s Dilemma Games on Structured Populations DAI Qiong-Lin, CHENG Hong-Yan, LI Hai-Hong, YANG Jun-Zhong 120301 Maximal Violation of Bell Inequality for Any Given Two-Qubit Pure State XIANG Yang 120302 Bidirectional Mapping between a Biphoton Polarization State and a Single-Photon Two-Qubit State LIN Qing 120303 The Time Division Multi-Channel Communication Model and the Correlative Protocol Based on Quantum Time Division Multi-Channel Communication LIU Xiao-Hui, PEI Chang-Xing, NIE Min 120304 Quantum Correlations Reduce Classical Correlations with Ancillary Systems LUO Shun-Long, LI Nan THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS 121201 Analytical Solution for the SU(2) Hedgehog Skyrmion and Static Properties of Nucleons JIA Duo-Jie, WANG Xiao-Wei, LIU Feng 121202 The Renormalized Equation of State and Quark Star LI Hua, LUO Xin-Lian, JIANG Yu, ZONG Hong-Shi 0 121301 Bs → φπ Decay in the Extra Down-Type Quark Model WANG Shuai-Wei, HUANG Jin-Shu, LU¨ Lin-Xia NUCLEAR PHYSICS 122101 Exotic Magnetic Rotation in 22F PENG Jing, YAO Jiang-Ming, ZHANG Shuang-Quan, MENG Jie ATOMIC AND MOLECULAR PHYSICS

123101 Density-Functional-Theory Studies of C20 in Femtosecond Laser Pulses WANG Zhi-Ping, BIAN Bao-An, WANG Li-Guang 123102 Ab initio Study of He Stability in hcp-Ti DAI Yun-Ya, YANG Li, PENG Shu-Ming, LONG Xing-Gui, GAO Fei, ZU Xiao-Tao 123103 Theoretical Study of Isotopic Effect of Oxygen Atom on the Stereodynamics for the O(3P ) +

D2 → OD + D Reaction LIU Shi-Li, SHI Ying 123201 Electron Correlation in Nonsequential Double Ionization of Helium by Two-Color Pulses ZHOU Yue-Ming, LIAO Qing, HUANG Cheng, TONG Ai-Hong, LU Pei-Xiang 123202 Analysis of Nanometer Structure for Chromium Atoms in Gauss Standing Laser Wave ZHANG Wen-Tao, ZHU Bao-Hua, XIONG Xian-Ming 123203 Strings of Ion Crystals in a Linear Trap for Quantum Information Processing ZHOU Fei, XIE Yi, XU You-Yang, HUANG Xue-Ren, FENG Mang 123401 Electron Capture Process in Collisions of Proton with Excited State of Helium WANG Xue-Rong, LIU Ling, WANG Jian-Guo

(Continued on inside back cover) 123701 Quantum Phase Transition of the Bosonic Atoms near the Feshbach Resonance in an Optical Lattice LI Ben, CHEN Jing-Biao FUNDAMENTAL AREAS OF PHENOMENOLOGY(INCLUDING APPLICATIONS) 124201 Coherent Beam Combining of Two Fiber Amplifiers Seeded by Multi-Wavelength Fiber Laser WANG Xiao-Lin, ZHOU Pu, MA Yan-Xing, MA Hao-Tong, XU Xiao-Jun, LIU Ze-Jin, ZHAO Yi-Jun 124202 Nonlinear Optical Properties and Ultrafast Dynamics of Undoped and Doped Bulk SiC DING Jin-Liang, WANG Yao-Chuan, ZHOU Hui, CHEN Qiang, QIAN Shi-Xiong, FENG Zhe-Chuan, LU Wei-Jie 124203 Highly Efficient Continuous-Wave Mid-Infrared Intracavity Singly Resonant Optical Paramet- ric Oscillator Based on MgO:PPLN YAN Bo-Xia, BI Yong, ZHOU Mi, WANG Dong-Dong, QI Yan, FANG Tao, WANG Bin, WANG Yan-Wei, ZHENG Guang, CHENG Hua 124204 Off-Resonant Third-Order Optical Nonlinearity of Au Nanoparticle Array by Femtosecond Z-scan Measurement WANG Kai, LONG Hua, FU Ming, YANG Guang, LU Pei-Xiang 124205 Stimulated Brillouin Scattering Damage of Large-Aperture Fused Silica Grating HAN Wei, HUANG Wan-Qing, LI Ke-Yu, WANG Fang, FENG Bin, JIA Huai-Ting, -Quan, XIANG Yong, JING Feng, ZHENG Wan-Guo 124206 Experiment of C-Band Wavelength Conversion in a Silicon Waveguide Pumped by Dispersed Femtosecond Laser Pulse GAO Shi-Ming, TIEN En-Kuang, SONG Qi, HUANG Yue-Wang, Salih Kagan KALYONCU 124207 Velocity Measurement Based on Laser Doppler Effect ZHANG Yan-Yan, HUO Yu-Jing, HE Shu-Fang, GONG Ke 124208 Correlated Photon Pair Generation in Silicon Wire Waveguides at 1.5 µm CHENG Jie-Rong, ZHANG Wei, ZHOU Qiang, FENG Xue, HUANG Yi-Dong, PENG Jiang-De 124209 High-Quality Continuous-Wave Imaging with a 2.53 THz Optical Pumped Terahertz Laser and a Pyroelectric Detector BING Pi-Bin, YAO Jian-Quan, XU De-Gang, XU Xiao-Yan, LI Zhong-Yang 124210 Pseudo-Rhombus-Shaped Subwavelength Crossed Gratings of GaAs for Broadband Antireflec- tion CHEN Xi, FAN Zhong-Chao, ZHANG Jing, SONG Guo-Feng, CHEN Liang-Hui 124211 Observation of Modulation Transfer Spectroscopy in the Deep Modulation Regime ZHOU Zi-Chao, WEI Rong, SHI Chun-Yan, WANG Yu-Zhu 124301 A Simple and Accurate Method for Calculating the Gaussian Beam Expansion Coefficients LIU Wei, YANG Jun 124302 Directivity of Spherical Polyhedron Sound Source Used in Near-Field HRTF Measurements YU Guang-Zheng, XIE Bo-Sun, RAO Dan 124401 Effects of Slip on Unsteady Mixed Convective Flow and Heat Transfer Past a Stretching Surface S. Mukhopadhyay 124501 Cluster Model for Wave-Like Motions of a 2D Vertically Vibrated Granular System CAI Hui, MIAO Guo-Qing 124502 Static Structure of Two-Dimensional Granular Chain WEN Ping-Ping, LI Liang-Sheng, ZHENG Ning, SHI Qing-Fan 124701 Can We Obtain a Fractional Lorenz System from a Physical Problem? YANG Fan, ZHU Ke-Qin 124702 Slip Magnetohydrodynamic Viscous Flow over a Permeable Shrinking Sheet FANG Tie-Gang, ZHANG Ji, YAO Shan-Shan PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES 125201 Quantum Effects on Rayleigh–Taylor Instability of Incompressible Plasma in a Vertical Mag- netic Field G. A. Hoshoudy 125202 Radiation Temperature Measurement of an Imploded X-Ray Source with a Filtered-Multi- Channel Pinhole Camera WANG Shou-Jun, DONG Quan-Li, , LI Yu-Tong, ZHANG Lei, Shinsuke Fujioka, Norimasa Yamamoto, Hiroaki Nishimura, ZHANG Jie 125203 Jet-Like Long Spike in Nonlinear Evolution of Ablative Rayleigh–Taylor Instability YE Wen-Hua, WANG Li-Feng, HE Xian-Tu CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMAL PROPERTIES 126101 Preparation of a Pd-Cu-Si Bulk Metallic Glass with a Diameter up to 11 mm DING Hong-Yu, LI Yang, YAO Ke-Fu 126201 Deflections of Nanowires with Consideration of Surface Effects LI He, YANG Zhou, ZHANG Yi-Min, WEN Bang-Chun 126401 Characterization and Magnetic Properties of Nickel Ferrite Nanoparticles Prepared by Ball Milling Technique G. Nabiyouni, M. Jafari Fesharaki, M. Mozafari, J. Amighian 126801 Strain-Engineered Low-Density InAs Bilayer Quantum Dots for Single Photon Emission LI Zhan-Guo, LIU Guo-Jun, LI Lin, FENG Ming, LI Mei, LU Peng, ZOU Yong-Gang, LI Lian-He, GAO Xin 126802 Effect of High Temperature Annealing on Conduction-Type ZnO Films Prepared by Direct- Current Magnetron Sputtering SUN Li-Jie, HE Dong-Kai, XU Xiao-Qiu, ZHONG Ze, WU Xiao-Peng, LIN Bi-Xia, FU Zhu-Xi CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES 127101 Effects of H on Electronic Structure and Ideal Tensile Strength of W: A First-Principles Calculation LIU Yue-Lin, ZHOU Hong-Bo, JIN Shuo, ZHANG Ying, LU Guang-Hong 127102 First-Principles Investigations of the Phase Transition and Optical Properties of Solid Oxygen -Hui, DUAN De-Fang, WANG Lian-Cheng, ZHU Chun-Ye, CUI Tian 127301 Annealing Effect on Photovoltages of Quartz Single Crystals TIAN Lu, ZHAO Song-Qing, ZHAO Kun

127302 Influence of O2 Flux on Compositions and Properties of ITO Films Deposited at Room Tem- perature by Direct-Current Pulse Magnetron Sputtering WANG Hua-Lin, DING Wan-Yu, LIU Chao-Qian, CHAI Wei-Ping 127303 GaN-Based Thin Film Vertical Structure Light Emitting Diodes Fabricated by a Modified Laser Lift-off Process and Transferred to Cu SUN Yong-Jian, YU Tong-Jun, JIA Chuan-Yu, CHEN Zhi-Zhong, TIAN Peng-Fei, KANG Xiang-Ning, LIAN Gui-Jun, HUANG Sen, ZHANG Guo-Yi

127304 Effect of Indium Ambient on Electrical Properties of Mg-Foped AlxGa1−xN XU Zheng-Yu, QIN Zhi-Xin, SANG Li-Wen, ZHANG Yan-Zhao, SHEN Bo, ZHANG Guo-Yi, ZHAO Lan, ZHANG Xiang-Feng, CHENG Cai-Jing, SUN Wei-Guo 127801 Optical and Structural Properties of Cr-Doped GaN Grown by HVPE Method YAN Huai-Yue, XIU Xiang-Qian, HUA Xue-Mei, LIU Zhan-Hui, ZHOU An, ZHANG Rong, XIE Zi-Li, HAN Ping, SHI Yi, ZHENG You-Dou 127802 Highly Efficient Simplified Organic Light-Emitting Diodes Utilizing F4-TCNQ as an Anode Buffer Layer DONG Mu-Sen, WU Xiao-Ming, HUA Yu-Lin, QI Qing-Jin, YIN Shou-Gen 127803 Quantum-Confined Stark Effect in Ensemble of Colloidal Semiconductor Quantum Dots WANG Zhi-Bing, ZHANG Hui-Chao, ZHANG Jia-Yu, Huaipeng Su, Y. Andrew Wang CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY

128101 Preparation, Morphology Transformation and Magnetic Behavior of Co3O4 Nano-Leaves MENG Ling-Rong, CHEN Wei-Meng, CHEN Chin-Ping, ZHOU He-Ping, PENG Qing 128501 Fabrication and Characteristics of AlInN/AlN/GaN MOS-HEMTs with Ultra-Thin Atomic

Layer Deposited Al2O3 Gate Dielectric MAO Wei, ZHANG Jin-Cheng, XUE Jun-Shuai, HAO Yao, MA Xiao-Hua, WANG Chong, LIU Hong-Xia, XU Sheng-Rui, YANG Lin-An, BI Zhi-Wei, LIANG Xiao-Zhen, ZHANG Jin-Feng, KUANG Xian-Wei 128502 Effect of Annealing on Microstructure and Electrical Characteristics of Doped Poly (3-Hexylthiophene) Films MA Liang 128503 A 10.7 µm InGaAs/InAlAs Quantum Cascade Detector KONG Ning, LIU Jun-Qi, LI Lu, LIU Feng-Qi, WANG Li-Jun, WANG Zhan-Guo, LU Wei 128504 ZnO-Based Transparent Thin-Film Transistors with MgO Gate Dielectric Grown by in-situ MOCVD ZHAO Wang, DONG Xin, ZHAO Long, SHI Zhi-Feng, WANG Jin, WANG Hui, XIA Xiao-Chuan, CHANG Yu-Chun, -Lin, DU Guo-Tong 128901 A Unifying Modularity in Networks HAO Jun-Jun, CAI Shui-Ming, HE Qin-Bin, LIU Zeng-Rong 128902 Nuclear Magnetic Resonance Measurements of Original Water Saturation and Mobile Water Saturation in Low Permeability Sandstone Gas GAO Shu-Sheng, YE Li-You, XIONG Wei, GUO He-Kun, HU Zhi-Ming GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS 129401 Chorus-Driven Outer Radiation Belt Electron Dynamics at Different L-Shells ZHANG Sai, XIAO Fu-Liang