Table of contents

16 November 2003, Volume 12 Issue 11 Previous issue    Next issue

GENERAL

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Heterogeneous anisotropic complex structure gradual model and constitutive relation Li Yong (李永), Song Jian (宋健), Zhang Zhi-Min (张志民) Chinese Physics, 2003, 12 (11):  1187-1193.  DOI: 10.1088/1009-1963/12/11/301 Abstract ( 923 )   PDF (398KB) ( 362 )   Four new gradually delaminate models of the three-dimensional macro-/mesoscopic structure and delamination of the heterogeneous anisotropic composite (HAC) are set up by conducting research into its structure and performance. A general theory, which demonstrates the three-dimensional constitutive relation of the macro-/mesoscopic performance of this structure is further developed. The macroscopic expression of HAC is presented in terms of a Tanigawa delaminate homogeneous equivalent approach, the mesoscopic problems are analysed utilizing Eshelby－Mori－Tanaka theory, with the introduction of the representative volume elements of monolayer single unit cell and interlaminar double unit cells. According to the gradual continuity of the structure as a whole, great attention is given to the modelling and research of the interlaminar macroscopic and mesoscopic problems of HAC structure. Comparison with the existing solutions is made through calculation of typical cases.

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An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system Zhang Yu-Feng (张玉峰) Chinese Physics, 2003, 12 (11):  1194-1201.  DOI: 10.1088/1009-1963/12/11/302 Abstract ( 1243 )   PDF (236KB) ( 541 )   A subalgebra of loop algebra $\tilde{A}_2$ is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz－Kaup－Newell－Segur (AKNS) hierarchy and a generalized standard form of the Schr?dinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra $\tilde{A}_2$ into loop algebra $\tilde{A}_1$. Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.

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New doubly periodic and multiple soliton solutions of the generalized (3+1)-dimensional KP equation with variable coefficients Chen Huai-Tang (陈怀堂), Zhang Hong-Qing (张鸿庆) Chinese Physics, 2003, 12 (11):  1202-1207.  DOI: 10.1088/1009-1963/12/11/303 Abstract ( 1274 )   PDF (221KB) ( 1335 )   A new generalized Jacobi elliptic function method is used to construct the exact travelling wave solutions of nonlinear partial differential equations (PDEs) in a unified way. The main idea of this method is to take full advantage of the elliptic equation which has more new solutions. More new doubly periodic and multiple soliton solutions are obtained for the generalized (3+1)-dimensional Kronig－Penny (KP) equation with variable coefficients. This method can be applied to other equations with variable coefficients.

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Effects of cross-correlated noises on the relaxation time of the bistable system Xie Chong-Wei (谢崇伟), Mei Dong-Cheng (梅冬成) Chinese Physics, 2003, 12 (11):  1208-1212.  DOI: 10.1088/1009-1963/12/11/304 Abstract ( 1130 )   PDF (218KB) ( 539 )   The stationary correlation function and the associated relaxation time for a general system driven by cross-correlated white noises are derived, by virtue of a Stratonovich-like ansatz. The effects of correlated noises on the relaxation time of a bistable kinetic model coupled to an additive and a multiplicative white noises are studied. It is proved that for small fluctuations the relaxation time $T_{\rm c}$ as a function of $\lambda$ (the correlated intensity between noises) exhibits very different behaviours for $\alpha < D$ and for $\alpha > D$ ($\alpha$ and $D$, respectively, stand for the ntensities of additive and multiplicative noises). When $\alpha < D$, $T_{\rm c}$ increases with increasing $\lambda$. But when $\alpha < D$, $T_{\rm c}$ increases with $\lambda$ for the case of weak correlated noises and sharply decreases with $\lambda$ for the case of strong correlated noises, and thus $T_{\rm c}$-$\lambda$ curve behaves with one extremum.

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Improving the prediction of chaotic time series Li Ke-Ping (李克平), Gao Zi-You (高自友), Chen Tian-Lun (陈天仑) Chinese Physics, 2003, 12 (11):  1213-1217.  DOI: 10.1088/1009-1963/12/11/305 Abstract ( 2058 )   PDF (227KB) ( 493 )   One of the features of deterministic chaos is sensitive to initial conditions. This feature limits the prediction horizons of many chaotic systems. In this paper, we propose a new prediction technique for chaotic time series. In our method, some neighbouring points of the predicted point, for which the corresponding local Lyapunov exponent is particularly large, would be discarded during estimating the local dynamics, and thus the error accumulated by the prediction algorithm is reduced. The model is tested for the convection amplitude of Lorenz systems. The simulation results indicate that the prediction technique can improve the prediction of chaotic time series.

ATOMIC AND MOLECULAR PHYSICS

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The characteristics of doubly and triply charged ions C2+2, C2-2 and C3+2 Zhu Zheng-He (朱正和), Wang Rong (汪蓉) Chinese Physics, 2003, 12 (11):  1218-1224.  DOI: 10.1088/1009-1963/12/11/306 Abstract ( 1100 )   PDF (258KB) ( 381 )   Based on group theory and atomic and molecular reaction statics, this paper derives the possible electronic states of C$^{2+}_2$, C$^{2-}_2$ and C$^{3+}_2$, and their reasonable dissociation limits and determines their ground electronic states C$^{2+}_2(X^3\Sigma_{\rm u})$, C$^{2-}_2(X^1\Sigma^+_{\rm g}$) and C$^{3+}_2(X^4\Sigma^-_{\rm u})$ using quantum mechanical calculations at the level of QCISD/6-311G$^*$. All the potential energy curves of their ground states have both a minimum and a maximum, which are the so-called "energy trapped" molecules. This sort of potential maximum is chiefly due to Coulomb repulsion. We propose the perturbation effect of ionic charges, which is used to explain why the orbital degeneracy of diatomic ions may be removed. The characteristics of potential curves for diatomic ions are briefly described.

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An abnormal phenomenon in the saturated absorption spectrum of 87Rb He Ling-Xiang (贺凌翔), Wang Yu-Zhu (王育竹) Chinese Physics, 2003, 12 (11):  1225-1228.  DOI: 10.1088/1009-1963/12/11/307 Abstract ( 1295 )   PDF (213KB) ( 455 )   A polarization-type saturated absorption spectrum set-up for ${}^{87}$Rb atom laser cooling and trapping has been established in our laboratory. By changing the polarization components and optical power, enhanced absorption of the transition 5S$_{1/2} F=2→5$P$_{3/2}F^′=3$ was observed with other transitions and cross-over resonance staying unvaried.

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High-intensity femtosecond laser absorption by rare-gas clusters Li Shao-Hui (李邵辉), Wang Cheng (王成), Liu Jian-Sheng (刘建胜), Wang Xiang-Xin (王向欣), Zhu Pin-Pin (朱频频), Li Ru-Xin (李儒新), Ni Guo-Quan (倪国权), Xu Zhi-Zhan (徐至展) Chinese Physics, 2003, 12 (11):  1229-1234.  DOI: 10.1088/1009-1963/12/11/308 Abstract ( 1153 )   PDF (233KB) ( 434 )   The energy absorption efficiency of high-intensity ($\sim10^{16}$W/cm^2) femtosecond laser pulses in a dense jet of large rare-gas clusters has been measured. Experimental results show that the energy absorption efficiency is strongly dependent on the cluster size and can be higher than 90%. The measurement of the ion energy indicates that the average ion energies of argon and xenon can be as high as 90 and 100keV, respectively. The dependence of the average energy of the ions on the cluster size is also measured. At comparatively low gas backing pressure, the average ion energies of argon and xenon increase with increasing gas backing pressure. The average ion energy of argon becomes saturated gradually with further increase of the gas backing pressure. For xenon, the average ion energy drops a little after the gas backing pressure exceeds 9 bar ($3.2\times 10^5$ atoms/cluster). The result showing the existence of a maximum average ion energy has been interpreted within the framework of the microplasma sphere model.

CLASSICAL AREAS OF PHENOMENOLOGY

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Three-dimensional nonlinear theory of travelling wave tubes and simulation Li Bin (李斌), Yang Zhong-Hai (杨中海) Chinese Physics, 2003, 12 (11):  1235-1240.  DOI: 10.1088/1009-1963/12/11/309 Abstract ( 1344 )   PDF (264KB) ( 503 )   A three-dimensional (3D) nonlinear theory of travelling wave tubes (TWTs) is developed, which includes a fundamental radio frequency (RF) and harmonics. When the instantaneous bandwidth exceeds an octave, the harmonic is generated and the mutual coupling between the harmonic and the fundamental RF can be observed in TWTs due to nonlinear interaction between the electron beam and the RF. At low frequencies the harmonic has an obvious effect. Based upon Tien's disc model, a plastic 3D super-particle model is proposed to improve the nonlinear analysis of TWTs. Numerical results employing a periodic magnetic focusing field are presented.

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Multilevel optical data recording methods on phase-change media Xiao Jia-Xi (肖家曦), Qi Guo-Sheng (齐国生), She Peng (佘鹏), Liu Rong (刘嵘), Xu Duan-Yi (徐端颐) Chinese Physics, 2003, 12 (11):  1241-1245.  DOI: 10.1088/1009-1963/12/11/310 Abstract ( 1144 )   PDF (275KB) ( 498 )   Multilevel data storage (ML) is a new method in the optical storage field, which is also a trend for improving the capability of future optical discs. This article introduces several ML methods based on phase-change media including pit depth modulation (PDM) and mark radial width modulation (MRWM). In addition, some disadvantages and advantages concerning the principle of these methods will be discussed. Finally, a new ML method will be advanced, through which the levels in one recording pit will be increased evidently.

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The mechanism research of the Cr4+Nd3+:YAG laser Chen Xiao-Bo (陈晓波), Wang Ling (王凌), Lü Lan-Bin (吕兰斌), Zhang Zhi-Guo (张治国), Zhu Jane G., Du Wei-Min (杜为民), Yang Guo-Zhen (杨国桢) Chinese Physics, 2003, 12 (11):  1246-1250.  DOI: 10.1088/1009-1963/12/11/311 Abstract ( 1392 )   PDF (229KB) ( 457 )   The 946nm diode-pump microchip self-Q-switched laser of a chromium and neodymium codoped yttrium aluminum garnet crystal material (Cr$^{4+}$Nd$^{3+}$:YAG) is studied, especially about its physical mechanism of operation. The ${}^4$F$_{3/2}→{}^4$I$_{9/2}$ transition of Nd$^{3+}$ ion is beneficial to achieving laser oscillation in a quasi-three-level system based on coating the cavity mirrors of the microchip with films that suppress the 1064nm operation and enhance the 946nm laser. The Cr$^{4+}$ ion is a saturable absorber. The initial loss $N_{\rm t1}$ is high, which acts as the threshold for laser oscillation. The stable loss $N_{\rm t2}$ is low because the Cr$^{4+}$ ion is acceleratively bleached by the fast enhancement of the oscillating laser. The high N$_{\rm t1}$, small $N_{\rm t2}$ and fast progresses permit the oscillating laser of the Cr$^{4+}$Nd$^{3+}$:YAG to have a good self-Q-switched property whose full width at half maximum is about 4.2ns. Its highest laser power is about 5.7mW. Its peak power is about 150W. Its good fundamental transverse TEM$_{00}$ mode results from the absorption bleaching established by both the pump and oscillating lasers, which suppress other transverse mode and allow the oscillation only in the fundamental transverse TEM$_{00}$ mode.

PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES

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Fokker-planck study of tokamak electron cyclotron resonance heating Shi Bing-Ren (石秉仁), Long Yong-Xing (龙永兴), Dong Jia-Qi (董家齐), Li Wen-Zhong (郦文忠), Jiao Yi-Ming (焦一鸣), Wang Ai-Ke (王爱科) Chinese Physics, 2003, 12 (11):  1251-1256.  DOI: 10.1088/1009-1963/12/11/312 Abstract ( 1384 )   PDF (341KB) ( 471 )   A Fokker－Planck study is carried out for tokamak electron cyclotron resonance heating by writing the quasi-linear diffusion operator into a form adaptive to the collision operator and considering the wave absorption characteristics of both the O-mode and the X-mode in different magnetic surfaces. Though the Fokker－Planck code is non-relativistic in nature, however, if the relativistic resonance condition for the nearly perpendicularly propagating waves is treated suitably, we can obtain correct results. The energy loss mechanism through anomalous transport is also modelled using a suitable loss term. In the heating phase, the electron distribution deviates from the Maxwellian distribution substantially, which leads to non-linear absorption characteristics. The wave damping rate is non-linear and changes with time. The electron pressure is usually anisotropic under different conditions: pe⊥/pe‖>2 for different D0 and $\tau_e$.

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Intertwisted fibrillar diamond-like carbon films prepared by electron cyclotron resonance microwave plasma enhanced chemical vapour deposition Yang Wu-Bao (杨武保), Wang Jiu-Li (王久丽), Zhang Gu-Ling (张谷令), Fan Song-Hua (范松华), Liu Chi-Zi (刘赤子), Yang Si-Ze (杨思泽) Chinese Physics, 2003, 12 (11):  1257-1260.  DOI: 10.1088/1009-1963/12/11/313 Abstract ( 1269 )   PDF (384KB) ( 563 )   In this paper, the structures, optical and mechanical properties of diamond-like carbon films are studied, which are prepared by a self-fabricated electron cyclotron resonance microwave plasma chemical vapour deposition method at room temperature in the ambient gases of mixed acetylene and nitrogen. The morphology and microstructure of the processed film are characterized by the atomic force microscope image, Raman spectra and middle Fourier transform infrared transmittance spectra, which reveal that there is an intertwisted fibrillar diamond-like structure in the film and the film is mainly composed of sp$^3$ CH, sp$^3$ C—C, sp$^2$ C═C, C═N and C$_{60}$. The film micro-hardness and bulk modulus are measured by a nano-indenter and the refractive constant and deposition rate are also calculated.

CONDENSED MATTER: STRUCTURAL, MECHANICAL, AND THERMAL PROPERTIES

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Ab-initio determination of the incommensurate modulated structure of Bi-2212 from x-ray powder diffraction data －a simulation Chen Jian-Rong (陈建荣), Gu Yuan-Xin (古元新), Zheng Chao-De (郑朝德), Fan Hai-Fu (范海福) Chinese Physics, 2003, 12 (11):  1261-1265.  DOI: 10.1088/1009-1963/12/11/314 Abstract ( 1069 )   PDF (465KB) ( 366 )   A set of x-ray powder diffraction data of the high-Tc superconductor Bi2Sr2Ca1Cu2Oy (Bi-2212) was simulated based on the experimental single-crystal diffraction data by merging together reflections with diffraction angles (2$\theta$) closer to each other than 0.04 degrees. There are three types of overlapping in the powder diffraction data, i.e. (i) overlapping of main reflections; (ii) overlapping of satellite reflections and (iii) overlapping of main and satellite reflections. The third type of overlapping was first separated into main and satellite components according to the ratio between the average intensity of that of types (i) and (ii). Then the overlapped reflections of main reflections and those of the satellites were uniformly partitioned. Heavy-atom sites in the basic/average structure were found using the uniformly decomposed main reflections by the conventional direct method. Phases of the satellites were derived by the multidimensional direct method. The resultant four-dimensional Fourier maps revealed correctly the essential feature of the modulation. No assumption on either the basic structure or the modulation is needed.

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Rapid eutectic growth in undercooled Al-Ge alloy under free fall condition Liu Xiang-Rong (刘向荣), Cao Chong-De (曹崇德), Wei Bing-Bo (魏炳波) Chinese Physics, 2003, 12 (11):  1266-1271.  DOI: 10.1088/1009-1963/12/11/315 Abstract ( 1338 )   PDF (298KB) ( 683 )   Eutectic growth in Al-51.6%wt Ge alloy has been investigated during free fall in a drop tube. With decreasing undercooling $\Delta T$, the microstructural evolution has shown a transition from lamellar eutectic to anomalous eutectic. A maximum cooling rate of $4.2\times10^4$K/s and undercooling of up to 240K (0.35$T_{\rm E}$) are obtained in the experiment. The eutectic coupled zone is calculated on the basis of current eutectic and dendritic growth theories, which covers a composition range from 48%－59% Ge and leans towards the Ge-rich side. The two critical undercoolings for the eutectic transition are $\Delta T_1^*$=101K and $\Delta_2^*$=178K. When $\Delta T≤\Delta T^*_1$, the microstructure for Al-51.6% Ge eutectic shows lamellar eutectic. If $\Delta T≥\Delta T^*_2$, the microstructure shows anomalous eutectic. In the intermediate range of $\Delta T^*_1<\Delta T<\Delta T^*_2$, the microstructure is the mixture of the above two types of eutectics.

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Microstructural evolution during containerless rapid solidification of Co-Si alloys Yao Wen-Jing (姚文静), Wei Bing-Bo (魏炳波) Chinese Physics, 2003, 12 (11):  1272-1282.  DOI: 10.1088/1009-1963/12/11/316 Abstract ( 1107 )   PDF (423KB) ( 562 )   The Co-12%Si hypoeutectic, Co-12.52%Si eutectic and Co-13%Si hypereutectic alloys are rapidly solidified in a containerless environment in a drop tube. Undercoolings up to 207K (0.14T_E) are obtained, which play a dominant role in dendritic and eutectic growth. The coupled zone around Co-12.52%Si eutectic alloy has been calculated, which covers a composition range from 11.6 to 12.7%Si. A microstructural transition from lamellar eutectic to divorced eutectic occurs to Co-12.52%Si eutectic droplets with increasing undercooling. The lamellar eutectic structure of the Co-12.52%Si alloy consists of $\varepsilon$Co and Co3Si phases at small undercooling. The Co3Si phase cannot decompose completely into $\varepsilon$Co and $\alpha$Co2Si phases. As undercooling becomes larger, the Co3Si phase grows very rapidly from the highly undercooled alloy melt to form a divorced eutectic. The structural morphology of the Co-12%Si alloy droplets transforms from $\varepsilon$Co primary phase plus lamellar eutectic to anomalous eutectic, whereas the microstructure of Co-13%Si alloy droplets experiences a `dendritic to equiaxed' structural transition. No matter how large the undercooling is, the εCo solid solution is the primary nucleation phase. In the highly undercooled alloy melts, the growth of $\varepsilon$Co and Co3Si phases is controlled by solutal diffusion.

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First-order Fréedericksz transition at the threshold point for weak anchoring nematic liquid crystal cell under external electric and magnetic fields Guan Rong-Hua (关荣华), Yang Guo-Chen (杨国琛) Chinese Physics, 2003, 12 (11):  1283-1290.  DOI: 10.1088/1009-1963/12/11/317 Abstract ( 1149 )   PDF (258KB) ( 404 )   Based on the modified formula of Rapini-Papoular, the equilibrium equation and boundary condition of the director have been obtained and the behaviour of the Fréedericksz transition at the threshold point has been studied for weak-anchoring nematic liquid crystal cells under external electric and magnetic fields with the methods of analytical derivation and numerical calculation. The results show that, except for the usual second-order transition, the first-order Fréedericksz transition can also be induced by a suitable surface anchoring technique for the liquid crystal cell given in the paper. The conditions for the existence of the first-order Fréedericksz transition are obtained. They are related to the material elastic coefficient $k_{11}$, $k_{33}$ the thickness of the liquid crystal cell, the external electric field and the strength of surface anchoring, etc.

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Combinative energy, oxygen deficiency and superconductivity in LnBa2Cu3O7-x(Ln=Nd, Er, Sm) Wang Shu-Xia (王淑霞), Liu Wen-Li (刘文利), Han Sheng-Hao (韩圣浩), Zhang Han (张酣) Chinese Physics, 2003, 12 (11):  1291-1295.  DOI: 10.1088/1009-1963/12/11/318 Abstract ( 1487 )   PDF (236KB) ( 470 )   The correlation among the combinative energy, superconductivity, oxygen content, the position of holes in different planes, and the position of holes in the Cu(2)-O plane in LnBa$_2$Cu$_3$O$_{7-x}$ (Ln= Nd, Er, and Sm) has been investigated on the basis of a block model. The results indicate that the combinative energy decreases with increasing $T_{\rm c}$ in all of these compounds. And also, the combinative energies are obviously different with holes at different positions in the Cu(2)-O plane when the oxygen deficiency is low. However, this difference becomes less with increasing the oxygen deficiency. The effect of the holes in different positions on the combinative energy supplies some clue to the understanding of an unresolved problem, i.e. whether the distribution of carriers in the CuO$_2$ plane is uniform or inhomogeneous. The results not only show that the structural characters, the combinative energy between two structural blocks and the superconductivity are closely interrelated in this class of compounds, but also reveal some differences among these systems. The relationship between the combinative energy and the $T_{\rm c}$ value in NdBa$_2$Cu$_3$O$_{7-x}$ shows some features different from the systems containing Er or Sm.

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES

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Electron self-energy and effective mass in a single heterostructure Hua Xiu-Kun (花修坤), Wu Yin-Zhong (吴银忠), Li Zhen-Ya (李振亚) Chinese Physics, 2003, 12 (11):  1296-1300.  DOI: 10.1088/1009-1963/12/11/319 Abstract ( 1291 )   PDF (248KB) ( 432 )   In this paper, we investigate the electron self-energy and effective mass in a single heterostructure using Green-function method. Numerical calculations of the electron self-energy and effective mass for GaAs/AlAs heterostructure are performed. The results show that the self-energy (effective mass) of electrons, which incorporate the energy of electron coupling to interface-optical phonons and half of the three-dimensional longitudinal optical phonons, increase (decrease) monotonically from that of interface polaron to that of the 3D bulk polaron with increasing the distance between the positions of the electron and interface.

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Evidence of phase separation in Nd1-xSrxMnO3(x=0.50, 0.51, 0.52, 0.53, 0.54, 0.55) Lu Yi (鲁毅), Li Qing-An (李庆安), Di Nai-Li (邸乃力), Li Run-Wei (李润伟), Ma Xiao (马骁), Kou Zhi-Qi (寇志起), Cheng Zhao-Hua (成昭华) Chinese Physics, 2003, 12 (11):  1301-1304.  DOI: 10.1088/1009-1963/12/11/320 Abstract ( 1359 )   PDF (225KB) ( 615 )   The temperature dependence of Mn ion magnetization and the conductivity difference versus temperature provide phase separation evidence in Nd$_{1-x}$Sr$_x$MnO$_3$($x$=0.50, 0.51, 0.52, 0.53, 0.54, 0.55). This is different from other reports and may suggest that a fully charge-ordered state exists in quite a narrow range; the sample preparation procedure affects the charge ordering. The polaron effect also plays an important role in explaining the conductivity.

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Domain structures of Nd13Fe80B7 magnets during HDDR process Zhang Zhen-Rong (张臻蓉), Pang Zhi-Yong (庞智勇), Zhao Yi-Min (赵义敏), Zhang Zheng-Yi (张正义), Han Sheng-Hao (韩圣浩), Han Bao-Shan (韩宝善) Chinese Physics, 2003, 12 (11):  1305-1309.  DOI: 10.1088/1009-1963/12/11/321 Abstract ( 1389 )   PDF (647KB) ( 600 )   The domain structures of Nd$_{13}$Fe$_{80}$B$_7$ alloy at different stages of the HDDR process have been revealed using a magnetic force microscope. In the as-cast samples, the columnar crystals with easy axis perpendicular to one another are clearly characterized by their different domain structures. For the insufficient and sufficient HD treatment, an obvious change of domain structure occurs, which is related to the variation of composition and crystalline microstructure during the HD process. And for the samples after sufficient DR processing, it is confirmed that the configuration of the columnar crystals is retained by the detected domain structures.

GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS

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Central black hole masses of galaxies Fan Jun-Hui (樊军辉) Chinese Physics, 2003, 12 (11):  1310-1316.  DOI: 10.1088/1009-1963/12/11/322 Abstract ( 1290 )   PDF (213KB) ( 451 )   In this paper, the stellar velocity dispersions in the host galaxies are used to estimate the central black hole masses for a sample of elliptical galaxies. We find that the central black hole masses are in the range of $10^{(5.5-9.5)}M$⊙. Based on the estimated masses in this paper and those by Woo & Urry (2002) and the measured host galaxy absolute magnitude, a relation, log ($M_{BH}/M$⊙) = $-(0.25\pm4.3\times10^{-3})M_{\rm R} + (2.98\pm0.208)$ is found for central black hole mass and the host galaxy magnitude. Some discussions are presented.

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Classical and quantum wormholes with tachyon field Gao Chang-Jun (高长军), Shen You-Gen (沈有根) Chinese Physics, 2003, 12 (11):  1317-1324.  DOI: 10.1088/1009-1963/12/11/323 Abstract ( 1099 )   PDF (331KB) ( 385 )   The wormhole equations are presented in the presence of tachyon field. Specializing at some values of $\omega$ (the ratio of pressure to energy density), we find a family of classical and quantum wormhole solutions.