变换参考系解二体问题

【题目】真空中质量分别为m₁和m₂的两个小球,只受万有引力作用,某个时刻两个小球相距l₀,小球1的速度为v₀,方向指向小球2,小球2的速度为v₀,速度方向垂直两球球心的连线,问若m₁=m₂=m₀,当速度v₀满足什么条件时,两小球的间距可以为无穷远?解法1:惯性参考系法（质心参考系）解题思路:寻找惯性参考系,分析两小球相对于惯性参考系的受力,确定两小球在惯性参考系的运动特点,直接运用牛顿运动定律或能量守恒求两小     

典型习题中的现代物理知识（第二讲） 黑洞,GM/c2还是2GM/c2?

 上一讲所说引力红移,即频率为v₀的光子从质量为M、半径为R的天体表面发射到无穷远,频率会变为v（v0）,引力为红移: Z=（v₀-v）/v=（1-GM/c²R）^-1-1…（1） 由此可知,一定质量的天体,其半径越小,周围的引力场越强,光的引力红移现象越显著。当R缩到R_g=GM/c²…（2）时,v将变为零（Z→∞）,这意味着光子的动能mc²=hv₀在引力场中全部耗尽（hv₀=GMm/R）;若Rg,光子的动能再也无法满足引力势能增加的需要,这时该天体的引力场就强到了连光子也不能从其表面逃逸的程度。 第五届全国中学生物理竞赛（1988年）决赛中就有一题涉及到这种天体:“有一种超高密度的天体,其巨大引力使得光子也无法逃逸出来,……,这种天体叫黑洞。     

二维Frenkel-Kontorova模型中六角对称结构的摩擦现象

基于二维 Frenkel-Kontorova 模型, 运用分子动力学模拟方法, 研究了具有六角对称结构的系统从 locked 态到 sliding 态的相变, 并数值分析了上层原子采用六角对称结构时, 系统的不同参数对静摩擦力F_s和动摩擦力F_c的影响. 关键词： 二维 Frenkel-Kontorova 模型 超润滑 分子动力学 摩擦     

气相CCl2自由基的发射光谱

在CCl₄蒸汽的直流放电中,观测了从520到630nm波长范围的荧光发射并确认为CCl₂自由基A→X跃迁。近400条谱带归属为³⁵Cl-C-³⁵Cl和³⁵Cl-C-³⁷Cl的(v₁′,v₂′,0)→(v₁″,v₂″,0)一系列前进带组,并排列成Deslandres表。光谱分析给出CCl₂在激发态和基态的v₁和v₂振动频率及同位素位移,其中气相CCl₂在基态的数据尚未见文献报道。 关键词：     

多基点同伦群理论

本文建立了一个定理:当拓扑空间V单连通时,N(≥1)基点n阶同伦类集合π_n(V;v₁,v₂,…,v_N)可被构造成同构于单基点n阶同伦群π_n(V)的群,叫做N基点n阶同伦群;并且V的单连通性一般是不可省略的条件。给出了一些推论。简要地触及了这个定理及其推论在铁磁状态拓扑分类中的应用问题。 关键词：     

原子间相互作用势对中Al浓度Ni75AlxV25-x合金沉淀序列的影响

基于微观相场模型与反演算法,研究了中Al浓度及温度对Ni₇₅Al_xV_25-x合金沉淀过程的影响：在相同浓度下,L1₂与DO₂₂结构的第一近邻原子间相互作用势随温度升高呈线性增加,两者呈正比的关系;但在同一温度下,L1₂（DO₂₂）结构的第一近邻原子间相互作用势随Al原子浓度的增加而增加（减少）.同时将反演得出的原子作用势代入微观相场模拟中,探讨中Al浓度合金沉淀序列与原子作用势的关系,即当L1₂的第一近邻原子间相互作用势大于（小于）DO₂₂时,L1₂（DO₂₂）优先析出;当L1₂和DO₂₂的第一近邻原子间相互作用势相等时,两者同时析出.特别地,当Al原子的浓度等于0.0589时,发现L1₂和DO₂₂同时析出.利用微观相场法反演原子间相互作用势,为判断中Al浓度合金的沉淀序列增加了可信度.     

石墨/Li（Ni1/3Co1/3Mn1/3）O2电池充放电过程中电极材料的XRD研究

利用XRD系统地研究了石墨/Li（Ni_1/3Co_1/3Mn_1/3）O₂ 18650型锂离子电池充放电过程中正负极活性材料的晶体结构和微结构的变化.已观测到,由于Li原子的脱嵌,使得LiMO₂点阵参数a缩小,c增大,微应变增大,衍射强度比I₁₀₄/I₁₀₁和I₀₁₂/I₁₀₁降低;此外,由于Li原子的嵌入,2H-石墨的点阵参数a和c,以及微应变ε和堆垛无序度P都增加.同时,讨论了活性材料Li（Ni_1/3Co_1/3Mn_1/3）O₂和石墨在电池充放电过程中的嵌脱锂的物理机理.在充电时,正极Li（Ni_1/3Co_1/3Mn_1/3）O₂中处于（000）位的Li原子优先脱离晶体点阵,继后才是位于（2/3 1/3 1/3）和（1/3 2/3 2/3）位的Li原子离开点阵.锂嵌入石墨,优先进入碳原子六方网格面间的间隙位置,当负极的堆垛无序度达到一定值后,3R相逐渐析出.当电池满充或过充时,在六方石墨中形成LiC₁₂和LiC₆相.放电时,与上述过程相反,但并非是完全可逆的. 关键词： 锂离子电池 微结构 X射线衍射 嵌脱锂物理机理     

超声射流冷却CCl2自由基的激光诱导荧光激发谱

在497—517nm波长范围观测超声射流冷却CCl₂自由基的激光诱导荧光激发谱。得到CCl₂在?←X跃迁中(v₁,v₂,0)←(0,0,0)振动带清晰的K结构。对近80个子谱带作了归属。振动分析给出较精确的v＇₁和v＇₂振动频率和非谐性常数;从K结构的分析得出(A＇-B＇)和(A″-B″)的值,文献未曾报道过。 关键词：     

IrB和IrB$lt;sub$gt;2$lt;/sub$gt;力学性质的第一性原理计算

采用平面波赝势密度泛函理论对0–100 GPa静水压下P₁ -IrB（空间群Pnma）和P₅ -IrB₂（空间群Pmmn）结构的平衡态晶格常数、弹性常数等性质进行了研究. 研究结果表明,P₁ -IrB不可压缩性随着压强的增加而增强;P₅ -IrB₂ 结构在0–100 GPa范围内弹性常数、体弹模量、剪切模量均呈现出有规律的变化,当所加压强为50 GPa时,杨氏模量和在b方向的晶格常数发生异常变化. 对零压下P₁ -IrB和P₅ -IrB₂ 的电子结构的研究发现,二者均没有一个明显的带隙,主要原因为Ir原子和B原子间的共价作用. 从P₁ -IrB和P₅ -IrB₂的能带结构和态密度图可以发现这两种结构均有金属性. 关键词： 1 -IrB')" href="#">P₁ -IrB 5-IrB₂')" href="#">P₅-IrB₂ 第一性原理 力学性质     

Ni75Cr25-xAlx合金中L12相和D022相形核孕育期的微观相场模拟

基于微观相场动力学模型,运用原子图像、平均长程序参数和平均成分偏离序参数,研究了Al浓度对Ni₇₅Cr_25-xAl_x合金中L1₂相和D0₂₂相形核孕育期的影响以及孕育期与沉淀相析出顺序之间的关系.结果表明,L1₂相和D0₂₂相的形核孕育期不仅与Al浓度有关,而且与两相析出的先后顺序密切相关.当Al浓度小于7.5%时,先析出相为D0₂₂相,随着Al浓度的增大,D0₂₂相形核孕育期延长,后析出的L1₂相的形核孕育期也延长,L1₂相的原子簇?聚?速度加快;当Al浓度大于7.5%时,先析出相为L1₂相,随着Al浓度的增大,L1₂相的形核孕育期缩短,后析出的D0₂₂相形核孕育期也缩短.当Al浓度为7.5%时,L1₂相和D0₂₂相几乎同时析出,两者的孕育期没有明显的差别. 关键词： 形核孕育期 序参数 沉淀 微观相场模拟     

Infrared combination bands of CH4 in crystal fields

Overtone and combination bands were observed between 1000 and 5000 cm^–1 for CH₄ molecules in several crystal fields: solid phases I and II, and diluted alloys with Kr. In the anisotropic field of phase II, a full (rotation/libration)-vibration spectrum was observed in the combination bands 2v ₄,v ₂+v ₄, andv ₃+v ₄. Here, all lines could be assigned to transitions of either D_2d or O_h molecules. The structure ofv ₃–v ₄,v ₂+2v ₄,v ₁+v ₄,v ₂+v ₃ and 3v ₄ bands could not be resolved. In phase I, all spectra show the characteristic features of rotational diffusion, while in a Kr matrix a rotovibrational structure could again be observed in the overtone and combination bands.     

零质量射流激励下诱发液体相变及其格子Boltzmann方法模拟

使用格子Boltzmann方法对零质量射流激励下液体的相变演化过程进行了数值模拟和分析.首先,提出了此特定零质量射流进出口边界的处理格式.然后,结合Shan和Doolen提出的单组分多相模型,模拟了方腔内液体受到此零质量射流激励而诱发产生空化的过程,着重分析了三个重要射流参数ε/T,T和v out/v in对方腔内液体相变的影响.分析表明:演化过程中方腔内气相节点数量在初始阶段急剧增长,然后经振荡趋于一个稳定值.由于ε/T和v out/v in可以反映射流在出入方腔两个过程间相互转换时的急剧变化,所以能够影响方腔中的液体相变的演化;而改变参数T并不影响射流速度的变化程度,所以T对液体相变的影响较弱.对于本文给定的参数取值,ε/T较小时,方腔内液体相变生成的孤立气泡脱离壁面;较大的ε/T下产生附着于方腔壁面的气泡,并且能够加速液体的相变进程;v out/v in的增加使方腔内相应的孤立气泡所覆盖的范围略有减小.研究结果揭示了零质量射流激励诱发的液体相变过程,为进一步探索液体空化的控制途径奠定了基础.     

Dilemma game in a cellular automaton model with a non-signalized intersection

We numerically study traffic flow, energy dissipation and social payoff in the Nagel-Schreckenberg model with a non-signalized intersection. In terms of game theory, we analyze dilemma game observed in some traffic states. There are four traffic phases: free-flow phase, phase-segregated 1, phase-segregated 2 and jammed phase in the case of v _max > 1. In phase-segregated 1, maximum traffic flow corresponds to minimal energy dissipation. Dilemma game is observed at the phase-segregated 1 in the case of v _max > 1, and phase segregation state when v _max = 1. Theoretical analyses give an agreement with numerical results.     

The magnetic structure of neodymium crystal below the néel temperature

The phase transition in neodymium crystal from the paramagnetic to the magnetically ordered phase at the temperature T _N = 19.9 K is discussed in the terms of Landau's theory. It is shown that the calculated six-dimensional active representations at the point q = vb ₁, with—0.5 < v < 0.5, v ≠ 0, induce transitions to “single-q”, “double-q” and “triple-q” spin structures. A general formula for the spin vector S(r) for T < T _N is derived, from which a variety of spin structures can be obtained depending on the choice of the free parameters. The spin structure calculated by Bak and Lebech (1978) is obtained from that formula as a particular solution which is compatible with Landau's theory. An antiferromagnetic spin structure which represents a modification of that given by Bak and Lebech (1978) is calculated. The corresponding cross-section for neutron elastic magnetic scattering fulfils the experimental conditions listed by Lebech (1981). It is shown that the magnetic satellites of the type (h ± v, 0, 0) cannot appear for any type of structure following from our general formula for S(r). These satellites therefore can be connected only with representations at the point q = μ₁ b ₁ + μ₂ b ₂,—0.5 < μ₁, μ₂ < 0.5, μ₁, μ₂ ≠ 0, μ₁ ≠ μ₂.     

Anharmonic resonances between the v 1 + v ±1 6, v 3 + v ±1 5 + 2v 0 6 and v 3 + v ∓1 5 + 2v ±2 6 vibrational bands of CH3 79Br

The 22 K sub-bands, belonging to the v ₁ + v ^±1 ₆, v ₃ + v ^± ₅ + 2v ⁰ ₆ and v ₃ + v ^±1 ₅ + 2v ^±2 ₆ rovibrational bands of monoisotopic CH₃ ⁷⁹Br, have been identified. An RMS standard deviation of about 0·020 cm^-1 has been achieved by a least squares fit over 619 lines belonging to these 22 sub-bands. For this purpose, a model taking into account (a) anharmonic resonances between (i) v ₁ + v ^±1 ₆, and v ₃ + v ^±1 ₅ + 2v ⁰ ₆, (ii) v ₁ + v ^±1 ₆ and v ₃ + v ^?1 ₅ + 2v ^±2 ₆ and (iii) v ₃ + v ^±1 ₅ + 2v ⁰ ₆ and v ₂ + 2v ₃ + v ^±1 ₅ and (b) Coriolis resonances between the v ₂ and v ₅ modes, was used.     

Vibrationally excited states of HC5N: millimeter-wave spectroscopy and coupled cluster calculations

The rotational spectrum of HC₅N has been investigated in the millimeter-wave region, from 60 to 290 GHz, for 15 vibrationally excited states which lie approximately between 500 and 860 cm⁻¹, namely (v₆ v₇ v₈ v₉ v₁₀ v₁₁) = (000005), (000006), (000007), (000008), (000020), (000030), (001000), (010000), (100000), (000021), (000101), (001001), (010001), (000110), and (001010). Gas-phase copyrolysis of pyridine and phosphorus trichloride or, alternatively, a dc discharge in a gaseous mixture of vinyl cyanide and acetylene were used to produce the semi-stable HC₅N molecule. A large number of vibrational and rovibrational interactions has been taken into account to fit properly the measured transition frequencies of the states investigated. The most important perturbations are caused by the high-order Coriolis resonances observed between the (v₈, v₁₀) and (v₈ − 1, v₁₀ + 2) states, and between the (v₇, v₁₀, v₁₁) and (v₇ − 1, v₁₀ + 3, v₁₁ − 1) states, and by the cubic anharmonic interactions which mix the (v₆, v₈, v₁₁) states with the (v₆ − 1, v₈ + 1, v₁₁ + 1) states, and the (v₆, v₁₀) states with the (v₆ − 1, v₁₀ + 2) states. The analysis of the spectra was facilitated by CCSD(T) calculations with the cc-pVQZ basis, which provided accurate predictions of a large variety of spectroscopic constants like harmonic vibrational wavenumbers, vibration–rotation coupling constants, l-type doubling constants, and normal-coordinate cubic force constants. Excellent agreement between experiment and theory was generally observed, even when the experimental data were strongly perturbed by resonance effects.     

Study of the spectrum of CH335C1 between 4250 and 4600 CM?1: thev2+v4 andv4+v5 rovibrational bands

About 2500 lines of CH₃ ³⁵Cl have been assigned. The strong xy Coriolis resonance between thev ₂ andv ₅ modes is quite visible between thev ₄+v ₄ ^±1 perpendicular band, centered around 4383 cm^–1, and thev ₄ ^±1 +v ₅ ^±1 perpendicular component, centered around 4475 cm^–1, with a crossing of upper energy levels allowing the observation of lines which are normally forbidden. Although not yet observed with certainty, because of the great density of lines of the spectrum, thev ₄ ¹ +v ₅ ^±1 parallel component is nevertheless detectable by its effects onv ₂+v ₄ ^±1 which is linked by Coriolis resonance to both components ofv ₄+v ₅. Moreover the spectrum is much complicated by many other resonances with weak bands which occur at level crossings: it is the case ofv ₂+3v ₆ ^±1 , connected tov ₂+v ₄ ^±1 by the well known Darling Dennison resonance which couplesv ₄ ^±1 and 3v ₆ ^±1 , and also ofv ₅ ^±1 +3v ₆ ^±1 connected tov ₄ ^±1 +v ₅ ^±1 by the same resonance; but this last case is complicated by an anharmonic resonance betweenv ₅ ^±1 +3v ₆ ^±1 and 2v ₃+3v ₆ ¹ . Two more perturbations occur on the K=–1 side ofv ₂+v ₄: a weak Coriolis resonance gives rise to one subband ofv ₁+v ₂ at a level crossing withv ₂+v ₄, and thev ₁+v ₅ band (linked of course tov ₁+v ₂ by the Coriolis resonance between thev ₂ andv ₅ modes) is quite visible and perturbs several subbands ofv ₂+v ₄ of high values of K through an anharmonic resonance. Moreover, the complex (3v ₅ ^±1 ,v+2v ₅ ⁰ , 2v ₂+v ₅ ^±1 , 3v ₂,v ₂+2v ₅ ^±2 , 3v ₅ ^±3 ) system of Coriolis-connected bands is linked to the bands studied in the present work by two Fermi resonances: one betweenv ₂+2v ₅ ⁰ andv ₁+v ₂, and the other one betweenv ₁+v ₅ and 3v ₅ ^±1 , whose several subbands have been observed on the low part of the spectrum. The values of all the band centres and of the different coupling constants have been estimated, but all these interactions make the line assignments and the interpretation of the spectrum very difficult.     

Line strength measurements of carbonyl sulfide (OCS) in the 2v3, v1+2v2+v3, and 4v2+v3 bands

To support planetary studies of the Venus atmosphere, we measured line strengths of the 2v₃, v₁+2v₂+v₃, and 4v₂+v₃ bands of the primary isotopologue of carbonyl sulfide (¹⁶O¹²C³²S), whose band centers are located at 4101.387, 3937.427, and 4141.212 cm⁻¹, respectively. For this, infrared absorption spectra in normal carbonyl sulfide (OCS) sample gas were recorded at an unapodized resolution of 0.0033 cm⁻¹ at ambient room temperatures using a Bruker Fourier transform spectrometer (FTS) at the Jet Propulsion Laboratory. The FTS instrumental line shape (ILS) function was investigated, which revealed no significant instrumental line broadening or distortions. Various custom-made short cells and a multi-pass White cell were employed to achieve optical densities sufficient to observe the strong 2v₃ and the weaker bands in the region. Gas sample impurities and the isotopic abundances were determined from mass spectrum analysis. Line strengths were retrieved spectrum by spectrum using a non-linear curve fitting algorithm adopting a standard Voigt line profile, from which Herman–Wallis factors were derived for the three bands. The band strengths of 2v₃, v₁+2v₂+v₃, and 4v₂+v₃ of ¹⁶O¹²C³²S (normalized at 100% of isotopologue) are observed to be 6.315(13)×10⁻¹⁹, 1.570(2)×10⁻²⁰, and 7.949(20)×10⁻²¹ cm⁻¹/molecule cm⁻², respectively, at 296 K. These results are compared with earlier measurements and the HITRAN 2004 database.     

The integrated number of vibrational states in acetylene (12C2H2, 13C2H2, 12C2D2)

A calculated exhaustive set of vibrational state energies in ¹²C₂H₂, ¹³C₂H₂ and ¹²C₂D₂ has been used to analyse the evolution of the integrated number of states with increasing vibrational energy N(E) up to 15000 cm^?1, 12000cm^?1 and 10000 cm^?1 in each isotopomer, respectively. The regular contribution to N(E) was modelled analytically and numerical parameters were fitted. The other expected contribution to N(E), which is of oscillatory nature, was quantified and is discussed using energyand time-dependent theories. Related periods of oscillation and temporal recurrences are interpreted consistently in terms of the constant of the motion N_r = 5v₂ + 3v₂ + 5v₃ + v₄ + v₅ and of an average vibrational quantum. More pragmatically, the vibrational dynamics appear to be dominated by the bending vibrations, i.e., by the slowest oscillators.     

The infra-red spectrum of 13CH3F and the general harmonic force field of methyl fluoride

Analyses of the v ₃, 2v ₃, and (predominantly) v ₁ parallel bands, and of the v ₄ and v ₆ perpendicular fundamentals have been made for ¹³CH₃F in terms of the rotational structure observed with a resolution of ～0·2 cm^-1. In addition, the band centres of the strongly Coriolis-interacting v ₂ and v ₅ fundamentals are accurately located. Some elucidation of the complex Fermi resonance interactions in the 3000 cm^-1 region is achieved through study of spectra of crystalline samples. This enables all three components of the v ₁, 2v ₂, 2v ₅ ⁰ triad to be observed for both ¹²CH₃F and ¹³CH₃F, and estimates to be made for the unperturbed vibration frequencies.

The ¹³C frequency shifts determined for all six fundamentals are used in conjunction with existing frequency, Coriolis ζ, and centrifugal distortion data for CH₃F, CD₃F, CHD₂F and CH₂DF, to determine the general harmonic force field for methyl fluoride. The extra shift data enable all 12 parameters of the force field to be fixed within narrow limits for the first time. The disagreement with predictions of the hybrid orbital model in the A₁ species can be attributed to the effect of trans repulsions arising from the fluorine lone-pair electrons, an effect which contributes to the longer CH bond in methyl fluoride compared with the other methyl halides.