The geometry of the SU(2) Kepler problem

The SU(2) Kepler problem is defined and analyzed, which is a Hamiltonian system reduced from the conformal Kepler problem on T^*( ⁸ − {0}) by the use of the symplectic SU(2) action lifted from the SU(2) left action on the SU(2) bundle ⁸ − {0} → ⁵ − {0}. This reduced system has a parameter μ ε su(2) coming from the value of the moment map associated with the symplectic SU(2) action. If μ ≠ 0, the phase space of this system have a bundle structure with base space T^*( ⁵ − {0}) and fibre S². The fibre, a (co)adjoint orbit through μ for SU(2), represents the internal degrees of freedom, called the isospin, of the particle of this system. The SU(2) Kepler problem with μ ≠ 0 is then interpreted as describing the motion of a classical particle with isospin in the Newtonian potential plus a specific repulsive potential together with a Yang-Mills field. This Yang-Mills field is to be referred to as BPST Yang's monopole field in ⁵ − {0};, since it becomes the Belavin-Polyakov-Schwartz-Tyupkin instanton, restricted on S⁴. If μ = 0, the SU(2) Kepler problem reduces to the ordinary Kepler problem. Like the ordinary Kepler problem, the Hamiltonian flows of the SU(2) Kepler problem of negative energy are all closed. It is shown in an explicit manner that the energy manifolds and isoenergetic orbit spaces for the SU(2) Kepler problem of negative energy are both homogeneous manifolds on which SU(4) acts transitively to the right; those homogeneous manifold are classified into two, according as the parameter μ is zero or not. For a certain value of μ, however, they contracts to the manifold which represents the set of all the equilibrium states. The isoenergetic orbit spaces are finally shown to be symplectomorphic to certain Kirillov-Konstant-Souriau coadjoint orbits for U(4), if μ is not the exceptional value mentioned above.     

X-ray and high resolution selenium-77 solid state NMR spectroscopy as complementary probes to structural studies of organophosphorus diselenides

high resolution solid state NMR spectroscopy was employed to study structural properties of bis(diisopropoxyphosphorothioyl) diselenide 1 and bis(dineopentoxyphosphorothioyl) diselenide 2. The principal elements T_ii of effective dipolar/chemical shift tensor were calculated from spinning sideband intensities employing the WIN-MAS program. The values of anisotropy and asymmetry parameters reflect the distortion of the selenium environment. It was found that the T₃₃ component mostly contributes to changes in the isotropic chemical shifts. CP/MAS experiments were used to decide the assignment of space group by counting the number of crystallographically unique selenium centers in the unit cell. Crystals of diselenide 1 are triclinic, space group P with a=8.485(3) Å, b=8.508(1) Å, c=8.511(2) Å, =98.835(15)°, β=111.653(24)°, γ=93.524(21)°, V=559.5(3) ų, D_c=1.544(2) g/cm³ and Z=1. Refinement using 2222 reflections for 157 variables gives R=0.037. Crystals of diselenide 2 are triclinic, space group P1 with a=9.1418(8) Å, b=9.1465(8) Å, c=9.9200(9) Å, =74.751(8)°, β=74.629(7)°, γ=82.216(7)°, V=769.7(1) ų, D_c=1.365(2) g/cm³ and Z=1. Refinement using 3316 reflections for 297 variables gives R=0.0272.     

Electron spectroscopy observation of the 1s2s2p(S) state in atomic fluorine by means of electron impact excitation of the hf molecule

The energy of the 1s2s²2p⁶²S) core excited state in atomic fluorine is determined by studying a satellite in the electron beam excited KVV Auger electron spectrum of HF. The satellite is assigned to an atomic autoionization transition 1s2(²2p⁶(²S) → 1ss²2p⁴ (¹D). The energy of the 1s2s²2P⁶(²s) initial state has been determined to be 676.5 eV. Transitions to the 1s²2s²2p⁴³p(³P)and 1s²2s²2P⁴(¹S) states are also observed. The intensity for these transitions is very low.     

Remarks on stochastic non-linear birth and death models

The first part of this paper is an attempt to formulate and motivate additional work on the important problem of obtaining global bounds applicable to the controlled truncation of the paper relates specifically to the linear birth, quadratic death model. Asymptotic results are given for the first finite difference ΔT_m where T_m is the exactly known mean time to extinction starting from state m (m= 0,1,…). These results are in terms of the environmental carrying capacity n^* taken to be large. For m near zero ΔT_me^n*/(n^*)²; whereas, for m near n^*ΔT_m ≈ (π/2)^1/2/(n^*)^3/2. This indicates the vastly different time scales in those two regions of state space - with considerably slower action near extinction than near n^*.     

Analytical result for dimensionally regularized massless master non-planar double box with one-leg off shell

The dimensionally regularized massless non-planar double box Feynman diagram with powers of propagators equal to one, one leg off the mass shell, i.e., with p₁²=q²≠0, and three legs on shell, p_i²=0, i=2,3,4, is analytically calculated for general values of q² and the Mandelstam variables s,t and u (not necessarily restricted by the physical condition s+t+u=q²). An explicit result is expressed through (generalized) polylogarithms, up to the fourth order, dependent on rational combinations of q²,s,t and u, and simple finite two- and three-fold Mellin–Barnes integrals of products of gamma functions which are easily numerically evaluated for arbitrary non-zero values of the arguments.     

Projective and affine connections on S and integrable systems

It is known that the Korteweg–de Vries (KdV) equation is a geodesic flow of an L² metric on the Bott–Virasoro group. This can also be interpreted as a flow on the space of projective connections on S¹. The space of differential operators Δ⁽ⁿ⁾=∂ⁿ+u₂∂ⁿ⁻²++u_n form the space of extended or generalized projective connections. If a projective connection is factorizable Δ⁽ⁿ⁾=(∂−((n+1)/2−1)p₁)(∂+(n−1)/2p_n) with respect to quasi primary fields p_i’s, then these fields satisfy ∑_i=1ⁿ((n+1)/2−i)p_i=0. In this paper we discuss the factorization of projective connection in terms of affine connections. It is shown that the Burgers equation and derivative non-linear Schrödinger (DNLS) equation or the Kaup–Newell equation is the Euler–Arnold flow on the space of affine connections.     

Ca~(2+)掺杂对SmFeO_3的介电、铁磁特性及磁相变温度的影响

采用固相反应法制备Sm_(1-x)Ca_xFeO_3(x=0,0.1,0.2,0.3)样品,研究Ca~(2+)掺杂对SmFeO_3介电性能、铁磁性及磁相变温度的影响.X射线衍射图谱分析表明:所有样品的主衍射峰与SmFe03相符合且具有良好的晶体结构.随着x的增加,SmFeO_3样品的晶粒尺寸由原来的0.5μm逐渐增大到2μm.当f=1 kHz时,Sm_(1-x)Ca_xFeO_3(x=0.1,0.2,0.3)样品的ε_r分别是SmFe03的5倍、3倍和2.6倍,而tgσ增大一个数量级.在3T磁场作用下,SmFe03样品的M-H呈线性,随着x的增加,M-H逐渐趋向饱和,Sm_(1-x)Ca_xFeO_3(x=0.1,0.2,0.3)样品的M_r分别是SmFeO_3的20倍、31倍和68倍.X射线光电子能谱分析表明:Fe~(2+)和Fe3+共存于Sm_(1-x)Ca_xFeO_3样品中,Fe~(2+)/Fe~(3+)比例随着x的增加而增大,证明Ca~(2+)掺杂增加了Fe~(2+)的含量,形成Fe~(2+)—O~(2-)—Fe~(3+)超交换作用,增强SmFe03的铁磁特性.测量了Sm_(1-x)Ca_xFeO_3样品在外加磁场为1000 Oe(1 Oe=79.5775 A/m)的M-T变化关系,观测到其自旋重组温度(T_(SR))和尼尔温度(T_N)分别为438 K和687 K,发现SmFe03样品的T_(SR)和T_N均随着x的增加向低温方向移动,当x=0.3时,自旋重组现象消失.这主要是SmFeO_3样品磁结构的稳定性和Fe~(3+)—O~(2-)—Fe~(3+)及Sm~(3+)—O~(2-)—Fe~(3+)超交换三者共同作用的结果.     

Positivity of energy in five dimensional classical unified field theories, II

Five dimensional classical unified field theories as well as Yang-Mills theory with gauge group U(1), are described in terms of a Lorentzian five dimensional space V₅ with metric tensor γβ which admits a space-like Killing vector ζ. It is assumed that: (1) V₅ has the topology of V₄ x S¹, S¹ is a circle and V₄ is a four dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor Γ_β of V₅ satisfies Γ_β Uυ^{β 0} where U and υ are future oriented time-like vectors with γ_βυζ^β = 0. The spinor approach of Witten [4], Nester [3] and Moreschi and Sparling [5] is used to show that the conserved five dimensional energymomentum vector P = ifΓ_β = 0 then V₅ must admit a time-like Killing vector. Lichnerowicz's results [1] then imply that V₅ must be flat. A lower bound for P⁴ (the mass) similar to that found by Gibbons and Hull [6] is obtained.     

Field-induced parameters of re-entrant magnets and concentrated spin glasses

We have used electron spin resonance measurements to derive the temperature and frequency dependences of the field-induced magnetization [M(T, f)] and anisotropy field [H_an (T)] in a number of amorphous alloys belonging to the series (Fe_pNi_1−p)₇₅P₁₆B₆Al₃. In re-entrant (p > p_c, the critical concentration for ferromagnetism) alloys at hi gh frequencies (f = 35 GHz, field ≈ 12 kOe) M reduces as T^3/2 at high T and as T below ≈ 40 K, the deviation from T^3/2 becoming more marked as p → p⁺_c. For p close to p_c, lowering the frequency first causes the T term to increase and ultimately ( ≈ 4 GHz) changes the variation of M with T to that discovered previously for concentrated spin glasses, namely M is constant at low T and drops linearly at high temperatures. For the re-entrants, the results are interpreted on the basis of a model which invokes an energy gap in the spin-wave spectrum, introduces a non-zero density of states of the gap energy and takes into consideration a low-q cut-off in the spin-wave integral in thelow-T (T) regime.In the concentrated spin glasses [M (0) - M (T)]/ M (0) is well represented by the function [exp (Δ / T) - 1]^-1, where Δ has values close to the corresponding Curie-Weiss temperatures θ_p but much larger than the respective spin glass transition temperatures T_SG. The temperature dependence of H_an is largely given by the function (1 - T/T^*), where T^* is equal to the zero-field freezing temperature for the re-entrants and T_SG for the spin glasses, respectively.     

具有Dzyaloshinskii-Moriya相互作用的XY模型的量子相干性

研究了具有Dzyaloshinskii-Moriya(DM)相互作用的一维横场XY自旋链的量子相变和量子相干性.采用约旦-维格纳变换严格求解了哈密顿量,并描绘了体系的关联函数和相图,相图包含反铁磁相、顺磁相和螺旋相.利用相对熵和Jensen-Shannon熵讨论了XY模型的量子相干性.研究发现,相对熵与Jensen-Shannon熵所表现的行为都可以很好地表征该模型的量子相变.非螺旋相中量子相干性不依赖DM相互作用,而在螺旋相DM相互作用对量子相干性有显著影响.此外,指出了在带有DM相互作用的这一类反射对称破缺体系中关联函数计算的常见问题.     

The investigation of the peculiarities of Cu states localized by Al ions in YBCO single crystals by EPR technique

A new EPR spectrum in YBCO single crystals doped by Al³⁺ ions have been revealed. The spectrum was assigned to trivalent Cu³⁺ (S=1) ions in the Cu(1) position and the spin-hamiltonian parameters describing the observed spectrum have been determined on the basis of computer calculation of the EPR spectrum. The theoretical consideration and numerical estimations carried out showed that the ground state of Cu³⁺ ions in the YBCO system might be a triplet and, thus, the possibility of the observation of its EPR spectrum has been confirmed. The analysis of the experimentally determined g-factor values showed the important role of the hybridization of 3d-copper and 2p-oxygen states and, in particular, the significant contribution of the Cu²⁺-O^- state in the ground one. The stabilization of Cu³⁺ states in YBCO single crystals by Al³⁺ ions is discussed. It is shown that the observed temperature dependence of the g_x-value was due to the displacement of the apex oxygen O(4) in the direction of the CuO₂ plane as T→T_c.     

N=2 6-dimensional supersymmetric E6 breaking

We study the N=2 supersymmetric E₆ models on the 6-dimensional space–time where the supersymmetry and gauge symmetry can be broken by the discrete symmetry. On the space–time M⁴×S¹/(Z₂×Z₂′)×S¹/(Z₂×Z₂′), for the zero modes, we obtain the 4-dimensional N=1 supersymmetric models with gauge groups SU(3)×SU(2)×SU(2)×U(1)², SU(4)×SU(2)×SU(2)×U(1), and SU(3)×SU(2)×U(1)³ with one extra pair of Higgs doublets from the vector multiplet. In addition, considering that the extra space manifold is the annulus A² and disc D², we list all the constraints on constructing the 4-dimensional N=1 supersymmetric SU(3)×SU(2)×U(1)³ models for the zero modes, and give the simplest model with Z₉ symmetry. We also comment on the extra gauge symmetry breaking and its generalization.     

Deuteron NMR spectra of ND4ClO4 single crystal at low temperatures

²H NMR spectra of ND₄ClO₄ single crystal were obtained at ν₀ = 44 MHz. Orientation and temperature (1.9–75 K) dependences were measured. Fitting the spectra gives the effective quadrupole coupling constants for all deuterons and the ground torsional level structure. The isotope reduction of the (A-T) and (A-E) tunnelling splittings, i.e., the ratios of the respective splittings for NH₄⁺ and ND₄⁺, were found to be different. The splittings at T = 24 K are about 60% of the helium temperature values. The spectrum undergoes intermediate narrowing by reorientations between 26 and 34 K and tunnelling related features in the spectra are eradicated. After reaching the extreme narrowing limit, a doublet with gradually decreasing separation was observed, what was attributed to averaging by torsional oscillations of increasing amplitude. At high temperatures (T > 75 K), the narrow spectrum reflects fast multiaxial reorientation of the ammonium ion.     

NaZnLa(PO4)2中Ce3+和Tb3+的发光

采用高温固相反应合成了NaZnLa(PO₄)₂中掺杂Ce³⁺、Tb³⁺的荧光体,对其晶体结构、发光行为进行了研究,并尝试对NaZnLa(PO₄)₂:Ce,Tb荧光体进行调制。NaZnLa(PO₄)₂是LaPO₄的同构物,为单斜晶系独居石结构,从XRD谱数据得到NaZnLa(PO₄)₂基质的晶胞参数为a=0.6823nm,b=0.7045nm,c=0.6497nm,β=1039°,v=0.303nm³,其晶胞参数与单斜LaPO₄的晶胞参数相似。在NaZnLa(PO₄)₂:Ce,Tb荧光体中,Ce³⁺对Tb³⁺有良好的敏化作用,掺杂适量的BO₃^3-、Al³⁺、Dy³⁺,可以增强发光亮度。     

Mössbauer investigations on Np3S5

The 59.5 keV Mössbauer resonance of ²³⁷Np has been measured in Np₃S₅ at 77 and 4.2 K. At 77 K, two different quadrupolar splittings are observed. The first one (δ₁ = −5(1) mm sec⁻¹/NpAl₂) is assigned to the Np⁴⁺ ions and the second one (δ₂ = 28(1) mm sec⁻¹/NpAl₂) is assigned to the Np³⁺ ions. At 4.2 K the spectrum gives a ratio of areas S(Np³⁺)/S(Np⁴⁺) of 2.0. These results clearly confirm the crystallographic data of this mixed-valence material.     

Interactions between a tetraanionic porphyrin and the methylviologen dication in methanol studied by fluorescence quenching reaction

Fluorescence decay curves for 5,10,15,20-tetrakis(4-sulfonatophenyl)-21H,23H-porphine tetraanion (TPPS⁴⁻) have been measured in the absence and presence of the methylviologen dication (MV²⁺) with various ionic strengths in methanol. In the presence of MV²⁺ the fluorescence decays can be expressed by a double exponential function, I(t = I₁exp(−t/τ₁) + I₂exp(−t/τ₂). The contribution by the faster decay component to the total fluorescence signal increases with increasing MV²⁺ concentration. The faster decay process is attributed to fluorescence from the excited state of a solvent-separated ion pair (SSIP) formed between TPPS⁴⁻ and MV²⁺, and the slower process is attributed to fluorescence from free TPPS⁴⁻ ions in the solution. Rate constants for the quenching of fluorescence from free TPPS⁴⁻ by MV²⁺ (k_q) and formation constants for the SSIP (K_SSIP) were calculated and both are found to decrease with increasing ionic strength. The decrease in k_q and K_SSIP values can be interpreted in terms of the shielding of electrostatic attraction between the ions.     

周期受击陀螺系统随时间演化波函数的多重分形

研究了周期受击陀螺系统波函数的多重分形. 发现: 1)在打击次数较小时, 周期受击陀螺系统波包的扩散速度、扩散方向与打击强度相关, 打击强度越大扩散越混乱、扩散速度也越大; 2)波函数在相空间的精细结构的分布范围随着打击强度的增大而扩大, 最后充满整个相空间; 3)局部分维a的分布范围对应波函数在相空间的分布, 规则态时a 的分布范围最宽, 过渡态的a的分布范围较窄, 而混沌态的a的分布范围则最狭窄且稳定.     

用于α和μ常数变化测量的碘离子光谱研究

分子离子I_2~+的禁戒跃迁光谱有可能用于测量α和μ常数的变化,并且具有增强的灵敏度.通过分析I_2~+在11860—13100 cm~(-1)范围内的转动光谱,拟合了A~2Π_(3/2)-X~2Π_(3/2)系统31个振转带的5759根吸收谱线,得到5个属于X~2Π_(3/2)态和9个属于A~2Π_(3/2)态的振动能级准确的转动光谱常数.在量子噪声极限和1 Hz跃迁线宽的条件下,计算得到X~2Π_(3/2)和X~2Π_(1/2)之间的禁戒跃迁对α和μ常数变化测量的灵敏度为δ_(α/α)≈2.37×10~(-19)a~(-1)和δ_(μ/μ)≈1.18×10~(-18)a~(-1).     

三种还原M6O19n-杂多酸阴离子的ESR谱

本文报道了三种具有Lindqvist结构的还原杂多阴离子ESR谱,在室温和适合的pH范围内,VMo₅O₁₉^4-的溶液ESR谱含有8条超精细线而V₂W₄O₁₉^5-和V₃W₃O₁₉^6-则具有15条超精细线,这种15条谱线的形状会逐渐变化。长期放置后它会变成8条线,我们用计算机成功地模拟了这些复杂的8+15条谱线,得到了一系列不同8线和15线比例的模拟谱。文中也测定了77K冷冻谱的ESR参数并计算了近似的K、α值,把所得的结果与Keggin结构杂多阴离子的K、α值进行比较后发现:Keggin结构的K,α值较小,原因是Keggin结构中存在有角共享的氧原子增加了未偶电子的额外离域性。