Thickness-shear modes and magnetoelastic waves in a longitudinally magnetized ferromagnetic plate

Magnetoelastic (ME) waves and thickness-shear modes in the ferromagnetic plate are studied. Coupled vibrations of magnetization and shear elastic deformations excited simultaneously by a variable magnetic field propagate in two mutually perpendicular directions: parallel and normal to a surface. For parameters characteristic of isotropic ferromagnet with the sample magnetization and Zeeman field parallel to the surface, resonant frequencies of shear modes are computed and their dispersion law is examined. It is shown that the dependence of dimensional resonances frequencies on wave number k_z of ME wave propagating along saturating field direction occurs. The possibility of excitation of ME waves with different k_z explains multimode character of thickness ME resonances.     

Spectral Estimates for Periodic Jacobi Matrices

 We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ²(ℤ) of the form (Hψ) _n =a _n−1 ψ _n−1 +b _nψ _n +a _nψ _n+1 , where a _n=a _n+q and b _n=b _n+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of H. Received: 17 April 2002 / Accepted: 1 October 2002 Published online: 13 January 2003 Communicated by B. Simon     

Nichtsenkrechte Interbandübergänge in Graphit durch unelastische Elektronenstreuung

Characteristic energy loss measurements are carried out on natural graphite single crystals. The frequency and wave vector dependent dielectric constant ?(ω,q) forq⊥c-axis is determined by Kramers Kronig Analysis. The results are interpreted in terms of non vertical interband transitionsE(k)→E(k+q) between the π-bands within the two dimensional approximation.     

Mass distribution in n—polymer stochastic aggregation and large—mass behaviour

The exact solutions of the rate equations of the n-polymer stochastic aggregation involving two types of clusters, active and passive for the kernel \dprⁿ_k=1s_(ik)(s_(ik)=i_k) and \dsumⁿ_k=1s_(ik)(s_(ik)=i_k), are obtained. The large-mass behaviours of the final mass distribution of the active and passive clusters have scaling-like forms, although the models exhibit different properties. Respectively, they have different decay exponents γ=\dfrac{2n+1}{2(n-1)} and γ=q+\dfrac{2n+1}{2(n-1)} for \dprⁿ_{k=1}s(ik)(s(ik)=ik) and γ=\dfrac 3{2(n-1)} and γ=q+\dfrac 3{2(n-1)} for \dsumⁿk=1}s(ik)(s(ik)=ik), which include exponents of two-polymer stochastic aggregation. We also find that gelation is suppressed for kernel \dprⁿk=1s(ik)(s(ik)=ik) which is different from the deterministic aggregation.}     

On the normalization of the Gamov resonant wave function in the configuration space

The regularization of the normalization integral for the resonant wave function, proposed by Zeldovich, is valid only when |Req _res| > |Imq _res|. A new normalization procedure is proposed and implemented, which is valid when this condition fails. First, an arbitrarily normalized vertex function g(k) is calculated using the formula with the potential V(r) in the integrand. This Fourier integral converges for a potential with the asymptotics V(r) → constr ^?n exp(?μr) if |Imq _res| < μ/2. Then the function g(k) is normalized using the generalized normalization rule, which is independent of the resonance pole position. The proposed method is approved by the example of calculation for a virtual triton.     

Field transformation in a multimode optical waveguide with random irregularities of the film surface

A self-consistent mathematical model for the transformation of the average intensity of the mode spectrum I(z) of a waveguide field in a multimode planar optical waveguide with a step profile and rough surface is developed. This model is based on the matrix model for multiple scattering of modes in an optical waveguide. The elements of the intermode scattering matrix are found, which describe the process of mutual transfer of the energy of modes along a waveguide and their transformation into radiation modes. The transformation of the I(z) modes in waveguides with large-and small-scale inhomogeneities is investigated. It is shown that the largest qualitative differences in the noted dependences manifest themselves only in the initial portions of the optical waveguide. The length z of these portions is much smaller than the characteristic scale length L _k at which the fundamental energy of the kth mode excited in the optical waveguide is renewed. The effect of self-filtration of the mode spectrum I(z) is described, as a result of which a stable (normalized), independent of distance z, distribution I* is formed. It is established that irregularities of the optical waveguide boundaries exert a depolarizing effect on a guided light beam. The specific features of the normalization of the radiative dissipation of a group of modes I_i(z) in an optical waveguide are investigated. It is ascertained that, in the case of small-scale irregularities, the attenuation coefficient is described by a nonlinear monotonic dependence α(z), which asymptotically converges to the value α*, characteristic of the normalized field I*. When the optical-waveguide film has large irregularities, the dependence α(z) is characterized by a pronounced maximum due to the formation of alternative channels of radiative dissipation of the energy of waveguide modes.     

Toroidal Braid Group Action and an Automorphism of Toroidal Algebra Uq(sln+1,tor) (n ≥ 2)

Utilizing an action of a modification of the double affine Hecke braid group of type gl_n+1, we obtain an automorphism of the toroidal algebra U_q(sl_n+1) (n 2), which maps two central elements C and k₀ · k_n to k₀ · k_n and C^–1, respectively.     

Role of metastable atoms and molecules of helium in excitation transfer to hydrogen atoms

The afterglow of a discharge in helium with a small admixture of hydrogen is studied spectroscopically (p=40 Torr, [e]≤10¹¹ cm^?3). The time-resolved measurements of intensities of the first four lines of the Balmer series are performed. The concentrations of metastable helium atoms and molecules are evaluated from the relative intensity of the absorption lines. The ratios of excitation transfer rates from atoms He(2 ³ S ₁) k ₁(n) and molecules of helium He₂(a 2sσ ³Σ _u ⁺ ) k ₂(n) to atomic hydrogen H*(n) are measured to be k ₁(n=3)/k ₂(n=3)=0.04±0.02 and k ₁(n=4)/k ₂(n=4)=0.01±0.02. The ratios of excitation rate constants k ₂(n) corresponding to different states H(n) are measured to be k ₁(n=4)/k ₂(n=3)=0.023±0.01; k ₁(n=5)/k ₂(n=3)≤0.013; and k ₁(n=6)/k ₂(n=3)≤0.007.     

From chiral to two-dimensional Wess-Zumino-Novikov-Witten model via quantum gauge group

We study the canonical quantization of the SU(n) WZNW model. Decoupling the chiral dynamics requires an extended state space including left and right monodromies as independent variables. In the simplest (n = 2) case we explicitly show that the zero modes of the monodromy extended SU(2) WZNW model give rise to a quantum group gauge theory in a finite-dimensional Fock space. We define the subspace of U_q(sl(2)) ⊗ U_q(sl(2))-invariant vectors on which the monodromy invariance is also restored and construct the physical space applying a generalized cohomology condition.     

Scaling function for order-parameter correlations in expansion to order 1/n: I. Short-range interactions

The order-parameter correlation function $\text{G}\text{?}\text{(}\text{q}\text{, ξ}{}_1\text{)}$ is calculated in the critical region of momentum space $\text{q}$ in terms of a second-moment correlation length ξ₁ by means of perturbation expansion to order 1/n, for an n-vector system with short-range interactions, in zero field above T_c, for 2 < d < 4. The scaling function of the $\text{q}$ dependence is obtained in closed form with a precisely identified cutoff-dependent factor which is the amplitude of the correlation-length dependence of the susceptibility. Both the exponents and the coefficients of the expansion for fixed q as t = (T?T_c)/T_c → 0 are given explicitly and the former are shown to be in accordance with the operator product expansion. The coefficients of order 1/n in the terms associated with a t^k(1?α) dependence of the energy density, for integer k ≥ 1, are expected to be explicitly cutoff-dependent and this is verified by the detailed calculations for k = 1. The behaviour for fixed t and q → 0 is shown to be markedly different from the Ornstein-Zernike approximation. Detailed comparison is provided with the scaling function of the t dependence of the correlations appearing in parallel work.     

Projective quantum spaces

Associated to the standard SU _q (n) R-matrices, we introduce quantum spheresS _q ^2n-1 , projective quantum spaces _q ^n-1 , and quantum Grassmann manifoldsG _k( _q ⁿ ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziski and S. Majid.     

Theory for the first-order spin-wave instability threshold in ferromagnetic insulating thin films

A theory for the first-order Suhl and the parallel pumping instability in thin films is presented. Significant differences for the critical microwave field and wave vector to former calculations occur, which discuss the problem in terms of bulk spin-waves neglecting boundary conditions. A coupling matrixC _kk′ is introduced, which describes the couplings between the modes and the driving microwave field. For bulk standing spin-wavesC _kk′ is always diagonal. For the true discrete standing modes of a thin filmC _kk′ changes only in case of 1. Suhl instability and if the wavevector has a non vanishing component perpendicular to the film plane. Here the diagonal bulk couplings have to be replaced in part by off diagonal terms, describing couplings between modes, which perpendicular wave vector componentk _⊥ differs byπ/d (d=film thickness). The decisive quantity, which decides if the finite thickness of the film is of importance or if the film can be treated as a bulk system, is the frequency difference δω _k of the coupled modes. For δω _k much smaller than the spin-wave damping η _k the bulk approximation is correct. For \(\delta \omega _k > > \eta _k \) two experimental situations for 1. Suhl instability are discussed: For a perpendicular to the film plane magnetized film the critical microwave field is by π/2 bigger than in the bulk case. In an in-plane magnetized film the critical spin-waves propagate always in the film plane, as only hereC _kk′ remains identical to the bulk case.     

Heterodyne coincidence spectroscopy

A new type of light-scattering experiment, which should measure directly the triple static structure factor S ⁽³⁾ (k, q) of a fluid, is proposed. S ⁽³⁾(k, q) is the full spatial Fourier transform of the equilibrium triplet distribution function g ⁽³⁾(r ₁, r ₂, r ₃). The experiment may also be used to study dynamic correlation functions of the form <a_k (t)a_q (t′)a_k_q(t″)> (where a_k () is the kth spatial Fourier component of the density), thereby giving new information on mode-mode coupling. The method obtains its information from triple correlations in the arrival of scattered photons at three detectors. The detectors must be operated in the heterodyne mode (i.e. with a local oscillator); the scattering volume must be much larger than the volume over which molecular positions are correlated. Comparison is made with previous analyses of other multi-detector experiments.     

Glauber Dynamics for the Mean-Field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q??3 states and show that it undergoes a critical slowdown at an inverse-temperature ?? _s (q) strictly lower than the critical ?? _c (q) for uniqueness of the thermodynamic limit. The dynamical critical ?? _s (q) is the spinodal point marking the onset of metastability. We prove that when ??<?? _s (q) the mixing time is asymptotically C(??,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At ??=?? _s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^4/3. For ??>?? _s (q) the mixing time is exponentially large in n. Furthermore, as ?????? _s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^?2/3) around ?? _s . These results form the first complete analysis of mixing around the critical dynamical temperature??including the critical power law??for a model with a first order phase transition.     

Impact of anisotropy on polar optical vibration in a wurtzite cylindrical nanowire with ring geometry

The quasi-confined (QC) phonon modes, surface optical (SO) phonon modes and corresponding Fröhlich-like Hamiltonian in a wurtzite cylindrical nanowire with ring geometry are investigated in the framework of the dielectric continuum model and Loudon’s uniaxial crystal model. Numerical calculations are focused on the dispersion relations of the SO phonons and the electron–SO phonon coupling strength. Results show that there are only two branches of SO phonon modes. The dispersions of the two branches of SO phonon modes are obvious when the phonon wave-number k_z or the azimuthal quantum number m is small. Typical degenerating behavior of the SO modes is evidenced due to the anisotropic effect of wurtzite crystal. Moreover, when k_z or m are large enough, the frequencies of the two branches of SO modes converge to a definite limiting frequency in single planar heterostructure. The calculations of the electron–SO phonon coupling strength reveal that the high-frequency SO modes (SO₊) play a more important role in the coupling strength than the low-frequency ones (SO₋). Furthermore, the long-wavelength SO phonons with small m are the main factor contributing to the electron–phonon interaction.     

Integrable Nonlinear Evolution Equations on a Finite Interval

Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n−N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂ⁿ_xq. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving q_t+q_xxx and q_t−q_xxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively. Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II equations. We first show that the unknown boundary values at each end (for example, q_x(0,t) and q_x(L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution.     

Laplace operator and hodge decomposition for quantum groups and quantum spaces

LetΓ=Γ _±,z be one of theN ²-dimensional bicovariant first order differential calculi for the quantum groups GL _q (N), SL _q (N), SO _q (N), or Sp _q (N), whereq is a transcendental complex number andz is a regular parameter. It is shown that the de Rham cohomology of Woronowicz’s external algebraΓ ^{^} coincides with the de Rham cohomologies of its leftinvariant, its right-invariant and its biinvariant subcomplexes. In the cases GL _q (N) and SL _q (N) the cohomology ring is isomorphic to the biinvariant external algebraΓ _inv ^{^} and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. It is also applicable for quantum Euclidean spheres. The eigenvalues of the Laplace-Beltrami operator in cases of the general linear quantum group, the orthogonal quantum group, and the quantum Euclidean spheres are given.     

Spin-wave resonance in [Co x Ni1 − x ]N and [Co x P1 − x ]N gradient films

Gradient films of ferromagnetic 3d metals with prescribed magnetic potential profile along the film thickness are obtained. It is found that the spin-wave resonance spectrum in these films is characterized by anomalous dependences of resonance fields of spin-wave modes H _r on the mode number: H _r(n) ～ n, H _r(n) ～ n ^2/3.     

Measurement of the surface density of states by field ionization

It is shown that recent measurements of field ion energy distributions from clean tungsten surfaces probe the density of metal states in the vicinity of the surface. We find j(ω) = (2π/kh)Σ_m| ∫ d³rψ_m(r)γ_z|²δ(ω??_m), where j(ω) is the ion current a ω, ψ_mand ?_m are electronic metal eigenfunctions and eigenvalues in the presence of the external electric field used in field ionization and γ(z) is a function which is large near the noble gas atom. An explicit expression for γ(z) is given in the text. It is estimated that tungsten metal states with values of k₆ at least as large as 0.5 Å^?1 make an appreciable contribution to j(ω) where k₆ is the electron momentum parallel to the surface.