Electron-cluster interactions

Beam depletion spectroscopy has been used to measure absolute total inelastic electron-sodium cluster collision cross sections in the energy range fromE0.1 toE6 eV. The investigation focused on the closed shell clusters Na₈, Na₂₀, and Na₄₀. The measured cross sections show an increase for the lowest collision energies where electron attachment is the primary scattering channel. The electron attachment cross section can be understood in terms of Langevin scattering, connecting this measurement with the polarizability of the cluster. For energies above the dissociation energy the measured electron-cluster cross section is energy independent, thus defining an electron-cluster interaction range. This interaction range increases with the cluster size.     

Sharp real-part theorems for high order derivatives

We obtain a representation for the sharp coefficient in an estimate of the modulus of the nth derivative of an analytic function in the upper half-plane \mathbbC₊ {\mathbb{C}_{+} } . It is assumed that the boundary value of the real part of the function on ?\mathbbC₊ \partial {\mathbb{C}_{+} } belongs to L ^p . This representation is specified for p = 1 and p = 2. For p = ∞ and for derivatives of odd order, an explicit formula for the sharp coefficient is found. A limit relation for the sharp coefficient in a pointwise estimate for the modulus of the n-th derivative of an analytic function in a disk is found as the point approaches the boundary circle. It is assumed that the boundary value of the real part of the function belongs to L ^p . The relation in question contains the sharp constant from the estimate of the modulus of the n-th derivative of an analytic function in \mathbbC₊ {\mathbb{C}_{+} } . As a corollary, a limit relation for the modulus of the n-th derivative of an analytic function with the bounded real part is obtained in a domain with smooth boundary. Bibliography: 8 titles.     

Criteria for Validity of the Maximum Norm Principle for Parabolic Systems

We consider systems of partial differential equations, which contain only second derivatives in the x variables and which are uniformly parabolic in the sense of Petrovskii. For such systems we obtain necessary and, separately, sufficient conditions for the maximum norm principle to hold in the layer Rn × ( 0,T ] and in the cylinder × ( 0,T], where is a bounded subdomain of Rn. In this paper the norm is understood in a generalized sense, i.e. as the Minkowski functional of a compact convex body in Rm containing the origin. The necessary and sufficient conditions coincide if the coefficients of the system do not depend on t. The criteria for validity of the maximum norm principle are formulated as a number of equivalent algebraic conditions describing the relation between the geometry of the unit sphere of the given norm and coefficients of the system under consideration. Simpler formulated criteria are given for certain classes of norms: for differentiable norms, p-norms ( 1 p ) in Rm, as well as for norms whose unit balls are m-pyramids, m-bipyramids, cylindrical bodies, m-parallelepipeds. The case m = 2 is studied separately.     

Inverse vibrational distributions of photofragments

X-ray diffraction data from a Cr₂(SO₄)₃ solution were examined. A peak at 3.25 Å in the correlation function reveals the formation of inner chromium-sulphate complexes. Least-squares refinements of the i(s) curve are consistent with a structural unit in which the CrOS angle is 136°. Each sulphate ion interacts with approximately eight water molecules.     

Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball

A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L ^p . This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L ^p -norm of its boundary values is attained at the radial direction. This confirms D. Khavinson’s conjecture for p = 1 and p = 2. Bibliography: 11 titles.     

Photoabsorption resonances of Na20, Na21Cl and Na22Cl2 clusters

Plasma resonance profiles in the visible part of the spectrum were measured by photoabsorption spectroscopy for the clusters Na₂₀, Na₂₁Cl and Na₂₂Cl₂ in a beam. The resonance positions in Na₂₀ and Na₂₁Cl are close, suggesting that the Cl^- ion does not locate at the center of the metallic droplet and does not strongly modify the effective valence electron density. The spectrum of Na₂₂Cl₂ is noticeably dissimilar to the other two, raising the possibility of structural differences and/or incomplete charge transfer.     

Double and triple ionization of silver clusters by electron impact

The production of silver cluster cations Ag(n)(2+) (for several selected sizes in the range n = 39-119) and Ag(n)(3+) (for n = 58, 61, 67) by electron impact ionization of neutral precursors has been studied. The scaling of appearance energies with cluster radius follows the metallic droplet model but, curiously, with a slope which is estimated to be quite different from the literature values for single ionization, Ag(n)(+), as well as for the appearance of smaller Ag(n)(2+) ions. It is also found that as the electron energy increases, the yield of high-charge cations grows faster than that of singly-charged Ag(n)(+). This behavior is consistent with the power-law dependence of post-threshold ionization. The mechanisms involved in multiple ionization phenomena in clusters of noble metals are not yet fully understood and call for further experimental and theoretical examination.     

Beam broadening of polar molecules and clusters in deflection experiments

A beam of rotating dipolar particles (molecules or clusters) will broaden when passed through an electric or magnetic field gradient region. This broadening, which is a common experimental observable, can be expressed in terms of the variance of the distribution of the resulting polarization orientation (the direction cosine). Here, the broadening for symmetric-top and linear rotors is discussed. These two types of rotors have qualitatively different low-field orientation distribution functions, but behave similarly in a strong field. While analytical expressions for the polarization variance can be derived from first-order perturbation theory, for experimental guidance it is important to identify the applicability and limitations of these expressions, and the general dependence of the broadening on the experimental parameters. For this purpose, the analytical results are compared with the full diagonalization of the rotational Stark-effect matrices. Conveniently for experimental estimations, it is found that for symmetric tops, the dependence of the broadening parameter on the rotational constant, the axial ratio, and the field strength remains similar to the analytical expression even outside of the perturbative regime. Also, it is observed that the shape envelope, the centroid, and the width of the orientation distribution function for a symmetric top are quite insensitive to the value of its rotational constant (except at low rotational temperatures).