Small Metal Spheres Research Articles

Determination of the microwave field strength by microwave-induced transitory oscillations in pulsed electron spin resonance

The accurate determination of the microwave field strength B1 at the sample position is often a prerequisite for successful ESR experiments. In particular in pulse experiments, such as in extended time excitation (I, 2), the rotation angles induced by the microwave pulses must be set precisely to avoid artifacts. It is however known to the experimentalist that a reliable measurement of B1 is often difficult. A new method is proposed which is suitable for the measurement of B, in pulse experiments and which does not require a special test sample. In continuous-wave ESR, a variety of methods for the measurement of B1 have been proposed. The standard technique is based on the measurement of the shift of the resonator frequency induced by a small perturbing metallic sphere placed inside the resonator. The frequency shift is dependent on BI and on some further measurable parameters such as the loaded quality factor (3,4). Another technique proposed recently considers the magnetization hysteresis of in-phase and out-of-phase ESR spectra (5). Further approaches make use of the &-dependent saturation behavior (6), line broadening (5, 7-9), or signal intensity (10) of ESR transitions of selected test samples. For a determination of absolute B, values, however, all these methods require an additional calibration procedure. A method for absolute measurements, adapted from NMR (11), is based on the slight shift of the sideband signal positions caused by the strong centerband field in a pulse modulation experiment. The measurement requires a test sample with linewidths much smaller than the separation of the sidebands (12, 13). This is again a serious disadvantage, since size, shape, homogeneity, and dielectric property of sample, sample holder, and glassware in the resonator may severely alter the microwave field (14). In pulsed ESR, the absolute B, field strength can be determined from the duration TO of a a/2 pulse, rB1 = a/(2~~), leading to maximum signal, or of a r pulse, -r& = 1r/r0, causing a zero signal. This requires, however, a linewidth that is much smaller than the microwave field strength, 2/Tf 4 ~3~. Related measurements can also be based on Torrey oscillations in the rotating frame (15). In most solid-state work, however, the inhomogeneous linewidth by far exceeds the maximum B, fields available

Multipolar response of small metallic spheres: Nonlocal theory

The multipolar response of a small metallic sphere is studied with use of a nonlocal dielectric function. Results obtained with the hydrodynamic and Lindhard-Mermin models are presented and compared to those given by the local Drude model. We find an enhancement of the imaginary part of the multipole polarizabilities at low frequencies and pole order l${l}_{c}$, where ${l}_{c}$ is a cutoff order that corresponds to excitations at the high-wave-vector edge of the electron-hole pair continuum. The absorption coefficient for two very close spheres is calculated and the effect of nonlocality on the number and position of the multipolar absorption peaks is discussed.

Size and shape dependence of the surface plasmon frequencies for supported metal particle systems

The effect of the size and shape of small particles on plasmon excitations has been investigated, as possible indicators for the physical properties of supported metal catalyst particles. The plasmon frequencies of a small metal sphere, which is covered with an oxide shell and is half embedded in some kind of support, are calculated using hydrodynamic theory as functions of the radius of the particles. The presence of the support introduces new plasmon frequencies. In addition to frequencies close to those for isolated particles, there are three more frequencies if the support is of the same metal as the particle, and one frequency is added if the support is an insulator. All the surface plasmon frequencies show the same type of increase for particle sizes smaller than about 4 nm, but the magnitude of the increase varies considerably. Classical dielectric theory has been used to calculate the surface plasmon excitation frequencies for the supported hemispheres. A general formula has been derived and applied to some cases of special geometry. It has been shown theoretically that only the two neighboring even and odd modes are coupled in the supported hemisphere system. A shift of 3 eV for the surface plasmon for the supported hemisphere metal particle (11 eV) with respect to that in the supported metal sphere particle (8 eV) has been predicted and observed in the specimen Al AlF 3 .

Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects

Time- and frequency-dependent measurements of optical phase conjugation in gold colloids are reported. It is shown that the nonlinear response is fast on a 5-psec time scale and can be assigned to the electrons of the small metal spheres. The frequency-dependent measurements were performed in the neighborhood of the surface-plasma resonance and give experimental evidence of the validity of our resonant enhancement model. We also report the first model calculation of the third-order Kerr susceptibility of small metal particles. The dominant terms of this electric-dipole contribution are emphasized and lead to an expression for χ(3) that varies roughly as the inverse third power of the radius of the particles. This model accounts for the observed anisotropy of this susceptibility and provides an estimate of its magnitude, in agreement with the measured values. The limited size effect on the linear susceptibility is also discussed.

Electromagnetic properties of small metallic spheres: Diffuse surface scattering.

The effect of diffuse surface scattering on the electromagnetic properties of small metallic spheres is analyzed. We find an enhancement of the optical absorption in the infrared region, and a large blue shift for the dipolar plasma resonance of normal metals. This blue shift is found to be negligible in the case of noble metals.

Optical absorption by clusters of small metallic spheres.

An ansatz for the distribution of depolarization factors, based on a theory of the optical response of nearly touching spheres, is used to calculate the optical absorption of a medium containing isolated clusters of small metallic spheres. The calculated absorption agrees qualitatively with experiments over the entire frequency range. In the far-infrared region we find enhanced absorption and predict a small deviation from an ${\ensuremath{\omega}}^{2}$ frequency dependence.

Size-dependent photoemission shifts in small metal clusters

Density-functional calculations of the change in self-consistent-field energy ( Delta SCF) type are reported for core-level photoemission shifts in small metal spheres. The results for the atom-in-jellium vacancy model show that the binding energies are increased from bulk-metal values, but the photoemission shifts show considerable oscillations as a function of cluster size.

Theory of second-harmonic generation by small metal spheres.

A small metal sphere interacting with an incident electromagnetic wave will produce second-harmonic radiation in a quadrupolar mode. A hydrodynamical model for the electron gas is employed and a Green-function formalism is used to solve for the cross section for producing second-harmonic generation. Numerical results are obtained for aluminum and silver.

Theoretical and experimental study of plasmon excitations in small metallic spheres

The excitation of bulk and surface plasmons in small metallic spheres is studied by means of electron loss spectroscopy in a scanning transmission electron microscope. The results are compared with theoretical calculations. The possibilities and limitations of the technique are discussed

Quantum size and nonlocal effects in the electromagnetic properties of small metallic spheres.

The electromagnetic properties of small spheres are analyzed by including quantum size effects and nonlocal properties of the response function. Our model neglects the diffuseness of the surface electron charge but includes effects associated with the surface roughness, which is assumed to randomize the electrons scattering off the surface. Quantum size and roughness effects are discussed and we analyze under what specific conditions they can appear.

Dynamical electronic polarizability and the force sum rule

It is shown that a dynamical force sum rule is useful in analyzing the response of a finite electronic system to a uniform electric field that oscillates harmonically in time. The sum rule, which has been used previously in calculations of the response of atoms to electric fields, provides a direct test of electronic polarizability calculations and may lead to useful approximations for the electronic polarizability of metallic microstructures. We apply the sum rule to the case of a small metal sphere in the jellium model, and show that it directly leads to an approximation that is equivalent to the surface-plasmon-pole approximation for the red-shift of the surface plasmon frequency.

Quantum-size effects in the electromagnetic response of small spheres

The response of small metallic spheres to an electromagnetic field has been calculated to all multipole orders in the electronic excitations. Quantum-size effects have been analysed and found important for spheres with radii less than 10 AA, where large absorption peaks are associated with electron-hole-pair excitation energies.

Substrate-related effects on the optical behavior of a granular surface: The Maxwell Garnett theory revisited

An approximate treatment, describing the influence of a dielectric substrate on the optical behavior of a granular surface, is reported. It shows that discrepancies between experimental results and predictions mainly based upon the Maxwell Garnett theory [Philos. Trans. R. Soc. London 203, 385 (1904); 205A, 237 (1906)] cannot be interpreted as substrate-related effects. The magnitudes and locations of the multiple images of the unperturbed dipole of a small metallic sphere have been carried out in an approximate, though reliable, way in the presence of polarizing fields both parallel and perpendicular to the substrate. The resulting dipole is introduced in a long-known optical model, describing a granular surface as a planar array of equal dipoles interacting with each other. Graphical results, showing the influence of substrates of various dielectric constants, are presented. A discussion of possible improvements of the available model is also reported.

Electrodynamic response of metal spheres

A collective dipole model for the electrodynamic response of small metal spheres is presented. Model calculations of the extinction and scattering cross sections of metal spheres as a function of particle radius are compared with the exact Mie formulas. The characteristic size dependences of the peak position, width, and height of the dipole surface plasmon in small metal particles are attributed to the collective nature of the motion.

Far-infrared optical absorption due to surface phonon excitations in small metal particles

In this paper, the role played by the excitation of surface phonons in the far infrared optical absorption of small metallic spheres is explored. We find that for frequencies ranging between 100 cm -3 and 200 cm -1 this absorption mechanism is dominating over the ones coming from considering electronic excitations or the Joule heating.

Quantum-mechanical calculation of the van der Waals interaction between small metallic spheres

We give the first microscopic calculation of the leading terms in the van der Waals interaction energy between two very small metallic particles, treating their electronic excitations in the quantum-mechanical Infinite Barrier Model and the Random Phase Approximation. The results are very close to those obtained for the corresponding classical spheres, provided that we describe them by their effective radii calculated to satisfy the electrostatic zero-force sum-rule. We examine the higher order terms and the exact results at short distances in the plasmon-pole approximation, and discuss the convergence of the multipole expansions.

Localization and absorption of waves in a weakly dissipative disordered medium.

The effect of a small imaginary part ${\ensuremath{\epsilon}}_{2}$ to the dielectric constant on the propagation of waves in a disordered medium near the Anderson localization transition is considered. The n\ensuremath{\rightarrow}0 replica-field representation of the averaged Green's function leads to a nonlinear \ensuremath{\sigma} model with a symmetry-breaking perturbation proportional to ${\ensuremath{\epsilon}}_{2}$. In d=2+\ensuremath{\epsilon}, the renormalized energy absorption coefficient is shown to increase anomalously with frequency \ensuremath{\omega} near the mobility edge ${\ensuremath{\omega}}^{\mathrm{*}}$ as \ensuremath{\alpha}\ensuremath{\sim}(${\ensuremath{\omega}}^{\mathrm{*}}$-\ensuremath{\omega}${)}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}2)\ensuremath{\nu}/2}$, \ensuremath{\nu}=1/\ensuremath{\epsilon}. It is shown that the wavelength ${\ensuremath{\lambda}}^{\mathrm{*}}$ below which localization occurs is related to the elastic mean free path l by (l/${\ensuremath{\lambda}}^{\mathrm{*}}$${)}^{d\mathrm{\ensuremath{-}}1}$\ensuremath{\sim}1/\ensuremath{\epsilon} (d>2). This may occur near the limit of attainable disorder from a quenched random array of small dielectric or metallic spheres.

Self-consistent calculation of the polarizability of small jellium spheres

A self-consistent, density-functional calculation of the static polarizability of small metal spheres is reported. The jellium model of a metal, in which the positive ions are replaced by a uniform positive background, is employed, and the modified Sternheimer equation is used to compute the static dipole polarizability. The computations are reported for the closed-shell configurations containing from 8 to 254 electrons for neutral and charged spheres.

Effective relaxation time in small spheres: Diffuse surface scattering

Effects of surface roughness are shown to be important in order to obtain the effective electron mean free path in small metal spheres. This leads to an increase in the optical absorption in agreement with experimental observations.

Non-Local Optical Effects at Metal Surfaces

The non-local response of an irradiated metal surface is discussed in terms of the induced density in response to an external field. The field and the induced charge near the surface can be determined by solving Maxwell equations in the surface region and match the solution to the incoming and reflected wave in vacuum and to the asymptotic solution in the bulk of the solid. For incoming fields of wave lengths long in comparison with the extension of the surface region a key quantity is the center of gravity of the screening charge. This complex function of the frequency is important for a number of properties such as the reflectivity, the surface plasmon dispersion, image fields, van der Waals interactions between a molecule and a metal surface, etc. Considerations similar to the ones for a planar surface are also applied to the surface properties of a small metallic sphere.