The Boltzmann transport equation (BTE) is widely used in modern physical kinetics [19, 20] even at nanoscales Etomoxir [21, 22]. In our previous work, we solved BTE and determined the response of the conduction electrons on an electromagnetic wave allowing for dependence of the distribution function on the wavenumber. Then,

we obtained the spatially dispersive permittivity of metal and applied a generalized Mie theory [23] to calculate spectra of light extinction by silver nanospheres. Because of excitation of the rotationless (longitudinal) plasmon-polariton waves, the frequency of the Fröhlich resonance ω=3.5 eV [24] was found to increase with decreasing the radius a, so ω=3.635 eV for particles with Selisistat in vitro a=1.5 nm and ω=3.73 eV at a=1 nm (see Figure two of [25]; a similar dependence of ω on a as well as the formulas of [23] is found in a recent paper [26]). The blueshift by 0.15 eV and the width of the plasmon resonance calculated for a particle beam created by Hilger, Tenfelde, and Kreibig [27] were

in excellent agreement with the click here experimental data. We concluded that silver clusters with a<1 nm do not contribute into the extinction spectra of the beams. However, it was not possible to establish whether the contribution of these ultrathin particles into the integral extinction spectrum vanished due to MIT. Energies of the conduction electrons It is a common practice to assume that the conduction electron

energy ε in metal is a function of the continuous quasi-momentum This approach can be used to model the properties of bulk metals Florfenicol in which an electron state s can be described by a set of four quantum numbers, where is a projection of the electron spin. However, if an electron moves inside a microscopic sphere, the vector p can no longer describe an electron state. The set p x ,p y , and p z has to be replaced by a set of the radial q, angular momentum l, and magnetic m quantum numbers. Then, integrals over p should be replaced by sums over the electron states s, according to the following rule: , where V is the volume of the body and h is the Plank constant. In this study, we used discrete sets of the electron energies ε s . Shape of metal nanoparticles Every set of the electron energies ε s was obtained at a certain number N of electrons confined in a potential well of an ideal spherical shape. We used the parameters of bulk silver and gold [28], namely, the depth of the potential well U=9.8 eV and the Wigner-Seitz radius r s =0.16 nm. Because of the spherical symmetry of the electronic wave functions, different physical processes have peculiar features at magic numbers of electrons. The magic numbers, which we determined for perfect noble metal spheres, are presented in the ‘Background’ Section of this paper.