1 Introduction

Among the particular properties of metal clusters (MCs) that are of great interest for a range of applications some are caused by a pronounced selective optical absorption in the visible region due to collective oscillations of electrons, including surface plasmons (SP) [1]. Another important area of the application of MCs is related to their pronounced catalytic properties [2]. In the present work, we shall study theoretically the results of experiments on optical absorption that are used in the investigation of free metal clusters containing less than 50 atoms. It should be mentioned that only few studies have been devoted to the experimental investigation of optical properties of small MCs that are not supported by any substrate. In [3, 4], photoabsorption cross sections of neutral sodium clusters composed of 2–40 atoms were measured by longitudinal-beam-depletion spectroscopy in the visible range. In [5], photoabsorption cross sections for free, singly charged sodium clusters (with 18–40 atoms) were measured by recording the light-induced evaporation.

In the simplest approximation, a MC is assumed to have a spherical shape with a size much smaller than the wavelength of the SP excitation. This corresponds to the limiting case of the widely used Mie theory [6], which for alkali MCs provides the following expression for the SP triply degenerate dipole oscillation frequency,

$$ \omega_{\rm Mie}=\frac{\omega_p}{\sqrt{1+2\varepsilon_a}}, $$
(1)

where ω 2 p  = 4π n e e 2/m is the plasma frequency in the bulk, n e is the electron density, m and e are the electronic mass and charge, respectively, and \(\varepsilon_a\) is the real part of the dielectric constant of the surrounding medium. We note that expression (1) does not include the contribution of d-electrons which is important in the case of noble metals.

However in reality, synthesized clusters have a more complicated, non-spherical shape, [3] which leads to a splitting of the threefold SP frequency of Eq. (1). Despite this, only few studies have taken this into account. Thus, in [3–5, 7] the relation between the number of atoms and the cluster structure, which leads to a strong correlation between the energy spectrum and the cluster size, was discussed. Also for other properties it may be important to take the precise shape of the MC into account. Thus, the cluster shape plays an essential role in the processes of SP spectral line formation, [8] oscillation decay and dephasing [9, 10]. One way of taking the precise structure of the cluster into account is through the use of time-dependent local-density approximation (TDLDA) calculations. Such an approach was applied for calculating the optical response of spheroidal clusters [11] and satisfactory agreement with the experimental data of [3] was obtained. Another, simpler, numerical approach was developed in [12] with the purpose of determining the complex susceptibility of small, non-spherical particles and, consequently, the absorption spectrum.

The purpose of the present paper is to develop a simple, phenomenological theory for alkali metal clusters, which allows to extract values of parameters that describe the overall shape of the clusters from spectral measurements. We shall give analytical expressions for the frequencies of collective dipole oscillation in sodium clusters as a function of deviations from spherical shape. Finally, the results are compared with data from numerical calculations and experiments.

2 Calculation of SP spectrum of large alkali metal clusters with non-spherical shape

We apply the electrostatic approximation to calculate the SP frequencies of a free \(\varepsilon_a=1\) alkali metal cluster. The shape of the particle is given by

$$ r(\theta,\varphi)=R[1+f(\theta,\varphi)], $$
(2)

where r, θ, and \(\varphi\) are the spherical coordinates, and the function \(f(\theta,\varphi)\) which is assumed to take values much smaller than 1, describes small deviations from a spherical shape. The overall radius of the particle, R, is assumed to be smaller than the SP wavelength, but is otherwise irrelevant for the results of the present study. The boundary conditions for the electric field potential and its normal derivative at \(r=r(\theta,\varphi)\) are

$$ \Upphi=\Uppsi, \quad {\partial \Upphi \over \partial n}= \varepsilon_p {\partial \Uppsi \over \partial n}, $$
(3)

where \(\Uppsi\) is the potential in the inner region of the particle, \(\Upphi\) is the potential outside the MC in the medium in which the MC is embedded. In Eq. (3) \(\varepsilon_p\) is the dielectric function describing Na clusters, which as it is discussed below, in case of small clusters differs from the bulk one. As shown by experiment [13], the static polarizability of a small cluster and, consequently, \(\varepsilon_p\) depends on the number of atoms in the cluster N which can be explained as being due to size effects. Though we use a phenomenological approach introducing dielectric functions, nevertheless we show in Sect. 3 that the dependence of \(\varepsilon_p\) of MC on N can be revealed from experimental data on SP frequencies of sodium clusters. Another important point is that as it follows from the experimental data of Ref. [5] the imaginary part of \(\varepsilon_p\) for small Na clusters is of the same order of magnitude as its real part. Nevertheless, as it is shown in Sect. 3 the contribution of imaginary part of \(\varepsilon_p\) into the SP resonance frequency does not exceed 1–2 %.

We expand the solutions \(\Upphi\) and \(\Uppsi\) in terms of spherical functions Y lm (θ, φ)

$$ \begin{aligned} \Upphi&=\sum_{lm}p_{lm}\Upphi_{lm}\\ \Uppsi&=\sum_{lm}q_{lm}\Uppsi_{lm} \end{aligned} $$
(4)

with

$$ \begin{aligned} \Upphi_{lm}&={Y_{lm}(\theta,\varphi) \over r^{l+1}}\\ \Uppsi_{lm}&=r^lY_{lm}(\theta,\varphi). \end{aligned} $$
(5)

Taking Eq. (2) for the normal derivative of the potential into account, we find

$$ \frac{\partial }{{\partial n}} =\vec n \vec \nabla, \quad \vec n \sim \vec \nabla [r-Rf(\theta,\varphi)]= \vec n_0-R \vec \nabla f (\theta,\varphi), $$
(6)

where \(\vec{n}_{0}\) is a unit vector along the normal to the sphere. Using Eqs. (4) and (6) the boundary conditions, Eq. (3), can be written as follows (omitting, for brevity, the angular variables):

$$ \begin{aligned} \sum_{lm}p_{lm}\Upphi_{lm} & =\sum_{lm}q_{lm}\Uppsi_{lm} \sum_{lm}p_{lm} {\partial \Upphi_{lm} \over \partial r}-\varepsilon_p \sum_{lm}q_{lm} {\partial \Uppsi_{lm} \over \partial r} \\ &=R\left[\sum_{lm}p_{lm}(\vec{\nabla} f \cdot \vec{\nabla} \Upphi_{lm}) -\varepsilon_p\sum_{lm}q_{lm}(\vec{\nabla} f \cdot \vec{\nabla} \Uppsi_{lm})\right]. \end{aligned} $$
(7)

From the derivatives

$$ \begin{aligned} {\partial \Upphi_{lm} \over \partial r}&=-(l+1){Y_{lm} \over r^{l+2}}\\ {\partial \Uppsi_{lm} \over \partial r}&=lr^{l-1}Y_{lm}, \end{aligned} $$
(8)

and by introducing the notation

$$ \begin{aligned} &{ p_{lm} \over R^{l+m}}=P_{lm}\\ & q_{lm}R^l=Q_{lm}\\ &\vec {L}=-i (\vec {r} \times \vec {\nabla}) \end{aligned} $$
(9)

we arrive at the following formulation of Eq. (7) (see Appendix 1)

$$ \begin{aligned} &\sum_{lm}P_{lm}{Y_{lm} \over (1+f)^{l+1}}=\sum_{lm}Q_{lm} (1+f)^l Y_{lm} \\ &\sum_{lm}(l+1)P_{lm} {Y_{lm} \over (1+f)^{l+1}} + {\varepsilon_p} \sum_{lm} lQ_{lm} (1+f)^l Y_{lm} \\ & \quad=\sum_{lm} P_{lm}{(\vec {L} f) (\vec {L} Y_{lm}) \over (1+f)^{l+2}}- {\varepsilon_p} \sum_{lm} Q_{lm} (1+f)^{l-1}(\vec {L} f) (\vec {L} Y_{lm}). \end{aligned} $$
(10)

As a check of these equations we consider the perfect sphere for which f = 0. Then, Eq. (10) gives the following equations

$$ \begin{aligned} &P_{lm}=Q_{lm} \\ &(l+1)P_{lm}+{\varepsilon_p}lQ_{lm}=0, \end{aligned} $$
(11)

which gives the well-known result

$$ \omega_l=\frac{\omega_p}{\sqrt{1+ {l+1 \over l}}}. $$
(12)

In Eq. (12), ω l is the (2l + 1)-fold degenerate SP frequencies.

When introducing deviations from a spherical symmetry, i.e., setting f≠ 0, the angular momentum in Eq. (10) is no longer a good quantum number. Consequently, the (2l + 1)-degenerate frequency of Eq. (12) is split into up to (2l + 1) different frequencies. However, for a sufficiently small f (i.e., small deviations from spherical structure), the splitting is smaller than the separation of frequencies with different l, so that l remains approximately a good quantum number, whereas m, excluding the case of axial symmetry, does not. Thus, we can apply zeroth-order perturbation theory to Eq. (10) in order to calculate the splitting of the SP frequencies. For fixed l, eliminating ∑ l lQ lm (1 + f)l Y lm , one can obtain from Eq. (10)

$$ \sum_{m} \left [ 1+ l \left(1+ {\varepsilon_p} \right) \right] P_{lm}Y_{lm}= \left(1- {\varepsilon_p} \right)\sum_{m}P_{lm} (\vec {L} f) (\vec {L} Y_{lm}). $$
(13)

Multiplying both sides of (13) with Y lm and integrating over the angles we arrive at the following result

$$ \sum_{m'}\int Y_{lm}^*(\vec {L} f) (\vec{L} Y_{lm'})P_{lm'} d\Upomega=\Uplambda P_{lm} $$
(14)

with

$$ \Uplambda = \frac{(l+1) +l\varepsilon_p}{1 - \varepsilon_p}. $$
(15)

Through integration by parts one may prove that the left-hand side of Eq. (14) is hermitian. Similarly, also through integration by parts one may obtain a more convenient form of Eq. (14), i.e.,

$$ \sum_{m'}\int Y_{lm}^*(\vec {L}^2 f) Y_{lm'}P_{lm'} d\Upomega=-2\Uplambda P_{lm}. $$
(16)

In order to obtain further information on the consequences of deviations from spherical symmetry, the function \(f(\theta,\varphi)\) could be expanded in terms of spherical functions \(Y_{lm}(\theta,\varphi)\). Since l > 1 modes only appear in the spectra for N > 100, in Eq. (16) we keep the terms with l = 1. It is more convenient to use a formulation in terms of the components of the unit vector \(\vec{n} = \vec{r}/r\) (see Appendix 2). Then

$$ f=f_0+\vec {f} \cdot\vec {n} + \sum_{\alpha\beta} f_{\alpha\beta} \left(n_\alpha n_\beta- {1\over 3} \delta_{\alpha\beta}\right) +\dots $$
(17)

where \(f,\vec{f}\) and f αβ are constants. Without loss of generality we can set f 0 = 0 whereby the average of the function \(f(\theta,\varphi)\) vanishes. Furthermore, for l = 1 the matrix elements of \(\vec{f}\cdot\vec{n}\) as well as the matrix elements of the omitted terms containing n α in powers higher than 2 in Eq. (16) vanish, which can be shown by using the properties of the spherical functions \(Y_{lm}(\theta,\varphi).\) Therefore, for our purposes it is sufficient to replace the expansion of Eq. (17) by the following expression

$$ f=\sum_{\alpha\beta} f_{\alpha\beta} \left(n_\alpha n_\beta- {1\over 3} \delta_{\alpha\beta}\right). $$
(18)

By diagonalizing the matrix f αβ we arrive at

$$ \begin{aligned} f=&f_{xx}n_x^2 + f_{yy}n_y^2 +f_{zz}n_z^2- {1\over 3} {\rm Tr} \hat f\\ {\rm Tr} \hat f&=f_{xx}+f_{yy}+f_{zz}. \end{aligned} $$
(19)

Using Eq. (19) we can determine the eigenvalues, \(\Uplambda, \) of Eq. (16) for l = 1 and find

$$ \Uplambda_\alpha=- {6\over 5}\left ( f_{\alpha\alpha}- {1 \over 3} {\rm Tr} \hat f \right ). $$
(20)

The eigenvalues of Eq. (16) quantify the deviations of spherical symmetry. Furthermore from Eq. (20) it follows that \(\sum_\alpha \Uplambda_\alpha=0, \) which will be used below for the comparison with experimental results.

Combining Eqs. (20) and (15) and adopting \(\varepsilon_p= 1- \frac{{\omega _{p}^{2} }}{{\omega ^{2} }}\) we obtain

$$ f_{\alpha\alpha}- {1 \over 3} {\rm Tr} \hat f = {5\over 2} {\omega_p^2 - 3\omega_\alpha^2 \over \omega_p^2}, $$
(21)

where ω α  = ω x , ω y , ω z are the observable frequencies. Substituting f from Eq. (18) into Eq. (2) leads then to the following formula for the lengths of the three main axes of the (approximately spherical) cluster

$$ d_\alpha = 2R \left ( 1+f_{\alpha\alpha}- {1 \over 3} \sum_\beta f_{\beta\beta} \right ), $$
(22)

which, along with Eq. (21), gives the aspect ratio η = d α /d β , 

$$ \eta_{\alpha \beta} = {11\omega_p^2 - 15\omega_\alpha^2 \over 11\omega_p^2 - 15\omega_\beta^2 }. $$
(23)

In total, measuring the SP oscillation frequencies ω α allows accordingly to estimate the parameters \(\Uplambda_\alpha\) that describe the deviations of the cluster structure from being spherical. We emphasize that Eq. (23) provides only an approximation to the SP frequencies of dipole oscillations in the particles that is most accurate for small deviations from spherical shape.

3 Comparison with jellium shell model, numerical calculations and experimental data on SP excitations

First, we study the accuracy of Eq. (23) by comparing with results obtained in the framework of the Mie theory for spheroids [14]. As it is demonstrated in Ref. [14] the SP frequencies of a spheroidal MC depend only on the aspect ratio \(\eta=d_>/d_< (d_>\) and \(d_<\) are the major and minor diameters of the spheroid, respectively). In order to study the accuracy of our results, we substitute the frequencies from Ref. [14] into our Eq. 23 and calculate the aspect ratios η αβ that are finally compared with the results following from the Mie theory.

At first, consider the simpler case of a spheroid with an aspect ratio close to unity. For example for an oblate spheroid with η = 1.1 the exact formulas of [14] give ω x,y  = 0.57 ω p and ω z  = 0.601 ω p , whereas our approach gives η αβ  = 1.104, i.e., a discrepancy between η and η αβ of less than 0.4 %.

In order to estimate the range of applicability of our approach we next consider the case of larger values of η. Numerical results for η = 1.5, both for prolate and oblate spheroids, are presented in Table 1. In the fourth and fifth columns the values of \(\Uplambda_{x,y}\) and \(\Uplambda_z,\) obtained by substituting the frequencies ω x,y and ω z from Table 1 into Eq. (23), are listed. The final results are presented in the last column. It is highly encouraging to observe that the discrepancy between η and η αβ does not exceed 6 % (second row). We emphasize that η = 1.5 is larger than what is found for almost all metal clusters. Thus, Eq. (23) that describes a smooth deviation from spherical symmetry provides satisfactory agreement with the exact results for spheroids [14] even when these have relatively large aspect ratios. This suggests that our approach also can be used in estimating the shape parameters of more complicated nanoparticles with reasonable accuracy.

Alkali metal particles are the simplest systems for theoretical determination of SP spectra based on the jellium shell model [16], as their dielectric functions are purely determined by s electrons. The resonance frequencies of sodium clusters ω α can be expressed in terms of the static electric polarizabilities [17] χ α

$$ \omega_\alpha^2={Ne^2 \over m \chi_\alpha}, $$
(24)

where N is the number of valence electrons in the cluster. In Ref. [4], the frequencies ω α were determined using experimental data for average polarizabilities and the values of axial ratios were then calculated by assuming an ellipsoidal shell model.

As can be seen from Table 2 of Ref. [4], the experimental per atom static polarizabilities of small clusters normalized to the measured polarizability of the Na atom depend strongly on the number of atoms N in the cluster, and vary from 0.803 to 0.63 as the number N increases from 2 to 40. Therefore, the frequency ω p should also depend on the number of atoms in the cluster. It turns out that this dependence can be revealed using our present approach. Indeed, by summing over α on both sides of Eq. (21) we obtain

$$ \sum_\alpha \omega_\alpha^2=\omega_p^2. $$
(25)

Since the observable SP frequencies ω α depend on the number of atoms N (Ref. [4]), obviously Eq. (25) determines the values of ω p for each N, that is

$$ \sum_\alpha \omega_\alpha^2(N)=\omega_p^2(N). $$
(26)

Substituting the values of ω α from Ref. [4] into Eq. (26) we obtain ω p (N) for each particle and can thereby calculate the aspect ratio of the clusters assuming an ellipsoidal shape. In Table 2, we present the maximum values of the aspect ratios obtained from Eq. (23) for relatively large clusters with different numbers of atoms and compare them with the results of [4]. This comparison demonstrates a good agreement with differences smaller than 4 %.

It is important to mention that by using measured values of static polarizability χ α (N) or ω α (N) and calculating ω p (N) according to Eq. (26) it becomes possible to determine the dielectric function of the cluster as a function of the number of atoms

$$ \varepsilon_p (\omega, N)= 1- {\omega_p^2(N) \over \omega^2}. $$
(27)

In Ref. [5], measured photoabsorption spectra of charged free Na clusters (N = 14 − 48) have been identified as the SP resonances, and their spectral shapes were interpreted in terms of non-spherical distortions of the clusters. The mean energies of the SP excitation were determined and applying the approximate expression obtained in [18] the aspect ratios of the clusters were calculated. In order to demonstrate the applicability of our approach, we have used formula (23) to the experimental SP frequencies of Ref. [5] to determine the aspect ratios of the clusters. Thereby, we have assumed the existence of axial symmetry, ω x  = ω y .

For the quantitative comparison with the experimental data of Ref. [5] first we estimate the magnitude of the shift \(\Updelta \omega_a\) of the SP resonance frequencies ω a caused by relaxation processes. For small clusters investigated in the experiment the main mechanism of line broadening is the scattering of electrons on the MC surface—the so-called 1/R damping [1]. Indeed, as the measurements show [5], with increase of cluster size there is an obvious trend of decreasing of the SP linewidth from 0.65 to 0.3 eV. We proceed from the standard expression for \(\Upomega\) accounting for the relaxation processes [1],

$$ \varepsilon_p = 1- {\omega_p^2 \over\omega(\omega-i\gamma)}, $$
(28)

where γ is the relaxation constant. Substituting expression (28) with \(\varepsilon_a=1\) into (15) for l = 1 we find the equation for determining the complex resonance SP frequency \(\Upomega\) as follows:

$$ 3\Upomega(\Upomega-i\gamma)-\omega_p^2 (1+\Uplambda)=0, $$
(29)

which gives the following expression for \(\Upomega\)

$$ \Upomega_a=-{i\gamma \over 2} + \omega_p\sqrt{{1+\Uplambda_a \over 3}}\cdot \sqrt{1-{3\over 4} {\gamma^2 \over \omega_p^2(1+\Uplambda_a)}}. $$
(30)

Taking into account the smallness of \(\Uplambda\) (see Table 1), one obtains from Eq. (30) the following expression for the relative shift of the SP resonance frequencies ω a caused by the relaxation processes:

$$ {\Updelta\omega_a \over \omega_a} ={3\gamma^2 \over 8\omega_p^2(1+\Uplambda_a)}. $$
(31)

Taking into account again the smallness of \(\Uplambda\) and adopting in expression (31) for γ and ω a the experimental data of [5], we apply expression (26) to find ω p (N). This procedure gives the values \(\Updelta \omega _{a} /\omega _{a} = 8.10^{{ - 3}}\) and \(\Updelta \omega _{a} /\omega _{a} = 2.69.10^{{ - 3}}\) for N = 14 and N = 25, respectively. It is easy to see that for other clusters investigated the relative shift of frequency do not exceed 1 %. Thus although the imaginary part of dielectric function Im\((\varepsilon_p)=\omega _{p}^{2} \gamma /\omega ^{3}\) is not small (e.g. Im\((\varepsilon_p)\approx1.2\) for a 14-atom cluster), nevertheless its contribution into the frequency shift is not essential. Note that in [5] the fitting procedure for the two peak structure of the photoabsorption spectra in the case of spheroidal clusters is carried out under the assumption of the equal widths. It is interesting to mention that the imaginary parts (γ) of two SP frequencies \(\Upomega_a\) (as it can be seen from (30)) indeed do not depend on parameter \(\Uplambda_a\) describing a weak deviation from the spherical shape.

Table 1 The values of η, ω x,y , and ω z from Ref. [14] are substituted into Eqs. (21), (22), and (23) which gives the values of \(\Uplambda_{x,y}, \Uplambda_z\) and η αβ listed in the last three columns
Full size table

The results of our calculations for the aspect ratios along with the results of [5] obtained using approximate formula of [17] are presented in Table 3. As can be seen, the agreement between the aspect ratios obtained from Eq. (23) and expressions (9)–(10) of Ref. [5] is satisfactory for both prolate and oblate clusters with the case of N = 14 being the single exception.

Table 2 The values of ω p (N) (N is the number of atoms in the cluster), calculated by substituting the semi-empirical data of Table 1 of Ref. [4] into Eq. (25). The values of η max in the third and fourth columns are calculated using Eq. (23) and Table 1 of Ref. [4], respectively
Full size table
Table 3 Comparison of axes ratios of a spheroid, μ = R ⊥/R z calculated in this study (t.s.) and in Ref. [5]. R z is the vertical, conjugate radius and R ⊥ is the horizontal, transverse radius at the equator of a spheroid. N in the first and fourth columns gives the cluster size
Full size table

4 Application of the method to more complicated shapes

In this section, we present applications of our approach to clusters with more complicated shapes. When the condition \(f(\theta,\varphi)\ll1\) is satisfied our approach can be used for non-spheroidal MCs [19, 20] using measured resonance frequencies. Indeed, from an iterative approach to solve Eqs. (10) beginning with a zeroth-order expression [see Eqs. (13)–(14)] and keeping higher order multipoles in expression (17) it is possible to extract aspect ratios of MCs. For instance, the main features of SP spectra of some symmetric structures can be interpreted in the frame of the present approach.

As an example we shall here discuss an icosahedral MC that often is occurring in experiments [15]. For an icosahedral MC with smoothed vertices and edges the function in (1) can be presented as follows:

$$ f(\theta,\varphi)={F(\theta,\varphi)-\langle F(\theta,\varphi)\rangle \over 1+\langle F(\theta,\varphi)\rangle}, $$
(32)

where

$$ \begin{aligned} F(\theta,\varphi) = {c \over 1 - a^2\cos^2(\theta )} + \sum_{m=0}^4 {c \over 1 - a^2\{b\cos(\theta ) + \sqrt{1 - b^2}\sin(\theta )[\cos(\varphi)\cos(2\pi m/5) + \sin(\varphi)\sin(2\pi m/5)]\}^2}, \end{aligned} $$
(33)

a and c are fitting parameters and the constant b equals

$$ b=\cos \left( {2\pi \over 5 }\right )\left [ 1-\cos \left ({2\pi \over 5} \right ) \right ]^{-1} $$
(34)

In Eq. (32), the quantity \(\left\langle {F(\theta,\varphi)} \right\rangle\) denotes the average of the function \(F(\theta,\varphi)\) over \((\theta,\varphi)\). Note that expression (33) includes the first two terms of the infinite series of the multipoles of Eq. (17). In order to develop successive approximations it is convenient to expand \(F(\theta,\varphi)\) in a Fourier series with respect to \(\varphi\)

$$ \begin{aligned} F(\theta,\varphi)&={ c \over 1-a^2\cos^2 (\theta)} + { 5c \over 2[\alpha^2(\theta)-\beta^2(\theta)]^{1/2} } + { 5c \over 2[\alpha{'}^{2}(\theta)-\beta{'}^{2}(\theta)]^{1/2} } \\ &\quad + 5c \sum_{n=1}^\infty \left \{ { A^{-5n}(\theta) \over [\alpha^2(\theta)-\beta^2(\theta)]^{1/2}} + { B^{-5n}(\theta) \over [\alpha{'}^2(\theta)-\beta{'}^2(\theta)]^{1/2}} \right\} \cos (5\eta \varphi ), \end{aligned} $$
(35)

where

$$ \begin{aligned} &\alpha(\theta)=1-ab\cos (\theta), \quad\beta(\theta)=-a\sqrt{1-b^2}\sin(\theta) , \\ &\alpha{'}(\theta)=1+ab\cos (\theta), \quad \beta'(\theta)= a\sqrt{1-b^2}\sin(\theta),\\ &A(\theta)= - {\alpha(\theta) \over \beta (\theta)} +\sqrt{{\alpha^2(\theta) \over \beta^2 (\theta)}-1 }, \quad B(\theta)= - {\alpha'(\theta) \over \beta' (\theta)} -\sqrt{{\alpha{'}^2(\theta) \over \beta{'}^2 (\theta)}-1 }. \end{aligned} $$
(36)

We have here oriented the system so that the z axis is a C 5 axis and, therefore, the Fourier expansion (33) contains only fivefold harmonics. The best fit in (35) to the icosahedral shape with smoothed vertices and edges is reached for the values of the fitting parameters a = 0.9 and c = 2. Furthermore, since b ≈ 0.447 [see (34)], for the minimal values of A(θ) and B(θ) (reached at θ = π/2) we obtain 1.98. Thus, the Fourier coefficients of the function \(F(\theta,\varphi).\) decrease rapidly with increasing n which allows to apply successive approximations keeping only terms with n = 0 and n = 1. Indeed the next term with n = 2 is of the order 2−10 and can be neglected. It is important to note that from Eq. (32) we obtain the limitation \(\left| {f(\theta ,\varphi )} \right| \le 0.15\) which quantifies the range of applicability of our approach.

The zeroth order term of the Fourier expansion Eq. (35) has the form

$$ F_0(\theta,\varphi)={ c \over 1-a^2 \cos^2 (\theta)} + { 5c \over 2[\alpha^2(\theta)-\beta^2(\theta)]^{1/2} } + { 5c \over 2[\alpha{'}^2(\theta)-\beta{'}^2(\theta)]^{1/2} }. $$
(37)

By averaging in Eq. (32) over the azimuth angle it is easy to demonstrate that the surface has a shape close to spheroidal. The aspect ratio η 0 of this surface is given by

$$ \eta_0= \left ( 1+ {c \over 1-a^2} + {5c \over 1-a^2 b^2} \right ) \left ( 1+c + {5c \over 1-a^2 + a^2 b^2} \right )^{-1}, $$
(38)

where the terms in the first and second brackets are proportional to the lengths of the long and short axis, respectively. Substituting values found above for a, b, and c we obtain η 0 = 1.18 which satisfies the requirement of small deviations from spherical symmetry.

The contribution of the next term with n = 1 in Eq. (35) leads to

$$ \begin{aligned} F_1(\theta,\varphi) & = F_0(\theta,\varphi) \\ & \quad +5c \left \{ { A^{-5}(\theta) \over [\alpha^2(\theta)-\beta^2(\theta)]^{1/2}} + { B^{-5}(\theta) \over [\alpha{'}^2(\theta)-\beta{'}^2(\theta)]^{1/2}} \right \} \cos(5\varphi), \end{aligned}$$
(39)

which represents the surface of a slightly distorted icosahedron that is prolate along the z axis and differing from the ideal one only by smoothed vertices and edges. By analogy with the case of an ideal icosahedron it is possible to introduce two aspect ratios for this surface as follows:

$$ \begin{aligned} \eta_1 & = {1+ F_0(0) \over 1+ F_0(\pi/2)}, \\ \eta'_1 & ={1+ F_0(0) \over 1+ F_0(\pi/2) +5c \left \{ { A^{-5}(\pi/2) \over [\alpha^2(\pi/2)-\beta^2(\pi/2)]^{1/2}} + { B^{-5}(\pi/2) \over [\alpha{'}^2(\pi/2)-\beta{'}^2(\pi/2)]^{1/2}} \right \} }. \end{aligned}$$
(40)

Calculations give for η 1 = 1.182 and for η′1 = 1.251 which are very close to the aspect ratios of an ideal icosahedron, i.e., 1.173 and 1.251, respectively. The small deviation is a consequence of a partial symmetry breaking, conditioned by neglecting the higher order Fourier components in Eq. (35). Thus, our proposed approximation for an analytical presentation of the structure of icosahedral MCs represents reasonably well the main peculiarities of the shape, i.e., symmetry and aspect ratios. This suggests that although the expression (35) does not describe the edges and vortices precisely, it can nevertheless be used to determine the resonance frequencies for a comparison with experimental data.

It is hopefully obvious that functions similar to Eq. (33) for the icosahedron, but for clusters of other shapes as well can be constructed. For instance, for a cuboctahedron we have

$$ \begin{aligned} F_{co}(\theta, \varphi)&={d \over 1-a^2 [x(\theta, \varphi)+y(\theta, \varphi)]^2}+ {d \over 1-a^2 [x(\theta, \varphi)-y(\theta, \varphi)]^2} \\ &\quad+{d \over 1-a^2 [x(\theta, \varphi)+z(\theta, \varphi)]^2} + {d \over 1-a^2 [x(\theta, \varphi)-z(\theta, \varphi)]^2} \\ &\quad+ {d \over 1-a^2 [y(\theta, \varphi)+z(\theta, \varphi)]^2}+{d \over 1-a^2 [y(\theta, \varphi)-z(\theta, \varphi)]^2} \end{aligned} $$
(41)

where d and a are constants and \(x(\theta,\varphi)\), \(y(\theta,\varphi)\) and \(z(\theta,\varphi)\) are Cartesian coordinates expressed through the spherical angles θ and φ.

5 Conclusion

In the present work we have shown that exploiting the concept of weak deviations from a spherical shape it is possible to calculate the shape parameters of Na clusters using measured SP frequencies. The developed approach makes it possible to describe not only ellipsoidal clusters but also clusters with arbitrary deviations from spherical shapes, as long as these deviations are relatively small. As an example we have shown how MCs with icosahedral shape can be described in the frame of the present approach. Results of experimental data as well as of TDLDA calculations for the SP resonance frequencies of the small sodium clusters are reproduced accurately through a quite simple formula which links the aspect ratio of the cluster to the observable SP frequencies. In addition, our approach allows revealing the dependence of the dielectric function of a metal on the number of atoms in the cluster. Thus, we hope and believe that using the present approach it becomes possible to extract more information from experimental SP spectra about the size and shape of simple metal clusters than what usually is done.