Near field enhancement due to the optical response of small nanoparticles
Mario Zapata-Herrera
In this work, we propose to describe the behavior of the optical response of small metal nanoparticles (particles below $10$ nm of radius) as a function of size. We use classical models as well as quantum approaches to study the nanoparticles dielectric function in the whole range of nanometers using two infinite confinements, cubic and spherical, which are simple enough to obtain information of the quantum size effects on the optical response, measured on the enhancement field factor and absorption spectra, to find the limit of pure classical models at the nano-scale finding out enhancement factors of the order of $10^3$ for small silver nanoparticles. We confirm the plasmon resonance frequency through the absorption spectra, besides that, we have studied the role played by Localized Surface Plasmon Resonance (LSPR) in small metal nanoparticles as a function of the particle size and the environment, which are in good agreement with reported experiments and have found a limit of the classical behavior of small nanoparticles.
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\begin{document}
\title{Near field enhancement due to the optical response of small nanoparticles}
 \author{Mario Zapata-Herrera}
 \affiliation{Departamento de F\'isica, Universidad de los Andes,
              A.A. 4976, Bogot\'a D.C., Colombia}
 \author{Jefferson Fl\'orez}
 \affiliation{Departamento de F\'isica, Universidad de los Andes,
              A.A. 4976, Bogot\'a D.C., Colombia}
 \author{\'Angela Camacho}
 \affiliation{Departamento de F\'isica, Universidad de los Andes,
              A.A. 4976, Bogot\'a D.C., Colombia}
%\author{Mario Zapata Herrera}
%\date{\today}

\begin{abstract}
In this work, we propose to describe the behavior of the optical response  of small metal nanoparticles (particles below $10$ nm of radius) as a function of size. We use classical models as well as quantum approaches to study the nanoparticles dielectric function in the whole range of nanometers using two infinite confinements, cubic and spherical, which are simple enough to obtain information of the quantum size effects on the optical response, measured  on the enhancement field factor and absorption spectra, to find the limit of pure classical models at the nano-scale finding out enhancement factors of the order of $10^3$ for small silver nanoparticles. We confirm the plasmon resonance frequency through the absorption spectra, besides that, we have studied the role played by Localized Surface Plasmon Resonance (LSPR) in small metal nanoparticles as a function of the particle size and the environment, which are in good agreement with reported experiments and have found a limit of the classical behavior of small nanoparticles.
\end{abstract}

\maketitle

\section{Introduction}
\textit{Nano-Plasmonics} studies the optical properties of nanostructured materials based on the study of the interaction of an external electric field or electron charged beams with an interface dielectric-metal. The response of the conduction electrons of a nano-material is a collective resonant motion, called Localized Surface Plasmon Resonance (LSPR) which depends both on geometrical and compositional factors as the size, composition, shape, number of nanoparticles and the dielectric environment\cite{Juan,Carsten,Wei,Murray} as well as the nature of the incident light, e.g. polarization, intensity and wavelenght. Although for nanoparticles of tens of nanometers in size, this behavior is well understood, for particle sizes smaller than $10$ nm, there is controversy about the origin of reported LSPR blueshifts  as the particle size decreases below 10 nm. Due to the length scale of nanometers, this is an interesting open problem that is worth to be carefully examined.  One possible explanation is quantum size effect, which takes into account the discretization of energy levels \cite{Garcia,Genzel,Scholl} and therefore improve the actual description of the optical response of small nanoparticles, making classical theories insufficient for explaining several phenomena that appear due to the nature of nanoscopic systems. The main interest in understanding the plasmonic resonances of small nanoparticles lies in the physics of the limit between classical and quantum descriptions and therefore in its applications in biology, catalysis and quantum optics. \\

The plasmonic effect on isolated nanoparticles under the influence of a longitudinal polarized electromagnetic wave, produces a strong electric near field enhancement, which can be related to the incident electric field,  known as the enhancement field effect \cite{Nordlander}, and gives information about the LSPR. Recently,  EELS experiments  \cite{Scholl}  on single metal nanoparticles, corroborate the blueshifts of the LSPR as the particle size decreases. \\

 In this work, we propose to improve the description of the optical response of the small nanoparticles, which determines the  values of the optical constants to be used in the solution of the Maxwell equations to obtain the near field enhancement and the optical absorption. In particular, particle sizes smaller than $10$ nm are of special interest due to their natural quantum features that make them an important subject to understand the limit of pure classical phenomena,  where it is found that the ratio surface-volume plays a crucial role.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Classical dielectric optical response}
The classical dielectric function of a metal, can be described by a simplified model where the conduction electrons are considered like a free electron gas with a number $n$ of charges per volume unit moving on a background of positive ion cores. Based on a classical motion equation for an electron in a plasma, under the influence of an external electric field and assuming a harmonic response, the dielectric function as a function of the oscilation frequency of the external field for a free electron gas is given by
\begin{equation}
\epsilon(\omega)=1-\fr{\omega_p^2}{\omega^2+i\gamma\omega},\label{drude}
\end{equation}
where $\omega_p$ represents the bulk plasma frequency defined by $\omega_p^2=ne^2/\epsilon_0m$. The term $\gamma=1/\tau$  corresponds to a damping collision frequency in terms of the relaxation time of the free electron gas $\tau$.
By means of the dielectric function defined in (\ref{drude}), one can obtain the refractive index $\eta$ (related with the polarization), and the extinction coeficient $\kappa$ of a system (related with the optical absorption) through the equations

\begin{equation}
\eta^2=\fr{\epsilon_1}{2}+\fr{1}{2}\sqrt{\epsilon_1^2+\epsilon_2^2}\label{eta}
\end{equation}
and

\begin{equation}
\kappa=\fr{\epsilon_2}{2n}\label{kappa},
\end{equation}
These last two quantities are of great importance, due that these have all the optical information we can include as input parameters in the subsequent simulations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Quantum approaches}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\label{<GenzelModel>}Free electron gas in a cubic infinite box}
When the size of a particle becomes smaller, the continuous electronic band of a nanoparticle should break up into observable discrete states. Such effects have been observed in experiments with  thin films and metallic nano-particles \cite{Garcia,Genzel,Scholl} , showing a  notably difference between the results obtained for light-mater interaction in the quasi-static regime (particles until 10-100 nm of radius) and the ‘quantum regime’ (particles below 10nm of radius). As a first approximation, to describe the optical response of a valence electron gas in a metal nanoparticle,  Genzel and Martin \cite{Genzel} showed a simplified quantum model where  the free (valence) electron gas is confined in a cubic potential well with infinite sides, where the dielectric function of a single metal nanoparticle is given by 
%where the  energy levels are given by
%\be
%E_{h,k,l}=E_0(h^2+k^2+l^2),
%\ee
%where, $E_0=\fr{\pi^2\hbar^2}{2mL^2}$  which can be writen in terms of the Fermi energy $E_F$  and the intersection point of the $l$ axis with the Fermi sphere $l_F$ as $E_0=E_F/l_F^2$, taking into account, polarization in the z-axis solely, without loss of generality. In this model,
\be
\epsilon(\omega)=\epsilon_{\infty}+\fr{\omega_p^2}{N}\sum_{i,f}\fr{s_{if}(F_i-F_f)}{\omega_{if}^2-\omega^2-i\omega \gamma_{if}},
\ee
%where $\epsilon_{\infty}$ corresponds to the interband contribution, and the plasmon frequency $\omega_p$ is given in terms of the electronic density $n$, the elemental electric charge $e$ and the effective mass $m^*$,
%\be
%\omega_p^2=\fr{4\pi ne^2}{m^*}.
%\ee
where the terms $s_{if}$, $\omega_{if}$ and $\gamma_{if}$ are respectively, the oscillator strenght, the eigen-frequency and the damping for the dipole transition from an initial state $i$ to another, final state $f$ and $F_i$ and $F_f$  are the values of the Fermi-Dirac distribution function for the initial and final states. The oscillator strenght $s_{if}$  corresponds to the dipolar form in the z-direction in terms of the initial $ |i\rangle$ and final $ |f\rangle$ states
\be
s_{if}=\fr{2m\omega_{if}}{\hbar}|\langle f|z|i \rangle |^2.
\label{strenght}
\ee
with the parameter $m=M/N$  in terms of the mass of electron M and the atom valence number $N$.
Using the orthogonallity of eigen-functions, the term $\omega_{if}$, can be written by 
 \be
\omega_{if}=\Delta l (2l+\Delta l)\fr{E_0}{\hbar}.
\ee 
If only transitions in the $z$ direction is considered, $s_{if}$ takes the form
\be
s_{if}^2=\fr{64}{\pi^2}\fr{l^2(l+\Delta l)^2}{\Delta l^3(l+\Delta l)^3}\delta(h_i,h_f)\delta(k_i,k_f),
\ee
where $\{h,k,l\}$ are the set of quantum numbers associated to each direction $\{x,y,z\}$ in the momenta space. Due to the form of the eigen-functions in this type of confinement potential, the number of dipole transitions in the sum of the dielectric function depends only on the initial and final values of the quantum number $l$ corresponding solely to initial states within the Fermi surface through final states just outside the Fermi surface, $l_F$. \\
The final expression for the dielectric function obtained in \cite{Genzel} (in which only transitions with $\Delta l=1$ are considered)  reads
\be
\epsilon(\omega)=\epsilon_{\infty}+\omega_p^2\sum_{\Delta l}^{1,3,...}\sum_{l}^{l_F}\fr{S_{l,\Delta l}}{\omega_{l,\Delta l}^2-\omega^2-i\omega \gamma_{l,\Delta l}},\label{epsiloncubico}
\ee
where
\be
S_{l,\Delta l}=\fr{32}{\pi l_F}\fr{l^2(l+\Delta l)^2}{\Delta l^2(l+\Delta l)^2}
\ee
Using eq.(\ref{epsiloncubico}), we plot the real  and imaginary parts of the dielectric function vs. the incident photon energy for nanoparticles with $L=1.6, 3.6, 4.8, 6.4, 8$ and $16$ nm (we can relate the lenght of the cube $L$ with the radius of a nanosphere by the expresion $L=(\fr{4\pi}{3})^{1/3}R$), comparing with the classic Drude approach given by eq.(\ref{drude}) in the case of $L=16$nm  (see Fig.\ref{DielectricaGenzel}). We observe that the dielectric functions predicted by the quantum model disagree notably with the classical results in the range studied. The main difference is that both $\Re [\epsilon]$ and $\Im [\epsilon]$ exhibit a nonmonotic behavior according to the quantum model, versus the  monotonic response using classical Drude theory.
% predicts a with absorption definitely bigger than in classical Drude model forat least three orders of magnitude.
 However, as the lenght reaches the limit of $16$ nm, the agreement with the  classical description is good.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.7]{dielectricgenzel.jpg}
\caption{a) Real  and b) imaginary  parts of the dielectric function for a silver nanoparticle of lenght $L=1.6, 3.6, 4.8, 6.4, 8$ and $16$ nm. using an infinite cubic confinement potential.}
\label{DielectricaGenzel}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{\label{<SchollModel>}Free electron gas in a spherical infinite well}
Considering the derivation made for  Scholl \textit{et al.}, where the conduction electrons are considered as a free electron gas constrained by an infinite spherical potential well,  the corresponding transition frequencies $\omega_{if}$ of conduction electrons from occupied states $i$ to unocupied states $f$ inmediately outside of it from eq.(\ref{strenght}) are defined by $\omega_{if}=\frac{E_f-E_i}{\hbar}$,  where $E_i$ and $E_f$ are the initial and final energies, respectively, with associated initial $\ket{i}$ and final $\ket{f}$ states.\\

Solving the corresponding Schr\"odinger equation, for an electron constrained to a spherical potential,  with the appropriate boundary conditions we obtain a set of wavefunctions of the form \cite{Arfken} 

\begin{equation}
\psi(r,\theta,\phi)=\fr{1}{|j_{l+1}(\alpha_{nl})|} \sqrt{ \fr{2}{R^3}}  j_l \left( \fr{\alpha_{nl}}{R}r \right)Y_l^m(\theta,\phi),
\label{Psi}
\end{equation}
where, $j_l$ represents the spherical Bessel functions, $Y_l^m$ the standard spherical harmonics, and $\alpha$ is a zero of $j_l$. The energy eigen-values $E$ for this problem results in a quantized form associated to the $l$th spherical Bessel function by the expression 
\begin{equation}
E=\frac{\hbar^2\alpha_{ln}^2}{2MR^2},
\label{E}
\end{equation}
where $\alpha_{ln}$ is the $n$th zero of $j_l$, i.e. $j_l(\alpha_{ln})=0$ for $n=0,1,2,\dots$.
 The oscillator strenght of eq.(\ref{strenght}) can be calculated, taking $z=r \cos\theta$ in the expression
\be
|\bra{f}z\ket{i}|=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{R}r^2\sin\theta dr d\theta d\phi\Psi_{n_f,l_f,m_f}^*(r,\theta,\phi)r \cos \theta \Psi_{n_i,l_i,m_i}(r,\theta,\phi),
\ee
where the angular integral has the form
\be
I_{ang}=\sqrt{\fr{(l_i+m_i+1)(l_i-m_i+1)}{(2l_i+1)(2l_i+3)}}\delta_{\Delta l,+1}+\sqrt{\fr{(l_i+m_i)(l_i-m_i)}{(2l_i+1)(2l_i-1)}}\delta_{\Delta l,-1}
\label{angular}
\ee
and, the radial term,
\be
I_{rad}=\fr{1}{|j_{l_f+1}(\alpha_{n_fl_f})|} \fr{1}{|j_{l_i+1}(\alpha_{n_il_i})|}\left( \fr{2}{R^3}\right)\int_{0}^{R}dr  j_{l_f} \left( \fr{\alpha_{n_fl_f}}{R}r \right)r^3  j_{l_i} \left( \fr{\alpha_{n_il_i}}{R}r \right).
\ee
Then, the integral to calculate is simply
\be
|\bra{f}z\ket{i}|=\fr{I_{ang}}{|j_{l_f+1}(\alpha_{n_fl_f})||j_{l_i+1}(\alpha_{n_il_i})|}\left( \fr{2}{R^3}\right)\int_{0}^{R}dr  j_{l_f} \left( \fr{\alpha_{n_fl_f}}{R}r \right)r^3  j_{l_i} \left( \fr{\alpha_{n_il_i}}{R}r \right),
\ee
which  is different to zero only for values between $\Delta l=l_f-l_i=\pm 1$.

One simple form to calculate the allowed energies in this problem \cite{Scholl} is to take the approximate solution for the Bessel zeros $\alpha$

\begin{align}
E&=\frac{\hbar^2\pi^2}{8MR^2}(2n+l+2)^2,\notag\\
&=\frac{\hbar^2}{2MR^2}\left[\pi\left(n+\frac{l}{2}+1\right)\right]^2,\label{ES}
\end{align}
and for the  wavefunctions, to employ the asymptotic approximation \cite{Arfken}
\begin{equation}
j_l(x)\approx \frac{1}{x}\cos\left[x-\frac{\pi}{2}(l+1)\right],
\label{approx}
\end{equation}
valid only for $x\gg l^2/2+l$.

\begin{table}[htdp]
\begin{center}
\begin{tabular}{|c|c |c|c | c |c| }\hline
& $n=0$ & $n=1$ & $n=2$ & $n=3$ & $n=4$\\\hline
$l=0$ & 376.03 & 1504.11 & 3384.25 & 6016.45 & 9400.71\\
& 376.03 & 1504.11 & 3384.25 & 6016.45 & 9400.71\\\hline
$l=1$ & 769.26 & 2273.77 & 4530.04 & 7538.31 & 1129.86\\
& 846.64 & 2350.18 & 4606.35 & 7614.57 & 11374.9 \\\hline
$l=2$ & 1265.57 & 3151.57 & 5785.61 & 9170.69 & 13307.4 \\
& 1504.11 & 3384.25 & 6016.45 & 9400.71 & 13537.0\\\hline
$l=3$ & 1860.45 & 4134.43 & 7148.86 & 10912.1 & 15426.0\\
& 2350.18 & 4606.35 & 7614.57 & 11374.9 & 15887.2\\\hline
$l=4$ & 2550.93 & 5219.83 & 8617.81 & 12760.9 & 17653.2\\
& 3384.25 & 6016.45 & 9400.71 & 13537.0 & 18425.4\\\hline
\end{tabular}
\end{center}
\caption{Comparison between energies given by eq.(\ref{E}) upper and by eq.(\ref{ES}) lower in units of meV when $R=1$ nm for various values of $l$ and $n$.}
\label{beta}
\end{table}

 In order to compare the exact expression with the approximation  given in reference \cite{Scholl}, we compare the energies given by eq. (\ref{E}) and eq.(\ref{ES}) for $R=1$nm in Table. \ref{beta},  where we observe a good agreement only for $l=0$, and a significant overestimation for $ l >0$. Besides this, in Figure \ref{j0} we show $j_l(kr)=j_l(\alpha_{ln}r/R)$ and its asymptotic approximation given by eq. (\ref{approx}) versus $r/R$ and $n=1$. As we can see, the asymptotic approximation and the Bessel functions are identical for $l=0$, but the approximation deviates remarkably from the actual function when $l=1, 2$, and $3$. Recall that this deviation occurs for $r/R<1$, i.e. inside the nanosphere. Therefore, we conclude that the asymptotic approximation in eq.(\ref{approx}) is not acceptable to describe the wavefunction of a free electron inside metallic nanospheres.\\
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.65]{bessel.jpg}
\caption{Comparison between $j_l$ and its asymptotic approximation (\ref{approx}) versus $r/R$ for different $l$'s and $n=1$.}
\label{j0}
\end{center}
\end{figure}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.65]{dielectricfunctionscholl.jpg}
\caption{\normalsize{a) Real  and b) imaginary  parts of the dielectric function for a silver nanoparticle of radius $R=1,2,3,4,5,$ and $10$nm using an infinite spherical confinement potential.}}
\label{DielectricaScholl}
\end{center}
\end{figure}

Using a similar treatment as in the infinite cubic confinement potential, we calculate the real  and imaginary parts of the dielectric function vs. the incident light energy for a nanosphere varying its radius from $1$ to $10$ nm (see Fig \ref{DielectricaScholl}) where  the dielectric functions predicted by the quantum model disagree notably with the classical results in the radius range studied (given by eq.(\ref{drude})). The main difference is that both $\Re[\epsilon]$ and  $\Im[\epsilon]$ exhibit a nonmonotic behavior according to the quantum model, as well as in the cubic case, due mainly to the natural chosen confinement potential and the number of states involved adequately.

\section{Enhancement Field Factor and Absorption spectra for small nanoparticles }

The interaction of metallic nanoparticles with an electromagnetic wave, produces an enormous near field, which can be calculated solving the Maxwell equations in matter. Generally in literature it is usually incorrect to use the bulk optical constants to define the nature of matter, at least for particles below $10$ nm of radius, when the quantum nature is evident. Accordingly,  to solve the problem of light-nanoparticle interaction, we use the analytical expressions of $\eta$ and $\kappa$ previously calculated  for each particle size and confinement potential to solve the vector Maxwell equations numerically, by means of a standar finite elements method (FEM). We define the enhancement field factor (EFF) as the norm of the ratio between the scatter electric near field just in the north pole of the spherical nanoparticle  $\vec{E}_{out}$ and an incident $z$-polarized electric field $\vec{E}_{inc}$ (see sketch in Figure \ref{EF}).
\be
EFF=\fr{|\vec{E}_{out}|^2}{|\vec{E}_{inc}|^2}
\ee

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.4]{model1.jpg}
\caption{Sketch of the calculation of the enhancement field factor.}
\label{EF}
\end{center}
\end{figure}
For an isolated silver particle with different sizes ($R=1,2,3,4,5$ and $10$ nm), we compare the solutions for three different sets of optical constants, which are taken as input parameters. Firstly, we use bulk optical constants taken from experiments of Johnson and Christie \cite{JandC}, where we observe the EFF increase with the nanoparticle size, due to the higher charge concentration, as usually in a classic framework without any resonance energy shift (Figure \ref{EFclasico}.a). Secondly, we show the EFF for a single silver nano-particle using the infinite cubic confinement potential (Figure \ref{EFclasico}.b), and finally (Figure \ref{EFclasico}.c) the  infinite spherical confinement potential. For the last two cases, we observe a fine structure in the plasmonic resonance  when the particle size decreases and the resonance shows a  blue shift around  of   $1-1.5$eV. This fine structure appereance is a consecuence of the transitions between discrete electronic states and the blue shift is due to strong near field effect, which increases as the particle decreases. This new feature indicates a more energetic plasmon for smaller particles, as seen in Figure \ref{EFclasico}.b. We also show the comparison of EFF for a single particle $R=10$ nm using Drude parameters and quantum parameters. The last one shows a slightly blue shift and EFF twice greater than in the former. However, for smaller particles, the EFF decreases and blue-shifts remarkably for the cubic confinement.\\ 

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.62]{efftodasn1.jpg}
\caption{Enhancement field factor (EFF) for a single metallic  nanosphere using the bulk optical constants \cite{JandC} in a), infinite cubic potential in b) and infinite spherical potential in c).  We compare in a) and b) with the  classical result (Drude). Inset in b): Quantum results for $R=1,2$ and $5$ nm. In all cases, we have a refraction index of the surrounding medium $n_m=1$. }
\label{EFclasico}
\end{center}
\end{figure}

The nanoparticle of $R=2$ nm under the quantum confinement  potential,  does not show a definite maximum, as expected, instead of it, appear many little transitions in the range from  $3.5$ to $5$ eV.  The $1$ nm particle has again a higher peak to $4$ eV but still has a minor peaks at $6$ and $7$ eV,  corresponding to  electronic transitions. The region between $R=3$ and $2$ nm is a frontier where the surface reaches the main importance and the optical response is dominated by the electronic transitions at the Fermi level, while for smaller particles the strong confinement makes again the core electrons to respond to the external perturbation, although in a weaker way due to the small number of charges present. 
For $R=2$ nm the competition between surface and volume is clear. Recently R. C. Monreal et. al. \cite{Monreal} discuss the influence of quantum size effect in small Ag nanoparticles as a competition to the spill out that can move the surface plasma energy toward the red or the blue. In our work we suppose infinite potential and therefore neglect the spill out effect, however we find the competition between surface and volume processes at very small nanoparticles. These spectra are in good agreement with the results found in reference \cite{Scholl}, although clearly we see that the use of an infinite spherical potential gives us a richer structure in both EFF and absorption spectra (see Figure \ref{absorption}), besides showing the blue shifts as a function of nanoparticle size. The use of a cubic potential confinement allows to know the experimental blue shift in the plasmon resonance, but does not have the same rich structure than the spherical. This can be interpreted as a consequence of the high degenerance in the cubic case. \\

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.67]{absorcion.jpg}
\caption {Absorption spectra for a silver nanoparticle using the optical constants  from the quantum model: Infinite cubic potential (a and b) and infinite spherical potential (c and d) for different particle sizes (L=1.6, 3.2, 4.8, 6.4, 8 and 16 nm and R=1, 2, 3, 4, 5 nm and 10 nm). The graphs a) and c) correspond to a refractive index of surrounding medium $n=1.0$ and the graphs b) and d) to $n=1.5$.}
\label{absorption}
\end{center}
\end{figure}

Figure \ref{absorption}, also shows the resonance value from each absorption spectra for both confinement potential (cubic and spherical), dependent on the surrounding environment: air, with refractive index $n=1.0$ and SiO (glass) with $n=1.5$, in which we observe that the LSPR in both spectra change significantly with this parameter, showing that in the glass case, the plasmon resonance values are more defined and follow a well marked tendency relative to real experiments (see \cite{Scholl} and \cite{Monreal}).\\

We conclude that independent from the confinement used to obtain the values of the dielectric function the size effect is important for plasmon formation in small nanoparticles making clear a frontier between $R\sim3,4$  nm. Another considerations related with the ratio Surface/Volume $(S/V)$ should bring understanding of the origin of the plasmon formation.  The $S/V$ ratio increases from $6\%$ ($R=10$nm) to $60\%$ ($ R=1$ nm). Between $R=2$ nm and $R=1$ nm this ratio doubles. In following table we see the relationship between R and the ratio $S/V$. The collective plasmonic behavior is seen for $S/V$ less than $15\%$. This ratio maintains under $50\%$ for $R < 4$ nm up to $R = 2$nm. In this range there is a competition between surface and volume effects, but for $R=1$ nm the percentage of surface electrons is $60\%$ and definitely the surface effects dominate. 


\begin{table}[htdp]
\begin{center}
\begin{tabular}{|c |c |c |c|c |c | c|}\hline
R(nm)& $1nm$& $2nm$ & $3nm$ & $4nm$ & $5nm$ &$10nm$ \\\hline
$S/V$&$60$& $30$ &$20$ & $15$ & $12.5$ &$6$ \\\hline
\end{tabular}
\end{center}
\caption{Percentage surface-volume ratio Vs. nanoparticle (sphere) radius.}
\label{alpha}
\end{table}


For $R= 1$ nm, where $60\%$ of the atoms are at the surface,  the response comes mainly from the surface, while for $R=2$nm at the surface are only $30\%$ and the electronic transitions are strong enough to absorb the energy of the field. \\
\section{Conclusions}

We present a mathematically more detailed quantum description of the dielectric response for small metal nanoparticles (less than $10$ nm) using two different  confinement potentials. We study the optical response of isolated  nanoparticles as a function of particle size, by obtaining the enhancement near field factor and absorption spectra. For small particles, we can see a richer spectrum in the optical response, that can be described as a consequence of the evident quantum nature at the nanoscale.\\

 We also present the localized surface plasmon resonance in small nanoparticles as a function of size,
% shape and environment,%
 carrying out a finite elements method to solve the complete vector Maxwell equations. We can conclude that the use of a cubic potential confinement, allows to explain the experimental blue shift in its plasmon resonance. Because we study spherical nanoparticles is physically meaningful the use of spherical confinement, so we found in this case a better description of the electronic transitions in addition to the blueshift. However, the spherical confinement should be more precise physically and therefore the appearance of electronic transitions become clear.\\

We obtain plasmon resonance energy as function of diameter of nanoparticles in good agreement with up to now reported experiments. Therefore our calculation shows that the simple models that take into account confinement explain satisfactorily the optical behavior of small nanoparticles, and allow us to fix a limit for diameters between 2 and 4 nm as the frontier between classical and quantum behavior. Effects such as spill out recently reported are important and we are now working on it.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{11}
\bibitem {Juan}{Arias, J. C.; Camacho, A. S., \textit{JEMAA}, 2011.}

\bibitem {Carsten}{Carsten Sonnichsen \textit{Ph. D Dissertation}, Munich University, 2001.}

\bibitem {Wei}{Wei, Q.-H; Su, K.-H.; Durant, S.; Zhang, X., \textit{Nano Lett}, \tb{4}, 1067, 2004.}

\bibitem {Murray}{Murray, W. A.; Barnes, W. L.  \textit{Adv. Mat},\tb{19}, 3771, 2007}.
\bibitem{Scholl}{J. Scholl, Ai Leen Koh and J. Dionne. \textit{Nature}, \tb{423}, 421, 2012.}
\bibitem{Genzel}{Genzel L., Martin T. P.. Kreibig U., \textit{Z. Physik B}, \tb{21}, 339, 1975.}
\bibitem{Garcia}{Garcia de Abajo, F. J.; \textit{Nature}, \tb{483}, 417, (2012)} 
\bibitem{Nordlander}{Nordlander, P.  et. al., Nano Lett. 5, 1569, 2005}
\bibitem{Arfken}{Mathematical Methods for Physicists, G. B. Arfken and H. J. Weber. 5th Ed., Harcourt/Academic Press, 2001.}
\bibitem{JandC}{Johnson and Christy, \textit{Phys. Rev. B}, \tb{6}, 12, 4370 (1975)} 
\bibitem{Monreal}{R. C. Monreal, T. J. Antosiewicz and S. P. Apell.; arXiv:1304.3023}
\end{thebibliography}

\end{document}
1307.1815.extracted_bib
\begin{thebibliography}{11}
\bibitem {Juan}{Arias, J. C.; Camacho, A. S., \textit{JEMAA}, 2011.}

\bibitem {Carsten}{Carsten Sonnichsen \textit{Ph. D Dissertation}, Munich University, 2001.}

\bibitem {Wei}{Wei, Q.-H; Su, K.-H.; Durant, S.; Zhang, X., \textit{Nano Lett}, \tb{4}, 1067, 2004.}

\bibitem {Murray}{Murray, W. A.; Barnes, W. L.  \textit{Adv. Mat},\tb{19}, 3771, 2007}.
\bibitem{Scholl}{J. Scholl, Ai Leen Koh and J. Dionne. \textit{Nature}, \tb{423}, 421, 2012.}
\bibitem{Genzel}{Genzel L., Martin T. P.. Kreibig U., \textit{Z. Physik B}, \tb{21}, 339, 1975.}
\bibitem{Garcia}{Garcia de Abajo, F. J.; \textit{Nature}, \tb{483}, 417, (2012)} 
\bibitem{Nordlander}{Nordlander, P.  et. al., Nano Lett. 5, 1569, 2005}
\bibitem{Arfken}{Mathematical Methods for Physicists, G. B. Arfken and H. J. Weber. 5th Ed., Harcourt/Academic Press, 2001.}
\bibitem{JandC}{Johnson and Christy, \textit{Phys. Rev. B}, \tb{6}, 12, 4370 (1975)} 
\bibitem{Monreal}{R. C. Monreal, T. J. Antosiewicz and S. P. Apell.; arXiv:1304.3023}
\end{thebibliography}