After ablation the weight loss of Au and Ag plate is measured to be 1 and 0.6 mg respectively. Regarding density of the metals, these values in 10 mL of pure water produce fractional volume of 5.7 × 10^–6 for Ag and 5.2 × 10^–6 for Au colloid; the fractional volumes are approximately equal. Because filling factor of the samples is very small, then we consider clusters, as separated single particles in full statistical disorder so, interaction of clusters has not any rule in absorption cross section. Figure 3 shows the TEM images for gold nanocolloid and related histogram and Figure 4 shows corresponding images for silver nanocolloid. We obtain the average diameter of nanoparticles by directly counting the number of nanoparticles insight on a given micrograph and measuring their diameter. Mean diameter of the gold nanoparticles is 40.30 nm and 18.10 nm before and after illumination respectively. Mean diameter of the silver nanoparticles is 18.68 nm and 8.08 nm before and after illumination respectively.

In extinction spectroscopy, a valuable parameter for investigation of size distribution is contrast, which is defined as the difference between the magnitude of absorbance at the peak and the valley divided by the magnitude of absorbance at the peak. Our data show that the contrast of Ag1 (after ablation) and Ag2 (after illumination) samples is equal to 90% and 91% respectively; which means that the contrast has increased by 1% due to illumination [12]. Our data for gold colloid show that the contrast of Au1 (after ablation) and Au2 (after illumination) are equal to 40% and 33% respectively, i.e., there is a 7% decrease due to illumination.

Silver nanoparticles become charged when the sample is subjected to laser pulses. Once nanoparticles accumulate sufficient charges, they may break up to form smaller nanoparticles [7].

The absorption spectra of colloidal samples are shown in Figure 5. The plasmon peak position of Au1 is located at 522 nm which after illumination is blue shifted to 523 nm and for Ag1 sample plasmon peak position is blue shifted from 412 to 408 nm after illumination. Ag and Au nanoparticles plasmon peak position have 4 and 1 nm blue shift respectively, but behavior of their absorbance in illumination process is opposite.

Concentrating on vertical axis in Figure 5 it is clear that the absorbance of silver is increased from 2.3 to about 3 under illumination, while for gold nanoparticles we see a decrease in absorbance from 1.7 to 1.4.

The Mie theory is based on the resolution of the Maxwell equations in spherical coordinates using the multi pole expansion of the electric and magnetic fields by taking into account the discontinuity of the dielectric constant between the sphere and the surrounding medium. For spherical particles, the extinction, scattering, and absorption cross sections are calculated from Mie theory by series expansion of the involved fields into partial waves of different spherical symmetries [8]:

where k, is the wave vector and a_n, b_n are scattering coefficients in terms of Ricatti-Bessel cylindrical functions. The Mie spectra depend on the radius of the sphere (R) and dielectric function. Bulk dielectric function [Ԑ_bulk(ω)] incorporates both, size independent conduction electron processes and interband transitions. We use Johnson and Christy data [13, 14] for the dielectric function of gold and silver nanoparticles. Size-dependent damping effects change the magnitude of the nanoparticle’s dielectric function. However, the damping related to the mean free path of free electrons, is strongly affected by the size. Dielectric function and damping constant for nanoparticles can be written as [15–17]:

where v_F is the Fermi velocity, A is the scattering constant that includes details of the scattering process and ω_P is the bulk plasma frequency given by ωP=ne2ε0m in which n is the density of free electrons in the metal, m electron mas and Ԑ₀ the permittivity of free space. Gold and silver used parameters in this work are summarized in Table 1. Mean free path of electrons differs from the reported values in Table 1, it is equal to l_eff and we use our reported method [17] for calculating this parameter.

We have developed a new method to calculate the mean free path of electrons in spherical nanoparticles and related Г(R) in the Drude model; accordingly, we have corrected ϵ(R,ω). In Figure 6 our new approximation is shown for calculation of l_eff. As seen in Figure 6, inside the sphere, the electron located at point da can move in all directions. If the path traveled in hitting the surface is greater than the length of the mean free path, we consider the length traveled to be l, with this condition numerical averaging are obtained over all points and all directions. Of course, due to the symmetry of spherical nanoparticles, the electrons are distinguished only in the radial direction relative to the surface, that is, if completely identical electrons are removed from the averaging, only the electrons that are in the radial direction and move outward or inward are averaged. Since we consider the electron distribution inside the nanoparticle to be uniform by averaging l over all angles from zero to π and radii from zero to R and applying the weight function for shell electrons with thickness da, we obtain the diagram in Figure 7 for the mean free path length of the electrons inside the nanoparticle. In this figure the red line shows the mean free path length in terms of radius by adding the term related to the surface scattering, including the scattering of electrons colliding the ions using Eq. 5 and the blue line shows the mean free path length according to the normally calculated mean free path.

In Figure 7 variation of effective mean free path against radius is shown in the range of 0–100 nm. For nanoparticles below 40 nm, there is a linear dependence for normal calculation while variation of effective mean free path is not linear in our calculation. Beyond 40 nm both effective mean free path length increase with different slopes.

When the slope of the graph is larger it means that the curve tends toward bulk values more rapidly. At last, in two separate 3D graphs (not shown here), we plotted the variation of wavelength and absorbance against radius and standard deviation to estimate the radius range for experimentally produced gold nanoparticles.

According to the values calculated for l_eff2 in the method mentioned above and using Eq. 4, the imaginary and real part of the dielectric function has been calculated and plotted in terms of radius and wavelength in a three-dimensional diagram. After plotting the real and imaginary part of ϵ(R,ω) in 3D graphs, C_ext and C_abs are plotted in other 3D graphs (not shown here), we show that the SPR’s positions are relatively constant for two samples; meanwhile, it is shown that the magnitude of absorbance’s have visible changes for these two samples.

According to the values of absorption and peak position of plasmon resonance for Au1 and Au2 samples that are given in previous section, we can estimate the radius and standard deviation of Au1 and Au2 samples. For the Au1 sample, the average radius is between 17 to 20 nm and the standard deviation is in the range of 6 to 9 nm, and in the Au2 sample, the radius and standard deviation values are in the range of 12 to 14 nm and 7 to 9 nm, respectively. Therefore, it cannot be accurately said that the spectrum is caused by an average radius and a unique distribution.

Absorbance of nanoparticles can be obtained from the following equation [15]:

where n_i is total number of nanoparticles in each size group, L is optical path length of samples and N is total number of nanoparticles. The absorbance of silver nanoparticles versus wavelength is simulated for 5, 15, 25 and 35 nm radiuses in Figure 8a where each sample of nanoparticles has the same amount of mass. As can be seen, maximum absorbance of the silver nanoparticles with the size 15 nm, is placed around the wavelength of 400 nm and has a magnitude larger than other radiuses. Therefore, in a sample with the same mass percent of nanoparticles, the maximum absorbance will be for 15 nm particles.

According to the histograms, it is clear that the number of smaller nanoparticles is more than the larger ones, while the absorption cross-section and the total volume of the smaller nanoparticles are smaller, so they do not have a significant effect on the absorption.

Very large nanoparticles are fewer so they have also an insignificant result. Also, in Figure 8b the absorbance is simulated versus wavelength for gold nanoparticles with radiuses of 5, 10, 20, 30 and 40 nm. In this case, the maximum value of the absorbance belongs to colloidal gold nanoparticles 20 nm and 30 nm radii in size. Comparing the two graphs, it is clear that silver nanoparticles have sharper absorbance, and their full width at half maximum (FWHM) is less than gold nanoparticles.

The magnitude of absorption in gold nanoparticles for different sizes follows a similar pattern, as a result, gold nanoparticles are less sensitive to size distribution than silver nanoparticles.

For simulation of absorbance according to size distribution, we fit a normal distribution to histograms, and then by programming in MATLAB software we reconstruct the absorbance of the samples. It is obvious that n_i is a fraction of the total nanoparticles of the sample and also, is proportional to the total mass of the ablated material in the water. It is natural that the sum of the coefficients of the total mass is equal to one.

Where y_i and x_i are a fraction of the total nanoparticles, according to size distribution and a fraction of the total mass in water respectively. M is the total ablated mass into the water and ρ is the density of matter. By combination of Eq. 7, 8 and 9, we will have:

By solving these equations, unknown parameters, namely N and the absorbance are determined. Total number of nanoparticles are estimated to be 6.9 × 10¹², 110.45 × 10¹², 1.6 × 10¹³ and 19.9 × 10¹³ for Ag1, Ag2, Au1 and Au2 samples respectively. Our estimation shows that after illumination, the number of Ag/Au nanoparticles is increased by a 16/12.4 fold multiplication factor. The ablated gold material is more than silver (1 mg to 0.6 mg), consequently under illumination, silver nanoparticles receive more energy than gold nanoparticles and melting/boiling point of silver is smaller than gold, so size reduction of silver nanoparticles is more. Figure 9 shows computationally reconstructed absorbance for the Ag1 sample along with experimental data. In this figure, we neglect size dependence and put A = 0 in Eq. 5.

The reconstructed graph does not match well with the experimental one because the dielectric function does not depend on the radius. Best A parameters have been studied in many articles [15]. We use A = 1 for silver nanoparticles and A = 0.33 for gold nanoparticles. Absorbance in Figure 10 and Figure 11 is simulated and compared with the empirical graph, in this case, there is a better agreement between experiment and theory.

The mean free path of electrons in gold nanoparticles is greater than the corresponding value in silver nanoparticles, so for different values ​​of A, no significant difference in the magnitude of absorption is observed in gold nanoparticles.

In this work, we used our idea for modeling dielectric function [17] and tried to reconstruct the absorbance-wavelength graphs for plasmonic nanoparticles by varying A parameter, and viewed a lot of differences between A = 0 and A = 1 mode. By this model, we can see a better agreement between experimentally produced UV/VIS absorption spectra and theoretically produced spectra.