2. Anharmonic cluster heating rate and expansion

Consider an atomic cluster with density $n_c$ and radius $r_{c0}$. The equilibrium free electron density of each cluster is specified by $n_{c0}$. A Gaussian laser beam of frequency $\omega$ and wavenumbers k propagates through the cluster. The electric field of the laser beam is given by

(2.1a,b)\begin{equation}\boldsymbol{E}_L = \hat{e}_yA_L(z,t)\,{\rm e}^{{\rm i}(kz-\omega t)} + {\rm c}{\rm .c}{\rm .,}\quad A_L^2 = A_{L0}^2 \,{\rm exp}\left( {-\displaystyle{{r^2} \over {r_0^2 }}} \right)\,{\rm exp}\left( {-\displaystyle{{{\left( {t-\displaystyle{z \over c}} \right)}^2} \over {\tau_L^2 }}} \right).\end{equation}

Under the electric field of the laser, the cluster electrons oscillate in the y direction, ${\rm \Delta }\hat{e}_y$. As the electron oscillates, a space charge is produced and a restoring force is applied to it. A schematic of the electron oscillation in the laser field is shown in figure 1. The third-order electric susceptibility of the plasma is totally assumed to result from the motion of a rigid cold core within the ion potential well. Until a uniform ion distribution will generate a harmonic potential and while the electron oscillations are small, compared with the cluster radius, the restoring force is ${-}m\omega _{pe0}^2 \varDelta {\rm /}3$. Any deviation from uniformity will present an anharmonic component to the electric potential. Because the cluster is symmetrically spherical, we can express the potential of the cluster as a polynomial expansion, i.e. $V(r) = {\textstyle{1 \over 2}}\omega _0^2 r^2 + {\textstyle{1 \over 4}}br^4 + \cdots$. In this case, the oscillation of electrons becomes comparable to the cluster radius, and restoring force (for ${r}^{\prime} > r_{c0}$) becomes (Kumar & Tripathi Reference Kumar and Tripathi2013)

(2.2)\begin{equation}\boldsymbol{F}_{{\rm res}} = {-}\displaystyle{{m\omega _{pe0}^2 } \over 3}\varDelta \left( {1-\displaystyle{9 \over {16}}\displaystyle{\varDelta \over {r_{c0}}}} \right)\hat{e}_y,\end{equation}

where $\omega _{pe0}^2 = 4{\rm \pi} n_{e0}e^2/m$, e and m are the electron plasma frequency, charge and mass of electron, respectively. Therefore, the motion equation of cluster electrons in the laser electric field and the restoring force of space charges is

(2.3)\begin{equation}\displaystyle{{{\rm d}^2\varDelta } \over {{\rm d}t^2}} = {-}\displaystyle{{\omega _{pe0}^2 } \over 3}\varDelta \left( {1-\displaystyle{9 \over {16}}\displaystyle{\varDelta \over {r_{c0}}}} \right)-\nu \displaystyle{{{\rm d}\varDelta } \over {{\rm d}t}}-\displaystyle{e \over m}E_L,\end{equation}

where $\nu$ is the electron–ion collision frequency. Using the well-known perturbative method, we solve (2.3) as

(2.4a,b)\begin{equation}\varDelta = \varDelta ^{(1)} + \varDelta ^{(2)} + \varDelta ^{(3)} + \cdots ,\quad \begin{cases}\varDelta^{(1)} = \Delta_1\,{\rm e}^{{\rm i}(kz-\omega t)} + {\rm c}{\rm .c}.\\ \varDelta^{(2)} = \Delta_2\,{\rm e}^{{\rm i}(2kz-2\omega t)} + {\rm c}{\rm .c}.\\ \varDelta^{(3)} = \Delta_3\,{\rm e}^{{\rm i}(3kz-3\omega t)} + {\rm c}{\rm .c} . \end{cases}\!\!\!\!\!\!,\end{equation}

Using (2.3), we can derive the first-, second- and third-order momentum equations as follows:

(2.5)\begin{gather}\displaystyle{{{\rm d}^2\varDelta ^{(1)}} \over {{\rm d}t^2}} + \displaystyle{{\omega _{pe0}^2 } \over 3}\varDelta ^{(1)} + \nu \displaystyle{{{\rm d}\varDelta ^{(1)}} \over {{\rm d}t}} = {-}\displaystyle{e \over m}(A_L\,{\rm e}^{{\rm i}(kz-\omega t)} + {\rm c}{\rm .c} .),\end{gather}

(2.6)\begin{gather}\displaystyle{{{\rm d}^2\varDelta ^{(2)}} \over {{\rm d}t^2}} + \displaystyle{{\omega _{pe0}^2 } \over 3}\varDelta ^{(2)} + \nu \displaystyle{{{\rm d}\varDelta ^{(2)}} \over {{\rm d}t}} = \displaystyle{3 \over {16}}\displaystyle{{\omega _{pe0}^2 } \over {r_{c0}}}\varDelta ^{(1)}\varDelta ^{(1)},\end{gather}

(2.7)\begin{gather}\displaystyle{{{\rm d}^2\varDelta ^{(3)}} \over {{\rm d}t^2}} + \displaystyle{{\omega _{pe0}^2 } \over 3}\varDelta ^{(3)} + \nu \displaystyle{{{\rm d}\varDelta ^{(3)}} \over {{\rm d}t}} = \displaystyle{3 \over 8}\displaystyle{{\omega _{pe0}^2 } \over {r_{c0}}}(\varDelta ^{(2)}\varDelta ^{(1)}),\end{gather}

so that the solutions of these equations become:

(2.8)\begin{gather}\varDelta ^{(1)} = \displaystyle{e \over m}\displaystyle{{A_L\,{\rm e}^{{\rm i}(kz-\omega t)}} \over {\left( {\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}\nu \omega } \right)}} + {\rm c}{\rm .c}{\rm .,}\end{gather}

(2.9)\begin{gather}\varDelta ^{(2)} = {-}\displaystyle{3 \over {16}}\displaystyle{{\omega _{pe0}^2 } \over {r_{c0}}}\displaystyle{{e^2} \over {m^2}}\displaystyle{{A_L^2 \,{\rm e}^{{\rm i}(2kz-2\omega t)}} \over {{\left( {\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}\nu \omega } \right)}^2\left( {4\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}2\nu \omega } \right)}} + {\rm c}{\rm .c}{\rm .,}\end{gather}

(2.10)\begin{gather}\varDelta ^{(3)} = \displaystyle{3 \over {16}}\displaystyle{{\omega _{pe0}^2 } \over {r_{c0}}}\displaystyle{{e^3} \over {m^3}}\displaystyle{{A_L^3 \left( {1-\displaystyle{{3\omega_{pe0}^2 } \over {16r_{c0}(4\omega^2-\omega_{pe0}^2 /3 + {\rm i}2\nu \omega )}}} \right){\rm e}^{{\rm i}(3kz-3\omega t)}} \over {{\left( {\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}\nu \omega } \right)}^3\left( {9\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}3\nu \omega } \right)}} + {\rm c}{\rm .c} .\end{gather}

Figure 1.

Schematic representation for the cluster electron oscillation in the laser field.

To compare the ratio of $\varDelta ^{(1)}$ to $r_{c0}$, variation of the first-order normalized oscillation amplitude $(\varDelta ^{(1)}/r_{c0})$ as a function of the normalized frequency is plotted in figure 2. As shown in figure 2, the oscillation amplitude is comparable to the cluster radius at the intensities $I\approx 10^{13}\;{\rm W}\;{\rm c}{\rm m}^{{-}2}$, $10^{14}\;{\rm W}\;{\rm c}{\rm m}^{{-}2}$ and $10^{15}\;{\rm W}\;{\rm c}{\rm m}^{{-}2}$. For $I > 10^{16}\;{\rm W}\;{\rm c}{\rm m}^{{-}2}$, the oscillation amplitude becomes larger than the cluster radius. In fact, for $I < 10^{16}\;{\rm W}\;{\rm c}{\rm m}^{{-}2}$, the amplitude of the oscillation is not large enough to cause the Coulomb explosion, thus the ions in the cluster can be assumed as a static background. The velocity of the oscillation of cluster electrons may be written as

(2.11)\begin{align} \begin{aligned}\boldsymbol{V}_{{\rm osc}}& = \displaystyle{{{\rm d}\boldsymbol{\varDelta }} \over {{\rm d}t}} = {\hat{e}}_y\left[ {\displaystyle{{e\omega A_L\,{\rm e}^{{\rm i}(kz-\omega t)}} \over {{\rm i}m\left( {\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}\nu \omega } \right)}}-\displaystyle{{3\omega \omega_{pe0}^2 e^2A_L^2 \,{\rm e}^{{\rm i}(2kz-2\omega t)}} \over {{\rm i}8r_{c0}m^2{\left( {\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}\nu \omega } \right)}^2\left( {4\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}2\nu \omega } \right)}}}\right.\\ &\quad \left.+\, \displaystyle{{9\omega \omega_{pe0}^2 e^3A_L^3 \left( {1-\displaystyle{{3\omega_{pe0}^2 } \over {16r_{c0}(4\omega^2-\omega_{pe0}^2 /3 + {\rm i}2\nu \omega )}}} \right)} \over {16{\rm i}r_{c0}m^3{\left( {\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}\nu \omega } \right)}^3\left({9\omega^2-\displaystyle{{\omega_{pe0}^2 } \over 3} + {\rm i}3\nu \omega } \right)}}\,{\rm e}^{{\rm i}(3kz-3\omega t)} + {\rm c}{\rm .c}.\right]\end{aligned} \end{align}

Figure 2.

Normalized oscillation amplitude as a function of the normalized frequency for different laser field intensities.

When resonance occurs, that is, when the electron density of the cluster triples the critical density, i.e. $\omega _{pe0} = \sqrt 3 \omega$, the rates of heating and expansion of the cluster increase. In the adiabatic condition, we have $T_eV^{\gamma -1} = {\rm const} .$, where V is the volume of the cluster equal to $4{\rm \pi} r_c^3 /3$ and $\gamma$ denotes the ratio of specific heats. The average heating rate of the cluster electrons is obtained under the influence of the laser electric field as well as the cluster expansion under adiabatic conditions as follows:

(2.12)\begin{equation}\displaystyle{{\rm d} \over {{\rm d}t}}(T_e) = {-}\displaystyle{e \over 3}\boldsymbol{E}_L^\ast \boldsymbol{\cdot }\boldsymbol{V}_{{\rm osc}}-\displaystyle{{3T_e(\gamma -1)} \over {r_c}}\displaystyle{{{\rm d}r_c} \over {{\rm d}t}},\end{equation}

where $\gamma = 5/3$, and r_c and T_e are the cluster radius and the temperature of cluster electron, respectively. As the radius of the cluster increases, its average density decreases, which keeps $\omega _{pe}^2 r_c^3 = {\rm constant}$, i.e. $\omega _{pe}^2 = \omega _{pe0}^2 (r_c/r_{c0})^{{-}3}$ (Tiwari & Tripathi Reference Tiwari and Tripathi2006), where $\omega _{pe0}$ and r_{c 0} are the initial cluster electron plasma frequency and the initial cluster radius, respectively. The collision frequency is related to the electron temperature as $\nu = \nu _0(T_e/T_0)^{-{3 / 2}}$ (Kumar & Tripathi Reference Kumar and Tripathi2009, Reference Kumar and Tripathi2011), where T ₀ is the electron temperature at t = 0. By defining normalized parameters as $\xi = T_e/T_0$, $b_1 = \omega ^2/\nu _0^2$, $d = \omega _{pe0}^2 /\omega ^2$ and $\eta = r_c/r_{c0}$, we replace these into (2.12) as

(2.13)\begin{align} \begin{aligned} \displaystyle{{{\rm d}\xi } \over {{\rm d}\tau }}& = \displaystyle{{\alpha ^2\,{\rm exp}\left( {-\displaystyle{{\tau^2} \over {\nu_0^2 \tau_L^2 }}} \right)} \over {3\left( {\xi^{3/2}{\left( {1-\displaystyle{d \over {3\eta^3}}} \right)}^2 + \displaystyle{1 \over {b_1\xi^{3/2}}}} \right)}} + \displaystyle{{{\rm d}V_{th}} \over {4r_{c0}\omega }}\displaystyle{{\alpha ^3{\left( {\exp \left( {-\displaystyle{{\tau^2} \over {\nu_0^2 \tau_L^2 }}} \right)} \right)}^{3/2}} \over {\eta ^3}}\\ &\quad \times \displaystyle{{\left( {\displaystyle{1 \over {b_1\xi^3}}} \right)-\left( {1-\displaystyle{d \over {3\eta^3}}} \right)\left( {5-\displaystyle{{2d} \over {3\eta^3}}} \right)\,{\rm cos(}kz-\omega t)} \over {\left[ {{\left( {1-\displaystyle{d \over {3\eta^3}}} \right)}^2 + {\left( {\displaystyle{1 \over {b_1\xi^3}}} \right)}^2\left( {\xi^{3/2}{\left( {4-\displaystyle{d \over {3\eta^3}}} \right)}^2 + \displaystyle{4 \over {b_1\xi^{3/2}}}} \right)} \right]}}\\ &\quad + \displaystyle{{{\rm d}V_{th}} \over {8r_{c0}\nu _0}}\displaystyle{{\alpha ^3{\left( {\exp \left( {-\displaystyle{{\tau^2} \over {\nu_0^2 \tau_L^2 }}} \right)} \right)}^{3/2}} \over {\eta ^3}}\displaystyle{{\left[ {\left( {4-\displaystyle{d \over {3\eta^3}}} \right){\left( {1-\displaystyle{d \over {3\eta^3}}} \right)}^2-\displaystyle{1 \over {b_1\xi^3}}\left( {8-\displaystyle{{5d} \over {3\eta^3}}} \right)} \right]\sin (kz-\omega t)} \over {\left[ {{\left( {1-\displaystyle{d \over {3\eta^3}}} \right)}^2 + {\left( {\displaystyle{1 \over {b_1\xi^3}}} \right)}^2\left( {{\left( {4-\displaystyle{d \over {3\eta^3}}} \right)}^2 + \displaystyle{4 \over {b_1\xi^3}}} \right)} \right]}}\\ &\quad-2\displaystyle{\xi \over \eta }\displaystyle{{{\rm d}\eta } \over {{\rm d}\tau }}, \end{aligned}\end{align}

where $\alpha = (eA_{L0}/\omega )(\sqrt {{\rm exp}[{-}r^2/r_0^2 ]/mT_0} = \alpha _0\sqrt {{\rm exp}[{-}r^2/r_0^2 ]}$, ${t}^{\prime} = t-z/c$ and $\tau = \nu _0{t}^{\prime}$. The radius of the cluster expands as

(2.14)\begin{equation}{\rm d}r_c/{\rm d}t = c_s = \sqrt {T_e/m_i} ,\end{equation}

where c_s is the velocity of sound, m_i is the ion mass and therefore we derive the time average of the cluster radius rate as

(2.15)\begin{equation}\displaystyle{{{\rm d}\eta } \over {{\rm d}\tau }} = \displaystyle{{c_{s0}} \over {\nu _0r_{c0}}}\displaystyle{{c_s} \over {c_{s0}}} = g\sqrt \xi ,\end{equation}

3. Bremsstrahlung X-ray emission

The electrons that acquire enough energy from the laser field and leave the cluster are called free electrons. These free electrons are adjacent to the ions and are absorbed by the ions in the nanoplasma medium. When these electrons, with an initial velocity V_e, pass in the vicinity of the ions with the ion charge state, z, and the impact parameter, b, they accelerate and radiate in the X-ray regime. The electron irradiated energy, using the Larmor formula (Jackson Reference Jackson1962), is

(3.1a,b)\begin{equation} U = \displaystyle{{2e^2a_c^2 \tau _c} \over {3c^3}} = \displaystyle{{4ze^6} \over {3m^2c^3b^3V_e}},\quad\begin{cases} a_c = \displaystyle{{ze^2} \over {b^2m}}\\ \tau_c = \displaystyle{{2b} \over {V_e}}\end{cases}\!\!\!\!\!\!,\end{equation}

The electron acceleration is a function of time in the X-ray regime; therefore, Fourier analysis of the acceleration is necessary. The electron acceleration and its Fourier transform are achieved as

(3.2a,b)\begin{equation} \begin{cases} a_{cz}(t) = \displaystyle{{ze^2} \over m}\displaystyle{b \over {{(b^2-{(V_et)}^2)}^{3/2}}}\\ a_{cy}(t) = \displaystyle{{ze^2} \over m}\displaystyle{{V_et} \over {{(b^2-{(V_et)}^2)}^{3/2}}} \end{cases}\!\!\!\!\!\!,\quad \begin{cases} a_{cz}(\omega ) = \displaystyle{{ze^2} \over {m{\rm \pi} }}\displaystyle{{2\omega K_1\left( {\displaystyle{{\omega b} \over {V_e}}} \right)} \over {V_e^2 }}\\ a_{cy}(\omega ) = \displaystyle{{ze^2} \over {m{\rm \pi} }}\displaystyle{{2{\rm i}\omega K_0\left( {\displaystyle{{\omega b} \over {V_e}}} \right)} \over {V_e^2 }}\end{cases}\!\!\!\!\!\!,\end{equation}

where $K_0(\omega b/V_e)$ and $K_1(\omega b/V_e)$ are the modified Bessel functions. According to the Parsval relation, the energy of a signal is equal to its Fourier transform energy, $\int_{-\infty }^\infty {{|{a_c(t)} |}^2\,{\rm d}t} = {\rm \pi}\int_0^\infty {{|{a_c(\omega )} |}^2\,{\rm d}\omega }$. As a result, the energy emitted by the electron becomes:

(3.3)\begin{align}\begin{aligned} U& = \displaystyle{{2e^2} \over {3c^3}}\int_{-\infty }^\infty {{|{a_c(t)} |}^2\,{\rm d}t} \displaystyle{{2{\rm \pi} e^2} \over {c^3}}\int_0^\infty {{|{a_c(\omega )} |}^2\,{\rm d}\omega}\\ &=\displaystyle{{{\rm \pi} e^6z^2} \over {12m^2c^3}}\displaystyle{1 \over {b^3V_e}}, \end{aligned}\end{align}

Thus, the radiation power per electron is

(3.4)\begin{equation}P = \int_{b_{\min }}^{b_{\max }} {n_iV_eU2{\rm \pi} b\,{\rm d}b} = \displaystyle{{n_i{\rm \pi} ^2z^2e^6} \over {6m^2c^3}}\left[ {\displaystyle{1 \over {b_{{\rm min}}}}-\displaystyle{1 \over {b_{{\rm max}}}}} \right],\end{equation}

where, using the law of conservation of angular momentum, $b_{\min } = \hbar /mV_e$, $b_{\max }\approx V_e/\omega$ (Krane Reference Krane1996) and n_i is the number of the electron–ion collisions per unit time in b to b + db. The frequency of ω corresponds to electrons having a velocity as $V_e > \sqrt {2\hbar \omega /m}$. Considering the Maxwellian velocity distribution function, the power radiated per unit frequency interval per unit becomes:

(3.5)\begin{equation}P_B(\omega ) = \int_{\sqrt {2\hbar \omega /m} }^\infty {Pf_e4{\rm \pi} V_e^2 \,{\rm d}V_e} = \int_{\sqrt {2\hbar \omega /m} }^\infty {Pn_e{\left( {\displaystyle{m \over {2{\rm \pi} T_e}}} \right)}^{3/2}\exp \left( {\displaystyle{{mV_e^2 } \over {2T_e}}} \right)4{\rm \pi} V_e^2 \,{\rm d}V_e} .\end{equation}

The total power radiated per unit frequency for the plasma channel of length L is obtained as

(3.6)\begin{equation}P_B^T (\omega ) = \int_0^\infty {\int_0^L {\displaystyle{{4{\rm \pi} n_cr_c^3 } \over 3}P_B(\omega )2{\rm \pi} r\,{\rm d}r\,{\rm d}z} } ,\end{equation}

After solving (3.5) and (3.6) for $n_i = n_e$, we obtain

(3.7)\begin{equation}P_B^T (\omega ) = \displaystyle{{\sqrt {\rm \pi}z^2n_cv_{th}Lm^2r_0^2 } \over {72\sqrt 2 n_ec^3\hbar }}\int {\omega _{pe}^6 r_c^3 } \left( {\displaystyle{r \over {r_0}}} \right)\xi ^{1/2}\left( {2 + \displaystyle{{\displaystyle{{\hbar \omega } \over {T_0}}} \over \xi }} \right)\exp \left[ {-\displaystyle{{\displaystyle{{\hbar \omega } \over {T_0}}} \over \xi }} \right]{\rm d}\left( {\displaystyle{r \over {r_0}}} \right),\end{equation}

where $v_{th}^2 = T_0/m$ is the electron thermal velocity. Using the normalization factor $P_B^0 = (m\omega _{pe0}^4 )/(n_{e0}c)$, we get

(3.8)\begin{equation}\bar{P}_B^T = \displaystyle{{P_B^T } \over {P_B^0 }} = \displaystyle{{{\rm \pi} \sqrt {2{\rm \pi} } z^2r_{c0}^3 n_c} \over {72}}\displaystyle{{v_{th}} \over c}\displaystyle{{Lm\omega _{pe0}^2 r_0^2 } \over {ch}}\int_0^\infty {} {\eta ^{{-}6}} \left( {\displaystyle{r \over {r_0}}} \right)\xi ^{1/2}\left( {2 + \displaystyle{{\displaystyle{{\hbar \omega } \over {T_0}}} \over \xi }} \right)\exp \left[ {-\displaystyle{{\displaystyle{{\hbar \omega } \over {T_0}}} \over \xi }} \right]{\rm d}\left( {\displaystyle{r \over {r_0}}} \right).\end{equation}

4. Numerical results

In this section, we study numerically the heating rate of the electrons and the expanding of the radius rate of the cluster by solving the coupled equations (2.13) and (2.15). In figure 3, the normalized electron heating rate of a deuterium cluster as a function of normalized time for different values of radial locations is plotted for n_c = 10¹³ cm⁻³, r_{c 0} = 10⁻⁶ cm located at $r/r_0 = 0.3$, 0.7 and 0.1, when the parameters are $I\approx 10^{14}\;{\rm W}\;{\rm c}{\rm m}^{{-}2}$, r ₀ = 10⁻² cm, g = 0.3, b ₁ = 10⁵, d = 5, $\omega _{pe0} = 5 \times 10^{15}\;{\rm rad}\;{\rm s}^{{-}1}$, $\hbar \omega /T_0 = 10$, $\alpha _0 = 0.57$ and $\nu _0\tau _L = 1$. In figure 3, a sharp peak at the resonance time is observed. Furthermore, the heating rate becomes large at a location closer to the radial axis of the laser.

Figure 3.

Normalized electron heating rate of cluster as a function of normalized time for different radial locations.

The variation of the electron heating rate of the cluster as a function of normalized time for the anharmonic and harmonic clusters is plotted in figure 4. At the resonant frequency, the temperature of the electrons in the anharmonic cluster is higher than that of them in the harmonic cluster (Kumar et al. Reference Kumar, Singh and Verma2014), owing to the nonlinear restoring force. Figure 5 shows the variation of the electron heating rate of the cluster as a function of normalized time for different values of laser field intensities. It is found that the electron heating rate of the cluster is larger at more intensities.

Figure 4.

Normalized electron heating rate of cluster as a function of normalized time for anharmonic and harmonic clusters.

Figure 5.

Normalized electron heating rate of cluster as a function of normalized time for different laser field intensities.

The three-dimensional (3-D) plot of the electron heating rate of the deuterium cluster as a function of the normalized time and the normalized radial locations is depicted in figure 6.

Figure 6.

Normalized electron heating rate of cluster as a function of normalized time and of normalized radial locations.

The variation of the normalized cluster radius for the deuterium cluster as a function of the normalized time is shown in figure 7. In this figure, before the resonance time, the cluster radius slowly expands; however, when resonance occurs, the radius of the cluster expands rapidly.

Figure 7.

Normalized average cluster radius as a function of normalized time for different radial locations.

The variation of the cluster radius as a function of the normalized time for the harmonic and anharmonic clusters is plotted in figure 8. As shown in figure 8, after the resonance time, the harmonic and anharmonic cluster radii expand gradually. In addition, the variation of the anharmonic cluster radius is almost identical to the harmonic cluster radius (Kumar et al. Reference Kumar, Singh and Verma2014).

Figure 8.

Normalized average cluster radius as a function of normalized time for harmonic and anharmonic clusters.

Figure 9 shows the variation of the normalized cluster radius as a function of the normalized time for various laser field intensities. It is found that by moving away from the radial axis of the laser, i.e. for large values of $r/r_0$, a reduction in the expansion of the cluster radius occurs.

Figure 9.

Normalized average cluster radius as a function of normalized time for different laser field intensities.

From figure 9, it is found that with increasing the intensity of the laser field, the radius of the cluster is increased. Additionally, the 3-D variation of the cluster radius as a function of the normalized time and of the normalized radial locations, $r/r_0$ is plotted in figure 10.

Figure 10.

Normalized average cluster radius as a function of normalized time and of normalized radial locations.

The normalized X-ray emission as a function of the normalized time for $\hbar \omega /T_0 = 12$ is drawn in figure 11. As shown in figure 11, the emission of X-rays has a sharp peak at the resonance time, then it slowly decreases, and at the plasma frequency of the cluster electrons less than the plasma resonance, the rate of Bremsstrahlung encounters is not sufficient to produce X-rays.

Figure 11.

Normalized X-ray emission as a function of the normalized time for the deuterium cluster.

A comparison of the spectra of Bremsstrahlung X-ray radiation of deuterium, helium, neon and argon clusters is presented in figure 12. In the laser–nanocluster interaction, a significantly higher X-ray emission peak occurs at the resonance frequency for larger clusters such as the argon cluster. It is observed that the intensities of X-ray radiation spectra of deuterium and helium are approximately 0.01 times the intensities of X-ray radiation spectra of neon and argon. We plot the normalized X-ray emission as a function of the normalized time for different values of $\hbar \omega /T_0 = 8$, 12 and 18 in figure 13. It is found that with increasing $\hbar \omega$_, with respect to the electron temperature, the intensity of the Bremsstrahlung X-ray radiation spectrum is reduced.

Figure 12.

Normalized X-ray emission as a function of the normalized time for the deuterium, helium, neon and argon clusters.

Figure 13.

Normalized X-ray emission as a function of normalized time for different values of $\hbar \omega /T_0 = 8$, 12 and 18 for the deuterium cluster.

The variation of intensity of the Bremsstrahlung X-ray radiation spectrum as a function of $\hbar \omega /T_0$ at $\nu _0{t}^{\prime} = 0.6$ for the deuterium cluster has been plotted in figure 14. It is revealed that the intensity of the Bremsstrahlung X-ray radiation spectrum exponentially decreases at frequencies $\hbar \omega \gg T_e$, and it increases at the resonance frequency, $\omega = \omega _{pe0}/\sqrt 3 \approx 0.57\omega _{pe0}$. The comparison among the spectra of X-ray radiation for the deuterium, helium, neon, and argon clusters as a function of $\hbar \omega /T_0$ is presented in figure 15. It is found that the spectrum of X-ray radiation of all these clusters exponentially decreases at frequencies $\hbar \omega \gg T_e$.

Figure 14.

Spectrum of Bremsstrahlung X-ray radiation for $\nu _0{t}^{\prime} = 0.6$ for the deuterium cluster.

Figure 15.

Spectrum of Bremsstrahlung X-ray radiation for the deuterium, helium, neon and argon clusters.