Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers

*Boucerredj Noureddine and Khaled Beggas*

## **Abstract**

The nanoplasma formations of Na and Kr clusters, which contained 2 <sup>10</sup><sup>7</sup> atoms per cluster, irradiated by intense femtosecond laser field have been predicted in detail within the framework of the modified nanoplasma model. Based on this modified model, ionization process, heating, expansion, and explosion of the cluster have been studied. When laser intensity is above a critical value, it blows off all of electrons from the cluster and forms a non-neutral ion cloud. The analytic calculation provides time evolution of radius of the cluster, internal and external fields, coulomb and hydrodynamic pressures, electron density, and ion and electron energy. During the coulomb explosion of the resulting highly ionized, hightemperature nanoplasma, ions acquire their energy. It is shown that ultrafast ions are produced in this comparative study (4.4 keV for Kr cluster and 2.2 keV for Na cluster), which can be the source of energetic ions. We have found that the coulomb pressure is little than the hydrodynamic pressure for both clusters.

**Keywords:** large clusters, nanoplasma, laser plasma interaction, laser-produced plasma, ion source, cluster explosion

## **1. Introduction**

During the last years, nanoplasma (nanometer scale plasma) formed from atomic clusters irradiated by high-power femtosecond lasers, capable of achieving a light intensity up to 1021–10<sup>22</sup> W/cm<sup>2</sup> , has attracted a great attention and has been studied by several groups [1–5]. This large interest is due to the high energetic ions and electrons produced when rare gas clusters explode [5, 6]. The intense femtosecond laser field ionize atoms of the cluster several times and ions reaches a high degree of ionization [7–9], and are very effective to produce strong X-ray radiation [10]. In addition, fusion reaction has been observed in laser deuterium cluster interaction indicating that nuclear fusion occurs after explosion of the cluster [11]. The measurements of the energy absorption by the formed plasma have shown that the plasma absorbs a large fraction of the incident laser energy [12]. The interaction of rare gas and metallic clusters with ultrafast lasers has become a topic of great interest; a new fundamental insight into ultrafast laser-driven excitation and decay dynamics of many particle systems can be gained [13]. Macroscopic and hydrodynamic impact energy transfer from nanoplasma electrons to ions in exploding cluster using molecular dynamics simulation has been studied recently [14]. When

the laser interacts with clusters, it changes the regime of electromagnetic wave propagation, and very efficient absorption of radiation has been demonstrated with a formation of very high temperature under dense plasma [15, 16]. The coulomb explosion plays an important role in the case of small-diameter clusters [17].

the cluster occurs through inverse bremsstrahlung collisions. The density of the

*Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers*

*∂E*int

In this model there are three main assumptions: First, we assume that λ > > R0, where λ and R0 are the laser wavelength and the initial cluster radius, respectively. Second, the electronic and ionic densities are distributed uniformly within the cluster. The third assumption is that the electrons are assumed to be instantaneously thermalized and their energy distribution is Maxwellian. The nanoplasma model takes into account the highly efficient collisional heating which is calculated in this model as the heating of a uniform dielectric sphere under the electric field of the

> 3 j j *ε* þ 2

where Eext is the external electric field, t is the time, and ε is the plasma dielectric

where w is the plasma frequency, ν is the total electron-ion collision frequency,

s

where ne is the electron density, ε<sup>0</sup> is the vacuum dielectric constant, e is the

ffiffiffiffiffiffiffiffi *nee*<sup>2</sup> *ε*0*m*

> ! *z*,

*τ* � �<sup>2</sup> � � *<sup>e</sup>*

> ffiffiffiffiffiffiffiffi 2*I*max *ε*0*c* q

> > *v*

! *z*,

*p*

*<sup>ε</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>w</sup>*<sup>2</sup>

*wp* ¼

*ext*ðÞ¼ *t E*<sup>0</sup> sin ð Þ *wt f t*ð Þ*e*

*ext*ðÞ¼ *<sup>t</sup> <sup>E</sup>*<sup>0</sup> exp �2 ln 2 *<sup>t</sup>*

where τ is the laser pulse duration at the full width at half maximum (FWHM), f(t) is a Gaussian distribution of laser profile, and E0 is the amplitude of the laser

The total electron-ion collision frequency including the electron-surface collision

*ν* ¼ *νei* þ *ν<sup>s</sup>* ¼ *νei* þ

where *v* is the velocity of the electrons, *R* is the cluster radius, and *νei* is the

*<sup>ν</sup>ei* <sup>¼</sup> *ni <sup>Z</sup>*<sup>2</sup> � �*e*<sup>4</sup> lnΛ<sup>1</sup> <sup>2</sup>ð Þ *πε*0*me* <sup>2</sup>

*<sup>∂</sup><sup>t</sup>* (1)

*Eext*ð Þ*t* , (2)

*w w*ð Þ <sup>þ</sup> *<sup>i</sup><sup>ν</sup>* , (3)

, (4)

, where c is the light velocity.

*<sup>R</sup>* , (5)

*<sup>v</sup>*<sup>3</sup> , (6)

*<sup>∂</sup><sup>t</sup>* in the cluster is given by [9, 25]: *∂u <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>ε</sup>E*int

*E*intðÞ¼ *t*

where ε is the plasma dielectric constant.

*DOI: http://dx.doi.org/10.5772/intechopen.90320*

laser. The internal electric field is given by [26].

and wp is the electronic plasma frequency given by

electron charge, and m is the electron mass. The external electric field is given by

> *E* !

> *E* !

field given as the maximum laser field, *E*max ¼

electron-ion collision frequency given by [27, 28]

frequency, *vs = v/R*, is given by the formula

**103**

constant given by a Drude model,

energy deposition *<sup>∂</sup><sup>u</sup>*

The interaction of intense femtosecond laser field with atomic cluster leads to the ionization of the parent atoms of the cluster. Ionization of the cluster begins on the rising edge of the laser field. Parent atoms absorb laser intensity and liberate a small number of electrons by direct optical ionization which produces a gas of electrons that forms the nanoplasma. The collisional ionization becomes important between electrons and ions leading to the rapid heating of the cluster to a nonequilibrium superheated state then the expansion of the cluster [18, 19].

The interaction of single clusters with intense femtosecond laser produces the hot electrons with energy up to 3 keV [20, 21]. After heating up the cluster, charge separation of the hot electrons and the repulsive coulomb force between the positive ions leads to the explosion of the cluster, and ions acquire their energy by the conversion of the energy deposited by the laser in the cluster to ion kinetic energy.

In this present research, we study the formation of the nanoscale plasma from the irradiated clusters by an intense femtosecond laser field, and we investigate a comparative behavior of the different physical parameters of the formed nanoplasma from rare gas (Kr) and metallic (Na) clusters within the framework of the modified nanoplasma model. After a general introduction on intense laser cluster interaction and plasma formation, we show in Section 2 a brief description of the nanoplasma model, followed by a discussion of the obtained results in Section 3. Finally, we report the obtained results in the conclusion section.

## **2. Nanoplasma model**

The development of the numerical model called nanoplasma model was made firstly by Ditmire et al. [9], and after that reformulated by Megi et al. [18] by the addition of the term of electron collisions with surface in the expression of the electron-ion collision frequency. This numerical model has proved its success in reproducing experimental results of the interaction of single cluster with intense femtosecond laser field [19] and describes the interaction in terms of the formation of high-density nanoplasma.

In this model, the expanding cluster was treated as a spherical nanoplasma, subjected to the intense laser field. Ionization process such as field ionization is described by the Ammosov, Delone, and Krainov model [22]. Tunnel and collisional ionization are described using the Lotz formula [23]. Collisional ionization in the nanoplasma leads to the production of highly charged ions.

#### **2.1 Ionization mechanisms**

The basic mechanisms in this model are that, the laser strips electrons from the parent atoms by direct optical ionization and collisional ionization. The direct optical ionization begins when laser pulse liberates a small number of electrons. Field ionization is described in this model by Ammosov et al. [18, 22, 24], and collisional ionization, which occurs from inelastic collisions between electrons and ions, is described by the Lotz formula [9, 18, 23]. Due to the high excitation and hot electrons and densities reached inside the cluster, collisional ionization is the dominant ionization mechanism in the nanoplasma leading to the production of highly charged ions. The laser deposits its energy into the free electrons inside the cluster. These electrons absorb the magnetic power and heating up the ions. The heating in

*Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers DOI: http://dx.doi.org/10.5772/intechopen.90320*

the cluster occurs through inverse bremsstrahlung collisions. The density of the energy deposition *<sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>t</sup>* in the cluster is given by [9, 25]:

$$\frac{\partial u}{\partial t} = \varepsilon E\_{\text{int}} \frac{\partial E\_{\text{int}}}{\partial t} \tag{1}$$

where ε is the plasma dielectric constant.

the laser interacts with clusters, it changes the regime of electromagnetic wave propagation, and very efficient absorption of radiation has been demonstrated with a formation of very high temperature under dense plasma [15, 16]. The coulomb explosion plays an important role in the case of small-diameter clusters [17].

The interaction of intense femtosecond laser field with atomic cluster leads to the ionization of the parent atoms of the cluster. Ionization of the cluster begins on the rising edge of the laser field. Parent atoms absorb laser intensity and liberate a small number of electrons by direct optical ionization which produces a gas of electrons that forms the nanoplasma. The collisional ionization becomes important between electrons and ions leading to the rapid heating of the cluster to a nonequilibrium superheated state then the expansion of the cluster [18, 19].

The interaction of single clusters with intense femtosecond laser produces the hot electrons with energy up to 3 keV [20, 21]. After heating up the cluster, charge separation of the hot electrons and the repulsive coulomb force between the positive ions leads to the explosion of the cluster, and ions acquire their energy by the conversion of the energy deposited by the laser in the cluster to ion kinetic energy. In this present research, we study the formation of the nanoscale plasma from the irradiated clusters by an intense femtosecond laser field, and we investigate a

nanoplasma from rare gas (Kr) and metallic (Na) clusters within the framework of the modified nanoplasma model. After a general introduction on intense laser cluster interaction and plasma formation, we show in Section 2 a brief description of the nanoplasma model, followed by a discussion of the obtained results in Section 3.

The development of the numerical model called nanoplasma model was made firstly by Ditmire et al. [9], and after that reformulated by Megi et al. [18] by the addition of the term of electron collisions with surface in the expression of the electron-ion collision frequency. This numerical model has proved its success in reproducing experimental results of the interaction of single cluster with intense femtosecond laser field [19] and describes the interaction in terms of the formation

In this model, the expanding cluster was treated as a spherical nanoplasma, subjected to the intense laser field. Ionization process such as field ionization is described by the Ammosov, Delone, and Krainov model [22]. Tunnel and collisional ionization are described using the Lotz formula [23]. Collisional ionization in the

The basic mechanisms in this model are that, the laser strips electrons from the

parent atoms by direct optical ionization and collisional ionization. The direct optical ionization begins when laser pulse liberates a small number of electrons. Field ionization is described in this model by Ammosov et al. [18, 22, 24], and collisional ionization, which occurs from inelastic collisions between electrons and ions, is described by the Lotz formula [9, 18, 23]. Due to the high excitation and hot electrons and densities reached inside the cluster, collisional ionization is the dominant ionization mechanism in the nanoplasma leading to the production of highly charged ions. The laser deposits its energy into the free electrons inside the cluster. These electrons absorb the magnetic power and heating up the ions. The heating in

comparative behavior of the different physical parameters of the formed

Finally, we report the obtained results in the conclusion section.

nanoplasma leads to the production of highly charged ions.

**2. Nanoplasma model**

*Novel Imaging and Spectroscopy*

of high-density nanoplasma.

**2.1 Ionization mechanisms**

**102**

In this model there are three main assumptions: First, we assume that λ > > R0, where λ and R0 are the laser wavelength and the initial cluster radius, respectively. Second, the electronic and ionic densities are distributed uniformly within the cluster. The third assumption is that the electrons are assumed to be instantaneously thermalized and their energy distribution is Maxwellian. The nanoplasma model takes into account the highly efficient collisional heating which is calculated in this model as the heating of a uniform dielectric sphere under the electric field of the laser. The internal electric field is given by [26].

$$E\_{\rm int}(t) = \frac{\mathfrak{J}}{|\varepsilon + \mathfrak{L}|} E\_{\rm ext}(t),\tag{2}$$

where Eext is the external electric field, t is the time, and ε is the plasma dielectric constant given by a Drude model,

$$\varepsilon = 1 - \frac{w\_p^2}{w(w + i\nu)},\tag{3}$$

where w is the plasma frequency, ν is the total electron-ion collision frequency, and wp is the electronic plasma frequency given by

$$w\_p = \sqrt{\frac{n\_\epsilon e^2}{\varepsilon\_0 m}},\tag{4}$$

where ne is the electron density, ε<sup>0</sup> is the vacuum dielectric constant, e is the electron charge, and m is the electron mass.

The external electric field is given by

$$\begin{aligned} \overrightarrow{E}\_{\text{ext}}(t) &= E\_0 \sin \left( wt \right) \overrightarrow{f}(t) \overrightarrow{e}\_x, \\ \overrightarrow{E}\_{\text{ext}}(t) &= E\_0 \exp \left( -2 \ln 2 \left( \frac{t}{\tau} \right)^2 \right) \overrightarrow{e}\_x, \end{aligned}$$

where τ is the laser pulse duration at the full width at half maximum (FWHM), f(t) is a Gaussian distribution of laser profile, and E0 is the amplitude of the laser field given as the maximum laser field, *E*max ¼ ffiffiffiffiffiffiffiffi 2*I*max *ε*0*c* q , where c is the light velocity.

The total electron-ion collision frequency including the electron-surface collision frequency, *vs = v/R*, is given by the formula

$$
\nu = \nu\_{\text{ei}} + \nu\_{\text{s}} = \nu\_{\text{ei}} + \frac{v}{R}, \tag{5}
$$

where *v* is the velocity of the electrons, *R* is the cluster radius, and *νei* is the electron-ion collision frequency given by [27, 28]

$$\nu\_{\epsilon i} = \frac{n\_i \langle Z^2 \rangle e^4 \ln \Lambda\_1}{2 (\pi \varepsilon\_0 m\_\epsilon)^2 v^3},\tag{6}$$

$$
\upsilon = \sqrt{\upsilon\_{th}^2 + \upsilon\_{osc}^2},
\tag{7}
$$

ion kinetic energy or through coulomb explosion. For a hydrodynamic expansion,

*Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers*

where <Z> is the mean ion charge state. The coulomb explosion is the main

The rate of the temperature decrease from the cluster expansion is given by [9]:

¼ �2 *Te R ∂R*

The transfer of thermal energy to the cloud ions in the cluster nanoplasma results from the coulomb collision of energetic electrons, which can be described by

> ¼ � *Te* � *Ti τeq*

where Te is the electron temperature, Ti ion temperature, and me and mi are the

The calculation was carried out with an intense femtosecond laser including a

*Calculated time evolution of the cluster radius R of the Na and Kr clusters contained 2* � *<sup>10</sup><sup>7</sup> atoms in cluster irradiated by an intense femtosecond laser with a pulse duration of 200 fs (FWHM), wavelength of 390 nm,*

*kBTe me* þ *kBTi mi*

� �<sup>3</sup>

, wave length of 390 nm, and pulse duration of

2

process by which ions gain kinetic energy in the intense laser field.

*∂Te ∂t* � � � � exp

*∂Te ∂t* � � � � *eq*

The electron-ion equilibration time τeq is given by [9, 30]

*<sup>τ</sup>eq* <sup>¼</sup> <sup>3</sup>*memi* 8 ffiffiffiffiffi

<sup>2</sup>*<sup>π</sup>* <sup>p</sup> *niz*<sup>2</sup>*e*<sup>4</sup> ln<sup>Λ</sup>

h i *Eion* ≈h i *Z kBTe*, (13)

*<sup>∂</sup><sup>t</sup>* (14)

(15)

(16)

the mean ion energy will be of order [24]

*DOI: http://dx.doi.org/10.5772/intechopen.90320*

**2.3 Electron-ion thermalization**

a thermal equilibration rate given by [9]

electron and ion mass, respectively.

**3. Results and discussion**

peak intensity of 10<sup>17</sup> W/cm<sup>2</sup>

**Figure 1.**

**105**

*and peak intensity of 10<sup>17</sup> W/cm<sup>2</sup>*

*.*

where *vth* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *kBTe=<sup>m</sup>* <sup>p</sup> is the thermal velocity and *vosc* <sup>¼</sup> *eE*int *me<sup>ω</sup>* is the oscillation velocity of the electron in the field, <Z > is the mean ion charge, and lnΛ<sup>1</sup> is the modified coulomb logarithm given by the expression.

$$\ln \Lambda\_1 = \begin{cases} \frac{1}{4} \ln^2 \left( \frac{1 + m\_\varepsilon v\_{oc}^2}{\hbar w} \right), \text{for } \hbar w \gg k\_B T\_\varepsilon\\ \frac{1}{4} \ln^2 \left( 1 + \frac{v\_{oc}^2}{v\_{th}^2} \right) + \ln \left( \frac{v\_{oc}^2}{v\_{th}^2} + \exp \left( \frac{1}{3} \sqrt{\frac{\pi}{2}} \right) \right) \ln \Lambda, \text{for } \hbar w \ll k\_B T\_\varepsilon. \end{cases} \tag{8}$$

where kB is the Boltzmann constant, Te is the electron temperature (see [9] and [18]), and lnΛ is the classical coulomb logarithm [29].

## **2.2 Cluster expansion**

The expansion of the nanoplasma is driven by two mechanisms. First, the hydrodynamic heating is driven by the hot electron pressure in the cluster. The heated electrons or the heated nanoplasma start to expand, then pull the heavy ions. The electron or hydrodynamic pressure has the expression

$$P\_h = n\_\epsilon k\_B T\_\epsilon,\tag{9}$$

where ne is the electron density in the cluster *ne* <sup>¼</sup> *Ne* <sup>4</sup> 3*πR*<sup>3</sup> and Ne is the number of electrons in the cluster.

Second, the coulomb explosion is driven by charge buildup on the cluster. After ionization, some hot electrons escape the coulomb barrier formed by ions, leaving a net positive charge on the cluster. The resulting repulsive coulomb force leads to the expansion of the cluster. The coulomb pressure is given by

$$P\_{Coul} = \frac{3Q^2\varepsilon^2}{32\pi^2\varepsilon\_0 R^4},\tag{10}$$

where Q is the built-up charge on the cluster due to electron escape (see [9] and [18]). The total pressure is given by

$$P = P\_h + P\_{Coul} \tag{11}$$

The coulomb pressure scales as (1/R<sup>4</sup> ) shows that it will be important for small clusters. The hydrodynamic pressure scales as (1/R<sup>3</sup> ) is therefore more important for large clusters. Then, the cluster radius equation is given by

$$\frac{\partial^2 R}{\partial t^2} = \mathbf{5} \frac{P}{n\_i m\_i} \frac{\mathbf{1}}{R} \,, \tag{12}$$

where ni and mi are the density and mass of ions in the cluster, respectively.

The ion energies from cluster explosions are due to the electron-ion collisions heating up the electrons, but not the ions. This can be seen from the electron-ion equilibration time. In the other hand, there is insufficient time for electron energy to be transferred to the ions through collisions. Instead, the ions gain energy in the hydrodynamic expansion, where the thermal energy of the electrons is converted to ion kinetic energy or through coulomb explosion. For a hydrodynamic expansion, the mean ion energy will be of order [24]

$$
\langle E\_{ion} \rangle \approx \langle Z \rangle k\_B T\_\epsilon,\tag{13}
$$

where <Z> is the mean ion charge state. The coulomb explosion is the main process by which ions gain kinetic energy in the intense laser field.

The rate of the temperature decrease from the cluster expansion is given by [9]:

$$\left.\frac{\partial T\_e}{\partial t}\right|\_{\text{exp}} = -2\frac{T\_e}{R}\frac{\partial R}{\partial t} \tag{14}$$

#### **2.3 Electron-ion thermalization**

*v* ¼

modified coulomb logarithm given by the expression.

*osc* ℏ*w* � �

[18]), and lnΛ is the classical coulomb logarithm [29].

The electron or hydrodynamic pressure has the expression

where ne is the electron density in the cluster *ne* <sup>¼</sup> *Ne* <sup>4</sup>

expansion of the cluster. The coulomb pressure is given by

<sup>þ</sup> ln *<sup>v</sup>*<sup>2</sup> *osc v*2 *th*

ln <sup>2</sup> <sup>1</sup> <sup>þ</sup> *mev*<sup>2</sup>

*v*2 *osc v*2 *th*

� �

ln <sup>2</sup> <sup>1</sup> <sup>þ</sup>

where *vth* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*Novel Imaging and Spectroscopy*

1 4

8 >>><

>>>:

**2.2 Cluster expansion**

electrons in the cluster.

**104**

[18]). The total pressure is given by

The coulomb pressure scales as (1/R<sup>4</sup>

clusters. The hydrodynamic pressure scales as (1/R<sup>3</sup>

for large clusters. Then, the cluster radius equation is given by

*∂*2 *R <sup>∂</sup>t*<sup>2</sup> <sup>¼</sup> <sup>5</sup>

1 4

lnΛ<sup>1</sup> ¼

q

*kBTe=<sup>m</sup>* <sup>p</sup> is the thermal velocity and *vosc* <sup>¼</sup> *eE*int

, for ℏ*w* ≫ *kBTe*

velocity of the electron in the field, <Z > is the mean ion charge, and lnΛ<sup>1</sup> is the

þ exp

where kB is the Boltzmann constant, Te is the electron temperature (see [9] and

The expansion of the nanoplasma is driven by two mechanisms. First, the hydrodynamic heating is driven by the hot electron pressure in the cluster. The heated electrons or the heated nanoplasma start to expand, then pull the heavy ions.

Second, the coulomb explosion is driven by charge buildup on the cluster. After ionization, some hot electrons escape the coulomb barrier formed by ions, leaving a net positive charge on the cluster. The resulting repulsive coulomb force leads to the

where Q is the built-up charge on the cluster due to electron escape (see [9] and

*P nimi* 1

where ni and mi are the density and mass of ions in the cluster, respectively. The ion energies from cluster explosions are due to the electron-ion collisions heating up the electrons, but not the ions. This can be seen from the electron-ion equilibration time. In the other hand, there is insufficient time for electron energy to be transferred to the ions through collisions. Instead, the ions gain energy in the hydrodynamic expansion, where the thermal energy of the electrons is converted to

*e*2

*PCoul* <sup>¼</sup> <sup>3</sup>*Q*<sup>2</sup>

� � � � r

1 3 ffiffiffi *π* 2

*Ph* ¼ *nekBTe*, (9)

<sup>32</sup>*π*<sup>2</sup>*ε*0*R*<sup>4</sup> , (10)

) shows that it will be important for small

) is therefore more important

*<sup>R</sup>* , (12)

*P* ¼ *Ph* þ *PCoul* (11)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *v*2 *th* þ *v*<sup>2</sup> *osc*

, (7)

lnΛ, for ℏ*w* ≪ *kBTe:*

3*πR*<sup>3</sup> and Ne is the number of

*me<sup>ω</sup>* is the oscillation

(8)

The transfer of thermal energy to the cloud ions in the cluster nanoplasma results from the coulomb collision of energetic electrons, which can be described by a thermal equilibration rate given by [9]

$$\left.\frac{\partial T\_e}{\partial t}\right|\_{eq} = -\frac{T\_e - T\_i}{\tau\_{eq}}\tag{15}$$

The electron-ion equilibration time τeq is given by [9, 30]

$$\pi\_{eq} = \frac{3m\_e m\_i}{8\sqrt{2}\pi m\_i x^2 e^4 \ln \Lambda} \left(\frac{k\_B T\_e}{m\_e} + \frac{k\_B T\_i}{m\_i}\right)^{\frac{3}{2}} \tag{16}$$

where Te is the electron temperature, Ti ion temperature, and me and mi are the electron and ion mass, respectively.

## **3. Results and discussion**

The calculation was carried out with an intense femtosecond laser including a peak intensity of 10<sup>17</sup> W/cm<sup>2</sup> , wave length of 390 nm, and pulse duration of

#### **Figure 1.**

*Calculated time evolution of the cluster radius R of the Na and Kr clusters contained 2* � *<sup>10</sup><sup>7</sup> atoms in cluster irradiated by an intense femtosecond laser with a pulse duration of 200 fs (FWHM), wavelength of 390 nm, and peak intensity of 10<sup>17</sup> W/cm<sup>2</sup> .*

τ = 200 fs (full width at half maximum (FWHM)) irradiating the Kr and Na clusters containing 2 � <sup>10</sup><sup>7</sup> atoms per cluster. Inside the cluster, the temperature, the density gradient, and the internal field are assumed to be uniform. We have used the modified nanoplasma model to study the temporal variation of the different physical parameters (radius, electron temperature, expansion velocity, electron density, coulomb and hydrodynamic pressures, ion and electron energy, etc.) of the formed nanoplasma.

Time dependence of the cluster radius R for the rare gas Kr and metallic Na clusters is illustrated in **Figure 1**. Time evolution shows clearly a rapid expansion of the metallic cluster (Na) than the rare gas cluster (Kr). In the case of the metallic cluster, it expands rapidly due to the rapid ionization of the cluster.

The time zero in the calculation is the peak intensity of the laser at around �255fs for Na cluster and �230fs for Kr cluster; a small number of electrons are created through optical ionization; these electrons form the nanoplasma. The inelastic collisions between electrons and ions increase the number of electrons and ions. When electron density rises to reach 3ncrit, where ncrit is the critical electron density given by the formula

$$m\_{crit} = \frac{\varepsilon\_0 m\_c w^2}{\varepsilon^2},\tag{17}$$

cluster; the expansion of the cluster lowers the electron density to bring the system into resonance with laser field. The electron density in the cluster drops to 3ncrit (**Figure 3**), and we have the second resonance; (when the ratio ne/ncrit = 3), at 200 fs and 47 fs for both clusters Na and Kr respectively. We see that times of resonances are different for both clusters, metallic cluster (Na) and rare gas cluster (Kr).

*Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers*

The field inside the cluster is strongly enhanced and reached the values 1.22 <sup>10</sup><sup>12</sup> V/m and 8.00 <sup>10</sup><sup>11</sup> V/m for Kr and Na clusters, respectively (**Figure 2**); then we have a very rapid deposition of the energy into the electrons; at this point, the ions very rapidly stripped the hot electrons. The internal field Eint(t) is almost equal

The calculated time evolutions of the hydrodynamic pressure and the coulomb

one are shown in **Figure 4** for both clusters. The total charge on the cluster increases, and the repulsive forces lead to the increase of the coulomb pressure to

*Calculated electron density ne normalized by the critical electron density ncrit as a function of time for 2*

*wavelength of 390 nm, and peak intensity of 1017 W/cm<sup>2</sup>*

*atoms Na and Kr clusters irradiated by an intense femtosecond laser with a pulse duration of 200 fs,*

*Calculated time variation of the coulomb and hydrodynamic pressures for Na and Kr clusters (contained 2 107 atoms) irradiated by an intense femtosecond laser with a pulse duration of 200 fs (FWHM), wavelength of*

*.*

*.*

to the external field when the electron density ne is very low.

*DOI: http://dx.doi.org/10.5772/intechopen.90320*

**Figure 3.**

**Figure 4.**

**107**

*390 nm, and a peak intensity of 1017 W/cm<sup>2</sup>*

*107*

where ε<sup>0</sup> is the vacuum permittivity and ω is the laser frequency.

The internal field is amplified and becomes greater than the external one (**Figure 2**), and we have a strong absorption of the laser energy by the cluster. The first resonance (between laser field and the formed nanoplasma) occurs at t ≈ �250 fs, in the Na cluster, but at �230 fs it occurs in Kr cluster (**Figure 3**); the cluster expansion velocity increases dramatically; and more electrons are liberated through tunnel, laser-driven, and thermal ionization. The hot electrons can leave the cluster to the 3ncrit resonance.

When the electron density ne is greater than 3ncrit, the field inside the cluster is smaller than the external one. The maximum value of the ration *ne ncrit* is about 3.5 for Na cluster than 7 for Kr cluster; this difference in the time of resonances and the maximum values of the ratio *ne ncrit* is caused by the number of electrons in such cluster. When the hot electrons leave the cluster and the cluster expands, the electron density starts to decrease at time �176 fs for Na cluster and �163 fs for Kr

#### **Figure 2.**

*Time dependence of the internal and external electric fields for the Na and Kr clusters contained 2* � *107 atoms per cluster. Same parameters of the laser are used as those in Figure 2.*

*Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers DOI: http://dx.doi.org/10.5772/intechopen.90320*

cluster; the expansion of the cluster lowers the electron density to bring the system into resonance with laser field. The electron density in the cluster drops to 3ncrit (**Figure 3**), and we have the second resonance; (when the ratio ne/ncrit = 3), at 200 fs and 47 fs for both clusters Na and Kr respectively. We see that times of resonances are different for both clusters, metallic cluster (Na) and rare gas cluster (Kr).

The field inside the cluster is strongly enhanced and reached the values 1.22 <sup>10</sup><sup>12</sup> V/m and 8.00 <sup>10</sup><sup>11</sup> V/m for Kr and Na clusters, respectively (**Figure 2**); then we have a very rapid deposition of the energy into the electrons; at this point, the ions very rapidly stripped the hot electrons. The internal field Eint(t) is almost equal to the external field when the electron density ne is very low.

The calculated time evolutions of the hydrodynamic pressure and the coulomb one are shown in **Figure 4** for both clusters. The total charge on the cluster increases, and the repulsive forces lead to the increase of the coulomb pressure to

#### **Figure 3.**

τ = 200 fs (full width at half maximum (FWHM)) irradiating the Kr and Na clusters containing 2 � <sup>10</sup><sup>7</sup> atoms per cluster. Inside the cluster, the temperature, the density gradient, and the internal field are assumed to be uniform. We have used the modified nanoplasma model to study the temporal variation of the different physical parameters (radius, electron temperature, expansion velocity, electron density, coulomb and hydrodynamic pressures, ion and electron energy, etc.) of the formed

Time dependence of the cluster radius R for the rare gas Kr and metallic Na clusters is illustrated in **Figure 1**. Time evolution shows clearly a rapid expansion of the metallic cluster (Na) than the rare gas cluster (Kr). In the case of the metallic

The time zero in the calculation is the peak intensity of the laser at around �255fs for Na cluster and �230fs for Kr cluster; a small number of electrons are created through optical ionization; these electrons form the nanoplasma. The inelastic collisions between electrons and ions increase the number of electrons and ions. When electron density rises to reach 3ncrit, where ncrit is the critical electron

*ncrit* <sup>¼</sup> *<sup>ε</sup>*0*mew*<sup>2</sup>

The internal field is amplified and becomes greater than the external one (**Figure 2**), and we have a strong absorption of the laser energy by the cluster. The

t ≈ �250 fs, in the Na cluster, but at �230 fs it occurs in Kr cluster (**Figure 3**); the cluster expansion velocity increases dramatically; and more electrons are liberated through tunnel, laser-driven, and thermal ionization. The hot electrons can leave

When the electron density ne is greater than 3ncrit, the field inside the cluster is

*ncrit* is caused by the number of electrons in such

Na cluster than 7 for Kr cluster; this difference in the time of resonances and the

*Time dependence of the internal and external electric fields for the Na and Kr clusters contained 2* � *107 atoms*

*per cluster. Same parameters of the laser are used as those in Figure 2.*

cluster. When the hot electrons leave the cluster and the cluster expands, the electron density starts to decrease at time �176 fs for Na cluster and �163 fs for Kr

where ε<sup>0</sup> is the vacuum permittivity and ω is the laser frequency.

first resonance (between laser field and the formed nanoplasma) occurs at

smaller than the external one. The maximum value of the ration *ne*

*<sup>e</sup>*<sup>2</sup> , (17)

*ncrit* is about 3.5 for

cluster, it expands rapidly due to the rapid ionization of the cluster.

nanoplasma.

density given by the formula

*Novel Imaging and Spectroscopy*

the cluster to the 3ncrit resonance.

maximum values of the ratio *ne*

**Figure 2.**

**106**

*Calculated electron density ne normalized by the critical electron density ncrit as a function of time for 2 107 atoms Na and Kr clusters irradiated by an intense femtosecond laser with a pulse duration of 200 fs, wavelength of 390 nm, and peak intensity of 1017 W/cm<sup>2</sup> .*

#### **Figure 4.**

*Calculated time variation of the coulomb and hydrodynamic pressures for Na and Kr clusters (contained 2 107 atoms) irradiated by an intense femtosecond laser with a pulse duration of 200 fs (FWHM), wavelength of 390 nm, and a peak intensity of 1017 W/cm<sup>2</sup> .*

2.0 <sup>10</sup><sup>12</sup> Bar and 9.9 <sup>10</sup><sup>11</sup> Bar for Kr and Na clusters, respectively. The values of the coulomb pressure for both clusters are small compared to the hydrodynamic pressure due to the hot electrons which are 1.5 <sup>10</sup><sup>13</sup> Bar for Kr cluster and 7.5 10<sup>12</sup> Bar for Na clusters. This pressure leads to the increase of the cluster expansion velocity. During the majority of time, the dominant pressure is the hydrodynamic pressure with a little contribution of the coulomb explosion force; then the hydrodynamic pressure is dominant in driving the explosion. It has been noticed that for small diameter clusters, the coulomb explosion force plays a key role [5]. Our study of the coulomb explosion of large Kr and Na clusters shows that the hydrodynamic pressure plays an important role than the coulomb one like the previous experimental results [28].

The coulomb pressure induced by this loss of the plasma neutrality combined with the hydrodynamic pressure leads to the expansion of the nanoplasma. The enhancement of the internal field (**Figure 2**) leads to a very efficient absorption of the laser energy, resulting to the production of high charge states and high energetic ions 4.4 keV for Kr cluster and 2.2 keV for Na cluster (**Figure 5**). Then the combined effect of free streaming and the coulomb and hydrodynamic pressures leads

*Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers*

A modified nanoplasma model has been contributed to examine the cluster explosion dynamics and scaling of ion energies. We found that the excitation of large atomic cluster produces a superheated plasma. We also compared the behavior of the nanoplasma in the Kr rare gas and Na metallic clusters irradiated by an intense femtosecond laser. A similar behavior was found for both cases metallic and rare gas clusters in the time evolution of the different physical parameters (the cluster radius, the electron density, the internal and external electric fields, the pressures, and ion energies). The formation of nanoplasma in the case of Na cluster is rapid than in the case of Kr cluster 255 fs for Na cluster and 230 fs for Kr cluster. We have found that the ion energies in the Kr and Na clusters are 4.4 and 2.2 keV, respectively, which may provide a new ultrahigh energy ion source. The ionization and expansion of the Na metallic cluster are faster than that of the Kr rare gas cluster. The hydrodynamic pressure plays an important role in the interaction with the laser field for both cases (metallic and rare gas clusters); the hydrodynamic pressure was found of order of 10 times than the coulomb pressure; then we conclude that the hydrodynamic pressure is responsible for the dynamic of ioniza-

to the final explosion of the cluster.

*DOI: http://dx.doi.org/10.5772/intechopen.90320*

tion, expansion, and explosion of the clusters.

Boucerredj Noureddine\* and Khaled Beggas

\*Address all correspondence to: boucerredj@yahoo.fr

Mokhtar Annaba University, Algéria

provided the original work is properly cited.

Semiconductor Laboratory, Physics Department, Faculty of Sciences, Badji

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**4. Conclusions**

**Author details**

**109**

When electrons gain energy through inverse Bremsstrahlung collisions, the time evolution of ion energy (**Figure 5**) roughly follows the evolution of the internal field. The increase of the electronic temperature leads to higher ionized states, and electron-free streaming rate increases sharply; then electrons with high energy can leave the cluster (**Figure 6**), as the maximum electron energy is above 3 and 6 keV for Na and Kr clusters, respectively.

**Figure 5.**

*Time evolution of ion energy for Na and Kr clusters. The laser parameters are the same as in Figure 4.*

#### **Figure 6.**

*Total electron energy as a function of time of Na and Kr clusters (contained 2 <sup>10</sup><sup>7</sup> atoms) irradiated by a femtosecond laser with a peak intensity of 1017 W/cm<sup>2</sup> , pulse duration of 200 fs, and wavelength of 390 nm.*

*Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers DOI: http://dx.doi.org/10.5772/intechopen.90320*

The coulomb pressure induced by this loss of the plasma neutrality combined with the hydrodynamic pressure leads to the expansion of the nanoplasma. The enhancement of the internal field (**Figure 2**) leads to a very efficient absorption of the laser energy, resulting to the production of high charge states and high energetic ions 4.4 keV for Kr cluster and 2.2 keV for Na cluster (**Figure 5**). Then the combined effect of free streaming and the coulomb and hydrodynamic pressures leads to the final explosion of the cluster.

## **4. Conclusions**

2.0 <sup>10</sup><sup>12</sup> Bar and 9.9 <sup>10</sup><sup>11</sup> Bar for Kr and Na clusters, respectively. The values of the coulomb pressure for both clusters are small compared to the hydrodynamic pressure due to the hot electrons which are 1.5 <sup>10</sup><sup>13</sup> Bar for Kr cluster and 7.5 10<sup>12</sup> Bar for Na clusters. This pressure leads to the increase of the cluster expansion velocity. During the majority of time, the dominant pressure is the hydrodynamic pressure with a little contribution of the coulomb explosion force; then the hydrodynamic pressure is dominant in driving the explosion. It has been noticed that for small diameter clusters, the coulomb explosion force plays a key role [5]. Our study of the coulomb explosion of large Kr and Na clusters shows that the hydrodynamic pressure plays an important role than the coulomb one like the previous experi-

When electrons gain energy through inverse Bremsstrahlung collisions, the time evolution of ion energy (**Figure 5**) roughly follows the evolution of the internal field. The increase of the electronic temperature leads to higher ionized states, and electron-free streaming rate increases sharply; then electrons with high energy can leave the cluster (**Figure 6**), as the maximum electron energy is above 3 and 6 keV

*Time evolution of ion energy for Na and Kr clusters. The laser parameters are the same as in Figure 4.*

*Total electron energy as a function of time of Na and Kr clusters (contained 2 <sup>10</sup><sup>7</sup> atoms) irradiated by a*

*, pulse duration of 200 fs, and wavelength of 390 nm.*

mental results [28].

*Novel Imaging and Spectroscopy*

**Figure 5.**

**Figure 6.**

**108**

*femtosecond laser with a peak intensity of 1017 W/cm<sup>2</sup>*

for Na and Kr clusters, respectively.

A modified nanoplasma model has been contributed to examine the cluster explosion dynamics and scaling of ion energies. We found that the excitation of large atomic cluster produces a superheated plasma. We also compared the behavior of the nanoplasma in the Kr rare gas and Na metallic clusters irradiated by an intense femtosecond laser. A similar behavior was found for both cases metallic and rare gas clusters in the time evolution of the different physical parameters (the cluster radius, the electron density, the internal and external electric fields, the pressures, and ion energies). The formation of nanoplasma in the case of Na cluster is rapid than in the case of Kr cluster 255 fs for Na cluster and 230 fs for Kr cluster. We have found that the ion energies in the Kr and Na clusters are 4.4 and 2.2 keV, respectively, which may provide a new ultrahigh energy ion source. The ionization and expansion of the Na metallic cluster are faster than that of the Kr rare gas cluster. The hydrodynamic pressure plays an important role in the interaction with the laser field for both cases (metallic and rare gas clusters); the hydrodynamic pressure was found of order of 10 times than the coulomb pressure; then we conclude that the hydrodynamic pressure is responsible for the dynamic of ionization, expansion, and explosion of the clusters.

## **Author details**

Boucerredj Noureddine\* and Khaled Beggas Semiconductor Laboratory, Physics Department, Faculty of Sciences, Badji Mokhtar Annaba University, Algéria

\*Address all correspondence to: boucerredj@yahoo.fr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

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[2] Nishihara K, Hamitani H, Murakami M, Bulanov SV, Esirkepov TZ. High energy ions generated by laser driven coulomb explosion of cluster. Nuclear Instruments and Methods in Physics Research. 2001;**464**:98-102

[3] Perry MD, Mourou G. Science. 1994; **32**:917

[4] Kundu M, Bauer D. Optimizing the ionization and energy absorption of laser-irradiated clusters. Physics of Plasma. 2008;**15**:033303. DOI: 10.1063/ 1.2896578

[5] Ditmire T, Tish JWG, Springate E, et al. Nature (London). 1994;**386**:54

[6] Springate E, Hay N, Tish JWG, Mason MB, Ditmire T, Hutchinson MHR, et al. Explosion of atomic clusters irradiated by high intensity laser pulses: Scaling of ion energies with cluster and laser parameters. Physical Review A. 2000; **61**:063201

[7] McPherson A, Thompson BD, Krainov AB, et al. Nature (London). 1994;**370**:631

[8] Lezius M, Dobosz S, Normand D, Schmidt M. Physical Review Letters. 1998;**80**:261

[9] Ditmire T, Donnelly T, Rubenchikh AM, et al. Physical Review A. 1996;**53**:3379

[10] Ditmire T, Zweiack J, Yonovsky VP, et al. Nature. 1999;**398**:489

[11] Ditmire T, Smith RA, Tish JWG, Hutchinson MHR. High intensity laser absorption by gases of atomic clusters. Physical Review Letters. 1997;**78**(6): 3121-3124

[20] Boucerredj N, Beggas K, Djabri I. Coulomb explosion of Kr clusters irradiated by intense laser field. In: 2nd International Meeting on Materials for Electronic-Applications (IMMEA) May 8–10. Tunisia: Hammamet; 2009

*DOI: http://dx.doi.org/10.5772/intechopen.90320*

*Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers*

[29] Johnston TW, Dawson JM. Correct values for high-frequency power

absorption by inverse bremsstrahlung in plasmas. Physics of Fluids. 1973;**16**:722

[30] Spitzer L. Physics of Fully Ionized Gases. New York: Interscience; 1967

[21] Boucerredj N, Beggas K, Djabri I. Study with modified nanoplasma model the interaction of intense laser with large metallic clusters of Pb. In: International Conference in

Nanomaterial and Renewable Energy (ICNMRE) July 5–8. Morocco: Esafi;

[22] Ammosov MV, Delone NB, Krainov VP. Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field. Soviet Physics - JETP. 1987;**64**:1191

[23] Lotz WZ. Electron-impact

ion emission, simulations and

[25] Landau LD, Lifshitz EM.

Oxford: Pergamon; 1984

[26] Jakhson JD. Classical

and Sons; 1975

4961-4968

**111**

ionization cross-sections and ionization rate coefficients for atoms and ions from hydrogen to calcium. Physica A. 1968;

[24] Tish JWG, Hay N, Mendham KJ, Springate E, Symes DR, Comley AJ, et al. Interaction of intense laser pulses with atomic clusters: Measurements of

applications. Nuclear Instruments and Methods in Physics Research. 2003;**205**:

Electrodynamics of Continuous Media.

Electrodynamics. New York: John Wiley

[27] Djaoui A, Offenberger AA. Heating of underdense plasmas by intense-short pulse lasers. Physical Review E. 1994;**50**:

[28] Polishchuk AY, Vehn JMT. Electron-ion relaxation in a plasma interacting with an intense laser field. Physical Review E. 1994;**49**:663-666

2010

**216**:241

310-323

[12] Heidenreich A, Last I, Jortner J. Nanoplasma dynamics in Xe clusters driven by ultraintense laser fields. European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics. 2008;**46**:195-202

[13] Arbeiter M, Fennel T. Rare-gas clusters in intense VUV XUV and soft X-ray pulses: Signature of the transition from nanoplasma-driven cluster expansion to coulomb explosion in ion and electron spectra. New Journal of Physics. 2011;**13**

[14] Last I. and Jortner J., Microscopic and hydrodynamic impact energy transfer from nanoplasma electrons to ions in exploding clusters, Molecular Physics. UK: Taylor and Francis; 2018; **116**(19-20):2461-2473. DOI: 10.1080/ 00268976.2018.14 78135

[15] Micheau S, Jouin H, Pons B. Modified nanoplasma model for laser cluster interaction. Physical Review A. 2008;**77**:053201

[16] Last I, Schek I, Jortner J. The Journal of Chemical Physics. 1997;**107**:6685

[17] Ditmire T. Simulation of exploding clusters ionized by high intensity femtosecond laser pulses. Physical Review A. 1998;**57**(6):R4094-R4097

[18] Megi F, Belkacem M, Bouchen MA, Suraud E. Zweicknagel, on the importance of damping phenomena in clusters irradiated by intense laser fields. Journal of Physics B: Atomic, Molecular and Optical Physics. 2003;**36**:273-282

[19] Ditmire T, Springate E, Tisch JWG, et al. Explosion of atomic clusters heated by high-intensity femtosecond laser pulse. Physical Review A. 1998;**57**(1):369 *Nanoplasma Formation From Atomic Clusters Irradiated by Intense Femtosecond Lasers DOI: http://dx.doi.org/10.5772/intechopen.90320*

[20] Boucerredj N, Beggas K, Djabri I. Coulomb explosion of Kr clusters irradiated by intense laser field. In: 2nd International Meeting on Materials for Electronic-Applications (IMMEA) May 8–10. Tunisia: Hammamet; 2009

**References**

**32**:917

1.2896578

**61**:063201

1994;**370**:631

1998;**80**:261

**110**

[1] Reinhart PG, Suraud E. Introduction

Physical Review Letters. 1997;**78**(6):

[12] Heidenreich A, Last I, Jortner J. Nanoplasma dynamics in Xe clusters driven by ultraintense laser fields. European Physical Journal D: Atomic, Molecular, Optical and Plasma Physics.

[13] Arbeiter M, Fennel T. Rare-gas clusters in intense VUV XUV and soft X-ray pulses: Signature of the transition

from nanoplasma-driven cluster expansion to coulomb explosion in ion and electron spectra. New Journal of

[14] Last I. and Jortner J., Microscopic and hydrodynamic impact energy transfer from nanoplasma electrons to ions in exploding clusters, Molecular Physics. UK: Taylor and Francis; 2018; **116**(19-20):2461-2473. DOI: 10.1080/

3121-3124

2008;**46**:195-202

Physics. 2011;**13**

00268976.2018.14 78135

2008;**77**:053201

[15] Micheau S, Jouin H, Pons B. Modified nanoplasma model for laser cluster interaction. Physical Review A.

[16] Last I, Schek I, Jortner J. The Journal of Chemical Physics. 1997;**107**:6685

[17] Ditmire T. Simulation of exploding clusters ionized by high intensity femtosecond laser pulses. Physical Review A. 1998;**57**(6):R4094-R4097

[18] Megi F, Belkacem M, Bouchen MA,

importance of damping phenomena in clusters irradiated by intense laser fields. Journal of Physics B: Atomic, Molecular and Optical Physics. 2003;**36**:273-282

[19] Ditmire T, Springate E, Tisch JWG, et al. Explosion of atomic clusters heated by high-intensity femtosecond laser pulse. Physical Review A. 1998;**57**(1):369

Suraud E. Zweicknagel, on the

Instruments and Methods in Physics

[3] Perry MD, Mourou G. Science. 1994;

[4] Kundu M, Bauer D. Optimizing the ionization and energy absorption of laser-irradiated clusters. Physics of Plasma. 2008;**15**:033303. DOI: 10.1063/

[5] Ditmire T, Tish JWG, Springate E, et al. Nature (London). 1994;**386**:54

[6] Springate E, Hay N, Tish JWG,

Hutchinson MHR, et al. Explosion of atomic clusters irradiated by high intensity laser pulses: Scaling of ion energies with cluster and laser

parameters. Physical Review A. 2000;

[7] McPherson A, Thompson BD, Krainov AB, et al. Nature (London).

[8] Lezius M, Dobosz S, Normand D, Schmidt M. Physical Review Letters.

[10] Ditmire T, Zweiack J, Yonovsky VP,

[11] Ditmire T, Smith RA, Tish JWG, Hutchinson MHR. High intensity laser absorption by gases of atomic clusters.

[9] Ditmire T, Donnelly T, Rubenchikh AM, et al. Physical

et al. Nature. 1999;**398**:489

Review A. 1996;**53**:3379

Mason MB, Ditmire T,

to Cluster Dynamics. Germany: Wiley-VCH Verlag Weinheim; 2004

*Novel Imaging and Spectroscopy*

[2] Nishihara K, Hamitani H, Murakami M, Bulanov SV, Esirkepov TZ. High energy ions generated by laser driven coulomb explosion of cluster. Nuclear

Research. 2001;**464**:98-102

[21] Boucerredj N, Beggas K, Djabri I. Study with modified nanoplasma model the interaction of intense laser with large metallic clusters of Pb. In: International Conference in Nanomaterial and Renewable Energy (ICNMRE) July 5–8. Morocco: Esafi; 2010

[22] Ammosov MV, Delone NB, Krainov VP. Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field. Soviet Physics - JETP. 1987;**64**:1191

[23] Lotz WZ. Electron-impact ionization cross-sections and ionization rate coefficients for atoms and ions from hydrogen to calcium. Physica A. 1968; **216**:241

[24] Tish JWG, Hay N, Mendham KJ, Springate E, Symes DR, Comley AJ, et al. Interaction of intense laser pulses with atomic clusters: Measurements of ion emission, simulations and applications. Nuclear Instruments and Methods in Physics Research. 2003;**205**: 310-323

[25] Landau LD, Lifshitz EM. Electrodynamics of Continuous Media. Oxford: Pergamon; 1984

[26] Jakhson JD. Classical Electrodynamics. New York: John Wiley and Sons; 1975

[27] Djaoui A, Offenberger AA. Heating of underdense plasmas by intense-short pulse lasers. Physical Review E. 1994;**50**: 4961-4968

[28] Polishchuk AY, Vehn JMT. Electron-ion relaxation in a plasma interacting with an intense laser field. Physical Review E. 1994;**49**:663-666

[29] Johnston TW, Dawson JM. Correct values for high-frequency power absorption by inverse bremsstrahlung in plasmas. Physics of Fluids. 1973;**16**:722

[30] Spitzer L. Physics of Fully Ionized Gases. New York: Interscience; 1967

**113**

**Chapter 8**

**Abstract**

**1. Introduction**

mations and tumors.

of certain contraindications.

*Haider N. Al-Tameemi*

Imaging in Low Back Pain

Medical imaging plays an important role in the evaluation of low back pain (LBP). The choice of certain radiological method over other depends on many factors like patient's presentation, presence of contraindication, availability, relative cost of the test, and the expected impact of the results on management. Radiological evaluation helps the physician reach the most likely cause of LBP, confirm the provisional diagnosis, provide alternative one, or narrow the differential diagnosis. Plain X-ray radiograph is useful in initial general assessment. Magnetic resonance imaging (MRI) is the imaging modality of choice in the evaluation of LBP because of elegant demonstration of anatomical details and many pathologies. Computerized tomography (CT) can provide high-resolution images of the bony structures and is particularly invaluable in trauma. Other imaging modalities are rarely used usually as problem-solving or in selected conditions. For example, sonography may have a role in the evaluation of soft tissue lesions and the sacroiliac joints. Angiography is useful for vascular evaluation. Isotope imaging may be used in the elucidation of of hidden cause of pain (tumors or fracture). Conventional myelography and discography are virtually obsolete in current clinical practice because of the presence of much safer and accurate new modalities. Finally, inter-

ventional radiology has an increasing role in treating certain conditions.

**Keywords:** imaging, radiology, X-ray, MRI, CT scan, sonography, angiography

The main role of imaging in patients with low back pain (LBP) is to help physicians reach the most likely cause of the pain. The use of one or more of the different radiological investigations in a suitable manner will enhance the detection of the underlying cause of the LBP in a timely way. Moreover, radiology has a role not only in the diagnosis but also in the treatment of some conditions that lead to LBP. The rapidly advancing interventional radiology is increasingly utilized as an adjuvant or sole therapeutic option of a variety of conditions like vascular malfor-

There are many imaging modalities that can be used in the evaluation of LBP. Selection of the appropriate modality depends on different factors, like the patient's conditions, clinical state, availability, and cost of the test and the presence

with the most to the least commonly and widely used ones.

We will discuss each imaging modality from different points of view starting

## **Chapter 8**
