1. INTRODUCTION
The exposure of large clusters comprising several thousand atoms or
molecules to the field of a short intense laser pulse (∼100 fs long
or some 30 periods of laser field) produces highly excited matter
(Sullivan et al., 1996; Boehly et al., 1997). The heating of
conduction electrons in the case of metal clusters or primary ionized
electrons in the case of clusters of inert gas atoms on the one hand
and the absence of a fast heat sink mechanism like that in an ordinary
plasma on the hand allow the achievement of a state of much greater
excitation of the electron subsystem as compared with the excitation of
isolated atoms and molecules. In this case, atomic ions remain
practically unheated. Following the fast initial multiple ionization,
for the rest of the duration of the laser pulse, the matter of the
cluster ion is an ideal plasma composed of electrons and multiple
charged atomic ions.
There are different mechanisms of expansion pertaining to clusters of
noble gases when exposed to a laser pulse (Lezius
et al., 1998). The energy distribution of electrons in
the exploding cluster includes electrons with kinetic energies up to 30
keV, which is several orders of magnitude greater than the energies
observed in the case of above-threshold ionization of individual atoms
or molecules (Delone & Krainov, 2000).
Experimental results are adequately explained by a theoretical model
that regards the cluster as a small plasma sphere. The cluster is
ionized by the laser field and internal collisions and then starts to
expand. As the cluster expands, the electron concentration decreases
and at the some point in time the frequency of the laser field comes
into resonance with the frequency of the surface Mie resonance. This
leads to a rapid transfer of electromagnetic (EM) energy to the
electrons, causing a sharp peak in the electron temperature. At this
instant the atomic ions of the cluster are vigorously stripped to high
charge numbers by the hot electrons and the cluster explodes. Such
collective phenomena are very important in the case of a cluster, while
being absent in the event of the interaction of a laser field with
individual atoms.
We consider here the models of the cluster plasma, which allows us to
analyze properties of this system in the process of the production of
the dense plasma by a laser pulse. These models describe the heating of
the cluster when the average electron energy (i.e., electron
temperature T) increases to a few kiloelectron volts and tens
of kiloelectron volts, and the collision processes dominate. The
electron–electron collisions establish the Maxwell distribution
with an electron temperature T that grows with time. Depending
on the laser intensity, the classical thermal electron energy
3T/2 can be larger or smaller than the average
ponderomotive electron energy Up =
F2/4ω2, where F and
ω are the laser field strength amplitude and laser frequency,
respectively. Here and below the atomic system of units e =
me = ħ = 1 is used. The electron
temperature T increases until the cluster ion begins
significantly to expand.
Besides the heating, some electrons are ejected from the cluster by
the laser field by means of the thermal evaporation from the surface of
the cluster and by other ways (see below). Accordingly, the charge
Z′ of the cluster ion increases with time during the
laser pulse. We find a simple relation between the growth of the
electron temperature T and the cluster ion charge
Z′.
Simple estimates confirm that the cluster plasma is ideal. The
condition for being an ideal plasma is NZ3 <<
T3, where N is the concentration of atoms
in the cluster, Z is the mean charge of the typical
multicharged atomic ion inside the cluster, and T is the
electron temperature.
This work is organized in three sections. In Section 2 the different
ionization mechanisms are studied and in Section 3 a summary and
conclusion is presented.
2. IONIZATION OF CLUSTERS
2.1. Internal ionization
For our analysis, we take a cluster consisting of n =
105 atoms. The kind of atoms is really unimportant at the
irradiation by intense laser pulse. Let us choose Na atoms (Z
= 11) and a linearly polarized laser pulse with the peak field
intensity of I = 1015 Wcm−2, light
frequency ω = 1 eV, and the pulse duration τ = 50 fs. Such a
cluster has the radius R = 100 × 10−8
cm.
At first, the laser pulse induces a dipole excitation that causes the
center of mass of the valance electron cloud to oscillate about the
ionic background. The oscillations are dominated by the Mie surface
plasmon whose period for Na clusters is about 1.5 fs. However, very
soon these collective oscillations get perturbed because of the cluster
atoms ionozation. The latter process is analogous to Landau damping,
that is, the attenuation of the plasmon in an electron gas. A typical
relaxation time due to Landau damping is 10 fs, roughly valid for
clusters with n = 50–1000 atoms. It shrinks again
∼n−1/3 for very large values of
n.
We shall assume that the collisions inside the cluster are high
enough that no thermal or other gradients exist within the cluster. The
electron temperature of the cluster is high during the laser pulse, of
the order of T = 1 keV and uniform due to high thermal
conductivity; the electron distribution is isotropie and described by a
classical Maxwell distribution. Here,
is the electron thermal velocity and vF =
F/w is the field velocity. The Maxwell
distribution is valid in the region of a weak laser field
and in the region vF >
ZvT of a very strong laser field (Chichkov et al., 1992).
The heating of an electron subsystem occurs mainly due to the elastic
electron–ion collisions (inverse bremsstrahlung) and because of
other processes as well. We will see that the atoms in the cluster are
fully ionized under such conditions due to the inelastic
electron–ion collisions. Then, the electron concentration is
Ne = ZN where the atomic
concentration for liquid Na metal is N = 2.44 ×
1022 cm−3.
The thermal equilibrium is realized because time
τee of electron–electron collisions is
small compared to the duration τ of the laser pulse. This time can
be estimated as (Silin, 1965)
Here, ln Λ ∼ 10 is the typical value of the Coulomb
logarithm. These collisions provide a further damping mechanism that
drives the electron cloud toward thermal equilibrium.
We can assume that the cluster is a small plasma sphere. Such an
assumption is reasonable as long as the cluster radius R is
large compared with the Debye radius rD,
that is, R >> rD, where the
Debye screening length at T = 1 keV is
Let us first consider the internal ionization, that is, the process
when electrons are removed from their host atoms but remain bound to
the cluster. In the clusters, the inner ionization is produced by
inelastic electron–ion collisions. The equilibrium concentration
NZ of different atomic ions with the
charge Z is given by the statistical Saha equation (when the
equilibrium is established only by collisions; Landau & Lifshitz, 1982):
Here, EZ−1 is the ionization
potential for the atomic ion with the charge Z − 1.
The condition of the total ionization of all atomic shells can be
approximated in the form NZ =
NZ−1 = N/2, where
Z is now the nuclear charge (Z = 11 in the case of
sodium atoms). Then, the hydrogen-like ionization potential
EZ−1 ≈
Z2/2, and the condition of total ionization of
an atom is of the form (Ne =
ZN)
In the case of Na atoms, we obtain from this equation T* =
0.34 keV. Hence, at the electron temperature T = 1 keV, we
have bare Na nuclei and electrons only. Here, T* is the
temperature required for complete internal ionization. In addition to
the inelastic electron–ion collisions, the internal ionization
may also be produced due to the above-barrier ionization of atomic ions
inside the cluster by an intense laser pulse (see below).
The results for the production of highly charged atomic states (e.g.,
Xe20+, Kr8+) resulting from the interaction of
intense laser fields (up to ∼ 1015
Wcm−2) with atomic clusters are given (Snyder et al., 1996).
2.2. Ionization at the inelastic electron–ion
collisions
Total internal ionization inside the cluster considered in the
previous subsection requires some time for its realization. This time
should be less than the duration of the ultrashort laser pulse. Let us
make in this subsection some estimates. The multiple ionization occurs
due to the inelastic collisions of hot electrons with multicharged
ions. The cross section for this process is given by the well-known
simple semiempirical Lotz formula (Landau &
Lifshitz, 1984):
Here, E is the kinetic energy of the incident electron,
Ez is the ionization potential of the
atomic ion, and q is the number of electrons in the atomic
shell being ionized. We use further the value of
Ez = Z2/2 for
hydrogen-like ions. Hence, the rate of the process in question is
obtained from Eq. (5) in the form
Here, v is the velocity of the incident electron,
Ne is the electron concentration, and
N is the concentration of atoms in the cluster (i.e., of bare
nuclei in the considered case of the total ionization). The electron
velocity has two components: the thermal velocity
and the induced field velocity F/ω. In more typical case
,
we should average Eq. (6) over a Maxwell distribution. Assuming T
<< Z2/2, we find
In the opposite limit
,
we can simply substitute v = F/ω into Eq. (6):
Naturally, we assumed here that the impact ionization takes place in
the vicinity of the maximum of the alternating field strength
F cos ωt. It should be noted that in this limit,
the ionization has a threshold, and the condition F/ω
> Z should be fulfilled. This condition is stronger than
the above condition
.
In the above example of T = 1 keV, laser intensity
I = 1015 Wcm−2 (the field strength
F = 0.166 a.u.), laser frequency ω = 1 eV, we find that
the ratio
.
But the value of F/ω = 4.5 < Z =
11; hence, below a threshold we have the only mechanism of thermal
ionization which is determined by Eq. (7). For total ionization of the
Na cluster from Eq. (7) we get w = 0.22 fs−1.
Thus, electron–ion collisions occur very often during the
ultrashort laser pulse.
2.3. Penetration of the laser field inside the cluster
The electromagnetic field of the laser radiation penetrates through
the cluster and is damped within it. The depth of penetration δ,
connected with the excitation of plasma oscillations, can be identified
with the imaginary part of the wavelength inside the cluster (Krainov, 2000):
where the dielectric permittivity is
In our case of the Na cluster for the visible light we have
|ε| = 370 and δ = 100 ×
10−8 cm, implying that this part of damping of the
laser radiation within the cluster is small. We conclude that plasma
oscillations cannot be excited as a rule (except for resonance
conditions).
Another part of the damping is connected with the electron–ion
collisions. In τ approximation, the stationary kinetic equation for
this process is of the form (at ω <<
νei)
Here, f′ is the small perturbation to the equilibrium
Maxwell distribution function f0 for electrons, and
F and ω are the laser field strength amplitude and laser
frequency, respectively. We have directed the field strength F
along the axis x (the laser radiation is assumed to be
linearly polarized). The rate νe of the elastic
electron–ion collisions is of the well-known form (Landau & Lifshitz, 1982; see also Eq. (1))
It is seen that this quantity is on order of the laser frequency
ω ∼ 2 fs−1.
It should be noted that this well-known expression is valid for rapid
electrons (T >> Z2). In the case of
slow electrons (T << Z2), the
situation is more complex (Silin, 1965). This
expression is valid under the condition Zω <<
T3/2 only. It corresponds to the scattering of
electrons on small angles. In the opposite case, Zω
>> T3/2, the rate of electron–ion
collisions depends on the laser frequency:
In this limit, the scattering of electrons on large angles is
important, and therefore the Coulomb logarithm is absent.
We derive first the averaged energy that an electron obtains from the
laser field per unit time at the condition ω <<
νei:
We substituted here 〈cos2 ωt〉 =
1/2. Deriving the simple integral with the Maxwell distribution
f0, we find the power density
Pa = NedE/dt absorbed per unit of the plasma volume:
The intensity of the laser radiation I =
cF2/8π decreases with the depth z
of its penetration inside the cluster. The corresponding equation is of
the form
Its solution is
where the depth l of penetration of laser radiation inside
the cluster is
In the case of an Na cluster and a temperature of T = 1 keV,
this depth is l = 1.7 × 10−8 cm. This
quantity is very small compared to the cluster radius R.
In the opposite limit, ω >> νei,
the power density that is absorbed within the cluster per unit of the
plasma volume is given by the well-known relation of the induced
bremsstrahlung at the electron–ion collisions (Fedorov, 1997):
It should be noted that the quantity
F2/ω2 is the energy that an
electron obtains from the electromagnetic field at each collision with
an ion. In this case, the penetration length of laser radiation inside
the cluster is
Thus, it is seen that in both limits, ω <<
νei and ω >>
νei, the penetration length of laser radiation
is very small compared to the size of the cluster. These results are
valid only for homogeneous cluster plasma.
Multiple internal ionization can also be produced by the conduction
electrons near the surface of the cluster which from the
electromagnetic field, obtain the energy of order of the ponderomotive
energy Up =
F2/4ω2. This mechanism is
effective in the case of long wavelength laser radiation (LaGattuta, 1998). Hence, we can conclude that the
laser radiation does not penetrate deep inside the cluster and is
absorbed in a thin layer on the surface of the cluster. Of course, most
of the laser radiation is reflected from the cluster. Absorption is
caused by the effective electron–ion collisions.
2.4. External ionization
Let us now consider the ionization of the cluster, that is, the
removal of electrons to infinity with the formation of a cluster ion.
In the case of the metallic clusters there are conduction electrons
inside the cluster. The first mechanism is the field (cold) ionization
of electrons from the surface of the cluster. This process ends when
the attraction of an electron to the cluster ion will be stronger than
the force from the external laser field. Thus, we find, according to
the Coulomb law,
Here, Z′ is the charge of the cluster ion. In the case
of laser intensity I = 1015 W/cm2
(field strength F = 8.5 × 108
Vcm−1) and R = 215 ×
10−8 cm, we find Z′ = 27,200. The
number of electrons released by means of this mechanism is very small
compared with total number of electrons in the hot sodium cluster (the
latter is Zn = 1.1 × 107). The mechanism of a
Xe cluster ionization with N = 1100 atoms in a laser field of
1016 Wcm−2 was studied in a one-electron,
one-dimensional approximation (Last & Joyner,
1998). The initial above-barrier ionization is realized up to
the total charge of Z′ = 1200. Then, another ionization
mechanism has to be responsible for the further loss of electrons. The
equilibrium state of the resulting cluster ion may be regarded as a
conducting sphere whose charge is located in a thin surface layer. The
strong Coulomb field in this layer knocks out additional electrons from
the atomic ions toward the cluster center. These electrons are
accelerated to a high velocity and then leave the cluster. The
Z′ = 2600 ionization becomes possible. This is the
so-called ignition mechanism of ionization (Rose-Petruck et al., 1997). For example,
the field at the surface of a cluster of 25 Ne atoms when all of the
atoms are singly ionized is on the order of F = 5 ×
1012 Vcm−1. This large field lowers the
ionization barrier and enables subsequent ionization events to occur,
which in turn further increases the field and lowers the ionization
barrier. Hence, the fields created by the initial events
“ignite” the cluster to undergo further ionization.
In the case of an inert gas cluster, the situation is rather similar
to that of the metallic clusters. Of course, no conduction electrons
exist within the cluster before the laser pulse. But the
barrier-suppression ionization of the rare gas atoms occurs quickly at
the surface of the cluster. The Bethe condition for this ionization is
(Bethe & Salpeter, 1977; Brunner, 1997)
where Ez is the atomic ionization
potential of an atomic ion and Z is the charge of this ion.
For example, for Ne atoms (Z = 10), we find
E1 = 21.5 eV, and single ionization of all atoms in
the cluster takes place when the field strength reaches F =
8.0 × 108 Vcm−1. Hence, such
ionization takes place very quickly (during the atomic time ∼0.01
fs). However, according to Eq. (22), the barrier-suppression ionization
of the resulting Ne+ ion by such a field is impossible,
because the ionization potential of this ion is E1
= 41.0 eV, and the required field strength should be F = 1.5
× 109 Vcm−1. Of course, the tunneling
ionization is possible in the latter case, but it takes a very long
time.
According to Eqs. (21) and (22) we find the external ionization
degree Z′ of the cluster through the ignition mechanism.
Eliminating the field strength F from these, one can find
relations:
In the example of the Xe cluster with 1000 atoms, we obtain
R = 25 × 10−8 cm and for total double
ionization of the Xe atoms we have Z′ = 330.
The next mechanism for the electron emission from the surface of the
cluster is the usual thermoemission. We use the well-known
Richardson–Dushman formula for the current density of the
thermoemission from the spherical surface of the cluster (Neuman, 1987; Hoyaux,
1968) in atomic units:
Here, A0 = 1/2π2 is the
Richardson parameter. The quantity
JZ′ =
Z′/R is the ionization potential for the
cluster ion with the charge Z′ >> 1, when the work
function may be disregarded.
Integrating Eq. (24) over the laser pulse τ = 50 fs, we obtain
the final charge Z′ of the cluster ion
(JZ′ >> T):
In the above case of the Na cluster, we find Z′ =
240,000. Thus, the thermoemission is a much more effective way to
ionize clusters than the field (cold) ionization. With a field
intensity I = 1015 Wcm−2, we find
that nearly 270,000 electrons are ejected from the Na cluster with
106 atoms (1.1 × 107 electrons) during the
laser pulse. This shows that the ionization degree of the cluster is
very low.
3. SUMMARY AND CONCLUSION
The present article deals with the large clusters exposed to intense
ultrashort laser pulses. The cluster is excited because of the
interaction of the electron subsystem with the field of the laser
pulse. Both experimental and theoretical studies of the evolution of
clusters exposed to the intense ultrashort laser pulse are currently
still at their initial stage.
Experimental results are adequately explained by a theoretical model
that considers the cluster as a small plasma sphere. The cluster is
ionized by the laser field and internal collisions, and then starts to
expand. As the cluster expands, the electron concentration decreases,
and at a special time the frequency of the laser field comes into
resonance with the frequency of the surface Mie resonance. This leads
to a rapid transfer of EM energy to the electrons, causing a sharp peak
in the electron temperature. At this moment, the atomic ions of the
cluster are stripped to high charge numbers by the hot electrons and
the cluster explodes. Such collective phenomena are very important in
the case of a cluster, while being absent in the event of the
interaction of a laser field with individual atoms.
The laser field does not penetrate into the cluster, being absorbed
on its surface. When the cluster explodes, however, its density is much
reduced and more of the laser radiation gets into the cluster. This
leads to additional internal ionization in the expanding cluster. The
explosion reveals the shell structure of the clusters, so the separate
layers of the cluster explode in succession. The first ions have rather
large kinetic energies because of Coulomb repulsion and the efficient
energy transfer from hot electrons to the outer layer of the
cluster.
Different mechanisms, responsible for cluster ionization, were
investigated. It was found that the ionization of large clusters,
irradiated by a strong intense ultrashort laser pulse, is realized by
means of the surface thermoemission. Furthermore, it was proved that
the degree of ionization is low. Moreover, electron evaporation from
the hot cluster surface was treated by the well-known Richardson
formula, disregarding the cluster charging in the course of
evaporation. In addition, the role of the cluster ion surface in the
electron heating was studied and it was shown that elastic reflection
of electrons takes place due to the large charge of the cluster ion,
caused by the strong external ionization. Further analysis showed that
the plasma oscillation cannot be excited as a rule except for resonance
conditions. Moreover, taking into account ionization mechanisms, it was
concluded that the laser radiation does not penetrate deep inside the
cluster due to the strong absorption at its surface in a thin layer.
This means that electrons absorb the electromagnetic energy in the
course of the elastic collisions with ions.
