Over-the-barrier ionization of H2O by intense ultrashort laser pulses

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application of the ultrashort laser pulses is in eye surgery, which allows to correct the near- and farsightedness by reshaping the cornea [6-7]. In the point of view ...
Over-the-barrier ionization of H2O by intense ultrashort laser pulses S. Borbélya, K.Tőkésib, L. Nagya and D. G. Arbóc a

Faculty of Physics, Babeș-Bolyai University, str. Kogălniceanu nr. 1, 400084 Cluj-Napoca, Romania b Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), P. O. Box 51, H-4001 Debrecen, Hungary c Institute for Astronomy and Space Physics, IAFE, CC 67, Suc. 28 (1428) Buenos Aires, Argentina Abstract. The ionization of the H2O molecule in intense ultrashort electric fields was studied theoretically. Ionization probability densities were calculated using classical and quantum mechanical approaches. The water molecule is treated within the hydrogenic approximation. Classical calculations were carried out using the classical trajectory Monte Carlo method, while quantum mechanical calculations are based on the sudden strong field approximation. A good agreement between classical and quantum calculations was found. Keywords: H2O, ionization, ultrashort pulses PACS: 32.80.Fb, 42.50.Hz, 33.20.Xx

INTRODUCTION In the last years advanced laser facilities have achieved intensities of the order of 1015 W/cm2 and pulse lengths of the order of 10 fs, which corresponds to few cycles of an oscillating electric field of 800 nm wavelength [1-2]. In the last years research activities have turned for investigations of the interaction between such short and strong pulses with matter. These processes are of considerable interest for both basic and applied science. From the fundamental point of view they might broaden our general understanding of the dynamics of atomic processes for laser-matter interactions. From the application point of view they can help us to find a way of controlling ultrashort quantum processes, which are important in a number of applications like in laser-driven fusion [3], in plasma heating [4], in controlling chemical reactions [6] and in the development of fast optical devices. With the possibility of fine manipulation of these ultrashort pulses they also became a very useful tool in various medical research and application. The ultrashort laser pulses can deliver a big amount of energy inside a living tissue into a very confined space, which makes possible the manipulation and destruction of tissues. One of the earliest and most successful application of the ultrashort laser pulses is in eye surgery, which allows to correct the near- and farsightedness by reshaping the cornea [6-7]. In the point of view of the fundamental medical research ultrashort laser pulses allow in vivo manipulation of living tissues and organisms at cellular level, like the disruption of the development in fly embryos [8], disrupting the neural pathways in C. elegans [9], subcellular nanosurgery of cell cytoskeletons [10] and of mitochondria [11], exciting neurons in brain slice [12], microstroke in rodents [13], etc. In order to achieve a better control over these processes a detailed understanding of the interaction between the human tissue and ultrashort laser pulses is needed. A major component (about 65%) of the human tissue is the H2O. Most of the processes induced by the ultrashort laser pulses in biological tissues can be explained based on the investigation of the interaction between the water molecules and the laser field. It was shown experimentally [14] that during the interaction the first step is the ionization of H2O followed by the Coulomb explosion of the remaining molecular ion. To our best knowledge there is no theoretical work, which describes the ionization of the H2O by ultrashort laser pulses in details. Several papers [15-18] presented the ionization cross sections for charged particle impact. According to these studies, it was shown that the simple methods like classical trajectory Monte-Carlo (CTMC) can provide reasonably accurate results within the hydrogenic approximation [15]. In the present work we studied the

ionization of the H2O molecule by intense half-cycle laser pulses in the over-the barrier regime applying also the hydrogenic approximation. We use classical (CTMC) and quantum mechanical (Volkov) models. Atomic units are used throughout the calculations.

THEORY In our investigations the single active electron approximation is used, and only the electrons from the two highest occupied molecular orbitals (HOMO) were taken into account. The 1B1 orbital has an ionization potential of 12.6 eV [15] and it has a predominantly pz2 character, while the 3A1 orbital has an ionization potential of 14.7 eV [15]. In the hydrogenic approximation the active electron is moving in a one center Coulomb potential generated by the core with effective charge of Zeff. The linearly polarized laser pulse is represented by its electric component ̂

sin

0                

2

2

sin

if

0,

, elsewhere

(1)

where is the polarization vector, is the frequency of the carrier wave and is the pulse duration. Here the dipole is neglected in approximation is implicitly applied, when the spatial dependence of the external electric field Eq.(1).

Quantum Mechanical Calculations Our quantum mechanical calculations are based on the approximate solution of the time dependent Schrödinger equation (TDSE) for a one active electron system in an external laser field: ̂ 2

Ψ

Ψ

,

(2)

where is the Coulomb potential between the active electron and the rest of the system and Ψ is the time dependent wave function of the system. In the present approach the wave function is searched in the following form ,

Ψ

ΨV

,

,

(3)

where ΨV , are the Volkov wave functions. The Volkov wave functions are the solution of the TDSE in dipole approximation for a free charged particle in a radiation field and they can be expressed as ΨV

,

exp

exp

2

.

(4)

In the above expression (5) is the vector potential of the electromagnetic field. By substituting the time dependent wave function from Eq. 3 into the TDSE given by Eq. 2 we obtained the following expression for the expansion coefficients: ,

2

 

, exp

2

2

2

The initial condition for Eq. 6 can be obtained from the continuity of the wave function at

 

exp 0 as follows

(6)

,

0

1 2

|

exp

,

where is the initial state. The transition probability from initial state defined momentum is given as (see for more details [19]) ,

2

(7) to a free final state

with a well

(8)

.

The simplest way of solving Eq. 6 is by neglecting the Coulomb potential between the active electron and the rest of the system ( 0). This zero order solution of Eq. 6, also called Volkov or sudden strong field approximation [20], provides good results only for the over-the-barrier ionization, at high laser field intensities with high momentum transfer and small duration. Using the initial condition given by Eq. 7 the Volkov model has the , 0 expansion coefficients. Taking following analytical solution, which can be expressed by the into account the character of the 1B1 and 3A1 orbitals we use 2p hydrogen-type atomic wave functions for the initial state. The effective charge of the hydrogen-type core was set in such way that the ionization energy for 2p electrons matches with the binding energy of 1B1 and 3A1 electrons i.e.,   1.92 and 2.07 for the 1B1 and 3A1 orbitals, respectively.

Classical Calculations In the present approach, Newton’s classical non-relativistic equations of motion are solved [21-22] numerically when an external field given by Eq. (1) is included. The initial position and momentum vectors of the active electrons are distributed according to the microcannonical distribution, ,

~

,

2

(9)

is the Coulomb potential with 1.6 [15] for both 1B1 and 3A1 where is the ionization potential and orbitals. For these initial parameters, the equations of motion were integrated with respect to time by standard Runge-Kutta method until the real exit channels were obtained. In this work the total number of primary histories was 500 000 for each collisions system. The single- and double-differential ionization probabilities ( ) were computed with the following formulas:



(10)

,

(11)

Δ ∆ The standard deviation for differential probabilities is defined through /

Δ

.

(12)

In Eqs. (11) – (14), is the total number of trajectories calculated for a given collision system and is the number of trajectories that satisfy the criteria for the ionization under consideration in the energy interval Δ or in the perpendicular (Δk ) and parallel Δk ) momentum intervals of the electron.

RESULTS AND DISCUSSIONS Calculations were performed for H2O molecule applying the hydrogenic approximation and using laser pulses with 1 a.u. The carrier wave frequency is 0.05 a.u., duration of 3 a.u., 5 a.u. and 10 a.u. at laser field intensity

which is close to the carrier wave frequency generated by the Ti-sapphire lasers. These pulse parameters lead to the value of 0.05 for the Keldysh parameter, which is a characteristic value for the over-the-barrier ionization regime. The double differential momentum distributions of electrons ejected from the H2O molecule are calculated for the 1B1 and 3A1 initial states using CTMC and Volkov models. They are presented on Fig. 1. as a function of the momentum components ( - parallel with the polarization vector ̂; - perpendicular to the polarization vector ̂) of the ejected electrons. At first sight both models predict the same probability densities. In each case the electrons are ejected with maximum probability with momentum value , which is the momentum gain from the external laser field. However, important differences can be observed in the maxima of the ionization probabilities. For the CTMC the maxima of the predicted probability densities are shifted toward smaller momenta. This shift is caused by the Coulomb attraction of the core, which was not taken into account in the Volkov model during and after the ionization. In the classical picture the electrons are decelerated by the Coulomb attraction. The obtained probability densities have a cylindrical symmetry around the polarization vector because of the used spherically symmetric hydrogenic model of the H2O, and do not depend on the orientation of the molecule. This would not be true, if a correct description of the molecule would be applied. Based on these arguments (strong approximation for the initial state and neglecting the multicenter character of the Coulomb potential) we note that the obtained angular resolved probability densities are not precise within the hydrogenic approximation. These deficiencies of the present approaches will be improved in a future paper by using more precise initial state wave functions, and by using a higher order solution of Eq. 6.

FIGURE 1. (color online) Two-dimensional ionization probability density as a function of the parallel and perpendicular momentum of the ejected electrons for ω = 0.05 a.u., τ = 3 a.u. and E0 = 1 a.u. First column: 1B1 orbital, Second column: 3A1 orbital. First row: CTMC. Second row: VOLKOV.

0.8

τ = 3 a.u.

C TM C - 1B 1 C TM C - 3A 1 VO LK O V - 1B 1 VO LK O V - 3A 1

0.7 0.6

dP/dE

0.5 0.4 0.3 0.2 0.1 0.0

0

1

2

3

4

5

Electron energy [a.u.] 0.5

τ = 5 a.u.

C TM C - 1B 1 C TM C - 3A 1 VO LK O V - 1B 1 VO LK O V - 3A 1

0.4

dP/dE

0.3

0.2

0.1

0.0

0

2

4

6

8

Electron energy [a.u.] 0 .2 4

0 .1 6

dP/dE

τ = 10 a.u.

C TM C - 1B 1 C TM C - 3A 1 VO LK O V - 1B 1 VO LK O V - 3A 1

0 .2 0

0 .1 2

0 .0 8

0 .0 4

0 .0 0

0

5

10

15

20

Electron energy [a.u.] FIGURE 2. (color online) Ionization probability densities as a function of electron energies for at different pulse durations.

= 0.05 a.u. and

= 1 a.u.

Single differential ionization probability as a function of the electron energy is plotted on Fig. 2. Here the shift of the maxima toward smaller energies in the CTMC probability density can be clearly seen. Beside this shift the probability densities obtained using CTMC and Volkov models are good agreement at high momentum transfer. In both models the 3A1 probability densities are wider than the 1B1 probability densities, and in the case of the CTMC results the 3A1 probabilities have their maxima at lower energies according to the higher ionization potential.

CONCLUSIONS The ionization of the H2O by intense ultrashort pulses in the over-the-barrier regime was theoretically studied. Within the hydrogenic approximation a good agreement between CTMC and Volkov results was found. It was showed that the hydrogenic approximation describes the angular resolved ionization probabilities inaccurately, however this imprecision will not appear in the angular integrated probability densities. At high momentum transfer we found that beside the shift caused by the neglect of the Coulomb potential in the case of Volkov model the agreement between the CTMC and Volkov calculations is good. In both models the 3A1 probability densities are wider than the 1A1 probability densities.

ACKNOWLEDGEMENTS This work was supported by the Romanian Academy of Sciences (grant no. 31/2008), the Romanian National Plan for Research (PNII) under contract No. ID 539, the European COST Action CM0702, the Hungarian-Argentine collaboration PA05-EIII/007, the Hungarian National Office for Research and Technology, the grant “Bolyai” from the Hungarian Academy of Sciences, and the Hungarian Research Found OTKA (K72172).

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