We studied theoretically the influence of relativistic effects on the energy distribution of electrons in the tunneling ionization of atoms by a field of linearly polarized super-intense laser radiation. It was shown that the energy distribution of ejected electrons is determined by relativistic law though the electron kinetic energy can be less than its rest energy. The relativistic probability of ionization along the field strength decreases exponentially with the electron kinetic energy, but more quickly than in the non-relativistic case.
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It is well known that, in the adiabatic approximation of non-relativistic quantum mechanics, the ionization rate can be calculated according to the Landau-Dykhne approach :
Here Ei is the binding energy of the initial atomic state, and Ef(t) is the energy of the final continuum state taking into account the field of laser radiation. Here and below, the atomic system of units is used, as a rule, where e = me = ħ = 1 and the speed of light is c = 137.02 . Finally, Ef(t) = -Ei.
Adiabatic approximation is valid when the energy of laser photon is small, i.e. ħω << Ei. Eq. (1) is correct as well as in the relativistic theory, since the classical action S = Pidxi is relativistic invariant quantity, and the coordinate part of this action influences only the pre-exponential factor in the ionization rate, Eq. (1). We neglect this factor both in the non-relativistic energy distribution of ejected electrons  and in the relativistic approach. Thus, we can apply Eq. (1) in the relativistic case using the relativistic expression for the energy Ef(t) of a free electron in the field of laser radiation.
Here F and ω are the amplitude and the frequency of laser field, respectively. We studied the influence of relativistic effects on the angular distribution of ejected electrons in the field of laser radiation in Refs. [5,6].
Now we apply this approach to derive the relativistic energy distribution of ejected electrons in the field of linearly polarized low-frequency laser radiation. We restrict ourselves by the case of electron ejection along the electric field strength vector only, since the most part of electrons are emitted along this direction in a strong laser field. Then the classical relativistic energy Ef(t) of an electron in laser field is given by well-known expression 
Here p = p(t) is the electron momentum for arbitrary time moment t, and p 0 = p(0) is the initial value of this momentum for t = 0. The value of p is determined from the cubic equation :
Substituting Eq. (3) into Eq. (1) and taking into account Eq. (4), we obtain the relativistic energy distribution of ejected electrons in the tunneling ionization of an atom (with exponential accuracy). We restrict ourselves by the moderate values of kinetic energies of ejected electrons
i.e. by the case when these energies are less than the rest energy of the electron. Then the calculation of the integral in Eq. (1) is simplified, and we find after simple, but cumbersome calculations
Here the quantity ω 0 is the non-relativistic total tunneling ionization rate which can be calculated, foe example, in the frames of ADK theory .
The first term in the exponent of the expression (6) is the non-relativistic energy distribution of ejected electrons in the linearly polarized electromagnetic field [2,9]. Thus, the non-relativistic rate of electron ejection with the kinetic energy Ee exponentially decreases with Ee. The non-relativistic width of this distribution is
It is seen from Eq. (6) that the relativistic effect is important under the condition
which does not contradict the above condition Ee < c 2 since γ 2 << 1.
Thus, it follows from Eq. (6) that the relativistic width of the energy distribution is
For example, at γ = 0.1 the conditions c 2 > Ee > γ 2 c 2 are required for 500 keV > Ee > 5 keV. According to (9) the value of ΔEe(rel) is equal to 20 keV for radiation of CO2 - laser.
In conclusion, it follows from the suggested approach that the radiation of linearly polarized CO2 - laser pulse with the intensity exceeding 1015 W/cm2 produces relativistic distribution of ejected electrons with keV-energies along the polarization axis for most kinds of atoms, though the kinetic energies of these electrons are less than their rest energy mc 2. The results of this work can be used in the analysis of experimental data of Rochester group .
This work was supported partially by Russian Council of Fundamental Research (grant N 96-02-18299) and by Soros Foundation (grant N 22p).
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