THE EFFECT OF MAGNETIC FIELD ON THE TUNNELING YIELD OF AMMONIA MOLECULES

We analyzed the influence of magnetic component of the laser field on the tunneling yield, in a strong near-relativistic field for a squared hyperbolic secant pulse distribution. The obtained results indicate that the inclusion of the magnetic component is necessary in the observed regime.


INTRODUCTION
Photoionization of atoms and molecules is the initial step of many interesting strongfield phenomena such as electron recollision and nonsequential double ionization (NDI) , molecular imaging by the recolliding electrons (MIRE) (PENG et al., 2019), highorder above-threshold ionization (HATI) (BRENNECKE and LEIN, 2018), high-order harmonic generation (HHG) (BRAY et al., 2019). Additionally, the electrons emitted by photoionization contain structural information suitable for orbital imaging (YUAN et al., 2017;CHANG et al., 2019) and nuclear dynamics, as well (ARNOLD et al., 2017). Because of all above mentioned, the ionization mechanism depending on the laser parameters, such as intensity, power, frequency, and wavelength has attracted considerable interests of both experimental and theoretical groups. The goal was to obtain an accurate theoretical model of atoms and molecules photoionization processes and as a result, multiple theoretical approaches (KELDYSH, 1965;KRAINOV, 1997;ZHAO et al., 2014) have been developed. In this paper we are interested in those which are dealing with molecules that are exposed to the external laser field.
During photoionization (PI), tunneling ionization (TI) can be considered as an initial key process. Therefore, a detailed understanding of TI is necessary for the further understanding of strong-field physics of atoms and molecules. According to (KELDYSH, 1965), the choice of the governing PI mechanism is dictated by the famous Keldysh parameter, . TI is a limiting case of PI for the value of the Keldysh parameter much less than unity, = √ 2 ≪ 1, where is ionization potential, frequency and the strength of the field. Although analysis and understanding of the physical processes associated with TI are progressing fast, complete, and detailed modeling of the mechanisms is still an open scientific topic. In the opposite limit, ≫ 1, the electron is ionized from the molecule by absorbing several photons and this process is known as a multiphoton ionization (MPI). The intermediate regime (~1) is intuitively considered as the borderline between the TI and MPI depending on laser field parameters.
Molecules are much more difficult systems for modeling than atoms due to their complex dynamic. Nevertheless, the values for TI rates of atoms and molecules with approximately the same ionization potential and nearly identical binding energies are usually close to each other (TONG et al., 2002). Tong and his coworkers in (TONG et al., 2002) investigated the ionization of molecules versus atoms while presenting a novel theoretical model (so-called MO-ADK) for calculating the ionization rates of diatomic molecules. According to (ZHAO et al., 2011;GONG et al., 2017), their model can be successfully extended to triatomic and polyatomic molecules.
In this paper, we investigated the photoionization rates and yields of ammonia molecule (NH3) using non-perturbative approach. We analyzed numerically and analytically the influence of the magnetic field component on the tunneling yield, in a strong near-relativistic field. We choose ammonia molecule because it is simple enough for our calculations, but also for following reasons. As demonstrated recently, its properties such as diffusivity, viscosity, and structure have a considerable impact on the internal evolution and magnetic field of giant planets (ROBINSON et al., 2017). The increase of ammonia molecule emissions in the atmosphere has negative influence on climate change, environmental and public health. This is why the number of research papers related to NH3 emissions into the atmosphere increased over the previous year (WANG et al., 2020; LI et al., 2020).

THEORETICAL BACKGROUND
The first attempt to determine molecule transition rate was classical, the adiabatic Ammosov-Delone-Krainov model (ADK) (AMMOSOV et al., 1986) which had limited success. The problem was that the ADK ionization model often overestimates the ionization rates thereby shifting saturation intensities to lower values for many molecules. That is why, sixteen years later, the ADK theory is upgraded for molecules, and the MO-ADK is obtained (TONG et al., 2002). The main improvement is the incorporation of molecular orbitals. The characteristic exponential dependence of transition rate on the field intensity, , and the ionization potential, is kept (TONG et al., 2002): where is the magnetic quantum number along the molecular axis, and * are the angular and the effective principal quantum numbers, * = (here denotes the effective Coulomb 2 . In addition, ′ , ( , , ) denotes Wigner's rotation matrix, where ( , , ) are the Euler angles. The rotation around the molecular axis is represented by , and represents the azimuthal angle, while the angle of rotation around the polarization vector is, , and for linearly polarized lasers is fixed to 0. The linearly polarized laser field is assumed. Atomic units, = = ℏ = 1, are used through the paper (SHULL and HALL, 1959).
In order to determine Wigner's rotation matrix, it is convenient to introduce rotation operator, ̂( ) = exp(−iαЈn), where is an arbitrary angle, and Јn = − is projection of angular momentum on arbitrary axis (YAMANI AND FISHMAN, 2008). For the Euler angles, rotation operator becomes (ZARE, 1988): Now, Eq. (2) can be rewritten as the following: By projecting the equation from the left with ⟨ ′ | and using the orthonormality of the angular momentum states, ⟨ | ⟩ = , we found an explicit expression for the Wigner's rotation matrix elements (ZARE, 1988): where the Wigner's matrix, ( , , ), is a matrix of dimension (2 + 1) × (2 + 1). Because we are interested in examining the ionization rate along the z-axis, the matrix elements presented using Eq. (4) can be found using the following expressions: Based on the results presented in (WIGNER, 1931) −matrix element ′ , ( ), can be derived using the following series expression: where ( + ) and ( − − ′ − ) are binomial coefficients. The summation over parameter is only restricted to the argument of any factorial which is non-negative.
Using Eq. (5) We made summation over and obtained: After some trigonometric transformation, as a result, the final expression is obtained: Following the same recipe, the other coefficients can be calculated: .
The abovementioned calculations of Wigner's rotation matrix provide a great accuracy for a wide range of atoms and molecules and are used to evaluate the ionization rates in the frame of both ADK and MO-ADK theories. To obtain a more accurate expression of the rate for the special case of ammonia molecule, we observed geometry optimization which leads to N-H bond length of 1.017 Å, while H-N-H bond angle is 107.8° (see (NILSSON, 2005)). In this case, as one can see from Figure 1, its molecule has a pyramidal shape forming an oscillating umbrella structure (See Fig. 1). In order to compare theoretical findings with experimental, it is convenient to calculate ionization yield, , because this quantity is usually measured in experiments. Its expression is directly related to the ionization rate, and dependent upon the laser electric field strength, as (MILADINOVIĆ and PETROVIĆ, 2014): where , , ′ ( ) = ∑ ( where is the relativistic kinetic energy of ejected photoelectrons defined as in (KRAINOV, 1998): Here denotes the speed of light in atomic units, = 137.02, and is the momentum of ejected photoelectrons. It is important to note that under the frame of tunneling theory, the magnetic effects set a lower limit on Keldysh adiabaticity parameter. According to (MILADINOVIĆ and PETROVIĆ, 2015) the condition → 0 represents an extreme relativistic limit. In that case, the relativistic Keldysh parameter, , must be introduced: , where is the ion charge. For purpose of incorporating the magnetic component of the laser field in the nearrelativistic transition rate, we shall extend Eq. (15) using the definition of the Lorentz ionization rate (ZHAKENOVICH et al., 2015): where is the electron velocity and is the stabilization factor. By implementing the equation for Lorentz transition rate (Eq. (16) . (17) Eq. (17) cannot be solved analytically due to the complex integral representation. Because of that, the ion yield obtained at near-relativistic field intensities while including the magnetic field component (Eq. (17)), must be solved numerically.

RESULTS AND DISCUSSION
We considered ammonia, 3 , molecule, in the linearly polarized laser field, with a hyperbolic-secant-squared time distribution, using the MO-ADK theoretical approach. We observed the ionization of ammonia through tunneling of electrons from the highest occupied molecular orbital, with in a range covering from 0.025 to 0.8. The laser field intensity varied within the range 10 13 < < 10 17 Wcm −2 . We used laser wavelenght = 800 nm.
First of all, we investigated the geometry-dependent ionization behavior of ammonia molecules. Analysis presented in Fig. 2 is especially important for the study of the nuclear motion in neutral ammonia (WILEY AND MCLAREN, 1955). In Fig. 2, we presented the orientation-dependent ion rate for ammonia for the pure nonrelativistic approach, Next, we observed ammonia ionization yield. In order to validate the analytical solution given in Eq. 14, Eq. 13 was integrated numerically using Wolfram Mathematica scripts (WOLFRAM, 1999). The integral in Eq. 13 was discretized using the Laplace transform, (LESKO and SMITH, 2003). Our analytical and numerical calculations are presented in Fig. 3 and one can see a good agreement between analytical and numerical curve. It is also important to stress that ionization yields presented in Fig. 3 are for weak or non-relativistic fields and because of that can be linked to one of the previous equations (Eq. 13 for numerical solution and Eq. 14 for analytical solution). In the near relativistic field intensities region, 10 16 < < 10 17 Wcm −2 , we investigated the influence of magnetic field component on the total ammonia yield. For this purpose, we compared results for relativistic ion yield without, ( , ) (Eq. 10), and with correction of the magnetic field, ( , , ) (Eq. 17). Results are presented in Fig. 4. Fig. 3, shows that for the lower laser field intensities region, both curves, the one which includes correction of the magnetic field ( , , ) and the other which neglect the magnetic field component ( , ), have almost the same flow. But with the intensity increasing, there is a deviation between curves. One can see that for < 0.5 × 10 16 Wcm −2 , both curves sharply rise together, but with field increase,

CONCLUSION
In summary, we have investigated the influence of the magnetic component of the laser field on the photoionization yield of the ammonia molecule. We showed that, in the nearrelativistic intensities, magnetic component has a strong contribution to the yield and should be taken into account. Our results indicate that this contribution increases with field intensities. Our findings are in a good agreement with available theoretical results.