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Ionization or ionisation is the process by which an atom or a molecule acquires a negative or positive Electric charge by gaining or losing , often in conjunction with other chemical changes. The resulting electrically charged atom or molecule is called an ion. Ionization can result from the loss of an electron after collisions with subatomic particles, collisions with other atoms, molecules, electrons, , (2025). 9783030738921, Springer. ISBN 9783030738921 , , and ions, (1997). 9783642083228, Springer-Verlag. ISBN 9783642083228 (1992). 9780198520207, Clarendon Press; Oxford University Press. ISBN 9780198520207 (1985). 9783642691973, Springer. ISBN 9783642691973 or through the interaction with electromagnetic radiation.P. Schmelcher (2025). 030645811X, Kulver Academic Publishers. 030645811X Heterolytic bond cleavage and heterolytic substitution reactions can result in the formation of ion pairs. Ionization can occur through radioactive decay by the internal conversion process, in which an excited nucleus transfers its energy to one of the inner-shell electrons causing it to be ejected.
Positively charged ions are produced by transferring an amount of energy to a bound electron in a collision with charged particles (e.g. ions, electrons or positrons) or with photons. The threshold amount of the required energy is known as ionization energy. The study of such collisions is of fundamental importance with regard to the Few-body systems, which is one of the major unsolved problems in physics. Kinematically complete experiments, i.e. experiments in which the complete momentum vector of all collision fragments (the scattered projectile, the recoiling target-ion, and the ejected electron) are determined, have contributed to major advances in the theoretical understanding of the few-body problem in recent years.
The Townsend discharge is a good example of the creation of positive ions and free electrons due to ion impact. It is a cascade reaction involving in a region with a sufficiently high electric field in a gaseous medium that can be ionized, such as air. Following an original ionization event, due to such as ionizing radiation, the positive ion drifts towards the cathode, while the free electron drifts towards the anode of the device. If the electric field is strong enough, the free electron gains sufficient energy to liberate a further electron when it next collides with another molecule. The two free electrons then travel towards the anode and gain sufficient energy from the electric field to cause impact ionization when the next collisions occur; and so on. This is effectively a chain reaction of electron generation, and is dependent on the free electrons gaining sufficient energy between collisions to sustain the avalanche.Glenn F Knoll. Radiation Detection and Measurement, third edition 2000. John Wiley and sons,
Ionization efficiency is the ratio of the number of ions formed to the number of electrons or photons used.
In general, the analytic solutions are not available, and the approximations required for manageable numerical calculations do not provide accurate enough results. However, when the laser intensity is sufficiently high, the detailed structure of the atom or molecule can be ignored and analytic solution for the ionization rate is possible.
Keldysh modeled the MPI process as a transition of the electron from the ground state of the atom to the Volkov states.Volkov D M 1934 Z. Phys. 94 250 In this model the perturbation of the ground state by the laser field is neglected and the details of atomic structure in determining the ionization probability are not taken into account. The major difficulty with Keldysh's model was its neglect of the effects of Coulomb interaction on the final state of the electron. As it is observed from figure, the Coulomb field is not very small in magnitude compared to the potential of the laser at larger distances from the nucleus. This is in contrast to the approximation made by neglecting the potential of the laser at regions near the nucleus. Perelomov et al. included the Coulomb interaction at larger internuclear distances. Their model (which we call the PPT model) was derived for short range potential and includes the effect of the long range Coulomb interaction through the first order correction in the quasi-classical action. Larochelle et al. have compared the theoretically predicted ion versus intensity curves of rare gas atoms interacting with a Ti:Sapphire laser with experimental measurement. They have shown that the total ionization rate predicted by the PPT model fit very well the experimental ion yields for all rare gases in the intermediate regime of the Keldysh parameter.
The rate of MPI on atom with an ionization potential in a linearly polarized laser with frequency is given by
\left|C_{n^* l^*}\right|^2 \sqrt{\frac{6}{\pi}} f_{lm}
E_i \left(\frac{2}{F} \left(2E_i\right)^{\frac{3}{2}}\right)^{2n^* - |m|- \frac{3}{2}}
\left(1 + \gamma^2\right)^{\left|\frac{m}{2}\right|+ \frac{3}{4}}
A_m (\omega, \gamma) e^{-\frac{2}{F}\left(2E_i\right)^{\frac{3}{2}} g\left(\gamma\right)}
where
f_{lm} &= \frac{(2l + 1)(l + |m|)!}{2^m |m|!(l - |m|)!} \\
g(\gamma) &= \frac{3}{2\gamma} \left(1 + \frac{1}{2\gamma^2} \sinh^{-1}(\gamma) - \frac{\sqrt{1 + \gamma^2}}{2\gamma}\right) \\
|C_{n^* l^*}|^2 &= \frac{2^{2n^*}}{n^* \Gamma(n^* + l^* + 1) \Gamma(n^* - l^*)}
\end{align}
The coefficient is given by
A_m (\omega, \gamma) =
\frac{4}{3\pi} \frac{1}{|m|!}
\frac{\gamma^2}{1 + \gamma^2}
\sum_{n>v}^\infty e^{-(n - v) \alpha(\gamma)}
w_m \left(\sqrt{\frac{2\gamma}{\sqrt{1 + \gamma^2}} (n - v)}\right)
where
w_m(x) &= e^{-x^2} \int_0^x (x^2 - y^2)^m e^{y^2}\,dy \\
\alpha(\gamma) &= 2\left(\sinh^{-1}(\gamma) - \frac{\gamma}{\sqrt{1 + \gamma^2}}\right) \\
v &= \frac{E_i}{\omega} \left(1 + \frac{2}{\gamma^2}\right)
\end{align}
\left|C_{n^* l^*}\right|^2 \sqrt{\frac{6}{\pi}}
f_{lm} E_i \left(\frac{2}{F} \left(2E_i\right)^{\frac{3}{2}}\right)^{2n^* - |m|- \frac{3}{2}}
e^{-\frac{2}{3F} \left(2E_i\right)^{\frac{3}{2}}}
As compared to the absence of summation over n, which represent different above threshold ionization (ATI) peaks, is remarkable.
\sum_{n=N}^{\infty} 2 \pi \omega^2 p \left(n - n_\mathrm{osc}\right)^2 \int \mathrm{d}\Omega
\left|FT \left(I_{KAR} \Psi \left(\mathbf{r}\right)\right)\right|^2
J_n^2 \left(n_f, \frac{n_\mathrm{osc}}{2}\right)
where:
where is the time-dependent energy difference between the two dressed states. In interaction with a short pulse, if the dynamic resonance is reached in the rising or the falling part of the pulse, the population practically remains in the ground state and the effect of multiphoton resonances may be neglected. However, if the states go onto resonance at the peak of the pulse, where , then the excited state is populated. After being populated, since the ionization potential of the excited state is small, it is expected that the electron will be instantly ionized.
In 1992, de Boer and Muller showed that Xe atoms subjected to short laser pulses could survive in the highly excited states 4f, 5f, and 6f. These states were believed to have been excited by the dynamic Stark shift of the levels into multiphoton resonance with the field during the rising part of the laser pulse. Subsequent evolution of the laser pulse did not completely ionize these states, leaving behind some highly excited atoms. We shall refer to this phenomenon as "population trapping".
We mention the theoretical calculation that incomplete ionization occurs whenever there is parallel resonant excitation into a common level with ionization loss. We consider a state such as 6f of Xe which consists of 7 quasi-degnerate levels in the range of the laser bandwidth. These levels along with the continuum constitute a lambda system. The mechanism of the lambda type trapping is schematically presented in figure. At the rising part of the pulse (a) the excited state (with two degenerate levels 1 and 2) are not in multiphoton resonance with the ground state. The electron is ionized through multiphoton coupling with the continuum. As the intensity of the pulse is increased the excited state and the continuum are shifted in energy due to the Stark shift. At the peak of the pulse (b) the excited states go into multiphoton resonance with the ground state. As the intensity starts to decrease (c), the two state are coupled through continuum and the population is trapped in a coherent superposition of the two states. Under subsequent action of the same pulse, due to interference in the transition amplitudes of the lambda system, the field cannot ionize the population completely and a fraction of the population will be trapped in a coherent superposition of the quasi degenerate levels. According to this explanation the states with higher angular momentum – with more sublevels – would have a higher probability of trapping the population. In general the strength of the trapping will be determined by the strength of the two photon coupling between the quasi-degenerate levels via the continuum. In 1996, using a very stable laser and by minimizing the masking effects of the focal region expansion with increasing intensity, Talebpour et al. observed structures on the curves of singly charged ions of Xe, Kr and Ar. These structures were attributed to electron trapping in the strong laser field. A more unambiguous demonstration of population trapping has been reported by T. Morishita and Chii-Dong Lin.
where is the rate of quasi-static tunneling to i'th charge state and are some constants depending on the wavelength of the laser (but not on the pulse duration).
Two models have been proposed to explain the non-sequential ionization; the shake-off model and electron re-scattering model. The shake-off (SO) model, first proposed by Fittinghoff et al., is adopted from the field of ionization of atoms by X rays and electron projectiles where the SO process is one of the major mechanisms responsible for the multiple ionization of atoms. The SO model describes the NSI process as a mechanism where one electron is ionized by the laser field and the departure of this electron is so rapid that the remaining electrons do not have enough time to adjust themselves to the new energy states. Therefore, there is a certain probability that, after the ionization of the first electron, a second electron is excited to states with higher energy (shake-up) or even ionized (shake-off). We should mention that, until now, there has been no quantitative calculation based on the SO model, and the model is still qualitative.
The electron rescattering model was independently developed by Kuchiev,[1] Schafer et al, Corkum, Becker and Faisal and Faisal and Becker.[2] The principal features of the model can be understood easily from Corkum's version. Corkum's model describes the NS ionization as a process whereby an electron is tunnel ionized. The electron then interacts with the laser field where it is accelerated away from the nuclear core. If the electron has been ionized at an appropriate phase of the field, it will pass by the position of the remaining ion half a cycle later, where it can free an additional electron by electron impact. Only half of the time the electron is released with the appropriate phase and the other half it never return to the nuclear core. The maximum kinetic energy that the returning electron can have is 3.17 times the ponderomotive potential () of the laser. Corkum's model places a cut-off limit on the minimum intensity ( is proportional to intensity) where ionization due to re-scattering can occur.
The re-scattering model in Kuchiev's version (Kuchiev's model) is quantum mechanical. The basic idea of the model is illustrated by Feynman diagrams in figure a. First both electrons are in the ground state of an atom. The lines marked a and b describe the corresponding atomic states. Then the electron a is ionized. The beginning of the ionization process is shown by the intersection with a sloped dashed line. where the MPI occurs. The propagation of the ionized electron in the laser field, during which it absorbs other photons (ATI), is shown by the full thick line. The collision of this electron with the parent atomic ion is shown by a vertical dotted line representing the Coulomb interaction between the electrons. The state marked with c describes the ion excitation to a discrete or continuum state. Figure b describes the exchange process. Kuchiev's model, contrary to Corkum's model, does not predict any threshold intensity for the occurrence of NS ionization.
Kuchiev did not include the Coulomb effects on the dynamics of the ionized electron. This resulted in the underestimation of the double ionization rate by a huge factor. Obviously, in the approach of Becker and Faisal (which is equivalent to Kuchiev's model in spirit), this drawback does not exist. In fact, their model is more exact and does not suffer from the large number of approximations made by Kuchiev. Their calculation results perfectly fit with the experimental results of Walker et al. Becker and Faisal have been able to fit the experimental results on the multiple NSI of rare gas atoms using their model. As a result, the electron re-scattering can be taken as the main mechanism for the occurrence of the NSI process.
The short pulse induced molecular fragmentation may be used as an ion source for high performance mass spectroscopy. The selectivity provided by a short pulse based source is superior to that expected when using the conventional electron ionization based sources, in particular when the identification of optical isomers is required.
The Kramers–Henneberger(KF) frame is the non-inertial frame moving with the free electron under the influence of the harmonic laser pulse, obtained by applying a translation to the laboratory frame equal to the quiver motion of a classical electron in the laboratory frame. In other words, in the Kramers–Henneberger frame the classical electron is at rest. Starting in the lab frame (velocity gauge), we may describe the electron with the Hamiltonian:
By applying a transformation to the laboratory frame equal to the quiver motion one moves to the ‘oscillating’ or ‘Kramers–Henneberger’ frame, in which the classical electron is at rest. By a phase factor transformation for convenience one obtains the ‘space-translated’ Hamiltonian, which is unitarily equivalent to the lab-frame Hamiltonian, which contains the original potential centered on the oscillating point :
The utility of the KH frame lies in the fact that in this frame the laser-atom interaction can be reduced to the form of an oscillating potential energy, where the natural parameters describing the electron dynamics are and (sometimes called the “excursion amplitude’, obtained from ).
From here one can apply Floquet theory to calculate quasi-stationary solutions of the TDSE. In high frequency Floquet theory, to lowest order in the system reduces to the so-called ‘structure equation’, which has the form of a typical energy-eigenvalue Schrödinger equation containing the ‘dressed potential’ (the cycle-average of the oscillating potential). The interpretation of the presence of is as follows: in the oscillating frame, the nucleus has an oscillatory motion of trajectory and can be seen as the potential of the smeared out nuclear charge along its trajectory.
The KH frame is thus employed in theoretical studies of strong-field ionization and atomic stabilization (a predicted phenomenon in which the ionization probability of an atom in a high-intensity, high-frequency field actually decreases for intensities above a certain threshold) in conjunction with high-frequency Floquet theory.Gavrila, Mihai. "Atomic structure and decay in high-frequency fields." ''Atoms in Intense Laser Fields,'' edited by Mihai Gavrila, Academic Press, Inc, 1992, pp. 435-508.
The KF frame was successfully applied for different problems as well e.g. for higher-hamonic generation from a metal surface in a powerful laser field
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