Editing Tunnel ionization (section)

== DC tunneling ionization ==
Tunneling ionization from the ground state of a [[hydrogen atom]] in an electrostatic (DC) field was solved schematically by [[Lev Landau]],<ref>L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965), 2nd ed., pg 276.</ref> using parabolic coordinates. This provides a simplified physical system that given it proper exponential dependence of the ionization rate on the applied external field. When {{tmath| E \ll E_a }}, the ionization rate for this system is given by:
: <math> w = 4 \omega_a \frac{E_a}{\left|E\right|} \exp\left[ -\frac{2}{3}\frac{E_a}{\left|E\right|} \right]</math>

Landau expressed this in [[atomic units]], where {{tmath|1= m_\text{e} = e = \hbar = 1 }}. In [[International System of Units|SI units]] the previous parameters can be expressed as:
: <math>E_a = \frac{m_\text{e}^2 e^5}{(4\pi \epsilon_0)^3 \hbar^4}  </math>,
: <math>\omega_a = \frac{m_\text{e} e^4}{(4\pi \epsilon_0)^2 \hbar^3}</math>.

The ionization rate is the total [[probability current]] through the outer classical turning point. This rate is found using the [[WKB approximation]] to match the ground state hydrogen wavefunction through the suppressed coulomb potential barrier.

A more physically meaningful form for the ionization rate above can be obtained by noting that the [[Bohr radius]] and hydrogen atom [[ionization energy]] are given by
: <math>a_0 = \frac{4\pi \epsilon_0 \hbar^2}{m_\text{e} e^2} </math>,
: <math>E_\text{ion}=R_\text{H} = \frac{m_\text{e} e^4}{8 \epsilon_0^2 h^2} </math>,
where <math>R_\text{H} \approx \mathrm{13.6\, eV} </math> is the [[Rydberg constant|Rydberg energy]]. Then, the parameters <math> E_a </math> and <math>\omega_a </math> can be written as
: <math>E_a = \frac{2 R_\text{H}}{e a_0} </math>, <math>\omega_a = \frac{2 R_\text{H}}{\hbar}</math>.
so that the total ionization rate can be rewritten
: <math> w = 8 \frac{R_\text{H}}{\hbar} \frac{2 R_\text{H}/a_0}{\left|e E\right|} \exp\left[ -\frac{4}{3}\frac{R_\text{H}/a_0}{\left|eE\right|} \right]</math>.

This form for the ionization rate <math> w </math> emphasizes that the characteristic electric field needed for ionization <math>E_a = {2 E_\text{ion}} / {e a_0} </math> is proportional to the ratio of the ionization energy  <math>E_\text{ion} </math> to the characteristic size of the electron's orbital {{tmath| a_0 }}. Thus, atoms with low ionization energy (such as [[alkali metal]]s) with electrons occupying orbitals with high principal quantum number <math> n </math> (i.e. far down the periodic table) ionize most easily under a DC field. Furthermore, for a [[hydrogenic atom]], the scaling of this characteristic ionization field goes as {{tmath| Z^3 }}, where <math> Z </math> is the nuclear charge. This scaling arises because the ionization energy scales as <math> \propto Z^2 </math> and the orbital radius as {{tmath| \propto Z^{-1} }}. More accurate and general formulas for the tunneling from Hydrogen orbitals can also be obtained.<ref>{{cite journal |last1=Yamabe |first1=Tokio |last2=Tachibana |first2=Akitomo |last3=Silverstone |first3=Harris J. |date=1977-09-01 |title=Theory of the ionization of the hydrogen atom by an external electrostatic field |journal=Physical Review A |volume=16 |issue= 3 |pages=877–890 |doi=10.1103/PhysRevA.16.877 }}</ref>

As an empirical point of reference, the characteristic electric field  <math> E_a </math> for the ordinary hydrogen atom is about {{val|51|ul=V|upl=Å}} (or {{val|5.1|e=3|u=MV/cm}}) and the characteristic frequency <math> \omega_a </math> is {{val|4.1|e=4|u=THz}}.