Bound and continuum energy distributions of nuclear fragments resulting from tunneling ionization of molecules

We present the theory of tunneling ionization of molecules with both electronic and nuclear motion treated quantum mechanically. The theory provides partial rates for ionization into the different final states of the molecular ion, including both bound vibrational and dissociative channels. The exact results obtained for a one-dimensional model of ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ are compared with two approximate approaches, the weak-field asymptotic theory and the Born-Oppenheimer approximation. The validity ranges and compatibility of the approaches are identified formally and illustrated by the calculations. The results quantify that at typical field strengths considered in strong-field physics, it is several orders of magnitude more likely to ionize into bound vibrational ionic channels than into the dissociative channel.

Figure 1

Illustration of the regions of applicability of the WFAT [Eq. (21)] and BOA [Eq. (36)] for the present model in the reduced nuclear mass $M$ (divided by the proton mass $m_p$) and field strength $F$ plane. The blue line shows the critical field $F_c$ above which over-the-barrier ionization becomes accessible; the WFAT is applicable in the region indicated by blue shading. The red line shows the field $F_{\text{BO}}$ below which retardation effects become essential [32]; the BOA is applicable in the region indicated by red shading. In the overlap region [Eq. (50)], both approaches apply. The black dots indicate two systems ($M/m_p=1/2$ and 1, which corresponds to ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$, respectively) at two field strengths ($F=0.026$ and 0.079) considered in the present calculations.

Figure 2

Solid lines show BO potentials for the molecular ion $U_{\text{ion}}\left(R\right)$ and neutral molecule $U_{\text{mol}}\left(R\right)$ for $F=0$ [Eq. (29)] in the present 1D model. Corresponding dashed lines show BO potentials for 3D $\mathrm{H}{{}_2}^{+}$ [49] and ${\mathrm{H}}_2$ [50]. The bound ${\chi }_v\left(R\right)$ and continuum $\chi (R,k)$ states of $\mathrm{H}{{}_2}^{+}$ defined by Eq. (4) are illustrated at their energies ${\varepsilon }_v$ and $\varepsilon \left(k\right)=k^2/2M$, respectively, in the $U_{\text{ion}}\left(R\right)$ curve. The shaded (gray) region near the bottom of $U_{\text{mol}}\left(R\right)$ illustrates the field-free ground-state nuclear wave function ${\mathrm{\Psi }}_0\left(R\right)$ of ${\mathrm{H}}_2$.

Figure 3

Squares of the asymptotic coefficients in Eq. (16) for ${\mathrm{H}}_2$ (upper panel) and ${\mathrm{D}}_2$ (lower panel) calculated using an accurate molecular bound-state eigenfunction ${\mathrm{\Psi }}_0(z,R)$ [Eqs. (17)], within the BOA [Eqs. (42)], and in the Franck-Condon approximation [Eqs. (46)]. The boxes (at $\varepsilon <0$) and lines (at $\varepsilon >0$) represent results corresponding to the discrete and continuous parts of the molecular ion spectrum, respectively. The lines are multiplied by factors indicated in the figure to make them visible on the same scale as the boxes. The dots on the $\varepsilon$ axes indicate energies ${\varepsilon }_v$ of the bound vibrational states of the ion. The vertical dashed line at $\varepsilon =U_{\text{ion}}\left(R_0\right)<0$ indicates an estimate of the location of the maximum of the distributions; see Sec. 2d1.

Figure 4

Solid lines show the real part of the BO potential in Eq. (24) (blue line, left axis) and the ionization rate (31) for fixed nuclei (red line, right axis) at field $F=0.053$. The dashed line (right axis) shows the rate obtained within the WFAT [Eq. (47)]. The shaded regions near the bottom of the BO potential illustrate the real (light blue) and imaginary (pink) parts of the nuclear wave function $\mathrm{\Psi }\left(R\right)$ for ${\mathrm{H}}_2$ at the same field; compare with Fig. 2.

Figure 5

Partial ionization rates (boxes at $\varepsilon <0$) and dissociation spectra (lines at $\varepsilon >0$) for ${\mathrm{H}}_2$ at two representative field strengths, $F=0.026$ (upper panel) and $F=0.079$ (lower panel), calculated using an accurate molecular SS eigenfunction $\mathrm{\Psi }(z,R)$ [Eqs. (11)], within the WFAT [Eqs. (18)], and within the BOA [Eqs. (41)]. The lines are multiplied by factors indicated in the figure to make them visible on the same scale as the boxes. The dots on the $\varepsilon$ axes indicate energies ${\varepsilon }_v$ of the bound states of the ion. The vertical dashed lines at $\varepsilon =U_{\text{ion}}\left(R_{\mathrm{\Gamma }}\right)<0$ indicate an estimate of the location of the maximum of the distributions within the BOA; see Sec. 2c.

Figure 6

Same as in Fig. 5, but for ${\mathrm{D}}_2$.

Figure 7

Total ionization rates (upper panel) and the same rates divided by the field factor (19) for the ground vibrational ionic channel with $\varkappa ={\varkappa }_0$ (lower panel) as functions of the field strength for ${\mathrm{H}}_2$. The exact results obtained from the SS eigenvalue (8) are compared with the predictions of WFAT [Eq. (20)] and BOA [Eq. (33)]. In addition, the sum on the right-hand side of Eq. (12) and separately the term describing dissociation [Eq. (13b)] are shown. The dot on the left axis in the lower panel indicates the value of the asymptotic coefficient squared $|g_0{|}^2$ to which the ratio shown in the figure converges as $F\to 0$.

Figure 8

Same as in Fig. 7, but for ${\mathrm{D}}_2$.

Figure 9

Ratio of the total ionization rates of ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ as a function of the field strength. The solid (black) line shows the results calculated using the exact rates from Eq. (8), the dashed (red) line is obtained using the WFAT rates from Eq. (20), and the dash-dotted (blue) line is obtained using the BOA rates from Eq. (33). The dotted (green) line shows predictions of the formula (57).

Figure 10

The normalized density in the continuum part of the SS eigenfunction for ${\mathrm{D}}_2$ at $F=0.079$; see Eqs. (58) and (59). The ridge represents the dissociating nuclei.