Nuclear excitation of ${}^{229}\mathrm{Th}$ induced by laser-driven electron recollision

Previously we proposed a new approach of exciting the ${}^{229}\mathrm{Th}$ nucleus using laser-driven electron recollision [Wang, Zhou, Liu, and Wang, Phys. Rev. Lett. 127, 052501 (2021)]. The current article is aimed at elaborating the method by explaining further theoretical details and presenting extended new results. The method has also been improved by adopting the electronic excitation cross sections calculated recently by Tkalya [Tkalya, Phys. Rev. Lett. 124, 242501 (2020)]. The new cross sections are obtained from Dirac distorted-wave calculations instead of from Dirac plane-wave calculations as we used previously. The distorted-wave cross sections are shown to be 5 to 6 orders of magnitude higher than the plane-wave results. With the excitation cross sections updated, the probability of isomeric excitation of ${}^{229}\mathrm{Th}$ from electron recollision is calculated to be on the order of ${10}^{-12}$ per nucleus per (femtosecond) laser pulse. The dependency of the excitation probability on various laser parameters is calculated and discussed, including the laser intensity, the laser wavelength, and the laser pulse duration.

Figure 1

Illustration of the RINE approach. An outer electron of the ${}^{229}\mathrm{Th}$ atom (or ion) is emitted into the continuum via tunneling ionization. It is later driven back by the oscillating laser electric field, recollides with its parent ion core, and excites the nucleus from the ground state to the isomeric state.

Figure 2

(a) Total ($E2+M1$) electronic excitation cross sections of ${}^{229}\mathrm{Th}$ from the ground state to the isomeric state. The black (upper two) curves are from distorted-wave calculations and the red (gray, lower two) curves are from plane-wave calculations. For each calculation, the solid curve uses set 1 [Eqs. (10) and (11)] of the reduced transition probabilities and the dashed curve uses set 2 [Eqs. (12) and (13)]. (b) Comparison between (the large component of) a Dirac plane wave and the corresponding distorted wave for electron energy 100 eV (with $l=0, j=1/2$). (c) Extension of panel (a) into a much larger energy range.

Figure 3

(a) Illustration of an electron collision trajectory. The electron has an asymptotic velocity $v_0$ and an impact parameter $b$. Its distance to the nucleus is denoted $R_0$ as the electron passes the $x=0$ plane. (b)–(d) The relationship between $R_0$ and $b$ for electron energies 10, 50, and 100 eV, as labeled on each figure. For each energy, four different ion-core charges are used. The ion-core potential is described by the GSZ potential [Eq. (14)].

Figure 4

Dependency of $|\tilde{V}\left({\omega }_0\right){|}^2$ on $b$ (left column) and on $R_0$ (right column), for different electron energies 10, 50, and 100 eV, as labeled on the figure. Each curve has been normalized to its own peak value. For each energy, four different ion-core charges are used.

Figure 5

(a) Relationship between the recollision time $t_r$ and the ionization time $t_i$. (b) Relationship between the recollision energy $E_r$ and the ionization time $t_i$. For both panels, the solid black curve is for the case where only the laser potential is considered, i.e., the Simpleman model, and the red dashed curve is for the case including the ion-core GSZ potential. The laser intensity used here is $5\times {10}^{13}$ $\mathrm{W}/{\mathrm{cm}}^2$ and the laser wavelength is 800 nm.

Figure 6

(a) Effective recollision flux density for the first-emitted electron (red, leftmost), the second-emitted electron (blue, middle), and the third-emitted electron (magenta, rightmost). The laser electric field is shown as the gray background. The pulse duration is 30 fs and the peak intensity is ${10}^{14} \mathrm{W}/{\mathrm{cm}}^2$. Note that the flux density is shown in atomic units ($1\mathrm{a}.\mathrm{u}.=1.48\times {10}^{37}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$). (b) The corresponding nuclear excitation probability. Both the total excitation probability and the individual contribution from each electron are shown, as labeled.

Figure 7

(a) Nuclear excitation probability as a function of laser peak intensity. The laser pulse has a wavelength of 800 nm and a pulse duration of 30 fs. (b) Accumulated excitation probability during the laser pulse for an intensity of $2\times {10}^{14} \mathrm{W}/{\mathrm{cm}}^2$. Contributions from each electron are also shown, as labeled. (c) Similar to panel (b), but for an intensity of $2\times {10}^{15} \mathrm{W}/{\mathrm{cm}}^2$.

Figure 8

(a) Nuclear excitation probability as a function of laser pulse duration, for two different peak intensities as labeled on the figure. The laser pulses have wavelengths of 800 nm. (b) Accumulated excitation probability during a laser pulse for an intensity of $1\times {10}^{14} \mathrm{W}/{\mathrm{cm}}^2$ and a pulse duration of 10 fs. (c) Similar to panel (b), but for a pulse duration of 20 fs.

Figure 9

(a) Nuclear excitation probability as a function of laser wavelength. The pulse duration is fixed at 30 fs. Two peak intensities are shown, as labeled on figure. (b) Accumulated excitation probability during a laser pulse with a wavelength of 400 nm and an intensity of $5\times {10}^{14} \mathrm{W}/{\mathrm{cm}}^2$. (c) Similar to panel (b), but for a wavelength of 1600 nm.

Figure 10

Nuclear excitation probability with (black solid circles) and without (blue empty diamands) the ion-core GSZ potential. The black curve with solid circles is the same as Fig. 7. The laser wavelength is 800 nm and the pulse duration is 30 fs.

Figure 11

Absorption potential felt by the recolliding electron as it travels through the electron cloud of ${}^{229}\mathrm{Th}^{+}$. The potential is calculated using the code elsepa with three different electron incoming energies as labeled. Note that the potential is shown in atomic units ($1\mathrm{a}.\mathrm{u}.=27.2$ eV).