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Density-functional study of multielectron ionization of sodium clusters by strong femtosecond laser pulses
Computer Physics Communications 147 (2002) 205–208 www.elsevier.com/locate/cpc
Density-functional study of multielectron ionization of sodium clusters by strong femtosecond laser pulses L.I. Kurkina Institute of Thermophysics SB RAS, Acad. Lavrentyev ave., 1, Novosibirsk 630090, Russia
Abstract Electronic excitations in small sodium clusters irradiated by strong laser pulses are studied within the time-dependent density-functional formalism and jellium model. Resonant and over-the-barrier mechanisms of ionization are considered. The dependence of the electronic response on cluster size, photon energy, intensity and time duration of light pulses is discussed. 2002 Elsevier Science B.V. All rights reserved. PACS: 36.40.-c; 36.40.Cg; 31.15.Ew Keywords: Clusters; Ionization; Intense light fields; Density-functional theory
1. Introduction In recent years, new experimental techniques have opened up the very interesting regime of strong electronic excitations in clusters which are induced by intense short laser pulses. The strong laser field can cause multielectron ionization of clusters resulting in their explosion and the emission of intense Xrays, highly charged ions, and particles with extremely high kinetic energies. Theoretical models developed for the strong laser-cluster interaction take two paths: the elucidation of the mechanism of the multielectron ionization and the simulation of the cluster explosion. In the present work we focus on singleparticle mechanisms of electron emission from laserirradiated sodium clusters. The computations are performed within the time-dependent density-functional theory (TDDFT) [1]. Sodium clusters consisting of 8, E-mail address: [email protected] (L.I. Kurkina).
20, and 40 atoms (experimentally observed “magic” numbers) are considered. The jellium model is used in which the clusters of “magic” size correspond to spheres with closed electronic shells. The spherical symmetry is retained in the time-dependent study, that limits the investigation to single-particle excitations separating them from the dipole surface plasma resonance (which is also excited in the visible part of the optical spectrum and can cause electronic autoionization). The dependence of the electron response on intensity, length, and frequency of light pulses as well as on cluster size is studied.
2. Details of calculations Much work has been done within the linearized version of the TDDFT on clusters in weak external fields [2]. However, the TDDFT in the general form provides a suitable nonlinear nonperturbative approach for investigations of intense electronic exci-
0010-4655/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 2 ) 0 0 2 4 6 - 1
206
L.I. Kurkina / Computer Physics Communications 147 (2002) 205–208
tations. In the present work the study of electronic excitations in small sodium clusters under a strong laser field is based on the direct numerical solution of the time-dependent Kohn–Sham equation (atomic units |e| = m = h¯ = 1 are used) ∂ ∇2 + V (r, t) ψj (r, t) i ψj (r, t) = − (1) ∂t 2 for a jellium sphere in an external potential Vext (r, t) caused by an electromagnetic field. The one-electron effective potential V (r, t) can be constructed as n(r , t) − n+ (r ) dr V (r, t) = Vext (r, t) + |r − r | + Vxc (r, t), (2) where n(r, t) is the electronic density of the jellium cluster (obtained from one-electron wave functions ψj (r, t)), n+ (r) is the density of the spherical uniform positive jellium background (it coincides in absolute value with the average density of valence electrons in bulk sodium). A background (cluster) radius is defined as R = N 1/3 rs (N is the number of atoms in the cluster, rs is the Wigner–Seitz radius; for sodium, rs = 3.98 a.u.). The exchange-correlation potential Vxc (r, t) was constructed in the local approximation of Vosko et al. [3]. Assuming that the cluster interacts with light pulses polarized linearly along the zdirection and using the dipole approximation, the external potential can be written as:
electrons escaping from a sphere of radius Rbox = 1.5R around the cluster: R box
Nesc (t) = 4π
n(r, 0) − n(r, t) r 2 dr.
0
3. Results The above scheme was used for the numerical study of electron escape from sodium clusters under Gaussian-type light pulses with half-height duration T = 10, 20, 40, 100, and 200 fs (full pulse widths are 25, 50, 100, 250, and 500 fs, respectively), peak intensities of I0 = 5 × 1012 –1 × 1014 W/cm2 , and photon energies ω of 1–3 eV (with a mesh width of 0.05 eV). Fig. 1 shows the change of Nesc during the interaction of Na8 with a 40-fs pulse at I0 = 2.5 × 1013 W/cm2 and different photon energies. For the characterization of the electron emission from clusters under a light pulse as a whole, Nesc was averaged over the last one third of the pulse. This quantity is denoted by Nesc . Fig. 2 presents Nesc for Na8 and Na40 under light pulses of different length and intensity. As would be expected, the increasing of
Vext (r, t) = zE(t) cos ωt. The field strength E(t) is taken to be a Gaussian form. To retain the spherical symmetry of the problem, we turn to spherical coordinates, represent z = r cos θ (where θ is the angle of an electron radius-vector r with an electrical vector E of the electromagnetic wave), and average Vext (r, t) over angular variables. This gives 2r E(t) cos ωt. (3) π Eq. (1) with the potential (2)–(3) reduces to the one-dimensional (with respect to spatial coordinates) Kohn–Sham equation which was solved with the mesh width for a time coordinate of 2.5 × 10−18 s. The initial state was taken from ground-state densityfunctional calculations [4]. The electron emission from clusters was evaluated through the number of Vext (r, t) =
Fig. 1. Time dependence of Nesc for Na8 : T = 40 fs (full length is 100 fs), I0 = 2.5 × 1013 W/cm2 , and photon energies as indicated. The pulse envelope is plotted by curve 1 (in arbitrary units).
L.I. Kurkina / Computer Physics Communications 147 (2002) 205–208
Fig. 2. Spectral dependence of Nesc for Na8 and Na40 : (1) I0 = 5 × 1012 W/cm2 , T = 20 fs; (2) I0 = 5 × 1012 W/cm2 , T = 200 fs; (3) I0 = 1 × 1013 W/cm2 , T = 100 fs; (4) I0 = 5 × 1013 W/cm2 , T = 20 fs; (5) I0 = 1 × 1014 W/cm2 , T = 10 fs. The pulses (2)–(5) carry the same energy.
pulse intensity at fixed length or pulse length at fixed intensity leads to the rise of the electron emission. The intensity dependence of Nesc is more essential. Really, a comparison of curves 2–5 in Fig. 2 has shown that the number of escaping electrons increases with intensity although the pulse energy is conserved through the respective decreasing of pulse duration. It has been found that ionization rises with increasing cluster size, too. In this case, when the spectra of Nesc are divided by cluster radii squared, at photon energies of up to 2 eV, the resulting plots for different clusters under the same light pulses are practically superimposed, in the 2–3-eV range, although the size dependence of the spectra persists (because of the size peculiarities of the cluster one-electron energy spectra), the maximum values of Nesc /R 2 come close together. Consider in more detail the variation of Nesc with cluster size, pulse intensity and length. At the minimum of the considered pulse intensities I0 = 5 × 1012 W/cm2 , the spectra of Nesc reflect the groundstate electronic structure of the clusters and, hence, can be formed by one-photon resonant absorption. So, for Na8 (the electronic configuration is 1s2 1p6 ), the enhancement of Nesc below 2 eV can be caused by transitions of 1p electrons to 1d and 2s levels (groundstate density-functional calculations predict energies
207
of these transitions to be equal to 1.23 and 1.90 eV, respectively). The single photon ionization threshold of the 1p shell is 3.23 eV. Between 2 and 3 eV the value of Nesc is close to zero for lack of allowed electronic transitions. For Na20 (1s2 1p6 1d10 2s2 ) and Na40 (1s2 1p6 1d10 2s2 1f14 2p6 ), bound–bound transitions from outer occupied shells also occur predominantly below 2 eV (the ground-state density-functional theory yields, for Na20 , 2s → 2p, 1.19 eV; 1d → 1f, 1.13 eV; 1d → 2p, 1.80 eV; for Na40 , 2p → 3s, 1.21 eV; 1f → 2d, 1.74 eV), and the single photon ionization thresholds for both clusters are about 2.7 eV. The peaks in Nesc at I0 = 5 × 1012 W/cm2 are slightly below the values mentioned last. This can be due to the polarization of clusters under the external electric field, which leads to the rise of one-electron energies, but weakly changes the distances between energy levels [4]. With increasing light intensity, the main peaks in Nesc are enhanced, extended, complicated, and blue-shifted. The broadening and displacement of the main ionization peaks are primarily related to the onset of over-the-barrier ionization (OBI) coupling with resonant absorption. Our calculations have shown that the conditions for the OBI of the clusters are realized at intensities I0 1 × 1013 W/cm2 . During the laser-cluster interaction, the OBI starts prior to the resonant ionization and leads to the fall of oneelectron energy levels in the cluster and to the increase of the distance between levels. As a result, the resonant absorption occurs at higher energies than follows from the unperturbed energy spectrum. In this case the greater the photon energy, the later the resonance happens. As is seen from Fig. 1, at ω = 1.80 eV the sharp increase of Nesc begins earlier than at ω = 2.25 eV. In both cases the electron escape is connected with the ionization decay of 1p electron excitation to unfilled electronic shells. At ω = 2.30 eV, the resonance is not observed, because the OBI has no time during the pulse to reduce cluster energy levels to the necessary value. With increasing light intensity the OBI grows, and one-electron energy levels are lowered. Because of this, the resonance peaks in Nesc are extended and blue-shifted. Furthermore, with increasing pulse intensity, the spectra of Nesc are complicated, which can be due to multiphoton absorption. The pulse duration has a weaker influence on the ionization spectra of clusters than the field intensity. The increase of pulse duration at fixed intensity affects
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L.I. Kurkina / Computer Physics Communications 147 (2002) 205–208
Fig. 3. Spectral dependence of Wkin for Na40 : (1) I0 = 5 × 1012 W/cm2 , T = 20 fs; (2) I0 = 1 × 1013 W/cm2 , T = 100 fs; (3) I0 = 5 × 1013 W/cm2 , T = 20 fs. The pulses (2) and (3) carry the same energy.
mainly the resonant (one- or multiphoton) absorption and manifests itself in the enhancement of resonant peaks. To estimate cluster heating by light pulses, the variation of the kinetic energy Wkin (t) of electrons kept in the cluster was calculated as the difference between the kinetic energy of electrons inside Rbox at time t and the kinetic energy of electrons in the nonperturbative cluster at t = 0. As for Nesc (t), Wkin (t) was averaged over the last one third of a pulse (the result is designated as Wkin ). It has been
found that the shape of Wkin (Fig. 3) is similar to that of Nesc (Fig. 2). The most efficient “pumping” of energy in clusters occurs in the process of the resonant one-photon absorption. It should be noted that the size-dependence of Wkin is steeper than that of Nesc , and in the average Wkin is proportional to the fourth power of the cluster radius. In summary, the calculations have shown that at high light intensity the single-particle mechanisms can cause the strong multielectron ionization and heating of metal clusters. The efficiency of the process is determined mainly by the field intensity rather than the total energy of a pulse.
Acknowledgements This work was supported by RFBR (Grants No. 9915-96028, 00-03-33043).
References [1] [2] [3] [4]
E.K.U. Gross, W. Kohn, Adv. Quant. Chem. 21 (1990) 255. M. Brack, Rev. Mod. Phys. 65 (1993) 677. S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. L.I. Kurkina, O.V. Farberovich, Z. Phys. D 37 (1996) 359.
Density-functional study of multielectron ionization of sodium clusters by strong femtosecond laser pulses L.I. Kurkina Institute of Thermophysics SB RAS, Acad. Lavrentyev ave., 1, Novosibirsk 630090, Russia
Abstract Electronic excitations in small sodium clusters irradiated by strong laser pulses are studied within the time-dependent density-functional formalism and jellium model. Resonant and over-the-barrier mechanisms of ionization are considered. The dependence of the electronic response on cluster size, photon energy, intensity and time duration of light pulses is discussed. 2002 Elsevier Science B.V. All rights reserved. PACS: 36.40.-c; 36.40.Cg; 31.15.Ew Keywords: Clusters; Ionization; Intense light fields; Density-functional theory
1. Introduction In recent years, new experimental techniques have opened up the very interesting regime of strong electronic excitations in clusters which are induced by intense short laser pulses. The strong laser field can cause multielectron ionization of clusters resulting in their explosion and the emission of intense Xrays, highly charged ions, and particles with extremely high kinetic energies. Theoretical models developed for the strong laser-cluster interaction take two paths: the elucidation of the mechanism of the multielectron ionization and the simulation of the cluster explosion. In the present work we focus on singleparticle mechanisms of electron emission from laserirradiated sodium clusters. The computations are performed within the time-dependent density-functional theory (TDDFT) [1]. Sodium clusters consisting of 8, E-mail address: [email protected] (L.I. Kurkina).
20, and 40 atoms (experimentally observed “magic” numbers) are considered. The jellium model is used in which the clusters of “magic” size correspond to spheres with closed electronic shells. The spherical symmetry is retained in the time-dependent study, that limits the investigation to single-particle excitations separating them from the dipole surface plasma resonance (which is also excited in the visible part of the optical spectrum and can cause electronic autoionization). The dependence of the electron response on intensity, length, and frequency of light pulses as well as on cluster size is studied.
2. Details of calculations Much work has been done within the linearized version of the TDDFT on clusters in weak external fields [2]. However, the TDDFT in the general form provides a suitable nonlinear nonperturbative approach for investigations of intense electronic exci-
0010-4655/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 2 ) 0 0 2 4 6 - 1
206
L.I. Kurkina / Computer Physics Communications 147 (2002) 205–208
tations. In the present work the study of electronic excitations in small sodium clusters under a strong laser field is based on the direct numerical solution of the time-dependent Kohn–Sham equation (atomic units |e| = m = h¯ = 1 are used) ∂ ∇2 + V (r, t) ψj (r, t) i ψj (r, t) = − (1) ∂t 2 for a jellium sphere in an external potential Vext (r, t) caused by an electromagnetic field. The one-electron effective potential V (r, t) can be constructed as n(r , t) − n+ (r ) dr V (r, t) = Vext (r, t) + |r − r | + Vxc (r, t), (2) where n(r, t) is the electronic density of the jellium cluster (obtained from one-electron wave functions ψj (r, t)), n+ (r) is the density of the spherical uniform positive jellium background (it coincides in absolute value with the average density of valence electrons in bulk sodium). A background (cluster) radius is defined as R = N 1/3 rs (N is the number of atoms in the cluster, rs is the Wigner–Seitz radius; for sodium, rs = 3.98 a.u.). The exchange-correlation potential Vxc (r, t) was constructed in the local approximation of Vosko et al. [3]. Assuming that the cluster interacts with light pulses polarized linearly along the zdirection and using the dipole approximation, the external potential can be written as:
electrons escaping from a sphere of radius Rbox = 1.5R around the cluster: R box
Nesc (t) = 4π
n(r, 0) − n(r, t) r 2 dr.
0
3. Results The above scheme was used for the numerical study of electron escape from sodium clusters under Gaussian-type light pulses with half-height duration T = 10, 20, 40, 100, and 200 fs (full pulse widths are 25, 50, 100, 250, and 500 fs, respectively), peak intensities of I0 = 5 × 1012 –1 × 1014 W/cm2 , and photon energies ω of 1–3 eV (with a mesh width of 0.05 eV). Fig. 1 shows the change of Nesc during the interaction of Na8 with a 40-fs pulse at I0 = 2.5 × 1013 W/cm2 and different photon energies. For the characterization of the electron emission from clusters under a light pulse as a whole, Nesc was averaged over the last one third of the pulse. This quantity is denoted by Nesc . Fig. 2 presents Nesc for Na8 and Na40 under light pulses of different length and intensity. As would be expected, the increasing of
Vext (r, t) = zE(t) cos ωt. The field strength E(t) is taken to be a Gaussian form. To retain the spherical symmetry of the problem, we turn to spherical coordinates, represent z = r cos θ (where θ is the angle of an electron radius-vector r with an electrical vector E of the electromagnetic wave), and average Vext (r, t) over angular variables. This gives 2r E(t) cos ωt. (3) π Eq. (1) with the potential (2)–(3) reduces to the one-dimensional (with respect to spatial coordinates) Kohn–Sham equation which was solved with the mesh width for a time coordinate of 2.5 × 10−18 s. The initial state was taken from ground-state densityfunctional calculations [4]. The electron emission from clusters was evaluated through the number of Vext (r, t) =
Fig. 1. Time dependence of Nesc for Na8 : T = 40 fs (full length is 100 fs), I0 = 2.5 × 1013 W/cm2 , and photon energies as indicated. The pulse envelope is plotted by curve 1 (in arbitrary units).
L.I. Kurkina / Computer Physics Communications 147 (2002) 205–208
Fig. 2. Spectral dependence of Nesc for Na8 and Na40 : (1) I0 = 5 × 1012 W/cm2 , T = 20 fs; (2) I0 = 5 × 1012 W/cm2 , T = 200 fs; (3) I0 = 1 × 1013 W/cm2 , T = 100 fs; (4) I0 = 5 × 1013 W/cm2 , T = 20 fs; (5) I0 = 1 × 1014 W/cm2 , T = 10 fs. The pulses (2)–(5) carry the same energy.
pulse intensity at fixed length or pulse length at fixed intensity leads to the rise of the electron emission. The intensity dependence of Nesc is more essential. Really, a comparison of curves 2–5 in Fig. 2 has shown that the number of escaping electrons increases with intensity although the pulse energy is conserved through the respective decreasing of pulse duration. It has been found that ionization rises with increasing cluster size, too. In this case, when the spectra of Nesc are divided by cluster radii squared, at photon energies of up to 2 eV, the resulting plots for different clusters under the same light pulses are practically superimposed, in the 2–3-eV range, although the size dependence of the spectra persists (because of the size peculiarities of the cluster one-electron energy spectra), the maximum values of Nesc /R 2 come close together. Consider in more detail the variation of Nesc with cluster size, pulse intensity and length. At the minimum of the considered pulse intensities I0 = 5 × 1012 W/cm2 , the spectra of Nesc reflect the groundstate electronic structure of the clusters and, hence, can be formed by one-photon resonant absorption. So, for Na8 (the electronic configuration is 1s2 1p6 ), the enhancement of Nesc below 2 eV can be caused by transitions of 1p electrons to 1d and 2s levels (groundstate density-functional calculations predict energies
207
of these transitions to be equal to 1.23 and 1.90 eV, respectively). The single photon ionization threshold of the 1p shell is 3.23 eV. Between 2 and 3 eV the value of Nesc is close to zero for lack of allowed electronic transitions. For Na20 (1s2 1p6 1d10 2s2 ) and Na40 (1s2 1p6 1d10 2s2 1f14 2p6 ), bound–bound transitions from outer occupied shells also occur predominantly below 2 eV (the ground-state density-functional theory yields, for Na20 , 2s → 2p, 1.19 eV; 1d → 1f, 1.13 eV; 1d → 2p, 1.80 eV; for Na40 , 2p → 3s, 1.21 eV; 1f → 2d, 1.74 eV), and the single photon ionization thresholds for both clusters are about 2.7 eV. The peaks in Nesc at I0 = 5 × 1012 W/cm2 are slightly below the values mentioned last. This can be due to the polarization of clusters under the external electric field, which leads to the rise of one-electron energies, but weakly changes the distances between energy levels [4]. With increasing light intensity, the main peaks in Nesc are enhanced, extended, complicated, and blue-shifted. The broadening and displacement of the main ionization peaks are primarily related to the onset of over-the-barrier ionization (OBI) coupling with resonant absorption. Our calculations have shown that the conditions for the OBI of the clusters are realized at intensities I0 1 × 1013 W/cm2 . During the laser-cluster interaction, the OBI starts prior to the resonant ionization and leads to the fall of oneelectron energy levels in the cluster and to the increase of the distance between levels. As a result, the resonant absorption occurs at higher energies than follows from the unperturbed energy spectrum. In this case the greater the photon energy, the later the resonance happens. As is seen from Fig. 1, at ω = 1.80 eV the sharp increase of Nesc begins earlier than at ω = 2.25 eV. In both cases the electron escape is connected with the ionization decay of 1p electron excitation to unfilled electronic shells. At ω = 2.30 eV, the resonance is not observed, because the OBI has no time during the pulse to reduce cluster energy levels to the necessary value. With increasing light intensity the OBI grows, and one-electron energy levels are lowered. Because of this, the resonance peaks in Nesc are extended and blue-shifted. Furthermore, with increasing pulse intensity, the spectra of Nesc are complicated, which can be due to multiphoton absorption. The pulse duration has a weaker influence on the ionization spectra of clusters than the field intensity. The increase of pulse duration at fixed intensity affects
208
L.I. Kurkina / Computer Physics Communications 147 (2002) 205–208
Fig. 3. Spectral dependence of Wkin for Na40 : (1) I0 = 5 × 1012 W/cm2 , T = 20 fs; (2) I0 = 1 × 1013 W/cm2 , T = 100 fs; (3) I0 = 5 × 1013 W/cm2 , T = 20 fs. The pulses (2) and (3) carry the same energy.
mainly the resonant (one- or multiphoton) absorption and manifests itself in the enhancement of resonant peaks. To estimate cluster heating by light pulses, the variation of the kinetic energy Wkin (t) of electrons kept in the cluster was calculated as the difference between the kinetic energy of electrons inside Rbox at time t and the kinetic energy of electrons in the nonperturbative cluster at t = 0. As for Nesc (t), Wkin (t) was averaged over the last one third of a pulse (the result is designated as Wkin ). It has been
found that the shape of Wkin (Fig. 3) is similar to that of Nesc (Fig. 2). The most efficient “pumping” of energy in clusters occurs in the process of the resonant one-photon absorption. It should be noted that the size-dependence of Wkin is steeper than that of Nesc , and in the average Wkin is proportional to the fourth power of the cluster radius. In summary, the calculations have shown that at high light intensity the single-particle mechanisms can cause the strong multielectron ionization and heating of metal clusters. The efficiency of the process is determined mainly by the field intensity rather than the total energy of a pulse.
Acknowledgements This work was supported by RFBR (Grants No. 9915-96028, 00-03-33043).
References [1] [2] [3] [4]
E.K.U. Gross, W. Kohn, Adv. Quant. Chem. 21 (1990) 255. M. Brack, Rev. Mod. Phys. 65 (1993) 677. S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. L.I. Kurkina, O.V. Farberovich, Z. Phys. D 37 (1996) 359.