stabilization of one-dimensional soft-core and singular

Eur. Phys. J. D 59, 321–327 (2010) DOI: 10.1140/epjd/e2010-00119-3

Stabilization of one-dimensional soft-coreand singular model atoms

T. Dziubak and J. Matulewski

Eur. Phys. J. D 59, 321–327 (2010)DOI: 10.1140/epjd/e2010-00119-3

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Stabilization of one-dimensional soft-coreand singular model atoms

T. Dziubak and J. Matulewskia

Instytut Fizyki, Uniwersytet Miko�laja Kopernika, ul. Grudziadzka 5, 87-100 Torun, Poland

Received 30 March 2009 / Received in final form 6 October 2009Published online 5 May 2010 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2010

Abstract. Ultrastrong-field one-electron atom photoionization in one dimension reexamined. Theapplicability of soft-core and “cusp” potentials as models of a 1D hydrogen atom is revised in the contextof the stabilization phenomenon. Our ab initio numerical simulations reveal that the results obtained usingthe 1D smoothed Coulomb potential and those obtained for Coulomb potential with a singularity stronglydisagree.

1 Introduction

Recent experimental realizations of attosecond soft X-raylaser pulses [1–3] brought the interest back to quantumoptics of super-strong lasers interacting with atomic sys-tems. With this new perspective in mind we would liketo return to the basic concept developed in the late 80s,which allowed for first numerical investigations of atomicsystems in the presence of ultra-strong laser fields, namelywe would like to reexamine the widely used soft-core modelatom, introduced more than twenty years ago in [4–6] andcalled sometimes the quasi-Coulomb or “soft” Coulombpotential. Using this potential, several very importantresults were obtained in late 80s and early 90s of thepast century by numerical simulations of the dynamicsof atomic systems interacting with laser pulses, especiallyin an ultra-strong field regime, in which the standard per-turbation theory completely fails. The soft-core potentialenabled 1D numerical studies to be performed already atthe same time at which first experiments were performed,i.e. at late 80s and early 90s [7], while the singular poten-tial (even in 1D) causes serious numerical inconveniencesup to these days.

The phenomenon, which was in the center of the com-munity’s interest, was adiabatic stabilization against pho-toionization: a decrease of the ionization probability foran increasing laser field intensity [8,9]. This counterintu-itive feature of strong field ionization occurs for the laserfields intensity of a few atomic units if the laser cycle fre-quency is about unity (soft X-ray radiation). The expla-nation of this phenomenon is usually based on the ideaof the Kramers-Henneberger (KH) well [8], which is themean potential of the oscillating nucleus in the electron’sreference frame. The importance of the so-called slow drift,i.e. long-time oscillations caused by an interaction of an

a e-mail: [email protected]

oscillating electron driven by the field with the originalpotential, should also be taken into account [10]. Alreadyin early 90s the community were aware of the possible dif-ficulties of observing the stabilization in the presence ofthe one-dimensional singular Coulomb potential. However,most calculations were done using the classical mechan-ics and thus the problem was analyzed in terms of phasespace and acceleration [11–13]. Such a classical approachdoes not treat purely quantum mechanical phenomena likepacket tearing, which appears essential in our results. Acomparison of the fully quantum dynamics for the singularand smoothed potentials was made by Schwengelbeck andFaisal already in 1994 [14], who observed that a Coulombatom was easier to ionize than a a soft-core one, but theydid not study the stabilization phenomenon.

The other phenomenon which should be mentioned inthis context is high harmonics generation (HHG). Its un-derstanding, developed to a large extent due to ab initionumerical simulations performed for the soft-core modelpotential, was essential for an experimental realization ofan attosecond laser pulse mentioned in the first sentence.The modification in HHG due to the presence of the sin-gularity in the potential has only recently been reportedby Gordon et al. [15,16] for values of the Keldysh param-eter comparable to those of our simulations. The resultsobtained by those authors suggest that the presence of thepotential singularity is even more important than the dif-ference of the dimensionality of the system. For examplethe HHG spectra for a one-dimensional system with theCoulomb potential are similar to those observed for three-dimensional system and differ much from ones obtained forone-dimensional systems with a soft-core potential. Thisis against the common belief, which is that the singularityof three-dimensional potential is ineffective due to easinessof passing it by. Thus there exists at least one situationin which the presence of a singularity in the binding po-tential essentially influences the results in a similar way in

322 The European Physical Journal D

one- and three-dimensional case. In their most recent pa-per Kaestner et al. [17] investigate photoionization prob-abilities of H+

2 and HHG spectra, but not stabilization,in ultrastrong fields for singular and smoothed potentialsand also find essential discrepancies between the resultsfor soft-core one-dimensional models and those for truethree-dimensional ones.

Therefore it is important to verify how such a sin-gularity influences the presence of stabilization againstultrastrong-field photoionization in systems of various di-mensions. Unfortunately fully three-dimensional numeri-cal calculations are still barely within the capabilities ofpresent computers if grids with a large number of nodesin all directions are required. Note that this is the case ofultra-strong field photoionization of models with singularbinding potentials for which a large amplitude of spatialoscillations of the wave-packet needs using a grid with anextensive spatial range while rapid changes of the poten-tial around the origin – a small spatial step. Thereforesimulations for an arbitrary laser polarization still requireextraordinary numerical capabilities. However, for an ax-ial symmetry of the problem, that is for linearly polarizedlaser electric field, one can reduce the problem to a two-dimensional one [18–24], which already may be treatednumerically with a reasonable effort. It is parallelizationof calculations which opens new perspectives, in particularthe CUDA technology [25] is very promising. An approachto simulating the ionization of H2 using parallelization formassively-parallel processors has been made by Taylor andco-workers [26].

We have recently initiated 3D calculations of photoion-ization with an axial symmetry. They are extremely timeconsuming, but first observations may already be made.They show that, in contrast to 1D case, in 3D using a soft-ened potential as a model of the real Coulomb potentialmay be justified if just qualitative results are expected.Moreover, as concerns the stabilization, in 3D the poten-tial shape does not influence the results that much, sincethe singularity may be easily passed by; this is not thecase as concerns HHG for which the shape of the poten-tial is more important than the dimensionality [15,16]. Forthe stabilization in 3D the shape of the pulse envelopebecomes much more important. Our preliminary resultsconfirm the common belief that the 1D smoothed poten-tial better imitates the 3D hydrogen atom, than does the1D Coulomb potential.

We are aware that one-dimensional studies may be lessinteresting for experimentalists than three-dimensionalones, but they should be treated as a complement of someinvestigations which were highly important not only for-merly, but also currently – there are many papers whichdeal with a one-dimensional Coulomb potential and itsunusual properties (we list them in [27]).

To summarize our motivation, the aim of this paperis to finally settle whether the discrepancy of the predic-tion of obtaining the stabilization phenomenon for softand singular one-dimensional model atom is confirmed inab initio quantum simulations. We have recently founda similar discrepancy of the results, concerning an ana-

logue of the stabilization phenomenon, namely in numer-ical ab initio simulations of recombination in strong-fieldattosecond (few-cycle) laser pulses performed for singu-lar and smoothed Coulomb potentials in various dimen-sions [27].

2 Numerical method

The soft-core potential is given as follows:

V (x) = − V0√a2 + x2

(1)

(V0 = 1 atomic unit (a.u.) = 27.2116 eV is the potentialstrength, a = 1 a.u. = 0.53 A – its width). The soft-core potential is devoid of a singularity; it allows for asignificant decrease of the number of spatial nodes of thegrid on which the calculations are performed. Despite thesmoothed core, it keeps the most important feature ofthe long-range Coulomb potential (a = 0 a.u.), i.e. theRydberg series of the bound states, including the near-threshold ones. Moreover, the model potential is regular atthe origin and even; thus energy and parity have commoneigenstates. In addition it is free of mathematical difficul-ties typical of the 1D Coulomb potential, which consist inenergy degeneracy of even and odd eigenstates and of thederivative of the former ones being discontinuous at theorigin [28–31]; those states are here taken into account onequal footing.

Another model of the Coulomb potential may be con-sidered, namely:

V (x) = − V0

|x| + |b| (2)

(cf. [28]). For b = 0 it reproduces the singular Coulombpotential, while for b > 0 it produces a “cusp” – the poten-tial which is not smooth at x = 0, but it is finite there. Amore detailed description of this potential and the resultsobtained using it are given in Section 3.2 below.

One-dimensional numerical time-dependent calcula-tions were performed using the Crank-Nicholson schemeimplemented in our original software package. For ourcalculation we have used computers with AMD Opteronprocessors. The simulations were performed in the dipoleapproximation in the velocity gauge (because of its numer-ical properties), i.e. the Hamiltonian of the system reads(in atomic units, used throughout the paper)

H = −12

∂2

∂x2− iA(t)

∂

∂x+

12A2(t) + V (x), (3)

where the vector potential A(t) is obtained by integrationof the laser electric field:

ε(t) = −ε0f(t) cos(ωt). (4)

Three kinds of the envelope function f(t) were used in thepresent calculations. The first one was the square envelopegiven by

f(t) = Θ(t)Θ(tf − t), (5)

T. Dziubak and J. Matulewski: Stabilization of one-dimensional soft-core and singular model atoms 323

where Θ(t) is the Heaviside step function. In this case thelaser acts during the time interval (0, tf ), namely the 1T ,3T , 8T , 16T and 32T . The laser field frequency ω has beentaken equal to 1 a.u. (which corresponds with XUV laserfrequency, i.e. f = 658 THz) and thus the optical period isT = 2π a.u. = 150 as. We have changed the laser field am-plitude ε0 in the range between 0.05 and 5 a.u. which cor-responds to laser intensities 1014−1018 W/cm2. The de-tails of the algorithm were described in references [10,27].

The first smooth pulse envelope also used in our sim-ulations was given by

f(t) =

⎧⎨

⎩

sin2(ωt/12), 0 < t < 3T1, 3T < t < 11Tsin2(ω(tf − t)/12), 11T < t < 14T.

(6)

The pulse duration is tf = 14T , the turn-on and turn-off intervals are equal to 3T and the plateau lasts for 8T .These parameters were very often used in papers preparedby Eberly, Su and their co-workers.

A more realistic pulse envelope shape is given byGaussian function i.e.

f(t) = exp(−((t − 7T )/2.5T )2). (7)

This pulse lasts for 14T with maximum at t = 7T . Thewidth of the pulse equal to 2.5T causes that the pulsevanishes for times when the simulation begins and whenit ends and significant values appear only during about5 optical cycles. Results for smooth pulse envelopes willbe discussed in Section 3.3.

The number of spatial nodes depends on the potentialused. For the 1D hydrogen atom potential with a singu-larity it is necessary to use almost 220 nodes in the spa-tial grid in order to cover the range (−1200:1200) a.u =(−635,635) A (the spatial grid step is equal to aboutΔx = 0.003 a.u. = 0.00159 A). We introduce an artificialpotential cut-off for its value at x = 0 equal to −1015 en-ergy atomic units, which in fact concerns only one node.The potential values at neighboring nodes are equal to1/Δx > −330 a.u. (−8.9 keV). For the singular potentialthe time step was equal to T/105 a.u., while for the modelatom it could be ten times larger. Also the number of thespatial nodes in latter case was smaller by a factor of afew tens and a several times smaller range was sufficient.The convergence of the results was carefully checked, es-pecially in the former case, including the convergence fordifferent values of the potential cut-off. The results werethe same for the cut-off levels smaller than −1010 and startto significantly diverge for values larger than −105 (whichis much lower than cut-off used in [14]). Our simulationfor both Coulomb singular potential and smoothed modelpotential are in agreement with the results presented inreference [14] for selected laser intensities calculated there.

The ground state of the appropriate potential (the oddone in the Coulomb potential) was taken as an initial stateand the time-dependent populations of the ground statesas well as of all other essential bound states were calcu-lated. The eigenstates of the one-dimensional singular hy-drogen atoms may be found analytically, however the dis-cussion lasts whether the states with even symmetry have

any physical meaning (cf. [28–31] and [27]). In order to ob-tain the bound states of the smoothed model atom we usedour implementation [32] of the Davidson method [33–35]to solve the stationary Schrodinger equation.

3 Results

3.1 Revision of smoothed model potential

The best illustration of the discrepancy between the re-sults in both cases mentioned above is Figure 1 whichshows the evolution of the wave-function (the black-ness is proportional to its square modulus). The evi-dence of the wave-packet tearing in the vicinity of theCoulomb potential’s singularity and of an escape of thefreed wave-function’s parts is clearly visible (upper plot).The same electric field, namely ε0 = 2.5 a.u. (I0 =2.19 × 1017 W/cm2), applied to the electron bound bythe soft-core potential (middle plot) does not cause such astrong deformation of the wave-function. Only relativelysmall parts are freed in the direction dependent on thecurrent phase of the slow drift, but the shape of the wave-function is kept during the whole simulation. This differ-ence is not caused by the different energies of the initialstates (the ground state of the Coulomb potential is equalto −0.5 a.u. = 13.6058 eV while of the soft-core potentialequal to −0.67 a.u. = 18.2318 eV). We have checked thatthe soft-core potential with V0 = 0.78 a.u. = 21.2250 eVand thus with its ground state raised to −0.5 a.u. =13.6058 eV does not change the general wave-function’sappearance (lower plot of Fig. 1). It is clear that it isthe the shape of the potential around x = 0 which isresponsible for this difference. Though the Keldysh pa-rameter [36,37] in both cases is the same (approximately1/ε0) the picture of photoionization for a potential with asmoothed core better resembles the over-the-barrier ion-ization (OTB) than that for the singular potential (at leastup to ε0 = 5 a.u. and I0 = 8.755×1017 W/cm2). As a con-sequence the quasi-Coulombic or “soft” Coulomb poten-tial may barely be considered a model of the true Coulombpotential in 1D. Thus a straightforward generalization ofthe results obtained using this model potential to the caseof the Coulomb potential cannot be fully justified. See alsothe discussion below concerning the cusp potential, whichadvises one against identifying lack of the singularity asthe only factor responsible for the presence of the stabi-lization. A finite, but not smooth potential may also leadto the total bound states’ population being a monotonicfunction of the laser field intensity.

Different pictures of the wave-function are reflectedon both the time (not shown) and the laser intensitydependence of total bound states’ population (the sur-vival probability). The latter is shown at Figure 2. In thecase of the Coulomb potential with a singularity the sta-bilization phenomenon, i.e. the non-monotonicity of thefinal total population of the bound states as a functionof the field amplitude, is not visible, but the small fluc-tuations are seen at the very beginning of the evolution(the solid curve for t = 1T ). The laser with an intensity


Fig. 1. The time-dependent wave-function of an electron inthe presence of the Coulomb potential (upper plot), the soft-core potential with V0 = 1 a.u. = 27.2116 eV (middle) and thesoft-core potential with V0 = 0.78 a.u. = 21.2250 eV (lower),all interacting with the square laser pulse of (ε0 = 2.5 a.u., I0 =2.19× 1017 W/cm2). The level of the blackness is proportionalto the value of the probability density of finding the electron.

ε0 > 1 a.u. (I0 > 3.51 × 1016 W/cm2) ionizes the atomalready during a few optical cycles. Of course in practiceone can sum up populations of only a finite number ofbound states, twenty in our calculations, but the high-est of them was already almost empty. For ε0 > 1 a.u.(I0 > 3.51 × 1016 W/cm2) the population of even andodd states in the case of ionization are comparable (notshown). This was also the case in the recombination [27].

The same quantity in the case of the soft-core potentialis shown in the lower plot of Figure 2. As seen in Figure 1,the wave-function is not torn apart, thus the bound states’total population after the laser pulse switch-off is muchhigher than in the previous case. Moreover, the plot provesthe existence of the stabilization phenomenon for 1.5 <ε0 < 3.5 a.u. (7.9 × 1017 < I0 < 4.3 × 1017 W/cm2)for t ≥ 3T , for pulses longer than the time of the wave-function’s adjusting to the presence of the laser field (orin other words to the new binding potential – the KHwell) [10]. Also in the case of the soft-core potential withV0 = 0.78 a.u. = 21.2250 eV the total population as afunction of ε0 (not shown) looks much the same as in thecase of V0 = 1 a.u. = 27.2116 eV.

The insets in the upper and lower plots of Figure 2show the probability of finding the electron in the neigh-borhood of the nucleus (the integral of the wave-function’smodulus square in the range containing both classical

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

0.0 0.4 1.4 3.2 5.6 8.8

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

0.4 1.4 3.2 5.6 8.8

0

0.2

0.4

0.6

0.8

1

0.0 0.4 1.4 3.2 5.6 8.8laser intensity Ι0 ( × 1017 W/cm2)

a = 0

0

0.2

0.4

0.6

0.8

1

0.0 0.4 1.4 3.2 5.6 8.8laser intensity Ι0 ( × 1017 W/cm2)

a = 0

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

popu

latio

n of

bou

nd s

tate

s

laser field amplitude ε0 (a.u.)

a = 1

Fig. 2. The total population of the bound states as a functionof the laser electric field ε0 for various lengths tf of a rect-angular pulse. The case of the Coulomb potential is shown inthe upper plot, while that of the soft-core potential – in thelower plot. The solid line corresponds to the value after 1T , thedashed line – 3T , the dotted line – 8T and dash-dotted – 16T .The insets present the population calculated as the probabilityof finding the electron in the vicinity of the nucleus, namely inthe range of (−5, 15) a.u. = (−2.65, 6.36) A containing bothwave-packet turning points.

turning points). Both quantities correspond to variousdefinitions of the ionization level and thus of stabiliza-tion given by Su et al. [38] and by Geltman [39]. How-ever, in the examined case both quantities are in a fairlygood agreement, which convinces us that the slow driftdoes not influence much the ionization process. Only inthe case of the soft-core potential for the ε0 > 4 a.u.(I0 > 5.62 × 1017 W/cm2) is its presence visible.

3.2 The cusp model potential

In addition to 1D hydrogen atom potential (a = 0) andsmoothed model atom potential (a = 1) we performedcalculations for another model potential given by equa-tion (2). We have checked two values of b: 0.05 and 0.5. Theshape of the potential in the first case, outside the smallvicinity of x = 0, is similar to the singular potential. Theground state energy is equal to −4.26 (the minimal valueof the potential is equal to −20), that is it is located muchdeeper than the twice degenerate ground states of theone-dimensional Coulomb potential. The excited states’energies, but not the ground state, fulfil approximately


0

0.2

0.4

0.6

0.8

10.0 0.4 1.4 3.2 5.6 8.8

laser intensity Ι0 ( × 1017 W/cm2)

b = 0.05

0

0.2

0.4

0.6

0.8

10.0 0.4 1.4 3.2 5.6 8.8


b = 0.05

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5

popu

latio

n of

bou

nd s

tate

s


b = 0.5

Fig. 3. The same as in Figure 2, but for cusp model potentialgiven by equation (2). The upper plot presents results for b =0.05, the lower – b = 0.5.

the dependence −1.83/n2 (the lowest exact values are:−0.431,−0.254, −0.116,−0.0864). For such a potential weperformed simulations of the wavefunction evolution for arectangular pulse shape and various laser intensities. Thedependence of the total population obtained in this caseis visible at the upper plot of Figure 3. Evidently no man-ifestation of a nonmonotonic dependence is visible. Thusit is not the completely impassable singularity which maybe a single cause responsible for shattering the wavefunc-tion and its dispersion. A finite, but narrow potential actswith a similar effectiveness. One should however note animportant difference between the cusp potential and thetrue Coulomb potential. In the latter case the probabilitydensity of finding the electron at x = 0 is exactly equal tozero – all the eigenfunctions (including those of the con-tinuum) take zero value at x = 0. Contrary to that, in thecase of the cusp potential the ground state’s eigenfunctionas well as those of the excited states with an even symme-try possess a maximum at x = 0. This is reflected in thedynamics of both systems: in the Coulomb potential it isextremely easy to tear and spread the whole wavepacketand thus to empty the ground state during the first opticalcycle, while in the case of the cusp potential the popula-tion of the ground state decays slowly and monotonically,which manifests itself by the presence of a high and gradu-ally decreasing peak in the wavefunction located near thenucleus.

On the other hand the cusp potential with the parame-ter b = 0.5 is much more shallow and thus more similar tothe smoothed model potential (a = 1). Nevertheless, thenew potential, even the shallow one, still possesses thecusp which tears the wavefunction much more efficientlythan the smooth one. However, the tearing is much weakerthan in the case of b = 0.05 and the main part of the wave-function remains in the vicinity of the nucleus, oscillatingin the rhythm of the laser field. The ground state energyis equal to −0.860, which is comparable with the groundstate energy of the model atom and of the Coulomb poten-tial. The energies of the lowest excited states are −0.245,−0.146, −0.0842, −0.0611, which more or less fits the de-pendence −1.07/n2. The plot of the total population as afunction of the laser intensity is presented in the lower partof Figure 3. In this case the stabilization window is wellvisible, but it is less pronounced compared with the caseof a smoothed potential due to packet’s tearing (cf. thelower plot of Fig. 2). One can thus conclude that not onlythe presence of the singularity in the potential, but alsoof a sufficiently deep “cusp” causes wavepacket’s tearingand dispersion, which prevents the stabilization.

3.3 An influence of a smooth pulse envelope

The ionization in short and intense laser fields dependsstrongly on the initial phase of the laser as well as onthe pulse envelope. Thus we decided to check whether thediscrepancy due to the presence of a singularity or of asufficiently deep cusp of the potential at the nucleus ap-pears also for more realistic smooth pulse envelopes givenby (6) and (7). The evolution of the wavefunction calcu-lated for such pulses is presented in Figures 4 and 5. Inboth cases one can see the same discrepancy as that vis-ible for a rectangular pulse envelope. This again reflectson the dependence of the total final population of boundstates on the electric field intensity shown in Figure 6 forboth cases of smooth pulse envelopes. They look much thesame as those for rectangular pulses.

4 Conclusions

To summarize, basing on ab initio quantum-mechanicalcalculations, we have presented the differences in theionization process, in particular in the stabilizationphenomenon, simulated for systems with the singularCoulomb potential and its two models with a removedsingularity: a “soft” Coulomb potential and a “cusp” po-tential. The most important observation is that stabiliza-tion is distinctly seen in the case of the binding potentialbeing smoothed or having a shallow cusp. A singularityor a sufficiently deep cusp leads to tearing the wavepacketinto small parts which never more return in the vicinityof the nucleus. In the latter case the bound-states’ popu-lation quickly drops and no stabilization is observed.

Our extensive comparison of ultra-strong laser fieldionization of one-dimensional atomic models clearlyreveals essential discrepancies between the results for


Fig. 4. The same as in Figure 1, but for a smooth pulse withthe envelope given by equation (6).

Fig. 5. The same as in Figures 1 and 4, but for the smoothpulse with the envelope given by equation (7).

0

0.2

0.4

0.6

0.8

10.00 0.35 1.40 3.16 5.62 8.78


a = 0

14T (odd only) (s)14T (s)

14T (odd only) (g)14T (g)

0

0.2

0.4

0.6

0.8

10.00 0.35 1.40 3.16 5.62 8.78


a = 0

14T (odd only) (s)14T (s)

14T (odd only) (g)14T (g)

0.1

0.3

0.5

0.7

0.9

0 1 2 3 4 5po

pula

tion

of b

ound

sta

tes


a = 1

14T (s)14T (g)

0.1

0.3

0.5

0.7

0.9

0 1 2 3 4 5po

pula

tion

of b

ound

sta

tes


a = 1

14T (s)14T (g)

Fig. 6. The same as in Figure 2, but for the smooth pulseswith envelopes given by (6) (dashed line) and (7) (solid line).

soft- and hard-core atoms. As to a generalization forthree-dimensional atoms, a common belief as well as ourpreliminary results suggest that photoionization dynam-ics for such systems is better imitated by soft-core one-dimensional models than by hard-core ones. However, suchcomparisons require further studies, because there are pro-cesses like, e.g. HHG, for which other authors conclude theopposite.

We would like to thank A. Raczynski and J. Zaremba for ex-tensive and fruitful discussions and a help in preparing thismanuscript. This work was partially supported by the Euro-pean Social Fund and the budget of Poland within the In-tegrated Regional Operational Programme, measure 2.6 “Re-gional Innovation Strategies and transfer of knowledge” underthe project of Kujawsko-Pomorskie voivodship “Scholarshipsfor Ph.D. students 2008/2009 – IROP”.

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