Spallative ablation of dielectrics by X-ray laser
N.A. Inogamov, V.V. Zhakhovsky, A.Ya. Faenov, V.A. Khokhlov, V.V. Shepelev, I.Yu. Skobelev, Y. Kato, M. Tanaka, T.A. Pikuz, M. Kishimoto, M. Ishino, M.Nishikino, Y.Fukuda, S.V.Bulanov, T.Kawachi, Yu.V.Petrov, S.I.Anisimov, V.E.Fortov
aa r X i v : . [ phy s i c s . op ti c s ] D ec Spallative ablation of dielectrics by X-ray laser
N. A. Inogamov ⋆ , V. V. Zhakhovsky , , A. Ya. Faenov , , V. A. Khokhlov , V. V. Shepelev ,I. Yu. Skobelev , Y. Kato , , M. Tanaka , T. A. Pikuz , , M. Kishimoto , M. Ishino ,M. Nishikino , Y. Fukuda , S. V. Bulanov , T. Kawachi , Yu. V. Petrov , S. I. Anisimov ,V. E. Fortov L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Chernogolovka 142432, Russia Department of Physics, University of South Florida, Tampa, Florida 33620-5700, USA Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow 125412, Russia Kansai Photon Science Institute, Japan Atomic Energy Agency, Kyoto 619-0215, Japan Institute for Computer Aided Design, Russian Academy of Sciences, Moscow 123056, Russia The Graduate School for the Creation of New Photonics Industries, Hamamatsu, Shizuoka 431-1202, JapanReceived: 27-Nov-2009 / Revised version: date
Abstract
Short laser pulse in wide range of wave-lengths, from infrared to X-ray, disturbs electron-ion equi-librium and rises pressure in a heated layer. The casewhere pulse duration τ L is shorter than acoustic relax-ation time t s is considered in the paper. It is shown thatthis short pulse may cause thermomechanical phenom-ena such as spallative ablation regardless to wavelength.While the physics of electron-ion relaxation strongly de-pends on wavelength and various electron spectra of sub-stances: there are spectra with an energy gap in semi-conductors and dielectrics opposed to gapless continuousspectra in metals. The paper describes entire sequenceof thermomechanical processes from expansion, nucle-ation, foaming, and nanostructuring to spallation withparticular attention to spallation by X-ray pulse. PACS:
There are many industrial applications using short pulselasers. New exciting possibilities are connected with de-velopment of the X-ray lasers. Ablations by the long andshort pulses differ qualitatively. The first - evaporate,boil, and, at higher fluences, move matter by ablativepressure created in hot plasma corona. While the re-lease of pressurized layer is the main process in the caseof short pulse. It is shown below that this is true for anylaser wavelength.Let’s consider short pulse. Irradiation with sufficientintensity transfers substance into warm dense matterstate. In condensed phase, the cohesive properties areimportant. The cohesion is result of interatomic attrac-tion. Due to stiff behavior of solids and liquids, theirexpansion in rarefaction wave is very different from ex-pansion of gas. Stiff means that the typical moderate ⋆ email: [email protected] (N. Inogamov) expansions ∆ρ/ρ ∼ (0 . − .
2) cause the order of magni-tude pressure drops and even change of sign of pressure -from compressed to stretched state, here ∆ρ is the den-sity drop due to expansion. Stretched metastable layerunder tensile stress appears as a result of the release ofthe warm dense matter with its stiff response.In metastable state substances are sensitive to tem-perature and degree of stretching. Nucleation probabil-ity exponentially depends on amplitude of negative pres-sure. Therefore, sharp nucleation threshold appears onthis amplitude. The threshold depends on temperature.Nucleation is followed by development of two-phase layercomposed of condensed phase and voids. If the laserheated layer melts, then nucleation takes place in liq-uid. In this case, expansion of the two-phase layer leadsto foaming. The foaming may cause formation of thesurface nanorelief [1].The metastability and nucleation are well-known forrelease initiated by shock coming to the surface fromthe bulk of a target. But in the case of shock sent bya long laser pulse, an ion beam, or by an explosion ofchemical explosive, near nucleation threshold the two-phase layer locates far from the target surface. In thecase of large scale, the development of the two-phaselayer cannot disturb the surface of a target because thenucleation layer and the target surface are far from eachother and are independent from each other. On the con-trary, the short pulse lasers initiate foaming very closeto the surface - since the attenuation depth d att forX-ray photons, or thickness of a skin layer δ skin , maybe as small as ten nanometers. In this case the foamingstrongly interacts with the surface, eventually produc-ing frozen surface structures. Therefore we can use suchterms as nanofoam, nanostructures, or nanospallation todescribe the situation with small depth d att . Photon absorption and collisional processes are de-fined by photon energy and electronic structure. Infrared(IR) and visible radiation excite valent electrons whereasX-rays are absorbed mainly by internal shells in the one-
Inogamov, Zhakhovsky, Faenov et al. photon interactions. This is why the X-ray absorption isqualitatively similar for metals, semiconductors and di-electrics. Action of quanta from the one electron-Voltrange of energy depends strongly on existence of the for-bidden gap ∆. In metals, they are absorbed mainly viainverse Bremsstrahlung in the skin layer. In cases withgap, the seed Keldysh ionization, inverse Bremsstrahlungheating of ionized electrons and electron avalanche con-trol the rise of number of conduction electrons n e ( x, t )[2,3,4,5]. Significant heating takes place when plasmafrequency for these electrons ω pl ( n e ) overcomes laserfrequency ω L during the laser pulse. During the rest ofthe pulse, the substance with a gap absorbs laser energyin a skin layer. This means that during the rest of thepulse the absorption becomes similar to the absorptionof metals. This greatly increases spatial density J · cm − of absorbed energy. Estimates show that, in these condi-tions, laser electric field strength is comparable to atomicfields of external electrons, their wave functions are dis-torted by electromagnetic wave, and probabilities of themultiphoton and tunnel ionizations are significant.Relaxation of electrons to equilibrium, after the endof a pulse, depends on a band structure. In metals, freeelectrons cool due to electron-ion energy transfer and dueto thermal conductivity. The same is true for semicon-ductors which pass to metallic state during their melting.In substances which keep the gap after a pulse, the con-centration of free electrons n e and their temperature T e decrease after the end of a heating laser pulse as a resultof recombination, diffusion of electrons and holes, andelectron heat conduction. Three-body recombination isusually more significant than radiative recombination.Relaxation time t eq is of the order of 1-10 ps for allthese cases.An acoustic response time t s is necessary to decreasepressure few times in the laser-heated layer with thick-ness d T . It equals to t s = d T /c s in bulk targets, or t s = d f /c s in foils with small thickness d f < d T , where c s is sound velocity.Electrons are light, their velocities are high and theelectronic thermalization time τ e is small - typicallyit is at the femtosecond range. In metals, this corre-sponds to rather high values of T e ∼ T e ∼
300 Kthe e-e relaxation in metals is slow, since e-e collisionfrequency ν ee is small. If τ e ∼ −
10 fs then at thepicosecond time scale we have two thermodynamic sub-systems: electrons and ions. Total pressure p = p e + p i iscomposed of partial pressures. The contribution p e > T e − T i >
0; in the one-temperature state wehave p e = 0 . In our conditions, typical absorbed energy is E abs ∼ (0 . − E coh ; where E coh is heat of sublimation; e.g.,for Al E coh ≈ T i ≈ E abs / · atom − ) kK , p i ≈
19 ( E abs / · atom − )( n/ × cm − ) GPa , (1)if heat capacity is ≈ k B , and Gruneisen parameter is Γ i ∼ . The parameter Γ = V ( ∂p/∂E ) V links pressurerise and fast absorption of energy at the isochoric stage.At the two-temperature stage t < t eq , electron tem-peratures T e are much higher than T i . For metals, inthe Fermi-gas approximation, we have T e ∼ p E e /γ,T e = 17 Z − / ( n/ × cm − ) / ( E e / · atom − ) / kKfor t e = T e /T F < , k B T F = E F . Here E e ∼ E abs is electron energy per atom, not per electron. Z is thenumber of electrons per ion, γ = π n e k B / E F is theelectron heat capacity constant written in the Fermi-gasapproximation.Electronic contribution p e is significant when T e ≫ T i , then T e − T i ≈ T e . In this case, in metals, p e p au = 25 Z nn au E F E au s t e π / t e /
41 + t e − ! , (2)where t e is normalized temperature T e , and p au =2 . × GPa, n au = 6 . × cm − , /n / au =0 .
53 ˚A(Bohr radius), E au = 27 . E F /E au = (1 / π Zn/n au ) / is Fermi energy. The interpolation (2) has the right limitsfor t e ≪ , t e ≫ .p e ≈ . Z / ( n/ × cm − ) / ( T e / GPa (3)if t e < . Fig. 1 (Color on line) Comparison of the ablation thresh-olds (for incident fluence) as function of λ L and τ L : threesquares [3], filled circle [23], two empty circles - this work.pallative ablation 3 We call a laser pulse short if τ L < t s . Usually thecondition t eq < t s holds. In such a case, the rarefactionrelease, driven by p e , envelopes smaller mass than the p i -release, and nucleation takes place in one-temperaturestate. In the case when t eq > t s expansion, subsequentstretching and nucleation are connected with p e . In thetwo-temperature state, the vapor-solid and vapor-liquidcoexistence curves are shifted in the direction toward thetwo-phase region. This is a result of a blow out by theelectronic pressure of atomic system, composed of atomswhich attract each other. At the same absorbed energy,the pressure p e is few times smaller than p i , since thedegenerate and classic gases of electrons are softer andtheir Gruneisen parameters Γ e = 2 / Γ i ≈ . − . τ L < t s . Other parameters, such aswavelength (from IR to X-ray) and spectra ( ∆ = 0 or ∆ = 0) , are less significant. Therefore, spallative abla-tion is an important mechanism for removal of material.For IR and visible quanta hν ∼ d spot ≪ d T are predicted [1,9].For hν ∼ E abs together with small absorp-tion depth, similar to skin depth in metals, are achievedabove threshold fluence F brkd for optical breakdown.At the same time, heating E abs is a very sharp func-tion of absorbed fluence F abs near this threshold [2,3,4,5]. Therefore, spallative ablation is limited to thenarrow region near F brkd , because the value of heat-ing is restricted E abs < E lim − up , E lim − up ∼ . E coh [15]. Above this limit, cohesive property is weak againststrong stretching in a hydrodynamic rarefaction wave -expansion proceeds similar to expansion of heated gas(that is, without spallative plate). The cohesive prop-erty is responsible for creation of a spallative plate andspallative cupola [16]. The cupola is necessary for theinterference which results in appearance of the varyingin time Newton rings [16].The second circumstance is connected with the widthof the gap ∆. The semiconductors such as Si and GaAs,with rather narrow ∆, metallize during melting. In themeanwhile molten dielectrics remain in dielectric state.In this case, the cupola is dielectric, and Newton inter-ference oscillations are weak (weak oscillations due topresence of oxide film have been detected in [13]).The Newton rings are bright manifestation of exis-tence of spallative ablation. Appearance of rings meansthat a light wave interferes between spallative cupola andthe rest of the target. This is a very surprising example of spallative plate so thin that it is even transparent (!)to light - the skin depth is of the order of 10 nm. Inmore customary spallation by a long laser pulse [17,18],the plate is much thicker.Contrary to the case with quanta hν ∼ τ L /t s influences the maximum pressure cre-ated by absorption of laser energy. Then, new resultsconcerning freezing of nanostructures at a late stage arepresented. After that, the theoretical model of spalla-tive X-ray ablation, and experimental findings connectedwith this model, are described. It is shown that as a re-sult of the conditions, presented in the following lines(i), (ii) and (iii), the threshold F abl for the X-ray spalla-tive ablation is extremely low, in comparison with othercases with different laser wavelength λ L and durations τ L . This is illustrated in Fig. 1. The three mentionedconditions are: (i) 100% absorption of X-rays (no re-flection); (ii) negligible diffusion and heat conductionloses out from the small attenuation depth d att ; and(iii) smaller energy densities necessary for spallative ab-lation, in comparison with evaporative ablation of equalamount of material. Fast laser pulse transfers matter into energy containingstate similar to a state of a chemical explosive behinda front of a detonation wave. This transfer is a base forhydrodynamic release, metastable decay and spallation.As was said, the pulse is short if its duration τ L is com-parable or shorter than acoustic time t s . In this case,a pathway at a thermodynamic phase plane consists oftwo parts. One is passed during a pulse τ L , while theother - during an acoustic response t s . The first partcorresponds to the heating along the isochor ρ = ρ initial with complications concerning the two-temperature de-tails. The second part is formed by an approximatelyisentropic release along an adiabatic curve. It intersectsthe coexisting curve and penetrates into the two-phaseregion.The states of material near the target surface, irra-diated by a long pulse τ L ≫ t s , are located near thecoexistence curve. The hotter state relates to the largerabsorbed laser intensity. Pressure created by a long pulseequals approximately to the pressure p sat − vap of satu-rated vapor. The later is limited by pressure p cr in the Inogamov, Zhakhovsky, Faenov et al. critical point, e.g., for Al p cr ∼ . T short | bin and T long | bin of material at the coexisting curve, in cases with shortand long pulses, the maximum pressure p short | max atthe short pulse thermodynamic pathway is much higher;here the subscript ”bin” marks the coexistence curve alsocalled the binodal. A short pulse pathway deviates sig-nificantly from the coexistence curve into the high pres-sure condensed phase region. This is why the pressure p short | max is higher. The T short | bin corresponds to in-tersection of the part 2 of the short pulse pathway withthe coexistence curve. The first paper, where the path-way has been used, was the paper [11]. Now this usefulconception is widespread [1,6,8,24].The temperature T short | bin | abl corresponding to spalla-tive ablation threshold is significantly below the criti-cal temperature T cr . Increase of the absorbed fluence F abs above F abl , increases T short | bin above T bin | abl . There are distinct fluence F ev and temperature T bin | ev above which the spallative layer disappears [16]. The ra-tio T cr /T bin | ev depends on material properties and de-tails of the two-temperature stage. It seems that it islarger for Au in comparison with Al.For F > F ev , material expands without spallationplate. This regime is different from the ”phase explosion”[25] by long pulse with T long | bin ≈ T cr . Release of highpressure p > , created isochorically by short pulse,produces expansion with high rate of stretching ∂u/∂x, where u and x are along expansion direction. After nu-cleation in metastable state inside two-phase region theinertia of expanding matter inflates bubbles. The inertiais significant since the rate ∂u/∂x is high. Whereas inthe case of the ”phase explosion”, the expansion of two-phase mixture is driven more slowly by weaker forcesconnected with pressure difference ∼ p sat − vap − p out be-tween pressure inside bubbles and pressure outside thetarget surface.Importance of the ratio τ L /t s is illustrated in Fig. 2.The acoustic time for this case is t s ≈
100 nm / (5 . / s) =20 ps. We see that positive and negative pressures beginto decrease in their amplitude when the ratio τ L /t s be-comes larger than unity. Distributions of pressure in thebulk Al target are shown at the instant t = 10 ps afterthe maximum of intensity I ∝ exp( − t /τ L ) of pumppulse. We use two-temperature hydrodynamic code de-scribed in Ref. [26]. Relaxation time t eq is of the orderof 3 ps. As previously stated, a short pulse initiates a sequenceof stages: (i) release, (ii) metastable state, (iii) nucle-ation, (iv) evolution of foam. Nucleation takes place ifabsorbed fluence is above nucleation threshold F nucl . It x , nm048 P, GPa t=10 ps, Al,F abs =65 mJ/cm t L = 0.1 ps t L = 1 ps t L = 3 ps t L = 10 ps t L = 30 ps t L = 100 ps I nitialsurface Fig. 2 (Color on line) Strong decrease of maximum pressurefor long pulses. is important that nucleation does not mean tacitly thatspallative plate will run away. The spallative ablationthreshold F abl and F nucl are separated. The separa-tion ( F abl − F nucl ) /F abl belongs to the few % range.As was shown in [1], during release, the foam inflates tothickness comparable to the distance between the foamand free surface of a target. Therefore it perturbs surfacecausing appearance of the surface nanorelief.Development of the nanorelief is a dynamical phe-nomenon, caused by deceleration of free surface, infla-tion of foam, and surface tension resistance to inflation.There is the appearance time t appr when it develops.This time is much larger than t s because fluence F is ≈ F abl , and the nanorelief appears near the stop-ping point of free surface. Expansion velocities at thisstage are small, and the nanoreleif develops slowly. For-mation of the nanorelief may be experimentally observedas changes in reflectivity. There are changes due to sur-face nanorelief and due to absorption in thick foam undersurface [27].In Al, and in many other materials, the ablationthreshold F abl is higher than the melting threshold F melt , and then at F > F nucl bubbles appears inside themolten layer. There are two cases with slow developmentof foam: one when F nucl < F abs < F abl and anotherwhen F abs is slightly above F abl . In the first case, thefoam remains closed under the surface. An example ofthis is shown in Fig. 3. In the second case, there is processof slow detachment of the spallative plate. The plate isconnected by the random net of the liquid filaments withfoam. The filaments are stretched and break off one af-ter another during detachment of the plate. During thisprocess, a nanobrush from a forest of standing filamentsappears.The subsequent evolution is material and target (bulktarget versus foil, spot radial size) dependent. Let’s com- pallative ablation 5 ment first the dependence on material properties. Inheavy metals late stages of foam development are espe-cially slow. This is significant because there is competi-tion between the late links of the foam development cas-cade and the rate of conductive cooling. E.g., density andmaterial strength ratios for Au vs. Al are 19.3/2.7 and20/13 [7], and gold foam moves slower. In Au, the layerbetween foam and melting front is thinner than in Al,freezing temperature is higher 1337/933, molten layeris thicker, and thermal conductivity 318/237 is higher;while heat capacity and surface temperature 2.5 kK atablation threshold are approximately equal.Cooling due to electron heat conduction is important.In liquid state, the foam under target surface
F < F abl and nanobrush
F > F abl finally disappears because thesurface tension collapses bubbles with low vapor pressurein the foam and smooths down the threads and surfacebumps. Bubbles collapses when supporting them ten-sile stress decreases to zero. Contrary to this smoothing,the undersurface bubbles
F < F abl and ”nano-brush”
F > F abl remain in the final relief if conductive losesare fast enough to freeze them. The nano-brush may beaccompanied with frozen bubbles under. Comparison ofAu vs. Al shows that in the case of Au, this is easier.This picture describes appearance of black gold in re-cent experiments [1,28]. Focusing of the laser beam andthickness of foil are also significant, since tight focusingimproves cooling, while in thinner foils the reservoir toadopt heat from molten layer is smaller.Molecular dynamic (MD) simulation of developmentof foam and its freezing requires huge computer powerand smart multi-processors algorithm, because foam oc-cupies large volume and inflation/freezing processes arevery slow. In the work [1] the first part concerning theinflation of foam has been done. Here, we present newresults describing freezing.In order to include the electron thermal conductivityin the MD code, the electrons as classical particles wereadded to atom subsystem. Each ”electron” is assumedto associate with its host atom/ion. An electron alwaysallocates the position of host, but it has its own veloc-ity. The neighboring atoms may exchange their electronswith some frequency, determined from known experi-mental thermal conductivity of liquid aluminum/gold atmelting point. The pair of atoms, where electron-electronexchange takes place, is chosen randomly from the listof neighbors of each atom. The list contains neighboringatoms within cutoff distance.In addition to the frequency of electron-electron ex-change between atoms, we adjusted the mass of ”elec-trons”, in order to obtain the characteristic time of electron-ion energy exchange in collisions. The electron-ion scat-tering depends on the velocity vectors of an electron withmass m as a projectile and target ion with mass m according to equations: v ′ = ( m v + m v + m v n ) / ( m + m ) (4) Fig. 3 (Color on line) Freezing of Al foam as result of con-ductive cooling. Green and red colors correspond to solid andliquid Al. The melting/recrystallization front moves very slowat the time interval to which the left picture t = 154 ps be-longs. At the right picture t = 468 ps the overcooled liquidlayer with T = 570 K is partly frozen, the underlying bubblesare strongly deformed but survived. v ′ = ( m v + m v − m v n ) / ( m + m ) (5)where v is an electron-ion relative speed, and v ′ and v ′ are the velocities of particles in the laboratory systemafter collision. The unknown unit vector n is assumedto be distributed isotropically and is produced by using agenerator of random directions. The Eqs. (4,5) conservethe momentum and total energy of electron and ion sub-systems. Moreover, such a combined Monte-Carlo-MDapproach guarantees conservation of local charge neu-trality.Our tests indicate, that the described above electron-electron exchange and electron-ion collision procedureslead to energy diffusion through dynamic net of atoms,simulated by MD, and give a correct solution of contin-uum heat conduction equation for liquid aluminum/gold.Two stages of evolution of Al nanobubble chain areshown in Fig. 3. Fluence in this case is slightly belowspallative ablation threshold F abl . Therefore spallativeplate keeps its connection to target after the process ofinflation of bubbles and freezing. Our approach differsfrom the approach applied in the important paper [29],where the one-dimensional (1D) finite-difference schemehas been used. The scheme takes into account 1D ther-mal conductivity and is solved parallel to MD simu-lation. The 1D scheme employs transversally averagedtemperature fields from the parallel MD simulation.It is important that in case of foam our method allowsto describe spatial separation of thermal fluxes pumpingheat through threads and walls with small cross-section.This greatly delays cooling through foam. Therefore thebase of the foam, which is located from the bulk side,cools faster than the film between the foam and free sur-face as it is shown in Fig. 3 (b). A liquid layer abovethe bubbles in Fig. 3 (a) has temperature gradient from1380 K on its free surface to 1340 K at its internal sur-face contacting with bubbles. The liquid layer shown inFig. 3 (b) is in supercooled state, comp. with [30]. Itstemperature is T ≈
570 K which is below melting point933 . Inogamov, Zhakhovsky, Faenov et al. least. The freezing process fixes this correlation withformation of surface nanostructures having correlationlength a few inter-bubble distance.The foam may be partially frozen from the bulk side,when, in the case with fluence above ablation thresh-old F abs > F abl , still liquid film between the foam andfree surface of target begins to transform to the spalla-tive plate and the process of breaking of the liquid-solid threads starts. In this case the forest of the frozenthreads appears (black gold [1,28]). MD simulation showsthat the gold foam has a thick multi-level structure withsignificant drop of temperature and expansion rate ∂u/∂x through levels from one level to another toward the bulkside, see movies [31]. Detailed description of the MD sim-ulation results, concerning foaming and freezing, is be-yond the scope of our paper. Here we have to show thata short X-ray laser pulse may cause similar thermome-chanical phenomena as an optical pulse. Thick foam isformed if the fluence F is significantly above the thresh-old F abl . The ”flashes” of the successive nucleations dur-ing a process of formation of thick foam take place whenpositive pressure p is lowered by the decrement p str [27], where p str is material strength. At acoustic stagepressure is halved from the initial value to the value cor-responding to the established compression wave [7]. If p str does not depend on temperature then the distancesbetween successive flashes are equal. In case p str ( T ) thedistances is growing toward the bulk since temperaturedecreases and p str ( T ) increases.Near threshold F abl the foam evolution time ∼ . − ∼ t eq ∼ −
10 ps for hν ∼ F abl is lower than the melting thresh-old F melt . The work [15] is devoted to comparison ofthe spallative ablations from liquid and solid states. Ifwe rise fluence up to F melt in LiF then deceleration offree surface to almost stopping point will be not possi-ble, because to slow down motion it is necessary to fulfillthe condition F ≈ F abl . This means that in LiF near F abl there are no melting/freezing processes describedabove. But other substances, e.g. Al, have F melt < F abl . In these cases X-ray laser will cause nanostructuring.
Dependence of the attenuation depth d att ( hν ) takenfrom [32] is shown in Fig. 4. The right arrow correspondsto the Ag X-ray laser described in next Section. Thedepth d att is large for high energy photons. Then theacoustic response t s = d att /c s corresponding to hardphotons is long. For energy ¯ hω L = 12 keV , which willbe achieved soon at XFEL [33], it is t s ∼
50 ns. This is laser wavelength, nm0.11101001000 attenuation depth, micronsAl, 1s r = 2.69 g/cc 89.3 eVL-shellK-shell12 keV Fig. 4
Variation of d att ( hν ) with photon energy hν. very large time in comparison with the picosecond timescale of the atomic processes.The estimate of the spallative ablation threshold forthese hard photons is F abl = ζ d att n at E coh ≈
200 J / cm , where ζ = 0 . − . E abl = ζ E coh , E abl is energy per atom at the thresh-old, n at is the atom concentration in solid state. FutureXFEL lasers [33] will have 0 . < ¯ hω L < τ L ∼
20 fs , and fluence up toseveral hundred J/cm . Irradiation by such pulse cancause spallative ablation of huge sub-millimeter piece ofcondensed target.Equations describing the electron-ion non-equilibriumstage and hydrodynamic motion are ∂x/∂t = u, ρ∂x = ρ o ∂x o , ρ o ∂u/∂t = − ∂p/∂x o , (6) ρ o ∂E sume /ρ∂t = − ∂q e ∂x o − p e ∂u∂x o − ρ o ρ ˙ E ei + ρ o ρ Q, (7) ρ o ∂E i /ρ∂t = − ∂q i ∂x o − p i ∂u∂x o + ρ o ρ ˙ E ei , (8) ρ o n e /ρ∂t = − ∂j∂x o + Qu i + ν imp n e − κ rec n e (9)This is system of equations in Lagrangian coordinate x o corresponding to an initial position of a material point.The initial density profile inside the target is ρ ( x, t = −∞ ) = ρ o . Values x, u, p = p e + p i , E sume = n e u i + E e , E i , n e in (6-9) are functions on variables x o , t ; n e is electron concentration n e in the conduction band;˙ E ei = α ( T e − T i ) is an electron-ion energy exchangerate; q e,i = − ( ρκ e,i /ρ o ) ∂T e,i /∂x o are heat fluxes; j = − ( ρ D/ρ o ) ∂n e /∂x o is diffusion flux of electrons; κ e,i areelectron and ion thermal conductivities.Primary electrons and primary holes are producedduring X-ray pulse acting on LiF. Their kinetic and pallative ablation 7 potential energies are E e | prim ∼ −
30 eV and u i ,E e | prim + u i = hν = 89 . T e and concentration n e . The non-equilibrium electrons and holes relax to ther-malized state with secondary electrons and holes throughAuger processes, impact ionization, and three-body re-combination. Relaxation time is τ rel ∼ τ rel is smaller than the pulse duration τ L = 7 ps. There-fore, if we exclude the short time lapse τ rel in the be-ginning of the laser pulse, then we can include contribu-tion of primary particles into electron energy budget (7)through energy conservation (radiative loses are small).It is supposed in equations (7) and (9) that the lasersource ( ρ o /ρ ) Q supplies energy directly to secondaryelectrons and produces secondary electrons at the rate Q/u i , where u i ≈ ∆ is an ionization potential of sec-ondary electrons, ∆ ≈
14 eV is width of forbidden gapin LiF.Every primary electron produces ( hν = 89 . / ( u i + E e ) ≈ τ L = 7 ps, F = 10 mJ/cm , LiF, d att =28 nm gives n e | max = (1 − n at , T e | max ∼ | max corresponds to the maximum values. Thesevalues are achieved at the end of X-ray laser pulse. Theparameters correspond to the experiment described innext Section. Electrons are classical since their kineticenergy (3 / k B T e ∼ E F ∼ . E ei = α ( T e − T i ) ≈ αT e = AE e , A = 2 α/ k B , since heating ofa lattice by electrons is significant only when electronsare much hotter than a lattice. Lattice temperature for F = 10 mJ/cm is T at | max ≈
700 K. Pressure waveinitiated by this fast increase of temperature is shownin Fig. 5. The wave travels to the bulk side of a target(to the right side in Figure) from the irradiated surface,while temperature profile remains ”frozen” into matter.The point x = 0 is an initial position of the surface. Theprofile of the wave contains tail with negative pressure p neg (tensile stress). The amplitude | p neg ( t ) | is enoughto cause spallative ablation. Crater depth is 40-50 nm.These findings agree with experimental results describedbelow. The experiment has been performed with the Ne-likeAg soft x-ray laser (XRL) facility at JAEA Kansai Pho-ton Science Institute, working at transient collisionalscheme [22,34,35]. The XRL beam with an energy ∼ × x, nm -0.8-0.400.40.8 P, GPa
P(x, t = 20 ps)Ti(x, t = 5 ps)Ti(x, t = 20 ps) T i , K Fig. 5 (Color on line) Pressure and temperature profiles inLiF irradiated by X-ray laser pulse. crystal of 2 mm thickness and 20 mm diameter, by usinga spherical Mo/Si multilayer mirror of 1050 mm radiusof curvature (see Fig. 6). The total energy on the LiFcrystal of the XRL beam after passing 200 nm Zr filterand reflecting from the focusing mirror was ∼
170 nanoJin a single shot. The luminescence of stable color centers(CCs) [36,37,38,39] formed by XRL radiation, was usedto measure the intensity distribution in the XRL laserfocal spot [22,35]. After irradiation of the LiF crystalwith the XRL, the photo-luminescence patterns from thecolor centers (CCs) in LiF (shown in Fig. 6) were ob-served by using a confocal fluorescence laser microscope(OLYMPUS model FV300). An OLYMPUS BX60 mi-croscope in visible differential mode and an atomic forcemicroscope (AFM, TOPOMETRIX Explorer), operatedin the tapping mode, have been used for measurementsof the size of ablative spot. As it was shown in our previ-ous experiments [35], only about 6% of full laser energyis concentrated to the best focus spot of ∼
200 squaremicrons. This corresponds to energy ∼ orlaser intensity ∼ × W/cm . Two types of experimental investigations of XRL ab-lation threshold of LiF crystals were done. In the first ex-periments the Zr filter has been removed, and the XRLbeam expands without attenuation by the filter. Thispowerful beam has been focused on the surface of LiFcrystal. In Fig. 7 the AFM image of the focal spot of thisXRL beam is presented. The image is obtained after asingle laser shot. Ablation of crystal is clearly seen at theAFM image and traces. The ablation threshold for LiFirradiated by a single shot is 10 . . From thetraces in Fig. 7, we could see that the ablation depthsvaried between 30 and 55 nm. These values are close tothe theoretical crater depths calculated above.In the second type of experiments, the Zr filter wassettled inside the propagation path of the XRL beam.Three shots have been done in the same focusing spot
Inogamov, Zhakhovsky, Faenov et al. X (cid:882) ray laser Spherical (cid:3) mirror (cid:39) Z LiF (cid:3) detector X (cid:882) ray (cid:3) laser a) b)
X ray (cid:3) laser LiF d t tAg target f (cid:3) = (cid:3) (cid:3) mmFilter E’ XRL = (cid:3) (cid:3) nJ (cid:39) Z (cid:39) Z (cid:39) X (cid:39) X LiF (cid:3) detectorAg (cid:3)(cid:3) target XZ (cid:3) stage Images (cid:3) of (cid:3) X (cid:882) ray (cid:3) laser (cid:3) spots (cid:3) at (cid:3) different (cid:3) Z (cid:3) positions: c)) (cid:882) (cid:3) (cid:3) (cid:3) (cid:882) (cid:882) (cid:3) Z (cid:3)(cid:3) , (cid:3) mm+ (cid:3) Fig. 6 (Color online) (a) The experimental set up for recording the Ag XRL beam patterns near the best focus position on aLiF crystal. (b) Sketch of motion of a LiF crystal during experiments. (c) The patterns of XRL beam focusing spots recordedon a LiF crystal at -3 mm to +4 mm from the best focus position. AFM image of focal spot after irradiation ep t h , n m AFM (cid:3) image (cid:3) of (cid:3) focal (cid:3) spot (cid:3) after (cid:3) irradiation (cid:3)(cid:3) by (cid:3) full (cid:3) (10.2 (cid:3) mJ/cm ) (cid:3) XRL (cid:3) energy ~ (cid:3) (cid:3) nm D n m ~ 55 nm D ep t h , (cid:3) (cid:3) nm Fig. 7 (Color on line) AFM image and traces, taken through orthogonal directions, of the ablative spot on LiF crystal,irradiated by single shot with full (10.2 mJ/cm2) laser intensity of XRL beam pulse. with fluence of 5 mJ/cm for each shot. We could see inFig. 8 that in this case, a crater, with an ablation depthof about 50 nm, appears on the surface of the crystal.This is similar to the first experiment with single moreintensive XRL shot.The ablation threshold obtained in our experimentsis much smaller in comparison with previous experiments,see Fig. 1. Our threshold is 3400, 300, and 10 timessmaller than thresholds for nanosecond and femtosec-ond Ti:sapphire lasers, and for nanosecond 46.9 nm softXRL, respectively. It is shown that short pulse of XRL causes thermome-chanical response as in cases with optical lasers. Thereis spallative ablation as result of appearance of tensilestress which overcomes material strength above abla-tion threshold. Near threshold stretching of melt may beaccompanied by nanostructuring. Nanostructures freezedown if conductive cooling is fast. Value of the X-raythreshold is small in comparison with irradiation by longerwavelengths and/or longer pulse.Work has been supported by the RFBR grant No.09-08-00969-a (NAI, VVZh, VAK, and YuVP). This re-search has been partially supported by the Japan Min- pallative ablation 9
AFM (cid:3) image (cid:3) trace (cid:3) laser (cid:3) shots (cid:3)(cid:3) E (cid:3) = (cid:3) (cid:3) mJ/cm , (cid:3) W/cm (cid:3) um (cid:3) x (cid:3) (cid:3) um300 (cid:3) um (cid:3) x (cid:3) (cid:3) um (cid:3) laser (cid:3) shots (cid:3)(cid:3) E (cid:3) = (cid:3) (cid:3) mJ/cm , (cid:3) W/cm t h , n m (cid:3) nm D ep t Luminescent (cid:3)(cid:3)(cid:3) image Luminescent (cid:3)(cid:3)(cid:3) image (cid:3) um (cid:3) x (cid:3) (cid:3) um50 (cid:3) um (cid:3) x (cid:3) (cid:3) um 25 (cid:3) um (cid:3) x (cid:3) (cid:3) um AFM (cid:3) imageVisible (cid:3) imageVisible (cid:3) image
Fig. 8 (Color on line) (a) The luminescence, visible and AFM images of XRL beam focusing spots on the surface of a LiFcrystal obtained after one and three shots with XRL laser intensity 5 mJ/cm . Trace was done for AFM image, obtained inthe case of three shot irradiation of a LiF crystal. istry of Education, Science, Sports and Culture, Grant-in-Aid for Kiban A No 20244065, Kiban B No. 21360364and by the RFBR grant No. 09-02-92482-MNKS-a (AYaF,IYuS, TAP).
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