Angulo Gareta, Julio

Garcia Saiz, Maria

Gregory, C.

Khattak, Fida

Kohanoff, Jorge

Naz, Ghulam Shabbir

Neely, D.

Notley, M.

Riley, David

Sahoo, Sudhakar

Shearer, Francesca

Thornton, K.A.

Woolsey, N.C.

In this paper we describe experimental results on angularly resolved x-ray scatter from a sample of warm dense aluminium that has been created by double sided laser-driven shock compression. The experiment was carried out on the Central Laser Facility of the Rutherford Appleton Laboratory, using the VULCAN laser. The form of the angularly resolved scatter cross-section was compared with predictions based on a series of molecular dynamics simulations with an embedded atom potential, a Yukakwa potential and a bare Coulomb potential. The importance of screening is evident from the comparison and the embedded atom model seems to match experiment better than the Yukawa potential

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Structure of warm dense matter via angularly resolved x-rayscatterRiley, D., Angulo Gareta, J., Garcia Saiz, M., Khattak, F., Kohanoff, J., Sahoo, S., ... Neely, D. (2009). Structureof warm dense matter via angularly resolved x-ray scatter. Plasma Physics and Controlled Fusion, 51(12),[124036]. DOI: 10.1088/0741-3335/51/12/124036Published in:Plasma Physics and Controlled FusionDocument Version:Publisher's PDF, also known as Version of recordQueen's University Belfast - Research Portal:Link to publication record in Queen's University Belfast Research PortalPublisher rights© 2009 IOP PublishingGeneral rightsCopyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or othercopyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associatedwith these rights.Take down policyThe Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made toensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in theResearch Portal that you believe breaches copyright or violates any law, please contact openaccess@qub.ac.uk.Download date:15. Feb. 2017Structure of warm dense matter via angularly resolved x-ray scatterThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2009 Plasma Phys. Control. Fusion 51 124036(http://iopscience.iop.org/0741-3335/51/12/124036)Download details:IP Address: 143.117.23.73The article was downloaded on 20/11/2009 at 16:01Please note that terms and conditions apply.The Table of Contents and more related content is availableHOME | SEARCH | PACS & MSC | JOURNALS | ABOUT | CONTACT USIOP PUBLISHING PLASMA PHYSICS AND CONTROLLED FUSIONPlasma Phys. Control. Fusion 51 (2009) 124036 (6pp) doi:10.1088/0741-3335/51/12/124036Structure of warm dense matter via angularly resolvedx-ray scatterD Riley1, J J Angulo Gareta1, E Garcı´a Saiz1, F Y Khattak1, J Kohanoff1,S Sahoo1, G Shabbir Naz1, S F C Shearer1, K A Thornton1, C Gregory2,N C Woolsey2, M Notley3 and D Neely31 School of Mathematics and Physics, Queens University Belfast, Belfast BT7 1NN, UK2 Department of Physics, University of York, Heslington, York YO10 5DD, UK3 Central Laser Facility, Rutherford Appleton-Laboratory OX11 OQX, UKE-mail: d.riley@qub.ac.ukReceived 19 June 2009, in final form 10 September 2009Published 11 November 2009Online at stacks.iop.org/PPCF/51/124036AbstractIn this paper we describe experimental results on angularly resolved x-rayscatter from a sample of warm dense aluminium that has been created bydouble sided laser-driven shock compression. The experiment was carried outon the Central Laser Facility of the Rutherford Appleton Laboratory, using theVULCAN laser. The form of the angularly resolved scatter cross-section wascompared with predictions based on a series of molecular dynamics simulationswith an embedded atom potential, a Yukakwa potential and a bare Coulombpotential. The importance of screening is evident from the comparison andthe embedded atom model seems to match experiment better than the Yukawapotential.(Some figures in this article are in colour only in the electronic version)1. IntroductionWarm dense matter (WDM) is a state of matter that is of significant scientific interest. Itis a state intermediate between the solid state, where Coulomb forces dominate over thermalmotion and classical plasmas where the converse is the case. Its physical characteristics includepartial electron degeneracy with T ∼ TF where TF is the Fermi temperature and strong ion–ioncoupling; quantified by the strong coupling parameter: = (Ze)2akBTi, (1)where a is the ion-sphere radius, Ti is the ion temperature and Z is the average ionization.This parameter is important in determining the microscopic arrangement of the ions, whichat values of 10 starts to resemble that of a liquid metal [1]. Such matter is of importance,0741-3335/09/124036+06$30.00 © 2009 IOP Publishing Ltd Printed in the UK 1Plasma Phys. Control. Fusion 51 (2009) 124036 D Riley et alFigure 1. Schematic of the experiment.primarily due to its relevance to planetary interiors [2] and inertial fusion experiments. Onemethod for diagnosing such matter is by x-ray scattering [3–5]. For a quasi-monochromaticx-ray scatter source we can expect a spectrally integrated scatter signal, I (q), of the formbelow [6, 7];I (q) = IT[(fi(q) + ρ(q))2Sii(q) + ZbSinc(q) + ZfSee(q)], (2)where q is the scatter momentum, IT(q) is the classical Thomson cross-section for an electron,fi(q) is the ionic form factor, ρ(q) is the electron–ion correlation term and Sii(q) is the staticion–ion structure factor. These terms come together to give us the quasi-coherent Rayleighscatter from the ions. The terms S inc(q) and See(q) refer, respectively, to the incoherent bound-free Compton scatter from Zb bound electrons and the static electron structure factor for scatterfrom Zf free electrons per atom. For a moderate Z plasma at modest temperature, the lattertwo terms are usually small compared with the Rayleigh scatter, e.g. for Al at the range ofconditions in the current experiment, the Thomas Fermi model leads us to expect between 10and 11 bound electrons on average compared with two or three free electrons. Since the ionicform factor factor scales as Z2, we can see how Rayleigh scatter dominates. Indeed, in anearlier paper [8] it was seen in high resolution scatter spectra that the Rayleigh scatter wasclearly dominant over incoherent (spectrally shifted) scatter.2. Experimental arrangementsThe experiment was performed at the Rutherford Appleton Laboratory using the multi-beamlaser facility VULCAN. A schematic of the experiment is shown in figure 1. The main sixbeams, delivering nominally 600 ps pulses of 200 J each in IR, were frequency doubled to527 nm (with up to 100 J per beam) and synchronously focused in two opposing groups ofthree onto the sample. The focusing optics made use of phased zone plates (PZPs) combined2Plasma Phys. Control. Fusion 51 (2009) 124036 D Riley et alFigure 2. Optical streak measurement of the temporal profile for the shock driving pulse.with f/10 lens to generate a 3 mm flat topped focal spot (with 5 mm offset from the best focusto reduce the level of the associated central spike). As can be seen from figure 1 the samplefoil was at 45◦ to the incident beams, meaning that they created an elliptical focal spot withmajor and minor diameters of 4.2 mm and 3 mm, respectively. The pulse shape in secondharmonic was measured with the use of an optical streak camera and can be seen in figure 2.The peak intensity achieved on target was ∼4.5 × 1012 W cm−2. By careful adjustment ofbeam energies, the balance between the opposing bunches was kept to within ∼10%. The useof three beams per side was useful in keeping the shock drives balanced as it smoothed outshot to shot fluctuations on individual beams.The target itself consisted of a CH/Al/CH sandwich with thicknesses 4.5/6/4.5 µm. Infigure 3 we can see a HYADES [9] simulation of the density and temperature for three differenttimes after the start of the pulse reaching the target surface. The slight asymmetry is a resultof 10% imbalance in the beams for the shot and illustrates the maximum asymmetry typicallyencountered. The graphs show only the conditions for the Al core of the sample. The simulationused multi-group radiation transfer with 35 groups logarithmically spaced from 0.01–15 keVand the SESAME equation of state tables [10]. In an earlier publication [8] we showed thatfor one sided irradiation of similar targets at similar pulse durations and intensities, the timefor shock breakout predicted by HYADES closely matched experimental measurements, theextent that we have enough confidence in the simulated plasma conditions to be confidentof conclusions drawn from the experimental scatter data. Two shorter pulse beams of 80 psduration were also frequency doubled and synchronously focused with f/10 lenses onto a3 µm thick Ti target to generate the backlighting x-ray beam. The intensity on target wascontrolled by changing the focal spot dimension. The back-lighter x-ray signal principallyconsisted of the He-α line (1s2–1s2p 1P and satellites) emission at 4.7–4.75 keV. This is by farthe dominant spectral feature in the few kiloelectronvolts spectral region and for the intensitiesused here has a duration similar to the optical pulse [11] thus allowing ∼100 ps resolution.The emission level was monitored with the help of two flat-crystal spectrometers coupledCCD systems, one in the rear employing Si (1 1 1) and one in the front of the target makinguse of Ge(2 2 0) crystal. After passing through an array of two pinholes, the probing x-rayback-lighter beam was restricted to a narrow cone which determines the angular resolution ofthe experiment. This narrow cone was then incident on the sample. However, high resolutionrestricts the incident signal to a lower signal level and a spread of 4◦ was employed as acompromise between resolution and signal level. This meant that the x-ray probe covered a3Plasma Phys. Control. Fusion 51 (2009) 124036 D Riley et al00.511.522.53-8 -4 0 4 8distance (microns)density (g/cc)(a)2ns1.5ns1ns00.511.522.5-8 -4 0 4 8distance (microns)Ion temperature (eV)(b)2ns1.5ns1nsFigure 3. (a) Mass density profile for double sided illumination at 5 TW cm−2 at 1, 1.5 and 2 nsafter the peak of the driving pulse. (b) Ion temperature for same conditions and timingsfocal spot ∼1×1.4 mm, somewhat smaller than the shocked area. By delaying the back-lighterbeams relative to the shock driving beams, the back-lighter x-rays could be made to probe thesample at a range of times relative to the shock compression. An Andor dual-chip x-ray CCDsystem, covering a range from 50◦ to 80◦ scattered angle, was used in the photon countingmode to detect the scattered signal. Appropriate shielding was in place to prevent x-rays fromthe back-lighter source from hitting the CCD directly. As for previous experiments, in orderto ensure that we have only scattering from the shock compressed region, we took several testshots where the entire assembly including the target holder was put into place but without theactual sample foil—the subsequent shots resulted in detected photon counts less than 5% ofthe typical scatter signals, thus demonstrating that our signals come from the shocked sample.The dual-chip CCD has two chips of 27.6 × 6.9 mm2 dimension sited side by side. Thechips were ∼12 cm from the sample. Each third of a chip thus subtended ∼4.5◦ thus matchingthe angular resolution of the probe cone of x-rays. We made histograms centred about fivepositions on each chip so as to ‘over-sample’. Filtering of 250 µm Be, 72 µm Mylar and 39µmAl allowed transmission of just under 6% of the Ti-He-α photons whilst severely suppressingthe kiloelectronvolt photon emission from the hot CH plasma on the surface of the samples.As in earlier, work, [12] we use single photon counting to measure the number of photonsscattered into each unit of solid angle. Although the crystal spectrometers monitoring thethroughput of the back-lighter source onto the target gave us shot to shot comparisons, thecrystals were not absolutely calibrated and so comparisons with theory in this paper are madewith respect to variation with angle rather than absolute cross-sections.3. Experimental resultsIn figure 4 we can see cross-section measurements taken at 1.5 ns and at 2 ns after the startof the shock driving beams with conditions similar to those illustrated in figure 3. The errorbars are based on counting statistics of the photons in each histogram. We have compared thedata with some simulations. Firstly, we can see that the one component plasma (OCP) model,which assumes that the ions are embedded in a uniform electron gas and interact via the bareCoulomb potential, predicts a distinctive peak. However, a molecular dynamics model basedon an embedded atom potential fits the shape of the experimental data very well. The embeddedatom model [14] was developed from condensed matter physics but has been applied to liquidmetals [15] and has been used to investigate phenomena such as melting and freezing [16]. Thistype of model effectively accounts for multi-body interactions and screening via a potentialthat includes direct pair potentials as well as the energy of embedding an atom in the local4Plasma Phys. Control. Fusion 51 (2009) 124036 D Riley et al024681040 50(a) (b)60 70 80Angle of scatter (º)Scatter (barns/Sr)Experiment 1.5nsOCPYukawa-MDEAM-MD024681040 50 60 70 80Angle of scatter (º)Scatter (barns/sr)Experiment 2nsOCPYukawa-MDEAM-MDFigure 4. Scatter cross-section at two different delay times compared with three models. Notethat, because we do not have an absolute calibration, the experimental data have been scaled tothe level of the EAM–MD model. However, comparison of the shape of the cross-section as afunction of angle leads to clear conclusions. The OCP model shows a distinctive peak, which isclearly reduced on introducing screening with a Yukawa type potential. However, it is clearly theembedded atom–MD model that fits quite well in both cases.electron density contributed by all neighbours. Its parameters are generally adjusted to matchexperimental values for a range of thermodynamic properties. We have used this type ofmodel previously [8, 16]. We do not have calibrations of the crystal spectra and CCD singlepixel counting efficiencies, although estimates of these yield cross-sections within a factor of2 of those expected. Accordingly, in the absence of accurate calibrations, we have scaled theexperimental data to best show the fit to the EAM–MD model in this case. We can readilyunderstand why the unscreened OCP does not work by considering the screening length, λ,expected for the electron gas to screen the ions (eg see [13]):λ−2 = 4e2meπh¯3∫fe(p) dp, (3)where fe(p) is the Fermi–Dirac distribution for electrons with momentum, p. By usingthis, we can calculate that for the conditions of figure 3, we have λ/a ∼ 0.3. We can notethat, in equation (1), we define the strong coupling parameter in terms of a bare Coulombinteraction between the ions, yielding values of  ∼ 60–80. If the interaction is screenedwith a screening length λ defined above, we can define an effective strong coupling parameter′ =  exp (−a/λ) and, with the screening calculated above, ′ ∼ 2–3. The electron–electronradius, rs in units of the Bohr radius is ∼2.7–3 for this data and the OCP is only expected toreally apply when rs < 1. With this in mind, we have also compared the data with a screenedCoulombic potential of the formV (r) = Ze24π0re−r/λ (4)with λ determined as in equation (3). We can see that, although the effects of screening areobvious from comparison with the OCP, the fit to the data is still not as good as for the EAM–MD simulation. We note, in passing, that for the data, it appears that the peak in scatter isat a higher angle for the later time, lower density case. This is counter-intuitive and may bea result of the broadness of the peak which, combined with statistical uncertainty (the errorbars represent only one standard deviation), serves to make the exact peak position less welldefined.5Plasma Phys. Control. Fusion 51 (2009) 124036 D Riley et al4. ConclusionsWe have made angularly resolved x-ray scatter measurements from a WDM sample. Theelectron density is not high enough to allow screening by electrons to be neglected. By usingan embedded atom model we have fitted the shape of the data well, although we do not haveabsolute calibration. A comparison with an unscreened simulation shows the degree to whichscreening makes a difference. Interestingly, a model using a linear screening model (Yukawapotential) did not match the data as well as the embedded atom approach. In future experimentsit may be possible to gather data from more uniform samples with smaller statistical error barsin order to make detailed investigations of the ion–ion structure factor and different approachesto its calculation.AcknowledgmentThis work was funded by the UK Engineering and Physical Sciences Research Council grantsEP/D031532/1 and GR/R09572/01.References[1] Hansen J-P and McDonald I R 2006 Theory of Simple Liquids 3rd edn (New York: Academic)[2] Ichimaru S 1982 Rev. Mod. Phys. 54 1017[3] Woolsey N C, Riley D and Nardi E 1998 Rev. Sci. Instrum. 69 418[4] Riley D et al 2000 Phys. Rev. Lett. 84 1704[5] Glenzer S H et al 2003 Phys. Rev. Lett. 90 175002[6] Chihara J 2000 J. Phys.: Condens. Matter 12 231[7] Nardi E 1991 Phys. Rev. A 43 1977[8] Garcia Saı´z E 2008 Phys. Rev. Lett. 101 075003[9] Larsen J T and Lane S M 1994 J. Quant. Spectrosc. Radiat. Transfer 51 179[10] Lyon S P and Johnson J D 1992 Group T-1, Los Alamos National Laboratory Technical Report LA-UR-92E3407[11] Riley D, Woolsey N C, McSherry D, Khattak F Y and Weaver 2002 Plasma Sources Sci. Technol. 11 484[12] Riley D, Woolsey N C, McSherry D and Nardi E 2000 J. Quant. Spectrosc. Radiat. Transfer 65 463[13] Wu¨nsch K, Vorberger J and Gericke D O 2009 Phys. Rev. E 79 010201 R[14] Daw M S and Baskes M I 1984 Phys. Rev. B 29 6443[15] Mei J, Davenport J W and Fernando G W 1990 Phys. Rev. B 43 4653[16] Mei J and Davenport J W 1992 Phys. Rev. B 46 216

Structure of warm dense matter via angularly resolved x-ray scatter

Structure of warm dense matter via angularly resolved x-ray scatter

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