ultrabright x-ray laser scattering for dynamic warm dense ... · max planck institute for the...

Correction notice Nature Photonics 9, 274–279 (2015)

Ultrabright X-ray laser scattering for dynamic warm dense matter physicsL. B. Fletcher, H. J. Lee, T. Döppner, E. Galtier, B. Nagler, P. Heimann, C. Fortmann, S. LePape, T. Ma, M. Millot, A. Pak, D. Turnbull, D. A. Chapman, D. O. Gericke, J. Vorberger, T. White, G. Gregori, M. Wei, B. Barbrel, R. W. Falcone, C.-C. Kao, H. Nuhn, J. Welch, U. Zastrau, P. Neumayer, J. B. Hastings and S. H. Glenzer

In the original version of Supplementary Fig. S3 the top panel was missing labels on the right-hand vertical axis and the vertical scaling of curves in the bottom panel was incorrect. Neither of these issues affect the analysis or conclusions of the Article and both were corrected on 30 July 2015.

© 2015 Macmillan Publishers Limited. All rights reserved

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHOTON.2015.41

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Supplementary information: Ultra-bright x-ray laser scattering for dynamic warm dense matter physics L. B. Fletcher1,2, H. J. Lee1, T. Döppner3, E. Galtier1, B. Nagler1, P. Heimann1, C. Fortmann4, S. LePape3, T. Ma3, M. Millot2,3, A. Pak3, D. Turnbull3, D. A. Chapman5,6, D. O. Gericke5, J. Vorberger7, T. White8, G. Gregori8, M. Wei9, B. Barbrel2, R. W. Falcone2, C.-C. Kao1, H. Nuhn1, J. Welch1, U. Zastrau1,10, P. Neumayer11, J. B. Hastings1 and S. H. Glenzer1

1SLAC National Accelerator Laboratory, 2575 Sand Hill Road, MS 19, Menlo Park, CA 94025. 2Physics Department, University of California Berkeley, Berkeley, CA 94709, USA. 3Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, USA. 4QuantumWise A/S, Lersoe Parkalle 107, 2100 Koebenhavn, Denmark. Plasma Physics Group, 5AWE plc, Aldermaston, Reading RG7 4PR, UK. 6Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry CV4 7AL, UK. 7Max Planck Institute for the Physics of Complex Systems, Noethnitzer Strasse 38, 01187 Dresden, Germany. 8University of Oxford, Parks Road, Oxford, OX1 3PU, UK. 9General Atomics, San Diego, CA. 10Institute for Optics and Quantum Electronics, Friedrich-Schiller-University 07743 Jena, Germany, 11 GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1 64291 Darmstadt, Germany.

In comparison with optical Thomson scatteringS1, the x-ray scattering processS2 can also access the single particle regime measuring electron velocity distribution functions or the collective scattering regime observing electron plasma (Langmuir) oscillations, ion acoustic waves and other collective phenomenaS2,S3. However, the energy of the incident x-ray photon with frequency ω0 is large enough to give a significant Compton shift to the frequency of the scattered radiation. During the scattering process, the incident photons transfer momentum ℏk and energy ℏ𝜔𝜔 = ℏ𝑘𝑘 ! ∕ 2𝑚𝑚! to the electrons. In warm dense matter, momentum and energy is primarily transferred to delocalized electrons or to those whose binding energy is less than the energy transferred to the electrons by Compton scattering. These electrons are considered weakly bound.

In this study, the forward x-ray scattering spectra from solid aluminium show a well-resolved plasmonS3 feature that is down-shifted in energy by 19 eV from the incident 8 keV elastic scattering feature. Measurements with the backward spectrometer in this energy range shows no scattering feature with the same shift providing strong evidence of a collective plasmon phenomenon as predicted for these conditions; bound-free scattering features are predicted to be negligible in this energy range and no feature has been observed with the Compton spectrometer. At larger energy shifts, the backscatter spectrometer observes the Compton scattering feature downshifted in energy by 250 eV; in our conditions the Compton scattering spectrum is not sensitive to the temperature reflecting the Fermi velocity distribution of the nearly degenerate state.

To resolve the plasmon spectrum our measurements require a seeded x-ray beam. In addition, the measurements use a highly efficient graphite crystal spectrometer (HAPG) with suitable resolution. The crystal in the forward scattering spectrometer is 40 µm thick while in backscattering we employed a 100 µm thick crystal giving rise to slightly different instrument functions and consequently slight differences in the spectral shape of the elastic scattering feature.

For the analysis of x-ray scattering experiments, we developed a comprehensive analysis toolS2,S4-S7 that is based on the Chihara formulaS8. The formula describes the contributions to elastic scattering and inelastic scattering from free (delocalized) and weakly bound electrons by the dynamic form factor

€

S k,ω( ) = f I k( ) + q k( )2Sii k( ) + Z f See k,ω( ) + Zc SCE k,ω −ω'( )SS k,ω'( )dω '∫ (1)

with Zf and ZC denoting the number of free and weakly bound core electrons, respectively.

The last term of Eq. (1) includes inelastic scattering by weakly bound electrons, which arises from bound–bound and bound– free transitions to the continuum of core electrons within an ion, SCE(k,ω), modulated by the self-motion of the ions, represented by SS(k,ω). The corresponding spectrum of the scattered radiation is that of a Raman-type band. Recently, there has been much activity to improve the


NATURE PHOTONICS | www.nature.com/naturephotonics 11

Supplementary information: Ultrabright X-ray laser scattering for dynamic warm dense matter physics L. B. Fletcher1,2, H. J. Lee1, T. Döppner3, E. Galtier1, B. Nagler1, P. Heimann1, C. Fortmann4, S. LePape3, T. Ma3, M. Millot2,3, A. Pak3, D. Turnbull3, D. A. Chapman5,6, D. O. Gericke5, J. Vorberger7, T. White8, G. Gregori8, M. Wei9, B. Barbrel2, R. W. Falcone2, C.-C. Kao1, H. Nuhn1, J. Welch1, U. Zastrau1,10, P. Neumayer11, J. B. Hastings1 and S. H. Glenzer1


In comparison with optical Thomson scatteringS1, the x-ray scattering processS2 can also access the single particle regime measuring electron velocity distribution functions or the collective scattering regime observing electron plasma (Langmuir) oscillations, ion acoustic waves and other collective phenomenaS2,S3. However, the energy of the incident x-ray photon with frequency ω0 is large enough to give a significant Compton shift to the frequency of the scattered radiation. During the scattering process, the incident photons transfer momentum ℏk and energy ℏ𝜔𝜔𝜔𝜔 = ℏ𝑘𝑘𝑘𝑘 ! ∕ 2𝑚𝑚𝑚𝑚! to the electrons. In warm dense matter, momentum and energy is primarily transferred to delocalized electrons or to those whose binding energy is less than the energy transferred to the electrons by Compton scattering. These electrons are considered weakly bound.




€




1

Supplementary information: Ultra-bright x-ray laser scattering for dynamic warm dense matter physics L. B. Fletcher1,2, H. J. Lee1, T. Döppner3, E. Galtier1, B. Nagler1, P. Heimann1, C. Fortmann4, S. LePape3, T. Ma3, M. Millot2,3, A. Pak3, D. Turnbull3, D. A. Chapman5,6, D. O. Gericke5, J. Vorberger7, T. White8, G. Gregori8, M. Wei9, B. Barbrel2, R. W. Falcone2, C.-C. Kao1, H. Nuhn1, J. Welch1, U. Zastrau1,10, P. Neumayer11, J. B. Hastings1 and S. H. Glenzer1


In comparison with optical Thomson scatteringS1, the x-ray scattering processS2 can also access the single particle regime measuring electron velocity distribution functions or the collective scattering regime observing electron plasma (Langmuir) oscillations, ion acoustic waves and other collective phenomenaS2,S3. However, the energy of the incident x-ray photon with frequency ω0 is large enough to give a significant Compton shift to the frequency of the scattered radiation. During the scattering process, the incident photons transfer momentum ℏk and energy ℏ𝜔𝜔𝜔𝜔 = ℏ𝑘𝑘𝑘𝑘 ! ∕ 2𝑚𝑚𝑚𝑚! to the electrons. In warm dense matter, momentum and energy is primarily transferred to delocalized electrons or to those whose binding energy is less than the energy transferred to the electrons by Compton scattering. These electrons are considered weakly bound.




€




© 2015 Macmillan Publishers Limited. All rights reserved© 2015 Macmillan Publishers Limited. All rights reserved

http://dx.doi.org/10.1038/nphoton.2015.41

2 NATURE PHOTONICS | www.nature.com/naturephotonics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHOTON.2015.41

2

theoretical description of this term, e.g., Refs. S9, S10. For the plasmon scattering spectrum measured in this study, we find that this term is not important compared to the free electron dynamic structure.

The second term describes the contribution to the scattering from free (delocalized) electrons that do not follow the ion motion. Here, See(k,ω) is the high frequency part of the electron-electron correlation function and it reduces to the usual electron feature in the case of an optical probe. Under the assumption that inter particle interactions are weak, so that the nonlinear interactions between different density fluctuations are negligible, this term is often calculated in the random phase approximation (RPA)S2,S4. In the classical limit, it reduces to the usual Vlasov equation. In the limit of the RPA, strong coupling effects are not accounted for, thus limiting the model validity. To expand the theoretical modeling into the warm dense matter regime, the Born-Mermin approximation (BMA) has been developedS11,S12. The theory was subsequently improved to allow for different models for the dynamic collision frequency and for local field corrections (LFC)S13. Its application to laser experiments on compressed boron has shown that the description of the plasmon dispersion is greatly improved with this theoryS14. However, at this time, the theoretical description of many-body effects in warm dense matter is not complete.

Here, we analyze the increase of the plasmon shift with material compression using the BMA-LFC. This effect relies on the relative increase in plasma frequency with density and is well understood compared to the absolute plasmon energy at arbitrary wavenumber. Figure S1 demonstrates that for our conditions the calculated plasmon shift is not sensitive to the choice of the model for See(k,ω). For a scattering angle of 13˚ the plasmon shift calculated with BMA and BMA-LFC agree. Only with increasing scattering angle do we observe discrepancies between these models.

Figure S1 | Plasmon frequency shift. Calculations are shown of the plasmon frequency in compressed

aluminium for three scattering angles, temperature of 1.75 eV, and three delocalized electrons. Calculations in the Born-Mermin Approximation with and without Local Field Correction agree for a small scattering

angle of 13˚ that was chosen in this experiment.

2 4 6 8 10 120

5

10

15

20

25

BMABMA + LFC

270

200

130

ρ [g/cm3]

Δω

[eV

]

3

Figure S2 | Theoretical fits of x-ray forward scattering spectra. (Left) Theoretical fits for 2θ = 13˚ using

BMA&LFC are shown for Ti = Te = 1.75±0.5 eV indicating the error bar of the temperature. (Right) Calculations with the BMA and BMA&LFC are shown for three temperatures demonstrating that these

models provide the same answer for our conditions.

For the scattering spectra at 13˚, we analyze the plasmon scattering spectra and the intensity ratio between elastic scattering and plasmon scattering by fitting theoretical x-ray scattering spectra to the experimental data. Figure S2 shows the fits for various temperatures. Also shown are calculations using BMA and BMA&LFC. We find no differences for our conditions, but MEC/LCLS provides a great opportunity for future investigations of these and related phenomenaS15,S16 when using larger scattering angles. For the purpose of the present analysis we conclude that the error in density is 5% and that differences in the theoretical approximation are negligible for our conditions.

Finally, the first term of Eq. (1) accounts for the density correlations of electrons that dynamically follow the ion motion. This includes both the core electrons, represented by the ion form factor f(k), and the screening cloud of free (and valence) electrons that surround the ion, represented by q(k). The ion-ion density correlation function, Sii(k,ω), reflects the thermal motion of the ions and/or the ion plasma frequency and is thus sensitive to ion temperature. Uncertainties in calculations of these terms primarily determine the error in temperature. We can make the following approximation Sii(k,ω) = Sii(k)δ(ω) and obtain the static structure factor for ion-ion correlations, Sii(k) from our experimental data as outlined in Figure 4 of the paper and further discussed below.

In many previous studies, the analysis of the elastic scattering amplitude alone was often not sufficient to provide information about the state of the dense plasma. This is due to the fact that although Sii(k) can be obtained from calculations using the Hyper-Netted Chain (HNC) approximation these calculations must assume an effective interaction potential. In this study, the observation of the fully wavenumber resolved scattering amplitude W(k) provides the important constraints on the theoretical modeling. In particular, Sii(k) is directly obtained from the measured wavenumber resolved scattering data using W(k)= Sii(k)(f(k)+q(k))2 with the atomic form factor f(k) and the screening function q(k) calculated from the number of bound core electrons, ZC =10 for aluminium.

Figure S3 shows various screening functions q(k) and atomic form factors f(k) together with our measurements of the wavenumber resolved scattering data, W(k), indicating the sensitivity to the choice of the function q(k). Good agreement is observed for the HNC calculations with a screened Coulomb potential using a Yukawa screening term (Y) together with a Short Range Repulsion (SRR) term, similar results can be obtained with a hard sphere model. However, it is apparent that a Coulomb potential or a screened Coulomb potential cannot account for the experimental observations or the results from the DFT-MD simulations, cf. Fig. 4. At small wavenumbers, we note that these models show different sensitivity to the temperature. This is due to the fact that the repulsive core part of the potential is temperature independent; the remaining sensitivity is due to the temperature dependence of the screening cloud. Here,

7940 7960 7980 8000 8020 8040

0

1

2

3 Compressed aluminium fit(Te=1.75 eV, ρ=2.32ρ

0)

Increased elasticscattering at 8 keV:1.75 eV temperature

Plasmon shift:2.3x compression

Solid density aluminium fit(ρ=ρ

0)

Energy (eV)

Inte

nsity

(A.U

.)

Multi-shot signal (700 average)

Single shot signal

Te=1.75 eV - 0.5 eVTe=1.75 eV + 0.5 eV

7940 7960 7980 8000 8020 8040

0

1

2

3

Energy (eV)

Inte

nsity

(A.U

.)

Te=1.75 eV (BMA)

Te=1.75 eV - 0.5 eV (BMA)

Te=1.75 eV + 0.5 eV (BMA)

ρ=2.32ρ0

Te=1.75 eV (BMA+LFC)

Te=1.75 eV - 0.5 eV (BMA+LFC)

Te=1.75 eV + 0.5 eV (BMA+LFC)





2

theoretical description of this term, e.g., Refs. S9, S10. For the plasmon scattering spectrum measured in this study, we find that this term is not important compared to the free electron dynamic structure.

The second term describes the contribution to the scattering from free (delocalized) electrons that do not follow the ion motion. Here, See(k,ω) is the high frequency part of the electron-electron correlation function and it reduces to the usual electron feature in the case of an optical probe. Under the assumption that inter particle interactions are weak, so that the nonlinear interactions between different density fluctuations are negligible, this term is often calculated in the random phase approximation (RPA)S2,S4. In the classical limit, it reduces to the usual Vlasov equation. In the limit of the RPA, strong coupling effects are not accounted for, thus limiting the model validity. To expand the theoretical modeling into the warm dense matter regime, the Born-Mermin approximation (BMA) has been developedS11,S12. The theory was subsequently improved to allow for different models for the dynamic collision frequency and for local field corrections (LFC)S13. Its application to laser experiments on compressed boron has shown that the description of the plasmon dispersion is greatly improved with this theoryS14. However, at this time, the theoretical description of many-body effects in warm dense matter is not complete.

Here, we analyze the increase of the plasmon shift with material compression using the BMA-LFC. This effect relies on the relative increase in plasma frequency with density and is well understood compared to the absolute plasmon energy at arbitrary wavenumber. Figure S1 demonstrates that for our conditions the calculated plasmon shift is not sensitive to the choice of the model for See(k,ω). For a scattering angle of 13˚ the plasmon shift calculated with BMA and BMA-LFC agree. Only with increasing scattering angle do we observe discrepancies between these models.

Figure S1 | Plasmon frequency shift. Calculations are shown of the plasmon frequency in compressed

aluminium for three scattering angles, temperature of 1.75 eV, and three delocalized electrons. Calculations in the Born-Mermin Approximation with and without Local Field Correction agree for a small scattering

angle of 13˚ that was chosen in this experiment.

2 4 6 8 10 120

5

10

15

20

25

BMABMA + LFC

270

200

130

ρ [g/cm3]

Δω

[eV

]

3

Figure S2 | Theoretical fits of x-ray forward scattering spectra. (Left) Theoretical fits for 2θ = 13˚ using

BMA&LFC are shown for Ti = Te = 1.75±0.5 eV indicating the error bar of the temperature. (Right) Calculations with the BMA and BMA&LFC are shown for three temperatures demonstrating that these

models provide the same answer for our conditions.

For the scattering spectra at 13˚, we analyze the plasmon scattering spectra and the intensity ratio between elastic scattering and plasmon scattering by fitting theoretical x-ray scattering spectra to the experimental data. Figure S2 shows the fits for various temperatures. Also shown are calculations using BMA and BMA&LFC. We find no differences for our conditions, but MEC/LCLS provides a great opportunity for future investigations of these and related phenomenaS15,S16 when using larger scattering angles. For the purpose of the present analysis we conclude that the error in density is 5% and that differences in the theoretical approximation are negligible for our conditions.

Finally, the first term of Eq. (1) accounts for the density correlations of electrons that dynamically follow the ion motion. This includes both the core electrons, represented by the ion form factor f(k), and the screening cloud of free (and valence) electrons that surround the ion, represented by q(k). The ion-ion density correlation function, Sii(k,ω), reflects the thermal motion of the ions and/or the ion plasma frequency and is thus sensitive to ion temperature. Uncertainties in calculations of these terms primarily determine the error in temperature. We can make the following approximation Sii(k,ω) = Sii(k)δ(ω) and obtain the static structure factor for ion-ion correlations, Sii(k) from our experimental data as outlined in Figure 4 of the paper and further discussed below.

In many previous studies, the analysis of the elastic scattering amplitude alone was often not sufficient to provide information about the state of the dense plasma. This is due to the fact that although Sii(k) can be obtained from calculations using the Hyper-Netted Chain (HNC) approximation these calculations must assume an effective interaction potential. In this study, the observation of the fully wavenumber resolved scattering amplitude W(k) provides the important constraints on the theoretical modeling. In particular, Sii(k) is directly obtained from the measured wavenumber resolved scattering data using W(k)= Sii(k)(f(k)+q(k))2 with the atomic form factor f(k) and the screening function q(k) calculated from the number of bound core electrons, ZC =10 for aluminium.

Figure S3 shows various screening functions q(k) and atomic form factors f(k) together with our measurements of the wavenumber resolved scattering data, W(k), indicating the sensitivity to the choice of the function q(k). Good agreement is observed for the HNC calculations with a screened Coulomb potential using a Yukawa screening term (Y) together with a Short Range Repulsion (SRR) term, similar results can be obtained with a hard sphere model. However, it is apparent that a Coulomb potential or a screened Coulomb potential cannot account for the experimental observations or the results from the DFT-MD simulations, cf. Fig. 4. At small wavenumbers, we note that these models show different sensitivity to the temperature. This is due to the fact that the repulsive core part of the potential is temperature independent; the remaining sensitivity is due to the temperature dependence of the screening cloud. Here,

7940 7960 7980 8000 8020 8040

0

1

2

3 Compressed aluminium fit(Te=1.75 eV, ρ=2.32ρ

0)

Increased elasticscattering at 8 keV:1.75 eV temperature

Plasmon shift:2.3x compression

Solid density aluminium fit(ρ=ρ

0)

Energy (eV)

Inte

nsity

(A.U

.)

Multi-shot signal (700 average)

Single shot signal

Te=1.75 eV - 0.5 eVTe=1.75 eV + 0.5 eV

7940 7960 7980 8000 8020 8040

0

1

2

3

Energy (eV)

Inte

nsity

(A.U

.)

Te=1.75 eV (BMA)

Te=1.75 eV - 0.5 eV (BMA)

Te=1.75 eV + 0.5 eV (BMA)

ρ=2.32ρ0

Te=1.75 eV (BMA+LFC)

Te=1.75 eV - 0.5 eV (BMA+LFC)

Te=1.75 eV + 0.5 eV (BMA+LFC)





4

we estimate an error in temperature of 0.5 eV including contributions from both uncertainties in the models and noise of the data.

Figure S3 | Interaction potential and elastic scattering amplitude. (Top) Three screening functions q(k) are shown for temperatures ranging from 1 eV to 10 eV along with the atomic form factors f(k). (Bottom) Calculations of the wavenumber resolved scattering amplitude are compared with the experimental data.

Results that include a Yukawa (Y) screened potential and short-range repulsion (SRR) or that assume a hard sphere agree with the experimental data while a screened Coulomb potential (C) shows significant

discrepancies. As comparison with the high-temperature HNC-C data we show recent results by Souza et al.S17.

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

q(k)

0 2 4 6 8 10 12 14 16 18 20

k [A − 1]

T = 1 eVT = 2 eVT = 5 eVT = 10 eV

HNC-Y+SRR

HNC-C(shifted+1)

Hard Sphere(shifted-1)-1.5

-2.0

f(k)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0

50

100

150

200

250

300

350

400

W(k)

0 1 2 3 4 5 6 7 8

k [A − 1]

T = 1 eVT = 2 eVT = 5 eVT = 10 eV

HNC-Y+SRR

HNC-C

Hard Sphere

Souza et al.(Te=10 ev, ρ=8.1 g/cm3)

5

The interaction potential, cf. Eq. (2) of the main manuscript, defines the shape of the wavenumber

resolved scattering data and the angle of the maximum scattering amplitude; the latter is sensitive to the density. Figure S4 compares the angle of maximum scattering amplitude versus density from plasmon data with theoretical approximations. The result show that using Eq. (2) for the calculation of the dynamic structure factor S(k,w) provides a model to determine density from wavenumber resolved scattering.

Figure S4 | Ion-ion correlation peak The measured wavenumber of the peak x-ray scattering amplitude

versus compression from plasmons (error bars are of the size of the symbols). The comparison with theory shows that including short-range repulsion correctly predicts the scattering angle.

The structure factors are important for calculations of physical properties. Our Equation (3) of the main manuscript derives from the following considerations. The internal energy U is the sum of kinetic and potential energies

𝑈𝑈 = 𝐾𝐾 + 𝑉𝑉 (2) The potential energy is given by

𝑉𝑉 = !!!!!

!𝒌𝒌!!ℏ ! 𝑆𝑆!" 𝑘𝑘 − 𝛿𝛿!" 𝑉𝑉!" 𝑘𝑘!" (3)

with the two-body potential Vab. The free energy and therefore all other thermodynamic quantities can be obtained via a charging

𝐹𝐹 𝑛𝑛,𝑇𝑇 = 𝐹𝐹!" 𝑛𝑛,𝑇𝑇 + !"!

!! 𝜆𝜆𝜆𝜆 (4)

In a classical system, the pressure can be derived to be 𝑝𝑝 = 𝑛𝑛!𝑘𝑘!! 𝑇𝑇 + 𝑝𝑝!"#!$$ (5)

𝑝𝑝!"#!$$ = − !!!!!

!𝒌𝒌!!ℏ ! 𝑆𝑆!" 𝑘𝑘 − 𝛿𝛿!" 𝑈𝑈!" 𝑘𝑘!" (6)

Here Uab(k) is the Fourier transform of the derivative of the interaction potential Vab(k) 𝑈𝑈!" 𝑘𝑘 = 𝐹𝐹𝐹𝐹 !

!!!"𝑉𝑉!" 𝑟𝑟 (7)

2.6

3.0

3.4

3.8

4.2

Cor

rela

tion

peak

posi

tion

(k0 [

A-1

])

Compression (ρ/ρ0)

Te = 1.75 eV, Zf = 3

Coulomb

Yukawa + SRRYukawa

Spe

ctra

lly re

solv

ed s

catte

ring

(pla

smon

dat

a)

Wavenumber resolvedscattering

0

1 2 3

Experimental data





4

we estimate an error in temperature of 0.5 eV including contributions from both uncertainties in the models and noise of the data.

Figure S3 | Interaction potential and elastic scattering amplitude. (Top) Three screening functions q(k) are shown for temperatures ranging from 1 eV to 10 eV along with the atomic form factors f(k). (Bottom) Calculations of the wavenumber resolved scattering amplitude are compared with the experimental data.

Results that include a Yukawa (Y) screened potential and short-range repulsion (SRR) or that assume a hard sphere agree with the experimental data while a screened Coulomb potential (C) shows significant

discrepancies. As comparison with the high-temperature HNC-C data we show recent results by Souza et al.S17.

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

q(k)

0 2 4 6 8 10 12 14 16 18 20

k [A − 1]

T = 1 eVT = 2 eVT = 5 eVT = 10 eV

HNC-Y+SRR

HNC-C(shifted+1)

Hard Sphere(shifted-1)-1.5

-2.0

f(k)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0

50

100

150

200

250

300

350

400

W(k)

0 1 2 3 4 5 6 7 8

k [A − 1]

T = 1 eVT = 2 eVT = 5 eVT = 10 eV

HNC-Y+SRR

HNC-C

Hard Sphere

Souza et al.(Te=10 ev, ρ=8.1 g/cm3)

5

The interaction potential, cf. Eq. (2) of the main manuscript, defines the shape of the wavenumber

resolved scattering data and the angle of the maximum scattering amplitude; the latter is sensitive to the density. Figure S4 compares the angle of maximum scattering amplitude versus density from plasmon data with theoretical approximations. The result show that using Eq. (2) for the calculation of the dynamic structure factor S(k,w) provides a model to determine density from wavenumber resolved scattering.

Figure S4 | Ion-ion correlation peak The measured wavenumber of the peak x-ray scattering amplitude

versus compression from plasmons (error bars are of the size of the symbols). The comparison with theory shows that including short-range repulsion correctly predicts the scattering angle.

The structure factors are important for calculations of physical properties. Our Equation (3) of the main manuscript derives from the following considerations. The internal energy U is the sum of kinetic and potential energies

𝑈𝑈 = 𝐾𝐾 + 𝑉𝑉 (2) The potential energy is given by

𝑉𝑉 = !!!!!

!𝒌𝒌!!ℏ ! 𝑆𝑆!" 𝑘𝑘 − 𝛿𝛿!" 𝑉𝑉!" 𝑘𝑘!" (3)

with the two-body potential Vab. The free energy and therefore all other thermodynamic quantities can be obtained via a charging

𝐹𝐹 𝑛𝑛,𝑇𝑇 = 𝐹𝐹!" 𝑛𝑛,𝑇𝑇 + !"!

!! 𝜆𝜆𝜆𝜆 (4)

In a classical system, the pressure can be derived to be 𝑝𝑝 = 𝑛𝑛!𝑘𝑘!! 𝑇𝑇 + 𝑝𝑝!"#!$$ (5)

𝑝𝑝!"#!$$ = − !!!!!

!𝒌𝒌!!ℏ ! 𝑆𝑆!" 𝑘𝑘 − 𝛿𝛿!" 𝑈𝑈!" 𝑘𝑘!" (6)

Here Uab(k) is the Fourier transform of the derivative of the interaction potential Vab(k) 𝑈𝑈!" 𝑘𝑘 = 𝐹𝐹𝐹𝐹 !

!!!"𝑉𝑉!" 𝑟𝑟 (7)

2.6

3.0

3.4

3.8

4.2C

orre

latio

npe

akpo

sitio

n (k

0 [A

-1])

Compression (ρ/ρ0)

Te = 1.75 eV, Zf = 3

Coulomb

Yukawa + SRRYukawa

Spe

ctra

lly re

solv

ed s

catte

ring

(pla

smon

dat

a)

Wavenumber resolvedscattering

0

1 2 3

Experimental data





6

In Eqs. (3) and (6), Sii(k) is directly obtained from the measured data. References to the Supplemental Information S1. Froula D. H., Glenzer S. H., Luhmann N. C., Sheffield J. Plasma Scattering of Electromagnetic Radiation by Sheffield, 2nd edition (Academic Press, 2010). S2. Glenzer S. H. & Redmer R. X-ray Thomson scattering in high energy density plasmas. Rev. Mod. Phys. 81 1625 (2009). S3. Tonks, L. & Langmuir, I. Oscillations in ionized gases. Phys. Rev. 33, 195-210 (1929). S4. Gregori G., Glenzer S. H., Rozmus W., Lee R. W. & Landen O. L. Theoretical model of x-ray

scattering as a dense matter probe, Phys. Rev. E. 67, 026412 (2003). S5. Gregori G., Glenzer S. H. & Landen O. L. Strong coupling corrections in the analysis of x-ray

Thomson scattering measurements, J. Phys. A 36, 5971 (2003). S6. Gregori G., Glenzer S. H., Chung H.-K., Froula D.H., Lee R.W., Meezan N.B., Moody J.D., Niemann C., Landen O.L., Holst B. & Redmer R. Measurement of carbon ionization balance in high-temperature plasma mixtures by temporally resolved X-ray scattering, Journal of Quantitative Spectroscopy and Radiative Transfer 99, 225 (2006). S7. Gregori G., Glenzer S. H. & Landen O. L. Generalized x-ray scattering cross section from non-equilibrium solids and plasmas, Phys. Rev. E 74 026402 (2006). S8. Chihara J. Difference in x--ray scattering between metallic and non--metallic liquids due to conduction electrons, J. Phys. F: Met. Phys. 17, 295 (1987). S9. Johnson W. R., Nilsen J. & Cheng K. T. Resonant bound-free contributions to Thomson scattering of X-rays by warm dense matter, High Energy Density Phys. 9, 407 (2013). S10. Fletcher L. B., Kritcher A.L., Pak A., Ma T., Döppner T., Fortmann C., Divol L., Jones O. S., Landen O. L., Scott H. A., Vorberger J., Chapman D. A., Gericke D. O., Mattern B. A., Seidler G. T., Gregori G., Falcone R. W. & Glenzer S. H. Observations of Continuum Depression in Warm Dense Matter with X-Ray Thomson Scattering, Phys. Rev. Lett. 112, 145004 (2014). S11. Redmer R., Reinholz H., Röpke G., Thiele R. & Höll A. Theory of X-Ray Thomson Scattering in Dense Plasmas, IEEE Trans. Plasma Sci. 33 77 (2005).

S12. Hoell A., Bornath Th., Cao L., T. Döppner T., Düsterer S., Förster E., Fortmann C., Glenzer S. H., Gregori G., Laarmann T., Meiwes-Broer K.-H., Przystawik A., Radcliffe P., Redmer R., Reinholz H., Röpke G., Thiele R., Tiggesbäumker J., Toleikis S., Truong N. X., Tschentscher T., Uschmann I. & Zastrau U. Thomson scattering from near-solid density plasmas using soft X-ray free electron lasers, High Energy Density Physics 3 120 (2007). S13. Fortmann C., Wierling A. and Röpke G. Influence of local-field corrections on Thomson scattering in collision dominated two-component plasmas. Phys. Rev. E 81, 026405 (2010). S14. Neumayer P. Fortmann C., Döppner T., Davis P., Falcone R. W., Kritcher A.L., Landen O. L., Lee H. J., Lee R. W., Niemann C., Le Pape S. & Glenzer S. H. Plasmons in Strongly Coupled Shock-Compressed Matter, Phys. Rev. Lett. 105, 075003 (2010).

7

S15. Faeustlin R. R., Bornath Th., Doeppner T., Duesterer S., Foerster E., Fortmann C., Glenzer S. H, Goede S., Gregori G., Irsig R., Laarmann T., Lee H. J., Li B., Meiwes-Broer K.-H., Mithen J., Nagler B., Przystawik A., Redlin H., Redmer R., Reinholz H., Roepke G., Tavella F., Thiele R., Tiggesbaeumker J., Toleikis S., Uschmann I., Vinko S. M., Whitcher T., Zastrau U., Ziaja B., and Tschentscher Th., Observation of Ultrafast Nonequilibrium Collective Dynamics in Warm Dense Hydrogen. Phys. Rev. Lett. 104, 125002 (2010). S16. Hau-Riege S. P., Graf A., Döppner T., London R. A., Krzywinski J., Fortmann C., Glenzer S. H., Frank M., Sokolowski-Tinten K., Messerschmidt M., Bostedt C., Schorb S., Bradley J. A., Lutman A., Rolles D., Rudenko A., and Rudek B. Ultrafast Transitions from Solid to Liquid and Plasma States of Graphite Induced by X-‐Ray Free Electron Laser Pulses. Phys. Rev. Lett. 108, 217402 (2012). S17. Souza A. N., Perkins D. J., Starrett C. E., Saumon D. & Hansen S. B. Predictions of x-ray scattering spectra for warm dense matter. Phys. Rev. E 89, 023108 (2014).



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