Pith. sign in

REVIEW 2 major objections 4 minor 90 references

Reconciling chemical models of X-ray Thomson Scattering with the Bethe $f$-sum rule

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Including bound-bound transitions restores the f-sum rule in X-ray Thomson scattering models.

desk verdict A solid, useful paper that shows the standard Chihara bound-electron treatment violates the Bethe f-sum rule and that adding exact hydrogenic bound-bound and bound-free terms fixes it; the core claim holds, but the numerical evidence needs a convergence statement and a caption fix. read the letter →

arxiv 2607.25481 v1 pith:66NVLWKJ submitted 2026-07-28 physics.plasm-ph physics.atom-phphysics.comp-ph

Maximilian P. Böhme , Paul Hamann , Veronika A. Kruse , Hannah M. Bellenbaum , Armin Bergermann , David T. Bishel , Thomas Gawne , Dirk O. Gericke
show 4 more authors
Zhandos A. Moldabekov Pontus Svensson Jan Vorberger Tobias Dornheim
This is my paper
classification physics.plasm-phphysics.atom-phphysics.comp-ph
keywords X-rayThomsonscatteringChiharadecompositionBethef-sumrulebound-boundtransitionsdynamicstructurefactorhydrogenicmatrixelementsimpulseapproximationwarmdensematter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to fix a long-standing inconsistency in the standard model used to interpret X-ray Thomson scattering (XRTS) experiments on warm dense matter. The Chihara decomposition, the field's default framework, violates the Bethe f-sum rule, a fundamental constraint that the first frequency moment of the spectrum must equal q²/2. The authors show the violation has two causes: bound-bound transitions (electron excitations between discrete atomic levels) are neglected, and bound-free transitions are treated with plane-wave final states (the impulse approximation). They construct a minimal analytical extension for hydrogenic ground states that includes both exact bound-bound matrix elements and exact Coulomb continuum final states, and demonstrate that the combined spectrum satisfies the sum rule over the full momentum range. This matters because sum-rule compliance is a prerequisite for quantitative normalization of XRTS data and for modern model-free analysis methods.

What carries the argument

The load-bearing ingredient is the analytic evaluation of hydrogenic transition matrix elements with a complete final-state basis: bound-bound matrix elements via parabolic coordinates and Laguerre-polynomial generating functions, and bound-free matrix elements via the analytic Coulomb (confluent-hypergeometric) continuum integral. Completeness of the final-state sum over discrete bound states plus continuum is what enforces the Bethe f-sum rule; the paper shows both pieces are required, and the expressions are fast enough for forward fitting.

What would settle it

Recompute the first frequency moment for the 1s state while increasing the truncation of the bound-state sum (n_max = 10, 20, 50, 100) for, say, q = 1 Å⁻¹ and q = 2 Å⁻¹; if the result does not converge to q²/2, or if the combined moment changes by more than the quoted precision as n_max grows, the central claim fails.

Watch

Extended reading notes

Core claim

For a hydrogenic atom in its ground state, the Bethe f-sum rule ∫dω ω S(q,ω) = q²/2 can be satisfied to arbitrary precision — and, to the authors' knowledge, for the first time in an implemented bound-state treatment within the Chihara decomposition — provided the inelastic spectrum includes both the full set of bound-bound transitions and an exact bound-free contribution built from Coulomb continuum wavefunctions rather than plane waves. The impulse approximation alone underestimates the first moment at small momentum transfer and overestimates it at larger q; adding bound-bound transitions to an impulse-approximation bound-free piece only partially repairs the discrepancy. Ray-tracing dete

Load-bearing premise

The paper's claim of exact sum-rule compliance rests on including 'enough' bound-bound transitions, but the maximum principal quantum number used in the verification is not reported; if omitted high-n states carry non-negligible weight in the shown q range, the claimed precision is not fully established.

Editorial extensions

If this is right

  • Chihara-model spectra for cold hydrogen now satisfy the Bethe f-sum rule, enabling sum-rule-based normalization and the use of imaginary-time correlation function methods that require exact frequency-moment relations.
  • Standard impulse-approximation codes misweight and misplace bound-free spectral features at small to intermediate momentum transfers; for carbon and aluminum these deviations persist into q ranges used in backscattering experiments.
  • Bound-bound transitions such as 1s→L-shell carry significant spectral weight in ground-state atomic hydrogen and should be included when interpreting XRTS from such targets.
  • The analytic matrix elements are computationally cheap (a speed-up of roughly three orders of magnitude over direct numerical integration), making the model practical for iterative forward fitting.
  • The framework is a foundation for a finite-temperature extension of Chihara models that would also, by construction, comply with the f-sum rule.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the completeness argument carries over to finite temperature, an analogous treatment with thermally occupied bound states should make warm-dense-matter Chihara models compliant with the f-sum rule, which would directly improve temperature and density inference from XRTS.
  • The same exact Coulomb final states could serve as a benchmark for average-atom and screened-hydrogenic codes, quantifying the error introduced by approximate continuum wavefunctions.
  • The paper's 'enough bound-bound transitions' criterion suggests a practical convergence test for any implementation: increase the maximum principal quantum number until the first moment converges to q²/2, and use that as a quality check in fitting routines.
  • The predicted visible excess spectral weight from 1s→L-shell transitions could be sought in a dedicated cold-hydrogen XRTS experiment; a null result would indicate that line-broadening or plasma-environment effects wash out the feature.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper addresses the violation of the Bethe f-sum rule (BFSR) in the standard Chihara decomposition used for X-ray Thomson scattering (XRTS). The authors argue that the violation arises from neglect of bound-bound transitions and from use of the impulse approximation (IA) for bound-free transitions. They present analytic expressions for hydrogenic bound-bound and bound-free dynamic structure factors (DSFs), based on parabolic-coordinate matrix elements and the Nordsieck/Bethe-Maximon integral, and show numerically that only the combination of exact bound-bound and bound-free contributions satisfies the BFSR. They also perform HEART ray-tracing simulations for cold atomic hydrogen, predicting experimentally visible deviations from the IA-based Chihara model. The model is intended for inclusion in the xDAVE code.

Significance. If correct, the paper provides a minimal, analytic, and computationally efficient implementation of bound-state transitions in a Chihara-type XRTS model that satisfies an exact sum rule. The analytic expressions offer speed-ups of about three orders of magnitude over direct numerical integration (Fig. 7), and the planned open-source release is a strength. The central mathematical idea is sound: completeness of the hydrogenic bound and continuum states guarantees the f-sum rule. The main limitation is that the numerical verification of the sum rule lacks convergence details, and one printed formula contains an apparent typo; these issues are fixable and do not invalidate the underlying construction. The detector simulation is explicitly a proof-of-concept with simplifying assumptions, which the authors acknowledge.

major comments (2)
  1. [Section II E, Eq. (19)] The bound-free DSF is written as S^bf_{nℓm}(q, ω) = 2πν ∫_{-1}^{1} dµ µ |M^bf_{k,nℓm}(q)|², with k = ν µ e_z. After enforcing energy conservation via the delta function, the angular integration measure is dµ dφ (with ∫ dφ = 2π and ∫ dµ), with no additional factor µ. The extra µ in the integrand is therefore erroneous and would alter the computed DSF. The correct expression should be 2πν ∫_{-1}^{1} dµ |M^bf|². Please correct this equation and confirm that the numerical results (e.g., Figs. 3 and 5) were obtained with the corrected form; the cross-check in Fig. 7 suggests this is the case, but the printed formula is misleading.
  2. [Section III A, Eq. (14)] The demonstration that the analytic model satisfies the Bethe f-sum rule to 'arbitrary precision' is incomplete. The paper does not report the maximum principal quantum number n_max used in the bound-bound sum, nor a convergence test, nor an error bound for the omitted high-n Rydberg states. The statement that 'enough bound-bound transitions' must be included, and the suggestion that the f-sum rule itself can be used to check whether enough transitions have been included, makes the verification circular when the claim is that the model satisfies the sum rule. Please report the residual |Ω^(1)(q) − q²/2| versus n_max for representative q, and provide an analytic or numerical estimate of the tail contribution (e.g., using the asymptotic decay of the bound-state matrix elements). This is needed to support the abstract's 'arbitrary precision' claim and to make the numerical demonstration rep
minor comments (4)
  1. [Section II E] The notation k = ν µ e_z in Eq. (19) is confusing. If µ = cos θ is the polar angle between k and q (taken along z), the vector is k = ν(µ e_z + √(1−µ²)(cos φ e_x + sin φ e_y)); the printed expression appears to write only its z-component.
  2. [Figure 7] The figure caption states the comparison is for the state |1,0,0⟩, while the text above it refers to |3,2,−1⟩. One of these is incorrect.
  3. [General] There are several typos: 'to the best of out knowledge' (Sec. II A), 'in principal' (Sec. II D), and 'remarks that electrons respond' (Sec. II A). Please proofread.
  4. [Section III B / Conclusion] The abstract and conclusion say the model 'will be made available' in xDAVE, but no repository link or version identifier is given. For reproducibility, please cite the exact code version or provide a DOI when the code is released.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Bethe f-sum rule is used as an external benchmark, not as a fitted input; the only concern is unreported truncation in the numerical check.

full rationale

The central derivation is self-contained: the bound-bound contribution (Eq. 14) and the exact bound-free contribution (Eqs. 16, 19) are computed from hydrogenic wavefunctions and Coulomb continuum states, and the Bethe f-sum rule (Eq. 6) is an independent, externally known identity. No parameter is fitted to the sum rule, and the agreement in Fig. 4 is a numerical verification rather than an enforced result. The analytic bound-free matrix element in Appendix D is based on Nordsieck's integral and Bethe–Maximon, i.e. external mathematical sources, not on an unverified self-citation. The heavy self-citation to xDAVE and imaginary-time methods is contextual and does not support the central claim. The only caveat is the passage in Section III A stating that perfect agreement requires 'enough bound-bound transitions' and proposing the sum rule as a convergence check, while no n_max or convergence bound is reported. This is a reproducibility/rigor limitation of the numerical demonstration, not a circular reduction: the underlying identity is exact and the matrix elements are not adjusted to force compliance. Had the truncation been used as the sole criterion for agreement, the verification would be weakened, but the paper's mathematical derivation remains independent of that numerical check.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The BFSR verification is free of fitted parameters: it follows from the complete hydrogenic basis and known analytic matrix elements from the literature. Hand-chosen inputs occur only in the illustrative ray-tracing section (Voigt sigma=gamma=0.5 eV, 7.4 keV beam, S_ii=1, unspecified target normalization) and in the untabulated truncation of the bound-bound sum. No new physical entities are postulated.

free parameters (4)
  • bound-state truncation (maximum principal quantum number for the bound-bound sum) = not specified
    The paper states BFSR compliance requires 'enough bound-bound transitions' but does not report the cutoff or a convergence criterion; this is a hand-chosen numerical truncation in the sum-rule demonstration.
  • Voigt profile widths sigma=gamma = 0.5 eV each
    Chosen for the source convolution in Figures 5 and 6; not fitted to data, but shapes the predicted detector signal.
  • incident X-ray beam energy = 7.4 keV
    Chosen for the HEART ray-tracing geometry at the European XFEL HED end station; affects the kinematics in Figure 6.
  • absolute detector normalization (target density, path length, incident flux) = not stated
    Figure 6 reports absolute event counts, but the text gives no target density, path length, or incident flux needed to absolute-scale the comparison.
assumptions (6)
  • standard math The hydrogenic eigenstates (discrete bound states plus continuum Coulomb scattering states) form a complete basis for the single-electron Coulomb Hamiltonian.
    Needed for the closure relation underlying the Bethe f-sum rule; invoked in Section II A and used to define S_i(q,omega).
  • standard math The Bethe f-sum rule, integral domega omega S(q,omega) = N q^2/2, is the exact first-moment constraint for the electronic dynamic structure factor.
    Used as the benchmark throughout Section III A; cited to Giuliani-Vignale and Schulke.
  • domain assumption The Chihara decomposition separates the DSF into elastic/inelastic and bound/free contributions and is the appropriate framework for XRTS analysis.
    This is the chemical-picture model being extended; Section II B.
  • domain assumption The target is in the electronic ground state at T=0 and can be treated as isolated hydrogen atoms, with no molecular bonding or band structure.
    Central scoping assumption stated in Section III B; enables the use of hydrogenic wavefunctions.
  • domain assumption Bound-bound transitions can be represented as zero-width delta lines because natural linewidths are negligible; the observed width comes from the source Voigt profile.
    Used for the DSF and detector simulations in Section III B; dense-plasma broadening is explicitly deferred.
  • ad hoc to paper In the detector simulation, the ion-ion structure factor S_ii(q)=1 and the free-electron screening cloud phi(q)=0.
    Explicitly chosen to isolate the bound-state effect in Figure 6; the paper acknowledges this may not be quantitatively accurate.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Reconciling chemical models of X-ray Thomson Scattering with the Bethe $f$-sum rule." pith.science (2026). https://pith.science/paper/66NVLWKJ

@misc{pith2026260725481,
  author       = {Pith},
  title        = {Pith review of: Reconciling chemical models of X-ray Thomson Scattering with the Bethe $f$-sum rule},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66NVLWKJ}},
  note         = {Machine review of arXiv:2607.25481}
}
abstract

X-ray Thomson scattering (XRTS) is a key diagnostic for high-energy-density plasmas, which can exhibit significant quantum effects even at elevated temperatures. XRTS experiments are commonly interpreted using the Chihara decomposition, that was derived in the chemical picture and, thus, separates contributions from bound and free electrons. Despite being the de-facto standard for analysing measurements, a well-known shortcoming is that the standard bound-state treatment in the form of the impulse approximation fails to satisfy fundamental theoretical constraints, most notably the Bethe $f$-sum rule (BFSR). The problem arises due to the usage of plane waves in the impulse approximation as well as non-negligible contributions from bound-bound transitions. In this work, we present a minimal analytical extension of the Chihara decomposition of the dynamic structure factor for matter in the ground state, using hydrogenic bound-free and bound-bound transitions. We demonstrate that compliance with the BFSR is only achieved when both bound-bound transitions are explicitly included and an exact treatment of the bound-free contribution is applied. Finally, detector ray-tracing simulations for atomic hydrogen demonstrate experimentally detectable deviations from the standard Chihara model. The model will be made available in the open source XRTS library xDAVE [Bellenbaum et al., Phys. Plasmas (in print), arxiv:2604.27237].

Figures

Figures reproduced from arXiv: 2607.25481 by the authors.

Figure 1
Figure 1. Violation of the Bethe f-sum rule in the impulse approximation (IA). Panel (a) depicts the first frequency mo￾ment of the IA divided by the nuclear charge state for the elements H (green), Be (red), C (magenta) and Al (turquoise) as a function of the wave number compared to the analytical dispersion relation of the Bethe f-sum rule (black dashed). Panel (b) illustrates the relative error of the f-sum rule as a funct… view at source ↗
Figure 2
Figure 2. Absolute value of scattering matrix elements [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. DSF using the Schumacher Impulse approximation [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: Inelastic DSF of ground-state atomic hydrogen. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 4
Figure 4. Figure 4: Contributions to the first moment of S(q, ω) for an electron in the 1s-orbital. Panel (a) shows the analytic result (black dashed), the bound-bound contribution (red) and the bound-free contribution (green). Panel (b) replace the exact bound-free treatment with the IA.…
Figure 6
Figure 6. Figure 6: Comparison of the simulated XRTS spectra ob [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Comparison of the bound-free dynamic structure factor of hydrogen in the ground state for the state [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

90 extracted references · 3 canonical work pages

  1. [1]

    C for a discussion of the selection rules)

    Dipole limit, TRK sum rule In the long-wavelength limit (q→0), the transition matrix elementM ij(q) reduces to that of the dipole op- erator |Mij(q)|2 =| ⟨i|eiqr |j⟩ |2 ≈δ ij +q 2| ⟨i|ˆr|j⟩| {z } dij |2.(8) While for any transitioni→j, there is always someq >0 at whichM ij assumes a finite value, the dipole matrix ele- mentd ij vanishes except for transit...

  2. [2]

    Vorberger, F

    J. Vorberger, F. Graziani, D. Riley, A. D. Baczewski, I. Baraffe, M. Bethkenhagen, S. Blouin, M. P. B¨ ohme, M. Bonitz, M. Bussmann, A. Casner, W. Cayzac, P. Cel- liers, G. Chabrier, N. Chamel, D. Chapman, M. Chen, J. Cl´ erouin, G. Collins, F. Coppari, T. D¨ oppner, T. Dorn- heim, L. B. Fletcher, D. O. Gericke, S. Glenzer, A. F. Goncharov, G. Gregori, S....

    arXiv 2025

  3. [3]

    Dornheim, H

    T. Dornheim, H. Bellenbaum, T. Gawne, J. Vorberger, and D. O. Gericke, Overview of x-ray thomson scatter- ing measurements of extreme states of matter (2026), arXiv:2604.23687 [physics.plasm-ph]

    arXiv 2026

  4. [4]

    W¨ unsch, J

    K. W¨ unsch, J. Vorberger, G. Gregori, and D. O. Gericke, Europhysics Letters94, 25001 (2011)

    2011

  5. [5]

    S. H. Glenzer and R. Redmer, Rev. Mod. Phys81, 1625 (2009)

    2009

  6. [6]

    Neumayer, C

    P. Neumayer, C. Fortmann, T. D¨ oppner, P. Davis, R. W. Falcone, A. L. Kritcher, O. L. Landen, H. J. Lee, R. W. Lee, C. Niemann, S. Le Pape, and S. H. Glenzer, Phys. Rev. Lett.105, 075003 (2010)

    2010

  7. [7]

    Sperling, E

    P. Sperling, E. J. Gamboa, H. J. Lee, H. K. Chung, E. Galtier, Y. Omarbakiyeva, H. Reinholz, G. R¨ opke, U. Zastrau, J. Hastings, L. B. Fletcher, and S. H. Glen- zer, Phys. Rev. Lett.115, 115001 (2015)

    2015

  8. [8]

    D. S. Bespalov, U. Zastrau, Z. A. Moldabekov, T. Gawne, T. Dornheim, M. Meshhal, A. Amouretti, M. Andrze- jewski, K. Appel, C. Baehtz, E. Brambrink, K. Buakor, C. Camarda, D. Chin, G. Collins, C. Cr´ episson, A. Descamps, J. Eggert, L. B. Fletcher, A. Forte, G. Gre- gori, M. Harmand, O. S. Humphries, H. H¨ oppner, J. Kuh- lke, W. Lynn, J. L¨ utgert, M. Ma...

    2026

Show all 90 references
  1. [9]

    Frydrych, J

    S. Frydrych, J. Vorberger, N. J. Hartley, A. K. Schuster, K. Ramakrishna, A. M. Saunders, T. van Driel, R. W. Falcone, L. B. Fletcher, E. Galtier, E. J. Gamboa, S. H. Glenzer, E. Granados, M. J. MacDonald, A. J. MacK- innon, E. E. McBride, I. Nam, P. Neumayer, A. Pak, K. Voigt...

    2020

  2. [10]

    D¨ oppner, M

    T. D¨ oppner, M. Bethkenhagen, D. Kraus, P. Neumayer, D. A. Chapman, B. Bachmann, R. A. Baggott, M. P. B¨ ohme, L. Divol, R. W. Falcone, L. B. Fletcher, O. L. Landen, M. J. MacDonald, A. M. Saunders, M. Sch¨ orner, P. A. Sterne, J. Vorberger, B. B. L. Witte, A. Yi, R. Red- mer...

    2023

  3. [11]

    Bergermann, U

    A. Bergermann, U. Kleinschmidt, S. H. Glenzer, and R. Redmer, Physics of Plasmas33, 023901 (2026)

    2026

  4. [12]

    M. P. Desjarlais, J. D. Kress, and L. A. Collins, Phys. Rev. E66, 025401(R) (2002)

    2002

  5. [13]

    Gregori, S

    G. Gregori, S. H. Glenzer, W. Rozmus, R. W. Lee, and O. L. Landen, Phys. Rev. E67, 026412 (2003)

    2003

  6. [14]

    A. D. Baczewski, L. Shulenburger, M. P. Desjarlais, S. B. Hansen, and R. J. Magyar, Phys. Rev. Lett.116, 115004 (2016)

    2016

  7. [15]

    Dornheim, T

    T. Dornheim, T. D¨ oppner, P. Tolias, M. P. B¨ ohme, L. B. Fletcher, T. Gawne, F. R. Graziani, D. Kraus, M. J. MacDonald, Z. A. Moldabekov, S. Schwalbe, D. O. Ger- icke, and J. Vorberger, Nature Communications16, 5103 (2025)

    2025

  8. [16]

    B¨ ohme, Z

    M. B¨ ohme, Z. A. Moldabekov, J. Vorberger, and T. Dorn- heim, Phys. Rev. Lett.129, 066402 (2022)

    2022

  9. [17]

    Z. A. Moldabekov, S. Schwalbe, U. H. Acosta, T. Gawne, J. Vorberger, M. Pavanello, and T. Dornheim, npj Com- putational Materials12, 168 (2026)

    2026

  10. [18]

    Dornheim, H

    T. Dornheim, H. M. Bellenbaum, M. Bethkenhagen, S. B. Hansen, M. P. B¨ ohme,et al., Physics of Plasmas32, 052712 (2025)

    2025

  11. [19]

    Sch¨ orner, M

    M. Sch¨ orner, M. Bethkenhagen, T. D¨ oppner, D. Kraus, L. B. Fletcher, S. H. Glenzer, and R. Redmer, Physical Review E107, 065207 (2023)

    2023

  12. [20]

    D. A. Chapman, J. Vorberger, K. W¨ unsch, and D. O. Gericke, High Energy Density Physics8, 175 (2012)

    2012

  13. [21]

    H. M. Bellenbaum, M. P. B¨ ohme, M. Bonitz, T. D¨ oppner, L. B. Fletcher, T. Gawne, D. Kraus, Z. A. Moldabekov, S. Schwalbe, J. Vorberger, and T. Dornheim, Phys. Rev. Res.7, 033016 (2025)

    2025

  14. [22]

    Chihara, Journal of Physics F: Metal Physics17, 295 (1987)

    J. Chihara, Journal of Physics F: Metal Physics17, 295 (1987)

    1987

  15. [23]

    Chihara, Journal of Physics: Condensed Matter12, 231 (2000)

    J. Chihara, Journal of Physics: Condensed Matter12, 231 (2000)

    2000

  16. [24]

    M. P. B¨ ohme, L. B. Fletcher, A. D. Baczewski, H. M. Bel- lenbaum, Z. A. Moldabekov, J. Vorberger, D. A. Chap- man, M. J. MacDonald, S. Schwalbe, T. R. Preston, D. Kraus, T. Gawne, F. R. Graziani, S. Hamel, T. D¨ opp- ner, and T. Dornheim, Contributions to Plasma Physics n/a, e70149

  17. [25]

    [3] for a recent overview of more than 90 WDM XRTS experiments

    as well as inertial confinement fusion facilities [10, 26]; see Ref. [3] for a recent overview of more than 90 WDM XRTS experiments. Several works have used the Chihara decomposition for synthetic data generation [27, 28], comparison against PIMC simulations [17] and in conjun...

    2026 arXiv

  18. [26]

    D. A. Chapman, J. Vorberger, L. B. Fletcher, R. A. Bag- gott, L. Divol, T. D¨ oppner, R. W. Falcone, S. H. Glenzer, G. Gregori, T. M. Guymer, A. L. Kritcher, O. L. Landen, T. Ma, A. E. Pak, and D. O. Gericke, Nature Communi- cations6, 10.1038/ncomms7839 (2015)

    2015 doi

  19. [27]

    Poole, D

    H. Poole, D. Cao, R. Epstein, I. Golovkin, T. Walton, S. X. Hu, M. Kasim, S. M. Vinko, J. R. Rygg, V. N. Gon- charov, G. Gregori, and S. P. Regan, Physics of Plasmas 29, 072703 (2022)

    2022

  20. [28]

    Poole, M

    H. Poole, M. K. Ginnane, M. Millot, H. M. Bellenbaum, G. W. Collins, S. X. Hu, D. Polsin, R. Saha, J. Topp- Mugglestone, T. G. White, D. A. Chapman, J. R. Rygg, S. P. Regan, and G. Gregori, Phys. Rev. Res.6, 023144 (2024)

    2024

  21. [29]

    L. B. Fletcher, H. J. Lee, T. D¨ oppner, E. Galtier, B. Na- gler, 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. Za...

    2015

  22. [30]

    A. L. Kritcher, D. C. Swift, T. D¨ oppner, B. Bachmann, L. X. Benedict, G. W. Collins, J. L. DuBois, F. El- sner, G. Fontaine, J. A. Gaffney, S. Hamel, A. Lazicki, W. R. Johnson, N. Kostinski, D. Kraus, M. J. MacDon- ald, B. Maddox, M. E. Martin, P. Neumayer, A. Nikroo, J. Nil...

    2020

  23. [31]

    Dornheim, M

    T. Dornheim, M. B¨ ohme, D. Kraus, T. D¨ oppner, T. R. Preston, Z. A. Moldabekov, and J. Vorberger, Nature Communications13, 7911 (2022)

    2022

  24. [32]

    Gawne, J

    T. Gawne, J. Vorberger, Z. Moldabekov, H. Bel- lenbaum, and T. Dornheim, Model-free interpreta- tion of x-ray thomson scattering measurements (2026), arXiv:2604.25735 [physics.plasm-ph]

    2026 arXiv

  25. [33]

    Gawne, S

    T. Gawne, S. Schwalbe, T. Chuna, U. Hernandez Acosta, T. R. Preston, and T. Dornheim, Computer Physics Communications318, 109878 (2026)

    2026

  26. [34]

    Gawne, H

    T. Gawne, H. Bellenbaum, L. B. Fletcher, K. Appel, 17 C. Baehtz, V. Bouffetier, E. Brambrink, D. Brown, A. Cangi, A. Descamps, S. Goede, N. J. Hartley, M.-L. Herbert, P. Hesselbach, H. H¨ oppner, O. S. Humphries, Z. Konˆ opkov´ a, A. Laso Garcia, B. Lindqvist, J. L¨ utgert, M....

    2024

  27. [35]

    Schwalbe, H

    S. Schwalbe, H. Bellenbaum, T. D¨ oppner, M. B¨ ohme, T. Gawne,et al., Static linear density response from x-ray Thomson scattering measurements: a case study of warm dense beryllium (2025), arXiv:2504.13611 [physics.plasm- ph]

    2025 arXiv

  28. [36]

    of an HED experiment. Chihara decomposition mod- els would be an excellent tool to drive the development of imaginary-time based inference techniques as they offer a fast and simple way to calculate a dynamic structure fac- tor suitable to approximately describing the plasma e...

  29. [37]

    Dornheim, M

    T. Dornheim, M. P. B¨ ohme, D. A. Chapman, D. Kraus, T. R. Preston, Z. A. Moldabekov, N. Schl¨ unzen, A. Cangi, T. D¨ oppner, and J. Vorberger, Physics of Plas- mas30, 042707 (2023)

    2023

  30. [38]

    Codes such as FEFF9 [65] have been used to generate the real-space Green’s functions approach in Ref

    that may also not fulfil the Bethef-sum rule due to the neglect of bound-bound transitions, while treat- ing the bound-free transitions coherently. Codes such as FEFF9 [65] have been used to generate the real-space Green’s functions approach in Ref. [50], but have unfor- tunat...

  31. [39]

    Dornheim, T

    T. Dornheim, T. D¨ oppner, A. D. Baczewski, P. Tolias, M. P. B¨ ohme, Z. A. Moldabekov, T. Gawne, D. Ran- jan, D. A. Chapman, M. J. MacDonald, T. R. Preston, D. Kraus, and J. Vorberger, Scientific Reports14, 14377 (2024)

    2024

  32. [40]

    M. P. B¨ ohme, W. M. Martin, H. M. Bellenbaum, M. Berrens, J. Vorberger, S. Schwalbe, Z. A. Mold- abekov, T. Gawne, S. Hamel, B. Aguilar-Solis, A. Sharma, F. R. Graziani, T. D¨ oppner, S. H. Glenzer, T. Dornheim, and D. T. Bishel, Physics of Plasmas33, 042701 (2026)

    2026

  33. [41]

    W¨ unsch, J

    K. W¨ unsch, J. Vorberger, and D. O. Gericke, Physical Review E79, 010201 (2009)

    2009

  34. [42]

    W. R. Johnson, J. Nilsen, and K. T. Cheng, Physical Review E86, 036410 (2012)

    2012

  35. [43]

    Gregori, S

    G. Gregori, S. H. Glenzer, F. J. Rogers, S. M. Pollaine, O. L. Landen, C. Blancard, G. Faussurier, P. Renaudin, S. Kuhlbrodt, and R. Redmer, Physics of Plasmas11, 2754 (2004)

    2004

  36. [44]

    H. M. Bellenbaum, B. Bachmann, D. Kraus, T. Gawne, M. P. B¨ ohme, T. D¨ oppner, L. B. Fletcher, M. J. Mac- Donald, Z. A. Moldabekov, T. R. Preston, J. Vorberger, and T. Dornheim, Applied Physics Letters126, 044104 (2025)

    2025

  37. [45]

    Sch¨ ulke, J

    W. Sch¨ ulke, J. R. Schmitz, H. Schulte-Schrepping, and A. Kaprolat, Physical Review B52, 11721 (1995)

    1995

  38. [46]

    Chuna, J

    T. Chuna, J. Vorberger, T. Gawne, T. Dornheim, and M. S. Murillo, Mermin’s dielectric function and the f-sum rule (2026), arXiv:2603.04054 [cond-mat.stat-mech]

    2026

  39. [47]

    D. A. Chapman, D. Kraus, A. L. Kritcher, B. Bachmann, G. W. Collins, R. W. Falcone, J. A. Gaffney, D. O. Gericke, S. H. Glenzer, T. M. Guymer, J. A. Hawre- liak, O. L. Landen, S. Le Pape, T. Ma, P. Neumayer, J. Nilsen, A. Pak, R. Redmer, D. C. Swift, J. Vorberger, and T. D¨ op...

    2014 doi

  40. [48]

    H. M. Bellenbaum, D. A. Chapman, M. P. B¨ ohme, T. Gawne, S. Schwalbe, W. M. Martin, M. Bussmann, D. O. Gericke, U. H. Acosta, J. Vorberger, and T. Dorn- heim, X-Ray Diagnostics Analysis Verification and Ex- ploration (xDA VE) Code for the Prediction and Inter- pretation of X-...

    2026

  41. [49]

    L¨ utgert, S

    J. L¨ utgert, S. Schumacher, J. Rips, C. Qu, T. D¨ oppner, and D. Kraus, Computer Physics Communications325, 110173 (2026)

    2026

  42. [50]

    A. D. Baczewski, T. Hentschel, A. Kononov, and S. B. Hansen, Predictions of bound-bound transi- tion signatures in x-ray Thomson scattering (2021), arXiv:2109.09576 [physics]

    2021 arXiv

  43. [51]

    Eisenberger and P

    P. Eisenberger and P. M. Platzman, Physical Review A 2, 415 (1970)

    1970

  44. [52]

    Giuliani and G

    G. Giuliani and G. Vignale,Quantum Theory of the Elec- tron Liquid(Cambridge University Press, Cambridge, 2008)

    2008

  45. [53]

    Sch¨ ulke,Electron dynamics by inelastic X-ray scat- tering, Vol

    W. Sch¨ ulke,Electron dynamics by inelastic X-ray scat- tering, Vol. 7 (OUP Oxford, 2007)

    2007

  46. [54]

    B. A. Mattern and G. T. Seidler, Physics of Plasmas20, 022706 (2013)

    2013

  47. [55]

    Reiche and W

    F. Reiche and W. Thomas, Zeitschrift f¨ ur Physik34, 510 (1925)

    1925

  48. [56]

    Kuhn, Zeitschrift f¨ ur Physik33, 408 (1925)

    W. Kuhn, Zeitschrift f¨ ur Physik33, 408 (1925)

    1925

  49. [57]

    Bethe, Zeitschrift f¨ ur Physik76, 293 (1932)

    H. Bethe, Zeitschrift f¨ ur Physik76, 293 (1932)

    1932

  50. [58]

    J. F. Ogilvie and G. J. Fee, European Journal of Physics 35, 025017 (2014)

    2014

  51. [59]

    Bethe and E

    H. Bethe and E. Salpeter,Quantum Mechanics of One- and Two-electron Atoms, Dover books on physics (Dover Publications, 2008)

    2008

  52. [60]

    Belkic, Journal of Physics B: Atomic and Molecular Physics14, 1907 (1981)

    D. Belkic, Journal of Physics B: Atomic and Molecular Physics14, 1907 (1981)

    1907

  53. [61]

    H. E. Moses and R. T. Prosser, Journal of Mathematical Physics35, 5660 (1994)

    1994

  54. [62]

    S. H. Glenzer and R. Redmer, Rev. Mod. Phys.81, 1625 (2009)

    2009

  55. [63]

    Dornheim, H

    T. Dornheim, H. M. Bellenbaum, M. Bethkenhagen, S. B. Hansen, M. P. B¨ ohme, T. D¨ oppner, L. B. Fletcher, T. Gawne, D. O. Gericke, S. Hamel, D. Kraus, M. J. Mac- Donald, Z. A. Moldabekov, T. R. Preston, R. Redmer, M. Sch¨ orner, S. Schwalbe, P. Tolias, and J. Vorberger, Physi...

    2025

  56. [64]

    Dornheim, A

    T. Dornheim, A. Cangi, K. Ramakrishna, M. B¨ ohme, S. Tanaka, and J. Vorberger, Physical Review Letters 125, 235001 (2020)

    2020

  57. [65]

    Dornheim, S

    T. Dornheim, S. Groth, J. Vorberger, and M. Bonitz, Phys. Rev. Lett.121, 255001 (2018)

    2018

  58. [66]

    S. H. Glenzer, O. L. Landen, P. Neumayer, R. W. Lee, K. Widmann, S. W. Pollaine, R. J. Wallace, G. Gre- gori, A. H¨ oll, T. Bornath, R. Thiele, V. Schwarz, W.- D. Kraeft, and R. Redmer, Phys. Rev. Lett.98, 065002 (2007)

    2007

  59. [67]

    Neumayer, C

    P. Neumayer, C. Fortmann, T. D¨ oppner, P. Davis, R. W. Falcone, A. L. Kritcher, O. L. Landen, H. J. Lee, R. W. Lee, C. Niemann, S. Le Pape, and S. H. Glenzer, Physi- cal Review Letters105, 10.1103/physrevlett.105.075003 (2010)

    2010 doi

  60. [68]

    Schumacher, F

    M. Schumacher, F. Smend, and I. Borchert, Journal of Physics B: Atomic and Molecular Physics8, 1428 (1975)

    1975

  61. [69]

    J. J. Rehr, J. J. Kas, F. D. Vila, M. P. Prange, and K. Jorissen, Physical Chemistry Chemical Physics12, 5503 (2010)

    2010

  62. [70]

    R. D. Cowan,The Theory of Atomic Structure and Spec- tra(University of California Press, Berkeley, 1981). 18

    1981

  63. [71]

    H. R. Griem,Principles of Plasma Spectroscopy, edited by M. G. Haines, K. I. Hopkraft, I. H. Hutchinson, C. M. Surko, and K. Schindler (Cambridge University Press, New York, 2005)

    2005

  64. [72]

    Volonte, Journal of Physics D: Applied Physics11, 1615 (1978)

    S. Volonte, Journal of Physics D: Applied Physics11, 1615 (1978)

    1978

  65. [73]

    R. M. More, J. Quant. Spectrosc. Radiat. Transfer27, 345 (1982)

    1982

  66. [74]

    Li and F

    X. Li and F. B. Rosmej, Europhysics Letters99, 33001 (2012)

    2012

  67. [75]

    T. A. Gomez, T. Nagayama, P. B. Cho, D. P. Kil- crease, C. J. Fontes, and M. C. Zammit, Journal of Physics B: Atomic, Molecular and Optical Physics55, 10.1088/1361-6455/ac4f31 (2022)

    2022 doi

  68. [76]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathe- matical Tables, ninth dover printing, tenth gpo printing ed. (Dover Publications, New York, 1964)

    1964

  69. [77]

    Pollock, Computer Physics Communications52, 49 (1988)

    E. Pollock, Computer Physics Communications52, 49 (1988)

    1988

  70. [78]

    Omidvar, H

    K. Omidvar, H. L. Kyle, and E. C. Sullivan, Physical Review A5, 1174 (1972)

    1972

  71. [79]

    Nordsieck, Physical Review93, 785 (1954)

    A. Nordsieck, Physical Review93, 785 (1954)

    1954

  72. [80]

    H. A. Bethe and L. C. Maximon, Physical Review93, 768 (1954)

    1954

  73. [81]

    Zastrau, K

    U. Zastrau, K. Appel, C. Baehtz, O. Baehr, L. Batchelor, A. Bergh¨ auser, M. Banjafar, E. Brambrink, V. Cerantola, T. E. Cowan, H. Damker, S. Dietrich, S. Di Dio Cafiso, J. Dreyer, H.-O. Engel, T. Feldmann, S. Findeisen, M. Foese, D. Fulla-Marsa, S. G¨ ode, M. Hassan, J. Hause...

    2021

  74. [82]

    Sikorski, M

    M. Sikorski, M. Ramilli, R. de Wijn, V. Hinger, A. Moz- zanica, B. Schmitt, H. Han, R. Bean, J. Bielecki, G. Bor- tel, T. Dietze, G. Faigel, K. Kharitonov, C. Kim, J. C. P. Koliyadu, F. H. M. Koua, R. Letrun, L. M. Lopez, N. Reimers, A. Round, A. Sarma, T. Sato, M. Tegze, and ...

    2023

  75. [83]

    H. M. Bellenbaum, M. P. B¨ ohme, and T. Gawne, X-ray diagnostics, analysis, verification and exploration code (2026)

    2026

  76. [84]

    G. F. Chew and G. C. Wick, Physical Review85, 636 (1952)

    1952

  77. [85]

    Schnaidt, Annalen der Physik413, 89 (1934)

    F. Schnaidt, Annalen der Physik413, 89 (1934)

    1934

  78. [86]

    L. D. Landau and E. M. Lifshits,Quantum Mechanics: Non-Relativistic Theory, Course of Theoretical Physics, Vol. v.3 (Butterworth-Heinemann, Oxford, 1991)

    1991

  79. [87]

    Ribaldone and J

    C. Ribaldone and J. K. Desmarais, The Journal of Chem- ical Physics163, 074102 (2025)

    2025

  80. [88]

    D. A. Varshalovich, A. N. Moskalev, and V. K. Kher- sonskii,Quantum theory of angular momentum(World Scientific, 1988)

    1988

  81. [89]

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gom- mers, P. Virtanen, D. Cournapeau, E. Wieser, J. Tay- lor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del R ´ ıo, M. Wiebe, P. Peterson, P. G´ erard-Marchant, K. She...

    2020

  82. [90]

    Meurer, C

    A. Meurer, C. P. Smith, M. Paprocki, O. ˇCert ´ ık, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johans- son, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Rouˇ cka,...

    2017

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.