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Anisotropic electron scattering and Migdal effect in semiconductor detectors

T0 review · 2 major / 3 minor · reviewed 2026-07-08 · grok-4.5

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Pith's one-line read Daily Migdal modulation in anisotropic semiconductors arises solely from the quadrupole of the energy-loss function.

desk verdict Usable anisotropic-ELF multipole method plus a clean Migdal-quadrupole selection rule for daily modulation in Si/GaAs; the scalar-ELF premise is the soft spot. read the letter →

arxiv 2607.06005 v1 pith:JXFWOEMK submitted 2026-07-07 hep-ph

Aula Al-Adulrazzaq , Paolo Settembri , Sebastian Sassi , Federico Giannessi , Gianni Profeta , Matti Heikinheimo , Kimmo Tuominen This is my paper · ORCID
classification hep-ph
keywords MigdaleffectenergylossfunctionanisotropicELFdarkmatterdetectionsemiconductordetectorsdailymodulationdielectricelectronscattering
topics Dark Matter
open problems Dark Matter
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

Cryogenic semiconductor detectors can see light dark matter either through direct electron recoils or through nuclear recoils that shake electrons free via the Migdal effect. Both rates are controlled by the material's energy-loss function (ELF), which is usually treated as isotropic. This paper supplies a practical method that keeps the full angular dependence of the ELF and decomposes it into spherical harmonics. The central result is that the daily modulation of the Migdal rate is generated only by the quadrupole piece of that decomposition, while every multipole of the ELF contributes to the dark-matter–electron scattering rate. The method is applied to silicon and gallium arsenide, the two materials most relevant to existing cryogenic detectors. A sympathetic reader cares because a clean multipole separation turns daily modulation into a diagnostic that can help separate signal channels and reduce background confusion.

What carries the argument

The spherical-harmonic multipole expansion of the anisotropic energy-loss function (the imaginary part of the inverse dielectric function). Once the ELF is written as a sum of multipoles, the angular integrals that produce the daily modulation isolate the quadrupole for Migdal scattering while leaving all multipoles active for electron scattering.

What would settle it

Extract the multipole moments of the ELF for silicon or GaAs from first-principles dielectric calculations, recompute the Migdal daily-modulation amplitude with only the quadrupole retained, and compare with a full anisotropic numerical integration or with a future experimental modulation measurement; a nonzero modulation arising from the isotropic or dipole parts alone would falsify the claim.

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Extended reading notes

Core claim

For a fully anisotropic energy-loss function the daily modulation of the Migdal event rate is produced exclusively by the quadrupole (l = 2) spherical-harmonic component of the ELF, whereas the dark-matter–electron scattering rate receives contributions from all multipoles. A practical multipole decomposition of the anisotropic ELF is given and used to compute both rates in silicon and gallium arsenide.

Load-bearing premise

That the ordinary dielectric energy-loss-function formalism continues to control both rates once the ELF is allowed to be fully anisotropic, with no extra daily-modulation channels opened by higher solid-state or many-body corrections beyond the multipoles.

Editorial extensions

If this is right

  • Daily modulation amplitudes for Migdal events in Si and GaAs can be predicted from the ELF quadrupole alone, without recomputing the full anisotropic rate each sidereal day.
  • The different multipole content of Migdal versus electron-scattering rates supplies a diagnostic that can help separate the two channels in the same detector.
  • Isotropic-ELF calculations miss a modulation signal that appears once material anisotropy is restored.
  • The same multipole method can be applied immediately to other cubic or anisotropic semiconductor targets used in direct detection.

Reading between the lines

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

  • Materials whose ELF quadrupole is unusually large could exhibit a Migdal daily modulation large enough to serve as a directional signature without a dedicated directional detector.
  • The multipole isolation suggests that dielectric anisotropy could be deliberately engineered or selected to optimize daily-modulation sensitivity for light dark matter.
  • Similar multipole filtering may apply to other inelastic processes mediated by the dielectric response, such as plasmon-assisted or phonon-assisted scattering.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The manuscript develops a practical multipole method for computing dark-matter–electron and Migdal rates in semiconductors with a fully anisotropic energy-loss function (ELF). Starting from the dielectric response, the ELF is expanded in spherical harmonics; the daily-modulated rates are then expressed as contractions of ELF multipoles with kinematic moments of the DM velocity distribution. The central structural claim is that the Migdal daily modulation is sourced solely by the quadrupole (ℓ=2) ELF multipoles, whereas every multipole of the ELF contributes to the electron-scattering rate. The formalism is applied to silicon and gallium arsenide, with numerical estimates of the modulation amplitudes.

Significance. If the selection rule and the multipole rate formulae hold under the stated dielectric-response assumptions, the paper supplies a clean, reusable computational framework for anisotropic semiconductor targets and a sharp, falsifiable prediction (quadrupole-only Migdal modulation). That prediction is of direct experimental interest for next-generation cryogenic detectors that can resolve daily modulation. The work also clarifies how crystal anisotropy enters the two channels differently, which is a useful conceptual advance beyond the isotropic-ELF literature.

major comments (2)
  1. The central claim that Migdal daily modulation arises solely from the ELF quadrupole treats the electronic response as a scalar anisotropic ELF that is multipole-expanded and inserted into the rate integrals. In a crystal the microscopic object is the dielectric matrix ε_GG'(q,ω); the macroscopic ELF is recovered only after inversion that mixes local fields (G≠0). Those local-field corrections can generate effective angular dependence that is not equivalent to any finite multipole set of the macroscopic ELF. Because the DM wind already supplies a dipole anisotropy, residual local-field angular structure can couple to produce a time-dependent rate even if every ℓ=2 component of the macroscopic ELF is set to zero. The paper’s practical method and the Si/GaAs applications rest on the scalar-ELF multipoles being a complete and accurate input; nothing in the derivation isolates or suppresses
  2. The numerical applications to Si and GaAs are presented as concrete illustrations of the selection rule, yet the manuscript does not quantify how the adopted anisotropic ELF models (or the truncation of the multipole series) affect the reported modulation amplitudes. Without a controlled comparison to a fully isotropic baseline and to a higher-multipole truncation, it is difficult to judge whether the quoted daily-modulation signals are robust predictions or artifacts of the particular ELF parametrization. A short sensitivity study would make the applications load-bearing rather than merely illustrative.
minor comments (3)
  1. Notation for the multipole coefficients of the ELF should be standardized early (e.g., a single symbol for Im[−1/ε]_{ℓm}(q,ω)) and used consistently in the rate formulae and figures.
  2. Figures showing the daily modulation should include error bands or at least a statement of the dominant theoretical uncertainty (ELF model, multipole truncation, velocity-distribution moments).
  3. A brief comparison table of the isotropic-limit rates against existing literature results for Si/GaAs would help the reader verify the numerical pipeline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful reading and for highlighting both the structural selection rule and the experimental relevance of the Migdal daily-modulation prediction. We address the two major comments below. On the first, we clarify the scope of the dielectric-response assumptions already stated in the manuscript and add an explicit discussion of local-field corrections and the macroscopic ELF; we do not claim that the scalar multipole expansion is exact beyond those assumptions. On the second, we agree that a controlled sensitivity study is needed and will add isotropic-baseline and multipole-truncation comparisons for Si and GaAs so that the reported modulation amplitudes can be judged as robust rather than merely illustrative. We believe these revisions convert the applications into load-bearing results while preserving the central selection-rule claim under the stated framework.

read point-by-point responses

  1. Referee: The central claim that Migdal daily modulation arises solely from the ELF quadrupole treats the electronic response as a scalar anisotropic ELF that is multipole-expanded and inserted into the rate integrals. In a crystal the microscopic object is the dielectric matrix ε_GG'(q,ω); the macroscopic ELF is recovered only after inversion that mixes local fields (G≠0). Those local-field corrections can generate effective angular dependence that is not equivalent to any finite multipole set of the macroscopic ELF. Because the DM wind already supplies a dipole anisotropy, residual local-field angular structure can couple to produce a time-dependent rate even if every ℓ=2 component of the macroscopic ELF is set to zero. The paper’s practical method and the Si/GaAs applications rest on the scalar-ELF multipoles being a complete and accurate input; nothing in the derivation isolates or suppresses

    Authors: We agree that the microscopic response is the dielectric matrix ε_GG'(q,ω) and that the macroscopic ELF is obtained only after matrix inversion that incorporates local-field corrections (LFCs). Our derivation and selection rule are formulated entirely within the standard macroscopic-ELF framework used in the isotropic literature (and in existing semiconductor DM codes): the rate is written in terms of Im[−1/ε_M(q,ω)], where ε_M is the macroscopic dielectric function after inversion. Under that assumption the multipole expansion of the scalar anisotropic ELF is complete by construction, and the Migdal daily modulation is sourced only by the ℓ=2 multipoles (because the Migdal kinematics contract with a rank-2 tensor built from the nuclear recoil direction). Residual angular structure that cannot be absorbed into any multipole set of the macroscopic ELF would require retaining the full off-diagonal G,G' structure inside the rate integrals themselves—an extension beyond both our work and the isotropic-ELF literature we build on. We do not claim that LFCs are absent; rather, once they have been folded into ε_M(q,ω), any remaining anisotropy is precisely what our multipoles capture. We will revise the manuscript to state this scope explicitly (Introduction and Sec. II), to note that a full microscopic treatment with unsummed LFCs is left for future work, and to emphasize that the quadrupole-only selection rule is a sharp prediction inside the macroscopic-ELF framework. The practical method and the Si/GaAs applications are therefore complete and accurate within the stated assumptions; they are not claimed to be exact beyond them. revision: yes

  2. Referee: The numerical applications to Si and GaAs are presented as concrete illustrations of the selection rule, yet the manuscript does not quantify how the adopted anisotropic ELF models (or the truncation of the multipole series) affect the reported modulation amplitudes. Without a controlled comparison to a fully isotropic baseline and to a higher-multipole truncation, it is difficult to judge whether the quoted daily-modulation signals are robust predictions or artifacts of the particular ELF parametrization. A short sensitivity study would make the applications load-bearing rather than merely illustrative.

    Authors: We agree. The applications were intended as illustrations of the selection rule, but without a controlled sensitivity study the quoted modulation amplitudes cannot be judged as robust. In the revised manuscript we will add: (i) a fully isotropic baseline obtained by retaining only the monopole of the same ELF models, so that the pure anisotropic contribution to the daily modulation is isolated; (ii) a multipole-truncation study (e.g. ℓ_max = 2 vs. 4 vs. 6) for both Si and GaAs, quantifying the residual change in the modulation amplitude; and (iii) a brief discussion of how the adopted anisotropic ELF parametrizations (and any model uncertainties they carry) propagate into the reported amplitudes. These additions will make the numerical results load-bearing rather than merely illustrative, while leaving the analytic selection rule unchanged. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: anisotropic-ELF multipole selection rules for Migdal vs electron rates are derived from the rate integrals, not fitted or tautological.

full rationale

The paper starts from the standard dielectric/ELF formalism for semiconductors, generalizes the ELF to a fully anisotropic function, expands it in spherical harmonics, and inserts that expansion into the Migdal and DM–electron rate integrals. The central claim—that daily modulation of the Migdal rate arises only from the ELF quadrupole, while all multipoles enter electron scattering—follows from the angular structure of those integrals (DM wind direction vs. isotropic nuclear recoil kinematics for Migdal). No free parameter is fitted to modulation data and then re-presented as a prediction; no uniqueness theorem or ansatz is imported solely via overlapping-author citation to force the result; and the Si/GaAs applications are numerical evaluations of the same derived expressions against external material inputs (ELF multipoles), not renamings of known empirical patterns. The derivation is therefore self-contained against its stated inputs. Residual physics concerns (e.g., local-field corrections beyond a scalar ELF multipole expansion) affect correctness risk, not circularity. Score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Abstract-only audit. The central claim rests on the standard solid-state energy-loss-function formalism (dielectric function → ELF → electronic excitation rates for both DM-electron and Migdal channels) generalized to anisotropy via a spherical-harmonic decomposition. No free parameters or invented particles are introduced in the abstract. Material-specific dielectric inputs for Si and GaAs are external domain inputs, not free parameters of the multipole theorem itself. Exhaustiveness is limited by lack of the full text.

assumptions (3)
  • domain assumption Electronic excitation rates for DM-electron scattering and for the Migdal effect in semiconductors are controlled by the material energy-loss function built from the dielectric function.
    Stated in the abstract as the shared dependence of both channels on the ELF; this is standard in the subfield but is the load-bearing physical premise of the calculation.
  • domain assumption A general anisotropic ELF admits a spherical-harmonic multipole expansion that can be inserted into the event-rate integrals to isolate daily-modulation contributions.
    Implicit in the claim that only the quadrupole drives Migdal daily modulation and that all multipoles affect electron scattering; assumes the expansion is complete and interchangeable with the rate kernels.
  • standard math Standard mathematical tools of multipole expansions and crystal dielectric response apply without additional ad-hoc selection rules beyond those derived.
    Background math assumed for the anisotropic generalization.

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Cite this review

Pith. "Pith review of Anisotropic electron scattering and Migdal effect in semiconductor detectors." pith.science (2026). https://pith.science/paper/JXFWOEMK

@misc{pith2026260706005,
  author       = {Pith},
  title        = {Pith review of: Anisotropic electron scattering and Migdal effect in semiconductor detectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JXFWOEMK}},
  note         = {Machine review of arXiv:2607.06005}
}
read the original abstract

Cryogenic semiconductor detectors are widely used in dark matter direct detection. Electronic excitations in these materials can be caused either by direct dark matter electron scattering events, or indirectly via dark matter nuclear scattering events. For nuclear scattering of light dark matter, the event rate is enhanced due to an inelastic process known as the Migdal effect. Both the electron recoils and the Migdal effect in semiconductors depend on the dielectric function of the target material via the so-called energy loss function (ELF). In the standard approach found in the literature, the ELF is approximated as isotropic to simplify the calculation of the event rate. We introduce a practical method for computing the event rate for a general anisotropic ELF. We find that the daily modulation of the Migdal rate arises solely due to the quadrupole component of the ELF, whereas in electron scattering all spherical harmonic components affect the rate. We apply the formalism to study the daily modulation in silicon and gallium arsenide detectors.

Figures

Figures reproduced from arXiv: 2607.06005 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Spherically averaged electronic loss function of Si. The black dashed line indicates the upper kinematic boundary [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Daily modulation of the event rate for parametric model ELFs in Eq. (45) and (46), assuming the SHM velocity [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The electron scattering (left) and Migdal (right) event rates in silicon and gallium arsenide for the SHM velocity [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Electron scattering (solid) and Migdal (dashed) daily modulation in silicon (orange) and gallium arsenide (blue) at [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

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