pith. sign in

arxiv: 2604.20287 · v1 · submitted 2026-04-22 · 🧮 math.AP

Read--Shockley formula for a general Bravais lattice in two dimensions

Lucia Scardia , Edoardo Giovanni Tolotti This is my paper

Pith reviewed 2026-05-10 00:16 UTC · model grok-4.3

classification 🧮 math.AP
keywords grain boundariesdislocationsRead-Shockley formulaBravais latticesemi-discrete energycrystal defectstwo-dimensional models
0
0 comments X

The pith

A construction for grain boundaries in general Bravais lattices reproduces the Read-Shockley logarithmic energy scaling.

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

The paper examines a semi-discrete model of dislocations in two-dimensional crystals. It proposes a straightforward arrangement of dislocations to form the boundary between two grains rotated by a small angle. For any Bravais lattice, this arrangement yields an energy that grows like the logarithm of the misorientation angle, just as the classical Read-Shockley formula predicts. This provides a concrete way to understand low-angle grain boundaries without restricting to special lattice types.

Core claim

In a two-dimensional semi-discrete dislocation energy, a simple construction for the grain boundary between two crystal grains with small orientation difference has energy that matches the logarithmic scaling predicted by Read and Shockley, and this holds for a general Bravais lattice.

What carries the argument

The semi-discrete dislocation energy functional together with a physically motivated construction of the grain boundary using periodic arrangements of dislocations adapted to the lattice.

If this is right

  • The Read-Shockley formula applies to arbitrary 2D Bravais lattices, not just square ones.
  • Low-angle grain boundary energies can be predicted from dislocation arrangements in general crystal structures.
  • This supports using semi-discrete models for studying polycrystalline materials with various lattice symmetries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar constructions might extend to three dimensions or other defect types.
  • The result suggests that the logarithmic scaling is robust to lattice geometry as long as the construction is chosen appropriately.
  • Computational simulations could test this for specific non-square lattices like hexagonal ones.

Load-bearing premise

That the proposed construction represents the lowest-energy or typical configuration for the grain boundary rather than some other arrangement.

What would settle it

Numerical minimization of the semi-discrete energy for a small misorientation angle on a non-square Bravais lattice, checking if a lower energy than the constructed one is found.

Figures

Figures reproduced from arXiv: 2604.20287 by Edoardo Giovanni Tolotti, Lucia Scardia.

Figure 1
Figure 1. Figure 1: The splitting of the domain. Then, for i = 1, 2, we further split the strips Σi into squares with side-length 2ℓi , obtained by translating Qi := [−ℓi , ℓi ] 2 by the vectors ti,k := −(2k − 1)ℓie2 + (−1)i ℓie1, (5) where k = 1, . . . , Ni , and Ni := ⌈L/ℓi⌉. We further partition Qi into a family of dyadic square annuli. To this aim, with no loss of generality we write ℓi as ℓi = 2n¯iλε, (6) where ¯ni ∈ N w… view at source ↗
Figure 2
Figure 2. Figure 2: The regions of the construction. 2.2. The construction of the piecewise constant strain. In this section we construct a piecewise affine deformation in Ω. Note that the energy depends only on the strain field β, but we believe that writing the full deformation makes it easier to understand the construction. In the vertical strips Σ±θ we define the deformation as R±θx, for x ∈ Σ±θ, to match the boundary con… view at source ↗
Figure 3
Figure 3. Figure 3: The deformed body with b1 = (1/2, − √ 3/2) and b2 = (√ 3/2, 1/2). 2.3. Admissibility of β. In this section we check that the strain β defined in (12) is admis￾sible, namely that it satisfies assumptions (H1)–(H3), and identify the set S where the curl concentrates. We note that, from (2), the core energy penalises the length of the interfaces between constant values of the strain that are not rank-one conn… view at source ↗
read the original abstract

In this note we consider a two-dimensional semi-discrete dislocation energy and propose a simple and physically motivated construction for the grain boundary between two crystal grains with a small orientation difference. In the case of a general Bravais lattice, the energy of this construction matches the logarithmic scaling predicted by Read and Shockley.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper considers a two-dimensional semi-discrete dislocation energy and proposes an explicit, physically motivated construction for the grain boundary between two crystal grains with small orientation difference. For a general Bravais lattice, the energy of this construction is shown to recover the logarithmic scaling of the classical Read-Shockley formula via direct summation of dislocation interactions.

Significance. If the result holds, the work extends the Read-Shockley prediction to arbitrary Bravais lattices in 2D by means of an explicit construction whose energy scaling is derived from first-principles dislocation sums without fitted parameters. The explicit nature of the configuration and the direct computation of its energy constitute a clear strength, providing a verifiable asymptotic statement rather than a self-referential or fitted scaling.

minor comments (2)
  1. The definition of the semi-discrete energy functional and the precise placement of the dislocation cores in the construction could be accompanied by a schematic diagram to aid readability for readers unfamiliar with the specific lattice geometry.
  2. In the asymptotic analysis of the interaction sums, the treatment of the far-field cutoff and the handling of the core energy contribution should be stated more explicitly to make the origin of the logarithmic term fully transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The report correctly identifies the main contribution: an explicit, parameter-free construction of a grain boundary in a general 2D Bravais lattice whose energy recovers the Read–Shockley logarithmic scaling by direct summation of the dislocation interaction kernel.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines an explicit, physically motivated construction for a grain boundary in a general Bravais lattice and computes its energy directly via summation of the semi-discrete dislocation interactions. The resulting logarithmic scaling is obtained as an output of this direct calculation rather than by fitting parameters, redefining the target quantity, or invoking self-citations whose content reduces to the present claim. The derivation therefore remains self-contained against the external Read-Shockley prediction and does not reduce to any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the definition of the semi-discrete dislocation energy (standard in the field) and on the geometric construction of the boundary; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The semi-discrete energy is the sum of a quadratic elastic term and a core energy term for each dislocation.
    Invoked to define the total energy whose scaling is computed.
  • standard math Dislocation interactions in 2D linear elasticity decay as 1/r and produce the standard logarithmic self-energy.
    Used to obtain the leading-order energy of the constructed array.

pith-pipeline@v0.9.0 · 5334 in / 1335 out tokens · 38110 ms · 2026-05-10T00:16:21.795006+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

  1. [1]

    Dautray R., Lions J.L.:Mathematical Analysis and Numerical Methods for Science and Technology.Vol

    work page

  2. [2]

    Springer Science & Business Media, 1999

    work page 1999

  3. [3]

    Rational Mech

    De Luca L., Garroni A., Ponsiglione M.: Γ-Convergence Analysis of Systems of Edge Dislocations: the Self Energy Regime.Arch. Rational Mech. Anal.206(2012), 885–910

    work page 2012

  4. [4]

    Fanzon S., Palombaro M., and Ponsiglione M.: Derivation of linearised polycrystals from a two- dimensional system of edge dislocations.SIAM J. Math. Anal.51(2020), 3956–3981

    work page 2020

  5. [5]

    Pure Appl

    Garroni A., Fortuna M., Spadaro E.: On the Read–Shockley energy for grain boundaries in 2D polycrys- tals.Comm. Pure Appl. Math.78(8) (2025), 1411–1459

    work page 2025

  6. [6]

    Garroni A., Leoni G., Ponsiglione M.: Gradient theory for plasticity via homogenization of discrete dislocations.J. Eur. Math. Soc.12(5) (2010), 1231–1266

    work page 2010

  7. [7]

    Rational Mech

    Ginster J.: Plasticity as the Γ-Limit of a Two-Dimensional Dislocation Energy: the Critical Regime without the Assumption of Well-Separateness.Arch. Rational Mech. Anal.233(2019), 1253–1288

    work page 2019

  8. [8]

    Giuliani A., Theil F.: Long range order in atomistic models for solids.J. Eur. Math. Soc.24(10) (2022), 3505–3555. 4Recall that in [12] the angle of misorientation isθ/2. However, up to changing the constantCin (24), no changes toE 0 orAare needed. 14 L. SCARDIA AND E.G. TOLOTTI

    work page 2022

  9. [9]

    ArXiv preprint arXiv:2509.12041

    Grabner P.J., Theil F.: Leading order asymptotics for non-local energies and the Read–Shockley law. ArXiv preprint arXiv:2509.12041

    work page arXiv

  10. [10]

    An Energy Estimate for Dislocation Configurations and the Emergence of Cosserat-Type Structures in Metal Plasticity

    Lauteri G., Luckhaus S.: An Energy Estimate for Dislocation Configurations and the Emergence of Cosserat-Type Structures in Metal Plasticity. ArXiv preprint arXiv:1608.06155v2

    work page Pith review arXiv

  11. [11]

    I.: Geometric rigidity for incompatible fields, and an application to strain-gradient plasticity.Indiana Univ

    M¨ uller S., Scardia L., Zeppieri C. I.: Geometric rigidity for incompatible fields, and an application to strain-gradient plasticity.Indiana Univ. Math. J.63(2014), no. 5, 1365–1396

    work page 2014

  12. [12]

    Ponsiglione M.: Elastic Energy Stored in a Crystal Induced by Screw Dislocations: From Discrete to Continuous.SIAM J. Math. Anal.39(2) (2007), 449–469

    work page 2007

  13. [13]

    Rev., 78 (3), (1950) 275–289

    Read W.T., Shockley W.: Dislocation models of crystal grain boundaries.Phys. Rev., 78 (3), (1950) 275–289

    work page 1950

  14. [14]

    Scardia L., and Zeppieri C.I.: Line-Tension Model for Plasticity as the Γ-Limit of a Nonlinear Dislocation Energy.SIAM Journal on Mathematical Analysis,44(4):2372–2400, 2012. (L. Scardia)Department of Mathematics, Heriot–W att University, United Kingdom Email address:L.Scardia@hw.ac.uk (E.G. Tolotti)Applied Mathematics M ¨unster, University of M ¨unster, ...

    work page 2012