# ⚛  L1 Principle — Electron Backscatter Diffraction (EBSD) — crystallographic orientation mapping

**ID:** `L1-089` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Electron Microscopy — *Kikuchi-pattern crystal orientation*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** euler angle field
> **📡 Carrier:** electron · **🌫 Noise:** shot poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554184

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## 🧠 1. Introduction

**Electron Backscatter Diffraction (EBSD) — crystallographic orientation mapping** is a **nonlinear inverse problem** whose unknown lives in **euler angle field** space, within the **Kikuchi-pattern crystal orientation** sub-domain of **Electron Microscopy**.

Measurements consist of electrons collected by an electron detector via a **kikuchi diffraction pattern** sensing mechanism.

The forward operator applies, in order: L · excitation · electron beam operator; L · diffraction kikuchi operator; L · pattern match operator; integration over the solid angle of incidence/emission.

Observations are corrupted by Poisson shot noise from quantum-limited detection. Existence of the recovered euler angle field is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 25); pattern_center_drift dominates the stability cliff; band_contrast_degradation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

## ⚙ 2. Forward Model

Physical chain: **x** → L · excitation · electron beam → L · diffraction kikuchi → L · pattern match → Angular integration → **y** (detector).

```
y = ∫dΩ `L.pattern_match` `L.diffraction_kikuchi` `L.excitation.electron_beam` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.excitation.electron_beam` | L · excitation · electron beam operator |
| `L.diffraction_kikuchi` | L · diffraction kikuchi operator |
| `L.pattern_match` | L · pattern match operator |
| `int.angular` | Integration over the solid angle of incidence/emission |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Electron Microscopy |
| Sub domain | Kikuchi-pattern crystal orientation |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | euler_angle_field |
| Noise model | shot_poisson |
| Integration axis | angular |
| Difficulty delta | 5 |
| L dag | 4 |

## 📡 4. Measurement Model

Existence of the recovered euler angle field is guaranteed within the declared Omega bounds. Uniqueness is local rather than global (non-convex landscape); convergence depends on initialisation and priors. Stability is moderately conditioned (kappa_eff ~= 25); pattern_center_drift dominates the stability cliff; band_contrast_degradation and the remaining mismatch parameters contribute higher-order bias terms. Photon-shot-noise-limited (poisson counting) sets the irreducible data-fidelity floor, while TV / wavelet-sparsity / deep priors stabilise recovery at the ill-conditioned end of Omega.

| Metric | Value |
|---|---|
| Metric | orientation_MAE_deg |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `ebsd_kikuchi` · **Forward operator:** `ebsd_kikuchi_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 1024 |
| W | px | 1024 |
| Kv | — | 20 |
| Step nm | nm | 100 |
| Peak electrons | — | 200 |
| Pattern center drift | — | 0 |
| Band contrast degradation | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 256 – 4096 |
| W | px | 256 – 4096 |
| Kv | — | 10 – 30 |
| Step nm | nm | 10 – 1000 |
| Peak electrons | — | 20 – 2000 |
| Pattern center drift | — | 0.0 – 0.1 |
| Grain boundary ambiguity | — | 0.0 – 0.3 |
| Band contrast degradation | — | 0.0 – 0.5 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.5

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 40.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **orientation_MAE_deg**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x552dd04e4f1654b0604973f09e411c7cb62855308209272563bafeac057db8f7`
- **Chain tx hash:** `0xf705d315ca1c7fac5e4e647d8af9337dbf77876231e98acf65c0e826e4a5f5da`
- **Chain block:** `41554184`

---

## File Mapping

This bundle consists of: `L1-089.md`, `L1-089.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-089.md` | Source of truth — edit this | Human or LLM |
| `L1-089.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-089`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._