HAR etch · Identifiability
The same marginals can tell different stories.
Show what separate angle and energy distributions leave unidentified.
Admissible interval width
Values & comparison details
Difference = comparison − reference. Rounded for display; original data retained.
- Marginals only
- 15 eV
- With one joint observation
- 6.25 eV
Energy per incident member. Width of admissible bounds, not a confidence interval or etch-rate improvement.
Open full figure THE QUESTION
How much bottom-delivered energy can be inferred from separate angle and energy distributions?
THE APPROACH
A static absorbing trench with depth-to-width ratio 10 uses three angle categories and three energy categories. Exact finite-table bounds compare all admissible joint distributions, first with marginals alone and then with one hypothetical joint-cell observation.
WHAT THE COMPARISON SHOWS
Three things to take away.
- 01
Both extreme joint tables preserve the same 56.25% direct-bottom count fraction and 60 eV mean incident energy. Their delivered energies still differ.
- 02
One hypothetical joint-cell observation narrows the admissible interval to 32.5–38.75 eV per incident population member.
- 03
The independence assumption gives 33.75 eV, but its joint table violates the added observation. A scalar value inside the interval does not make the underlying model admissible.
Evidence, assumptions & the next question
What was numerically checked
Six stored optimization extrema across three finite cases agree with independent exact-rational vertex enumeration, an analytic control and primal/dual certificates.
How far the result goes
Synthetic prescribed probabilities and static first-contact geometry. These are admissible bounds, not confidence intervals. Delivered projectile energy is not etch rate, deposited heat, flux or material yield.
What a physical study would need next
Acquire joint angle-energy information and qualify reflection, charging, chemistry and evolving geometry before making material-specific etch predictions.
SA-PROC-ETCH-HAR-011 · Angle-energy joint identification · 2026-09-27
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