The Math of Liberation

 

The Math of Liberation

Notes on probability density, the Beta distribution, Andrews–Mika diagram, and breakage types.

What is Probability Density?

Probability density is a way to show how common items are along a scale. If the horizontal axis is particle type (left = pure waste, middle = mixed, right = pure mineral), the vertical axis (probability density) shows how many particles fall at each point.

Remember: higher curve → more particles there; lower curve → fewer particles.

1. U-shaped Beta Distribution (Why Beta?)

The Beta distribution is used because it is flexible and can make a U-shape. A well-ground ore often yields:

  • Many pure waste particles (left tail)
  • Many pure mineral particles (right tail)
  • Few mixed (middling) particles (middle)


Quick gist: Left = L0 (liberated gangue), Right = L1 (liberated mineral), Middle = middlings.

The Four Parameters (short):

  1. Average grade — overall metal fraction in the rock.
  2. Variance — how much grades vary (spread).
  3. L0 — fraction of pure waste (liberated gangue).
  4. L1 — fraction of pure valuable mineral (liberated mineral).

2. The Andrews–Mika Diagram (The Map)

The Andrews–Mika diagram shows the attainable region — what particle sizes and grades are physically possible after breaking a parent particle. It enforces conservation: you cannot create a particle richer or larger than allowed by the parent rock's internal distribution.If the parent contains only a tiny speck of mineral, you cannot create a large pure mineral particle out of thin air. The diagram mathematically blocks impossible outcomes.

3. Breaking Bad — Random vs Preferential Fracture

Two basic fracture behaviours:

  • Random breakage — fractures occur unpredictably (many mixed particles).
  • Preferential breakage — fractures follow weak mineral boundaries (better liberation, less grinding).Bottom line: Preferential breakage is great — you get freed minerals with less energy.

Quick Summary Checklist

  • Liberation: break rocks until mineral phases are free from gangue.
  • Grain size rule: grind finer than the characteristic mineral grain size to increase liberation.
  • Stereology: use 2D images (microscope) to estimate 3D textures.
  • Beta distribution: a flexible curve (often U-shaped) that models liberated + unliberated particle fractions (L0, L1).
  • Andrews–Mika: diagram that defines physically attainable particle grades/sizes after breakage.



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