Fraction of Alpha in the Eutectic Microstructure Calculator
Use the lever rule at the eutectic temperature to compute alpha and beta fractions inside the eutectic constituent, and optionally estimate total phase fractions just below eutectic temperature.
Results
Enter compositions and click Calculate Fractions.
How to Calculate Fraction of Alpha in the Eutectic Microstructure
In binary alloy design, one of the most practical questions is: how much alpha phase exists inside the eutectic microstructure? This value controls hardness, ductility, machinability, and often corrosion behavior. If you work with common eutectic systems such as Pb-Sn, Al-Si, Ag-Cu, or other engineering pairs, understanding this calculation helps you move from a qualitative phase diagram reading to quantitative process decisions.
The good news is that the method is straightforward. You use the lever rule on the eutectic tie line at the eutectic temperature. The required values are the alpha composition at eutectic temperature (CαE), beta composition at eutectic temperature (CβE), and the eutectic liquid composition (CE). When those are known, you can directly compute the fraction of alpha and beta inside the eutectic constituent.
Core Equation (Lever Rule in the Eutectic)
If composition is expressed as concentration of the same component on one axis (for example wt% Sn, wt% Si, or wt% Ag), then the alpha fraction within eutectic is:
fα(eutectic) = (CβE – CE) / (CβE – CαE)
and beta fraction in eutectic is:
fβ(eutectic) = 1 – fα(eutectic)
This works as long as CE lies between CαE and CβE. The equation remains valid even if the alpha side has the higher numerical value, because both numerator and denominator switch sign together.
Why This Matters in Real Manufacturing
- In solder alloys, eutectic colony balance affects wetting, joint reliability, and thermal fatigue life.
- In cast Al-Si, alpha fraction in eutectic relates strongly to castability and post solidification response.
- In precision castings, eutectic morphology and phase ratio influence shrinkage behavior and crack sensitivity.
- In quality control, measured microconstituent percentages are often cross checked against lever rule calculations for process validation.
Step by Step Workflow
- Read CαE and CβE from the eutectic tie line of the phase diagram at eutectic temperature.
- Read CE at the eutectic point.
- Substitute into the lever rule equation.
- Convert to percent by multiplying by 100.
- Optionally compute total alpha in the full alloy if composition is not exactly eutectic.
For non eutectic alloy compositions (hypoeutectic or hypereutectic), alpha inside eutectic is still computed from the same eutectic tie line. Then total alpha in alloy is obtained by combining primary phase plus eutectic contribution.
Reference Data for Common Eutectic Systems
The following values are commonly cited in materials textbooks and engineering references. Minor variation can occur by database source and impurity level, but these values are useful for design screening and calculator checks.
| System | Eutectic Temperature | CE | CαE | CβE | Calculated fα in Eutectic | Calculated fβ in Eutectic |
|---|---|---|---|---|---|---|
| Pb-Sn (composition in wt% Sn) | 183 C | 61.9 | 18.3 | 97.8 | 45.16% | 54.84% |
| Al-Si (composition in wt% Si) | 577 C | 12.6 | 1.65 | 100.0 | 88.87% | 11.13% |
| Ag-Cu (composition in wt% Ag) | 779 C | 71.9 | 91.2 | 8.0 | 76.80% | 23.20% |
Interpreting the Table
Notice how the eutectic alpha fraction can vary dramatically across systems. In Al-Si, alpha (Al-rich) dominates the eutectic mass fraction because eutectic composition lies much closer to the beta boundary in wt% Si coordinates. In Pb-Sn, the split is more balanced. This is why simply saying “eutectic structure formed” is never enough for engineering decisions; you need phase ratios.
Worked Example: Pb-Sn Eutectic Alpha Fraction
Suppose you need alpha in the eutectic of Pb-Sn solder. Using typical values: CE = 61.9 wt% Sn, CαE = 18.3 wt% Sn, CβE = 97.8 wt% Sn.
fα = (97.8 – 61.9) / (97.8 – 18.3) = 35.9 / 79.5 = 0.4516
Therefore alpha in the eutectic microstructure is 45.16%, while beta is 54.84%. If image analysis from metallography gives a very different value, investigate etching contrast, thresholding, or whether your alloy is not exactly at eutectic conditions.
From Eutectic Fraction to Total Alloy Fraction
Engineers often need total alpha fraction in the full microstructure, not only inside eutectic colonies. For that, include overall composition C0:
- Hypoeutectic (C0 < CE): primary alpha forms first, then remaining liquid transforms eutectically.
- Hypereutectic (C0 > CE): primary beta forms first, then remaining liquid transforms eutectically.
For hypoeutectic alloys:
fprimary-alpha = (CE – C0) / (CE – CαE)
feutectic = 1 – fprimary-alpha
ftotal-alpha = fprimary-alpha + feutectic x fα(eutectic)
For hypereutectic alloys:
fprimary-beta = (C0 – CE) / (CβE – CE)
feutectic = 1 – fprimary-beta
ftotal-alpha = feutectic x fα(eutectic)
Sensitivity to Composition Error
Even small composition reading errors can shift phase fraction predictions. The table below shows a practical sensitivity check using Pb-Sn eutectic values with CE varied by measurement uncertainty.
| Case | CE used (wt% Sn) | fα in Eutectic | Change from Baseline |
|---|---|---|---|
| Baseline | 61.9 | 45.16% | 0.00% |
| CE low by 0.5 | 61.4 | 45.79% | +0.63% |
| CE high by 0.5 | 62.4 | 44.53% | -0.63% |
This sensitivity looks small in absolute terms, but in high volume quality control and property models, a one percent shift in phase fraction can be significant. That is why consistent composition basis and accurate diagram data are essential.
Common Mistakes and How to Avoid Them
- Mixing wt% and at% in one calculation.
- Reading tie line endpoints from a wrong temperature instead of eutectic isotherm.
- Using overall alloy C0 in place of CE when calculating eutectic constituent fractions.
- Assuming micrograph area fraction equals mass fraction without density correction in high contrast density systems.
- Ignoring metastable transformation products after non equilibrium cooling.
If your cooling is rapid, diffusion limits can shift effective compositions away from equilibrium. In that case, use this method as an equilibrium benchmark, then compare to observed microstructure for process calibration.
Best Practices for Advanced Users
- Standardize one composition convention (single component axis, same unit basis).
- Archive exact source of phase diagram values for traceability.
- Run uncertainty bands for CE, CαE, and CβE instead of a single point estimate.
- Pair calculations with image analysis and DSC where possible.
- Document whether reported values are fraction in eutectic or fraction in total alloy.
In production environments, a controlled calculation template like this calculator reduces interpretation drift between teams and supports repeatable metallurgical decision making.