Isotope Mixing Model Simulator: Binary & Ternary End-Member Analysis

simulator intermediate ~10 min
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50:50 mixing — midpoint on the mixing line

Equal mixing of end-members A (δX = −25, Ry = 0.704) and B (δX = −5, Ry = 0.735) produces a mixture at δX = −15, Ry = 0.720, plotting at the midpoint of the binary mixing line.

Formula

δ_mix = f × δ_B + (1 − f) × δ_A (linear mixing for ratios)
R_mix = (f × C_B × R_B + (1−f) × C_A × R_A) / (f × C_B + (1−f) × C_A)
Curvature: r = C_A / C_B (r = 1 gives straight line, r ≠ 1 gives hyperbola)

Tracing Sources with Isotopes

One of the most powerful applications of isotope geochemistry is identifying and quantifying the sources of geological materials. When a basaltic magma assimilates continental crust on its way to the surface, the mixture inherits isotopic signatures from both the mantle source and the crustal contaminant. Plotting the isotope ratios of multiple samples reveals a systematic trend — a mixing line — whose end-members reveal the original sources.

Binary Mixing in Isotope Space

For two end-members A and B, the mixture M plots along a line in ratio-vs-ratio space. The position of M along the line is controlled by the mixing fraction f and the concentration ratio of the element in each end-member. When concentrations are equal, the mixing line is straight and f maps linearly to position. When concentrations differ, the line curves — a hyperbola whose curvature reveals the concentration contrast.

Ternary Mixing and Beyond

With three end-members, data plot within a triangle in isotope space. Each vertex represents a pure end-member, and any interior point can be decomposed into unique proportions of all three. Adding more isotope systems (Sr, Nd, Pb, Hf) over-constrains the problem, allowing statistical tests of whether the assumed end-members are correct. This simulation lets you visualize how mixing fraction and end-member positions control the mixture composition.

Applications in Magma Genesis

The Sr-Nd isotope diagram is the classic tool for mantle geochemistry. Depleted MORB-source mantle plots with low ⁸⁷Sr/⁸⁶Sr and high ¹⁴³Nd/¹⁴⁴Nd, while old continental crust plots in the opposite corner. Ocean island basalts scatter between several identified mantle end-members (DMM, HIMU, EM-I, EM-II), revealing the isotopic heterogeneity of the deep mantle and the recycling of crustal material through plate tectonics.

FAQ

What is isotopic mixing?

Isotopic mixing occurs when two or more reservoirs with distinct isotopic compositions combine. The resulting mixture plots along a line (binary) or within a triangle (ternary) in isotope ratio space. The position along the mixing line reveals the proportional contribution of each source.

Why are mixing lines sometimes curved?

When the element concentrations differ between end-members, mixing lines in ratio-ratio space become hyperbolic. The curvature parameter r = C_A/C_B controls the curvature. Only when concentrations are equal (r = 1) is the mixing line straight.

How are mixing models used in geochemistry?

Mixing models quantify magma contamination by continental crust, trace groundwater sources and pollution, identify sediment provenance, and determine the relative contributions of mantle and crustal sources in volcanic rocks. Sr-Nd-Pb isotope arrays are the primary tools.

What limits mixing model accuracy?

Uncertainties in end-member compositions, the assumption of only two or three sources (real systems may have more), and post-mixing processes like fractional crystallization or weathering can all distort the mixing signal and lead to erroneous source proportions.

Sources

Embed

<iframe src="https://homo-deus.com/lab/isotope-geochemistry/isotope-mixing/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub