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Equation 11 · Radioactive Clocks: How Nuclear Physics Gave Evolution Its Time

What does this equation mean?

Rt=R0+N87N86(eλt−1),R_t = R_0 + \frac{N_{87}}{N_{86}}\left(e^{\lambda t} - 1\right),

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Start withN_87
Divide byN_86
This relates toR_t
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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RtR_t

Symbol R_t

each mineral’s measured present-day strontium-87 to strontium-86 ratio.

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R0R_0

Symbol R_0

the shared initial ratio recovered as the line’s intercept.

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N87N_{87}

Symbol N_87

N8N_87 occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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N86N_{86}

Symbol N_86

each mineral’s measured rubidium-87 to strontium-86 ratio [ 18 ].

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eλte^{\lambda t}

Symbol e^λ t

e^λ t is one of the signed contributions combined to compute the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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addition

addition

Add the term after the plus sign to the term or group before it.

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subtraction

subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

The uranium-lead and argon-argon methods both need to know, or be able to ignore, how much daughter isotope was already present when the mineral formed. The isochron method, used most often with the rubidium-strontium and samarium-neodymium systems, solves this by measuring several minerals from the same rock, each of which crystallized from the same well-mixed magma at the same time and so began with the same strontium or neodymium isotopic composition, but which incorporated different amounts of the radioactive parent isotope depending on their individual chemistry. Plotting each mineral’s present-day ratio of radiogenic daughter to stable reference isotope against its ratio of parent to…
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The uranium-lead and argon-argon methods both need to know, or be able to ignore, how much daughter isotope was already present when the mineral formed. The isochron method, used most often with the rubidium-strontium and samarium-neodymium systems, solves this by measuring several minerals from the same rock, each of which crystallized from the same well-mixed magma at the same time and so began with the same strontium or neodymium isotopic composition, but which incorporated different amounts of the radioactive parent isotope depending on their individual chemistry. Plotting each mineral’s present-day ratio of radiogenic daughter to stable reference isotope against its ratio of parent to reference isotope produces a straight line whose slope gives the age and whose intercept gives the initial composition directly: Rt=R0+N87N86(eλt−1)R_t = R_0 + \frac{N_{87}}{N_{86}}\left(e^{\lambda t} - 1\right). where RtR_t is each mineral’s measured present-day strontium-87 to strontium-86 ratio, R0R_0 is the shared initial ratio recovered as the line’s intercept, and N87N_{87}/N86N_{86} is each mineral’s measured rubidium-87 to strontium-86 ratio [ 18 ] . The initial composition is no longer an assumption fed into the calculation; it is an output the data themselves supply, and a poor fit to a straight line is itself a warning that the minerals did not, in fact, share a common origin or have remained closed systems since.

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