"s" is the sensitivity (e.g. in [Asec/mol])

And finally:

Note: We assume the sensitivity for two isotopic species of the same element as being equal

Let us assume that we have loaded n_{1} resp. n_{2} moles of two isotopic species (1,2), i.e. a total of n = n_{1} + n_{2} moles.

The 'observed isotope ratio', i.e. the ratio of the two relevant ion currents is, therefore, given by

Note: The ratio of the mol fractions is the isotope ratio in the yet solid phase of the sample, which can be computed, using Rayleigh's law.

Their masses are *m*_{1} < m_{2} < m_{3}, i.e., in our example,both ratios always have one species with the same mass in common (*m*_{1}, m_{2}, or m_{3}).

Let us, as an example, assume that the number

The strict application of these equations , hence, sets some restrictions for the definition of the

involved ratios and requires different algorithms for the four legal ratio definitions.

Such restrictions can be circumvented by the application of the third Rayleigh equation, which

relates fractionation to the total amount of sample on the filament. However, the elimination of the

mol fraction q does not result in an exlicite expression for *R*_{T}. Therefore, *R*_{T} must be calculated by

the application of, either, the '*regula falsi (secant method)*' , or, of '*Newton's algorithm (tangent *

Both algorithms normally converge within a few iteration loops (less than 10), except for very noisy

data, which may cause difficulties.

In such cases, one better switches to one of the empiric fractionation laws, which are described in

the following. (If the iteration does not converge, your data are, for sure, so noisy that virtually any

systematic error of the computed result cannot be pushed over to the particular fractionation

correction algorithm).

If the lower masses of the known and the unknown ratios are equal, the non-iterative equation

delivers the accurate Rayleigh normalization value.

Let be

the (time dependent) mole fractions of these species and

and

the mole fraction of the whole sample on the filament.

Then, the *Rayleigh Law* describes the time dependence of the measured ratio (r_{abs}) as follows (**see footnote 1**):

or

or

During a thermal ionization (TIMS) isotope ratio measurement, everybody can easily observe, that isotope ratios systematically vary with time. This effect is called *fractionation*.

It has a simple reason: For the lighter isotopic species of an isotope ratio, the rate of evaporation is higher than for the heavier species. Hence, the lighter isotope is depleted with time in the not yet evaporated fraction of the sample. And, vica versa, it is, at any time, enriched in the vapor phase, as compared to the remaining solid phase.

Thus, in order to describe fractionation mathematically, it suggests itself, to relate fractionation and sample consumption. This is what the__Lord Rayleigh's 'law' (1896)__ does: It describes the time dependence of the isotope ratio in the *not yet evaporated fraction of the sample*.

In a thermal ionization source, the ion current*'i' *reflects the rate of flow (dn/dt) of those particals which hit the hot surface.

It has a simple reason: For the lighter isotopic species of an isotope ratio, the rate of evaporation is higher than for the heavier species. Hence, the lighter isotope is depleted with time in the not yet evaporated fraction of the sample. And, vica versa, it is, at any time, enriched in the vapor phase, as compared to the remaining solid phase.

Thus, in order to describe fractionation mathematically, it suggests itself, to relate fractionation and sample consumption. This is what the

In a thermal ionization source, the ion current

Inserting *n*_{2}(t)/n_{2 }into the linearized approximation results in

A very good (linearized) approximation of this law is given by

Hence, we may expect roughly linear decay (or increase, if b<1 ) of the fractionation pattern with time for an exponentially decaying ion current.

Other than exponential emission profiles produce other, sometimes strange, fractionation patterns. Oftenly, the fractionation pattern also reflects a particular evaporation behaviour, like varying multiple species emission and so on.

Let us now consider the Rayleigh equations of two isotope ratios (e.g. the unknown and the known ratio in a three isotope system).

Other than exponential emission profiles produce other, sometimes strange, fractionation patterns. Oftenly, the fractionation pattern also reflects a particular evaporation behaviour, like varying multiple species emission and so on.

Let us now consider the Rayleigh equations of two isotope ratios (e.g. the unknown and the known ratio in a three isotope system).

and

;

Let be

the mole fraction of this species. We thus get two equations

and

with

and

(i=1,2)

This above equation describes a relation, which is called '__fractionation line__' in its graphical representation.

In its numeric version, it is used to calculate the true isotope ratio*R*_{T1} , given the measured ratios *r*_{obs1} and *r*_{obs2}, and the true ratio *R*_{T2} of the internal reference.

For the assignment of the common species in the two relevant ratios we appearantly have to distinguish between 4 cases (given that both ratios are defined with either the light or the heavy species in the nominator, respectively):

(A): m_{1} is common to both ratios,

(B): m_{2} is in the nominator of *r*_{1} and in the denominator of *r*_{2},

(C): m_{2} is in the denominator of *r*_{1} and in the nominator of *r*_{2},

(D): m_{3} is common to both ratios.

Hence, the exponent*X* in the above equation is, respectively, given by:

In its numeric version, it is used to calculate the true isotope ratio

For the assignment of the common species in the two relevant ratios we appearantly have to distinguish between 4 cases (given that both ratios are defined with either the light or the heavy species in the nominator, respectively):

(A): m

(B): m

(C): m

(D): m

Hence, the exponent