A
Primer on
Isotope Notation
The isotope ratio:
The relative abundance of two isotopes a and b of element X can be expressed as an isotopic ratio, aR:
By convention, the more abundant isotope is placed in the denominator. This notation is not particularly helpful in itself however, as changes in R could result from changes in either aX or bX. It is however the basis for several expressions that are very useful.
Fractional abundance, aF:
This
expression is particularly useful in artificially enriched systems where the
ratio of aX to bX is increased by the intentional
addition of a pure source of one of the two isotopes (usually the heavy
isotope). The pure source is referred to as a spike, the process of adding
the source as "spiking" a sample. Spikes allow us to monitor the movement of
isotopes from one reservoir to another in response to one of more chemical
reactions.
Relationship between aR and aF:
or alternatively,
The delta notation, daX.
In many natural systems, the isotopic ratio aR exhibits variance in the range of the third to fifth decimal place. Numbers this small are best presented in terms of per mil, or parts per thousand (‰). This variance can be expressed relative to the isotopic ratio of a standard aRstd using a difference relationship knows as the delta notation (daX):
or alternatively, 
Change of
isotopic reference scale:
This relationship can be used to convert an isotopic value from one reference scale to another reference scale.
Measures of fractionation: a, 1000ln(a), e, and D
We are interested in measuring the isotopic offset between substances. Such offsets arise from the expression of an isotope effect due to equilibrium or kinetic fractionation during a physical process or chemical reaction. The size of this isotopic fractionation can be expressed in several ways.
The
fractionation factor
a is defined as:
where K is the equilibrium
constant for the associated reaction and n is the number of atoms exchanged.
For simplicity, isotope exchange equations are usually written such that n=1
so that
a =K. It is worth noting
that most values of a are close to
1, with variability in the third through fifth decimal place. We can see
that this is the case if we express a
in terms of the delta notation
Because a is close to unity, it is convenient to express fractionation in ways that accentuate the differences between dA and dB. This can be done in one of three ways which yield approximately the same value for the per mil fractionation.
Natural log notation, 1000ln(a):
One would
think that this notation would have its own symbol, but surprisingly, it
does not! We can approximate the fractionation in per mil from the
fractionation factor a by taking the natural log of a and multiplying it by
1000. This is written simply as 1000ln(a). Mathematically, this works
because we can think of a as being composed of 1 + e, a small deviation. Now
if we take the natural log of 1+ e, the result can be expanded as a type of
infinite series known as a Maclaurin series in which the first term is e, is
the largest and thus serves as a reasonable approximation of the natural log
of a. If we retain more terms, we would obtain a more accurate result, but
by convention, only the first term is retained:
Multiplying by 1000
yields the result in per mil. But since we are interested in
e anyway, there
is an easier way to express the per mil fractionation.
The
epsilon notation,
e:
The epsilon notation has the advantage over the 1000ln(a)
notation in that it is an exact expression of the per mil fractionation.
There is a final way of determining the per mil fractionation. This last
method is the least accurate, but most commonly applied because of its
simplicity.
The capital delta notation, D:
This method is least accurate because the errors in the two isotopic measurements do not cancel as is the case for random errors when calculating a ratio. In most cases, however, it is sufficiently accurate.