Instead
of trying to precisely measure the water isotopes in
house standards against the VSMOWSLAP
scale, you can use a set of 3 appropriate house
standards to mix your own VSMOW or SLAP standards and
make sure, you get precisely the same readings as you
get in the current international reference standards
(VSMOW2, SLAP2), or mixtures of the two, no matter how
good your calibration is.
Since
the conversion from
concentrations to the delta notation is linear,
this is as easy as mixing waters of different
concentrations. The only complication is that there is
no water with zero concentrations of ^{18}O or
^{2}H. Even the strongly depleted SLAP standard
with δ^{18}O
=  55.5 ‰ is equivalent to an ^{18}O
concentration of 1893.91 ppm and its δ
^{2}H =  427.5 ‰ is equivalent to a ^{2}H
concentration of 89.173 ppm.
Given
you have standards A, B and C with the water
isotopes δ^{18}O_{A},
δ^{2}H_{A},
δ^{18}O_{B},
δ^{2}H_{B}, δ^{18}O_{C},
δ^{2}H_{C},
you may mix it in the fractions a, b and c to get a
desired composition δ^{18}O_{aim},
δ^{2}H_{aim}.
using these equations:
.......
conditions :
(1)
a x δ^{18}O_{A}
+ b
x δ^{18}O_{B}
+ c
x δ^{18}O_{C}
= δ^{18}O_{aim}
(2)
a x δ^{
2}H_{A} + b
x δ^{
2}H_{B} + c
x δ^{
2}H_{C} = δ
^{2}H_{aim}
(3)
a + b + c = 1
.......
auxiliary equations :
(4
) j = δ^{
2}H_{A}  δ^{
2}H_{B}
_{ }(5)
m = (
δ^{18}O_{aim}
 δ^{18}O_{A}
) / (
δ^{18}O_{B}
 δ^{18}O_{A}
)
(6)
n
= ( δ^{18}O_{C}
 δ^{18}O_{A}
) / (δ
^{18}O_{B}  δ^{18}O_{A}
)
.........
result :
(6)
c = ( δ^{
2}H_{aim}  δ^{
2}H_{A} + n x j
) / (δ^{
2}H_{C}  δ^{
2}H_{A} + n x j )
(7)
b = m  n x c
(8)
a = 1  b  c
If
you use the right selection of house standards, there is
a possible solution, that gives you positive fractions
for all three house standards. Obviously one has to be
higher and one lower than the standard you are aiming to
mix (otherwise you get negative fractions, which are
impossible). And the third one is used to adjust the deuterium
excess (d = δ^{2}H
 8 x δ^{18}O)
 meaning you have to choose an appropriate one, so it
is actually elevating or decreasing your deuterium
excess.
The
attractive thing about this scheme is, that it is not
necessary to have a precisely calibrated machine, you
just need to know the exact reading for your house
standards on your machine with your current calibration
(or just uncalibrated machine readings), and the exact
reading of the standard you are aiming to "copy".
Example:
You
have
solution A: δ^{18}O_{A},=
+ 2.1 ‰, δ^{2}H_{A
}= + 13 ‰, ( d_{A}
=  3.8 ‰ )
solution B: δ^{18}O_{B}
=  6.7 ‰, δ^{2}H_{B}
=  50 ‰,
( d_{B} = + 3.6 ‰ )
solution C: δ^{18}O_{C}
=  13.0 ‰ δ^{2}H_{C}
=  71 ‰, ( d_{C}
= + 33 ‰ )
you
want to create VSMOW which shows in your machine with
the settings you used for the other three solutions:
δ^{18}O_{aim}
= + 0.2 ‰ δ^{2}H_{aim}
= +1.4 ‰
you
need a mix of 84.36 % solution 1, 7.34 %
solution 2 and 8.31 % solution 3 !
δ^{18}O_{aim}
= 0.8436 x 2.1‰ + 0.0734 x
(6.7 ‰ ) + 0.0831 x (13 ‰ ) = + 0.2 ‰
δ^{2}H_{aim}
= 0.8436 x 13 ‰ +
0.0734 x (50 ‰ ) + 0.0831 x (71 ‰
) = + 1.4 ‰
The
other advantage is, that you are not relying on house
standards, that might carry an error depending on the
quality and linearity of the machine calibration. You
simply try to get as close to the reference material as
possible, regardless of its calibration. We had problems
trying to use secondary standards that other labs had
measured for us. We only found out our machine is
actually very linear, when we prepared mixtures of the
original standards VSMOW2 and SLAP2, which then have
precisely known expected concentrations (at least to the
precision you can trust your pipetting skills).
original time stamp 15.Feb 2018, changes 20Jan19
