New Statistical Algorithms for the Analysis of Mass - FU Berlin, FB MI ...

Figure 3.2.3: This
shows the definition of
peak width: the width
of a peak is defined to
be the width the peak
has at its half maximum
(Full Width At Half Maximum,
FWHM).
28 CHAPTER 3. MATHEMATICAL MODELING AND ALGORITHMS
this yields:
T OF = t0 + ta + tD + td
Modern TOF MS machines usually ensure that vD ≫ v0 and t0 and td are as
small as possible. Nevertheless, these small contributions to the overall TOF
have an impact on the spectra as we will see later. However, following from
equations 3.2.1 and 3.2.2 we can state that:
�
m
T OF ∝
(3.2.3)
z
This exhibits the actual relation between mass and time-of-flight:
m
z = a · T OF 2 + b (3.2.4)
a being a proportionality constant that can be shown to be
a = 2 · sa · e · UD
(2 · sa + D) 2
and b another constant modeling the influence of t0 and td (and potentially
others). Of course, equations 3.2.3 and 3.2.4 only hold if all accelerations are
constant during measurement.
The beauty of these constants is that they can be used to calibrate a mass
spectrometer by determining their (machine dependent) values from the times
of flight of some known m/z values.
Resolution Issues
The resolution of a mass spectrometer states the ability to separate ions of
similar mass-to-charge ratio. It is defined as m
z
△ m where
z
m
z is the value of
interest and △ m
m
z the width of a peak at this z value at half maximum height
(full width at half maximum height, FWHM, see Figure 3.2.3). Intuitively,
it is clear that the narrower a peak for ions of a particular m
z ratio, the better
the resolution of the machine. To show this let us start with Equation 3.2.4:
m
z
= a · t2
Differentiation with respect to time yields:
d m
z
dt
For a finite interval this becomes:
and hence
△ m
z
△ m
z
= 2 · a · t
= 2 · a · t · △t
· t = 2 · m
z
m
z
△ m
z
= t
2 · △t
· △t
The final equation clearly shows that resolution mostly dependends on the
difference in the measured flight times for ions of similar mass or m/z values.