whereL,v, andVare the ion drift length, the ion velocity, and the accelerating
potential, respectively.
Despite the simplicity of the equation, an ion’s initial spatial and angular
kinetic energy spread complicate the precise determination ofm/z. These issues
are addressed to improvem/zaccuracies in high performance time-of-flight
mass spectrometers (Guilhaus et al., 2003).
11.2.3.2 Sector Magnets In magnetic sector instruments, ions accelerated
from an ionization source are deflected by a magnetic field and adopt a
constant trajectory of radius (r) around the center of the field. For a fixed
magnetic field strength (B) and a fixed accelerating potential (V), only ions with
a certain momentum-to-charge ratio will pass through slits placed after the
magnetic field (Glish and Vachet, 2003). The equation that characterizes the
radius of curvature as a function of them/zratio is given by the following:
m=z¼r^2 B^2
2 V
A mass-to-charge spectrum can be obtained by scanningB(at constantV)orV
(at constantB), so that ions of differentm/zcan travel through the slits with a
typical resolution of a few hundred. The low resolution is primarily due to the ions’
velocity spread. To improve the resolving power, an electrostatic analyzer (ESA) is
placed in the ion optic pathway before or after the magnet sector. The
combination of an ESA and a magnetic sector provides both directional and
energy focusing (double focusing) thus improving them/zaccuracies significantly.
The resolution of a double-focusing instrument is typically in the range of 10^4 –10^5.
11.2.3.3 Quadrupole Mass Filters Quadrupole mass analyzers consist of
four electrodes, ideally of hyperbolic rods, that are accurately positioned in a
radial array. For practical as well as economic reasons, most quadrupole mass
filters have employed electrodes of circular cross section. A potential is applied
to one pair of diagonally opposite rods consisting of a DC voltage and an rf
voltage. To the other pair of rods, a DC voltage of opposite polarity and an rf
voltage with a 180phase shift are applied. The ion motion under the influence
of this two-dimensional (2D) field can be described mathematically by the
solutions to the second-order linear differential equation, known as Mathieu
equation, from which the Mathieu parameters,auandqu, can be derived as
au¼ax¼ay¼
4 zU
mr^20 v^2qu¼qx¼qy¼2 zV
mr^20 v^2wherem/zis the mass-to-charge ratio of the ion,Uis the DC voltage,Vis the rf
amplitude, vis the rf frequency, andr 0 is half the distance between two
diagonally opposite rods.
326 APPLICATION OF LIQUID CHROMATOGRAPHY/MASS SPECTROMETRY