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STUDIA UNIVERSITATIS BABES-BOLYAI, PHYSICA, SPECIAL ISSUE, 2003
POTPOURRI OF ION OPTICS
Damaschin Ioanoviciu
National Institute for Research - Development of Isotopic
and Molecular Technologies Cluj-Napoca ROMANIA
This overview deals with some special cases of angular, double and time
ion focusing analyzers. These include inhomogeneous magnetic field
sectors perfectly focusing wide ion beams, Wien filters with
inhomogeneous fields operated independently or in double focusing
configurations, analyzers with high mass dispersion using electric prisms
and oblique incidence magnets. Flight time analyzers of various kinds are
also reviewed as classical reflectrons with homogeneous fields but
accounting for oblique incidence, cylindrical reflectrons, perfect velocity
focusing in time by quadrupole trap configuration source combined with a
field free space. Quadrupole filters and traps are just mentioned for some
calculation procedures. The use of ion optics to describe metastable peak
shapes and peak tails in mass spectra closes the review.
Introduction
There are some ion optical solutions which were not applied to large scale or
which, from various reasons, are still in a development stage, or are waiting for
some technological refinement to become fully operational. Most of the ion optical
solutions from this potpourri belong to these categories.
Magnetic sectors focusing wide ion beams
The inhomogeneous magnetic fields decreasing with the radius with an index n=1
can focus perfectly wide ion beams if the sector limits are cut by circular contours.
This happens when the ion main path is circular [1] and also when the main path is
a logarithmic spiral [2],[3].
The logarithmic spiral main path
introduces an additional parameter
helping to obtain better mass
dispersion and then resolution. The
main path written in polar cordinates
has the form: r = r0e where  =
2mU/(eB12r12) – 1 where m is the ion
mass, U its energy, e its charge, B1 the
magnetic field induction at r1 from thr
origin. The resolution of such a sector
can be approximated with the formula:
 = l2(e-1)/(2s), where s is the
Figure 1. Perfect angular focussing in
source slit width.
an r-1 magnetic analyzer
The “wedge” magnetic field, created
by two plane pole faces, by its
DAMASCHIN IOANOVICIU
additional parameter K offers the possibility to better focus with respect to the
homogeneous magnetic field. The field intensity changes with the distance r, to the
axis z, the virtual intersection of the two pole face planes: B = B1r1/r. The main
path is given in a parametric form: r = rIeK cos 
z = - riKeK cos cosd . Here K = (2mU/e)/(B1r1) ,  the angle between the ion
velocity and the z axis.
A complete matrix description of
this kind of magnetic sectors,
including the fringing effects was
detailed in ref. 4. The outstanding
focusing properties of the 180o
deflecting sectors, known from
ref. 5 were incorporated in the
instrument of ref. 6. It attained a
resolution of 1340 on m=2u peak,
and an ion current of 9x10-12A.
It is worth to mention the
outstanding properties of this
geometry: radial angular focusing
Figure 2. “Wedge” field 180o
of second order (possibly of third
deflecting analyzer
order after some sources),
stigmatic focusing with both radial and axial unit magnification. This layout is
suitable for low mass isotopic analyzers, especially for deuterium as the sensitivity
is extremely high. Such a magnetic sector was proposed to be used in a proposal
for time resolved ion momentum spectrometry [7], having obvious advantages
compared with homogeneous magnetic sectors initially used with this purpose [8].
The limited use of inhomogeneous magnetic fields in mass spectrometry originates
from the saturation effects impeding their use at high masses in instruments at
convenient costs.
Homogeneous magnetic sectors
seem to have no more focusing
misteries at the present time.
However some doubt persists
concerning
fringing
field
accounting calculations by using
fringing field integrals [9]. The
second order radial angular
aberration coefficient of a
symmetric, oblique incidence,
magnetic sector can be obtained
in two ways: a) by multiplying
Figure 3. Symbols for two symetric halves of a
the transfer matrices of the
homogeneous, oblique incidence, magnetic field
analyzer parts located between
analyzer
the collector and the analyzer’s
POTPOURRI OF ION OPTICS
middle, and expressing this by the matrix elements of this second half As or b)
by multiplying all the transfer matrices from the collector to the ion source A  .
As = 2GxxG2 Obviously, the difference  = As - A  should vanish. Instead 
= Lt(1+Lt/2) - (1+Lt)/2 with t = tan
 = 2I1t2/c2,  the incidence (emergence) angle, I1 the fringing field integral. .
Electrostatic and crossed field mass analyzers
Not only magnetic field sectors can handle curved axis beams. The spiral cylinder
plate condensers can focus beams curved on axes of same shape [10].
The second order focusing theory being developed in ref. 11. The main path inside
this kind of deflector is xpressed
by: r = r0e with  =  2U/(eE0r0)1 with E0 the r component of the
electric field at r0.
The Wien filter, the oldest particle
analyzer, invented before 1897
[12], was resurected by studies
from ref 13 and the theory
developed, including fringing field
effects in ref. 14. Experimentally a
Wien filter attained a resolution of
Figure 4. Spirally shaped electrode condenser
4100 on the C2H4+-N2+ doublet, at
a current of 1.2810-16A[15]
The length of the field free spaces, in front of and after the filter have the length L
(symmetric case): L = 1/[Lktan(kZ/2) with k = 1/(1/re – 1/re + 1/ ) ,
=2mU/(eB0) , Z the filter effective length , re and rm the axial radius of curvature
of the electric potential surface and of the magnetic force line at the ion main path.
Figure 5. Inhomogeneous field Wien filter
DAMASCHIN IOANOVICIU
The resolution can be estimated with the simple formula: =1/(k2s). The basic
data of tht Wien filter mass spectrometer were: L = 41 cm, rm =10 cm and Z = 30
cm. The greatest , of about 4 m length, was working at Michigan State University .
The study of Wien filter was extended to electric and magnetic crossed field
sectors, including fringing field effects in ref. 16. By this last we steped in the area
of double focusing mass spectrometry. The Wien filter can be associated with an
electrostatic condenser to ensure double, angular radial and energy focusing
simultaneously as theory shows [17] and as it was incorporated in real world
instruments [18], [19].
Figure 6. Wien filter – electrostatic condenser double focussing geometry
An estimative formula for the resolution is:  = r0/(2 – c)(1/Mes – 1)/s. The used
symbols are: c = re/r0, Mes the magnification of the electric sector. The basic
parameters of the spectrometer described in ref. 18 are: r0 = 20 cm, e = 31.80, l1 =
40.2 cm  = 5600 on the C2H4-N2 doublet. Double focusing can be obtained also
by using part of the gap of a magnetic sector to create a Wien filter there.
Figure 7. Compact Wien filter – homogeneous magnetic
field double focussing mass spectrometer
For homogeneous fields the theory was developed in ref. 20. The resolution may
be calculated with the following formula:  = r0/(s+r0 + A2 + 2A+ …). r0
is the main path radius inside the magnetic field, Aij are aberration coefficients. An
instrument which keeps half Dempster’s spectrometer geometry for magnetic
POTPOURRI OF ION OPTICS
deflection has the theoretical resolution:  = 1/(s/r0 + 2 + 4 + 32/4). Such an
instrument was operated at resolution 170, for 56 mm main path radius, ion current
of 1.6x10-8 A, in perfect agreement with the instrument and beam parameters [21].
Partial pressure gauges were proposed based on the theory developed for Wien
filters created and combined with wedge field sectors to create high sensitivity
stigmatic, double focusing (triple focusing) geometries [22]. Compact
oversimplified geometries including Wien filters and magnetic sectors in the same
gap were analyzed from the aberration point of view [23].
Figure 8. Oversimplified Wien filter – homogeneous
magnet double focussing geometry
An attempt to create double focusing mass spectrometers of high resolution by
increasing mass dispersion was made by using Kelman’s electric prisms [24]. The
theory developed in ref. 25 was incorporated in a mass spectrometer attaining a
resolution of 11000 with a remarcably high sensitivity [26].
Figure 9. High mass dispersion double focussing mass spectrometer
including an electric prism
The field free spaces, one of them including also the electric prism electrodes can
be calculated with the formula: L/r0 = (t-M)/(1-t2) and to estimate the resolution we
can use the formula:  = r0(1 – M)/[2s(1-t)] The parameters of the constructed
DAMASCHIN IOANOVICIU
instrument were: = 31.730, =900, electric deflection in the prism 54.30, M = 0.287, mass dispersion D = 42.91 cm for r0 = 25 cm.
A word on quadrupoles
Dynamic mass analyzers as quadrupoles and quadrupole traps represented an
occasion for new theoretical approaches: a direct calculation method for ion
trajectories using only the initial
conditions for the filter [27] and a peak
shape/resolution calculation for the trap
[28].
More about time-of-flight mass
analyzers
Time-of-flight mass spectrometry is not
only a fashion but , among many others,
it allowed Tanaka to accede to the
Chemistry Nobel Prize of 2002. Much
Figure 10. Peak shape of axially
effort was spent to improve this branch of
ejected ions from a qudrupole trap
mass spectrometric instrumentation and
the ion optical solutions with this purpose
were reviewed successively in 1994 [29], 1995 [30], 1998 [31] and 2001 [32].The
theory of already routinely used homogeneous electric field mirror mass
spectrometers with one or two stages was refined to account even for third order
contributions and the detector position effect in resolution formulas [33] Matrix
formalism was adopted to describe the single and double stage mirror time-offlight mass spectrometer performance, having oblique packet incidence [34]., [35]
This method was already used to describe flight time through various other ion
optical elements as electrostatic deflectors, magnetic sectors, electric and magnetic
quadrupoles in a second order approximation [36]. Such combined transversallongitudinal (flight time) transfer matrices were reviewed also in ref. 37.
The resolution and mass scale modification in post source focusing applied to
linear time-of-flight mass spectrometers was analyzed in ref. 38.
Initial velocity ion focusing in time is mandatory for ions resulted from matrix
assisted laser desorption/ionization. The delayed ion extraction from the source
used at the present time could be replaced by a perfect velocity focusing procedure.
The ion source shaped as a Paul trap is fed by high voltage pulses correlated with
the ionizing laser pulses [39], [40]. This configuration ensures perfect velocity
focusing for ions created on the end cap tip, after a flight over a field free space of
appropriate length. Resolutions estimated to about 50,000 could be obtained by this
procedure for high mass ions [41].
Metastable peak shapes
Ion optics is also a way to describe the shape of the metastable peaks resulted with
internal energy release. Metastable peaks of gaussian, flat topped and dish topped
POTPOURRI OF ION OPTICS
shape result from the energy released and the instrumental parameters in single and
double focusing static mass spectrometers [41], [42].
Figure 11. Gaussian metastable peak in a
double focussing mass spectrometer
Figure 12. Flat topped peak in a single
focussing mass spectrometer
Figure 13. Dish topped metastable peak
Figure 14. Gaussian metastable
peak in a TOF mass spectrometer.
Figure 15. TOF mass spectrometer detected
metastable peak with a small flat top
DAMASCHIN IOANOVICIU
A sample of the formulas describing the metastable peak shape is given next: i a =
I0l/2[ 2-(+v)ln(+v)-(-v)ln(-v)], the symbols used being  = /(Dxl), v
=/(Dxl) =m3T/(m2U) for the disintegration m1+  m2+ + m3 . ia is the ion
current detected for the metastable peak, Io is the parent ion current, l the length of
the path part from that the metastable ions are collected,  half the detector slit
width, T is the energy released during the process,  the coordinate of the beam
axis on the collector. In time-of-flight mass spectrometers with single stage
reflectrons the metastable peak shapes can be related also to the irespective
geometic parameters and to the released energy [43]. Two specific metastable peak
shapes for reflectrons are given in Fig. 14 and Fig. 15.
Peak tails in mass spectra
Of major importance for isotopic analysis are the peak tails produced by elastic
scattering of ions on residual gas molecules. An ion optical procedure first used by
Menat [44] to calculate such tails in electromagnetic separators was generalised for
static sectors and other ion optical elements operated independently or in tandem
[45], [46]. The general form of the tail current i+ depends of the scattering cross
section o, on the current which generates the tail I+, on the residual gas molecules
per unit volume nr, on the slit width , and on the position of the slit with respect to
the main path of the scattered beam, measured in relative mass difference m/m:
i+ = oI+nr(m/m)1.69C For the Mattauch-Herzog double focusing mass
spectrometers, the coefficient C has the explicit form: C = 7.16(1.11 + f/r0e +
0.6r0e/rom) Besides the main path radii of the electric and magnetic deflection, it
depends on the distance between the deflectors f. For the Nier-Johnson geometry,
another very popular double focusing mass spectrometer configuration C= 19.6.
Figure 16. Peak tail shapes in linear and reflectron time-of-flight
mass spectrometers
POTPOURRI OF ION OPTICS
Obviously this kind of interactions blures the ion packet contour in time-of-flight
mass spectrometers. These effects were estimated in ref. 47. The shapes of tails in
linear and reflectron time-of-flight mass spectrometers are given in Fig. 16.
Conclusion
Ion optics has an undeniable role to play in the future development of mass
spectrometry. As in the field of engineering, from a lot of patents only some arrive
to be applied and even fewer become of general notoriety. So happens to ion
optical solutions too.
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