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
Figure 1. Perfect angular focussing in an r -1 magnetic analyzer main path written in polar cordinates has the form: r = r
0 e

where

=

2mU/(eB
1
2 r
1
2 ) – 1 where m is the ion mass, U its energy, e its charge, B
1
the magnetic field induction at r
1
from thr origin. The resolution of such a sector can be approximated with the formula:

= l
2
(e

-1)/(2
 s), where s is the source slit width.
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 = B
1 r
1
/r. The main path is given in a parametric form: r = r
I e z = - r i
K
 e K cos

K cos
 cos
 d

. Here K = (

2mU/e)/(B
1 r
1
) ,

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 180 o 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 -12 A.
It is worth to mention the outstanding properties of this
Figure 2. “Wedge” field 180 o deflecting analyzer geometry: radial angular focusing of second order (possibly of third 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
Figure 3. Symbols for two symetric halves of a homogeneous, oblique incidence, magnetic field analyzer second order radial angular aberration coefficient of a symmetric, oblique incidence, magnetic sector can be obtained in two ways: a) by multiplying the transfer matrices of the analyzer parts located between 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
= 2G xx
G 
2 Obviously, the difference

= A  s
- A  should vanish. Instead

= Lt(1+Lt/2)

- (1+Lt)

/2 with t = tan


= 2I
1 t 2 /c 2,

the incidence (emergence) angle, I
1
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 = r
0 e

with

=

2U/(eE
0 r
0
)-
1 with E
0
the r component of the electric field at r
0
.
The Wien filter, the oldest particle analyzer, invented before 1897
Figure 4. Spirally shaped electrode condenser
[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
4100 on the C
2
H
4
+ -N
2
+ doublet, at a current of 1.28

10 -16 A[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 =

=

2mU/(eB

1/

(1/r e
– 1/r e
+ 1/

) ,
0
) , Z the filter effective length , r e
and r m
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/(k 2
 s). The basic data of tht Wien filter mass spectrometer were: L = 41 cm, r m
=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:

= r
0
/(2 – c)(1/M es
– 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: r
0
= 20 cm,
40.2 cm

= 5600 on the C
2
H
4
-N
2
 e
= 31.8
0 , l
1
=
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:

= r
0
/(s+r
0

+ A 

2 + 2A 
 + …). r
0 is the main path radius inside the magnetic field, A ij
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/r
0
+

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/r
0
= (t-M)/(1-t 2 can use the formula:

= r
0
) and to estimate the resolution we
(1 – M)/[2s(1-t)] The parameters of the constructed

DAMASCHIN IOANOVICIU instrument were:

= 31.73
0 ,

=90 0 , electric deflection in the prism 54.3
0 , M = -
0.287, mass dispersion D  = 42.91 cm for r
0
= 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].
Figure 10. Peak shape of axially ejected ions from a qudrupole trap
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 effort was spent to improve this branch of 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
=
I
0
 l/2[ 2

-(

+v)ln(

+v)-(

-v)ln(

-v)], the symbols used being

=

/(

D x l), v
=

/(

D x l)

=
 m
3
T/(m
2
U) for the disintegration m
1
+

m
2
+ + m
3
. ia is the ion current detected for the metastable peak, I o
is the parent ion current, l the length of the path part from that the metastable ions are collected, width, T is the energy released during the process,


half the detector slit
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 n r
, 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 + =
 o
I + n r

(m/
 m) 1.69
C For the Mattauch-Herzog double focusing mass spectrometers, the coefficient C has the explicit form: C = 7.16(1.11 +
 f/r
0e
+
0.6r
0e
/r om
) 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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