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INTRODUCTION

Fourier Transform (FT) mass spectrometers

have ultimate resolutions exceeding those of

conventional Time-of-flight (TOF) mass

spectrometers. However, in order to achieve

this, long acquisition times are required (of up

to several seconds in some cases). In contrast,

high-performance TOF mass spectrometers have

typical flight times of tens of microseconds. This

allows them, for example, to profile mobility

separations lasting only a few milliseconds.

In this poster we present a theoretical

treatment of a novel orthogonal acceleration

(oa)-TOF geometry, utilising the quadro-

logarithmic potential familiar from Orbitrap

mass spectrometry, which could combine the

high resolution of FT instruments with the speed

of TOF analysis.

AN INVESTIGATION OF THE SPACE OF TRAJECTORIES IN A NOVEL OA-TOF GEOMETRY

Keith Richardson and John Hoyes Waters Corporation, Wilmslow, UK

METHODS The cylindrically symmetric quadro-logarithmic potential takes

the form:

This potential produces independent motion in the radial

(trapping) and axial (mass analysis) directions. The axial

component is quadratic, leading to simple harmonic motion

with a frequency that is m/z dependent but energy

independent. In the Orbitrap1 mass spectrometer, the chosen

shape of the electrodes fixes the ratio between the logarithmic

(b0) and quadratic (k) parts of the potential. We propose

instead to use a segmented construction allowing freedom of

choice of this ratio. The method of construction (Figures 1 and

2) consists of two concentric sets of N rings to which a set of

voltages Vo(n) and Vi(n) are applied. A typical device geometry

would be a length of around 1m with an outer radius of 10cm

and an inner radius of 2cm.

References

1. A. Makarov, Anal. Chem 72 (6) pp 1156-1162 (2000).

2. J.M. Brown, A.J. Gilbert, J.B. Hoyes, D.J. Langridge, and J.L. Wildgoose, WO 2011/154731 A1.

3. See, for example, H. Goldstein, Classical Mechanics, Addison Wesley, (1980).

The parameter space (R0,b0) of trajectories for a device with

k=8x104Vm-2 and ΔU=1000V is shown in Figure 4 along with

examples of effective potentials corresponding to the points

labeled a)-f). To obtain a stable bound orbit, the starting

(injection radius) must lie in the range RE < R0 < RS+ where RE

is the only non-trivial solution of U(RE)=U(RS+ ). This

corresponds to the blue shaded region on the large plot.

Above the straight red line the effective potential has a

minimum, while trajectories lying along the brown quadratic

curve are circular: R=RS- (stable) or R=RS+ (unstable). Above

this curve, the trajectory starts at a radial maximum R0=Rmax.

In Figure 5, further constraints on this space are explored.

1) Rmin<RD<Rmax so that ions can repeatedly miss then

ultimately hit the detector.

2) Rmin>Rin so that ions do not hit the inner electrodes.

CONCLUSION Following axial beam expansion, the quadro-

logarithmic potential can be utilized for multi

pass oa-TOF mass analysis.

The parameter space has been classified for a

chosen geometry, and a solution with an effective

TOF path length of over 10m has been identified.

Many technical challenges remain to be

addressed before such a device could be

constructed.

DISCUSSION In this poster we have briefly examined the theoretical

properties of a novel multi pass orthogonal time of flight mass

analyser utilizing a quadro-logarithmic potential. Many

practical problems (both electrical and mechanical) would have

to be overcome before such a device could be constructed:

each of the ring electrodes in the assembly must be carefully

machined and positioned, to achieve orthogonal acceleration

voltages must be pulsed accurately an rapidly from their initial

values to their final values, the detector must be placed within

the analyser field without destroying isochronicity and the

number of passes that can be utilized is limited by the initial

radial spread of the ions. Nevertheless with appropriate beam

conditioning, it is anticipated that mass resolutions of several

hundred thousand should be achievable in a relatively compact

geometry.

Figure 1. Construction of the quadro-logarithmic oa-TOF with equipotential lines for an Orbitrap superimposed for purposes of illustration only. The segmentation of the inner electrodes is not shown.

Figure 2. Ions are introduced through an aperture in the outer electrodes and a switched electric sector helps to accomplish stable circular trajecto-ries. The annular detector is shown in green.

Figure 5. Constraints: k=8x104Vm-2, ΔU=1000V, RD=3.9cm, Rin=2cm. The white region corresponds to potentially useful trajectories with

Rmin<RD<Rmax. This roughly corresponds to 2.6<ρ<5.2.

The sequence of events for a single injection is as follows:

1. Axial (z) beam expansion2 prior to injection to reduce turn-around time in low extraction field ~20 kV/m.

2. Ion injection into stable circular orbits with r=R0 and k=0

(Figure 2).

3. Orthogonal and radial acceleration (k > 0 and increased b0). 4. Unique path, multipass time of flight in eccentric orbits be-

tween radii r=Rmax and r=Rmin with simple harmonic axial

motion (Figure 3). Ions repeatedly miss an annular detec-

tor situated at the isochronous plane (z=0). 5. Ions strike the detector on the final pass.

In order to produce this behaviour, it is important to control

the interplay between axial and orbital motion. The equations of motion are derived below starting from the Lagrangian3

written in cylindrical polar coordinates:

The three Euler-Lagrange equations are

where qi stands in turn for the coordinates r,ϕ and z:

The second equation simply expresses conservation of the z-

component of angular momentum Lz, and the first equation

may therefore be rewritten in the form of motion in an effec-tive potential containing a “centrifugal” term:

For ions injected into circular orbits radius R0 through a poten-

tial drop ΔU, Lz2 = 2mq R0

2 ΔU and we find

(1)

(2)

This potential has extrema at RS– and RS+ satisfying

RESULTS The initial stable circular orbits are obtained by setting k=0

and b0=2ΔU. Following orthogonal acceleration, Eq. (1) pro-

duces simple harmonic axial motion with period Tz=2π / ωz

where ωz= √(qk/m).

From Eq. (2) the period of the radial motion is Tr=2√(m/q)τ where

Rmax

Rmin

3) Ions will hit the detector when r=Rmin and z=0 simultane-

ously. This occurs for pairs of positive integers (i,j) satisfying

We therefore look for points near which condition 3) is satisfied

for j=N but away from where it is satisfied for 1≤j<N.

Conditions 1) and 2) correspond roughly to 2.6<ρ<5.2, but

most useful solutions lie towards the lower end of this range.

The example in Figure 3 corresponds to the solution (i,j)=

(19,7) or ρ=2.85 for which ions hit the detector after 18½ ra-

dial periods and 6 detector passes. The time of flight for an

ion of m/z 500 is just under 165μs and the effective TOF path

length is just over 10m. Figure 3. Trajectory with k=8x104Vm-2, ΔU=1000V , R0=0.065m, b0=3700V. The dot indicates the start of the trajectory. The detector extends from r=0.02m to RD=0.0385m.

r =R0=Rmax

r =Rmin

RD

RD

a)

Ions escape

b)

f) e) d)

c)

Ions escape

Bound orbit

Stable circular orbit Bound orbit Ions escape

Figure 4. Trajectory classification and examples of effective potentials for k=8x104Vm-2, ΔU=1000V: a) R0=0.1m, b0=1300V, b) R0=0.1m, b0=1900V c) R0=0.1m, b0=1990V, d) R0=0.1m, b0=2400V, e) R0=0.1m, b0=3590V, f) R0=0.1m, b0=3590V. The shaded region corresponds to bound trajectories. Orbits satisfying the requirements of the analyser discussed here are of types c) and e).