1. Introduction
Electrons and ions placed in the electromagnetic trap are a unique object which gives a possibility to perform a wide range of experiments in the area of fundamental and applied physics [1]. The two-dimensional field of the quadrupole capacitor allows us to realize the radio-frequency sensitive mass-spectrometer [2, 3].
In this study we investigate the motion of particles in a quadrupole capacitor and magnetic field. Application of the homogeneous magnetic field leads to a new regime of work of such a device. It is shown that the two-dimensional trap with magnetic field can be used for separation of particles beam according to the specific charge and mass of the ions. Such device can be particularly attractive for fabrication of a high-resolution mass-spectrometer.
The application of the magnetic field leads to the splitting of ion states with different values of specific charge in the space of system parameters.
2. Electromagnetic field of the system
Let us consider the electromagnetic system -- the quadrupole capacitor placed in the static magnetic field (Fig. 1). The electrostatic field of the quadrupole in vicinity of the z-axis (with the potential ) determines equipotential hyperbolic surfaces expressed by the following formulae: y2=x2+2R2 and y2=x2-2R2; V0 -- electrostatic potential applied to the electrodes, R -- space separation of the electrodes.
Fig.1. The experimental setup. The quadrupole capacitor with the length L is situated along the z-axis. Its beginning and end correspond to z=0 and z=L, respectively. The source of accelerated ions and the detector are localized on the z-axis at certain z<0 (source) and z>L (detector). The applied field is directed along the y-axis.
The volume restricted by electrodes between the planes z=0 and z=L, is a waveguide or an open resonator. In such system the basic transversal electromagnetic wave can propagate with a critical frequency equal to zero. This wave excited by the source can propagate for any frequency . A unique feature of such an open system is that the structure of a basic wave field is analogous to the structure of electric and magnetic fields in the static case.
The solution of Maxwell's equations, satisfying the boundary conditions on
the surface of the electrodes are as follows:
Let us limit the radio-frequency range to . In such case
and
and the electromagnetic field of
such system has the following form:
3. Solution of equations of ion's motion
Let us consider the motion of nonrelativistic ions with a mass m and
charge e in a quadrupole trap with fields expressed by the formula (1).
Taking into account the electric and magnetic fields (1) we obtain the
following equations of motion:
The first integral is defined by Eq. (4). Substituting
4. Conditions for confined motion of ions in xy-plane
Let us consider the system shown in Fig. 1. We would like to formulate conditions for the passing of the ions through the quadrupole without collision with the electrodes and conditions for splitting the beam in dependence on mass and charge of the ions.
It is well known that the Mathieu's equation has two classes of solutions -- stable and unstable ones. Regions of stability of solutions of Eq. (8) in ()-plane (available zones) are restricted by curves (Fig. 2).
Fig.2. The Mathieu's equation has two classes of solutions -- stable and unstable ones. The regions corresponding to the stable solutions in the plane () are limited by the lines .
We are interested in values of the parameters for which the solutions of both Eqs. (6) and (7) are stable.
Let us consider a beam, containing ions with a charge ei and mass mi (i=1,2) entering into the system. From the formula (5) it is seen that n=a/b=2V0/V and this ratio does not depend on the ion mass and the frequency . When the frequency changes, as long as V0 and V change in the same way, n remains constant and then the parameters and q=-b in Eq. (6) fall into the line (where q>0 if ei<0 and q<0 if ei>0). Let us assume My[-a,-b] is the stable solution of Eq. (6) for q=-b in the range of [-yR,-yL].
It is seen that the values of yR and yL do not depend on the ratio
ei/mi and that they are determined by the value of n only and
configuration of available zones. The parameters and q=b in
Eq. (7) lies on the lines
The magnetic field removes space degeneracy with respect to the specific charge. In the domain of stability the solution of Eq. (7) expressed by Mx(i)(a+g,b) for b taken from [xRi,xLi] interval, corresponds to different values of frequency at the fixed n. The motion in the (x,y)-plane is confined to the values b from the intersection of [qLi,qRi], [yL,yR] and [xLi,xRi].
An analysis of the ion movement can be simplified by introducing a central-symmetrical reflection in ()-plane. In this way we change a stable area of Eq. (8) from the third quarter to the first with a modified Mathieu diagram form.
Let us consider the dependence of ion separation on the specific charge. The ions with different ei/mi values should fulfill the stability conditions described by Eqs. (6) and (7), respectively. Every point on ()-plane represents a single state of a particle. By increasing the frequency the points can be moved along the straight lines and towards the origin. The slope of can be controlled by changing induction of the magnetic field.
In this study we will consider the separation of single charged ions with masses m1 and m2: m2>m1. We assume that V>0, V0>0. For proper work of the device within the stables areas the following parameters should be chosen , |q|<4.
The region of stability of the solutions of Eq. (7) is denoted in Fig. 2 with the symbol ``I''. In the next step we will consider in detail the case for one n (n0<n<n1) only. The limits of the interval [yL,yR] are determined by the intersection of line with the and curves. The limits of [xLi,xRi] intervals are determined by the intersection of lines (9) and and curves. The stable area of Eqs. (6) and (7) is defined by
For
, which is the root of equation
b(m2)=yR, ions with a mass m2 fall within the stable area. The
parameter b(m2) takes the value from the [yR,yL2] interval if the
frequency changes as follows:
For
we have b(m1)>yR. For such case only
ions with a mass m2 fall within the collector. The parameter b(m1)
takes the values [yR,xLi] when the frequency is changing within the
interval
Assuming that m1=m,
the ratio
is obtained, where Q is the mass resolution of the system.
For we obtain
Based on the analysis of the data presented in Fig. 1 it is possible to show that the changes of B give the possibilities to extract optimal areas when . Such an optimal range is shown in the figure by the rectangle ABCD.
One should add that for large values of n the regime of separation of ions is realized for , |q|>4 (it is not shown in Fig. 2).
Paul has considered the regions , |q|<1 for the ion separation depending on the mass. The application of the external static magnetic field improves the mass-spectrometer sensitivity.