ANALYTICAL ION-OPTIC

AND NUMERICAL

METHODS

SYSTEMS OF A QUADRUPOLE

FOR MODELING MASS

SPECTROMETER

V. V. Titov

UDC 621.384.8

Extensive use of quadrupole mass spectrometers in nuclear power and technology is significantly impeded by the stringent requirements imposed on the stability of power supplies and the precision in the manufacturing of the mass analyzers. In this paper analytical and numerical computational methods are developed.

F O R M U L A T I O N OF T H E P R O B L E M AND ASSUMPTIONS The principal aim of this paper is to model mathematically ion separation by mass number in a quadrupole mass analyzer with a distorted field in order to determine the dependence of the transmission on the distortion parameters. In constructing a mathematical model the analytical method of [1] for solving the problem is elaborated and the following assumptions are adopted. In order to study the characteristic features of ion separation in a quadrupote mass analyzer with a distorted field the electric field potential can be given in the form [2] O(x, y, z, t) = (x 2 - yZ)(U - Vcos cot) [1 +

(Er/2)G(z)]2/~,

(1)

where x, y, and z are mutually orthogonal coordinates; U and V are the constant and variable components of the voltage supplied to the quadrupole mass analyzer; co = 2~rf is the angular frequency of the variable component; 2r o is the shortest distance between the field-setting electrodes; the function G(z) characterizes the type of the distortions; and, E r = 2Aro/r o is a small parameter, characterizing the relative geometric distortions. It follows from Eq. (1) that second-order approximation of the field and three types of distortions are studied [2]: distortions caused by the nonparallelism of the field-setting electrodes of length L, and then for them G(z) = +z/L; distortions caused by the nonrectilinearity of the field-setting electrodes, and then G(z) = 2(z/L)(z/L - - 1); and, edge distortions of the quadrupole mass analyzer, and then G(z) = (zo - - z)/r o, where zo is the coordinate of output diaphragm of the ion source with aperture 2Ito. In the latter case the small parameter is (zo - - z)/2r o < < I, E r = - 1. The maximum distance over which the fringing field extends is chosen to be 2r0, i.e., the field potential is zero at z o = 2r 0. The validity of this approximation was confirmed by point-by-point calculations of the ion trajectories in the complete spatial distribution of the fringing field [2, 3]. In all cases the maximum distortions, i.e., the mutual nonparallelism and nonrectilinearity of the electrodes, and also the edge distortions of the mass analyzer, are taken. The total distortions are determined as dispersions. In the numerical modeling of ion-optic systems engineering methods are being developed for calculating the transport of ion beams as a whole from the source through the fringing fields in the mass analyzer, taking into account the space charge. The characteristic features of the source together with the focusing optics are as follows: the ion gun is made in the form of immersion lenses and the monochromator is made in the form of a spherical capacitor with beam-turning angle of 90 ° [3]. The following assumptions are used in order to construct the mathematical model. The process of ion production is not studied. It is of a probabilistic character and is determined by the ionization cross section. The ion current at the exit of the source depends on the ionization efficiency and the transmission of the source [4]. VNIITFA. Translated from Atomnaya ]~nergiya, Vol. 75, No. 2, pp. 109-121, August, 1993. Original article submitted March 9, 1993. 618

1063-4258/93/7502-0618512.50 ©1994 Plenum Publishing Corporation

Only transmission is calculated in the model. All parameters are optimized with respect to the transmission. Two distributions are taken as the initial ion velocity (angle) and coordinate distributions: with constant density and normal density [5]:

=

v i - o'ooi ;

=

where o = u and cr = t~ are deflections along the Cartesian coordinates x and y and the corresponding velocity (angle); e is the emittance of the beam from the source or the acceptance of the beam in the fringing field of the mass analyzer (in our case it is normalized, i.e., e = 1). The initial radial u0 and angular t~0 envelopes of the beam are determined by the source aperture and the ion energy spread. The energy spread is determined by the thermal energy (0.1-1 eV [4]) and the energy of fragments produced as a result of dissociation of molecules (0.2-4 eV [6]). For this reason, the maximum values of the beam envelopes are taken as u o and %. In the development of the mathematical model of the source Poisson's equation is solved numerically by the method of successive approximations in an iteration procedure, starting with Laplace's equation, when one of the electrodes is the ion beam itself in the "transparent" form [3]. In the general case Liouville's theorem is not applicable for interacting particles and the phase volume in such a system is not conserved. However, this theorem can be applied for systems with space charge whose density decreases rapidly. The validity of this assumption was partially confirmed with the help of point-by-point calculations of the ion trajectories, taking into account the space charge and measurements of beam characteristics according to the probe method [3]. In the mathematical model of the quadrupole mass analyzer with scattering of ions the linear approximation of the fringing field is studied i7]: qb(x, y, z)

=

t~0(x2

-

y2)z/*{.

(2)

Then the equations of motion of the ions can be written in the form of the following system [3]: /gl- [a-2qcos2(~-~o)]ulu3/ro=O,

i = 1,2;

(3)

'u'a + [a - 2q cos 2(~ -. ~0)] ( u2 - u2)/ro = O,

(4)

where u 1 = x; U 2 = y; U 3 = Z ; [iI are the second derivatives of the corresponding coordinates with respect to ~; 2~ = cot, a = 8eU/(mro2%2), q = 4eV/(mro2c%2) is the parameter in Mathieu's equation [7]; ro is the radius of the electric field; and Go is the initial phase at injection. The field of the form (2) can be modeled with the help of electrodes with rounded tips. This approximation is valid for a fringing field extending over the distance ro [7]. The numerical solutions of the nonlinear differential equations of motion of ions in the fringing field (Hill's equations) are found with the help of approximating linear matrix transformations. The validity of this approximation has been confirmed by calculations of ion trajectories in the complete spatial distribution of the fringing field [7, 8].

ANALYTICAL T H E O R Y Analysis of ion trajectories in the distorted fields of the mass analyzer. By analyzing beam dynamics in phase space it can be shown that the trap mechanism of ion separation by mass number in the distorted field of a quadrupole mass analyzer is determined by the properties of the solutions of Hill's differential equation [9]. For a potential field of the form (1) the equations of motion of the ions reduce [2] to a system of Hill's differential equations of the Mathieu form with a small inhomogeneous part: '" --- (a - 2v cos ~ ) u l = +_. [coes u l a(u3) ] (a - 2q cos 2~), u t

i"a

=

_

iWz)(

_ Zq

--" u )aC/d. 3,

1 = 1, 2;

(5)

where dG/du 3 is the z derivative of the function G(z). 619

-I

I

-!

0

x/c~

Fig. 1. Profile of ion beam in the form of a region of stable trajectories through a quadrupole mass analyzer in the (x/r o, y/r o) plane for two injection phases. It is evident from the system (5) that the principle of independence of ion oscillations as a function of the coordinates breaks down in a distorted field. Then the solution of these equations can be reduced to an approximate solution in the form of the general solution of the homogeneous Mathieu equation with combined coefficients, taking into account the inhomogeneous part, with whose help the beam trapping zone in phase space can be determined [2]: u=

~

(6)

[Ce~k)nosin (fit+ SO)+ Se~o cos SO] ,

where tg~ = 3'1/72; 3"1 = Cqo + x~20; 3'2 = ~20-- XCqo;Z =if)q20 +X~o +)~0; Cello, Seato arethe even and odd Mathieu functions of integer order 2n with eigenvalues a k and bk; the suffix 0 means that ( = (0; the coefficie~s are defined as ~1o = (UoS'e3o - Uo Se3o)/Wo' a20 = (uoCe/~o - uode~o )/WO; Ce3o = ~ c2n cos (2n + fl)~o;

Se3o = ~ C2n sin (2n + fl)~o

is the Mathieu function of fractional order/3;/3 is the stability parameter of Mathieu's equation (for u = x:/3 = 1 --/3x; for u = y:/3 =/3y; the distortion parameters, defined as XlO = Er ~O~V21/(4LWo), •20 = [2Er QIzO~2/(Wo L) ] X [k0~0/(3L) - 1/2 ] n XaO=[f21~O/(4Wo) ] [2zo/ro- Zo~o/ro +- (ff22/ff21)(Ro/ro)21, characterize the degree to which they influence the ion trajectory; to

the Wronskian determinant is Wo= Ce/~(~o)S'e3(~o) - Ce3(~o)Se/~(~o); the harmonic coefficients are ~~l = 2a - ~ C~n(2n + ~

--to

O0

fl)2/( E C2n); t/~--

if22 = a 2 + q 2 +

(a + q)(a + f~l)"

. The relative deviation of the working point in Mathieu's diagram from

oO

the vertex of the stability triangle can be taken as .the criterion for estimating the computational error in the distorted fields. The expressions for the ion trajectories in the general case of distortions, including distortion of the supply voltage, x rp =3/(1 + ZxO)(2aZlxO + a~xo) (Ce~12osin SOx+ Se0)2nocos SOx) -- on the boundary of_ the region of stability; x = f r o + ZxO)(alxO2 2 + a~xo) (Ce~12osin [(1 - flx)~ + SOx] + Se~12ocos SOx); -- near the boundary; 2 7c,~(0) yrp ='V(1 + Z~,O)(a2yO+ a 2yOJ v~2nO sin SOy -- on the boundary; and, x o)0 sin (fly ~ + SOy) y ='1/(1 + X~o)(a~y0 + a2y0)Ce~n

620

-- near the boundary,

show that the distortions in the analyzer increase the amplitude of oscillations by a factor of -41 + Xo2 [2]. Since the distortion coefficient Xo in the expression (6) depends on the phase at injection, the distortion-induced increment to the amplitude of the ion oscillations has a phase dependence. For zero phase Xo = 0, and the distortions do not affect the ion trajectory. For Go = 7r/2, Xo assumes its maximum value, and the amplitude of the oscillations increases so that part of the beam is cut off by the aperture of the mass analyzer. This decreases the transmission of the analyzer. Ion beam dynamics in the phase space of a distorted fields. In the mathematical modeling of ion separation in distorted fields a phase space is introduced on the basis of the principles of statistical mechanics. In this phase space the law of conservation of phase volume -- Liouville's theorem -- holds. For stable ion trajectories (6) the boundary of the beam in the phase plane (u, ti) is described by a system of ellipses with the quadratic form [2] (7)

r u 2 + 2Am~ + Bu 2 = e

and the parameters oo

r = (E c,.)(1 + Zo>(de o + n~--O0 O0

A = - ( ~ C2n)2(1 + Xo)(Ceflodeflo + SepoS'e~o)/W~; n=-¢¢

B = (• C2n)2(1 + ~2)(Ce/~0 + Se~o)/Wo2. n~---'--O0

This ellipse determines the coordinates and velocities with which the ions will pass through the mass analyzer. Ions with values outside the ellipse do not pass through the analyzer. Thus the area of the ellipse is proportional to the acceptance, i.e. the trapping area, corresponding to transmission of the beam. The region of intersection of the ellipses (inscribed Floquet ellipse) for different phases determines the boundary of initial conditions for 100% transmission. It can be shown that the parameters of the ellipse A, B, and P depend on the injection phase G0, while the acceptance e, equal to the area of the ellipse divided by 7r, is an invariant. This follows from Liouville's theorem for ion motion in the phase plane. This theorem can be formulated thus: The size of the phase volume (or area) occupied by a groop of ions cannot change. In [2] it was shown, by analyzing the dynamics of an ion beam in phase space, that a beam passing through a quadrupole mass analyzer has a profile with pulsating density and cross section and formed by the equipotential lines of the distorted field (Fig. 1). For this reason, the ion current at the exit of the mass analyzer is modulated in amplitude with the frequency of the alternating component of the field. The maximum current is observed with zero phase. The degree of modulation is proportional to the distortion of the field. Transmission and optimal injection of ions in an analyzer with a distorted field. The analytical expression obtained in [2] for determining the transmission of a quadrupole mass analyzer with a distorted field shows that it is equal to, to within a proportionality constant, the product of the acceptances (areas of the inscribed Floquet ellipses, formed by cutting out the central part of the phase plane in the coordinates x and y):

r = LxlxLyly,

(8)

where

7-----WO/[ (E C 2 n ) 2 " V ~ ] ; Z =

%/'

2n

[ t/=--

+ Z2pg)

nC2n )2 + 2fl

;

L

621

r/r~ gs e,1

Iz

i

2...~

Fig. 2. Ratio of the transmission of a quadrupole mass analyzer with distorted and ideal fields versus the resolution for distortions of 1 (1) and 10 gm (2) (in a logarithmic scale).

~/

rel. units

2

3

I

a,

e'

Z/-

Fig. 3. Transmission versus the phase of transit through the flinging field between the ion source and the preliminary filter (1), the preliminary filter and mass analyzer with the voltage on the electrodes of the preliminary filter changing from Vp = 0 (2) to Vp = V (3) with a step of Vp/10V. 1 , ~x - 0.727/x/P,, By -- 0.513/x/R; R = M/AM is the expansion; Xopt is obtained by substituting into XlO, x20, and X30 the optimal values of the injection phases for all types of distortions. It is evident from the relation (8) that the transmission is directly proportional to the product ~y(1 -- ~x), because the Wronskian determinant W 0 is proportional to ~ [7], i.e., the transmission is inversely proportional to the expansion, and field distortion decreases transmission by a factor of (1 + Xx2)(1 + Xy2). The approximation function [2] for the transmission losses in a distorted field as a function of expansion and the relative distortion shows that in a logarithmic scale the transmission is inversely proportional to the relative distortion raised to the power 1.18 (Fig. 2):

lg(T/To) =

(Ar0/r0) -1'18 [1 -- 0,038(Ar0/r0)°'6561gR I.

(9)

As the expansion and distortion increase, the losses increase by a factor of 30. These losses can be reduced to a minimum by gatir~ the ion-beam with zero injection phase in each cycle of the alternating component of the field. As a result the transmission grows exponentially by a factor of 30 to 100 as a function of the distortion and resolution (Fig. 3). Investigation of the &ef..fectof the phase characteristics of ion injection in a quadrupole mass analyzer on the transmission and theoretical analysis-~ the results established [2] that by gating the beam the transmission of the mass analyzer can be increased for a nonparallel beam of ions at the entrance. Instrumental--methodological gating of an ion beam is achieved bY applying a voltage on the quadrupole mass analyzer in the form of a nonuniform standing wave with linearly distributed amplitude along the ion 622

transmission axis. Semiempirical relations have been obtained for the transmission as a function of the injection phase. These relations are of a resonant character with an optimum at a definite injection phase (see Fig. 3): T = 0,875 exp [ - 0,066(~ o - ~oP~ 2 ] + 0,126.

(i0)

NUMERICAL METHODS Ion-optic system of the source. In the mathematical model of the source the law of conservation of energy relates the velocity t) of ions with mass number M to the potential difference (~1 - - ~2) [5]:

.z)/M,

;~ = V 2 ( , ~ , 1 -

and Maupertius' principle of least action, similar to Fermat's well-known principle in geometric optics, according to which A

light in a nonuniform medium propagates in a maimer so that the optical path length is minimum [5]: 6 B f n(s)ds = 0, where

B ~B is the variation symbol and n(s) is the index of refraction, gives a representation of the electric field in the form of a system of equipotentials with variable index of refraction. In the ion-optic system of the source numerical solutions of Poisson's 3 equation for the potential of the field AO = ~ d2~/du~ = - 4srp, are studied by the method of iterations with the help o~ l=1

Liebman's transformation [3]

Oil k = 0,25(cbi+l] k + dPi_l] k + CPi]+lk + (1)/]_lk), where k is the number of iterations, qsijk is the potential at a point with the coordinates (z i, uj), and the differential equations of motion of the ions [3] are

k" = - (1/M)dCb/dul;

l = 1, 2, 3,

where u 1 = x; U 2 = y; u 3 = z; i1l is the second derivative of the corresponding coordinate with respect to time, and M = m/q is the mass number, and by the method of matrix transformations [5]

Ulnj

n

[mll m12]

where M r =i=17111v//]= [ rn21 m2zJ

=

MT

(11)

L•,oj ,

is the transfer matrix for the elementary transformation matrices, determined according

to the law of refraction [3]:

M;j =

[_ 1/I,'

;q,

hi] = 1 + cli]/fil; zi+ I = ~'i + h i + / ; zi = zi-1 + hi;

ail = 2hi/(~l--Oil/Oi_1il

+ I); 1/fq = (2hi/L)(~f~i/qbi_lil

_ 1).

623

The physical meaning of the coefficients mkl (k, l = 1, 2) of the transfer matrix M T is as follows: the matrix coefficients m22 and mll are the angular and linear magnifications, respectively. One or another type of focusing is realized in a definite transverse plane of the ion-optic system, depending on the conditions imposed on the elements of the transfer matrix MT: for mll = 0 a parallel beam converges to a point, m12 = 0 corresponds to transformation of a point source into a point image, for m21 = 0 a parallel beam remains parallel, and for m22 = 0 a beam from a point source becomes a parallel beam. Ion-optic system of the fringing field of a quadrupole mass analyzer. In the mathematical modeling of a quadrupole mass analyzer taking into account the fringing fields, numerical solutions of the differential equations of motion of the ions (3)(4) in the fringing field (Hill's equation) are studied with the help of Galerkin--Slezkin integral averaging [3] in the iteration procedure:

n= (~f-- ~s)-l f [a -- 2q cos

2(~ -- ~0)] (u 2 --

u2)rold~,

where ~s and ~f are the initial and final phases of transit of the ions through the fringing field. With the help of the variable H 2 = ( ~ ; - ~s) - l f

[a - 2q cos 2(8 - ~0)] u3(~)roidld

Hill's equation (3) is reduced to Meissner's equations [7]:

u"l +. H2Ul = 0;

l = 1, 2.

The solution of these equations with the help of the transformation matrices [10]

rM3 = [

LUJ

[

cos [H(~ - ~o) ]

{sin [H(~ - ~50)1 } l #

" H sin [H(~ - ~0) ]

cos [H(~ - ~0) ]

[ ch [ H ( ~ - ~0)1 {sh [ H ( ~ - ~0)]}]H] L

f o r H 2 _> 0;

for H 2 < 0

J

can be written in a matrix form (11), where the transfer matrix MT [10] is

ulo, ulo, u20, f120 are the initial coordinates of the ion in the phase planes (u 1, ul) and (U2, [12), respectively, determined by the output maximum radial and angular envelopes of the beam from the source; cosh(~) and sinh(~) are hyperbolic functions. The properties of Twiss's matrices for periodic systems give a restriction on stability: if ] Sp(MT) ] = ] mlt + m22 ] = I 2cos0rfi) [ < 2, the trajectories of the ions are stable, where Sp(M T) is the trace of the transfer matrix of the ions and is Mathieu's stability parameter; if I mll + m22 ] -> 2, then the trajectories are unstable [7]. Ion-beam dynamics in phase space. In the mathematical model of beam transport as a whole, a phase space where Liouville's theorem holds (the law of conservation of phase volume) is introduced. A consequence of this theorem is that if the determinant of an elementary transfer matrix is constant and equal to unity, then the determinant of the resulting matrix determining the transformation of the parameters of the particle trajectories of the entire ion-optic system is also equal to unity [5]: detM T = mllm22 -- m12m21 = 1. For stable ion motion the boundary of the beam in the phase plane is described by a family of ellipses with the quadratic form (7), where m22 !] =

624

-

m12m22 2 m12

- 2m22m21

m21

mllm22 + ml2rn21

- mllrn2] 2

- - 2t~l 12 m 11

roll

IF°1 A0 . B0

(12)

4/r~I

ges

g,2

L

_.t___

b

4s

-J,I o

i0

u//-,

Fig. 4. (a) Ion-beam dynamics in the phase plane of the emittance of the source with inductively coupled plasma at the exit of the skimmer (1), stopper (2), extractor (3), collimator (4), focusing electrode (5), and monochromator (6) with a step of 1/5 of the accelerating gap and (b) acceptance of the fringing field of the quadrupole mass analyzer for relative transit time through the fringing field from 0 (1) to 37r (2) with a step 1/5 of the hf period. If the coefficients of the quadratic form (7) are put into a matrix A

-

FB

A z

-

A

'

625

then the transformation (12) can be written in the compact form A = M T A 0 I~T, where the tilde indicates transposition of the matrix. The coefficients on the principal diagonal of this matrix determine the linear and angular envelopes of the particle beam: Uraax--.ffe B/(FB - AZ); //rnax='~/e F/(FB - A2); the coefficients on the other diagonal determine the degree to which the ellipse deviates from the canonical position: tg(20) = - 2 A / ( B -- F), where 0 is the angle between the principal axis of the ellipse and the abscissa of the phase space (u, u). By virtue of Liouville's theorem the area of the ellipse S = 7re/~/FB -- A ~, where e is the emittance o f the source or the acceptance of the fringing field of the quadrupole mass analyzer, i.e., under linear unimodular transformation of the phase space the area of emission or trapping of the beam in the phase space is conserved (detMw = 1). The parameters of the emittance and acceptance ellipses are found in terms of the coefficients of the transfer matrix for an arbitrary set of initial conditions. The angular and radial profiles of the ion beam are found from the initial distributions for constant density and normal distribution in terms of the angular and radial envelopes.

COMPUTATIONAL RESULTS AND DISCUSSION The computational results are displayed in Fig. 1 in the form of the region of stable trajectories in the plane (x/ro, Y/r0). This region is formed by the maximum possible deflections with which ions pass through the quadrupole mass analyzer. Ions moving along a trajectory within this region (for the corresponding set of initial conditions) pass through the mass analyzer, while ions moving on trajectories outside this region do not pass through the analyzer. It is obvious that the distortions lead to transformation oi' the region of stable trajectories from curve 1 with zero phase to curve 2 with 40 = 7r/2. For this reason, the ion beam passing through the mass analyzer has the form of a profile with pulsating cross section and density and formed by the equipotentials of the distorted field. The ion current at the exit of the analyzer is amplitude-modulated with the frequency of the alternating component of the field. Maximum current is observed at zero phase. The degree of modulation, determined by the ratio of the areas within curves 1 and 2, increases with the degree of distortion. The dependence of the relative transit losses (ratio of the transmission of the quadrupole mass analyzer with distorted and ideal field) on the resolution, as displayed in Fig. 2, is exponential. As the resolution increases, the transmission losses increase. This is especially noticeable with increasing distortions of the field, where the transmission decreases by a factor of 30 at R = 1000 (see curve 2). The curves shown in Fig. 3 are of a resonance character with an optimum at a definite injection phase. As the amplitude of the voltage on the prefilter increases from Vp -- 0 up tO V the optimal phase decreases from ~o°pt = 1.8~- to 0.37r. This means that the usual prefilter [7] does not make it possible to avoid transmission losses due to the differences between the optimal injection phase from the source into the prefilter (~o°pt = 0; see curve 1 in Fig. 3) and from the prefilter into the analyzer (~0°pt = 0.3~r; see curve 3). In a prefitter with a linear distribution of the voltage amplitude along the axis the injection phase is always close to the optimal phase (~o°pt = 0); these losses can be avoided and a 30-fold gain in tranmission can be achieved. According to the results of calculations of the transformation of the phase regions, bounded by ellipses, in an ion-optic system of the source with an inductively coupled plasma and the fringing field of the quadrupole mass analyzer, maximum transmission is observed for linearly increasing distribution of the potentials on the source electrodes. The emittance ellipse in Fig. 4a at first stretches and rotates into the region of divergence at the exit of the extractor and stopper, stretches and rotates into the region of convergence at the exit of the focusing electrode, and contracts and rotates in the direction of the canonical arrangement at the exit of the collimator. The area of the ellipses remains constant, according to Liouville's theorem, as the ellipses transform. Figure 4b displays the dynamics of the acceptance ellipse in the fringing field of the quadrupole mass analyzer for the phase plane (u/r 0, t~/ro). The initial angular and radial characteristics of the beam correspond to maximum values of the characteristics of the emittance ellipse of the ion source: u/R 0 = 0.357 and u/R 0 = 0.075 (the divergence angle 2c~ = 4°). The parameters a and q of Mathieu's equation were chosen for resolution R = 100. At first, the ellipse stretches out; this corresponds to transformation of the transfer matrix for relative transit time of the ions through the fringing field 4o = 37r..Next, as the phase decreases, the acceptance ellipse starts to rotate into the region of canonical arrangement. The minimum of the radial envelope of the beam is observed for the phase 40 = 1.87r; the minimum of the angular envelope is 626

fl/~/

\ \\\i,,

.3

1 g,5

I,#

1,,~

u&,

Fig. 5. Radial profiles of a beam in an ion-optic system of a source with an inductively coupled plasma, taking into account the space charge (-) and neglecting the space charge ( - - - ) for normal distribution of ions in the beam (the numbering of the curves follows Fig. 4a). observed for (o = 7r. As the phase decreases from 7r to zero the ellipse stretches out and rotates into the region of beam divergence. In the process, both the angular and radial envelopes increase. However, the area of the ellipse remains constant in accordance with Liouville's theorem. It follows from the ion-beam profile shown in Fig. 5 that the limiting radial envelope of the beam remains less than unity, i.e., less than the size of the apertures of the electrodes Of the system, and at the exit -- less than the size of the aperture of the collimator. The space charge causes the ion-beam profile to have an annular structure along the radius at the exit of the skimmer, extractor, and stopper. The annular structure vanishes at the exit of the ion-optic system, when the space charge becomes negligibly small. It can be conjectured that such a structure is associated with the collective acceleration and selffocusing of ions and electrons in the expansion chamber, i.e., in the zone of action of the space charge. It is obvious that the density of the space charge drops exponentially immediately behind the skimmer; this confirms the validity of the assumptions made above. The curves in Fig. 6 are of an exponentially harmonic character. As the divergence angle increases from 0.3 to 27 ° (see Fig. 6a), the optimal phase decreases exponentially; as the exit aperture of the source increases from 0,03r o to 0.5r o the oscillations decrease, and the optimal phase increases and approaches 1.7%r. As the exit aperture of the source increases from 0.03r 0 to 0.5r 0 (see Fig. 6b) the optimal phase increases exponentially; as the divergence angle increases from 0.3 to 17 ° the optimal phase decreases and approaches 1.7%r. The harmonic oscillations of the dependence are maximum for the initiaI radial envelope from uo/ro = 0.03 to 0.3 and initial angular envelope from ~ / r 0 (divergence angle 3 °) to 0.2 (11°). Matching the emittance of the beam from the ion source and the acceptance in the fringing field of the quadrupole mass analyzer showed that the resolution and transmission are affected mainly by the beam divergence. Analysis of beam dynamics in phase space shows that the transmission depends exponentially on the squared phase of the ejection, and the optimal phase at injection is not constant and depends on the initial angular and radial envelopes according to an exponentially harmonic law (see Eq. (10), Fig. 6): ~opt = au x exp (-- euho/ro) cos (c u zc ho/ro) + b u ~;

~opt= at~gg exp (-- eiLuo/ro) U

where

+ b;~ ~,

a i = 50,832(ho/ro)4 - 0,478; bh= O,02/(ho/ro) + 1,846;

ek = 14,617 exp ( - 30hO/ro) + 2,475; au= - 508 exp ( - 15Uo/ro) + 0,417; bu= 0,322uo/r o + 1,608; e u = - 17,463 exp ( - 4uo/ro) + 17,932; c. = - 0,09 exp ( - 8uo/ro) + 5,977.

627

Z,!

z,o! /,3:

/,i I

o

,,

Z,./

Zo t

ii Z,7

~5o

.. l,~

~,z

~s'

~

,,/,.,

Fig. 6. (a) Optimal phase of transit through the fringing field versus the initial angular envelopes of the beam with the exit aperture of the source increasing from u0 = 0.03r 0 (1) to 0.5r 0 (2) with a step of 1/16 uo and (b) radialenvelopes with the divergence angle increasing from 0.3 (1) to 9 (3) and 17 ° (2) with a step of 1/16 of uo/r0 . The engineering methods of computation developed in this work for ion-optic systems of a quadrupole mass spectrometer have made it possible to increase the transmission by a factor of 2-30, depending on the resolution, by matching the emittance of the beam from the source with an inductively coupled plasma and the beam acceptance in the fringing field of the analyzer. An experimental check gave good agreement (within 1%) with the computed values of the parameters.

CONCLUSIONS 1. Mathematical modeling of ion separation by mass number in a distorted field of a quadrupole mass analyzer, constructed on the basis of the principles of statistical mechanics, made it possible to obtain the phase-space picture of the dynamics of the beam and develop new concepts in the theory. This made it possible to develop experimentally tested instrumental-methodological devices for eliminating transmission losses due to field distortion and reduce by a factor of 10 the requirements imposed on the precision of manufacturing of the quadrupole mass analyzer without transmission loss. 2. Analysis of ion-beam dynamics in phase space in a quadrupole mass analyzer with a distorted field showed that the influence of the distortions can be reduced to a minimum by gating the beam with zero phase in each cycle of the alternating component of the field with the help of a nonuniform standing voltage wave applied to the mass analyzer. This made it possible to increase the transmission from 30 to 100 times, depending on the resolution. Semiempirical relations for the transmission as a function of the injection phase were obtained by investigating the effectiveness of ion injection with beam gating and the 628

advantages of beam gating in a quadrupole mass analyzer for solving the problem of fundamental reduction of the detection thresholds were substantiated.

REFERENCES ,

2. . .

5. . .

8. . 10.

G. I. Slobodenyuk, Quadrupole Mass Spectrometers [in Russian], Atomizdat, Moscow (1974). V. V. Titov, "Theoretical aspects of the operation of a quadrupole mass spectrometer with a distorted analyzing field," At. l~nerg., 61, No. 2, 103-110 (1986). A. N. Matantsev and V. V. Titov, "Working models of an ion-optic system of a quadrupole mass spectrometric apparatus," Voprosy atomnoi nauki i tekhniki. Ser. Radiatsionnaya tekhnika, No. 1(32), 74-78 (1986). L. Valy, Atom and Ion Sources, Akademiai Kiado, Budapest (1977). V. P. Kartashev and V. I. Kotov, Principles of Magnetic Optics of Charged-Particle Beams [in Russian], t~nergoatomizdat, Moscow (1984). G. I. Kir'yanov and V. V. Titov, "Estimate of the intensity of spurious peaks in a quadrupole mass spectrometer," Voprosy atomnoi nauki i tekhniki. Ser. Radiatsionnaya tekhnika, No. 1(29), 50-54 (1985). P. Dawson (ed.), Quadrupole Mass Spectrometry and Its Application, Elsevier, New York (1976). P. Dawson and Yu. Binggi, "An approximate method for determining the emittance of ion source in quadrupole mass spectrometry," Int. J. Mass Spectrom. Ion Process., 54, 159-167 (1983). 1~. P. Sheretov, "Hyperboloid mass spectrometers," Izmereniya, kontro' i avtomatizatsiya, No. 11-12 (33-34), 29-43 (1980). M. Baril, "Etude des proprietes fondamentales de l'equation de Hill pour le Dessin de filtre quadrupolaire," Int. J. Mass Spectrom. Ion Plays., 35, 179-200 (1980).

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