Chapter 7 Mass Analyzers With Combined Electrostatic and Magnetic Fields

Chapter

7 Mass Analyzers With Combined Electrostatic and Magnetic Fields

Contents

7.1. Sector Field Mass Analyzers with Energy Focusing 7.2. Wien Filter 7.3. Penning Traps

260 265 271

Ion mass analyzers performing spatial separation of particle beams and using combined electrostatic and magnetostatic fields either consist of several consecutive electrostatic and magnetic stages or alternatively use superimposed electrostatic and magnetostatic fields. The first (staged) approach is used to achieve energy focusing or, in other words, to eliminate energy dispersion and thus to increase the mass resolving power of the analyzer. Analyzers with energy focusing are considered in Section 7.1. Among analyzers with superimposed fields only one, the Wien filter considered in Section 7.2, is of widespread use as a unique dispersive device with the straight optic axis. The analyzers with curved optic axes and crossed electrostatic and magnetostatic fields, while thoroughly studied theoretically (see, for example, Ioanoviciu, 1974; Toyoda et al., 1995), have not yet become popular in practice due to their complexity. A combination of static electric and magnetic fields also can be used to trap and store ions in a restricted volume. This combination, known as the Penning trap or ion cyclotron resonance (ICR) cell and considered in Section 7.3, allows mass analysis of ions based on recording and measuring frequencies of their rotation in the trap.

Advances in Imaging and Electron Physics, Volume 157 ISSN 1076-5670, DOI: 10.1016/S1076-5670(09)01607-3

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2009 Elsevier Inc. All rights reserved.

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Mass Analyzers With Combined Electrostatic and Magnetic Fields

7.1. SECTOR FIELD MASS ANALYZERS WITH ENERGY FOCUSING 7.1.1. Integral Relation for the Rigidity Dispersion in Multistage Analyzers Comprising Electrostatic and Magnetic Sector Fields As shown in Sections 5.2 and 6.1.4, the general integral relation of Eq. (2.54) for the rigidity dispersion is considerably simplified for charged particle motion in magnetic or electrostatic sector fields. For multistage magnetic sector fields the corresponding relation for the energy and mass dispersions is given by Eq. (5.20), and for multistage electrostatic sector fields the relation for the energy dispersion is given by Eq. (6.23). When N sector magnetic fields and M sector electrostatic fields are placed in series with no ion beam acceleration or deceleration between these fields, the relation for the energy dispersion of the resulting multistage system takes the following form: ðMÞ ðEÞ N M DK 1 X Sn 1 X Sm ¼  ; Mx 2ðDa0 Þ n ¼ 1 rðnMÞ ðDa0 Þ m ¼ 1 rðmEÞ ðMÞ

(7.1)

where Sn is the area illuminated inside the nth magnetic field by the ðMÞ beam emitted from a point object with the angular spread ðDa0 Þ, rn is ðEÞ the curvature radius of the optic axis in this field, Sm is the area illumiðEÞ nated inside the mth electrostatic field, and rm is the curvature radius of the optic axis in this field. At the same time, the relation for the mass dispersion retains the form of Eq. (5.20), because electrostatic fields possess no mass dispersion. As in cases of pure magnetic or electrostatic multistage systems, the areas in Eq. (7.1) change sign after each Gaussian image plane, and the signs of the curvarure radii depend on the direction of the beam deflection. From Eq. (7.1) it is evident that proper illumination of magnetic and electrostatic sector fields can eliminate the energy dispersion DK in the system, whereas the mass dispersion remains nonvanishing. For a system consisting of only two fields, one electrostatic and one magnetic, the energy dispersion vanishes when the ratio SðMÞ =rðMÞ is twice as large as the ratio SðEÞ =rðEÞ and two these ratios have the opposite signs. This means that either (i) the directions of deflection in both fields must coincide and there exists a Gaussian image plane between the fields or (ii) there is no intermediate Gaussian plane but the directions of deflection in the two fields are opposite. Static mass analyzers in which the angular-focusing ½ðx j aÞ ¼ 0 and the energy-focusing ½ðx j dÞ ¼ 0 are achieved simultaneously at the final image plane are often called double focusing. If the spatial focusing also is stigmatic ½ðy j bÞ ¼ 0, they are referred to as triple focusing.

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7.1.2. Mass Analyzers with Electrostatic and Magnetic Sectors Deflecting in One Direction According to Eq. (2.12a) or (2.12c), elimination of the energy dispersion in a mass analyzer allows a considerable increase in the mass resolving power of the analyzer for a large energy spread created in the ion source; magnetic analyzers with supplementary electrostatic deflectors appeared soon after electrostatic sector field analyzers were introduced (Bainbridge and Jordan, 1936). The next natural step to raise the resolving power of the two-stage energy-focusing magnetic analyzer working in the spectrometric mode (that is, with scanning ions through a fixed narrow slit by ramping the magnetic field) was elimination of the image aberrations created by the analyzer at the slit position. Studies of various geometries of two-stage mass analyzers with energy focusing (Johnson and Nier, 1953; Hintenberger and Ko¨nig, 1957) have shown that some secondorder aberrations, in particular the angular aberration ðx j aaÞa20 in the dispersion plane, can be eliminated in these analyzers by a proper choice of their configuration. A well-known example of a sector field mass spectrometer with energy focusing and elimination of the second-order angular aberration ðx j aaÞa20 is the Nier–Johnson analyzer with the 90-degree deflecting cylindrical electrostatic sector field and the 60-degree deflecting homogeneous magnetic sector field (Johnson and Nier, 1953). The scheme of this analyzer with the same direction of ion beam deflection in the electrostatic and the magnetic field is shown in Figure 146. The absence of focusing in the y-direction, perpendicular to the dispersion plane xz, in the Nier–Johnson analyzer restricts the beam transmission in this direction and leads to a large angular aberration ðx j bbÞb20 . One way to eliminate this drawback is to use fields varying in the y-direction. Since creating inhomogeneous conical magnetic sector fields with high accuracy is technically difficult, a preferable method is using toroidal electrostatic fields. A combination of a toroidal electrostatic sector field with a homogeneous magnetic field allows double-focusing sector field mass analyzers to eliminate most important geometric and chromatic second-order image aberrations ðx j aaÞa20 , ðx j adÞa0 d0 , ðx j ddÞd20 , ðx j bbÞb20 , ðx j yyÞy20 (Matsuda, 1974; Taya et al., 1978). However, toroidal sector deflectors were not widely used in double-focusing mass analyzers because the precise assembly of toroidal electrodes often leads to asymmetry of the deflector field with respect to the median plane y ¼ 0 of the analyzer, and this asymmetry is a known source of a parasitic ion beam distortion that is difficult to correct (Matsuda, Matsuo, and Takahashi, 1977; Yavor and Berdnikov, 1993; Yavor, Berdnikov, and Wollnik, 1997). Focusing in the y-direction in sector field mass analyzers can be alternatively achieved by implementing quadrupole lenses in devices consisting of a cylindrical electrostatic deflector and a homogeneous

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(a)

Energy-selecting aperture

90-degree cylindrical electrostatic deflector

60-degree homogeneous magnet

(b)

x lm

FIGURE 146 (a) Trajectories of ions with different energies and initial directions in the dispersion plane of the Nier–Johnson mass analyzer (Johnson and Nier, 1953) with double focusing; at the intermediate Gaussian image plane an aperture can be placed to restrict the energy spread accepted by the analyzer. (b) Trajectories of ions with different masses and initial directions in the same analyzer; the points of the final images form the ‘‘angular’’ mass focal line inclined with respect to the profile plane by the angle lm ¼ 62:9 degrees.

deflecting magnet. Combining one or multiple quadrupole lenses with electrostatic and magnetic sector fields became the most successful way of designing high-performance magnetic mass spectrometers with corrected second-order or even third-order aberrations (Matsuda, 1974, 1981, 1985, 1987, 1990; Matsuo, Sakurai, and Ishihara, 1990). An additional problem exists in mass analyzers working in the spectrographic mode without magnetic field ramping—that is, recording mass spectra either with the aid of multicollector devices like positionsensitive detectors or by using a movable exit slit. Assume that for a certain reference ion mass the double focusing is achieved: ðx j aÞ ¼ 0 and ðx j dÞ ¼ 0 at the final image point. Then, however, for other ion masses the points of the angular and energy focusing may not coincide. The inclination angle lm with respect to the profile plane of the ‘‘angular’’ mass focal line, which is the set of points where angular focusing occurs for ions of different masses, was given by Eq. (2.59b). Similarly, the

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x ~ lm

FIGURE 147 Trajectories of ions with different energies and masses in the analyzer of Figure 146; energy-focusing points form the ‘‘energy’’ mass focal line inclined with respect to the profile plane by the angle ~lm ¼ 60:7 degrees.

~m of the ‘‘energy’’ mass focal line, which is the set of inclination angle l points where energy focusing occurs for ions of different masses, is given by the formula ðx j dgÞ tan ~ lm ¼  ða j dÞDm

(7.2)

lm ; that is, the ‘‘angular’’ mass focal line and (Figure 147). In general, lm 6¼ ~ the ‘‘energy’’ mass focal line intersect at the angle lm  ~lm . Divergence of the ‘‘angular’’ and ‘‘energy’’ mass focal lines for side masses reduces the range of masses for which a high resolving power can be achieved. Early studies of optimization of two-stage energy-focusing mass spectrographs, aimed at the coincidence of the two just-mentioned mass focal lines, were performed by Hintenberger and Ko¨nig (1958). Later it was discovered (Matsuda and Wollnik, 1989) that implementing a quadrupole lens between the electrostatic and magnetic sector fields allows the design of mass spectrographs with double focusing in a wide mass range and with small image aberrations.

7.1.3. Mattauch–Herzog Mass Analyzer Although double-focusing analyzers with electrostatic and magnetic sector fields deflecting in opposite directions generally have worse ionoptical quality compared with analyzers with both fields deflecting in the same direction, they are still used because of their compact sizes (Matsuda, 1989). A special geometry, which to date remains the most widely used among the considered type of mass analyzers, was proposed by Mattauch and Herzog (1934). This geometry consists of a cylindrical electrostatic sector analyzer that forms a parallel beam of ions initially

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45⬚

FIGURE 148 Ion trajectories with three different initial directions in the dispersion plane and three different masses in a Mattauch–Herzog-type mass analyzer.

diverging from a point object at the optic axis, and the 90-degree deflecting homogeneous sector magnet with the straight exit boundary inclined under the angle –45 degrees with respect to the profile plane normal to the optic axis (Figure 148). Because of such configuration, after passing through the sector magnet ions of all masses are focused at the position of the exit effective boundary of this magnet, thus providing a straight mass focal line in the entire mass range. Moreover, the ratio SðMÞ =rðMÞ of the area illuminated by the ion beam inside the magnet to the deflection radius does not depend on this radius and thus on the ion mass. This means, that by a proper choice of the ratio SðEÞ =rðEÞ in the electrostatic sector field ½SðEÞ =rðEÞ ¼ ðSðMÞ =rðMÞ Þ=2, the analyzer can be made double focusing in the entire mass range ðlm ¼ ~ lm ¼ 45 Þ. Thus, the Mattauch– Herzog mass analyzer is the ‘‘ideal’’ mass spectrograph accepting a wide mass range (limited only by the dipole magnet size) with the straight coinciding ‘‘angular’’ and ‘‘energy’’ mass focal lines. The quality of the Mattauch–Herzog analyzer is limited by secondorder image aberrations. However, at least some of these aberrations can be reduced by design optimization. It is not possible to completely eliminate the second-order angular aberration ðx j aaÞa20 along the entire focal line, but by optimization of geometric parameters this aberration can be canceled at one point (for one ion mass) so that it remains small in the wide vicinity of this point. The ‘‘classical’’ design of the Mattauch–Herzog mass spectrograph suffers from a large angular aberration ðx j bbÞb20 arising in the exit fringing field of the dipole magnet. This aberration can be considerably reduced by extending magnetic poles and forming the focal line inside the magnetic field, but this approach is technically inconvenient because it requires placing a detector between the magnetic poles; thus, it usually is preferable to shorten the magnetic poles slightly to form the focal line in the field-free space. Alternatively, curved entrance magnet boundaries can be used, which also helps to reduce chromatic aberrations (Robinson, 1957). However, probably the most efficient way is to use ion beam focusing in the y-direction perpendicular to the dispersion plane

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265

to reduce the image size in this direction at the detector plane. Such focusing can be achieved either by implementing a toroidal deflector instead of a cylindrical one, or, perhaps even better, by implementing a quadrupole lens in the analyzer.

7.2. WIEN FILTER The Wien filter is a charged particle analyzer with crossed electrostatic and magnetic fields in which the electrostatic and the magnetostatic force counterbalance each other at the optic axis so that this axis is straight. However, this counterbalancing is violated for particles with masses or kinetic energies different from the reference ones, so that a Wien filter has nonzero spatial mass and energy dispersions. Wien filters are used in cases where in-line geometry of charged particle beams is advantageous (Teodoro et al., 1993; Tsuno, 1994; Liu and Tang, 1995; Marx, Gerheim, and Scho¨nhense, 1997; Niimi et al., 2005), a large accepted mass range is important (Aberth and Wollnik, 1990), or velocity filtering is required (Anne and Mueller, 1992).

7.2.1. Paraxial Optical Properties and Aberrations of a Wien Filter The Wien filter (Wien, 1897) is a charged particle analyzer with superimposed electrostatic and magnetic fields. Both fields are 2D, independent of the coordinate z. The plane y ¼ 0 is the plane of symmetry for the electrostatic field and the plane of anti-symmetry for the magnetic field. Examples of shapes of electrodes and pole pieces forming these fields are shown in Figure 149, although more complicated shapes have been also proposed (Kato and Tsuno, 1990) for reducing image aberrations. The equations of charged particle trajectories inside a Wien filter have the form of Eqs. (2.66). In order for the z-axis (at which the electrostatic ~ ¼ 0) to be the straight optic axis of the potential is assumed to be zero: U filter, the right-hand side of Eq. (2.66a) should vanish at this axis for the particle of the nominal energy ðd0 ¼ 0Þ and mass ðg ¼ 0Þ: ~ ~ y ¼ 0. This means that the values E0 and B0 of the electrostatic @ U=@x B field strength and magnetic flux density at the z-axis must be related as Qðref Þ E0 ðref Þ

2K0

Qðref Þ B0 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ; r0 ðref Þ 2mðref Þ K0

(7.3)

where r0 is some ‘‘characteristic radius.’’ It is convenient to represent trajectory equations in a Wien filter in the dimensionless coordinates x ¼ x=r0 ,  ¼ y=r0 and z ¼ z=r0 , at the same time expressing the magnetic

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Mass Analyzers With Combined Electrostatic and Magnetic Fields

(a)

y

(b)

y Pole piece

Pole piece Electrode Electrode

Electrode

x

Pole piece

Electrode

x

Pole piece

FIGURE 149 Section by a plane z ¼ const through electrodes and pole pieces of Wien filters with (a) homogeneous and (b) inhomogeneous electrostatic and magnetic fields. Homogeniety of the electrostatic field in the presence of conducting pole pieces is improved in case (a) by adding shims at the electrode edges (equipotential lines of electrostatic field are shown).

~ ~ x ¼ @ W=@x, flux density in terms of the scalar magnetic potential B ~ ~ By ¼ @ W=@y: 0 0 ~ 1 þ x 2 þ  2 @U x ¼ ~ þ d0 @x 1  2q U ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

00

( ) 0 ~ ~ 1 þ x 2 þ 0 2 0 2 @W 0 0 @W x þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ x Þ ; @ @x ~ þ d0 Þð1 þ gÞ ð1  2U

(7.4a)

0 0 ~ 1 þ x 2 þ  2 @U  ¼ ~ þ d0 @ 1  2q U ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

00

( ) 0 ~ ~ 1 þ x 2 þ 0 2 0 0 0 @W 2 @W þx ; þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ  Þ @x @ ~ þ d0 Þð1 þ gÞ ð1  2U

(7.4b)

~ ~ ~ x , @ W=@ ~ y and prime denotes derivative ¼ r0 B where @ W=@x ¼ r0 B with respect to z. The normalized electrostatic field potential and magnetic flux density components can be expressed in form of Taylor expansions in the vicinity of the optic axis x ¼ y ¼ 0. Taking into account the symmetry conditions at the plane y ¼ 0, these expansions have the form ~ Þ ¼ uð10Þ x þ uð20Þ x2 þ uð02Þ 2 þ uð30Þ x3 þ uð12Þ x2 þ . . .; Uðx; ~ Wðx; Þ ¼ wð01Þ  þ wð11Þ x þ . . .;

(7.5a) (7.5b)

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267

where uð10Þ ¼ 1 and wð01Þ ¼ 1 according to Eq. (7.3). In the linear (firstorder) approximation Eqs. (7.4) thus read as x00 þ ð1  nÞx ¼

d0  g ; 2

(7.6a)

and 00 þ n ¼ 0;

(7.6b)

where n ¼ 2uð02Þ þ wð11Þ (here we took into account that uð20Þ ¼ uð02Þ , which is the consequence of the Laplace equation for the electrostatic field potential). The solutions of Eqs. (7.6), being rewritten in the nonnormalized coordinates, have the following form:     pffiffiffiffiffiffiffiffiffiffiffi z pffiffiffiffiffiffiffiffiffiffiffi z r0 þ a0 pffiffiffiffiffiffiffiffiffiffiffi sin 1n 1n xðzÞ ¼ x0 cos r0 1  n  r0  pffiffiffiffiffiffiffiffiffiffiffi z r0 ðd0  gÞ; 1  cos 1n þ (7.7a) 2ð1  nÞ r0  yðzÞ ¼ y0 cos

pffiffiffi z n r0



  pffiffiffi z r0 þ b0 pffiffiffi sin : n r0 n

(7.7b)

Note that the energy and mass dispersions in a Wien filter have opposite signs. For a homogeneous Wien filter shown in Figure 149a, uð20Þ ¼ wð11Þ ¼ 0 and thus n ¼ 0. Such filter is not focusing particles in the y-direction. If 0 < n < 1, the filter is focusing in both x- and y-directions. When n ¼ 1, the Wien filter is not focusing in the x-direction. With n < 0 the filter is defocusing particles in the y-direction, and trigonometric functions in Eq. (7.7b) turn to hyperbolic ones. Finally, with n > 1 the filter is defocusing charged particles in the x-direction, and trigonometric functions in Eq. (7.7a) turn to hyperbolic ones. Note that the same linear optic properties of the Wien filter can be obtained with different relations between inhomogeniety coefficients uð20Þ and wð11Þ of the electrostatic and the magnetic field, provided that the parameter n remains unchanged. Note also that in the linear approximation the relative deviation of the particle velocity with respect to the nominal one is expressed as qffiffiffiffiffiffiffiffiffi ðref Þ  2K d0  g mðref Þ qffiffiffiffiffiffiffiffiffi :  2 2Kðref Þ

qffiffiffiffi ðref Þ

vv vðref Þ

¼

2K m

(7.8)

mðref Þ

Thus, in the linear approximation two particles with equal velocities are not separated by a Wien filter independent of their kinetic energies

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Mass Analyzers With Combined Electrostatic and Magnetic Fields

x z

FIGURE 150 Trajectories of ions of three different masses in a Wien filter of Figure 149a. The second-order angular aberration and the inclination of the mass focal line are clearly seen.

and masses. In other words, a Wien filter separates charged particles according to their velocities. For this reason, the Wien filter is often referred to as a velocity filter. In general, velocity filters can be designed not only with superimposed electrostatic and magnetic fields but also with consecutively arranged pure electrostatic and pure magnetic deflectors (Mu¨nzenberg et al., 1979). A Wien filter possesses all aberrations typical of optic systems with one plane of symmetry (see Section 2.3.2): second-order geometric and chromatic ones, and so on. Figure 150 shows an example of focusing and dispersion of charged particles in a Wien filter. Linear optic properties and second-order aberrations of homogeneous and inhomogeneous Wien filters have been studied in details by Ioanoviciu (1973). Using multipole electrostatic and magnetic fields to reduce Wien filter aberrations was considered in a few publications (see, for example, Kato and Tsuno, 1990; Martinez and Tsuno, 2004; Niimi et al., 2005, 2007). It was also shown that the aberrations of the Wien filter can be reduced using a ‘‘double’’ Wien filter with an intermediate image point (Ioanoviciu, Tsuno, and Martinez, 2004; Tsuno, Ioanoviciu, and Martinez, 2005).

7.2.2. Integral Relation for the Rigidity Dispersion in a Wien Filter The method of deriving the general integral relation for the rigidity dispersion (described in Section 2.2.6) cannot be directly applied to systems with a spatial dispersion and infinite curvature radius of the optic axis. However, only minor modifications of the calculation method are required in this case. Assume we have a system in which the electrostatic and the magnetostatic force counterbalance each other at the optic axis, so that this axis remains straight even if each field varies as happens, for example, in the fringing fields of a Wien filter. This means that at each point of this axis, Eq. (7.3) holds with r0 dependent on the position of this point: r0 ¼ r0 ðzÞ. From Eq. (7.7a) we can see that the coordinate and angular deflection of a charged particle from the optic axis due to the

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269

rigidity difference from the rigidity of the reference particle at a small interval ds of length along the optic axis is    pffiffiffiffiffiffiffiffiffiffiffi ds r0 ðsÞ ðdsÞ2 ðd0  gÞ  ðd0  gÞ; (7.9a) 1  cos 1n ðdxÞ ¼ 4r0 ðsÞ 2ð1  nÞ r0 ðsÞ   pffiffiffiffiffiffiffiffiffiffiffi ds 1 ðdsÞ ðd0  gÞ  ðd0  gÞ: ðdaÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi sin 1n r0 ðsÞ 2r0 ðsÞ 2 1n

(7.9b)

Neglecting the squared small value (ds)2 in Eq. (7.9a), we can represent a momentary diversity between the trajectory of a reference particle and a trajectory of an arbitrary particle in the vector form of Eq. (2.45), in which CðsÞ ¼ ðd0  gÞ=½2r0 ðsÞ. From this point on, the consideration of Section 2.2.6 can be applied without any changes, so that finally we come to the relation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðsf ðDxÞ DK Dm d0  g ðxs j aÞ Kðref Þ ðsÞ ds: (7.10) ¼ d0 þ g¼ ðref Þ Mx Mx Mx 2 r0 ðsÞ K s0

0

For a constant kinetic energy at the optic axis in the sharp-cutoff approximation ðr0 ¼ constÞ, we obtain for the energy and mass dispersions of a Wien filter the following relations:

and

DK 1 S ; ¼ Mx 2r0 ðDa0 Þ

(7.11a)

Dm 1 S ; ¼ Mx 2r0 ðDa0 Þ

(7.11b)

  S LE  RK ½ðDx0 ÞðDa0 Þ ¼ Lm  Rm ½ðDx0 ÞðDa0 Þ ¼  2r

0

  ; 

(7.12)

where S is the area illuminated inside the Wien filter field by a beam emitted from a point object with the angular spread ðDa0 Þ. Combining a Wien filter with an electrostatic sector analyzer can compensate for the energy dispersion of the Wien filter by the energy dispersion of the sector field. In this way, one can achieve energy focusing and thus design a double-focusing mass analyzer (Ioanoviciu and Cuna, 1974) with the mass dispersion given by Eq. (7.11b). Alternatively, it is possible to compensate for the energy dispersion of a Wien filter by combining it with a magnetic sector field. In this case, the quality factor of the two-stage spectrometer is a sum of the quality factors of the Wien

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Mass Analyzers With Combined Electrostatic and Magnetic Fields

filter and the magnetic sector field, if the signs of the corresponding area to radius ratios are the same:   ðW Þ S SðMÞ   (7.13) Lm  Rm ½ðDx0 ÞðDa0 Þ ¼  ðW Þ þ ðMÞ ; r0 r0 where the superscripts (W) and (M) correspond to the Wien filter and to the sector magnet, respectively.

7.2.3. Fringing Field Effects in a Wien Filter Fringing field effects in a Wien filter can be investigated using the fringing field integral method of Section 2.4.2. Below we consider in more detail the case of entrance fringing fields; the effects in the exit fringing fields are analogous. We introduce in the fringing field region the normalized coordinates X ¼ x=g0 , Y ¼ y=g0 , and Z ¼ z=g0 , where g0 is some characteristic size of the fringing field region—for example, the half gap between the magnetic poles. It is advantageous to introduce effective boundaries of the electrostatic and of magnetic fields separately. ðEÞ ðMÞ The positions Z0 and Z0 of the entrance effective boundaries can be defined by the conditions that read similar to the condition introduced in Section 2.4.2: ðEÞ Z0

Z ð2

¼ Z2 

eðZÞdZ;

(7.14a)

Z1

ðMÞ Z0

Z ð2

¼ Z2 

bðZÞdZ;

(7.14b)

Z1

where eðZÞ ¼ Eðg0 zÞ=E0 is the electrostatic field strength distribution along the optic axis normalized by the value of this strength well inside the Wien filter, bðZÞ ¼ Bðg0 zÞ=B0 is the magnetic flux density distribution along the optic axis normalized by the value of this flux density well inside the Wien filter, the point Z1 is located in the field-free space outside the filter, and the point Z2 is located well inside the filter where the values E0 and B0 are both achieved. If the field distributions eðZÞ and bðZÞ coincide, the optic axis of a charged particle beam remains straight in the fringing field. However, in practice it is impossible to achieve coincidence of these two distributions, so that the optic axis in the fringing field regions is generally curved in the xz-plane. The fringing field integral method applied to the Wien filter

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271

(Ioanoviciu, 1973) gives the following expression for the total bend Da of the optic axis in the entrance fringing field region in the x-direction: Z ð2

½bðZÞ  eðZÞdZ:

Da ¼

(7.15)

Z1

From Eq. (7.15) it follows that in order for the optic axis to experience no bend in the fringing field of a Wien filter, the positions of the effective boundaries of the electrostatic and magnetic fields must coincide: ðEÞ ðMÞ Z0 ¼ Z0 . In this case, the parallel shift Dx of the optic axis in the field-free space and inside the Wien filter is given by the formula Z ð2 ðZ

½eðzÞ  bðzÞdB dZ;

Dx ¼ e r0 2

(7.16)

Z 1 Z1

where e ¼ g0 =r0 . Similar to the case of an electrostatic sector deflector, the fringing fields of the Wien filter act as weak lenses defocusing the charged particle beam in the x-direction. In the entrance fringing field of the filter with coinciding effective boundary positions for electrostatic and magnetic fields this effect, concentrated at the effective boundary, can be expressed as the following relation between the angular parameters aðÞ and aðþÞ of the effective trajectories at the outer and inner sides of the effective boundary: 2 3 Z Z ð2 ð2 16 7 aðþÞ ¼ aðÞ þ e 42 eðZÞ½1  eðZÞdZ þ eðZÞbðZÞdZ  Z2 5xðÞ : (7.17) r0 Z1

Z1

The defocusing effect is generally weaker than in an electrostatic sector with the same values of the parameters E and r0 and the same distribution of the function eðzÞ. In particular, in case of coinciding functions eðzÞ and bðzÞ, Eq. (7.17) shows that the optical power of the fringing field lens is twice as small as it is in the electrostatic deflector [see Eq. (2.100b)].

7.3. PENNING TRAPS 7.3.1. Fourier Transform Mass Detection Mass analyzers that separate ions spatially or in time can typically reach maximal mass resolving power values of several hundred thousands. The limit is set by both optical reasons (spatial or TOF aberrations) and

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Mass Analyzers With Combined Electrostatic and Magnetic Fields

electronic noise due to power supply instabilities. An alternative way to determine an ion mass is to measure the so-called cyclotron frequency of its rotation in a homogeneous magnetic field. According to Eq. (1.38), the angular frequency oc of the circular ion motion in the plane perpendicular to the magnetic flux density vector B of the magnetic field ðoc ¼ 2pfc , where fc is the cyclotron frequency of ion rotation) is v QB oc ¼ ¼ ; (7.18a) r m where v, m, and Q are the ion velocity, mass, and charge, respectively, and r is the radius of rotation. In SI units, Eq. (7.18a) can be rewritten in the form fc ½MHz ffi 15:356

ZB½T ; m½a:m:u:

(7.18b)

where Z is the ion charge number. So, for example, in the field with 7 T magnetic flux density, a singly charged ion of the mass m ¼ 1000 a.m.u. rotates with the frequency of fc  107.5 kHz. Note that the cyclotron frequency does not depend on the ion velocity and radius of motion but is defined only by the ion mass to charge ratio and the magnetic field strength. The simplest way to measure the cyclotron frequency is to detect a current originated from mirror charges induced on a pair of plates placed on both sides of the region of the ion motion. For a single ion, this current is harmonic with the frequency fc. If several ions of different masses are present, the signal will be a superposition of harmonic functions with different frequencies. Thus, one can define the mass spectrum of ions by performing the Fourier transformation of the detected current. Note that detection of the induced current requires that ions of each mass move coherently, in a well-confined bunch. The attractive feature of measurement of the cyclotron frequency is that this measurement can be done quite precisely, if the time of detection is long enough. In general, the precision of the frequency (and thus the mass) measurement is proportional to the number of cycles ions make in the field. This means that for a fixed detection time, the precision of the mass measurement is inversely proportional to the ion mass, since the heavier the ion is, the less turns it makes during this time. For the detection time of 1 s in the field with B ¼ 7 T the mass resolving power for the mass 100 a.m.u. can exceed 107. Note that the higher the field, the larger is the cyclotron frequency and thus the higher resolving power can be reached for the same detection time. That is one reason why strong magnetic fields, created by superconducting solenoid magnets, are typically used in Fourier transform (FT) mass spectrometers, which are also called ion cyclotron resonance (ICR) spectrometers.

Mass Analyzers With Combined Electrostatic and Magnetic Fields

273

In principle, detection of the frequency of a cyclic ion motion also can be used for mass measurements in electrostatic fields. Simple electrostatic ion traps, in which ions oscillate between two mirrors (Benner, 1997; Rockwood, 1999; Schmidt et al., 2001) are not generally suitable for this purpose, because ion bunches gradually disperse in time. Although a well-confined ion motion can be observed in such traps (Pedersen et al., 2002), it is caused by space charge effects and is thus dependent on the number of ions in bunches. The only electrostatic device successfully used up to now for FT-ICR mass spectrometry is the Orbitrap mass analyzer (Makarov 2000; Hu et al., 2005). This analyzer uses the ion motion in the hyperlogarithmic potential of Eq. (8.25), created between two rotationally symmetric electrodes (see Figure 140 for the electrode shape). This motion is harmonic in the direction z of the rotational symmetry axis, and the frequency of this harmonic motion does not depend on the ion kinetic energy or on the direction of motion. However, the mass-resolving power of Orbitrap, though exceeding 200,000, is limited, in particular by instabilities of electrostatic power supplies and by imperfections of the field structure caused by complex electrode geometry. From the point of view of accuracy and stability, magnetic fields created by superconducting solenoid magnets have definite advantages compared with electrostatic fields. The main reason is that the electric current flowing in superconducting coils is closed in the loop and is practically not influenced by external power supplies during mass measurements. The relative inhomogeneity and instability of the magnetic field of a superconducting solenoid in the volume of 1 cm3 during the time period of 1 s is typically less than 1012. That is why the FT method became an attractive tool for extremely high-precision mass measurements in nuclear physics (Kluge and Bollen, 1992), although since the appearance of the first FT-ICR mass spectrometer (Comisarow and Marshall, 1974) this tool has also become extensively used in analytical chemistry. The principles and various aspects of operation of FT-ICR mass spectrometers are very well covered in details in the up-to-date literature (see Brown and Gabrielse, 1986; Ghosh, 1995; Marshall, Hendrickson, and Jackson, 1998; and references therein). For this reason, only the main ion-optical features of FT mass analyzers are briefly reviewed below.

7.3.2. Ion Motion in Penning Traps The ion motion in a homogeneous magnetic field is confined radially, in the xy-plane perpendicular to the direction of the magnetic flux density vector, but not in the axial direction z of this vector. In order to keep ions moving in a restricted volume long enough, a weak electrostatic field is added to the homogeneous magnetic field. The idea of trapping ions in a combination of a homogeneous magnetic field and an electrostatic field

274

Mass Analyzers With Combined Electrostatic and Magnetic Fields

(a) B

(b) V0

z

z

z0 –V0

–V0

V0

r0 z

Cap electrode

r

r

V0

Vcomp –V0 Vcomp

V0

Ring electrode

Ion trajectories

FIGURE 151 (a) Section through a meridianal plane and 3D view (with a part of the pffiffiffi electrodes cut out) of a Penning trap with hyperbolic electrodes. In this trap z0 ¼ r0 = 2. Narrow holes in the cap electrodes are made for ion injection and ejection. (b) Coaxial cylindrical electrodes of an ion cyclotron resonant cell (ICR) with the ‘‘open’’ geometry. Compensation electrodes with the potentials Vcomp serve to achieve the optimal approximation of the hyperbolic field in the central region of the trap.

was first used by Penning (1937). The rotationally symmetric electrostatic field creating a potential well in the z-direction is the 3D quadrupole field with the potential   2V0 r2 ; (7.19) Uðr; zÞ ¼ 2 z2  2 r0 where r2 ¼ x2 þ y2 is the radial coordinate in the xy-plane, r0 is the radial dimension of the trap, and –V0 is the electrostatic potential at the point {r ¼ r0, z ¼ 0}. The 3D quadrupole field is formed by a set of three rotationally symmetric hyperbolic electrodes, the ring one and two cap electrodes as shown in Figure 151a. The electrostatic potential applied to

Mass Analyzers With Combined Electrostatic and Magnetic Fields

275

pffiffiffi the cap electrodes (with the tip points {r ¼ 0, z ¼ z0 ¼ r0 = 2}) has the same value V0 but the opposite sign compared with the potential at the ring electrode. In practice, the distribution of Eq. (7.19) is usually shifted by the value V0, so that the ring electrode is grounded and the voltage 2V0 is applied to both cap electrodes. Technically this simplifies use of the ring electrode for measurement of induced charges and for excitation of the ion motion in the trap (see Section 7.3.3). Although the first FT-ICR traps (also called FT-ICR cells) used hyperbolic electrodes (actually, the name Penning trap in chemical literature now refers exclusively to the traps with hyperbolic electrodes, although in nuclear physics all FT-ICR cells are called ‘‘Penning traps’’), it was soon understood that the field structure of Eq. (7.19) can be approximated with good accuracy in a small volume by sets of electrode arrangements of much simpler shapes such as cubes or cylinders (Gabrielse and Mackintosh, 1984; Vartanian, Anderson, and Laude, 1995). Actually, even hyperbolic electrodes do not form a perfect hyperbolic field because of manufacturing and truncation errors, as well as the presence of holes for ion injection and ejection, and to improve the field quality one needs to implement auxiliary electrodes to create a compensated Penning trap (Gabrielse, 1983). Now most popular are open trap geometries (with end-caps replaced by large-aperture electrodes, see an example in Figure 151b), which allow easy external ion injection and ejection (Gabrielse, Haarsma, and Rolston, 1989). Various FT-ICR cell shapes are reviewed by Guan and Marshall (1995). In the combination of a homogeneous magnetic field with the flux density B and the electrostatic quadrupole field of Eq. (7.19), the equations of ion motion have the form 1 x¨  oc y  o2z x ¼ 0; 2

(7.20a)

1 y¨ þ oc x  o2z y ¼ 0; 2

(7.20b)

z¨ þ o2z z ¼ 0;

(7.20c)

where 2 oz ¼ 2pfz ¼ r0

rffiffiffiffiffiffiffiffiffiffi QV0 m

(7.21a)

is the axial angular frequency (and fz is the axial frequency). In SI units, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3:1267 ZV0 ½V : (7.21b) fz ½MHz ffi r0 ½mm m½a:m:u

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Mass Analyzers With Combined Electrostatic and Magnetic Fields

In particular, with V0 ¼ 1 V, r0 ¼ 10 mm a singly charged ion of the mass 1000 a.m.u. oscillates with the axial frequency fz  9.9 kHz. Note that the axial frequency of Eqs. (7.21) is usually much lower than the cyclotron frequency of Eqs. (7.18). The axial ion motion, defined by Eq. (7.20c), is independent of the radial ion motion and is expressed as zðtÞ ¼ z0 cosðoz t þ ’z Þ;

(7.22)

where z0 and ’z are constants defined by the initial ion position and velocity at t ¼ 0. The solutions of Eqs. (7.20a) and (7.20b) for the radial ion motion are as follows: xðtÞ ¼ xþ ðtÞ þ x ðtÞ;

(7.23a)

yðtÞ ¼ yþ ðtÞ þ y ðtÞ;

(7.23b)

xþ ðtÞ ¼ rþ cosðoþ t þ ’þ Þ;

(7.24a)

yþ ðtÞ ¼ rþ sinðoþ t þ ’þ Þ;

(7.24b)

x ðtÞ ¼ r cosðo t þ ’ Þ;

(7.25a)

y ðtÞ ¼ r sinðo t þ ’ Þ;

(7.25b)

where

the parameters rþ, r–, ’þ, and ’– are defined by the initial ion position and velocity at t ¼ 0, and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oc o2c o2z þ  ; (7.26a) oþ ¼ 2 4 2 oc o ¼  2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2c o2z  4 2

(7.26b)

Thus, the projection of the ion motion to the xy-plane is a superposition of two rotations: the rotation described by Eqs. (7.24) with the reduced cyclotron frequency fþ ¼ oþ =ð2pÞ, and the rotation described by Eqs. (7.25) with the much lower magnetron frequency f ¼ o =ð2pÞ. Examples of ion trajectories in a Penning trap for two different ratios between the radius rc of the modified cyclotron motion and the radius rm of the magnetron motion are shown in Figure 152. Note that all three fundamental frequencies of the ion motion in a Penning trap are independent of the ion velocity (and of the amplitude of the motion) and are determined only by the ion mass to charge ratio and

Mass Analyzers With Combined Electrostatic and Magnetic Fields

(a)

277

(b) r

r

z

z r

r

FIGURE 152 Radial projection and 3D view of an ion trajectory in case of rc rm (a) and rc  rm (b).

by the field strengths. In particular, even if the amplitude of the magnetron motion is zero, the reduced cyclotron frequency is still smaller compared with the cyclotron frequency fc in a pure magnetic field. This is explained by the fact that while the electrostatic field forms a potential well in the z-direction, it also forms a potential ‘‘hill’’ in the xy-plane with the center at r ¼ 0. Therefore, the electrostatic field creates a force acting outward of the z-axis in the radial direction, and this force resists the magnetic field, which turns the ion toward the optic axis. With increasing the the trapping field potential V0, p ffiffiffi angular frequency oz of the axial motion grows, and with oz ¼ oc = 2 the electrostatic radial force counterbalances the deflecting force of the magnetic field and the ion motion becomes unstable. The latter condition sets the upper limit of the mass to charge ratio for ions stable in the trap:   m B2 r20 ¼ : (7.27) Q max 8V0 Reduction of the cyclotron frequency must be considered in mass measurements, because directly detected by recording the current of the induced mirror charges is the reduced cyclotron frequency of Eq. (7.26a) and not the ‘‘ideal’’ one of Eq. (7.18a). From Eqs. (7.26a) and (7.21a) the following relation is obtained: m B 2V0 ¼  2 2: Q oþ oþ r0

(7.28)

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Mass Analyzers With Combined Electrostatic and Magnetic Fields

In the typical case of a weak electrostatic field in the trap, the relation o oz oþ is valid. Then Eq. (7.26b) can be rewritten as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oc oc o2 o2 2V0  1  z2  z ¼ 2 : (7.29) o ¼ 2 2 2oc 2oc r0 B Thus, whereas the cyclotron frequency is inversely proportional to the ion mass and the axial frequency is inversely proportional to the square root of this mass, the magnetron frequency is almost independent of the ion mass as long as this frequency is much lower than the cyclotron one.

7.3.3. Excitation of Ion Motion in Penning Traps To reliably measure the frequency of the radial ion motion in a Penning trap, ions of one mass must travel as a confined bunch at a sufficiently large radius to produce a recordable mirror charge current at the detector electrodes. To form the required ion bunches, an ion excitation technique is used (Schweikhard and Marshall, 1993). The simplest way of exciting the ion motion is to apply a dipole RF electric field Ex ¼ Exd sinðotÞ to the ions in the trap. In case the frequency of this electric field matches the frequency of the reduced cyclotron motion for some ion mass ðo ¼ oþ Þ, ions of this mass come into the resonance with the electric field. In particular, an ion that was initially in the rest state at the optic axis (x ¼ y ¼ z ¼ 0) moves along a spiral trajectory in the xy-plane with the radius linearly increasing in time. Thus, by applying the excitation during a fixed time interval, it is possible to form a coherent ion bunch with a certain radius of rotation. Typically, the excitation field is created by RF voltages applied in opposite phases to parts of trap ring electrodes (Figure 153a), although this field is not exactly the dipole one. Also, the magnetron ion motion can be excited by applying the electric field in the xy-plane with the angular frequency o ¼ o . The axial motion of ions can be excited if an axial dipole electric field Ez ¼ Ezd sinðoz tÞ with the axial angular frequency oz is created, usually by applying RF voltages in opposite phases to the end-cap electrodes. Dipole excitation affects the cyclotron, magnetron, and axial motions independently. Alternatively, to excite the radial ion motion, a quadrupole field can be created by applying RF voltages of opposite signs to the pairs of trap ring electrodes (Figure 153b). The field with the angular frequency o ¼ 2oþ excites the reduced cyclotron motion, whereas the field with the angular frequency o ¼ 2o excites the magnetron ion motion. Both these excitations are independent. Moreover, it can be shown that by applying a quadrupole excitation with the angular frequency o ¼ oc ¼ oþ þ o of

Mass Analyzers With Combined Electrostatic and Magnetic Fields

(a)

(b)

279

−U cos(w t)

−U cos(w t)

U cos(w t)

−U cos(w t)

U cos(w t)

U cos(w t)

FIGURE 153 Applying of auxiliary harmonic voltages to the sectors of the ring electrode in case of the (a) dipolar and (b) quadrupolar excitations of the ion motion.

the cyclotron motion it is possible to influence both cyclotron and magnetron motions. More precisely, under the influence of the considered quadrupole excitation the energy of the magnetron motion is periodically transformed to the energy of the cyclotron motion and back. As mentioned above, excitation of the reduced cyclotron motion is mass selective. When it is necessary to analyze simultaneously ions in a wide range of masses, a broadband RF signal is used for excitation, for example by sweeping the frequency of excitation (Guan and Marshall, 1996b). For ions created inside a sufficiently large volume in the trap (say, by the electron impact ionization) or injected into the trap from outside, the ion cloud must be cooled before excitation to reduce the cloud dimensions and to concentrate it around the center of the trap. De-excitation (also called axialization) by applying a RF field in the phase opposite to the phase of excitation is possible only for a well-bunched ion cloud but not for ions distributed randomly in the trap volume. So, the simplest way to cool ions is to let them collide with the buffer gas at low pressure. The mechanism of collisional cooling is the same as used in RF multipole guides (see Section 4.4). There is, however, one special feature of collisional cooling in Penning traps. With decreasing ion kinetic energy the amplitude of the axial motion decreases, because the ion is confined in an electrostatic potential well. The radius of the cyclotron motion also decreases, as follows from Eq. (1.38). However, the magnetron motion, which is actually the motion around the ‘‘potential hill’’ of the electrostatic field in the radial direction, increases its radius with decreasing ion kinetic energy, because the magnetic field force (proportional to the ion velocity) resists the weaker radial electrostatic field force acting

280

Mass Analyzers With Combined Electrostatic and Magnetic Fields

outward of the z-axis of the cell. Thus, ion collisions with a buffer gas suppress ion axial and cyclotron motion but increase the radius of the magnetron rotation. In order to prevent this effect, collisional cooling is used together with quadrupole excitation of ions at the cyclotron angular frequency oc (Savard et al., 1991; Guan and Marshall, 1993). The quadrupole excitation transmits the energy of the magnetron motion into the energy of the cyclotron motion, and collisional cooling cools this cyclotron motion (as well as the axial motion). As a result, the overall radial amplitude of ion motion decreases. Note that after the ions in the trap are cooled and concentrated in the vicinity of the z-axis, excitation by a broadband electric signal, in which frequencies in the vicinity of some specific one are absent, allows the excitation of the radial motion of all ions except those of a specific mass and even the removal of these ions from the trap by increasing their amplitudes of motion to the values exceeding the radial dimension of the trap, leaving only the ions of a selected mass inside the trap. This process is called ion isolation. Isolated ions can be then extracted from the trap—for example, through a hole in the end-cap electrode by applying an electric pulse to this electrode.

7.3.4. Ion Injection into Penning Traps Although analyzed ions can be produced inside the Penning traps (e.g., with ionization of gas molecules by an electron beam or a laser shot), most mass spectrometers based on Penning traps use external ion sources. Delivery of ions from a source to the trap is performed along the axis of rotational symmetry of the solenoid magnetic field. The fringing field of a superconducting solenoid magnet forms a very strong nonlinear lens that can even reject ions injected into the trap with offsets from the solenoid axis, the effect most noticeable for ions with small mass to charge ratios. To prevent ion losses, it is advantageous to focus the injected ion beam onto the region of the magnet fringing field. Even more efficient is using RF ion guides (McIver, 1985; Nikolaev and Franzen, 2008) extended through the magnetic fringing field and confining ions in the vicinity of the solenoid axis in the fringing field region. Initial trapping of injected ions inside the trap volume can be achieved in several ways. First, ions can be injected into the trap through a hole, preferably made in the end-cap electrode. Because the trapping fields are static, ions can be injected with very low energies (of only a few electronvolts) because they do not need to overcome a pseudopotential barrier as in case of 3D Paul traps (see Section 9.3.2). However, even in this case ions will not be trapped without forcing them to lose their energy inside the trap. Trapping of externally injected ions can be accomplished by filling the trap with a buffer gas (at the pressure of several millitorrs), so that ions

Mass Analyzers With Combined Electrostatic and Magnetic Fields

281

injected into the trap cannot, after several collisions with gas molecules, escape the potential well in the axial direction. With this method it is possible to trap ions from a continuous beam. Alternatively, the potentials at the trap electrodes can be switched to create a potential well in which ions are trapped. This approach to capturing ions requires prebunching of the ion beam to be injected into the trap. These two methods can be combined to arrange a two-stage Penning trap (Schnatz et al., 1986). The first trap is gas filled and traps the entire set of externally created ions, with their subsequent cooling and isolation of the ions of a desired mass. The latter ions are then extracted through a differential pumping system into the second vacuum Penning trap, where ions are captured by switching electrostatic potentials at the electrodes.