InternationalJOUrnalof Mass Spectrometry and Icn Physics
ElsevierPubIishing Company,Amsterdam - Printedin theNetherlands
NON-LINEAR
RESONANCES
ETERS DUE TO IMPERFECT
P. H. DAWSON
AND
Ku’. R.
IN
QUADRUPOLE
MASS
FIELDS I. THE QUADRUPOLE
45
SPJZTROMION TRAP
WI3EI-IEN
General Eiectric Research and Dertlopment Center, Schenectady, N. Y. (U.S.A.)
(ReceivedAugust
9th,
1968)
ABSTRACT
Peak-shape distortions and peak-splitting are comn~only observed phenomena with all members of the q-uadrupole family of mass spectrometers. Errors in electrode shape, or spacing, or harmonics in the r-f_ field may produce non-linear resonances. The resonances result in peak splitting by causing some ion trajectories to become unstabIe that etherwise, in a perfect field, would be stabfe and limited in amplitude. A theory of non-linear resonances in quadrupole fields was applied to the three-dimensionai rotationally symmetric quadrupoie ion trap to predict resonance locations in the MaSeu stability diagram. Numerical computation of trajectories was used to derive peak shapes for various antounts of distortion and to determine the exterlt of peak-splitting. The calculations were used to relate the performance to the spacing errors in assembling the electrode structure. The occurrence of resonances, in agreement with this theory, was experimentally demonstrated. A method is presented for obtaining good peak &apes in spite of the presence of field distor’,rons, but with some loss in sensitivity. This method was investigated both experimentally and theoretically_
IXTl-RODLJC-I-ION
Yen Busch and Paul’ have discussed the eifects of fidd errors in the quadrupole mass filter. They used analytical methods to predict the occurrence of nonlinear resonances. The resonances cause ion paths to be unstable under some conditions where, in a perfect field, the pat& would be stable and limited in amplitude. It was possible to determine the positions in the Mathieu a-q diagram where resonance “lines” occur for different types of field faults, but not to relate -the extent of field distortion to the device performance. Von Busch and Paul gave someexperimental data for a quadrupole 3 meter in length to support their findings. In the present work, numerical computations of ion trajedtorieswere carried out, making it possrble to de&r&e both the position and extent of the non-linear J. Mas.~ Spectrome~
and Ion Physics, 2 (1969)45-59
P. H. DAWSOX,
46
N. R. WHETTEN
resonances for particular kinds of field distortion. Results of general validity are obtained for estimating the degree of field perfection required for a particular performance. This pe’sfz (I) extends the analytical treatment to the three-dimensional quadrupole ion trap 2--5_Resufts of the numerica computations are presented and the theory is compared \withexperiment. A method is suggestLd for avoiding the effects cf resonances, although sacrficing some sensitivity, even if the field is imperfe& A later paper (Part XI) will concern the application of the numerical compu’r;rtion of trajectories to field errors in quadrupole mass filters and monopole mass spectrometers.
THEORY
In this section, a summary of the theory of resonances in quadrupole fields is presenkd. A more detailed description is given in the L4ppendix. The potential in a perfect rotationally symmetric three-dimcnsionai quadrupole field can be expressed in the form @ = $-$r’
-22)(U+
YCOSG r)
where zo is haif the minimum distance between the end caps, t and r are coordinates along and perpendicular to the z-axis, and U and V are the d-c. and r-f. potentials applied to the ring electrode with respect to the end caps (Fig. 1). References 2, 3, 5, and 6 discuss ion motion in perfect quadrupo!e fields.
fig. 1. Ma*&eu diagram for the rhree4metiond quadrupole ion trap. The shaded area is the region of stable (bounded) ion ua+toties, iu the absence of resonances. T@e resonance instability lhcs arc shown for the w of thkd- and fourth-order distortions of the field. The insert shows the pmand defines the coodi~~cs of the ion trap. The hyperbolic surf2ces ue rotatio&y symsetric 2bouttheZaxis. J- Mass Specrrotne~
and Ion Physks- 2 (1969) 45-59
THE QUADRUPOLE
47
ION TRAP
Third-crder distortions
in this section, harmonics in the r.f. vcitage are not considered. We can then express disto,rtions in the field as a series with both second- and higher-order terms in the expression fcr @_ Only terms +Gth rotational symmetry are retained! The third-order term corresponding to a three-Gmensional hexapole is of the form @,
= -& =o
$
(3r”z-223)(Uf
vcosa
t)
A represents the relative amount of distortion or distortion weighting factor. The coefhcients in this equation have been chosen to satisfy Laplace’s equation. This distortion is asymmetric. For example, an error in the spacing of one cap electrode with respect to the ring introduces a large third-order distortion. Resonan-:e lines in the Mathieu a-q stabiiity diagram resulting from thirdorder distortions would be expected to occur aIon% the lines /3,= 2/3 and /%++I% = 1. These lines are shown in Fig. 1, with the shaded area repesentin; the conditions for stsbIe solutions to the Mathieu equations of motion. The sum resonances occur because of a coupling of motion in the r and z directions. Fourth-order distortions %I GIariy, the fourth-order (octopole) A’ (r”t2z4-88rzZz)(Ut 4z02 zoz
@4 = -
1
distortion term is given by: V cosS2 ;)
-4’ is the fcurth-order distortion weighting factor. The fourth-order distortion is symmetric. Fourth-order resonance lines xcur at j?, = 3, &+j3, = 1, and j3= = 8_ These resonance Iines aIso are shown in Fig. 1. A symmetrical error in the spacing of the two end caps would introduce a large fourth-order distortion. Since resonance lines are found to intersect the apex of the stabiIity diagram, it would be expected that peak splitting could result when the mass scan line passes throu& this apex. Peak splitting in the quadrupole ion trap resulting from these resonances may be more pronounced than in the mass filter, because with ion storage operation the number of cycles an ion spends in the field may be considerably greater.
TRAJECTORY
COMF’L-TATIOX
Method and ussrrnptions The third-order distortion, as expressed by the equations of motion (1) and (2) in the appendix, was treatd in detail. Trajectory computations were made bv numerical integration as described previously3, but in this case simultaneous J. Mols Spectramerry and Ion Phyzic& 2 (1969) 45 -59
48
P. H. DAWSOS,
N. R. WHETTEN
computatians for both t and z directions were necessary. It was ako necessary to assume initial conditions for both r and z directions, and stability criteria for both directions. The assumptions made were = 0.001 m
‘inicti dr cZ
ziaith:
0.001m
=
dz
)
initial
=
O
(z > =
0
initial
Ions were assumed lost if their amplitude exceeded 0.01 m in either the T or z direction. A further simpiikation in computation was to consider ions formed at only one phase of the r-f_ field (ref. 3 discusses ^Lhe si,@ficance of these as-
--II I I I T i
I
T
I f
Fw2. Computed peak shapes for an ion storage t&c of 200/z field cycles in the presence of a tbird-adcrdisto~nwith aweightingfactorof A = lOzd3. The peak shaps are for the folbwing q/u scan lines: (a) Z-O=, Cb) 2.0, tc) 1.9875. Cd) 1975,
SWonwjp
ad
Ion Pl~ysks, 2 (1969) 45-59
THE QUADRUPOLE
49
ION TRAP
In a pure quadrupole calculation everything can be staled. Doubling the initial displacement doubles the trajectory amplitude at all points. This is not true when distortions are present. However, a calculation for a 0.001 M initial displacement and a 0.01 M limit will apply for a 0.002 M displacement and a 0.02 M Emit in the presence of distortions if the distortion weighting factor A is held constant and the size of the device, given by zO, is doubled. t
I
v-
L
I
1
I
II
.
,
I
1.20
I.25
qz
T i i
-r i
, US
jd
i I&
ii0 %
-
?Fig.3. Computed peak shapes for an ion storage time of 2oL)ljrfield cycles with a third-order distortiotkweighting factor A =4&3. The q/a scan lines are @I i-975, fb) 1.35, (c) 1.925,and (d) 1.90. J. Muss Spectrometnyand Ion Physics, 2 (1969) 45-59
so
P. H. DA wVSON, N. R. WHETTEN
For the examin ation of peak shapes, the maximum ion storage time within the field was taken as 2OOjlr cycles. By computing ion trajectories for successive points along a number of (a, q) scan lines, the effects of various degrees of distortion at different mass resolutions were investigated. Peak shapes Figure 2(a) to (f) illustrates the peak shapes computed for a series of mass, approaching successively closer to the tip of the stability diagram. A was taken equal to XOq,/3. This is a large distortion as will be illustrated later. Resonance dips occur when some ions exceed the stability limits within this storage time of 200.15~ cycles. Severrrf features are of interest. The resonance dip becomes greater the cioser the scan line is to the stability apex- that is, the effects are worse at high resolution. Apart from the major dip evident at Iow resohttion, compfex multiple peak splitting occurs at higher resolution. Figure 3(a) to (d) shows peak shapes for a distortion with a weighting (A) of 4zJ3. The effects ar3 Iess severe and the device can be operated at lower qja values than before. IMultiple splitting is less important. At lligh resolutions the resonance appears as a shoulder on the high-mass side_ scan
iines
-0-6’ r
-0-7
1.0
I 1.1
I 1.2
1.3 % Fig. 4. The tip of the a-q stabi!ity dkligramcomputed for various amounts of third-order distortion. The stabiiity boundaries aretkoseat half maximum peak height. Line (a) is the stxbility boundary with no distortion; (b) correspondsto a distortion -whereA = l&,/3; (c) A = 41~13; (d) A = IOz,J3. Resonance ties Cc’) and (d3 are obtained from the computedpeak shapesof Figs. 2 and 3. The width of the resonancedips is indicatedby the cross-batchingof the resonan= lines for the variousscau ties. J. Ma.ss Sperrromeby
and lo/x Physics,
2 (1%9)
45-59
THE
QUADRUPOLE
Resonance
LON
TRAP
51
lines
The numerically computed stability regions and resonances iliustrated in Figs. 2 a__d 3 can be plotted on the (a, 4) diagram as shown in Fig. 4. Data for a resonance of weight l-62,/3 and for no distortion are alsc included. The stability boundaries are those at hsIf the peak height (IOO/IZcycies). There is a cIose simiIarity to the resonance lines predicted by the theory. However, there is a slight change in the position of the resonance iine with the degree of dlstortion. There is also a narrowing of the stable region due to a shift in the z stability boundary with increasing distortion. Large distortions would make it experimentally difficult to attain high resolution, owing to the change in shape (“blunting”) of the stability tip. Performance
and_fieid perfectiorz
The results described above can be interpreted in terms of the mechanical perfection needed in electrode assembly in the three-dimensional qoadrupole to achieve a given performance. Some potential contours for a case with no distortion, and with the distortion A = lot,/3 are shown superimposed in Fig. 5. The COEtours are for the zero potentiai and for 4@zo”/(U+ V cosD t) = +0.0002. The latter was chosen to approximately conform with the assumption that the maximum allowable amplitude was 0.01 M so that the potential contours can be visual-
Fig. 5. Potential contours throcgh a cross-section of the device. The solid lines are contours with no distortion, and the dashed lines with a thirdsrder distortion present. The contours are far 4@zO’/(Ut Y co&! Z) = 0 and = 0.0002. The third-order distortion has a weighting factor A = lOzd3. J- Mass Spcctiurnetry and Zon PhFsks, 2 (1969) 45-59
B. H. DAWSON,
52
N. R. WHETTEN
ized as the electrodes_ &4iikeiy mechanical error is in the spacing of the end cap electrodes relative to tke ring electrode. An error suck that one cap was closer than the other wouId incIuda a large amount of third-order distortion. A fourth-order distortion result if both of the end-caps are symmetrically mispiaceti with respect to the ring electrode. By examination of Fig. 5 an error in z spacing of about 4.5 *A, for a device of 0.025 m total radius would give rise to the third-order weighting factor 102,/3. The percentage emor is proportional to A, and so approximate estimates can be made of the required accuracy if the maximum A value that can be tolerated for a given performance is available.
Fig. 6. A log=Iog plot of the number of field cycles needed before the stability limit is exceeded at the (0, q) position of the third-order resonance versus the weighting factor. The lines shown are for scan lineswith nominal (withoutdistortion) resolutions of (a) 10, (b) 26, and (CC) 52. Some information on this subject is given in Fig. 6. The number of cycles an ‘3n requires to exceed the stabiiity criteria a’i the minimum of the resonance dip has been plotted on a log-log scale against the weighting factor A_ The three lines correspond to tkree different scan lines of differing nominal (i.e. with no distortion) resoIution. For practical purposes the straight lines are sufficiently good appro_ximations to use for extrapoiation and interpolation. The scatter of points for tke high-resolution line is probably due to the difficuIty of determining the dip when it is at the edge of tke peak_ The position of the lines in the A direction for a given number of cycles in the field is very rougkly proportionai to tke nominal resolution. This figure therefore provides the means for estimating the permissible tolerances in mechauical spacing for a giben resolution and storage time if no resonance dips are to be present in the mass pea&. For example, for a resolution J_ Mii
Soeczmmct7y and Ior: Physfcs, 2 (1969) 45-59
THE
QUADRUPOLE
ION
TRAP
53
of about 50 and a storage time of NO/n cycles the spacing error must be Iess than about 0.35 %, or 8 x 10S5 m for the device described above. Higher resolutions require smaller -x-rors as wodd longer storage times. High resolution requires sufficient storage time to sort the ions. In the mass filter, for example ~/AEz N(43.5)’ where n is the number of r-f. cycles the ion is in the field. Consequently the attainment of very high resolution is a garticuiarly demanding requirement. However, the iact that the resonance dip occurs on the shoulder of the peak at high resolution, may make some third-order distortion tolerable.
EXPEFUME1UiAL
OBSERVATION
OF FiJ%ONA?4CES
Dawson and Whetten have reported peak shapes with multiple spIitting ul z three-dimensional quadrupole device. The results reported here were obtained with similar instruments. The same resonances have been found with severai units of two different sizes, one with z. = 8 x 10m3 m and the other9 with z. = 4 x 10e3 M. The effects seemed more severe in the smaller size.
Peak shapes Figure ‘7 shows a series of m/e = 28 peak shapes for the smaller device as the scan line was changed to increase the resolution. The resonance dips were large and very narrow and an extremely slow scan was required to prevent their extent from being masked by the response of the recording system. The ion storage 0
e
b
d
h
Fig. 7, Experimcntdly obxrved peak shapes. The ion storage time was 1W field cycles. The peak was m/e=28 from the residualgases. The carbon monoxide partial pressure was about 10-O torrThe Q/Qmlue decreasesfrom (a) to (h). The resoh~tionat half-height is about 18 for (II). J- Mass Spectromeby ati Ton Pfzysi”, 2 (1969) 43-59
P. H. DAWSON,
54
N. R. WHETTEN
time was 0-i seconds and the frequency of the r-f_ supply was 1 -MHz, so that some ions were required to stay 10’ cycks within the field. There are three major resonances at low resohrtion, and some minor dips are ako evident.
2Zesonmrcelines The digs and the boundaries of the peaks in Fig. 7 have been plotted on a 1Mathieu diagram in Fig. 8. The coordinates of the diagram are in experimental units r&ted to the applied r-F. and d-c. vohages, but the scale can be judged from the separation of the Tao z stability boundaries. The resonances consider& major
Fig. 8. Resonance lines in the (0, q) diagram derived from Fig. 7. The major resonances are plotted as solid circks, the &or resonancesas OFCYI circles.The SC&Sare ia arbitrary units, but can be estimated from the seDaration of the z St2Mity line. The major resonance lines dppear to correspond well with those predicted in Fig. 1, for 8, = 0.67; pv,+/3, = 1; and fl,+/?J2 = 1. have been plotted as solid points, the minor ones as open circles- The major re~~orznces occur as resonance lines. The third-order lines &t+& = 1 and and the fourth-order line &-I-#?= = 1 all appear to be present_ The /?== 8, = 5, + lint, if present, seems to be less important_ These data are mainly for resolutions lower than those considered in the section “Trajectory computatiorl” but there is seen to be good correlation between the theory, *thecomputer simulation, and the experiment_ Sources offield
error
It was generally assumed in the preceding section that the source of the fieId error was likely to be a mis-spacing of the electrodes. This seems to be the 3. Mass S@ectrometry rurdion Ptr)-sks,2 (1969)
45-59
THE QUADRUPOLE
55
ION TRAP
most !ikely systematic error to produce the strong resonance lines observed expetimentaliy, particularly since the same lines have been observed with a number of different devices. However, other errors occur and some have been experimental?y investigated._ The holes in the electrodes. particularly in the drawout cap, may produce distortions. This is under investigation, but so far there has been no observed correlation between size and number of drawout holes and the peak splitting4. The presence of charging on the ceramic spacers may distort the fields, but again4 diSerent spacing techniques have nr,t altered the resonance effects. The presence of the electron beam in the tra;; distorts the Geld and is another possible but unproven source of error. Finally, two different r.f. supplies at 1 MHz and 1.8 MHz have been used. The latter used frequency doubling’ while the former did not. There was no si,ticant difference between the two, ahhough the /I = + positions were not closely examined.
BIAS, PE.4K SHAPE, AX3
PERFORMANCE
Dawson and Whetten discovered empirically that the application of a small bias voltage between the two end-cap electrodes, usually chosen as a few per cent of the bc. voltage between the ring and the caps, gave a much better peak shape. This was, however, accompanied by some loss of sensitivity. A tentative explanation was offered in ternzs of non-linear resonances and a shift in c
RESC!NANCE LINES
\
Fig. 9. Experimental observations on the influence of a bias voltage between the end-caps on the stability region and the resonances. The boundaries a, b, and c are those found for 0, 2-5, and 10 y0 bias. The corresponding resonance positions are not great!y changed- The scales are in arbitrary units. For 10 % bias, (c), the apex of thestability diagram is seen to be well removed from the corresponding resonance lines. 1. Mass Spectromefry and Jon P~sks,
2 (1969) 45-59
P. H_ DXWSOK,
56
N. R. WHETTEN
the z stability bo*undarycaused by the bias voltage. The stability tip was thought to be moved away from the region where the resonances occur. This technique for accepting the presence of distortions in the field and still achieving good peak shapes has been investigated further.
The results were obtained with the z. = 8 x 10m3 nr device operating with the I.3 MHz supply. Fi_eure9 shows the stability boundaries and the resonance dips for 0,2.5 znd 10 % bias. The bias produces a shift in the /I= = 0 stability boundary to higher Q values and very little shift in the other boundary or in the position of the resonance dips. By applying sufficient bias a region of operation at the stability tip free from major resonance effects can be obtained. Less bias is required the higher the chosen resolution. Pteoreticid
The use of the bias voltage improved the peak shapes but reduced the sensitivity. This biassing effect was e.xamined by the computation of trajectories. The caiculated stability boundar-es for various percentages of bias are shown in Fig. 10. Cn.Iy the ions formed at the zero phase (see ref. 3) of the r-f. field were considered and only ALhz & = 0 stability boundary_ The boundary shift observed experimcnta.Uyis predicted but there ti also a broadening of the boundary, its position deI
t
d
r STAi3lLlTY LlYlTS GT,Pr?ASE
. \
AT ION
FORMATIONI
/' IOPHASE qTFORMATlON) 1
I
1.24
I
1.26
%-
I
128
Compurcd~ationsbip between thzbii voItageandthestability tip.The amountsof bias arc (a) 43) I %. (c) 2.5 %, (d) 5 ye. The three lines in each case from left to right, are for ions forixd at 10-i M, 0 and - lW” JR above the center point of the device towards the drawout cap Fig. 10.
or ekcfmde.
The I lir5ils have bwz dexriibcd cariict.
b. Muss Specrroinezry
andlon
PlluJic-r, 2 (1969)
45-59
THE
QUADRUPOLE
ION
57
TRAP
pending where the ion was initially formed within the device. Considering onIy the possible positions of ion formation as a factor in determining sensitivity and resolution and assuming ali tiitial values of r but only initial va’rclesup to lo- ’ m for z are possible (as in Fig. 12 (b) of ref. 3), one can make some assessment of the relative loss in sensitivity due to use of the bias. Fig. 11 shows some sensitivity versus resolution curves derived on t&s basis. It must be eimphasizcd that these curves describe effects due to the spread in position of ion formation only. Another IO<
I
I
I
i
2bo
i
RECOLUTIOR
I
I
I
I I 600 AT HALF HEIGHT
4th
I
I
I 800
Fig. 11. Computation of peak height vs. sensitivity to show the decrease in sensitivity when a bias voltage is used. These results are for cornparing different amounts of bias only. A major loss in sensitivity with resolution is due to the field phase? which has not been considered here. Only the spread in position of ion formation has been considered.
factor in the loss in sensitivity at increased resolution is the loss of ions formed at different phases of the r-f. field. However, Fig. I1 gives a qualitative picture of the likely deterioration in sensitivity if it is necessary to use a bias between the endcap electrodes.
APPENDIX
Following expressed as:
von Busch and Paul’,
the potentiai
in the mass filter may be
J. Mass Spectrometry and Ion Physics, 2 (1969) 45-59
58
I’. H. DAWSON,
E;. R. WHETTEN
where r and rp are polar coordinates, U and Y are the d.c. and r-f. potentials, and fb is the angular r-f. frequent:+_ For the ideal quadrupole field, X = n = 2. Terms with n > 2 represent harmonics in the r-f_ voltage. Terms with N > 2 represent mecha.nisal errors in the quaclrupole field geometry. Only slight deviations from a perfect field are considered, so that a,- << a2 _ b,, << b,, and terms with both N and n f 2 can be neglected. The ideal quadrupole field, expressed in Cartesian coordinates, is then, taking a, = I: @ = -$
(_J?-J-‘)[U+b,VcosQ
(r-fZ)]
E -$
(2-yZ)[P]
d&ining P by this equation_ The third-order term with N = 3, n = 2 represents a p-ure hexapole field
In Cartesian coordinates, considerkg
only one of the two
possibIe hexapoies, it i-ontributes to the potential a distortion term
Lsplace’s equation is stiI1 obeyed_ The higher-order terms (N f nances give2 by
2, II- 2) in the potential produce su 1 reso-
where K can have the values AT,N-2, N-4 _ _ _ _ Since the resonance lines of the form /I,+ (N-2)@, pass through the a?ex of *be stability triangle of the mass fitter, they can result in “doublets” mass lines (or peak splitting) in the mass spectrum_ The extent of this splitting will be discussed in the second paper (Part II) A three-dimensional rotationally symmetric quadrupole field is used in the quadrupole ion trap’ and the mass spectrometric devices related to it3*4*a. If
perfect, the potentiai with respect to the center is
Here ti and Y are the potentiak applied to the rin g electrode_ The caps are grounded_ 0zm1.I~ distortions which retain the rotational
symmetry wilI be considered so that x and J directions are still equiva!ent_ Terms with n # 2 will not be included; that is, onIy geometrical errors are included_ The third-order distortion (threedimensional he.xapoIe) and fourth-order distortion terms have been given in section II, along with the corresponding resonance lines which they produce. The sum resonances correspond to a coupling of the motion in the r and z directions The equation of motion in the r direction for tie third-order distortion komes L Mffzs Spcctromutry anfflon PAysirs, 2 (1%9) G-59
THE QUADRUPOLE
d'r
hE+f(%
iON
TRAP
f 2q, cos2<)(r+-
59 3A zo
rz) = 0
(1)
where C = $SZt. For :he z direction d%
~i(a=f2q
cos2<)(z--
3A 4Z,
wherea= = -24
4= = -2s;
r 2L55zz) .
= 0
4ZO
and a, q are the constants in the Mathieu equation.
REFERENCES 1 2 3 4
5 6 7 8 9
F. vos Busai XSD W. PAUL, 2. Physik, 164 (1951) X38. E. FISCHER,2. Physik, 156 (1959) 26. P. H. DA-X AWD N. R. WHEITEV J. Vacuum Sci. Teclinoi., 5 (1965) 1. P. H. D~wsos ASD N. R_ WHETIT_i’ , J. Vacuum Sri. Technol., 5 (1968) 11. \v.PxU-.,H.P. &IQlARD A?CC U.voS ZAHS, Z. i’/zysik, 152 (19%)143. R. F. LEVER, IBM J. Rex Derelop.. 10 (1966) 26. N. W. MCLACE~E;, Theory and Application of Mathieu Functions, Oxford Univ. Press, London, 1951. G. brr'isc~~~~, Z. Angew. P&z, 22 (1967) 321. P. H. DAV.%S am N_ R. W -, to be published. J. Mass Spectrometry and ion Physics, 2 (1969) 45-59