EP0459602B1 - Mass spectrometric high-frequency quadrupole cage with superposed multipole fields - Google Patents

Description

Mass spectrometric high-frequency quadrupole cage with superimposed multipole fields

The invention relates to an ion cage mass spectrometer, a quistor, an ion trap or the like according to the preamble of patent claims 1 and 6.

From DE-PS 944 900 a mass spectrometer is known in which the electrodes are arranged so that the surfaces of the ring electrode and the end cap electrodes form a one-piece rotary hyperboloid or a two-part rotary hyperboloid, the end cap electrodes being conductively connected to one another and between the ring electrode and a time-varying voltage is applied to the end cap electrodes. If there is a potential U + V between the ring electrode and the end cap electrodes ^. sin (ωt) is generated, ions remain whose specific charge e / m lies in a certain range between the electrodes, while the others hit the electrodes. The superimposition of DC and radio frequency fields in such mass spectrometers is called a quadrupole memory field. To a good approximation, the ion movement forms a spatial overlay of two independent harmonic oscillators. The forces of the storage field, which acts on the ions, oscillate in the ion cage formed in this way. The force integrated over half of the so-called secular period approximately fulfills the condition of a harmonic oscillator, so that such a system is also called a pseudo-harmonic oscillator. Two such pseudoharmonic oscillator systems form the aforementioned ion cage, which is also referred to as a quistor or ion trap (for terminology: Dawson, "Quadrupole Mass Spectrometry", Elsevier, Amsterdam, 1976; Mahrs / Hughes "Quadrupole Storage Mass Spectrometry", John Wiley & Sons, New York, 1989). The two pseudoharmonic oscillator systems of the quistor consist of a cylinder-symmetrical system, which shows the same behavior regardless of the coordinate in the direction of the cylinder axis (z-axis), and a plane system, the behavior of which is independent of the distance r from the cylinder axis.

In both pseudo-harmonic oscillator systems, ie in the r and z directions, the ions vibrate with so-called "secular frequencies", which are completely independent of one another. The secular frequencies can be determined using known formulas. Since the secular frequencies in the r and z directions and the storage frequency only have a common divisor in rare situations, the movement patterns of the ions are usually very complicated.

An ion cage can be used as a mass spectrometer. The well-known basic principle of mass spectrometry consists in the proportions of ions with different masses relative to one another ascertain. For this purpose, so-called scan methods are used, which carry out the measurement of the different types of ions one after the other by varying measurement or filter conditions. Various scanning methods are known for the ion cage.

At this point, however, only the method of mass-selective ejection of the ions from the cage is interesting. For this purpose, the ions of successive masses are ejected sequentially in time from the cage and fed to a detection system, so that the measurement signals of the ions can be processed to a mass spectrum in a known manner.

As previously known, mass-selective ejection can be carried out in three different ways. First, the ions can be ejected by changing the storage conditions in the ion cage so that the ions move mass by mass beyond the edge of the stability range, become unstable, and leave the ion cage ("Mass-selective instability scan", US Pat. No. 4,440 884). Secondly, the secular frequency of successive ion masses can be excited by an external high-frequency voltage so that they absorb kinetic energy in resonance and thus leave the cage ("mass-selective resonance scan by excitation frequency", US Pat. No. 4,736,101). And thirdly, the ions can be introduced into a device-specific non-linear resonance condition in which they absorb kinetic energy and leave the cage ("mass-selective scan by means of non-linear device resonance", US Pat. No. 4,882,484 or EP-A- 0 336 990).

In all applications of the ion cage as a mass spectrometer, it is desirable that the ejection process of non-specific ions takes place as quickly as possible.

A mass spectrometer is already known from US Pat. No. 4,882,484 Generic type known, in which the non-linear resonances of an octupole field superimposed on the quadrupole field are used to accelerate the creation of the mass spectrum. A general reference to the structure and shape of the multipole field overlay of the quadrupole field cannot be found in this document.

The known quadrupole cage can not only be used to identify individually supplied substances on the basis of their primary spectra, but can also be used to identify mixture components by tandem mass spectrometry, whereby daughter ion spectra are generated. First, an ion type, the parent ion, is selected; all other types of ions are removed from the cage. Then the parent ion is fragmented by collision with a gas introduced into the cage. To do this, the parent ion must be accelerated to increase the collision energy above the fragmentation threshold. The simplest way is to excite ion oscillation in the z direction by an alternating voltage between the end cap electrodes which is in resonance with the corresponding secular frequency.

However, the excitation is critical in the known quadrupole cages. In the quadrupole field, the amplitude of the secular motion increases linearly with time, and eventually the ions will collide with the end cap electrodes. Fine tuning is required between a low excitation voltage and a high collision gas density, and a yield of about 30 to 50% of daughter ions can be achieved; the rest of the parent ions are lost.

The invention is therefore based on the object of developing the generic mass spectrometer in such a way that in order to increase the ability and the detection power while further - resolving - accelerating the measurement of the mass spectrum a general rule for the appropriate type of multi-field overlay is given, the ion losses from the spectrometer being reduced by unwanted resonances in use for tandem mass spectrometry and the yield should be increased in the case of shock-induced fragmentation.

This object is achieved by a mass spectrometer of the type mentioned by the features listed in the characterizing part of claim 1. Particularly advantageous embodiments of the invention according to claim 1 are the subject of the dependent claims.

The invention is based on the surprising finding that, with a multipole overlay according to the invention, be it in a mathematically exact description or in accordance with the approximate formula of claim 6, the time smearing of the ejection process can be tightened, thereby accelerating the creation of the mass spectrum. Furthermore, ion losses are reduced and the yield of daughter ions is improved. The superimposition of z-asymmetrical multipole fields intensifies the ejection by the non-linear resonance effects that then occur.

It has been found that it is generally not necessary to superimpose higher order multipole fields than octupole fields on the base quadrupole field, although this is fundamentally possible and is within the spirit of the invention. It should be mentioned that the occurrence of nonlinear resonances and their subsequent effects are described by Fv Busch and W. Paul in the "Zeitschrift für Physik" 164, pages 588 to 594 (1961). It is found here that the non-linear resonances in the mass spectrometer caused by field errors are so weak that they do not impair its functionality, only to split mass lines can lead in the spectrum. Advantageous effects of the non-linear resonances are not recognized, so that it cannot be derived from this document how these could lead to an improvement in the properties of the mass spectrometer.

The surface shape of the electrodes is chosen in the invention so that the effect of the desired multipole field overlay results. In the mathematically exact embodiments of the invention, the exact dimensions of the electrodes are determined by the relative strength A₃ of the sextupole field or the relative strength A₄ of the octupole field in relation to the strength A₂ of the quadrupole field. The strengths of the sextupole field or the octupole field in relation to the quadrupole field can be between approximately 0% and 20%, it being particularly advantageous if the proportion of the overlaid fields is between 0.5% and 4.5%; the proportion is particularly preferably between 1% and 3%.

According to the formulas specified according to the invention, the electrodes can easily be shaped in such a way that mathematically exact superimpositions of the quadrupole field with predetermined contributions of the octupole field or the sextupole field are obtained. The deviations due to the overlaid fields are mainly noticeable in the outer areas of the spectrometer area, while an almost exact quadrupole field is present in the area of the center.

It should be noted that the manufacture of electrodes according to the regulation of the invention in one embodiment, as is the subject of claim 6, is carried out by successively adding higher-order thermal springs in w, once the measure p 1 for the proportion of the octupole field, the measure p₂ are specified for the portion of the sextupole field or the correction portion p₃ of the octupole field. It is again advantageous if p₁, p₂ and p₃ are between 0% and 20%, these variables should not, however, at the same time take the value 0, so that in any case a superimposed heat contributes.

Fig. 1

den Längsschnitt durch Elektrodenanordnung eines Massenspektrometers gemäß der vorliegenden Erfindung, wobei ein Oktupolfeld als Multipolfeld höherer Ordnung einem Basis-Quadrupolfeld überlagert ist,

Fig. 2

einen Längsschnitt durch die Elektrodenanordnung, wobei ein Sextupolfeld überlagert ist, und

Fig. 3

einen Längsschnitt durch die Elektrodenanordnung, wobei sowohl ein Oktupol- als auch ein Sextupolfeld überlagert sind.

The invention is explained in detail below using exemplary embodiments with reference to the schematic drawing. It shows:

Fig. 1

2 shows the longitudinal section through the electrode arrangement of a mass spectrometer according to the present invention, an octupole field as a higher order multipole field being superimposed on a base quadrupole field,

Fig. 2

a longitudinal section through the electrode arrangement, wherein a sextupole field is superimposed, and

Fig. 3

a longitudinal section through the electrode arrangement, both an octupole and a sextupole field are superimposed.

ist. Das durch die Elektrodenform erzeugte Oktupolfeld hat eine Stärke A₄/A₂ von 2 %, gemessen in der Äquatorebene 4 an der Ringelektrode 3. Durch das überlagerte Feld werden nichtlineare Kräfte sowohl in z-Richtung als auch in Abhängigkeit von r, dem Abstand von der z-Achse, erzeugt. Dadurch werden die Säkularfrequenzen von den Säkularamplituden abhängig und nehmen entweder zu oder ab. In beiden Fällen jedoch wird eine Resonanzkatastrophe der Säkularamplitude verhindert. Durch das Oktupolfeld verschiebt sich die anwachsende Säkularschwingung in der Frequenz und in der Phase und erreicht eine maximale Amplitude, wenn die Phasenverschiebung 90° beträgt, danach nimmt die Amplitude wieder ab. Daher übt das Oktupolfeld wie alle anderen "geradzahligen" Multipolfelder eine überraschend positive Wirkung aus. Nahezu alle Ionenverluste durch Resonanzeffekte werden verhindert, was auch immer die Resonanz verursacht haben mag.Figure 1 shows the arrangement of two end cap electrodes 1, 2, which are each arranged at a distance z₀ from the equatorial plane 4. The descriptive coordinate system is chosen so that the equatorial plane 4 coincides with the coordinate plane z = 0. Between the end cap electrodes 1, 2 there is a ring electrode 3 such that the entire arrangement of the electrodes 1, 2, 3 is axially symmetrical, the axis of symmetry coinciding with the z axis of the coordinate system. The distance of the ring electrode 3 from the center z = 0 in the equatorial plane 4 is denoted by r₀. The electrode arrangement is chosen so that $${\text{r}}_{\text{0}}{\text{/ z}}_{\text{0}}\text{=}\sqrt{\text{1.8}}$$

is. The octupole field generated by the electrode shape has a strength A₄ / A₂ of 2%, measured in the equatorial plane 4 at the ring electrode 3. The overlaid field causes non-linear forces both in the z direction and as a function of r, the distance from the z -Axis generated. As a result, the secular frequencies become dependent on the secular amplitudes and either increase or at. In both cases, however, a resonance catastrophe of the secular amplitude is prevented. The increasing secular oscillation shifts in frequency and phase through the octupole field and reaches a maximum amplitude when the phase shift is 90 °, after which the amplitude decreases again. Therefore, like all other "even" multipole fields, the octupole field has a surprisingly positive effect. Almost all ion losses due to resonance effects are prevented, whatever the cause of the resonance.

• (1) Resonanzen zwischen den Endkappenelektroden 1, 2, die durch eine Anregungsfrequenz hervorgerufen werden,
• (2) nichtlineare Resonanzen aus Überlagerungen von gegenüber der Speicherfrequenz verschobenen Frequenzen oder hervorgerufen durch Multipolfelder, die durch ungenaue Anordnung der Elektroden erzeugt werden oder auch durch Oberflächenladungen auf den Elektroden.

Normally disturbing resonances can be

• (1) resonances between the end cap electrodes 1, 2 caused by an excitation frequency,
• (2) non-linear resonances from superpositions of frequencies shifted with respect to the storage frequency or caused by multipole fields which are generated by inaccurate arrangement of the electrodes or also by surface charges on the electrodes.

The only exception is the so-called octupole sum resonance, in which the ion absorbs energy in both the r direction and the z direction.

With the electrode arrangement from FIG. 1, it is also possible to avoid the disadvantages of the prior art with regard to the generation of daughter ions. If an octupole field is superimposed on the base quadrupole field, the excitation voltage can be selected such that the parent ions never reach the end cap electrodes 1, 2. Yields of daughter ions in the order of 80 to 100% of the parent ions are possible.

An octupole field that normally blocks the resonance reactions of ions can still have a positive effect on the Have resonance response during a scan. When the secular frequency reaches the external excitation frequency, due to the coupling of the secular frequency and the secular amplitude, the effects of the increase in the sampling frequency and the decrease in the amplitude are compensated, whereby the ion is expelled from the mass spectrometer.

Durch die punktierten Linien 5, 6 wird die entsprechende Elektrodenstruktur angedeutet, bei der ein reines Quadrupolfeld vorliegen würde. Es zeigt sich, daß Abweichungen nur in den Außenbereichen der Elektrodenanordnung auftreten, während sich im Inneren ein nahezu exaktes Quadrupolfeld ergibt. Durch die Überlagerung des Sextupolfeldes bleibt die Säkularfrequenz in z-Richtung im wesentlichen unverändert, während in r-Richtung eine Frequenzaufspaltung erfolgt. Das Sextupolfeld erzeugt eine starke nichtlineare Resonanz bei einer Frequenz, die bei exakt einem Drittel der Speicherfrequenz liegt. Wenn nun eine Anregungsspannung phasengerecht und mit dieser Frequenz aufgegeben wird, wird die Ionenoszillation zunächst durch die diese Anregungsspannung vergrößert, was zu einem linearen Ansteigen der Säkularamplitude führt, dann wird die Oszillation exponentiell durch die Sextupolresonanz ansteigen. Die Hexapolresonanz kann daher für ein massenselektives Ausstoßen des Ions verwendet werden. Durch das Überlagern des Sextupolfeldes wird daher der Ejektionsprozeß verschärft. Gute Ergebnisse werden dabei erreicht, wenn der Anteil A₃ des überlagernden Sextupolfeldes 2 % des Quadrupolfeldes beträgt.FIG. 2 shows an electrode arrangement comprising end cap electrodes 1, 2 and ring electrode 3, in which the electrodes are shaped in such a way that a sextupole field is superimposed on the base quadrupole field. The dimensioning of the electrodes is otherwise the same as that of FIG. 1, in particular is again $${\text{r}}_{\text{0}}{\text{/ z}}_{\text{0}}\text{=}\sqrt{\text{1.8}}\text{.}$$

The dotted lines 5, 6 indicate the corresponding electrode structure in which a pure quadrupole field would be present. It can be seen that deviations only occur in the outer regions of the electrode arrangement, while an almost exact quadrupole field results in the interior. By overlaying the sextupole field, the secular frequency remains essentially unchanged in the z direction, while frequency splitting takes place in the r direction. The sextupole field produces a strong nonlinear resonance at a frequency that is exactly one third of the storage frequency. If an excitation voltage is now applied in phase and at this frequency, the ion oscillation is first increased by this excitation voltage, which leads to a linear increase in the secular amplitude, then the oscillation will increase exponentially due to the sextupole resonance. The hexapole resonance can therefore be used for mass-selective ejection of the ion. The ejection process is therefore exacerbated by overlaying the sextupole field. Good results are achieved when the proportion A₃ of the overlying sextupole field is 2% of the quadrupole field.

FIG. 3 shows an electrode arrangement in which both a superimposed octupole field and a superimposed one Sextupol field have been generated, the octupole portion is 2% and the sextupole portion is 6%. The combination of the two superimposed fields has the result that the advantages of both systems are realized in the arrangement. The loss of ions is reduced by the octupole effect, the non-linear resonance of the sextupole field promotes the ejection of the ions and sharpens the ejection process. It has been found that the best results are achieved if the proportion A₃ of the overlaid sextupole field is twice as large as the proportion A₄ of the overlaid octupole field.

The features of the invention disclosed in the above description, in the drawing and in the claims can be essential both individually and in any combination for realizing the invention in its various embodiments.

REFERENCE SIGN LIST

• 1 end cap electrode
• 2 end cap electrodes
• 3 ring electrode
• 4 equatorial plane
• 5 dotted line (quadrupole structure)
• 6 dotted line (quadrupole structure)

Claims (6)

1. An ion cage mass spectrometer, also called a quistor or ion trap, comprising two end-cap electrodes (1, 2) and an annular electrode (3) disposed between the end-cap electrodes (1, 2) so that the entire arrangement of electrodes (1, 2, 3) has axial symmetry, the axis of symmetry defining the z axis and the point z = 0 being equidistant from the two end-cap electrodes, voltage supplies for generating an ionstoring HF quadrupole field, means for generating ions of the substances to be investigated by mass spectrometry inside or outside the ion cage, means if required for introducing the ions into the ion cage, and means for detecting the aforementioned ions coming out of the ion cage, characterised in that a sextupole potential $${\text{P}}_{\text{s}}{\text{= (A}}_{\text{3}}{\text{/4z}}_{\text{o}}{}^{\text{3}}{\text{* (3r}}^{\text{2}}{\text{z-2z}}^{\text{3}}\text{) * [U - Vcos(ωt)],}$$

   

   or an octupole potential $${\text{P}}_{\text{o}}{\text{= (A}}_{\text{4}}{\text{/4z}}_{\text{o}}{}^{\text{4}}{\text{) * (r}}^{\text{1}}{\text{+8z}}^{\text{4}}{\text{/3-8r}}^{\text{2}}\text{z2) * [U - Vcos(ωt)],}$$

   

   or a linear combination of the two is superposed on the exact quadrupole potential $${\text{P}}_{\text{q}}{\text{= (A}}_{\text{2}}{\text{/4z}}_{\text{o}}{}^{\text{2}}{\text{) * (r}}^{\text{2}}{\text{-2z}}^{\text{2}}\text{) * [U - Vcos(ωt)],}$$

   

   with

   r =   distance from the z-axis,

   z =   distance from the plane z = 0,

   z_o =   distance of an end cap from the centre z = 0,

   A₂ =   strength of the quadrupole field,

   A₃ =   strength of the sextupole field,

   A₄ =   strength of the octupole field,

   U =   value of the DC voltage,

   V =   peak value of the AC voltage,
   = angular frequency of the AC voltage and

   t =   time.
2. A mass spectrometer according to claim 1, characterised in that superposition of exact sextupole and octupole fields is brought about by a surface shape of the end-cap electrodes (1, 2) r_k(z) and of the annular electrode (3) r_r(z) in accordance with the equations $${\text{r}}_{\text{k}}\left(\text{z}\right)\text{=}\sqrt{\text{(d-}\sqrt{{\text{(d}}^{\text{2}}{\text{+e}}_{\text{k}}\text{))}}}$$

   

   and $${\text{r}}_{\text{r}}\left(\text{z}\right)\text{=}\sqrt{\text{(d-}\sqrt{{\text{(d}}^{\text{2}}{\text{+e}}_{\text{r}}\text{))}}}\text{,}$$

   

   with the abbreviations $${\text{d = 4*z}}^{\text{2}}{\text{- (3A}}_{\text{3}}{\text{/2A}}_{\text{4}}{\text{)*z*z}}_{\text{o}}{\text{- (A}}_{\text{2}}{\text{/2A}}_{\text{4}}{\text{)*z}}_{\text{o}}{}^{\text{2}}\text{,}$$

   

   $${\text{e}}_{\text{k}}{\text{= (2A}}_{\text{2}}{\text{/A}}_{\text{4}}{\text{)*z}}_{\text{o}}{}^{\text{2}}{\text{*z}}^{\text{2}}{\text{+ (2A}}_{\text{3}}{\text{/A}}_{\text{4}}{\text{)*z}}_{\text{o}}{\text{*z}}^{\text{3}}{\text{- (8/3)*z}}^{\text{4}}{\text{+ P}}_{\text{k}}\text{,}$$

   

   $${\text{e}}_{\text{r}}{\text{= (2A}}_{\text{2}}{\text{/A}}_{\text{4}}{\text{)*z}}_{\text{o}}{}^{\text{2}}{\text{*z}}^{\text{2}}{\text{+ (2A}}_{\text{3}}{\text{/A}}_{\text{4}}{\text{)*z}}_{\text{o}}{\text{*z}}^{\text{3}}{\text{- (8/3)*z}}^{\text{4}}{\text{+ P}}_{\text{r}}\text{,}$$

   

   where P_k and P_r are proportional to the desired peak AC potentials at the electrodes (1, 2 and 3).
3. A mass spectrometer according to claim 2, characterised in that the surface shapes of the end-cap electrodes (1, 2) r_k(z) and of the annular electrode (3) r_r(z) are determined by the equations $${\text{r}}_{\text{k}}\left(\text{z}\right)\text{=}\sqrt{\text{(d-}\sqrt{{\text{(d}}^{\text{2}}{\text{+f}}_{\text{k}}\text{))}}}$$

   

   and $${\text{r}}_{\text{r}}\left(\text{z}\right)\text{=}\sqrt{\text{(d-}\sqrt{{\text{(d}}^{\text{2}}{\text{+f}}_{\text{r}}\text{))}}}\text{,}$$

   

   with the abbreviations $${\text{d = 4*z}}^{\text{2}}{\text{- (3A}}_{\text{3}}{\text{/2A}}_{\text{4}}{\text{)*z*z}}_{\text{o}}{\text{- (A}}_{\text{2}}{\text{/2A}}_{\text{4}}{\text{)*z}}_{\text{o}}{}^{\text{2}}\text{,}$$

   

   $${\text{f}}_{\text{k}}{\text{= (2A}}_{\text{2}}{\text{/A}}_{\text{4}}{\text{)*z}}_{\text{o}}{}^{\text{2}}{\text{*(z}}^{\text{2}}{\text{-z}}_{\text{o}}{}^{\text{2}}{\text{) + (2A}}_{\text{3}}{\text{/A}}_{\text{4}}{\text{)*z}}_{\text{o}}{\text{*(z}}^{\text{3}}{\text{-z}}_{\text{o}}{}^{\text{3}}{\text{- (8/3)*(z}}^{\text{4}}{\text{-z}}_{\text{o}}{}^{\text{4}}\text{),}$$

   

   and $${\text{f}}_{\text{r}}{\text{= (2A}}_{\text{2}}{\text{/A}}_{\text{4}}{\text{)*z}}_{\text{o}}{}^{\text{2}}{\text{*(z}}^{\text{2}}{\text{+z}}_{\text{o}}{}^{\text{2}}{\text{) + (2A}}_{\text{3}}{\text{/A}}_{\text{4}}{\text{)*z}}_{\text{o}}{\text{*(z}}^{\text{3}}{\text{+z}}_{\text{o}}{}^{\text{3}}{\text{) - (8/3)*(z}}^{\text{4}}{\text{+z}}_{\text{o}}{}^{\text{4}}\text{).}$$

   
4. A mass spectrometer according to claim 2 or 3, characterised in that $${\text{0.002 <= A}}_{\text{4}}{\text{/A}}_{\text{2}}\text{<= 0.08}$$

   

   and $${\text{0 <= A}}_{\text{3}}{\text{/A}}_{\text{2}}\text{<= 0.16.}$$

   
5. A mass spectrometer with a superposed exact sextupole field according to claim 1, characterised in that the surface shapes of the end-cap electrodes (1, 2) r_k(z) and of the annular electrodes (3) r_r(z) are determined by the equations: $${\text{r}}_{\text{k}}\text{=}\sqrt{{\text{(2z}}^{\text{2}}{\text{-2z}}_{\text{o}}{}^{\text{2}}\text{*g}\left(\text{z}\right)\text{)}}$$

   

   and $${\text{r}}_{\text{r}}\text{=}\sqrt{{\text{(2z}}^{\text{2}}{\text{-2z}}_{\text{o}}{}^{\text{2}}\text{*g}\left(\text{z}\right)\text{)}}\text{,}$$

   

   with $${\text{g(z) = (A}}_{\text{2}}{\text{+A}}_{\text{3}}{\text{)/(A}}_{\text{2}}{\text{+3*A}}_{\text{3}}{\text{*z/z}}_{\text{o}}\text{),}$$

   

   and $${\text{0.001 <= A}}_{\text{3}}{\text{/A}}_{\text{2}}\text{<= 0.2.}$$

   
6. A mass spectrometer according to claim 1 with approximate sextupole and octupole fields, characterised in that the multipole fields are determined by surface shapes of the electrodes (1, 2, 3) in accordance with the equations $${\text{z}}_{\text{r}}{\text{(r) = (w}}_{\text{r}}{\text{+ (p}}_{\text{1}}{\text{*w}}_{\text{r}}{\text{) + (p}}_{\text{2}}{\text{*w}}_{\text{r}}{}^{\text{2}}{\text{) + (p}}_{\text{3}}{\text{*w}}_{\text{r}}{}^{\text{3}}\text{)),}$$

   

   $${\text{z}}_{\text{k}}\text{(r)}\mathit{\text{=}}{\text{(w}}_{\text{k}}{\text{+ (p}}_{\text{1}}{\text{*w}}_{\text{k}}{\text{) + (p}}_{\text{2}}{\text{*w}}_{\text{k}}{}^{\text{2}}{\text{) + (p}}_{\text{3}}{\text{*w}}_{\text{k}}{}^{\text{3}}\text{)),}$$

   

   with $${\text{w}}_{\text{r}}{\text{= w}}_{\text{r}}{\text{(r) = ((r}}^{\text{2}}{\text{- r}}_{\text{o}}{}^{\text{2}}\text{)/2,}$$

   

   $${\text{w}}_{\text{k}}{\text{= w}}_{\text{k}}{\text{(r) = ((r}}^{\text{2}}{\text{- r}}_{\text{o}}{}^{\text{2}}\text{)/2,}$$

   

   and $${\text{0 <= p}}_{\text{1}}\text{<= 0.2}$$

   

   (roughly approximate octupole component) or $${\text{0 <= p}}_{\text{2}}\text{<= 0.2}$$

   

   (approximate sextupole component) and/or $${\text{0 <= p}}_{\text{3}}\text{<= 0.2}$$

   

   (more closely approximated octupole component), but p₁, p₂ and p₃ do not vanish simultaneously.

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