Hoffman D.M., Singh B., Thomas J.H. (Eds). Handbook of Vacuum Science and Technology

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M d^x

'Q~dt^~

'Q~di^~

M d^z

'a'dt^'

^0^

H-

Vcos

(Ot)

-^{U-^

Vcos (ot)

310 Chapter 3.2: Mass Analysis and Partial Pressure Measurement

The equations of motions are determined by the constant U DC potential and

Vcosojt

RP potentials applied with 180-degree phase difference to the two oppos-

ing pairs of electrically connected rods (of hyperbolic cross section ideally, but

often round in practice).

(2)

(3)

(4)

These equations describe oscillatory motions under the influence of Vcosiot

superimposed on the focusing effect of Urn x direction and defocusing effect in y

direction, (ro is the pitch circle radius of the rods,

is the circular frequency of

the RP voltage, and z axis is parallel with the axis of the cylinders.) The relation-

ship between M/Q and the scanning parameter (voltage) is linear, resulting in a

linear mass axis. (The space between peaks is independent from M/Q.)

The motion of ions is independent in the x and y direction. In x direction Equa-

tion (2) offers the simple explanation that the DC component in itself would gen-

erate a simple harmonic motion, while the RF component in itself would generate

a forced oscillation. The superposition of the two forces will result in trajectories

similar to a resonance system excited "off resonance"—oscillation with increas-

ing, then decreasing amplitudes

[119].

In j direction the DC component defocus-

ing action is counterbalanced only by the focusing effect of the RP field. Since, as

direction, the RP component in itself would generate simple harmonic motion

where the displacement will be 180 degrees out of phase of the force, the addition

of the DC component will cause oscillations around

path that will be increasingly

farther apart from the z axis. However, the more the mean of the oscillation devi-

ates from zero, the larger is the difference between the force at maximum displace-

ment (being out of phase—pointing toward the z axis) and at minimum displace-

ment (pointing away from the z axis). This difference, which results in a net force

toward the z axis, creates the focusing effect of RP for the motion in y direction.

By substitution, the equations of motions in the x and y direction can be brought

to a single form—to a Mathieu-type equation

[120].

Solutions to the Mathieu

equation give ion trajectories as a function of operating parameters, U, V, to, and

mass-to-charge ratio. Those combinations that result in stable trajectories define

the "stability region" (can be expressed in terms of two dimensionless variables

a = WlmrQ-o)'^ and q =

AVImrQ-o)'^

where m = M/Q). In the a-q plane an area

resembling a triangle contains all those a, q pairs that correspond to stable trajec-

tories.

(The three comer points are: a,q (0,0); a,q (0.237,

0.706)

apex; and a,q

(0,

0.91).)

The curve connecting the first two points marks the border of the insta-

bility in the y direction, all points to the left being unstable in y direction. The line

3.2.4 Ion Separation and Analyzers

311

connecting the last two points marks the border of the x instability region, all

points to the right being unstable in x direction. At the apex of the triangle only

one mass number (m) have stable trajectories (infinite resolution), while on a line

segment within the triangle a range of mass numbers are "stable." The high end

of this mass range is called "high pass critical mass," ions with higher mass num-

ber will have unstable trajectories in the y direction. Ions with lower mass number

than the "low pass critical mass" will have unstable trajectories in the x direction

[121].

Examples of ion trajectories and the a-q stability chart has been published

in an earlier volume

[122].

Instead of the a-q stability chart, we will discuss sta-

bility criteria and scanning in the U-V plane (Figure 3). As seen above, three cor-

ner points mark the triangle-like area of a-q pairs which assure stable trajectories.

In a given instrument (given ro) operating at a certain frequency (given

(D)

a and q

are proportional to U/m and V/m respectively. Therefore in the U-V plane each

mass has its own stability triangle. Scanning is realized by simultaneously in-

creasing the DC and RF voltage from an initial DC value

{UQ)

along the "scan

line."

Ions with higher mass numbers (m) than the "high pass critical mass" will

have unstable trajectories in y direction, while ions with lower mass numbers than

the "low pass critical mass" have unstable trajectories in

the

x direction

[121].

Ex-

amples of ion trajectories can be found in Methods of Experimental Physics, Vol-

ume 14

[122],

and a sketch of the stability diagram is shown in Figure 3.

Fig.

RF Voltoge

Sketch of ion path stability diagrams (quadrupole instruments). X and

directions perpendic-

ular to the axis of the analyzer.

312 Chapter 3.2: Mass Analysis and Partial Pressure Measurement

Point a in Figure 3 on the scan line corresponds to the

and

pair where

ions of mass to charge ratio equal to

become stable not only in x direction but

also in y direction. Ions of higher mass number are unstable in y direction, as

shown by the "7unstable" curve of

(nii > mi), which crosses the scan line at

point c. Ions of much lower mass number than

are unstable in x direction. The

"x unstable line" for mass m^ crosses the scan line at point 6, while for m2 it

crosses at point d. For the

U^,

voltage pair the

"JC

unstable" line would cross the

scan line for a somewhat lower mass than nii at point a. (This mass number will

be called

— Am.) It means that ions with mass number higher than m\ -Am but

lower than

would exhibit stable trajectories both in x direction and y direction

when the voltage pair

Ur^,

is selected. The width of this window. Am, is clearly

related to the slope of the scan

line

—

close to the tip of the stability regions the

bandwidth of the filter will be narrow.

If U and V are increased simultaneously along the scan line, eventually point b

is reached. It is the point where the x trajectories of ions of mass

become un-

stable. The y unstable line for a somewhat higher mass number (mi -hAw/) would

cross the scan line at the same point b. Therefore, as U and V are increased simul-

taneously along the scan line, a mass window of width Am is moved from lower

mass numbers to higher mass numbers (scanning).

The overall result is that the RF field without the DC component will create a

"high pass" filter that allows ions above a critical M/Q ratio to pass while the

addition of the DC component will create a "low pass" filter. The magnitude of

the DC voltage determines the "low pass" critical mass — if it is lower than

the "high pass" critical mass, no ions can reach the detector; if it is higher than the

"high pass" critical mass, a window is created.

Mass scan is accomplished by moving the "window" (Am) by changing the

amplitude of the RF (and the magnitude of the DC) voltage. When DC/RF ratio is

low (<0.1), the width of the window is large, resulting in low resolution. When

increasing the DC/RF ratio, the resolution increases on the expense of the trans-

mission efficacy. If DC/RF ratio approaches about 0.168, the apex of the stability

region is reached where the transmission drops to zero at infinitely high resolving

power. Experimentally this critical DC/RF ratio has been found to depend weakly

on the excitation frequency

[123].

The excitation frequency,

(o,

also affects the ul-

timate obtainable resolving power (M/iiM 2-3 times higher at 1.66 MHz than at

0.87 MHz).

While the resolving power will change as the mass number is varied in a typi-

cal instrument, two special cases can be identified. If the DC/RF ratio is kept con-

stant along the mass sweep (U/V independent from M/Q), then this represents a

"constant resolving power mode" of operation (constant

MIHsM).

In this case the

peak width will be proprtional to the mass number—resulting in narrow peaks at

low masses and wider peaks at high masses. Another special case is when the

DC/RF ratio is varied as the mass number is increased in such a way that the width

of the peak (AM) is kept constant. This is referred to as the "constant peak width

3.2.4 Ion Separation and Analyzers 313

mode" of operation. The resolving power {MI\M) will be consequendy propor-

tional to the mass number.

In practice the magnitude of an "offset" DC voltage (L'i), "resolution low") and

the magnitude of the slope of the DC/RF function (approximately UIV at high

masses, "resolution high") can be varied separately to adjust resolution at low and

high MIQ. The magnitude of the RF and DC voltage is increased to shift the

"window" along the mass scale from low MIQ toward high MIQ. A convenient

and often attempted setup is to keep the peak width (AM) more or less constant

along the entire mass range. A so-called integral spectrum is obtained easily by

switching off

the

DC component and allowing all ions above MIQ to reach the de-

tector as RF {V) is swept and the spectrum is recorded. As Equation (4) shows,

the initial axial velocity remains unchanged in the quadrupole field.

Besides the so-called first-stability regions shown on Figure 3, there are several

more predicted by theory as the F-unstable and X-unstable curves cross each other

again. These are of limited practical value due to decreased transmission and

usable mass range, but some applications may benefit from the much improved

abundance sensitivity (such as sepaiating He from D2) [124,1251.

Even though commercial instruments have been marketed using the ideal hy-

perbolic cross section electrodes (Pt-coated ceramics)

[126],

the majority of prac-

tical analyzers employ metal rods of circular cross section. The benefit of hyper-

bolical, coated ceramic electrodes is not only the more perfect quadrupole field

but rather the ability to produce partial pressure information at elevated tempera-

tures up to 400°C (750°F). Operating the analyzer at elevated temperatures is help-

ful when quantitative measurement of adsorbing components such as moisture is

attempted

[127].

The optimum radius for cylindrical electrodes has been deter-

mined theoretically (r = 1.148

TQ)

[128].

The effect of replacing the hyperbolic

cross section with a circular cross section has been studied extensively. The equa-

tion of motion includes terms reflecting coupling of the x and y fields. This results

in a "blunted tip" and a slight shift to the right of the stability diagram, limiting

the theoretically achieved maximum resolution [129,130]. The effect of the ra-

dius of the electrode relative to the optimum is important because of the increase

in the required driving power to achieve the same resolving power and sensitivity

(a factor of two, if r/ro is 1% higher than optimum at M/AM=400). At constant

driving power, the result is lower resolving power and sensitivity

[129].

Although

hyperbolic rods exceed round rods in performance (maximum achievable resolv-

ing power), increased driving frequency can make up for the difference (round

rod quadrupole driven at 1.7 MHz performs similarly to a same-size hyperbolic

driven at 0.7 MHz)

[131].

The sharp decrease in sensitivity of quadrupoles with

circular rods at a resolving power lower than a quadrupole with hyperbolic rods is

primarily due to the decrease in the effective instrument radius. The perturbations

in the electric fields caused by the circular shapes limit the maximum excursion

of the oscillating ions from the z axis to a radius significantly smaller than

TQ.

The transmission efficacy—which is the ratio of current collected to that en-

314

Chapter 3.2: Mass Analysis and Partial Pressure Measurement

tering the quadrupole can be found by measuring the step corresponding to the

selected mass number in integral mode (DC off) and standard mode (DC on)

when using a conventional ion source. Plotting the transmission against the DC/

RF ratio (which determines the resolving power) one will typically find transmis-

sion exceeding 80% up to about 0.9 times the DC/RF ratio corresponding to the

apex of the stability diagram (about DC/RF=0.168)

[123].

(Transmission was

measured as a ratio of the ion current collected

vs.

the current entering the quadru-

pole at a given MIQ. The step in the "AC only" or total pressure mode was used

as a measure of the current entering the filter. Overall transmission of the system,

i.e., the ratio of ions collected to ions generated in the source (for a given MIQ

when the filter is tuned to this M/Q), is typically

10~

-10~^.) Using a narrow ion

beam, well-defined energy source movable under vacuum (such as an alkali ion

source Na"^,

K"*",

etc.) enables one to measure the transmission efficacy as a func-

tion of initial displacement, direction and velocity

[132].

From these measure-

ments empirical transmission laws [133] and the acceptance eUipses of a particu-

lar quadrupole configuration can be derived.

Examining the transmission as a function of resolving power (Figure 4) at a

given ion energy, three different regions can be distinguished. Since quadrupoles

Fig.

I-

0.8

0.6

0.4

0.2

0.0

iso

dc/rf ratio curves

iso

Ion

energy curves

Peak Width AM(amu)

Dependence of quadrupole analyzer transmission on resolution measured as the width of a

peak (at

10%

height).

3.2.4 Ion Separation and Analyzers 315

in practice most often operate in constant peak width rather than in constant reso-

lution mode, in the following discussion we use peak width as a measure of reso-

lution. (Note that peak width increases as resolving power decreases.)

If at a given ion energy setting the resolution is low (peak width high), the

transmission will be found to be practically a slow function of resolution (such as

the quasi-horizontal part of line B on Figure

4).

This is the "source-limited" mode

of operation; the acceptance of the analyzer approaches or exceeds the emittance

of the source. Stability and reproducibility are determined primarily by changes

in the source. Transmission typically increases with initial velocity of ions (ion

energy) due to the fringing fields effects (see section on "coupling of the ionizer

to the analyzer," later). The peaks are "flat-topped" — trapezoidal.

Increasing the resolution (decrease in peak width) will take us to the region

where the acceptance of the analyzer will be considerably smaller than the emit-

tance of the source. This is the "analyzer limited" mode of operation, where the

transmission efficiency will be strongly dependent on the entrance conditions.

The latter is primarily due to the transient momentum impulse the ions receive

at the entrance of the analyzer characterized by imperfect and highly complex

quadrupole fields. It is well established that fringing fields introduce a depen-

dence of the acceptance on axial ion velocity. In the fringing fields, ions are stable

in the xz plane but unstable in the yz plane [134,135]. Combined acceptances

{xz and yz planes) are at the maximum at axial velocities where the ions spend

about two cycles in the fringing fields, and it falls off sharply for slower ions.

(Combined acceptance decreases by a factor of 10 if the time spent in the fringing

fields increases to four cycles [135].) This effect shows up as a decrease in sensi-

tivity with decreasing ion energy at a given resolution for a given mass or as mass

discrimination for ions injected at constant ion energy. (Heavier ions are slower

at constant ion energy, resulting in lower transmission as they spend more time

in the fringing fields.) Transmission in the "analyzer-limited" mode of operation

will be inversely proportional to the square of resolving power as a general rule.

Peak shape is triangular. (Between 0.7 and 1.5 Am on line B.)

Further increase in resolution at the same ion energy will result in very sharp

decrease in transmission, indicating that the "length" limited mode of operation is

reached. Here the analyzer is increasingly unable to distinguish an ion with stable

trajectory from an ion with unstable trajectory, which is related to the number of

RF cycles the ion takes to pass through

[135].

Peaks are sharp, triangular but their

height decreases much more rapidly than their width as resolution is increased.

If these transmission-resolution (or peak width) curves are taken at all pos-

sible ion energies, the resulting envelope of all these curves defines the optimum

performance curve. This curve connects the points corresponding to the highest

possible transmission (or sensitivity) at each resolution (or peak width). Note that

a different pair of DC/RF ratio and ion energy setting will realize maximum sen-

sitivity at each resolution. Even though operating on this curve provides the max-

imum performance, in some cases it may be justifiable to operate below this curve

316 Chapter 3.2: Mass Analysis and Partial Pressure Measurement

(such as in a source-limited mode to avoid that change in DC/RP ratio affects

peak height.) The falloff of this optimum performance curve for a particular ana-

lyzer represents the limitations of the instrument

itself.

Factors that determine the

maximum attainable resolution (besides finite length) are imperfections such as

circular rods instead of hyperbolic, deviations from the perfect symmetrical geom-

etry (displacement of

rods,

mechanical distortions due to baking), contamination

of surfaces, surface charges

[136].

Alignment of the ion source, imbalance of

electric fields (DC/RF ratio) and spurious ripple superimposed on the RF can all

be further limiting factors.

A further aspect of looking at transmission as a function of resolution is that

only the points along the optimum performance curve are defined by a unique

pair of ion energy and DC/RF ratio. All other transmission-resolution combina-

tions below that can be realized by two pairs of ion energy DC/RF setting. One

pair will be closer to the "source limited mode" (such as the intersection of lines

b and B on Figure 4), while the other will be closer to the "length-limited" mode

(such as the intersection of lines a and A). This is the result of the fact that both

transmission and resolution are functions of the two variables: ion energy and

DC/RF setting, thus defining two surfaces. The intersection of these two surfaces

is the curve called the "optimum performance curve."

The optimum performance curve is not independent of mass-to-charge ratio.

Since ions of higher masses spend longer time in the analyzer (at the same ion en-

ergy),

the length-limited region will shift toward the left (if peak width is used as

a measure of resolution—higher resolving power, i.e., lower Am). On the other

hand, high mass ions will spend more time in the fringing fields also, resulting in

a downward shift of the source-limited region. The overall effect is that the opti-

mum performance curve shifts to the left and downward as mass-to-charge ratio

increased.

Coupling of

the

ionizer to the analyzer is an important topic especially in the

case of quadrupole mass filters. If the substance to be analyzed fills the ionization

chamber volume uniformly and is in thermal equilibrium with the walls, ions are

created in the space determined by the ionizing beam. Consider first the much

simpler one-dimensional case, where ions are generated along a line between two

points determined by the width of

the

ionizing beam (Figure

5).

Plotting the trans-

verse initial velocity of the ions exiting the source against their location results in

an ellipse that contains all the possible combinations. This is the emittance of the

source (phase space) at the exit. On the other hand, the analyzer will accept ions

depending on the position as well as transverse initial velocity of the ions. The el-

lipse that shows ions of what initial conditions will pass the analyzer is called the

acceptance

ellipse.

(This ellipse represents the upper boundary for initial velocity/

initial displacement combinations in the phase space that enables an ion to pass

the analyzer tuned to its particular MIQ ratio.) The overlap of these two ellipses

is the extraction. Depending on which ellipse is smaller, and so which one deter-

3.2.4

Ion

Separation

and

Analyzers

317

Fig.

nitial transverse velocity

Analyzer acceptance

nitial pes

Source emittance

Cross section

source

analyzer

nterface

Illustration of the ion source emittance and quadrupole analyzer acceptance ellipses.

mines extraction, there are source-limited and acceptance-limited modes of opera-

tion

[137].

In a practical case, a source can be characterized by a five-dimensional

phase space—two directions perpendicular to the axis of symmetry along with

two velocities in these directions and the velocity along the axis. (The acceptance

and emittance can be characterized in more detail by a series of ellipses corre-

sponding to various percentages of transmission and emission probabilities.) The

electric fields between the source and the analyzer play a crucial role in determin-

ing transmission by altering the emittance of the source through the stray, so-

called fringing fields [138,139]. Fringing fields are also responsible for mass dis-

crimination; slow ions (at a constant ion energy, high-mass ions) have less chance

of passing this region than faster ones.

As seen from the preceding discussion, the entrance and exit regions of the

analyzer, where the DC field gradually approaches its final value (fringing fields),

play a major role in analyzer overall transmittance. In the stability diagram, the

fringing fields corresponds to the section of the scan line before it reaches the sta-

bility region (e.g., on Figure 3 the

U^,

voltage corresponds to the state when the

filter is tuned to about mass number mj. However, as an ion (mass number mi) is

traveling from the source to the area between the quadrupole rods, it is experienc-

ing a gradual "buildup" of the electric quadrupole field. This corresponds to be-

ing exposed to the portion of the scan line between

and point a that is beyond

318 Chapter 3.2: Mass Analysis and Partial Pressure Measurement

the stability region for an ion of mass number

i.) As a result, the ion trajectories

are unstable leading to high losses. A way to avoid this is to "bend" the scan line

so that it lies entirely within the stability region. This means that RF has to be in-

creased first, DC following only later on. This is called the "delayed DC ramp"

[140],

which can be realized by using a set of four additional electrodes between

the source and analyzer (and possibly also between the analyzer and detector)

driven by RF only. Several commercial partial pressure analyzers apply this

technique.

Variations of the quadrupole analyzer from its standard version as described

earlier can be grouped as variations in the driving voltage and variations in the

geometry of the filter.

A possible way to avoid losses and mass discrimination of high masses associ-

ated with fringing fields is not to use DC voltage at all. This so-called AC-only

mode of a quadrupole is essentially a high-pass filter, providing an integral spec-

trum. To obtain a regular spectrum, a "retarding field" or energy filter is applied

at the exit of the analyzer, which works based on the observation that ions that are

close to the border of the stability region (corresponding to the step in integral

mode) tend to be defocused, filling the aperture at the exit. Since the fringing

fields (from the RF) increase in field strength as the displacement from the z axis

increases, these ions gain more energy from the fields than others

[141].

A further

clear advantage is the elimination of the need to keep a constant DC/RF ratio. In-

deed, instruments with low mechanical tolerances behave in a superior way in this

mode of operation, but there is little difference if the instruments have been man-

ufactured accurately. Disadvantages are the difficulties at low M/Q (below 20)

and high-ion energy (above 4eV) related to the increased number of RF cycles an

ion must spend in the filter to obtain the same resolution as compared with the case

where the focusing/defocusing action of DC voltage is also present [142,143,144].

The removal of stringent control of DC/RF ratio can alternatively be achieved

by driving the rods with a rectangularly time-varying voltage waveform

[145],

with the promise of better stability, but this variation has not yet been adapted in

commercial instruments.

Variations in

thQ

filter

geometry have been investigated, such as concave shaped

[146],

flat electrodes

[147],

coated ceramics

[148],

and electrodes made of

X 3)

wires to approximate quadrupole fields or to eliminate higher-order terms

[149].

All these are still using functionally four electrodes.

It has been recognized very early (1963) [150] that it is not necessary to use

four electrodes to employ the mass analyzer properties of quadrupole fields and

Equations (2) to (4). The combination of one circular (or hyperbolic) and a V-

shaped electrode (the circular rod being in the 90-degree opening of the V) can

simulate a quarter of the quadrupole field. This arrangement is the ''monopole''

mass spectrometer that has also been manufactured commercially. Other propos-

als—which have not been commercialized—are the ''dipole'' mass spectrometer

composed of two circular rods on one side of a plate electrode creating

"half"

3.2.4 Ion Separation and Analyzers

319

quadrupole field

[151].

A theoretical investigation of'multipole'' devices (hexa-

pole,

octapole) as a generalization of the geometry of the quadrupole has already

been carried out

[152];

and some of the geometries and means to realize it (such

as shaped glass) are patented.

Shorter but higher number of rods open up the way to increase the maximum

allowable operating pressure. Since the mean free path is inversely proportional

to the pressure, ion molecule collisions will affect linearity above 10"^ ton*

(~10"-^ Pa) (see Section 3.2.1.2). Shorter rods reduce ion-molecule collisions,

but in order to maintain resolution the frequency of the RF voltage has to be in-

creased. (Resolution is related to the number of cycles the ions spend in the

quadrupole field.) Using the same range of U and V (limited by arcouts) the pitch

circle radius of the rods (ro) has to be decreased proportionally to the increase

0).

(See Section 3.2.4.1 The variables defining the stability region —

and q—

are function of the product of

and

co).

The smaller ro in turn will reduce the ac-

ceptance area of the analyzer which can be offset by using an array of quadru-

poles acting parallel. A practical device using a 3X3 quadrupole array (16 rods

forming 9 quadrupole analyzers) has been described

[153].

While the smaller size

of the analyzer head and the higher operating pressure limit are clear advantages,

resolution overall and power required to drive the analyzer are less favorable than

in the case of regular quadrupoles.

If the cross section of the quadrupole (four circles or hyperbolae) is rotated

around an axis passing through the center of a pair of opposing rods (e.g., the

axis) a ring and two end-caps are formed. This structure is a "three dimensional

quadrupole" or the quadrupole ion trap. The electrons may enter via a center hole

in one of the end-caps while the other may provide an exit for the ions formed

(and stored) in the space confined by the ring electrode and the two end caps. Ions

formed by electron impact are analyzed by increasing the amplitude of the RF

voltage on the ring electrode. This results in destabilization of the ion trajectories

for higher and higher masses in axial direction. The ejected ions are detected by

an external detector. Trapped ions can be measured also by power absorption of a

supplementary excitation signal or by measuring the differential charge on the

end-caps induced by ion motion

[154].

Quadrupole ion traps found application as

detectors of field apparatus used for environmental characterization or monitor-

ing

[155].

3.2.4.2 Magnetic Sector Analyzers

As Equation (1) shows, ions with a velocity v in a perpendicular magnetic field B

will accelerate in a direction perpendicular to both v and B, thus describing a cir-

cular path of radius r:

M v^

77—= vfi (5)

Q r