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7/29/2019 qmstheory.doc

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Quadrupole Mass Spectrometry

The Physics of Quadrupole Mass Spectrometry

The essential physical components of a quadrupole mass spectrometer consist of an

ionization chamber, accelerating electrodes, a physical apparatus consisting of four longrods (colored green in the drawing below) with hyperbolic cross section, a geometrical

arrangement as shown in the figures below and a detector.

Figure 1a Figure 1b

Suppose that at the surface of the upper and lower hyperbolic rods the electrical potential is given by

[ cos( )] / 2U V t = when 2 20y x r= + ,

and at the oppositely charged, left and right hyperbolic rods the potential is given by

[ cos( )] / 2U V t = when 2 20x y r= + ,

where Uis a constant i.e. time-independent voltage, V is the maximum value of a time-

varying RF sinusoidal voltage and 2 f = where f is the frequency. In the evacuatedregion between the rods (colored white) the electrical potential is given by the Poissonequation

2 4 = ,

where is the charge density between the rods. Since there is no charge in that region,the electrical potential satisfies the equation


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2 0 = .

The general solution to this equation is2 2( , )x y a x b y = + with 0a b+ = i.e. the sum of

the constants ,a b must add up to zero. The andx y coordinates are independent of each

other and can take on any values (within the white area) between the rods. We have

therefore

2 2( , ) ( )x y a x y = ,

where the constant a can be determined from the boundary conditions given above. Thatis to say, if the y values are those along the surface of the upper and lower hyperbolae

then

2 2

0y x r= + and [ cos( )] / 2U V t = .

We get 20[ cos( )] / 2 ( )U V t a r = and therefore the electrical potential is uniquelydetermined by the relation

2 2

2

( )( , ) [ cos( )]

2 o

x yx y U V t

r

= . Eq. (1)

If on the other hand, if the x values are given by

2 2

0x y r= + and [ cos( )] / 2U V t = ,

along the surfaces of the left and right hyperbolae. Under this condition one has2[ cos( )] / 2 ( )oU V t a r = i.e. we get the same result as given above in Eq. (1). See

Appendix B for a further description of the electrical potential and its effect on the

motion. If a charged particle (atom, molecule) with position ,x y is injected into the

evacuated region between the four rods and propelled down the z- axis (i.e.

perpendicular to the ,x y plane) it will be subjected to a force F given by

= eF .

From this and Eq. (1) we get Newtons equations of motion for a particle with chargee

and mass m i.e.

2

2 2

0

( )[ cos( )]

d x ym e U V t

dt r

=

r i j,

or in terms of this vector equations components


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2

2 2

0

2

2 2

0

2

2

[ cos( )],

[ cos( )],

0.

d x e U V t x

dt m r

d y e U V t y

dt m r d z

dt

=

=

=

Notice that the particles motion along the z axis is not affected by the electric fieldsand its speed along that axis is not changed.

Mathematical Detail

The differential equations which describe the motion of the charged particle along

either the x or the y-axis can be written as the single equation i.e.

2

2[ 2 cos(2 )]

d fa q f

d

= , Eq. (2)

where

2 2 2 2

0 0

4 2, , / 2

= = =

e ea U q V t

r m r m. Eq. (2.1)

If f x= the constants ,a q are defined above. On the other hand, if f y= then , anda q are replaced by , anda q . The quantity is a scaled time. Equation (2) had beensolved by Mathieu in 1868 in connection with the oscillations of an elliptically shaped

membrane. That is, the equation was known much earlier then the invention of the

quadrupole mass spectrometer by W. Paul in 1953. In order to understand thefunctioning of a quadrupole mass spectrometer, one has to examine the solutions of the

Mathieu equation.

The general solution of the Mathieu equation can be shown to have the form

( ) ( )f Ae F Be F = + , Eq. (2.2)

where ,A B are constants of integration. These constants are determined by the initial

values of the andx y coordinates and their corresponding velocity components.

Furthermore, the quantity is a function of the constants , anda q alone and not the


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initial conditions. The functionFis a periodic function of the scaled time. The term inEq. (2.2) is the all-important quantity. In general it can be a complex number i.e.

( , )a q i = + , Eq.(2.3)

where i is the imaginary number and, , are real functions of ,anda q . One can showthat stable solutions (i.e. ones which remain finite as increases) exist only if the real

part of i.e. 0 = and the imaginary part of i.e. is not equalto an integer. Thus,

stable states exist only for certain special combinations of anda q . In all other cases,

the x and/or the y coordinates of the charge become very large as time increases.

Physically this means that in the stable cases, the particle oscillates in the x and y

directions within the free space between the quadrupoles and eventually exits that regionto be counted i.e. detected. In the unstable cases, the particle veers off in the x and/or y

direction, hits a quadrupole rod or exits laterally and is lost.

The Mathieu Functions

The case of the unstable oscillations, i.e. those which arise when is equal to an

integer, is an interesting one in that, these produce curves in the ,a q plane along which

stable solutions do not exist. In fact these values of anda q i.e. these curves ( )a q

establish boundaries between regions where stable solutions do exist.

If Eq. (2) is rewritten in the following way i.e.

2

22 cos(2 )

dq f a f

d

+ =

, Eq. ( 2.4)

we see that the Mathieu equation can be viewed as an eigenvalue problem. That is to say,

for a given value of q , only special values of a are allowed. These are called theeigenvalues or characteristic values of the operator within the curly brackets in the

equation above. For each value of the eigenvalue a , there is a corresponding function f

called an eigenfunction.

Since the cosine term on the left-hand side of the equation is periodic, and an even

function in , the eigenfunctions must be either even or odd functions of the variable .

In fact one can show that in this situation i.e. where f is an eigenfunction, the quantity

in Eq. (2.3) equals 0, or1 . Hence, abecomes a function of q .

Furthermore, in the simplest case i.e. where 0q = , the eigenfunctions which are justcos( )r and sin( )r are seen to be periodic with periods or 2 and have eigenvalues

2a r= where, ris an integer i.e. 0, 1, 2, 3 .


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To summarize for 0q = ,

2cos( )

,sin( )

r

rf a r

r

= =

.

In the cases where 0q , the even eigenfunctions of the Mathieu equation are denoted by( , )rce q . These have period if ris an even integer, and have period 2 if ris

an odd integer. The eigenvalues a associated with the ( , )rce q functions (these

correspond to the 0 = case) are designated by ra . The odd eigenfunctions( , )rse q (these correspond to the 1 = case) have period , if ris an even

integer and have period 2 if ris odd. The eigenvalues associated with the

( , )rse q functions are designated by rb .

In summary we have

( ) ( ), ( )

( ) ( )

r r

r

r r

ce a qf a q

se b q

= =

The first few eigenvalues and eigenfunctions have been displayed in the appendix of this

section. Mathieu called these eigenfunctions, elliptical cosine and elliptical sine functions

respectively and were seen as generalizations of the cosine and sine functions appearing in

the 0q = case. See Appendix A for a look at the first few of these eigenfunctions andeigenvalues.

These eigenvalues increase in value (for a given value of q ) in the order

0 1 1 2 2 3a b a b a b< < < < < L

Because of the different (even and odd) symmetries of the two solutions i.e. ( , )rce q

and ( , )rse q one can show that the curves 1( ), ( )r ra q b q+ cannot cross each other no matter

what the value of q as seen in the Figure (2) below. One can also show that the shaded

areas (here is fractional) in the diagram correspond to the values of ,a and q which

permit stable oscillations to occur for the x coordinate. The 1( ), ( )r ra q b q+ curves form the

boundaries between the stable and unstable regions.


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Figure (2)

The regions of the ,a q (stability) plane, which correspond tosimultaneous stability for

both the x, and the y coordinates are shown as the shaded areas in the Figure (3) below.

Figure (3)

In the diagram

Figure (4), we have

enlarged and coloredpink thefirst stable

region i.e. thesection of the ,a q

plane, which

corresponds tosimultaneously

stability of the x, and

the y coordinates.


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Figure (4)

The value of q which corresponds to the maximum value of 0 1, anda b in the pink areais =qmax 0.701. This is determined by finding the point at which the 0 1, anda b

curves cross. The value of either ( )0 10.701 , or (0.701)a b at this point is 0.24276.

The Mass Filter

In order to discuss the mass filtering ability of the quadrupole apparatus we will

consider the top part of the first stability region in the diagram above (i.e. the top part ofthe pink area). This expanded view is displayed in the Figure (5). In addition, we note

that the ratio of / 2 /a q U V = . A line called the control line or the operating lineshown in the diagram is the straight line /a q const = , (here 1/ 3;const ). Note that thisline passes through the origin and cuts through the upper portion of the pink area. This

cutting of the 0 1,a b curves defines a range of q values that the system takes on as it


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moves along the control line. This set of q values corresponds to a collection of stable

trajectories for the charged particle.

Figure (5)

[Notice that the values for the x coordinate and y coordinates are in the range0 1< < within the triangular (stable) area.]

Since movement along the /a q const = line can be physically accomplished by varyingV along the line ( / 2)U const V = , the system can occupy a series of states, which allowthe particle to pass through the apparatus and be detected. These states will be described

below.

We begin this discussion of mass selection by stating that in a given experiment,

0 and r are assumed to be fixed. Solving Eqs. (2.1) for V and U along a stability

boundary we get

2 2

0

2 2

0

,8

( ).4

r mV q

e

r mU a q

e

=

=

Eq. (3)


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We see that for a given mass to charge ratio, the ( ),a q versus q stability boundary of the

first stable region can be mapped onto the ,V Uplane as shown in Figure (6) below. This

has been done for 3 different mass to charge ratios, all three curves having beendisplayed on the diagram below.

Figure (6)

Given the control line shown in the diagram, we see that species which have mass tocharge ratios of 28, 69 and 219 respectively, will pass through the quadrupole and be

detected as V increases along the line. In the first case, masses around 28 will pass ifV

is in the range of 37 to 68 volts. In the second case, masses around 69 will pass ifV is in

the range100 to 137 volts, and masses around 219 will pass if V is in the range 310 to

440 volts. In the figure under the one given above, we see that the line width of thechange current intensity versus time ( m V t i.e. the RF voltage is varied linearly with

time) is dependent on the width of the line segment cut by the control line across thestable region. Note also that the line width is proportional to the mass of the particle. Ifthe slope of the control line is increased, the line width decreases.


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A Global View of Stability

Seen in another way, we can think of the m/z ( =z e) curves given above as crosssections of a surface in , ,U V m space with the control line being a plane in that space.

We can make this apparent as follows. If we introduce the constant K, which is defined

as2 2

0

8

rK

e

= ,

and the scaled versions of andV Udefined as

/

/ 2 ,

V K

U K

V = ,

U =

equations (3) take on the forms given in the relations

,

( ).

mq

maq=V =

U

Solving the first of these equations for q and substituting this into the second, we get

aU(m,V) =m (V/m),

where it is clear that U is a function of the particles mass m and its scaled RF voltage

V as seen in Figure (7).

Figure (7)


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If a control line (here seen as a plane) is passed through this surface we get the plot given

below in Figure(8)

Figure (8)

If now cross sections of the plot in Figure (7) are displayed in the figures below, we get aseries of plots.

In the first case, a plot of U vs. V for various values of the mass is given below inFigure (9).

Figure (9)


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We can see that the position of the maxima move to the right as the mass increases and

the height of the max points also increases with the mass. In fact, one can easily show

that maxV and maxU are proportional to the particle mass i.e. 0.7060m=maxV0.23698m=maxU . The ratio of these two values determines the slope of the control line,

which passes through the origin, and the tips of each of the curves.

Figure (10)

That is, 0.33567U = Vis the equation of the control curve, which passes through the tips

of all of the mass curves cf. Figure (10). A scaled RF voltage ( 1)mmaxV corresponding

to the location of the peak in the m1 curve and with constant voltage ( 1)mmaxU will

uniquely pass a particle with that value of the mass and no other, at those values of the

two voltages.

In another cross section of the surface shown in Figure (11) we plot U vs. m for variousvalues ofV .

Figure (11)


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One can see that the control plane (here seen as a horizontal line corresponding to the

value of 1V =V) intersects theU versus m curve. That is to say, for a given V there is a

range of masses that will be permitted to pass through the quadrupole apparatus.

In order to narrow down the uncertainty in the mass; it is the usual custom for quadrupole

mass spectrometers to be operated with a control line of high slope as shown inFigures (5) or (10). That is, the control line is chosen so that it cuts across the tip of the

vs.U V curves, thereby allowing for high resolution of the mass spectrum.

In general its really not possible to pick a control line that is linear and also just cut

through the tip of the vs.U V curves. This is due to imperfections in the fields and the

use of round rods rather than hyperbolically shaped ones. In modern instruments, an

ideal curve scan function of vs.U V is used instead of a straight line. This can be seen

in Figure (12) below.

Figure (12)

References

Dawson, P. H. Quadrupole Mass Spectrometry and its Applications, Elsevier Press,

Amsterdam, 1976. Also see a later edition of this text published by the AmericanInstitute of Physics in 1995.

McLachlan, N. W. Theory and Application of Mathieu Functions, Dover Publications,1964.


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Appendix A

Notation

The symbol for the Laplacian operator 2 in two dimensions has the usual meaning i.e.

2 22

2 2x y

= +

.

Similarly, the gradient vector operator is defined by

x y z

= + +

i j kr r r

,

where i, j, k are the unit vectors pointing along the x, y, and z-axes respectively.

Formulae for the First Few Eigenvalues and Eigenfunctions of the Mathieu

Equation ( 1q< )

2 4 6 8

0

1

2

7 29 68687

128 2304 18874368a q q q q= + + +L

2 30

1

32

1 1cos(2 )

2 128

1cos(4 ) cos(6 ) 7 cos(2 )

9( ) 1 q q qce + = + L

2 3 4 5 6 7 8

1

1 1 1 49 55 265

8 64 1536 589824 9437184 113246208

11

368641b q q q q q q q q= + + +L

( )21 1 1 1sin( ) sin(3 ) sin(3 ) sin(5 )8 64 3

( ) q qse + += L


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Appendix B

In the figures above we have plotted the electrical potential ( , )x y at two differenttimes. That is to say the surface oscillates in time between these two shapes. Note that the


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surface contains a saddle point. One can see that if a particle were near the saddle point

and if the surface oscillated at just the right frequency it might be trapped i.e. execute

stable motion within a small region near the saddle point. If on the other hand it did notoscillate at the correct frequency it would slide off the surface towards infinity. That is to

say, the motion would not be stable in time. This is the physical basis of the quadrupole

mass spectrometer.

The actual path of the changed particle through the mass spectrometer for a stable motion

is pictured below.

In a similar way, the path of a charged particle with unstable motion is show as spiralingout of control.

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