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NATIONAL BUREAU OF STANDARDS REPORT

8064

A VELOCITY MODULATED TIME-OF-FLIGHT MASS SPECTROMETER

By

R. M. Mills

/c?. a

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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Printing Office, Washington, D.C., 20402.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1002-11-10421 August 12, 1963 8064

1002 - 11-10121

A VELOCITY MODULATED TIME-OF -FLIGHT MASS SPECTROMETER

By

R. M. Mills

A large portion of the funds for support of

the work reported were provided by

Bureau of Ships

Department of the Navy

Code 638

Project No. SF150703, Task No. 3352

IMPORTANT NOTICE

NATIONAL BUREAU OF ST

intended for use within the

to additional evaluation and

listing of this Report, either the (Mice of the Director, N

however, by the Government

to reproduce additional copl

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST)

on October 9, 2015.

progress accounting documents

rmatiy published it is subjected

, reproduction, or open-literature sion is obtained in writing from

Such permission is not needed,

prepared if that agency wishes

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

A VELOCITY MODULATED TIME -OF -FLIGHT MASS SPECTROMETER*

By

R. M. Mills

ABSTRACT

This paper discusses the possibility of

using velocity modulation to improve a timeof -flight mass spectrometer. Calculations

show that the resolution can be greatly increased while keeping the unusually high duty

cycle. Previously experienced difficulties

with harmonics are expected to be significantly

reduced by the use of velocity modulation.

The resolution has been calculated for

three experimental arrangements. The simplest yields a resolution of 650 at 80% duty

cycle. If a special "correction wave" generator is added, a resolution of 13,000 at

86% duty cycle seems possible. The third

arrangement is a variation of the first which

avoids certain grounding difficulties. It

is shown that this gives approximately the

same resolution and duty cycle as does the

first.

Equations are derived giving general

expressions for resolution and duty cycle

for the simplest velocity modulated arrangement. Finally, a rough experimental verification of the velocity/ modulated scheme

is given.

1. Introduction

A time-of-f light mass spectrometer has been built which

is capable of operating at an unusually high duty cyle [l].

Ions are first accelerated through a constant d-c voltage

and then pass into a cylindrical drift tube about a meter

long. A radio frequency (rf) sine voltage is applied at

the exit and entrance gaps of the drift tube in such a way

that the field at the entrance is the negative of that at

* The work reported is also being submitted to the George

Washington University in partial fulfillment of the requirements for a Master of Engineering Science Degree.
2

the exit. Ions of mass m which travel through the drift tube

in any whole number of rf cycles, n, therefore experience no

net change in energy. These ions which experience no energy

change are referred to as resonant ions. Mass discrimination

is achieved by measuring the number of resonant ions that

arrive at the collector located after the drift tube. It is

assumed that both gap widths are small so that the transittime through the rf fields is taken to be zero [l].

Besides high duty cycle, this instrument offers advantages

of simplicity and light weight. However, it has two serious

limitations. Its resolution is moderately low (about *+0), and

the presence of harmonics (for a given mass, there are peaks

for every whole number value of n) makes the instrument most

useful in applications requiring the analysis of a simple mass

spectrum.

This scheme assumes that the velocity of the ions passing

through the drift tube depends only on the mass of the ions

and the d-c voltage through which the ions are accelerated before entering the drift tube. It is shown in [l] that the

resolution is limited by the influence which the rf voltage at

the entrance gap has on the velocity of the ions as they travel

through the drift tube, referred to as velocity modulation. The

flight time of the ions through the drift tube depends on the

instantaneous value of the rf field at the entrance gap to a

significant degree, even though the peak to peak value of the

rf voltage is only a few per cent of the d-c accelerating

voltage.

This paper suggests a scheme whereby velocity modulation

can be used to improve resolution while keeping the high duty

cycle, and to reduce the cluttering due to the presence of the

harmonics

2. The Velocity Modulation Scheme

Figure 1 is a simplified schematic diagram of the proposed

velocity modulated time-of -flight mass spectrometer. As before,

ions accelerate through a d-c voltage, V, , and then pass

through the drift tube entrance gap, (a), across which a radio

frequency saw tooth voltage, V , , is applied. The saw tooth

voltage applied at both the entrance (a) and exit (c) gaps

rises linearly with time until the instantaneous flyback. To

the first approximation, ions entering the drift tube at phase

cot can be made to bunch at the midpoint of the drift tube, and
- 3 -

continue their flight to exit through gap (c) at phase -cot

by choosing the proper ratio of V , /V, . See Figure 2. This

means that the last ions of a particular rf cycle to enter

gap (a) are the first to exit through gap (c). Referring to

Figure 2, it is seen that the ions receive V , (t/T) electron

volts at (a) from the rf wave and -V , (t/T) at (c). Here t

is the time at which the ions traverse the entrance gap and T

is the period of the rf voltage, t is defined only when -T/2

< t < T/2 and is equal to zero when the r.f. voltage is zero

during the rising portion of the saw tooth wave. As before, the

resonant ions neither gain nor lose energy from the applied rf

fields and the same detection system as described in tl] which

measures the number of ions experiencing no net energy change

can be used. This measurement is made by the retarding field

method [2],

3 ]. Since the discussion which follows depends on

an understanding of the retarding field method, a review from

reference [1] will be given.

After leaving the drift tube through gap (c), the ions

are decelerated through a potential of -V. . They then

approach an electrical barrier placed immeSiately in front of

the Faraday cage collector. For a given barrier potential, E,

measured with respect to the ion source, only ions with energies equal to E electron volts or greater can pass. The relationship between S(E), defined as the number of ions per unit

of time that penetrate a barrier of height E, and N(e), defined

as the number of ions approaching the barrier per unit time

with energy between e and e+de at the barrier, can be expressed as:

Since the detection system must measure the number of ions

which have had their energies unchanged by the rf fields and

since these particular ions arrive at the barrier with the

same energy that they had at the source, N(e) at the point

where e = 0 is the quantity sought. N(e) is obtained from the

above equation by differentiating with respect to E:

00

S(E) = f N(e) de

dS(E)

dE

the point where e E
- 4 - The differentiation is approximated in the instrument by introducing a variation in E, AE, about the point E = 0. S(E) is

proportional to the collector current. The variation in collector current, or the a-c component, is therefore approximately

proportional to -N (e) for a given AE. Using AE of finite

size means that an average value of -N(e) is actually measured

over ^ the range of e = 0 ± AE. The variation in ion current is

amplified by a-c amplifiers and fed to synchronous detectors,

AE supplying the reference signal used in the detector.

Returning to the discussion of the velocity modulation

scheme, two modifications of the arrangement shown in Figure 1

have been studied. It will be shown that an even greater improvement in resolution results if an additional rf wave, called

the correction wave (e ), is applied to the exit gap (c) as

shown in Figure 3«

The schemes pictured in Figures 1 and 3 bave the practical

disadvantage that one, but not both of the end grids, shown as

dotted lines, can be at rf ground potential. Therefore, either

the ion source or collector must follow the rf voltage in order

to insure that the ions receive energy from the rf voltage only

at gaps (a) and (c). To avoid this complication, the scheme

shown in Figure 4 was studied. Two separate, but synchronized,

sawtooth generators are employed, one producing a linearly rising

voltage and the other a linearly falling voltage, the latter

generator being connected with its polarity reversed. The end

grids at gaps (a) and (c) are at rf ground. It can be seen in

Figure 4 that the resulting fields at gaps (a) and (c), going

from left to right, are linearly rising. This is a necessary

condition for operation of a velocity modulated mass spectrometer. All resonant ions are bunched when they reach gap (b)

at a time when the rf voltage goes through zero. The wave

form at (b) therefore, does not influence the resonant ions.

A glossary of symbols is included at the back of this

paper.

The kinetic energy may be equated to the potential energy

through which an ion has fallen and solved for transit time:

3« Calculations

3a. RF Wave Shape

rf

(1)
~ 5 - nere ‘t

( ac ) j_ s ^he time taken by ions to travel between (a)

and (c), L/ n is the distance between (a) and (c), m is the

mass of thea 5ons, t , n is the flight time of an ion when t

- 0 between (a) and0

(§7, and q is the charge of the ion. As

explained in Section 2, the device depends on the rf voltage

at -cor being the negative of that at +cor. The desired rf wave

form therefore has odd symmetry, and the desired flight time

for resonance is (see figure 2):

hac) = Vac) “ -°-* < "/T < °‘5 (2)

Note that ions which enter late in the cycle (large t) are

subjected to a higher accelerating potential and have a shorter

transit time, t/ \. Combining equations (1) and (2), and solving for the rf v voltage:

In the above equation, t must be much smaller than t / \ so

that the second order term (and higher) can be neglected.

Otherwise the expression does not have odd symmetry and equation (2) is not valid. This requirement means that higher harmonics must be used. If t « t / \ equation (3) reduces to

an expression showing that V « should be a linear function of

entrance time, t. It is thus 1 shown that a sawtooth voltage is

the proper wave form to produce, to the first approximation,

the desired velocity modulation in Figure 1.

3b. Peak Shapes of Scheme in Figure 1.

Define e « as the net energy, in electron volts, which

an ion receives from the entrance and exit rf fields, e « is equal to zero for resonant ions, e ~ can be expressed1

as

a function of the difference between two numbers, nn /.^and

m§<iS5lted i£ftc io

is the number of rf c/cles taken by velocity

m

^o(ac)

travel through the drift tube and is equal to

(ac) "t ^ie num^ er cycles taken by velocity

ated r sonant ions and is equal to t'

/T is a whole number. 'o(ae).. - 2 where

From the definitions above, it is clear that if

( ac ) = n2(ac)’ e rf ke e Qua l to -zero. However, if the numblr'of

recycles taken by a velocity modulated ion is different from

n

2l ) by some fraction of a cycle, e « will be equal to the

peaK^rf voltage times that fraction. rFor example, if an ion
- 6

takes 1/4 of a cycle longer than n

2( ac ) to Pass between gaps

(a) and (c), e : , will be equal to V

gtA. Since the rf voltage

rises linearly with time, e r £

can be expressed as follows over

most of the possible values of t: __ e rf

~~ V

st ! n

l(ac) n

2(ac)J (4)

n

l(ac) is determined physically by the mass of the ion. It is

possible to use a number of values for n

2( ac ) in equation (4)

corresponding to the whole number values of th

ac

j/T. However,

it is understood that n

2( ac ) will be chosen so as to minimize

the absolute value of . Difficulties may arise

^ n

l(ac) n2(ac)-^

if t is near ±0.5 or if the absolute value of r „

-\ n

l(ac)

" n

2(ac) J is large. See appendix.

Substituting for values of n

q( ac ) and using equation (1) to express t^

ac ) ^o(ac)

T W 1 + A

and n

2(ac)

w

t’ / x -

o (ac ) T

in equation j

As a special case of n

j_( ac )? define n

0 ( ac ) as the number of

of rf cycles taken by any ion passing through gar (a) at t - 0.

n

o(ac) can a ^ so de axPre ssed as t

Q ( ac

)/T* Likewise, define as a

special case of n

2( ac ) the new variable, n

o( ac )f equal to the

number of rf cycles taken by a resonant ion which passes through

gap (a) at t = 0. ^(ac ) is equal to t^ ac) .

• T

The above equation can now be written as:

rf = no(ac

- . n

o(ac) + 2s

n

V

st firm J

(5) —

y

where s - t/T and A = st A is to be considered constant. Its

V

dc

proper value for obtaining the desired velocity modulation is calculated as follows. When t = T/2, = 1/2V Therefore,
- 7

- neglecting higher order terms in equation ( 3 ):

V

st = 2T

2V.„ t dc o(acj

At resonance, t o(ae) - t’ (ac) = n^ (ac)

T , so:

n

dc o(ac) (6)

It will be useful to plot e f/V , as a function of s. using

the above value of A in equation^

SLThe task is simplified by

first plotting er f/Vs ^ for n£( ac ) and then. finding the other curves by using the approximation;

small distance away, A £r©m the original Independent

variable, is equal t© ^rf^st at the original point plug

& n

0 ( aQ j. In ether words, equation (7) says that the ehangt in

dependent variable is equal to the change in independent variable

whieh is true ifi d

whieh is vdt t ease of s (A = 1/10).

Figure $ is a set ©f t

r jAgt curves for n^g) equal to 3?

through 4l. A was taken as 4/nJ. . a i/iQ, The non-iero values of for the n^^) E 40 curve when s / 0 are the re- sult of neglecting higher order terms in equation (3)»

From equation ($) it is seen thati
- 8 - The duty cycle can be found from Figure 5 by noting what

fraction of a cycle e^/V ^ stays within arbitrarily chosen

limits. In figure 6, the peak shapes are plotted for 0.00 < e

rf/"^st < 0.026. It is seen that the peak heights decrease as

n

o(ac) ciiang ed from a value of b/A. Define n

Q( ac ) as that

particular value of n* , which is equal to:b/A and produces the

highest duty cycle peak.

The presence of multiple harmonics (discussed in Reference

1) is reduced, but not eliminated. Here, though, there is

much more space between successive harmonic peaks, thus reducing

the problem of cluttering in a complex mass spectrum. For the

case shown, the resolution appears to be about 650 with a duty

cycle of 80$.

3c. Peak Shapes of Scheme in Figure 3

; The ideal curve shape for e^/V ^ versus s is a horizontal

line for the nj^

ac ) harmonic and vertical lines corresponding

to all other values of n

0 ( ac )* One way to approach this condition is to apply an additional rf voltage, called the correction wave (e ), to the final gap, (c) . This correctional wave

should be equal to the negative of the e r f/^ s ^ versus exit time

curve. The transformation of Figure 5 from entrance to exit

time does not change the curve shape to any great degree. It

thus seems possible that the proper e wave shape can be approximated by a full wave rectification or a sine wave which has a

period double that of or 2T. Calculations were done to

find what the curves in Figure 5 would look like if an e of

this form is added to the last gap. Rather than transforming

all of the curves in Figure (5) to exit time, it was easier to

change e to entrance time. A new set of curves corresponding

to Figure 5 was drawn (see Figure 7) and from these, the peak

shapes were found as before. The expression corresponding to

equation (5) is:

e ~ _ n

rf - o

V

st

= n

o( ac ) + 2s * G (1= COS Tfs)

+
, 9 - where C is the amplitude of e

c

* Therefore, the approximation

f

iven in equation (7) still holds to the same degree of valdity. Figure 8 shove the peaks for values of n’ (ac) from

38 to kl with the arbitrary limit of the differentiating voltage chosen to be -0.00? < + e

g <0. The resolution in

V

st

this case appears to be about 13*000 with a duty cycle of 86$.

3d. Peak Shapes of Scheme in Figure k

It will be shown that the final epf for this case is, to a

good approximation, the same as the case shown in Figure 1. In

order- to simplify nomenclature, define e

rf/^ s ^ for Figure 1

scheme as F. Also let the subscripts (a) and (c) denote a quantity at or between the appropriate gaps. For example,

t^ ac ) is

the flight time between (a) and (c), and e^^) is the rf energy

in electron volts received at (b), the center gap.

As was mentioned earlier, the resonant ions are bunched at

the center gap. If the frequency of the rf voltage is such that

n

o(ac) an eyenJ ^ole number, the bunch of resonant ions will

traverse gap (b) when the rf voltage is zero and erf ^) is zero.

If the number, n

j_( a]3)* rf cycles taken by an ion to travel

between (a) and (b) is different from the number n

2(ab)> taken

by a resonant ion to travel the same distance,

(^)

is not zero.

Note that the rf Voltage at (b),is twice that at (a) and (c), arid

has the opposite sign. It can be seen that;

erf(b)

“ ~ 2V

st^

n

l(ab)

~n

2(ab) ^ , The net energy received at (a) and (c) is as before:”

erf(ac) ” v

st ^ n

l(ac) ~ ^(ac)^

The total energy received from the rf fields by an ion passing through the three gaps is:

erf(2)

= - 2V

st

(nl(ab)

" n

2(ab) } + V

st

(nl(ac)"n

2 (ac) )
10 - In the above equation n

1(ab) , n

1(ac) , n2(ac) , and n

2(ab) can be

expanded as follows:

n

l(ab)

" t Cab) = L

Cab)

Or:

n

l(ab)

= n

o(ab)

n

l(ac)

- b (ac

m

1

2qVdc '1 + St T

V

dc T

Or:

n

l(ac)

= n

o(ab)

4

TTTs

+ n

o(bc)

I "+ As + Aerf (b)

V

st

n

2(ac) t o(ac) 2 T

T

n

2(ab)

Since t

= t oCab)

o(ac)

= 2i: o(ab) ^is isnT ^ true t different V^

Q

can be

applied to the first and second drift tubes so that t / \ is o(,ac;

twiee ^(ab) 55

n

2(ac)

= 2n2(ab)

= 2n2(bc)

n

l(ac) can now 136 exPresse<i as:

n,/.-,) = n_,_^ 1 + n. l(ac) o(ab)

^ i + As o(ab)

I

1 + As + Ae

V

£f.q>) . st
11

can be written as:

r

erf(S) 2no(ab) - 2n2(ab) +j

n

o(ab) + n

o (ab)

V

st

|X + AS —

jjl’ + As "

, L. \

1 + As + Aerf(b 5 V

dc

-2n2 (

or:

V

*rf (2) = n

st

o (ab) 1 1

V

st

mcr ( 8 )

Expanding equation (8) using the first two terms of the binomial

expansion:

! rf(Z) '= n

o(ab)

st

V

1 - As - A

erf (b) -1 + As

2 V

st

e Ae. rf(Z) = - n

o(ab) ^rfCb)

T

at

2 V

st

But

erf (b) = " 2V

st f

n

l(ab)

_ n2(ab)

erf(b) = -V

gt ^2

n

j_( ab ) - 2n2(abl

And A = VnS(ac) = 2/n^ (ab)

Therefore:

erf (Z) = n

o(ab) F = n

o(ac) F

V

st

no(ab) no(ac)

( 9 )
12 - The net rf energy which the ions receive is therefore

the same as with the scheme shown in figure 1 when n

G ( ac ) is near n

Q( ac )? with one important exception. If n

0 ( ac ) = n

o( ac ) is an °dd number, n

0 ( at))

will not be a whole number*

This means that the resonant ion bunch goes through (b) at

a time when the rf voltage is at a maximum giving the ion

between ± .V ^ electron volts. The exact result is hard to

predict because of the discontinuity in the rf voltage at

corr = 7r, but it should be safe to assume that the odd harmonics

(i.e. odd values of n

Q( ac ) will be greatly reduced, if not

eliminated* The absence of the odd harmonics reduces further

the cluttering problem mentioned in [1].

When n

Q ( ac ) is not equal to ^o(ac )> e

rf(Z) is changed by

a factor of n / n/ti" / n . The -effect on the peak shapes is o(ac) oCac) , * a more rapid disappearance of harmonics as n / x is increased

O \ aC y above n

o( ac ) and a more gradual disappearance of the harmonics

for n / \ less than n"/ x. oCac; o(ac)

The actual resolution of the n

Q( ac ) harmonic for the scheme

shown in Figure b was calculated without using the binomial expansion approximation [i.e. by using equation (8)3* The results, resolution of 660 at 75% maximum duty cycle j are nearly

the same as those given in section B (resolution of 6?0 at 80%

maximum duty cycle), thereby reaffirming the conclusion that

the binomial expansion approximation results in the same expressions for e ^ as the scheme shown in figure 1.

3e. Maximum Duty Cycle

One of the principal attributes of the mass spectrometer

described in [l] is its high duty cycle. It has been shown

that velocity modulation can be used to achieve a considerable

increase in resolution. An approximate expression for maximum duty cycle for the velocity modulated instrument can be

derived to show that the increase in resolution has not been

made at the expense of the duty cycle.
- 13 - The derivation is based on equation (5)> which was derived for the Figure X scheme. If conditions are such that the

absolute value of both /As + A e

r f c^) tend As are small comV

st

pared to unity so equation (8) can be represented accurately

by the first two terms of the binomial expansion, the following

derivation applies to the schemes of both Figures 1 and 4 since

the maximum duty cycle occurs when ^

Q ^ 8iC ) = ^(ac ) = no(ac)‘

Equation (5) can be expanded as follows:

V

rf - n

st

o(ac) 1 - As + 3/8 A

2

s2

- 5/16 A

3

s 3 +

2

Since A is equal to 4/n^

ac ^

V

st

r~

no(ac)~no(ac)

+ 2 1- no(ac)

L J

Pi t f n

o(ac)

—

+ 6

n

o (ac.

s' n

o(ac)

•no (ac)

( 10 )

+2s

-20 n

o (ac)

l o(al

( 11 )

n'

The maximum duty cycle occurs when n = n'

0 ( ac ) = n

o(ac) “ ^(ac)*

Making this substitution and neglecting the last term, equation (11) becomes:

e ~ 'v rf - 6s

Vst n"1

o (ac

( 12 )

The minimum of the e

r p/v

s ^ curve for the n”/ \ o(ac' harmonic

is at the origin (See Fig. 5)* The diff erentiating voltage

used for detection should therefore swing between zero and

some arbitrarily chosen upper limit, say °(. The maximum

duty cycle is found by solving equation (12) for s with

equal to and multiplying by two in order to account

for the positive and negative values of s. This quantity is

the fraction of the rf cycle for which e^/V ^ is between 0

and °( and thus is the desired expression:

Maximum Duty Cycle = 2

°( n

M

/ x o(ac) (13)
- Ik - 3f , Approximate Resolution of the n” Harmonic

An approximate expression for the resolution when n

G ( ac ) = n

o(ac) can a ^- so ^e derived from equation (12). As before, the

derivation applies to the scheme in Figure 1 and under some

conditions to the Figure k scheme as well. By definition intro

duced earlier:

^o(ac)

Therefore for a small change in m, Am:

m = 1/2 n

o (ac

Am A n / x (!*+) o(ac)

= n

o (ac) T = L (ac)y 2qVdc

The estimates of resolution which have been given are based

on equation (l4) , A ^

0 ( ac ) being measured graphically from

curves similar to Figures 5 and 7-

Define A+ and A_ such that their sum is the peak width

at half peak height point in figure 9* In order to measure

the duty cycle at n

0 ( ac ) - n

o( ac ) + ^he curve corresponding to fto( ac ) + A

+ can be drawn in above the curve

corresponding to n

Q( ac ) in Figure 5, and the points at which

this new curve leaves the differentiating region can be determined as before. If equation (7) can be assumed to be valid,

the new curve corresponding to n

Q( ac ) + A+

is parallel to the

n

o(ac) curve and shifted above the n”^

ac ^ curve by an amount

A+

. Therefore, instead of drawing a new curve above the one

corresponding to n

o( ac )? the n

g(ac) curve can t> e used if it

is imagined that the dj ffarentiating curve is shifted down

by an amount A+

. The differentiation zone actually has lower

and upper limits of e

rf^s -i ~ 0 and erf/V

st

= c< - But if it

undergoes an imaginary shift down to new limits of : -A

+ and erf/V

st

= « - A+» the n

o(ac) ourve amerges from the

zone only at the upper boundry.

4»
- 15 - To find the value of s where the curve emerges from the

differentiation zone, consider equation (11). In this case,

the simplification of equation (12) cannot he made without

justification since n

0 ( ac ) / n

o( ac )* However, the needed

justification can he had from equations (7) and (12). Equation (12) shows that, to a good approximation, e^/V ^ varies

as the second power of s for the resonance curve. And in equation (7) it is seen that, again to a good approximation, the

e -/V , curve which is different from the resonance curve hy

rf s t

An / n is equal to the resonance curve plus the constant An / n o(ac; H oCac;

If this approximation is to he compatible with equation (11),

the first and third terms on the right side of equation (11)

must he the significant terms, since the first term in equation (11) is the same as the constant in (7)« Shifting the

differentiation zone is a way of accouting for the constant

term in equation (11). Therefore, equation (12) can he used

when n / \ = n"/ n + A^ if it is changed to read: o(ac; o(ac; +

2 _ r n / \ s - 6 o (ac

"/2

o (ac (12a)

Solve equation (12a) for s at the houndry of the shifted

diff erentiation zone where e^/V ^ - °< - A

+ : ro

S =

(« - A

+

) n

6 n

I I VJ o (ac

(12b)

o(ac)

If equation (12b) is multiplied hy 2 to account for the

positive and negative values of s, the duty cycle is obtaire d

as before:

Duty Cycle At (n" + A.) = 2

~ ^

n

o(ac) (12 c)

6 Mac)

The duty cycle at n

0 ( ac ) = no( ac ) + is also known hy definition to he 1/2 the maximum expressed in equation (13):

1/2 Maximum Duty Cycle

l*

n

o(ac) (13a)
~ 16 - Equating ( 13a) and (12c):

\

* n

o(ac) = 2

and solving for A

If 4 n", x » °C

o (ac ) /v

A

+

= 3/4 cc

[°( - A,)n"/ \ + o (ac)

’ 6 n

o(ac) .

= 2

6( “S(.c>*V

^n

o(ac) q(

4 o(

o (ac

(15)

In the same way A_ can be found by making an imaginary

shift of the differentiation zone up by A_. The n

Q ( ac ) = n

o(ac) curve now usually leaves the zone at both the upper

(i.e. e

rf/Vs

-£ = °< + A_) and lower ( e

rf/v

s t = A_) limits.

Thus the portion of the s scale which contributes to the

output signal will be twice the difference between two terms,

one corresponding to upper limit and the other to the lower

limit. Again using equation ( 13 ) multiplied by 1/2:

« n", x o(ac) ^2

n

o(ac)

Solving for A_

If n” / \-A o(ac;

(°c + A ) n" n o(ac) L n" , 2

, ^ n

o(ac)

\ = cC if

1 ^ n

o(ac)

0 (ac ) n

o(ac)

l6\^o(ac) J

is approximately equal . to unity:

+1-1/2/ no(ac)

A = _JL. °< 16

I

(16)
- 17 -

It is possible that the n", curve leaves the differentiation zone only at the lower limit after the imaginary shift

up. If

(c< + A- ) n

o(ac) > 3/2

n",_ n-A

this condition exists. In this case, the derivation of A

starts with: —

°< n

M

/ \ o(ac)

rsJ= 2 0.5 - A n

M

/ x

I - o(acJ

6 L 1

^ n

o(ac) ^

However, in practice, °c and A_ will almost always be small

enough so that equation (16) can be used. It will be assumed

that this is the case.

Since An/ \ = A +A_L = 21 , equation (14) can be otacj - + yz"“ 1 written, as:

m - 8 no (ac

Am 21 °(

4. Practical Considerations

Three assumptions are made in the above derivations.

First, it is assumed that the initial ion energy spread can

be neglected. In practice, the energy spread can be made

insignificant by using a large V^

c

. It is shown in [4] that

the limit in resolution is:

m _ V

Am 2V

o

where V is the total energy of an ion in the drift tube and

V

q

is its .initial energy. Thus, V

Q

determines the lowest

value of which can be used to obtain the desired resolution. Second, it is assumed that a satisfactory saw tooth

wave generator can be built. That is, the wave must have the

necessary linearity at the required frequency and amplitude.

The fly back time does- not need to be instantaneous since

Figure 5 shows that ions which pass gap (a) near t = ± 0.5T
- 18 - are not within the differentiating region anyway* • And third,

the electrical harrier is assumed to have the necessary energy resolution. Rough estimates of these three assumptions

indicate that their validity is good enough for the instrument to operate, hut perhaps at a lower duty cycle and resolu

tion than calculated.

Possible parameter values are as follows!

V

dc

V

st

L

(ac) - Saw Tooth Frequency - Corresponding Mass Range

3000 volts

300 volts

1 meter

1 to 10 me

mass 8 to 73^

Since the dimensions of the instrument are a few per

cent of the fundamental bonier component wave length at

10 me, dispersion of the wave shape may become a problem,

requiring short transmission lines. In addition to wave

dispersion, there will be a phase difference between the

waves at gaps (a) and (c). However, this is not expected

to cause trouble. Consider Figure 4. If there is a high

frequency sine wave on the synchronizing line, there will

be a phase difference between the two ends of the line.

Express this difference as t /seconds. Equation (2) should

now be changed to p

t (ac)

= ^o(ac)

+ V 2t (2a)

Combining (2a) and (1) as before and solving for V

rf/Vdc : V

rf ZZ 1 -1

V

dc I -

2

t t

I 0 0

t / — 2t

If is small compared to unity, the expression for

t

o

Vrf/Vdc can be approximated by a binomial series, neglecting
- 19 - all terms of the second or higher powers”.

V

rf

V

dc

= H- -n

1

o (ac)

- 1/2 tL

= A - 1/2

Or:

V

rf

= V

st - 1/2

T

This plots as a linearily increasing voltage with increasing

t. The line is shifted down below the origin by an amount

V

st

z 6 Since the total voltage at the drift tube is

2T

V

dc

+

^rf* same can 1° e achieved by leaving V

r£

symmetrical about the origin as before , and reducing V,

r- A 4- “1 ac

to V,dc

change.

1

- A t

4

2T

-]• Normally this will be an insignificant

The phase difference between the gaps will also

manifest itself by increasing n” / n from a whole number

. o \,ac

by an amount .

A rough verification of the velocity modulation ideas

has been made with the instrument described in [1]. By

expanding a sine wave in a power series, it is seen that

sin x is a linear wave for small values of x. . And since

sin (x + tt) = - sin x, sin x is also a linear wave having

a negative slope when x is near tt. The apparatus has the

property that the rf voltage applied at gap (a) is the negative of that at gap (c). Thus resonance can occur using the

velocity modulation scheme with a sine wave if n

Q( ac ) is a

whole number plus 1/2. The proper peak to peak sine voltage,

still denoted as V , , can be derived by first expanding

equation (1) in a 'Binomial series:

(ac) ’o(ac) X

- V

rf

2V

+

dc

If Vpf/V^Q is small:

^(ac) ^o(ac) 1 - st W

sin cot (a)

dc

(18)
is t .(> V.ine fit vmxc.-.‘. an 10a

(a). Unlike c values from 0 b<

c. '.rou' _a gap

let denote

c/:e time at which the ion passes gap \c) - Then

'(c) "(a)

' ^(ac)

Cl s ing e qua t i on ( 1

) i T

(c) T

(a)

r ^o(ac) L

o ( a c ) s in ore (a)

It is shown in simple klystron theory [Jp that bunching

occurs when

isl 0.

This means that there is no change in ion arrival tine per

change in starting time. In the scheme being discussed here,

it is required that!

a t

LslL -i

d T i(a)

(20)

In other words, the change in arrival tine must be equal

to the negative of the change in starting tine. An ion

starting late by a certain amount must arrive early by the

same amount. Differentiating equation (19)*

d t

d t

Co) - 1 - i Vac) ° CGS f0T (a)

T

(a)

d t

d t

(a)

(c) - 1 -

x o(ac). cos cor (a) (21)

.aquation (21) equals -1 if t ^

is zero and whet

krr no(ac) - 2

Or i

tt no(ac
21

it n

12.000 if V

dc = 3,000 volts.

7r n

'o(ac)

o (ac)

^ r ’ ^st 3^20 volts, peak to peak n

o(ac)

Figure 10 shows some velocity modulated peaks obtained

from the reference [l] instrument using a sine rf voltage,

for Argon. The velocity modulated peaks are the negative

peaks which begin to appear at about n \ = 23*5 and ino V a c ) crease in intensity until the 26.5 or 27*5 harmonic. The

peak to peak sine voltage was set at roughly l*+5 volts using

an oscilloscope. However, the amplitude can not be maintained

exactly at this value as the frequency is swept with the

present equipment.

The negative direction of the velocity modulated peaks

indicate a 180° change between the reference, differentiating voltage and the signal from the Faraday cage. The

differentiating voltage is actually applied to the source

and the barrier which is in front of the Faraday cage is

at a constant potential. As the ions leave the source, they

are energy modulated in phase with the reference voltage.

When they reach gap (a), they are velocity modulated so that

T, the first ions to leave (a) are the last to arrive at (c). n The velocity modulation is then neutralized at gap (c). The

velocity modulation at gap (a) therefore accounts for the

phase shift of the resonant ions between the Faraday signal

and reference signal.

No further study has been made of velocity modulation

using sine voltages since it is felt that an instrument

using a saw tooth wave will be greatly superior, especially

in duty cycle. However, Figure 10 and the theoretical discussions on the use of velocity modulation in a time -of

flight mass spectrometer demonstrates the feasibility of

building an instrument which has high resolution and high

duty cycle.
22

4. References

[1] R.M. Mills, "Journal of Research", National Bureau

of Standards, Section C, Volume 670? Number Oct. -Dec.

[2] R.E. Fox, W.M. Hickam, D.J. Grove and T .Kjeldass, Jr

"Rev. Sci. Instr." 26, 1101, (1955).

[3I L.B. Leder and J.A. Simpson, "Rev. Sci. Instr." 29 . 571, (1958).

[41 H.S. Katzenstein and S.S. Friedland, "Rev. Sci.

Instr.", 26, 324, (1955)

[5] "Klystrons and Microwave Triodes", hy Donald R.

Hamilton, Julian K. Knipp, and J.B. Horner Kuper,

M.I.T. Radiation Laboratory Series No. 7, McGrawHill Book Co. Inc., New York, (1948), p.206.
23 -

5* Glossary of Symbols

ACe - c

erf'

erf2-

FL

(ac)

-

»o(ac)-

n

1 / \ o (ac ) -

i »> II / \ O RC) ~ n

l (ac) - n

2 (ac ) -

1“

sTj;(ac)-

t o (ac) - ^o (ac)

“ Vdc~

VrfV f /V,

st dc

Amplitude of the correction wave, e

Q

Instantaneous correction wave value

Net energy, in electron volts, an ion receives from the

rf fields.

Net energy, in electron volts, an ion receives from the

thre^ gaps in the scheme shown in Figure 4

e rf/ v st for the case shown in Figure 1

Distance between gaps (a) and (c)

Mass of ion

Number of rf cycles taken by an unmodulated ion to

travel between gaps (a) and (c). Equal to

b o(ac)/T

Number of rf cycles taken by an unmodulated resonant ion

to travel between gaps (a) and (c). Prime denotes a

whole number of cyles. , That particular value of n

o(ac) equal to 4/A. A should

be adjusted so that no(ac) I s a whole number.

Number of rf cycles taken by any velocity modulated ion

to travel between gaps (a) and (c). Equal to t( ac )/T « Number of rf cycles taken by velocity modulated,

resonant ions. Equal to t / n - 2t ^ o (ac JT

Charge of the ion

t/T

Period of applied rf voltage

Travel time of an ion between (a) and (c).

Unmodulated flight time between gaps (a) and (c).

Unmodulated flight time of a resonant ion between gaps

(a) and (c). Equal to ni( ac )t.

d-c voltage thio ugh which ions are accelerated.

Instantaneous value of the rf voltage.

A+ - A

nrco

o(-

Peak to peak value of the rf voltage.

n n -nl / n when the duty cycle is half its maximum

ovacy o\,acy o n / \ value, on the high o(ac) side of the peak.

n % - nl/ \ when the duty cycle is half its maximum

ovacy o lac ; n . \ value on the low ~o(ac) side of the peak.

Time at which an ion passes the entrance gap, (a); considered to be the independent time variable.

Angular frequency of the applied rf voltage.

Arbitrarily chosen upper limit of the differentiating

voltage in units of a

r:f*/V .
APPENDIX

The Validity of Equation (4)

Strictly speaking, equation (4) is not valid for all

ions because of the discontinuity in the rf voltage. Consider Figure 11, where one cycle during which ions pass

through the entrance gap and one cycle during which they

leave the drift tube through the exit gap are divided into

three periods. The widths of both periods I and II are

( n

l( ac ) ~ n2(ac)^ T * The exPress ^- on f°r e

rf

Tn equation^)

is derived by comparing the flight time of the ion in question with a resonant ion. If a resonant ion passes the entrance gap during period I, it will exit during period III.

If a non-resonant ion enters during period I, and if it is

lighter than the resonant ion such that n

j_( ac ) “ n

2(ac)

<"^

> the ion exits during period II and equation (4) is valid.

The trouble comes when a non-resonant ion leaving during I,

is heavier than the resonant ion so that n1 , \ - n0 / ^>0. Mac; 2Cac;

In this case, the non-resonant ion emerges after the beginning of the next cycle (i.e. after the r.f. voltage fly back).

This ion has passed through negative rf fields at both the

entrance and exit gaps and therefore experiences a large

deceleration instead of the moderate acceleration it would

have received if it had started during II or III, In the

same way, a non-resonant ion which is lighter than the resonant ion starting during period III experience accelerations

at both the entrance and exit gaps, again making e

r^ larger

than predicted in equation (4). In summary, equation (4)

is valid when

- 0 . 5+ (nl(ac)“n

2(ae) ) < t/T <0.5

if n

j_( ac ) ~ n

2(ac) P°s ^tive an(^ when

- 0.5 < t/T < 0.5+ ^ n

l(ac) ^(ac))

if n

j_( ac ) ~ n

2(ac) neSative. This means that equation

(4) is valid when ions heavier than resonant ions leave

during periods II and III and when lighter than resonant

ions leave during periods I and II. In all other cases

the absolute value of e

p

^. is much larger than that given

by equation (4). When the above conditions do not hold

equation (4) can be made valid again by changing n

2(ac)

t0 n

2(ac) + 1 when n

I(ac)

“ n

2(ac) ls Positive or to

n

2(ac) - 1 when ^j_( ac ) - n

2(ac) is ne§a

'tive -
- 25 -

The above complications are ignored in the derivations that follow. This simplification can be justified

by examining the consequences (1) at resonance, (2) near

resonance, and ( 3 ) far from resonance. (1) At resonance,

n

l(ac)

” n

2(ac)

= so ^at the exceptions do not apply.

(2) Near resonance the effect of using n

2(ac) ins tead °F the

correct value of n0 / n + 1 or nn / % - 1 is to make e ~ 2(ac; 2Cacj rf

larger than predicted, so that ions which are close enough

to resonant ions to be normally represented in the detection

system's output signal are excluded from the signal during

a fraction of each cycle equal to n

q( ac ) “ n

2(ac) # °^er

words, the actual duty cycle is reduced by a factor of

n

l(ac)

~ n

2(ac) ^rom the calculated value. Note that this

effect does not change the peak height at its maximum and

causes a more rapid decline as n

0 ( ar>)

is changed from resonance. Thus, the net effect is a sharpening of the peak

shapes. However, this effect is small for case 2 (near

resonance) since n

q( ac ) ~ n

2(ac) neapiy equal to zero.

( 3 ) Far from resonance the use of n0 / % instead of

n

2(ac)

+ i or n

2(ac) Pesuits again in a larger absolute

value of e

r £

than calculated, although the increase is not

as large as in (2). If the increase in is large enough

the ions are not represented in the output signal. In this

case the reduction in duty cycle will be larger since

n

l(ac)

“ n

2(ac) iapg er * However, far from resonance the

calculated duty cycle is near zero (see Figure 6) and a

further reduction is not significant.

USCOMM-NBS-DC

EXlT(c)

FIELD

ENTRANCE

(a)FIELD

AVI AV

DRIFT TUBE

(a)

r r A/W AA/V

(c)

SAW TOOTH

GENERATOR

FIG. I

(/)

Ui

tO

CO

£

7

oII

TIME t

LX

-T —

r

4- ll

t(oc)_

1

TIME r

cn

LU

CO

CO

£

FIG.

o
FIG. 3

u

DRIP

(o) SAW TOOTH

GENERATOR

in

: TUBE

(b) | SAW TOOTH (c)

GENERATOR

L.

/ SYNCHRONIZING LINE

** /.«IW,

< 1/

1?

TIME

FIG. 4
CD

Hh

u UJ

2

UJ

<

q:

h"

z

UJ

I

FIG. 5
6

0

T

C\J O

6

v

>i v < O

CD N; H> o o o

31DAO Aina

FIG.

6-

PEAK

SHAPES

USING

SIMPLE

VELOCITY

MODULATION

SCHEME
re>

u.
313A0 Aina

I

00

6

PEAK

SHAPES

USING

CORRECTION

WAVE
EXIT

FIELD

ENTRANCE

FIELD

H

XO

LUX

FIG. 9

TIME - n

'o(oc)

TTT

TIME

HihiHlh

i

FIG. II

U. S. DEPARTMENT OK COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A.*V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

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USCOMM—NBS-DC