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NOV 2 7 1970

NBS TECHNICAL NOTE 388

U.S.

ARTMENT

OF

)MMERCE

National

Bureau

of

Standards

The Iron Storage Technique

for Application in Frequency

Standards
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NBS TECHNICAL NOTE 388

ISSUED NOVEMBER 1970

Nat. Bur. Stand. (U.S.), Tech. Note 388, 23 pages (Nov. 1970)

CODEN: NBTNA

The Ion Storage Technique for Application

in Frequency Standards

Helmut Hellwig

Atomic Frequency and Time Standards Section

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

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Table of Contents

THE ION STORAGE TECHNIQUE FOR APPLICATION

IN FREQUENCY STANDARDS

Helmut Hellwig

Page

Abstract m ^ v Key Words m ^ v

1. INTRODUCTION 1

2. THE ION STORAGE TECHNIQUE 2

3. STORAGE CAPACITY 7

4. COLLISIONS, HEATING, AND ESCAPE FROM THE TRAP 8

5. EXAMPLE OF AN ION STORAGE DEVICE ......... H

6. FREQUENCY ACCURACY ................. 13

6. 1. Stark Effect . 13

6.2. Ion-Ion Collisions (Pressure Shift) 14

6. 3. Second-Order Doppler Shift . . . 15

6.4. Shifts Due to Magnetic Fields 15

7. FREQUENCY STABILITY 16

8. CONCLUSIONS. is

9. REFERENCES 22

List of Figures

Figure 1 Schematic configuration of an ion trap 2

Figure 2 Potential curves of the ion trap (cylindrical

coordinates) 21

111
THE ION STORAGE TECHNIQUE FOR APPLICATION

IN FREQUENCY STANDARDS

Helmut Hellwig

Atomic Frequency and Time Standards Section

National Bureau of Standards

Boulder, Colorado 80302 USA

Abstract

The basic technique of trapping ions in a quadrupole

configuration driven by a radio frequency field is described,

following closely a discussion originally given by H. G.

Dehmelt. Relationships between the trap size, ion oscillation frequency, frequency and amplitude of the rf driving

field, depth of the potential well, and the storage capacity

are given, and a possible trap configuration is numerically

evaluated. The effects of collisions, rf heating, and the

loss of ions from the trap are briefly discussed. The

second-order Doppler effect is identified as the most

serious accuracy limitation, whereas electric and magnetic

fields as well as ion collisions are not expected to impose

—14

any accuracy limitation worse than 10 . The frequency

stability performance of an ion storage device is estimated

to be of the same order as for conventional atomic frequency

standards.

The ion storage technique thus is seen as a promising

candidate for a primary frequency standard if further experimental research leads to a control of the second-order

Doppler effect. Technical difficulties are anticipated also

in the continuous generation of ions and in their preparation

for interrogation.

Key Words : Figure of merit; Frequency stability;

Frequency standard; Ion collisions;

Ion storage; Ion trapping.

IV
THE ION STORAGE TECHNIQUE FOR APPLICATION

IN FREQUENCY STANDARDS

Helmut Hellwig

Atomic Frequency and Time Standards Section

National Bureau of Standards

Boulder, Colorado 80302 USA

1. INTRODUCTION

At present the most accurate frequency standard is the cesium

beam tube. This fact is widely recognized, and is reflected in the

international agreement of 1967 which bases the definition of the second

on the transition frequency of cesium. Naturally there is a constant

demand for a more accurate realization of this definition. In addition

frequency standards of extreme precision are needed in applications

such as navigation and communication systems, air traffic control

systems, and various special military systems.

Looking ahead it therefore appears both necessary and desirable

to find ways and means to meet these demands. The cesium beam tube

is already at a very advanced stage of technical development. An

accuracy capability of the order of 1 13 seems to be a limit for cesium

beam tubes which will prove difficult to surpass.

Alternative approaches are being sought which hold the promise

of significantly exceeding the accuracy performance of cesium beam

tubes (Hellwig, to be published 1970). They include the hydrogen maser

(Kleppner, et al. , 1965), atomic and molecular beam tubes such as thallium (Beehler and Glaze, 1966) and hydrogen storage (Hellwig, 1970),

infrared and optical devices such as the methane saturated absorption

(Barger and Hall, 1969), and the ion storage technique (Dehmelt, 1967).

The purpose of this report is to outline the basic concept of the

ion storage technique, its shortcomings and advantages, and its potential

in terms of frequency stability and accuracy.
2. THE ION STORAGE TECHNIQUE

An atomic frequency standard is based ideally on the electric

or magnetic dipole transition of an unperturbed free atom observed for

an infinite length of time. The atomic cesium beam tube and the atomic

hydrogen maser represent attempts toward realization of these ideal

conditions. In the case of the cesium beam, the ideal of the free atom is

realized to a high degree; however, the observation time, i.e. , the time

of flight through the microwave cavity, is fairly short. Interestingly, in

the case of the hydrogen maser, the reverse is true. These two devices

represent different compromises.

In contrast to this the ion storage technique holds the promise of

combining the virtues of both the cesium tube and the hydrogen maser,

and even exceeding their performances with respect to approaching the

ideal conditions. Trapping times far in excess of one second have been

achieved experimentally as reported by Fortson, Major, and Dehmelt

(1966) and by Dehmelt (1967). In the following equations (1) through (24)

of this paper a discussion of the trapping technique is given which relies

strongly on the survey article by Dehmelt (1967) in connection with

papers by Fischer (1959) and Wuerker, Shelton, and Langmuir (1959).

The notation of Dehmelt (1967) is used. For a more detailed derivation

of some of the following equations the reader is referred to Dehmelt'

paper (1967).

An ion with mass m and charge e, placed in an oscillating,

homogeneous electric field of amplitude E will perform harmonic

oscillations with the angular frequency Q, of the field

m'z = eE cos fit . (l)
A solution

z(t) = z + £(t) (2)

may be found where "z is the center of motion, and £(t) is determined by

eE

< C(t) = -C COS Ot with c = -Jr . (3)

The ions are confined to a region of dimension Jf if their center

of motion does not move. This form of trapping is unstable; any initial

velocity remains as a drift, and spatial confinement is not achieved.

However, if the oscillating homogeneous field is replaced by an

oscillating inhomogeneous field, we can create conditions which allow

spatial trapping due to the fact that a resultant force F will act on the

center of motion. Without proof it is noted that the force F may be

derived from an artificially defined pseudo-potential j/j |F = -grad 0, (4)

where relates to the oscillating field by

eE^(x,y,z)

*

= ^ ' (5)

In order to effect trapping, we must require a force which always points

to some center of confinement. Recalling the problems encountered in

dipole optics, e.g., in the beam optics of an ammonia maser, we are led

to a spatial quadrupole configuration as shown in figure 1. A hyperbolic

ring R of radius r is closed at both sides by hyperbolic end caps C and

C separated by a distance 2 z . The center of the trap is given by

(r, z) = (0, 0) in cylindrical coordinates. The electric potential of this
trap is described by

= A(r3

- 2z2

), (6)

where the oscillating field causes

A = A cos Qt . (7)

The electric field is -grad 0, and using eq (5) one obtains at the center

of motion (z, r")

eA2 0(r,z) ^-j^- (r

2 + 4z 2

). (8)

The ions are confined with their center of motion in a potential well

ip(r, z) as shown in figure 2. The depth of the potential well in the z direction (r = 0) is

D = 4z old^ (9)

and in the r direction (z = 0)

"'- 5

feJ •

From figure 2 and eqs (8) to (10) it is apparent that the center of motion

z~ itself will not be at rest. It will oscillate harmonically in the r and z directions with the frequencies Zo and oO according to the potential

curves of figure 2. Inspection of eq (10) shows that the wells will have
identical depth for the configuration r = 2 z . This case shall be

assumed for all further discussion/'4 It warrants Zo = 60 , and thus

z r prevents transfer of energy between the r-motion and the z-motion

which could lead to escape from the trap (collision with the electrodes).

The oscillation of the center of motion may be written

z - z cos 60 t (11)

z

where co can be determined from

z

eD = jmz2

co 2

. (12)

2 z

The actual motion of the ion is a superposition of two oscillations,

(a) the harmonic oscillation of z as described by eq (11), and (b) the

oscillation £(t) according to eq (3) but with the important addition that

the amplitude is no longer a constant but £ = £_(z) = £n

(t). We may

combine these equations with eq (6) to determine the electric field

strength. We obtain for the complete motion (in one dimension)

z(t) = z + £(t)

60 z = (1 + Jz^-pr cos Glt)~z n cos co t . (13)

&« z

Without proof we note (Dehmelt, 1967) that these results (the validity of

the pseudopotential approach) depend on the condition

60 << Q . (14) z

"Effective symmetrizing may also be achieved in the general case

r d 2 z by simply adding a dc bias to the oscillating voltage.
Wuerker (1959) has noted that trapping is possible as far down as

Cl > 2 u) , however , it is advisable to chose values of Cl > 1 60 and to

z z avoid any harmonic relationship.

Of considerable importance for the actual experiment is the

determination of the required rf voltage amplitude V . The amplitude

may be determined from eq (6)

.2 , o „2 v

o = |0(z = V r = 0) " 0(z = 0,r =ro

} ' = A

o

( o +2

o

)

" (15)

With A from eq (9) and W — from eq (12) we may write

V 2 <-Kr J z — S-D . 60 (16)

For r = 2 z eq (16) simplifies to

V = 3J2v -*¥- D

60 (17)

Assuming fi~ 10 60 the required rf voltage amplitude is about 40 times

the depth of the potential well.
3. STORAGE CAPACITY

The effective storage volume V may be approximated by a sphere

of radius z

V = 1„ Z3. (18 )

The maximum rpossible charge density n can be found from the & y pmax

Laplace equation

Albr

= 4v n . (19) 'rnax

Using eq (8) we calculate for the maximum ion density, n = — n » ° n max e 'rnax

n =

a ^ D

3 . (20) max 47Te z^

The total number of ions which can ultimately be stored is then

z D

N = -2— . (21) max e

For a typical trap D « 10 V and z « 1 cm we obtain N ~ 108

. max
4. COLLISIONS, HEATING, AND ESCAPE FROM THE TRAP

The movement of the trapped ions is not of the random. Brownian

nature but is ordered according to eq (13), resembling spatial Lissajou

patterns (Wuerker, et al. , 1959). However, the distribution of the

amplitudes of the individual ion motions, e. g. , their energies, will be

random. The energy of one ion is given by the time average of its

,1.2 —

kinetic energy (— mz ) . The mean energy W of one ion in the trapped

ion cloud corresponds to an effective ion temperature T. which may be

defined by (k = Boltzmann constant)

3kT. = W. (22)

i

We note that the formal temperature T. depends on various processes

related to the trapping and may greatly differ from room temperature.

The ions will collide with each other transferring motional energy.

The time between collisions T can be obtained from a formula for a

c uniform plasma given by McDonald (1957)

k T.

3 ^3

(^)

T

c

= —^n (23)

n Z In A

where the constant C has the value 0. 78 X 1 7 —tt^ r- . The

V

3/s cm3

symbols have the following meaning: A and Z = atomic weight and ion

charge, n = ion density, and A = ratio of the cutoff distance for Coulomb

interaction to that of closest approach. With A ~ Z = 1 (protons),

n = n , and A = z /r «-. z n

1 /3

one obtains for a typical trap max mm max

(D « 10 V, z r=s 1 cm) and for T.?« room temperature the value r «* 1 ms."

l c

',% Note that according to eq (23) for Coulomb type interaction, t decreases

as the temperature decreases.

8
This result indicates that ion-ion collisions occur rather frequently

in a trap filled to its capacity. Certainly j is much shorter than the

storage time of ions.

We therefore assume that the ion energies are distributed

according to a Maxwell function. Ions in the high energy "tail" of the

Maxwell distribution are capable of leaving the trap. The loss rate

l/r of ions can be calculated to be 1 B

To 2T VkT. B c 1

- a - e D/k T.

e D \ ' i (24)

This loss of ions represents not only a decrease in ion density but also

a heat loss to the trapped ion cloud such that the mean ion energy decays

as exp(-t/r ). This type of cooling is, of course, not very practical. A very

promising method of extracting energy from the ions has been applied

successfully by Church and Dehmelt (1969). The method follows directly

from eq (13). An external lossy resonance circuit is coupled to the

micromotion frequency 60 of the ions thereby damping their oscillation.

Such cooling can be made very effective but is in competition with

significant heating processes of which the predominant one is collision

heating. Ions will collide with parent atoms or background atoms and

transfer energy to them which is subsequently dissipated in randomizing

collisions to the ion cloud. The ion, supposed to be at rest after such a collision (as an approximation), will pick up kinetic energy according to

eq (13) and figure 2. This energy has to be supplied by the trapping rf

field. The net effect is an energy delivery from the rf field to the ion

cloud. In the case of collisions with parent atoms, charge exchange also

has to be considered.
This collision heating can of course be avoided by working in a very good vacuum, or it can be reduced and even reversed (cooling) by

selecting heavy ions and a background gas of light atoms in which case

the light atoms will quickly transport energy away from the ion cloud.

In order to get a feeling for the vacuum which is necessary to avoid

collisions we may write down approximate, numerical relationships

between the collision time t and the particle density n and the pressure p

r

r = C^; n= C

2 p , (25)

where the coefficients are approximately C « 10~° Ns/m2

, Cp ?« 3X10 ° N~ m~ . For example a vacuum of

10~8 N/m2

\ 1 N/m2 = 7. 5 X 10~ 3 torrj corresponds to a particle density

of 3X10 particle s/m3 and a collision time of 103

s, the actual values

depending somewhat on the kind of gas under consideration. To obtain

efficient cooling from a background gas of light atoms we must require

that the collision time is much shorter than the storage time. This

condition somewhat contradicts the free particle approach because a more efficient collisional cooling can be expected to introduce also

larger perturbations.

10
5. EXAMPLE OF AN ION STORAGE DEVICE

For the design of an ion storage device the following parameters

may be regarded as given: the ratio Cl/oj , the dimensions r and z and the maximum number of ions to be stored N . One has to know

max

the required depth D of the potential well, the amplitude of the rf

voltage, V , and the oscillation frequency oJ . \J z As before we shall assume r = 2 z . We then obtain from

eqs (12), (17), and (21) the following relationships

— eN

max

V = lyflSr- D , (26;

00 z

00 z * m z ^

or, rewritten in terms of the given parameters,

eN ~ max D = , z

o

V

Q .3^e^^, (27)

z

i m

max

CO e|- —;— 3 z 1 m /z

11
It is most interesting to note that the voltage amplitude V required to

store a given number of ions decreases as the trap size increases.'1"

The same is valid for D and t3 . This is rather important because

z lower voltages and larger trap sizes appear desirable from the

engineering point of view. An actual trap configuration shall be calculated using eq (27). The parameters are chosen as follows:

ft _. .„ .,8 10; z n = — = 10 cm; N = 10 . CO 2 max

This leads (for He4 ions) to

_ w

D = 1.4V; V = 60 V; -~ « 80 kHz.

2tt

For a given ion density n , Vn increases with the square of the trap

max u size.

12
6. FREQUENCY ACCURACY

Since no actual device is discussed in this report, all possible

uncertainties due to the method of interrogating the stored ions are

excluded, such as cavity pulling, external magnetic fields, microwave

spectrum, etc. Only those effects will be discussed which relate

directly to the storage process.

6. 1. Stark Effect

The Stark effect on a hydrogen-like atom has been calculated

by C. Schwartz (1959). The fractional frequency shift of the hyperfine

transition is given by

Au 14- 91 e

2 a4 - = - ?

E* , (28)

where E = electric field, a = Bohr radius, e = electron charge, and

£ - absolute permittivity. We may rewrite eq (28)

=¥- = CE2

, (29)

V

with C «* -3. 5X10 cm2 /V 2

. For the previously discussed trap

configuration, where electric fields of the order of 10 V/cm are used,

the corresponding Stark effect is clearly negligible. This feature, the

near absence of shifts due to the containing fields, is the most interesting advantage of the ion storage method. Other known storage

methods such as the coated storage bulb (hydrogen) or the buffer gas

cell (rubidium) introduce fractional frequency shifts of 10~ and more,

13
6.2. Ion-Ion Collisions (Pressure Shift)

The repulsive electrostatic force of the ions provides for

collision parameters which are large enough to prevent van der Waals

interaction. The only remaining interaction is a Coulomb interaction

leading to Stark effect perturbation of the energy levels analogous to

the one described in the preceding section 6. 1. A rough estimate of

the effect can be given by calculating the electric field experienced by

one ion at the distance r from another similar ion,' where r is the

mean separation of ions in the trap. We can calculated r from eq (20)

using the relation

r « n ' . 30 max

The corresponding average electric field seen by the ions is

E = ~ « en2/3 . (31)

r max

With densities of the order of 10 ions/cm3 we obtain E « 2 X 10" V/cm.

According to eq (2 9) such fields will cause totally negligible effect.

14
6. 3. Second-Order Doppler Shift

The second-order Doppler frequency shift is given by

V m'

where v = ion velocity and c = velocity of light. This frequency shift

can not be compensated, it can be reduced only by lowering the ion

velocity, that is, by ion cooling. However, we are not primarily concerned with the magnitude of the total shift, but with the uncertainty

associated with this correction. The frequency accuracy of the ion

storage device therefore depends critically on the ability to control

and/or measure the mean ion velocity.

6. 4. Shifts Due to Magnetic Fields

Various sources for magnetic fields exist in the ion storage

device. (1) The driving electric rf field has a magnetic component.

(2) The movement of ions corresponds to electric currents which cause

magnetic fields. (3) An ion moving through the containing electric field

experiences a motional magnetic field which is proportional to the

vector product of the velocity vector with the electric field vector.

All these frequency shifts shall not be discussed in detail.

They can be expected to cause only very small effects. In fact, as H. Schixssler and his colleagues have concluded (Schussler, Fortson,

and Dehmelt, 1969), they give rise to an uncertainty of less than 10~ 15

.

15
7. FREQUENCY STABILITY

The fractional frequency stability of a quantum electronic

frequency standard may conveniently be written as

K 1 a " = 7T (33)

The symbols in eq (33) have the following meaning: j is the measurement time interval, the constant K contains the device characteristics,

and M is a basic figure of merit'1' given by

M = /n~ v T (34)

s r

where T = relaxation time, and n = flux of signal atoms. As an

r s example we have for a beam tube (Cutler and Searle, 1966; Lacey,

Helgesson, and Holloway, 1966)

0.3 88

T Jlttv

and for a maser oscillator (Shimoda, Wang, and Townes, 1956; Cutler

and Searle, 1 966)

Km = ZJTltV V

hV

This illustrates the fact that for the same figure of merit M the operation of a device as a beam tube is superior to its use as a maser by the

factor** (Hellwig, 1970)

/kT

'"This M differs from the usual definition in the literature of a ''figure

of merit. " **This is only valid for frequencies y < kT/h.

16
The flux of signal atoms and the relaxation time which make

up our basic figure of merit are not the only important parameters in

determining frequency stability. The principle which is used in the

interrogation of the atoms is also important and is reflected in the value

of K. Using the same value of M, the ion storage device might be operated as a beam tube, as a maser, or using some other method. Its

actual stability performance will then depend on the corresponding value

of K.

Nevertheless it is of interest to compare the basic figure of merit

of the ion storage device with those of existing devices. In order to do

this, eq (34) will be rewritten introducing 77 = n T , the number of signal

atoms during the period T .

M = 7??T

r

. (35)

We shall now compare the cesium beam tube, a hydrogen storage device,

and the methane saturated absorption device with the ion storage device.

Two alternative assumptions are made: The optimistic assumption is

based on a 100 percent use of the stored ions (every one of the stored

ions can be excited and interrogated), the pessimistic one on values

actually realized in the experiment.

Table I Figures of Merit for Different Methods

Methane

Cesium Hydrogen saturated Ion storage

beam storage absorption Optimistic Pessimistic

l(f 10

5 108 103

1 10"5 102 10

10

6

1 105 102

17

V I0b T (s) r M(s^2

)

10"a

10

2
It can be concluded that the basic stability performance of the ion

storage device may be expected to be somewhere between the cesium

beam and the hydrogen storage devices.

8. CONCLUSIONS

We discussed the principles of ion trapping and some of the

problem areas involved. It was concluded that the trapping process

itself does not contribute significantly to frequency uncertainties. In

fact the total influence of the involved static and dynamic electric and

magnetic fields on the transition frequency was predicted to cause not

more than a fractional frequency bias of 10

It was shown that the action of the trapping rf field can be

described by a potential well with a finite depth and capacity. Typically

the maximum capacity corresponds to a trapping of 1 8 ions. However,

in order to avoid "boiling over" with its associated loss of ions and

change of temperature, the trap should not be filled to capacity. An

interesting feature is the decrease in the required driving voltage with

an increase in trap size for a constant total number of trapped ions.

The trap size is, of course, limited due to other requirements related

to the microwave interrogation of the trapped ions.

The movement of the trapped ions which is correlated with the

trapping rf field causes a second-order Doppler shift of the frequency

which depends on the mean square velocity of the ions. Cooling of the

ions, e. g. , by coupling their motion to an external lossy circuit, is

possible; however, various heating processes occur in the trap which

are not well understood. The lowest effective temperature attained so far

in an experiment was 900 K (Church and Dehmelt, 1969). The constancy

of this temperature is poor, and its value cannot yet be measured very

precisely. An optimistic assumption for the accuracy in temperature

determination would be ±50 K. This leads to a second-order Doppler

shift uncertainty of approximately ±2 X 10

-11 for helium ions and ±1 X 10"

for mercuryy ions.

18
In order to reduce the main source of heating, the collision

processes, an excellent vacuum has to be maintained, preferably below

10~8 N/m2 (& I0~

10 torr). This somewhat contradicts the requirement

for efficient ionization of atoms and preparation of the ions for interrogation where higher particle densities would be desirable.

A distinct problem area is the interrogation of the ions, or in

other words, the design of an operating frequency standard (Dehmelt,

1969). The discussion of these aspects could be the subject of a separate report. Here we shall only mention that present techniques of

ionization and excitation are based on collisions with electrons and a polarized atomic beam (e.g., cesium), respectively. Obviously these

techniques allow only a pulsed operation. Although this does not preclude a working frequency standard, continuous operation is highly

desirable. A solution might be optical pumping for the excitation and

chemical reactions for the ionization where neutral particles drift into

the trap region with subsequent reactions leading to ions. However the

efficiency of such methods appears to be fairly low.

In summary, one concludes that the main advantage of ion storage

is the ability to realize long interaction times without perturbing the

microwave transition significantly as compared to other storage techniques like the coated bulb for atomic hydrogen where a moderate

amount of perturbation occurs. As was pointed out in the preceding

section 7, however, the short term frequency stability cannot be

expected to be very high; but good slave oscillators might render this

aspect unimportant. The most severe problems are those of the

determination of the effective temperature (second- order Doppler shift)

and its constancy, and the techniques to be used for creating and

interrogating the ions.

Solutions are to be found in fairly extensive research programs,

the outcome of which can not yet be predicted.

19
-o o

V cosAt

Figure 1 Schematic configuration of an ion trap

20
D'

y

-r„

Figure 2 Potential curves of the ion trap (cylindrical coordinates)

21
9. REFERENCES

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