CC uARDED BY

NBS MONOGRAPH DO y 5 q. A

Electron Optical Studies

Of Low-Pressure GaseSriBRAf^'^

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

1
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UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Electron Optical Studies

of Low-Pressure Gases

L. Marton, David C. Schubert, and S. R. Mielczarek

National Bureau of Standards Monograph 66

Issued August 16, 1963

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 35 cents
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Library of Congress Catalog Card Number: 63-60061
Preface

This project had its origin in a proposal which Dr. L. Marton, the principal

investigator on the project, made to the Office of Naval Research in June 1954.

The proposal was based on earUer experiments on electron optical schheren

methods which had been carried out under Dr. Marton's direction in the Electron Physics Section of the National Bureau of Standards.

Realizing the importance of this new approach to low-pressure research,

the OflBce of Naval Research activated the project on Februar}- 1, 1955. Since

that time, a nmnber of people besides the principal investigator have worked on

the project. Chief among these were Dr. David C. Schubert, who served as project leader throughout the history of the project, and Stanley R. Mielczarek.

Space does not permit mention of all the other members of the Electron Physics

Section who assisted in the work from time to time, but recognition must be

given to the contribution of John Kneisly III, who designed the CassegTainian

system used to provide the synchronizing hght pulses ia the stroboscopic

operation. In addition, we thank August B. Dauses and Lytle E. Mann of

the Electron Physics Section instrument shop, whose careful, patient work

made the experimental part of the program possible. Our thanks are also due

to the various personnel of the Office of Naval Research and especially to Ralph Cooper who served as Contract Officer during most of the life of the

project and whose interest in the project and understanding of the difficulties involved made the task much more pleasant than it might otherwise have been.

John A. Simpson, Chief

Electron Physics Section

m
Contents

Preface Page

1. The importance and significance of low-pressure gas studies 1

2. Existing techniques for mapping low-pressure

gas flow 2

3. The principles of electron optical schlieren 3

4. Realization of the electron optical schlieren method 6

4.1. Electron optics 6

a. Optics of the beam-generating sec- tion 6

b. Optics of the test section 6

c. Optics of the detection section 8

5. Gas kinetics ..- 11

5.1. Beam source. _ : 12

5.2. Chopped beams 13

6. Stroboscopic mechanisms 14

6.1. Optics 14

6.2. Electronics 16

7. High-vacuum section 16

Page

8. Intermediate-vacuum section 19

9. Vacuum pump system 20

10. Auxiliary equipment 20

11. Experimental results 21

12. Interpretation of the data 21

12.1. Scanning of the negatives 22

12.2. Computation of velocity distributions, 24

12.3. Reliability of the iteration process 25

13. Appendix 27

13.1. The photographic camera 27

13.2. Earth's field neutralization 29

13.3. Variation of the compensating field with position along the ?-axis 31

13.4. Relative positions of the chopper disks. 32

13.5. Computation of the initial distribution function 34

13.6. Simplified approximate computation of the molecular density distribution.. 35

14. References 36

IV
Electron Optical Studies of Low-Pressure Gases

L. Marton, David C. Schubert, and S. R. Mielczarek

This document is the final report of research carried on in the Electron Physics Section

of the National Bureau of Standards during the period from February 1, 1955 to March of 1962 in developing an electron optical method for the visualization of low-pressure gas flow. The goal of this work was to develop and demonstrate the suitability of electron optical

techniques for recording the spatial distribution of gas molecules at fixed times with suffi- cient accuracy that the velocity distribution could be derived. The work concentrated on

the properties of chopped molecular beams of cadmium at extremely low pressures where the

densities of the particles in the molecular beam pulse correspond to a pressure of less than

10-" Torr. The project was successful. During the course of this program, equipment was built, methods of operation were developed, the theory of the electron optical schlieren was developed in some detail, and ways of efficiently converting the data obtained into

velocity distributions of the gas molecule were investigated.

1. The Importance and Significance of Low-Pressure Gas Studies

Low-pressure gas studies are concerned with

the motions of gas molecules under conditions

such that the mean-free paths of the molecules

are significantly larger than the mean interaction

distances, and therefore the equations of con- tinuum aerodynamics are no longer applicable. In the limit of approaching zero pressure, gas

molecules behave entirely independently [1, 2],' mo\nng in straight lines except when they strike

solid or liquid surfaces where they may be absorbed or reflected and may exchange momentum

and energy with the surface.

Early in the history of the gas flow problem,

researchers assumed that molecules were reflected

from a solid surface with an angular distribution

proportional to the cosine of the angle between

the reflected beam and the normal to the surface

[3], and that the velocity distribution corresponded

to the temperature of the reflecting surface [4]. Measurements of heat transfer to surfaces, however, showed evidence that the temperature of

gas molecules lay somewhere between the temperature of the incident gas atoms and that of the

surface. On this empirical basis, the concept of the accommodation coefficient a was defined as the ratio between the eflFective temperature change

imposed upon the gas molecules striking the surface

and the initial temperature difference between the

solid surface and the gas [5]. Thus,

where T,, Tr, and Ts are tlie temperatures of the

incident beam, the reflected beam, and the surface,

respectively. When the so-called cosine law

distribution for reflected gas molecules was used

to predict the tangential forces on the boundary

surfaces in gas viscosity experiments, the predicted

forces consistently exceeded the measured forces. To reconcile tljeory with experiment, it was as- sumed that the reflected molecules retain a

portion of their mean incident momentum components tangent to the surface. This empirical

I Figures in brackets indicate the literature references at the end of this Monograph.

property is expressed quantitatively by the viscous

slip coefficient, defined as

V=Pr/Pi, (1.2)

where and Pr are tlie average components of particle momentum tangent to the surface before

and after reflection. These are the relationships which should be

applied to the boundary conditions of ordinary

problems in continuum aerodynamics. One can

expect that both the viscous slip and accommodation coefficients will be functions of the particular

gas, the surface material, and the condition of

the surface. Unfortunately, existing data con- cerning these coefficients are rather scanty and

somewhat conflicting [6]. Moreover, calculations

of microscopic interactions of atoms impinging on surfaces are still incomplete [7, 8]. Without dependable knowledge of the boundary

conditions on model surfaces in a wind tunnel, one may expect that difficulty arises in relating

accurately the observed wind tunnel phenomena

to the basic physical properties of the flow. Determination of these coefficients for a single

substance suitable for use in a wind tunnel would

be considered a major breakthrough [9]. These

coefficients are readily computed from velocity

distributions of reflected molecules before and

after impact with a solid surface. Availability

of experimental velocity distribution data for reflected atoms would also be a powerful tool fo- investigating the physics of surface interactions,

which are now incompletely understood.

In the interaction of the upper atmosphere wit i missiles and satellites (a field of study which came

into prominence during the course of this project),

conditions approaching the limit of low-density

flow exist. The mean-free path of the molecules

becomes large in comparison with the dimensions

of the vehicle. Consequently, continuum effects are absent and transfer of energy and momentum

between the gas and the vehicle reduces to singleparticle interaction. The computation of heating

and drag effects at high altitudes thus depends

decisively upon knowledge of molecular velocity

distributions after reflection [10].

1
Besides these two more obvious uses for such

velocity field data, there are useful applications

in continuum aerodynamics, for at a short distance

from a sharp leading edge in In^personic flow the

rarefied state may occur locally, although the

greater part of the gas stream is nmch more dense

[11]. Whether conditions such as these are of

interest from the pure physics standpoint or for the more applied needs of aerodynamics does not

matter. There exists a great need for methods

of determining precisely the velocity distribution

of gas atoms. The problem of obtaining these

data interested a large number of investigators

in the past and imdoubtedly will continue to do

so in the future. It is probably worthwliile at

this point to review some of the methods that have

been applied in the past and to show where they

fail to give the necessary precision or sensitivity.

One may hope—by means of the electron optical

schlieren technique—to obtain more dependable

data whicli are valid in the lowest pressure regions.

2. Existing Techniques for Mapping Low-Pressure Gas Flow

The methods of mapping gas flow are naturally

divided into two groups, those useful at gas

pressures of around n atm and those that have

been developed for pressures below In some

cases the same method has been extended by

careful work to cover both regions.

One method, highly developed recently, for work at around atmospheric pressure is the hot

wire detector [12, 13]. In this technique the hot

wire detector measures the temperature of a wire with. knowTi geometrNJ- and heating power

supported in the gas stream. The gas stream

cools the wire, which in a manner similar to a Pirani thermocouple vacuum gage gives evidence

as to the conditions of flow at that point. Unfortunately, the response is really that of a flowmeter.

It measures a combination of pressure and

velocity. With careful design, the response can be made surprisingly rapid. Fluctuation fre- quencies up to lOVsec have been measured in

this wslj at atmospheric pressure [14]. At low

pressures, however, the response is correspondingly more sluggish, and it is thus unsuited to velocity

measurements at low pressures. It has been

highl}' developed over the years and remains the

standard ^vind tunnel tool of today.

In the higher pressure range, perhaps the

most elegant, most used, and most accurate

method of mapping gas flow is the optical interferometer [15]. In this case, light is passed

through the gas; differences in optical path length

show up as light and dark fringes in an interferometric display. The space between each two

successive dark fringes corresponds to one wavelength in path difference. This method maps an

entire flow in two dimensions, since the optical

path length is a function of the integral of density

along the path. It is not useful in the case of a trulj^ three-dimensional flow, but then only

point-bj^-point techniques are. Moreover, being

a continuum device, it fails from lack of sensitivity

when the refractive index of the rarefied gas

becomes indistinguishable from that of free space. At the present time, with the present equipment,

the interferometer is too insensitive for studies

of rarefied gases. How^ever, in the range where

it is useful it remains perhaps the best of all the current methods.

Anotlier optical method which shares many of

the advantages and the disadvantages of the

optical interferometer is that of optical schlieren [16]. This method also makes use of the

continuum properties of the gas. A light beam

is passed through the flow and observed in sucli a manner that the prismatic effects of gradient of density are observed. The word "schlieren" is a German expression meaning "striation," and it describes the early application of the method to observe striations in the flow patterns of heated

gases. The fact that the density gradients

(ratlier than the density itself) are observed has

both advantages and disadvantages. For sharply

changing flows the gradient of density is some- what easier to detect than the density itself but, on the other hand, if one is interested in mapping

the entire flow, the gradient is one step further

from the desired results than in the case of the

interferometric method. Although highly developed, the optical schlieren method fails in the

low-density range, as does the optical interferometer, because the index of refraction approaches

unity thi'oughout the region of gas flow. This

method, however, does serve as the inspiration for the electron optical schlieren method and the

theory will be developed in some detail in a later section.

In view of the importance of the problem, it is not surprising that many efforts have been made

to develop techniques which are workable at much lower pressures. One general method,

which although tedious and awkward in application is frequently used, is the capture of individual

molecules or groups of molecules as they arrive

at a target normal to the direction of flow. Several

variations have been tried, all of them with some

success. First in this class is the condensation

target [1, 17]. The beam material is collected on

a target which may be at a low temperature or react chemically with the atomic species. With

careful development techniques [18, 19], or Geiger counters, and radioactive tracing [20, 21],

it is possible to get extremely high sensitivity,

detecting the arrival of only a few or of a few

thousand atoms. It is a slow, cumbersome

method not much used and certainly not easily

adaptable for velocity determination.

The arrival of atoms can also be detected by the

surface ionization gage [22, 23]. This technique

makes use of the fact that atoms striking a hot

tungsten wire may be ionized if any of the work

functions of the tungsten exceed the ionization

potential of the atom. Using tungsten, it is

2
possible to get nearly 100 percent efficiency of

detection for the alkali metals. This method

can be adapted to give time resolution and will serve as a point-by-point detector to map a field of flow. Again, it is a point-by-point detector,

and hence it is a time-consuming, difficult problem

to map a flow of any extent. But it can be and

has been done.

A detector which offers some promise is the

so-called electron bombardment detector [24]. In

this case, the molecular beam is bombarded with

electrons which ionize about one atom in 10*. The ions pass through a mass spectrograph, and

in this manner the effect of most background gases can be removed. With the use of time-resolved mass spectrometry, this method could also give

time-resolved information. However, again the

difficulties inherent in such a method make it seem

rather unattractive.

A more obvious and straightforward approach is to adapt the techniques used in continuum aerodynamics to the low-pressure region by using radiation which interacts more strongly with the gas

atoms. In this effort, the absorption of ultraviolet light and of X-radiation has been used [25,

26]. At great expense and great effort, this method can be made to extend the range of the

usual optical techniques several orders of magnitude lower. This technique requires, however, the

use of either reflecting or quartz transmission

optics, and in the long run seems to be rather

limited.

Another technique which has been tried makes

use of the afterglow [27] of nitrogen (following RF

excitation at a point upstream from the test area)

.

This afterglow is sufficiently bright to be detected

and gives a response proportional to the gas density. This ingenious technique suffers from the

fact that now one is dealing with an excited and

partially ionized gas whose properties differ from

those of the neutral beams one started out to study. In many cases the focus of interest is on

just those properties which are different in the

neutral molecule from what they are in an excited

molecule or ion, and so again this technique has

its limitations.

In the search for radiation which interacts more

strongly with the gas atoms, the interaction of

electrons with atoms was not overlooked. The

direct application of this effect yields the method

which is known as attenuation by scattering [28]. This method, which has been used to measure the

pressure variations behind shock waves, employs

a thin, pencil-shaped electron beam which passes

through the gas and out a small aperture. Electrons scattered to a sufficient angle miss the hole

and thus reduce the current measure. Careful

current measurements therefore give a measure of gas density along the beams, and if an area is swept out by the pencil, then the flow characteristics over this area may be deduced. This

method may, in fact, be considered a special case

of the electron optical schlieren method, and is called bright field schlieren. It suffers from the

intrinsic difficulty that one is attempting to measure a small increment of scattered current in the presence of the large unscattered current. Moreover, in the sinaple form it is a point-by-point

method and hence the task of mapping a gas flow

of any extent is rather tedious. However, it does

lead us to the electron optical schlieren method,

which is the principal concern of this report.

Historically, the first electron optical schlieren

work was done by Boersch [29]. He used it to photograph carbon tetrachloride vapor jet at a pressure of about 1 Torr. Later, the method was

used by L. Marton and his collaborators to observe

small electric and magnetic fields of crystals [30,

31] and space charge in the cutoff magnetron [32]. Also, the first stroboscopic observations were made

in connection with this latter work.

3. The Principles of Electron Optical Schlieren

AH the methods of detecting gas flow by means

of electron beams depend upon the fact that

electrons are scattered by gas molecules. This

scattering has been recognized as a, minor annoyance in electronic devices from the earliest days

of the electron optical art. The total cross section

for scattering falls in the neighborhood of 10~'^ cm^. It is worth noting that the existence of the

total cross section implies that some of the elec- trons passing through a moderately rarefied aggregation of gas molecules will not be perturbed in any way by the presence of the molecules. This

result contrasts sharply with the results that occur when an electron beam passes through a

cloud of charged particles. The charged particles

produce a macroscopic field which affects every

electron trajectory [33]. For low, uniform gas

densities the total probability that an electron

wiU be scattered is given by

P,=<jIN, (3.1)

where a is the scattering cross section, I is the

electron path length, and N is the -number of

molecules per unit volume. A rule-of-thumb measure of the total scattering probability is the

fact that at 1 Torr and zero degrees Celcius

the electron mean-free path is about 0.3 mm [34]. Thus, whenever the product of pressure in Torr

and the electron path length through the gas is greater than 0.3 mm^, the electrons undergo on

the average more than one collision passing

through the gas. In this case, almost every

electron carries some information in its altered

trajectory. Devising the most efficient methods

for gathering and interpreting this wealth of data

constitutes a major challenge which we must

decline in order to concentrate on the study of extremely rarefied gases.

If the effect of the pressure of the gas under

study is lowered several orders of magnitude,

that is, to 10~* mm or below, the mean-free path

of the electron through the gas is raised to the

3
order of meters. In most practical systems, then,

the majority of the incident electrons must pass

through the region of rarefied gas without undergoing any scattering whatsoever. Study of the

undeflected electrons cannot provide any information not already available from the study of the much less numerous scattered electrons.

Even with idealized equipment in which the

ELECTRON GUN SOURCE

ATOMIC SEAM PHOTOGRAPHIC PLATE

TOP VIEW OF A BEAM FOCUSING LENS Figure 1. The basic electron optical schlieren technique

einployed in this experiment. An electron beam produced by the gun on the left is directed to cross the path of a molecular beam propagating in a direction perpendicular to the

electron beam. In this interaction region, some (1 percent at 10-> Torr) of the

electrons will be scattered through small angles by the gas molecules. The

electron lens is adjusted to focus the primary beam of electrons onto a small

stop. Some of the electrons whose paths have been altered will miss this stop and pass on to be registered on a photographic plate. The electron lens

is arranged so that the mid-plane of the molecular beam is conjugate to that

of the photographic plate thus producing an image in focus. The density at a given point on the image is proportional to the average gas particle density along the electron trajectory.

electron beam intensity fluctuations and the variation of the detector efficiency are at a minimum,

statistical mechanics requires that the presence

of useless electrons represent noise to the detector

and tends to mask the data [35]. It is clearly

useful to have a discriminator which eliminates all electrons which have not been scattered at least once in the test region, while allowing the scattered

electrons to pass without interference. The

classic dark-field schlieren arrangement fiUs this requirement.

If a beam of electrons or of light consisting of precisely parallel rays or of rays diverging from

a point source, as illustrated in figure 1 , is allowed

to pass through an ideal converging lens, all of the rays must pass through a common point, the

focus, after going through the lens. A solid stop

of infinitesimal dimension located at this common

point will capture or reflect all undisturbed rays

passing through the lens. Any perturbation of a ray before it reaches the lens so that the position

and slope of the ray that enters the lens does not

coincide with the position and slope of any of the

rays in the original bundle will cause the perturbed

ray to miss the stop. The magnitude of the smallest perturbation which will cause a ray to miss

the stop is proportional to the ratio of the stop

diameter to the focal length. The minimum

usable stop diameter in turn depends upon deviation of the beam from the idealized properties

and upon the lens aberrations.

HELIUM TARGET

BEAM SOURCE

DARK FIELD SCHILIEREN TECHNIQUE

Figure 2. Overall schematic of the stroboscopic electron optical schlieren apparatus.

This figure illustrates two principles of experimentation utilized in this experiment, namely, the electron schlieren optics and the differential pumping scheme.

Notice first that the electron Ijeam generating section is drawn at an angle with respect to the rest of the apparatus. The angle which actually is only 1°, has been

exaggerated here for clarity. The light from the incandescent electron emitter was prevented from reaching and exposing the photographic plate in the camera by

the condenser and schlieren apertures, which are very small and are far apart. The electron schlieren optical scheme shown here is slightly different from that in fig. 1, although the principle is the same. As can be seen, the electron

optical scheme consists of three parts located in sections /?, A, and C. In section B is the beam generator or electron gun, a condenser lens (fig. 4), and an aperture. A limiting aperture, alwut 0.937 cm in diameter, located near the gun, serves to limit the diameter of the primary electron beam. The outermost electrons

are thus prevented from striking the walls of the drift tubes and introducing inhomogeneities in the electron beam. The optics in section A begin at the aperture and terminate at the schlieren stop. The object strived for in this section is to give the beam the required

parallelism in the region of interaction with the molecular beam. It is also necessary that this region should be reasonably field-free. The principles of the design of the lens that accomplish this are described in detail starting on page 6. The third and last section of the electron optics is the detection section. It begins with the schlieren assembly anil ends at the photographic plate. This section contains the filter-accelerator lens. The filter section eliminates the slow

electrons originating at or eoming through the annular region of the schlieren a.sscmbly. The acceleration section raises the energy level of the signal electrons and focuses them on the photographic plate. The description of the optics in this region begins on page 6. The differential pumping scheme is also illustrated in this figure. Sections B, D, and C contain gassy components, viz, a hot electron gun in B, a hot oven whose internal pressure is about 10-' Torr and an electric motor with oiled bearings, all in section D, and a magazine loaded with photographic plates in section

C. Therefore, each of these sections must be isolated from the vacuum point of view, as much as is possible from .section A. This is accomplished by allowing only the smallest practicable openings between them. Each .section is pumped individually and section A contains in addition a liquid helium trap. A complete

description of the diflerential pumping system is found starting on page 6. 4
From a theoretical light optical point of view

[16], there is no great difficulty in designing lenses

which can produce focal spots a few wavelengths

in diameter. Problems of execution, such as producing or supporting a small stop, lens focusing

and alinement, and mechanical perfection of the

lens set the ultimate limit in the sensitivity of light optical schlieren. High sensitivity is needed

in these instruments because the angular perturbation of the rays is quite small. The scattering

of photons from individual molecules does not

occur with usable probabihty; thus the perturbation of the light rays depends upon the gradient

of the refractive index of the gas, considered as a continuous medium.

In the case of air, a pressure variation of 1 Torr in the distance of 1 mm normal to the light

ray causes the light ray to bend with a 2.5-km

radius of curvature. Thus, the light optical

schlieren observation is impractical because of

the extreme precision of the required instrumentation in gaseous systems operating at pressures

much below 1 Torr. The situation is much more favorable in the case

of electron optical schlieren. If an electron is scattered at all by the gas under study, it is usually scattered through a much larger angle

than the available deflection in the light optical

case. For example, in a rather unfavorable case where a 2}2-kev electron beam is scattered by

hydrogen atoms, simple classical theory, based on elastic scattering from a Thomas-Fermi atomic

model, predicts that for about 50 percent of the

scattered electrons the deflection angle exceeds

0.02 radian [36]. A discriminator which responds

to this scattering angle will thus provide 50 percent

of the maximum possible detection sensitivity,

and there is no practical advantage in making this

discriminator appreciably more sensitive.

At this point, perhaps it is well to emphasize

the essential difference between the light optical

schlieren method and the electron schlieren

method. The light optical method depends on

the gradient of the density whereas the electron

optical schlieren is a measure directly of the

density, since the scattering probablity depends on

the density of the gas. Hence, although our

electron optical schlieren photographs are produced by a true schlieren technique, their analysis

is exactly the same as that of interferograms of light optics. When a parallel beam of electrons

is brought to a focus onto the center of a stop by

an ideal electron lens, those which were scattered

through angles greater than the stop radius over

the lens focal length will miss the stop, while

the remaining electrons are captured. A reasonable size for this schlieren stop is something like

//lOO (see fig. 1). In this simple theory we have considered the

electron paths initially to be exactly parallel.

This, of course, is an idealization; and it is a failure

of this idealization which gives rise to some inconvenient variations of detection sensitivity. If the

initial rays are not all parallel, the lens treats the

rays as if they had already been scattered through

Figure 3. (a) Schematic drawing of the differential scattering cross

section <r id). (b) Polar coordinate diagram of the information shown in

fig. 3a for the illustration of the role of aberrations in the optical system.

The information given in figure 3a is presented liere in polar form. Scatter- ing witli a polar angle e and the azimuth angle 4> is represented by a vector, drawn from the origin with length 9 and direction at the angle (j> measured

from the horizontal. The solid concentric circles represent lines of equal

scattering cross section. The two dotted circles represent the region in scattering space resulting in capture by the schlieren stop. Curve a repre- sents the situation which exists if the unscattered ray hits the center of the

schlieren stop. In this case, electrons strike the stop or proceed to the photo- graphic plate according as 6 is less than or greater than da. Curve b represents the corresponding situation for rays which, although

undisturbed, strike the stop at the distance 9i/from center. Here / is the

focal length of the focusing lens. The displacement is the result of non-ideal

properties of the electron source and the focusing lenses. In this case, the

critical value of 9 for discrimination is a function of The electrons lost to the

schlieren on the left side of the displaced circle are more numerous than those gained on the right side. Thus, the response of the photographic plate is slightly greater in case b than ease a. For small values of Si, however, the

difference need not be great, since fli=0 represents an e.xtremum of the probability of capture by the schlieren stop.

small angles, and hence the illuminating condition

plays a role in determining the ultimate sensitivity.

Moreover, if the lens lacks ideal focusing properties, as all electron lenses do, the parallel beams of electrons do not produce a point image. These

deviations, known as lens aberrations, are more

easily compensated for in light optics than in electron optics and indeed turn out to be one of the major limitations of the method. The roles of aberrations and imperfections in the optical

system are illustrated in figure 3.

685125 O—63 2
4. Realization of the Electron Optical Schlieren Method

The efforts of the project divided naturally into

two sections. The first, completed in about 1957, was the realization of an instrument capable of visualizing a continuous molecular beam, at an

equiiibrium pressure of perhaps 10~^ Torr, existing

in a background one order of magnitude lower.

The second phase, which extended from this date

until the end of the project, was that of demonstrating the feasibility of measuring the velocity

distribution of a chopped molecular beam as it progressed across the field of view in the vacuum. The problems in the two stages are essentially

similar except that in the chopped-beam measure- ments one is working with an intensity some orders

of magnitude lower, and the vacuum problem is complicated by the necessity of rotating mechanism, motors, etc., within the vacuum. The two

will be discussed as a single problem until the

problems special to the chopped beam are considered in a later section.

4.1. Electron Optics

The electron optics in both systems must per- form three functions. First, an electron beam

with reasonably well-defined properties is generated. The electron beam is propagated through

the test region, where it is subject to the perturbation of the object space caused by the presence of

the molecular beam. Finally, the properties of

the beam emerging from the test region are measured. Each of these regions has its separate

properties and will be dealt with in a separate

section.

a. optics of the Beam-Generating Section

The beam-generating section, shown schematically in figure 2, consists of an electron gun, a condensing lens, and a condenser aperture. The

electron gun is required to give a uniform illumination which is without structure. It would be

helpful if the electron gun produced only electrons

and no other source of radiation. All electron guns

operating with hot emitting elements, of course,

Figure 4. Iron-clad condenser lens. 6

emit visible light. One of the problems which was met and overcome was that of preventing this light from reaching the photographic plate. For

this purpose a bend was introduced into the

electron optical column and apertures were used

to eliminate straight paths through the system.

In the first system, which used no small apertures

except in the plane of the schlieren stop, a 4° bend was required and corresponding deflection of the

electron beam with a magnetic yoke was essential. Two small apertures were employed in the second

instrument, which required only a 1° bend.

Although they were sometimes used, the magnetic

yokes were not essential in this instrument. Operation without the correcting deflection merely

shifted the exploring beam slightly sidewaj^s and

provided no special problems. Since deflections

tended to introduce beam structure, the deflection

yokes were left de-energized for precise work.

They were useful, however, for swift alinement of

the instrument when precise results were not

needed. A small deflection angle, 1°, and a small

condenser aperture, 1 mm, used simultaneously were sufficient. The electron gun used for most of the work was

a telefocus gun of the Siemens type [37]. This gun

provides a well-defined crossover a short distance

in front of the gun. This crossover was focused

by the condenser lens into a small spot of higher

convergence at the entrance to the test section.

The existence of this small spot at the entrance to the test section made it possible to enhance the

vacuum in the test section by means of differential

pumping. The electron beam entered the test section through an aperture only 1 mm in diameter.

The lens which was adopted for this purpose was

a conventional iron-clad magnetic lens designed

roughly according to the criteria of Liebmann [38] (see fig. 4).

Throughout the course of the investigation we

were plagued by minor inhomogeneities in the

electron beam. Many of these were caused by

small deviations from cylindrical symmetry in the

vicinity of the gun cathode. Small irregularities at the condenser crossover and at the schlieren

stop also contributed to these disturbances. In most cases the beam inhomogeneities are of too

slight contrast to be observed on a fluorescent

screen. Thus it is not surprising that they have

been ignored in the literature of electron optics.

b. Optics of the Test Section

The small electron spot is placed at the focal

point of the first of two principal lenses which

straddle the interaction space. It is the purpose

of the first lens to give the required parallelism

to the beam at the point of interaction with the

atomic beam. The second coil focuses the beam

on the schlieren stop. It is also required that the

atomic beam be in a reasonably field-free region.

Hence, these lenses are double-coil lenses so arranged that the field has a stationary value of
zero at the mid-plane of the molecular beam.

The design problem in these lenses, which have

reasonably long focal lengths is to place the center

of this region of nearly zero field at an acceptable

position for the object plane of the second lens. The following paragraph gives the design of these

lenses in some detail. The coil system was designed around the re- quirement that the magnetic field strength should

be as near as possible to zero throughout the

region occupied by the molecules under study in order to provide straight electron trajectories

through the test region. With two coils of equal

diameter, acting in opposite directions, we were

able to produce a lens field with a peak axial field strength more than 300 times greater than the

strongest axial magnetic field in the region

occupied by the molecular beam. The method

used was to derive a set of required relationships

between the various coil parameters such that at the mid-plane of the atomic beam, i.e., at the

object plane, the axial magnetic field H{z) and its

first derivative H'{z) should be zero. The axial magnetic field produced by a coil of n turns of radius a carrying a current i expressed

in magnetic units is

2Tmia^

/ z^\2\3/2

For less cumbersome notation, let us write z in units of the coil radius a. Then

Ho

H'(z) =

and

(1 +22)3/2

dz {l+z^yi^'

H'jz) ^ -3

H(z) 2+1/2'

(4.2)

(4.3)

(4.4)

Let the distance from the object plane to the

respective coU planes be Zia and 22a, respectively.

For the field and its first derivative to vanish

simultaneously, we must have H{z2) = —H(zi)

SindH'{22) = -H'iz,). Hence

H'{z,) _H'{z{)

m^2) H{z,)

(4.5)

Using the formula for H'/H, we find that this result yields 21+1/21 = 22+1/22. Solutions are

2i = 22 and 21 = 1/22. (4.6)

The first solution is trivial, as the coils then

coincide and produce equal and opposite fields everywhere. The second solution gives a family

of possible useful solutions. The condition

H(z2) = —H(zi) is met by running currents through

the coUs in opposite senses and correctly choosing

the turns ratio R=712/111. Thus

1 R R z\R

[l+zlf^ [I+2IP 3/2 [l+22]3/2

(4.7)

Without further consideration, the choice 2i =

1.25, 22=0.8, i?=0.512 was made as the normalized distance between coils (21 — 22=0.45) and the

normalized distance from the object plane to the nearest lens (22=0.8) were convenient. A 10-in. mean coil diameter was then selected as the largest

compatible with the design of the main chamber.

(The coils had to be able to slide up to the outer

wall of the chamber without being stopped by the

top flange of that chamber.)

The stray axial field at a distance 2 from the

object plane is given by

H,,,^[H"{z,)-\-H"{z,)]~, (4. With

2a2

this becomes

H{z,)=HJK{z,),

H,,,=HAK' ' (21) -RK' ' (22) ] ~ (4.9) 2a?

(4.1) By simple differentiation,

K"{z)-- 3(42^-1)

'(1 +

2^)^/2" (4.10)

With the adopted values of 2i, 22, and R, we

2^ then have ^?8t,=0.0805 H^—^- With o = 10 in. w- and 2=0.5 in. at the extremity of the molecular

beam, ilgtr reduces to 0.00201 Hq, while the peak

field occurs at 2i^—0.15 and has the value 0.643

Rough computations for the spherical aberration

constant of this coil system, used as a lens, gave

5^eph=0.0107/cml With a beam_ radius of 1.25

cm, this leads to a circle of confusion [39]

Ar/=7 S'ephr3= 0.0052 cm. (4.11a)

In the symmetrical arrangement of two sets of

lenses, as used in this experiment, the Ar's of the

two lenses add, and the circle of confusion due to the spherical aberration of the condenser lens, as seen at the condenser aperture, is imaged at unity

magnification. The Seidel coefficient for this lens was computed to be (SBph=0.0184. The beam

radius in passing through this lens is 0.625 cm, and

the radius of the circle of confusion produced by

the condenser lens is thus

Ar<,=0.0011 cm. (4.11b)

The circle of confusion on the schlieren stop is then

Ar,=2Ar/+Ar,=0.0115 cm. (4.11c)
Our practical experience in manipulating the focus

in the vicinity of the condenser aperture indicated

that the crossover diameter in that plane was

about 0.05 cm. We believe that the discrepanc}^

between this result and Ar^ above is due to the

finite source size in the electron gun and spherical

aberration in the gun, although no figures on these

aspects of gun performance were available.

c. Optics of the Detection Section

Figure 5 is a photograph of the schlieren stop

used. In this case, a scroll cup about 2 mm in diameter was made from a platinum sheet and

spot-welded onto a heavy platinum wire. In

earlier work the stop was made from sheet molybdenum spot-welded onto a tungsten rod. The

platinum cup had the advantage that it shielded

the electron beam from the ever-present brown

deposit that forms whenever an electron beam

spot hits a surface in a dynamically pumped

system. This deposit is capable of storing a charge and therefore of affecting the electron

image on the plate. Further, the platinum stop can very easily be cleaned by simply heating to a red glow.

It can be shown that a high photographic

efficiency and a high scattering cross section are mutually contradictory requirements. Scattering

cross section falls rapidly and photographic effi- ciency increases rapidly as the electron energy is increased. The resulting overall response of a simple electron schlieren camera is rather poor at

all beam energies. Under typical conditions a 30-min exposure is required for a barely satis- factory photograph of a molecular beam with

Figure 5. Two views of the schlieren stop and aperture.

Dimensions of parts are cup diameter, 1 mm; aperture diameter, 3.9 mm;

and sheet thiclcness 0.38 mm. 8

density corresponding to lO""* Torr pressure. To

secure the advantages of both high and low beam

energies, an electron accelerator lens is inserted

immediately following the schlieren stop. The

beam energy in the scattering region may thus be

kept conveniently low for high scattering cross

section, while energies of 10 kev or more are employed to expose the photographic plate.

If the beam energy in the scattering region is made extremely low, however, the experimentalist

is plagued by a variety of annoyances. Stray

electrostatic and magnetic fields deflect the beam

to an increasing extent, and space charge limits

the available beam current. The magnetic field of the earth is opposed by an equal field generated

by compensating coils (see sees. 13.2 and 13.3),

but additional magnetic fields are introduced by

ferrous materials used in the vacuum pumps and

electronic equipment. Since these materials have

high magnetic permeability, their internal fields are negligible in comparison with the external

field. Consequently, the scalar magnetic potential on the surface of a ferromagnetic object

is essentially constant. In the case of a sphere

of volume V, in the presence of the earth's field He, this condition is equivalent to a dipole of

strength M= VHe/Sir at the center of the sphere.

Thus at a distance I from the center of the sphere,

the disturbing field varies from VHe/Sw P to VHel^TT V, depending upon direction of the dis- placement. For objects of different shapes, this formula for the stray field is inexact, but it does

give the correct order of magnitude. On this basis we estimate that non-uniform fields of the

order of 10 mgauss were present in the scattering area of our instrument.

Electrostatic fields may be produced by charging

of nonconducting impurities on metal surfaces or by differences of surface potential on different

crystal planes of the metal. Such effects are not

too well understood and certainly not easy to control. Thus, while meticulous attention to stray electrostatic and magnetic fields would

probably provide satisfactory optical conditions

at considerably lower voltages than we used, it was more practical in this experiment to raise the

operating voltage to a level at which these dis- turbances could be tolerated. By trial and error, a beam energy of about 2 kev in the scattering area was found to work best in our instrument.

Somewhat less satisfactory operation was possible at voltages ranging dowmward to 1000 v. At

voltages below this minimum operating point,

image distortions and alinement difficulties were

extreme. The acceleration of the beam from 2 kev in the scattering region to 12 kev at the photographic plate reduces the required exposure time

from 30 rain to a few seconds. The improvement,

which indeed is vital to the success of the method,

is not without its inherent difficulties. An ac- celerating field in the region of the schlieren stop

picks up electrons which arise from the intense

beam striking the schlieren stop, accelerates them,

and forms an intense inhomogeneous background

on the photographic plate. To avoid the back-
ground effect, we designed the lens to have a

decelerating or a filtering stage before the ac- celerating stage. Since most of the electrons

arising at the stop are either secondaries or elec- trons which have transferred energy to the stop,

they are of significantly lower energy than the

primary electrons and may be removed by modest

energy filtration. Because of the importance of

this stage, its design is given in detail. The mechanical features of the post-acceleration

lens are shown in figure 6. A series of 17 copper

apertures supported on three glass insulating rods

provides controlled potentials on the boundary of

the region in which the acceleration takes place.

The first aperture plate sits directly in front of the

schlieren stop and is at ground potential, while

the last plate is attached to a tube extending into

the drift space leading to the image plane. It is at the potential of the camera. A short glass

tube insulates the camera assembly from the

main instrument housing, which is at ground potential. By applying various potentials on the

aperture plates, a wide variety of axial potential

distributions may be produced. Electrode potentials which are positive with respect to ground

are produced by a resistance bleeder connected

to a 30-kv positive supply and fed to the plates

through kovar seals. Six negative potentials

are derived from a bleeder consisting of wirewound, 1 -megohm rheostats connected to a negative 150-v regulated supply. The outputs

of both supplies were adjustable from zero to the

maximum voltage. The resistors in each leg of

the bleeder chain were pairs of 2-w carbon resistors. One resistor of each pair was chosen to be just

short of the desired resistance and the other,

much smaller resistor was selected to bring the

combination to the desired resistance. The

resistance values were checked on a precision

Wheatstone bridge. Thus we constructed an

inexpensive voltage divider of better than 1 percent initial precision from resistors of 10

percent manufacturing tolerance.

We chose the electrode potentials in a manner

designed to simulate an axial potential distribution

with desired focusing properties. Since the elec- tron beam is most easily disturbed in the lowvoltage portion, we devoted more apertures to control of the low-voltage portion of the field than to the high-voltage portion. Thus, instead

of being spaced uniformly, the plates were spaced

at exponentially increasing distances from one

another. The hole diameter increased uniformly

from plate to plate.

The field was required to accelerate the electrons

which missed the stop to the desired final potential

and to reject any secondary electrons produced at the stop. In addition, it was required to provide

a focused image of the object plane with minimal

distortion at the fixed image plane. With the

schlieren lens spacing and focal length, as employed

in the instrument, the lens had to work with a

virtual object approximately 38 cm before the

first plate. The image plane was 9.05 cm beyond

the camera flange entrance.

FiGUEE 6. Exploded view of the filter accelerator.

At left is the housing which has provision for 20 lead-in connections. Next

is the disk assembly. On the right is the glass insulator section. The

glass cylinder is made of Pyrex and sealed into the brass rings with an

epoxy resin. The camera bolts onto the fliange on the right.

Experience with the first instrument had shown

us clearly that if the lens fields are strong enough

to form an intermediate image in addition to the

final image, intolerable pincushion distortion is produced. If only one image is formed, the

pincushion distortion is of suitable size to be

balanced out by the barrel distortion produced by

the schlieren lens coil. In operating the lens

without any intermediate images, the ratio of the

final electron energy to the initial energy is limited

by the need to keep the lens weak. Since photographic efficiency increases rapidly with voltage

ratio, it is clear that we must seek the maximum

voltage ratio consistent with the required object

and image positions, thus permitting us to attain

focus at the highest possible accelerating voltage.

From the point of view of simple lens theory, the

axial potential distribution <A=<^)oe*^ appears superficially to be very attractive. The modified

Gaussian ray solutions u=r{(t)/<j)o) ^'^ are the sine

and cosine of — kz. With these facts, it is easy

to show that if the exponential solution is fitted to the initial and final potentials at the extremities

of the lens, focusing occurs at a voltage ratio of

20.4. The field is, however, quite unusable, since no provision is made for the rejection of secondary

electrons; and the sudden termination of the field at the plane of the camera fiange would require a small aperture at that position. To reject

secondary electrons we lower the axial potential

slightly in the region just following the stop.

Lowering the potential by only a few percent is sufficient to stop all of the secondary electrons,

since they leave the stop without forward momentum. It is desirable, however, that the field should

vanish in the vicinity of the schlieren stop. The

simplest field form with this property is

4>(2) = <Po~az' +b^. (4.12)

The general shape of this field in the neighborhood

of the schlieren stop is shown in figure 7. For

9
% 1.2 -

0,9 L

ACCELERATOR PLATE

NO. 7

Figure 7. Decelerating and transition field, as given by

eg (4.12).

<t>=(f>o exp- (4.14)

where r is the radius of the tube and Ui is the

first root of the zero-order Bessel function Jo{x).

The field given by our family of potential

distributions falls off too slowly at large values

of 2 to match the exponential decline of a field inside a conducting cylinder. To make our model realistic, we modified the field to the form

0=</)oe j^k

J*

exp^—^^rfxj (4.15)

in which

adequate control of the contours of this field as

far as the point where 0=0o with 0'(2)>O, we

assigned the first six electrodes to this section of

the field. For the remainder of the field, we

wanted </)70 to build up to a maximum and then

fall asj'mptotically back to zero. The asymptotic

return of <^70 to zero should match the roughly

exponential falloff of field in the drift tube beyond

the camera flange.

A field for which the Gaussian trajectories are known analytically and which essentially satisfies these requirements is given by

which provides an acceleration voltage ratio

W<^i=exp [^]- (4.13b)

This field has the property that

(4.13c)

where u =r^'^, and r{z) is any paraxial ray solution.

The solution for the magnetic lens with —r"lr

given as the same function of z has been found

analytically by Glaser [40]. Experimenting by

trial and error with various values of k^, a, and the

position of the origin in the lens field, we were

able to choose the most suitable potential distri- bution from this family of potential curves. To

be acceptable, a potential curve had to provide an image of the object in the image plane, had to match the potential and field of an acceptable

solution (4.11) at the seventh electrode, and had

to approach uniform potential in a realistic manner

in the drift tube leading to the image plane.

The axial potential in a region surrounded by a conducting tube is approximated by [41]

The focusing power thus has a bell-shaped

distribution similar to (4.12, 4.13), but fades out

much more rapidly at high values of z. This

field thus readily fits the boundary condition in the drift tube. If this field is fitted to match

(4.12) in the neighborhood of 2 =0, we have a

realistic field which can be matched at the lowvoltage end by a suitable polynomial function

which provides the retarding potential. The

focusing characteristics of this "error-function"

field differ only slightly from the characteristics

of the matching accelerating field. A rough check on the change in focusing characteristics is obtained by computing trajectories for a field of form

(4.12) which is terminated at the points where

<^'2/0^ in the error-function field falls to half of the

corresponding value in (4.13). The focusing

effects of the retarding field given by the polynomial function prove to be so slight that they may

be neglected in the design calculations.

Having developed a piecewise-analytic expression for the axial potential distribution, we are ready to compute the potentials to be applied to the plates. For this purpose, we utilize the series expansion

<t>{r,z)=<i>{Q,z)+^,<l>" {z)+^,4>'''{z)+

(4.17)

which gives the radial variation of 0 as required

by the Laplace equation. The required electrode

potentials are thus found to differ slightly from

the potential desired at the axial position at the same value of z. The corrections are, however,

so small that only the first two terms of the series are needed. Because of the small degree of correction needed, we are justified in assuming

that the actual 4>(z) on the axis deviates from the

planned (t>{z) by only a few percent. The properties of the designed field, together with the

electrode dimensions, positions, and potentials are given in the following table: 10
Electrode No.

Radius z position 0/^0 on

Electrode

cm cm

1 0. 318 0. 000 1.000

2 .456 .318 0. 995

3 .596 .684 . 975

4 . 734 1. 107 .923

5 .872 1.591 .910

6 1.012 2. 143 .925

7 1. 160 2.788 1.000

8 1.289 3. 518 1. 096

9 1.427 3.722 1. 240

10 1. 568 5.360 1.493

11 1.707 6.490 1.879

12 1.847 7. 788 2. 466

13 1.982 9. 282 3. 434 14 2.124 11.005 4. 846

15 2. 262 12. 990 6. 677 16 2.396 15. 276 8.204

17 2.540 17. 816 10.000

, (z-6.14)g ^=0.830e 22.4 (4.18)

Axial potential:

-^=0.966 exp r r g-^^^lr^'^^l (2^2.79)

<l>o LJ2.79 J

(4.19a)

-^=1-0.07622+0.0258 2^ (2^2.79) (4.19b)

^0

5. Gas

The electron optical schlieren system, as described in the preceding sections, is a sensitive

device for mapping gas density distributions at extremely low pressures. Exposures of 2 or 3 sees are sufficient to provide normal exposures

in the presence of gas densities lower than 10~® Torr. The chief barrier to operation at pressures

four orders of magnitude lower is simply the need

to keep the background vacuum at that level. Furthermore, the schlieren camera can be used

with a shutter speed of a microsecond or less if required.

To test the spatial resolution capabilities of

this instrument, one needs a sharply defined static

formation of gas molecules. The conventional

molecular beam is ideally suited for this purpose.

Molecules escaping into a vacuum from a small

opening in the side of a chamber containing gas

at a moderately low pressure travel in straight

lines from the opening, provided that the pressure

inside the source chamber is sufficiently low. The

majority of the atoms will leave a slit of widtb

w and travel in a straight path until they hit a

solid surface, provided that the mean free path

X, corresponding to the ambient pressure within

the source chamber, is large in comparison with

the slit width 10 w [42]. A second slit parallel

to the source slit captures all molecules whose

paths do not lie within a ribbon-shaped region

limited by the two sets of slits. If precautions

are taken so that molecules completing their trip

In actual operation the post acceleration potential is adjusted as needed to secure exact focus.

The actual potential distribution then no longer

belongs to the family of distributions which formed

the basis of our calculations. The design can be

considered satisfactory, however, as long as the

actual voltage ratio for the in-focus condition is a major fraction of the calculated ratio. This

condition was fulfilled. Terminating the accelerator stage is the camera

housing. The requirement that the interaction

space be at ground potential necessitates that the

final stage of the accelerator and the camera be

at 12 kv above ground. The insulation is achieved

by inserting a glass section in the vacuum wall. The camera housing is provided with a protective

shield to prevent accidental contact by the operator, and a safety interlock to shut off the high

voltage when the camera is to be operated. The

camera serves two functions: it contains and

manipulates the photographic plates for recording

the data, and it provides a fluorescent screen on which the operator can view the image. The

photographic material used in this device was

process plates, 6.5X9 cm. Construction details

and photos of this camera are given in section

13.2.

Kinetics

are not allowed to return to the interaction space,

the resulting molecular beam is sharply defined

in space with an easily calculated density distri- bution. The sharp density gradient at the edges

of the beam provides a critical test of the resolving

power of any detection system for rarefied gases. For testing the time-resolving ability of the

schlieren system, we need to vary the gas formation rapidly with time in a reproducible manner.

If the beam is divided into segments by means of a rapidly rotating wheel with radial slots in the

beam path, the segments will move in the direction of beam propagation. The modified Maxwellian velocity distribution of the individual

molecules will, however, cause the molecular

beam segments to spread almost as rapidly as they propagate. A second slotted disk rotated

at the same speed as the first, but with the slots phased to pass through the beam at an appropriate later time, will limit the beam pulses to those molecules whose velocity was just right to pass from one disk plane to the other in the chosen

delay time. By control of this delay time and the

width of the chopper slits, any mean velocity

present in the original beam and any velocity

spread within reason may be selected. The

restriction of the velocity spread to a narrow

range entails a corresponding loss of beam in- tensity. The geometric spreading of a beam as

it leaves the source slit causes the density without

chopping to fall to a value somewhat lower than

11
ijw times the density at the source sUt, where I is the distance from the source sht to the point

in question. It is thus evident that when the

density is further reduced by approximately

bvjv, where v and bv are the mean value and spread

of the pulse velocity, the maximum possible

density of atoms in the pulse corresponds to a mean free path

\=lQl-vlbv. (5.1)

Since cross sections for electron scattering are an order of magnitude lower than those for molecular collisions, the proportion of electrons scat- tered in passing through a chopped beam is only

under ideal conditions. Inserting typical experimental conditions, viz, iJ=3-10* cm/sec, 5zj=1-10*

cm/sec, A2=2.5 cm, t=4 cm, yields o-= 1/480,

and it is thus evident that every effort must be

made to maintain high intensity in the chopped

,beam.

Figure 8. Photograph of components suspended from the top of the main chamber.

Starting from left to right one can see first the mirror assembly of the Cassegrainian light optical triggering system. Ne.xt is the target assembly, and behind that a liquid nitrogen trap can be seen on the bottom. The

septum has been removed from this picture to reveal the set of chopping

disks, and the components located between the disks, namely the molecular beam collimator slits and mirror A/i of the Cassegrainian optics system. Behind the first chopping disk the oven with its shield and supports can be

seen. Above the top plate are the control rods mounted on stainless steel bellows and to the left of those are the various structures of the light optical system. The mirror Mi is identified in figure 10.

If the beam is finally used to study accommodation and viscous slip coefficients by reflection

from solid targets, the relatively compact beam

before impact is spread out over most of the field of view and another factor of 20 to 200 in beam

intensity is lost.

5.1. Beam Source

The cadmium molecular beam had the shape

of a ribbon running horizontally across the elec- tron path. Its cross-sectional measurements were

1.3 mm X 35 mm, with the long dimension in the

direction of electron flow. A metal septum

fitted with slits very slightly wider than the

molecular beam thickness separated the highvacuum interaction space from the relatively

gassy molecular-beam-generating components,

which were separately pumped. The molecular

beam was generated in a conventional oven with

two sets of collimating slits. The oven was machined from OFHC copper into a hollow rec- tangular cube having a removable top to allow

loading of the oven. A slit opening 3.3 mm X 25.4

mm was machined and a set of adjustable colli- mating slits was mounted over it. The slit opening was set to be 0.5 mm X 25 mm. The

oven was heated by a set of tungsten coils en- cased in alumina tubes to provide insulation.

The heaters were inserted into holes drilled into

the oven located along the outer edges of four

sides and parallel to the slit. A re-entrant tube into the oven chamber was provided to contain a thermocouple used for temperature

control and measurement. The oven was heated

by unfiltered rectified a-c current controlled by a commercial automatic temperature regulator.

The oven was supported kinematically on top

of three rounded stainless steel screws which bore

into V slits milled across the bottom of the oven. A spring pressure contact on top of the oven

insured that the oven would bear firmly against

its supports. Adjustments of oven position and

tilt were made by turning the supporting screws. Since contact of the oven with its surroundings was limited to seven point contacts with stainless

steel, which is a poor conductor, conductive heat

loss was low. To reduce the heat loss further, a

stainless steel shield was built around the oven, and the base for the mounting screws and stainless

steel shield was mounted on thin stainless steel tubes, which are good insulators (see fig. 8). The dimensions pertinent to the formation of

the molecular beam and a beam packet are as foUows: oven slits, 1 mm X 25 mm; first chopper

blade slits, 2 mm X 22 mm and about 5.1 mm from

oven slits; collimating slits, 0.5 mm X 25 mm

about 20 mm from first chopper blade; second

chopper blade slits, 0.5 mm X 25 mm set as close as possible to, but not touching, the collimator

(estimated 0.5 mm) ; and finally, the septum slits, 0.5 mm X 25 mm about 2.5 mm from second

chopper blade. In the continuous-beam work,

the dimensions are the same but chopper disks are removed.

12
For the bulk of the experimental work the material used in this oven was cadmium. The con- densation properties of this material were both an advantage in lowering the residual background

pressure and a handicap, in that the more sophisticated ovens [43] using parallel channel flow as developed by numerous workers were unsuitable

for us because of condensation of the cadmium

vapor in the long narrow channels. With our

oven, we obtained a molecular beam which was of

sufficient density to be useful in the steady state and chopped conditions.

In addition to the work wijth the cadmium

vapor, preliminary experiments were run using

carbon dioxide and air from a similar source. Once again, the technique was successful in detecting the presence of these continuous molecular

beams.

5.2. Chopped Beams

At the temperatures at which the beam-source oven is operated, the mean velocity of the molecular beam is approximately 300 m/sec, with a

velocity spread characteristic of the oven temperature of approximately 320 °C. In order to do

significant time-resolved work, it is necessary not

only to pulse the beam, but also to monochromatize the beam to such a degree that a limited range

of velocities may be assigned to the members of

the packet. The velocity selection is done in the

classic manner by using two offset slits on the

periphery of a pair of rotating disks. In order

that a reasonable ratio of geometrical pulse length

Figure 9. The septum, chopping disks, and driving gear, mounted on the three-bar suspension system. The septum was made from an aluminum alloy and silver plated in the

region that is exposed to the electron beam. This is necessary since the oxide

layer on aluminum becomes charged when electrons strike it, thus building up

an electrostatic charge which of course affects the electron beam streaming by

parallel and immediately adjacent to the septum. The three-bar suspension system can also be seen in this photograph. The

double set of bars was made from 1.91 cm stainless steel rods, whereas the

single bar was 2.5i cm in diameter. All the clamps used were made from aluminum alloy.

to pulse width may be attained, these slits must

be rotated at the highest possible speed. In our

case they are at the edge of a 20-cm disk revolving

at 8,000 rpm giving a surface cutting speed of

about 75 m/sec. This results in a molecular

beam pulse which is, as it enters our field of view,

2 mm long and whose velocity spread is approximately ±15 percent of its mean velocity. As

each pulse moves across the interaction region, it

is viewed at a selected instant by pulsing the elec- tron beam giving a stroboscopic or time-freezing

result. A timing reference pulse is generated by means of a light beam which passes through the

chopper slit to a photomultiplier (see fig. 10).

Figure 9 shows an overall view of the chopper

assembly with its driving motor. The two chopper disks are clearly recognizable on the end of

the motor shaft. These disks are mounted with

the series of alining pins onto the rotating shaft

so that various energies may be selected by changing the slit offset according to the system in section

13.4. The entire chopper assembly, the .septum

between the two vacuum chambers, the collimating slits, and the Cassegrainian light optical

synchronization system are all suspended from

the top of the chamber on a three-bar optical

bench-type mount. This arrangement allows

the various elements to be alined mechanically

outside of the vacuum and then clamped rigorously in place, where they will remain despite the

inevitable vibration due to the motor. The

chopper shaft is carried on ball bearings which

are lubricated by diffusion-pump vacuum oil. Experience with special vacuum ball bearings,

silver lubricated, showed that the standard precision ball bearings, lubricated with vacuum oil, ran much more smoothly and had longer life than

the more complicated special-purpose bearings.

The slight increase in background pressure, due

to the vapor pressure of the oil, seemed to be

negligible. The chopper motor is a 400-cycle,

three-phase synchronous motor operating from

an individual power supply. This motor has

proved reliable in operation and is lubricated by the same low-pressure vacuum oil used for the bearings. It is otherwise unmodified and operated

in the form that it was received from the manufacturer.

An adjustable mount for a solid target was

provided in the interaction space. Two degrees

of freedom, permitting the target to be rotated

to vary the incidence angle and to line up the

target plane with the electron beam axis, were

available without breaking vacuum (see fig. 8).

The motion was provided by rods driven axially

through stainless steel bellows. Aluminum gear

linkages translated the axial motion into rotary

motion. Although the target mount was not used

during the work comprising this report, it was

considered necessary to design and build the target

mechanism and the synchronization system to- gether so that the separate systems would be

geometrically compatible.

13

685125 0—63 3
\J s

LIGHT SOURCE

SOURCE APERTURE

EXIT SLIT PLANE MIRROR WINDOW IN VACUUM WALL

PLANE MIRROR CONVEX MIRROR CONCAVE MIRROR CONCAVE MIRROR PHOTOMULTIPLIER TUBE CHOPPER DISK

Figure 10a. nizing the packet.

Schematic of the light optics used for sijnchroelectron beam pulse with the molecular beam

SEPTUM

TO

CASSEGRAINIAN ASSEMBLY

SEPTUM SLIT

Figure 10b. Magnified schematic of the light paths in the vicinity of the chopper blade. At the instant illustrated in the figure, the atomic beam is still cut off by

the chopper, but light is being transmitted at full intensity. Atomic beam

paths, as they will occur a few microseconds later, are shown in broljen lines. The light cone is inclined at a 29" angie to Iceep the Cassegrainian mirror system (not sho«Ti) and the spherical mirror (in the upper right hand corner) out of the atomic beam path. Because of inclination and spread, tlie light beam cannot pass through the septum slit and the collimator slit if both are arranged as simple rectangular apertures in plane surfaces. A circular hole

in the lower jaw of the septum slit permits passage of light through that

surface. The lower jaw of the collimator slit is turned around so that the beam limiting edge of the lower jaw is farther from the chopper disk than is the corresponding edge of the upper jaw. In this way ample passage for the

light beam is provided, while the atomic beam is restricted to a width of

1 mm.

6. Stroboscopic Mechanisms

6.1. Optics

In order that the beam source shall show no

significant blurring on the plate, it is essential

that the electron beam have an on-time in the

neighborhood of 1 to 5 ^isec and be generated at

a time in the range from 10 to 100 ;usec after the

time the molecular beam leaves the last chopper

blade. When the molecular beam leaves the last chopper blade, a large electrical synchronizing

pulse with an adequate time derivative must be

supplied.

A light beam modulated by the second chopper

disk and fed into a 931 A photomultiplier provides

this timing element. Attempts to use a slit other

than the one in the process of cutting the molecular

beam proved impractical. Thus we were required

to send the light beam and the molecular beam

simultaneously through the same sHt of the second

chopper disk and through the collimating slit. 14

To accomplish proper synchronization requires some sophisticated optics, since our chopper

blades cut the beam at about 75 m/sec and a timing

accuracy of about 0.2 usee is required. The

light signal must increase through most of its range while the chopper blades are advancing only

about 0.015 mm. The optics of the system are showm in figures 8 and 10. The requirement that the light signal should

increase substantially while the chopper blade moves 0.015 mm is roughly equivalent to restrict- ing the light which finally enters the photomultiplier to a slit about 0.03 mm wide in the plane

of the chopper slit. The problem of simply

restricting the incident light to a crossover of that

size, without regard for intensity, is by no means

trivial. The fact that the illuminated area is approximately 0.01 mm^, however, makes the

intensity problem also critical. Under ideal conditions, the intensity at the exit of an optical

system is /<=/o sin^a^, where Ig and /< are the
light intensities in the object and image planes

and a, is the convergence angle of the light from

the last optical element.

Since the light output of a tungsten filament at 2000 °K is about 250 lumens/cm^, and 5-10~^

lumens are needed to provide a 1-v signal from the

photomultiplier, the minimum usable convergence

angle is 2-10"^ radians, and a larger angle is to be

preferred. Since we cannot place physical mirrors

or lenses in the molecular beam path, the light

beam must be inclined at a substantial angle with

respect to the molecular beam. For obvious

reasons, there cannot be a physical aperture limiting the beam spread in the plane of the chopper

slit. Thus, the image of the light source or of a suitable set of limiting slits must be focused in a plane which intersects the chopper slit plane at the desired reference position. After the light has

passed through the chopper slit into the space between chopper disks, it is in a very inconvenient

place for mounting a photomiiltiplier. Consequently, it must be transmitted to a more suit- able location. One must also avoid contamination

of the vacuum in the interaction space by a gassy

light som"ce and the organic binders of commercial

compound lenses. Finally, stray light which

might expose the photograph is to be a^'oided, and

the optical elements must not be too sensitive to coating by stray cadmium atoms.

To divorce the problem of adequate illumination

in the plane of the chopper slit from that of re- stricting the area through which the light must

pass, we arranged to have an equivalent lens form

a magnified image of the chopper slit jaw in a convenient image plane outside the vacuum.

Large slits then limit the region for light accept- ance as needed. When good focus exists, a sharp

magnified image of the chopper blade edge sweeps

pass the selector slit. The desired performance

is thus obtainable if the resolution of the magnifying system is better than 0.015 mm. A con- densing system to provide illumination at the

required brightness is still necessary, but stray

light in the neighborhood of the desired trans- mission area does not seriously degrade the out- put pulse. To avoid contamination of the vacuum

and loss of performance due to coating of lens

elements with stray cadmium, we mounted the

source and the photomultiplier outside the vacuum

chamber and used mirror optics exclusively within

the chamber. Instead of being reflected to a separate exit after passing through the chopper

slit, the light was returned approximately along

its original path by a spherical mirror located between the spinning chopper disks.

The convergent, inclined light beam cannot

pass through three parallel horizontal plane slits which Imiit the molecular beam, since the product

of the minimum practicable distance between slit planes and the tangent of the slope angle greatly

exceeds the desired molecular beam width. To

bypass this difficulty, we set the reference position of the chopper blade transmitting the light

pulse slightly below the position for full trans- mission of the molecular beam pvdse. The light

beam passes through a circular hole in the lower

jaw of the septum slit and forms a focused source image in the best plane for passage through the

chopper slit. Because the chopper slit is then

about 0.050 cm below alinement with the collim- ator slit, the light beam easily clears the upper

jaw of the collimator slit. The lower jaw of the

collimator is set back about 0.5 cm from the chopper slit plane. Thus, when the light reaches the

plane of the lower jaw it has risen weU above the

jaw and has successfully run the gauntlet of

limiting slit systems. The geometry of arrangement of light beam and slits is shown schematically in figures 10a and b. The object plane and image plane for the spherical mirror is a plane normal to the axis of the light

beam, passing through the intersection of the axis

with the plane of the chopper disk. The mirror

forms an inverted image of the incident illumination pattern at unity magnification. A light ray

may be interrupted by the chopper blade either

as it arrives at an object point or as it leaves at a corresponding image point. Thus, no light is re- tm-ned to the magnifying lens until the chopper

jaw is clear of the light axis. The region of inci- dence for which light transmission occurs then

expands in both directions from the axis as the

chopper jaw recedes from the axis. The magnifying lens thus sees two shadow images of the

chopper slit jaw receding from each other. A

magnified image of this phenomenon is projected

onto an entrance slit which precedes the photomultiplier. If the system is properly focused, the

illumination begins as a line along the middle of

this slit and widens until the entire slit is illuminated. The slit jaws thus merely limit the time

in which the buildup of light intensity is linear and

give the light pulse a flat top after the linear

buildup. For maximum freedom from synchronization jitter, the voltage level for triggering the

commercial pulse generator should lie at the center

of the linear buildup of light intensity. A video

amplifier raised the 1-v signal level from the photomultiplier to 10 V, which is more appropriate for triggering the pulser.

The success of this system depends critically

upon the imaging properties of the magnifying

lens, which also serves to form an image of the source in the plane of the slit jaws. For this pm*-

pose, we used a pair of concentric spherical mirrors

arranged as a Cassegrainian telescope objective.

Optically, they are equivalent to a strong thin lens located at their common center of curvature. This enables us to obtain a lens of large aperture

and high magnification in a position that would

not be available for a physical lens. The efi^ective lens position at the center of curvature of the

spheres is directly in the path of the electron beam ; but the electrons and molecules are not disturbed

by a virtual lens. As part of the condenser sys- tem, it provides good incident Ulumination. As

part of the viewing system, it provides a highly

magnified high resolution image of the chopper

slit edge for convenient control of the pulse rise time. By proper choice of the ratio mirror radii,

15
we are able to correct spherical aberration through

third order. In an exact ray trace, the deviation

from Gaussian image position of a ray converging

at 15° was only 0.025 mm in the image plane.

The remaining details of the optical system are shown in figure 10a. Light emitted by a coiled

tungsten filament passes through a limiting

aperture Ai and the plane glass window W into

the vacuum system. An image of the filament is projected onto the plane of the chopper slit by the

Cassegrainian objective consisting of mirrors M2

and M3. An angular bend in the optical path is secured by plane mirror Mj. After reflection by

the spherical mirror, the light is again focused by

the Cassegrainian objective and reflected by Mi

to pass out of the vacuum system through the

plane glass vacuum window W. The mirror

is misalined just enough, however, that instead of

returning to silt Ai, the light strikes the small

plane mirror M5 and forms an image of the chopper

blades at the exit slits A2. The Cassegrainian

objective magnifies about 16 times, thus making

the adjustment of slits A^ and A2 relatively easy. Diverting the returning light to a displaced set of

slits was possible because the light to and from the

filament subtends a small aperture. Thus, the

incoming and outgoing light bundles may be separated in space by a small misalinement. This

small aperture is in turn compatible with the

requirement of a large aperture for adequate

illumination in the chopping plane because of the

large aperture ratio (equal to the image magnification) provided by the Cassegrainian objective.

After passing through slit A2, which limits the

height of the light pulse, the light is directed to the photomultiplier by an uncorrected lens. Since

this lens is required only to direct all of the light

passing through slit A2 to the multiplier cathode,

it is easily focused and alined. The use of this lens permits us to locate the photomultiplier and

associated video amplifier conveniently close to the pulser.

6.2. Electronics

The photomultiplier and preamplifier gave

pulses of about 10-v amplitude. This was more

than sufficient to trigger a delayed pulse generator

of commercial manufacture which in turn was

coupled by a high-voltage condenser to a diode

clamping circuit which controlled the base voltage

of the cutoff pulse. A block diagram showing the

functions of the various elements in the synchroPHOIO KULIIPLIEB

IICHI BE>H

PPEAKPllflER

1« 10 lOv

PHOIO

KULTIPIIER SUPPlt

OEUt

PUtSER 1 OSCILIOSCOPE ©

mClROKCUK -iFiGURE 11. Block diagram of synchronizing elements.

nizing chain is given in figure 11. Although it is convenient and straightforward, this method of

turning the electron beam on and off caused some

mechanical difficulty in the gun. In order to

relieve the pulse generator from having to drive

the excessive capacitance of the standard gun, a new gun was constructed in which the control grid was a small element controlled by a conductor

running through a separate seal through the side

of the gun insulator. In this way, by keeping the

grid electrode small and maximizing the clearance

between it and the electrodes at fixed potentials, we were able to keep the total capacitance of the

control electrode and its lead-in connection to less than 10 pf, which is quite acceptable. Prior to the use of this system, extensive experiments were

performed in which the beam was deflected elec- trostatically out of the limiting aperture. For

reasons which are still not understood, all these

methods gave rise to a residual beam in the off condition which added to the background in the

plate. With the electron gun biased below cutoff,

there are no electrons anywhere in the optical sys- tem and residual effects do not occur. In order

to view the molecular beam pulse as it traverses across the viewing space, it is necessary to delay

the strobe electron beam by periods of 10 to 100

/usee after the chopper blade intercepts the light

beam. The commercial equipment proved quite

satisfactory for this task, and as the figures show

reproducible patterns could be obtained over thousands of exposures where the jitter on the leading

edge of the pulse may be deduced to lie well below

2 Msec.

7. High-Vacuum Section

The success of the experiment depends criti- cally upon our ability to maintain a lower density

of background gas molecules than the density of chopped-beam molecules in the interaction chamber. We have shown that the signal-to-noise

ratio on the photographic plate is roughly the

molecular density ratio multiplied by the ratio

of electron patli lengths in beam and background.

By employing a small aperture at the entrance to

the main chamber, we shorten somewhat the ef- fective electron-path length through the background vacuum. An electron may miss the

schlieren stop only if the scattering takes place

in the region between the entrance aperture and

the schlieren stop. If scattering takes place before the electron reaches the entrance aperture,

but the electron still goes through that aperture,

the electron is definitely captured by the stop.

16
This is true because the optical system forms an image of the entrance aperture, at imity magnification, on the schlieren stop, which is twice as large as the entrance aperture. Regardless of

any previous scattering it may have undergone, an electron which passes through the entrance

aperture is headed for this image on the schlieren

stop. Similarly, scattering which takes place

after the electrons have passed the schlieren stop

is of little consequence, provided that the majority

of the electrons is not scattered in that region.

The region in which a high vacuum is of crucial

importance to the experiment is thus confined to the 75 cm between the entrance aperture and the

schlieren stop.

Conventional vacuum techniques used in construction of electron microscopes are sufficient to provide a vacuum of about 10~^Torr. Higher vacua

ranging from 10~* Torr to 10"'° Torr are possible

today b}* rigorous exclusion of organic substances

from the vacuum chamber, driving off adsorbed

gases through baking, and gettering, or cryogenic

pumping [44]. The design adopted in this experiment was a compromise between the need for high vacuum and reasonable economy in construction. The optical sections containing the electron

gun and the photographic plates were separated

from the interaction space by airtight walls

pierced only by the 1-mm entrance aperture and

the 3-mm aperture which siirrounds the schlieren

stop. The molecular beam source, the chopper

motor, and the chopper disks are also inherently

gassy. The need to minimize distance between

the chopper and the field of view made it impractical to put these elements in a separate

chamber from the interaction space. Instead, a

thin, stainless steel septum was used to divide

the chamber into two separately pumped sections

(see fig. 9). Adjustable slits on the septum

reduced the area for molecular communication

between the two subchambers to the minimum

that could be achieved without having the septum

slits define the molecular beam. The arrangements for support and removal of the septum were

not considered sufficiently reliable to depend upon

the septum slits for beam collimation. The sep- tum slit width was thus set at 1 mm to give

reasonable running clearance around the molecidar

beam.

The apparatus is thus divided into four separately pumped chambers, A, B, C, and D, as shown

in figure 2. Sections A and B are housed in a

rectangular, stainless steel chamber 25.4 cm wide

and deep by 51 cm long, divided into two sections

by the septum which slides into close-fitting

grooves in the walls of the chamber. By providing a series of grooves, we made the geometry

of the chambers easy to vary. All of the apparatus

required in these two sections is hung from a three-bar optical bench attached to the roof

I)late, wliich is removable. Thus, optical alinement of the various parts may be carried out in the atmosphere, in full view, without the spatial

restrictions of the chamber walls. The alined

sj^stem is then simply lowered into the rest of the chamber, which merely serves as the pumping

orifice and vacuum envelope. The two sections

of the chamber are pumped by separate 6-in. oil-diffusion pumps with individual speeds of

900 liters/sec. Liquid-nitrogen-cooled baffles keep

to a minimum the amount of oil vapor diffusing

back into the chambers. Drift tubes 20 cm long

to the entrance aperture and the schlieren stop

form an integral part of the main chamber A.

All of the removable flanges connected to the

main chamber employ flat, soft aluminum gaskets.

A seal is accomplished by opposing V-shaped,

stainless steel ridges which bite into the metal

from both sides. The ridges are integral parts of

the respective mating flanges. Two sets of ridges are employed, one inside the other, so that air must diffuse through leaks in both sets of ridges

-1' k II

TO VACUUM PUMP

HELIARC WELO

SOFT A« GASKET

' WITH PUMPING HOLE

- CHAMBER WALL

3Z1 1

' A' W -r

STANOARD FLANGE

BAKEABLE ALL METAL VACUUM SEAL

Figure 12. Cross-sectional view of the all-metal bakeable seal used on all parts of the apparatus that were heated.

The figure on the right shows the dimensions that -were used. The dimensions are not critical and small variations are tolerable. We found that one strin- gent requirement on this type of seal is that the 0.00.5-in. flat on the ridges must be completely Iree from all scratches, nicks, or other rough spots. The principle behind this type of seal is that as the ridge is slowly squeezed into the gasket material the metal in the gasket is smoothly displaced about the ridge. On the spot where a nick or scratch is present, the metal will not displace smoothly but will instead tear leaving a minute hole which acts as a very slow

leak. We used soft aluminum gaskets, but copper or other soft bakeable metals are also suitable. 17
to enter the vacuum system (see fig. 12). The

space between the inner and outer seals is evac- uated to about 100 M pressure by being connected

to the fore-pump hne. Thus, a perfect seal is not required to keep the leak rate well within our pumping capacity. In practice, however,

these seals held so weU that only a slight increase

of pressure was encountered if the space between

seals was opened to the atmosphere. Thus, the

extra seal, with pumping between seals, insured

that the leak rate past the seals had to be a

negligible source of gas in the interaction chamber.

In the event of a suspected gasket leak, the space

between seals was filled with helium and the

vacuum leak tested with a commercial helium

leak detector, thus providing a quick means of

finding the offending flanges.

With all of these precautions and with all of

the gas sources removed, however, the equilibrium

pressure of the interaction chamber A registered

about 310~^ Torr, which is definitely not good

enough for chopped-beam work. To give the

chamber A a more decisive advantage, vacuumwise, over the other chambers, we resorted to bakeout at temperatures from 150 to 300 °C,

thus removing adsorbed material from the walls

of the chamber. To insure the persistence of high pumping speed, untroubled by back diffusion at the lowest pressures, we provided for sealing

off the oil pump with a 6-in. gate valve and added

a cryogenic trap. With the pump sealed off.

liquid hydrogen or liquid helium is added to the

trap, and all of the heavier gases are condensed

and frozen on the cold surfaces. Because of its relatively low heat of vaporization and high cost, liquid helium cannot be economically used in Dewars where the cold surfaces are exposed to room temperature thermal radiation. Thus, communication between the liquid helium tank and

the chamber to be pumped must pass through

a one-hit baffle, and all straight-line optical paths

between the tank and surfaces above liquid

helium temperature must be eliminated.

The design of our cryogenic trap is shown in

figiire 13. A C-shaped liquid nitrogen tank

surrounds the cryogenic section, and a single row

of copper Venetian blinds at liquid nitrogen

temperature is arranged so that all incoming

radiation paths miss the inner tank, while the

pumping impedance is held to a minimum. The

evaporated helium or hydrogen passes through

a loosely wound spiral coil surrounding the inner

tank before being conducted to the atmosphere.

From the side facing the opening of the C, gas

molecules can pass between the coils to reach the

tank, and they can also reach the tank in a straight

line path from the bottom of the cold shield. On the sides facing the solid wall at liquid nitrogen

temperatiu-e, however, the space between coils

is filled by a mantle made of aluminum foil to which the coils are attached.

I OF 3 TUBES SUPPORTING

THE LIQUID N2 SHIELD

LIQUID HELIUM FILLING TUBE

COIL EXHAUST

-FILLING VENT

LIQUID HELIUM TRAP

18

FiGTJHE 13. Drawing of the liquid helium trap.
The coils and the mantle are cooled to a temperature well below that of liquid nitrogen by the

cold gas evaporating from the liquid state. This

is possible because, in the case of helium, the heat

required to raise the gas temperature by even

10 °C is far greater than the heat of vaporization

from the liquid state. Thus without appreciably

lowering the pumping speed of the Dewar, the

coil and mantle reduce still fiu"ther the thermal

radiation available to evaporate the helium or hydrogen. In this connection it is worth noting

that lowering the temperatm-e of a surface by a

factor of 2 lowers the rate of blackbody radiation

by 2*, which is a good order of magnitude. About

2.1 liters of liquid helium are used in filling the

Dewar, which has a capacity of about 1 liter. One filling is sufficient for an 8-hr day of operation.

With either liquid hydrogen or liquid helium, the

Dewar was successful in producing a 10"^ Torr vacumn with the apparatus actually taking pic- tures of chopped molecular beams. Hydrogen

was shghtlj" favored because of its lower cost.

The chamber housing sections A and B are fabricated by shielded arc welding of %-in. plate

of 304 stainless steel. Strip heaters are bolted

onto the outside of the walls, and the chamber is surrounded by asbestos sheets wrapped in aluminum foil for bakeout. The bakeout power

consumption was 2800 w. The focusing coils were slid away from the main chamber for bakeout. Other precautions were to siu-round the oven and the chopper motor with copper shields

which could be cooled to liquid nitrogen temperatiu-e. Thus, it was possible to bake the chamber

at temperatures up to 300 °C without overheating

the chopper motor or evaporating the cadmium

from the oven. Nevertheless, the motor was

the delicate component whose presence limited

baking temperatures. Since this apparatus was

built, motors with the ability to withstand

higher baking temperatures have been produced

commercially. With the higher bakeout temperature and with an extra stage of differential

pumping, it is believed that a lO"-"" Torr vacuum

should be possible in instruments of the future.

8. Intermediate-Vacuum Section

The motor coils, motor insulating materials,

bearing oils, and the hot surfaces of the oven, its shields and mounts, together with the air dis- solved in the cadmium, provide .copious sources of gas in chamber D. The tight-fitting septum with

a total area open to section A of less than 0.5 cm

and the narrow clearance between the chopper

disk and the septum combine to reduce effective

pvunping speed between the two chambers to a few liters/sec. In the present design, the approach

to luniting the resulting throughput of gas into the

main chamber was limited to brute force pmnping

with a 900 liters/sec oil-diffusion pump. For ultrahigh vacua, it is recommended that this section be

modified to provide a separately pumped intermediate chamber between the gassy components

and the interaction chamber A. This would re- quire moving the oven about 3 cm back from the

chopper disks and inserting a second septum into

the main chamber. The onlj gassy component

left in the intermediate chamber would then be

the bearing for the chopper drive shaft. By means

of close clearances between a rotating sleeve at- tached to the chopper disk and the main support

shaft, the rate of leakage to the intermediate

chamber could be kept low.

The electron gun, section B, and the accelerator- camera, section C, are likewise rather gassy. The

hot filament gun warms adjacent parts to a point

where they give off adsorbed gas continuously,

but not to a point where the evolution of gas is completed in any reasonable time. In the present

experiment, the 1 mm connecting aperture between this chamber and chamber A, limiting

pumping conductance to less than 0.1 liter/sec, was sufficient to make chamber B a negligible source of gas in the main chamber. Brass construction with silver-soldered joints and elastomer

0-ring seals was adequate for this section, and

also for section D, since the vacuum requirements

are not stringent. Both of these systems were

pimiped by MC-275 oil-diffusion pumps with a rated speed of 270 Hters/sec. These 4-in. pumps

were suspended underneath 8-in. baffle chambers

which provided a 4-in. cylindrical passage with

cold walls. At the entrance to this liquid nitrogen

cooled passage, a cooled one-hit baffle consisting

of truncated conical shells of copper provided a

barrier to oil molecules with minimum pumping

impedance. The precaution of providing a ring

aperture with a sharp edge just below the cryogenic tank appears to have been effective in pre- venting creep of oil up the walls of the baffle. No

traces of oil were ever observed in either the gun

section or the camera section.

In the camera, section C, the moisture and adsorbed gas on the emulsions provide serious contamination. The annular space between the

schlieren stop and the surrounding aperture has

five times the area of the entrance aperture. Thus,

the flux of gas from the camera section to section

A is probably an important source of the background gas in that chamber. Since there is no way to eliminate effusion of gas from the photographic plates, it appears that the best way to reduce this gas source is to install an extra stage

of differential pumping. A drift tube with an at- tached pump, a magnetic lens, and an aperture of

the same size as the schlieren aperture would be

the extra equipment required. The intermediate

stage could then also be subjected to baking if desired. With this added stage of differential

pumping, leakage into the main chamber could be

reduced by a factor of at least 100. 19
9. Vacuum Pump System

The four diflfusion pumps pumping the separate

regions are connected to a common manifold and

to a 600-Hters/sec booster pump. The fore vacuum for this pump is provided by a 7.1- Hters/sec niechanical pump. In order to isolate

its vibration, the mechanical pump is placed in an adjoining room. Throughout the course of

the project, this pumping system has proved

reasonably trouble-free and, in general, satis- factory.

10. Auxiliary Equipment

Other prominent features in the photograph

of the overall experimental arrangement, figure

14, are the four rectangular coils used to counteract

the earth's magnetic field. These coils, generating

two components separately variable to counteract the earth's field, are in the form of elongated

Helmholtz coils and give a compensation which is accurate to perhaps 10 percent. The details of

this field's compensation are given in section 13.2. These coils, together with the three magnetic

lenses in the system, are energized from elec- tronically regulated, constant-voltage sources.

It was found that the drifts due to the change of resistance of the coils were small enough that the

added expense and complexity of constant current

supplies were unnecessary. The following table

gives the characteristics of the power supplies

used to excite these various coils and to provide

the electrostatic potentials needed for the electron

optics.

Figure 15. Three electron schlieren images of continuous atomic beams. To get tlie proper perspective, one must visualize the electron beam propagating from behind the paper and forming an image in the plane of the photograph. The molecular beam originates to the left of the image and propagates from left to right. Unlike the other images that follow, in this set, the field of view is so wide as to include the final set of collimator slits and target. In the left image, the molecular beam was cut off by a shutter. The image then is that due to the scattering of the electrons by the molecules that make up the residual vacuum. The middle image shows the cadmium beam striking a surface which was maintained at 77 °K. At this temperature almost all of the cadmium condenses immediately on the target. In the image on the right, the target was at room temperatitfe and the cadmium beam was reflected from the target. Multiple scattering of the molecules in this region gives the appearance of an increase in background. Because of contrast variations in the reproducing and producing processes this effect may not be visible in the printed copies.

Figure 14. Overall view of the assembled apparatus. The electron optical axis is from left to right, whereas the molecular beam

axis is directed out of the paper toward the reader. All of the components

are described in the text except the three-way glass valve which is attached to the camera. This is a commercial object which was used to pre-pump the camera chamber and seal it off before it was opened to the vai'uum in the apparatus. One can see seme of the strip heaters attached to tlic cryotrap and main chamber in preparation for a baking cycle.

Characteristics of power supplies

Function Potential Current Regulation*

ma %

Gun 0 to 6 kv 0 to 100 1 Condenser 0 to 500 V 0 to ."iOO 0. 01 Focusing coils 0 to 300 V 0 to 300 0.02 Post acceleration.. 0 to .30 kv 0 to 10 Unregulated

Post deceleration 0 to 300 V 0 to 100 1 Magnetic compensation twin

supplies 0 to 100 v 0 to 100 0.1 Pulse amplifier. 0 to 300 v 0 to 150 0.02

*A11 regulated supplies supply constant voltage.

20
FiGUBE 16. Stroboscopic images of chopped atomic beams.

Set of Images produced by operating the instrument stroboscopically. In contrast with the previous figure, the field of view here is restricted to exclude the collimator and target areas. Starting with a 15 jjsec delay from

the time the pulse is generated until the on time for the electron beam pulse, the first image shows the molecular beam packet entering the field of view. With each successive image representing an additional 15 ^sec delay, one can

see the molecular packet progressing across and out of the field of view. If one examines the form of the packet, one sees a taper in the trailing end, i.e., the left end. This shape is a consequence of the fact that the molecules

continue to stream past the edge of the chopper slit as it is in the process of cutting ofl the beam segment from one side. A similar taper should appear

in the leading end. The visibility of both of these tapers is blurred, however, by the spread in molecular velocities over a 3 to 2 range. In the trailing edge of the pulse,

this velocity spread is effective for a shorter period of time than in the leading

edge. Hence, the effect of the spread almost completely washes out the beam taper in the leading edge, while this taper is clearly visible in the trail- ing edge. The slope of this taper is the ratio of chopper cutting speed to mean beam velocity, i.e., 1 to 4. The 3 to 2 spread in molecular velocities produces some increase in the length of the packet as it proceeds across the

field of view. These images have a time resolution, i.e., and electron pulse width of 3

/isec. The maximum particle density is about 2X10'<'/cm3.cm3. The irregular bright spot in the center of the schlieren image is an imperfection of the electron optical imaging technique.

11. Experimental Results

The preceding three figures give the kind of

pictiires which may be obtained with this equipment. Figures 15a, 15b, and 15c are photographs

of a continuous cadmium beam reflected from a

brass surface and diffusing into space. The beam

contains a significant density of multiply reflected

atoms. The latter two figures show the main

results with a sharp cadmium beam. Figure 16

shows the spread of the cadmium beam pulse

due to the finite energy resolution of the chopper

sHt as the pulse progresses across the observation

chamber. In all cases, one can notice certain

artifacts in the background which are described

in the caption. By the end of the program, the

sources of most of these were understood and the

Figure 17. A series of microdensitometer traces across

the image labeled SO iisec. The vertical scale represents the position along the length of the packet

in mm units. The lowest trace is that of the background at the edge of the

field of view on the left side of the image. The successive traces show the

rapid build up and one can also see the taper and spreading in the packet. By subtracting the background in the image, it appears that the photographic

density difference can be read with an accuracy better than 10 percent, and

spatial resolution better than 0.1 mm on the photographic plate.

most serious of these disturbing factors removed.

How these factors affect the available contrast is shown in figure 17, which presents densitometer

traces under various conditions of alinement.

12. Interpretation of the Data

We have shown that a practical electron optical

schlieren apparatus does produce photographs of

low-density gas formations with a spatial resolution of about 0.1 mm and a time resolution of about

1 Aisec. The principle of velocity analysis, using

data from these photographs, is essentially that if a group of molecules is in a fixed point at time ti, then their spatial distribution at time ^2 is readily

translated into the velocity distribution. More

precisely, if we choose our coordinate system so that all molecules are at the origin at time ^i, and their spatial distribution at time <2 is given by

N{x,y), then the velocity distribution is given by

i(V,,Vy)-{t2-t,y-N(VS2-t,], V,[t,-t,]). (12.1)

Experimentally, however, it is impractical to use a molecular beam packet whose initial dimensions are negligible in comparison with its final dimensions. As the packet spreads in space, its 21
mean densit}'^ decreases correspondingly until it

is masked by the background gas density. Whatever velocity distribution information is to be

obtained must be extracted from photographs in

which the mean spread due to the different

velocities amounts to only a few times the original

beam dimensions.

The original distribution of the beam packet

may take either of two forms, depending upon the

purpose of the experiment. We shall treat both

initial forms by means of a single mathematical

formalism. The first case to be considered is the

determination of velocity distribution from two

photographs of the same beam packet at different

times. We will concern ourselves only with the

special case in which the velocity distributions of

molecules at different positions in the earlier

picture are assumed identical. This condition

is approached in the case of a beam packet which

has just left the second chopper. We then have an

initial spatial distribution Nt^{x,y).

The second case which we shall consider is the

determination of the velocity distribution of

molecules which have been reflected from a solid

target. For this case we choose the Cartesian

coordinate system so that y=0 on the target

surface. We can then write a space-time density

function No(x,t) describing the arrival of molecules

at the target. The form of A^o might be determined

from a series of stroboscopic pictures showing the

progress of the incident beam toward the target

surface or, alternatively, it might be determined

by calculation from the known geometry of the

experiment. In section 13.4 we discuss the calculation of Nt^{x,y). and of No{x,t).

Assuming that this initial solution is available, one has first the task of determining the spatial

distribution Nt^ix, y) of the molecules from the

density pattern on the photograph and then the

task of deducing the velocity distribution from

the initial and final molecidar distributions.

12.1. Scanning of the Negatives

A distribution Ne(Xi, yi) of electrons on the

photographic plate gives rise to a photographic

density distribution D{Xi, yt) which may be read

with a photodensitometer. Here, in accordance

with usage in photographic work, D is the base

10 logarithm of the ratio of light transmitted

through the developed negative to that incident

upon the negative at each point. Present-day

evidence is that for electrons, D is proportional

to Ne at low exposure levels, although the curve is certainly nonlinear at higher levels. This behavior contrasts sharply with the situation in light optics, where D is usually plotted against

the logarithm of the intensity. In that case, a

threshold level of illumination is found, below

which the response is relatively poor. The absence of a threshold in the case of ex- posure by electrons may be understood from the

fact that each 10-kev electron has sufficient

energy to expose several grains of the emulsion,

while several light photons are required to expose

a single grain. At low exposure levels, the

majority of photons may be involved in grain

excitations which did not produce exposure,

and the response at low levels is thus poor. In

electron exposure, however, the first electron is as

effective in exposing grains as any other electron.

Each developed grain of silver absorbs a fraction

of the light incident on it which is proportional

to its thickness. When a light ray passes through

a series of developed grains, the successive losses

by absorption produce an exponential dependence

of transmitted light on the number of grains

encountered. The photographic density, which

is proportional to the logarithm of the transmitted

light, is thus directly proportional to the spatial

density of developed grains.

If the electron beam energy is sufficient to penetrate the emulsion with most of the incident

energy intact, the, probability that a silver grain

will be developed is independent of its depth in the emulsion. The probability that a silver grain

will escape excitation then becomes a decreasing

exponential function of the number of incident

electrons per unit area in the vicinity of the grain.

Combining these results, we conclude that the

photographic density must be related to the number of electrons per unit area by an equation of

the form

where Z?max and Aq are constants characteristic

of the emulsion, the developer, the development

time, and the development temperature.

Instead of depending upon the accuracy of this

formula, however, it is best to calibrate the plate

response against known relative electron densities.

We are not much concerned with absolute electron

densities, since we intend to calculate only a

relative density pattern for the molecular beam

packet.

To calibrate the emulsion, following a method

widely used for calibration of spectroscopic plates

[45], we set up our apparatus in the schlieren condition and deliberately bias the electron gun to provide a non-uniform illumination pattern. The

situation is ideal if the beam intensity fades

gradually to zero at the edge of the field of view.

A series of plates is then exposed with linearly

increasing exposure times, and a densitometer

trace is taken along a common line on each exposure. The instrument must be adjusted to give

all of these traces with respect to a common base

line. When these traces are superimposed, we

obtain a picture something like figure 18. Starting at any convenient point on the lowest

curve, we define a unit of electron density as the

density required to produce the photographic density of that point on the lowest curve. The photographic density corresponding to 2, 3, 4 . . .units

of electron density may then be obtained by drawing a vertical line through the upper curves and

picking off the ordinates. We plot the restdting

22
in

EXPOSURE CHARGE DENSITY, ARBITRARY UNITS

a

POSITION, SCALE IS ARBITRARY

Figure 18. Illustration of the method for calibrating photographic plate response.

To illustrate the method, we assumed a hypothetical photographic response curve, shown as the solid curve in figure 18a. Aside from being monotonic,

this curve has no special properties. It was produced simply as several segments from a draftsman's French curve. A hypothetical photographic density plot, shown as curve a in figure ISb was drawn by the same technique. The curve was made to approach the base line (corresponding to the residual photographic

density on an unexposed developed plate) asymptotically, but the two halves were made deliberately asymmetric. Curve a thus represents a plausible densitometer trace of a schlieren exposure made under conditions of non-uniform illumination. Five units of exposure time were arbitrarily associated with

this curve. We assumed that this density trace was produced by film having the response shown in figure 18a. By straightforward graphical construction, we

used the two curves to generate additional hypothetical density traces corresponding to 4, 3, 2, and 1 unit of exposure time. These traces are shown as curves

b, c, d, and e, respectively, in figure 18b. The set of five curves thus constitutes a possible set of density traces consistent with the.given plate response curve. Imagining the curves to have been obtained experimentally and the response curve to be unknown, we require a practical procedure for extracting the

response curve from the given data. In figure 17b, a vertical line was drawn through an arbitrary point near the top of curve a. At the intersections of this

vertical line with curves b, c, d, and e, we draw horizontal lines, and at the intersections of these horizontal lines with any of the curves, \\ e draw additional

verticals. Using the fact that the intersections of the five curves with a given vertical represent photographic responses to exposures in the ratio 1:2:3:4:5, we

develop a consistent set of relative exposures which cover practically the whole usable range. The unit of exposure was defined arbitrarily by assigning 10 units

to the exposure producing the highest response in this scheme of intersections. When ordinates were plotted against exposiu'e in these arbitrary units, the

e.xperimental points shown in figure 18a were obtained. The absence of significant systematic error and the small range of scatter about the original response curve show that the method can produce accurate calibrations, even when the plate response is highly nonlinear. The horizontal scale used in the plotting of points in figiire 18a depends upon the arbitrary definition of the exposure unit, and our choice was such as to . make the scale for the points agree with that for the original curve within 1 percent. If a difTerent unit size had been chosen, however, the resulting calibration • curve would have been jast as suitable for analysis of stroboscopic pictures. Use of the same calibration scheme on opposite sides of the hypothetical densitometer traces gives two independent values of photographic density at each exposure value. In this idealized example, however, most corresponding points were

too nearly coincident to be resolved as separate values.

densities against units of electron density. Now

from the intersections of the vertical line with the

upper curves, we draw horizontal lines and note

where these intersect any of the curves. Wherever

this happens, we can draw another vertical line and find the density readings corresponding to another series of integral multiples of the unit of electron density. All of the information thus

obtainable by continuing the series of horizontal

and vertical lines and deducing the number of exposure units that should be associated with each

intersection of a vertical line with a curve is plotted on a single graph. The points thus obtained

should include most of the integral multiples of the

basic exposure unit.

Next, we can start with another arbitrary point

and construct a graph of photographic density

versus integral multiples of the second arbitrary

exposure unit. After drawing smooth curves through the two sets of points, we then. attempt

to reconcile the exposures which produce equal

densities. This process gives a highly redundant

way to compute the ratio of the two arbitrary

exposure units. The two curves may then be

combined into a single curve with twice as many

experimental points. Continuing in this manner, we can obtain as many points as desired for a

precise calibration curve. The technique works

equally well whether the response is linear or not. We can use the calibration curve for our emulsion to translate photographic density patterns

into corresponding electron density patterns

through point-by-point conversion.

Suppose now that we have two electron sclilieren photographs taken vmder identical conditions

except for the time delay between the light signal

23
and the electron beam pulse. In the first exposure, the time delay is set at perhaps 1000 usee, so that the molecular beam pulse, whether direct or reflected, has been completely dispersed and

trapped. This picture gives a map of the product

of incident electron current density and background gas density. Since we expect the background gas density at this late point in the duty

cycle to be essentially uniform, we can attribute

aU of the density variation to irregularities in incident current density and in instrument re- sponse (due to lens aberrations, nonparallel beam,

etc.). In the second exposure, we have the same

background plus the atomic beam packet at the

time which we have selected.

If the two sets of exposure conditions were

exactly the same, we can write the molecular density function as seen in the image plane

D(xi,yi)-- 'Ni{xi,yi)

•1, (12.3)

where Ni and A^2 are the observed electron densities in the respective exposures. If through imperfect control of exposure conditions, the integral

of gun current times time in the two exposures is not the same, the constant in eq (12.3) will differ from one. In the absence of any better criterion,

we would be forced to set this constant in the manner which gives the most plausible spatial

distribution of the chopped pulse.

Finally, the density function computed in the

image plane is transferred to the object plane by

substituting for (xt, yt) . The density function thus computed is expressed in units of the

background density multiplied by the ratio of

effective electron path lengths in the background

and in the beam.

12.2. Computation of Velocity Distributions

The method of computation employed in this

section is similar to an iteration method used by

Bennett and Estermann [46] in a one-dimensional

study of the velocity distribution in a molecular

beam. In the present work, this method is adapted to a different form of input data and

extended to cover two-dimensional distributions

in velocity space. We assume that by means of the densitometer

the photographic density distribution D{xt, y^ has

been determined and that the procedure of section

12.1 has been carried out, yielding the spatial density function 1'{xq, yo), which is the line integral

of the molecular density function along the elec- tron path through (xq, yo). For simplicity, in this

section, we drop the subscript o, which has been

used to indicate merely that the coordinates are measured in the object plane. It is important at

this point that the orientation of the coordinate

axes with respect to the molecular beam and to the target plane should be specified unambiguously. In the work which follows, the a;-axis is

parallel to the target plane and directed upward,

while the y-axis is directed horizontally in the di- rection of propagation of the atomic beam. The

origin is located in the horizontal mid-plane of the

atomic beam at its intersection with the plane of the final beam chopper. We designate the observed spatial density distribution at the known

time t by l',{x, y).

From the geometry and timing characteristics

of the apparatus we have knowledge of the molecular beam distribution at an earlier stage of its evolution. This information is described by a

function hy{x, t) which describes the rate of arrival

of molecules at a given plane normal to the direction of beam propagation. If we are studying

direct beams, i.e., those which have not interacted

with any object in their path, the function hy{x, t) is referred to arrival at the plane of the final chopper. In reflected beam studies, this function

describes arrival at the target plane. The computation of hy{x, t) is described in section 13.5 and

a simplified approximate computation is given in

13.6. In this section, hy{x, t) is considered to be

known.

The coordinates in x—t space together with the

velocity components Vx, Vy serve to designate all information of interest about a given molecule.

The packet of molecules may be defined by a density function p(x, t, Vx, Vy), where

dp=h(x, t)N{vx, Vy)dxdt dv^^dvy. (12.4)

Molecules leaving the region of x—t space of

dimensions dx, dt with the exact velocity components Vx, Vy are found at time t to occupy a parallelogram in x—y space of base dx and altitude Vydt.

Conversely, the molecules with velocity components {vx, Vy) which are observed in the region of area dxdy must have originated in an area of x—t

space equal to dxdt=^^- (12.5)

The total number of molecules observed in the area dxdy is an integral of eq (12.4) with the

substitution of expression (12.5) for dxdt. Thus we obtain

dK{x,y)J^^- N{v'x,v'y)dv'xdv'y, (12.6)

where x and t may be determined in terms x, y,

'^x, K the equations of motion at constant

velocity. It is most convenient in the work which

foUows to work with a related distribution function

di{vx, Vy)=fdl'(x, y), (12.7)

where Vx=xlt, Vy=ylt. The new distribution function is the velocity distribution that we would

compute if we assumed that all of the molecules

had been at position x=0, y=0 at time t =0. This approximate starting condition is made as

realistic as possible by choosing the origin in 24
x—t space at the centroid of the distribution

h{x, t). The experimentally determined function

I (vx, Vy) is thus a first-order approximation to the

true velocity distribution function N(Vj:, Vy). Substituting eq (12.7) into eq (12.6) we find that

I {Vx, Vy) is related to the true distribution by the

integral equation:

iivx, 0=JJI Kx, t)N{v'x, v'y)dv'xdv'y. (12.8)

Now if we represent the actual molecular velocity components (v'x, v'y) as the sum of the apparent

velocity components and a correction:

Vx=Vx+v'x'; V'y=Vy+V'y

eq (12.8) may be rewritten as

(12.9)

ii'Ox, Vy)= \\ ^ n Kx, t)N(Vx+v'x', Vy-\-v'y')dVxdv'y'

.

(12.10)

With the equations of constant velocity motion,

i.e., x=VxT=Xo+Vx{T-t) (12.11a)

y=v^T=yo+v'y{T-t), (12.11b)

and a bit of algebra, we obtain

:

-=r(<-|±|-<) (12.12a)

77, (12.12b)

Vy +Vy

and

I (VZ, Vy)=jjfiVx, Vy, V'x', Vy)N(Vx+Vx, Vy +Vy)

dv'x'dvy, (12.13)

where

ffe,..,.;',.;')=^*(r[<|±|-4

3:) '^^-"^

Since /is a known function of its four variables,

the double integration becomes a linear operator,

and in operational notation, eq (12.13) becomes

simply

l{Vx, Vy)=L[N{Vx, Vy)]. (12.15)

If h had been a delta function of {x, t) , then I and TV would be identical. If h{x, t) is sharply

peaked about the origin, I must be a reasonable

first approximation to N. Thus we write:

and the operational equation becomes

l-Ll=LNi. (12.17)

The function li{Vx, Vy) = l—L I may be computed in a straightforward manner. Equation (12.17)

above thus becomes

U(Vx,Vy)=LN,(Vx,Vy). (12.18)

This equation has the same form as the original

operational eq (12.15), with a new pair of known

and unknown functions. We may thus proceed

with additional degrees of approximation by

repeating the operation with which li was derived from I. Thus by induction we arrive at the

result

mvx,vy)=i+j::a„ (12.19)

where

l,= l,-i-Ll,_,. (12.20)

12.3. Reliability of the Iteration Process

The process of successive approximations which

has just been derived is an inversion of the double

integration process

I (Vx, Vy)=L- NiVx, Vy) = Sfj{Vx, Vy, v'x, Vy)N(Vx

+v'x',Vy+v'y')dv'x'dyy'. (12.21)

The value of I at any fixed velocity (Vx, Vy) is a weighted integral of the value of A^in the neighborhood of the given velocity. This averaging process

tends to round off any prominent features in

N{Vx^, Vyo), making i an inherently smoother function than N. A Fourier transform of i is thus

relatively lacking in high-frequency components,

as compared with the corresponding transform of

N. The iteration process which we have develops! restores the high-frequency components

which were attenuated by the weighted

integration.

The way in which this process works may be

illustrated by applying our iteration method in a thought experiment to the inversion of a convolution integral in one dimension. The resulting

conclusions must be applied with caution to oiu" problem, since the double integration process

L • N{Vx, Vy) is not exactly a convolution process

.

It differs from a convolution process in that the

multiplier / is a function of the fixed variables

Vx, Vy as well as of the dummy variables v'x, v'y. Thus, we limit the use of convolution theorems

to prediction of experimental error, rather than

attempting to use convolution to find N(Vx,Vy). In our experiment N(y) is an unknown function

to be determined by experiment and calculation.

The apparatus is designed to yield the output

function

l(x)=L-Nix)^ r f(y) N(x+y)dy, (12.22a)

\J — CO

^(t'x, Vy)= l(vx, Vy)+Ni{Vx, Vy), (12.16) whcTC /(?/) is a known normalized function

25
(J" /(?/)#= l). (12.22b)

Because of various experimental errors, the ap- paratus yields an output function

lm{x) = lix) + le(x), (12.23)

where le{x) represents experimental error. If we

succeed in exactly inverting the integral equation,

we obtain an experimentally measured value of N,

which differs from the true value by an error function

NM)=mx)+Ne(x). (12.24)

By definition, Nm{x) must satisfy the relationship

i^{x)=L-Nr,{x), (12.25)

and hence, by subtraction

le{x)=L-NXx). (12.26)

If we take the Fourier transforms of both sides

of these equations, i.e., y(w)=—L^ j g{x)e~^"Hx,

-y 27r J —ca the convolution theorem [47] provides the relationships : ~ (12.27a)

(12.27b)

L(a;)=7(co).iV,(c.) (12.27c)

where the bar over a function signifies the Fourier

transform of the corresponding function of x. The apparatus thus reacts to the unknown input

function with a frequency response /(w) . Since

the integral C,(w) representing the convolution is a weighted average of values of N(x) at neighboring

values of x, we can expect precise reproduction

of slowly varying functions of x and extreme

attenuation of rapid fluctuations. Thus /(w)

approaches unity at low values of w and approaches

zero at high values of w. The transforms of

N{x), Nm{x), and Ne{x) may be obtained by

inverting the above equations:

lv„(co)=i^

/(co)

iV(co)=l^

iv,(co)=iM.

/(a.)

(12.28a)

(12.28b)

(12.28c)

The fact that /(w) approaches zero at large

values of w provides no special difficulty, since

2,(w) has been forced to be correspondingly small

26

by the response of the instrument. The measure- ment error ie{x) is not necessarily so deficient in high-frequency components, however, and it is entirely possible that NXoi) may increase

indefinitely as w approaches infinity. In fact,

the random character of grain noise on photographic emulsions suggests that te(w) should

approach a constant value as w approaches

infinity. A complete correction for the smoothing

effect of the finite instrumental frequency response

is thus rendered impossible by the presence of

high-frequency noise in the measured response

function.

It is instructive, now, to observe the frequency

responses of the successive approximations

obtained by applying our iteration process to the conversion of experimental data. The itera- tion process yields the series of functions of

X} Im.aix), Im.iix), im.2{x) . . . Satisfying the rela- tions

lm.k(x) = l m. (*-l)(x) -Lirr.. ik-l){x) (12.29a)

lmM)=L-N^{x). (12.29b)

Successive approximations to Nm{x) are NmAx) = y ] 'i=nt m.k{x).

Applying the Fourier transform to these condition yields

rTM,*(w)=rTM.(,_,)(co)[i-7(co)]. (12.30)

Hence, by induction

TmA<^) = lm.o(cS)[l-J(co)f (12.31a)

and

=NM •7(co)z;?=o[i-7(w)]^ (12.31b)

Now if we let g{cii) = l—f{w), the equation

becomes

N^.,(o^)=NM[i-g(o^)]I2Uo'gio^y (12.32)

or

,(a,) =NM[1 -?+n'^)]- (12.33)

Now fir(co) is close to zero at low values of w and

approaches unity from below as w approaches

infinity. The behavior of ^*(w) for several values

of k is illustrated in figure 19a.

The result of a finite number of iterations is related to the limiting result of an infinite number

of iterations by the transform equation

where

(12.34)

A(a,) = l-2*+'("). (12.35)

The effect of a few iterations is thus to raise the
frequency response nearly to unity in the range

of frequencies where an appreciable response was

originally present, while providing only a small

increase of response in regions where the original

response was low. This is illustrated in figure 19b.

In the limit as k approaches infinity, the effective

frequency response approaches unity at all frequencies.

If the characteristics of the unknown function

N{x) are well enough known in advance to yield

the general shape of A^(w) and hence of {-(w), and

the noise spectrum is known, we can use this information to choose k for optimum conversion

of the data. The ideal data function E (co) can be

expected to be larger than ieij^) for frequencies

below a certain value u,. and to be smaller than

ieW) for higher frequencies. The response to any frequency below thus yields more signal

than noise and vice versa. The maximum

fidelity to meaningful signal consistent with

adequate noise rejection will occur if the steepest

slope in occurs in the neighborhood of co,.. The transition from nearly unity response to nearly zero response thus occurs in the neighborhood of the lowest frequency at which noise

exceeds signal.

In practical use of the iteration, one would not

attempt to match /^(w) to estimated signal and

noise spectra. Instead, one would simply carry

out successive iterations and pick out the most

convincing solution on the basis of expected

properties of the unknown function, just as a microscopist carries out through-focus studies

and selects the most convincing picture. The

above thought experiment serves, however, to indicate strongly that the result of such an iteration process is likely to extract close to the

maximum of useful information from the experimental data.

In this section, we have outhned the principles

to be employed in analysis of the data, where a complete velocity distribution is desired. In

many cases, the desired information may be less detailed and considerable simplification of the

procedure may be feasible. Such simplifications

are most efficiently made by the person wishing

to obtain the velocity information, and were,

therefore, considered beyond the scope of this project.

g((ij

Figure 19a. Variation of the function g^ioi) for successive values of k.

In this illustrative example, <;(„) was chosen as l-e-"^. The function

g^iso) may be regarded as a measure of frequency distortion in the result of the l£th iteration in the process of computing N(i).

Figure 19b. Variation of the frequency response function

fjs (oj) after several successive iterations. As the iteration process proceeds, the frequency response is made to approach unity in regions where it was initially high, while considerably

smaller gains occur in regions where the initial response was poor. The

initial response function was chosen as in consistency with figure a. Figures 19a and 19b are mirror inversions of each other. From this figure we

see that the process of successive approximations produces a region of excel- lent response in the low frequency region and gradually extends the high

response region to higher frequencies. The response to the relatively low

frequency signal spectrum is thus improved far more than the response to high frequency noise is increased.

13. Appendix

13.1. The Photographic Camera

There are several methods with which one can

detect electron beams. When the desired information is in the form of an image, e.g., electron

microscope images, the simplest and most reUable

method is to use photographic plates. The

plates provide a compact permanent record,

which may be analyzed in whatever detail seems

initially suitable without jeopardizing the possibihty of more detailed analysis if desired at a

later time. Since our recording problem is quite

similar to that in the electron microscope, we

decided to use photographic plates to record the

electron optical images.

The technique of using photographic plates in electron optical work differs in only two ways

from that of the light optical analogue. First,

the electrons can have no obstruction in their path

to the plate; second, the exposure must be made

under hard vacuum condition lest the electron

27
Figure 20. Top view of camera exterior.

Figure 21. Center section of camera as seen from side facinq the operator.

image become diffused by scattering from the

residual gas molecules. The processing of the

plates is identical to that used in the light optical

analogue.

The requirements on the camera were such that

it was necessary to design and build this unit in our own laboratory. In addition to storing and

manipulating the sensitized plates, the camera

had to be capable of being evacuated to a pressure

of 10~^ Torr or better. Also, to enable the operator to see and control the image produced by the

apparatus, a phosphorescent viewing screen had

to be built into the device in such a way that the

screen could be replaced with a photographic

plate when an exposure was to be made. In

addition, it is very desirable to provide for re- moving the exposed plates and reloading the camera without disturbing the vacuum in the

main chamber. The camera described below fulfills all these requirements.

The construction materials used were all nonmagnetic; namely, brass, aluminum alloy, phosphor bronze, stainless steel type 304, glass, and

an epoxy resin to avoid interfering with the

magnetic fields used to control the optics of the

electron beam.

The camera consists of three main parts bolted

together, as can be seen in figure 20. The center

section, as seen in figures 21 and 22, contains the

mechanisms for manipulating and storing the

plates. By studying figures 21 and 22 one can

see that by turning the knurled knob, one causes

the parallel clamp or tongs to pick up a plate from

the magazine and carry it to the area immediately

in front of the viewing screen where the exposure

may be made. When the motion is reversed, the

plate is deposited in the magazine. If the reverse motion is continued, overriding "the stop, the

magazine is advanced to the next position,

bringing the next plate in line for the next exposure. The advance mechanism employs a simple

ratchet system where the short tong, as seen in figure 22, presses against two lever arms. One

arm unlocks the magazine while the other causes

28
the magazine to advance one notch. On reverse motion of the tong, a spring, seen in figure 22, forces the alining wedge to engage and lock the

magazine in place. Simultaneously the ratchet cam advances to the next groove, leaving the

magazine in line and locked in position for the next cycle.

On the front section, as seen in figures 20 and

23, are the viewing screen-vacuum seal device

and the port through which the film magazine is loaded into the camera. The vacuum seal on this port opening is simple and seU-explanatory. The

screen-vacuum seal port is simply a metal tube

in which a thick glass disk was sealed in place near one end with epoxy resin. On the end of the tube a recess was made to hold an 0-ring

which seals off the camera chamber when the tube

is advanced into contact with the back plate of

the camera. To make the vacuum seal between

the cylinder wall and that of the camera, an

0-ring groove was machined in the front plate to form a "WHson seal," thus allowing the cylinder

to move along its axis and rotate. With the

diameter of the viewing port being about 3 in. and with the camera pumped to a high vacuum,

the pressure on this surface was slightly over a hundred pounds. This, of course, makes it difficult to pull open the valve. Although it is not shown in the photograph, the solution to this problem simply consisted of adding a fine threaded

section to the inner cylinder and attaching a "nut" to the outer one. Thus, by rotating the

cylinder, one could force the inner cylinder to move along its axis. The back section consists of a flat plate with two

ports. One port is connected to a vacuum pmnp

for prepumping the camera chamber before it is opened to the main section of the apparatus. The

other is in line with the viewing screen valve

located on the front plate. It is attached to the

accelerator by a conventional 0-ring groove flange.

This camera proved to be reliable and was used

throughout the work.

13.2. Earth's Field Neutralization

The most direct way to ehminate the misalinement caused by the action of the earth's magnetic

field is to produce an equal and opposite field. A

field of sufficient strength is readily produced by a

configuration of wire carrying an electric current. In general, such a field is non-uniform and thus

provides only partial compensation which is restricted to a limited region in space.

It is possible, however, to enclose certain spaces

with a continuous distribution of circulating cur- rent such that the induced magnetic field within

the space is uniform. For example, a sphere may

have surface currents in circular paths about its 2-axis with surface current density proportional

to cos d, where 6 is the angle between the axis and

a radius to any point on the sphere. Similarly, an infinitely long cylinder may have surface currents in longitudinal paths with current density

proportional to cos 6, where 6 is again measured

between the field direction and a radius to the

point in question. These continuous current

distributions may be effectively simulated by

configurations consisting of a finite number of conductors. The resulting fields are quite smooth

near the center of the enclosed space, but depart

progressively from the uniform central field as the

boundaries are approached. It is thus necessary

for the entire region traversed by the electrons to be within a sphere or cylinder of much smaller

diameter than that of the surface containing the

conductors if a high degree of field compensation

is to be obtained.

If the spherical geometry were to be used, this

requirement would mean that the diameters of the

compensating coils would have to be large in comparison with the length of the optical path

from the source to the image plane. The impracticality of such an arrangement requires no

elaboration. With cylindrical geometry, the variation in directions transverse to the optic axis

will be small, as long as the radius of the cylinder

carrying the conductors is several times greater

than the largest distance from the axis permitted

for an electron. In mounting conductors outside the walls of the vacuimi system, this requirement is almost

automatically fulfilled. Field variations as a function of the z coordinate are more difficult to

control, but even here the cylindrical geometry

presents less difficulty than the spherical case. The conductors parallel to the axis must be terminated at some reasonable distance beyond the

Umits of the optical paths, and connecting wires

across the ends of the imaginary cylinder complete

the current loops. For good field imiformity

along the 2-axis, the extension of the conductors

beyond the optical limits should be large in comparison with the cylinder radius. If this requirement is carried out, the observer watching the

fluorescent screen must sit inside the current loops.

We shall show presently, however, that field compensation within 5 percent of the earth's

field can be obtained without accepting such

awkwardly long coils. In general, a set of coils compensates for only one component of the earth's field. Since the

experimentalist generally likes to choose the

orientation of his apparatus in accordance with

mechanical convenience, it is to be expected that

both horizontal and vertical transverse field components will be present if the optical axis is horizontal. It is possible for a single set of coils to balance out the two transverse field components

if it is properly oriented by rotation about the

optic axis. It is more convenient, however, to adjust for precise compensation by varying the

currents in two sets of coils rotated at right angles

to each other. Thus, the mechanical supports of the coils can be fixed and the adjustments may be

made by setting a potentiometer.

The above arrangement makes no provision for balancing out the axial component of the earth's

field. Instead of deflecting the electron paths as the other two components do, this field acts like 29
a continuous distribution of weak magnetic lenses. The focal power of this distributed lens becomes

important in distances of the order of

-=2c

/2m

H H

cm,

where Vis the accelerating potential of the electron

beam in volts and H is the earth's magnetic field in oersteds. If we attempted to work with elec- tron beams with only a few volts of accelerating

potential, this focusing effect would become fairly

important; but a variety of annoyances, which

are easy to accept when working with beams at higher potentials, would complicate operations

much more seriously than this focusing effect. Consequently, the electron beam is never operated

at voltages low enough to provide serious focusing

effects from this source; and there is no practical

need to cancel out the axial component of the

earth's field. Returning to the problem of cancelling out the

transverse components of the earth's field, there

remains the question of the number of compensating coils to be arranged about the imaginary

cylinder. Neglecting end effects, each long rec- tangular coil may be thought of as a pair of wires

carrying currents in opposite directions parallel to the axis. Imagining the coUs to be in horizontal

planes (thus providing a vertical compensating

TOP VIEW

30-

30'

CROSS

SECTION

Figure 24. Top view and cross-sectional view of optimum

pair of elongated rectangular coils for field compensation.

field component), conditions of symmetry require

that each coil above the x axis should be matched

with a duplicate coil at the same distance below

the axis. The induced vertical field on the y axis

is then an even function of y. If H(y) is expanded as a Taylor series, the result

is Hiy)=Ho+H2y'_+H^y'-{- .... If only one compensating coil is used, the first unwanted term

will be H2y'^. If two coils are used, we can ar- range the coil positions such that H =Q and the

first unwanted term is Hj?/^. Similarly, as additional coils are used, one more term can be eliminated from the series for each additional coil. Unless accurate compensation is needed over a wide range of y values, a very small number of

coils proves to be adequate. In the present ex- periment, two coils were used for each transverse

field component. The use of two coils provides more accurate field compensation than one, but

the more obvious reason for not using a single coil

is the fact that one coil in the horizontal plane

would get in the way of an observer looking at the

fluorescent screen. Thus, the practical choice is whether to use two or four coUs to neutralize each

field component. In the two-coil arrangement,

the wires and current directions are arranged as shown in figure 24. The vertical component of the magnetic field 2% due to one of the wires is given by Hn=— cos where r„ is the distance from the point in question

to the wire, dn is the slope angle of the line con- necting the point to the wire, and i is the current

in abamperes or one-tenth the current in amperes.

Thus, if the vertical distance between the coils is 2d, the vertical field strength along the axis

becomes

rr( . J^Id^ r 11 1

10 lM'-\-y'-\-2dz'^W+y'-2dzJ

0.8V3

(13.1)

y where <^=^' ^ current in amperes.

The variation of field strength along the y axis

is then approximately

AH

(13.2)

thus providing field compensation within 0.4 per- cent for values of \y/d\ up to 0.5. Within the range

\r\<Cd/2we are assured that the field is comparably uniform. This can be seen when the scalar potential is written as a series of spherical harmonics:

m=-^\r cos e-g^ r^P, (cos 0)+ . . . ],

(13.3)

30
• N

•

Figure 25. Pair of Helmholtz coil sets for compensating

both transverse components of the earth's magnetic field. With conventional currents in directions Indicated, horizontal colls produce

a field directed upward and coils in vertical plane produce a field directed to the left. In practice, current directions are readily checked with a hand

compass.

where r and 6 are polar coordinates, and the

P„(cos d) are Legendre polynomials. Thus, the

Hne integral of AH/H horn the plane y=0 to any

point on the sphere of radius d/2 does not exceed

A^^ d d

1280

(13.4)

When two sets of coils are used to compensate

for both the horizontal and the vertical component

of the earth's field, the configm-ation of wires

becomes that shown in figure 25. In the present

experiment, however, it was felt that the inconvenience of having coils run almost directly above

and below the optical system was not justifiable In most of the United States, the earth's magnetic

field is inchned at an angle of 60° or more to the

horizontal. The horizontal component of the

earth's field is thus roughly half of the vertical

component, and the portion of that component

which is transverse to the optic axis is somewhat

smaller, depending upon the orientation of the

instrument. Thus, the horizontal field to be

neutralized is smaller than the vertical field, and

the required degree of compensation is corre- spondingly less vigorous. It is convenient to build

the compensating coils in the form of a box, as shown in figure 26. The horizontal compensating field is given

approximately by

Hix) =- 0.8/

d

3a'

1 +

a'

(13.5)

where o-=a;/c?. The presence of terms proportional

to <r^ in the denominator limits the range within

which field compensation is as good as at the end

VERTICAL COILS

Figure 26. Practical configuration of compensating coils as used in this apparatus.

points for the vertical field to |x/c?|<0.0376, as compared with \y/d\<C0.5. The difference between

these two ranges illustrates strikingly the importance of adhering to the geometry which gives

maximum field compensation.

The above calculations were based on the

assumption of infinitely thin single conductors.

Actual coils consist of a large number of turns of

wire in series, and the total cross section of wire,

insulation, and interstices between wires may

amount to several square centimeters. It is thus proper to raise a question as to the validity

of the approximations made in the calculations.

The presence of n conductors instead of one merely requires that the coil current should be

The replacement of a filamentary current with

a distributed current causes some modification

in the field. It is worth noting, however, that

a uniform current distribution in a circular con- ductor of any size produces the same external

magnetic field outside the conductor as is produced

by a filamentary current. Thus, if we draw a

circle of equal cross-sectional area about the

current-carrying cross section, the deviation from

results for a filamentary conductor may be

thought of as the resultant of equal and opposite

currents in the areas not common to both the

circle and the actual current-carrying cross section.

If the largest dimension of the current-carrying

cross section is co, and the fraction of cross section

not in the circle is rj, it is easy to show that an upper limit for the relative deviation of the field at the center of the Helmholtz coil structure from

the calculated value is rju'^/Kjd', a number which is less than 0.01 for all practical values of u/d.

13.3. Variation of the Compensating Field

With Position Along the z-Axis

In order to compute the end effects for a pair

of Helmholtz coils of finite length, we use the

formula for the vertical component of field at any

point due to current in a straight wire of finite length

- Hy=~ (sin 01+sin 62) cos </>, (13.6)

31
y

X

z

Figure 27. Diagram giving nomenclature for compensating

field calculations.

The wire segment BC is horizontal, and the plane OAD is normal to the

wire. The point 0 is the point at which the vertical field component is to be computed.

at 0-^=1.207, and then falls to zero at (Tj=0. At

(T^=5, however, the value of / is only 1.07. In a long coil with ai+o-2»l, the term in which is large is very close to 1, and the fractional deviation of the field from the asymptotic value is thus

just half of the difference between unity and/(cr;)

for the small value of a. We can thus maintain

a neutralizing field of uniform strength within 18

percent for all points up to positions closer to the

end of the coil than 1.2d and within 3.5 percent

for positions more than 5d distant from the ends

of the coil. For the sake of completeness, we now give the

expression for Hx when the same geometry is used

to generate a horizontal field

:

Hr=2i9(<^0+9(<^2)], (13.9)

where

(13.10)

Again, the difference between the actual field and

the field of infinitely long coils is small wherever

o"! and 0-2 are greater than 5. Since values of a

this large are needed for good field compensation

and the excess length of coil needed is thus 5d,

it is a good general rule that the lateral dimension

of the compensating coil should be as smaU as practicable for clearance around the vacuum walls

of the system.

where r, di, 62 are shown in figure 27. The plane OAD is drawn perpendicular to the

straight wire BC and through the point of interest

0. The line OD is horizontal. By simple geometry, sin 61= .—!-=:> and sin 62= , — In the

case of longitudinal wires, li and 1 2 are distances

from the point in question to opposite ends of the

coil, r is equal to 2d, and 4> is always 30° for the

coils producing a vertical field. The electrical circuit is completed at each end

of each coil by transverse wires for which C 1 = f, 2 ^3

^d-sjd, r=-\ Az^+d"^, and cos 4>= , ^> where

z is the distance to the end of the coU. If dis- tances to opposite ends of the coil are written as

Asi, A22, and for convenience

total field on the axis at any point becomes

H,=^'-^Ui<^r)+Ji<r2)], (13.7)

where

The function /(cr_,) approaches unity as crj approaches infinity. As cr_; is progressively lowered,

the function rises to a maximum value of 1.359

13.4. Relative Positions of the Chopper Disks

The mean velocity of the molecular beam packet

selected by the chopper disks is determined by the

distance between the planes of the disks and the

delay between the times at which the two slits are centered with respect to the molecular beam.

More precisely, a range of velocities is transmitted,

as determined by the slopes of lines drawn between slit jaws in figure 31b. The widths of the

two sets of chopper slits thus enter into the range

of velocities transmitted.

Since it is easier to vary the time delay between

the two chopper disks than to cut new sUts on a chopper disk, it is worthwhile to plot the velocity

ranges that will be transmitted for different

relative displacements of the disks. This is done for a particular set of slit widths in figure 28.

We can pick the range of velocities corresponding to any relative displacement on the table by

displacing the second chopper disk by a suitable

angle with respect to the first in the direction

opposite to the rotation of the pair. The required

displacement angle is given by dd=d/rM, where d

is the displacement distance and is the distance

from the rotation axis to the middle of the slit. We designed the chopper disks with a simple

locating mechanism for setting the disks at any

desired orientation relative to each other.

The second chopper disk fitted onto a hub which was an integral part of the first disk as shown in

32
900

800

700

600

500

400

300

200

100

VELOCITY LIMITS OF TRANSMISSION FOR CHOPPER SLITS .25 AND 2.00 mm WIDE

RETAINING SCREW DOWEL

PINS

.25 .500 .75 1.0 125 MEDIAN DISPLACEMENT BETWEEN SLITS, cm Figure 28. Nominal and limiting transmission velocities for slits as used in velocity selector. Middle curve is nominal velocity based on 8.9 cm mean chopper radius, 8000 rpm rotational velocity, and 2.5 cm distance between chopper planes. Neighboring curves give range oi velocities within which ma.ximum

transmission will occur with slit widths of 2.0 mm and 0.25 mm. Outer

curves give velocity limits outside of which no beam transmission occurs. Taper from maximum transmission to zero transmission is approximately

linear.

figure 29. On circles of appropriate diameter on each disk, holes were drilled normal to the

disk plane. Dowel pins passing through a solid

washer, into which appropriate mating holes were drilled, secured the two disks in a fixed

orientation with respect to each other. Two

pins at diametrically opposite positions were

required to secure each disk to the washer. The

angular orientation of the two disks with respect

to each other thus depended upon the angular

positions of the alining holes in each of the disks

and on the two corresponding circles on the washer.

By drilling a number of holes at selected angular

spacings on both disks and on the corresponding

hole circles of the washer, we can greatly increase

the number of orientations available. The angular spacings of the holes in each set are shown

in figure 30. The holes are unevenly spaced in each hole circle except the inner circle of the guide ring. The

separate possible orientations of the disks with

respect to the guide ring are so chosen that 600

different combinations are possible, giving 300

different relative positions of the chopper disks

with respect to each other. Thus, the range of

relative orientations from —30° to +30° is covered in 0.2° intervals. The method for finding the

correct hole combination for a given displacement

is given below.

The holes are numbered in each circle for con- venience in determining proper settings. On the

guide ring a double scribe line runs through a set

of four holes. The centers of these four holes lie on a common line through the center of the ring.

A letter L determines the left side of the ring in the preferred orientation. With the row of holes

horizontal and the L on the left, we number the

hole pairs 1, 2, 3, 4, 5, starting with the pair on

the right within the lines and proceeding in the

counterclockwise direction. Likewise, on the disks

RETAINING WASHER

GUIDE RING

BALL BEARINGDRIVE SHAFT

Figure 29. Cross-sectional view of the chopper disks and

their connection to the driving shaft.

disc: inner circle gives outer circle gives

V*— 35.8 — ^^'^

^

Xn 0 0 0\

RING

INNER CIRCLE GIVES A OUTER CIRCLE GIVES B

\0 0 0 Q/

Figure 30. Angular spacing of holes in the chopper disks and in the guide rings for provision of known angular

spacings between the two chopper disks. we number the holes consecutively in the counterclockwise direction from a recognizable number

one hole. In each disk the number one hole is precisely in line with a slit. When the number

one hole is facing to the right, the hole-to-hole

spacing just below it is noticeably larger than the

corresponding spacing above it. Thus, the

interval 1-2 is always smaller than the interval

1 1-12. The above information should be sufficient 33
to locate the number one hole on each disk and

to determine the number associated with each

hole. If any difficulty is encountered, this same

information, along with the actual spacings of

holes in degrees, is given in the drawing.

The chopper rotates in the counterclockwise

direction, as seen from the target, and we want the

disk nearest to the source to lead the other one. In the following table, the lead is given in degrees

for various hole combinations. A positive number

indicates lead in the sense given above, while a negative number indicates lag. The number of

the desired holes in each circle is given by an

appropriate letter.

Displacement table {degrees advance in counterclockwise rotatiort)

a 6 e d a 6 c d

* 1 2 3 4 5 6 1 2 3 4 5 6

1 0 +28 -4 +24 -8 +20 1 3 ±30 -2 26 -6 22 -10

2 -24 4 -28 0 28 -4 2 3 6 -26 2 ±30 -2 26

3 12 -20 8 -24 4 -28 3 3 -18 10 -22 6 -26 2

4 -12 16 -16 12 -20 8 4 3 18 -14 14 -18 10 -22

5 24 -8 20 -12 16 -16 5 3 -6 22 -10 18 -14 14

1 2 29 -3 25 -7 21 -11 1 4 -1 27 -5 23 -9 19 2 2 5 -27 1 29 -3 25 2 4 35 3 -29 -1 27 -5

3 2 -19 9 -23 5 -27 1 3 4 11 -21 7 -25 3 -29

4 2 17 -15 13 -19 9 -23 4 4 -13 15 -17 11 -21 7

5 2 -7 21 -11 17 -15 13 5 4 23 -9 19 -13 15 -17

•The value of 6 is 1 for integral degrees. Reducing both a and 6 by 1 leav- ing c and d constant reduces the displacement by 0.2°. Do not carry t.hi<! process beyond a reduction of O H". The designations of letters and the hole circle to which each applies are as foUows:

Letter Circle a inner circle of guide ring

b outer circle of guide ring

c inner disk circle, i.e., disk nearest source oven

d outer disk circle, i.e., disk nearest target.

Thus a=l, 6 = 1, c=3, d=2 would mean that hole

1 of the inner guide circle is to be pinned to hole

3 of the inner disk circle, while hole 1 of the outer

guide circle is to be pinned to hole 2 of the outer

disk circle. From the table, one reads the dis- placement for this combination as —2°. The

table gives the values of a, c, and d for integral

numbers of degrees of relative advance. The

value of b is always 1 for integral numbers of degrees. The values of the advance on the chart

may be reduced by multiples of 0.2° up to and

including a reduction of 0.8° by reducing both

a and b by one unit for each 0.2° of reduction in

angle, while leaving c and d unchanged. Since

the designation of holes on the guide ring is 1, 2,

3, 4, 5, 1, 2, . . . in an unending cycle, the value

of a or 6 immediately below 1 is always 5.

13.5. Computation of the Initial Distribution

Function

The molecular density function N(,{x, t) on a given target surface or Nti-":, v) at a given time is the resultant of a series of elimination processes

which have restricted the class of molecules

which are permitted to enter the interaction space.

Within the molecular beam source, the molecular

density is described by a density function of the

three space coordinates, the three velocity components, and the time. Except for slight density

deficiencies due to the constant loss of molecules

through the source slit and their replacement by

evaporation (or in the case of a permanent gas, by diffusion from a higher pressure source), the

density is independent of position and time.

The velocity dependence is a normalized function of the form

771 „ ^ ^ -r («2+«^+0-

e

^

Of all the molecules described by this density

function, we want to know how many will be

found in the area dxdy or dxdt encompassing a given space-time position {x, y, t). We are not

concerned with the normalization factors in the source density function, since we want only

relative values of the final density A^o or Ni at

different points.

In order for a molecule to have the coordinate

i^i, Vi, 2i) at time ti, the equations describing its motion must describe a line passing through the

given time ti. These may be written

x=xi+x'{y—yi) (13.11a)

z=2,+z'iy-y,) (13.11b)

t=t^M'{y-yi), (13.11c)

where x'=dx/dy, z'—dzldy, and t'=dt/dy.

The density function is then an integral in a four-dimensional space consisting of Zi, x' , z'

,

and t' of a density function in this space. This

density function is easily determined from the

density function describing the molecules within

the source chamber. The limits of integration are, however, somewhat difficult to state in simple

form. At the source opening y=ys, the allowed

region is the rectangular source opening. Substituting y=ys into eq (13.11) translates this

condition into a rectangular allowed region in

x' —z' space. The position, but not the area of the

rectangle, depends upon Zi. Similarly, the

positions of the slit jaws, limiting x and z position

as the particle goes through the collimator

assembly, provide a second rectangular area in

x'—z' space. The allowed region is the region,

if any, which is common to both rectangles.

The chopper slits impose additional limitations on the allowed region for integration.

If thne i =0 is chosen as the time when the axis

of the second chopper slit is exactly alined with

the collimator slit, the collimator slit allows

passage at time t of those molecules for which

[2+?'Af] tan oit ^^<a;<[2+''ivf] tan at-l cos cof cos at

(13.12)

where w is the angular velocity of the chopper

34
disk, 202 is the -wddth of the chopper slit, and Tm is the radial distance from the chopper's axis of

rotation to the plane 2=0. A similar condition

for passage exists in the plane of the first chopper

disk with ut+d replacing ut and replacing a2. The width of the first chopper slit is 2ai, and the

lead angle between the axis of the two chopper

shts is 6. As in the work with the source and collimator,

this inequality may be translated into relations

in x' — z' space by substituting the value of y in the chopper slit plane into the inequalities of

eq (13.12). In this case, however, the resulting

relations depend upon the values of both t' and Zi, which are intended as variables of integration.

The two sets of requirements in x' — z' space form

a diamond-shaped pattern when plotted. The

area common to both the diamond and the rec- tangle (at fixed values of z^ and t') represents

the allowed range of coordinates in x' — z' space

for which a molecule can reach the desired point

in space time without being intercepted by any

aperture. As z^ and t' are varied, the positions

of the diamond and of the rectangle shift some- what with relatively Httle change in volume.

For sufficiently large or small values of either z^ or

t' , the two quadrilaterals do not overlap at all. The region in which an integration can be

carried out is thus bounded by a closed figure in Zi—t' space. This region is subdivided into

smaller regions characterized by differing topological properties. For example, in one region of z-^-

SLIT _^2o^|,_

Figure 31 (a). Ray diagram for spatial collimation of the molecular beam, (b) Corresponding ray diagram in t-y space for chopping and monochromatizing the molecular beam.

t' space, the region of overlap in x'—z' space

might be a triangle involving two sides of the diamond and one of the rectangle. In an adjoining

z-^—i' region, the overlap in x'—z' might be a quadrilateral involving the previous three sides and an

adjacent side of the rectangle.

The various topological possibilities of overlap

involving different combinations of sides of the

two quadrilaterals thus constitute a series of boundary conditions of algebraically different

form, although the equations defining the sides of

the quadrilaterals are fairly simple. In perhaps a dozen or more regions of Zi—t' space, the boundaries of the allowed region in x' — z' space can be

stated as a group of straight line segments whose

constants are certain functions of Zi and t' . In

each region, then, it is possible to integrate first over x' — z' space, with z^ and t' carried as parameters and then over Zi—t' space, and the

resulting quantities may be added. We have

then computed the molecular density at the single

point (xi, yi,

In view of the extreme complication involved in

calculating the molecular density at a given point,

the prospect of computing an exact piecewise

analytic density function No{x, t) or Nt{x, y)

appears hopeless. The problem does lend itself to programmed automatic computation, however,

and it is to be hoped that a practical computer

program wiU be produced by future investigators.

13.6. Simplified Approximate Computation of

the Molecular Density Distribution

Much of the complication in the preceding

treatment of the formation of a molecular beam

packet is due to the fact that the chopper blades

do not remain parallel to the collimator slit axis

as they cut the molecular beam. The effective

cutting velocity as a function of position in the

beam, is equal to u^z+Tm). Since the slit widths

are independent of z, this means that the trans- mission on time for each chopper is greater at points close to the chopper axis than at points near the rim. In our apparatus the hmits of

z are ±1.25 cm, whUe Tm is 4.375 cm, leading to a variation in cutting velocities from 6/7 to 8/7

of the mean velocity.

To first order, the effects of a higher velocity

for a positive value of z are cancelled out by the

lower velocity at the corresponding negative,value

of 2. Thus, if we neglect this effect altogether,

we can expect an error of second order in z/Tm or, in this experiment, an error of up to 5 percent.

With this simplification, we become no longer

concerned with coordinates and slopes in the

2-direction. The integral in four-dimensional

space in the last section thus reduces to one in

two-dimensional x'—t' space. Furthermore, the

boundaries ,bf the volume integral, which were

extremely complicated in the last section, now

reduce to a simple parallelogram in x'—t' space. An aid in finding the boundary conditions for the integration and one which also provides some

physical insight is the pair of ray diagrams in figures 31a and b. These figures are the projec35
tions of limiting molecular paths in x—y space

and in t—y space. In x—y space the range of paths is limited by the source slit and the collimator slit. In t—y space, the fu-st and second

chopper slits perform this function. Straight lines

connecting the slit edges divide the accessible areas

of x—y space into regions a, b, and c and divide

t—y space into regions a, /S, js', and 7. In finding

the molecular density associated with the point

(xi, yi, ti), we first note whether the given point

lies in region a, b, or c. If the point lies in region b, the limits of integration are

X} ds / , ^Xi~\-(ls

In region a, the upper limit is replaced by

and in region c, the lower limit is replaced by Xi —Oc

2/1

On the boundaries between regions, the alternate

limits applying for the adjacent regions are equal.

The diagram in y—t space can be constructed only when the particle path in x—y

space is known. The y planes of the two chopper

slits are, of course, simply the positions of the

disks. The widths of the slits in units of time

are their geometrical widths divided by the mean

cutting velocity wrM. The center of each slit is displaced by the x value of the particle path, as

it crosses that slit plane, divided again by cor^/. An additional displacement d/w is imposed upon the

first chopper s^it. Thus, we are able to construct

figure 31b.

For a given point of observation (ti, yi), the

limits of integration in t' depend upon the region

a, /3, /3', or 7 in which the point lies. It is not

difficult to show that the limits of integration for

t' are as given in the following table

:

Limits of integration in t' as determined by location in figure Sib

Region Lower limit of t' Upper limit of t'

— (2/i +«^2 — 2/2)2;' —a2+rj|f« X\ — (2/1 — 2/2)2;' + ai a

w7-ji/(2/i + c^2 — 2/2) oiTMiyi — yt)

/J Xi - (2/1 + d2 — y2)x' — a2 + rud Xi — {y\ + d2 — y2)x' + a2+rM9

WAf(2/i + cf2 — 2/2) wj/(2/i + (i2 — 2/2)

Xi -(2/1 — 2/2)2;' — ai X\ -(2/1 — 2/2)2;' + ai 0' oirM{yi — y2) wrjif (2/1 — 2/2)

Xi — (2/1 — 2/2)3;' — ai Xi - (2/1 + <^2 — 2/2)2;' + 02 +rjfS y cor if (2/1 — 2/2) tarM{y\ +d2 — y2)

The location of the point under consideration

(in space a, b, or c, and a, /3, 0', or 7) is determined

by its coordinates {xi,yy,ti) and the parameters

of the experiment. When an entire distribution,

N({xi,yi) or Ny(xi,ti), is to be mapped by integrations at a series of points, some points will be found

in aU three of the areas in x-y space and in at least

three of the four areas in t-y space. Thus, the

expression for is piecewise-analytic in its dependent variables, and separate expressions must

be written for each of the nine or more analytic

segments.

14. References

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Pres.s, Cambridge, England, 1931).

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[4] E. H. Kennard, Kinetic Theory of Gases (McGrawHill Book Co., Inc., New York, 1938).

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[12] J. Statder, G. Goodwin and M. O. Creuyer, NACA

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(1956).

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37

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