Group Title: scattering of low velocity neutral particles
Title: The Scattering of low velocity neutral particles
Full Citation
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 Material Information
Title: The Scattering of low velocity neutral particles The Cs-Ar and Cs-N2 interactions
Physical Description: vii, 143 l. : illus. ; 28 cm.
Language: English
Creator: Imeson, Thomas Cole, 1938-
Publication Date: 1965
Copyright Date: 1965
Subject: Scattering (Physics)   ( lcsh )
Particles (Nuclear physics)   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida, 1965.
Bibliography: Bibliography: l. 139-142.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097903
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000565765
oclc - 13583760
notis - ACZ2184


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December, 1965


The author wishes to express his appreciation to

Dr. S. O. Colgate, Chairman of his supervisory committee,

whose able assistance and guidance made this investigation

possible, and to E. C. Whitehead for his assistance in the

design and construction of the apparatus. He is also

grateful to the members of his supervisory committee for

their comments and suggestions concerning this dissertation.

Finally, appreciation is extended to the Department of

Health, Education and Welfare and the National Science

Foundation for their support and financial assistance.



. . . . . ii

. . . . . . vi







A. Apparatus . . . . .

1. Velocity selector . .

2. Molecular beam apparatus.

3. Beam formation . . .

4. Scattering chamber . .

5. Detector. . . . .

B. Procedure . . . . .

. . . 1
. . . 8

. . . 8

..... 27

. . . 32

..... 39

..... 43

. . .. 46

1. Measurement of transmission
characteristics of the velocity
selector . . . . . .

2. Measurement of total cross-sections

III. THEORY. . . . . . . . . .

A. Transmission Characteristics of a
Parallel Wall, Spiral Groove
Velocity Selector . . . ..

B. Collision Theory. . . . . . .




Section Page


A. Measurement of Transmission. . . . 97

B. Cross-Sections . . . . . .. 117

C. Concluding Remarks . . . . . 126

APPENDICES .. . . . . . . ..... 128

REFERENCES . . . . . . . . . . 139

VITA . . . . . . . . . . . . 143


Table Page

1. Operating Conditions for Measurement of
Transmission of a Maxwellian Beam of Cesium
Atoms . . . . . . . . . . 49

2. Typical Operation Conditions for Measurement
of Cross-Sections of Cesium with Nitrogen and
Argon . . . . . . . . .. . 57

3. Velocity Selector Parameters. . . . . 60

4. Interatomic Potential Constants for Cesium (in
Units of 10-58 erg. cm6). . . . . . 125


Figure Page

1. Photograph of apparatus . . . .. .. 10

2. Trajectory of a point moving across rotating
plane . . * * * * . . 12

3. Orientation of grooves in velocity selector 15

4. Geometries of possible velocity selecting
grooves . . . . . . ... ... 17

5. Velocity selector . . . . . . . 19

6. Photograph of machine set-up for cutting
parallel wall spiral grooves. . . . . 22

7. Suspension system for velocity selector . 26

8. Cross-sectional view of apparatus . ... 30

9. Cross-sectional view of oven assembly . .. 34

10. Photograph of oven and scattering chamber
assembly. . . . . . . . . . 38

11. Scattering chamber. . . . . . . 41

12. Schematic diagram of detector . . ... 45

13. Coordinate system for transmission analysis 62

14. Transmitted velocities. . . . . . 68

15. Characteristic transmission profile . . 72

16. Time dependent distribution of transmitted
velocities at scattering zone . . . . 75

17. Time dependent transmitted intensities at
scattering zone . . . . . . . 77

18. Coordinate system for off normal transmission 80

Figure Page

19. Orbit of classically scattered particle . 87

20. Transmission of cesium beam at 405K. . . 101

21. Narrow groove transmission. . . . . 106

22. Determination of attenuation correction . 108

23. Incident angular dependent theoretical
transmission. . . . . . . . . 111

24. Comparison of experimental and theoretical
transmission as a function of y . . . 114

25. Incident angular dependent resolution . 116

26. Average relative velocity dependent total
cross-sections for Cs-Ar. . . . ... 119

27. Average relative velocity dependent total
cross-sections for Cs-N2. . . . . . 121

28. Relative scattering gas density along central
beam axis . . . . . . . . 135



All chemical and physical properties of matter in

bulk are manifestations of the forces which exist between

atoms. The various types of interactions between ions,

atoms, and molecules in their ground states can be

generally classed into those that act over short or long

distances. The short-range, or valance, forces arise

from the interaction of electrons associated with the

participating molecules;2 they are repulsive and, as a

rule, highly directional. Long-range interactions are

attractive and usually less pronounced.

Forces between molecules at small distances are

rather difficult to treat generally on a theoretical basis.

Much that is known about them has come from calculations

involving specific interactions. In most circumstances

these forces are considered to vary exponentially with

intermolecular separation. I. Amdur and co-workers,3-5

using high energy molecular beam techniques, have conducted

extensive investigations into the nature of short-range


Long-range forces are usually divided into three

categories: electrostatic, induction, and dispersion. An


inverse relationship to powers of the intermolecular sepa-

ration can be attributed to all forms of these forces.

Explanations of the first two types depend entirely on

electrostatic principles; the basis for dispersion inter-

actions, however, is found only in quantum mechanics.

Electrostatic forces constitute the simplest

contribution to long-range interactions. These result

from the interaction of electric moments which may be

present in molecules.

Induction forces emanate from the interaction of a

molecule possessing a permanent electric moment (ion, di-

pole, etc.) and another molecule. This interaction arises

from the moments induced in one molecule by the permanent

charge distribution in the other molecule. In all cases

the force varies directly with the polarizability of the

molecule in which the moment is induced.

Although the classical significance of dispersion

forces6 is not completely apparent from their quantum

mechanical derivation, a plausible explanation based on the

following reasoning has been offered. At any instant the

electronic configuration of a molecule will be such as to

generate an instantaneous dipole moment (although the

molecule may be completely neutral). This instantaneous

dipole will then induce a moment in an adjacent molecule

thus giving rise to an attractive force between them. The


dispersion force then results from the instantaneous force

of attraction averaged over all instantaneous electronic

configurations of the molecule.

Several statistical theories have been developed

which relate the intermolecular potential energy to various

macroscopic properties of matter; these properties include:

viral coefficients of the equation of state, Joule-Thomson

coefficient, and low density transport effects. If the

intermolecular potential energy for a system is known, then

the macroscopic properties listed above can be calculated,

or conversely a representative form of the potential may be

obtained from these measured properties.

A problem arises in that detailed quantum mechanical

calculations concerning the interaction potential can be

made only for the simplest cases; namely, the interactions

of H atoms, He atoms, and H2 molecules.7 This difficulty

results from an inability to treat many-body systems with

any degree of mathematical accuracy. Therefore, inter-

molecular potentials must generally be obtained experi-


The procedure for arriving at the potential involves

choosing an arbitrary analytical form with adjustable para-

meters, then fitting this function to the experimental

data in order to obtain the best fit. With the exception

of mutual-diffusion coefficients for unlike molecules, the

macroscopic properties of matter listed above are rather

insensitive to the type of potential form used. This fact

results in potentials of quite different character appearing

to fit the experimental data with more or less equal pre-

cision.8 As can be seen a more sensitive technique is

required for determination of the potential with appreciable


In general a direct and, in principle, simple means

of obtaining information concerning the intermolecular

potential involves the technique of corpuscular beam

scattering.' The unique suitability of beam experiments

for investigation of particle interaction results from

their production of controlled collisions.

A molecular beam can be defined as a stream of mole-

cules in a highly evacuated region moving practically

collision free in straight and almost parallel trajectories

within the confines of the geometrically defined beam. A

beam may be formed by effusive flow of particles through

an orifice with subsequent definition by one or more other

apertures in line with the first. A molecular beam

apparatus, therefore, consists fundamentally of a source

chamber or oven, a series of collinear slits to define the

beam, and a detector.

The first molecular beam of neutral particles moving

in straight lines with thermal velocities was generated in

the laboratory of L. Dunoyer.lO The importance of molecular

beams as a research tool was first recognized by 0. Stern11

in 1919. He used this technique to obtain the first direct

measurement of the average velocity of silver atoms in the

vapor state.

The utility of molecular beam scattering techniques

in determining intermolecular potentials originates in the

relationship between the potential and the angle through

which the relative velocity vector of the colliding particles

is turned during a collision. A number of investigations

have been carried out using molecular beams to furnish
information concerning interaction potentials.12

Massey and Mohr18 were the first to give an approxi-

mate quantum mechanical relationship between the relative

velocity dependent collision cross-section and the inter-

molecular potential. Experimentally the cross-section is

determined by the attenuation of a molecular beam passing

through a gas filled region. Factors effecting the

attenuation of a molecular beam are: energy of the beam,

molecular masses, temperature and density of the scattering

gas, scattering path length, aperture of the detector, and

the nature of the interaction between the beam and target


Several difficulties arise in the determination of

relative velocity dependent cross-sections in the low

velocity region. Because of the Maxwellian or near

Maxwellian velocity distribution present in the beam source,

all molecular beams have rather large velocity spreads (the

exception being beams formed at low temperature). Thus for

precise measurements some type of velocity selection is

generally required. For this and other purposes a variety

of velocity selector types has been developed. These

include: gravitation deflection, diffraction from crystal

gratings, mechanical selectors involving rotating slotted

devices, and magnetic deflection.19-29 The latter two

types are those in general use at the present.

Scattering experiments conducted with velocity

selected beams traversing a gas filled region require

averaging over all velocities present in the scattering

gas to arrive at the velocity dependent collision cross-

section. A more precise determination of the cross-section

may be obtained if this averaging is eliminated. This may

be accomplished by the use of a second velocity selected

beam intersecting the original one. In this way highly

controlled pairwise collisions can be created.

Use of two velocity selectors creates two rather

serious problems. The first concerns the reduction in

intensity that must always accompany velocity selection.

This problem can be dealt with by improving detection

techniques and increasing the original beam intensity.30'31

The second problem results from use of two independent

rotating slotted types of velocity selectors. When selec-

tors of this type are used the transmitted beam is pulsed,

and the pulses from the two beams should be phased in such

a way that they arrive at the impact zone simultaneously.

Of course the magnetic deflection type of velocity selector

does not suffer from this difficulty, however, it is

specific to particles that possess a magnetic moment. The

phasing problem can be resolved by the use of a slotted

rotating velocity selector which acts on both beams

simultaneously. Design and development of velocity selec-

tors of this type has progressed in various laboratories.32

The development of a double beam velocity selector of

practical geometry and proven reliability for molecular

beam scattering experiments will be reported here.


A. Apparatus

The molecular beam apparatus (Figure 1) to be des-

cribed here allows the simultaneous velocity selection of

two molecular beams. This selection technique facilitates

production of highly controlled pairwise molecular col-

lisions, thus allowing the interactions of atomic or

molecular species to be investigated with a high degree of

precision. The unique feature of this apparatus is the

velocity selector; therefore it is appropriate to describe

it before other aspects of the instrument are discussed.

1. Velocity selector

The trajectory of a point, moving with constant

velocity v in the positive direction along the x axis,

projected onto a coordinate system which is allowed to

rotate about an axis through its origin with a constant

angular speed w is a spiral of Archimedes given by x1 =

r=vG/w A representation of this situation is shown in

Figure 2. The trajectory is seen to be a function of the

velocity and it is this fact which allows velocity selection

to be effected. In order to fabricate a velocity selector,

it is necessary to machine grooves on the surface of a

Fig. 1.-Photograph of apparatus during final
stages of completion.



: l r -

Fig. 2.-Trajectory of point moving with constant
velocity v projected on plane rotating
about the origin with angular velocity w.

- -

circular disk; the groove width being such as to accommodate

only trajectories of a narrow band of velocities. The

simultaneous velocity selection of two molecular beams is

accomplished by positioning grooves in precise relationships

to one another and directing the two beams through the

selector so they intersect at the axis of rotation. Figure

3 illustrates the spacings (900) between related grooves in

the present selector.

For use in the first selector three different groove

geometries (Figure 4) were considered with regard to ease

of fabrication and transmission characteristics. The ideal

groove subtends the same angle from the center of the

selector at any position along the groove. It possesses

excellent transmission characteristics but offers con-

siderable difficulties in construction. The linear groove

is the simplest to fabricate but possesses poor trans-

mission properties. The parallel wall groove can be

fabricated with some difficulty but has rather good trans-

mission characteristics. It is for these reasons that the

latter groove geometry was chosen.

The velocity selector was machined from a three inch

thick 7075-T651 aluminum plate, ultrasonically tested for

uniformity and furnished by the Aluminum Company of America.

A 15 in. diameter rotor blank was cut whose over-all-thick-

ness was 2.950 in. (Figure 5). After thorough testing as

Fig. 3.-Orientation of grooves for simultaneous
velocity selection of perpendicular



Fig. 4.-Geometries of some possible velocity
selecting grooves.



L. U)\ \
U) I \\\~

Fig. 5.-Drawing of aluminum velocity selector
showing geometry of the four linear
optical alignment grooves.



to its dynamic and static balance the blank was prepared for

grooving. The arrangement used to machine the grooves is

shown in Figure 6. The step and four threaded holes at the

top of the blank selector (Figure 5) were used to position

and attach it to an indexing rotary table mounted concen-

trically on a Troyke rotary table which in turn was clamped

to the table of a South Bend vertical milling machine. Rigid

support of the rotary fixture was achieved by the use of a

large thrust and double row ball bearing combination. One

hundred sixty equally spaced positions were obtained by

locating forty bored and reamed holes in the plate at the

bottom of the fixture with four equally spaced indexing

holes in an aluminum ring clamped to the rotary table. The

handle of the commercial rotary table was replaced by a

pinion which engaged a rack mounted on the longitudinal

table of the milling machine. The grooves were cut with

two-flute high speed steel end mills under a spray of

kerosene from a high-pressure mist coolant system. The

cutters were driven in a Precise high speed grinder miller

mounted on the spindle of the milling machine. The spindle

of the miller grinder was indicated into a position di-

rectly over the center of the system of rotary tables.

With the rack and pinion engaged any linear motion of the

transverse milling machine table caused the system of

rotary tables to move through an angle proportional to the

Fig. 6.-Photograph of machine set-up for cut-
ting parallel wall, spiral grooves.



linear displacement, the proportionality being determined

by the pitch diameter of the pinion gear. The path taken

by the tool through the velocity selector, mounted in the

prescribed arrangement, is a spiral of the desired type.

With the grinder miller running at a speed of 36,000

RPM the grooves were cut to a depth of 0.245 in. (in five

steps) and a width of 0.055 in. A finishing cut was taken

on each groove with a 0.069 in. diameter tool set at a

depth of 0.250 in. The roughing and finishing operations,

although conducted separately, were carried out on a

particular groove and then on the groove diametric to it.

This procedure assures that minimum imbalance is imparted

to the selector by tool wear. Upon completion of the

grooving operation the pinion was decoupled from the rack

to allow machining of four straight optical alignment

grooves (Figure 5). To give the straight grooves an

effective slit width of .010 in., a 0.125 in. diameter end

mill, offset from the center by 0.0575 in., was usea to

machine half the groove length with the remaining half be-

ing cut after offsetting the tool the same distance from

center in the opposite direction. By means of these grooves

the beam sources were set to perpendicularity. A total of

148 spiral grooves were cut in the selector, twelve grooves

(three in each quadrant) were omitted to avoid their

intersection with the straight grooves.

The device through which the velocity selector is

coupled to its drive motor33 (Figure 7) allows the selector

to be suspended by means of a flexible shaft directly from

the vertical shaft of the motor. The flexible shaft makes

it unnecessary to perform the difficult task of dynamically

balancing the coupled rotor. The drive bushing D, construc-

ted of 304 stainless steel, has an outside nonretaining

taper which matches exactly the inside taper of the 305

stainless steel cap C. This taper effects accurate posi-

tioning of the drive bushing; application of torque is

through four cleats on the bottom of D which fit into slots

milled in the top of the velocity selector (Figure 5). The

flexible shaft F is machined from a single length of 304

stainless steel. Its lower end is shrunk fit into the

central bored hole in D and welded at W. The upper portion

of F threads loosely into RH 10-32 internal threads in the

motor shaft. Both F and S have matching 600 tapers which

position the shaft F so that it is precisely concentric

with the motor shaft. The selector is driven clockwise;

consequently positive positioning of F with respect to S

is continually maintained. However, a LH 1/2 20 lock nut

Y is provided to bind the flexible shaft against a pre-

cision lock washer L to insure against loosening of the

internal threads (particularly during deceleration). The

diameter of the present flexible shaft is 0.100 in. and its

exposed length is 1.5 in.

Fig. 7.-Diagram of velocity selector suspension
system: S-drive motor shaft; N-left-
hand lock nut; L-lock washer; F-flexible
shaft; B-taper head bolt; D-tapered
drive bushing; W-point where shaft is
welded to drive bushing; C-stainless
steel cap.





The static balance of the velocity selector was

maintained to as high a degree as possible during all

machining operations. This was done in order to reduce the

amount of flexing the shaft must undergo during operation

so as to minimize the occurrence of a shaft failure. All

curved surfaces were maintained concentric to within

+ 0.0001 in. with all horizontal surfaces maintained

parallel within the same limit. In addition the four

equally spaced threaded holes used to attach the cap C to

the velocity selector were drilled and bored to the same

geometry, and the threads rolled rather than tapped. This

procedure does not remove any material (as does a cutting

tap) and thus helps maintain balance. The bolts, B (Fig.

7), were constructed from 303 stainless steel with sixteen

full 5/16-18 threads and drilled through their length to

provide pump relief. The bolt heads were cut with a 3

retaining taper in order to maintain symmetry. Special

sockets with matching 3 tapers were used for their instal-

lation and removal. The weight of all bolts was held to

within + 0.2 mg. With these precautions the static balance

of the selector was highly preserved.

2. Molecular beam apparatus

The machine constructed to receive the velocity

selector consists of three regions; the first two are

chambers in which beams are produced and collimated, the

third is the chamber which houses the velocity selector and

detector. In the view of the apparatus shown in Figure 8

the second beam chamber is not seen; it is located directly

behind the velocity selector chamber perpendicular to the

beam chamber shown. With the exception of pumps and

baffles all the main components of the apparatus are con-

structed from stainless steel with heliarc welded joints.

Each region is evacuated by its own diffusion pump; a

manifold connecting the three chambers is provided for the

evacuation of any chamber by pumps of any other. All

gaskets used are teflon, neoprene, or silicone rubber 0-

rings. A collar, shock mounted from the frame, located

under the top flange of the baffle chamber serves to sup-

port the machine. The frame is coupled to the floor by

means of bonded rubber shock mounts, and a system of jack

screws allows leveling of the instrument. Vibration

elimination is further achieved by the use of bronze bellows

in the lines connecting the diffusion pumps to their

mechanical vacuum pumps. All other connections to the

machine are made through small diameter (3/8 in. and less)

soft copper tubing; this tubing is sufficiently flexible to

require no additional vibration elimination.

The beam chambers are constructed from 6 in. diameter

Ladish tees. Each is evacuated with its own 6 in. diameter

Fig. 8.-Cross-sectional view of two of the
three main chambers of the apparatus
(the second beam chamber being
perpendicular to the one shown and
located directly behind the velocity
selector chamber).


oil diffusion pump backed by a 13 c.f.m. mechanical vacuum

pump. Each has an air cooled circular chevron baffle4

to retard the migration of oil from the diffusion pump into

the high vacuum region. A Veeco RG-75 ionization gauge and

a port that communicates with a McLeod gauge are provided

in each chamber for the measurement of pressure.

The central chamber (Figure 8) of the apparatus con-

tains, in addition to the velocity selector, the detector

and mechanisms for the interruption of molecular beams

(beam flags). Above this chamber is a smaller enclosure

which contains the velocity selector drive motor. Both of

these regions are evacuated by a CVC model MCF-1400 oil

diffusion pump backed by a CVC E-70 mechanical vacuum pump.

The velocity selector region is separated from the dif-

fusion pump by a freon cooled chevron baffle. The line

through which the motor chamber is evacuated communicates

with the diffusion pump below the baffle; consequently,

the gases given off from the motor are, to a great extent,

not allowed tc enter the velocity selector region. There

are four main ports in the central chamber; two of these

communicate with the beam chambers, the other two are 6.5

in. in diameter thus permitting ready accessibility to any

portion of the chamber. The latter ports when fitted with

lucite covers allow the operation of the velocity selector

to be observed while the machine is under vacuum. Pressure

in the velocity selector region is monitored by a Veeco

RG-75 ionization gauge and may be measured by a McLeod

gauge. The two beam flags present in the chamber are so

constructed that in their raised position the beams are

interrupted. These flags are operated by independent

mechanisms which convert rotary motion of a shaft through

a Wilson seal in the chamber wall to linear motion of the

flags. Bulkheads attached to the beam chamber ports, in

addition to isolating the beam chambers, contain collima-

ting slits which help define the molecular beam.

3. Beam formation

A cross-sectional view of the oven assembly used to

generate a beam of alkali metal atoms is shown in Figure 9.

This assembly is mounted on a flange which bolts onto the

rear of a beam chamber. The oven is designed to accept

glass ampules of alkali metal which are broken after the

entire apparatus has been evacuated. It has rectangular

geometry and is constructed from oxygen-free, high-

conductivity copper. The opening in the top of the oven,

through which the glass ampule is introduced, is sealed

with a tapered plug, C, fabricated from the same material

used for the oven. A modified 10-32 stainless steel allen

cap screw, V, is provided to break the glass ampule which

is located in the oven well. This screw is driven by means

of an allen wrench attached to a 1/4 in. diameter shaft, S,

Fig. 9.-Oven assembly: A-alignment plug, B-
double row ball bearing, C-oven cap,
F-supporting flange, G-gear, H-holes
to receive heaters, K-knees, L-lever,
P-alignment port cap, R-ceramic ring,
S-shaft, T-thermocouple location, V-
ampule breaking screw, W-Wilson seal.


s J=*


j 119 ^ ^ 1 _ _
^^^^~l-!^^ t^^
^^\\\\^ ^j ^^t

i i ^/

through a flexible wire; this shaft exits the vacuum system

through a Wilson seal, W. Engagement of the allen wrench

and screw V is assured during the ampule breaking operation

by the following means. A portion of the shaft S outside

the vacuum chamber is provided with 1/4 32 external

threads; a fixture with internal matching threads is then

screwed onto the shaft and attached securely to the com-

pression cap of the Wilson seal after engagement of the

allen wrench, thus fixing the position of the allen wrench

with respect to the screw V. After the glass ampule is

broken the fixture is released from the compression cap and

used to jack the allen wrench away from the screw thus

removing from the oven a possible heat sink. Razor blades35

are used as slit jaws for the oven; their edges are main-

tained parallel by location of the blades with reference

to the walls of a slot milled into the face of the oven.

The width of the milled slot allows slit widths ranging from

0 to 0.020 in. to be used. The plug A is provided for

optical alignment purposes; it is stainless steel and has

1/4 28 threads for insertion. Optical alignment of the

oven is accomplished with the plug removed from the oven.

Heat is supplied to the oven by means of four 1/8 in.

diameter tubular heaters36 located in holes H. These

heaters are arranged and powered so as to maintain the slit

region at a slightly higher temperature than the rest of the

oven. The temperature of the oven is monitored by two

iron-constantan thermocouples located at positions T.

By virtue of its mount, the oven has two degrees of

freedom; these are: rotation about the beam axis and move-

ment perpendicular to the plane of the beam (pivoting).

Both of these actions are controlled from outside the

vacuum chamber. The entire oven assembly is supported by

the inner race of the double row ball bearing B (Figure 9),

the outer race of the bearing being securely affixed to the

supporting flange. Rotation is achieved by the action of

a spur gear on the oven mount gear G. The spur gear is

attached to a 1/4 in. diameter shaft which extends through

the supporting flange by means of a Wilson seal. The oven

is supported on four stainless steel tubular stilts mounted

on a ceramic ring, R (Figures 9 and 10). Between the ring

and gear G are located two knees, K, which enable the oven

to be pivoted. Pivoting is produced by the action of a

shaft (entering the vacuum system through a Wilson seal) on

a lever L attached to the ceramic ring (Figures 9 and 10).

This shaft is threaded through a support affixed to the

flange, and the lever is held against it by means of a

tension spring.

A liquid nitrogen cooled shield (Figure 10) is lo-

cated directly in front of the oven. Practically all the

alkali metal effusing from the oven that does not constitute

Fig. 10.-Photograph of the oven and scatter-
ing chamber assembly with the liquid
nitrogen trap and cold shield in


the beam is removed by condensation on the shield. In

addition the shield reduces the background pressure by

means of cryogenic pumping.

4. Scattering chamber

Experiments conducted thus far have involved the use

of one beam with a scattering chamber located between the

beam oven and the velocity selector. These experiments

have been conducted in order to obtain data to which the

results of future double beam experiments may be compared.

The scattering chamber is mounted directly to the

oven assembly (Figure 10). Two of the four stilts that

support the oven are replaced by 1/4 in. diameter precision

shafting. These shafts are attached to the ceramic ring of

the oven assembly and located by means of reamed holes in

the plate supporting tne oven; they pass along the side of

the over and extend beyond it. Two 1/4 in. diameter reamed

holes, H (Figure 11), in the scattering chamber are pre-

cisely located so as to accept these shafts and align the

-catzering chamber with the oven.

The scattering chamber is constructed of aluminum.

A well is provided (Figure 11) to accept a modified RCA

model 1946 thermocouple gauge (Appendix I). This gauge is

constructed using only the wires from the RCA tube. These

wires are attached to a support system that enables them to

be introduced into the 1/2 in. diameter well. An iron-

Fig. 11.-Diagram of the scattering chamber:
S-body of chamber, TCG-thermocouple
gauge, G-teflon gaskets, P-plugs
with 0.004 x 1/L in. slits, L-blank
off cap.


gas In
^ ^ \ I [\^\\^^'N

constantan thermocouple is attached to the support flange

of the gauge to allow monitoring of the scattering chamber

temperature. A port which communicates with the thermo-

couple well is provided for introduction of the scattering

gas through a 1/4 in. copper tube extending to the outside

of the vacuum system. The scattering path length is de-

termined by plugs inserted into a 1/2 in. diameter reamed

hole whose axis corresponds to the central beam axis (when

the chamber is on its mount). The plugs have a wall thick-

ness of 0.002 in. at their ends; they are provided with

0.004 x 1/4 in. slits cut by a 0.004 in. slotting saw.

Teflon rings with inner diameters slightly less than 1/2 in.

are placed in steps at either end of the 1/2 in. hole; it

is these rings which effect a seal about the plugs and cause

them to remain in position. Each plug's open end is slotted

so that the position of its slits may be precisely located

by the alignment of these slots with corresponding ones

milled into the body of the scattering chamber. The region

between the plugs communicates with the thermocouple well

by means of a 1/4 in. milled slot. The distance between

the ends of the plugs is 0.001 in. + 0.0005 in.

For calibration purposes the plugs in the scattering

chamber are replaced by a 0.498 in. diameter shaft which

passes through both teflon rings and is held in place by

compression caps. This arrangement seals the chamber and

in addition serves to extrude the teflon rings so they

maintain their ability to position the plugs.

5. Detector

The beam of alkali atoms is detected on a tungsten

surface ionization (Langmuir-Taylor) detector.37'8 The

alkali metal atom striking the hot tungsten surface is ad-

sorbed; it may then give up an electron and evaporate as a

positive ion. All experiments thus far have involved the

use of cesium beams; the ionization efficiency for cesium

on a tungsten surface at 1300K or higher is 1.00.39

The detector is mounted from a spiroid gear whose

axis of rotation coincides with that of the velocity se-

lector. This allows the detector to be moved about the

scattering zone. The spiroid gear is driven from outside

the vacuum system through a modified Wilson seal. A full

rotation of the selector drive shaft produces movement of

the detector through 1 deg. of arc.

A schematic diagram of the detector is shown in

Figure 12. Beam atoms reach the hot tungsten filament D

by passing through slits in electrodes C and G. The 0.003

in. diameter tungsten filament40 is spot welded at the top

to a 0.040 in. diameter support rod and at the bottom to a

0.010 in. diameter tungsten spring, which in turn is spot

welded to the lower support. Power to heat the filament is

Fig. 12.-Schematic diagram of detector: D-
heated tungsten filament, P-repeller,
G-guard electrode, Fl and F2-focusing
electrodes, C-collector electrode.

oc (9


supplied by a 6 volt lead storage battery. The filament

and repeller P are at a positive potential with respect to

ground with the repeller being at a slightly higher po-

tential than the filament. The repeller has a grid covered

window to allow optical alignment. The guard electrode G

has a slit 3/8 in. long and .063 in. wide; it is employed

to collect all ions from those parts of the filament not

exposed to the beam. The potential on G is negative with

respect to the filament. Those positive ions resulting

from beam atoms striking the filament are accelerated

through the slit in G and focused by Fl and F2 to strike

to the right or left of the 0.010 in. slit in the collector

electrode C (located at ground). These ions striking C are

read as a current with the aid of a Cary model 31 vibrating

reed electrometer. All potentials to the electrodes and

filament are furnished by lead storage batteries, with the

repeller being 48 volts off ground.

B. Procedure

The procedure for the investigation of transmission

characteristics of the velocity selector and the measurement

of cross-sections will be outlined.

1. Measurement of transmission characteristics of the

velocity selector

The positions of the oven, scattering chamber, and

detector are determined by the location of the velocity

selector within the apparatus; consequently, the alignment

of the apparatus must be checked. Provision is made for the

accurate location of the motor (from which the selector is

suspended) in regard to the top flange of the selector

chamber (Figure 8). This provision and the ability to level

the apparatus, thus placing the motor shaft in a vertical

position, enable the velocity selector's position with

respect to the apparatus to be reproducible. During the

introduction of the velocity selector into the machine the

flexible shaft (Figure 7) is sometimes bent slightly. The

condition of the shaft is determined (after leveling the

machine) by viewing the lower edge of the selector tele-

scopically through the front port of the apparatus. If the

shaft is bent, the lower edge will rise and fall as the

selector is turned slowly. This occurrence is corrected by

lifting the rim of the selector gently by hand at its low

point, the test being repeated until the flat surface of

-he rotor is horizontal. After the selector has been

positioned in this manner the alignment of the oven,

scattering chamber, and detector with the selector is

checked optically. Checks such as described above have

never revealed any deviations from the initial alignment

of the apparatus.

The measurement of transmission characteristics does

not necessitate the use of a scattering chamber; conse-

quently, this component along with the cold shield (Figure

10) were not present in the apparatus. The slit located

in the bulkhead separating the beam chamber from the

velocity selector chamber was set at 0.020 in. and used to

collimate the beam. A shield was provided just below the

inner rim of the velocity selector to mask out all beam

particles that did not traverse the entire selector. The

experimental constants are listed in Table 1. With these

modifications accomplished, the transmission characteristics

of the selector were determined.

After the alignment of the apparatus is checked, a

glass ampule of cesium is introduced into the oven and the

assembly placed in the beam chamber. The apparatus is then

initially evacuated by a small mechanical vacuum pump.

After the initial pump down, the machine must be re-leveled

to compensate for the external atmospheric pressure exerted

on evacuated bellows in the fore line. The diffusion pumps

then reduce the pressure to operating conditions (5 x 106

mm Hg for the beam chamber, and 1 x 10-6 mm Hg for the

velocity selector chamber). Under vacuum the oven is out-

gassed by heating to a temperature approximately 200C above

that at which it will be operated, and the filament aged at

0.7 amp. for several hours.

Table l.--Operating Conditions for Measurement of Trans-
mission of a Maxwellian Beam of Cesium Atoms.

Oven slit width---------------------------------- 0.004 in.

Collimating slit width--------------------------- 0.020 in.

Detector width (tungsten wire)------------------- 0.003 in.

Beam height------------------------------------ 2.5 in.

Length of beam path in collimating chamber, Il--- 4.00 in.

Length of beam path in scattering chamber, 2----12.375 in.

Oven temperature----------------------------- ---- 405K
Pressure in beam chamber-------------------- 1 x 10- Torr.
Pressure in velocity selector chamber------- 8 x 10-7 Torr.

Radial position of detector, Rd------------------ 1.875 in.

Approximately two hours before the measurements are

to begin the oven is turned on and allowed to reach operating

temperature (1300C). At 1300C and with 0.004 in. wide oven

slits the mean free path of the Cs in the oven is approxi-

mately three times the slit width. During the warm up time,

the velocity selector is accelerated to operating speed.

The selector motor is driven by a variable frequency two

phase electronic power supply through two ganged 8 amp.

variacs. During acceleration the rotor passes through

regions of relative instability before the shaft is flexed

sufficiently to permit the selector to spin about its true

inertial center. The speed of the velocity selector is

determined stroboscopically; the output of the stroboscope

being registered on an electronic counter whose count period

is determined by another counter monitoring the 60 cps line

frequency. Also, during the oven warm up period, the de-

tector wire is flashed for 1 min. at 1 amp.; this cleans

the filament and reduces the background.

After the oven is stabilized at the desired tempera-

ture, it and the detector are positioned to allow maximum

transmission at the selector speed for which this condition

applies when the detector is in the beam axis (Section III,

A). With the oven and detector properly aligned, measure-

ments are ready to be made.

The selector is accelerated to the highest speed for

which measurements are to be made. A beam profile is scanned

at this speed by moving the detector completely through the

beam while indicating every 1/4 deg. of arc. These and

following measurements are made using the electrometer (in

the current sensitive mode) whose output is recorded on the

chart of a Leeds and Northrup Speedomax H recorder. The

velocity selector motor is then decoupled from its power

supply and the rotor allowed to coast. Alternately the

transmitted intensity and the background (beam interrupted

by flag) are measured. During each of these periods the

oven temperature and selector speed (at a particular time)

are determined. The oven temperature is determined by

reading the emf generated by the iron-constantan thermo-

couple attached to the oven with a Leeds and Northrup type

X-2 potentiometer. The speed of the selector is determined

stroboscopically by setting the stroboscope to a speed

slightly below that of the selector and then waiting for

the stroboscopic image to stabilize stroboscopee and selec-

tor synchronized) at which moment a timing mark is placed

on the recorder chart. Therefore, the speed of the selector

at the .-me of the mark was made is that for which the

stroboscope is then set; the output of the stroboscope is

then determined by the electronic counting method described

above and recorded. Since the time between successive

measurements is short, a linear interpolation between

adjacent measured intensities is acceptable; thus the

transmitted intensity is proportional to the distance be-

tween the interpolated intensity at a particular selector

speed and the background trace on the chart. During the

course of these measurements the amplification factor of

the electrometer is changed; consequently it is necessary

to normalize all the data to conform to a particular

amplification factor (in this case the 300 my scale of the

electrometer). The above procedure was followed for

velocity selector speeds from 8000-1800 RPPH.

2. Measurement of total cross-sections

All components described in the first section of

chapter are present in the apparatus. The mask described

previously is removed from under the velocity selector and

placed across the bulkhead slit at a height which corres-

ponds to the bottom of the selector. The electrometer

used to amplify the detector signal is operated in the

rate of change mode with the beam intensity being pro-

portional to the slope of the recorder trace. Modifications

made to the recorder enable it to trip three switches as

the pen moves up scale. The first and third switches are

set at a precise distance (3.000 in.) apart and used to

start and stop an electronic counter. This counter re-

ceives a 1000 cycle signal from a low frequency oscillator.

Thus, the precise time the recorder pen takes to move a

specific distance is obtained. The ratio of the counter

reading to the distance the recorder pen moves is pro-

portional to the slope of the recorder trace and hence the

beam intensity. The second switch actuated by the recorder

is located midway between the first and third; it places a

small blip on the recorder chart indicating the position

at which the slope of the trace is taken. The other two

switches also place blips on the recorder chart to indicate

initiation and termination of the timing operation. A

fourth blip is placed on the recorder chart by a switch

located directly in front of the apparatus; this blip is

used to mark the point at which the selector speed is

determined. All mechanisms that place blips on the recorder

chart also cause a time recorder to print out the time to

the nearest 0.01 sec.

Before cross-sections measurements can be made, the

thermocouple gauge in the scattering chamber must be cali-

brated. The calibration plug is inserted in scattering

chamber, and this chamber along with the oven is placed in

the beam chamber. The entire apparatus is pumped down to

simulate operating conditions. Power is supplied to the

heater of the gauge by means of the 2 volt storage battery.

The current drawn by the heater is held to approximately

59 ma. by inclusion of appropriate fixed resistors. The

gauge heater is operated for at least half a day to allow

time for it to stabilize. The oven is heated to operating

temperature and liquid nitrogen added to the trap; this

allows the temperature of the scattering chamber to

stabilize at approximately 50C (for an oven temperature of

1600C). Calibration of the gauge may now begin. With the

gauge under high vacuum (pumped on by the other beam

chamber's diffusion pump) a reading of its output E is

made with the K-2 potentiometer. Pumping on the gauge is

then terminated, and gas is introduced into the scattering

chamber through a leak valve with a 1000:1 taper until the

desired pressure is obtained, at which time the flow of gas

is interrupted (pressure is prevented from building on the

low pressure side of leak valve by engaging an oil diffusion

pump located in the gas handling system associated with the

machine). After the pressure in the system has equilibrated,

a portion of the gas is trapped in the McLeod gauge and the

thermocouple gauge output, E, is read. The thermocouple

gauge is pumped down and three successive readings of the

former pressure are made on the trapped gas in the McLeod

gauge (which is read with a cathatometer). While the McLeod

gauge is being read, E is again determined. The temperature

of the McLeod gauge region and that of the scattering

chamber is also recorded. This procedure is followed over

the pressure range 0.5 to 5.0 microns in 0.5 micron incre-


After calibration of the thermocouple has been

accomplished the machine is vented with dry nitrogen and the

oven and scattering chamber removed. It should be noted at

this point that, as stated previously, the alignment of the

machine has never been observed to be affected by removal of

either the oven assembly or velocity selector from the

machine; consequently, the alignment checking procedure is

no longer repeated after removal of either of these com-

ponents. The oven is loaded with a glass ampule of Cs and

replaced in the machine. The same procedure (prior to

actual measurements) as outlined previously is now followed,

the one exception being that liquid nitrogen is added to the

trap in the beam chamber before the Cs ampule is broken.

Once liquid nitrogen has been added it is important that the

trap never be allowed to become empty during a run; this

requires refilling the trap at least every half hour.

Prior to the actual taking of data the pressure in

the scattering chamber must be adjusted (by means of the

above mentioned leak valve) to produce a beam attenuation

of about 20 per cent at the highest selector speeds (8000

RPM). This operation must, of course, be carried out at

constant velocity selector speed. Having established the

appropriate pressure in the scattering chamber, measurements

may now be taken.

The beam flag is raised to interrupt the beam and

the background is measured. An amplification factor for the

electrometer (in the rate of change mode) is chosen such

that the slope of the recorder trace is not more than 1.5.

The velocity selector is decoupled from its power supply

and allowed to decelerate. Alternately the unscattered

beam intensity I0 (gas out of scattering chamber) and

scattered beam intensity I (gas in) are measured. These

measurements are obtained in the manner previously des-

cribed for the electrometer in the rate of change mode.

During the time I0 is being traced by the recorder the out-

put, E of the thermocouple gauge and the temperature of

the oven are read on the potentiometer. Between the

measurement of IO and I the speed of the selector at a

particular time is determined; the procedure used is the

same as has been described for the transmission character-

istic determination. After sufficient time has elapsed for

the equilibration of gas in the scattering chamber, I is

determined; at the same time E for the thermocouple gauge

and the temperature of the scattering chamber are read with

the potentiometer. This procedure is followed until the

selector has reached a speed of 1000 RPM (usually in 14 or

15 hours). Typical operation conditions under which these
measurements are taken are listed in Table 2.

Table 2.--Typical Operation Conditions for Measurement of
Cross-Sections of Cesium with Nitrogen and Argon.

Oven slit width---------------------------------- 0.002 in.

Scattering chamber slit width-------------------- 0.004 in.

Scattering chamber length------------------------ 0.100 in.

Detector width (tungsten wire)------------------- 0.003 in.

Beam height-------------------------------------- 0.250 in.

Distance between oven and scattering chamber----- 4.95 in.

Distance between scattering chamber and velocity
selector--------------------------------------12.00 in.

Pressure in beam chamber (gas out of scattering
chamber)-------------------------------- 6 x 10-7 Torr.

Pressure in beam chamber (gas in scattering
chamber)-------------------------------- 9 x 10-7 Torr.
Pressure in velocity selector chamber---- 2.0 x 10-6 Torr.

Radical position of detector--------------------- 1.875 in.

Angular resolving power-------------------------- 3 min.

The time interval between successive measurements

is short allowing linear interpolations to be made between

successive IO's and I's; in addition this type of interpo-

lation may be used for the determination of selector speeds

between timing marks. Consequently, it is possible to

obtain two cross-section measurements from the determination

of one pair of intensities; these measurements are made at

the selector speed corresponding to the time at which the

slope of the particular trace (I or IO) is measured. It is

from the scattering chamber pressure and the ratio of Io/I

(at a particular velocity selector speed) that the cross-

sections are calculated.


A. Transmission Characteristics of a Parallel Wall, Spiral

Groove Velocity Selector41

Parameters of the velocity selector are given in

Table 3. All graphical representations of this section are

for the specific geometry of this selector. The coordinate

system to be used in this analysis is shown in Figure 15.

The projected trajectory of a point moving with velocity

vM corresponds to the center line of a groove when the

selector rotates with an angular speed of w; this tra-

jectory is defined by the polar angle 9 and the radial

distance r from the center of the selector. The equation

of the center line of a groove is

M = (w/vM) (R1 r) = (90/2) (R1 r) (1)

where the substitution w/vM = 0/2 has been used. Equation

(1) is obtained by equating (R1 r)/vM the time required

to move a radial distance R1 r, to Q/w the time for

rotation of the selector through the angle 9. The equations

for the walls of the groove are obtained in like manner with

the exception that provisions must be made for the walls to

remain parallel to the center line; the quantity sin-1 (a/r)

accomplishes this purpose, and the equations are


Table 3.--Velocity Selector Parameters.

R--------------------------------------------- 6.500 in.

R2--------------------------------------------- 3.000 in.

S--------------------------------------------- 3.500 in.
-------------------------------------------- 0.1297 rad.

a ----------------------------------------------0.0345 in.

Groove depth------------------------------------ 0.250 in.

Number of grooves, n----------------------------- 148

Trmin(y= 0)------------------------------------ 0.319
Tr,max(y 0)------------------------------------ 0.681

Resolution-------------------------------------- 17.3%

Fig. 13.-Coordinate system for velocity
selecting grooves.


Upper wall: U (@o/)( R2-a2/ -/r~-a2) + sin-la/r)

Lower wall: GL = (o/9)(R2-a2 -r ) sin- (a/r)

Several approximations may be made to simplify Eqn. (2).
For r >, R2 (which is always the case during selection)
the groove width, a, is much less than r; consequently the
small angle approximation for the sine function can be
used. In addition the quantities (R12-a2)1/2 and (r2a2)1/2
may be approximated by R1 and r, respectively. Upon
introducing these approximations and requiring GL(R1) = 0
when t = 0, Eqns. (1) and (2) become

9M = (Qo/Q)(R1-r) + a/RI +wt
QU = GM + a/r (5)
=L OM a/r
The outer limit of the groove substends an angle
2a/R1 from the center of the selector. Therefore, the time
interval during which a narrow beam of molecules, incident
normal to the selector, can enter the groove is 0 < t <
- 2a/R1w If a reduced time tr R1ct/2a is defined, the
admittance interval becomes 0 < tr < 1. Molecules with
velocity v entering the selector during this time interval
will have projected trajectories given by

9 (w/v)(Ri-r) (R1-r)/(RVr) (4)

where a reduced velocity Vr v/oR1 has been defined.

At time tr only those molecules which enter the
groove with reduced velocities in the range Vr (tr,max)

Vr < Vr(t,min) will be transmitted. Those molecules with
velocities outside this range will undergo collisions with
the groove wall and be taken out of the beam. The posi-
tions at which these beam molecules strike the wall are
given by

rU = (R1/2Lv) {V L+ a(2tr-1) +

C(LV + 2atr-a)2 4aLv]1/2} (5)

rL = (R1/2Lv) {LV + a(2tr-1) +

[(LV + 2atr-a)2 + 4aLy]1/2 (6)
LV = R12 w (1/v-l/vM) = R1/Vr R12o/ (7)

Those molecules for which Vr < Vr(t,min) will
strike the upper wall. Equations (5) and (6) are obtained
by solving Eqns. (4) with (2). Beam molecules that are
transmitted by the selector follow one of two types of tra-
jectories. The first or oblique trajectory, is one which
moves continuously toward, or away from, a wall of the
groove; the second or tangential trajectory, exhibits what
may be called a turning point in that it moves first toward
a groove wall and then away from it. It is this latter
type which usually determines the velocity range trans-
mitted by the groove. Molecules entering the groove at

time tr with velocity Vr will be transmitted provided
Eqns. (5) and (6) have no real roots in the range R1 >-r >
R2. Those paths for which there exist double roots in
this range will make grazing collisions with one of the
groove walls thereby determining the maximum and minimum
transmitted reduced velocities. It is for these latter
cases that the discriminants of Eqns. (5) and (6) are zero.
Therefore, the limiting L 's may be obtained as

L ,min,tan = a {l+[2(1-tr)]1/2}2 R > r,tan >R2 (8)

LV,max,tan -a[1+ (2t )1/2 2 R1 rL,tan > R2 (9)


rU,tan = 1/ 1 [2(1-tr)]1/2} (10)

rL,tan R1/1 + (2tr) /2] (11)
The trajectories will be tangential for rU,tan or rL,tan >
R2. In most cases rU,tan or rLtan become less than R2
during the interval 0 s< tr < 1 so that a transition occurs
between tangential and oblique trajectories; consequently
some velocities are limited by collisions at the inner
edge of the groove walls. The time Tr at which the transi-
tion occurs is found by setting Eqns. (10) and (11) equal
to R2'

Tr,max 1 Tr,min 2/2R2 (12)

In order to arrive at the oblique limited velocities
the limiting L 's must first be obtained. This is
accomplished by setting rU and rL (Eqns. (5) and (6)) equal
to R2'

LV,min,ob = (R1 a/)[C(R1+R2)/R2-2tr tr < Tmin (15)

LV,max,ob -(R1 a/9)( /R2+2tr) tr > r,max (14)
The limiting oblique reduced velocities are obtained by
equating Eqns. (13) and (14) to (6) and the limiting
tangential reduced velocities by equating Eqns. (8) and (9)
to (6); so that there results

Yr,min,ob K[R.9o+a(Rl+2)/R2-2atr]-1 0T< tr< Tr,min
Vr,min,tan {RIgo + (a9/R1)[l + (2-2tr)1/2 2}-1

Tr,min tr'< 1 (16)

Vr,max,tan Rlo (a9/R1)l[ + (2tr)l/2 2}-1

0 < tr Tr,max (17)

Vrmax,ob (R1Qo a 2/R2 2atr)-l

Tr,max tr< 1 (18)
Figure 14 shows a shaded area on the Vr, tr plane which
represents the transmissions of the present selector. There
exist slight discontinuities at tr = Tr,mn and Tr,max;
however, the scale of the plot is such as to render these

Fig. 14.-Plot of transmitted reduced velocities
versus reduced time of arrival at edge
of velocity selector.




,,, "//

4.1 //

3.9 -

3O 02 0.4 Q6 0.8 1.0

discontinuities non-evident. The length of time a groove
will be capable of transmitting molecules with a particular
reduced velocity, Vr, is found from the length of the
horizontal line through Vr that spans the shaded area. As
can be seen, the groove is completely transparent to re-
duced velocities in the range 4.08 to 4.24. The fraction
of total time f(V ) a groove will transmit a particular
reduced velocity can be obtained by solving Eqns. (15) -
(18) for tr; thus

ft(Vr) = tr = Rl/2a (R1-R2)/2R2 J/2aVr

V1 < Vr V2 (19)

ft(Vr) tr = (1/2)[1 (R1/a)(l/Vr-RIo/2 )o

+ [(R1/a)(1/Vr-RlQo/j )1/2 (20)

V2 Vr V3

ft(Vr) 1 V3 Vr < V4 (21)

ft(Vr) 1 tr = (1/2)[1 + (R1/a)(1/Vr-R1@o/? )]

+ [(R1/a)(R10/9 -1/vr)11/2 (22)

V4< Vr V5

ft(Vr) (1 tr) = -Rl1o/2a + (R1+R2)/2R2 + J/2aVr'


V5 Vr V6

If the above equations are plotted (Figure 15) a character-
istic transmission profile is obtained for the particular
selector. The reduced velocity distribution function for a
transmitted pulse is

F(Vr)dVr ft(Vr)I(Vr)dVr (24)

where I(V ) is the reduced velocity distribution function
in the beam before velocity selection takes place. The
transmitted intensity of a single pulse can be obtained, if
I(Vr) is known, by

i f f(Vr)I(Vr)dVr (25)

The total transmitted intensity then is made up from
contributions of all pulses and is given by

I i nw/2n (26)

where n is the number of grooves in the selector and w
is the angular speed of the selector in rad./sec.
In order to glean as much information as possible
from scattering experiments performed using this type of
velocity selector, it is necessary to know the intensity
and velocity distribution at the scattering zone (r = 0)
as a function of time. This may be accomplished by con-
sidering a molecule with velocity v arriving at the outer
edge of the selector at time t; the time of arrival at the

Fig. 15.-Characteristic transmission profile
showing fractional reduced time ft
that a groove will transmit mole-
cules arriving at the selector with
normal incidence.




scattering zone is t + R1/w (and in terms of reduced

variables, tr R1/2aVr). The transformation tr-+tr -

R1/2aVr acting on the reduced velocities to be transmitted

taken as a function of the reduced time of arrival at the

edge of the selector generates the reduced velocity distri-

bution function in terms of the reduced time of arrival at

the scattering zone. This transformation is illustrated

in Figure 16. A vertical line between ABC and ADC and

through tr indicates the range of reduced velocities arriving

at the scattering zone at time tr. The locus of points

described by AB and CD are hyperbolas given by

AB: Vr = R1/2atr (27)

CD: Vr = R1/2a(l tr) (28)

The sides BC and DA are obtained by substitution of tr =

tr R1/2aVr into Eqns. (15) (18). The velocity spread,

AVr, at the scattering zone as a function of reduced time

is shown in Figure 17. Dashed lines indicate discreet

pulses; as can be seen for the present selector there is an

overlap of transmitted pulses at the scattering center.

This overlap creates a ripple superimposed on a constant

transmitted intensity. At the scattering zone the resultant

time dependent intensity is indicated by the solid line.

That portion of the transmitted pulse indicated in Figure

17 by AB is determined entirely by the velocity distribution

Fig. 16.-Time dependent distribution of trans-
mitted velocities at scattering zone.

O o






Fig. 17.-Time dependent transmitted intensi-
ties at scattering zone.


function, I(Vr), of the unselected beam; in the present

case I(V ) is taken to be that which exists in a Maxwellian

beam. The transmitted intensity at the scattering center

may be found as

it f I(V)dVr (29)


In terms of Figure 15 a characteristic resolution,p ,
may be defined as

p = AVrft = 1/2)/VrM = R1Ar,1/2/ (30)

This treatment of transmission characteristics

should be extended further to encompass those particles

which arrive at the outer edge of the selector along paths

other than normal. This situation will occur in wide or
divergent beams. The coordinate system which represents
"off normal" transmission is given in Figure 18. Where

the angle of incidence of a molecule is y its trajectory

projected onto the velocity selector is given by

9 ( /v)[(l2-h2) 1/2 (r2h2)1/2] + sin(h/r)-y ;

the angle Y is to be taken as positive or negative as the
molecule approaches above or below the horizontal line of
Figure 18. The usual approximations may be made in Eqn.

Fig. 18.-Coordinates for transmission of mole-
cules arriving at selector along path
striking wheel at angle Y with respect
to normal. Oy is angle through which
narrow detector must be moved from
beam axis to intercept transmitted




(31) since the y's are small and h << R2, consequently

y = (w/v)(R1-r) + (y/r)(R1-r) (32)

y0 is the angle through which a narrow detector placed at
radial distance rd must be moved to intercept the molecule.
The transmission equations considering the y
dependence are derived in the same manner as those for
normal incidence ( = 0). These equations are listed below
and have been given the number corresponding to the
analogous equation previously given for Y = 0.

rU = (R1/2Lv) Lv + a(2tr-1) R1 + C(Lv + 2atr-a-yR1)2

-4 Lv(a- yR)] 1/2) (Sa)

rL = (R1/2Ly){ Lv + a(2tr-l) yR1 [(Lv + 2atr-a-y R)2

+4 L (a+ RY)11/2} (6a)

Lv,min,tan a(3 2tr)-YR1 + 2[2a(l-tr)(a-Rl)] 1/2

R1 -rU,tan ,R2 (8a)

Lv,max:,tan a(2tr + 1) + yR1 + 2[2atr(a+yR1)]1/2}

R1 >rL,tan >, R2 (9a)

rU,tan R1 {a -yR1 + [2a(l-tr)(a- yR1)1/2}

{a(5-2tr) yR + 2[2a(l-tr)(a-yR1)]1/2 (10a)

rL,tan R1 a + yR1 + [2atr(a+ R)]1/2} /
{a(2tr+l) + yR1 + 2[2atr(a+ yR1)]1/2} (1la)

rmax (a+ yR1) / 2aR2 (12a)

Lv,min,ob = (R1 a/2 ) [(R1+R2) / R2 2tr YR1/R21

tr < r,min (13a)

Lv,max,ob = -(R1 a/ R ) [ 2/R2 + 2tr+yR1/R2]

tr > Tr,max (14a)

Vr,minob = R1 + a(R+R2)/R2 2atr 2ra R1/R2]-1
0 < tr < Tr,min (15a)

r,min,tan = R1 (9/RI)(a(-2tr) -yR1 +

2[2a(l-tr)(a-yR1)]1/2)} -1 Tr,min< tr < 1 (16a)

Vr,max,tan n oR1 (9/R1)(a(2tr+1) + yR1 +
2[2atr(a- yR1)]1/2 -1 0< < Tr,max (17a)

Vrmaxob = ~ R1 a9/R2 2atr -yaR1/R2-1

Tr,max +< tr 0< (18a)
ft(Vr) tr = Rlo/2a + (Rl+R2)/2R2 -YR1/R2 /2aVr

V1 < Vr < V2


ft(Vr) r tr = (1/2)[1 + R1/a (Rl/a)(l/Vr R1@0/ )] +

C(l+ yR1/a)(R1/a)(l/Vr R1o/ ) 1/2

V2 V Vr V3 (20a)

ft(Vr) = (1-tr) = (1/2)[1 + 3 yR1/a + (R1/a)(1/Vr -

Rlgo/1 )] + {(1 +yR/ a) (R0o/1 1/Vr) +

2yR1/a]) 1/2 V4 < Vr < V5 (22a)

ft(Vr) = (1-tr) = -Rl9o/2a + R/2aVr + (R1+R2)/2R2 +

,yR1/2R2 V5 Vr V6 (23a)

The total transmitted intensity of a molecular beam
effusing through a narrow slit and incident on a narrow
detector at a position y is
I = n /2n f y (Vr)I (Vr)dVr .. (26a)

B. Collision Theory42-46

Before describing the quantum mechanical treatment
of elastic atomic collisions, the classical representation
should be considered in some detail. This will afford
certain concepts to which aspects of the quantum treatment
can be referred.

The assumption will be made that molecules can be

represented as point centers of force and, in addition,

that they interact in pairs with conservative forces along

lines joining their centers. For the present, consideration

will be given to collisions that occur between a constant

speed molecular beam and a fixed target molecule.

Generalizations to actual situations will be discussed later.

Consider a homogeneous beam of particles moving

along paths that are parallel but otherwise distributed at

random. Let this beam pass over a particle held in a fixed

position. If attention is now drawn to the fraction of

particles that undergo a particular deflection by the

target particle, then a quantity G may be defined such

that the fraction of incident particles which are deflected

into a solid angle do is given by Gdw G is called the

scattering coefficient or differential scattering cross-

section. It will, in general, vary with the direction of

scattering and, consequently, should be denoted in polar

coordinates as G(9,0).

As a rule a more useful quantity is the polar

scattering coefficient, F(0). This quantity represents

the fraction of beam particles incident on a unit area

which are scattered by one scattering center through angles

9 and 9 + dG. Consequently all particles scattered (by a

spherically symmetric potential) into the range dO are

represented by G(@)dw = 2t G(G)sin QdQ = F(Q)dG .

The total collision cross-section, S, is the total

fraction of beam particles per unit area removed from the

beam by collision with one scattering center. Therefore

S(Q) = F(Q)dQG 2i fG(G)sin QdG (33)
0 o

The quantity measured experimentally is given by

S(G') f F(Q)dG = 2n i G(G)sin QdQ (34)

This last equation results from the fact that all detectors

must have a finite width; thereby not allowing detection of

scattering through angles less than 9' .

The classical scattering coefficient may now be

used to calculate the collision cross-section for any

spherically symmetrical type of force between beam particles

and a fixed scatterer. Because of the interaction force the

beam molecule will be deflected so as to move along a plane

curve (Figure 19) (this case being for a repulsive force;

for an attractive force the orbit would curve the other way).

The collision parameter, b, is merely the distance of

closest approach assuming no interaction takes place. The

net deflection is denoted by the angle 9 and the distance

of closest approach by ro. At any instant the position of

the molecule is given by the polar angle, 0, and the

distance from the scattering center, r .

Fig. 19.-Classical orbit of a particle moving
with velocity v through an isotropic
repulsive force field originating at
a fixed point: b-impact parameter, r
and 0 coordinates of particle, r -
distance of closest approach, Q -
deflection angle.



\ 0

-- I-

Let m be the mass of the molecule so that at any
instant its angular momentum and kinetic energy are given
by mr2 and m(r2 + r2 2)/2 respectively.$7 The initial

values of the latter quantities are mvb and mv2/2; so that

by conservation of angular momentum and energy there
results: ,

mvb = mr 2 (35)

mv2/ 2 = ( + r2 2)/2 + V(r) (36)

where V(r) is the potential energy between the interacting

particles. Solving the above two equations for < and i ,
one obtains

dk/dr = b[ 1 2V(r)/mv2 (b/r)2-/2/r (37)

The total change in ( during the collision is n G, or
since the orbit is symmetrical about the point of closest

t -Q 2 f (d

This allows calculation of the net deflection 9, as a
function of b,
oo -1/2
G(b) = n 2b o r-1 (1-2V(r)/mv2)r2-b2} dr
0 (39)

Paths of approach taken by particles in the beam on
the average are directed uniformly over a plane perpendicular

to their direction. Consequently the fraction of molecules

in the beam with impact parameters between b and b + db

are scattered into a range d9; so that de = 9(b)db and

F(G) = 2tb/9(b). This then allows the calculation of the

total scattering cross-section, for a given scattering

potential, from Eqn. (33).

The general treatment of scattering must consider

the motion of the target particles and the distribution of

velocities present in the beam. The physical situation

can be related to the discussion by dividing this effect

into two parts: one arising from the initial motion of the

scattering particle, and the second from the acceleration

of the scattering molecule during collision. The first

effect is dealt with by considering the relative motions

of the interacting particles and performing an appropriate

averaging over all the relative motions. The second effect

is understood by considering the fact that when two particles

move under the action of mutual forces, the motion of one

relative to the other is the same as would be its actual

motion if the second one were held fixed and the first ones

mass reduced to mlm2/(ml+m2).

In a number of experimental conditions a problem

arises with the use of classical mechanics to describe the

differential scattering cross-section. This results from

the divergence of G(G) for all force fields which extend to

infinity. For such fields as these there is no upper limit

to the impact parameter. All deflections are counted no

matter how large the impact parameter. This generates an

effective scattering area for a particle which is infinite.

If quantum mechanics is used to describe the scattering

process, there will exist a range beyond which the deflection

of a scattered particle will be less than the uncertainty

of its position and momentum; consequently the particle

will not be considered to have undergone any interaction

with the scattering center. It is the problem, divergence

of the classical differential scattering cross-section for

small angles, that requires a quantum mechanical treatment

to describe the scattering process in most experimental


In order to treat the scattering problem quantum

mechanically the fact that the motion of particles can not

be described with complete accuracy must be taken into

account. Wave packets are used from whose average co-

ordinates arise the classical orbits. The determination of

collision cross-sections must resolve itself into finding

solutions to the Schroedinger wave equation,

-(2/2u) V2 + V(r)* = E* (40)

This equation represents a particle of mass u and kinetic

energy E moving through a scattering potential of V(r).

Equation (40) may be transformed into one involving the

wave number k = ,v/h of the particle undergoing scattering,

V2, + (k2-U) = 0 (41)

U = 2V(r)/h .

Scattering is determined by the asymptotic form of
the wave function,

'(r,G) = A[exp(ikz) + f(9)exp(ikr)/r] (42)
r- DO

which is a solution to the above wave Eqn. (33) and assumes

the action of a spherically symmetrical potential with the
z axis as the direction of incidence of the beam. Here A

is a normalization constant. The term A[exp(ikz)] repre-

sents a plane wave incident on the scattering center, while

the second term in (42) represents an outgoing spherical

wave; for a spherically symmetric potential the amplitude

of the scattered wave depends only on the polar angle 9

and is inversely proportional to r The incident beam
flux is given by viAl2 whereas the scattered flux along an

outward radius is given by vAl 21f(Q) 2/r2 Then by

definition the differential scattering cross-section is:

G(G) = lf(9)12


Although the asymptotic behavior of the wave function
determines the scattering cross-section, it can not be found

unless the wave equation is solved for all space. There are

two generally accepted means for obtaining solutions: the

method of partial waves and an integral equation method.

The former method will be discussed here.

The general solution to the wave Eqn. (33) consists
of an infinite sum of Legendre polynomials

=(r, ) = T R (r)P (cos 9) (44)

where j is the angular momentum quantum number. Provided

the potential energy V(r) has a more rapid decrease than

1/r the general solution can take the asymptotic form,

k(r,9) ---(kr)-1 T (22 + l)exp[i( E /2 + 8 )]P
^r-.. =o

(cos G)sin(kr- ~E/2 + S ) (45)

The angle 8n is called the phase shift of the J, partial

wave. It is the difference in phase between the actual

radial function R (r) and the radial wave function in the

absence of a scattering potential (V(r)=O). A repulsive

potential implies a decrease in the relative velocity of the
interacting particles thus increasing the wavelength; there-

fore, the scattered wave is "pushed out" relative to that

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