,^T °F **

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NBS TECHNICAL NOTE 871

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

Geometrical Calibration

of the NBS

ttering Apparatus
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

the Institute for Computer Sciences and Technology, and the Office for Information Programs.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent system of physical measurement; coordinates that system

with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of a Center for Radiation Research, an Office of Measurement Services and the following divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Nuclear

Sciences 2 — Applied Radiation - — Quantum Electronics 3 — Electromagnetics 3 — Time

and Frequency " — Laboratory Astrophysics " — Cryogenics 3

. THE INSTITUTE FOR' MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized

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THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

accessibility of scientific information generated within NBS and other agencies of the Federal Government; promotes the development of the National Standard Reference Data System and

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organizational units: Office of Standard Reference Data — Office of Information Activities — Office of Technical

Publications — Library — Office of International Relations.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

3 Part of the Center for Radiation Research.

3 Located at Boulder, Colorado 80302.

* Part of the Center for Building Technology.
3 IS?? Geometrical Calibration of the NBS

Electron Scattering Apparatus

S. Penner, S. P. Fivozinsky,

and J. W. Lightbody, Jr.

Nuclear Sciences Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

L. S. Cardman

University of Illinois Champaign, Illinois 61801

W. P Trower

Virginia Polytechnic Institute and

State University

Blacksburg, Virginia 24061

Sponsored by

Research Corporation of New York

405 Lexington Avenue

IJew York, New York 10017 and

U.S. Energy Research and Development Administration

Washington, D.C. 20545

AT ^M& *k i w \

§

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton. Secretary

V^* NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued June 1975
National Bureau of Standards Technical Note 871

Nat. Bur. Stand. (U.S.), Tech. Note 871, 79 pages (June 1975) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1975

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. C13.46:871). Price $1.40 (Add 25 percent additional for other than U.S. mailing.)
CONTENTS

Page

Contents Ill

Illustrations IV

I . Introduction 1

II. Description of Apparatus 2

A. Spectrometer 4

B. Aperture Slits 7

C. Target Support Assembly 12

D. Beam Profile Scanners 21

III . System Geometry 27

IV. Target Assembly 36

V. Realization of the Baseline 39

VI . Scattering Angle 43

A. Theodolite Alignment 43

B

.

Azimuthal Dial Gauge Calibration 46

C. Radial Dial Gauge Correction 47

D. Fiducial Angle Measurements 48

E. Target Pedestal Dial Gauge Effect 49

F. Fiducial Angle Results 51

G

.

Scattering Angle Equation 52

VII. Solid Angle 52

VIII. Target Angle Calibration 58

IX. Summary 62

X. Acknowledgements 65

References 66

Tables 67

III
ILLUSTRATIONS

Figure Title Page

1 Plan view of electron scattering apparatus 3

2 Electron scattering spectrometer side view 5

3 Transverse motion of spectrometer aperture accompanying radial motion 6

4 Photograph of Aperture Slit Assembly 8

5 Illustration of slit-edge penetration effects 10

6 Target Pedestal Assembly 13

7 Target Holder Blank 16

8 Procedure for aligning the indicator ring 17

9 Location of the indicator ring dial gauges A, B, C, and D 19

10 Indicator ring dial gauge readings as a function of spectrometer angle 20

11 Operation of beam profile scanners 22

12 Upstream beam profile scanner 23

13 Target location beam scanners 25

14 Coordinate system used in this report, in plan view 28

15 Azimuthial dial gauge mounted on the rotating base of the spectrometer 30

16 Coordinate system used in this report in side view 31

17 Definition of quantities measured relative to the base line 33

18 Target ladder 37

K 19 Method of surveying the baseline 40

20 Setup for aligning the theodolite on the axis of rotation 44

21 Differences of pairs of scattering angle measurements plotted against the

difference of the corresponding (D-B) measurements 50

22 Method for measuring L - L 54 ° x y

2

3

Setup for measuring L 55

24 Target angle measuring circuit ; . .

.

59

IV
Geometrical Calibration of the NBS Electron Scattering Apparatus

S. Penner, S. P. Fivozinsky , J. W. Lightbody , Jr.,

L. S. Cardman , and W. P. Trower

A comprehensive calibration of the geometry of the NBS

electron scattering apparatus is described. A complete set of

measured parameters is tabulated in this report. Combining

these parameters with observed values of certain variables as described herein permits the accurate determination of the

solid angle, scattering angle, and target angle for each cross section measurement made with the apparatus. The uncertainty

in a cross section measurement due to the imprecision of the

geometry calibrations is less than one part in lCr .

Key Words: Absolute cross section; beam profile; electron

scattering; scattering angle; solid angle; spectrometer.

I . Introduction

The ability to determine absolute elastic electron scattering cross sections to very

high accuracy is being developed by the NBS electron scattering group for several reasons:

to establish the charge distribution of the proton, a fundamental object,

to establish precise cross sections for carbon which are widely used as a

standard in relative measurements of other electron scattering cross sections,

both elastic and inelastic, and

to determine the importance of dispersive effects in electron scattering.

To make significant contributions to the solution of these problems , an overall

cross section measurement accuracy well below one percent must be achieved. A careful

examination of all the factors influencing the overall accuracy of cross section measurements has been made, and we have concluded that the desired accuracy is obtainable [l].

Among the many experimental factors which must be understood are those related to the

geometry of the apparatus

.

In the following sections of this report, we describe those parts of the NBS electron

scattering apparatus [2] which determine the geometry, define the coordinate systems used

and the quantities to be measured, describe the measurements which are made, and discuss

Figures in brackets indicate the literature references on page 66.
the sources of error which affect the accuracy of measured cross sections.

II. Description of Apparatus

The NBS electron scattering apparatus is shown in figure 1. The detector system and

shielding platform which are mounted to the upper half of the spectrometer magnet are

omitted from this figure for clarity.

The energy-resolved electron beam is incident from the right in figure 1, in a

4- inch diameter vacuum pipe. The beam is approximately one-half inch in diameter when it

enters the magnetic field of the quadrupoles. The beam is focused by the quadrupoles to

a spot of less than 1 mm diameter at the target. After leaving the quadrupoles, the beam

passes through the upstream beam profile scanners, whose function is to measure the size

and position of the beam at this point. The beam then passes through a non- intercepting

current monitor which is part of the charge measuring system [3], The beam pipe diameter

is reduced to two inches between the current monitor and the scattering chamber. In

order to be able to determine the beam direction at the target, the region between the

upstream beam scanners and the target is kept free of magnetic fields by wrapping the

pipe with magnetic shielding. The scattering chamber is an 18- inch diameter vertical

aluminum cylinder. Ports on the scattering chamber at 17.5° intervals between 40 and

162.5° allow direct connection of the beam-line and the spectrometer vacuum systems.

The scattering, chamber which is mounted on the target support assembly contains the

target-location beam profile scanners. Downstream of the target chamber, the beam

passes through a short section of ^-inch vacuum pipe followed bv a long section of 12-inch

pipe before entering the Faraday Cup. Scattered electrons pass through one of the scattering chamber ports into the spectrometer. The aperture slits, which are rigidly attached to the spectrometer magnet, define the scattering angle and the acceptance solid

angle of the spectrometer. The high-resolution spectrometer magnet is mounted on a

carriage which permits radial motion of the magnet. This carriage is mounted on the rotating base which permits changing the scattering angle. The spectrometer, aperture

slits, target support assembly, and upstream beam scanners define the system geometry.

We now describe these components in detail.
Figure 1. Plan view of electron scattering apparatus,
A. Spectrometer

The spectrometer magnet [4], shown in figure 2, has a bending radius of 76 cm, a

-4- deflection angle of 169.8°, and a resolution capability of 2 x 10 • . The detection system

for scattered electrons is rigidly attached to the spectrometer exit flange [5]. Since the

spectrometer acceptance angles are completely determined by the aperture slits , the magnet

has no direct effect on the experimental geometry. Because the aperture slit box is

rigidly attached to the magnet, motion of the magnet does affect the geometry. The magnet,

which is fixed to its carriage by two main support bolts and the shielding supports . is not

completely rigid. The effect of this lack of rigidity on the reproducibility of scattering

angle and solid angle calibrations is small, and has been taken into account in the error

analysis

.

Radial motion of the spectrometer is needed in order to connect the spectrometer vacuum

system to the appropriate port of the scattering chamber. The two main beams of the

carriage ride on the carriage support beams through several sets of roller bearings. The

support beams are an integral part of the rotating base. A hand-operated drive screw which

connects the carriage and the rotating base provides radial motion. The radial position of

the spectrometer is measured by means of a dial gauge mounted on the rotating base with its

plunger engaging one of the cross members of the carriage. Like all the dial gauges used

in the apparatus, the radial dial gauge has a nominal travel of one inch, a least count of

2 0.001 inch, and a standard deviation of about 0.0002 inches. The gauge is located about

two feet below and to the side of the center of the aperture, with its plunger aligned in

the direction of carriage motion.

In addition to the roller bearings which support the carriage, there are roller

bearings which engage the sides of the support beams to prevent sideways motion of the

carriage on its support. This arrangement is not completely satisfactory. Small, nonreproducible motions of the spectrometer on the base are found to accompany radial displacement of the carriage. In addition, there is a small transverse component of the

spectrometer motion coupled to the radial motion. Measurements of the transverse position

of the aperture slit box as a function of radial spectrometer position are shown in

figure 3.

2 Most of the dimensions given in this report will be quoted in customary units, since the

instruments used for the calibrations all read out directly in inches.
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RADIAL POSITION, /?, INCHES

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Figure 3. Transverse motion of spectrometer aperture accompanying radial motion. The

solid curve indicates the motion predicted from the change of scattering anele with radial position.
The variations in transverse position at a given radial position are of order 0.001 inches,

corresponding to a variation in scattering angle of 0.002 degrees. The systematic change in

transverse position agrees with independent observations of the variation of scattering

angle with radial position. The latter measurements will be discussed later. Because of

these effects, a radial dial gauge reading is needed for each setting of the scattering

angle in order to determine the exact scattering angle.

The rotating base of the spectrometer rides on the fixed base by means of a tapered

roller bearing approximately six feet in diameter. A motor-driven ring and pinion gear

arrangement rotates the spectrometer to change the scattering angle. Small rotations can

i be made manually by turning one of the gears in the speed reduction drive from the motor.

The fixed portion of the spectrometer base rests on a concrete pedestal and is attached by

a number of large bolts. During installation, the base was carefully leveled on steel

wedges and then grouted in place. The concrete pedestal which rests directly on the fourfoot thick reinforced concrete sub flooring provides rigidity for the entire structure.

The spectrometer does not execute a pure rotation about a fixed axis because of imperfections in the main bearing, carriage support, and mounting of the spectrometer to the

carriage. The geometry of the apparatus can nonetheless be completely determined using

i fixed reference points, arbitrarily defined, which are reproducible and accessible to

measurement. The structural imperfections do cause motions of the apparatus which are not

completely reproducible. Thus, a major part of the geometry calibration is the determination of the size of these "random" effects. The random motions are found to be small

! enough so that the geometrical calibrations are meaningful at the desired level of accuracy.

j B. Aperture Slits

A sophisticated aperture slit system has been developed to obtain the desired solid

angle accuracy. The spatial extent of the electron beam at the target, penetration of slit

edges by the scattered electrons , and the desire to have a variable aperture in order to

optimize count rates were important factors in the design of the aperture system. Figure 4

is a photograph of the slit box showing the four tungsten cylinders which define the

aperture

.

The circular cross section of the defining elements minimizes slit edge penetration
Figure 4. Photograph of Aperture Slit Assembly.
and makes the penetration depth independent of the direction of arrival of the electrons.

In figure 5, line A-A represents an electron which is tangent to the slit edge. Line B-B

represents an electron which penetrates the slit by a perpendicular distance e, passing

through a thickness t(e) of tungsten. For small values of e, the circular cross section

results in larger values of t than any sharp-edged slit geometry. Furthermore, the value

of e needed to obtain a given value of t is independent of the direction of electron

travel (in the plane, at least), which would certainly not be the case in any slit geometry

with plane surfaces

.

Since the resolution of our spectrometer is better than 100 keV for electron energies

to 120 MeV, it is safe to assume that an electron which loses more than one MeV by

mization in the slit edges will not be counted in the elastic scattering peak. In

lgsten, this requires t (e) - 0.086 cm. The cylinders have, a radius R = 0.875 inches,

-3

so that e = 10 mm for one MeV energy loss. Multiple scattering will further reduce the

effective value of e, and therefore slit edge penetration is always negligible relative

to other uncertainties in establishing the true solid angle.

The vertical acceptance angle of the spectrometer is defined by a pair of cylinders

whose axes are horizontal and parallel to each other. The plane defined by these axes is

perpendicular to the spectrometer optic axis. The cylinders are moved independently on

precise ball-bushing guides by stepping motors, in a direction perpendicular to their

axes and in their original plane.

The horizontal acceptance angle is defined by the second pair of cylinders, whose

axes are vertical and motion horizontal. The plane defined by the axes of this pair is

accurately parallel to the plane defined by the first pair.

The construction accuracy of the slits has been carefully checked using gauge blocks

\ and a reference right circular cylinder. Because the tungsten slits are very heavy, all

measurements were made with the slit box in the same orientation as that in which it is

-4

used. Each cylinder of a pair remains parallel to the other within 10 radians throughout their entire range of motion. The two pairs of cylinders are orthogonal to each other

to similar angular accuracy, and the planes defined by the two pairs are also parallel to

similar accuracy.
B A

B A

Figure 5. Illustration of slit-edge penetration effects,

10
The opening of each slit pair is monitored by a Moire fringe counter which consists of

a precision-ruled grating attached to one cylinder and a readout head attached to the

other [6]. The readout head counts the rulings which pass it with a least count of 0.0001

inches. The accuracy is - 0.0001 inches, established by comparing the fringe counter

readout with gauge block measurements . The Moire" fringe counters measure the slit openings

to within a constant since they have arbitrary, resettable zeros. Reference positions

for each slit are established by electrically sensed limit switches whose positions correspond to apertures of approximately 0.020 inches horizontally, and 0.030 inches vertically. Reproducibility of the reference positions is established by setting the counter

readings to zero with the slits at their reference positions. The slits are opened,

adjusted a number of times, and eventually returned to the reference positions. The resulting readout is nearly always within - 0.0002 inches of zero, even after several days.

To determine the true aperture openings , the openings which correspond to the reference positions of the slits must be measured. We do this by measuring the counting rate

for elastic electron scattering as a function of slit opening, as given by the fringe

counters. The openings at the reference positions are determined by extrapolating to

the counter reading which produces zero count rate. With counting-rate loss and small

background corrections directly measured by the detector system, the extrapolation is

found to be linear provided that, for the horizontal aperture, the slits are opened

symmetrically about the reference position. A typical single determination of a reference opening to a standard deviation of 0.0003 inches can be made in about 15

minutes.

Finally, we note that an optical determination of the scattering geometry is valid

only in the absence of the stray magnetic field in the region between the target and the

aperture slits. Our aperture slits are well outside the spectrometer magnetic field.

_3

The stray field near the slits is -^ 3 x 10 of the central field in the magnet. Based

on this value and taking account of the symmetry of the field about the spectrometer midplane, we can show that the stray field effect on scattering angle and solid angle is

-4 -3

completely negligible: less than 10 on solid angle; less than 10 degrees on scattering angle.

11
C. Target Support Assembly

The target support assembly shown in figure 6 serves many functions besides simply

holding targets in the beam near the axis of rotation of the spectrometer. Ten inches of

vertical motion is available so that several targets can be placed in the beam alternately

without opening the vacuum system. Vertical oscillation with typically 1/4 inch travel,

or horizontal oscillation with 1/2 inch travel and variable speed, or both simultaneously,

is available to reduce local heating of targets by the beam. The horizontal drive

mechanism also operates the target location beam profile scanners. Rotational motion,

which is fitted with a high-precision readout, allows selection of target angles.

The target support base is bolted and grouted to the floor of the room independently

from the spectrometer. Leveling and translating adjustments are provided in order to

bring the rotation axis of the target assembly into coincidence with the spectrometer

rotation axis. The devices which make these adjustments are omitted from figure 6 for

the sake of clarity. The base, the scattering chamber baseplate, the two vertical support columns which connect them, and four guide rods constitute the stationary part of

the target assembly. A stepping motor drives a jackscrew to provide vertical motion,

and a multiturn potentiometer which serves as the sensing element of the serve system

that controls vertical target motion. This servo provides vertical positioning reproducible to better than 0.004 inches.

The movable part of the assembly rides on ball bushings which run on the guide rods.

These ball bushings are fixed to the jacking flange and the bellows flange shown in

figure 6. These two members are rigidly connected to form a stable, vertically moving

platform for the rest of the assembly. The motors, gear trains, potentiometer, limit

switches, etc., which provide the rotation and horizontal oscillate motions are omitted

from figure 6 for clarity; however, the driveshafts for these motions can be seen protruding below the bellows flange. Vacuum-tight connection between the moving and stationary parts of the target assembly is provided by a large bellows. Rotation and oscillation drives are carried into the evacuated part of the system by a pair of rotary feedthroughs

.

12
Figure 6. Target Pedestal Assembly. The components are: (1) base, (2) support column,

(3) guide rods, (4) scattering chamber baseplate, (5) jackscrew, (6) jacking

flange, (7) bellows flange, (8) bellows, (9) rotary feed-thrus, (10 rotation

driveshaft, (11) oscillation driveshaft, (12) turntable bearing, (13) turntable,

(14) linear cam, (15) idler wheel, (16) translation stage, (17) target platform,

(18) target ladder roads, (19) indicator ring.

13
A pinion gear driven by one of the rotary feedthroughs engages a ring gear fixed to

the rotating turntable. The rotational motion is guided by a high quality ball bearing.

Great precision is needed in all components of the rotation mechanism to insure that the

external readout, a ten- turn potentiometer, tracks the angle of rotation of the target.

The standard deviation of the readout is 0.08 degrees.

The second rotary feedthrough drives the horizontal translational motion through a

reduction gear pair. The driven gear is fixed to the lower end of a shaft which passes

through the rotating turntable on its axis of rotation. This shaft is guided by two

ball bearings (not shown in figure 6) mounted in the turntable. The upper end of the

shaft holds a linear cam which pushes against an idler wheel mounted to the moving part

of the translation stage. The remainder of the translation mechanism, which is omitted

from figure 6 for clarity, consists of two parallel rods fixed to the turntable, and two

ball bushings which are fixed to the translation stage and ride on each of the rods to

provide a smooth horizontal motion so that the target platform remains at a fixed angle

during horizontal oscillation. A spring is used to keep the idler wheel in contact

with the linear motion cam. Note that when the turntable rotates, the idler wheel moves

around the cam. Thus in normal use, horizontal translational motion accompanies rotation.

A purely rotational movement of the target platform is obtained by disengaging the idler

wheel from the cam and clamping the translation stage to the turntable. The horizontal

translation motion is driven by a stepping motor. Exactly 4-00 steps are needed to produce one full revolution of the cam. Two hundred steps drive the translation stage from

one extreme of its travel to the other and the next 200 steps return the stage to the

original position. Except near the turn-around points, the motion is extremely linear,

producing a displacement of (0.00283 - .00003) inches per step. The motion is also

highly reproducible, such .that upon repeated traverses through 400-step cycles, the

ladder returns to the same position within 0.0005 inches standard deviation. The

horizontal translation mechanism provides the motion needed for the target-location

beam-profile scanners described in section II D.

The upp'ermost portion of the target support assembly consists of the target platform

and target ladder rods. The rods are precision-ground 1/2- inch diameter stainless steel,

extending eleven inches above the platform. Targets mounted in holders are stacked on

14
e ladder rods. The holders, shown in figure 7, position thin foil targets within 0.001

ches of the target plane , defined to be the plane of the axes of the two ladder rods

.

The ladder rods must be coplanar, but they need not be accurately parallel within this

plane. It is highly desirable that the target plane contains the axis of rotation of the

turntable, and that this axis be vertical. More precisely stated, the axis of rotation

should be accurately parallel to the direction of "vertical" motion of the target ladder.

By means of a series of measurements and adjustments, we have been able to achieve these

conditions to a high degree of accuracy. The measurements are described in section IV.

After final adjustment, all critical parts are held in precise alignment by ground steel

dowel pins. Thus disassembly for servicing is possible without changing the alignment.

The target support assembly is not completely rigid when subjected to horizontal

forces, which occur during vacuum system pumpdown, changing of beam line plumbing (other

experiments use the same beam line), and coupling or uncoupling of the spectrometer to

the scattering chamber ports. To facilitate repositioning the target support assembly

when it does move, an indicator ring is fitted to the underside of the scattering chamber

baseplate. This ring consists of an 18-inch diameter cylindrical disc whose outer sur-

' face is, for all practical purposes, a perfect circle. The ring was initially aligned

concentrically with the target platform axis of rotation by the method illustrated in

figure 8. A dial gauge is rigidly supported from the target platform so that its plunger

points radially inward, engaging the outer surface of the indicator ring. Dial gauge

i

I

readings are taken while the target platform is rotated. The indicator ring is repositioned and the readings repeated until the dial gauge reading does not change with rotai tion. The indicator ring is then pinned in position and a final set of dial gauge readings as a function of angle of rotation are taken. We achieved a total indicator reading

(difference between largest and smallest value) of 0.0010 inches. Because the target

\ axis of rotation is not perfectly parallel to the direction of vertical target motion,

this concentricity holds exactly only when the vertical target position is at a particular

reference setting. The reference setting used for this measurement, referred to as target

position No. 5, places a standard BeO beam- viewing bullseye at beam height. Note that

these concentricity measurements can only be made when the target translation stage motion

is disengaged.

15
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of the beam. The size and shape of the hole, and dimension x depend on the dimensions of the target foil. A retaining frame (not shown) presses the target

foil against the flat surface of the blank. The frame, foil, and holder are assembled with small screws around the periphery of the foil.

16
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17
The indicator ring, first used to place the target rotation axis in coincidence with

the spectrometer rotation axis, allows the target assembly to be returned to the same

location when necessary. The method employed to measure indicator ring location is illustrated in figure 9. Four dial gauges (A, B, C, D in the figure) contact the indicating

ring radially at 90° intervals around its periphery. These gauges are supported on a

common rigid framework fastened to heavy webs which are integral members of the rotating

part of the spectrometer base. Thus, these dial gauges rotate with the spectrometer,

but do not share its radial motion. The target support assembly is placed at the center

of rotation of the spectrometer by translating the support assembly in order to minimize

the changes of readings of all four dial gauges with respect to spectrometer angle

changes. The readings taken at the final position of the target assembly as a function

of spectrometer angle are displayed in figure 10 and Table I. In this figure and table,

only the difference readings between diametrically opposed pairs of gauges are given,

since any motion of the indicating ring relative to the dial gauges produces equal magnitude, opposite sign changes in the readings of the two opposed gauges. The use of

opposed pairs reduces measurement errors caused by effects like imperfections in the

indicator ring surface, changes in dimensions of the structure due to thermal expansion,

etc. Note that if the indicator ring is moved a distance X along the D-B axis (for

example), the value of (D-B) will change by 2X. In figure 10, the values of (D-B) and

(A-C) readings are plotted. Five complete independent sets of data were taken, allowing

a determination of the variance of the measurements. Since the average values for the

data at each angle have standard deviations of 0.0011 inches, the angle dependences

shown in figure 10 are real effects. If these dependences were due to the indicator

ring not being concentric with the axis of rotation, the curves in figure 10 would be

sine waves with a period of 360°, identical amplitudes, and a 90° phase difference

between D-B and A-C readings. This is clearly not the case. The most reasonable explanation of the observed angular dependences is that they are caused by imperfections

in the main bearing which guides the spectrometer rotation. These effects do not influence our ability to determine the scattering geometry, provided that the position

of the spectrometer in space at any given scattering angle is reproducible. The

indicator ring dial gauge measurements show that the axis of rotation of the spectrometer,
Figure 9. Plan view showing the location of the indicator ring dial gauges A, B, C, and D.

19
DIFFERENCE READING, INCHES

Figure 10. Indicator ring dial gauge readings as a function of spectrometer angle.

Circles are values of D-B, triangles are values of A-C. Data are from Table I, with the average value subtracted. The standard deviation for each plotted

point is indicated by the flags on the D-B point at 110 degrees.

20
at a fixed scattering angle is reproducible to - 1.1 x 10 inches (standard deviation) in

each horizontal direction. The influence of this variation on the accuracy of scattering

angle determinations will be discussed further in section VI.

D. Beam Profile Scanners

Four profile scanners are needed to measure the spatial and angular distribution of

the electron beam incident on a scattering target. The scanners all operate on the same

principle, as illustrated in figure 11, but differ from each other in detail. A horizontal

beam profile is obtained by having a vertical sensing rod move across the beam in the

horizontal direction as shown in figure 11a. The sensing rod can be any device which produces an output signal proportional to the current, I, striking the rod. The profile is

this current as a function of the rod position, x, as shown in figure lib. The scale

factor and initial position of the rod must be known. Fine details of the beam profile

are not needed in order to measure cross sections to the desired accuracy. It is sufficient to determine the beam centroid location, x, and root-mean-square width w, which are

defined by

x = /I(x)xdx//I(x)dx (1)

and

w = C/I(x)(x - x)

2 dx//I(x)dx]ls

. (2)

Note that the current I(x) is an integral of the beam current density J(x,y) over the

transverse direction. That is

I(x) = /J(x,y)dy . (3)

It is clear from the form of eq (l) and (2) that the constant of proportionality between

the sensing signal and actual current is irrelevant. A reasonably linear response is

needed.

The upstream beam scanners are needed to determine the direction and angular size

of the beam incident on the target. Two scanning devices, for measuring horizontal and

vertical profiles, are mounted in a common chamber. Figure 12 is a cross sectional view

of the horizontal upstream scanner. The sensing element is a 1/8-inch diameter glass

rod, used as a Cerenkov detector. Light generated by the electron beam is guided along

21
SENSING ROD

BEAM

X

DIRECTION

OF MOTION

X

Figure 11. Operation of beam profile scanners

a. Cross sectional view of beam with sensing rod. _ _

b. Current striking the sensing rod as a function of its position x. x beam centroid location.

22
WINDOW

GLASS

ROD

RADIUS

ARM

AXIS

OF

ROTATION

PHOTOMULTIPLIER

ELECTRON

BEAM

ROTARY

VACUUM

SEAL

^STEPPING

MOTOR

Figure 12. Upstream beam profile scanner.

23
the rod and emitted from its end, in the direction of the photomultiplier tube. The rod

is bent so that its free end is on the axis of rotation, in order that the light intensity

seen by the phototube be independent of the position of the rod. The sensing rod is

driven by a stepping motor to produce a rotary motion with radius p. The step size is

(1/200) of a revolution. Thus, the transverse coordinate, x , of the rod is given by

X

R

= P Sin

100

°

' (4)

where N is the number of steps taken. N defines the starting position so that x = on

the axis of rotation. The stepping motor is driven in synchronism with the linac pulse

repetition frequency, one step per beam pulse. The photomultiplier output signal due to

each beam pulse is integrated, digitized, and stored in the on-line computer as a function

of the step number N. After one revolution, the computer gates off the pulse train which

drives the stepping motor, and analyzes the spectrum obtained. This spectrum has two

peaks; one obtained when the sensing rod passed through the beam moving toward the right,

the other when the rod was moving toward the left. Both peaks are needed in the analysis

because the starting position, N = (50 - 5) steps, is not accurately known. Computer

analysis of the two-peaked spectrum eliminates the dependence on N , and yields x and w

employing eq(l) and (2) (converted to discrete sums instead of integrals). A single

measurement of this type locates the beam centroid with a standard deviation of about

+ -4

0.4 steps, corresponding to an uncertainty in beam direction of - 0.9 x 10 radians

(- 0.005 degrees).

The current sensing element of the target location beam scanner is a thin wire

operating as a secondary emission monitor. The wire is mounted in a target holder frame as

shown in figure 13. A flexible lead connects the sensing wire to a feedthrough mounted

in one of the scattering chamber ports. The signal is amplified by a charge-sensitive

amplifier and is read into the on-line computer system through an analog-to-digital converter, as described above for the upstream scanners. The same readout system is used

for both horizontal and vertical profile measurements, the only difference being that the

target ladder height is changed to put the appropriate sensing wire at beam elevation.

Both horizontal and vertical profile measurements are obtained by horizontal oscillation

of the target holder. As in the case of the upstream scanners, motion is produced by a

24
u

I

Figure 13. Target location beam scanners.

25
stepping motor driven synchronously with the beam repetition frequency, the only differences being that in this case the motion is linear rather than sinusoidal and 400 steps

are required to complete one cycle of motion. The position scans are analyzed by the

computer in the manner previously described, again using the left-going and right-going

portions of the scan to eliminate the need for accurate knowledge of the starting position.

The position of the horizontal scanning wire in the direction transverse to the beam

axis is

x = [x + (N - N )A] cos 6_ - z sin 0_, (5)

B ° o 1 o -1

where is the target angle,

A is the step size (.00283 - .00003 in.),

and x and z are the coordinates of the wire in the holder,

o o Here we use N - 100 steps. Since x and z are not accurately known, a priori,

o

r o o

J —

they must be determined by calibration. The calibration procedure consists of measuring

a beam centroid position at several values of 9 . A typical set of measurements consisting of four scans at each of three angles, 9„,, will determine x , x , and z to

r d o o within standard deviations of .0016, .008, and .0025 inches, respectively. After

calibration, subsequent determinations of x can be made without changing 9 . Note that

B T this procedure determines the transverse beam centroid position with respect to the

axis of rotation of the target assembly.

Vertical beam profile measurements use the inclined sensing wire shown in figure 13.

The height, h, at which this wire intercepts the horizontal centroid of the beam (x ) is

is given by

h + S o

X

R'-' + z tan 9 m - (N - N )A cos 9 m v T o' (6)

since this sensing wire lies in a plane which is parallel to the plane defined by the

centers of the target ladder rods , and displaced from it by a distance z . The symbols

in eq (6) not previously defined are S (= 0.25), the slope of the inclined wire to the

horizontal direction and the constant h . The value of z can be determined by

o v

J measurements at different settings of 9 , as in the case of the horizontal scanner, but

there is no analogous "internal" method for determining h . Fortunately, the beam height

26
can be adequately determined by using the bullseye beam viewer (estimated accurate to <0.02

inches. )The vertical beam scanner is used only to measure changes in beam height occurring

during data taking, and the beam size. The standard deviation of a single measurement of

relative beam height is typically 0.002 inches.

III. System Geometry

In this section we will define the quantities which specify the scattering geometry.

We employ a coordinate system whose X and Z axes, and origin "0," are shown in figure 14.

The Y axis of this coordinate system points out of the plane of the figure, and iyertically

upward in space. This coordinate system rotates with the spectrometer, since its Z axis

bisects the line joining the centers of the cylinders which define the horizontal spectrometer aperture (at point Q).

The origin point (0 in figure 14) is chosen to be a fixed point in space, although

neither the target support assembly nor the spectrometer axis is rigidly fixed in space.

The point is established by finding a point on the axis of rotation of the target

turntable, at nominal beam height, with the target support assembly in its reference

position. The reference position is obtained at the vertical target setting #5 defined

previously, and with the indicator ring dial gauges at the values listed in Table I. During the calibration, the location of point in the horizontal plane was established to

_3

well under 10 inches. However, since the target assembly was moved, the original position can be reproduced only to within - 0.0011 inches in each horizontal direction as

already discussed in section II

.

We next define a baseline, shown in figure 14 by line 0B, along the incident beam

direction. The second fixed point needed to define the baseline is the geometrical

center of the exit flange of the vacuum pipe fixed in the quadrupole magnet shown in

figure 1. This point is only needed to define the baseline direction in the X-Z plane,

since this plane is horizontal by assumption.

Once the baseline has been established, the calibration of the scattering angle

scale consists of determining the angle, 9 , between the baseline and the Z axis. We

| measure at nine reproducible positions, established by fiducial blocks permanently

mounted on the fixed base of the spectrometer. One of the fiducial positions corresponds

27
Figure 14. Coordinate system used in this report, in plan view. The shaded circles represent the horizontal aperture slits. The shaded rectangle represents a target

oriented in transmission geometry. The line B indicated the beam direction.

28
to 6 = 180°; that is, the point Q in figure 14 is (very nearly) on the baseline OB. The

other eight fiducial positions correspond to scattering angles at which the spectrometer

aperture is aligned with the scattering chamber ports. The angular position of the ro-

-4

tating spectrometer base can be set at any of the fiducial positions, to within 3 x 10

degrees, by observing the reading of a dial gauge mounted on the rotating base, as shown

in figure 15. Scattering angles differing from a fiducial position by <0 . 4° are determined with the aid of the reading of this azimuthal dial gauge. In addition to depending

on the azimuthal dial gauge reading, the scattering angle is a function of the spectrometer's radial position because of the transverse motion accompanying radial motion, as

described in section II and displayed in figure 3.

In order to locate the spectrometer aperture in the horizontal plane, the distance

OQ, the aperture, W, and the angle, y , must be known. The distance OQ, hereafter called

L , is a function of the spectrometer's radial position and depends weakly on because

the spectrometer motion about is not a pure rotation, y , defined as the angle between

the line joining the centers of the aperture cylinders and the Z axis, is positive in the

direction shown in figure 14. W, measured in the direction joining the centers of the

aperture cylinders, is perpendicular to the Z axis only if y =0. As described in

section IIB, W = is defined from an extrapolation to zero count rate which, in

general, does not exactly coincide with the point at which the cylinders make physical

contact.

The location of the vertical spectrometer aperture shown in figure 16 is defined by

the distance L , opening H, and angle y , analogous to L , W, y for the horizontal case, y y xx

Additionally, the vertical distance y must be known.

The point at which the incident electron beam strikes the target is measured by the

profile scanners relative to the fixed point P and angle , indicated in figure 14. P

is the intersection of the axis of rotation of the target pedestal and the X-Z plane.

The angle, 9 , between the baseline direction B and the normal, n, to the target plane,

is positive in the sense shown in figure 14.

and P, indicated in figure 14 to be different points, were coincident when the

geometry calibration was performed. P is slightly displaced from if the vertical target position is different from its reference position or if the indicator ring, and

29
Figure 15. Azimuthial dial gauge mounted on the rotating base of the spectrometer and

indicating on one of the fiducial angle blocks. The dial gauge and its

mounting block are removed from the rotating base to allow rotation of the

spectrometer. Alignment pins insure accurate repositioning.

30
4

Figure 16. Coordinate system used in this report. In this view the X-axis is perpendicular to the plane of the figure. The circles at the left side of the figure

represent the vertical aperture slits.

31
therefore the target assembly, is not at the position defined by the values of D-B and A-C

in Table I.

In section II, we described the measurement of beam position relative to the axis of

rotation by means of the beam profile scanners. Note that x , defined by eq (5) is measured perpendicular to the baseline OB, which is not in the X direction of the coordinate

system defined by figure 14 unless 6 =0. The target point coordinates x and z are

given by

X

B *r

= ^Tl^ cos (9 o

" V + n sin (9 o" V + V

and (7) X

B

zt

=

" ^Te^ sin (e o- V + n cos (6

o

" V + zp

•

In eq (7), x^ and z are the coordinates of point P, and r| is the displacement of the

target point from the target plane (which passes through P). The positive direction of

n is in the direction of the normal n. For target holders constructed as shown in

figure 7, the maximum value of n is the target-foil thickness.

The vertical distance, h, of the target point from the origin, 0, can normally be

neglected since |h| % 0.02 in., but the vertical displacement of the slit assembly, y ,

cannot

The last quantity needed to define the scattering geometry is the actual direction of

the incident electron beam. The projection into the horizontal plane of the angle between

the baseline and the actual beam line is shown as a in figure 17. In this figure, the

A points and P, and the axes labeled B and Z are the same as in figure 14, and T is the

point where the beam intersects the target plane. In terms of measurable quantities,

x

B

+ x'

p

- (xR

+ x

c

) a

X

~ B„

"" (8)

5

'c Some target holders have been prepared with a recess which limits n to - (one-half

of the target foil thickness), and makes the average value of n zero.

32
o

QC

x

-

or

'iJ z.<

Q_

ILdO

tr

<

UJ

CD

x

CD

§ S

UJ

OUJ^

<CO_l

Figure 17. Definition of quantities measured relative to the baseline. Plan view. The

electron beam goes from right to left in this figure along the line marked "B"

33
where x is the transverse coordinate of the axis of the horizontal upstream beam scanner,

x' is the projection, in the horizontal direction perpendicular to the baseline, of

the distance OP , and

B is the distance between the axis of the horizontal upstream beam scanner and the

X origin point 0.

The small angle approximation used here is adequate since | ot

| £ 10 radians and the

A projection on the baseline of the distance OT is negligible ('vO.l in. compared to B ^

A 100 in.). All quantities on the righthand side of eq (8) are positive in the directions

indicated in figure 17, except for x' , which is negative as shown. For completeness, we

also define the angle, a of the beam line projected into the vertical plane, although its

effect on a cross section determination is extremely small. The appropriate analog of

eq (8) is

h - (y R

a

Y B^

+ V ' m(9)

where y , y , and B are exact analogs of x , x , and B , respectively. The distances h, R C Y K L X

y , y are measured from the horizontal plane passing through 0, with the positive direcR C tion upward.

With the geometry fully defined, we can now explicitly state how the effective solid

angle Q

e«> and the effective scattering angle, 0, are determined. An electron incident

on the target along the direction defined by a and av scatters from a target nucleus loA I cated at (x , h, z ) through polar angle 9, at an azimuthal angle <f>. The solid-angle

integral of eq (l) is to be performed over the range of (6 ,4>) allowed by the positions of

the aperture slits. The integration limits are thus functions of x , h, z , a v , a , 1 1 X Y

6 , W, H, L.,, L , y„, Y , y„, and R„ .* Of these, all but the first five are constants

o' X' y ' X y

J A S for any one cross section measurement. However, x , h, z , a , and a are distributed

i i x y variables because of the angular and spatial spread of the beam as well as the range of

scattering centers in the target, represented by the variable n • We can write

* There is also a dependence on the angle of multiple scattering in the target which

will not be discussed here except, to note that this effect can be accounted for by modifying a and a appropriately

34
/a(E,6)dfi = a(E,0)-fi

aperture eff ' (10)

where the integration limits depend on all of the geometrical parameters, including appropriate averaging over the values of distributed variables. To obtain a Taylor's series

expansion of the cross section, we define

1

6 o(E,6 ) Sn

( V

= n!o(E,8 ) 6(cos 0) n

(11)

Then

tt __ = Ys (9-,) f ( cose " cos 6 n ) n

dfi eff £_, n 1 1 aperture

n=o

We have written a computer subroutine which computes

(12)

p (ej = ; (cos e - cos e n ) dn , n 1 1 aperture

(13)

for as many values of n as needed, for any value of , when the 14 geometrical parameters''

listed above are supplied as input. Averaging over the distributed variables is performed

by Monte-Carlo methods. A convenient definition of F is that value of which makes P

identically zero. For reasons of numerical accuracy, we obtain from the relation

cos = cos

o

+ wP (14)

The experimental geometry also enters into the determination of W , the number of target

nuclei per unit area normal to the beam direction. For a uniform-thickness plane target

j only , a , and a are needed to determine W . The dependence on ex is negligible, and

1 a y o y the dependence on a is always slight.

" The number of input parameters is reduced to 13 since h and y enter only in the

combination h - y = y . A 1

35
IV. Target Assembly

In this section we begin a description of the alignment and calibration of the apparatus, the order being approximately chronological so the dependence of each step on the

preceding ones can be seen. The first step is to align and calibrate the target support

assembly so that its axis of rotation can be utilized as the geometric origin.

The first step in aligning the target pedestal is to establish the relative orientation of the target plane, axis of rotation, and the directions of vertical and horizontal

motion of the target platform. The measurements about to be described had to be repeated

several times, adjusting the orientation of the components each time, until the alignment

was judged to be satisfactory. Only the final set of measurements will be described.

Figure 18 shows how the measurements are performed. A target holder blank mounted on the

ladder rods provides a solid surface in the target plane, R . (By definition, the target

plane is the plane containing the axes of the ladder rods.) A dial gauge, mounted at

nominal beam height, has its plunger in contact with this surface. The line through

labeled "A" is the axis of rotation of the target turntable. The angle between R and A

is defined to be 3. Note that R does not necessarily pass through 0; in fact, the horizontal distance , p, between and R is the quantity measured by the dial gauge. Rotating

the turntable through 180°, (i.e., 6 -> 6 + 180°), moves the target plane to the position

indicated by the line R , still at angle g with respect to A, but in the opposite direction. In order to measure p in this orientation, the target blank must be reversed on

the rods. The direction of vertical motion of the target platform is indicated by the

line "Y" in figure 18. The projection of the angle between Y and A into the plane of the

figure is 6 . Lowering the target pedestal a distance V from the reference position causes

the ladder plane to lie along the line R , which is parallel to R . In order to remain

at the same height in the room the target blank must be moved up the ladder rods by the

same amount (V), thus moving the blank toward the dial gauge and increasing p. The dial

gauge can be mounted at any azimuthal angle 6 . Final data were taken at 6 = 180° and

270°. For each of these values of 8 , we measured p at seven values of V over a range

o of about 7 inches, both with mTo= 6 and with 6,= 6 + 180°. A least-squares fit to

I o the data with the equation

36
R2Y A R, R3

Figure 18. Target ladder, illustrating the method for measuring the alignment of the

ladder with a dial gauge. A is the axis of rotation of the turntable. R-^ is

the plane of the target ladder rods. A 180° rotation of the turntable puts

this plane in the direction indicated by Ro. Y is the direction of "vertical"

motion of the target pedestal. A vertical motion through a distance V as indicated translates the plane of the target ladder rods from Ri to Ro. Determination of the angles [3 and ', is discussed in the text.

37
p = (p + gV) cos (G_ - 8 ) + 6(9 )V (15)

o To o

yields p , 8, 6, and their standard deviations. The results are given in Table VI.

The misalignments and uncertainties represented by p , 8 , and 6 affect the determination of the scattering geometry by way of n , x , z and x' . Aside from finite target

thickness effects, n is determined entirely by p; that is

n = p + BV + n', (16)

o

where n'is measured from the target plane. We retain p in eq (16) even though its value

is zero, for purposes of error propagation. The effects of imperfect target pedestal

alignment (p f 0) on the scattering angle and solid angle are always very small, less than

-2 4 . 10 degrees and 3 parts in 10 , respectively, for V up to ten inches.

Since the targets oscillate horizontally, the measurements described above were

checked while the target was oscillated. Changes in dial gauge readings were less than

-4

0.2 x 10 inches from the average value, and the effect is negligible.

Horizontal target motion was calibrated with a dial gauge whose plunger is aligned

with the oscillation direction. The total travel is 0.461 inches. Through the central

0.3 inches of this range, the motion is a linear function of the number of pulses provided by the stepping motor drive. The average slope is A = (2.83 - 0.03) x 10 inches/

step, and is not measurably different for the two directions of motion. Individual position measurements scatter about the best fit straight line with a standard deviation of

_3 about 0.5 x 10 inches.

The next step in the target assembly alignment is to measure the location of the

assembly relative to the spectrometer. The procedure is described in section II and the

results presented in figure 10 and Table I. By definition, the axis of rotation of the

target pedestal passed through the origin when the calibration was performed, except for

the effect of 6(9 )V in eq (15). In section II, we discussed the lack of rigidity of the

target support assembly, and the necessity of returning the assembly to its original position, represented by the values of D-B and A-C in Table I. In practice it is difficult to

return the pedestal to exactly the desired point. Instead, we correct the values of x ,

z , and z' for the differences between the pedestal dial gauge readings and their

38
original values. Since the indicator ring is 9 1/4- inches below nominal beam height, and

the motion involved is not a pure translation of the pedestal, we measured the tilting

motion that accompanies horizontal motion by autocollimation techniques employing a mirror

mounted in the target ladder. These measurements show that the target axis at beam height

moves S = 1.156 - .005 times as far as the indicator ring for all small horizontal mov tions , independently of the direction of motion. Combining this effect with the effect

represented by eq (15), we obtain

x = V[cos Q 6(270°) r- sin 9 6(180°)] - (S /2 ) [(A - C) - (A - C) 1, (17) p o o v F

z = -V[sin 6 6(270°) + cos 6(180°)] + (S /2 ) C(D - B) - (D - B)], (18)

p o o v F

and

p ~v' ' [}'" ~'F J

~ ~o x'_ = V 6(270°) - (S„/2)|[(A - C) - (A - C)^] cos

+ [(D - B) - (D - B)_] sin 6 \, (19)

F o )

where (D - B) and (A - C) are the differences in readings of the appropriate dial gauge

pairs. The unsubscripted values refer to any given experimental setup, and the values with

subscript F are the fiducial values listed in Table I.

In addition to the translational motion, twisting of the target pedestal is also observed. The main source of twisting is a torque accidentally applied to the scattering

chamber when it is connected to the spectrometer flange. Reasonable care in making this

-2

connection will cause errors estimated to be less than 10 degrees in target angle, less

4 than 2 parts in 10 in sclid angle, and no effect on the scattering angle.

V. Realization of the Baseline

The baseline is defined to be the horizontal line through the origin point and the

center of the quadrupole exit flange, P, as shown in figure 19. A theodolite placed on this

line is used to locate the axes of the upstream beam saanners and the spectrometer horizontal aperture slit center relative to the baseline. The aperture center is located only

for a reference spectrometer angle, 6 % 180°. Subsequently, all other values of G are

calibrated relative to the reference angle.

A reference point at the origin (i.e., on the axis of rotation) is realized by a

39
Figure 19. Method of surveying the baseline with a theodolite located at T. The points

and P define the baseline, and Q represents any other object to be located

relative to the baseline which is indicated by the line labelled "B". 40
fixture in the shape of a right circular cylinder surmounted by a cone, machined so that

the apex of the cone is accurately on the axis of the cylinder, with the axis

perpendicular to the base of the cone. The fixture is placed on the target turntable,

near its center, with the ladder rods removed and the horizontal oscillation cam disengaged. A dial gauge is mounted to the support assembly, with its plunger contacting the

cylindrical surface of the fixture. The turntable is rotated and variations in the dial

gauge readings noted. The position of the fixture on the turntable is adjusted until the

dial gauge reading is invariant under rotation. The fixture and thus the cone apex is

then accurately on the axis of rotation. A second dial gauge, mounted at right angles to

the first, is used simultaneously for improved speed and accuracy. The fixture can be

centered well enough so that neither dial gauge deviates from its average reading by more

-4

than 2 x 10 inches during a full 360° rotation. When properly positioned, the cone apex

appears to be stationary when it is sighted from the theodolite 18 feet away and the target pedestal is rotated.

The sighting marker at the quadrupole exit flange is a tightly fitting lucite disk,

embedded with fine wire cross hairs passing through its center. When the disk is rotated

in the flange, or removed and repositioned, no motion of the cross hair center can be

seen from the theodolite.

The theodolite is placed within .02 inches of the baseline, and its position measured by triangulation, as indicated in figure 19. Distances along the baseline are measured with a steel tape ruler to an accuracy of 0.1 inch , resulting in essentially nc

error in locating the components relative to the baseline. A set of measurements of angles

( d> ,

ct> , <(>_ ) and distances (D , D , D^) as indicated in figure 19 yield the transverse

, o p Q o p Q

offset, y , of any component, Q. Since the angular difference between any pair of theodolite settings is typically 10 arc seconds for these measurements, the limiting factor

in locating the components along the baseline is the visibility of the components and

reference markers rather than the quality of the instrument. Systematic effects are reduced by illuminating all viewed objects symmetrically and approaching the sight line

from opposite directions . The standard deviation of individual angle measurements is

about 2 arc seconds (10 5 radians).

The 180° reference position of the spectrometer is established as indicated above.

41
The displacement, y , of the center of the horizontal aperture slits, with the slits

closed to the limit positions, is determined by the method described. The distance, L , A from the origin to the aperture slit center, together with y , determine the offset of

the spectrometer angle from 180°. In addition to L and y , the spectrometer angle is a

X Q

function of the radial dial gauge reading, R, the azimuthal dial gauge reading, V , and of

D-B and A-C. The corrections for R, V , and the target pedestal dial gauges are discussed

in section VI. It should be noted that the correction for the target pedestal dial

gauges is different from the one described by eq (17) - (19). That correction accounts

for motion of the target pedestal; the correction needed here is for the undesired motions

in the spectrometer support assembly to be discussed in section VI. During the measurement process described here the target pedestal was not disturbed.

The 180° reference position was measured independently three times. Averaging the

three sets of data gives the fiducial angle 6 (180), and an estimate of the measurement

error, which are given in Table II.

The center of rotation of the upstream horizontal beam scanner, x , is also measured

relative to the baseline by triangulation. The theodolite is sighted on the lucite rod

of the scanner, with the rod near its center of travel, i.e. x„ ^ 0, by setting N ^ N . R — J & — o A second sighting is made after rotating the scanner by exactly 100 steps. Since the

average of these two values of x is zero, the average of the two sighting directions

passes through the center of rotation.

Vertical measurements are made in a manner similar to the horizontal ones , except

that only one fixed point is needed since the baseline is defined to be horizontal in

space. The leveling bubble of the theodolite determines the horizontal direction. Forward and reverse transit readings are taken in order to minimize theodolite errors. The

heights measured are the standard Bullseye center, the center of the vertical aperture

slits, and the axis of rotation of the upstream vertical beam scanner. Defining the bullseye height as y = 0, the latter two heights are y and y , respectively.

A C Table II summarizes the results of all measurements described in this section.

Errors listed for 6 x , y and y are standard deviations of the means, based on the

± L. A C variance of the individual measurements. Values of B anc

X eq (8) and (9), respectively, are also given in the table,

variance of the individual measurements. Values of B„ and B , which are defined under

X y

42
VI . Scattering Angle

Having established the fiducial angle 6 (180), all other spectrometer angles can be

measured relative to (180) by means of a theodolite located on the axis of rotation.

For each nominal scattering angle, a theodolite measurement of the angular difference

from 8 (180) is made. In all cases the object sighted is the center of the horizontal

aperture slits, with the slits set at their limit positions. With each angle measurement, a set of dial gauge readings is taken, since the spectrometer angle is strongly correlated with the azimuthal gauge reading, V, and weakly correlated with the radial reading,

R. Auxiliary measurements made by changing V and R establish these dependences. All

Spectrometer angle measurements can thus be reduced to a set of fiducial angles 8_ corresponding to arbitrary fiducial values of V and R.

In the course of these measurements, "random" motions of the spectrometer relative to

a fixed coordinate system were discovered. The target pedestal dial gauge readings were

found to be correlated with the random motions , and were used to perform some corrections

to the 9 calibrations. The errors in this correction procedure can be estimated with

sufficient accuracy to establish that the calibration concept is valid. Thus, 9 can be

o obtained from a set of measured quantities to the desired accuracy of < 0.01°, without

having to resurvey the apparatus each time the scattering angle is changed.

A. Theodolite Alignment

In order to use a theodolite to measure the scattering angles, it is essential to locate the theodolite at the chosen origin, that is, on the axis of rotation. Since the

sighting distance is only about 24- inches, the accuracy required is about 5 x 10 inches

-3 to limit the angular error to 10 degrees. This accuracy was achieved with the use of

auxiliary dial gauges by the method illustrated in figure 20. The theodolite is set up

as close as possible to the axis of rotation of the target pedestal and sighted on a convenient marker (seme 15 ft. distant). Two dial gauges are mounted from the target pedestal

baseplate with their plungers contacting the body of the theodolite. Next the target

pedestal is rotated through approximately 90° , while the theodolite is rotated through

exactly the same angle in the opposite direction by continuing to sight the original

43
DIAL GAUGES

THEODOLITE

l/W

TARGET

TURNTABLE

Figure 20. Setup for aligning the theodolite on the axis of rotation.

44
marker. Any changes in dial gauge readings are noted. The procedure is repeated until a

full rotation has been made. Both dial gauge readings will remain constant under this

procedure if and only if the axes of rotation of the target pedestal and of the theodolite

coincide. In addition 'to the usual leveling screws for tilt adjustment, the theodolite is

equipped with a screw-operated translation stage so that it can be adjusted in successive

approximations until the coincidence of axes is obtained. After final adjustment, the dial

gauge at the elevation of the theodolite eyepiece (i.e., nominal beam height) does not

-4

deviate more than 1.5 x 10 inches from its average reading at any point in the rotation.

-4

The attendant residual error in scattering angle is less than 4 x 10 degrees. The

second dial gauge, located about 5 inches below nominal beam height deviates by no more than

-4 . 7 x 10 inches from its mean reading. Thus the axis of rotation and the theodolite ver-

-4 . . tical axis are parallel to within 1.4 x 10 radians and the resulting error in scattering

angle is negligible since all sightings are made in the horizontal plane.

If the optic axis of the theodolite does not pass through its axis of rotation, no

error in scattering angle would result, provided that the sighting distance remained

constant as the spectrometer is rotated. However, the pivot-to-slit distance, L , does

change slightly with angle. Since the maximum variation in L is less than 0.05 inches,

-5

the angular error due to the location of the optic axis is less than 8 x 10 radians

per inch offset. The offset in the instrument is thought to be less than 0.001 inches, but

in order to use the theodolite at 24-inch sighting distance, we added a close-up lens in

front of the theodolite objective, which could shift the optic axis. By sighting the same

object with and without the close-up lens (this is possible at one sighting distance --

about six feet), we measured the offset to be certainly less than 0.05 inches, which gives

a negligible error in the scattering angle.

The physical height of the theodolite required that the target pedestal be lowered by

4 3/8 inches from the reference target position to place the theodolite eyepiece at

nominal beam height. Because of the misalignment of the axis of rotation relative to the

direction of vertical pedestal motion discussed in section IV, the theodolite was not at

the defined origin point 0, but at a point analogous to P in figure 14. For each value of

measured with the theodolite we must correct the observation for this offset. Using

o

to

eq (17) and (18), with the terms dependent on A, B, C, and D set to zero, the correction is

45
X

q _ a — P measured o L - z

x p

Since the z term has a completely negligible effect, we apply the correction in the form

9=6 . - -^- [cos G 6(270) - sin 6 6(180)] . (20)

o measurt L o o

x

This correction, which has been applied in obtaining the values of 6 listed in Table I,

-3 never exceeds 5x10 degrees . The error in this correction due to uncertainties in

6(270) and 6(180) is a systematic function of angle. The error in the angular difference

between 6 and the reference angle (180°) is

E(6 - 180°) = -I [(i + cos ) 2e

2 (270) + sin2

9 e 2 (180)]

1/2

, (21)

o L o o X

where e(9) is the error in 6(9).

The reproducibility of the angle measurements is deduced by comparing sets of angle

readings taken without moving the spectrometer. Averaging over all such sets of data, the

-Li- standard deviation of a single measurement is about 2 arcseconds (6 x 10 degrees). This

reproducibility takes accou. t of theodolite motion between readings , our ability to center

the theodolite sighting at the slit center, and the readability of the theodolite scales

(the least count is one arcsecond)

B . Azimuthal Dial Gauge Calibration

The azimuthal dial gauge measures we difference between the spectrometer angle

and a fiducial angle 9 . Our definition of 9 includes the requirement that the azimuthal

gauge reading is V = 0.5000 inches, except at 9 (180) where V = 0.1+902 inches. To determine the dependence of 9 on V, we set V to 0.9000 inches and to 0.1000 inches by rotating

the spectrometer and measuring the angular motion of the spectrometer aperture slits with

the theodolite mounted at the axis of rotation. The result for the average of two independent determinations is

^ = -1.2566 - 0.0004 degrees/inch , (22)

46
Aft from which we can compute 6 from 9 with an error in 9 due to the uncertainty in —~ plus

higher order geometrical effects (see figure 15 for the actual geometry) of less than

2.5 x 10 degrees for any value of V between 0.0 and 1.0 inch.

C. Radial Dial Gauge Correction

In connecting the spectrometer vacuum chamber to a scattering port, the spectrometer

is moved radially so that the target pedestal dial gauges remain at their correct readings;

thus, the radial position of the spectrometer is not a free variable. In fact, when the

spectrometer angle calibration was made we did not know the required radial position to

better than - 0.03 inches. Because transverse motion accompanies radial motion we need to

know the change in 9 caused by changing the radial position, as measured by R, in order

to compute 9 . The calibration is made by measuring the change in the angle to the aperture slits, as seen by the theodolite at the axis of rotation, as the spectrometer is

moved radially inward (i.e., increasing R). The direction is specified because there

appears to be a small hysteresis between inward and outward travel. In connecting the

vacuum system, the spectrometer is normally moved inward to its final position. A set of

such measurements taken with the spectrometer near the 180° position gave A9/AR = 0.0172

+ desrees

- 0.0002 —rS-r , for 0.3 <R <0.8 inches. No deviation from linearity was observab.

e

inch

in this data set.. The dependence of A9/AR on spectrometer position was checked at a

nominal scattering angle of 93° with the result that A9/AR = 0.035 - 0.003 degrees/inch,

in poor agreement with the 180° result. Due to an error in data reduction, this discrepancy was not noticed in time to study the effect further. We therefore adopt the

careful measurement as our best value of the effect, ignore any dependence on spectrometer angle, and adopt the difference between the two values as a measure of the error:

£L= (0.017 t 0.012)

^^ , (23) AR inch

which is the solid line in figure 3. Fiducial angles are defined to be the values of 9

that occur when R = 0.700 inches and V is at its fiducial value, V . The spectrometer

angle measurements were made with R between 0.689 and 0.700 inches; thus, the largest

+ -4

correction applied for the A9/AR effect was (5.2 - 1.3) x 10 degrees. Subsequently,

we have found that the spectrometer always connects to the scattering chamber with R in

47
+ -3

the range 0.67 to 0.73 inches. Thus, the radial correction never exceeds - 1,1 x 10

"

degrees, and the uncertainty in scattering angle due to the estimated error in A6/AR

+ -4 never exceeds - 4 x 10 degrees.

D. Fiducial Angle Measurements

The chronology of the fiducial scattering angle calibration was:

1. The baseline was surveyed.

2. The spectrometer was moved to 6 (180) and surveyed relative to the baseline.

3. The theodolite was aligned on the axis of rotation.

4. The theodolite angle to the horizontal aperture slits was measured.

5. The spectrometer was rotated successively to the other nominal angles,

168° , 145° 58° , 40° , and the theodolite angle to the aperture

slits measured at least twice at each angle.

6. The spectrometer was moved to about 38°, then back to 40°.

7. The theodolite angle to the slits was remeasured at 40°, and successively

at the other nominal angles, ending with a second measurement at 9 (180).

8. The theodolite was removed from the target pedestal and placed on the

baseline

9. The baseline was resurveyed including the aperture slit location needed

to determine 6 (180).

The two full sets of measurements of angular differences between each fiducial angle

and 6 (180), were found to be systematically disparate, indicating a correlation with

the direction of rotation of the spectrometer. With each theodolite angle measurement

we have a full set of dial gauge readings. Most of the data was taken with V - 0.5000

inches , although a few points were taken at other values to calibrate the azimuthal dial

gauge. The radial spectrometer position was not adjusted, but the radial gauge reading, R,

was recorded each time. A range of variation of 0.011 inches was observed, presumably due

to slight non-rigidity of the spectrometer support system coupled, perhaps, with the main

bearing not being absolutely horizontal. (The bearing could be out of horizontal by

5 x 10 radians. Since the total weight is about 5 x 10^ kg, there could be a 25 kg

force in the plane of the bearing.) Fortunately, we also recorded the target pedestal

48
dial gauge readings (A, B, C, D) with each angle measurement, which helped greatly in understanding the systematic differences in the measurements.

E. Target Pedestal Dial Gauge Effect

_3 A systematic difference, averaging 3.2 x 10 degrees, was detected between the angle

measurements made while rotating the spectrometer toward smaller scattering angle and

those made while rotating in the reverse direction. The anticipated sources of errors,

which have been discussed above, are too small to explain the observed effect. (The corrections which are systematic functions of scattering angle have no effect, of course.)

The angular differences between forward and reverse measurements are strongly correlated

with differences in D-B values taken at the same time, as shown in figure 21. The variation in spectrometer angle with D-B reading is

a ,|^ = 2.52 - 0.44 degrees/inch. (24)

The quoted error is the standard deviation obtained from a least squares fit to the data

points. There is no statistically significant variation of with (A-C). If we assume

r} ft 4- (^ ft + we find , . = 1.5 - 2.2 and , .. = 2.8 - 0.6 degrees per inch, but the quality of

a

\

A-C) o

(

D-B

)

fit is somewhat poorer than for the case where a dependence on D-B only is assumed. We

therefore adopt the simpler hypothesis that the deviation is a function of D-B only.

It is easy to show that the observed effect is not due to real motion of the target

pedestal alone. If the target pedestal moved, the change in scattering angle could be computed from the dial gauge reading changes, knowing the target-to-aperture distance. This

J\ A JN A effect would be „, . n

. = -1.2 degrees per inch, , , = (to first order), in disagreeo {. A-C ) a \ D-d )

'• ment with the observed variation.

We conclude that the spectrometer angle is not a unique function of V and R as expected.

The main spectrometer bearing could have a looseness which permits motion of the rotating

part of the bearing relative to the stationary par/t . If (A-C) and V are constant, this is

a rotation about the point of contact of the V dial gauge. However, if this were a rigid

49
\°\ o

T \

\ °

\ °

o

o

oo O

CD

O O

CM

S33U93Q \ 01 *0V

o

CM

I

O

CM

o cn - Ld

X

O

GO

i Q

<

O

l

Figure 21. Differences of pairs of scattering angle measurements plotted against the

difference of the corresponding (D-B) measurements. The solid line is the

best fit to the data 36 8(D-B)

2.52 - 0.44 degrees per inch.

50
body rotation, the effect on the sighting angle to the slits would be only 0.6 degrees per

inch change in (D-B).

To handle this problem, a variation of the actual scattering angle

which correlates with D-B according to eq (24) is assumed and the calibration data is corrected for this effect. Once the calibration was completed, real motion of the target

pedestal was permitted and we can no longer distinguish between real pedestal motion and

the apparent motion. Therefore, each time a new scattering angle is set up there is an

uncertainty in the target location equal to the standard deviation of the target pedestal

dial gauge readings observed during calibration and an uncertainty in scattering angle,

measured from the defined origin, 0, equal to the variation in the pedestal gauge readings

multiplied by the correlation factor of 2.52 degrees/inch. During the calibration, five

independent sets of pedestal dial gauge readings were taken: two with scattering angle

data, two with the solid angle data, and one with the target angle data. The average

values of (D-B) and (A-C) are listed in Table I and plotted in figure 10. The standard

_3 deviation of the mean at each nominal angle is 1.1 x 10 inches, and the standard deviation

-3 -3

of a single value is 2.2 x 10 inches. Thus, there is an uncertainty of 1.1 x 10 inches

each in target coordinates x,^ and z . There is an additional uncertainty in scattering

-3 -3

angle, 6 , of (2.52 deg./in) x (2.2 x 10 in) = 5.5 x 10 degrees. Since x is in the

direction measured by (A-C) and only (D-B) motion changes 6 , these two uncertainties can

be treated as independent. The uncertainty in 6 is correlated with the change in solid

angle due to change in target-to-aperture distance. The solid angle effect, 1 part in

4 4 10 , is much smaller than the effect of 3 to 6 parts in 10 on the calculated cross

section.

F. Fiducial Angle Results

The angles , , measured by the theodolite at the axis of rotation are converted to m

fiducial angles, , by combining the corrections expressed by eq (20), and (22) through

(24) to obtain

0=0 -~ [cos 6(270) - sin 6(180)] + 1.2566 (P-tL)

f m L o o F x

- 0.047 (R-R ) - 2.52 [(D-B) - (D-B) ] + G , (25)

51
where the constant G is needed because the theodolite has an arbitrary zero angle direction.

G is obtained from the 180° measurements, since in that case 6 (180) is known from the

base line calibration. Unsubscripted variables V, R, (D-B), are observed values. The corresponding variables with subscript F are the fiducial values.

The values of 8 given in Table I are the averages of two determinations, except at

180° and 14-5° where three determinations are used. The standard deviation of a single

_3

measurement is 1.5 x 10 degrees. Therefore, the random error associated with each 6 -3 -3

is 1.0 x 10 degrees. The random error in 8 (180) of 3.3 x 10 degrees is a common error

for all other fiducial angles since this error is associated with the term G in eq (25).

The sources of error in 8 are summarized in Table III.

G. Scattering Angle Equation

The scattering angle, 8 , is calculated from

e o

=

9 f

+ w {V ~ V + § (R " V

= - 1.2566(0 - 0.5000) + 0.047(R - 0.7) . (26)

The values of 6 are given in Table I, and V and R are dial gauge readings in inches.

There are no terms in eq (26) involving the target pedestal gauges or the pedestal axis

misalignment because of the definition of 8 we have chosen. These sources of error affect

only the target location (Xp, z ). The total error in scattering angle includes uncertainties in the terms of eq (26) as well as uncertainties in the target location and beam direction. The largest sources of error are the uncertainty in the fiducial angles,

- 3 - 3

<_ 4.1 x 10 degrees, and the target pedestal dial gauge effect, 5.5 x 10 degrees. However, the uncertainty in (D-B) also affects the scattering angle through a , and the solid

angle through z^. Similarly, other sources of error (e.g., (A-C), 6(270), 6(180), etc.)

enter into the total cross section uncertainty by several routes.

VII. Solid Angle

The acceptance solid angle of the spectrometer is defined by the aperture slits. In

this section we describe the measurements which determine the distances L and L and the

x y angles y and y . All other quantities entering into the determination of solid angle

x y 52
have already been discussed, i.e., the target coordinates (x , y , z ), the aperture openings H and W, and the slit edge radii R .

The difference between L and L is measured with a micrometer depth gauge as illusy x trated in figure 22. A ground flat steel plate is placed against the horizontal aperture

defining cylinders and a depth gauge is placed in contact with the vertical aperture defining cylinders. Since all cylinders have equal diameters, the desired distance is simply

the depth gauge reading plus the plate thickness,

AL - L - L = 2.648 - 0.001 in. (27) x y

Figure 23 illustrates the method by which L was measured. A precision ground steel

sphere, 1.0 inch in diameter, is located on the axis of rotation of the target pedestal

employing the technique described in section V for centering the conical marker. A measuring rod of length L consisting of an aluminum rod capped with a second precision ground

K steel sphere, is positioned with the spherical end contacting the vertical aperture cylinders. The spectrometer is moved radially inward until the end of the measuring rod contacts the first sphere. The reading, R, of the radial dial gauge is noted at the point

of contact. A simple geometry calculation yields the result

L (R) = L n + r - r n + ~\ /2R r + r n 2

- HR - H

2 /4 . (28)

y R 2 1 y s 1 1 s

Equation (2 8) is valid only if the rod is held horizontal, with its spherical end centered left to right along the aperture-defining cylinders. The rod was held horizontal to

within 1/16 inch and centered to within 1/8 inch, thus causing an error in L of less than

y -4

5 x 10 inches. The rod length, measured at thermal equilibrium in a temperature controlled room, was L =19.9955 ± 0.0002 inches at 73°F. The temperature in the spectrometer

room was 5.8° lower when the calibration was made, requiring a correction of -0.001<4 inches

_ C _ "I for thermal contraction of the aluminum rod (12.4- x 10 °F ). Both steel spheres have

radii 0.5000 - 0.0002 inches, and the one at the axis of rotation was centered to within

+ -4- +

- 2 x 10 inches. The aperture cylinders have R .= 0.8750 - 0.0005. The aperture H,

measured for the present purpose with a feeler gauge, was 0.0302 - 0.0005 inches. The

fringe counter system (see section II) was checked during the calibration to be sure

that the aperture was not changed by the pressure of the measuring rod against the slits

,

53
Figure 22. Method for measuring L^ - L

54
DC

(MH

Figure 23. Setup for measuring L , side view.

55
-4

and a change of 1 x 10 inches was noted. These uncertainties are propagated through

_i+ eq (28) and combined in quadrature to obtain an uncertainty of 7 x 10 inches which we

take to be the systematic error in L . y

To obtain the origin-to-slit distance as a function of the radial position of the

spectrometer, a calibration of the form

(29)

is needed, where R is chosen to be the same value as for the 6 calibration. The fiducial

length L is a function of angle because the spectrometer does not execute a pure rotation

about the origin. The slope, 3L /3R, is expected to be -1.0 because the radial dial gauge

sensing direction is parallel to the radial spectrometer motion and the gauge reading increases as the spectrometer moves forward. Because of the imperfections in the spectrometer motion, we measured the slope at each nominal scattering angle, by adding caps to the

measuring rod which increases its length by about 0.1 and 0.2 inches. The slope varied

from -0.998 (at 163°) to -1.006 (at 110°) with an average value of -1.0027. The validity

of this value is questionable since the average value would be exactly -1.0 if there

were an error of 5 x 10 inches in the thickness of the 0.2 inch cap. We therefore adopt

the value of 3L /3R - -1.0 and assume conservatively that its accuracy is better than

- 6 x 10 , corresponding to the largest observed deviation. In the final calibration,

with no caps on the measuring rod, the values of R at the contact point were between

0.716 and 0.752 inches. The error in L due to the uncertainty in the slope is thus alo

J

ways less than 3 x 10 inches. The calibration consisted of measuring the value of R

at which contact is made with the spectrometer at each fiducial angle (V - 0.500 in.).

Two values were obtained at each angle, one while rotating forward (163° * 40°), and

again during the reverse rotation. For each measurement we obtain a value of L (9 ) from

eq (29), along with target pedestal dial gauge readings (D-B) and (A-C) . A correlation

between changes in L and changes in D-B was observed, given by

3L -0.94 - 0.30 (30)

3 (D-B)

A rigid-body motion of the entire rotatable structure would give 3L /3(D-B) = -1/2. As

o

56
expected, there is no significant dependence on (A-C). Here, as in the case of the scattering angle, we conclude that some non-rigid motion can occur during rotation of the

spectrometer, but we can correct the calibration data for its effect. Thus, for each

measured R,

L (6 r

) = L t (J? - 0.700) - 0.94 [(D-B) - (D-B)l , (31)

o F y F

with L obtained from eq (28). The two values of L for each 6^ are averaged to obtain the

y o F values listed in Table I. From the variance of the individual measurements, the random

-4 error of each L value in Table I is 5 x 10 inches. The largest correction made for the

last term in eq (31) was 0.0024 inches, with a corresponding uncertainty of 8 x 10

inches. The errors in the L determination are summarized in Table IV.

o To calculate L and L for a given experimental setup, eq (27) and (29) are used with

3L /8R = -1.0 and L (6) taken from Table I. No correction is to be made for the pedestal

y o F dial gauge effect, for the same reasons given at the end of section VI. The uncertainty

_3

in each L and L value due to this effect is 2.1 x 10 inches, which is obtained from

y x

_3

eq (30) and the 2.2 x 10 inch standard deviation of D-B. Note that no correction is

made for the azimuthal dial gauge, i.e., for 6 ± 8 . This is valid since L (9 ) changes

at most by 9 x 10 inches between adjacent fiducial angles, whereas |9 - 8 | <0.6

_4

degrees. Thus we expect the 6 dependence of L to be less than 3 x 10 inches.

o y The aperture slit system is bolted to the entrance flange of the spectrometer vacuum

chamber. This flange is not precisely perpendicular to the optic axis of the spectrometer.

The solid angle depends (weakly) on the angle between the normal to the plane of the

aperture slits and the Z axis. Optical autocollimation is used to measure this angle.

The theodolite is mounted on the baseline, as described in section V, some 18 feet upstream of the target location, and sighted on the axis of rotation. A front-surfaced

plane mirror, mounted in a target holder so that its reflecting surface is in the target

-4

plane (vertical within 4 x 10 radians), is rotated until the theodolite is sighted on the

center of the aperture slits. Next, a second front-surface mirror made from an optical flat

is held flat against the aperture slits. If the plane of the aperture were perpendicular

to the Z axis, one would then see the objective lens of the theodolite centered in the

field of view. The displacements, horizontally and vertically, of the theodolite in the

57
field of view measure y and y , respectively. The method is not highly accurate (the x y

_3

standard deviation is about 0.5 x 10 radians), but is sufficient since the solid angle is

very weakly dependent on y and y . The same result occurs whether the mirror is held

against the horizontal slits or the vertical slits, confirming the parallelism of the

planes of the two pairs of slits. Measurements with the spectrometer at 9 = 40° and 93°

-3 -3

give & the same result for y within 1 x 10 radians, and for y within 2 x 10 radians.

x y The differences with scattering angle are presumably due to the spectrometer symmetry

plane not pointing precisely toward the axis of rotation and the axis of rotation of the

spectrometer not being exactly vertical. The variations of y and y with scattering

x y

angle are negligible, so we adopt the average value of the measurements taken as being

applicable to all scattering angles. The result is

Y = (5-1) x 10~ radians , x and (32)

+ -3

Y

= (-5 - 1) x 10 radians

y

The effect of y and y on the cross section would be negligible if the target were lox y cated at the origin, but with the target point being as much as 0.2 inches from the origin

(note, e.g., y = 0.17 in. in Table II), assuming y and y are zero could lead to cross

A x y

bi- section errors of a few parts in 10 .

VIII. Target Angle Calibration

The target turntable is driven by a ring-and-pinion gear arrangement having an 8 to 1

reduction ratio. A high-quality ten- turn potentiometer connected to the driving shaft

constitutes two arms of the bridge circuit shown in figure 24. The other two arms of the

bridge are formed by a 40 turn potentiometer in the target controller chassis in the counting room. The position of the movable contact of the 10 turn pot, and therefore the angle

of the target turntable, is compared with the position of the 40 turn pot by a sensitive

null circuit. A reading of the turn counter connected to the latter potentiometer thus

determines the target angle. We expect the turn counter to have a sensitivity of about

360°/(8 x 4) = 11.25 degrees/turn, so that the turn counter least count, 2 x 10 turns,

corresponds to about 0.0225 degrees.

58
'.2 o

»j

m

CO

i±j

S2o

N

?2

i

. CD

1 1

1

1

1

1

hoog

Figure 24. Target angle measuring circuit.

59
For scattering angles from 40 to 14-5 degrees, we normally perform experiments with the

target bisecting the scattering angle, i.e., with M) /2 . Our target angle calibration

consists of finding the turn counter reading corresponding to each bisecting target angle

for these seven values of 6 . For each there are two values of , the second being obtained from the first by rotating the target turntable through 180° (and reversing the direction of the mirror in the target holder). The calibration is performed by optical

sighting, using a setup similar to the one described in section VII for determining the

angles y and y . The theodolite is set up on the baseline, pointing at the axis of rox y tation. The mirror mounted in a target holder is rotated, with the target turntable,

until the theodolite cross hairs are aligned with the center of the horizontal aperture

slits, with these slits closed to their limit position. For each , four independent

measurements are made, two with the mirror aligned "up," and two with the mirror aligned

"down;" the difference between up and down being a rotation of 180° about the normal to

the mirror. Averaging up and down readings corrects for any angular error in locating the

mirror in the target plane, but not for a translational misalignment. In addition to the

values of corresponding to the bisector angles described above, calibrations of the

turn counter by autocollimation were obtained at ^ 0° and 180° —actually at = -0.005

degrees because the theodolite was not precisely on the baseline for this calibration.

All other calibrations are therefore at angles given by

T

= (6 F

- .005)/2

or

= (0 - .005)/2 + 180 degrees ,

since the calibrations were performed with the spectrometer set at the fiducial angles

A systematic difference between up and down mirror orientation of 1.2 - 0.1 turn

_2

counter dial divisions was observed. A dial division is 10 turns, so the systematic

difference corresponds to an angular misalignment of about 0.07 degrees. This misalignment does not affect the calibration because of the averaging of up and down readings.

Table V gives the target angle calibration in the form of the turn counter readings

corresponding to each measured value of 6 . Based on the reproducibility of individual

measurements, the standard deviation of each dial setting should be about 0.5 divisions

60
(0.06 degrees). To check this error estimate, we make a least squares fit of the data in

Table V to

T = a^ + a

2

6 T

, (33)

where T is the turn counter reading in units of dial divisions. We find

a = 823.5 - 0.6 divisions ,

and (34)

a = 8.752 - 0.004 divisions/degree .

The standard deviation of each measurement for this fit is 1.3 divisions. If the enlarged

deviations were caused by a translational misalignment, the effect would be

an

T(e

T

+ 180) - t(o

t )« —— sin e T (35) x

where n is the displacement of the mirror from the target plane. Since no correlation of

this form is seen in the data, there is no significant translational error in mirror positioning. Further study of the deviations of the data from the straightline fit of

eq (33) reveals a periodic variation which can be expressed by

T = a + a 6 - 1.62 cos 8(0 + 5°) , (36)

with a and a„ having the values given by eq (34) . The fit of the data to eq (36) is much

better than to eq (33), implying a standard deviation of 0.68 divisions or 0.08 degrees

for each value of T in Table V, in reasonable agreement with our original estimate.

The oscillatory term in eq (36) seems to have its origin in the 8 to 1 gear reduction

of the turntable drive. An oscillation of the output angle of the vacuum feedthru

in the rotation drive about the input angle with a 1.5° amplitude could produce the

observed effect.

To use the target angle calibration the turn counter is set at one of the values

given in Table V and the target pedestal rotated to obtain a null on the balancing circuit.

This places the target at the corresponding angle to within 0.08 degrees standard deviation. If angles other than those listed in the table are required, or if the turn counter

61
is accidentally set at a value not given in Table V, the target angle can be calculated

from eq (34) and (36). In this case the error can be taken to be 0.08 degrees combined.

in quadrature with a small error due to the uncertainty in a .

IX. Summary

The experimental geometry is completely defined by the thirteen parameters 9 , x ,

y m , zm , o ,a ,W,H,L ,L , y , y , and R . In addition, 9 m is needed to calculate some

T T x y x y 'x' 'y s T of the above parameters and to obtain the effective target thickness W from the target

foil thickness, t, from

W = t/cos (6„ + a ) cos a . (37)

o T x y

The scattering angle, 9 , is given by eq (26). The numerical constants (A6/AP), V ,

(A9/AR), and R are summarized in Table VI, and the fiducial angles, 9 , are listed in

Table I. Determination of the dial gauge readings V and R are made for each experimental

setup

The target angle 9 can be obtained from eq (36). It is best to use the fiducial

target angles, 9 , and corresponding turn counter dial readings, T , listed in Table V.

Performing a Taylor's series expansion on the small oscillatory term in eq (36), we find

T - T

9 T

= 6 TF

+ a

[1 " a

3 sin 8(9TF

+ 5°)1 ' < 38 )

which is accurate to better than 10~ degrees for |t - T | < 2 divisions. The numerical

constants in eq (38) are given in Table VI.

The target coordinates x,^ and z are obtained from measured quantities by eq (7).

Expressions for x , n , x and z , which are needed to evaluate eq (7), are given by

B p p eq (5), (16), (17), and (18), respectively. The vertical coordinate of the target is

given by

yT

= h -yA , (39)

after translating the coordinate origin to the height of the center of the Y slits. This

translation is in accordance with the definition of the Y coordinate used in the computer

program we use to calculate the solid angle and mean scattering angle. The constants

62
p , 3, 6(180), 6(270), A, and S needed to evaluate eq (7) and (39) are listed in Table VI.

The fiducial values (A-C) and (D-B) are given in Table I, and y is given in Table II.

r r /\ The parameters x and z are obtained for each experimental setup by the procedure described after eq (5) in section II. The step count (N - N ) is obtained from the online

computer output of the beam scanners. Subscript T here specifies the horizontal scanner

at the target location, for which N = 100. The remaining variables V, n'j (A-C) and (D-B)

must be measured for each experimental setup. Equation (6) cannot be used to determine h

because of the unknown value of h , although this equation is useful for determining

changes in h during data taking. We can determine h to sufficient accuracy (- 1/32" or

less) by visual observation of the beam spot on the standard bullseye, whose center is

at h = 0.

The beam angles relative to the baseline are given by eq (8) and (9). Expressions

for x , x , h, and x " are given by eq (4), (5), (6), and (19), respectively, and y * is

K Jj p R given by the analog of eq (4) for the vertical coordinates. The constants x , y , B , and

B are given in Table II

.

y The origin-to-aperture distances are obtained from eq (27) and (29). The constants

3L /3R, R„, and AL are listed in Table VI, and the fiducial values L (9 F) are given in

Table I.

The aperture width and height, W and H, are simply the respective fringe counter

readings, described in section II, minus the extrapolated readings at which the counting

-4- rates become zero. The standard deviations of the readings are 10 inches, and the

uncertainty of the extrapolated- to-zero readings are statistical ones associated with the

extrapolation.

I

i The final parameter of the set is R which is listed in Table VI.

Constants listed in the tables have their corresponding uncertainty given, if that

uncertainty affects the accuracy of the computed solid angle or mean scattering angle. All

errors should be interpreted as one standard deviation. The propagation of the measurement errors into the uncertainty of a calculated cross section at a specified mean scattering angle is complicated because some of the measurement errors affect the final result

in several ways. For example, an error in affects the calculated values of x , z , and

a . The uncertainties given in Tables I, II, V, and VI are systematic in the sense that

the actual (unknown) error affects all calculated cross sections, although not necessarily

63
by the same amount. In addition, there are a number of sources of error which have random

effects on individual cross section measurements. An intermediate class of uncertainties

affect all cross sections measured at one nominal scattering angle systematically, but

randomly for different scattering angles. For example, 8 contains all three types of

error (see eq 38): the uncertainty obtained by propagating the error in a is systematic

to all measurements; the uncertainty due to the calibration of fiducial values 6 (T ) are

systematic to all cross sections measured with a given 8 but independent for different

values of 8 ; and the error due to the settability of T(- 0.2 divisions) is random each

time the target angle is set. Because of these complications, we cannot give overall error

estimates here. We intend to study the effects of all sources of errors by numerical

methods, using the computer programs which calculate the system geometry. In order to do

this study, we need estimates of the random errors in the variables entering into the

geometry parameters as well as the errors in the constants of Tables I, II, V, and VI. In

the cases of x , z , the values of N occurring in eq (4-), (5), and (6), and the aperture

slit zero extrapolation, error estimates are obtained from the variance of the measurements. For the variables V, R, T, V, (A-C), (D-B), h, W, and H, our estimates of the

associated random errors for a single measurement are summarized in Table VII.

The most important source of error is the irreproducible motion of the spectrometer

on its carriage, discussed in section VII. The effect of this source of error on 8 , L , 39 8L

o

° X and L is described by the measured correlations _ , ^ „. and „ > ^ „ s

. The same source of

y 3(D-B) 3(D-B) error affects x through eq (7) and (17), z through eq (7) and (18), and a through eq (8)

and (19). The target pedestal dial gauge effect influences the latter variables because

we set the target pedestal at the position defined by the fiducial values of A-C and D-B,

but these fiducial values as read on the gauges may be in error due to the irreproducible

motion. The influence on x , z , and a can be obtained by differentiating the equations

for these variables with respect to (A-C) and (D-B) . It is important to note that the

effect of this source of uncertainty on the several variables is completely correlated.

In Table VIII, where we summarize these effects, we allow for this correlation by indicating the sign of each error term. The uncertainties in 8 , L . L , il, zm , and a for

o ' x y T T x each setup of the spectrometer is obtained by multiplying the coefficients given in

Table VIII by the standard deviation of a single setting of (D-B) or (A-C) . All terms

64
arising from (D-B) are to be added algebraically, and then combined in quadrature with the

terms arising from (A-C) . Thus, the total effect on the scattering angle of this source

of error is given by

AG = d<

39 I

2 x

)XT

3a

3 (D-B) 3 (D-B),'JLx 3(A-C)

F 3(A-C),

1/2

(40)

-3 -3

where d = 2.2 x 10 inches. At 9 = 90°, using L = L (93° ) + AL , A9 = 6.7 x 10 degrees,

o x o The effect of this error on the quoted value of a measured cross section is approximately

3 parts in 10

4

. The effect on cross section ... entering through the solid angle calculation

is much smaller because the error entering the solid angle through z is of opposite sign

to that entering through L or L . to b x y

X. Acknowledgements

The precision mechanical devices needed to make the calibrations described in this

report valid were designed and installed by the Mechanical Instrumentation Group of the

NBS Nuclear Instrumentation Section. We wish to acknowledge the invaluable contribution

of the members of this group: John Lantz, Dave Mohr, Joe Bradley, and Cliff Watkins

.

I Without their dedicated efforts, this project could not have been completed successfully.

65
References

[1] "Experimental Apparatus Requirements for Accurate Electron Scattering Experiments,"

S. Penner and J. W. Lightbody, Jr., Internal Report No. 369, Nuclear Sciences Division,

NBS, 1972 (unpublished).

[2] A more complete description of the entire system is given in "Experimental Techniques

for Electron Scattering Investigations," Samuel Penner, NBS Technical Note 523 (1970),

and references cited therein.

[3] J. S. Pruitt, Nucl. Instr. and Methods 92_, 285 (1970).

J. S. Pruitt, Nucl. Instr. and Methods 100 , 433 (1972).

[4] S. Penner and J. W. Lightbody, Jr., Proc. Int. Symposium on Magnet Technology, 154.

Stanford Linear Accelerator Center, Stanford, California, CONF-650922 (1965).

[5] J. K. Whittaker, IEEE Trans, on Nucl. Sci. NS-19 , 3, 444 (1972). A complete description of the performance of the detector system is being prepared.

[6] The device used is available commercially. Details of the operation are regarded as

proprietary by the manufacturer.

66
Table I

Angle- Dependent Parameters Affecting the Scattering Angle and Solid Angle

Nominal Dial Guage Differences

Angle Fiducial Values , Inches

Fiducial Angle 6 Degrees

Degrees (D-B)

F (A-C),

Fiducial Aperture

Distance L (6 ) o F Inches

M-0 0.0300 0.0119

58 0.0337 0.0071

75 0.0334 0.0055

93 0.0321 0.0096

110 0.0309 0.0150

128 0.0305 0.0192

145 0.0311 0.0157

163 0.0297 0.0073

180 0.0259 -0.0032

40.3491 - 0.0041

57.6031 - 0.0040

75.2602 - 0.0040

92.7753 - 0.0039

110.2659 - 0.0038

127.8827 - 0.0037

145.3855 - 0.0037

163.2745 - 0.0036

180.0061 - 0.0033

21.0612 - 0.0010

21.0681 - 0.0009

21.0763 - 0.0008

21.0839 - 0.0007

21.0891 - 0.0007

21.0916 - 0.00(

21.0904 - 0.0009

21.0829 - 0.0010

The quoted uncertainties, which are standard deviations, are highly correlated

for the various angles. The largest contribution to the total error is the

error in the value of 9 (180°), which is a common error for all other angles.

The quoted uncertainties, which are standard deviations, contain a random

measurement error of 0.0005 inches in each value. The remaining errors

for L at various angles are correlated, with the largest contribution

being 0.0004 inches due to the uncertainty in the measuring rod length

(L in eq. [28]).

67
Table II

Quantities Measured Relative to Baseline

Standard

Quantity Value Deviation Units

e F (iso) a 180.0061 ± .0033 degrees

x

c

- 0.006 ±0.008 inches

y c 0.022 ±0.009 inches

y A 0.170 ±0.012 inches

Bx

107.40 ±0.05 inches

By 114.36 ±0.05 inches

Measured with R = 0.700 inches, V = 0.4902 inches,

and (D-B) and (A-C) values given in Table I.

68
Table III

Sources of Error in Fiducial Angle Values

Source Magnitude

(degrees

)

6 F (180), From Table II

9 (180), From Effects Listed Below

Theodolite Alignment

Theodolite Axis Offset

Uncertainties in 6(270) and 6(180)

Azimuthal Dial Gauge

Radial Dial Gauge Correction

Target Pedestal Dial Gauge Correction

Variance of Measurements

Total (combined in quadrature)

3.3x10

i.o

b

0.4

0.1

-3

51.

<0 1

±0 3

1

4.0x10

Does not apply to 6 (180).

These errors are common to all fiducial angles.

From eq (21). Largest value of interest occurs at 0-4-0°,

No error since all data is taken at V=V„.

69
Table IV

Errors in Origin to Aperture Slit Distance

Source

Standard Deviation

(inches

)

Effects entering through eq (28) .0007

Uncertainty in 3L /3R <.0003

y

Target pedestal dial gauge effect <_. 0008

Random measurement error .0005

Total (combined in quadrature) <_.0012

Error common to all values of L ( 6 )

.

o i b . . . There is some correlation in the errors at various 6

from these effects, but it can be neglected.

70
Table V

Results of Target Angle Calibration

Nominal Tp

Scattering 6 Degrees Dial Divisions T for 6 + 180.0

Angle

,

a Degrees

163

40

58

75

93

110

128

145

8.363 751.7

0.005° 822.8

20.178 1001.7

28.805 1075.4

37.634 1152.3

46.391 1227.8

55.137 1307.0

63.945 1384.2

72.696 1460.0

2327.2

2396.7

2576.4

2652.1

2726.8

2804.0

2882.0

2960.6

3034.4

The listed values of 9 orient the target for transmission scattering at these scattering angles, except as noted.

Target is oriented for backscattering at a scattering angle of 163°.

The listed target angle makes the path length in the target of incoming and scattered electrons equal.

This target angle is used to calibrate the beam scanner as described

in section II D.

71
Table VI Miscellaneous Constants

Standard Section Where

Quantity Units Value Deviation Defined

Ue\ degrees -1.2566 0.0004 VI B M inch

V

T inches 0.5

3 — VI B

( A9

1 degrees 0.047 0.012 VI C AR 1 inch

*F

inches 0.7 — VI C

a

2 (deg)"

1 8.752 0.004 VIII

0.02585 __ IX

a

3

p 10

_3

in 0.2 IV

o

e 10

-1+rad 0.8 0.5 IV

6(180) 10

_4

rad 0.9 0.7 IV

6(270) 10

_4

rad 4.4 0.8 IV

A 10" 3 in 2.83 0.03 IV

P inch 1.25 0.03 II D

f 3L -1.0 0.006 VII

3R

AL inches 2.648 0.001 VII

inches 0.8750 0.0005 VIII

10~ 3rad 5.0 1.0 VII

10" 3rad -5.0 1.0 VII — 1.156 .005 IV

3 At

F

= 180°, V

?

= 0.4902 inches,

72
Table VII

Estimated Errors (Inches) in Determination

Of Variable Parameters

Variable Standard Deviation Section Where

Defined

V 0.0002 VI B

R 0.0005 VI C

T 0.68 VIII

V 0.25 . IV

A-C 0.0003

D-B 0.0003

h 0.02 II D

W 0.0001 aII B

H 0.0001

a

II B

This error is in addition to, and independent of,

the error in determining the reference opening of

the slits. The latter error is obtained from the

extrapolation to the zero- count -rate opening, discussed in section II B.

73
Table VIII

Errors Due to Target Pedestal Dial Gauge Effect

The use of the terms listed in obtaining the errors

in the experiment is described in section IX.

Variable

Affected Function Value

x

x„

a

x

!!°_ 2.52 ± 0.44 £&

3(D-B)

L

3Lo - 0.94 ± 0.30

3(D-B)

3L x L o 0.94 ± 0.30

7 3(D-B)

3X

T

S v 0.578 - 0.003

3(A-C) 2

ZT 3(D-B) " " 2

9Z

T_ _ S v -0.578 ± 0.003

F

3ax

S v + -3 -1

cos 9 (5.38 - 0.02) x 10 cosS in

3(D-B) 2B ^" o

V "" MU "•"" "

*w ~~ o

F x

3a X

SV / r- on + oo\ n~3on• o • ""I ^— sin G -(5.38 - 0.02) x 10 sine in

3(A-C) ~ 2B °"""o w -° u «."*/ ~ *« —qJ F x

74
RBS-H4A<fME*. 7-731

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-8712. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Geometrical Calibration of the NBS Electron Scattering

Apparatus

5. Publication Date June 1975

6. Performing Organization Code

7. author(S) s. Penner, S. P. Fivozinsky, J. W. Lightbody, Jr.,

L. S. Cardman. and W. P. Trower

8. Performing Organ. Report No.

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DEPARTMENT OF COMMERCE

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12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

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16. ABSTRACT (A 200- word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.)

A comprehensive calibration of the geometry of the NBS

electron scattering apparatus is described. A complete set

of measured parameters is tabulated in this report. Combining

these parameters with observed values of certain variables as described herein permits the accurate determination of the

solid angle, scattering angle, and target angle for each cross section measurement made with the apparatus. The uncertainty

in a cross section measurement due to the imprecision of the

geometry calibrations is less than one part in 10 .

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Absolute cross section; beam profile; electron scattering; scattering

angle; solid angle; spectrometer

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