ABS TRACT
AN EXPERIMENTAL MEASUREMENT

OF COSMIC-RAY MUON SCATTERING BY ALUMINUM,
IN THE MOMENTUM REGION OF 1.60 Bev/c

by Ronald Wilbur Beery

An account is presented of an investigation into the
scattering behavior of (1.60 + 0.05, - 0.04) Bev/c p+ mesons
from cosmic rays. The muons were allowed to traverse a
spark chamber triggered by a counter telescope. The alums
inum plates of the spark chamber provided the scattering
material for the incoming muons. A description of circuits
and equipment developed for this experiment is presented.

‘Momentum selection was made by using a lead absorber
to slow muons having the above momentum, stopping them in a
scintillation counter. Final selection of events required
observation of a proper‘p-e decay signature in oscilloscope
photographs made of the signal from the stopping scintil-
lation counter. The oscilloscope display was also arranged
to distinguish between protons, pions, and the desired muons.
This served as a method of eliminating the former two types
of events from the muon data. As a further verification
of the particle selection, the apparent muon lifetime was
measured and was in excellent agreement with the accepted
value. The narrow limits of momentum obtained and the posi-
tive particle-identification procedure represent an im—
provement over previous cosmic-ray experiments of this kind.

The track photographs obtained were measured on

semi-automatic scanning equipment, the output of which was
analyzed by a computer program developed for this experiment
to determine the scattering angle of the particle in the
chamber. Details of this analysis program are presented.
The results of the scattering measurements, in the form of
an integral distribution of projected angles, are compared
with the distributions predicted by the theories of Moliere,
of Cooper and Rainwater, and of Drell and Schwartz. Details
of the computer calculations required for the theoretical
predictions are also presented.

The results of the present experiment demonstrate
agreement with both the theory of Cooper and Rainwater and
of Drell and Schwartz throughout the range of scattering
angles observed: 0 - 4.5 degrees. This range corresponds
to momentum transfers up to 190 Mev/c. Good agreement with
Cooper-Rainwater and somewhat better agreement with Drell-
Schwartz rather than with Moliere is clear. This experiment
does not display the excessive appearance of high—angle
scattering reported in early cloud chamber experiments with
cosmic-ray muons. This experiment, using cosmic-ray muons,
corroborates the results of the recent experiment performed
by Masek et al., with 2 Bev/c muons obtained from the

Bevatron.

AN EXPERIMENTAL MEASUREMENT
OF COSMIC RAY MUON SCATTERING BY ALUMINUM,

IN THE M MENTUM REGION OF 1.60 Ecv/c

By

Ronald Wilbur Bcery

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

1966

ACKNOWLEDGMENT

I wish to acknowledge my appreciation to Professor
Joseph Ballam for his suggestion of the problem and direction
in the early stages of the work. I am grateful also for his
interest and willingness to continue the responsibility of
seeing the work completed even after his transfer to
Stanford University.

The valuable suggestions and advice of Dr. John
Scandrett during the construction of the spark chamber
equipment are acknowledged. I am also grateful to the
many professors and staff of the Department of Physics
who have given encouragement during the long—term construc-
tion and operation schedule of a cosmic—ray experiment such
as this. With regard to construction assistance I am
indebted to the late Mr. Charles Kingston and to Mr. Richard
B. Hoskins as well as the entire staff of the machine shop
for their counsel and assistance in the design and mechan-
ical construction of the experimental apparatus. I am also
indebted to Mr. Ernest F. Brandt and the staff of the elec-
tronics shop for their assistance in constructing the electronic
equipment necessary to the experiment.

My thanks go to Mr. William M. Patmos and to Professor
David Marker for their assistance in the early stages of the
computer programming for the data analysis and the theory
calculations respectively. 'My thanks also go to Mr. Michael
Yarnold who, as an undergraduate, headed up my film-scanning
staff, and to the other students on the scanning staff for

their interest and help in the work.
ii

I wish to express finally my appreciation to my
wife Alice for her patience and encouragement throughout
the work and for her efforts in typing the manuscript and
final copies of this work.

The construction and operation of the apparatus for

this experiment were made possible by a grant supported by

the National Science Foundation.

iii

TABLE OF CONTENTS

ACKNOWLEDGMENT
TABLE OF FIGURES
I. INTRODUCTION

A. Background of muon investigations
B. Motivation for the present experiment
0. Approach of this experiment

II. EXPERIMENTAL.APPARATUS

A. General description

B. Spark-chamber system

C. Counter—telescope system

D. Automatic photographic system
E. Stopping-pulse display system
F..Analysis system

III. ANALYSIS PROCEDURE

A. Analysis of oscilloscope photographs
1. Types of events identified
2. Lifetime study

B. Analysis of track photographs

C. Data reduction

IV. THEORETICAL PREDICTIONS
A. The Cooper-Rainwater calculation
B. The Moliere calculation
C. The Drell-Schwartz calculation
V. RESULTS AND CONCLUSIONS
A. Discussion of experimental uncertainties
B. Comparison of observed scattering with
predicted values
C. Summary and conclusions
APPENDICES
A. Spark—chamber system

B. Counter—telescope system

C. Modification of Tektronix 517 oscilloscope
for 1ij-e display

iv

Page
ii

vi

oo O\O\l—' I—‘

78
84

89

Page

D. The automatic-camera system 95
E. Momentum determination 101
F. Program "Spark" 105

G. Calculation of theoretical multiple-
scattering distributions 126
REFERENCES 155

TABLE OF FIGURES

Figure

11-1

II-la
11-2
11-3
II-n

II-5

III-l

V-l
V-2

A-l
B-l
B—la

D-l

Block diagram of experimental
arrangement

Detail of scattering apparatus
Overall view of apparatus
Close-up of spark chamber

Close-up of coincidence circuit

Characteristic Oscilloscope trace,

showing‘p-e decay signature

Normalized plot of lifetimes for
1280 events

Results from 100 "doubles" events

Normalized integral scattering
distributions

Spark-chamber circuit diagram

Coincidence-circuit diagrams

Page

10
ll
12
16
20

26

#1
71

75
83
87

Coincidence-circuit diagrams (cont'd) 88

Circuit diagram of primary

modifications to 517 oscilloscope

Camera-circuit diagram and diagram
of optical geometry of experiment

vi

9h

98

I. INTRODUCTION

A. Background of muon investigations
The characteristic properties of muons have in
recent years been the subject of considerable interest
among those working in particle physics. The following
characteristics measured in these muon investigations
are of interest here:

1) Muon mass found to be 206.767 1- 0.00321 times
that of the electron.

2) Muon mean lifetime found to be 2.2001 1 0.008
usecondszz.

3) Yb - 7e distinction. In an experiment by Danby,
et al.,23 it was demonstrated that the muon has
its own neutrino, quite apart from that assoc-
iated with the electron. This was concluded from
the fact that neutrinos, produced by"nfip decays
and allowed to interact subsequently with protons,
caused the appearance of numerous muons but no
observed electrons. This provided an answer to the
problem of the forbiddenness of the decay mode
.P+‘—I e+ +I',

Of particular interest are other characteristics in which
the muon displays a high degree of similarity to the elec-
tron. The following are of interest here:

1) Muon charge found to be 1.00002 1
0.0000412292l times that of the electron.

2) Muon-electron similarity in weak interactions.
1

2

A value for the P- + p capture rate has been

calculated24 using the observed rate of neutron

decay (an electron-producing process) to evaluate

the required constants. This calculated rate is in

good agreement with the capture rates measured with

both bubble chamberszsa26 and counters27928.

3) Muon-electron similarity in electromagnetic

interactions.

a) Measured value of gP = 2(1.001162 i

b)

0.00005)192:29. This measured value gives

a comparison ratio gP/ge = 1.000001 1
0.000005. Thus the muon is a spin %
particle, obeying Fermi-Dirac statistics

as does the electron.

Comparison of e-p ande-p scattering.
Extensive work on electron scattering by
protons as well as by various nuclei has
been carried out by R. Hofstadter30, D. G.
Ravenshall3l, L. N. Hand et al.,32 and J. R.
Dunning, Jr. et al.,33 among others. Work
witth-p scattering is more recent, having
been carried out notably by H. F. Davis et
al.,3h and R. Cool et al.35 The latter work
presents a detailed comparison between e-p
and‘p-p scattering, with the conclusion that
there is no observable difference throughout

momentum transfers ranging from 450-1100 Mev/c.

3

It has seemed on the basis of these results that
the muon may be described by essentially the same theo-
retical formulations as have been developed for the
electron. Results of scattering experiments of muons upon
atomic nuclei, however, have not always been in agreement
with the above results, particularly when compared with the
e-p,‘p-p similarity cited above. Discrepancies were first
noted in the behavior of muons obtained from cosmic rays,
which, of course, were the original sources for these
particles. For higher energies, where the momentum exceeds
600 Mev/c, excessive scattering contributions-—particu1arly
at higher angles-—have been reported by a number of cosmic—
ray researchers, beginning with the first systematic search
for this effect by Amaldi and Fidecaro in 19503. Outstanding
among the experiments since 1950 is the work of Whittemore
and ShuttL’ in 1952 with 0.3—3 Bev/c muons, McDiarmid5 in
1954 with 0.2-4 Bev/c muons, Lloyd and‘Wolfendale6 in 1955
with 0.6-10 Bev/c muons, and Lloyd, Rossle and‘Wolfendale7
in 1957 with 5—.50 Bev/c muons. Without exception their
results show high-angle scattering in excess of that pre-
dicted by theories based on a nuclear-charge distribution
of finite extent. The fact that the muon-scattering results
between 1950-58 seemed to agree more nearly with the
Molierelo "point-charge nucleus" theory than with the
"finite-nucleus" theories advanced during that time was
not taken to imply that the nucleus was acting in fact as

a POint charge, but rather as suggesting either a deficiency

4
in the theory or that the muon has some mode of interaction

not covered by the "finite-nucleus" theories. An increasing
anomaly for greater momenta was not found to occur in an
experiment by Lloyd and'Wolfendale7. Also a short-range
interaction with nucleons was not indicated, since the
amount of large-angle scattering in light elements is much
less than would be expected from the cross section per
nucleon obtained from muon-scattering results for leadz.
"Finite-nucleus" theories with which the experimental
results have been compared include primarily those of
Olbert8, of Cooper and Rainwaterg, and of Drell and
Schwartzl7. Olbert's theory is a modified version of the
Molierelo theory, where the finite extent of the nucleus
is accounted for by forcing the single-scattering law to
be 55 0 for scattering angles greater than a cutoff angle
0,. The Olbert theory provides an underestimation of
large-angle scattering whereas the Moliere theory provides,
of course, an overestimation of scattering in this region.
The Cooper-Rainwater theory using the same general approach
represents a more accurate treatment of the effects on the
scattering to all angles due to a nucleus of finite extent.
The Drell-Schwartz theory, however, approaches the problem
by developing a sum rule which yields the scattering dis-
tribution resulting from both inelastic and elastic scat-
tering. This sum rule is particularly useful since it does
not require knowledge of the complicated final-state wave
functions associated with the final excited states in which

the scattering nucleus may be found. A knowledge of the

5

ground-state wave function is sufficient.

The experiments conducted previous to 1959 were
plagued by several serious limitations, typical of cosmic-
ray work during that period, which included:

1) poor statistics at large angles, caused by

the slow data-gathering capability of cloud
chambers

2) measurement errors in determining particle

momentum and/or deflection angles

3) contamination of results by events involving

particles other than muons, pions being the

major concern.
An experiment by Fukui, Kitamura, and Watasell in 1959
with 1 Bev/c muons gave results consistent with the
Cooper-Rainwater theory. This experiment though showing
a degree of improvement in some of the above areas was
shown to be inconclusive by Lloyd and Wolfendale12 in
1960, primarily because of uncertainties in momentum
determination and in particle identification.

To fill the need for a conclusive experiment on
this subject Masek, Heggie, Kim and‘Williams13 performed
an experiment with the Bevatron at Berkeley using 2 Bev/c
muons obtained from decay of the pion beam. Scattering
was done both in lead with momentum transfers up to 265
Mev/c, and in carbon with momentum transfers up to #00
Mev/c. .A total muon flux of 2.5 x 107 particles was obtained

in the experiment. Mbmentum spread was not more than i 3.5

6

percent, and the fractional contamination by pions was of
the order of h.9 x 10‘5. The scattering in lead was com-
pared with the predictions of the Cooper-Rainwater theory
and excellent agreement was found. The scattering in
carbon was compared with the predictions of the Drell-
Schwartz theory and again excellent agreement was found.
Thus the question appeared settled, for machine—made muons

at least, that no anomalous scattering exists.

B. Mbtivation for the present experiment

There remained a need, however, for a cosmic-ray
experiment to demonstrate conclusively whether the anom—
alous scattering observed by so many cosmic-ray workers
was indeed due solely to the inaccuracies encountered in
the three categories listed above. Such an experiment
should provide significant improvements in each of these
three difficult areas. The present experiment was designed
to fill this need by applying the capabilities of a number
of recently—developed research tools to a cosmic—ray

scattering investigation.

C. Approach of this experiment

The approach of this experiment was first of all to
utilize a spark chamber to replace the cloud chambers used
almost exclusively for cosmic-ray work in the past. This
allowed a greater rate of data collection and greater
accuracy of angle measurEment. Plastic scintillation
cubunters were used in place of the trays of Geiger—

Phleller tubes common to cosmic-ray work. These provided

7

a significant improvement in the momentum determination

and made possible the type of particle identification
technique developed for this experiment. This technique
provided for, among other things, direct observation and
removal from the data of pion events seen. Also computer-
programmed scattering-angle calculations developed in this
experiment replaced the hand deflection-angle measurement
methods previously employed. Thus significant improvements
have resulted in areas (2) and (3) above. Some improvements
were made in (l), with further improvement available in this

area upon continued pursuit of the method of this experiment.

II. EXPERIMENTAL.APPARATUS
A. General description

The apparatus for this experiment consisted of an
aluminum-plate spark chamber located in a cosmic-ray
counter telescope, which was composed of three scintil-
lation counters: an aperture counter, a stopping counter,
and an anticoincidence counter. The three-channel coin-
cidence system provided a mode of event selection which
required double coincidence between the first two counters
and anticoincidence with the third. The output of the
coincidence circuit then triggered the spark chamber.

The visible track formed in the spark chamber was
recorded on 35mm film by a motor-driven camera which
automatically advanced to the next frame after each
spark~chamber event. In addition, the pulse from the
stopping counter was displayed on a fast oscilloscope and
photographed by a second automatic camera, synchronized to
the first. This provided a simultaneous photograph of the
stopping pulse corresponding to each chamber event selected
by the coincidence system.

In view of the low rate of data collection inherent
iJI this experiment, the apparatus was designed to be fully
au1tomatic in its operation. Use of lOO—foot rolls of film
III each camera allowed eight-hour periods of operation

before film changes were required.
8

9

The data was analyzed with the help of the follow-

ing basic items of equipment:

1) A Gilliland motorized film-projection machine

on which the oscilloscope film was scanned for

event selection,

2) A Hydel digitized film scanner tied to an IBM 526

3)

summary card punch by which the coordinates of
each spark in the chamber photograph of a given
event were digitized and punched on IBM cards
for processing,

The Michigan State University Control Data
Corporation--3600 computer by which all

calculations upon the raw data were performed.

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Figure II-la. Detail of scattering apparatus.

12

 

Figure 11-2. Overall view of experimental apparatus.

B. Spark chamber system

The spark chamber used in the present experiment
was divided into three sections: (1) the main chamber,
in which the usable events took place, and (2) an upper
and (3) lower thin—plate chamber used for more accurate
determination of the incoming and outgoing directions of
the particle. The main chamber section is a laminated
assembly of lO%—inch by lO%—inch aluminum plates 5/32-
inch thick, separated by clear plexiglas spacing strips
l/h-inch thick placed around the perimeter of each pair
of plates, to form.a series of closed cells. Twenty-two
such aluminum plates and plexiglas spacers comprise the
main chamber. The aluminum.plates themselves were used as
the scattering material. The upper and lower chambers are
constructed of 3/64-inch aluminum plates assembled into
milled slots in the sides of a closed plexiglas shell, with
the gap spacing being identical to that of the main chamber.

To provide a helium atmosphere within the chamber
for formation of the ion trail, a series of small holes
were drilled through the right face of the chamber (one
through the center of each cell spacer). Into each of
these holes was cemented the free end of a short length of
soft plastic surgical tubing branching from a manifold
block mounted on the side of the chamber. To complete the
flow system, a similar manifold was used to collect the
exhaust gas into a single tube, for flow control purposes,

which is vented to the atmosphere. Tanks of ordinary
l3

l4

welder's helium were used as a gas supply. The helium
passed first through a liquid-air cold trap and on to the
flow control panel. The cold trap was necessary because
considerable water and traces of oil were found in the
helium. Removal of these impurities improved the chamber
performance in terms of the gap firing efficiency and
spark uniformity and also allowed chamber operation over
a longer period before disassembly and cleaning of the
plates in caustic potassium hydroxide solution became
necessary.

The flow control panel consisted of a branching
valve arrangement with a separate flow gauge and metering
valve in each branch. Helium passing through the main branch
proceeded directly to the chamber distribution manifold. A
flow rate of 215 liters per minute was maintained in this
branch for best operation. Helium passing through the re—
maining branch was made to bubble through a glass tube con-
taining 2.5 inches of ethyl alcohol. The helium flow rate
through this branch was kept at the low rate of one bubble
per second from an orifice l/32-inch in diameter. This
minute quantity of alcohol dissolved in the helium and
served as both a quenching agent for the ionization process
and, to some extent, as a wave-length shifter to bring the
wave-length of the light emitted from the spark discharge
nearer the blue region where the film sensitivity is higher.

To supply the high-voltage pulse necessary to fire

the chamber, a thyratron pulsing unit was used which was

16

 

Figure 11-3. Close-up of spark chamber.

C. Counter telescope system

A vertical counter telescope system subtending a
solid angle of 0.15 steradian was constructed as shown in
'Fig. II-l,2. This solid angle allowed reception of approx-
imately 5 percent of the available cosmic-ray intensity.
Plastic scintillation counters viewed by 6810A photomulti—
plier tubes were constructed for each position in the
telescope system. The aperture counter at the top consisted
of a circular block of scintillating plastic 9 inches in
diameter by 3 inches thick with a flat region cut on one
side to which the photo tube was attached. The stopping
counter consisted of a block of scintillating plastic 24
inches by 24 inches by 7 inches with each of the four
corners cut off at 45 degrees so that phototubes may be
attached at those positions. The corners are cut in such a
way that as many as four phototubes could be attached at
each corner.

'For reasons of cost only two phototubes were actually
installed, giving a total of eight in all for this counter.
This provided a ratio of photo-cathode area to total counter
surface area of 0.96 percent, which is considerably smaller
than recommended by the experience of others.

The "anti" counter consisted of another block of
scintillating plastic 24 inches by 24 inches by 2 inches
with one phototube on each corner mounted as in the above
case. In this counter the ratio of photo-cathode area to

surface area was 0.65 percent, notably smaller than the
17

18

value for the stopping counter. Indeed it was observed
that the anti counter was less efficient than was the
stopping counter as it was not able to produce the expected
reduction in count rate compared to the doubles rate (i.e.,
integral count rate). Fortunately this deficiency only
caused film to be wasted, since the burden of the momentum
determination as well as particle identification rests upon
the film record of the stopping pulse with its required decay
signature. For this reason the cathode-area fraction was
allowed to remain at the above value throughout the experiment.

The power for the phototubes was derived from two
Hamner N-4035 power supplies. The aperture counter and the
anti counter were operated from one of these supplies and
the eight tubes on the stopping counter were operated from
the second supply. In addition the voltage on each tube
in the entire system was adjustable within +0 percent, -20
percent of the power supply output in 16 steps for the
purpose of matching the sensitivity of all the phototubes
in each counter by equalizing pulse height or count rate.

The signals from the three counters were fed to the
appropriate coincidence or anti coincidence input in the
tunnel diode coincidence unit which was also built during
the course of the present research work. This unit used
gallium arsenide tunnel diodes throughout as monostable
multivibrators, each triggered by the previous stage, and
the first triggered by its respective scintillation counter

signal. The primary features of this design were its

19
narrow coincidence resolution time (2 to 3 nanoseconds),
short dead time, and reliability for input signals ranging
in amplitude from 50 millivolts to 50 volts approximately.
This latter feature is particularly important in working
with relatively large scintillation counters. A detailed
circuit description and schematic of this unit is given
in Appendix B. A delay cable 60 feet long had to be
inserted in each coincidence channel to assure the condition
that the anti signal, whenever present, arrived at the
coincidence circuit before the coincidence signals from the
aperture counter and stopping counter. The tunnel diode
coincidence unit has an output circuit consisting of a
transistor amplification stage and individual emitter
follower output stages to supply isolated coincidence output
signals to the spark chamber, the 517A oscilloscope and a

count rate monitoring scaler.

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D. Automatic photographic system

The photographic aspects of this experiment required
two cameras. One photographed the particle tracks in the
spark chamber; this was done three-dimensionally by means of
a mirror to provide a stereo view of the chamber. The
remaining camera photographed the traces appearing on the
screen of the 517A oscilloscope. These traces displayed
the pulse signature produced in the stopping counter by the
particle making the track in the chamber.

1. The optical arrangement

The optical schematic of the chamber camera arrange-
ment is shown in Figure D-l. A large chamber-to-camera
distance was chosen in order to assure an adequate depth
of field and to compensate for the fact that the spark chamber
plates were parallel. These requirements were made more
demanding by the use of a 45 degree mirror to project a
90 degree steroscopic view of the chamber onto the film.

As can be seen from the figure, a useful depth of field
twice the chamber depth is required for this type of view
since the two images are not located at equal distances
from the lens. This results in a minimum f number of 5.6
at a distance of 20 feet. In the room which housed the
experiment, space considerations required the use of an
additional plane mirror to fold the viewing beam back upon
itself, allowing the chamber camera to be mounted directly
above the chamber.

As is also apparent from the figure, the image of
21

22

the stereo view will be smaller than that of the direct
view by the ratio of their distances. Correction for this
effect as well as that of parallax was made in the analysis
procedure. A separate section of the analysis program
performed this task.
2. The operational system

The camera system was designed for fully automatic
operation. This was accomplished by using camera units
built to expose 1200 frames per 100 foot roll of 35 mm film
and by the construction of a camera control unit capable of
automatic sequencing and control of all film advance functions.
The following basic functions are performed by the camera
control unit (in order of occurrence):

a. Blank the grigger circuit of spark
chamber. Blanking is accomplished by relay interruption of
the signal line to the grid of each 5022 thyratron.

b. Start film drive motor in each camera.
The sprocket motor in each camera operates on 24 volts AC.

c. Advance frame number in each camera.
Solenoid-activated six-digit counters are used for the
purpose of registering on the film the frame number of
each event.

d. Apply braking to film drive motors.

The discharge of 250 mfd at 30 volts into the motor coils
is used to provide the DC braking necessary to obtain
positive frame position control.

e. Illuminate, for a set duration, the

23

chamber itself and the frame numbers thus recording these
on the film prior to the next event.

f. Unblank the trigger circuit of spark
chamber to accept the next event.

These operations were performed in a properly timed
sequence by the use of appropriate relays controlled by the
R-C timing circuits, as described in greater detail in
Appendix D. The complete film advance sequency requires
approximately 0.85 seconds. ‘With an event rate of 2.5-3
per minute the dead time due to this operation was of the

order of 5 percent.

E. Stopping-pulse display system

A positive particle identification was sought for
this experiment in an attempt to avoid some of the prob-
lems inherent in the earlier work with cosmic-ray muons
as mentioned previously. Photographic recording of the
pulses from the stopping counter provided, in principle,
the means necessary to make positive particle identifi-
cation. The muons may be distinguished from stable pro-
tons or electrons and from pions by the number of pulses
observed and from other singly-decaying particles on a sta-
tistical basis by a check of the characteristic lifetime
of the events accepted. Thus a satisfactory mechanism of
selection is available if the pulses registered in the
stopping counter are suitably displayed for photography.
A specialized means for doing this has been developed in
the present experiment. Of the multiply-decaying particles
the most significant constituent in the cosmic-rays is, of
course, the pion. It was the large time scale difference
between the two steps in the decay of a pion via a muon to
an electron that made the development of a satisfactory
display method such a challenge.

An attempt was made to record stopping 77'+
mesons as well asdp+'s. This was done by a complex
scan pattern for the 517A oscilloscope which worked by
following a single fast sweep (6 pion lifetimes long)

with a displaced slow sweep (5 muon lifetimes long).

24

25

A Z-shaped trace was decided upon, of which the top
trace was set at 25 nsec/cm for a total sweeptime of
140 nsec to resolve‘flEp-e decays occurring in the
stopping counter. Next the sweep rate was changed
automatically to 2 Psec/cm and the beam retraced ready
for the second trace at this slower rate to display

‘pne decay pulses. The change of sweep rate was effected
by connecting a new timing capacitor corresponding to
the slower rate in parallel with the one corresponding
to the initial fast sweep. This connection was made by
a triggered thyratron just as the first trace was com-
pleted. The thyratron was extinguished again upon
completion of the slow trace ready for the next event.
Proper retrace between the fast and slow traces was

an automatic result of the above procedure since the
sweep system in the 517A consisted simply of a timing
capacitor whose voltage rose linearly with time, and

was applied to the deflection plates via a DC amplifier.
Thus the voltage accumulated upon the first timing
capacitor was immediately reduced, upon connection of
the new uncharged timing capacitor, to l per cent (by
the ratio of sweep rates) of its value at the end of

the first trace. The sole reason the beam retraced
between times was that the second capacitor was
initially uncharged and was larger by a factor of 100
than the first. Thus the beam should return essentially

to its starting point on the screen. Retrace instead to

26

      
  

muon stopping_

pulse

 

 

 

  
   
 

e stopping sweep marker
ulse ulse

 

 

Figure 11-5. Characteristic oscilloscope trace,

showing )J-e decay signature.

27

approximately 10 per cent of the full scale value was the
practical result, due to stray capacitance in the sweep
amplifier and deflection plate leads. The retrace re-
quired approximately 0.80 psec, thus placing the begin-
ning of the second trace at about 1 psec after the
passage of the particle through the apparatus.

Two additional modifications were added for
analysis convenience. A negative step pulse was
generated and applied to the vertical input along
with the regular signal to displace the second trace
to a position below the initial fast trace. The
particular portion of the trace (three in all, in-
cluding the retrace) to which each observed pulse
belongs could in principle be determined from the
apparent pulse width, since all pulses are essentially
the same width in reality. It is, however, much more
satisfactory to have the traces separated into a
Z-pattern as mentioned above. The other feature was
the addition of a negative marker pulse introduced
in the vertical input at a time of 10 psec after the
top trace. This was included for purposes of sweep
calibration monitoring. The position and height of
this pulse at the end of the second trace in every
photograph gave immediate indication of any trouble
developing in either the sweep timing or vertical
amplifier sections of the oscilloscope.

The above predetermined sweep sequence was

28
iniated by a single external trigger pulse from the

coincidence circuit. The delay encountered in initiating
conduction in the 5022 thyratrons that fire the spark
chamber was z 300 nsec so that the first fast trace of
the oscilloscope was completed before large amounts of RF
from the chamber discharge could appear in the scope picture.
It is, of course, in the top trace where problems of pulse
identification and resolution are most difficult. Slight
timing adjustment to assure this desirable condition was
made by adding 20 feet of cable to the chamber trigger line.
The scope display system had a stopping pulse width
of z 40 nsec width at half maximum. This is somewhat more
than desired as only pions persisting for more than two
lifetimes can be expected to display thedp pulse in a
777P'e decay. Efforts to materially improve this condition
have thus far been unsuccessful, but a major effort at such
improvement is scheduled as a primary task preceeding

further research to be done with the apparatus.

F. Analysis system

The analysis system consisted primarily of two
film scanning machines, one equipped with an IBM 526
key punch, which were used to prepare the data for use
on the Michigan State University computer (a C.D.C. 3600)
on which the computations were carried out.

The first of the two scanners is a Gilliland
vertical projection model modified for continuous film
traversal by means of a foot-operated speed control.
This facilitated the scanning of the oscilloscope film
for suitable events. The corresponding chamber film
photographs of events judged as having been muons that
came to rest in the stopping counter were measured on
the second scanning machine, a Hydel unit. This unit
equipped with a Datex digital encoder system and an
IBM 526 summary key punch was used to provide on IBM
cards data describing the x, y coordinates on the film
of each spark in the particle track. These coordinate
positions were punched on the cards as five-digit
numbers which were linearly related to position on
the film. The scale is arbitrary and must be converted
to position in real space; this is done in the geometry
subroutine of the program.

All computation related to track behavior in each
event was carried out in the SCOPE program especially
written for this experiment. This program performs the

following major tasks for each event:
29

30

l) Sorting and identification of each event

2) Geometry conversion to real-space coordinates

3) Storage of all converted real—space data on
magnetic tape

4) Elimination of events having an insufficient
number of data points present

5) Calculation of least-square line fits to
spark positions for incoming and outgoing directions
in both the direct and stereo views

6) Calculation of the total deflection angle
for each event from the line fits in the direct and
stereo views

7) Partitioning of the data with respect to how
many events in a given run display a track deflection
angle in each 0.25 degree increment from 0 degrees to
7 degrees.

The partitioning supplies the necessary information
from which the differential scattering distribution may
immediately be plotted. This same partitioning operation
is performed upon the calculated values of azimuth angle
for each of the track segments as well.

A more detailed description is presented in

Section III-B and a program listing is given in Appendix F.

III. ANALYSIS PROCEDURE

A. Analysis of oscilloscope photographs
1. Types of events identified
The types of events identified by analysis of the
oscilloscope photographs made of the signal emitted by the
stopping counter may be categorized according to the nume
ber of individual stopping pulses observed for each event.
An interpretation of each such category follows:

a) One-pulse event: Such an event would
have been caused by a particle which either was not stopped
within the counter or did not undergo a subsequent decay
within the observation time of the oscilloscope (which
includes stable particles, of course). In either case
all events of this type were rejected as not representing
muon events. Since the observation time of the oscilloscope
was ll‘Pseconds, i.e., five muon lifetimes, less than 0.7
percent of the real muon events were lost by rejecting
such one-pulse events.

b) Two-pulse event: Such an event was
taken to be due to a muon which was stopped in the counter
and subsequently decayed into an electron and two neu—
trinos. The electron, being given an initial kinetic
energy as a result of the decay process, produces another
stopping pulse in the scope trace, identifying the time
of the decay. All such events were marked for subsequent
track measurement. As described in section (2) below,

a random sample from events in this category yielded a
31

32
measured mean lifetime in close agreement with the

accepted value for muons.

A false two-pulse event could have been caused by
a single pulse event together with a second pulse caused
by the appearance in the counter of another cosmic ray
particle within the oscilloscope observation time. As is
shown in the next section, the fractional occurrence rate
for this is of the order of 3 x 10"+ for muons. Thus out
of the 2,266 events used in the final results of the exper-
iment only of the order of 0.7 such spurious muon events
would be expected. As a practical matter this must be
increased by a factor of 2 due to the fact that in most
of the oscilloscope photographs there is a second trace
triggered by persistent transients in the apparatus. In
the upper and middle portions of the trace the real and
spurious traces are easily distinguishable by the presence
of the initial stopping pulse and the residual radio-
frequency pickup in their respective portions of the
real trace. In the lower slow trace, however, these
two traces are indistinguishable and thus provide a dup-
licate time interval in which a pulse from the cosmic ray
background could appear. It is asserted that the presence
of this second trace in the oscilloscope photographs does
not affect the reliability of identifying two-pulser-e
events. This is because the occurrence of any false’p-e
event having its second pulse in the lower trace of the

spurious sweep would be governed by the above mentioned

33

expectation of 0.7 such occasions out of 2,266 events.
Even this total of 1.4 possible false p-e events causes a
negligible effect on the resulting scattering distribution
of the experiment. The pee lifetime plot shown in part 2.,
of this section was made assuming all two—pulse events to have
occurred within the original trace. No inconsistency with
this assumption is apparent in the lifetime results.

The contribution of spurious two-pulse events in
which one of the particles was a proton is expected only
in the ratio that protons are present with muons in the
cosmic rays. This ratio of protons to muons in the cosmic
rays above 0.7 Bev/c is less than 1 percent according to
Rossils. Thus only (10‘2)(3 x 10'“)(2,266) = 6.8 x 10"3
events out of 2,266 events are expected to be of this type.
Contamination of the data due to protons may thus also be
neglected.

c) Three pulse event: Such an event was

taken to be one of the following: (1) a flfipne event,
(2) an event in which another charged particle entered the
counter within the observation time of the scope trace in
addition to a bona fide pee event, or (3) an event in which
two additional charged particles entered the stopping
counter following an initial single pulse event. As can
be seen from the several possibilities mentioned, the
rejection of all observed three-pulse events constitutes a
more than sufficient means by which to eliminate from the
scattering data such pion events as produced a flap-e

decay in the stopping counter.

34
Events involving pions which traverse the chamber

but decay into muons in flight before reaching the stop-
ping counter cannot be eliminated from the data by the
above selection method if the muon formed in flight sub-
sequently decays in the normal manner in the counter.
However, because of the relatively short time of flight
at relativistic velocities for the distance of 4.5 feet
from spark chamber to stopping counter, together with
the relativistic lifetime dilatation existing through
the greater portion of this distance, the number of

pion events not identified because of in-flight decay
was considered negligible.

The relative number of three-pulse events caused
by mechanism (2) and (3) may be estimated in the following
manner. First of all the total muon intensity must be
calculated for the apparatus of the experiment. The
integral muon intensity, Iv, at sea level due to the
hard component of the cosmic rays is given as 0.008/cm2-
sterad-sec by Rossi16. The total count rate may be ob-
tained from this by integrating over the area and solid

angle of the counter.
6....

an
The count rate .3721: [yo/(1', (of 9).“) 90/90/975
0

Taking Iv = 0.008/cm2-sterad-sec, A = 2600 cm2 for the
stopping counter, and gmax. = 700 from vertical due to
cutoff by depression of basement laboratory below sur-

rounding terrain, gives

70‘
jig=a7TIy[% “5" 5123/7 = war/5..

35
This is only a rough estimate since no account was taken

of the shielding effect of the lead pile above the st0p~
ping counter.

A more careful estimation is next obtained by con-
sidering the total solid angle in three zones: 05 86. 9°;
904 92270; and 27°< 95 70°. The value of 90 was chosen
because it represents the half angle of the cone subtended
at the stopping counter by the area of the top of the lead
pile. The next interval of 180 was chosen to be equal to
the full angle of the central cone for symmetry purposes.
For each zone a different value of IV was taken according
to the approximate amount of effective absorbing material
present in each particular zone.

dN/dt = (77/3)g[2(.00u5)(930) + (.0045)(2600~930)

+ (.0062)(2600-930)] [1-<.987)3] +

+ £2(.0053)(930)+(.0053)(2600~930)+(.OO62)0

(2600-930)_][ <.987)3-(.888)§ + [2(.0062)-

(930)+( .0053) (2600-930)+( .0073) (2600-930)] -

[C.888)3-<.3u2)3]}
This yields dN/dt = 31.5 incident muons/sec. From this
value of dN/dt the relative probability for appearance of
an additional pulse on the oscilloscope within the 10,psec
observation time may be obtained. This relative probabil-
ity is equal to the ratio of the observation time to the
average time between background muons. Thus

Prob. = Tobs/Tbackground = Tobs(dN/dt)=(10 x 10‘6seo)°

'(3l.5/sec)

3.15 x 10““

36
Thus for three-pulse events having another muon

in addition to a p-e decay the above calculated prob-
ability gives an expectation of five events out of 8,000.
These false three-pulse events will appear with one pulse
in the top trace and the remaining two somewhere in the
middle or lower traces. There are other higher order
possibilities by which three-pulse events of this type
may occur. These include P—e, p; )1, p, )J;}J, )1, p. The
relative probabilities for these mechanisms are smaller
than the above by factors of 0.01, 3.15 x 10‘“, and
3.15 x 10"6 respectively.

Since the sweep duration of the fast upper trace
is nearly six pion lifetimes (mean pion lifetime taken
as 25.5 x 10"8 sec21), more than 99 percent of all real
7T-p-e events should appear with two pulses in the top
trace. In contrast, however, the probability for a second

muon pulse to appear in the top trace is
Prob. = 140 x 10"9 sec (31.5/sec) = 4.41 x 10-5.

This gives an expectation of 0.035 occurrences of an
additional pulse in the top trace out of 8,000 events.

The appearance of a p-e decay in the first trace with

a subsequent additional muon pulse appearing in the

slow bottom trace is the only remaining first order
mechanism for false three-pulse events having two pulses in
the top trace. The probability of having a P—e decay
within 140 mp-seconds is

37

—F50 x 10-9 - 140 x 10-9)
Noe (2.2 x 10-6 - Noe (2.2 x IO‘5

No

 

 

-2.27 x 10-3 -6.36 x 10‘2

= e -e = 0.059
This probability must further be multiplied by 3.15 x 10‘“,
the probability obtained above for having an additional
muon within the 10 p-sec observation time, yielding
1.86 x 10‘s. This gives an expectation of 0.15 occur-
rences out of 8,000 for three-pulse events having a P-e
in the top trace followed by an additional pulse in the
lower trace. There is a total expectation in 8,000 events
of 0.035 + 0.15 = 0.185 false occurrences of three-pulse
events having two pulses in the top trace, forming an
apparent 1ij-e event. This is still, of course, a very
small expectation, implying that if any decays are seen in
the top trace with a third pulse later, they should be con-
sidered bona fide 171p-e events. Of the 6 three-pulse events
observed, none displayed two pulses in the top trace.

The fact that no three-pulse events qualifying as
‘flfiP-e events were observed must be compared with the ex-
pected pion flux at sea level. This flux is, however, not
very well known as yet. The best estimate which seems to
be available at present is due to B. Peters and Y. Pal of
the Niels Bohr Institute in Copenhagen37. They estimate the

integral pion flux to be of the order of 10 percent as great

38
as that of protons in the momentum region of this experiment.

The proton to muon ratio in turn is less than one percent
according to Rossil6. Thus out of 8,000 muon events, some-
thing less than eight 1T-).1—e events should be expected.

Not all pion decay events occurring in the stopping
counter can be observed in the oscilloscope trace due to
the width of the stopping pulse. The minimum resolving
time is 50 mpsec or 2 pion lifetimes. Assuming a pion life-

time of 25.5 mpsec21

only the fraction 0.135 remain of the
original number present. Thus out of the eight pions sug-
gested above, at most one observed case should be expected.
Indeed the figure of eight expected pions out of 8,000 muon
events may well be excessive due to the fact that in looking
for pion decays, one is concerned with the differential
rather than the integral flux. The relative differential
pion flux in the region of 1.6 Bev/c may well be notably
smaller than the relative integral flux as there is indi-
cation from cosmic-ray results37 that the differential pion
flux actually decreases with momentum below several Bev/c.
Since out of 8,000 events no 7ij-e events were
observed in the decay time range of 50-140 mysec, an upper
expected limit to the observed pion incidence for this
experiment of l in 8,000 may be set. Since only 0.135 of
the total incident pions have their decay observable in the
time range of 50—140 mpsec, this corresponds to an observed
total upper limit of 7 pions in 8,000 events. This value is

consistent with the value of 8 pions in 8,000 events

39
calculated above. The value of 7 in 8,000 corresponds to a

maximum expectation of 2 pions in the 2,266 events used in
the final results of the experiment. It must further be
borne in mind that both the calculated value of 8 in 8,000
and the observed value of 7 in 8,000 are upper limits. As
indicated above there is reason to believe that both the
calculated and observed figures for the number of pions

included in the data of the experiment are high.

2. Lifetime study

As a check on the acceptance criteria for events chosen
for measurement as muon events having the proper momentum, the
muon lifetime was calculated from a sample of 1,280 events
chosen at random from the data of this experiment. These were
selected by choosing some 50 rolls of oscilloscope film dis—
tributed uniformly throughout the total film data and meas-
uring the delay time to the second pulse of each two—pulse
event in that film roll. Thus the sample was also taken
without regard to whether that event was subsequently
measured for scattering or rejected due to poor track quality.
As it turned out 676 events of the 1,280 had acceptable
tracks for measurement, 604 events did not. Fig. III-l shows
a plot of N(t)/No versus distance along the oscilloscope
trace measured in centimeters on the scanning table. From
this distance in centimeters the corresponding time t may be
determined from the geometry of the system. Taking d as the
distance of pulse separation on the scanning table, the con-

version to time is as follows:

40

5.3 cm on scope (9.3 psec)
39.8 cm on scanner ( 47cm )

[0. 310 nsec/cm] d

The solid line of Fig. III-1 is the theoretical decay curve
for muons. It was obtained from the decay law

N(t) = Noe“'}‘t = Noe-t/pc
Differentiating gives

-t = ln (N(t)) = N
__. Tfi“'7 In 10 log (t)

 

’C o No

Or 1n 10 log (No > z = (0.301 psechm)d
TNT-{3) t/T 2.20 JJSEC

2.302 log (No/N(t)) = (l.403/cm)d

Choosing d = 20 cm for convenience, a ratio of No/N(t) =
16.60 satisfies this equation. The solid line in Fig. III—1
is drawn with this slope. Slight departure from a straight
line at each extreme of the plot is due to nonlinearities

in the oscilloscope sweep rate at the extremes of the trace.
This came about in connection with the response time of the
circuit causing the automatic change in sweep speed after
the first fast trace. A best fit to the data in Fig. III-l
yields a muon mean life 0.6 percent less than the accepted
value of 2.201Psec21 used above in forming the theoretical
curve. This is well within the estimated 1 percent sweep
calibration tolerance that it was found possible to main-
tain over the long term duration of the experiment. In

view of the good agreement of the measured mean life with
the accepted value for muons, the particle selection procedure

used in this experiment is established as a satisfactory

criterion for muon selection.

__10-1

L'7

5
l l

 

Figure III-1.

’C‘ =2.20pseo/'

41

     

(conversion to time is 0.310psec/cm.)

10 15 20 25cm.

l l I l 1 l J l

j

Normalized lifetime plot of 1280 events
selected at random from 8000 two-pulse
events.

B. Analysis of track photographs

The muon track photographs taken in the spark
chamber were measured on a Hydel scanning machine and
the digitized output was punched out on IBM cards by
the IBM 526 Summary punch attached to the Hydel. With
this scanning-punching chain the x-y coordinate position
of any point on the film frame may be recorded on IBM
cards in b-digit numbers having an arbitrary scale
factor but nearing a relationship to position on the
film which is linear to rive significant figures.

Each data card contains a ten-column heading
which records the film frame number being measured,
the view number ("1” for direct, "2" for stereo),
card number within a given view measurement (1—5 to
complete the measurements in each View), and the
operator number. The remaining seventy columns are
available for punching data. Since both the x-y
coordinates require 5 columns on the card, totalling
ten columns per point on the film, each card carries
the coordinate data of seven points along the track.
The position of the center of the spark in each gap
along the track was recorded in this manner; there
are 30 such points to be recorded in each view. In
addition to these 30 points the upper and lower front
corners of the main chamber, which are visible in both
views, were chosen as tiducial marks. These riducial

marks were used as a means of establishing both the
42

43
vertical Z axis reference direction and the scale factor
pertaining to real distances in each view. This makes a
total of 32 points in each view, requiring five cards per
View or ten cards in all, per event. From these recorded
points the complete track can be reconstructed and the
desired quantities calculated using a computer program

‘written for this experiment.

C. Data reduction

The computer program, for which a listing is given
in Appendix F, performed the following tasks upon the raw
track data obtained on the Hydel scanner.

1. Sorting and geometric reconstruction

a. The cards are processed ten at a time
and are checked for identical frame number and sequential
View and card numbers. If cards are missing or out of
order, the program will cause an appropriate diagnostic
to be printed, scrap that particular event and move on
to the next.

b. The position data of the 32 points is
first converted to centimeters measured in real space.
Then the coordinate points are corrected for camera
parallax, using the known optical geometry of the
experimental set-up.

c. These data are then written on tape
for permament storage. This phase of the calculations will
thus never have to be repeated for subsequent runs with
the same data. Nor will the original data cards again be
needed, as future calculations can be made directly from
the tape.

2. Performing the necessary calculations upon
the data tape

a. First the appropriate track points are

taken from memory, from which line fits are formed

44

45
to the incoming and outgoing directions in both chamber

views. There are often gaps in the chamber which do not
fire in a given photograph, leaving points missing from
the track data. A minimum of two points in the four gaps
or the thin plate chamber and two in the first tour gaps
of the main chamber is set as the criteria for acceptance
of a given event for processing. A maximum of six points
would be available for the line fit if all gaps in the
thin plate chambers fired properly.

b. Having met this criteria, a least
squares line fit is made to each of the projected track
segments. This line fit process yields values for the
slope with reapect to the u a: is and the intercept in
the x-y plane at Z = 0.

c. A statistical parameter called R2 is

calculated for each of the four line fits done above.

This q aantity defined as

_e a «t- a
(m- OVEN... ,ZXL-Mr)” 2, Z. ‘MGUD

 

 

 

 

#1-'1
where lg
. M —.—= 2*
X-‘fiélxk ‘Rd 2 mks!

has values between 0 - 1.0 and is a measure of how
nearly the observed points fall exactly on the least
squares line. A cutoff value of R2 = 0.960 was estab-

lished by correlating the values obtained for R2 with

46
the appearance of the track as viewed on the scanning screen.

The event was rejected if any one of the four line fits
yielded a value of R2 less than 0.960.

Causes of this could be either operator error or
individual spark displacements. The former was effectively
eliminated by remeasuring all events rejected for too low a
value of R2. A portion of the rejected events were recovered
in this way. The latter are assumed to be caused by delta
rays formed in a given gap. These events which pass the R2
test have their line fit parameters made available to the
space angle and projected angle calculation procedUre
that follows.

d. Another subroutine uses the values obtained
in step b. above for the slope and intercept of each track
segment, to determine the azimuth angle O of both the
incoming and outgoing tracks projected on the x-y plane.

From the signs of the slopes, the quadrant in which each
track segment lies around the Z axis in the upper hemisphere
is determined. These quadrant values are in turn used in
assigning the correct value of 4) to the arctan¢ , yielding
values ranging from 0 degrees to 360 degrees measured
counterclockwise from the positive X axis.

e. The value of the apparent total deflection
angle 9 present in the track is calculated in a function

subroutine, by a process derived upon direction cosines.

47
f. Next the projected angle with respect

to the + Z axis of each track segment is calculated from
the corresponding slope parameter. A modified value of
the two projected angles in the direct view is formed by
correcting for the inclination angle in the y-z plane
made by the top track. This is done in order than when
the projected deflection angle in the direct view is
calculated from the value of the projected incoming and
outgoing directions, it will be indeed the deflection
angle projected in every case on a plane containing the
incoming direction of the particle. This formality is
dictated by the nature of the statistical scattering
theories with which the experimental results are to be
compared. In principle this is quite different from the
deflection angle projected upon an arbitrary plane, i.e.,
that of the front face of the chamber, which does not
necessarily contain the direction of the incoming par-
ticles coming in from random directions within the
vertical acceptance cone of the counter telescope
system. Yet since this cone has a maximum opening

angle of 15 degrees or less, the correction to the
calculated deflection angle required for track in~
clination amounts to only a few percent. The values

of incoming and outgoing projected angles for both
views are printed out on the Projected Angle Data

page. Along with this are printed the calculated

values of projected deflection angle in the direct

48
and stereo views and the projected deflection angle in

the direct view, corrected as described above.

g. A calculation of the zenith angle (with
respect to the +Z axis) of the incoming track was per-
formed at this point from the values found for the slope
of the incoming track in each view. Vector addition of the
tangents in each view was used to find the tangent of the.

zenith angle.

3. Summary functions performed by the computer

program

a. A partitioning operation was performed
upon the resulting values of ¢in, ¢out, the total
deflection angle 6, the zenith angle gin, the incoming
direction in the direct view (projected) Gin, and the
corrected projected deflection angle in direct View
9 dir. This operation consists of having the computer
take successive small increments of the quantity in
question, counting and storing for each interval the
total number of cases, out of each group of fifty events,
that yielded a value of that quantity within each such
interval. This is a valuable aid to subsequent plotting
of the results in the form of distribution functions for each
of the above quantities. Each interval is defined to
be single-ended so as to avoid ambiguity for any

resulting value of the angle. In the case of ¢ in,

49
for example, the first interval is formed by the relation

02 ¢ C 10°; the second is 10 Q ¢< 20’, et cetera, up
to 360 degrees. The interval size chosen for the above
quantities is 10 degrees, 10 degrees, 0.25 degrees,
0.20 degrees, 0.20 degrees, and 0.25 degrees respectively.
The number of events occurring in each interval is cum~
ulative for each successive group of fifty events through-
out the entire run.

b. There is assembled at the end of each
group of fifty events a list of frame numbers for all
events rejected for having an insufficient number of
spark points from which to form a line fit.

c. There is also assembled at this point
a list of those events rejected for R2 less than 0.960.
Recovery of these events was attempted by remeasuring
each such rejected event. Some 15-20 percent of such
remeasured events then passed the R2 test.

4. Print-out of results
The following information was printed out by the
program in this order.

a. Information printed out at beginning
of run

(1). List of events stored on tape.
After storage of the data on tape, a complete list of
frame numbers of all such events is printed out in
groups of fifty.

(2). List of frame numbers for

events not placed on tape due to cards out of order

50
Within event, wrong event number on a card, et cetera.

Along with each frame number will be an appropriate
diagnostic statement describing the reason for deletion.

b. Information printed after processing
of each group of fifty events throughout the run

(1). Line data page. The slopes,
intercepts, correlation factors, and number of points
used are listed for each of the four line fits asso-
ciated with each event number. Two pages are required
to list this information for fifty events.

(2). Space angle data page. On this
page the computed values of azimuth angle ¢ and corre-
sponding quadrant are printed out for both the incoming
and outgoing track. Along with this the calculated value
of the projected incoming direction a‘in and of the total
deflection angle 9is listed. A final column relists all
values of 9 which were 1.000 degrees or more, with zeros
listed for all events having Bless than 1.000 degrees.
The events whose value of 9 is listed in this column
have their spark position data duplicated on another tape
for separate storage of all events giving high deflection
angles.

(3). Projected angle data page. On
this page are printed the calculated values of the in-
coming and outgoing projected angles for both views, the
values of the projected deflection angle in each view,

and the projected deflection angle in the direct view,

51
corrected for track slant as described previously.

(4). First list of rejected events.
A list of event numbers for events having less than the
minimum number of spark points present is printed on
the page following the projected angle data.

(5). Second list of rejected events.
A list of event numbers for events not passing the 2
0.960 test on R2 is printed on the next page. Events
appearing in this list were remeasured and rerun as a
group.

c. Information printed at the end of

an entire run

(1). The last page number used in
the run.

(2). The total number of events
processed to date.

(3). The total number of events
"saved" (i.e., having 9 2. 1.000 degree) to date. This
information is printed out so that a total count of these
quantities may be accumulated over any number of runs.
This is accomplished by reading these numbers back in on
a data card for the next run.

(4). Also the number of events saved
during this run is listed.

(5). List of the frame numbers of
every event having a calculated track deflection angle

9 of 1.000 degree or greater.

52

(6). Results of the partitioning
subroutine upon P in, ¢ out, 9 , 9 dir. (cor-
rected),akf,, 0:16 in as defined above are printed out
last, using a separate page for each.

A complete listing of Program Spark is presented

in Appendix F.

IV. THEORETICAL PREDICTIONS

The data of this experiment are to be compared with
the scattering theories set forth by L. Cooper and J. Rain-
water9 and by s. D. Drell and c. L. Schwartzl7 for multiple
scattering of fast particles by extended nuclei. These
methods deal directly with the case where the scattering
centers are nuclei of a known charge distribution of finite
extent. This is in contrast to previous treatments of the
problem, notably the Molierelo theory which assumes the
nucleus to be a point charge and the Olbert8 theory which,
as a first approximation to treating the extended nucleus
case, assumes that the single scattering law becomes iden-
tically zero for angles greater than a certain maximum
angle Cg calculated from the properties of the scattering
material in question. As can be inferred from the above
listed assumptions, the former theory overestimates the
multiple scattering at higher angles, while the latter

underestimates predicted scattering in this region.

A. The Cooper-Rainwater calculation
The Cooper-Rainwater theory begins as does that of
Moliere and of Olbert with a single scattering law which is
the Rutherford scattering law, modified to account for
shielding by atomic electrons. This law has the following
form:
tumor“) = 0/2(c92 +CPm2)‘3/2

where C? is the projected angle

(Fm is the equivalent screening angle
53

54
and Q = 477 (Nt/A)[(Z/p )(eZ/mecz)(0.51l/pc‘)7a

eZ/mec2 = re = 2.82 x 10'13 cm
N = Avagadro's number
Z = proton number of the scattering material

A = atomic number of the scattering material

(1’
ll

thickness of the scattering material

9 = v/c :5l.for'the relativistic particles

dealt with in this experiment

pc = momentum in Mev.
This basic single scattering law is taken as is, in the
Moliere theory. In the Olbert treatment f1(4’,4’m) is
taken as above for angles CPsuch than OéIU/flq’al but
is set E 0 for ICPI>C,9,. The cutoff angle CPO is deter-
mined from 69m by the relationship

(1.67 x lQE)rg§:}{i
a/rn = roAIIJ

(fb/CPm

_ (1.67 x 10“)(2.81a;x 10-13)
' 1.1 x lO-I3IAZL/3

 

_ 1 3 2
where CPm.- lslfleiflacz) Z / 1.13 + 3.76(z/137.€)2 l/
137pc

z x/a = ’hc/pca

and £90 =‘fic/pcrn ='hc/pc(l.l x 10"13)A1/3
The Cooper-Rainwater theory, however, employs the above
single scattering law for all angles, modified by the
appearance of a form factor which also is a function of
angle, thus:

fIC'RM) = 0/2<a>2 + CPm2)'3/2 fiNW/aa

where 5FN(CP/ZPO) = 5FN(y) is defined by

5fn<y> = FNC<y> + FNI<y)

55 ~
where FNC(y) is the term contributed by elastic

scattering.

In fact (FNC(y))% is the Fourier transform of the
nuclear charge distribution. FNI(y) is the term contrib-
uted by inelastic scattering. FNI can be written approx-
imately as 1/2 (l~FNC)(Fp) where Fp is the form factor for
scattering by a single proton. This, however, can be taken
as 5:51 for momentum transfers 2?. 200 Mev/c. Thus its
effect can be neglected for all present purposes.

The exact form taken by Cooper-Rainwater for FNC,
chosen in such a way as not to underestimate the effects
due to multiple scattering, is defined by a curve drawn
through a set of points for which values of FNC were
given and then attached to an asymptotic formula

for y = 0, l, 2, 3, FNC is taken as follows:

FNC = 1.00, 0.82, 0.50 and 0.15.

For yZ u, FNC = 12/y‘+.

This form for FNC amounts to taking up a position between
the results of assuming a uniform charge distribution in
the nucleus, which gives a value of 9/y4 for large values
of y, and the results set forth by Williams which give a
value of 16/y4 for large values of y.

This choice of FNC(y) results in an fi$fi(y) such that
5TN(Y) = l for small values of y = I‘Q/CPOI
and becomes Z‘1 for large values of y = I‘p/Wo’

with the rapid change occurring around y 252. This is an

improvement over Olbert's tacit assumption of:

In

56
FN(y) = l for yg l

and E 0 for y > 1
In the present theory §N(y)becomes Z":L instead of zero
for large values of y and the functional form is known by
which it reaches this value from its value of l for y = 0.
The Fourier representation of the single scattering

law can be written thus

act-3(4) = iii/ECUeV‘? «N

I’ .
and g0) =-/‘fiC‘R(CP)e“]-7Q (14?
From this a value for the probability of a final projected
angle after any number of scatterings is given by
11(0)ch = da/27[21w e g<7>"g(°)df

The quantity g(T) - g(0) = Q/2 (e170,?"1) 31K d'flflfldfl'
_.. (0'2 +4m2)3/2

For the sake of convenience, the variables are transformed,
following Moliere, as follows:

x = (200)-249, n = (zcofif

and G = -g- 1n[( XZ/e)CPm2/2GQ]

2, 24/3 A"1 t
._ 5.66 + 1.2L» log 10 [1.139 + 3.76CZ/137)Z

Thus gm - em) = g<7(/<260>%> - gm E 300

Whence 8(n) = l/2E//’

a
M(x) then becomes
'

e/einx esc’oan

After suitable integration and expansion by powers

 

of l/2G as described in the Cooper-Rainwater paper, the

following form of MCX) is obtained.

57

MC'RCX) = e'xz/W/‘f [1 + q(1,x)/t+0] + l/4GVTN(L,X)
q(L,x) = 2(2x2-1) [1n(L/1.26) +

{2%osth)‘1/t)dt] + 6x2 -(i/L%)<cosh (2139-1)
-(2x/L) sinh (21x)

where L = 1/4 and

2L
/ DZ’(cosh(t)---j/E7 dt fi<2Lx)2/2.2 +
0

+ (21.x)l4/L‘.I+: + ‘ . . . + (2LX)n/n(n:)

for values of n = 2, 4, 6, 8, . . . .

av
NCL,x) =/ >\-3 T(x’,)\)d)~
L

T(x,A) = exp [;(x +ql)%7 + exp [;(x-A)E]-2 expC-xz)

B. The Moliere calculation
The Moliere multiple scattering distribution which
assumes a point nucleus,is latent in the expression:

‘
MC-R(x) = 1/21r/ei7tx e301) c171

-’

an
- jI‘JCX'/X )(cosQLx')-l)dx'
where 3(a) - l/2G 0 (x'3+ Xm2)3/§—

The single scattering law used by Moliere was

far) = 0/2 (4’2 + 42112er

This differs from the one used by COOpernflainwater
only by the omission of the factor 5f3(dth). Thus the
Moliere scattering distribution may be obtained from the

Cooper-Rainwater calculation by merely setting fNCy) = 1

—‘

58
in the above expression for M(x). The machinery set up
to calculate M(x) from the final form shown in part A
above can thus be used to also calculate MM01'(X). The
results of this Moliere calculation are used for com-

parison purposes in discussing the results of the present

experiment.

C. The Drell-Schwartz calculation

The data of this experiment are further to be
compared with the scattering distribution predicted by
the method set forth by Drell and Schwartzl7. In this
method the total differential scattering cross section
(elastic plus inelastic) is constructed from a knowledge
of the nuclear ground state wave functions. This method
considers the specific case of high energy electrons
scattering from light nuclei so that the interaction
may be treated in the first Born approximation. Only
values of the momentum transfer [if] 5. 200 Mev/C are
considered so that the nucleon recoil velocities Vrec
are kept well in the nonrelativistic range ( Free ,6 1/5).
In this region of momentum transfer values the dominant
factors in the scattering are the Coulomb forces and
the magnetic forces, viz., single-particle magnetic
moment interactions.

The form for the low resolution (all final
energies accepted at each specified angle) differential
cross section df/dn, given as equation 19 in the Drell-

Schwartz paper, is
dU-__ a Z 1 2 9/ —1 o
__. 2%”)fi‘flfjfii’ +59%, + (25c: ( a) )

d.m
[GAME/M4 T> " 7%" (2%,” W *
+13 gig 4 53' '7'} MA; "I" ("i ’ 14499.7},

,3 me (05 (94) i
The term d (To/€111 = Aka slh’fl/a) ko

 

59

60

2.818 x 10'l3cm (classical electron radius)

re

me 0.511 Mev (mass of electron in Mev units)
K = PC (in Mev) where P is the final momentum of
the particle after scattering
K0 = P00 (in Mev) where po is the initial momentum

of the particle before scattering

fl = total Spatial scattering angle (in radians)

Note here that only through the factor reme do the
properties of the electron, which was taken to be the
incident particle in the Drell-Schwartz development, appear.
Since, however, the classical electron radius re is

defined as
re = ez/meC2 = 2.818 x 10‘13cm

the product reme = e2/C2 is actually independent of
electron characteristics. Thus for scattering of muons

no change is required in d Vo/dn. since
rpmp = (eZ/mPCZ)mP = e2/C2 = reme.
From the kinematics of the problem the relationship

between K and K0 is

l

 

K/KO = _
1 + 2Ko/AM sin2 0/2

This ratio is essentially unity for the present experiment
since the quantity (2Ko sin2 5/2)/AM is 2.69 x 10"3 even

for 9 = 17 degrees, the largest value encountered any

61
place in the calculation.

The term f2(q), the proton form factor given by
Hofstadter18 as
2 = - 2

The momentum transfer q, from the kinematics has the form
. i
q = 2Ko 310(9/2)[l + 2Ko/AM sin2(9/23‘2

= 2Ko sin( 6/2) V K/Ko

Since K/Ko was shown to be essentially unity above, q is
replaced by qo, the momentum transfer as a function of 6

for purely elastic scattering
qO = 2K0 SID 9/2

This value of qo was used in the present calculation
wherever q appears in the expression for da'/d;¢ above.
For use in calculating the proton form factor above, qo
must be in units of f‘l. This requirement is satisfied
by using .K0 in units of PC/‘FtC where PC is in Mev and
11C is in Mev—f.

The term lglq) is related to the form factor for

two body correlations and is given by Drell-Schwartz as
* o4 A ‘5)
j(q)=Z+Z(Z-1)/¢po elq'h‘l-rzqgo’r

where FCQ) is the form factor for two body correlations.

In an electron scattering experiment by Chollet, Bounin

62

and Bishop19 performed at the Linear Accelerator Labor-
atory of the University of Paris at Orsay, the form of
F(q) for aluminum in the momentum transfer region of this

experiment was found (in terms of x) to be

Feed [1 -(16/39)x +(5/1559x2] e-(x/2)

where x = (aq)2/2 with the quantity a, the

nuclear size parameter, = 1.86 x 10'13cm according to

their results. For the present experiment q may be replaced
by qo in the above expressions. The terms following fan)
in the above expression for the cross section represent
scattering contributions due to magnetic forces and are
denoted herein by finagmo). The terms infinagmo) were
evaluated for the present experiment using Z = 13, A = 27,

M = l3(938.256) + 14(939.552) = 938.8 Mev, q -_-’: qo
27

s<TT> (the average ground state kinetic energy per nucleon)
= 30 Mev, ,pp = 2.793 nm, andlpn = -l.9l nm. The final
term in the 5Fhag(qo), however, was taken as negligible
since it is not expected to make any larger relative
contribution than was the case in the work of Masek, et
31.? for carbon.

The differential cross section obtained as above
was converted to a differential scattering distribution

by the following defining relationship.20

40' __ 111’.)
Iii-(ma New

63
where N0 = total number of incident particles

()n = density of nuclei (nuclei per unit volume)
t = thickness of target in cm.

Also dn. is defined 27 sin 9d? , whence

d 9/ d1; l/2T/sin 0

Using this in the above relationship yields
d T/dn(NofJnt) = (dN/daxda/dm) = (dN/da)(1/2p'sin9)

Thus (on/d6), = 277‘ sin9€nt(d0’/d}t.)g
Now En can be written as

P“ - ‘“°‘/m%;:;:“;:.:ia°m "wast: - M A
where NA is Avagadro's number, (niis the mass
density, and A is the atomic weight.

The quantity No, the total number of incident
particles, was set equal to 1.0, with the result that
the quantity (dN/d 9), d9 gives the probability for a
given particle to be scattered to the element of angle
d9 at 9 . This facilitates the normalization of all
distributions to unity, which has been the practice in
this work. ‘

The above expression for (dN/da )9 is written
in terms of total spatial scattering angle rather than
projected angles. Since, however, both the Cooper-

Rainwater and Moliere multiple scattering theories,

64
described in parts A. and B. of this section, are written

in terms of projected angles, a method was developed for
converting the Drell-Schwartz single scattering distribution
to projected angles. This method is based on taking a
conical view of the path of an incoming particle. The
point of scattering defines the apex of the cone formed
by a figure of revolution done upon the deflected track
about an axis along the incoming direction. Azimuthal
symme:ry about the incoming direction is assumed throughout.
Taking such a conical view of the scattering geometry, a
plane may be imagined at an arbitrary distance R0 from the
scattering site along the incoming direction and perpen-
dicular to it. On this plane may be imagined concentric
circles about the point where the incoming direction vector
pierces the plane. These concentric circles are constructed
such that each i-th one has an equal incremental radius
increase dri over the previous one of Rod 6? .

The relative number of projected angle events
occurring within d (.7 at C? were found by establishing
a plane of projection containing the incoming direction
vector. The contributions to the projected angle scattering
distribution from the various spatial deflection angles were
found by calculating the fractional area of each successive
concentric ring which lies between the parallel lines in
the plane of the circles, perpendicular to the plane of
projection, that bound the element of projected angle d6? .
For each element d6? at projected angle 47the fractional

area contributions for all 97 (P weighted by the

65

scattering distribution dN/d9 evaluated ate , are summed
to form the total value of (IN/d6? evaluated at (P .

The resulting conversion relationship is as follows

.0

(dN/dqlhy = g}, exam/<19 )9, where the sum-
mation is incremented by an amount A 9 . At a given
value of 9 the beginning point of the summation is at

9io = 4+ A9. Thereafter the successive terms in the
sum are evaluated at 01 = C? +(i - i0 + 1)49 , where
using 459 is taken to be numerically equivalent to using
the quantity 4 47. The Ci coefficients represent the
fraction of each concentric ring which falls within the
parallel lines bounding the zone of width RodC? at angle 6? .
The first coefficient C10 arises from an area which is a
segment of a circle and is bounded at 91=CP tnra chord
of the circle as it bounds the element A4) and at I9 =

C? +467 by the arc segment which is tangential to the

remaining boundary of dfl. The area of such a segment

is found from geometry to be given by
310 = %r210 (2 Yb" sinZ [10)

where {10 is the angle in the plane of the rings measured
in either direction from the projection plane around to the
intersection of the lower boundary of 44’ with the outside

of the io-th ring. From the above definition Yio is

given by
I (up _ )9, may _ 10.0
3 -arcos r4; - arcos Fraud: - arcos true,

‘64»?

‘t«(&P+A€P)

3‘ BFCDS

66
There are two identical area contributions such as that

above, symmetrically placed about the incoming direction

vector. Thus the fractional area coefficient Cio =
34;, (ii/F WCJQ‘SMJX‘SZJ
A..- ”1.0-1 7[’:‘03 _ 6:. '1 a]

tan"1 9;, (.1 )2, ~5/n 1);")
7666151494" ‘ t4”’€;.1)

 

 

 

.1

To form the remaining terms in the summation Xio 1s

generalized to

Y1 = arcos 61) = arcosaznni.) = arcosftdhgo/
a I t‘h(294(zage04uéy

In all cases of 1) 10 the area contributions di were

 

approximated by a parallelogram whose area is given by

a. c[cascP(R¢)AC}_’]L-€r—j—:5: (R6; 49)]

where R4: RO/cosd, Roi = Ro/cos 91, and [49’ is

taken equal to [4?] . Thus (11 becomes
__ MACH:
4" " sin )5;

It is apparent that rather than two there are
four such area segments in the i-th ring symmetrically
placed around the incoming direction vector. The

fractional area coefficient Ci then is
.1
44, _ 4140/ R3
’FL- Fla-1 “mange”, tang) 5;?

 

67
The values of Ci thus calculated, together with the values

of dN/da , were then used to form the sum defined above
for each value of (,7 at which a value of (dN/dCP )6, was
desired. The upper limit of the summation is set at 0° ;
in practice, however, this series converges very rapidly
since (dN/da )9 decreases by five orders of magnitude by
about 14 degrees. 01 also decreases monotonically for
larger angles.

The set of values for (dN/da)“; obtained as
described above constitutes a series of points on the
differential scattering distribution for projected
angle single scattering. In order to be directly com-
parable with the results of the Cooper—Rainwater cal—
culation and those of the experiment, it is necessary
finally to make a multiple scattering correction to
this single scattering theory. This was done by cal-
culating the function g(y), the differential distribu-
tion for single scattering from an extended nucleus

developed in the Cooper-Rainwater paper9.
: .J_- 2 2 '3 2
saw) 230' + ym ) / fimy)

where B = Q/ 002, y = (“P/CPO or x/xo, and ym =Ctom/CPO.
0, ($0, 61m, x, and :T&(y) were defined in part A., of
this section. A ratio of multiple to single scattering
probability from Cooper-Rainwater was then formed by
dividing'MC’R(y) by g(y). This ratio was formed for
each of the 260 values of y for which values of MC-R(y)

and g(y) are obtained during the course of the computer

68
calculation. This set of ratios was then employed

as a correction factor to the Drell-Schwartz projected
angle distribution obtained as described above. The
Drell-Schwartz distribution, corrected for multiple
scattering is plotted in integral form in Fig. I"?
along with the theoretical distributions of Moliere
and Cooper-Rainwater and the results measured in
the present experiment.

Appendix G gives a review of the computer
program, written in CDC-3600 Fortran, by which each
of the theoretical predictions described above was

calculated.

V. RESULTS AND CONCLUSIONS

A. Discussion of experimental uncertainties

Experimental uncertainties are present in the meas-

ured values of the scattering angles determined for each
muon event represented in the data here reported. These
arise in two general areas:

1) failure of the track, made visible by the series
of spark discharges in the chamber, to lie exactly
along the original path of the particle

2) inability to transfer with complete exactness the
coordinate positions of the series of sparks com-
prising the visible track to IBM data cards to be
used by the analysis program.

The major cause of (l) is taken to be a random displacement
of individual sparks in the track due to the secondary ion-
ization effects in the helium atmosphere of knock-on elect—
rons. This effect results in a "scattering” of the points
along the track which in turn causes an apparent scattering
angle in the observed track not caused by any real physical
scattering phenomenon. This apparent scattering is often
referred to as "noise level scattering". The uncertainty
of (2) is caused by the inability of the scanning operator
to position the reference indicator on the Hydel Scanner
over the exact center of each spark in the track for each
event. The random distribution of measured positions about

the actual positions on the film introduces an additional

69

70

”scattering” of the points along the track beyond that of
(1) above.

The overall error distribution resulting from both
of the above causes of false scattering can be treated by
the method developed by McDiarmidl“ for anal m3 is of cloud
chamber tracks. From his method the standard deviation
can be obtained for the overall noise level scattering
from all such causes as the above. The following scattering
distribution is developed in his paper for the case of

pep—9e», i. e., negligible real scattering:
2
2:: (< r¢>M): [_ (c ¢>,.)]
P(C ¢7M) =F:) (4)/ 3 XP 3' a- 1

For the case n = 2 which pertains to the present experiment

 

 

 

as described below, this reduces to

__J-(< ¢>4)e ((46);!)
PC< $7.1): a" x’o-a‘l'

 

 

Setting the derivative cf this expression equal to zero
defines the location of the peak of the <¢ >n distri-

bution in terms of 0'

Nil-I

(C ¢> n)2 most prob. 0" 2

This relationship may then be inverted to yield a value for
0" once the value of (<¢>n)(most prgb.) is identified

from a plot of the experimental results. In this manner

0" = V3- (4¢>n)most prob.

71
Numer -——-1

of events I
- 20 V4::), I

/
l—- '18

l 11
l
l

 

 

 

l

I

l

I

| l
_J

16 -—-— 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
'2
I
I

I
c¢ >n (in degrees) \\ l
l

2: 1. 6 as 1.0 1 a:

l 1

Figure V-l. Plot of.100.fideublesv events- Two groupings
or the data are shown..Sal£d curve represents
theoretical predictions for noise-level scat-
tering.

 

I

 

71
NMth ———9
of events I— I

— 20 I
I
, I
l
I

/ l

I" [16 -——-I I

 

 

_ '13

 

1l

 

 

 

 

 

 

 

 

 

—_——l

 

 

 

 

I
I2
I
I

 

 

I
4¢ >n (in degrees) \\ I
2 u 6 s 10 1 1:1;

J l I l l I

Figure V-l. Plot of 100 "doublesv events. Two groupings
cf the data are shown- Saltchuree represents
theoretical predictions for noise-level scat-
tering.

 

72
The identification of (< ¢>n)(most prob.) is

carried out here as in McDiarmid's paper by measuring
events obtained under a "doubles” counter criteria in

the apparatus. That is, these were obtained with the
anticoincidence counter disabled, thus accepting particles
having any momentum greater than 1.56 Dev/c in order to re-
duce the contribution from real scattering to a low value.

A plot is given in Fig. V-l for 100 such events upon which
two measurements of the scattering angle were made, corre-
sponding to n = 2 in the above expressions. A separate
histogram for each of two groupings of the data is given.
From these histograms the most probable apparent scattering,
(< ¢> n)(most prob.) is 0.37 degrees at the location of
the peak of the histogram. The solid curve in Fig. V-l is

a plot of the distribution P(<: ¢>n> above for peg—9 0°
and n = 2. The fact that this curve is a good approximation
to the data establishes the assumption that the data used
here had in fact negligible real scattering and thus pro-
vides a valid means of measuring 0'. The value of 0.525
obtained for CF was then used in the determination of the

theoretical distributions presented in Fig. V-2.

B. Comparison of observed scattering with predicted
values
In the present experiment a total of 2,266 usable
events identified as muon events were obtained upon which
track measurements and deflection angle calculations were

performed. Many events beyond this number were identified

a:

tc

Dri

73

as muon events but were not usable due to subminimal track
quality of one type or another. Most of the events re-
jected in this way either failed to pass through the entire
chamber (i.e., entering or leaving through one side), had
too few sparks present to obtain an accurate line fit, or
had too many displaced sparks due to knock-on electrons

set in motion along the track.

Figure V-2 shows the resulting integral scattering
distribution for the 2,266 accepted events. The curve is
normalized to unity. This is a plot of the relative number
of events displaying a measured projected scattering angle
equal to C? or greater. These measured projected scattering
angles were calculated from.the data for each event in the
manner described previously in Section III—C and.Appsndix F.
.Also shown in Fig. V-2 by solid lines are the predicted
integral scattering distributions for this experiment
which follow from the theories of Cooper and Rainwaterg,
of Drell and Schwartzl7, and of Molierelo. These curves
were obtained from calculations done as described previously
in Section IV and in.Appendix G. The error bars shown on
the experimental points are statistical only. The effect
of other measurement uncertainties, discussed in Section VeA,
are accounted for in the calculation of the predicted dis-
tributions.

The contribution to the scattering distributions due
to multiple scattering are as follows (evaluated for the
Drell-Schwartz theory):

74

 

[Scattering angle ] l.0°l 2.00 [3.00 l4.00|5.oo j
[Multiple scattering]69% l27.4% llZ.6%|9.5% l8.2%]

 

In comparing the experimental results with the
theories as shown in Fig. V-Z, it is apparent that the
data are in best agreement with the predictions of the
Drell-Schwartz calculation within the statistical uncer—
tainties shown. This calculation gives predicted values
about 10 percent higher than those of Cooper-Rainwater for
angles greater than 2 degrees. The Drell-Schwartz calcu-
lation represents a slightly better fit to the experimental
data than does that of Cooper—Rainwater.

The Moliere distribution, while differing from that
of Cooper-Rainwater only by a factor of 3.2 at the highest
angle use, nevertheless does not represent a suitable fit
to the data. A x 2 test was made on the data with respect
to each of the theories from the relationship

exp
M: N1 ‘ - N1
X2=Z

i=1 Vfiifififiir

This gives values of 4.76, n.92 and ll.#0 respectively for

Th. 2

 

the Drell—Schwartz, Cooper—Rainwater, and Moliere theories.
The X2 tables38 give the following predictions for these
values. The probability is approximately 92.5 percent that
the data are consistent with the first two theories, as com-
pared to a no percent probability that the data are consis-
tent with the Moliere theory.

75
1.0

L-2

a 10-1

\\ Moliere

N
gall-Schwartz \
e \

(d in degrees)
.0
l

-— 2 Cooper-Rainwater

r 10-3

 

0.5 l 1.5 2.0 2.5 3.0 3. ll» 0
I l I L l

b

 

Figure V-2. Normalized plot of experimental and theo-
retical integral scattering distributions.

C. Summary and conclusions

The work presented herein was an experimental
investigation of the scattering behavior of 1.60 Bev/c
muons experiencing momentum.transfers up to 190 Nev/c
in the aluminum.plates of a spark chamber. Cosmic rays
were utilized as the source of these muons. Details of
the experimental equipment constructed for operation of
the spark chamber, particle detection and identification,
and automatic track photography have been presented along
with details of the computer programs developed for data
analysis and theoretical predictions.

The muon track data obtained was analyzed to yield
measured values of projected angle scattering. It was
found that within the uncertainty limits shown in Fig. V-2
the observed scattering was in good agreement with the
extended-nucleus theories of Cooper and Rainwater9 and
of Drell and Schwartzl7, rather than with the point-
nucleus theory of Molierelo. Thus the results of the
present experiment are in good agreement with the results

13 and of Fukui, et al.1]- .At the same

of Mbsek, et al.,
time the data and results of the present experiment are
given with somewhat firmer bases, with respect to improved
momentum determination and particle identification as
described earlier, than those upon which the results of

the latter were felt by some12 to have rested.

76

77

Having observed no identifiable ”anomalous" scat-
tering in an experiment where both the momentum determina-
tion and particle identification were good, it is concluded
that upon careful investigation the similarity of muons and
electrons with respect to electromagnetic scattering indeed
holds for cosmic ray muons as has been shown for machine
made muonsl3. The present experiment represents the best
cosmic ray muon scattering experiment to date with respect

to the above critical features.

APPENDIX A: Spark chamber system

The spark chamber system consists of three prin-
cipal divisions: (l) the spark chamber structure itself,
(2) the helium gas flow system, and (3) the thyratron
pulsing unit.

1) Spark chamber

As mentioned in the text and as shown in Fig. 11-3,
the spark chamber used in this experiment is constructed
in three sections: the upper, main, and lower chambers.
The main chamber is composed of S/32-inch aluminum plates
10 l/2-inches square with a tab on one corner of each for
electrical connection. These plates were assembled with
tabs alternated with respect to location at front and
rear of the right-hand chamber face to facilitate the
connection of alternate plates to the thyratron pulser
and the remaining ones to ground. To handle the high
current pulses without excessive voltage drop, oneuinch
copper straps were used to connect the chamber plates
together, and one-inch braid carried the current from
the thyratron circuit.

The gap spacing between plates was formed by
plexiglas spacing strips sliced from 3/4-inch sheet
stock and machined to 0.25-inch within a ten thousandth.
A close gap tolerance was necessary in an attempt to
obtain uniformly high spark efficiency in each gap. The
spacing strips were assembled to form a square around the

outside of each successive plate. The next plate was
78

79

placed upon the spacing strips forming a closed cell,
and then another set of spacing strips was placed upon
it in the same manner. Twenty-two such gaps form the
main chamber. The entire assembly including the upper
and lower chambers is held together by tensioning bolts
at each corner bearing upon a l/2-inch thick aluminum
plate on top and bottom.

The upper and lower chambers were each constructed
by milling l/l6—inch grooves in plexiglas pieces at
0.25-inch spacing in such a manner that the pieces are
assembled into a closed box having continuous grooves
around the inside wall at every 0.25-inch increment.
The 3/64-inch thick aluminum plates used for these
chambers were inserted in the box grooves just before
the final side was mounted in place. Both the upper
and lower chambers were built with four gaps in pairs
of two, separated by a distance of l l/l6-inch. Elec-
trical connections to the plates were brought out in a
similar manner to that described above for the main
chamber.

2) The helium gas flow system

The gaseous medium for spark production decided
upon for this experiment was helium with a trace of
ethyl alcohol vapor as an additive. Of the gasses
commonly used for spark chamber work, helium was
chosen since it is the only one economically feasible

in a free flow system which exhausts into the atmosphere.

80

This flow method was chosen because it is mechanically
impractical to attempt to seal for evacuation a chamber
constructed as described above.

Helium, of the grade most readily available from
welding suppliers, was first run through a cold trap
containing liquid air. This procedure was adopted when
after only a few months' operation a noticeable deter-
ioration of gap efficiency was observed. It was
determined that a trace of both oil and water as well
as some hydrogen was present in the welder's helium.
The latter could not conveniently be eliminated, but
the liquid air trap effectively removed the oil and
water before it entered the system. The result was
a considerably longer time period before disassembly
and cleaning of the chamber again became necessary to
restore gap efficiency.

The helium flow control panel consists of two
Hoke metering valve and flow gauge units, one on each
branch of a two-way flow path. Most of the helium
passed through the main branch and on to the distri-
bution manifold on the spark chamber. A flow rate of
2.5 liters per minute was maintained in this branch.

A small amount of helium, however, was diverted into
the second branch through a second metering valve and
flow gauge to a l/32~inch orifice at the base of a
glass tube containing ethyl alcohol. The tube was

filled to a depth of 2.5 inches and helium was bubbled

81

through the alcohol at a rate of one bubble per second.
The purpose of the alcohol is described in the main text.

The gas flowing in the two branches was again com-
bined and sent to the chamber. Each of the three sections
of the chamber was fed from a separate flow control valve
to its own manifold block. At this point individual flow
control to each of these divisions was necessary in order
to keep the helium flow rate per cell equalized among the
three sections, since each contained a different number
of cells.

3) The high voltage pulsing unit

As described in the main text the spark chamber
was fired by discharging into the chamber a bank of
capacitors charged to 15 KV. The discharge of each
capacitor bank was controlled by a separate 5022
hydrogen thyratron being triggered ultimately by the
coincidence circuit. The Amperex 6279/5022 version
was chosen because of its superior rise time charac-
teristics and higher voltage handling capabilities.
Biasing the grid of each thyratron at +15 V (adjustable
to best operating point) further decreased the rise
time into conduction. An Amperex EFP6O secondary emission
tube, used in a single-shot blocking oscillator circuit,
delivered a fast-rise 150 V pulse to the grid of the
6279/5022. The EFP6O was triggered by the positive
pulse delivered to it from its respective 6688 cathode

follower stage. The three cathode followers, each

82

'driving its own EFP6O and 6279, received their signals
from the coincidence circuit via a 6688 input amplifier-
inverter stage. A negative pulse having an amplitude

of 0.1 V or greater triggered the spark chamber discharge,
but a larger input pulse was desirable for reducing the
rise time to a minimum. A negative input pulse of one
volt or more produces an overall spark discharge delay
time of :3 200 nsec. The circuit of the thyratron unit
is shown in Fig. A-l. This unit was constructed in 1961
following the general plan of a similar unit just com-
pleted at that time by Prof. D. Meyer and Prof. K.

Terwilliger of the University of Michigan, Ann Arbor.

83

 

 
     

 

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SPARK CHAMBER PULSE CIRCUIT
FIGURE A-l.

APPENDIX B: Circuit description of the counter telescope

system

The counter telescope system consists of two
divisions: (l) the scintillation counters with their
circuit boxes and power supplies, and (2) the tunnel
diode coincidence circuit.

1) Counters and phototubes

The three scintillation counters in the telescope
system are fitted with one, eight, and four 6810A photo-
multiplier tubes corresponding to the aperture, stopping,
and anti counters respectively. Each phototube was
wired for negative pulse output according to a suggested
circuit found in the data sheet supplied with each tube,
except that the bleeder resistances were multiplied by
a factor of 6.5 to conserve power supply current. This
was feasible only because of the low average count rate
involved.

The DC operating voltage for each phototube circuit
was supplied via a distribution panel from two Hamner
N—h035 power supplies. One of these power supplies was
associated with the eight tubes on the stopping counter;
the remaining one supplied both the aperture and anti
counters, via a second distribution panel. Each phototube

voltage could be adjusted to a desired level below that
of the power supply voltage. A voltage drop ranging to

minus 20 per cent in 16 steps was provided by this method.

84

f3

85
The purpose of this arrangement was to allow independent
voltage adjustment of each tube to equalize pulse heights
among the several tubes on the same counter.

The aluminum mounting sleeves of each phototube
were connected to the power supply at negative cathode
potential to reduce dielectric stress on the glass walls
of the phototube. It has been the experience of many
that failure to do this causes noise in the phototube
output.

2) Coincidence unit

The tunnel diode coincidence circuit constructed
for this experiment (see Fig. B-l,hD was designed from
the basic circuit given by Whetstone and Kounosua‘.
Basically it consisted of an input module for each
scintillation counter with a number of tunnel diode
univibrators cascaded to provide a high degree of
isolation from the input signal as well as standard-
ization of signal shape. Once the input signal is
standardized in this manner, the coincidence function
may then be performed by merely feeding the signal from
each coincidence channel into a current adder, which
in turn triggers another univibrator whenever there is
a signal present in all inputs at the same time.

Anticoincidence operation is also available by
arranging that the signal from the "anti" input module
be used to gate the coincidence channel at a subsequent

stage in the circuit. It is necessary in this mode of

86

operation that the ”anti" signal arrive in the circuit
before the coincidence signal. For this reason the
signals from the aperture and stopping counters were
each delayed 60 nsec by means of additional cable length.

As shown in Fig. 11-4 the entire unit was con-
structed in a copper chassis to minimize troublesome
effects arising from pickup of RF radiation from the
spark chamber. Use of silverplated finger stock on
the tight-fitting bottom plate provided a highly
effective closed conducting shell for shielding of
the circuit. Within the chassis are copper dividers
to decrease signal pickup from one point in the circuit
to another. The necessity of this measure was evident
from experience with a pilot model in which the degree
of singles rejection was limited by dependence on the
amplitude of the input signal. Partitioning the

chassis effectively eliminated this problem.

 

 

 

 

 

 

 

 

 

 

 

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Figure B—l.

Coincidence circuit diagrams.

 

 

 

 

 

 

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Figure B-la. Coincidence circuit diagrams, cont'd.

APPENDIX C: Modification of Tektronix 517 oscilloscope

for 7T—y-e d isplay

As described in the main text, more than one
method of identifying 77-}1~e decay pulses on the 517
was attempted. The one which proved most reliable in
the presence of large quantities of RF radiation from
the spark discharge consisted essentially of programming
the sweep of the 517 to produce a characteristic Z-trace
as described in the main text.

To accomplish this several rather extensive
modifications were made to the sweep gate and sweep
amplifier circuits of the 517. A diagram of the most
crucial modifications is given in Fig. C-l. These are
the circuits responsible for (l) automatic sweep speed
change after the first trace and (2) trace separation
and marker pulse presentation.

As may be seen in Fig. C-l the automatic sweep
speed change was accomplished by using a 5727 thyratron
to switch in a new timing capacitor (1200 pr) for the
slow sweep, after completion of the fast trace. The
original timing capacitor 0129J is adjustable to cal-
ibrate the fast trace, while the 25K rheostat shown
serves to calibrate the slower trace. The 150 ”pf
capacitor serves for the duration of the first trace
to effectively short the two resistances across which
it is connected, but becomes of negligible effect in

comparison with 1200 Rpf by the time the second trace
89

90
is fully underway. The use of this 150 ppf capacitor
proved to be an effective means of providing the proper
charging resistance for each timing capacitor, in order
to maintain a linear sweep.

The delay required before switching in the 1200
uuf capacitor for slow sweep was provided simply by the
inherent delay involved in establishment of conduction
in the thyratron, following application of plate
potential. The plate potential is simply the potential
appearing across C129J as it charges during the first
sweep. The firing delay is a function of the bias
potential present on the control grid of the 5727. Thus
a variable bias potential was provided by the 10K rheo-
stat by which it is possible to calibrate the firing of
the thyratron to occur precisely at the end of the first
trace.

When set for double sweep operation, the action
started from the initial trigger pulse proceeds on its
own, including horizontal retrace between sweeps. Auto-
matic horizontal retrace is a direct consequence of the
fact that in the 517 the sweep voltage appearing on the
deflection plates is simply an amplified version of the
instantaneous potential across the timing capacitor
0129J as it charges with time. Thus when the uncharged
1200 upf capacitor is switched across the fully charged
0129J at the end of the first trace, the potential across

it is reduced to approximately 1 per cent of its "full-

91
sweep" value. This action is passed along by the DC

sweep amplifiers to the deflection plates as a very
effective retrace (limited slightly by stray capacitance

in the amplifier and deflection plates). The second trace
begins immediately after retrace since the lZOO‘upf begins
to charge immediately upon being connected by the thyratron.

A ramp and marker generator was designed to produce
a step voltage by which to separate the slower trace to a
position on the screen below that of the fast trace. A
schematic for this circuit is given in Fig. C-l. The rise
time of this negative step function produces the vertical
movement in the cross bar of the Z-trace as the beam is
lowered after the first trace at the same time that the
change of the timing capacitors is causing horizontal
retrace. By the time the sweep speed change and retrace
are complete the negative step function will also have
reached its full amplitude where it remains throughout
the slow trace.

The negative step function for sweep separation was
obtained from the square negative pulse present at the
first triode section of the 6D38. The leading edge of
this wave form was delayed by a two-section RC delay
network and then fed to the base of 2N501 transistor.
TThis transistor was connected as an emitter-follower
to provide a low impedance output to drive the vertical
input of the scope.

The purpose of the 1N67 is to clip the signal into

92
the 2N501 at ~10V. This method of signal limiting retains

the short rise time associated with the high amplitude
signal from the 6DJ8 circuit without overdriving the 2NSOl.
The purpose of the 1N39 is to block passage of the negative
step function back into the differentiation circuit attached
to pin 6 of the 6DJ8.

A negative marker spike was obtained by differen~
tiating the negative-going trailing edge of the wave form
generated by the 6DJ8 monostable multivibrator circuit.

The 2.5K trimpot which controls the period of the circuit
was used to set the time of the marker pulse to be pre~
cisely lO‘psec.after the arrival of each phototube pulse
accepted by the coincidence circuit. The appearance of
this marker pulse in each scope picture served as a con—
tinual calibration monitor on overall sweep time.

In addition to the modifications described above,
several other incidental changes and additions were made
in the 517. The two 6AG7 clamping tubes V112 and V113
used to discharge the timing capacitor following each
trace were replaced by eight 6CL6 tubes for greater
discharge current capacity and hence faster retrace.

Two additional 6CL6 clamping tubes were inserted between
pin 8 of V117 and ground to more quickly discharge the
effective shunt capacitance of the positive deflection
jplate during retrace. This treatment should in principle
Thave also been applied to the negative deflection plate.

Trhis was not done since sufficient benefit was obtained

93
from clamping the positive deflection plate as to make

it no longer mandatory that it be done on the negative
plate as well. Also because of the reversed polarity on
the negative plate, clamping is difficult without arranging
a more sophisticated circuit to control conduction in the
clamping tubes than was required for the ones that were
installed.

The R126 and the L105 in the trigger amplifier
were increased by a factor of 5, to 160012 and 604ph
respectively, to increase the trigger signal for more
stable triggering of the sweep gate circuit of V111
and V119.

The circuits added required power supply capa-
bilities exceeding that of which the 517 power unit was
capable. For this reason a supplemental power supply
was constructed and installed in the scope to supply
6.3 V AC at 12 amperes, 6.3 V AC at 1 ampere, and
~90 V DC regulated with 9001 VR tube.

All of the above equipment was constructed on
chassis plates and installed within the case of the 517
indicator unit. Subsequently, however, it became apparent
that the 180 V DC regulated supply within the scope was
being overloaded by portions of the modification circuits.
Therefore an additional 180 V DC supply was used to supply
the new circuits. No further overload symptoms in the

main power supply were encountered.

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APPENDIX D: The automatic camera system

To make continuous automatic operation of the
spark chamber possible an automatic film advance camera
system was designed and constructed for this experiment.
The cameras used were Graflex units accepting 100 foot
rolls of 35 mm film. These were motorized by the addition
of a 120 rpm gear motor on each to drive the sprocket
directly, with a cam attached to the sprocket shaft to
operate a microswitch upon each one-third turn of the
shaft. Short framing was employed in order to obtain as
many photographs on a 100 foot roll of film as possible.
For each frame the film was advanced approximately one
inch by a 120 degree rotation of the sprocket shaft.

This allows 1200 frames per roll instead of 800 for the
standard one and one-half inch frame spacing used for
35 mm film.

The camera motors were controlled by a circuit
for which a schematic diagram is given in Fig. D-l. Each
motor was a two pole induction type operating on 24 V AC,
which can be very effectively brought to an abrupt stop
by substitution of 30 V DC for the AC driving voltage.
This DC braking procedure made possible accurate frame
spacing as indexed by the cam and microswitch assembly
on each motor. The central motor control circuit also
controlled the synchronized advance of frame indexes
placed in view of the cameras. To simplify spark gap

identification during analysis, illuminating lights were
95

96
also controlled by this circuit to expose the frame count
on the film in each camera, and to expose the front edges
of the spark chamber plates upon the film of the chamber
camera.

A resume of the sequence of camera advance operations
is as follows: In the spark chamber high voltage pulse
chassis was located a sensing circuit to trigger the
camera circuit upon each spark chamber discharge. This
was accomplished by looping a current pick up around
the plate lead of one of the 6279 thyratrons. The signal
induced in this lead upon discharge triggers a 6DJ8 mono-
stable multivibrator which in turn energizes a 60h triode
having the coil of a DPST relay in its cathode. The
closing of this relay completes the 2A V AC circuit to
start the camera motors and switches the 125 V DC to the
sequencing relays that control the remainder of the camera
and lighting functions.

The first step in the sequence that follows is that
simultaneous with the starting of the camera motors is the
activation of the blanking relays: three in the chamber
circuit (one for each 6279 to prevent discharge during the
camera advance operation, see Fig. A-l), and one in the
coincidence circuit (see Fig. B-LD. Each of these relays
has its own timing capacitor to determine the hold time
after the 125 V DC sequence-starting voltage has been
removed. Also the starting of the camera motors activates

relay RL2 through which their AC current flows. This

97
3PDT relay connects the two 10 Pf capacitors in the
light timing circuits to 125 V DC for charging while
the motors are advancing the film. The third pole on
RL2 serves to activate another relay which controls the
frame indexes, advancing them one count. The duration of
the voltage pulse delivered to these indexes was leng-
thened by the timing capacitor on this index advancing
relay, for more reliable index response. When the film
has been advanced one frame, the motors are stopped by
the DC braking current. Following automatic termination
of the braking current relay R12 releases. Release of
RL2 connects each of the light timing capacitors to its
respective light relay which in turn illuminates the
chamber and scope frame indexes for the respective
durations for which 31A and 31B are set. The timing range
available for these lights is 1/2, 1/u, 1/8, 1/16 second,
with the setting chosen for suitable exposure of the frame
numbers in each case.

The control of the motors for proper frame spacing
occurs as follows: Upon initiation of the camera advance
cycle, relay RLq receives 125 V DC as do the blanking
relays, etc., as described above. The adjustable series
resistance and shunt capacitor preceding this relay
provide a delay in the operation of RLl sufficient to
assure that both camera motors have turned far enough
from their rest positions to release the microswitch

from its "index" position to the "run" position. In this

98

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

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99

run condition the motor receives 2“ V AC directly. The time
delay on RL4 is such that the large contactor RLl is acti-
vated soon after the time when the microswitch in both
cameras has been placed in the run position. At this

time the index lead to the motors is converted by RLl

from 2“ V AC to 30 V DC. Thus whenever each motor again
reaches its index position at the next frame location,

its microswitch transfers the motor coil to the index

line where the 30 V DC present there provides sufficient
DC current to effect complete braking in less than 1/30

of a revolution of the sprocket (based on a maximum of

1/2 revolution of the motor shaft and 10:1 gear reduction
to the sprocket shaft). This is equivalent to frame
indexing to better than 10 per cent in less than 0.2
second motor travel time per frame. This short camera
cycling time was desirable to minimize the dead time of
the apparatus.

The cycle terminating relay RL5 served to assure
proper termination of the cycle after a predetermined
period for protection of the motor coils which at the
end of their travel carry a high value of DC braking
current. Relay RL5 served to interrupt the 125 V DC
to all sequencing relays, each of which in turn released
at a time governed by its own timing capacitor as
described above. A further purpose served by RL5 was
the uniform termination time it provided for the entire

sequence, especially on manual camera cycling, regardless

100
of how long the manual switch may be depressed. Following

termination by RL5 the main motor relay RLl is deactivated
at a time controlled by RLu such that both cameras have
certainly completed their framing and have come to rest.
The blanking relays in the chamber and coincidence
circuits (mentioned above) were set to drop out last of
all, making the apparatus sensitive once again to the
transit of the next acceptable particle. The coincidence
circuit blanking relay is in fact the last to return to
normal operating position to prevent the arcing of any
other motor relay from inducing a spurious signal in the
coincidence unit, as this would have resulted in con-
tinuous cycling of the camera system.

The camera cycling system described above proved
to be a very reliable one. There was a fairly regular
mortality rate on the microswitch operated by each mOtor
cam and this was due to a calculated risk taken on the
current rating of the type of microswitch chosen. The
choice was dictated by the desired physical properties
of size and mounting type. But this system, which
included several improvements to cure early operational
ills, performed in excess of 500,000 camera cycles without

serious maintenance problems.

APPENDIX E: Momentum determination

The momentum of the muons studied was determined
using the total depth of stopping material penetrated
following observation in the spark chamber until coming
to rest in the stopping counter. Referring to Fig. II-l
for vertical dimensions of the apparatus, there existed,
in addition to the hO-inch lead pile, a l/2-inch thick
aluminum plate upon which the chamber is mounted and a
l/h-inch thick steel plate upon which the lead pile rests.

The total stopping thickness in gm/cm2 was found from the
formula
:2 91d) e;

R (HM/'- 03) "

where (91 is the density of the i-th type of stopping
material; di is the total depth in centimeters of the
i-th type of material; and I€i_is the relative effi-
ciency of this material as an absorber compared to Pb.
The average density of the lead bricks used in this
experiment was determined to be 11.08 gm/cm3, while the
remaining densities were taken from the handbook. The
E i values for aluminum and iron were taken from infor-

mation contained in Cosmic Ray Physics15 by Dr. D. J.

 

Montgomery.
Using these values
R = (11.08 gm/cm3)(uo in)(2.5u cm/in)(l) + (2.70
gm/cm3) ' (% in)(2.5h cm/in)(1.#6) + (7.6
gm/cm3)(% in) - (2.54)(1.25) = 1137 gm/cm2

(Eb equivalent)
101

102
This corresponds to 1.56 Bev/c15 for the momentum of the
muons passing through the chamber. This value of the
momentum is the minimum value a particle may have if it
enters the apparatus from a vertical direction. Muons
entering from directions lying at some non-zero angle
with respect to the vertical will have a somewhat larger
depth of stopping material through which to pass in
reaching the stopping counter. The maximum value this
may have is determined by the maximum angle about the
vertical which can be accepted by the geometry of the
counter telescope arrangement. It was determined from
the tabulated experimental measurements of the incoming
track direction with respect to the vertical Z axis,
that 95 percent of all accepted events entered the
apparatus within a vertical cone extending 15.4 degrees
from the vertical. Using this angle the corresponding

maximum path length in lead was calculated from right

 

triangle geometry to be (40") ¢E7+’tan2 (15.4631: 41.5".
The corresponding maximum range in lead is 1178 gm/cmz.
In calculating the range for the fastest particles ac-
cepted by the apparatus, the effective stopping power of
the stopping counter must be included. This counter has
a usable thickness of 6-inches, a density of 0.9 gm/cm3
and an absorbing efficiency of 1.90 from the same curves
used abovels. This yields a range contribution of

26.0 gm/cmz. The maximum total range of an acceptable

particle (i.e., that stopped in the stepping counter)

103

is then 1204 gm/cmz. This corresponds to a maximum
momentum of 1.65 Bev/c. The median momentum for this
experiment was taken as 1.60 Bev/c; this is slightly
less than the arithmetic mean of 1.605 Bev/c. This
choice is based on the fact that the relative muon
incidence rate decreases as the cosine2 of QjN
measured from the vertical, the effect of which can be
seen in the data for incoming tract direction. Thus the
momentum of the muons in this experiment was taken to be
1.60 Bev/c + 0.05, - 0.00 or approximately 1.60 Bev/c

f 2.8%.

In view of the narrow momentum limits held in
this experiment, it was unnecessary to integrate over
the muon intensity distribution within the accepted
range interval. This results from the fact that the
i 2.8% spread in momentum stated above causes an
expected change of only i 1.8% in the muon incidence
rate as determined from the plot of the differential
muon range spectrum found in the cosmic ray survey
article by B. Rossi16. Further, it can be seen from
the equations of Section IV. A. that the major effect
of the value of pc upon the predicted scattering dis-
tribution is its effect on the conversion factor between
the variable x used in the calculation and the projected
angle variable CT. Specifically C7 is proportional to
x/pc, where x is the reduced scattering variable used

in the theory. Thus a a 2.8% spread in momentum causes

104

only a i 2.77% change in the value of the scattering
distribution in its region of steepest slope (in the
vicinity of 1.0 degree) and less change for other

values of angle CP. The uncertainty resulting from

assuming the momentum to be a constant is therefore

sufficiently small to be neglected.

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117

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mm u HMQHZ

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119

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120

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.. 1 III!

‘IIIIIIIII‘I

APPENDIX G: Calculation of theoretical multiple scattering

distributions

1. Calculation of the Cooper-Rainwater distribution
A. The differential scattering distribution M(x)
As given in Section IV. B., the form of the dif-
ferential distribution that describes large angles as

well as small is

—X
MC-R<x) =i—flf—[1 + 193+ +467: ML)”

The parameters and functions in this expression
for M(x) are as defined in the main text.

In performing the calculation of M(x) the program
written for the task was arranged in the following manner:
Subroutine COOP calculates the quantities G,Q, x0, (Pb,

Cqu t (in gm/cmz), (2"GQ)%, all of which are needed
later on in the calculation. In addition to these quan-
tities, the form factors FNC(y) and 13%(y) are calculated
in this subroutine.

To perform this, a table of 21 values of FNC(y)
for values of y from y = O to y = h increments of 0.25
is entered in an array in the machine memory. These

C

values of FN were obtained from plotting the four values

4
for y = 0, l, 2, 3, that Cooper and Rainwater suggest,
which is then to be matched to the function FNC =12/y“
for y 2 l-I. A function routine FORM (y) interpolates

this table of FNC values or uses the assmptotic value

of FNC(y) for any required vigée of y. The values of

_II

127
FNC(y) thus generated are used in COOP by which to obtain

the’corresponding values of %N(Y) from the relation

91mm = FNC(y) + <1 - FIICIy)>/z.

The values of J%h(y) obtained in this manner are stored
in an array under Labelled Common for later use.
Subroutine RAIN, called successively for each
value of x, controls the remaining operations involved
in calculating the differential scattering distribution
M(x). Function TXIAM is called a number of times to
furnish, for each value of x, an array of Values of the
function T(x,)\) as defined above. Function QLX is
called to evaluate the function Q(L, x) as defined above.
The integral represented by N(L, g) is evaluated by means
of‘Weddle's rule in Function WEDL.' The value of MIX) then
is merely the sum of each of the terms whose value was
just calculated. Each value of Mix) thus obtained is
entered in an array in Labelled common containing the set
of.MKx) values that correspond to the set of required
values of x. MIX) is also printed out at this point along
with the corresponding value of 6? and.x. Printed out also
is the value of Q(L, x) and the number of terms found

necessary for convergence of the series used to calculate

(KL, 3:)?

w

B. The corrected differential scattering
distribution
The next step in the calculation is to obtain
a differential scattering distribution corrected for
noise level scattering. This is done as discussed in

in the main text by evaluating the following integral

.0 -x—X’)a
My) 5 JV CJX

 

g(X') :
.1-77'0'
-00
where g(XI) is the correfteq distribution, x' =
. ox’
measured deflection, c. ‘r" is the normal distri-

bution function used as a weighting factor upon M(X)
as a method of obtaining the expected value of the
differential distribution function at x' due to the
total of all the contributions from deflections
having all values of x. This integral is evaluated
by numerical integration in Function GPTHTA, which
in turn calls Function FNTGRL to do the actual inte-
gration using an extended five-point Newton-Cotes
quadrature method.

A set of g(xt) values is obtained by repeating
the above procedure for various values of x' by which
a corrected differential distribution g(xt) for each
x' needed is made available. These values of g(xv)
are printed out at this point together with their

corresponding (4" and x' values.

128

C. The integral scattering distribution
From the set of g(x') values obtained in
Function GPTHTA an integral distribution is obtained
by numerical integration of these values in Function

GTHETA. This is done to evaluate the integral

'9
Gag) "y/zjo") “(X

For purposes of normalization this may be

rewritten as

1 w
I :: —-——— I dX/
G, 09) 6(0) Z7“)

The integration of this expression is performed by

the same Newton-Cotes function routine mentioned above.
Normalization is performed after integration by dividing
each value of G(XL') by C(O) to obtain GN(XL') for each
value of XL' to be used. The values of XL: were gen-
erated stepwise in the machine upon reading in values

of the step size and upper limit on XL'. The calcu-
lated values of C(XL') and GN(XL') were printed out

at this point together with the corresponding values

of 6? IL and x'L.

129

II. Calculation of the Moliere scattering distribu-

tion

The Moliere differential scattering distribution
was calculated using the same program as described in
Section I. above, making the proper identification
concerning .3& in the calculation of M(x). (FNC)%
is the Fourier transform of the nuclear charge dis-
tribution. Thus the point charge nucleus which is
assumed in the Moliere theory will, since it has
negligible extent, have a corresponding FNC-—+ 1.00.
From the expression for .3% = FNC + —%— (1 - FNC)
we see that J$fi -——9 1.00 also. Thus to calculate
the Moliere differential distribution, the integrand
of NCL, N) was merely rewritten with the fill} factor
omitted--effective1y setting ffik = 1.00.

The Moliere corrected differential distribution
and the Moliere corrected integral distribution were
then calculated in exactly the same manner as

described in B. and C. above.

130

III. Calculation of the Drell-Schwartz scattering

distribution

To obtain the Drell-Schwartz predicted distribution,
a subroutine DRELL was written in a straight-forward manner
to calculate the function JN/Jo developed in section IV.-C.
Values of this function were obtained at 260 points in the
range of the experiment. Printed out at each of the 260
points were the following quantities: 6 , qo(in Mev/c),
f2(qo), Fqu): quO)’ jmagmo), dd‘o/dfi. , dr/dJL, dN/dg.

A second subroutine MSCOR was written to calculate
the multiple scattering correction factor required to
correct the single scattering law of Drell-Schwartz to one
which includes the effects of multiple scattering. This
was done by calculating the ratio of the multiple to
single scattering laws of Cooper-Rainwater at each point.
This ratio was then stored in the memory until after the
following step.

Next a third subroutine DRELPR was written to cal-
culate dN/dCP from the appropriate linear combination of
the dN/dO values, as developed in Section IV.-C. Values
for dN/dCP were obtained at each of the 260 values of d
as above. These values were then corrected by using the
correction factors generated in subroutine MSCOR, giving
a differential distribution for multiple scattering based
on the Drell-Schwartz single scattering results. Printed
out at this point were the following quantities: 6? , x,

“HQ/(14%, (dN/d (WM- , 7oMS. The terms (dN/d ens and
131

132
(dN/dCTXM, are respectively the single scattering distri-

bution of Drell-Schwartz converted to projected angles and
this distribution corrected for multiple scattering.

The set of values obtained for (dN/d<?)M were then
operated on by function GPTHTA and then by function GTHETA
whose purposes and printed results are described in sections
I.-B. and I.-C. of this appendix.

In addition both (dN/d 5?)8 and (dN/d LP)M were
operated on directly by function GTHETA. By comparing the
resulting distributions printed out from this step, an
estimate of the precent difference in the integral dis—
tribution due to multiple scattering was made.

A listing of the computer program described above,

written in CDC-3600 Fortran, follows:

133

.MADm 23m AAMMQ OZHmD

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.mmDQ<> mmx mo

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mmDA<> X mo Bmm ZHIQ<MM mom Axvz mo ZOHB¢ADUA<U see A ZOHBmO

WAZO mOOU MZHBDOMmDm ZH

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131-I

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136

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OHNz.Hux O OO
H+x<zzumHNz

0.0 n 23m

O.H u HH + x<zzvo
HzmHOHmmmou amOH mma 9mm
mszezoo

Hmuxvo u Hxvo

x<zz O u x H OO

BmOH mma BOO HH< Ham

o.~ u HHVU
o.m u Hmvo
O.H u Hmvo
0.0 u Havo
O.H u Amvo
o.m u Amvo
O.H u HHVD

mszmHoHOOOOO zm>mm HmmHO OOH 9mm
HHOOOxm .HHOOOO zOHmzmzHO
HHOO.x¢zz.xOO HOmz oneOzOO

sz

150

OHsz.x.OO.Oxz«.x.mmx.H.NOO HszO
mHH.OON.mHHHm.HImOxVOH
OON.OO~.HONHO.OImOxOOH
OOH.OON.HONHm.:ImOxOOH
:ON.OO~.HONHO.OImOxOOH

mHH OH OO

NHO OH OO

~O~.OON.HONHm.HImOxOOH

mHH OH OO
.OHsz OzHHszO Hzm>mmm HHHB Ommm OOHmOmzH mHH OH OO OzHOamx HZOZOHOHO
HHOOHsz u OHsz

Hxvxz< u Oxz<

HOOIOHOHOIHxsz< u HHVOHsz
HOZOHOIOZOHOI.NO\NIIHH + mmxv n mm
H + H2 n x

H + Hm I meO\zHHO u Hz

HuHm

Hz.OH n H OH OO

H u OH

OH OH OO

HOzHHzOO

xHOOIx u x

O.O n HOOOHsz

OH OH OO

O u OH

N.H.OHOOOIOOOHH
HOZOHOIOZOHOI.NO\NIIHH + mOxV u Om
Hz.H n O O OO

O.O.OHH.OOOIHOOOOH
HOZOHOIOZOHOI.NV\NIIHH + szO u HOO
HOzOHmIHHOI.HVOHmOnV\.H u zmozx

NHO
wON
OON
:ON
NON
HON
OON

NH
OH

OZ¢ZGMBZH HEB mm¢mmmm

meO\zHHO u Hz
zHHO u x
HOOmOxz< \O\ zozzou

O

HO.H0.00H.OOO I OOOOH

HOZOHOIOZOHO¥.OO\OIIHO I OOOO u OO

H:.HHO. uHHOOHszOO.xO.OH.
. onvszOH.xO.O.OO. ume.x0.0.00.

Oz.OH u H OOH OO

Hz u OH

OOzHHZOO

xHOO + x u x

nme.xO.:.HHO. "OOOO.OO.:.HHOH
"OOOOO.OO.OH. "HOO. OHOHOZOOO

OHsz.O.OO.Oxz<.x.OOx.O.OOO HzHOO

.OHZHN OZHHZHmm Bzm>mmm AAH3 mamm OMHMMWZH

151

OOH.OOO.OOHHO.HIOOOOOH
OOO.OOO.HOOHO.OIOOOOOH
OOO.OOO.HOOHO.:IOOOOOH
:OO.OOO.HOOHO.OIOOOOOH
OOH OH OO

OHO OH OO
OOO.OOO.HOOHO.HIOOOVOH
OOH OH OO
OOH OH OO OzHOOOO HZOEOHOHO
HOOOHsz u OHsz
Hxvxz< u Oxz<

H + H2 I O n O

O.O u HOOOHsz

OO OH OO

O u OH

HO.HO.OOHOOO I OOOOH

HOzOHOIOzOHOI.OO\OI«Hx I OOOO n ma

H:.HHO. ”HHOOHszOO.xO.OH.
. "HOOOZOOH.OO.O.OO. ume.xO.O.OO.

Oz.Hz n O OO OO
OO.OO.OOHOOO I HOOOOH
H + Hz u Hz

A<EOHm«<SOHm¥.NV mmNOMmN u 8mm

A + quo ZHAD + HZ u NZ
0.0 n X
MDZHHZOU

xAMQ I X n X mHH

"OOO.xO.:.HHO. "OOOO.HO.:.HHOH

"OOOOO.HO.OH. "HOO. OHOHOzOOO OOO

mm
mm
mN
wNH

MOO
mam
wOm
mom
now
Now
HOM
OOm

ON
ON

mN

OH
ma

0

152

OZOOOOHzH OOH OOOOOOO
HHOOO HOOOH OOHOZ OOO .OOOO ..OHOHOHO OzHOOHHOOO HOOOOHzH OO .OHOO
HxOOzH.z:z.zHHO.OzOHO.xHOO.2HHOO OHOOHO onHOZOO

OZO
ZOOHOO OOOH
«HOHOO .OOO .OOOHOOH .HOO HzHOO OOH
OOH.OOH.OOOHHOIOO.OI<HOHOOOOH
H:.HHO. "HOHOOHOOOOOHH.H
Om .0.0m.u OOO H.O.HOOOH.xO .O.OO.uH.OOO zHOOOIOHOOHmOH. OHOHOZOOO HOO
HO«O«.OOOHOOO*HOOO.OOVOOOx u OOOHOOH
HOHsz.xHOO.OzOHOOHZOOZOOzx n «HOHOO OOOH
OOzHHZOO OOH
OHHOO+OHHOHOO
H:.HHO. "HHOOHszOO.xO.OH. "OOO.OO.:.HHO. "OOOO.OO.O.HHOH
. "HHOOXOOH.OO.O.OO. ume.xO.O.OO. umOxO:.xO.OH. "HOO. OHOHOZOOO :OO
OHsz.H.OO.Ox2<.x.OOx.H.3OO HzHOO :HO
OO.OOO.OOHO.OIOOOOOH OOO
OOO.OOO.HOOHO.OIOOOOOH OOO
OOO.OOO.HOOHO.:IOOOOOH.OOO
OOO.OOO.HOOHO.OIOOOOOH OOO
OO OH OO Ho:
:HO OH OO OOO
OOO.OOO.HO¢HO.HIOOOOOH
OO OH OO

.UBZHX OZHBZHNE BZm>m~E 41:3 mmmm OmflmmmZH Om OH. 00 UZHDOHMM BZMSHH¢Hm

HHOOHsz u OHsz
HHOxz< n Osz
HOOIOOOOOOHHOxEO nHHOOHsz
H + H2 I H u H OO
OOOH OH OO
0.0 n HOOOHsz OO
Oz.H u O OO OO HO

153

O0.00.00 HzIzOOH
OIOOOI.OH+HHHIOOO+HOIOOOO*.OO+HHOVO+HOIOOOOO.H+OuO
n2
:.z.OuO OO OO
0.0uO
HOOO onOzmzHO
HzHOO HOz OOOO HO OOHOHOOOH OZOOOOHzH u H
OHzHOO HOz zmmzHOO HzszOOzH u OHHOO
HOOOOOOIOHHOOOOOO OHzHOO HOz OO OOOzOz HOHOH u z
znsz<z oz ... OHzHOO O>HO OO zszsz OOOHOOOO
HOHZOZOHOO OOOOOOIOHOOOO OOH HOOOO
OOOHOOOOOO OOHOOIZOHBOz HzHOOIO>HO OOOZOHOO
HO.<HHOO.zOHmOHzm ZOHHOZOO

ON

OUDUUUU

MZHHDOMODW ZOHH<mOMHZH H<0HmmZDZ HMUHZMU

HO.<HHOO.zOHOOHzm ZOHHOZOO

Ozm
ZOOHOO

HHxOOzHOO.xHOO.H+xOOzHIzVHOOH2H u «HOOHO
OOzHHZOO

HHOOEO u HHOO

z.xOOzH u H OO OO

OO OH OO

OOzHHZOO

HHOOBHO u HHOO

z.xOOzH u H OH OO

2Oz.H:.O.:.O.O .OOOH OO

zHHO u x

H + xHOO\zHHO u xOOzH

.H+xHOm\zHHOuz

HOOOOOHHO \O zozzOO

HOOOOsz \O\ ZOEZOO

szz .O .O .HHOOOzm .HOO .HO \O\ zozzOO
HOOOOO onOzmzHO

Om
ON

3
OH
O

154

Ozm
szHOO

O.OO\OI<HHOOuHOOHzm OO
HOIOOOH

HOHOOO.OIHOIOOOOOOHO.O+HOIOOOOOO.OIHHIOOOIOHOH.OO+HOOOOOHO:O.H+OuO OO
2.2uO OO OO

H+Zu2 mN

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