5
”7 ,
7“
‘
,7
7 7
5
‘
7 _7 ,
, 7
7"
77
g
51

 

POLARIZATION OF POSITIVE MUONS
IN A FREON BUBBLE CHAMBER

Thesis for the Degree of M. 5.
MICHIGAN STATE UNIVERSITY

Wilbur Reed Langford
1960

IIIIIIIIIIIIIIIIIII

3 12930

 

L I B R A R Y
Michigan State
University

 

‘I. FL? «J.-

 

POLARIZATION 0F POSITIVE MUONS

IN A FREON BUBBLE CHAMBER

by
Wilbur Reed Langford

AN ABSTRACT

Submitted to the College of Science and Arts

Michigan State University of Agriculture and
Applied Science in partial fulfillment of

the requirements for the degree of

MASTER OF SCIENCE
Department of Physics

1960

Approved

 

 

 

ABSTRACT

The depolarization of positive mu mesons, decaying at rest,
in a freon bubble chamber, has been measured by observing the
decay asymmetry of the ensuing electrons. The mu mesons come from
the decay of positive pions produced in a synchrocyclotron. From
1328 mu decays having electrons of energy $6.9 Nev the decay
asymmetry parameter, cowas found to be 0.165 : 0.0h8 which
corresponds to a depolarization of SO 1 1h%. It thus appears
that freon.may be used as an analyzer for mu mesons in experiments
where large numbers of these mesons are observed. An analysis
of a sample of electrons with energy;11.2 Hey shared an increase

of polarization with energy as predicted by Lee and Yang.

POLARIZATION OF POSITIVE MUONS

IN A FREON BUBBLE CHAMBER

by
Wilbur Reed Langford

A THESIS

Submitted to the College of Science and Arts
Michigan State University of Agriculture and
Applied Science in partial fulfillmnt of
the requirements for the degree of

MASTER OF SCIENCE

Departnent of Physics
1960

ACKNOWIEDGMENT

I am most grateful to Prof. J. Ballam for suggesting and sponsoring

this research problem. His assistance during the course of the

experiment, and especially with the final manuscript, is greatly

appreciated. I wish to adtmwledge Dr. R. Crittenden for the use of

film for scanning, and also John Scandrett, James Parker, and John

Boyd for their consultation. I an indeed obliged to Justin Huang for

the method of detemining the space angle as appears in the appendix.
Special thanks go to Jerry Lassa for his help in seaming a large
portion of the film and to Jean Hill for typing the final copy of the

manuscript.

ii

TABLE OF CONTENTS

I. Introduction . . . . . . . . .

II. Description of the Experiment

A. Film Exposures e e e e e e

B. Measurement of Events . .

III. calcu18t10n8 e e e e e e e o e

A. Required Parameters . . .

B. Range. . . . . . . . . . .
C. Least Squares Solution for

D. Depolarization due to an External Magnetic

IV. Results and Discussion . . . .
AppendiXIeooeeeeoeeee

Appendix II

Bibliography 0 e e e e e e e e e e

111

O

the Asymmetry

.12

.32

.b0

I. INTRODmTION

In 1956, Yang and Lee‘ predicted that the muon, in the decay
«+ -v p‘ *1/ , should be completely polarized on the basis of none

conservation of parity and the two-component theory of the neutrino.

This polarization consists of the orientation of the spin of the muon

along the direction of its linear momentum. Yang and Lee also predicted

that with the subsequent decay of the muon, If -v e‘ +3) +7, a forward-

backward asymetry of the positive electron with respect to the direction

of the muon should be observed. Confirmation of this was subsequently

given by T. Coffin et al.2 who found an asymmetry with an energ
dependence consistent with the two-component prediction for the complete

polarization of the muon.
However, the mount of asymmetry observed in any medium depends on,

the extent to which the muon is depolarized, before decay, by external

fields. These fields may either be residual magnetic fields or internal

atomic fields. In addition, the formation of muonium (a bound state

of a positive muon with a negative electron) may also depolarize the

muon. The amount of this depolarization will depend on the medium in

which the decay occurs. For example, this depolarization is approxi-

mately 0% in graphite and 96% in silicon dioxide.3
The purpose of the present work is to measure the amount of

depolarization that results in freon-1381 (CBrF3 or bromotrifluoro-
methane), considering the positive muon to be 1C0$ polarized initially.
Freon, which has a short radiation length, is now rather commonly used
in bubble chambers whenever an investigation of processes involving

These gammas usually come from «’ decay and

There is a class of strange particles

gamma rays is desired.

bremsstrahlung of electrons.
which have alternate decays into muons or electrons. Freon chambers

are used to investigate these branching ratios; for example, the
following decays might be observed:
K. -+ PI, + of. +2/
4 3* + 1" +7/

K *p-+«

N VII

0
+

*e-+y¢

K'4p*+w

* -
49-4.11...

+

ti

EI 31

'6

it up

Nap”?

98-4.-

M
t
t
e
D
+

+
+

Z'+P'+n+

{I II

+e'+n+

+e*+Ԥ+y

Z +P++n+V
+e*efi+‘i/
it’ll—IF";
ee’O-fi-O'Z-j

It might turn out to be interesting to look at the p polarization in

all of the above decay reactions and for this purpose a measurement

of the amount of depolarization due to the freon itself is important.
Lee and fangs found the normalized electron distribution in the

decay, ’1’ -> e" +7/+77, to be

 

cm- 2 x3 [(3-2x) + (1-2203 cos 91%?—
withx-L and:_ a, “flurry“
FE (max) vaI2 + IfAI’

where I'D-e— is the electron momentum and 1“,, and fA represent the usual

vector and axial vector coupling constants. 6, wherever it appears,

is the angle between the muon and electron direction vectors.

Integration of the equation for dN over all 1: gives

 

an 1
I411 cm -1--3——§cos6.

For fr - IA, 3" has a maximum value of 1 so that

1
1"“;- C086.

 

In:
dfl.

However, in an eXperiment one measures

Im -1-thos9

 

dN
(in
where a, - 0.33P and P is the polarization of the muons. Since

experimental numbers for the decay of muons in carbon and hydrogen give
a. 0.26 1' 0.023, it can be assmned that the muons are approximately

100% polarized in n -}1 decay. Thus, measurement of a in this

experiment gives the depolarization of the muons by freon. The results

on 1328 decays give a- 0.165 t 0.0I.8 and therefore P - 0.1.953 0.11.1.

which indicates that the muons are 501 depolarized by freon. For the

26h events of higher energy with the asymmetry¢z - 0.3131 0.107,

Tho¢depolarization.is only 6%. This value shows the predicted increase

of the asymmetry with muon energy. However, the depolarization due

to the freon is the same, within one standard deviation, for both

energy groups.

II. DESCRIPTION OF THE EXPERIMENT
A. Film Exposures

Photographs of the meson decay events were obtained from exposures
made in August 1958 and February 1959 at the Carnegie Institute of
Technology synchrocylotron.6 Pi mesons were produced from the internal
proton beam of the accelerator and then passed through the field of a
bending and.analyzing magnet. The emerging beam.was moderated by a
copper absorber of sufficient thickness so that pions would stop
near the center of the bubble chamber. Stereo pictures were taken for
each chamber expansion with the camera lenses at an angular deviation
of 8 1/3‘ from.an axis extending from the bubble chamber to a central
point between the lenses. In figure I, the general physical arrange-

ment at the cyclotron site is shown.

B. Measurement of Events

Six sets of data were taken with the following number of events

in each set:

Tam Events
Data 1 251
Data 2 269
Data 3 267
Data h 266
Data 5 275
Data 6 265

Total 1592.

Figure 1. Experimental arrangement at the synchrocylotron

at the Carnegie Institute of Technology.

 

 

SYNCHROCYCLOTRON

 
     

INTERNAL CARBON
PROTON * TARGET
BEAM

 

 

 

 

 

\
\
CYCLOT RON
SHIELOING

BEN DING AND

 

 

 

 

 

 

 

 

ANALYZING
‘ MAGNET
4
/V
17* sEAIII-/ I
I
I
,. _ | _
men: __J I
I L I ~\\
I
cu
nasoeeen
FREON BUBBLE
I I CHAMBER

 

 

Figure 1e

 

Scanning of the film was accomplished by projecting the stereo
views (which are side by side on the film) onto a translucent screen
that could be viewed fromnthe opposite side from.the projector for
convenience. By means of the 10 fiducial.marks (see Figure 2),
uniform magnification could be achieved with the proper adjustment of
the angle of the screen. The fiducial.marks were also used to
determine the magnification involved.

An arbitrary coordinate system was set up in the scanner plane
with axes parallel to each of the sets of x and z fiducial marks on the
scanning screen. The center fiducial mark on the front bubble chamber
glass, designated by the subscript 0, was used as the reference point.
The position of this reference is used later in the computer program.
All distances in the scanner plane were measured to the nearest .Smm.
with the aid of two perpendicular plastic scales mounted on the arm of
a drafting machine. Angles in the scanner plane were read to within
.5’ with the aid of a magnifier on the protractor of the drafting
machine. The angles were measured with respect to the direction of
the a coordinate axis of the scanner plane (which is oppositely
directed to the pion beam) in a counter-clockwise direction from 0'
to 360'. Angles of exactly 0’ or 180‘ were avoided to prevent infinite

values for the cotangent. The orientation of the arbitrary coordinate
system was such that all parameters had positive values only. This

eliminated any sign errors.

Figure 2. An example of the decays
11“}: .2}
P++e*+y +2.7

as was seen in the scanner plane.

 

10

 

Figure 2.

11

Directions of muon paths of 3 mm. in length in the scanner plane
(maximum length determined in computations section) are usually easily
measured. However, if the path length is less than 1 mm., its
direction cannot be measured accurately since this approaches
the value of its width. Also , low energ pos itrons from mu decays have
an isotrOpic distribution7; therefore, only those events were measured
in which the positron path length was 5 cm. (2.78 cm. in the bubble
chamber) or greater in the scarmer plane. The values for the muon
path length of 1 mm. to 3 mm. and the minimum value of S cm. for the

positron path served as limits for choosing events for measurement.

With these considerations, values for the parameters
ZOL’ ‘oa’ ”ow 92L! e31.
z1L’ ‘IR’ Z1R’ 923: 93a

were measured (as required for space angle calculations in appendix 1)

where represents the coordinate of the left fiducial mark, 1:

20L on

and ion are the coordinates of the right fiducial mark, an, xm, and

211% are the coordinates of the muon decay point, 621. and 6211 are the
angles for the muon direction in the left and right views, and 631.

and 633 are the corresponding angles for the positron direction.

12

III. CALCULATIONS

A. Required Parameters

The needed parameters for computation of the space angle in
appendix I are the magnification factors m and M (defined below) and

the quantities from the tvm views

9

21.’ 03L and

left: 20L, 2.1L, z2L’ 2.3L;

right: ‘on’ 1“on; ‘IR’ ”135 ‘23, z211‘ ‘31v “3&3 6212’ 0311

where 32L, xZR’ ”ZR are the coordinates of the projected point along
the muon, path and ‘BL’ :33, 53R are the coordinates of a projected
point along the initial direction of the positron path.

The six parameters

ZZL’ 331.3 “‘23, ‘33? ”2w 23a

are to be evaluated in terms of the scanning data from the two stereo
views. These x and a'quantitiea can be evaluated from projections
along the um and positron directions as follows. For the first

quadrant, it is seen from the diagrams below

13

x—e»

 

 

 

 

 

 

that

0
‘2L - x3 cot 62L and s

' I
2R 1: COt 92R

so that the following equations can be written directly:
I ' I
‘21. Z11. * ‘2L zIL * ”‘3 °°t 92L

. I
231. 211. + 231. ‘11.. + x, cot 0

3L
x21. " "2R' "IR * x3
‘31. ' ‘33 ' x112 ’ ‘3

z2R ' ”m " z212' " “In * ”‘2 °°'° 92R

0 - ’
ZBR 31R + :33 21R +x3 cot 63R

XZL - x2R and x3L - xBR since they are equivalent vertical distances

in the two views. The above six equations are valid in all four

quadrants only when the following sign conventions are used:

Quadrant

1‘1 3-13
1 + e
2 + -
3 - -
h - +

1h

The magnification factors are given by the ratios

unit in bubble chamber
unit in i

m C
u . unit in film .
unitih scanner SIR;-
All of these paraneters were used to find the space angle shown in
appendix I. The actual space angles for the data were determined with

the Mistic computer using the computer program shown in appendix II.

B. £29.32
Considering the rest-mass energies of 139.6 Mev and 105.7 Hev
for the pi and mu mesons respectively, the kinetic energy and
subsequently the range of the muon may be determined. It is easily
seen from the mass defect, that the combined kinetic energy of the
muon plus the neutrino is 33.9 Mev. The kinetic energy of the muon is
then found to be II.1 Mev from a two-body decay calculation. 'lhe
range of a 14.1 Nev muon in freon 1331, as projected in the scanner
plane, ' is approximately 3 mm. as determined from range energy curves.
From the rest-mass energy of a positron, considering the muon to
have decayed from rest, the two body decay calculation gives the maximum
kinetic energ of the emitted positron as 514.? Mev with a corresponding

range of 33.5 cm. in the scanner plane (18.6 cm. in the bubble chamber).

C. Least Squares Solution for the Asymmetry;

The angular distribution of the decayed positrons for each event

3
i may be defined by f (xi) - 1 + an:1 where x1 - cos 91. The

equation is normalized by integration over the solid angle, 0’ ,

1
fridw-zu (1+axi)dx1-hn

For a nomalization to 1, the new equation summed over all 1 becomes

1
Ifi .147?“ *a'xi).

A solution for the most probable value of the asymetry a. may be

0
derived from a. method of least squares.’ Consider the error equation

.2 - a - 1 - 0.x 2
n {(51) q”, 11?. TL")
where
1 0.1
J1 (fi-fi-Tl”)?‘0
is the difference equation between the observed values f1 and the

theoretical values I}; (1 + 62. 1:1). The error equation may be minimized
for the best value of 4 by taking the partial derivative with respect

 

toav,1.8.,
2
«932 fixi *1 a”‘1
I- +
5,3, fw‘im in?”
where
M -o
30.

since f i is a measured quantity in the error equation and hence

dependent only upon the data and not upon a. From the minimized

 

equation
a.h:§fixi-ixi . huffixi
ix: 1"?
noting that 2 x1 . 0 in the interval ~15 x1_<_1. The error or

i
deviation in a, is found from the following equations by using an

error equation10 for (JGY‘:

86L
(Jar - i («N-,- 6M)? 2: (June i (fir

but
" j xi Z X. h" f! XI

 

a.(... , f1, see)-'

 

I x12 f ’12
and from an equation below f1 - 2? fi—i-f
hence,
29% ' a“; ‘T: " a??? ‘ m‘:
such that for A x1 - .1
51- ”I.

which when normalized for 20 intervals along 1|:i gives

J2.

l . “3—4 I 3:... 2'"
- Ni \fi x1

and (W1 is the number of events occuring with an angular

 

zlf'

He

where (f Ni -

distribution on X1 or) T51 - 1—355-9- - 66.34 for the 1323 events or

171 - g—g—li - 13.2 for the set of 261: events. Though f1 is a measmd

quantity, the following analysis is essential for determining its
value from the results of the computations. The differential number

of events, dN, may be defined as (11'! - f(6) 2n sin 9 d 6 or

AN, .1.i_3"_1_

1
Efi 2'I'I'f'Acosi '2? 6x1

For differential values of the cosine of the space angle equal to

01 (A11 ' s1)

10
C — " ess
E ’1 2s [5 N1 * A N2 1

1?

which when normalized for N events gives

f _10 AN,

ifi'u

Values for fi‘ are obtained, in this manner, for substitution into the
equation for 0.. The average statistical variation in N1 may be found

from the square root of the mean, which when normalized gives

Jug-[:1

“1

D. Depolarization due to an External Majgrgtic Field

The depolarization of the muon due to its precession in a magnetic
field may be determined from the following analysis. Consider the

precessional frequency of an electron
p. H
O.) e - W

where

e3 _ e13
we'ch 8111 PO mec .

 

 

Analogously, assuming the gyromagnetic ratio g of the muon to be the

same as that for an electron11 , the precessional frequency of the

 

muon is
H
w - P’
P 2 m r)
)1
which gives an angular deviation in time At of
H
A 0 - L—’ I t

2%

where m is in terms of me’ The above equations are only valid for a field

perpendicular to the magnetic moment of the muon. From the diagram, assuming

18

WWi-Dps

 

 

the correct equation fu' AG is seen to be
pa H cos 6
A9 - At ,
2 m g
p
The average angular deviation is given by this last equation if the

average value for the cos 6 is determined. An average value is found
from the integral of the cosine divided by the range of the limits, i.e.,

{wkcosede

. .636
"/2

 

The average angular deviation for the muon with a mean life of 2.2x10"
sec. in a field of about I: gauss ( at the bubble chamber) is found to
be .12 radian. This is only about a factor of two greater than the

maximum error in angle measurement and is deemed negligible.

19

IV. RESULTS AND DISCUSSION

The results of the first five sets of data for 1328 events gave
an absolute value of (a! I 0.1653 0.01:8. The additional set of 261:
events in data 6 was scanned for positrons with a track length equal
to or yeater than 8 cm. in the scanner plane (b.1411 cm. in the freon
bubble chamber.) This last set of data showed a larger value of
(CL! I 0.313 3 0.107 for the higher energy particles.

The equation for the normalized electron distribution was integrated
to correspond to the energy range of the measured positrons. With

the electron energy Be and

the equation for the positron track of length, L.>.. 2.78 cm., and

8, became

energy, E _>_ 6.9 Hev
be $7: I .996 - .33h3ccs 6

and for the positron with track length, L2 h.lIlI cm., and energy,

E 211.2 Mev, it bacame

luv %- .986 - .3373P cos a

where the maximum energy was considered to be 55.2 Hev. This variance

from the equation

hw%-1-}§cose

20

dN

Figure 3. The spacial distribution of N' I
N d(cos e)

x 10"

for 1328 events plotted against

cos 9 where E's?" 6.9 Nev.

 

o..- o;
a I

 

A

()

21

c mooxovo. u no... -_

0.

ON

on

on

00

Oh

 

22

Figure u. The special distribution of N' - Wi‘g'fi)‘ 10"
for 2614 events plotted against cos 9

where Fe 2. 11 .2 Mev.

 

 

 

m mooCOT H m_m.v .._

 

a:

 

 

40V

.. 00.

 

 

2b

is insignificant with respect to the error of this eXperiment.
However, the increase in a. with increased electron momentum is in
accord with the predictions of Lee and lang.s This increase in the
asynnnetry with increased electron momentmn, for the energies used

here, is also seen from the equation

hw% -1-a.cose

when it is noted that A Ni increases with increased electron momentum
as shown in the electron spectrum for )1+ decay.12

The errors in G, occuring from the measurements, are of a
purely statistical nature. Any systanatic change, due to the external
magnetic field, has been neglected as being small.

As a result or the foregoing measurements, it seems that the
depolarization effects in freon are approximately the same as in
nuclear emulsions.3 This means that a considerable nunber of events
involving muon decays must be measured before a 10% measurement on the
asymetry can be achieved. However, with the advent of large chambers
and intense beams of strange particles, freon is not an impractical

medium for observing these particles.
a.

Unfortunately very little can be said regarding the detailed
mechanism by which the muons are depolarized. Present evidence
shows a large range of polarizations which depend on purity of material

as well as the material itself.3

25

APPENDIX 1'3

DETERMINATION OF THE SPACE ANGLE

A. Coordinate Transfomatigg
1. Measurements made with respect to the scanning coordinate
axes may be transformed to a coordinate system with respect to the

front fiducial mark by the equations,

ziL 0L ‘ z1L

' - -

x1R xiR ‘03
. - -

ziR 203 Z11%,

where i - 1 indicates the mu decay point, i - 2 indicates an arbitrary
point along the projected mu direction, and i . 3 indicates an arbitrary
point along the electron path or the projected electron path whenever
the path does not continue on a straight course from its original
direction. The sign of the primed coordinates is dictated by the

pion beam direction and the center fiducial mark (+) as shown below

in the left and right scanning views designated by the subscripts

 

 

 

 

L and R.
a + x': + x'
t +z':.z| + +z'2-z' 8'
.---+(xOL, 20L) x “"+(XOR’ 80R) ‘—
+ s' : - z' - x': - x' (beam direction)
; + z': - 2'
(030%.. 8 (0,0)R 2

Figure Se

26

Diagram.of space coordinate systems and planes
for the calculation of the space angle (between
the muon and positron direction vectors) from
measurements in the two stereo projections.

This diagram, however, exemplifies the coordinate

systems with respect to the optical axis of the

camera for the right view.

27

p(O,—L,O)

  

Coordinate System
wnh respect to

Fiducial Mark

0.0.20.)

 

 

 

 

 

 

 

 

 

 

 

 

o,o,zo')
—>+Z
AXIS
Front Surfac Plane
1%
& +Y (Optical Axis)
A X I S (x"o'z')
+x/
AXIS
(X,Y,Z)

 

Figure 5.

 

28

2. Using the magnification factor M, which is equal to the ratio
of a unit in the film plane to a unit in the scanner plane, the three
preceeding primed paraueters transfer to a coordinate system in the
film plane with respect t6 the optical axis of the cameras as follows:

a . n _ n
31L ‘OL an

a . u
2in “‘13

3 I
m ’hm °

2in
The signs in these equations are consistent with the sign convention
illustrated in the diagram on page 25.
3. The transformation of the parameters, with respect to the
optical axis, into the plane of the front surface of the bubble chamber
II ll 8!
gives the parameters 211. , XiR , and ZiR in terms of the double
primed coordinate parameters. This transformation is made with the
magnification factor m, which is the ratio of a unit in the bubble

chamber to a unit in the film plane, and the aid of the triangles below.

 

 

 

 

 

 

A A
B C(mx") B C(ms")
Z ”’

D E(X”) D E(Z”)
AE-L AB-L-mz"sin¢
AC-L-mz"sin¢ AD'L
BC.- mx" BC - mz"cos 95

DE - x" DE ' Z”

29

Fran similar triangles ABC and ADE of the left diagram it is seen that
L _ X“
L -mz"sin¢ mx"

or for X"
x" . u”
(1 - ms"sin fl/L)

Likewise from the triangles ABC and ADE of the right diagram

mz'cosg¢ .

(1 - mz"sin.¢/L)

Z" .

These last two equations may be expressed in a simpler form by using
a binomial expansion. Thus, considering the left and right stereo

views, the expression:

21L" - minL"cos ¢L(1 + mLsin ¢inLu/LL)
xiRll . wjgn(1 + “Rain ¢R21Rtl)
21R" - mRziR” cos ¢R(1 0 mRsin ¢R21R”/LR)

are obtained.
h. To refer X" and Z” (which are with respect to the optical

axis) back to the coordinate system with respect to the front fiducial

mark (ZOL', ZOR'), the following equations are used:

t . _ u
ZiL Z0L ziL

XiR' ' 1‘13"

I. .
ZiR' ' ziR " ZOR °

30

S. The final transformation to space coordinates within the

bubble chamber can be made considering the following pyramidal figure.

   

 

 

‘5 ' LR
AE - LR +Yi/n
/ : 6 BC - 2m” - 20R. + ziR'
D 1 x”, o, z") E“ ' Zea. * Zi
l/A! a 4" a DB - xiR" - xiR‘
W ”"1
‘1

x , Y

F H 1 1’1

From triangles ABC and am,

Zn t I
LR .iR _ Zoe * ZiR

t I
LR + Yi/n ZOR + Z1 20R 4- 21

and similarly for the corresponding left view,

 

It . .
LL 21L 20:. + ZiL

U I
LL + Ii/n ZOL + zi ZOL + Z1

it is possible to obtain the space coordimte in the Y direction,

I _ t
Y ZiL ZiR ,
. “M
i ZiR 21L

and in the Z direction,

It - l
21 - (1 4- Yi/nLRfl-ifi 20R ’

where Z R. and ZOL' are the measured distances from the camera axis
0

to the front fiducial mark for the two views. From the triangles

31

ABD and AEP‘,

so that the X coordinate is found to be,

x1 - (1 * Yi/nLR)xiR. e

The equations of section A give Xi, Y1, and Z in terms of the

1
quantities 20L, XOR, zOR’ an, at“, and sin and the constants mLsinBL/LL,

mL cos 6L, mRsin OR/LL’ chos 9R, mR, M, ZOL.’ ZOR" ZOL" ZOR.’ 1/nLL,

and 1/nLR.

3. Cos of the Space Anglg

The muon and electron direction vectors are defined by

r2 - r2(X2 - X1, Y2 - Y1, 22 - Z1)
r3 - r3(X3 - X1, Y3 - Y1, 23 - 21)
respectively. The cosine of the space angle 923 is then

23'..._,’ ' +.-_#
(r2)(r3)U/ZX2'X1)2 * (YZ'Y1)2 * (22-21):/(X3-x1)2 + (33341)2 (43 2;)2:

 

cos a

 

The space angle may easily be determined from trigonometry tables

for values obtained by evaluating the equations of section A and B.

A.

32

APPENDIX II
COMPUTER PROGRAM

Computer Tape Sequence

Tape Order Input (Library, DOI)

Preset Parameters (00 3X)

Sb Constants

SS Sin-Cos Subroutine (Library, T5)

86 Square Root Subroutine (Library, R1)

S7 Cot Subroutine

38 Print Out Subroutine (Library, P17)

S9 Temporary Storage (allocation, not on tape)
SK Data Input Subroutine (Library, N2)

SS Address Change Constants

Heading (tape print out)

Master Program

Black Switch Transfer Order (2h h7ON)

83 Data (separate Tape)

White Switch Transfer Orders (OF F 26 1S9L)

83 Data (additional data tapes)

B. Scaling of Parameters

1 e 53 Dat‘

0)
1)
2)
3)
b)
S)
6)
7)
8)
9)
1o)

-4
‘OEX10

-4
21Lx10

-3
92Lx10

-3
933x10

.4
10Rx10

.‘
20Rx10

x1Rr10“
s1Rx10"
62Rr10'3
93Rr10'3
Film Number

2. Sb Constants
(Description)

0)
1)
2)
3)
b)
S)
6)
7)
8)
9)

2.512.8
“L” inGI/I'L

-1
mLcosGLs1O

“R’mR/La

chosORJdO'1
me1O-3

MxlO

n -3
”CL 110

1 -3
50R x10

1 -a
ZOL x10

33

10)
11)
12)
13)
1h)
15)
16)
17)
18)

'19)

20)
21)
22)
23)
2h)
25)
26)
27)
28)

1 -a
ZOR x10

1/nLL
1/aLR

1x10'1
1x10“2
1x10"3

0
1o3/ax180
5/8

180x1 0.3/71
8

x10'3

8

2
2..
103/28
8x10‘1
8x10‘3
8x1 0-‘
90x1 0‘3
180x10-3
360x10'3

C. Memory Storggg

1.

2.

Preset Parameters

s3)
Sb)
85)
s6)
s7)
58)
S9)
SK)
SS)

00F 00 hOOF
00F 00 20F
00F 00 70F
00F 00 91F
00F 00 100F
00F 00 IhOF
00F 00 200F
00F 00 280F
00F 00 306F

Sh Constants
(Decimal Form)

0)
1)
2)
3)
h)
S)
6)
7)
8)
9)
10)
11)
12)

0.009765625
0.1018
0.9923
0.09895
0.9821
0.1038
0.5382
0.000620h
0.0006116
0.000655
0.000673
0.02369
0.02h0h

3h

13)
1b)
15)
16)
17)
18)
19)
20)
21)
22)
23)
2h)
25)
26)
27)
28)

0.1

0.01

0.001
0.0.......0
0.69h.....0
0.625
0.05729
0.256
0.00390625
0.076525
0.8

0.008
0.0008
0.09

0.18

0.36

3. SS Constants
(Address Changes)

0)

1)
2)

003F 00F
00F OOBF

00F 006F

D. S? Cot Subroutine

0)
1)
2)
3)
h)
S)
6)
-7)
8)
9)
10)
11)
12)
13)
1h)
15)
16)
17)
18)
19)
20)
21)
22)
23)
2h)

ho 23L
h2 19L
L0 283h
so 23L
00 3F

26 55
LJ ZLL
26 55
b0 26L
L2 25L
L5 26L
66 25L
ho 27L
7J zosh
F5 1hL
19 36F
F5 29L
LO 28L
L1 1ssh
LS 21L
h1 29L
26 19L

K5
L5
32
75
b0
50
ho
50
10
L7
32
50
7a
50
ho
ho
ho
ho
36
22
h2
NO
00
00
00

23L
19L
17Sh
2hL

25L
7L
8F
26L
19L
16$h
23Sh
27L
s9
1hL
28L
29L
20L

1hL
S9

35

25)
26)
27)
28)
29)

53 EB 53 E3 EB

E. Heading

0)
1)
2)
3)
b)
S)
6)
7)
8)
9)
10)
11)
12)
13)
1h)
15)
16)
17)
18)

92 131?
92 2F
92 962F
92 258F
92 899F
92 322F
92 5781“
92 135?
92 898?
92 962F
92 963F
92 S78F
92 6h3F
92 3F
SO 53
26 SK
50 109
26 88
92 515F

~n in vs vs

53 £3 53 EB EB

92 259F
92 S78F
92 387?
92 S1hF
92 387?
92 Sth
92 770B
92 S15F
92 S1hF
92 6b3F
92 770?
92 707F
92 131F
92 3F
50 1hL
LS 1053
50 16L
92 1359
26 SOOF

~m ~n '4 1n ~e

F.

Master Program

0)
1)
2)
3)
h)
S)
6)
7)
8)
9)
10)
11)
12)
13)
1h)
15)
16)
17)
18)
19)
20)
21)
22)
23)
2h)

L1 53
L5 233
26 S7
LS 233
36 BL
Lb 183
LS 663
to 889
LS 183
ho 759
L0 ZSSh
LS 853
26 S7
L5 353
32 16L
Lh 753
26 18L
L0 139
L5 333
26 57
L5 333
36 25L
Lh 133
LS 633
b0 1159

36
50
32
L0
L5
to
Lb
26
L0
L5
to
50
32
L0
L5
ho
L5
ho
50
32
L0
L5
ho
Lh
26

156L
1L
157L
27Sh
S9
789
ZSSh
11L
59
633
859
11L
1S7L
27Sh
159
959
753
939
13L
157L
27Sh
289
1059
253k
28L

36

25)
26)
27)
28)
29)
30)
31)
32)
33)
3h)
35)
36)
37)
38)
39)

,h0)

h1)
h2)
h3)
uh)
h5)
h6)
h7)
LB)
h9)
50)

L5 153
no 1059
L0 256k
LS 953
26 57
L5 953
32 33L
Lh 783
26 35L
L0 389
L5 153
LS 653
L5 753
L5 53
ha 1359
79 65h
hO 1hS9
7J 15h
b0 1589
7J 1hs9
50 1659
50 163k
ho 17S9
79 225h
to 1859
L0 639

L5
to
50
32

L5
ho
LS
ho
ho
ho
ho

L5
ho

259
633
1159
28L
157L
27Sh
359
1289
783
1259
hS9
SS9
639
LS9
1359
75h
1hS9
1SSh
1559
1659
25h
10F
1759
96h
583
1959

51)
52)
53)
Sh)
55)
56)
57)
58)
59)
60)
61)
62)
63)
6h)
65)
66)
67)
68)
69)
7o)
71)
72)
73)
7h)
75)
76)
77)

50

50

50
hO
7J

50
ho

L5
ho
7J
50
Lo
7J

50
50
SS
LS
ho
7J
SO

50

1989

2059
1§sh
2159
2259
hSh
10?
2359
2&59
1859
589
2659
65b
2759
2859
55h
10F
2959
16st

95h
3189
11st
2h59
3259
2539

7J
ho
7J
ho
7J
50

to
79

ho

50
ho
7J
50
50
b0
7J

to

50
ho
79
ho
7J

37

78)
79)
80)
81)
82)
83)
8b)
85)
86)
87)
88)
89)
90)
91)
92)
93)
9h)
95)
96)
97)
98)
99)
100)
101)
102
109)
10h)

168k

128k
3539
3059

10F
3659
3859
135h
3559
16st
3759
225k
hOS9

155

SOL

SOL

66
ho

50

ho
7J
50
ho
79

50

L5
ho

L5
ho

L5
to

F5

26

L5
ho

3359
3h59
1§sh
3559
165k
3659
228k
3hS9
3959
2h59
10?
3759
1OSb
38L
38L
55
62L
62L
55
86L
86L

1S7L
15$
38L-
h1s9
8859

L5
ho

L5
to

L5
ho
7J

ho
7J

26
50
ho
7J
50

N0

26 ‘

50
50
55
50
ho
7J

h289
h9S9
h059
hhS9
5159
3959
L659
5359
h889
h9S9
5589
5089
ShS9

S6
5159
S739
5259
S359
5889

5939
165k

h859
6189

5239

L5
ho

L5
to

ho
7J
50

50
ho
7J
50
b0
7J

E1

ho
7J

to
7d

ho

38

3959
D359
5059
3859
hSS9
5259
h059
h859
5hS9
h959
5059
5559
117L
S689
5159
5259
5859
5359
5759
12hL
5959
5659
133h
6089
5159
8959
6289

132)
133)
13h)
135)
136)
137)
138)
139)
1h0)
1L1)
1h2

1&3)
1th)
1h5)
1L6)
1h?)
1h8)
189)
150)
151)
152)
153)
15k)
155)
156)
157)
158)

50

50
S5
50
26
F5

26
92
L5
D2
n0
L5
no
L5
n0
L5
L5
hz
n0
L5
50
26
or

5059
6259
165k

105
38
158L
255
1h2L
1355
1h3L
38L
659
1th
559
1h6L
3859
1h8L
1S1L
86L
hos9
152L
33
5x

N

to

92
ho
32
L1
92
N0
no
NO
D6
no
h6
no
86

no
no
h6
50
26
26

39

159) 50 53 50 159L
160) 26 ‘ 5x 26 L
161) 2h h70n

1.

2.

9.

10.

11.

12.

13.

ho

BIBLICBRAPHY

c. n. Yang and T. D. Lee, Phys, Rev. 10h, 258 (1956).

D. Barley, T. Coffin L. Garwin, L. M. Lederman, and M. Heinrich,
Phys. Rev. 199, 835 (1957).

R. A. Swanson, Phys. Rev. _1_1_2_, S80 (1958).
J. Ballam. Private Communication.
5. D. Lee and c. n. Yang, Phys. Rev. 105, 1671 (1957).

R. R. Crittenden W. D. Walker, and J. Ballam, Phys. Rev.
(to be published)

J. Rainwater, Ann. Rev. Nuclear Sci. 1, 1 (1957).

B. Rossi, High-Emery Particles (Prentice-Hall Book Company, Inc.,
New York, 1951) p. h}

C. R. Wylie Jr. , Advanced Engineering Mathematics (MoGraw-Hill
Book Camparw, Inc., New York, 1951) p. 527.

J. Scandrett, Private Communication.

T. Coffin, R. L. Garwin, L. M. Lederman, S. Penman, and A. M.
Sachs, Phys. Rev. 106, 1108 (1957).

R. E. Marshak, Meson Physics (Dover Publications, Inc., New York,
1952) p. 207

J. Huang, Private Communication.

HICHIGQN STQTE UNIV. LIBRRRIES

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