MON K’ROEEUCTIGN EN
NEUTRON - PROTON CGLLISEONS
IN THE N'ElGHB-QRHOQD
(3F [.6 EEV/C

BY
JAMES E. PARKER

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93 01764 0115

This is to certify that the

thesis entitled

PION PRODUCTION IN NEUTRON-PROTON
COLLISIONS IN THE NEIGHBORHOOD
OF 1.67 BEV/ C
presented by

James Evans Parker

has been accepted towards fulfillment
of the requirements for

BhD _ degree in _Eh;LSJ_c.S_'

:- g ’ Major professor

Date to!“ H], / gt 3

0-169

LIBRARY

Michigan State
niversity

 

 

Edward F. Grover Co.

 

Washington, D. C.

 

 

ABSTRACT

PION PRODUCTION IN NEUTRON-PROTON
COLLISIONS IN THE NEIGHBORHOOD
OF 1.6 BEV/C

by James E. Parker

A study has been made of single pion production by neutrons
on protons in a hydrogen bubble chamber. The momentum spectrum of
incident neutrons has a peak at 1.6 bev/c with a full width of 0.7
bev/c in the laboratory system. The angular and laboratory kinetic
energy distributions of the final state nucleons show the peripheral
nature of the interaction. Good agreement was obtained when the data
were compared with the peripheral interaction theory of Selleri and
Ferrari using a cutoff function. The effective mass distributions
show the presence of the 3,3 isobar and indicate that using experi-
mental cnoss-sections and angular dependence for the scattering
vertex gives a slightly better agreement than using a simple Breit-
Wigner resonance. Laboratory kinetic energy distributions show that a
cutoff function which reduces the cross-section at high momentum

transfer is necessary.

Approved: Q72 [/4 flafléZ/N
fl 7

PION PRODUCTION IN NEUTRONAPROTON
COLLISIONS IN THE NEIGHBORHOOD
OF 1.67BEV/C

BY

James E: Parker

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

1962

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42’
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ACKNOWLEDGEMENTS

The author would like to express his gratitude to Professor
J. Ballam and to J. H. Scandrett for their help and guidance during
the experimental work and during the writing of the thesis; to Drs.
R. K. Adair, L. B. Leipuner, w. Chinowsky, and other menbers of the
bubble chamber group for the use of the chamber and the genereous
loan of their film; to Dr. G. B. Collins and the members of the
Cosmotron staff for the operation Of the accelerator; to Robert
Lawrence, Jennifer Lee, and Nancy Sweet and others for assistance
in scanning and measuring the film and drawing many of the figures;
to Drs. G. Kalbfleish and M. Horowitz for giving him copies of the
ATHOS program; to Leonard Harding for handling the computational
material at the University of Michigan; and to Dr. F. Selleri for
his comments and suggestions with regard to the theoretical curves.
He would eSpecially like to express his thanks to the National

Science Foundation for financial support of the work.

ii

TABLE OF CONTENTS
Abstract
Acknowledgements
I. Introduction
II. Description of the experiment
III. Measurement and selection of events
IV. Results and interpretation
A. The ppTF'events
B. The four body events
V. Figures
VI. Appendix: description of programs
A. Track fitting program
B. Kinematic testing program
C. The peripheral model program

References

ii

16
22
24

112
118
120
127

LIST OF FIGURES

l. Diagrams representing pion production in nucleon-nucleon
collisions.

2. Beam design.

3. Typical picture.

4. Neutron beam direction cosines.

5. - 8. Standard deviation of circle fit.

9. Measurement of back fiducial mark.

10. Mbmentum spectrum of fir beam particles.

11. meentum spectrum of .45 bev kinetic energy positive pions.

12. Mbmentum spectrum of .60 bev kinetic energy positive pions.

13. - 22. Initial Q value.

23. Fractional change in momentum made by QEVAL.

24. Neutron lab momentum of ppnr events.

25. Neutron center of mass momentum of ppnr events.

26. Scatter plot of Y and Z direction cosines of the beam particle
of apparent ppnF events whose adjusted beam angle was greater
than 5°.

27. - 31. Proton center of mass momentum.

32. - 36. Proton lab kinetic energy.

37. — 41. Proton center of mass angular distribution.

42. — 46. M* of pn’ combinations.

47. — 51. Angular distribution of pion in M* system.

52. - 56. Pion lab kinetic energy.

57. - 61. Pion center of mass momentum.

62. - 66. Pion center of mass angular distribution.

67. Elastic and charge exchange scattering cross-section for

n’p scattering as a function of the invariant mass of the pair.

68. Peripheral diagrams for the ppflr events.

69. Contributions to the kinetic energy spectrum.

70. Comparison of cross-section of the reactions n+p—9 p+p+nF
and p+p-9 p+p+nP at .97 bev.

71. Comparison of M* spectrum from peripheral model and from phase
space with resonance.

72. Incident lab momentum of pnnfnr events.

73. - 75. Nucleon lab momentum.

76. - 78. Nucleon center of mass angular distribution.

24
25
26
27
28
32
33
34
35
36
46
47
48

49
SO
55
6O
65
7O
75
8O
85

9O
91
92

93

94
95
96
99

79.
85.
88.

A1.

A2.

- 84. M* of nucleon—pion combinations.
- 87. M* of pion-pion combination.

Incident lab neutron momentum of pp‘rfiTo events.

Geometry for finding points in the chamber from measured
points.

Geometry of the circle fitting proceedure.

102
108
111

116
117

I. INTRODUCTION
Although much work has been done on pion production in p-p

interactions (1'7)

, not much has been done with n-p interactions
except those involving incident protons on neutrons in complex nuclei.
Probably the greatest reason for this is the inherent difficulties

in working with a neutral beam of particles. Those experiments which

(8_10) have resulted in

have dealt with incident neutrons on protons
few events and the energies of the incident neutrons of these events
were spread over a rather wide range. Hence the detailed analysis of
momentum and angular distributions of the final particles at a given
energy were statistically inconclusive. In this experiment we have
obtained a larger number of events and are able to sub-divide the
events into slices of incident energy and still maintain statistical
significance.

The earliest theory dealing quantitatively with pion production

(11) In this model he considers

is the statistical model due to Fermi.
that since the interaction is so strong the incident particles come
together inside an interaction volume and quickly attain a sort of
thermodynamic equilibrium between the various possible final states.
Such final states are limited by the conservation rules such as
conservation of energy, momentum, charge, baryons, etc. The system
then proceeds into one of the possible outgoing channels with a
probability proportional to the available volume in phase space.
Branching ratios of the possible final states are predicted as well

as the spectra of the variables characterizing the final particles.

Put another way, the model assumes that the matrix element coupling
initial and final states for a given reaction is constant. This theory
is called the statistical model and the spectra of variables predicted
by it are referred to as "phase space" curves.

As early experimental information was obtained it became
apparent that the statistical model as it stood was an oversimplification.
Irregularities in the momentum and energy spectra of final particles
were observed which were not predicted. Even the multiplicities of
final state particles were not accurately predicted. It became

apparent that final state interactions must taken into account.

2

In the early and middle 1950's there was considerable effort
to incorporate the effects of final state interactions into the

theory starting with the work of Watson (12)'

The assumptions

generally made in this early theory were that the final state

forces play no important role in the primary interaction which

causes the interaction between the incident particles, but the

reaction is then distorted by the short range forces between the

final particles before they leave the region of their mutual inter-

action. Introducing into the theory the principles of conservation

of isotopic Spin and of parity and applying the rules of conservation

of angular momentum and of the Pauli Principle placed severe restrictions

on the final states allowed. The effects of the known forces between

nucleons and the very strong T = 3/2, J = 3/2 resonance between pion

and nucleon were introduced into the theory and the result was to

effectively distort the phase Space available to final particles.

Reviews of the development and application of this early theory are

contained in articles by Rosenfeld and by Gell Mann and Watson (13).
The development of theory in this manner finally led to the

"isobar" model of Lindenbaum and Sternheimer (14). In this theory

the interaction is assumed to proceed as a two step process: the

formation of one or two isobars followed by the subsequent decay of

the isobar into pion and nucleon. See figure 1a. Hence they use a

two body phase space factor multiplied by the cross-section for

formation of the isobar and a factor determined by the kinematics

of the decay of the isobar. The isobar predominately responsible

for pion production at moderately low energies (< 1.5 bev) is the

isobar with effective mass 1.237 bev corresponding to the 3/2,3/2

resonance in pion-nucleon scattering. The angular distribution of

isobar formation was considered in two cases: isotrOpic and

only forward and backward in the center of mass system. The pions

were assumed to be emitted isotropically in the isobar center of mass

systems As better resolution along with additional data at higher

energies were obtained, two additional resonances were observed in

the T = 1/2 state in pion-nucleon scattering. These resonances at

about 600 mev and 880 mev are not as strong and are at a much higher

energy than the 3/2 resonance at about 190 mev. The effects of these

higher resonances have been incorporated into the isobar model, and in

3

an attempt to account for multiple pion production it is assumed

that the higher isobars can decay into one of the lower isobars by
emitting a pion as well as by decaying directly to pion and nucleon.
The model has been extended to include pion-nucleon interactions as
well. Discussion of the application of the isobar model is contained

(15)

extensive bibliography of recent work is contained in Fickinger, et al

in many articles of the past few years. In particular a fairly
of this reference.
Although the isobar model has been fairly successful in predicting
some of the features of pion production; certain of the features, such
as strong forward-backward peaking of the nucleons and a strong tendency

toward small momentum transfer to the nucleons,(5'7’16’17)

suggests a
more direct interaction model, namely the peripheral model. Several
authors have written theoretical papers or discussed experimental results
(18-24) The work of Selleri

is of particular interest. The

in connection with such a peripheral model.

(25)

model which he proposes is a direct interaction model in that in the

in collaboration with several people

inelastic scattering of any particle by a nucleon, the process is
dominated by the exchange of a single virtual particle from the cloud
of the nucleon and the scattering of the incident particle by this
virtual particle. See figure 1b. We are interested here in pion
production in n-p interactions where the exchanged particle is a pion,
although other processes may be treated as well. The symbols labeling
the diagram in figure 1b will be used to represent the 4-momentum of the
particles, but when the energy component is used explicitly an additional
zero subscript indicates energy and the symbol in the diagram represents
the 3-momentum. In terms of the S matrix Selleri writes:

Sfi ~ <q2]j(0)lp2> A;(A2)<q3qilj(0)lp1)
where the Operator j(O) is the current operator for the virtual particle
defined by the equation of its field 45 (x): (El—5L2) qb(x) = j(x), and
Ifl;is its propagation function. The vertex function <q2'j(0)jpg> is a
function only of the 4~momentum transfer; A? = (pg-q2)2, and represents
the emission of the pion from the lower vertex. The vertex function
(qaqllj(0)’p%> is proportional to the analytic continuation of the
matrix element for the process k+p1 —9 q3+q1, and represents the inter-
action between the incident particle and the virtual particle at the

upper vertex.

4

In more explicit terms Selleri has developed from the theory

the general partial cross-section:

 

 

62 5+ (m 4’2): K2(A’)}1”2(A‘) co‘ P0 060' Q .
O" _—_- __ 1 I 9/ A r
d (117le (A2 +44“)? ' E? 9-}: (CU, 6&5 ’ )

.140, +3) 435/132 4% .
5min) 630

’/

where G2 is a coupling constant, F: {Ip1.p2)2 -M§2Nhf21, aJis the
invariant mass of particles ql and q3, the superscript Q implies
evaluation in the q1,q3 center of mass system, and 3%; is the
differential cross-section for the process k+p1 -’ q3+q1. The
factors K and K' are the form factors for the lower vertex and of
the propagator defined so that K( .13) = 10H?) = 1. This is a
rigorous formula but cannot be applied without the dependence upon
A? of the form factor terms K and K' and the partial cross-section
term 3%: . The approximation valid for small A2 (pole approximation)
consists of putting in A2: 1? in K and K' and in Egg . In 5%-
the (F dependence enters practically only through the angle. The

expression then reduces to:

 

 

4c,- . 62 A2+(mz-M.l2 601E 571320“ wag)-
(277)“: WWW 53" n

47?; ~ 8) 43.1 42': 1%:
3,, 520 630

In the simplest application the pole approximation is used. Several
comparisons of the predictions of this theory with experiment have

been made (26)

and the results have been remarkably good. The more
general theory not limited to small momentum transfer has been applied
by Ferrari and Selleri and indications are that the one pion exchange
model is valid at large momentum transfers. They have also attempted
to extract a tentative behavior of the pionic form factor. See the

second reference in (20).

5

A further advantage to the theory of Selleri is that it
predicts absolute cross-sections. In other theories there are
adjustable parameters such as the interaction volume in the
statistical model and the isobar model. In the isobar model
there are other parameters which have to be fixed from phenomenolOgical
arguments, such as the angular dependence of isobar production and
the probabilities of formation and decay of the higher isobars. In
the Selleri theory the partial cross-section entering into the
upper vertex can be measured by other means in the case of N-N
interactions. The model is also based on a very descriptive
picture of the interaction.

Until now only the pap experiments have been compared with this
theory. The n-p experiments have resulted in too little data to be
able to subdivide the events into small slices of incident energy and
still maintain statistical significance. There are enough data in
this experiment to allow such subdivision. We have been able to
break the bulk of our data into 50 mev/c slices of incident center

of mass momentum and still have over 200 events in each slice.

II. DESCRIPTION OF THE EXPERIMENT

The experimental design has been described by Adair (26)
will be sketched here also. See figure 2. The external proton beam
of the Cosmotron was brought to a focus by three 8" x 32" quadrupole
magnets and then refocussed by two 12" x 40" quadrupoles onto a steel
target (pion target). Negative pions at about 4° from the proton beam
were focussed by a series of quadrupoles and bending magnets onto
a polyethylene target (theta target). The beam was originally set up
to investigate K2 interactions and decays. At the theta target
interactions such as 7T'+ (n or p)-—> (/V or 2:) + K° + pions give
some neutral K's going toward the chamber. The K°'s produced at
about 6° from the pion beam reached the chamber located at ten feet
from the theta target. At this distance practically all of the K2
part of the K°'s has decayed out of the beam leaving K2's.. Inherent
in a neutral beam of this sort is a rather large background of neutrons
as can be seen from the large number of proton recoils in a typical
picture. See figure 3. These neutrons are produced by the pion beam
in the theta target principally by pi-nucleon elastic and inelastic
collisions. The neutrons produced will have a spread in momentum due
to the spread in pion momentum and the Fermi momentum of the target
nucleons as well as from the inelastic scatters.

Average external protons were about 7x109 per pulse giving about
2.5x105 pions at 1.6 bev/c t 7 percent on an area of about 50 cm2 at the
theta target. From the number of decays of K2 into 3 pions Adair<26’27)
estimates the number of Kg's in the chamber to be about 1.5 per pulse.
The target is located in the entrance of a large sweeping magnet to
divert charged particles from the chamber and about 40 gm/cm2 of lead
were placed behind the theta target to convert gamma rays and sweep out
the resulting showers.

The types of events to be expected from the K2 interactions

are as follows:

Eo+p—9A +(1,2)7r

” 22 + (1,2)v

- +

~9K +p+7T

K°+paK++p+IT-'
K°+p-*K°+p

'7

From the neutrons in the beam one would expect to see
interactions of the type:
n+p—r n+n+pions
-r n + p + pions
._; p + p + pions
Of these types those with only one charged track Cannot be analyzed.
' Since no five pronged events (ppninfn7)'were observed in the scanning,
it is assumed that very few if any of the three pronged events were
. three pion production. This leaves us with the reactions:
h n+p—,p+p+rr- A '
._, p + p +77“ +'n°
_.. p+n+TT++1T°
All of these types will be three pronged stars in the bubble chamber.
The detector was Adair's 14 inch hydrogen bubble chamber which
has a depth of 8 inches. There are two fiducial marks on the inside
surface of the front window 10 inches apart and in line with the beam
direction. Also there is one such mark in the center of the inside
surface of the back window. The chamber was viewed by three cameras
equally spaced around a circle of 13 inches diameter 36.5 inches from
the front window. The demagnification of the camera system was about 11.
The chamber was illuminated by a flash tube at the oppOsite end of the
apparatus and the light was condensed by a large lens behind the rear
window to a point in the plane of the cameras so that only scattered
light reached the cameras. Thus the tracks are seen as bright tracks
on a dark field. The chamber was in a magnetic field of 17.5 kilogauss.
Magnetic measurements of the field without the chamber in place were made,
and the shape of the field was fitted by the function:
B: B0 +Bz(z-zo)2 -BI.(r-ro)2
where Bo'= 17.5, BZ = 0.0062319, and Bra: 0.0035486. zo and r0 are
the geometrical center of the bubble chamber.
About 39000 pictures were scanned for three pronged stars
whose vertices were within a fiducial region of the chamber. The
fiducial region was chosen to exclude the edges of the chamber so that
events with short tracks which leave the illuminated region of the
chamber would not bias the results. A track may also be short because
it stops in the chamber. All events that originated within this volume

were measured and accounted for. Events that proved to originate

8

within two centimeters of either window were discarded. In the
pictures there are a large number of obvious p-p elastic scatters
and some'n-p scatters. These events could be interpreted as a
three pronged event with the negative track going backwards in the
lab. In general p-p scatters can be recognized by three heavily
ionized tracks and a 90° opening angle in the outgoing state. If
there was any doubt that the event was such a scatter it was
measured and interpreted anyhow. Such scattering events are easily
distinguished after the measurement has been done by the existence

of a fairly large momentum going backwards in the lab.

III. MEASUREMENT AND SELECTION OF EVENTS

When the film was scanned events were sketched on a Special
sheet and track numbers were assigned to the three tracks. A note
was made if the track either stopped in the chamber or left the
front or back window. The film and picture number of the event was
also noted on the sheet. Events were then measured on a Hydel film
reader, which is a digitized projector. In this machine the film is
held flat under vacuum on a glass plate mounted in a heavy base upon
which is mounted a Gaertner stage. The stage carries the projector
lenses and three small projectors (one for each View) which superimpose
a light reticle on the picture. The lead screws of the stage are
coupled to Datex digitizers which ultimately cause an I.B.M. card punch
to record the measurement. By turning cranks under the viewing screen
one can position the stage so that the reticle is superimposed over
the point to be measured. The least significant digit of the digitizers
represents one micron on the film and the over-all measurement accuracy
of the stage is about two microns.

The events were measured in each of the three views. For each
view the front fiducial marks were measured on a separate card. Then
each track was measured at five roughly equally Spaced points starting
with the vertex of the event and ending with the end of the track. In
certain circumstances however the measurement did not extend to the
end of the track. If a track stopped in the chamber its momentum
changed appreciably over the length of the track. By measuring many
such stopping tracks we found that if about half the range of the
track were measured the calculated value of momentum agreed quite well
with the range-momentum curves. Thus for stopping tracks we measured
only the first half of the track; the whole track was then measured
as an extra track in order to measure the range.

It was found that for long tracks with appreciable curvature
the error function from the fitting procedure was excessive. This
error function represents the scatter of the measured points from the
fitted circle. This is caused by the fact that such tracks may be
slowing down somewhat over their range and by the fact that the
magnetic field is not constant throughout the chamber. W6 found that
by limiting the sagitta of such tracks we were able to improve this

situation and still maintain a good measure of momentum. These tracks

9

10

were then remeasured in order to measure the visible range.

A final reason for treating tracks in a special manner came
from tracks that were too short to give accurate measurements. In
order for this not to introduce bias, we measured tracks in the normal
fashion if the total range as seen on the viewing screen was larger
than a fixed amount. If the track was shorter than that amount it was
measured backwards. The reason for this lies in the manner in which
first and last points are treated by the computer program. By
measuring the track backwards a good measurement was obtained of the
endpoint so that the range and angles could be calculated by hand.
Mbst of these events turned out to originate within two centimeters
of one of the windows and hence the event was discarded. Mbst of
the events remaining in this category had short stopping tracks and
hence the momentum could be read from the range-momentum curves.

After measurement the data was reduced by a computer program
written by us for the I.B.M. 709 at the University of Michigan

Computing Center. (28)

This program fitted the measured points for
each track to the best helix in the chamber by a least squares method
using data from all three views simultaneously. An error function
was derived from the scatter of the measured points from the fitted
circle. Angles were computed in a straightforward fashion and
momentum was calculated from the radius of the helix and the average
magnetic field along the track.

After the helix fitting takes place the program proceeds to
attempt a kinematic fit of the event to the possible interpretations
having three particles in the final state. The procedure used by
this program is to adjust the measured momenta in order to balance
energy in such a manner that an error function, (the sum of the
squared deviations of the momenta from their original values measured
in terms of the probable error in momentum), was minimized. This
function was called "chisquare". If this chisquare was not too
large and the energy equation was made to balance within one mev,
the computer continued with the interpretation and computed center
of mass momentum values for the event. If chisquare got too large

before the energy equation could be balanced, the interpretation was

11

deemed incorrect. Up to five iterations of the momentum adjustment
were allowed although one was in general enough to either cause
chisquare to be too large or to balance energy.

In the case of the four body final state interpretations with
an outgoing neutral particle, a special program was written to
calculate the momentum of this neutral particle using the equations
of energy and momentum balance. In this case less information is
available and it is necessary to assume that the beam direction is
fixed.

After the results of the track fitting were back from the
computer, the film was then rescanned with the computer information
at hand. Using the measured values of momenta and the appearance
of the tracks of the film, a judgement of the relative ionization of
the tracks was made to attempt to determine the identification of
the particles. Due to the quality of the pictures the range of
ionization was limited so that most of the time one could say only
that a track was or was not a pion. Usually this was enough
information along with the results of the kinematic test to confidently
identify the event as one of the several types. There still remained
some events that could not be so identified and several tests were
made to help in pinning these down.

In order to determine the source of the incident neutrons we
made a scatter plot of the apparent beam direction as measured by
the program for good ppTF events. The program calculates the direction
cosines of the beam particle directly from the vector sum of the
outgoing particle momenta. The events chosen in this plot were those
which fit the pp“? criteria of kinematics and ionization regardless
of beam angle. Figure 4 shows this scatter plot for the region near
zero angle. No accumulation of points was observed at any position
except as shown which is the direction of the theta target. One
might expect the possibility of some neutrons coming through the
shielding directly from the pion target but no evidence of this was
seen. Using this information we required that the beam particle be
within 5° of the x-axis to be acceptable. This criterion may exclude
some pp F events but should not introduce bias into the resulting
sample. Although the geometry implies a beam Spread of about 3°,

50 was chosen to allow for some random measuring error.

12

In order to determine the over-all accuracy of the measuring
and fitting procedure we studied the variable which I shall call here
§;g, On the computer pages this is called "standard deviation of circle
fit". It is obtained from the scatter of measured points from the
fitted circle. We made a histogram of Sig for all tracks regardless
of other variables (See figure 5.) and separate histograms for tracks
of different ranges (See figures 6 - 8.). We found that Sig typically
was in the neighborhood of 0.01 cm and its Spread varied slightly with
range. In order to reduce measurement errors we put an upper limit on
Sig of 0.03 cm, and if it was larger than that amount the track was
remeasured. Except in a few cases we were always able to get a satis-
factory measurement. In those cases where we failed we could ascribe
the difficulty to poor tracks or an unfortunate overlay of some back-
ground track or to a slight track deflection. For a track with a
sagitta of 0.5 cm and a Sig of 0.01 cm one would find a 2 percent
probable error in momentum. One can see that for relatively Short
stiff tracks the relative error in momentum would be larger.

As an internal check of the measuring and calculation procedure
we made 200 measurements of the three fiducial marks and calculated
the position of the back fiducial mark and the position where a ray
going from the back mark would pierce the front window. See figure 9.
Of particular interest is the z-coordinate of the back fiducial mark.
This measurement gives an average measurement of 20.324 cm where the
8 inch depth is 20.320 cm. The half width of the distribution of
about 0.025 cm which agrees with the estimated uncertainty of measure—
ment in the z direction. The corresponding uncertainty in the x and y
measurement is about one fifth that of the z measurement or 0.005 cm.

To check the measurement and calculation of momentum directly

we made measurements of some beam tracks of charged particles. We

(29)

--L n.

and some positive pions loaned by" H. J. Martin and E. D. Alyea of Indiana. .

Negative pions were at 1. 89 bev/c as quoted by Walker and our measurement

measured some negative pion pictures kindly loaned to us by Walker

was 1.925 bev/c or about 1.8 percent high. See figure 10. The spread
in the measurement was about that quoted by Walker. This momentum value
is quite high and the tracks are quite stiff. The positive pions from
Indiana were at 0.6 bev/c and 0.725 bev/c and the results of the

measurement are Shown in figures 11 and 12 and agree quite well with

13

the known values.

The kinematic test program is supposed to balance energy and
momentum at the same time for the possible mass assignments. Now if
one were starting with perfect values of the momenta and angles there
would be one and only one satisfactory choice of masses, namely the
correct ones. However, since we do not have in each event perfect
values, the program adjusts the values of momentum in the best way
to balance energy for each mass assignment in turn. One might ask
just how sensitive the problem is to discrimination between the
various assignments. In order to get some feeling for this we
plotted the initial Q values for each mass assignment for those events
which were identified as good ppn’ events. The Q value here is
defined as the total incident energy in the lab minus the total
outgoing energy. In other words these Q values are for the initial
values of momentum before any adjustment. See figures 13 - 17.
we see that for the correct identification (figure 13) the
distribution centers about 5 mev with a Spread of about 17.5 mev
on either side. The displacement from zero of 5 mev represents
our total systematic error. The Q distributions for the pK+fl—
identifications are shown in figures 14 and 15. These are Spread
out over much larger values and Shifted to higher Q values. No
negative values were found in the sample taken. One can See that
there is only a small overlap of these curves over the similar
curve for the ppn’ identification. The similar curves for the pK-IT+
identifications (figures 16 and 17) Show that the distribution
is again very wide but covers the zero region and overlaps the
ppnr curve entirely. Fortunately this identification can generally
be ruled out from track ionization since there must be a positive
pion and the negative track must not be a pion. The important
point of these curves is that in all cases the wrong identification
gives a Q value generally far from.zero. Similar distributions
were made for those events which were identified as pnTFn-. See
figures 18 - 22. Here we see again that the Q values for the
3-body identifications are spread out over a wide range although

they overlap the zero region.

 

14

A final test of the kinematic fitting program was to examine
for a sample of the ppnr events the relative adjustment of the momen-
tum done by the routine. That is (change of momentum)/(initial momen—
tum). This is shown in figure 23 and indicates that the adjustment
is symmetrical about zero and averages about 2 percent.

The criteria for accepting events into one of the various event
types were that the ionization of the tracks was consistent with the
particle identification, the respective kinematic test indicated
consistency with the type, and in the case of the 34body types the
beam angle as calculated from the adjusted momenta was within 5° of
the direction of the theta target. As has been mentioned before, the
range of ionization on the film was not large and it was not always
possible to uniquely identify all of the particles by ionization.

Mbst of the time it was possible to distinguish between pions and

other particles. From about .3 bev/c to .95 bev/c the proton ioniza-
tion is twice minimum or better while the pion ionization is almost
minimum. Below .3 bev/c the range—momentum relation begins to be
useful in identifying the tracks. In this respect it would not have
been possible to always distinguish between the ppn' events and one of
the several pKn types had not the incident momentum of the K°'S been
(26) estimated the momentum of the K°'S to be 1050 r/-
150 mev/c, whereas the incident neutron momuntum ranged from 1000 mev/c
to above 2000 mev/c. See figure 24. Mbst of the events found had

an incident momentum larger than possible for a K° and could hence

known. Adair

be called neutron events. The contamination from.the K°'S in the
beam will be insignificant in the neutron events and present only
at the lowest energies. However the contaminatial by neutrons in a
study of the K°'S may be considerable since the neutron Spectrum
overlaps the K° Spectrum and the number of the neutron interactions
is approximately 100 times larger.

Tests were made by running some of the ppnr events through
the program for the four body types. we found that about four tenths
of the three body events would give reasonable incident momenta for
the pnntn’ identifications but only one in a sample of one hundred
fit the ppnrnP identification. Fortunately the pnnfnr identification
can more often be distinguished by ionization. we therefore feel that

any contamination between the types of event is not too serious.

15

0f the events scanned we selected 3441 to analyze for this
paper. From these we identified 2012 as ppn’, 633 as pnINIF} and
65 as perfiP. This left 791 not counted. In studying these remaining
events we found that about 245 of them appeared to be real ppnr events
which were excluded due to a calculated beam'angle of larger than 5°.
We made a scatter plot of the direction cosines of the beam particle
of these events to see if they might indicate a source of neutrons
other than the theta target, but they do not. See figure 26. They
seem to Spray uniformly about the direction of the theta target in
azmuth but they do tend to be going slightly upward entering the
chamber. 0f the remaining 487 events 97 appeared to be pp"? events
which barely failed the chisquare test that we imposed. 0f the
remaining 390 events 134 were not identified due to errors in measure-
ment. Of these about 60 percent had tracks that were very Short due to
leaving the chamber and about 40 percent had tracks that were either
fuzzy or obscured by some background track or otherwise hard to measure.
Repeated measurement of these events did not improve them. This left
254 events unidentified principally because tHe ionization was not

consistent with any of these types. Table 1 summarizes these results.

Table 1. Classification of the events.

 

 

4 Event type Number Percent
pprr: 2012 58.4
pnrrrr 633 18.4
pprr 65 1.9
ppnr‘but angle larger than 5° 245 7.1
ppnF but chisquare larger than 7.8 97 2.8
Errors prevented identification 134 3.9
Otherwise not identified 255 7.6

 

 

 

 

 

Total 3441

 

IV. RESULTS AND INTERPRETATION
A. THE ppn’ EVENTS
In order to compare the results of this experiment with the
peripheral model we have written a computer program to calculate the
expected spectra of various variables. The calculation is based on

(21). In

the very detailed theoretical report by Ferrari and Selleri
this report cross-sections and Spectra are presented in a general
form and in the "pole approximation" which means when the transfered
particle is considered to be real. ‘We have written the program.using
the pole approximation and have not included any interference terms.

The differential cross-section in all final variables reduces to:

 

 

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where G is a coupling constant, F ={(p1-P1)z " MIZMZZ}: A is the
4-momentum transfer (see figure 2), M, m, and Here the masses of
incident, outgoing, and transfered particles,co is the invariant mass
of the q1q3 system, the Q superscripts imply evaluation in the rest
system of q; and q3, the S—function brings about over—all 4~momentum
conservation, and GT is the cross-section for the process:

P1+1C " Q1 + Q3 -
In order to evaluate the Spectrum of any variable one must integrate
this equation over the proper variables. A change of variables is made
to a set of variables which includes the one whose Spectrum is desired,
then the resulting equation is integrated over all other variables.
Ferrari and Selleri hare given expressions for the Spectrum of lab and
center of mass energy and of center of mass angular distribution of
particles q; and q; which are in a form for easy computation.

In the above expression for the differential cross-section
notice that all of the quantities are straightforward kinematical
quantities except g. Our computer program is written so that the
part which calculates this factor is a separate subroutine and can

easily be changed. We have written several forms of this subroutine:

16

\

17

one inserts for the cross-section a Breit Wigner type of resonance
multiplied when necessary by an angular part (eg. 1 + 30050) which
would use in the case of n-p inelastic scatter the value and width
of the resonance in pi-nucleon scattering. we have also written
a subroutine which does a linear interpolation in energy on tables

of the cross-section or the coeficients in the power series of cose
up to cos50 for the differential cross-section. The values for the
coeficients as a function of energy were obtained from a thesis by
Wood (30) and a few low energy points were obtained from Goodwin,

et al, (31).

peripheral model using the actual experimental cross-section for the

Hence we are able to calculate the Spectra from the

upper vertex as a function of the energy and angle.

We have also written a routine which inserts a cutoff function
of the form e'A'into the calculation in order to compare the pole
approximation with a non-pole approximation. Spectra will be presented
using the pole approximation with the experimental partial cross-section,
the cutoff function with the experimental partial cross-section, and
the cutoff function with a Breit Wigner resonance for the partial
cross-section.

To facilitate comparison with this theory we have subdivided
our experimental data into slices of incident energy. The subdivision
was made in 50 mev/c Slices of incident center of mass momentum as

Shown in table 2.

Table 2. Subdivision of the ppTF data.

 

 

 

 

 

 

 

 

 

 

 

Slice Lab Mbmentum. C. of M. Momentum No. of Events
From To Center From To Center
1 1.28 1.43 1.35 0.55 0.60 0.575 361
2 1.43 1.58 1.50 0.60 0.65 0.625 339
3 1.58 1.75 1.67 0.65 0.70 0.675 378
4 1.75 1.92 1.84 0.70 0.75 0.725 294
5 1.92 2.10 2.01 0.75 0.80 0.775 152
otal 1524

 

Momenta are

in bev/c.

 

18

The total elastic and charge exchange cross-sections for
nTp scattering are shown in figure 67. The upper limits of M*
for the five Slices of the data are Shown and indicate that only
the lowest peak correSponding to the 3,3 resonance in the pi-nucleon
system is involved in these data. There may be some effect due to
the broader peak in the elastic cross-section at 1.5 bev in the
higher slices but it will not be discernible as such from the general
Shape of the curves Since it is so broad and the higher slices
only approach this peak.

In the pp"? final state the protons of course are indis-
tinguishable. Also either the proton or the neutron can scatter
from the pion in the cloud of the other nucleon. Hence there are
two possible diagrams for this reaction (See figure 68.), and in
both cases either proton must be counted. In diagram 1 the
probability of the emission of the 71° is half that of the w’ in
diagram 2, but the charge exchange cross-section is approximately
twice the elastic cross-section so that both diagrams contribute
about equally.

we have also used the ATHOS program from Berkeley, Calif.
to calculate invariant phase Space curves for comparison. In all
of the curves to be presented for the ppnF data the peripheral
model curves are drawn with a solid line and the phase space curves
are shown dashed. In some cases the phase Space curves alone
are Shown, in which case they are still dashed. Ordinates on the
graphs are twice the number of events for all curves except those
describing the pion distributions, in which case ordinates are the
number of events.

Center of mass momentum distributions of the protons are
Shown in figures 27 - 31 and compared with the peripheral model
and the statistical model. One can see that the agreement with
the peripheral model is fairly good from slice to slice with the
exception of Slice 1. This trend will be seen in other variables
also and might have been anticipated since the kinematical limit
of MN for slice 1 is barely over the peak in the cross-sections
as seen in figure 67. The Spread in incident energy within the

Slice may even affect the Shape of this curve.

19

Lab kinetic energy distributions of the protons are Shown in
figures 32 - 36. Again comparison between the data, the peripheral
model, and phase space is shown. The general features of the peripheral
model are low energy peaking and a rather broad flat distribution
toward higher values before dropping down to zero at the upper limit.
The low energy peak comes from pa in diagram 1 and from p; in diagram 2.
See figure 69. The proton p2 is the proton resulting from the station-
ary proton in the bubble chamber which has emitted a pion and recoils.
Its low kinetic energy distribution indicates the predominance of
low momentum transfer in these peripheral interactions. The proton
p1 here results from the pion in the cloud of the passing nucleon
scattering the stationary bubble chamber proton. Its low energy
distribution Simply reflects the difference in mass between the pion
and the proton. The distribution from proton pa in diagram 2
contributes a broad distribution which causes the shoulder at the
high end of the curves. This also indicates low momentum transfer
to the Spectator nucleon. Proton p; in diagram 1 contributes a broad
lump which fills in the center of the distribution. AS before, Slice 1
does not Show enough structure to warrant discussion but the agreement
with the peripheral model is seen in the higher slices as the departure
from the statistical model becomes more pronounced.

Center of mass angular distributions of the protons are shown
in figures 37 - 41. Comparison with the peripheral model 21 shows
(excellent agreement for all Slices.

M The invariant mass, M*, of the pn’ pairs is Shown in figures 42 - 46.
M? is the total energy of the two particles in the rest system of the
pair. The effect of the 3,3 resonance between the pion and the proton
can be seen clearly in all slices as a peak at M*== 1.237 bev. Again
the agreement with the peripheral model is good.

Figures 47 - 51 Show the angular distribution of the pion
in the rest system of the pnr pair. The 9 = 0 direction of this
system is the direction of motion of the isobar from the center of
mass system of the event. This angular distribution does not reflect
directly the angular distribution of the four pronged vertex in
diagrams 1 and 2. In the experimental distributions no structure is

seen except an isotropic distribution.

20

For completeness, distributions of lab kinetic energy of
the pious are shown in figures 52 - 56. Also pion center of mass
momentum distributions are Shown in figures 57 - 61, and center
of mass angular distributions are shown in figures 62 - 66. Phase
space curves are also Shown for the lab and center of mass energy
distributions. The angular distributions Show an isotropic distribution.

Finally we compared the results of this experiment with n+p
interactions with the p+p interactions of Barnes, et al (6) at 970 mev.
In the present experiment there is no good way of estimating the
total flux of neutrons in the chamber in order to get absolute
cross-section values. However in the p+p experiment the total flux
of protons was known and such cross-section values were obtained for
the final state pmnP, which is similar to our ppnr state. Now the
incident p+p system is always in a T = 1 state while the n+p system
is in the T = 1 state half the time and the T = 0 state half the time.
Assuming that in both cases the reaction N+N -’ N+N+fl’goes by way
of the 3,3 resonance in the pi-nucleon system, isotOpic Spin argu-
ments predict that the ratios of the final states from these reactions

will be as follows:

n+p —' n+(p+fl°) = n+(n+7;r) = p+(n+“°) = p+(p+rr-) ::
2/12 1/12 2/12 : 1/12 ;

pane?) : p+(p+rP) ..
1/12 . 2/12

and p+p -+ n+(p+r#)
9/12

Noting that the final state ppnP Should occur twice as often as the
ppnr state by these arguments, we normalized our ppTF data to the
total p+p cross-section and then compared our data with one half of
the ppm? Spectrum of lab kinetic energy. See figure 70. The Simi-
larity of magnitude of the two histograms merely indicates that the
ppnP cross-section is the proper fraction of the total cross-section,
but the Similarity of shapes of the two histograms is good evidence
for the validity of charge independence.

It is also interesting to note the similarity in the shape of
the curves of MT obtained from the peripheral model and from phase

space with a resonance between pion and nucleon folded in. In the

21

ATHOS program there is a subroutine, ARAMIS, which does the phase
Space with resonance. We have used this routine, and figure 71
Shows the comparison with the peripheral model. The curves Shown
are for an incident momentum in the lab of 1.67 bev/c. Comparison
at other energies Show a similar picture. Both models are roughly
similar, as would be expected, but the peaks of the two curves
occur at Slightly different points and the peripheral model gives
a little higher Shoulder at the upper end of the Spectrum.

B. THE FOUR-BODY EVENTS.

In the case of four particles in the final state with one
of them neutral the problem of balancing energy and momentum is not
overdetermined, as it is with the three body events. In order to
find the missing momentum, we had to assume the direction of the
incident particle, assign all masses, and find the incident momentum
as well as the component of momentum of the missing particle parallel
to the incident direction. The transverse component of the missing
momentum is equal and opposite to the transverse component of the sum
of the three visible momenta. The two remaining unknown quantities,
the parallel component of the missing momentum and the incident momentum,
can be solved from the remaining momentum equation and the energy
equation. The method used was an iterative one. The first guess
for the parallel component of missing momentum was zero, and the
resulting energy unbalance was calculated. If the energy unbalance
was larger than 1 mev, Newton's method was used to reduce this
unbalance. A maximum of ten passes through the loop was imposed,
and in all cases where this maximum was exceeded the values of
incident momentum were too large to be printed.

The spectrum of incident momentum for the pnnin- events is
Shown in figure 72. Now if the neutrons incident upon the chamber
are caused by the pions in the beam interacting with nucleons in
the theta target, then this spectrum extends much too high. Accounting
for the Spread of seven percent in the pion momentum and the Fermi
momentum of nucleons in the target, the maximum incident momentum
should be about 2.1 bev/c. Figure 72 Shows that the Spectrum Spills
over to well above this maximum. I feel that this is mainly due to
the restrictions in the kinematic fitting problem coupled with the
uncertainty on the measured momenta. If one were to plot the
unbalance of energy as a function of incident momentum, he would
find that at high values of momentum the curve levels and approaches
a constant value. The important difference is that the Slope of the
curve is smaller on the high side of any zero point than it is on the
low side. The effect is that random errors in the measured quantities
will tend to distort the shape of the momentum Spectrum to the high

side.

22

23

In figures 73 - 75 we have the nucleon lab momentum Spectra.
Here we see that there is a strong tendency to have fast protons and
slow neutrons in the lab. This will be reflected also as a strong
assymmetry in the center of mass angular distributions of protons
and neutrons as seen in figures 76 - 78. The very high piling up
of neutrons in the backward direction is not understood.

Plots of the invariant mass of pairs of particles are shown
in figures 79 - 87. we see that there is no evidence for any reson—
ance effects at all, neither in the nucleon-pion pairs nor in the
pion-pion pairs. Thresholds for 0, 1, and 2 isobar production are
about 1.22, 1.66, and 2.12 bev/c incident momentum; hence there
should not be any double isobar production, but could be some single
isobar production. Nevertheless no evidence of the isobar is seen.

Several attempts were made to Select events from the total
sample of pnnin— events in order to find any hidden evidence of
the isobar. Events with nucleon angles very close to forward or
backward were culled out of the sample, but no isobar was seen.

Also no isobar was seen looking at charged tracks only, which should
eliminate any effect of the fitting procedure which might mask the
isobar.

Since there were so few ppnfin° events, we could not subdivide
them into reasonable Slices of energy. The Spectrum of incident '
momentum is shown in figure 88, and shows the same tendency to spill
over to high values as does the pnninr momentum spectrum. Again no

evidence of isobar formation was seen for these events.

In all probability the difficulty with these data is due to
the fact that the problem is just barely determined, and the data
are not too useful. We present it here for completeness, and it is

not to be taken as evidence for the lack of isobar formation.

24

 

Figure la. Diagram.representing pion production in nucleon-nucleon

collisions in the isobar model.

pa 1.

Figure 1b. Diagram representing pion production in nucleon-nucleon
collisions in the peripheral model.

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Figure 24. Neutron lab momentum of pprr- events.

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49

 

Scatter plot of Y and Z direction cosines of the beam
particle of apparent ppnr events whose adjusted beam

angle was greater than 5°. The circle represents 5°.
The minus Y direction is up, and the plus Z direction

is toward the back window.

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361 ppnr_events with 1.28<< Po < 1.431bev/c.

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a. Pole approx. with experimental 0;.
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Figure 57. Pion center of mass momentum;
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.— — — — phase Space .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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———— I phase Space

 

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Figure 59. Pion center of mass momentum;

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— — - — phase space

 

 

 

 

 

 

 

 

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Figure 60. Pion center of mass momentum;

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84

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Figure 61. Pion center of mass momentum;

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1|.OOH

 

 

 

 

 

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Ao\>mo.4 4554844402

m.N .. O.N m.H
4 4 4 4 4 4 4 4 4 4 4 4

 

 

 

 

 

 

 

 

mezmgm

 

 

 

 

 

 

 

 

 

 

 

 

1. OOH

 

 

96

 

 

 

 

 

     

 

 

 

 

 

 

 

 

20—
15“—
EVENTS -—
10"— 1"
P-
5—
C 1 1 1 1 1 1 1 1 1 1 4
.5 1.0 1.5
Proton lab momentum (bev/c)
20'—
C FLT
15—
EVENTS -
10—
5—
l l l 1 1 .4J
.5 1.0 1.5

Neutron lab momentum (bev/c)

Figure 73. Nucleon lab momentum: 184 pn7T+TT- events with

97

 

 

 

 

 

 

 

 

 

 

 

 

15C-
EVENTS : f4
10—
5_ 11-—1
_1111111111111111—1
.5 1.0 1.5
Proton lab momentum (bev/c)
201—
1_-
1.—
15—
4
EVENTS 4
1.0-—
E
—— r‘
54
”1111111111111 [—1
.5 1.0 1.5

Neutron lab momentum (bev/c)

Figure 74. Nucleon lab momentum: 174 pnTr+Tr" events with

98

 

 

 

 

 

 

 

 

 

 

 

 

 

15 [-
EVENTS :
10 —
5 _—
1 1 1 1 1 1 1 1 1 1 1 151
.5 1.0 1.5
Proton lab momentum (bev/c)
o
2 E
15—
EVENTS 4
10—
: 1A
54
__ 1 1"1 1 1 1 1 1 _ 1 1 F7_l
.5 1.0 1.5

Neutron lab momentum (bev/c)

Figure 75. Nucleon lab momentum: 128 pnn+fl- events with
2.1 < Po < 2.3 bev/c.

30
EVENTS
2O

10

99

 

 

 

 

 

 

 

 

l l l l 1 1 1 l l

 

 

 

 

 

 

 

 

1 O -1
008
Protons
60" '—
501- '—
40__ .4
EVENTS
30" “
2O ‘ '-
1 L 1 1 1 1 l 1 l
1 O -1
Cos
Neutrons
Figure ’NS. Nucleon center of mass angular distribution:

184 pnufn' events with 1.7 < Po <Il.9 bev/c.

301—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘1
EVENTS
20- 4
.444r-m
1 1 l 1 1 C1) 1 1 1 1
Cos
Protons
7OF- 4.
60- 4.
50“- .4
40r- 4
EVENTS

30—- 4.
20- 41

10

T—1__J
1 1 44 1 I 1 4 1
1 O -l
Cos
Neutrons

Figure '77. Nucleon center of mass angular distribution:
1'72. pnw+rf events with 1.9 < Po < 2.1 bev/c.

lOl

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30—- -1
EVENTS
2134 4
104- 4
1 O -1
Cos
Protons
70-— "
60-— "
SOt— '“
40- _
EVENTS
30- —‘
20r- "
lO- “
W 1 1 1 r— [—41
. O —1
Cos
Neutrons

Figure 78. Nucleon center of mass angular distribution:

128 pnTr+TT- events with 2.1 < Po < 2.3 bev/c.

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VI. APPENDIX: Description of programs.
A. TRACK FITTING PROGRAM.

The track fitting program calculates momenta, angles, range,
coordinates, etc. of up to five tracks per event by fitting five
measured points on the tracks in each of three views to a helix
whose axis is parallel to the z axis and the magnetic field. The
program was designed especially for R. K. Adair's hydrogen bubble
chamber which is 14 inches in diameter and 8 inches deep. There are
three cameras located on a circle of 13 inches diameter looking
perpendicularly at the chamber from a distance of 36.5 inches from
the front window. The front window has two fiducial marks on its
inside surface ten inches apart and in line with the beam.direction.
There is a magnetic field in the chamber along the z axis of 17.5
kilogauss.

The input data to the program is on standard I.B.M. punched
cards. Data for an event consists of a measurement of the front
fiducial marks for a view and measurements of five roughly equally
Spaced points on each track in the view starting with the origin of
the track and ending at the end of the track. This is done for each
View. There are identifying numbers in the first columns of each
card giving the event number and track and View numbers. This data
is read into the computer and reconstructed so that the points are
effectively reprojected onto the front window plane. The coordinate
system used here has its origin at one of the fiducial marks, the x
axis goes through the other front fiducial mark, and the z axis goes
from the front window to the back window.

Since the first points measured on a track were at the origin
of the track, the points measured were actually corresponding points
on the track. The calculation of the xyz coordinates of these corres-
ponding points is a simple matter. Refer to figure A1. The equations
for the x, y, and z coordinates of the point are:

x - (U1 - A1) z/D = U1

y - (V1 - B1) Z/D
x - (U2 — A2) z/D

y -'(Vé - Ba) z/D

X - (U3 - A3) Z/D 2 Us
Y - (V3 - Ba) Z/D = V3 -

II H II
5 E 5‘

112

113

These are six equations for the three unknowns x, y, and 2. To
combine these by the method of least squares, multiply each term
of each equation by its coeficient of x and add the six equations

together. Repeat this with the coeficients of y and then 2 to
obtain the three equations:

3
3x - Z(u.-—A.~) z/D = Z a.
£13!

3

33 -Z(V;'Bgl 53/0 = ZV;

(:I

:(“VAJX + :(W‘lej —Z3[(U.'Az)z+(‘/r3.)z]z/D

it! 2:]

3
: ZL (Mi—Ag) u; + (Vi‘BL)\/£

Now dividing each of these equations by 3 the sums reduce to averages,

and the result can be expressed more simply as:

x —(L7—/3)Z/D =5?
:3 -— ($7"£§.) EE,/I7 ‘3 I;

,\
£l

1
bl

\_/

><

+

/-\

<1

1

031

\J
(i:

l

r--1
E

l
3:
+
2‘

t
m ‘
k1.

L__4
S
b

where the bars imply averages.

114

This reduces to:

whereR = W-K*E+B*v_‘§*V+U'2 —U +v2 -V2

Having obtained the coordinates of the origin of the track
we go about finding the coordinates of the last point measured.
Since corresponding points were not measured for the last point
we have to interpolate along the track as measured in two of the
views in order to find corresponding points. The slopes of the
tracks in the x-y plane are tested to see which view pairs are
most favorable for this interpolating. Circles are fitted through
the last three points in two of the views,and the intersections of
these circles and a line parallel to the positions of the camera
axes going through the final point of the third view are taken as
points corresponding to the point in the third view. Now having
corresponding points for the end of the track the same procedure
as above may be applied to obtain xyz coordinates of the endpoint.

Now these two points will define a family of helixes with
axes parallel to the z axis. If the projection of the helix
representing the track is found,then the helix is determined.
Coordinates are rotated to a system such that the origin of the
track lies at the origin and the endpoint is on the positive x axis.
Refer to figure A2. The point d' is found by intersecting the line
p1'-p5' and the line parallel to the y axis through p3'. Then projecting
from the point A,B through d' down to the x axis we find the point d
which is also the x coordinate of the point p3. Finally the line
from A,B projected through p3' to the line parallel to the y axis
through d intersects at p3 giving both x and y coordinates of p3.
This is done for each of the middle points of the track in all views.
The result is an array of points closely fitting a circle. By least
squares a circle is fit to these points which goes through the endpoints

and whose center lies on the perpendicular bisector of the line

115

segment between the endpoints. This is a one parameter fit. The
value of the radius of the circle is then taken as the average of
the distances between the center of this circle and the 15 data
points. The deviations of these distances from the mean serve as
a measure of the accuracy of the fit.

Now knowing the helix which describes the track,the direction
cosines of the track are calculated in a straightforward fashion as
well as the range of the track in the chamber. In order to obtain
the momentum of the track from the radius of the helix and the magnetic
field in the chamber, the average value of the magnetic field along
the track is obtained by integrating the field along the track. By
measurements of the magnetic field without the bubble chamber in
place we found that a good representation of the field is:

3= Bo +B(z — Zo)2-B(I' -ro)2
where 30’ 17.5,13 = 0.0062319, B = 0.0035486, 20 and ro are the
center of the chamber, and r is the radial position in the chamber.

The average magnetic field is obtained by a straightforward integration:

B _ fB-ds
" Ids , where ds is an element of path length.

 

The momentum is then obtained from the well known formula:
0.3 * B * R

 

P = .
Slnlf

where V is the dip angle of the track. Units are mev/c, kilogauss,

and centimeters.

116

rel

AJJL

 

 

A2)Bz '

 

4‘4
?

/ (’92)

 

 

 

(u. 0)
‘ (A) " D)
x l
l + + ’
P732: {1:7}; 20in :1 on)

Figure Al. Geometry for finding points in the chamber from measured
points. The upper figure represents the arrangement of the cameras

with reSpect to the coordinate system used. Here the z axis is into

the page. The lower figure represents the geometry in the x-z plane

for relating points in the front window plane to the coordinates of

the point in the chamber. A & B are the coordinates of the camera
positions, U & V are the coordinates of the measured point as projected
onto the front window. D is the effective distance of the camera lenses
from the front window. D = n(d+5/n*), where d is the actual air distance
between the front glass and the lens, 5 is the total glass thickness, and
n and n* are the indices of refraction of hydrogen and glass respectively.

117

(A,B)

P5

8

 

 

 

F: 9‘ P5

Figure A2. Geometry of the circle fitting proceedure.

The points p1 and p5 represent the first and last point of the
track in the chamber, while pl'and p5' represent the correSponding
points from this view as projected onto the front window. Points

p3 and p3' have the same meaning for some middle point.

B. KINEMATIC TESTING PROGRAM.

The subroutine QEVAL was written to attempt to balance energy
and momentum of the events assuming certain mass assingments of the
incident and outgoing particles by adjusting the magnitudes of the
outgoing momenta without changing the directions. The mass assignments
correspond to the possible three body final states of the experiment:

n + p -ev p + p + n’

K°+ p ——-> p + K++ Tr"-

K°+p_, p+TT++K-.
The method is to adjust the outgoing momenta in the best manner to
eliminate the unbalance of energy while keeping the chisquare function
representing the adjustment of momenta a minimum. In mathematical
terms the problem is to minimize chisquare subject to the condition
that the change in Q is equal to -Q, where Q = Ein - Eout'

The incident momentum, P0, is always equal to the vector sum
of the outgoing momenta. Let awe , X be the direction cosines of the
tracks at the vertex of the event. The apparent unbalance of energy

then is given by:

={(Pd +501 we a.) +(flfl +afluéfl.)
+(FY+FY2+F3X3)Z +mo:}a +mP

_ f 32W} _ gang? jz - {gz+m;)2l/z

The chisquare function we wish to minimize is:

=: (z Z..)/G§f

1”

where z = 1/p. z is used rather than p since it is proportional to
curvature, not radius of curvature, and should be more normally

distributed. The condition on the minimum of chisquare is:
aa 4
_Za J24ZaafflJzz—Q,
faJ

where

jgél 2 faxéfig +'¢ayfiz “t AQIJ: __ _5;

3?; E", A"

Jp-___r‘
7i?” ’0‘

i

h

and

118

119

For simplicity let us represent the equations for X2 and dQ as:
X2: ZBth‘z
e: 2AM- +69 :0

To minimize X2 subject to G, form:
yxxxz-ua :0

where J. is an arbitrary constant. This leads to:

Z (Bl-x, ~AA,‘)XX" = 0

or B- x. ~ AA,“ = 0 {or .1/{i ,

C (

Substitute this result back into G to solve for A:

AZ’f—izww 2

l l

50 A

~'6{//<25-/E:i3g )

1A3 a

am! X- = -—~—-———-—- -

J 58 :2:/4://E2

H

Now substituting for the x's, A's, and B's we obtain the increments

to make in the 2's:

0'?
12,. = r “ .

jg: ( zkjgihjfL

The problem was done for each mass assignment in turn. Up to

OJ
E)

I

0.1
N

. 5?
446.?

q

five iterations of the data adjustment were allowed although rarely
more than one was necessary. If after an iteration the chisquare
function was larger than 7.8 the program stopped further calculation.
If the value of Q was made less than 1 mev and the chisquare function
was still less than 7.8, the program continued to calculate the
momenta, direction cosines, and energies of all tracks in the center

of mass of the system.

C. THE PERIPHERAL MODEL PROGRAM.

The theory and development of the equation for the calculations
done by the SELLERI program are set down in a paper by E. Ferrari and

F. Selleri (21). This report is very comprehensive and the reader is

referred to it for a complete discussion of the problem. It contains
a brief resume of the status of the peripheral model, some general
considerations of S-matrix theory, a thorough discussion of the
peripheral S—matrix in particular, as well as the equations for
calculation of various cross-sections. The description of the model
has already been presented on page 3, and the general equation for

the differential cross-section on page 15. The problem is to integrate
this equation with reSpect to the prOper variables in order to

obtain the spectra desired.

Before discussing the cross-section equations it will be well
to review some of the nomenclature of the problem. Referring to
figure 1b, the incident particle,p1, interacts with a meson, k, in
the cloud of the stationary particle, p2, which recoils as qz.
p1 scatters from k and goes out as particles q; and q3. The
incident particle masses are M1 and M2, the outgoing particle masses
are m1, m2, and m3, and the rest mass of the transferred meson k

islp. Some important invariant quantities used in the evaluation are:

W2 = —(p1 + pz)2
42 = (Q2 - P2)2
t2 = (Q1 - P1)2
«3 =: -(q1 + Q3)2
112 = '(Q2 + Q3)2
2? : (Q2 - 131)2
E2 = (Q1 - P2)2
r2 = (Q3 - p1)2
f2 3? (Q3 - P2)2

2 = -(q1 + qz)2

Four different coordinate systems are used in the evaluation of
certain expressions: the B system is the over-all rest system, the
L system is the lab system, the Q system is the rest system of the
pair of particles ql and q3, and the P system is the rest system of
the pair of particles qz and q3. The evaluation of any variable in

one of these systems is indicated by a superscript B, L, Q, or P.

q1, q2, etc. imply 4~momentum until the energy is used explicitly.

120

121

An additional zero subscript implies the energy of the particle.
Hence qzoL is the energy of the particle q; in the lab, and q1Q is
the magnitude of the momentum of particle ql evaluated in the Q
system. Explicit expressions for all kinematic variables are
given for each coordinate system as functions of the above invar-
iant quantities and masses, so that one can evaluate the equations
with ease.

Ferrari and Selleri have derived the formulas for evaluating
the spectra of certain variables pertaining to particles q; and q2.
Starting with the general expression for the cross-section as a
function of all final variables, the procedure is to change variables
to a set which includes the variable whose spectrum is desired and
then integrate over all other variables.

For particle qz all the information needed is contained in

g§%§i} . Quantities like d3q2/q20 are invariant and can be evaluated
in any coordinate system. Let
3 5 B 8 5
”£4.53 = 32 43“ 0160592 1%
10

The integral over dsz gives 2n for unpolarized initial particles.

To make the coordinate transformation d3q2/q20 —? dAzda}:
4510045055923 = I'A’A‘ .1602

where the Jacobian ! I

After integrating over the 5-function:

j1’5,1’5L (We-6) fl} J45: ”(mg/41960-
61°53.» ‘0

Putting these relations into the general differential cross-section:

 

 

”(17 ._ G1, (3.0600700) Az+(m2’M‘4)_z .
a(AfAQu‘ gfl’F' (4y!+,42)z

Now to evaluate a particular spectrum there are simple relations
for changing variables fromA2 orw2 to leave the variable of interest.
The minimum/maximum value ofA2 is given by

1 _ 5 (3 7- K B
A (WM/m -- 23.. I?» we 442 *4 2 zip.

I ..\l11l|

122

In order to evaluate the spectrum of lab kinetic energy,

TzL is expressed as a function ofA2 in the relation:
L l 2
7:, = 3&1[A‘+(M.~m.)] -

Then the partial cross-section as a function of T2L iS°

 

50:.”(4‘)
do: = ,2ng . £3494: W2 xw‘fi‘w orfw),
,(7; JA‘ 47F (A‘Hu’)2

(m, I'm»):
where “5;“(23) is given by:

60:“,(4‘) :: WZJ-m: - M5{A2+Mzz+m;) +
/"7,2

1.

To evaluate certain of the variables in these equations two functions

are defined:

(MAME/147‘; 31;, (la/2+ "72441)
I/

7. 2
{DOA/,mz/Mz] : 5;; {wt—.2 mmum + WW)

These functions than represent the center of mass energy and 3-momen-
tum of a particle with mass m colliding with a particle of mass Miat

a total center of mass energy W} The unknown variables above are

expressed as:

3% P(w2 MIZJIAz)
605 :: 5(W) M32) M11)
P,“ r P (WW/z, Mi) .

123

For the center of mass angular distribution dA2 can be

transformed to dcosezB through the relation:

3 l3
€05.92, 22’0‘368 {At +3442: 4M: ”1,010 610}

In this equation all terms except A2 itself are functions ofcu2 alone.

 

Hence dA2 = 2p§quB dcostB. Using the relation F: W plB we get:

 

 

(W'mz)
a(0” .. (;Z 1 a 5 f -
- 4.4.... clad flu) 1 Az+m1ML) 07(0)),
0‘9”“? ‘mflfiwt 5 (4 +‘1’“)
(%W*”%)2

where

The center of mass energy distribution can be obtained from

the relation: 8
$30 = £F<WAK"”§fi/“Jz)

In this case the integration over 132 can be carried out analytically:

 

 

a
4¢=£=2w£2r= 6'2 113600700)-
A 7;“ X32.“ dw" 4 W H“

 

/, :EJWW __ [#‘-(M.-m.)‘][z1:,.,(w/ -A’..,,.,,(w)]
Am“) “ ’“ [Aim (WWI 4?... (w) + N]

The distribution of the invariant mass,aa, and its square,cd2,

is obtained directly from the expression for the center of mass energy

distribution.

For particle q; the situation is much more complicated since
q; is involved in the upper vertex and the cross-section function, 61.
All of the information required is present in the expression for 2%EE;;. '
First the expressions linking this partial cross-section with the
spectra of interest will be given, and then the evaluation of the
partial cross-section will be discussed.

The spectrum of T1L is obtained from the relation:

"= 5;?{521‘Mz-WJ2}

 

 

where

3
“in: ([2) : W2+m’z- “:48: {[24147’1-JM21) +
2

[Ll-2+ ("7’ ”fling—2+ (m. Hwy} 2.

 

The angular distribution of q; in the center of mass is

obtained from the expression:

£05913 : ’Bzzf{+’”7/+Mz "‘W}
JZfib 6,8

 

giving (Warn) 2
.1 0‘ 2 a 5 126 , M2
X405 638 fl ( 5 é’u {Xi
m2. * ms)!

The center of mass energy distribution of q1 is obtained
from the expression:

6:03 : E (WJW’12u-L)

2

 

 

giving {WV (74]
4.9: = X 0" = .2 W 312' : 02W .HZ 95’0"
0(778 ”(6MB 1“: szo/{L
{twin (1‘)
where

{MIX (20) :- £60850! '7’1,2~M’z +2 glsflfl

‘7‘ 5 B x z 5 B,
imp; (74) " “gig/o {:0 —777,—/W, “”253 fl .

125

4%

'14.

Now to evaluate the partial cross-section

. J‘ov .
procedure 18 followed as for .ZZZZ;;°

 

[3 g __ ’34 B 019,131 B '4‘: JuzJéz :
Za’ " ‘3. 5M 4 (p’ 2’:-

JW §4(&_g) '3'”, £1“:— fX/JSQPJQUZP a
3.. 5.. u

and the link between the integration variables and A2 andzoz, upon
which the integrand essentially depends, must be found. 112 can be

expressed as a function of uz, t2, and cosezP through

P _ I F f’ z z z
C0592 -6283133P {1 32’ g.) ”m: ‘M2 ‘4} '

, first one connects the angle ezp to 92p, and up through the

 

For L02

trigonometric identity
P . P
Cos €,F : 6&5“wang + ilhpr W 9, 5052/3

where 6213 is the scattering angle of qz with reSpect to p1:

P I P P z- 1. 7-
6056, = 2 ,, P {a 5,0 fie -104. M I}
53!”,

and up is the angle between p1 and pg:

1 z
p : P F M ”(z—'Wf‘
: -—-____~ 2 4 ’ 3'
605M riff/Q? M law

 

These relations can be solved for 42 and 5’2, and finally «22 is

obtained from the relation

2

001: wZ—m; -M,Z -M,"-—A’ ~41“ .

Finally putting these quantities into the expression for the

partial cross-section we get:

I
2

 

J10» _ I 62 53F [my ’7 41+ (”Z‘MJ)
42.2w " W F‘ 22’ ’ WW)

:7

11%sz ”'5 5% (W,..5..,&) -

 

126

There are a similar set of equations for the diagram where the
incident particle p1 acts as the spectator and a meson from the cloud
of p; scatters particle p2. They are obtained from the above formulas
by interchanging pl'eo p2, which means also the interchange of barred
quantities and unbarred ones, and M1 H M2; 91PH Glp, OZQHGZQ.
The connection between physical quantities and invariants remains
unchanged.

The program is broken into several subroutines, one major one
to calculate each of the above mentioned Spectra. By appropriate
indices punched on one of the data cards, one may call upon any or
all of the subroutines to calculate one or more spectra. That is,
one may call for the Spectra of TZL, TlL from the first diagram or
also from the second diagram; four Spectra in all. One may also
ask for only the first Spectrum, TzL. In addition to the main
subroutines there are a few smaller subroutines to calculate
certain factors common to all of the equations. One of them
calculates the factor 53%): another calculates the partial
cross-section WE}, , another calculates the function 5%:(60/ 6059].
Since these are separate subroutines, it is an easy matter to write
different versions of them, as has been done for the partial
cross-section CF. ‘we have written one such version which inserts
a Breit4Wigner resonance multiplied by an angular dependence factor,
and also we have one which interpolates tables of the actual cross-
sections for the reaction k + p; -+. q1 + q3 containing energy and
angular dependence. The values used for this subroutine were
obtained from published experimental data.

The integrations are done by Simpson's rule and the number of
points in each level of integration is variable. This is done by
Specifying the number of points in each level by indices on one of

the data cards.

 

 

REFERENCES

1.

' 2.

10.

11.

12.

13.

14.

H. A. Bethe, F. de Hoffmann, Mesons and Fields, Vol. 2,

Row, Peterson & Co. (1955).

A. H. Rosenfield, Phys. Rev. 26, 1, 139, (1954). This work
contains a review of the information available on pion production
from N-N collisions at cyclotron energies.

A. P. Batson, B. B. Culwick, J. G. Hill, L. Riddiford, Proc.

Roy. Soc. 251, 218, (1959).

. G. B. Chadwick, G. B. Collins, C. E. Swartz, A. Roberts, S.

DeBenedetti, N. C. Hien, P, J, Duke, Phys. Rev. Letters 3, 12,
611, (1960).

. E. Pickup, D. K. Robinson, E. O. Salant, Phys. Rev. Letters 2;

5, 192, (1961). w. J. Fickinger, E. Pickup, D. K. Robinson,

E. 0. Salant, Phys. Rev. Letters 2, 5, 196, (1961).

V. Barnes, D. V. Bugg, W. P. Dodd, J. B. Kinson, L. Riddeford,
Phys. Rev. Letters 1, 7, 285, (1961).

G. Da Prato, Nuovo Cim. 22, 1, (1961).

W. A. Wallenmeyer, Phys. Rev. 195, 3, 1058, (1957).

W. B. Fowler, R. P. Shutt, A. M. Thorndike, W. L. Whittemore,
Phys. Rev. 95, 4, 1026, (1954).

F. N. Holmquist, PhD Thesis, Univ. of Cal., Berkeley, UCRL-8559,
(1958).

E. Fermi, Prog. Theor. Phys. (Japan) 5, 570, (1950);

Phys. Rev. 9_2_, 452, (1953); Phys. Rev. 93, 1434, (1954);

Phys. Rev. 81, 683, (1951).

K. M. Wetson, K. A. Brueckner, Phys. Rev. 8g, 1, 1, (1951);
Watson, Phys. Rev. 86, 6, 852, (1952);

Brueckner and Watson, Phys. Rev. 86, 6, 923, (1952);

Watson, Phys. Rev. 88, 5, 1163, (1952).

See reference 2 and M; Gell Mann, K. M. Watson, Ann. Rev.
Nuclear Science 5, 219, (1954).

S. J. Lindenbaum, R. M. Sternheimer, Phys. Rev. 195, 6, 1874, (1957);
Phys. Rev. 196, 5, 1107, (1957); Phys. Rev. 332, 5, 1723, (1958);
Phys. Rev. 123, 1, 333, (1061); Phys. Rev. Letters _5_, 1, 24, (1960).

127

15.

16.
17.
18.
19.
20.

21.

22.

23.

24.
25.

26.

27.

28.
29.

30.
31.

128

See references 3, 4, 14, 17, 19, and I. Derado, N. Schmitz,
Phys. Rev. 1555 1, 309, (1960); V. P. Kenney, W. D. Shephard,

c. D. Gall, Phys. Rev. 126, 2, 736, (1962); E. L. Hart,

R, I, Louttit, D. Leurs, T. W. Morris, W. J. Willis, S. S.
Yamamoto, Phys. Rev. 125. 2, 747, (1962); w. J. Fickinger,

E. Pickup, D. K. Robinson, E. O. Salant, Phys. Rev. ;25, 6,

2082, (1962), E. Pickup, D. K. Robinson, E. O. Salant,

Phys. Rev. gag, 6, 2091, (1962).

E. M. Friedlander, Phys. Rev. Letters 5, 5, 212, (1960).

I. Derado, Nuovo Cim. 15, 5, 853, (1960).

G. F. Chew, F. E. Low, Phys. Rev. 11;, 6, 1640, (1959).

F. Bonsignori, F. Selleri, Nuovo Cim. 55, 3, 465, (1960).

F. Selleri, Phys. Rev. Letters 5, 2, 64, (1960).

E. Ferrari, F. Selleri, Phys. Rev. Letters 2, 10, 387, (1961).

E. Ferrari, F. Selleri, CERN report 2385/TH.227 (1961);

CERN report 2845/TH. 243 (1962).

J. Izuka, A. Klein, Phys. Rev. 12;, 2, 669, (1961);

Prog. Theor. Phys. 25, 1018, (1961).

s. D. Drell, K. Hiidak, Phys. Rev. Letters 2, 5, 199, (1961).

s. D. Drell, Rev. Mod. Phys. 52, 3, 458, (1961).

In reference 21 the second report contains an explicit derivation
of the problem as well as formulae for the various cross—sections
and Spectra.

R. K. Adair, Rev. Mbd. Phys. 35, 3, 406, (1961).

H. J. Martin, L. B. Leipuner, W. Chinowsky, F. T. Shively,

R. K. Adair, Phys. Rev. Letters 6, 6, 283, (1961).

For a description of the computer codes see the appendix.

A. R. Erwin, R. March, W. D. Walker, E. West, Phys. Rev.

Letters 5, 11, 628, (1961). ’

C. D. WOOd, PhD Thesis, Univ. of Cal., Berkeley, UCRL 9507, (1961).
L. K. Goodwin, R. W. Kenney, V. PerezéMendez, Phys. Rev. Letters
5, 11, 522, (1959).

 

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