Measurement  of  the  angle  θ           For  better  understanding  I  am  showing  you  a  different  particle  track  diagram  bellow.  Where  at   point  C  particle  𝜋!  𝑎𝑛𝑑  Σ!  are  created  and  the  Σ!  decays  into  𝜋∓  𝑎𝑛𝑑  K!    particles   The  angle  θ  between  the  π−  and  Σ−  momentum  vectors  can  be  determined  by  drawing  tangents   to  the  π−  and  Σ−  tracks  at  the  point  of  the  Σ−  decay.  We  can  then  measure  the  angle  between   the  tangents  using  a  protractor.  Alternative  method  which  does  not  require  a  protractor  is  also   possible.  Let  AC  and  BC  be  the  tangents  to  the  π−  and  Σ−  tracks  respectively.  Drop  a   perpendicular  (AB)  and  measure  the  distances  AB  and  BC.  The  ratio  AB/BC  gives  the  tangent  of   the  angle180◦−θ.  It  should  be  noted  that  only  some  of  the  time  will  the  angle  θ  exceed  90◦  as   shown  here.   Determining  the  uncertainty  of  Measurements   In  part  B,  It  is  asked  to  estimate  the  uncertainty  of  your  measurements  of  𝜃  and  r.     Uncertainty  of  measurement  is  the  doubt  that  exists  about  the  result  of  any  measurement.  You   might  think  that  well-­‐made  rulers,  clocks  and  thermometers  should  be  trustworthy,  and  give   the  right  answers.  But  for  every  measurement  -­‐  even  the  most  careful  -­‐  there  is  always  a  margin   of  doubt.     It  is  important  not  to  confuse  the  terms  ‘error’  and  ‘uncertainty’.       Error  is  the  difference  between  the  measured  value  and  the  ‘true  value’  of  the  thing  being   measured.   Uncertainty  is  a  quantification  of  the  doubt  about  the  measurement  result   Since  there  is  always  a  margin  of  doubt  about  any  measurement,  we  need  to  ask  ‘How  big  is  the   margin?’  and  ‘How  bad  is  the  doubt?’  Thus,  two  numbers  are  really  needed  in  order  to  quantify   an  uncertainty.  One  is  the  width  of  the  margin,  or  interval.  The  other  is  a  confidence  level,  and   states  how  sure  we  are  that  the  ‘true  value’  is  within  that  margin.   You  can  increase  the  amount  of  information  you  get  from  your  measurements  by  taking  a   number  of  readings  and  carrying  out

1. Measurement
of
the
angle
θ
For
better
understanding
I
am
showing
you
a
different
particle
track
diagram
bellow.
Where
at
point
C
particle
�! ��� Σ!

2. are
created
and
the
Σ!
decays
into
�∓ ��� K!
particles
The
angle
θ
between
the
π−
and
Σ−
momentum
vectors
can
be
determined
by
drawing
tangents
to
the
π−
and
Σ−
tracks
at

3. the
point
of
the
Σ−
decay.
We
can
then
measure
the
angle
between
the
tangents
using
a
protractor.
Alternative
method
which
does
not
require
a
protractor
is
also
possible.
Let
AC
and
BC
be

4. the
tangents
to
the
π−
and
Σ−
tracks
respectively.
Drop
a
perpendicular
(AB)
and
measure
the
distances
AB
and
BC.
The
ratio
AB/BC
gives
the
tangent
of
the
angle180◦−θ.
It
should
be
noted
that

5. only
some
of
the
time
will
the
angle
θ
exceed
90◦
as
shown
here.
Determining
the
uncertainty
of
Measurements
In
part
B,
It
is

6. asked
to
estimate
the
uncertainty
of
your
measurements
of
�
and
r.
Uncertainty
of
measurement
is
the
doubt
that
exists
about
the
result
of
any
measurement.
You
might
think
that
well-­‐made
rulers,

7. clocks
and
thermometers
should
be
trustworthy,
and
give
the
right
answers.
But
for
every
measurement
-­‐
even
the
most
careful
-­‐
there
is
always
a
margin
of
doubt.
It
is
important

8. not
to
confuse
the
terms
‘error’
and
‘uncertainty’.
Error
is
the
difference
between
the
measured
value
and
the
‘true
value’
of
the
thing
being
measured.
Uncertainty
is
a
quantification

9. of
the
doubt
about
the
measurement
result
Since
there
is
always
a
margin
of
doubt
about
any
measurement,
we
need
to
ask
‘How
big
is
the
margin?’
and
‘How
bad
is
the
doubt?’

10. Thus,
two
numbers
are
really
needed
in
order
to
quantify
an
uncertainty.
One
is
the
width
of
the
margin,
or
interval.
The
other
is
a
confidence
level,
and
states
how
sure
we
are
that

11. the
‘true
value’
is
within
that
margin.
You
can
increase
the
amount
of
information
you
get
from
your
measurements
by
taking
a
number
of
readings
and
carrying
out
some
basic
statistical
calculations.
The

12. two
most
important
statistical
calculations
are
to
find
the
average
or
arithmetic
mean,
and
the
standard
deviation
for
a
set
of
numbers.
The
‘true’
value
for
the
standard
deviation
can
only
be

13. found
from
a
very
large
(infinite)
set
of
readings.
From
a
moderate
number
of
values,
only
an
estimate
of
the
standard
deviation
can
be
found.
The
symbol
s
is
usually
used
for
the
estimated

14. standard
deviation.
Suppose
you
have
a
set
of
n
readings.
Start
by
finding
the
average:
For
the
set
of
readings
x={16,
19,
18,
16,
17,
19,
20,
15,
17
and

15. 13},
the
average
is
� = !!
!
=
17.
Next
find
(�! − �)!
Then
� =
(!!!!)!
!
!!!
!!!
= 2.21
To
calculating
standard
uncertainty
u
when
a

16. set
of
several
repeated
readings
has
been
taken,
use
� =
�
�
(The
standard
uncertainty
of
the
mean
has
historically
also
been
called

17. the
standard
deviation
of
the
mean,
or
the
standard
error
of
the
mean.)
Lifetime
calculation

18. In
part
C
you
are
asked
to
determine
the
life
time
of
the
neutral
particles
from
their
momentums.
The
Σ−
lifetime
can
be
approximately
determined
using
the
measured
values
of
the
Σ−

19. track
lengths.
The
average
momentum
of
the
Σ−
particle
can
be
found
from
its
initial
and
final
values:
�! =
1
2
(�!! + �!!)
Where
�!! ��� �!!
are
initial
and
final
momentums

20. of
Σ
particle.
And
can
be
found
using
the
measured
track
length
�!.
The
length
of
time
that
the
Σ−lives(the
time
between
its
creation
and
decay)
is
� =
�!
�
���������� ������ �� ���� �� �ℎ� ���

21. � =
�!
�
1 −
�!
�!
���������� ������ ������ ���ℎ
��������
� =
�!�!
�!�
1
PHY 4822L (Advanced Laboratory):
Analysis of a bubble chamber picture

22. Introduction
In this experiment you will study a reaction between
“elementary particles” by analyzing their
tracks in a bubble chamber. Such particles are everywhere
around us [1,2]. Apart from the standard
matter particles proton, neutron and electron, hundreds of other
particles have been found [3,4],
produced in cosmic ray interactions in the atmosphere or by
accelerators. Hundreds of charged
particles traverse our bodies per second, and some will damage
our DNA, one of the reasons for the
necessity of a sophisticated DNA repair mechanism in the cell.
2
Figure 1: Photograph of the interaction between a high-energy
π--meson from the Berkeley
Bevatron accelerator and a proton in a liquid hydrogen bubble
chamber, which produces two neutral
short-lived particles Λ0 and K0 which decay into charged
particles a bit further.
Figure 2: illustration of the interaction, and identification of
bubble trails and variables to be
measured in the photograph in Figures 3 and 4.

23. The data for this experiment is in the form of a bubble chamber
photograph which shows bubble
tracks made by elementary particles as they traverse liquid
hydrogen. In the experiment under
study, a beam of low-energy negative pions (π- beam) hits a
hydrogen target in a bubble chamber.
A bubble chamber [5] is essentially a container with a liquid
kept just below its boiling point (T=20
K for hydrogen). A piston allows expanding the inside volume,
thus lowering the pressure inside
the bubblechamber. When the beam particles enter the
detector a piston slightly decompresses the
liquid so it becomes "super-critical'' and starts boiling, and
bubbles form, first at the ionization
trails left by the charged particles traversing the liquid.
The reaction shown in Figure 1 shows the production of a pair
of neutral particles (that do not leave
a ionized trail in their wake), which after a short while decay
into pairs of charged particles:
π - + p → Λo + Ko,
3
where the neutral particles Λo and Ko decay as follows:
Λo → p + π-, Ko → π+ + π-.
In this experiment, we assume the masses of the proton (mp =

24. 938.3 MeV/c2) and the pions (mπ+ =
mπ- = 139.4 MeV/c2) to be known precisely, and we will
determine the masses of the Λ0 and the K0,
also in these mass energy units.
Momentum measurement
In order to “reconstruct” the interaction completely, one uses
the conservation laws of (relativistic)
momentum and energy, plus the knowledge of the initial pion
beam parameters (mass and
momentum). In order to measure momenta of the produced
charged particles, the bubble chamber is
located inside a magnet that bends the charged particles in
helical paths. The 1.5 T magnetic field is
directed up out of the photograph. The momentum p of each
particle is directly proportional to the
radius of curvature R, which in turn can be calculated from a
measurement of the “chord length” L
and sagitta s as:
r = [L2/(8s)] + [s/2] ,
Note that the above is strictly true only if all momenta are
perfectly in the plane of the photograph;
in actual experiments stereo photographs of the interaction are
taken so that a reconstruction in all
three dimensions can be done. The interaction in this
photograph was specially selected for its
planarity.
In the reproduced photograph the actual radius of curvature R of
the track in the bubble chamber is
multiplied by the magnification factor g, r = gR. For the
reproduction in Figure 3, g = height of
photograph (in mm) divided by 173 mm.

25. The momentum p of the particles is proportional to their radius
of curvature R in the chamber. To
derive this relationship for relativistic particles we begin with
Newton's law in the form:
F = dp/dt = e v×B (Lorentz force).
Here the momentum (p) is the relativistic momentum m v γ,
where the relativistic γ-factor is defined
in the usual way
γ = [√(1- v 2/c2)]-1.
Thus, because the speed v is constant:
F = dp/dt = d(mvγ)/dt = mγ dv/dt = mγ (v2/R)(-r) = e v B (-r) ,
where r is the unit vector in the radial direction. Division by v
on both sides of the last equality
finally yields:
mγv /R = p /R = e B ,
identical to the non-relativistic result! In “particle physics
units” we find:
p c (in eV) = c R B , (1)
4

26. thus p (in MeV/c) = 2.998•108 R B •10-6 = 300 R (in m) B
(in T)
Measurement of angles
Draw straight lines from the point of primary interaction to the
points where the Λ0 and the K0
decay. Extend the lines beyond the decay vertices. Draw
tangents to the four decay product tracks at
the two vertices. (Take care drawing these tangents, as doing it
carelessly is a source of large
errors.) Use a protractor to measure the angles of the decay
product tracks relative to the parent
directions (use Fig. 3 or 4 for measurements and Fig. 2 for
definitions).
Note: You can achieve much better precision if you use graphics
software to make the
measurements, rather than ruler and protractor on paper.
Examples of suitable programs are
GIMP or GoogleSketchUp, both of which can be obtained for
free.
Analysis
The laws of relativistic kinematics relevant to this calculation
are written below. We use the
subscripts zero, plus, and minus to refer to the charges of the
decaying particles and the decay
products.
p+sinθ+ = p-sinθ-
(2)

27. p0 = p+cosθ+ + p-cosθ-
(3)
E0 = E+ + E-,
where E+ = √(p+2c2 + m+2c4) , and E- = √(p-2c2 + m-
2c4)
m0c2 = √(E02 - p02c2)
Note that there is a redundancy here. That is, if p+, p-, θ+, and
θ- are all known, equation (2) is not
needed to find m0. In our two-dimensional case we have two
equations (2 and 3), and only one
unknown quantity m0, and the system is over-determined. This
is fortunate, because sometimes (as
here) one of the four measured quantities will have a large
experimental error. When this is the case,
it is usually advantageous to use only three of the variables and
to use equation (2) to calculate the
fourth. Alternatively, one may use the over-determination to
"fit'' m0, which allows to determine it
more precisely.
A. K0 decay
1. Measure three of the quantities r+, r-, θ+, and θ-. Omit the
one which you believe would
introduce the largest experimental error if used to determine
mK. Estimate the uncertainty of
your measurements.

28. 2. Use the magnification factor g to calculate the actual radii R
and equation (1) to calculate the
momenta (in MeV/c) of one or both pions.
3. Use the equations above to determine the rest mass (in
MeV/c2) of the Ko.
4. Estimate the error in your result from the errors in the
measured quantities.
5.
5
Β. Λo decay:
1. The proton track is too straight to be well measured in
curvature. Note that θ+ is small and
difficult to measure, and the value of mΛ is quite sensitive to
this measurement. Measure θ+, r-
and θ-. Estimate the uncertainty on your measurements.
2. Calculate mΛ and its error the same way as for the Ko.
3. Estimate the error in your result from the errors in the
measured quantities.
4. Finally, compare your values with the accepted mass values
(the world average) [3], and
discuss.
C. Lifetimes:
Measure the distance traveled by both neutral particles and
calculate their speed from their

29. momenta, and hence determine the lifetimes, both in the
laboratory, and in their own rest-frames.
Compare the latter with the accepted values [3]. Estimate the
probability of finding a lifetime value
equal or larger than the one you found.
References:
[1] G.D. Coughlan and J.E. Dodd: “The ideas of particle
physics”, Cambridge Univ. Press,
Cambridge 1991
[2] “The Particle Adventure”, http://particleadventure.org/
[3] Review of Particle Physics, by the Particle Data Group,
Physics Letters B 592, 1-1109
(2004) (latest edition available on WWW: http://pdg.lbl.gov )
[4] Kenneth Krane: Modern Physics, 2nd ed.; John Wiley &
Sons, New York 1996
[5] see, e.g. K. Kleinknecht: “Detectors for Particle Radiation”,
Cambridge University Press,
Cambridge 1986;
R. Fernow: “Introduction to Experimental Particle Physics”,
Cambridge University Press,
Cambridge 1986;
W. Leo: Techniques for Nuclear and Particle Physics
Experiments :
A How-To Approach; Springer Verlag, New York
1994 (2nd ed.)

30. Note: Experiment adapted from PHY 251 lab at SUNY at Stony
Brook (Michael Rijssenbeek)
6
Figure 3 : Photograph of the interaction between a high-energy
π--meson from the Berkeley
Bevatron accelerator and a proton in a liquid hydrogen bubble
chamber. The interaction produces
two neutral particles Λ0 and K0, which are short-lived and
decay into charged particles a bit further.
The photo covers an area (H•W) of 173 mm • 138 mm of the
bubble chamber. In this enlargement,
the magnification factor g = (height (in mm) of the photograph
)/173 mm.
7
Fig. 4: Negative of picture shown in Fig. 3