use “overleaf”to write the report. I give the format in the files. (
https://www.overleaf.com/project/5d703830619cc0000131534e
) this is the link for the format .
I already give some part of it, I need the code of overleaf. You should log in to overleaf and you could just fix in the sample I give to u.
1
PHY 4822L (Advanced Laboratory):
Analysis of a bubble chamber picture
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.
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 = 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.
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
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)
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.
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
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.)
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
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
some
basic
statistical
calculations.
The
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
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
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
13},
the
average
is
𝑥 = !!
!
=
17.
Next
find
(𝑥! − 𝑥)!
Then
𝑠 =
(!!!!)!
!
!!!
!!!
= 2.21
To
calculating
standard
uncertainty
u
when
a
set
of
several
repeated
readings
has
been
taken,
use
𝑢 =
𝑠
𝑛
(The
standard
uncertainty
of
the
mean
has
historically
also
been
called
the
standard
deviation
of
the
mean,
or
the
standard
error
of
the
mean)
Lifetime
calculation
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−
track
lengths.
The
average
momentum
of
the
Σ−
particle
can
be
found
from
its
initial
and
final
values:
𝑝! =
1
2
(𝑝!! + 𝑝!!)
Where
𝑝!! 𝑎𝑛𝑑 𝑝!!
are
initial
and
final
momentums
of
Σ
particle.
And
can
be
found
using
the
measured
track
length
𝑙!.
The
length
of
time
that
the
Σ−lives(the
time
between
its
creation
and
decay)
is
𝑡 =
𝑙!
𝑣
𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑎𝑡 𝑟𝑒𝑠𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑎𝑏
𝑡 =
𝑙!
𝑣
1 −
𝑣!
𝑐!
𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑚𝑜𝑣𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒
𝑡 =
𝑚!𝑙!
𝑝!𝑐
COLLEGE
PHYSICS LAB REPORT
STUDENTS NAME
ANALYSIS OF A BUBBLE CHAMBER PICTURE
SUPERVISED BY:
19/05/2020
1. Introduction
A bubble chamber is a vessel filled with a superheated transparent liquid (most often liquid hydrogen) used to detect electrically charged particles moving through it. It was invented in 1952 by Donald A. Glaser, for which he was awarded the 1960 Nobel Prize in Physics.
A convenient way to study the properties of the fundamental subatomic particles is through observation of their bubble trails, or tracks, in a bubble chamber. Using measurements made directly on a bubble chamber photograph, we can often identify the particles from their tracks and calculate their masses and other properties. In a typical experiment, a beam of a particular type of particle is sent from an accelerator into a bubble chamber, which is a large liquid-filled vessel. To simplify the analysis of the data, the liquid used is often hydrogen, the simplest element. The use of liquid hydrogen, while it simplifies the analysis, complicates the experiment itself, since hydrogen, a gas at room temperature, liquefies only when cooled to -246◦C. For charged particles to leave tracks in passing through the chamber, the liquid must be in a “super-heated” state, in which the slightest disturbance causes boiling to occur. In practice, this is accomplished by expanding the vapor above the liquid with a piston a few thousandths of a second before the particles enter the chamber.
2. Methods
2.1 Materials needed:
1. student worksheet per student
2. Ruler
3. Scissors
4. Glue stick
5. Pocket calculator
2.2 Procedures
2.2.1 Calculation of the X Particle’s Mass.
Make measurements on each of the photographs. In particular, for each of the circled events measure these four quantities:
· `Σ – The length of the Σ track,
· θ – the angle between the Σ− and π− track,
· s – the sagitta of the π− track,
· `π – The chord length of the π− track.
Your values for the event should be close to those given in the sample input. Run the program using each set of measurements, and tabulate the computed X0 mass from each event. Compute an average of the calculated masses and find the average deviation, expressing your result as Mx ±∆Mx.
Compare your final result with some known neutral particles listed below and identify the X0 particle based on this comparison.
Particle mass (in MeV/c2)
π0 135
K0 498
n 940
Λ0 1116
Σ0 1192
Ξ0 1315
2.2.2 Determination of the Angle θ
.
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. We can show an alternative method which does not require a protractor. 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 angle 180◦ −θ. It should be noted that only some of the time will the angle θ exceed 90◦.
2.2.3 Determination of Momentum from Range.
We cannot determine the momentum of the Σ− track from its radius of curvature because the track is much too short. Instead, we make use of the known way that a particle loses momentum as a function of the distance it travels. In each event in which the K− comes to rest before interacting, energy conservation applied to the process requires the Σ− particle to have a specific momentum of 174 MeV/c. The relatively massive Σ− particle loses energy rapidly, so its momentum at the point of its decay is appreciably less than 174 MeV/c even though it travels only a short distance. It is known that a charged particle’s range, d, which is the distance it traveled before coming to rest, is approximately proportional to the fourth power of its initial momentum, i.e., d ∝ p4. For a Σ− particle traveling in liquid hydrogen, the constant of proportionality is such that a particle of initial momentum 174 MeV/c has a range of 0.597cm. Next we find the “residual range,” which is the difference between the maximum range d0 and the Σ− track length `Σ. Note that when `Σ = 0.597 you get pΣ = 0 as you would expect.
2.2.4 Calculation of the Sigma Lifetime.
The Σ− lifetime can be approximately determined using the measured values of the Σ− track lengths. The average momentum of the Σ− particle can be found from its initial and final values:
p ¯ Σ = 1/2(174 + pΣ), where pΣ is found from , using the measured track length `Σ.
The length of time that the Σ− lives (the time between its creation and decay) is t = `Σ /v , where `Σ is the length of the Σ− track and v is the average velocity of the Σ− particle.
Write a program that calculates the amount of time that each Σ− lives, and determine an average lifetime. The accepted value is 1.49×10−10 seconds. Since all our photographs are less than life-size, the computed times must be multiplied by the scale factor 1.71. This method of finding the Σ− lifetime can only be expected to give a very approximate result because (a) only four events are used and (b) we have ignored the exponential character of particle decay.
3. Results and Discussion
The bubble chamber is similar to a cloud chamber, both in application and in basic principle. It is normally made by filling a large cylinder with a liquid heated to just below its boiling point. As particles enter the chamber, a piston suddenly decreases its pressure, and the liquid enters into a superheated, metastable phase. Charged particles create an ionization track, around which the liquid vaporizes, forming microscopic bubbles. Bubble density around a track is proportional to a particle’s energy loss.
Bubbles grow in size as the chamber expands, until they are large enough to be seen or photographed. Several cameras are mounted around it, allowing a three-dimensional image of an event to be captured. Bubble chambers with resolutions down to a few micrometers (μm) have been operated.
The entire chamber is subject to a constant magnetic field, which causes charged particles to travel in helical paths whose radius is determined by their charge-to-mass ratios and their velocities. Since the magnitude of the charge of all known charged, long-lived subatomic particles is the same as that of an electron, their radius of curvature must be proportional to their momentum. Thus, by measuring their radius of curvature, their momentum can be determined. A careful quantitative analysis of measurements made on tracks in bubble chamber photographs can reveal much more than can a simple visual inspection of the photographs.
First, while reactions can often be unambiguously identified by their topology, such identification can be confirmed if we make measurements of the length, direction, and curvature of each track, and then analyze these data by computer. Second, through such a procedure we can determine whether an unseen neutral particle was present. Third, we can determine properties of an unseen neutral particle. In each of the photographs there are one or more events. The circled event in each photograph is the one of particular interest because all of its tracks lie very nearly in the plane of the photograph and this considerably simplifies the analysis. Such “almost coplanar” events are a rare occurrence since all directions are possible for the particles involved. For events that are more non-coplanar we must analyze at least two stereoscopic photographs of each event in order to completely describe its three dimensional kinematics. For each of the circled events we will first determine three quantities:
(a) The momentum of the π− particle (pπ);
(b) The momentum of the Σ− particle (pΣ) at the point of its decay; and
(c) The angle θ between the π− and Σ− tracks at the point of decay
On their way through the liquid, particles constantly loose energy because of the ionization processes and Bremsstrahlung. A lower momentum corresponds to a smaller track radius in a magnetic field: In the bubble chamber; The Lorentz force
𝐹𝐿 = 𝑞 ∙ 𝑣 ∙ 𝐵
acts as centripetal force 𝐹𝑐 = 𝛾 ∙ 𝑚 ∙ 𝑣 2 𝑟 (with 𝛾 = 1 √1− 𝑣 2 𝑐 2 for relativistic particles).
Therefore: 𝑞 ∙ 𝑣 ∙ 𝐵 = 𝛾 ∙ 𝑚 ∙ 𝑣 2 𝑟 ⇒ 𝑟 = 𝛾∙𝑚∙𝑣 𝑞∙𝐵 = 𝑝 𝑞∙𝐵 or 𝑝 = 𝑞 ∙ 𝑟 ∙ 𝐵.
Where; q= electric charge of particle in C 𝑣= speed of particle in m/s
𝑐=speed of light in vacuum in m/s 𝐵= Magnetic field strength in T
𝑚=mass of particle in kg 𝑟= radius of curvature of the particle track in m
𝑝= (relativistic) momentum of the particle
3.1 Energy loss
A fast charged particle traversing the bubble chamber liquid loses continuously energy by interactions with the atoms of the medium, which become ionized. At low momenta the losses are large and have a dependence of the type 1/v2; the losses for v=c tend to a constant value of about 0.27 MeV/cm in liquid hydrogen. The losses are furthermore proportional to the square of the particle charges; all elementary particles have charge +1 or -1 times the proton charge. An electron with more than few MeV has always a velocity close to the velocity of light; it loses a relatively small energy by ionization and more by radiating photons (the bremsstrahlung process). A fast electron thus yields a track with slightly less than 10 bubbles per centimeter and which spirals because of the large energy loss by radiation.
How can we be sure that the event is really an elastic event? This can be done by checking if energy and momentum are conserved.
I) Conservation of energy: (Total energy of the K+) + (mass energy of a stationary proton) is equal to (Total energy of the outgoing K+) + (total energy of the outgoing proton)
II) Conservation of linear momentum. (Momentum of incident K+) + (0) is equal to (momentum of outgoing K+) + (momentum of outgoing proton)
Although bubble chambers were very successful in the past, they are of limited use in modern very-high-energy experiments for a variety of reasons:
· The need for a photographic readout rather than three-dimensional electronic data makes it less convenient, especially in experiments which must be reset, repeated and analyzed many times.
· The superheated phase must be ready at the precise moment of collision, which complicates the detection of short-lived particles.
· Bubble chambers are neither large nor massive enough to analyze high-energy collisions, where all products should be contained inside the detector.
· The high-energy particles may have path radii too large to be accurately measured in a relatively small chamber, thereby hindering precise estimation of momentum.
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