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Introduction to basic particle physics

Editing: Tsourlidaki

CERN

Let’s take a look at one of the largest labs in the world. (http://www.youtube.com/watch?feature=player_embedded&v=2jup2R9Jtnc)

The Large Hadron Collider (LHC) is a giant scientific facility located 100m under the Earth’s surface, near Geneva on the border of France and Switzerland. It is the most advanced particle accelerator in CERN and is used by scientists in order to study the basic particles of matter.

(http://www.youtube.com/watch?v=roorfpw5tBA&feature=player_embedded)

ATLAS is a very big basic particle physics experiment that is performed in the LHC at CERN.

The ATLAS detector studies the proton collisions in extremely high energies, in order to collect information about the fundamental forces of the Universe which have governed it since the very first moment of its existence. Among the mysteries studied within ATLAS are the identification of the origins of mass, the existence of more dimensions, the unification of the fundamental forces as well as proof of the existence of dark matter.

(http://www.youtube.com/watch?v=zJj8u1Fxn9o&feature=player_embedded)

Basic particles

Molecules consist of atoms, which are the smallest unit of matter with characteristic properties, and are the chemical elements. Atoms in their turn consist of protons, neutrons and electrons.

Protons and neutrons consist of other smaller particles, called quarks. Up to now, leptons (one of them is the electron) and quarks (it is thought there are six of them) are considered to be the basic particles of matter.

Each kind of lepton and quark has its corresponding anti-particle: a particle of equal mass but opposite charge and spin.

Leptons

Electron(e-) muon (μ-) Tau (τ-)

Electron neutrino (νe) Muon neutrino (νμ) Tau neutrino (ντ)

Related questions

1. Does momentum depend on the direction of speed?2. What is an insulated system?3. What does “conservation of momentum” mean in reality?4. During a collision is the kinetic energy maintained?5. How are basic particles categorised?6. Are new particles produced during a particle collision?7. What kind of research is done at CERN?8. What is the objective of the ATLAS experiment?

Introduction to basic particle


Vector addition

Scalar and vector sizes

Scalar sizes can be fully described only by measuring them. For vector sizes to be fully identified, we need to know both their extent and direction. To perform mathematical operations with vectors their direction must be taken into account.

Vector addition

In order to add two or more vectors, we must place them sequentially so that the end of the first one rests on top of the next one, being careful not to alter the vector’s angle. The beginning of the total vector is the beginning of the first vector and its end is the end of the last one.

If all vectors have a common starting point, then we can use the parallelogram rule to add them:

From the ending point of each vector we draw a parallel to the second vector. The point where the two parallels intersect is the end of the sum vector. This method is suitable for adding only two vectors at a time.

Vector analysis

In physics there is often a need to resolve a vector into orthogonal components. To do this, we implement the following procedure. We draw two dotted lines from the end of the vector, parallel to the axis x'x and y'y respectively. The point where the two dotted lines intersect is the end of the corresponding component.

The experiment

Open the ΗΥΡΑΤΙΑ program and follow the steps below to perform the exercise.

1. Select the file “JiveXML_5104_20655.xml” (use the buttons “Previous Event” and “Next Event”) to see the data on the collision we want to study. If the file is not displayed in the fact list, save it on your local computer from “Reading and Assignments”. Select “File” → “Read event locally”, find the file and open it.

2. In “Track Momenta Window” select the “Simulated” card to see the simulations of all the trajectories you will study.

3. Draw the momentum vectors for each particle. First, find the angle and the magnitude based on the data of the table in the “Track Momenta Window”:

a. Convert the angle (φ) for each particle from radians to degrees and enter them in the corresponding column of the following table.

b. If the magnitudes of all momentum vectors (column P[GeV]) are too big, you have to draw them all to scale. In order to normalise your values, divide them all by the smallest value. Add the results in the “Students worksheet” file that you will find in “readings and assignments”.

4. Based on your calculations, draw the corresponding vectors.5. Draw the vector of the resultant momentum according to your estimations, as well as the vector of the

remaining momentum (non-accurate solution).

Analysis-Discussion

1. Why did you normalise the momentum vector magnitudes? Why did you divide them by the smallest momentum value? Would there be any difference if you divided by another number? How does this normalisation affect your results?

2. Based on the vector analysis, calculate the magnitude of the complete momentum as well as the corresponding vector angle. Make sure you use the initial values and not the normalised ones.



3. What error sources are there?4. Is the complete momentum you calculated for the level x-y zero? If no, why not?5. Is the principle of momentum maintenance valid? If yes, why did you come up with a non-zero

momentum?6. Does the vector you drew for the neutron fit with the corresponding vector in the detector

simulation?7. Based on the exercise you did and the answers you gave to the previous questions, write a short

report following the form provided.

About

Brief description: Students will determine the resultant momentum of all particles detected during a hadron collision and will calculate the magnitude and momentum of the residual vector.

Connection with the curriculum:

Greece: 1st Grade Upper Secondary School Physics, Chapters 1 3, §1.2 and § 3.2 of the text book.

Teaching targets:

1. Learn the principle of the conservation of momentum.2. Practise adding vectors.3. Measure vector angles and convert radians to angle degrees.4. Learn about research in the field of basic particle physics.

Age: 15 – 18 – Time required: 2½ teaching units

Technical requirements:

1. Computers with an Internet connection2. Data analysis tool, HYPATIA

a. Save HYPATIA-v4 from the webpageb. Decompress the saved file and save the extracted file directly onto your computer’s hard disk

in C:\c. Double click on the “Hypatia 4.jar” file to open the program.

Note: This program requires installing the Java Runtime Environment (version 1.4 or newer) software, which you may find here: http://www.oracle.com/technetwork/indexes/downloads/index.html

Preparing students: Mass, speed, acceleration, power, energy and the way all these are connected, i.e. Newton’s laws, should have already been discussed in class.

Students also learn the difference between scalar and vector sizes.

Note: The theory presented in “Theory behind the experiment” concerns the relevant exercise in the guide and does not constitute a full theoretical approach.

Key words: mass, speed, acceleration, energy, collision, principle of the conservation of momentum, radians, degrees, vector

Author(s): Thanos Leontios



Additional information

Description of equipment1. Open the program and show its basic functions.

Image 1. The HYPATIA data analysis tool

a. Inertial masses windowb. List of trajectories included in the present HYPATIA filec. Canvas window

Use the magnifying glass in each part to magnify the image. When a trajectory is selected, it becomes white.

d. Name of file which appearse. Transverse view of the detector showing all trajectoriesf. Longitudinal view of the detector showing all trajectoriesg. 3D energy diagram for the x-y planeh. Detected trajectories window

Contains all the recorded data for the trajectories detected.i. File navigation optionj. List of recorded trajectories with the corresponding datak. Control window

You can change the view settings for a file or add filters to the projection of trajectories.



2. Describe the different parts of the ATLAS detector to your students

Image 2. An example of how the various particles are detected in different parts of the detector. Note that the neutral particles leave their mark only in the calorimeters. Therefore, their trajectories appear only in the red and green parts and not in the inner part of the detector.

Image 3. Representation of an ATLAS detector

The various layers are positioned concentrically, circling the area of the collision.

Starting from the interaction point (the point where the protons and antiprotons collide with each other) and moving outward, the parts of the ATLAS detector are as follows:

Trajectory detector (or inner detector) (green, brown): It is the innermost part of ATLAS and consists of three sub-detectors designed to detect charged particles. The neutral particles (e.g. photons) pass through this region without being detected. All the charged particles interact with the detector but they pass through theoretically without any change in their direction or their energy.

Image 4. The inner detector.



Calorimeters: When a particle (charged or not) enters the calorimeter it collides with the detector’s dense material. This collision gives rise to a series of other particles and almost all the energy of the original particle is absorbed by the calorimeter. Because of this, the calorimeter is inserted after the internal probe, so as to record the trajectory of the particle before it is absorbed. The calorimeters measure energy and have two different parts:

• The electromagnetic calorimeter (grey/green): measures the total energy of e+, e- and photons. Therefore, if one is looking for electrons, their trajectory stops in the calorimeters.

Image 5. The electromagnetic calorimeter and the hadronic calorimeter.

• The hadronic calorimeter (red) measures the total energy of hadrons (such as protons and neutrons).

The only particles with the ability to penetrate the detectors and the calorimeters and continue towards the muon detector are muons and neutrinos.

Detectors / muon spectrometers: This is the outer layer of the detector (blue). Muons are the only charged particles that penetrate the hadron calorimeter almost unaffected and reach the muon detector. Their trajectories are the only ones recorded in the outermost layer of the muon detector.

Particles that are not detected: Neutrinos interact very weakly with matter, thus are not detected at all. Their presence can be confirmed by measuring the “lost” momentum.

Image 6. The muon detector.

Residual energy / momentum: This is the energy and the momentum needed for the principles of maintaining momentum and energy to apply.



In the LHC the initial momentum along the beams is unknown because the hadrons’ energy is continuously exchanged between the particles, and therefore the residual energy cannot be measured. However, the initial momentum vertically to the beam distribution line is zero. Therefore, the existence of a non-zero momentum implies the existence of residual momentum and energy (Etmiss). The residual momentum is represented in the image of the detector with a dotted line indicating the direction of the “lost” momentum.

Magnets: Τhe ATLAS detector is located in a powerful magnetic field which bends the trajectories of charged particles. The fields are generated by four kinds of magnets – three toroidal shaped ones and a tubular one (not shown in the representation). The positively and negatively charged particles are directed in opposite directions by the same magnetic field. The curvature and direction of the particle’s trajectory are used to determine the momentum and charge of a particle.

(Source: http://hypatia.phys.uoa.gr/Simplified_Basics/)

Form and guidelines for the project writing

Name(s), class, section

Summary The abstract should summarise briefly and clearly the

project’s content. It should clearly explain to the reader what information can be drawn from the project. The most important elements of the summary are the presentation of the problem and the project’s contribution. It should be between 70 and 120 words.

Introduction – Description of

the problem

The introduction must consist of two paragraphs: in the

first one the general problem should be presented and

commented on. The second should illustrate the focus of

the project. Typically the second paragraph should start

with a phrase like “The aim of this project is to…”.

Hypothesis – Initial idea In the hypothesis and original concept section preliminary

assumptions and projections based on the current knowledge on the topic should be presented. The basic

Part Description



concepts and definitions that are essential for the understanding of the problem should also be analysed.

Experimental setup In this section, the experimental setup and everything

concerning materials, equipment or software used for the experiment should be described.

Performing the

experimentHere the experiment performed is presented in detail.

Authors should provide a detailed description of every

step of the experiment, including the measurements they

undertook and the way they did it.



Data analysis In the analysis data section, the data deduced from the

experiment performed are presented and all calculations based on them are carried out. All results must be necessarily accompanied by a percentage error between the experimental and the theoretical value.

Commenting on the

resultsThe authors present their observations on the results

obtained. They comment on their findings. If some of the

results are incorrect they should indicate the possible

sources of error.


Conclusion The conclusion should briefly state the original problem and the general

content of the project. It should be a standalone piece, that is, by reading this alone, the reader should be able to get the gist of the project’s main idea without having to read it all. Usually, the conclusion ends with a paragraph describing possible extensions of this project and presents future relevant studies.

Sources At the end of the project, reference should be made to all sources of

information. If the source is a website, the link must be given. If the source is a book the title, the author and the publishing house should be

named.

Lab information:

In order to conduct this exercise, the data analysis tool HYPATIA will be used. It has been created solely for educational purposes by the University of Athens and the Institute of Physics of Belgrade. HYPATIA has been designed to analyse real data from the ATLAS experiment conducted at the Large Hadron Collider at CERN. Students will measure the momentum of various particles and by using the conservation of momentum principle they will discover the existence of particles whose trajectory has not been detected.

Further information: http://hypatia.phys.uoa.gr

Lab exercise #1: Conservation of momentum during particle collision

A. General information

Brief description: Students will determine the resultant momentum of all particles detected during a hadron collision and will calculate the size and momentum of the residual vector.


Connection with the curriculum:

Greece: 1st Grade of High School Physics, Chapters 1 3, § 1.2 and § 3.2 of the textbook.

Teaching targets:

1. Learn about the conservation of momentum principle.

2. Practise adding vectors.

3. Measure vector angles and convert radians to angle degrees.

4. Learn about research in the field of basic particle physics.

Age: 15-18

Time required: 2½ teaching units

Technical requirements:

1. Computers with an Internet connection2. HYPATIA data analysis tool -

- save version HYPATIA-v4 from the webpage

http://hypatia.phys.uoa.gr/Downloads/

- Decompress the file you saved and then save the extracted file directly to your computer hard disk in C:\

- Double click on the “Hypatia 4.jar” file to open the program.

Notice: The program requires installing the Java Runtime Environment software (version 1.4 or newer). which can be found here: http://www.oracle.com/technetwork/indexes/downloads/index.html


Students’ preparation: Mass, speed, acceleration, power, energy and the way all these are connected, i.e. Newton’s laws, should have already been discussed in class.

Students also learn the difference between scalar and vector sizes.

Note: The theory presented in “Theory behind the experiment” concerns the relevant exercise in the guide and does not constitute a full theoretical approach.

Key words: mass, speed, acceleration, energy, collision, principle of the conservation of momentum, radians, degrees, vector

B. Activity description

B.1 Activities for eliciting questions

Interest stimulation

You could begin your lesson by discussing with your students about CERN and the experiments conducted there. You could focus on the following two subjects, in order to attract their attention:

a. The Large Hadron Collider (LHC@CERN)

A gigantic scale scientific tool located 100 metres below the surface of the Earth near Geneva, Switzerland, on the premises of CERN. It extends beyond the border between Switzerland and France. It is a particle accelerator – particles are the building blocks of the material world. It is expected to revolutionise the way we understand nature, from the microcosm to the infinite universe.

Two beams of subatomic particles (hadrons) – either protons or lead ions – travel in opposite directions inside the circular accelerator, gaining energy in each round. Physicists use the LHC to reproduce the conditions that existed in the universe immediately after the big bang, causing the head-on collision of the two beams. Teams of physicists from around the world analyse the particles produced by the collision, using special detectors in a number of experiments conducted with the LHC.

Show the following videos to your students:

- CERN in 3 minutes

- LHC in 10 minutes

- ATLAS – From dream to reality


The above videos can be found on the “Learning with ATLAS@CERN” homepage, (http://www.learningwithatlas.eu/)

b. Simulation of particle collision (proton-antiproton collision)

Use the following videos to explain to your students how particle collisions happen in the LHC and why scientists conduct these experiments.

- http://hands-on-cern.physto.se/ani/acc_lhc_atlas/lhc_atlas.swf

- http://www.youtube.com/watch?v=k64s4Ho-8-I

The LHC is the largest and most powerful particle accelerator in the world and the most recent addition to the accelerator system at CERN. It consists of superconducting magnets and a number of systems which accelerate particles, maximising their energy as they move within the 27km accelerator ring.

Within the accelerator two particle beams travel, in opposite directions to each other, at a speed close to the speed of light. These beams travel in different rings which are completely empty and as they travel in these they accelerate while their energy increases through the application of a strong electromagnetic field generated by superconducting magnets. These magnets are made of special materials, suitable for use in such conditions without causing resistances or losses in energy. In order to minimise energy losses, the magnets are cooled at -271oC, i.e. near absolute zero! For this reason most of the accelerator is connected to a cooling system with liquid helium for cooling the magnets and the peripherals.

To properly direct beams of particles through the accelerator, thousands of different kinds of magnets are used. These include 1232 dipolar magnets 15m long, used to bend the beams and 392 four-pole magnets 5 to 7 m long, used for focusing the beams. Shortly before the collision, other types of magnets are used to make the beams get closer to each other, so as to increase the likelihood of a collision. The particles are so small that the aim of causing them to collide is tantamount to throwing needles from two opposite directions from a 10km distance with such precision as to make them meet halfway!

The accelerator control centre, the technical support and all the infrastructures are located in the control centre of CERN. From there, the beams inside the LHC are driven so as to collide at four different points along the accelerator ring. These points correspond to the locations of the four particle detectors.

(Source: http://public.web.cern.ch/public/en/LHC/HowLHC-en.html)


Eliciting questions based on existing knowledge

The natural laws which students will study are Newton’s laws, the principle of conservation of energy and momentum. Students need to know vector addition and analysis. Make an introduction asking students the following questions to see what they know so far.

1. Does momentum depend on the direction of speed?2. What is an insulated system?3. What does “conservation of momentum” mean in reality?4. During a collision is the kinetic energy maintained?5. How are basic particles categorised?6. Are new particles produced during a particle collision?7. What kind of research is done at CERN?8. What is the objective of the ATLAS experiment?

Relevant theory

Newton’s laws

Newton’s first law is called the Law of Inertia. According to this, the kinetic state of any body in an inertial system does not change, when either no forces are exerted on it or their resultant is zero.

The above relationship is bidirectional, which means that the opposite is also true. If, in other words, a body is

moving at a constant speed towards the inertial observer, then the resultant of all the forces exerted on it is

zero.

Newton’s second law is known as the fundamental law of mechanics. According to this, the speed of a body depends on the resultant force exerted on it. The resultant force acting on a body is equal to the rate of the momentum change. For a body with a constant mass, the resultant force is equal to the product of mass and

acceleration of the body: F m a

Newton’s third law is also known as the action-reaction law. According to it, when a body exerts force on another one, then the second exerts force of equal magnitude and opposite direction on the first one.

F F1 ,2 2 ,1


Principle of conservation of momentum

Momentum is defined as the product of a body’s mass and velocity.

The principle of energy conservation is one of the fundamental laws in physics: In an insulated body system momentum is always kept constant. That is, momentum is not created nor destroyed but only transferred through exertion of forces.

Momentum is simultaneously maintained in all three dimensions and is a vector. P m u

The principle of conservation of energy

The energy in an insulated system may change form (kinetic, dynamic, heat, etc.) but can never be destroyed or created from scratch. The total energy of an insulated system always remains constant.

Scalar and vector sizes

Scalar sizes can be fully described only by measuring them. For vector sizes to be fully identified, we need to know both their extent and direction. To perform mathematical operations with vectors their direction must be taken into account.

Vector addition

In order to add two or more vectors, we must place them sequentially so that the end of the first one rests on top of the next one, being careful not to alter the vector’s angle. The beginning of the total vector is the beginning of the first vector and its end is the end of the last one.

A B R A B C D R

Image 7. Addition of two vectors Image 8. Addition of four vectors


If all vectors have a common starting point, then we can use the parallelogram rule to add them:

From the ending point of each vector we draw a parallel to the second vector. The point where the two parallels intersect is the end of the sum vector. This method is suitable for adding only two vectors at a time.

A B RA B C

R

Image 9. Addition of two vectors Image 10. Addition of three vectors. First A and Bare added to R' . Then, R' is added to C .

Vector analysis

In physics there is often a need to resolve a vector into orthogonal components. To do this, we implement the following procedure. We draw two dotted lines from the end of the vector, parallel to the axis x'x and y'y respectively. The point where the two dotted lines intersect is the end of the corresponding component.

Image 11. Vector analysis


Basic particles

Image 12. The atom structure

considered to be the basic particles of matter.

Since the exercise is about collisions of elementary particles, it would be helpful if students knew some basics about the fundamental building blocks in nature. After you have finished introducing the above laws of nature, present the basic elementary particles.

Molecules consist of atoms, which are the smallest unit of matter having characteristic properties, the chemical elements. The atom, in turn, is composed of protons, neutrons and electrons.

Protons and neutrons are composed of other smaller particles, quarks. Up to now leptons (one of them is the electron) and quarks (it is thought there are six of them) are

Every type of lepton and quark has its corresponding anti-particle: a particle of equal mass but opposite charge and spin.

Leptons

Electron(e-) muon (μ-) Tau (τ-)

Electron neutrino (νe) Muon neutrino (νμ) Tau neutrino (ντ)

Quarks

Up (u) Charm (c) Top (t)

Down (d) Strange (s) Bottom (b)

Image 13. Leptons and quarks existing in nature


B.2 Active exploration

Initial hypotheses or predictions suggested

Any experiment is carried out because of the need to investigate an initial hypothesis. Thus, students should make their own initial predictions on detecting particles and then conduct a relevant experiment.

Ask students to make predictions about the following issues:

- The principle of conservation of momentum during the collision on the axis which is perpendicular to the propagation direction of the beams (plane x-y).

- How can we measure the total momentum in the plane x-y?

- If you count the total momentum at x-y, what would you expect to find?

Make sure to write down the students’ predictions so as to be able to recall them in subsequent stages for students to reassess them.

Planning and guiding the research

The central idea of the exercise is for students to discover a particle that has not been detected using the principle of conservation of momentum and the data analysis tool HYPATIA. In any such particle collision the total momentum in the x-y plane must be zero. Students will be asked to measure the total momentum and confirm this fact. However, the momentum that will be measured will actually be non-zero. This fact should lead them to the conclusion that there must be another particle whose trajectory was not recorded by the detectors. In order to maintain the overall momentum the momentum of the particle should be equal in magnitude and of opposite direction to the total momentum calculated by students initially.


Description of equipment

1. Present ΗΥΡΑΤΙΑ to the students. Open the program and show its basic functions.

Image 14. The HYPATIA data analysis tool

a. Inertial masses windowb. List of trajectories included in this HYPATIA filec. Canvas windowUse the magnifying glass in each part to magnify the image. When a trajectory is selected, it appears white.d. Name of the file that appearse. Transverse view of the detector showing all trajectories.


f. Longitudinal view of the detector showing all trajectories.g. 3D energy diagram for the x-y plane.h. Detected trajectories windowContains all the recorded data for the trajectories detected.i. File navigation optionj. List of recorded trajectories with the corresponding datak. Control window

You can change the display settings of a file, or add filters to the projection of trajectories.

2. Describe the different parts of the ATLAS detector to your students

Image 15. An example of how various particles are detected indifferent parts of the detector. Note that the neutral particles leave traces only in the calorimeters. Therefore, the trajectories appear only in the red and green parts and not in the inner part of the detector.

Image 16. Representation of the ATLASdetector


The various layers are positioned concentrically, circling the area of the collision.

Starting from the interaction point (the point where the protons and antiprotons collide with each other) and moving outwards, the parts of the ATLAS detector are as follows:

Track detector (or inner detector) (green, brown): it is the innermost part of ATLAS and consists of three sub-detectors designed to detect charged particles. The neutral particles (e.g. photons) pass through this region without being detected. All charged particles interact with the detector but pass through theoretically without any change in their direction or their energy.

Calorimeters: When a particle (charged or not) enters the calorimeter, it collides with the detector’s dense material. This collision gives rise to a series of other particles and almost all the energy of the original particle is absorbed by the calorimeter. Because of this, the calorimeter is inserted after the internal probe, so as to record the trajectory of the particle before it is absorbed. The calorimeters measure energy and have two different parts:

• The electromagnetic calorimeter (grey/green): measures the total energy of e+, e- and photons. Therefore, if one is looking for electrons, their trajectory stops in the calorimeters.

• The hadronic calorimeter (red) measures the total energy of hadrons (such as protons and neutrons).

The only particles with the ability to penetrate the detectors and the calorimeters and continue towards the muon detector are muons and neutrinos.

Image 17. The inner detector.

Image 18. The electromagnetic calorimeter and the hadronic calorimeter.


Detectors / muon spectrometers: This is the outer layer of the detector (blue). Muons are the only charged particles that penetrate the hadron calorimeter almost unaffected and reach the muon detector. Their trajectories are the only ones recorded in the outer layers of the muon detector.

Image 19. The muon detector.

Particles that are not detected: Neutrinos interact very weakly with matter, thus are not detected at all. Their presence can be confirmed by measuring the “lost” momentum.

Residual energy / momentum: This is the energy and momentum necessary for the principles of momentum and energy conservation to apply. In the LHC, the initial momentum along the beams is unknown due to the fact that the hadrons’ energy is continuously exchanged between particles, making it impossible thus for the residual energy to be measured. However, vertically to the beam propagation line, the initial momentum is zero. Therefore, the existence of a non-zero momentum implies the existence of residual momentum and energy (Etmiss). The residual momentum is represented in the image of the probe with a dotted line indicating the direction of the “lost” momentum.

Magnets: The ATLAS detector is located in a powerful magnetic field which bends the trajectories of charged particles. The fields are generated by four kinds of magnets – three toroidal shaped ones and a tubular one (not shown in the representation). The positively and negatively charged particles are directed in opposite directions by the same magnetic field. The curvature and direction of the particle ’s trajectory are used to determine the momentum and charge of a charge.

(Source: http://hypatia.phys.uoa.gr/Simplified_Basics/)

3. Explain the idea of this exercise to students based on the laboratory instructions they have. It is stated that the main objective is to measure the resultant momentum. Discuss how to measure the momentum and how the principle of conservation of momentum is applied in this case.


In order to help students, explain how momentum is represented using vectors and measurements from ΗΥΡΑΤΙΑ:

Each trajectory belongs to a particle whose identity is given in column “Type”. Based on the magnitudes in columns “Pt (GeV)” and “f” (radians) (the measures of momentum and direction respectively) the corresponding vector for the momentum of each particle can be drawn.

Note: Momentum is measured in GeV because we assume that the speed of light is equal to 1:

E mc 2 E m c c E p c

Caution: Do not make any comments about the expected result.

B.3 Creation

Gathering data through observation

It is recommended to divide the class in groups of 3 or 4 students before starting this exercise.

Ask your students to open the ΗΥΡΑΤΙΑ program and follow the steps below (they are also included in their lab guide) to perform the exercise.

1. Select the “JiveXML_5104_20655.xml” file (use the buttons “Previous Event” and “Next Event”) to see the data from the collision to be studied. If the file does not appear in the list of events, select “File” → “Read event locally” find this file and open it.

2. In the “Track Momenta Window” select the “Simulated” card to see representations of all the trajectories that you will study.

1


3. Draw the momentum vectors for each particle. First, find the angle and the magnitude based on the data of the table in the “Track Momenta Window”:- Convert the angle (phi) for each particle from radians to degrees and enter the values in the corresponding column of the table below.- If the magnitudes of momentum vectors (column P [GeV]) are too big, you must draw them to scale. To normalise your values you divide them all by the smaller value. Insert the results in the corresponding column.

Name of trajectory Angle in degree Normalised measure

SimChargedTrack0 (P1) 15.98552 10.23485


SimChargedTrack3 (P3) 254.2214 1


4. Based on your calculations draw the corresponding vectors.


Total residual momentum

2 identicaltrajectories

2 identicaltrajectories

Total momentum

Total momentum

5. Draw the vector of the resultant momentum according to your estimates and the vector of the residual momentum (not exact solution)


B.4 Discussion

Explanation of results based on the data collected

Ask your students to answer the following questions in order to deduce conclusions.

1. Why did you normalise the vectors’ momentum magnitudes? Why did you divide them by the lowest value of momentum? Would there be a difference if you had divided by another number? How does this normalisation affect your results?If we drew vectors based on their actual magnitudes, the arrows would be very large. So we normalise values, to get a diagram normal in size. This is why all values were divided by the same number, which we chose to be the smallest momentum value that came up from our measurements, for reasons of convenience. If we divided by any other number, it wouldn’t make any difference as long as we used the same number for all our values. Normalisation does not affect our calculations at all, since we normalised all values using the same number.

2. Based on vector analysis, calculate the magnitude of the total momentum and the corresponding angle of the vector. Make sure you use the original values and not the normalised ones.

P

P

t


Px P συν ( 15, 98) Px 12, 98759 GeV

1 1 1

Px P συν ( 15, 98) Px 12, 98759 GeV

2 2 2

Px P συν ( 254, 22) Px 0, 35894 GeV

3 3 3

Px P συν ( 16, 21) Px 10, 84091 GeV

4 4 4

x tot

36, 45714 GeV

Py P ημ( 15, 98) Py 3, 720579 GeV1 1 1

Py P ημ( 15, 98) Py 3, 720579 GeV

2 2 2

Py P ημ( 254, 22) Py 1, 27026 GeV

3 3 3

Py P ημ( 16, 21) Py 3,152592 GeV

4 4 4

y tot

9, 323489 GeV

Ptot tot Py

Ptot 37, 63 GeV

Py Py εφ tot

Px 1

tot 14, 66o

Pxtot tot

2 2


3. What sources of error are there?


Checking the event file, it can be seen that apart from the four main trajectories there are hundreds of other much smaller ones. In order to make it possible for the calculations to take place, we ignored them because of their very small size. However, their omission is a small source of error which affects the final value of the total momentum. But even if they had been included in the calculation, the total momentum would still be non-zero.

4. Is the total momentum you calculated for plane x-y zero? If not, why not?The total momentum is not zero as expected. This is because of the existence of a particle whose trajectory has not been detected. This missing particle is a neutrino. Neutrinos are very small and interact with matter very weakly. Thus their existence was not detected by the probe.

5. Does the principle of momentum conservation apply? If yes, why did you come up with non-zero momentum?The principle of conservation of momentum applies, as for any kind of collision. The total momentum calculated is non-zero because of the missing particle. Thus, we conclude that the “lost” neutrino momentum must have a momentum of equal magnitude and opposite direction to the total momentum calculated.

6. Does the vector you drew for the neutrino fit with the corresponding vector in the representation of the detector?In the detector, the “lost” momentum is represented by a dotted line. The direction of the two vectors should be identical.

7. Based on the exercise you performed and the answers you gave to the above questions, write a short report using the form provided.

Examination of other possible interpretations

Ask the groups to present their results based on their report. Compare the results of the groups and discuss any deviations.

Discuss why the total momentum calculated was non-zero and what other possible explanations may exist.


B.5 Feedback

Interpretation of the presentation

Ask students to report their findings and how they deduced them. Also discuss with them which parts of the exercise they found more difficult as well as the similarities of the exercise they conducted in comparison with actual equivalent studies at CERN.

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