Exercises on High Energy Physics

                                                          by Dieter.Bartneck@sz-online.de

Contents

Constants and Formula used in High Energy Physics

Exercise 1 - Atoms

Exercise 2 - Nuclei

Exercise 3 - Particle Waves and Relativity

Exercise 4 - Virtual Particles

Exercise 5 - Accelerators

Exercise 6 – Particle Beam

Constants and Formula used in High Energy Physics

Constants

contents

Formula

contents

Exercise 1 - Atoms

Try to estimate the radius of one gold atom!

Hints

1.     Take a cube with sides of one centimetre. Divide the cube into smaller cubes, one for each gold atom.

2.     You’ll need Avogadro’s constant, the density of gold and the mass of one gold atom.

Solution

atomic weight and density of gold:

 

There are 6^.10²² atoms in the cube.

 

One atom of gold has a diameter of 3^.10^-10 m, which gives about 10^-10 m for the radius of an atom.

 

contents

Exercise 2 - Nuclei

Try to estimate the radius of the nucleus of a gold atom!

Hints

1.     Rutherford observed in his famous scattering experiments with 6 MeV a-particles, that some of the particles were very strongly deflected, although very few were flying back to the direction of the source.

2.     The experiment shows that the nucleus has a very strong electric field concentrated almost in one point. The repulsive Coulomb potential the a-particles feel, stops them if they are moving in the direction of the centre of mass, at a certain distance from the nucleus.

3.     Assume that the a-particles stop when all their kinetic energy has been converted into electrostatic potential energy.

Solution

The radius of the nucleus of gold should be in the order of 10^-14 m.

N.B. This calculation assumes that the gold nuclei do not recoil from the a-particles. This is a good approximation as the mass of a gold nuclei is much larger than that of an a-particle.

contents

Exercise 3 - Particle Waves and Relativity

What energy particles (e.g. electrons) are needed to investigate the inner structure of

·        atoms,

·        protons and neutrons?

Hints

1.     As in a microscope, the resolution is determined by the wavelength.

2.     At higher energies, particles become relativistic.

Solution

·        To investigate the crystal structure of a material you will need a resolution of about 1^.10^-10 m.


Electrons are non-relativistic at this energy because it is much less than their rest energy of  0.5 MeV.

·        The dimension of the nucleons is smaller than 1^.10^-15 m.
                  

Electrons are relativistic at this energy. You don’t have to consider their rest mass.

contents

Exercise 4 - Virtual Particles

Knowing the dimension of neutrons and protons, you can estimate the mass of the particle, which is responsible for the interaction between the nucleons.

Hints

1.     The interaction between charged particles is carried by photons. Because the range of this interaction is infinite, the rest mass of a photon has to be zero. The interaction between nucleons is limited to a range of about 10^-15 m.

2.     The Heisenberg uncertainty relation allows fluctuations of energy for a very short time, so "virtual" particles can be created, which are responsible for the interaction between particles.

Solution

The uncertainty relation for a distance from 10^-15 m allows a maximum energy deviation of about 200 MeV. A particle which is responsible for the interaction of two nucleons should have a mass in the order of  200 MeV but can only exist for about 10^-24 s. According to the prediction of Yukawa (Nobel prize in 1949) Powell and collaborators found the p-meson in 1947 with a mass of about 140 MeV.

 

contents

Exercise 5 - Accelerators

What dimensions (radius, magnetic field, revolution frequency) should an accelerator have to produce Z and W bosons, which are responsible for the weak interaction between particles? Is it possible to build an accelerator that replicates the maximum energies of cosmic ray on Earth?

Hints

1.     The Large Electron Positron Collider (LEP) a storage ring at CERN, has a length of about 27 kilometres. The electrons and positrons are moving at ultra-relativistic speed.

2.     The mass of the Z boson produced in LEP1 (to 1995) is about 91 GeV. The mass of the W boson is about 80 GeV; the LEP2 (to 2000) runs at an energy sufficient to produce W's in pairs. Using a collider you have twice the energy of the beam to create new particles.

3.     The highest energy of cosmic ray ever recorded is over 20 TeV.

Solution

For the calculation of the magnetic field for relativistic particles one can use:

LEP: r_eff = 3 km     Only 2/3 of the 27 km are used for bending magnets.                 

        B = 0.1 T for Z's and B = 0.18 T for W's in pairs, f_rev = 11 kHz

To get the energy of 20 TeV at a magnetic field of 10 T (maximum with superconducting magnets) you have to build a tunnel, which has twice the radius of the LEP.

N.B. The reason for the big radius of the LEP tunnel is the synchrotron radiation of the accelerated electrons and positrons, which energy is proportional to g³/r.  

 

contents

Exercise 6 – Particle Beam

Compare the density of gold and oxygen to the density of a particle beam in an accelerator.

Hints

The particle beam is divided in so-called bunches in an accelerator flying around like cigars at a certain distance. One bunch for the Large Hadron Collider (LHC) at CERN has a length of about 15 cm and a radius of about 17 mm, filled with 10¹¹ protons.

  Solution

 

The density of the beam bunch is between the density of a solid and a gas.

contents