* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Exercises Introduction 1.1 F.H What is the de Broglie
Survey
Document related concepts
Introduction to gauge theory wikipedia , lookup
Electromagnetism wikipedia , lookup
Density of states wikipedia , lookup
Work (physics) wikipedia , lookup
Conservation of energy wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
History of subatomic physics wikipedia , lookup
Elementary particle wikipedia , lookup
Atomic nucleus wikipedia , lookup
Nuclear physics wikipedia , lookup
Atomic theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Transcript
Exercises Introduction 1.1 F.H What is the de Broglie-wavelength h/p for a particle with momentum 1.2 TeV/c 1.2 F.H. A particle has a momentum 3500 MeV/c. What values will the quanties β, γ, T (kinetic energy) and E (total energy) have for an electron (m0 = 0.511 MeV/c2), a proton (m0 = 938.272 MeV/c2) and a deuteron (m0 = 1875.613 MeV/c2) 1.3 F.H. Calculate the minimum beam energy for the production of π0-meson (m0 = 134.98 MeV/c2) in the collision of an accelerated proton on a proton at rest (m0 = 938.272 MeV/c2). What would be the beam energy needed in a colliding beam experiment (both protons accelerated to the same energy). X 1.4 F.H. Protons are accelerated in a synchrotron. At injection the kinetic energy amounts to 40 MeV, at extraction it is 2500 MeV. The particle orbit has a length of 183.472 m. What are the momentum, velocity and orbital frequency of the particles at injection and extraction. What does this imply for the magnetic field and the frequency of the accelerating voltage. X 1.5 F.H. With which accuracy the velocity of a proton v = βc should be measured in order to determine its momentum with an accuracy 10-3. Determine the relation between ∆β/β and ∆p/p. 1.6 F.H. Determine the radius of curvature of an electron with a kinetic energy of 10 keV in the earth magnetic field (B = 3.1 x 10-5 T) 1.7 F.H. What is electric rigidity |E| ρ of an α-particle with an energy of 5.4 MeV 1.8 F.H. What is the Bρ value of a proton beam with a kinetic energy of 100 MeV. 1.9 S.B. The radius of a nucleus with mass A is given by r = 1.2 x10-15 A1/3 m. The nuclear interaction has a very short range, it is reasonable to assume that it only comes into play when nuclei “touch” eachother. Calculate the minimum energy of an α-particle needed to allow a nuclear reaction on 14N and 208Pb in a classical approach. Hint: What is the potential energy in the electrostatic field when both nuclei touch each other. 1.10 H.W. Derive relativistic expressions for p(β, E), p(Ekin) and Ekin(γ). Simplify these expressions for large γ. Derive the classical non-relativistic expressions from the relativistic ones. 1.11 H.W. Protons are accelerated to a kinetic energy of 100 MeV. Calculate their total energy, momentum and velocity in terms of the velocity of light. 1.12 H.W. Electrons are accelerated in a 3 km long linear accelerator with an effective gradient of 20 MV/m. At the entrance of the accelerator the electrons have a velocity v = c/2. What is the length of the accelerator in the restframe of these electrons. What are the energy and velocity of the electrons at the exit of the accelerator. After passing through the accelerator they travel down a 3 km long beamline. How long does this tube appear for the electrons. created modified 13/11/07 23:07 13/11/07 23:07 1/3 ExercisesIntroduction.doc X 1.13 H.W. A charged π-meson has a mass of 139.568 MeV/c2; its halflife is 26.029 ns. What is the halflife observed in the laboratory for π-meson with a kinetic energy of 20 and 100 MeV. At which distance from the source will the intensity of a π-meson beam have dropped to 50 % of the initial value for 20 and 100 MeV kinetic energy. 1.14 H.W. A proton with a kinetic energy of 1 eV is emitted parallel to the surface of the earth. What is the bendig radius due to the gravitational force. What are the electric and magnetic fields needed to obtain the same bending radius. What is the ratio between the electric and magnetic fields. Is this ratio different for a proton with a kinetic energy of 1 GeV. (the gravitational constant is 6.671 x1015 Nm2/kg2, the gravitational acceleration at the surface of the earth is 9.85 m/s2). 1.15 H.W. A circular accelerator with a circumference of 300 m contains a uniform distribution of single charged particles orbiting with the speed of light. If the circulating current is 1 A, how many particles are orbiting. We instantly turn on an ejection magnet, so that all particles leave the accelerator during the time of one revolution. What is the peak current at the ejection point. What is the duration of the pulse. If the accelerator is a synchrotron accelerating particles at a rate of 100 acceleration cycles per second, what is the average ejected particle current. 1.16 H.W. Prove that the electric and magnetic fields generated by a uniform cylindrical particle beam with charge density ρ0 within a radius R is given by Er(r) = ρ0/2ε0 r and Bϕ(r) = µ0 /2 ρ0 β c r. What are the fields for radii r > R. X X Consider a highly relativistic beam of electrons (E=500 GeV). The beam bunches contain 1010 electrons each; the have a radius of 0.2 µm and a length of 1 mm. What are the electric and magnetic field at the surface of the beam. What is peak current of the bunches. If in a collider two such bunches pass eachother at a distance of 10 µm, what is the deflection angle of each beam due to the other. 1.17 H.W. The beam bunch of the previous problem is traveling through free space. Determine the distance from the axis as a function of the distance traveled for a particle initially at a distance σ from the axis. 1.18 H.W. Particles in an accelerator undergo elastic collisions with the atoms in residual gas. The rms scattering angle in these collions is given by σϑ = 20 [MeV/c] Z/βp (s/λ)1/2 [rad], where Z the charge of the particles (in units of the electron charge), p the momentum of the particles, s the distance traveled and λ the radiation length of the material at atmospheric pressure in air λ = 500 m). Derive an expression for the beam radius as a function of s. What is the approximate tolerable gas pressure in a proton storage ring if the beam is supposed to orbit for 20 hours with an increase in beam radius of at most a factor two. 1.19 E.W. A betatron has a beam radius of 0.1 m and is powered from the 50 Hz mains. Its peak guide field is 1 T while the flux linking the orbit is twice that which would result from a uniform field of this value. What will be the peak energy of the electrons it accelerates. created modified 13/11/07 23:07 13/11/07 23:07 2/3 ExercisesIntroduction.doc X 1.20 S.B A classical cyclotron has a radius of 0.2 m and a field of 1 T. What is the orbital frequency of protons in this field. What is the acceleration voltage needed to ensure that the protons reach the outer radius of the cyclotron 1.21 S.B. Analyze for which values of the accelerating phase focussing exists in a Widerö linear accelerator. Hint: at what phase particles with a higher energy than the reference particle arrive in the acceleration gap; what effect does this have on their energygain in the gap. Why does this lead to transverse defocussing that has to be compensated by focussing elements in the drifttubes (Hint: What happens with the radial force on the particle while it traverses the gap; start from the electrostatic case) Determine the length of a Widerö linear accelerator accelerating protons from 100 keV to 10 MeV using an accelerating voltage of 200 kV at a frequency of 20 MHz 1.22 S.B. In a betatron or Widerö “ray transformer” the radius of the orbit is stable when ∂ Bguide 1 ∂ ∫ Bacc da = the following condition is met , where Bguide is the field at ∂t 2 ∂t πr 2 the location of the orbit and Bacc the magnetic field in the region enclosed by the orbit. Derive this condition 1.23 H.W. Calculate the minimum power rating for the motor driving the charging belt of a van de Graaff acclerator producing a charge current of 1 A at 5 MV 1.24 H.W. Calculate the length of the first four drift tubes of a Widerö linear accelerator for the following parameters: • starting energy 100 keV • energy gain per gap 1 MeV • frequency 7 MHz • particle: proton (mc2 = 938.27 MeV) and electron (mc2 = 0.511 MeV) 1.25 H.W. Consider a betatron operating at 60 Hz. The orbit radius is 1.23 m and the maximum magnetic field at the orbit 0.81 T. Electrons are injected at 50 keV; calculate the magnetic field at injection and the energy gain per turn for the first turn and at the time the electron has gained 20 MeV. Discuss the reason for the difference in energy gain. 1.26 S.B. What are the energies that can be achieved for protons and electrons in the betatron of 1.26 Excercises marked with a cross in the last column are assignments that can be handed in for credit. Assignments should be handed in at the latest tuesday 20 november at the beginning of the lecture. created modified 13/11/07 23:07 13/11/07 23:07 3/3 ExercisesIntroduction.doc X X