Download Exercises Introduction 1.1 F.H What is the de Broglie

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).
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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.
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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.
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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.
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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.
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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.
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