Download Exercise Sheet 1 to Particle Physics I

Exercise Sheet 1 to Particle Physics I
1) Use the Particle Data Group (PDG) webpage (or other sources of information) to express
the following quantities in the elementary particle physics natural units (i.e. in proper eV units
using h̄ = c = 1): atomic radius (1 Å), nucleon radius (1 fm = typical size of atomic nuclei) classic
electron radius re , reduced Compton wave length of the electron λe /2π, graviational acceleration
at the earths surface gN .
[Find all relevant numbers from PDG webpage: click on the “Physical Constants” link.]
2) Determine, as for exercise 1, in natural units the inverse mean life times (also called decay width
Γ = τ −1 ) of the neutron (n), muon (µ− ), the pions (π +,−,0 ) and the rho meson (ρ). Search through
the PDG particle listings to get the information. Compare the results with the masses of these
particles. Think of what one may learn from the numbers?
3) Yukawa hypothesized a so-called meson particle being responsible for the strong force acting
inside the atomic nuclei (“Atomkern”). The idea was that the exchange of the meson creates a
potential of the form V (r) ∝ 1r e−rmY (called Yukawa potential today) inside the nucleus that causes
its formation. Estimate the mass mY of the meson Yukawa postulated using the assumption that
the interaction range of the potential is of the size of the nucleon radius. [Comment: To start you
need to figure out how to estimate the interaction range of the potential. There are different ways
to do so.]
4) Estimate the mass mY of Yukawa’s meson in a different way by assuming that the exchange
of mesons between the nucleons (the protons and neutraon inside the atomic nucleus) causes a
momentum uncertainty of the nucleons of the order of mY and that the corresponding maximal
uncertainty in the spatial location of the nucleons (based on Heisenberg’s uncertainty principle)is
of order the nucleon radius. Discuss the assumptions made in exercises 3 and 4 with your fellow
students and in class. What do exercises 3 and 4 teach us?
5) This calculations is what J.J. Thomson used to determine the charge-mass ratio of the electron:
If you set up a homogeneous and time-independent magnetic field perpendicular to the direction of
flight of a charged particle and - perpendicular to the magentic field and to the direction of flight
of the charged particle - a homogeneous and time-indpendent electric field, it is possible that the
charge particles flight direction is just straight without any deflection. (a) Which velocity does the
charged particle need to have for a given E and B fields in this situation? (b) Now, switch off the
electric field, and you observe that the particle travels along a circle with radius r. Determine that
charge-to-mass ratio q/m for the charged particle.
6) Prior to the discovery of the neutron, there was the model that atomic nuclei consist of (positively
charged) protons and (negatively charged) electrons, where the atomic number Z is equal to the (in
comparison to the electrons) exceeding number of protons. The idea does not seem unreasonable,
because electrons are produced in the nuclear β-decay of neutrons (n → p + e− + ν̄e ). But there is a
problem that makes this idea look quite unnatural for the experienced particle physicists brain due
to a mismatch in the expected and experimentally observed typical energy scales. The exercises
task is to make you see this problem as well: First estimate, using the uncertainty relation for
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location and momentum, the minimal momentum the electrons typically have in the atomic nuclei.
Is the electron non-relativistic? (You might have to consult some relativistic formulae in case it
is not obvious to you.) Now determine the typical energy of electrons radiated in the β-decay
within the model using the assumption that it is about half the size of the energy the electron
had inside the nucleus prior to the decay. Compare the result with the experimentally measured
electron energy spectrum from the tritium β-decay. You can easily find the spectrum using google
in connection with the KATRIN experiment. Have a closer look on the energy distribution of the
emitted electrons observed. Which energies do the electrons have? What is the maximal energy?
Why is there something unnatural - if at all?
Discuss in class and with your fellow students why we one can discuss this problem in such general
terms without having to consider a specific model. Which kind of questions cannot be answered in
that way?
The issue is that the model cannot be proven to be wrong by the unnaturalness argument. However,
one needs to invent another model for some additional mechanism to reconcile the the original model
work properly. This is not the property of a good model. In any case we know today the the model
is simply crap.
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