492 9 Particle Physics – I
9.26 Heavy mesons,M= 950 meproduced in nuclear interactions initiated by a
more energetic beam ofπ-meson, have an energy of 50 MeV. Their tracks up
to the point of decay have a mean length of 1.7 m. Calculate their mean life
time.
[University of Durham 1960]
9.27 A pion beam from an accelerator target has momentum 10 GeV/c. What frac-
tion of the particles will not have decayed into muons in a pathlength of
100 m?
Out of the pions decays one muon of a 8 GeV/c and a neutrino are produced
at the beginning of the flight path. Assuming that the decay particles follow
the same path, calculate the difference in arrival times at the end of the path.
(Pion mass= 139 .6 MeV, muon mass= 105 .7 MeV. Mean life time of pion
= 2. 6 × 10 −^8 s,c= 3 × 108 ms−^1 )
[University of Durham 1972]
9.28 Pions in a beam of energy 5 GeV decay in flight. What are the maximum and
minimum energies of the muons from these decays? (mπ = 139 .5MeV/c^2 ;
mμ= 105 .7MeV/c^2 )
9.29 Assuming that a drop in intensity by a factor less than 10 is tolerable, show
that a 1 GeV/cK+meson beam can be transported over 10 m without a serious
loss of intensity due to decay, while aΛ-hyperon beam of the same momen-
tum after the same distance will not have useful intensity (Take masses ofK+
meson andΛ-hyperon to be 0.5 and 1 GeV/c^2 , respectively and their lifetimes
10 −^8 and 2. 5 × 10 −^10 s respectively.
[University of Durham 1970]
9.30 If a particle has rest mass m 0 and momentump, show that the distance traveled
in one lifetime isd =pT 0 /m 0 whereT 0 is the life time in the frame of
reference in which the particle is at rest.
[University of Dublin 1968]
9.31 A beam of muon neutrinos is produced from the decay of charged pions of
Eπ=20 GeV. Show that the relationship between the neutrino energy in the
laboratory frame,Eν, and its angle relative to the pion beamθ, for smallθ,is
Eν=Eπ
(1+γ^2 θ^2 ){
1 −mμ^2 /mπ^2}
whereEπis the energy of the pion andγ=Eπ/mπ>> 1
[University of Cambridge, Tripos 2004]9.32 It is intended to use a charged mono-energetic hyperon beam to perform scat-
tering experiments off liquid hydrogen. Assuming that the beam transport sys-
tem must have a minimum length of 20 m, calculate the minimum momentum
of aΣbeam such that 1% of the hyperons accepted by the transport system
arrive at the hydrogen target (τ= 0. 8 × 10 −^10 s,mΣ= 1 .19 GeV/c^2 ). What