Question
In 2016 the T2k experiment won the Breakthrough Prize in Fundamental Physics for research in neutrino oscillations: the ability of the almost-massless particles called neutrinos (symbol: v) to oscillate between different "flavors" while in flight. The experiment relied on a neutrino beam that was generated on one side of Japan and was directed at a large water Cherenkov neutrino detector on the other side of Japan. So how does one create a neutrino beam in the first place? a) First, we begin with a high energy proton beam. The protons in the beam are accelerated to speeds that contract their individual "length" (size along direction of propagation) by a factor of 50. Calculate alpha_p = 1 - v_p/c (subscript p stands for "proton"), which is a much more convenient way of expressing the speed than something v_p = 0.99999999999999999998 c (and I'm not saying that's the correct value!) b) Next, the protons are slammed into a solid target, producing a forward-directed beam of pions out the back of the target. Each proton produces several pions, and each pion decays into a muon and a neutrino while it is in flight. The lifetime of a pion at rest is 26 ns. Let's consider a pion that is traveling in the forward direction at a speed described by alpha = 5.0 times 10^-3. How long will the pion take to decay in the stationary laboratory reference frame? c) How far will the pion travel in the stationary laboratory frame before it decays? d) In the inertial rest frame of the pion, i.e. a frame traveling forward at the constant speed of the pion, the muon and the neutrino are emitted back-to-back when the pion finally decays. In other words, the muon and the neutrino velocities have directions separated by 180deg. But in the rest frame of the pion, the back-to-back muon-neutrino velocity axis could point in any direction with equal probability. If the muon happens to be emitted in the forward direction with a speed v = 0.10c as measured in the rest frame of the pion, what will be the muon's speed in the stationary lab frame? Write the answer as alpha_v = 1 - v /c. e) The muon itself will eventually decay into an electron and two neutrinos after 2.2 mu s. This gives us a total of 3 neutrinos for every original pion that was produced in the proton target, and that's how you make a neutrino bean. How far will the forward-emitted muon described above travel from the pion decay position before the muon itself decays? f) We need to make a vacuum pipe that is at least long enough to allow the full decay process (pion rightarrow neutrino + muon rightarrow electron + 2 neutrinos) to occur before it comes to an end.
Explanation / Answer
a. let original length of protons be lo
then contracted length l
so from the given data
l/lo is 1/50
but l is lo*sqroot[1-[vp/c]*[vp/c]]
so, 1/2500 is 1 - vp*vp/c*c
so vp/c is 0.9996
alpha , 1 - vp/c is 0.0004
b. rest lifetime of pion is to, 26 ns
alpha is 0.005 for the pion
decay time is t
then t is to/sqroot[1 - v*v/c*c]
1 - v/c is alpha
so, v/c is 0.995
t is 26/sqroot[1 - v*v/c*c]
t is 260.32 ns
c. distance travelled by pion
d is v*t
v is 0.995c
d is 0.995c*260.32/10[exp[9]]
d is 77.70552 m
d. given speed of muon in pions frame of reference v' is 0.1c
so speed of muon in lab frame , v" is v'/sqroot[1-v*v/c*c]
so v" is 0.1c*sqroot[1 - 0.995*0.995]
1 - v"/c is 0.99001
