expansion of the full transformation 2.2 Special relativity and colliders (a) Th

Question

expansion of the full transformation 2.2 Special relativity and colliders (a) The Large Hadron Collider was designed to collide protons together at 14 TeV center-of-mass energy. How many kilometers per hour less than the speed of light are the protons moving? (b) How fast is one proton moving with respect to the other? 2.3 The GZK bound. In 1966 Greisen, Zatsepin and Kuzmin argued that we should not see cosmic rays (high-energy protons hitting the atmosphere from outer space) above a certain energy, due to interactions of these rays with the cosmic microwave background. (a) The universe is a blackbody at 2.73 K. What is the average energy of the photons in outer space (in electronvolts)? (b) How much energy would a proton(p+) need to collide with a photon () in outer space to convert it to a 135 MeV pion (")? That is, what is the energy threshold for p++1p++10? (c) How much energy does the outgoing proton have after this reaction? This GZK bound was finally confirmed experimentally 40 years after it was conjec- tured [Abbasi et al., 2008] Is the transformation Y : (t, r, y, z) (t,a,--y, z) a Lorentz transformation? If so, why is it not considered with P and T as a discrete Lorentz transformation? If not, why not? 2.4

Explanation / Answer

a. given universe is a blackbody of temperature T = 2.73 K

then average wavelength of the photon in this universe be lambda

from weins displacement law

T = b/lambda

where b = 2.89*10^-3

hence lambda = 2.89*10^-3/2.73 = 1.0586*10^-3 m

hence average energy of a photon , E = hc/lambda = 6.63*10^-34*c/lambda = 1.8788*10^-22 J

Eav = 1.1743*10^-3 eV

b. let minimum KE required by proton be KE

then for the reaction to happen

KE + Eav = 135*10^6 eV

KE = 134,999,999.99882587 eV

c. after this reaction the outgoing protonj has 0 KE

now rest mass of proton = mo = 1.6*10^-27 kg

so energy of photon = moc^2 = 1.6*10^-27*c^2 = 1.44*10^-10 J

in eV

Eo = 900 MeV

so the proton leaves with rest energy 900 MeV and 0 Kinetic energy after the reaction

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