ph135c_hw7

ph135c_hw7 - Physics 135c H. W. Assignment 7 1.) Bertulani...

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Unformatted text preview: Physics 135c H. W. Assignment 7 1.) Bertulani Probs. 8.7, 8.8 & 8.9 2.) For this problem you will use Fermi’s Golden Rule dΓ = 2π 2 |Af i | ρf h ¯ to calculate the energy spectrum of electrons emitted in neutron β -decay n → p + e− + νe . ¯ Ignoring the proton recoil and Coulomb effects, the phase space factor ρf is given by ρf = d3 pe d3 pν δ (Ee + Eν − ∆) (2π ¯ )3 (2π ¯ )3 h h where the δ (Ee + Eν − ∆) factor ensures energy conservation in the final state with ∆ = Mn − Mp . Calculate the energy spectrum of emitted electrons, dΓ/dEe , assuming the neutrino mass is zero, by integrating over the neutrino phase space and plot the 2 (unnormalized) spectrum vs the kinetic energy of the emitted electron. Note: the |Af i | factor is essentially independent of energy (see Prob. 3) and the electron and neutrino are obviously relativistic. 3.) The matrix element in Prob. 2 can be written as 2 2 2 2 |Af i | = G2 gV |MF | + gA |MGT | F where 1 |MF | = 2Ji + 1 2 2 2 ± τ ˆ f i , and n,p fi 2 1 |MGT | = 2Ji + 1 2 σj τ ˆˆ f j fi ± i , n,p and the f i sums over initial and final spin directions, the j sums over the components of the σ operator while the n,p sums over all nucleons in a nucleus. There is also a term ˆ that depends on pe × pν that we ignore (it disappears when integrating over the neutrino 2 2 momentum). Evaluate |MF | and |MGT | explicitly for free neutron decay. 4.) Use the results from Prob. 2 and 3 to calculate the mean lifetime of the neutron, τn = 1/Γ with Γ = (dΓ/dEe )dEe . You can use the measured values for GF = 1.166 × 10−11 MeV−2 (¯ c)3 , gV = 1.0 and gA = 1.27. Compare your result with the measured value h of τn = 886 ± 1 sec. 5.) The dominant decay mode of the charged pion is to a muon and neutrino, eg. π + → µ+ + νµ such that a precise measurement of the momentum of the µ+ from a π + decaying at rest allows a measurement of the mass of the νµ . If you can measure the muon momentum with a fractional uncertainty of δp = 4 × 10−6 and including the uncertainties in the mass p of the muon and pion (see Particle Data Group web page), estimate your sensitivity to the neutrino mass (eg. estimate the smallest value for the neutrino mass that you could reliably measure). ...
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This note was uploaded on 12/04/2011 for the course PHY 7070 taught by Professor Smith during the Spring '11 term at Wisconsin.

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