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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
|Af i | ρf
¯ 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 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
(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
3.) The matrix element in Prob. 2 can be written as
2 2 2
|Af i | = G2 gV |MF | + gA |MGT |
|MF | =
2Ji + 1 2 2 2 ± τ
ˆ f i , and n,p fi 2 1
|MGT | =
2Ji + 1
2 σj τ
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
that depends on pe × pν that we ignore (it disappears when integrating over the neutrino
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
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
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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