36
1.4
1. The Particles and Forces of the Standard Model
at rest). Show that if the electrons are highly relativistic then
2
q = −4EE
′ sin
2 θ/2, where θ is the scattering angle in this frame.
Deduce that for elastic scattering E
′ and θ are related by
/ (
)
E
′ = E
1 +
2E sin
2 θ/2 .
M
(b) Electrons of energy 4.879 GeV scatter elastically from protons, with
θ = 10
◦ . What is the observed value of E
′ ?
(c) In the scattering of these electrons, at 10
◦ , it is found that there is
a peak of events at E
′ = 4.2 GeV; what is the invariant mass of the
produced hadronic state (in MeV)?
(d) Calculate the value of E
′ at which the ‘quasi-elastic peak’ will be
observed, when electrons of energy 400 MeV scatter at an angle
θ = 45
◦ from a He nucleus, assuming that the struck nucleon is at
rest inside the nucleus. Estimate the broadening of this final peak
caused by the fact that the struck nucleon has, in fact, a momentum
distribution by virtue of being localized within the nuclear size.
(a) In a simple non-relativistic model of a hydrogen-like atom, the energy levels are given by
−α
2 Z
2 μ
E n = 2n 2
where Z is the nuclear charge and μ is the reduced mass of the
electron and nucleus. Calculate the splitting in eV between the
−
n = 1 and n = 2 states in positronium, which is an e
+ e bound
state, assuming this model holds.
(b) In this model, the e
+ e
− potential is the simple Coulomb one
2
e
α
−
= − .
4π∈ 0 r
r
Suppose that the potential between a heavy quark Q and an antiquark ¯
Q was
α s
− r
where α s is a ‘strong fine structure constant’. Calculate values of
α s (different in (i) and (ii)) corresponding to the information (the
quark masses are phenomenological ‘quark model’ masses)
(i) the splitting between the n = 2 and n = 1 states in charmonium
(c¯ c) is 588 MeV, and m c = 1870 MeV;
1.4
1. The Particles and Forces of the Standard Model
at rest). Show that if the electrons are highly relativistic then
2
q = −4EE
′ sin
2 θ/2, where θ is the scattering angle in this frame.
Deduce that for elastic scattering E
′ and θ are related by
/ (
)
E
′ = E
1 +
2E sin
2 θ/2 .
M
(b) Electrons of energy 4.879 GeV scatter elastically from protons, with
θ = 10
◦ . What is the observed value of E
′ ?
(c) In the scattering of these electrons, at 10
◦ , it is found that there is
a peak of events at E
′ = 4.2 GeV; what is the invariant mass of the
produced hadronic state (in MeV)?
(d) Calculate the value of E
′ at which the ‘quasi-elastic peak’ will be
observed, when electrons of energy 400 MeV scatter at an angle
θ = 45
◦ from a He nucleus, assuming that the struck nucleon is at
rest inside the nucleus. Estimate the broadening of this final peak
caused by the fact that the struck nucleon has, in fact, a momentum
distribution by virtue of being localized within the nuclear size.
(a) In a simple non-relativistic model of a hydrogen-like atom, the energy levels are given by
−α
2 Z
2 μ
E n = 2n 2
where Z is the nuclear charge and μ is the reduced mass of the
electron and nucleus. Calculate the splitting in eV between the
−
n = 1 and n = 2 states in positronium, which is an e
+ e bound
state, assuming this model holds.
(b) In this model, the e
+ e
− potential is the simple Coulomb one
2
e
α
−
= − .
4π∈ 0 r
r
Suppose that the potential between a heavy quark Q and an antiquark ¯
Q was
α s
− r
where α s is a ‘strong fine structure constant’. Calculate values of
α s (different in (i) and (ii)) corresponding to the information (the
quark masses are phenomenological ‘quark model’ masses)
(i) the splitting between the n = 2 and n = 1 states in charmonium
(c¯ c) is 588 MeV, and m c = 1870 MeV;