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Question 2
Consider the following kaon-plus decay,
6 pts
.
Show that the decay is allowed by considering the various conservation rules. State any
conditions on the decay.
Draw a Feynman diagram for the decay. Be sure to label the hadrons, quarks and leptons, and
gauge bosons.
Label each particle so that the decay has a consistent color scheme.
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Question 3
What is the maximum energy of the electron emitted in the
in MeV units.
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decay of
? Give your answer
Question 4
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A semiconductor has a band gap of 0.85 eV. If the semiconductor is initially at room
temperature, what temperature change would increase the conductivity by 50%?
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Question 5
Show that the kinetic energy operator
eigenstates.
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and linear momentum operator
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Question 6
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The work needed to pull quarks apart is so large that hadrons are created in the process. As
such, quarks cannot be observed in isolation. This inability to isolate quarks is called color
confinement.
The mass of the lightest hadron, the
meson, is 135 MeV/c2. Estimate the magnitude of the
strong force within a nucleon, which is ~1 fm in size. Express your answer in newtons. Show all
unit conversions.
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Question 7
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Spectroscopic notation for an electronic configuration is 2S+1LJ, where S is the total spin
quantum number, L is the total orbital angular momentum quantum number, and J is the
quantum number for the angular momentum
. The convention is to use letters S, P,
D, etc. for L = 0, 1, 2, etc.
What spectroscopic states are possible for excited-state helium having electrons in the 1s and
2p electronic states? Show how the allowed values for S, L, and J are worked out.
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Question 8
22 pts
The Fermi gas model.
The asymmetry term in the semi-empirical mass formula (Equation 9.12) describes how the
binding energy is reduced (i.e., the energy of the nucleus is increased) as a nucleus moves
away from the N = Z condition. In deriving the asymmetry term, we ignore electrostatic repulsion
and assume the neutrons and protons are each low-temperature Fermi gases.
The move away from N = Z is visualized as the shifting of neutrons over to the proton column (or
vice versa) in Figure 9.7 or in the figure shown. Due to the exclusion principle, this leads to an
increase in the total energy of the system.
This “shifting” is at the heart of the derivation shown in Section 9.2 of the text, although the
steps there are somewhat involved. Here, you will go through a simplified derivation of the
asymmetry term. As always, be sure your reasoning is clear for each section of the problem.
a) Assume energy levels in the nucleus are equally spaced, separated by an amount
. If N0 =
Z0 (i.e., N = Z initially) and neutrons are “shifted” over to become protons, show that the
increase in total energy of the nucleus is
. (Hint: it may be easier to consider only
even .)
b) Express in terms of the new N and Z of the asymmetric (N ≠ Z) nucleus.
c) Assume
, where is the atomic mass number. Show that this, combined with the
results of parts a) and b), leads to the asymmetry term in Equation 9.12.
d) Finally, justify the
assumption used in c). To do this, show that to a good
approximation, the Fermi energy for neutrons and the Fermi energy for protons are equal AND
have the same value for all nuclei. If energy levels in the nucleus are equally spaced, show that
.
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Question 9
17 pts
Probability of electron capture.
Because the nucleus is finite in size, there is a non-zero probability of finding the electron in the
nucleus. In fact, electron capture is an important nuclear process.
For this problem, do not use an integral table; show all steps in the integration. Cancellation of
terms in any expansion must be clear. As always, reasoning must be clear.
a) Show that for an electron in the ground state of hydrogen, the probability of being found in the
nucleus is
,
where
,
is the radius of the nucleus, and
is the Bohr radius. The hydrogen atom
wave functions are found in Chapter 6.
b) Use a series expansion to show that for
, the probability density (probability of capture
per volume) is constant. That is, show that for a small nucleus, the probability of electron
capture doubles when the volume of the nucleus doubles.
The following is helpful.
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