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MARKUS KLUTE: Welcome
back to 8.701.

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So in this video, we'll talk
about the nuclear shell model.

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We've already seen an
interesting empirical model

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to describe nuclear binding
energies-- the liquid drop

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model.

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But it comes short
in the description

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of all aspects of the nucleus.

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So let's see what
we can find here.

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First of all, you probably
remember shell models

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from atomic physics.

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And shell models are very
successful in describing

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hydrogen, for example.

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The question is, can this
also work for the nucleus?

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After all, the nucleus
is a many-body system,

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compared to hydrogen, where you
have a proton and an electron

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circling around.

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There's no analytic solutions,
like the Schrodinger equation.

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There is no dominant center
for a long-range force,

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like the proton has been
the dominant center.

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And we have short-range
forces with many pairs

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of interacting nucleons.

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And I can continue the
list of difficulties.

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On the other hand, the
interactions kind of

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average out and result
in a potential which

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depends only on the
position, but not

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on the timing of the nucleus.

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And that leads us, then, to what
we call a nuclear mean field.

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So on average, our proton and
our neutron inside the nucleus

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sees a specific potential.

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And we can use that, then,
parameterized as potential

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with a harmonic oscillator,
and use that model, then,

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in order to describe
our nucleus.

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So this works, actually,
surprisingly well.

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But before we go there, we'll
look at experimental evidence

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for closed nuclear shells.

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So again, here is our plot
of the binding energy.

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And you see that there
are those areas here

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that seem to be some sort
of higher binding energies.

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And it turns out those happen
at so-called magic numbers.

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Magic numbers are 2,
8, 20, 28, 50, and 126.

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So the question now is,
how can we explain this?

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Where does this come from?

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So again, the experimental
evidence is numerous.

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We find that the number of
stable isotopes or isotones

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is significantly higher
for nuclei with a proton--

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or neutron, or both--

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numbers equal to one
of those magic numbers.

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The nuclear capture
cross-section,

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meaning the likelihood to
capture a proton or a neutron,

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are high for nuclei where
exactly one nucleon is

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missing from a magic number.

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But it's significantly
lower for nuclei

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with number of nucleons
equal to the magic number,

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meaning that there is this
concept of a closed shell.

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We either just add a
nucleon to close it

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or you have to pay
a higher price.

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The energy of excited states for
nuclei with a proton or neutron

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number equal to the magic
number are significantly higher

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than for other nuclei.

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And these are all
experimental observations.

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And the excitation probabilities
of the first excited states are

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low for nuclei with a
proton-- or neutron, or both--

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numbers equal to
the magic numbers.

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Quadruple moments-- we haven't
discussed those at length,

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but you can think about them
as deformations of the nuclei.

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They almost vanish for nuclei
with proton or neutron numbers

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equal to the magic numbers.

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So those are more kind of
sphere kind of objects.

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Here's a plot which
shows or points

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out the double magic numbers--

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as in, those are a nucleus
where both the proton

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number and the neutron number
are laying on the magic number.

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So calcium here
has two of those,

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with 20 protons and 20 neutrons,
or 20 protons and 28 neutrons.

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And there's alphas.

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Those are specifically
interesting object of research.

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There was some historic
confusion in this,

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and it came from the fact that
while the experimental data

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pointed to nuclear magic numbers
of 2, 8, 20, 28, 50, and 126,

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if you just think about a
flat bottom potential, just

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a flat potential, you
find magic numbers which

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are 2, 8, 20, 40, 70, and 112.

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And those are typically
not in agreement.

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So therefore, it seemed like
that this shell model kind of

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worked, but not really.

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We found agreement here,
but then disagreement

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in the higher part
of the magic numbers.

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So something was missing.

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And so what was missing
was the spin-orbit part

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of the discussion.

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We alluded to this
in the nuclear force.

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What you have to do is, beyond
three-dimensional harmonic

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oscillator, you have to
add the spin-orbit coupling

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to the Hamiltonian.

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And when you do that,
you change the orbit such

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that the magic numbers agree
with the experimental data.

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So you see here, the
potentials for proton,

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which has also the
Coulomb repulsion added,

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and the nuclear
potential, and then you

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see that the spin-orbit coupling
slightly changes the potential.

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All right.

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As a comparison here,
the nuclear and atomic

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shell models, just
for an example.

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And you see we call them shells
because we see that the energy

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gaps between individual
shells are quite large, much

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larger than within the shell.

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And the same-- this is
for the atomic model.

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And for the nucleus,
you see very similar.

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So it's not that extended, but
still larger gaps in energy

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when you go from one
state to the next.