M2L8m.txt

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Formally, you can define the quadrupole moment
by the expectation operator.
You take the nucleus with maximum M I,
and now you calculate the expectation value
of this operator.
This is, of course, motivated by just electrostatics.
If you take an expansion of the classical electrostatic energy
into multipoles, you find the quadrupole configuration
to be related to a quadrupole moment.
Quadrupole moments couple to the derivative of electric fields.
And then, in this purely classical description,
you have this term where beta is the angle between two symmetry
axes, namely between the symmetry
axis of the electric field gradient
and the quadrupole tensor-- just the classical quadrupole
tensor, as it comes out of Jackson.
Yes, so you can see the quantum mechanical definition
of the quadrupole moment.
Or more generally, so this is quadrupole moment,
if you a moment with L, the operator.
Which tells you whether you have a non-vanishing moment.
A non-vanishing deformation is actually a spherical tensor--
TLM.

And what you see above is a spherical tensor--
T 2 0 L equals 2 M equals 0.
And those spherical tensors are defined by the fact
that they transform as spherical harmonics.
And now you sort of realize what it means.
If you want a magnetic or electric moment with L,
the operator for the moment transforms
like angular momentum L. And now you
realize that you have the triangle rule.
If you want a matrix element where I and L overlap with I,
you want to make sure that I L and I couple.
And you have a triangle rule.
So therefore, if you want a magnetic moment,
or electric moment of L, and you evaluate this expectation
value, well a least the triangle rule
can only be justified like this.
Or in other words, you can only get a non-vanishing moment
if L is smaller than 2 I. And this
is what we discussed in the clicker question
for the two cases of L equals 1 and L equals 2.
So ultimately, it's a selection rule,
which is related to the triangle rule for the additional angular
momenta.

But I like much better the argument,
how many orientations do you need to find out
that something is elliptical?
But it's formalized here.
All right, let's just spend one more minute
on the quadrupolar structure.
So based on the expansion of the electrostatic energy, what
determines the quadrupolar structure
is this angle-- this cosine angle.
Which is the angle between the axis of the nucleus
and the axis of an electric field gradient.
And that means it is the angle-- it's
the cosine of the angle between J, the outer environment,
and I, the axis of the nucleus.
So therefore, when we would derive-- I'm not deriving it,
but if you would derive-- an expression for quadrupolar
structure, the quadrupolar structure would
be proportional to a quantity, C,
which is nothing else than the dot product of I and J.
And it is in my notes now, I and J have units of hbar.
So I'm dividing it out here.
As you know I dot J can be expressed by quantum numbers F
F plus 1 minus I I plus 1 minus J J plus 1.
So therefore, the quadrupolar energies, E2, L equals 2,
E L equals 2, involve the quasi--
the classical expression at cosine square.
So therefore, you would expect there is a quadrupole constant,
and then it is cosine square.
But well, usually quantum mechanically,
when we have the square of a quantity,
we have to write it as quantity times quantity plus 1.
So this is the quadrupolar structure.
And to remind you, we just discussed
that for the hydrogen atom.
The magnetic hyperfine structure had
involved the same product of I dot J, but in a linear way.
So the reason why I'm not discussing quadrupolar
structure in more detail is the hyperfine structure associated
with quadrupole moments is much, much smaller
than the hyperfine structure associated
with magnetic moment, typically by a factor of 100.
The only exceptions are molecules,
because molecules can have-- because
of molecular binding mechanisms-- a much, much
larger electric field gradient.
So therefore in molecules, quadrupolar structure is more important than in atoms