MARKUS KLUTE: Hello.
Welcome back to 8.701.
In this lecture, this little video,
we're going to look at reactions and how
they relate to cross-sections.
So we start or continue the discussion
of how we can relate experimentally
measured determined properties to the forces involved.
The last lecture, we looked at decay rates and the rates
of unstable particles.
This time, we're going to look at the reaction rates expressed
as cross-sections.
We can start doing this by looking at this simplified
picture.
So the reaction rate is related to the rate
of the beam, so how many particles per second
are available for the interactions times the number
density of the particle in the target.
So you have your target here.
And clearly, the number of reactions depend on how dense
your target is, the thickness-- so you go along how the thicker
the material the target is, the more likely it is for reactions
to occur--
and then by the actual physics--
by the likelihood of a collision to occur.
And this likelihood is called a cross-section.
And we can think about this cross-section
as a geometrical area.
All right?
So let's look at this a little bit more.
We can stay with a very classical model,
a model in which we have two billiard balls--
and light one, a small one, with radius r1,
and a larger one with the radius r2.
Clearly, a collision occurs when the impact parameter
b here between those two billiard balls
is smaller than the sum of the radii.
OK?
So now we can analyze this reaction a little bit more
and look at angular distributions.
We find that the cross-section differential distribution
is given as a function of sine theta.
We can also express this using the azimuth or as a solid angle
here.
As a reminder, the solid angle theta
is equal to sine theta d theta d phi.
All right?
So for this specific problem here,
we talk about an isotropic reaction,
because by definition the cross-section per solid angle
is independent of theta and phi.
The mapping between sine theta and theta is not trivial.
That's why you see this shape of the distribution.
Yeah.
But the d sigma d cosine theta is
[INAUDIBLE] if you look at this as a function of cosine theta.
All right.
So this is just a classical picture
of what we're going to do later in the class.
It's using quantum field theory or Feynman rules
in order to calculate cross-sections, or the decay
rates.
But this classical picture is really
something I would like you to keep in mind--
the idea of the likelihood of the collision to occur
is in units of an area and can be
seen as a geometrical cross-section is kind of a very
nice picture to keep in mind.
And it also helps estimating orders of magnitude of rates
of collisions to occur.