MARKUS KLUTE: Welcome back to 8.20, special relativity.
In the previous section, we have seen
how we can look at energy and momentum of particles
in a decay.
Here we now want to, in collisions of particles,
create new particles.
The example, the first example here,
is the collision of two protons to create a proton, a neutron,
and a charged pion.
The masses are given there.
So the question now is, what is the minimal energy
needed in order for this process to occur
in a fixed-target experiment?
Fixed-target experiment is we have an accelerated proton
and another proton at rest.
This might just be a hydrogen target just sitting there.
So the question is, how much energy--
how much do we have to accelerate
the proton for this process to be possible?
Now, again, stop the video, and try to work this out.
The important part here is to realize
that minimal energy here means that, after the decay
or the decay after the process occurred,
all the new particles need to be addressed.
That is when the process requires minimal energy.
So, instead of analyzing this in the laboratory frame,
we want to analyze this in the center-of-mass frame.
All right, the momentum has to be
conserved in this discussion.
So there needs to be some sort of momentum.
But, in the center-of-mass frame, that's not required.
So, in that frame, the momentum of all outgoing particles
can be 0.
And that's how we start the discussion here.
So, in this S prime frame--
here S prime is the center-of-mass frame--
the energy, the minimal energy required, is 2 times
the mass of the proton times gamma.
So here, two protons are colliding
with the same velocity.
And that's then equal to the energy
after this process, c squared times the sum of the masses,
the sum of the mass of the proton, the neutron,
and the charged pion.
And then you just have to solve this for gamma
to find gamma equal to 1.08 or beta in this frame of 0.37.
Note, this is the gamma, relativistic gamma,
or the velocity beta of the protons, two protons
in the center-of-mass frame.
So we're not quite there yet with our answer.
The answer then needs to be boosted back
into the laboratory frame.
And we have seen how we can do this for beta or velocities
in general.
We find beta in the laboratory frame is 2 times--
or just result, 0.37, over 1 plus 0.37
squared, which is 0.65.
That velocity, we can then take and calculate
the gamma factor of the proton in the fixed-target experiment.
All right, so we analyzed this situation
in the center-of-mass frame and then
did a Lorentz transformation by just looking at the velocity
into the fixed-target frame.
So this means now, numerically, that the proton colliding
with the proton at rest has a total energy of this one
proton of gamma m0 c squared, which is 0.32 times 938 MeV
over c--
MeV.
And so that results in 1.238 GeV.
But we're interested in the kinetic energy.
So the kinetic energy here is given
by gamma minus 1 m0 c squared, which is 300 MeV.
So we have to accelerate a proton to 300 MeV
in order to be able to have this process to occur.
All right, very similar problem now, but here
we want to produce anti-matter.
So we have a process of proton plus proton into three protons
and an antiproton.
Charge is conserved.
In the initial state, the charge was 2.
In the final stage, the charge was plus 2 as well.
OK, this works very similar as in the previous problem.
But what we want to do here is compare the fixed target
with symmetric collisions.
OK, so, again, the question is, what is the minimal energy
needed in order to produce antiprotons
in proton-proton collisions?
OK, so, exactly following the same procedure as before,
in the center-of-mass energy, the energy is 2 times
the mass of the protons times gamma times c squared.
And that's 4 times the mass of the proton.
OK, gamma prime, so the gamma factor
in the center-of-mass frame is 2.
Beta is 0.75.
And then we just do the very same thing again.
We calculate the velocity in the fixed-target frame.
And we find the velocity of beta of 0.96 and gamma of 3.57.
So, if we compare this now, we need a pair of 1 GeV--
remember, gamma minus 1 is the kinetic energy--
protons in a collider experiment or 2.57 GeV
protons in a fixed-target experiment.
OK, so you see that, in fixed-target experiment,
in order to produce new particles,
the energy has to be much larger, a factor of 2.5
here in this example, than a colliding experiment.
And that explains why we use collider experiments in order
to test the energy frontier, in order to produce the largest
possible energies.
And the LHC is one example where we
have proton-proton collisions in a circular ring where
those protons are brought together
in symmetrical collisions.