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MARKUS KLUTE: Welcome back
to 8.20, special relativity.

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In the previous
section, we have seen

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how we can look at energy
and momentum of particles

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in a decay.

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Here we now want to, in
collisions of particles,

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create new particles.

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The example, the
first example here,

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is the collision of two protons
to create a proton, a neutron,

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and a charged pion.

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The masses are given there.

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So the question now is,
what is the minimal energy

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needed in order for
this process to occur

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in a fixed-target experiment?

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Fixed-target experiment is
we have an accelerated proton

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and another proton at rest.

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This might just be a hydrogen
target just sitting there.

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So the question is,
how much energy--

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how much do we
have to accelerate

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the proton for this
process to be possible?

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Now, again, stop the video,
and try to work this out.

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The important part
here is to realize

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that minimal energy here
means that, after the decay

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or the decay after
the process occurred,

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all the new particles
need to be addressed.

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That is when the process
requires minimal energy.

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So, instead of analyzing
this in the laboratory frame,

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we want to analyze this in
the center-of-mass frame.

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All right, the
momentum has to be

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conserved in this discussion.

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So there needs to be
some sort of momentum.

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But, in the center-of-mass
frame, that's not required.

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So, in that frame, the momentum
of all outgoing particles

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can be 0.

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And that's how we start
the discussion here.

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So, in this S prime frame--

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here S prime is the
center-of-mass frame--

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the energy, the minimal
energy required, is 2 times

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the mass of the
proton times gamma.

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So here, two protons
are colliding

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with the same velocity.

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And that's then
equal to the energy

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after this process, c squared
times the sum of the masses,

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the sum of the mass of
the proton, the neutron,

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and the charged pion.

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And then you just have
to solve this for gamma

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to find gamma equal to 1.08
or beta in this frame of 0.37.

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Note, this is the gamma,
relativistic gamma,

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or the velocity beta of
the protons, two protons

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in the center-of-mass frame.

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So we're not quite there
yet with our answer.

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The answer then needs
to be boosted back

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into the laboratory frame.

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And we have seen how we can
do this for beta or velocities

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in general.

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We find beta in the
laboratory frame is 2 times--

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or just result, 0.37,
over 1 plus 0.37

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squared, which is 0.65.

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That velocity, we can
then take and calculate

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the gamma factor of the proton
in the fixed-target experiment.

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All right, so we
analyzed this situation

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in the center-of-mass
frame and then

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did a Lorentz transformation
by just looking at the velocity

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into the fixed-target frame.

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So this means now, numerically,
that the proton colliding

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with the proton at rest has
a total energy of this one

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proton of gamma m0 c squared,
which is 0.32 times 938 MeV

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over c--

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MeV.

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And so that results
in 1.238 GeV.

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But we're interested
in the kinetic energy.

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So the kinetic
energy here is given

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by gamma minus 1 m0 c
squared, which is 300 MeV.

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So we have to accelerate
a proton to 300 MeV

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in order to be able to
have this process to occur.

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All right, very similar
problem now, but here

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we want to produce anti-matter.

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So we have a process of proton
plus proton into three protons

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and an antiproton.

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Charge is conserved.

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In the initial state,
the charge was 2.

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In the final stage, the
charge was plus 2 as well.

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OK, this works very similar
as in the previous problem.

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But what we want to do here
is compare the fixed target

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with symmetric collisions.

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OK, so, again, the question
is, what is the minimal energy

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needed in order to
produce antiprotons

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in proton-proton collisions?

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OK, so, exactly following
the same procedure as before,

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in the center-of-mass
energy, the energy is 2 times

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the mass of the protons
times gamma times c squared.

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And that's 4 times the
mass of the proton.

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OK, gamma prime,
so the gamma factor

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in the center-of-mass
frame is 2.

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Beta is 0.75.

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And then we just do the
very same thing again.

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We calculate the velocity
in the fixed-target frame.

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And we find the velocity of
beta of 0.96 and gamma of 3.57.

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So, if we compare this now,
we need a pair of 1 GeV--

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remember, gamma minus 1
is the kinetic energy--

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protons in a collider
experiment or 2.57 GeV

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protons in a
fixed-target experiment.

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OK, so you see that, in
fixed-target experiment,

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in order to produce
new particles,

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the energy has to be much
larger, a factor of 2.5

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here in this example, than
a colliding experiment.

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And that explains why we use
collider experiments in order

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to test the energy frontier,
in order to produce the largest

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possible energies.

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And the LHC is one
example where we

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have proton-proton collisions
in a circular ring where

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those protons are
brought together

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in symmetrical collisions.