Experimenty v c

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Experimenty v cásticové fyzice
Jirí Dolejší, Olga Kotrbová, Charles University
Prague
We have standard ways to discover properties of
things around us and to look to the inside of
objects (like the small alarm clock on the
photograph). But these methods may not be
applicable to atom and its components we do not
have any sufficiently small screwdriver, we even
cannot look at it with sufficient resolution.
With the best current microscopes we can only see
the individual atoms like on this picture from
the tunneling microscope, but not the inside of
them.
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But already at the beginning of the 20th century
E. Rutherford and his collaborators developed a
novel method to study the inside of atoms. They
shot a-particles towards a thin gold foil and
discovered that the idea best corresponding to
the experimental data is the idea of atom being
almost empty with small heavy nucleus and
electrons flying around. Today gold nuclei are
collided at RHIC (Relativistic Heavy Ion
Collider) today and we expect to learn more about
the matter at collision.
How methods based on collisons can work? Let us
try!
A first attempt I will try to use collision
instead of a screwdriver
I succeeded in getting inside! But not
sufficiently deep into the structure. Maybe I
need higher energy Probably it would be better
to start with a simpler object
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Atoms, nuclei, particles Implicitly we look at
them as small balls. Instead of playing with
balls I suggest to flick coins and study their
collisions. Put one coin on a smooth surface,
flick another coin against it, observe the
result.
Disclaimer The collisions of euro and dollar
have neither political nor economical meaning.
After playing a while with different coins you
will get some experience. Look at the following
situations and decide, which of the coins is
heavier (arrows show velocities).
B
C
A
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I hope you answers are correct both coins have
the same mass in B, blue one is lighter in A and
heavier in C. You can repeat the experiment with
coins and doublecoins glued together with
doublesided tape
What about describing the scattering of coins in
a manner usual in mechanics? Relevant variables
are the masses m1, m2, velocities before and
after collision v1 , v2 and v1', v2. Important
variables are energy E and momentum p.
v1'
m1
v1
Both energy and momentum in an isolated system
are conserved
a
b
The red coin is at rest at the beginning
m2
v2'
v1 0
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Let us work only with momenta energy
conservation... and conservation of both
momentum components
Let us play with momenta We would like to get
rid of p2 and b and it is appealing to use for
that purpose the relation
We just rewrite the equations, make a square of
each one and sum them
we can insert into energy equation
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The last equation
is a the quadratic equation for
The discriminant is
and solutions
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Let us remind that we are looking for real
nonnegative solutions. The simplest case is the
case of equal masses m1 m2
So in collision of two coins with equal masses
the coin cannot be scattered backwards. It either
continues forwards, maybe deflected, or stops. We
can see from the conservation equations that if
the coin stops, the target coin takes over the
whole energy and momentum. Try it on a
billiard!!!
Instead of continuing the detailed discussion, we
will plot the depen-dence of momentum after
collision on the scattering angle
1projectile 2target
Only lighter particle can be scattered backwards
The deflection of the heavier particle is limited
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Although we have played with coins, we used only
the most general mechanical laws. Our results
holds for collisions of any objects, including
subatomic particles. The only complication is
related to the pleasant fact, that particles can
be accelerated to speeds close to the speed of
light. The effects well described or predicted by
the special theory of relativity appear -
particles have bigger mass, unstable particles
live longer. We can quite easily modify our
calculation
Instead of counting kinetic energy
we should deal with total energy
we will insert into energy conservation
equation (all masses here are rest masses)
The calculation needs more time, more paper and
more patience. We will only show you the result
of the simplest case m1 m2 m
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The difference of nonrelativistic and
relativistic calculation is visible on the graph
Nonrelativistic case
The quantitative understanding of kinematics of
collisions enables us to compare masses of coins
or particles by just colliding them.
First success! Collisions are good at least for
something, not only for destroying everything ...
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The formula for relativistic energy can be
rewritten in the form
Energy and momentum have different values in
different reference frames - a bottle in my hand
in the train has no kinetic energy with reference
to the train but it may have quite significant
energy referred to the ground. But the special
expression above made from E and p equals always
(independently of the reference frame) the square
of the particle rest mass times c4, i.e. is
constant. This feature offers a surprisingly
simple way to measure the mass of an unstable
particle
Measure energies and momenta of decay products
and , then calculate
Unstable particle with unknown mass
You have got the mass M !
You will find words like energy-momentum
fourvector, invariant mass etc. in advanced
textbooks. They refer to the same things as
above, you can learn more ...
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To measure the masses of particles is clearly not
enough. How to get some deeper insight, how to
understand structure, interactions etc.? Maybe
the way is to look what everything can happen and
how probably. We can start again with our
macroworld.
What is the probability that I will catch a ball
shot at me? The best solution to answer this
question is to make an experiment. Being exposed
to randomly placed moderate shots I caught all
the yellow shots. After several repetition of
this experiment I succeeded catching everything
inside the area with the yellow boundary.
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My ability to catch ball shots is characterized
by the yellow area - it is about 2 m2 and it
means that from the flow of shots with a
density 10 shots per square meter I expect to
have 20 catches.
My catching ability is characterized by an
effective area which I can cover. Physicists
use a special name for this quantity - the cross
section and typically use the letter s for
it. If the flow of incoming particles with the
density j hits the target, then the number of
interesting events N with the cross section s is
The standard unit for the cross section is 1
barn 1 b 10-28 m2
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In terms of cross-section we can express a lot of
information. For example the shots I am catching
can vary in hardness - in energy of the ball. I
can easily catch the slow balls but I will
probably try to hide myself from hard shots.
So for hard shots my cross section of catching
will be zero and the cross section of a ball
hitting me will be close to the area of my
silhouette ( the band around accounting for the
diameter of the ball). One may consider the
special (or partial) cross section for a ball
breaking my glasses etc. All the possible
processes can be summarized in the total cross
section.
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sball-me
The energy dependence of different cross-section
relating to the interaction of me and the ball
could look like this graph (and betray a lot
about me )
At this energy I try to avoid ball hit
At this energy I am frozen from the fear
total
At this energy I am catching best
catch
hit
injury
death
Energy of the ball in apropriate units
The cross-sections for proton-proton interactions
are displayed on this plot
Elastic cross section - colliding particles stay
intact, they only change the direction of their
flight
inelastic
At this energy the colliding protons have enough
energy to create a new particle - pion. This is
one example of a process contributing to the
inelastic cross section stotal - selastic
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To be continued