10.3 The Cross Section Educational Proposal
Cross section is an important concept in classical physics, it is central in the study of
the interactions between elementary particles in nuclear physics at both low and high
energies in atomic collisions and becomes essential in quantum mechanics where it
is not possible to define the trajectory of a particle but only the probability of finding
it in a certain space. We start in preparing this approach long time ago (Corni et al.
1996), starting from one-dimensional collisions and the basic elements about plane
collisions for rigid bodies, using the conservation principles in classical physics to
analyze the different cases and showing at the same time which associations can be
derived from it.
We suggest to begin and go over a geometrical interpretation of the cross section,
which is consistent only with the classical case of rigid spheres, to introduce its
probabilistic meaning highlighting general aspects and showing how to relate
measurements for interpretations, independently from the traditional force/equation
model of motion/trajectory scheme.
In the elastic collision of two rigid spheres (1 and 2), with respect to the center of
mass reference frame, in the case of infinite rigidity normal collision of identical
spheres, we derive that the rebound of the two spheres is symmetrical with respect to
the line through their centers. The interaction during the collision is characterized by
the impulse vector I acting between the two spheres (i.e., from 2 to 1) and the
comparison between the initial and final states of the system gives us all possible
information about what happened during the collision. The scattering angle, θ, is
connected to the distance b between the lines on which the centers of the two spheres
were moving before the collision (impact parameter), as derived from the following
equation: b ¼ (R 1 + R 2 ) cos(θ/2). In more complex cases, this straightforward
procedure fails. The dependence of the scattering angle θ from the impact parameter
b becomes more difficult, when we replace sphere 2 with an irregular shaped object.
Another cause of unpredictability is the finite precision with which the impact
parameter is determined. Final states obtained with many collisions changing initial
conditions can be significantly different.
To understand how the scattering angle can offer information about the collision
or the geometry of the two bodies, we discuss the asymmetry of results of the
experiment of Fig. 10.1, where data on the scattering angles distributed over the
range (À160
, 160
) are collected for a certain number of impacts (400 events). This
relation between distribution asymmetry and asymmetrical shape gives a qualitative
example of how to extract the basic properties of the collision phenomenon from the
scattering angle distribution. In general, a complete characterization of the collision
requires the knowledge of the probability P i for any given measurement outcome S i .
For N tot observations carried out, these probabilities P i can be obtained from the
averages P i ¼ hN i /N tot i, where N i is the number of outcomes S i .
In this way, it is clear that the probability, which is the element which is more
related to the interaction, depends on the details of the measuring procedure. As in
106
M. Michelini
Cross section is an important concept in classical physics, it is central in the study of
the interactions between elementary particles in nuclear physics at both low and high
energies in atomic collisions and becomes essential in quantum mechanics where it
is not possible to define the trajectory of a particle but only the probability of finding
it in a certain space. We start in preparing this approach long time ago (Corni et al.
1996), starting from one-dimensional collisions and the basic elements about plane
collisions for rigid bodies, using the conservation principles in classical physics to
analyze the different cases and showing at the same time which associations can be
derived from it.
We suggest to begin and go over a geometrical interpretation of the cross section,
which is consistent only with the classical case of rigid spheres, to introduce its
probabilistic meaning highlighting general aspects and showing how to relate
measurements for interpretations, independently from the traditional force/equation
model of motion/trajectory scheme.
In the elastic collision of two rigid spheres (1 and 2), with respect to the center of
mass reference frame, in the case of infinite rigidity normal collision of identical
spheres, we derive that the rebound of the two spheres is symmetrical with respect to
the line through their centers. The interaction during the collision is characterized by
the impulse vector I acting between the two spheres (i.e., from 2 to 1) and the
comparison between the initial and final states of the system gives us all possible
information about what happened during the collision. The scattering angle, θ, is
connected to the distance b between the lines on which the centers of the two spheres
were moving before the collision (impact parameter), as derived from the following
equation: b ¼ (R 1 + R 2 ) cos(θ/2). In more complex cases, this straightforward
procedure fails. The dependence of the scattering angle θ from the impact parameter
b becomes more difficult, when we replace sphere 2 with an irregular shaped object.
Another cause of unpredictability is the finite precision with which the impact
parameter is determined. Final states obtained with many collisions changing initial
conditions can be significantly different.
To understand how the scattering angle can offer information about the collision
or the geometry of the two bodies, we discuss the asymmetry of results of the
experiment of Fig. 10.1, where data on the scattering angles distributed over the
range (À160
, 160
) are collected for a certain number of impacts (400 events). This
relation between distribution asymmetry and asymmetrical shape gives a qualitative
example of how to extract the basic properties of the collision phenomenon from the
scattering angle distribution. In general, a complete characterization of the collision
requires the knowledge of the probability P i for any given measurement outcome S i .
For N tot observations carried out, these probabilities P i can be obtained from the
averages P i ¼ hN i /N tot i, where N i is the number of outcomes S i .
In this way, it is clear that the probability, which is the element which is more
related to the interaction, depends on the details of the measuring procedure. As in
106
M. Michelini