Chapter 14 - Collision Theory
Section 14.1 - The Scattering Angle and Impact Parameter
Definition. The angle between incoming and outgoing velocities is the scattering angle \(\theta\). Note that \(\theta = 0\) corresponds to no scattering and \(\theta = \pi\) is a maximal value for \(\theta\).
Definition. The impact parameter \(b\) is the perpendicular distance from the projectile's incoming path to a parallel axis through the center of the target. \(b = 0\) implies a head-on collision. Note that \(\theta = \theta(b)\).
Section 14.2 - The Collision Cross Section
Definition. Consider multiple targets. Then, the target density \(n_{tar}\) is the number of targets per unit area as viewed from the incident direction.
If \(A\) is the total area of the target assembly, the total number of targets becomes \(n_{tar}A\).
Definition. The cross sectional area or cross-section is defined as \(\sigma = \pi R^2\), where \(R\) is the radius of one target as seen from the front.
Now, the total area of all targets is \(n_{tar}A\sigma\). Then, we can see the probability of a hit is simply the area of all targets divided by area, or \(n_{tar} \sigma\). Naturally, if we send a test beam with \(N_{inc}\) particles, we expect some fraction \(N_{sc}\) to be scattered. Then,
If we then let \(R_{inc} = N_{inc} / \Delta t\) for some time \(\Delta t\), we find the rate of incoming particles per unit time. We can do the same to \(N_{sc}\) to see that
Definition. Typical nuclear dimensions are about \(10^{-14} \text{m}\), so the cross-sections are measured in units of \(10^{-28} \text{m}^2\). This unit is known as \(1 \text{barn}\).
Section 14.3 - Generalizations of the Cross Section
Consider an incident sphere with radius \(R_1\) and a target sphere of radius \(R_2\). Then, we only care for \(b \leq R_1 + R_2\). We know that \(A = \pi(R_1 + R_2)^2\). Then, \(\sigma = A = \pi(R_1 + R_2)^2\) (as any interaction in the area results in a collision). So, \(N_{sc} = N_{inc} n_{tar} \sigma\)
Now, consider an example in which the particle may be captured or absorbed as well. Then, we can repeat the previous logic to see that \(N_{cap} = N_{inc} n_{tar} \sigma\). If a target can both deflect and capture particles, we see both \(N_{cap}\) and \(N_{sc}\), where \(\sigma_{cap} + \sigma_{sc} = \sigma_{tot}\).
Definition. \(\sigma_{cap}\) and \(\sigma_{sc}\) are the capture cross section and scattering cross section respectively. Additionally, we can define the ionization cross section \(\sigma_{ion}\) as the effective area of the target atom for an ionizing electron, and the fission cross section \(\sigma_{ris}\) as the effective area of a \(U^235\) nucleus for fission by neutron bombardment.
Definition. A collision is said to be elastic if the internal motion of the target is left unchanged. Otherwise, the collision is elastic.
Definition. The ground state of an atom is its lowest possible energy level. If an incident electron scatters elastically, it will leave the target in its ground state. Otherwise, atomic excitation will be seen.
Note that we can differentiate the types of collisions as \(\sigma_{sc} = \sigma_{el} + \sigma_{inel}\). Then, the total cross section \(\sigma_{tot} = \sigma_{sc} + \sigma_{cap} + \sigma_{ion}\), which is the total cross section for any interaction with the target particle.
Section 14.4 - The Differential Scattering Cross Section
Definition. For a cylinder on a circle with radius \(r\), with an arc length of \(s\), we define the angle \(\delta \theta = s/r\), which comes from the definition of radians.
Definition. For a cone on a sphere with radius \(r\) and area \(A\), we define the solid angle \(\delta \Omega = A / r^2\), with units called steradians (abbreviated as sr), and ranges from \(0\) to \(4\pi\) (due to the maximum surface area of a sphere). Note this works for any shape of cone (eg. cones with non-rectangular bases).
We will work in modified spherical polar coordinates, with the target on the origin and \(z = \rho\). For a cone in the range \(\theta\) to \(\theta + d\theta\) and \(\phi\) to \(\phi + d\phi\), that is, cones with a rectangular base, we see that based on \(A = r^2 \sin \theta d\theta + d\phi\),
We now define
Definition. Here, \(d\simga = \frac{d\sigma}{d\Omega} d\Omega\), where we define the differential scattering cross section as \(d\sigma / d\Omega\). This lets us say
We then see that
Section 14.5 - Calculating the Differential Cross Section
Consider the case of axial symmetry, that is, the differential cross section is independent of \(\phi\). Then, we can see that for \(\theta = \theta(b)\), we can consider particles approaching in the range \(b\) to \(b + db\). The annulus created by this has a cross-sectional area of \(d\sigma = 2\pi b db\).
We then see that the particles emerge between angles \(\theta\) and \(\theta + d\theta\) with solid angle \(d\Omega = 2\pi \sin \theta d\theta\)
We can thus compute the differential cross section as
Section 14.6 - Rutherford Scattering
Consider scattering electrons of off nuclei. Then, we know that
The rest of this section is complicated.
Section 14.7 - Cross Sections in Various Frames
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Section 14.8 - Relation of the CM and Lab Scattering Angles
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