2. Scattering, cross sections and decays
Learning Objective: understand scattering, and the role of form factors, being able to
calculate the form factor for simple charge distributions
2.1 Cross sections
The cross section is a measurable quantity related to a process in particle (and nuclear) physics.
By measuring cross sections theory can be compared to experiment. It is the job of the
experimental physicists to measure cross sections, and that of the theorists to use models to
calculate them. We will use this concept throughout the course, so I introduce it at very
beginning, rather than leaving it for later, as some textbooks do.
The idea of cross-section arises from the simplest model of a nucleus (or some other particle)
as a completely absorbing sphere of cross-sectional area σ. Consider a uniform beam of
J particles per second per unit area incident on a thin sheet of material, in which there are 𝑁𝑁
nuclei illuminated by the beam. The effective nuclear area for absorption is then σ𝑁𝑁.
The rate at which particles interact via a specific process indicated by the subscript 𝑟𝑟 is then
𝑊𝑊𝑟𝑟 = J 𝑁𝑁 𝜎𝜎𝑟𝑟
From an experimental point of view, the quantity 𝐽𝐽𝐽𝐽 is defined by the design parameters of the
experiment. This quantity is referred to as the “luminosity” 𝐿𝐿 of the experiment. 𝑊𝑊𝑟𝑟 is measured
experimentally, as the observed rate of interaction. From the equation above, we can measure
𝜎𝜎𝑟𝑟 , which is a quantity that holds the basic physics of the interaction and can be calculated from
theory.
The unit of cross-section used in nuclear and particle interactions is the barn, b, equal to
10–28 m2. In interactions between high energy particles, smaller units such as the millibarn
(mb=10–3 b) or even picobarn (pb =10–12 b) are often used.
Example 2.1: Alpha particle cross sections on gold
A gold atom has a nuclear radius of 7 fm. Calculate its geometrical cross section.
Compare with the cross sections for alpha particles of kinetic energy of 6 MeV scattering
above 30o, which is 157.3 b
Solution:
The geometrical cross section is simply 𝜋𝜋𝑅𝑅2 = 154 × 10−30 m2 = 1.54 b
2.2 Partial and differential cross sections
In most cases there are several possible reactions between the incident and target particles, and
the cross-section for each will be different. These individual cross-sections are known as
partial cross-sections, and their overall sum is the total cross-section. If 𝑟𝑟 indicate a specific
interaction process, then the total cross section is written as
𝜎𝜎 = � 𝜎𝜎𝑟𝑟
𝑟𝑟
15
, After a reaction or scattering has occurred the outgoing particles often have an anisotropic
distribution, with different rates observed in different angular directions. The observation of
the interaction rate as a function of the scattering angles (𝜃𝜃, 𝜙𝜙) is used to measure a differential
cross section. Then the number of particles scattered per second into solid angle dΩ at (θ,φ)
with respect to the incoming beam is written
𝑑𝑑𝜎𝜎𝑟𝑟 (𝜃𝜃 , 𝜙𝜙)
𝑑𝑑𝑑𝑑(𝜃𝜃, 𝜙𝜙) = 𝐽𝐽𝐽𝐽 𝑑𝑑𝑑𝑑
𝑑𝑑Ω
𝑑𝑑𝜎𝜎 (𝜃𝜃 ,𝜙𝜙)
where 𝑟𝑟 is the differential cross section for the process. The measurement of the
𝑑𝑑Ω
differential cross section will provide further information on the physics processes involved in
a specific interaction.
The partial cross-section for a specific process can be obtained by integrating the differential
cross-section over all solid angles.
2𝜋𝜋 𝜋𝜋
𝑑𝑑𝑑𝑑
𝜎𝜎 = � � sin(𝜃𝜃) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
0 0 𝑑𝑑Ω
Very often there is no dependence on φ and the integral reduces to
𝜋𝜋
𝑑𝑑𝑑𝑑
𝜎𝜎 = 2𝜋𝜋 � sin(𝜃𝜃) 𝑑𝑑𝑑𝑑
0 𝑑𝑑Ω
2.3 Scattering definitions
Consider the interaction
𝑎𝑎 + 𝐴𝐴 → 𝐴𝐴 + 𝑎𝑎
where a beam of particles of type a strikes a nucleon 𝐴𝐴. In this section we will discuss
the situation when the scattering is “elastic”. This means that both particles will emerge
from the collision as particle of the same time. In other words the nucleon and the
particle a are not excited by the collision.
The interaction rate per unit target particle (A), is 𝑊𝑊 = 𝐽𝐽𝐽𝐽, where 𝐽𝐽 is the flux of incident
particles (a) per unit incident area and unit time, as defined earlier.
Consider an incident beam with a density of 𝑛𝑛𝑎𝑎 particles per unit volume traveling with
a speed 𝑣𝑣𝑖𝑖 towards the target. The number of particles that cross an area S per unit
time Δ𝑡𝑡 is given by
𝑁𝑁𝑎𝑎 = 𝑛𝑛𝑎𝑎 𝑆𝑆 𝑣𝑣𝑖𝑖 Δ𝑡𝑡
Which is the number of particles contained in a solid with base of area S and height
𝑣𝑣𝑖𝑖 Δ𝑡𝑡. Therefore we can rewrite the flux as number of particles per unit area and unit
time as
𝑁𝑁𝑎𝑎
J=
𝑆𝑆 Δ𝑡𝑡
16
Learning Objective: understand scattering, and the role of form factors, being able to
calculate the form factor for simple charge distributions
2.1 Cross sections
The cross section is a measurable quantity related to a process in particle (and nuclear) physics.
By measuring cross sections theory can be compared to experiment. It is the job of the
experimental physicists to measure cross sections, and that of the theorists to use models to
calculate them. We will use this concept throughout the course, so I introduce it at very
beginning, rather than leaving it for later, as some textbooks do.
The idea of cross-section arises from the simplest model of a nucleus (or some other particle)
as a completely absorbing sphere of cross-sectional area σ. Consider a uniform beam of
J particles per second per unit area incident on a thin sheet of material, in which there are 𝑁𝑁
nuclei illuminated by the beam. The effective nuclear area for absorption is then σ𝑁𝑁.
The rate at which particles interact via a specific process indicated by the subscript 𝑟𝑟 is then
𝑊𝑊𝑟𝑟 = J 𝑁𝑁 𝜎𝜎𝑟𝑟
From an experimental point of view, the quantity 𝐽𝐽𝐽𝐽 is defined by the design parameters of the
experiment. This quantity is referred to as the “luminosity” 𝐿𝐿 of the experiment. 𝑊𝑊𝑟𝑟 is measured
experimentally, as the observed rate of interaction. From the equation above, we can measure
𝜎𝜎𝑟𝑟 , which is a quantity that holds the basic physics of the interaction and can be calculated from
theory.
The unit of cross-section used in nuclear and particle interactions is the barn, b, equal to
10–28 m2. In interactions between high energy particles, smaller units such as the millibarn
(mb=10–3 b) or even picobarn (pb =10–12 b) are often used.
Example 2.1: Alpha particle cross sections on gold
A gold atom has a nuclear radius of 7 fm. Calculate its geometrical cross section.
Compare with the cross sections for alpha particles of kinetic energy of 6 MeV scattering
above 30o, which is 157.3 b
Solution:
The geometrical cross section is simply 𝜋𝜋𝑅𝑅2 = 154 × 10−30 m2 = 1.54 b
2.2 Partial and differential cross sections
In most cases there are several possible reactions between the incident and target particles, and
the cross-section for each will be different. These individual cross-sections are known as
partial cross-sections, and their overall sum is the total cross-section. If 𝑟𝑟 indicate a specific
interaction process, then the total cross section is written as
𝜎𝜎 = � 𝜎𝜎𝑟𝑟
𝑟𝑟
15
, After a reaction or scattering has occurred the outgoing particles often have an anisotropic
distribution, with different rates observed in different angular directions. The observation of
the interaction rate as a function of the scattering angles (𝜃𝜃, 𝜙𝜙) is used to measure a differential
cross section. Then the number of particles scattered per second into solid angle dΩ at (θ,φ)
with respect to the incoming beam is written
𝑑𝑑𝜎𝜎𝑟𝑟 (𝜃𝜃 , 𝜙𝜙)
𝑑𝑑𝑑𝑑(𝜃𝜃, 𝜙𝜙) = 𝐽𝐽𝐽𝐽 𝑑𝑑𝑑𝑑
𝑑𝑑Ω
𝑑𝑑𝜎𝜎 (𝜃𝜃 ,𝜙𝜙)
where 𝑟𝑟 is the differential cross section for the process. The measurement of the
𝑑𝑑Ω
differential cross section will provide further information on the physics processes involved in
a specific interaction.
The partial cross-section for a specific process can be obtained by integrating the differential
cross-section over all solid angles.
2𝜋𝜋 𝜋𝜋
𝑑𝑑𝑑𝑑
𝜎𝜎 = � � sin(𝜃𝜃) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
0 0 𝑑𝑑Ω
Very often there is no dependence on φ and the integral reduces to
𝜋𝜋
𝑑𝑑𝑑𝑑
𝜎𝜎 = 2𝜋𝜋 � sin(𝜃𝜃) 𝑑𝑑𝑑𝑑
0 𝑑𝑑Ω
2.3 Scattering definitions
Consider the interaction
𝑎𝑎 + 𝐴𝐴 → 𝐴𝐴 + 𝑎𝑎
where a beam of particles of type a strikes a nucleon 𝐴𝐴. In this section we will discuss
the situation when the scattering is “elastic”. This means that both particles will emerge
from the collision as particle of the same time. In other words the nucleon and the
particle a are not excited by the collision.
The interaction rate per unit target particle (A), is 𝑊𝑊 = 𝐽𝐽𝐽𝐽, where 𝐽𝐽 is the flux of incident
particles (a) per unit incident area and unit time, as defined earlier.
Consider an incident beam with a density of 𝑛𝑛𝑎𝑎 particles per unit volume traveling with
a speed 𝑣𝑣𝑖𝑖 towards the target. The number of particles that cross an area S per unit
time Δ𝑡𝑡 is given by
𝑁𝑁𝑎𝑎 = 𝑛𝑛𝑎𝑎 𝑆𝑆 𝑣𝑣𝑖𝑖 Δ𝑡𝑡
Which is the number of particles contained in a solid with base of area S and height
𝑣𝑣𝑖𝑖 Δ𝑡𝑡. Therefore we can rewrite the flux as number of particles per unit area and unit
time as
𝑁𝑁𝑎𝑎
J=
𝑆𝑆 Δ𝑡𝑡
16
