Show that the intensities of a molecular beam before and after passing through a chamber of length / containing inert scattering atoms are related by \(I=\mathrm{I}_{0} \mathrm{e}^{-\mathrm{N} \sigma \mathrm{l}}\), where \(\sigma\) is the collision cross-section and \(N\) the number density of scattering atoms.

The concept of a collision cross-section is central to understanding interactions between particles, such as those in a molecular beam experiment. Imagine each scattering atom as a target with a certain area; this area is the collision cross-section, denoted as \( \sigma \). It represents the effective area that a moving molecule can collide with. The larger the collision cross-section, the more likely collisions are to occur. A high \( \sigma \) value indicates that molecules are more effective at scattering other particles.

In practical terms, collision cross-sections are quintessential in predicting how particles scatter when they encounter one another in a controlled environment, such as a gas in a chamber or collider experiments in particle physics. Different factors, such as the energy of the particles and the intermolecular forces, can affect the size of the cross-section, and therefore the number of collisions happening in a system.

Number density, expressed as \( N \) in scientific equations, is a crucial concept describing how crowded a space is with particles. Specified as the number of particles per unit volume, number density is a key factor in calculating the frequency of interactions, like collisions, within a given volume of material. The higher the number density, the more particles there are in a space, and consequently, the higher the chance for collisions.

Application in Molecular Beam Scattering

In the context of molecular beam scattering, the number density of scattering atoms dictates the number of potential collisions. The product of the number density \( N \) and the collision cross-section \( \sigma \) gives us insight into the likelihood of beam particles being deflected or absorbed as they travel through a scattering medium. By multiplying these two values, we can assess the impact on beam intensity over a certain distance.

Differential equations are mathematical tools used to describe changes in systems, often representing physical phenomena, by showing how a function's rate of change depends on the function itself. These equations are essential in various branches of science and engineering, encompassing everything from dynamics in physics to the rates of reaction in chemistry.

In molecular beam scattering, a differential equation is formulated to reflect the rate at which beam intensity decreases as the beam progresses through a scattering medium.

Link to Scattering

The differential equation \( \frac{dI}{dx} = -IN\sigma \) is obtained by considering the change of intensity \( I \) with respect to a small change in position \( x \). Solving this equation provides us with a mathematical relationship between the initial and final intensities of the beam, describing the beam's attenuation profile as it interacts with the scattering atoms.

The Beer-Lambert law is a widely used relationship that describes how the absorption of light is dependent on the properties of the material through which the light is travelling. It states that the absorbance of light passing through a medium is directly proportional to the concentration of the absorbing substance and the path length of the medium.

This principle is perfectly analogous to the intensity decrease of a molecular beam in our context. In a similar fashion, the Beer-Lambert law is expressed mathematically as \( I = I_0 e^{-N\sigma l} \), showing that the transmitted intensity \( I \) decreases exponentially with the product of number density \( N \) and collision cross-section \( \sigma \), as well as the length of the medium \( l \). Thus, we see this law exemplified in the relationship between the initial \( I_0 \) intensity and final intensity \( I \) of a molecular beam that has passed through a chamber containing inert scattering atoms.