17. Ion-Nuclear Interactions I — Scattering and Stopping Power Derivation, Ion Range

The paragraph introduces the topic of photon interactions, specifically the photoelectric effect, Compton scattering, and pair production. It emphasizes the importance of understanding these phenomena for interpreting detector spectra and knowing their cross-sections relative to photon energy and material composition. The professor also discusses the significance of the map of effects based on energy and atomic number (z). The context of the discussion is a review session for a lab that involves analyzing banana data. Additionally, the professor mentions the impact of listening to Russian presentations on their ability to spell technical terms and the structure of scientific presentations in Russian. The paragraph concludes with an invitation for students to contribute to the discussion about the cross-sections of the mentioned effects.

This section delves into the specifics of the cross-sections for pair production, Compton scattering, and the photoelectric effect. It discusses the conditions under which these interactions occur, such as the minimum energy required for pair production being 1.022 MeV. The professor outlines the dependence of these interactions on atomic number (z) and energy, providing insights into the likelihood of each effect based on these factors. The paragraph also touches on the concept of stopping power in the context of ion-nuclear interactions and introduces the mathematical derivation of stopping power, which will be covered in the following paragraphs.

The professor begins the derivation of the formula for stopping power, which is a measure of energy loss of a charged particle as it passes through a medium. The explanation involves the concept of the impact parameter (b), which describes the distance of closest approach between the charged particle and an electron in the medium. The paragraph discusses the forces involved in the interaction and how they result in a deflection of the particle's trajectory. The integral of the force over the distance of interaction is identified as the change in momentum, which is related to the stopping power. The professor also addresses the assumption that the velocity of the particle can be treated as constant for the purpose of this derivation.

The paragraph continues the mathematical derivation of stopping power, integrating the force exerted on the charged particle over its path through the medium. The professor simplifies the integral by separating the constants from the variable and applying a variable change to facilitate the integration. The resulting expression for stopping power is then related to the change in momentum and subsequently to the change in kinetic energy of the particle. The discussion highlights the assumptions made in the derivation, such as the negligible change in the particle's velocity due to a single collision.

The professor transitions from the abstract derivation to considering the charged particle's interaction with a real material, taking into account the electron density of the medium. The paragraph discusses how the stopping power is influenced by the number density of atoms in the material, the charge per atom (Z), and the volume of the cylindrical shell through which the particle passes. The integral for the energy loss is then expressed in terms of these material properties, leading to a more comprehensive understanding of how stopping power is affected by the medium's composition and structure.

The paragraph addresses the limitations of the stopping power formula, particularly at low energies where the natural logarithm term in the formula could become negative, leading to unphysical results. The professor discusses the De Broglie wavelength of the electron, which provides a minimum impact parameter (b min) that cannot be integrated over due to the uncertainty principle. The maximum impact parameter (b max) is associated with the mean ionization potential of the material. The resulting stopping power formula is then expressed in terms of these limits, and its physical implications are explored, including the behavior of the formula at low and high energies.

The professor describes the graphical representation of stopping power as a function of kinetic energy, highlighting the characteristic curve that increases with energy and includes a maximum. The concept of the range of a particle is introduced, which is the distance it travels before coming to a stop. The relationship between stopping power and range is explored, with the professor explaining that the range increases with the square of the particle's energy. The paragraph also touches on the practical applications of this knowledge, such as in medical treatments like proton therapy.

The discussion shifts to the application of stopping power in proton therapy for cancer treatment. The professor contrasts the uniform dose distribution of X-ray therapy with the localized energy deposition of proton therapy, which minimizes damage to surrounding healthy tissue. The paragraph explains how the energy of the proton beam can be adjusted to maximize the energy release at the tumor site, resulting in a higher effectiveness of treatment. The question of how to change the range of protons without altering their energy is posed, leading to a consideration of various methods to modulate the proton beam's interaction with the patient's tissue.

The final paragraph involves a Q&A session where the professor addresses questions from students regarding the stopping power concept. Topics include the role of the charge per atom (big Z) and the charge on the particle (little z), the absence of the particle's mass in the stopping power formula, and the implications of these factors on the energy transfer process during particle interactions with the medium. The professor emphasizes the importance of understanding the derivation and the physical meaning behind the stopping power formula, as well as its limitations and areas of applicability.