8.1 Scattering cross-sections and phase shifts
4 min read•july 30, 2024
Scattering cross-sections and are key concepts in understanding particle interactions. They help us quantify how particles scatter off each other and reveal important information about the underlying forces at play.
These ideas are crucial for analyzing experimental data and making predictions in quantum mechanics. By studying cross-sections and phase shifts, we can uncover the nature of potentials and gain insights into fundamental particle properties.
Scattering Cross-Sections and Amplitudes
Defining Differential and Total Scattering Cross-Sections
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- The (dσ/dΩ) represents the probability of an incident particle scattering into a solid angle dΩ per unit time, normalized by the incident flux
- Integrate the differential over all solid angles to obtain the (σ), which gives the total probability of scattering in any direction
- Example: In a scattering experiment with alpha particles and gold foil (), the differential cross-section is larger for small scattering angles, indicating a higher probability of forward scattering
Relating Scattering Amplitudes to Cross-Sections
- The (f(θ, φ)) is a complex function describing the amplitude and phase of the scattered wave relative to the incident wave
- The differential scattering cross-section is related to the scattering amplitude by dσ/dΩ = |f(θ, φ)|^2
- The connects the total scattering cross-section to the imaginary part of the forward scattering amplitude: σ = (4π/k) Im[f(0)], where k is the wave number of the incident particle
- Example: In , the scattering amplitude depends on the spin states of the particles, leading to different cross-sections for singlet and triplet states
Phase Shifts in Scattering
Physical Interpretation of Phase Shifts
- Phase shifts (δₗ) represent the change in phase of the scattered wave relative to the incident wave for a given angular momentum quantum number ℓ
- A positive indicates an attractive interaction between the incident particle and the , while a negative phase shift suggests a repulsive interaction
- The phase shift measures the time delay experienced by the scattered wave due to its interaction with the scattering potential
- Example: In low-energy electron scattering from atoms (), the phase shift for the s-wave (ℓ = 0) can become negative, leading to a minimum in the total cross-section
Energy Dependence and Partial Wave Expansion
- The energy dependence of the phase shifts provides information about the nature of the scattering potential, such as the presence of bound states or resonances
- The phase shifts are related to the scattering amplitude through the : f(θ) = (1/k) Σ (2ℓ+1) exp(iδₗ) sin(δₗ) Pₗ(cos θ), where Pₗ are the Legendre polynomials
- Example: In resonant scattering (), the phase shift for a particular partial wave exhibits a rapid change of π radians near the energy, leading to a peak in the cross-section
Calculating Cross-Sections with Phase Shifts
Determining Phase Shifts for Central Potentials
- Calculate the scattering cross-section from the phase shifts using the partial wave expansion of the scattering amplitude
- For a central potential V(r), determine the phase shifts by solving the radial Schrödinger equation with appropriate boundary conditions
- The phase shifts for a hard-sphere potential of radius R are given by δₗ = -kR for ℓ = 0 and δₗ = 0 for ℓ > 0, resulting in a total scattering cross-section of σ = 4πR^2
- For a square-well potential of depth V₀ and radius R, obtain the phase shifts by matching the solutions of the radial Schrödinger equation inside and outside the well, leading to a transcendental equation for δₗ
Approximation Methods for Weak Potentials
- Use the to calculate the phase shifts for weak potentials, giving δₗ ≈ -(mk/ℏ^2) ∫₀^∞ r^2 V(r) jₗ(kr)^2 dr, where m is the reduced mass and jₗ are the
- Example: In the scattering of low-energy electrons by atoms, the Born approximation provides a good estimate of the phase shifts and cross-sections for weak atomic potentials
Asymptotic Behavior of Scattering Wave Functions
Asymptotic Form and Phase Shifts
- The scattering wave function (ψ(r, θ)) describes the state of the scattered particle and has a specific asymptotic form at large distances from the scattering center
- In the asymptotic region, express the scattering wave function as a sum of an incident plane wave and an outgoing spherical wave: ψ(r, θ) ≈ exp(ikz) + [f(θ)/r] exp(ikr)
- The phase shifts appear in the asymptotic form of the partial wave components of the scattering wave function: ψₗ(r) ≈ sin(kr - ℓπ/2 + δₗ) for r → ∞
- The phase shifts determine the asymptotic phase of the scattered wave relative to the incident wave, and their energy dependence relates to the time delay experienced by the scattered particle
Levinson Theorem and Bound States
- The connects the phase shifts at zero energy to the number of bound states in the potential: δₗ(0) = nₗπ, where nₗ is the number of bound states with angular momentum ℓ
- Example: In the scattering of neutrons by a square-well potential, the presence of a bound state in the s-wave (ℓ = 0) leads to a phase shift of π at zero energy, as predicted by the Levinson theorem
Key Terms to Review (20)
Born approximation: The Born approximation is a fundamental concept in quantum mechanics that simplifies the treatment of scattering problems by approximating the scattered wave function as a linear response to the incoming wave. This approach is particularly useful when the interaction potential is weak, allowing for an analytical solution to complex scattering processes, such as atomic transitions and interactions between particles. By using this approximation, one can relate physical observables like cross-sections and phase shifts to the potential governing the scattering event.
Breit-Wigner Resonance: Breit-Wigner Resonance refers to a phenomenon in quantum mechanics where the scattering cross-section exhibits a peak at a specific energy due to the presence of a short-lived intermediate state. This resonance occurs when the energy of the incoming particles matches the energy of the virtual particle created during scattering, leading to an enhancement in the probability of scattering events. The shape of the resonance is characterized by a Lorentzian function, which describes how the cross-section varies with energy.
Differential scattering cross-section: The differential scattering cross-section is a measure of the likelihood of scattering events occurring at specific angles and energy levels when a particle interacts with a target. It provides a detailed description of how the intensity of scattered particles varies as a function of scattering angle, allowing physicists to analyze scattering processes such as atomic transitions and collisions. This concept is crucial for understanding the underlying mechanisms of particle interactions and the corresponding changes in the states of the particles involved.
Elastic Scattering: Elastic scattering is a process in which particles collide and scatter off each other without any change in their kinetic energy. This means that the total energy of the system is conserved, although the direction of the particles may change. In the context of particle interactions, elastic scattering is important for understanding scattering cross-sections and phase shifts, as well as how different partial waves contribute to the overall scattering amplitudes.
Inelastic Scattering: Inelastic scattering occurs when particles collide and exchange energy, resulting in a change in the internal state of one or both of the particles involved. This process is crucial for understanding how energy is transferred in interactions and can lead to changes in momentum and kinetic energy. Inelastic scattering plays an important role in determining scattering cross-sections and can affect phase shifts, as well as provide insight into the properties of the interacting particles through partial wave analysis.
Levinson Theorem: Levinson Theorem is a fundamental result in quantum mechanics that relates the scattering phase shifts of a potential to the properties of bound states. It establishes a connection between the number of bound states in a quantum system and the phase shifts induced by the interaction potential, providing insights into scattering phenomena and resonance behavior.
Neutron-proton scattering: Neutron-proton scattering is the process in which a neutron collides with a proton, resulting in an exchange of energy and momentum. This interaction is fundamental in nuclear physics as it helps in understanding the strong force that holds atomic nuclei together, and it can reveal important information about the structure and behavior of nucleons.
Optical Theorem: The optical theorem is a fundamental principle in scattering theory that relates the total cross-section of a scattering process to the forward scattering amplitude. It states that the imaginary part of the forward scattering amplitude is directly proportional to the total cross-section, linking the probability of scattering to the phase shifts experienced by waves interacting with a potential. This theorem plays a crucial role in analyzing scattering processes and is particularly important in contexts involving phase shifts and approximations in quantum mechanics.
Partial wave expansion: Partial wave expansion is a mathematical technique used in quantum mechanics to describe scattering processes by breaking down a wave function into contributions from individual angular momentum states. This method simplifies the analysis of interactions by representing the total scattering amplitude as a sum over these partial waves, each characterized by its own phase shift. By relating the contributions of different angular momentum states, partial wave expansion connects to important concepts such as scattering cross-sections and scattering amplitudes.
Phase Shift: A phase shift is a change in the phase of a wave, typically measured in degrees or radians, that occurs when a wave interacts with a potential or obstacle during scattering processes. This shift can alter the interference patterns of the waves involved, and is crucial for understanding scattering phenomena, as it connects to how particles behave when they encounter each other or a barrier. The phase shift can significantly affect the scattering amplitude and ultimately influences the cross-section measurements used in quantum mechanics.
Phase Shifts: Phase shifts refer to the changes in the phase of a wave as it interacts with a potential barrier or scatterer. These shifts can affect how waves, such as those in quantum mechanics, are scattered and ultimately lead to observable phenomena like interference patterns. Understanding phase shifts is crucial in analyzing scattering processes and determining scattering cross-sections, which describe the likelihood of particles interacting in a certain way.
Potential Barrier: A potential barrier is a region in quantum mechanics where the potential energy of a particle is higher than its total energy, creating a 'wall' that can affect the particle's motion. This concept is essential in understanding how particles scatter and how they can tunnel through barriers that classical mechanics would deem insurmountable. The behavior of particles at potential barriers leads to various phenomena, such as reflection and transmission, which are crucial for analyzing scattering processes and understanding phase shifts.
Ramsauer-Townsend Effect: The Ramsauer-Townsend effect refers to the phenomenon where the probability of scattering of electrons by noble gas atoms, such as argon or neon, decreases at certain energies, leading to a minimum in the scattering cross-section. This effect is significant in understanding how electrons interact with matter and is closely linked to the concepts of scattering cross-sections and phase shifts, which describe how particles change direction after colliding.
Resonance: Resonance refers to the phenomenon where a system oscillates at maximum amplitude at certain frequencies, known as the resonant frequencies. This concept is particularly significant in scattering processes, where the interaction between particles can lead to pronounced effects in cross-sections and phase shifts. When a system reaches resonance, even a small input of energy can cause large responses, making it essential for understanding behaviors in quantum mechanics.
Rutherford Scattering: Rutherford scattering refers to the experimental observation of the scattering of alpha particles by a thin foil of gold, which provided critical evidence for the nuclear model of the atom. This phenomenon demonstrated that atoms have a small, dense nucleus that contains most of the mass, while the rest of the atom is mostly empty space. The insights gained from this scattering experiment laid the groundwork for understanding atomic structure and interactions at a fundamental level.
Scattering amplitude: Scattering amplitude is a complex quantity that describes the probability amplitude for a scattering process, which reflects how likely particles are to scatter off one another during interactions. It plays a crucial role in predicting the outcomes of scattering events, including atomic transitions, and provides insight into the underlying physics by connecting with measurable quantities like cross-sections and phase shifts.
Scattering cross-section: The scattering cross-section is a measure of the probability of scattering events occurring between particles, often expressed in terms of an effective area that quantifies how likely an incoming particle will interact with a target particle. This concept is crucial for understanding interactions at the quantum level, particularly when discussing atomic transitions and the influence of phase shifts on scattering processes. The larger the cross-section, the higher the likelihood that scattering will occur.
Scattering potential: Scattering potential refers to the potential energy landscape that an incoming particle interacts with during a scattering event. This interaction can lead to changes in the direction and energy of the particle, which is crucial for understanding the fundamental behavior of particles in quantum mechanics, particularly in how they scatter off targets. The characteristics of the scattering potential directly influence the calculations of scattering cross-sections and phase shifts, allowing physicists to predict outcomes of scattering processes.
Spherical Bessel Functions: Spherical Bessel functions are special functions that arise in solving problems involving spherical symmetry in quantum mechanics, particularly in the context of wave equations. They are closely related to the regular Bessel functions but are specifically adapted for three-dimensional spherical coordinates, playing a crucial role in understanding scattering phenomena and calculating phase shifts when particles interact with potential fields.
Total Scattering Cross-Section: The total scattering cross-section is a measure of the probability that a scattering event will occur between particles, effectively quantifying the target area that a particle presents to an incoming particle. It combines contributions from all possible scattering angles and mechanisms, offering insights into atomic transitions and interactions, as well as the underlying phase shifts that influence the scattering process.