[Paper Review] Bremsstrahlung in alpha-decay: angular analysis of spectra

Sergei P. Maydanyuk, Sergei V. Belchikov|ArXiv.org|Apr 6, 2004

Nuclear physics research studies6 references3 citations

TL;DR

This paper presents a quantum mechanical model for calculating bremsstrahlung spectra in alpha-decay that explicitly accounts for the angle between the alpha-particle emission direction and photon emission. By incorporating the angle-dependent phase factor $ e^{-ikr\cos\beta} $ and using multipolar expansion, the method derives an analytical expression showing how the spectrum varies with emission angle, enabling direct comparison with experimental data collected at different angles.

ABSTRACT

A quantum mechanical method of calculation of bremsstrahlung spectra in alpha-decay of heavy nuclei with taking into account an angle between directions of the alpha-particle motion and the photon emission is presented. Dependence between the bremsstrahlung spectrum and the angle value is obtained in a simple analytical form. The method can be used for a comparative analysis of experimental data, obtained at different angles.

Motivation & Objective

• To develop a theoretical framework that accounts for the angular dependence of bremsstrahlung spectra in alpha-decay, addressing discrepancies in experimental data collected at different angles.
• To provide a non-stationary, quantum electrodynamical model that avoids semi-classical approximations and additional fitting parameters used in classical models.
• To enable comparative analysis of experimental bremsstrahlung spectra from different groups (e.g., Russian-Italian and Japanese teams) that report differing results for $^{210}\text{Po}$.
• To support the study of subbarrier bremsstrahlung and tunneling times in alpha-decay by offering a physically grounded, angle-sensitive spectral model.
• To improve the description of bremsstrahlung by emphasizing the dominant role of E1 multipole transitions in the angular dependence of the spectrum.

Proposed method

• Uses a multipolar expansion of the electromagnetic vector potential in quantum electrodynamics to model photon emission during alpha-decay.
• Derives the matrix element $ p(k_i, k_f) $ using the wave functions of the initial and final states, incorporating the radial and angular dependence of the system.
• Introduces the angle $ \beta $ between the alpha-particle motion and photon emission direction via the phase factor $ e^{-ikr\cos\beta} $, making the spectrum angularly dependent.
• Applies Clebsch-Gordan coefficients and spherical harmonics to couple the initial $ l=0, m=0 $ state with final $ l=1, m=\mu $ states, isolating the dominant E1 contribution.
• Expresses the photon emission probability $ dW/d\Omega_\nu $ in terms of $ |p(k_i, k_f)|^2 $, with explicit angular dependence through the exponential phase factor.
• Simplifies the angular integral using orthogonality of spherical harmonics and polarization vector decomposition into circular states $ \xi_{\pm1} $, yielding a compact analytical form.

Experimental results

Research questions

• RQ1 How does the bremsstrahlung spectrum in alpha-decay depend on the emission angle between the alpha-particle and the photon?
• RQ2 Can a quantum mechanical model without semi-classical approximations accurately describe the angular dependence of bremsstrahlung spectra?
• RQ3 Why do experimental spectra for $^{210}\text{Po}$ differ between groups measuring at $ 90^\circ $ and $ 25^\circ $, and can this be explained by angular dependence?
• RQ4 What is the dominant multipole contribution to bremsstrahlung in alpha-decay, and how does it vary with emission angle?
• RQ5 Can this model be used to extract tunneling times from experimental bremsstrahlung spectra via angular analysis?

Key findings

• The bremsstrahlung spectrum exhibits a clear dependence on the emission angle $ \beta $, with the phase factor $ e^{-ikr\cos\beta} $ introducing angular modulation into the spectrum.
• The E1 multipole transition provides the dominant contribution to the bremsstrahlung spectrum, as confirmed by the $ l=1 $ term in the multipolar expansion.
• The model yields a simple analytical expression for the angular dependence of the spectrum, making it suitable for direct comparison with experimental data collected at different angles.
• The method avoids additional fitting parameters by relying on first-principles quantum electrodynamics, improving theoretical consistency over classical models.
• The approach allows for a more accurate description of subbarrier bremsstrahlung and supports the determination of tunneling times in alpha-decay.
• The model improves upon previous quantum mechanical approaches by explicitly incorporating the angle between particle and photon emission, resolving inconsistencies in prior isotropic models.