فرمول‌بندی اشباع در پدیده مالتی‌پکتینگ با رویکردی تحلیلی

برای مطالعه دقیق پدیده مالتی‌پکتینگ، باید اثر بار فضایی را در نظر گرفت. شبیه‌سازی این پدیده با در نظر گرفتن اثر بار فضایی از نظر محاسباتی هزینه‌بر است. در این مقاله، روشی برای فرمول‌بندی رشد ذرات در فرآیند مالتی‌پکتینگ ارائه شده است که می‌تواند در شبیه‌سازی مالتی‌پکتینگ با اثرات بار فضایی به کاهش هزینه‌های محاسباتی در مطالعات آتی این پدیده کمک کند. نتایج نشان می‌دهد که در مطالعه‌ی مالتی‌پکتینگ، استفاده از تابع لجستیک برای پیش‌بینی تعداد ذرات در اشباع مالتی‌پکتینگ توسط بارفضایی، تا 75٪ زمان شبیه‌سازی را کاهش می‌دهد. همچنین نتایج نشان می‌دهد که محاسبه‌یِ تعداد ذرات در حالت اشباع با تابع پیشنهادی با نتایج شبیه‌سازی هماهنگی خوبی دارد. در این پژوهش هر دو نوع مالتی‌پکتینگ تک سطحی و دوسطحی با شرایط فیزیکی مختلف شامل سطوح مختلف میدان رادیوفرکانسی و بسامدهای معمول مورد استفاده در ساختارهای رادیوفرکانسی  مورد مطالعه قرار گرفته و نتایج بدست آمده قابل تعمیم به بسیاری ساختارهای رادیوفرکانسی  کاربردی از جمله موجبرها، کاواک‌ها و پنجره های رادیوفرکانسی است.

formulation of saturation in multipacting phenomenon using analytical approach

1. introductionmultipacting is a resonant electron multiplication phenomenon that occurs in rf vacuum structures. it begins when an initial electron, accelerated by the oscillating rf electric field, collides with a surface and generates secondary electrons. if the secondary electron yield (sey) exceeds unity, these emitted electrons can be further accelerated and continue to impact surfaces, initiating an avalanche of electron multiplication. this process can cause significant degradation in the performance of rf components such as accelerating cavities, waveguides, and rf windows. the sey is a function of both the incident electron energy and the surface material. for most materials, sey becomes greater than one within a certain energy range. when the conditions fall within this range, sustained multipacting becomes possible. however, as the electron population grows, mutual repulsion between electrons—known as the space charge effect—gradually limits the growth rate. eventually, this leads to a saturation state where the number of electrons reaches a steady value. accurate prediction of this saturation behavior is essential for the design and optimization of rf devices. while full-scale numerical simulations using particle-in-cell (pic) methods—such as those implemented in cst studio suite—can accurately depict the effects of space charge, they are computationally expensive. this is primarily due to the need to resolve particle-particle interactions and maintain fine temporal and spatial resolution. in this study, we propose a simplified analytical approach to model the saturation behavior of multipacting. the method is based on the logistic growth function, which effectively describes the nonlinear evolution of the electron population under the influence of space charge. by using this model, it is possible to estimate the saturation characteristics without relying on time-consuming pic simulations. the approach is validated for both single-sided and two-sided multipacting scenarios, under various rf field amplitudes and frequencies. due to its general formulation, the model can be applied to a broad class of rf systems, including cavities, waveguides, and rf windows. 2. methodologyto address the high computational cost of simulating multipacting with space charge effects, we proposed an analytical approach based on a logistic growth model. first, we examined the temporal evolution of particle number in single- and two-sided multipacting configurations using cst particle-in-cell (pic) simulations. key particle growth features—such as the saturation plateau and inflection point—were extracted by analyzing the first and second derivatives of the growth curves. based on these analyses, a logistic function was formulated to represent particle population dynamics. this model was then validated across various material properties, rf field amplitudes, and frequencies commonly used in rf structures. to ensure robustness, raw simulation data were post-processed using local averaging techniques to minimize numerical fluctuations and highlight the fundamental growth trend. 3. results and discussionour results indicate that the particle growth behavior in multipacting with space charge closely follows a logistic profile. the proposed logistic function successfully predicted both the growth rate and the saturation particle number (ns) with high accuracy, showing relative errors as low as 0.05% in two-sided configurations and under 10% in single-sided cases. by identifying the critical turning point (tc), where the particle growth rate peaks, we demonstrated that ns can be accurately estimated without requiring a full simulation to saturation. this method led to a significant reduction in simulation time, achieving up to 75%-time savings across tested models. for example, in one single-sided case, the runtime dropped from 40 minutes to 10 minutes. the good agreement between the analytical predictions and cst simulation results confirms that the logistic function effectively shows the physical behavior of the system. furthermore, the formulation is shown to be broadly applicable to a variety of rf components, including waveguides, cavities, and rf windows. 4. conclusionthis study introduces a logistic-based analytical model to predict particle saturation in multipacting phenomena under the influence of space charge. the proposed formulation significantly reduces simulation time while maintaining high prediction accuracy. the method was successfully applied to multiple scenarios with different rf field strengths, frequencies, and material properties. future research may explore the application of this method to more complex 3d geometries and transient field conditions, as well as integrate it with machine learning techniques for real-time prediction and control of multipacting in practical systems.