In this paper, we introduce SpINR, a novel framework for volumetric reconstruction using Frequency-Modulated Continuous-Wave (FMCW) radar data. Traditional radar imaging techniques, such as backprojection, often assume ideal signal models and require dense aperture sampling, leading to limitations in resolution and generalization. To address these challenges, SpINR integrates a fully differentiable forward model that operates natively in the frequency domain with implicit neural representations (INRs). This integration leverages the linear relationship between beat frequency and scatterer distance inherent in FMCW radar systems, facilitating more efficient and accurate learning of scene geometry. Additionally, by computing outputs for only the relevant frequency bins, our forward model achieves greater computational efficiency compared to time-domain approaches that process the entire signal before transformation. Through extensive experiments, we demonstrate that SpINR significantly outperforms classical backprojection methods and existing learning-based approaches, achieving higher resolution and more accurate reconstructions of complex scenes. This work represents the first application of neural volumetic reconstruction in the radar domain, offering a promising direction for future research in radar-based imaging and perception systems.