Reconstructed proton charge structure: Insights from Patterson function analysis

Reconstructed proton charge structure: Insights from Patterson function analysis | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Reconstructed proton charge structure: Insights from Patterson function analysis Pui Sum Yuen This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7435823/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We present an empirical, approximate, and graphical method to reconstruct the proton’s internal charge structure using Patterson function analysis. The resulting model is non-negative and compact, with a radius of half-maximum Patterson charge density of 0.09 fm—identical to the experimental value. The radius of half-maximum charge density of the reconstructed proton structure is 0.051 fm, which lies within the upper bound for non-negative radial Patterson charge density. This approach offers a novel perspective on proton structure and complements existing theoretical models. Nuclear Physics Proton structure Patterson function form factor deconvolution nuclear physics charge density distribution Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 1. Introduction In Yuen (Preprint 2025 I), we use the Patterson function of the proton charge density to demonstrate that the proton charge density is non-negative. From this, we establish a lower bound for the proton radius at 0.09 fm, offering a potential resolution to the proton radius puzzle. Similarly, in Yuen (Preprint 2025 II), the Patterson function of the proton magnetization density reveals that the magnetization density is also non-negative, with a magnetic radius limit of 0.08 fm—an order of magnitude smaller than values commonly reported in the literature. For the neutron, another fundamental nucleon, the Patterson function of its charge density shows that the distribution cannot be classified as simply non-negative or non-positive (Yuen, Preprint 2025 III). A model consisting of a hard-core positive charge center surrounded by a soft negative shell may apply to the neutron as well. Our findings suggest that the positive core may be more compact than previously proposed. Although the neutron magnetization density has received less attention, Yuen (Preprint 2025 IV) investigates its magnetic structure. The results mirror those of the neutron charge density, indicating a hard positive core and a soft negative shell. With these studies (Yuen, Preprint 2025 I–IV), the first stage of our project—examining the electric and magnetic structures of the proton and neutron via Patterson function analysis—is complete. The next stage involves reconstructing the explicit charge and magnetization density functions. In this article, we present the results for what is likely the most critical of these: the proton charge density function. Determining the proton charge density from its Patterson counterpart is a deconvolution problem, constrained by the requirement of non-negativity. Rather than relying solely on mathematical formalism, we adopt an empirical, approximate, and graphical approach to explore the proton’s internal structure. 2. Proton charge distribution in the literature Several studies have investigated the proton’s charge distribution using diverse theoretical and experimental approaches. Table 1 summarizes key contributions. Table 1 Reported proton charge distribution. Study Method / Approach Kelly (2002) Sachs form factors LaViolette (2006) A Turing Wave Pattern Sick (2012) Electron scattering data Csörgõ et al. (2020) Lévy imaging of pp elastic scattering Gramolin & Russell (2021) Elastic electron-proton scattering, transverse charge density Freese & Miller (2023) Light front synchronization Won et al. (2024) Energy momentum tensor form factors 3. Methodology In Yuen (Preprint 2025 I), we found that the half-maximum of the proton Patterson charge density occurs at r = 0.09 fm, and the first zero appears at r = 0.37 fm. Based on this observation, we hypothesize that the actual proton charge density may resemble the Patterson profile—characterized by a sharp central peak followed by a gradual decline to zero. If this assumption holds, the Patterson charge distribution can serve as a suitable trial object. The first image is constructed by enclosing the Patterson charge density within a sphere of radius ≤ 0.37 fm. The radial Patterson charge density derived from this image is shown in Figure 1 , with a half-maximum radius of 0.15 fm. To align the half-maximum positions with the original Patterson profile, we scale the coordinates of the first image by a factor of 0.09 / 0.15, yielding a smaller second image . The radial Patterson charge density of the second image is presented in Figure 2 . It is non-negative, and its half-maximum radius matches the original value of 0.09 fm. Figure 3 compares the normalized radial Patterson charge density of the proton (Yuen, (Preprint 2025 I), Figures 1 and 4) with that of the second image. For r ≦ 0.12 fm, the two curves overlap closely. We therefore adopt the second image as our reconstructed proton charge density. The charge density of the second image has been deposited in Zenodo. Figure 4 displays the radial charge density ρ(r) derived from the second image, with a half-maximum radius of 0.051 fm. Following the methodology of Kelly (2002), we present the reconstructed radial profile r²ρ(r) in Figure 5 , which highlights the spatial concentration of the proton’s charge density. For calculations at z = 0 fm, a grid resolution of 100 × 100 is used, as shown in Figure 6 . At a resolution of 100 × 100 × 100, the minimum charge density within the unit cell is −0.0991 fm⁻³, while the maximum reaches 1338 fm⁻³. 4. Discussion and conclusion In Section 3 , we constructed a second image whose radial Patterson charge density closely matches the experimental profile of the proton (Yuen, (Preprint 2025 I)), as shown in Figs. 2 and 3 . Both exhibit a radius of half-maximum Patterson charge density of 0.09 fm. This alignment supports the selection of the second image as a valid representation of the proton’s internal charge structure. The radial charge density derived from this second image (Fig. 4 ) yields a half-maximum radius of 0.051 fm, which is smaller than the 0.09 fm Patterson half-maximum radius. This result adheres to the upper bound established in Yuen, (Preprint 2025 I) for non-negative radial Patterson charge densities. Although this reconstructed radius is an order of magnitude smaller than the conventional proton charge radius range of 0.84–0.90 fm (Yuen, (Preprint 2025 I)), it reflects the compactness of the central charge concentration. Figure 6 further illustrates the charge density distribution at z = 0 fm, revealing a sharp central peak followed by a rapid decline to near-zero values. This core-dominated structure is consistent with Rutherford’s early findings and aligns with most models listed in Table 1 , with the exception of Csörgő et al. (2020), whose approach emphasizes spatial hollowness. Together, Figs. 4 and 6 support the hypothesis of a highly compact, positive core within the proton, reconstructed through empirical Patterson function analysis. This empirical Patterson function approach complements existing theoretical frameworks and may inform future studies of nucleon substructure. Declarations Data availability The charge density of the second image, structure factor magnitudes and phases, and the charge density at z=0 fm are available on Zenodo at https://doi.org/10.5281/zenodo.16849345 Funding information The author has not received funding from any organization. Competing interests The author declares no competing interests. References Csörgõ, T., Pasechnik, R. & Ster, A. Proton structure and hollowness from Lévy imaging of pp elastic scattering, Eur. Phys. J. C 80 , 126 (2020). Freese, A. & Miller, G. A. Light front synchronization and rest frame densities of the proton: Electromagnetic densities, Phys. Rev. D 107 , 074036 (2023) Gramolin, A. V. & Russell, R. L. Transverse charge density and the radius of the proton, Phys. Rev. D , 105 , 054004 (2022). Kelly, J. J. Nucleon Charge and Magnetization Densities from Sachs Form Factors, Phys. Rev. C , 66 , 065203 (2002). LaViolette, P. A. The Electric Charge and Magnetization Distribution of the Nucleon: A Turing Wave Pattern? vixra.org, Jan. 2006. Sick, I. Problems with proton radii, Prog. Part. Nucl. Phys. 67 , 473-478 (2012). Won, H.-Y., Kim, H.-C. & Kim, J.-Y. Flavor structure of the energy momentum tensor form factors of the proton. Phys. Lett. B 850 , 138489 (2024). Yuen, P. S. Proton charge structure revisited: Insights from Patterson function analysis. (Preprint 2025 I) OSF Preprintshttps://doi.org/10.31219/osf.io/6jsd8_v1 Notable result: The radius at half-maximum Patterson charge density is 0.09 fm. Yuen, P. S. Proton magnetic structure revisited: Insights from Patterson function analysis. (Preprint 2025 II) Research Square https://doi.org/10.21203/rs.3.rs-7408118/v1 Yuen, P. S. Neutron charge structure revisited: Insights from Patterson function analysis. (Preprint 2025 III) OSF Preprints https://doi.org/10.31219/osf.io/eqjcs_v1 Yuen, P. S. Neutron magnetic structure revisited: Insights from Patterson function analysis. (Preprint 2025 IV) Research Square https://doi.org/10.21203/rs.3.rs-7422459/v1 Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7435823","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":504297315,"identity":"3d3784a9-0f17-4eb2-8b33-de80268ec043","order_by":0,"name":"Pui Sum Yuen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAUlEQVRIiWNgGAWjYFACxgYog/kAiOQhRQtbAgNDAkgLM9HW8RiAtDAQ1GIukdz8mTfncGK/dM+3Bx9/HJaRdz9/8HEFg52cbgN2LZYzEtukebcdTpw55+x2wxkJh3kMzyQzG55hSDY2O4Bdi8GNxDZmkJYNN3K3SfMkpPEYNiSzSTYwHEjchlsL0GFALftv5DyT/gPS0v+YoJYGsMM2SOSwSTMk2PDISxCwxbLnYZvk3G3pxjNupJlJ9qTZ8BhIPDY2bDDA7Rdz9vTHH95us5btn5H8TOKHjYS9fH/iw4cNFXZyOL0PoZqRRA4giePRUocQkW/ArXoUjIJRMApGJgAATONdnW9BtzkAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0002-2352-3993","institution":"Nil. I am an independent researcher","correspondingAuthor":true,"prefix":"","firstName":"Pui","middleName":"Sum","lastName":"Yuen","suffix":""}],"badges":[],"createdAt":"2025-08-22 15:25:23","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7435823/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7435823/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89786764,"identity":"c80dc51e-b9c7-443f-843a-43458a7873d7","added_by":"auto","created_at":"2025-08-25 04:31:59","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":32945,"visible":true,"origin":"","legend":"\u003cp\u003eRadial Patterson charge density of proton from the first image.\u003c/p\u003e\n\u003cp\u003eThe graph displays the radial Patterson charge density as a function of radial distance r (in femtometers). Blue data points form a curve that begins at approximately 250 fm⁻⁶ at r=0, sharply decreases to near-zero values by r=0.4 fm, and remains close to zero beyond that. This profile suggests a compact central charge concentration.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7435823/v1/f323bb23ff4b77d2844ec923.png"},{"id":89786571,"identity":"d9123280-c264-49e9-9ed8-489836d5fe12","added_by":"auto","created_at":"2025-08-25 04:23:59","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":32494,"visible":true,"origin":"","legend":"\u003cp\u003eRadial Patterson charge density of the proton (second image).\u003c/p\u003e\n\u003cp\u003eThis graph presents the radial Patterson charge density of the proton derived from the second reconstructed image. The curve begins with a sharp central peak of approximately 1200 fm⁻⁶ at r=0, followed by a rapid decline to near-zero values by r=0.3 fm, and remains close to zero up to r=0.8 fm. This profile suggests a highly compact positive core and supports the hypothesis of a dense central charge concentration within the proton. The second image serves as a refined approximation of the proton’s internal charge structure, emphasizing the steep gradient and spatial confinement of the central charge density.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7435823/v1/b807de73961feb2665816a0d.png"},{"id":89787720,"identity":"6f56b8e2-a157-4acb-a73a-c30de195a3ed","added_by":"auto","created_at":"2025-08-25 04:40:00","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":42767,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of normalized radial Patterson charge densities of the proton.\u003c/p\u003e\n\u003cp\u003eThis plot contrasts the normalized radial Patterson charge density of the proton (green) with that derived from its second reconstructed image (blue). Both curves exhibit a sharp central peak at r=0, followed by a rapid decline toward zero by r≈0.3 fm. The close overlap in the inner region suggests that the second image effectively captures the compact core structure of the proton. The normalization highlights the relative spatial distribution of charge density, supporting the hypothesis of a dense central positive core with minimal peripheral extension.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7435823/v1/d44d039e4dc1f75e61a4a220.png"},{"id":89786584,"identity":"ac163f62-66b5-4245-a076-5d5f2bb08024","added_by":"auto","created_at":"2025-08-25 04:24:00","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":32236,"visible":true,"origin":"","legend":"\u003cp\u003eRadial charge density of the proton derived from the second image.\u003c/p\u003e\n\u003cp\u003eThis graph shows a sharply peaked central charge density of approximately 1400 fm⁻³ at r = 0, which declines steeply to near-zero by r = 0.2 fm and remains negligible up to r = 0.8 fm. The pronounced central peak and rapid spatial decay suggest a highly compact positive core with minimal peripheral extension.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7435823/v1/069e2032bcfa96fdb0c768bb.png"},{"id":89786573,"identity":"86fc22a4-7173-4166-ab33-297d1c54f30f","added_by":"auto","created_at":"2025-08-25 04:23:59","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":29964,"visible":true,"origin":"","legend":"\u003cp\u003eRadial distribution of r²ρ for the proton, reconstructed from the second image using Patterson function analysis.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7435823/v1/42bf19a964f010e659c2f6cc.png"},{"id":89786574,"identity":"f75e23f2-5b51-4684-b686-51f6a75057ba","added_by":"auto","created_at":"2025-08-25 04:24:00","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":104411,"visible":true,"origin":"","legend":"\u003cp\u003eCharge density distribution of the proton at z = 0 fm, derived from the second image.\u003c/p\u003e\n\u003cp\u003eColour codes:\u003c/p\u003e\n\u003cp\u003eOrange: Peak charge density = 1338 fm\u003csup\u003e-3\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eLight Blue: 668.5 \u0026lt; ρ ≤ 1258 fm\u003csup\u003e-3\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eBlack: 334.25 \u0026lt; ρ ≤ 668.5 fm\u003csup\u003e-3\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eYellow: 0 \u0026lt; ρ ≤ 334.25 fm\u003csup\u003e-3\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eGrey: -0.098 ≤ ρ ≤ 0 fm\u003csup\u003e-3\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-7435823/v1/d48456aee6f9d488494933fe.png"},{"id":89787944,"identity":"172ffe7c-0d18-46c8-b97e-5e7d0ff0e272","added_by":"auto","created_at":"2025-08-25 04:48:00","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":570317,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7435823/v1/dfe2c2d6-0e1f-42f9-bd4b-da616382a60e.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eReconstructed proton charge structure: Insights from Patterson function analysis\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn Yuen (Preprint 2025 I), we use the Patterson function of the proton charge density to demonstrate that the proton charge density is non-negative. From this, we establish a lower bound for the proton radius at 0.09 fm, offering a potential resolution to the proton radius puzzle. Similarly, in Yuen (Preprint 2025 II), the Patterson function of the proton magnetization density reveals that the magnetization density is also non-negative, with a magnetic radius limit of 0.08 fm\u0026mdash;an order of magnitude smaller than values commonly reported in the literature.\u003c/p\u003e\u003cp\u003eFor the neutron, another fundamental nucleon, the Patterson function of its charge density shows that the distribution cannot be classified as simply non-negative or non-positive (Yuen, Preprint 2025 III). A model consisting of a hard-core positive charge center surrounded by a soft negative shell may apply to the neutron as well. Our findings suggest that the positive core may be more compact than previously proposed. Although the neutron magnetization density has received less attention, Yuen (Preprint 2025 IV) investigates its magnetic structure. The results mirror those of the neutron charge density, indicating a hard positive core and a soft negative shell.\u003c/p\u003e\u003cp\u003eWith these studies (Yuen, Preprint 2025 I\u0026ndash;IV), the first stage of our project\u0026mdash;examining the electric and magnetic structures of the proton and neutron via Patterson function analysis\u0026mdash;is complete. The next stage involves reconstructing the explicit charge and magnetization density functions. In this article, we present the results for what is likely the most critical of these: the proton charge density function. Determining the proton charge density from its Patterson counterpart is a deconvolution problem, constrained by the requirement of non-negativity. Rather than relying solely on mathematical formalism, we adopt an empirical, approximate, and graphical approach to explore the proton\u0026rsquo;s internal structure.\u003c/p\u003e"},{"header":"2. Proton charge distribution in the literature","content":"\u003cp\u003eSeveral studies have investigated the proton\u0026rsquo;s charge distribution using diverse theoretical and experimental approaches. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e summarizes key contributions.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eReported proton charge distribution.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eStudy\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eMethod / Approach\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eKelly (2002)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eSachs form factors\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eLaViolette (2006)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eA Turing Wave Pattern\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSick (2012)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eElectron scattering data\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eCs\u0026ouml;rg\u0026otilde; et al. (2020)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eL\u0026eacute;vy imaging of pp elastic scattering\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eGramolin \u0026amp; Russell (2021)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eElastic electron-proton scattering,\u003c/p\u003e\u003cp\u003etransverse charge density\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eFreese \u0026amp; Miller (2023)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eLight front synchronization\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eWon et al. (2024)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003eEnergy momentum tensor form factors\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"3. Methodology","content":"\u003cp\u003eIn Yuen (Preprint 2025 I), we found that the half-maximum of the proton Patterson charge density occurs at r = 0.09 fm, and the first zero appears at r = 0.37 fm. Based on this observation, we hypothesize that the actual proton charge density may resemble the Patterson profile\u0026mdash;characterized by a sharp central peak followed by a gradual decline to zero. If this assumption holds, the Patterson charge distribution can serve as a suitable trial object. The \u003cstrong\u003efirst image\u003c/strong\u003e\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003eis constructed by enclosing the Patterson charge density within a sphere of radius \u0026le; 0.37 fm. The radial Patterson charge density derived from this image is shown in \u003cstrong\u003eFigure 1\u003c/strong\u003e, with a half-maximum radius of 0.15 fm. To align the half-maximum positions with the original Patterson profile, we scale the coordinates of the first image by a factor of 0.09 / 0.15, yielding a smaller \u003cstrong\u003esecond image\u003c/strong\u003e\u003cstrong\u003e.\u0026nbsp;\u003c/strong\u003eThe radial Patterson charge density of the second image is presented in \u003cstrong\u003eFigure 2\u003c/strong\u003e\u003cstrong\u003e.\u003c/strong\u003e It is non-negative, and its half-maximum radius matches the original value of 0.09 fm. \u003cstrong\u003eFigure 3\u003c/strong\u003e\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003ecompares the normalized radial Patterson charge density of the proton (Yuen, (Preprint 2025 I), Figures 1 and 4) with that of the second image. For r ≦ 0.12 fm, the two curves overlap closely. We therefore adopt the second image as our reconstructed proton charge density.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003eThe charge density of the second image has been deposited in Zenodo.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFigure 4\u003c/strong\u003e displays the radial charge density \u0026rho;(r) derived from the second image, with a half-maximum radius of 0.051 fm. Following the methodology of Kelly (2002), we present the reconstructed radial profile r\u0026sup2;\u0026rho;(r) in\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003eFigure 5\u003c/strong\u003e\u003cstrong\u003e,\u0026nbsp;\u003c/strong\u003ewhich highlights the spatial concentration of the proton\u0026rsquo;s charge density. For calculations at z = 0 fm, a grid resolution of 100 \u0026times; 100 is used, as shown in\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003eFigure 6\u003c/strong\u003e\u003cstrong\u003e.\u0026nbsp;\u003c/strong\u003eAt a resolution of 100 \u0026times; 100 \u0026times; 100, the minimum charge density within the unit cell is \u0026minus;0.0991 fm⁻\u0026sup3;, while the maximum reaches 1338 fm⁻\u0026sup3;.\u003c/p\u003e"},{"header":"4. Discussion and conclusion","content":"\u003cp\u003eIn Section \u003cspan refid=\"Sec3\" class=\"InternalRef\"\u003e3\u003c/span\u003e, we constructed a second image whose radial Patterson charge density closely matches the experimental profile of the proton (Yuen, (Preprint 2025 I)), as shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e and \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. Both exhibit a radius of half-maximum Patterson charge density of 0.09 fm. This alignment supports the selection of the second image as a valid representation of the proton\u0026rsquo;s internal charge structure. The radial charge density derived from this second image (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) yields a half-maximum radius of 0.051 fm, which is smaller than the 0.09 fm Patterson half-maximum radius. This result adheres to the upper bound established in Yuen, (Preprint 2025 I) for non-negative radial Patterson charge densities. Although this reconstructed radius is an order of magnitude smaller than the conventional proton charge radius range of 0.84\u0026ndash;0.90 fm (Yuen, (Preprint 2025 I)), it reflects the compactness of the central charge concentration. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e further illustrates the charge density distribution at z\u0026thinsp;=\u0026thinsp;0 fm, revealing a sharp central peak followed by a rapid decline to near-zero values. This core-dominated structure is consistent with Rutherford\u0026rsquo;s early findings and aligns with most models listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, with the exception of Cs\u0026ouml;rgő et al. (2020), whose approach emphasizes spatial hollowness. Together, Figs.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e support the hypothesis of a highly compact, positive core within the proton, reconstructed through empirical Patterson function analysis. This empirical Patterson function approach complements existing theoretical frameworks and may inform future studies of nucleon substructure.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e The charge density of the second image, structure factor magnitudes and phases, and the charge density at z=0 fm are available on Zenodo at\u003c/p\u003e\n\u003cp\u003ehttps://doi.org/10.5281/zenodo.16849345\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding information\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author has not received funding from any organization.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e The author declares no competing interests.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eCs\u0026ouml;rg\u0026otilde;, T., Pasechnik, R. \u0026amp; Ster, A. Proton structure and hollowness from L\u0026eacute;vy imaging of pp elastic scattering, \u003cem\u003eEur. Phys. J. C\u003c/em\u003e \u003cstrong\u003e80\u003c/strong\u003e, 126 (2020).\u003c/li\u003e\n\u003cli\u003eFreese, A. \u0026amp; Miller, G. A. Light front synchronization and rest frame densities of the proton: Electromagnetic densities, \u003cem\u003ePhys. Rev. D\u003c/em\u003e \u003cstrong\u003e107\u003c/strong\u003e, 074036 (2023)\u003c/li\u003e\n\u003cli\u003eGramolin, A. V. \u0026amp; Russell, R. L. Transverse charge density and the radius of the proton, \u003cem\u003ePhys. Rev. D\u003c/em\u003e, \u003cstrong\u003e105\u003c/strong\u003e, 054004 (2022). \u003c/li\u003e\n\u003cli\u003eKelly, J. J. Nucleon Charge and Magnetization Densities from Sachs Form Factors, \u003cem\u003ePhys. Rev. C\u003c/em\u003e, \u003cstrong\u003e66\u003c/strong\u003e, 065203 (2002).\u003c/li\u003e\n\u003cli\u003eLaViolette, P. A. The Electric Charge and Magnetization Distribution of the Nucleon: A Turing Wave Pattern? vixra.org, Jan. 2006.\u003c/li\u003e\n\u003cli\u003eSick, I. Problems with proton radii, \u003cem\u003eProg. Part. Nucl. Phys.\u003c/em\u003e \u003cstrong\u003e67\u003c/strong\u003e, 473-478 (2012).\u003c/li\u003e\n\u003cli\u003eWon, H.-Y., Kim, H.-C. \u0026amp; Kim, J.-Y. Flavor structure of the energy momentum tensor form factors of the proton. \u003cem\u003ePhys. Lett. B\u003c/em\u003e \u003cstrong\u003e850\u003c/strong\u003e, 138489 (2024).\u003c/li\u003e\n\u003cli\u003eYuen, P. S. Proton charge structure revisited: Insights from Patterson function analysis. (Preprint 2025 I) OSF Preprintshttps://doi.org/10.31219/osf.io/6jsd8_v1\u003c/li\u003e\n\u003cli\u003eNotable result: The radius at half-maximum Patterson charge density is 0.09 fm.\u003c/li\u003e\n\u003cli\u003eYuen, P. S. Proton magnetic structure revisited: Insights from Patterson function analysis. (Preprint 2025 II) Research Square https://doi.org/10.21203/rs.3.rs-7408118/v1\u003c/li\u003e\n\u003cli\u003eYuen, P. S. Neutron charge structure revisited: Insights from Patterson function analysis. (Preprint 2025 III) OSF Preprints https://doi.org/10.31219/osf.io/eqjcs_v1\u003c/li\u003e\n\u003cli\u003eYuen, P. S. Neutron magnetic structure revisited: Insights from Patterson function analysis. (Preprint 2025 IV) Research Square https://doi.org/10.21203/rs.3.rs-7422459/v1\u003c/li\u003e\n \u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Nil. I am an independent researcher","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Proton structure, Patterson function, form factor, deconvolution, nuclear physics, charge density distribution","lastPublishedDoi":"10.21203/rs.3.rs-7435823/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7435823/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe present an empirical, approximate, and graphical method to reconstruct the proton\u0026rsquo;s internal charge structure using Patterson function analysis. The resulting model is non-negative and compact, with a radius of half-maximum Patterson charge density of 0.09 fm\u0026mdash;identical to the experimental value. The radius of half-maximum charge density of the reconstructed proton structure is 0.051 fm, which lies within the upper bound for non-negative radial Patterson charge density. This approach offers a novel perspective on proton structure and complements existing theoretical models.\u003c/p\u003e","manuscriptTitle":"Reconstructed proton charge structure: Insights from Patterson function analysis","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-25 04:23:55","doi":"10.21203/rs.3.rs-7435823/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"09309fde-96f3-4fc2-a837-e3b9286d5b58","owner":[],"postedDate":"August 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":53581596,"name":"Nuclear Physics"}],"tags":[],"updatedAt":"2025-08-25T04:23:55+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-25 04:23:55","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7435823","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7435823","identity":"rs-7435823","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}