[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-664715-105":59,"doc-detail-664715-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","lattice-design-in-particle-accelerators-lattice-design-in-particle-accelerators","Lattice Design in Particle Accelerators - Lattice Design in Particle Accelerators","","Lattice design principles for particle accelerators are presented, focusing on both the “non-problem” and the “problem” of low-energy and space-charge dominated beams. The discussion covers LHC and storage-ring lattice structures, dipole and quadrupole roles, beam rigidity, ring geometry, and the governing equations of motion using Hill’s equation. It explains betatron tune and beta functions, phase-space ellipse concepts, emittance invariance via Liouville’s theorem, and matrix-based propagation of optical parameters through lattice elements, with emphasis on how to optimize focusing properties and begin practical lattice design.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/lattice-design-in-particle-accelerators-lattice-design-in-particle-accelerators/664715/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/lattice-design-in-particle-accelerators-lattice-design-in-particle-accelerators/664715.png","ImageObject",300,407,{"name":92,"@type":93},"putri.kecil17","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why are dipole magnets needed in a circular accelerator lattice?","Question",{"text":112,"@type":113},"Because the beam orbit must be bent into a closed ring. The bending angle over one revolution must sum to 2π, linking the dipole field integral to ring circumference and particle rigidity.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does Hill’s equation relate to transverse particle motion in the lattice?",{"text":117,"@type":113},"It governs the transverse oscillations under periodic focusing from quadrupoles. The general solution introduces functions like β(s) and the phase advance ψ(s), which determine amplitude and tune per turn.",{"name":119,"@type":110,"acceptedAnswer":120},"What determines beam size and how is it connected to the beta function?",{"text":121,"@type":113},"The beta function β(s) sets the envelope of particle trajectories and therefore the beam size. Amplitudes are expressed as x(s)=ε·β(s)·cos(ψ(s)+ϕ), showing that β(s) controls growth and focusing-induced variation.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},664715,1793268642,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},962094509873,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Lattice Design in Particle Accelerators  \nBernhard Holzer  \n... in the context of space charge dominated beams  \nThe Non-Problem  \nLHC lattice structure  \n768 x 768 colliding bunches  \ngoal: 2808 bunches  \nNp =1.2*1011 per bunch  \nx ′′ + K(s)x = 0  \nx ′′ + (Kq (s)x + Ksc (s)) x = 0  \nKsc = ea−22c(r0βIγ)3  \nHills equation without ...  \n... and with space charge  \nβ = 1, γ ≈ 7500 Ksc ≈ 3.6 * 10−11  \nKq ≈ 9 * 10−3  \nI like high energy accelerators !!!  \nThe Problem: low energy part  \nAlvarez structure „Drift Tube Linac“(GSI Unilac)  \nBunch Splitting in the PS  \nCERN: Linac 2 injection into PSB  \nNp ≈ 1.5*1013 protons per bunch, Einj = 50 MeV  \nβ = 0.31 γ = 1.05  \n| circular machines: somewhere in the lattice we need a number of dipole magnets, that are bending the design orbit to a closed ring\u003Cbr>Geometry of the ring:\u003Cbr>centrifugal force = Lorentz force\u003Cbr>|  |\n| --- | --- |\n| | \u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>\u003Cbr>p = momentum of the particle,ρ = curvature radius\u003Cbr>Bρ= beam rigidity |\n| typical low energy storage ring (TSR)\u003Cbr>8 dipole magnets of equal bending strength |  |\n\nρ  \nα = dsρ ≈ dlρ = BBdρl  \nThe angle swept out in one revolution must be 2π, so  \nfield map of a storage ring dipole magnet  \n…for a full circle  \nBdl = 2π ~~ ~~pq  \n∫  \nNota bene:  \nis  \nusually required !!  \nEkin = 60MeVE 2 = p2c 2 + m 2c 4  \n…8 dipoles, l=1m B=0.95T  \n\n| \u003Cbr>\u003Cbr>hor. plane vert. plane\u003Cbr>\u003Cbr>\u003Cbr>quadrupole magnet  ~~ ~~\u003Cbr>\u003Cbr>\u003Cbr>Example: HERA Ring:\u003Cbr>Bending radius: ρ = 580 m\u003Cbr>Quadrupol Gradient: g = 110 T/m\u003Cbr>k = 33.64*10-3 /m2\u003Cbr>1/ρ2 = 2.97 *10-6 /m2\u003Cbr>dipole magnet\u003Cbr>\u003Cbr>\u003Cbr>For estimates in large accelerators the weak focusing term 1/ρ2 can in general be neglected |  |\n| --- | --- |\n| Solution for a focusing magnet | |\n\n\n| Or written more convenient in matrix form:\u003Cbr>Hor. focusing Quadrupole Magnet\u003Cbr>Hor. defocusing Quadrupole Magnet |  | 􀀢 x 􀀥 􀀢 x 􀀥 􀀢 m11 m12 􀀥 􀀢 x 􀀥\u003Cbr>􀀤 ' 􀀧 = M * 􀀤 ' 􀀧 = 􀀤 􀀧 * 􀀤 ' 􀀧\u003Cbr>􀀣x 􀀦 s 􀀣x 􀀦 0 􀀣 m21 m22 􀀦 􀀣x 􀀦 0\u003Cbr>\u003Cbr>|\n| --- | --- | --- |\n| Drift space | |  |\n| |  |  |\n\n\n| Transformation through a system of lattice elements\u003Cbr>combine the single element solutions by multiplication of the matrices |  |\n| --- | --- |\n| 􀀢 x 􀀥 􀀢 x 􀀥􀀤 ' 􀀧 = M(s2 , s1) * 􀀤 ' 􀀧􀀣x 􀀦 s2 􀀣x 􀀦 s1 | \u003Cbr>focusing lens dipole magnet\u003Cbr>defocusing le\u003Cbr>court. K. Wille\u003Cbr>|\n| in each accelerator element the particle trajectory corresponds to the movement of a\u003Cbr>harmonic oscillator „ |  |\n| x(s)\u003Cbr>0\u003Cbr>typical values in a strong foc. machine:\u003Cbr>x ≈ mm, x´ ≤ mrad\u003Cbr>|  |\n\n\n|  Question: what will happen, if the particle performs a second turn ? \u003Cbr>  or a third one or   1010 turns |\n| --- |\n| x\u003Cbr>0\u003Cbr>\u003Cbr>s |\n\nGeneral solution of Hill´s equation:  \n(i)    \nε, Φ = integration constants determined by initial conditions  \nβ(s) periodic function given by focusing properties of the lattice ↔ quadrupoles  \nInserting (i) into the equation of motion…  \nΨ(s) = „phase advance“ of the oscillation between point „0“ and „s“ in the lattice.  \nFor one complete revolution: number of oscillations per turn „Tune“  \nThe Beta Function  \nAmplitude of a particle trajectory:  \nx(s) = ε * β(s) * cos(ψ(s) + ϕ)  \nMaximum size of a particle amplitude  \nβ determines the beam size  \n( ... the envelope of all particle trajectories at a given position“s” in the storage ring.  \nIt reflects the periodicity of the magnet structure.  \n| general solution of Hill equation\u003Cbr>| | |\n| --- | --- | --- |\n| from (1) we get\u003Cbr>\u003Cbr>Insert into (2) and solve for ε |  |  |\n| \u003Cbr>* ε is a constant of the motion …it is independent of „s“\u003Cbr>* parametric representation of an ellipse in the x x‘ space\u003Cbr>* shape and orientation of ellipse are given by α, β, γ |  |  |\n\nBeam Emittance and Phase Space Ellipse  \ns  \nLiouville: in reasonable storage rings  \narea in phase space is constant.  \nA = π*ε=const  \nε beam emittance = woozilycity of the particle ensemble, intrinsic beam parameter,  \ncannot be changed by the [foc. properties](foc. properties).  \nScientifiquely speaking: area covered in t","cbCaiu992YNphqEy","https://ap.wps.com/l/cbCaiu992YNphqEy","pdf",7274618,14,"English","# Lattice design: non-problem vs low-energy problem\n## Dipole magnets and storage ring geometry\n## Quadrupole focusing and matrix optics\n## Hill’s equation, tune, and beta function\n## Phase-space ellipse, emittance, and Liouville invariance\n## Transforming optical parameters through lattice cells\n## Lattice cell types for space-charge dominated beams","[{\"question\":\"Why are dipole magnets needed in a circular accelerator lattice?\",\"answer\":\"Because the beam orbit must be bent into a closed ring. The bending angle over one revolution must sum to 2π, linking the dipole field integral to ring circumference and particle rigidity.\"},{\"question\":\"How does Hill’s equation relate to transverse particle motion in the lattice?\",\"answer\":\"It governs the transverse oscillations under periodic focusing from quadrupoles. The general solution introduces functions like β(s) and the phase advance ψ(s), which determine amplitude and tune per turn.\"},{\"question\":\"What determines beam size and how is it connected to the beta function?\",\"answer\":\"The beta function β(s) sets the envelope of particle trajectories and therefore the beam size. Amplitudes are expressed as x(s)=ε·β(s)·cos(ψ(s)+ϕ), showing that β(s) controls growth and focusing-induced variation.\"}]","Lattice Design in Particle Accelerators - Lattice Design in Particle Accelerators | PDF",35]