[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665324-105":59,"doc-detail-665324-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","the-band-structure-of-the-electrons-section-51-introduction","The Band Structure of the Electrons - Section 5.1 Introduction","","The document explains how the electronic band structure arises in the Fourier representation using Kohn–Sham plane waves and reciprocal lattice vectors. It derives the Hamiltonian matrix elements and the nearly diagonal behavior for large reciprocal vector differences, then formulates free-electron dispersions and energy bands within the first Brillouin zone. A hexagonal lattice example introduces special k-points and band degeneracies, followed by how small nonzero crystal potentials couple plane waves, producing energy shifts near Brillouin-zone boundaries. It also relates band occupation at T=0 to electron filling of bands and links the electron count and band structure to solid electrical and optical properties.",{"@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/the-band-structure-of-the-electrons-section-51-introduction/665324/",{"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/the-band-structure-of-the-electrons-section-51-introduction/665324.png","ImageObject",300,407,{"name":92,"@type":93},"4398046744996","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How is the Hamilton operator written in the Fourier representation for Kohn–Sham plane waves?","Question",{"text":112,"@type":113},"In the Fourier basis of plane waves, the Kohn–Sham Hamiltonian becomes a matrix with diagonal kinetic-energy terms and off-diagonal elements given by the effective crystal potential veff(Gn−Gm).","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What determines whether the Hamilton matrix is approximately diagonal?",{"text":117,"@type":113},"The potential veff(Gn−Gm) decreases as the magnitude |Gn−Gm| increases, so for large |Gn−Gm| the off-diagonal coupling becomes small and the Hamilton matrix is essentially diagonal.",{"name":119,"@type":110,"acceptedAnswer":120},"How does a weak crystal potential modify the free-electron band structure?",{"text":121,"@type":113},"When the potential deviates slightly from zero, plane waves with different reciprocal vectors are coupled under the Bragg condition, causing hybridization and energy splitting near the Brillouin-zone boundaries, with larger changes for degenerate states.","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},665324,1793298201,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":66,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":138,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143},4398046744996,"5 The Band Structure of the Electrons  \n5.1 Introduction  \nIn the Fourier representation, i.e., in the basis of plane waves  \nχn (k) = !1Vg~~ ~~ ei (k+Gn)r , (5.1)  \nthe Hamilton operator of the Kohn-Sham equation has the form:  \nhn,m = 2!2m (k + Gn)2 δn,m + veff(Gn − Gm) . (5.2)  \nveff(Gn − Gm) decreases with increasing length |Gn − Gm |, and thus hn,m is basically diagonal for large |Gn − Gm | . The corresponding eigenvalue equation enables the calculation of the energies and eigenfunctions for given k.  \n| |  |  |\n| --- | --- | --- |\n| | | |\n\nFigure 5 . 1: Fourier representation of the atomic potentials of Al, Si and Ag.  \nIn Fig. 5. 1 we show the function veff(G) for three examples: Al, Si, and Ag. In fact, the figure shows the potential 1 of atoms and therefore, veff is defined for continuous values of |G| . For a solid we have roughly  \nM  \nveff(r) =\"veff−atom(r − RI) ,  \nI=1  \n1 To be precise, Fig. 5. 1 shows the effective potential for atomic pseudopotentials. The definition of pseudopotentials is given in Section 5.5.1 below.  \nand then, in the Fourier representation veff−atom is only needed at discrete reciprocal lattice vectors. These are determined by the lattice structure, and we have: ~~2π~~a = 1.55 Å−1 (Al), 1 . 16 Å−1 (Si), and 1 .54 Å−1 (Ag) .  \nIf all Fourier components veff(Gl) are small, we obtain the dispersion of free electrons:  \nϵn (k) = 2!2m (k + Gn)2 . (5.3)  \nϵn (k) as a function of k is called the energy band n. For the one-dimensional case we have shown this dispersion before (cf. Fig. 4.13) . However, the one-dimensional case is not typical, because in this case at most two-fold degeneracy can be present. Therefore, we will now discuss a two-dimensional example, which shows all the important characteristics of a band structure – even those of three-dimensional structures. We will investigate a hexagonal lattice. The 1st Brillouin zone and the labels of special k-points are shown in Fig. 5.2. If we now evaluate Eq. (5.3), we obtain the band structure of free electrons, as shown in Fig. 5.3, for G0 = (0 , 0) = Γ , G 1 = ~~2π~~a (0 , 1) , G2 = ~~2π~~a (cos30◦ , sin30◦ ) = ~~2π~~a ( ~~ ~~√2~~3~~ , ~~1~~2) , G3 = ~~2π~~a (cos30◦ , − sin30◦ ) = ~~2π~~a ( ~~ ~~√2~~3~~ , − ~~1~~2) , etc. Here, we restrict ourselves to the boundary of the so-called irreducible wedge, which is hatched in Fig. 5.2. By reflection and rotation of this wedge the full 1st Brillouin zone can be obtained.  \nFigure 5 .2: Brillouin zone of a hexagonal cell. Special k-points are shown. The hatched area is the irreducible part.  \nThe point K is at ~~1~~3 (G2 +G3 ) = ~~2π~~a ( ~~ ~~3 , 0), and along the ΓK-direction we have k = (kx , 0) . We obtain the results:  \nn = 0 , i.e. , G0 : ϵ0 (k) = ~~ ~~2!2mk2x  \nϵ0 (Γ) = 0  \nϵ0 (K) = ~~ ~~2!2m ( ~~2π~~a 3 )2  \nn = 1 , i.e. , G 1 : ϵ 1 (k) = ~~ ~~2!2m (k2x +( ~~2π~~a )2 )ϵ 1 (Γ) = ~~ ~~2!2m ( ~~2π~~a )2  \nϵ 1 (K) = ~~ ~~2!2m ( ~~1~~3 +1) · ( ~~2π~~a )2  \nFigure 5 .3: Band structure of free electrons in a hexagonal lattice. The numbers in brackets give the degeneracy.  \nFigure 5 .4: The band structure of nearly free electrons in a hexagonal lattice for two different systems and two different Fermi energies: ϵ1F for 1 electron per cell and ϵ2F for 2 electrons per cell.  \nn = 2 , i.e. , G2 : ϵ2 (k) = ~~ ~~2!2m \\# $kx + ~~2π~~a √2~~3~~ %2 +( ~~2π~~a )2 · ~~1~~4 &  \nϵ2 (Γ) = ~~ ~~2!2m ( ~~2π~~a )2 ( ~~3~~4 + ~~1~~4)  \n= ~~ ~~2!2m ( ~~2π~~a )2 = ϵ1 (Γ)ϵ2 (K) = ~~ ~~2!2m ( ~~2π~~a )2 · ~~7~~3  \netc.  \nThe result for ϵn (k) is shown in Fig. 5.3. Even for free electrons the band structure – just because of the reduction to the 1st Brillouin zone – looks rather complicated.  \nWhich modifications can we expect if the potential – which has been zero so far – slightly differs from zero? This will be investigated in more detail now.  \nPlane waves with different Gn are coupled by the crystal potential (Bragg-condition,  \n− π  \na  \nΓ + ~~π~~a  \n\n| \u003Cbr>\u003Cbr>| \u003Cbr>\u003Cbr>| \u003Cbr>\u003Cbr>| \u003Cbr>\u003Cbr>| \u003Cbr>\u003Cbr>|\n| --- | --- | --- | --- | ---","cbCaihF0NPHP7PQS","https://ap.wps.com/l/cbCaihF0NPHP7PQS","pdf",1180307,25,"English","# 5 The Band Structure of the Electrons\n## 5.1 Introduction\n## What Can We Learn from a Band Structure?","[{\"question\":\"How is the Hamilton operator written in the Fourier representation for Kohn–Sham plane waves?\",\"answer\":\"In the Fourier basis of plane waves, the Kohn–Sham Hamiltonian becomes a matrix with diagonal kinetic-energy terms and off-diagonal elements given by the effective crystal potential veff(Gn−Gm).\"},{\"question\":\"What determines whether the Hamilton matrix is approximately diagonal?\",\"answer\":\"The potential veff(Gn−Gm) decreases as the magnitude |Gn−Gm| increases, so for large |Gn−Gm| the off-diagonal coupling becomes small and the Hamilton matrix is essentially diagonal.\"},{\"question\":\"How does a weak crystal potential modify the free-electron band structure?\",\"answer\":\"When the potential deviates slightly from zero, plane waves with different reciprocal vectors are coupled under the Bragg condition, causing hybridization and energy splitting near the Brillouin-zone boundaries, with larger changes for degenerate states.\"}]","The Band Structure of the Electrons - Section 5.1 Introduction | PDF",63]