[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-179851-105":59,"doc-detail-179851-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","cluster-approaches-to-random-alloys-an-appraisal","Cluster approaches to random alloys - An appraisal","","The paper critically evaluates cluster extensions of the coherent potential approximation (CPA) within effective medium theory, focusing on the decoupling schemes that determine computational feasibility. It shows that fully self-consistent calculations in three-dimensional systems are extremely difficult. A self-consistent pair-CPA treatment is presented for simple-cubic lattices with both diagonal and off-diagonal disorder. The study establishes that non-analyticities are a general feature of CPA extensions in multiple scattering frameworks and qualitatively explains how off-diagonal randomness affects Anderson localization via the density of states.",{"@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/cluster-approaches-to-random-alloys-an-appraisal/179851/",{"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/cluster-approaches-to-random-alloys-an-appraisal/179851.png","ImageObject",300,407,{"name":92,"@type":93},"Theodora","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-09-02",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the main focus of the paper on cluster extensions of CPA?","Question",{"text":112,"@type":113},"The paper studies how different decoupling schemes in CPA-based cluster extensions affect self-consistent calculations and the resulting density of states for random alloys.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why are self-consistent CPA cluster calculations difficult in three-dimensional systems?",{"text":117,"@type":113},"The paper explains that the cluster generalisations require solving overly complex simultaneous conditions, making numerical self-consistent computation immensely difficult to perform in three dimensions.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the paper conclude about non-analyticities in CPA extensions?",{"text":121,"@type":113},"It shows non-analyticities are a general and inevitable feature of CPA extensions within the multiple scattering framework, contradicting earlier beliefs about certain extensions being analytic.","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},179851,1788338471,{"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":52,"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,"preview_page_count":4},687197207919,"https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552","# Cluster approaches to random alloys:An appraisal\n\nVIPIN SRIVASTAVA*,MEENA CHATURVEDI and SK JOSHIDepartment of Physics,University of Roorkee,Roorkee 247672*Department of Physics,Heriot-Watt University,Edinburgh,UK  \nMS received 13 February 1978;revised 26 June 1978  \nAbstract.Some of the cluster extensions of the coherent potential approximation(CPA)based on the effective medium theory have been critically studied with res-pect to the decoupling schemes involved in them.Their computational tractability hasbeen examined and it has been found that the self-consistent calculations in three-di-mensional systems are immensely difficult to perform.A self-consistent calculationhas been reported for simple cubic lattices with diagonal and off-diagonal disorderusing a pair-CPA method.A significant finding of the paper is that it has beenshown that non-analyticities are a general feature of extensions of CPA within mul-tiple scattering framework.The non-analyticities were reported several times but ageneral proof of their existence was not noticed.It was also believed that the so-called molecular—CPA is analytic,this has been shown to be wrong here.The den-sity of states results with off-diagonal randomness have been qualitatively understoodto yield some information about the influence of off-diagonal randomness on Ander-son iocalisation of an electron.  \nKeywords.Density of states;off-diagonal randomness;analyticity;Anderson-Iocalisation.  \n## 1.Introduction\n\nIt is now known that the generalisation of the coherent potential approximation(CPA)(Soven 1967)to include the cluster effects in disordered alloys is non-unique,and presents a difficult problem when one tries to calculate the distribution ofeigenvalues.  \nIn this paper we present a self-consistent calculation of electronic density of statesin a simple-cubic binary alloy using a pair—CPA.We discuss the numerous difficultiesthat one encounters with to perform a self-consistent calculation,and show how aseries of approximations has to be made in order to make the numerical computationfeasible.A comprehensive review of the attempts made to generalise CPA has beengiven by Elliott et al(1974a),more recent approaches have been discussed byEhrenreich and Schwartz(1976)and Kumar and Joshi(1977),and most recent at-tempt,not included in these references,is by Bloom and Mattis(1977).The numericalresults reported so far are mainly for one-dimensional systems.Except the self-consistent results of Moorjani et al(1974a),the rest are obtained either throughnon-self-consistent methods,employing iterative methods like average t-matrixapproximation,or through the methods that deal with scattering from a single siteand include in some approximate way the local environmental effects.  \nThe development of cluster-CPA has taken two directions.One of them,to betermed as'single-site framework’,imbeds a pair of sites or triplet of sites,etc.,in aneffective medium and treats the scattering from it exactly.The other one,to be termed  \nas‘cluster framework’,instead treats exactly the scattering from a cluster of nearestneighbours,and next nearest neighbours,etc.The two frameworks incorporate twodifferent decoupling schemes,and yield different effective mediums after calculation.The latter can be reduced to the former after some approximations,but the formercannot be generalised to yield the self-consistent conditions of the cluster framework.We have discussed the subtleties involved in the two kinds of decoupling schemes.  \nA big problem with the cluster generalisations of CPA,which has been reportedoften but never investigated deeply,is that the Green's function becomes non-analytic for certain energies for certain sets of parameters.We have analysed thisproblem here and have found that the generalisations of CPA that involve aneffective medium denoted by a self-energy matrix should inevitably suffer from thisproblem.  \nIn §2 we categorise some of the effective medium approaches in two classes termedas sin","cbCaigGjGdLxce42","https://ap.wps.com/l/cbCaigGjGdLxce42","pdf",967680,23,"English","# 1. Introduction\n## 2. Cluster generalisation of CPA\n## 3. Non-analyticity problem\n## 4. Numerical results for density of states","[{\"question\":\"What is the main focus of the paper on cluster extensions of CPA?\",\"answer\":\"The paper studies how different decoupling schemes in CPA-based cluster extensions affect self-consistent calculations and the resulting density of states for random alloys.\"},{\"question\":\"Why are self-consistent CPA cluster calculations difficult in three-dimensional systems?\",\"answer\":\"The paper explains that the cluster generalisations require solving overly complex simultaneous conditions, making numerical self-consistent computation immensely difficult to perform in three dimensions.\"},{\"question\":\"What does the paper conclude about non-analyticities in CPA extensions?\",\"answer\":\"It shows non-analyticities are a general and inevitable feature of CPA extensions within the multiple scattering framework, contradicting earlier beliefs about certain extensions being analytic.\"}]","Cluster approaches to random alloys - An appraisal | PDF",58]