[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-665620-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-665620-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","alkemade-lines-binary-joins-alkemade-theorem-and-composition-triangles","Alkemade Lines - Binary Joins, Alkemade Theorem, and Composition Triangles","","Explains Alkemade lines in ternary phase diagrams and how they simplify interpretation of ternary phase relations. Defines a binary join and an Alkemade line based on whether two regions with a common boundary share primary crystal compositions. Describes the Alkemade theorem linking intersections between boundary lines and corresponding Alkemade lines to temperature maxima and minima, and uses it to determine slopes toward eutectic points. Introduces composition triangles formed by three non-crossing Alkemade lines, with an example workflow for labeling primary phase fields and drawing Alkemade lines.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/alkemade-lines-binary-joins-alkemade-theorem-and-composition-triangles/665620/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/alkemade-lines-binary-joins-alkemade-theorem-and-composition-triangles/665620.png","ImageObject",300,407,{"name":42,"@type":43},"River Wang","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What distinguishes a binary join from an Alkemade line in a ternary system?","Question",{"text":62,"@type":63},"A binary join connects compositions representing crystal phases. An Alkemade line is a join between the compositions of primary crystals of two areas that share a common boundary line, which makes the join satisfy the adjacency condition.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does the Alkemade theorem relate boundary lines to temperature behavior?",{"text":67,"@type":63},"It states that the intersection of a boundary line (or its extension) with the corresponding Alkemade line (or its extension) indicates a temperature maximum on the boundary line and a temperature minimum on the Alkemade line.",{"name":69,"@type":60,"acceptedAnswer":70},"What is a composition triangle, and how is it constructed?",{"text":71,"@type":63},"In a ternary diagram, Alkemade lines divide the diagram into composition triangles. Each composition triangle is formed by three Alkemade lines, and Alkemade lines never cross each other.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},665620,1793305257,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":33,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":30,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":79,"read_time":111},1099514067438,"https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542","2-4-2 Alkemade Lines:  \nA straight line connecting the points representing the compositions of the crystals in the ternary system is called a binary join. A join connecting the compositions of the primary crystals of two areas having a common boundary line is called an Alkemade line. The use of Alkemade lines simplifies the analysis and interpretation of ternary phase relations. The following example should help to clarify the distinction between joins and Alkemade lines.  \nIn the system shown in fig.(35), there is an intermediate binary compound BC which has a primary phase field labeled bc. The line A-BC is a join connecting the compositions of the primary crystals of two areas having a common boundary line, ED, and therefore, the line A-BC is an Alkemade line. Lines AC, AB, B-BC, and BC-C are also Alkemade lines. Line BC, although a join, does not fulfil1 the condition of the common boundary line because areas c and b are not adjacent; consequently, BC isnot an Alkemade line.  \nFig (35) Join BC is not an Alkemade line.  \n1  \nThe Alkemade theorem states that the intersection of a boundary line (or boundary line extended) with its corresponding Alkemade line (or Alkemadeline extended) represents a temperature maximum on that boundary line anda temperature minimum on the Alkemade line. The Alkemade theorem permits one to analyze with respect to temperature the direction of slope of boundary lines.  \nIn fig.(36) the slopes of the boundary lines for the system A-B-C have been determined by applying the Alkemade theorem. Three boundary lines slope to a common junction at E and at D. These points are eutectic points.  \nFig. (36) Application ofthe Alkemade theorem to determine slopes of  \nboundary lines.  \n2-4-2 Composition Triangles:  \nAlkemade lines divide a ternary diagram into composition triangles. In the diagram of fig.(35), triangles A-BC-B and A-BC-C are composition triangles. A composition triangle is always made up of three Alkemade lines.  \n2  \nIt should be noted that Alkemade lines never cross one another.  \nExample: For fig. (37), label primary phase fields and construct Alkemade lines.  \nFig. (37) Ternary systems for example.  \n3","cbCaifV2FtIF1eXe","https://ap.wps.com/l/cbCaifV2FtIF1eXe","pdf",270462,"English","# 2-4-2 Alkemade Lines\n## Binary joins vs. Alkemade lines\n## Alkemade theorem and slope analysis\n# 2-4-2 Composition Triangles\n## Composition triangles from Alkemade lines\n## Non-crossing property\n## Example: labeling phase fields and constructing lines","[{\"question\":\"What distinguishes a binary join from an Alkemade line in a ternary system?\",\"answer\":\"A binary join connects compositions representing crystal phases. An Alkemade line is a join between the compositions of primary crystals of two areas that share a common boundary line, which makes the join satisfy the adjacency condition.\"},{\"question\":\"How does the Alkemade theorem relate boundary lines to temperature behavior?\",\"answer\":\"It states that the intersection of a boundary line (or its extension) with the corresponding Alkemade line (or its extension) indicates a temperature maximum on the boundary line and a temperature minimum on the Alkemade line.\"},{\"question\":\"What is a composition triangle, and how is it constructed?\",\"answer\":\"In a ternary diagram, Alkemade lines divide the diagram into composition triangles. Each composition triangle is formed by three Alkemade lines, and Alkemade lines never cross each other.\"}]","Alkemade Lines - Binary Joins, Alkemade Theorem, and Composition Triangles | PDF"]