{"id":1354,"date":"2014-09-15T12:15:04","date_gmt":"2014-09-15T03:15:04","guid":{"rendered":"http:\/\/www.lookingatnothing.com\/?p=1354"},"modified":"2015-03-09T14:02:49","modified_gmt":"2015-03-09T05:02:49","slug":"guinier-in-polydisperse-systems-revisited","status":"publish","type":"post","link":"https:\/\/lookingatnothing.com\/index.php\/archives\/1354","title":{"rendered":"Guinier in polydisperse systems revisited"},"content":{"rendered":"<p>Last week, I was contacted by Christian Gollwitzer (author of <a href=\"http:\/\/dx.doi.org\/10.1107\/S1600576713031981\">this excellent paper<\/a>) about something posted <a href=\"http:\/\/www.lookingatnothing.com\/index.php\/archives\/201\">here<\/a> (and in particular detailed <a href=\"http:\/\/www.lookingatnothing.com\/wp-content\/uploads\/2010\/12\/Guinier_short.pdf\">in this document<\/a>) a while ago. It concerns the behaviour of the Guinier approxiation for polydisperse systems, and it looks like I made a mistake when writing that document.<!--more-->The document states that the radius of gyration used in the Guinier method for polydisperse systems will be the volume-squared-weighted radius of gyration. However, Christian heard a slightly different message from his colleagues. Time to investigate.<\/p>\n<p>The easiest and most fun way to investigate this is to redo the simulations and fits. It is easy to simulate scattering patterns from polydisperse spheres with a variety of polydispersities, and to then fit these to a particular model to see how well the model works.<\/p>\n<p>For those interested, the derivation and some more background information and references on the true result is jotted down in this document: <a href=\"http:\/\/www.lookingatnothing.com\/wp-content\/uploads\/2014\/09\/Guinier_limits_revisited.pdf\">Guinier_limits_revisited<\/a>. The Python code used for the simulations and plotting can be downloaded from this link: <a href=\"http:\/\/www.lookingatnothing.com\/wp-content\/uploads\/2014\/09\/guinierfit.zip\">guinierfit<\/a> (warning: the code doesn&#8217;t look pretty at all!).<\/p>\n<p>In summary, the literature is correct, and the radius of gyration is the square root of that value I derived three years ago: it is the square-root of the volume-squared weighted average radius. However, this is not the same as the volume-weighted or intensity weighted value. Literature references to the correct equation can be found in Otto Glatter&#8217;s 1982 book and in (f.ex.) <a href=\"http:\/\/dx.doi.org\/10.1107\/S0021889804008969\">this 2004 paper by Gr\u00e9gory Beaucage<\/a>. Derivations for Guinier in polyidsperse systems appear to date back to at least 1964, in <a href=\"http:\/\/dx.doi.org\/10.1016\/0001-6160(64)90132-4\">this paper by R. Baur and V. Gerold<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<div class=\"mh-excerpt\"><p>Last week, I was contacted by Christian Gollwitzer (author of this excellent paper) about something posted here (and in particular detailed in this document) a <a class=\"mh-excerpt-more\" href=\"https:\/\/lookingatnothing.com\/index.php\/archives\/1354\" title=\"Guinier in polydisperse systems revisited\">[&#8230;]<\/a><\/p>\n<\/div>","protected":false},"author":2,"featured_media":1357,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"activitypub_content_warning":"","activitypub_content_visibility":"","activitypub_max_image_attachments":4,"activitypub_interaction_policy_quote":"anyone","activitypub_status":"","footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1,52,85],"tags":[],"class_list":["post-1354","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized","category-lit","category-looking-into-something"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/lookingatnothing.com\/wp-content\/uploads\/2014\/09\/Rg_comparisons.png","jetpack_shortlink":"https:\/\/wp.me\/p1gZ2v-lQ","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/posts\/1354","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/comments?post=1354"}],"version-history":[{"count":2,"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/posts\/1354\/revisions"}],"predecessor-version":[{"id":1359,"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/posts\/1354\/revisions\/1359"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/media\/1357"}],"wp:attachment":[{"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/media?parent=1354"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/categories?post=1354"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lookingatnothing.com\/index.php\/wp-json\/wp\/v2\/tags?post=1354"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}