<\/p>\n\n\n\n
On consid\u00e8re un cristal \u00e9clair\u00e9 par un rayonnement X incident caract\u00e9ris\u00e9 par une onde plane monochromatique, de vecteur d’onde ![\"\overrightarrow{q}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-e89732e8f9bc07ead0b9ef5b14dff557_l3.png\" "\\"Rendered")
et d’amplitude ![\"A_q\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-d3eef49b81799711f09913aa9ef665e0_l3.png\" "\\"Rendered")
. Le facteur de diffusion atomique ![\"f_Y(\overrightarrow{q},\overrightarrow{q}')\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-b1bcda8f34d508796f93fcdf4c1314ad_l3.png\" "\\"Rendered")
d’un \u00e9l\u00e9ment chimique ![\"Y\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png\" "\\"Rendered")
est d\u00e9fini par ![\"A_{q'} = f_Y(\overrightarrow{q},\overrightarrow{q}')A_{q}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-e3be68331a17f9b1e7a2bb58e9def2bf_l3.png\" "\\"Rendered")
. Dans la suite, on consid\u00e8rera que la diffusion est \u00e9lastique de sorte que la diffraction par le cristal se ram\u00e8ne \u00e0 l’\u00e9tude d’interf\u00e9rence entre les ondes planes monochromatiques diffus\u00e9es par les atomes de la structure cristalline.<\/p>\n\n\n\n
D\u00e9phasage entre deux ondes diffus\u00e9es :<\/strong> Sur le sch\u00e9ma ci-contre, on a:<\/p>\n\n\n\n La diff\u00e9rence de marche est donc \u00e9gale \u00e0 La diff\u00e9rence de marche entre les deux ondes diffus\u00e9es par <\/a> (1) <\/span> <\/span> Or, on sait que <\/a> (2) <\/span> <\/span> On note Diffraction par un cristal parfait :<\/strong> <\/a> (3) <\/span> <\/span> On rappelle que l’intensit\u00e9 totale est <\/a> (4) <\/span> <\/span> En effet, dans l’expression de l’amplitude, le premier facteur caract\u00e9rise le motif, et on peut donc poser:<\/p>\n\n\n\n <\/a> (5) <\/span> <\/span> De plus, l’autre membre de l’expression de l’amplitude d\u00e9pend uniquement du r\u00e9seau de Bravais. On \u00e9crira alors:<\/p>\n\n\n\n (6) <\/span> <\/span> On en d\u00e9duit alors imm\u00e9diatement que l’intensit\u00e9 est maximale lorsque <\/a> (7) <\/span> <\/span> Conclusion :<\/strong> On consid\u00e8re un cristal \u00e9clair\u00e9 par un rayonnement X incident caract\u00e9ris\u00e9 par une onde plane monochromatique…<\/p>\n","protected":false},"author":1,"featured_media":490,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"rop_custom_images_group":[],"rop_custom_messages_group":[],"rop_publish_now":"initial","rop_publish_now_accounts":[],"rop_publish_now_history":[],"rop_publish_now_status":"pending","_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"categories":[3,4],"tags":[58,59,61,60,63,27,62,57],"class_list":["post-498","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-education","category-physique","tag-bravais","tag-cristal","tag-diffraction","tag-matiere-condensee","tag-ondes","tag-physique","tag-rayons-x","tag-reseau"],"_links":{"self":[{"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/posts\/498","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/comments?post=498"}],"version-history":[{"count":27,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/posts\/498\/revisions"}],"predecessor-version":[{"id":551,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/posts\/498\/revisions\/551"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/media\/490"}],"wp:attachment":[{"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/media?parent=498"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/categories?post=498"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.leotheodon.com\/V4\/wp-json\/wp\/v2\/tags?post=498"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
On consid\u00e8re deux atomes situ\u00e9s en ![\"O\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-41136ed7463f7254f4e6df131f8be490_l3.png\" "\\"Rendered")
et ![\"P\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" "\\"Rendered")
, interagissant avec l’onde incident de vecteur d’onde et \u00e9mettant chacun une onde de vecteur d’onde ![\"\overrightarrow{q}'\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-7f9225f7e16531ff73264e92308e2984_l3.png\" "\\"Rendered")
.
<\/p>\n\n\n\n\n\n![\"(AP)\perp(AO)\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-1fa6c539f21cbbc3aee8994f559a49d3_l3.png\" "\\"Rendered")
<\/li>![\"(BP)\perp(BO)\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-fefa5061a667cfd8d4e150cb3816ef9e_l3.png\" "\\"Rendered")
<\/li><\/ul>\n\n\n\n![\"AP + PB\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-8edfefdc7f3e57a1311e605069034b1e_l3.png\" "\\"Rendered")
.<\/p>\n<\/div>\n\n\n\n\n![\"Rendered](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-ea08e8225ab1a4b6faaec3cb68436943_l3.png\" "\\"Rendered")
<\/p><\/p>\n<\/div>\n<\/div>\n\n\n\n et est ![\"\delta = AP + PB\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-d82c47f4b64ff4d67740094d14869a8b_l3.png\" "\\"Rendered")
. On a alors la relation suivante:<\/p>\n\n\n\n![\"\begin{equation*}](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-8bf6210bc676e9ebc85972c1f8f79b80_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\Vert \overrightarrow{q} \Vert = \Vert \overrightarrow{q}' \Vert = \frac{2\pi}{\lambda}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-cd24d0a2517a6a441bd1507ddef6a89f_l3.png\" "\\"Rendered")
o\u00f9 ![\"\lambda\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-2b5c45836864531b8e37025dabadd24a_l3.png\" "\\"Rendered")
est la longueur d’onde. On en d\u00e9duit l’expression du d\u00e9phasage:<\/p>\n\n\n\n![\"\begin{equation*} \Delta(\varphi) = \Vert \overrightarrow{q} \Vert \delta = \overrightarrow{OP}.(\overrightarrow{q}-\overrightarrow{q}') = \overrightarrow{OP}.\overrightarrow{K} \end{equation*}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-ac624255ae4fd7b44a30bf198846a9fa_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\overrightarrow{K} = \overrightarrow{q}-\overrightarrow{q}'\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-183271785f533e7cd84b95c37880424e_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\n
On consid\u00e8re un cristal caract\u00e9ris\u00e9 par un motif contenant ![\"M\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-10ebb71bad275c1815a8f2a8c5dea0be_l3.png\" "\\"Rendered")
atomes, dont les positions sont d\u00e9termin\u00e9es par ![\"\overrightarrow{OP} = \overrightarrow{T} + \overrightarrow{\rho} = \sum_i n_i \overrightarrow{a_i} + (\alpha \overrightarrow{a_1} + \beta \overrightarrow{a_2} + \gamma \overrightarrow{a_3})\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-4cfb71410ad1ba7aa9dd150574080f3f_l3.png\" "\\"Rendered")
, o\u00f9 les ![\"\overrightarrow{a_i}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-cf9708807eb79603c953d9b7c9332c72_l3.png\" "\\"Rendered")
forment une base adapt\u00e9e au r\u00e9seau de Bravais, les ![\"n_i\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-1ffdddde68452977938ddffe328f4b78_l3.png\" "\\"Rendered")
sont des coefficients entiers, et ![\"\alpha\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-8f0b6b1a01f8fcc2f95be0364c090397_l3.png\" "\\"Rendered")
, ![\"\beta\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-b6a7605b1bcca8f1b416eaf733f34e08_l3.png\" "\\"Rendered")
et ![\"\gamma\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-4de02fc502ed5dbd15f371728ea270a3_l3.png\" "\\"Rendered")
sont des coefficients r\u00e9els. Le cristal est de dimension ![\"N_i\vert \overrightarrow{a_i}\vert\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-28dddb56c41c418afff1699c3d4a3762_l3.png\" "\\"Rendered")
pour ![\"i = 1,2,3\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-3d4782ed4b5fe15db9eb6febb90340b6_l3.png\" "\\"Rendered")
. L’amplitude totale diffract\u00e9e par le cristal est la somme des amplitudes des ondes \u00e9mises par chaque atome du cristal.<\/p>\n\n\n\n![\"\begin{equation*}A_{tot}(\overrightarrow{K}) = A_q \displaystyle \sum_{j=1}^M f_{Y_j}(\overrightarrow{q},\overrightarrow{q}')e^{i\overrightarrow{\rho}_j.\overrightarrow{K}}\sum_{n_1=1}^{N_1} \sum_{n_2=1}^{N_2} \sum_{n_3=1}^{N_3} e^{i(n_1 \overrightarrow{a_1}.\overrightarrow{K} + n_2 \overrightarrow{a_2}.\overrightarrow{K} + n_3 \overrightarrow{a_3}.\overrightarrow{K})}\end{equation*}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-66f7956371ee81193e6f2238f9703f55_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"I_{tot} = A_{tot}.A^*_{tot}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-2f42b66c08cc827b0a1fdd0f008853a0_l3.png\" "\\"Rendered")
. D’apr\u00e8s ce qui pr\u00e9c\u00e8de, on voit que l’intensit\u00e9 totale peut s’\u00e9crire sous la forme suivante:<\/p>\n\n\n\n![\"\begin{equation*} I_{tot}(\overrightarrow{K}) = I_{motif}(\overrightarrow{K}).I_{reseau}(\overrightarrow{K}) \end{equation*}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-1d7f4acb9ec2b62d0439db52d031f1ab_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\begin{equation*}](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-bebbea8177a77861f5dc2fab7b93c53d_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n![\"\begin{align*}](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-4bd6262190ac39e55e6a1ac7d2ea9b13_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"\overrightarrow{a_i}.\overrightarrow{K} = 2n\pi, \forall i =1..3\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-8cc89304fd490de98aeef991db9c52b5_l3.png\" "\\"Rendered")
, avec ![\"n\in\mathbb{N}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-588616762f1acc1b153991dd6fa4f83c_l3.png\" "\\"Rendered")
(condition de <\/strong>Laue<\/strong><\/em>). En particulier, cette condition peut s’\u00e9crire sous la forme suivante:<\/p>\n\n\n\n![\"\begin{equation*}](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-d9f9be13bf9e4e383244b6dfd9c62292_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n
Ce dernier r\u00e9sultat signifie que l’intensit\u00e9 est maximale lorsque ![\"\overrightarrow{K}\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-7821fac019ecc8ea16a4ea614f93c518_l3.png\" "\\"Rendered")
appartient au r\u00e9seau r\u00e9ciproque du cristal. De plus, on retrouve bien le fait que l’intensit\u00e9 est maximale lorsque ![\"\overrightarrow{q} = \overrightarrow{q}'\"](\"https:\/\/www.leotheodon.com\/V4\/wp-content\/ql-cache\/quicklatex.com-01bc9840302568d60ea4e1411bbab107_l3.png\" "\\"Rendered")
(c’est-\u00e0-dire que l’on observe l’onde en sortie du cristal dans la m\u00eame direction que l’onde incidente).<\/p>\n\n\n\nT\u00e9l\u00e9charger au format pdf :<\/a>T\u00e9l\u00e9charger<\/a><\/div>\n\n\n\n
<\/p>\n","protected":false},"excerpt":{"rendered":"