{"id":66,"date":"2017-02-16T09:32:01","date_gmt":"2017-02-16T09:32:01","guid":{"rendered":"http:\/\/users.utu.fi\/sajuhak\/?page_id=66"},"modified":"2018-04-14T08:52:45","modified_gmt":"2018-04-14T06:52:45","slug":"cpg","status":"publish","type":"page","link":"https:\/\/users.utu.fi\/sajuhak\/cpg\/","title":{"rendered":"Crystallographic Point Groups"},"content":{"rendered":"

Point groups are symmetry operations with a common point that form a group. Point groups have an order number associated with them, with 1 being the least symmetric and then the number increases with increasing symmetry.<\/p>\n

On this page, all highest order symmetry operations in each crystallographic point group are demonstrated on objects which have these symmetries. Crystallographic point groups are those 3-dimensional point groups which have translational symmetry, so the only permitted rotation symmetries are 1, 2, 3, 4, and 6-fold rotations. There are 32 crystallographic point groups in total with order numbers from 1 to 48. Point group names are given in Sch\u00f6nflies notation<\/a> and Hermann-Mauguin notation<\/a>.<\/p>\n

Triclinic<\/h1>\n

Triclinic point groups contain only identity and inversion.<\/p>\n

Only identity:<\/strong>
\nSch\u00f6nflies: \"C_1\"
\nHermann-Mauguin:\u00a0\"\mathbf{1}\"
\nPoint group order: 1<\/p>\n

\nhttp:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/triclinic-C_1-0000-0030.webm<\/a><\/video><\/div>\n

Inversion:<\/strong>
\nSch\u00f6nflies: \"C_i\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{1}}\"
\nPoint group order: 2<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/triclinic-C_i-0000-0060.webm<\/a><\/video><\/div>\n

Monoclinic<\/h1>\n

Monoclinic point groups have one 2-fold rotation axis.<\/p>\n

One 2-fold axis of rotation or a mirror plane:<\/strong>
\nSch\u00f6nflies: \"C_2\"
\nHermann-Mauguin:\u00a0\"\mathbf{2}\"
\nPoint group order: 2<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/monoclinic-C_2-0000-0030.webm<\/a><\/video><\/div>\n

One 2-fold axis of rotoinversion:<\/strong>
\nSch\u00f6nflies: \"C_{1h}\" or \"C_s\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{2}}\" or \"\mathbf{m}\"
\nPoint group order: 2<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/monoclinic-C_1h-0000-0090.webm<\/a><\/video><\/div>\n

One 2-fold axis of rotation and a mirror plane perpendicular to it:<\/strong>
\nSch\u00f6nflies: \"C_{2h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\frac{2}{m}}\"
\nPoint group order: 4<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/monoclinic-C_2h-0000-0060.webm<\/a><\/video><\/div>\n

Orthorhombic<\/h1>\n

Orthorhombic point groups have three 2-fold axes of rotation or two mirror planes and one 2-fold axis of rotation.<\/p>\n

Three 2-fold axes of rotation:<\/strong>
\nSch\u00f6nflies: \"D_{2}\" or \"V\"
\nHermann-Mauguin:\u00a0\"\mathbf{222}\"
\nPoint group order: 4<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/orthorhombic-D_2-0000-0090.webm<\/a><\/video><\/div>\n

Two mirror planes and one 2-fold axis of rotation parallel to both of them:<\/strong>
\nSch\u00f6nflies: \"C_{2v}\"
\nHermann-Mauguin:\u00a0\"\mathbf{mm2}\"
\nPoint group order: 4<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/orthorhombic-C_2v-0000-0090.webm<\/a><\/video><\/div>\n

Three 2-fold axes of rotation with mirror planes perpendicular to each of them:<\/strong>
\nSch\u00f6nflies: \"D_{2h}\" or \"V_{h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\frac{2}{m}\frac{2}{m}\frac{2}{m}}\" or \"\mathbf{mmm}\"
\nPoint group order: 8<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/orthorhombic-D_2h-0000-0180.webm<\/a><\/video><\/div>\n

Trigonal<\/h1>\n

Trigonal point groups have one 3-fold axis of rotation.<\/p>\n

One 3-fold axis of rotation:<\/strong>
\nSch\u00f6nflies: \"C_{3}\"
\nHermann-Mauguin:\u00a0\"\mathbf{3}\"
\nPoint group order: 3<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/trigonal-C_3-0000-0030.webm<\/a><\/video><\/div>\n

One 3-fold and one 2-fold axis of rotation:<\/strong>
\nSch\u00f6nflies: \"D_{3}\"
\nHermann-Mauguin:\u00a0\"\mathbf{32}\"
\nPoint group order: 6<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/trigonal-D_3-0000-0060.webm<\/a><\/video><\/div>\n

One 3-fold axis of rotation and a mirror plane parallel to it:<\/strong>
\nSch\u00f6nflies: \"C_{3v}\"
\nHermann-Mauguin:\u00a0\"\mathbf{3m}\"
\nPoint group order: 6<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/trigonal-C_3v-0000-0060.webm<\/a><\/video><\/div>\n

One 3-fold axis of rotoinversion:<\/strong>
\nSch\u00f6nflies: \"S_{6}\" or \"C_{3i}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{3}}\"
\nPoint group order: 6<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/trigonal-S_6-0000-0060.webm<\/a><\/video><\/div>\n

One 3-fold axis of rotation, inversion, and one 2-fold axis of rotation with a mirror plane perpendicular to it:<\/strong>
\nSch\u00f6nflies: \"D_{3d}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{3}\frac{2}{m}}\" or \"\mathbf{\bar{3}m}\"
\nPoint group order: 12<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/trigonal-D_3d-0000-0090.webm<\/a><\/video><\/div>\n

Tetragonal<\/h1>\n

Tetragonal point groups have one 4-fold axis of rotation.<\/p>\n

One 4-fold axis of rotation:<\/strong>
\nSch\u00f6nflies: \"C_{4}\"
\nHermann-Mauguin:\u00a0\"\mathbf{4}\"
\nPoint group order: 4<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/tetragonal-C_4-0000-0030.webm<\/a><\/video><\/div>\n

One 4-fold axis of rotoinversion:<\/strong>
\nSch\u00f6nflies: \"S_{4}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{4}}\"
\nPoint group order: 4<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/tetragonal-S_4-0000-0060.webm<\/a><\/video><\/div>\n

One 4-fold axis of rotation and two 2-fold axes of rotation:<\/strong>
\nSch\u00f6nflies: \"D_{4}\"
\nHermann-Mauguin:\u00a0\"\mathbf{422}\"
\nPoint group order: 8<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/tetragonal-D_4-0000-0090.webm<\/a><\/video><\/div>\n

One 4-fold axis of rotoinversion, one 2-fold axis of rotation and one mirror plane:<\/strong>
\nSch\u00f6nflies: \"D_{2d}\" or \"V_{d}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{4}2m}\"
\nPoint group order: 8<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/tetragonal-D_2d-0000-0120.webm<\/a><\/video><\/div>\n

One 4-fold axis of rotation with two mirror planes parallel to it:<\/strong>
\nSch\u00f6nflies: \"C_{4v}\"
\nHermann-Mauguin:\u00a0\"\mathbf{4mm}\"
\nPoint group order: 8<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/tetragonal-C_4v-0000-0090.webm<\/a><\/video><\/div>\n

One 4-fold and two 2-fold axes of rotation with mirror planes perpendicular to each of them:<\/strong>
\nSch\u00f6nflies: \"D_{4h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\frac{4}{m}\frac{2}{m}\frac{2}{m}}\" or \"\mathbf{\frac{4}{m}mm}\"
\nPoint group order: 8<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/tetragonal-D_4h-0000-0180.webm<\/a><\/video><\/div>\n

One 4-fold axis of rotation with a mirror plane perpendicular to it:<\/strong>
\nSch\u00f6nflies: \"C_{4h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\frac{4}{m}}\"
\nPoint group order: 16<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/tetragonal-C_4h-0000-0060.webm<\/a><\/video><\/div>\n

Hexagonal<\/h1>\n

Hexagonal point groups have one 6-fold axis of rotation.<\/p>\n

One 6-fold axis of rotation:<\/strong>
\nSch\u00f6nflies: \"C_{6}\"
\nHermann-Mauguin:\u00a0\"\mathbf{6}\"
\nPoint group order: 6<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/hexagonal-C_6-0000-0030.webm<\/a><\/video><\/div>\n

One 6-fold axis of rotoinversion or one 3-fold axis of rotation with a mirror plane perpendicular to it:<\/strong>
\nSch\u00f6nflies: \"C_{3h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{6}}\"
\nPoint group order: 6<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/hexagonal-C_3h-0000-0120.webm<\/a><\/video><\/div>\n

One 6-fold and two 2-fold axes of rotation:<\/strong>
\nSch\u00f6nflies: \"D_{6}\"
\nHermann-Mauguin:\u00a0\"\mathbf{622}\"
\nPoint group order: 12<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/hexagonal-D_6-0000-0090.webm<\/a><\/video><\/div>\n

One 6-fold axis of rotoinversion, one mirror plane, and one 2-fold axis of rotation:<\/strong>
\nSch\u00f6nflies: \"D_{3h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{6}m2}\"
\nPoint group order: 12<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/hexagonal-D_3h-0000-0120.webm<\/a><\/video><\/div>\n

One 6-fold axis of rotation with two mirror planes parallel to it:<\/strong>
\nSch\u00f6nflies: \"C_{6v}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{6}mm}\"
\nPoint group order: 12<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/hexagonal-C_6v-0000-0090.webm<\/a><\/video><\/div>\n

One 6-fold axis of rotation with a mirror plane perpendicular to it:<\/strong>
\nSch\u00f6nflies: \"C_{6h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\frac{6}{m}}\"
\nPoint group order: 12<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/hexagonal-C_6h-0000-0060.webm<\/a><\/video><\/div>\n

One 6-fold and two 2-fold axes of rotation with mirror planes perpendicular to each of them:<\/strong>
\nSch\u00f6nflies: \"D_{6h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\frac{6}{m}mm}\" or \"\mathbf{\frac{6}{m}\frac{2}{m}\frac{2}{m}}\"
\nPoint group order: 24<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/hexagonal-D_6h-0000-0180.webm<\/a><\/video><\/div>\n

Cubic<\/h1>\n

Cubic point groups have four 3-fold axes of rotation.<\/p>\n

One 2-fold and one 3-fold axis of rotation:<\/strong>
\nSch\u00f6nflies: \"T\"
\nHermann-Mauguin:\u00a0\"\mathbf{23}\"
\nPoint group order: 12<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/cubic-T-0000-0060.webm<\/a><\/video><\/div>\n

One 4-fold axis of rotoinversion, one 3-fold axis of rotation, and one mirror plane:<\/strong>
\nSch\u00f6nflies: \"T_{d}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{4}3m}\"
\nPoint group order: 24<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/cubic-T_d-0000-0120.webm<\/a><\/video><\/div>\n

One 2-fold axis of rotation with a mirror plane perpendicular to it and one 3-fold axis of rotation:<\/strong>
\nSch\u00f6nflies: \"T_{h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\frac{2}{m}\bar{3}}\" or \"\mathbf{m\bar{3}}\"
\nPoint group order: 24<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/cubic-T_h-0000-0120.webm<\/a><\/video><\/div>\n

One 4-fold, one 2-fold, and one 3-fold axis of rotation and inversion:<\/strong>
\nSch\u00f6nflies: \"O\"
\nHermann-Mauguin:\u00a0\"\mathbf{\bar{432}}\"
\nPoint group order: 24<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/cubic-O-0000-0090.webm<\/a><\/video><\/div>\n

One 4-fold axis of rotation with a mirror plane perpendicular to it, one 3-fold axis of rotation, inversion, and one 2-fold axis of rotation with a mirror plane perpendicular to it:<\/strong>
\nSch\u00f6nflies: \"O_{h}\"
\nHermann-Mauguin:\u00a0\"\mathbf{\frac{4}{m}\bar{3}\frac{2}{m}}\" or \"\mathbf{m\bar{3}m}\"
\nPoint group order: 48<\/p>\n

http:\/\/users.utu.fi\/sajuhak\/wp-content\/uploads\/sites\/177\/2017\/02\/cubic-Oh-0000-0180.webm<\/a><\/video><\/div>\n","protected":false},"excerpt":{"rendered":"

Point groups are symmetry operations with a common point that form a group. Point groups have an order number associated with them, with 1 being the least symmetric and then the number increases with increasing symmetry. On this page, all highest order symmetry operations in each crystallographic point group are demonstrated on objects which have … Continue reading “Crystallographic Point Groups”<\/span><\/a><\/p>\n","protected":false},"author":186,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-66","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/users.utu.fi\/sajuhak\/wp-json\/wp\/v2\/pages\/66","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/users.utu.fi\/sajuhak\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.utu.fi\/sajuhak\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.utu.fi\/sajuhak\/wp-json\/wp\/v2\/users\/186"}],"replies":[{"embeddable":true,"href":"https:\/\/users.utu.fi\/sajuhak\/wp-json\/wp\/v2\/comments?post=66"}],"version-history":[{"count":7,"href":"https:\/\/users.utu.fi\/sajuhak\/wp-json\/wp\/v2\/pages\/66\/revisions"}],"predecessor-version":[{"id":334,"href":"https:\/\/users.utu.fi\/sajuhak\/wp-json\/wp\/v2\/pages\/66\/revisions\/334"}],"wp:attachment":[{"href":"https:\/\/users.utu.fi\/sajuhak\/wp-json\/wp\/v2\/media?parent=66"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}