空間群と反射条件 (出現則/消滅則)

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 回折とは、結晶に対して特定の条件(ブラッグ条件)で波を入射したとき散乱波が強め合う現象のことです。ただし、ブラッグ条件を満たしたからと言って、必ず回折が起きるわけではありません。結晶が複合格子並進、らせん、映進などの対称操作を有している場合は、たとえブラッグ条件を満たしても、特定の面指数の回折波の強度は消滅1します。このページでは、回折が観測された結晶面指数のリストから空間群を絞り込むための方法を提供します。

 以下に示す表は、回折によって出現する面指数と空間群の関係一覧を示します。消滅する面指数でないことにご注意ください。

  • 反射条件(Reflection conditions)は面指数のタイプによって分類されており、たとえば上段に \(h0l\) とある場合は 「\(h\) と \(l\) は任意だが \(k\) が \(0\) である結晶面」が条件の対象となります。 \(hkl\) と書かれていたらすべての結晶面が対象となりますし、\(hhl\) と書かれていたら\(h=k\) であるような結晶面が対象です。
  • 条件中の \(h, k, l, h+k\) などはその値が偶数のときに出現することを意味しています。\(h,l\) などカンマで区切られている場合は「 \(h\) と \(l\) がともに偶数」が出現条件であることを意味します。偶数以外の出現条件は \(=3n,\,\, =4n,\,\, =6n\) などと明記されています。空欄の場合は、その面指数タイプに対して出現規則がないことを意味します。
  • 空間群の表記は、標準表記でないものもありますのでご注意ください。空間群表記に続くカッコ内の数値はInternational Tables for Crystallography, Vol A. に記載されている空間群番号を示しています。

Triclinic 三斜晶系

Reflection conditionsExtinction
symbol
Point group
\(1\)\(\bar{1}\)
None\(P – \)\(P1\, (1)\)\(P\bar{1}\, (2)\)

Monoclinic 単斜晶系

Unique axis b

Reflection conditionsExtinction
symbol
Laue class \(1 2/m 1\)
Point group
\(hkl, 0kl, hk0\)\(h0l, h00, 00l\)\(0k0\)\(2\)\(m\)\(2/m\)
\(P1 – 1\)\(P121\, (3)\)\(P1m1\, (6)\)\(P1 2/m 1\, (10)\)
\(k\)\(P12_11\)\(P12_11\, (4)\)\(P1 2_1/m 1\, (11)\)
\(h\)\(P1a1\)\(P1a1\, (7)\)\(P1 2/a 1\, (13)\)
\(h\)\(k\)\(P1 2_1/a 1\)\(P1 2_1/a 1\, (14)\)
\(l\)\(P1c1\)\(P1c1\, (7)\)\(P1 2/c 1\, (13)\)
\(l\)\(k\)\(P1 2_1/c 1\)\(P1 2_1/c 1\, (14)\)
\(h + l\)\(P1n1\)\(P1n1\, (7)\)\(P1 2/n 1\, (13)\)
\(h + l\)\(k\)\(P1 2_1/n 1\)\(P1 2_1/n 1\, (14)\)
\(h + k\)\(h\)\(k\)\(C1 – 1\)\(C121\, (5)\)\(C1m1\, (8)\)\(C1 2/m 1\, (12)\)
\(h + k\)\(h, l\)\(k\)\(C1c1\)\(C1c1\, (9)\)\(C1 2/c 1\, (15)\)
\(h + l\)\(l\)\(k\)\(A1 – 1\)\(A121\, (5)\)\(A1m1\, (8)\)\(A1 2/m 1\, (12)\)
\(h + l\)\(h, l\)\(k\)\(A1n1\)\(A1n1\, (9)\)\(A1 2/n 1\, (15)\)
\(h + k + l\)\(h + l\)\(k\)\(I1 – 1\)\(I121\, (5)\)\(I1m1\, (8)\)\(I1 2/m 1\, (12)\)
\(h + k + l\)\(h, l\)\(k\)\(I1a1\)\(I1a1\, (9)\)\(I1 2/a 1\, (15)\)

Unique axis c

Reflection conditionsExtinction
symbol
Laue class \(1 1 2/m\)
Point group
\(hkl, 0kl, h0l\)\(hk0, h00, 0k0\)\(00l\)\(2\)\(m\)\(2/m\)
\(P11 – \)\(P112\, (3)\)\(P11 m\, (6)\)\(P11 2/m\, (10)\)
\(l\)\(P12_11\)\(P112_1\, (4)\)\(P112_1/m\, (11)\)
\(h\)\(P11a\)\(P11a\, (7)\)\(P11 2/a\, (13)\)
\(h\)\(l\)\(P11 2_1/a\)\(P11 2_1/a\, (14)\)
\(k\)\(P11b\)\(P11b\, (7)\)\(P11 2/b\, (13)\)
\(k\)\(l\)\(P11 2_1/b\)\(P11 2_1/b\, (14)\)
\(h + k\)\(P11n\)\(P11n\, (7)\)\(P11 2/n\, (13)\)
\(h + k\)\(l\)\(P11 2_1/n\)\(P1 1 2_1/n\, (14)\)
\(h + l\)\(h\)\(l\)\(B11 – \)\(C112\, (5)\)\(C11m\, (8)\)\(C11 2/m\, (12)\)
\(h + l\)\(h, k\)\(l\)\(B11n\)\(C11c\, (9)\)\(C11 2/c\, (15)\)
\(k + l\)\(k\)\(l\)\(A11 – \)\(A112\, (5)\)\(A11m\, (8)\)\(A11 2/m\, (12)\)
\(k + l\)\(h, k\)\(l\)\(A11a\)\(A11n\, (9)\)\(A11 2/n\, (15)\)
\(h + k + l\)\(h + k\)\(l\)\(I11 – \)\(I112\, (5)\)\(I11m\, (8)\)\(I11 2/m\, (12)\)
\(h + k + l\)\(h, k\)\(l\)\(I11b\)\(I11b\, (9)\)\(I11 2/b\, (15)\)

Unique axis a

Reflection conditionsExtinction
symbol
Laue class \(2/m 1 1\)
Point group
\(hkl, h0l, hk0\)\(0kl, 0k0, 00l\)\(h00\)\(2\)\(m\)\(2/m\)
\(P – 11\)\(P211\, (3)\)\(Pm11\, (6)\)\(P2/m 11\, (10)\)
\(h\)\(P2_111\)\(P2_111\, (4)\)\(P2_1/m 11\, (11)\)
\(k\)\(Pb11\)\(P a 11\, (7)\)\(P2/b 11\, (13)\)
\(k\)\(h\)\(P2_1/b 11\)\(P2_1/b 11\, (14)\)
\(l\)\(Pc11\)\(Pc11\, (7)\)\(P2/c 11\, (13)\)
\(l\)\(h\)\(P2_1/c 11\)\(P2_1/c 11\, (14)\)
\(k + l\)\(Pn11\)\(Pn11\, (7)\)\(P2/n 11\, (13)\)
\(k + l\)\(h\)\(P2_1/n 11\)\(P2_1/n 11\, (14)\)
\(h + k\)\(k\)\(h\)\(C – 11\)\(C211\, (5)\)\(Cm11\, (8)\)\(C2/m 11\, (12)\)
\(h + k\)\(k, l\)\(h\)\(Cn11\)\(Cn11\, (9)\)\(C2/n 11\, (15)\)
\(h + l\)\(l\)\(h\)\(B – 11\)\(B211\, (5)\)\(Bm11\, (8)\)\(B2/m 11\, (12)\)
\(h + l\)\(k, l\)\(h\)\(Bb11\)\(Bb11\, (9)\)\(B2/b 11\, (15)\)
\(h + k + l\)\(k + l\)\(h\)\(I – 11\)\(I211\, (5)\)\(Im11\, (8)\)\(I2/m 11\, (12)\)
\(h + k + l\)\(k, l\)\(h\)\(Ic11\)\(Ic11\, (9)\)\(I2/c 11\, (15)\)

Orthorhombic 直方晶系

Reflection conditionsExtinction
symbol
Laue class mmm \((2/m\, 2/m\, 2/m)\)
Point group
\(hkl\)\(0kl\)\(h0l\)\(hk0\)\(h00\)\(0k0\)\(00l\)\(222\)\(mm2, m2m, 2mm\)\(mmm\)
\(P – – – \)\(P222\, (16)\)\(Pmm2\, (25)\)
\(Pm2m\, (25)\)
\(P2mm\, (25)\)
\(Pmmm\, (47)\)
\(l\)\(P – – 2_1\)\(P222_1\, (17)\)
\(k\)\(P – 2_1 – \)\(P22_12\, (17)\)
\(k\)\(l\)\(P – 2_12_1\)\(P22_12_1\, (18)\)
\(h\)\(P2_1 – – \)\(P2_122\, (17)\)
\(h\)\(l\)\(P2_1 – 2_1\)\(P2_122_1\, (18)\)
\(h\)\(k\)\(P2_12_1 – \)\(P2_12_12\, (18)\)
\(h\)\(k\)\(l\)\(P2_12_12_1\)\(P2_12_12_1\, (19)\)
\(h\)\(h\)\(P – – a\)\(Pm2a\, (28)\)
\( P2_1ma\, (26)\)
\(Pmma\, (51)\)
\(k\)\(k\)\(P – – b\)\(Pm2_1b\, (26)\)
\( P2_1mb\, (28)\)
\(Pmmb\, (51)\)
\(h + k\)\(h\)\(k\)\(P – – n\)\(Pm2_1n\, (31)\)
\( P2_1mn\, (31)\)
\(Pmmn\, (59)\)
\(h\)\(h\)\(P – a – \)\(Pma2\, (28)\)
\( P2_1am\, (26)\)
\(Pmam\, (51)\)
\(h\)\(h\)\(h\)\(P – aa\)\(P2aa\, (27)\)\(Pmaa\, (49)\)
\(h\)\(k\)\(h\)\(k\)\(P – ab\)\(P2_1ab\, (29)\)\(Pmab\, (57)\)
\(h\)\(h + k\)\(h\)\(k\)\(P – an\)\(P2_1an\, (30)\)\(Pman\, (53)\)
\(l\)\(l\)\(P – c – \)\(Pmc2_1\, (26)\)
\( P2cm\, (28)\)
\(Pmcm\, (51)\)
\(l\)\(h\)\(h\)\(l\)\(P – ca\)\(P2_1ca\, (29)\)\(Pmca\, (57)\)
\(l\)\(k\)\(k\)\(l\)\(P – cb\)\(P2cb\, (32)\)\(Pmcb\, (55)\)
\(l\)\(h + k\)\(h\)\(k\)\(l\)\(P – cn\)\(P2_1cn\, (33)\)\(Pmcn\, (62)\)
\(h + l\)\(h\)\(l\)\(P – n – \)\(Pmn2_1\, (31)\)
\( P2_1nm\, (31)\)
\(Pmnm\, (59)\)
\(h + l\)\(h\)\(h\)\(l\)\(P – na\)\(P2na\, (30)\)\(Pmna\, (53)\)
\(h + l\)\(k\)\(h\)\(k\)\(l\)\(P – nb\)\(P2_1nb\, (33)\)\(Pmnb\, (62)\)
\(h + l\)\(h + k\)\(h\)\(k\)\(l\)\(P – nn\)\(P2nn\, (34)\)\(Pmnn\, (58)\)
\(k\)\(k\)\(Pb – – \)\(Pbm2\, (28)\)
\( Pb2_1m\, (26)\)
\(Pbmm\, (51)\)
\(k\)\(h\)\(h\)\(k\)\(Pb – a\)\(Pb2_1a\, (29)\)\(Pbma\, (57)\)
\(k\)\(k\)\(k\)\(Pb – b\)\(Pb2b\, (27)\)\(Pbmb\, (49)\)
\(k\)\(h + k\)\(h\)\(k\)\(Pb – n\)\(Pb2n\, (30)\)\(Pbmn\, (53)\)
\(k\)\(h\)\(h\)\(k\)\(Pba – \)\(Pba2\, (32)\)\(Pbam\, (55)\)
\(k\)\(h\)\(h\)\(h\)\(k\)\(Pbaa\)\(Pbaa\, (54)\)
\(k\)\(h\)\(k\)\(h\)\(k\)\(Pbab\)\(Pbab\, (54)\)
\(k\)\(h\)\(h + k\)\(h\)\(k\)\(Pban\)\(Pban\, (50)\)
\(k\)\(l\)\(k\)\(l\)\(Pbc – \)\(Pbc2_1\, (29)\)\(Pbcm\, (57)\)
\(k\)\(l\)\(h\)\(h\)\(k\)\(l\)\(Pbca\)\(Pbca\, (61)\)
\(k\)\(l\)\(k\)\(k\)\(l\)\(Pbcb\)\(Pbcb\, (54)\)
\(k\)\(l\)\(h + k\)\(h\)\(k\)\(l\)\(Pbcn\)\(Pbcn\, (60)\)
\(k\)\(h + l\)\(h\)\(k\)\(l\)\(Pbn – \)\(Pbn2_1\, (33)\)\(Pbnm\, (62)\)
\(k\)\(h + l\)\(h\)\(h\)\(k\)\(l\)\(Pbna\)\(Pbna\, (60)\)
\(k\)\(h + l\)\(k\)\(h\)\(k\)\(l\)\(Pbnb\)\(Pbnb\, (56)\)
\(k\)\(h + l\)\(h + k\)\(h\)\(k\)\(l\)\(Pbnn\)\(Pbnn\, (52)\)
\(l\)\(l\)\(Pc – – \)\(Pcm2_1\, (26)\)
\( Pc2m\, (28)\)
\(Pcmm\, (51)\)
\(l\)\(h\)\(h\)\(l\)\(Pc – a\)\(Pc2a\, (32)\)\(Pcma\, (55)\)
\(l\)\(k\)\(k\)\(l\)\(Pc – b\)\(Pc2_1b\, (29)\)\(Pcmb\, (57)\)
\(l\)\(h + k\)\(h\)\(k\)\(l\)\(Pc – n\)\(Pc2_1n\, (33)\)\(Pcmn\, (62)\)
\(l\)\(h\)\(h\)\(l\)\(Pca – \)\(Pca2_1\, (29)\)\(Pcam\, (57)\)
\(l\)\(h\)\(h\)\(h\)\(l\)\(Pcaa\)\(Pcaa\, (54)\)
\(l\)\(h\)\(k\)\(h\)\(k\)\(l\)\(Pcab\)\(Pcab\, (61)\)
\(l\)\(h\)\(h + k\)\(h\)\(k\)\(l\)\(Pcan\)\(Pcan\, (60)\)
\(l\)\(l\)\(l\)\(Pcc – \)\(Pcc2\, (27)\)\(Pccm\, (49)\)
\(l\)\(l\)\(h\)\(h\)\(l\)\(Pcca\)\(Pcca\, (54)\)
\(l\)\(l\)\(k\)\(k\)\(l\)\(Pccb\)\(Pccb\, (54)\)
\(l\)\(l\)\(h + k\)\(h\)\(k\)\(l\)\(Pccn\)\(Pccn\, (56)\)
\(l\)\(h + l\)\(h\)\(l\)\(Pcn – \)\(Pcn2\, (30)\)\(Pcnm\, (53)\)
\(l\)\(h + l\)\(h\)\(h\)\(l\)\(Pcna\)\(Pcna\, (50)\)
\(l\)\(h + l\)\(k\)\(h\)\(k\)\(l\)\(Pcnb\)\(Pcnb\, (60)\)
\(l\)\(h + l\)\(h + k\)\(h\)\(k\)\(l\)\(Pcnn\)\(Pcnn\, (52)\)
\(k + l\)\(k\)\(l\)\(Pn – – \)\(Pnm2_1\, (31)\)
\( Pn2_1m\, (31)\)
\(Pnmm\, (59)\)
\(k + l\)\(h\)\(h\)\(k\)\(l\)\(Pn – a\)\(Pn2_1a\, (33)\)\(Pnma\, (62)\)
\(k + l\)\(k\)\(k\)\(l\)\(Pn – b\)\(Pn2b\, (30)\)\(Pnmb\, (53)\)
\(k + l\)\(h + k\)\(h\)\(k\)\(l\)\(Pn – n\)\(Pn2n\, (34)\)\(Pnmn\, (58)\)
\(k + l\)\(h\)\(h\)\(k\)\(l\)\(Pna – \)\(Pna2_1\, (33)\)\(Pnam\, (62)\)
\(k + l\)\(h\)\(h\)\(h\)\(k\)\(l\)\(Pnaa\)\(Pnaa\, (56)\)
\(k + l\)\(h\)\(k\)\(h\)\(k\)\(l\)\(Pnab\)\(Pnab\, (60)\)
\(k + l\)\(h\)\(h + k\)\(h\)\(k\)\(l\)\(Pnan\)\(Pnan\, (52)\)
\(k + l\)\(l\)\(k\)\(l\)\(Pnc – \)\(Pnc2\, (30)\)\(Pncm\, (53)\)
\(k + l\)\(l\)\(h\)\(h\)\(k\)\(l\)\(Pnca\)\(Pnca\, (60)\)
\(k + l\)\(l\)\(k\)\(k\)\(l\)\(Pncb\)\(Pncb\, (50)\)
\(k + l\)\(l\)\(h + k\)\(h\)\(k\)\(l\)\(Pncn\)\(Pncn\, (52)\)
\(k + l\)\(h + l\)\(h\)\(k\)\(l\)\(Pnn – \)\(Pnn2(34)\)\(Pnnm\, (58)\)
\(k + l\)\(h + l\)\(h\)\(h\)\(k\)\(l\)\(Pnna\)\(Pnna\, (52)\)
\(k + l\)\(h + l\)\(k\)\(h\)\(k\)\(l\)\(Pnnb\)\(Pnnb\, (52)\)
\(k + l\)\(h + l\)\(h + k\)\(h\)\(k\)\(l\)\(Pnnn\)\(Pnnn\, (48)\)
Reflection conditionsExtinction
symbol
Laue class mmm \((2/m\, 2/m\, 2/m)\)
Point group
\(hkl\)\(0kl\)\(h0l\)\(hk0\)\(h00\)\(0k0\)\(00l\)\(222\)\(mm2, m2m, 2mm\)\(mmm\)
\(h + k\)\(k\)\(h\)\(h + k\)\(h\)\(k\)\(C – – – \)\(C222\, (21)\)\(Cmm2\, (35)\)
\( Cm2m\, (38)\)
\( C2mm\, (38)\)
\(Cmmm\, (65)\)
\(h + k\)\(k\)\(h\)\(h + k\)\(h\)\(k\)\(l\)\(C – – 2_1\)\(C222_1\, (20)\)
\(h + k\)\(k\)\(h\)\(h, k\)\(h\)\(k\)\(C – – (ab)\)\(Cm2e\, (39)\)
\( C2me\, (39)\)
\(Cmme\, (67)\)
\(h + k\)\(k\)\(h, l\)\(h + k\)\(h\)\(k\)\(l\)\(C – c – \)\(Cmc2_1\, (36)\)
\( C2cm\, (40)\)
\(Cmcm\, (63)\)
\(h + k\)\(k\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(C – c(ab)\)\(C2ce\, (41)\)\(Cmce\, (64)\)
\(h + k\)\(k. l\)\(h\)\(h + k\)\(h\)\(k\)\(l\)\(Cc – – \)\(Ccm2_1\, (36)\)
\( Cc2m\, (40)\)
\(Ccmm\, (63)\)
\(h + k\)\(k, l\)\(h\)\(h, k\)\(h\)\(k\)\(l\)\(Cc – (ab)\)\(Cc2e\, (41)\)\(Ccme\, (64)\)
\(h + k\)\(k. l\)\(h, l\)\(h + k\)\(h\)\(k\)\(l\)\(Ccc – \)\(Ccc2\, (37)\)\(Cccm\, (66)\)
\(h + k\)\(k. l\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(Ccc(ab)\)\(Ccce\, (68)\)
\(h + l\)\(l\)\(h + l\)\(h\)\(h\)\(l\)\(B – – – \)\(B222\, (21)\)\(Bmm2\, (38)\)
\(Bm2m\, (35)\)
\(B2mm\, (38)\)
\(Bmmm\, (65)\)
\(h + l\)\(l\)\(h + l\)\(h\)\(h\)\(k\)\(l\)\(B – 2_1 – \)\(B22_12\, (20)\)
\(h + l\)\(l\)\(h + l\)\(h, k\)\(h\)\(k\)\(l\)\(B – – b\)\(Bm2_1b\, (36)\)
\(B2mb\, (40)\)
\(Bmmb\, (63)\)
\(h + l\)\(l\)\(h, l\)\(h\)\(h\)\(l\)\(B – (ac) – \)\(Bme2\, (39)\)
\(B2em\, (39)\)
\(Bmem\, (67)\)
\(h + l\)\(l\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(B – (ac)b\)\(B2eb\, (41)\)\(Bmeb\, (64)\)
\(h + l\)\(k. l\)\(h + l\)\(h\)\(h\)\(k\)\(l\)\(Bb – – \)\(Bbm2\, (40)\)
\( Bb2_1m\, (36)\)
\(Bbmm\, (63)\)
\(h + l\)\(k. l\)\(h + l\)\(h, k\)\(h\)\(k\)\(l\)\(Bb – b\)\(Bb2b\, (37)\)\(Bbmb\, (66)\)
\(h + l\)\(k, l\)\(h. l\)\(h\)\(h\)\(k\)\(l\)\(Bb(ac) – \)\(Bbe2\, (41)\)\(Bbem\, (64)\)
\(h + l\)\(k. l\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(Bb(ac)b\)\(Bbeb\, (68)\)
\(k + l\)\(k + l\)\(l\)\(k\)\(k\)\(l\)\(A – – – \)\(A222\, (21)\)\(Amm2\, (38)\)
\(Am2m\, (38)\)
\(A2mm\, (35)\)
\(Ammm\, (65)\)
\(k + l\)\(k + l\)\(l\)\(k\)\(h\)\(k\)\(l\)\(A2_1 – – \)\(A2_122\, (20)\)
\(k + l\)\(k + l\)\(l\)\(h, k\)\(h\)\(k\)\(l\)\(A – – a\)\(Am2a\, (40)\)
\(A2_1ma\, (36)\)
\(Amma\, (63)\)
\(k + l\)\(k + l\)\(h, l\)\(k\)\(h\)\(k\)\(l\)\(A – a – \)\(Ama2\, (40)\)
\(A2_1am\, (36)\)
\(Amam\, (63)\)
\(k + l\)\(k + l\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(A – aa\)\(A2aa\, (37)\)\(Amaa\, (66)\)
\(k + l\)\(k. l\)\(l\)\(k\)\(k\)\(l\)\(A(bc) – – \)\(Aem2\, (39)\)
\(Ae2m\, (39)\)
\(Aemm\, (67)\)
\(k + l\)\(k. l\)\(l\)\(h, k\)\(h\)\(k\)\(l\)\(A(bc) – a\)\(Ae2a\, (41)\)\(Aema\, (64)\)
\(k + l\)\(k. l\)\(h, l\)\(k\)\(h\)\(k\)\(l\)\(A(bc)a – \)\(Aea2\, (41)\)\(Aeam\, (64)\)
\(k + l\)\(k, l\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(A(bc)aa\)\(Aeaa\, (68)\)
\(h + k + l\)\(k + l\)\(h + l\)\(h + k\)\(h\)\(k\)\(l\)\(I – – – \)\(I222\, (23)\)
\( I2_12_12_1\, (24)\)
\(Imm2\, (44)\)
\( Im2m\, (44)\)
\( I2mm\, (44)\)
\(Immm\, (71)\)
\(h + k + l\)\(k + l\)\(h + l\)\(h, k\)\(h\)\(k\)\(l\)\(I – – (ab)\)\(Im2a\, (46)\)
\( I2mb\, (46)\)
\(Imma\, (74)\)
\( Immb\, (74)\)
\(h + k + l\)\(k + l\)\(h, l\)\(h + k\)\(h\)\(k\)\(l\)\(I – (ac) – \)\(Ima2\, (46)\)
\( I2cm\, (46)\)
\(Imam\, (74)\)
\( Imcm\, (74)\)
\(h + k + l\)\(k + l\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(I – cb\)\(I2cb\, (45)\)\(Imcb\, (72)\)
\(h + k + l\)\(k, l\)\(h + l\)\(h + k\)\(h\)\(k\)\(l\)\(I(bc) – – \)\(Iem2\, (46)\)
\( Ie2m\, (46)\)
\(Iemm\, (74)\)
\(h + k + l\)\(k, l\)\(h + l\)\(h, k\)\(h\)\(k\)\(l\)\(Ic – a\)\(Ic2a\, (45)\)\(Icma\, (72)\)
\(h + k + l\)\(k, l\)\(h, l\)\(h + k\)\(h\)\(k\)\(l\)\(Iba – \)\(Iba2\, (45)\)\(Ibam\, (72)\)
\(h + k + l\)\(k, l\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(Ibca\)\(Ibca\, (73)\)
\(Icab\, (73)\)
\(h + k, h + l, \)
\(k + l\)
\(k, l\)\(h, l\)\(h, k\)\(h\)\(k\)\(l\)\(F – – – \)\(F222\, (22)\)\(Fmm2\, (42)\)
\(Fm2m\, (42)\)
\( F2mm\, (42)\)
\(Fmmm\, (69)\)
\(h + k, h + l,\)
\( k + l\)
\(k, l\)\(h+l=4n;\)
\( h, l\)
\(h+k=4n;\)
\( h, k\)
\(h = 4n\)\(k = 4n\)\(l = 4n\)\(F – dd\)\(F2dd\, (43)\)
\(h + k, h + l, \)
\(k + l\)
\(k+l=4n; \)
\(k, l\)
\(h, l\)\(h+k = 4n;\)
\( h, k\)
\(h = 4n\)\(k = 4n\)\(l = 4n\)\(Fd – d\)\(Fd2d\, (43)\)
\(h + k, h + l, \)
\(k + l\)
\(k+l=4n; \)
\(k, l\)
\(h+l=4n;\)
\( h, l\)
\(h, k\)\(h = 4n\)\(k = 4n\)\(l = 4n\)\(Fdd – \)\(Fdd2\, (43)\)
\(h + k, h + l, \)
\(k + l\)
\(k+l=4n; \)
\(k, l\)
\(h+l=4n; \)
\(h, l\)
\(h+k=4n;\)
\( h, k\)
\(h = 4n\)\(k = 4n\)\(l = 4n\)\(Fddd\)\(Fddd\, (70)\)

Tetragonal 正方晶系

Reflection conditionsExtinction
symbol
Laue class
\(4/m\)\(4/mmm\)
Point group
\(hkl\)\(hk0\)\(0kl\)\(hhl\)\(00l\)\(0kl\)\(hh0\)\(4\)\(\bar{4}\)\(4/m\)\(422\)\(4mm\)\(\bar{4}2m, \bar{4}m2\)\(4/mmm\)
\(P – – – \)\(P4\, (75)\)\(P\bar{4}\, (81)\)\(P4/m\, (83)\)\(P422\, (89)\)\(P4mm\, (99)\)\(P\bar{4}2m\, (111)\)
\(P\bar{4}m2\, (115)\)
\(P4/mnm\, (123)\)
\(k\)\(P – 2_1 – \)\(P42_12\, (90)\)\(P\bar{4}2_1m\, (113)\)
\(l\)\(P4_2 – – \)\(P4_2\, (77)\)\(P4_2/m\, (84)\)\(P4_222\, (93)\)
\(l\)\(k\)\(P4_22_1 – \)\(P4_22_12\, (94)\)
\(l=\)
\(4n\)
\(P4_1 – – \)\(P4_1\, (76)\)
\( P4_3\, (78)\)
\(P4_122\, (91)\)
\( P4_322\, (95)\)
\(l=\)
\(4n\)
\(k\)\(P412_1 – \)\(P4_12_12\, (92)\)
\( P4_32_12\, (96)\)
\(l\)\(l\)\(P – – c\)\(P4_2mc\, (105)\)\(P\bar{4}2c\, (112)\)\(P4_2/mmc\, (131)\)
\(l\)\(l\)\(k\)\(P – 2_1c\)\(P\bar{4}2_1c\, (114)\)
\(k\)\(k\)\(P – b – \)\(P4bm\, (100)\)\(P\bar{4}b2\, (117)\)\(P4/mbm\, (127)\)
\(k\)\(l\)\(l\)\(k\)\(P – be\)\(P4_2bc\, (106)\)\(P4_2/mbc\, ( 135)\)
\(l\)\(l\)\(P – c – \)\(P4_2cm\, (101)\)\(P\bar{4}c2\, (116)\)\(P4_2/mcm\, ( 132)\)
\(l\)\(l\)\(l\)\(P – cc\)\(P4cc\, (103)\)\(P4/mcc\, (124)\)
\(k+l\)\(l\)\(k\)\(P – n – \)\(P4_2nm\, (102)\)\(P\bar{4}n2\, (118)\)\(P4_2/mnm\, (136)\)
\(k+l\)\(l\)\(l\)\(k\)\(P – nc\)\(P4nc\, (104)\)\(P4/mnc\, (128)\)
\(h+k\)\(k\)\(Pn – – \)\(P4/n\, (85)\)\(P4/nmm\, (129)\)
\(h+k\)\(l\)\(k\)\(P4_2/n\)\(P4_2/n\, (86)\)
\(h+k\)\(l\)\(l\)\(k\)\(Pn – c\)\(P4_2/mnc\, (137)\)
\(h+k\)\(k\)\(k\)\(Pnb – \)\(P4/nbm\, (125)\)
\(h+k\)\(k\)\(l\)\(l\)\(k\)\(Pnbc\)\(P4_2/nbc\, (133)\)
\(h+k\)\(l\)\(l\)\(l\)\(k\)\(Pnc – \)\(P4_2/ncm\, (138)\)
\(h+k\)\(l\)\(l\)\(l\)\(k\)\(Pncc\)\(P4/ncc\, (130)\)
\(h+k\)\(k+l\)\(l\)\(k\)\(Pnn – \)\(P4_2/nnm\, (134)\)
\(h+k\)\(k+l\)\(l\)\(l\)\(k\)\(Pnnc\)\(P4/nnc\, (126)\)
\(h+k+l\)\(h+k\)\(k+l\)\(l\)\(l\)\(k\)\(I – – – \)\(I4\, (79)\)\(I\bar{4}\, (82)\)\(I4/m\, (87)\)\(I422\, (97)\)\(I4mm\, (107)\)\(I\bar{4}2m\, (121)\)
\(I\bar{4}m2\, (119)\)
\(I4/mmm\, (139)\)
\(h+k+l\)\(h+k\)\(k+l\)\(l\)\(l=\)
\(4n\)
\(k\)\(I4_1 – – \)\(I4_1\, (80)\)\(I4_122\, (98)\)
\(h+k+l\)\(h+k\)\(k+l\)\(l=\)
\(4n\)
\(k\)\(h\)\(I – – d\)\(I4_1md\, (109)\)\(I\bar{4}2d\, (122)\)
\(h+k+l\)\(h+k\)\(k, l\)\(l\)\(l\)\(k\)\(I – c – \)\(I4cm\, ( 108)\)\(I\bar{4}c2\, (120)\)\(I4/mcm\, (140)\)
\(h+k+l\)\(h+k\)\(k, l\)\(l=\)
\(4n\)
\(k\)\(h\)\(I – cd\)\(I4_1cd\, (110)\)
\(h+k+l\)\(h, k\)\(k+l\)\(l\)\(l=\)
\(4n\)
\(k\)\(I4_1/a – – \)\(I4_1/a\, (88)\)
\(h+k+l\)\(h, k\)\(k+l\)\(l=\)
\(4n\)
\(k\)\(h\)\(Ia – d\)\(I4_1/amd\, (141)\)
\(h+k+l\)\(h, k\)\(k, l\)\(l=\)
\(4n\)
\(k\)\(h\)\(Iacd\)\(I4_1/acd\, (142)\)

Trigonal 三方晶系

Hexagonal axes

Reflection conditionsExtinction
symbol
Laue class
\(\bar{3}\)\(\bar{3}m1\, (=\bar{3}\,2/m\,1), \bar{3}m\)\(\bar{3}1m\, (=\bar{3}\,1\, 2/m)\)
Point group
\(hkil\)\(hh0l\)\(h h \overline{2h} l\)\(000l\)\(3\)\(\bar{3}\)\(321, 32\)\(3m1, 3m\)\(\bar{3}m1, \bar{3}m\)\(312\)\(31m\)\(\bar{3}1m\)
\(P – – – \)\(P3\, (143)\)\(P\bar{3}\, (147)\)\(P321\, (150)\)\(P3m1\, (156)\)\(P\bar{3}m1\, (164)\)\(P312\, (149)\)\(P31m\, (157)\)\(P\bar{3}1m\, (162)\)
\(l\)
\(=3n\)
\(P3_1 – – – \)\(P3_1\, (144)\)
\(P3_2\, (145)\)
\(P3_121\, (152)\)
\( P3_221\, (154)\)
\(P3_112(151)\)
\( P3_212(153)\)
\(l\)\(l\)\(P – – c\)\(P31c\, (159)\)\(P\bar{3}1c\, (163)\)
\(l\)\(l\)\(P – c – \)\(P3c1\, (158)\)\(P\bar{3}c1\, (165)\)
\( -h+k+l\)
\(=3n\)
\(h+l\)
\(=3n\)
\(l\)
\(=3n\)
\(l\)
\(=3n\)
\(R(obv) – – \)\(R3\, (146)\)\(R\bar{3}\, (148)\)\(R32(155)\)\(R3m\, (160)\)\(R\bar{3}m\, (166)\)
\( -h+k+l\)
\(=3n\)
\(h+l\)
\(=3n; l\)
\(l\)
\(=3n\)
\(l\)
\(=6n\)
\(R(obv) – c\)\(R3c\, (161)\)\(R\bar{3}c\, (167)\)
\(h- k+l\)
\(=3n\)
\( -h+l\)
\(=3n\)
\(l\)
\(=3n\)
\(l\)
\(=3n\)
\(R(rev) – – \)\(R3\, (146)\)\(R\bar{3}\, (148)\)\(R32(155)\)\(R3m\, (160)\)\(R\bar{3}m\, (166)\)
\(h- k+l\)
\(=3n\)
\( -h+l\)
\(=3n; l\)
\(l\)
\(=3n\)
\(l\)
\(=3n\)
\(R(rev) – c\)\(R3c\, (161)\)\(R\bar{3}c\, (167)\)

Rhombohedral axes

Reflection conditionsExtinction
symbol
Point group
\(hkl\)\(hhl\)\(hhh\)\(3\)\(\bar{3}\)\(32\)\(3m\)\(\bar{3}m\)
\(R – – \)\(R3\, (146)\)\(R\bar{3}\, (148)\)\(R32(155)\)\(R3m\, (160)\)\(R\bar{3}m\, (166)\)
\(l\)\(h\)\(R – c – \)\(R3c\, (161)\)\(R\bar{3}c\, (167)\)

Hexagonal 六方晶系

Reflection conditionsExtinction
symbol
Laue class
\(6/m\)\(6/mmm\)
Point group
\(hh0l\)\(hh2hl\)\(000l\)\(6\)\(\bar{6}\)\(6/m\)\(622\)\(6mm\)\(\bar{6}2m\)\(6/mmm\)
\(\bar{6}m2\)
\(P – – – \)\(P6\, (168)\)\(P\bar{6}\, (174)\)\(P6/m\, (175)\)\(P622\, (177)\)\(P6mm\, (183)\)\(P\bar{6}2m\, (189)\)\(P6/mmm\, (191)\)
\(P\bar{6}m2\, (187)\)
\(l\)\(P6_3 – – \)\(P6_3\, (173)\)\(P6_3/m\, (176)\)\(P6_322\, (182)\)
\(l=3n\)\(P6_2 – – \)\(P6_2\, (171)\)\(P6_222\, (180)\)
\(P6_4\, (172)\)\(P6_422\, (181)\)
\(l=6n\)\(P6_1 – – \)\(P6_1\, (169)\)\(P6_122\, (178)\)
\(P6_5\, (170)\)\(P6_522\, (179)\)
\(l\)\(l\)\(P – – c\)\(P6_3mc\, (186)\)\(P\bar{6}2c\, (190)\)\(P6_3/nmc\, (194)\)
\(l\)\(l\)\(P – c – \)\(P6_3cm\, (185)\)\(P\bar{6}c2\, (188)\)\(P6_3/mcm\, (193)\)
\(l\)\(l\)\(l\)\(P – cc\)\(P6cc\, (184)\)\(P6/mcc\, (192)\)

Cubic 立方晶系

Reflection conditions Extinction
symbol
Laue class
\(m3\)\(m3m\)
Point group
\(hkl\)\(0kl\)\(hhl\)\(00l\)\(23\)\(m\bar{3}\)\(432\)\(\bar{4}3m\)\(m\bar{3}m\)
\(P – – – \) \(P23\, (195)\) \(Pm\bar{3}\, (200)\) \(P432\, (207)\) \(P\bar{4}3m\, (215)\) \(Pm\bar{3}m\, (221)\)
\(l\) \(P2_1 – -\)
\(P4_2 – -\)
\(P2_13\, (198)\) \(P4_232\, (208)\)
\(l=4n\) \(P4_1 – – \) \(P4_132\, (213) \)
\(P4_332\, (212)\)
\(l\)\(l\) \(P – – n\) \(P\bar{4}3n\, (218)\) \(Pm\bar{3}n\, (223)\)
\(k\)\(l\) \(Pa – – \) \(Pa\bar{3}\, (205)\)
\(k+l\)\(l\) \(Pn – – \) \(Pn\bar{3}\, (201)\) \(Pn\bar{3}m\, (224)\)
\(k+l\)\(l\)\(l\) \(Pn – n\) \(Pn\bar{3}n\, (222)\)
\(h+k+l\)\(k+l\)\(l\)\(l\) \(I – – – \) \(I23\, (197)\)
\(I2_13\, (199)\)
\(Im\bar{3}\, (204)\) \(I432\, (211)\) \(I\bar{4}3m\, (217)\) \(Im\bar{3}m\, (229)\)
\(h+k+l\)\(k+l\)\(l\)\(l=4n\) \(I4_1- -\) \(I4_132\, (214)\)
\(h+k+l\)\(k+l\)\(2h+l=4n, l\)\(l=4n\) \(I – – d\) \(I\bar{4}3d\, (220)\)
\(h+k+l\)\(k, l\)\(l\)\(l\) \(la – – \) \(Ia\bar{3}\, (206)\)
\(h+k+l\) \(k, l\) \(2h+l=4n, l\)\(l=4n\) \(Ia – d\) \(Ia\bar{3}d\, (230)\)
\(h+k, h+l,\)
\( k+l\)
\(k, l\)\(h+l\)\(l\)\(F\) \(F23\, (196)\) \(Fm\bar{3}\, (202)\) \(F432\, (209)\) \(F\bar{4}3m\, (216)\) \(Fm\bar{3}m\, (225)\)
\(h+k, h+l,\)
\( k+l\)
\(k, l\)\(h+l\)\(l=4n\) \(F4_1 – – \) \(F4_132\, (210)\)
\(h+k, h+l,\)
\( k+l\)
\(k, l\)\(h, l\)\(l\) \(F – – c\) \(F\bar{4}3c\, (219)\) \(Fm\bar{3}c\, (226)\)
\(h+k, h+l, \)
\(k+l\)
\(k+l=4n;\)
\(k, l\)
\(h+l\)\(l=4n\) \(Fd – – \) \(Fd\bar{3}\, (203)\) \(Fd\bar{3}m\, (227)\)
\(h+k, h+l, \)
\(k+l\)
\(k+l=4n;\)
\(k, l\)
\(h, l\)\(l=4n\)\(Fd-c\) \(Fd\bar{3}c\, (228)\)
  1. これはX線や中性子線のような、散乱能が小さく試料中で高々1回程度しか散乱が起きないような入射波を使った場合に成立します。電子線のような散乱能の大きい波を用いた場合、複合格子並進に由来する出現則/消滅則は満たすものの、らせん・映進に由来する出現則/消滅則は満たされません。 ↩︎

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Stupidity has a certain charm; ignorance does not. Frank Zappa