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<title>The k-vector Types of Space Groups</title>
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<a href="http://www.cryst.ehu.es/">Bilbao Crystallographic Server</a> <img src="Fd3m%20kvec_files/a.gif" alt="arrow"> k-vector types and Brillouin zones
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<h2 align="center">The k-vector types of space group 227 [<i>F</i><i>d</i>-3<i>m</i>]</h2><center>
<h3>(Table for arithmetic crystal class m -3 mF)</h3><h3></h3><h3>Fm-3m-O<sub>h</sub><sup>5</sup> (225) to Fd-3c- O<sub>h</sub><sup>8</sup>(228)</h3><h3>Reciprocal space group (Im-3m)<sup>*</sup>, No.229</h3><br><a href="http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-kv-list?gnum=227&fig=fm3qmf">Brillouin zone</a><br><br><br><table align="center" border="1" cellpadding="5" cellspacing="0" width="75%"><tbody><tr><th colspan="3" bgcolor="#e3e4fa">k-vector description</th><th colspan="3" bgcolor="#e3e4fa">Wyckoff Position</th><th colspan="1" bgcolor="#e3e4fa">ITA description</th></tr><tr><td colspan="2" align="center" bgcolor="#e3e4fa"><b>CDML*</b></td><td rowspan="2" align="center" bgcolor="#e3e4fa"><b>Conventional-ITA</b></td><td rowspan="2" colspan="3" align="center" bgcolor="#e3e4fa"><b>ITA</b></td><td rowspan="2" align="center" bgcolor="#e3e4fa"><b>Coordinates</b></td></tr><tr><td align="center" bgcolor="#e3e4f4"><b>Label<b></b></b></td><td align="center" bgcolor="#e3e4f4"><b>Primitive<b></b></b></td></tr><tr><td align="center" bgcolor="#ffffff" width="15%">GM</td><td align="center" bgcolor="#ffffff">0,0,0</td><td align="center" bgcolor="#ffffff">0,0,0</td><td align="center" bgcolor="#ffffff">a</td><td align="center" bgcolor="#ffffff">2</td><td align="center" bgcolor="#ffffff">m-3m</td><td align="center" bgcolor="#ffffff">
0,0,0
</td></tr>
<tr><td align="center" bgcolor="#f5f5f5" width="15%">X</td><td align="center" bgcolor="#f5f5f5">1/2,0,1/2</td><td align="center" bgcolor="#f5f5f5">0,1,0</td><td align="center" bgcolor="#f5f5f5">b</td><td align="center" bgcolor="#f5f5f5">6</td><td align="center" bgcolor="#f5f5f5">4/mm.m</td><td align="center" bgcolor="#f5f5f5">
0,1/2,0
</td></tr>
<tr><td align="center" bgcolor="#ffffff" width="15%">L</td><td align="center" bgcolor="#ffffff">1/2,1/2,1/2</td><td align="center" bgcolor="#ffffff">1/2,1/2,1/2</td><td align="center" bgcolor="#ffffff">c</td><td align="center" bgcolor="#ffffff">8</td><td align="center" bgcolor="#ffffff">.-3m</td><td align="center" bgcolor="#ffffff">
1/4,1/4,1/4
</td></tr>
<tr><td align="center" bgcolor="#f5f5f5" width="15%">W</td><td align="center" bgcolor="#f5f5f5">1/2,1/4,3/4</td><td align="center" bgcolor="#f5f5f5">1/2,1,0</td><td align="center" bgcolor="#f5f5f5">d</td><td align="center" bgcolor="#f5f5f5">12</td><td align="center" bgcolor="#f5f5f5">-4m.2</td><td align="center" bgcolor="#f5f5f5">
1/4,1/2,0
</td></tr>
<tr><td align="center" bgcolor="#ffffff" width="15%">DT</td><td align="center" bgcolor="#ffffff">u,0,u</td><td align="center" bgcolor="#ffffff">0,2u,0</td><td align="center" bgcolor="#ffffff">e</td><td align="center" bgcolor="#ffffff">12</td><td align="center" bgcolor="#ffffff">4m.m</td><td align="center" bgcolor="#ffffff">
0,y,0
: 0 < y < 1/2</td></tr>
<tr><td align="center" bgcolor="#f5f5f5" width="15%">LD</td><td align="center" bgcolor="#f5f5f5">u,u,u</td><td align="center" bgcolor="#f5f5f5">u,u,u</td><td align="center" bgcolor="#f5f5f5">f</td><td align="center" bgcolor="#f5f5f5">16</td><td align="center" bgcolor="#f5f5f5">.3m</td><td align="center" bgcolor="#f5f5f5">
x,x,x
: 0 < x < 1/4</td></tr>
<tr><td align="center" bgcolor="#ffffff" width="15%">V</td><td align="center" bgcolor="#ffffff">1/2,u,1/2+u</td><td align="center" bgcolor="#ffffff">2u,1,0</td><td align="center" bgcolor="#ffffff">g</td><td align="center" bgcolor="#ffffff">24</td><td align="center" bgcolor="#ffffff">mm2..</td><td align="center" bgcolor="#ffffff">
x,1/2,0
: 0 < x < 1/4</td></tr>
<tr><td align="center" bgcolor="#f5f5f5" width="15%">SM</td><td align="center" bgcolor="#f5f5f5">u,u,2u ex</td><td align="center" bgcolor="#f5f5f5">2u,2u,0</td><td align="center" bgcolor="#f5f5f5">h</td><td align="center" bgcolor="#f5f5f5">24</td><td align="center" bgcolor="#f5f5f5">m.m2</td><td align="center" bgcolor="#f5f5f5">
x,x,0
: 0 < x <= 3/8</td></tr>
<tr><td align="center" bgcolor="#f5f5f5" width="15%">S</td><td align="center" bgcolor="#f5f5f5">1/2+u,2u,1/2+u ex</td><td align="center" bgcolor="#f5f5f5">2u,1,2u</td><td align="center" bgcolor="#f5f5f5">h</td><td align="center" bgcolor="#f5f5f5">24</td><td align="center" bgcolor="#f5f5f5">m.m2</td><td align="center" bgcolor="#f5f5f5">
x,1/2,x
: 0 < x < 1/8</td></tr>
<tr><td colspan="3" align="left" bgcolor="#f5f5f5">S~SM<sub>1</sub>=[K M]</td><td align="center" bgcolor="#f5f5f5">h</td><td align="center" bgcolor="#f5f5f5">24</td><td align="center" bgcolor="#f5f5f5">m.m2</td><td align="center" bgcolor="#f5f5f5">
x,x,0
: 3/8 < x < 1/2</td></tr>
<tr><td colspan="3" align="left" bgcolor="#f5f5f5">SM SM<sub>1</sub>=[GM M]</td><td align="center" bgcolor="#f5f5f5">h</td><td align="center" bgcolor="#f5f5f5">24</td><td align="center" bgcolor="#f5f5f5">m.m2</td><td align="center" bgcolor="#f5f5f5">
x,x,0
: 0 < x < 1/2</td></tr>
<tr><td align="center" bgcolor="#ffffff" width="15%">Q</td><td align="center" bgcolor="#ffffff">1/2,1/4+u,3/4-u</td><td align="center" bgcolor="#ffffff">1/2,1-2u,2u</td><td align="center" bgcolor="#ffffff">i</td><td align="center" bgcolor="#ffffff">48</td><td align="center" bgcolor="#ffffff">..2</td><td align="center" bgcolor="#ffffff">
1/4,1/2-y,y
: 0 < y < 1/4</td></tr>
<tr><td align="center" bgcolor="#f5f5f5" width="15%">A</td><td align="center" bgcolor="#f5f5f5">u,-u+v,v ex</td><td align="center" bgcolor="#f5f5f5">-2u+2v,2u,0</td><td align="center" bgcolor="#f5f5f5">j</td><td align="center" bgcolor="#f5f5f5">48</td><td align="center" bgcolor="#f5f5f5">m..</td><td align="center" bgcolor="#f5f5f5">
x,y,0
: 0 < x < y <= 3/8 U<br>U
x,y,0
: 0 < x < 3/4-y < y < 1/2 </td></tr>
<tr><td align="center" bgcolor="#f5f5f5" width="15%">B</td><td align="center" bgcolor="#f5f5f5">1/2+u,u+v,1/2+v ex</td><td align="center" bgcolor="#f5f5f5">2v,1,2u</td><td align="center" bgcolor="#f5f5f5">j</td><td align="center" bgcolor="#f5f5f5">48</td><td align="center" bgcolor="#f5f5f5">m..</td><td align="center" bgcolor="#f5f5f5">
x,1/2,z
: 0 < z < x <= 1/4-z</td></tr>
<tr><td colspan="3" align="left" bgcolor="#f5f5f5">B~B<sub>1</sub>=[K M W]</td><td align="center" bgcolor="#f5f5f5">j</td><td align="center" bgcolor="#f5f5f5">48</td><td align="center" bgcolor="#f5f5f5">m..</td><td align="center" bgcolor="#f5f5f5">
x,y,0
: 3/4-y <= x < y < 1/2</td></tr>
<tr><td colspan="3" align="left" bgcolor="#f5f5f5">A B<sub>1</sub>=[GM M X]</td><td align="center" bgcolor="#f5f5f5">j</td><td align="center" bgcolor="#f5f5f5">48</td><td align="center" bgcolor="#f5f5f5">m..</td><td align="center" bgcolor="#f5f5f5">
x,y,0
: 0 < x < y < 1/2</td></tr>
<tr><td align="center" bgcolor="#ffffff" width="15%">C</td><td align="center" bgcolor="#ffffff">u,u,v ex</td><td align="center" bgcolor="#ffffff">v,v,-v+2u</td><td align="center" bgcolor="#ffffff">k</td><td align="center" bgcolor="#ffffff">48</td><td align="center" bgcolor="#ffffff">..m</td><td align="center" bgcolor="#ffffff">
x,x,z
: 0 < z < x <= 3/8-z/2</td></tr>
<tr><td align="center" bgcolor="#ffffff" width="15%">J</td><td align="center" bgcolor="#ffffff">u,v,u[GMXUL] ex</td><td align="center" bgcolor="#ffffff">v,-v+2u,v</td><td align="center" bgcolor="#ffffff">k</td><td align="center" bgcolor="#ffffff">48</td><td align="center" bgcolor="#ffffff">..m</td><td align="center" bgcolor="#ffffff">
x,y,x
: 0 < x < y <= 1/2-x U<br>U
x,y,x
: 1/4 < y < 1/2, 1/2-y < x < 3/8-y/2 </td></tr>
<tr><td colspan="3" align="left" bgcolor="#ffffff">J~J<sub>1</sub>=[GM L X<sub>3</sub>] + [L K M]</td><td align="center" bgcolor="#ffffff">k</td><td align="center" bgcolor="#ffffff">48</td><td align="center" bgcolor="#ffffff">..m</td><td align="center" bgcolor="#ffffff">
x,x,z
: 0 < x < z <= 1/2-x U<br>U
x,x,z
: 0 < z < 1/4, 3/8-z/2 < x < 1/2-z </td></tr>
<tr><td colspan="3" align="left" bgcolor="#ffffff">C + J<sub>1</sub>=[GM M X<sub>3</sub>] \ [GM L]</td><td align="center" bgcolor="#ffffff">k</td><td align="center" bgcolor="#ffffff">48</td><td align="center" bgcolor="#ffffff">..m</td><td align="center" bgcolor="#ffffff">
x,x,z
: 0 < z < 1/2 -x < 1/2, x!= z</td></tr>
<tr><td align="center" bgcolor="#f5f5f5" width="15%">GP</td><td align="center" bgcolor="#f5f5f5">u,v,w</td><td align="center" bgcolor="#f5f5f5">-u+w+v,u+w-v,u-w+v</td><td align="center" bgcolor="#f5f5f5">l</td><td align="center" bgcolor="#f5f5f5">96</td><td align="center" bgcolor="#f5f5f5">1</td><td align="center" bgcolor="#f5f5f5">
x,y,z
: 0 < z < x < y < 1/2-x U<br>U
x,y,z
: 0 < z < 1/2-y < x < y < 1/2 U<br>U
x,y,1/2-y
: 1/4 < y < 1/2; 1/2-y < x < 1/4. </td></tr>
</tbody></table><br><br><br><br><center>* Cracknell, A. P., Davies,
B.L., Miller, S. C., and Love, W. F. (1979). Kronecker Product Tables.
Vol. 1. General Introduction and Tables of Irreducible Representations
of Space Groups. New York: IFI/Plenum.</center><center></center><br><center> The asymmetric unit of ITA is obtained from that used in these tables by </center><br><center> reflectionthrough the plane x,x,z . </center><br><center> The asymmetric unit is obtained from the representation domain of CDML by the equivalence </center><br><center> [L K W M] ~[L U W X] through the two-fold rotation around the axis Q. </center><br><center> Wing: [GM L X<sub>3</sub>] x,x,z: 0 < x < z < 1/2-x </center><br><center> </center><br><center>The transformation matrix that relates the primitive (CDML) base with the conventional-ITA is -<b>a</b>+<b>b</b>+<b>c</b>,<b> </b><b>a</b>-<b>b</b>+<b>c</b>,<b> </b><b>a</b>+<b>b</b>-<b>c</b></center><br><br><br><form method="post" action="/cgi-bin/cryst/programs//nph-kv-list"><table align="center" cellpadding="10" cellspacing="6" width="35%"><tbody><tr><td bgcolor="#a9a9f5">If you want to identify a <b>k</b>-vector you have to introduce:</td></tr><tr> <td bgcolor="#cecef6">
1. The reciprocal bases:
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primitive (CDML)
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conventional dual (ITA)
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<td>2. The <b>k</b>-vector: </td>
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k<sub>x</sub>
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k<sub>y</sub>
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<input gtbfieldid="20" name="k1" value="" size="6" type="text">
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k<sub>z</sub>
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<input gtbfieldid="21" name="k2" value="" size="6" type="text">
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<input name="identify" value="identify" type="submit">
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<input name="gnum" value="227" type="hidden">
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