<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; ">OK. Let me try to shed some light on this unexpectedly obscure point.<DIV><BR><DIV><DIV>On May 11, 2006, at 6:41 AM, Huiqun Zhou wrote:</DIV><BR class="Apple-interchange-newline"><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Hmm ..., because there are 4 lattice points, that are (0,0,0) and 3 end points of the</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">basis vectors as stated, in a diamond structure, and there are two primitive elements</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">which are located at (0,0,0) and (1/4,1/4,1/4) for this structure, so there should be</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">4 x 2 = 8 atoms in a conventional unit cell of diamond structure.</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>I am sorry, Huiqun, but I am afraid that this argument is wrong and confusing. What do you mean by "4 lattice points"? Of course, every triplet of vectors originating at (0,0,0) has one common point (the origin), and 3 end points. AND SO WHAT?</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">But because of periodicity, theoretically you can choose either only the primitive</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">elements (2 atoms) in an unit cell with appropriatelly selected large number of k points</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">for calculation, or the whole member (8 atoms) in an unit cell with less k points for</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">calculation. I may be wrong, please correct me.</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>Although it is certainly true that, for a same system, doing a calculation with a larger conventional unit cell would require a smaller number of k points, I feel that it is confusing to try to explain a property of the real-space lattice with something that has only to do with the way calculations are technically done (the number of k points).</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Let me try to clarify the matter as much as I can.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>1) The Bravais lattice of the diamond structure is face-centered cubic (FCC). A basis for this Bravais lattice is, e.g., (1/2,1/2,0), (1/2,0,1/2), (0,1/2,1/2).</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>2) Each elementary cell of this FCC lattice contains two equivalent atoms, located at (000)+R(l,m,n) and (1/4,1/4,1/4)+R(l,m,n), where R(l,m,n) = l*(1/2,1/2,0) + m*(1/2,0,1/2) + n*(0,1/2,1/2) is a generic point of the Bravais lattice (l,m,n being integer numbers).</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>3) The above basis for the Bravais lattice defines a unit cell of *minimum volume*. Of ourse, one is free to choose as a basis any triplet of integer linear (and linearly independent) combinations of the minimum basis. For instance, one could choose: (1,0,0) = (1/2,1/2,0)+(1/2,0,1/2)-(0,1/2,1/2), (0,1,0)= [guess what?], and (0,0,1)=[guess what?]. These three vectors form a basis for the simple cubic (SC) Bravais lattice. Calculate the volume of the original FCC unit cell (defined at point 1 above) and compare it with the volume of the SC unit cell considered here (hint: the volume of a parallelepiped is the absolute value of the determinant of the 3x3 matrix whose columns are the coordinates of the three vectors which form the edges of the parallelepiped). You will find that the the volume of the SC cell is 4 times larger than the volume of the FCC cell. This means that if you choose to describe the diamond lattice as SC instead of FCC (which you are free to do), the unit cell will contain 4x2 instead of 2 atoms. You see where the factor 4 comes from? Nothing to do with the "number of lattice points" (whatever these lattice points may be).</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>4) Now, the volume of the Brillouin zone (BZ) is (2\pi)^3 divided by the volume of the unit cell. Hence, the volume of the SC BZ is 1/4 the volume of the FCC BZ. That's why, in order to sample a SC BZ with the same accuracy of an FCC cell you will need 1/4 as many k points. But this is another story which has little to do with the main point of this thread.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Hope this clarifies a little bit the muddy waters</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Stefano Baroni</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Huiqun Zhou</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">----- Original Message ----- From: "Eyvaz Isaev" <<A href="mailto:eyvaz_isaev@yahoo.com">eyvaz_isaev@yahoo.com</A>></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">To: <<A href="mailto:pw_forum@pwscf.org">pw_forum@pwscf.org</A>></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Sent: Thursday, May 11, 2006 2:51 AM</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Subject: Re: [Pw_forum] Confused on nat definition</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> <BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Hi,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> <BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">This may seem silly, but I'm confused as to what</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">exactly the number of</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">atoms in a unit cell is (nat).<SPAN class="Apple-converted-space"> </SPAN>For example, the</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Silicon example says</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">that nat is only two, yet a diamond structure such</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">as this should have</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">much more than two atoms per unit cell.</DIV> </BLOCKQUOTE><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Let us consider the diamond case. If you choose as</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">basis vectors next 3 vectors (which are the standard</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">choice)</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">1/2, 1/2, 0</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">1/2, 0<SPAN class="Apple-converted-space"> </SPAN>, 1/2</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">0<SPAN class="Apple-converted-space"> </SPAN>, 1/2, 1/2</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">you have only 2 atoms in the unit cell</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">(parallelepiped) spanned by these vectors:</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">0, <SPAN class="Apple-converted-space"> </SPAN>0, 0</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">1/4, 1/4, 1/4</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">If you decide to choose as basis vectors next 3 ones</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">1 0 0</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">0 1 0</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">0 0 1</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">you have 8 atoms in the unit cell which is now a cub.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">If your choice is the latter<SPAN class="Apple-converted-space"> </SPAN>for CaF2 structure you</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">will have 12 atoms, but using the former - only 3.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">So, number of atoms (nat) in a unit cell depends on</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">your unit cell choice defined by 3 basis vectors.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> <BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Is the definition of nat the<SPAN class="Apple-converted-space"> </SPAN>number of basis</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">vectors?</DIV> </BLOCKQUOTE><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">To me it is not so clear, but see above.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Bests,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Eyvaz.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> <BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">_______________________________________________</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Pw_forum mailing list</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="http://www.democritos.it/mailman/listinfo/pw_forum">http://www.democritos.it/mailman/listinfo/pw_forum</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> </BLOCKQUOTE><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">__________________________________________________</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Do You Yahoo!?</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Tired of spam?<SPAN class="Apple-converted-space"> </SPAN>Yahoo! Mail has the best spam protection around</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="http://mail.yahoo.com">http://mail.yahoo.com</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">_______________________________________________</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Pw_forum mailing list</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="http://www.democritos.it/mailman/listinfo/pw_forum">http://www.democritos.it/mailman/listinfo/pw_forum</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> </BLOCKQUOTE><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">_______________________________________________</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Pw_forum mailing list</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="http://www.democritos.it/mailman/listinfo/pw_forum">http://www.democritos.it/mailman/listinfo/pw_forum</A></DIV> </BLOCKQUOTE></DIV><BR><DIV> <SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">---</SPAN></SPAN></SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">Stefano Baroni - SISSA</SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">&</SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">DEMOCRITOS National Simulation Center - Trieste</SPAN></SPAN></SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">[+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)</SPAN></SPAN></SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; ; font-size: 12px; "><BR style="; font-size: 12px; "></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">Please, if possible, don't</SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">send me MS Word or PowerPoint attachments</SPAN></SPAN></SPAN></FONT></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">Why? See:</SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></SPAN></FONT><A href="http://www.gnu.org/philosophy/no-word-attachments.html"><FONT class="Apple-style-span" color="#0023E9" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;; color: rgb(0, 35, 233); -khtml-text-decorations-in-effect: underline; "><SPAN class="Apple-style-span" style="color: rgb(0, 35, 233); font-size: 12px; -khtml-text-decorations-in-effect: underline; "><SPAN class="Apple-style-span" style="color: rgb(0, 35, 233); font-size: 12px; -khtml-text-decorations-in-effect: underline; ">http://www.gnu.org/philosophy/no-word-attachments.html</SPAN></SPAN></SPAN></FONT></A></DIV><DIV><BR class="khtml-block-placeholder"></DIV><BR class="Apple-interchange-newline"></SPAN></SPAN></SPAN> </DIV><BR></DIV></BODY></HTML>