Thank you very much. I had indeed lost track of the orientation of the reciprocal lattice vectors (switched them with the real lattice vectors). For the current simulations, then indeed kmag gives me a nice cone when plotted versus k1 and k2 (transformed to an orthogonal basis of course). Unfortunately it means I will have to rerun the simulations.
Thank you very much! :-) On Tue, 2009-12-15 at 17:56 -0500, Steven G. Johnson wrote: > On Dec 15, 2009, at 3:52 PM, Oliver Willekens wrote: > > I wanted to plot the lightcone by plotting kmag versus k1 and k2 in a > > triangular lattice. However, it does not look like a cone(segment). > > Plotting it versus sqrt(k1^2+k2^2) works of course. > > kmag is not sqrt(k1^2 + k2^2), except in a square lattice. The reason > is that k1 and k2 are in the reciprocal basis which is non-Cartesian > in a non-square lattice, and hence the Pythagorean theorem does not > apply. > In MPB, kmag is (vector3-norm (reciprocal->cartesian k)) > > > k1 k2 k3 kmag/2pi (comment) > > 0.3333 0.3333 0 0.3849 (K-point) > > 0.5000 0 0 0.5774 (M-point) > > > > However, in this basis, the magnitude of the vector Gamma-K should be > > 1/sqrt(3), no? > > No. The correct answer is 2/3 / sqrt(3) = 0.384900179459751. > > The reciprocal lattice vectors for your lattice are 2pi/a (1/sqrt(3), > +/- 1). Hence the K point, in Cartesian coordinates, is the sum of > these two vectors times 1/3, or 2pi/a (2/sqrt(3), 0) / 3. Hence the > length is 2/3 / sqrt(3) in units of 2pi/a. > > See appendix B of our book if you need more explanation about > reciprocal lattices (http://ab-initio.mit.edu/book). > > In general, when you plot kmag over the irreducible Brillouin zone > boundaries, the only portions that will be straight lines ("look like > a cone") will be the portions that correspond to straight lines from > the origin (Gamma). > > Steven > > _______________________________________________ > mpb-discuss mailing list > [email protected] > http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/mpb-discuss _______________________________________________ mpb-discuss mailing list [email protected] http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/mpb-discuss
