Brillouin zone square lattices

The pictures in Fig. 10.12 give an impression of how s orbitals interact with each other in a square lattice. Depending on the k values, i.e. for different points in the Brillouin zone, different kinds of interactions result. Between adjacent atoms there are only bonding... 

Fig. 2.9. Ground state for a square lattice of dipoles a. Orientations of dipole moments b. wave vectors of the structure in the first Brillouin zone c. and d. orientations of dipole moments in infinitely small and large external electric fields, respectively.

The first Brillouin zone, formed by the bisector lines between the center and the nearest lattice points in reciprocal space, is a square bounded with lines h = ula and ky - nia. The lowest Fourier components of the sum of the local density of states (LDOS) over a range of energy A should have the form ... 

See Scanning tunneling spectroscopy Superconductors 332—334 Surface Brillouin zone 92 hexagonal lattice 133 one-dimensional lattice 123, 128 square lattice 129 Surface chemistry 334—338 hydrogen on silicon 336 oxygen on silicon 334 Surface electronic structures 117 Surface energy 96 Surface potential 93 Surface resonance 91 Surface states 91, 98—107 concept 98... 

Figure 16.13. Brillouin zone of the reciprocal lattice of the strictly 2-D square lattice, demonstrating the construction of 1 k = ki, C2Zki = k2, C zki = k3, Q"zki = k4.

Figure 1. The dispersion of spin excitations near Q along the edge of the Brillouin zone. The dispersion was calculated in a 20x20 lattice for x = 0.06 and T = 17 K (filled squares, the solid line is the fit with u>k = [wq + c2(k — Q)2]1/2). Open squares are the experimental dispersion [1] of the maximum in the frequency dependence of the odd x"(qw), q = k — Q, in YBa2Cua06.5 (x 0.075 [13]) at T = 5 K.

Figure 4 The Brillouin zone of the square lattice. Solid curves are two of four ellipses forming the Fermi surface at small x. Dashed lines are the Fermi surface contours shifted by (k — Q). Regions of k and k + k contributing to the damping (14) are shaded.

Brillouin zones for a square-planar Bravais lattice. The small circles indicate reciprocal lattice points. The first three Brillouin zones lie entire-lywithinthesquare of side2fa each of them has area fa2. The first Brillouin zone, indicated by "1", is centered atthe origin and includes the origin point. The second Brillouin zone is indicated as "2", etc. the third as "3", etc. The diagonal and horizontal lines indicate Bragg "planes" (which must be lines in 2D). Zones 4, 5 (not shown), and 6 (not shown) lie partially outside the square of side 2b. Adapted from Ashcroft and Mermin [4]. 

If we wish to study a stale at the face of the Jones Zone, we must consider not only the plane wave with wave number at that face, say k,, o = [110]27c/a, and that at the opposite face, /states differs from the others by a lattice wave number, so that if the free-clectron bands were plotted in the reduced-zone scheme, they would all be at the same point, the point [001]27c/fl, which is at the center of one of the square faces of the Brillouin Zone, for example, the point X in... 

Fig. 14. Reciprocal lattice and 1st Brillouin zone for the square lattice (upper part) and triangular lattice (lower part). The c(2x2) structure is described by the single wavcvcctor q0 id reciprocal space, while the (2x1) structure on the square lattice is described by a star (q, q2), as well as the V3xV R30o structure on the triangular lattice. The star of the (2x2) structure on the triangular lattice contains three members q, q 2 and q 3.

The following simple example should illustrate the foregoing results. We consider a simple two-dimensional square lattice with unit distance t, the Brillouin zone of w hich is also a square lattice with unit distance 2ir/f. The point group operators of C4, are E, 2C4, C 2, 2general point fei is now chosen (see Figure 10.la) and the system subjected to the operators of Ct. The origin of the coordinates is chosen as fe(0, 0), the point with fei = fe.> = 0. Since fe-i does not lie on asymmetry axis or symmetry plane the point is shifted by all the point group operators except E. For example, C ki = fej, Ctki = k, and so on (see Fig. 

Figure 3.3. Illustration of the constmction of Brillouin Zones in a two-dimensional crystal with ai = X, E2 = y. The first two sets of reciprocal lattice vectors (G = 2 rx, 2 ry and G = 2 r( x y)) are shown, along with the Bragg planes that bisect them. The hrst BZ, shown in white and labeled 1, is the central square the second BZ, shown hatched and labeled 2, is composed of the four triangles around the central square the third BZ, shown in lighter shade and labeled 3, is composed of the eight smaller triangles around the second BZ.

Figure 3.6. Left the first six Brillouin Zones of the two-dimensional square lattice. The position of the Fermi sphere (Fermi surface) for a crystal with Z = 4 electrons per unit cell is indicated by a shaded circle. R ht the shape of occupied portions of the various Brillouin Zones for the 2D square lattice with Z = 4 electrons per unit cell.

Figure 3.7. Left the symmetry operations of the two-dimensional square lattice the thick straight lines indicate the reflections (labeled ax,ay,ai,as) and the curves with arrows the rotations (labeled C4, C2, C4). Right the Irreducible Brillouin Zone for the two-dimensional square lattice with full symmetry the special points are labeled F, S, A, M, Z, X.

Draw the occupied Brillouin Zones for the 2D square lattice with n = 2 electrons per unit cell, by analogy to Fig. 3.6. 

Fig. 4 1st Brillouin zone and occupied states (shaded region) for the square planar lattice. 

Illustration of an N-process in which the collision is elastic and a U-process in which the collision is inelastic. If the resulting vector from a phonon-phonon collision falls outside the first Brillouin zone (the square box), the momentum is Bragg reflected back into the first Brillouin zone with a transfer oi G1r momentum to the lattice in the form of crystal momentum. 

Figure 2.5 (a) The method for construeting the Brillouin zone for a square planar lattice and the first Brillouin zone for (b) a faee-eentered eubic crystal and (c) a body-centered cubic crystal. For the fee erystal the diamond-shaped faees of the Brillouin zone are along cube axes, [100]-type direetions, while the hexagonal faees are along [111] cube diagonals. For discussion of the [100], [111], and other erystal indiees, see Chapter 4. 

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