Brillouin Zones & Reciprocal Lattices ayer for the SC, BCC & FCC Lattices Brillouin Zones The First Brillouin Zone (BZ) is defined as the Wigner- Seitz primitive cell in the reciprocal lattice. It gives a geometric interpretation, in the reciprocal lattice, of the diffraction condition. * The Brillouin construction exhibits o 2 all wavevectors k that can be Bragg reflected “e by the crystal. * The constructions divide the , reciprocal space into fragments, out © of which The First Brillouin Zone |* The electronic number density n(r) in the crystal glccPlayer periodic in space: n(r) = n(r +T), with T equal to a Direct Lattice Translation Vector: T=n,a,+n,a,+ na, * So, n(r) can be expressed as a (spatial) Fourier series expansion. So, for a one-dimensional model crystal, n(x) can be represented as where the p’s are integers and the Fourier coefficient of the number density can be written as:* Only The Set of Reciprocal Lattice Vectors Gtifatyer satisfy both (1) & (3) (previous slide) lead to an electronic number density n(r) that is invariant under lattice translations. * It’s not too hard to show that the set of G’s that meet this requirement are of the form G = yb, +v2b> + v3b3! where v,, 0, & v3 are integers & the b,’s are vectors which are defined as: 123,02 b,-a ; = 273, * The a,’s are the primitive lattice vectors for the crystal structure. It also can be shown that The Set of Reciprocal Lattice Vectors¢ In 3 Dimensions, the Fourier coefficient of thbloyer number density has the form: Vn (rye OF The vectors G are called Reciprocal Lattice Vectors * As we said, the electronic density n(r) is required to be invariant (periodic) under lattice translations: n(r) = n(r +T) (2) * That is, it must satisfy: nets =r) when eo ll PPE e772)The Diffraction Condition (Brage’s”” Law) in the Reciprocal Lattice «An X-Ray diffraction pattern of the lattice Can be interpreted as a map of the reciprocal lattice of the crystal. This statement is consistent with the following theorem: The Set of Reciprocal Lattice Vectors G determines seabibie eared*Now, look at this condition for elastic scattering (specular reflection): G = Ak ol’ k | 3k GG" =f This result is called The Laue Condition. It’s not too difficult to show that it is 100% equivalent to The Bragg Law!The Laue Condition is: “2 °\\Pyer G = Ak =k'-k > 2k-G+G" =0 This is 100% equivalent to The Bragg Law! The result that k = G can also be expressed to give the relations that are called The Laue Relations or the Laue Equations. These are obtained by taking the dot product of both Ak & G with a,, a, & a;. The Laue Equations are:| Ak-a; =2nv;. 1 =1.2.3 By changing 0, X-Ray diffraction e can be used to map all x e° Reciprocal Lattice Vectors. A e This geometric method of finding Reciprocal Lattice Vectors is called The Ewald ConstructionFirst 4 Brillouin Zones: Square Lattice” First Four Brillouin Zones for a Square LatticeReciprocal Lattices for the SC, FCC & BCC Lattices Direct Lattice Reciprocal Lattice Volume = a) ie IO e =x b, =(20/ ads . ao =ay bs =(2n/a)y (Qn/ay b3=(2n/ a falx+y by ="A(y+2) ay =4hal-x+y+z) bo ="2(x+2) 4(2n/ a): a3 =4o(x-y+z2) bs =—"(x+y)Fourier Analysis of a Basis! *\\«"er Structure Factor & Atomic Form Factor. + Earlier, we said that the scattering amplitude for a unit cell has the form: F = [dVa(re"C* = NSc cell The quantity S¢ is called The Structure Factor. It is defined as an integral over a single unit cell. If the electron density is written as a superposition of the electron densities in the cell, taking into account the # of atoms in the basis, it has the form:° If (& only if!) the electron densi ihe”ver atom is spherically-symmetric, the atomic form factor f; can be written: f= wart (rr? This means that when the electronic density has the form n(r) = Zg(r), then f;= CZ (C =constant, Z = atomic charge.) That is, in this case, JS-is the ratio of radiation amplitude scattered by the electron distribution to that scattered by a localized electron.Examples of Structure Factor Calculations” BCC lattice In a BCC lattice, there are 2 atoms per cubic unit cell located at (000) & (A444). The structure factor is then: = eh +3) This structure factor has its Maximum, Se = at when the sum of the indices is even. That is when: v,+ vy+ 0,= 2n (n = integer) This structure factor Vanishes, S, = 0 when the sum of the indices is odd. That is when: +y.tv,= Int] (=iExamples of Structure Factor Calculations” FCC lattice For a FCC lattice, there 4 atoms per cubic unit cell located at (000), (02%), (‘4 012) & ('‘2%40). The structure factor is then: eH) —ni(v2+V3) ae This structure factor has its Maximum, S,, = 4f when all indices v,, 02, 03 are even or odd, This structure factor Vanishes, S = 0 when the indices are partially even & partially odd. To summarize, in a FCC lattice, no reflections occur n_the indi re partiall n_and partially odd. _From Ch. 1: Ca SlicePlayer Lattice Planes & Miller Indices A Lattice Plane is any plane containing at least three non-colinear Bravais lattice points. Generally, a lattice plane is described by giving a vector normal to that plane, & there happens to be reciprocal lattice vectors normal to any lattice plane. Choose the the shortest such reciprocal lattice vector to arrive at the Miller indices of the plane. Thus a plane with Miller indices h, k, /, is normal to the reciprocal lattice vector hb, + kb, + Ib.SXUVICVY UE LYESEIUE BEIUIATDS WCUMUMUCUW as Recall That All planes in a set are identical. The perpendicular distance between pairs of adjacent planes is the d-spacing. Miller Indices define the orientation of the plane within the unit cell. The Miller Indices define a set of planes parallel to one another: (002) planes are parallel to (001) planes, and so on. So, in analyzing crystal diffraction results, the lattice planes are labeled with their Miller ea °d - Spacing Formula [2 sivcr d = Distance between Parallel Lattice Planes with Miller Indices (hkl) Its not difficult to prove the following: Orthogonal Cubic Crystal Systems Crystal Systems (a =B=y=90°) (a= Oa A cubic crystal system is a special case of anThe Laue Condition is:. (3 °\"yer This is 100% equivalent to The Bragg Law! The result that k = G can also be expressed to give the relations that are called The Laue Relations or the Laue Equations. These are obtained by taking the dot product of both Ak & G with a, a, & a3. The Laue Equations are:| Ak-a, =2nv,, 7 =1,2.3 By changing 0, X-Ray diffraction can be used to map all Reciprocal Lattice Vectors. This geometric method of finding Reciprocal Lattice Vectors is called The Ewald Construction