Point Lattices: Bravais Lattices

Transcription

1 Physics for Solid Stte Applictions Februry 18, 2004 Lecture 7: Periodic Structures (cont.) Outline Review 2D & 3D Periodic Crystl Structures: Mthemtics X-Ry Diffrction: Observing Reciprocl Spce Point Lttices: Brvis Lttices Brvis lttices re point lttices tht re clssified topologiclly ccording to the symmetry properties under rottion nd reflection, without regrd to the bsolute length of the unit vectors. A more intuitive definition: At every point in Brvis lttice the world looks the sme. 1D: Only one Brvis Lttice

2 Wigner-Sietz Cell 1. Choose one point s the origin nd drw lines from the origin to ech of the other lttice points. 2. Bisect ech of the drwn lines with plnes norml to the line. 3. Mrk the spce bout the origin bounded by the first set of plnes tht re encountered. The bounded spce is the Wigner-Sietz unit cell. 1D Periodic Crystl Structures M(x) S (x) (1) (1) (1) (1) (1) M (x) x = x x M(q) S (q) M (q) 4π 2π 0 2π = 4π 2π 0 2π 4π q 4π 2π 0 2π

3 Reciprocl Lttice Vectors 1. The Fourier trnsform in q-spce is lso lttice 2. This lttice is clled the reciprocl lttice 3. The lttice constnt is 2 π / 4. The Reciprocl Lttice Vectors re K -2 K -1 K 0 K 1 K 2 q Atomic Form Fctors & Geometricl Structure Fctors M (x) d 1 d 2 Different Atoms (1) nd (2) x d 1 M (x) d 2 Sme toms t d 1 nd d x

4 2D & 3D Crystl Structures A. Orthogonl Primitive Lttices Vectors Then choose primitive lttice vectors long the x, y nd z xes The 3D smpling function is Fourier Trnsform of Smpling Function This is product of three independent 1D smpling functions

5 Therefore, Reciprocl Lttice The reciprocl lttice vectors re with primitive lttice vectors 3D Periodic Functions A periodic function cn be written s convolution nd

6 Reciprocl Spce Representtion The convolution in rel spce becomes product in reciprocl spce since S(k) is series of delt functions t the reciprocl lttice vectors, Mp(k) only needs to be evluted t reciprocl lttice vectors: Here PC men to integrte over one primitive cell, such s the Wigner-Seitz cell Therefore, M(q) is crystl structure in q-spce Generlized Fourier Trnsform Therefore this is generlized Fourier Trnsform in 2D nd 3D with

7 Form fctors nd Structure Fctors Mp(q) is the FT of toms in one primitive cell. Let ech tom j of n n tom bsis hve density function f j (r) nd be locted position d j. The Fourier components tht re needed for Mp re where the tomic form fctor is f j (K). If the bsis consist of ll the sme type of toms then, Atomic form fctor Geometricl Structure Fctor Oblique Primitive Lttice Vectors

8 Periodic functions with oblique lttice vectors Periodic function Therefore need to find set {K i } of ll possible vectors q such tht Reciprocl Lttice Vectors for Oblique vectors Let And ssume tht {K j } lso forms lttice with primitive vectors b k Then to hve we need By construction, the primitive reciprocl lttice vectors re

9 Smpling function for oblique vectors Write ll vectors in rel spce in terms of the i s nd write ll vectors in reciprocl spce in terms of the b j s Then the dot product is simply And s proven in the notes, even for the oblique vectors, Rectngulr Lttice The primitive lttice vectors re orthogonl in this cse. The primitive reciprocl lttice vectors re lso define rectngulr lttice, but rescled inversely.

10 BCC FCC BCC in rel spce hs n FCC reciprocl lttice with FCC in rel spce hs BCC reciprocl lttice with

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12 X-Ry Diffrction k Observtion point r Plne wve e ik.r Outgoing sphericl wve scttering point r The totl mplitude of the wve rriving t r from ll the r points in the smple locted with density n(r) is

13 Scttering in the Fr-field In the fr-field region, r - r >> L nd we cn use the pproximtion so tht Becuse the density is periodic function, So tht Therefore, the mplitude is zero unless k = k K j. The Brgg Condition Squring the condition k = k Kj gives X-ry diffrction is elstic, it does not chnge the mgnitude of the wve vector, so tht k = k, which gives the Brgg Condition unit vector

14 Brgg Condition This sweeps out plne t the perpendiculr bisector of K. These re the sme plnes tht define the Wigner-Sietz cell construction, nd for this reciprocl spce define the First Brillioun zone. K/2 K/2 0 k k K Brgg Condition d sin θ θ Lttice plnes Or