Bravais Gitter Structure

SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I. Pascual Structure of solids

Struktur von EinkristallenIdealer Kristall: unendliche Wiederholung von identischen Strukturelementen Struktureinheit wird als Basis bezeichnet und die Vorschrift fr die Aneinanderreihungresultiert in einem Raumgitter Kristall = Gitter + Basis

Das Bravais-GitterUnendliches Gitter von Raumpunkten mit einer Anordnung und Orientierung, die exakt gleich aussieht, egal von welchem Gitterpunkt wir das Gitter betrachten

R = n1a1 +n2 a22-dimensional

R = n1a1 + n2 a2 + n3 a33-dimensional

n1, n2, n3 : ganze Zahlena1, a2, a3: Basisvektoren (Gitterkonstanten)

T = n1a1 + n2 a2 + n3 a3

Bravais-Gitter invariant gegenber diskretenTranslationen um den Translationsvektor


Crystal structure in 2D

BravaisBravais lattice in two dimensionslattice in two dimensionsKristall = Gitter + Basis


Crystal structure in 2D

Not an unit cell

The primitive unit cell

Wigner-Seitz- unit cellIs a primitive cell


Bravais-Gitter7 Kristallsysteme mit 14 Bravaisgittern

Experimentalphysik IV; SoSe2008; Petra Tegeder


Simple cubic The simple cubic structure is a Bravais lattice. The basis is one atom. So there is one atom per unit cell.

a1 a2

a3 A simple cubic structure is not a good idea for

packing spheres (they occupy only 52% of the total volume).

Only two elements crystallise in the simple cubic structure (F and O).

1/8 atom


hcpABABAB...

fccABCABCABC...

The face-centred cubic (fcc) and hexagonal close-packed (hcp) structure have the same packing fraction

Close-packed structures: fcc and hcp


Kristallstrukturen


kubisch raumzentriertes Gitter (bcc-Struktur) bcc: body centered cubic

kubisch flchenzentriertes Gitter(fcc-Struktur) fcc: face centered cubic

Festkrper, deren Basis nur aus einem Atom besteht

z.B. Alkalimetalle, Metalle(Wolfram Tantal, Molybdn) z.B. Edelmetall, Edelgase

Jedes Atom hat 8 nchste Nachbarn(Koordinationszahl 8)

Jedes Atom hat 12 nchste Nachbarn(Koordinationszahl 12)


Close-packed structures: hcp

The hcp lattice is NOT a Bravais lattice. It can be constructed from a Bravais lattice with a basis containing two atoms.

the packing efficiency is of course exactly the same as for the fcc structure (74 % of space occupied).


Kristallstrukturen


Natriumchlorid-Struktur Csiumchlorid-Struktur

Beispiele: Beispiele:

Festkrper, deren Basis nur aus zwei Atomen besteht

Cl-


Diamant-Struktur Zinkblende-Struktur

Kristallstrukturen


Beispiele: Beispiele:


Gitterstrukturen der chemischen Elemente


dhcp-Struktur (double hexagonal closed packed)Stapelfolge ABACABAC


X-ray diffraction

The atomic structure of crystals cannot be determined by optical microscopy because the wavelength of the photons is much too long (400 nm or so).Thecrystals can be used to diffract X-rays (von Laue, 1912).


Experimentelle Methoden zur Strukturbestimmung

Beugung von Wellen an periodischen Strukturen

Die Bragg-Bedingung:

2d sin = n n= 1, 2, 3, . . . .

William Henry Bragg (1862-1942) William Lawrence Bragg (1890-1971)Nobelpreis f. Physik (1915)


Labelling crystal planes (Miller indices)1. determine the interceptswith the axes in units of the lattice vectors

2. take the reciprocal of each number

3. reduce the numbers to the smallest set of integers having the same ratio. These are then called the Miller indices.

step 1: (2,1,2)step 2: ((1/2),1,(1,2))

step 3: (1,2,1)

S1 = m1 aS2 = m2 b

S3 = m3 c

h = p/m1k = p/m2l = p/m3

Miller Indices: Triple (hkl) integer numbers

p: smallest integer numberleading to integer hklnumbers


Example


The reciprocal lattice

for a given Bravais lattice

The reciprocal lattice is also a Bravais lattice

the reciprocal lattice is defined as the set of vectors G for which

a useful relation is



if we have

The vectors G of the reciprocal lattice giveplane waves with the periodicity of the lattice.

In this case G is the wave vector and 2pi/|G| the wavelength.

then we can write


The reciprocal latticeexample 1: in two dimensions

|a1|=a|a2|=b

|b2|=2pi/b

|b1|=2pi/a


Lattice waves

real space reciprocal space

a

b

2pi/a

2pi/b(0,0)



example 2:

The fcc lattice is the reciprocal of the bcc lattice and vice versa.


The reciprocal of the reciprocal latticeis again the real lattice

|a1|=a|a2|=b

|b2|=2pi/b

|b1|=2pi/a


Other applications of the reciprocal lattice

example: charge density in the chain

Fourier series

alternatively


Other applications of the reciprocal lattice

1D reciprocal lattice


Die van Laue-Beugung


Max von Laue (1879-1960)Nobelpreis fr Physik 1914

van Laue-Bedingung:

k k = k = G

Bei der Streuung an einem Kristall wird nurdann konstruktive berlappung der Teilwellen erhalten, wenn die nderung k = k- k des Wellenvektors gleich einem Gittervektor G des reziproken Gitters ist

Der Satz G der reziproken Gittervektoren bestimmt die mglichen Beugungsreflexe. Messanordnung Laue-Beugung:

kontinuierliches Rntgenspektrum (variables );Beugung am Einkristall


The Ewald construction

Draw (cut through) the reciprocal lattice.

Draw a k vector corresponding to the incoming x-rays which ends in a reciprocal lattice point.

Draw a circle around the origin of the k vector.

The Laue condition is fulfilled for all vectors k for which the circle hits a reciprocal lattice point.

Laue condition if G is a rec. lat. vec.Elastic scattering

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Bravais Gitter Structure