Crystal Structure

Definisi Kristal Kristal merupakan zat padat akan tetapi zat padat tidak selalu berstruktur kristal Zat padat dikatakan berstruktur kristal jika atom-atom penyusunnya tertata secara teratur dan periodik Ilustrasi struktur kristal dalam gambaran dua dimensi T merupakan Vektor Translasi A, B, dan C adalah atom Penyusun kristal a1 adalah jarak antara atom

Kisi Kristal Sebuah kristal memiliki simetri translasi-menurut definisi. Jika r (r) adalah rapat elektron dalam kristal di r maka ada 3 vektor a, b & c  sehingga: r (r) = r (r + u · a + v · b + w · c) dengan u, v & w integer. Setiap bentukan identik dinamakan unit cell. a, b & c = vektor sel satuan. Panjang vektor sel satuan a = |a|, b = |b|, c = |c|. α, β & γ = sudut-sudut antara vektor 2 sel satuan. Right handed coordinate system.

Koordinat Fraksional Sembarang posisi di dalam kristal dapat dinyatakan: r = (u + x)· a + (v + y)· b + (w + z)· c dengan u, v & w integer & 0 < x, y, z < 1. x, y & z disebut “fractional coordinates” & menyatakan posisi di dalam sel satuan. a b c

Sistem Kristal Struktur kristal dapat digambarkan dalam bentuk kisi, dimana: Setiap titik kisi akan ditempati oleh atom atau sekumpulan atom Kisi kristal memiliki sifat geometri yang sama seperti kristal Kisi yang memiliki titik-titik kisi yang ekuivalen disebut kisi Bravais sehingga titik-titik kisi tersebut dalam kristal akan ditempati oleh atom-atom yang sejenis Titik A, B dan C adalah ekuivalen satu sama lain Titik A dan A1 tidak ekivalen (non-Bravais)

Point, Plane and Space Groups Point Groups = Collection (group) of symmetry operators that all pass through the same point. The group must be closed, have an identity element, and every element must have an inverse. There are 32 unique ways in which lattice points can be arranged in space. Plane Groups = Group of symmetry operators that are compatible with two-dimensional symmetry in a plane. Space Groups = Collections of symmetry operators that are compatible with three-dimensional crystallographic (i.e. translational) symmetry. There are 230 space groups. Because protein and nucleic acid molecules are chiral, there are only 65 “biological” space groups.

Sistem Kristal Titik-titik kisi Bravais dapat ditempati oleh atom atau sekumpulan atom yang disebut basis Kisi Sekumpulan titik-titik yang tersusun secara periodik dalam ruang Basis Atom atau sekumpulan atom Sehingga apabila atom atau sekumpulan atom tersebut menempati titik-titik kisi maka akan membentuk suatu struktur kristal

Bravais Lattices 7 UNIT CELL TYPES + 4 LATTICE TYPES = 14 BRAVAIS LATTICES

Lattice Planes Useful concept for crystallography & diffraction Think of sets of planes in lattice - each plane in set parallel to all others in set. All planes in set equidistant from one another Infinite number of set of planes in lattice d d-interplanar spacing

Lattice Planes Keep track of sets of planes by giving them names - Miller indices (hkl)

Miller Indices (hkl) Choose cell, cell origin, cell axes: origin b a

Miller Indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest: origin b a

Miller Indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin: origin b a

Miller Indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin Find intercepts on cell axes: 1,1,∞ origin b 1 a 1

Miller Indices (hkl) Choose cell, cell origin, cell axes Draw set of planes of interest Choose plane nearest origin Find intercepts on cell axes 1,1,∞ Invert these to get (hkl) (110) origin b 1 a 1

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Indeks Bidang: Kristal Kubik

Bidang 001

Bidang-bidang 111

Bidang-bidang 110

Jarak Antar Bidang Kristal Kubik

Jarak Antar Bidang

Volume Sel Satuan

Sudut Antar Bidang

Arah Bidang Kristal Kubik

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Exercises

Lattice Planes Two things characterize a set of lattice planes: interplanar spacing (d) orientation (defined by normal)

A1 (FCC/CCP, Struktur Cu) Berapa: Jumlah atom terdekat? Fraksi okupansi atom

A2 (BCC, Struktur W) Berapa: Jumlah atom terdekat? Fraksi okupansi atom

A3 (HCP, Struktur Mg) Berapa: Jumlah atom terdekat? Fraksi okupansi atom

B1 (Struktur Halite)

C2 (Struktur Rutile)

Latihan Gambarkan struktur kristal fluorite.

Strange Indices For hexagonal lattices - sometimes see 4-index notation for planes (hkil) where i = - h - k a3 a1 a2 (110) (1120)

Zones 2 intersecting lattice planes form a zone zone axis zone axis zone axis [uvw] is ui + vj + wk i j k h1 k1 l1 h2 k2 l2 plane (hkl) belongs to zone [uvw] if hu + kv + lw = 0 if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then (h1+h2 k1+k2 l1+l2 ) also in same zone.

zone axis [uvw] is ui + vj + wk Zones Example: zone axis for (111) & (100) - [011] zone axis [uvw] is ui + vj + wk i j k h1 k1 l1 h2 k2 l2 i j k 1 1 1 1 0 0 (100) (111) [011] (011) in same zone? hu + kv + lw = 0 0·0 + 1·1 - 1·1 = 0 if (h1 k1 l1) and (h2 k2 l2 ) in same zone, then (h1+h2 k1+k2 l1+l2 ) also in same zone.

Reciprocal lattice Real space lattice

Real space lattice - basis vectors Reciprocal lattice Real space lattice - basis vectors a a

Real space lattice - choose set of planes Reciprocal lattice Real space lattice - choose set of planes (100) planes n100

Real space lattice - interplanar spacing d Reciprocal lattice Real space lattice - interplanar spacing d (100) planes d100 1/d100 n100

Real space lattice  the (100) reciprocal lattice pt planes d100 n100 (100)

Reciprocal lattice The (010) recip lattice pt n010 (100) planes d010

The (020) reciprocal lattice point planes d020 (010) (020) (100)

More reciprocal lattice points (010) (020) (100)

The (110) reciprocal lattice point (100) planes n110 d110 (010) (020) (100) (110)

Still more reciprocal lattice points (100) planes (010) (020) (100) the reciprocal lattice (230)

Reciprocal lattice Reciprocal lattice notation

Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

Reciprocal lattice Reciprocal lattice for hexagonal real space lattice

Reciprocal lattice

Contoh Soal 1. Wolfram membentuk kristal kubus berpusat badan . Dari fakta bahwa rapat massa wolfram 19,3 gr/cm3. Hitung a. Panjangnya sisi sel satuan b. Jarak antar bidang 222

2. Insulin membentuk kristal dari jenis orthorombik dengan dimensi sel satuan 13 x 7,48 x 3,09 nm. Bila rapat massa kristal adalah 1,315 g/cm3 dan terdapat enam molekul insulin per sel satuan, berapa massa molar dari protein insulin. 3. Molibdenum membentuk kristal kubus berpusat badan dan pd suhu 20° C rapat massanya 10,3 gr/cm3. Hitung jarak antara pusat-pusat atom molibdenum yg berdekatan.