Introduction Example of structure Miller index

Chap 1

Dept of Phys

M.C. Chang Introduction Example of crystal structure Miller index

Bulk structure of matter

Crystalline solid Non-crystalline solid Quasi-crystal (1982 ) (amorphous, glass)

Ordered, but not periodic

2011 Shechtman

www.examplesof.net/2017/08/example-of-solids-crystalline-solids-amorphous-solids.html#.XRHQxuszaJA nmi3.eu/news-and-media/the-contribution-of-neutrons-to-the-study-of-.html Introduction Example of crystal structure Miller index

• A simple lattice (in 3D) = a set of points with positions at

r = n1a1+n2a2+n3a3 (n1, n2, n3 cover all integers ),

a1, a 2, and a3 are called primitive vectors (原始向量), and

a1, a 2, and a3 are called lattice constants ( 晶格常數).

Alternative definition: • A simple lattice = a set of points in which every point has exactly the same environment.

Note: A simple lattice is often called a . From now on we’ll use the latter term ( not used by Kittel). scienceline.ucsb.edu/getkey.php?key=4630 Miller index Miller s. 基元 (of (of ) is basis (one , or a group of atoms) is a physics physics is a term. 晶體 crystal Example of crystal structure ofcrystal Example lattice (How repeat)(How to (What to repeat) is a math term, while while term,math is a 晶格 Lattice Introduction Note: Note: Crystal structure = structure Crystal Bravais+ lattice bas Decomposition: • Bravaiscorresponding a has crystal Every basi and a lattice Introduction Example of crystal structure Miller index

Bravais lattice non-Bravais lattice = Bravais + p-point basis (p>1)

Triangular (or hexagonal) lattice Honeycomb lattice

Primitive vectors

a2 basis

a1

Honeycomb lattice = triangular lattice + 2-point basis (i.e. overlap of 2 triangular lattices)

A simple way to determine the number p of basis: Just look around the environment of atoms. Introduction Example of crystal structure Miller index

Unit cell ( 晶胞) primitive cell (原始晶胞 ) non-primitive cell

Primitive vectors

• A primitive cell contains only 1 lattice point. • A non-primitive cell contains 2 or more lattice points. – For Bravais lattice, a unit cell can be primitive or non-primitive – For non-Bravais lattice, a unit cell is always non-primitive Introduction Example of crystal structure Miller index

A special primitive cell: Wigner-Seitz (WS) cell • Method of construction (works in 1D, 2D, 3D)

• The WS cell enclosing a lattice point is the region of space that is closer to that lattice point than to any others.

• Advantage of the WS cell: It has the same symmetry as the lattice (symmetry here means translation, inversion, and rotation ) Introduction Example of crystal structure Miller index

Crystal structures of elements

ccp (cubic close packing) = fcc

Note: It’s rare to see simple cubic lattice.

chemicalstructure.net/portfolio/magnesium-and-aluminum/ Introduction Example of crystal structure Miller index

1). bcc lattice (Li, Na, K, Rb, Cs… etc )

One possible choice of primitive vectors a lattice d a constant a=() xyzˆ +ˆ − ˆ , 1 2 d a a=() −+ xyzˆ ˆ + ˆ , conventional unit cell 2 2 (慣用晶胞 , non-primitive ) d a a=() xyzˆ − ˆ + ˆ . 3 2

• A bcc lattice is a Bravais lattice. • For convenience of description, we can also treat it as a simple cubic lattice with 2-point basis . Introduction Example of crystal structure Miller index

2). fcc lattice (Ne, Ar, Kr, Xe, Al, Cu, Ag, Au… etc )

One possible choice of primitive vectors a d a a=() xˆ + yˆ , 1 2 d a a=() yˆ + zˆ , 2 2 A primitive d a a=() zˆ + xˆ . Solid Ar at -189.3 C unit cell A conventional 3 2 unit cell (non- primitive)

• A fcc lattice is also a Bravais lattice. • It can also be seen as a simple cubic lattice with 4-point basis . Introduction Example of crystal structure Miller index

3). hcp structure (Be, Mg… etc )

2 overlapping “simple 2-point basis hexagonal lattices”

• hcp structure = simple hexagonal lattice + 2-point basis

• Primitive vectors: a1, a2, c [ c=2 a√ (2/3) for hcp, see Prob.3] • The 2 atoms of the basis are located at

d1=0, d2 = (2/3) a1+ (1/3) a2+(1/2) c Introduction Example of crystal structure Miller index

The tightest way to pack spheres in 3D:

– ABCABC …= fcc – ABAB …= hcp – Other close packed structures: ABABCAB… etc. Introduction Example of crystal structure Miller index

Viewing from different angles

• (配位數 ) = 12, packing fraction ～ 74% (Cf: bcc, coordination number = 8, packing fraction ～ 68%)

Wiki: close packing of equal spheres ) 18 √ /

π Nature, 3 July 2003 Miller index Miller ≦ s in 3-dim s 3-dim in (the value of fccvalue (the and hcp) Example of crystal structure ofcrystal Example can show that the crystalline form has the lowest lowest the energy. form has show crystalline that the can nobody Introduction Kepler’s conjecture (1611): The packing of packing sphere Thefraction Kepler’s conjecture (1611): (By way, the Introduction Example of crystal structure Miller index

4). structure (C, Si, Ge… etc )

= 2 overlapping fcc lattices (one is displaced along the main diagonal by 1/4)

= fcc lattice + 2-point basis, d1=0, d2=(a/4)( x+y+z)

• Very low packing fraction ( ～36%!) • If the two atoms on the basis are different, then it is called Zincblend (閃鋅 ) structure (eg. GaAs, ZnS… etc), which is a familiar structure with an unfamiliar name. Introduction Example of crystal structure Miller index

Indexing crystal planes: Miller index ( h,k,l )

An Indexed PbSO 4 Crystal

primitive vectors of Bravais lattice (but not always the simplest one, e.g, z=1 The rules: the simple cubic lattice of fcc lattice)

1. 取截距 (以a1, a2, a3為單位) a3 得 (x, y, z) a2 y=3 a1 2. 取倒數 (1 /x, 1/y, 1/z ) x=2 3. 通分成互質整數 (h,k,l ) (h,k,l )=(3,2,6) Introduction Example of crystal structure Miller index

For example, cubic (including bcc, fcc… etc)

[1,1,1]

• Square bracket [ h,k,l ] refers to the “direction” ha1+ka2+la3, instead of crystal planes! • For cubic crystals, [ h,k,l ] direction ⊥ (h,k,l ) planes Introduction Example of crystal structure Miller index

Diamond structure (eg. C, Si or Ge ) Termination of 3 low-index surfaces:

•{h,k,l } = ( h,k,l )- + those equivalent to it by crystal symmetry • = [ h,k,l ]-direction + those equivalent to it by crystal symmetry Introduction Example of crystal structure Miller index Microscopic symmetry ↔ Macroscopic property • appearance

Prob.2.9: Ha üy’s law regarding the Al melt on Si {111} angles between crystal faces surface reveals the symmetry of the diamond lattice

• (eg. ) • optical activity 旋光性 雙折射 方解石