phys 624: crystal structures and symmetry 1 crystal...

PHYS 624: Crystal Structures and Symmetry 1

Crystal Structures andSymmetry

Introduction to Solid State Physics

http://www.physics.udel.edu/∼bnikolic/teaching/phys624/phys624.html


Translational Invariance

• The translationally invariant nature of the periodic solid and the fact thatthe core electrons are very tightly bound at each site (so we may ignoretheir dynamics) makes approximate solutions to many-body problem ≈ 1021

atoms/cm3 (essentially, a thermodynamic limit) possible.

Figure 1: The simplest model of a solid is a periodic array of valance orbitalsembedded in a matrix of atomic cores. Solving the problem in one of theirreducible elements of the periodic solid (e.g., one of the spheres in theFigure), is often equivalent to solving the whole system.


From atomic orbitals to solid-state bands

• If two orbitals are far apart, each orbital has a Hamiltonian H0 = εn,where n is the orbital occupancy ⇐ Ignoring the effects of electronic corre-lations (which would contribute terms proportional to n↑n↓).

+ ++ + . . . = Band

E

Figure 2: If we bring many orbitals into proximity so that they may exchangeelectrons (hybridize), then a band is formed centered around the locationof the isolated orbital, and with width proportional to the strength of thehybridization


From atomic orbitals to solid-state bands

• Real Life: Solids are composed of elements with multiple orbitals thatproduce multiple bonds. Now imagine what happens if we have severalorbitals on each site (ie s,p, etc.), as we reduce the separation between theorbitals and increase their overlap, these bonds increase in width and mayeventually overlap, forming bands.

↓• The valance orbitals, which generally have a greater spatial extent, willoverlap more so their bands will broaden more.

↓• Eventually we will stop gaining energy from bringing the atoms closertogether, due to overlap of the cores ⇒ Once we have reached the optimalpoint we fill the states 2 particles per, until we run out of electrons.

↑• Electronic correlations complicate this simple picture of band forma-tion since they strive to keep the orbitals from being multiply occupied.


Band developments and their fillingquantum numbers nl elemental solid

1s H,He

2s Li,Be

2p B→Ne

3s Na,Mg

3p Al→Ar

4s K,Ca

3d transition metals Sc→Zn

4p Ga→Kr

5s Rb,Sr

4d transition metals Y→Cd

5p In-Xe

6s Cs,Ba

4f Rare Earths (Lanthanides) Ce→Lu

5d Transition metals La→Hg

6p Tl→Rn


• For large n, the orbitals do not fill up simply as a function of n as we would expect

from a simple Hydrogenic model with En = mZ2e4

2h2n2 ⇒ Z → Znl

z/r

1s

2sp2s

3spd

1s

3s3p

3d4spdf4s

4p5spdf5s

4d5p 6spdf6s4f5d

2p

+

s

datom

4f

Ce Valence Shell

6s

5d

VPr)

VPr) + C l Pl+1)/r 2

Figure 3: Level crossings due to atomic screening. The potential felt by states with large

l are screened since they cannot access the nucleus. Thus, orbitals of different principle

quantum numbers can be close in energy. I.e., in elemental Ce, (4f15d16s2) both the 5d

and 4f orbitals may be considered to be in the valence shell, and form metallic bands.

However, the 5d orbitals are much larger and of higher symmetry than the 4f ones.

Thus, electrons tend to hybridize (move on or off) with the 5d orbitals more effectively.

The Coulomb repulsion between electrons on the same 4f orbital will be strong, so these

electrons on these orbitals tend to form magnetic moments.


Different Types of Chemical Bonds

• The overlap of the orbitals is bonding:

Bond Overlap Lattice constituents

Ionic very small (< a) closest unfrustrated dissimilar

packing

Covalent small (∼ a) determined by the similar

structure of the orbitals

Metallic very large (� a) closest packed unfilled valence

orbitals

Table 1: The type of bond that forms between two orbitals is dictated largelyby the amount that these orbitals overlap relative to their separation a.


Covalent Bonding

• The pile-up of charge which is inherent to the covalent bond is important for the

lattice symmetry. The reason is that the covalent bond is sensitive to the orientation of

the orbitals.

S P

+

-S

P+ -

No bonding Bonding

Figure 4: A bond between an S and a P orbital can only happen if the P-orbital is oriented with either its plus or minus lobe closer to the S-orbital.I.e., covalent bonds are directional!


Ionic Bonding

• The ionic bond occurs by charge transfer between dissimilar atoms which initially have

open electronic shells and closed shells afterwards. Bonding then occurs by Coulomb

attraction between the ions.

+e-

Na+Na

Cl +e-

Cl-

Na+ Cl-+ Na+ Cl- + 7.9 eV

+ 3.61 eV

+ 5.14 eV

r

r

Cl

Na

= 1.81

= 0.97

Figure 5: The energy per molecule of a crystal of sodium chloride is (7.9-5.1+3.6) eV=6.4eV lower than the energy of the separated neutral atoms.The cohesive energy with respect to separated ions is 7.9eV per molecularunit. All values on the figure are experimental.


Metallic Bonding

• Metallic bonding is characterized by at least some long ranged and non-directional

bonds (typically between s-orbitals), closest packed lattice structures and partially filled

valence bands.

3d x - y 2 2

4S

Figure 6: In metallic Ni (FCC, 3d84s2), the 4s- and 3d-bands (orbitals) are almost

degenerate and thus, both participate in the bonding. However, the 4s-orbitals are so

large compared to the 3d-orbitals that they encompass many other lattice sites, forming

non-directional bonds. In addition, they hybridize weakly with the d-orbitals (the different

symmetries of the orbitals causes their overlap to almost cancel) which in turn hybridize

weakly with each other. Thus, whereas the s-orbitals form a broad metallic band, the

d-orbitals form a narrow one.


Discrete translations symmetry

• Translational symmetry of the lattice: There exist a set of basis vec-tors (�a,�b,�c) such that the atomic structure remains invariant under transla-tions through any vector �rn = n1�a+n2

�b+n3�c where n1, n2, n3 are integers.

a

b

Figure 7: One may go from any location in the lattice to an identical locationby following path composed of integral multiples of the vectors �a and �b.

• Note that basic building blocks of periodic structures can be more complicated than a

single atom. For example in NaCl, the basic building block is composed of one Na and

one Cl ion which is repeated in a cubic pattern to make the NaCl structure.


Lattice types and symmetry

• A collection of points in which the neighborhood of each point is the sameas the neighborhood of every other point under some translation is calledBravais lattice.

• The primitive unit cell is the parallel piped (in 3D) formed by the prim-itive lattice vectors which are defined as the lattice vectors which producethe primitive cell with the smallest volume �a · (�b × �c).

• There are many different primitive unit cells—common features: each cellhas the same volume and contains only one site of Bravais lattice (Wigner-Seitz cell → site is in the center of the cell).

• Non-primitive unit cell: Minimal region (which can contain severalparticles) of a crystal that has the same Point Group symmetry as thecrystal itself and that produces the full crystal upon repetition.

• SPACE GROUP: The complete set of rigid body motions that takecrystal into itself G = T�rn

+ R(θ, θ)


Example: 2D Bravais lattices

|a| = |b|, = /2

Square

|a| = |b|, = /2

Rectangular

Hexangonal

|a| = |b|, = /3

Centered

a

b

ab

ab

ab

Figure 8: Two dimensional lattice types of higher symmetry. These have higher sym-

metry since some are invariant under rotations of 2π/3, or 2π/6, or 2π/4, etc. The

centered lattice is special since it may also be considered as lattice composed of a two-

component basis, and a rectangular unit cell (shown with a dashed rectangle).


Example: 3D Bravais lattices

• The situation in three-dimensional lattices can be more complicated: thereare 14 lattice Bravais lattices (for example there are 3 cubic structures,shown in the Figure).

a = xb = yc = z

a = Px+y-z)/2b = P-x+y+z)/2c = Px-y+z)/2

a = Px+y)/2b = Px+z)/2c = Py+z)/2

Cubic Body Centered Cubic

Face CenteredCubic

ab

c

ab

c

Figure 9: Three-dimensional cubic lattices. Note that the primitive cells ofthe centered lattice is not the unit cell commonly drawn.


Lattice decorated with a basis

• To account for more complex structures like molecular solids, salts, etc., one also allows

each lattice point to have structure in the form of a basis. A good example of this in

two dimensions is the CuO2 planes which characterize the cuprate high temperature

superconductors. Here the basis is composed of two oxygens and one copper atom laid

down on a simple square lattice with the Cu atom centered on the lattice points.

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O

Cu O

O Basis

b

a

Primitivecell and latticevectors

Figure 10: A square lattice with a complex basis composed of one Cu andtwo O atoms as in cuprate high-temperature superconductors.


Primitive vs. Non-primitive unit cell

• Crystal structure with primitive unit cell, whose atoms are put in the sitesof Bravais lattice, overlaps with the Bravais lattice itself. However:

Primitive Unit Cell

Non�primitive Unit Cell

• Pay attention to 45◦ rotation around axis passing through the yellow atom!


Point GROUP symmetry: C3v example


Symmetry transformation form GROUPS

→ A group S is defined as a set {E, A, B, C . . .} which is closed under abinary operation ◦ (i.e., A ◦ B ∈ S) and satisfies the following axioms:

• the binary operation is associative (A ◦ B) ◦ C = A ◦ (B ◦ C)

• there exists an identity E ∈ S: E ◦ A = A ◦ E = A

• for each A ∈ S, there exists an A−1 ∈ S : A ◦ A−1 = A−1 ◦ A = E

• In the point group context, the operations are: inversions, reflections,rotations, and improper rotations (inversion rotations).

• The binary operation is any combination of these; i.e. inversion followedby a rotation.


Group Designations

• Schonflies point group symbol—These give the classification according to

rotation axes and principle mirror planes. In addition, their are suffixes for mirror planes

(h-horizontal=perpendicular to the rotation axis, v-vertical=parallel to the main rotation

axis in the plane, d-diagonal=parallel to the main rotation axis in the plane bisecting the

two-fold rotation axes):

Symbol Meaning

Cj (j=2,3,4, 6) j-fold rotation axis

Sj j-fold rotation-inversion axis

Dj j 2-fold rotation axes ⊥ to a j-fold principle rotation axis

T 4 three-and 3 two-fold rotation axes, as in a tetrahedron

O 4 three-and 3 four-fold rotation axes, as in a octahedron

Ci a center of inversion

Cs a mirror plane


Reduction of quantum complexity via symmetry

• If a Hamiltonian is invariant under certain symmetry operations, then wemay choose to classify the eigenstates as states of the symmetry operationand H will not connect states of different symmetry.

• Symmetry operation R leaves H invariant: RHR−1 = H ⇒ [H, R] = 0

• If |j〉 are the eigenstates of R|j〉 = Rj |j〉, then I =∑

j |j〉〈j| is the identityoperator.

• Expand HR = RH and examine its elements:

∑

k

〈i|R|k〉〈k|H|j〉 =∑

k

〈i|H |k〉〈k|R|j〉 ≡ (Rii − Rjj) Hij = 0

• Hij = 0 if Ri and Rj are different eigenvalues of R → when the statesare classified by their symmetry, the Hamiltonian matrix becomes blockdiagonal, so that each block may be separately diagonalized.


Face-centered cubic (FCC) lattice

a = Px+y)/2b = Px+z)/2c = Py+z)/2

Face Centered Cubic PFCC)

a

b c

Principle lattice vectorsClose-packed planes

3-fold axes

4-fold axes

x

y

z

Figure 11: The Bravais lattice of a face-centered cubic (FCC) structure. As shown

on the left, the FCC structure is composed of parallel planes of atoms, with each atom

surrounded by 6 others in the plane. The total coordination number (the number of

nearest neighbors) is 12. The principle lattice vectors (center) each have length 1/√

2

of the unit cell length. The lattice has four 3-fold axes, and three 4-fold axes as shown

on the right. In addition, each plane shown on the left has the principle 6-fold rotation

axis ⊥ to it, but since the planes are shifted relative to one another, they do not share

6-fold axes. Thus, four-fold axes are the principle axes, and since they each have a

perpendicular mirror plane, the point group for the FCC lattice is Oh.


Hexagonal close packed (HCP) Lattice

mirror plane

3-foldaxis

three 2-foldaxes in plane

Figure 12: The symmetry of the HCP lattice. The principle rotation axis isperpendicular to the two-dimensional hexagonal lattices which are stacked toform the HCP structure. In addition, there is a mirror plane centered withinone of these hexagonal 2d structures, which contains three 2-fold axes. Thusthe point group is D3h.

• The HCP structure is similar to the FCC structure, but it does not correspond to a

Bravais lattice (there are five cubic point groups, but only three cubic Bravais lattices).


FCC vs. HCP lattice

A A AA A

A A AA A

A A AA A

FCC HCP

These spaces unfilled

B B BB B

B B BB B

B B BB B

C C CC C

C C CC C

C C CC C

A A AA A

A A AA A

A A AA A

B B BB B

B B BB B

B B BB B

Figure 13: A comparison of the FCC (left) and HCP (right) close packedstructures. The HCP structure does not have a simple Bravais unit cell,but may be constructed by alternately stacking two-dimensional hexagonallattices. In contract, the FCC structure may be constructed by sequentiallystacking three shifted hexagonal two-dimensional lattices.


Body-centered cubic (BCC) lattice

• Just like the simple cubic and FCC lattices, the body-centered cubic(BCC) lattice has: four 3-fold axes, three 4-fold axes, with mirror planesperpendicular to the 4-fold axes, and therefore belongs to the Oh group.

2s,2p

RPr)

fcc

bcc

12 6 24

8 6 12

1s

0 1 2 3

1

2

rPA)o

Figure 14: Absolute square of the radial part of the electronic wave function.For the BCC lattice, both the 8 nearest, and 6 next nearest neighbors lie ina region of relatively high electronic density. This favors the formation of aBCC over FCC lattice for some elemental metals.


Conclusion: Classification of Crystal Symmetries

• 7 Crystal Systems (possible Point Groups for Bravais lattices in 3D):cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, rhom-bohedral.

• 14 Bravais lattices.

• 32 Point Groups for lattices decorated with a basis.

• 230 = 73 symorphic (put objects of some point group symmetryon the lattice sites) + 157 non-symorphic (translation + rotationleave lattice invariant, but neither the translation nor rotation appliedindependently are symmetry of the lattice).

• 1651 Magnetic groups after lattice points are decorated with quantum-mechanical spin-1/2.

phys 624: crystal structures and symmetry 1 crystal...

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