References Space Groups for Solid State Scientists , G. Burns and A . M. Glazer

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References Space Groups for Solid State Scientists , G. Burns and A . M. Glazer

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References Space Groups for Solid State Scientists , G. Burns and A . M. Glazer

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Physics 590

“International Tables of Crystallography”

“Everything you wanted to know about beautiful flies, but were afraid to ask.”

Schönflies

Gordie Miller (321 Spedding)

Proposed Plan

What basic information is found on the space grouppages…

Stoichiometry of the unit cell (Wyckoff sites)

Site (point) symmetry of atoms in solids

Solid-solid phase transitions (group-subgroup relationships)

Diffraction conditions – what to expect in a XRD powder pattern.

References

Space Groups for Solid State Scientists, G. Burns and A. M. Glazer

(Little mathematical formalism; prose style)

International Tables for Crystallography (http://it.iucr.org/)

Bilbao Crystallographic Server (www.cryst.ehu.es)

(Comprehensive resources for all space groups)

What can we learn from the International Tables?

BaFe2As2

Space Group: I4/mmm

Lattice Constants:a = 3.9630 Å

c = 13.0462 Å

Asymmetric Unit:

Ba (2a):000

Fe (4d):½0¼

As (4e):000.3544

c

Intensity

(Arb. Units)

(hkl) Indices

h + k + l= even integer (2n)

(013)

(116)

(200)

(112)

(213)

b

a

(015)

(004)

(028)

(215)

(011)

(002)

2θ (Cu Kα)

Typical Space Group Pages…

Symbolism

Diffraction

Extinction

Conditions

Point Symmetry Features

Stoichiometry

Structure of Unit Cell

Subgroup/Supergroup

Relationships

Symbolism

Point Group of

the Space Group

Crystal

System

Space Group

Molecules

Solids

S2= h C2

NOTE:

,

(z)

y

In Schönflies notation, what does the symbol S2 mean?

(x,y,z)

C2 rotation followed by shC2 axis

S2= inversion

x

In International notation, what does the symbol mean?

(z)

y

2-fold (C2) rotation followed by inversion ( )

(x,y,z)

,

Why are the symbols S2 and not used?

x

Symbolism: Crystal Systems

What rotational symmetries are consistent with a lattice (translational symmetry)?

C1C2 (2π/2)C3 (2π/3)C4 (2π/4)C6 (2π/6)

- = angle between b and c
- = angle between a and c
- = angle between a and b

c

a

b

Symbolism: Bravais Lattices

7 Crystal Systems = 7 Primitive Lattices (Unit Cells): P

(rhombohedral)

“Centered Lattices”

- = angle between b and c
- = angle between a and c
- = angle between a and b

?

c

a

b

I

F

C

B

A

Body-

(All) Face-

Base-

Symbolism: Point Groups

- Schönflies
- Notation

Symbolism: Crystallographic Point Groups

32 Point Groups

Allowed Rotations = C1C2C3C4C6

Yes:

Laue Groups

b

c

- m2m or mm2

a

b

c

c

a

b

a+b

a–b

Questions for Friday…

For the following space group symbol

What is the crystal class?

What is the lattice type?

Which face(s) are centered?

(c)What is the point group of the space group using Schönflies notation?

(d)Does the point group contain the inversion operation?

Cmm2

Orthorhombic

Base (C)-centered

ab-faces

C2v

No

Symbolism: Crystallographic Point Groups (cont.)

c

a

b

- 31m

- 312

c

a

b

a

b

c

Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m)

Improper Rotations

Proper Rotations

–

+

4 / m mm

,

,

–

+

,

,

–

–

+

+

33 Matrices:

,

,

–

–

+

+

–

+

y

,

,

x

–

+

(Determinant = +1)

(Determinant = -1)

Symmorphic Space Groups

General

Position

Special

Positions

Point Group= {Symmetry operations intersecting in one point} (32)

Space Group = {Essential Symmetry Operations} {Bravais Lattice} (230)

Symmorphic Space Groups

Ba (2a):

4/mmm (D4h)

Fe (4d):

m2(D2d)

As (4e):

4mm (C4v)

General

Position

Special

Positions

“Ba2Fe4As4”

Z = 2

Space Group: I4/mmm

Lattice Constants:a = 3.9630 Å c= 13.0462 Å

Asymmetric Unit:

Ba (2a):000

Fe (4d):½0¼

As (4e):000.3544

BaFe2As2

Space Groups (230)

SymmorphicSpace Groups (73): {Essential Symmetry Operations} is a group.

Point Group of the Space Group

Nonsymmorphic Space Groups (157): {Essential Symmetry Operations} is a not a group.

Space Group Operations: Screw Rotations and Glide Reflections

Screw Rotations:

Rotation by 2/n (Cn) then

Displacement j/n lattice vector || Cn axis

(allowed integers j = 1,…, n–1)

Symbol = nj

21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65

I41/amd

4/mmm

Point Group of the Space Group

Glide Reflections:

Reflection then

Displacement 1/2 lattice vector || reflection plane

Axial:a, b, c (lattice vectors = a, b, c)

Diagonal:n (vectors = a+b, a+c, b+c)

Diamond: d (vectors = (a+b+c)/2, (a+b)/2, (b+c)/2,

(b+c)/2)

P42/ncm

Nonsymmorphic Space Groups (157)

The Origin!

Si: 0, 0, 0