Chapter B1: Crystal Structures and Symmetries Georg Roth Institute of Crystallography RWTH Aachen University

http://www.xtal.rwth-aachen.de http://www.frm2.tum.de

The „crystal palace“ in Aachen-Burtscheid Outstation of the IfK at FRM II in Garching

Symmetry Principles are widely used in Solid State Science Mathematics, Physics, Chemistry, Crystallography, … Why do Crystallographers use symmetry? A: To understand the direction dependence of macroscopic physical properties: Anisotropy B: To write down crystal structures in a concise manner 1 cm3 of matter consists of (roughly) 1023 atoms. Write down 3x1023 atom-coordinates? Or better use the symmetry concept and write down only very few atoms …one to a few hundred at best…

1 Outline: • Crystal lattices The lattice concept: points, directions, planes • Crystallographic coordinate systems Unit cell, origin choice, 7 crystal systems, 14 Bravais-lattices • Symmetry operations and symmetry elements Translation, rotation, roto-inversion, screw-axes, glide-planes • Crystallographic point groups and space groups 32 point groups, symmetry directions, Hermann-Mauguin symbols, 230 space groups in 3D • Quasicrystals Ordered solids without translational symmetry • Application of symmetry:

Crystal structure of YBa2Cu3O7

2-dim. periodic pattern of snowflakes

Hexagonal symmetry 60° 60° 60° 60° 60° 60°

b4 Unit cell b2 a4

a b a a 2 b b3

Choice of origin: point of highest symmetry (6-fold rotation point)

2 2-dim symmetry: 6-fold, 3-fold & 2-fold rotation axes:

60° a b

120° 180°

Crystal = Lattice  Motive

Motive Basis vectors, translational symmetry

a b 3a 2b

=3a+2b(+0c)

general translation vector : =ua+vb+wc; u, v, w  Z (3dim.)

3 How to describe the contents of the unit cell? …usually: atoms, here: snowflakes…

a b y b r1 1 x1a

r1 = x1a + y1b (0  x, y < 1)

Positions of atoms in the unit cell, expressed as fractional coordinates:

Positional vector rj = xja + yjb + zjc (0  x, y, z < 1) (3D, atom label: j)

3 dim. periodicity of crystals  crystal lattice 3 non-linear basis vectors a, b and c define a parallelepiped, called unit cell of the crystal lattice and the crystallographic coordinate system with its origin

angles between basis vectors: angle (a,b) =  angle (b,c) =  angle (c,a) =  faces of unit cell: face (a,b) = C face (b,c) = A face (c,a) = B

any lattice point (point lattice) is given by a vector  = ua + vb + wc (u, v, w  )  is also known as translation vector

4 Lattice points and lattice directions according to the translation vector  = ua + vb + wc (u, v, w  ) • lattice points are indicated by the corresponding integers → uvw • lattice directions or lattice rows by → [uvw] [square brackets] [001]

[-2-3-1] or [2 3 1 ]

[231]

[010]

[100]

Lattice planes (crystal faces are special lattice planes) 3 non-collinear lattice points define a lattice plane: • interceptions of a lattice plane with the axes X (a), Y (b) and Z (c): ma, nb, oc • reciprocal values: 1/m, 1/n, 1/o (with smallest common denominator no/mno, mo/mno, mn/mno) • Miller indices: h = no, k = mo, l = mn  Miller indices (hkl) describe a set of equally spaced lattice planes.

I: 1 1 1 1/1 1/1 1/1 h=1 k=1 l=1  (111) II: 1 2 2 1/1 1/2 1/2 h=2 k=1 l=1  (211)

5 Projection of the lattice of graphite (hexagonal) down the Z-axis on the XY-plane to illustrate the decrease of the d(hkl)-spacing between lattice planes (hk0) as their indices h and k increase:

Y (100)

X d(100)

Hexagonal crystal system a=b, c, ==90°, =120° 1 d(hkl) = 4 h2 +k 2 +hk + l 2 3 a2 c2

Lattice spacings are directly accessible by experiments: Diffraction of X-rays, neutrons, electrons, ... = coherent elastic scattering For a crystal: Interference between waves scattered from a set of lattice planes (hkl)  Bragg equation for the reflection condition 2d(hkl)·sin(hkl) =  d(hkl): interplanar distance of a set of lattice planes (hkl) (hkl): scattering angle, angle between the incident beam and the lattice plane (hkl) : wavelength of the radiation

[hkl]

(hkl) (hkl)

(hkl) d(hkl)

6 Coordinate systems of crystals = 7 crystal systems in 3 dim.

Name of system Minimum symmetry Conventional unit cell

triclinic 1 or -1 a  b  c;     

monoclinic one diad – 2 or m (‖Y) a  b  c; ==90°, >90°

three mutually perpendicular orthorhombic a  b  c; ===90° diads – 2 or m (‖X, Y and Z)

tetragonal one tetrad – 4 (‖Z) a = b  c; ===90°

trigonal one triad – 3 or -3 (‖Z) a = b  c; ==90°, =120° (hexagonal cell)

hexagonal one hexad – 6 or -6 (‖Z) a = b  c; ==90°, =120°

four triads – 3 or -3 cubic a = b = c; ===90° (‖space diagonals of cube)

|| means: parallel to, diad means: 2-fold rotation, triad: 3-fold etc.

Are the 7 crystal systems in 3D, corresponding to the 7 significantly different, symmetry adapted coordinate systems all we need?

Not quite:

There are good arguments (again based on symmetry) to define 7 additional lattices with more than one lattice point per unit cell  The 7 non primitive “centered” lattices

 Altogether: The 14 Bravais-lattices

7 The 14 Bravais lattices (represented by their unit cells)

triclinic P monoclinic P monoclinic A orthorhombic P monoclinic axis‖c (0,0,0 + 0, ½, ½)

• orthorhombic I orthorhombic C orthorhombic F tetragonal P (0,0,0 + ½, ½, ½) (0,0,0 + ½, ½,0) (0,0,0 + ½, ½,0 ½,0, ½ + 0, ½, ½)

The 14 Bravais lattices (cont.)

tetragonal I Trigonal/hexagonal P hexagonal/ cubic P rhombohedral 14 Bravais lattices: 7 primitive lattices P for the 7 crystal systems with only one lattice point per unit cell + 7 centered (multiple) lattices cubic I cubic F A, B, C, I, R and F with 2, 3 and 4 lattice points per unit cell

8 Diffraction geometry: Concept of the reciprocal lattice:

Reminder: The crystal lattice 'direct lattice' is composed of the set of all lattice vectors generated by the linear combination of the basis

vectors a1, a2, a3 with coefficients u, v, and w (positive or negative integers, incl. 0).

a = u a1 + v a2 + w a3. The Fourier-transform which is occurring during a diffraction experiment, transforms this direct lattice into the so called

’reciprocal lattice’, with basis vectors 1, 2, 3 and the integer Miller indices h,k,l as the coefficients.

Diffracted intensity I() is only observed at the nodes of this reciprocal lattice addressed by the vectors:

 = h 1 + k 2 + l 3 of the reciprocal lattice.

Example: Monoclinic cell a1, a2, a3,  > 90°

. Direct and reciprocal basis d001 vectors satisfy the following a3 conditions:

3 1a1 = 2a2 = 3a3 = 1 . d100 This means that |ai| = 1 / |i|  and

* 1a2 = 1a3 = 2a1 = ... = 0 1 a2||2 a1 This means that each i is perpendicular to aj and ak: * = * =  =  = 90°  = (a a )/V * = 180°–  i j k c with V = a (a a ) as the d : lattice spacing for the c 1 2 3 hkl volume of the direct cell set of lattice planes (hkl)

9 Crystallographic symmetry operations All types listed systematically: 1. Translations  = ua + vb + wc (u, v, w  ) properties: no fixed point, shift of entire point lattice 2. Rotations: 1 (identity), 2 (rotation angle 180°), 3 (120°), 4 (90°), 6 (60°) properties: line of fixed points which is called the rotation axis 3. Rotoinversions (combination of n-fold rotations and inversion): 1(inversion), 2 = m (reflection), 3 , 4 , 6 properties: (exactly one) fixed point

4. Screw rotations nm (combination of n-fold rotations with m/n· translations ‖ to rotation axis) properties: no fixed point 5. Glide reflections a, b, c, n, d (combination of reflection through a plane (glide plane) and translation by glide vectors a/2, b/2, c/2, (a + b)/2, ..., (a  b  c)/4 ‖ to this plane) properties: no fixed point Rotations and rotoinversions are called point symmetry operations because they leave at least one point fixed.

Point symmetry operations

rotations rotoinversions

1=identity inversion

2-fold = 180°-rotation 2-fold rotation combined with inversion = reflection

10 1 3

How many distinct combinations of point symmetry operations (rotation & roto-inversion) are possible in 3 D?

The answer is: 32

32 crystallographic point groups (“crystal classes”)

The point group symmetries determine the anisotropic (macroscopic) physical properties of crystals:

Mechanical, Electrical, Optical, Thermal, ... Tensorial Crystal Physics

11 Hermann-Mauguin symbols of point groups e.g. m m 2 and symmetry directions Basis vectors are conventionally chosen parallel to important symmetry directions of the crystal system. Example: In the cubic lattice, a, b and c are parallel to the 4-fold rotation axes.

A maximum of 3 independent main symmetry directions (“Blickrichtungen”) are sufficient to describe the complete point group symmetry of a crystal. These symmetry directions are specific for each of the 7 crystal systems and are essential to understand the Hermann-Mauguin symbol

symmetry directions in the orthorhombic lattice a  b  c;  =  =  = 90°

z z z

y y y x x x

[100] [010] [001] 2 m

12 symmetry directions in the tetragonal lattice a = b  c;  =  =  = 90° z z z

y y y

x x x

[001] [100] [110] 4 2 m m

symmetry directions in the cubic lattice a = b = c;  =  =  = 90°

z z z

y y y x x x

[100] [111] [110] 4 m 3

13 All possible combinations of point symmetry operations in 3 dim. lead to 32 crystallographic point groups (crystal classes)

Plotted: „stereographic projections“  : Point on upper hemisphere  : Point on lower hemisphere

Nomenclature:

Left: Schoenfliess- Symbol

Right: Hermann- Mauguin-Symbol

14 Example: Orthorhombic system mb Crystallographic point group: mm2

2 ‖ to [001]: b symmetry operation  − x −1 0 0 x ma      represented by − y =  0 −1 0 y      a rotation matrix  z   0 0 1 z 

-x,-y,z a b x a y x,y,z

− x −1 0 0 x m ⊥ b:  x  1 0 0 x ma ⊥ a:      b       y  =  0 1 0 y − y = 0 −1 0 y            z   0 0 1 z   z  0 0 1 z 

Crystallographic point groups which have a centre of symmetry  11 Laue classes

Crystal systems (7) Laue classes (11) triclinic -1 monoclinic 1 2/m 1 = 2/m orthorhombic 2/m 2/m 2/m = 2/m m m 4/m tetragonal 4/m 2/m 2/m = 4/m m m -3 trigonal -3 2/m = -3 m 6/m hexagonal 6/m 2/m 2/m = 6/m m m 2/m -3 = m -3 cubic 4/m -3 2/m = m -3 m

By diffraction methods, only the 11 Laue classes can be distinguished and not all the 32 crystal classes. The diffraction experiment – by its nature - always adds a centre of symmetry!

15 Crystallographic symmetry operations are isometric movements in crystals: 1. Translations  = ua + vb + wc (u, v, w  ) properties: no fixed point, shift of entire point lattice 2. Rotations: 1 (identity), 2 (rotation angle 180°), 3 (120°), 4 (90°), 6 (60°) properties: line of fixed points which is called the rotation axis 3. Rotoinversions (combination of n-fold rotations and inversion): 1(inversion), =2 m (reflection), , , 3 4 6 properties: exacly one fixed point

Screw rotations nm = n + m/n·

120°



1/3

31 = 3 + 1/3 

+ 42, 43 and 65

16 Crystallographic symmetry operations are isometric movements in crystals: 1. Translations  = ua + vb + wc (u, v, w  ) properties: no fixed point, shift of entire point lattice 2. Rotations: 1 (identity), 2 (rotation angle 180°), 3 (120°), 4 (90°), 6 (60°) properties: line of fixed points which is called the rotation axis 3. Rotoinversions (combination of n-fold rotations and inversion): 1(inversion), =2 m (reflection), , , 3 4 6 properties: exacly one fixed point

 m

reflection: mirror plane m ⊥ image plane

a a a/2

glide reflection: glide plane a ⊥ with glide vector a/2

17 In 3 dimensions: All possible combinations of point symmetries of the 32 crystallographic point groups with lattice translations (the 14 Bravais lattices) and symmetry elements with a translational component (glide planes, screw axes) lead to exactly

230 crystallographic space groups

 International Tables for Crystallography Vol. A (Theo Hahn, Ed.)

Conventional graphic symbols for symmetry elements: • symmetry axes (a) perpendicular, (b) parallel, and (c) inclined to the image plane • symmetry planes (d) perpendicular and (e) parallel to the image plane

18 Summary (intermediate):

To describe the anisotropy of macroscopic physical properties, we need: • The point group symmetry (1 out of 32)

To describe a crystal structure, we need: • Choice of unit cell with basis vectors • The space group symmetry (1 out of 230) • The atomic positions in the unit cell

…see example (YBa2Cu3O7) below…

…just as a reminder: The “magic” crystallographic numbers in 3D space:

7: Crystal systems (triclinic, monoclinic…) 14: Bravais lattices (P, C, A, B, I, R, F) 32: Crystal classes (point groups) 11: Laue classes (point groups with inversion center) 230: Space groups (all useful combinations of point group symmetry with translational symmetry)

…is this system closed and final…?... …not really!

19 Quasicrystals: Nobel-Prize in physics 2011 awarded to Dan Shechtman “for the discovery of quasi crystals”

HoMgZn: Icosahedral HoMgZn: Electron diffraction quasi crystal pattern taken along the -5 axis

Symmetry of quasi crystals? No translation lattice! A new ordered ground state of solid matter: 2D-analog: Crystal: Quasi crystal:

• long range order • long range order • (3D) periodic • aperiodic • unit cell (repeat unit) • no unit cell (in 3D) • translational-symmetry • no translational symmetry

20 Diffraction experiment  Crystal structure

T Example: YBa2Cu3O7- C  • Ceramic high-TC superconductor with TC = 92 K

• Technical application with liquid N2 cooling is possible

Atom positions in YBa2Cu3O6.96 orthorhombic, space group P 2/m 2/m 2/m a = 3.858 Å, b = 3.846 Å, c = 11.680 Å (at room temperature) atom/ion multiplicity site symmetry x y z

Cu1/Cu2+ 1 2/m 2/m 2/m 0 0 0

Cu2/Cu2+ 2 m m 2 0 0 0.35513(4)

Y/Y3+ 1 2/m 2/m 2/m ½ ½ ½

Ba/Ba2+ 2 m m 2 ½ ½ 0.18420(6)

O1/O2- 2 m m 2 0 0 0.15863(5)

O2/O2- 2 m m 2 0 ½ 0.37831(2)

O3/O2- 2 m m 2 ½ 0 0.37631(2)

O4/O2- 1 2/m 2/m 2/m 0 ½ 0

21 crystal system

P Bravais lattice, symmetry directions: 2/m‖[100], 2/m‖[010], 2/m‖[001] b c

3 different projections of the symmetry

a a

Space group symmetry P m m m operating on a general point x,y,z gives a total of 8 symmetry equivalent points

From: International Tables for Crystallography Th. Hahn (ed.)

YBa2Cu3O7- 

22 Y1

O1 Cu2 YBa2Cu3O7- 

Cu1

Chapter B1: Crystal Structures and Symmetries Georg Roth Institute of Crystallography RWTH Aachen University

Thank you for your attention!