chapter b1: crystal structures and symmetries

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Chapter B1: Crystal Structures and Symmetries

Georg RothInstitute of CrystallographyRWTH Aachen University

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

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

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…

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

http://www.frm2.tum.de/

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Outline:

• Crystal lattices The lattice concept: points, directions, planes

• Crystallographic coordinate systemsUnit 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

• QuasicrystalsOrdered solids without translational symmetry

• Application of symmetry: Crystal structure of YBa2Cu3O7

Hexagonal symmetry

60°

60°

60°60°

60°

60°

Unit cell

a b

a ba2

b2

a3

b3

a4

b4

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

a b

2-dim. periodic pattern of snowflakes

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120° 180°

60°

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

a b

a b

Basis vectors, translational symmetry

3a 2b

=3a+2b(+0c)

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

Motive

MotiveCrystal = Lattice

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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)

a br1 x1ay1b

r1 = x1a + y1b(0 x, y < 1)

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

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

any lattice point (point lattice) is given by a vector

= ua + vb + wc (u, v, w )

is also known as translation vector

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

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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]

[010]

[001]

[100]

[231]

[-2-3-1]or [ ]132

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 11/1 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)

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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:

(100)

d(100)

X

Y

Hexagonal crystal system a=b, c, ==90°, =120°

2

2

2

22

34

1)(

c

l

a

hkkhhkld

+=

++

For a crystal: Interference between waves scattered froma 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 beamand the lattice plane (hkl)

: wavelength of the radiation

[hkl]

(hkl)

d(hkl)(hkl)

(hkl)

Lattice spacings are directly accessible by experiments:Diffraction of X-rays, neutrons, electrons, ...

= coherent elastic scattering

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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°

orthorhombicthree mutually perpendicular diads – 2 or m (‖X, Y and Z)

a b c; ===90°

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

trigonal(hexagonal cell)

one triad – 3 or -3 (‖Z) a = b c; ==90°, =120°

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

cubicfour triads – 3 or -3

(‖space diagonals of cube)a = b = c; ===90°

|| 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

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The 14 Bravais lattices (represented by their unit cells)

triclinic P monoclinic Pmonoclinic axis‖c

monoclinic A(0,0,0 + 0, ½, ½)

orthorhombic P

orthorhombic I(0,0,0 + ½, ½, ½)

orthorhombic C(0,0,0 + ½, ½,0)

orthorhombic F(0,0,0 + ½, ½,0

½,0, ½ + 0, ½, ½)

tetragonal P•

tetragonal I Trigonal/hexagonal P hexagonal/rhombohedral

cubic P

cubic I cubic F

The 14 Bravais lattices (cont.)

14 Bravais lattices:

7 primitive lattices P forthe 7 crystal systems with onlyone lattice point per unit cell

+7 centered (multiple) lattices

A, B, C, I, R and F with 2, 3 and 4 lattice points per unit cell

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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.

*

3

1 a2||2

a3

a1

. d100

.

d001

* = * = = = 90°

* = 180°–

dhkl: lattice spacing for the set of lattice planes (hkl)

Direct and reciprocal basis

vectors satisfy the following

conditions:

1a1 = 2a2 = 3a3 = 1

This means that |ai| = 1 / |i|

and

1a2 = 1a3 = 2a1 = ... = 0

This means that each i is

perpendicular to aj and ak:

i = (aj ak)/Vc

with Vc = a1(a2a3) as the

volume of the direct cell

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

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Crystallographic symmetry operationsAll 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):(inversion), = m (reflection), , ,

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.

1 2 3 4 6

Point symmetry operations

rotations rotoinversions

1=identity

2-fold = 180°-rotation 2-fold rotation combinedwith inversion = reflection

inversion

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1

2

3

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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

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Hermann-Mauguin symbols of point groupse.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 theHermann-Mauguin symbol

symmetry directions in the orthorhombic lattice

a b c; = = = 90°

x

y

z

[100]

m2

[010]

m2

[001]

m2

x

y

z

x

y

z

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symmetry directions in the tetragonal lattice

a = b c; = = = 90°

x

y

z

[001]

m4

[100]

m2

[110]

m2

x

y

z

x

y

z

symmetry directions in the cubic lattice

a = b = c; = = = 90°

y

x

[100]

m4

[111]

3

[110]

m2

x

y

z

x

y

zz

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

Nomenclature:

Left: Schoenfliess-Symbol

Right: Hermann-Mauguin-Symbol

Plotted:„stereographicprojections“ : Point on upperhemisphere : Point on lower hemisphere

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Example: Orthorhombic system

Crystallographic point group: mm2

a

b

−

−

=

−

−

z

y

x

z

y

x

100

010

001symmetry operation

represented by

a rotation matrix

2 ‖ to [001]:

a

b

x

y x,y,z

-x,-y,z

ma

ma ⊥ a:

−

=

−

z

y

x

z

y

x

100

010

001

mb

mb ⊥ b:

−=

−

z

y

x

z

y

x

100

010

001

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

tetragonal4/m

4/m 2/m 2/m = 4/m m m

trigonal-3

-3 2/m = -3 m

hexagonal6/m

6/m 2/m 2/m = 6/m m m

cubic2/m -3 = m -3

4/m -3 2/m = m -3 m

By diffraction methods, only the 11 Laue classes can be distinguishedand not all the 32 crystal classes.

The diffraction experiment – by its nature - always adds a centre of symmetry!

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Crystallographic symmetry operationsare isometric movements in crystals:




properties: exacly one fixed point




1 2 3 4 6

120°

1/3

31 = 3 + 1/3

Screw rotations nm = n + m/n·

+ 42, 43 and 65

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Crystallographic symmetry operationsare isometric movements in crystals:




properties: exacly one fixed point




1 2 3 4 6

m

reflection: mirror plane m ⊥ image plane

a

a/2

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

a

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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

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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!

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Quasicrystals:

Nobel-Prize in physics 2011 awarded to Dan Shechtman

“for the discovery of quasi crystals”

HoMgZn: Icosahedral quasi crystal

HoMgZn: Electron diffraction pattern taken along the -5 axis

Symmetry of quasi crystals? No translation lattice!

A new ordered ground state of solid matter:

Crystal: Quasi crystal:

• long range order

• (3D) periodic

• unit cell (repeat unit)

• translational-symmetry

• long range order

• aperiodic

• no unit cell (in 3D)

• no translational symmetry

2D-analog:

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Diffraction experiment Crystal structure

• Ceramic high-TC superconductor with TC = 92 K

• Technical application with liquid N2 cooling is possible

,

TCExample: YBa2Cu3O7-

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

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a

b

a

c

c

b

crystal system

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

3 different projectionsof the symmetry

Space group symmetry P m m moperating on a general point x,y,z

gives a total of 8 symmetryequivalent points

From: International Tables for Crystallography

Th. Hahn (ed.)

YBa2Cu3O7-

Ba

O3

O2

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YBa2Cu3O7- O1

O4

Cu2

Cu1

Y1

Thank you for your attention!

Chapter B1: Crystal Structures and Symmetries

Georg RothInstitute of CrystallographyRWTH Aachen University

chapter b1: crystal structures and symmetries

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