Single Crystals, Powders and

Twins

Master of Crystallography and Crystallization – 2013

T01 – Mathematical, Physical and Chemical basis of Crystallography

Solıd MaterıalTypes

Crystallıne

Single Crystal

PolycrystallıneAmorphous

(Non-Crystalline)

Crystalline Solids

• A Crystalline Solid is the solid form of a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimensions.

• Single Crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material.

“A material is a crystal if it has essentially a sharp diffraction pattern"Acta Cryst. (1992), A48, 928 Terms of reference of the IUCr commission on aperiodic crystals

Crystallography

Crystallography ≡ The branch of science that deals with the geometric description of crystals & their internal arrangements. It is the science of

crystals & the math used to describe them. It is a VERY OLD field which pre-

dates Solid State Physics by about a century! So much of the terminology (& theory notation) of Solid State Physics originated in crystallography.

A Single Crystal has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry.

Single Crystals

Single PyriteCrystal

AmorphousSolid

Polycrystalline Solids

A Polycrystalline Solid is made up of an aggregate of many small single

crystals (crystallites or grains). Polycrystalline materials have a high degree of order over many atomic or molecular dimensions. These ordered regions, or single crystal regions, vary in size & orientation with respect to

one another. These regions are called grains (or domains) & are separated

from one another by grain boundaries.

The atomic order can vary from one domain to the next. The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains that are < 10 nm in diameter are called nanocrystallites.

PolycrystallinePyriteGrain

Amorphous Solids

Amorphous (Non-crystalline) Solids are composed of randomly orientated atoms, ions, or molecules that do not form defined patterns or lattice structures. Amorphous materials have order only within a few atomic or molecular dimensions. They do not have any long-range order, but they have varying degrees of short-range order. Examples of amorphous material include amorphoussilicon, plastics, & glasses.

Departures From the “Perfect Crystal”

A “Perfect Crystal” is an idealization that does not exist in nature. In some ways, even a crystal surface is animperfection, because the periodicity is interrupted there.

Each atom undergoes thermal vibrations around their equilibrium positions for temperatures T >= 0K. These can also be viewed as “imperfections”.

• Also, Real Crystals always have

foreign atoms (impurities), missing

atoms (vacancies), & atoms in

between lattice sites (interstitials) where they should not be. Each of these spoils the perfect crystal structure.

Crystal LatticeCrystallography focuses on the geometric properties of crystals. So, we imagine each atom replaced by a mathematical point at the equilibrium position of that

atom. A Crystal Lattice (or a Crystal) ≡ An idealized description of the

geometry of a crystalline material. A Crystal ≡ A 3-dimensional periodic

array of atoms. Usually, we’ll only consider ideal crystals. “Ideal” means one

with no defects, as already mentioned. That is, no missing atoms, no atoms off of the lattice sites where we expect them to be, no impurities,…Clearly, such an ideal

crystal never occurs in nature. Yet, 85-90% of experimental observations on crystalline materials is accounted for by considering only ideal crystals!

Platinum(Scanning Tunneling Microscope)

Platinum Surface Crystal Lattice &Structure of Platinum

Single CrystalsFor single crystals, we see the individual reciprocal lattice points projected onto the detector and we can determine the values of (hkl) readily, measure the intensities of each peak and assign the associated diffraction angles q.

This information can allow us to determine the electron density within a unit cell using the methods we have seen earlier in this course. In a crystalline powder, the situation is modified significantly

PowdersIf a powder is composed of numerous crystallites that are randomly oriented (“all” possible orientations are present), the reciprocal lattice points for each of the crystallites combine to form spheres of reflection. Since these spheres will always intersect the Ewald sphere, at the detector, we observe these as circles and we can only measure the diffraction angle q of the rings.

We lose a considerable amount of information because the 3D (hkl) data for each of the single crystal reflections is effectively reduced to 1D (q) data in the powder XRD spectrum. Furthermore, the overlap of peaks (because of necessary or accidentally similar d-spacings) makes the interpretation of the intensity data more complicated.

In a modern XRD experiment, the diffraction patterns shown on the right are integrated to provide plots of the type shown on the left. Although much of the information that we would prefer to have for structure solution is “gone”, XRD can provide much information and, in some cases, structure solution is possible.

Despite the apparent limitations, XRD is very useful for the identification of compounds for which structural data has already been obtained. For example, applications in WinGX (LAZY Pulverix), the ICSD, Mercury, Diamond and PowderCell will calculate the XRD pattern for any known structures. Below is the calculated powder XRD pattern for the [nBu4N][MeB(C6F5)3] structure we examined in the refinement part of the course. Rietveld Análisis of these patterns allows to refine structural models or XRD quantitative analysis.

This assigned XRD pattern was calculated using PowderCell, which is a particularly useful program for any of you who wish to use our powder diffractometer. The program is also useful because it can be used to examine and predict how a diffraction pattern is altered when unit cell parameters or fractional coordinates are changed (such modifications can occur when the temperature of an experiment is changed).

Wherefore Powder Diffraction?

Single crystal data provides– Iobs for each observed

diffraction spot– Nice method for subtracting

background– Tried and true methods for

structure determination• Though not always easy!

http://lmb.bion.ox.ac.uk/www/lj2001/garman/garman_01.html

Or: you can’t get a nice

single/non-twinned/

low mosaicity/

highly diffracting crystal!

0

2000

4000

6000

8000

10000

12000

14000

2 3 4 5 6 7 8 9 10 11 12

2 theta

In

ten

sit

y

Powder Data– CCD / Image plate detectors (shown)– 3D / 2D– Overlap problem

• All Freidel pairs exactly overlapped• Identical d-spacings• Higher 2q• Larger structures= proteins!

Single crystal data 2D location of

points, 3D when you add angle

Precession. Powder data is only

2D

Powder methods

• Mimic single crystal diffraction techniques– End goal - solve unknown structures

– Rietveld refinement

• Frontier goals– Find the ‘limits’ for size and resolution

– Synchrotron data• Better signal to noise (generally)

• Better resolution (generally)

• Thinner peaks- less overlaps

• Wavelength choice

Resolution here refers

to maximum d-spacing

resolved -- max 2q

Appropriate Starting Model

• find a suitable starting model– Human lysozyme

• 1.5 Å structure pdb code: 1LZ1– Single crystal data

• P212121 spacegroup• 130 residues

– Hen egg white lysozyme• “1.9 Å data”, 5 data sets• P43212 spacegroup• 129 residues

gap

~60% identity

The ‘hard part’ is over…

• Globally the structure is in the right spot

– Mol rep solution with modified model

– Use this as a basis for Rietveld refinement

• Extend refinement region to 2-12o 2q

– Low angle shifts- solvent model

– High angle - overlaps

• Rigorous method for structure optimization

– Rigid body and restrained refinement (ccp4i- Refmac)

– Only works up to 3 Å (12o)

– Overlaps hurt

Goal: best

‘false’ minima

Quartz

Fluorite

Twins

Feldspar

Rutile

Twins

Aggregates formed of individual crystals of the same species,

grown together with well definedorientations

Contact twins Penetration twins

Polysynthetic twins Cyclic twins

Origin:

- Growth

- Deformation

- Transformation

Example: Quartz Penetration TwinsD

aup

hin

e Brazil

Optical appearance of twins

Reentrant angles

Non-uniforme optical extinction

Twin boundaries

Twins in diffraction experimentsIhkl

twin = vIIhklI + vII Ihkl

II + vIIIIhklIII + vIVIhkl

IV +...n

= Σ vn Ihkln

i=1

Refinement: Structure of the individualTwin laws volume fractions (additional parameters)

m´

m´

m

m

Unidentifiedtwins in

structuredetermination

Hugedisplacementparameters

split atompositions

short interatomic

distances

Merohedral Twins• The twin element belongs to the holohedry of the lattice, but

not to the point group of the crystal.

• The reciprocal lattices of all twin domains superimpose exactly.

• In the triclinic, monoclinic and orthorhombic crystal systems, the merohedral twins can always be described as inversion twins.

Non-merohedral Twins• Twin operation does not belong to the Laue group or point

group of the crystal.

• In practice there are three types of reflections:

– Reflections belonging to only one lattice.

– Completely overlapping reflections belonging to both lattices.

– Partially overlapping refections belonging to both lattices.

Non-merohedral Twins

Warning signs

• The Rint value for the higher-symmetry Laue group is only slightly higher than for the lower-symmetry Laue group

• The mean value for |E2-1| is much lower than the expected value of 0.736

• The space group appears to be trigonal or hexagonal

• The apparent systematic absences are not consistent with any known space group

• For all of the most disagreeable reflections Fo is much greater than Fc

(Herbst-Irmer & Sheldrick, 1998)

Classification of Merohedral Twins

• Twinning by Merohedry

• Twinning by Pseudo-Merohedry

• Merohedry of the lattice

• “reticular merohedry”

Twinning by MerohedryType I:

Inversion twins: The operation of twinning is part of the Laue Group of the individual, but not of the point group

Example: Symmetry of an individual: PmLaue Group: 2/mPoint group: m a → -a (a,b,c)=(a,b,c) -1 0 0Twin operation inversion center b → -b 0 -1 0 Symmetry of the twin: P2´/m c → -c 0 0 -1

( )

R

a

c

a

c

a

c

a

c

Twin Matrix

Pnna

< Bi-O > distance: 2.23Å (Bi 4+)

Semiconductor and diamagnetic

Example: Merohedry Type Ia=5.975(1), b=6.311(1), c=9.563(2)Å

J. Solid State Chem. 147,1999, 117; Solid State Sciences 8, 2006, 267.

Pnn2Inversion twin 0.5:0.5

< Bi(1)–O > distance: 2.31 Å (Bi3+)< Bi(2)–O > distance: 2.15 Å (Bi5+)

Ag4Bi2O6

Twinning by MerohedryType II:

The twin operation forms part of the holoedry of the lattice, but not of the Laue Group of the individual

Example: Symmetry of an individual: P4/mLaue Group: 4/mHoloedry of the tetragonal lattice: 4/m2/m2/m

Twin operations: 2 || a, m a, 2 || [110], m [110]Symmetry of the twin: 4/m 2´/m´ 2´/m´

R

a

b

a

b

aa

b b

Maximumsymmetry

Example: Merohedry Type IIa=9.4309(4) Å, c=38.4799(17) Å

space group P32

Symmetry of twinP322´1

Twin operation: 2-fold rotation around the a (b) –axis

-1 -1 00 1 00 0 -1( )

Farran, Alvarez-Larena, Piniella, Capparelli & Friese (2008), Acta Cryst. C64, o257.

C14H14I2O2Te·0.5C2H6OS

Twinning by Pseudo-merohedryThe twin operation corresponds to a

pseudosymmetry element of the lattice

Example: specialized metric, monoclinic lattice with β ≈ 90º Symmetry of the individual: monoclinic P12/m1Symmetry of the lattice: ≈ orthorhombic ≈ 2/m2/m2/mTwin operation: 2 || [100], m [100], 2 || [001], m [001]Symmetry of the twin: P2´/m´2/m2´/m´

a

c

a

c

a

c

a

c

R

Example: Twinning by Pseudo-Merohedry

Structural phase transition

350K: P3m1 293K: C121

K. Friese, G. Madariaga & T. Breczewski (1999), Acta Cryst. C55, 1753-1755K. Friese, M. I. Aroyo, C.L. Folcia, G. Madariaga, T. Breczewski (2001), Acta Cryst. B57, 142-150

Tl2MoO4

Point group 3m P3m1 a=6.266(1) Å, c=8.103(2)Å

12 symmetry

operations a´= - a1 + a2

b´= - a1 – b2

c´= c

t=6

Point group 2 C121 a´=10.565 Å

2 symmetry b´=6.418Å

operations c´=8.039Å

ß=91.05°

Example: Twinning by Pseudo-Merohedry

Group/Subgroup RelationshipsC. Hermann (1929), Z. Kristallogr. 69, 533-555

Number of symmetry operations of the point group of G Number of symmetry operations of the point group of U

t = translationengleich =

Tl2MoO4

a1

a2

a’b’

Example: Twinning by Pseudo-Merohedry Tl2MoO4

Trigonal metrics:a=6.266(1) Å, c=8.103(2)Å

Idealized orthorhombic metrics:a=10.853 Åb=6.266 Åc=8.103Å

Monoclinic metrics:a´=10.565 Å b´=6.418 Åc´=8.039Å ß=91.05°

´Obverse / Reverse twins´ in rhombohedral lattices

Merohedry of the lattice

Obverse: 0, 0, 0; Reverse: 0, 0, 0;2/3, 1/3, 1/3; 1/3, 2/3, 1/3; 1/3, 2/3, 2/3 2/3, 1/3, 2/3

Individual I Individual IITwin

0

1/3 2/3 2/3 1/3

00

1/3,2/3 1/3,2/3a1 a1a2 a2

Example: Merohedry of the lattice

a=5.595(15) Å, b=5.595(15) Å, c=11.193(2) Å, γ =120 º

K. Friese, L. Kienle, V. Duppel, H. M. Luo & C. T. Lin (2003), Acta Cryst. B59, 182-189

Sr3(Ru0.33,Pt0.66)CuO6

Obverse/reverse Twin = 180° rotation around c*+

3-fold axis of the trigonal system

0.093 0.337 0.090

180º 120º

1 0 0 -1 0 0 0 –1 0

0 1 0 0 -1 0 1 –1 0

0 0 1 0 0 1 0 0 1

0.150 0.217 0.113

240º 300º 60º

-1 1 0 0 1 0 1 0 0

-1 0 0 -1 1 0 1 –1 0

0 0 1 0 0 1 0 0 1

( ( (

( ( (

) ))

) ) )

Example: Merohedry of the latticeSr3(Ru0.33,Pt0.66)CuO6

Twinning by reticular merohedry

Part of the lattice points of the individuals overlap, another part corresponds to points of one individual only

a

c

a

c

a

c c

a

Special relationship hidden within the metrics

Example: Reticular Merohedry

a=12.8018(4)b=9.1842(9) c=23.690(3)Å ß=105.458(8)º

P21/c

a,b,2c→ orthorhombic cella´= 9.18b´= 12.80c´= 45.67

S. Schlecht & K. Friese (2003), Eur. J. Inorg. Chem. 2003, 1411-1415

GeTe4C24H20: (PhTe)4Ge(Germanium tellurolate)

→ orthorhombic Cella,b,2c→ orthorhombic Cella,b,2c

a´= -b = 9.18 Åb´= a = 12.80 Åc´= a + 2c = 45.67 Åα’ = 89.78 ºβ’ = 90.0 ºγ’= 90.0 º

a=12.8018(4) Åb=9.1842(9) Åc=23.690(3) Å ß=105.458(8) º

P21/c

m’

h0l

Reciprocal spaceExample:(PhTe)4Ge

Individual 1 Individual 2

wrong structure:unidentified twin

Splitting of reflections

Specialized metrics

Huge unit cells

Large percentage of unobserved reflections

Non-standard extinction rules

Reentrant anglesNon-uniforme optical extinction

Large difference between internal R-value and final agreement factors

Large displacement parameters

Unreasonable interatomic distances

Phase transitions

Pseudosymmetry

Identification of twins

Re

cip

roca

lsp

ace

Single Crystals, Powders and

TwinsMaster of Crystallography and Crystallization – 2013

T01 – Mathematical, Physical and Chemical basis of Crystallography

END