Single Crystals, Powders and
Twins
Master of Crystallography and Crystallization – 2013
T01 – Mathematical, Physical and Chemical basis of
Crystallography
Solıd Materıal
Types
Crystallıne
Single Crystal
Polycrystallıne
Amorphous
(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
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 predates Solid State Physics by about a century! So much of the
Crystallography ≡
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 Pyrite
Crystal
Amorphous
Solid
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.
Polycrystalline
Pyrite
Grain
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 shortrange order. Examples of amorphous material include amorphous
silicon, 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 an
imperfection, 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 Lattice
Crystallography 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
Platinum Surface
(Scanning Tunneling Microscope)
Crystal Lattice &
Structure of Platinum
Single Crystals
For 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
Powders
If 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!
Or: you can’t get a nice
single/non-twinned/
low mosaicity/
highly diffracting crystal!
http://lmb.bion.ox.ac.uk/www/lj2001/garman/garman_01.html
Powder Data
– CCD / Image plate detectors (shown)
– 3D / 2D
– Overlap problem
•
•
•
•
All Freidel pairs exactly overlapped
Identical d-spacings
Higher 2q
Larger structures= proteins!
14000
12000
10000
Intensity
Single crystal data 2D location of
points, 3D when you add angle
Precession. Powder data is only
2D
8000
6000
4000
2000
0
2
3
4
5
6
7
2 theta
8
9
10
11
12
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
gap
– 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
~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
Twins
Twins
Quartz
Aggregates formed of individual
crystals of the same species,
grown together with well defined
orientations
Fluorite
Rutile
Feldspar
Contact twins
Penetration twins
Origin:
- Growth
- Deformation
- Transformation
Polysynthetic twins
Cyclic twins
Brazil
Dauphine
Example: Quartz Penetration Twins
Optical appearance of twins
Reentrant angles
Non-uniforme optical extinction
Twin boundaries
Twins in diffraction experiments
Ihkltwin = vIIhklI + vII IhklII + vIIIIhklIII + vIVIhklIV +...
n
=
Σ
vn Ihkln
i=1
Refinement: Structure of the individual
Twin laws
volume fractions (additional parameters)
m´
m
m
m´
Unidentified
twins in
structure
determination
Huge
displacement
parameters
split atom
positions
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 Merohedry
Type I:
Inversion twins: The operation of twinning is part of the Laue Group
of the individual, but not of the point group
R
a
a
c
c
c
a
c
a
Example: Symmetry of an individual: Pm
Twin Matrix
Laue Group:
2/m
Point group:
m
a → -a (a,b,c)=(a,b,c) -1 0 0
Twin operation
inversion center
b → -b
0 -1 0
Symmetry of the twin: P2´/m
c → -c
0 0 -1
(
)
Ag4Bi2O6
Example: Merohedry Type I
a=5.975(1), b=6.311(1), c=9.563(2)Å
Pnna
< Bi-O > distance: 2.23Å (Bi 4+)
Semiconductor and diamagnetic
Pnn2
Inversion twin 0.5:0.5
< Bi(1)–O > distance: 2.31 Å (Bi3+)
< Bi(2)–O > distance: 2.15 Å (Bi5+)
J. Solid State Chem. 147,1999, 117; Solid State Sciences 8, 2006, 267.
Twinning by Merohedry
Type II:
Maximum
symmetry
The twin operation forms part of the holoedry of the
lattice, but not of the Laue Group of the individual
R
b
a
b
b
a
Example: Symmetry of an individual:
Laue Group:
Holoedry of the tetragonal lattice:
Twin operations:
Symmetry of the twin:
a
b
a
P4/m
4/m
4/m2/m2/m
2 || a, m  a, 2 || [110], m  [110]
4/m 2´/m´ 2´/m´
C14H14I2O2Te·0.5C2H6OS
Example: Merohedry Type II
a=9.4309(4) Å, c=38.4799(17) Å
space group P32
Symmetry of twin
P322´1
Twin operation:
2-fold rotation around the a (b) –
axis
(
-1 -1 0
0 1 0
0 0 -1
)
Farran, Alvarez-Larena, Piniella, Capparelli & Friese (2008), Acta Cryst. C64, o257.
Twinning by Pseudo-merohedry
The twin operation corresponds to a
pseudosymmetry element of the lattice
R
c
c
a
a
c
c
a
a
Example: specialized metric, monoclinic lattice with β ≈ 90º
Symmetry of the individual: monoclinic
P12/m1
Symmetry of the lattice:
≈ orthorhombic ≈ 2/m2/m2/m
Twin operation:
2 || [100], m  [100], 2 || [001], m  [001]
Symmetry of the twin:
P2´/m´2/m2´/m´
Example: Twinning by Pseudo-Merohedry
Tl2MoO4
Structural phase transition
350K: P3m1
293K: C121
K. Friese, G. Madariaga & T. Breczewski (1999), Acta Cryst. C55, 1753-1755
K. Friese, M. I. Aroyo, C.L. Folcia, G. Madariaga, T. Breczewski (2001), Acta Cryst. B57, 142-150
Example: Twinning by Pseudo-Merohedry
Tl2MoO4
Point group 3m
12 symmetry
operations
P3m1
a=6.266(1) Å, c=8.103(2)Å
a´= - a1 + a2
b´= - a1 – b2
c´= c
b’
a’
t=6
Point group 2
2 symmetry
operations
a2
C121
a1
a´=10.565 Å
b´=6.418Å
c´=8.039Å
ß=91.05°
Group/Subgroup Relationships
C. Hermann (1929), Z. Kristallogr. 69, 533-555
t = translationengleich =
Number of symmetry operations of the point group of G
Number of symmetry operations of the point group of U
Tl2MoO4
Example: Twinning by Pseudo-Merohedry
Trigonal
metrics:
a=6.266(1) Å,
c=8.103(2)Å
Idealized
orthorhombic
metrics:
a=10.853 Å
b=6.266 Å
c=8.103Å
Monoclinic
metrics:
a´=10.565 Å
b´=6.418 Å
c´=8.039Å
ß=91.05°
Merohedry of the lattice
´Obverse / Reverse twins´ in rhombohedral lattices
Obverse: 0, 0, 0;
2/3, 1/3, 1/3;
1/3, 2/3, 2/3
Reverse: 0, 0, 0;
1/3, 2/3, 1/3;
2/3, 1/3, 2/3
0
a1
1/3
Individual I
0
0
2/3
a2
1/3,2/3
Twin
1/3,2/3
a1
2/3
1/3
Individual II
a2
Sr3(Ru0.33,Pt0.66)CuO6
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
Example: Merohedry of the lattice
Sr3(Ru0.33,Pt0.66)CuO6
Obverse/reverse Twin = 180° rotation around c*
+
3-fold axis of the trigonal system
0.093
100
010
001
0.337
180º
-1 0 0
0 -1 0
0 01
0.150
240º
-1 1 0
-1 0 0
0 01
0.217
300º
0 1 0
-1 1 0
0 01
0.090
120º
0 –1 0
1 –1 0
0 01
( ) ( ) ( )
0.113
60º
1 00
1 –1 0
0 01
( )( ) ( )
Twinning by reticular merohedry
Part of the lattice points of the individuals overlap, another
part corresponds to points of one individual only
c
c
a
a
c
c
a
Special relationship hidden within the metrics
a
GeTe4C24H20: (PhTe)4Ge
(Germanium tellurolate)
Example: Reticular Merohedry
a=12.8018(4)
b=9.1842(9)
c=23.690(3)Å
ß=105.458(8)º
P21/c
S. Schlecht & K. Friese (2003), Eur. J. Inorg. Chem. 2003, 1411-1415
a,b,2c
→
orthorhombic
cell
a´= 9.18
b´= 12.80
c´= 45.67
m’
→ orthorhombic Cell
a,b,2c
a=12.8018(4) Å
b=9.1842(9) Å
c=23.690(3) Å
ß=105.458(8) º
P21/c
a´= -b = 9.18 Å
b´= a = 12.80 Å
c´= a + 2c = 45.67 Å
α’ = 89.78 º
β’ = 90.0 º
γ’= 90.0 º
Example:
(PhTe)4Ge
Reciprocal space
h0l
wrong structure:
unidentified twin
Individual 1
Individual 2
Identification of twins
Reentrant angles
Non-uniforme optical extinction
Reciprocal space
Splitting of reflections
Specialized metrics
Huge unit cells
Large percentage of unobserved reflections
Non-standard extinction rules
Phase transitions
Pseudosymmetry
Large difference between internal R-value and final agreement factors
Large displacement parameters
Unreasonable interatomic distances
Single Crystals, Powders and
Twins
Master of Crystallography and Crystallization – 2013
T01 – Mathematical, Physical and Chemical basis of
Crystallography
END