Alsoinitial Steps in the Crystal Structural Determination - PowerPoint PPT Presentation

Title: Alsoinitial Steps in the Crystal Structural Determination


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Key things to know to describe a crystal
Also---initial Steps in the Crystal Structural
Determination
(1) Unit Cell Parameters (a, b, c, a, b, g)
(2) Lattice Types (P, I, F, C,)
(3) Space Groups
(4) Positions of atoms not related by symmetry in
unit cell ---asymmetric unit
A space group consists of a set of symmetry
elements that completely describes the symmetry
of a crystal.
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230 Space Groups
14 Bravais lattices 32 point groups screw
axes glide planes
The symmetry of a crystal is completed
described by a space group
International Tables for Crystallography, Vol.
A has complete tabulations of all 230 space
groups.
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A Review of Point Group Symmetry Elements
Crystallography Spectroscopy
Rotation around an axis
n (2, 3, 4, 6) Cn
Improper Rotation (rotation-reflection)
Sn
Hermann Maugin
Schoenflies
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Symmetry in Crystals
A crystal lattice is infinite and has
translational symmetry that a finite-sized
molecule doesnt have.
In terms of the number of symmetry elements,
the translational symmetry has two opposite
effects.
(1) It generates some new symmetry elements such
as screw axes and glide planes
(2) It reduces the number of point symmetry
elements.
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Rotation Lattice Translation (? to rotation
axis)
1) In terms of symmetry operations rotation
lattice translation lattice point RL L
2) Choose unit cell so that all vectors
describing the lattice translations are integral
repeats of cell edges, (distance along a one
integral translation along a, etc.)then R must
be an integer matrix and its trace must integer
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Rotation Lattice Translation (? to rotation
axis)
cos ? sin ? 0 - sin ? cos ?
0 0 0 1
Rotation operation through angle ?
Trace ( 1 2 cos ?) integer n
n-1 2
cos ?
and ? 0, 60, 90, 120, or 180
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Constraint of Translational Symmetry on Point
Group Symmetry in Crystals
Only 2, 3, 4, 6-fold rotation axes possible in
crystals.
4-fold
3-fold
2-fold
6-fold
8-fold also impossible
5-fold impossible to fill space
Note that a molecule such as ferrocene can
have 5-fold rotation
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32 Point Groups
A point group is a set of symmetry elements
that describe the symmetry of a crystal.
For crystals, there are only 32 unique ways to
combine point group symmetry elements.
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Relationship between point groups and physical
properties
SHG the doubling of light frequency Piezoelectr
icity generation of dipole from mechanical
stress or conversely
change of shape in an electric field. Ferroelectr
icity in which there are oriented electric
dipoles.can only be observed in polar classes.
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2-D Plane Groups --- Escher
Homework Exercise 3.40 Huheey
What symmetry elements can you find?
Answer
4 different mirror planes
Mirror plane periodic translation, t,
generates mirror plane at t/2
?All four mirrors are generated by two
perpendicular mirrors (mm) two periodic
translations, a and b
Plane group pmm
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2-fold rotation with ? translation
4 non-equivalent 2-fold axes
Space group P2
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Rotation Perpendicular Translation
Combination of a rotation, A? with a translation
t?
A? t ? B?
In general, a rotation about an axis A through
an angle, ? , followed by a translation perpendic
ular to the axis, is equivalent to a rotation
through the same angle, ? , in the same sense,
but about an axis B situated on the
perpendicular bisector of AA' and at a distance
(AA'/2)cot ?/2 from AA'.
A? causes P1 ? P2 then,
t causes P2 ? P3 ? A? t causes P1 ?
P3 which is equivalent to a rotation operation
B? about the axis B
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P2
2b 001 2b at (0 y 1/2)
P 1 2 1
No. 3
Monoclinic 2
c
b
a
a
2b 101 2b at (1/2 y 1/2)
Origin on 2 unique axis b
2nd Setting
Number of positions, Wyckoff notation and site
symmetry
Co-ordinates of equivalent positions
What if there is only one molecule/unit cell???
Molecule must have 2-fold symmetry With its
2-fold coincident with one of the crystal 2-fold
axes!!
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Molecular or Formula Weight, Number of formula
units, Density
Z number of formula units (molecules) per unit
cell
Generally, Z is the number of equivalent
positions for the space group, but may be some
simple fraction (special positions) or simple
multiple of that number
Z can be derived if the density of crystal is
determined
M atomic mass of molecule in amu V volume of
unit cell in Å3
where M is the formula or molecular weight
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Depth by Escher 3-D arrayof fish
Each fish is found at the intersection of three
lines of fish, all of which cross each other at
right angles.
This gives unit cells, each of which contains one
molecule (fish). If the eyes of the fish are
ignored, each fish, and ? each unit cell, has C2v
(mm2) symmetry. The space group would then be
Pmm2
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some matrix notation
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Space Group P2/m
four equivalent positions asymmetric unit
1/4th of the unit cell
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Symmetry in Crystals
A crystal lattice is infinite and has
translational symmetry that a finite-sized
molecule doesnt have.
In terms of the number of symmetry elements,
the translational symmetry has two opposite
effects.
(1) It generates some new symmetry elements such
as screw axes and glide planes
(2) It reduces the number of point symmetry
elements.
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Additional Symmetry Elements for Crystals
The point group does not completely describe
the symmetry of a crystal because it does not
take into consideration translational symmetry
In crystals, additional symmetry elements are
generated by combining point group symmetry
elements with translational symmetry
Glide planes
Screw axes
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Screw Axes example
21 axis, passing through origin, parallel to b
axis
2b(0 1/2 0) (x, y, z) (?x, 1/2 y, ?z)
?x, 1/2 y, ?z
1/2 t
(x, y, z)
2b(0 1/2 0) operation on a cat
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Screw Axes example
21 axis, passing through origin, parallel to b
axis
Can abbreviate this
2b(0 1/2 0) (x, y, z) (?x, 1/2 y, ?z)
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COMBINATIONS OF ROTATIONS WITH PARALLEL
TRANSLATIONS
allowed values of the pitch interval
where i integer
for 2-fold screw axis t 0, t/2, t
for 6-fold screw axis t 0, t/6, t/3, t/2,
2t/3, 5t/6, t
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Screw axes
Types of Screw Axes
2-fold screw axis 21 3-fold screw axis 31,
32 4-fold screw axis 41, 42, 43 6-fold screw
axis 61, 62, 63, 64, 65
Definition of Screw Axes
A screw axis, nm, is defined as a rotation around
the n-fold axis, followed by a translation of m/n
along the direction of the axis.
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Examples of Screw Axes
32
31
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the repetition of a point by the possible screw
axes
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Combinations of mirror planes with parallel
translations
consider
glide plane
m
t
?

• generally 1/2 t
• 1/4 t for some d glides

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Glide Planes
A glide plane is a reflection, followed by a
translation in a direction parallel to the plane
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Space Group P21/c
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Space Group No. 14
b
a
c
c
b
a
screw axis
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Homework Exercise Huheey 3.41
Among the thirteen possible monoclinic space
groups are P21, P21/m, and P21/c. Compare these
space groups by listing the symmetry elements for
each.
These are all primitive space groups with a two
fold screw axis (21). In P21/m there is a mirror
plane perpendicular to 21. In P21/c there is a
glide plane perpendicular to that axis with the
glide translation parallel to the crystal c axis.
As for the point group twofold axis, a 21 with a
mirror plane or a glide plane perpendicular to
it creates a center of inversion.
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Space Group No. 14, continued
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Space Group No. 15
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Notes on Space Group No. 15, C2/c
2b( 0 ) mb(0 1/2) ?1(000)
1
2
3
2b(0 0 1/2) mb(0 0 1/2) ?1(000)
4
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2b(0 0 1/2) 1(1/2 1/2 0) 2b(1/2 1/2 1/2)
21 at x z 1/4
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mb(0 0 1/2) 1(1/2 1/2 0) mb(1/2 1/2 1/2)
n-glide at y 1/4
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2b(0 0 1/2) mb(1/2 1/2 1/2) ?1(1/2 1/2 0)
?1 at x y 1/4
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230 Standard Space Group Symbols
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Matrix Representation
In actual computation, the symmetry operation
is done through matrix multiplication.
Each symmetry element is represented as a 3 x
3 matrix, plus a translational vector.
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Symmetry and Asymmetric Unit
A knowledge about the space group of a crystal
greatly simplifies the structural analysis of a
crystal.
Because of the symmetry elements, only a
fraction of a unit cell (can be as small as
1/192) is unique. Other parts of the unit cell
can be generated through symmetry.
This unique part of a unit cell is called the
asymmetric unit.
In a crystal structure determination, it is
only necessary to find atoms in the asymmetric
unit.
For example, for space group P-1, the
asymmetric unit is 0 x 1/2, 0 y 1, 0
z 1 Only 1/2 of unit cell is unique. The other
half can be generated.
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Even though we see objects with 8-fold rotation
symmetry almost daily, such a symmetry element
is prohibited in a crystal.