Constructing crystals in 1D, 2D

1
LET US MAKE SOME CRYSTALS

• Constructing crystals in 1D, 2D 3D
• Understanding them using the language of ?
  Lattices ? Symmetry

Additional consultations
http//cst-www.nrl.navy.mil/lattice/index.html
2
Spend some time in lower dimensional (1D 2D)
crystals before going to 3D crystals. This will
help in the understanding of the concepts
involved.

• In this section we will deal with some model
  examples. These examples have sometimes been
  chosen to give a startling effect (so that the
  essential points are driven home).

1D
3
Making a 1D Crystal

• Some of the concepts are best illustrated in
  lower dimensions ? hence we shall construct some
  1D and 2D crystals before jumping into 3D.
• A strict 1D crystal 1D lattice 1D motif.
  There is only one kind of a 1D lattice.
• The only kind of 1D entity, which can contribute
  to a motif is the line segment (or a collection
  of line segments). Though in principle a
  collection of points can be included.
• There are only two types of crystals in 1D (for
  now restricting ourselves to Euclidean space)
  (true 1D where the lattice and motif both are
  1D).? Crystal-1 with m symmetry? Crystal-2
  with only t. (as in upcoming slide)

Lattice

Motif

Crystal
It has been shown that in real systems 1D
crystals cannot be stable!!
4

• Other ways of making the same crystal
• We had mentioned before that motifs need not sit
  on the lattice point- they are merely associated
  with a lattice point.
• Here is an example

• There will always be strange ways of associating
  motif with lattice ? usually we chose the most
  natural way.

A natural way
Other strange ways of associating
motif with lattice point
Has been associated with
This lattice point
This motif
5
Crystals in 1D

• There are only two types of crystals in 1D (note
  that classification of crystals is based on
  symmetry) 1, m? crystal with m ( ofcourse
  t) (in effect there are two mirrors m0 m½) ?
  t m1 m2 (t m0 m½)? crystal with only t (i.e.
  only 1-fold) (Basically no symmetry apart from
  t) ? t

Note that the motif has a mirror plane in the
middle
Crystal-1
Motif
Two mirror points (extended for better
visibility-planes become points in 1D !!)
m
m1
m2
m0
m½
Note the mirror plane of the lattice has been
made to coincide with that of the motif (this
is a preferred way of doing things!!).
This motif does not have a vertical mirror
Motif
Crystal-2
1 (with only t)
Note that in 1D the action of a mirror is
equivalent to that of a inversion centre is
equivalent to that of a 2-fold axis.
Click here to know more
6

• Note
• For illustration purposes we will often relax
  this strict requirement of a 1D motif.? We will
  put 2D motifs on 1D lattice to get many of the
  useful concepts across.
• These are called frieze patterns (note the
  spelling of frieze). Note the periodicity is
  still in 1D.
• We could also use 3D entities as a motif to
  decorate the 1D lattice.
• The periodicity in all these cases (where the
  lattice is 1D) remain in one dimension.

1D lattice 2D Motif
Each of these atoms contributes half-atom to
the unit cell
An example of a frieze pattern artisans have
used such patters in various architectures.
The lattice points are at the centres of the red
patterns (i.e. alternate black dot).
We will see more such examples (from a symmetry
point of view) in the coming slides (albeit not
as creative as the ones made by artisans)
the circles look like 3D due to the shading!
needless to say these are not atoms
7
Time to brush-up some symmetry concepts before
going ahead
Lattices have the highest symmetry (Which is
allowed for it)? Crystals based on the lattice
can have lower symmetry
In the coming slides we will understand this
IMPORTANT point
If any of the coming 7 slides make you a little
uncomfortable you can skip them (however, they
might look difficult but they are actually easy)
8

• As we had pointed out we can understand some of
  the concepts of crystallography better by
  putting 2D motifs on a 1D lattice. These kinds
  of patterns are called Frieze groups and there
  are 7 types of them (based on symmetry).

Progressive lowering of symmetry in an 1D
lattice? illustration using the frieze groups
Consider a 1D lattice with lattice parameter a
Asymmetric Unit
Unit cell
a

• Asymmetric Unit is that part of the structure
  (region of space), which in combination with the
  symmetries (Space Group) of the lattice/crystal
  gives the complete structure (either the lattice
  or the crystal)
• (though typically the concept is used for
  crystals only)

The concept of the Asymmetric Unit will become
clear in the coming slides
The unit cell is a line segment in 1D ? shown
with a finite y-direction extent for clarity
and for understating some of the crystals which
are coming-up
9

• This 1D lattice has some symmetries apart from
  translation. The complete set is
• Translation (t)
• Horizontal Mirror (mh)
• Vertical Mirror at Lattice Points (mv1)
• Vertical Mirror between Lattice Points (mv2)

• Note
• The symmetry operators (t, mv1, mv2) are enough
  to generate the lattice
• But, there are some redundant symmetry operators
  which develop due to their operation
• In this example they are 2-fold axis or
  Inversion Centres (and for that matter mh)

t mh mv1 mv2
mmm
Or more concisely
mh
mmm
The intersection points of the mirror planesgive
rise to redundant inversion centres (i)
Three mirror planes
mv1
mv2
mirror
10
Note of Redundant Symmetry Operators
t
mmm
Three mirror planes
Redundant inversion centres
Redundant 2-fold axes

• It is true that some basic set of symmetry
  operators (set-1) can generate the structure
  (lattice or crystal)
• It is also true that some more symmetry operators
  can be identified which were not envisaged in the
  basic set ? (called redundant)
• But then, we could have started with different
  set of operators (set-2) and call some of the
  operators used in set-1 as redundant
• ? the lattice has some symmetries ? which we call
  basic and which we call redundant is up to us!

How do these symmetries create this lattice?
mirror
Click here to see how symmetry operators generate
the 1D lattice
11
Asymmetric Unit

• We have already seen that Unit Cell is the least
  part of the structure which can be used to
  construct the structure using translations
  (only).
• Asymmetric Unit is that part of the structure
  (usually a region of space), which in combination
  with the symmetries (Space Group) of the
  lattice/crystal gives the complete structure
  (either the lattice or the crystal) (though
  typically the concept is used for crystals only)
• Simpler phrasing It is the least part of the
  structure (region of space) which can be used to
  build the structure using the symmetry elements
  in the structure (Space Group)

Asymmetric Unit

mv2

mh
Lattice point
Which is theUnit Cell
Unit cell
If we had started with the asymmetric unit of a
crystal then we would have obtained a crystal
instead of a lattice

t
Lattice
12
Decoration of the lattice with a motif ? may
reduce the symmetry of the crystal
The crystals obtained by the decoration of a 1D
lattice (i.e. with 1D periodicity) with a 2D
motif gives rise to patterns, which are
historically known as frieze patterns.
t
1
mmm
Decoration with a sufficiently symmetric motif
does not reduce the symmetry of the
lattice Instead of the double headed arrow we
could have used a circle (most symmetrical object
possible in 2D)
t
2
mm
Decoration with a motif which is a single headed
arrow will lead to the loss of 1 mirror plane
mirror
13
t
3
mg
Presence of 1 mirror plane and 1 glide reflection
plane, with a redundant inversion centrethe
translational symmetry has been reduced to 2a
t
ii
4
2 inversion centres
glide reflection
mirror
14
t
5
m
1 mirror plane
t
g
6
1 glide reflection translational symmetry of 2a
t
7
No symmetry except translation
glide reflection
mirror
15
2D
Video Making 2D crystal using discs
16
Making a 2D Crystal

• Some aspects we have already seen in 1D ? but 2D
  many more concepts can be understood in 2D.
• 2D crystal 2D lattice 2D motif.
• As before we can relax this requirement and put
  1D or 3D motifs!

• We shall make various crystals starting with a 2D
  lattice and putting motifs and we shall analyze
  the crystal which has thus been created.
• In many of the examples which follow we will use
  the square lattice as an illustrative starting
  point. We will progressively decorate the lattice
  with motifs of lower and lower symmetry.
• The square lattice has 4mm symmetry.
• The important point to be noted (which can be a
  cause for considerable confusion) is that the
  adjective/term in front of a lattice or a crystal
  (e.g. a square lattice or a rectangle crystal),
  denotes the symmetry of the structure and not any
  shape or geometry (i.e. not the shape of the unit
  cell).? I.e. the term square refers to the
  presence of at least a 4-fold axis and the term
  rectangle refers to the presence of m or 2mm
  symmetry.
• Based on symmetry there are 10 types of crystals
  possible in 2D Square (4mm, 4) Rectangle (2mm,
  m) Hexagonal (6mm, 6, 3m, 3) Parallelogram (2,
  1).Note that 3 3m point groups fall under the
  hexagonal class.

Continued
17
Example-1
Square Lattice
Note that this is a patch of an infinite lattice
Circle Motif


Square Crystal
Mirrors not included in the figure
Note that this is a patch of an infinite crystal
Continued
18
Square Lattice
Circle has infinity (?) symmetry at the centre
Circle Motif


Square Crystal
4mm
Symmetry of the lattice and crystal identical ?
Square Crystal
Including mirrors
4mm
Continued
19
Important Note
gt
Symmetry of the Motif
Symmetry of the lattice
Hence Symmetry of the lattice and Crystal
identical (symmetry of the lattice is
preserved) ? Square Crystal
Symmetry of the Motif

• Any fold rotational axis allowed! (through the
  centre of the circle)
• Mirror in any orientation passing through the
  centre allowed!
• Centre of inversion at the centre of the circle

20

• What do the adjectives like square mean in the
  context of the lattice, crystal etc?

Funda Check

• Let us consider the square lattice and square
  crystal as before.
• In the case of the square lattice ? the word
  square refers to the symmetry of the lattice (and
  not the geometry of the unit cell!).
• In the case of the square crystal ? the word
  square refers to the symmetry of the crystal (and
  not the geometry of the unit cell!)

21
Example-2

Square Motif

Square Lattice
Square Crystal
Unlike the case of the circle, we have to specify
the orientation of the square (when used as a
motif)
Continued
22
Important Note

Symmetry of the Motif
Symmetry of the lattice
Hence Symmetry of the lattice and Crystal
identical ? Square Crystal
4mm
4 mv md
mv
md
Symmetry of the Motif

• 4mm symmetry

If the Symmetry of the Motif ? Symmetry of the
Lattice The Symmetry of the lattice and the
Crystal are identical
Important Rule
i.e. Symmetry of the lattice is NOT lowered ? but
is preserved
Common surviving symmetry determines the crystal
system
This could be phrased in the language of subsets
23
Example-3

• In a the above example we are assuming that the
  square is favourably oriented. And that there are
  symmetry elements common to the lattice and the
  motif.
• In the current example, we rotate the square
  (motif) by an arbitrary angle before decorating
  the square lattice.
• The lattice and motif both have 4mm symmetry, but
  due to the rotation of the square only the 4-fold
  axes survive. Due to the presence of this 4-fold
  this kind of crystal is classfied under the
  square class.

Square Crystal

Square Motif

Square Lattice
Rotated
4
Rotated by an arbitrary angle (not 45? or 90?)
24

• How do we understand the crystal made out of
  rotated squares?

Funda Check

• Is the lattice square ? YES (it has 4mm symmetry)
• Is the crystal square ? YES (but it has 4
  symmetry ? since it has at least a 4-fold
  rotation axis- we classify it under square
  crystal- we could have called it a square
  crystal or something else as well!)
• Is the preferred unit cell square ? YES (it has
  square geometry)
• Is the motif a square ? YES (just so happens in
  this example- though rotated wrt to the lattice)

Infinite other choices of unit cells are possible
? click here to know more
25
Example-4
Note the motif is not just the shape here, but
also its orientation (w.r.t the lattice).
Square Lattice

Triangle Motif

Square Crystal
Rectangle Crystal
Symmetry of the lattice and crystal different ?
NOT a Square Crystal
Isnt this amazing ? square lattice with a
triangle motif giving a rectangle crystal!
m
Here the word square does not imply the shape in
the usual sense
This can also be called pseudo-square crystal
(as the geometry is square, but the symmetry is
rectangular. (Similarly in 3D we can have
pseudo-cubic, etc.)
Continued
26
Symmetry of the structure
Only one set of parallel mirrors left
This crystal does NOT have a centre of
inversion! For this crystal the up direction is
not the same as the down direction. (0 1 ? 0
?1).
m
m0
m½
Actually, there are two set of mirrors m½ m0.
27
Important Note
lt
Symmetry of the Motif
Symmetry of the lattice

• Mirror
• 3-fold

The symmetry of the motif determines the symmetry
of the crystal ? it is lowered to match the
symmetry of the motif (common symmetry elements
between the lattice and motif ? which survive)
(i.e. the crystal structure has only the symmetry
of the motif left even though the lattice
started of with a higher symmetry) ? Rectangle
Crystal (has no 4-folds but has mirror)
Note that the word Rectangle denotes the
symmetry of the crystal and NOT the shape of the
UC
If the Symmetry of the Motif lt Symmetry of the
Lattice The Symmetry of the lattice and the
Crystal are NOT identical
Important Rule
i.e. Symmetry of the lattice is lowered ? with
only common symmetry elements
28

• How do we understand the crystal made out of
  triangles?

Funda Check

• Is the lattice square ? YES (it has 4mm
  symmetry).
• Is the crystal square ? NO (it has only m
  symmetry ? hence it is a rectangle crystal).
• Is the unit cell square ? YES (it has square
  geometry the unit cell by itself has 4mm
  symmetry) (we have already noted that other
  shapes of unit cells are also possible) (Lattice
  parameters ab, ?90?).
• Is the motif a square ? NO (it is a triangle!).

• Is the underlying lattice really a square
  lattice?

Funda Check

• Well! This is a tough one!
• In reality the underlying lattice cannot be
  viewed in isolation and should be looked at in
  the context of the crystal being considered.
• Viewed in isolation it is a square lattice, in
  the context it may be called a rectangle lattice!

29
Example-5

Triangle Motif

Square Lattice
Parallelogram Crystal
Rotated
Also called Oblique Crystal
Rotated by an arbitrary angle (not 45? or 90?)
Crystal has No symmetry except translational
symmetry as there are no symmetry elements common
to the lattice and the motif (given its
orientation)
As before this is a pseudo-square crystal (as
the geometry of the unit cell is square, but the
symmetry is that belonging to the parallogram
class.
30
Some more twists
31
Example-6
Square Lattice
Random shaped Motif

In Single Orientation

Square Crystal
Parallelogram (Oblique) Crystal
Symmetry of the lattice and crystal different ?
NOT Square Crystal
No Symmetry
Except translation
32
Example-7
Square Lattice
Random shaped Object

Randomly oriented at each point

Square Crystal
Amorphous Material(Glass)
Symmetry of the lattice and crystal different ?
NOT even a Crystal
No Symmetry
Note the orientational disorder
33

• Is there not some kind of order visible in the
  amorphous structure considered before? How can
  understand this structure then?

Funda Check

• YES, there is positional order but no
  orientational order.
• If we ignore the orientational order (e.g. if the
  entities are rotating constantly- and the above
  picture is a time snapshot- then the time
  average of the motif is like a circle). (This
  is like a fan with three blades, but when
  rotating fast it looks like a circle? the
  identity of the blades is lost in the blurred
  picture).

• Hence, this structure can be considered to be a
  crystal with positional order, but without
  orientational order!

Click here to know more
34
Summary of 2D Crystals
Click here to see a summary of 2D lattices that
these crystals are built on
Crystal Order of the point group is in the brackets Highest Symmetry Possible Other symmetries possible At least (symmetry) Lattice Parameters(of conventional unit cell)
1. Square 4mm 8 4 4 4 (a b , ? 90?)
2. Rectangle (Rectangular) 2mm 4 m 2 m (but no higher than 2-fold) (a ? b, ? 90?)
3. 120? Rhombus (Hexagonal) 6mm 12 6 6, 3m 6, 3 3 3 (a b, ? 120?)
4. Parallelogram (Oblique) 2 1 - (a ? b, ? general value)
Note 3 is part of hexagonal
Point Group Symmetry Present Point Group Symmetry Present Lattice Type Unit Cell Shape
Single Combinations
1, 2 Parallelogram Parallelogram
m 2mm Rectangle Rectangle
m 2mm Centred Rectangle Rectangle
4 4mm Square Square
3, 6 3m, 6mm 120? Rhombus 120? Rhombus
4 crystal systems in 2D
35
Q A
Make a crystal having only 2-fold symmetry.

• Let us consider an example of two options to
  understand the underlying concepts.
• Option-1 Take a (i) square or and decorate it
  with a rotated rectangle as the motif. (i.e.
  take a high symmetry lattice and lower the
  symmetry by using the motif). (ii) rectangle
  lattice and decorate it with a rotated
  rectangle as the motif. (i.e. start with a
  lower symmetry lattice and use a motif so as to
  retain only the2-fold).
• Option-2 Take a parallelogram lattice and
  decorate it with a circle. (i.e. take a low
  symmetry lattice (only with a 2-fold) and retain
  its symmetry with a high symmetry motif).

1(i)
1(ii)
2
36
Q A
Describe the crystal in the figure below. The
lattice parameters are a b, ? 90?.

• In the absence of the motif the lattice is a
  square lattice (simple square lattice, UC at
  45?).
• The crystal has only one set of vertical mirrors
  and hence is a rectangle crystal.
• Now comes the tricky point? How do I describe
  the crystal?.
• The crystal has to be described as
• Lattice Centred rectangle latticeMotif

UC of just the lattice
UC of the crystal
Hence, the UC of the crystal has to be used for
the lattice in this context.
The vertical mirrors in the crystal
The crystal
The underlying lattice
37
Give examples of crystals belonging to all the
ten 2D point groups.? As an illustration use
square and 120? rhombus lattices as starting
points.
Q A
Symmetry of the crystal
4
4mm
2mm
Rectangle
Square
m
Note the unit cells are squares or 120? rhombi
Hexagonal
3
3m
6
6mm
Parallelogram/Oblique
2
1
38
Give examples of crystals belonging to all the
ten 2D point groups.? Use conventional unit
cells this time around.
Q A
Crystal
Symmetry of the underlying lattice
Crystal
Symmetry of the underlying lattice
6mm
4mm
4
4mm
Square
6
2mm
6mm
3m
m
2mm
Rectangle
3
2
Note the symmetry of the lattice defines the
Crystal System
2
Parallelogram
1
Hexagonal
4 crystal systems in 2D
39
How to go from a square crystal to a rectangle
crystal? (Illustrate with examples).
Q A
Square lattice Square motif Square crystal
Square lattice Rectangle motif Rectangle
crystal
2
2mm
4mm
1
Rectangle lattice Square motif Rectangle
crystal
2mm
3
Rectangle lattice Rectangle motif Rectangle
crystal
2mm
40
Funda Check
How come both the below crystals (in Fig.1) have
2 symmetry?

• In the first case the shape and orientation of
  the motif leads to the lowering of the symmetry
  of the underlying lattice from 4mm to 2.
• In the second case the lattice itself has only a
  2 symmetry and using a motif of higher symmetry
  (the well oriented yellow rectangle motif has 2mm
  symmetry) cannot increase the symmetry of the
  structure and hence the crystal has only 2
  symmetry.
• The location of these symmetry elements is given
  in the figure to the right (Click here to know
  more about space groups in 2D (Click here to know
  more about space groups in 2D).

2
p2
2
Fig.1
2
41
Funda Check
Can the symmetry of the lattice be increased by
using a higher symmetry motif?

• No.
• The maximum symmetry one can get is that of the
  lattice.
• As an example let us consider a rectangle lattice
  (whose symmetry is 2mm). Let us make a crystal by
  associating with each lattice point a square
  (with a higher 4mm symmetry). However, the
  crystal thus obtained still has a 2mm symmetry!
• We could use a circle (the object with the
  highest symmetry possible in 2D) and still we
  will get a crystal with 2mm symmetry (only).

In spite of decoration with a motif with 4mm
symmetry (that too a well aligned one) the
symmetry of the crystal obtained is still 2mm
Lattice with 2mm symmetry
42
From the previous slides you must have seen that
crystals have
CRYSTALS
Orientational Order
Positional Order
Later on we shall discuss that motifs can be
(Motifs)
MOTIFS
Geometrical entities
Physical Property

• In practice some of the strict conditions imposed
  might be relaxed and we might call a something a
  crystal even if
• Orientational order is missing
• There is only average orientational or
  positional order
• Only the geometrical entity has been considered
  in the definition of the crystal and not the
  physical property

43
3D
44
Making a 3D Crystal

• A strict 3D crystal 3D lattice 3D motif.
• We have 14 3D Bravais lattices to choose from.
• As an intellectual exercise we can put 1D or 2D
  motifs in a 3D lattice as well.(we could also
  try putting higher dimensional motifs like 4D
  motifs!!).
• We will illustrate some examples to understand
  some of the basic concepts (most of which we
  have already explained in 1D and 2D).

In this chapter we only consider an outline of
the crystal structures like CCP, HCP BCC
crystals. More details can be found in the link
below.
Chapter_4a_Structure_of_Solids_Metallic
45
Simple Cubic (SC) Lattice
Sphere Motif

Graded Shading to give 3D effect
Note SC is a lattice when we decorate it with
a single sphere motif (e.g. Polonium atom), then
we get a crystal, which is in common usage
language referred to as a SC crystal.
Simple Cubic Crystal
Conventional Unit cell of the SC lattice

• If these spheres were spherical atoms then the
  atoms would be touching each other.
• The kind of model shown is known as the Ball and
  Stick Model.
• In the true unit cell 1/8th of the atom is within
  the unit cell.

46
To know more about Close Packed Crystals click
here
Sphere Motif

Body Centred Cubic (BCC) Lattice
Atom at (½, ½, ½)
Body Centred Cubic Crystal
Atom at (0, 0, 0)

Unit cell of the BCC lattice
Space filling model
Central atom is coloured differently for better
visibility
So when one usually talks about a BCC crystal
what is meant is a BCC lattice decorated with a
mono-atomic motif
Note BCC is a lattice and not a crystal
47
Sphere Motif

Face Centred Cubic (FCC) Lattice
Close Packed implies CLOSEST PACKED
Cubic Close Packed Crystal(Sometimes casually
called the FCC crystal)
Point at (½, ½, 0)
Point at (0, 0, 0)
Unit cell of the FCC lattice

Space filling model
So when one talks about a FCC crystal what is
meant is a FCC lattice decorated with a
mono-atomic motif
Note FCC is a lattice and not a crystal
48
More views
All atoms are identical- coloured differently for
better visibility
49
Two Carbon atom Motif(0,0,0) (¼, ¼, ¼)

Face Centred Cubic (FCC) Lattice
Diamond Cubic Crystal

Tetrahedral bonding of C (sp3 hybridized)
It requires a little thinking to convince
yourself that the two atom motif actually sits at
all lattice points!
Note This is not a close packed crystal
There are no close packed directions in this
crystal either!
50
Two Ion Motif

Face Centred Cubic (FCC) Lattice
(0,0,0)
(½,0,0)
NaCl Crystal
Cl? Ion at (0, 0, 0)

Na Ion at (½, 0, 0)
The Na ions sit in the positions (but not
inside) of the octahedral voids in an CCP crystal
? click here to know more
Solved Example
Note This is not a close packed crystal
Has a packing fraction of 0.67
51
NaCl crystal further points
Click here Ordered Crystals
This crystal can be considered as two
interpenetrating FCC sublattices decorated with
Na and Cl? respectively
Inter-penetration of just 2 UC are shown here
More views
Coordination around Na and Cl? ions
52

• Now we present 3D analogues of the 2D cases
  considered beforethose with only translational
  symmetry and those without any symmetry.

The blue outline is NO longer a Unit Cell!!
Triclinic Crystal(having only translational
symmetry)
Amorphous Material (Glass) (having no symmetry
what so ever)
53
Making Some Molecular Crystals

• We have seen that the symmetry (and positioning)
  of the motif plays an important role in the
  symmetry of the crystal.
• Let us now consider some examples of Molecular
  Crystals to see practical examples of symmetry of
  the motif vis a vis the symmetry of the
  crystal.(click here to know more about molecular
  crystals ? Molecular Crystals)
• It is seen that there is no simple relationship
  between the symmetry of the molecule and the
  symmetry of the crystal structure. As noted
  before? Symmetry of the molecule may be
  retained in crystal packing (example of
  hexamethylenetetramine) or? May be lowered
  (example of Benzene)

54
Q A
Give an example of a real crystal wherein the
motif leads to a lowering the symmetry of the
lattice (on the formation of a crystal)?

• The FCC lattice has a true 4-fold axis. On the
  formation of a diamond crystal the symmetry along
  lt100gt is lowered to a 2-fold.
• Note the DC structure has a 41 screw axis (along
  lt100gt).

Funda Check

• From reading some of the material presented in
  the chapter one might get a feeling that there is
  no connection between geometry and symmetry.
  I.e. a crystal made out of lattice with square
  geometry can have any (given set) of symmetries.
• In atomic systems (crystals made of atomic
  entities) we expect that these two aspects are
  connected (and not arbitrary). The hyperlink
  below explains this aspect.

Click here ? connection between geometry and
symmetry
55
Q A
How do we handle the case wherein the symmetry of
the lattice and motif do not coincide (i.e.
shifted by a translation vector)?

• Let us consider the example as below (F1 and F2).
• In F1 the symmetries of the lattice and motif
  coincide while in F2 there is a relative shift
  (red vector).
• In the second case (F2) we have to ignore the
  lattice and overlay the symmetry operators on the
  crystal.

Lattice point
Relative shift between the 4-folds of the lattice
and motif
F1
F2