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Introduction to crystals symmetry

Symmetry in 3D

5/1/2013 1 L. Viciu| AC II | Symmetry in 3D

Outlook

• Symmetry elements in 3D: rotoinversion, screw axes

• Combining symmetry elements with the lattice

• Space group symbols/representation

• Wyckoff positions

• Crystallographic conventions

• Symmetry in crystal systems


Definitions - a remainder

• Unit cell: The smallest volume that can generate the entire crystal structure only by means of translation in three dimensions.

• Lattice: A rule of translation.


Bravais lattice

Basis/ Motif +

Crystal structure

1. single atom: Au, Al, Cu, Pt 2. molecule: solidCH4 3. ion pairs: Na+/Cl- 4. atom pairs: Carbon, Si, Ge

1. FCC 2. FCC 3. Rock salt 4. diamond

•A crystal system is described only in terms of the unit cell geometry, i.e. cubic, tetragonal, etc •A crystal structure is described by both the geometry of, and atomic arrangements within, the unit cell, i.e. face centered cubic, body centered cubic, etc.

+


The Bravais lattices


From 2D to 3D

• Bravais lattices may be seen as build up layers of the five plane lattices:

cubic and tetragonal stacking of square lattice layers

orthorhombic P and I stacking of rectangular layers

orthorhombic C and F stacking of rectangular centered layers

rhombohedral stacking of hexagonal layers

hexagonal stacking of hexagonal layers

monoclinic stacking of oblique layers

triclinic stacking of oblique layers


Combining symmetry elements

For two dimensions

– 5 lattices

– 10 point groups

– 17 plane groups

For three dimensions

– 14 Bravais lattices

– 32 point groups

– 230 space groups


• Symmetry operations in 2D*: 1. translation

2. rotations

3. reflections

4. glide reflections

• Symmetry operations in 3D: the same as in 2D

+

inversion center, rotoinversions and screw axes

* Besides identity


Symmetry elements in 3D

• In three dimensions we have additional symmetry operators:

1. inversion center = symmetry center

2. inversion axis = rotoinversion or improper rotation axis

3. screw axes

• In addition, the mirror line becomes a mirror plane, and the glides become glide planes.


1. The center of symmetry/inversion center

• The symmetry center: the intersection point of a mirror plane and a 2-fold axis

1Symbol: “one bar”; Graphical:

),,( zyx(x,y,z)

Inversion center

• It is always present in one of the situations:

P2/m 3P


if an inversion axis with odd multiplicity is present

a rotation axis with even multiplicity and a reflection to it is present

• An inversion center requires a 2-fold axis and a mirror plane to it.

• Therefore, an even rotation with a reflection perpendicular to it will give an inversion center.

A 1 2

3

i

1 has the coordinates (x y z)

2 has the coordinates ( )


zyx

zyx

A= inversion, 1

*The order of the operation is not important

321 180 reflectionbyrotation

inversion


• The point groups that contain an inversion center are called Laue groups.

• If you take away the translational part of the space group symmetry and add an inversion center, you end up with the Laue group.

• The Laue groups are used in diffraction: describe the symmetry of the diffraction pattern.


Centrosymmetric structures

• If an inversion center is present then the structure is centrosymmetric

• In a centrosymmetric structure, the unit cell is chosen so that the origin lays on the inversion center

82% of inorganic crystals are centrosymmetric because Inversion center leads to equal forces in opposing directions favoring stability

• In a non-centrosymmetric space group, the origin is chosen at a highest symmetry point.


George M. Sheldrick.

32 point groups

11 centrosymmetric point group +

432 class*

20 non-centrosymmetric point groups:

piezoelectrics

10 point groups with no unique

polar axis

10 point groups with unique polar axis:

Pyroelectric + ferroelectrics

• Polar direction = a crystal direction that is not related by symmetry to the opposed direction

*432 class has a polar axis but no piezoelectricity


Space groups and enantiomorphous molecules

Klockmanns Lehrbuch der Mineralogie, 16. Auflage, Enke VerlagStuttgart 1978

Enantiomorphous molecules crystallise in space groups without inversion centre , i, and without reflection planes, m. Altogether there are 11 Point groups possible for enantiomorphous molecules. (“Biological Point Groups”)

Enantiomorphous crystals of tartaric acid (monoclinic structure, space group P21)

The most common chiral space groups are P212121, P21, P1 and C2221.

15% of all crystals are enantiomorphic and potentially optically active 5/1/2013 16 L. Viciu| AC II | Symmetry in 3D

2. Axes of inversion/rotoinversion

• Rotation + center of symmetry = inversion axis, , pronounced n bar

• It is also called improper rotation axes (Schoenflies symbols, Sn)

• Two objects related by an operation of inversion axis are enantiomorphous

n

m21. 2.

3.

m

4.

atom inversion

axis 180ᵒ

6

4

3

2

1

m

Symbol/graphic


144,133 if Therefore 4 bar is a distinct operation

Spinoid (like a squashed Td)


The rotation is combined with a translation in a defined crystallographic direction by a defined step (half a unit cell for twofold screw axes, a third of a unit cell for threefold screw axes, etc

counter clockwise rotation with 360ᵒ/n

Translation with m/n (m<n)

+

Srew axis nm

Rotation + Translation in axial direction

3. Screw axes: nm

c. Hammond, the basics of crystallography and diffraction


3. Screw axes: nm

(Fig. 140) Inorganic structural chemistry, U. Mueller

Rotation + Translation in axial direction:

Not shown

21: ½ displacement on c

31: 1/3 displacemnt on c; 32: 2/3 displacement on c;

41: ¼ displacemnt on c; 42 2/4 = ½ displacement on c; 43: ¾ displacement on c

61: 1/6 displacement on c; 62: 2/6= 1/3 displacement on c; 63: 3/6= ½ displacement on c, etc


Screw axis of order 3: 31; 32 31

rotation by a 3-fold axis and then translation by 1/3

T

T is translation periodicity Is translation component = 1/3

Red color is used only to show the objects obtained by translation periodicity

32 rotation by a 3-fold axis and

then translation by 2/3


Screw axis of order 3: 31; 32 31

rotation by a 3-fold axis and then translation by 1/3

T

32 rotation by a 3-fold axis and

then translation by 2/3

31 gives an anticlockwise spiral 32 gives a clockwise spiral


T

42

T

41

T

43

41 gives an anticlockwise spiral

43 gives a clockwise spiral Color is used only to show the objects

obtained by translation periodicity; otherwise the color represent the same motif 5/1/2013 23 L. Viciu| AC II | Symmetry in 3D

T

42

42 gives two motifs at every level and they are 180

away from each other

42 gives two spirals - one left handed (anticlockwise)

and one right handed (clockwise) – in the same

pattern


Quartz symmetry

View along c axis

P3121 and P3221 P6221 and P6421

- quartz (hexagonal) - quartz (trigonal) C573


Glide planes in Space Groups

• In 2D the gliding (translation) is in the direction of the dashes:

1

g2 g1 2 3

• In 3D we can look at a glide from 4 different directions:

Down from the top (1)

Edge on and to the gliding direction (2)

Edge on and to the gliding direction (3)

Edge on and in between normal to and parallel to the gliding direction (4)

* - gliding direction

*

(2) (4)

(3)

(1)


(1) Looking down from top of a glide use or or

a glide

b

a

a

= ½ of a axis

b glide

= ½ (a+b) axis

b

a

= ½ of b axis

n glide = diagonal glide

b

a

Ex: n-glide

c glide is represented in plane as well and the translation is ½ of c axis 5/1/2013 27 L. Viciu| AC II | Symmetry in 3D

(2) Looking normal to the gliding direction use a dashed line

a

b

(3) Looking in the direction of gliding use a dotted line

a

b

(4) Looking not parallel not perpendicular to the direction of gliding use a mixed line

a

b

Ex: b glide

Ex: b glide

Ex: b glide


• Found in centered cells only (not in primitive lattices)

• It originates from the diamond structure where the diamond glide has been first observed.

• The translation is ½ of the ½(a+b) axis

d-glide = diamond glide

a+b

½ (a+b)

½ (½ (a+b))


Graphical symbols for symmetry elements

Massa, Werner. Crystal Structure Determination

motif in a general position in the plane

motif at arbitrary height above the plane

Enantiomorph of motif in the plane

Enantiomorph of motif at arbitrary height

above the plane

+

+

Equivalent position diagrams:

Yellow box: Rotation and inversion axes Blue box: screw axes Green box: inversion center Red box: glide planes Cyan box: mirror planes

n - diagonal glide d- diamond glide in centered cells only

* = axis in the plane; # = glide in the screen plane; & = perpendicular to the screen plane

either of the two directions one direction only 5/1/2013 30 L. Viciu| AC II | Symmetry in 3D

Direction of the 2-fold rotation axes when looked at by the side

Direction of the 2-fold screw axes when looked at by the side

Only 2-fold rotations and 2-fold screw axes are shown graphically!


Combining symmetry operations in 3D 1. Combining rotations Two crystallographic rotations (1-,2-, 3-, 4- and 6-fold) at an intersection point will give a 3rd rotation which must be crystallographic (1-,2-, 3-, 4- and 6-fold).

B A = C

A

C B

1

2

3

11 axial combinations: 1, 2, 3, 4, 6, 222, 322, 422, 622 (dihedral 23, 432 (in Schoenfliess notation 23 is the Td and 432 is the Oh)

Euller constrictions in 3D: there are 11 combinations for the crystallographic rotations

321 BbyrotationAbyrotation

Rotation by C

5/1/2013

32 L. Viciu| AC II | Symmetry in 3D

3. A rotation and reflection in 3D

1A=2

A rotation, , followed by a reflection with a mirror parallel to it and at an angle half of the rotation angle, /2, gives another reflection.

2mm

A

1

If the mirror is perpendicular to the rotation, it does not create another reflection! In this case the resulted point group is labeled 2/m

Ex: 2-fold rotation + reflection

*The two mirrors will contain the rotation

2


A= inversion,

1 has the coordinates (x y z)



zyx

zyx

1

A 1 2

3

A 1 2

3

m

2

(inversion, mirror, 2-fold identity)

A rotation followed by a reflection with a mirror perpendicular to it gives an inversion.

321 reflectionrotation

Inversion


Crystallographic Conventions: unit cell and unique axes

• Unit cell: Right handed system a, b, c, α, β, γ

• Unique axis: the direction with highest rotation symmetry

The monoclinic angle is β with β≥ 90°. This makes b the unique axis in the monoclinic system

In the tetragonal, trigonal and hexagonal systems, c is the unique axis.


Symmetry directions • How the symmetry elements are oriented with respect to the axes of

the unit cell, a, b and c ( the places in the Hermann-Mauguin symbol for point group)

Crystal system Order of direction

Triclinic Monoclinic Orthorhombic Tetragonal Trigonal Hexagonal cubic

- b a, b, c c, a c, a c, a c

P2/m P2

Ex: monoclinic


[100] – Axis or plane to the x-axis .

[010] – Axis or plane to the y-axis.

[001] – Axis or plane to the z-axis.

[110] – Axis or plane to the line running at 45° to the x and y axes (face diagonal).

[120] – Axis or plane to the line running on the a or on the b axis in the hexagonal cell.

[111] – Axis parallel or plane perpendicular to the body diagonal.

Crystal System Symmetry Direction

Primary Secondary Tertiary

Triclinic None

Monoclinic [010] b

Orthorhombic [100] a [010] b [001] c

Tetragonal [001] c [100]/[010] a/b [110]

Hexagonal/Trigonal [001] c [100]/[010] a/b [120]

Cubic [100]/[010]/[001] a/b/c [111] [110]


Space group symbol rules/meaning 1. First letter: P, A, B, C, F, I or R translation symmetry + type of centering

ex: P 4mmm; C mm2; F mmm

2. The orientation of the symmetry elements: to coordinate system x, y and z.

The highest multiplicity axis or if only one symmetry axis present they are on z

Ex: P 21/c: 21 axis in the z direction

If highest multiplicity axis is 2-fold the sequence is x-y-z

ex: Cmm2: 2-fold axis on z; or Cm2m the2-fold axis on y

The highest symmetry axis is mentioned first

ex: I 4/mcm: 4-fold axis on z and two 2-fold axes on a and b

3. An inversion center is mentioned only if it is the only symmetry element

ex:

4. A reflection plane to a symmetry axis is designated by a fraction bar ”/”

ex: P 2/mmm: mirror palne on 2-fold axis exception m to odd rotation axis: 3/m 6 the inversion axis used instead

1P


Triclinic system

11 PandP

No symmetry elements or only inversion center present:

a

b

a

b

Note: the projection of objects in an oblique lattice should be made parallel with the translation otherwise the number of the obtained points will be different from the number of lattice points.


We consider two unit cells with one lattice point inside the cell (primitive) stacked on top of each other

The projection of the atoms into the basal plane makes the lattice points be different from a primitive cell.

c

a

b

c

a

b

The projection of the atoms into the basal plane gives one point in agreement with a primitive cell.

Projecting Projecting


In a monoclinic system there are possible two settings

1. c is to the (ab) plane and the angle , between a and b, the general angle

2. b is to the (ca) plane and the angle , between c and a, is general angle

90

90

a

b

c

1st setting

90

90

c

a

b

2nd setting

Monoclinic System

the unique axis (either b or c) the 2-fold axis parallel to it and/or the mirror plane to it;


Monoclinic space group P 2 1st setting: Unique axis c

o b

a

o c

ap

o a

c

o b

cp

2nd setting: Unique axis b

c is the direction of 2-fold axis (c (ab)) b is the direction of 2-fold axis (b(ac))

The direction of the 2-fold axis when looking on the side (an arrow is the symbol for a 2-fold axis along on the page)

and

This is the symbol for the 2-fold axis to the plane


P2 – unique axis c

Atoms at + z (the atom in the cell gives the atom outside by rotation with a 2-

fold axis)


Space group Pm: unique axis c (mirror c)

Mirror plane viewed directly from above (the drawn plane is the mirror plane)

‘ + - , Two atoms superimposed in projection: one at +z and one to –z The right side of the symbol says that the atom is at +z The left side of the symbol says that an enantiomorph sits at -z

Above view

Side view-along the c axis (parallel to the mirror plane)

Side view : to the mirror plane


Monoclinic System Either a two fold axis (2) or a mirror plane (m) or both (2/m) • 2implies that the 2-fold axis is parallel to the unique axis (b or c depending on

the setting) • m implies a mirror plane to the unique axis It can also have glide planes or screw axis (i.e. Cc, P2, P21/n

2 (P 2) 2-fold axis to b 2/m (P 2/m) [unique axis c] 2/m = reflection plane to the 2-fold axis

The reflection plane is to the paper/figure Inversion center because we have a 2-fold and a mirror plane

b axis to the plane*

b axis to the plane*

*Please note the a, b, and c axes labeling: in the left the b axis is to the plane in the right, the b axis is to the plane

2-fold axis to c ( to the plane)*

2-fold axis to c ( to the plane)*


Unconventional Lattices

Monoclinic A (B or C ), can always be transformed into monoclinic P (red cell), with

half the unit cell volume.

That is why the list of the 14 Bravais lattices does not include monoclinic B nor monoclinic I nor several other unconventional lattices.

Monoclinic I, can always be transformed into monoclinic B (A or C) -dashed cell, with the same unit cell volume

a b

c


What information is included in the space group C2/m?

C- side centered cell 2/m – 2-fold axis along b and mirror plane to it (or on b axis)

Ex: Na0.5CoO2 – monoclinic structure in space group C2/m with

a = 4.9043(2), b = 2.8275(1), c = 5.7097(3) and = 106.052(3)


Orthorhombic system

m m 2 (P m m 2)

three 2-fold axes parallel to the cell edge three mirror planes parallel to the faces

The symmetry elements are along all three directions, a, b and c

(i.e. Pnma, Cmc21, Pnc2)

In an orthorhombic symmetry no direction is special than any other.

The magnitude of the translations is:

cab 5/1/2013 49 L. Viciu| AC II | Symmetry in 3D

Space group symbol for orthorhombic

cplane

caxis

bplane

baxis

aplane

aaxis

Symbol for the lattice type

•If there is a 2-fold and a 2-fold screw symmetry along the same axis, the 2-fold is preferred in the symbol •If there are more mirror planes perpendicular to the same axis, the preferred order is m>a>b>c>n>d


What information is included in the space group symbol Pna21?

P primitive lattice 21 orthorhombic system with screw axis 21 along (parallel) c n diagonal glide plane on a (moves the motif ½ of the diagonal of b and c) a axial glide plane on b (moves the motif ½ a - to a)

Ex: LiB3O5


Tetragonal symbol

• 1st the 4-fold symmetry along c direction

• 2nd symmetry element along a (which must apply to b as well)

• 3rd symmetry element along the a,b diagonal

4/mmm

4/m m m (P 4/m m m)

The primary symmetry symbol will always be either 4, (-4), 41, 42 or 43 (i.e.P41212, I4/m, P4/mcc)


What information is included in the space group symbol I41cd?

I body centered lattice 41 tetragonal system with screw axis 41 along (parallel) c c glide plane on a and b because a = b (moves the motif ½ c - to c) d diamond glide plane on [110] direction (the diagonal of xy plane)


What information is included in the space group symbol P4mm?

P primitive lattice 4 tetragonal system with 4-fold axis to c m mirror plane on a and b m mirror plane on the ab diagonal [110]

Ex: PuS2


Trigonal and hexagonal systems

• 1st the symmetry element along c direction:

- 3-fold axes (rotation and screw axis) in trigonal

i.e P31m, R3, R3c, P312)

- 6-fold axes in hexagonal (i.e. P6mm, P63/mcm)

• 2nd symmetry element with respect to a/b (a = b)

• 3rd symmetry element with respect to a, b diagonal (s)

Trigonal Hexagonal

3 (P 3) )13(13 mPm 6 m m (P 6 m m) 5/1/2013 55 L. Viciu| AC II | Symmetry in 3D

1/3, 2/3)

2/3, 1/3)

For trigonal lattice there are two possibilities: 1. The third translation to a1 and a2 2. The third translation inclined so that the projection falls either on (1/3, 2/3) or (2/3,

1/3). The two situations are actually the same

Rhombohedral lattice

The “double body centered cell” is then described by a primitive cell taking as one base the point at 2/3, 1/3, 1/3 and the points at the corner of the triangle of which the point is related

(2/3, 1/3, 1/3)

(1/3, 2/3, 2/3) Hexagonal/trigonal lattice


3

1

3

1

3

2

3

2

3

2

3

1

The rhombohedral R cell expressed in hexagonal axes

one hexagonal cell three hexagonal cells with additional nodes

Rhombohedral R cell (red) inscribed in hexagonal

The hexagonal cell is three times larger than the rhombohedral and has additional nodes at ),,(),,( 3

23

23

13

13

13

2 and 57 L. Viciu| AC II | Symmetry in 3D

The relation between rhombohedral R cell (brown) and hexagonal cell (black)

Rhombohedral R cell (brown) inscribed in an F-centered cubic cell (black)

Blue - the rhombohedron projection on ab plane of the hexagonal cell

In the rhombohedral (R) lattice, the first character (3 or 3 bar) denotes the unique space diagonal of the cell and the next defines the directions that are perpendicular to the 3-fold axis.


• 1st symmetry element along c direction (either a 2-fold axis or a 4-fold axis parallel to, and/or a plane to, c direction)

• 2nd 3-fold axis or 3-fold inversion axis along the body diagonals

• 3rd symmetry element along the face diagonals (either 2-fold axis parallel to, or mirror plane to, the face diagonals)

mmmm 3/23/4

4/m m in cubic symmetry because the four 3-fold axes and the nine mirror planes automatically generates the three 4-fold axes, six 2-fold axes and a center of symmetry

The secondary symmetry symbol will always be either 3 or –3 (i.e. Ia3, Pm3m, Fd3m)

Cubic System

Isometric system = the translations are identical in all 3

directions (a=b=c)

the direction of the 4-fold axis is always along the edge of the unit cell


Cubic symmetry: axes

4 fold rotation axes

(passing through

pairs of opposite

face centres,

parallel to cell axes)

TOTAL = 3

3-fold rotation axes

(passing through cube

body diagonals)

TOTAL = 4

2-fold rotation axes

(passing through

diagonal edge centers)

TOTAL = 6


3 equivalent planes in a cube

6 equivalent

planes in a cube

Cubic symmetry: planes


- F: face centered cubic - m: mirror ( 4/m) on c axis - 3: 3-fold rotoinversion on the body diagonal - m: mirrors on the face diagonal

What information is included in the space group Fm3m?

Ex: NaCl


Essential Symmetry

System Essential Symmetry Symmetry axes

Cubic four 3-fold axes along the body diagonals

Tetragonal one 4-fold axis parallel to c, in the centre of ab

Orthorhombic three mirrors or three 2-fold axes

perpendicular to each other

Hexagonal one 6-fold axis down c

Trigonal (R) one 3-fold axis down the long diagonal

Monoclinic one 2-fold axis down the “unique” axis

Triclinic no symmetry

Essential symmetry is that which defines the crystal system (i.e. is unique to that shape).


Herman Mauguin symbol

Long notation •C 1 2/m 1 •P 21 21 2 •P 2/m 2/n 21/a •I 41/a 2/m 2/d •P 63/m 2/m 2/c •F 4/m (-3) 2/m

Short notation •C2/m •P21212 •Pmna •I41/amd •P63/mmc •Fm(-3)m

Rotation/screw axes + mirror/glide planes to them

Only mirror/glide planes (in their absence rotation/screw axes)*

*In monoclinic, tetragonal and hexagonal both rotation/screw axis and the mirror/glide plane for the primary direction only is retained in the short H.M. symbol!


Occurrence of crystal systems

Inorganic materials Organic materials

1. Cubic 2. Orthorhombic 3. Monoclinic

1. Monoclinic 2. Orthorhombic 3. Tetragonal 4. triclinic


Reduction in symmetry

Cubic Tetragonal

Three 4-fold axes One 4-fold axis

Four 3-fold axes No 3-fold axes

Six 2-fold axes Two 2-fold axes

Nine mirrors Five mirrors


cubic (1) tetragonal (2) distortion goes with an off-centre displacement of Ti4+

and the dipoles are pointing along c axis tetragonal BaTiO3 is ferroelectric

BaTiO3 (1) At temp. >120ᵒC : cubic perovskite structure (a=4.018Å) (2) At temp.< 120ᵒC : tetragonal structure (a=3.997Å, c=4.031 Å)

(1) (2)

c

Views on the [100] direction = a axis


crystal structure phase I: cubic, space group –Pm3m (paraelectric phase) phase II: tetragonal, space group – P4mm (ferroelectrical phase) phase III: orthorhombic, space group – Bmm2 (ferroelectric phase) phase IV: rhombohedral, space group – R3m (ferroelectric phase)

Changing the symmetry it changes the properties

KNbO3

tetragonal cubic hexagonal

435ᵒC 225ᵒC -10ᵒC

orthorhombic

cooling

tetragonal cubic rhombohedral

120ᵒC 5ᵒC -90ᵒC

orthorhombic

cooling

BaTiO3


General and special positions: Wyckoff positions

General position = no symmetry elements lying on it

the number of points in general position = the number of symmetry operations

Special position = symmetry element(s) lying on the position point

Ex: space group Pm: monoclinic with two mirror planes on b axis

Multiplicity Wyckoff Letter

Site Symmetry

Coordinates

2 c 1 (1) x,y,z (2) x,-y,z

1 b m x, ½ , z

1 a m x,0,z

Alphabetic order from bottom up (no physical meaning

Number of created

positions by symmetry

½

b

a

o


Generating crystal structure from crystallographic description

SrTiO3

Space group: ; a = 3.90 Å mPm3

Atom Wyckoff

sites Symmetry x y z

Ti 1a 0 0 0

Sr 1b 0.5 0.5 0.5

O 3d 4/m m m 0.5 0 0

mm3

mm3


Generating crystal structure from crystallographic description

SrTiO3

Space group: ; a = 3.90 Å mPm3

Atom Wyckoff

sites Symmetry x y z

Ti 1a 0 0 0

Sr 1b 0.5 0.5 0.5

O 3d 4/m m m 0.5 0 0

mm3

mm3

O: 0.5 0 0; 0 0.5 0; and 0 0 0.5

Z=1


Example of crystal with P4mm: PuS2

Lattice parameters: a = b = 3.943Å; c = 7.962Å

Multiplicity And

Wyckoff letter

Atom Site Symmetry

Coordinates

x y z

1a Pu1 4mm 0 0 0

1b Pu2 4mm ½ ½ 0.464

1a S1 4mm 0 0 0.367

1b S2 4mm ½ ½ 0.097

2c S3 2mm ½ 0 0.732

Coordinates

2c 2mm ( ½ , 0, z) and (0, ½ z)


Wyckoff positions of space group P4mm


Crystal structure of PuS2

S2 ( ½ , ½ 0.097)

S1 (0, 0, 0.367)

S2 ( ½ , 0, 0.732)

0, 0.367,1 0.732

0.097, ½


Atom Wyckoff position

x y z Occ

Co 2a 0 0 0 1

Na 4i 0.806(2) 0 0.491(2) 0.25

O 4i 0.3871(3) 0 0.1740(4) 1

What information is included in the space group C2/m?

C- side centered cell 2/m – 2-fold axis along b and mirror plane to it (or on b axis)

Ex: Na0.5CoO2 – monoclinic structure in space group C2/m with

a = 4.9043(2), b = 2.8275(1), c = 5.7097(3) and = 106.052(3)

Coordinates

(0,0,0) + (1/2,1/2,0) +

4i m (x,0,z) and (-x, 0, -z) 5/1/2013 75 L. Viciu| AC II | Symmetry in 3D

Wyckoff positions of space group C2/m [unique axis b]


Generating crystal structure of NaCl

NaCl: unit cell, a = 5.652Å; space group Fm3m

Multiplicity And

Wyckoff letter

Atom Site Symmetry

Coordinates

x y z

4a Cl m-3m 0 0 0

4b Na m-3m ½ ½ ½


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