prof.p. ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/cmp/symmetry-structure1.pdfmiller indices - a...

P.Ravindran, PHY075- Condensed Matter Physics, Spring 2015: Structure & Symmetry in Solids

http://folk.uio.no/ravi/CMP2015

Prof.P. Ravindran, Department of Physics, Central University of Tamil

Nadu, India

Structure & Symmetry in Solids

1


Crystals

Atoms that are bound together, do so in a way that

minimizes their energy.

This most often leads to a periodic arrangement of the

atoms in space.

If the arrangement is purely periodic we say that it is

crystalline.

2


3


Write the prefix that represents each number below:

1

2

3

4

5

mono

di

tri

tetra

penta

6

7

8

9

10

hexa

hepta

octa

nona

deca

4


Symmetry Operations

Translational

Reflection at a plane

Rotation about an axis

– Inversion through a point

Glide (=reflection + translation)

Screw (=rotation + translation)

5


Rotations

6


Mirror Symmetry

7


Inversion

8


2D - Point Groups

9


3D Crystal Lattice10


3D Lattice

7 crystal systems

14 Bravais lattices

230 Space groups

11


32 point symmetries

– 2 triclinic

– 3 monoclinic

– 3 orthorhombic

– 7 tetragonal

– 5 cubic

– 5 trigonal

– 7 hexagonal

Point groups & Space Groups

12


13

Each of the unit cells of the 14 Bravais lattices has oneor more types of symmetry properties, such asinversion, reflection or rotation,etc.

SYMMETRY

INVERSION REFLECTION ROTATION

Elements Of Symmetry

P.Ravindran, PHY075- Condensed Matter Physics, Spring 2015: Structure & Symmetry in Solids 14

Lattice goes into itself through

Symmetry without translation

Operation Element

Inversion Point

Reflection Plane

Rotation Axis

Rotoinversion Axes


15

Inversion Center

A center of symmetry: A point at the center of the molecule.

(x,y,z) --> (-x,-y,-z)

Center of inversion can only be in a molecule. It is notnecessary to have an atom in the center (benzene, ethane).Tetrahedral, triangles, pentagons don't have a center ofinversion symmetry.

Mo(CO)6


16

Reflection Plane

A plane in a cell such that, when a mirror reflection in

this plane is performed, the cell remains invariant.


17

Examples

Triclinic has no reflection plane.

Monoclinic has one plane midway between andparallel to the bases, and so forth.


We can not find a lattice that goes into itself

under other rotations

• A single molecule can have any degree of rotational

symmetry, but an infinite periodic lattice – can not.

Rotation Symmetry


19

Rotation Axis

This is an axis such that, if the cell is rotated around itthrough some angles, the cell remains invariant.

The axis is called n-fold if the angle of rotation is 2π/n.

90°

120° 180°


Axis of Rotation20


Axis of Rotation


Can not be combined with translational periodicity!

5-fold symmetry


23

5-fold symmetry

Kepler wondered why snowflakes have 6 corners,never 5 or 7.By considering the packing of polygons in2 dimensions, demonstrate why pentagons andheptagons shouldn’t occur.

Empty space not

allowed


24

90°

Examples

Triclinic has no axis of rotation.

Monoclinic has 2-fold axis (θ= 2π/2 =π) normal to the

base.


Crystallographic Points, Directions, and

Planes

It is necessary to specify a particular

point/location/atom/direction/plane in a unit cell

We need some labeling convention. Simplest way is to

use a 3-D system, where every location can be

expressed using three numbers or indices.

– a, b, c and α, β, γ

x

y

z

βα

γ

25


ISSUES TO ADDRESS...

• How do atoms assemble into solid structures?

(for now, focus on metals)

• How does the density of a material depend on

its structure?

• When do material properties vary with the

sample (i.e., part) orientation?

1

Why do we care about crystal structures,

directions, planes ?

Physical properties of materials depend on the geometry of crystals

26


Crystallographic Points, Directions, and Planes

Crystallographic direction is a vector [uvw]

– Always passes thru origin 000

– Measured in terms of unit cell dimensions a, b, and c

– Smallest integer values

Planes with Miller Indices (hkl)

– If plane passes thru origin, translate

– Length of each planar intercept in terms of the lattice parameters a, b, and c.

– Reciprocals are taken

– If needed multiply by a common factor for integer representation

27


Miller indices - A shorthand notation to describe certain

crystallographic directions and planes in a material.

Denoted by [ ], <>, ( ) brackets. A negative number is

represented by a bar over the number.

Points, Directions and Planes in the Unit

Cell

28


• Coordinates of selected points in the unit cell.

• The number refers to the distance from the origin in terms of

lattice parameters.

Point Coordinates29


30

Point Coordinates – Atom position

Point coordinates for unit cell center are

a/2, b/2, c/2 ½ ½ ½

Point coordinates for unit cell corner are 111

Translation: integer multiple of lattice constants identical position in another unit cell

z

x

y

a b

c

000

111

y

z

2c

b

b


Crystallographic Directions

1. Vector repositioned (if necessary) to

pass

through origin.

2. Read off projections in terms of

unit cell dimensions a, b, and c

3. Adjust to smallest integer values

4. Enclose in square brackets, no commas

[uvw]

ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]

-1, 1, 1

families of directions <uvw>

z

x

Algorithm

where overbar represents a negative index[ 111 ]=>

y

31


Crystal Directions

Fig. Shows

[111] direction

We choose one lattice point on the line as anorigin, say the point O. Choice of origin iscompletely arbitrary, since every lattice pointis identical.

Then we choose the lattice vector joining O toany point on the line, say point T. This vectorcan be written as;

R = n1 a + n2 b + n3c

To distinguish a lattice direction from a latticepoint, the triple is enclosed in square brackets[ ...] is used.[n1n2n3]

[n1n2n3] is the smallest integer of the samerelative ratios.


For some crystal structures, several nonparallel

directions with different indices are

crystallographically equivalent; this means that

atom spacing along each direction is the same.

33

Families of Directions <uvw>


Crystal Directions

]100[],010[],001[],001[],010[],100[ 100

34


Miller Index

Step 1 : Identify the intercepts on the x- , y- and z- axes.

Step 2 : Specify the intercepts in fractional co-ordinates

Step 3 : Take the reciprocals of the fractional intercepts

(i) in some instances the Miller indices are best multiplied or divided

through by a common number in order to simplify them by, for

example, removing a common factor. This operation of multiplication

simply generates a parallel plane which is at a different distance from

the origin of the particular unit cell being considered.

e.g. (200) is transformed to (100) by dividing through by 2 .

(ii) if any of the intercepts are at negative values on the axes then the

negative sign will carry through into the Miller indices; in such cases

the negative sign is actually denoted by overstriking the relevant

number.

e.g. (00 -1) is instead denoted by 100

35


Miller index is used to describe directions and planes in a

crystal.

Directions - written as [u v w] where u, v, w.

Integers u, v, w represent coordinates of the vector in

real space.

A family of directions which are equivalent due to

symmetry operations is written as <u v w>

Planes: Written as (h k l).

Integers h, k, and l represent the intercept of the plane

with x-, y-, and z- axes, respectively.

Equivalent planes represented by {h k l}.

Miller Index For Cubic Structures36


x y z

[1] Draw a vector and take components 0 2a 2a

[2] Reduce to simplest integers 0 1 1

[3] Enclose the number in square brackets [0 1 1]

z

y

x

Miller Indices: Directions37


z

y

x

x y z

[1] Draw a vector and take components 0 -a 2a

[2] Reduce to simplest integers 0 -1 2

[3] Enclose the number in square brackets 210

Negative Directions38


Miller Indices: Equivalent Directions

z

y

x

1

2

3

1: [100]

2: [010]

3: [001]

Equivalent directions due to crystal symmetry:

Notation <100> used to denote all directions equivalent to [100]

39


Directions

40


41

210

X = 1 , Y = ½ , Z = 0

[1 ½ 0] [2 1 0]

X = ½ , Y = ½ , Z = 1

[½ ½ 1] [1 1 2]

Examples


42

Negative directions

When we write the direction

[n1n2n3] depend on the origin,

negative directions can be

written as

R = n1 a + n2 b + n3c

Direction must be

smallest integers.

Y direction

(origin) O

- Y direction

X direction

- X direction

Z direction

- Z direction

][ 321 nnn

][ 321 nnn


Directions of the form <110> in cubic

systems

43


44

X = -1 , Y = -1 , Z = 0 [110]

Examples of crystal directions

X = 1 , Y = 0 , Z = 0 [1 0 0]


45

Examples

X =-1 , Y = 1 , Z = -1/6

[-1 1 -1/6] [6 6 1]

We can move vector to the origin.


(A) Determining crystal structure

Diffraction methods directly measure the distance between parallel planes of lattice

points. This information is used to determine the lattice parameters in a crystal

and measure the angles between lattice planes.

(B) Plastic deformation

Plastic (permanent) deformation in metals occurs by the slip of atoms past each

other in the crystal. This slip tends to occur preferentially along specific lattice

planes in the crystal. Which planes slip depends on the crystal structure of the

material.

(C) Transport Properties

In certain materials, the atomic structure in certain planes causes the transport of

electrons and/or heat to be particularly rapid in that plane, and relatively slow away

from the plane.

Example: Graphite

Conduction of heat is more rapid in the sp2 covalently bonded lattice planes than in

the direction perpendicular to those planes.

Example: YBa2Cu3O7 superconductors

Some lattice planes contain only Cu and O. These planes conduct pairs of

electrons (called Cooper pairs) that are responsible for superconductivity. These

superconductors are electrically insulating in directions perpendicular to the Cu-O

lattice planes.

Why are planes in a lattice important? 46


Atomic planes in BCC

{001} {011} {111}

47


Atomic planes in FCC

{001} {011} {111}

48


(GPa)

Anisotropy49


50

Crystal Planes

Within a crystal lattice it is possible to identify sets of

equally spaced parallel planes. These are called lattice

planes.

In the figure density of lattice points on each plane of a set

is the same and all lattice points are contained on each set

of planes.

b

a

b

a

The set of

planes in

2D lattice.


51

Miller Indices of plane

Miller Indices are a symbolic vector representation for the orientationof an atomic plane in a crystal lattice and are defined as thereciprocals of the fractional intercepts which the plane makes with thecrystallographic axes.

To determine Miller indices of a plane, take the following steps;

1) Determine the intercepts of the plane along each of the three crystallographic directions

2) Take the reciprocals of the intercepts

3) If fractions result, multiply each by the denominator of the smallest fraction


The intercepts of a crystal plane with the axis defined by a set of

unit vectors are at 2a, -3b and 4c. Find the corresponding Miller

indices of this and all other crystal planes parallel to this plane.

The Miller indices are obtained in the following three steps:

1. Identify the intersections with the axis, namely 2, -3 and

4.

2. Calculate the inverse of each of those intercepts,

resulting in 1/2, -1/3 and 1/4.

3. Find the smallest integers proportional to the inverse of

the intercepts. Multiplying each fraction with the

product of each of the intercepts (24 = 2 x 3 x 4) does

result in integers, but not always the smallest integers.

4. These are obtained in this case by multiplying each

fraction by 12.

5. Resulting Miller indices is

6. Negative index indicated by a bar on top. 346

52


Miller Indices of plane

Reciprocal numbers are: 2

1 ,

2

1 ,

3

1

Plane intercepts axes at cba 2 ,2 ,3

Indices of the plane (Miller): (2,3,3)

(100)

(200)

(110)(111)

(100)

Indices of the direction: [2,3,3]a

3

2

2

bc

[2,3,3]


55

Family of Planes

Planes that are crystallographically equivalent have the

same atomic packing.

Also, in cubic systems only, planes having the same indices,

regardless of order and sign, are equivalent.

Ex: {111}

= (111), (111), (111), (111), (111), (111), (111), (111)

(001)(010), (100), (010),(001),Ex: {100} = (100),

_ __ __ _ __ _ _ __


56

Axis X Y Z

Intercept

points 1 ∞ ∞

Reciprocals 1/1 1/ ∞ 1/ ∞Smallest

Ratio 1 0 0

Miller İndices (100)

Example-1

(1,0,0)


Axis X Y Z

Intercept

points 1 1 ∞

Reciprocals 1/1 1/ 1 1/ ∞Smallest

Ratio 1 1 0


Example-2

(1,0,0)

(0,1,0)

57


Axis X Y Z

Intercept

points 1 1 1

Reciprocals 1/1 1/ 1 1/ 1Smallest

Ratio 1 1 1

Miller İndices (111)(1,0,0)

(0,1,0)

(0,0,1)

Example-3


Axis X Y Z

Intercept

points 1/2 1 ∞

Reciprocals 1/(½) 1/ 1 1/ ∞Smallest

Ratio 2 1 0

Miller İndices (210)(1/2, 0, 0)

(0,1,0)

Example-4


Axis a b c

Intercept

points 1 ∞ ½

Reciprocals 1/1 1/ ∞ 1/(½)

Smallest

Ratio 1 0 2


Example-5


Axis a b c

Intercept

points -1 ∞ ½

Reciprocals 1/-1 1/ ∞ 1/(½)

Smallest

Ratio -1 0 2


Example-6


Example-7


63

Indices of a Family or Form

Sometimes when the unit cell has rotational symmetry,

several nonparallel planes may be equivalent by virtue of

this symmetry, in which case it is convenient to lump all

these planes in the same Miller Indices, but with curly

brackets.

Thus indices {h,k,l} represent all the planes equivalent to

the plane (hkl) through rotational symmetry.

)111(),111(),111(),111(),111(),111(),111(),111(}111{

)001(),100(),010(),001(),010(),100(}100{


z

y

x

z=

y=

x=a

x y z

[1] Determine intercept of plane with each axis a ∞ ∞

[2] Invert the intercept values 1/a 1/∞ 1/∞

[3] Convert to the smallest integers 1 0 0

[4] Enclose the number in round brackets (1 0 0)

Miller Indices of Planes64


z

y

x

x y z

[1] Determine intercept of plane with each axis 2a 2a 2a

[2] Invert the intercept values 1/2a 1/2a 1/2a

[3] Convert to the smallest integers 1 1 1

[4] Enclose the number in round brackets (1 1 1)

Miller Indices of Planes

65


z

y

x

Planes with Negative Indices

x y z

[1] Determine intercept of plane with each axis a -a a

[2] Invert the intercept values 1/a -1/a 1/a

[3] Convert to the smallest integers 1 -1 1

[4] Enclose the number in round brackets 111

66


• Planes (100), (010), (001), (100), (010), (001) are

equivalent planes. Denoted by {1 0 0}.

• Atomic density and arrangement as well as electrical,

optical, physical properties are also equivalent.

z

y

x

(100)

plane

(010)

plane

(001) planeEquivalent Planes

67


68


(0 1 1)

69


The (110) surface

Assignment

Intercepts : a , a ,

Fractional intercepts : 1 , 1 ,

Miller Indices : (110)

The (100), (110) and (111) surfaces considered above are

the so-called low index surfaces of a cubic crystal system

(the "low" refers to the Miller indices being small numbers -

0 or 1 in this case).

70


Miller Indices (hkl)

reciprocals

Crystallographic Planes71


Crystallographic Planes

_

(1 1 1)

72


In the cubic system the (hkl) plane and the vector

[hkl], defined in the normal fashion with respect

to the origin, are normal to one another but this

characteristic is unique to the cubic crystal

system and does not apply to crystal systems of

lower symmetry

73


Symmetry-equivalent surfaces

The three highlighted surfaces are

related by the symmetry elements of

the cubic crystal - they are entirely

equivalent.

In fact there are a total of 6 faces related by the symmetry

elements and equivalent to the (100) surface - any surface

belonging to this set of symmetry related surfaces may be

denoted by the more general notation {100} where the Miller

indices of one of the surfaces is instead enclosed in curly-

brackets.

74


In case of a cubic structure, the Miller index of a plane, in

parentheses such as (100), are also the coordinates of the direction

of a plane normal. It stands for a vector perpendicular to the family

of planes, with a length of d-1, where d is the inter-plane spacing.

Due to the symmetries of cubic crystals, it is possible to change the

place and sign of the integers and have equivalent directions and

planes:

•Coordinates in angle brackets or chevrons such as <100>

denote a family of directions which are equivalent due to

symmetry operations. If it refers to a cubic system, this

example could mean [100], [010], [001] or the negative of any

of those directions.

•Coordinates in curly brackets or braces such as {100} denote

a family of plane normals which are equivalent due to

symmetry operations, much the way angle brackets denote a

family of directions.

75


Distance between two planes (d spacing)76


In the cubic crystal system, a plane and the direction normal to

it have the same indices.

[101] direction is normal to the plane (101)

Distance (d) separating adjacent planes

(hkl) of a cubic crystal of lattice constant

(a) is:

222 lkh

ad

Angle () between

directions [h1 k1 l1]

and [h2 k2 l2] of a

cubic crystal is:

))(()cos(

2

2

2

2

2

2

2

1

2

1

2

1

212121

lkhlkh

llkkhh

77


d spacing

2

2

2

2

2

2

c

l

b

k

a

h

ndhkl

Example

The lattice constant for aluminum is 4.041

angstroms. What is d220?

Answer

Aluminum has an fcc structure, so a = b = 4.041 Å

angstroms 43.1

041.4

22

11

2

22

2

22

a

kh

dhkl

78


Miller-Bravais Indices

• For hcp crystal structure

• Planes (h k i l), directions [h k i l]

• Sum of the first three indices h + k + i = 0

79


With hexagonal and rhombohedral crystal systems, it is possible

to use the Bravais-Miller index which has 4 numbers (h k i l)

i = -h-k

where h, k and l are identical to the Miller index.

The (100) plane has a 3-fold symmetry, it remains unchanged by a

rotation of 1/3 (2π/3 rad, 30°). The [100], [010] and

the directions are similar. If S is the intercept of the plane

with the axis, then

i = 1/S

i is redundant and not necessary.

80


In the hcp crystal system there are four principal axes; this leads

to four Miller Indices e.g. you may see articles referring to an

hcp (0001) surface. It is worth noting, however, that the

intercepts on the first three axes are necessarily related and not

completely independent; consequently the values of the first

three Miller indices are also linked by a simple mathematical

relationship.

81


To determine the Miller-Bravais indices of

the crystallographic direction indicated on

the a1-a2-a3 plane of an h.c.p. unit cell (ie.

c = 0), the vector must be projected onto

each of the three axes to find the

corresponding component, such that:

82


83


ex: linear density of Al in [110]

direction

a = 0.405 nm

Linear Density

Linear Density of Atoms LD =

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD -==

84


Crystallographic Planes

We want to examine the atomic packing of

crystallographic planes

Iron foil can be used as a catalyst. The atomic

packing of the exposed planes is important.

a) Draw (100) and (111) crystallographic planes

for Fe.

b) Calculate the planar density for each of these

planes.

85


86

Planar Density of (100) Iron

Solution: At T < 912ºC iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R3

34a

2D repeat unit

= Planar Density =

a 2

1

atoms

2D repeat unit

= nm2

atoms12.1

m2

atoms= 1.2 x 1019

1

2

R3

34area

2D repeat unit


87

Planar Density of (111) Iron

Solution (cont): (111) plane1 atom in plane/ unit surface cell

333

2

2

R3

16R

3

42a3ah2area

atoms in plane

atoms above plane

atoms below plane

ah2

3

a2

1= =

nm2

atoms7.0

m2

atoms0.70 x 1019

32R

3

16

Planar Density =

atoms

2D repeat unit

area

2D repeat unit


88

HCP Crystallographic Directions

Hexagonal Crystals

– 4 parameter Miller-Bravais lattice coordinates are

related to the direction indices (i.e., u'v'w') as

follows.

'ww

t

v

u

)vu( +-

)'u'v2(3

1-

)'v'u2(3

1-

]uvtw[]'w'v'u[

-

a3

a1

a2

z


89

HCP Crystallographic Directions

1. Vector repositioned (if necessary) to

pass through origin.

2. Read off projections in terms of unit

cell dimensions a1, a2, a3, or c

3. Adjust to smallest integer values

4. Enclose in square brackets, no

commas [uvtw]

[ 1120 ]ex: ½, ½, -1, 0 =>

dashed red lines indicate

projections onto a1 and a2 axesa1

a2

a3

-a32a2

2a1

-a3

a1

a2

z Algorithm


Procedure for finding Miller indices for hcp lattice in four-index notation:

1. Find the intersections, r and s, of the plane with any two of the basal plane axes.

2. Find the intersection t of the plane with the c axis.

3. Evaluate the reciprocals 1/r, 1/s, and 1/t.

4. Convert the reciprocals to smallest set of integers that are in the same ratio.

5. Use the relation i = -(h+ k), where h is associated with a1, k is associated with a2,

and i is associated with a3.

6. Enclose all four indices in parentheses: (h k i l)

90


Example: What is the designation, using four

Miller indices, of the lattice plane shaded pink

in the figure?

• The plane intercepts the a1, a3, and c axes at r =

1, u = 1, and t = ∞, respectively.

• The reciprocals are

•1/r = 1

•1/u = 1

•1/t = 0

• These are already in integer form.

• We can write down the Miller indices as (h k i

l) = (1 k 1 0), where the k index is undetermined

so far.

• Use i = (h + k). That is, k = 2.

• The designation of this plane is (h k i l) =

)1021(

91


HCP and FCC structures

The hexagonal-closed packed (HCP) and FCC structures both have the ideal packing fraction of 0.74 (Kepler figured this out hundreds of years ago)

The ideal ratio of c/a for hcppacking is (8/3)1/2 = 1.633

Crystal c/a

He 1.633

Be 1.581

Mg 1.623

Ti 1.586

Zn 1.861

Cd 1.886

Co 1.622

Y 1.570

Zr 1.594

Gd 1.592

Lu 1.586

92

P.Ravindran, PHY075- Condensed Matter Physics, Spring 2015: Structure & Symmetry in SolidsChapter 3: The Structure of Crystalline Solids 93

Crystallographic Planes continue.....

Hexagonal Crystal system

prof.p. ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/cmp/symmetry-structure1.pdfmiller indices - a...

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