ucsd nano106 - 03 - lattice directions and planes, reciprocal lattice and coordinate transformations

Lattice Directions and Planes, Reciprocal Lattice and Coordinate Transformations

Shyue Ping OngDepartment of NanoEngineeringUniversity of California, San Diego

Lattice planes and directions

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2

Readings¡Chapter 5 of Structure of Materials


Lattice Directions¡ Directions in a lattice is denoted by

¡ E.g., denotes the direction parallel to the a-axis in any lattice.

¡ Negative numbers are denoted with a bar above the number, e.g.,


uvw[ ]

100[ ]

112!" #$≡ a−b+ 2c

Lattice Planes¡ A lattice plane of a given Bravais lattice is a plane (or

family of parallel planes) whose intersections with the lattice are periodic (i.e., are described by 2d Bravaislattices) and intersect the Bravais lattice; equivalently, a lattice plane is any plane containing at least three noncollinear Bravais lattice points.


Miller indices¡ Lattice planes are represented by Miller indices, denoted

as , where h, k and l are integers. Note the use of the round brackets instead of the square brackets used for lattice directions.

¡ The procedure for determining the Miller indices of a plane is best illustrated using an example.


hkl( )

Procedure for determining the Miller indices

¡ Let’s say we have a plane in the lattice specified by a, b and c.

1. If the plane pass through the origin, displace the plane by an arbitrary amount so that it does not pass through the origin (not required for worked example).

2. Determine the intercepts of plane with three lattice vectors, in units of the lattice vector length. If the plane is parallel to one or more of the axes, this corresponding number is ∞. In the example, these are 1:2:3.

3. Invert all three numbers. If the plane is parallel to one or more of the axes, this corresponding number is 1/∞ = 0. For the example, we get 1:½:1/3.

4. Reduce the numbers to the nearest integers (known as the relative primes). We get (632).


a

b

c

2b

3c

Example¡Determine the Miller indices of the following

planes


ac

b

ac

b

0.5

¡ When the lattice has symmetry (i.e., non-triclinic), certain planes are equivalent to each other under symmetry. Such families of planes are represented with a curly brackets, i.e. {hkl}.

¡ Similarly, families of directions are denoted by <uvw>.

Families of planes or directions


{110} family of planes in a cubic lattice<111> family of directions in a cubic lattice

Permutations of Miller indices¡ From the cubic example, we may observe that all planes

in the {110} family is given by permutations of the indices and their negatives:

¡ For lower symmetry systems, families are still given by permutations, though not all permutations belong to the same family.¡ Rhombohedral:¡ Orthorhombic:


110{ }= 110( ), 110( ), 110( ), 101( ), 101( ), 101( ), 011( ), 011( ), 011( ){ }

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

Miller-Bravais Indices of Hexagonal Crystal System¡ Hexagonal system is defined by four basis vectors, three of which are

co-planar.

¡ Miller-Bravais indices are given by intercepts with all four basis vectors . i is a redundant index and is given by i = -(h+k)

¡ The four-index representation allows families of planes for hexagonal systems to be represented as permutations, e.g.,


1122( )

hkil( )

1120{ }= 1120( ), 1210( ), 2110( ), 1120( ), 1210( ), 2110( ){ }

Miller-Bravais Indices for Directions in Hexagonal System¡ Denoted as [uvtw].

¡ By convention, t = -(u+v), similar to indices for planes.

¡ It can be shown that the relationship between a three-Miller index [u’v’w’] and the corresponding four-Miller index [uvtw] is given by (please review proof on your own accord):


u = 13(2u '− v ')

v = 13(2v '−u ')

t = −(u+ v)w = w '

u ' = 2u+ vv ' = 2v+uw ' = w

u, v, t, w èu’, v’, w’ u’, v’, w’ èu, v, t, w

Examples¡Determine the Miller indices of the following

planes.


0.5

The reciprocal lattice


Readings¡Chapter 6 of Structure of Materials


Definition of the Reciprocal Lattice¡ For a lattice given by basis vectors a, b and c, the

reciprocal lattice is given basis vectors a*, b* and c* where:


a* = b× ca.(b× c)

b* = a× ca.(b× c)

c* = a×ba.(b× c)

Properties of the Reciprocal Lattice Basis Vectors1. a*.a = 1 (similarly for b* and c*)2. a* is perpendicular to both b and c,

i.e., a*.b = a*.c = 0 (similarly for b*and c*)

3. Using an alternative notation where a1, a2 and a3 represent the three lattice vectors,

ai*.aj=δijwhere δij is the Kronecker delta.

Important Note: In solid-state physics, the reciprocal lattice vectors have a factor of 2π.

Reciprocal Lattice and Lattice Planes¡ The reciprocal lattice is a lattice, just like the real space lattice.

¡ While real space vectors are represented by (u,v,w), reciprocal vectors are customarily represented by (h, k, l) (we will see the reason for this notation in a moment)

¡ Consider all real space vectors that are perpendicular to the reciprocal space vector above:


g = ha* + kb* + lc*

r = xa + yb+ zc

r.g = xa + yb+ zc( ). ha* + kb* + lc*( )= hx + ky+ lz = 0

Equation of plane passing through origin

Reciprocal Lattice and Lattice Planes, contd.¡ What is the equation of lattice planes with Miller indices (hkl)?

Remember that h, k and l are the reciprocals of the intercepts with the intercepts with the three axes:

¡ Key result: The reciprocal lattice vector g with components (h, k, l) is perpendicular to lattice planes with Miller indices (hkl) and is usually denoted as . The reciprocal lattice therefore lets us describe plane normals with simple integers.


ghkl

x1/ h

+y1/ k

+z1/ l

= N

Value of N determines distance of plane to origin

Relationship between reciprocal lattice vector length and interplanar spacing

¡ We have seen earlier that is perpendicular to (hkl).

¡ Consider an arbitrary point in the plane (hkl) given by vector t from the origin. The distance from the plane to the origin is given by:


ghkl

a

b

c

2b

3c

t

g632

dhkl =1ghkl

Inter-layer spacing is given by reciprocal of the reciprocal lattice vector length

Blackboard proof

Reciprocal lattice is just like any other lattice¡ You can similarly define a reciprocal metric tensor

¡ And distances between points in the reciprocal space is given by the same relations as the crystal lattice.

¡ It can also be shown that the reciprocal and crystal metric tensors are inverses of each other, i.e.,


g* =a*.a* a*.b* a*.c*

b*.a* b*.b* b*.c*

c*.a* c*.b* c*.c*

!

"

####

$

%

&&&&

g* = g−1

Coordinate Transformations


Coordinate transformations¡ Sometimes, we want to choose a different unit cell or set

of basis vectors for a lattice. For example, we may want to use the primitive basis vectors instead of the conventional one, or vice versa.

¡ How do we derive geometric quantities such as positions, lengths, etc. in the new basis?


Alternative notation for lattice vectors

¡Thus far, we have denoted the three lattice vectors as

¡We will here introduce an alternative indicial notation which will make it easier to represent certain kinds of operations, especially summations and matrix multiplications.


a,b,c

a1, a2, a3

Notation¡ Instead of using the conventional to represent basis vectors, let

us choose to represent them as .

¡ Let’s say the new basis vectors are . This is known as a change of reference frame.

¡ In general, the relationship between the new basis vectors and original lattice vectors can be represented by a linear transformation:


a, b, c{ }a1, a2, a3{ }

a1!1, a2!, a3!{ }

a1!1'

a!2

a!3

"

#

$$$$$

%

&

'''''

=

c11 c12 c13

c21 c22 c23

c31 c32 c33

"

#

$$$$

%

&

'''' a1

a2

a33

"

#

$$$$

%

&

'''' or

A' = CA

Note that in this case, the lattice vectors are written as rows in the matrix A and A’.

How do you determine the transformation matrix?¡ It can be shown that

¡Usually, it is far simpler to do an “inspection” to determine the matrix. Example, how do you express each ai’ in terms of the original vectors ai for the lattice shown here?


cij = ai!a j* Blackboard proof

Transformation Relations for Direct Positions / Vectors


Blackboard derivation

Transformation Relations for Metric Tensor


Blackboard derivation

Special case – the reciprocal lattice

¡What if our new coordinate system is the reciprocal lattice?

¡Recall that we proved that

¡We simply have


cij = ai!a j*

cij = ai*a j

* = gij* = gij

−1, i.e.,

C = g−1

p1* p2

* p3*( ) = p1 p2 p3( )

a1 ⋅a1 a1 ⋅a2 a1 ⋅a3a2 ⋅a1 a2 ⋅a2 a2 ⋅a3a3 ⋅a1 a3 ⋅a2 a3 ⋅a3

"

#

$$$$

%

&

''''

Summary of Coordinate Transformations


Transformation Relationship

Position/vector to position/vector

Position/vector to reciprocal position/vector

Reciprocal position/vector to Reciprocalposition/vector

p1' p2

' p3'( ) = p1 p2 p3( )C−1

p1 p2 p3( ) = p1' p2

' p3'( )C

p1*'

p2*'

p3*'

!

"

####

$

%

&&&&

= C

p1*

p2*

p3*

!

"

####

$

%

&&&&

p1*

p2*

p3*

!

"

####

$

%

&&&&

= C−1

p1*'

p2*'

p3*'

!

"

####

$

%

&&&&

p1* p2

* p3*( ) = p1 p2 p3( )g

p1 p2 p3( ) = p1* p2

* p3*( )g−1

Summary of Coordinate Transformations


Transformation Relationship

Metric Tensor g' =CgCT

g =C−1g' CT( )−1

Expected knowledge and tips¡ It is not expected that you memorize how to

derive all the coordinate transformation relations. The derivation was shown to give you a deeper appreciation of how it all works.

¡You only need to know how to apply the relations in doing transformations.

¡Be very very careful in noting whether it is a row x matrix or a matrix x column multiplication. The two are not the same!


Worked example: Conventional to Primitive Transformation of FCC¡ What is the relationship between the primitive basis

vectors and the conventional basis vectors?


Blackboard

Worked example: Conventional to Primitive Transformation of FCC¡ How does the [101] direction in the cubic frame

transform in the rhombohedral frame?

¡ How does the (110) plane in the cubic frame transform in the rhombohedral frame?


Blackboard

Worked example: C-centered Orthorhombic to Primitive¡ What is the relationship between the primitive basis

vectors and the conventional basis vectors?


Blackboard

Worked example: C-centered Orthorhombic to Primitive¡ How does the [111] direction in the conventional frame

transform in the primitive frame?

¡ How does the (110) plane in the conventional frame transform in the primitive frame?


Blackboard

ucsd nano106 - 03 - lattice directions and planes, reciprocal lattice and coordinate transformations

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