Structures and Reciprocal Lattice

Perovskite Structure and Derivatives

There are many ABO3 compounds for which the ideal cubic structure is distorted to a lower symmetry (e.g. tetragonal, orthorhombic, etc.)

SrTiO3 - Cubic Perovskites

General formula: ABO3 where A and B are cations. The easiest way to visualize the structure is in terms of the BO6 octahedra which share corners infinitely in all 3 dimensions.

The A cations occupy every hole which is created by 8 BO6 octahedra, giving the A cation a 12-fold oxygen coordination, and the B-cation a 6-fold oxygen coordination. In the example shown here, (SrTiO3) the Sr atoms sit in the 12 coordinate A site, while the Ti atoms occupy the 6 coordinate B site.

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Spinels

Stoichiometry = AM2X4Space Group = Fd3m (227)They take their name from MgAl2O4

The spinel structure is a rather complex arrangement based upon a cubic close packed anion array, with 1/2 of the octahedral sites filled and 1/8 of the tetrahedral sites filled. In each cell, there are 64 tetrahedral sites and 32 octahedral sites (the same as the number of oxygen per unit cell, that is, 32).

It is one of the most prevalent structure types for ternary oxides and sulfides.

A polyhedral based view of the structure reveals chains of edge-sharing octahedra, together with tetrahedra that are isolated from each other

A M X

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spinel

Spinels

Spinels can be classified as either normal or inverse, depending upon the cation distribution

Normal spinels are contain A cations on the tetrahedral sites and M cations on the octahedral sites, as depicted in the description below

In contrast, the tetrahedral sites in an inverse spinel are occupied by the M cations and the octahedral sites by a 50:50 mixture of A and M cations.

One of the most intensively studied spinel compounds is magnetite, Fe3O4. Magnetite is an inverse spinel, due to the fact that the tetrahedral sites are occupied by Fe3+, while the octahedral sites are occupied by 50:50 mixture of Fe2+ and Fe3+.

The structure of magnetite can be approximated as a cubic unit cell with composition (Fe3+)8[Fe2+1/2 Fe3+1/2]16 O32

Total n. of Fe atoms per cell = 8+16=24, Fe:O ratio = 24/32 = 3/4 Therefore we write Fe3O4

Crystallogrpahic computing

Vector in a coordinate system 0, , , :

X: coordinate matrix A: matrix of the basis vectors

scalar product between two vectors and

cross product between two vectors and

Some definitions:

product (rows by columns) between two matrixes and

transposed matrix of (i.e. exchange with )

0, , , a coordinate system in which the origin and three basis vectors are fixed

In a coordinate system 0, , , the scalar products between the basis vectors are:

cos cos cos

In a cartesian coordinate system (where 90:

0 0 0

In the case of crystal systems, were a, b, c represent the basis vectors of theunit cell, the above equalities ( 0, etc.) apply only to tetragonal,orthorombic and cubic systems!

5

Let us now consider the scalar product :

This can be written as:

In matrix notation:

G is the metric tensor. Its element define the moduli of a, b and c and the angles between them

cos

cos cos

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Some insights into the metric tensor G

In cartesian coordinates the metric tensor becomes:

0 00 00

And indeed, in cartesian coordinates:

The metric tensor G can be considered as a machine with two slots: one for a row vector, one for a colum vector. Enter these vectors in the slots, and you will get their scalar product!

row vector

column vector

Metric tensor Scalar product

= a scalar (i.e a number)!

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If , then

= 2cos 2cos 2cos

Which gives the square modulus of a vector

Make it a row vectorcolumn vector Fill the slots of the Metric tensor

= square modulus (i.e a number)!

In a diagramatic form: 1) take a column vector; 2) make a row vector from it; 3) enter both in the slots of G; 4) get the square modulus!

Remember: without a metric tensor, you cannot calculate things like scalar products or moluli!!!!

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More insights into the metric tensor G

cos cos

cos cos cos cos

The value of the determinant is:

det 1 cos cos cos 2coscoscos

One can demonstrate that is equal to , that is, the square volume of the celldefined by a, b and c

Scalar triple product:

Lets figure out a geometric meaning of this. Lets look at the cell with the b and c axis in the plane

sinb

c

sin

So, is a vector with modulus equal to the area of the A face of the cell. Its direction is perpendicular to the A face:

If we divide by its modulus, we get:

This is a unit vector with the same direction as

Nowletusconsider:

The volume V of the parallelepided (i.e. the unit cell) is then given by:

For a scalar triple product the following rule holds:

Which basically tells us that we can choose any of the faces (A, B, C) as the base for the parallelepid, and then consider the normal to it as the direction of the heigth. We will always get the same value for the volume of the cell.

This gives then the projection of the basis vector a on the direction defined by the unit vector . It is therefore the heigth of the parallelepidep with the face A as a base.

The reciprocal lattice

Let a, b, c be the elementary translations of a space lattice, which we call the direct lattice

We define now a second lattice, called the reciprocal lattice, which is defined by translations a*, b* and c* satisfying the following relations:

1

The first equation tells us that is normal to the plane , , is normal to the plane , . For example for :

The second equation fixes the modules of , and

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Since is normal to the plane , , it must be proportional to the cross product

1But then, since the following relation holds

This can be rewritten as

1

1

Therefore:

1

1

1

In terms of moduli:

1

sin

1

sin

1

sin

...And obviously the reciprocal of the reciprocal lattice is the direct lattice:

1

1

1

...But, what is it exacly a reciprocal lattice? Let us look at the simple case of an orthorombic lattice (Ratios a:b:c unrestricted, Angles = = =90)

1

sin

1

1

1

1

a

bc

a*=1/a

b*=1/bc*=1/c

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: volume of the unit cell in the reciprocal lattice

In other cases however, where the angles differ from 90, we would have something like this, for example:

a

bc

1

sin

1

sin

1

sin

Black dots: direct latticeYellow dots: its reciprocal lattice 15

It also exemplifies a lot the crystallographic calculations. Example:

Let us consider their scalar product, which takes a very simple form:

and are two vectors of the direct and reciprocal space, respectively

XX

X X

This property will be very useful in the theory of diffraction16

Consider the vector of the reciprocal space

Consider also the family of planes (hkl) in the direct space

We recall the plane of the family (hkl) closest to the origin:

A

B

C

N.

Consider the vectors B-A, C-A and C-B

A

B

C

B A

C A

C B

B A 0

C A 0

C B 0

is then perpendicular to the plane, therefore to the

(hkl) family 17

is perpendicular to the (hkl) family

N.

(hkl) family of planes

We have the noteworthy result that

Consider .This is the interplanar distance, or equivalently state the distance from the origin of the plane nearest to it.

1

1

can be calculated by the scalar product of with the unit vector defined along the direction of

, which is

. Therefore:

1

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How can we make use of this important relation?

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We can use them to build a reciprocal lattice starting from the direct lattice. Example:

Consider the direct lattice

b

0 a


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a

b

0

Take the 010 plane closest to the origin

010


21


b

0

Draw the vector of the reciprocal lattice normal to it

010

010

This identifies the 010 lattice point of the reciprocal lattice

a


22


b

0

Do the same for the (110) plane

110

010


110

a


23


b

0


010


110

210

210

a


24


b

0


010


110

310

210310

And now we start seeing a row in the reciprocal lattice....

a


25


b

0

Do the same for other planes....

010110

210310

010100

200300

a


26


a

b

0

010110

210310

Do the same for other planes....

0010

2030

010100

200300


27

So that the whole reciprocal lattice is built

a

b

0 a*

b* *

For a 2D lattice:b* aa* bV=absin

1

1

sin

1

1

sin

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An interactive example of a 2D reciprocal lattice

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Reciprocal lattice demo

Reciprocal lattice

Consider the following 2D lattice

a*

b*

...and its reciprocal lattice

b

0 a

Reciprocal lattice

Forget for a moment about the reciprocal lattice

b

0 a

Reciprocal lattice

Now add one lattice poit at the center of each cell in the 2D lattice

b

0 a

Reciprocal lattice

To stay with a primitive cell, we need to take a different choice for the lattice vectors, for instance:

0 a

b

Reciprocal lattice

However, this forces us to make a new choice also for the primitive cell of the reciprocal lattice

0 a

b

b*

For a 2D lattice:b* aa* bV=absin

1

1

sin

1

1

sin

a*

Reciprocal lattice

The reciprocal lattice is then....

0 a

b b*

a*

*

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A comparison

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FCC lattice

The primitive translational vectors of the FCC lattice are:

1

2

1

2

1

2

, , :orhogonal unit vectors (cartesian unit vectors)

Volume of the primitive cell:

1

4

Relationship between the fcc and the primitive rhombohedral unit cell

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BCC lattice

Relationship between the bcc and the primitive rhombohedral unit cell

1

2

1

2

1

2

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Let us calculate the primitive translational vector of the reciprocal lattice:

=

1

1

1

similarly:

1

1

1

2

1

2

1

2

Reciprocal lattice of the FCC lattice

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The primitive translational vectors of the lattice reciprocal to the FCC lattice are therefore:

1

1

1

As can be seen below, these (apart from a scale factor) are primitive translational vectors of a BCC lattice:

Therefore, the reciprocal of a FCC lattice is a BCC lattice!

Conversely, the reciprocal of a BCC lattice is a FCC lattice!

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Structures and Reciprocal Lattice