Reciprocal Lattice

• Points to discuss • Reciprocal lattice • Definition and examples • First Brillouin Zone • Lattice planes • Miller Indices Reciprocal Lattice

• A diffraction pattern is not a direct representation of the crystal lattice • The diffraction pattern is a representation of the reciprocal lattice. • But what is a reciprocal lattice? Reciprocal Lattice

• For every real lattice there is an equivalent reciprocal lattice. A two dimension (2‐D) real lattice is defined by two unit cell vectors, say and inclined at an angle. The equivalent reciprocal lattice in reciprocal space is defined by two reciprocal vectors, say and . • Each point in the reciprocal lattice represents a set of planes. Reciprocal Lattice

• The set of all wave vectors that yield plane waves with the periodicity of a given Bravais lattice is known as its reciprocal lattice. • Analytically, belongs to the reciprocal lattice of a Bravais lattice of points , provided that the relation holds for any and for all in the Bravais lattice. • Factoring out we can characterize the reciprocal lattice as the set of wave vectors satisfying =1 for all in Bravais lattice.

Reciprocal Lattice

• The Bravais lattice that determines a given reciprocal lattice is often referred to as the direct lattice when viewed in relation to its reciprocal lattice. • Reciprocal lattice is a Bravais lattice. We shall prove it in next slides. • Now question is that how reciprocal lattice vectors can be chosen. Reciprocal Lattice vectors in 2D

Reciprocal lattice Real lattice b1 a1

a2 b2

Reciprocal lattice vectors can also be denoted by a*,b* etc. or sometime by g1, g2 etc. Defining the reciprocal lattice vector

Take two sets of 2D planes: Draw directions normal: These lines define the

orientation but not the length a1 b1

1 a2 We use to define the lengths d b2 These are called reciprocal lattice vectors b1 and b2 Dimensions = 1/length Reciprocal Lattice vectors • The reciprocal vectors are defined as follows:

• is of magnitude 1/d1 where d1 is the spacing of the vertical planes, and is perpendicular to .

• is of magnitude 1/d2 where d2 is the spacing of the horizontal planes, and is perpendicular to . • A reciprocal lattice can be built using reciprocal vectors. Both the real and reciprocal constructions show the same lattice, using different but equivalent descriptions. • Consider the following animation.

The animation can be envisaged as follows:

• The real lattice is described at the left, the reciprocal lattice is described at the right. • g is the reciprocal lattice vector. • The absolute value of g is equal to and the direction is that of the normal N to the appropriate set of parallel atomic planes of the real lattice separated by distance d. The animation can be envisaged as follows: • Imagine yourself to be an atom. • When you fly around the crystal formed from two parallel 7 x 7 planes, you would see the picture at the bottom. • If someone watches you from above and tries to figure out what you are seeing, he would draw the pictures at the top of the screen. • One picture is a the real image, the other is an imaginary picture. Reciprocal lattice vectors

• From examples it is obvious that for 3D .

i.e. is perpendicular to both and . Similarly is perpendicular to both and and is perpendicular to both and

. Reciprocal lattice vectors

• The cross product defines a vector parallel to with modulus of the area defined by and . • The volume of the unit cell is thus given by

• We can define the reciprocal lattice vectors , and in terms of direct lattice vectors , and as follows.

Reciprocal lattice vectors Reciprocal Lattice is a Bravais Lattice

• Reciprocal lattice vectors satisfy,

Where

can be written as a linear combination of Reciprocal Lattice is a Bravais Lattice

• If is any direct lattice vector, then

and thus,

Since is an integer, are also integers. Thus is times an integer. Thus reciprocal lattice is a Bravais lattice. The Reciprocal Of The Reciprocal Lattice • Since Reciprocal lattice is itself a Bravais lattice, one can construct its reciprocal lattice. • Let reciprocal of the reciprocal lattice is the set of all vectors G satisfying for all in the reciprocal lattice. • Since any direct lattice vector has this property, all direct lattice vectors are in the lattice reciprocal to the reciprocal lattice. The Reciprocal Of The Reciprocal Lattice • No other vectors can be, for a vector not in the direct lattice has the form,

with at least one non-integral for that value of , and the above condition is violated for the reciprocal lattice vectors . Reciprocal Lattice Of Simple Cubic Bravais Lattice • The simple cubic Bravais lattice with cubic primitive cell of side has as its reciprocal lattice a simple cubic lattice with cubic primitive cell of side . Where is the crystallographer's definition. • The cubic lattice is therefore said to be self‐ dual, having the same symmetry in reciprocal space as in real space. Reciprocal Lattice Of SC Bravais Lattice

If then Reciprocal Lattice Of fcc Bravais Lattice

• The reciprocal lattice to a fcc lattice is the bcc lattice. • Consider a fcc compound unit cell. • Locate a primitive unit cell of the fcc, i.e., a unit cell with one lattice point. • Take one of the vertices of the primitive unit cell as the origin. • Give the basis vectors of the real lattice. Reciprocal Lattice Of fcc Bravais Lattice • Then from the known formulae you can calculate the basis vectors of the reciprocal lattice. • These reciprocal lattice vectors of the fcc represent the basis vectors of a bcc real lattice. • Note that the basis vectors of a real bcc lattice and the reciprocal lattice of an fcc resemble each other in direction but not in magnitude. Reciprocal Lattice Of fcc Bravais Lattice

• The fcc Bravais lattice with conventional cubic cell of side has as its reciprocal a bcc lattice with conventional cubic cell of side . i.e. This has precisely the form of the bcc primitive vectors provided that the side of the cubic cell is taken to be . Reciprocal Lattice Of bcc Bravais Lattice • The reciprocal lattice to a bcc lattice is the fcc lattice. • Only the Bravais lattices which have 90 degrees between (cubic, tetragonal, orthorhombic) have parallel to their real‐space vectors. Reciprocal Lattice Of bcc Bravais Lattice • The bcc Bravais lattice with conventional cubic cell of side has as its reciprocal a fcc lattice with conventional cubic cell of side . i.e. • Reciprocal of bcc is fcc and reciprocal of fcc is bcc this proves that the reciprocal of the reciprocal is the original lattice. Reciprocal Of Simple Hexagonal Bravais lattice • A simple Hexagonal Bravais lattice with lattice constants c and a has its reciprocal another simple Hexagonal lattice with lattice constants and rotated through about the c-axis with respect to the direct lattice. Reciprocal Of Simple Hexagonal Bravais lattice

Reciprocal Lattices to SC

Direct lattice Reciprocal lattice Volume

 /2 a3 Reciprocal Lattices to fcc

Direct lattice Reciprocal lattice

Volume  /22 a3 Reciprocal Lattices to bcc

Direct lattice Reciprocal lattice

Volume

 /24 a3 Volume Of The Reciprocal Lattice Primitive Cell • If is the volume of the primitive cell with side in the direct lattice the primitive cell of the reciprocal lattice has a volume . • The simple cubic Bravais lattice, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side ( in the crystallographer's definition). The cubic lattice is therefore said to be self‐ dual, having the same symmetry in reciprocal space as in real space. Brillouin zone

• In the propagation of any type of wave motion through a crystal lattice, the frequency is a periodic function of wave vector k. • In order to simplify the treatment of wave motion in a crystal, a zone in k‐space is defined which forms the fundamental periodic region, such that the frequency or energy for a k outside this region may be determined from one of those in it. Brillouin zone

• This region is known as the Brillouin zone sometimes called the first or the central Brillouin zone. • It is usually possible to restrict attention to k values inside the zone. • Discontinuities occur only on the boundaries. • The central Brillouin zone for a particular solid type is a solid which has the same volume as the primitive unit cell in reciprocal space. Construction of first Brillouin zone

Draw lines connecting the origin point to its nearest neighbors.

Draw perpendicular bisectors to these lines. These perpendicular bisectors are Bragg Planes.

Taking the smallest polyhedron containing the point bounded by these planes is first Brillouin zone.. First Brillouin Zone

Higher Brillouin Zones • The second Brillouin zone is the set of points that can be reached from the first zone by crossing only one Bragg plane. • The (n + 1)th Brillouin zone is the set of points not in the (n ‐ 1)th zone that can be reached from the nth zone by crossing n ‐ 1 Bragg planes. • The nth Brillouin zone can be defined as the set of points that can be reached from the origin by crossing n ‐ 1 Bragg planes, but no fewer. The locus of points in reciprocal space that have no Bragg Planes between them and the origin defines the first Brillouin Zone. It is equivalent to the Wigner‐Seitz unit cell of the reciprocal lattice. Small black dots represent point of intersection of Bragg planes The second Brillouin Zone is the region of reciprocal space in which a point has one Bragg Plane between it and the origin. This area is shaded yellow in the picture below. Note that the areas of the first and second Brillouin Zones are the same. Small black dots represent point of intersection of Bragg planes The construction can quite rapidly become complicated as you move beyond the first few zones, and it is important to be systematic so as to avoid missing out important Bragg Planes. Small black dots represent point of intersection of Bragg planes http://www.doitpoms.ac.uk/tlplib/brillouin_zones/printall.php First Brillouin zone of bcc lattice • The reciprocal of the bcc lattice is the fcc lattice. The first Brillouin zone of the bcc lattice is just the fcc Wigner Seitz cell.

First Brillouin zone of bcc lattice

First Brillouin zone of fcc lattice

• The first Brillouin zone of the fcc lattice is just the bcc Wigner Seitz cell. First Three Brillouin Zones Of bcc and fcc lattices Lattice planes

• Any plane containing at least three non‐ collinear Bravais lattice points. • Because of the translational symmetry of the Bravais lattice, any such plane will actually contain infinitely many lattice points which form a 2D Bravais lattice within the plane.

Lattice Planes Of simple Cubic Bravais Lattice Family Of Lattice Planes

• A set of parallel equally spaced lattice planes, which together contains all the points of the three dimensional Bravais lattice. • Any lattice plane is a member of such family. • Resolution of a Bravais lattice into a family of lattice planes is not unique. Theorem of possible families of lattice planes • For any family of lattice planes separated by a distance d, there are reciprocal lattice vectors perpendicular to the planes the shortest of which have a length of . • Conversely, for any reciprocal lattice vector there is a family of lattice planes normal to and separated by a distance d, where is the length of the shortest reciprocal lattice vector parallel to . Proof Of First Part Of Theorem • Given a family of lattice planes. • be a unit vector normal to the planes. • is a reciprocal lattice vector. • The plane wave is constant in planes perpendicular to and has the same value in planes separated by . • One of the lattice planes contains the Bravais lattice point , must be unity for any point r in any of the planes. Proof Of First Part Of Theorem • The planes contain all Bravais lattice points =1 for all , so that is indeed a reciprocal lattice vector. is the shortest reciprocal lattice vector normal to the planes. • For any wave vector shorter than will give a plane wave with wave length greater than . • Such a plane wave cannot have the same value on all planes in the family, and cannot give a plane wave that is unity at all Bravais lattice points.

Proof Of Second Part Of Theorem • Given a reciprocal lattice vector. • Let be the shortest parallel reciprocal lattice vector. • Consider a set of real space planes on which the plane wave has the value unity. • These planes are perpendicular to and separated by a distance . • All Bravais lattice vectors satisfy for any reciprocal lattice vector they must all lie within these planes. Proof Of Second Part Of Theorem • The spacing between the lattice planes is also d but not an integral multiple of d, for if only every nth plane in the family contained Bravais lattice points. • Then according to the first part of the theorem, the vector normal to the planes of length i.e. the vector , would be a reciprocal lattice vector. • This would contradict our original assumption that no reciprocal lattice vector parallel to is shorter than . Miller Indices Of lattice Planes • The correspondence between reciprocal lattice vectors and families of lattice planes provides a convenient way to specify the orientation of a lattice plane. • In general we describe the orientation of a lattice plane by giving a vector normal to that plane. • There are reciprocal lattice vectors normal to any family of planes, we pick a reciprocal lattice vector to represent the normal. Miller Indices Of lattice Planes • To make the choice we should use the shortest such reciprocal lattice vector. In this way we arrive at the Miller indices of the plane. • Miller indices of a lattice plane are the coordinates of the shortest reciprocal lattice vectors normal to that plane, with respect to the specified set of primitive reciprocal lattice vectors. • A plane with Miller indices is normal to the reciprocal lattice vector . • are integers. • They have no common factor. • They depend on particular choice of primitive vectors. Miller Indices Of lattice Planes • A set of three integers that designate crystallographic planes, as determined from reciprocals of fractional axial intercepts. • In any kind of repeating pattern, it is useful to have a convenient way of specifying the orientation of elements relative to the unit cell. This is done by assigning to each such element a set of integer numbers known as its Miller index. Miller Indices Of Cubic Bravais Lattice • In simple cubic Bravais lattice the reciprocal lattice is also simple cubic and the Miller indices are the coordinates of a vector normal to the plane in the obvious cubic coordinate system. • Fcc and bcc Bravais lattices are described in terms of a conventional cubic cell .Any lattice plane in a fcc or bcc lattice is also a lattice plane in the underlying simple cubic lattice, the same elementary cubic indexing can be used to specify lattice planes. Miller Indices Of Cubic Bravais Lattice • It is only in the description of non‐cubic crystal that we must remember that the Miller indices are the coordinates of the normal in a system given by the reciprocal lattice, rather than the direct lattice. • A lattice plane with Miller indices is perpendicular to the reciprocal lattice vector = , it will be contained in the continuous plane for suitable choice of the constant A. Miller Indices • This plane intersects the axis determined by the direct lattice primitive vectors , at the points and , where the are determined by the coordination that indeed satisfy the equation of the plane . • Since and , it follows that, Miller Indices • Intercepts with the crystal axes of a lattice plane are inversely proportional to the Miller indices of the plane. • Miller indices is a set of integers with no common factors, inversely proportional to the intercepts of the crystal plane along the crystal axes.

Figure shows Miller indices of a lattice plane. The shaded plane can be a portion of the continuous planes in which the points of the lattice plane lie. The Miller indices are inversely proportional

to the xi. Examples of Miller Indices Indexing planes in three‐dimensions • We proceed in exactly the same way, except that we now have 3‐digit Miller indices corresponding to the axes a, b and c.

The indices may denote a single plane or a set of parallel planes.

Some examples of planes Some More Examples Some More Examples

Lattice planes are usually specified by giving their Miller indices in parentheses (h,k,l). In cubic system a plane with a normal (4,-2,1) is called a (4,-2,1) plane. The commas are eliminated without confusion by writing n instead of –n, simplifying the description to (421)

Conventions For Specifying Directions • A similar convention is used to specify directions in the direct lattice, but to avoid confusion with the Miller indices (directions in the reciprocal lattice) square brackets are used instead of parenthesis. Some directions Notation of specifying family of planes • (100), (010) and (001) planes are equivalent in a cubic crystal. • We refer to them collectively as the {100} planes, and in general we use {hkl} to refer to the (hkl) planes and all those that are equivalent to them by virtue of the crystal symmetry. Notation of specifying family of directions

• The [100], [010], [001], [100], [010] and [001] directions in the cubic crystal are referred to , collectively, as the <100> directions. Symmetry Operations

3-tetrad axes 4-triad axes

6-diad axes