Lecture 3 Part 1 - Crystal Structure - 2 (Unit Cell, Lattice, Crystal)

Summary

This lecture delves into the concept of unit cells in crystallography, explaining how they can fill space without voids. It clarifies the difference between a lattice—a periodic arrangement of points—and a crystal, which includes atoms or molecules at these points. The talk covers the parameters needed to define unit cells in 1D, 2D, and 3D, highlighting the importance of choosing the smallest unit cell with the highest symmetry. The discussion also touches on primitive and non-primitive unit cells, emphasizing that the term 'unit cell' refers to the basic repeating unit, not necessarily a unit of size one.

Takeaways

• 📐 The concept of a unit cell is fundamental in understanding lattice structures, where an infinite number of possible unit cells can fill the entire space.
• 🔍 Unit cells can have various geometries, but the most common in 2D are parallelograms, while in 3D, parallelepipeds are typical.
• 📏 In 2D, a unit cell is defined by two vectors, 'a' and 'b', and the angle 'alpha' between them, which can lead to different shapes like squares or rectangles depending on their values.
• 🚫 A triangular unit cell cannot fill space by translation alone without leaving voids, and thus is not a qualified unit cell for space filling.
• 🧊 In 1D, the unit cell is simply a line, which can be repeated to construct the entire lattice.
• 💎 The difference between a lattice and a crystal is that a lattice is an arrangement of points, while a crystal is a lattice with atoms, ions, or molecules occupying these points.
• 🔑 The term 'primitive unit cell' refers to the smallest unit cell with the highest symmetry, typically containing one lattice point or one motif.
• 🔄 While the guideline is to choose the smallest possible unit cell, it's possible to choose any number of unit cells to fill the space.
• 📈 The parameters required to define a unit cell in 3D are a, b, c, alpha, beta, and gamma, which are six lattice parameters.
• 🔢 The size of the unit cell is not necessarily 'unit' in measurement; it refers to the basic repeating unit in the structure.
• 💠 The concept of a crystal is essentially a lattice plus a motif or basis, which defines what sits on the lattice points and how to repeat them to construct the crystal structure.

Q & A

What is the concept of a unit cell in the context of a lattice?

A unit cell is the smallest repeating unit in a lattice that can be used to fill the entire space without leaving any gaps. It is defined by its geometry and can be translated along the lattice vectors to construct the lattice.

Can a unit cell be any shape, such as a triangle or a parallelogram?

In 2D, a unit cell can be a parallelogram. While a triangle might seem like a candidate, it cannot fill the entire space by translation alone without leaving voids, hence it is not a qualified unit cell for filling space.

What are the parameters required to describe a unit cell in 2D?

In 2D, the parameters required to describe a unit cell are two vectors, one in the x direction (denoted as 'a') and another in the y direction (denoted as 'b'), along with the angle between them (alpha).

What is the significance of the angle alpha in a 2D unit cell?

The angle alpha is the included angle between the two vectors a and b. It, along with the lengths of a and b, determines the shape and orientation of the unit cell.

What is the difference between a primitive and a non-primitive unit cell?

A primitive unit cell has one lattice point per unit cell, effectively containing one atom, molecule, or ion if it's a crystal. A non-primitive unit cell has more than one lattice point per unit cell, meaning it contains multiple atoms, molecules, or ions.

Why is a parallelogram considered a qualified unit cell for a 2D lattice?

A parallelogram is considered a qualified unit cell for a 2D lattice because it can be translated along the x and y directions without leaving any voids, thus filling the entire space.

What is the simplest form of a unit cell in 1D?

In 1D, the simplest form of a unit cell is a line segment, which represents the basic unit being repeated along the lattice.

What is the difference between a lattice and a crystal?

A lattice is a translationally periodic arrangement of points in space, while a crystal is a lattice where these points are occupied by atoms, ions, or molecules, forming a solid material with a periodic structure.

How many lattice parameters are required to define a unit cell in 3D?

In 3D, six lattice parameters are required to define a unit cell: three edge lengths (a, b, c) and three angles (alpha, beta, gamma).

What does the term 'motif' or 'basis' refer to in the context of a crystal?

In the context of a crystal, 'motif' or 'basis' refers to the arrangement of atoms, ions, or molecules that occupy the lattice points, defining the crystal's structure.

Why is the size of a unit cell not necessarily related to the number '1'?

The size of a unit cell is not necessarily related to the number '1' because the term 'unit cell' refers to the basic unit that is repeated in the lattice to construct the entire structure, rather than its size being a unit length.

Outlines

00:00

📐 Geometry of Unit Cells in Crystal Lattices

The first paragraph delves into the concept of unit cells within a lattice structure. It explains that while there can be an infinite number of possible unit cells, the geometry of a unit cell is crucial for filling space without leaving voids. The paragraph discusses the parameters needed to describe a unit cell in 2D, such as vectors in the x and y directions and the angle between them. It further explores the conditions for different types of unit cells, including square, rectangular, and the limitations of using a triangle as a unit cell due to its inability to fill space without rotation. The concept of primitive and non-primitive unit cells is introduced, with the former being the smallest unit cell with the highest symmetry and effectively one lattice point per unit cell. The paragraph concludes with a discussion on choosing the smallest possible unit cell as a guideline but acknowledges that other unit cells can also fill the space.

05:00

🔍 Distinguishing Lattices from Crystals

The second paragraph focuses on the distinction between a lattice and a crystal. A lattice is defined as a translationally periodic arrangement of points in space, while a crystal is a lattice where the points are occupied by atoms, ions, or molecules. The paragraph explains that a crystal is essentially a lattice plus a motif or basis, which is the actual substance sitting on the lattice points. It reinforces the idea that the lattice dictates how to repeat points, and the basis or motif dictates what to repeat. The summary also touches on the parameters required to define a unit cell in 2D and 3D, emphasizing that the unit cell size is not necessarily unitary but represents the basic repeating unit. The paragraph concludes by highlighting the difference between a lattice and a crystal, with the latter being a filled lattice with a specific motif.

10:01

💎 Constructing Crystals from Lattices

The third paragraph builds upon the previous discussion by illustrating how to construct a crystal from a lattice. It uses the analogy of placing different motifs, such as hearts or footballs, at each lattice point to create various types of crystals. The paragraph emphasizes the infinite possibilities for unit cells in 2D and 3D, while also posing questions to the audience about the number of primitive unit cells that can be defined in 1D and 2D. The summary highlights the creative aspect of defining unit cells and the fundamental principles that govern the construction of crystals from lattices.

Keywords

💡Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that can be translated to fill the entire space without leaving voids. It is fundamental to the video's theme as it discusses the geometry and conditions of unit cells in different dimensions. For instance, the script mentions that in 2D, a unit cell can be a parallelogram with parameters a, b, and alpha, which define its shape and orientation.

💡Lattice

A lattice is a translationally periodic arrangement of points in space, which serves as the foundation for the formation of crystals. The video script uses the concept of a lattice to explain how unit cells are defined and how they can be translated to fill space. The script also distinguishes between a lattice and a crystal, where the latter includes a motif or basis occupying the lattice points.

💡Lattice Vector

A lattice vector is a vector that can be used to translate a unit cell within the crystal lattice. It is crucial for understanding how the unit cells fill space, as mentioned in the script when discussing the translation of the unit cell in the x and y directions in 2D.

💡Primitive Unit Cell

A primitive unit cell is defined as the smallest unit cell with the highest symmetry, containing effectively one lattice point. The script explains that it is a guideline to choose the smallest possible unit cell, which is often the primitive unit cell, but it is not a strict requirement.

💡Non-Primitive Unit Cell

A non-primitive unit cell contains more than one lattice point per unit cell, as opposed to the primitive unit cell. The script provides an example where a non-primitive unit cell has two lattice points, making it larger and less symmetric than a primitive unit cell.

💡Crystal

A crystal is a solid material whose constituents, such as atoms, ions, or molecules, are arranged in a highly ordered, repeating pattern extending in all three spatial dimensions. The script clarifies the difference between a lattice and a crystal by stating that a crystal is a lattice plus a motif or basis, which is the object occupying the lattice points.

💡Motive or Basis

The term 'motive' or 'basis' refers to the object, such as an atom, ion, or molecule, that occupies the lattice points in a crystal. The script uses this concept to illustrate how a crystal is formed by adding a motive to a lattice, turning a set of points into a material structure.

💡Translational Symmetry

Translational symmetry is a property of a lattice where the pattern repeats identically when translated by a certain vector. The script emphasizes this property when discussing how unit cells can be translated to fill the entire space without leaving voids, which is essential for the formation of a crystal.

💡Lattice Parameters

Lattice parameters are the geometric parameters that define the shape and size of a unit cell in a crystal lattice. The script mentions that in 2D, three parameters (a, b, and alpha) are needed, while in 3D, six parameters (a, b, c, alpha, beta, and gamma) are required to define the unit cell.

💡Tiling

Tiling refers to the process of covering a surface or space with flat shapes without gaps or overlaps. In the context of the script, tiling is used to describe how unit cells can fill space in 2D and 3D, and how different shapes, such as triangles, cannot fill space without leaving voids.

💡Dimensionality

Dimensionality refers to the number of independent spatial dimensions in a system. The script discusses unit cells and lattices in 1D, 2D, and 3D, highlighting how the geometry and the number of unit cells that can fill space vary with dimensionality. For example, in 1D, the unit cell is a line, while in 2D, it can be a parallelogram.

Highlights

The concept of a unit cell in a lattice and the infinite possibilities of unit cells that can fill the entire space.

Defining the geometry of a unit cell, including conditions for a square, rectangular, or parallelogram unit cell.

The necessity of translation and rotation for unit cells to fill space without leaving voids.

The impossibility of using a triangle as a unit cell for space-filling by translation alone.

The distinction between a parallelogram and parallelepiped as qualified unit cells in 2D and 3D respectively.

The definition of a unit cell in 1D as a line.

The explanation of primitive and non-primitive unit cells and their significance in crystal structure.

The guideline to choose the smallest possible unit cell for lattice analysis.

The identification of different unit cells in a rectangular lattice and the impossibility of a square unit cell due to lattice point arrangement.

The concept of a crystal as a translationally periodic arrangement of atoms, ions, or molecules.

The difference between a lattice and a crystal, emphasizing the role of the motif or basis in defining a crystal.

The parameters required to define a unit cell in 2D and 3D: a, b, alpha for 2D and a, b, c, alpha, beta, gamma for 3D.

The clarification that the unit cell size does not necessarily have to be 1 and represents the basic repeating unit.

The idea that any parallelogram with lattice points as vertices can be a unit cell in 2D.

The discussion on the infinite possibilities for unit cells in 1D and the concept of primitive unit cells.

The exploration of different kinds of unit cells in 2D and the theoretical considerations for their selection.

Transcripts

So, in the last class we have looked at the concept of unit cell, and if you are having

a lattice you can in principle identify infinite number of unit cells -- possible unit cells

that can actually fill the entire space.

But then the question is what should be the geometry, alright?

What should be the geometry of that unit cell?

Should it always be a parallelogram or should -- can it be -- for instance, in 2D, should

it always be a parallelogram?

Can we not have a triangular unit cell and so on, right?

So, the -- the way that the unit cell is defined is the following.

So, unit cell is an entity which when translated along the lattice vector or along the coordinate

direction.

So, for instance if you are talking about 2D what are the parameters that are required

to describe a unit cell?

For instance, one vector in x direction and another vector in y direction.

So, this is vector let us say a.

The distance is a and the distance is b and the angle alpha, right?

Now, you can have -- so this -- this side is a, that side is b and they are at an angle

-- a and b have an included angle of alpha.

So, now you can define as many number of unit cells as you want.

So, you can have conditions like a equal to b, and alpha equal to 90 degrees, then you

will have a square unit cell.

If a is not equal to b and alpha equal to 90, then you will have a rectangular unit

cell, and so on, right?

So, when you have this small geometry, and if you take this geometry and translate the

geometry in the lattice directions -- lattice vector direction.

This is what is called lattice vector and you take this geometry and translate it both

in x direction as well -- as well as in y direction for 2D, then it should fill the

entire space.

It should not leave behind any voids.

Only then such a geometry qualifies as unit cell.

So, now, imagine having a triangle, right?

Can you fill the entire space of this particular lattice using a triangular unit cell of your

choice without leaving behind any voids?

Is it possible or not?

Just by translation it is not possible.

You will have to rotate the unit cell.

And hence your triangle cannot be a qualified unit cell for -- for filling the space, right?

So, in general in 2D, a parallelogram will be a qualified unit cell, and in 3D a parallelepiped

will be a qualified unit cell, right?

What will be a unit cell in 1D?

. It will be a line.

Line will be the unit cell in 1D.

And then we have discussed the concept of primitive unit cell and non-primitive unit

cell, right?

What is the concept of -- what is the meaning of primitive unit cell?

Primitive unit cell is the one which has effectively one lattice point associated with the unit

cell and if it is a crystal then effectively it will have 1 atom or 1 molecule or 1 ion

depending upon what is the point that is sitting -- we will talk about it in a moment ok?

So, here we have -- this is the smallest unit cell right with highest symmetry.

So, usually you choose -- the guideline is you choose the smallest possible unit cell

as your unit cell of the lattice that you are looking at, but it is not a requirement.

You can actually -- in principle you can choose infinite number of unit cells as we have shown

and then fill the space, ok?

Alright.

So, again previously we have seen a rectangular unit cell and hence you have a different set

of unit cells that you would get.

Here, it is not possible to get square as a unit cell because the way the lattice points

are arranged it is not possible to get square as a unit cell.

So, you will find something else as your appropriate unit cells and here you have seen this is

a primitive unit cell, this is a non-primitive unit cell, right?

Why is it a non-primitive unit cell?

Because effectively it has 2 lattice points per unit cell, and hence it is a non-primitive

unit cell.

So, you can also choose non-primitive unit cells to fill the space.

There is no problem with that.

Alright.

So, this -- I think we have looked at this concept.

So, you have this tiling and then 2-dimensional map and you are trying to see different kinds

of -- what are the different unit cells that are possible.

So, we -- we have already identified this as a one possible unit cell and then that

geometry as another possible unit cell and that geometry as another possible unit cell.

Alright.

So now, we have only looked at the definition of lattice, right?

What do you mean by crystal?

What is the difference between crystal and a lattice?

So, lattice by definition is a translationally periodic arrangement of points in 3D.

That is what we have seen, right?

When we saw the Cambridge wall example, what we did was we have identified bricks that

looked identical, right?

-- and then it is a 2 dimensional space and we have identified those points and then removed

the brick -- brick wall and then the set of points is what we call lattice.

So, it is basically a translationally periodic arrangement of points in 3D in general.

But, if you are defining a 2D lattice, that will be in 2D.

If you are defining 1D, then it will be 1D, clear?

Now, what is crystal?

So, we only said lattice -- when we are defining lattice we only said points, right?

-- periodic arrangement of points.

These points can actually be occupied by some real stuff.

If they are occupied by atoms, ions or molecules, then that becomes a crystal, right?

So, a translationally periodic arrangement of atoms is what is called -- in 3D is what

is called a crystal.

Sometimes, a crystal need not be of a pure metal, but it can be of an alloy.

So, that means, instead of having -- if you look at iron crystal, so only iron will be

there, but if you want to look at iron oxide Fe2O3 will be there at the lattice point that

is the idea, ok?

So, to reinforce this idea of the correspondence between lattice and the crystal, a crystal

is nothing but, lattice plus a motif or basis.

That means, whatever sits on the lattice points is what is called motif, then the moment you

add motif then that becomes a crystal of that motif, right?

So, what lattice tells you is how to repeat the points, right?

So, the moment you remove you have -- we have removed the brick wall, you have got the points.

Now, once you have the points, you can actually construct the entire brick wall because it

-- it told you -- or it showed you how to repeat these points.

So, lattice tells us how to repeat, and what does basis or motif tells us -- is what to

repeat.

So, in 2D we have seen -- how many lattice parameters are required in 2 dimensional situation

to define a lattice or to define a unit cell?

I am sorry to define a unit cell?

Three: a, b and alpha.

In 3D, you have 3 edges and 3 angles.

So, a, b, c, alpha, beta and gamma, right?

You have these 6 lattice parameters and then you get the cell of the lattice, that is the

unit cell.

Please note that unit cell has nothing to do with the size being unit, ok?

So, unit cell size need not be 1.

It is the basic unit that is being repeated, right?

It is not about the number 1, it is the basic unit that is repeated, that is why it is call

unit cell, ok?

And then, that cell of the -- you have the cell of the lattice, and then from there you

can actually construct the lattice, but that is -- that will not be called a crystal -- the

lattice cannot be called a crystal.

You can only call lattice as crystal when you add a motif to that, ok?

So, how do we understand that?

So, let us say these are your set of points, lattice points, and then you have your heart

as motif, ok?

And you put heart at each and every position and then you get something called love crystal

or heart crystal.

You replace that with football then you will get football crystal.

You replace that with hatred, you will get hatred crystal.

You can construct any crystal, right?

So, that is the idea.

So, now, you replace that and then you know what clearly the difference between how do

you distinguish the definition of a lattice and a crystal, right?

Alright.

So, in summary, in 2D any parallelogram whose vertices are lattice points.

In a unit cell, in 2D you will have any parallelogram whose vertices are lattice points.

If it is 3D, it should be a parallelepiped, and you -- we have discussed that in principle

you can have infinite possibilities for unit cells in 2D and 3D.

How many possibilities are there in 1D?

How many kinds of unit cells can you generate in 1D?

Infinity.

How?

Very good.

How many primitive lattice -- primitive unit cells can you define in 1D?

How many?

. In 2D?

Think about it, ok?