The primitive, body-centred and face-centred cubic unit cells

Summary

This video explores the structure of crystalline solids, focusing on how particles like atoms, ions, or molecules arrange in a regular array. It explains unit cells, the smallest portions that define crystal structures, and discusses the three types of cubic unit cells: simple, body-centered, and face-centered. Each cell type differs in particle arrangement and coordination number. The video highlights how stacking these unit cells leads to different packing efficiencies, which explains why certain crystal structures, like those of sodium chloride and zinc sulfide, are more common in nature due to their efficient space utilization.

Takeaways

• 🔍 The particles in a crystal, such as atoms, ions, or molecules, are arranged in a regular, close-packed array.
• 🧱 A unit cell is the smallest portion of a crystal that defines its structure, and repeating unit cells form the entire crystal.
• 🔲 Simple cubic unit cells have particles only at the corners, and the particles touch along the edges but not along the diagonals.
• 🧩 Each simple cubic unit cell contains one complete particle when considering particle portions from neighboring unit cells.
• 🔢 In a simple cubic structure, each particle has six nearest neighbors – one above, below, left, right, front, and back.
• 💠 Body-centered cubic unit cells have particles at each corner and one particle in the center, providing a total of two particles per unit cell.
• 🔺 In a body-centered cubic structure, each particle has eight nearest neighbors due to the particle in the center touching the corner particles.
• 🏛 Face-centered cubic unit cells have particles at each corner and in the center of each face, resulting in a total of four particles per unit cell.
• 🔗 The face-centered cubic structure gives each particle 12 nearest neighbors, four above, below, and around.
• 🌐 Cubic closest packing, based on the face-centered cubic structure, is the most efficient stacking, occupying 74% of available volume.

Q & A

What is a unit cell in a crystal structure?

A unit cell is the smallest portion of a crystal structure that defines its overall structure. By stacking unit cells in all three directions, the entire crystal structure is formed.

What are the three types of cubic unit cells discussed in the script?

The three types of cubic unit cells are: simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC).

How do particles in a simple cubic unit cell interact?

In a simple cubic unit cell, particles are located at the corners of the cube. They touch along the edges of the cube but not diagonally across the faces or through the body of the cube.

What is the coordination number in a simple cubic unit cell?

The coordination number in a simple cubic unit cell is 6, meaning each particle has six nearest neighbors: one above, one below, one to the left, one to the right, one in front, and one in back.

What distinguishes a body-centered cubic (BCC) unit cell from a simple cubic unit cell?

A body-centered cubic (BCC) unit cell has particles at each corner and one particle in the center of the cube. In contrast, the simple cubic unit cell only has particles at the corners.

What is the coordination number in a body-centered cubic (BCC) unit cell?

In a body-centered cubic (BCC) unit cell, the coordination number is 8, meaning each particle has four nearest neighbors above and four below it.

What is the particle arrangement in a face-centered cubic (FCC) unit cell?

In a face-centered cubic (FCC) unit cell, particles are located at each corner of the cube and in the center of each face, but there is no particle in the center of the cube.

What is the coordination number in a face-centered cubic (FCC) unit cell?

The coordination number in a face-centered cubic (FCC) unit cell is 12, meaning each particle has 12 nearest neighbors: four around it, four above, and four below.

Why is the simple cubic unit cell rarely found in nature?

The simple cubic unit cell is rarely found in nature because it uses space inefficiently, with only 52% of the available volume occupied by particles. More efficient structures like body-centered cubic (BCC) and face-centered cubic (FCC) are more common.

What structure does sodium chloride (NaCl) adopt, and how are the ions arranged?

Sodium chloride (NaCl) adopts the rock salt structure, where face-centered cubic arrays of chloride ions and sodium ions interpenetrate. The smaller sodium ions fit into the spaces between the larger chloride ions.

Outlines

00:00

🔬 Understanding Crystal Structures and Unit Cells

The first paragraph introduces the concept of crystalline solids, where particles (atoms, ions, or molecules) are arranged in a regular array. It explains that the unit cell is the smallest portion of a crystal that defines its structure, and by stacking unit cells in all directions, a crystal is formed. The focus is on cubic crystal systems, which are common in many elements and simple compounds. The types of cubic unit cells—simple cubic, body-centered cubic, and face-centered cubic—are explored, emphasizing how particles are arranged at the corners of these cells. The concept of coordination number, which is the number of nearest neighbors a particle has, is introduced for each type of cubic unit cell.

05:01

🔶 Efficiency of Stacking in Crystals

The second paragraph discusses different ways of stacking particles to form crystal structures. It begins with the simple cubic unit cell, which has an inefficient use of space, occupying only 52% of the available volume. Next, the body-centered cubic unit cell is introduced, where better stacking leads to 68% volume occupancy, commonly seen in alkali metals. Finally, the most efficient stacking, cubic closest packing (based on the face-centered cubic unit cell), occupies 74% of the volume. This structure is adopted by many elements, compounds, and ionic compounds, such as sodium chloride and zinc sulfide. The paragraph explains how smaller ions fit into the gaps between larger ions, forming structures like sodium chloride (rock salt) and zinc blend, where ions are tetrahedrally arranged.

Keywords

💡Crystalline Solid

A crystalline solid is a material in which the particles (atoms, ions, or molecules) are arranged in a highly ordered, repeating pattern called a crystal lattice. In the video, crystalline solids are discussed in relation to how their particles are organized in a regular array, which is essential to the formation of various types of cubic unit cells.

💡Unit Cell

A unit cell is the smallest repeating portion of a crystal structure that defines the arrangement of particles. The video uses unit cells to explain how these basic building blocks stack together in three directions to form a crystal. Several types of unit cells, such as simple cubic and body-centered cubic, are described to illustrate this concept.

💡Cubic Crystal System

The cubic crystal system refers to a category of crystal structures where the unit cell forms a cube. The video focuses on the three types of cubic unit cells: simple cubic, body-centered cubic, and face-centered cubic, each with a unique arrangement of particles at the corners, faces, and center of the cube.

💡Simple Cubic Unit Cell

A simple cubic unit cell has particles located only at the corners of the cube. The video explains that in this structure, the particles touch along the cube edges but not diagonally. Despite its simplicity, the simple cubic arrangement is relatively rare in nature due to inefficient space usage.

💡Body-Centered Cubic Unit Cell

In a body-centered cubic unit cell, particles are located at each corner and one in the center of the cube. The video highlights that this arrangement allows each corner particle to touch the center particle, resulting in a higher coordination number compared to the simple cubic structure. Many metals adopt this arrangement due to its greater packing efficiency.

💡Face-Centered Cubic Unit Cell

A face-centered cubic unit cell contains particles at each corner and in the centers of each face of the cube. This structure is the most efficient of the cubic systems, with 74% of the available volume occupied by particles. The video mentions that many elements and compounds, such as sodium chloride, adopt this arrangement.

💡Coordination Number

The coordination number refers to the number of nearest neighbors surrounding a particle within a crystal structure. In the video, the simple cubic unit cell has a coordination number of six, while the body-centered cubic unit cell has eight, and the face-centered cubic unit cell has twelve. This concept is crucial for understanding the stability and arrangement of particles in a crystal.

💡Packing Efficiency

Packing efficiency is the percentage of available space within a crystal that is actually occupied by particles. The video contrasts the packing efficiency of different cubic unit cells, with the simple cubic being least efficient (52%), body-centered cubic more efficient (68%), and face-centered cubic being the most efficient (74%).

💡Cubic Closest Packing

Cubic closest packing is a highly efficient arrangement where spheres are stacked to minimize unused space. The video describes how this structure is based on the face-centered cubic unit cell and allows 74% of the available volume to be filled. Many elements and compounds, including sodium chloride, adopt this packing structure.

💡Sodium Chloride Structure

The sodium chloride structure, also called the rock salt structure, involves face-centered cubic arrays of chloride and sodium ions. In the video, this structure is shown as an example of how smaller sodium ions fit into the spaces between larger chloride ions, creating a stable arrangement. This structure is common in many alkali halides and ionic compounds.

Highlights

Particles in a crystal structure are arranged in a regular, repeating array, though the spaces between particles are greatly exaggerated in illustrations.

The unit cell is the smallest portion of a crystal structure that defines its overall arrangement, with unit cells stacking in three dimensions to form the structure.

Many elements and compounds have unit cells derived from the cubic crystal system, which includes simple cubic, body-centered cubic, and face-centered cubic unit cells.

A simple (or primitive) cubic unit cell has particles only at the corners, with particles touching along the cube's edges but not along diagonals.

The simple cubic unit cell contains one particle in total, derived from the eight corners of the cube, each contributing 1/8 of a particle.

The coordination number in a simple cubic structure is six, meaning each particle has six nearest neighbors—one above, one below, one to the left, right, front, and back.

In the body-centered cubic unit cell, there is a particle at each corner and one in the center, which increases the number of particles per unit cell to two.

The coordination number in a body-centered cubic structure is eight, as each particle has four nearest neighbors above and four below.

A face-centered cubic unit cell has particles at each corner and in each face, resulting in a total of four particles per unit cell.

The coordination number in a face-centered cubic structure is twelve, as each particle has four nearest neighbors in three different layers (above, below, and surrounding).

The face-centered cubic structure is more efficient than the simple cubic and body-centered cubic, with particles occupying 74% of the available volume.

The simple cubic structure only occupies 52% of the available volume, making it an inefficient and rarely observed arrangement in nature.

The body-centered cubic arrangement improves efficiency with 68% of the available volume occupied, adopted by many metals including alkali metals.

Cubic closest packing, based on the face-centered cubic unit cell, provides the most efficient use of space, with spheres filling 74% of the volume.

Several compounds adopt the cubic closest packing structure, including sodium chloride (rock salt structure) and zinc sulfide (zinc blende structure).

Transcripts

if you could travel within a crystallin

solid you would see the particles atoms

ions or molecules arranged in a regular

array here the spaces are greatly

exaggerated but in reality the particles

are packed close together the unit cell

of a crystal structure is the smallest

portion that defines the structure

stacking unit cells next to each other

in all three directions gives the

structure many elements and simple

compounds have unit cells from the cubic

Crystal system let's examine the three

types of cubic unit

cells all cubic unit cells have

particles at the corners of a cube the

simple or primitive cubic unit cell has

particles at the corners

only in reality the particles lie as

close to each other as possible

note that the particles touch along the

cube edges but not along a diagonal in

the face or along a diagonal through the

body by slicing away parts that belong

to neighboring unit cells we see that

the actual unit cell consists of

portions of the

particles when the cells pack next to

each other in all three dimensions we

obtain the

crystal if we fade the others out you

can see the original group of eight

particles within the array and the unit

cell within that group we find the

number of particles in one unit cell by

combining all the particles portions in

the simple cubic unit cell eight Corners

Each of which is 1/8 of a particle

combined to give one

particle a key feature of a crystal

structure is its coordination number the

number of the nearest neighbors

surrounding each particle in a simple

cubic array any given particle has a

neighboring particle above below to the

right to the left in front and in back

of it for a total of six nearest

neighbors the body centered cubic unit

cell has a particle at Each corner and

one in the center which is colored pink

to make it easier to see with fullsize

spheres you can see that the particles

don't touch along the edges of the cube

but Each corner particle does touch the

one in the center the actual unit cell

consists of portions of the corner

particles and the whole one in the

center 88s give one particle and the one

in the center gives another for a total

of two

particles in this tiny portion of a body

centered cubic array you can see that

any given particle has four nearest

neighbors above and four below for a

total of eight nearest neighbors the

face centered cubic unit cell has a

particle at Each corner and in each face

which are colored yellow here but none

in the center the corner particles don't

touch each other but Each corner does

touch a particle in the face and those

in the faces touch each other as well

the actual unit cell consists of

portions of particles at the corners and

in the faces 88 at the corners gives one

particle and half a particle in each of

six phases gives three more for a total

of four

particles in this tiny portion of a face

centered cubic array notice that a given

particle has four nearest neighbors

around it four more above and four more

below for a total of 12 nearest

neighbors stacking spher shows how the

three cubic unit cells arise arrange a

layer of spheres in horizontal and

vertical rows note the large diamond

shaped space among the particles placing

the next layer directly over the first

gives a structure based on the simple

cubic unit cell those larger spaces mean

an inefficient use of space in fact only

52% of the available volume is actually

occupied by

spheres because of this inefficiency the

simple cubic unit cell is is seen rarely

in

nature a more efficient stacking occurs

if we place the second layer over the

spaces formed by the first layer and the

third layer over the space is formed by

the second that simple change leads to

68% of the available volume occupied by

the Spheres and a structure based on the

body centered cubic unit cell many

metals including all the alkali metals

adopt this

arrangement for the most efficient

stacking shift every other Row in the

first layer so the large diamond shaped

spaces become smaller triangular spaces

and place the second layer over them

then the third layer goes over the holes

visible through the first and second

layers in this Arrangement called cubic

closest packing spheres occupy 74% of

the volume note that it is based on the

face centered CBE unit cell many

elements calent compounds and as you'll

see in the next two examples ionic

compounds adopt cubic closest packing

sodium chloride adopts the sodium

chloride or rock salt structure as do

many other Alkali halides alkaline earth

oxides and sulfides and other ionic

compounds picture separate face centered

cubic arrays of chloride ions and sodium

ions ions as they approach and

interpenetrate each other the smaller

sodium ions fit in the holes between the

larger chloride ions and the NAC unit

cell zinc sulfide adopts the zinc blend

structure as do the copper one halides

and several other compounds if face

centered cubic arrays of zinc ions and

sulfide ions approach and interpenetrate

slightly offset from each other each ion

becomes surrounded tetrahedrally by four

of the other ions note the blinking zinc

ion and the four sulfides you can see

the relative positions in this slightly

expanded view of the zinc blend unit

cell