# Crystal Lattice Structure
## Crystal Structure

A crystal structure is defined by both the geometry of the repeating unit cell and the specific arrangement of atoms within that cell. Common examples include face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP). Each structure determines the physical properties of the material by specifying not only the shape and size of the unit cell but also how atoms are ordered within it.[^1]

### Common Crystal Structures

- **Simple Cubic (SC)**: Atoms at the corners of a cube.[^1]
- **Body-Centered Cubic (BCC)**: Atoms at all eight corners and a single atom at the center of the cube.[^1]
- **Face-Centered Cubic (FCC)**: Atoms at corners and at the center of each face of the cube.[^1]
- **Hexagonal Close-Packed (HCP)**: Layers of atoms arranged in hexagons, stacked in a specific sequence.[^1]
<img src=https://www.shutterstock.com/image-illustration/unit-cells-crystal-lattices-600nw-347363240.jpg>
### SC, BCC, FCC, HCP Coordination Number

| Structure | Coordination Number |
| :-- | :-- |
| SC | 6 |
| BCC | 8 |
| FCC | 12 |
| HCP | 12 |

[^1]

- **SC (Simple Cubic):** Each atom is in contact with 6 neighbors, one along each axis direction.
- **BCC (Body-Centered Cubic):** Each atom touches 8 others, positioned at cube corners relative to a body center atom.
- **FCC (Face-Centered Cubic):** Each atom contacts 12 other atoms—4 in the same layer, 4 above, and 4 below.
- **HCP (Hexagonal Close-Packed):** Each atom also touches 12 neighbors, due to the dense stacking pattern.

The coordination number directly impacts the material's packing efficiency and properties, with higher numbers generally implying tighter atomic bonding and more stability.[^1]
## The Seven Crystal Systems

A crystal system classifies unit cells according to their geometric properties—lengths of edges and angles between them—without regard to atomic arrangement. There are seven main crystal systems, each supporting multiple crystal structures.[^1]

| Crystal System | Unit Cell Axes \& Angles |
| :-- | :-- |
| Cubic | $a = b = c;\ \alpha = \beta = \gamma = 90^\circ$[^1] |
| Tetragonal | $a = b \neq c;\ \alpha = \beta = \gamma = 90^\circ$[^1] |
| Orthorhombic | $a \neq b \neq c;\ \alpha = \beta = \gamma = 90^\circ$[^1] |
| Rhombohedral (Trigonal) | $a = b = c;\ \alpha = \beta = \gamma \neq 90^\circ$[^1] |
| Monoclinic | $a \neq b \neq c;\ \alpha = \gamma = 90^\circ,\ \beta \neq 90^\circ$[^1] |
| Triclinic | $a \neq b \neq c;\ \alpha \neq \beta \neq \gamma \neq 90^\circ$[^1] |
| Hexagonal | $a = b \neq c;\ \alpha = \beta = 90^\circ,\ \gamma = 120^\circ$[^1] |

<img src=https://cdn.britannica.com/73/105673-050-78E58296/crystal-systems.jpg>
[^1]: concept_check.pdf

[^2]: https://www.sciencelearn.org.nz/images/5537-seven-crystal-systems

### Dense Packing and Non-Dense Packing

Dense packing and non-dense packing refer to the arrangement of atoms in a crystal structure, significantly influencing the material's energy and stability. Dense packing results in closer atomic arrangements, reducing energy, whereas non-dense packing leaves more empty space, which raises the internal energy.

#### Dense Packing

- Dense packing occurs in structures like face-centered cubic (FCC) and hexagonal close-packed (HCP), where atoms are tightly arranged with minimal empty space.[^1]
- This minimizes the number of unsatisfied chemical bonds at the surface, resulting in lower surface energy and greater thermodynamic stability.[^1]
- Dense-packed materials are generally stronger and more stable because the tightly arranged atoms share more bonds with neighbors, lowering the system's overall potential energy.[^1]

#### Non-Dense Packing

- Non-dense packing is found in structures like body-centered cubic (BCC) and simple cubic, where atoms are further apart and there are more voids between them.[^1]
- These arrangements have higher internal energy due to more unsatisfied or "dangling" bonds on the surface, resulting in higher surface energy and less stability.[^1]
- Materials with non-dense packing are typically less tightly bonded and may be less dense and have somewhat higher reactivity compared to densely packed ones.[^1]

#### Energy Perspective

- As the planar (surface) density increases (i.e., more dense packing), the number of satisfied atomic bonds increases and the number of unsatisfied bonds decreases.
- This reduced number of unsatisfied surface bonds leads to a lower surface energy, making the material energetically favorable.[^1]
- Conversely, lower planar density (non-dense packing) increases the number of unsatisfied bonds and results in higher surface energy, making the structure less stable.[^1]

**Summary Table:**

| Packing Type | Example Structure | Atomic Arrangement | Surface Energy | Stability |
| :-- | :-- | :-- | :-- | :-- |
| Dense Packing | FCC, HCP | Very close-packed | Low | High |
| Non-Dense Packing | BCC, Simple Cubic | Looser, more voids | High | Lower compared to dense |

[^1]
<img src=https://cdn1.byjus.com/wp-content/uploads/2022/03/crystal-structure-packing-and-its-effect-on-bonding-energies.png>
Dense packing is energetically preferred because it allows more atoms to achieve stable, low-energy bonding, while non-dense packing leaves more high-energy, unsatisfied bonds, making the material less stable.[^1]
## Atomic Packing Factor(APF)

APF and how to calculate their APF

The Atomic Packing Factor (APF) quantifies how densely atoms pack in a crystal structure. Each structure—Simple Cubic (SC), Body-Centered Cubic (BCC), Face-Centered Cubic (FCC), and Hexagonal Close-Packed (HCP)—has a characteristic APF.

### How to Calculate APF

APF is calculated as:

$$
\text{APF} = \frac{\text{Volume of atoms in the unit cell}}{\text{Total unit cell volume}}
$$

### APF Values \& Formulas

| Structure | Number of Atoms per Unit Cell | APF Formula | APF Value |
| :-- | :-- | :-- | :-- |
| SC | 1 | $\frac{\frac{4}{3}\pi r^3}{a^3}$ where $a=2r$ | 0.52 |
| BCC | 2 | $\frac{2 \times \frac{4}{3}\pi r^3}{a^3}$ where $a=\frac{4r}{\sqrt{3}}$ | 0.68 |
| FCC | 4 | $\frac{4 \times \frac{4}{3}\pi r^3}{a^3}$ where $a=2\sqrt{2}r$ | 0.74 |
| HCP | 6 | $\frac{6 \times \frac{4}{3}\pi r^3}{\text{Volume}_{\text{HCP cell}}}$ | 0.74 |

- **SC:** Atoms touch along the edge, so $a = 2r$.
- **BCC:** Atoms touch along the body diagonal, so $a = \frac{4r}{\sqrt{3}}$.
- **FCC:** Atoms touch along the face diagonal, so $a = 2\sqrt{2}r$.
- **HCP:** HCP calculation is more complex, but its APF is mathematically identical to FCC.[^1]

### Summary

- **SC:** Sparse packing, APF = 0.52
- **BCC:** Moderate packing, APF = 0.68
- **FCC \& HCP:** Closest possible packing for equal spheres, APF = 0.74

APF measures how efficiently atoms fill space in each crystal system—the denser the packing, the higher the APF, imparting greater material density and stability.[^1]
## Theoretical density of metals

The theoretical density of metals measures how densely atoms are packed in a metal's crystal structure. It is calculated using the following formula:

$$
\rho = \frac{n \cdot A}{N_A \cdot V_c}
$$

Where:

- $\rho$ = theoretical density (g/cm³)
- $n$ = number of atoms per unit cell
- $A$ = atomic weight (g/mol)
- $N_A$ = Avogadro's number ($6.022 \times 10^{23}$ atoms/mol)
- $V_c$ = volume of the unit cell (cm³)[^1]

### Example: Theoretical Density of Iron (BCC)

- Iron: BCC structure ($n = 2$), atomic weight ($A = 55.85$ g/mol), unit cell edge ($a = 0.2866$ nm = $2.866 \times 10^{-8}$ cm)
- $V_c = a^3 = (2.866 \times 10^{-8})^3 = 2.35 \times 10^{-23}$ cm³

$$
\rho = \frac{2 \cdot 55.85}{6.022 \times 10^{23} \cdot 2.35 \times 10^{-23}} = 7.87\ \text{g/cm}^3
$$

This value agrees with experimentally observed values for iron, illustrating the close relationship between atomic structure and density in metals.[^1]

# Crystalline and Non-Crystalline

Crystalline and non-crystalline (amorphous) materials differ fundamentally in atomic arrangement, physical properties, and behavior.

## Crystalline Materials

- Atoms are arranged in a highly ordered, repeating 3D structure called a crystal lattice.[^1]
- They have long-range order, meaning patterns of atomic arrangement repeat over large distances.[^1]
- Exhibit sharp melting points because their atoms break free from the lattice at a specific temperature.[^1]
- Show phenomena like polymorphism or allotropy (the ability to exist in more than one crystal structure).[^1]
- Examples: metals, ceramics, some polymers, and most minerals.[^1]

## Non-Crystalline (Amorphous) Materials

- Atoms are arranged randomly, so there is no long-range periodic order.[^1]
- Only have short-range order, meaning consistent atomic arrangement extends only to nearest neighbors.[^1]
- Soften gradually over a broad temperature range, showing no sharp melting point.[^1]
- Do not display polymorphism or allotropy because there is no defined crystal structure.[^1]
- Examples: glass, many polymers, some ceramics.[^1]

## Comparison Table

| Property | Crystalline Materials | Non-Crystalline Materials |
| :-- | :-- | :-- |
| Atomic Arrangement | Highly ordered, 3D lattice | Random, no long-range order |
| Structure Range | Long-range order | Short-range order |
| Melting Point | Sharp, well-defined | Broad temperature range |
| Polymorphism/Allotropy | Possible | Not possible |
| Examples | Metals, minerals, some ceramics | Glass, plastics, some ceramics |

Crystalline materials have long-range periodicity, sharp transitions, and defined structures, while non-crystalline materials lack long-range order and transition more gradually.[^1]
# Isotropy and Anisotropy
Anisotropy and isotropy describe how a material’s properties vary or remain constant with direction.

## Isotropy

- **Isotropy** is when a material exhibits the same physical properties in all directions.[^1]
- This means mechanical, thermal, electrical, or other characteristics do not depend on the measurement direction within the material.
- Most common in polycrystalline materials with randomly oriented grains and in noncrystalline (amorphous) substances, where there is no preferred structure direction.[^1]

## Anisotropy

- **Anisotropy** is when a material’s physical properties vary with direction.[^1]
- It results from an internal structure where atoms or molecules are arranged differently along different axes, such as in single crystals or textured polycrystals.[^1]
- For example, the speed of sound, thermal conductivity, or strength might be greater along one crystallographic direction than another.

In summary, isotropic materials behave identically in all directions, while anisotropic materials show direction-dependent behavior due to their internal structure.[^1]
# Next Part : Miller Indices and Diffraction
https://hackmd.io/r94Crq_XTTOnLkiKPXr4vg

[^1]: concept_check.pdf
[^1]: concept_check.pdf

[^2]: https://www.sciencelearn.org.nz/images/5537-seven-crystal-systems