Laboratory G3 – Crystallography Aims • Develop ski...

Abstract

This laboratory explored how crystalline materials can be described in terms of lattices, unit cells and common structure types. In two dimensions, close packing of identical spheres was investigated using ping-pong balls, and appropriate 2D Bravais lattices and unit cells were identified. In three dimensions, simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC) unit cells were constructed as space-filling models, and the number of atoms per unit cell and the locations of voids were determined. Using ball-and-stick model kits, the structures of common ionic and covalent solids—sodium chloride (rock salt), fluorite/anti-fluorite, sphalerite and diamond—were built and examined in terms of coordination, packing and symmetry. Finally, an edge dislocation was modelled to visualise a common crystal defect. The experimentally built models were compared to idealised online structures and theoretical expectations. Overall, the experiment demonstrated how an apparently complex three-dimensional crystal can be reduced to a simple repeating unit cell, how packing efficiency relates to structure type, and how defects distort local atomic arrangements.

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1. Introduction

Many technologically important materials—metals, ceramics, semiconductors and salts—are crystalline. In a crystal, atoms, ions or molecules are arranged periodically in three dimensions. This periodicity can be described by:

• A lattice: an infinite array of equivalent points in space.
• A basis: the group of atoms associated with each lattice point.
• A unit cell: a volume defined by three primitive vectors that, when translated along those vectors, tiles space without gaps or overlaps.

Different choices of primitive vectors give different but equivalent unit cells. In two dimensions these fall into a finite set of families called the 2D Bravais lattices; in three dimensions there are 14 3D Bravais lattices, grouped into crystal systems (cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic and triclinic).

Because atoms experience strong short-range repulsion and longer-range attraction, many crystalline solids adopt close-packed arrangements that maximise packing density. In 2D this corresponds to dense packings of disks (our ping-pong balls); in 3D, the most common close-packed arrangements are hexagonal close packing (HCP) and cubic close packing (CCP, equivalent to the FCC lattice).

Real materials are seldom perfect. Ions of different sizes arrange into interpenetrating sub-lattices (as in NaCl, fluorite or sphalerite), and real crystals contain defects, such as vacancies and dislocations. Edge dislocations, corresponding to an extra half-plane of atoms inserted into the crystal, play a crucial role in plastic deformation of metals and ceramics.

Aims and objectives

The overall aim of this laboratory was to develop an intuitive and practical understanding of crystallography by building and analysing physical models. Specifically, the objectives were to:

1. Identify unit cells and assign 2D Bravais lattice types in tessellating images.
2. Construct and sketch simple cubic, body-centred cubic and face-centred cubic unit cells using space-filling (ping-pong ball) models and determine the number of atoms per unit cell and location of voids.
3. Determine and compare 2D packing densities for square and hexagonal close-packing of identical spheres.
4. Construct 3D ball-and-stick models of common crystal structure types (NaCl, fluorite/anti-fluorite, sphalerite and diamond) and relate them to unit cells, coordination numbers and voids.
5. Build a model of an edge dislocation and interpret the resulting distortion in terms of crystal structure.

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2. Experimental Methods

2.1 2D close packing and Bravais lattices

Ping-pong balls were used as hard-sphere models for atoms in two dimensions. On a flat surface, balls were arranged in different tessellating patterns. Two main arrangements were explored:

• Square packing: balls arranged on a square grid, each ball touching four neighbours (up, down, left, right).
• Hexagonal (triangular) packing: rows of balls in which each ball in one row sits in the “gap” between two balls in the adjacent rows, giving six nearest neighbours.

The patterns were inspected visually to identify:

• A convenient 2D unit cell (parallelogram that tiles the plane).
• The corresponding 2D Bravais lattice type (e.g. square, rectangular, hexagonal).

For each packing, the packing density (fraction of area occupied by the circles) was calculated by relating the area of the chosen unit cell to the area of the circles within it.

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2.2 Construction of cubic unit cells with ping-pong balls

2.2.1 Simple cubic (SC)

• Four ping-pong balls were glued together in a square, forming a layer of 2 × 2 spheres.
• A second identical layer was prepared.
• The two layers were glued directly on top of each other to form a 2 × 2 × 2 cube of spheres.
• Once set, the cube was examined and sketched, highlighting edges of the unit cell drawn between the centres of corner atoms.
• The number of atoms “belonging” to the unit cell (accounting for corner sharing) was calculated.

2.2.2 Body-centred cubic (BCC)

• Three balls were aligned in a straight line and glued.
• Two additional balls were glued at right angles to the central ball to form a cross-shaped arrangement of five balls in a single plane.
• A ball was glued above the central ball, forming a triangle when viewed from the side.
• After drying, the structure was inverted and completed symmetrically to form a cube with one ball at the body centre.
• The resulting arrangement was adjusted to make all edges equal and cube-like.
• A sketch was made marking the unit cell and the body-centred atom.
• The number of atoms per unit cell was calculated.

2.2.3 Face-centred cubic (FCC)

• Two cross-shaped “BCC-like” 5-ball units (central + 4 in a cross) were prepared as for the intermediate stage of the BCC.
• Four balls were arranged and glued in a square to form a 2 × 2 face.
• One 5-ball unit was glued underneath this square and one on top, aligned so that each face of the resulting cube contained a ball at its centre.
• The model was checked to ensure a cubic shape with equal edges and rotational symmetry by 90° about each axis.
• Voids in the structure were located visually (tetrahedral and octahedral gaps), and the number of atoms per unit cell was determined.

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2.3 Ionic and covalent crystal structures with ball-and-stick models

A Minit Lattices–type model kit was used to build ball-and-stick models.

2.3.1 Sodium chloride (rock salt)

• Green and grey octahedral atomic centres were chosen to represent Cl⁻ and Na⁺, respectively (or vice versa, as long as consistent).
• Short straws were used as bonds. Rows of three centres were built in alternating colour patterns (green–grey–green and grey–green–grey).
• Nine rows were assembled into a 3 × 3 grid, ensuring each green centre was bonded only to greys and vice versa.
• The pattern was extended in three dimensions to at least 3 × 3 × 3 unit cells, maintaining six-fold (octahedral) coordination.
• The structure was compared to a standard NaCl model from online sources for correctness.

2.3.2 Fluorite / anti-fluorite

• 8-coordinate centres were used for the cations (Ca²⁺-like) and tetrahedral centres for anions (F⁻-like), or vice versa for anti-fluorite.
• Four of the shortest connectors were attached to each tetrahedral atom, then four such tetrahedral units were attached to a single 8-coordinate centre on one side to reproduce the basic fluorite unit.
• These units were connected and replicated to generate a 3 × 3 × 3 array of unit cells.

2.3.3 Sphalerite (ZnS)

• Equal numbers of tetrahedral and 8-coordinate centres were selected, again with short connectors.
• Only half of the available tetrahedral sites in the FCC anion framework were occupied, reproducing the sphalerite structure.
• The arrangement was built to 3 × 3 × 3 unit cells.

2.3.4 Diamond

• Only tetrahedral centres and short connectors were used.
• A diamond-type network was assembled by ensuring each atom had four tetrahedrally arranged neighbours.
• The structure was extended to approximately 20 × 20 × 20 cm, limited by the number of connectors, to reveal repeating units and planes.

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2.4 Edge dislocation

• An 8 × 3 grid of simple cubic unit cells was constructed as a base using identical connectors and atomic centres.
• A second identical layer was added on top.
• For subsequent layers, two 3 × 3 blocks were built with a 2 × 3 gap in the middle, representing a missing row of unit cells.
• This pattern was repeated for several layers until the 3 × 3 blocks could be connected across the gap with a connector the same length as the unit cell edge, introducing an extra half-plane of atoms.
• Connectors were adjusted (cut where necessary) to stabilise the structure; the final model resembled an extra half-plane terminating inside the crystal.

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3. Results

3.1 2D Bravais lattice identification and packing densities

From the provided tessellating image (Figure 1 in the script), a repeating pattern of equivalent points (e.g. intersection of three petals of the same colour) was identified. Connecting nearest equivalent points defined a primitive hexagonal lattice, consistent with a 2D Bravais lattice exhibiting six-fold rotational symmetry.

Using ping-pong balls on a flat surface, two distinct close-packing arrangements were identified:

1. Square packing (square lattice)

   • Lattice constant: $a = 2r$, where $r$ is the ball radius.
   • Area of unit cell: $$A_{\text{square}} = a^2 = (2r)^2 = 4r^2$$
   • Each cell contains one full circle (four quarters from four corners).
   • Area occupied by spheres: $A_{\text{atoms}} = \pi r^2$.
   • Packing density: $$\eta_{\text{square}} = \frac{A_{\text{atoms}}}{A_{\text{square}}} = \frac{\pi r^2}{4r^2} = \frac{\pi}{4} \approx 0.79$$

2. Hexagonal (triangular) packing (hexagonal lattice)

   • Centres of spheres form a triangular lattice with nearest-neighbour distance $a = 2r$.
   • A convenient primitive cell is a rhombus with side $a$ and internal angle 60°.
   • Area of unit cell: $$A_{\text{hex}} = a^2 \sin 60^\circ = (2r)^2 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}\,r^2$$
   • Each primitive cell contains one full circle.
   • Packing density: $$\eta_{\text{hex}} = \frac{\pi r^2}{2\sqrt{3}\,r^2} = \frac{\pi}{2\sqrt{3}} \approx 0.91$$

Thus, both the square and hexagonal 2D Bravais lattices can be arranged so that each circle contacts its neighbours, but the hexagonal packing is more efficient, giving the highest packing density.

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3.2 Space-filling cubic unit cells

From visual inspection and counting of equivalently shared atoms:

• Simple cubic (SC)

  • Atoms at all eight corners of the cube.
  • Each corner atom is shared among 8 neighbouring unit cells.
  • Number of atoms per unit cell: $$N_{\text{SC}} = 8 \times \frac{1}{8} = 1$$

• Body-centred cubic (BCC)

  • Atoms at eight corners and one at the body centre.
  • Number of atoms per unit cell: $$N_{\text{BCC}} = 8 \times \frac{1}{8} + 1 = 2$$

• Face-centred cubic (FCC)

  • Atoms at eight corners and one at the centre of each of the six faces.
  • Each face-centred atom is shared between two unit cells.
  • Number of atoms per unit cell: $$N_{\text{FCC}} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4$$

In the FCC model, visual inspection revealed both tetrahedral voids (small gaps between four spheres forming a tetrahedron) and octahedral voids (gaps between six spheres forming an octahedron). These voids correspond to potential interstitial sites in real crystals.

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3.3 Structures of ionic and covalent lattices

From the ball-and-stick models:

• NaCl (rock salt)

  • One ion type (e.g. Cl⁻) forms an FCC lattice.
  • The other ion type (Na⁺) occupies all octahedral voids.
  • Each Na⁺ is surrounded by six Cl⁻ and vice versa (6:6 coordination).
  • Unit cell contents: 4 Na⁺ and 4 Cl⁻ (overall 4 “NaCl” formula units per unit cell).

• Fluorite / anti-fluorite

  • In fluorite (e.g. CaF₂), Ca²⁺ forms an FCC lattice and F⁻ ions occupy all tetrahedral sites.
  • In anti-fluorite, the roles of cations and anions are reversed.
  • Coordination: cations are 8-coordinate; anions are 4-coordinate.

• Sphalerite (ZnS)

  • One species (e.g. S²⁻) forms an FCC lattice.
  • The other species (Zn²⁺) occupies exactly half of the tetrahedral holes.
  • Coordination is 4:4 with tetrahedral geometry around each atom.

• Diamond

  • All atoms are equivalent and tetrahedrally coordinated (sp³-like).
  • Structure can be viewed as two interpenetrating FCC lattices offset along a body diagonal.
  • The extended model displayed clear {111} close-packed planes and repeating tetrahedral motifs.

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3.4 Edge dislocation

The edge dislocation model appeared as a simple cubic lattice in which an extra half-plane of atoms terminated within the crystal. Locally:

• Above and below the inserted half-plane, bond lengths and angles were distorted relative to the ideal SC lattice.
• Far from the dislocation line, the lattice reverted to an undistorted regular array.

The dislocation line itself ran along the edge where the extra half-plane ended, visualising the core of an edge dislocation.

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4. Discussion

4.1 2D packing and its implications

The calculation of packing densities showed that:

• Square packing has a density of $\eta_{\text{square}} = \pi/4 \approx 0.79$.
• Hexagonal packing has a higher density of $\eta_{\text{hex}} = \pi/(2\sqrt{3}) \approx 0.91$.

This confirms that the hexagonal (triangular) Bravais lattice provides the densest 2D packing of identical circles. This result mirrors close packing in real materials: many metals adopt close-packed arrangements (HCP or CCP/FCC) in 3D because these maximise the number of nearest neighbours and minimise empty space, leading to lower potential energy and higher density.

The observation that different valid unit cells can be drawn on the same pattern (e.g. primitive vs centred choices) reinforces a central idea of crystallography: the unit cell is not unique. However, “conventional” cells are often chosen to reflect the maximum symmetry, which is why hexagonal or cubic cells are typically preferred for highly symmetric lattices.

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4.2 Comparison of cubic unit cells

The SC, BCC and FCC ping-pong ball models highlighted how the number of atoms per unit cell and packing efficiency increase from SC to FCC:

• SC has only one atom per unit cell and low coordination (each atom has 6 nearest neighbours).
• BCC has two atoms per unit cell and a higher coordination number (8).
• FCC has four atoms per unit cell and the highest coordination number of the three (12).

Although explicit 3D packing densities were not calculated in this lab, known results show that FCC (and HCP) are more densely packed than BCC, which in turn is denser than SC. The presence of well-defined tetrahedral and octahedral voids in FCC explains why many ionic solids can be understood as “host” FCC lattices with the other ion species occupying interstitial sites (as in NaCl, fluorite and sphalerite).

Physically handling the models helped to link abstract crystallographic descriptions to 3D geometry: for example, rotating the FCC model made it easier to see different families of planes ({100}, {110}, {111}) as closely packed or more open planes.

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4.3 Relationship between structure type and bonding

The ionic structures (NaCl, fluorite, sphalerite) and covalent diamond structure illustrate how bonding type influences structure:

• NaCl: The 6:6 coordinated rock salt structure balances electrostatic attraction between oppositely charged ions, while keeping like-charged ions as far apart as possible. The FCC–octahedral void picture shows how both sub-lattices interpenetrate with high symmetry.

• Fluorite / anti-fluorite: The 8:4 coordination pattern reflects a different size and charge ratio between cations and anions. Occupation of all tetrahedral sites by the smaller species allows efficient packing while maintaining charge neutrality.

• Sphalerite: The 4:4 tetrahedral coordination is characteristic of both covalent (diamond) and some ionic/covalent mixed structures. Sphalerite can be viewed as an “ionic cousin” of diamond, with Zn–S bonds in place of C–C bonds.

• Diamond: Purely covalent sp³ hybridisation leads to a very rigid 3D network. The tetrahedral coordination and relatively low packing density (compared with close-packed metals) contribute to diamond’s extreme hardness but also its brittleness.

By rotating and inspecting these models, it became clearer how coordination numbers, void types and sub-lattices emerge naturally from the underlying simple lattices (typically FCC or simple cubic).

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4.4 Edge dislocations and their structural consequences

The edge dislocation model provided a tangible picture of a line defect in a crystal. The extra half-plane of atoms produced:

• Compressive regions above the dislocation core (atoms pushed closer together).
• Tensile regions below the core (atoms further apart).

Although the experiment did not directly measure mechanical properties, the model highlights why dislocations greatly influence plastic deformation:

• Dislocations allow planes of atoms to “slip” past one another progressively without requiring all bonds across a plane to break simultaneously.
• The motion of such defects under applied stress is easier than breaking the perfect lattice at once, lowering the yield stress of the material.

Understanding where unit cells “break down” near defects is important for interpreting real crystallographic data (e.g. broadening in X-ray diffraction peaks) and for designing stronger materials through dislocation pinning.

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5. Conclusions

1. A repeating pattern of equivalent points in 2D can be described in terms of a 2D Bravais lattice and unit cell; in the supplied figure, a primitive hexagonal lattice was an appropriate description.

2. Two key 2D close-packed arrangements of identical spheres—square and hexagonal packings—were constructed and analysed. The calculated packing densities, $\eta_{\text{square}} = \pi/4 \approx 0.79$ and $\eta_{\text{hex}} = \pi/(2\sqrt{3}) \approx 0.91$, showed that hexagonal packing is the most efficient.

3. Space-filling models of simple cubic, body-centred cubic and face-centred cubic unit cells demonstrated how the number of atoms per unit cell increases from 1 (SC) to 2 (BCC) to 4 (FCC), and how FCC contains distinct tetrahedral and octahedral voids that are crucial for ionic structures.

4. Ball-and-stick models of NaCl, fluorite/anti-fluorite, sphalerite and diamond connected theoretical structure descriptions to tangible 3D arrangements, highlighting the relationship between lattice type, coordination number, void occupation and bonding.

5. The edge dislocation model illustrated how an extra half-plane of atoms distorts the ideal crystal lattice locally and provided a physical picture of a line defect important for mechanical behaviour.

Overall, the laboratory successfully met its aims of developing practical skills in representing and interpreting crystal structures, deepening understanding of unit cells and Bravais lattices, and linking microscopic structural features to macroscopic material properties.

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