Experiment 1Bench 1MZU-PE / GON-1Ready

Exploring Crystal Structures

Crystallographic Model Stage

Mount a lattice model on the goniometer stage, index it along the principal directions and measure its packing.

Duration: 3 hours
Kittel: Chapter 1
Topics: Crystal latticeUnit cellBravais latticesAPF

Specimen viewing chamber

Goniometer stage · specimen: Polonium (Po)

ContactOverlap

Ready

Drag to rotate · scroll to zoomViewing Free

— a— b— c

Cell metrology

Direct readout from the mounted specimen

Atoms per cell

Coordination number

Sphere radius r/a

Volume filled

OVL

Lattice parameter a

Density

g/cm³

Specimen note

Lattice points at the cube corners only. Each corner atom is shared between eight cells, so the cell contains a single atom. Polonium is the only element that adopts it.

Nearest neighbour

3.359Å

Formula units / cell

Density

9157kg/m³

Model stage control

MZU-PE / GON-1

Ready

Sample carousel

Sphere size

Radius r/a0.3000

ContactOverlap

Hold Shift while dragging for the fine vernier.

Index along

Chamber

ONOFF

Cell edges

ONOFF

Bonds

ONOFF

Spin

Reset view

Sphere size

Key Equations

A crystal structure is a Bravais lattice plus a basis. The unit cell is the smallest volume that repeats to fill space.

\mathrm{APF}=\frac{n\cdot\tfrac{4}{3}\pi r^{3}}{a^{3}}

The stage reads out the filled volume fraction for whatever sphere radius you set. It equals the packing factor only when the spheres are just touching — that is what the Contact lamp is for.

\rho=\frac{n\,M_r}{N_A\,a^{3}}

Contact conditions: SC $2r=a$ , BCC $4r=a\sqrt{3}$ , FCC $4r=a\sqrt{2}$ , diamond $8r=a\sqrt{3}$ .

Procedure

0/4

ACompare all five structuresTurn the sample carousel through SC, BCC, FCC, NaCl and diamond. For each, wind the radius knob up until the Contact lamp lights, then record the reading. Blue spheres are the corner / A-type atoms; orange are the additional basis atoms.
BIndex the FCC cellLoad FCC and press the [111] index button, then [100]. Note the shape the face-centre atoms make when you look down the body diagonal, and how many atoms you can see on the square face.
CVerify the BCC packing factorWith BCC mounted, find the contact radius on the stage and check it against 4r = a√3. Then work the packing factor out by hand and compare it with the volume-filled readout.
DRock salt and diamondLoad NaCl: identify which colour is the smaller Na⁺ ion and count each ion's neighbours. Then load diamond and look at the tetrahedral geometry with the bonds switched on.

Table 1 — crystal structure properties

0 entries

Bring the spheres into contact before recording, so the volume-filled reading is the true packing factor.

#	Structure	Atoms/cell	CN	r/a	Filled fraction	a/ Å	Density/ kg/m³	Specimen
No readings yet — set the controls, then press “Record reading”.

Analysis & Reflection

Analysis questions

From your table, which structure has the highest packing factor and which the lowest? What does a higher packing factor tell you about how efficiently the atoms fill space?
FCC has CN = 12 while SC has CN = 6. How does coordination number relate to the ductility of a metal? Copper (FCC) is very ductile, tungsten (BCC) is more brittle — explain that trend.
Silicon has the diamond structure with CN = 4 and is a semiconductor; copper is FCC with CN = 12 and is a metal. What does the coordination number alone tell you about the type of bonding?
The stage gives a filled fraction of about 0.65 for rock salt when the ions touch, but 0.52 if you force both ions to the same size. Why does the ionic radius ratio matter so much?

Physics problems

A metal crystallises FCC with $a = 3.61\ \text{Å}$ and has a density of $8960\ \text{kg}\,\text{m}^{-3}$ . Identify it using $\rho = nM_r/(N_A a^3)$ , then check your answer against the stage readout for copper.
Explain what the basis is in the rock salt structure, and why NaCl is described as two interpenetrating FCC lattices.
In diamond each atom forms four covalent bonds. Switch the bonds on and describe how that tetrahedral geometry follows from the (¼,¼,¼) displacement of the second atom in the basis.
Why does the stage draw the BCC body-centre atom in a different colour from the corner atoms, even though every atom in pure iron is chemically identical?