Crystalline Solids and Lattice Energy
Crystalline Solids
Solids are rigid substances with definite shape and volume whose constituent atoms, ions, or molecules are closely packed and held together by strong cohesive forces. Based on the regularity of their internal arrangement, solids are classified into crystalline solids and amorphous solids.
Crystalline Solids: Solids in which atoms, ions, or molecules are arranged in a definite, repeating three-dimensional geometric pattern that extends throughout the entire crystal.
Amorphous Solids: Solids whose constituent particles lack a regular orderly arrangement. Glass, plastics, rubber, and glue are examples. They may contain small ordered regions called crystallites but lack long-range regularity.
Formation of Amorphous Solids: Many crystalline solids can be converted to amorphous solids by melting and then rapidly cooling the molten mass, so the constituent particles do not find time to arrange themselves.
Crystalline solids display a set of distinctive properties that arise directly from their ordered internal arrangement. These properties are useful for identifying and classifying crystalline materials.
Geometrical Shape: All crystalline solids have a definite, distinctive shape due to the orderly three-dimensional arrangement of particles. The interfacial angles at which crystal surfaces intersect are always the same, regardless of the crystal's overall size. Even when ground to powder, the characteristic angles persist.
Sharp Melting Points: Crystalline solids melt at a specific, well-defined temperature. Unlike amorphous solids, which soften over a temperature range and lack definite heats of fusion, crystalline solids transition sharply from solid to liquid.
Cleavage Planes: When crystalline solids are broken, they fracture along definite planes called cleavage planes. These planes are inclined to one another at characteristic angles that vary from one solid to another.
[Anisotropy]: Some crystalline solids show variation in physical properties depending on the direction of measurement. Properties like refractive index, thermal expansion, and electrical and thermal conductivities can differ along different crystal axes. For example, graphite conducts electricity far better parallel to its layers than perpendicular to them.
Symmetry: The repetition of faces, angles, or edges when a crystal is rotated through 360° along its axis. Symmetry elements include the center of symmetry, plane of symmetry, and axis of symmetry.
Crystal Habit: The shape in which a crystal usually grows. It depends on growth conditions — a cubic NaCl crystal becomes needle-like when 10% urea is present as an impurity.
Allotropy — Elements in Multiple Crystalline Forms
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Sulphur (S) — rhombic and monoclinic forms
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Carbon (C) — cubic (diamond) and hexagonal (graphite)
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Tin (Sn) — grey tin (cubic) and white tin (tetragonal)
Transition Temperature Examples
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Grey tin (cubic) ⇌ White tin (tetragonal)
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Sulphur S₈ (rhombic) ⇌ Sulphur S₈ (monoclinic)
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KNO₃ (orthorhombic) ⇌ KNO₃ (rhombohedral)
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Na₂SO₄·10H₂O (hydrated) ⇌ Na₂SO₄ (anhydrous) + 10 H₂O
Crystal Lattice and Unit Cell
A crystal lattice (or space lattice) is the three-dimensional array of points in a crystal, where each point represents the position of an atom, ion, or molecule. These positions are called lattice points or lattice sites. Every lattice point in a crystal has an identical environment.
The unit cell is the smallest repeating block of the crystal lattice that contains all the characteristic structural features of the entire crystal. If you know the exact arrangement of particles in a unit cell, you know their arrangement throughout the whole crystal. The unit cell is specified by six parameters called crystallographic elements: three edge lengths and three inter-axial angles.
$$a,\; b,\; c \quad \alpha,\; \beta,\; \gamma$$
The six crystallographic elements that define the size and shape of a unit cell
$a,\; b,\; c$=Lengths of the three edges along the x, y, and z axes(pm or Å)
$\alpha$=Angle between edges $b$ and $c$(degrees (°))
$\beta$=Angle between edges $a$ and $c$(degrees (°))
$\gamma$=Angle between edges $a$ and $b$(degrees (°))
Corner Sharing: A particle at a corner is shared by 8 adjacent unit cells, contributing $\frac{1}{8}$ per unit cell.
Edge Sharing: A particle on an edge is shared by 4 unit cells, contributing $\frac{1}{4}$ per unit cell.
Face Sharing: A particle on a face is shared by 2 unit cells, contributing $\frac{1}{2}$ per unit cell.
Body Centre: A particle at the body centre belongs entirely to one unit cell, contributing 1.
Crystals are classified into seven crystal systems based on the dimensions and angles of their unit cells. Each system has a unique combination of edge lengths ($a$, $b$, $c$) and inter-axial angles ($\alpha$, $\beta$, $\gamma$).
The Seven Crystal Systems
1
Cubic — $a=b=c$, $\alpha=\beta=\gamma=90°$ — Fe, Cu, Ag, Au, NaCl, NaBr, Diamond
2
Tetragonal — $a=b \neq c$, $\alpha=\beta=\gamma=90°$ — SnO₂, MnO₂
3
Orthorhombic — $a \neq b \neq c$, $\alpha=\beta=\gamma=90°$ — Rhombic sulphur, BaSO₄, K₂SO₄
4
Monoclinic — $a \neq b \neq c$, $\alpha=\gamma=90°$, $\beta \neq 90°$ — Borax, Na₂SO₄·10H₂O
5
Hexagonal — $a=b \neq c$, $\alpha=\beta=90°$, $\gamma=120°$ — Graphite, ZnO, CdS, Ice, Zn, Cd
6
Rhombohedral (Trigonal) — $a=b=c$, $\alpha=\beta=\gamma \neq 90°$ (<120°) — Bi, Al₂O₃, NaNO₃, KNO₃
7
Triclinic — $a \neq b \neq c$, $\alpha \neq \beta \neq \gamma \neq 90°$ — H₃BO₃, CuSO₄·5H₂O
Ionic Crystals
Ionic solids are crystalline solids in which positively charged cations and negatively charged anions are held together by strong electrostatic forces of attraction called ionic bonds. Examples include NaCl, KBr, MgO, and CsF. Ionic crystals do not exist as individual neutral molecules — the entire crystal is one continuous lattice of alternating cations and anions.
Hardness and Stability: Very high energy is required to separate cations from anions against the electrostatic forces, making ionic crystals very hard and stable.
High Melting and Boiling Points: The strong non-directional ionic bonds result in high melting and boiling points and low volatility.
Non-Directional Bonding: Since ionic bonds are non-directional, ions pack closely in a systematic alternating arrangement that minimises space.
Formula Mass, Not Molecular Mass: Because ionic solids do not exist as discrete molecules, we refer to their formula mass rather than molecular mass.
The structure and geometry of an ionic solid depend on the radius ratio of the cation to the anion. This ratio determines how ions can pack together and fixes the coordination number — the number of nearest neighbours of opposite charge surrounding each ion.
$$\text{Radius ratio} = \frac{r^+}{r^-}$$
The ratio of the cation radius to the anion radius, which determines the geometry and coordination number of the ionic crystal
$r^+$=Radius of the cation(pm or Å)
$r^-$=Radius of the anion(pm or Å)
$0.414 < \frac{r^+}{r^-} < 0.732$
→Octahedral geometry with coordination number 6 (e.g., NaCl)
$\frac{r^+}{r^-} \geq 0.732$
→Cubic geometry with coordination number 8 (e.g., CsCl)
Shape Determination: The geometry of an ionic crystal is independent of the chemical nature of the atoms and depends only on the number of ions and their radius ratio. NaCl and CsF have the same geometry because their radius ratios are the same.
Coordination Number in NaCl: Each Na⁺ ion is surrounded by six Cl⁻ ions at the corners of a regular octahedron, so the coordination number is 6. Each Cl⁻ is similarly surrounded by six Na⁺ ions.
Ionic Distance in NaCl: The distance between two nearest Cl⁻ ions is 5.63 Å, so the distance between adjacent Na⁺ and Cl⁻ ions is $\frac{5.63}{2} = 2.815$ Å.
Sodium chloride (NaCl) has a face-centred cubic structure. The unit cell contains exactly 4 NaCl formula units. Each Na⁺ is bonded equally to all six surrounding Cl⁻ ions — there are no discrete NaCl molecules in the solid.
Chloride Ion Count: 8 Cl⁻ at corners (each shared by 8 cells: $\frac{1}{8}$ each) plus 6 Cl⁻ at face centres (each shared by 2 cells: $\frac{1}{2}$ each): $\frac{8}{8} + \frac{6}{2} = 1 + 3 = 4$ Cl⁻ per unit cell.
Sodium Ion Count: 12 Na⁺ at edge centres (each shared by 4 cells: $\frac{1}{4}$ each) plus 1 Na⁺ at the body centre (fully owned): $\frac{12}{4} + 1 = 3 + 1 = 4$ Na⁺ per unit cell.
No Discrete Molecules: All six Cl⁻ ions around each Na⁺ are at the same distance. NaCl exists as independent molecules only in the vapour phase; in the solid it has formula units of NaCl.
Ionic solids have several characteristic physical properties that follow directly from their structure and bonding.
Electrical Conductivity: Ionic crystals do not conduct electricity in the solid state because ions remain tightly held in fixed positions. They conduct only when molten or dissolved, where ions become free to move.
Brittleness: Ionic crystals are highly brittle. They consist of parallel layers of alternating cations and anions. When external force causes one layer to slide, like charges align face-to-face and repel, causing the crystal to fracture.
Solubility: Ionic solids dissolve in polar solvents (e.g., water) where they dissociate into ions and undergo fast ionic reactions.
High Density: Due to close packing of ions, ionic solids generally have high density.
Molecular Crystals
Molecular solids are crystalline solids whose constituent particles are individual polar or non-polar molecules (or atoms, as in solidified noble gases). The particles are held together by weak intermolecular forces — dipole-dipole interactions, van der Waals (London dispersion) forces, or hydrogen bonding — which are much weaker than ionic, covalent, or metallic bonds.
Polar Molecular Solids: Crystals of polar molecules such as ice (H₂O) and sugar. They generally have higher melting and boiling points than non-polar molecular solids.
Non-Polar Molecular Solids: Crystals of non-polar molecules such as iodine (I₂), sulphur (S₈), phosphorus, and solid CO₂ (dry ice). They have the lowest melting and boiling points among all crystalline solid types.
Ionic vs Molecular Crystals — Key Differences
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Particles: Ions (ionic) vs molecules or atoms (molecular)
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Binding forces: Strong electrostatic (ionic) vs weak intermolecular (molecular)
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Hardness: Hard and rigid (ionic) vs soft and compressible (molecular)
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Melting point: High (ionic) vs low (molecular)
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Volatility: Low (ionic) vs high (molecular)
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Electrical conductivity: Conducts when molten or dissolved (ionic) vs non-conductor (molecular)
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Density: High (ionic) vs low (molecular)
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Examples: NaCl, MgO, KBr (ionic) vs I₂, CO₂, ice, sugar (molecular)
The weak intermolecular forces in molecular solids give them a distinct set of physical properties that contrast sharply with ionic and covalent crystals.
Softness and Compressibility: Molecular crystals are soft and easily compressible because the intermolecular forces holding molecules together are very weak.
Low Melting and Boiling Points: Minimal energy is needed to overcome the weak intermolecular attractions, so molecular solids are volatile and melt or boil at low temperatures.
Electrical Insulation: No free electrons or mobile ions are present, making molecular solids poor conductors of electricity.
Solubility: Polar molecular crystals dissolve in polar solvents (sugar in water); non-polar molecular crystals dissolve in non-polar solvents (iodine in CCl₄).
Transparency: Many molecular crystals are transparent to light.
Lattice Energy
Lattice energy is the energy released when one mole of an ionic crystal is formed from its constituent gaseous ions. Equivalently, it is the energy required to separate one mole of a solid ionic compound into isolated gaseous ions. It is expressed in kJ mol⁻¹.
$$Na^+(g) + Cl^-(g) \rightarrow NaCl(s) \quad \Delta H = -787 \; \text{kJ mol}^{-1}$$
The lattice energy of NaCl — energy released when one mole of solid NaCl forms from gaseous Na⁺ and Cl⁻ ions
$Na^+(g)$=Gaseous sodium ion(kJ mol⁻¹)
$Cl^-(g)$=Gaseous chloride ion(kJ mol⁻¹)
$\Delta H$=Enthalpy change (lattice energy)(kJ mol⁻¹)
Formation (gaseous ions → solid)
→$\Delta H$ is negative (exothermic, energy is released)
Dissociation (solid → gaseous ions)
→$\Delta H$ is positive (endothermic, energy must be absorbed)
The magnitude of lattice energy depends on the sizes and charges of the ions. Understanding these trends allows you to predict and compare lattice energies across different compounds.
Cation Size Effect: Keeping the anion constant, lattice energy decreases (becomes less negative) as the cation size increases. Larger cations reduce the tightness of packing, weakening electrostatic attraction.
Anion Size Effect: Keeping the cation constant, lattice energy decreases as the anion size increases. NaCl (−787) > NaBr (−728) > NaI (−690) because Cl⁻ < Br⁻ < I⁻ in ionic radius.
Charge Effect: Ions with higher charges produce much stronger electrostatic attraction. MgO (Mg²⁺, O²⁻) has a far greater lattice energy than NaCl (Na⁺, Cl⁻) because the charges are doubled.
Lattice Energies of Selected Ionic Compounds
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LiCl: −833 kJ mol⁻¹
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NaF: −895 kJ mol⁻¹
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NaCl: −787 kJ mol⁻¹
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NaBr: −728 kJ mol⁻¹
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NaI: −690 kJ mol⁻¹
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KCl: −690 kJ mol⁻¹
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KBr: −665 kJ mol⁻¹
A more negative lattice energy indicates stronger ionic bonding and a more stable crystal. Compounds with high lattice energy tend to have higher melting points, greater hardness, and lower solubility.
Application Strategy
Worked Examples
Problem Statement
In the NaCl unit cell, Cl⁻ ions occupy the 8 corners and 6 face centres, while Na⁺ ions occupy the 12 edge centres and 1 body centre. Calculate the number of NaCl formula units per unit cell.
unit cellionic solidcoordination number
Strategic Toolkit
$$n = \frac{N_{c}}{8} + \frac{N_{e}}{4} + \frac{N_{f}}{2} + N_{b}$$
Step 1
Count effective Cl⁻ ions
Corner Cl⁻ ions are each shared by 8 unit cells, and face-centre Cl⁻ ions are each shared by 2 unit cells. Add their contributions.
$$n_{Cl^-} = \frac{8}{8} + \frac{6}{2} = 1 + 3 = 4$$
Step 2
Count effective Na⁺ ions
Edge-centre Na⁺ ions are each shared by 4 unit cells, and the body-centre Na⁺ belongs entirely to this cell.
$$n_{Na^+} = \frac{12}{4} + 1 = 3 + 1 = 4$$
Step 3
Determine formula units
Since there are equal numbers of Na⁺ and Cl⁻ ions, the unit cell contains 4 NaCl formula units.
$$\text{NaCl formula units per unit cell} = 4$$
Problem Statement
Arrange NaF, NaCl, NaBr, and NaI in order of increasing magnitude of lattice energy (most negative first). Explain the trend using ionic sizes.
lattice energyionic solid
Strategic Toolkit
$$U \propto \frac{1}{r^+ + r^-}$$
Step 1
Identify the variable factor
All four compounds share the same cation (Na⁺). The anions increase in size from F⁻ to Cl⁻ to Br⁻ to I⁻.
$$r_{F^-} < r_{Cl^-} < r_{Br^-} < r_{I^-}$$
Step 2
Apply the size-energy relationship
As anion size increases, the interionic distance increases, packing becomes less tight, and electrostatic attraction weakens. Lattice energy decreases (becomes less negative).
$$\text{Lattice energy: } NaF > NaCl > NaBr > NaI$$
Step 3
Verify with experimental data
The experimental lattice energies confirm this trend exactly.
$$\boxed{-895 > -787 > -728 > -690 \; \text{kJ mol}^{-1}}$$
Problem Statement
The ionic radius of Na⁺ is 102 pm and that of Cl⁻ is 181 pm. Calculate the radius ratio and predict the coordination number and geometry of NaCl.
coordination numberionic solidcrystal lattice
Strategic Toolkit
$$\text{Radius ratio} = \frac{r^+}{r^-}$$
Step 1
Calculate the radius ratio
Divide the cation radius by the anion radius to find the radius ratio.
$$\frac{r^+}{r^-} = \frac{102}{181} = 0.564$$
Step 2
Determine the geometry
The ratio 0.564 falls in the range 0.414–0.732, which corresponds to octahedral geometry.
$$0.414 < 0.564 < 0.732 \implies \text{Octahedral, CN} = 6$$
Step 3
Confirm with NaCl structure
This prediction is consistent with the known NaCl structure, where each Na⁺ is surrounded by six Cl⁻ ions in an octahedral arrangement, and each Cl⁻ is surrounded by six Na⁺ ions.
$$\text{NaCl: octahedral geometry, coordination number } = 6$$
Memorize formulas
Lattice Energy (Formation Reaction)
Energy released when 1 mole of ionic solid forms from gaseous ions; always exothermic (negative); units: kJ mol⁻¹; NaCl = −787 kJ mol⁻¹; more negative = stronger ionic bonding
Formula
$$M^+(g) + X^-(g) \rightarrow MX(s)$$
Radius Ratio
Determines ionic crystal geometry and coordination number; r⁺/r⁻ = 0.414–0.732 gives octahedral (CN = 6, e.g., NaCl); r⁺/r⁻ ≥ 0.732 gives cubic (CN = 8, e.g., CsCl)
Formula
$$\frac{r^+}{r^-}$$
Unit Cell Ion Counting
Corner ions shared by 8 cells (÷8), edge ions by 4 (÷4), face ions by 2 (÷2), body ion unshared (÷1); NaCl unit cell = 4 formula units
Formula
$$n = \frac{N_{c}}{8} + \frac{N_{e}}{4} + \frac{N_{f}}{2} + N_{b}$$
Lattice Energy Trend
Lattice energy increases (more negative) with smaller ionic radii and higher ionic charges; compare by keeping one ion constant
Formula
$$U \propto \frac{z^+ \cdot z^-}{r^+ + r^-}$$
Review
1
Which of the following is a defining characteristic of crystalline solids?
Ans
Their constituent particles are arranged in a regular, repeating three-dimensional pattern
Review
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Recall definition of crystalline solid
A crystalline solid has its constituent particles arranged in a definite geometric pattern that repeats throughout the entire solid
Regular, repeating 3D arrangement
2
The property by which crystalline solids show different physical properties along different directions is called:
Ans
Anisotropy
Review
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Identify the property
Directional dependence of physical properties in a crystal arises from its non-uniform internal structure along different axes
Anisotropy
3
What is the difference between a crystal lattice and a unit cell?
Ans
A crystal lattice is the repeating pattern of points in space, while a unit cell is the smallest repeating unit that generates the entire lattice
Review
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Define crystal lattice
A crystal lattice is the geometric arrangement of lattice points in three-dimensional space
Array of points in space
Define unit cell
The unit cell is the smallest repeating structural unit of the crystal that contains all symmetry elements of the lattice
Smallest repeating unit of the lattice
4
A lattice point in a crystal represents:
Ans
The position of an atom, ion, or molecule (or group of these) in the crystal structure
Review
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Understand lattice point meaning
Each lattice point marks the location of one or more atoms, ions, or molecules that repeat throughout the crystal
Position of atom/ion/molecule (or group)
5
The coordination number of a particle in a crystal is defined as:
Ans
The number of nearest neighbour particles surrounding a given particle
Review
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Define coordination number
The coordination number counts the number of immediately adjacent particles (nearest neighbours) to any given particle in the crystal
Number of nearest neighbours
6
In a NaCl crystal, each unit cell contains 4 Na⁺ ions and 4 Cl⁻ ions. If Cl⁻ ions occupy the corner and face-centre positions while Na⁺ ions occupy the edge-centre and body-centre positions, how many formula units of NaCl does one unit cell contain?
Ans
4
Review
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Apply the ion counting formula
$n = \frac{N_c}{8} + \frac{N_e}{4} + \frac{N_f}{2} + N_b$ where $N_c$ = corner, $N_e$ = edge-centre, $N_f$ = face-centre, $N_b$ = body-centre
$$n = \frac{N_c}{8} + \frac{N_e}{4} + \frac{N_f}{2} + N_b$$
Count Cl⁻ ions
8 corners × $\frac{1}{8}$ each + 6 face-centres × $\frac{1}{2}$ each = 1 + 3 = 4 Cl⁻ ions
$$4 \text{ Cl}^- \text{ ions}$$
Determine formula units
With 4 Na⁺ and 4 Cl⁻ per unit cell, the unit cell contains 4 NaCl formula units
4 formula units of NaCl
7
In a CsCl crystal, each Cs⁺ ion is surrounded by 8 Cl⁻ ions at the corners of a cube. What is the coordination number of this structure?
Ans
8
Review
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Visualise the CsCl structure
Cs⁺ is at the body centre; Cl⁻ ions are at all 8 corners of the cubic unit cell
8 nearest neighbours
State coordination number
Each ion is surrounded by 8 ions of opposite charge
Coordination number = 8
8
Which of the following best describes molecular crystals?
Ans
They are composed of neutral molecules held together by intermolecular forces such as van der Waals forces or hydrogen bonding
Review
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Identify molecular crystal bonding
Molecular crystals have strong covalent bonds within each molecule but weak intermolecular forces between molecules
Neutral molecules held by intermolecular forces
9
For a cation of radius 58 pm and an anion of radius 220 pm, the radius ratio $\frac{r^+}{r^-}$ is approximately 0.26. What is the predicted coordination number and geometry of the resulting ionic crystal?
Ans
Coordination number 4, tetrahedral
Review
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Calculate the radius ratio
$\frac{r^+}{r^-} = \frac{58}{220} = 0.264$
$$\frac{r^+}{r^-} \approx 0.26$$
Apply the radius ratio rule
0.225–0.414 → tetrahedral (CN = 4); 0.414–0.732 → octahedral (CN = 6); 0.732–1.0 → body-centred cubic (CN = 8)
0.26 falls in the 0.225–0.414 range → tetrahedral, CN = 4
10
Given the lattice energy relationship $U \propto \frac{z^+ \cdot z^-}{r^+ + r^-}$, which of the following ionic compounds would have the greatest lattice energy?
Ans
MgO
Review
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Identify ionic charges
MgO: $z^+ = 2$, $z^- = 2$; NaCl, KBr, CsI: $z^+ = 1$, $z^- = 1$
$$z^+ \cdot z^- = 4 \text{ (MgO) vs. } 1 \text{ (others)}$$
Compare ionic radii
Mg²⁺ (72 pm) + O²⁻ (140 pm) is much smaller than the sum of radii for NaCl, KBr, or CsI
$$r^+ + r^- \text{ is smallest for MgO}$$
Apply the lattice energy relationship
Largest $z^+ \cdot z^-$ and smallest $r^+ + r^-$ both favour the greatest lattice energy for MgO
MgO has the greatest lattice energy
11
Lattice energy of an ionic compound is defined as the energy change that occurs when:
Ans
One mole of gaseous ions combine to form one mole of the solid ionic crystal
Review
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State the lattice energy reaction
Lattice energy corresponds to the formation of the solid crystal from its constituent gaseous ions
$$M^+(g) + X^-(g) \rightarrow MX(s)$$
12
Which statement correctly distinguishes ionic crystals from molecular crystals?
Ans
Ionic crystals are composed of cations and anions held by electrostatic forces; molecular crystals consist of neutral molecules held by intermolecular forces
Review
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Compare the two crystal types
Ionic crystals: cations + anions, electrostatic forces, high melting point. Molecular crystals: neutral molecules, intermolecular forces, low melting point
Key difference is the nature of constituent particles and bonding forces
13
Using the lattice energy trend $U \propto \frac{z^+ \cdot z^-}{r^+ + r^-}$, the lattice energy of NaF is 910 kJ/mol. If the ionic radii of Li⁺ and F⁻ are such that $r_{Li^+} + r_{F^-}$ is approximately 20% smaller than $r_{Na^+} + r_{F^-}$, and both have $z^+ = z^- = 1$, what is the approximate lattice energy of LiF?
Ans
1138 kJ/mol
Review
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Set up the proportion
Since charges are identical, $\frac{U_{LiF}}{U_{NaF}} = \frac{r_{Na^+} + r_{F^-}}{r_{Li^+} + r_{F^-}}$
$$\frac{U_{LiF}}{910} = \frac{1}{0.8}$$
Calculate the factor
The inverse of 0.80 is 1.25, meaning LiF's lattice energy is 1.25 times that of NaF
$$U_{LiF} = 910 \times 1.25$$
Compute the value
$910 \times 1.25 = 1137.5$ kJ/mol
$$U_{LiF} \approx 1138 \text{ kJ/mol}$$
14
A solid has atoms located at each corner of a cube. No atoms are present at any face centre, edge centre, or body centre. How many atoms does this unit cell contain?
Ans
1
Review
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Apply the ion counting formula
$n = \frac{N_c}{8} + \frac{N_e}{4} + \frac{N_f}{2} + N_b$ with $N_c = 8$, $N_e = 0$, $N_f = 0$, $N_b = 0$
$$n = \frac{8}{8} + 0 + 0 + 0 = 1$$
15
A unit cell has atoms at all 8 corners and at the centre of each of the 6 faces. Using the ion counting formula, how many atoms does this unit cell effectively contain?
Ans
4
Review
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Count corner contributions
8 corners × $\frac{1}{8}$ per corner = 1
$$\frac{8}{8} = 1$$
Count face-centre contributions
6 faces × $\frac{1}{2}$ per face = 3
$$\frac{6}{2} = 3$$
Sum the contributions
$n = 1 + 3 = 4$ atoms per unit cell
$$n = 4$$
Question 1 of 15
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1
Which of the following is a defining characteristic of crystalline solids?