1. What holds a crystal together?

Question 1

What are the 5 types of interactions that hold crystals together?

Answer 1

• Van der Waals interactions
• Ionic bonding
• Covalent bonding
• Metallic bonding
• Hydrogen bonding

Question 2

What are Van der Waals interactions?

Answer 2

The temporary, attractive interaction between two (electrically) neutral atoms. This occurs when one atom forms an instantaneous dipole which induces a dipole in another atom, lowering the potential energy of the atom pair.

Question 3

Why can neutral atoms form instantaneous dipoles?

Answer 3

They all have a rapidly fluctuating dipole moment which forms an instantaneous dipole at any moment in time.

Question 4

Give the equation for the electric field produced by an instantaneous dipole

Answer 4

E(R) = electric field
p₁ = instantaneous dipole
R = position

Question 5

Give the equation for the potential energy of Van der Waals interactions

Answer 5

U(R) = potential energy
α = polarisability
p₁ = instantaneous dipole
R = position
A = constant

Question 6

State the Pauli exclusion principle

Answer 6

Two or more electrons may not occupy the same quantum state.

Question 7

When is it possible for electron clouds to overlap?

Answer 7

When some electrons are promoted to higher quantum states, meaning that the Pauli exclusions principle is still obeyed.

Question 8

Give the equation for the repulsive force between atoms due to the Pauli exclusion principle

Answer 8

U(R) = potential energy
R = position
B = constant

Question 9

Define the Lennard-Jones potential

Answer 9

The total energy of interaction between two atoms, equal to the sum of the attractive (Van der Waals) and repulsive (Pauli exclusion) potentials.

Question 10

Give the equation for the Lennard-Jones potential in terms of A and B

Answer 10

U(R) = potential energy
R = position
A = constant
B = constant

Question 11

Give the equation for the Lennard-Jones potential in terms of ε and σ

Answer 11

U(R) = potential energy
R = position
ε = energy parameter
σ = range parameter

Question 12

What does ε represent in the Lennard-Jones potential?

Answer 12

The binding energy. It indicates the strength of the interaction between the atoms (depth of the potential energy well).

Question 13

What does σ represent in the Lennard-Jones potential?

Answer 13

The separation at which U(R) = 0. It indicates the approximate size of the atom (the radius of the ‘repulsive core’).

Question 14

What is the cohesive energy of a Van der Waals crystal?

Answer 14

The potential energy of the whole crystal, found by summing the Lennard-Jones potential over all atom pairs.

Question 15

Give the equation for the total cohesive energy of a Van der Waals crystal

Answer 15

U = cohesive energy
R = nearest neighbour distance
ε = energy parameter
σ = range parameter
N = number of atoms
p_ij = distance between reference atom i and any other atom j

Question 16

How can the equilibrium separation for two atoms experiencing Van der Waals interaction be found?

Answer 16

By differentiating the cohesive energy to find the force and setting the equation equal to 0 before rearranging for the separation distance.

Question 17

What are the 4 quantum numbers that are used to describe an atom?

Answer 17

n = principle quantum number (integer numbers)
l = angular momentum quantum number (n - 1)
mₗ = magnetic quantum number (-l, …, 0, …, +l)
mₛ = spin quantum number (± 1/2)

Question 18

Orbitals are filled in the order of increasing ___.

Answer 18

n + l

Question 19

Where two orbitals have the same value of ___, they are filled in order of increasing _.

Answer 19

n + l
n

Question 20

Define ionic bonding

Answer 20

A result of the electrostatic interaction between oppositely charged ions.

Question 21

Give 1 example of an ionically bonded molecule

Answer 21

Sodium chloride

Question 22

Give the equation the potential energy of electrostatic interactions

Answer 22

U = attractive potential energy
q = charge
R = position

Question 23

Give the equation for the total interaction potential of an ion pair

Answer 23

U(R) = total potential
R = position
B = constant
q = charge

Question 24

What is the Madelung constant?

Answer 24

A parameter that considers the position of all neighbouring ions in a crystal to work out an approximation of the cohesive energy of ionic crystals.

Question 25

Give the equation for the total interaction potential of an ion pair in terms of the Madelung constant

Answer 25

U(R) = total potential
R = position
B = constant
α = Madelung constant
q = charge

Question 26

How can the binding energy and the equilibrium separation of two ions be found graphically?

Answer 26

The local minimum of the graph represents the binding energy on the x-axis and the separation distance on the y-axis.

Question 27

How is the equilibrium separation calculated for 2 ionically bonded atoms?

Answer 27

By differentiating the total energy to find the force and setting the equation equal to 0 before rearranging for the separation distance.

Question 28

Define covalent bonding

Answer 28

A chemical bond that forms due to the sharing of electrons between atoms, forming new molecular orbitals.

Question 29

Give 5 examples of covalently bonded crystals

Answer 29

• Diamond
• Zinc-blende
• Graphite
• Silicon
• GaAs

Question 30

What is LCAO

Answer 30

Linear Combination of Atomic Orbitals: it is the method for describing orbitals.

Question 31

State the LCAO equation

Answer 31

ψ_i = molecular orbital
n = number of atomic orbitals
φ_n = atomic orbitals
c = weighted coefficient

Question 32

The number of molecular orbitals formed in a covalent bond is always _____ to the number of atomic orbitals involved.

Answer 32

Equal

Question 33

Describe the crystal structure of diamond

Answer 33

Each atom is bonded to 4 others in a tetrahedral geometry

Question 34

Describe the spatial distribution of molecular orbitals in diamond

Answer 34

Forms an sp³ molecular orbital
4 valence electrons (hybridisation) are bonded

1s² 2s² 2p² —> 1s² 2s¹ 2p³

Question 35

Describe the crystal structure of graphite

Answer 35

Question 36

Describe the spatial distribution of molecular orbitals in graphite

Answer 36

Forms an sp² molecular orbital
3 valence electrons (hybridisation) are bonded and 1 free electron (dangling bond)

1s² 2s² 2p² —> 1s² 2s¹ 2p³

Question 37

Define metallic bonding

Answer 37

Bonding between metal atoms in which the valence electrons are removed from the ion cores and are free to move about the crystal due to this delocalisation.

Question 38

What causes the cohesion of solids?

Answer 38

The attractive interaction between the electrons and the nuclei of atoms that make up a solid. The attraction is due to a specific type of interaction.