1. What holds a crystal together? Flashcards
What are the 5 types of interactions that hold crystals together?
- Van der Waals interactions
- Ionic bonding
- Covalent bonding
- Metallic bonding
- Hydrogen bonding
What are Van der Waals interactions?
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.
Why can neutral atoms form instantaneous dipoles?
They all have a rapidly fluctuating dipole moment which forms an instantaneous dipole at any moment in time.
Give the equation for the electric field produced by an instantaneous dipole
E(R) = electric field
p₁ = instantaneous dipole
R = position
Give the equation for the potential energy of Van der Waals interactions
U(R) = potential energy
α = polarisability
p₁ = instantaneous dipole
R = position
A = constant
State the Pauli exclusion principle
Two or more electrons may not occupy the same quantum state.
When is it possible for electron clouds to overlap?
When some electrons are promoted to higher quantum states, meaning that the Pauli exclusions principle is still obeyed.
Give the equation for the repulsive force between atoms due to the Pauli exclusion principle
U(R) = potential energy
R = position
B = constant
Define the Lennard-Jones potential
The total energy of interaction between two atoms, equal to the sum of the attractive (Van der Waals) and repulsive (Pauli exclusion) potentials.
Give the equation for the Lennard-Jones potential in terms of A and B
U(R) = potential energy
R = position
A = constant
B = constant
Give the equation for the Lennard-Jones potential in terms of ε and σ
U(R) = potential energy
R = position
ε = energy parameter
σ = range parameter
What does ε represent in the Lennard-Jones potential?
The binding energy. It indicates the strength of the interaction between the atoms (depth of the potential energy well).
What does σ represent in the Lennard-Jones potential?
The separation at which U(R) = 0. It indicates the approximate size of the atom (the radius of the ‘repulsive core’).
What is the cohesive energy of a Van der Waals crystal?
The potential energy of the whole crystal, found by summing the Lennard-Jones potential over all atom pairs.
Give the equation for the total cohesive energy of a Van der Waals crystal
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
How can the equilibrium separation for two atoms experiencing Van der Waals interaction be found?
By differentiating the cohesive energy to find the force and setting the equation equal to 0 before rearranging for the separation distance.
What are the 4 quantum numbers that are used to describe an atom?
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)
Orbitals are filled in the order of increasing ___.
n + l
Where two orbitals have the same value of ___, they are filled in order of increasing _.
n + l
n
Define ionic bonding
A result of the electrostatic interaction between oppositely charged ions.
Give 1 example of an ionically bonded molecule
Sodium chloride
Give the equation the potential energy of electrostatic interactions
U = attractive potential energy
q = charge
R = position
Give the equation for the total interaction potential of an ion pair
U(R) = total potential
R = position
B = constant
q = charge
What is the Madelung constant?
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.
Give the equation for the total interaction potential of an ion pair in terms of the Madelung constant
U(R) = total potential
R = position
B = constant
α = Madelung constant
q = charge
How can the binding energy and the equilibrium separation of two ions be found graphically?
The local minimum of the graph represents the binding energy on the x-axis and the separation distance on the y-axis.
How is the equilibrium separation calculated for 2 ionically bonded atoms?
By differentiating the total energy to find the force and setting the equation equal to 0 before rearranging for the separation distance.
Define covalent bonding
A chemical bond that forms due to the sharing of electrons between atoms, forming new molecular orbitals.
Give 5 examples of covalently bonded crystals
- Diamond
- Zinc-blende
- Graphite
- Silicon
- GaAs
What is LCAO
Linear Combination of Atomic Orbitals: it is the method for describing orbitals.
State the LCAO equation
ψ_i = molecular orbital
n = number of atomic orbitals
φ_n = atomic orbitals
c = weighted coefficient
The number of molecular orbitals formed in a covalent bond is always _____ to the number of atomic orbitals involved.
Equal
Describe the crystal structure of diamond
Each atom is bonded to 4 others in a tetrahedral geometry
Describe the spatial distribution of molecular orbitals in diamond
Forms an sp³ molecular orbital
4 valence electrons (hybridisation) are bonded
1s² 2s² 2p² —> 1s² 2s¹ 2p³
Describe the crystal structure of graphite
Describe the spatial distribution of molecular orbitals in graphite
Forms an sp² molecular orbital
3 valence electrons (hybridisation) are bonded and 1 free electron (dangling bond)
1s² 2s² 2p² —> 1s² 2s¹ 2p³
Define metallic bonding
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.
What causes the cohesion of solids?
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.
