Cmp Pt 2: Lectures 10-19 :(EXAM QUESTIONS
WITH CORRECT VERIFIED SOLUTIONS
2025/2026)
what is the significance of full fermi gas degeneracy in metals at room
temp? - ANS>this explains the observed contribution of electrons to the
heat capacity of a metal
what is heat capacity? - ANS>the amount of energy needed to raise the
temperature of a body by one degree.
what is wrong with the classical interpretation of electron heat
capacity? - ANS>classically:
- electron thermal energy E = (3/2)kBT
- thus total electron gas thermal energy U = n(3/2)kBT
- this electron gas would then have to turn into a significant
contribution to the heat capacity of the metal it is in
- in reality, this contribution is much smaller and metals don't have that
much heat capacity
how does the free-electron gas contribute to a metal's heat capacity? -
ANS>at room temp, the free electron gas of a metal is practically fully
degenerate; the fully degenerate electrons are not able to absorb any
more energy and hence they cannot contribute to the heat capacity of
the metal. only the few electrons close to Ef are able to absorb thermal
energy and jump to higher states, and hence contribute to the heat
capacity of the metal.
,what is the zone/band theory of solids? what does it help to define? -
ANS>the band theory of metals is based on the valence band and
conduction band, and it defines conductors, semiconductors and
insulators very clearly.
it includes a periodic electric potential that represents the potential
that the electrons 'feel' as it travels across the crystal lattice of positive
ions what is the valence band? - ANS>the valence band is made up of
the valence shell orbitals which have electrons in them. eg. sodium
valence band is made up of 3s1 orbital (the last orbital)
what is the conduction band? - ANS>the conduction band is made up of
those orbitals which are unoccupied by electrons either in the valence
shell or higher unoccupied shells. thus, the orbitals of the conduction
band are empty. eg. sodium's valence band is 3s1, and the next orbital
3p is empty so it forms a conduction band
what assumptions do we need to make in order to build the model of
the wavefunction of an electrion in a periodic potential? - ANS>1.
positive ions in the lattice are completely immobile, moving electrons
don't transfer momentum to them and thermal vibrations of the lattice
are not present
2. the ions are the sources for the periodic electrostatic force that acts
on the mobile, delocalised electrons
3. the ions are sitting precisely on the vertices of the crystal lattice
, 4. consider each separate electron as a single electron moving in a
periodic potential
what is the impact of the periodic potential on the schrodinger
equation? - ANS>the hamiltonian contains the potential term, which
adds a potential term to the schrodinger equation which is periodic
what is bloch's theorem? - ANS>a theorem that specifies the form of
the wave functions that characterise electron energy levels in a periodic
crystal
what is q? what is ℏq? - ANS>q = the bloch wave vector = crystal wave
vector
ℏq = bloch momentum = crystal momentum
what is the kronig-penney model? - ANS>the simplest model of the
motion of a delocalized electron through a periodic potential. consists
of two regions: area I (potential well) and area II (potential hill)
what is area I of the kronig-penney model? what does it do to SE? -
ANS>area I: regions where V(x) = 0, ie. potential wells. the schrodinger
equation is just without the V(x) term
WITH CORRECT VERIFIED SOLUTIONS
2025/2026)
what is the significance of full fermi gas degeneracy in metals at room
temp? - ANS>this explains the observed contribution of electrons to the
heat capacity of a metal
what is heat capacity? - ANS>the amount of energy needed to raise the
temperature of a body by one degree.
what is wrong with the classical interpretation of electron heat
capacity? - ANS>classically:
- electron thermal energy E = (3/2)kBT
- thus total electron gas thermal energy U = n(3/2)kBT
- this electron gas would then have to turn into a significant
contribution to the heat capacity of the metal it is in
- in reality, this contribution is much smaller and metals don't have that
much heat capacity
how does the free-electron gas contribute to a metal's heat capacity? -
ANS>at room temp, the free electron gas of a metal is practically fully
degenerate; the fully degenerate electrons are not able to absorb any
more energy and hence they cannot contribute to the heat capacity of
the metal. only the few electrons close to Ef are able to absorb thermal
energy and jump to higher states, and hence contribute to the heat
capacity of the metal.
,what is the zone/band theory of solids? what does it help to define? -
ANS>the band theory of metals is based on the valence band and
conduction band, and it defines conductors, semiconductors and
insulators very clearly.
it includes a periodic electric potential that represents the potential
that the electrons 'feel' as it travels across the crystal lattice of positive
ions what is the valence band? - ANS>the valence band is made up of
the valence shell orbitals which have electrons in them. eg. sodium
valence band is made up of 3s1 orbital (the last orbital)
what is the conduction band? - ANS>the conduction band is made up of
those orbitals which are unoccupied by electrons either in the valence
shell or higher unoccupied shells. thus, the orbitals of the conduction
band are empty. eg. sodium's valence band is 3s1, and the next orbital
3p is empty so it forms a conduction band
what assumptions do we need to make in order to build the model of
the wavefunction of an electrion in a periodic potential? - ANS>1.
positive ions in the lattice are completely immobile, moving electrons
don't transfer momentum to them and thermal vibrations of the lattice
are not present
2. the ions are the sources for the periodic electrostatic force that acts
on the mobile, delocalised electrons
3. the ions are sitting precisely on the vertices of the crystal lattice
, 4. consider each separate electron as a single electron moving in a
periodic potential
what is the impact of the periodic potential on the schrodinger
equation? - ANS>the hamiltonian contains the potential term, which
adds a potential term to the schrodinger equation which is periodic
what is bloch's theorem? - ANS>a theorem that specifies the form of
the wave functions that characterise electron energy levels in a periodic
crystal
what is q? what is ℏq? - ANS>q = the bloch wave vector = crystal wave
vector
ℏq = bloch momentum = crystal momentum
what is the kronig-penney model? - ANS>the simplest model of the
motion of a delocalized electron through a periodic potential. consists
of two regions: area I (potential well) and area II (potential hill)
what is area I of the kronig-penney model? what does it do to SE? -
ANS>area I: regions where V(x) = 0, ie. potential wells. the schrodinger
equation is just without the V(x) term
