Spring Flashcards
In 1D, how is wavenumber related to wavelength?
For a particle in a box, how is energy level spacing related to L (length of sides of box)?
Energy level spacing decreases as L increases (levels get closer together)
Roughly, how is density of states found in 1D?
By finding the number of k states between k and k+Δk.
- by dividing width from k to k+Δk by the gap between each state
The taking this number of states and dividing it by the gap Δk
How is partition function found using density of k states?
For 2D and 3D what does this represent?
Number of intersections of equally spaced lines which correspond to an allowed energy level
What is E(k)? What varies between dimensions?
How is the number of states between k and k+Δk related to the number of states between E and E+ΔE?
What is the formula for D(E)?
For typical E, which dimensions have D(E) independent of E?
2D only
D(E) for 3D depends on sqrt E
D(E) for 1D is proportional to 1/sqrt E
Is D(E) always the same for each dimension?
No, D(E) depends on E’s dependence on k so varies e.g. for an electron in a graphene sheet (E=ℏsk)
What is the equation for Z without E explicitly subbed in?
How can Z be evaluated using D(E)?
Subbing in D(E) and evaluating over dE
In words, what is energy density of states?
The number of states with energy between E and E+ΔE, divided by ΔE
Do both D(E) and D(k) change for materials?
No, D(k) always the same for each dimension (1D, 2D, 3D) but D(E) can change for non-parabolic relations between E and k
What does the Boltzmann probability distribution tell us? What is the equation?
The probability that any one of the particles occupies a particular quantum state of energy E
How can the Boltzmann probability distribution be used to find average number of particles in each state? What are we assuming here?
Where Na is amount of particles in whole box.
We assume na(E) «1 so we don’t worry about several particles on one level
From the average number of particles in a state, na, how can average number of particles with energy between E and E+dE be found?
Z is?
The partition function
na(E) is ?
Number of particles in a particular energy state
What is λD? Why do we introduce it?
Thermal deBroglie Wavelength corresponding to a particle whose total KE is kBT.
Introduced to simplify Z
How does energy distribution relate to volume of gas?
It’s independent.
What is α(E)? (in words and as a product of other values)
Number of particles per unit energy between E and E+dE.
= naD(E)
dN=α(E)dE
How does α(E) (particles per unit energy) depend on E?
Product of an exponential term and a sqrt dependence so gives rise to a peak in dependence.
On a graph of α(E) against E, roughly what value does the peak take?
3/2KBT
When deriving the Maxwell speed distribution, what energy equation is used? What kind of energy is this?
E=1/2mu2
Just kinetic
When deriving the maxwell speed distribution, we start with dN and rearrange the dE term to what?
dE = mudu where u is speed
What is n(u)?
Number of particles, per unit speed, with speed between u and u+du
Why is the maxwell speed distribution indpendent of ℏ?
Because the density of states used to derive it assumed that quantized energy levels are so close they act as a continuum and therefore effects of discrete energy levels are lost
What is n(u)du? What is du?
Number of particles in the speed interval u -> u + du where du is that difference between either side of the interval
What exactly is λD?
For a quantity, A(u), that depends on speed, u, how is the average of that quantity found? Assuming it obeys Maxwell-Boltzmann statistics.
What is the integral of n(u)du from 0 to infinity equal to?
Na total number of particles
What integral would you evauluate to find mean magnitude of the speed of a particle in a Maxwellian gas?
What integral would you evaluate to find mean square speed of a particle in a Maxwellian gas?
For root mean square speed, what order do the mean, square, and root take place in?
square, mean, root.
a mean is taken of u2 and then this is rooted
Does root mean square speed equal mean speed?
No
What question are we attempting to answer with ‘‘molecular beams’’?
speed distribution of a gas emerging through a hole in a container wall
What happens in the process of MBE?
Molecular Beam Epitaxy: atoms are heated in ovens and fired at something
For the problem of a particle escaping through a hole we consider slightly different ranges for spherical co-ordinates, what are these and why?
θ from 0 to pi/2 (compared to normal up to pi, because we don’t want the particle going backwards)
φ from 0 to 2pi (same as normal)
How do we find what fraction of particles in a box volume, V, would escape in a given time, t ?
All particles within a particular cylinder of length ut (have to travel the length in time) and cross section Acosθ will make it out. (A is area of hole at a slant of angle θ)
therefore
To find particles leaving a hole:
Once we have found the fraction of particles that would lie in a particular cylinder, what is done next and how?
We find the number of cylinders that end in the hole Acosθ on the wall. This is done using small changes in spherical co-ords.
Once we have found this value, how do we find total number of atoms to escape in time t?
How would we find flux of atoms escaping through a hole (number of atoms per time per unit area of the hole)
How is a black body defined?
A black body absorbs all electromagnetic radiation that is incident on it, none is reflected.
As a black body gets hotter what happens to the photons and energy it emits?
1) emitted photons have more energy, hence lower wavelength
2) it emits more total energy (across all wavelengths)
How can overall fraction of photon’s escaping a black body be found (explanation and integral)?
- model cylinders that end in exit area A, find fraction of velocities that end in this A point. Integrate over all u
For a black body, from the fraction of photons that can escape a given hole in time t, how do we find energy emitted in time t?
Multiplying the fraction Ftot by total energy of radiation in the cavity.
What is rate of emission / power for a black body? Define terms
dQ/dt = cuA/4
c = speed of light
u = energy per unit vol
A = area of hole
How can we adjust energy per unit volume u to a continuum for a range of wavelengths?
What is u(λ)?
Spectral density
How is energy per unit volume related to spectral density?
u=energy per unit vol
u(λ)=spectral density
How does u(λ) relate to wavelength and T (temp)
u(λ), for a given temp T, peaks at a particular wavelength, this wavelength decreases as T increases
How does rate of energy emission dQ/dt relate to spectral energy density? What law is this?
What is Wein’s scaling law?
That the curve of wavelength against spectral density can be made universal by plotting u(λ)/T5 against λT
What did Rayleigh and Jeans assume?
That energy per radiation mode is equal to the average energy of each particle oscillating in the wall
When deriving density of modes, D(λ)dλ is equal to what expression in k?
2D(k)dk
-then we can get dk in terms of dλ and use D(k) eq from formula sheet
What happens to radiation incident on a black body?
It is absorbed and re-emitted.
By Wein’s law, plotting what two variables gives a universal curve?
u(λ)/T5 against λT
What are the two main problems with the Rayleigh jeans model?
1) infinite power output
2) spectral density tends to infinity as λ tends to 0
How is u(λ)dλ related to E(λ) generally speaking?
How did Planck explain energy for each EM mode to explain blackbody radiation?
Assumed energy for each mode came from simple harmonic motion at a set frequency w and radiation is generated by a transition between these levels.
When deriving energy of particles for a harmonic oscillator, what do we assume about En?
ignoring 0 point energy -
The harmonic oscillator is a rare case we can calculate what?
How would we find average energy of an oscillating particle with frequency w? (steps)
1) Start with equation for calculating partition function, z, from En (shown here)
2) Use En = nh(bar)w
3) use equation for energy from partition function (formula sheet)
What does Planck’s model for radiation from an object correctly predict?
Shape of the emission spectrum
Total power emitted
How do we find peak wavelength of spectra density under Planck’s distribution?
Differentiate equation for u(λ) wrt λ and set to 0.
Split into two halves, plot and look for intersection
What logic is the classical model of specific heat capacity based on? When does it hold true?
- each oscillation has energy kBT
- a solid has 3 directions of oscillation
- total internal energy = Etot=3NkBT
- therefore CV =
which gives 3R for 1 mole and holds above room temp
When does the classical model of specific heat capacity agree with experiments?
T larger than or equal to room temp
What turned out to be wrong with Einstein’s model for specific heat?
It assumed all particles oscillate at the same frequency (Einstein frequency)
What does Einstein’s model for specific heat capacity incorrectly predict?
That CV would tend to 0 exponentially not proportional to T3
What is the Dulong-Petit law?
That around room temp and above, CV tends to 3R (molar gas constant)
What is the foundation of Debye’s model of specific heat?
That atoms couple to each other (coupled to all others) and oscillate coherently - propagating sound waves
The sound waves are quantized and are transverse or longitudinal
What is the difference between longitudinal and transverse phonons?
Longitudinal phonons only oscillate in 1 direction (quantization direction)
Transverse propagating perpendicular to quantization direction so one of two ways (2 polarizations)
How is angular frequency related to speed, v, and wave vector k?
w = vk
What is the density of modes for all 3 types of phonon?
If N oscillators are coupled together, how many possible modes of vibration do they have? How many total modes is this?
N possible modes of vibration
And 3 directions, so 3N total phonon modes
Define the 3 Debye quantities? (other than wavelength)
Why does Debye’s model fail at T <10K?
It neglects contribution of conduction electrons to internal energy and specific heat
From Clausius’s principle, the entropy of an isolated system…….?
tends to a maximum
Is entropy extensive or intensive?
Extensive, proportional to number of particles
What is the chemical potential for a system with N particles?
Physically / qualitatively, what is chemical potential ?
It measures the rate at which a system’s entropy changes as particles are added.
How does particle behaviour depend on chemical potential?
Particles flow away from the system with the highest chemical potential
If particles are reacting to make new particles, how are their chem potentials related?
The sum of the chem potentials of the reacting particles equals the chemical potential of the particles produced
Entropy _________ on the approach to equilibrium.
Increases
What is the general condition for equilibrium in chemical reactions?
In first term we saw, knowing S also depends on N, how can we adjust this?
What are two alternative definitions for chemical potential involving E and F?
What is W in the Bolztmann distribution?
Number of ways in which the particles of a system can be distributed amongst the energy levels of the system
How can we write chemical potential, utilising the Boltzmann distribution?
What sign does chemical potential take?
Always negative
How is F related to Z? What are F and Z?
Helmholtz free energy and Partition function
How do we convert from Z formula for distinguishable particles to indistinguishable particles?
divide by N!, where N is number of particles, to account for the amount of states that are the same when particles are indistinguishable
Degeneracy, g, equates to?
g distinct quantum states
How is chemical potential related to z?
How does the partition function for N distinguishable and N indistinguishable particles vary?
What has to be the case for this to hold?
Only hold when prob of two particles on the same level is low e.g high temp
What must be true about probability density for indistinguishable particles?
Probability density must stay the same when particles are swapped
What is ‘phase factor’, K, what purpose does it serve?
The factor that relates two wavefunctions when particles are swapped - disappears when prob density is taken
Wavefunctions with phase factors +1 or -1 are called?
symmetrical (K=+1)
anti-symmetrical (K=-1)
Particles with symmetrical wavefunctions (K=+1) are called..?
Bosons
What kind of spin quantum numbers do bosons have?
Integer spin quantum numbers, s=0,1,2,3,…..
Particles with antisymmetric wavefunctions (k=-1) are called?
Fermions
What kind of spin quantum numbers do fermions have?
half-integer s=1/2,3/2….
What are some examples of Bosons?
Photons, phonons, mesons..
What are some examples of fermions?
electrons, protons, neutrons
If a particle has total spin number s, how many different quantum states does each allowed energy level correspond to?
2s+1
When two indistinguishable particles are swapped, the wavefunction changes, for this module, how does it change?
either the same, or negative
For n bosons, how many terms do we need in a wavefunction to maintain symmetry?
n!
What derivation leads to the pauli-exclusion principle?
Can two fermions have the same energy?
Yes but not the same spin too, if two fermions are in the same energy level they must have different spins.
What is the grand canonical ensemble?
The set of all possible quantum states of a total system = system A + large reservoir/heat bath
For a system, A, connected to a large reservoir - what is the probability of system A being in state B?
Equates on the formula sheet to ‘particle number not fixed’
Break down the terms in the grand partition function.
How can we rearrange entropy to be in terms of the grand partition function for systems with variable particle number?
What is grand potential, qualitively?
an analogue of Free energy, F=E-TS for systems with variable particle amounts.
How are the 3 thermodynamic variables derived from the equation from the grand potential?
What units does the grand potential have?
Energy
What is the fermi-dirac distribution, qualitatively?
The probability that a single particle state with energy ei is occupied for a given T and μ (where only 1 particle can occupy, e.g. fermions)
What is the Bose-Einstein distribution, qualitatively?
The average number of bosons in a single particle state with energy ei
Define terms in the grand potential equation:
How is the probability of a fermion being up spin in an energy level related to its probability of being down in the same level?
To find total number of fermions across states, we sum probability for each energy across all energies. We then multiply by 2, why?
To account for each energy level having two states, spin up and spin down
Why are boson/fermion distinctions not significant at high temps?
At high enough temps, properties of a gas are independent of whether it’s made from bosons or fermions
In a high T limit, why does the probability for the ground state affect the Fermi-Dirac distribution?
At high T, probability of lowest state being occupied is much smaller than 1, so the exp on the bottom of the probability equation is much larger than 1. this makes the +1 in the fermi dirac distribution negligible
At high T, if we analyse a Fermi gas for pressure, what do we recover?
The ideal gas law
Define fermi energy.
EF = the energy of the highest occupied state when T=0 (all levels below this will be full)
What can be notes about energy levels with E<EF?
They are full as they are below the fermi energy
Define Fermi wavevector.
kF = magnitude of the wavevector of the highest occupied state at T=0 (State where E=EF
Define fermi temperature.
TF=EF/KB
Define fermi velocity.
vF= h(bar)KF/ m
where KF is the fermi wavevector
Why at T=absolute zero, is the pressure (quantum pressure) independent of T?
Because no more fermions can be ‘squished’ into the states below EF
What is the qualitative meaning of TF?
Fermi gases at temperature much lower than the fermi temperature are said to be degenerate and can be used to act like they’re at T=0 with a very small amount of particles just above EF (about KBT over)
What are conduction electrons?
The few electrons excited to levels within KBT of EF which contribute to specific heat at low T at a different proportion to other particles
What do conduction electrons explain?
The deviation of Debye’s theory from real data on heat capacity as electrons dominate heat capacity at very low Ts
What is the biggest difference between Bosons and Fermions at low T? What does this cause?
No limit on number of Bosons in an energy level so we expect a lot to fall into the lowest energy level - Bose Einstein condensation
What is the critical temperature TC?
The temp, where higher than this no bosons will occupy the lowest energy state
