Statistical Physics Pt.II Flashcards
1
Q
What is the partition function for a particle in a 1D box? Why do we need to we need to find an integral for the partition function for a large box? What happens as L increases in terms of the density of states?
A
Too difficult to evaluate. As the width of the box increases, the energy level spacing decreases.
2
Q
In a 1D box, what is the Number of states in each block?
A
3
Q
What is the density of states in k-space for a 1D box?
A
4
Q
Using the density of states in k-space, find the partition function as an integral.
A
5
Q
Evaluate the following integral to find 1D particle in a partition function.
A
6
Q
What is the equation for energy levels of a particle in a 2D box?
A
7
Q
In a 2D box, what is the Number of states in each block?
If need help look at image.
A
8
Q
What is the density of states in k-space for a 2D box?
A
9
Q
Given the density of states in k-space, find the partition function for a 2D particle in a box, and evaluate it.
A
10
Q
What is the equation for energy levels of a particle in a 3D box?
A
11
Q
In a 3D box, what is the Number of states in each block?
A
12
Q
What is the density of states in k-space for a 3D box?
A
13
Q
Given the density of states in k-space, find the partition function for a 2D particle in a box, and evaluate it.
A
14
Q
State the density of states in kspace in 1D, 2D, 3D.
A
15
Q
A
16
Q
- State ‘image text’
2.Relate the wavevector range to the corresponding energy range.
A
Remember energy for particle in box = a*k^2–What is being referred to by ‘corresponding energy’. Therefore
N(k–> k + Delta K) = N(E–> E + Delta E) are the same, k and E are just scaled relative to each other.
17
Q
What is the Number of states in an energy interval (E–> E+ Delta E)? State in terms of k-space density, and dE/dk
A
18
Q
State the energy density of states.
Hint: Image
A
19
Q
Try this
A
Try this
20
Q
Try this
A
Try this. Need to get energy equation first!
21
Q
What is the Energy density?
A
22
Q
Starting with a gas consisting of (N)a atoms each of mass m, confined within a 3D box of volume V= Lx Ly Lz find the average number of particles that occupy each state.
Hint: Boltzmann probability distribution
A
23
Q
Given eq(3.2) find an expression for the total number of states with Energy between E + dE.
A
N(E–> E+dE) = D(E) dE (The number of STATES with energy between E and E+dE)
Avg Number of Particles with energy between E and E+dE =
avg number of particles per state x number of states.
24
Q
What is the relationship between the Partition function and Thermal de Broglie wavelength
A
25
Given these 2 equations, re-write them in terms of the De Broglie wavelength to find the number of particles whose energies lie in the range of E to E+dE.
26
27
Find n(u), the number of particles per unit speed with speeds between u + du by using equation 3.5.
Hint:
Relate speed range to corresponding energy range using KE equation.
28
29
Summarise how the maxwell speed distribution is found.
1. Consider density of states with energies between E and E+dE.
2. Find the number of particles with energies between E and E + dE by using the Boltzmann probability and energy density eq.
3. Re-write in terms of de Broglie Wavelength.
4. Relate speed to Energy by using kinetic energy eq. Also replacing dE with du.
5. You have found u(n)
30
READ for context for mean quantitites of a gas obeying Maxwell-Boltzmann statisitics.
31
What is the mean Magnitude of the speed of a particle in a Maxwellian gas?
Hint:
1.Integral/flashcard 30
2.Integral of n(u) du = Na
32
What is the mean square speed of a particle in a Maxwellian gas?
Hint:
1.Integral/flashcard 30
2.Integral of n(u) du = Na
33
Read
Molecular Beam epitaxy- Read
34
Need for later card (36)
Need for later card (36)
35
Need for card (36)
Need for card (36)
36
Find the total number of atoms to escape an oven in time t. Oven has hole area A.
Hints: Cards 34, 35, Lecture 4; Molecular Beams.
File to access is on desktop.1. Find the fraction of atoms with velcocity vecotor U that lie within cylinder with hole area A. Don't forget to find the cross section (cos or sin theta?) since it is slanted.
2. Gas may contain many different velocity vectors, consider a cluster of velocity vectors near U on the surface of a sphere. Find cluster- which is a fraction of all velocity vectors, which equals the area that their tips occupy/total surface of sphere.
37
Using eq(4.10) find the total flux of atoms escaping through a hole in a box.
38
State the flux of the atoms escaping through a hole/the maxwell speed distribution weighted by speed.
39
(a) more particles will travel a further distance in a time t, therefore, it is possible for more particles to escape.
40
What does it mean for/if a blackbody is in thermal eq. with its surroundings?
What happens as a body gets hotter?
41
A model for a black body.
Find the overall fraction of photons/flux of photons escaping for this black body; Lecture 5a.
Flux is number per unit time per unit area
Hints:
Similar to question/Card 36 except u(n)[the maxwell speed distribution] is replaced with a constant c, the speed of the photon. File to access is on desktop.
1. Find the fraction of atoms with velcocity vecotor U that lie within cylinder with hole area A. Don't forget to find the cross section (cos or sin theta?) since it is slanted.
2. Gas may contain many different velocity vectors, consider a cluster of velocity vectors near U on the surface of a sphere. Find cluster- which is a fraction of all velocity vectors, which equals the area that their tips occupy/total surface of sphere.
42
Given the flux/total fraction of photons that escape a box/cavity in time t is [eq 5.3] find energy emitted in time t, using the total energy of the radiation of the cavity and total fraction of photons/flux of photons that are emitted in a time t.
Hence, find the rate of energy emission (power).
Hint:
1. To find the total energy of radiation, multiply the total energy of the cavity/oven/box by the fraction/flux of photons which are escaping(the radiation is the photons which are escaping, so if we multiply the fraction which are leaving by the total energy of the system we find the energy which is being radiated/total energy of photons escaping).
2. What is the total energy; consists of energy density, u, and Volume, V. Note that u is no longer velocity but energy density.
43
What is the integral that gives the total energy density/energy per unit volume in terms of the spectral energy density?
44
Theory developed of Blackbody spectrum had to fit the following experimental observations.
45
Find the density of modes inside an oven/blackbody cavity.
Hint: Similar to density of states in a 3D box, but also taking into account potential polarization/orientation of modes.
46
Given the density of states in k space/density of modes, find the density of states/modes in the wavelength interval (lambda--> lambda+ d_lambda).
Hence find the spectral energy density by assuming a particular average energy of each oscillating particle in the oven wall.
47
READ this
- What do experiments show about radiation emitted from a black body?
- What is the failure of the Rayleigh-Jeans model, show this using the equations provided for both spectral density u(Lambda) and Power output dQ/dt.
Hint: For spectral density integrate eq(6.1) from 0 to infinity or integrate without limits and then show lim(lambda to 0) u(lambda) = 0.
48
What was Planck's idea for modelling the blackbody spectrum?
Are the oscillators in Thermal equilibrium?
-Planck's idea was to assume that the energy within each EM/Standing wave mode of frequency w, originates from particles performing Simple Harmonic motion of the same frequency.
Energy is quantized into discrete energy levlels.
-The harmonic oscillators are in thermal equilibrium.
49
What does it mean for the Simple Harmonic Oscillators and the radiation they produce to be in thermal equilibrium?
Since the Oscillators are in equilibrium, the energy within each mode equals the average energy of each oscillating particle.
50
READ THIS AND NEXT SLIDE
51
What is the partition function for the energy levels within a Harmonic Oscillator?
Evaluate this using a known geometric summation.
52
Given 6.5, state Planck's distribution for a blackbody spectrum.
53
54
Try to find the peak of Planck's distribution, the answer is shown here:
Using a computer, we can find the points of intersection, and hence the wavelength at which these points meet. This wavelength is the PEAK wavelength, which agrees with Wien's Law.
This is one result that agrees with experiment (observation 1), unlike the Rayleigh-Jeans model.
55
The second observation- Wien's scaling law fits to Planck's model, another correct prediction which corresponds to experiment (observation2).
Derivations for cards 54 and 56.
56
Given the partition function for the harmonic oscillator, what is the average energy of a single particle of a particle in the system.
57
Show that Planck's distribution correctly leads to Stefan's Law (another expermental observation, 3).
58
Hints:
(1) Consider summation form of average energy (2) (Energy i * probablility of energy i) continuing from above.
(3) Look for potential chain rule.
(4) Cancel Ei terms
(5) Look for chain rule between numerator and denominator.
1/z * dZ/dT = d(ln Z)/dZ * dz/dT = d(ln Z)/dT
You do not need to substitute beta, you could just see that d(exp(terms))/dT =
-Ei *- 1/(kT^2)* exp(terms). Because you want to write in terms of a derivative you must integrate, hence divide by the coefficients of the differential. I.e. exp(kx) = d/dx[1/k*exp(kx)].
59
Eq 6.3- Partition function for Simple Harmonic Oscillator.
60
In a classical picture, what is the total energy and Cv for a solid containing N atoms?
CAN OSCILLATE 3 WAYS, HENCE 3*Kb T for 1 atom, 3*N*Kb T for N atoms.
Info on lower T on next cards.
61
What was Einstein's model for specific heat?
**Not the specific heat itself, the assumptions he made for his model/calculations.
What did he neglect in his model?
Einstein assumed that each atom performs SHM in each direction with quantised energy levels. He assumed that they all oscillate at the same frequency, the EINSTEIN frequency.
He neglected the interactions/COUPLING between atoms and assumed that they acted independent of one another. Coupling is a more appropriate word.
62
Use Einstien's model to calculate the specific heat. Use the equation of the partition function of a 1D oscillator.
63
Having found Einstein's equation for Cv and by considering limits for low and high temperatures, what is the problem?
64
What was DEBYE'S model of specific heat?
What are quantised sound waves called?
What types of phonons are there? (Hint: 2 types).
Debye's model: Atoms couple to one another and oscillate coherently to create propagating sound waves. Atoms oscillate with simple harmonic motion.
These sound waves are quantised (called phonons).
Phonons exist over a range of frequencies.
65
Image: 2 types of phonon illustrated.
How many polarizations do transverse phonons have?
Taking into account polariztion, how many modes of oscillation/ how many phonons are there?
Transverse phonons have 2 modes of oscillation. Since they oscillate PERPENDICULAR to the quantiztion direction, they can oscillate on either of the 2 axes perpendicular to quantiztion direction. I.e. If wave is travelling in x direction, can oscillate in and out of screen (y-direction) and down and up/vertically (z-direction).
66
Considering the modes of oscillation, find the phonon density of states for a cube with side lengths L.
Next, find the angular frequency density --> in the form D(w) dw = ...
67
Find the internal energy contribution for one mode of oscillation for DEBYE'S model.
Hint:
SHO partition function formula for energy.
Density of states dw.
68
Find the TOTAL internal energy for DEBYE's Model
Hint:
SHO partition function formula for energy.
Density of states dw.
(continutation from card 67, but now consider the contribution of multiple modes)
69
Find the Debye frequency, wd.
Hint: Consider N oscillators coupled, and the number of modes of phonons. Then think of how to find N.
Not what is needed but of same form to find N--> D(k) = N(k---> k + dk)/ dk
70
71
Cv takes ages so you may want to avoid that one.
72
Read, important to describe systems with a variable number of particles.
73
What is Clausius' principle?
What are the conditions for chemical equilibrium? (Related to entropy)
What is an extensive quantity and is entropy one?
State the change in entropy, ds, as particles transfer between 2 systems A and B.
Hint: Entropy is dependent on numer of particles, so is a function of A and B, S(Na, Nb).
For chemical equilibrium, the entropy of 2 systems A and B is at a maximum.
Extensive = Proportional to number of particles, entropy is extensive.
S = S(Na, Nb)
dS = 9s/9Na dNa + 9s/9Nb dNb
dS = dNa * (9s/9Na dNa - 9s/9Nb)
74
What is the chemical potential of a system with N particles?
What is the Physical meaning of the Chemical potential (what does it measure?)
Physically -the Chemical potential is a measurement of the rate at which a systems entropy changes as particles are added.
75
Using the equations that relate the entropy to the number of particles in systems A and B, and the formula for the chemical potential, state a formula relating entropy to the chemical potentials of the system.
What is the relation between Chemical potential for systems A and B at equilibrium?
Remember Na + Nb = const -->
dNa + dNb = 0
dNa = -dNb
76
What is the rate of change of entrop (dS/dt) as a system APPROACHES chemical equilibrium (no yet reached it)?
77
Given the equation for the rate of change of entropy as chemical equilibrium is approached:
What does it mean physically for-
(a) Chem Potential of sytem A>Sytem B
(b) Chem Potential of system B > System A
78
What does it mean for the total number of particles to not be conserved?
If the total number of particles is not conserved, Na and Nb are independent quantities, this means that any small change Nb causes no change to Na, hence dNb/dNa = 0.
79
What is the change in entropy, ds for a system with a non-conserved number of particles?
What are the chemical potentials equal to at EQUILIBRIUM?
Hint: consider dNa/dNb and dNb/dNa
dNa/dNb and dNb/dNa = 0 when number of particles not conserved.
80
Give the reate of chage of dN_RG in terms of dN_R and dN_G.
81
Considering the entropy is an extensive property:
State for this system;
- dS
- dS in terms of chemical potentials
- Relation between chemical potentials at equilibrium for this system.
82
State the rate of change of entropy as equilibrium is approached for this system
83
State the rate of change of entropy as equilibrium is approached for this system.
What does it mean physically for-
(a) Chem Potential of particles R and G > Chem potential of particle RG
(b) Chem Potential of Particle RG > particles R and G
84
85
(2) To answer you must state the relationship between the chemical potentials.
IMPORTANT to THINK about what an increase in dNa for particle A means for how much change has occured for other particles.
IMPORTANT to THINK about what an increase in dNa for particle A means for how much change has occured for other particles.
I.e. for an increase in dNa, there would be a third of an increase for dNb, or in other words, and increase in 1 for B, means an increase of 3 for A.
86
What is the differential form of the first law when also taking into account the number of particles, N?
(make dS the subject i.e. dS = ?)
What is (partial) dS/dE? dS/dV? dS/dN?
(make dS the subject i.e. dS = ?)
87
What is the differential form of the first law when also taking into account the number of particles, N?
Write in terms of dE. Next substitute the chemical potenial for one of the terms.
Next, state the free energy, taking into account the number of particles, and hence, the chemical potential.
88
State 3 forms of the chemical potential (Hint: partial derivatives)
89
Read:
What equation would you use for a heat bath, where the internal energy is changing?
90
State the expression for the chemical potential where the internal energy and volume are constant.
Now state in terms of the number of microstates W.
91
92
93
State the Partition function for 3 distinguishable particles.
State the general form for N distininguishable particles.
(Both in terms of the partition function for 1 particle)
94
State the Partition function for 3 indistinguishable particles.
State the general form for N indistininguishable particles.
(Both in terms of the partition function for 1 particle)
95
Given that the degeneracy of the particle is g, give the partition function for a single particle.
State for N indistinguishable particles, and the chemical potential.
Chem. Pot = -T dS/dN
96
Find the chemical potential, now in terms of Z1 instead of Zn (use stirling's approx).
Also find Z1 as an integral of density of states.
97
Derive this chemical potential for a hot low density gas.
Ans shown across cards 95 to 96.
98
Using the chemical potential for a hot low-density gas find the chemical potential for a free proton and electron and a Hydrogen atom.
Don't look at card until after completion.
Find the condition for equilibrium in the sun given e + p <-----> H
99
What is the relationship between free energy and partition function?
F = Kb * T * ln(z)
100
What is the statistical definition of entropy?
Hint: Probabilities
Derive the free energy equation in terms of the partition function.
Image is something different to questions but still important to understand.
Can use AI to explain rather than searching for it.
101
