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PHYS 420 function

创建日期：2024-06-07 17:34:56

所属分类：物理Physical

标签： PHYS 420

function

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PHYS 420

Midterm 2 Take home
• Time: Start from 5 pm EST on Nov. 10th (Friday), 2023. Due at 11:59pm
EST on Nov. 11th (Saturday), 2023. Submit on canvas.
• Please keep the writing clean and clear to reduce the work of our TA.
• You are free to use any software to do the integrals and summation in this exam.
Please write clearly what is the integral or summation that you want to
do and then give the results. You do not have to show the work for integrals and
summation. You do need to show your work for calculations other than the
integral and summation.
• Please feel free to use the textbook, the lecture notes, and the homework solutions.
You should NOT consult any other resources.
• You should NOT communicate with anyone (other than the TA and the
instructor) about the exam during the exam.
• Please sign below to acknowledge the Academic Integrity code at the Pennsylvania
State University. Violations of the Academic Integrity code could result in
nullification of your exam.
Signature:
——————————————————————————
1
1 Surface absorption (again) (14 points)
In this problem, we will deal with a realistic model of surface absorption. Consider a ideal
gas consist of N indistinguishable particles of mass m at temperature τ inside a cubic tank
of size V = L× L× L. The total number of particles is fixed in this box.
The particles can either move freely in the volume V , or be absorbed on the surface. A
particle absorbed on the surface can still move freely on the 2d surface. The bulk of the
system and the surface can be viewed as two systems in thermal and diffusive contact.
As a result, in equilibrium, they share the same temperature and chemical potential.
The energy of a particle moving in the bulk is given by Eb =
P2
2m
, where P is the 3-
dimensional momentum of the particle. The energy of a particle on the surface is given by
Es =
p2
2m
− , where p is the 2-dimensional momentum of the particle on the surface; is a
positive constant, which represents the binding energy.
(a) (4 points) Show that the single particle partition function of surface parti-
cles in the high temperature limit is given by
Z
(s)
1 = e
/τ
( mτ
2pi~2
)
A, (1)
where A = 6L2 is the surface area (remember to include all 6 faces of the cube). (Hint: In
this problem, we consider the high temperature limit so that the discrete sum in the partition
function can be approximated by a Gaussian integral. )
(b) (5 points) Show the grand partition function of the surface system is given by
Z = eλsZ(s)1 , (2)
where λs = e
µs/τ , µs is the chemical potential on the surface, and Z
(s)
1 is the surface single
particle partition function. (Hint: here we will need to use the partition function of N -
particle system is given by ZN =
ZN1
N !
.) Use this grand partition function to calculate the
average number of particles on the surface, labeled by Ns. Ns should be a function of τ , µs,
and A. In the end, the solve the µs in terms of Ns, τ , and A.
(c) (2 points) Write down the expression for the chemical potential of the ideal gas in
the bulk, labeled by µb. (Hint: You don’t need to derive this again, just quote results from
the 3d ideal gas model we discussed in class.)
(d) (3 points) In equilibrium, the chemical potentials of the bulk and the surface should
be equal to each other, namely µs = µb. Use this condition to work out the average particle
density on the surface. Express your result in terms of the pressure p of the bulk ideal
gas, the binding energy , and the temperature τ .
2
2 A variant of the 2d photon gas (14 points)
Let us consider a 2-dimensional system of size V = L × L. Inside there are certain type
of particles that do not have conserved particle number, similar to the photon
discussed in class.
The single particle eigen-states (also referred as modes/orbitals) are plane-waves, labeled
by the 2-dimensional wave-vector k. The 2-dimensional wave-vector, k, is in the form k =
pi
L
(nx, ny), where nx and ny are positive integers, namely nx, ny = 1, 2, 3, ...,∞. Assume there
is only one polarization for this particle. The particle in consideration has an interesting
dispersion relation, ε(k) = ~
√
v2xk
2
x + v
2
yk
2
y. For this dispersion relation, the equal energy
contour in k-space is ellipse as shown in Fig. 1.
Figure 1: (Left) The dispersion of the particle. (Right) The constant energy contour of the
dispersion relation in k-space.
(a)(4 points) Based on the dispersion relation described above, show the density of
state function D(ε) (defined as number of states per unit energy) for this type of particle
is given by
D(ε) =
V
2pivxvy~2
ε (3)
(Hint: the area of ellipse is piab where a and b are length of the two axis.)
(b)(3 points) Assume the particles also obey the Planck distribution. Based on this,
calculate the the total energy and heat capacity of the system at temperature τ .
(c)(4 points) Calculate the free energy of the system at temperature τ . (As photon
system has zero chemical potential, we can use F = −τ log(Z), where Z is the grand partition
function for the photon system). Then calculate the entropy as function of temperature τ
and volume V .
(d)(3 points) If the system goes through a reversible isentropic expansion(a process
that preserves the entropy) from an initial state with temperature τ0 and volume V0 to a
final state with volume V1, what is the final temperature τ1?
3
3 Pressure of a solid (12 points)
Consider a piece of solid material. The total energy of a solid is then given by
U = U0 + Up,
(4)
where U0 is the energy of a solid when all the atoms are at rest in their equilibrium positions,
and Up is the energy of the lattice vibrations, namely the phonon gas. We assume U0 only
depends on the volume of the solid, denoted by V . The total energy of the phonon gas, Up,
is given by
Up =
∑
i∈all modes
si~ωi, (5)
where si is the occupation number of the mode i, and ωi is the angular frequency of that
mode. The frequencies of the phonon modes are dependent on the volume of the solid (but
not on the temperature). We make the following assumption:
dωi
dV
= −γωi
V
for all i, (6)
where γ is a constant.
(a)(4 points) Use the following definition for pressure,
p = −
(
∂U
∂V
)
σ
, (7)
to show that the pressure of the solid is given by
p = −dΦ
dV
+ γ
Up
V
(8)
(In this problem, you don’t need to calculate the explicit form for Up. )
(b)(5 points) Prove the following relation(
∂Up
∂V
)
τ
= γ
CV τ
V
− γUp
V
(9)
where CV is the heat capacity of the solid at constant volume. (Hint: You need to start from
Eq. (5). Take the derivative of Up with respect to temperature, you will be able to get the
expression for the heat capacity. Then you will need to take the derivative of Eq. (5) with
respect to volume V and try to prove the equation we want. Remember the frequencies are
volume dependent as given in Eq. (6). In the process, you will also need to use the Planck
distribution for si, namely 〈si〉 = 1e~ωi/τ−1 . )
4
(c)(3 points) Let us assume U0 has the following simple form
Φ(V ) =
(V − V0)2
2κ0
, (10)
where V0 and κ0 are constants. Let us define the compressibility of the solid at constant
temperature as the following
κτ = − 1
V
(
∂V
∂p
)
τ
. (11)
Physically, compressibility quantifies the volume reduction of a solid in response to applied
pressure. Use the results in (a) and (b) to show κτ for the solid is given by
κτ =
1
V

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