1. No category

NPRE-529/CSE-529
Interaction of Radiation with Matter II
– Multiple Events and Computational Methods
Homework Assignments
Prof. Yang Zhang
Department of Nuclear, Plasma, and Radiological Engineering
Department of Materials Science and Engineering
Program of Computational Science and Engineering
University of Illinois at Urbana-Champaign
[email protected]
http://zhang.npre.illinois.edu
October 27, 2014
NPRE-529, Fall 2014
1
PS-1, due on 09/23/2014
Problem Set 1
Readings:
Chapters 1, M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
1.1
The purpose of these questions is to identify students’ research interest and divide students into
groups for the computational projects.
1. Write a one paragraph synopsis of your current research. (10 points)
2. Write a one paragraph proposal for your computational project. (10 points)
1.2
Kardar: Page 32, Problem 8 hard core gas (5 + 5 + 5 + 5 points)
1.3
Kardar: Page 31, Problem 6 glass (5 + 5 + 5 + 5 + 5 + 5 points)
1
NPRE-529, Fall 2014
2
PS-2, due on 10/07/2014
Problem Set 2
Readings:
Chapters 4, M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
2.1
Kardar: Page 29, Problem 1 surface tension (5 + 5 + 5 points)
2.2
Kardar: Page 120, Problem 1 classical harmonic oscillator (10 + 5 + 10 points)
2
NPRE-529, Fall 2014
3
PS-3, due on 10/28/2014
Problem Set 3
Readings:
Chapters 4, M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
3.1
Kardar: Page 121, Chapter 4, Problem 5 Non-harmonic gas (10 + 5 + 10 + 10 points)
3.2
Kardar: Page 122, Chapter 4, Problem 6 Surfactant adsorption (5 + 10 + 5 points)
3.3
Computational Project White Paper
Each group (containing 3 students) should finalize the ideas for the computational project, and
submit a white paper (proposal). The white paper should contain the following five components:
1. Introduction: What is the background of the research project? Why is the research project
interesting? What is known?
2. Problem, question or challenge: What is the major problem/question/challenge? Why couldn’t
people solve it previously? What is the specific problem you will address?
3. Approach: How will you solve the problem? What’s new?
4. Objective: What are you going to achieve?
5. Significance: Why is the result important? Why should we care? What are the implications
In a broad context?
Format requirements:
1. Two pages. Figures are allowed, but are subject to the two-page limit. References do not
count towards the page limit.
2. Font: Times New Roman (Word) or Computer Modern family (LATEX) at 11 points
3. Line Spacing: Single
4. Margins in all directions should be one inch.
Each student should choose a different color when working on the white paper together, so that
your contributions are clear.
3
NPRE-529, Fall 2014
4
PS-4 (midterm), due on 11/04/2014
Problem Set 4 (midterm)
Readings:
Chapters 7 and 8, D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University
Press (1987).
4.1
An elastic scattering experiment.
PN
1. Let ρ(r) = l=1 δ(r − rl ), show that ρ(2) (rj , rk ) = hρ(rj )ρ(rk )i only depends on rj − rk , and
show that consequently we can define
Z
1
ρ(2) (r) =
drk ρ(2) (r + rk , rk )
V
and the pair distribution function
1
1
g(r) = 2 ρ(2) (r) =
ρ
ρN
*
X
+
δ[r − (rl − rl0 )]
l6=l0
(5 points)
2. Show that the structure factor S(k) = N1 hρk ρ∗k i can be expressed as the Fourier transform of
the pair distribution function g(r)
Z
S 0 (k) = 1 + ρ dr e−ik·r [g(r) − 1] = S(k) − (2π)3 ρδ(k)
(5 points)
3. For an isotropic system, show that
S 0 (k) = 1 + 4πρ
where j0 (x) =
sin(x)
x
Z
dr r2 j0 (kr)[g(r) − 1]
is the spherical Bessel function of the first kind. (5 points)
4. Describe how to measure the structure factor S(k) by elastic scattering (diffraction) experiments. (5 points)
4.2
An inelastic neutron scattering experiment.
1. Starting from the general double differential cross section
Z ∞
D
E
0
d2 σ
kf 1 X
=
bl bl 0
dt e−iωt e−ik·rl (0) e−ik·rl (t)
dΩdω
ki 2π¯h 0
−∞
l,l
where bl is the bound scattering length of atom l, show the separation of incoherent and
coherent double differential cross sections due to spin average. (5 points)
4
NPRE-529, Fall 2014
PS-4 (midterm), due on 11/04/2014
2. Compute the coherent σinc and incoherent σcoh scattering cross sections for light (H) and
heavy (D) hydrogen. (5 points)
3. Explain why light water is especially important for nuclear science and engineering. (5 points)
4. Explain how an inelastic scattering experiment is related to the correlation function and linear
response function of the measured system. (5 points)
5