January 12, 2015
- Consider a system of atoms which held together by a pairwise potential
of the form
where r ≡ |ri−rj| denotes the distance between site atoms at site i and site j given in normalized length units. ϕ is given in normalized energy units. For each of the systems given below, answer the following 3 subquestions:ϕ(r)=e−r ⎛
⎝1 r12− 1 r6⎞
⎠, - Enumerate the number and separation of the nearest neighbor and next-nearest-neighbor separations as multiples of r.
- Find the interaction energy per particle of the system as a function of r including only the nearest and next-nearest-neighbor contributions.
- Find the numerical value of the nearest neighbor separation of particles at equilibrium including only the nearest and next-nearest-neighbor contributions.
- System composed of two particles only, separated by a distance r.
- System composed of an infinite lattice of particles in a two-dimensional
hexagonal lattice
where each particle position is located at the point
whereri = n1i a1 + n2i a2
and where n1i and n2i are integers.a1 = r ^xand a2 = r ⎛
⎝1 2^x+ √3 2^y⎞
⎠, - System composed of an infinite lattice of particles in a two-dimensional
honeycomb lattice
configuration where each particle position is located at the point
whereri = n1i a1 + n2i a2 or ri = r ^x+ n1i a1 + n2i a2
and where n1i and n2i are integers. (Note that the honeycomb lattice is similar to the hexagonal lattice with an atom missing from the center of each hexagon.)a1 = √3 r ⎛
⎝√3 2^x+ 1 2^y⎞
⎠and a2 = √3 r ⎛
⎝√3 2^x− 1 2^y⎞
⎠,
File translated from TEX by TTH, version 4.01.
On 11 Jan 2015, 20:19.
PHY 752 -- Assignment #2
Jan. 14, 2015
Continue reading Chapters 1 & 2 in Marder.
- Consider the 6 by 6 group multiplication table given above.
- For each element of the group, find its inverse.
- Find the subgroups of the 6-dimensional group.
- Find the classes of the 6-dimensional group.
January 21, 2015
- Consider a group of 4 elements described by the multiplication table
Find the irreducible representations of this group.E A B C E E A B C A A E C B B B C E A C C B A E
File translated from TEX by TTH, version 4.01.
On 20 Jan 2015, 23:57.
PHY 752 -- Assignment #4
Jan. 23, 2015
Continue reading Chapters 1 & 2 in Marder.
- Consider the set of 8 processes based on transformations of a general
point in space (x,y,z) given below.
- Find the multiplication table for these processes
- For each process, find its inverse.
- Find the characters of this group.
- (x,y,z)
- (-x,y,z)
- (x,-y,z)
- (-x,-y,z)
- (x,y,-z)
- (-x,y,-z)
- (x,-y,-z)
- (-x,-y,-z)
PHY 752 -- Assignment #5
Jan. 26, 2015
Read Section 7.3 in Marder.
- Consider the set of 8 processes based on transformations of a general
point in space (x,y,z) given below.
- Find the multiplication table for these processes
- Find the characters of this group.
- Compare your results with the character table given as Table II in the BSW paper and associate your characters with the designated symmetry labels.
- (x,y,z)
- (x,-y,z)
- (x,y,-z)
- (x,-y,-z)
- (x,z,y)
- (x,-z,y)
- (x,z,-y)
- (x,-z,-y)
PHY 752 -- Assignment #6
Jan. 28, 2015
Read Section 6 in Marder.
- Consider the following two-dimensional electron systems, finding
their densities of states and their Fermi levels (at temperature 0 K).
Assume that in both cases, there is 1 electron within the unit cell area
A.
Your answer should depend on A and on the parameter X.
- First consider the system where the electron energy E as a function of the radial wavevector k has the dispersion E(k)=X k2.
- Second consider the system where the electron energy E as a function of the radial wavevector k has the dispersion E(k)=X k.
PHY 752 -- Assignment #7
Jan. 30, 2015
Read Chapter 7 in Marder.
- Work out some of the missing details of the Kronig-Penney model potential discussed in Lecture 7.
February 2, 2015
- Consider a one-dimensional tight-binding model system described by
a tridiagonal Hamiltonian which has non-trivial
matrix elements Hn n′ of the form:
for all site indices n, where α and β are real energy parameters.Hn n = α and Hn (n±1) = β - Consider the case where the site indices n,n′ take the values 1, 2, 3 exclusively and find the eigenvalues.
- Consider the case where the site indices n,n′ take the values 1 ... 8 exclusively and find the eigenvalues.
- Consider the case where the site indices n,n′ have an infinite range (−∞ ≤ n,n′ ≤ ∞). Compare the energy range for this system with that of the previous two samples.
File translated from TEX by TTH, version 4.01.
On 1 Feb 2015, 19:18.
PHY 752 -- Assignment #9
Feb. 4, 2015
Read Chapter 8 in Marder.
- Find the tight binding expression for the band dispersion of a face-centered-cubic lattice in terms of an onsite matrix element α and nearest neighbor matrix element β (of the form ssσ in the Slater-Koster terminology). Plot the band structure along at least one direction in the fcc Brillouin zone.
PHY 752 -- Assignment #10
Feb. 6, 2015
Read Chapter 8 in Marder.
- The problem concerns the tight binding band structure of the π bands
of a graphene sheet. Using the notation given on slide 21 of lecture 10,
find the tight binding form of the matrix elements of the Hamiltonian
Hrr(k), Hbb(k),
Hrb(k), and Hbr(k). Express your
answer in terms of the (pp π)1 and (pp π)2
nearest neighbor and next nearest neighbor interaction integrals.
- Find the eigenvalues as a function of k.
- Plot the bands along at least 1 direction of the hexagonal Brillouin zone.
February 9, 2015
- This problem is concerned with variationally estimating the
ground state electronic energy of a two-electron atom with nuclear
charge Ze in the Hartree-Fock approximation. The Hamiltonian
for the two-electron system is
Assume that the spatial part of the two-electron wavefunction can be written in the formH(r1,r2) = − ħ2 2m( ∇21 + ∇22)−Ze2 ⎛
⎝1 r1+ 1 r2⎞
⎠+ e2 |r1 − r2|.
whereΨ(r1,r2)=ϕ(r1) ϕ(r2),
where N is the normalization factor, a = ħ2/(me2) is the Bohr radius, and α is a variational parameter.ϕ(r) = N e−αr/a, - Show that
Hint: Show that the two electron term involves the integralE(α) ≡ 〈Ψ| H | Ψ〉 〈Ψ| Ψ〉= ħ2 2m a2⎛
⎝α2 − 2 α ⎛
⎝Z− 5 8⎞
⎠⎞
⎠. ⌠
⌡∞
0dr r2 e−2 αr/a ⎛
⎝1 r⌠
⌡r
0dr′r′2 e−2 αr′/a + ⌠
⌡∞
rdr′r′e−2 αr′/a ⎞
⎠= 2 ⌠
⌡∞
0dr r e−2 αr/a ⌠
⌡r
0dr′r′2 e−2 αr′/a. - Find the value of α that minimizes the Hartree Fock energy E(α) and the corresponding estimate of the ground state energy of the two-electron system.
- Show that
File translated from TEX by TTH, version 4.01.
On 8 Feb 2015, 01:52.
PHY 752 -- Assignment #12
Feb. 11, 2015
Continue reading Chapter 9 in Marder. Also read original papers by Hohenberg and Kohn and Kohn and Sham.
- Supply the detailed steps needed to derive Eq. 9.50 in Marder.
- Noting that, in terms of the electron density n(r), kF=( 3 π2 n ) 1/3. Find the functional derivative of Eq. 9.50 with respect to n(r).
PHY 752 -- Assignment #13
Feb. 13, 2015
Continue reading Chapter 9 in Marder.
- Work problem #4 on page 259 (Chapter 9) of Marder's text. (Extra credit for also working #5.)
February 16, 2015
- This problem involves finding the functional form of an exchange
potential by evaluating the functional derivative of the exchange
energy expression with respect to the density. In class we noted that
for
the corresponding potential is given byExc = ⌠
⌡d3r f(n(r, |∇n(r)|),
SupposeVxc(r) = ∂f(n(r, |∇n(r)|) ∂n− ∇· ⎛
⎝∂f(n(r), |∇n(r)|) ∂|∇n|∇n |∇n|⎞
⎠.
Here β represents a given constant. Also suppose that the system is spherically symmetric so that n(r) = n(r). Find the expression for Vxc(r) in terms of n(r) and its radial derivatives.f(n(r, |∇n(r)|) = − 3 e2 4 π(3 π2)1/3 ( n(r) )4/3 ( 1 + β|∇n(r)|2 ).
File translated from TEX by TTH, version 4.01.
On 15 Feb 2015, 15:33.
PHY 752 -- Assignment #15
Feb. 18, 2015
Continue reading Chapter 10 in Marder.
- This problem concerns the second-order and Numerov methods of solving a two-point differential equation as shown on slide 10 of Lecture 14. Verify the results given on this slide and try one other choice of N.
PHY 752 -- Assignment #16
Feb. 20, 2015
Continue reading Chapter 10 in Marder.
- Run the program graphatom for the atom of your choice. Obtain plots of the radial wavefunctions and of the core and valence electron densities. List the Kohn-Sham energies.
PHY 752 -- Assignment #17
Feb. 23, 2015
Continue reading Chapter 10 in Marder.
- Work problem 2a at the end of Chapter 10 in Marder's text.
PHY 752 -- Assignment #18
Feb. 25, 2015
Continue reading Chapter 10 in Marder.
- Run the program atompaw for the atom of your choice. Obtain plots of the basis and projector functions and of the logarithmic derivatives. Allow "world" access directory so that your instructor can check the results.
PHY 752 -- Assignment #19
Mar. 16, 2015
Review the mid-term exam in general and rework Problem 3, especially part d. It may be convenient to refer to the Brillouin zone diagram in Fig. 7.10 of your textbook.
- Examine the form of the band structure of graphene near the K point of the Brillouin zone. Show that for a state near the K point (k=kK+κ), where κ is assumed to be small, the band dispersion is approximately linear in κ.
PHY 752 -- Assignment #20
Mar. 18, 2015
Read Chapter 16 of Marder
- Work Problem #3 at the end of Chapter 16 (page 477) of Marder.
PHY 752 -- Assignment #21
Mar. 20, 2015
Read Chapter 16 of Marder
- Work Problem #5 at the end of Chapter 16 (page 478) of Marder.
PHY 752 -- Assignment #22
Mar. 23, 2015
Read Chapter 17 of Marder
- Work Problem #9 at the end of Chapter 17 (page 519) of Marder.
PHY 752 -- Assignment #23
Mar. 27, 2015
Read Chapter 18 & 19 of Marder
- Using the approximations discussed in class, find the expression for the binding energy of an electron or hole bound to a P impurity in an otherwise perfect lattice of Si. Use literature sources to determine the appropriate dielectric constant and electron or hole effective masses. Also estimate the effective bohr radius of the electron or hole.
PHY 752 -- Assignment #24
Mar. 30, 2015
Finish reading Chapter 19 in Marder
- Work Problem #4 at the end of Chapter 19 in Marder (page 607).
PHY 752 -- Assignment #25
Apr. 1, 2015
Start reading Chapter 20 in Marder
- Work Problem #1 at the end of Chapter 20 in Marder (page 628). Note: It is not necessary to do parts a-d, but you can evaluate the real part of the dielectric function at ω=0 from the form given in equation 20.85. You may earn extra credit for evaluating the real part of the dielectric function at larger ω.
PHY 752 -- Assignment #26
Apr. 6, 2015
Start reading Chapter 21 in Marder and also the pdf file from Bassani's book.
- Consider the scalar potential formuation of the electric field effects discussed in class as possibility #2 and show that provided that the crystal potential U(r) is local, that the transition probability is equivalent to Eq. 5.9 in Bassani's pdf file. Note that you may use the commutator relationship that [r, H0]=i ℏp/m.
PHY 752 -- Assignment #27
Apr. 8, 2015
Start reading Chapter 22 in Marder and also the pdf file "Maximally localized Wannier functions: Theory and applications" from the Review of Modern physics PDF
-
From Eq. 10 and 11 of the article, show that the Wannier functions
are normalized according to
<Rn|R'm>= δR R' δn m .
PHY 752 -- Assignment #28
Apr. 10, 2015
-
In class, we discussed a model of the attraction of an electron a normal
z outside a surface by the effective image potential for z > 0:
V(z) = -e2/(4z).
Assume that for z < 0 , V= ∞ . Find the eigenstates of an electron in the model Hamiltonian.
PHY 752 -- Assignment #29
Apr. 13, 2015
- Consider an fcc crystal of Cu with a lattice constant of a=3.61505 Angstroms. Construct a "super" unit cell with sides of 2a along the x-direction, a along the y-direction, and a along the z-direction. Find the corresponding d-spacings and estimate the X-ray diffraction intensities from the values given for the conventional unit cell.
PHY 752 -- Assignment #30
Apr. 15, 2015
Read Hubbard model section of Chapter 26 of Marder.
- Consider the two-site Hubbard model. Find all of the eigenvalues and eigenfunctions corresponding to two electrons with total spin 1.