Related questions
Question
![For a simple cubic structure with lattice parameter a, the energy dispersion in the tight-binding
(TB) approximation is obtained by
E(k) = Eo – a – 27 [cos(ka) + cos(kya) + cos(k,a)] ,
where a and y are overlap integrals of the TB approximation.
) Expand the energy E(k) near the bottom of the band (i.e., near the I point, |k| = k = 0) where
= Eo – a – 6y, and effective
ka < 1 to the second order in k. You may define for simplicity Er
2² E`
-1
()
electron mass m = h
Use Er and m, in the energy relation.
) Expand E(k) near the first Brillouin zone boundary at the W point (k = ky = kz = ")
in
%3D
terms of dk = |8k]. Sk « 1, where k; =
- Ski, and i = x, y, z. Give the result using Er and
a
m defined above.](https://content.bartleby.com/qna-images/question/99d1fc48-d7b2-428e-9847-83620ea7fe5b/892d6dfc-1c92-44f3-ad8b-94ee52bebefe/z7582c_processed.jpeg)
Transcribed Image Text:For a simple cubic structure with lattice parameter a, the energy dispersion in the tight-binding
(TB) approximation is obtained by
E(k) = Eo – a – 27 [cos(ka) + cos(kya) + cos(k,a)] ,
where a and y are overlap integrals of the TB approximation.
) Expand the energy E(k) near the bottom of the band (i.e., near the I point, |k| = k = 0) where
= Eo – a – 6y, and effective
ka < 1 to the second order in k. You may define for simplicity Er
2² E`
-1
()
electron mass m = h
Use Er and m, in the energy relation.
) Expand E(k) near the first Brillouin zone boundary at the W point (k = ky = kz = ")
in
%3D
terms of dk = |8k]. Sk « 1, where k; =
- Ski, and i = x, y, z. Give the result using Er and
a
m defined above.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 2 steps with 2 images
Knowledge Booster
Similar questions
- A CO molecule is initially in the n = 2 vibrational level. If this molecule loses both vibrational and rotational energy and emits a photon, what are the photon wavelength and frequency if the initial angular momentum quantum number is l = 3?arrow_forwardFor 3D free electron gas, the density of states counts the number of degenerate electron states dn per energy interval dE around a given energy E as g(E): = dn dE 3 (2m₂)2V 1 E2 2π²ħ³ At absolute zero temperature, N electrons can fill up all low lying energy levels (following Pauli exclusion principle) up to a given energy level E called Fermi energy. From the density of states, what is the relation between the total electron states N below a given energy E? Use this result to show that the Fermi energy EF is given by - - 2010 (307² M)³ ħ² 3π²N\3 EF 2me Varrow_forwardA diatomic molecule is modeled as a Morse oscillator, and one finds that its energy level differences decrease from E2 - E₁ = 1374.2 cm-¹ to E7 - E6 = 1139.6 cm ¹. (a) Use this information and the quantum energy levels for the Morse oscillator to find the harmonic angular frequency, w, in cm-¹. (b) What is the dissociation energy, D (in kJ/mole) for this Morse oscillator? (Note that the energy units, 11.9627 J/mole = 1 cm-¹.)arrow_forward
- P9E.11 (a) For a linear conjugated polyene with each of N carbon atoms contributing an electron in a 2p orbital, the energies E, of the resulting A molecular orbitals are given by: E, =a+2B cos- N+1 k=1, 2,.,N Use this expression to make a reasonable empirical estimate of the resonance integral B for the homologous series consisting of ethene, butadiene, hexatriene, and octatetraene given that t-n ultraviolet absorptions from the HOMO to the LUMO occur at 61 500, 46 080, 39 750, and 32 900 cm", respectively. (b) Calculate the T-electron delocalization energy, Egdo:= E, - n(a+ B), of octatetraene, where E, is the total T-electron binding energy and n is the total number of T-electrons. (c) In the context of this Hückel model, the molecular orbitals are written as linear combinations of the carbon 2p orbitals. The coefficient of the jth atomic orbital in the kth molecular orbital is given by: cN sin j=1,2.N jkn j=1, 2,.,N Evaluate the coefficients of each of the six 2p orbitals in each…arrow_forwardConsider a rigid lattice of distinguishable spin 1/2 atoms in a magnetic field. The spins have two states, with energies -HọH and +µoH for spins up (1) and down (4), respectively, relative to H. The system is at temperature Т. (a) Determine the canonical partition function z for this system. (b) Determine the total magnetic moment M = µo(N+ – N-) of the %3D system. (c) Determine the entropy of the system.arrow_forwardShown below Is a portlon of the rotational-vibrational IR-spectrum of DCI (where D H), with 2. the posltlon of each peak Identifled by wavenumber (cm), 0.6 0.5 04 0.3 2090 2110 Wavenumber (cm) Absorbance 2.060.3arrow_forward
arrow_back_ios
arrow_forward_ios