Problems4.1. Using the nearly free-electron approximation for a one-dimensional (1-D)crystal lattice and assuming that the only nonvanishing Fourier coefficientsof the crystal potential are v(7/a) and v(-A/a) in (4.73), show that near theband edge at k = 0, the dependence of electron energy on the wave vectork is given byEk = Eo +2m*where m* = mo[1 – (32m,a*/h*n*)v(n/a)²]¬l is the effective mass of the-electron at k = 0.4.2. The E-k relation of a simple cubic lattice given by (4.79) is derived fromthe tight-binding approximation. Show that near k 0 this relation can beexpressed byEx == Eno +2m*where m* = h? /2B,a?.And for k T/a, show that the E-k relation is given byE = Eno +2m*where m* == -n² /2B,a?.

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Problems 4.1. Using the nearly free-electron approximation for a one-dimensional (1-D) crystal lattice and assuming that the only nonvanishing Fourier coefficients of the crystal potential are v(7/a) and v(-n/a) in (4.73), show that near the band edge at k = 0, the dependence of electron energy on the wave vector k is given by Ek = Eo + 2m* where m* = mo[l – (32mža*/h*n*)v(7/a)²]¬l is the effective mass of the electron at k = 0. %3D - %3D 4.2. The E-k relation of a simple cubic lattice given by (4.79) is derived from the tight-binding approximation. Show that near k 0 this relation can be expressed by ħ?k? Ek = Eno + 2m* where m* = h2 /2B,a². And for k 7/a, show that the E-k relation is given by Ek = Eno + 2m* where m* = %3D

Problems 4.1. Using the nearly free-electron approximation for a one-dimensional (1-D) crystal lattice and assuming that the only nonvanishing Fourier coefficients of the crystal potential are v(7/a) and v(-n/a) in (4.73), show that near the band edge at k = 0, the dependence of electron energy on the wave vector k is given by Ek = Eo + 2m* where m* = mo[l – (32mža/hn)v(7/a)²]¬l is the effective mass of the electron at k = 0. %3D - %3D 4.2. The E-k relation of a simple cubic lattice given by (4.79) is derived from the tight-binding approximation. Show that near k 0 this relation can be expressed by ħ?k? Ek = Eno + 2m where m* = h2 /2B,a². And for k 7/a, show that the E-k relation is given by Ek = Eno + 2m* where m* = %3D

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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Question

Problems
4.1. Using the nearly free-electron approximation for a one-dimensional (1-D)
crystal lattice and assuming that the only nonvanishing Fourier coefficients
of the crystal potential are v(7/a) and v(-A/a) in (4.73), show that near the
band edge at k = 0, the dependence of electron energy on the wave vector
k is given by
Ek = Eo +
2m*
where m* = mo[1 – (32m,a*/h*n*)v(n/a)²]¬l is the effective mass of the
-
electron at k = 0.
4.2. The E-k relation of a simple cubic lattice given by (4.79) is derived from
the tight-binding approximation. Show that near k 0 this relation can be
expressed by
Ex =
= Eno +
2m*
where m* = h? /2B,a?.
And for k T/a, show that the E-k relation is given by
E = Eno +
2m*
where m* =
= -n² /2B,a?.

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