5.7 Consider the dispersion relation (4.4) for a one-dimensional monatomic lattice of N atoms. Show that...
15. In BZ(I) of the ID monatomic lattice (a) the number of normal modes is equal to the number of the allowed values of (b) the group velocity increases with increasing a (c) the vibrational motions of the atoms are equivalent (a) , and -correspond to the same normal mode 16. A transverse elastic wave is travelling in a continuous medium in 1D. For this wave (a) the velocity depends on Young's modulus (b) the density of states is constant...
Condense Matter Physics 12. a) Using the fact that the allowed values of k in a one-dimensional lattice are given by k- n(2T/L), show that the density of electron states in the lattice, for a lattice of unit length, is given by g(E) = b) Evaluate this density of states in the TB model, and plot a(E) versus E. 12. a) Using the fact that the allowed values of k in a one-dimensional lattice are given by k- n(2T/L), show...
help with the second part please for 7 marks k Using the Debye approximation for a one-dimensional monatomic lattice with atomic spacing a and sound speed o, show that 18+71 2π k, k Using the Debye approximation for a one-dimensional monatomic lattice with atomic spacing a and sound speed o, show that 18+71 2π k,
Consider a one-dimensional tight binding model of electrons hopping between atoms. Let the distance between atoms be called a, and here let us label the atomic orbital on atom ln) for n-1,..,N (you may assume orthonormality of orbitals, ie., (1m)- nm). n as Suppose there is an on-site energy e and a hopping matrix element -t. In other words, suppose (IH|m) = E for n-m and (1비m)=-t for n=m±1. (a) Derive and sketch the dispersion curve for electrons. (b) How...
Assume there is a 2-dimensional crystalline sample, in which the ions can vibrate within the crystal plane (but not out of plane). It is a square lattice with lattice constant a. The sample itself is a square with side length L. There are N=L2/a2 atoms in the sample. (a) Derive the form of the density of states D(w). (The answer can be expressed in terms of dw/dk.) (b) Assume the phonon dispersion relation is w= vk, calculate the density of...
2. The dispersion relation for oscillation of a string (It has a length L with linear mass density μ and under tension T is given by: α is a positive constant. The string is fixed at x-0 and x-L. At t:0 The sting displacement is given by: (a) (10 points) Find the phase velocity (b) (10 points) Find the group velocity (c)(7 points) What are the frequencies of the normal modes (d) (3 points) at what time t will the...
5) Consider an oligomer with N 3 bonds occupying four lattice sites on a two dimensional square lattice with lattice constant b. One end of the oligomer is fixed at the origin of the lattice. (a) How many different conformations would such an oligomer have it if can occupy the same lattice site many times (simple random walk)? (b) How many different conformation would such an oligomer have it it cannot occupy the same lattice site (self-avoiding walk)? (c) Find...
Consider a one-dimensional tight binding model of electrons hopping between atoms. Let the distance between atoms be called a, and here let us label the atomic orbital on atom n as In) for n-1,..N (you may assume orthonormality of orbitals, i.e, (n|m) -8nm) Suppose there is an on-site energy є and a hopping matrix element-t. In other words, suppose 〈nlH1m)=ε for n-mand (IH1m)=-t for n-m±1. (d) What is the density of states? Consider a one-dimensional tight binding model of electrons...
Consider a one-dimensional tight binding model of electrons hopping between atoms. Let the distance between atoms be called a, and here let us label the atomic orbital on atom n as In) for n-1...N (you may assume orthonormality of orbitals, ie., (nlm)- m) Suppose there is an on-site energy and a hopping matrix element-t. In other words for n=m and for n=m±1. suppose(n[HIm) = ε <n m)=-t (a) Derive and sketch the dispersion curve for electrons. Consider a one-dimensional tight...
Band structure Consider a one-dimensional semiconductor crystal consisting of 11 atoms with nearest- neighbor atoms separated by a 5 . The band structure for electrons in the conduction band is given by Ec(k) = 101(k-0.2n)2-A(k-02n)"] + 2.25 [eV] and the band structure for holes in the valence band is given by where the wavevector k s in units ofA-1. The allowed wavevectors are--< k 즈 al (a) Is this a direct or indirect gap semiconductor? What is the energy gap...