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Problem 4. Low-dimensional materials play an important role in nanotechnology. Consider a two-dimensional Fermi gas of...
Consider a two-dimensional non-interacting and non-relativistic gas of N spin-1/2 fermions at T 0 in a box of area A. (a) Find the Fermi energy εF. (b) Show that the total energy is given by E- NE. 2
Statistical_Mechanics 2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D. It is contained in an area of dimensions L x L. The particle mass is m. (a) Find the density of states D(e) N/L2 (b) Find the Fermi energy as a function of the particle density n = (c) Find the total energy as a function of the Fermi energy ef. (d) Find the chemical potential u as a function of n and T....
Consider a two-dimensional (2D) Bose gas at finite but low T confined in a square box potential with side lengths L and area A = L^2. 2. Consider a two-dimensional (2D) Bose gas at finite but low T confined in a square box potential with side lengths L and area A = L2. Using the density of states function as you found above, derive an expression for the 2D phase space density and argue why Bose-Einstein condensation does not occur...
Consider N non-interacting electrons confined to a two-dimensional square well of dimensions a × a. Derive an expression for the Fermi energy of this system in terms of the areal density σ = N/a2 and calculate the corresponding density of states. Show all steps.
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and...
a) Consider the above equation, 2T obtained for three dimensions. Following the same lines of reasoning above, show that in two dimensions, this equation becomes: g,A where A is the two-dimensional area of the two-dimensional sample b) Find kF as a function of N and A for such a two-dimensional system e) Find the Fermi energy for such a two-dimensional system d) At nonzero temperature, we want to compute the chemical potential (T). By imposing that the average number of...
a) Consider the above equation, 2T obtained for three dimensions. Following the same lines of reasoning above, show that in two dimensions, this equation becomes: g,A where A is the two-dimensional area of the two-dimensional sample b) Find kF as a function of N and A for such a two-dimensional system e) Find the Fermi energy for such a two-dimensional system d) At nonzero temperature, we want to compute the chemical potential (T). By imposing that the average number of...
Graphene is a single graphitic layer of carbon atoms, which has the remarkable feature that electrons in graphene behave as two dimensional massless relativistic fermions with a "speed of light, graphene 《 c. Andre Geim and Konstantin Novoselov were awarded the 2010 Nobel Prize in Physics "for groundbreaking experiments regarding the two-dimensional material graphene". a) ) Calculate the density of states as a function of energy for massless relativistic electrons in graphene with for a system of size L (note...
Question 4: (i) Write down the form of the Bose-Einstein distribution and discuss the phenomenon of Bose-Einstein condensation for a boson gas in three dimensions. In particular, carefully explain why the chemical potential becomes very close to the energy of the lowest single- particle state at sufficiently low temperature and describe how that changes the usual approach of replacing a discrete sum over energies with a continuum integral. Discuss how the occupation of the lowest single-particle state changes as a...
4. A particle moves in a periodic one-dimensional potential, V(x a)-V(x); physically, this may represent the motion of non-interacting electrons in a crys- tal lattice. Let us call n), n - 0, +1, t2, particle located at site n, with (n'In) -Sn,Let H be the system Hamiltonian and U(a) the discrete translation operator: U(a)|n) - [n +1). In the tight- binding approximation, one neglects the overlap of electron states separated by a distance larger than a, so that where is...