For a finite system of Fermions where the density of states increases with energy, th chemical potential (a) Decreases with temperature (b) Increases with temperature (c) Does not vary with temperature (d) Corresponds to the energy where the occupation probability is 0.5
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- For 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 VSuppose that an energy level (j) includes 6 states (gj =6) and 3 particles (Nj =3). What will be the possible distributions of the particles among the states according to Fermi Dirac statistics * ?- Dirac statisticsAssume that the chances of the patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?
- For a spin 2 Fermi gas em! Ve de N= 2 whene flE) is the Fermi- Diracd dhistribution function. (a) What i's the definitonof fermi enengy(EF), and fermi temperature (TF). (b) Find EF in terms of N an V (C) what is Ff for Aluminium. (d) whati is the toba enenpy o4 ther termi' goes ot T=0 k, le) what i's the pressure of the Fermi's at T=oと.Consider n-doped Ge with N=10¹8 cm-³ with mobility 300 cm²/Vs at room temperature T=300 K. Calculate approximate thermoelectric transport coefficients: 1) mean free path, 2) resistivity, 3) Seebeck coefficient, 4) Peltier coefficient, and 5) electronic thermal conductivity. Use materials parameters for the effective density of states in the conduction band N=10¹⁹ cm³³.Calculate the partition function of a two-level system at 25 °C with an energy gap of 10-2¹ J, assuming: a) Both states are non-degenerate. b) The ground state is non-degenerate, and the excited state is 3-fold degenerate.
- (d) Fermions are represented by Dirac spinors and obey the Dirac equation. The Dirac equation is (i", - m)=0 where, in the so-called chiral basis, the gamma matrices are: x=(-15) - where i runs over 1,2,3 and o are the Pauli spin matrices. i. In this basis, calculate the 'fifth' gamma matrix 75 = iyºy¹z²z³. ii. Determine the result of the projection operator (1+75) acting on the spinor - (3). X =In Fermi-Dirac Statistics, one energy state can be occupied by more than one particle. a) True b) FalsePlot the Fermi-Dirac probability of occupation function fFD(E) for T = 0, 10, 100, 200, 300 and 400K.