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2. a) Draw the free electron energy levels in the first Brillouin zone of a BCC crystal, from the zone center to the middle of one of the zone faces. (This is called the N direction.) Go up to energy \(20(\frac{\pi}{a})^2(\frac{\hbar^2}{2m})\). In a BCC crystal with “nearly” free electrons, the actual energy levels should be only slightly perturbed from this ideal free electron situation. b) On your graph, indicate the lowest-energy level crossing that is not at the edge of the Brillouin zone. What is the degeneracy of the crossing, including the degeneracy within the individual curves but for now ignoring spin? (I.e., what is the size of the matrix that you would have to diagonalize to find the energies in the presence of a weak potential, if the Hamiltonian is independent of spin?) List any three independent matrix elements of the potential that matter for this crossing. c) Repeat part b) for the lowest non-zero energy at \(k = 0\).

          2. a) Draw the free electron energy levels in the first Brillouin zone of a BCC crystal, from the zone center to the middle of one of the zone faces. (This is called the N direction.) Go up to energy \(20(\frac{\pi}{a})^2(\frac{\hbar^2}{2m})\). In a BCC crystal with “nearly” free electrons, the actual energy levels should be only slightly perturbed from this ideal free electron situation.
b) On your graph, indicate the lowest-energy level crossing that is not at the edge of the Brillouin zone. What is the degeneracy of the crossing, including the degeneracy within the individual curves but for now ignoring spin? (I.e., what is the size of the matrix that you would have to diagonalize to find the energies in the presence of a weak potential, if the Hamiltonian is independent of spin?) List any three independent matrix elements of the potential that matter for this crossing.
c) Repeat part b) for the lowest non-zero energy at \(k = 0\).

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2. a) Draw the free electron energy levels in the first Brillouin zone of a BCC crystal, from the zone center to the middle of one of the zone faces. (This is called the N direction.) Go up to energy 20((π)/(a))^2((ħ^2)/(2m)). In a BCC crystal with “nearly” free electrons, the actual energy levels should be only slightly perturbed from this ideal free electron situation.
b) On your graph, indicate the lowest-energy level crossing that is not at the edge of the Brillouin zone. What is the degeneracy of the crossing, including the degeneracy within the individual curves but for now ignoring spin? (I.e., what is the size of the matrix that you would have to diagonalize to find the energies in the presence of a weak potential, if the Hamiltonian is independent of spin?) List any three independent matrix elements of the potential that matter for this crossing.
c) Repeat part b) for the lowest non-zero energy at k = 0.

Added by Willie H.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Madhur L

06/19/2023

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The energy levels should go up to 20((\pi)/(a))^2((\hbar^2)/(2m)), where a is the lattice constant, \hbar is the reduced Planck's constant, and m is the electron mass. Show more…

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a) Draw the free electron energy levels in the first Brillouin zone of a BCC crystal, from the zone center to the middle of one of the zone faces. (This is called the N direction.) Go up to energy 20((pi )/(a))^(2)((ℏ^(2))/(2m)). In a BCC crystal with "nearly" free electrons, the actual energy levels should be only slightly perturbed from this ideal free electron situation. b) On your graph, indicate the lowest-energy level crossing that is not at the edge of the Brillouin zone. What is the degeneracy of the crossing, including the degeneracy within the individual curves but for now ignoring spin? (I.e., what is the size of the matrix that you would have to diagonalize to find the energies in the presence of a weak potential, if the Hamiltonian is independent of spin?) List any three independent matrix elements of the potential that matter for this crossing. c) Repeat part b) for the lowest non-zero energy at k=0. 2. a) Draw the free electron energy levels in the first Brillouin zone of a BCC crystal, from the zone cen ter to the middle of one of the zone faces. (This is called the N direction.) Go up to energy 20()2(2m). In a BCC crystal with "nearly free electrons, the actual energy levels should be only slightly perturbed from this ideal free electron situation. b) On your graph, indicate the lowest-energy level crossing that is not at the edge of the Brillouin zone. What is the degeneracy of the crossing, including the degeneracy within the individual curves but for now ignoring spin? (I.e., what is the size of the matrix that you would have to diagonalize to find the energies in the presence of a weak potential, if the Hamiltonian is independent of spin?) List any three independent matrix elements of the potential that matter for this crossing. c) Repeat part b) for the lowest non-zero energy at k = 0.

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Transcript

00:01 Hi, in this question we have to discuss the 24 possible configuration, so possible configuration for the system with total energy.

00:11 So, for e total is equal to 106 electron volts.

00:16 So, here we need to distribute 20 electron among the energy levels, energy levels by pauli's exclusion principle.

00:28 So, here the energy levels are 0 .5, 1 .5, so these are the energy levels that is 0 .5, 1 .5, 2 .5, 3 .5, 4 .5, 5 .5 etc.

00:43 So, all are in electron volts.

00:45 Then, so then we have to determine the energies e1, e2, e3, e4, e5, e6 corresponding to the energy levels this one.

00:55 Then, we have to represent the, so represent the each configuration using the notation filled and empty.

01:02 So, here figure 1 can be represented by filled, filled, empty, so empty, empty, empty, empty, empty, so empty, empty, empty, empty, empty, so this is the representation.

01:25 Similarly, we have to determine the 24 possible configuration.

01:31 So, let us describe all those configuration.

01:36 So, these are the 20 combination which has to be noted.

01:39 So, these are the remaining 4 combinations, so that we can have 24 contributions.

01:44 Then, coming to part 2, we have the probability of finding the electron in the 24 combination which are likely to occur.

01:50 So, here they are obtained by dividing the total number of energy levels by total number of combinations...

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