Question

Notify Me Bookmark

5. It is stated that the chemical potential of a phonon gas is zero, and that hence it is possible to obtain the average number of phonons in a crystal with frequency ω by substituting µ = 0 into 〈N〉 = k 1 e β(ϵk−µ) − 1 Explain how it is possible that µ = 0, if the free energy of an Einstein solid or the Debye solid explicitly depends on N ?

5. It is stated that the chemical potential of a phonon gas is zero, and that hence it is possible to obtain the average number of phonons in a crystal with frequency ω by substituting µ = 0 into ⟨N⟩ = k 1 e β(ϵk−µ) − 1 Explain how it is possible that µ = 0, if the free energy of an Einstein solid or the Debye solid explicitly depends on N ?

Image text
  1. It is stated that the chemical potential of a phonon gas is zero, and that hence it is possible to obtain the average number of phonons in a crystal with frequency ω by substituting μ = 0 into
⟨ N ⟩ = ∑ k 1 e β ( ϵ k − μ ) − 1
Explain how it is possible that μ = 0 , if the free energy of an Einstein solid or the Debye solid explicitly depends on N ?

Detailed Answer

0 0

Please enter your email here. You will get email when this question is answered by our tutor.

Doubtrix Logo

Problem is not yet solved!

Click on Notify me to get email when question is answered.

Join Doubtrix with 7-days free trial

  • Icon Correct & Verified Answers
  • Icon No AI-Generated Answers
  • Icon Video Solution for Difficult Questions
  • Icon Work With Experts to Reach at Correct Answers
Notify me Login Sign Up

Similar/Related Questions

For phonon gas, we have introduced two models to describe its statistical properties. Namely they are the Einstein model and the Debye model. We have learnt that for non-interacting systems, the statistical properties (i.e. the partition function) are mainly controlled by the dispersion relations and the density of states. The latter also strongly depends on the dimension of the system. (a) Discuss about the dispersion relations and the density of states of the two phonon models. Make plots to show their differences. (b) For phonons, the total number of modes are fixed. Note this does not imply the total number of phonons is fixed. Each mode is treated as a quantum harmonic oscillator, so it is a state that can be occupied by any integer number of phonons (since phonons are bosons). We are thus looking at partition functions of the "grand canonical ensemble" where number of phonons are not fixed. Write down the partition function of the Einstein model and the Debye model in three dimensions, and explain why we do not have a chemical potential in the partition function. (c) Derive the specific heat capacity for the Debye model from the partition function above. You will encounter an integration that is difficult to do, and you can leave it so. Try to do a change of variable for the integration, so the integrand is dimensionless. Explain why the integration is still dependent on temperature. (a mistake in the lecture notes was corrected). (d) Can you also derive the specific heat capacity in two-dimension using the Debye model? Note the total number of modes also depend on the spatial dimension, which will affect the Debye frequency.

For phonon gas, we have introduced two models to describe its statistical properties. Namely they are the Einstein model and the Debye model. We have learnt that for non-interacting systems, the statistical properties (i.e. the partition function) are mainly controlled by the dispersion relations and the density of states. The latter also strongly depends on the dimension of the system. (a) Discuss about the dispersion relations and the density of states of the two phonon models. Make plots to show their differences. (b) For phonons, the total number of modes are fixed. Note this does not imply the total number of phonons is fixed. Each mode is treated as a quantum harmonic oscillator, so it is a state that can be occupied by any integer number of phonons (since phonons are bosons). We are thus looking at partition functions of the "grand canonical ensemble" where number of phonons are not fixed. Write down the partition function of the Einstein model and the Debye model in three dimensions, and explain why we do not have a chemical potential in the partition function. (c) Derive the specific heat capacity for the Debye model from the partition function above. You will encounter an integration that is difficult to do, and you can leave it so. Try to do a change of variable for the integration, so the integrand is dimensionless. Explain why the integration is still dependent on temperature. (a mistake in the lecture notes was corrected). (d) Can you also derive the specific heat capacity in two-dimension using the Debye model? Note the total number of modes also depend on the spatial dimension, which will affect the Debye frequency.

Solution
0 0

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim. In a manual on how to have a number one song, it is stated that a song must be no longer than 210 seconds. A simple random sample of 40 current hit songs results in a mean length of 236.8sec and a standard deviation of 56.81sec. Use a 0.05 significance level and the accompanying Minitab display to test the claim that the sample is from a population of songs with a mean greater than 210sec. What do these results suggest about the advice given in the manual? 3 Click the icon to view the Minitab display. What are the hypotheses? A. H0 : µ = 210sec H1 : µ ≤ 210sec C. H0 : µ = 210sec H1 : µ > 210sec B. H0 : µ < 210sec H1 : µ > 210sec D. H0 : µ > 210sec H1 : µ ≤ 210sec Identify the test statistic. t = (Round to two decimal places as needed.) Identify the P-value. The P-value is (Round to three decimal places as needed.) State the final conclusion that addresses the original claim. Choose the correct answer below. A. Fail to reject H0. There is insufficient evidence to support the claim that the sample is from a population of songs with a mean length greater than 210sec. B. Fail to reject H0. There is sufficient evidence to support the claim that the sample is from a population of songs with a mean length greater than 210sec. C. Reject H0. There is insufficient evidence to support the claim that the sample is from a population of songs with a mean length greater than 210sec. D. Reject H0. There is sufficient evidence to support the claim that the sample is from a population of songs with a mean length greater than 210sec. What do the results suggest about the advice given in the manual? A. The results suggest that 236.8 seconds is the best song length. B. The results suggest that the advice of writing a song that must be no longer than 210 seconds is not sound advice. C. The results do not suggest that the advice of writing a song that must be no longer than 210 seconds is not sound advice. D. The results are inconclusive because the average length of a hit song is constantly changing. 3: Minitab display One-Sample T Test of mu = 210 vs > 210 N Mean StDev SE Mean Bound T P 40 236.80 56.81 8.98 221.67 2.98 0.002
Solution
0 0

A MOS capacitor has an n+ polysilicon gate with EF = EC+0.15 eV in the polysilicon, a 50 nm thick SiO2 gate dielectric with Qss = 0, and a p-type silicon substrate with Na = x.5×1016 /cm3, where x is the last digit of your student number. For silicon at room temperature ni = 1010 /cm3, Nv = 1.8×1019 /cm3, Nc = 3.2×1019 /cm3, and ϵSi = 11.7ϵ0. For SiO2 ϵSiO2 = 3.9ϵ0. a. Find the threshold voltage for this MOS capacitor. b. If the capacitor is made into a MOSFET with L = 2 μm, W = 200 μm, and a channel field effect mobility of 450 cm2/V⋅s, find ID for VGS − VT = 2 V and VDS = 1 V. c. Suppose it is desired to have a depletion mode MOSFET and that any gate oxide thickness is possible, but no other parameters can be changed. What is the most negative threshold voltage that is possible for the MOS capacitor with your substrate doping? d. Suppose it is desired to have a MOSFET with a threshold voltage of +0.5 V and again any gate oxide thickness is possible, but no other parameters can be changed. What oxide thickness will give VT = 0.5 V ? e. Suppose the source, drain, and body of the MOSFET are all grounded and the capacitance-voltage characteristics are measured. Sketch the high and low frequency capacitance-voltage characteristics and label significant features.
Solution
0 0