Question

A zone-boundary phonon mode (K3) in an improper ferroelectric material YMnO3 involves the displacement of atoms, as shown in the figure. In the paraelectric phase, the phonon mode vibrates in a harmonic potential, while in the ferroelectric phase, it vibrates in quartic potential (parameters are given below). We have to solve for the Schrodinger wave equation to find the ground state energy (E0) and the energy of five other levels just above E0. [-?²/2m ?²/?x² + V] ? = E? Harmonic potential parameters V = K0 + K2x² K2 = 1.4494 eV/Å² K0 = 0 eV Quartic potential parameters V = K0 + K2x² + K4x? K4 = 1.0342 eV/Å? K2 = -1.4494 eV/Å² K0 = 0 eV 1) Solve for harmonic and quartic potential and calculate the spacing between various energy levels. Choose -1 < x < 1 Angstrom where x denotes the magnitude K3 distortion in Angstrom. The effective mass m of mode is m = 266.10 amu. [50 points] 2) For both harmonic and quartic potential, plot the first four eigenvectors. [50 points]

          A zone-boundary phonon mode (K3) in an improper ferroelectric material YMnO3 involves the displacement of atoms, as shown in the figure. In the paraelectric phase, the phonon mode vibrates in a harmonic potential, while in the ferroelectric phase, it vibrates in quartic potential (parameters are given below). We have to solve for the Schrodinger wave equation to find the ground state energy (E0) and the energy of five other levels just above E0.

[-?²/2m ?²/?x² + V] ? = E?

Harmonic potential parameters
V = K0 + K2x²
K2 = 1.4494 eV/Å²
K0 = 0 eV

Quartic potential parameters
V = K0 + K2x² + K4x?
K4 = 1.0342 eV/Å?
K2 = -1.4494 eV/Å²
K0 = 0 eV

1) Solve for harmonic and quartic potential and calculate the spacing between various energy levels. Choose -1 < x < 1 Angstrom where x denotes the magnitude K3 distortion in Angstrom. The effective mass m of mode is m = 266.10 amu. [50 points]
2) For both harmonic and quartic potential, plot the first four eigenvectors. [50 points]

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A zone-boundary phonon mode (K3) in an improper ferroelectric material YMnO3 involves the displacement of atoms, as shown in the figure. In the paraelectric phase, the phonon mode vibrates in a harmonic potential, while in the ferroelectric phase, it vibrates in quartic potential (parameters are given below). We have to solve for the Schrodinger wave equation to find the ground state energy (E0) and the energy of five other levels just above E0.

[-?²/2m ?²/?x² + V] ? = E?

Harmonic potential parameters
V = K0 + K2x²
K2 = 1.4494 eV/Å²
K0 = 0 eV

Quartic potential parameters
V = K0 + K2x² + K4x?
K4 = 1.0342 eV/Å?
K2 = -1.4494 eV/Å²
K0 = 0 eV

1) Solve for harmonic and quartic potential and calculate the spacing between various energy levels. Choose -1 < x < 1 Angstrom where x denotes the magnitude K3 distortion in Angstrom. The effective mass m of mode is m = 266.10 amu. [50 points]
2) For both harmonic and quartic potential, plot the first four eigenvectors. [50 points]

Added by Aaron P.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

10/10/2022

Step 1

441 \times 10^{-24}\) grams and \(\omega = 22.912 \times 10^2\) rad/s. Show more…

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A zone-boundary phonon mode (K3) in an improper ferroelectric material YMnO3 involves the displacement of atoms, as shown in the figure. In the paraelectric phase, the phonon mode vibrates in a harmonic potential, while in the ferroelectric phase, it vibrates in quartic potential (parameters are given below). We have to solve for the Schrodinger wave equation to find the ground state energy (E0) and the energy of five other levels just above E0: [-ħ²/2m ∂²/∂x² + V] ψ = Eψ Harmonic potential parameters: V = K0 + K2x² K2 = 1.4494 eV/Å² K0 = 0 eV Quartic potential parameters: V = K0 + K2x² + K4x⁴ K4 = 1.0342 eV/Å⁴ K2 = -1.4494 eV/Å² K0 = 0 eV 1) Solve for harmonic and quartic potential and calculate the spacing between various energy levels. Choose -1 < x < 1 Angstrom where x denotes the magnitude K3 distortion in Angstrom. The effective mass m of mode is m = 266.10 amu. [50 points] 2) For both harmonic and quartic potential, plot the first four eigenvectors. [50 points]

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Transcript

00:01 Hello student answer to the question is as follows for harmonic.

00:06 First as v equals to k0 plus k to x squared so k not equals to 0 v equals to k2 x square equals to half k x square omega square m equals to k by k to m x square equals to half m omega square x square so k equals to k2 equals to half m omega square so m omega square equals to half m omega square so w omega 2 will be equals to w by or omega y under root 2 so we know for harmonic potential v is equal to half m omega square x square so here e n will be n plus half equals to omega hashtag omega and here en equals to n plus half omega square omega two under root 2 so omega 2 will be equal to under root k2 by m so k2 will be equals to 1 .449 4 e v by am strong not 2 that will be equals to 1 .44 .94 multiplied by 1 .6 into 10 to per minus 19 j by m strong square equals to 2 .319 into 10 is to per minus 19 j by a j by m strong square hence 1 .66 into 10 is to per minus 24 gram mass or m will be equals to 2 .66 in 0 .1 multiplied by 1 .6 x6 into 10 is to power minus 24 which will be equals to 4 .41 .726 into 10 is to power minus 24 grams here omega 2 will be equals to under root 0 .00525 multiplied by 10 is to power 5 which will be equal to under root 0 .0 525 into tennis to part 2 equals to 0 .2 to 9 .1 8 into 10 is to part 2 equals to 22 .910.

03:11 So from here e0 is equals to under root 2 by 2 omega 2 equals to under root 2 by 2, omega 2 equals to under root 2 by 2 into 1 .05 multiplied by 10 is to par minus 34 multiplied by 22 .912 equals to 17 .0138 multiplied by 10 is to per minus 34 joule.

03:47 So, e1 equals to 3 by under root 2, w2 equals to 51 .0414.

03:58 Into 10 is to per minus 34 joules whereas e2 equals to 5t1 omega 2 by under root 2 equals to 85 .069 into 10 is to minus 34 jules so delta e will be equals to e n minus e n minus that will be equals to 24 .057 into 10 for minus 34 joules.

04:37 Now the four eigenitates will be as follows...

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