For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbours”â€”above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ε ) for the particular state of the 4 × 4 square lattice shown in Figure 8.4? Figure 8.4. One particular state of an Ising model on a 4 × 4 square lattice (Problem 8.15).

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Answer 1

Textbook Question

Chapter 8.2, Problem 15P

For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbours”â€”above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ε ) for the particular state of the 4 × 4 square lattice shown in Figure 8.4?

Chapter 8.2, Problem 15P, For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four

Figure 8.4. One particular state of an Ising model on a 4 × 4 square lattice (Problem 8.15).

Expert Solution & Answer

To determine

Total energy in terms of ε .

Explanation of Solution

Introduction:

Draw a diagram to show one Ising model on (4×4) lattice

An Introduction to Thermal Physics, Chapter 8.2, Problem 15P

Here, each dipole has four nearest neighboring dipoles except dipoles on edges.

The lattice in above diagram has 14 nearest neighboring dipoles in between parallel dipoles as well as 19 neighboring dipoles in between anti-parallel dipoles in total.

Write the expression of total interaction energy U of the lattice

U=(10)(ε)+(14)(−ε)

Simplify the above expression

U=−4ε

Conclusion:

Thus, the total energy is −4ε in terms of ε .

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A particle of mass M moves in a periodic potential with the form, V Vo [15 2 16 2πx 1 - cos + 16 : (1 6πx - COS where Vo is the depth of the lattice and d is a constant with units of length. The potential, represented in Figure 1, has minima at x = nd where n is an integer. V 0 d 2d 3d 4d 5d 6d x Figure 1: For use with Q11 In this question, you may find the following approximation useful: cos(0) 102/2, if is small. (a) Show that close to x = 0 this potential can be approximated by that of a harmonic oscillator. Derive its angular frequency w in terms of Vo, d and M and use this expresssion to determine the energy eigenfunction of the ground state for this potential. (b) Use the tight-binding approximation to write down energy eigenfunctions for a particle moving in this potential. Use Bloch's theorem to show that the probability density associated with these eigenfunctions has the periodicity of the lattice.

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Q3.1 Please answer the following question throughly and detailed. Need to understand the concept.

Answer 2

Textbook Question

Chapter 8.2, Problem 15P

For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbours”â€”above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ε ) for the particular state of the 4 × 4 square lattice shown in Figure 8.4?

Chapter 8.2, Problem 15P, For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four

Figure 8.4. One particular state of an Ising model on a 4 × 4 square lattice (Problem 8.15).

Expert Solution & Answer

To determine

Total energy in terms of ε .

Explanation of Solution

Introduction:

Draw a diagram to show one Ising model on (4×4) lattice

An Introduction to Thermal Physics, Chapter 8.2, Problem 15P

Here, each dipole has four nearest neighboring dipoles except dipoles on edges.

The lattice in above diagram has 14 nearest neighboring dipoles in between parallel dipoles as well as 19 neighboring dipoles in between anti-parallel dipoles in total.

Write the expression of total interaction energy U of the lattice

U=(10)(ε)+(14)(−ε)

Simplify the above expression

U=−4ε

Conclusion:

Thus, the total energy is −4ε in terms of ε .

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Textbook Question

Chapter 8.2, Problem 15P

For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbours”â€”above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ε ) for the particular state of the 4 × 4 square lattice shown in Figure 8.4?

Chapter 8.2, Problem 15P, For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four

Figure 8.4. One particular state of an Ising model on a 4 × 4 square lattice (Problem 8.15).

Expert Solution & Answer

To determine

Total energy in terms of ε .

Explanation of Solution

Introduction:

Draw a diagram to show one Ising model on (4×4) lattice

An Introduction to Thermal Physics, Chapter 8.2, Problem 15P

Here, each dipole has four nearest neighboring dipoles except dipoles on edges.

The lattice in above diagram has 14 nearest neighboring dipoles in between parallel dipoles as well as 19 neighboring dipoles in between anti-parallel dipoles in total.

Write the expression of total interaction energy U of the lattice

U=(10)(ε)+(14)(−ε)

Simplify the above expression

U=−4ε

Conclusion:

Thus, the total energy is −4ε in terms of ε .

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Textbook Question

Chapter 8.2, Problem 15P

For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbours”â€”above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ε ) for the particular state of the 4 × 4 square lattice shown in Figure 8.4?

Chapter 8.2, Problem 15P, For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four

Figure 8.4. One particular state of an Ising model on a 4 × 4 square lattice (Problem 8.15).

Expert Solution & Answer

To determine

Total energy in terms of ε .

Explanation of Solution

Introduction:

Draw a diagram to show one Ising model on (4×4) lattice

An Introduction to Thermal Physics, Chapter 8.2, Problem 15P

Here, each dipole has four nearest neighboring dipoles except dipoles on edges.

The lattice in above diagram has 14 nearest neighboring dipoles in between parallel dipoles as well as 19 neighboring dipoles in between anti-parallel dipoles in total.

Write the expression of total interaction energy U of the lattice

U=(10)(ε)+(14)(−ε)

Simplify the above expression

U=−4ε

Conclusion:

Thus, the total energy is −4ε in terms of ε .

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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

Total energy in terms of ε .

Explanation of Solution

Introduction:

Draw a diagram to show one Ising model on (4×4) lattice

An Introduction to Thermal Physics, Chapter 8.2, Problem 15P

Here, each dipole has four nearest neighboring dipoles except dipoles on edges.

The lattice in above diagram has 14 nearest neighboring dipoles in between parallel dipoles as well as 19 neighboring dipoles in between anti-parallel dipoles in total.

Write the expression of total interaction energy U of the lattice

U=(10)(ε)+(14)(−ε)

Simplify the above expression

U=−4ε

Conclusion:

Thus, the total energy is −4ε in terms of ε .

Answer 8

Expert Solution & Answer

To determine

Total energy in terms of ε .

Explanation of Solution

Introduction:

Draw a diagram to show one Ising model on (4×4) lattice

An Introduction to Thermal Physics, Chapter 8.2, Problem 15P

Here, each dipole has four nearest neighboring dipoles except dipoles on edges.

The lattice in above diagram has 14 nearest neighboring dipoles in between parallel dipoles as well as 19 neighboring dipoles in between anti-parallel dipoles in total.

Write the expression of total interaction energy U of the lattice

U=(10)(ε)+(14)(−ε)

Simplify the above expression

U=−4ε

Conclusion:

Thus, the total energy is −4ε in terms of ε .

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

Total energy in terms of ε .

Answer 12

Explanation of Solution

Introduction:

Draw a diagram to show one Ising model on (4×4) lattice

An Introduction to Thermal Physics, Chapter 8.2, Problem 15P

Here, each dipole has four nearest neighboring dipoles except dipoles on edges.

The lattice in above diagram has 14 nearest neighboring dipoles in between parallel dipoles as well as 19 neighboring dipoles in between anti-parallel dipoles in total.

Write the expression of total interaction energy U of the lattice

U=(10)(ε)+(14)(−ε)

Simplify the above expression

U=−4ε

Conclusion:

Thus, the total energy is −4ε in terms of ε .

Answer 13

Ch. 8.1 - Prob. 2P Ch. 8.1 - Prob. 4P Ch. 8.1 - Prob. 9P Ch. 8.1 - Prob. 10P Ch. 8.1 - Prob. 11P Ch. 8.1 - Prob. 12P Ch. 8.1 - Prob. 13P Ch. 8.2 - For a two-dimensional Ising model on a square...Ch. 8.2 - Prob. 16P Ch. 8.2 - Prob. 17P

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