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Q. 8.19

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An Introduction to Thermal Physics
Found in: Page 345
An Introduction to Thermal Physics

An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

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Short Answer

The critical temperature of iron is 1043K. Use this value to make a rough estimate of the dipole-dipole interaction energy ε, in electron-volts.

The dipole interaction energy of iron = 0.011eV

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1. Given information

Formula for critical temperature,

kTc=nε

Here,

k =Boltzmann constant,

n= number of nearest neighboring dipole

ε= interaction energy.

Step 2. To find the dipole interaction energy of iron

We have,

ε=kTcn

Iron is BCC structure, and the value of n is 8.0,

Substituting the value of 8.617×10-5eV/k = k,1043k for critical temperature of iron Tc and 8.0 for n

ε=8.617×10-5eV/k(1043k)(8)

=0.011eV

The dipole interaction energy of iron = =0.011eV

Most popular questions for Physics Textbooks

For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four "neighbors"-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).

Modify the ising program to compute the average energy of the system over all iterations. To do this, first add code to the initialise subroutine compute the initial energy of the lattice; then, whenever a dipole is flipped, change the energy variable by the appropriate amount. When computing the average energy, be sure to average over all iterations, not just those iterations in which a dipole is actually flipped (why?). Run the program for a 5 x 5 lattice for T values from 4 down to l in reasonably small intervals, then plot the average energy as a function of T. Also plot the heat capacity. Use at least 1000 iterations per dipole for each run, preferably more. If your computer is fast enough, repeat for a 10x 10 lattice and for a 20 x 20 lattice. Discuss the results. (Hint: Rather than starting over at each temperature with a random initial state, you can save time by starting with the final state generated at the previous, nearby temperature. For the larger lattices you may wish to save time by considering only a smaller temperature interval, perhaps from 3 down to 1.5.)

Show that the Lennard-Jones potential reaches its minimum value at r=r0, and that its value at this minimum is -u0. At what value of r does the potential equal zero?

Draw all the diagrams, connected or disconnected, representing terms in the configuration integral with four factors of fij. You should find 11 diagrams in total, of which five are connected.

Show that, if you don't make too many approximations, the exponential series in equation 8.22 includes the three-dot diagram in equation 8.18. There will be some leftover terms; show that these vanish in the thermodynamic limit.

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