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Chapter 10: 1CQ (page 413)

The left diagram in FIGURE 10.1 might represent a two atom crystal with two bands. Basing your argument on the kinetic energy inside either individual well, explain why both energies in the lower band should be roughly equal to that of the n=1 atomic state and why both energies in the upper should roughly equal that of the n=2 atomic state

Short Answer

Expert verified

Both energies in the lower band should be roughly equal to that of n=1atomic state because the two n=2 atomic states have the same kinetic energy, the linear combination of these states will also have the same energy as of n=2 state.

Step by step solution

01

Understanding the Concept

For an individual well, the kinetic energy of an electron for n=1 atomic state or n=2 atomic state is a fixed number. When there are two wells, there are two atomic states at n=1 or n=2.

02

Understanding Why Kinetic Energy is the same

An electron at n=2 molecular state has large separation, and so converges to the addition or subtraction of the two n=2atomic states. Since the two n=2 atomic states have the same kinetic energy, the linear combination of these two states will also have the same energy as that of n=2 atomic state.

Hence, an electron at n=1 molecular state has small separation, and so the energy of the molecular state is only roughly equal to energy for n=1 atomic state.

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Most popular questions from this chapter

A string wrapped around a hub of radius Rpulls with force FT on an object that rolls without slipping along horizontal rails on "wheels" of radiusr<R. Assume a massmand rotational inertia I.

(a) Prove that the ratio ofFTto the object's acceleration is negative. (Note: This object can't roll without slipping unless there is friction.) You can do this by actually calculating the acceleration from the translational and rotational second la was of motion, but it is possible to answer this part without such a "real" calculation.

(b) Verify thatFTtimes the speed at which the string moves in the direction of FT (i.e., the power delivered byFT) equals the rate at which the translational and rotational kinetic energies increase. That is.FTdoes all the work in this system, while the "internal" force does none. (c) Briefly discuss how parts (a) and (b) correspond to behaviors when an external electric field is applied to a semiconductor.

In Figure 10.24, the band n = 1 ends at k=4πL, while in Figure 10.27 it ends atπa. Are these compatible? If so, how?

Assuming an interatomic spacing of 0.15 nm, obtain a rough value for the width (in eV) of then=2 band in a one-dimensional crystal.

The accompanying diagram shows resistivity (reciprocal of conductivity) data for four solid materials from 77Kto 273K. scaled so that the maximum value plotted for each material is 1. Two are metals, one of which undergoes a transition between ordered and disordered spins in this temperature range. Speculate as to which plots correspond to these two metals and what the other two materials might be. Explain your reasoning.

In a buckyball, three of the bonds around each hexagon are so called double-bonds. They result from adjacent atoms sharing a state that does not participate in the sp2 bonding. Which state is it, and what is this extra bond σ-bond or a π-bond? Explain.
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