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By the end of this section, you will be able to:
  • Describe two main approaches to determining the energy levels of an electron in a crystal
  • Explain the presence of energy bands and gaps in the energy structure of a crystal
  • Explain why some materials are good conductors and others are good insulators
  • Differentiate between an insulator and a semiconductor

The free electron model explains many important properties of conductors but is weak in at least two areas. First, it assumes a constant potential energy within the solid. (Recall that a constant potential energy is associated with no forces.) [link] compares the assumption of a constant potential energy (dotted line) with the periodic Coulomb potential, which drops as −1 / r at each lattice point, where r is the distance from the ion core (solid line). Second, the free electron model assumes an impenetrable barrier at the surface. This assumption is not valid, because under certain conditions, electrons can escape the surface—such as in the photoelectric effect. In addition to these assumptions, the free electron model does not explain the dramatic differences in electronic properties of conductors, semiconductors, and insulators. Therefore, a more complete model is needed.

Figure shows three inverted U shaped structures in a row and two incomplete ones on either side of the row. There are red dots at the bottom between two consecutive figures, with plus signs below them. The distance between two consecutive dots is a. The shapes are labeled minus 1 by r. There is a dotted just above the shapes. This is labeled approximate constant potential energy.
The periodic potential used to model electrons in a conductor. Each ion in the solid is the source of a Coulomb potential. Notice that the free electron model is productive because the average of this field is approximately constant.

We can produce an improved model by solving Schrödinger’s equation for the periodic potential shown in [link] . However, the solution requires technical mathematics far beyond our scope. We again seek a qualitative argument based on quantum mechanics to find a way forward.

We first review the argument used to explain the energy structure of a covalent bond. Consider two identical hydrogen atoms so far apart that there is no interaction whatsoever between them. Further suppose that the electron in each atom is in the same ground state: a 1 s electron with an energy of −13.6 eV (ignore spin). When the hydrogen atoms are brought closer together, the individual wave functions of the electrons overlap and, by the exclusion principle, can no longer be in the same quantum state, which splits the original equivalent energy levels into two different energy levels. The energies of these levels depend on the interatomic distance, α ( [link] ).

If four hydrogen atoms are brought together, four levels are formed from the four possible symmetries—a single sine wave “hump” in each well, alternating up and down, and so on. In the limit of a very large number N of atoms, we expect a spread of nearly continuous bands of electronic energy levels in a solid (see [link] (c)). Each of these bands is known as an energy band    . (The allowed states of energy and wave number are still technically quantized, but for large numbers of atoms, these states are so close together that they are consider to be continuous or “in the continuum.”)

Energy bands differ in the number of electrons they hold. In the 1 s and 2 s energy bands, each energy level holds up to two electrons (spin up and spin down), so this band has a maximum occupancy of 2 N electrons. In the 2 p energy band, each energy level holds up to six electrons, so this band has a maximum occupancy of 6 N electrons ( [link] ).

Questions & Answers

what is energy
Isiguzo Reply
friction ka direction Kaise pata karte hai
Rahul Reply
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
Varia Reply
fundamental note of a vibrating string
fasoyin Reply
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
mallam Reply
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
Mathematical expression of principle of relativity
Nasir Reply
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
obia Reply
What is the importance of de-broglie's wavelength?
Mukulika Reply
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
how those weight effect a stable motion at equilibrium
Nonso Reply
how do I differentiate this equation- A sinwt with respect to t
Evans Reply
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry
de broglie wave equation
LoNE Reply
vy beautiful equation
chandrasekhar
what is electro statics
fitsum Reply
when you consider systems consisting of fixed charges
Sherly
Diagram of the derive rotational analog equation of v= u+at
Nnamnso Reply
what is carat
Arnulfo Reply
a unit of weight for precious stones and pearls, now equivalent to 200 milligrams.
LoNE
a science that deals with the composition, structure, and properties of substances and with the transformations that they undergo.
LoNE
what is chemistry
Mrs Reply
what chemistry ?
Abakar
Practice Key Terms 5

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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