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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 force
Afework Reply
The different examples for collision
Afework
What is polarization and there are type
Muhammed Reply
Polarization is the process of transforming unpolarized light into polarized light. types of polarization 1. linear polarization. 2. circular polarization. 3. elliptical polarization.
Eze
Describe what you would see when looking at a body whose temperature is increased from 1000 K to 1,000,000 K
Aishwarya Reply
how is tan ninety minus an angle equals to cot an angle?
Niicommey Reply
please I don't understand all about this things going on here
Jeremiah Reply
What is torque?
Matthew Reply
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study.
Teka
Torque refers to the rotational force. i.e Torque = Force × radius.
Arun
Torque is the rotational equivalent of force . Specifically, it is a force exerted at a distance from an object's axis of rotation. In the same way that a force applied to an object will cause it to move linearly, a torque applied to an object will cause it to rotate around a pivot point.
Teka
Torque is the rotational equivalence of force . So, a net torque will cause an object to rotate with an angular acceleration. Because all rotational motions have an axis of rotation, a torque must be defined about a rotational axis. A torque is a force applied to a point on an object about the axis
Teka
When a missle is shot from one spaceship towards another, it leaves the first at 0.950c and approaches the other at 0.750c. what is the relative velocity of the two shipd
Marifel Reply
how to convert:m^3/s^2 all divided by kg to cm^3/s^2
Thibaza Reply
Is there any proof of existence of luminiferious aether ?
Zero Reply
mass conversion of 58.73kg =mg
Proactive Reply
is Space time fabric real
Godawari Reply
What's the relationship between the work function and the cut off frequency in the diagram above?
frankline Reply
due to the upthrust weight of the object varise with force in which the body fall into the water pendincular with the reflection of light with it
Gift
n=I/r
Gift
can someone explain what is going on here
falanga
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falanga
what is meant by fluctuated
Olasukanmi Reply
If n=cv then how v=cn? and if n=c/v then how v=cn?
Natanim
convert feet to metre
Mbah Reply
what is electrolysis
Mbah
Electrolysis is the chemical decomposition of electrolyte either in molten state or solution to conduct electricity
Ayomide
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Ayesha Reply
can someone help explain why v2/c2 is =1/2 Using The Lorentz Transformation For Time Spacecraft S′ is on its way to Alpha Centauri when Spacecraft S passes it at relative speed c /2. The captain of S′ sends a radio signal that lasts 1.2 s according to that ship’s clock. Use the Lorentz transformati
Jennifer
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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