5(10pts)A simple model of a metal consists of electrons moving in a one dimensio

Question

5(10pts)A simple model of a metal consists of electrons moving in a one dimensional lattice with narrow deep potential wells at each atom. The resulting allowed electron energies consist of bands of allowed energies separated by gaps. If there are N atoms in the metal then there are qN electrons, where q is the number of valence electrons. There are N possible energy states in each allowed band. A conductor is a material with a partially filled band and an insulator is a material with only filled or empty bands. a) Taking accounts the electron spin and the exclusion principle how many electrons are b) Which represent conductors; qeven or q odd and why? a) In this theory which of the following materials is conductors (circle your answers). required to fill a band? Explain 3Li 80 11Na 15P 18Ar 22Ti 26Fei 37Rb

Explanation / Answer

a) One band is due to one or more atomic energy levels. The minimum number of states in a band is double the number of atoms in that material. According to Paulis exclusion principle no two electrons in an atom can have the same four quantum numbers. Hence a single energy level can have two electrons with opposite spin.

The answer is minimum two electrons are needed to fill a band.

b) Both q even and q odd can be conductors.

The reason is if q is odd the band will be partially filled (e.g. in the case of q=1). It will have available energy levels at slightly higher energies. The electrons can move to the unoccupied higher energy levels and contribute to the conductivity.

In the case of q= even, it may be conducting if the filled band overlaps with an empty bands where the electrons can move in an applied electric field and can act as conductor.

c) Let us calculate the valence electrons:

3Li (1s22s1): number of valence electrons= 1

8O (1s22s22p4): number of valence electrons= 6

11Na(1s22s22p63s1): number of valence electrons= 1

15P(1s22s22p63s23p3): number of valence electrons= 5

18Ar (1s22s22p63s23p6): number of valence electrons= 0

22Ti ([Ar]3d24s2): number of valence electrons= 2

26Fe ([Ar]3d64s2): number of valence electrons= 2

37Rb ([Ar]5s1): number of valence electrons= 1

According to the theory Li, Na and Rb which has one valence electrons (partially filled band) must be conductors

Ar has completely filled band and hence it is not a conductor.

Ti and Fe is also having partially filled band due to the 2 valence electrons in 4s level. So they are also conductors.

P has half filed p-energy levels which are free to move in conduction band. It behave as asemiconductor.

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