I was solving the following problem the following way:(adsbygoogle = window.adsbygoogle || []).push({});

In a solid the atoms are regularly arranged in space. The potential seen by an electron is thus periodic and the energy levels are arranged in bands in which the energy of a state k is given by:

e_{k}= E_{at}+ 2t cos(ka)

The relevant band structure of the metal Silver (Ag) can be roughly modelled with a band of that form with t= 4 eV and a= 0.409 nm. Assume Ag has one conduction electron per atom.

a) Calculate the maximum velocity of an electron.

e_{k}= E_{at}+ 2t cos(ka)

with:

a= 0.409* 10^{-9}m

t= 4 eV= 4* 1.6*10^{-19}= 6.4*10^{-19}J

Therefore:

e_{k}= E_{at}+ 1.28*10^{-18}cos(0.409* 10^{-9}k)

v_{g}= 1/h_{with line}* (d e_{k}/dk)

d e_{k}/dk= -1.28*10^{-18}*0.409* 10^{-9}sin(0.409* 10^{-9}k)

v_{g}=-4.96*10^{6}sin(0.409* 10^{-9}k)

The velocity is maximal when the sine is maximal, therefore when the sine=1. In that case the velocity is -4.96*10^{6}m/s.

b) Calculate the drift velocity (average velocity) of electrons in a silver wire of 1mm^{2}cross-sectional area through which a current of 1A is flowing, knowing v_{drift}= I/nAq and n= 5.85*10^{28}electrons per m^{3}.

v_{drift}= I/nAq

v_{drift}= 1/5.85*10^{28}* 1*10^{-6}* 1.6*10^{-19}= -1.069*10^{-4}m/s

Now my question is: why are the velocities calculated in a and b so different from each other?!

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# QM- Drift & Maximum velocity of an electron

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