Silver is a monovalent metal. Calculate (a) the number density of conduction electrons, (b) the Fermi energy, (c) the Fermi speed and (d) the de Broglie wavelength corresponding to this electron speed. See Appendix F for the needed data on silver.
a) The number density of conduction electrons in silver is .
b) The Fermi energy of silver metal is .
c) The Fermi speed of silver metal is .
d) The de-Broglie wavelength corresponding to this electron speed is .
a) The monovalent element silver is given.
b) Molar mass of silver (Appendix F), A = 107.870 g/mol
c) Density of silver (Appendix F), d = 10.49 g/
Using the given formula for the number of atoms per unit volume, we can get the required value of the conduction electrons per unit volume considering the metal is monovalent. Now using this value in the equation of Fermi energy, we calculated the energy of the metal. Now, using this value of Fermi energy, we can get the speed of the electron in conduction. Further, we can use this value of speed in the de-Broglie wavelength relation; we can get the value of the wavelength of the electron.
d= density of the atom, M = mass of a single atom
where n is the number of conduction electrons per unit volume, m is the mass of an electron and h is Planck’s constant.
Here, is the fermi speed.
Since each atom contributes one conduction electron, using equation (i) and equation (ii), we can get the number density of the conduction electrons in silver as follows:
Hence, the value of the number density is .
Using the given data and the above number density value in equation (iii), the Fermi energy of silver metal can be calculated as follows:
Hence, the value of Fermi energy is 5.49 eV.
Using the above Fermi energy value and equation (iv), we can get the speed of the conducting electron in Fermi level of silver as follows:
Hence, the value of Fermi speed is .
Using the above speed value, the de-Broglie wavelength corresponding to the speed of the conducting electron can be given using equation (v) as follows:
Hence, the value of de-Broglie wavelength is .
In a simplified model of an undoped semiconductor, the actual distribution of energy states may be replaced by one in which there are states in the valence band, all having the same energy , and states in the conduction band all these states having the same energy . The number of electrons in the conduction band equals the number of holes in the valence band.
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