Home work

How would such de Broglie waves interact with the ions of the crystal lattice? An ideal crystal can be thought of as consisting of a sequence of parallel planes, each with a two-dimensional periodic arrangement of ions. It can be shown that the de Broglie waves can travel between such crystal planes without a loss in intensity. Consequently one would not expect a perfect crystal to have any electrical resistance.

A real metal crystal, on the other hand, always exhibits an electrical resistance in the normal conducting state. This indicates that the de Broglie waves, of the conduction electrons are scattered. In general a wave is scattered at obstacles whose dimensions are greater than or comparable to the wavelength. (This effect is easy to perceive in the case of light waves. The headlight beams of an automobile, for instance, are not scattered, on a clear night because the air molecules are small compared with the wavelength of light. On a foggy night, however, the situation changes: since the fog droplets can have a diameter greater than or comparable to the wavelength of the light, the headlight beams are scattered and may interfere with the driver's vision.)

In a normal metal dynamic and static imperfections in the crystal lattice are responsible for the scattering of de Broglie waves and hence for the electrical resistance. At room temperature by far the largest component of the resistance is the result of thermal effects, which cause the ions to oscillate around their ideal lattice positions. Only at very low temperatures is electrical resistance mainly the result of static defects such as impurity atoms and vacant lattice sites.