188 11.Electrical Properties of Materials FIGURE 11.4.Schematic representation of an electron path through a conductor (contain- ing vacancies,impu- rity atoms,and a grain boundary)un- der the influence of an electric field.This classical description does not completely describe the resistance in materials. temperatures.At near-zero temperatures,the electrical resistance does not completely vanish,however (except in superconduc- tors).There always remains a residual resistivity,pres (Figure 11.3),which is thought to be caused by "collisions"of electrons (i.e.,by electrostatic interactions)with imperfections in the crys- tal (such as impurities,vacancies,grain boundaries,or disloca- tions),as explained in Chapters 3 and 6.The residual resistivity is essentially temperature-independent. On the other hand,one may describe the electrons to have a wave nature.The matter waves may be thought to be scattered by lattice atoms.Scattering is the dissipation of radiation on small particles in all directions.The atoms absorb the energy of an incoming wave and thus become oscillators.These oscillators in turn re-emit the energy in the form of spherical waves.If two or more atoms are involved,the phase relationship between the individual re-emitted waves has to be taken into consideration. A calculation!shows that for a periodic crystal structure the in- dividual waves in the forward direction are in-phase,and thus interfere constructively.As a result,a wave which propagates through an ideal crystal (having periodically arranged atoms) does not suffer any change in intensity or direction (only its ve- locity is modified).This mechanism is called coherent scattering. If,however,the scattering centers are not periodically arranged (impurity atoms,vacancies,grain boundaries,thermal vibration of atoms,etc.),the scattered waves have no set phase relation- ship and the wave is said to be incoherently scattered.The energy of incoherently scattered waves is smaller in the forward direc- tion.This energy loss qualitatively explains the resistance.In L.Brillouin,Wave Propagation in Periodic Structures,Dover,New York (1953).188 11 • Electrical Properties of Materials temperatures. At near-zero temperatures, the electrical resistance does not completely vanish, however (except in superconduc￾tors). There always remains a residual resistivity, res (Figure 11.3), which is thought to be caused by “collisions” of electrons (i.e., by electrostatic interactions) with imperfections in the crys￾tal (such as impurities, vacancies, grain boundaries, or disloca￾tions), as explained in Chapters 3 and 6. The residual resistivity is essentially temperature-independent. On the other hand, one may describe the electrons to have a wave nature. The matter waves may be thought to be scattered by lattice atoms. Scattering is the dissipation of radiation on small particles in all directions. The atoms absorb the energy of an incoming wave and thus become oscillators. These oscillators in turn re-emit the energy in the form of spherical waves. If two or more atoms are involved, the phase relationship between the individual re-emitted waves has to be taken into consideration. A calculation1 shows that for a periodic crystal structure the in￾dividual waves in the forward direction are in-phase, and thus interfere constructively. As a result, a wave which propagates through an ideal crystal (having periodically arranged atoms) does not suffer any change in intensity or direction (only its ve￾locity is modified). This mechanism is called coherent scattering. If, however, the scattering centers are not periodically arranged (impurity atoms, vacancies, grain boundaries, thermal vibration of atoms, etc.), the scattered waves have no set phase relation￾ship and the wave is said to be incoherently scattered. The energy of incoherently scattered waves is smaller in the forward direc￾tion. This energy loss qualitatively explains the resistance. In FIGURE 11.4. Schematic representation of an electron path through a conductor (contain￾ing vacancies, impu￾rity atoms, and a grain boundary) un￾der the influence of an electric field. This classical description does not completely describe the resistance in materials. 1L. Brillouin, Wave Propagation in Periodic Structures, Dover, New York (1953)