11.1.Conductivity and Resistivity of Metals 189 short,the wave picture provides a deeper understanding of the electrical resistance in metals and alloys. According to a rule proposed by Matthiessen,the total resis- tivity arises from independent mechanisms,as just described, which are additive,i.e.: p=Pth+Pimp Pdef=Pth+Pres. (11.8) The thermally induced part of the resistivity Ph is called the ideal resistivity,whereas the resistivity that has its origin in impuri- ties (pimp)and defects(pder)is summed up in the residual resis- tivity (pres).The number of impurity atoms is generally constant in a given metal or alloy.The number of vacancies,or grain boundaries,however,can be changed by various heat treatments. For example,if a metal is annealed at temperatures close to its melting point and then rapidly quenched into water of room tem- perature,its room temperature resistivity increases noticeably due to quenched-in vacancies,as already explained in Chapter 6. Frequently,this resistance increase diminishes during room tem- perature aging or annealing at slightly elevated temperatures due to the annihilation of these vacancies.Likewise,work hardening, recrystallization,grain growth,and many other metallurgical processes change the resistivity of metals.As a consequence of this,and due to its simple measurement,the resistivity has been one of the most widely studied properties in materials research. Free Electrons The conductivity of metals can be calculated(as P.Drude did at the turn to the 20th century)by simply postulating that the elec- tric force,e.provided by an electric field (Figure 11.2),ac- celerates the electrons (having a charge -e)from the cathode to the anode.The drift of the electrons was thought by Drude to be counteracted by collisions with certain atoms as described above. The Newtonian-type equation(force equals mass times acceler- ation)of this free electron model m变+w=e:g (11.9) leads,after a string of mathematical manipulations,to the con- ductivity: 0、 r.e2.T (11.10) m where v is the drift velocity of the electrons,m is the electron mass,y is a constant which takes the electron/atom collisions into consideration (called damping strength),T=m/y is the average time between two consecutive collisions (called the relaxation time),and Nr is the number of free electrons per cubic meter inshort, the wave picture provides a deeper understanding of the electrical resistance in metals and alloys. According to a rule proposed by Matthiessen, the total resis￾tivity arises from independent mechanisms, as just described, which are additive, i.e.: 
th imp def
th res. (11.8) The thermally induced part of the resistivity th is called the ideal resistivity, whereas the resistivity that has its origin in impuri￾ties (imp) and defects (def) is summed up in the residual resis￾tivity (res). The number of impurity atoms is generally constant in a given metal or alloy. The number of vacancies, or grain boundaries, however, can be changed by various heat treatments. For example, if a metal is annealed at temperatures close to its melting point and then rapidly quenched into water of room tem￾perature, its room temperature resistivity increases noticeably due to quenched-in vacancies, as already explained in Chapter 6. Frequently, this resistance increase diminishes during room tem￾perature aging or annealing at slightly elevated temperatures due to the annihilation of these vacancies. Likewise, work hardening, recrystallization, grain growth, and many other metallurgical processes change the resistivity of metals. As a consequence of this, and due to its simple measurement, the resistivity has been one of the most widely studied properties in materials research. The conductivity of metals can be calculated (as P. Drude did at the turn to the 20th century) by simply postulating that the elec￾tric force, e 
, provided by an electric field (Figure 11.2), ac￾celerates the electrons (having a charge e) from the cathode to the anode. The drift of the electrons was thought by Drude to be counteracted by collisions with certain atoms as described above. The Newtonian-type equation (force equals mass times acceler￾ation) of this free electron model m d d v t v
e 
(11.9) leads, after a string of mathematical manipulations, to the con￾ductivity:
Nf  m e2   , (11.10) where v is the drift velocity of the electrons, m is the electron mass,  is a constant which takes the electron/atom collisions into consideration (called damping strength), 
m/ is the average time between two consecutive collisions (called the relaxation time), and Nf is the number of free electrons per cubic meter in 11.1 • Conductivity and Resistivity of Metals 189 Free Electrons