6.1.Lattice Defects and Diffusion 103 where ns is the number of regular lattice sites per unit volume,kB is the Boltzmann constant(see Appendix II),and Ef is the energy that is needed to form a vacant lattice site in a perfect crystal. As an example,at room temperature,n for copper is about 108 vacancies per cm3,which is equivalent to one vacancy for every 1015 lattice atoms.If copper is held instead near its melt- ing point,the vacancy concentration is about 1019 cm-3,or one vacancy for every 10,000 lattice atoms.It is possible to increase the number of vacancies at room temperature by quenching a material from high temperatures to the ambient,that is,by freez- ing-in the high temperature disorder,or to some degree also by plastic deformation. Other treatments by which a large number of vacancies can be introduced into a solid involve its bombardment with neutrons or other high energetic particles as they exist,for example,in nuclear reactors(radiation damage)or by ion implantation.These high en- ergetic particles knock out a cascade of lattice atoms from their po- sitions and deposit them between regular lattice sites (see below). It has been estimated that each fast neutron may create between 100 and 200 vacancies.At the endpoint of a primary particle,a de- pleted zone about 1 nm in diameter (several atomic distances)may be formed which is characterized by a large number of vacancies. Among other point defects are the interstitials.They involve for- eign,often smaller,atoms(such as carbon,nitrogen,hydrogen,oxy- gen)which are squeezed in between regular lattice sites.The less common self-interstitials(sometimes,and probably not correctly, called interstitialcies)are atoms of the same species as the matrix that occupy interlattice positions.Self-interstitials cause a sub- stantial distortion of the lattice.In a dumbbell,two equivalent atoms share one regular lattice site.Frenkel defects are vacancy/inter- stitial pairs.Schottky defects are formed in ionic crystals when, for example,an anion as well as a cation of the same absolute va- lency are missing (to preserve charge neutrality).Dislocations are one-dimensional defects(Figure 3.20).Two-dimensional defects are formed by grain boundaries(Figure 3.15)and free surfaces at which the continuity of the lattice and therefore the atomic bonding are disturbed.We shall elaborate on these defects when the need arises. 6.1.2 Vacancies provide,to a large extent,the basis for diffusion,that Diffusion is,the movement of atoms in materials.Specifically,an atom may move into an empty lattice site.Concomitantly,a vacancy migrates Mechanisms in the opposite direction,as depicted in Figure 6.1.The prerequi- site for the jump of an atom into a vacancy is,however,that the Diffusion by atom possesses enough energy (for example,thermal energy)to squeeze by its neighbors and thus causes the lattice to expand Vacancies momentarily and locally,involving what is called elastic strainwhere ns is the number of regular lattice sites per unit volume, kB is the Boltzmann constant (see Appendix II), and Ef is the energy that is needed to form a vacant lattice site in a perfect crystal. As an example, at room temperature, nv for copper is about 108 vacancies per cm3, which is equivalent to one vacancy for every 1015 lattice atoms. If copper is held instead near its melt￾ing point, the vacancy concentration is about 1019 cm3, or one vacancy for every 10,000 lattice atoms. It is possible to increase the number of vacancies at room temperature by quenching a material from high temperatures to the ambient, that is, by freez￾ing-in the high temperature disorder, or to some degree also by plastic deformation. Other treatments by which a large number of vacancies can be introduced into a solid involve its bombardment with neutrons or other high energetic particles as they exist, for example, in nuclear reactors (radiation damage) or by ion implantation. These high en￾ergetic particles knock out a cascade of lattice atoms from their po￾sitions and deposit them between regular lattice sites (see below). It has been estimated that each fast neutron may create between 100 and 200 vacancies. At the endpoint of a primary particle, a de￾pleted zone about 1 nm in diameter (several atomic distances) may be formed which is characterized by a large number of vacancies. Among other point defects are the interstitials. They involve for￾eign, often smaller, atoms (such as carbon, nitrogen, hydrogen, oxy￾gen) which are squeezed in between regular lattice sites. The less common self-interstitials (sometimes, and probably not correctly, called interstitialcies) are atoms of the same species as the matrix that occupy interlattice positions. Self-interstitials cause a sub￾stantial distortion of the lattice. In a dumbbell, two equivalent atoms share one regular lattice site. Frenkel defects are vacancy/inter￾stitial pairs. Schottky defects are formed in ionic crystals when, for example, an anion as well as a cation of the same absolute va￾lency are missing (to preserve charge neutrality). Dislocations are one-dimensional defects (Figure 3.20). Two-dimensional defects are formed by grain boundaries (Figure 3.15) and free surfaces at which the continuity of the lattice and therefore the atomic bonding are disturbed. We shall elaborate on these defects when the need arises. Vacancies provide, to a large extent, the basis for diffusion, that is, the movement of atoms in materials. Specifically, an atom may move into an empty lattice site. Concomitantly, a vacancy migrates in the opposite direction, as depicted in Figure 6.1. The prerequi￾site for the jump of an atom into a vacancy is, however, that the atom possesses enough energy (for example, thermal energy) to squeeze by its neighbors and thus causes the lattice to expand momentarily and locally, involving what is called elastic strain 6.1.2 Diffusion Mechanisms Diffusion by Vacancies 6.1 • Lattice Defects and Diffusion 103