of thermally activated electrons can be thermally excited to move from the valence to the conduction band and that under the influence of an external field they will be impelled to move in the direction of the field, colliding with the lattice of the crystalline dielectric and dissipating their energy by phonon interactions [Bartnikas and Eichhorn, 1983]. Accordingly, breakdown is said to occur when the average rate of energy gain by the electrons, A(E, T, Te 5), exceeds that lost in collisions with the lattice, B(T, Te, E). Hence, the breakdown criterion can be stated as A(E, T, Te, s)=B(T, Te, s) (55.16) where E is the applied field, T the lattice temperature, T the electron temperature, and E an energy distribution constant. Thus in qualitative terms as the temperature is increased gradually, the breakdown voltage rises because the interaction between the electrons and the lattice is enhanced as a result of the increased thermal vibrations of the lattice. Ultimately, a critical temperature is attained where the electron-electron interactions surpass in importance those between the electrons and the lattice, and the breakdown strength commences a monotonic decline with temperature; this behavior is borne out in NaCl crystals, as is apparent from Fig. 55.5 Ivon Hippel and Lee, 1941]. However, with amorphous or partially crystalline polyethylene, the maximum in breakdown strength is seen to be absent and only a decrease is observed [Oakes, 1949; as the crystalline content is increased in amorphous-crystalline solids, the breakdown strength is reduced. The electron avalanche concept has also been applied to explain breakdown in solids, in particular to account for the observed decrease in breakdown trength with insulation thickness. Since breakdown F lue to electron avalanches involves the formation of space charge, space charges will tend to modify the 2 conditions for breakdown. Any destabilization of the s trapping and detrapping process, such as may be i caused by a perturbation of the electrical field, will initiate the breakdown event [LeGressus and blaise 992]. The detrapping of mobile charge carriers will be accompanied by photon emission and formation of the plasma breakdown channel, resulting in the dissi- 200150-10050050100200 pation of polarization energy. If dipole interaction is neglected, the polarization energy due to a trapped figure 55.5 Dielectric breakdown characteristics of charge is of the order of 5x eV, where x is the dielectric sodium chloride (von Hippel and Lee, 1941] and polyethylene usceptibility. The release of the polarization energy [Oakes, 1949 will be accompanied by electrical tree growth in and melting of the polymer. The breakdown process in gases is relatively well understood and is explained in terms of the avalanche theory. A free electron, occurring in a gas due to cosmic radiation, will be accelerated in a field and upon collision with neutral molecules in its trajectory will eject, if its energy is sufficient, other electrons that will in urn undergo additional collisions resulting in a production of more free electrons. If the electric field is sufficiently high, the number of free electrons will increase exponentially along the collision route until ulti mately an electron avalanche will form. As the fast-moving electrons in the gap disappear into the anode, they leave behind the slower-moving ions, which gradually drift to the cathode where they liberate further electrons with a probability y. When the height of the positive ion avalanche becomes sufficiently large to lead to a regeneration of a starting electron, the discharge mechanism becomes self-sustaining and a spark bridges the two electrodes. The condition for the Townsend breakdown in a short gap is given by ylexp(ad-1=1 where d is the distance between the electrodes and a represents the number of ionizing impacts per electron per unit distance. The value of y is also enhanced by photoemission at the cathode and photon radiation in c 2000 by CRC Press LLC© 2000 by CRC Press LLC of thermally activated electrons can be thermally excited to move from the valence to the conduction band and that under the influence of an external field they will be impelled to move in the direction of the field, colliding with the lattice of the crystalline dielectric and dissipating their energy by phonon interactions [Bartnikas and Eichhorn, 1983]. Accordingly, breakdown is said to occur when the average rate of energy gain by the electrons, A(E, T, Te, j), exceeds that lost in collisions with the lattice, B(T, Te, j). Hence, the breakdown criterion can be stated as A(E, T, Te, j) = B(T, Te, j) (55.16) where E is the applied field, T the lattice temperature, Te the electron temperature, and j an energy distribution constant. Thus in qualitative terms as the temperature is increased gradually, the breakdown voltage rises because the interaction between the electrons and the lattice is enhanced as a result of the increased thermal vibrations of the lattice. Ultimately, a critical temperature is attained where the electron–electron interactions surpass in importance those between the electrons and the lattice, and the breakdown strength commences a monotonic decline with temperature; this behavior is borne out in NaCl crystals, as is apparent from Fig. 55.5 [von Hippel and Lee, 1941]. However, with amorphous or partially crystalline polymers, as for example with polyethylene, the maximum in breakdown strength is seen to be absent and only a decrease is observed [Oakes, 1949]; as the crystalline content is increased in amorphous-crystalline solids, the breakdown strength is reduced. The electron avalanche concept has also been applied to explain breakdown in solids, in particular to account for the observed decrease in breakdown strength with insulation thickness. Since breakdown due to electron avalanches involves the formation of space charge, space charges will tend to modify the conditions for breakdown. Any destabilization of the trapping and detrapping process, such as may be caused by a perturbation of the electrical field, will initiate the breakdown event [LeGressus and Blaise, 1992]. The detrapping of mobile charge carriers will be accompanied by photon emission and formation of the plasma breakdown channel, resulting in the dissi￾pation of polarization energy. If dipole interaction is neglected, the polarization energy due to a trapped charge is of the order of 5c eV, where c is the dielectric susceptibility. The release of the polarization energy will be accompanied by electrical tree growth in and melting of the polymer. The breakdown process in gases is relatively well understood and is explained in terms of the avalanche theory. A free electron, occurring in a gas due to cosmic radiation, will be accelerated in a field and upon collision with neutral molecules in its trajectory will eject, if its energy is sufficient, other electrons that will in turn undergo additional collisions resulting in a production of more free electrons. If the electric field is sufficiently high, the number of free electrons will increase exponentially along the collision route until ulti￾mately an electron avalanche will form. As the fast-moving electrons in the gap disappear into the anode, they leave behind the slower-moving ions, which gradually drift to the cathode where they liberate further electrons with a probability g. When the height of the positive ion avalanche becomes sufficiently large to lead to a regeneration of a starting electron, the discharge mechanism becomes self-sustaining and a spark bridges the two electrodes. The condition for the Townsend breakdown in a short gap is given by g[exp(ad) 21] = 1 (55.17) where d is the distance between the electrodes and a represents the number of ionizing impacts per electron per unit distance. The value of g is also enhanced by photoemission at the cathode and photon radiation in FIGURE 55.5 Dielectric breakdown characteristics of sodium chloride [von Hippel and Lee, 1941] and polyethylene [Oakes, 1949]