where the Wagner absorption factor K is given by (T1+t2-τ)t (55.13) t1t2 where T, and T2 are the relaxation times of the two contiguous layers or strata of respective thicknesses d, and d2; T is the overall relaxation time of the two-layer combination and is defined by t=(ef d2+e2 di)(o,d2 +odi), where E, E2, On and o2 are the respective real permittivity and conductivity parameters of the two discrete layers. Note that since ef and e? are temperature- and frequency-dependent and o, and o, are, in addition, lso voltage-dependent, the values of T and e will in turn also be influenced by these three variables. Space charge processes involving electrons are more effectively analyzed, using dc measurement techniques. If retrap- ping of electrons in polymers is neglected, then the decay current as a function of time t, arising from detrapped electrons, assumes the form [ Watson, 1995] dr)=mE (55.14) where n(E)is the trap density and v is the attempt jump frequency of the electrons. The electron current displays the usual tI dependence and the plot of i(t)t versus kTIn(vt) yields the distribution of trap depths. g.(55.14)represents an approximation, which underestimates the current associated with the shallow traps and overcompensates for the current due to the deep traps. The mobility of the free charge carriers is determined by the depth of the traps, the field resulting from the trapped charges, and the temperature. As elevated mperatures and low space charge fields, the mobility is proportional to exp(-AH/kT] and at low temperatures to(T)4 [LeGressus and Blaise, 1992]. A high trapped charge density will create fields. which will in urn exert a controlling influence on the mobility and the charge distribution profile In polymers, shallow traps are of the order of 0.5 to 0.9 eV and deep traps are ca 1.0 to 1.5 ev, while the activation energies of dipole orientation and ionic conduction in solid and liquid dielectrics fall within the same range. It has been known that most charge trapping in the volume occurs in the vicinity of the electrodes; this can now be confirmed by measurement, using thermal and electrically stimulated acoustical pulse methods [Bernstein, 1992]. In the latter method this involves the application of a rapid voltage pulse across a dielectric specimen. The resulting stress ve propagates at the velocity of sound and is detected by a piezoelectric transducer. This wave is assumed ot to disturb the trapped charge; the received electrical signal is then correlated with the acoustical wave to determine the profile of the trapped charge. Errors in the measurement would appear to be principally caused by the electrode surface charge effects and the inability to distinguish between the polarization of polar dipoles and that of the trapped charges [wintle, Temperature influences the real value of the permittivity or dielectric constant e insofar as it affects the density of the dielectric material. As the density diminishes with temperature, e falls with temperature in accordance with the Clausius-Mossotti equation [P]= (e′-1)M (55.15) (E′+2)d where [P] represents the polarization per mole, M the molar mass, d, the density at a given temperature, and E'=E Equation(55. 14)is equally valid, if the substitution E=(n) is made; here nis the real value of the index of refraction. In fact, the latter provides a direct connection with the dielectric behavior at optical ith the complex permittivity, the index of refraction is also a complex quantity, and its imaginary value n"exhibits a loss peak at the absorption frequencies; in contrast with the evalue which an only fall with frequency, the real index of refraction n'exhibits an inflection-like behavior at the absorption frequency. This is illustrated schematically in Fig. 55.2, which depicts the knor n"and n-l values as a function of frequency over the optical frequency regime. The absorption in the infrared results from atomic c 2000 by CRC Press LLC© 2000 by CRC Press LLC where the Wagner absorption factor K is given by (55.13) where t1 and t2 are the relaxation times of the two contiguous layers or strata of respective thicknesses d1 and d2; t is the overall relaxation time of the two-layer combination and is defined by t = (e¢ 1 d21e¢ 2 d1)/(s1d21s2d1), where e¢ 1, e¢ 2 , s1, and s2 are the respective real permittivity and conductivity parameters of the two discrete layers. Note that since e¢ 1 and e¢ 2 are temperature- and frequency-dependent and s1 and s2 are, in addition, also voltage-dependent, the values of t and e² will in turn also be influenced by these three variables. Space charge processes involving electrons are more effectively analyzed, using dc measurement techniques. If retrap￾ping of electrons in polymers is neglected, then the decay current as a function of time t, arising from detrapped electrons, assumes the form [Watson, 1995] (55.14) where n(E) is the trap density and n is the attempt jump frequency of the electrons. The electron current displays the usual t–1 dependence and the plot of i(t)t versus kTln(nt) yields the distribution of trap depths. Eq. (55.14) represents an approximation, which underestimates the current associated with the shallow traps and overcompensates for the current due to the deep traps. The mobility of the free charge carriers is determined by the depth of the traps, the field resulting from the trapped charges, and the temperature. As elevated temperatures and low space charge fields, the mobility is proportional to exp[–DH/kT] and at low temperatures to (T)1/4 [LeGressus and Blaise, 1992]. A high trapped charge density will create intense fields, which will in turn exert a controlling influence on the mobility and the charge distribution profile. In polymers, shallow traps are of the order of 0.5 to 0.9 eV and deep traps are ca. 1.0 to 1.5 eV, while the activation energies of dipole orientation and ionic conduction in solid and liquid dielectrics fall within the same range. It has been known that most charge trapping in the volume occurs in the vicinity of the electrodes; this can now be confirmed by measurement, using thermal and electrically stimulated acoustical pulse methods [Bernstein, 1992]. In the latter method this involves the application of a rapid voltage pulse across a dielectric specimen. The resulting stress wave propagates at the velocity of sound and is detected by a piezoelectric transducer. This wave is assumed not to disturb the trapped charge; the received electrical signal is then correlated with the acoustical wave to determine the profile of the trapped charge. Errors in the measurement would appear to be principally caused by the electrode surface charge effects and the inability to distinguish between the polarization of polar dipoles and that of the trapped charges [Wintle, 1990]. Temperature influences the real value of the permittivity or dielectric constant e¢ insofar as it affects the density of the dielectric material. As the density diminishes with temperature, e¢ falls with temperature in accordance with the Clausius-Mossotti equation (55.15) where [P] represents the polarization per mole, M the molar mass, do the density at a given temperature, and e¢ = es. Equation (55.14) is equally valid, if the substitution e¢ = (n¢)2 is made; here n¢ is the real value of the index of refraction. In fact, the latter provides a direct connection with the dielectric behavior at optical frequencies. In analogy with the complex permittivity, the index of refraction is also a complex quantity, and its imaginary value n² exhibits a loss peak at the absorption frequencies; in contrast with the e¢ value which can only fall with frequency, the real index of refraction n¢ exhibits an inflection-like behavior at the absorption frequency. This is illustrated schematically in Fig. 55.2, which depicts the kn¢ or n² and n¢ 21 values as a function of frequency over the optical frequency regime. The absorption in the infrared results from atomic K = ( ) t t tt tt +- - t t 1 2 12 1 2 i t kT vt ( ) = n E( ) [ ] ( ) ( ) P M do = ¢ - ¢ + e e 1 2