On the Theory of the Breakdown of the Electrical Strength of Ionic Crystals

V. A. Chuenkov

The question of electrical strength is one of the urgent problems of the physics of dielectrics. The study of the processes occurring in solid dielectrics placed in a strong electric field is of great interest from both the practical and the theoretical points of view.

The existing experimental data [1] show that electrical breakdown of solid dielectrics occurs as a result of ionization of the atoms of the dielectric by electron impact and the formation of an electron avalanche.

B. M. Vul showed [2] that in the phenomenon of electrical breakdown of solid dielectrics it is necessary to distinguish two stages, namely, the stage of breakdown of electrical strength (the breakdown of the stationary regime of electrical conductivity) and the stage of destruction of the material. The loss of electrical strength is the beginning of breakdown and at the same time the cause determining all subsequent processes. Therefore the first and main task of the theory of electrical breakdown is to determine the conditions for the breakdown of electrical strength. Analysis of modern theories shows [1] that the most rigorous criterion for breakdown of electrical strength can be obtained on the basis of the corresponding Boltzmann integro-differential equation.

Let us consider a homogeneous dielectric with an infinitely wide conduction band. Let the electric field be directed along the \(z\)-axis. The integro-differential equation for determining the distribution function of conduction electrons will in this case be written in the form [3]:

\[ \frac{\partial f}{\partial t}+eE\frac{\partial f}{\partial p_z}=b-a, \tag{1} \]

where \(f\) is the distribution function, \(e\) is the electron charge, \(p_z\) is the component of the electron momentum along the \(z\)-axis, \(E\) is the intensity of the external electric field, \(t\) is time, and \(b-a\) is the change in the distribution function per unit time due to collisions. In solids, electrons may undergo the following types of collisions: scattering by lattice vibrations, ionization collisions, recombination collisions, and collisions associated with the occurrence of excited states.

In ionic crystals there is usually only one excitation level, differing only slightly from the ionization potential (the width of the forbidden band). For example, in NaCl the ionization potential is \(I=9.6\ \text{eV}\), and the excitation potential is \(I_{в}=7.8\ \text{eV}\); in KBr we have \(I=8.3\ \text{eV}\), \(I_{в}=6.55\ \text{eV}\) [4]. If, moreover, it is assumed, by analogy with gases, that the probability of excitation amounts to only a few percent of the probability of ionization [5], then collisions leading to the occurrence of excit—

... states in a solid may be neglected in comparison with ionizational ones.

Electron recombination plays a substantial role only in a small neighborhood of zero energy in phase space. Indeed [6, 7], localization of an electron in the lattice can occur only in the case when the time of flight of the electron past a defect is greater than the period of lattice vibrations, i.e., for \(\dfrac{l}{v} > \dfrac{2\pi}{\omega}\), where \(l\) is the size of the defect, \(v\) is the electron velocity, and \(\omega\) is the frequency of lattice vibrations. For \(\omega \sim 10^{13}\ \mathrm{sec}^{-1}\), \(l \sim 10a\) (\(a\) is the lattice constant), we find that electron recombination is possible at energies

\[ \varepsilon \lesssim 10^{-4}\ \mathrm{eV} \ll \frac{h}{2\pi}\omega . \]

Consequently, everywhere, except for a small neighborhood of zero energy, the recombination term in the kinetic equation may be neglected. The principal process that prevents electrons in a strong electric field from accumulating the energy necessary for ionization is scattering by lattice vibrations; we now turn to its consideration. For crystals of the alkali-halide type, the displacement of ions from the equilibrium position in an oscillation with wave number \(\overline{L}\)¹ may be written in the form [3] (pp. 435 and 438)

\[ \overline{u}_{+,\overline{L}} = \frac{1}{\sqrt{M_{+}N}} \left[ a_{\overline{L}}(t)\,\overline{\zeta}_{+,\overline{L}}\,e^{2\pi i(\overline{L}\overline{r})} + a_{\overline{L}}^{*}(t)\,\overline{\zeta}_{+,\overline{L}}^{*}\,e^{-2\pi i(\overline{L}\overline{r})} \right], \tag{2} \]

\[ \overline{u}_{-,\overline{L}} = \frac{1}{\sqrt{M_{-}N}} \left[ a_{\overline{L}}(t)\,\overline{\zeta}_{-,\overline{L}}\,e^{2\pi i(\overline{L}\overline{r})} + a_{\overline{L}}^{*}(t)\,\overline{\zeta}_{-,\overline{L}}^{*}\,e^{-2\pi i(\overline{L}\overline{r})} \right], \tag{3} \]

where \(M_{+}\) and \(M_{-}\) are the masses, respectively, of the positive and negative ions, \(N\) is the number of elementary cells in the crystal, and \(\overline{\zeta}_{+,\overline{L}}\) and \(\overline{\zeta}_{-,\overline{L}}\) are amplitudes; the asterisk denotes conjugate quantities. The polarization arising in this case is equal to

\[ \overline{P}_{\overline{L}} = \frac{e_{\overline{L}}^{*}}{2a^{3}} \left( \overline{u}_{+,\overline{L}} - \overline{u}_{-,\overline{L}} \right) = \frac{e_{\overline{L}}^{*}}{2a^{3}\sqrt{N}} \left[ \frac{a_{\overline{L}}(t)}{\sqrt{M_{\overline{L}}}}\,e^{-2\pi i(\overline{L}\overline{r})} + \frac{a_{\overline{L}}^{-}(t)}{\sqrt{M_{\overline{L}}^{*}}}\,e^{-2\pi i(\overline{L}\overline{r})} \right], \tag{4} \]

where \(a\) is the lattice constant, and \(e_{\overline{L}}^{*}\) is the effective charge of the ion [1]

\[ \frac{1}{\sqrt{M_{\overline{L}}}} = \frac{\overline{\zeta}_{+,\overline{L}}}{\sqrt{M_{+}}} - \frac{\overline{\zeta}_{-,\overline{L}}}{\sqrt{M_{-}}}. \tag{5} \]

The energy of interaction of the electron with the polarization waves is equal to

\[ V=e\sum_{\overline{L}} V_{\overline{L}}, \tag{6} \]

where the quantity \(V_{\overline{L}}\), determined from Poisson’s equation,

\[ \nabla^{2}V_{\overline{L}}=4\pi\,\operatorname{div}\overline{P}_{\overline{L}}, \tag{7} \]

¹ Letters with a bold bar above them denote vector quantities.

is equal to

\[ V_{\overline L}= \frac{e_{\overline L}^{*}}{a_3 i\overline L\sqrt N} \left[ \frac{a_{\overline L}(t)}{\sqrt{M_{\overline L}}} e^{2\pi i(\overline L\,\overline r)} - \frac{a_{\overline L}^{*}(t)}{\sqrt{M_{\overline L}^{*}}} e^{-2\pi i(\overline L\,\overline r)} \right]. \tag{8} \]

It is seen from (8) that scattering by transverse waves is impossible \((\overline\zeta_{+,\overline L}\) and \(\overline\zeta_{-,\overline L}\perp \overline L)\).

Writing the Hamiltonian of the entire system and the wave function of the unperturbed system, respectively, in the form

\[ H=H_{\mathrm{osc}}+\varepsilon+V, \tag{9} \]

\[ \Psi=\psi(\overline r,\overline k)\cdot \prod_{\overline L} y(b_{\overline L},n_{\overline L})\cdot e^{-\frac{U}{h/2\pi}t}, \tag{10} \]

where

\[ H_{\mathrm{osc}}= \frac{1}{2}\sum_{\overline L} \left(b_{\overline L}^{*2}+\omega_{\overline L}^{2}b_{\overline L}^{2}\right); \quad b_{\overline L}(t)= \frac{a_{\overline L}(t)+a_{\overline L}^{*}(t)}{\sqrt 2}; \]

\[ b_{\overline L}^{*}(t)= \frac{a_{\overline L}(t)-a_{\overline L}^{*}(t)}{i\sqrt 2}; \quad \varepsilon=\frac{P^2}{2m^{*}}; \]

\[ U=\sum_{\overline L} \left(n_{\overline L}+\frac{1}{2}\right) \frac{h}{2\pi}\omega_{\overline L} +\varepsilon_{\overline K}; \quad \psi(r,\overline K)= \frac{1}{2\pi^{3/2}}e^{2\pi i(\overline K\,\overline r)}, \]

\(y(b_{\overline L},h_{\overline L})\) is the wave function of an oscillator with quantum number \(n_{\overline L}\), \(m^{*}\) is the effective mass of the electron, and, regarding \(V\) as a small perturbation, for the probability of absorption \(P_{\overline L}^{a}\) and the probability of emission \(P_{\overline L}^{e}\) per unit time by an electron of a phonon with energy \(\dfrac{h}{2\pi}\omega_{\overline L}\) and momentum \(h\overline L\), we obtain the following expressions:

\[ P_{\overline L}^{a}= \frac{e^2 e_{\overline L}^{*2} n_{\overline L}} {N a^6 \dfrac{h}{2\pi}\omega_{\overline L} L^2 M_{\overline L}} \frac{\sin \zeta_a t}{\zeta_a} \delta(\overline K-\overline K'\pm \overline L), \]

\[ P_{\overline L}^{e}= \frac{e^2}{N a^6} \frac{e_{\overline L}^{*2}(n_{\overline L}+1)} {\dfrac{h}{2\pi}\omega_{\overline L} L^2 M_{\overline L}} \frac{\sin \zeta_e t}{\zeta_e} \delta(\overline K-\overline K'\pm \overline L). \tag{11} \]

Here

\[ n_L= \frac{1} {e^{\frac{h}{2\pi}\frac{\omega_{\overline L}}{\kappa T}}-1}, \quad \frac{h}{2\pi}\zeta_e= \varepsilon'-\varepsilon+\frac{h}{2\pi}\omega_{\overline L}, \quad \frac{h}{2\pi}\zeta_a= \varepsilon'-\varepsilon-\frac{h}{2\pi}\omega_{\overline L}, \tag{12} \]

where \(\varepsilon\) and \(\varepsilon'\) are, respectively, the energy of the electron before and after scattering.

The quantity \((b-a)\) in the case of scattering by lattice vibrations of the alkali-halide type (in what follows only these crystals will be discussed) is equal to (we take into account acoustic and optical longitudinal vibrations):

\[ (b-a)^P = 2a^3 N \int (b-a)^P_{\bar L}\, L^2\, dL \,\sin\alpha\, d\alpha\, d\varphi , \tag{13} \]

where

\[ \begin{aligned} (b-a)^P_{\bar L} ={}& f(\varepsilon',\theta')\cdot P^e_{\bar L\,op} + f(\varepsilon'_1,\theta'_1)\cdot P^e_{\bar L\,ak} + f(\varepsilon'',\theta''_1)\cdot P^a_{\bar L\,op} \\ &+ f(\varepsilon''_1,\theta''_1)\, P^a_{\bar L\,ak} - f(\varepsilon,\theta)\left(P^e_{\bar L\,op}+P^e_{\bar L\,ak}+P^a_{\bar L\,op}+P^a_{\bar L\,ak}\right), \end{aligned} \tag{14} \]

\[ \begin{gathered} \varepsilon'=\varepsilon+\frac{h}{2\pi}\omega^{op}_{\bar L} +\frac{h}{2\pi}\zeta^{op}_{e}, \qquad \varepsilon'_1=\varepsilon+\frac{h}{2\pi}\omega^{ak}_{\bar L} +\frac{h}{2\pi}\zeta^{ak}_{e}, \\[4pt] \varepsilon''=\varepsilon-\frac{h}{2\pi}\omega^{op}_{\bar L} +\frac{h}{2\pi}\zeta^{op}_{a}, \qquad \varepsilon''_1=\varepsilon-\frac{h}{2\pi}\omega^{ak}_{\bar L} +\frac{h}{2\pi}\zeta^{ak}_{a}. \end{gathered} \tag{15} \]

Here \(2a^3N\) is the density of states in the space of wave numbers; \(\theta,\theta',\theta'_1,\theta'',\theta''_1\) are the angles between the direction of the electric field and the electron velocities \(\bar v,\bar v',\bar v'_1,\bar v'',\bar v''_1\); \(\alpha\) is the angle between \(\bar L\) and \(\bar v\) (the direction of \(\bar v\) is taken as the polar axis). The difference between \(P_{\bar L\,op}\) and \(P_{\bar L\,ak}\) consists in the fact that the quantities \(e^*_{\bar L}\), \(\omega_{\bar L}\), \(n_{\bar L}\), \(M_{\bar L}\), \(\zeta_a\), and \(\zeta_e\) depend differently on \(\bar L\) in the case of optical and acoustic vibrations. Integration over \(\alpha\) in (13) is easily carried out if one takes into account that, for large \(t\),

\[ \frac{\sin \zeta t}{\zeta} \simeq \pi\delta(\zeta), \tag{16} \]

where \(\delta(\zeta)\) is the delta function, and that

\[ \sin\alpha\, d\alpha = \pm \frac{d\zeta}{2\pi v L}, \tag{17} \]

where the plus sign corresponds to the electron emitting energy, and the minus sign to absorption. Relation (17) is easily obtained from the laws of conservation of energy and momentum,

\[ \varepsilon'=\varepsilon \pm \frac{h}{2\pi}\omega+\frac{h}{2\pi}\zeta, \tag{18} \]

\[ \bar K'=\bar K \pm \bar L. \tag{19} \]

After integration of expression (13) over \(\alpha\), the quantities \(\varepsilon'\), \(\theta'\), \(\varepsilon'_1\), \(\theta'_1\), \(\varepsilon''\), \(\theta''\), \(\varepsilon''_1\), \(\theta''_1\) will be equal, respectively, to \(\varepsilon'_0\), \(\theta'_0\), \(\varepsilon'_{10}\), \(\theta'_{10}\), \(\varepsilon''_0\), \(\theta''_0\), \(\varepsilon''_{10}\), \(\theta''_{10}\), which corresponds to \(\zeta^{op}_a=\zeta^{ak}_a=\zeta^{op}_e=\zeta^{ak}_e=0\).

According to one of the formulas of spherical trigonometry,

\[ \cos\theta'_0=\cos\theta\cos\beta'_0+\sin\theta\sin\beta'_0\cos\varphi, \tag{20} \]

where \(\beta_0\) is the angle between \(\bar v\) and \(\bar v'\), whose cosine, for \(\varepsilon \gg \dfrac{h}{2\pi}\omega\), is, according to formula (19), equal to

\[ \cos\beta'_0 \simeq 1-\frac{\left(\dfrac{h^2}{2\pi}\right)^2 L^2}{2m^{*2}v^2}. \tag{21} \]

Analogous expressions are obtained for the angles \(\theta'_{10}, \theta''_{0}, \theta''_{10}, \beta'_{10}, \beta''_{0}, \beta''_{10}\). Representing \(f\) as a series in Legendre polynomials,

\[ f(\varepsilon,\theta)=f_0(\varepsilon)+f_1(\varepsilon)\cos\theta+ f_2(\varepsilon)\left(\frac{3}{2}\cos^2\theta-\frac{1}{2}\right)+\ldots \tag{22} \]

and carrying out in (13) the integration over \(\alpha\) and \(\varphi\), taking (20) and (21) into account, and assuming that \(\varepsilon \gg \dfrac{h}{2\pi}\omega\), we obtain, instead of (13),

\[ (\varepsilon-a_i)p \int_{L_{\min}}^{L_{\max}} \left\{ \frac{2\pi e^2}{a^3 vL} \left( \frac{e_{\overline L\,\mathrm{op}}^{\,2}}{M_{\overline L}^{\mathrm{op}}} + \frac{e_{\overline L\,\mathrm{ac}}^{\,2}}{M_{\overline L}^{\mathrm{ac}}} \right) \frac{df_0}{d\varepsilon} + \right. \]

\[ \left. + \frac{\pi e^2\dfrac{h}{2\pi}}{a^3 vL} \left[ \frac{(2n_{\overline L}^{\mathrm{op}}+1)\omega_{\overline L}^{\mathrm{op}} e_{\overline L\,\mathrm{op}}^{\,2}}{M_{\overline L}^{\mathrm{op}}} + \frac{(2n_{\overline L}^{\mathrm{ac}}+1)\omega_{\overline L}^{\mathrm{ac}} e_{\overline L\,\mathrm{ac}}^{\,2}}{M_{\overline L}^{\mathrm{ac}}} \right] \frac{d^2 f_0}{d\varepsilon^2} \right. \]

\[ \left. - \frac{2\pi^2 e^2 hL}{a^3 m^{*2}v^3} \left[ \frac{(2n_{\overline L}^{\mathrm{op}}+1)e_{\overline L\,\mathrm{op}}^{\,2}}{M_{\overline L}^{\mathrm{op}}\omega_{\overline L}^{\mathrm{op}}} + \frac{(2n_{\overline L}^{\mathrm{ac}}+1)e_{\overline L\,\mathrm{ac}}^{\,2}}{M_{\overline L}^{\mathrm{ac}}\omega_{\overline L}^{\mathrm{ac}}} \right] f_1(\varepsilon)\cos\theta - \frac{\pi^2 e^2 hL}{a^3 m^{*2}v^3} \right. \]

\[ \left. \times \left[ \frac{(2n_{\overline L}^{\mathrm{op}}+1)e_{\overline L\,\mathrm{op}}^{\,2}}{M_{\overline L}^{\mathrm{op}}\omega_{\overline L}^{\mathrm{op}}} + \frac{(2n_{\overline L}^{\mathrm{ac}}+1)e_{\overline L\,\mathrm{ac}}^{\,2}}{M_{\overline L}^{\mathrm{ac}}\omega_{\overline L}^{\mathrm{ac}}} \right] f_2(\varepsilon) \left(\frac{3}{2}\cos^2\theta-\frac{1}{2}\right) \right\}\,dL . \tag{23} \]

Let us note that the coefficient of \(f_2\) has been calculated approximately, since in what follows we shall need only its order of magnitude. Integration over \(L\) in (23) is impossible, since the dependence of the quantities \(n_{\overline L}^{\mathrm{op}}, n_{\overline L}^{\mathrm{ac}}, \omega_{\overline L}^{\mathrm{op}}, \omega_{\overline L}^{\mathrm{ac}}, M_{\overline L}^{\mathrm{op}}, M_{\overline L}^{\mathrm{ac}}, e_{\overline L}^{\mathrm{op}}, e_{\overline L}^{\mathrm{ac}}\) on \(\overline L\) is in general unknown. However, this integration can be carried out approximately on the basis of the following considerations. Let us assume [1] that for \(L \ll \dfrac{1}{10a}\), \(\omega_{\overline L}=\omega_0\), \(e_{\overline L}=e^*\), and for \(L \gg \dfrac{1}{10a}\), \(\omega_{\overline L}=\omega_t\), \(e_{\overline L}=e\), where \(\omega_t\) is the experimentally determined frequency of transverse optical vibrations, coinciding with the frequency of short-wavelength longitudinal optical vibrations; \(\omega_0\) and \(e^*\) are determined from the formulas

\[ \omega_0^2=\frac{2\pi e^{*2}}{Ma^3}\,\frac{\chi_0\chi}{\chi-\chi_0}, \tag{24} \]

\[ \frac{\omega_0^2}{\omega_t^2}=\frac{\chi}{\chi_0}; \tag{25} \]

the frequency of long-wavelength longitudinal optical vibrations and the effective charge of the ion, respectively; \(M\) is the reduced mass of the ions, \(\chi_0\) is the square of the refractive index of light, and \(\chi\) is the static dielectric constant. We shall next consider the longitudinal vibrations of a linear chain of ions located at equal spacing \(a\). We shall take the masses of the ions to be

consider alternately equal to \(M_+\) and \(M_-\). In this case [8] the frequencies and amplitudes of the vibrations are determined from the equations

\[ \omega_L^2=\frac{q_0}{M_+M_-}\left[M_+ + M_- \pm \sqrt{(M_+ + M_-)^2 - 4M_+M_- \sin^2 2\pi La}\right], \tag{26} \]

\[ \frac{\zeta_{-,L}}{\zeta_{+,L}} = -\frac{2q_0\sqrt{\dfrac{M_-}{M_+}}\cos 2\pi La}{M_- \omega_L^2 - 2q_0}, \tag{27} \]

\[ \zeta_{+,L}^2+\zeta_{-,L}^2=1, \tag{28} \]

where \(q_0\) is the elastic constant of the bond acting between two neighboring particles. In equation (26) the plus sign corresponds to optical vibrations, the minus sign to acoustic ones; equation (28) expresses the normalization condition. From equations (26)—(28) it follows that the quantities \(\dfrac{1}{M_L^{op}}\) and \(\dfrac{1}{M_L^{ak}}\) (see (5)) are equal to

\[ \frac{1}{M_L^{op}}= \begin{cases} \dfrac{1}{M}, & \text{for } L \ll \dfrac{1}{2\pi a},\\[6pt] \dfrac{1}{M_+}, & \text{for } L \to L_0=\dfrac{1}{4a}, \end{cases} \tag{29} \]

\[ \frac{1}{M_L^{ak}}= \begin{cases} 0, & \text{for } L \ll \dfrac{1}{2\pi a},\\[6pt] \dfrac{1}{M}, & \text{for } L \to L_0=\dfrac{1}{4a}, \end{cases} \]

\[ \frac{\omega_t}{\omega_{ak}^{max}}=\sqrt{\frac{M_-}{M_+}}, \]

where \(L_0\) is the maximum wave number corresponding to the vibration with the smallest wavelength for the given chain of atoms, and \(\omega_{ak}^{max}\) is the maximum frequency of acoustic vibrations. The limiting relations (29) are also valid in the case of a three-dimensional model of the crystal.

Determining from formulas (18) and (19) the quantities \(L_{max}\) and \(L_{min}\), which for

\[ \varepsilon \gg \frac{h}{2\pi}\,\omega \]

are equal to

\[ L_{max}\simeq 2K, \tag{30} \]

\[ L_{min}\simeq \frac{K\dfrac{h}{2\pi}\,\omega_0}{2\varepsilon}, \]

and carrying out, with the aid of relations (24), (25), (29), the approximate integration over \(L\) in (23), we obtain

\[ (b-a)^p = \frac{a_1}{\varepsilon^{1/2}}\frac{df_0}{d\varepsilon} + \frac{b_1}{\varepsilon^{1/2}}\frac{d^2 f_0}{d\varepsilon^2} - \frac{r_1}{\varepsilon^{1/2}} f_1 \cos\theta - \frac{3r_1}{\varepsilon^{1/2}} f_2\left(\frac{3}{2}\cos^2\theta-\frac{1}{2}\right), \tag{31} \]

where

\[ a_2=\frac{\pi \sqrt{2m^*}\,e^2}{Ma^3} \left( e^{*2}\ln \frac{2\pi\sqrt{2\varepsilon}}{10a\omega_0\sqrt{m^*}} + e^2\ln 10 \sqrt[3]{\frac{3}{4\pi}} \right), \]

\[ b_2= \frac{ \pi e^2 \dfrac{h}{2\pi}\sqrt{m^*} }{ \sqrt{2}\,Ma^3 } \left( e^{*2}\omega_0 \frac{ e^{\frac{\frac{h}{2\pi}\omega_0}{kT}}+1 }{ e^{\frac{\frac{h}{2\pi}\omega_0}{kT}}-1 } \ln \frac{2\pi\sqrt{2\varepsilon}}{10a\omega_0\sqrt{m^*}} + \right. \]

\[ \left. + e^2\omega_t \frac{ e^{\frac{\frac{h}{2\pi}\omega_t}{kT}}+1 }{ e^{\frac{\frac{h}{2\pi}\omega_t}{kT}}-1 } \ln 10 \sqrt[3]{\frac{3}{4\pi}} \right), \]

\[ r_2= \frac{ \pi^2\left(\dfrac{3}{4\pi}\right)^{1/3} e^4 h }{ 2\sqrt{2m^*}\,Ma^5\omega_t } \frac{ e^{\frac{\frac{h}{2\pi}\omega_t}{kT}}+1 }{ e^{\frac{\frac{h}{2\pi}\omega_t}{kT}}-1 }. \tag{38} \]

The change in the distribution function as a result of ionization collisions can be written in the form

\[ (b-a)^{\mathrm{ion}}_1 = 2\beta_1N_1Q \left( I+ 2\varepsilon \right) \sqrt{ \frac{2(I+2\varepsilon)}{m^*} } \int_0^\pi f_3(I+2\varepsilon,\theta)\sin\theta\,d\theta \quad \text{for } \varepsilon\le I, \tag{39} \]

\[ (b-a)^{\mathrm{ion}}_3 = 2\beta_1N_1Q(I+2\varepsilon) \sqrt{ \frac{2(I+2\varepsilon)}{m^*} } \int_0^\pi f_3(I+2\varepsilon,\theta)\sin\theta\,d\theta - \]

\[ -\beta_1N_1Q(\varepsilon) \sqrt{\frac{2\varepsilon}{m^*}}\, f^3(\varepsilon,\theta) \quad \text{for } \varepsilon\ge I, \tag{40} \]

where \(f^3(\varepsilon,\theta)\) is the distribution function for \(\varepsilon>I\), \(Q(\varepsilon)\) is the total effective ionization cross section, \(\beta_1\) is the number of electrons in the outer shell of the ionized atom, and \(N_1\) is the number of ionizable atoms per unit volume. In writing (39) and (40) it was assumed [9] (we have in mind energies of the order of the ionization potential) that the effective ionization cross section is isotropic both with respect to the direction of motion of the ejected electron and with respect to the direction of motion of the ionizing electron after the act of ionization. In addition, it was assumed that after the act of ionization the ejected and ionizing electrons have the same energy. The latter assumption is quite plausible for \(\varepsilon\sim I\). In the subsequent calculations it will also be assumed [5] that

\[ Q(\varepsilon)=S_0(\varepsilon-I), \tag{41} \]

where \(S_0=\mathrm{const}\).

We now turn to the solution of the kinetic equation. We shall divide the entire set of values of \(\varepsilon\) into four regions.

\[ 1.\quad 0\le \varepsilon \le \frac{h}{2\pi}\omega_0. \]

where

\[ a_2=\frac{\pi \sqrt{2m^*}\,e^2}{Ma^3} \left( e^{*2}\ln\frac{2\pi\sqrt{2\varepsilon}}{10a\omega_0\sqrt{m^*}} +e^2\ln 10\,\sqrt[3]{\frac{3}{4\pi}} \right), \]

\[ b_2= \frac{\pi e^2 \frac{h}{2\pi}\sqrt{m^*}}{\sqrt{2}\,Ma^3} \left( e^{*2}\omega_0 \frac{ e^{\frac{\frac{h}{2\pi}\omega_0}{kT}}+1 }{ e^{\frac{\frac{h}{2\pi}\omega_0}{kT}}-1 } \ln\frac{2\pi\sqrt{2\varepsilon}}{10a\omega_0\sqrt{m^*}} + e^2\omega_t \frac{ e^{\frac{\frac{h}{2\pi}\omega_t}{kT}}+1 }{ e^{\frac{\frac{h}{2\pi}\omega_t}{kT}}-1 } \ln 10\,\sqrt[3]{\frac{3}{4\pi}} \right), \]

\[ r_2= \frac{ \pi^2\left(\frac{3}{4\pi}\right)^{1/3}e^4h }{ 2\sqrt{2m^*}\,Ma^5\omega_t } \frac{ e^{\frac{\frac{h}{2\pi}\omega_t}{kT}}+1 }{ e^{\frac{\frac{h}{2\pi}\omega_t}{kT}}-1 }. \tag{38} \]

The change in the distribution function due to ionizing collisions can be written in the form

\[ (b-a)_1^{\mathrm{ion}} = 2\beta_1N_1Q \left( I+ 2\varepsilon \right) \sqrt{\frac{2(I+2\varepsilon)}{m^*}} \int_0^\pi f_3(I+2\varepsilon,\theta)\sin\theta\,d\theta \quad \text{for } \varepsilon\le I, \tag{39} \]

\[ (b-a)_3^{\mathrm{ion}} = 2\beta_1N_1Q(I+2\varepsilon) \sqrt{\frac{2(I+2\varepsilon)}{m^*}} \int_0^\pi f_3(I+2\varepsilon,\theta)\sin\theta\,d\theta - \beta_1N_1Q(\varepsilon) \sqrt{\frac{2\varepsilon}{m^*}}\, f^3(\varepsilon,\theta) \quad \text{for } \varepsilon\ge I, \tag{40} \]

where \(f^3(\varepsilon,\theta)\) is the distribution function for \(\varepsilon>I\), \(Q(\varepsilon)\) is the total effective ionization cross section, \(\beta_1\) is the number of electrons in the outer shell of the atom being ionized, and \(N_1\) is the number of ionized atoms per unit volume. In writing (39) and (40) it was assumed [9] (we have in mind energies of the order of the ionization potential) that the effective ionization cross section is isotropic both with respect to the direction of motion of the ejected electron and with respect to the direction of motion of the ionizing electron after the ionization event. In addition, it was assumed that after the ionization event the ejected and the ionizing electrons have the same energy. The latter assumption is quite plausible for \(\varepsilon\sim I\). In the subsequent calculations the assumption [5] will also be made that

\[ Q(\varepsilon)=S_0(\varepsilon-I), \tag{41} \]

where \(S_0=\mathrm{const}\).

We now proceed to the solution of the kinetic equation. We divide the entire set of values of \(\varepsilon\) into four regions.

\[ 1.\quad 0\le \varepsilon \le \frac{h}{2\pi}\omega_0. \]

Solving the equation in this region is difficult because expansion in a small parameter is impossible and there is no expression for the effective recombination cross section. This solution will not be needed by us in what follows.

1. \(\dfrac{h}{2\pi}\omega_0 \ll \varepsilon \ll \varepsilon'\).

We shall denote the distribution function in this region by \(f^1(\varepsilon,\theta)\). Substituting (22), (31), and (39) into equation (1), then multiplying the resulting equation successively by \(\sin\theta\,d\theta\), \(\cos\theta\sin\theta\,d\theta\), \(\left(\dfrac{3}{2}\cos^2\theta-\dfrac{1}{2}\right)\sin\theta\,d\theta\), and carrying out the integration over \(\theta\), we obtain, in the stationary case, the following three equations for determining \(f_0^1\), \(f_1^1\), \(f_2^1\):

\[ -\frac{d}{d\varepsilon}\left[ \frac{1}{3}\left(\frac{2}{m^*}\right)^{1/2} eE\varepsilon f_1^1 -a_1 f_0^1 -b_1\frac{df_0^1}{d\varepsilon} \right] = \varepsilon^{1/2}(b-a)_1^{\mathrm{ion}}, \qquad (42\text{–}1) \]

\[ \left(\frac{2}{m^*}\right)^{1/2} eE\varepsilon^{1/2}\frac{df_0^1}{d\varepsilon} +\frac{2}{5}\left(\frac{2}{m^*}\right)^{1/2} eE\varepsilon^{1/2}\frac{df_2^1}{d\varepsilon} + \frac{6}{5}\left(\frac{1}{2m^*}\right)^{1/2}\frac{eE}{\varepsilon^{1/2}} f_2^1 = -\frac{r_1}{\varepsilon^{1/2}} f_1^1, \qquad (42\text{–}2) \]

\[ \frac{2}{3}\left(\frac{2}{m^*}\right)^{1/2} eE\varepsilon^{1/2}\frac{df_1^1}{d\varepsilon} - \frac{2}{3}\left(\frac{1}{2m^*}\right)^{1/2}\frac{eE}{\varepsilon^{1/2}} f_1^1 = -\frac{3r_1}{\varepsilon^{1/2}} f_2^1. \qquad (42\text{–}3) \]

Discarding in the left-hand side of equation (42–2) the last two terms and then solving equations (42–1) and (42–2), we obtain (we have neglected the weak dependence of the coefficients \(a_1\) and \(b_1\) on \(\varepsilon\))

\[ f_0^1 = C_1+C_2y_1(\varepsilon) -\frac{1}{a_1}\int \varepsilon^{1/2}(b-a)_1^{\mathrm{ion}}\,d\varepsilon + \frac{y_1(\varepsilon)}{a_1} \int \frac{\varepsilon^{1/2}(b-a)_1^{\mathrm{ion}}}{y_1(\varepsilon)} \,d\varepsilon, \qquad (43) \]

\[ f_1^1 = -\frac{e\sqrt{\dfrac{2}{m^*}}\,E}{r_1}\, \varepsilon\,\frac{df_0^1}{d\varepsilon}, \qquad (44) \]

where

\[ y_1(\varepsilon) = \exp\left( -\int_0^\varepsilon \frac{a_1\,d\varepsilon}{b_1+p_1E^2\varepsilon^2} \right), \qquad (45) \]

\[ p_1=\frac{2}{3}\frac{e^2}{m^*r_1}. \qquad (46) \]

Let us estimate the limit of applicability of our approximation. Noting that, in order of magnitude,
\(\dfrac{df_1^1}{d\varepsilon}\simeq\dfrac{f_1^1}{\varepsilon}\) and
\(\dfrac{df_2^1}{d\varepsilon}\simeq\dfrac{f_2^1}{\varepsilon}\), we find that our approximation is valid when

\[ |f_2^1|\ll \left|\varepsilon\,\frac{df_0^1}{d\varepsilon}\right|. \qquad (47) \]

From (42–3) and (44) we obtain

\[ \left| f_2^1 \right| \approx \left| \frac{2 e^2 E^2 \varepsilon}{9 m^* r_1^2}\,\frac{d f_0^1}{d\varepsilon} \right|, \]

after which condition (47) is written in the form

\[ E \ll \left| \frac{3\sqrt{m^*}\,r_1}{\sqrt{2}\,e} \right|, \tag{48} \]

which gives for NaCl, for example, \(E \ll 3.4 \cdot 10^7\ \dfrac{\mathrm{V}}{\mathrm{cm}}\).

3. \(\varepsilon'' \ll \varepsilon \ll I\).

Let us assume that the distribution function \(f^3(\varepsilon,\theta)\) decreases so sharply with increasing difference \(\varepsilon - I\) that the term \((b-a)_1^{\text{ion}}\) in the region \(\varepsilon'' \ll \varepsilon \ll I\) may be neglected (\(\varepsilon''\) for NaCl is equal to \(1.83\ \mathrm{eV}\)). Solving, under this assumption, equation (1) in the stationary case with allowance for (22) and (37), we obtain (we denote the distribution function in the region \(\varepsilon'' \ll \varepsilon \ll I\) by \(f^2(\varepsilon,\omega)\))

\[ f_0^2 = -\frac{j}{2\pi(2m^*)^{3/2}Q_2} + C_3 y_2(\varepsilon), \tag{49} \]

\[ f_1^2 = -\left(\frac{2}{m^*}\right)^{1/2} \frac{eE}{r_2}\, \varepsilon^2 \frac{d f_0^2}{d\varepsilon}, \tag{50} \]

where

\[ j = \frac{8\pi}{3}m^* eE\varepsilon f_1^2 - 2\pi(2m^*)^{3/2} \left( a_2 f_0^2 + b_2 \frac{d f_0^2}{d\varepsilon} \right) = \mathrm{const} \tag{51} \]

is the electron flux in energy space,

\[ y_2(\varepsilon) = \exp\left( -\int_{\varepsilon''}^{\varepsilon} \frac{a_2\,d\varepsilon}{b_2+p_2E^2\varepsilon^3} \right), \qquad p_2=\frac{2e^2}{3m^*r_2}. \tag{52} \]

Equations (49) and (50) are valid for

\[ E \ll \frac{3\sqrt{m^*}\,r_2}{\sqrt{2}\,e\varepsilon}, \tag{53} \]

which for NaCl, even at \(\varepsilon=I\), gives \(E \ll 6.3\cdot 10^6\ \dfrac{\mathrm{V}}{\mathrm{cm}}\).

4. \(\varepsilon \gg I\).

In this range of values of \(\varepsilon\) one may neglect collisions of electrons with lattice vibrations in comparison with ionization collisions,\(^1\) and also, owing to the sharp decrease of \(f^3(\varepsilon,\theta)\) with increasing \(\varepsilon\) (as will be shown below), that part of the ionization term (40) which takes into account the arrival of electrons from regions of higher values \(\varepsilon \gg 2I\).

\(^1\) The number of ionization collisions per unit time, for example for NaCl, is

\[ \beta_1 N_1 v Q(\varepsilon) \simeq 2\cdot 10^{14} \]

at

\[ \beta_1=6,\qquad N_1=\frac{1}{2a^3}=2.25\cdot 10^{22}\ \mathrm{cm}^{-3},\qquad v=\sqrt{\frac{2I}{m}}=1.88\cdot 10^8\ \frac{\mathrm{cm}}{\mathrm{sec}}, \]

\[ S_0=2.4\cdot 10^{-6}\ \frac{\mathrm{sec}^2}{\mathrm{cm}^2},\qquad \varepsilon-I=2\ \mathrm{eV}, \]

whereas the number of collisions with lattice vibrations at \(\varepsilon \simeq I\) is

\[ \frac{1}{\tau}=\frac{r_2}{I^{3/2}}\simeq 4\cdot 10^{13} \]

((38), (41), and Table 1).

Consequently, equation (1), for \(\dfrac{\partial f^3}{\partial t}=0\), is written in this case approximately in the form (see (40) and (41))

\[ eE\frac{\partial f^3}{\partial p_z} = -\beta_1 N_1 S_0 \sqrt{\frac{2\varepsilon}{m^*}}\, (\varepsilon-I)\, f^3(\varepsilon,\vartheta) \tag{54} \]

or

\[ \begin{aligned} eE\sqrt{\frac{2}{m^*}}\cdot \varepsilon^{1/2}x\,\frac{\partial f^3}{\partial \varepsilon} &+ \sqrt{\frac{1}{2m^*}}\,eE\,\frac{1-x^2}{\varepsilon^{1/2}}\, \frac{\partial f^3}{\partial x} \\ &= -\beta_1 N_1 S_0 \sqrt{\frac{2\varepsilon}{m^*}}\, (\varepsilon-I)\, f^3(\varepsilon,\vartheta), \end{aligned} \tag{55} \]

where \(x=\cos\vartheta\).

Solving the partial differential equation (55) under the assumption that, for \(\varepsilon=I\), \(f^3(\varepsilon,\vartheta)=f^2(\varepsilon,\vartheta)\), we obtain

\[ \begin{aligned} f^3(\varepsilon,x) &= f^2\left(I,\sqrt{1-\frac{\varepsilon}{I}(1-x^2)}\right) \\ &\quad \times \exp\left\{ -\frac{2q}{E} \left[ y(\varepsilon,x) - y\left(I,\sqrt{1-\frac{\varepsilon}{I}(1-x^2)}\right) \right] \right\}, \end{aligned} \tag{56} \]

where

\[ \begin{aligned} y(\varepsilon,x) &= \frac{1}{2}\varepsilon^2 x + \frac{3}{4}\varepsilon^2 x(1-x^2) - I\varepsilon x \\ &\quad + \frac{3}{8}\varepsilon^2(1-x^2)^2 \ln\frac{1+x}{1-x} - \frac{I\varepsilon}{2}(1-x^2) \ln\frac{1+x}{1-x}, \end{aligned} \tag{57} \]

\[ q=\frac{\beta_1 N_1 S_0}{2e}, \tag{58} \]

\[ \begin{aligned} y\left(I,\sqrt{1-\frac{\varepsilon}{I}(1-x^2)}\right) &= -\frac{1}{2}I^2\sqrt{1-\frac{\varepsilon}{I}(1-x^2)} \\ &\quad -\frac{3}{4}\varepsilon I(1-x^2) \sqrt{1-\frac{\varepsilon}{I}(1-x^2)} + I^2\sqrt{1-\frac{\varepsilon}{I}(1-x^2)} \\ &\quad +\frac{3}{4}\varepsilon^2(1-x^2)^2 \ln \frac{ 1-\sqrt{1-\frac{\varepsilon}{I}(1-x^2)} }{ 1+\sqrt{1-\frac{\varepsilon}{I}(1-x^2)} } \\ &\quad -\frac{\varepsilon I}{2}(1-x^2) \ln \frac{ 1-\sqrt{1-\frac{\varepsilon}{I}(1-x^2)} }{ 1+\sqrt{1-\frac{\varepsilon}{I}(1-x^2)} }. \end{aligned} \tag{59} \]

From (49) and (50) we have

\[ f^2\left(I,\sqrt{1-\frac{\varepsilon}{I}(1-x^2)}\right) = -\frac{j}{2\pi(2m^*)^{3/2}a^2} + \]

\[ + C_3 y_2(I)\left[1- \sqrt{\frac{\frac{2}{m^*}a^2 e E I^2}{r_2\left(b_2+p_2E^{2/3}\right)}} -\sqrt{1-\frac{\varepsilon}{I}(1-x^2)}\right]. \tag{60} \]

Let us note that \(\sqrt{1-\dfrac{\varepsilon}{I}(1-x^2)}\) must be taken with the plus sign for holes and with the minus sign for electrons. Formulas (59) and (60) are written for electrons.

It follows from the solution (56) that the values of the angles \(\theta\) between the direction of the velocity and the direction of \(e\vec E\) for electrons with a given energy \(\varepsilon\) are contained in the intervals \(0\leq \theta\leq \theta_m\) and \(\pi-\theta_m\leq \theta\leq \pi\), where

\[ \theta_m=\arc\sin \sqrt{\frac{I}{\varepsilon}}; \tag{61} \]

since, for angles not falling within these intervals, the solution becomes imaginary. Moreover, the first of these intervals for electrons (the second for holes) likewise should not be taken into account, since in this case the solution (56) diverges as \(\varepsilon\to\infty\). Thus, for \(\varepsilon>I\), electrons can move only at acute angles to the direction of action of the force \(e\vec E\), which agrees with our assumptions concerning the absence, in this range of \(\varepsilon\) values, of collisions redistributing the electrons over angles.

Thus, we have determined the distribution function of the conduction electrons for all values of \(\varepsilon\), with the exception of \(\varepsilon\sim 0\). However, the values of the constants \(C_1\), \(C_2\), \(C_3\), and \(j\) are unknown to us. The values of three of these constants can be determined from the conditions of matching the functions \(f_0^1\) and \(f_0^2\) and their first derivatives at \(\varepsilon=\varepsilon'\), and from the condition that, in the stationary case, the flux of electrons \(j\) in energy space through the boundary \(\varepsilon=I\) is equal to the number of ionizations per unit time \(N_i\), which gives, if (32), (38), (41), (43), (45), (46), (49), (52) are taken into account and the last two terms in (43) are neglected at \(\varepsilon=\varepsilon''\),

\[ C_1=-\frac{j}{2\pi(2m^*)^{3/2}a_2}, \tag{62} \]

\[ C_3=C_2y_1(\varepsilon''), \tag{63} \]

\[ j(I)=N_i=\beta_1N_1S_0\int \sqrt{\frac{2}{m^*}\varepsilon}\,(\varepsilon-I)f^3(\varepsilon,\theta)\,dp_x\,dp_y\,dp_z = \]

\[ =4\pi m^*\beta_1N_1S_0 \int_{\pi-\theta_m}^{\pi}\int_I^\infty \varepsilon(\varepsilon-I)f^3(\varepsilon,\theta)\,d\varepsilon\,\sin\theta\,d\theta. \tag{64} \]

Let us now note that for \(S_0\simeq 2\cdot10^{-6}\,{\rm sec}^2/{\rm g}^2\) (Table 1) and \(\beta_1=6\), \(N_1\simeq 3\cdot10^{22}\,{\rm cm}^{-3}\), the function \(f^3(\varepsilon,\theta)\) is practically equal to zero already at \(\varepsilon\simeq 1.5I\). Let us also note that as \(\theta\to\pi\) (see (56)—(60)),

\[ y(\varepsilon,x)-y\left(I,\sqrt{1-\frac{\varepsilon}{I}(1-x^2)}\right)\simeq -\frac{(\varepsilon-I)^2}{2}. \tag{65} \]

Taking these circumstances into account, the integral in (64) can be represented with a sufficient degree of accuracy in the form\(^1\) (the quantity \(q\) is negative for electrons)

\[ \int_{\pi-\theta_m}^{\pi} \int_I^{\infty} \varepsilon(\varepsilon-I) f^3(\varepsilon,\theta)\, d\varepsilon \sin\theta\, d\theta = \]

\[ = \left[ C_1 + C_2 y_1(\varepsilon'') y_2(I) \right] \int_I^{\infty} \varepsilon(\varepsilon-I) \left( 1-\sqrt{1-\frac{I}{1.3\,\varepsilon}}\, e^{-\frac{q}{E}(\varepsilon-I)^2} \right) d\varepsilon - \]

\[ - C_2 \frac{ \sqrt{\frac{2}{m^*}}\, a_2 I^2 y_2(I)\, eE\, y_1(\varepsilon'') }{ r_2\left(b_2+p_2 I^3 E^2\right) } \int_I^{\infty} \varepsilon(\varepsilon-I)\times \]

\[ \times \left( \frac{1}{2} + \frac{1}{2}\, \frac{1.3\,\varepsilon-I}{\sqrt{1.3\,\varepsilon I}}\, \ln \frac{ 1-\sqrt{\frac{I}{1.3\,\varepsilon}} }{ \sqrt{1-\frac{I}{1.3\,\varepsilon}} } \right) e^{-\frac{q}{E}(\varepsilon-I)^2}\, d\varepsilon . \tag{67} \]

Expanding in (67) the logarithmic factor and \(\sqrt{1-\dfrac{I}{1.3\,\varepsilon}}\) in a series in \(\dfrac{I}{1.3\,\varepsilon}\) and restricting ourselves to the first two terms of the expansion, we obtain after integration with respect to \(\varepsilon\)

\[ \int_{\pi-\theta_m}^{\pi} \int_I^{\infty} \varepsilon(\varepsilon-I) f^3(\varepsilon,\theta)\, d\varepsilon \sin\theta\, d\theta = \]

\[ = -\frac{IE}{5.2\,q} \left[ C_1 + C_2 y_1(\varepsilon'')y_2(I) - C_2 y_2(I)\, \frac{ \sqrt{\frac{2}{m^*}}\, a_2 I^2 eE\, y_1(\varepsilon'') }{ r_2\left(b_2+p_2 I^3 E^2\right) } \right]. \tag{68} \]

The combination of (62), (64), and (68) gives

\[ C_1 = C_2 \frac{ eIE\, y_1(\varepsilon'') y_2(I) }{ 2.6\,(2m^*)^{1/2} a_2 - eIE } \left[ 1 - \frac{ \sqrt{\frac{2}{m^*}}\, a_2 I^2 eE }{ r_2\left(b_2+p_2 I^3 E^2\right) } \right]. \tag{69} \]

Having determined the function \(f^3(\varepsilon,\theta)\) and the relations between the constants, we can estimate the magnitude of the last two terms in (43). Using (39),

\(^1\) We have applied formula (65) for all possible values of the angles, thereby overestimating the result of integration with respect to \(\theta\), since as the deviation of \(\theta\) from \(\pi\) increases the function \(f^3\) decreases. However, we compensated for this overestimate by reducing the interval of possible values of \(\theta\), replacing formula (61) by formula (66)

\[ \theta_m = \arcsin \sqrt{\frac{I}{1.3\,\varepsilon}} . \]

Numerical integration justifies such an approximation.

(41), (43), (45), and carrying out the integration over \(\theta\) in the same way as in the calculation of the number of ionizations, we obtain for the first of these terms a value equal to

\[ \frac{1}{a_1}\int_0^\varepsilon \varepsilon^{1/2}(b-a)_1^{\text{ion}}\,d\varepsilon \simeq \]

\[ \simeq C_2\frac{2eIE\,y_1(\varepsilon'')\,y_2(I)} {2.6\sqrt{2m^*}\,a_1} \left[ 1-\frac{\sqrt{\dfrac{2}{m^*}a_2 I^2 eE}} {r_2(b_2+p_2 I^3E^2)} \right]\times \]

\[ \times\left( 1+\frac{eIE}{2.6\sqrt{2m^*}\,a_2-eIE} \right) \sqrt{\frac{\varepsilon}{I+2\varepsilon}}\, e^{-\frac{4q}{E}\varepsilon^2}, \tag{70} \]

which, in absolute value, for \(S_0=2\cdot10^{-6}\,\dfrac{\text{sec}^2}{\text{g}}\), \(\beta_1=6\), \(N_1\simeq3\cdot10^{22}\,\text{cm}^{-3}\), and \(E\simeq10^6\,\dfrac{\text{V}}{\text{cm}}\), amounts to about \(0.2\) and is smaller than the first term in (43), and an even smaller fraction of the second term for all values of \(\varepsilon\). Thus, the influence of the third term in (43) on the distribution function is relatively small. The fourth term is also relatively small, since its upper limit is equal to

\[ \frac{1}{a_1}\int \varepsilon^{1/2}(b-a)_1^{\text{ion}}\,d\varepsilon, \]

i.e. to the third term. Let us note, moreover, that the third and fourth terms have different signs. The estimate given above shows that neglecting the term \((b-a)_1^{\text{ion}}\) in the region \(\varepsilon''\leq \varepsilon\leq I\) is quite justified (the integral in (70) decreases sharply with increasing \(\varepsilon\)).

Taking into account (62), (63), and (69), we obtain, up to the normalizing constant \(C_2\), the following expressions for the functions \(f^1\), \(f^2\), \(f^3\) (see (43), (44), (49), (50), (56), (60)):

\[ f^1=C_2\left\{ \frac{eIE\,y_1(\varepsilon'')\,y_2(I)} {2.6(2m^*)^{1/2}a_2-eIE} \left[ 1-\frac{\sqrt{\dfrac{2}{m^*}a_2I^2eE}} {r_2(b_2+p_2I^3E^2)} \right] +\right. \]

\[ \left. +y_1(\varepsilon)\left[ 1+\frac{\sqrt{\dfrac{2}{m^*}a_1eE\varepsilon}} {r_1(b_1+p_1E^2\varepsilon^2)} \cos\theta \right]\right\} - \]

\[ -\frac{1}{a_1}\int \varepsilon^{1/2}(b-a)_1^{\text{ion}}\,d\varepsilon +\frac{y_1(\varepsilon)}{a_1} \int \frac{\varepsilon^{1/2}(b-a)_1^{\text{ion}}}{y_1(\varepsilon)}\,d\varepsilon, \tag{71} \]

\[ f^2=C_2\left\{ \frac{eIE\,y_1(\varepsilon'')\,y_2(I)} {2.6(m^*)^{1/2}a_2-eIE} \left[ 1-\frac{\sqrt{\dfrac{2}{m^*}a_2I^2eE}} {r_2(b_2+p_2I^3E^2)} \right] +\right. \]

\[ \left. +y_1(\varepsilon'')\,y_2(\varepsilon) \left[ 1+\frac{\sqrt{\dfrac{2}{m^*}a_2eE\varepsilon^2}} {r_2(b_2+p_2\varepsilon^3E^2)} \cos\theta \right]\right\}, \tag{72} \]

\[ \begin{aligned} f^3={}& C_2 y_1(\varepsilon'')y_2(I) \left\{1+\frac{eIE}{2{,}6(2m^*)^{1/2}a_2-eIE}\right. \\ &\left.\times\left[+1\sqrt{\frac{2}{m^*}}\, \frac{a_2 I^2 eE}{r_2(b_2+p_2 I^3E^2)}\right] -\frac{\sqrt{\frac{2}{m^*}}\,a_2 I^2 eE}{r_2(b_2+p_2 I^3E^2)} \sqrt{1-\frac{1{,}3\varepsilon}{I}(1-x^2)} \right\} e^{-\frac{q}{E}(\varepsilon-I)^2}. \end{aligned} \tag{73} \]

The relations (71)—(73) determine the distribution function in the stationary case.

To find the field strength \(E=E_{\mathrm{br}}\) at which a stationary solution is impossible, we shall apply the criterion first formulated in [10]. The essence of this criterion is as follows.

Let us introduce two characteristic energy values: 1) \(\varepsilon_{1/2}\)—the value of the energy such that equal numbers of electrons, after ionization, fall in the regions above and below it; 2) \(\varepsilon_p\)—the value of the energy at which the electron is, on the average, in equilibrium, i.e., where there is a balance between the energy given up per unit time to the lattice and the energy received from the field.

Starting from the definition of the value \(\varepsilon_{1/2}\), it can be found from the equation

\[ \begin{aligned} &\int_0^{\varepsilon_{1/2}}\int_{\pi-\Theta_m}^{\pi}\int_0^{2\pi} \beta_1N_1\sqrt{\frac{2(I+2\varepsilon)}{m^*}}\,Q(I+2\varepsilon) f^3(I+2\varepsilon,\theta)\, \frac{1}{2}(2m^*)^{3/2}\sqrt{I+2\varepsilon}\, d(I+2\varepsilon)\sin\theta\,d\theta\,d\varphi \\ &= \int_{\varepsilon_{1/2}}^{\infty}\int_{\pi-\Theta_m}^{\pi}\int_0^{2\pi} \beta_1N_1\sqrt{\frac{2(I+2\varepsilon)}{m^*}}\,Q(I+2\varepsilon) f^3(I+2\varepsilon,\theta)\, \frac{1}{2}(2m^*)^{3/2}\sqrt{I+2\varepsilon}\, d(I+2\varepsilon)\sin\theta\,d\theta\,d\varphi . \end{aligned} \tag{74} \]

Taking into account (41), (66), and (73), and carrying out the integration over all three variables, we obtain

\[ \varepsilon_{1/2}=\sqrt{\frac{\ln 2\cdot |e|\cdot E}{2\beta_1N_1\cdot S_0}}. \tag{75} \]

The quantity \(\varepsilon_p\) can be determined from the equation

\[ B(\varepsilon_p)=eE\,v_z(\varepsilon_p) =eE\, \frac{\displaystyle\int_0^{\pi} f(\varepsilon_p,\theta)\,v_p\cos\theta\sin\theta\,d\theta} {\displaystyle\int_0^{\pi} f(\varepsilon_p,\theta)\sin\theta\,d\theta}, \tag{76} \]

\({}^{1}\) In writing expression (73), the remarks made in the note on p. 57 were taken into account.

where \(B(\varepsilon_p)\) is the mean energy loss per unit time for electrons with energy \(\varepsilon_p\), determined by the formula (see (11), (16), (17), etc.)

\[ B(\varepsilon_p)=4\pi a^3 N \int \frac{h}{2\pi}\,\omega\left(p_L^e-p_L^a\right)L^2\,dL\,\sin\alpha\,d\alpha . \tag{77} \]

Carrying out the integration in (77) and using (71) and (72), we obtain, instead of (76),

\[ \frac{ \dfrac{2}{3}\,\dfrac{e^2 E^2 \varepsilon_p^2}{m^*} }{ \left(r_1 b_1+\dfrac{2}{3}\,\dfrac{e^2 E^2 \varepsilon_p^2}{m^*}\right) \left(1-\dfrac{p_0}{y_1(\varepsilon_p)}\right) } =1 \quad \text{for } \varepsilon_p<\varepsilon'', \tag{78} \]

\[ \frac{ \dfrac{2}{3}\,\dfrac{e^2 E^2 \varepsilon_p^3}{m^*} }{ \left(r_2 b_2+\dfrac{2}{3}\,\dfrac{e^2 E^2 \varepsilon_p^3}{m^*}\right) \left(1-\dfrac{p_0}{y_1(\varepsilon'')y_2(\varepsilon_p)}\right) } =1 \quad \text{for } \varepsilon''<\varepsilon_p<I, \tag{79} \]

where

\[ p_0= \frac{|e|IE\,y_1(\varepsilon'')y_2(I)} {2.6(2m^*)^{1/2}a_2+|e|IE} \left[ 1+ \frac{ \sqrt{\dfrac{2}{m^*}}\,a_2 I^2 |e|E }{ r_2(b_2+p_2 I^3 E^3) } \right], \tag{80} \]

\(a_1, b_1, r_1, y_1(\varepsilon), a_2, b_2, r_2, y_2(\varepsilon)\) are determined from formulas (32), (38), (45), and (52). In writing (78), in accordance with the estimates given above, we have discarded the last two terms in (71).

Calculation shows that \(\varepsilon_{1/2}\) increases, while \(\varepsilon_p\) decreases with increasing field strength \(E\); moreover, all electrons with energy \(\varepsilon>\varepsilon_p\) are accelerated, and electrons with energy \(\varepsilon<\varepsilon_p\) are decelerated.

Let us now note that, as a result of ionization, \(2N_i\) electrons enter the region \(\varepsilon\le I\) per unit time\(^1\) (\(N_i\) is the number of ionizations per unit time), while \(j=N_i\) electrons leave it. Consequently, a stationary state is possible only when, in the region \(\varepsilon\le I\), \(N_i\) electrons recombine per unit time (the number of ionizations must equal the number of recombinations). It may be asserted that the latter condition is necessarily fulfilled if \(\varepsilon_{1/2}<\varepsilon_p\). Indeed, in this case an ionization event leads, on average, to the appearance of less than one electron capable of being accelerated by the field and of causing a new ionization (Fig. 1a).

Under these circumstances, of the \(2N_i\) electrons appearing in the region \(0\le\varepsilon\le I\) as a result of ionization, on average more than \(N_i\) electrons are braked by the lattice and have the possibility of entering the region of small \(\varepsilon\), where recombination is possible. Consequently, in the present case recombination can compensate ionization. The electron flux \(j\) and the number of ionizations \(N_i\) for \(\varepsilon_p>\varepsilon_{1/2}\) are determined by the number of fluctuation-induced transfers of electrons from the region \(\varepsilon<\varepsilon_p\) into the region \(\varepsilon>\varepsilon_p\), and are constant quantities for the given field and given temperature. Thus, for \(\varepsilon_p>\varepsilon_{1/2}\), disruption of the stationary regime of electrical conduction is impossible.

For \(\varepsilon_p<\varepsilon_{1/2}\), an ionization event leads on average to the appearance of more than,

\(^1\) The ionizing and ejected electrons after ionization enter the region \(\varepsilon\le I\), since, owing to the rapid decrease of the function \(f^3(\varepsilon,\theta)\) with increasing \(\varepsilon\), the number of electrons with energies \(\varepsilon\ge 3I\) is negligibly small.

of one electron capable of being accelerated by the field and causing new ionization (Fig. 16). In this case the electrons are retarded by the lattice and have an opportunity to recombine with, on the average, fewer than \(N_i\) electrons, and recombination cannot compensate ionization. The flux of electrons through the boundary \(\varepsilon = I\), and the number of ionizations, continuously increase and acquire a chain character. Consequently, in this case a stationary state of electrical conductivity is impossible.

From consideration of these two cases it follows that the condition for the breakdown of electric strength is the fulfillment of the equality

\[ \varepsilon_{1/2}(E_{np})=\varepsilon_p(E_{np}). \tag{81} \]

The combination of equations (75), (78), (79), and (81) makes it possible to determine the value of the breakdown field strength of the crystal. Let us apply the results obtained to alkali-halide crystals, for which the most detailed experimental material on electrical breakdown is available.

Fig. 1a and b. One half of the electrons produced as a result of ionization has energy \(\varepsilon>\varepsilon_{1/2}\), and the other half has energy \(\varepsilon<\varepsilon_{1/2}\). Electrons with energy \(\varepsilon>\varepsilon_p\) are accelerated; electrons with energy \(\varepsilon<\varepsilon_p\) are retarded.

With the exception of the parameter \(S_0\), which enters into the expression for the effective ionization cross section, all quantities in equations (75), (78), (79) can be calculated from formulas (24), (25), (32), (35), (38), (45), (52), (80) by substituting into these formulas the experimental values of the reduced mass \(M\), the lattice constant \(a\), the dielectric constant \(\varkappa\), the square of the refractive index of light \(\varkappa_0\), the residual-radiation frequency \(\omega_t\), and the ionization potential \(I\) (Table 1\(^1\)).

Knowing the experimental values of the breakdown field strength at room temperature, we can determine from the above equations the constant \(S_0\) and the effective ionization cross section. The results of the calculations are given in Table 1. From Table 1 it is seen that the effective ionization cross sections for alkali-halide crystals are close to one another and are, in order of magnitude, \(10^{-16}\ \mathrm{cm}^2\). We note that this value is very close to the values of the effective ionization cross sections in gases at the same values of \(\varepsilon\) [12] as are indicated in Table 1. Therefore the value obtained by us for the effective ionization cross section appears quite reasonable. The results of the theoretical calculation appear still more encouraging if one takes into account that, when \(S_0\) is increased or decreased by an order of magnitude from the value indicated in Table 1, the magnitude \(E_{np}\) increases or decreases, respectively, by less than a factor of 2 in comparison with its experimental value. Consequently, the theory gives the correct order of magnitude of the breakdown field strength for alkali-halide crystals when the effective ionization cross section lies in the interval \(10^{-15}\)–\(10^{-17}\ \mathrm{cm}^2\). The calculation shows that the breakdown field strength increases with increasing effective ionization cross section. Physically this is clear. Indeed, the larger the effective ionization cross section, the more sharply, with increasing \(\varepsilon\), does the electron distribution function decrease in the region \(\varepsilon \gg I\) (see (73)); the smaller is the mean energy of the electrons after ionization, the

\(^1\) The values of \(\dfrac{h}{2\pi}\omega_t\), \(E_{np}\) were taken from [11], and the values \(\varkappa\), \(\varkappa_0\) from [4] (p. 21). The values of the ionization potentials \(I\) of the crystals NaCl, KCl, RbCl, NaBr, KBr, RbBr were determined from the position of the second maximum of the absorption band [4] (p. 116), while the values of the ionization potentials of the crystals LiF, NaF, KF, NaI, KI, RbI were determined from the position of the first maximum of the absorption band [1].

Table 1

Theoretical values of the constant \(S_0\) and of the effective ionization cross section for alkali-halide crystals

Crystals	\(M\), in \(g \cdot 10^{24}\)	\(a\), in Å	\(z\)	\(z_0\)	\(\dfrac{h}{2\pi}\omega_t\), in eV	\(\varepsilon''\), in eV	\(I\), in eV	\(N_1=\dfrac{1}{2a^3}\), in \(cm^{-3}\cdot 10^{-22}\)	\(E_{np}\left(\dfrac{\text{experiment at }T=300^\circ K}{\,}\right)\), in \(\dfrac{V}{cm}\cdot 10^{-6}\)	\(\varepsilon_{1/2}(E_{np})=\varepsilon_p(E_{np})\)	\(S_0\), in \(\dfrac{sec^2}{g}\)	\(Q(\varepsilon)\cdot 10^{16}\) at \(\varepsilon-I=50\) eV
LiF	10.15	2.008	9.27	1.92	0.072	3.59	11.5	6.18	3.1	2.05	\(0.43\cdot 10^{-6}\)	0.34
NaF	17.26	2.312	6.0	1.74	0.033	2.71	9.2	4.05	2.4	1.38	\(1.12\ 10^{-6}\)	0.90
KF	21.23	2.669	6.05	1.85	0.03	2.03	10.4	2.63	1.8	1.13	\(1.91\cdot 10^{-6}\)	1.53
NaCl	23.2	2.81	5.62	2.25	0.024	1.83	9.6	2.25	1.5	1.0	\(2.4\ 10^{-6}\)	1.92
KCl	30.9	3.14	4.68	2.13	0.019	1.47	9.4	1.61	1.0	1.03	\(2.1\cdot 10^{-6}\)	1.68
RbCl	41.7	3.27	5.0	2.19	0.017	1.35	8.9	1.43	0.83	0.94	\(2.36\cdot 10^{-6}\)	1.89
NaBr	29.7	2.97	5.99	2.62	0.018	1.64	8.5	1.91	0.81	1.8	\(0.47\cdot 10^{-6}\)	0.38
KBr	43.5	3.29	4.78	2.33	0.015	1.34	8.4	1.40	0.7	1.15	\(1.36\cdot 10^{-6}\)	1.09
RbBr	68.9	3.42	5.0	2.33	0.011	1.24	7.9	1.25	0.63	0.92	\(2.13\cdot 10^{-6}\)	1.71
NaJ	32.32	3.225	6.60	2.91	0.016	1.39	5.37	1.49	0.7	1.58	\(0.67\cdot 10^{-6}\)	0.54
KJ	49.63	3.522	4.94	2.69	0.013	1.17	5.6	1.14	0.6	1.06	\(1.68\cdot 10^{-6}\)	1.35
RbJ	84.66	3.664	5.0	2.63	0.0105	1.08	5.5	1.02	0.5	0.76	\(3.03\cdot 10^{-6}\)	2.43

more difficult to accelerate anew the electrons produced as a result of ionization; hence the higher is \(E_{\mathrm{br}}\). However, as was said above, the dependence of \(E_{\mathrm{br}}\) on the effective ionization cross section is very weak. The latter circumstance is confirmed experimentally by the fact that the most diverse dielectrics, consisting of atoms with different effective ionization cross sections, have the same order of magnitude of \(E_{\mathrm{br}}\). The temperature dependence of \(E_{\mathrm{br}}\) for KBr, calculated according to the theory for \(S_0 = 1.36 \cdot 10^{-6}\ \frac{\mathrm{sec}^2}{\mathrm{g}}\) (see Table 1), is given in Fig. 2. The temperature dependence of \(E_{\mathrm{br}}\) has the same character for other alkali-halide crystals as well. In the same Fig. 2 the temperature dependence of \(E_{\mathrm{br}}\) for KBr is given, obtained by Hippel and Alger \([13]\) experimentally in tests in the pulsed regime with a duration of \(10^{-6}\ \mathrm{sec}\). As is seen from Fig. 2, the theory agrees quite satisfactorily with the experimental data obtained on pulses of duration \(10^{-6}\ \mathrm{sec}\), when electrical breakdown is not complicated by side effects of the type of heating and high-voltage polarization. The dependence of the breakdown field strength on the lattice parameters is characterized by the following data.

For example, the value of \(E_{\mathrm{br}}\) for KBr increases by a factor of 1.64 when the reduced mass \(M\) is decreased by a factor of two, increases by a factor of 6.14 when the lattice constant is decreased by a factor of two, and decreases by a factor of 2.3 when the lattice vibration frequency is increased by a factor of 3. Such a dependence of \(E_{\mathrm{br}}\) on the lattice parameters is quite physically understandable. Indeed, the smaller \(M\), the more energy the electron gives to the lattice at each collision, the more difficult it is to accelerate it to the ionization potential, and the higher is \(E_{\mathrm{br}}\); the smaller \(a\), the shorter the mean free path, and again the more difficult it is to accelerate the electron to the ionization potential, so the higher is \(E_{\mathrm{br}}\); the larger \(\omega\), the smaller the number of vibrations excited at a given temperature, the weaker the scattering of electrons by lattice vibrations, the more easily the electron is accelerated, and the lower, consequently, is \(E_{\mathrm{br}}\).

Fig. 2. Experimental and theoretical curves of the temperature dependence of \(E_{\mathrm{br}}\).

1—experimental; 2—theoretical.

The dependence of the breakdown field strength on the ionization potential in our case is difficult to determine, since we do not know the dependence of \(S_0\) on \(I\). If, however, one assumes that \(S_0\) does not depend on \(I\), then the dependence of \(E_{\mathrm{br}}\) on \(I\) is practically absent.

CONCLUSION

In the present work an attempt has been made to construct a theory of the breakdown of the electrical strength of ionic crystals on the basis of the integro-differential Boltzmann equation. A criterion for the breakdown of electrical strength has been obtained.

The theoretical values of the breakdown field strength obtained with the aid of this criterion are in satisfactory agreement with experi-

then, for reasonable values of the effective ionization cross section. The theory gives a temperature dependence of \(E_{np}\) that agrees with experiment and makes it possible to draw conclusions about the dependence of \(E_{np}\) on various lattice parameters. The theory set forth above has the following limits of applicability:

1. The electric field must be homogeneous.
2. The thickness of the dielectric \(L\) must be much greater than the length \(l^*\), after traversing which the electron acquires the energy necessary for ionization. This is required for the formation of a sufficiently powerful electron avalanche capable of causing destruction of the material. The length \(l^*\) can be determined from the following considerations.

From formulas (44) and (50), which may be rewritten in the form

\[ f_1^1=-eE l_0^1(\varepsilon)\frac{df_0^1}{d\varepsilon},\qquad f_1^2=-eE l_0^2(\varepsilon)\frac{df_0^2}{d\varepsilon}, \tag{82} \]

it follows that the electron mean free path is

\[ l_0(\varepsilon)=\frac{\sqrt{\dfrac{2}{m^*}\varepsilon}}{r_1}\quad \text{for } \varepsilon<\varepsilon'',\qquad l_0(\varepsilon)=\frac{\sqrt{\dfrac{2}{m^*}\varepsilon^2}}{r_2}\quad \text{for } \varepsilon''<\varepsilon<I. \tag{83} \]

Averaged over the interval from \(\varepsilon_1\) to \(\varepsilon_2\), the mean free path assumes the value

\[ \overline{l_0}(\varepsilon_1,\varepsilon_2)= \frac{\displaystyle\int_{\varepsilon_1}^{\varepsilon_2} l_0(\varepsilon)\,d\varepsilon} {\varepsilon_2-\varepsilon_1}. \tag{84} \]

The probability that an electron will traverse without collisions a length \(l\), different from \(\overline{l_0}\), is equal to

\[ W=e^{-\frac{l}{\overline{l_0}}}. \tag{85} \]

Moving without collisions in the field \(E\), the electron acquires energy equal to the difference \(\varepsilon_2-\varepsilon_1\) over the length

\[ l_1(\varepsilon_2,\varepsilon_1)=\frac{\varepsilon_2-\varepsilon_1}{|e|E}. \tag{86} \]

From (85) and (86) it follows that, taking collisions into account, the electron will acquire the same energy over the length

\[ l_2(\varepsilon_2,\varepsilon_1)=l_1 e^{\frac{l_1}{\overline{l_0}}}. \tag{87} \]

It was shown above that, when electric strength is lost, the majority of electrons perform ionization at an energy \(\varepsilon\simeq I+2\varepsilon_p\), and after ionization have energy \(\varepsilon\simeq\varepsilon_p=\varepsilon_{1/2}\). Consequently, the length \(l^*\), after traversing which under conditions of loss of electric strength the electron acquires the energy necessary for ionization, is equal to

\[ l^*=l_2(I+2\varepsilon_p,\varepsilon_p)\simeq \frac{I+\varepsilon_p}{|e|E_{np}}\, \exp\left[ -\frac{3r_2\sqrt{\dfrac{m^*}{2}(I+\varepsilon_p)^2}} {(I+2\varepsilon_p)^3|e|E_{np}} \right], \tag{88} \]

which gives for NaCl (Table 1) the value \(l^* \simeq 10^{-4}\) cm. Consequently, the theory is applicable for dielectric thicknesses \(L \gg 10^{-4}\) cm. For values of \(L\) comparable with \(l^*\), an increase in the breakdown voltage with decreasing \(L\), i.e., strengthening of the dielectrics, should apparently be observed. Indeed, experimentally, strengthening of dielectrics is observed at \(L \simeq 10^{-3}—10^{-4}\) cm [14].

1. The time \(t\) for which the voltage is applied must be no less than the time for propagation of the electron avalanche from one electrode to the other. The propagation time of the electron avalanche \(t_\lambda\) can be determined from the formula

\[ t_\lambda = t_0 e^{\frac{t_0}{\bar{t}^*}}, \tag{89} \]

where

\[ t_0 = \sqrt{\frac{2m^*L}{|e|E_{pr}}} \tag{90} \]

is the time required for an electron in the field \(E = E_{pr}\) to traverse the path \(L\) in the absence of collisions,

\[ t^* = \bar{\tau} e^{-\frac{t_u}{\bar{\tau}}} \tag{91} \]

is the time between two ionizing collisions when scattering by lattice vibrations is taken into account,

\[ t_u = \frac{\sqrt{2(I+2\varepsilon_p)m^*}}{|e|E_{pr}} \tag{92} \]

is the time between two ionizing collisions without taking account of scattering by lattice vibrations,

\[ \bar{\tau} \simeq \frac{2(I+2\varepsilon_p)^{5/2}}{5r_2(I+\varepsilon_p)} \tag{93} \]

is the mean time (in the interval from \(\varepsilon_p\) to \(I+2\varepsilon_p\)) between two collisions with lattice vibrations. For \(L \simeq 0.1\) cm for NaCl we obtain \(t_\lambda = 1.56 \cdot 10^{-10}\) sec. For \(t < t_\lambda\), an increase in breakdown voltage with decreasing \(t\) should be observed\(^1\)). Experimentally, a rise in breakdown voltage with decreasing \(t\) is observed at \(t = 10^{-8}\) sec [13]. The discrepancy between the theoretical estimate and experiment in this case may be explained as follows. First, for the completion of breakdown not one avalanche but several may be required, which will naturally lead to bringing the theoretical estimate closer to experiment. Second, the magnitude of \(t_\lambda\) depends very substantially on \(\bar{\tau}\). Suffice it to say that if, instead of \(\bar{\tau} = 1.43 \cdot 10^{-14}\) sec, obtained from formula (93), one takes \(\bar{\tau} = 2 \cdot 10^{-14}\) sec, then for NaCl at \(L \simeq 0.1\) cm one obtains \(t_\lambda = 1.48 \cdot 10^{-7}\) sec. Consequently, a slight error in the estimate of \(\bar{\tau}\) leads to a very considerable change in the value of \(t_\lambda\). Taking this into account, one may regard the theoretical estimate of \(t_\lambda\) as satisfactorily agreeing with experiment.

When the above three conditions are fulfilled, the breakdown voltage coincides with that field strength at which the disruption of electrical strength occurs, i.e., the disruption of the stationary

\(^1\)) Experiments carried out by E. A. Konorova at the P. N. Lebedev Physical Institute of the Academy of Sciences of the USSR showed that the breakdown voltage of glass increases by approximately a factor of 1.5 when \(L\) is changed from \(10^{-3}\) to \(10^{-4}\) cm.

of the electrical-conduction regime, while the criterion for breakdown is the criterion formulated above for the loss of electric strength. For sufficiently small dielectric thicknesses (comparable with \(l^*\)) and sufficiently short times of voltage application (less than \(10^{-8}\) sec), an electron avalanche powerful enough to cause destruction of the dielectric may fail to arise even when the condition for loss of electric strength is fulfilled. In this case, destruction of the dielectric may require a somewhat larger field than that at which the loss of electric strength occurs. Consequently, for small dielectric thicknesses and short times of voltage application, the breakdown criterion may be different from the criterion for loss of electric strength obtained above. The theory set forth above does not give a quantitative description of the physical processes occurring in the second stage of breakdown—the destruction of the dielectric. At the same time, a number of qualitative statements about the course of the second stage of breakdown can be made here. It seems to us that destruction of the dielectric is inseparably connected with the preceding process of loss of electric strength and with the conditions for the passage of an electron avalanche in a solid. One may suppose that the conditions for the passage of an electron avalanche in solids are very similar to those that occur in the breakdown of gases at high pressures [15] (it is known that compressed gases and solids have the same order of magnitude of breakdown field strength; for example, Table 1 in [1]).

An electron avalanche advancing from the cathode to the anode leaves behind it a positive space charge of immobile ions which, increasing in magnitude simultaneously with the growth of the electron avalanche, ultimately halts its further development. However, this same positive space charge, concentrated mainly near the head of the retarded electron avalanche, promotes the formation at the cathode of a new electron avalanche (the field created by the space charge is added to the external field in the direction of the cathode, and subtracted from it in the direction of the anode). The second avalanche, approaching the positive space charge from the cathode side and reaching the dimensions of the first avalanche, compensates it and enables the first electron avalanche to advance farther toward the anode. At the same time, from this moment on the dimensions of the first electron avalanche moving toward the anode remain unchanged because of the retarding action of the positive space charge continuously “pursuing” it.

The second electron avalanche, in turn, leaves behind it a positive space charge, which is compensated by a third avalanche, and so on. This process leads to the formation of a highly conducting channel growing both in the direction of the anode and in the direction of the cathode. The transverse cross section of the channel is determined by the dimensions of the head of the first electron avalanche moving toward the anode.

When the highly conducting channel reaches the anode and the cathode, a short circuit takes place, as a result of which there occurs a rapid transition of the dielectric into a highly conducting state, accompanied by the release of large amounts of energy in small volumes and over short times. One may think that the true cause of destruction of the dielectric is connected precisely with these processes, which lead to the appearance of a shock wave, analogous to what occurs in the breakdown of gases at high pressures. The shock wave “blows out” portions of the dielectric where the bonds between atoms (ions) have been weakened as a result of the passage of the electron avalanche, so that after breakdown a narrow channel is formed, penetrating the entire thickness of the dielectric from one electrode to the other.

Some authors (for example, Franz [16]) believe that in the final (second) stage of electrical breakdown of solid dielectrics the crystal is ordinarily melted by the electron current. According to Franz, electrical breakdown occurs when the electron current heats the crystal to the melting temperature.

In view of the fact that the final stage of electrical breakdown of solid dielectrics has not been studied at all, it is difficult to give preference to one mechanism or another. We shall note only that Franz’s work contains a number of inaccuracies. First, in Franz’s treatment the change in the distribution function due to scattering by lattice vibrations is expressed through the symmetric part of the distribution function and its derivative.

In a correct calculation, the symmetric part of the distribution function itself should not enter the expression for the change in the distribution function due to collisions with lattice vibrations (only its derivatives enter; see (31), (37)). Consequently, there is reason to think that Franz’s calculation of the electron distribution function was carried out incorrectly, which may have a very substantial effect on all the results. Moreover, if the breakdown criterion is based on its final stage, as Franz did, then the theory must deal with that part of the dielectric volume which is enclosed in the breakdown channel, explain the formation of this channel and its dimensions. This is essential, since in breakdown only an insignificant part of the total volume of the dielectric, having quite definite dimensions, is destroyed. Franz’s theory does not take into account the finite dimensions of the breakdown channel. This circumstance may also lead to errors.

In conclusion, we note that for the further development of the theory of electrical breakdown it is necessary to calculate the effective cross section for ionization by electron impact in a solid, to calculate the effective recombination cross section, and to take into account the scattering of slow electrons by lattice vibrations. Thus, the further development of the theory of electrical breakdown will require the solution of a number of general problems in solid-state physics.

I take this opportunity to express my gratitude to E. I. Adirovich for a number of suggestions and for discussion of the results.

REFERENCES

1. V. A. Chuenkov. UFN, LIV, 185, 1954.
2. B. M. Vul. ZhTF, 2, 1, 1932.
3. F. Seitz. The Modern Theory of Solids, 1949.
4. Mott and Gurney. Electronic Processes in Ionic Crystals, pp. 116–117, 1950.
5. N. A. Kaptsov. Electrical Phenomena in Gases and Vacuum, pp. 195–196, 1947.
6. D. I. Blokhintsev. ZhETF, 6, 1053, 1936.
7. E. I. Adirovich. Some Problems in the Theory of Luminescence of Crystals, p. 297, 1951.
8. M. Born and M. Göppert-Mayer. Theory of Solids, 1938.
9. L. Landau and E. Lifshitz. Quantum Mechanics, Part I, pp. 496–497, 1948.
10. V. A. Chuenkov. Dissertation, FIAN, Chapter II, 1953.
11. A. A. Vorob’ev. ZhTF, 10, 1183, 1940.
12. Taro Kinara. Rev. of Modern Physics, 24, 45, 1952.
13. A. Hippel and P. S. Alger. Phys. Rev., 76, 127, 1949.
14. A. E. W. Austen and S. Whitehead. Proc. Roy. Soc., A 176, 44, 1940.
15. L. Loeb. Basic Processes of Electrical Discharges in Gases, 1950.
16. W. Franz. Ergebn. Exakt. Naturwiss., 27, 1–55, 1953.

Moscow, P. N. Lebedev Physical Institute
Academy of Sciences of the USSR.

Errata

Page	Line	Printed	Should read
5	15 from top	Zinerman	Zingerman
9	2 from bottom	breakdown voltage	breakdown field strength
13	4 from top	$\varepsilon_m \simeq 4\,\dfrac{h}{2\pi}$	$\varepsilon \simeq 4\,\dfrac{h}{2\pi}\,\omega$
24	caption under figure	Fig.	Fig. 9.
50	14 from top	$\dfrac{1}{M_L^{ak}}=\left\{\dfrac{1}{M}$	$\dfrac{1}{M_L^{ak}}=\left\{\dfrac{1}{M_-}$
53	10 from top	$\dfrac{df^{1/2}}{d\varepsilon}$	$\dfrac{df^{1/2}}{d\varepsilon}$
57	5 from top	$\left(1-\sqrt{1-\dfrac{1}{1.3\,\varepsilon}}\right)e^{-\frac{q}{E}(\varepsilon-I)^2}$	$\left(1-\sqrt{1-\dfrac{1}{1.3\,\varepsilon}}\right)e^{\frac{q}{E}(\varepsilon-I)^2}$
64	Equation (87)	$l_2(\varepsilon_2,\varepsilon_1)=l_1 e^{-\frac{1}{l_0}}$	$l_2(\varepsilon_2,\varepsilon_1)=l_1 e^{-\frac{l_1}{l_0}}$
65	18 from bottom	$I \div 2\varepsilon_p$	$I+2\varepsilon_p$
85	7 from bottom	$10^{14}\cdot\dfrac{\pi\,0.52\cdot10^{-2}}{4}\cdot10^{-}$	$10^{14}\cdot\dfrac{\pi\,0.52\cdot10^{-2}}{4}\,10^{-2}$
111	2 from top	production	conductivity
111	8 and 10 from top	activation	activation
120	5 from top	strength of chemical	strength from chemical
135	literature 6	p. ...	p. 113
147	11 from bottom	$\varepsilon=1000$	$\varepsilon\simeq1000$
148	caption to Fig. 4	$i_{\mathrm{breakd.}}\, i_{\mathrm{test.}}$	$i_{\mathrm{breakd.}}/i_{\mathrm{test.}}$
204	Fig. 1	KBr	RbBr
253	caption to Fig. 6	$2-f=4\cdot10^3$ Hz; $3-f=10^2$ Hz	$2-f=4\cdot10^2$ Hz; $3-f=10^3$ Hz
273	7 from top	KCJ	KCl
280	caption to Fig. 4	$210^5$	$2\cdot10^5$
294	1 from bottom	from 10	from $10^{-4}$ at $f=2\cdot10^7$ to $5\,10$
282		table	table 2
324	3 from top	p. 269	p. 251
336	16 from bottom	[3,4]	[3]
337	10 from top	[5]	[4]
337	4 from bottom	[6]	[5]
337	1 from bottom	[7]	—
338	2 from top	[8]	[6]
338	4 and 5 from top	[9,10]	[7,8]
339	8 from bottom	[11,12]	[9,10]
340	14 from top	[13]	[11]
347	23 from bottom	[14]	[12]
347	3,4 from bottom	[15]	—
348	17 from bottom	[16]	[13]
359	4 from top	$\psi_e[M]x=\mathrm{Const}$	$\psi_e[M]x=\mathrm{Const}$
451	1 from bottom	mechanical	technical