ULTRAFINE METAL PARTICLES INSIDE A CRYSTAL LATTICE

M. V. Savostyanova, Leningrad

III. MECHANISM OF FORMATION OF METAL PARTICLES IN A CRYSTAL LATTICE1

§ 6. Transition of different states of distribution into one another

Metal particles that are in one or another state inside a crystal lattice may, under the action of various factors, undergo diverse changes.

Restricting our consideration of the process of formation of metal particles exclusively to a crystalline medium, we can indicate in advance which external factors exert the strongest influence on the coloring centers embedded in the lattice.

The connection of the centers with the lattice is effected by electrons; therefore, those external factors will act on an atomic center which are capable of disrupting the bond of the electron with the center and with the lattice.

Such action will be exerted, first, by a quantum of light, by the impact of which the electron may be temporarily or completely torn away from the ion, and, second, by the thermal vibrations of the lattice. In the latter case, reversible as well as irreversible processes may also occur. The reversible change in absorption (ionic and atomic) with a change in temperature was discussed above. Of interest to us is the irreversible case of complete detachment of the electron.

Deformation of the lattice can also play an essential role in transition processes.

Let us consider separately the influence of the above-mentioned factors on the ionic (I), atomic (A), and colloidal (K) distributions. We shall judge the presence of a transition from one phase to another, again, by the change in the optical properties of the crystal, chiefly by the absorption spectrum.

a) Transition \(I \to A\) from ionic distribution to atomic.

As was indicated above, this transition occurs for alkali-halide salts, pure and with an admixture of heavy metals (phosphors

centers) under the action of light and under the action of an electric field (with the introduction of external electrons). In one case the transition \(I \to A\) is brought about by heating (the thermal method); this occurs for crystals containing hydrogen ions. When such a crystal is heated above \(400^\circ\), \(F\)-centers appear and, at the same time, the absorption band caused by hydrogen ions (the \(U\)-band) diminishes. The \(U\)-centers are converted into \(F\)-centers (in this case they are called \(F_U\)-centers).

Let us consider some quantitative relations accompanying the formation of atomic centers.

The thermal equilibrium between \(U\)- and \(F_U\)-centers is governed by the relation

\[ \frac{N_F}{N_U}=Ae^{-\frac{E'}{kT}}, \]

where \(E'\) is of the order of \(1.1\ \mathrm{eV}\).

If this process is regarded as a bimolecular reaction, then \(E'\) is half the heat of reaction (for more detail see \(^{76}\)).

The formation of \(F\)-centers under the action of light (the photochemical method) is characterized by the quantum yield and by the limiting concentration of color centers \(\eta\). The quantum yield depends strongly on temperature; see, for example, Fig. 22, which corresponds to the photochemical transition from \(U\)-centers (hydrogen ions) to \(F_U\)-centers for a KBr crystal. This dependence is of very great importance for elucidating the mechanism of liberation of color centers.

The limiting value of the quantum yield in this process is of the order of 1, as illustrated by Fig. 23, which shows the formation of \(F_U\)-centers from \(U\)-centers. The shaded area of the \(F_U\)-band is equal to the shaded area of the \(U\)-band at whose expense it grows; the ratio

Fig. 22. Quantum yield of the formation of \(F\)-centers from \(U\)-centers at different temperatures

Fig. 23

concentration of \(U\)- and \(F\nu\)-centers (\(N_U\) and \(N_F\)) within the limits of measurement accuracy is equal to unity.

The number of centers released under the action of light is proportional to the number of absorbed quanta only in the first seconds; a state of “saturation” sets in very rapidly. The limiting concentration increases as the quality of the crystal deteriorates (for example, under deformation; see also the works of Brilliantov and Morgenstern\({}^{4}\)). Usually the limiting concentration \(N\) is of the order of \(10^{13}\)—\(10^{15}\).

Fig. 24

In Fig. 24 are brought together averaged values of \(\eta\) as a function of temperature for artificial crystals of KCl and NaCl according to the data of Scheitberger\({}^{57}\) (the latter curve corresponds to Fig. 22) and of a natural NaCl crystal. These curves can be represented by an interpolation formula of the form

\[ \eta = 1 - \left[1 - e^{-\frac{E}{kT}}\right]^A \sim A e^{-\frac{E}{kT}} + \ldots \]

where the values of \(A\) and \(E\) for different salts are as follows:

\[ \begin{aligned} \mathrm{KBr}\quad & A = 7, & E &= 0.085\,\mathrm{eV}^{24} \\ \mathrm{KCl}\quad & A = 4.4, & E &= 0.125\,\mathrm{eV}^{24} \\ \text{Rock salt}\quad & A = 5.5, & E &= 0.119\,\mathrm{eV}^{24} \\ \mathrm{NaCl}\quad & A = 6.3, & E &= 0.129\,\mathrm{eV}^{24} \end{aligned} \]

The question of the limiting concentration is most closely connected with the question of the lifetime of \(F\)-centers. Some data in this direction were obtained by Gilsch\({}^{20}\) on crystals with \(U\)-centers. The thermal equilibrium that has just been discussed can be disturbed by illumination from the region of the \(U\)-band. The additional centers, however, are not stable, and their concentration rapidly falls to the equilibrium value. In Fig. 25 a series of curves is presented (on a logarithmic scale) for the decrease of the concentration of \(F\)-centers with time, beginning from a concentration \(\sim 10^{15}\). At high temperatures the dependence is expressed by an exponential curve; at low temperatures deviations are observed, the cause of which is still unclear. If by \(\frac{1}{\alpha}\) we denote the lifetime of an \(F\)-center, i.e., the time during which the concentration falls to \(\frac{1}{e}\) of its initial value, then

\[ \alpha = S e^{-\frac{1\,\mathrm{eV}}{kT}}. \]

When a crystal is colored by the electrochemical (additive) method, there is likewise a limiting concentration of coloring centers. Mollwo[^36] and Regener[^53], having improved Roze’s method of coloring crystals in vapors, were able to vary the temperature of the object independently of the temperature, and consequently also of the pressure, of the surrounding vapor. By applying such a technique it was possible to establish that the number of \(F\)-centers \(N_F\) formed in \(1\ \mathrm{cm}^3\) is proportional to the number of metal atoms in the vapor surrounding the crystal, \(N\). The coefficient of proportionality

\[ a=\frac{N_F}{N} \]

is always greater than unity and decreases as the temperature is raised (Fig. 26); in other words, at high temperatures the limiting concentration approaches the concentration of atoms in the vapor (of the order of \(10^{18}\)).1 Extrapolating the curves of Fig. 26, Pohl[^46] obtains, at \(t=20^\circ\), \(a_{\mathrm{KBr}}=3.6\cdot10^4\) and \(a_{\mathrm{KCl}}=9\cdot10^4\); also extrapolating the vapor-pressure curve for K \((N=7.5\cdot10^8)\), he obtains for the equilibrium concentration at \(t=20^\circ\).2

Fig. 25

Fig. 26

$$ C_{\mathrm{KBr}} = 3 \cdot 10^{13}, \quad C_{\mathrm{KCl}} = 7 \cdot 10^{10}. $$

The concentration values here are of the same order as in photochemical coloration.

Let us now consider the course of the process $I \to A$ in phosphors.

The scanty material concerning the liberation of color centers has at present been obtained only for silver phosphors. As already mentioned, in these objects absorption of the “atomic” type is observed in the region around $300 \ \mathrm{m}\mu$.

The bands arise both upon the introduction of external electrons$^{68,69}$ and upon illumination of the phosphor with ultraviolet rays.

When atomic centers are liberated in phosphors by the electrochemical method, a limiting concentration is observed. However, in contrast to pure alkali-halide salts, the number of atomic silver centers at one and the same concentration of Ag in the melt (from which the phosphor is prepared) does not depend on the temperature at which electrons are introduced into the crystal$^{69}$ (Table 4).

Table 4

Concentration in the melt	Temperature of the experiment, °C	Concentration of atomic centers $\times 10^{-16}$
KBr + 0.001 mol. % Ag	550	2.6
KBr + 0.001 mol. % Ag	640	2.55
KBr + 0.001 mol. % Ag	720	2.46
KBr + 0.002 mol. % Ag	500	3.84
KBr + 0.002 mol. % Ag	600	3.60
KBr + 0.002 mol. % Ag	720	3.36

At large concentrations of the foreign metal, both upon illumination with ultraviolet rays and upon the introduction of external electrons, colloidal centers are formed along with atomic ones$^{1}$). The limiting concentration of silver in potassium salts is of the order of 0.001–0.002 mol. % Ag in the melt ($2.5$–$3.5 \cdot 10^{16}$ atomic centers in $1 \ \mathrm{cm}^{3}$ of crystal); for sodium salts the limiting concentration is much lower: in the presence of 0.0022 mol. % Ag in the NaCl and NaBr lattices, colloidal particles are formed immediately upon penetration of electrons.

Changes affecting atomic centers lead, on the one hand, to the transition $A \to I$, and on the other to their combination into larger complexes—colloidal ones (transition $A \to K$).

$^{1}$) For this reason, the study of the simultaneous liberation of atomic and colloidal centers is possible only in silver phosphors, in which the colloidal absorption band has the appearance of a narrow maximum. In other metals (Cu, Au, Tl, Pb, etc.) colloidal absorption occurs chiefly in the ultraviolet region.

b) Transition \(A \to I\)

\(F\)-centers obtained by different methods (thermal, photochemical, and electrochemical) behave differently in the transition \(A \to I\).

\(F_U\)-centers formed by the thermal method begin to transform into \(U\)-centers if, at a given temperature, the concentration exceeds the value of the equilibrium concentration.

\(F\)-centers obtained photochemically, upon illumination of the crystal by ultraviolet rays from the region of intrinsic absorption, by \(\gamma\)-rays, or by X-rays, are very unstable: they are destroyed under the action of: 1) heating, 2) light (from the region of the \(F\)-band).

The conditions of decay and the lifetime of \(F\)-centers have not been studied in detail. It is known only that the quality of the crystal is of great importance; thus, natural crystals of rock salt are bleached at \(\sim 100^\circ\), whereas in artificially colored objects (KCl, KBr, etc.) the coloration disappears already at room temperature within several minutes.

Atomic centers obtained by introducing external electrons (the additive method), on the contrary, are very stable: crystals with such centers are bleached with great difficulty and only at a sufficiently high (\(>600^\circ\)) temperature. Bleaching is accelerated under the action of light and occurs very rapidly if the heated crystal is placed in an electric field (see Part I).

The same may be said about the behavior of the \(F\)-centers of phosphors, with the difference that in an electric field the centers are immobile.

We may dwell in somewhat greater detail on the action of light on \(F\)-centers. Here, generally speaking, two processes take place: bleaching and the appearance of new centers with a new absorption band \(F'\), partly superposed on the \(F\)-band. The change in the absorption curve in this case for KCl is shown in Fig. 27. The initial curve \(A\) is resolved into two: \(F\) (lowered, since part of the \(F\)-centers has been irreversibly knocked out) and \(B\) (belonging to the \(F'\)-centers). At ordinary (room) temperatures, bleaching under the action of light is observed only in crystals colored photochemically. Additively colored crystals are bleached only upon heating.

The action of light on \(F\)-centers is accompanied by luminescence and by an internal photoeffect.

The first process is of decisive importance in elucidating the mechanism of light absorption in \(F\)-centers; a whole series of works is devoted to it \(^{29, 7, 8, 28, 45}\).

The second process, the presence of which testifies to the appearance of free electrons inside the crystal, is more interesting in the study of the mechanism of formation of the solid phase.

A very detailed exposition of the works in this field can be found in Tartakovsky’s book \(^{67}\) on photoconductivity; here we shall recall only certain facts needed for the subsequent exposition.

If an additional velocity in a definite direction is imparted by means of an electric field to the photoelectrons produced in the crystal under the action of light, then an instrument connected in series with the crystal will indicate a current.

Fig. 28 shows the variation with time of this current in the crystal at different temperatures (the left-hand part of the figure). At low temperatures the current is quite inertia-free and does not change with time.

An electron torn away by light may, on its path, be detained at one or another point of the crystal; the distance from the ion from which it was torn away to its first detention, calculated per unit field, i.e. the path, we shall call, following Pohl, \(w\) (displacement, Schubweg).

Fig. 27

Quantitatively, the mean displacement is understood to mean the component of the electron trajectory directed toward the anode, along the continuation of which the number of observed electrons (\(F\)-centers) falls to \(1/e\) of its initial value.

It is quite obvious that, in a real lattice, the thermal vibrations of the lattice elements will have a noticeable influence on the paths in the direction of increasing them, and thereby the current will also be increased. Here we can no longer expect complete absence of inertia; this is indeed seen from the curves of Fig. 28 corresponding to higher temperatures. The “thermal” part of the photocurrent is marked by hatching; this current also occurs after the illumination is switched off and may continue for tens of minutes.

The current detected by the electrometer is

\[ i = ne \frac{w}{d}, \tag{1} \]

where \(n\) is the number of electrons transported, \(e\) is their charge, \(w\) is the mean displacement, and \(d\) is the thickness of the crystal. If \(w > d\), we have the saturation current; in this case

\[ i_{\max} = \frac{1}{2}ne. \]

The quantum yield of the internal photoeffect is, by definition, equal to

\[ \eta = \frac{n_{\mathrm{э}}}{n_{\mathrm{кв}}}. \]

Then from (1) we obtain

\[ \eta \frac{w}{\varphi}=\frac{i}{I}\cdot \frac{d}{\varphi}\cdot P, \tag{2} \]

where \(I\) is the energy of the absorbed light in watts, \(P=\frac{h\nu}{e}\) volts is the dimensional factor, and \(\varphi\) is the potential difference.

This expression can be applied to cases I, II and partly to III of Fig. 28 (to those cases in which the current may be considered constant). In case IV one has to use the expression

\[ \eta \frac{w}{\varphi}=\frac{\int i\,dt}{E_{abs}}\cdot \frac{d}{\varphi}\cdot P, \tag{3} \]

where \(E_{abs}\) is the total energy of the absorbed light in watt-seconds.

Fig. 28. NaCl crystal with a concentration of \(F\)-centers \(8\cdot 10^{15}\)

Glaser\(^{12,13}\), in Pohl’s laboratory, carried out a whole series of measurements of the photocurrent on various crystals at different temperatures; from the data for \(i\) and \(I\) (or \(E\)) he calculated the values of \(\eta \frac{w}{\varphi}\) of the current by formula (2), and of the inertial current by formula (3).

Typical curves of the temperature dependence of the quantity \(\eta \frac{w}{\varphi}\) for a KBr crystal are shown in Fig. 29; in the upper part of the figure the type of current variation for the given temperature interval is indicated in accordance with Fig. 28.

For an inertia-free current in the temperature interval near \(-150^\circ\) we have an almost constant value of \(\eta \frac{w}{\varphi}\); lower down there is a sharp fall. Higher, at about the temperature \(-130^\circ\), an increase of the quantity \(\eta \frac{w}{\varphi}\) begins. If it is assumed that in this temperature interval \(\eta\) has a constant value equal to 1 (see below), then the rise of the curve should be attributed to an increase in the magnitude of the shift under the influence of thermal motion\(^{1}\).

The magnitude of these shifts—we shall call them primary and secondary\(^{2}\)—depends on certain external factors, and in different ways.

\(^{1}\) For AgCl, Arsen’eva\(^{1a}\) showed the independence of the shift from temperature down to \(t=-170^\circ\).

\(^{2}\) In Pohl’s terminology: Schubweg and thermisch vergrösserter Weg.

Fig. 29. KBr crystal. Concentration of \(F\)-centers \(3 \cdot 10^{16}\). 1 — coloration in potassium vapor, 2 — photochemical coloration (from \(U\)-centers).

Let us note first of all the presence in the crystal of “impurities” (in the broadest sense of the word), primarily colloidal particles. In Fig. 30, c and d, two curves \(A'\) and \(B'\) are presented, corresponding to one and the same KCl crystal (concentration \(3.8 \cdot 10^{16}\) in case c, and \(16.6 \cdot 10^{16}\) in case d), when it is illuminated

Fig. 30. a, c — KCl. Concentration of \(F\)-centers \(3.8 \cdot 10^{16}\), \(\lambda = 555\ \mathrm{m\mu}\); b, d — KCl. Concentration of \(F\)-centers \(16.6 \cdot 10^{16}\), \(\lambda = 555\ \mathrm{m\mu}\).

under light of wavelength \(\lambda = 555\ m\mu\); in Fig. 30 \(a\) and \(b\) the absorption curves corresponding to these cases are drawn. From the latter figures it is seen that curve \(A\) refers to a crystal with \(F\)-centers; absorption curve \(B\) (maximum \(\sim 850\ m\mu\)) indicates that a considerable part of the \(F\)-centers, after illumination (by a voltaic arc), formed colloidal particles. The magnitudes of the shifts in these two cases are sharply different; both the primary and the secondary shifts in the second case are approximately a hundred times smaller; the colloidal particles act as “traps” for electrons.

The magnitude of the shift depends on the method by which the crystal is colored. In Fig. 29 curves are shown for two crystals, one of them (points 1) colored additively in potassium vapor, and the other—photochemically (from \(U\) centers, points 2). As we see, the method of coloring has no effect at all on the primary shift, but strongly affects the secondary one: for the additively colored crystal the ratio

\[ \frac{w_2}{w_1} \]

is of the order

\[ \frac{10^{-10}}{5\cdot 10^{-13}} \sim 200, \]

whereas for the photochemically colored one it is only \(\sim 10\). The same picture is also observed for crystals colored by X-rays. Hence one may conclude that thermal diffusion is greatly reduced when foreign molecules are present in the crystal (hydrogen ions or KH molecules in the first case, bromine molecules in the second).

Finally, let us note a very substantial fact: the decrease in the magnitude of the shift with increasing concentration of atomic centers. In this case the magnitudes of the primary shift lie quite satisfactorily on a straight line, i.e. the dependence on concentration is linear; for secondary shifts, on the contrary, the expression

\[ \eta \frac{w}{\varphi} \sim c^{-0.5}. \]

fits better.

One should also note the influence of the quality of the crystal. Table 5 gives the maximum values of the shift in a unit field, reduced to a concentration of \(10^{16}\) centers/\(cm^3\) (see Glaser\(^{13}\)).

As we see, rock salt differs from NaCl and other salts. The values corresponding to KI are underestimated. This circumstance, however, is attributed to the fact that in KI crystals it is difficult to obtain atomic centers alone, without colloidal ones.

A few words about \(F'\)-centers. They are also very unstable and are destroyed upon illumination in the region of the \(F'\)-band or upon heating; in some cases they can be observed only at temperatures below \(-50^\circ\). The action of light on \(F'\)-centers is also accompanied by a current (inertial) (the region shaded in the cell in the right-hand part of Fig. 28).

Quantitative measurements in the transition of \(F\)-centers into \(F'\)-centers and back make sense only in additively colored crystals, where reversible bleaching does not occur.

In Fig. 31 a curve is plotted of the relative decrease of \(F\)-centers

\[ \frac{\Delta F}{F} \]

in a KCl crystal with changing temperature (according to data—

Table 5

Crystal	\(\dfrac{w_1}{\varphi}\), maximum, in \(m/V/m\), \(10^{12}\)	\(\dfrac{w_2}{\varphi}\), maximum, in \(m/V/m\), additive	\(\dfrac{w_2}{\varphi}\), maximum, in \(m/V/m\), photochemical
Rock salt	10	—	—
NaCl, artificial crystals	5	—	—
KCl, artificial crystals	2.5	\(5\cdot 10^{-10}\)	\(5\cdot 10^{-12}\)
RbCl, artificial crystals	1	\(1\cdot 10^{-10}\)	—
KBr, artificial crystals	1.5	\(1\cdot 10^{-10}\)	\(5\cdot 10^{-12}\)
RbBr, artificial crystals	2	\(5\cdot 10^{-12}\)	—
KI, artificial crystals	0.6	—	\(6\cdot 10^{-13}\)
Average for artificial crystals	\(2\cdot 10^{-12}\ m/V/m\)	—	—
AgCl \((-170^\circ)^2\)	\(4\cdot 10^{-8}\ m/V/m\)	\(12\cdot 10^{-8}\ m/V/m^1\)
\(\lambda = 546\ m\mu\)	—	—	—
TlBr \((-70^\circ)^2\)	\(2.2\cdot 10^{-8}\)	—
\(\lambda = 546\ m\mu\)	—	—

\(^1\) “Secondary current.”
\(^2\) Given for comparison according to Lehfeldt \(^{32}\).

Fig. 31

Fig. 32

by Pick) \(^{49}\); these data show that: 1) upon absorption of light in the \(F\)-band only a certain fraction of the \(F\)-centers disappears (not more than 75%), 2) the magnitude \(\dfrac{\Delta F}{F}\) depends very strongly on temperature. In Fig. 32 the curve of the quantum yield of the transition \(F \to F'\) for KCl is shown; this curve resembles the curve of the quantum yield upon colora-

linking. It is essential to note that the maximum value of \(\eta\) is equal to 2.

For the quantum yield of the reverse transition \(F \to F\), which occurs alongside the transition \(F \to F'\), there are as yet no data. From the course of the curve in Fig. 31, however, one may conclude that the quantum yield of the transition \(F'—F\) does not depend on temperature, as a result of which, when the temperature is lowered, the equilibrium between \(F\)- and \(F'\)-centers is disturbed in favor of the former1.

c) The \(A—K\) Process

The formation of colloidal particles from atomic ones is the central question of the entire range of phenomena considered by us. Unfortunately, there are very few quantitative data here.

We shall first note some qualitative data (established by ultramicroscopic observations) concerning the character of the precipitation of colloidal particles in the crystal lattice; they also apply to the case described above of the photochemical precipitation of silver in silver halide salts.

A characteristic feature of all cases of precipitation of colloidal metal is the extremely nonuniform distribution of colloidal particles inside the crystal. Especially illustrative results are obtained in silver halide salts, where the precipitation of ultramicrons under the action of light occurs literally “before one’s eyes”; for this reason crystals of such salts are especially suitable for studying the conditions of precipitation of colloidal particles. The general character of this process is illustrated by a series of photographs (Fig. 33)

Fig. 33a                Fig. 33b

Fig. 34. Ultramicroscopic photograph of an exposed crystalline AgCl film. The ultramicrons of silver separated out along cracks. Along the large crack the crystal is optically empty

for AgCl and (Fig. 34) for AgBr. One may note different types of ultramicron separation. In the case of silver-halide crystals, which for ultramicroscopic investigations can be prepared in the form of thin crystalline films[^30], the ultramicrons separate out chiefly on (more exactly, under) the upper and lower surfaces of the crystals. Bulk separation is observed to a much smaller degree. Both in surface and in bulk separation the ultramicrons are located chiefly along cracks and the boundaries of individual subindividuals, near the outer upper surface of the crystalline film (see, for example, Fig. 33a, which shows the surface of a crystalline film with small needle-like crystals of the same salt adhering to it, and Fig. 33b, where the same area after exposure to “active” light is shown). There are also observed “optically empty” regions (Fig. 34), where even under the strongest illumination no separation of ultramicrons is observed.

The magnitude of the colloidal particles formed is just as varied; alongside areas strewn with white (i.e. very large) ultramicrons, we have (in AgCl) bright-green regions (the finest particles; absorption and scattering maxima near \(500\,m\mu\)), within which individual particles are not visible (colored background).

Observations with an ocular Nicol analyzer show[^78] that the particles are in most cases anisotropic; they may be imagined as elongated ellipsoids arranged completely at random.

An uneven distribution of ultramicrons is also observed in rock salt (see, for example, Fig. 37).

The formation of colloidal particles often occurs alongside the formation of atomic centers; sometimes the appearance of the atomic phase cannot be traced.

In pure form such a case may occur for gold phosphors[^35], in which the separation of colloidal particles in a crystal to whose melt a gold salt has been added takes place already at temperatures of \(300^\circ\) and below, without any external influences.

The same phenomenon also takes place when ions penetrate into crystals by diffusion; it is observed, again, in gold and, in some cases, in nickel^47.

The most interesting cases are those of the photochemical separation of colloidal particles when the crystal is illuminated by wavelengths from the region of its intrinsic absorption. In alkali-halide salts such a phenomenon under ordinary conditions is observed only in LiF; according to Schneider^56, a lithium fluoride crystal, under prolonged (35 h) illumination by an intense hydrogen tube, and also under electron bombardment, acquires an orange or red coloration of a typically colloidal appearance. X-rays give only an atomic band at about \(300\,m\mu\).

Silver-halide salts are a particularly important case; we have already spoken of them. An intermediate case is found in phosphors: at small concentrations, in their behavior they approach alkali-halide salts; at large concentrations (and, for gold, at all concentrations), they approach silver-halide salts.

In the process of formation of colloidal particles, temperature is a very important factor.

Silver-halide salts are of special interest for investigation in this respect, since here we have the possibility of carrying out parallel studies on photographic layers.

Lüppo^33 points out that not only “good,” pure AgCl crystals, but also deformed crystals, are not colored at the temperature of liquid air. The separation of coloring centers occurs, however, here as well, if the crystal has first been exposed to a small number of quanta at room temperature. Very interesting, and at the same time quantitative, results were obtained by Webb and Evans^74 for photographic plates, which they cooled in a special apparatus down to the temperature of liquid air. As is known, the photosensitivity of a photographic layer is characterized by the density of blackening of the developed layer. The dependence of the density of blackening on the number of quanta incident on the plate is given by the so-called characteristic curve (Fig. 35). Webb and Evans showed that, when the temperature is lowered, the photosensitivity falls sharply (the characteristic curve is lowered).

If, however, the exposure is made in several stages (for example, 10 or 20 times for 10 or, respectively, 5 sec), warming the plate in the intervals at room temperature, then the sensitivity increases, and with a large number of interruptions approaches that which the plate had at room temperature. Berg and Mendelssohn^2 continued these investigations down to the temperature of liquid helium (\(4^\circ\mathrm{K}\))^2a; it turned out that at temperatures below \(90^\circ\mathrm{K}\) (liquid air) the decrease in sensitivity is very slight. If by sensitivity \(S\) one understands the quantity reciprocal to the exposure necessary to obtain a density of 0.1 above fog, then, according to the data of Berg and Mendelssohn, \(S\) at \(90^\circ\mathrm{K}\) is \(7\%\), and at \(20^\circ\mathrm{K}\)—\(4\%\) of \(S\) at room temperature.

Let us now consider the formation of colloidal particles from atomic centers. The behavior of crystals with atomic centers varies depending on the method by which these centers are obtained. These peculiarities are also manifested in the formation of colloidal particles.

Fig. 35. Emulsion from pure AgBr.

Effect of temperature. Curve A at a development temperature of 20°C; curves B, C, D, E at a temperature of −186°. Development times for A and B: 150 sec.; C—2 times for 80 sec.; D—4 times for 40 sec.; E—8 times for 20 sec. For curves C, D, E the emulsion was heated to 29°C in the intervals between exposures.

Let us begin with additively colored crystals. The process of formation of colloidal particles depends on the temperature at which the crystal containing \(F\)-centers is kept.

\(\alpha)\) The crystal is colored at a high (\(>400^\circ\)) temperature and then cooled. It may be expected that, upon cooling of the crystal, because of the existence of a limiting concentration for each given temperature, supersaturation and formation of colloidal particles will occur. This is indeed observed, and especially sharply in crystals of natural rock salt. The particles obtained in this case are very large: an additively colored rock-salt crystal, on cooling, appears dirty gray (curve I, Fig. 36). It is very difficult to avoid the formation of colloidal particles during cooling of additively colored crystals, by rapid cooling (“quenching”) of the crystal. However, even with this method one can never be sure of the complete absence of colloidal centers. The presence of colloidal particles invisible to the eye can be verified by examining the photocurrent curves for the given object: the rise of the curve at low temperatures (see, for example, Fig. 29), attributed to the photoeffect from colloidal particles, is absent in photochemically colored crystals and exists in additively colored ones\(^{13}\) (at higher temperatures this weak effect is masked by the considerably stronger photoeffect from atomic centers).

\(\beta)\) The crystal is “quenched” and then heated. Upon heating, colloidal particles of one size or another may be obtained depending on the heating temperature and the conditions (rate) of quenching.

Similar experiments were carried out by the author\(^{54}\) for one and the same rock-salt crystal. The results of the experiments are summarized in Table 6 and Fig. 36.

PARTICLES OF METAL WITHIN A CRYSTALLINE LATTICE

Table 6¹)

Crystal \(d = 1.55\) mm.

No. in order	Observation	Heating temperature, °C	Heating time	Position of max. in \(m\mu\)	\(2\rho\)	\(Kd\) (absorption at max.)	Color of crystal	Note
1	207	—	—	638	60	2.91	Blue	Rapid cooling
2	208	450	1 min.	550	5	2.36	Red-violet	Rapid cooling
3	210	300	6 sec.	555	5	10.40	Red-violet	Slow cooling
4	211	300	2 min.	564	10	0.51	Faintly rose	Slow cooling
5	212	400	8 »	577	20	4.11	Cherry-red

The first row of Table 6 and curve \(I\) of Fig. 36 correspond to the crystal immediately after coloring and slow cooling. After the first experiment the crystal contains rather large particles \((2\rho \sim 60m\mu)\). Heating for 1 min. at \(450^\circ\) produced a small quantity of fine colloidal particles \((2\rho \sim 5m\mu)\) [and \(F\)-centers (curve \(II\))]. After the 3rd experiment there formed in the crystal exclusively colloidal particles, arising at the expense of atomic ones. With one more heating at the same temperature these particles almost all disappeared (No. 4), evidently passing into \(U\)-centers, and again appeared, though somewhat coarsened (No. 5), during the final heating to \(400^\circ\) (curve \(III\)).

Fig. 36. Absorption curves for an additively colored NaCl crystal.

In the photographs of Fig. 37 (according to Rexer)⁵² one can see the change in the ultramicroscopic picture in cases of this kind. A quantitative treatment of the process described above, of the transition of atomic centers into colloidal ones and back, is possible only if the presence of hydrogen ions is taken into account; these take upon themselves part of the electrons from the \(F\)-centers; as mentioned above, in additive coloration they are always present.

Let us now consider additively colored phosphors. Here we must likewise reckon with the presence of \(U\)-centers; however, apparently,

¹) The estimate of the particle diameter \(2\rho\) was made by comparing the experimental absorption curves with the theoretical ones (Fig. 17).

they are a poor competitor for heavy ions, whose bond with electrons proves stronger. Therefore here the picture of the transition of atomic centers into colloidal ones appears in a purer form. We cite some data (from unpublished measurements made by I. I. Breĭdo) for phosphorus KJ—0.05 mol. % Ag, colored by electrons; the \(F\)-centers in some cases (KJ, KCl crystal 3e) were driven back out, while in others (KCl, crystals 2c and 2D) they remained (Table 7). The transition of atomic centers into colloidal ones can be judged from the yellowing of the crystal.

It is very difficult to determine the exact transition temperature: in places yellowing begins (for KJ) already at \(100^\circ\). By an approximate estimate, the temperature of “mass” formation of particles for KJ is of the order of \(300^\circ\), and for KCl much higher, not below \(550^\circ\).

Let us now consider the formation of colloidal particles in photochemically colored crystals. The \(F\)-centers are very unstable; they disappear (irreversibly) upon heating to \(\sim 100^\circ\); upon illumination with wavelengths from the region of the \(F\)-band they partly disappear, partly are transformed into new \(F'\)-centers (which are not colloidal).

There is, however, the possibility of converting centers into colloidal ones under the simultaneous action of a) light and heat (careful heat-

Fig. 37. Tyndall cone of an untempered colored NaCl crystal: a—after coloring in vapors and ordinary cooling; б—after heating to \(450^\circ\) and “quenching,” в—after “annealing” and “quenching,” г—after repeated heating to \(400^\circ\) and “quenching,” д—after heating to \(\sim 700^\circ\) and “quenching.”

METAL PARTICLES WITHIN THE CRYSTAL LATTICE

Table 7

KJ + 0.05 mol. % Ag

No. in order	Crystal No.	Appearance of the crystal before heating	Heating temperature in °C	Heating time in min.	Appearance of the crystal after heating
1	11a	Colorless	110	50	Yellowish in places
2	11a	Yellowish in places	420	20	Yellowing
3	9a	Colorless	200	35	Yellowish in places
4	9a	Yellowish in places	340	15	Yellowing

KCl + 0.005 mol. % Ag

No. in order	Crystal No.	Appearance of the crystal before heating	Heating temperature in °C	Heating time in min.	Appearance of the crystal after heating
1	2e	Violet with a yellow rim	400—520	50	The crystal became discolored
2	2e	Colorless	500—620	25	Yellowing
3	3e	Colorless	400—520	50	No changes
4	3e	»	500—620	25	Yellowing

KCl + 0.05 mol. % Ag

No. in order	Crystal No.	Appearance of the crystal before heating	Heating temperature in °C	Heating time in min.	Appearance of the crystal after heating
1	2D	Dark violet	620—680	35	Light violet
2	2D	Light violet	680	180	Green (violet-yellow)

heating not above 100°); β) by light and deformation: a yellow rock-salt crystal, when brought into the light, instantaneously turns blue\(^{54,*}\); its absorption spectrum indicates the colloidal nature of this coloration. In both cases the particles obtained are very large and of different sizes (broad absorption curves).

If atomic centers have been produced in the crystal by the action of light from the region of the \(U\)-band, then it is possible to convert \(F\)-centers into colloidal ones by heating alone\(^{1}\). For natural rock salt this phenomenon occurs already at temperatures from 350°. In greater detail

... study of this process has shown that, apparently, a necessary condition for the formation of particles is the presence in the crystal of at least a small number of colloidal centers. This actually occurs if the $F$-centers, formed simultaneously with the $U$-centers when electrons are introduced from outside, have not been expelled back. Otherwise the phenomenon described is not observed; the crystal becomes completely decolorized upon heating. Nor can the formation of particles be observed in artificial crystals of NaCl, KCl, etc., probably because here, as indicated, a considerable part of the $F$-centers passes into $U$-centers. For natural rock salt, on the contrary, very sharp transitions are observed, illustrated by the curves of Fig. 38. Here the initial material was a crystal of rock salt, colored additively and slowly cooled; to the eye it appeared gray and contained particles of all possible diameters (curve 1, Fig. 38). After illumination with an aluminum spark, atomic centers appear (at the expense of the $U$-centers) (curve 2, Fig. 38). Upon heating to $350^\circ$ a colloidal absorption band arises with a maximum at $570\,m\mu$; the crystal becomes deep violet (curve 3). Since in this process the principal role is played by $U$-centers, quantitative counts are possible only when their presence is taken into account. It is possible that, when the crystal is heated, simultaneously with the formation of colloidal particles there also occurs a thermal process of transition of $U$-centers into $F$-centers.

Fig. 38

Summarizing what has been said about the formation of colloidal centers, we note the following variants of this process: a) “supersaturation” upon cooling of an additively colored crystal with a large number of atomic centers; b) thermal “coagulation”; c) a photochemical process (LiF, AgCl and AgBr, phosphors); d) coagulation from atomic centers under the action of light from the region of the $F$-band (under the necessary condition—the deformation of the crystal).

г) Changes in the colloidal distribution

Two processes are possible: a) a decrease in the sizes of the colloidal centers, up to their complete disappearance (processes $K \to A$ or $K \to I$), and b) coarsening of the colloidal centers. Both of these processes are actually observed both upon heating and under the action of light.

Colloidal particles obtained thermally in additively colored “pure” crystals disintegrate into atomic ones upon heating above the temperature which corresponds to supersaturation. Conversely, in additively colored phosphors (Ag, Cu)

colloidal particles are extremely stable up to the melting temperature of the crystal, when the metal precipitates from the melt in pure form.

Gold phosphors behave quite differently: as we have mentioned, here the colloidal particles can exist only at temperatures below 300°. On heating above this temperature the crystal again becomes decolorized. This phenomenon has not yet been studied in greater detail.

As regards photochemical action, the colloidal particles obtained by the additive method at room temperature are quite stable with respect to it. At high temperature ($\sim 300^\circ$) and in an electric field, Pohl[^23] observed, under illumination, the disappearance of colloidal particles at the cathode.

Turning to the question of the change of colloidal particles obtained by the photochemical method, let us first consider an intermediate case, when the colloidal particles are obtained by the thermal method, but the atomic centers serving as their material were separated as a result of a photochemical process. We have in mind the case, described on pp. 185, 186, of the transition of $F_U \to K_U$ centers. These centers are very unstable with respect to ultraviolet light (from the region of the $U$ band): upon illumination of the violet crystal (curve 3 in Fig. 38) with an aluminum spark the colloidal particles disappear completely or in part and the $F$ band is obtained anew (curve 4 in Fig. 38); in the present case the crystal was subjected to the action of the $\gamma$-rays of radium. The process of formation of colloidal particles can be repeated an unlimited number of times; at the same time, along with a decrease of colloidal absorption at the maximum, there is sometimes observed an increase of absorption in the long-wave part of the spectrum, which indicates a coarsening of the particles1. A quantitative interpretation of these processes is possible only by taking into account the presence of $U$ centers.

Let us now consider those cases in which the colloidal particles themselves were obtained by the direct action of light.

In a series of alkali-halide salts (except LiF) such centers, as we have mentioned, are obtained only under the combined action of illumination and pressure (or slight heating). These crystals are decolorized very easily, already at temperatures up to 200°. Naturally colored blue rock salt behaves in exactly the same way: on heating to a temperature above 400° it is decolorized within a few seconds; at lower temperatures the process proceeds more slowly, and it is possible to follow the change of color through violet and red, corresponding to a gradual decrease in the particle sizes2. Naturally colored crystals prove insensitive to light.

In phosphors and in silver-halide salts, changes in colloidal particles occur both under the action of heating and under the action of light; they disappear a) upon heating to a temperature on the order of \(100^\circ\), and b) upon illumination by wavelengths from the absorption region of the colloidal particles. In silver-halide salts, moreover, a number of features of great importance have been found. Namely, the bleaching of such a crystal consists not in a gradual lowering, down to disappearance, of the entire absorption band, but, on the contrary, in the formation of a “dip” in that portion of the curve which corresponds to the wavelength of the incident light (Figs. 39 and 40). A decrease in the absorption of light by any medium testifies above all to the destruction of the coloring centers, i.e., in the present case, of the colloidal particles. The fact of selective absorption becomes entirely understandable if one takes into account the non-monodispersity of the colloidal particles; under the action of light of a given wavelength, particles of such sizes disappear whose absorption maximum corresponds to the wavelength incident on the crystal.

The same phenomenon is observed also in a photographic layer; it consists in the disappearance of the latent image under the action of red and infrared rays; in the photographic literature it is known under the name of the Herschel effect.

Webb and Evans\(^{74}\) investigated the temperature dependence (for photographic layers) of this phenomenon and found that at the temperature of liquid air it does not occur.

Fig. 39 and 40. \(K_1\) and \(K_2\)—absorption curves of AgCl before and after secondary exposure. \(D\)—experimental curves of dichroism, \(D_m\)—calculated.

In isolated crystalline films of silver halide salts, the Herschel phenomenon can be traced ultramicroscopically. The disappearance and resorption of the particles occur very rapidly, before one’s eyes.^72

Interesting phenomena occur when a colored AgCl crystal is illuminated with polarized red light. Optical observations show that such a crystal becomes dichroic; an analogous phenomenon on a photographic plate was discovered and studied by Weigert. This phenomenon is explained and calculated by taking into account the departure of the particles from a spheroidal form. In Fig. 40 a calculated curve of dichroism is given for the Ag—AgCl system for elongated ellipsoids; in the same figure, and in Fig. 39, are the experimental curves under exposure to wavelengths of \(500\,m\mu\) (Fig. 40) and \(600\,m\mu\) (Fig. 39).^1) The parallelism of the experimental and theoretical curves is striking; the lack of coincidence in their position in the spectrum may, of course, be ascribed to the considerable dimensions of the particles in the experiment (the theoretical curves refer to the finest colloid).

Comparing Fig. 21 with Figs. 39 and 40, Cherdyntsev^78 draws certain conclusions about the character of the phenomenon. We have already mentioned that, when an AgCl or AgBr film is exposed to light from the region of intrinsic absorption, particles of different form and size, and also differently oriented particles, are separated out; very schematically each particle may be represented in the form of an ellipsoid. As indicated above, the absorption of such ellipsoids depends on the direction of the electric vector and on the eccentricity of the ellipsoid; when working with unpolarized light and with particles of one size and eccentricity, oriented in all directions, we would have (according to Gans) two absorption maxima. In reality we have particles of different size and eccentricity, in all probability grouped around some mean direction; the superposition of the absorption curves gives one continuous broad curve, which is indeed observed experimentally.

In Fig. 41, in the upper part, the absorption of a crystal containing particles of two eccentricities is schematically shown, the axes of which (for simplicity) we take to be located in two mutually perpendicular directions. In the lower part of the figure there is also shown, very schematically, the expected picture of the change of particles under the action of polarized light (the electric vector is directed horizontally) of different wavelength (\(\sim 600\) and \(\sim 500\,m\mu\)).

This change of the particles, obviously, will have the same character as under exposure to natural light (the Herschel phenomenon), i.e. it will consist in a diminution of the particles; this phenomenon will

^1) The arrow in the upper part of the figure denotes the width of the monochromatic region used in the exposure.

is most strongly expressed in those particles whose absorption maximum coincides with the wavelength of the incident light.

Under the action of red rays only ellipsoids 2 diminish; after exposure, predominantly ellipsoids of type 1 remain (flattened in the direction of the electric vector). With the other (vertical) direction of the electric vector the picture will be the reverse. The dichroism curve, giving the difference of the absorption curves, will have a maximum in the red region. Under the action of short-wave (green) light the dichroism curve will run in the opposite sense, as is readily seen from consideration of the corresponding scheme.

Fig. 41. On the action of polarized light on the system Ag—AgCl (photochemically colored AgCl)

In the figure: direction of the electric vector; scheme of the absorption curves of small ellipsoidal Ag particles in AgCl; action of red light; action of green light; before illumination; after illumination; dichroism; \(600\,m\mu\); \(500\,m\mu\); \(\lambda\).

The second process of change in the colloidal distribution is an increase in particle size. This process may occur both spontaneously (unpublished experiments of P. V. Meiklyar) and under the action of light (for AgCl at \(\lambda < 470\,m\mu^{78}\)); we still have insufficient data about it.

d) Photocurrent in colloidally colored crystals

The internal photocurrent in colloidally colored crystals possesses certain special features.

As Fig. 42\(^{13}\) shows, the photocurrent curves of crystals with colloidal centers differ from the analogous curves for atomically colored crystals. In the left part of the figure are given the absorption curves \((A)\), and the quantities \(\eta \dfrac{w}{\varphi}\), referred to a unit of incident energy, as a function of wavelength \((B)\), and the dependences of \(\eta \dfrac{w}{\varphi}\) on the temperature \(C\) for a KCl crystal containing atomic centers (illumination \(\lambda = 555\,m\mu\)); the curves in the right-hand part of the figure refer to the same crystal after coagulation of the \(F\)-centers (fine colloid).

Fig. 42. At top left—the concentration of \(F\)-centers, \(6.5\cdot 10^{16}\); right-hand part—colloid

In Fig. 42 analogous data are given for a KCl crystal with larger particles; curve \(B\) now gives the value \(\eta \dfrac{w}{\varphi}\) per unit of absorbed energy. The maximum at \(560\,m\mu\) is caused by the small number of \(F\)-centers present in the crystal together with the colloidal ones. In the upper part of Figs. 42c and 43c the type of photocurrent is schematically indicated (corresponding to Fig. 28).

Analysis of the curves indicates the following characteristic features of the photoconductivity of colloidally colored crystals:

a) As is seen from Fig. 42, the secondary shift in the presence of atomic centers alone is approximately 200 times greater (at room temperature) than the same quantity in the case where all the atomic centers have coagulated into larger particles.

b) For atomic centers the curve of the spectral distribution of the photocurrent, referred to a unit of incident energy, runs paral-

ly to the absorption curve, which indicates the constancy of the quantum yield for different wavelengths (see, for example, Bur, p. 204). In the case of colloidal particles the photocurrent curve is not in any correspondence with the absorption curve (Fig. 43, and also ¹⁶).

Fig. 44 shows the results of Groshev’s experiments ¹⁶, who investigated colloidally distributed copper in NaCl. Since in these experiments the specimen was so strongly colored that for the region \(\lambda < 580\,m\mu\) the absorption was practically complete, the photocurrent (as also in Fig. 43) is referred to the unit of absorbed energy. The rise of the curve toward shorter waves (see also the curves of Hilsch and Ottmer ²¹ for the photocurrent in blue rock salt) indicates an increase in the quantum yield as the wavelength decreases. The same picture is observed in the “normal” photoeffect, with the difference that here the curves are strongly shifted toward longer waves.

Groshev ¹⁶ showed that the new limit of the photoeffect can be calculated from the relation

\[ \lambda_0=\frac{\lambda}{\sqrt{\varepsilon}}, \]

where \(\varepsilon\) is the dielectric constant of the medium and \(\lambda_0\) is the limit in vacuum. Taking for NaCl \(\varepsilon=5.6\) and taking from his observations \(\lambda=640\,m\mu\), Groshev obtained for the system Cu—NaCl

\[ \lambda_0=\frac{640}{\sqrt{5.6}}=278\,m\mu; \]

this number lies within the limits \(260\text{–}300\,m\mu\) of the red limit of the photoeffect of copper in vacuum.

Fig. 43. KCl, coarse colloid

c) A characteristic feature of the temperature dependence of the quantity \(\eta \frac{w}{\varphi}\) in the photoeffect with colloidal particles is the absence of a drop in the curve at low temperatures. On the contrary, one may note a certain rise of the curve (see also Fig. 29). Another feature is the appearance of inertial currents already at temperatures of about \(-150^\circ\) (see the schematic representation of the character of the current).

§ 7. On the mechanism of formation of atomic and colloidal centers in the crystal lattice

The material presented in the preceding paragraph is far from sufficient for solving the problem of the mechanism of formation of the smallest particles of metal; attempts at a theoretical interpretation of these results are also incomplete and are in their first stage. Nevertheless, in the last 2–3 years certain milestones have already become apparent, along which, evidently, the further development of the question will proceed.

Fig. 44. Absorption of light \(K\) and photocurrent \(I\) per unit of incident energy in an NaCl crystal containing colloidal copper

We have already repeatedly pointed out that in all processes of formation and destruction of atomic and colloidal centers in a crystal the principal role is played by electrons. Therefore it is obvious that all attempts to approach the study of the mechanism of these processes must be based on determining the energy states of electrons in the crystal lattice. This question constitutes one of the fundamental parts of the modern quantum-mechanical theory of the solid body, which we shall briefly recall.

a) Fundamental propositions of the modern theory of the solid body1

The fundamental propositions of Pauli—Sommerfeld—Bloch are as follows.

1. To the valence electrons in a crystal are assigned definite discrete values of energy, the so-called energy levels. These levels are grouped nonuniformly: at some values of the energy they are very close to one another and form a practically continuous band—a zone. Some values of energy an electron cannot assume at all; these are the so-called

forbidden regions lying between the zones. Sometimes zones may overlap one another.

1. Each zone is a set of discrete levels according to the number \(N\) of atoms in the portion of the crystal under consideration (in the so-called main region). Since, for the corresponding nondegenerate states in the atom, two electrons can simultaneously occupy each level (according to the Pauli principle), then, when the total number of valence electrons is odd, in the uppermost of the zones occupied by electrons half the levels remain empty, which makes it possible for the electrons to acquire greater energy (to be accelerated) when an electric field (of ordinary strength) is applied. Such a solid is a conductor.

When the number of valence electrons in the atom is even, when all the levels in the zone indicated above are occupied, the electrons cannot be accelerated by an electric field; in this case we are dealing with an insulator.

1. Above the uppermost of the occupied zones (the so-called main one) there is situated the next empty zone of permissible values of the electron energy. If, in some way, the electron energy increases so much that it corresponds to this zone, then the electron can be accelerated by an electric field; in the insulator electronic conductivity will be observed. Therefore this zone, empty under ordinary conditions, bears the name of the conduction zone.

One may try to compare the energy zones of a solid just considered with the energy levels of free ions: to each zone there corresponds in the ion a certain discrete level. Slater and Shockley \(^{63}\) followed the formation of zones from these discrete levels, gradually decreasing the distance between ions \((\mathrm{Na}^{-}\) and \(\mathrm{Cl}^{+})\) and thus taking into account their ever-increasing interaction; for an \(\mathrm{NaCl}\) crystal they obtained a superposition of two systems of zones corresponding to the systems of discrete levels of two ions, \(\mathrm{Na}\) and \(\mathrm{Cl}\). The highest of the occupied zones (the so-called main one) in the crystal corresponds to the \(3P\) shell of the \(\mathrm{Cl}\) ions, while the conduction zone corresponds to the \(3S\) level of the electron of the \(\mathrm{Na}\) atom.

Such are the qualitative conclusions of the new quantum-mechanical theory of the solid.

The complete solution of the problem, consisting in finding the eigenfunctions and eigenvalues of the Schrödinger equation,

\[ \Delta^{2}\psi + \left(8\pi^{2}\frac{m}{h^{2}}\right)(E - V)\psi = 0, \]

where \(V\) is the periodic field of the lattice, is extremely complicated.

At the present time we have only the first elementary attempts at its solution. Thus, Mott \(^{40}\) gives the values of the function \(\psi\) for weakly bound electrons (Fig. 45) in the conduction band for an \(\mathrm{NaCl}\) crystal; the dashed line denotes \(\psi\) for a free \(\mathrm{Na}\) atom (the radii of alkali-metal atoms are greater than the radii of their ions). The solid curves depict the function \(\psi\) in the conduction band according to

lines connecting two neighboring ions of the same and of opposite signs. Consideration of the last curve shows that with greatest probability the electron is located near the Na ion and with least probability near the Cl ion.

This figure permits one to draw certain conclusions (Mott^40) about the displacement of an electron in the conduction band. An electron introduced from outside or raised from the fundamental level will circulate about the sodium ions; since the probability of finding the electron between two neighboring Na ions along the diagonal is sufficiently great, the electron will pass freely from one sodium ion to another, almost as freely as in a metal.

Fig. 45

The Bloch theory set forth above has certain inaccuracies, essential when it is applied to dielectrics, to which attention has been drawn by various authors^62.

Thus, Frenkel (1932)^10,11 pointed out that if some additional energy is imparted to an electron, then, moving away from the ionic residue, it must interact with the “positive hole”^1) remaining in the fundamental band, in whose field it is all the time located.

For such a state of the electron, at Frenkel’s suggestion, the term “excited state” of the electron became established, in contrast to the state of ionization (conduction band), corresponding to the complete liberation of the electron from the influence of the hole. For the system: excited electron—positive hole, Frenkel proposed the term “exciton,” which has gained wide currency; with the exciton is associated an “excitation wave” propagating through the crystal.

This conception of excited states in the crystal lies at the basis of the most recent work of a number of investigators—Mott^38–43, Seitz^59–61,65, as well as some others (Franck and Teller^9, Wannier^73). Frenkel’s exciton is a fundamental concept in the modern theory of the photochemistry of crystals.

^1) In the case of NaCl the “hole” will be the chlorine atom. If at some place in the crystal a negative ion is absent, or if an electron has been removed from a negative ion, then this circumstance will be sensed by the whole system as a deficiency of negative charge, or as an excess of positive charge. Therefore the atom formed from a negative ion upon its loss of an electron will be a positive “hole.”

Let us dwell briefly on Mott’s views, developed by him in a number of papers. A hole and an electron attract one another with a force equal to \(\dfrac{e^2}{kr^2}\), if \(r\) is large in comparison with interatomic distances (\(k\) is the dielectric constant of the crystal). There is a series of energy states of the electron, converging to the “series limit,” in exactly the same way as this occurs for electrons in the field of a proton.

Mott’s basic conception concerning the photochemical process consists in the fact that in the process of absorption of light—whether in the intrinsic absorption band (the first peak) or in the \(F\)-band—what takes place is not photoionization, but only excitation.

Mott admits the possibility also of complete detachment of the electron, but only under the action of large quanta, corresponding to the “series limit”; for intrinsic absorption he correlates this process with the second peak.

Mott attributes the first peak to the formation of an exciton; from the classical point of view this corresponds to the transition of an electron from a chlorine ion into the sphere of action of a sodium ion, i.e. to that picture of the photochemical process which a number of authors have long maintained.

The electron and the positive hole forming the exciton, with time, either move apart, and for this the expenditure of a certain activation energy is required (thermal impacts), or they recombine with emission or by giving up the energy in the form of heat (inactive absorption) \(^{10, 11, 44}\). This last case corresponds to the damping of the “excitation wave.”

Mott calculates the probability of both processes as a function of temperature; it turns out that in the first case the probability falls sharply with decreasing temperature, while in the second (recombination with liberation of heat) it falls with increasing temperature (the calculation for liberation of an electron upon absorption of light in the \(F\)-band will be given below).

In what follows we shall be interested only in the first case of the decomposition of an exciton into a free electron and a positive residue (a halide atom in intrinsic absorption or a metal ion in absorption in the \(F\)-band).

b) Formation of atomic centers

The most essential point in the process of formation of \(F\)-centers is the fixing of the electron at one or another place in the lattice.

Certain conclusions about the binding energy of the electron inside the lattice may be drawn on the basis of the above-mentioned experiments of Mollwo and Rehner on the equilibrium concentration. According to these experiments, the concentration of atomic \(F\)-centers \(C_{\text{cryst}}\) is always greater than the concentration of atoms in the surrounding vapor \(C_{\text{vap}}\). Considering formally this coloration process as dissolution of the metal in the crystal and

applying the law of mass action in the form, for example, \(\mathrm{Na}_{\text{vap}} \rightleftarrows \mathrm{Na}^{+}_{\text{cryst}}\), Pohl\(^{46}\) obtains from the expression

\[ \frac{C_{\text{cryst}}}{C_{\text{vap}}} = \alpha = \mathrm{const}\cdot e^{-\frac{W}{kT}}, \]

where \(W\) is the “heat of dissolution” (values of \(W\) of the order of \(-0.25\) eV for KBr, \(-0.1\) eV for KCl and \(-0.21\) eV for KI)\(^{1}\). These numbers, together with that, characterize the work of binding of the electron in the \(F\)-center; it is negative, and therefore the atomic centers in the crystal possess a lower potential energy than in the free state in the vapor. The cause producing this decrease in energy is the environment of the atomic center—the bond with the neighboring ions of the lattice.

The question of the stabilization of electrons in the lattice reduces to calculating those conditions under which the electrons would possess a minimum of potential energy, or, in the language of quantum mechanics, to determining the lowest possible energy levels for the electrons forming \(F\)-centers.

At present there are no quantitative data on this question; we shall now set forth some qualitative considerations. Let us consider the most general case, when in the crystal there are free electrons (moving in the conduction band) of one origin or another.

On becoming bound, the electron must pass to some level lying in the interval between the conduction and the fundamental zones. The outer electrons cannot pass to the latter, since all its levels are occupied. There may be a whole series of binding levels, corresponding to a series of local possibilities for the electron to become bound at one or another place in the lattice: traps for electrons may be all kinds of inhomogeneities of the lattice, caused, first, by missing or wandering ions (according to Schottky or Frenkel) (Fig. 46), secondly, by foreign ions or, in general, extraneous inclusions, and finally simply by cracks or internal surfaces.

Any theory of electron binding must take into account the fact emphasized above of the exponential increase in the concentration of coloring centers with temperature.

For this reason, for example, de Boer’s\(^{5}\) “adsorption” picture (set forth by him in his book, Ch. X) was abandoned by the author himself. At present de Boer\(^{6}\) bases himself on Schottky’s theory, according to which the electron is bound near a positive hole arising as a result of the absence of an ion; indeed, as we have already mentioned, the number of vacant sites that determine the ionic electrical conductivity of the crystal increases with rising temperature according to the same law.

\(^{1}\) Mott\(^{15}\) calculated the magnitude of \(W\), using a certain circular process, and obtained values of the same order as those given above.

To all these pictures, based on one or another disturbance of the lattice, Gippel^25, ^26 adds yet another one, according to which the potential well into which the electron “falls” is created by the electron itself during its motion through the crystal, as a result of the dissipation of its energy in the lattice and the inevitable accompanying rearrangement of the lattice. The point is the delay time of the electron; if it is so large that the ions have time to rearrange into new equilibrium positions, the electron will prove to be trapped.

Fig. 46. KBr crystal [110] according to Schottky

All the enumerated pictures of trapping appear a priori possible; their validity is determined by the binding energy of the electron (the “depth of the potential well”).

We shall not prejudge the question of in which of the above cases the electron will be most strongly trapped (its level will be lowest); it is possible that, according to de Boer’s latest ideas, the deepest potential wells will be the regions mentioned above near “holes” at the sites of missing ions. It is undoubted, however, that if the number of free electrons in the crystal for some reason exceeds the number of possible sites, the electrons will have to be trapped where their binding is weaker (the potential level is higher). We shall return to this question below when considering the mechanism of formation of colloidal particles.

The process of electron trapping itself Mott^40 considers, proceeding from the ideas outlined above about excitation levels, on one particular example of the de Boer–Schottky picture mentioned above. A point at which a negative ion is absent acts as a positive hole, whose field extends beyond its limits and also encompasses six neighboring ions, on one of which the electron that has entered this field may settle.

Until now, in considering the conditions for trapping an electron, we have made no distinction between electrons introduced from outside (in additive coloration) and those obtained as a result of a photochemical act. In order to consider the specificity of the process of electron trapping in this latter case, it is necessary to dwell in more detail on the scheme of energy levels in the crystal.

The first attempt at a qualitative construction of a level scheme in crystals of alkali-halide salts on the basis of experimental

of the data belongs to P. S. Tartakovskii; this scheme is described in detail by him in his book on photoconductivity[^67]. Tartakovskii based himself on the Bloch—Wilson theory and took into account only two bands—the fundamental band and the conduction band.

We have attempted, taking Tartakovskii’s scheme as a basis and using Schottky’s calculations (see below), to visualize the relative arrangement of the levels, taking account also of excitation levels; in Fig. 47, which gives a scheme of levels for NaCl and AgCl, the fundamental band and the conduction band are denoted by the letters \(O\) and \(P\), the excitation level by the letter \(B\); in the left-hand part the \(F\)-levels and the corresponding excitation levels of the \(F\)-centers, \(BF\), are marked.

Dotted lines denote separate (localized at individual points of the crystals)1 levels of electrons situated in inhomogeneities, cracks, etc., i.e. points in which their excitation work is smaller[^17]. We have already said that regions of this kind are considered responsible for the long-wavelength tail of the intrinsic-absorption band, extending far toward the red end of the spectrum. (This is evident from the absorption of AgCl (Fig. 2); for alkali-halide salts Hilsch and Pohl[^22] showed photoelectrically the presence of \(F\)-centers even when crystals are illuminated with visible light.)

Fig. 47

In constructing the scheme of levels in a real crystal this fact must be taken into account.

The distance between \(O\) and \(P\) for NaCl we take, following Mott[^42], to be equal to \(h\nu\) of the “series limit” (9.64 eV). To determine the position of the level \(B\) one must know the work of optical (or thermal) activation. Let us dwell on these calculations.

According to the above-stated views of Mott, the work of optical activation for the intrinsic absorption of alkali-halide salts is given directly as the difference of the quantum energies in the first peak and at the “series limit”; for NaCl Mott takes this number to be

\[ h\nu_{2p} - h\nu_{1} = 9.64 - 7.8 = 1.84 \text{ eV}. \]

For silver salts, which have a very diffuse absorption band, \(h\nu\) of the first peak for AgCl is 4.88, but the “series limits” cannot be determined; nevertheless one can approximately determine the activation work on the basis of the following considerations (Mott \(^{39}\)): an electron and a positive hole attract one another with a force \(\dfrac{e^2}{k_0 r^2}\); if they are in the stationary state of principal number \(n\) and high azimuthal quantum number, then the expression for the energy of their separation may be written in the form of a hydrogen-like formula

\[ W=\frac{m_0 e^4}{2h^2 k_0^2 n^2}, \]

where \(m_0=\dfrac{m_1\cdot m_2}{m_1+m_2}\), and \(m_1\) and \(m_2\) are the effective masses of the electron and the positive hole. Suppose that \(m_0\) is equal to the mass \(m\) of a free electron; then

\[ W=\frac{13.5}{n^2 k_0^2}\,\mathrm{eV}. \tag{*} \]

It follows from expression (*) that for silver halide salts the quantity \(W\) must be considerably smaller than for alkali halides; thus for AgCl \(k_0=\mu^2=4.8\), whereas for NaCl \(k_0=2.33\) (\(\mu\) is the refractive index); whence

\[ \frac{W_{\mathrm{AgCl}}}{W_{\mathrm{NaCl}}} = \frac{2.33^2}{4.8^2} \sim \frac{1}{4.24}; \quad W_{\mathrm{AgCl}}\sim 0.42\,\mathrm{eV}^{1}). \]

For the work of dissociation of an \(F\)-center, one may, following Mott (see below), take for NaCl a number of the order of \(0.3\,\mathrm{eV}\); the \(h\nu F\)-band for NaCl at \(0^\circ\mathrm{K}\) is equal to \(2.73\,\mathrm{eV}\); therefore the \(F\)-level must be situated below the conduction band by \(\sim 3.0\,\mathrm{eV}\). For AgCl, reasoning as above, we have \(W_0\sim 0.07\,\mathrm{eV}\); assuming, in accordance with Toporets (see above), that the \(F\)-band could lie at \(\lambda\sim 520\,m\mu\) (\(h\nu=2.4\,\mathrm{eV}\)), we obtain that the \(F\)-level must be placed below the conduction band by \(2.47\,\mathrm{eV}\).

In order that an electron excited by light can become fixed with the formation of an \(F\)-center, it must pass to the \(F\)-level. This process can be effected either directly—we see that the \(F\)-level lies below the \(B\)-level—or with a “transfer,” in which the electron first, under the action of thermal impacts, passes into the conduction band and only then falls to the \(F\)-level.

It is possible that both processes occur simultaneously; however, the formation of stable \(F\)-centers by the first process is improbable. Indeed, the transition of an electron from the excitation level directly to the \(F\)-level means that when the excitation wave in its motion through the crystal reaches a positive hole, formed, for example, by a missing ion, the field of the latter

\(^{1}\) Thus for AgCl the distance between the ground and conduction bands is equal to \(4.88\,\mathrm{eV}+0.42\,\mathrm{eV}\approx 5.3\,\mathrm{eV}\).

will prove stronger than the field of the positive hole caused by the halogen atom; the electron will now move about another center. But the halogen atom remains nearby and will be able, at the first favorable moment (thermal collisions), to pull the electron back to itself again. One would therefore expect that, with increasing temperature, the number of \(F\)-centers would become smaller and smaller. Experiment, however, gives a different picture.

In the second process (when the electron reaches the \(F\)-level through the conduction band) this objection no longer applies, since the electron and the halogen atom turn out to be locally separated; the electron in this case is in no way different from electrons introduced from outside in additive coloration. Necessary consequences of this picture are: 1) the presence of a photocurrent, 2) temperature dependence.

Indeed, Tartakovskii \(^{67}\) and his collaborators (Podlubnyi) observed a photocurrent under the action of ultraviolet rays, though one much weaker than might have been expected.

The second consequence finds firm experimental confirmation in the facts mentioned above—the dependence of the quantum yield on temperature and the presence of a saturation state. Indeed, the factor \(b \sim 0.1\ \mathrm{eV}\) in the semiempirical expression \(\eta =\)

\[ = a \cdot e^{-\frac{b}{kT}} \]

is obviously nothing other than the energy of thermal activation.

Mott \(^{41}\) attempts to calculate the course of the quantity \(\eta\) as a function of temperature on the basis of theoretical ideas about the probability of one or another stage of the photochemical act.

Let us denote by \(A\) the probability of recombination of an exciton in the time \(dt\); then \(\frac{1}{A}\) is the mean lifetime of the exciton.

Let further \(B\,dt\) be the probability that the partners will separate \(^{1}\); then the probability of this process per absorbed quantum will be equal to

\[ \eta = \frac{B}{A+B} = \frac{1}{1+\frac{A}{B}} . \]

Mott assumes that \(A\) and \(B\) depend on temperature, namely:

\[ A = a + a^{1} e^{-\frac{E}{kT}}, \]

\[ B = b + e^{-\frac{W}{kT}} . \]

All experiments refer to a temperature interval sufficiently far

\(^{1}\) Mott has in mind a concrete case—the diffusion of hydrogen atoms out of the crystal.

of the absolute-zero temperature. In the expression \(A\) the second term is of principal importance, and therefore we have

\[ \tau_1=\frac{1}{\left|1+\frac{a'}{b} e^{-\frac{W-E}{kT}}\right|}, \]

where \(W-E\), evidently, is the work of thermal activation \(W_m\). It may be said in advance that this quantity \(W_m\) will be smaller than the work of optical activation \(W_0\); this follows directly from the Franck–Condon principle: according to this principle the process of optical dissociation occurs so rapidly that the ions do not have time to move into new equilibrium positions. On the contrary, in thermal dissociation the rearrangement of the lattice has a noticeable effect, reducing the work of dissociation.

This also follows from the expression

\[ W=\frac{13.5}{n^2 k^2} \]

(p. 200), since the values of the static dielectric constant \(k\) corresponding to thermal activation are considerably larger than the value \(k_0=\mu^2\). Thus, for NaCl \(k_0=2.33\), \(k=5.3\), and therefore

\[ \frac{W_0}{W_m}=\frac{5.3}{2.33}=4.1. \]

For AgCl

\[ \frac{W_0}{W_m}=\frac{11}{4.8}=5.3. \]

Hence we obtain for NaCl

\[ W_m=\frac{1.8\ \mathrm{eV}}{4.1}\sim 0.44\ \mathrm{eV}^{1}). \]

\[ \mathrm{AgCl}\quad W_m=\frac{0.42}{5.3}\sim 0.08\ \mathrm{eV}. \]

We have attempted to trace the fate of the electron of the halide ion under the action of light; the initial moment is the appearance of an exciton, followed by recombination or decay. Quantitatively, the process of photochemical coloration is regulated by the ratio between the number of electrons liberated in the decay of the exciton and the number of electrons fixed in \(F\)-centers. If these numbers are equal, then one \(F\)-center corresponds to each absorbed quantum. Experiment shows that we approach such conditions only at temperatures above \(20^\circ\) (\(i_1=80\%\)), and moreover only in the first—

\({}^{1}\) In comparing the values of \(W\), the experimental (\(\sim 0.1\ \mathrm{eV}\)) and the calculated (\(\sim 0.5\ \mathrm{eV}\)), one should take into account the fact that they correspond to different photochemical acts; the experiment, as indicated, refers to absorption in \(U\)-centers (hydrogen ions), whereas Mott’s theoretical calculations refer to intrinsic absorption (halide ions).

moments of illumination. The violation of equality between the number of absorbed quanta and the number of centers that have separated out may be due to two causes: 1) an increase in the probability of recombination and a decrease in the probability of exciton decay—of this we have just spoken, and 2) a deficiency of trapping points (i.e., \(F\)-levels) in comparison with the number of already free electrons, which, in this way, must return to their initial position (the basic level). Indeed, as we have already said, the number of \(F\)-centers is determined by the number of vacant sites; according to Pohl’s calculations, the concentration of \(F\)-centers under additive coloration at \(t = 20^\circ\mathrm{C}\) (if such a process could take place) would be of the same order (\(10^{15}\)) as is usually observed in photochemical coloration. Thus the state of saturation may be explained.

c) Destruction of atomic centers

We have already seen that atomic centers, generally speaking, are very unstable and readily dissociate into a free electron and a metal ion; the free electrons cause the appearance of electronic conductivity. Dissociation may be thermal and optical; in the latter case we are dealing with photoconductivity.

According to Mott, in this case as well the absorption of light corresponds only to excitation of the electron; complete liberation is caused by thermal fluctuations.

A necessary consequence of such a picture is a temperature dependence of the photocurrent on temperature; this fact does indeed occur (see above). Assuming that the sharp fall of the curve \(\eta, \frac{w}{\varphi}\) (Fig. 29) is caused by a decrease not of the shift, but of the quantum yield\(^1\), Mott \(^{15}\) attempts to find this dependence theoretically. We present his reasoning briefly. Under the action of a light quantum from the region of the \(F\)-band, the electron passes into an excited state; for its transition into the conduction band a certain additional energy (activation energy) is required, which we shall denote by \(\varepsilon_m\). The probability \(pdt\) that this process will occur during the time \(dt\) may be represented as

\[ pdt = \nu e^{-\frac{\varepsilon}{kT}},\quad \text{where } \nu \sim 5\cdot 10^{12}\ \text{sec.}^{-1} \]

(of the order of the frequency of the natural vibrations of the lattice in the infrared region). If \(A\,dt\) is the probability that during the time \(dt\) the excit—

\(^1\) This assertion is somewhat arbitrary, since we have only one equation relating the current strength and the quantity of absorbed light energy; therefore we cannot determine separately the values of \(\eta\) and \(\frac{w}{\varphi}\). Other experiments on the determination of the temperature dependence of \(\eta\) in the formation of \(P\)- and \(F\)-centers also give a sharp fall of \(\eta\) when the temperature is lowered; therefore there is reason to suppose that here, too, the main factor is the quantum yield, whereas the shift is almost independent of temperature.

if the excited electron returns to the normal state, then the probability that this electron will be free is equal to

\[ \frac{P}{P + A} = \frac{1}{1 + \left(\frac{A}{\nu}\right)e^{\frac{\varepsilon}{kT}}}. \tag{*} \]

Mott assumes that this factor gives the quantum yield \(\eta\), and that \(p=\eta^w\), where \(w\) is the displacement of the electron in a unit field; moreover, at temperatures above \(-150^\circ\mathrm{C}\), \(\eta=1\), so that the segment of the curve in this region gives the change of \(w\). Extrapolating the curve to low temperatures and multiplying the values obtained by the factor \((*)\), Mott obtained a curve (the solid curve in Fig. 48) having the same character as the experimental curves.

Comparison of the theoretical curve with the experimental one for NaCl enables Mott to give an order of magnitude for the activation work \(\varepsilon_m\): for the curves to coincide it is necessary to take \(\varepsilon/k\) in expression \((*)\) equal to \(860^\circ\), i.e. \(\varepsilon_m = 0.075\ \mathrm{eV}\), and \(A/\nu = 0.0033\). This gives for \(1/A\), i.e. for the mean lifetime in the excited state, \(\sim 0.7 \cdot 10^{-10}\ \mathrm{sec}\).

Fig. 48

Hence it is easy to calculate the total energy for removing an electron from an \(F\)-center (into the conduction band) optically: it evidently consists of the energy of the quantum causing the excitation and the energy

\[ \varepsilon_0 = \varepsilon_m \frac{R^2}{k_0^2}. \]

For NaCl we have the following values: the energy at the maximum of the \(F\)-band at \(0^\circ\mathrm{K}\) (observed), \(2.73\ \mathrm{eV}\); the energy of transfer of the electron from the excited to the free state \(\varepsilon_0 = 0.30\ \mathrm{eV}^{1)}\); the total energy \(R_0 = 3.03\ \mathrm{eV}\).

As regards the work of thermal dissociation of \(F\)-centers, it, as was indicated, must be smaller than in the case of optical dissociation.

In calculating the magnitude of the thermal dissociation, Mott\(^{15}\) bases himself on the data of Polya and his collaborators on the mobility of \(F\)-centers. As we have already noted, the mobility is

\[ V = V_0 e^{-\frac{U}{kT}}, \quad \text{where } U \leq 1\ \mathrm{eV}. \]

The mobility is proportional to the concentration of electrons \(n\) in the conduction band at any moment, so that

\[ n = n_0 e^{-\frac{U}{kT}}, \tag{**} \]

\(^{1)}\) Mott substitutes here \(\varepsilon_m = 0.07\ \mathrm{eV}\), which clearly contradicts all his preceding reasoning.

where \(n_0\) is a constant. Mott calculates this value theoretically: the number of electrons entering the conduction band per unit time will be proportional to

\[ e^{-\frac{R_m}{kT}}, \]

where \(R_m\) \(^1\) is the required energy of thermal transfer of an electron into the conduction band. The number of electrons leaving the conduction band (recombining with ions) is proportional to the product of \(n\) and the number of vacant sites in the lattice; this latter consists of \(n\) (the number of decayed \(F\)-centers) and \(N\), where \(N\) is the number of vacant sites in the lattice that are in thermal equilibrium, independently of the presence of \(F\)-centers. According to the calculations of Mott and Littleton \(^{42}\), \(N\) proves to be of the order of \(10^{22}\cdot e^{-\frac{1}{2}\frac{E}{kT}}\), where \(E=W_0^+ + W_0^- - W_1\) (the works of removal of negative and positive ions and the lattice energy \(W_1\)). For NaCl, \(\frac{1}{2}E \sim 0.95\ \mathrm{eV}\). Since here \(U=0.94\ \mathrm{eV}\), and \(n_0\) is of the order of the number of \(F\)-centers per unit volume (\(10^{14}\)), \(N \gg n\), and we find that the number of electrons leaving the conduction band per unit time is proportional to

\[ ne^{-\frac{1}{2}\frac{E}{kT}}. \]

Equating the number of electrons entering and leaving the conduction band, we obtain

\[ e^{-\frac{R_m}{kT}}=\mathrm{const}\, ne^{-\frac{E}{2kT}} \]

or

\[ n=\mathrm{const}\, e^{-\frac{\left(R_m-\frac{1}{2}E\right)}{kT}}. \]

Comparing with \((**)\), we obtain

\[ R_m=U+\frac{1}{2}E. \]

Some values of \(R_m\), in electron-volts, are given in Table 8.

Table 8

	NaCl	KCl	KBr
\(U\) observed	0.94	1.00	0.84
\(\frac{1}{2}E\) calculated (see \(^{43}\))	0.95	0.95	0.94
\(R_m=\)	1.89	1.95	1.78

\(^1\) The quantity \(R_m\) in Mott’s notation corresponds to \(W\).

As we see, \(R_m\) is considerably smaller than \(R_0\) for optical dissociation.

The liberated electrons must ultimately pass from the conduction band to one of the lower levels—become fixed in the crystal. In photochemically colored crystals the most probable transition is to the ground level, from which the electrons were torn away in the photochemical act, i.e., in other words, a return to the halide (or hydrogen) atoms; each such act of recombination is evidently manifested as the disappearance of one \(F\)-center and is accompanied by luminescence. We shall discuss another possibility below.

In additively colored crystals the main band is occupied; therefore here decoloration occurs only when the electron, by diffusion (remaining in the conduction band), reaches the surface of the crystal. In the process of diffusion we directly observe the displacement of the front of coloring centers through the crystal; therefore diffusion evidently consists in the successive fixation of the electron in an \(F\)-center and its liberation during thermal dissociation. Indeed, the rate of diffusion, as we indicated above, depends strongly on temperature.

The quantities characterizing the displacement of \(F\)-centers in the lattice are: a) the rate of thermal diffusion, b) the mobility \(V\) of electrons in an electric field, and c) the “shift” of electrons \(w\) in the photoeffect. The difference between processes b) and c) is only that in the first of them the electrons are liberated thermally, and in the second—optically. Obviously, only quantities of secondary shift can be compared with \(V\).

We have seen that both quantities depend on a number of circumstances. Leaving aside for the moment the temperature dependence, we note that \(w\) decreases (by \(\sim 100\) times) in the presence of colloidal particles (Fig. 42) and with an increase in the concentration of coloring centers; for the same reason the mobility of electrons also decreases.

The influence of foreign ions or impurities should be noted especially. Thus, according to Hecht \(^{18}\), \(w\) in AgCl decreases strongly upon the addition of only \(0.05\%\) AgCl.

Toporets’ experiments indicate that the silver ions present in the crystal divert to themselves part of the electrons from the \(F\)-centers.

Closely connected with questions of mobility or of the magnitude of the shift is the question of the residence time of the electron at one ion or another, i.e., the question of the stability of \(F\)-centers. Some data can be obtained by studying the velocity of the coloring centers (electrons) in crystals both in an electric field and without it (thermal dissociation). As we indicated above, this velocity can be represented by the expression \(V = V_0 e^{-\frac{U}{kT}}\), where \(U \sim 1\ \mathrm{eV}\), and \(V_0\) is the limiting velocity, corresponding, evidently, to the case when the electron is practically no longer fixed at intermediate ions (is in the conduction band); it is of the order of the velocity of electrons

in metals. The delay time obeys Boltzmann’s law (Smakula ^64). This fact may be interpreted in the following way ^3, ^26: the electron moves freely over a distance of the mean displacement \(w\) during the time \(t_1\) and is held up in the lattice for the time \(t_2\).

Then the mean velocity of the \(F\)-center in the direction of the field is

\[ C=\frac{w}{t_1+t_2}. \]

The delay time \(t_2\) is equal to infinity at absolute zero and will be comparable with the period of the lattice’s own thermal vibrations \(\tau\), if the entire activation energy \(U\) is drawn from the lattice’s vibrational energy. Between these limiting cases there will occur a distribution according to Boltzmann’s law

\[ \tau=\tau_0 e^{\frac{U}{kT}}, \]

where \(\tau_0=\frac{1}{\nu_m}\), \(\nu_{\max}\) is the vibration frequency. If \(w\) is not very large, as is the case for alkali-halide salts, \(t_1 \ll t_2\), and

\[ C=w\cdot \nu_{\max} e^{-\frac{U}{kT}}, \]

the smaller the possible magnitude of the displacement, i.e. the longer the delay time, the more stable the \(F\)-centers, and conversely. Thus, in AgCl (according to Lefeld) at \(-170^\circ\), \(w\) is of the order of \(4.0\cdot 10^{-4}\ \mathrm{cm/V/cm}\), whereas for alkali-halide salts (Table 6) we have, for the secondary displacement in photochemical coloration, numbers of the order of \(5\cdot 10^{-8}\ \mathrm{cm/V/cm}\); hence it follows that in silver salts the \(F\)-centers must be extremely unstable.

г) Formation of colloidal particles

As already mentioned, examination of ultramicroscopic photographs both for alkali-halide and for silver-halide salts (Figs. 33, 34, 37) shows that colloidal particles are deposited not according to the laws of chance, but that in the lattice there exist regions where they are deposited densely and, conversely, regions where they are not deposited at all; here, as in the case of liquid solutions, it is evidently necessary to assume the presence of certain condensation centers.

Let us first consider the mechanism of condensation of the metal around these centers. Condensation of the kind that occurs, for example, in the formation of colloidal particles in a liquid and gaseous medium, i.e. the direct transfer of metal atoms, here appears at first sight to be unlikely.

Indeed, we have already mentioned that, upon illumination of AgCl or AgBr crystals, silver particles with diameters up to \(100\,m\mu\) are deposited—and, moreover, with extraordinary rapidity—containing, consequently, up to 15 million atoms. In alkali-halide salts the greatest concentration of atomic centers in photochemical coloration is of the order of \(10^{-5}\); if we adopt this value also for the silver-halide salts, then we obtain that \(15\cdot 10^6\) atoms must be drawn to one center from a volume with a radius of about \(2.5\cdot 10^3\)—along this length there fit up to \(5\cdot 10^3\) elementary cells.

In considering this process, we must, first of all, recall the fact of the thermal diffusion of atomic centers—we have seen that this phenomenon was connected with electron jumps, with electronic conductivity; one may imagine that here too we are dealing with the same kind of diffusion, directed toward individual points—the crystallization centers.

As has been mentioned more than once, a necessary condition for the thermal diffusion of electrons is the presence of conductivity leading to the equalization of charges. Therefore the formation of colloidal particles likewise should be observed only when this condition is met.

A typical example of this kind is provided by silver-halide salts—good ionic conductors at room temperature; when the temperature is lowered the conductivity drops sharply; we have seen that the photosensitivity of photographic plates also falls.

Silver phosphors behave similarly to silver salts (at high concentration); we cited above (Part I) the point of view according to which, in a phosphor, we have islands of a halide salt embedded in the main lattice, with an altered period.

Alkali-halide salts at ordinary temperature possess very low ionic conductivity, acquiring it only at a temperature of 400–500°; at the same time, at ordinary temperatures, the formation of colloidal particles is not observed in them either under the action of ultraviolet rays or under the action of visible rays (from the region of the \(F\)-band). The only exception is LiF\(^1\)); it should be noted, however, that its electrical conductivity is about 100 times greater than that of potassium salts. This phenomenon can, however, be observed in deformed crystals under the action of visible light on \(F\)-centers, i.e., upon their optical dissociation. Here too we must note that deterioration in the quality of the crystal is accompanied by an increase in its ionic conductivity.

Since the mechanism of ionic conductivity in alkali-halide and silver-halide salts is different—in the former the halogen ions move, and in the latter the metal ions—then, evidently, the mechanism of charge equalization must also be different.

Gurney and Mott, considering the formation of colloidal silver particles during the photolysis of silver-halide salt crystals, believe that the electrons produced in the crystal under the action of light,

\(^1\)) There are grounds to suppose, however, that this difference in the behavior of LiF and other salts is only quantitative: indeed, as Schneider\(^56\) indicates, LiF acquires a noticeable colloidal coloration only after 35 hours of irradiation with a water lamp; it is possible that other salts under the same conditions would also behave similarly. The chief quantitative difference is caused by the optical properties of colloidal lithium; as was indicated above, the absorption intensity here must be several times greater than for the Na—NaCl system, which is why even the weakest traces of colloidal particles will already be noticeable.

accumulate at the center of crystallization and attract to themselves the silver ions located in the interstices of the lattice (according to Frenkel).

In the alkali-halide salts we must, obviously, allow for the possibility that cations leave the points of electron accumulation.

In both these cases, however different they may be, what is common is the idea that it is not neutral atoms as such that move in the crystal, but electrons and ions. There are, however, along with this, a number of other facts that do not fit into our scheme. These are the formation of colloidal silver particles upon heating phosphorus of low concentration, containing only atomic centers (Breido’s experiments). As we have said, these atomic centers of Ag do not move in the field, i.e. they do not dissociate. Thus here it is necessary to allow for the possibility of migration of these atomic centers as such and of their coalescence with the formation of colloidal particles.

By analogy we may allow for the existence of the same process also upon heating pure alkali-halide crystals containing centers of the alkali metal, although in this case their dissociation may also take place.

Let us now consider the question of condensation centers, restricting ourselves to the case of diffusion of dissociated \(F\)-centers, i.e. electrons. Evidently, as in the formation of \(F\)-centers, we must be dealing with sufficiently low energy levels.

Gurney and Mott\(^{14}\) assume that such condensation centers may be Ag and Ag\(_2\)S particles; the level corresponding to such a particle (the Fermi level of the metal) lies considerably below the conduction band of the salt. The experimental facts concerning the magnitude of the “shift” of the electron \(w\) in the photoelectric effect do indeed confirm the circumstance that colloidal particles can serve as electron traps.

This picture, however, is based on an assumption that is certainly not fulfilled in a number of cases: if one can admit the presence of metallic grains (specks) in emulsion crystals and, perhaps, even in isolated crystals of a halide salt, then their appearance, for example, in a NaCl crystal upon its deformation seems quite improbable. The conception of Gurney and Mott\(^{1}\) is essentially

\(^{1}\) In evaluating this article one should bear in mind that it is chronologically the first in a series of Mott’s works on questions of the photochemistry of crystals; the points of view developed in these works and used by us in the present exposition are outlined here, in Gurney and Mott’s work, only partially; its conclusions therefore cannot be regarded as definitive.\(^{2a}\)

The most serious shortcoming of this work of Gurney and Mott is, however, that they consider here the processes in silver salts in isolation, whereas, as we have tried to show in the present review, the photolysis of silver salts is only a particular case of the whole variety of phenomena of metal separation in the crystal lattice. The study of the question of photolysis can lead to valuable results only in the case of a parallel investigation of all phenomena both in silver-halide salts and in alkali-halide salts; Mott and Gurney have already taken this path, as is evidenced by their numerous works during 1938.

\(^{*}\) Advances in Physical Sciences, vol. XXII, issue 2.

relates only to the second phase of formation of colloidal particles—their growth, perhaps at the expense of smaller particles (see above, Meiklar’s experiments), but leaves entirely without explanation the most interesting and important first phase of the process of formation of such a particle, which can already be regarded as a particle of metal.

This question is directly connected with the question of \(F\)-centers; it is therefore quite logical here as well to apply Mott’s reasoning concerning the mechanism of their formation. Thus, Mott \(^{40}\) assumes that the nuclei for \(F\)-centers are vacant sites left by missing halide ions; the field of such a positive hole also embraces neighboring metal ions. An electron drawn into this field may therefore settle on one of these ions.

In our case we may reason in a similar way, with the only difference that here it will not be a matter of one, but of several electrons drawn into one field and settling on neighboring ions.

Such a picture is not difficult to imagine if one takes into account the possibility of the formation of double, triple, etc. vacant sites, with a correspondingly doubled, etc. field: indeed, according to the pictures of Schottky and Frenkel, the number of missing ions responsible for electrical conductivity increases with temperature—since the temperature dependence of the electrical conductivity is the reverse, one may expect that the number of holes decreases when the temperature is lowered. Unfortunately, we have no indications whatever concerning the mechanism of this phenomenon, except for Frenkel’s remark that, on cooling the crystal, vacant sites may coalesce and even form cavities. These will be positive holes with double, triple, etc. fields. The atomic centers around them will be situated close to one another. Since the radius of a sodium atom, \(1.85\ \text{Å}\), is much larger than the radius of a sodium ion, \(0.96\ \text{Å}\), and even than that of a chlorine ion, the question here may already be one of a collectivization of electrons, i.e. the beginning of a metallic crystal lattice is being laid. How will such an arrangement of atomic centers affect the energy levels of the electrons, i.e. the absorption band? One should expect that the absorption of two adjacent atoms will be somewhat different from the absorption of the same atoms when they do not influence one another. Indeed, we have seen that, under certain circumstances, when the crystal is illuminated with light, from the region of the \(F\)-band there is observed the appearance of a new \(F'\)-band, which should be attributed to new centers. The quantum yield of the \(F \to F'\) transition can give some indications concerning the properties of these centers: as we have seen, its limiting value is equal to 2.

Pohl \(^{46}\) interprets this fact in the following way. Each quantum tears away only one electron; thus one \(F\)-center is knocked out. This electron attaches itself to an ion neighboring one of the \(F\)-centers present in the crystal; each of them, owing to mutual interaction, now possesses a somewhat changed

spectrum \(F'\) ^1); thus in this process we have one more vanished \(F\)-center and two new \(F'\)-centers. In this way the quantum yield, equal to two, is explained for this process.

On complexes of three and more atomic centers (\(F''\), \(F'''\), etc.) we can as yet say nothing from optical observations; as is readily understood, the influence of each succeeding center will become less and less.

The electron level in an \(F'\)-center, as we see, proves to be higher than in an \(F\)-center. The same will also hold for \(F''\), \(F'''\)-centers, etc.; but as soon as such a particle acquires the properties of a metal, its level will at once become lower, and it will already be a center of particle growth, attracting to itself electrons from \(F\)-centers.

Above, on the basis of the experimental material, we outlined three types of separation of colloidal particles: thermal coagulation, “supersaturation,” and, finally, various cases of photochemical processes. We spoke of thermal coagulation a few lines above. The explanation of the process of formation of colloidal particles upon cooling a crystal containing \(F\)-centers in limiting concentration likewise presents no difficulty: upon cooling, the number of possible points for the fixation of \(F\)-centers (i.e., missing halide ions) decreases; a supersaturation of electrons is obtained, which, evidently, must fill higher levels (\(F'\), etc.), in accordance only with the picture described.

In the photochemical process of formation of colloidal particles, the experimental results may be divided into two groups of phenomena: a) colloidal particles appear at once and under the action of light from the region of intrinsic absorption—silver halide salts, phosphors, LiF; b) the material for colloidal particles is \(F\)-centers, upon which light from the region of the \(F\)-band acts—alkali-halide salts deformed by alkali.

Such a division, however, is artificial. On the contrary, there are grounds for supposing that, in all the examples listed, when colloidal particles would seem to be separated directly under the action of light from the region of intrinsic absorption, the primary product of the reaction is nevertheless \(F\)-centers. Thus, in phosphors and in LiF the illumination was carried out with ultraviolet undecomposed light; in this way, along with the formation of \(F\)-centers under the action of ultraviolet rays, we could also have their coagulation under the action of rays from the region of the \(F\)-band; indeed, measurement of the absorption in these cases gives simultaneously the absorption bands of both phases. As for the silver halide salts, here, as has repeatedly been pointed out, the atomic phase has not yet been experimentally isolated; according to calculations

^1) It cannot be asserted that the centers mentioned above near double holes will be identical with these \(F'\)-centers—the surrounding lattice in the two cases possesses somewhat different properties. Perhaps they can be identified with the \(F_1\)-centers of Tartakowski ^67, observed by him in the surface layers of crystals.

Toporets’s \(F\)-band should lie near \(520\,m\mu\) and, in this way, may overlap with the region of intrinsic absorption extending far into the region of visible rays.

Consequently, here too it is highly probable that two processes proceed simultaneously: the formation of \(F\)-centers and their coagulation, and moreover under the action of the same rays.

Finally, a priori one may also admit a third possibility: the simultaneous formation of both atomic and colloidal centers; this is indicated by the presence of a photographic current upon exposure of silver-halide crystals \(^{32}\).

In judging the correctness of the pictures set forth above, some assistance may be provided by considering the possible energy levels of the electrons; we have in mind the electrons appearing in the first case as a result of exciton decay, and in the second case upon the decay of \(F\)-centers.

In the first case, by the beginning of the action of ultraviolet light, levels lying below the \(F\)-level (metal particles?) must be present in the crystal, since otherwise the electrons will first fill these latter. We have already spoken of the small probability of such a fact—in any case in phosphors and in NaCl.

The same consideration applies also to the second case; and indeed, usually in alkali-halide salts (photochemically colored), under the action of light from the region of the \(F\)-band, bleaching is observed, i.e. the transition of electrons to the very lowest (ground) level; \(F\)-centers with a higher level are unstable.

In deformed crystals the entire course of the process is clear: the electrons torn by light from the \(F\)-centers do not reach the principal band, but settle somewhere higher. What additional conditions that facilitate this process are created by deformation of the lattice is, for the present, difficult to say.

The difficulties connected with the necessity of assuming the presence of levels lying below the \(F\)-level can be avoided by accepting the third possibility. Let us consider the case of silver-halide salts and silver phosphors with a high concentration of Ag (we have agreed to regard such phosphors as a special case of silver-halide salts). Here, as we said at the beginning of the article, there are some grounds for expecting the transition of the ionic bond into an atomic one, i.e. the formation of \(F\)-centers and, perhaps, even atomic pairs \((F')\) (see Toporets’s considerations). Further, as we mentioned, the thermal activation work necessary for the dissociation of the exciton and the liberation of the electron is here very small; therefore one may expect that the number of electrons resulting from the primary photochemical act will be much greater than the number of possible fixation points for them at the given temperature; in other words, we shall have a supersaturation of electrons which, having filled all the \(F\)-levels, will also occupy the higher-lying \(F'\)-levels.

This point of view finds some confirmation in the observation mentioned above, according to which at low temperature

AgCl acquires a colloidal coloration only in the case of preliminary (weak) exposure at room temperature: at low temperatures the probability of liberation of electrons upon decay of the exciton decreases (and, correspondingly, the probability of recombination increases), their concentration decreases, so that only \(F\)-centers are separated out, which we still do not know how to observe (we add that Lefeld\({}^{32}\) observed a photocurrent in AgCl only at sufficiently low temperatures). If, however, colloidal particles were present in the crystal from the preceding exposure, further growth (in the spirit of Gurney and Mott’s picture) is also possible at low temperature.

Let us now return to the question of the displacement of electrons and ions during the growth of a colloidal particle. Considering this process as directed thermal diffusion and bearing in mind the views set forth above by Mott on the trapping of electrons in \(F\)-centers, it is quite logical to imagine that the electrons, drawn into the field of the seed, will settle on metal ions directly adjacent to the seed—whether atomic pairs, etc., or larger particles—gradually neutralizing them. The particle will thus grow at the expense of metal ions in its immediate surroundings. Of course, charge equalization must take place in this process; in alkali-halide salts this is accomplished by the departure of halide ions, and in silver halides by the arrival of silver ions (according to Gurney and Mott); their motion, naturally, proceeds at a lower velocity than the motion of the electrons.

Gurney and Mott, in the article already cited, give another picture, according to which the electrons settle directly on the seed (an Ag or Ag\(_2\)S particle), charging it and forming an electron cloud around it. This cloud attracts to itself the metal ions located in the interstices (according to Frenkel), which then adhere to the seed. Thus here the growth of the particle occurs exclusively at the expense of metal ions wandering in the lattice and distant from the seed. Here the fate of the halide is unclear; evidently, the growing particle must somehow displace all neighboring ions, which is very difficult to imagine.

In summary, let us note that the principal feature of the picture we have sketched is that the colloidal particles are formed from dissociated \(F\)-centers, by means of their thermal diffusion; in this lies its difference from the picture of Gurney and Mott, who try to dispense with an atomic phase. The logical necessity of its existence, at least at low temperatures, is clear, incidentally, to these authors as well; they introduce the concept of certain metastable states of the electron (\({}^{14}\), § 2), differing in no essential way from \(F\)-centers. In Berg’s last work\({}^{2a}\), undertaken for the purpose of confirming the theory of Gurney and Mott, there are, on the contrary, indications against their basic conception: Berg believes that at low temperatures the electrons are trapped not at seed levels (specks), but at certain other levels (shallow traps), very unstable; when the temperature is raised, liberation occurs—

liberation of electrons and, first, their recombination with halide ions “born” in the lattice, and second, deposition at traps. This explains both the decrease in the sensitivity of photographic layers with lowering temperature and its certain finite value even at the temperature of liquid helium (4° K): the blackening of the plate in this case is caused exclusively by a process taking place inside the crystallites during their warming to room temperature.

A few words about the mechanism of the disintegration or resorption of colloidal particles. The most natural assumption is that here, as in the case of the bleaching of crystals with \(F\)-centers, the primary action of light consists in tearing electrons away from the coloring centers. In the present case we would have an external photoeffect from colloidal metal particles inside the crystal—indeed, we have already mentioned above its presence in crystals of alkali-halide salts. This conception, however, is still at the very first stage of development: a number of questions (for example, concerning the spectral distribution of the Herschel effect, on the one hand, and of the photocurrent, on the other) require further refinement and clarification.

The photoeffect here too is only the primary act; it is followed by a secondary one—the disintegration of the colloidal particle; concerning this process we also still have very little data, apart from the observation by Pohl, noted above, on colloidal sodium particles in NaCl1. Thus here, in the disintegration of colloidal particles, we encounter the same condition that is necessary for their formation—the possibility of displacement of ions. Silver-halide salts satisfy this condition, and in them the phenomena of resorption of colloidal particles in the absence of light are indeed observed very distinctly (the phenomena of Herschel and Weigert), ceasing at low temperatures.

In all the views outlined above on the mechanism of formation (and resorption) of colloidal particles in the crystalline lattice there are at present still very many obscurities and internal contradictions, chiefly because of the lack of sufficiently complete experimental data; the principal path of research has by now already become quite definite, and one may hope that in the very near future it will lead to final results.

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1. When the crystal was illuminated in an electric field, simultaneously with the photoconductivity, a displacement of colloidal particles toward the cathode took place; however, only in the case when the temperature of the crystal was so high (~300°) that displacement of ions could occur. ↩↩↩↩↩↩↩

2. Lirman and Rexer[^34] state that, during decolorization of blue salt, they succeeded in observing atomic coloration as well; this observation, however, seems doubtful. ↩↩