Application of Radioactive Isotopes to the Study of Diffusion in Solids

A. A. L’vov

Diffusion processes play an extremely important role in a number of branches of technology, in particular in metallurgy, and, consequently, knowledge of the rate of diffusion under various conditions helps metallurgists create new types of alloys with the required properties.

Diffusion from an external medium into a metal is associated with cementation, nitriding, cyaniding, and so forth. Processes opposite in direction to diffusion include the decarburization of steels, the evaporation of zinc from brass, and others. The processes of transformation of a eutectoid into a solid solution (for example, austenite–pearlite transformations in steels), sintering of powders (powder metallurgy, in particular the production of superhard alloys), welding and soldering of metals, and many other processes are also, to one degree or another, connected with diffusion.

On the other hand, knowledge of the diffusion rate and of the influence upon it of such factors as temperature, the size of the crystal grain, the concentration of various kinds of atoms (for example, the influence of nickel atoms on the diffusion rate of carbon in steel and vice versa), the ratio of the atomic radii of the diffusing element and the element of the medium, the type of crystal lattice (method of packing of atoms, density), crystallographic direction (anisotropy), mosaicity, zones distorted by plastic deformation, and the like, makes it possible for physicists to determine the nature of solids, both crystalline and amorphous.

The simplest type of diffusion is self-diffusion, i.e., the displacement of atoms of some substance in the crystalline lattice (in the case of crystalline bodies) of that same substance. This is the most “pure” case of diffusion. Therefore, the study of self-diffusion is of considerable theoretical interest for clarifying the mechanism of diffusion. Incidentally, self-diffusion is also of technical value, since it underlies such phenomena as recrystallization and creep.

Diffusion can occur (Fig. 1) both along the surface of a solid body (surface diffusion) and in its volume (volume diffusion). In the latter case, diffusion is possible both through the lattice of a crystallite (grain) and along the boundaries separating crystallites (in the case of polycrystalline bodies, which include all metals). At constant temperature, surface diffusion has the greatest rate, and diffusion through the grain the least.

Fig. 1. Possible diffusion paths in a polycrystalline body.

The rate of diffusion of a given type of atoms in a given medium at a given temperature is determined by the magnitude of the diffusion coefficient \(D\).

At temperatures far from the melting point, diffusion in solids is an extremely slow process. Raising the temperature sharply accelerates diffusion—this occurs as a result of an increase in the amplitude of atomic vibrations. The dependence of the diffusion coefficient \(D\) on the absolute temperature \(T\) is expressed by the relation:

\[ D = Ae^{-\frac{Q}{RT}}, \tag{1} \]

where \(Q\) is the activation energy (“loosening”), \(R\) is the universal gas constant, and \(A\) is a certain constant quantity (equal to \(D\) at \(T=\infty\)). This dependence was first found on the basis of experimental data, and later was obtained by Ya. I. Frenkel from the most general considerations of the theory. To find the values of \(A\) and \(Q\) from experimental data it is convenient to use the following method. Taking the logarithm of expression (1), we have:

\[ \ln D = \ln A - \frac{Q}{R}\cdot \frac{1}{T}. \tag{2} \]

If we now construct, on the basis of experimental data, the dependence of \(\ln D\) on \(\frac{1}{T}\), then the intercept cut off on the ordinate axis by the straight line expressing this dependence is equal to \(\ln A\), while the tangent of the angle of inclination of this straight line is the quantity \(\frac{Q}{R}\). One can find \(Q\) in another way as well, knowing two values of the diffusion coefficient \(D_1\) and \(D_2\) at different temperatures \(T_1\) and \(T_2\). Indeed,

\[ \frac{D_2}{D_1} = \frac{e^{-\frac{Q}{RT_2}}}{e^{-\frac{Q}{RT_1}}}, \]

whence

\[ Q=-\frac{\lg \dfrac{D_2}{D_1}}{0.218\left(\dfrac{1}{T_1}-\dfrac{1}{T_2}\right)}. \tag{3} \]

All methods for determining diffusion coefficients are based on measuring the concentration of diffused atoms in various parts of the specimen. In the case of diffusion of foreign atoms this is not associated with any fundamental difficulties and can be carried out by methods of microchemistry, spectral analysis, X-ray structural analysis, optical methods, and certain other methods. In the case of self-diffusion, all these methods are unsuitable, since self-diffusion does not disturb the homogeneity of the substance.

Attempts were made to find the rate of self-diffusion on the assumption that the rate of diffusion of atoms of a given element in a metal is close to the rate of diffusion in it of atoms of certain other elements ^42–45. Thus, for example, attempts were made to find the coefficient of self-diffusion of copper using atoms of Be, Sn, Zn, Si (Fig. 10). This led to incorrect results, since the rate of self-diffusion is less than the rate of diffusion of foreign atoms, owing to the fact that the lattice around a foreign atom is distorted and, consequently, its activation energy is smaller than in the case of self-diffusion. For example ^42, at 165° C \(D_{\mathrm{Au\ in\ Pb}} = 5 \cdot 10^{-3}\ \mathrm{cm^2/day}\), whereas \(D_{\mathrm{Pb\ in\ Pb}} = 10^{-9}\ \mathrm{cm^2/day}\).

In the most recent years, works have been carried out ^58,59 on determining the coefficient of self-diffusion of copper by studying the change in shape of individual spherical copper particles (500–600 microns in diameter) during their sintering, based on volume diffusion, as a result of holding this powder at high temperature (700–900° C) for 40–50 hours. By measuring the surface radii of the “sintered” particles at different heating times and temperatures, the coefficients of self-diffusion of copper were determined. The theory of this phenomenon was given by Ya. I. Frenkel ^57.

However, the most radical solution to the difficulties arising in the study of self-diffusion had to be sought in the development of a method for distinguishing the diffusing atoms from the atoms of the medium in which self-diffusion occurs. Such a possibility is provided by the use of the so-called method of “labeled” atoms. Beginning in the 1920s, the method of “labeled” atoms, based on the use of radioactive isotopes, began firmly to win a place for itself among other methods of investigation.

The discovery of the phenomenon of artificial radioactivity gave a powerful impetus to its further development, and it is now widely used in biology, medicine, physics, geology, etc. In this article we shall discuss only the application of radioactive indicators

for determining the rate of diffusion and, in particular, the rate of self-diffusion in solids. The method of stable isotopes was not used for studying diffusion in solids.

Artificial radioactive isotopes for the study of diffusion were first applied by A. M. Zagrubskii\(^{1,2}\), who used the radioactive isotope of gold to study self-diffusion in gold. G. Hevesy’s use of natural radioactivity is considerably less valuable because of the limited number of naturally radioactive elements. Almost the only radioactive elements with which such measurements could be made proved to be ThB, RaD, and also Po.

The general principle of using radioactive indicators for studying the rate of diffusion is as follows. Atoms of the radioactive isotope of the substance whose diffusion rate is to be determined are deposited on a specimen of the substance under study. The specimen prepared in this way is subjected to prolonged heating at high temperature; after this, using a radiation detector, the distribution of radioactive atoms in the specimen is studied, and, knowing the distribution, the diffusion coefficient \(D\) is found. Geiger–Müller counters, ZnS screens, photographic plates, etc., can be used as radiation detectors. At present three basic methods are clearly distinguished for determining diffusion coefficients by means of radioactive indicators; these will be discussed below.

1. METHOD OF REMOVING LAYERS\(^{1-11}\)

The method is based on the successive removal of layers from the surface of the specimen after a diffusion process that has proceeded at a certain temperature, and on determining the average concentration of the radioactive material in a layer as a function of the location of this layer in the specimen.

To determine the coefficient of self-diffusion of gold, A. M. Zagrubskii\(^{1,2}\) used the radioactive isotope \(\mathrm{Au}^{198}_{79}\). (Half-life \(T_{1/2}=65\) hr.) The isotope was obtained by bombarding stable gold with neutrons from a radon–beryllium source by the reaction:

\[ \mathrm{Au}^{197}_{79}+{}^{1}_{0}\mathrm{n}\leftrightarrow \mathrm{Au}^{198}_{79}+\gamma . \]

A layer of ordinary gold was deposited electrolytically on a specimen of radioactive gold (the specimen being rotated in order to obtain a uniform layer of stable gold). After this the specimen, consisting of two parts—radioactive and stable—was placed in a furnace and held for some time under conditions of constant high temperature, at which the diffusion process took place. To measure the concentration of radioactive gold

in the inactive one at different distances from the interface, three layers were removed by successive dissolution in acids. Then partial evaporation of the solutions was carried out, and gold was precipitated from their residues with the aid of FeSO$_4$. The thickness of the removed layers was calculated by weight, which is easy to do, knowing the area of the plate and its specific weight.

As a result, three filters with precipitates were obtained, whose activity had to be measured. The relative activity $\dfrac{C}{C_0}$ was determined, since absolute measurements of activities with a Geiger–Müller counter are difficult to perform. For measurement, the filter with the precipitate was wrapped directly around the counter; therefore, in each case the geometrical conditions of the experiment were reproduced exactly. The number of pulses per minute, as is known, is proportional to the number of radioactive atoms in the precipitate.

Fig. 2. Plot of the dependence of $\dfrac{C}{C_0}$ on $\dfrac{x}{h}$ for various $K$.

The calculation was performed in the following way. The solution of the diffusion equation under the corresponding boundary conditions gives:

$$ \frac{C}{C_0} = 1-\frac{1}{2} \left\{ \int_{0}^{\frac{h-x}{2\sqrt{Dt}}} e^{-y^2}\,dy + \int_{0}^{\frac{h+x}{2\sqrt{Dt}}} e^{-y^2}\,dy \right\}, \tag{4} $$

where $C$ is the concentration of the diffused atoms (proportional to the activity), $h$ is the total thickness of the layer of nonradioactive gold, and $x$ is the depth of occurrence of the etched-off layer.

On the basis of equation (4), a plot (Fig. 2) was constructed of the dependence of $\dfrac{C}{C_0}$ on $\dfrac{x}{h}$ for various $K$, where

$$ K=\frac{h}{2\sqrt{Dt}}. \tag{5} $$

With the aid of this graph, from the measured values of \(\dfrac{C}{C_0}\) and \(\dfrac{x}{h}\), \(K\) was found, and, knowing \(h\), as well as the time \(t\) during which the diffusion process proceeded, \(D\) was calculated from relation (5). The error due to the finite thickness of the layer being removed was \(1\%\), for all the formulas given above are valid for layers of infinitely small thickness. In this way diffusion coefficients were found for several specimens held at different temperatures.

As was to be expected according to equation (2), all the experimental points lay on a straight line expressing the dependence of \(\ln D\) on \(\dfrac{1}{T}\), whence it was already easy to find \(A\) and \(Q\) by the method indicated above. As a result, the following was obtained:

\[ D_{\mathrm{Au\ in\ Au}}=1.36\cdot 10^{4} e^{-\frac{53000}{RT}}\ \mathrm{cm}^{2}/\mathrm{day}. \]

The method of removing layers, in a somewhat different variant, was applied to determine the coefficient of self-diffusion of silver.^6 Radioactive silver, consisting of a mixture of two isotopes

\[ \mathrm{Ag}^{105}_{47}\left(T_{1/2}=45\ \text{days}\right) \quad \text{and} \quad \mathrm{Ag}^{106}_{47}\left(T_{1/2}=8.2\ \text{days}\right) \]

was obtained by bombarding palladium with protons in a cyclotron, according to the reactions:

\[ \mathrm{Pd}^{105}_{46}+\mathrm{H}^{1}_{1}\to \mathrm{Ag}^{105}_{47}+\mathrm{n}^{1}_{0}; \]

\[ \mathrm{Pd}^{106}_{46}+\mathrm{H}^{1}_{1}\to \mathrm{Ag}^{106}_{47}+\mathrm{n}^{1}_{0}. \]

The radioactive silver was chemically separated from the palladium, and from it an electrolytic solution was prepared; then, using ordinary electroplating technique, a layer of radioactive silver \(0.0004\ \mathrm{cm}\) thick was deposited on one side of a silver disk. The disk was prepared from chemically pure silver (the impurity present in the greatest concentration was copper—\(0.04\%\)). A second identical disk was pressed against the active side of the first disk, and then the specimen thus prepared was placed in a quartz tube (from which the air had been thoroughly evacuated) and introduced into an electric furnace for diffusion heating, which lasted, for example, at a temperature of \(876^\circ\mathrm{C}\) for 4.78 days.

After heating, layers parallel to the radioactive interlayer were removed (with the aid of a lathe). These removed layers, \(0.05\) to \(0.13\ \mathrm{mm}\) thick, were dissolved in \(\mathrm{HNO}_3\), and the radioactivity of the solutions was measured with a liquid glass Geiger–Müller counter.

The experimental errors were small. The temperature was measured with an accuracy of up to \(1^\circ\mathrm{C}\), and the error in measuring the temperature caused an error in determining the value of \(D\) of about \(2\%\). Errors in determining distances likewise amounted to no more than \(2\%\). The error in measuring the activity with the counter did not exceed \(1\text{–}2\%\).

All these errors together caused an uncertainty in the determination of \(D\) of no more than 3–4%.

The layer of radioactive silver deposited on the disk was so thin that the solution of the diffusion equation satisfying the corresponding boundary conditions could be written in the form

\[ C=\frac{C_{0}}{2\sqrt{\pi Dt}}\,e^{-\frac{x^{2}}{4Dt}}, \tag{6} \]

where \(x\) is the distance from the radioactive interlayer.

The error due to neglecting the finite thickness of the layer of radioactive silver was less than 1%. Of course, in this work, as in all others, the decrease in activity owing to radioactive decay was taken into account. In Fig. 3 the circles show the results of the measurements. The solid curve corresponds to equation (6) with the appropriate value of the diffusion coefficient. The agreement between the experimental data and the theoretical curve indicates that the diffusion equation describes well the phenomenon of self-diffusion in a solid. The fact that both sides of the curve are symmetric with respect to the initial plane shows that the two disks were well joined to one another and that the diffusion process in them proceeded symmetrically. The supposition that a portion of the radioactive atoms remains “stuck” to the initial plane is incorrect.

Fig. 3. Distribution of radioactive silver atoms in a specimen after heating at \(876^\circ\mathrm{C}\) for 4.78 days.

The curve shown in Fig. 3 is inconvenient for calculating the diffusion coefficient \(D\). Therefore, taking the logarithm of expression (6), we find:

\[ \lg C=\lg \frac{C_{0}}{2\sqrt{\pi Dt}}-0.4343\,\frac{x^{2}}{4Dt}, \]

or

\[ \lg C=\lg K-0.1086\,\frac{x^{2}}{Dt}. \tag{7} \]

If \(\lg C\) is plotted as a function of the square of the distance from the radioactive interlayer, then the experimental points must

● Points from the right branch of the curve in Fig. 3
○ Points from the left branch of the curve in Fig. 3

Fig. 4. Dependence of the logarithm of the concentration of radioactive silver atoms on the square of the depth of their penetration.

Fig. 5. Temperature dependence of the coefficient of self-diffusion of silver.

be arranged along a straight line, as was indeed obtained (Fig. 4). The tangent of the angle of inclination of this straight line, according to (7), is equal to

\[ \tan \alpha = - \frac{0.1086}{Dt}, \tag{8} \]

and, since the time \(t\) is known, \(D\) can easily be calculated. But the diffusion coefficient calculated at a given temperature is greater than the true one (at \(950^\circ\mathrm{C}\) by 3.5%, and at \(725^\circ\mathrm{C}\) by 2.6%) because of the thermal expansion of silver and, consequently, the increase in the value of \(x^2\); this must be taken into account. Next, to find \(A\) and \(Q\), one proceeds as described above. The dependence of \(\ln D\) on \(\frac{1000}{T}\) is shown in Fig. 5. The circles correspond to the experimental data. As a result of this work it was established that, for silver,

\[ D_{\mathrm{Ag\ in\ Ag}} = 0.895\, e^{-\frac{45\,950}{RT}}\ \mathrm{cm^2/sec}. \]

The sectioning method was also used in the first work of G. Hevesy¹ on determining the self-diffusion coefficient of lead, with the aid of its natural radioactive isotope RaD (\(T_{1/2}=22\) years). The specimen consisted of two parts: one of RaD and the other of ordinary lead. After heating at \(280^\circ\mathrm{C}\) for 14 months, the specimen was cut into four parts, each of which was rolled into foil, and their activities were measured with an electroscope. In this work it was only established that

\[ D_{\mathrm{RaD\ in\ Pb}} < 0.0001\ \mathrm{cm^2/day}. \]

2. ABSORPTION METHOD

If the law of absorption of the indicator radiation in the material of the specimen is known, then the diffusion coefficient can be determined without destroying the specimen. The idea of the method is that, as the indicator atoms diffuse into the body of the specimen, the average path length that the radiation of these atoms must traverse in the material of the specimen before it reaches the detector changes. This causes a change in the total activity of the specimen, on the basis of which the diffusion coefficient can be found.

Since the absorption laws for \(\alpha\)- and \(\beta\)-radiation in matter are different, the practical ways of carrying out the absorption method are accordingly somewhat different. Let us consider both cases separately.

a) Case of a \(\beta\)-active indicator¹²–¹⁵

The absorption method, based on knowledge of the absorption coefficient of the \(\beta\)-radiation of the indicator in the material of the specimen, was applied to determine the self-diffusion coefficient of copper¹³.

Radioactive copper \( \mathrm{Cu}^{64}_{29}\) (\(T_{1/2}=12.8\) hours) was obtained from zinc by bombarding the latter with neutrons from a radon–beryllium source, according to the reaction:

\[ \mathrm{Zn}^{64}_{30}+\mathrm{n}^{1}_{0}\to \mathrm{Cu}^{64}_{29}+\mathrm{H}^{1}_{1}. \]

The irradiated specimen was dissolved in concentrated hydrochloric acid, and the radioactive copper was deposited, by means of ordinary electroplating, onto a rotating disk of inactive copper. The disk was prepared from spectroscopically pure copper (of the impurities, the largest fraction was zinc, of which there was less than \(0.01\%\)). To take account of the effect of the decrease in activity due to natural radioactive decay, a control specimen was left without heating. The activity of this control specimen was checked before and after measuring the activity of the specimen subjected to heating. The accuracy of counting was \(3\%\).

Fig. 6. Possible arrangement of the specimen and counter.

The absorption coefficient of copper \(\mu\) for \(\beta\)-particles emitted by \(\mathrm{Cu}^{64}_{29}\) was determined in the following way. Between the active side of the specimen and the counter, copper foils of various thicknesses were placed, and the number of pulses per minute was measured for the various thicknesses of these foils. As a result of the data obtained, it turned out that when a copper foil \(0.025\ \mathrm{mm}\) thick was present between the active side of the sector and the counter, the number of pulses per minute decreased by a factor of 2; hence, according to the equation

\[ N=N_{0}e^{-\mu x}, \tag{9} \]

where \(N\) is the number of pulses per minute and \(x\) is the thickness of the foil, it follows that \(\mu=276\ \mathrm{cm}^{-1}\).

After measuring the absorption coefficient, the initial activity \(N_{1}\) of the specimens was measured. The specimens were then heated at the required temperatures in quartz tubes, from which (to avoid oxidation of the copper) air was continuously pumped out during the entire heating, maintaining the pressure in them at a level of \(10^{-4}\ \mathrm{mm}\) Hg. To reduce evaporation of the radioactive copper, two specimens were placed in each quartz tube in such a way that their active sides were pressed against each other. To stop the process of intensive diffusion at high temperature at the required moment, the quartz tubes with the specimens were immersed in a tank of cold water, after which

APPLICATION OF RADIOACTIVE ISOTOPES

the samples were removed from the tubes and their new activity \(N_2\) was measured. The calculation of \(D\) was carried out as follows. The ratio of the number of pulses per minute after diffusion, \(N_2\), to the number of pulses per minute before the start of diffusion, \(N_1\), is a function of the diffusion coefficient \(D\), the absorption coefficient \(\mu\), the thickness of the plate \(l\), and the time \(t\) during which the diffusion process took place; but the time, absorption coefficient, and plate thickness are known, and therefore

\[ \frac{N_2}{N_1}=f(D,\mu,l,t)=f(D), \tag{10} \]

i.e., knowing \(\frac{N_2}{N_1}\), we can also find \(D\). The form of the dependence of \(\frac{N_2}{N_1}\) on \(D\), in the case of the arrangement of the sample and counter shown in

Fig. 7. Dependence of the relative activity on \(\mu^2Dt\) in case \(a\) (Fig. 6).

Fig. 8. Dependence of the relative activity on \(\mu^2Dt\) in case \(b\) (Fig. 6).

Fig. 6, \(a\), for a considerable thickness of the sample (sufficient so that the diffusing radioactive atoms do not reach the opposite side—practically \(l>3\) mm) will be as follows:

\[ \frac{N_2}{N_1} = e^{\mu^2Dt} \left[ 1-\frac{2}{\sqrt{\pi}} \int_0^{\sqrt{\mu^2Dt}} e^{-q^2}\,dq \right]. \tag{11} \]

Its graph is given in Fig. 7, where, in addition to the curve for \(l=\infty\), curves for finite values of \(l\) are also plotted \((l_3<l_2<l_1)\).

With the arrangement of the sample and counter shown in Fig. 6, \(b\), the dependence of \(\frac{N_2}{N_1}\) on \(D\) will be different; for it only a graph is given (Fig. 8). In this case it is necessary that the thickness

the sample was small; therefore we do not see on the graph the curve corresponding to the value \(l=\infty\).

\(D\) can also be found in another way. For this purpose, graphs are constructed of the dependence of the activity of the sample on the time of its heating at constant temperature (i.e., at a constant diffusion coefficient)

\[ \frac{N_2}{N_1}=f(t). \tag{12} \]

This graph, according to equation (11), must have almost the same form as the graph of the dependence \(f(D)\) in Fig. 7. Several such curves \(f(t)\) were constructed for various \(D\).

Fig. 9. Dependence of the relative activity on time for several diffusion coefficients.

From measurements of the activity of a sample held for various times at constant temperature, a number of values of

\[ \frac{N_2}{N_1}=f(t), \]

were obtained and plotted on the same graph. Then that curve \(f(t)\) was selected with respect to which the experimental points were located more symmetrically, and from \(f(t)\) the diffusion coefficient \(D\) was determined. The self-diffusion coefficient of copper was found in this way as well.

The experiments were carried out on samples of the same thickness at temperatures of 750, 850, and \(950^\circ\)C. Fig. 9 presents all the experimental results and nine curves \(f(t)\) corresponding to different \(D\). The selected curves are shown by solid lines. It turned out that the diffusion coefficients corresponding to these curves ensure a straight-line dependence of \(\ln D\) on \(\frac{1}{T}\) (Fig. 10).

As a result, it was obtained that:

\[ D_{\mathrm{Cu\ in\ Cu}}=11e^{-\frac{57200}{RT}}\ \mathrm{cm^2/sec}. \]

By means of this method the self-diffusion coefficients of iron15 in ferrite (α-phase) and austenite (γ-phase) were determined through the use of radioactive iron atoms \({}^{59}_{26}\mathrm{Fe}\) (\(T_{1/2}=47\) days). The measurements were carried out in the range from 751 to 887°C in the α-phase and from 935 to 1112°C in the γ-phase. When both curves were extrapolated to the ferrite-to-austenite transformation temperature (910°C), it turned out that the rate of self-diffusion in the α-phase is greater than in the γ-phase by approximately 150 times. In calculating \(D\), the thermal expansion of the specimen was also taken into account, which changed the value of \(D\) by 3—4%. The results of preliminary investigations may be written in the form of the relations:

Fig. 10. Comparative data on the temperature dependence of the self-diffusion coefficient of copper and of the diffusion coefficients of Be, Sn, Zn, Si in copper.

\[ D_{\mathrm{Fe\ in\ \alpha\text{-}Fe}}=340000e^{-\frac{77200}{RT}}\ \mathrm{cm^2/sec}; \]

\[ D_{\mathrm{Fe\ in\ \gamma\text{-}Fe}}=0.00104e^{-\frac{48000}{RT}}\ \mathrm{cm^2/sec}. \]

b) Case of an α-active indicator16—29.

As the measure of the absorptive capacity of a body for the α-radiation of an indicator one should choose the range of α-particles in the given body. The point is that α-particles in a solid body, and also in a gas (at the corresponding constant pressure), have a constant range (see the table on p. 430), and, consequently, if an atom, while diffusing, penetrates from the surface to a depth greater than the free path length, then the α-particle emitted by such an atom will be unable to reach the surface and will not be registered.

Table I

On the specimen (in particular, of PbCl₂), for which the diffusion coefficient \(D_{\mathrm{Pb\ in\ PbCl_2}}\) was to be measured,\(^{16,17}\) a radioactive isotope of lead, ThB (\(T_{1/2}=10.6\) hours), was deposited; on decaying, it gives the short-lived α-active ThC (\(T_{1/2}=61\) min.). After the establishment of radioactive equilibrium between ThB and ThC, the intensity of the α-particles was measured with an electrometer or a ZnS screen (by counting the number of scintillations). The specimen was then kept for some time at a high temperature, and the intensity measurement was again carried out. Knowing the range of the α-particles in the specimen and in air, and also the heating time \(t\), the diffusion coefficient can be determined from the decrease in intensity \(\dfrac{N_2}{N_1}\). The α-particle method made it possible to determine \(D\) of the order of \(10^{-8}\ \mathrm{cm^2/day}\).

Another variant of the use of an α-active indicator is based on the use of recoil atoms (in the case of ThC—ThC″ atoms). The exceptionally small range of the recoil atoms in PbCl₂ (see Table I), equal to only 100 atomic layers, makes it possible to determine exceedingly small diffusion coefficients, of the order of \(10^{-13}\ \mathrm{cm^2/day}\). The intensity of the emission of the recoil atoms can be judged from the γ- and β-activity of a negatively charged (to −220 V) copper plate located on the active side of the specimen. The collection of ThC″ atoms on this plate was begun after radioactive equilibrium had been reached among ThB, ThC, and ThC″. Measurement of the γ- and β-activity of ThC″ (\(T_{1/2}=3\) min.) was carried out with an electrometer. Knowing the range of the recoil atoms in the specimen and the heating time \(t\), the diffusion coefficient \(D\) can be calculated from the decrease in intensity \(\dfrac{N_2}{N_1}\).

The course of the calculations is as follows.\(^{16,18}\) If the activity of the ThC″ atoms before the experiment is equal to unity, and then became equal to \(\dfrac{N_2}{N_1}\), then \(1-\dfrac{N_2}{N_1}\) characterizes the number

atoms of ThC that have penetrated into the specimen in such a way that their recoil atoms cannot leave the specimen.

If the range of a recoil atom in the material of the specimen is denoted by \(a\), then, taking into account the randomness of the direction of their emission, we have:

\[ \frac{N_2}{N_1} = \int_0^a \frac{1}{\sqrt{\pi Dt}} \left(1-\frac{x}{a}\right) e^{-\frac{x^2}{4Dt}}\,dx, \tag{13} \]

where \(t\) is the time and \(x\) is the distance of the ThC atom from the surface.

Taking

\[ \xi=\frac{a}{2\sqrt{Dt}}, \tag{14} \]

we obtain:

\[ \frac{N_2}{N_1} = \frac{2}{\sqrt{\pi}} \int_0^\xi e^{-u^2}\,du - \frac{1}{\xi\sqrt{\pi}} \left(1-e^{-\xi^2}\right). \tag{15} \]

Equation (15) is solved graphically, and from it \(\xi\) is found, and then, with the aid of (14), \(D\).

When recording \(\alpha\)-particles, one must also take into account the circumstance that the ionizing effect produced by an \(\alpha\)-particle emitted from the surface will be greater than the ionizing effect of an \(\alpha\)-particle emitted from the specimen at a distance \(x\) from the surface.

Fig. 11. Temperature dependence of the diffusion coefficient of Pb in PbCl₂ and PbI₂.

With the aid of the method of \(\alpha\)-activity and recoil-atom activity, G. Hevesy and his co-workers determined the diffusion coefficients of Pb in PbCl₂ and Pb in PbI₂ \(^{16,17}\). As a result it was obtained:

\[ D_{\mathrm{Pb\ in\ PbCl_2}} = 1.06\cdot 10^7 e^{-\frac{38120}{RT}} \ \mathrm{cm^2/day}; \]

\[ D_{\mathrm{Pb\ in\ PbI_2}} = 3.43\cdot 10^4 e^{-\frac{30000}{RT}} \ \mathrm{cm^2/day}. \]

The discrepancy of the straight lines (Fig. 11) obtained from measurements of \(\alpha\)-activity and recoil-atom activity is connected with the inaccurate measurement of the range of the recoil atoms PbJ₂.

The absorption method with an \(\alpha\)-active indicator was also applied to the study of the self-diffusion of lead \(^{21,22}\) with the aid of the same ThB.

It was obtained:

\[ D_{\mathrm{Pb\ in\ Pb}} = 5.76\cdot 10^5 e^{-\frac{28050}{RT}} \ \mathrm{cm^2/day}. \]

It follows from this that at room temperature lead atoms exchange places once per day. If two pieces of lead were in close contact for 1600 million years at room temperature, then mutual penetration could be to a depth of no more than \(0.1\) mm.

3. METHOD OF THE LONGITUDINAL SECTION

To determine the distribution of indicator atoms in the body of a specimen after diffusion, alongside the method of successive layer removal and the absorption method, the method of the longitudinal section is also used. The essence of this method is as follows. One side of a rectangular specimen several centimeters long is covered with a thin layer of radioactive material whose diffusion coefficient in the specimen is to be measured. The specimen is subjected to diffusion annealing for a time \(t\). Then

Fig. 12. Arrangement of the specimen, screen, and counter in the case of the integral method.

the specimen is cut in a plane perpendicular to the surface on which the radioactive atoms were deposited, and with the aid of a detector the distribution of activity over the cut surface is investigated. Instead of cutting the specimen, one sometimes merely removes the surface layer from one of the lateral surfaces of the specimen. Removal of the surface layer is necessary, since it is known that in the surface layer the diffusion process proceeds more rapidly than inside the specimen.

To study the distribution of activity on the cut surface, two procedures are used—the differential and the integral. In the first, the activity of individual regions of the cut surface is measured directly. In the second, it is calculated on the basis of data on the change in the total activity of the cut surface as it is gradually covered by a screen opaque to radiation.

a) Integral method30. The specimen is placed in a lead box with the etched surface facing upward. Above the specimen there is a Geiger–Müller counter. Between the counter and the specimen a lead screen is placed, covering the specimen (Fig. 12).

The screen is then gradually shifted, exposing an ever larger portion of the section surface, and the activity is measured as a function of the position of the screen (the distance \(x\) from its edge to the boundary on which the radioactive layer has been deposited). In practice, the following procedure is used to determine \(D\).

First the screen is shifted so that a portion of the section surface of some width \(x_1\) is open (on the side of the radioactive layer), and the total activity \(N_1\) is measured (Fig. 12, a). Then the screen is displaced in the opposite direction (Fig. 12, b) so that the activity \(N_2\) of the open portion of the section surface (on the side opposite to the radioactive layer) would be equal to \(N_1\), and the width \(x_2\) of the section surface covered by the screen is measured. It can be shown that

\[ \left. \begin{aligned} N_1 &= \frac{J_0}{\sqrt{\pi}} \int_0^{\frac{x_1}{2\sqrt{Dt}}} e^{-z^2}\,dz,\\[6pt] N_2 &= \frac{J_0}{2}\left(1-\frac{2}{\sqrt{\pi}}\int_0^{\frac{x_2}{2\sqrt{Dt}}} e^{-z^2}\,dz\right), \end{aligned} \right\} \tag{16} \]

where \(J_0\) is a function of the initial concentration of radioactive atoms deposited on the surface of the specimen, the width of the specimen, and the absorption coefficient. Equating \(N_1=N_2\), we have:

\[ \frac{1}{2} = \frac{1}{\sqrt{\pi}}\int_0^{y_1} e^{-z^2}\,dz + \frac{1}{\sqrt{\pi}}\int_0^{y_2} e^{-z^2}\,dz, \tag{17} \]

where

\[ y_1=\frac{x_1}{2\sqrt{Dt}};\qquad y_2=\frac{x_2}{2\sqrt{Dt}}. \tag{18} \]

Expressing \(y_2\) in (17) through \(y_1\) with the aid of (18), we obtain:

\[ \frac{1}{2} = \frac{1}{\sqrt{\pi}}\int_0^{y_1} e^{-z^2}\,dz + \frac{1}{\sqrt{\pi}}\int_0^{\frac{x_2}{x_1}y_1} e^{-z^2}\,dz, \tag{19} \]

whence, by graphical methods, \(y_1=f\left(\frac{x_2}{x_1}\right)\) is found for the known \(\frac{x_2}{x_1}\). Knowing \(y_1\) and the annealing time \(t\), \(D\) is calculated from expression (18).

b) Differential method31–39. For direct measurement of the activity at each point of the cut surface, it is convenient to use, as a detector, a photographic plate exposed by simply placing the cut surface on the emulsion.

In this way the rate of volume diffusion of Na\(_{11}^{24}\) atoms (\(T_{1/2}=14.8\) hours) in glass was determined31. A radioactive isotope of sodium was applied to the end of a glass rod. Then this rod was heated for a certain interval of time, cooled, and a longitudinal cut was made. A photographic plate was placed against the plane of the cut, and after exposure and development its relative blackening was studied along the trace of the rod. Using this technique, it proved possible to construct a curve of the dependence of the logarithm of the relative concentration of diffusing atoms

Fig. 13. Dependence of the logarithm of the relative concentration of Na\(_{11}^{24}\) atoms on the square of the depth of their penetration into glass.

Na\(_{11}^{24}\) on the square of the depth of their penetration (Fig. 13). In Fig. 13, for comparison, a curve obtained by the method of removing layers and measuring their activity with a counter is also given. The identical slope of the two straight lines \(A\) and \(B\) indicates that the data for determining the diffusion rate of Na\(_{11}^{24}\) in glass at one and the same temperature by means of two different methods coincided.

The differential method using a photographic plate as detector has also found application in the study of surface diffusion or, in other words, the surface “creep” of atoms32–39. Thus, for example, the diffusion of α-active Po\(_{84}^{210}\) (\(T_{1/2}=138\) days) on silver was investigated32. After the end of a silver foil had been dipped into a solution containing polonium, the foil

was kept for 3 hours at \(450^\circ\text{C}\); as a result, measurement of the activity already showed appreciable uniformity. With a 45-hour hold at \(500^\circ\text{C}\), the entire silver foil was already uniformly covered with a layer of polonium, although the polonium had partly evaporated. In this work it was also established that polonium begins to evaporate at \(350^\circ\text{C}\), surface diffusion is already noticeable at \(300^\circ\text{C}\), and volume diffusion is not observed up to \(500^\circ\text{C}\), which may serve as an interesting illustration for comparing surface and volume diffusion. To determine the rate of surface diffusion of Po on silver, another method was also applied. A drop of a solution containing Po was deposited on silver foil, and the increase (as a result of heating) in the diameter of the spot formed on a photographic plate during contact exposure was measured.

Fig. 14. Dependence of the penetration of Po atoms in silver on time.

Using silver wire\(^{33}\) instead of foil and passing a direct current through it to maintain the required temperature, the time dependence of the penetration of the front of Po atoms toward the anode was obtained (Fig. 14). It turned out that the penetration rate in a field of \(1\ \dfrac{\text{volt}}{\text{cm}}\) is equal to: \(1.7\ \dfrac{\text{cm}}{\text{hour}}\) at \(350^\circ\) and \(3.2\ \dfrac{\text{cm}}{\text{hour}}\) at \(400^\circ\).

CONCLUSION

It now remains to compare these three methods.

The first method requires the removal of strictly parallel layers, and their thicknesses must be measured precisely. The thinner the layer removed, the more accurately \(D\) can be calculated, but the smaller the activity of the layer and, consequently, the lower the accuracy of its determination. Essential advantages of this method are its accuracy (provided all required conditions are observed), its universality with respect to the choice of indicators (\(\alpha\)-, \(\beta\)-, and \(\gamma\)-active), and the possibility of checking the correctness of the measurements of the activity and thicknesses of the removed layers by the correspondence of the experimental points to the linear dependence of \(\lg C\) on \(x^2\) (Fig. 4).

The absorption method does not require the rather laborious removal of layers and measurement of their activity several times—this is its advantage—but it requires knowledge of the coefficient

absorption coefficient \(\mu\) in the case of a \(\beta\)-active tracer, or the range of the particles in the case of an \(\alpha\)-active tracer. The latter, however, is not always obtainable with sufficient accuracy (which is bad, since \(\mu\) enters equation (11) squared). The use of \(\gamma\)-active tracers is undesirable because of the great penetrating power of \(\gamma\)-rays, as a result of which the diffusion of \(\gamma\)-active atoms into the depth of the specimen does not lead to a noticeable decrease in activity. As an advantage of the method based on the use of an \(\alpha\)-active tracer, one should note the possibility of measuring extremely small diffusion coefficients.

The longitudinal-section method requires neither measurement of the absorption coefficient nor the removal of layers, i.e., it eliminates the errors associated with these operations—this is its advantage. In integral form it has as yet practically not been applied anywhere, but it is clear that it can give good results only if the following conditions are observed:

a) the radioactive layer must be thin;

b) the time of diffusion annealing must be very long, in order to ensure appreciable penetration of the radioactive atoms into the depth;

c) the distances \(x_1\) and \(x_2\) (see Fig. 12) must be measured with the greatest accuracy (to \(0.001\ \text{cm}\), or even more accurately);

d) the length of the specimen must be large in comparison with \(x_1\) and \(x_2\).

The differential method of longitudinal section, using a photographic plate as detector, has all the shortcomings of photographic photometry, in particular laboriousness, low accuracy of measurements, and the necessity of a careful sensitometric study of the photographic emulsion. Other radiation detectors have not yet been used in the case of the differential method.

It should be supposed that the method of successive removal of layers will prove most convenient for the study of volume diffusion.

The radioactive-tracer method, as applied to the study of diffusion and especially self-diffusion, is undoubtedly beyond any competition with other methods (chemical, spectral, etc.). It is clear that the possibility of studying the rate of self-diffusion has directed the work of researchers chiefly in this direction, although the radioactive-tracer method can be applied just as successfully to the study of the diffusion of foreign atoms.

Among the shortcomings of the method it must be noted that not all radioactive isotopes can be used for tracer purposes, but only those that have a half-life long enough so that after diffusion annealing, which sometimes lasts several days, the activity would still be appreciable.

It should also be noted that the difference in the masses of the radioactive and stable isotopes and the effect of the radiation on the crystal lattice somewhat change the value of the diffusion coefficient. The absence of strict proportionality between the concentration of indicator atoms and the number of measured pulses also entails errors in the measurement of \(D\). However, the direct measurement errors usually completely mask the influence of the factors indicated.

Table II gives data on self-diffusion obtained with the aid of radioactive indicators, as well as by other methods.

Table II

Continuation of Table IV

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