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Tracer atoms

Radioactive tracer atoms have been used to study chemical reaction mechanisms in the production of TNT, RDX/HMX and NC. The results of these three independent investigations at FicArsn are summarized below ... [Pg.393]

For example, suppose a planar layer of N tracer atoms is the starting point, and suppose that each atom diffuses from the interface by a random walk in a direction perpendicular to the interface, in what is effectively one-dimensional diffusion. The probability of a jump to the right is taken to be equal to the probability of a jump to the left, and each is equal to 0.5. The random-walk model leads to the following result ... [Pg.213]

The path that the diffusing atom takes will depend upon the structure of the crystal. For example, the 100 planes of the face-centered cubic structure of elements such as copper are identical to that drawn in Figure 5.7. Direct diffusion of a tracer atom along the cubic axes by vacancy diffusion will require that the moving atom must squeeze between two other atoms. It is more likely that the actual path will be a dog-leg, in <110> directions, shown as a dashed line on Figure 5.7. [Pg.217]

Figure 5.17 Correlated motion during vacancy diffusion (a) vacancy can jump to any surrounding position and its motion follows a random walk (b, c) the motion of a tracer atom is correlated, as a jump into a vacancy (b) is most likely to be followed by a jump back again (c). Figure 5.17 Correlated motion during vacancy diffusion (a) vacancy can jump to any surrounding position and its motion follows a random walk (b, c) the motion of a tracer atom is correlated, as a jump into a vacancy (b) is most likely to be followed by a jump back again (c).
However, the diffusion of a tracer atom by the mechanism of vacancy diffusion is different. A tracer can only move if it is next to a vacancy, and in this case, the tracer can only jump to the vacancy (Fig. 5.17b). The possibility of any other jump is excluded. Similarly, when the tracer has made the jump, then it is equally clear that the most likely jump for the tracer is back to the vacancy (Fig. 5.17c). The tracer can only jump to a new position after the vacancy has diffused to an alternative neighboring position. [Pg.229]

In the case of interstitial diffusion in which we have only a few diffusing interstitial atoms and many available empty interstitial sites, random-walk equations would be accurate, and a correlation factor of 1.0 would be expected. This will be so whether the interstitial is a native atom or a tracer atom. When tracer diffusion by a colinear intersticialcy mechanism is considered, this will not be true and the situation is analogous to that of vacancy diffusion. Consider a tracer atom in an interstitial position (Fig. 5.18a). An initial jump can be in any random direction in the structure. Suppose that the jump shown in Figure 5.18b occurs, leading to the situation in Figure 5.18c. The most likely next jump of the tracer, which must be back to an interstitial site, will be a return jump (Fig. 5.18c/). Once again the diffusion of the interstitial is different from that of a completely random walk, and once again a correlation factor, / is needed to compare the two situations. [Pg.229]

Suppose that a very thin planar layer of radioactive Au tracer atoms is placed between two bars of Au to produce a thin source of diffusant as illustrated in Fig. 5.8. A diffusion anneal will cause the tracer atoms to spread by self-diffusion as illustrated in Fig. 5.3. (A mathematical treatment of this spreading out is presented in Section 4.2.3.) Suppose that the diffusion ex-... [Pg.116]

Figure 5.8 Thin planar tracer-atom source between two long bars. Figure 5.8 Thin planar tracer-atom source between two long bars.
The tracer atoms will spread out as they would in the absence of current however, they will also be translated bodily by the distance Ax = (VA)t relative to an embedded inert marker as illustrated in Fig. 5.9. [Pg.117]

Figure 5.9 (a) The initially thin distribution of tracer atoms that, subsequently, will... [Pg.117]

Equation 8.19 contains the correlation factor, f, which in this case is not unity since the self-diffusion of tracer atoms by the vacancy mechanism involves correlation. Correlation is present because the jumping sequence of each tracer atom produced by atom-vacancy exchanges is not a random walk. This may be seen by... [Pg.171]

A rough estimate for f can be obtained based on the number of nearest-neighbors and the probability that a tracer atom which has just jumped and vacated a site will return to the vacant site on the vacancy s next jump. A vacancy jumps randomly into its nearest-neighbor sites, and the probability that the return will occur is 1 jz. This event will then occur on average once during every 2 jumps of an atom. For each return jump, two atom jumps are effectively eliminated by cancellation, and the overall number of tracer-atom jumps that contribute to diffusion is reduced by the fraction 2/z. According to Eq. 8.3, D is proportional to the product Tf, and since the number of effective jumps is reduced by 2/z, f can be assigned the value f 1 - 2/z = 0.83 for f.c.c. crystals. More accurate calculations (see below) show that f = 0.78. [Pg.172]

In Section 3.1.1, self-diffusion was analyzed by studying the diffusion of radioactive tracer atoms, which were isotopes of the inert host atoms, thereby eliminating any chemical differences. Possible effects of a small difference between the masses of the two species were not considered. However, this difference has been found to have a small effect, which is known as the isotope effect. Differences in atomic masses result in differences of atomic vibrational frequencies, and as a result, the heavier isotope generally diffuses more slowly than the lighter. This effect can—if migration is approximated as a single-particle process—be predicted from the mass differences and Eq. 7.14. If mi and m2 are the atomic masses of two isotopes of the same component, Eqs. 7.13 and 7.52 predict the jump-rate ratio,... [Pg.174]

Calculate the correlation factor for tracer self-diffusion by the vacancy mechanism in the two-dimensional close-packed lattice illustrated in Fig. 8.22. The tracer atom at site 7 has just exchanged with the vacancy, which is now at site 6. Following Shewmon [4], let p. be the probability that the tracer will make its next jump to its kth nearest-neighbor (i.e., a 7 — k jump). 6k is the angle between the initial 6 —> 7 jump and the 7 —> k jump. The average of the cosines of the angles between successive tracer jumps is then... [Pg.195]

Figure 8.22 Two-dimensional close-packed lattice. The tracer atom at 7 has just... Figure 8.22 Two-dimensional close-packed lattice. The tracer atom at 7 has just...
Figure 9.7 (a) Grain boundary moving with constant velocity v. Tracer atoms are... [Pg.217]

Fisher has produced a relatively simple solution for a specimen geometry that is convenient for experimentalists and which has been widely used in the study of boundary self-diffusion by making several approximations which are justified over a range of conditions [9, 10]. The geometry is shown in Fig. 9.8 it is assumed that the specimen is semi-infinite in the y direction and that the boundary is stationary. The boundary condition at the surface corresponds to constant unit tracer concentration, and the initial condition specifies zero tracer concentration within the specimen. Rapid diffusion then occurs down the boundary slab along y, while tracer atoms simultaneously leak into the grains transversely along x by means of crystal diffusion. The diffusion equation in the boundary slab then has the form... [Pg.218]

Figure 9.8 Isolated-boundary (Type-B) self-diffusion associated with a stationary grain boundary, (a) Grain boundary of width 6 extending downward from the free surface at y = 0. The surface feeds tracer atoms into the grain boundary and maintains the diffusant concentration at the grain boundary s intersection with the surface at the value cB(y = 0, t) = 1. Diffusant penetrates the boundary along y and simultaneously diffuses transversely into the grain interiors along x. (b) Diffusant distribution as a function of scaled transverse distance, xi, from the boundary at scaled depth, yx, from the surface. Penetration distance in grains is assumed large relative to 5. Figure 9.8 Isolated-boundary (Type-B) self-diffusion associated with a stationary grain boundary, (a) Grain boundary of width 6 extending downward from the free surface at y = 0. The surface feeds tracer atoms into the grain boundary and maintains the diffusant concentration at the grain boundary s intersection with the surface at the value cB(y = 0, t) = 1. Diffusant penetrates the boundary along y and simultaneously diffuses transversely into the grain interiors along x. (b) Diffusant distribution as a function of scaled transverse distance, xi, from the boundary at scaled depth, yx, from the surface. Penetration distance in grains is assumed large relative to 5.
Fick s first law, in the form given by Equation 5.12, allows us to define the tracer- and the self-diffusion coefficients. Diffusion of a tracer isotope is the case when a diffusing atom, which is marked by their radioactivity of by their isotopic mass (see Figure 5.1) [7], is introduced in an extremely dilute concentration in an otherwise homogeneous crystal with no driving force [4], In this case, the tracer gradient of concentration will give rise to a net flow of tracer atoms. Consequently,... [Pg.222]

Two alternative approaches exist. The first one involves significantly lowering the temperature to values where the diffusion of vacancies can be observed with a technique like STM. At lower temperatures a surface vacancy can then be artificially created by ion bombardment or direct removal of an atom by the tip. This approach has been applied successfully to several semiconductor surfaces [29-31]. For metal surfaces, although vacancy creation at a step by direct tip manipulation of the surface has been demonstrated [32], to our knowledge, no studies have been published where the diffusion of artificially created vacancies in a terrace has successfully been measured. The second approach involves the addition of small amounts of appropriate impurities that serve as tracer atoms in the first layer of the surface [20-24]. The presence and passage of a surface vacancy is indirectly revealed by the motion of these embedded atoms. If one seeks to measure both the formation energy and the diffusion barrier of surface vacancies explicitly, a combination of these two approaches is needed. [Pg.353]

The problem of vacancy-mediated tracer diffusion in two dimensions has been investigated for a long time [40-44] and several different methods (simulation, analytical models, enumeration of trajectories, etc.) can be used to address it. The mathematics of this type of diffusion was solved first for the simplest case [41], when the diffusion of the vacancy is unbiased (all diffusion barriers are equal the tracer atom is identical to the other atoms), the lattice is two-dimensional and infinite. There is a single vacancy present that makes a nearest-neighbor move in a random direction at regular time intervals and has an infinite lifetime, as there are no traps. The solution is constructed by separating the motion of the tracer and that of the vacancy. The correlation between the moves of the tracer atom is calculated from the probability that the vacancy returns to the tracer from a direction, which is equal, perpendicular or opposite to its previous departure. The probability density distribution of the tracer atom spreads with... [Pg.357]

In a very recent study the lattice calculations have been generalized to biased diffusion [44]. The difference between the tracer atom and the substrate atoms was taken into account by having different vacancy-tracer and vacancy-substrate exchange probabilities, while the rate of vacancy moves was kept constant. A repulsive interaction reduces, while a moderately attractive interaction increases the spreading of the tracer distribution. [Pg.358]

In this subsection we describe a discrete model for vacancy-mediated diffusion of embedded atoms, solve it numerically for the case of In/Cu(0 0 1), and present the results. Our model is defined on a two-dimensional simple square lattice of size / x / (typically, l = 401) centered around the origin. This corresponds to the top layer of a terrace of the Cu(00 1) surface, with borders representing steps. The role of steps in the creation/annihilation of vacancies will be discussed in more detail in the next section. All sites but two are occupied by substrate atoms. At zero time the two remaining sites are the impurity (or tracer) atom, located at the origin, and a vacancy at position (1,0). This corresponds to the situation immediately after the impurity atom has changed places with the vacancy. [Pg.358]

The only allowed motion is the exchange of the vacancy with one of its neighboring atoms. The exchange rate depends on the local environment, i.e. on the relative position of the vacancy and the impurity atom. This takes into account the effect of the lattice stress induced by the tracer atom on the energy landscape observed by the vacancy. Each rate is simply proportional to the Boltzmann factor e-AE/kBT wjjere is the activation energy for the considered diffusional move and kBTis the thermal... [Pg.358]

When treating the above model numerically, we separate the motion of the vacancy and the tracer atom, as has been performed also in some of the analytical treatments referred to in Section 3.1. In our case of a finite lattice, this separation introduces an approximation, which is valid only if the tracer atom is relatively close to the middle of the lattice. First, we calculate the probabilities that the vacancy, released at one atomic spacing from the tracer, returns the first time to the tracer from equal (/ eq), perpendicular (/ perp) or opposite (popp) directions we also calculate the probability of its recombination (Prec) at the perimeter instead of returning to the tracer. Knowing these return and recombination probabilities, we calculate the statistics of the motion of the tracer atom, which performs a biased random walk of finite length. The probability distribution of the direction of each move with respect to the previous one, and the probability that a move was the last one, are obtained from the return and the recombination probabilities. [Pg.359]

In practice, both the return probabilities of the vacancy and the displacement distribution of the tracer atom are obtained via direct evaluation of probabilities (enumeration of trajectories), which has better convergence properties than Monte-Carlo-type methods. [Pg.359]

Using this, the tracer atom is described as if it forms a pair with the vacancy on one of the bonds adjacent to its original site, it walks on the bond lattice, and at the end of the walk (which happens after each move with probability prec) it is released with equal probability at either end of the last visited bond. Results for the probabilities of the different jump lengths (beginning-to-end vectors of these trajectories) are shown in Fig. 9. Note, that the model calculations in Fig. 9 contain no adjustable parameters. [Pg.361]

Figure 9 The probabilities of the jump lengths of the tracer atom for T = 320 K and l = 401 lattice spacings. Filled circles correspond to experimental values (measured at this temperature and terrace size), open circles are from the model described in the text, and the solid curve is the continuum solution described in Section 3.3. The distribution depends only on the magnitude of the jump lengths with no directional dependence. (Each dataset is normalized separately such that the probabilities corresponding to a subset of the jump vectors, 1 < r < 6, add up to unity. These are the probabilities that are determined with good accuracy in the experiment.)... Figure 9 The probabilities of the jump lengths of the tracer atom for T = 320 K and l = 401 lattice spacings. Filled circles correspond to experimental values (measured at this temperature and terrace size), open circles are from the model described in the text, and the solid curve is the continuum solution described in Section 3.3. The distribution depends only on the magnitude of the jump lengths with no directional dependence. (Each dataset is normalized separately such that the probabilities corresponding to a subset of the jump vectors, 1 < r < 6, add up to unity. These are the probabilities that are determined with good accuracy in the experiment.)...
A general advantage of numerical modeling is that we have access to quantities which are difficult or impossible to measure experimentally. One example in our calculation is the probability that a tracer atom had an encounter with a vacancy, but its net displacement was zero. Although this value is non-zero, its temperature dependence is weak, which means that it can be incorporated in the constant jump-rate prefactor. This justifies the simplifying approach in the analysis of the experimental measurements to associate In-vacancy encounters with detectable (non-zero) jumps of the indium atom. [Pg.362]

The final distribution of the tracer atom after the vacancy has recombined is obtained by the integration of the loss term in Eq. (4) ... [Pg.362]

If we assume that the steps are indeed the sources and sinks for surface vacancies and we confine ourselves to the simplest case where there is no interaction between the vacancy and the tracer atom, the recombination probability of a vacancy, prec, introduced in the previous section, will decrease with increasing distance from a step. This is schematically illustrated in Fig. 10. This decrease in prec with distance from a step allows us to experimentally verify whether the steps are indeed the sole sources and sinks for vacancies. The experimental verification consists of the following. Assume that we are tracking the motion of an embedded atom somewhere in a terrace, a given distance away from a step. Once a vacancy has formed at the step, has diffused to the embedded atom, and has had an initial exchange with the atom (i.e. in our measurements we observe the embedded atom to make a jump), the probability for it to have further encounters with the same vacancy is determined by the value 1 — prec. Since prec decreases with increasing distance to the step, the vacancy will on average have more encounters with the tracer atom the further it is away from the step. This will cause the atom... [Pg.363]

Figure 10 A schematic illustration of the effect of the presence of a step on the diffusion of a surface vacancy, (a) Schematic topography, with a step in the middle, (b) The recombination probability depends logarithmically on the distance, (c) Random walks that bring the vacancy far from the step will result on average in a much larger number of encounters with a tracer atom on the terrace than shorter random walks. Figure 10 A schematic illustration of the effect of the presence of a step on the diffusion of a surface vacancy, (a) Schematic topography, with a step in the middle, (b) The recombination probability depends logarithmically on the distance, (c) Random walks that bring the vacancy far from the step will result on average in a much larger number of encounters with a tracer atom on the terrace than shorter random walks.
In this section, we discuss the energetics of the vacancy-mediated diffusion of tracer atoms in a surface. In particular, we focus on the energetics and the profound differences that are encountered in interpreting the results from the Pd/Cu(0 01) and the In/Cu(0 01) experiments. [Pg.364]


See other pages where Tracer atoms is mentioned: [Pg.390]    [Pg.578]    [Pg.581]    [Pg.206]    [Pg.42]    [Pg.117]    [Pg.172]    [Pg.200]    [Pg.200]    [Pg.200]    [Pg.215]    [Pg.94]    [Pg.123]    [Pg.358]    [Pg.359]    [Pg.360]    [Pg.360]   
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