Reaction depth

When, in NRA, resonances are used, and the depth profile of an isotope A is obtained from the excitation curve N(Eq), the reaction depth x is given by the requirement that projectiles incident with energy Eq are slowed down to the resonance energy Fr at X, which leads to ... 

As it will be shown in Chapter 6, for large reaction depths (times), many-particle effects begin to play an important role. This means that our simple estimates presented here are no longer valid. Since our derivation assumed that 0 > 1, the question of the range of applicability of these estimates will be discussed anew. 

All the above-said demonstrates well that there are arguments for and against applicability of the superposition approximation in the kinetics of bimolecular reactions. Because of the absence of exactly solvable problems, it is computer simulation only which can give a final answer. Note at once some peculiarities of such computer simulations. The largest deviations from the standard chemical kinetics could be expected at long t (large ). Unlike computer simulations of equilibrium phenomena [4] where the particle density is constant, in the kinetics problems particle density n(t) decays in time which puts natural limits on time of reaction. An increase of the standard deviation at small values of N(t) = (N) when calculating the mean concentration in computer simulations compel us to interrupt simulations at the reaction depth r = Io 3, where... 

Computer simulations of bimolecular reactions for a system of immobile particles (incorporating their production) has a long history see, e.g., [18-22]. For the first time computer simulation as a test of analytical methods in the reaction kinetics was carried out by Zhdanov [23, 24] for d, = 3. Despite the fact that his simulations were performed up to rather small reaction depths, To < 1, it was established that of all empirical equations presented for the tunnelling recombination kinetics (those of linear approximation - (4.1.42) or (4.1.43)) turned out to be mostly correct (note that equations (5.1.14) to (5.1.16) of the complete superposition approximation were not considered.) On the other hand, irrespective of the initial reactant densities and space dimension d for reaction depths T To his theoretical curves deviate from those computer simulated by 10%. Accuracy of the superposition approximation in d = 3 case was first questioned by Kuzovkov [25], it was also... 

Note that if primitive approach (5.1.64) is valid at t < 1 only (small reaction depths r < 0.5), the linear approximation has the greater applicability range, r < 1 whereas at r > 1 the linear approximation begins to deviate considerably from computer simulations. 

For unequal concentrations, n (t) < ns(t), the reaction depth r < T0 = 3 reached in computer simulations is not enough for finding asymptotic laws but still permits to estimate qualitatively the accuracy of the superposition approximation. In Figs 5.7 and 5.8 numerical solution of the relevant kinetic equations is compared with computer simulations. To make situation more transparent, the linear approximation results are plotted in curves (d) for a single choice of initial concentrations only. 

An increase of the standard deviation at r 3 due to small number of survived particles, demonstrates a limited possibility of the direct statistical simulations for a system with a variable number of particles. However, certain conclusions could be drawn even for such limited statistical information. Say, if for equal concentrations the analytical theory based on the superposition approximation seems to be quite adequate, for unequal concentrations its deviation from the computer simulations greatly increases in time. The superposition approximation gives the lower bound estimate of the actual kinetic curves tia( ) but if for d = 2 shown in Fig. 5.8 the deviation is considerable, for d, = 1 (Fig. 5.7) it is not observed, at least for the reaction depths considered. 

More concrete conclusions could be drawn for the linear approximation applicability it is adequate for small reaction depths r < 1, whereas at r > 1 it is in serious error. In its turn, errors of the superposition approximation are essentially less, the relevant lower bound estimate is quite acceptable to fit theoretical parameters to the experimental curves. 

As it took place for the tunnelling recombination, divergence in results is not large. It will be shown in Chapter 6 that the reaction depths studied here are enough to establish appearance of the new asymptotic kinetic laws. The superposition approximation giving a lower bound estimate of the kinetics, reproduces correctly the kinetics at long times. Results of the linear approximation are not plotted since they diverge considerably from the statistical simulations. 

Summing up, note that the direct statistical (computer) simulation does not demonstrate serious errors of the superposition approximation for equal reactant concentrations. Divergence begins first of all for unequal concentrations for not very large reaction depths T 2 it is almost negligible, at T > 2 and especially asymptotically (as t —> oo) it becomes important, but the complete quantitative analysis cannot be done due to unreliable statistics of results. In this Section we have restricted ourselves to the A + B —> 0 reaction without particle generation. Testing of the superposition approximation accuracy for the case of particle creation will be done in Chapter 7. 

The value of R = 1 [by the definition of R and the dimensionless concentration n(R), such that n(0) = 1] corresponds to the case when the reaction has destroyed pairs AB separated by the relative distances r less or equal to the mean distance between particles. In other words, according to equation (6.1.73), a new asymptotic law with a = 1/2 occurs already at very small reaction depths, r 0.5 ... 

Direct establishment of the asymptotic reaction law (2.1.78) requires performance of computer simulations up to certain reaction depths r, equation (5.1.60). In general, it depends on the initial concentrations of reactants. Since both computer simulations and real experiments are limited in time, it is important to clarify which values of the intermediate asymptotic exponents a(t), equation (4.1.68), could indeed be observed for, say, r 3. The relevant results for the black sphere model (3.2.16) obtained in [25, 26] are plotted in Figs 6.21 to 6.23. The illustrative results for the linear approximation are also presented there. 

A comparison with the correlation dynamics of the A + B —> 0 reaction, equations (5.1.33) to (5.1.35), shows their similarity, except that now several terms containing functionals J[Z have changed their signs and several singular correlation sources emerged. The accuracy of the superposition approximation in the diffusion-controlled and static reactions was recently confirmed by means of large-scale computer simulations [28]. It was shown to be quite correct up to large reaction depths r = 3 studied. 

The reaction depth correlates to the electron donor ability of Red and the stability degree of M2+(Ox ) complex. The complexation causes an anodic shift of metal redox potentials, which reaches almost 100 mV for transition metal cations (Maletin et al. 1979, 1980, 1983). 

Fig. 31. Plot of interfacial width w vs D for a blend of olefinic copolymers d75/h66 (cf.text) at T0=356 K, extracted from nuclear reaction depth profiling experiments, based on the reaction 3He+2H 4He+1H+18.35 MeV and backward angle detection of 1H. Note that the spatial resolution is optimal near the air surface (4 nm) but quickly deteriorates for large distances from the air surface. Therefore, the error bar on w/2 increases strongly with increasing D. Full and dotted curves represent the approximate asymptotic formula w2 = Wq + D / 4 (a factor 7tco/(l+co/2) in Eq. (127) being approximated as unity), choosing w0= b> and b=l 1 8 nm or b=10.6 nm, respectively. From Kerle et al. [84]...

J.A. Sawicki, J. Roth, L.M. Howe, Thermal release of tritium implanted in graphite studied by T(d,a)n nuclear reaction depth profiling analysis, J. Nucl. Mater. 162-164 (1989) 1019... 

If for equal particle concentrations the reaction depth D 3 is enough to confirm the asymptotic reaction law (it is well demonstrated in [21] for d 2), it is no longer true for unequal concentrations, np, t) < In... 

A type of angle-dependent x-ray photoemission spectroscopy was used to investigate the molecular orientation at the surface of sulfonated polystyrene as a function of reaction depth. A model based on these measurements indicates that at a critical sulfonation depth the aliphatic hydrocarbon backbone becomes exposed preferentially at the surface. These results are consistent with surface energy and tribo-electric charging measurements, which also reveal the effects of associative interactions in the form of conversion dependencies. 

In 1922, T. De Donde (1870-1957) proposed to use the derivative of free enthalpy over the reaction depth as parameter of chemical affinity. Its value shows the change in maximum useful work with each 1 mole step of a reaction and is measured in joules per 1 mole, i.e., it shows the propellant force of the reaction per 1 mole of the substance (Figure 1.9). Positive values of chemical affinity A. indicate the excess of reactants and the reactions direction from left to right, negative ones, the excess of products and reactions direction right to left in the reactions equation. In the process of relaxation A tends to 0. 

From equation (1.85) follows that the distance, which remains for the reaction to proceed to the end may be evaluated from the reaction depth ... 

There are additional considerations for CPAA. The requirement for an accelerator essentially limits the use to some substantial facilities the primary applications of which are in other research fields, and CPAA must obtain a share of the beam time. Generally, only one beam is available, i.e., only one sample or one comparator standard is irradiated at a time. These disadvantages maybe outweighed by combining the quantitative information with spatial and depth information, which can be obtained using focused particle beams and modulation of particle energy to change reaction depth. CPAA has been discussed in more detail by Strijckmans (1994). 

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