Clearly, in the case of non-symmetrical periodic inhomogeneities, including two-dimensional (that is, depending on two spatial coordinates), the maximal reduc­tion in the energy deposition must be manifested when there is a rectangular profile of these non-uniformities. Let us examine the influence of the presence of craters with vertical walls and an arbitrary base (Fig. 7.8) on the efficiency of the energy deposition. Let us introduce the dimensionless parameters of the inhomogeneities:

Let D1(8+ + 😎 be the depth of the craters; a the area of their base; and S the equivalent area of the surface of the elementary cell of the active layer containing one crater, defined as the ratio of the total surface of the active layer to the total number of craters. As in the one-dimensional model, we shall assume that the maximal deviation of the thickness of the active layer from its average value does not exceed the range R1 by an order of magnitude (that is, 8±D1 < 1), and the distance between the nearest craters markedly exceeds the fragment range in the layer material:

S1/2 » R1. (7.16)

From the conservation of the average thickness of the active layer (S — a)8+D1 = a8-D1 it follows that:

8+ = 2 S 8; 8- = 2 ^ 8.

If D — = 0, that is, the depth of the craters reaches the substrate, then 8- = 1. Hence

in accordance with the second equation (7.17) we find that for a given ratio of areas ff/S, the value of the parameter S cannot exceed

and the minimal value which the ratio ff/S can assume for a given S is

The effective energy deposition in the two-dimensional case is determined by an equation analogous to Eq. (7.14), in which integration is performed over the area of the equivalent cell active layer surface S. For the crater model in question, this equation boils down to the simple expression

Figure 7.9 shows the results of calculations for the model of craters with vertical walls when D0 = 0.5 in a system comprised of two plane parallel plates.

Let us transform Eq. (7.19) using Eqs. (7.15) and (7.17):

є = є^-)[1 — 2(1 — ff/S)S]ff + є^+)(1 + 2Sff/S)(1 — ff/S).

S

Substituting the value of Smax from Eq. (7.18), which clearly corresponds to the minimal possible value of є, we get emin = ^D+). Thus the limit minimally possible value of є for a non-uniform layer is equal to the value of є which has a uniform layer with a normalized thickness equal to the maximal thickness of the non-uniform layer. Such correlations are also not hard to write for the model of

Fig. 7.9 Dependence of the share of energy of fission fragments efficiently transmitted to the gas on the relative area of the craters: (1)D1 = 0.5; (2)D1 = 1; the solid line is S = 0.5; the dotted line is S= 1

Fig. 7.10 Cross section of the layer of uranium oxide — protoxide: (1) substrate of aluminum; (2) layer of uranium oxide-protoxide; (3) epoxide resin

active layers with two-dimensional projections having vertical walls, which give results analogous to the crater model.

Of course, the models we have discussed are only an illustration of the signif­icant influence of inhomogeneities; they cannot serve as a method providing precise results of calculation of e, because the spectrum of possible configurations of real inhomogeneities, even for one specific layer, is hard to convey mathematically. Figure 7.10 shows photograph [24] enlarged roughly 1,000 times of the cross section of one of the layers of the uranium oxide-protoxide used in NPL experi­ments at VNIIEF. The best method of allowing for the influence of inhomogeneities in determining e is evidently to introduce the corrective multiplier e = fie(pi), which is found from a comparison of the experimentally determined efficiency of energy deposition to the gas with the computed efficiency e(D), obtained on the assumption of uniformity of the active layer, D1 = D1.

The circumstance that a maximal reduction in the energy deposition occurs for inhomogeneities having a rectangular profile in the cross section of the active layer makes it possible to determine the lower limit of the possible drop in the energy deposition to the gas, and to select acceptable values of the amplitude of the inhomogeneities. Analysis conducted on the results of calculations showed that if the amplitude of fluctuations of the inhomogeneities does not exceed one half the thickness of the active layer, then the maximal possible reduction in the energy deposition will be no more than 10 %. However, it should be noted that this conclusion is valid provided that the conditions (7.12) and (7.13), or (7.16), are met. Otherwise, a reduction in the energy deposition can prove more significant. The reason for a marked reduction in the energy deposition could be, for example, screening of the escape of fragments to the gas by the inhomogeneities themselves, when h < R1.

In study [21], in several specific examples, the influence of periodic one-dimensional inhomogeneities of the active layer, determined by inhomogene­ities of the surface of the substrate on which this layer was deposited, was examined computationally. The inhomogeneities had a rectangular profile. The width of the projections of the non-uniformities is equal to their height, while the period of the non-uniformities is equal to twice their height.

The surface of the uranium turned toward the gas was considered ideally smooth. Calculations of the energy deposition of the fragments were carried out by the Monte Carlo method, which automatically led to allowance for the screening factor of substrate projections in the active layer. As the material of the active layer, uranium dioxide UO2 was considered (R1 ~ 8 qm), and the substrate material was Al and W. The height of the inhomogeneities was varied within the limits of 2-10 qm. The results of these calculations agreed with the results discussed above: with the growth in amplitude of the inhomogeneities, the efficiency of the energy deposition drops, and when the amplitude corresponds to S = 1 in Eq. (7.11), it decreases by ~50 %. This last evidently testifies to a relatively weak influence of screening (at least for the examples considered).

In contrast to direct numerical calculations for a specific geometric profile of the active layer and its composition, the methods of (7.14) and (7.19) in the limits of the constraints (7.12) and (7.13) have greater universality. These methods used the formulas (7.11) and (7.14)-(7.19) and the methods of calculations from studies [14—16], in which the efficiency of the energy deposition of both Є and e(D1) are determined by means of the normalized thickness of the active layer D1 = SU/R1. Because the range R1 is inversely proportional to the density of the decelerating medium, in the utilized approach, the inhomogeneities of the normalized thickness of active layers include both the inhomogeneities of the real thickness, provoked by roughness of the geometric surfaces bounding these layers, and the inhomogeneities of density of the layer material.