One way to raise the total efficiency of nuclear-laser facilities is to increase the efficiency of use of the nuclear fission energy generated in the uranium-containing layers of the laser channels. This efficiency depends on the thickness, composition, and degree of uniformity of the uranium-containing layers, the geometry of the lasers themselves, and the composition and density of the laser-active gas.

The basic problem of optimization is to force the fission fragments to inject as much energy as possible into the gas mixture. The energy absorbed in the gas is proportional to D1e(D0, D1) (see Sect. 7.3 of this chapter). With an increase in D1, the value of D1e(D0, D1) at first increases nearly linearly (see Fig. 7.5), but as D1 continues to increase, the rate of increase of D1e(D0, D1) decreases, because segments of the uranium layer far from the gas inject more and more energy into the layer itself, and less and less into the gas. When D1 = 1, D1e(D0, D1) reaches a maximal value fission fragments from the most remote layers of uranium do not reach the gas at all. Saturation begins with D1« 0.5. The energy intensity in the substrates with the uranium layer with an increase in thickness of this layer grows linearly, while the energy injection in the gas, starting with D1 > 0.5, grows insignificantly (<6 %). Consequently, the thickness of the uranium layer D1 ~0.5 can be considered optimal.

The value of є falls monotonously with an increase in D1. The drop continues even with D1 > 1. Thus with regard to D1, the functions єф0, D1) and D^(D0, D1) behave in the opposite manner: the thicker the active layer, the less efficiently it is used, but the greater the total energy deposition to the gas.

With regard to the normalized transverse dimension of the laser channel D0, the functions єф0, D1) and D^(D0, D1) behave identically (see Fig. 7.5, for example), because both are proportional to the total energy absorbed in the gas. For low values of D0, which correspond to small gas densities, the functions єф0, D1) and D^(D0, D1) are small, because fragments intersect the laser channel with small energy losses. As D0 increases, the functions in question also increase and for D0 ~ 1 they reach saturation—then fragments injected in the gas transfer nearly all of the energy to it.

The characteristics of NPLs excited by fission fragments are determined by many factors, which also include the formation and development of optical inho­mogeneities of the medium, that is, inhomogeneities of the gas density. Next we will examine the link between the efficiency of the energy deposition of fragments and the growing optical inhomogeneities. We shall limit ourselves to the mode of short irradiation of the gas in the laser channel, which is of practical importance, that is, to pulsed excitation of the cells or excitation of a portion of the gas when there is continuous flowing of the gas medium through the laser channel under the conditions of stationary irradiation with neutrons. The characteristic time of equal­ization of the gas temperature in the laser cell is tt~ d2/a ~ 1 s. If the excitation time of the gas in the channel is sufficiently small (t1/2 < 10~2 s), then heat exchange has a marked effect only on the thin wall layer of gas with a thickness of ~1 mm [19, 20, 25, 27, 38, 39]. A “passive zone,” with a large positive density gradient, is formed here which is not part of the volume involved in lasing. In the remaining part of the channel, the processes of heat conductivity do not have time to markedly influence the distribution of gas density arising in the course of irradiation (see Chap. 8). The distribution of the gas temperature and density is determined by the distribution

function of the specific fragment energy deposition F ^ r ^. Because this function

drops with distance from the channel walls [14], the gas density increases with approach to the channel axis; a focusing gas lens is formed. During lasing, light beams both from the axial and from the peripheral regions of the active part of the laser volume oscillate around the optical axis of the system during their multiple passage of the laser cavity. Consequently, the axial region can have the greatest influence on the parameters of the laser radiation.

The degree of non-uniformity of the energy deposition of fission fragment can be determined by the ratio

F1 = F(0)/F1,

where F(0) is the specific deposition at the center of the channel (minimal value), and F1 is the maximal value of the specific energy deposition to the gas (directly at the surface of the uranium layer).

Fig. 7.15 Dependence of the coefficient of non-uniformity of the energy deposition i (a) and i (b) on the normalized parameters Dj and D0: Curves 1-5 correspond to values of D0 equal to: 1.0; 0.8; 0.6; 0.4; 0.2

The relative share of the energy deposited by fragments in the central part of the channel is characterized by the expression

I = F(0)/(F>, (7.24)

where (F> is the average value of the specific energy deposition along the channel section. Figure 7.15 shows the dependencies i and calculated from the correla­tions of studies [14, 16] for a system formed by two parallel infinitely extended flat uranium layers.

For all the clarity of the coefficient i1 for optimization of the energy deposition, it is more convenient to use the parameter i. The fact is that the function F ^ r ^ has

the greatest gradient and reaches the greatest value on the channel wall; however, it is in the wall region that the “passive zone” is formed such that light beams are bent on the channel wall.

The value of i with growth in the relative width of the channel D0 decreases monotonously, while the efficiency є of the energy deposition grows monotonously, and starts to reach saturation in the region of D0 > 0.4 [14]. Because the total energy deposited in the gas by the fragments is proportional to D^(D0, D1), the energy released by them in the axis of the laser channel can be characterized by the dimensionless parameter

m = DJє|I. (7.25)

Figure 7.16 shows the dependence of the parameter m on the normalized values D0 and D1 for a system of two plane-parallel infinitely extended layers. In the region of D0 ~ 0.4, the parameter m has a clearly pronounced maximum. With an increase in the thickness D1 of the active layer, the parameter m grows monotonously, reaching saturation at D1 = 1. This increase is similar to the behavior of the dependence of the product D^(D0, D1) on D1. Saturation starts in the region of D1 ~0.5. Thus the

co Fig. 7.16 Dependence of optimization parameter w on the normalized width of the gas interval: Curves 1-7 correspond to values of D1 equal to 1.0; 0.5; 0.4; 0.3; 0.2; 0.1; 0.01

behavior of the parameter w also leads to the conclusion that an active layer thickness of D1~0.5 can be considered optimal. Data of the dependence for w, like the curves in Fig. 7.15, were calculated for uniform gas distribution in the cell and a square law of fragment deceleration, that is for n = 2 in Eq. (7.8).

In Fig. 7.17, the broken curve shows the dependence w(D0), obtained by calculation by the method [14], for a system formed by two parallel, infinitely extended flat uranium layers for D1 = 0.4 (which corresponds to a uranium oxide — protoxide layer with a thickness <5u = 3.2 mg/cm2, used on the LUNA-2M setup [40—43]). Correlations from [15, 16] make it possible to allow for the influence of edge effects arising due to the finite length b of the uranium layers in the direction perpendicular to the optical axis of the laser cell. In the experiments shown here, b = 6 cm, and the distance between the uranium layers is d = 2 cm. The results of calculation of w with allowance for edge effects are shown in Fig. 7.17 by a solid curve; in addition, experimental values of laser output powers are marked on the graph for various mixtures [40-45]. For convenience in comparison, the values of the parameter w, and laser output power J are calculated relative to their maximal values wmax and Jmax.

Deviations of experimental results from the curve w/wmax can be explained by differences in the kinetics of plasma-chemical processes and in the formation of an inverse population when there are variations in the partial composition of gas mixtures. The w parameter itself is entirely independent of these processes. The correlation of its behavior to the experimental dependence J(D0) testifies to the significant influence not only of the indicated processes, but also of the regularities of the energy transfer from charged particles to the gas, thermo — and gas-dynamic processes, and geometry of the system. In addition, this parameter does not allow for more subtle effects. Thus, for example, the lower the initial pressure of the gas, which is equivalent to a lower value of D0, the greater the volume of the near-wall

m. J ®max Jmax Fig. 7.17 Dependencies of the parameter w of energy deposition optimization and the laser output power J on the normalized transverse dimension of the laser cell: the broken curve is the calculation for infinitely extended uranium layers; the solid curve is calculation for layers with a finite transverse dimension b = 6 cm; filled triangle (He-Xe) [40, 44]; open triangle (He-Kr) [41, 45]; open square (Ar-Xe) [42]; filled square (pure Xe) [43] experiment

passive zone (see Chap. 8), which at the moment of end of the neutron pulse can amount to ~30 % for D0 ~ 0.1. Consequently, the volume of the active lasing region and the output laser energy will be less. Allowance for this factor would lead to improvement of the correlation between the curve w/wmax and the experimental points J/Jmax for D0 < 0.4.

Apart from calculations of the parameters p, ^1, and w cited above for the “square law” of fragment deceleration, corresponding to n = 2 (see Sect. 7.2 of this chapter), for a deeper investigation analogous calculations were carried out for the “linear deceleration law” (n = 1) and the n = 3/2 law. These dependences behave similarly to the dependencies obtained for n = 2. Figure 7.18 shows the curves for each of the deceleration laws, reflecting the change in the optimal normalized transverse dimension D0max, that is, the dimension corresponding to the maximum of the function w, depending on the normalized thickness D1 of the active layer for a system of two infinitely extended flat layers. A comparison of these curves with the experimental results in Fig. 7.17, according to which D0max ~ 0.4-0.45, testifies more in favor of the “square law” of deceleration.

Investigations of an analogous type were also conducted for a cylindrical geometry of laser cells. As was to be expected, qualitatively the results in no way differ from the corresponding results for a plane geometry. Figure 7.19 provides a comparison of the dependencies of optimal transverse dimension D0max for cylin­drical and infinitely extended flat geometries calculated using the “square law” of deceleration.

Fig. 7.18 Dependence of optimal normalized width for a system formed by two parallel infinitely extended flat uranium layers, on the thickness of the active layer for three deceleration laws: (1) “linear”; (2) “3/2 law”; (3) “square law”

A) max

0.8 I—————————————-

0,7

0.6 ‘ * “ ‘………………………………………………………………………………………………………………………….

0.5

0.4 — ————————————————-

0.3

0.2 ———- —————- ————-

o0 0,2 0.4 0.6 0.8 1

A

The calculations shown above for the optimization parameter m were carried out with the assumption of a uniform gas distribution. For a closed system, formed by two infinite flat layers of uranium, the possible redistribution of the gas density in the course of irradiation, owing to the non-uniformity of the specific energy deposition function and under the influence of heat transfer processes, will not lead to a change either in the dependence itself, m (D0, D1), or in the optimal value D0max. Indeed, in such a system, the gas density is a function only of the transverse

coordinate. The distribution of the specific sources F^r, , fixed in Lagrangian

coordinates, does not depend on redistribution of the density, and is equal to its value in a non-perturbed medium [16, 22, 36]. For a closed cylindrical cell, this distribution of sources for D0 < 0.5, fixed in Lagrangian coordinates, also differs little from the true one [36]. The same also occurs for D0 > 0.5, for irradiating times T1/2 < 3 ms [16].

Matters are a little more complicated when the cell has large buffer volumes, into which a considerable quantity of gas may flow in the course of irradiation when

there are high power depositions. The gas flow leads to a reduction in its average density in the active part of the cell, which is equivalent to a reduction in D0; thus the parameter m varies in the course of irradiation. In this case, the question is at what moment of the excitation pulse is it necessary to achieve an optimal energy deposition? The very process of optimization requires performance of gas-dynamic calculations to determine the transformation in time of the specific energy deposi­tion function F^r, t^ over the entire length of the excitation pulse. However, if

distortions of density through the cell volume are not too large and the quantity of gas flowing to the buffer volumes is not great, as occurred in experiments [40—43], optimization for non-perturbed gas as a reasonable zero-order approximation is reasonably accurate. This also applies to laser channels with continuous gas pumping, in which a constant mass expense of gas mixture is realized.

In conclusion of this section, it should be pointed out that other authors have also sought the criteria for energy deposition optimization. An essentially very similar method was used in study [46]: “Since in order to raise the efficiency of fission fragments, it is necessary to ensure the greatest escape of fragments from the fuel to the gas with the minimal non-uniformity of the energy release profile with respect to the thickness of the gas channel, it is possible to adopt as the channel optimiza­tion criterion” m’ = eF(0)/(F), that is, m’ = e^. At the same time, the “minimal energy intensity which facilitates cooling of the channel is reached in the case when” the value m’ of [46] “is maximal.” This approach, in agreement with the results discussed above, shows that the optimal width of the gas channel formed by two infinitely extended plane-parallel uranium layers is equal to D0max « 0.5 for n = 3/2 and D0max « 0.4 for n = 2. However, “with an increase in the thickness of the fuel, the maximal value of m’ decreases quickly.” Graphs of the dependencies m'(D0, Dj), similar in form to the graphs of m in Fig. 7.16, are arranged in order opposite that of m in the indicated figure [46]. The parameter m’ climbs steeply with a reduction in D1, reaching a maximal value for D1 = 0, which physically corre­sponds to a zero energy deposition because of the absence of uranium. The authors [46] did not allow for one significant detail. The coefficient e is a relative value, reflecting the share of energy deposited into the gas relatively to the total energy of the fission fragments released in the active layer. In reality, the energy character­istics of the laser are determined by the energy deposition, which is characterized by the product D1e(D0, D1), rather than e(D0, D1). Therefore, for optimization it is necessary to use the parameter m, defined by Eq. (7.25), rather than the product e^. The approach used in study [46] allowed its authors to optimize with respect to D0 (the gas pressure), but they could not optimize with respect to D1, because the coefficient of efficiency e^ introduced in this way reaches a maximal value for D1 = 0. From the academic standpoint, the material of the active layer is indeed utilized most fully in infinitely thin layers. However, the chief practical goal is still not to maximize some parameters that could be interpreted as the “fission fragment efficiency,” but to inject as much energy as possible in the laser-active gas, and to use it to pump the laser.

We note that study [47] proposed using, in the capacity of optimization criterion, the product at" = eF(0)/Fj = The results of calculation for the “square law” of

deceleration show a maximum of at" in the region of D0 « 0.3-0.4. This is some­what below the value of D0max, determined by the aggregate of experimental results on the laser output power (see Fig. 7.17).