The majority of studies dedicated to examining the characteristics of nuclear — excited plasma are computational-theoretical, and the data presented previously are essentially a brief review of them. The number of experimental studies in this area is very small, especially as regards experiments performed with the help of reactors.

Experiments with stationary reactors were devoted primarily to measuring of electrons and the electron temperature using microwave diagnostics [28, 66, 67] and electrical probes [68—70]. Measurements were carried out for 4He [66]; 3He [69]; Ne [67, 70]; Ar and Xe [70]; and for the Ne-Ar mixture [28, 68, 70] at ionization of gas media directly by n, y radiation of a reactor [66, 67], as well as by uranium fission fragments [28, 68, 70] and products of the nuclear reaction 3He (n, p)3H [69].

The most interesting are the optical methods of plasma diagnostics, which do not introduce any perturbations in the characteristics to be measured. Information about the properties of the plasma in this case can be obtained from investigation of the characteristics of its radiation or absorption—the intensity and wavelength of individual lines, the width and shape of contours of the lines, etc. [71, 72]. We know of only one study [73] in which, during reactor experiments, the concentra­tions of metastable atoms in He, Ne, and the He-Ne mixture were measured using the optical method. Excitation of gas media at pressures of up to 1.2 atm was carried out by products of the nuclear reaction 10B(n, a)7Li in the process of irradiation of a gas cell having a thin layer of 10B by a neutron beam of a TRIGA-pulsed reactor. To determine the concentrations of metastable atoms He*(21S), He*(23S), Ne*(3s [1/2]0), and Ne*(3s[3/2]2), absorption of radiation by these atoms at transitions

with wavelengths of 501.6 nm; 388.9 nm; 626.6 nm and 621.7 nm, respectively, was used. The concentrations of metastable atoms, depending on the pressure of the gas media and the neutron flux density, can be significant. Thus for helium at PHe = 0.2-0.5 atm and a neutron flux density Ф = 4 x 1013 cm-2 s-1, the concen­tration of atoms He*(23S) was (2-4) x 1011 cm-3, which is comparable in size to the concentration of electrons (~1012 cm-3).

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).

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

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

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).

A scheme for shaping an incoherent radiation field may contain a set of laser cavities with subsequent periscopic addition of radiation into a piecewise continu­ous (mosaic) field. The field is then telescoped and focused onto the target.

The radiation of N laser channels may be added into a single (incoherent) mosaic beam using periscopes (Fig. 10.6). If divergence of laser beam from one channel is equal to ~5 x 10~4 rad, after telescoping a united light beam up to the dimensions of the transverse section of the RL, the radiation divergence may be <5 x 10~5 rad.

For systems with a large number of identical laser channels it is expedient to use serial, parallel, or mixed addition of radiation from individual channels to increase the intensity of the laser radiation. Figure 10.7 gives the principal schematics of this addition for three channels. As shown by the calculations, the total length of a “multipart” cavity with a high Q-factor may be 30-50 m (~10 channels). With parallel addition, the number of channels are limited by the losses in the semi-

transparent plates and also may be more than 10; with mixed (serial-parallel), respectively, more than 100 channels. In this case, the laser radiation is withdrawn from the RL by several (from 2-4 to 20-30) beams with a laser power of ~100- 200 kW each. These beams may either be added by external systems (including coherent), or each is directed to its target independently.

VNIIEF researchers are currently studying methods of “direct” addition (parallel and serial) in different laser setups (LUNA-2M, LM-4, and LM-8). The first results from these studies are given in studies [29—32]. Direct addition methods are preferable when system size increases. In this case, difficulties arise with the application of methods using a common mirror. On the other hand, direct addition methods introduce increased demands for optical quality (as to coefficients of refraction and transmittance) and for optical uniformity in the active laser medium.

With nuclear pumping, startup of the individual lasers of a multi-channel system occurs in a random way. Use of the direct addition method permits the first laser that starts lasing to operate in “master oscillator” mode. Thus, it is possible to achieve complete phasing of all lasers in a set. In this case, the radiation from all channels will have identical polarization, phase, and frequency.

Outside of Russia, lasers operating on transitions of rare gas atoms were first pumped by nuclear radiation in 1975 at LANL in the United States, with excitation of the mixture He-Xe (A = 3.51 pm) by uranium fission fragments [38]. Further investigations were aimed at finding and studying parameters of NPLs operating on various transitions of Xe, Kr, and Ar atoms. The basic results of the experiments are shown in Table 3.5.

It is shown that in these studies, carried out mainly in the United States, lasing was obtained on the same transitions nd-(n + 1)p of the Xe, Kr, and Ar atoms (except for the 3.65 pm line of the Xe atom and 1.27 pm line of the Ar atom). Maximal output powers up to 1 kW were registered in the experiments [51] with the

APRF fast pulsed reactor (Фтах = 4.3 x 1016 cm-2 s-1) for a laser with the mixture 3He-Ar (A = 1.79 ^m) when a multiple-path laser cell with a volume of 3.6 l was used (see Chap. 2, Sect. 2.1). However, the laser efficiency was not great, and amounted to ~0.01 %.

In studies of American researchers published prior to 1989, the efficiency for IR lasers operating on the transitions of Xe, Kr, and Ar atoms did not exceed 0.1 %, which may be explained by the design features of the laser cells, which were sealed at the ends by windows arranged at the Brewster angle. In these windows, owing to the reduction of light transmission under the effects of the reactor n, y-radiation, additional losses occur inside the resonator (see Chap. 2, Sect. 2.2). In addition, the presence of Brewster windows prevented lasing at the intensive 2.63 and 2.65 ^m lines of the Xe atom, because of absorption of radiation in atmospheric water vapor in the cavity areas located between the windows and the mirrors.

Significantly higher values of щ = 3.3 and 5.6 % were obtained at the Sandia Laboratories for lasers using the mixtures He-Ar-Xe (A = 2.03 ^m) and Ar-Xe (A = 1.73 ^m) when cells with windows made from radiation-resistant optical materials were used, and with use of comparatively low specific power depositions q < 10 W/cm3 [45]. At higher specific power depositions and specific absorbed energies, there was a reduction in efficiency, and lasing was observed at the leading edge of the pumping pulse, which could be caused by the significant deteriorations of optical uniformity of the medium (especially in the Ar-Xe mixture) or by plasma effects. Of the research done on the xenon NPL at Sandia, one should note the studies [46, 55], which examined the influence of helium additions on the charac­teristics of an Ar-Xe laser. The introduction of helium to the Ar-Xe mixture leads to a change in the laser spectrum (instead of the 1.73 ^m line, the 2.03 ^m line appears), and to an increase in the laser pulse duration.

Sandia carried out research into the amplification properties of NPL active media operating on transitions of the atoms Xe [45, 46, 56, 57] and Ar [53, 54]. It should be noted that the first experiments [58] to determine the gain of NPLs operating on transitions of the Xe atom were carried out in 1981 for a 3He-Xe- laser at the 2.65 ^m line (in study [58], the 2.63 ^m line was mistakenly given as the lasing line). In the studies [45, 53, 54], the Rigrod theory [25] was used to determine the small-signal gain, while in the other studies, direct measurements were made by the “oscillator-amplifier” scheme using cw tunable color-center lasers, or low-pressure gas-discharge lasers using an He-Xe mixture.

Of the studies in the United States, one should mention [42], where the influence of additives of 235UF6 on the characteristics of the Ar-Xe laser (A = 2.6 ^m) was investigated. These experiments were aimed at determining the possibility of using the gaseous compound 235UF6 as a volume source of fission fragments. It was shown that reduction in the energy parameters is observed at a low concentration of uranium hexafluoride of ~0.5 %, which testifies to the effective “quenching” of the excited atoms Xe* by the molecules of UF6. An analogous conclusion follows from calculations of characteristics of a laser based on the mixture 235UF6-Xe [59] and luminescence investigations [60], which detected a more intense “quenching” of the ArI lines in the 235UF6-Ar mixture than in 3He-Ar. Thus at present there are no data that would allow us to hope for the possibility of creating lasers based on 235UF6 as the buffer gas.

Visible-range NPLs (A = 585.3, 703.2, and 724.5 nm) on 3p-3s transitions of the Ne atom were studied at VNIIEF, VNIITF, and the Sandia Laboratories (see Chap. 3, Sect. 3.2). In contrast to NPLs operating on the nd-(n + 1)p transitions of Xe, Kr, and Ar atoms, in neon NPLs the main processes leading to the formation of an inverse population of levels were established rather reliably [4, 83—92]. Already in the first studies of neon lasers excited by nuclear radiation and electron beams there are nearly identical notions of the lasing mechanism, which may be explained by the extensive spectroscopic information about the processes of populating and relaxation of 3p and 3s states of the Ne atom (see [10], for example). In particular, populating of the level 3p'[1/2]0 owing to dissociative recombination of molecular ions Ne+ with electrons was proven earlier in study [93], where additional Doppler broadening of the 583.3-nm line in the afterglow phase was observed, and the suggestion of using the Penning reaction to depopulate the lower laser states was demonstrated as early as 1970 (for example, see [1]). Because the lower 3s states of the Ne atom are depopulated in the Penning reactions

Ne * (3s) + M — M+ + e + Ne,

sometimes such lasers are called Penning lasers. When nuclear pumping was used, the admixtures M = Ar, Kr, Xe, and H2 were used to depopulate the 3s levels.

During the development of theoretical models, fundamental attention was paid to the transition 3p'[1/2]0-3s'[1/2]10 of the Ne atom (A = 585.3 nm), where lasing was most efficient when using the ternary mixture He-Ne-M (a diagram of Ne atom levels with lasing transitions is shown in Fig. 3.3). Formation of an inverse population in the He-Ne-M mixture occurs as a result of the following sequence of basic plasma processes:

In this sequence there are three stages at which process competition takes place that influences the population of the upper lasing level:

(a) Competition of charge-transfer processes

He) + Ne — Ne++ 2He, (5.10)

He) + M — M++ 2He; (5.11)

(b) Competition of dissociative recombination

Ne+ + e — Ne * (3p) + Ne (5.12)

and the charge-transfer process

Ne+ + M — M++ 2Ne; (5.13)

(c) Competition of radiative transition at the 585.3-nm line and the quenching processes of level 3p'[1/2]0 owing to the Penning reaction, as well as in collisions with He and Ne atoms. Therefore the populating efficiency of level 3p'[1/2]0 will depend on the type of quenching admixtures M, the pressure and composition of the mixture, and the electron concentration (specific power deposition).

If a rare gas (M = Ar, Kr, or Xe) is used as the quenching admixture, then in the second phase the losses are not significant, because the processes (5.13) for these atoms have low rate constants (< 10~13 cm3/s [94]). Replacement of Ar(Kr, Xe) by H2 leads to a marked competition of the processes in the second phase, because the rate constant of the process (5.13) for H2 molecules is quite high—4.2 x 10~n cm3/ s [95]. Therefore in NPLs based on 3p-3s transitions of the Ne atom, it is expedient to use hydrogen as the quenching admixture only at high specific power depositions
(q > 1 kW/cm3) [96]. We note that in the model [90, 91], the rate constant of process (5.13) with participation of the H2 molecule, determined during the process of numerical modeling, proved to be three orders of magnitude less (1.3 x 10-13 cm3/s). This low value contradicts the above data of [95] and the results of measurement [84] of the luminescence intensity at X = 585.3 nm in the He-Ne-Ar mixture as a function of the concentration of Н2. As the data in [84] show, for q « 1 kW/cm3 a twofold reduction in intensity (reduction in population of the upper lasing level by a factor of 2) occurs at H2 pressure of about 1.5 Torr, while the influence of H2 is determined precisely by process (5.13). The rate constant of this process, estimated from the conditions of the experiment [84], is no less than 5 x 10~n cm3/s, which agrees with the data of study [95].

In studies [89, 90], the models included the ternary charge-transfer reaction, which for M = Ar has the view:

Ne+ + Ar + R! Ar+ + 2Ne + R, (5.14)

where R is the third particle (in this case, He, Ne, or Ar). The rate constant of this reaction in model [90] for R = Ne was taken as equal to 5 x 10~32 cm6/s. The model [89] used a significantly higher value of this constant (3.5 x 10~30 cm6/s), which agrees with the data of [97]. The question of the need to include process (5.14) in the models remains open for now because the results of luminescence investiga­tions [84] testify to an insignificant influence of this process on the concentration of Ne2+ ions.

To calculate lasing characteristics, it is necessary to determine the populations of the 3p levels of the Ne atom. To this end, usually a system of kinetic equations is solved which constitutes a balance of the rates of populating and relaxation processes for each of ten 3p levels. Of the basic processes, it is necessary to allow for populating of these levels through reaction (5.12) and the collisional intra-multiplet transitions between them, the radiative and collisional relaxation of the 3p levels during collisions with the He and Ne atoms, as well as the Penning reaction for each of these levels with the participation of M atoms.

Reaction (5.12) is the main populating process of 3p levels, and in most of the studies it was assumed that all of these levels are populated during recombination of molecular ions Ne+ in the ground vibrational state. A somewhat different mecha­nism was considered in studies [87, 88], where it was assumed that level 3p'[1/2]0 (upper among the 3p levels) is populated with the participation of vibrationally excited ions Ne+. The percentage of dissociative recombination flux of (5.12) reaching level 3p'[1/2]0, according to various information, is around 7 % at a low neon pressure [10], 3-17 % when neon pressure is reduced from 760 to 10 Torr [87, 88], and 7-10 % for 20-100 Torr [98]. In ternary mixtures He-Ne-M with a large content of helium, the efficiency of populating level 3p0[1/2]0 is increased [87, 88], which the authors explain by the formation of vibrationally excited ions Ne2+ in the processes:

Ne + 2He! HeNe+ + He,
HeNe+ + Ne! Ne+ (o > 0) + He.

The relationship between the processes of dissociative recombination of rare gas molecular ions with the kinetics of populating of their vibrational levels was studied in the afterglow of a pulsed gas discharge (see [10], for example).

Lasing was also observed in neon NPLs at the 703.2- and 724.5-nm lines, which start from level 3p[1/2]1. This level is effectively populated both through process (5.12), and as a result of intramultiplet collisional transitions from the higher 3p — states.

One more mechanism of populating of 3p levels, proposed in study [53] and based on the investigation of the kinetics of stopping fission fragments in neon and a He-Ne mixture, consists of direct excitation of these levels by nuclear particles. However, as was demonstrated by experimental data [89], for He-Ne-Ar and Ne-Ar mixtures excited by an electron beam with a short duration of 3 ns, pulses of luminescent and laser radiation appear with a delay of up to 200 ns with respect to the pumping pulse, while its value is inversely proportional to the helium pressure. This delay can arise only in the case when laser states are populated not as a result of direct excitation, but owing to subsequent plasma processes. The authors [89] explain the onset of delay to electron “cooling” processes to temper­atures at which the rate of the recombination process (5.12) becomes significant and sufficient to achieve the laser threshold.

The population of 3p levels of the Ne atom is markedly influenced by collisional intra — and intermultiplet transitions during collisions with Ne [53, 99, 100] and He [101, 102] atoms. It is noted in studies [99, 100] that inter-multiplet transitions predominate for the 3p[1/2]1, which is the lowest among all the 3p states. At high specific power depositions, the processes of collisional relaxation of 3p levels during collisions with plasma electrons are added to these processes. For neon NPLs, the latter are less significant than, for example, for NPLs based on transitions 5d-6p of the Xe atom. In study [89] the conclusion was drawn that the effect of collisional mixing of the levels by electrons in a neon laser is not significant up to electron concentrations of 6 x 1015 cm~3.

Populating of the lower lasing level 3 s'[1/2]10 occurs through radiative and collisional intermultiplet transitions from 3p states. Depopulating of level 3 s’ [1/2]10 and three other 3s levels is carried out mainly through the Penning reaction (5.9), as well as through the reactions of associative ionization, intra-multiplet relaxation, and three-body processes with the formation of excimer molecules Ne2*. The processes of depopulating of 3s levels were considered most fully in studies [103, 104]. Table 5.9 shows the rate constants of Penning processes for all four 3s levels. We note for comparison that the rate constants of Penning processes for the atoms Ne*(3p'[1/2]0) with the participation of Ar and H2 are 5.3 x 10~n and 4.6 x 10~n cm3/s [105], respectively.

The most detailed kinetic models for NPLs based on 3p-3s transitions of the Ne atom [88—90] include up to 450 plasmochemical reactions [90]. For example, model [90] considered the atoms, molecules, and ions He*, Ne*, Ar*, Ar**, Ne2*, Ne2**, He2*, HeNe*, Ar2*, He+, Ne+, Ar+, He+, Ne+, Ar+, He+, Ne+, Ar^, and HeNe+, as well as the kinetics of population and relaxation of individual levels belonging to the groups of states Ne*(3 s,3 s’), Ne*(3p,3p’), Ne*(4 s), and Ne (5 s). The calculations were primarily carried out for the transition 3p'[1/2]0- 3 s’ [1/2]10 of the Ne atom (A = 585.3 nm). The kinetics of the neon NPL at the 703.2- and 724.5-nm lines (Ne-Kr mixture) were considered, evidently, only in studies [87, 88]. Experimental data obtained in pumping of a neon laser with nuclear radiation and electron beams were used for testing of the models.

The main differences of the models lie not in the quantity of plasma processes included in them, but in the use or absence of certain ones (for example, reaction (5.14)), differences in the rate constants for a number of important processes, and the probabilities of population of 3p levels through the reaction of dissociative recombination (5.12). One of the basic results of numerical modeling was the conclusion that the maximal efficiency of neon NPLs is no more than 0.5 % [90, 91].

As in studies [35, 36], the aforementioned laser’s cavity stability was analyzed in [34] in a paraxial approximation involving the use of the ray matrix technique. It was assumed that the axis of a system with two-dimensional transverse inhomoge­neities is located in a single plane. In this instance, as is generally known (see, e. g., study [29]), the conversion of the projections of light rays in the mutually perpen­dicular meridional planes can be independently examined within the framework of Gaussian optics. For the two-mirror linear cavity of the laser type under consider-

which the fission fragment energy is deposited the matrix of beam propagation in the active volume (M2); the matrix of propagation in a homogeneous medium between the other active volume boundary and the second mirror (M3); the matrix of beam reflection on the second mirror (M4); then the M3, M2, and M1 matrices again; and, finally, the M0 matrix, which describes beam reflection on the first mirror. A cavity is stable if |m| < 1, where m = (A + D)/2.

For an active volume with a one-dimensional quadratic transverse refractive index distribution,

the stability parameter takes the form [35, 36]:

my = C1 cos2/ — C2( cos 2/ + 1) — (C3y2 + C4y—2)( cos2/ — 1) — -(C5Y + C6r—1) sin 2y,

where

It is not difficult to demonstrate that for a laser in the active medium of which the refractive index distribution takes the form

the stability parameter is

mx = Cich2<9 — C2(ch20 + 1) — (C302 + C40-2) (ch20 — 1)+
+ (C50 — C60-1 )sh20,

where

while the C1-6 coefficients are in agreement with the coefficients in equation (8.39).

Experiments [2] aimed at studying optical inhomogeneities in a laser with planar uranium layers (see Chap. 7, Sect. 7.4, and Sect. 8.2 of this chapter) revealed that the refractive index distribution in the x = const planes is well described by parabola (8.38), while in the y = const planes, it is fairly well depicted by parabola (8.41). An analysis of the experiments and calculations using the procedure set forth in Sect. 8.2 of this chapter demonstrated that the refractive index distribution in the transverse cross-section of the laser cell’s active volume can be described with an accuracy of up to a few percents by the dependence

n(x, y, t) = n0(t) + 2p(t)x2 — 2a( ()y2. (8.44)

In a paraxial approximation, geometric optics equations (8.17) take the form [30, 33]

For gases, Дn = n — n0 << 1, which when taking dependence (8.44) into account, makes it possible to present these equations in the form of two independent equations for the x — and y-components,

d2x dAn(x)

0 dz2 dx ’

d2y dAn(y)

n0 dZ2 = ~~dT ’

i. e., to independently examine the passage of the light beams and the fulfillment of the stability criterion in the x, z, and у, z planes. It is obvious that a cavity of this type will be stable if two conditions are simultaneously observed: Шх| < 1 and Шу| < 1.

If the cavity geometry remains unchanged, the stability parameters my and mx (see (8.39) and (8.42)) will then only be dependent upon the corresponding dimen­sionless inhomogeneity parameters у and 0, which are determined by equations (8.40) and (8.43). Thus, if the у values vary over time, then my will periodically pass through the I my I = 1 values, which is explained by the presence of the trigono­metric sines and cosines in expression (8.39). In this instance, if the 0 values vary over time, then mx can also intersect the | mx| = 1 values. During the transition of the my (or mx = 1) values to | my| = 1 (or | mx| = 1), the alteration of cavity stability is observed. The у and 0 values at which such a transition occurs are critical values of the dimensionless inhomogeneity parameters. They are designated below as ycr and 0cr.

Over the course of pumping, the inhomogeneities characterized by the dimen­sionless у and 0 parameters monotonically increase [2, 23, 35, 36, 45]. The depen­dence of the my stability parameter upon the value of the у dimensionless inhomogeneity parameter over the variation range of real interest is presented in Fig. 8.17. The corresponding dependence of the mx stability parameter upon the 0

dimensionless inhomogeneity parameter is shown in Fig. 8.18. During irradiation time, the my parameter periodically intersects the | my| = 1 value in making the transition from the stable region to the instable one, then again returns to a stable state. The instability region sections in Fig. 8.17 are numbered in order of growth inhomogeneity parameter values: the first-order instability region comes first, then the second-order region, etc.

Changes in the stability of a laser with one-dimensional inhomogeneities of the type described in (8.38) were calculated in [35, 36], which revealed the cavity stability losses and restorations occurring over the course of a pulse that correspond in time to the moments of the total cessation and restoration of lasing observed during the experiments. Over the course of these calculations, the refractive index distribution was approximated by parabola (8.38), which was common over the entire volume; therefore, the loss of stability corresponded to the total cessation of lasing.

It was previously mentioned that during a number of experiments involving planar and cylindrical uranium layers, both fluctuations in laser power and their total cessation were observed. The reason for the fluctuations can be sought in the temporary departures of only a portion of the cavity’s volume from the stable state. So, direct calculations demonstrated that during the approximation of inhomoge­neities using correlation (8.44) based on their values in the peripheral regions of the transverse cross-section of the cell’s active volume, higher a and p coefficients are generally obtained than during an approximation based on the central regions. Apparently, stability is first lost in the peripheral sections, while the central sections remain stable. An artificial technique was used in study [34] to verify this proposition. During the investigation of stability in the y and z planes, the active section of the cavity was evenly split into separate regions along the x axis that fully
occupied the gap between the uranium layers, d. An a(t) parabola coefficient was selected for each region, i, at a relevant moment in time based on the results of density distribution calculations and it was used to compute the myi stability parameter. The transverse areas of the regions for which | myi | < 1 were then added. If the result of this addition is assigned to the overall area of the cell’s transverse cross-section, we then obtain a characteristic that is dependent upon time, Sy(t), which is found in an approximation of the total absence of a relationship between the individual regions.

We note that during this approach, the stability status of each individual region is examined outside its relationship with the status of the other regions, i. e., as if each region is isolated from the remaining ones. This technique a priori precludes the possibility of the origination of a stable (or quasistable) state between different regions, which is far from obvious. The use of this technique can be justified by the fact that during passage through the inhomogeneous media under consideration, each light beam is deflected by a slight angle so that when the transverse dimensions of a region are not too small, the bulk of the beams remain within the confines of this cavity section if its status is stable. Thus, the region Дxi dimension must be limited from below, because it is apparent that the smaller the transverse dimension, the more numbers of the beams can travel outside the confines of this region to the neighboring ones. As a criterion, it was assumed that a region’s size must not be smaller than a beam’s displacement after one Cavity pass: Дхг — > Lф, where L is the distance between the mirrors and ф is the beam deflection angle. The deflection angle was estimated from the correlation ф ~L (3n3x)~Lpmx0, where x0 is half the dimension of the uranium layer in the direction of the х axis, and pm is the maximum p coefficient in the region most far removed from the center at the time of pumping pulse termination.

A similar approach was also used when calculating the stability in the x and z planes. The maximum number of splits in all the calculations presented below did not exceed N = 30. During the calculations for a cylindrical cell, splitting into coaxial cylindrical regions was carried out. The presence of axial symmetry was assumed.

Once again, we stress that the technique used is artificial and cannot be rigor­ously substantiated. It a priori rules out the possibility of the origination of a stable state between different regions, which is far from obvious.

Laser pumping is not the only application for NOCs. For example, they were considered as a possible means for initialization chemical processes and for photo-dissociation of water or carbon dioxide. In this case, the full efficiency for the conversion of nuclear energy into the end products of these reactions may reach 10 % when a NOC based on Хе2* (^max = 172 nm) molecules is used [43, 44].

One more possible application of the NOC is the conversion of light emission into electricity. In this case, the efficiency for the conversion of nuclear energy into electrical energy is 30-40 % using dielectric transformers with a wide band gap

(for example, A1N, diamond or corundum) and the appropriate choice for the NOC emission spectrum [43].

Another variant for photoelectric conversion of NOC radiation has been pro­posed [57]. In this case, the study examines the potential for using radioactive waste as a source of energy for pumping NOCs with subsequent conversion of the optical radiation into electrical energy using photoelectric transformers. The main source of ionizing radiations from the spent fuel of nuclear reactors is the y-ray 137Cs (the half-life is 10.7 years, the energy of у-quanta is 662 keV). As an active medium for NOCs, an Ar-N2 mixture has been proposed that primarily radiates in the ranges X = 350-410 and 750-1,050 nm on the vibrational transitions of the N2 molecule. The output characteristics were evaluated for a device consisting of 200 radioactive waste containers, 0.5 m in diameter and 1-m high, placed in three tiers. The expected specific electrical power is about 1 W/kg, and the full electrical power of this device with a 40 m radius (200 containers) may equal to 1 MW.

NOC radiation may also be used for photosynthesis of biological products, for example, microalgae chlorella. For large-scale photosynthesis, a paper [58] pro­poses to develop a NOC whose central part is a solution reactor-breeder on thermal neutrons with uranium-thorium fuel, and the cover consists of a luminescent material. In this case, biomass production may be about 1,000 g/(m2/day). This significantly exceeds the analogous value for solar illumination <150 g/(m2/day).

And, finally, based on the NOC, it is possible to create neutron detectors for operating control of pulsed as well as stationary nuclear reactors. In these detectors, a luminescent emission that has an intensity linearly related to the neutron flux density is withdrawn beyond the biological reactor shielding using light-guide fibers and recorded using photodetectors [59, 60]. Neutron detectors based on NOCs have a series of advantages as compared to ionization chambers: the lack of a power supply; a low sensitivity to y-ray emission; the potential to produce a small NOC for intrareactor measurements; etc.

Lasers operating on transitions of the I atom (A = 1.315 ^m) excited as a result of photodissociation of the molecules CF3I and C3F7I, or by transfer of energy to iodine atoms from metastable molecules of oxygen O2*(1A) + I(2P3/2)! O2*(3X) + I*(2P1/2), are well known as some of the most powerful sources of laser IR radiation [164, 165]. The literature reviewed several variations of excitation of the iodine laser by nuclear radiation: (a) formation of metastable molecules O2* (1Д) with subsequent energy transfer to iodine atoms [166, 167]; (b) formation of vibrationally excited molecules H2*(v > 2) in the mixture Ar-H2-I2 with subsequent transfer of energy to iodine atoms [168]; use of radioluminescent radiation of excimer molecules (KrF, XeBr, etc.) for photodissociation of molecules CF3I or C3F7I [169, 170].

Studies [166, 167] provide the results of experiments and theoretical investiga­tions into excitation by nuclear radiation of mixtures of He, Ne, and Ar with O2, aimed at determining the concentrations of O2*(^) necessary to achieve the laser threshold. Formation of O2*(^) can occur both during direct excitation of the oxygen-containing mixtures, and during photolytic decomposition of O3 by the radiation of excimer molecules, for example KrF. Pumping of an oxygen-iodine laser, or an Ar-H2-I2 laser [168], does not exist as yet.

A method of pumping an iodine laser using radioluminescent radiation of excimers is based on high conversion efficiencies of luminescence, which for many excimers is 20-30 % [170]. When the 3He-Xe-CHBr3 mixture is irradiated with the pulsed neutron flux of the TRIGA reactor, intensive radiation of the XeBr* molecules occurred (A = 282 nm), which then was used to photodissociate C3F7I [171, 172]. The conversion efficiency of luminescence for the XeBr* molecules (A = 282 nm) was around 1 %. The design of the laser cell is shown in Fig. 3.7. The power of the laser radiation at the line 1.315 ^m was not great: ~20 mW.

The design of the LIRA setup [32, 33] proposed by VNIITF associates is one version of a high-powered pulsed nuclear-laser facilities. It is presumed that the LIRA setup (like the OKUYaN) consists of two blocks: a rather large subcritical block made from laser elements in the form of long thin-walled pipes, and two relatively small “initiating” pulsed reactors disposed inside this block (Fig. 6.16).

The reactor block contains two DRAKON pulsed solution reactors operating synchronously (half-width of pulse duration is 2 ms, energy release in the core is 30 MJ). The ELIR reactor [29], which was operated for many years at VNIITF, was

used as the basis of this reactor. The basic calculated characteristics of the DRAKON reactor are given in study [33].

The LIRA setup laser block consists of seven identical amplification laser assemblies, each of which consists of 19 laser elements. An eighth assembly includes eight laser elements, one of which is a master oscillator. The laser elements are enclosed in an aluminum tube 300-mm diameter, and are arranged it at intervals of 60 mm. The pipe is filled with water and surrounded by graphite, forming a laser assembly (Fig. 6.17). The outside dimensions of the assembly are 43 x 43 cm in cross section, with a length of 3.8 m. On the outside, the laser assemblies are surrounded by a neutron polyethylene moderator 10-cm thick.

Each laser element consists of three concentric tubes. A 6 ^m thick layer of metallic 235U is deposited to the inside surface of the first, innermost, tube, which is 48 mm in diameter. The two outer aluminum tubes have a 12 ^m layer of 235U between them and constitute a simulator, which is necessary to obtain a neutron multiplication coefficient in the laser block of kef < 0.9. The total mass of the uranium in the laser setup is around 20 kg.

To optimize the laser block design, calculations were carried out to study the effect on energy release of different neutron moderators in the assemblies (poly­ethylene, graphite) and water in the space between the laser elements and the absorbent screens. Calculations were also used to determine the temperature of the uranium layers and the gas pressure in the laser element after the pulse.

The optical circuit of the LIRA setup is shown in Fig. 6.18. One of the laser elements of the eighth laser assembly is a master oscillator. Laser radiation from the master oscillator is split into seven beams of identical power and directed to the other seven laser elements of this same assembly for preliminary amplification. Then the laser radiation is expanded using telescopes to a diameter of 300 mm and

goes to the other amplifier assemblies. After amplification, the beam aperture is reduced to 100 mm and output past withdrawn beyond the biological shielding.

In the opinion of the authors of [32, 33], when a He-Ar-Xe mixture (A = 2.03 ^m) is used as the laser medium, the specific energy deposition to the gas medium is about 1 J/cm3, while the specific energy deposition is ~8 mJ/cm3 for a lasing pulse duration of 5 ms. Under these conditions, the full energy of laser radiation of the LIRA setup can reach ~4.5 kJ.

In study [34], experimental investigations of a multichannel laser assembly on the BARS-5 pulsed reactor [29] were carried out (Figs. 6.19 and 6.20). The assembly consists of a package of 19 stainless steel tubes with an internal diameter of 15 mm and wall thickness of 0.2 mm. Each of the tubes held thin-walled aluminum tubes with a layer of 235U3O8 (layer thickness 2.5 mg/cm2, layer length 1,050 mm). The specific power deposition of the gas mixtures He-Ar-Xe (A = 2.03 ^m) and He-Ar (A = 1.79 ^m) reached 5 kW/cm3 at a helium pressure of 3 atm. The half-width pulse duration of the thermal neutrons was about 130 ^s.

Fig. 6.20 Photo of end of laser assembly [34]

Under these conditions, laser radiation energy from the entire assembly was 9.9 J (2.03 ^m) and 10.3 J (1.78 ^m). The output power was around 30 kW.

Higher energy parameters were recently obtained in experiments [35] when a multichannel laser assembly was irradiated by neutrons using the reactor facility BARS-5 + RUN-2 which consists of BARS-5 pulsed reactor and RUN-2 neutron amplifier (see reference [29]). One end of laser assembly was located between the two cores of the BARS-5 reactor and the other one was located inside the RUN-2 core (see Fig. 6.21).

The laser assembly in reference [35] consists of the 37 stainless steel tubes with a wall thickness of about 0.1 mm and a length of 1.5 m. The end of the laser assembly had approximately the same design as shown in Fig. 6.20. Inside the each tube, the ten aluminum cylinders were placed with the inner diameter of 19.5 mm which were covered by thin layers of 235U3O8 (with a layer thickness of 2.5 mg/cm2). To

enhance the thermal-neutron flux, this laser assembly was located inside a cylin­drical polyethylene neutron moderator with the wall thickness of 60 mm.

As described in reference [35], investigations were carried out using a He-Ar-Xe (700:700:1) mixture at 4 atm pressure. Lasing occurs on a transition in the Xe atom (A = 2.03 pm). As an example of the result, Fig. 6.22 shows the oscillogram of one experiment. These experiments demonstrated record energy parameters for NPLs excited with help of pulsed reactors: laser output energy of 520 J with a laser pulse duration of about 400 ps (laser power output is 1.3 MW, n ~3 %). Such high laser energy parameters are attributed to the large active laser volume (16l) and high specific input power deposition (~1.5 kW/cm3) which was distributed rather uni­formly along the length of laser tubes.

Study [36] assessed the maximal possible energy parameters of pulsed nuclear — laser devices designed by an “initiating reactor-subcritical laser block” scheme. With allowance for the laser media efficiencies obtained up to now of ni ~ 1 % (for example, for xenon NPLs) and the limitations associated with the operating modes of the laser element and the specific features of neutron pulse in such a system, it is possible to obtain a full laser radiation energy of ~2.5 MJ for a pulse duration of ~5 ms. Characteristic dimensions of a laser block that includes 104-105 laser elements with a total active volume of ~250 m3 is 7-10 m. Three pulsed TRIGA — or ACPR-type reactors [7] with an energy release in the reactor core of 100-200 MJ can be used as “initiating” reactors.

During the design of a particular laser system, the need may arise for the prelim­inary estimation and optimization of the spatial distribution of gas density in the laser channel’s active region. The behavior of a gas in systems where the density deviations are comparable to, or exceed, their average values over the channel’s volume can only be correctly calculated using two-dimensional numerical pro­grams similar to that of the Idaho National Engineering Laboratory [44]. In ref. [45], a simplified technique was proposed for the direct numerical calculation of spatial inhomogeneities in the channel of a gas-flowing NPL. The only example of a similar calculation available at the time of publication performed using a simplified technique for determining the specific energy deposition of fission fragments in a channel of finite width with a plane-parallel uranium layer arrangement was also presented in ref. [45] by way of an illustration. A comparison demonstrated that the
difference in the results of density distribution calculations using the procedure described in ref. [45] and the calculations performed using the two-dimensional gasdynamic program described in ref. [44] does not exceed ~2 %.

The fundamentals of the procedure described below [46] consist of the follow­ing. As a gas flows through a plane-parallel channel, the length of which, b, is comparable to its transverse dimension, d, the pressure in the laser channel’s transverse cross-section is almost homogeneous (see the first section of this chapter). The typical gas temperature equalization time in the channel comes to tt ~ d /fo, where a is the thermal diffusivity coefficient of the gas. For the gas mixtures used in NPLs, a < 1 cm2/s; hence, it follows that, at d ~ 1 cm, the temperature equalization time in the channel is tt > 1 s. The time during which a portion of the gas traverses the channel in the x direction is determined by the obvious correlation т ~ b/U. At gas-flow velocities of U > 1 m/s, the average time that the portion of gas is present in the channel comes to т < 0.1 s. We will now apply the technique under consideration for the two-dimensional calculation of the gas flow in a channel to the gas flow velocities at which the time that the portion of gas is present in the channel, т, is much shorter than the typical heat exchange time, tt. In this instance, the heat exchange processes in the bulk of the gas volume can be ignored during approximate calcula­tions. The only exception consists of the thin near-wall passive zone layers, the method for considering the effect of which will be discussed next.

In the first approximation of the proposed technique, the flow lines, and accord­ingly, the gas jets separated along each line are assumed to be rectilinear, while the spatial distribution of fragment energy deposition is calculated under the assump­tion that the gas density distribution is homogeneous throughout the channel volume. The dependence of gas temperature upon the x coordinate is then calcu­lated for each flow line, by means of which, using a state equation and proceeding on the basis of the condition that the gas pressure is almost homogeneous through­out the entire channel, the spatial gas density distribution is calculated. Thereafter, pursuant to the results obtained in refs. [44, 50], according to which the gas’s longitudinal velocity profile, Un(x), is transversely homogeneous (with the excep­tion of the narrow near-wall viscous boundary layer), and based on the requirements of mass flow constancy both within the channel as a whole and in each of the jets separated, the gas velocity and the variation in the transverse dimensions of the jets as a function of the x coordinate are calculated. The latter makes it possible to determine the curvature of the flow lines, and using them to correct the spatial distribution of gas density. During the next step, the specific energy deposition distributions of the fragments are calculated based on the density distribution obtained during the previous step, then the procedure for finding the density distribution is repeated using the arrangement described above, etc.

When determining the basic computational correlations of the stationary prob­lem, it was assumed that the gas’s density, velocity, and temperature distributions, as well as the specific intensity of uranium nuclear fission, are not dependent upon the z coordinate directed along the system’s optical axis (in Fig. 9.1, the z axis is perpendicular to the figure’s plane and coincides with the direction of the laser’s optical axis), i. e., a purely two-dimensional problem was examined. The gas flow in
the planar channel was represented by superimposing the independent jets, the camber and thickness of which can vary with the distance traversed, x. The effect of the viscosity within the jets and the friction between adjacent jets was ignored. Each jet was tagged with an Eulerian coordinate at the inlet, y0, and its width was assumed to be so small that the density, velocity, and specific energy deposition within the confines of any of its transverse cross-sections could be regarded as homogeneous. Heat exchange between the jets was ignored. Then in the active region outside the confines of the passive zone by virtue of pressure homogeneity for a given jet, from the energy balance equation, we get

U(x yo)p(x, yo)T(x, У0)ср5у(Х; Уо) — UoPoToCp8у0

where T(x, y0) is the gas temperature in the jet under consideration at a point with a coordinate of x; U(x, У0) is the gas velocity in the jet under consideration at a point with a coordinate of x; U0 is the gas velocity at the channel inlet; w(x, y0) is the specific power deposition at a point with a coordinate of x for a jet marked with a coordinate of y0; Sy(x, y0) is the thickness of the jet under consideration at a point with a coordinate of x; £y0 is the thickness of the jet under consideration at the channel inlet; p(x, y0) is the gas density in the jet at a point with a coordinate of x; and p0 is the gas density at the channel inlet.

From the condition of mass flow constancy in each jet, we get

U(x, y0)p(x, y0)<5y(x, У0) = ^Po^ (9-143)

then, according to the previous equation,

In the main body of the flow (outside the confines of the viscous boundary layers), the gas’s longitudinal velocity profile, as previously mentioned, is transversely homogeneous (U(x, y0) = Un(x)) and Eq. (9.144) is simplified

For rare gases, the relationship between density, temperature, and pressure is described with good accuracy by the ideal gas state equation

P = cv(r — 1)pT.

Thus, for the gas density in each jet, we get

P 1

— ——— dx

Cp U(x, y-)p(X; Уо)

0

According to Eq. (9.143), the relationship between the y0 coordinate and the y Eulerian coordinate takes the form

Уо 0

The dependence of the near-wall passive zone’s transverse dimension upon time is described by an equation of the (9.142) type. Direct calculations (see Chap. 8) revealed that the proportionality factor for sealed lasers is A « 3 and is slightly dependent upon cell dimensions, neutron pulse shape and intensity, and uranium layer thickness, as well as upon the gas mixture’s thermophysical properties. A passive zone is also formed in the channels of NPLs with gas pumping, the wall temperature of which is lower than the gas flow temperature. Its size is a function of many parameters that reflect the geometry and physical conditions of a flow, especially including viscosity and heat conduction. In the general case, it is not possible to derive an explicit analytical expression for the dependence of passive zone size upon the longitudinal x coordinate. However, since the time required for a portion of the gas to traverse a distance of x in the main section of the channel (outside the viscous layer) is. X

dx

UM

0

it is then proposed that the transverse dimension of the near-wall passive zone at this distance be roughly determined using the correlation

As was shown in the preceding section (also see ref. [51]), even in the simplest case of Un(x) = U0 and w/p = const, the A factor is dependent upon the Prandtl number, Pr = v/a, and may lie within limits of 1.4 < A < 1.9.

During the calculation procedure used in ref. [46], temperature distribution in the passive zone was described by the parabolic approximation

Tp{x, y) = a0 (x) + ai (x)y + a2(x)y2.

The a, coefficients are determined from the conditions at the passive zone boundary and at the channel wall. So, for example, when a constant wall temperature is present, the following conditions are used:

Tp(x, la) = T(x, la); dTp(x, la)/dy = 0; Tp(x, 0) = To.

If the velocity in the flow core is also not dependent upon the longitudinal x coordinate (i. e., Un = U0) then the thickness of the viscous layer is determined by the correlation [2]

lv(x)= . (9.147)

The situation is more complicated when Un is a function of the longitudinal x coordinate. Extant literature source contains no direct recommendations on how to proceed in this instance. Therefore, since the time required for a portion of the gas to traverse a distance of x within the channel outside the viscous layer is determined by Eq. (9.145), it is proposed that the transverse dimension of the viscous layer at this distance be approximately determined using the correlation

The following approximation was used to describe velocity distribution in a viscous layer at y0 < lv(x)

In this case, according to the impulse theorem, R = 4.64 [2]. Such a distribution duplicates the true velocity distribution [2] near a planar surface washed around by an incompressible fluid that has a constant viscosity and a constant velocity outside the confines of the viscous layer with an error of less than 1 %.

Thus, an artificial technique was used to describe the width of the passive zone and the viscous layer, as well as temperature and velocity distribution within their confines. Nonetheless, at sufficiently high velocities, U0, the dimensions of these structures are small; therefore, it was anticipated that the proposed approach would not lead to noticeable errors during the calculation of density, temperature, and velocity distributions in the active zone.

The spatial distribution of the specific power deposition due to fission fragment slowing-down was calculated using the procedure described in ref. [49].

During the third iteration step, the gas density distribution differs from the distribution obtained at the end of the first step by not more than 1 %, which is indicative of the good precision of the iterative scheme used.

A comparison demonstrated that the difference between the density distribution calculation results obtained using the procedure described in ref. [46] and similar calculations [44] performed using a two-dimensional gasdynamic program like the one employed during the calculations in ref. [45] based on a simplified procedure for determining the specific energy deposition of fission fragments, does not exceed

2 %. In addition, it was learned that the variation of the A factor value from 1.4 to

3 only results in perceptible density changes during calculations of the same problem alternative in the passive region; for the active region, these changes do not exceed ~1 %. At a gas-flow velocity of U0 = 4.5 m/s, A factor variability leads to relative fluctuations in the active region volume of ~10 %. These fluctuations diminish with an increase in gas-flow velocity; so, at U0 = 9 m/s, they come to ~6 %. Nonetheless, at all A values, the nature of the behavior of all the dependences cited in this work is preserved.

The channel studied during the calculations, which has dimensions of d = 2 cm and b = 6 cm, has open layers (without a protective film) that consist of metallic uranium with a thickness of <5u = 2 .78 ^m (a normalized thickness of D1 = 0.5). It was assumed that uranium fission intensity in the active layers comes to q = 2 x 1016 cm~3 x s_1. The spatial distributions of density and specific energy deposition in active volume of the gas were calculated for a wide range of helium and argon pressure and velocity variations. It was shown that in the central regions of the gas volume (at distances from the channel inlet and outlet on the order of the third fission fragment range in the gas), the dependence of these parameters upon the longitudinal coordinate parallel to the gas flow direction is almost linear in nature, while their dependence upon the transverse coordinate is close to parabolic. The results cited below correspond to a value of A = 3 (apparently the least probable value [51]), which corresponds to the smallest active region volume.

An important parameter that characterizes a real laser channel with a finite dimension of b along the gas flow direction is the efficiency, e, of total energy deposition in a laser-active region not involved in near-wall passive zones. It is defined as the ratio of the total fission-fragment energy deposition in the active region solely to the total fragment energy released in the uranium layers. It is useful to compare this parameter to the efficiency of energy deposition, e0, in an unperturbed gas in a channel without gas pumping (U0 = 0) that has a homogeneous density distribution (there are no near-wall passive zones). Below, the former will be called the real energy deposition efficiency, while the latter will be called the ideal energy deposition efficiency of an unperturbed gas.

In ref. [37] (also see Sect. 7.5), an energy deposition optimization parameter was introduced for NPLs without gas pumping, excited by fairly short neutron pulses with a duration in the millisecond range,

«0 = Dieo-

where D1 = SU/R1 is the normalized uranium layer thickness; R1 is the range of an average fission fragment in this layer; w0 is the specific power deposition in the center of the laser channel; and (w) is the average value of the specific power deposition over the channel cross-section.

In laser channels with gas pumping that operate in the stationary mode, a near­wall passive zone can reach appreciable dimensions at the channel outlet in the presence of relatively low gas-flow velocities. In addition, gas density steadily decreases in the longitudinal direction. For these reasons, real energy deposition efficiency, e, can differ conspicuously from unperturbed channel energy deposition efficiency, e0; thus, a more rigorous statement of the optimization parameter should be introduced:

where

b

w (x, 0) dx,

while w(b/2, 0) is the specific energy deposition value at a point that corresponds to the geometric center of the channel.

The «1 and «2 parameters are distinguished from «0 by the fact that the calculated real energy deposition efficiency, e, is used therein instead of unperturbed channel energy deposition efficiency, e0. In the first of these, the fraction characterizes the ratio of the specific energy deposition averaged over the channel’s entire central plane, y = 0, to the specific energy deposition averaged over the channel’s cross-section. In the second, it characterizes the ratio of specific energy deposition in the center of the channel to the specific energy deposition averaged over the channel’s cross-section.

The dependences of the «1 and «2 optimization parameters upon argon pressure in the channel at U0 = 4.5 m/s are shown in Fig. 9.22. Here, too, a similar

dependence for the ideal parameter, rn0 (at U0 = 0), is shown. The calculated dependences of the optimum (as far as energy deposition) argon and helium pressures upon gas-flow velocity are presented in Fig. 9.23. We note that the optimum argon pressure in a channel with gas pumping over the velocity interval studied, U0, may exceed the optimum pressure for a sealed unpumped laser operating in a comparatively short (<10 ms) pulse lasing mode (P « 0.6 atm) by

1.5 times (up to P = 0.9 atm), whereas the optimum helium pressure in a channel with gas pumping differs little (not more than 15 %) from the optimum pressure for a sealed unpumped laser. The spatial distributions of argon density at an optimum pressure of P = 0.9 atm and a velocity of U0 = 4.5 m/s are presented in Figs. 9.24 and 9.25 for T0 = 293°K.

The calculations made it possible to find out a quite important fact. At the optimum pressure precisely, the distribution of specific energy deposition in the central plane, y = 0, and in the adjacent planes, y = const, is symmetrized relative to the central cross-section, x = b/2 (see Fig. 9.26b), equalizing in a considerable

portion of the gas volume. At pressures lower that the optimum value in a gas region that stands away from the inlet and outlet sections by a distance on the order of one-third of the fragment range in the gas, specific energy deposition steadily decreases with an increase in x for any y = const plane (Fig. 9.26а). At pressures higher than the optimum value, a reverse pattern emerges (Fig. 9.26c), with the exception of the near-wall regions, where the decrease in energy deposition with an increase in x is retained at all pressure values.

It follows from flow calculations in real NPL channels that the dependence of the main flow’s velocity upon the coordinate is close to linear (this fact is in full agreement with the results of the calculations performed in ref. [44]):

Un(x) « U0 + Bx. (9.149)

In this case, according to Eq. (9.146), we get:

l.(_,) » + bSL

It follows from a comparison of the Eq. la(t) « Afat and relation (9.150) that the time required for a portion of the gas to traverse a distance of x from the channel inlet comes to

t

The latter is in agreement with the results obtained in ref. [43], in which

2E0ei^uq Т0Р0 cpd

where E0 is the kinetic energy of the uranium fission fragments, and e, is a certain effective value of the share of energy of the fission fragments that transmit to the gas. An analysis of the calculation results demonstrated that e, is slightly smaller than the ideal energy deposition efficiency, e0; however, the deviation of e, from e0 at all pressure and velocity values does not exceed 20 %. In the presence of sufficiently high velocities and weak fission intensities, when Bx/U0 << 1, we get a simple dependence

la(x) — A