The schematic for shaping laser radiation to a significant degree is determined by the purpose of the RL. Practice frequently requires that light beams be integrated from individual cells of the RL to a united light field. All methods of radiation adding can be divided into two major groups: incoherent and coherent. The first are primarily used where the quality parameters of the laser beam do not play a leading role, but the following are required: reservation of radiation sources; obtaining more output power at a given wavelength, which technically cannot be obtained on a single laser; forming a spatially limited optical field (beam) with different wavelengths, etc. Coherent addition of beam radiation in multi-channel laser facility repeatedly increases the intensity and brightness of the laser radiation and reduces its divergence. This is important for transportation of the radiation over a large distance. Coherent addition of N light beams, each of which has a beam divergence close to the diffraction limit, increases beam intensity in the far-field by N2 times. Coherent, serial addition of light beams, as a rule, requires quick-acting adaptive optics that correct the phase of radiation in all laser channels. Implemen­tation of this operation mode for RL is a complicated task.

Investigations of NPL problems were begun at VNIITF in 1979, and in 1981 the first experiments were carried out on pumping a He-Xe mixture with uranium fission fragments [31]. In contrast to the majority of experimental studies, in which the laser cells were placed close to the cores of the pulsed reactors, the VNIITF experiments used the specialized EBR-L setup, which included a fast neutron pulsed reactor and laser cell (see Chap. 2, Sect. 2.5).

The EBR-L setup was used to research IR lasers operating on transitions of the Xe and Ar atoms, the basic results of which are shown in Table 3.4.

Table 3.4 Results of studies of NPLs operating on IR transitions of rare-gas atoms (VNIITF) Atom Mixture A, pm Ф, к x 10-14, cm-2 s-1 Wout, W ni, % Works cited Xe He-Xe 1.73 50 300 0.2 [32, 33] Ar-Xe 1.73 0.6 430 2.1 [33] He-Ar-Xe 1.73 1.0 2,600 1.7 [32, 33] Ne-Ar-Xe 1.73 1.0 700 1.7 [32, 33] He-Xe 2.03 19 430 0.3 [32, 33] He-Ar-Xe 2.03 0.5 2,000 1.5 [32, 33] He-Ar-Xe 2.03 — 1.3 x 106 3 [34] He-Xe 2.65 25 170 0.15 [32, 33] He-Ar-Xe 2.63; 2.65 40 1,000 0.9 [32, 33] He-Ar-Xe 2.48 30 15 ~0.01 [33, 35] Ar He-Ar 1.79 3.2 — 1.2 [36, 37]

The high efficiency of the Xe-laser, confirmation of the results of experiments at VNIIEF, as well as the ni > 1 % obtained for the He-Ar laser (A = 1.79 pm), must be considered among the most important results. One should also note the study [37], in which experiments using the EBR-L and IGRIK pulsed reactors determined the energy characteristics for lasers using the mixtures He-Ar (A = 1.79 pm) and He-Ar — Xe (A = 2.03 pm), at high specific energy depositions of up to 2.5 J/cm3. As a result of the investigations, quite high specific output energies were obtained—7.5 J/l at A = 1.79 pm and 9 J/l at A = 2.03 pm.

Some important experiments were recently carried out at VNIITF using the He — Ar-Xe mixture (A = 2.03 pm). These experiments (discussed in reference [34]) demonstrated the maximal energy parameters for NPLs excited using pulsed reactors. The laser output energy was 520 J (output power of 1.3 MW) with an efficiency ni = 3 %. More detailed information is presented later in Chap. 6, Sect. 6.4.

In order to calculate the characteristics of any laser, it is necessary to know the populating rates of the upper and lower laser levels, Rj and Rt; the lifetimes of these levels Tj and Tj, with allowance for the different processes of their collisional quenching; the transition probability Aji; and the line width Дрс. The rates of populating Rj and Ri are determined from solving the system of kinetic equations which constitute a balance of the rates of formation and decay of individual components of the plasma (see Chap. 4, Sect. 4.3), with allowance for the processes of populating of the laser levels. If the above parameters of gas laser media are known, it is possible to determine the small-signal gain a0, saturation intensity Is, and output laser intensity Iout (for example, see [35, 36]):

where gj and gi are degeneracies of the upper and lower laser levels, h is Plank constant, c is light velocity, в is distributed losses, Дрс is line width, / is length of active medium, and r1 and r2 are reflectivities of cavity mirrors.

For NPLs operating on the IR transitions of Xe, Kr, and Ar atoms, the poor accuracy in determining the rate constants of a number of basic plasma processes is a serious problem when computing Rj and Ri. In some cases, such constants from different literature sources differ a few times, and sometimes they differ by an order of magnitude. Approximately the same difficulties take place for Aji, Tj, Ti, and Дрс. Let us examine the problem in more detail using the example of a xenon NPL.

Transition Probability Aji Virtually all kinetic models of xenon NPLs use calculated data on the probabilities of radiative transitions 5d-6p of the Xe atom (see Table 5.4). In a recently published study [65], an attempt was made to determine Aji from measured intensities of spontaneous radiation for the 1.73, 2.03, and 2.65 pm lines during excitation of pure xenon (PXe = 5 Torr) by a nanosecond electron beam. The precision in determining Aji was not great—values of Aji for the 1.73, 2.03, and 2.65 pm lines were equal to 4 x 105, 3.4 x 106, and 7.9 x 105 s_1, respectively.

Lifetimes Tj and Ti of Laser Levels To determine Tj and Ti, it is necessary to know the radiative lifetimes of the levels (see Table 5.4) and the rate constants of the processes of collisional quenching. The rate constants of the quenching processes of the lower 6p levels by different atoms are rather well known (see Table 5.5). For the upper 5d levels, such data are limited and frequently are estimates obtained as a result of numerical modeling. Table 5.6 provides information for the 5d[3/2]10 level, obtained on the basis of the best match of calculation results with experi­mental data [21, 23, 25, 32, 36] or from analysis of plasma processes in the

Table 5.6 Rate constants for the processes of collisional quenching of the 5d[3/2]10 level of the Xe atom by rare gas atoms Quenching atom He Ne Ar Xe Rate constant, 10-11 cm3/s (calculated estimate) 0.01 [32]; 0.2 [36];1.5 [66] _ 0.1 [21]; 0.25 [23]; < 0.1 [25]; 0.1 [32]; 3 [67] 30 [21]; 10 [25]; 4 [32]; 20 [36]; 1 [66] Rate constant, 10-11 cm3/s (measurement) <0.06 [51] ~0.1 [51] 5.5 ± 0.5 [51] Note: Revised data of study [36] are shown, which were obtained using more reliable data on Дус

afterglow of a gas-discharge plasma [66, 67]. Study [51] was probably the only experimental study that determined a number of rate constants for quenching processes of the 5d[3/2]10 level by Xe, Ne, and Ar atoms and by certain molecules (N2, CO, H2, etc.), based on measurements of the intensity of resonance vacuum UV line 119.2 nm (transition 5d[3/2]10 ! 1S0). The results of this research for Xe, Ne, and Ar atoms are also shown in Table 5.6.

At high specific power depositions (q > 1 kW/cm3), when the concentration of electrons ne > 1013 cm-3, there may be a reduction in the lifetime and change in the populations of the excited states owing to the processes of their quenching and mixing in collisions with plasma electrons. The rate constants of such processes with the participation of excited states of the Xe atom (6 s, 6p, 5d, etc.), obtained by calculation, are contained in studies [32, 68], for example.

Line Width Дис For high-pressure gas lasers (including gas NPLs), the broaden­ing of the luminescent line is homogeneous and due primarily to collisions with gas atoms. Calculations of the parameters of xenon NPLs were carried out using values of Дрс obtained as a result of selection in the course of numerical modeling of laser characteristics [29], from calculations using the Lennard-Jones potential [69], or from the results of measurement [70, 71] (Table 5.7).

Then let us briefly examine some of the most well-developed kinetic models of NPLs operating on IR transitions of Xe, Kr, and Ar atoms, which were tested from experimental results obtained in the pumping of active media both by nuclear radiation and fast electron beams.

VNIIEF Kinetic Models VNIIEF developed “small” stationary models [33—39] to determine the lasing mechanisms of NPLs based on IR transitions of Xe, Kr, and Ar atoms and preliminary calculations of laser characteristics. As demonstrated by analysis of kinetic plasma processes, for an adequate description of the operation of lasers operating on binary mixtures He-B (B = Xe, Kr, or Ar) or Ar-Xe, it suffices to include 10-15 basic reactions in the model, which were selected beforehand as a result of analysis of the characteristic times of plasma processes (see Tables 4.10 and 4.11).

It was presumed that populating of upper laser nd levels of Xe, Kr, and Ar atoms occurs selectively, solely through reactions of dissociative recombination of molec­ular ions B2l with electrons. The percentage of recombination flux populating these

Table 5.7 Collisional broadening of the 1.73, 2.03, and 2.65 pm lines of the Xe atom by He, He, Ar, and Xe atoms (half-width of the luminescent line Дус, GHz/atm) Quenching atom X, pm He Ne Ar Xe 1.73 48 [29]; 15 [70] 1.0 [70] 48 [29]; 72 [69]; 15 [70]; 15 [71] 80 [29] 2.03 65 [29] — 65 [29]; 63 [69]; 75 [71] 90 [29] 2.65 38 [29] — 82 [29]; 43 [69] ■ 29]

levels was determined in the process of modeling, and was taken as 100, 50, and 40 % respectively for the Xe^, KrJ, and ArJ ions. Apart from the reactions cited in Tables 4.10 and 4.11, the calculation models included the processes of collisional quenching of nd levels. For level 5d[3/2]10 of the Xe atom, the constants of the processes of collisional “quenching” by Xe, Ar, and He atoms according to the

~i /л — іл 7 О

revised data are 2 x 10- , 1 x 10- , and 2 x 10- cm /s, respectively.

The validity of the model was tested by calculations of the energy and threshold characteristics of NPLs using the mixtures He-Xe (A = 2.65 pm), Ar-Xe (A = 1.73 pm) [34—37], He-Kr (A = 2.52 pm), and He-Ar (A = 1.79 pm) [38, 39], excited by uranium fission fragments, products of the nuclear reaction 3He(n, p)3H, and electron beams as a function of pressure, mixture composition, and power deposition. The calculated curves for energy and threshold characteristics agree satisfactorily with experimental data. Table 5.8 provides a comparison of the results of calculations and experimental data in optimal modes with respect to pressure and composition.

Kinetic Models of the University of Illinois Initially the model was developed for an Ar-Xe laser (A = 1.73, 2.03, 2.63, 2.65 pm) and included around 100 plasmochemical reactions [21]. To test the model, data obtained from the pumping of Ar-Xe laser with electron beam [69, 78] and uranium fission fragments [79] were used. Later on this model was improved and was used to calculate the characteristics of NPLs using mixtures of He (Ne)-Ar-Xe (A = 1.73, 2.03, 2.63, 3.37 pm) excited by uranium fission fragments [80, 81].

A rather complex two-stage process was proposed as the main mechanism for populating 5d levels of the Xe atom. First, as a result of dissociative recombination ArXe+ + e, the states 7p and 7s of the Xe atom are populated, and then Xe atoms in the states 5d[3/2]10 and 5d[5/2]20 are formed in collisional processes Xe*(7p,7 s) + Ar(Xe). In these models it is assumed that level 5d [3/2]10 is populated during collisions with Ar atoms with an efficiency of 70 %, and level 5d [5/2]20—with Xe atoms with an efficiency of 50 % [21].

In these models, much attention is attended to processes of the so-called colli — sional mixing of levels 6 s, 6p, and 5d by plasma electrons, which reduce the population inversion of laser transitions. Calculations have shown that such pro­cesses lead to a cessation of lasing if the degree of plasma ionization exceeds some critical value (0.8-1.0) x 10-5 [82]. At atmospheric pressure, this corresponds to an electron density of about 2 x 1014 cm-3. In the opinion of the authors [21, 82], it is

Fig. 5.2 (1) Specific power deposition (1) and output power at the 1.73 pm line (2, 3) at excitation of an Ar-Xe mixture (0.68 atm; Ar:Xe = 99.5:0.5) by uranium fission fragments;

(2) experiment [79];

(3) calculation [21] these processes that cause the early cessation of lasing observed in experiments [79] (Fig. 5.2).

Associates of the University of Illinois also examined the lasing mechanism of NPLs operating on the transition of an Ar atom with X = 1.79 pm and carried out calculations of some of its characteristics [22]. It was proposed that the main process of populating of the upper laser level 3d[1/2]1 of the Ar atom in the mixture He-Ar is dissociative recombination of heteronuclear ions HeAr+ with electrons, while the processes of collisional destruction of HeAr+ ions in collisions with atoms were not included in the model. Earlier it was shown that such a populating mechanism is most likely mistaken, because of the low bond energy of the HeAr+ ion (0.027 eV [44]), which roughly coincides with the energy of thermal motion of atoms at room temperature. Therefore, HeAr+ ions will be effectively destroyed in the collisions (5.5) and (5.6).

IOFAN Kinetic Models The non-stationary models of xenon lasers [27—30, 32] developed at IOFAN are the most detailed. Their most recent modifications [29, 30, 32] include several hundred plasmochemical reactions with the participation of the ions He+, He^, He^, Ar+, ArJ, ArXe+, Xe+, XeJ, the excited atomic and molecular states He*, He2*, Ar*, Ar**, ArXe*, Ar2*, Xe2*, and all of the excited states 5d, 6p, 6 s, and 6 s’ of the Xe* atom. The numerical modeling was carried out for the mixtures He-Xe, Ar-Xe, and He-Ar-Xe, and pure Xe. As the authors of [32] state, they were able to implement complete optimization of laser characteristics of when simultaneous lasing on six lines belonging to the transitions 5d-6p occurred.

Calculations under the conditions of pumping of active media with an electron beam showed that for all laser lines, the highest energy characteristics are observed for the Ar-Xe mixture, while the maximal efficiencies (4.5 %) and specific energy
release (19 J/l) are achieved at the 1.73 ^m line. The intervals of the parameters at which such high energy characteristics can be obtained are determined.

As was shown in Table 5.3, the authors of these models assume that virtually all plasma processes ever discussed participate in populating the upper laser levels. Thus, to populate the level 5d[3/2]10, 40 % and 15 % of the fluxes of dissociative recombination of molecular ions, respectively, ArXe+ and Xe^, 20 % of excited atoms formed as a result of the Ar* + Xe reaction, and about 40 % of the reaction of three-body recombination of the atomic Xe+ ions are expended.

IOFAN associates also modeled an argon laser at the 1.27, 1.79, and 2.40 ^m lines [31]. The model was tested with experimental data on pumping a He-Ar mixture laser (A = 1.79 ^m) with electron beams (FIAN) and uranium fission fragments (VNIIEF, VNIITF, Sandia). In the opinion of the authors of [31], populating of the upper laser level occurs by means of two processes: dissociative recombination of HeAr+ ions (15-25 % of the total pumping flux) and ArJ ions (5­20 % of the total pumping flux). The model also considered processes associated with the presence of the impurities N2, O2, H2, and H2O in the He-Ar mixture, which makes it possible to more precisely consider the real experimental condi­tions. The inclusion of these impurities in the model was necessary to the authors of [31], in order to explain the behavior of experimental curves [77] in the region of partial pressures of argon < 5 Torr.

In NPLs based on mixtures of rare gases excited by the fission fragments of uranium nuclei, the shape of the lasing pulse is usually close to that of the neutron pulse that initiates the fission reaction [4—8, 49]. However, during a number of experiments involving mixtures, the bulk of which were comprised of gases with high atomic numbers (for example, Ar and Kr), the lasing pulse consisted of an irregular series of small peaks on the crest of a common pulse; the total suppression of lasing followed by its restoration was sometimes observed [4, 34—36].

Various explanations exist for the irregular behavior of lasing with an increase in power deposition. For example, in study [50], lasing in an Ar-Xe mixture (A = 1.73 pm) only occurred on the leading edge of the pumping pulse, which in the opinion of the authors of [51], was the result of Xe atom 6s, 6p, and 5d level collisional mixing by the plasma electrons.

Other attempts at explaining this effect are based on the hypothesis of the appearance of refractive index discontinuities that originate within low-intensity longitudinal sound and shock waves in the presence of comparatively short neutron pulses (т ~ 10~3 s) due to a pumping inhomogeneity in the direction of the optical axis. In particular, the cause of these inhomogeneities may be the presence of gas regions directly adjacent to a laser cell’s end windows that are not excited by the fragments. However, the absence of a correlation between the duration of the aforementioned micropeaks, the time intervals that separate them, the geometric dimensions of the cell, and sound velocity in the gas, as well as, more importantly, the disappearance of these peaks with a decrease in gas mixture density when other conditions remain unchanged, have given rise to doubts concerning the correctness of the proposed interpretation.

On the other hand, the peculiarities of the appearance and development of the transverse optical inhomogeneities that lead to the alteration of cavity stability can explain the absence of a time correlation between lasing and pumping pulses. A comparison of the calculated and experimental data cited in [35, 36] confirms this proposition. During the experiments, a rectangular laser cell was used that had two parallel planar uranium layers located close to one another. The calculation procedure was based on the use of the ray matrix technique to analyze the stability of a laser cavity filled with a medium that had a time-varying parabolic refractive index profile.

In studies [35, 36], during gasdynamic calculations of gas density redistribution in a laser cell that were performed while ignoring heat transfer processes, it was assumed that a gas motion only occurs in the direction normal to the uranium layer planes. Thus, the dynamics of altering cavity stability were examined in an approx­imation of the exclusive existence of one-dimensional inhomogeneities that were symmetrical relative to the cell’s central plane. The inhomogeneity of density distribution (and, accordingly, the refractive index) obtained by means of calcula­tions at each studied moment in time were approximated by a polynomial with a finite number of members. This polynomial’s first two coefficients were used for the parabolic representation of the refractive index profile in the entire volume of the cell’s active section. The calculations made it possible to identify the moments in time at which the loss of cavity stability and its subsequent restoration occur. They are in quite accurate agreement with the experimentally observed lasing cessation and resumption times. It is more difficult to carry out the analysis of laser cavity stability described in Chap. 7, Sect. 7.4 (planar uranium layers) and in the second section of this chapter, because it is impossible to regard the inhomogeneity distribution in such a system as homogeneous.

For condensed laser media, pumping with the help of NOCs was more successful than direct pumping of these media with nuclear radiation, as was considered in the previous section. There are two experimental studies which show that neodymium lasers may be pumped using NOCs based on xenon plasma [41] and the phosphor CsI(Tl) [55]. This method was also used successfully in experiments on the TRIGA reactor, in which a photo-dissociation iodine laser was pumped with the lumines­cent emission of XeBr* molecules (see Chap. 3, Sect. 6).

Experiments that pumped solid-state lasers with NOC emission were carried out on pulsed reactors [40, 41, 55].

Fig. 11.5 The spectra of the luminescent crystal CsI (0.5 % Tl) at different points of the pulsed irradiation on the VIR-2 reactor at Dy = 3.5 x 103 Gr and Ф = 4 x 1014cm~2. The insert shows the time dependence of the dose rate PY (upper curve) and the luminescent power density IY (lower curve) and indicates the moment in time (1, 2, 3, 4, 5, 6) when the spectra were recorded [2]

A series of experiments [40] on the TRIGA reactor used gas NOCs based on 3He-Kr(Ar, Xe) mixtures with excitation by the products of the nuclear reaction 3He (n, p)3H. The spectral-luminescent characteristics of these mixtures are given in two works [40, 52]. The specific power deposition at the maximum of a 12-ms reactor pulse was about 20 W/cm3. A schematic of the experiment is shown in Fig. 11.6. The light radiation from the NOC, which was placed near the reactor core, was transmitted to the active laser element Y3Al5O12:Nd3+, 3 mm in diameter and 62.5­mm long, using a bundle of ~2,000 quartz light-guide fibers that were 5-m long. To pump the volume of the laser element uniformly, the end faces of the fibers were uniformly distributed about its surface (Fig. 11.7). Laser action was not observed in these experiments. In study [40] absence of laser action is explained by the low specific power deposition in the laser medium due to loses during transportation of the light radiation and poor correlation between the plasma luminescence spectrum and the absorption spectrum of the Y3Al5O12:Nd3+ crystal.

Experiments performed in VNIIEF on the VIR-2 M reactor were more success­ful. As part of these experiments, fiber neodymium lasers were pumped with the light radiation of xenon plasma [41]. Prior studies have shown that the spectrum of the plasma radiation that appears when xenon at 1 atm is excited with uranium fission fragments (the specific energy deposition is ~5 J/cm3) is close to the emission spectrum of an absolutely black body at temperatures <6,000 K [41].

In experiments on the VIR-2 M reactor, the laser active element was placed inside the NOC between two plates with uranium layers (Fig. 11.8). The 10 x 6 cm plates were placed parallel to each other 2 cm apart. Their internal surfaces have a deposited layer of uranium oxide-protoxide. The thickness of the metallic 235U layer was about 5 mg/cm2. A fiber laser 1-4-m long with a fiber diameter of 40-50 ^m was wound around a frame and placed symmetrically between the

uranium layers. Irradiation of the NOC was done at the bottom of the central reactor channel. This series of experiments obtained laser action when using fiber neodym­ium lasers based on silicate and phosphate glasses. The results of one experiment on a neodymium laser based on phosphate glass with the NOC filled with xenon at 0.5 atm (plasma temperature 4,700 K) are shown in Fig. 11.9. The maximum energy of the laser radiation over a single pulse was 2.5 J.

Besides using gas NOCs to pump neodymium lasers, in VNIIEF also used NOCs based on solid-state scintillator. A brief overview [55] of these experiments reported that lasing was obtained when pumping a laser element based on neodym­ium glass with radioluminescent radiation from the CsI(Tl) crystal. The laser element and CsI(Tl) crystal were irradiated in the central channel of the BR-1 pulsed reactor [16]. The maximum laser power was equal to 4 kW at a laser pulse of approximately 60 ^s duration.

In conclusion, a recently published study [56] should be noted where pumping a laser based on the crystal YAP:Tm3+ (A = 1.85-2.0 ^m) was fulfilled using gas NPLs at the transitions of Ar (A = 1.79 ^m) and Xe (A = 1.73 ^m) atoms as a NOC. The active region of the laser crystal was 5-mm long with a volume of about 4 mm3. From the oscilloscope traces in Fig. 11.10, it can be supposed that the experiments were performed on the EBR-L pulsed reactor [16]. When a xenon NPL (A = 1.73 ^m) was used as the source for pumping, the threshold specific power deposition was about 35 kW/cm3, and the conversion efficiency of the absorbed optical radiation into laser radiation was 4 %. Such low laser efficiency is rather

unexpected since prior executed studies [56] showed that the conversion efficiency of laser radiation on lines 1.73 and 1.79 ^m into luminescent radiation was equal to 54 and 76 %, respectively.

Excimer lasers using molecular halogenides of rare gases RX (R = Ar, Kr, Xe; X = F, Cl, Br), excited by electron beams and in a gas discharge, are powerful sources of laser UV radiation [155, 156]. To pump these lasers, the radiation of nuclear explosions was also used (see Chap. 12). Excimer lasers operate most effectively in pulsed mode at high specific pumping powers of q > 0.1 MW/cm3, which is explained by the great radiation line width.

Excimer lasers can operate in quasi-CW mode, because after photon radiating, excimer molecules fall into the lower repulsive or weakly bound state. In this connection, some studies were carried out on the possibility of pumping excimer laser media with nuclear radiation using pulsed reactors as the neutron sources.

The largest number of investigations were devoted to the XeF-laser (A = 351 and 353 nm), which has the lowest laser threshold [60, 157—162]. In experiments [157] with the TRIGA reactor, a small-signal gain was registered of ~10-4 cm-1 at line 351 nm of the XeF molecule at a low specific power deposition of q « 40 W/cm3. Experiments performed with the SPR-III reactor showed that for the mixture 3He — Xe-NF3, the gain at the 351-nm line is around 7 x 10-3 cm-1 (q«5 kW/cm3) [158], while for pumping the mixture Ne(Ar)-Xe-NF3 with uranium fission frag­ments, a gain of ~2 x 10-3 cm-1 (q « 2 kW/cm3) was registered [48, 159]. Exper­iments using pulsed reactors with active media based on the molecule XeF [48,160] and KrF (A = 248 nm) [60], aimed at obtaining lasing, yielded no positive results. These experiments used the mixtures 3He-Xe-NF3, Ne-Xe-NF3 [48], Ne(Ar)-Xe — NF3(SF6) [160], and 3He-Kr-NF3 [60].

Calculations showed that for the mixture Ne-Xe-NF3, it is possible to obtain Пі ~ 1 % for q ~ 100 W/cm3 [161], and 500 W/cm3 [162]. The possibility of creating a nuclear-pumped XeF laser is determined primarily by the absorption coefficient in the active medium, the uncertainty of which greatly influences the results of calculations [161].

In conclusion, we note that study [163] reports achieving laser action at the transition of the XeCl molecule (A = 308 nm) when pumping the mixture Ar-Xe — CCl4(HCl) with uranium fission fragments in experiments with the EBR-L reactor (q ~1 kW/cm3). Although the authors assert that lasing was observed in the experiments, the cited data were insufficient for this conclusion.

In contrast to VNIIEF, where the concept of an autonomous stationary RL is being developed, since 1986, FEI (Obninsk) has been examining the concept of a high — power pulsed laser system based on an optical nuclear-pumped amplifier (OKUYaN) [23]. The “master oscillator-two-pass amplifier with phase conjuga­tion” configuration underlay the work of the OKUYaN-based laser system [23, 24] (Fig. 6.12). The OKUYaN includes reactor and laser blocks. The reactor, which is

Fig. 6.12 Optical scheme of laser system based on OKUYaN [23, 24]:

(1) master laser-oscillator; (2, 4) telescope systems;

(3) optical nuclear-pumped amplifier (OKUYaN);

(5) pulsed reactor core;

(6) Faraday cell; (7) phase conjugation cell;

(8) polarizer

sometimes called an “initiating” reactor, presupposes use of a multicore fast — neutron pulsed reactor. The laser block constitutes a booster subcritical core in which neutrons are multiplied. We note that the first proposal to develop such devices appeared more than 20 years ago (see, for example, review [25]). Require­ments for “initiating” reactors, their arrangement, and the results of calculation of the neutron physical characteristics, are considered in study [26].

At present there is an OKUYaN demonstration model operating at FEI, “Stand B” [23, 24, 27, 28]. It consists of two parts: the so-called “first workplace,” consisting of a two-core BARS-6 reactor and an experimental section for studying the characteristics of individual lasing elements (cells), and the “second work­place.” Organization of experiments for the “first workplace” was examined in Sect. 6.2.

The other part of Stand B (the “second workplace”), which includes the two-core BARS-6 reactor and laser block with a volume of around 2.5 m3, was put in operation in 1999. The design and the neutron-physical characteristics of the BARS-6 reactor were considered in some detail in studies [23, 28]. The BARS-6 reactor, built on the basis of the BARS-5 reactor [7, 29] from the design documen­tation and with the technical assistance of VNIITF (Snezhinsk), has two reactor cores, which are arranged on a platform and can be moved along a rail track to one of the two “workplaces” or to the biological shield, into which the core of the reactor is moved for the period of preparation for experiments.

The relative arrangement of the BARS-6 reactor and laser block is shown in Fig. 6.13. The laser block (shown in Fig. 6.14) is a cylindrical structure with a diameter of 1.7 m and length of 2.5 m, with a longitudinal axial cavity for placement of two BARS-6 reactor cores. It consists of up to 800 laser elements, their simulators, and elements of the neutron reflector, made in the form of tubes 49-mm in diameter and 2.5-m long, filled with polyethylene. The space between the tubes of the laser block contains around 760 neutron moderator elements made from shaped polyethylene. The laser block, in neutron-physical relations, constitutes a deep subcritical core with a neutron multiplication coefficient kef < 0.9. Its neutron — physical characteristics were optimized in study [30].

Fig. 6.13 Arrangement of the BARS-6 reactor and laser block at the “second workplace” [23, 24]: (1) two-core BARS-6 reactor; (2) laser block; (3) laser elements and its simulators; (4) outside neutron reflector; (5) inside neutron reflector

The laser element (Fig. 6.15a) is a stainless-steel tube with a diameter of 50 mm, wall thickness of 0.5 mm, and length of 2.5 m, which is coated on the inside with a 5 ^m thick layer of metallic 235U. The laser element simulator is in the form of two annular aluminum tubes, with the small space between them filled with 235U3O8 (Fig. 6.15b). It is a copy of a laser element in neutron-physical respects and contains the same amount of 235U (32 g).

The specific features of neutron pulse generation in the reactor with the laser block are discussed in studies [23, 28]. The half-width neutron pulse duration for the BARS-6 reactor without the laser block is around 100 ^s, while in the reactor with the laser block it is much more, 2 ms when there is an internal reflector present, and 20 ms in its absence. At the same time, the energy release in each reactor core of the BARS-6 is equal to 2 x 1017 fissions, and in the laser block, 2.2 x 1017 fissions.

For the configuration “reactor + laser block” with an internal reflector, experi­ments were carried out with the use of laser elements as part of a laser block [23, 31]. The main goal of the experiments is to study the operating modes of the “master oscillator-single-pass amplifier” configuration. In contrast to the amplifier
element, the master oscillator has windows at the ends arranged at a Brewster angle, and a stable laser cavity consisting of spherical and flat dielectric mirrors. A telescope was used to allow passage of the beam of the master oscillator through the amplifier without losses. The laser medium was a He-Ar-Xe mixture (A = 2.03 pm) at a pressure of 1.05 atm. The specific power deposition, averaged over the active volume, at the maximum of the neutron pulse for the amplifier element and the master oscillator were 40 W/cm3 and 80 W/cm3, respectively, at a half-width neuron pulse duration of 1.8 ms.

From the energy parameters of the laser radiation measured at the amplifier input and output, the small-signal gain (a0) and saturation intensity (Is) of the laser medium were determined: a0 = 8.1 x 10~3 cm-1; Is = 92 W/cm2. Using these parameters, it is possible to determine the laser power efficiency qI = «о Is/ q ~ 2 %, which agrees with the results obtained when NPLs were studied (see Table 3.5, and Chap. 3, Sect. 3.1). The results of these investigations allow one to hope for high energy parameters when there is full-scale operation of the entire OKUYaN laser block.

A circuit with continuous transverse gas circulation was used in the LM-4 four — channel laser module operating together with BIGR reactor [41], within which a CW lasing mode was first achieved for NPLs [26] (see Sect. 6.1). The lasing lasted

1.5 s and ended with exciting neutron pulse termination.

The LM-4 apparatus consisted of four laser channels (Fig. 9.15) that were incorporated into a common gas loop and were separated from one another by plate-type radiators. The active length of the channels, determined by the dimension of the uranium layers along the optical axis, was LA = 1 m. The dimension of each channel in the direction perpendicular to the gas flow was d = 2 cm, while in the direction parallel to the flow, it was b = 6 cm (see Fig. 9.1). The average thickness of the uranium layer was <5u = 5 mg/cm2. This layer was covered with a protective aluminum film that had a thickness of SA = 0.5 mg/cm2. The LM-4 apparatus was irradiated by a neutron flux from a BIGR pulse reactor [42] with a duration of ~1.5 s. Thermal neutron flux density in the channel location at the pumping pulse peak reached 3.5 x 1014 cm~2 x s_1. The gas circulation system, based on motion of the hydropneumatic piston with a rectangular cross-section [41], ensured a gas velocity that was homogeneous along the length of the channel. It was switched on roughly ~0.3 s before the neutron pulse start and brought the gas velocity up to its maximum value of 4.5 m/s over this time interval. Gas circulation ceased immedi­ately upon neutron pulse termination. Under such conditions, after the lasing threshold is exceeded by roughly ~2-3 times, the shape of the laser pulse quite closely duplicates that of the pumping pulse.

Fig. 9.15 Transverse section of an LM-4 laser module together with a BIGR reactor: (1) aluminum substrate with a uranium layer; (2) radiator; (3) graphite; (4) casing; (5) gas pipeline

In order to investigate the effect of the gas circulation mode on the shaping of a laser pulse, two experiments were carried out in ref. [26] that involved an Ar-Xe mixture (70:1) at a pressure of 0.35 atm. During the first of these tests, the pumping pulse’s leading edge noticeably advanced the start time of gas circulation (delay of circulation start time); during the second test, the circulation system was activated long before the start of the neutron pulse and cut off the gas feed at the pumping pulse’s leading edge (premature switching of gas circulation). These experiments demonstrated that lasing does not occur until gas circulation begins, and con­versely, that lasing is terminated when circulation stops.

The physical model proposed in ref. [43], which describes the dependence of lasing active volume upon gas mixture composition and density, gas flow velocity, and specific power deposition, provides entirely satisfactory agreement with the experimental time characteristics of laser pulses. According to this model, over the period of time required for a portion of the gas to traverse a laser channel, if gas velocity at the inlet and fission intensity in the uranium layers are negligibly altered,

AUo(t) << Uo, Aqc(t) << qc, (9.90)

then the gas mass flow at each given moment in time can be roughly regarded as constant:

PoUo(to) ~ P (x, t) x U(x, t), (9.91)

where U(x, t), p (x, t) is the gas velocity and density values in the laser channel in a plane with a coordinate of x at a moment in time of t, averaged over the cross­section; Uo is the gas velocity at the channel inlet (which is homogeneous through­out the cross-section); to is the moment in time that the portion of gas under consideration enters the channel; and po is gas density at the channel inlet (which is kept constant). The average velocity and density values are determined by the equations

It is assumed that gas density, velocity, and temperature distribution will not be dependent upon the z coordinate, which is oriented along the system’s optical axis. The approach used reduces the description of the gas flow in the channel to a one-dimensional model.

During a gas flow in a plane-parallel channel, the length of which, b, is comparable to its transverse dimension, d, the pressure differential, ДP, between the channel inlet and outlet should be small (ДP/P << 1 (see the first section of this chapter)). Thus, from the condition of approximate pressure equality at any given point in the channel, P(x) ~ const, and the state equation for an ideal gas, we get

P0T0 = p(x, t)- T(x, t). (9.93)

Here, T0 is the temperature at the channel inlet, and T(x, t) is the temperature value in a plane with a coordinate of x at a moment in time of t, averaged over the channel cross-section, which, when small temperature deviations, AT(x, y, t) << T(x, t), are present within the confines of the transverse cross-section under consideration, is determined by a formula similar to Eq. (9.92).

It follows from correlations Eqs. (9.91) and (9.93) that

U(x, t)= ^ U0(t0). (9.94)

T 0

By virtue of the conditions in Eq. (9.90), the following heat balance correlation for a laser channel at each given moment in time should roughly hold true

x

where qS is the surface power density of the sources of energy deposition in the gas from both uranium layers. With allowance for Eq. (9.95), correlation Eq. (9.94) takes the form

U(x, t)

The energy deposition distribution of surface fission fragment sources is quite homogeneous, with the exception of the edge parts at the channel inlet and outlet. Ignoring the edge effects, we can assume that qS = const. Also, bearing in mind that U(x, t) = dx/dt, then from the latter equation, we get an equation that reflects, in a one-dimensional model, the relationship between the path traversed by a portion of the gas in the channel and the time over the course of which this path is traversed:

dx/dt = U0(t0) + Bx,

where B = (T0P0Cpd) 1q5.

At the initial condition of x(t0) = 0, the solution to this equation for the time it takes a portion of the gas to traverse a distance of x from the channel inlet, т = t — t0, within which the Eq. (9.92) yield the average density and velocity value distribu­tion, takes the form

In the special case of sufficiently large pumping velocities,

Bx/U0 < < 1, (9.97)

we get a simple dependence for the travel time

т = x/U0. (9.98)

When a laminar gas flow is present, its heat exchange with the laser cell walls occurs by means of conventional molecular heat conduction. If the o т time value is small enough that the condition l << d is satisfied for the transverse dimension of the near-wall region involved in heat removal, then from Eqs. (8.3) and (9.98) when the condition (9.97) is satisfied, we get

l(x) ~ v/ox/U0. (9.99)

If it is assumed that the development of the heat removal zone at the channel wall provided gas circulation occurs in accordance with the same regularities as in a cell that contains a stagnant gas, then in the general case, the laser active region volume per unit of laser channel length (along the z-axis) equals

b

0

where yA(x) = d/2 — l(x), while the relationship between the x coordinate and travel time, т, is determined by Eq. (9.96).

It is not difficult to show that in the special case when the l << d and inequality (9.97) are satisfied for the entire range of x, the laser active region volume is determined by the equation

where Ai is the proportionality factor in correlation (9.99). In the case of sealed (without gas circulation) laser cells, the formula l(t) = A^/at, which corresponds to correlation (9.99) written in the form l(x) = A^Jax/u0, as a rule satisfactorily describes the yA(t) dependence obtained from an accurate gasdynamic calculation over an interval of 3d/8 < y < d/2 (y varies over limits of 0 to d/2). However, for the specific alternative of a planar cell filled with Ar at P0 = 0.35 atm and T0 = 293 K, it allows the calculated dependence of yA(t) over an interval of d/4 < y < d/2 to be reproduced with an accuracy better than 6 %. Here, Ai« 3 for sealed cells (see Sect. 8.3), while the A factor can take on values one-and-a-half to two times smaller than for cells with gas circulation.

The model described was used in ref. [43] to calculate the dynamics of the development of a laser region’s active volume during experiments involving artificial desynchronization between the start of neutron pulse and gas-circulation system switching. This volume was found from Eq. (9.100) using the results of the calculation of active region size variation with the passage of time in an identical sealed cell. The relationship between the x coordinate in expression (9.100) and travel time, t, which corresponds to the duration of irradiation in a sealed cell, was determined using Eq. (9.96). The dependences of the neutron flux and gas flow velocity upon time needed for the calculations were assigned based on experimental data.

When the lasing threshold is exceeded, the active volume and the specific power deposition in the gas, which is proportional to the neutron flux, determine the laser output power at each given moment in time. The experimental dependences of neutron flux density, gas velocity, laser output power, as well as the calculated dependence of the product of active region volume times neutron flux density, upon time relative to their maximum values are presented in Fig. 9.16 for an alternative with delay of circulation start time. The results of the calculation of similar dependences for an alternative with premature switching of gas circulation are reflected in Fig. 9.17. The calculated cross-sections of the active volume (the clear area) and the zone involved in heat removal (the dark area) are shown in Fig. 9.18 for several successive moments in time that correspond to Figs. 9.16 and 9.17.

Fig. 9.16 Dependences of neutron flux density (1), gas velocity (2), laser output (3), and the product of gas active volume and neutron flux density (4) upon time for alternative with delay of circulation start time

a R. u. b R. u.

Fig. 9.17 Dependences of neutron flux density (1), gas velocity (2), lasing output (3), and the product of gas active volume and neutron flux density (4) upon time for alternative with premature switching of gas circulation

b

Fig. 9.18 Time dependences of the transverse cross-sections of the laser-channel active volume (clear area) and the zone involved in heat removal (dark area) for two alternatives: (a) delay of circulation start time, and (b) premature switching of gas circulation

Entirely satisfactory agreement is observed between the behavior of the exper­imental dependences of laser output (3) upon time and the curves (4) that reflect the value of the product of the neutron flux and the calculated active region volume values at each given moment in time. The difference in the slopes of the corresponding curves at the leading edge (Fig. 9.17) is explained by the fact that the lasing start has a threshold nature, unlike the process of the formation of a near­wall heat removal zone. Apparently, the medium small-signal gain is minimal at the moment of lasing start and begins to increase with an increase in power deposition. Therefore, the experimental curve of the dependence of laser output upon time runs
noticeably steeper. During these experiments, thermal neutron flux density at the lasing threshold was Ф « 9 x 1013 cm~2 x s_1, which corresponds to «0.26 r. u. (see Fig. 9.17).

Peculiarities of the Gas Flow in Flowing NPLs. Two-dimensional calculations of spatial inhomogeneities in NPL flowing channels were performed for the first time at the Idaho National Engineering Laboratory [44] using a specially developed gasdynamic program. The results of the calculations performed in ref. [44] were later repeated at the VNIIEF [45]. Because the VNIIEF did not have a gasdynamic program similar to the one used in ref. [44], a simpler calculation procedure was proposed that was based on a series of reasonably simplified physical models [46] (see also the subsection entitled “Calculations of Spatial Inhomogeneities in Flowing NPLs”). The difference in the density distribution calculation results obtained using this procedure and those of the similar calculations performed using the two-dimensional program [44], which makes strict allowance for gasdynamic and heat exchange processes in the entire gas volume, do not exceed ~2 % [45, 46].

One of the principal aspects of this procedure consists of the use of a physical model according to which the heat exchange processes in the bulk of the gas volume can be ignored. This model holds true, provided that

U0 > > ab/d2,

where a is the thermal diffusivity coefficient of the gas. Because the gas-flow rate in the NPLs under consideration comes to U0~10 m/s, then at typical channel transverse dimensions, b~6 cm and d~2 cm, as well as a rare gas thermal diffusivity of a ~0.2-0.6 cm2/s, the condition written above is virtually ensured.

In ref. [46], calculations were performed for a wide range of working gas pressure and velocity variations, which made it possible to optimize both the total energy deposition in the gas and the degree of specific energy deposition inhomo­geneity through the channel volume.

A weak facet of the calculation procedure used consists of its passive zone model, i. e., the near-wall region that does not take part in lasing. Numerical and theoretical investigations carried out during 1979-1984, which are described in detail in Chap. 8, revealed that the temperature profile has a clearly pronounced maximum and the density profile has a clearly pronounced minimum in sealed NPLs at times of t < 0.1 s. The latter is explained by the fact that in direct proximity to the laser channel side wall to which a uranium layer is deposited, a gas zone is formed from which the heat is intensely removed to the wall. This is called the passive (dead) zone.

It is also obvious that the same passive zone formation effect must take place in NPL flowing channels (where, as is generally known, specific energy release increases from the center to the side surfaces), the wall temperature of which is lower than the gas flow temperature. Accordingly, a proposition was advanced in ref. [43] that the development of a heat removal zone at the channel wall during gas flowing occurs qualitatively under the same laws as in a cell without gas flowing, i. e., the existence of a passive zone is proposed, the boundary of which is
determined by the density minimum and the temperature maximum. The relation­ship between temperature and density is unequivocally determined by a state equation. Because the pressure in the laser channel’s transverse cross-section is almost homogeneous (P « const), then in order to fully describe both the density and the temperature profile, it is sufficient to examine the dependence upon the coordinate of one of these parameters.

However, in an actual situation involving a gas flow, a boundary layer with a thickness of lv is formed near the wall, within which velocity increases from zero at the wall itself to a value equaling that in the flow core. Over the course of temperature equalization between the wall and the heated gas, hydrodynamic effects and heat conduction effects have a strong influence on one another [2].

For a compressible gas, temperature and velocity distribution near a planar wall in the presence of stationary motion is described by an equation system [2] that includes heat conduction, motion, and continuity equations

as well as a state equation

T = P/(y — 1)cvp.

Here, k is the heat conductivity coefficient; u is the longitudinal velocity compo­nent; v is the transverse velocity component; w is the specific power of the internal energy sources; and n is the dynamic viscosity coefficient.

The heat conductivity and dynamic viscosity coefficients of rare gases have a power dependence upon temperature [34]:

k = k0Tn; n = n0Tn. (9.106)

For these gases, n « 0.7 [34]. Even within the viscous and thermal boundary layers (as well as the passive zone, if it has not been eliminated), under conditions of actual powerful flowing NPLs, the absolute temperature values vary by roughly two times both in the longitudinal and the transverse directions. According to the relation (9.106), temperature differentials of this type will lead to variations of
~60 % in the heat conductivity and dynamic viscosity coefficients. Therefore, in order to find the dependence of viscous boundary layer thickness and the velocity distribution within its confines under actual NPL operating regimes, upon gas mixture flowing conditions and parameters, it is necessary to solve the system of essentially nonlinear differential equations in partial derivatives (9.102)-(9.106). Here, in addition to the boundary condition for velocity at the wall itself, it is correct to give the boundary condition far from the wall not at infinity (y = 1), as is frequently done in order to simplify certain problems, but rather in the plane of symmetry, i. e., at a finite distance of y = d/2. The same thing also pertains to temperature. At the wall itself, a Derichlet boundary condition, T(x, 0) = f1(x), or a Neumann boundary condition, dT(x, 0)/dy = f2(x), is given for temperature.

Thus, the solutions for the effective thickness of a viscous boundary layer, lv, and the velocity distribution therein, as well as for the width of the passive zone, la, and the temperature distribution therein, are functionals of the type

where T0 is the temperature value at the channel inlet (in these functionals, depending upon the conditions assigned at the wall, the temperature T(x,0) may be present instead of the derivative 3T(x,0)/3y. In addition, for simplicity’s sake, the dependence upon thermophysical parameters, especially upon the Prandtl number, is not reflected here). In point of fact, the problem is even more complex, because the distribution of specific sources in the gas will be dependent not only upon the fission intensity of the uranium nuclei in the uranium layers deposited to the channels walls, but also upon velocity and temperature distribution. A problem of this type for actual NPL modes can only be solved by means of numerical techniques involving the use of two-dimensional gasdynamic programs.

Of course, it may also be useful to a certain extent to obtain analytical solutions by introducing various simplifying assumptions, for example, of the T(x, y) — T0 << T0, v = const, and w = const types, etc. This helps mark the paths for optimizing energy depositions in the gas by means of gas pressure and velocity variations, an even to estimate share of passive zone volume (however small). But it must be said that any viscous boundary layer and passive zone models not based on the results of rigorous two-dimensional numerical calculations or direct experiments for NPLs operating at real power depositions cannot be regarded as correct.

From equation system (9.102)-(9.106), it is not difficult to obtain an integral equation for the passive zone. Actually, from continuity Eq. (9.104) and state Eq. (9.105), we get

dT

u + v
dx

Placing this result into Eq. (9.102) and integrating from zero to la with allowance for the fact that v(x,0) = 0 and that pT = p0T0 at P = const, we obtain

Possibility of Passive Zone Elimination

We will now examine the conditions under which a passive zone may be elimi­nated. Let us suppose that la = 0; it then follows from Eq. (9.108) that dT(x,0)/dy = 0, i. e., there is no heat flux at the wall. On the other hand, setting the heat flux at the wall to equal zero, then from Eq. (9.108), we get

du (x, y) dv (x, y) = w (x, y)

dx dy cpPfT 0

The latter equation means that when there is no heat flux at the wall, a passive zone can only exist if the specific heat source power distribution profile is similar to the sum of profiles of the derivative for x from the longitudinal velocity component and the derivative for y from the transverse component. It is obvious that the simulta­neous existence of such a coincidence over the entire extent of a flow in an NPL channel under normal operating conditions is unrealistic.

Thus, a necessary and sufficient condition for passive zone elimination consists of the heat flux at the gas and channel wall boundary equaling zero. Moreover, there will be no passive zone if the heat flux is directed from the wall to the gas (dT(x, 0)/dy > 0). This result makes it possible to estimate the channel wall heating needed for passive zone elimination. We will then perform an upper estimate. To this end, we will assume that the gas within a channel part from the inlet with a length of x over a time interval with a duration of t is in a state of rest. Here, its density is homogeneous and equals the gas density at the inlet, p0, while the specific energy deposition in the entire volume under consideration coincides with that in an infinitely expanded planar channel. In this model, the specific energy deposition near the channel wall can only be overstated, because in an actual instance when there is no passive zone, dp/dy < 0 at any y value. Consequently, specific energy deposition near the wall in an actual situation will also be lower than in the model under consideration. Furthermore, in a real case, gas density steadily decreases in the direction of x, dropping roughly half in size near outlet (see refs. [44, 46]), which also leads to a decrease in specific energy deposition.

During gas pumping through any given cross-section of x = const at any given instant in time, a new portion of gas flows in that is heated to a lesser extent than in the case when this gas remains motionless. The gas velocity in each cross-section increases from the wall to the center from zero to U(x). Therefore, the transverse temperature gradients when pumping is absent do not at least exceed the absolute value of the similar gradients when gas pumping is present. Consequently, the heat flux from the wall to the center in an unpumped model does not exceed the corresponding fluxes in a flowing system. Thus, in the nonflowing model under consideration, the wall temperature can only be overstated as compared to a flowing system.

A comparison of an actual flowing system and a model nonflowing system in a plane at a distance of x from the inlet can be made over a time interval of t = x/U (x), where U(x) is the average gas velocity over the transverse cross-section. It steadily increases with an increase in the distance from the inlet [44, 46]. However, the t time value can be determined in a simpler manner:

t = x/U0, (9.109)

knowingly overstating it in this instance, and accordingly, overstating the warm-up temperature.

So, ignoring longitudinal heat conduction, we get the following boundary value problem for the gas temperature distribution in a nonflowing model:

The boundary condition in Eq. (9.111) reflects the fact established above that a necessary and sufficient condition for the absence of a passive zone consists of the heat flux at the gas and solid wall boundary equaling zero.

Making the upper-bound estimate even more rigorous, we will now assume that the heat conductivity coefficient is equal to its minimum value everywhere, which corresponds to a value at the inlet of k(x, y, t) = k0, i. e., we ignore its dependence upon temperature in Eq. (9.106), thereby understating heat removal from the wall to the center of the channel. In this instance, Eq. (9.110) takes the form:

(9.116)

The upper temperature values of the channel wall at its contact surface with the gas that are needed for passive zone elimination were calculated using Eq. (9.116). The temperature values obtained for uranium fission intensity in the active layers, q = 2 x 1016 cm-3 x s-1, at an active layer thickness of Sv = 2.78 x 10—4 cm, as well as channel transverse dimensions of b = 6 cm and d = 2 cm, are presented in Fig. 9.19.

The specific power deposition in a gas was calculated using the results obtained in ref. [49]. The gas temperature at the channel inlet was given as equaling T0 = 293 K.

By way of illustrating the depth of heat exchange zone penetration into the gas, the dependences of the relative temperature increments, в = (T(x, y) — T0)/(T(x,0) — T0), and specific power deposition, m = w(x, y)W(x,0), upon the transverse coordinate, calculated at a point of x = b = 6 cm for helium at a velocity of u0 = 4.5 m/s, are presented in Fig. 9.20.

In order to achieve the stationary mode, it is necessary to ensure wall cooling through the use of a liquid heat-transfer agent, such as water, for heat exchange.

At P = 1 atm, the boiling point of water is TK = 100 °C. Water boiling in the external cooling channel can lead not only to the noticeable worsening of heat exchange, but also to vibrations and acoustic perturbations on the outer cooled surfaces of the laser channels that occur both during the formation and during the detachment of steam bubbles. The latter must inevitably lead to the transmission of the aforemen­tioned perturbations through the channel wall to the laser gas. Consequently, small- scale density inhomogeneities will occur, the result of which will be a deterioration of laser characteristics, or perhaps, simply lasing failure.

For a planar wall in the stationary mode, the temperature differential between its limiting surfaces equals

AT = QSw/kw, (9.117)

where Q is the density of the heat flux through the wall; Sw is the wall thickness; and kw is the wall heat conductivity coefficient. A uranium layer is deposited to the wall surface that adjoins the gas (see Fig. 9.1). The fission fragments carry a portion of the energy released in the uranium layer, which equals є, into the gas. The remaining portion of the energy (1 — e) in the form of heat is released within the layer itself. If the requirement of passive zone elimination, dT(x,0)/dy = 0, is satisfied—i. e., the heat flux from the wall (or more precisely, from the uranium layer) into the gas equals zero—the excess portion of the heat (1 — e) must then flow through the wall to the heat-transfer agent. In this instance, the heat flux density will be

Q = (1 — e)E0q8v, (9.118)

where E0 is the energy released in the single fission.

As in the preceding calculations, let us assume that q = 2 x 1016 cm—3 x s—1. When the argon velocity at the inlet is U0 = 4.5 m/s, then based on the previously cited upper-bound estimates, the wall temperature at the contact boundary with the gas will equal Tw(b) = 1,055 K. In order to avoid heavy water boiling near the outer wall
surface, the temperature differential at the wall must exceed ДT = Tw(b) — TK = 682 K. Pursuant to Eqs.(9.117) and (9.118)

КДТ

(1 — e)EQq8U at є«0.1 and for the thickness of a wall made from zirconium (kw = 0.21 W/cm x K), we get Sw = 1 cm. Similarly, for U0 = 9 m/s (Tw(b) = 696 K), we obtain 8w « 0.5 cm.

Cooling with an External Heat-Transfer Agent. In the stationary operating mode, if all the energy released in the laser channel is removed exclusively through the gas by means of its pumping, then at a thermal neutron flux density of Ф ~1014 — 1015 cm—2 s—1, even during transverse pumping, the requisite velocity, as follows from a balance equation of the (9.4) type, must be U0~100 m/s. However, as previously stated, heat removal through the outer surface of the laser channel substrate can be accomplished concurrently with gas pumping using the heat-transfer agent that washes around this substrate. But in the quasi-stationary mode, an appreciable portion of the heat released in the uranium layer can be absorbed in the substrate itself, as is the case in the LM-4 apparatus [41], or in the material mass directly adjacent to the substrate.

If the directions of movement of the external heat-transfer agent and the gas in the laser channel coincide, it is not difficult to establish the correlation between the geometric and thermophysical parameters of a system, during which the projection of the refractive index gradient in the y direction perpendicular to the uranium layer’s plane is (Vn)y < 0 in the entire gas volume, i. e., there is no near-wall passive zone. This can take place provided that (VT)y > 0 everywhere, including near the uranium layer surface. This means that the heat can only flow from the substrate to the gas. In order to implement this condition, it is essential that the heat fraction carried out of the channel by the gas is not less than є.

The total amount of energy carried away by the gas per unit of time through the cross-section of a laser channel with a coordinate of x per unit of channel length in the z direction equals

Q1 = dU0P0cP(r(x)— T0) • (9.119)

It is obvious that the realization of the condition (VT)y > 0 must satisfy the inequality

0

where qS is the power surface density of energy deposition in the gas from both of the uranium layers that is directly due to the escaping fission fragments,

qs = 2EqNv( оф Sus, (9.121)

where the angle brackets denote averaging over the neutron spectrum.

The total amount of energy per unit length in the z direction that the heat-transfer agent carries away per unit of time through the cross-section with a coordinate of x for each of the two external channels intended for the heat-transfer agent equals

Qi = d, UIoplocpl(T,(x, t) — To), (9.122)

Here, the l index indicates that the corresponding parameters are related to the external heat-transfer agent.

The amount of energy carried away per unit of time from a unit of uranium layer surface through the substrate to the external heat-transfer agent that is required in order to implement the condition (VT)y > 0 must not exceed

qi = HfEoNu(cfФ)Яи(1 — є), (9.123)

i. e.,

x

Here, ^f is the ratio of the thermal energy released directly into the uranium layer during the fission of a single uranium nucleus to the total kinetic energy of the fragments, E0.

Combining Eqs. (9.119)-(9.124) and assuming in this instance that the uranium layer and the fission density within the confines of the channel’s expanse along the direction of the x-axis are homogeneous, we obtain

Here, allowance is made for the fact that the gas density, p(x), can be perceptibly altered downstream; therefore, є can be dependent upon x.

When an oncoming or transverse gas and heat-transfer agent flow is present, the situation becomes considerably more complex. So, for example, if the heat flux at the gas and uranium layer contact boundary equals zero during an oncoming flow in a given transverse cross-section plane of the system under consideration, the heat flux density vector on different sides of the that plane will have an opposite direction.

Passive Zone Transverse Dimension. As previously indicated, it is not possible to obtain a precise solution for the width of a passive zone in a gas-flowing channel under actual pumping conditions. Nonetheless, even an approximate estimative description of its behavior would be of interest, which can be obtained using a series of simplifying propositions. As far as order of magnitude, the average travel time of a gas particle in a channel comes to т ~ b/U0. If S is flow line displacement in the transverse direction, the transverse velocity component is then v ~ S/т. We will assume that flow line displacement within the confines of a passive zone in the presence of the flows originating in NPLs with gas pumping is much smaller than this zone’s thickness (S << la), i. e., v << laU0/b. The transverse temperature gradient in the zone under consideration is estimated as THa therefore, as far as order of magnitude, the second addend in the braces on the left side of Eq. (9.102) comes to v(dT/dy) << U0T/b. For the first addend in these same braces, we get u(dT/dx)~U0T/b. Thus, the second addend in the braces can be ignored. In ref. [46] (see the ensuing subsection), the quasi-Lagrangian coordinates associated with the flow lines in the active zone, x and y0, are introduced. They are expressed through Eulerian coordinates by the correlation dy0U0p0 = U(x, y0)p(x, y0)dy(x, y0), where U is the gas velocity in a transverse cross-section with a coordinate of x for a flow line that has a Eulerian coordinate of y = y0 at the channel inlet. By analogy, introducing a simpler relationship between the Eulerian x, y coordinates and the new x, y’ coordinates,

dy(x y’) = PodУ’/p(x, y’),

and taking into account the neglect of the second addend on the left side of Eq. (9.102), we obtain

We will now examine the behavior of a gas near a planar wall, assuming for simplicity’s sake that the channel’s opposite wall stands an infinitely great distance away. The conditions on approaching the wall’s leading edge are:

T(0,y’) = T0; u(0,y’) = U0; p(0,y’) = p0. (9.128)

The boundary conditions in the gas itself at the passive zone boundary, where the temperature reaches a peak, are:

dT(x, la)/dy1 = 0. (9.129)

On the wall itself, we will assume that

T(x, 0) = 7o. (9.130)

Let us suppose that source power, W, per unit mass is constant:

W = w(x, y’)/p(x, y’) = const, (9.131)

while the gas velocity outside the confines of the viscous boundary layer is constant and equals Uq:

u(x, y’ > lv) = Uq. (9.132)

According to the model being described, because the heat exchange processes in the bulk of the gas volume—i. e., outside the confines of the passive zone—can be ignored, the temperature of each flow line before it intersects that passive zone will then equal

We will determine the thickness of the viscous boundary layer near the surface of a planar plate using the correlation [2]

lv = R^J vx/Uq, (9.134)

where R is a numerical coefficient, and v is the kinematic viscosity. According to one of the viscous boundary layer models most frequently used in practice, the velocity profile therein is presented in the form of a linear dependence [2]:

my — 0 s y s l-(x);

Uq, y > lv(x).

In this instance, pursuant to the impulse theorem, R = 3.46 [2].

We will now examine the successive intersection of the viscous boundary layer and the thermal passive zone under consideration by a given flow line (Fig. 9.21). We will avail ourselves of a model according to which the zone in question and the viscous layer have a clearly pronounced boundary. Let us assume that la is the coordinate of the boundary of the zone under consideration in a cross-section with a coordinate of xa. If lv > la, then a liquid jet with a transverse coordinate of y’ all the way up to the intersection with the viscous layer’s boundary will have a velocity of U0. Pursuant to Eq. (9.134), starting with a longitudinal coordinate of xv = y2Uq/R2v, the jet flows within a region where the velocity, u(x, y’), is deter­mined by the linear dependence of Eq. (9.135) upon y. For a similar flow line at

p = p0, Eq. (9.133) can be presented as (for simplicity’s sake, we will omit the prime marks associated with the y coordinate below):

We will employ the hypothesis of the similarity of viscous and thermal boundary layers, extending it to the passive zone as well. A comparison of calculations and experiments [43] confirm the latter in an entirely satisfactory manner. Let us suppose that

la (x) = Aolv(x). (9.137)

From expression (9.136) for y = la under the condition la < lv, we get

W A o

We will seek a solution to the problem using the following separation of variables for the transverse derivative from temperature:

dT (x y)/dy = f(xMx, y).

Pursuant to the boundary condition in Eq. (9.129), the ф function becomes zero at the passive zone boundary. At the wall itself, when y = 0, it can be assumed that ф = 1. We will present ф(х, y) as a function of a single combined variable, Y = y/la(x); i. e., ф^, y) = ф^). Then, pursuant to Eq. (9.139)

From Eqs. (9.138) and (9.140), we get

whence, according to Eq. (9.139),

Equation (9.127) is written in the new coordinates in the following manner:

p(x, Y)k(x, Y)d Y) = p20cpll(x)U(x, Y)d-TxYl — p20Wl2a(x).

Integrating it from zero to unity with allowance for the conditions (p(Y = 0) = 1 and q>(Y = 1) = 0, as well as the dependence u(x, Y) = A0U0Y, for the velocity within the confines of the passive zone, we get

where a = k(x,0)/cpp0. We introduce 1

Y2ip(Y)dY

0 then using Eqs. (9.134), (9.137), and (9.138) for A1 , we obtain an equation for the A0 coefficient:

— a0 +1= 0,

where Pr = v/a is the Prandtl number.

Analysis demonstrates that at any uniform dependence, q>(Y), that satisfies the conditions q>(Y = 0) = 1 and q>(Y = 1) = 0, Eq. (9.141) has two pairs of complex conjugate roots and two real positive roots, the first of which is smaller than unity, while the second is greater than unity. The first of the real roots, A0 < 1, has physical significance by virtue of the assumption made that la < ly. However, if we proceed under the assumption that la > ly, then operating under this same premise, we obtain A0 < 1 for Pr > 0.3 (Pr > 0.6 for rare gases), i. e., la < ly, which contradicts the basic assumption.

An analysis performed for the Prandtl numbers that correspond to rare gases revealed that the A0 coefficient may be found within limits of 0.5 < A0 < 0.67. Thus, according to the correlations used, Eqs. (9.134) and (9.137), the coefficient A = A0 x R x /¥r in a dependence of

la = Afaxjv.

falls within limits of 1.4 < A < 1.9. Therefore, a conclusion can be reached: unlike sealed lasers, the size of a passive zone in gas-flowing lasers is dependent upon the Prandtl number, i. e., it is determined not only by heat conduction, but also by viscosity, and is roughly one-and-a-half to two times smaller than its value for sealed lasers without gas circulation.

Paper [25] proposes a scheme for an experiment (Fig. 12.8), which focuses the beam of ten identical conical laser modules on a target using elliptical mirrors to initiate a thermonuclear reaction. Each conical module is separated from the vacuum light guide by a quartz plate. The axes of the all laser modules and light guides are arranged on two conical surfaces whose axes are aligned and pass through the source of the y-radiation and target. The tops of the conical surfaces coincide with the source of the y-radiation and target. Due to the identical geometry of the arrangement of all the schematic elements, radiation from the conical amplifiers falls on the target simultaneously.

Thus, the use of the most powerful energy sources—nuclear explosive devices for gas-laser pumping would solve one of the main problems of the ICF: to determine the energy level that laser drivers need to reach one or another target gain.

In the optical materials that are used as substrates for mirrors and laser output windows, under the influence of the reactor radiation, additional losses of light can occur, associated with the reduction in their transparency. As was demonstrated by the results of studies [40—42], the absorption coefficient depends on the type of optical material and the concentration of impurities in it, the temperature of the sample, the wavelength of the light radiation, the absorbed dose, and absorbed dose rate. Absorption of light radiation in optical materials occurs as a result of forma­tion of color centers, which arise with the capture of charge carriers (electrons and holes) on structural defects of the material (for example, see [43, 44]). At the same time, the color centers are formed both as a result of the change of state of already existing defects and as a result of the onset of new defects.

In the majority of studies, materials were irradiated for a long time using stationary reactors, high power isotopic y-irradiating sources, and their coefficients
of absorption were measured before and after irradiation. The data obtained from stationary irradiation cannot be used to predict the value of the induced coefficient of absorption in the process of pulsed irradiation [41, 42], since frequently a large contribution to absorption is made by the color centers with short lifetimes. The coefficients of induced absorption for these two variants of irradiation can differ by a factor of 10 [42].

At VNIIEF, studies into the radiation resistance of optical materials to pulsed radiation have been carried out since the early 1970s. The methods of measuring induced absorption coefficients and radioluminescence of optical materials under the effects of reactor radiations are cited in study [40]. Basic attention was given to the change in optical properties of materials as a function of time during the reactor pulse.

The methods [40—42] used to measure the induced absorption coefficient in the process of pulsed irradiation (and for any time interval after it) are based on a very simple principle of measurement of the intensity of light radiation passed through the specimen prior to irradiation and any subsequent moment of time. CW lasers were used as the sources of light radiation; for example, a helium-neon or helium — cadmium laser, radiating at individual lines in the visible and IR regions of the spectrum, or a lamp with a continuous spectrum. In the latter case, light filters or a monochromator were used to isolate the probe light radiation in the narrow spectral range. To extract the useful signal against the background of various types of noise (including radiation noise in the photodetectors), the probe light signal was modulated.

Such methods make it possible to perform measurements of absorption coeffi­cients simultaneously in several wavelengths. For example, Figure 2.4 shows an oscillogram of one of the experiments with the VIR-2 reactor to measure induced absorption coefficients simultaneously at three wavelengths [40]. The absorbed dose rate of у radiation at the maximum of the reactor pulse was around 1 x 106 Gy/s. The contribution of the neutron radiation to the absorbed dose did not exceed 10 %. Induced absorption coefficients at wavelengths 633, 1,150, and

Fig. 2.4 Change in the transmission coefficient of a BaF2 crystal 10 mm thick when irradiated by a pulse of n, Y radiation of a VIR-2 reactor [40]: (1) reactor pulse, (2-4) signals of modulated probe light at wavelengths of 3,390, 1,150 and 633 nm, respectively. Scale division is 5 ms

3,390 nm in the pulse maximum were 0.65, 0.27, and <0.01 cm-1 respectively. It is clear from the data of Fig. 2.4 that the induced absorption coefficient decreases with an increase in light wavelength. This principle is observed for all optical materials.

The results of measuring induced absorption coefficients for certain optical materials obtained in experiments [42] with the pulsed TRIGA reactor are shown in Table 2.2.

Experimental investigations of various optical materials showed that the greatest radiation resistance is possessed by silica glass. Detailed research on the common types of silica glass KU-1, KV, and KI were carried out in experiments with the VIR-2 reactor [41]. In the absorption spectra measured 1 h after pulse irradiation, all the types of quartz showed the band with a maximum at the wavelength of 215 nm, which is characteristic for SiO2. Apart from the indicated band, the KI quartz (which lacks the hydroxyl group OH) had broad bands with maximums at wavelengths of 300 and 550 nm, while the KV quartz had a band with a maximum at a wavelength of 300 nm. The induced absorption coefficients measured in study [41] are shown in Table 2.3.

The value of the induced absorption coefficient depends greatly on the concen­tration of coloring impurities and hydroxyl OH. For the purest commercial glass, KU-1 (concentration of coloring impurities ~10 ppm, hydroxyl concentration <2,000 ppm), the induced absorption coefficient is minimal, which may be explained not only by the low concentration of coloring impurities, but also by the protective properties of the ОН+ ion.

An additional effect that can influence the recording instruments and the accu­racy of measurement of the light intensity is the radioluminescence of optical

Table 2.2 Maximum induced absorption coefficients (cm ‘) at an absorbed dose rate of 7 x 104 Gy/s [42] Wavelength of probe light, nm Material 325 633 3,390 Fused quartz (Corning 7940) 0.0073 — — Pyrex (Corning 7740) — 0.58 — CsI (monocrystal) — 0.067 <0.009 Sapphire — 0.014 <0.008 Spinel — 0.58 0.011 ALON — 4.9 0.05

Table 2.3 Maximum induced absorption coefficients (cm ‘) at absorbed dose rate of around 1 x 106 Gy/s [41] Wavelength of probe light, nm Type of silica glass 400 500 600-700 900 1,150 KI 0.6 — 0.28 0.022 0.015 KV 0.36 — 0.18 <0.01 <0.01 KU-1 0.07 0.045 <0.003 <0.005 <0.005

7 x 104 Gy/s [42]

materials [40, 42]. The intensity of radioluminescent radiation depends on the type of optical material, its temperature, and the spectral range. Table 2.4 shows the specific powers of radioluminescent radiation for certain materials [42].