In quasi-CW NPLs operating on transitions nd-(n +1) p of Xe, Kr and Ar atoms, the depopulating of the lower (n +1) p levels occurs as a result of collisional quenching in collisions with atoms. The efficiency of quenching, which depends on the pressure and composition of the gas mixtures, significantly influences not only the energy characteristics of the NPLs, but also the laser spectrum. The laser characteristics can be influenced not only by the quenching processes of the nd and (n +1) p levels by atoms of the medium, but also by the quenching and mixing of these levels by plasma electrons, the influence of the latter being especially appreciable for q > 103 W/cm3.

Let us examine the spectral characteristics of the most studied NPLs operating on transitions of the Xe atom (see Fig. 3.1a). Experiments showed that the laser spectra depend not only on the parameters of the cavity mirrors and the specific power depositions, but also on the type of buffer gas, the pressure, and the ratio of mix components. The list of laser lines and transitions for the different mixtures compiled on the basis of Sect. 3.1 in Chap. 3 is given in Table 5.4. The spectra of the Xe laser in experiments with electron beams were basically similar [57]; in addi­tion, the 3.43 pm laser line, belonging to the transition 7p [5/2]2-7 s[3/2]10 [24], was observed.

The level 5d [3/2]10, from which the most intensive laser lines 1.73; 2.03 and

2.65 pm begin, is a resonance level. The radiation time of the isolated Xe atom in this state is equal to 0.58 ns [55], but with taking into account the imprisonment of the resonance radiation, about 200 ns [36].

Table 5.4 Laser transitions in xenon NPLs Laser transition X, ЦШ r т ns r т, ns Ayi x 106, s-1 Mixes in which lasing was observed 5d[3/2h0-6p[5/2]2 1.73 0.58 33.0 0.30 [55]; 0.48 [56] He-Xe, Ne-Xe, Ar-Xe, He-Ar-Xe, Ne-Ar-Xe 5d[3/2]j0-6p[3/2] j 2.03 0.58 29.9 2.46 [55]; 2.4 [56] He-Xe, Ne-Xe, Ar-Xe, He-Ar-Xe, Ne-Ar-Xe 5d[5/2]30-6p[5/2]3 2.48 565 25.5 0.48 [55] He-Xe, Ar-Xe 5 d[5/2] 20-6p [5/2]2 2.63 667 33.0 0.74 [55]; 0.62 [56] Ar-Xe, Kr-Xe 5d[3/2]!°-6p[1/2]0 2.65 0.58 22.4 1.27 [55]; 1.6 [56] He-Xe, Ne-Xe, Ar-Xe, He-Ar-Xe, Ne-Ar-Xe 5 d[5/2] 20-6p [5/2]3 2.81 667 25.5 0.034 [55] Kr-Xe 5d[5/2]20-6p[3/2h 3.37 667 29.9 0.68 [55]; 0.59 [56] Ar-Xe 5d[7/2]30-6p[5/2]2 3.51 1,190 33.0 0.74 [55]; 0.61 [56] Xe, He-Xe 7p[1/2]r7 s[3/2]20 3.65 500 49.5 1.8 [55] He-Xe Note: Bold print shows the media in which these laser lines were predominant. Aj, is the probability of a radiative transition [55, 56]; тj and т, are the radiative lifetimes of the upper and lower laser levels [55]

From the data shown in Tables 3.1, 3.4, and 3.5 (see Chap. 3, Sect. 3.1), it follows that when a He-Xe mixture is used, the 1.73, 2.03, 2.65, and 3.65 цш lines are the most intensive. In the Ar-Xe mixture, lasing occurs primarily at the 1.73 and

2.65 цш lines, in the mixture Kr-Xe at the 2.63 цш line, and in pure Xe at the 3.51 цш line. All of the laser lines noted above (except for 3.65 цш) belong to the 5d-6p transitions. In the Ne-Xe mixture, low-power lasing was observed at the 1.73, 2.03, and 2.65 цш lines [58].

According to the model proposed in study [33], regardless of the type of buffer gas, the level 5d[3/2]10 of the Xe atom is populated initially; it is the upper level for the three most intensive lasing lines: 1.73, 2.03, and 2.65 цш. Populating of this level occurs selectively with an efficiency close to 100 %, through the dissociative recombination process Xe2+ + e. The absence of 1.73, 2.03, and 2.65 цш lines in the Kr-Xe mixture and pure Хе is explained by the high rates of collisional quenching of the 5d[3/2]10 level by Kr and Xe atoms.

The weaker 2.48, 2.63, 3.37, and 3.51 цш laser lines are also present in Не-Хе, Ar-Xe, and Kr-Xe mixtures and pure Хе; these begin with the levels 5d[5/2]30, 5d [5/2]20, and 5d[7/2]30.

Populating of these levels occurs as a result of collisional intra-multiplet transi­tions during collisions with He, Ar, Kr, and Xe atoms in ground states. In the He-Xe mixture, the 3.65 цш laser line, belonging to one of the 7p-7s transitions, was also observed [59]. It is possible that the appearance of this line is related to populating

Table 5.5 Rate constants of processes of collisional quenching of 6p levels of the Xe atom (in units of 10~n cm3/s) Quenching atom Level Xe Kr Ar Ne He 6p[1/2]0 0.58 [60]; 0.59 [61] 11.0 [63] 14 [60]; 20 [61] 3.4 [64] 2.0 [64] 6p[3/2]2 8.2 [60]; 10.1 [61]; 8.7 [62] 2.2 [63] 4.7 [60]; 4.0 [61]; 3.7 [62] 0.019 [62]; 0.05 [64] 0.17 [62]; 0.17 [64] 6p[3/2]1 12.8 [60]; 11.5 [62] 1.2 [63] 0.2 [62]; 1.0 [63] 0.033[62]; 0.039 [64] 7.4 [62]; 7.5 [64] 6p[5/2]3 5.3 [60]; 7.6 [62] 4.0 [63] 5.1 [62]; 2.5 [63] 0.22 [62]; 0.3 [64] 1.0 [62]; 0.7 [64] 6p[5/2]2 9.6 [60]; 11.6 [61]; 10.1 [62] 4.5 [63] 8.2 [60]; 8.2 [61]; 8.0 [62]; 8.6 [63] 0.57[62]; 0.80 [64] 0.95 [62]; 0.90 [64] 6p[1/2]1 13.3 [60]; 18.1 [62] 4.0 [63] 0.74 [62]; 0.6 [63] 0.035 [62]; 0.3 [64] 4.0 [62]; <0.2 [64]

of the 7p level as a result of the processes of collisional-radiative (three-body) recombination and subsequent cascade transitions. Indeed, experimental data [59] show that the optimal concentration of Xe for this line is much lower than for the 2.03 and 3.51 ^m lines. The absence of the 3.65 ^m line in mixtures with heavy buffer gases (Ar, Kr) may be explained by the fact that in this case, processes of the three-body recombination were suppressed because of the high electron temperature.

There are no reliable data in the literature on the rate constants for processes of collisional quenching of the 5d levels of the Xe atom, which makes analysis of experimental results and calculation of laser characteristics difficult. Such infor­mation exists for lower 6p levels (Table 5.5).

Laser spectra depend on the rates of processes of collisional quenching of lower 6p levels. From a comparison of the NPL energy characteristics shown in Sect. 3.1 of Chap. 3 and the data of Table 5.5, it is clear that the most powerful lasing is observed for lines with high rate constants of the processes of collisional quenching of lower 6p levels: the 2.03 and 2.65 ^m lines for the He-Xe mixture and 1.73 and

2.65 ^m lines for the Ar-Xe mixture. The addition of small amounts of He to the Ar-Xe mixture, which is of little consequence on the specific power deposition, leads to the appearance of powerful lasing at the 2.03 ^m line and elimination of competition of the 1.73 and 2.03 ^m lines (see Fig. 3.2), which may be explained by efficient quenching of the 6p[3/2]1 level by He atoms.

In the Ne-Xe mixture, almost all of the 6p levels were quenched with inadequate efficiency, so the energy characteristics for NPLs operating on this mixture are not great. As was shown in study [58], small additions of He or Ar to the Ne-Xe mixture led to a sharp increase in output power at the 2.03 and 2.65 ^m lines.

In study [48], it is proposed that cell wall heating be carried out at the same time as laser pumping in order to diminish, or totally eliminate, a passive zone. The simplest means for putting this process into effect is to inject 235U nuclei into the
cell walls: during the irradiation of a laser of this type by a neutron flux, cell body heating due to the nuclear fission of the uranium injected into its walls occurs synchronously with the escape of fragments from uranium-containing layer into the gas.

It is convenient to examine the synchronous heating process based on the example of a cylindrical laser cell with an active uranium layer that is directly deposited to its inner wall surface. Let us assume that the concentration of uranium nuclei in the cell wall material is such that, over the course of a neutron flux rise time that exceeds the typical temperature equalization time of the gas in the cell, tt, determined by means of correlation (8.33), there is no noticeable flow of thermal energy from the gas to the wall and vice versa. Thus, the correlation that defines the optimum concentration of uranium nuclei in the wall directly follows from gas, uranium layer, and wall temperature equality requirements at any given time during irradiation, as well as from the absence of a heat flow from the gas to the wall:

(8.37)

2

_____________ CvP0rU_____________

CvP0rU + CUPu(r2 — rUl) + CwPw(r2 — r) where Cu, cw are the specific heat capacities of the uranium-containing layer and the cell wall material; rU, r1 are the internal and external uranium layer radius values; r2 is the external radius of the tube wall; £w is the fraction of energy that the fission fragments transfer to the gas from all the energy released in the cell, including the energy released within the wall; and pU, pw are the uranium-containing layer and cell wall density density values.

In fact, when the above conditional requirements are present

(cUVUpU E CwVwpw)(T T0) (1 £w)Ew,

CvP0V(T — T0) = £wEw,

£wEw — £Eu .

Here, Ew is the total energy released in the uranium layer and the cell wall; EU is the energy released in the uranium layer; and V, VU, Vw are the volumes of the gas, the uranium layer, and the tube wall, respectively. Taking into account the fact that energy release in the wall and the uranium layer is proportional to the concentration of uranium nuclei therein, it is not difficult to derive correlation (8.37) from the latter three equations.

In order to verify the effect of the uranium, series of calculations were performed that involved different uranium concentration values, Nw, in the wall for different neutron flux rise times, starting with т — 4 ms or higher. These calculations revealed that at Nw — Nw0, the passive zone completely disappears at any of the neutron flux rise times studied. The temperature profiles for several successive moments in time

Fig. 8.16 Dependences of the radial coordinate of the active region’s outer boundary, the fraction of the energy absorbed in this region, and the refractive index gradient in this same region upon the concentration of uranium nuclei in the cell wall: (1) rA/rj; (2) Qg(tA)/Qg(r1); (3) Дn; the broken line is Nw = Nw0

at Nw = Nw0 = 4.95 x 1021 cm—3 are designated by the broken line in Fig. 8.13. The calculated dependences of the radial coordinate of the active region’s outer bound­ary and the fraction of fission fragment energy absorbed in this region, Qg(tA)Qg (r1), upon the total energy absorbed in the gas, as well as of the refractive index gradient in this region upon the Nw concentration value at D0 = 0.5 a neutron flux rise time of t = 0.04 s, are presented in Fig. 8.16.

The calculations demonstrated that, despite a positive effect, the synchronous wall heating process has a specific drawback: over a time frame of t < 0.1 s (see Fig. 8.13), the cell wall and the gas are heated by more than ДT = 1,000 °K, which limits opportunities for using the process.

We note that injecting uranium nuclei into the cell wall material is virtually equivalent to simply increasing the uranium mass in the uranium layer by a value that equals its mass in the wall. In Fig. 8.13, Nowк5 x 1021 cm—3, which at r1 = 1 cm and in the presence of a wall thickness of Sw = 1 mm, is equivalent to increasing the thickness of a metallic uranium layer by ~40 times. In the presence of such an increase, the energy that the fragments carry into the gas (D1 is increased from 0.5 to 20), which is proportional to D1e(D1) (see Chap. 7, Sect. 7.3), increases by 7 %. The energy (1 — e) that the fragments do not carry into the gas is released directly onto the inner surface of the cell wall; thus, just as when uranium is placed in the wall material, energy release therein is more homogeneous. In the example under consideration, the wall is made from zirconium, the thermal diffusivity of which is aw к 0.12 cm2/s; the typical temperature equalization time for a wall thickness of 8w = 1 mm comes to tw ~ S2w /aw ~0.1 s, which is an order larger than the neutron pulse rise time. Therefore, when all the necessary uranium is concentrated in the active uranium-containing layer, the cell wall’s inner surface will be heated quite a bit more intensely than during uranium blending, which will lead to a decrease in the permissible fluence values and accordingly in irradiation time. However, it follows from the foregoing that in order to achieve the desired effect over fairly short rise time of т < tw when concentrating all the uranium in the active layer, its total quantity may be smaller than during its blending. It cannot be ruled out that under specific conditions, the version when all the uranium is concentrated in the uranium-containing layer may prove to be preferable. The question of the optimum placement location and the total quantity of the additional uranium must be separately considered in each given case based on actual structural solutions and the requirements imposed on laser characteristics.

When using NOCs, the laser element may be removed from the source of the nuclear energy. This reduces the need for radiation resistance of the laser medium. For example, some publications [39, 40] report on experiments using the TRIGA pulsed reactor. In these experiments, light radiation from the NOC was transmitted using 5-m long bundles of light-guide fibers beyond the biological shielding and directed to the Y3Al5O12:Nd3+ crystal. Gas as well as condensed media can be used as NOCs.

Gas NOCs may be divided by emission spectrum into two types: sources with a continuous spectrum close to the spectrum of black-body radiation and sources with a line spectrum consisting of separate luminescent lines or bands. In the first case, the source of the light radiation is nuclear-excited plasma heated to sufficiently high temperatures. As the data show [41, 42], when gas media based on rare gases (krypton, xenon, etc.) containing the isotopes 3He, 10B, or 235U at atmospheric pressure are irradiated with neutron fluxes from pulsed reactors with specific energy depositions up to ~10 J/cm3, the media temperature can reach 104K. In this case, the spectra of the luminescence closely correspond to the emission spectrum of an absolutely black body. Intensity lines belonging to the atom transitions of rare gases in the 800-1,000 nm spectral range are sometimes superimposed on these spectra.

With specific energy depositions <1 J/cm3, the main contribution to the emis­sion spectra at X < 1 pm is made by luminescent radiation. Naturally, in this case, gas media with maximum conversion efficiencies should be used for NOCs. In the UV spectral range, these media are mixtures based on the excimer molecules of the rare gases R2* (R = Xe, Kr, Ar) and the halides of rare gases RX* (X = F, Cl, Br, I) [40, 43—47]. With the use of excimer molecules, there is no radiation reabsorption because when a photon is emitted, the excimer molecules transit to a lower dissociable state or a weakly linked ground state.

A simple expression was used to evaluate the maximum values for the conver­sion efficiency nX (Table 11.4): nX max = hv/wa, where hv is the photon energy of a luminescent transition, and wa is the average energy consumed for the formation of one primary active particle (an ion or excited atom of a buffer gas). During subsequent relaxation processes, this active particle can produce maximum of one excited excimer molecule. Using Platzman’s formula (see Chap. 4, Sect. 2), a link may be obtained between the wa and ionization potential of the buffer gas V;: wa к 1.2V;-. It follows from Table 11.4 that the conversion efficiency for some excimer molecules can reach 50 %. The bottom line of the table gives experimental data for some excimer molecules at excitation of the gas media by proton [48] and electron [49] beams, and also nuclear radiations [50, 51].

Table 11.4 The emission wavelength and conversion efficiency of several excimer molecules in the UV spectrum Excimer molecule Ar2* Kr2* Xe2* KrCl* KrF* XeBr* XeCl* XeF* X, (nm) 129 147 172 222 248 282 308 351 Buffer gas Ar Kr Xe Ar Ar Ar Ar Ar Maximum calculated conversion efficiency, (%) 51 50 50 30 26 23 21 19 Experimental data on conversion efficiency, (%) 29 [48] 30 [49] 12 [48] 30 [49] 68 [50] — 23 [50] ~10 [51] ~10 [51] 14 [50]

For effective pumping of active laser elements based on the condensed media, their absorption spectra must correspond to the emission spectra of the NOC. The most intense absorption bands of ruby and neodymium lasers are located in the visible and near IR spectra. These laser media can be pumped using NOCs based on excimer alkaline molecules [40, 44—46] or rare gases [40, 46, 52, 53]. According to estimates [40, 45, 46], the maximum conversion efficiency for alkaline molecules Li2* (Amax = 458 nm), Na2* (A^ = 436 nm), K2* (Amax = 575 nm), Rb2* (Amax = 601, 603, 606 nm), and Cs2* (Amax = 703, 713, 718 nm) may reach 40­50 %.

Experiments [40, 52, 53] have shown that in NOCs based on mixtures of rare gases, almost all of the most intense lines are located in the near-IR range of the spectrum, 800-1,000 nm, and belong to the transitions (n + 1)p — (n +1)s of the Xe, Kr, and Ar atoms (n = 5,4,3 with respect to Xe, Kr, Ar). Table 11.5 gives the results of a study of the luminescent characteristics of He, Ne, Ar, Kr, Xe, and their binary mixtures in the spectral region 740-1,100 nm when excited by uranium fission fragments [53]. The VIR-2 M pulsed reactor with a pulse duration of approximately 3 ms was used as the neutron source [16]. For the most intense lines, 912.3 and 965.8 nm (Arl) in the He-Ar mixture, 892.9 nm (KrI) in the He-Kr mixture, and 980.0 nm (Xel) in the Ar-Xe mixture, the conversion efficiency was 0.1-0.15 %. With excitation of 3He-Ar(Kr, Xe) mixtures with the products of the nuclear reac­tion 3He(n, p)3H (q = 20 W/cm3), similar values qA < 0.5 % were obtained for the spectral range 740-840 nm [52].

The active substance for NOCs based on condensed media is phosphor as liquid or solid. The characteristics of some liquid luminescent media are considered in the previous section. Of the other liquid media for NOCs, one study [54] proposes a saline solution of the isotope 245Cm in heavy water. The luminescence of this solution when excited by a-particles occurs in the 560-620 nm region at a com­paratively low nA ~ 0.1 %.

In VNIIEF, some characteristics of alkali halide crystals and luminescent plas­tics were studied in experiment [2] at the VIR-2 and at the BIR-2 pulsed reactors

[16] . The CsI(Tl), CsI(Na), and NaI(Tl) crystals were considered in the most detail. Earlier these crystals were mainly used at low excitation levels in the nuclear particle counting mode. The VNIIEF experiments showed that the luminescence spectra of these crystals change insignificantly at the absorbed doses of the

Table 11.5 The luminescent characteristics of rare gases and their binary mixtures with excitation by uranium fission fragments Gas medium Pressure (atm) q (W/cm3) wt (mW/cm3) Пі (%) He 1.8 60 85 0.15 Ne 0.64 34 94 0.28 Ar 0.45 47 85 0.18 Kr 0.32 46 120 0.26 Xe 0.23 38 120 0.32 He-Ne (0.56 % Ne) 1.8 43 51 0.12 He-Ar (0.56 % Ar) 1.8 46 180 0.39 He-Kr (0.56 % Kr) 1.8 46 260 0.57 He-Xe (0.56 % Xe) 1.8 40 230 0.57 Ne-Ar (1.7 % Ar) 0.64 34 260 0.76 Ne-Kr (1.7 % Kr) 0.64 37 210 0.57 Ne-Xe (1.7 % Xe) 0.64 34 240 0.71 Ar-Kr (2.2 % Kr) 0.45 47 85 0.18 Ar-Xe (2.2 % Xe) 0.45 44 260 0.59 Kr-Xe (3.1 % Xe) 0.32 52 150 0.29 Note: q is the specific deposition power; wt and Пі is the specific power of the luminescence and the conversion efficiency within the range 740-1,100 nm

y-radiation DY < 5 x 103 Gr and the neutron fluences F < 5 x 1014 cm~2 (Fig. ). Other important parameters for the CsI(Tl), CsI(Na), and NaI(Tl) crystals were studied as a function of the activator concentration and the dose rate of y-radiation: the specific output of luminescent radiation, and the conversion efficiency. It is possible to conclude from the results that the light output, for example, of the CsI (Tl) crystal increases nearly linearly with the growth in the absorbed dose up to

4.5 x 103 Gr. At this maximum dose, the specific light output is 1.1 J/cm3, and the conversion efficiency is about 6 %.

The possibility of pumping molecular CO2 lasers (X = 10.6 qm) with nuclear radiation was considered in many studies in the early stages of NPL research (see Chap. 1, Sect. 1.1). Despite the numerous experiments, attempts to pump CO2 lasers with nuclear radiation yielded negative results. Experiments [152] to excite a 3He-N2-CO2 mixture with nuclear reaction products 3He(n, p)3H showed the absence of gain at the 10.6 qm line in a wide range of changes in the total pressure (0.26-0.8 atm) and composition of the mixture. In these experiments, absorption of radiation of the probe laser with X = 10.6 qm was observed, which testifies to the predominant populating of the lower laser level when a CO2 laser is pumped with high-energy charged particles. Calculations [153] of kinetic processes in the He-N2- CO2 plasma also confirm the ineffectiveness of direct pumping of CO2 lasers with nuclear radiation.

Another variation of pumping the CO2 laser with preliminary excitation of a nitrogen molecule and subsequent transfer of energy from these molecules to CO2 molecules was implemented in experiments reported in reviews [126, 154]. A diagram of the experiments is shown in Fig. 3.6. Nitrogen passing through a tube with a layer of 10B at a velocity of 7.6 x 10-4 m3 s-1 was excited by the nuclear reaction products 10B(n, a)7Li in the neutron flux of a pulse reactor with a pulse duration of about 200 qs (Фтах = 5 x 1016 cm-2 s-1). Then the excited nitrogen was mixed in the laser cavity with CO2 and He. The laser radiation arose within 30 ms after the reactor pulse and had a power of about 100 W with a laser pulse duration of about 1 ms.

Fig. 3.6 Diagram of experiments on pumping a СО2 laser with nuclear radiation [126]: (1) tube with 10B layer; (2) channel to measure nitrogen pressure; (3) connector socket; (4) glass nozzle; (5) inlet tube; (6) high-reflectivity mirror; (7) pin electrode; (8) exit mirror

At present VNIIEF is developing a nuclear physics facility—a physical model of a stationary reactor-laser (RL) with transverse flowing of the laser medium [16—19]. The facility includes an IKAR-500 pulsed reactor and LM-16 laser module.

The IKAR-500 core is a graphite matrix (a cube with 2,400-mm sides) with nine reach-through holes that are 500 x 500 mm in cross section, which accommodate the reactor modules. In the graphite matrix (between the modules) at the top and on the sides, zirconium channels are mounted to accommodate the reactor control systems. An LM-16 laser module can be placed in one of the reach-through holes instead of a reactor module. The planned energy release in the reactor core for a startup with a duration from fractions of a second to tens of seconds is 500 MJ; the expected energy of laser radiation is >20 kJ.

Calculations of the nuclear physics characteristics of the IKAR-500 type facil­ities are a very difficult problem, because the reactor core has significant dimen­sions, high porosity due to the presence of a large number of laser channels, and a clearly pronounced anisotropy of the leakage neutron field, which is not typical for traditional reactor designs. The calculations of reactor core characteristics of such facilities must therefore be confirmed by direct physical experiments. In this regard,

VNIIEF has developed the IKAR-S critical stand (Figs. 6.8 and 6.9) for experi­mental investigation of the nuclear physics characteristics of the IKAR-500 reactor [18,19]. The basic difference between the IKAR-S critical stand and the IKAR-500 reactor is the absence of full-fledged control elements.

Absorbing rods made of boron carbide, B4C, are used as control elements for reactivity adjustment on the critical stand. Each of the control elements includes two absorbing rods. The reactivity control elements are functionally divided into four groups: two groups of reactivity control rods and two groups of emergency shutdown rods. The KNK-4 and KNK-15 chambers are used to register neutrons. Two plutonium-beryllium neutron sources are included as standard in the design of the critical stand.

Each reactor module consists of two fuel sections, which constitute a set of alternating layers of graphite and dispersed uranium-aluminum fuel elements (72 fuel elements in each section). A fuel element is a 5 x 60 x 900 mm uranium — aluminum plate containing 2.5 % uranium (90 % enriched in 235U), confined in a vacuum-sealed case made of Zr-Nb alloy with walls 0.5-mm thick.

Recently performed experimental and computational investigations [19—21] of the nuclear-physical characteristics of the IKAR-S stand made it possible to determine the effective neutron multiplication factor at various stages of core assembling, and showed the possibility of forecasting the critical mass parameters of the RL.

It was initially assumed that the IKAR-S would consist of sections in accordance with version 1 (Fig. 6.10a), which most fully models the IKAR-500 reactor core,

8

9

Fig. 6.9 Structural diagram of IKAR-S critical stand [19]: (1) reactor module of two fuel sections or LM-16 laser module; (2) graphite assembly; (3) neutron detector channels; (4) neutron source mechanisms; (5) control rods; (6) emergency shield rods; (7) horizontal channels; (8, 9) shielding gates reproducing the structure of the laser modules. The results of preliminary estimates using Nuclear Data Library (BAS) neutron constants for version 1 yielded an effective neutron multiplication factor kef = 1.076. The actually obtained value of kef = 0.906.

To bring IKAR-S to a critical state, the decision was made to increase the content of graphite in the sections while keeping the same number of fuel elements, which was supposed to lead to a shift of the neutron spectrum to the thermal range and a reduction in the core porosity, and accordingly, to a growth in kef. The corresponding calculations of kef were made using ENDF-B5 neutron constants, which were normalized to the experimental results obtained when IKAR-S was assembled per version 1. A diagram of the reactor sections, with a 46 % increase in the graphite content (version 2), is shown in Fig. 6.10b. In this case kef = 0.988 was obtained.

A further increase in reactivity was obtained when sections of the core’s lower row were modified in accordance with Fig. 6.10c (version 3) by raising the fuel elements to the upper part of the section (closer to the center of the reactor core). According to calculations, the expected increase in kef was 0.012, while the value obtained experimentally was 0.004; that is, as a result of reconfiguration of sections in the module of the lower row, kef = 0.992 was obtained.

Some discrepancy of calculated and experimental results may be explained by inadequate allowance for the specific features of the reactor core in the computa­tional model. In particular, when different versions of the reactor core were calculated, the significant influence of the quantity of aluminum (alloys) in the sections on the reactivity was noted; an increase in the quantity of aluminum (all other conditions remaining equal) led to a reduction in reactivity.

485

490

Fig. 6.10 Diagram of reactor section: (a) per version 1; (b) per version 2; (c) per version 3; (d) per version 4 of the studies and was examined in further experiments

In order to study this influence and achieve a critical state of the reactor core, the sections of version 2 of the central module were replaced with sections of version 4 (Fig. 6.10d), in which the aluminum details simulating radiators for cooling of the gas mixture were removed. The quantity of fuel elements and zirconium in the sections was not changed, while the quantity of graphite was increased by roughly 2 %. A value of kef = 0.998 was obtained.

After removal of the central source mechanism, a massive component of alumi­num alloy, from the reactor core, the system achieved a critical state; moreover, the reactivity margin of the reactor core was 0.62^eff. To reduce the reactivity margin of the reactor core, a mockup of the temperature sensor was additionally (in mirror- image symmetry) placed in the horizontal channel of the core where the sensor is placed. As a result, the margin reactivity amounted to 0.35^eff. The reactivity was determined from the reactor acceleration period, which was 14 s. Figure 6.11 shows the configuration of the reactor core that was assembled as a result of the studies and was examined in further experiments.

Fig. 6.11 Diagram of IKAR-S stand with nine modules in a critical state

In the course of the studies on physical startup of the critical stand, the charac­teristics (kef) were determined for roughly 40 different multiplying systems. The studies looked at how reactivity was affected by a reduction in reactor core porosity (by increasing the graphite content in the sections), an increase in the number of fuel elements in the central module, and a reduction in the quantity of aluminum alloy modeling the cooling radiators for the gas medium in the RL. Selection of the critical configuration of the critical stand core was accompanied by Monte Carlo calculations of kef. As new experimental data were obtained, the computational model was adjusted, which ultimately made it possible to quite accurately describe the multiplying properties of an IKAR type reactor core.

It was proposed that the IKAR-500 reactor be built on the basis of the IKAR-S critical stand. For this it was necessary to:

• select the reactor section configuration that would assure the required reactivity margin (~3^eff);

• ensure obtaining of a neutron pulse of the required (rectangular) shape;

• ensure safety of reactor operation.

The results of the physical startup showed that the proposed IKAR-500 reactor core configuration with sections per version 1 (Fig. 6.10a) did not assure the required reactivity margin when the existing fuel elements, with a content of 2.5 % uranium in a uranium-aluminum alloy, were used. One of the options to resolve the problem is to use “bare” fuel cores when the entire fuel section is sealed, which greatly reduces the mass of the zirconium alloy in the reactor core.

To assure the required pulse shape, it is proposed that a so-called reactivity modulator be used. This is a structure made of absorbent rods arranged in horizontal channels and controlled by linear stepper motors.

For fast transition of the reactor to a subcritical state, a system of fast-acting emergency shielding was developed based on the control elements existing on the IKAR-S stand. The absorbent rods are dropped to the lower position using compressed-air “guns” which reduces the reactor shutdown time to 0.2 s.

The basic problem in ensuring the safety of the IKAR-500 reactor is the lack of a temperature coefficient for reactivity damping when the characteristic pulse times are ~1 s. To ensure internal emergency shielding of the reactor, a system was developed that assures poisoning of the reactor core with a neutron-absorbing gas (3Не or BF3) [22]. Another option is to use uranium-graphite fuel similar to the fuel of the IGR reactor [7] in one or several reactor modules. This ensures the presence of an “instantaneous” temperature coefficient for reactivity damping.

It is proposed that all of the listed developments be tested on the critical stand of the IKAR-S. Thus the developed critical stand makes it possible to determine the parameters of the IKAR-500 reactor and to develop the real design of its primary assemblies and systems.

The shapes, dimensions, and structures of radiators may differ depending upon the operating mode, power density, and location of the laser system relative to the irradiation source. During operation in the pulse and quasi-pulse (ти~0.1-10 s) modes, a radiator may take the form of a simple porous structure that functions based on the principle of heat absorption by virtue of intrinsic heat capacity. It is obvious that in the stationary mode, an external heat-transfer agent should cool a radiator. The heat must be removed from this radiator by means of the heat-transfer agent washing over its end surfaces, which constitute a continuation of the outer surfaces of the uranium layer substrates. A radiator should consist of a set of plates

Fig. 9.1 Laser channel cross-section: (1) metal substrate; (2) uranium layer; (3) radiator

Heat-transfer agent

Fig. 9.2 Plate-type radiator directly adjacent to a laser channel and cooled by a liquid heat- transfer agent

(Fig. 9.2) or tubes with microchannels for flowing the cooling heat-transfer agent. The choice of potential engineering solutions is quite broad and will depend upon the conditions under which a specific laser system must operate.

It is not difficult to estimate the efficiency of the radiator under consideration using the results obtained from solving the problem of gas heat exchange with the channel walls. If the wall temperature of a channel as a portion of the gas passes through it can be considered constant, and if the flow itself is laminar, the depen­dence of the average gas temperature over the channel’s transverse cross-section upon the distance traversed, x, then takes the form [25]:

T(x)= TR + (To — Tr)J2 Ai exp — дЦ, (9.32)

where a is the thermal diffusivity coefficient of the gas; de is the equivalent diameter of the channel; U is the average gas velocity over the channel cross­section; T0 is the gas temperature at the channel inlet; and TR is the channel wall temperature. The A, and ft, values are presented in Table 9.2.

When the pressure of a helium-based gas mixture is P = 2 atm, a « 1 cm2/s. Let us assume that U = 10 m/s, while the distance between the plates is

S0 = Sr = 0.5 mm, which corresponds to an equivalent diameter of de = 1 mm (in the radiator channel, Re = U0de/v ~ 150, i. e., the flow is laminar). An estimate based on formula (9.32) demonstrates that in the presence of a plate length as low as x = bR = 3 cm, the difference between the initial gas temperature and the temper­ature of the radiator at its outlet drops 70 times.

The amount of heat that fission fragments transmit to the gas in a laser channel over a time frame of t comes to

G = q cLcdbLt,

where Lc is the length of the laser channel in the direction of the optical axis. The amount of heat absorbed by the radiator that is not cooled by the heat-transfer agent during heating from a temperature of TR to T0 is

G = mRLcdbRPRcR(T 0 — Tr)

(we assume that the height of the radiative cooler’s plates equals the height of the laser channel, d, in the direction of the y axis). Here, cR is the specific heat capacity of the radiator’s plate material; pR is this material’s density; and mR = Sr/(S0 + Sr ) is the radiator’s intermittency factor. It follows from these two equations that the radiator’s effective operating time is

RbRpRCR(TQ — Tr )

q cbL

For radiator made of aluminum plates at qc = 10 W/cm3, bR = 3 cm; bL = 6 cm, and (T0 — TR) = 100°K, operating time is t~6 s. These parameter values proved to be entirely acceptable for the radiators of the LM-4 laser module. The neutron excitation time during its operation in conjunction with a BIGR reactor was ~1.5 s [26].

A plate-type radiator is one alternative for the porous system used to intensify heat exchange processes [27—31]. The equations that describe temperature distri­bution in the components of a porous system in the stationary mode take the form [27—30]:

Gcp = hv(T — t), dx

d2T d2T

k + з/ — h*(T — T), (9-34)

where G is the specific mass flow of the component being flowed; cP is this component’s specific heat capacity; т is the gas temperature; hv is the volumetric inter-pore heat exchange intensity; T is the temperature of a solid porous compo­nent; and k is the heat conductivity coefficient of the solid component.

We will demonstrate that the temperature distribution in a plate-type radiator is described by equations similar to Eqs. (9.33) and (9.34).

Gas Cooling in a Radiator. We will select the x axis direction so that it coincides with the gas flow direction (see Figs. 9.1 and 9.2). Under actual conditions, the average gas temperature over the transverse cross-section in the direction of the z axis can change perceptibly at distances of not less than a few dozen radiator plate thickness values; therefore, the dependence of the parameters examined below upon the z coordinate is negligible. We will isolate a gas mass element, m, between the plates:

m = p(x, y, t)Al(x, y, t)Ad(x, y, t)80 = p0Al0Ad080 = const, (9.35)

where p(x, y, t) is the gas density at a point with coordinates of x and y at a moment in time of t; Al(x, y, t) is the length of the gas mass element; Ad(x, y, t) is its height; S0 is the distance between the radiator’s plates; Al0 is the length of the element at the radiator inlet; and Ad0 is its height at the inlet.

For most gases, including rare gases, the enthalpy of a unit mass of gas is determined as

w = cP jT,

where cP1 is the heat capacity at a constant pressure; therefore, at a point with coordinates of x and y for a moment in time of t, the enthalpy of the element under consideration is

w(x, y, t) = cpj Al(x, y, t)Ad(x, y, t)§0p(x, y, t)T(x, y, t).

Accordingly, the increase therein over a time frame of dt is

dT

dw = cPj A l(x, y, t) Ad (x, y, t) S0p (x, y, t) dt. (9.36)

The temperature of the gas mass element under consideration can change as a result of heat transmission to the preceding or subsequent elements, or as a result of heat exchange with the radiator’s plates. The effective distance over which heat propagates directly through a gas during a time interval of t is estimated by the correlation

l

where lg is the effective heat propagation distance and ag is the thermal diffusivity coefficient of the gas. The maximum time of gas residence in a radiator is

tm ^ L/u,

where L is the radiator dimension in the x direction and u is gas velocity. Over this time frame, the heat from the element under consideration propagates through the gas a distance of

lg (9.37)

At optimum pressures (P ~2 atm for helium and P ~0.5-0.9 atm for argon), ag < 1 cm2/s, L is on the order of a few centimeters [22], and, u > 10s_1. Placing these values into (9.37), we obtain lg < 10 2 cm, which is appreciably smaller than the longitudinal dimension of the gas in the radiator; i. e., longitudinal heat transfer in the gas should not play a noticeable role. However, the heat exchange of the mass element under consideration with the radiator’s plates is convective in nature, which, as is generally known, is considerably more efficient than conventional heat conduction.

The pressure drop, ДP, between the inlet and outlet of a channel formed by two parallel planes can be estimated using the formula [4]

Д P = 12rjUbR/8l, (9.38)

(n is the dynamic viscosity coefficient), according to which ДP ~ 10—3 atm for the helium cooling example considered above, which is much smaller than its absolute value of P = 2 atm. Therefore, the enthalpy decrease (—dw) should equal the amount of heat that the mass element under consideration transmits to two adjacent radiator plates. The amount of this heat is determined by the equation

dQ = Дl(x, y, t)Дd(x, y, t)q(x, y, t)dt, (9.39)

where q is the bilateral heat flux density at the gas contact boundary with the surfaces of the two radiator plates at a point with coordinates of x and y. During convective heat exchange,

q(x, y, t) = «{t(x, y, t) — Ts(x, y, t)g, (9.40)

where a is the heat transfer coefficient in a planar slit and Ts is the radiator’s plate surface temperature.

Using a minus sign (as the gas enthalpy decrease) to set Eq. (9.36) equal to expression (9.39) and substituting the result in Eq. (9.40), we obtain

Viewing the gas flow between the plates as the aggregate of the motion of jets with a height of Ad and a width of S0, then from the condition for mass flow maintenance, we write

u(x, y, t)p(x, y, t)Ad(x, y, t)S0 = MQP0 AdQSQ,

where u is the gas velocity component in the direction of the x axis and u0 is the gas velocity at the radiator inlet. With allowance for this correlation, Eq. (9.41) takes the form:

dx дт дт дт

= + u + V,

dt д t д x д y

where v is the gas velocity component in the direction of the y axis. An operating mode of practical interest is the one during which the following inequation is satisfied

дт дт

u » .

dx dt

Expression (9.44) means that the rate of change in the gas temperature as its location changes noticeably exceeds the rate of change in its temperature at any fixed point of a radiator with the passage of time. Furthermore, due to displacement restraint in the direction of the y axis by the radiator’s transverse dimension (see Fig. 9.2), gas jet shifting in this direction is small, so that

Taking this into account, Eq. (9.42) is written as дт

dx = —0{t(x, y, t) — Ts(x, y, t)g, (9.46)

where a0 is the thermal interaction coefficient of the gas with the radiator:

a

CP1P0 U0 80

Temperature Field in a Radiator Plate. The heat conduction equation for a plate takes the form:

where a is the plate’s thermal diffusivity coefficient, which can be regarded as almost constant for the temperature variation range in which we are interested.

We will introduce the average temperature over the transverse cross-section of a planar plate

where 8r is plate thickness. Integrating Eq. (9.48) for dz from zero to 8R/2, we obtain

From the condition of heat fluxes equality at the plate and gas contact boundary, we get

kR^T 0 = a1[T (x, y, t) — T(x, y, 8r /2, t)]. (9.51)

Here, kR is the plate’s heat conductivity coefficient and a1 is the gas’s heat transfer coefficient during heat exchange with a single planar surface. The latter is linked to the a coefficient by the correlation

ai = a/2. (9.52)

The plate’s maximum temperature differential in the transverse direction can be estimated as

dT(x, y, d/2, t) 8r dz 2

If this differential is much smaller than the difference in the gas temperature and the plate surface temperature

the T(x, y, d/2, t) temperature value on the right side of (9.51) can then be replaced with T (x, y, t). In this instance, placing Eqs. (9.51) into (9.50), we obtain

Here, taking Eq. (9.52) into account

It follows from a comparison of Eqs. (9.51) and (9.53) that Eq. (9.54) holds true given the condition that

SR < —. (9.56)

a

Accordingly, Eq. (9.46) can also be written in the form

^ ?) = — a0{T(x, y, t)— T(x, y, t)}. (9.57)

For the stationary case of dT/dt = 0, Eqs. (9.54) and (9.57) in their form fully agree with Eqs. (9.33) and (9.34), which describe heat exchange in a porous system (for simplicity’s sake, we will hereinafter omit the bar over the T, which denotes averaging Eq. (9.49)).

Boundary Conditions. Directly at the inlet end, as is also the case in a porous system [27], heat exchange between the radiator and the oncoming gas flow is virtually absent due to the latter’s negligible heat conduction; consequently,

The gas temperature at the inlet (x = 0) is a function of the y coordinate

t(0, y) = f (y).

During gas propagation along a radiator that is cooled by a liquid heat-transfer agent, its temperature can only decrease:

Ф > а У) < /max

where /max is the maximum gas temperature value at the radiator inlet. It is also obvious that the temperature of the radiator plates cannot exceed /max:

T(X > 0, y) < /max. (9.61)

At y = 0, symmetry condition must be satisfied

Convective heat exchange with the liquid heat-transfer agent takes place at the y = d/2 + Sw boundary:

T(d/2 + Sw) — Ti = q,/ah

where Sw is the thickness of the radiator’s substrate; Tl is the temperature of the liquid heat-transfer agent; ql is the heat flux density at the y = d/2 + Sw boundary; and al is the heat transfer coefficient at the y = d/2 + Sw boundary.

The temperature differential throughout the thickness of the radiator’s substrate (the radiator’s substrate and its plates may be made from different materials) equals

T(d/2) — T(d/2 + Sw) = qSw, (9.64)

kw

where kw is the heat conductivity coefficient of the radiator’s substrate. The heat flux at the radiator’s plate and substrate contact boundary is

k dT(d/2)

qi = — kR dy

The boundary condition at the radiator’s plate contact boundary in the presence of y = d/2 follows from Eqs. (9.63)-(9.65)

where

Approximation o/ an Ideal Radiator. The simplest and apparently the most conve­nient for elementary estimates and purely qualitative arguments is an “ideal”
radiator approximation: gas cooling is examined in a situation when the plate temperature variations of the radiator itself are sufficiently small that their effect can be ignored in Eq. (9.57); i. e., the temperature, T, can be regarded as constant and homogeneous. In the stationary case, when the heat transfer coefficient, a, is not dependent upon time, from Eq. (9.57), we get

X

ao(x)dx

0

The gas-flow velocity in the laser channel is U ~ 10 m/s and the width of the gap between the radiator’s plates, like the thickness of the plates themselves, is S0~ SR ~1 mm. Consequently, the gas velocity in the radiator is u ~20 m/s. The Reynolds number for a flow of this type is

ude

Re = — — 700,

V

where de = 2£0 is the equivalent diameter of the gap between the plates and v is the kinematic viscosity of the gas (v ~0.6 cm2/s). This is appreciably lower than the critical value of Rec ~ 2,300, above which the flow is not laminar, but turbulent. The heat transfer coefficient a is determined by way of the Nusselt number:

Nu

a = kg— , (9.69)

de

where kg is the heat conductivity coefficient of the gas. In the presence of a laminar flow, the Nu value in the initial part of the flow decreases with distance from the inlet into the slit between the plates to a constant value in the stabilized section [25, 32]. For a planar slit formed by two parallel planes, the distance within which the stabilized heat transfer part begins is determined by the inequality [25]

x > ud2e/70ag. (9.70)

Besides, if the channel wall temperature is constant, then Nu = 7.5 within the stabilized part. A power law dependence is recommended for the initial part [25]

Nu = 1.85(ud2/agx)1/3. (9.71)

A calculation of the & = (t—t) parameter (Curve 1) using the formula (9.86) with the recommended Nu(x) dependence [25] is presented in Fig. 9.3. There, too, a similar calculation (Curve 2) using formula (9.32) is offered. The calculations were performed for helium at P = 2 atm and t = 400 K; the gas velocity was u0 = 10 m/s,

Fig. 9.3 Dependence of the reduced gas temperature upon the distance traversed within a radiator: (1) a calculation using formula (9.68) in the presence of a variable Nu value; (2) a calculation using formula (9.32); (3) a calculation using formula (9.68) at Nu = 7.5

while the distance between the plates and their thickness came to S0 = Sr = 0.5 mm, which corresponds to an equivalent diameter of de = 1 mm. For comparison, a similar dependence (Curve 3), obtained using formula (9.68) under the assumption that the Nu number is constant and equals 7.5 over the entire expanse of the radiator, is shown in Fig. 9.3. According to Eq. (9.70), the expanse of the stabilization part is ~2 mm. It is obvious that gas cooling efficiency can be calculated with quite good accuracy using a constant Nusselt number to determine the heat transfer coefficient a.

Real Radiator. We will now examine the stationary problem дт/dt = dT/dt = 0. It is not difficult to reduce equation system (9.54) and (9.57), with boundary conditions (9.58)-(9.62) and (9.66), to a single linear equation in third-order partial derivatives

d3r d3 r d2r d2r dr

dx3 + dxd + ao dx2 + ao dy2 — Hdx = 0

and boundary conditions

where

r(x, y) = t(x, y) — T,. (9.78)

The solution to problem (9.72)-(9.78) takes the form:

1 Cn /2в

r(x, y) = n {s2„exp(s1„x) — S1„exp(s2„x)} cos ny. (9.79)

n=1 S2n — S1n d

Here, the range of problem eigenvalues is determined by the transcendental equation

hd

tg P — = ; n

2Pn

while the s1n and s1n are the negative roots of the characteristic equation

s3 + a0s2 — (Xn + H)s — a0Xn = 0, (9.80)

where

4en

d2 (Generally speaking, Eq. (9.80) has three different real roots. For the actual thermophysical parameter values of the radiator system under consideration, two of them are always negative, while one is positive and increases without limit as the n parameter value increases).

The Cn coefficients take the form:

Here, ||Фп ||2 = d + 8^T sin2вп is the square of the norm.

Thus, according to Eq. (9.78), gas temperature distribution takes the form

1′ cn (2в

t(x, y) = T, + — {s2nexp(s1nx) — S1nexp(s2nx)} cos — ny, (9.81)

n=1 S2n — S1n d

consequently, the temperature distribution in the radiator’s plates is determined by the equation

T(x, y) = t(x, y) + — C”S1”S2” {exp(s1„x) — exp(s2„x)} cos 2впу. (9.82)

«0 n=1 s2n s1n d

The series in Eqs. (9.81) and (9.82) quickly converge. So, the results of a calcula­tion using ten terms of sum in these equations differ by at least 0.4 % from the results obtained with allowance for 20 terms.

By way of illustration, the first of the ten eigenvalues of fin = d^[X~n/2 and the sin roots of Eq. (9.80) that correspond to them are presented in Table 9.3. The values cited correspond to helium at p0 = 4.6 x 10-4 g/cm3 and P = 2.8 atm, pumped at a rate of u0 = 9 m/s through an aluminum radiator with plate dimensions of d x SR = 2 x 0.05 cm. The thickness of the gap between the plates is S0 = 0.05 cm; the radiator’s substrate (also aluminum) has a thickness of Sw = 0.16 cm. The heat transfer coefficient of the liquid heat-transfer agent is at = 1.7 W/cm2 x K.

The calculations results for the aluminum plates of a radiator with a thickness of SR = 0.5 mm are presented in Figs. 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, and 9.10. The width of the slits between the plates is S0 = 0.5 mm. It was assumed that the laser channel within which gas mixture excitation by fission fragments occurs has transverse dimensions of d x b = 2 x 6 cm, that the thickness of the uranium layers is 5U = 2.78 x 10-4 cm, that the intensity of uranium nuclear fission is qU = 2 x 1016 cm-2 x s-1, and that the gas temperature at the laser channel inlet is Tg0 = 293 K. During the calculations, it was assumed that the heat-transfer agent’s heat transfer coefficient is a = 1.6 W/cm2 K, which corresponds to water at T; = 293 K, pumped at a rate of 5 m/s through an external cooling channel (see Fig. 9.2), the transverse dimension of which comes to 3 cm. The thickness of the radiator’s aluminum substrate was taken to equal = 1.6 mm.

Figures 9.4 and 9.5 reflect the calculation results for the argon temperature distribution within a radiator in the transverse and longitudinal directions following gas transition from a laser channel within which the gas pressure amounted to

0. 9 atm, while gas-flow velocity at this channel’s inlet was U0 = 4.5 m/s. At the outlet from the laser channel, gas-flow velocity as a result of heating by the fission fragments equals 10.78 m/s, which, in the presence of the SR and S0 dimensions indicated above, corresponds to an initial gas velocity in the radiator of u0 = 21.56 m/s. The gas temperature distribution at the radiator inlet (the channel outlet), t(0, y) = f(y), corresponds to Curve 1 in Fig. 9.4.

The heat transfer coefficient, a, determined by Eq. (9.69), will be dependent upon the gas heat conductivity coefficient, kg, which is a function of temperature:

For helium, k0 = 0.159 W/m x K, Tk = 320 K, and n « 0.69, while for argon, k0 = 0.0177 W/m x K, Tk = 300 K, and n« 0.71 [33—35]. Therefore, the heat transfer coefficient is reduced when the gas is cooled as it passes through the radiative cooler, leading to a slight decrease in the thermal interaction coefficient,

n	Pn	s1n	s2n	s3n
1	0.7677	-2.137	-0.336	1.115
2	3.3589	-3.685	-1.184	3.511
3	6.3986	-6.540	-1.310	6.492
4	9.5026	-9.591	-1.336	9.569
5	12.6250	-12.689	-1.346	12.677
6	15.7550	-15.805	-1.350	15.797
7	18.8888	-18.930	-1.353	18.925
8	22.0248	-22.060	-1.354	22.056
9	25.1622	-25.193	-1.355	25.190
10	28.3005	-28.327	-1.356	28.325

Table 9.3 Eigenvalues of and the roots of Eq. (9.80) that correspond to them

Fig. 9.5 Argon temperature distribution in the longitudinal direction at u0 = 21.56 m/s (U0 = 4.5 m/s): (1) у = 1 cm; (2) у = 0.5 cm;

(3) у = 0; solid lines are a real radiator; broken lines are an ideal radiator

a0, and the H parameter. During the performance of the calculations described above, a heat conductivity coefficient value was used that corresponded to the gas temperature at the radiator inlet, averaged over the transverse cross-section.

Comparative calculations were performed in order to check the extent of the influence of the thermal conductivity coefficient decrease, during which the heat conductivity coefficient was taken for the average gas temperature value (over the transverse cross-section) in a cross-section of x = 3 cm. The largest deviations occur in the case of argon; however, even they do not exceed в ~ 30 % when the relative deviations are calculated using the formula

1,100 1,000 900 800 700 600 500 400 300

=

T1 — Ti

where t1 is the temperature calculation at kg, determined by means of the average temperature at the inlet, and t2 is the temperature calculation at kg, determined by means of the average temperature in a cross-section of x = 3 cm.

The differences between the gas temperature and the radiator plate temperature for helium and argon are shown in Figs. 9.6 and 9.7. As is apparent from these graphs, the differences reach hundreds of degrees at distance of about 1-2 cm from the inlet, which testifies the invalidity of using the local thermal equilibrium model,

t, K

t «T, for the radiators under consideration, that is suggested in certain cases involving the calculation of heat exchange in porous systems [27].

The intensity of gas cooling will be substantively dependent upon the thickness of the slit gap between the radiator’s plates. The calculation results for longitudinal argon temperature distribution at S0 = 0.2 and 0.5 mm (Sr = 0.8 mm and 0.5 mm, respectively) are shown in Fig. 9.8 for comparison.

Presented in Fig. 9.9 is the dependence of the argon temperature at a distance of x = 3 cm from the radiator’s inlet upon slit width, S0, for the plate lattice spacing, Л, determined as

Л = <50 + Sr, (9.83)

Figure 9.9 demonstrates that it is possible to select a radiator length and slit width such that the temperature at the outlet will be homogeneous.

The calculated dependence of argon temperature upon the radiator spacing, Л, at a distance of x = 3 cm (for y = 0 cm) from the radiator inlet is shown in Fig. 9.10. The dependence presented suggests the possibility of performing radiator optimi­zation on the lattice spacing and the width of the gap between the plates.

Yet another possibility for the enhancement of radiator efficiency consists of expanding its transverse dimension along the y-axis immediately behind the inlet into the radiator itself. Indeed, the gas mass flow in the stationary mode equals G = p0u050d, so it follows from Eqs. (9.47) and (9.69) that

kgNu d 2cpG S0

Here, we took into account that de = 250 for a slit. By leaving the mass flow, G, and the slit width, 50, unchanged, but increasing the d dimension after the inlet, we achieve an increase in the thermal interaction coefficient, a0.

Sequential Circuit of Laser Channels and Radiators. In prospective multichannel laser systems that consist of a gas path made up of a sequential circuit of alternating laser channels and radiators (on the order of 25-50 channels and radiators), the longitudinal dimension of the radiators must be limited. The calcu­lations presented in the preceding subsection revealed that when radiators with a length of L = 3 cm (identical to those employed in the LM-4 quasi-pulse laser setup) are used in stationary multichannel laser systems, the gas temperature increase attained in the laser channel, t(0, y) — 7), is reduced by a total of two to three times. In this regard, a question arises: What temperature will the gas have after traversing a sequential circuit composed of N laser channels and N radiators?

As the calculation results demonstrated (see above), at distances of x > 3 cm from the inlet to the radiators, gas temperature distribution in the transverse direction is close to homogeneous; thus, for the sake of simplifying the description, we will assume that the aforementioned gas temperature distribution at the outlet for radiators with a length of L > 3 cm is scarcely dependent at all on the y coordi­nate. It follows from the solution to Eqn. (9.81) and the formula for Cn coefficients that the gas temperature at the first radiator’s outlet will be

tri(L) « Tl + <p(L) — и(Ь)Гі. (9.85)

Here, the following designations are introduced:

d/2

f (£) cos 2dn £ d£

1 d 2 в

Ф (L)~ 0 2 cos — — y {s2nexp(sinL) — sinexp(s2nL)};

n=1 II Фп Ir(s2n — sin) d

H (V)K, 2 2 n cos ddiy {s2nexp(sinL) s1nexp(s2nL)}

2-=i fink Фn Ir(s2n — sin) d

Let us suppose that the gas temperature at the first laser channel’s inlet equals t0, while at its outlet (at the first radiator’s inlet), ti(0, y) = f(y); i. e., the temperature increment after the laser channel has been traversed comes to

My) = f (y) — t0.

If we assume that all the channels in a common laser-radiator circuit are identical and that they operate under identical conditions, fission fragment energy deposition in the gas within each channel will then be identical, and accordingly, the temper­ature increments, Дт(y), will also be roughly identical.

Behind the first radiator, the gas temperature increment relative to the first channel’s inlet temperature, t0, is

Atri = T1 — T0 = Ti + ф — yTi — T0.

Because the gas temperature at the second channel’s inlet equals t0 + Atr1, while the temperature increment in each channel is identical, the temperature distribution at the second radiator’s inlet takes the form

f2(y) = f (y) + Атяь

then in accordance with the solution to Eqs. (9.81) and (9.85), at the second radiator’ s outlet, we get

тr2 = Ti + Ф — yTi + y Atr = (1 — y2) Ti + (1 — y) ф — уто,

i. e., the temperature increment at the second radiator’s outlet relative to the initial temperature, t0, is

AtR2 = (1 — y2)Ti + (1 + у)ф — (1 + y)T0.

For the outlet from the Nth radiator, we obtain

N

AtRn = l1 — yN)Ti + (Ф — T0) ym.

m=0

The sum on the right side of Eq. (9.86) is the sum of a decreasing geometric progression with a geometric ratio of y. Therefore, taking Eq. (9.85) into account, we get

The limiting value of AtRn (at N! 1) is

Atri = 1—yATR1. (9.88)

In the case of argon for a radiator with a length of L = 3 cm at a plate thickness of S0 = 0.5 mm and a gap thickness between plates of SR = 0.5 mm, ц = 0.358, while for helium, ц = 0.348. The limiting values of the e1 = AtRi/Atr1 ratios for argon and helium equal 1.558 and 1.534, respectively. The eN = Atrn/Atr1 ratios for these gases at N = 4 come to e4Ar = 1.532 and e4He = 1.511, which differ from the limiting value by a total of 1.5 %. Thus, starting with the fifth channel, all the lasers and radiators will operate under virtually identical conditions in the stationary mode.

It is useful to note that at S0 = 1 mm and SR = 1 mm (L = 3 cm), a similar calculation yields elAr = 3.497 and elHe = 2.513. Here, for N = 8, e8Ar = 3.26 and e8He = 2.469, which differ from the limiting value for argon by 6.8 % and for helium by 1.7 %. Technologically, it is quite a bit simpler to fabricate radiators with a plate thickness of S0 = 1 mm and a gap between plates of SR = 1 mm; therefore, in a multichannel apparatus with a large number (~50) laser channels sequentially connected in the gas path, it would perhaps be reasonable to give preference to radiators of this type.

Flow Behind a Radiator. The gas flow behind a radiator’s outlet was studied on the LUNA-2P setup [23, 24], the gas circuit of which included two laser cells with planar uranium layers (see Fig. 2.13). Three identical heat exchangers (radiators) were located on both sides of the cells: the first was at the first cell’s inlet; the second was between the cells; and the third was at the second cell’s outlet. The gas circulation system ensured the successive passage of the gas through the radiators and the laser cells. Laser cell active volume was limited by two planar aluminum plates with dimensions of 100 x 6 cm, which were positioned parallel to one another at a distance of d = 2 cm. Thin 235U oxide layers with a thickness 2.8 mg/cm2 were deposited to the inner surfaces of the plates. The radiators were made from rectangular aluminum plates with a thickness of S = 0.5 mm and a length along the gas flow of 3 cm. The distance between the plates came to 0.5 mm, while the height of the gas flow in the radiators was 2 cm. A laser channel scheme is presented in Fig. 9.1.

The LUNA-2P setup was irradiated by a pulsed thermal neutral flux from a VIR-2 M reactor that had a half-height pulse duration of ~3 ms. The thermal neutron flux density at the pulse peak, averaged over the laser cell’s active length (1 m), was 2.2 x 1015 cm~2 x s_1. The conditions and design of the experiments were the same as in ref. [36], where optical inhomogeneities in NPLs without gas circulation were studied (see Sects. 7.4 and 8.1). Specific energy deposition at the optimum gas pressures (PHe = 2 atm and PAr = 0.5 atm) came to 0.067 J/cm3.

Optical inhomogeneity measurements in a laser channel with gas circulation were performed using a Mach-Zehnder interferometer at a He-Ne laser wavelength of X = 0.63 pm under conditions of a reactor radiation background and slight mechanical fluctuations in the interferometer’s mirrors. In order to increase the measurement accuracy to 0.1 X, a comparative channel was used in the interferom­eter’s optical circuit. The fringe pattern was recorded using a standard SFR-2 M camera with a pulse feed system developed by the authors of refs. [23, 24], which expanded the recording time range to 0.1 s with a resolution no worse than 50 ps. Investigations were performed with helium, argon and air in the presence of a gas flow rate of up to 12 m/s in the laser channel, both at atmospheric pressure and at pressures selected in such a manner that the relative width of the gas gap between the uranium layers, D = d/R (where R is the range of an average uranium fission fragment in a gas of a given type) was close to the optimum value, D « 0.45 [37] (see Sect. 7.5).

A typical interferogram of the optical inhomogeneities in a laser channel at the time of neutron pulse termination, as well as the results of its processing, are

presented in Figs. 9.11 and 9.12. The interferogram was divided into three sections: the operating channel is in the center; and the comparative channels are above and below. It was experimentally shown in ref. [36] (see also Sects. 7.4 and 8.1) that when such an excitation mode is used, gas density redistribution in a laser is of the quasi-equilibrium type. Here, the acoustic waves that are usually observed in pulsed lasers are not present in the medium.

In sealed lasers (without gas circulation) during the heating of the active medium, its expansion into the buffer volume occurs, in addition to which a density profile is formed that corresponds to an energy deposition inhomogeneity. The gas circulation mode contributes its own specific features to the nature of the optical

inhomogeneities that originate. In the cross-section perpendicular to the uranium plate plane (Fig. 9.11), as in the case of a gas at rest, a positive lens originates that has a close-to-parabolic profile. When gas circulation is used, the parabola’s steepness increases downstream.

The distribution of refractive index deviations from the initial value, n0, in the cross-section parallel to the plate plane (along the x-axis in the plane of y = 0) is shown in Fig. 9.12. During exposure to an exciting pulse, the gas in the channel at a given velocity is shifted a total of 3 cm from the inlet radiator. Within this part (0 < х < 3 cm), the variation of the refractive index along the х-axis is almost linear in nature. The remaining portion of the gas in the region of х > 3 cm is present in the channel over the entire course of irradiation; therefore, as it approaches the outlet radiator, the dependence of the refractive index upon х becomes increasingly flat and reaches a constant value roughly ~2 cm before the outlet.

We note that the interferogram in Fig. 9.11 reflects a transient process in gas density redistribution, because the duration of the exciting pulse in these experi­ments was comparable to the time required for a portion of the gas to pass through the excitation zone.

In refs. [38, 39], calculations demonstrated that gas flow perturbation must occur at the outlet of a plate-type radiator, during which a pair of adjacent symmetrical stable vortex structures will be formed for each rectangular plate. The lateral dimen­sions of these vortices will decrease downstream and their dimension relative to plate thickness will be proportional to the Reynolds number. If the distance between the plates equals their thickness, the dimension will then be determined by the correlation

xw « 0.09 x Цк. (9,89)

The circulating motion of the gas in the vortices will lead to convective mixing, which will necessarily bring about the intensification of heat exchange processes as compared to conventional molecular heat conduction.

The laser sounding of an active medium in the gas flowing mode revealed that the fringe pattern near the radiator outlet disappeared due to sounding beam refraction on the radiator’s plates. The size of this region (~1 cm) and the large refractive index gradient value (~5 x 10~5 cm-1) confirm the proposition concerning increased thermal diffusivity near a radiator’s outlet. The dimension of the region described, xw, should increase with an increase in the plates’ trans­verse dimension and in gas flow velocity, and will be dependent upon the gas type.

In order to determine the thermal diffusivity associated with the origination of vortex structures at a radiator outlet, investigations were performed during which a nickel-chromium filament with a diameter of 0.2 mm and a length of 25 cm was placed in a laser channel at a distance of 3 mm from a radiator outlet. The filament was heated by an electric current, thereby creating a local thermal perturbation that extended downstream in the steady-state gas flow. The experimental procedure was based on the measurement of the temperature fields created in the cell’s volume using an interferometer. Interferograms were recorded using a He-Ne (LG-38) laser

Fig. 9.13 Interference fringe displacement as a function of height during filament heating: (1) x = 1 cm; (2) x = 3 cm; (3) x = 4 cm; (4) x = 5 cm

radiation and a Michelson interferometer. An SFR-2 M high-speed camera regis­tered the fringe patterns. The nature of heat exchange between the gas flow and the uranium layer substrate was studied separately. To this end, one substrate with a length of 25 cm was heated by AT = 40 °C using hot water that flowed inside it. The experiments described were carried out without neutron irradiation, i. e., there was no energy release in the uranium layers themselves.

A typical interferogram of the gas density perturbations that originate during filament heating in a helium flow (U = 8 m/s, Р = 2.2 atm, and S = 0.5 mm), together with the results of its processing, are shown in Fig. 9.13. The thermal perturbation had a Gaussian shape and expanded downstream, during which its peak value was reduced. Similar data that correspond to the same gas flow param­eters for an experiment involving substrate heating are presented in Fig. 9.14.

The experimental results made it possible to estimate the variation of effective thermal diffusivity coefficient along the gas flow. At the radiator outlet, it exceeds the molecular heat diffusivity coefficient by roughly one order and is reduced twofold at a distance of ~xw from the radiator outlet. This constitutes confirmation of the existence of vortex structures behind the radiator outlet, the transverse dimension of which is decreased downstream.

Fig. 9.14 Interference fringe displacement as a function of length during substrate heating: (1) y = —9.5 mm; (2) y = — 8 mm; (3) y = —6.5 mm

Based on the experimental results, an attempt was made to verify the hypothesis that the flow behind a radiator is turbulent in nature. The numerical analysis of a flow of this type did not present particular difficulties, because the temperature perturbation distribution behind the heated filament obtained during the experi­ments had a Gaussian shape. At first glance, this circumstance corresponds to the behavior of the heat wake from a thin linear source in a stationary turbulent flow created by artificially introducing a turbulizing wire grating at a certain distance ahead of the filament [40]. In this instance, thermal perturbation washout with distance from the source occurs in such a way that the maximum value and the half­height width of the perturbation peak are unequivocally linked to the turbulent heat diffusivity coefficient. The numerical analysis results demonstrated that when such allowance is made, the turbulent heat diffusivity coefficients, measured in cm2/s, are decreased downstream from a few tenths to order-of-unity values at the channel outlet. They will be dependent upon gas velocity and the transverse dimensions of the radiator plates. However, the coefficients obtained from an analysis of the downstream rate of variation in the maximum local thermal perturbation value proved to be several times higher than the similar coefficients found from this perturbation’s peak half-width washout rates. This fact suggests that it is impossible to regard the flow behind a radiator as a steady-state turbulent flow, and that the thermal diffusivity coefficient should be determined experimentally.

Experiments with a heated substrate revealed that the gas region near the uranium layers involved in heat exchange with the wall expands downstream (Fig. 9.14). It transverse dimension is roughly proportional to ~ ^/x (x is the distance from the cell inlet) and is dependent upon gas velocity, density and type. Depending upon the conditions in place, the full volume of the heat exchange zone in the cell may involve 10-30 % of the gas’s active volume. In ignoring the viscous boundary layer’s influence, it is not difficult to explain this dependence. Here, it can be roughly assumed that the gas velocity is homogeneous throughout the channel cross-section and equals U. We will isolate a plane with a coordinate of x in the transverse cross-section of the gas. Let us suppose that each such plane up to a moment in time of t = 0 is situated outside the channel’s confines. The time that this plane is located within the channel itself, including the region involved in heat conduction, equals to x/U. Over this time frame, the thickness of the region involved in near-wall heat exchange in the plane under consideration pursuant to Eq. (8.3) reaches a dimension of

where aeff is effective thermal diffusivity near the channel wall.

Thus, the significant role of vortex structures at the radiator outlet during heat conduction processes within a laser cell was demonstrated during the experiments. The most negative aspect consists of the considerable gas layer thickness near the substrate involved in the heat transfer process, as a result of which the lasing region
can only occupy a portion of the cell’s cross-section. In order to diminish this effect, it is necessary to reduce the size and intensity of the vortex structures in the gas at the radiator outlet by tapering its edges and bringing the temperature of the gas exiting the radiator as close as possible to that of the uranium layer.

The features of experiments using nuclear explosive devices which relate to the destruction of the laser device require careful preparation, in particular, of a detailed simulation of the physical processes in the active laser media and prelim­inary calculation of laser characteristics. These calculations were performed for lasers based on the transitions of the excimer molecules: XeF [23] and KrF [25].

Calculations [23] of experimental conditions [21—23] used a kinetic model of the XeF laser, which determined the laser characteristics on two spectral lines: 351 and 353 nm. Resulting from the solution of the radiative transfer equation, the total power density on each line was determined as a function of the distance to the coupling aperture at the top of the cone (Fig. 12.7). The radiation power at the surface of the focusing mirror was InR2 = 7.2 x 1010 W (R = 40 cm). This is approximately two times less than the power obtained experimentally. The calcu­lated maximum efficiency of the amplifier (2 %) was also lower than the experi­mental value (3 %).

A paper [25] analyzed the functioning of a laser device consisting of a cylindri­cal former and a conical amplifier with y-ray, travelling-wave pumping. A gas medium based on KrF (A = 249 nm) excimer molecules was considered as the active media. Special attention was paid to an analysis of the conditions for forming the directed radiation. Calculations showed that the efficiency of energy extraction from the conical amplifier as a function of its height was 20-50 %; the beam divergence at the output of the cone was 0«d0/L = 10~4 rad (where d0 is the diameter of the coupling aperture, and L is the length of the cone). These

Fig. 12.7 Dependence of the total power density of the 351 and 352 nm lines on the distance to the coupling aperture. The initial power densities at the input of the conic amplifiers were 1, 102, 104, and 106 W/cm2 (from bottom to top) [23]

calculations also showed that amplification of the spontaneous emission may influence the directivity of the useful radiation from the cone if the intensity of the spontaneous noise near the top of the cone is comparable to the saturation intensity of the active medium.

In experiments with pulsed nuclear reactors, the laser cell and the measurement devices are in the zone of intensive radiation. This imposes a constraint on the choice of structural and optical materials and photodetectors, and also makes it necessary to take special steps to test the functionality of individual unit assemblies at the time they are affected by reactor radiation, and afterwards. It is precisely the inadequate radiation resistance that is the basic reason for the absence of lasing when solid and liquid lasers were pumped by nuclear radiation (see Chap. 11, Sect. 11.1).

All materials and devices used in experiments with NPLs must satisfy the requirements of radiation resistance. Special attention must be paid to the cavity mirrors and windows for output of laser radiation, which in contrast to photodetectors and other measurement devices, cannot be removed from the zone of intensive irradiation. Owing to this, in parallel with the studies of the NPLs themselves, the radiation resistance of their elements and auxiliary devices were also studied [18,40].

The basic plasma processes occurring in a single-component gas mixture were examined previously in this chapter. Most often binary mixtures A-B are used as the active medium, where A is a buffer gas with a high ionization and excitation potential, and B is a laser additive with lower ionization and excitation potential. A diagram of the main plasma processes in a binary mixture is shown in Fig. 4.16.

The basic channels to transfer energy from ions and atoms of the buffer gas A to the atoms of the additive B are the charge-transfer process A+(Aj)+B! (B+) *+A (2A), the Penning reaction A* + B! (B+)*+ A + e (if the energy of the excited atom A* is greater than the ionization potential of atom B), or transfer of excitation A* + B! B*+A.

In high-pressure plasma, the molecular ions and Aj, Bj AB+ are the main ion types. These are formed as a result of three-body processes A+(B+) + 2A(A, B)! Aj (Bj, AB+)+A. Neutralization of plasma occurs as a result of recombination pro­cesses, among which, depending on the specific conditions, either the processes of three-body recombination of atomic ions A+(B+) + 2e(e, A)! A*(B*) + e(A), or the processes of dissociative recombination Aj(Bj, AB+) + e! A*(B*)+ A(B, A) predominate.

Kinetic equations, which represent the balance of rates of formation and decay of individual plasma components, are used to calculate the plasma parameters, and subsequently, the laser characteristics. In certain models, the number of considered plasmochemical reactions reaches several hundred (see [51], for example). For a satisfactory description of plasma phenomena relating to calculation of the charac­teristics of a specific laser, it is entirely sufficient to use 10-15 main processes for binary mixtures. In this regard, for the calculations it is sometimes helpful to use so-called “small” models, in which only the basic plasma processes are included. The results of calculation of the plasma parameters for mixtures He-Xe and Ar-Xe using “small” models are provided next as an example [52].

Fig. 4.16 Diagram of main plasma processes in the binary mixture A-B

Table 4.10 Basic plasma processes in a He-Xe mixture Process number Process Rate constant Works cited 1 He*(23S) + Xe! Xe+ + He + e 4.4 x 10~10 cm3/s [55] 2 He + 2He! HeJ + He 6.4 x 10~32 cm6/s [44] 3 He+ +Xe! Xe+ + 2He 4.7 x 10-10 cm3/s [56] 4 Xe+ + Xe + He! Xe+ +He 1.1 x 10~31 cm6/s [57] 5 Xe+ + e + e! Xe* + e 4.0 x 10~9Te~45 cm6/s [46] 6 Xe+ + e + He! Xe* + He 1.7 x 10~21Te~25 cm6/s [47] 7 Xe+ + e! Xe* + Xe 8.1 x 10~5Te~06 cm3/s [48] Note: Rate constants of processes are shown for Tg = 300 K, electron temperature Te, K

Table 4.11 Basic plasma processes in Ar-Xe mixture Process number Process Rate constant Works cited 1 Ar+ + 2Ar! ArJ + Ar 2.5 x 10~31 cm6/s [44] 2 Ar+ +Xe! Xe+ + 2Ar 1.2 x 10~9 cm3/s [58] 3 ArJ + e! Ar* + Ar 3.9 x 10~5Te~067 cm3/s [48] 4 Xe+ + Xe + Ar! Xe+ + Ar 2.0 x 10~31 cm6/s [59] 5 Ar* + Xe! Xe* + Ar 6.8 x 10-10 cm3/s [60] 6 Xe+ + e! Xe* + Xe 8.1 x 10~5Te~06cm3/s [48] 7 Xe+ + 2Ar! ArXe+ + Ar 1.0 x 10~31 cm6/s [59] 8 Ar+ +Xe! ArXe+ + Ar 2.0 x 10~10 cm3/s [61] 9 ArXe+ + Xe! XeJ +Ar 7.0 x 10~10 cm3/s [61] 10 ArXe+ + Ar! Xe+ + 2Ar 5.0 x 10-11 cm3/s [61] 11 ArXe+ + e! Xe* + Ar 1.7 x 10~5Te~05cm3/s [61] Note: Rate constants of processes are shown for Tg = 300 K, electron temperature Te, K

Calculations of the concentrations of electrons and ions and the electron tem­perature depending on the partial pressure of Xe are performed for the experimental conditions [53, 54], in which He-Xe and Ar-Xe NPLs excited by uranium fission fragments with a neutron pulse duration of 4 ms were studied. The specific power deposition at the neutron pulse maximum for PHe = 2 atm and PAr = 0.5 atm was around 20 W/cm3. The plasma processes that were included in the kinetic models are shown in Tables 4.10 and 4.11. The processes were selected as a result of preliminary analysis of the characteristic times of several tens of plasmochemical reactions with the participation of atomic and molecular ions of rare gases, as well as excited atoms and molecules.

The kinetic equations were supplemented with electron energy balance equa­tions. For the He-Xe mixture, this equation included the following processes of electron energy change: (a) ionization of the gas, as a result of which subthreshold electrons with an energy ee = 0.31V; = 7.6 eV arise at a rate off = q/w;; (b) the Penning process (1), as a result of which electrons with an energy of 8.5 eV are produced; (c) inelastic collisions of electrons with molecular ions HeJ, which can transfer a maximal energy of 2.4 eV to the electrons [62]; (d) elastic collisions of

Fig. 4.17 Dependencies of plasma parameters on partial pressure of Xe for He-Xe and Ar-Xe mixtures [52]: (a) He-Xe [(1) ne; (2) [Xe+]; (3) [XeJ]; (4) [He+]; (5) [HeJ]; (6) Te; (b) mixture Ar-Xe [(1) ne; (2) [Xe+]; (3) [XeJ]; (4) [ArXe+]; (5) [Ar+]; (6) [ArJ]; (7) Te

electrons with atoms of helium, as a result of which electron thermalization occurs. For the Ar-Xe mixture, the electron energy balance equation includes the processes similar to those cited in points (a), (c), and (d) for the mixture He-Xe.

Characteristic times of all plasma processes are much shorter than the pumping pulse durations, so a quasistationary mode is established in the plasma. The results of calculations of ion and electron concentrations and the electron temperature for the mixtures He-Xe and Ar-Xe at the neutron pulse maximum are shown in Fig. 4.17. With the increase in Xe partial pressure, there is a change in the ion composition of the plasma: concentrations of atomic ions and molecular ions of the buffer gas (HeJ, ArJ) decrease, while the concentration of the molecular ions XeJ increases. The low concentration of heteronuclear ions ArXe+ (Fig. 4.17b) is explained by their effective destruction as a result of collisions with atoms of Ar and Xe. Partial pressures of Xe for the mixtures He-Xe and Ar-Xe, at which the maximal output laser powers are achieved, are 1-2 Torr [53, 54]. In this case, as follows from Fig. 4.17 data, the process of dissociative recombination of molecular ions XeJ with electrons is the basic channel of formation of excited atoms Xe*.

In calculating plasma characteristics for a He-Xe mixture, the processes of formation and decay of heteronuclear ions HeXe+ were not taken into account. The dissociation energy of heteronuclear ions decreases with an increase in the difference in masses of the atoms making them up. Thus for the ions ArXe+ and HeAr+ the dissociation energy is 0.18 eV and 0.026 eV, respectively [63], while for the ion HeXe+ it is <0.02 eV. Consequently, HeXe+ ions are effectively destroyed in collisions with atoms, and their equilibrium concentration is insignificant. Mass — spectrometry measurements [64], which registered different heteronuclear ions except for the ion HeXe+, can serve as a confirmation of this.

Analogous calculations of plasma parameters using the “small” model were carried out for He-Ar and He-Kr mixtures [65]. As in the case of the He-Xe mixture, the processes of formation and decay of heteronuclear ions HeAr+ and HeKr+ were not taken into account. One of the significant differences between mixtures based on helium and mixtures in which the buffer gases are Ne, Ar, or Kr, is the appreciable role of the processes of collisional-radiative and three-body recombi­nation of atomic ions in the processes of plasma neutralization and formation of excited atoms (Fig. 4.17a). In mixtures based on Ne, Ar, or Kr, these processes are virtually entirely suppressed because of the high electron temperature.

The main problem that has to be resolved as a result of analysis of plasma processes in NPL gas media is identification of channels for the population of upper and lower laser levels, as well as determination of their populating rates.

Next experimental measurements of the energy deposition to the gas obtained in investigation of optical inhomogeneities in NPLs [27] will be analyzed. In this study, in each variant of filling of the laser cells with a gas mixture, the average specific energy deposition was determined, but issues related to the efficiency of the energy deposition and its dependence on various conditions were not examined. Later on that analysis of the data [27] was done in studies [25, 35]. The results yielded much better agreement with theory than prior studies [30, 31], and matched the results mentioned in [32].

Cylindrical Layers. In the first series of experiments [27], the cell was formed by a cylindrical pipe with an internal diameter of 35 mm and length of 77 cm. It held an aluminum tube 57-cm long, with an internal diameter of 28 mm and wall thickness of 3 mm. A layer of uranium oxide-protoxide (90 % enriched 235U) 5.9-mg/cm2 thick was applied to the internal surface of the aluminum tube. The cell was filled with rare gases (He, Ne, Ar) and irradiated with a pulsed thermal neutron flux from the VIR-2M reactor with a half-height pulse duration of t1/2 k 3 ms. The distribu­tion of the number of fissions per pulse along the length of the uranium layer, measured by the activation method using copper and uranium indicators, was found to be symmetrically relative to the cell center. In a reactor pulse, the average number of fissions along the length of the uranium layer, calculated per unit of uranium mass, was 1.5 x 1013 fissions/g. A change in the pressure in the course of the irradiation was measured using a DMI-6-2 inductive differential pressure sensor. The shape of the irradiating pulse and dependence of the pressure change on the time given are shown in Fig. 7.11.

The characteristic time of equalization of the pressure in the gas filling the cell is тр ~Lc’2ns, where us is the speed of sound in the gas; uHe к 103 m/s; uNe к 4.5 x 102 m/s; uAr к 3 x 102 m/s. Thus even for argon, the characteristic time of pressure relaxation тр ~ 1 ms is less than the half-width of the excitation pulse t1/2. Therefore, the pressure in the cell in the course of irradiation by the

pulsed neutron flux with a duration of tj/2 can be assumed in a first approximation to be uniform and it may be considered that redistribution of the gas density is in equilibrium.

This was confirmed by experiments [27], in which neither the interference method nor the pressure sensor registered acoustic oscillations in gas density and pressure (Fig. 7.11). Assuming that the gas is ideal, the pressure at each moment of time is uniform over the cell volume, and its walls are absolutely rigid, and using the energy conservation equation (with allowance for heat transfer, but ignoring the effect of viscosity), the constitutive equation, and the continuity equation, one can show (see Chap. 8, Sect. 8.2) that pressure at any moment of time may be described by the correlation:

n(t)= kgVT f, t df,

where df is an element of area of the internal surface of the cell; kg is the coefficient of heat conductivity of the gas; P0 is the initial pressure; Q is the intensity of energy release in the entire volume of uranium layers; V is the total gas volume in the cell; y is the adiabatic index; and П is the total heat flux through the surface bounding the cell volume.

Equation (7.20) essentially is a reflection of the energy conservation law and coincides with the correlation for an average pressure in the cell obtained in [34] for the more general case. If heat transfer through the cell walls can be ignored, then as in [22] we have

P(t)=P0 + (y — 1)qc(t), (7.21)

where qc is the specific energy deposition to the gas averaged over the cell volume.

Based on the maximal pressure jump, Eq. (7.21) was used in experiments [27] to determine the average specific energy deposition for the cell volume, qc, from which the specific energy deposition in the active portion of the gas volume VU (that is, the part of the cell volume directly bounded by the uranium layer) was then calculated using the formula qA = qcVVU. From the formula

4cV = qAVU

qU qU

(qU is the total energy release in the uranium layers), it is not difficult to calculate the share of fission fragment energy transmitted to the gas. The results of

Table 7.2 Experimental data and results of calculation of e for a cell with a cylindrical uranium layer Gas He Ne Ar P0, atm 1 2 3 5 1 0.25 0.5 1 qA, J/cm3 0.07 0.14 0.16 0.18 0.15 0.07 0.13 0.16 D0 0.243 0.486 0.729 1.217 0.700 0.254 0.508 1.018 e 0.026 0.054 0.055 0.067 0.057 0.026 0.047 0.058

calculations are given in Table 7.2. This table also provides the corresponding values of the specific energy deposition qA, the initial pressures in the cell [27], and the equivalent diameters D0 = d/R0 (d is the internal diameter of the active layer).

Figure 7.12 shows the experimental dependence e(D0) and the calculated depen­dence obtained by the methods of [14—16]. As is evident from the figure, the calculation and experiment differ roughly by a factor of 1.5 in the high value range of D0, and only for D0 « 0.25 does the difference reach ~2.

Flat Layers. In the second series of experiments with helium and argon [27], an interference method was used to measure the optical inhomogeneities in a cell with plane-parallel arrangement of uranium layers as a part of the LUNA-2M studies (see Chap. 2, Sect. 2.4). In this experiment, the pressure was not measured directly. The presence of a large buffer volume in the cell led to marked gas displacement from the active region in the course of excitation. According to the reduction in gas density measured by the interference method in the pumping region by the methods of [27], the average change in gas temperature over the entire cell was determined. Then using the formula qc = cV x AT (where cV is the specific heat capacity of the gas with a constant volume) and with allowance for the correlation of the active and total volumes, the value of the average energy deposition for the active volume was determined.

The cell constituted an aluminum pipe 220-cm long and 8 cm in diameter, in which two flat aluminum plates (2,000 x 60 x 2 mm) were placed in parallel with one another with a separation of 2 cm, with layers of uranium oxide-protoxide (90 % enriched 235U) 3.2 mg/cm2 thick deposited on the inside surfaces. The VIR-2M reactor was also used as the neutron source. The distribution of fission reactions along the length of the uranium layer was measured by the same method as previously. Table 7.3 provides experimental values of the specific energy deposition in the volume between the uranium layers. The relative thickness of the gas region is defined as D0 = d/R0, where d is the distance between the uranium — containing layers.

Figure 7.13 provides experimental and calculated dependencies of e(D0) for flat uranium-containing layers. The calculated dependence, as in [30], was obtained in an approximation of infinitely extended layers. In the case in point, the calculation and experiment differ roughly by a factor of 1.3-1.4, except for the region of small gas pressures (D0 « 0.08), where the relative difference reaches ~2.

Somewhat later, VNIIEF researchers conducted experiments [29] in which the efficiency of the energy deposition in a cell with flat uranium layers was measured

Gas	Не	Ar
P0, atm	0.5	2	5	0.25	0.5
qA, J/cm3	0.026	0.09	0.1	0.07	0.08
D0	0.087	0.347	0.87	0.181	0.363
e	0.031	0.108	0.120	0.084	0.096

Table 7.3 Experimental data and results of calculation of e for a cell with flat uranium layers

from the pressure jump. In these experiments, flat uranium layers with an area of 20 x 6 cm2 were arranged in a rectangular cell with a volume of 20 x 6 x 2 cm3, irradiated by a neutron pulse of the VIR-2M reactor (t1/2 ~ 3 ms). The cell was filled with helium. When the relative specific energy depositions were measured, a
reduction was found in the experimental values in comparison with the theoretical ones, roughly by a factor of 1.5 for D0 > 0.6 and by roughly double for D0 < 0.4. The average specific energy deposition to the gas was ~0.25 J/cm3 for D0 = 0.5. Direct determination of the average specific energy deposition was done from the maximal pressure jump measured using a DMI-6 sensor. Apart from this, in the same series of experiments, the total energy output from the layer by fission fragments was determined under vacuum conditions using thermoresistors in the form of nickel wires arranged at a distance of 5 mm from the uranium layer. The results demonstrated fair agreement with theory: the measured output energy was just 11 % lower than calculated. This difference is comparable in size with the measurement error of the method utilized.

Analysis of Experimental Results. The experimental results [27, 29] obtained by recalculation using the formula (7.21) do not allow for the influences of heat outflow to the aluminum substrate and the cell walls. The deposition of this factor, estimated in [27] from the pressure drop in the cell after the end of the neutron pulse, can amount to 15-25 %. It was noted above that calculations [19, 20] of thermal and gas dynamic processes carried out for experiments [27] with the assumption of uniformity of fission density along the length of the cell showed that by the end of the irradiating pulse, the relative reduction in pressure in the cell owing to heat removal to the aluminum substrate (the third member in the right part of Eq. 7.20) reaches 20-35 % and depends on the type of gas used and its initial density. The maximal deviation of calculated pressures from experimental did not exceed 15 %.

Another factor which can lead to a reduction in the experimental energy depo­sition is the non-uniformity of the uranium-containing layer. Layers viewed in their cross sections are reminiscent in shape to something halfway between rectangular projections and a sinusoid (see Fig. 7.10). A study of photographs of the cross sections of uranium layers showed that the deviations in thickness of the layers from their average value could be 50-70 %. Figure 7.14 shows the results of calculation of dependence є for the cylindrical layers used in experiments [27] on the dimensionless amplitude of inhomogeneities S for two of the one-dimensional models shown in Sect. 7.3 of this chapter (“a” and “b”). Calculations show that for the above shape and amplitude in the thicknesses deviation of the uranium layers from an average value, these inhomogeneities can lead to a reduction of є of 6­19 %.

Thus allowance for the heat outflow and the influence of the inhomogeneities in the uranium layers can markedly reduce or possibly even nullify the discrepancy between experimental and calculated values of specific energy deposition of fission fragments to the gas. We stress that in [27, 29], the energy deposition was found directly from the pressure jump, in contrast to [30, 31], where the energy deposition was determined by comparing the experimental curves of pressure oscillations with analogous oscillations curves obtained from calculating the dynamics of gas motion in the cell, while simultaneously selecting the specific energy deposition. The method used here could be implemented in view of the comparatively long duration of the irradiating pulse of t1/2~3 ms. With this long duration, the pressure in the

Fig. 7.14 Dependence of £

the efficiency of the energy deposition for a cylindrical uranium-containing layer on the amplitude of its inhomogeneities: (1)

D0 = 1; (2) D0 = 0.5; solid curves are sinusoidal inhomogeneities; broken curves are inhomogeneities in the form of rectangular teeth

cell in the course of irradiation changes quasistatically, and at each moment is virtually uniform for the entire gas volume.

The increase in the discrepancy (with a reduction in the initial gas density) between the calculated efficiencies of the energy deposition and their values determined by recalculation using formula (7.21) with pressure measurement data

can be explained using the following simple correlations. Let F^r, be the specific power of the sources of energy release in the gas due to the deceleration of fission fragments at the point r at the moment t. "Then the temperature increment at this point in the time St can be defined as ST^r, t^ ~ F^r, tjSt/pcn, where cn is

the specific heat capacity of the polytropic process which occurs at the point r when the gas is heated by fission fragments. For very low energy deposition, the change in pressure in the cell can virtually be ignored; then cn « cp.

At high values of energy deposition, when starting at a certain moment the pressure markedly exceeds the initial pressure, and heat removal from the gas is negligibly small, allowing a stationary density profile to be established in the cell. This profile remains unchanged with a further delivery of energy to the gas [22, 36]. In this case, cn« cv.

Because specific energy losses of fission fragments per unit of range are propor­tional to the density of the nuclei of the decelerating medium [14], one can consider that the specific power deposition for vTriable gas density in the cell is approxi­mately described by the equation F ^ r, = F0 ^ r, p ^ r, /p0, where F0 ^ r, tj

is the specific power deposition at the point with coordinate r at the time t for a uniform non-perturbed distribution of gas density p0.

The source function can be represented as F0 = Wr(^r’jФ(/), where Wr(^r^ is the spatial distribution of specific sources in a uniform gas, calculated relative to a

unit of neutron flux. Then the gas temperature at the point r at the time t, with weak influence of heat conductivity is

The heat flow from the gas to the cell wall through a unit surface is equal to gT = —kg x grad T. Calculations [19, 20] and experiments [27] showed that a narrow gas region forms close to the cell walls with the deposited uranium layer, from which heat is intensively removed into the wall. The temperature of the latter varies weakly and remains virtually equal to the initial gas temperature T0. The charac­teristic dimension I of this region is determined by the correlation I ~ fat, where a = kg’p0cp is the coefficient of thermal diffusivity of the gas. In the remaining part of the cell volume, the gas temperature increases in the course of irradiation, and its profile is very close to that of Eq. (7.22). Based on this, we can estimate the temperature gradient near the cell wall as

T 7m, t — T0 grad T ~ .

at

where r m is the coordinate corresponding to the maximal temperature close to the wall.

Because the wall region of intensive heat removal is rather narrow, we can consider the energy deposition wj^r m^ in Eq. (7.22) to be approximately equal to its value directly on the wall, Ww. Thus,

To determine the qualitative connection of the considered parameters, it suffices to represent the time dependency of the pulsed neutron flux in the form

Фт?/ти,

Ф() = фт(2 — t/r„), тn < t < 2т,

t > 2t„.

As a result, for the period of time when the gas is still cooling, we find

G(t) ^ f (t),

p0cn

where

0.2(t/Tn)3/2, 0 < t < Tn;

f (t)= 1.07 — 0.2(t/Tn)5/2 + 1.33(t/Tn)3/2 — 2 (t/Tn)1 z12, Tn < t < 2Tn;

[ 2(t/Tn)1/2 — 1:95, t > 2Tn:

Considering that the neutron fluence during the pulse is equal to ф = Фоттп, by the moment of the pulse end and after it ceases, the total quantity of heat passing through a unit surface area may be approximately represented by

G(t)^kgffif (2p — 1.95vT;). (7.23)

cnyfp~0

This dependence well reflects the experimental curves of the pressure drop in the cell after the pulse. It also shows that with a reduction in the initial density (or D0, because D0 = d/R0~p0), the influence of the heat removal both on the pressure measured in experiments [27, 29] and the average gas temperature, and on the formation of acoustic oscillations in studies [30, 31] must grow. It should be noted that the value of Ww also depends on p0 and consequently on D0. However, this is a comparatively weak dependence and manifests itself only in the narrow region of 0 < D0 < 0.4, where the function Ww increases roughly by a factor of 2 with a decrease in D0. For a relatively thick gas interval of D0 > 0.4, the value of Ww remains practically unchanged. Thus, using the results of [15, 16], it is not hard to show that when the uranium layers are in a plane-parallel arrangement, Ww for 0 < D0 < 1 can be approximately described by the correlation Ww ~ W0(1 — 0.5D0 + D01nD0), where W0 is equal to the value of Ww for D0 = 0.

Both the data of experiments [27, 29] on the deviation of є from the theoretical value, and the results of direct calculation of G(t) in direct calculations of gas behavior [20] confirm that the dependence of the heat removed through the cell walls on the initial gas density is indeed close to Eq. (7.23).

Further, when the energy deposition from flat uranium layers is calculated, one should allow for the influence of edge effects. Direct measurements [29] confirmed the presence of these effects. A marked reduction in the energy flux carried away by fragments is perceptible at a distance ~R0/3 from the edge of the layer and at the
edge itself is equal to half the value of the energy removed in the case of an infinitely extended layer.

One of the reasons for the discrepancy between the calculated (obtained for a non-perturbed medium) and experimental values of the efficiency of the energy deposition may be spatial redistribution of the gas density arising in the course of irradiation owing to the spatial non-uniformity of the energy deposition by fission fragments. In addition, the influence of dynamic and thermal relaxation of the gas may play a role. Thus the calculations carried out [30] showed that the values of gas densities in different points in the cell by the end of irradiation for a specific energy deposition of 0.58 J/cm3 differ two times over. To check these assumptions, a series of experiments were carried out using a rectangular cell with a volume of 20 x 6 x 2 cm3 with the same configuration as in [29] (the same neutron source and same gas), but with a somewhat greater distance from the source. The latter circumstance made it possible to reduce the average specific energy deposition from 0.25 to 0.03 J/cm3, and accordingly to reduce the density differentials through the cell volume. Measurements showed that the absolute discrepancy between the calculation for a non-perturbed medium in this experiment decreases by a factor of 1.5. This fact clearly testifies to a definite influence of the specific energy deposi­tions on the experimental results.

Possibly a marked role is played by the effect, discussed in [28, 33], of removal of some of the energy injected into the gas, in the form of UV radiation of excimer molecules. According to [28, 33], this effect is manifested in pure gases which do not contain impurities. When insignificant concentrations (~1 %) of impurities (nitrogen, hydrogen, etc.) are added in the gas, the experimentally measured efficiency may increase by 40 %. When the concentrations of impurities are >1 %, an increase in its quantity has little influence on the energy deposition. In this regard, it should be noted that in experiments [27], the criterion for gas purity was only the requirement that the presence of impurities not affect the value of the energy deposition, and the presence of impurities in a quantity <1 % was not tested. Because additions of an active laser component (xenon, for example) in NPL mixtures based on rare gases as a rule exceed 1 %, there should not be a marked drop in the efficiency of fragment energy deposition through the formation of excimer molecules.

Study [37] identified the important role of rapidly cooled small regions of the cell, residual volumes of the gas filling lines, the gaps between the optical and structural elements, and so forth. These regions are at first glance of insignificant volume, and for that reason their effects usually were not considered when results were processed. The authors of [37] showed, in the example of a rectangular cell of the type of [30], irradiated with a ~10 ms pulse, that correct allowance for the influence of rapidly cooled regions (in this cell their total volume is 15 %) leads to a reduction in the calculated pressure jumps in the cell by roughly a factor of 2 in comparison with analogous jumps in calculations that do not allow for the influence of these regions. In addition, in contrast to the results without allowance for the effects of small regions, calculations of the dependence of pressure on time in the updated scheme agree well with experiment [37].

Thus the method of indirect determination of the energy deposition by means of recalculating the data on measurement of pressure jumps [27, 29—31] does not allow for the marked influence of rapidly cooled small regions, and partial removal of energy to the walls due to heat conductivity, which greatly depends on the initial gas density. Moreover, in computer simulation (in a one-dimensional approxima­tion) of the oscillations of the gas pressure in the cell after excitation by short neutron pulses of duration тщ ~ 0.15 ms [30] and тщ ~ 0.4 ms [31], the comparison was done only with regard to the amplitude of the first peak of the pressure dependence on time. But in the behavior of the peaks following the first, as the data cited in [30] shows, there are significant differences between calculation and experiment. Allowance for heat removal (15-35 %), the influence of small, rapidly cooled regions (up to ~50 %), as well as the removal, by the excimer molecules of rare gases, of some of the energy injected by the fragments, in experiments with highly purified single-component media (up to ~40 %), allows us to say that there is no contradiction between the experimental and calculated values of energy depo­sition efficiency.

Thus according to presently existing data, evidently only the inhomogeneities of the uranium layers lead to a real reduction in the energy deposition of fission fragments to mixtures based on rare gases. Direct measurements [29] of the energy carried away by fragments from uranium layers under vacuum conditions conform within the limits of error to the corresponding calculated values of this parameter.