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.