Numerical investigations of the dynamics of the development of inhomogeneities during NPL excitation by pulsed neutron fluxes from a TIBR reactor [21] with a duration (at half the exciting pulse height) of t1/2 = 0.8 ms were among the first performed. During experiments [4] on this reactor, the aluminum cylindrical cell employed had an inside diameter of 2.7 cm and an effective length of 57 cm. A 235U oxide-protoxide layer with a thickness of 9 mg/cm2 was applied to the cell’s inner surface. The cell wall thickness was 2 mm, while the distance between the cavity mirrors equaled 1 m, and the average thermal neutron fluence per pulse over the cell’s effective length was 4.2 x 1013 cm~2. Curve 1 in Fig. 8.1 represents the time dependence of the thermal neutron flux density. Figure 8.2а reflects the results of computerized numerical calculations of the time evolution of the gas density distribution in a cell filled with an He-Xe mixture (a component ratio of 200:1) in the presence of an initial pressure of P0 = 1 atm and an initial temperature of T0 = 300° К [4]. These calculations demonstrate that the spatial distribution of temperature is inversely proportional to density distribution, while the pressure at any given moment in time is virtually uniform throughout the cell’s cross-section. This behavior is expected, because the typical distribution time scale for the perturbation in the cell, тр ~ r1 /us (us is sound velocity in the gas, and r1 is the cell’s inside radius), is significantly shorter than exciting pulse duration.

The results of numerical investigations [22, 23] revealed that the gas volume is divided into two parts. The equation dp(r, t)/dr = 0 defines the rA(t) interface. In the active region, 0 < r < rA(t), the gas density gradient, and consequently the refractive index gradient, are negative. At rA(t) < r < r1 (the passive zone), the gradient is positive and far exceeds the absolute value of the gradient in the active region. A passive zone originates where heat removal to the cell wall plays a leading role,

Ф, r. u. t / T„ Fig. 8.1 Thermal neutron flux density time dependence: (1) pulse with a half-height duration of Tn = 0.8 ms (та = 1 ms); (2) pulse with a duration of т„ и 1 s (та = 0.3 s)

Fig. 8.2 Gas density distribution throughout the cross-section of a laser cell: (a) a pulse with a duration of t1/2 = 0.8 ms: (1) t = 1 ms; (2) t = 1.4 ms; (3) t = 1.8 ms; and, (4) t = 2.2 ms; (b) a pulse with a duration of tj/2 и 1 s: (1) t = 0.03 s; (2) t = 0.06 s; (3) t = 0.1 s; and, (4) t = 1 s; (c) the stationary irradiation mode: (1) coolant velocity, u = 0.8 m/s; (2) u = 2 m/s; and, (3) u = 8 m/s

because the heat capacity per unit volume appreciably exceeds the gas unit volume heat capacity. At I = r1 — rA << r1, the time dependence of this zone’s transverse dimension is well described by the correlation [22, 23]

i ~ fat;

where a represents gas thermal diffusivity. When long-duration pulses are present, the passive zone involves the entire gas volume over a time frame of t ~0.1s.

The results of similar calculations for a gas mixture of the same composition during excitation by a pulse with a duration of t1/2 « 1 s, a fluence of 4 x 1014 cm-2, and a close-to-rectangular shape (see Curve 2, Fig. 8.1) are presented in Fig. 8.2b. The passive zone encompasses the entire gas volume in a few tenths of a millisecond.

Stationary Mode. Under stationary conditions, heat removal through the outer surface of a sealed cell must be arranged. Heat removal of this type can be realized by means of a gaseous or liquid heat-transfer agent washing over the outer surfaces of systems with internal heat release. As is generally known, a liquid heat-transfer agent is more efficient than a gas, so it is preferred.

Unlike the pulsed alternatives described above, a precise analytical solution to the problem of gas temperature and density distribution exists for the stationary mode. Under the conditions being examined, the temperature distribution along the cylindrical cell’s tube wall is defined by the expression

where kw is the thermal conductivity coefficient of the tube material; qw is the thermal flux density at the interface of the tube and the gas filling it; r2 is the radius of the tube’s inner surface that the heat-transfer agent washes over; Tw (r) is the tube wall temperature at a point with a coordinate of r; T, is the temperature of the external heat-transfer agent; and a, is the coefficient of heat transfer from the tube’s outer surface to the heat-transfer agent. We derive equation (8.4) by means of solving a thermal conductivity equation for a geometry with cylindrical symmetry

1 3 ( dTw _ dTw awr dr r dr dt ’

and with boundary and initial conditions of the type

(8.6)

where aw is the thermal diffusivity of the tube material.

System (8.5) and (8.6) describes the temperature space-time distribution in a cylindrical layer, r1 < r < r2, on the outer surface of which, with a coordinate of r = r2, convective heat exchange occurs. On the layer’s inner surface, at r = r1, a thermal flux of qw(t) is specified.

When the aw and kw coefficients are constant, the solution to (8.5) and (8.6) takes the form:

T / (0)exp{awX„ (t — 0) }d0 — Tw0exp{ — awX„t}}.

(8.7)

The eigenvalues of Xn are roots of the equation

kwVK{N1 (v^nr 1)J1 (ffiAnr2) — J1 (v^nr 1)N1 (лДйr2)} = a/jN Xnr^/0 Kr2) — J1 ^nr^N0 ХпГ2 ,

while the eigenfunctions that correspond to them are given by

Фп(г) = N^/k^r^j — *(0^).

Here Jv, Nv are the (first-order) Bessel and v-order Neumann functions. It is not difficult to see that in the stationary case, i. e., at dqw/dt = dT//dt = 0 and t! 1, and the solution to (8.7) takes on the simple form of (8.4).

In the stationary mode, the equation for heat balance in the gas on a surface with a coordinate of r in the presence of problem cylindrical symmetry is written as

where, as before, F is specific power deposition of gas; kg is the thermal conduc­tivity coefficient of the gas, which is very slightly dependent upon density for rare gases [24]. However, its dependence upon temperature is communicated with the same degree of accuracy as during the polynomial representation by the equation [24, 25]

kg(T) = kg0(T/Tk0f. (8.9)

The fragments transfer a portion of the uranium nucleus fission energy released in the uranium layer to the gas, while a portion (1 — e) remains in the layer itself.

Due to this layer’s low thickness (5U~ 1-10 ^m), the layer can simply be regarded as a surface heat source with a surface power density of SU when solving a thermal conductivity problem. In this case, the following boundary conditions must be satisfied at the gas and cell wall interface when r = r1:

Source surface power density equals (see the designations in Chap. 7, Sect. 7.1)

Su = (1 — e)NUOf Ф5цЄ0.

The balance of the energy released in a cylindrical gas volume of unit length and a radius of r1 is written as

Integrating equation (8.8) with allowance for equation (8.9) and using the conditions of (8.10)-(8.12), we obtain

Tw(r) = T, + —- — ^ln — , r1 < r < r2.

1 — Є —,r2 kw r2,

The heat transfer coefficient, a,, can be determined using the formula [26, 27]

Nu = 0,023Re0,8 x Pr04 (8.15)

through a Nusselt number of Nu = — ,de /k,, a Prandtl number of Pr = v, /a,, and a Reynolds number of Re = ude /v,, where de is the equivalent diameter of the heat — transfer agent channel, k, is the thermal conductivity coefficient of the heat-transfer agent, u is its velocity, and v, is the heat-transfer agent’s kinematic viscosity.

Using the expressions obtained, the temperature spatial distribution was calcu­lated in a gas mixture of the same composition as in the previous section at three liquid heat-transfer agent circulation velocities. Water at a temperature of T, = 323 K was selected as the heat-transfer agent. The equivalent diameter of the
heat-transfer agent channel was de = 2 cm. Gas density spatial distributions were determined based on the temperature fields found using a state equation. All the calculations were performed for a thermal neutron flux density of Ф = 4 x 1014 cm-2 s-1. The calculation results (Fig. 8.2c) demonstrate that the density and refractive index gradients are positive in the entire gas volume in the stationary mode.

Evaluation of Optical Characteristics. Reducing of radiation angular divergence and bringing it down to the diffraction limit insofar as possible is one of the most urgent problems in laser physics. Its solution is linked to the problem of the electromagnetic field distribution within a cavity. This field depends on the type of cavity used, the design and type of the laser itself, and the optical quality of the active medium filling the cavity (determined by refractive index spatial distribution, as well as this medium’s gain and absorption factors). In the general case, wave optics techniques should be used to calculate the characteristics of a cavity filled with an active medium. However, the solution of a wave equation is difficult for a wide range of cavity configurations involving an inhomogeneous medium [28, 29]. Still, a number of important characteristics, especially when high Fresnel number values, Nf = a^fkLf (ar is the transverse dimension of the cavity Lf is the distance between its mirrors, and X is the radiation wavelength) are present, can also be determined using ray optics techniques [28—30]. For example, a formula in a paraxial approximation is derived in [31] for estimating the laser beam divergence with an inhomogeneous transverse refractive index distribution of laser medium

& « 2v/2An, (8.16)

Here An is the refractive index variation in the transverse direction within the confines of the lasing zone. This procedure provides quit good agreement with experiments [31, 32].

Moreover, for optical systems that can be represented in the form of a series of mirrors and elements equivalent to thin lenses, or for systems with a quadratic transverse refractive index dependence, and for combinations of these systems (see study [30] for an example), it has been shown that a ray approximation is in full agreement with the results provided by a Gaussian beam wave optics approxima­tion. Gaussian beams play a significant role in laser optics: at Nf >> 1, they quite accurately describe the radiation field of a broad range of lasers with a stable cavity.

An important advantage of ray optics is visualization and the possibility of the comparatively simple qualitative and quantitative interpretations of the results obtained. In a geometric optics approximation, the equation that defines the shape of light rays takes the form [30, 33]:
(8.17)

where n is the refractive index; q is the length of the ray arc from a certain fixed point to a point that is characterized by the radius vector, “r(q); and, q = dr /dq is a unit vector that is tangential to the ray trajectory; it determines the direction of light beam propagation, i. e., it coincides with the normal to the electromagnetic wave front.

Here, it is appropriate to note a peculiarity. Let us assume that ez is the unit vector in the direction of the optical axis, z. For the variation of the scalar product of ez x n q along the light ray, we have

The right-hand part of an equation based on (8.17) can be rewritten in the form of ez • Vn. If Vn is perpendicular to the z axis everywhere, then from the latter equation we get ez x nq = const; i. e., the scalar product of ez x nq remains unchanged along the ray. Hence it follows that

where n0 is the refractive index at the point where the beam enters the gas volume under consideration; &0 is the angle between the z axis and the direction of the incoming ray and & is the angle between the tangent to the beam at this point and the z axis.

Let us suppose that a medium’s refractive index spatial distribution has a cylindrical or planar symmetry, and is only dependent on a coordinate perpendic­ular to the axis (or plane) of symmetry. Assume that in the region where the refractive index gradient is negative, a light ray passes parallel to the axis (or plane) of symmetry at a certain point at a distance of r0 from it, i. e., at this point, cos&0 = 1. In this instance, whatever trajectory a ray traces as a result of the

effect of the refractive index gradient, n ^ r ^, may not be less than n0 = n (r0), since cos& < 1 always, ray distance from the axis (or plane) of symmetry must not exceed Г0.

If the laser beam divergence with a gas density distribution that is only depen­dent upon a single transverse coordinate, as is the case in [31], is estimated as double the value of the maximum angle by which a ray intersecting an active lasing region can be deviated during propagation, formula (8.16) for estimating this divergence then follows from equation (8.18) in the & << 1 and n — 1 << 1 approximations.

In the paraxial approximation, & << 1, equation (8.17) for a cylindrically sym­metric system with a transverse refractive index dependence takes the form [30]:

d2r 1 dn

dz2 n dr

If a parabolic dependence approximates refractive index distribution

OV) ;

which quite accurately communicates the real refractive index distribution in the bulk of an NPL’s gas volume [2, 34—36], then from (8.19) at n — 1 << 1 and in the presence of the boundary conditions

we have [30]

Hence, in particular, it follows that the rays in a cavity formed by planar mirrors and filled with the inhomogeneous medium under consideration oscillate around the optical axis with a period of z0 = 2пД/an1.

Similarly, if the refractive index can be approximated by the dependence

n(r) = n„( 1 + Opr2) ;

in the same approximations from equation (8.19), we have

&0

r = r0ch(v«n2z) + —f=s

an2

In the case of irradiation by a pulse with a duration of t1/2 = 0.8 ms at a moment in time of t = 2.2 ms (Curve 4, Fig. 8.2а) in an active region of r < rA, then according to the well-known dependence n — 1 = pCn (Cn is the constant for a gas
of this type), the gas refractive index distribution can be approximated by a parabola with a coefficient of an1« 2.4 x 10~6 cm~2, which yields z0 ~ 40 m for the ray oscillation period. The displacement of a beam with initial parameters of $o = 0, z = 0, and r0 = 1 cm after transmission the active section of a cell with a length of Lc = 57 cm equals Дr ~ 10~6 cm. In the passive region, r > rA, so a rough approximation yields an2 « 0.77 x 10~4 cm~2. Given this an2 value, following a single pass with a distance of Lc = 57 cm along the cell axis, rays with initial parameters of &0 = 0, z = 0, and r0 > 1.2 cm reach its wall. Furthermore, following several cavity passes, almost all rays with parameters of &0 called = 0, z = 0, and r0 > rA reach the cell wall; therefore, the passive zone may also be a “dead zone.” The divergence of output laser beam from an active region, r < rA, estimated using formula (8.16), comes to & ~3 x 10~3 rad.

For the quasistatic irradiation mode, starting at t« 0.1 s, and for the stationary mode (Fig. 8.2b, c), the refractive index spatial distribution at r > 0.6 cm can be roughly approximated by a parabola with a coefficient of an2 > 0.5 x 10~4 cm~2. At this an2 value, a light ray that has initial parameters of &0 = 0, z = 0, and r0 > 0.6 cm is displaced toward the cell wall (r1 = 1.35 cm), traversing a distance of z < 2 m, which almost corresponds to a single round trip of a cavity with planar mirrors.

Thus, in sealed gas lasers with fission fragment pumping in the stationary excitation mode, conditions created in the bulk of the gas volume that are not favorable for lasing. The same thing also occurs in a quasi-stationary excitation mode, starting at a time of t~10-1 s. Under these conditions, lasing can only originate in the region adjacent to the cell axis.

In pulsed excitation modes with a duration of t1/2 < 10~2 s, the refractive index gradients in the bulk of the gas volume are negative and small, which provides the conditions needed to achieve lasing.

Comparison with Experiments. Already the results of the initial numerical and theoretical investigations of the inhomogeneities originating in NPLs have attracted much attention. Experiments using the interference procedure were carried out in 1982 to study inhomogeneities. The results of these investigations were later published in the unclassified press [2]. In the wake of these experiments, calcula­tions [23] were performed that modeled them with the maximum possible approx­imation of actual conditions using the procedure described at the start of this chapter. The set of gases, their initial pressure and temperature values, and the irradiation intensity time dependence embedded in the calculations fully duplicated the corresponding experimental characteristics.

The conditions under which the experiments [2] were performed, and the parameters of the cell with a cylindrical uranium layer that was used, are presented in Sect. 7.4 of Chap. 7. The longitudinal cross-section of a cell with the plane in which the axis of symmetry is situated along with the relative distribution of the thermal neutron flux is presented in Fig. 8.3. The full internal volume of the cell (excluding the space occupied by the aluminum cylindrical substrate) was 561 cm3. The volume of the cell’s active section (i. e., the internal volume of the substrate tube, bound by its end faces) equaled 351 cm3. We will call the difference in these

two volumes (i. e., the space between the windows in the cell’s outer body and the end faces of the aluminum substrate, as well as the gap between this substrate and the cell’s inner cylindrical surface), the “buffer” volume.

During the calculations, all physical parameters, including the distribution of specific energy sources in the gas and the uranium layer, were assumed to be dependent on a single coordinate, r, alone. The approach used corresponds to a one-dimensional calculation of the processes in a certain equivalent cell with an inside radius, r1, and active volume equal to the inside radius and full volume of the real cell. An uranium layer of the same thickness and composition as those of the real cell stretches over the entire length of the equivalent cell, which equals the ratio of the real cell’s full volume to the cross-sectional area of the active volume’s gas column. Here, it is assumed that the thermal neutron flux distribution is not dependent on the coordinates but equals the average flux value over the length of the equivalent cell. This value is normalized in such a way that the total fissions occurring in the real and equivalent cells were the same.

When using an approach of this type no allowance is made for the possibility of the longitudinal motion of the gas and its partial seepage into the buffer volume. Also, certain density and temperature distributions averaged over the length of the cell are obtained as a result of the calculations. However, it should be remembered that over the course of experiments [2] aimed at investigating changes in density distributions along a cell’s cross-section, an interference procedure was used. During this measurement the variation in the optical path of an He-Ne laser probing beam over the entire length of a cell, also including the end buffer volumes, is recorded. Experimental results obtained in this manner only make it possible to approximately evaluate the gas density changes averaged over the entire length of a cell [2]. Of course, it is impossible to regard a density distribution “averaged” along a beam’s optical path in this manner as fully equivalent to the distribution obtained
during calculations. Nevertheless, a comparison of the results of such calculations with a similar experiment permits a judgment to be made concerning the accuracy of the concepts advanced for the physics of the origination and development of optical inhomogeneities.

It was shown in Chap. 7 that uranium-containing layer inhomogeneities can lead to a decrease in the efficiency of fission fragment energy deposition into a gas. A series of photographs of the transverse sections of the layers used in the experiments (similar to the photos in Fig. 7.10) reveal that the thickness deviations in these layers from the average value are 50-70 %, while the shape of the inhomogeneities is reminiscent of something halfway between rectangular and sinusoidal protuberances. As analysis demonstrates (see Chap. 7, Sect. 7.7), such fluctuations necessarily lead to a 5-18 % decrease in energy deposition efficiency. During the calculations, an average value of 11 % was selected for this decrease, i. e., Є = 0.89є, where є is the energy deposition efficiency for an ideal uranium layer. This correction factor was taken into account in the subsequent calculation of specific energy depositions.

Over the course of the experiments [2], changes were recorded in the pressure, ДP(t) = P(t) — P0 (P0 is the initial gas pressure in the cell), and the optical path of the He-Ne probing laser beam. The density changes of the media under investiga­tion were recalculated using optical measurement data, which are unequivocally related to the changes in the optical path.

The dependences of pressure variation within the cell upon time and density spatial distribution at different moments in time obtained from the calculations are in good agreement with the experimental results. Some calculated (the solid lines) and experimental (the broken lines) dependences of pressure, ДР(0, and relative density distribution, x(r, t) = p(r, f)p0, over the radius of a cell, as well as changes in relative density at the cell axis over time, are shown in Figs. 8.4, 8.5, 8.6, and 8.7. In addition, the shape of the neutron pulse is presented in Fig. 8.4.

ДP, atm; Ф, r. u.

The characteristics obtained experimentally and as a result of calculations are presented in Table 8.1 (F(1)/F(0) is the ratio of the specific energy deposition into the gas at a distance of r = 1 cm from the cell axis to its value at r = 0; APmax is the maximum value of the pressure increase within the cell; and x(0, t) is the relative density value at the cell axis at a moment in time of t = 10 ms.) With the exception of two calculations involving argon at pressures of P0 = 0.25 and 0.5 atm, wherein the deviation in calculated relative density from the experimental value reached 20 %, this deviation was less than 2.5 % in all of the remaining calculations. The fact that the calculated relative density, x(0, t), consistently exceeds the experimen­tal value stands out. This is probably explained by the fact that no allowance was made in the calculations for the gas seepage into the buffer volume during the course of irradiation. However, the overrun is not large and does not lead to noticeable differences in the density profile’s spatial distribution.

An estimate of the effect of gas fraction seepage into the buffer volume can be obtained from quite simple considerations. It was noted in Sect. 7.4 of Chap. 7

Table 8.1 Comparison of experimental and calculated characteristics Gas He Ar Ne P0, atm 1 2 3 5 0.25 0.5 1 1 F(1)F(0) Exper. 1.11 1.33 1.43 2.5 1.11 1.43 2.5 1.67 Calcul. 1.10 1.21 1.45 3.1 1.10 1.22 2.15 1.41 ДР, atm Exper. 0.30 0.51 — 0.69 — — 0.74 0.61 Calcul. 0.34 0.58 0.68 0.65 0.41 0.68 0.83 0.72 x(0, t) Exper. 0.939 0.969 0.980 1.026 0.790 0.870 1.16a 0.945 t = 10 ms Calcul. 0.959 0.991 1.013 1.044 0.970 1.043 1.19 0.959 Note: aOnly the x(0, t) dependence up until a moment in time of t и 9 ms is presented in study [2] for Ar (Р0 = 1 atm). A value of x(0, t) = 1.16 was obtained by extrapolating the experimental results up until a moment in time of t = 10 ms

that the pressure within a cell over the course of irradiation by a pulsed neutron flux with a duration of t1/2 > 3 ms can, as a first approximation, be assumed to be homogeneous at any irradiation pulse moment, and that gas density redistribution can be regarded as having an equilibrium nature. In this instance, the gas density distribution averaged over the cross-section along the z coordinate, which coincides with the cell’s optical axis, can be presented in the form

where T(z, t) is the effective average (for the cell cross-section) temperature value, determined by the equation

at which time formula (8.20) holds true.

From the condition of gas mass constancy within a cell, we obtain

In this case it is noted that energy release distribution along the z axis is symmetrical relative to the geometric center of the cell. Thus,

For simplicity’s sake, we assume that the T(z, t) dependence takes the form

where LA is the length of the real cell’s active volume. Such a dependence must approximately correspond to the real temperature distribution, as can be concluded from the type of thermal neutron flux distribution (see Fig. 8.3). Placing the distribution from (8.23) into (8.22), we obtain

From (8.23) and (8.24) for the relative variation of average density, we have

^ ^ — 1 =—(L — La) ^ . (8.25)

p0 p0 L — La 1 — TB

The temperature determined using formula (8.21) can be used as the tempera­ture, TA(t), for estimates. In this instance, the T(r, t) dependence is taken from a calculation using the procedure described above.

For the approximate determination of the gas temperature in the buffer volume, it can be assumed that its compression over the course of gas expansion from the active volume occurs adiabatically, i. e.,

/—1

T«(t) = To( £)’

where P0, T0 are the initial pressure and temperature values in the cell. Such an approach is equivalent to ignoring heat removal on the cell wall within the buffer volume. In approximate calculations, this is warranted by the fact that, even in the active volume, ~30 % of the energy absorbed in the gas is carried away by means of similar heat removal. However, the temperature gradients in the buffer volume must be smaller than in the active volume, and accordingly, smaller than the share of energy carried off by means of heat transfer.

The results of a ApA/p0 calculation for all the gas pressures that were studied in the experiments [2] are presented in Table 8.2. The moment in time tm in the table corresponds to the irradiation pulse peak, while the moment in time tc corresponds to the pulse termination.

The calculated and experimental results revealed that for the rare-gas NPLs the lower density difference values, and consequently the lower refractive index values take plact for gases with a lower atomic numbers. The difference values decrease as the initial pressure drops. The experiments confirmed that a narrow region with a large positive density gradient originates near a substrate with a uranium-containing layer. This region grows over time. A comparison of the calculated and experimen­tal results makes it possible to reach a conclusion concerning the applicability of the proposed calculation model to the investigation of the optical properties of NPLs.

The experimental data [2] were later examined in study [37]. The calculations described above [23] were performed in accordance with a full gasdynamic pro­gram that quite correctly reflected heat transfer processes but made no allowance for gas seepage into the buffer volumes. A simplified analytical model was devel­oped in [37] within which an attempt was made to take gas seepage into account while simultaneously introducing a correction factor for heat exchange between the gas and the cell’s metal wall. The authors [37] proposed that viscosity and thermal conductivity are absent in the main part of the cell (not adjacent to the wall), that the gas is ideal, and that the gas pressure is homogeneous at any given moment of time. The thickness of the heat exchange zone adjacent to the wall, which is called the “passive” or “dead” zone, was taken to equal I = fat. Using the latter equation, the heat flux in the active and buffer parts of the cell wall was found, and then the correction factor for the calculated pressure was determined, which consists of the last addend in equation (7.20). The transverse specific energy deposition profile was
assumed to be constant in Lagrangian coordinates. In the longitudinal direction within the active part of the cell, the specific energy deposition was regarded at homogeneous, while in the buffer part, it was considered to equal zero.

A comparison of calculation results [37] and experimental data [2] revealed that a somewhat better agreement was achieved between the calculated and experimen­tal values in five of the six cases examined. Here, the maximum discrepancy between the calculation results from studies [23] and [37] is exhibited during exciting pulse termination. As far as density, the greatest discrepancy in these results (~5 %) is observed for neon. In the remaining cases, these discrepancies are 0.3-2.7 %. All these figures are perceptibly lower than the estimated value of the effect of neglecting gas seepage into the buffer volumes cited in Table 8.2.

In conclusion, the numerical and theoretical investigations of other authors concerning the development of transverse optical inhomogeneities should be mentioned [18, 38—40]. The authors [18, 38, 39] completely ignore heat exchange. Furthermore, in [39], the specific energy deposition profile, tied to Eulerian coordi­nates, is assumed to be constant over the entire duration of the excitation pulse. In studies [18, 38], above profile is tied to Lagrangian coordinates, which more properly corresponds with the physics of the processes occurring. In study [40], the depen­dence of the specific energy deposition profile upon variable density is precisely defined; however, simplified gasdynamic equations are used.