In the y = const planes, gas density, and accordingly, the refractive index in the bulk of a flowing laser’s gas volume decrease in an almost linear manner with an increase in x; only within the fairly short channel inlet and outlet parts near to edge of a uranium layer and further outside its confines the gas density not dependent upon the x coordinate. The latter results from a noticeable decrease in power deposition in direct proximity to the edges of the uranium layers and outside their confines. In any x = const plane, refractive index distribution (with the excep­tion of a near-wall passive zone) is satisfactorily described by a negative parabola. The parabola’s curvature increases in a linear manner with the x distance value. It reaches a maximum value near the edge of a uranium layer, x = b, and subsequently remains unchanged.

A conventional two-mirror cavity was used in the LM-4 apparatus [26, 41] that consisted of planar and spherical (with a radius of curvature of 20 m) mirrors. However, the two-dimensional stable-unstable cavity proposed in refs. [22, 24] is more convenient for a similar NPL with gas pumping. It should be unstable in the longitudinal direction in the y = const planes and stable in the transverse direction in the x = const planes. The latter can be achieved, for example, using mirrors with a predetermined radius of curvature in these planes.

The distinctive features noted above for gas density distribution in gas flowing NPLs do not make it possible to obtain a solution to the equation describing a light beam propagation that is suitable for studying cavity stability using the ray matrix technique in a channel’s entire volume, as was done for a sealed NPL in refs. [56, 57]. However, an obviously simplified approach can be used to derive stationary mode criterion for estimating the permissible gas heating from an analysis of stability in the x = const plane. So, the infringement of the stability condition in the space immediately past the right edge of a uranium layer is examined in ref. [58] (the gas is pumped in the direction of the x axis). In this region, the refractive index distribution has a parabolic shape

n(y) = no0) — 2«(b)y2, (9.rn)

where the parabola coefficient, a, is maximal and equals the value reached in direct proximity to the edge of the uranium layer. This coefficient increases with an increase in the power deposition in the gas, and accordingly, its overheating. If it exceeds the value at which the cavity becomes unstable in the x = const plane in the x > b region, it is then natural to assume that the instability in the transverse direction relative to x will be shifted into the x < b region with a further increase in the power deposition.

As previously demonstrated, with the exception of the thin near-wall passive zone layers, heat exchange in the bulk of the gas volume can be ignored. Because the pressure in the channel is almost homogeneous, density and temperature distribution is determined by the distribution of specific fission fragment power deposition. The fission fragment energy losses per unit path are proportional to the medium’s density [59]. We will ignore the edge effects, which have a compara­tively minor influence on integral energy deposition in the gas as it passes through the channel when the dimension of the uranium layers is sufficient. Therefore, along each flow line that a particular gas particle traverses, the power deposition per unit mass of gas, f, can be roughly regarded as constant. This approach is equivalent to the one used in refs. [57, 60] to represent the distribution of specific power deposition in the form F(r) = F0(r)p(r)/p0, where F0(r) is the specific power depo­sition at a point, r, in the presence of a homogeneous unperturbed gas density distribution, p0. For an infinitely expanded planar geometry that is characterized by dependence upon a single y coordinate alone, this equality is strict due to the absence of edge effects [61].

For the gas velocities at which inequality (9.97) holds true, the temperature increment of a particular gas particle of unit mass after it has traversed a distance of x, according to the approximations used, equals

Because the gas pressure differential in the channel is small, (ДP/P) << 1, then pursuant to the ideal gas state equation, p(x, y) = T0p0/T(x, y). Thus,

The surface power density of the fission fragments from the uranium layers into the gas can be expressed by way of the specific power deposition of the gas averaged over the transverse cross-section, qs = pfd. Thus, condition (9.97), in the presence of which correlation (9.98), used to derive Eq. (9.152), holds true, is equivalent to the condition (fxCpU0T0) << 1. Therefore, expanding (9.152) into a series and selecting the first-order terms, we obtain

With a plane-parallel uranium layer arrangement, the specific power deposition in a unit mass of gas in the coordinate system shown in Fig. 9.1, ignoring the edge effects, is determined by the equations [61]

1 R

f (y) = — E0qR-I(Y); P0 R0

— G 20 + Y + D1 , (9.154)

where Y = y/R0; R0, R1 are the average range of a fission fragment in the gas and in the uranium layer material, respectively; D0 = d/R0 is the normalized gas channel width; and D1 = 8R1 is the normalized uranium layer thickness, while the G function is determined by the correlation

G(£)= 2(1 — + 2^ln^). (9.155)

Following the appropriate transformations and the expansion of the addends containing the logarithms into a series for the small parameter 2YD0, with an accuracy up to terms of the second-order of smallness for central regions located near the channel’s plane of symmetry, we obtain

where

From the heat balance condition in the stationary circulation mode

X

0

in the approximation qS(x) = const for overheating averaged throughout the chan­nel’s cross-section, we get

ДT = T(b) — T0 = qsb. (9.157)

cpp0du0

We will express the surface power density of the fission fragments from the uranium layers into the gas, qS, in terms of the total energy of the uranium nuclear fission fragments, E0, the intensity of uranium fissions per unit volume, q, uranium layer thickness, £u, and energy deposition efficiency, є (the share of fission frag­ment energy transmitted to the gas):

qs = 2E0qSu£. (9.158)

Determining the refractive index based on its unperturbed value at the inlet, n00,

n(x, y) = 1 + (П00 — 1)p(X;y),

P0

then from Eqs. (9.153), (9.154), and (9.156) with allowance for the relationship of Eqs. (9.157) and (9.158) between q and ДT at x = b, we obtain Eq. (9.151), in which

The dimensionless у parameter determined by formula (8.40) was introduced in ref. [56] (see Chap. 8.5). Under a given spatial distribution of specific sources, it increases with an increase in energy deposition. The value of this parameter at which the infringement of the stability condition m < 1 begins is called the critical value. The ycr parameter reaches its highest values in the case of a cavity made up of

planar mirrors (with radii of curvature of r1 = r2 = 1), as well as a cavity with codirectional mirrors of identical curvature (r1 = —r2). In the case of other mirror curvature combinations, the ycr value does not exceed 1.5. The dependence of ycr upon the relative length of the clearance between a mirror and the active volume boundary, L1/La, calculated using the technique described in ref. [56], is presented in Fig. 9.31 for several radius of curvature values, r1, of one of the mirrors. It was assumed that L1 = L2, while the radius of curvature of the second mirror was Г 2 = 1.

If it is assumed that the stability condition should not be infringed in the x = const planes within the gas regions of x > b, where energy deposition reaches the maximum value, then from the requirement у < ycr, as well as Eqs. (8.40) and (9.159), we obtain

__ < n0(D0 + 2Р1)^0еГ0yKp < 4(n00 — 1)LA

During medium overheating, _T < 104 K (n0 — 1) << 1; therefore, in Eq. (9.160), n0 can be replaced with unity. Thus, according to the criterion under consideration, permissible gas overheating through values of D0, D1, R0, e, and La is determined by gas mixture density, the geometric dimensions of the active gas volume, and uranium layer thickness, while through ycr, it is determined by resonator mirror curvature and the distances from the mirrors to the active volume.

The L1 and L2 distance values are selected in such a way that the uranium nuclear fission fragments do not reach the cavity mirrors. In actual designs, they usually

come to L1 = L2 « 10 cm; thus, at LA ~ 1 m, it can be roughly assumed for estimates that ycr < 1.5. For the laser channel of the LM-4 apparatus (d = 2 cm, 5U = 2.7 x 10~4 cm, and LA = 1 m) during the pumping of a gas mixture, the basic component of which is helium (P0 = 2 atm) at T0 = 300 K, an estimate using formula (9.160) yields AT < 1,900 K, while for an argon-based mixture at P0 = 0.5 atm and T0 = 300 K, AT < 860 K. According to Eq. (9.160), a twofold increase in active length, LA, while the other parameters remain unchanged, as exemplified by an argon-based mixture, will result in the stability limitation becoming more rigorous (AT < 230 K.)