While conducting investigations associated with development of NPLs and the study of their characteristics, it becomes necessary to determine the total energy absorbed in the gas mixture, and the spatial distribution of the energy deposition through the gas volume. Such data make it possible to assess and to optimize the physical characteristics, including the laser efficiency, the threshold pumping power, the optimal pressure for the given geometry and composition of the gas mixture, the space-time structure of the optical inhomogeneities, and so forth. Both the total energy absorbed in the gas mixture and the specific energy deposition in each point depend on the geometry of the laser channel, the composition and density of the mixture, and the composition and thickness of the active layer containing the fissionable material. Usually lasers excited by fission fragments have the shape of an elongated circular cylinder or rectangular channel (see Chap. 2). In the first case, the active fissionable material is applied in a thin layer (5U = 1-10 qm) to the internal surface of the cylindrical laser body; in the second, to two opposing parallel internal surfaces of the rectangular channel.

NPLs are characterized by transverse dimensions comparable to the range of the fragment in the gas laser mixture. Therefore, the profile of distribution of the specific energy deposition in the transverse cross section of the laser markedly depends on the transverse coordinates. The dimensions of the lasers in the direction of the optical axis of the system exceed many times over the range of the fission fragments in the utilized mixtures. This means that in the overwhelming portion of the volume of the laser channel, the longitudinal distribution of the specific energy deposition is determined only by the shape of the longitudinal distributions of the gas density and neutron flux irradiating the laser.

One of the chief deficiencies of early methods [8—14] of calculating the energy deposition of heavy ions, including fission fragments, was that the density of the excited gas was assumed to be uniform. In principle, the Monte Carlo method makes it possible to calculate the energy deposition with allowance for the non-uniformity of spatial distribution of the medium density.

Study [12] employs an alternate “two region” approach to address this problem. However, the spatial distribution of the energy deposition is given in histogram form [14] (calculation of the energy deposition is done for a finite selected volume in the neighborhood of the given point), which creates evident difficulties for restoration of a continuous dependence of the target value on coordinates. Breaking up each volume element into smaller ones, with the goal of clarifying the behavior of the spatial dependence, requires a significant increase in the calculating time and operational memory of the computer to retain the desired statistical accuracy of the calculation.

Matyev [15—17] developed an analytical method of calculating the energy deposition to media with an arbitrary distribution of density of the excited gas.

The space-time distribution of the specific power deposition to the gas F r. t is determined as in [15, 16]

, ч, 4 R P Г. t, 4

f(7’1) = . OR * p0 ‘(?4 (u)

Here q is the average fission power for the uranium-containing active layer per unit of volume of this layer; R0, R; are the mean range of a fission fragment in the gas

and in the material of the uranium-containing layer respectively; p ^ r, t^ is the density of the gas at the point in question at the time t; p0 is the average (non-perturbed) gas density; and Z^r, t^ is a dimensionless functional, defined by the distribution of gas density and the geometry of the cell. The correlations for r, t^ obtained in [15, 16] come to operations involving double integration.

Although they have a very cumbersome appearance, it then becomes possible with comparatively little effort to conduct a precise calculation of the energy deposition in non-uniform media with a finite transverse dimension.

As in other studies [1—3, 8—14] dedicated to calculations of the energy deposition when there is slowing down of heavy ions, including fission fragments, in the gas, the dependence of the energy E, of a decelerating particle of given mass on the range x in the model [15—17] is presented in the form

Ei = Е,- o(1 — x)n. (7.8)

The exponent n determines the “slowing down law”: for n = 1, one speaks of a “linear law” for slowing down; for n = 2, a “square law,” for n = 3/2, a “3/2 law.” The linear law is usually used to describe the deceleration of light ions—protons and a-particles [1]. It is rarely applied to fission fragments [10]. Usually n = 2 and n = 3/2 are used to describe the deceleration of fission fragments. Both values of the exponent n were discussed in study [18]. The square law of deceleration [18], which is also the one used in most studies, is considered the most adequate for fission fragments.

A comparison of results of calculations by the methods of [15—17] with the published results of calculations by the Monte Carlo method [14] shows their close identity. Figures 7.3 and 7.4 show the results of calculation of the efficiency for absorption of the kinetic energy of fission fragments in a gas medium. Results are for both cylindrical and flat uranium layers. The figures include the designations: e, the ratio of fission fragment kinetic energy absorbed in the gas medium to their total kinetic energy, released in the uranium layer; D0 = d/R0, the normalized transverse dimension of the gas volume (d is the diameter in the case of a cylindrical uranium layer or thickness of gas in the case of a flat layer); and D; = 8U/R1, the normalized thickness of the uranium-containing layer.

Є, %

50

40 30 20 10 0

The ranges of the uranium fission fragments in certain materials are provided in Table 7.1. To describe the mass spectrum of fission fragments, frequently an approximation is used, which assumes that when a 235U nucleus is split by thermal neutrons, two fragments are formed. These are a light and heavy one, which have average masses of 96 amu and 140 amu, and kinetic energies of 100 MeV and

68 MeV respectively. An empirical relationship [14] can be used to determine the mass ranges R (mg/cm2) of fission fragments in various media:

R = (0.0391 + 0.0202As/Z]/2) Е3/3, (7.9)

where As and Zs are the mass number and the charge of atoms of the medium; Ef, MeV is the initial kinetic energy of the fragment. Sometimes a simpler relationship is used, which makes it possible to determine the so-called mean mass range R without dividing the fragments into light and heavy [14]:

R = 0.755 + 0.388AS x Z;1/2. (7.10)

Prior to development of the computational methods [15—17] during the early computational investigations of the dynamics of development of optical inhomo­geneities in sealed laser cells, the influence of changes in gas density on the profile of distribution of the energy deposition into the gas was taken into account by tying the specific energy deposition sources to Lagrangian coordinates [19—21]. But the distribution itself of these sources was determined by the method described in [14]. In this case, the power deposition of the decelerating fragments in a gas portion of fixed mass, calculated relative to the intensity of uranium nuclear fissions, does not depend on the time, and remains equal to its initial value in a non-perturbed gas. That approach provides a precise description of the spatial distribution of the energy deposition only for an infinitely extended flat geometry [16, 22]. For other types of geometry of the cell, in particular for cylindrical or rectangular laser channels, this technique does not adequately allow for the influ­ence of redistribution of density on the energy deposition. Thus, in the case of cylindrical cells with an axially symmetrical distribution of medium density, in which the diameter is equal to the mean range of a fragment in a non-perturbed gas, or exceeds it, with an increase in the energy deposition, the gas particles are concentrated toward the axis of the channel, where when there are very large energy deposition (significantly exceeding those realistically attainable even in the most powerful pulsed reactors), a region inaccessible to the fragments may form [23]. In [23] this phenomenon is called “axial screening.” Nonetheless, special investigations showed that for cylindrical cells, with pumping durations of t1/2 < 3 ms, fixation of the energy deposition in Lagrangian coordinates makes it possible to obtain gas temperature and density deviations for any point of the volume which are no more than 10 % of the corresponding values obtained in calculations with precise allowance for the influence of redistribution of density on the specific energy deposition [16, 21].