Some Features of Boron Isotopes Separation by Laser-Assisted Retardation of Condensation Method

Abstract

Boron isotopes have many applications in industry: medicine, semiconductor, and solar energy. Especially massive demand is for boron-10 isotopes in nuclear industry for nuclear reactors shielding and control. Various aspects of laser-assisted boron isotope separation by retardation of condensation method, such as irradiation conditions and laser and vacuum system design, have been considered. Irradiation conditions include interaction scheme of laser radiation and supersonic beam, dependence of efficiency of excitation on gas flow temperature and pressure. Basic physical constraints on laser intensity and its spectral properties have been discussed. The relation of gas flow properties, nozzle design, and vacuuming rate has been elucidated as well.

1. Introduction

Most important applications of boron isotopes are related to boron additives widely employed in nuclear plants, as boron carbide used in control rods, or as boric acid solution used as a chemical shim in the Pressurized Water Reactors(PWR) and semiconductor industry. Due to the much larger thermal neutron absorption cross-section of boron-10 (${\sigma }_n=3837\text{barn}$) than for boron-11 (${\sigma }_n=0.005\text{barn}$) the use of enriched ${\mathrm{H}}_3{\mathrm{BO}}_3$ allows to reduce the total amount of boron-based poison material in the primary reactor coolant system and, therefore, to reduce corrosion and wear on the other components of the reactor core [1]. Boron-10 enriched compounds are also used to increase the efficiency of nuclear reactor emergency shutdown systems and for nuclear fuel transportation. In semiconductor industry, boron is routinely used for producing p-type domains in silica. Decreasing the size of electronic devices makes the problem of heat removal more and more important. Its solution can be using isotopically pure boron, which provides minimal distortion of the crystal structure of silica matrix and minimizes the thickness of boron acceptor layers, and, therefore, increases the heat conductivity of the acceptor layer and the transistor switching power [2]. Boron-11 can be used in this case, because protection from cosmic and other kinds of radiation becomes important with advancing miniaturization of electronic devices and in solar panels [3, 4]. Moreover, boron isotope enriched boron nitride used in nanotubes and nanoribbons has high potential for applications in nanotechnology [5, 6, 7]. As an example, in spacecraft semiconductors boron-10 enriched $\mathrm{BN}$ nanotubes, can be used for radiation shielding [8]. Boron-10 can be also used as a neutron-detecting component in self-powered solid-state neutron detectors [9], used in nuclear materials studies and in well logging. Boron-10 in health care is applied in boron neutron capture cancer therapy [10, 11, 12], and in studies of food properties aimed at preventing cancer and other diseases [13].

There are three methods for boron isotopes separation commonly used in industry. These methods are based on the chemical exchange reaction [14], low-temperature fractional distillation [15], and gas centrifuging. In these methods, ${\mathrm{BF}}_3$ is used. In gas centrifuging, ${\mathrm{BCl}}_3$ due to its high vapor pressure at room temperature, can be also used, but it is far less efficient, because of presence of three chlorine isotopes. However, using ${\mathrm{BF}}_3$ is not economically justified, because 86% of price for high purity ${}^{10}\mathrm{B}$ comes from powder extraction from ${\mathrm{BF}}_3$. Hence, other methods, which rely on using ${\mathrm{BCl}}_3$, are needed.

Laser-assisted methods comprise Atomic Vapor Laser-assisted Isotope Separation (AVLIS) [2], and Molecular Laser Isotope Separation (MLIS) methods, also known as Selection of Isotopes by Laser EXcitation (SILEX) [16, 17, 18], http://www.silex.com.au/businesses/silex. MLIS methods can be classified as following: Chemical Reaction by Isotope Selective Laser Activation (CRISLA) [19, 20, 21], Condensation Repression by Isotope Selective Laser Activation (CRISLA-2) scheme (this scheme is also known as Separation of Isotopes by Laser-Assisted Retardation of Condensation (SILARC)), and Cold Walls Laser-Assisted Selective Condensation (CWLARC) scheme (this scheme is also known as SILARC-2 [22]). Laser-assisted methods are based on the selective excitation of target isotopes in atomic or molecular form by laser radiation. Selectivity is expressed via specific for different target isotopes in atoms or molecules resonant-like photon energy dependence of photoabsorption cross section. In case if multiphoton dissociation is used as isotope selection mechanism, frequency of infrared (IR) laser radiation should be red-shifted [21]. Two lasers (IR + UV) can be used for dissociation or ionization of molecules. In contrast to high selectivity, subsequent chemical reactions in CRISLA method can significantly deteriorate efficiency of the process. CWLARC method has three major disadvantages: Firstly, coaxial nozzle throughout is very low, which makes it only attractive for medicine applications, secondly, wall temperature should be kept at the same low-temperature level in quite narrow interval due to the specific temperature dependence of enrichment factor, and, thirdly, isotope harvesting success strongly depends on the symmetry of selectively excited molecules. As a general advantage of SILARC methods is that only a few photons are needed for selective excitation instead of several dozens of photons required in the methods based on multiphoton dissociation (CRISLA) or ionization(AVLIS). In contrast to popular laser-assisted methods, such as Molecule Obliteration Laser Isotope Separation(MOLIS) and Atomic Vapor Laser Isotope Separation (AVLIS), large laser fluence is not required and even harmful for isotopes harvesting in SILARC methods, due to destructive resonant interaction of strong electric fields with target molecules. Ideally, laser intensity should be just high enough for molecular excitation, and, therefore, able to guide molecular dynamics in needed direction. Moreover, SILARC methods are more efficient because of typically two orders of magnitude higher linear photo-absorption cross section, than nonlinear multiphoton absorption cross section [23]. Moreover, the controllability of sequential isotope scrambling effects is more easier. The total energy consumed by vacuum pumps to provide optimal pressure level in discharge chamber (${10}^{-2}-1\text{torr}$), is normally significantly smaller in MLIS methods than in AVLIS, where requirements for vacuum level in discharge chamber are exceptionally high (${10}^{-6}-{10}^{-7}\text{torr}$). Therefore, only SILARC method deserves more detailed analysis.

SILARC method conceived by Y.T. Lee in Ref. [24], and developed by Jozef Eerkens in [22, 25]. In this method isotopes harvesting is based upon well established mass separation effect in overcooled supersonic gas flow: monomers escape gas flow core at higher rate than van der Waals clusters (dimers and higher oligomers). The larger mass difference the more separation effect is pronounced. In order to produce gas flow with uniform pressure distribution, specially profiled supersonic nozzle should be designed. Formed oligomers can be drawn from the flow either by some cold surfaces (wavy plates or walls as in SILARC-2) [17, 26], or by skimmer blade as in SILARC scheme [26]. Viability of this method was originally demonstrated on example of sulfur isotope separation in ${\mathrm{SF}}_6$ target gas mixed with argon [27, 28].

At the first glance, the isotope production rate should increase with increasing nozzle throughput, which can be achieved by increasing nozzle dimensions, number of nozzles, or gas flow density. In the latter case, applying the laser radiation to a dense gas flow is not able to change mass distribution of formed clusters, because irreversible cluster growth sets in (in majority of clusters are over-critical [29]), and, therefore, monomer molar fraction available for excitation is rapidly decreasing. If gas flow density is not too high, then quantum optimal control methods can be helpful [30]. Too high gas density corresponds to the case when cluster loading is larger than the critical one, so-called over-critical loading. In this case, clusters are normally ionized by electromagnetic radiation but not decouple [31, 32]. However, if the gas flow dilution is high enough, its temperature is low, and transition time of molecules across irradiation cell (IC) is not too long, then the population of under-critical clusters is decreasing much slower. The upper limit for IC length is given by the condition, that the fraction of under-critical clusters in the gas flow is vanishingly small because most of them escaped gas flow. Apparently, only in the case, when under-critical clusters are still in the gas flow irradiation zone, selective resonant-like absorption of laser photons by target gas molecules can lead to retardation of their further binding with surrounding carrier gas molecules. Here, it will be considered only such gas flow pressures and temperatures that gas flow is mostly represented by monomers and dimers. We demonstrated, that increasing nozzle dimensions will decrease product cut and overall performance of isotope separation process. Therefore, only increasing the number of nozzles is the only option left to control.

Since enrichment factor and product cut are small, isotopes should be extracted by many recirculations of the gas flow. Calculations of the product cut, describing degree of gas flow separation, and enrichment factor, describing isotope content in the separated part of the gas flow, were carried out on the basis of static approximation of the transport model developed in Ref. [26].

In Ref. [33] we developed an iterative scheme for one-stage cascade and in Ref. [34] for two-stage cascade. Each cycle in this scheme corresponds to transition time of gas molecule through IC.

Configuration of the skimmer inlet should correspond to population profile of excited molecules across the separation cell cross section at the end of IC. In order to keep at minimum gas flow disturbance induced by skimmer inlet, it should be also carefully designed [35, 36].

Optimal pressure and temperature fields inside the gas flow can be provided by adequate nozzle wall shape, IC and skimmer chamber geometry, and proper choice of core and rim gas evacuation rates. Also, due to specific pattern of velocity field, yielded by the given nozzle design, the efficiency of isotope extraction is directly related to the skimmer inlet configuration and its distance from the nozzle outlet.

Since optimal enrichment facility design should be a compromise between selectivity (or time spent for extraction of a given amount of isotopes) and related energy consumptions (should be made as small as possible), we suggest that adequate criterion for efficiency of separation should correspond to the minimum of the objective function, which has a meaning of average over time of total energy consumed per unit isotope recovery [37].

In Refs. [38, 39, 40], it has been demonstrated that using an irradiation cell for isotope separation in SILARC scheme as an absorber part of CO₂ laser resonator can substantially cut electricity consumption. The idea to use a resonator as a multi-pass cavity for isotopes separation is not a novel one. For instance, as shown in Ref. [41], the use of resonator allows to increase production of selectively ionized ${}^{168}\mathrm{Yb}$ by the order of magnitude. Moreover, in Ref. [42], it has been shown that the selectivity of dissociation of ${\mathrm{CF}}_2\mathrm{HCl}$ molecules by CRISLA method can be significantly increased, if they are introduced inside resonator cavity of CO2 laser.

In case of overcooled gas flow, photo-absorption spectra are very narrow (only collisional broadening is dominant). Hence, the proper choice of laser pulse spectrum is crucial. In the case of CW-mode irradiation, hitting the target isotopologues is especially problematic for ${\mathrm{CO}}_2$ laser, because it’s emission spectrum is only discretely tunable, while the laser emission line should coincide with center of photo-absorption line with accuracy not worse than its full width at half of maximum (FWHM). At room temperature, the fundamental vibrational mode ${\nu }_3$ of ${\mathrm{BCl}}_3$ can be excited by ${\mathrm{CO}}_2$ laser pulses, having emission lines 10P(8)-10P(28) in their spectrum. At laser intensities lower than saturation limit, 10P(8)-10P(16) lines are better absorbed [43], while at larger laser intensities, where multiphoton processes start to play essential role, photo-absorption maximums are more red-shifted 10P(24)-10P(28) [19, 44, 45]. By increasing pressure in the laser medium, frequency mismatches can be compensated due to emission line broadening effects. However, due to high collision rate at room temperature, the rate of excitation loss equals to or even lower than the rate of excitation gain. Hence, in order to diminish it, the gas should be sufficiently dilute and cold. These conditions both can be fulfilled by supersonically expanding rarefied gas flow.

This chapter is of the following structure. In Section 2, the transpart model to describe main features of the SILARC method is presented. In subsection 2.1, some details of evaluation of spectral properties of photo-absorption cross section are given. In subsection 2.2, definitions and results of calculation of product cut and enrichment factor are given. In Section 3, the operational principles of boron isotopes separation facility and physical processes, they are based on, are elucidated. In Section 4, gas flow irradiation conditions, such as laser spectrum choice and beam radius, are given. In subsection 4.1, basic constraints on laser intensity variation range are discussed. In Section 5, gas flow properties and requirements applied on vacuum system to provide their optimal choice are given. In subsection 5.1 examples of calculation of optimal nozzle wall shape are given. In subsection 5.2, the calculation of minimal required vacuum pump rate related to the given nozzle profile is given schematically.

2. Transport model

Excitation dynamics of the overcooled gas flow, controlled by selectively tuned laser field for this target gas, can be modeled by the transport equations, describing population dynamics of four characteristic groups of species, presented in dilute enough overcooled gas flow [26]. Molar fractions of these characteristic groups fulfill the following material balance equation:

$$f_{i\mathrm{m}}+f_{i\ast }+f_{i!}+f_{i\mathrm{d}}=1,$$E1

where $f_{i\mathrm{m}}$, $f_{i\ast }$, $f_{i!}$, and $f_{i\mathrm{d}}$ are molar fraction of monomers, excited monomers, epithermal, and dimers of $i$th isotopologue of the target gas molecule, respectively. Population dynamics of these characteristic groups are described by the following system of equations

$$\left\{\begin{array}{l}\frac{{\mathit{df}}_{i\ast }}{\mathit{dt}}=k_{\mathrm{A}}{f}_{i\mathrm{m}}-f_{i\ast }\left\{\left(1-e_{\ast }\right)\left(k_{\mathrm{df}}+k_{\mathrm{VV}}+k_{\mathrm{VT}}+k_{\mathrm{se}}\right)+e_{\ast }{k}_{\mathrm{W}}\right\},\\ \frac{{\mathit{df}}_{i!}}{\mathit{dt}}=\left(1-e_{\ast }\right)\left(k_{\mathrm{df}}+k_{\mathrm{VT}}\right)f_{i\ast }-\left\{\left(1-e_1\right)k_{\mathrm{th}}+e_1{k}_{\mathrm{W}1}\right\}{f}_{i!},\\ \frac{{\mathit{df}}_{i\mathrm{d}}}{\mathit{dt}}=k_{\mathrm{df}}{f}_{i\mathrm{m}}-k_{\mathrm{dd}}{f}_{i\mathrm{d}}.\end{array}\right.$$E2

Here, $k_{\mathrm{df}}$ and $k_{\mathrm{dd}}$ are dimer formation and dissociation rates, $k_{\mathrm{VT}}$ and $k_{\mathrm{VV}}$ are vibration-translational and vibration-vibrational relaxation rates, $k_{\mathrm{se}}$ is photon spontaneous emission rate, $k_{\mathrm{th}}$ is epithermals thermalization rate, $k_{\mathrm{W}1}$ is “wall” escape rate for epithermals, $e_{\ast }$ and $e_1$ are to-the-wall survival probabilities for excited monomers and epithermals, respectively. In our calculations, it was taken into account, that transport coefficients depend on the respective isotopologue mass (index $i$ will be omitted below when no confusion occurs). In our case with sufficiently low gas flow pressure and target gas molar fraction, vibration-translational $k_{\mathrm{VT}}$, vibration-vibrational $k_{\mathrm{VV}}$ , and photon spontaneous emission rate $k_{\mathrm{se}}$ are very small and can be neglected. By assuming steady gas flow and continuous wave irradiation, the system transport equations are reduced to the system of algebraic equations, which was solved in Ref. [26].

Laser excitation rate is given by:

$$k_{\mathrm{A}}=\frac{{\sigma }_{\mathit{abs}}\left({\nu }_{\mathrm{ex}}\right)}{h{\nu }_{\mathrm{ex}}}\frac{t_{\mathrm{int}}}{t_{\mathrm{tr}}}{\varphi }_L,$$E3

where $t_{\mathrm{int}}=\frac{2R_L}{\overline{U}}$ is gas flow transition time across laser beam in case of continuous wave irradiation and $t_{\mathrm{int}}={\nu }_p{\tau }_p{t}_{\mathrm{tr}}$ in case of pulsed irradiation, and ${\varphi }_L$ is the laser field distribution, which is assumed to be pencil-like with radius $R_L$, which corresponds to full cross-wise overlap with planar gas flow(for higher nozzle throughput):

$${\varphi }_L=\frac{I_{\mathrm{las}}}{\pi R_L^2},$$E4

where $I_{\mathrm{las}}=I_{\mathrm{CW}}$ in case of CW-lasing, and $I_{\mathrm{las}}=I_p$ in the case of pulsed lasing.

2.1 Evaluation of photo-absorption cross section

According to Refs. [44, 46, 47], ${\mathrm{BCl}}_3$ molecules photo-absorption spectra in gaseous state are regularly blue-shifted by $\sim 8-9{\mathrm{cm}}^{-1}$ in respect to their values for ${\mathrm{BCl}}_3$ isolated in argon matrix [48]. Since ${\mathrm{BCl}}_3$ belongs to one of two symmetry groups, $D_{3h}$ or $C_{2v}$, depending on whether the masses of chlorine isotopes are the same or not, the lines, corresponding to ${\tilde{\lambda }}_1$ and ${\tilde{\lambda }}_4$ are contributed by both, so that respective absorption peak intensities are proportional to the following probabilities¹:

$$\begin{array}{r}{p}^{\left(1\right)}=P\left({\left(35\right)}^3\right)+1.5P\left({\left(35\right)}^2\left(37\right)\right)=0.64;p^{\left(4\right)}=P\left({\left(37\right)}^3\right)+1.5P\left({\left(37\right)}^2\left(35\right)\right)=0.08;\\ p^{\left(2\right)}=1.5P\left({\left(35\right)}^2\left(37\right)\right)=0.21;p^{\left(3\right)}=1.5P\left({\left(37\right)}^2\left(35\right)\right)=0.06.\end{array}$$E5

Measured distances between neighbor peaks are

$$\mathrm{\Delta }{\tilde{\lambda }}_{12}=1.41{\mathrm{cm}}^{-1};\mathrm{\Delta }{\tilde{\lambda }}_{23}=1.189{\mathrm{cm}}^{-1};\mathrm{\Delta }{\tilde{\lambda }}_{34}=1.301{\mathrm{cm}}^{-1}.$$E6

Linear photo-absorption cross sections for each line can be extracted from Naeperian infrared intensity

$$\begin{array}{l}{I}_{\mathit{IR}}=\frac{2}{\pi }\frac{1}{27.648}\int {\sigma }_{\mathit{abs}}\left(\tilde{\lambda },{\tilde{\lambda }}_k\right)d\tilde{\lambda },\\ {\sigma }_{\mathit{abs}}\left(\tilde{\lambda },{\tilde{\lambda }}_k\right)={\sigma }_{\mathit{abs}}\left({\tilde{\lambda }}_k\right)\frac{\mathrm{\Delta }\tilde{\lambda }}{4{\left(\tilde{\lambda }-{\tilde{\lambda }}_k\right)}^2+\mathrm{\Delta }{\tilde{\lambda }}^2}.\end{array}$$E7

According to experimental data for ${}^{11}{\nu }_3$ absorption band from [47]: $I_{\mathit{IR}}=231.3\mathrm{km}/\text{mole}$.

Since overcooled gas flow is irradiated cross-wise in opposite directions, Doppler broadenings in respect to laser beam outgoing from laser source and reflected from retro-mirror will compensate each other. Hence, photo-absorption cross section can be approximated by Lorentzian, having FWHM

$$2\mathrm{\Delta }{\nu }_P=\frac{4p_{\mathit{tot}}{\sigma }_{\mathit{QG}}}{\pi \sqrt{\pi k_B{T}_{\mathit{\text{flow}}}{M}_Q}c}=c\mathrm{\Delta }\tilde{\lambda }=2.31\mathrm{MHz}.$$E8

Thus, the total cross-section of isotopologues of sort $k$ is given by:

$$\begin{array}{c}{\sigma }_{\mathit{abs}}^{\left(k\right)}={\sigma }_{\max }^{\left(k\right)}\frac{\mathrm{\Delta }{\nu }_P^2}{{\left(\nu -{\nu }_k\right)}^2+\mathrm{\Delta }{\nu }_P^2},\\ {\sigma }_{\max }^{\left(k\right)}=\frac{27.648I_{\mathit{IR}}}{\mathrm{\Delta }\tilde{\lambda }}{p}^{\left(k\right)},\end{array}$$E9

where

$$\left\{{\sigma }_{\max }^{\left(1\right)},{\sigma }_{\max }^{\left(2\right)},{\sigma }_{\max }^{\left(3\right)},{\sigma }_{\max }^{\left(4\right)}\right\}=\left\{17.72;5.74;1.84;2.23\right\}\times {10}^{-16}{\mathrm{m}}^2.$$E10

2.2 Product cut and enrichment factor evaluation

Product cut can be evaluated as a fraction of escaped target gas molecules and oligomers from the gas flow core

$$\theta =\frac{Q_{\mathrm{esc}}}{Q_{\text{feed}}},$$E11

where $Q_{\text{feed}}=y_Q{N}_{\text{feed}},N_{\text{feed}}=Q_f{t}_{\mathrm{tr}}$ is the number of target gas molecules in the feeding gas flow and $Q_{\mathrm{esc}}$ is the number of escaped specied. The product cut can be transformed to $\theta =\mathfrak{a}+\mathfrak{b},$ where

$$\begin{array}{r}\mathfrak{a}=\sum _{\stackrel{k=1}{k\ne i}}^2{x}_k{\alpha }_k,{\alpha }_k=\left(1-f_{k\mathrm{d}}\right)\mathrm{\Theta }+f_{k\mathrm{d}}{\mathrm{\Theta }}_d;\\ \mathfrak{b}=x_i{\alpha }_i,{\alpha }_i=\left(1-f_{i!}-f_{i\mathrm{d}}\right)\mathrm{\Theta }+f_{i!}{\mathrm{\Theta }}_1+f_{i\mathrm{d}}{\mathrm{\Theta }}_d,\end{array}$$E12

relative isotope abundance of excited $i$th isotope is denoted as $x$ (for definiteness we assume that only ${}^{11}{\mathrm{BCl}}_3$ isotopologues are excited, i.e., $i=2$), and

$$\mathrm{\Theta }=1-e^{-k_{\mathrm{W}}{t}_{\mathrm{tr}}}$$E13

is a fraction of escaped from gas flow core monomers,

$${\mathrm{\Theta }}_1=1-e^{-{\mu }_1{k}_{\mathrm{W}}{t}_{\mathrm{tr}}}$$E14

is a fraction of escaped epithermals, and

$${\mathrm{\Theta }}_d=1-e^{-{\psi }_d{k}_{\mathrm{W}}{t}_{\mathrm{tr}}}$$E15

is fraction of escaped dimers, where

$${\psi }_d=\frac{{\sigma }_{\mathit{cQ}/G}{M}_{Q/G}^{1/2}}{{\sigma }_{\mathit{cQG}/G}{M}_{\mathit{QG}/G}^{1/2}}.$$E16

Enrichment factor by $i$th isotope is given by

$$\beta =\frac{{}^i{Q}_{\mathrm{esc}}/Q_{\mathrm{esc}}}{{}^i{Q}_{\text{feed}}/Q_{\text{feed}}}=\frac{{\alpha }_i}{\theta },$$E17

where ${}^i{Q}_{\text{feed}}/Q_{\text{feed}}$ and ${}^i{Q}_{\mathrm{esc}}/Q_{\mathrm{esc}}$ are relative abundances of $i$th isotope in the escaped and feed gas flows, respectively.

Product cut and enrichment factor as functions of gas flow core temperature (other parameters are fixed to their optimal values) were calculated in Refs. [37, 49]. Significant descrepancy between the maximal values of enrichment factors 1.0043 ($p_{\mathit{tot}}=10\text{mTorr},T=24\mathrm{K},{\sigma }_{\mathit{abs}}=13.75\mathrm{Mb},{\varphi }_L=39.31\mathrm{W}/{\mathrm{cm}}^2,L_{\mathit{IC}}=67\mathrm{cm}$) and 1.25 ($p_{\mathit{tot}}=10\text{mTorr},T=25\mathrm{K},{\sigma }_{\mathit{abs}}=0.071\mathrm{Mb},{\varphi }_L=10\mathrm{kW}/{\mathrm{cm}}^2,L_{\mathit{IC}}=20\mathrm{cm}$), obtained in [37, 49] respectively, can be explained by that the formula (32) from [26] should be multiplied by the vibration-translational(VT) transition probability, as it should be in order to get proper equilibrium dimer concentration (Eq. (124) in [25]). Here, $L_{\mathit{IC}}$ is gas flow length from nozzle outlet to skimmer inlet.

3. Operational principles

At the beginning of operation, the mixing tank is occupied only by carrier gas, while target gas molecules, having natural relative isotope abundance, are stored in the feed chamber. Then, the target gas is injected into the mixing tank. Target gas is seeded at a very low molar fraction($y_Q\sim 0.02$) into carrier gas, in order to minimize nearly resonant VV excitation loss caused by collisions among ${\mathrm{BCl}}_3$ molecules. Gas flow should be diluted as well in order to minimize nearly resonant excitation loss due to VT (vibration-translational) relaxation. In order to provide pressure $p_{\mathit{\text{flow}}}\approx 10\text{mtorr}$ and temperature $T_{\mathit{\text{flow}}}\approx 24\mathrm{K}$, that correspond to maximum of enrichment factor, provided Ar is chosen as a carrier gas, the total pressure in the mixing tank should be $P_0\approx 6\text{torr}$ [38], provided expansion remains isentropic at least along the gas flow core axis.

The scheme shown in Figure 1 allows multifrequency boron isotopes excitation in the absorber part of ${\mathrm{CO}}_2$ laser resonator (in Ref. [39] the same scheme was proposed, but for orthogonal gas flow irradiation). From the mixing tank gas expands continuously into the separation chamber, where it is irradiated cross-wise by the laser beam confined between mirror strips placed on opposite walls of resonator, three (one flat and two concave) mirrors outside the resonator cavity and grating.

Figure 1.

The principal scheme of boron isotope enrichment setup (edges, links, and details, that are invisible from the frontal surface, are displayed by dashed lines). The laser beam passes through a semi-transparent window and cross-wise impinges the gas flow. In order to increase excitation rate, the laser beam oscillates within a resonant multi-pass cavity between opposite mirror strips in order to interact with whole length of supersonic jet. Core gas is boron-10 enriched by selective laser evaporation of ${}^{11}{\mathrm{BCl}}_3$ isotopologues. It is circulated from the mixing tank and back in order to compensate pressure loss and to achieve a higher degree of target isotope recovery.

Multiple reflections from mirrors may lead, however, to the instability of laser power and frequency². As shown in Refs. [38, 39], the use of resonant absorber cavity for isotope separation will lead to 4–6.5 times increase of laser pulse peak intensity.

In order to provide the largest gas flow irradiation volume, having the optimal static pressure and temperature distributions, a uniform velocity profile can be provided by slit nozzle design, which is discussed in more detail in subsection 5.1.

Skimmer blade divides the gas flow into target isotope (boron-11) enriched (rim) and desired isotope (boron-10) enriched (core) fractions by the end of chamber. Core and rim gas flow fractions, having laser-controlled target isotopologue populations, are evacuated separately by two separate vacuum pumps, having appropriate pumping down rates. Before redirecting them back into mixing tank or exhausting into atmosphere, they are captured in respective Zeolite cold traps as shown in Figure 1. Roughly speaking, pumping out speeds should be chosen proportional to corresponding stagnation pressures. The larger the initial pressure loss, the less demanding requirements for vacuum pumps. Material, momentum, and energy balance equations for skimmer partition have been solved in Section 5.2. Besides evaluating required minimal levels for pumping down rates for core and rim gas flows, it helps to clarify contributions and relations between various parameters, affecting gas flow deceleration, and, therefore, stagnation pressures, corresponding to each vacuum pump.

4. Irradiation conditions

According to experimental data, the gaseous state ${}^{11}{\mathrm{BCl}}_3$ photo-absorption line center position ranges from ${\tilde{\lambda }}_{\max }^{\left(1\right)}=957.67{\mathrm{cm}}^{-1}$, as of [44], to ${\tilde{\lambda }}_{\min }^{\left(1\right)}=954.2{\mathrm{cm}}^{-1}$, as of [47], according to different physical conditions (more red-shifted spectra correspond to higher gas temperatures). The value $944.194{\mathrm{cm}}^{-1}$, measured in Ref. [19], corresponds to multiphoton absorption.

If photoabsorption takes place at physical conditions, corresponding to ${\tilde{\lambda }}_{\max }^{\left(1\right)}$, then FWHM, corresponding to 10P(4)-10P(10) emission lines, should be:

$$\mathrm{\Delta }{\tilde{\lambda }}_{\text{FWHM}}^{\left(k\right)}=\left\{-0.26;0.142;1.05;1.77\right\}{\mathrm{cm}}^{-1},$$E18

where 10P4 line (1-st entry) is almost vanishing, and line 10P6(2-nd entry) is very weak, so only line 10P8 can be used for one ($\mathrm{\Delta }{\tilde{\lambda }}_{\text{FWHM}}^{\max }=1.05{\mathrm{cm}}^{-1}$) or two ($\mathrm{\Delta }{\tilde{\lambda }}_{\text{FWHM}}^{\max }=1.31{\mathrm{cm}}^{-1}$) isotopologues excitation. In case, if photoabsorption takes place at physical conditions, corresponding to ${\tilde{\lambda }}_{\min }^{\left(1\right)}$, then FWHM, corresponding to 10P(8)-10P(14) emission lines, should be:

$$\mathrm{\Delta }{\tilde{\lambda }}_{\text{FWHM}}^{\left(k\right)}=\left\{-0.69;-0.19;0.81;1.632\right\}{\mathrm{cm}}^{-1}.$$E19

10P(8)-10P(12) lines can be used for simultaneous excitation of all three most abundant chlorine isotopologues, provided laser medium pressure is 5.75 bars, which corresponds to $\mathrm{\Delta }{\tilde{\lambda }}_{\text{FWHM}}^{\max }=0.81{\mathrm{cm}}^{-1}$ and laser pulse width ${\tau }_p=334\mathrm{ps}$.

4.1 Admissible laser intensity variation range

Absorbed energy, corresponding to pulsed excitation of chlorine isotopologues of ${}^{11}{\mathrm{BCl}}_3$, can be estimated from the formula for excitation rate, deduced in Ref. [39]:

$$k_A\left(t\right)=\frac{1}{2{\pi }^2{R}_L^2}\sum _{k=1}^{N_l}\underset{-\infty }{\overset{\infty }{\int }}\frac{\mathit{d\omega }}{\mathrm{\hslash }\omega }{e}^{\mathit{I\omega t}}\frac{{\mathit{dE}}_A^{\left(k\right)}}{\mathit{d\omega }},$$E20

where it was assumed, that laser beam is pencil-like ($R_L=2.8\mathrm{mm}$ is laser beam radius, corresponding to the thickness of gas flow core with optimal pressure and temperature inside). It can be represented as:

$$k_A\left(t\right)=\frac{1}{\pi R_L^2}\sum _{i=1}^{N_l}\sum _{j=1}^{N_l}{F}^{-1}\left[\frac{d{\mathcal{\gimel }}^{\left(\mathit{ij}\right)}}{\mathit{d\omega }}\right],$$E21

where

$$\frac{d{\mathcal{\gimel }}^{\left(\mathit{ij}\right)}}{\mathit{d\omega }}=\frac{1}{\mathrm{\hslash }\omega }\frac{{\mathit{dE}}_A^{\left(\mathit{ij}\right)}}{\mathit{d\omega }}$$E22

is absorbed photons spectral density. As shown in Ref. [39], laser pulse spectral density can be represented as:

$$\frac{{\mathit{dE}}_A^{\left(\mathit{ij}\right)}}{\mathit{d\omega }}=E_0^{\left(i\right)}{E}_0^{\left(j\right)}{\aleph }^{\left(i\right)}\left(t\right){\sigma }^{\left(j\right)}\left(\omega \right){\mathcal{\daleth }}^{\left(j\right)}\left(\omega \right).$$E23

Let us see what is the average number of resonantly absorbed photons per ${\mathrm{BCl}}_3$ molecule, provided laser pulse intensity $I_p^{\mathit{ref}}$ corresponds to some reference CW-mode intensity $I_{\mathit{CW}}^{\mathit{ref}}=1\mathrm{W}$ and ${\mathrm{BCl}}_3$ absorption spectrum, corresponding to ${\tilde{\lambda }}_{\min }^{\left(1\right)}$, was chosen. According to the formula, derived in Ref. [39], average numbers of photons in pulsed mode, absorbed by each isotopologue excitable by different lines, are:

$$\left\{N_{\mathrm{ph}}^{\left(1\right)},N_{\mathrm{ph}}^{\left(2\right)},N_{\mathrm{ph}}^{\left(3\right)},N_{\mathrm{ph}}^{\left(4\right)}\right\}=\left\{4.5,77.8,0.26,0.57\right\}\times {10}^{-3}.$$E24

It is seen, that majority of photons are absorbed by ${\nu }_2$ line, which is in the closest proximity to 10P8. However, the number of photons, absorbed within the excitation loss characteristic time interval, should be less than $N_{\mathit{\text{crit}}}^{\mathit{max}}=34-39$ [50], in order to avoid multiphoton dissociation. Excitation loss time coincides with gas flow transition time across laser beam, because all other relaxation times, such as dimer formation, vibration-vibrational, and vibration-translational transfers as well as spontaneous emission, are bigger [38]. Thus, the upper limit for laser intensity should be

$$I_{\max }=I_{\mathrm{ref}}\frac{N_{\mathit{\text{crit}}}^{\mathit{max}}}{\mathit{Max}\left\{N_{\mathrm{ph}}^{\left(k\right)}\right\}}=437-501\mathrm{W}.$$E25

The lower limit for applicability of SILARC method is, apparently, given by the following condition:

$$I_{\min }=\frac{I_{\mathrm{ref}}}{\mathit{Max}\left\{N_{\mathrm{ph}}^{\left(k\right)}\right\}}=12.8\mathrm{W}.$$E26

In the case, when ${\mathrm{BCl}}_3$ absorption spectrum, corresponding to ${\tilde{\lambda }}_{\max }^{\left(1\right)}$, was chosen, the average number of photons, absorbed by each absorption line, from 10P8 emission line are:

$$\left\{N_{\mathrm{ph}}^{\left(1\right)},N_{\mathrm{ph}}^{\left(2\right)},N_{\mathrm{ph}}^{\left(3\right)},N_{\mathrm{ph}}^{\left(4\right)}\right\}=\left\{0.02,0.1,0.41,0.29\right\}\times {10}^{-3}.$$E27

Thus, the upper and lower limits for laser intensity should be $I_{\max }=83.2-94.4\mathrm{kW},I_{\min }=2.45\mathrm{kW}.$

5. Gas flow properties and vacuum system design

Apparently, the main goal in vacuum system design is to provide the largest overlap of laser beam with gas flow core, having optimal pressure and temperature for isotope separation. This condition in the case of cross-wise irradiation can be fulfilled, provided gas flow is planar (interaction region with cross-wise directed laser beam is large), and preserves its shape over all its extension. This can be implemented, if the gas flow is perfectly expanded. Thus, the vacuum chamber pumping down speed should be carefully chosen. The pumping out speed is provided by two vacuum pumps. High-rate one evacuates the central part of the gas flow, and low-rate one evacuates the peripheral gas flow. The main gas flow stream is divided into two parts by the wedge-like inlet of the skimmer chamber, which is placed on the opposite side of the irradiation cell.

5.1 Nozzle design

Nozzle wall shape was found as a friction-free(isentropic) region corrected by the boundary layer. According to experimental data from Ref. [51], gas flow deceleration(full pressure drop) across the nozzle practically does not depend on angle of nozzle inlet, if ${\theta }_0<{85}^0$. The larger radius of the nozzle throat curvature $R_2$, the larger gas flow uniformity, and, therefore, the smaller deceleration. According to boundary layer theory, in order to take into account dissipative effects, nozzle contour, corresponding to isentropic gas flow expansion, should be corrected by displacement layer thickness (DLT).

In order to figure out which gas flow regime takes place-laminar or turbulent, one needs to know Reynolds number evolution along the nozzle axis. Since boundary layers from opposite walls do not interfere with each other, Reynolds number to characterize the transition from laminar to turbulent flow can be introduced as:

$${\mathit{Re}}_x=\frac{\mathit{\rho vx}}{\mu }.$$E28

If ${\mathit{Re}}_x>5\times {10}^5$, then gas flow gets certainly turbulent for most commercial surfaces. Therefore, for our nozzle design, it should be definitely stable, since ${\mathit{Re}}_x\left(X_{\mathit{fin}}\right)=2615$.

The boundary layer corrected nozzle profile for argon used as a carrier gas and break down parameter is shown in Figure 2a.

Figure 2.

(a) Cross-section of nozzle profile in zx plane (gas flow temperature at the nozzle outlet along its axis (it’s minimal value) corresponds to optimal value for boron isotopes separation) and (b) break down parameter distribution along the nozzle axis for boron isotopes separation.

Calculations of continuous gas flow are only valid until breakdown parameter $B=M\sqrt{\frac{\pi \overline{\gamma }}{8}}\frac{\lambda }{\rho }\left|\frac{\mathit{d\rho }}{\mathit{dz}}\right|$ is lower than its critical value $B_{\mathit{\text{crit}}}=0.05$. As seen in Figure 2b, gas flow departs from continuous regime practically overall nozzle extension. Therefore, caution must be exercised relative to obtained results on nozzle profile.

5.2 The model for estimation of minimal requirements for vacuuming rate

Influence of gas flow deceleration, caused by interaction with ambient gas and irradiation chamber-vacuum pump inlet tract, on temperature distribution in gas flow can be tolerated, if pressure builds up at vacuum pump inlet still provides an acceptable variation of static pressure for isotope separation in the discharge chamber. This condition can be fulfilled if vacuum tract resistance to the gas flow is reduced to a minimum and pumping out speed is accurately chosen. As well known, during gas flow transition across vacuum tract, gas pressure rises due to friction with walls, or, in the case of gas flow injected in the pipe of much larger diameter, with ambient gas, due to shock waves caused by hydrodynamic discontinuities.

Let us consider subsequent stages of gas flow evolution. In the beginning, gas chamber is pumped down until some pressure level, which is smaller than the pressure at nozzle outlet. In this case, gas flow gets under-expanded, and occupies the space between irradiation chamber walls, according to pressure of residual gas. On the next stage, ambient gas pressure increases accordingly to pumping down rate applied, so that angle between gas flow direction and oblique shock at nozzle outlet decreases. Since we need perfectly expanded gas flow, vacuum pump characteristics should be such that this angle vanishes after some time and then does not change.

As pointed out in Ref. [36], the large decrease in stagnation pressure across a normal shock wave may be reduced by decelerating the free-stream flow by means of one or more oblique shock waves, followed by a weak normal shock wave, produced by central body, placed at skimmer inlet. By employing that principle, an efficient external compression of the gas flow is achieved before the gas flows into a subsonic internal compression diffuser (skimmer partition of discharge chamber). Larger stagnation pressure recoveries are obtainable if several successive weak shock waves instead of one relatively strong conical shock wave are utilized for decelerating the gas flow.

To find gas density, one needs to solve a system of material, momentum, and energy conservation equations. For sake of simplicity, this system of equations can be transformed, according to hydraulic approximation, to the system of balance equations, where taking an average of conservation equations is carried out over cross sections at the characteristic intermediate stations $i$: The first station($i=1$) corresponds to inlets of the pipes, connected to the port of rim or core gas flow evacuating vacuum pumps, the second one($i=2$) corresponds to core gas flow evacuating vacuum pump inlet or to two pipe into one pipe connection, assigned for rim gas flow evacuation, the third one($i=3$) corresponds to rim gas flow evacuating vacuum pump inlet. This system of balance equations is following:

$$\left\{\begin{array}{l}\frac{d{\mathcal{N}}^{a;i}}{\mathit{dt}}+{\rho }_{\mathit{\text{in}}}^{a;i}{v}_{\mathit{\text{in}}}^{a;i}{A}_{\mathit{\text{in}}}^{a;i}={\rho }_{\mathit{\text{out}}}^{a;i}{v}_{\mathit{\text{out}}}^{a;i}{A}_{\mathit{\text{out}}}^{a;i},\\ \frac{d{\mathcal{P}}^{a;i}}{\mathit{dt}}+A_{\mathit{\text{in}}}^{a;i}\left(P_{\mathit{\text{in}}}^{a;i}+m_0{\rho }_{\mathit{\text{in}}}^{a;i}{\left[v_{\mathit{\text{in}}}^{a;i}\right]}^2\right)+\delta P^{a;i}=A_{\mathit{\text{out}}}^{a;i}\left(P_{\mathit{\text{out}}}^{a;i}+m_0{\rho }_{\mathit{\text{out}}}^{a;i}{\left[v_{\mathit{\text{out}}}^{a;i}\right]}^2\right),\\ \frac{d{\mathrm{ℰ}}^{a;i}}{\mathit{dt}}+{\rho }_{\mathit{\text{in}}}^{a;i}{v}_{\mathit{\text{in}}}^{a;i}{A}_{\mathit{\text{in}}}^{a;i}\left(\frac{m_0{\left[v_{\mathit{\text{in}}}^{a;i}\right]}^2}{2}+C_p{T}_{\mathit{\text{in}}}^{a;i}\right)+\delta Q^{a;i}={\rho }_{\mathit{\text{out}}}^{a;i}{v}_{\mathit{\text{out}}}^{a;i}{A}_{\mathit{\text{out}}}^{a;i}\left(\frac{m_0{\left[v_{\mathit{\text{out}}}^{a;i}\right]}^2}{2}+C_p{T}_{\mathit{\text{out}}}^{a;i}\right),\\ P_{\mathit{\text{out}}}^{a;i}={\rho }_{\mathit{\text{out}}}^{a;i}{k}_B{T}_{\mathit{\text{out}}}^{a;i}.\end{array}\right.$$E29

In order to close this system of equations, it should be supplemented by the ideal gas equation of state

$$P_a^{\mathit{\text{out}};i}={\rho }_a^{\mathit{\text{out}};i}{k}_B{T}_a^{\mathit{\text{out}};i},$$E30

and by equation for friction coefficient $f_a^i$. We assume that gas friction with pipe wall surface is laminar

$${\mathit{Re}}_D^{a;i}=\frac{{\rho }_a^{\mathit{\text{out}};i}{v}_a^{\mathit{\text{out}};i}{D}_i}{\mu }<2300,$$E31

where $D_i$ is pipe diameter. Hence, the friction coefficient can be estimated as:

$$f_a^i=\frac{64}{{\mathit{Re}}_D^{a;i}}.$$E32

The total number of molecules in rim and core gas flow fractions, accumulated over its transition time across the irradiation chamber are:

$$Q_{\mathit{\text{core}}}=\left(1-\mathit{\mu \theta }\right)Q_f$$E33

and

$$Q_{\mathit{rim}}=\mathit{\mu \theta }{Q}_f$$E34

respectively, where $\mu =0.02$ is target gas(${\mathrm{BCl}}_3$) molar fraction. Product cut, or fraction of molecules escaped the gas flow core, was calculated on the basis of the transport model, developed in Ref. [26]. It’s value, corresponding to optimal conditions $T_a^{\mathit{\text{in}};0}=25\mathrm{K},P_a^{\mathit{\text{in}};0}=10\text{mtorr}$, calculated in Ref. [38], is $\theta \approx 0.2$.

Heat transfer intensity can be phenomenologically approximated as:

$$q=h\mid T_w-T\mid ,$$E35

where wall temperature is fixed at $T_w=T_{\mathit{atm}}=300\mathrm{K}$, and $T$ is average temperature over gas flow volume. A coefficient of proportionality $h$ is introduced as $h=\rho {\mathit{vC}}_p{N}_{\mathit{St}},$ where Stanton number $N_{\mathit{St}}$ is related to momentum transfer due to friction by Reynolds analogy, which, however, yields satisfactory results only for gases:

$$N_{\mathit{St}}=\frac{1}{2}\mathit{Prf}$$E36

Friction factor $f$ is introduced as:

$$f=\frac{\mathit{\text{wall shear stress}}}{\frac{1}{2}\rho v^2}.$$E37

The total heat transfer can be approximated as:

$$\delta Q_a^i=\frac{{\overline{f}}_a^i}{2}\mathit{Pr}\left(T\right){\rho }_a{v}_a{C}_p\mid T_w-T\mid S_a.$$E38

Average values are approximated as $\overline{o}=\frac{o_{\mathit{\text{in}}}+o_{\mathit{\text{out}}}}{2}.$ We assume, that momentum transfer to the wall surface can be approximated by the formula for gas flow pressure build-up due to friction over planar surface:

$$\delta P_a^i=\frac{f_a^i}{8}{m}_0{\overline{\rho }}_a{{\overline{v}}_a}^2{S}_a$$E39

Since the efficiency of pumping down speed depends on the gas density at the vacuum pump inlet, one needs to know which model of vacuum pump is able to provide the required pumping out speed. Then, given that what gas is pumped down and provided it’s hold at room temperature, this dependence can be transformed into the following relation to inlet density:

$$\frac{{\mathit{dU}}^a}{\mathit{dt}}=\kappa f_{\mathit{\text{pump}}}\left({\rho }^a\right),$$E40

where factor $\kappa$ fits reference vacuum pump interpolated dependence on gas density to desired pumping down speed at a given density. We have used pressure dependence of pumping speed of the turbo-molecular pump (TMP) TURBOVAC MAG 3200C [52], as a reference, which is shown in Figure 3a, where different gases at room temperature are pumped down. Thus, the performance curve for pumping out of the air can be taken from interpolation:

Figure 3.

(a) TURBOVAC MAG 3200C pumping speed as a function of inlet gas pressure, [52], (b) Core gas flow density evolution, provided TMP-3403LMC with flange VG300 is used, [53], and (c) rim gas flow density evolution, provided HiPace 300 DN100 is used.

$$f_{\mathit{\text{pump}}}=3.07-3.09\times {10}^{-21}\rho \left(t\right)+9.5\times {10}^{-43}\rho {\left(t\right)}^2.$$E41

Moreover, it can be estimated pressure evolution within the pumped-down vessel. According to the Reynolds transport theorem, particle balance condition leads to the following equation for average gas density ${\rho }_a$ in the inter-chamber connector:

$$\left\{\begin{array}{l}{V}_{\mathit{ch}}\frac{d{\rho }_a\left(t\right)}{\mathit{dt}}=\frac{{\mathit{dN}}_a}{\mathit{dt}}-{\rho }_a\left(t\right)\frac{{\mathit{dU}}_a}{\mathit{dt}},\\ {\rho }_a\left(0\right)={\rho }_{\mathit{\text{in}}},\\ {\rho }_{\mathit{\text{in}}}=\frac{p_{\mathit{\text{in}}}}{k_B{T}_0}.\end{array}\right.$$E42

The vacuum pump should provide such pumping speed, that asymptotic gas density ${\rho }_{\mathit{as}}=\underset{t\to \infty }{lim}\rho \left(t\right)$ is within the range:

$${\rho }_{\mathit{\text{out}}}<{\rho }_{\mathit{as}}<{\rho }_{\mathit{\text{in}}},$$E43

where $T_0$ is room temperature.

Solution of the balance equation at the initial condition, that stagnant gas resides in the feed chamber at room temperature and pressure 10 mtorr, for core gas flow is shown in Figure 3b and for rim gas in Figure 3c. It is seen from this figure, that after rather short period of time stationary condition is reached (it depends on the volume of intermediate chamber). Pumping speeds were chosen from the condition that asymptotic density is ranged as (43).

Inter-chamber connector geometry and pipe length and diameter for core and rim gas flows respectively should be chosen so, that separated species are not driven due to high pressure back into irradiation chamber: $P_{\mathit{\text{out}}}^a<P_{\mathit{\text{in}}}$. We have the following solutions of the system of Eqs. (29)$P_{\mathit{\text{out}}}^{\mathit{\text{core}}}=0.27\mathrm{Pa},$$T_{\mathit{\text{out}}}^{\mathit{\text{core}}}=187.7\mathrm{K},$$v_{\mathit{\text{out}}}^{\mathit{\text{core}}}=44.68\mathrm{m}/\mathrm{s},$${\rho }_{\mathit{\text{out}}}^{\mathit{\text{core}}}=3.34\times {10}^{19}{\mathrm{m}}^{-3},$$P_{\mathit{\text{out}}}^{\mathit{rim}}=1.95\mathrm{Pa},$$T_{\mathit{\text{out}}}^{\mathit{rim}}=96.43\mathrm{K},$$v_{\mathit{\text{out}}}^{\mathit{rim}}=11.32\mathrm{m}/\mathrm{s},$${\rho }_{\mathit{\text{out}}}^{\mathit{rim}}=7.49\times {10}^{20}{\mathrm{m}}^{-3}$, corresponding to given geometry for core and rim gas flows. For core and rim gas flows one obtains ${\mathit{Re}}_D^{\mathit{\text{core}};1}=69$ and ${\mathit{Re}}_D^{\mathit{rim},1}=37$ , respectively. Thus, $f_{\mathit{rim}}^1=0.3$ and $f_{\mathit{\text{core}}}^1=0.2$.

Thus, we obtain the following values for pumping down speeds for core and rim gas flows $\frac{{\mathit{dU}}_{\mathit{\text{core}}}}{\mathit{dt}}=3300\mathrm{l}/\mathrm{s},\frac{{\mathit{dU}}_{\mathit{rim}}}{\mathit{dt}}=310\mathrm{l}/\mathrm{s}$.

Since the system of Eqs. (29) along with approximation for heat transfer (38), momentum loss (39), and friction coefficient (37) is valid only if gas flow is continuous along the wall surface and shock-free, while in real situations gas flow is rather injected in the pipe due to its significantly smaller cross section, comparing to the pipe area, especially in the case of rim gas flow, found pipe geometry is expected to be just a lower limit (pipe length can be larger), the same concerns to found pumping speeds for rim and core gas flows evacuation, which in turn can serve as an upper limit for pumping speed requirements.

6. Conclusions

In this chapter, physical constraints such as gas flow core temperature and pressure distributions, oblique shocks configuration in the course of gas evacuation from discharge chamber, condensation rate, pressure in ${\mathrm{CO}}_2$ laser medium, and spectral properties of ${\mathrm{BCl}}_3$ at different temperatures and pressures on the efficiency of SILARC method have been discussed. In particular, it has demonstrated the possiblity to extract boron isotopes from ${\mathrm{BCl}}_3$ gas flow more efficiently by applying a three-mirros scheme for multiline excitation. Recommendations on laser system design, optimal nozzle design, and parameters of the corresponding vacuuming system are given.