What is uranium enrichment with laser? Can it be used for peaceful development or is it only for military use? What are its technical solutions and technology?

Section A: Possible Lasers for Third Generation Laser Enrichment For any laser to be usable with third generation laser enrichment technology, it must emit light at either 16 or 5.3 µm. There are other performance characteristics, however, that must be considered when assessing the effectiveness some laser systems, and how accessible certain modifications to these systems may be to make them usable. These include the laser’s pulse repetition rate, the linewidth of the emitted pulses, how easily the central peak of the emitted pulse may be tuned to the desired wavelength, the duration of the pulse, and the energy fluence in each pulse. While there may be certain challenges with different systems in achieving what is considered ideal for uranium enrichment, some level of enrichment is still possible even if not all ideal performance characteristics are met, and cascading to 90 percent HEU may still be desirable if either obtaining more lasers, spending more money, using more space, or taking more time is possible and tolerable for the proliferator.Raman-shifted Transversely-Excited Atmospheric (TEA) CO2 Laser at 16 µm It is widely speculated that a Raman-shifted TEA CO2 laser is to be employed with SILEX. Such lasers operate by applying a pulsed RF discharge transversely across the resonator tube containing CO2 gas at a pressure of at least 1 atm, but it is more likely that a higher pressure (∼ 5−8 atm) is necessary for successful SILEX operation. What is necessary to understand, however, is the equipment and performance characteristics needed to generate pulses of ∼ 10 mJ (see section “Laser performance characteristics” in the main section of the paper) that allow for high selectivity (narrow laser linewidth) at the precisely tuned wavelength: 15.916 µm for the ν3 mode of 235UF6 and only slightly different at 15.931 µm for 238UF6. Given that the vibrational and rotational modes of a CO2 molecule emit light between 9−11 µm, CO2 laser light must be down-shifted to the 235UF6 ν3 line by Raman scattering. Raman Scattering Raman scattering is defined as the inelastic scattering of photons, and therefore involves either giving up (Stokes wave) or acquiring more (anti-Stokes wave) energy. For SILEX, a Stokes wave must be created to allow the generation of 15.916 µm light from a CO2 laser. In this case, Raman scattering involves the pumping of para-hydrogen (p-H2) gas to a virtual state with the decay to an excited rotational state, S(0) at 354.33 cm−1 . With this quantum of energy absorbed, an initial pumping wavelength of 10.177 µm is necessary so that exactly 15.916 µm is emitted (scattered) when the S(0) level is left excited.1 In identifying proliferation risks, the question is how a Stokes wave of 15.916 µm light is obtained.A Raman-shifting cell with hydrogen gas will have a higher fraction of p-H2 (spins oppositely aligned) at cooler temperatures. A cell cooled with liquid nitrogen (77 K) will have more p-H2 than if it was left at room temperature (300 K), and using liquid H2 (20 K) would have even more. The tradeoff is that using cooler temperatures is more expensive and marginally more complex, but that a higher threshold power per pulse is necessary at higher temperatures to generate any Raman scattered light. Once the required threshold is met, light can be passed multiple times through a Raman-shifter cell to amplify the gain by stimulated emission. W. R. Trutna and R. L. Byer2 calculated that the Raman gain coefficient α for each pass through a shifter cell of the Stokes wave Ps0 (amplified according to the expression Ps = Ps0 exp(α)) can be expressed as α = 4PpG λp + λs tan−1 ( L b ) (A.1) where PP is the pump power, L is the length of the cell (taken to be 3.77 m in this analysis), λp and λs are the respective pump and Stokes wavelengths, and b is a confocal parameter defined by b = 2πω2 p0/λp (A.2) where ω 2 p0 is the minimum pump spot size at the focus (usually referred to as the beam waist). G is the Stokes plane-wave gain coefficient of the pump intensity Ip and is given by G = 4λ 2 s∆N n2 sℏωp∆ωR ( dσ dΩ ) (A.3) where ∆N in the population density, ns is the index of refraction at the Stokes wavelength λs, ωp in the pump angular frequency, ∆ωR is the linewidth at the full width at half-maxium, and dσ/dΩ is the differential cross-section for Raman scattering from the S(0) state of p-H2 by CO2 laser pump photons. The use of circularly polarized light will increase the Raman cross section by 50 percent by suppressing anti-Stokes emission.3 A more useful expression for G that is proportional to the Stokes frequency ωs ∝ λ −1 s is G = 2ωsχ ′′ R nsnpc 2ϵ0 (A.4) where χ ′′ R is the on-resonance Raman susceptibility (related to the cross section), and ns and np are the indices of refraction of the Stokes and pump frequencies ωs and ωp. 4 If the mirror reflectance inside the Raman-shifting cell is R, the net gain after n transits through the cell is Ps0/Ps = R n exp[α(1 + R + R 2 + ...Rn )] = exp(αn) (A.5) where αn = nα = α ( 1 − Rn 1 − R ) + ln R. (A.6) Thus the higher R is for mirrors used in a multiple-pass cell, the higher the net gain Ps0/Ps will be of the emitted Stokes wave.5 The threshold pumping power is defined as the gain required to amplify spontaneous Stokes power to the 1-kW level,6 and the first proliferation concern is what is required to make this possible at a temperature 300 K. Spontaneous here refers to emission due to the Heisenberg uncertainty principle, ∆E∆t ≥ ℏ 2 . Trutna and Byer’s model7 is used to first calculate that the Raman gain required for Stokes threshold generation is αn = 44. Scaling the gain coefficient G(cell pressure = 3 atm) = 0.5 × 10−3 cm/MW for a pump wavelength of 1.064 µm in Equation A.4 (proportional to λ −1 s ) to 10.6 µm8 gives G(3 atm) = 3.4 × 10−5 cm/MW. At a pumping power Pp = 1 MW, the power gain coefficient per pass in a Raman cell is α = 0.08. For a 25-transit cell with 20 effective passes9 the net gain coefficient is αn = 1.60/MW. The pump power to achieve a threshold (α = 44) is then 27.5 MW or 1.9 J in 70 ns. This should be considered only an initial CO2 laser requirement without any advanced techniques to achieve a threshold 1 kW output at 15.916 µm. This pump power can be decreased by a factor of 1.6 from 27.5 MW if mirrors with R = 99 percent are used and 40 passes (effectively 33 for R = 100 percent) are made through the Raman cell. Circularly polarized light will reduce the pump power by a factor of 1.5, and cooling the p-H2 with liquid N2 (77 K) allows for an additional reduction factor of 2.4. These three techniques reduce the pump power by a factor of 5.76, or to a moderate resultant pump power of 4.7 MW (0.3 J for a 70 ns pulse). This should be interpreted as follows: if a minimum of 4.7 MW of peak power in a pulse from a CO2 laser is possible, at least 1 kW of Raman-shifted light is emitted. To produce a 15.916 µm pulse of at least this power, Raman scattering a CO2 laser pulse with 1 J of energy with a pulsed duration of 200 ns would accomplish it. If the pulse durations are higher than 200 ns, the CO2 lasers are not export controlled by the Nuclear Suppliers Group,10 but such lasers with pulses > 200 ns will still enrich uranium by the SILEX process as long as the energy per pulse increases accordingly to keep the peak power at 4.7 MW. This is why lasers with short pulses are preferred: the peak pulse power is easier to obtain. A summary of these possible adjustments and factors by which the threshold power would be reduced are provided in Table A.1. The use of other Raman-shifting techniques such as four-wave mixing with a 1.06 µm Nd:YAG laser12 or a Stokes seed laser13 would require an even smaller threshold pump power, as would cooling the p-H2 down to 20 K with LH2. This would have the effect of making other laser systems with less advanced designs or less burdensome combinations (power per pulse < 4.7 MW) capable for enrichment by SILEX. The question for a proliferator would be what lasers do they know about and how accessible are they given their knowledge about how to control the required pump power to produce 15.916 µm photons. Of potential serious concern in the future in the use of a quantum cascade laser (see later section) at this wavelength as a seed laser, which would drastically lower the required threshold pump power and greatly increase the number and kind of lasers for possible SILEX use, as well as the number of potential scientists and engineers who could be enlisted in the effort. A seed laser would be capable of easy power amplification, as its initial power in a Raman cell would be much greater than that from only spontaneous emission.Transversely Excited Atmospheric (TEA) CO2 Laser Transversely-Excited Atmospheric (TEA) lasers at high pressures have advantages over other systems because they are continuously tunable over a wide range of wavelengths and have very fast rise and fall times that give pulses of short duration. High pressures also make high power possible,but what is important is the peak pulse power for reaching the required Raman threshold, not the average power over a longer time interval. It is this feature which seems to be the dominant advantage over other lasers, as one of the problems discussed with TEA CO2 lasers is that the pulse repetition rate is low (about 1 kHz in commercially available lasers, but reported as high as 2 kHz),14 which leaves a high fraction of UF6 in the feed stream unirradiated. Other lasers have much higher repetition rates, but lack the capability to reach the peak power required for Raman scattering. In addition, while low repetition rates are an issue if desiring to keep capital costs low in a commercialization project, it is not a significant technical challenge for a proliferator willing to tolerate higher costs to interleave pulses from multiple lasers to increase the repetition rate. Numerous techniques are available to interleave pulses from multiple lasers,15 as well as newly developed techniques utilizing one laser with interleaver chips to multiply the repetition rate by several factors.16 The high running pressures (∼ 5−8 atm), however, come with perhaps the greatest technical challenge in SILEX operation, which is managing the stability of the high pressure gas as it is being pulsed at high frequencies.17 The design and manufacture of a pressure vessel requires special codes, very high electrical voltages are needed for the discharge to be triggered (∼ 10 kV/cm electrode gap per atmospheric pressure), the laser gas mixture must be transported through the cavity for high repetition frequencies, and the peak energies inside the laser cavity make the lifetime of the partial reflector relatively short.18 Such challenges seem to the author more of a technical obstacle than tuning or line-narrowing the beam, which should be possible by anyone skilled with lasers and the knowledge of diffraction gratings, etalons, piezo-electric mirrors, and many other widely published and documented techniques. Some of these techniques must be used to obtain the precise wavelength before Raman scattering, as the output from a high-pressure TEA laser likely needs tuning from its emitted wavelength.19 High pressures make this process easier due to the overlap of the pressure-broadened transitions over four ranges (10P, 10R, 9P, and 9R) between 9.2−10.8 µm.20 Commercially available TEA CO2 lasers advertise pulse repetition rates of 1 kHz, but they can be designed and sold commercially at higher ones. Such lasers also advertise a very stable resonator cavity that does not require any adjustment after the initial alignment, lessening the most challenging technical burden in the opinion of the author. For one commercially available TEA CO2 laser with a 1−2 kHz repetition rate, the author has been quoted a price by a supplier between $200,000−$250,000. For 500 Hz repetition rate laser, the price would still likely exceed $100,000. These lasers also have pulse energies and durations that exceed the threshold pump power for Raman scattering, assuming that widely reported and accessible techniques are applied. If 2 kHz repetition rates are possible, assuming the highest performance in the public domain claimed in 1991,21 and if the fractions of 235UF6 already dimerized along sections of an expanding free jet were not intolerably high prior to laser irradiation, three mirror bounces could increase a 2 kHz rate to 8 kHz. Adding three more lasers in a similar fashion with interleaved pulses would create a 32 kHz repetition rate, more than the 30 kHz rate needed to irradiate all uranium. Such an arrangement would likely need to involve a number of beam telescopes (two concave lenses) to maintain a desirable amount of beam collimation along the depth of one product stream in a three-up, two-down cascade to 90 percent HEU (see Section C of this online supplement, Figure C.1). Yet with only three such streams, a total of 12 lasers would be required that today are commercially available. At $250,000 per laser, the total laser price tag would be $3,000,000. If it was tolerable to use twice as much time to cascade to 90 percent HEU, only 6 such lasers would be needed to acquire the same material. If dimer formation was too high in certain irradiated areas of the free jet, more lasers could be added, or perhaps a nozzle with a longer neck could be designed There exist many websites that describe in great detail how to build TEA lasers. This would obviously come with the challenges of managing the gas discharge, but if this is manageable, pulsing at a higher frequency to generate a repetition rate higher than 1 kHz is possible. It appears that around ∼ 2 kHz appears to be the optimum rate to balance capital and operating costs,22 but more advanced technical skills could design higher performing lasers than what is commercially available, and there is information online that discusses the engineering. The challenges of high pressure gas management may even be lessened by using different isotopic mixtures of CO2 so that the laser remains highly tunable at lower pressures.23 If the main technical challenge is managing the high pressure gas, such a technique may be worth utilizing even if it requires more lasers to reach the desired performance characteristics. The risk with this technology it that there appear to be lots of options that a determined proliferator with advanced technical skills could choose for indigenous laser construction. There are a number of applications with TEA lasers that may complicate identifying enrichment activities. These applications include non-destructive testing (NDT) of materials, light detection and ranging (LIDAR), differential absorption and ranging LIDAR (DIAL) to measure the concentration of gases, extreme ultraviolet (EUV) generation, many kinds of laser marking (a broad category that includes laser engraving), pulse amplification, high energy physics, and pump sources for spectroscopy.24 These applications not only complicate identifying the intended purpose of a purchaser, but they allow knowledge about the workings of TEA lasers to spread at a rate that is proportional to the number of different applications and the scale of their use.Other Pulsed CO2 Lasers There are other CO2 lasers that pose proliferation risks, as their performance capabilities make enrichment by the SILEX process possible. These lasers would still need to be Raman-shifted, however, as the output of a CO2 laser is between 9−11 µm.25 An adequate rule of thumb provided in the section above on Raman scattering is that CO2 laser pulses need to reach a power threshold of 4.7 MW (assuming LN2 cooling to 77 K , circular polarization, 3 atm cell pressure, and mirrors of 99 percent reflectivity) to obtain peak pulse powers of 1 kW for Raman-shifted light. This is more easily accomplished with shorter pulses, and advancements and techniques that do shorten pulse times if applied to CO2 lasers should be monitored. However, what ultimately matters is whether 4.7 MW is attainable and if a proliferator will tolerate combinations of currently available lasers to attain this threshold. There exist today commercially available sealed, CO2 lasers that do not require the external gas flow of TEA lasers. There is also no challenging management of high-pressure gas. These lasers are available at around 10.2 µm, and could be tuneable to 10.177 µm at the operating pressures of 150 torr using techniques known by many scientists who work with lasers. Line narrowing the output to the desired bandwidth should be possible for those familiar with lasers as well. These lasers can emit greater than 750 W of average power at a range of repetition rates up to 200 kHz using a pulsed RF discharge with pulse durations between 2 and 1000 µs. It would still take a large number of lasers to reach 4.7 MW threshold power per pulse at a 10 kHz repetition rate, but suppose the threshold power was decreased to 2 MW with more cooling, four-wave mixing, or a seed laser (possibly a QCL at 15.91µm). Under this threshold, if lasers were ever developed that produced 4.5 kW of average power with pulse durations of 50 ns at a 10 kHz repetition rate, only about 9 of these lasers would be needed to produce in excess of a SQ of 90 percent HEU in one year in a clandestine facility as described in Figure 3 of the paper’s main section.26 This is only ∼ 6 times the power output of commercially available lasers now. If pulse durations were shortened by a factor of 40 from 2 µs to 50 ns, then 54 lasers would be needed in a clandestine facility if each emitted 750 W. The author is unaware of the performance limits that physical constraints place on these systems, but the potential for future developments in this area would require only the technical expertise of someone skilled with lasers for help in using the SILEX process. Currently, unless a proliferator wishes to use hundreds of commercially available low-pressure gas lasers, a scientist skilled in the use of managing high gas pressures would be useful to increase the pulse peak power closer to the 4.7 MW threshold. There are numerous applications for these commercially available lasers, but if the pulse durations are longer than 200 ns, they are not export controlled. In the opinion of the author, this should be changed. If a laser had 1.5 kW of average power with a 1 kHz repetition rate and pulses of 200 ns in duration, its uranium enrichment capabilities would exceed that of all commercially available TEA CO2 lasers. The author is not aware of such a laser, but is also unaware of what limitations exist in such a system being constructed with low pressure gases (100−150 torr). This is only a factor of 2 higher in average power and a factor of 10 shorter in pulse duration than what can be purchased today. It remains important to be aware that combining the powers of multiple laser beams is possible and should be considered accessible by anyone skilled in working with lasers and optics.27 The risk that a proliferator may attempt such techniques with numerous lasers depends upon the tolerance of the proliferator. What is required is determination to be successful, not sophisticated technical training measured by the types of academic degrees conferred upon the proliferators or a similar assessment of the country they are from.Carbon Monoxide (CO) Laser at 5.3 µm There are suggestions that a carbon monoxide (CO) laser should replace the CO2 Ramanshifted system.28 The advantages of this choice include the capability of irradiating all UF6 in a cross-axial free jet29 and a 1.8 cm−1 isotope shift with the 3ν3 vibrational band that is three times greater than that from the ν3 transition. This could allow for higher selectivity depending upon the narrowing of the ν3 absorption bands or the laser linewidths of the 16 µm system. In addition, with 5.3 µm photons having three times the energy of 16 µm light, three times more collisions with 238 UF6 are needed to deexcite the 3ν3 mode, and this allows the CO laser to irradiate at a higher temperature (∼ 150 K) where the fraction 235UF6 monomers found in dimers is very small and more uranium is accessible per laser pulse. The main challenge with a CO laser, however, is that the cross section for 3ν3 excitation is ∼ 10−22 cm2 , or roughly 5,000−10,000 times smaller than for the ν3 mode. This requires that this many more photons be available per unit area, or in each pulse if its duration is short.30 There are reportedly two ways to improve this small cross section: a continuous-wave (CW) CO laser and a mode-locked CO laser that uses a RF discharge in a supersonic stream. Both employ intracavity laser irradiation to more efficiently use energy and thereby compensate for the small 3ν3 cross section. Both systems will use a very long separation unit length (14−15 m)31 to limit the disadvantage from the small cross section (the author suggested 10 m with the 16 µm system), and bidirectional mirrors to redirect light back and forth for multiple passes with additional stimulated emission and amplification from the CO medium. This is not possible with the 16 µm system due to the Raman cell. In addition, the CO laser light is supposedly easier to keep collimated over long distances with the creation of Bessel waves from large radius end mirrors in the laser cavity.32 This is more challenging with 16 µm light due to the diffraction-limited effects at longer wavelengths, but constraints exist at all wavelengths. Continuous-Wave (CW) CO Laser at 5.3 µm A continuous-wave (CW) CO laser is pumped with an electric discharge that allows for the excitation of vibrational energy levels followed by their relaxation and stimulated emission of a spectrum over a broad range of wavelengths. A diffraction grating must be set at an angle to tune the laser light to a band that overlaps with the 3ν3 transition of UF6, and resonator mirrors with very high reflectivity must be used to limit cavity losses. For a cavity 15 m in length, a commercially available CW CO laser emitting 100 W of power would provide 66 kW/cm2 when adding together the multiple passes for a free jet through a 1-cm beam diameter.33 It is claimed that such lasers tuned to a single line (5.3 µm) can deliver 3 to 10 kW/cm2 . 34 The challenge with this design is the potential losses on the mirrors and diffraction grating, but these are skills most scientists knowledgable about lasers have or at least understand conceptually and could acquire. The efficiency is certainly limited by the need to operate on a single wavelength, so competing economically with the 16 µm system seems challenging. This depends upon the electro-optical efficiency and design of the Raman-shifting system (between 8 percent−14 percent total efficiency by the author’s calculation), but single wavelength operation with a CW CO laser should not expect an efficiency of more than a few percent.35 The most important concern regarding proliferation, however, is whether this system is accessible to a proliferator and not whether it can be operated at a lower cost. The design does seem simpler than the 16 µm system (fewer components), and the electronics controlling the system are almost certainly simpler because the pressure of the gas mixture is low compared to a TEA CO2 laser and there is no pulsed RF discharge or Q-switching. There may be more advanced tuning required with mirrors in the CW CO system, but this system seems to require fewer advanced skills and less sophisticated technical knowledge given that a Raman-shifting system is the equivalent of an additional laser. Knowing that these systems were sold to Iran in the 1970’s to help with laser enrichment,36 they should be considered a proliferation risk. In addition, CW CO lasers are not listed on the Nuclear Suppliers Group list of controlled equipment. Only CO lasers with repetition rates greater than 250 Hz appear.37 Broad spectrum CW lasers that are commercially available for around $100,00038 do have to be tuned to a single wavelength, but given the widespread knowledge about how to do this, this omission appears to be a serious oversight. Pulsed CO Laser at 5.3 µm There is a suggestion that a pulsed CO laser could perform more efficiently than both the 16 µm Raman-shifted and a CW CO lasers.39 Such a system would easily irradiate all uranium at a 10 MHz repetition rate, and would use a pulsed electric discharge designed to use energy as efficiently as a free-running CW CO laser that emits light within an interval of ∼ 4.9-6 µm. With the separation unit located between two bidirectional mirrors, 10 MHz pulses are continually propagating in both directions through the free jet containing UF6. For light traveling at 3 × 108 m/s in a cavity 15 m long, each pulse will pass the same location along the separation unit on average every 5 × 10−8 s. For UF6 traveling at 3×104 cm/s, ∼ 110K pulses will irradiate all UF6 molecules traveling through a 1 cm laser diameter. For a beam with 3 kW of average power with 3 ns pulses,this is 33 J of energy available to excite the 3ν3 mode. Compared to the ∼ 10 mJ required for the ν3 mode, this 3300 factor increase is close to the ∼ 5,000−10,000 times greater ν3 cross section. If the average laser power is as high as 10 kW, the amount of energy available is a factor of 11,000 times greater. This laser works by applying a RF discharge to a supersonic cross-axially flowing CO:N2 medium to pump CO molecules up to a laser level that corresponds to 5.3 µm. This requires timing the pulses to allow sufficient filling of this level and a laser cavity that selects this frequency. The duration of the RF pulse discharge must be short compared to the VV-exchange to the desired level, about 10−7 seconds. The pulse must also be long compared to the energy exchanges between rotational levels to allow for the entire rotational level population to contribute to laser action on the desired transition.40 About 60 W/cm3 is the optimum power input for the CO:N2 medium to allow VV-exchange from V=9 to the V=10 level and then emission on the vibrational-rotation transition of 10-9(7) corresponding to 5.3 µm light.41 This translates into a pulse repetition rate of 10 MHz and a laser electro-optical efficiency of ∼ 20 percent. The laser is mode-locked on 5.3µm with the use of an acousto-optical modulator inside the laser cavity. The exact frequency corresponding to the 3ν3 band of 235UF6 can be fine-tuned by a number of techniques including piezo-driven micro vibrations of resonator mirrors, diffraction gratings, etalons, and others. Any broadening of the linewidth from the optical power can be adjusted with line narrowing techniques, which also include the use of etalons and diffraction gratings. There are CO transitions that allow access to 3ν3 lines of 235UF6, and the large isotope shift of 1.8 cm−1 should allow for some uranium enrichment even if the output wavelength is not an exact match. Whether this issue would prevent the CO laser from attaining a better energy efficiency than a Raman-shifted system is unclear to the author. However, all these listed techniques are accessible to those skilled with lasers, so this should be considered another pathway to weapons material production. Additional photon efficiency is gained by using a separation unit ∼ 14 m long and a gas pressure of 1 torr at the location of laser irradiation. The gas pressure at the irradiation zone for 16 µm light is ∼ 0.02 torr.42 The means more UF6 can be irradiated for every CO laser pulse.A further risk with this technology is the large number of industrial applications that make identifying activities intended for uranium enrichment difficult. High-power CO lasers can be used in the the following applications: glass or ceramic cutting and welding, metal processing, surgical tissue cutting or skin resurfacing, laser sintering, and drilling multilayer boards (MLBs) and high-density interconnect structures (HDISs) for smartphones and tablets. In addition, the low absorption of 5 µm light in chalcogenide fibers opens up the possibility for delivering laser light by fiber, which is important in telecommunications technology. It should be anticipated that the number and scale of CO laser applications will grow in the future, further complicating the identification of enrichment activities. Which laser system will ultimately prove the most economical for SILEX is uncertain. The unfortunate news is that these two systems provide routes to highly-enriched uranium that are accessible by a larger (and growing) number of scientists than if only one laser system could be utilized.

Section B: Dimer Formation, Nucleation, and Particle Growth The first experimental evidence that lasers could suppress dimer formation was published by H. VandenBergh in 1985, and described the irradiation of supersonically expanded free jets of SF6/Ar mixtures with CO2 laser photons and the enrichments in SF6 isotopes that were observed in the rims of free jets.46 The result was surprising because it confirmed that dimerization occurred more quickly than previously thought possible, and could only be explained if it was dominated by lowvelocity, two-body collisions. The conventional theory requiring that three-body collisions were necessary for dimer formation was updated,47 and is in agreement with observations. The updated model allowed for the conversion of translational kinetic energy of one colliding atom or molecule to the internal vibrational or rotational energy of the other molecule absorbing the collision. In the case of forming a UF6:G dimer, the initial kinetic energy of G must be on the order of the energy of an excited vibrational mode of UF6. This is only possible for slow-moving molecules, and makes dimer formation more likely at low temperatures.48 Laser suppression of dimer formation also aids in another process that increases the enrichment of 235UF6 in the product stream. As 235UF∗ 6 forms dimers with G that quickly dissociate, unexcited 238UF6 forms heavy dimers (238UF6:G or 238UF6:UF6) that do not. This means that 235UF6 differs in mass by ∆M = MG + 3, where MG is the mass of the carrier gas, from 238UF6:G or by ∆M = 355 from 238UF6:UF6 and will have much larger rates of escape from the free-jet core due to the radial pressure gradient, as the escape rates are proportional to ∆M. This effect was first disclosed in Becker’s studies mentioned in the introduction49 and confirmed and further elucidated by A. A. Bochkarev, et al. with supersonic jets of argon and helium mixtures in 1970.50 The acquired translational recoil energy in the dissociation of a 235UF6 dimer only adds to this escape rate. The choice of a carrier gas G to mix with UF6 is determined by a number of competing factors. As just mentioned, the rate of radial escape due to the pressure gradient is proportional to ∆M, so a heavier carrier gas G will allow for higher separation. Upon the dissociation of 235UF∗ 6 :G, a heavier G will also result in 235UF6 acquiring more translational recoil velocity from the conservation of momentum and increase separation. Choosing a carrier gas G with its own vibrational mode, however, would allow some conversion into vibrational instead of translational energy and lower the separation with slower rates of escape. A heavier carrier gas G will also not form dimers as easily since the larger kinetic energy of G will need to be absorbed by vibrational or rotational modes of the UF6 molecule. The most important constraint is that a carrier gas G is chosen whose gas constant γ = cp/cv is not too low to allow acceptable adiabatic cooling. Equation 9 of the main section of the paper provides the relationship that determines what the temperature T will be at some position downstream of the nozzle, and G must be chosen so that there is adequate cooling to separate the absorption bands of the UF6 isotopes for high selectivity of 235UF6. Monatomic gasses with γ = 1.67 such as Xe and Ar both appear suitable, as does SF6 (γ ≈ 1.3). If vibrational to vibrational transfers between the ν3 mode of UF6 (628 cm−1 ) and the ν4 mode of SF6 (616 cm−1 ) are a minor factor, SF6 may be provide the highest enrichments and be the preferred choice.51 Successful condensation repression depends upon the free jet remaining in the vapor phase so that 235UF6 molecules may migrate out of the jet core for collection by the skimmer. The necessity of forming dimers is in tension here, as dimer formation is the first step in condensation. As the UF6/G gas is cooled by nozzle expansion, it will remain in the vapor phase well above the equilibrium vapor pressure pe(T) for some time due to the high curvature of small, spherical droplets (Kelvin effect) and the reduced binding energy per monomer of small particle clusters.52 As dimers, trimers, and other oligomers form, a “critical embryo” size is reached after which irreversible particle growth is possible. At a pressure pd(T) well above pe(T), these “critical embryos” will irreversibly grow (“nucleation”) by sweeping to their surface the condensable supply of UF6. 53 The time required for 20 percent of the UF6 to reach “critical embryo” size is tc, and this must be longer than the transit time ttr to collection.54 Beyond this time tc, significant cluster growth will impede the migration of 235UF6 to the rim and adversely affect the enrichment factor. Section C: Cascading to 90 Percent Highly Enriched Uranium The arrangement of multiple separation units in a cascade for the purpose of manufacturing 90 percent HEU with third generation enrichment technology is more complex than that used for centrifuges. Constructing a cascade for centrifuges has the design advantage in that product and tails streams are sent forward and back by only one stage for further processing. Such a cascade is called ‘symmetric’, and the cut θ in such an arrangement will be slightly under one-half.55 Third generation enrichment systems, however, use a smaller θ and therefore ‘asymmetrical’ amounts of enrichment and depletion in the two output streams. This requires that these streams be sent different numbers of stages forward or backward making the cascade design more complex. Such a design that minimizes the ratio of separative work to product produced is called ‘ideal’, and thereby ensures that streams of different isotopic concentrations are never mixed.This section will consider an ideal asymmetric cascade for the manufacture of 90 percent HEU using the performance characteristics of third generation enrichment technology. The most ideal cascade, however, is not necessarily that which uses the lowest amount of separative work and may differ from what is presented here. The reason is that separative work is closely related to the rate of material flow through a separation element and is a good measure of the energy consumption of an enrichment plant only if the two can be related to each another. The is not the case with laser isotope separation. The electricity consumed depends on the minimum number of photons needed to excite the 235UF6 in the feed, and thus does not depend on the rate of material flow but instead on the concentration of 235UF6 in the flow. This explains the interest that Global Laser Enrichment (GLE) has in obtaining access to the tailings of the gaseous diffusion processes from the plant in Paducah, Kentucky, for enrichment with SILEX.56 The energy expended in further stripping of existing tails stockpiles will be of little penalty compared to using natural uranium as feedstock.57 For the purposes of future proliferation assessments of SILEX technology or of other laser isotope separation techniques, the “ideal” cascade will balance the capital costs (believed to be laser-dominated) against power costs for the amount of enriched product.Results from J. W. Eerkens and the author are displayed in Table D.1. Limitations of the model in Equation D.6 are revealed here, with two different pressures for SF6, 0.026 torr and 0.003 torr, listed by the author. J. W. Eerkens does not accurately model the parameter changes as the free jet expands in the separation unit, as the model requires a constant temperature and pressure as inputs. However, an accurate model of these parameter changes in an expanding free jet is necessary to accurately calculate β. The author chose 0.026 torr at the location of laser irradiation to provide a sufficiently high β, but 0.003 torr where product was collected at the skimmer to allow a sufficiently high θ. The concept upon which this enrichment process is based is that as many 235UF6 monomers as possible need to be available for laser excitation, followed by the formation of the maximum number of dimers which then quickly dissociate. This can only happen at a temperature significantly higher than 42 K (Table D.1), where according to the author’s version of the model 10 percent of 235UF6 molecules already exist in dimers. If laser irradiation occurred at the more desirable temperature of 100 K, almost no 235UF6 molecules are found in dimers. As the free jet cools from this temperature, laser-excited 235UF∗ 6 will form more 235UF∗ 6 :G dimers than if laser excitation had occurred at 42 K. This allows the term f ! 235Θ! in Equation D.6 to increase, as more epithermal molecules will be present in the jet. As the jet continues to cool and the pressure lowers, all cuts (Θm, Θ! , and Θd ) will grow, and with Θ! > Θm > Θd it follows that β will as well. This evolving dynamic is not accounted for in the above model and is why β ≈ 2 appears to be an underestimate. If β is indeed higher, third generation laser enrichment technology will require less space and use energy more efficiently than what is estimated in the main section of this paper.