A HIGH SELECTIVITY, HIGH DISSOCIATION SIMPLE AND EFFICIENT SYSTEM FOR THE LASER SEPARATION OF THE UF6 ISOTOPES AND OTHER HEXAFLUORIDES

Abstract

The discovery of a method and the invention of a system for obtaining very high selectivityand dissociation of the desired .sup.235UF.sub.6 isotope in the Molecular Laser Isotope Separation (MLIS) process of the Uranium Hexafluoride (UF.sub.6) isotopes, in a single highly selective step, is described. The principle of the process and the concept of the invention are very simple: At temperatures below 100 K., and. preferably in the region of 60 K, nearly all the molecules of the expansion supercooled UE.sub.6 gas are in the ground state enabling the principles of the invention to be practically applied without-any interference from other inherent, processes. Then the frequency of the selecting laser must be at 628.527 cm.sup.1, or very close to it, for a three-photon absorption resonance with the [m(A.sub.2):(3V.sub.3)] sublevel of the third energy excitation state of the desired .sup.235UF.sub.6 isotope. The fixing of the frequency of the selecting laser is the first basic step of the invention. The second basic step is to increase the pumping intensity of the selecting laser to a. level at which thethree-photon absorption resonance with the [m(A.sub.2):(3V.sub.3)]sublevel, of the desired .sup.235UF.sub.6 isotope is established, elevating the molecules of the desired isotope .sup.235UF.sub.6 to the third energy excitation state. This is achieved through the power broadening at the fundamental and the second energy excitation level as the pumping intensity of the selecting laser is increased and as a consequence of the proximity of these levels to the pumping frequency. There is an intensity range for the selecting laser within which the molecules of the desired .sup.235UF.sub.6 isotope can be selectively elevated to the third energy level through the establishment of a three-photon absorption resonance without disturbing the molecules of the unwanted, isotope .sup.238UF.sub.6, leaving them unexcited. The selectively excited molecules of the desired .sup.235UF.sub.6 isotope are then driven to dissociation through, the higher vibrational levels of the v.sub.3-vibrational mode and. the quasicontinuum of energy states, by a simultaneously applied dissociating laser whose exact intensity and optimum frequency can again be experimentally determined, or by any other dissociation or separation-process following the original excitation of the .sup.235UF.sub.6 molecules to the third energy excitation state (3v.sub.3) through three-photon resonance with the [m(A.sub.2):(3V.sub.3)] sublevel. The process is unique in that it can be applied, to the treatment and separation of the desired .sup.235UF.sub.6 isotope from the Tails percentages of any isotope separation process. The method may also be. applicable to the SILEX system for enhancing the selectivity and efficiency of the process. The simplicity and versatility of the method enables: it to be applied to the separation of other hexafluoride isotopes or similar polyatomic molecules.

Claims

1. A method of preferentially exciting and selectively dissociating the molecules of the desired .sup.235UF.sub.6 isotope, in a supercooled UF.sub.6 gas mixture at low temperature such that the sublevels of the energy excitation states of the UF.sub.6 N3-vibrational mode are distinct and clear, by irradiating the UF.sub.6 gas with a narrow bandwidth laser beam whose frequency is in three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the third energy excitation state (3.sub.3) of the desired .sup.235UF.sub.6 isotope at 628.527 cm.sup.1, or a frequency sufficiently near to it for three photon resonance to be established with the [m(A.sub.2):(3.sub.3)] sublevel, the three-photon resonance being achieved through the adjustment of the pumping intensity of the selecting laser beam at 628.527 cm.sup.1 to a required specific intensity range whereby it is sufficiently intense to establish three photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel but it is kept below the intensity level of establishing multiphoton absorption with the higher levels, and the subsequent dissociation of the .sup.235UF.sub.6 molecules through the simultaneous or slightly adjustable time delayed application of other infrared or ultraviolet laser beams, or by any other dissociation or separation process following the original excitation of the .sup.235UF.sub.6 molecules to the third energy excitation state (3.sub.3) through three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel.

2. A highly selective dissociation method of claim 1 wherein the intensity of the selecting laser beam at 628.527 cm.sup.1 in three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the desired .sup.235UF.sub.6 isotope, or at a nearby frequency sufficiently close to it for three photon resonance to be established with the [m(A.sub.2):(3.sub.3)] sublevel, is adjusted at a pumping level within an intensity range at which the molecules of the desired isotope .sup.235UF.sub.6 are selectively elevated to the third energy excitation state.

3. The method of claim 1 of preferentially elevating the molecules of the desired .sup.235UF.sub.6 isotopic species to the [m(A.sub.2):(3.sub.3)] sublevel of the third energy excitation state (3.sub.3) whereby the appropriate pumping intensity level ensuring the simultaneous validity of inequalities (66) and the validity of inequalities (68), (69) and (70), is achieved by adjusting the pumping power level and the time duration of the applied selecting beam in the frequency region of three-photon absorption resonance with the [m(A.sub.2):(3.sub.3)] sublevel at 628.527 cm.sup.1.

4. The method of claim 2 of preferentially elevating the molecules of the desired .sup.235UF.sub.6 isotopic species to the [m(A.sub.2):(3.sub.3)] sub level of the third energy excitation state (3.sub.3) whereby the intensity of the selecting laser beam at 628.527 cm.sup.1, or at a nearby frequency sufficiently close to it for three photon resonance to be established with the [m(A.sub.2):(3.sub.3)] sublevel, is limited to intensity levels below which no other processes leading to the absorption of radiation by the unwanted isotope .sup.238UF.sub.6 can take place or drive the molecules of the desired isotope .sup.235UF.sub.6 through the quasicontinuum stages.

5. A method of preferentially exciting and selectively dissociating the molecules of the desired .sup.235UF.sub.6 isotope in the Molecular Laser Isotope Separation (MLIS) process according to claim 1 wherein the selectively excited .sup.235UF.sub.6 molecules to the third energy level are driven to dissociation through the simultaneous application, or with a small adjustable time delay, of an additional powerful infrared beam or any other beams driving the molecules of the desired .sup.235UF.sub.6 isotope through the quasicontinuum of energy states to dissociation.

6. A selective excitation and dissociation method of preferentially exciting and selectively dissociating the molecules of the desired .sup.235UF.sub.6 isotope in the Molecular Laser Isotope Separation (MLIS) process of claim 5 wherein a powerful dissociating infrared beam whose frequency closely matches the energy level differences between most of the levels from the third to the eighth energy states of the .sub.3-vibrational mode of the .sup.235UF.sub.6 isotope, is simultaneously applied to the supercooled molecular gas.

7. The method of claim 5, wherein the frequency of the dissociating powerful infrared laser beam lies in the range from 618 cm.sup.1 to 623.3 cm.sup.1 and its intensity is adjusted for the optimum dissociation of the selectively excited .sup.235UF.sub.6 molecules, and wherein additional infrared or ultraviolet beams can be simultaneously applied to the expansion supercooled UF.sub.6 gas mixture in order to further enhance the selective dissociation of the desired .sup.235UF.sub.6 isotope.

8. The method of claim 1, wherein the selectively elevated molecules of the desired .sup.235UF.sub.6 isotope to the [m(A.sub.2):(3.sub.3)] sublevel of the (3.sub.3) energy excitation state of the .sub.3-vibrational mode can be selectively separated from the molecular gas by any procedure, whether dissociation, radiational, chemical, mechanical or any other process.

9. A highly selective dissociation method of preferentially exciting and elevating the molecules of the desired .sup.235UF.sub.6 to the third energy excitation level [m(A.sub.2):(3.sub.3)] in the Molecular Laser Isotope Separation (MLIS) process, according to claim 1, wherein the selective three-photon absorption resonance frequency at 628.527 cm.sup.1 can be finely tuned between 628.45 cm.sup.1 and 628.6 cm.sup.1 for selecting the most optimum frequency for the best selectivity results.

10. A method of preferentially elevating the molecules of the desired .sup.235UF.sub.6 isotope to the third energy excitation level [m(A.sub.2):(3.sub.3)] in the Molecular Laser Isotope Separation (MLIS) process, according to claim 1 wherein the intensity of the selecting laser at the three-photon resonance frequency of 628.527 cm.sup.1 can be adjusted between 410.sup.9 W/m.sup.2 and 4010.sup.9 W/m.sup.2 and its pulse duration from 1010.sup.9 s to 4010.sup.9 s for obtaining the optimum operating conditions for maximum selectivity.

11. A method of preferentially exciting the molecules of the desired isotope .sup.235UF.sub.6 to the sublevel [m(A.sub.2):(3.sub.3)] of the third energy excitation state (3.sub.3) of the .sub.3-vibrational mode according to claim 1, wherein the duration of the selecting pumping beam in three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel at 628.527 cm.sup.1 is adjusted in conjunction with the frequency deviations of the intermediate levels .sub.1, and .sub.2 to facilitate the ready establishment of three-photon resonance and the selective excitation of the molecules of the desired isotope .sup.235UF.sub.6 to that level.

12. Apparatus for preferentially exciting and selectively dissociating the molecules of the desired .sup.235UF.sub.6 isotope in the molecular laser isotope separation process, comprising the design of an expansion nozzle capable of producing a supercooled molecular UF.sub.6 gas mixture at temperatures below 100 K, preferably in the region of 60 K, wherein the selecting laser beam and the dissociating laser of the method of claim 1 can be applied to the molecular UF.sub.6 gas at very high pulse repetition rates capable of irradiating all the molecules of the expansion supercooled gas.

13. Apparatus for the selective dissociation of the molecules of the desired isotope .sup.235UF.sub.6 in a supercooled UF.sub.6 gas mixture, comprising the design of an expansion nozzle of claim 12 whereby the selective and dissociating beams can be applied collinearly and where two, three or more expansion nozzles can be placed in series for a more efficient separation process in a single pumping step.

14. A selective excitation and separation method for preferentially exciting and selectively dissociating or separating the molecules of the desired .sup.235UF.sub.6 isotope, according to claim 1, wherein different UF.sub.6 gas assays can be employed as well as assays corresponding to the Tails percentages from other separation processes, or assays for higher enrichment separation factors, and where the pumping intensity parameters and the frequency parameters can be slightly adjusted to obtain the optimum efficiency for the isotope separation process.

15. A high selectivity isotope separation process for preferentially exciting and selectively dissociating or separating the molecules of the desired .sup.235UF.sub.6 isotope, of claim 14, wherein the Feed percentage of the desired isotope .sup.235UF.sub.6 can vary from high values to low values and where the isotope separation process described herein can be applied to produce more highly enriched Uranium Hexafluoride or applied to the treatment and separation of the desired isotope .sup.235UF.sub.6 of the Tails percentages, or to the treatment and separation of the desired isotope .sup.235UF.sub.6 of the Tails percentages of other separation processes, or to any other treatment of low percentage depleted Uranium.

16. A selective excitation method for the molecular laser isotope separation process according to claim 1, wherein the process is applied to the isotope separation of any other hexafluoride molecule wherein the frequency of the selecting laser corresponds to the frequency of three photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the third energy excitation state (3.sub.3) of the said other hexafluoride molecule, and which has similar energy structure for the .sub.3-vibrational mode, with its pumping intensity being adjusted within a specific intensity range set by the interaction parameters of the said molecules to fulfil the conditions described in the specification for the said molecules, the process being capable of application to any other polyatomic molecule with similar vibrational structure.

17. A selective excitation and separation process according to claim 1, wherein the method is applied to the SILEX process (a Separation of Isotopes by Laser Excitation (SILEX) process for the enhancement of the selective separation of the desired .sup.235UF.sub.6 isotope and the efficiency of the said process.

18. A method of preferentially exciting and selectively dissociating or separating the desired .sup.235UF.sub.6 isotope when used in the separation or enrichment of the Uranium Hexafluoride isotopes, or of any other hexafluoride molecules or similar isotopic species of claim 16.

19. A method of preferentially exciting and selectively dissociating or separating the desired isotope in a molecular laser isotope separation or enrichment process substantially as described herein and with reference to FIG. 9, FIG. 8(b), FIG. 10, FIG. 11, or FIGS. 9, 8(b), 10, 11 and 12(c) of the accompanying drawings.

Description

[0003] FIG. 1 shows the measured and calculated (from the analysis of the Schrdinger equation using the equivalent Morse potential) anharmonicity constants of the Hexafluorides. It is clear that as we move towards the heavy Hexafluorides the two values become identical demonstrating the equivalence of the vibrational ladder of the heavy Hexafluorides to that of an ideal harmonic oscillator.

[0004] FIG. 2 shows the absorption cross section of UF.sub.6 as a function of Pumping Fluence on a logarithmic scale at a gas temperature of 105 K and frequency of 627.6 cm.sup.1 It demonstrates that when the graph of the experimental measurements obtained using a diode laser are extended towards very low pumping fluences they are in very good agreement With the calculated theoretical results and the results from low fluence spectrophotometer experiments. This supports the consistency of the theoretical expressions given in the text and demonstrates that the vibrational ladder of the heavy Hexafluorides is a very dose match to that of an ideal harmonic oscillator.

[0005] FIG. 3 shows the power broadened absorption probabilities of the fundamental for the two Uranium isotopes .sup.238UF.sub.6 and .sup.235UF.sub.6, for four different pumping beam intensities ranging from 1010.sup.9 W/m.sup.2 to 6010.sup.9 W/m.sup.2, graphs (a) to (d). The peaks of the curves are located at the center of the absorption lines. We see that as we increase the pumping intensity the widths of tire absorption curves increase rapidly. Our interest in this invention lies mostly in the 510.sup.9 W/m.sup.2 to 3010.sup.2 W/m.sup.2 region.

[0006] FIG. 4 shows the power broadening curves for the consecutive transitions of the first eight vibrational levels of the UF.sub.6 .sub.3-vibrational ladder, for four different pumping intensities ranging from 1010.sup.9 W/m.sup.2 to 6010.sup.9 W/m.sup.2. The power broadening of the transitions increases progressively with higher vibrational levels as well as with higher pumping intensity. Only the lower levels are of particular interest in the invention.

[0007] FIG. 5: (a) The number of photons

[00001]${\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}$

interacting with the quasicontinuum of states is proportional to the interaction rate R.sub.int when the number of absorbed photons is 170<n.sub.qu=255. In this case n,

[00002]${\left({}_{\mathrm{IR}}\right)}^{\frac{2}{3}}\left[i.e.{\left({}_{\mathrm{IR}}\right)}^{-\frac{1}{3}}\right]$

as observed theoretically and experimentally; (b) The number of photons n interacting with the quasicontinuum of states is found not to be proportional to the interaction rate R.sub.int within the same range of absorbed photons. The dotted vertical lines on both graphs correspond to an applied number of photons of approximately 170 and 255 interacting with the quasicontinuum of energy states and corresponds to the interval in which

[00003]$R_{\mathrm{im}}{\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}.$

[0008] FIG. 6 shows the variation of the population of the UF.sub.6 gas in the ground state as a function of temperature. It is clear that at a temperature of around 60 K more than 85% of the molecules are in the ground state. Notice the dramatic drop of the ground state population of the UF.sub.6 gas down to 30% at a temperature of 110%.

[0009] FIG. 7 depicts a gas dynamic expansion through a two dimensional nozzle, having a slit throat.

[0010] FIG. 8: (a) Resonances of the first four energy levels for the two UF.sub.6 isotopes when pumping at the frequency of the fundamental at 628.306 cm.sup.1 at a pumping intensity of 3010.sup.9 W/m.sup.2; broken line curves correspond to the .sup.238UF.sub.6 isotope and the solid line curves correspond to the .sup.235UF.sub.6 isotope. (b) Resonances of the first four energy levels for the two UF.sub.6 isotopes when pumping at the three-photon resonance frequency with the [m(.sub.2):(3.sub.3)] sublevel of the third energy excitation state of the desired .sup.235UF.sub.6 isotope at 628.527 cm.sup.1 at a pumping intensity of 3010.sup.9 W/m.sup.2; broken line curves correspond to the .sup.238UF.sub.6 isotope and the solid line curves correspond to the .sup.235UF.sub.6 isotope. Comparison of FIG. 8(a) with FIG. 8(b) demonstrates the enormous difference in selectively elevating the molecules of the desired isotope .sup.235UF.sub.6 to the third energy excitation level when pumping at the three-photon resonance frequency of 628.527 cm.sup.1 as compared with pumping at the frequency of the fundamental at 628.306 cm.sup.1.

[0011] FIG. 9 depicts the selectivity of the desired isotope .sup.235UF.sub.6 to the third energy excitation state through the power broadening of the lower vibrational levels and pumping frequencies near the three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the state, at 628.527 cm. The pumping intensity is set at 20 GW/m.sup.2. The solid line curves correspond to the desired .sup.235UF.sub.6 isotope whilst the broken line curves correspond to the unwanted .sup.238UF.sub.6 isotope. The black solid line corresponds to the exact three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel at 628.527 cm.sup.1. The other frequencies correspond to the lines indicated on the figure.

[0012] FIG. 10 depicts the entire selectivity and dissociation process for a selective beam frequency of 628.527 cm.sup.1 (continuous black line in the first three energy excitation states) at a pumping intensity of 20 GW/m.sup.2 and a dissociating beam at a frequency of 620.6 cm.sup.1 (thick broken line from the third to the eighth energy excitation state) and pumping intensity of 80 GW/m.sup.2. The solid line curves correspond to the desired .sup.235UF.sub.6 isotope whilst the broken line curves correspond to the unwanted .sup.238UF.sub.6 isotope. We see that the dissociating frequency is in direct resonance with three of the five levels keeping the excitation energy within the .sub.3-vibrational mode up to the eighth energy excitation state of the desired isotope .sup.235UF.sub.6 without affecting or resonating with any of the levels of the unwanted isotope .sup.238UF.sub.6, which in any case are not populated at all at the low temperatures of the UF.sub.6 gas. Note the importance of not elevating molecules of the unwanted .sup.238UF.sub.6 isotope to the third energy excitation level. The actual pumping intensities can be smaller or higher than those depicted in the figure.

[0013] FIG. 11 shows a typical graph (one of hundreds drawn) for obtaining the selectivity between the two isotopes at the various pumping frequencies and intensities. All four graphs are drawn for a pumping intensity of 20 GW/m.sup.2. The abscissa |c(t)|.sup.2 provides information for the absorption probability of the corresponding frequency on the ordinate. From the analysis of such graphs we have calculated the relative selectivities to the third energy state for the two isotopes .sup.235UF.sub.6 and .sup.238UF.sub.6. The fourth energy level is included as an indication for qualitative comparisons. Note that the calculation of the selectivity and the graphic representations are relative and approximate but they give a good practical indication of the trends and the limits for the intensities, pulse durations and frequencies which must be applied to the molecular gas for obtaining outstanding selectivity.

[0014] FIG. 12 depicts the three-photon absorption resonance process. (a) All pathways contribute to the third order term in the interaction Hamiltonian during the three-photon absorption process; (b) The situation when the three photons are the same i.e. .sub.k=.sub.k=.sub.k. The intermediate states |.sub.i and |.sub.m although imaginary they nevertheless constitute solutions to the atomic Schrdinger equation; (c) The situation when the intermediate states |.sub.i and |.sub.m are real atomic or molecular states as in the case of a vibrational ladder where their position may differ slightly from exact resonance. This is the case of three-photon resonance with the third energy excitation level [m(A.sub.2):(3.sub.3)] of the .sup.235UF.sub.6 isotope when the vibrational ladder interacts with a one frequency pumping beam i.e. with reference to Figs. (a), (b) we set .sub.k=.sub.k=.sub.k, n.sub.k=n.sub.k=n.sub.k, m.sub.1=m.sub.2=m.sub.3=m, l.sub.1=l.sub.2=l.sub.3=l.

[0015] FIG. 13: The three-photon transition rate plotted against intensity for six different pumping frequencies. The broken vertical lines indicate the intensity level at which the three-photon transition rate to the third energy excitation level, exceeds the equivalent two-level transition rate with the same transition parameters. The dotted vertical lines on the graphs indicate the intensity level at which the selectivity of the desired isotope .sup.235UF.sub.6 to the third energy excitation level remains very high. It is evident that at the pumping frequency of the fundamental (628.306 cm.sup.1, graph (a)) it is extremely difficult to achieve any considerable selectivity of the desired isotope to the third energy excitation level. The optimum frequency range for achieving high selectivity of the desired isotope .sup.235UF.sub.6 is between 628.45 cm.sup.1 and 628.527 cm.sup.1 both from the point of view of the effective transition rate to the third energy level and the intensity range over which high selectivity to the third energy excitation level can be achieved 510.sup.9 W/m.sup.2 to 3010.sup.9 W/m.sup.2 (graphs (b), (c) and (d)). It is clear that whilst the induced transition rate for the desired isotope .sup.235UF.sub.6 takes off to enormous values as the intensity is increased the corresponding transition rate for the unwanted isotope remains extremely low.

[0016] FIG. 14: shows the selectivity to the third energy excitation level for the beam and gas parameters shown on the graphs, for various pumping intensities. The broken vertical lines indicate the intervals over which eq. (71) and conditions (68)-(70) are strictly valid. The first graph (a) (dotted background) corresponds to pumping intensity levels (<4.510.sup.9 W/m.sup.2) at which difficulties may arise in establishing three-photon resonance with the third energy excitation level. The last three graphs (c), (d) and (e) correspond to pumping intensity levels (>4.510.sup.9 W/m.sup.2) at which the three-photon absorption resonance, with the same pumping parameters, can easily be established. It can be seen that the selectivity drops with increasing pulse duration and also with increasing intensity levels. For the selective excitation of the molecules of the desired isotope .sup.235UF.sub.6 to the third energy excitation level [m(A.sub.2):(3.sub.3)] the optimum pumping intensities should be in the region 5-15 GW/m.sup.2 with pulse durations of less than 3010.sup.9 s. With these pumping parameters all the molecules of the desired isotope .sup.235UF.sub.6 are selectively elevated to the third energy excitation level. Note that these results have been obtained under the strict application of eq. (71) and the conditions (68)-(70), which are the strictest minimum conditions set in the calculations for achieving very high selectivity to the third energy level. The actual experimental conditions can be much more flexible and in practice much higher pumping intensities, exceeding 3010.sup.9 W/m.sup.2 can be applied whilst preserving very high selectivity since three-photon absorption with the unwanted isotope .sup.238UF.sub.6 is very difficult to establish at the pumping frequency of 628.527 cm.sup.1.

[0017] FIG. 15: (a) The percentage selectivity to the third energy excitation level as a function of the pumping intensity for various pumping pulse durations. The vertical broken lines denote the maximum interval over which eq. (71) and conditions (68)-(70) remain strictly valid. The other vertical lines are explained in the text. It can be seen that the shorter the pulse duration the higher the selectivity of the desired isotope. The optimum pumping intensity levels can be seen to be in the interval (4-15)10.sup.9 W/m.sup.2 with pumping pulse durations between (10-30)10.sup.9 s; (b) The total number of excited molecules to the third energy excitation level as a function of pumping intensity for various pulse durations, corresponding to the graphs of figure (a). The beam and gas parameters are shown on the graph and they are the same as those in figure (a). All the vertical lines on the graphs denote the same parameter limits as in figure (a). Since all the molecules of the desired isotope .sup.238UF.sub.6 are excited to the third energy level the increase in the number of excited molecules is due to the excitation of the molecules of the unwanted isotope. Subsequently shorter pulse durations with high intensity are preferable for the preservation of high selectivity to the third energy level. Note that these are the worst scenario cases since, in practice, three photon absorption with the unwanted isotope .sup.238UF.sub.6 is very difficult to achieve at the pumping frequency of 628.527 cm.sup.1. Much higher intensities can be applied whilst preserving high selectivity. All calculations have been performed under the strictest limiting conditions for the application of the theoretical calculations.

[0018] FIG. 16 shows the curves for the total number of excited molecules to the third energy level when the diameter of the pumping beam is increased to 0,012 m. The selectivity curves are the same as those of the smaller diameter beams (0.008 m in FIG. 15(a)) but the number of excited molecules is now more than doubled. The comparison with FIG. 15(b) indicates how the design of the expansion nozzle in effectively accommodating larger diameter beams is of paramount importance to the efficiency of the system.

[0019] FIG. 17 shows the curves for the total number of excited molecules to the third energy excitation level as a function of pumping intensity for various pulse durations as in FIG. 15(a) but for smaller gas density parameters N.sup.233=0.4255710.sup.21 molecules/m.sup.3, N.sup.233=0.304310.sup.19 molecules/m.sup.3. All the interaction parameters as well as the percentage selectivity to the third energy level in the graphs of FIG. 15(a) remain the same. The only results that change are those for the total number of excited molecules to the third energy level and these are to be compared with those of FIGS. 15(b) and 16. Again all the available molecules of the desired isotope .sup.235UF.sub.6 are elevated to the third energy excitation level. Note that the percentage selectivity does not change with gas density provided the ratio of the number of molecules of the two isotopes in the gas is the same.

[0020] FIG. 18: (a) The selectivity graphs as a function of pumping intensity for various pumping pulse durations for a Tails assay of the desired isotope (.sup.238UF.sub.6 (=99.75% .sup.235UF.sub.6=0.25%) at a temperature of 60 K; (b) The corresponding graphs for the total number of molecules elevated to the third energy level. The fact that all the available molecules of the desired isotope are elevated to the third energy excitation level makes the shapes of the resulting curves similar to those in FIGS. 15(a) and 15(b). The capability of treating the Tails is one of the most important aspects of the present invention.

[0021] FIG. 19 shows the variation of the selectivity as a function of pumping frequency and pumping pulse duration, for a pumping intensity of 810.sup.9 W/m.sup.2. We see that for this pumping intensity the frequency region for which eq. (71) is satisfied and conditions (68)-(70) remain strictly valid is between 628.45 cm.sup.1 and 628.56 cm.sup.1. Similar results are obtained for lower expansion supercooled gas assays. The results for Tails assays of the irradiated gas show similar characteristics but with slightly lower percentage selectivities to the third energy excitation level. This corresponds again to strictly limiting cases. Much higher pumping intensities can be employed without significant loss of selectivity.

[0022] It is to be emphasized that the results in FIGS. (14) to (19) have been obtained under the strict application of eq. (71) and the conditions (68) to (70). They give, however, a very good indication as to the trends, the intensity levels and the pulse durations for which the present invention can be applied. The actual experimental conditions are much more flexible and higher pumping intensities can be applied to the molecular gas. Note that, in practice, the elevation of the molecules of the unwanted isotope .sup.238UF.sub.6 is smaller than the one depicted in the graphs since for this isotope it is difficult to establish three photon absorption at the pumping frequency of 628.527 cm.sup.1 rendering the separation process much more favourable.

[0023] As with all the Hexafluorides the molecular population is by far greater in the Q-branch of the spectrum, so excitation in the .sub.3-vibrational mode of the Q-branch is the desired mode of excitation. Expansion supercooling of the UF.sub.6 enables the absorption bands in this region of the spectrum to be distinct and clear. The difference in the Q-branch absorption bands of .sup.235UF.sub.6 from .sup.238UF.sub.6 has been well established to be 0.604 cm.sup.1 and the ratio of the Q-branch peak heights for a sample containing the natural mixture of Uranium isotopes (0.71% in .sup.235UF.sub.6) is about 140 to 1. Because the integrated absorption coefficient is proportional to the number of molecules per unit volume, any isotope separation scheme will rely heavily on distinguishing between the Q-branches of the two isotopes.

[0024] The selective excitation of the .sup.235UF.sub.6 molecules has always been carried out through the application of a pumping beam whose frequency matches the ground to first level absorption line 628.306 cm.sup.1. Other beams were simultaneously applied to the supercooled molecular UF.sub.6 gas to enhance the dissociation of the molecules.

[0025] The process of the selective dissociation of polyatomic molecules is the absorption under collisionless conditions of many infrared photons of the same frequency by a single molecule, by exciting successively higher vibrational states of the molecule until its dissociation is reached. For Uranium Hexafluoride (UF.sub.6) the molecule must be driven through the energy levels to the dissociation energy of 2.95 eV (23800 cm.sup.1). An important factor in the enhancement of the multiphoton absorption is resonance at the fundamental, either on its own or for the enhancement of higher order absorption processes. Without it, absorption would be limited from the point of view of absorption cross section and the molecule having to be lifted through the vibrational ladder obeying the quantum mechanical selection rules.

[0026] The practical aspects affecting the selectivity and dissociation process in polyatomic molecules depend mainly on: (a) The case with which the molecules of the desired isotope .sup.235UF.sub.6 are selectively driven through the lower vibrational levels. (b) the level up to which the excitation energy remains within the same vibrational mode before being able of escaping to other vibrational modes in the quasicontinuum of states.

[0027] As the intensity of the pumping beam increases power broadening of the first few energy excitation levels occurs. In the past the magnitude of the power broadening of the fundamental transition has been grossly overestimated and the proposed schemes wrongly considered that selectivity was affected right from the interaction at the fundamental even at low pumping energies (D. Andreou, UK Patent GB 2256079B dated May 10, 1994 and U.S. Pat. No. 5,591,947 dated Jul. 1, 1997). Furthermore, the induced differential polarizability in the vibrational ladder, as well as other features in the interaction process were overestimated, rendering the selectivity process practically inapplicable.

[0028] To a first approximation the correct expression for the power broadening of a spectral line is

[00004]$\begin{array}{cc}v={\left[v_o^2+\frac{{}^2{E}_o^2}{{}^2{}^2}\right]}^{\frac{1}{2}}& \left(1\right)\end{array}$

where is the dipole moment of the transition, E.sub.o is the electric field of the applied laser beam and .sub.o is the natural linewidth of the transition. On substituting the values of the parameters for the ground to first energy excitation level of UF.sub.6 1.28510.sup.30 C.Math.m, .sub.o0.197 cm.sup.1=5.90610.sup.9 s.sup.1, we obtain that even for electric fields as low as 210.sup.6 V/m the second term dominates the value in the brackets

[00005]$\left(\frac{E}{}=0.259{\mathrm{cm}}^{-1}\right)$

compared with .sub.o=0.197 cm.sup.1. The power broadening of the transition thus becomes the dominant factor in the absorption process.

[0029] For a beam with intensity 4010.sup.9 W/m.sup.2 (100 m/within a beam radius r=410.sup.3 m and a pulse duration r=5010.sup.9 s) the electric field is 5.510.sup.6 V/m and we obtain (/2)=0.356 cm.sup.1 as the power broadened Full Width at Half-Maximum of the main .sup.238UF.sub.6 band. The frequency difference between the ground states of the two isotopes is 0.604 cm.sup.1 and even at these very high intensity levels the .sup.238UF.sub.6 molecules seem to be safe from absorption. The power broadening of the lower vibrational levels can, however, be properly exploited for the selective elevation of the molecules of the desired .sup.235UF.sub.6 isotope up the vibrational ladder.

[0030] To selectively excite large numbers of molecules of the desired isotope .sup.235UF.sub.6 and lead them efficiently to dissociation we must exploit the properties of the distinct levels and sublevels of the vibrational ladder and its interaction with the electromagnetic beams at specific frequencies and intensities.

[0031] The principles of the process are very delicately hidden under some of the fundamental concepts of the interaction of electromagnetic radiation with a vibrational ladder whose lower levels are a very close match to those of a harmonic oscillator. This is why the molecular gas should be expansion supercooled to very low temperatures below 100 K, and preferably to around 60 K, at which nearly all the molecules are in the ground state and the principles of the invention can be practically applied without any interference from other inherent processes. Then the invention of the method and its practical applicability is very simple: The frequency of the selecting laser must be at 628.527 cm.sup.1, or very close to it, for a three-photon absorption resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the third energy excitation state of the desired .sup.235UF.sub.6 isotope. Having defined the first basic step which is the fixing of the frequency of the selecting laser, the second basic step is to increase the pumping intensity of the selecting laser to a level at which the three-photon absorption resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the desired .sup.235UF.sub.6 isotope is established, elevating all the molecules of the desired isotope .sup.235UF.sub.6 to the third energy excitation state. This is achieved through the power broadening at the fundamental and the second energy excitation level as the pumping intensity of the selecting laser is increased. Here lies one of the most delicate points of the invention: if the pumping intensity of the selecting laser is low, three-photon resonance with the third energy level will be difficult to establish due to the lack of any resonance at the fundamental and the second energy excitation levels. On the other hand, if the pumping intensity of the selecting laser is very high, and because the quasicontinuum of energy states for the UF.sub.6 molecule can start at the third energy level for very high intensities, the selectively excited molecules could escape to other vibrational modes and also resonances can set in with the higher energy states of the molecules of the unwanted isotope. There is, however, an intensity range for which all the molecules of the desired .sup.238UF.sub.6 isotope can be selectively elevated to the third energy level through the establishment of a three-photon absorption resonance, without in any way disturbing the molecules of the unwanted isotope .sup.238UF.sub.6, leaving them unexcited.

[0032] To understand the basic process and the principles of the invention we first summarise some of the important properties of the Hexafluorides which have not been analysed before. The molecular Dissociation energy is equivalent to the binding energy of the .sub.3-vibrational mode of the molecule which is the energy required for breaking the first XFs-F bond. We have tabulated the Dissociation energies for the .sub.3-vibrational modes of the hexafluorides gathered from various references in the literature. For the UF.sub.6 they are perfectly compatible with those given by Jensen et al, Los Alamos Science Vol. 3, pp. 2-33, (1982) and Gilbert et al, SPIE Vol. 669, pp 10-17 Laser Applications in Chemistry (1986). For the UF.sub.6 molecule the dissociation energy .sub.diss=2.951 eV=23800 cm.sup.1 corresponding to an equivalent number of Dissociation photons =38 required for dissociating each particular molecule. The values of the other hexafluorides were calculated from the symmetry of their ground electronic states (.sub.1g), their common chemical characteristics and their spectra. The estimated values are all compatible amongst themselves.

[0033] Transitions up the vibrational ladder of an infrared active mode are generally governed by the angular momentum quantum mechanical selection rule l=+1. For higher vibrational levels with n>2 it is possible in practice to have transitions where this selection rule is violated. For n>5 selection rules become very loose. As a result of this selection rule there is a complete absence of all the first overtones (2.sub.3) in the infrared spectra of the molecules belonging to the O.sub.h group since they are forbidden.

[0034] The most important factor in the multiphoton absorption process and the selective dissociation of polyatomic molecules is the structure of the n.sub.3 vibrational ladder. Knowledge of the structure of this ladder provides the information needed to picture the possible pathways through which the photon energy can be selectively absorbed by the desired isotope. Taking the ground state of vibration as the zero reference point of the .sub.i-mode of vibration, the actual harmonic frequencies of the i.sup.th-mode of vibration can be obtained. The anharmonicity constants X.sub.ii (cm.sup.1) are related to the manifold origin of the levels of the .sub.i vibrational mode. We analysed the structure of the .sub.3-mode vibrational ladder in this convention from the theoretical results described by Krohn et al, Journal of Molecular Spectroscopy, Vol. 132, pp. 285-309, eq. (7), (1988) and Herzberg G., Molecular Spectra and Molecular Structure, Vol. II, Krieger Publishing Co, p. 211, (1991), with the various constants governing the .sub.3-vibrational ladder defined by: Frequencies (cm.sup.1): .sub.3.sup.o is the effective harmonic frequency; (.sub.3) is the manifold origin of the fundamental, (.sub.3) being the manifold origin of the higher vibrational levels; .sub.3 is the pure vibrational energy of the fundamental excited level. It is this quantity which is used in the determination of the exact position of the levels of the symmetry structure of the higher vibrational states .sub.3 (eigenstates of the vibrational manifolds); m(F.sub.1) is the centre of the absorption band of the fundamental (observed frequency) when taking into account the Coriolis shift. Constants (cm.sup.1): B.sub.3 is the rotational constant of 3.sub.3; B.sub.0 is the rotational constant of the ground state (=0); B.sub.3 is the Coriolis shift for .sub.3 to the band origin; X.sub.33 is the anharmonicity constant related to the manifold origins of the levels of the .sub.3 vibrational mode; G.sub.33 is the anharmonicity coefficient related to the vibrational angular momentum .sup.2; T.sub.33 is the anharmonicity constant related to the state of vibration. On imposing constraint conditions on the particular state of vibration, the relations 2T.sub.33+G.sub.330 and X.sub.336T.sub.33 generally hold for the .sub.3-vibrational mode of the heavy Hexafluorides. As the molecules become lighter then G.sub.332T.sub.33 (T.sub.33 negative).

[0035] For the first energy excitation level of the UF.sub.6 vibrational modes with =1 the degeneracies g.sub..sub.i of the various modes of vibration .sub.i (i=1, 2, 3, 4, 5, 6).

[00006]$\begin{array}{cc}{g}_{v_1}=1,g_{v_2}=2,g_{v_3}=g_{v_4}=g_{v_5}=g_{v_6}=3& \left(2\right)\end{array}$

[0036] For the first few vibrational levels we arrived at the following relations

[00007]$\begin{array}{cc}{G}_{33}=-\frac{1}{2}\left[\left(v_3\right)-v_3\right],X_{33}-G_{33}=\frac{1}{6}\left[\left(3v_3\right)-3v_3\right]& \left(3\right)\end{array}$

[0037] The frequencies and constants of the .sub.3-vibrational ladder of the hexafluorides bear the following relations amongst themselves: (i) The anharmonicity constant for the manifold origins X.sub.33 in cm.sup.1 is always negative; (ii) The anharmonicity constant related to the state of vibration T.sub.33 in cm.sup.1 is always negative; (iii) The anharmonicity coefficient related to the vibrational angular momentum G.sub.33 in cm.sup.1 is always positive.

[0038] The effective harmonic frequency .sub.3.sup.o is always greater than either (.sub.3) (the manifold origin of the fundamental in cm.sup.1) or .sub.3 (the frequency of the pure vibrational energy of the fundamental in cm.sup.1). Also since G.sub.33 is always positive .sub.3 is always greater than (.sub.3). Thus

[00008]$\begin{array}{cc}{}_3^o>v_3,{}_3^o>\left(v_3\right),v_3>\left(v_3\right),G_{33}-\frac{1}{3}{X}_{33}& \left(4\right)\end{array}$

where the last relation results from the constraint relations above and holds for the heavy hexafluorides if non-bonding interactions are ignored. Detailed analysis of the above structure of the .sub.3-vibrational ladder of the hexafluorides results in a set of limiting values of the anharmonicity constant X.sub.33 (cm.sup.1) and the effective harmonic frequency .sub.3.sup.o (cm.sup.1).

[00009]$\begin{array}{cc}{X}_{33}=\frac{1}{9}\left[\left(3v_3\right)-3{}_3^o\right]\frac{1}{8}\left[\left(3v_3\right)-3v_3\right]& \left(5\right)\end{array}$$\begin{array}{cc}{}_3^o\frac{1}{3}\left(3v_3\right)-\frac{3}{8}\left[\left(3v_3\right)-3v_3\right]& \left(6\right)\end{array}$

[0039] Note that both conditions (5) and (6) are expressed in terms of the frequency parameters of the .sub.3-vibrational ladder defined above. Inequality (5) defines the minimum value the unharmonicity constant of the manifold origins X.sub.33 can have, and it is always negative. Inequality (6) defines the maximum value of the effective harmonic frequency .sub.3.sup.o in cm.sup.1. For heavy hexafluorides, such as UF.sub.6, PuF.sub.6, and WF.sub.6, these limiting values are extremely close to the actually observed values. As we move to the lighter hexafluorides, such as SF.sub.6, the discrepancy of the actual measured experimental values differ from the limiting values defined by inequalities (5) and (6), but not substantially.

[0040] It is thus possible to obtain very good values for the vibrational constants of the hexafluorides from simple spectroscopic measurements and recordings. In Table 1 a comparison of the limiting values of X.sub.33 and .sub.3.sup.o dictated by the frequency conditions (5) and (6) with their measured values is made. We notice that for the heavy hexafluorides the calculated values using various methods are extremely close to the measured values. The order of the hexafluorides

TABLE-US-00001 TABLE 1 Comparison of the limiting values of X.sub.33 and .sub.3.sup.o dictated by the frequency conditions with their measured values Hexa- fluoride At. No v.sub.3 (cm.sup.1) (3v.sub.3) (cm.sup.1) Number of Dissociation photons [00010]$X_{33}\frac{1}{3}\left[\left(3v_3\right)-3v_3\right]$ (cm.sup.1) X.sub.33 (cm.sup.1) [00011]$\text{?}\frac{1}{3}\left(3v_3\right)-\frac{3}{8}$ [(3v.sub.3) 3v.sub.3] (cm.sup.1) .sub.3.sup.o (calculated) .sub.3.sup.o (measured) .sup.239PuF.sub.6 94 620 1854.5(3) ~38 0.6875 0.683(10)* 620.23 620.22(3) .sup.238UF.sub.6 92 627.724 1877.532(10) 38 0.705 0.700(10) 627.96 627.944(4) .sup.235UF.sub.6 92 628.328 1879.344(10) 38 0.705 0.700(10) 628.563 628.548(4) .sup.195PtF.sub.6 78 705 2108.5(2) ~40 0.8125 0.805(10)* 705.271 705.25(4) .sup.186WF.sub.6 74 713.286 2133.0(3) ~41 0.857 0.790(10)* 713.572 713.37(4) .sup.98MoF.sub.6 42 743 2223.0(3) 44 0.75 0.720(20)* 743.25 743.15(5) .sup.130TeF.sub.6 52 753.128 2252.0(3) ~41 0.923 0.890(20)* 753.436 753.337(10) .sup.80SeF.sub.6 34 780.072 2332.0(5) 40 1.027 0.952(10)* 780.41 780.19(4) .sup.32SF.sub.6 16 948.103 2828.341(1) 34 1.99605 1.74256(15) 948.7684 948.008(4) .sup.34SF.sub.6 16 930.702 2776.138(1) 34 1.99605 1.74256(15) 931.378 930.6072(4) [00012]$\text{?}\text{indicates text missing or illegible when filed}$
is from the heaviest to the lightest taking them according to their central atoms in groups of (a) The inner transition metals (b) The transition metals (c) The non-metals. Numbers given in parentheses are the estimated error limits in units of the last figure quoted. Values marked * are calculated values obtained through the application of the Morse potential, the frequency conditions, the available published spectroscopic data and the straight line graph of X.sub.33 against (.sub.e.sup.2/D.sub.diss).

[0041] Extending the analysis of the basic equations further and employing eqs. (3)-(6) we can calculate all the anharmonic constants for the hexafluorides X.sub.33, G.sub.33 and T.sub.33. For the heavy hexafluorides UF.sub.6, PuF.sub.6 and WF.sub.6 these calculated values are extremely close to the experimentally measured values. It is not possible to give a complete theoretical derivation of the procedure, but for the sake of complicity the results are summarized in Table 2 for the UF.sub.6 molecule. Many more calculations were carried out and the results were checked with the most reliable experimental values available. For the heavy hexafluorides they were in extremely good accordance. As we move towards the lighter hexafluorides the agreement between the calculated and experimental values diminishes. In Table 2 we summarise the best available vibrational constants of the .sub.3-vibrational ladder of the two Uranium Hexafluoride isotopes .sup.238UF.sub.6 and .sup.235UF.sub.6 obtained through our calculations and the best available experimental measurements. Similar tables have been constructed for all the hexafluorides.

TABLE-US-00002 TABLE 2 The vibrational constants of the UF.sub.6 molecule Observed Q-branch frequency [00013]$\{\begin{array}{cccc}{}^{238}m\left(F_1\right)& =& 627.7019& {\mathrm{cm}}^{-1}\\ {}^{235}m\left(F_1\right)& =& 628.306& {\mathrm{cm}}^{-1}\end{array}$ Effective harmonic frequency [00014]$\{\begin{array}{ccccc}{}^{238}{}_3^o& =& 627.944& \left(3\right)& {\mathrm{cm}}^{-1}\\ {}^{235}{}_3^o& =& 628.548& \left(3\right)& {\mathrm{cm}}^{-1}\end{array}$ Frequency of pure vibrational energy of fundamental [00015]$\{\begin{array}{ccccc}{}^{238}{v}_3& =& 627.724& \left(3\right)& {\mathrm{cm}}^{-1}\\ {}^{235}{v}_3& =& 628.328& \left(3\right)& {\mathrm{cm}}^{-1}\end{array}$ Frequency of the origin of the fundamental vibrational level [00016]$\{\begin{array}{ccccc}{}^{238}\left(v_3\right)& =& 627.244& \left(5\right)& {\mathrm{cm}}^{-1}\\ {}^{235}\left(v_3\right)& =& 627.848& \left(5\right)& {\mathrm{cm}}^{-1}\end{array}$ Frequency of the origin of the third vibrational energy level [00017]$\{\begin{array}{ccccc}{}^{238}\left(3v_3\right)& =& 1877.53& \left(8\right)& {\mathrm{cm}}^{-1}\\ {}^{235}\left(3v_3\right)& =& 1879.34& \left(8\right)& {\mathrm{cm}}^{-1}\end{array}$ Unharmonicity constant of the manifold origins X.sub.33 = 0.70 (1) cm.sup.1 Unharmonicity coefficient of the vibrational angular momentum G.sub.33 = 0.24 (1) cm.sup.1 Unharmonicity constant related to the state of vibration T.sub.33 = 0.140 (5) cm.sup.1 Coriolis shift B.sub.3 = 0.01105 (5) cm.sup.1 Coriolis constant .sub.3 = 0.1986 (7) Rotational constant B.sub.3 0.05563 (4) cm.sup.1 B.sub.o 0.05567 (4) cm.sup.1 (The numbers given in parentheses are the estimated error limits in units of the last figure quoted. They have been estimated from the best measurements of the available references in the literature.)

[0042] The unharmonicity constants , , in the Cartesian representation are related to the unharmonicity constants in the Polar representation X.sub.33, G.sub.33 and T.sub.33, by [Harzer et al, Journal of Molecular Spectroscopy, Vol. 132, pp. 310-322, eqs. (2a,b,c), (1988)]

[00018]$\begin{array}{cc}\begin{array}{ccc}=X_{33}+4T_{33},& =2X_{33}-8T_{33}+2G_{33},& =-2T_{33}-G_{33}\end{array}& \left(7\right)\end{array}$

[0043] The manifold band structures originating from the pure vibrational energies of the levels from =0 to =4 are listed in Tables 3 and 4 in this notation. [Akulin et al, Soviet Physics, JEPT 45, pp 47-52, (1977)] These tables define the precise positions of the energy states and their sublevels of the .sub.3-vibrational ladder of the two UF.sub.6 isotopes which will be very distinct and clear when the UF.sub.6 gas is supercooled to very low temperatures. It is the differences in the frequencies of the lower energy states of the two UF.sub.6 isotopes which we must exploit for the selective dissociation of the desired isotope .sup.238UF.sub.6. Data for the structure of the energy states of the UF.sub.6 from =5 to =8 are also available in the literature. We have constructed similar tables for all the hexafluorides and compared and analysed their vibrational ladders.

[0044] We now proceed to demonstrate that for the heavy hexafluorides the properties of their lower states are a very close match to those of an ideal harmonic oscillator. The .sub.3-vibrational mode of the heavy hexafluorides, such as UF.sub.6 and PuF.sub.6, vibrates in a similar way to the asymmetric stretching mode of a linear molecule of the XY.sub.2 type (such as CO.sub.2), with the amplitude of the motion of the central atom being very small compared to that of the two axial F-atoms, and the amplitudes of the four equatorial F-atoms being virtually negligible by comparison to those of the two axial F-atoms. This type of vibration has six anharmonic constants X.sub.ij which are very nearly equal, with the magnitude of each of the anharmonic constants being equal to of that of the anharmonic constant of the equivalent diatomic molecule having the same vibrating frequency [Herzberg G., Molecular Spectra and Molecular Structure, Krieger Publishing Co, Vol. II, p. 206, (1991)]. The vibrational constant can be written as

[00019]$\begin{array}{cc}{X}_{33}-\frac{1}{6}{\stackrel{\_}{}}_e{x}_e& \left(8\right)\end{array}$

where .sub.e (cm.sup.1) [.sub.e.sub.3] is the Morse frequency and x.sub.e is a dimensionless constant to be defined below. To check its practical validity we solve the Schrdinger equation for a diatomic molecule using the Morse potential following the procedure of [Pauling L. and Wilson E. B., Introduction to Quantum Mechanics, p. 274, McGraw Hill, (1935)]. With the energies of the vibrational levels in reciprocal centimeters (cm.sup.1) the following relations are obtained:

[00020]$\begin{array}{cc}{\stackrel{\_}{}}_e=\frac{{}_M}{2c}\sqrt{\frac{2D}{}},x_e=\frac{{}_M}{2}\sqrt{\frac{1}{2D}}=\frac{h{\stackrel{\_}{}}_{e}c}{4D},B_e=\frac{h}{8{}^2{I}_{e}c},{\stackrel{\_}{}}_e{x}_e=\frac{{}_M^{2}h}{8{}^{2}c}& \left(9\right)\end{array}$

where the subscript e denotes equilibrium values, .sub.e.sub.3 (cm.sup.1) is the Morse frequency, D (ergs) is the dissociation energy, .sub.M (cm.sup.1) is the Morse constant, (gm) is the reduced mass, x.sub.e is dimensionless, B.sub.e (cm.sup.1) is the rotational constant and I.sub.e=r.sub.e.sup.2 is the equilibrium moment of inertia of the molecule. The product .sub.ex.sub.e (cm.sup.1) given by the last of the relations (9) is called the anharmonicity constant. From relations (8) and (9), the spectroscopic data and the Dissociation energy it is possible to determine the vibrational constants of the heavy Hexafluorides to a very high degree of accuracy. It is not possible to give the complete analysis in the short space of a patent application, but from the relations (9) we see that the unharmonicity constant of diatomic molecules, and subsequently of all molecules whose vibrational modes exhibit similar characteristics such as the heavy hexafluorides, is

[00021]$\begin{array}{cc}{\stackrel{\_}{}}_e{x}_e\frac{{\stackrel{\_}{}}_e^2}{D}& \left(10\right)\end{array}$

[0045] This proportionality is the result of introducing the Morse potential into the radial part of the Schrodinger equation and from relations (9) we see that it is independent of the Morse constant .sub.M. Thus, from the relations (8) and (10), for the .sub.3-vibrational mode of polyatomic molecules, the vibrational constant should be proportional to

[00022]$\begin{array}{cc}{X}_{33}\frac{{\stackrel{\_}{}}_e^2}{D}& \left(11\right)\end{array}$

[0046] For the UF.sub.6 molecule all the parameters for the .sub.3-vibrational mode in (11) have been very accurately measured. For the lightest of the hexafluorides SF.sub.6, again all the parameters for the

TABLE-US-00003 TABLE 3 The Manifold Symmetry Structure Of The Vibrational Ladder Of .sup.238UF.sub.6 (bracketed numbers in the (v.sub.3) column are the uncertainties in the manifold origins of the levels) (v.sub.3) Symmetry Coriolis (cm.sup.1) structure l QN Eigenvalue shift Ground (0v.sub.3) = A.sub.1 0 000 0 0 Level 0.00 1.sup.st Energy (1v.sub.3) = F.sub.1 1 100 1v.sub.3 = 627.724 cm.sup.1 2B.sub.3 Excitation 627.244 m(F.sub.1) = 627.7019 cm.sup.1 Level (0.02) 2.sup.nd Energy (2v.sub.3) = E 2 200 2v.sub.3 + 2 2 = 1252,848 cm.sup.1 0 Excitation 1253.088 m(E) = 1252.85 cm.sup.1 Level (0.03) 0 A.sub.1 0 200 2v.sub.3 +2 + 4 = 1253.088 cm.sup.1 m(A.sub.1) = 1253.09 cm.sup.1 F.sub.2 2 110 2v.sub.3 + = 1255.648 cm.sup.1 +2B.sub.3 m(F.sub.2) = 1255.67 cm.sup.1 3.sup.rd Energy Excitation Level (3v.sub.3) = 1877.532 (0.12) F.sub.1 1 300 [00023]$3v_3+4++-{\left[{\left(+-2\right)}^2+24{}^2\right]}^{\frac{1}{2}}=1875.6{\mathrm{cm}}^{-1}$$m\left(F_1\right)=1875.6{\mathrm{cm}}^{-1}$ 0 F.sub.2 3 210 3v.sub.3 + 2 + 2 2 = 1880.972 cm.sup.1 0 m(F.sub.2) = 1880.97 cm.sup.1 F.sub.1 3 210 [00024]$3v_3+4+++{\left[{\left(+-2\right)}^2+24{}^2\right]}^{\frac{1}{2}}=1881.14{\mathrm{cm}}^{-1}$$m\left(F_1\right)=1881.14{\mathrm{cm}}^{-1}$ 0 A.sub.2 3 111 3v.sub.3 + 3 = 1883.772 cm.sup.1 0 m(A.sub.2) = 1883.77 cm.sup.1 4.sup.th Energy Excitation Level (4v.sub.3) = 2500.576 (0.27) E 2 400 [00025]$4v_3+8+2--{\left[{\left(2-4-\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2495.772{\mathrm{cm}}^{-1}$$m\left(E\right)=2495.77{\mathrm{cm}}^{-1}$ 0 A.sub.1 0 400 [00026]$4v_3+8+2+2-{2\left[{\left(+-2\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2495.762{\mathrm{cm}}^{-1}$$m\left(A_1\right)=2495.76{\mathrm{cm}}^{-1}$ 0 F.sub.1 4 310 4v.sub.3 + 6 + 3 6 = 2503.696 cm.sup.1 0 m(F.sub.1) = 2503.70 cm.sup.1 F.sub.2 2 310 [00027]$4v_3+4+4+3-{\left[{\left(2-+3\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2504.169{\mathrm{cm}}^{-1}$$m\left(F_2\right)=2504.17{\mathrm{cm}}^{-1}$ 0 E 4 220 [00028]$4v_3+8+2-+{\left[{\left(2-4-\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2506.5796{\mathrm{cm}}^{-1}$$m\left(E\right)=2506.58{\mathrm{cm}}^{-1}$ 0 A.sub.1 4 220 [00029]$4v_3+8+2+2+{2\left[{\left(+-2\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2506.83{\mathrm{cm}}^{-1}$$m\left(A_1\right)=2506.83{\mathrm{cm}}^{-1}$ 0 F.sub.2 4 211 [00030]$4v_3+4+4+3+{\left[{\left(2-+3\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2509.383{\mathrm{cm}}^{-1}$$m\left(F_2\right)=2509.38{\mathrm{cm}}^{-1}$ 0

TABLE-US-00004 TABLE 4 The Manifold Symmetry Structure Of The Vibrational Ladder Of .sup.235UF.sub.6 (bracketed numbers in the (v.sub.3) column are the uncertainties in the manifold origins of the levels) (v.sub.3) Symmertry Coriolis (cm.sup.1) structure l QN Eigenvalue shift Ground (0v.sub.3) = A.sub.1 0 000 0 0 Level 0.00 1.sup.st Energy (1v.sub.3) = F.sub.1 1 100 1v.sub.3 = 628.328 cm.sup.1 2B.sub.3 Excitation 627.848 m(F.sub.1) = 628.306 cm.sup.1 Level (0.02) 2.sup.nd Energy (2v.sub.3) = E 2 200 2v.sub.3 + 2 2 = 1254.056 cm.sup.1 0 Excitation 1254.296 m(E) = 1254.06 cm.sup.1 Level (0.03) A.sub.1 0 200 2v.sub.3 + 2 + 4 = 1254.296 cm.sup.1 0 m(A.sub.1) = 1254.30 cm.sup.1 F.sub.2 2 110 2v.sub.3 + = 1256.856 cm.sup.1 +2B.sub.3 m(F.sub.2) = 1256.88 cm.sup.1 3.sup.rd Energy Excitation Level (3v.sub.3) = 1879.344 (0.12) F.sub.1 1 300 [00031]$3v_3+4++-{\left[{\left(+-2\right)}^2+24{}^2\right]}^{\frac{1}{2}}=1877.42{\mathrm{cm}}^{-1}$$m\left(F_1\right)=1877.42{\mathrm{cm}}^{-1}$ 0 F.sub.2 3 210 3v.sub.3 + 2 + 2 2 = 1882.784 cm.sup.1 0 m(F.sub.2) = 1882.78 cm.sup.1 F.sub.1 3 210 [00032]$3v_3+4+++{\left[{\left(+-2\right)}^2+24{}^2\right]}^{\frac{1}{2}}=1882.95{\mathrm{cm}}^{-1}$$m\left(F_1\right)=1882.95{\mathrm{cm}}^{-1}$ 0 A.sub.2 3 111 3v.sub.3 + 3 = 1885.584 cm.sup.1 0 m(A.sub.2) = 1885.58 cm.sup.1 4.sup.th Energy Excitation Level (4v.sub.3) = 2502.992 (0.27) E 2 400 [00033]$4v_3+8+2--{\left[{\left(2-4-\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2498.188{\mathrm{cm}}^{-1}$$m\left(E\right)=2498.19{\mathrm{cm}}^{-1}$ 0 A.sub.1 0 400 [00034]$4v_3+8+2+2-{2\left[{\left(+-2\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2498.178{\mathrm{cm}}^{-1}$$m\left(A_1\right)=2498.18{\mathrm{cm}}^{-1}$ 0 F.sub.1 4 310 4v.sub.3 + 6 + 3 6 = 2506.112 cm.sup.1 0 m(F.sub.1) = 2506.112 cm.sup.1 F.sub.2 2. 310 [00035]$4v_3+4+4+3-{\left[{\left(2a-+3\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2506.585{\mathrm{cm}}^{-1}$$m\left(F_2\right)=2506.58{\mathrm{cm}}^{-1}$ 0 E 4 220 [00036]$4v_3+8+2-+{\left[{\left(2-4-\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2508.9955{\mathrm{cm}}^{-1}$$m\left(E\right)=2508.{\mathrm{cm}}^{-1}$ 0 A.sub.1 4 220 [00037]$4v_3+8+2+2+{2\left[{\left(+-2\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2509.246{\mathrm{cm}}^{-1}$$m\left(A_1\right)=2509.25{\mathrm{cm}}^{-1}$ 0 F.sub.2 4 211 [00038]$4v_3+4+4+3+{\left[{\left(2-+3\right)}^2+24{}^2\right]}^{\frac{1}{2}}=2511.97{\mathrm{cm}}^{-1}$$m\left(F_2\right)=2511.97{\mathrm{cm}}^{-1}$ 0

TABLE-US-00005 TABLE 5 Vibrational parameters of the UF.sub.6 and SF.sub.6 molecules Parameter .sup.238UF.sub.6 .sup.32SF.sub.6 v.sub.3 627.724 (cm.sup.1 ) 948.103 (cm.sup.1) D 4.727593612 10.sup.12 6.396156063 10.sup.12 (ergs) (ergs) X.sub.33 0.70 0.05 (cm.sup.1) 1.74256 0.00015 (cm.sup.1 ) [00039]$\text{?}/D$ 8.334841201 10.sup.16 14.05374243 10.sup.16 (cm.sup.2/ergs) (cm.sup.2/ergs) indicates data missing or illegible when filed
.sub.3-vibrational mode in the proportionality (11) have been very accurately measured. These are tabulated in Table 5. A straight line going through the points (X.sub.33, .sub.e.sup.2/D) for UF.sub.6 and SF.sub.6 will specify the line on which all the points of the other hexafluorides should lie, according to the proportionality relation (11). Since the values of .sub.e.sup.2/D are available for all the hexafluorides, the values of X.sub.33 for the other hexafluorides can be obtained from the graph, to a high degree of accuracy for all practical applications.

[0047] The values of X.sub.33 thus obtained will then have to be compatible with those obtained from eq. (8) through the application of the equivalent Morse potential. They also have to be compatible with the frequency conditions imposed by the analysis of the structure of the vibrational ladder described above. The values of X.sub.33 obtained through the application of the equivalent Morse potential are expected to deviate more as we move onto lighter hexafluorides and for SF.sub.6 the deviation is expected to be substantial. Subsequently, the values of all the unharmonicity constants of the hexafluorides and the vibrational frequencies can be determined to a degree of accuracy which is more than sufficient for all practical applications.

[0048] FIG. 1 shows the graph of X.sub.33 against .sub.e.sup.2/D. The fundamental frequencies .sub.e.sub.3 (cm.sup.1) and the Dissociation energies D are those found in Table 5. The straight line graph for X.sub.33 is defined by the accurately measured values for UF.sub.6 and SF.sub.6 (black line). The other line (broken line) corresponds to the equivalent Morse unharmonicity constant ().sub.ex.sub.e. We see that for the heavy hexafluorides the two values are virtually identical. As we move onto lighter hexafluorides the deviation between the two values generally increases. MoF.sub.6 constitutes a slight exception due to its very high dissociation energy.

[0049] From the graphs in FIG. 1 the values of X.sub.33 for the other hexafluorides can be calculated to a high degree of accuracy which is more than sufficient for any practical application. Only hexafluorides with totally symmetric electronic ground states (A.sub.1g) have been included in the graph. The graphs of FIG. 1 are self evident, in particular for the heavy hexafluorides UF.sub.6 and PuF.sub.6, and no more elaborate analysis can be included in the short space of a patent application. The conclusion is clear that the vibration of the .sub.3-vibrational mode of the UF.sub.6 molecule is extremely close to that of an ideal harmonic oscillator. We will not provide any further analysis on this topic except pointing out the fact that the frequency spread of the manifold structures of the vibrational ladders of the hexafluorides (Tables 3 and 4 for the UF.sub.6 molecule) plotted against the square of the vibrational quantum number are straight lines. This testifies a very high accuracy to the frequency values of the sublevels in the manifolds given in tables 3 and 4.

[0050] The absorption of light with intensity I.sub.o propagating through an absorbing medium is governed by the equation I.sub.=I.sub.oe.sup.()1 with ()=S.sub.() where ()(m.sup.1) is the absorption coefficient, (mole/m.sup.3) is the density (concentration), S.sub.() (m.sup.2/mole) is the vibrational band strength at frequency , I (m) is the path length, (m.sup.1) is the wavenumber (frequency) in (m.sup.1). Then

[00040]$\begin{array}{cc}{S}_{\text{?}}=\underset{\mathrm{band}}{}{S}_{\text{?}}\left(v\right)\mathrm{dv}=\frac{1}{pl}\underset{\mathrm{band}}{}\ln \left(\frac{I_{\text{?}}}{I_{\text{?}}}\right)\mathrm{dv}& \left(12\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where S.sub.o is the vibrational band strength in m/mole and the integration is carried over the entire vibrational band, including all hot bands. The vibrational band strength S.sub. is obtained by frequency scanning the entire band according to eq. (12). Once the vibrational band strength S.sub. of a hot absorption band is found we proceed to obtain the dipole moment of a particular absorption band through the Integrated Absorption Coefficient .sub.if.sup.(t)(m.sup.2).

[0051] Further analysis results in the vibrational band strength in terms of the transition dipole moment

[00041]$\begin{array}{cc}{S}_{\text{?}}=\frac{8{}^3{N}_{\text{?}}{v}_{\text{?}}}{ch}\frac{1}{4{}_{\text{?}}}{.Math.{}_{\text{?}}^{\text{?}}.Math.}^2& \left(13\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where N.sub.A (mole.sup.1) is Avogadro's number, .sub.o.sub.fi is the transition frequency in (m.sup.1) (which is the average frequency over which the entire absorption band is measured), .sub.o (F/m) is the permittivity of free space, .sub.fi=.sub.f||.sub.i=e.sub.f|r|.sub.i (C.Math.m) is the dipole moment of the transition between states |.sub.j and |.sub.i, .sub.fi.sup.(z), (C.Math.m) is the induced dipole moment approximated to the statistical average of the z-component of .sub.fi (parallel to the applied electric field) and e (C) is the electronic charge.

[0052] The dipole moments for absorption at the fundamental vibration .sub.3 and absorption at the third energy excitation level 3.sub.3, are then given by

[00042]$\begin{array}{cc}{\left\{{.Math.{}_{\text{?}}^{\text{?}}.Math.}^2\right\}}_{\text{?}}=\frac{4{}_{\text{?}}\mathrm{ch}}{8{}^3{N}_{\text{?}}}\frac{{\left(S_{\text{?}}\right)}_{\text{?}}}{{\stackrel{\text{?}}{v}}_{\text{?}}}=\frac{1}{6.758359883{10}^{\text{?}}}\frac{{\left(S_{\text{?}}\right)}_{\text{?}}}{{\stackrel{\text{?}}{v}}_{\text{?}}}& \left(14\right)\end{array}$$\begin{array}{cc}{\left\{{.Math.{}_{\text{?}}^{\text{?}}.Math.}^2\right\}}_{\text{?}}=\frac{4{}_{\text{?}}\mathrm{ch}}{8{}^3{N}_{\text{?}}}\frac{{\left(S_{\text{?}}\right)}_{\text{?}}}{{\left({\stackrel{\text{?}}{v}}_{\text{?}}\right)}_{\text{?}}}=\frac{1}{6.758359883{10}^{\text{?}}}\frac{{\left(S_{\text{?}}\right)}_{\text{?}}}{{\left({\stackrel{\text{?}}{v}}_{\text{?}}\right)}_{\text{?}}}& \left(15\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0053] Eqs. (14) and (15) are similar to those derived in the literature [Fox and Person, Journal of Chemical Physics, Vol. 64 (12), pp. 5218-5221, (1976); Kim et al, Chemical Physics Letters, Vol. 104 (1), pp. 79-82, (1984)] but all three equations (13)-(15) are here derived and expressed in SI units. They facilitate the determination of the dipole moments through direct experimental observation. After searching most of the experimental results in the literature the most reliable values for the vibrational band strengths for the three hexafluorides UF.sub.6, SF.sub.6 and PuF.sub.6, are summarized in Table 6. We have carried out many comparisons and calculations between theory and experiments and established that these values conform with our equations and the available spectroscopic data but these are outside the scope of the present account.

[0054] From eqs. (14) and (15), and the vibrational band strengths from Table 6 we calculate the dipole moments of the transitions. These are summarized in Table 7. All the available experimental results on the vibrational band strengths and the dipole moments have been analysed and compared to the theoretical calculations. There is very close agreement for the heavy

TABLE-US-00006 TABLE 6 Vibrational Band Strengths Hexafluoride .sup.32SF.sub.6 .sup.2388UF.sub.6 .sup.239PuF.sub.6 Absorption v.sub.o m(F.sub.1).sub.(v.sub.3.sub.) = 947.9768 cm.sup.1 v.sub.o m(F.sub.1).sub.(v.sub.3.sub.) = 627.702 cm.sup.1 v.sub.o m(F.sub.1).sub.(v.sub.3.sub.) = 620 cm.sup.1 frequencies (v.sub.o).sub.(3v.sub.3.sub.) m(A.sub.2).sub.(3v.sub.3.sub.) = 2845.28 cm.sup.1 (v.sub.o).sub.(3v.sub.3.sub.) m(A.sub.2).sub.(3v.sub.3.sub.) = 1883.77 cm.sup.1 (v.sub.o).sub.(3v.sub.3.sub.) m(A.sub.2).sub.(3v.sub.3.sub.) = 1859.9 cm.sup.1 (S.sub.).sub.v.sub.3 (km mole.sup.1) 1067 702 (20) 524 (20) (S.sub.).sub.3v.sub.3 (km mole.sup.1) 0.048 0.017 (0.002) 0.022 (0.003) [00043]$\frac{{\left(S_{}\right)}_{3v_3}}{{\left(S_{}\right)}_{v_3}}$ 4.498 10.sup.5 2.242 10.sup.5 4.1985 10.sup.5

[0055] Hexafluorides and the discrepancies for the lighter hexafluorides have been accurately accounted for, but the detailed calculations are outside the scope of the present account.

[0056] We have analysed a large amount of information on the vibrational spectra of the Uranium Hexafluoride molecule at low temperatures available in the literature [for example, Aldridge et al, Journal of Chemical Physics, Vol. 83 (1), pp. 34-48, (1985); Krohn et al, Journal of

TABLE-US-00007 TABLE 7 Dipole Moments (Units: 1 Debye = 3.3333 10.sup.30 C .Math. m = 10.sup.18 esu .Math. cm) Hexafluoride .sup.32SF.sub.6 .sup.238UF.sub.6 .sup.239PuF.sub.6 Absorption v.sub.o m(F.sub.1).sub.(v.sub.3.sub.) = 947.9768 cm.sup.1 v.sub.o m(F.sub.1).sub.(v.sub.3.sub.) = 627.702 cm.sup.1 v.sub.o m(F.sub.1).sub.(v.sub.3.sub.) = 620 cm.sup.1 frequencies v.sub.o m(A.sub.2).sub.(3v.sub.3.sub.) = 2845.28 cm.sup.1 v.sub.o m(A.sub.2).sub.(3v.sub.3.sub.) = 1883.77 cm.sup.1 v.sub.o m(A.sub.2).sub.(3v.sub.3.sub.) = 1859.9 cm.sup.1 |.sub.v.sub.3.sup.(z)| C .Math. m 1.2835 10.sup.30 1.2864 10.sup.30 1.1183 10.sup.30 Debye 0.385 0.386 0.3355 |.sub.3v.sub.3.sup.(z)| C .Math. m 5 10.sup.33 3.654 10.sup.33 4.18356 10.sup.33 Debye 0.0015 0.0011 0.00126 [00044]$\frac{.Math.{}_{3v_3}^{\left(z\right)}.Math.}{.Math.{}_{v_3}^{\left(z\right)}.Math.}$ 3.874 10.sup.3 2.84 10.sup.3 3.741 10.sup.3
Molecular Spectroscopy, Vol. 132, pp. 285-309, (1988)]. At very low temperatures, below 80 K the Q.sub.A subbandhead becomes dominant with the Q.sub.A(J) components spreading towards the higher frequencies. On the lower frequency side the position of the Q.sub.G subbandhead is clearly discernible. The spread of the whole Q-branch subbandheads represents the width of one single quantum state available for a transition. Table 8 lists the wavenumbers at the edges of the subbandheads Q.sub.A-Q.sub.G for the .sub.3 fundamental band of the .sup.238UF.sub.6 molecule. The frequency difference between the Q.sub.A(62) line and the Q.sub.G line is 0.197 cm.sup.1 and for all practical purposes this can be taken to be the width of one single quantum state.

[0057] The 0.fwdarw.3.sub.3 overtone of .sup.238UF.sub.6 was also studied extensively. Both, the absorption spectra of the .sub.3-fundamental band and 3.sub.3-fundamental band are dominated by the seven strong absorption lines labelled A-G, whose spacing increases toward lower wavenumbers. There is a striking similarity between the spectra of the Q-branches of the .sub.3 and 3.sub.3 bands of the UF.sub.6 molecule. For the 3.sub.3 overtone band, however, the separation between the B and E edges is

TABLE-US-00008 TABLE 8 Subbandhead frequencies (cm.sup.1) for the .sub.3-vibrational mode Q-branch of the .sup.238UF.sub.6 and .sup.32SF.sub.6 molecules Subband .sup.238UF.sub.6 .sup.32SF.sub.6 head (cm.sup.1) (cm.sup.1) Q.sub.A(66) 627.766 Q.sub.A(62) 627.758 Q.sub.A(58) 627.750 Q.sub.A(54) 627.743 Q.sub.A(50) 627.736 Q.sub.A(46) 627.730 Q.sub.A(42) 627.724 Q.sub.A(38) 627.718 Q.sub.A(30) 627.712 Q.sub.A(22) 627.707 Q.sub.A 627.701 947.975 Q.sub.B 627.697 947.969 Q.sub.C 627.685 947.953 Q.sub.D 627.667 947.930 Q.sub.E 627.640 947.899 Q.sub.F 627.605 947.858 Q.sub.G 627.561 947.807 Q.sub.H 947.754

TABLE-US-00009 TABLE 9 Wavenumbers and Assignments for the UF.sub.6 3.sub.3 absorption overtone Subband- head line 3.sub.3 (cm.sup.1) Identification Q.sub.A(8), Q.sub.A(10) 1875.743 Q.sub.A(54) Q.sub.A(35), Q.sub.A(52) 1875.720 Q.sub.A(50) Q.sub.A(54) 1875.700 Q.sub.A(46) Q.sub.A(26), Q.sub.A(34) 1875.681 Q.sub.A(42) Q.sub.A(56) 1875.665 Q.sub.A(38) Q.sub.A(38) 1875.628 Q.sub.A(56) Q.sub.A(42) 1875.624 Q.sub.A(26), Q.sub.A(34) Q.sub.A(46) 1875.621 Q.sub.A(54) Q.sub.A(50) 1875.615 Q.sub.A(35), Q.sub.A(52) Q.sub.A 1875.598 Q.sub.A(8), Q.sub.A(10) Q.sub.B 1875.594 Q.sub.B(23), Q.sub.B(44) Q.sub.B 1875.589 Q.sub.B(13), Q.sub.B(15) Q.sub.C 1875.570 Q(24), Q(26) Q.sub.D 1875.542 Q(33), Q(35) Q.sub.E 1875.503 Q(44), Q(46) Q.sub.F 1875.453 Q(55), Q(57) Q.sub.G 1875.387 Q(72), Q(74)
0.086 cm.sup.1 as compared to 0.56 cm.sup.1 for the .sub.3 band. The band origin occurs near the peak A. At very low temperatures the Q.sub.A subbandhead becomes dominant, with the Q.sub.A(J) components spreading towards the higher frequencies. On the higher frequency side the spread of the Q.sub.A subbandhead gives discernible peaks up to, and even beyond, J=56. Table 9 lists all the wavenumbers and assignments for the 3.sub.3 absorption overtone. The total spread of the Q-branch subbandheads can be considered to be between the [Q.sub.A(56)Q.sub.G].sub.3.sub.3 peaks. This spread is 0.278 cm.sup.1. We also consider it to be valid for a three-photon step wise absorption where the selection rule l=1 such as is the case in multiphoton interaction processes.

[0058] In Table 10 the spread of the subbandheads from Q.sub.A to Q.sub.G for the .sub.3-fundamental and the 3.sub.3 overtone, for the three Hexafluorides .sup.238UF.sub.6, .sup.100MoF.sub.6 and .sup.32SF.sub.6 have been summarized. All the values listed are those obtained from the available experimental data. The spread of the subbandheads (Q.sub.A-Q.sub.G).sub..sub.3 for the .sub.3-fundamental is seen to increase as we move from the heavier to the lighter hexafluorides, whilst for the 3.sub.3-overtone it is seen to increase by even larger amounts. At low temperatures the effective subbandhead spread (Q.sub.A-Q.sub.G), including small considerations for the Q.sub.A subbandhead spread towards higher frequencies, can be taken to represent the FWHM of the lineshape factor g(.sub.f) of a normalized Lorentzian distribution centred near the Q.sub.A to Q.sub.B subbandheads.

[0059] From the recordings of the absorption spectra and Table 10 we can obtain very good estimates for the normalized lineshape function g(.sub.f) given by

[00045]$\begin{array}{cc}g.Math.v_f)=\frac{2}{v_f},g\left(v_o\right)=\frac{2}{v_o}& \left(16\right)\end{array}$

where .sub.f is the overall envelope of the distribution consisting of the series of subbandheads of the recorded spectra. The subscript f signifies the final level and the subscript o signifies the fundamental vibration. The value given by eq. (16) is approximate as we have taken it to represent the maximum of a normalized Lorentzian distribution at the centre of the prominent

TABLE-US-00010 TABLE 10 Spread of the subbandheads from (Q.sub.A-Q.sub.G) for the .sub.3-fundamental and the 3.sub.3 overtone, for the three Hexafluorides .sup.238UF.sub.6, .sup.100MoF.sub.6, and .sup.32SF.sub.6 .sub.3 - fundamental (.sub.o).sub..sub.3 = 3.sub.3-overtone (Q.sub.A-Q.sub.G).sub..sub.3 (.sub.o).sub.3.sub.3 = Hexa- (effective) (Q.sub.A-Q.sub.G).sub.3.sub.3 fluoride cm.sup.1 cm.sup.1 .sup.238UF.sub.6 0.197 0.278 .sup.100MoF.sub.6 ~0.2 .sup.32SF.sub.6 0.23 ~0.54
subbandheads (Q.sub.A) in the case of supercooled UF.sub.6 gas), equal to the frequency difference between the extended spread of the Q.sub.A and the Q.sub.G edges. This spread of the whole Q-branch subbandheads represents the width of one single quantum state available for a transition. For all practical purposes, it gives a very sound value for experimental calculations. In the cases of higher order electromagnetic interactions, it can also set a lower practical limit for the value of the resonant denominators which is now determined by the value of g(.sub.f) corresponding to the smearing out of the final energy state.

[0060] By considering the spontaneous transition coefficient A and the induced transition coefficient B [Shimoda K., Introduction to Laser Physics, second edition, Springer-Verlag, pp. 78-84, eqs. (4.34) and (4.37), (1986); Weissbluth M., Photon-Atom Interactions, Academic Press, pp. 226-232, eqs. (5.154) and (5.177), (1989)] we have expressed the induced dipole moment for a two-level quantum system .sub.10.sup.(z) in terms of its characteristic parameters only, i.e. the frequency .sub.k and the spontaneous lifetime of the upper level t.sub.spon (in SI units):

[00046]$\begin{array}{cc}{.Math.{}_{\text{?}}^{\text{?}}.Math.}^2=\left(\frac{g_{\text{?}}}{g_{\text{?}}}\right)\frac{{}_{\text{?}}{c}^3}{{}_k^3{}^{\text{?}}{t}_{\text{?}}}& \left(17\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where g.sub.1 and g.sub.0 are the degeneracies of the two levels and as in all our expressions for the dipole moment and the spontaneous lifetime we include a power of the refractive index in accordance with the corresponding power of the speed of light c (=1). From eq. (17) we note that the lineshape factor g(.sub.o) does not enter into the expression for the dipole moment.

[0061] The induced absorption transition rate W.sub.01 for a two-level quantum system is given by [Yariv A., Quantum Electronics, second edition, John Wiley & Sons, pp. 162-165, eqs. (8.5-15), (1975)], (in SI units)

[00047]$\begin{array}{cc}{W}_{01}=W_{10}\left(\frac{g_{\text{?}}}{g_{\text{?}}}\right)=\left(\frac{g_{\text{?}}}{g_{\text{?}}}\right)\frac{{}^2}{8\mathrm{hv}{}^2}\frac{1}{t_{\mathrm{span}}}g\left(v_{\text{?}}\right)I_{\text{?}}& \left(18\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where (m) is the wavelength, (s.sup.1) is the frequency of the applied radiation, t.sub.spon (s) is the spontaneous lifetime of the upper level, is the refractive index and

[00048]$I_{\text{?}}\left(J/m^{2}s\right)$$\text{?}\text{indicates text missing or illegible when filed}$

is the intensity of the applied beam. g(.sub.o) is the lineshape function in (s) resulting from the smearing out of the position of the final energy state , with .sub.o being the width at half the maximum of the absorption spectrum of the level in (s.sup.1). For an equivalent two-level system of the ground to first energy excitation level of Uranium Hexafluoride (this would be for example the case for a weak probe beam used for absorption such as a diode laser) we have from Table 10 that =0.197 cm.sup.1, giving g(.sub.o)=1.0779365410.sup.10 s. Using the other fundamental constants for the UF.sub.6 molecule .sub.k=1.18350958210.sup.14 s.sup.1, t.sub.spon=0.087 s in eq. (27) we obtain (W.sub.01).sub.UF.sub.6=3.0017 I.sub..sub.k. This would be the transition rate for absorption between the ground and the first excitation level of the UF.sub.6 molecule had there been no further excitation up the .sub.3 mode vibrational ladder. Introducing into eq. (18) the equivalent parameters for the SF.sub.6 molecule we obtain (W.sub.01)=1.85058 I.sub..sub.k from which we see that the equivalent transition rate at the fundamental would be smaller for the lighter hexafluoride than for the heavier one.

[0062] On applying elementary quantum mechanical principles to the theory of the harmonic oscillator we have obtained a simple expression for the transition dipole moment

[00049]${}_{10}^{\left(z\right)}$

of the fundamental of a molecular system:

[00050]$\begin{array}{cc}{}_{10}^{\left(z\right)}=\frac{1}{\sqrt{3}}.Math.{}_{10}.Math.=\frac{e}{\sqrt{3}}.Math.{}_{\text{?}}.Math.r.Math.{}_{\text{?}}.Math.=\frac{e}{\sqrt{3}}{r}_{10}=\frac{e}{16\sqrt{3}}{\left(\frac{}{2m{}_k}\right)}^{1/2}& \left(19\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0063] This is a very simple expression for the dipole moments of molecules whose vibrational ladder structure is very close to that of a harmonic oscillator. It gives remarkably accurate values to the experimental ones obtained from the measurement of their vibrational band strengths (eq. (12)). Thus, the dipole moments of the .sub.3 vibrational mode of heavy hexafluorides can be calculated from knowledge only of the vibrational frequency of the fundamental and the three fundamental constants e, m and . All dipole moments are of the same magnitude. We have constructed tables for the hexafluorides comparing the results obtained from eq. (19) with the experimental values quoted in the literature.

[0064] For heavy hexafluorides, particularly for UF.sub.6, the agreement is extremely good. In particular, the value of

[00051]${}_{10}^{\left(z\right)}$

for the UF.sub.6 molecule calculated from eq. (19) is extremely close to the experimental value in Table 7 obtained from the measurement of the vibrational band strength through eq. (14). On the contrary, for the lightest of the hexafluorides SF.sub.6 there is a much larger discrepancy between the experimental and calculated values from eq. (19).

[0065] We derived two different expressions, eqs. (17) and (19), for the induced dipole moment .sub.01.sup.(z) of the fundamental transition of a harmonic oscillator. The first one was derived through the spontaneous and induced transition coefficients A and B and the second through the application of elementary harmonic oscillator theory. These two expressions, however, must be equivalent. On comparing the two expressions we obtain

[00052]$\begin{array}{cc}{t}_{\mathrm{span}}=\left(\frac{g_{\text{?}}}{g_{\text{?}}}\right)\frac{1536{}^{\text{?}}m{}_{\text{?}}{c}^3}{e^2{}_k^2{}^3}& \left(20\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

from which we see that for a molecular system whose energy levels are a very close match to those of a harmonic oscillator the spontaneous lifetime of the first excited level depends only on the inverse of the square of the frequency difference .sub.k between the two levels, and the degeneracy of the levels. For the .sup.238UF.sub.6 molecule the frequency of the fundamental .sub.k has been very accurately measured to be .sub.k=627.7019 cm.sup.1=1.18237166810.sup.14 s.sup.1. The degeneracy of the upper level (=1) of the .sub.3 vibrational mode is g.sub.1=3 and thus (g.sub.1/g.sub.o)=3. Substituting these values in eqs. (19) and (20) we obtain

[00053]$.Math.{}_{10}^{\left(z\right)}.Math.=1.2876{10}^{-30}Cm$

and t.sub.spon=0.086523 s.

[0066] The dipole moment and the spontaneous lifetime of the first energy excitation level of UF.sub.6 have been accurately measured to be

[00054]$.Math.{}_{10}^{\left(z\right)}.Math.=\left(1.280.012\right)10^{-30}Cm$

and t.sub.spon=(0.0860.003) s [Kim K. C. and Person W. B., Journal of Chemical Physics, Vol. 74(1), pp. 171-178, (1981)] and we see that the values obtained from eqs. (19) and (20) are in very good agreement with the measured experimental values. This further demonstrates the closeness of the vibrational ladder of the UF.sub.6 molecule to that of an ideal harmonic oscillator. As we move to the molecules of lighter hexafluorides the discrepancies between the measured experimental values and those calculated increase. The above equations for the dipole moments pertain only to molecules whose vibrational ladder is a very close match to that of a harmonic oscillator such as the very heavy hexafluorides.

[0067] The fundamental quantity which determines the strength of the interaction between matter and radiation is the fine structure constant .sub.fs. From eq. (20) we can obtain an expression for the spontaneous lifetime of an excited quantum system whose energy levels are a very close match to a harmonic oscillator in terms of the fine structure constant:

[00055]$\begin{array}{cc}\frac{1}{t_{\mathrm{span}}}=\frac{4{}_{\text{?}}{}^2}{c^2}\frac{{}_k^3}{e^2}\left(\frac{g_{\text{?}}}{g_{\text{?}}}\right){.Math.{}_{10}^{\left(z\right)}.Math.}^2& \left(21\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0068] This is in perfect agreement with that given in the theoretical literature [Weissbluth M., Photon-Atom Interactions, Academic Press, p. 232, eq. (5.177), (1989)]. It corroborates that all calculations leading up to equations (17) to (21) for the dipole moment and the spontaneous lifetime of the levels, are valid and consistent with one another.

[0069] An atomic state cannot be an infinitely sharp state but must have a finite energy spread which will be limited by a level width =2 where is the spectral width of the spontaneous emission line of the transition and is interpreted as the level width. It is the minimum possible width assuming all other broadening mechanisms have been eliminated. We have investigated all other broadening mechanisms of the energy levels but they are all negligible compared to the power broadening of the transitions occurring during the interaction in isotope separation processes.

[0070] From eqs. (20) and (21) we see that for a harmonic oscillator model

[00056]${}_k^3{t}_{\mathrm{span}}=\mathrm{constant},{}_k^3{t}_{\mathrm{span}}{.Math.{}_{10}^{\left(z\right)}.Math.}^2=\mathrm{constant}\mathrm{and}{.Math.{}_{10}^{\left(z\right)}.Math.}^2=\mathrm{constant},$

the last relation resulting from the correspondence relation between classical quantities and the quantum mechanical quantities, and all three relations are perfectly compatible amongst themselves. We have elaborated on the above relations and in conjunction with the available experimental values of the hexafluorides whose values have been very accurately measured we obtained the possible experimental values of the spontaneous lifetimes and the dipole moments of the other hexafluorides. These are in very good agreement with the values available in the literature but further details are outside the scope of the present patent application.

[0071] We have derived the absorption coefficient of a classical harmonic oscillator in terms of the fundamental physical constants and also the absorption coefficient of a quantum mechanical oscillator in terms of the dipole moment of system. The two expressions must be completely equivalent and on comparing the two we can establish the correspondence of the classical quantity (e.sup.2/m) to its quantum mechanical equivalent in terms of the dipole moment. The value of the classical lifetime of a perfect harmonic oscillator was found to be

[00057]$\begin{array}{cc}\frac{1}{t_{\mathrm{class}}}=2.fwdarw.\frac{1}{3t_{\mathrm{span}}}=\frac{2}{3}\frac{1}{4{}_e}\frac{{}_o^2{}^3}{c^3}\frac{g_o}{g_1}\frac{2{}_o}{3}{.Math.{}_{10}.Math.}^2=\frac{g_o}{g_1}\frac{{}_o^2{e}^2{}^3}{4608{}^3{}_{o}m{c}^3}& \left(22\right)\end{array}$

[0072] Consider the case of the .sub.3-vibrational mode of the Uranium Hexafluoride molecule. If we substitute .sub.o=1.18237185610.sup.14 s.sup.1 and the numerical value of the degeneracies of the two levels (g.sub.o/g.sub.1)=, we obtain 1/t.sub.class=3.852528 s.sup.1 or t.sub.spon=0.086523 s, a value in perfect agreement with that obtained above through the application of eq. (20). This value for t.sub.spon is in very good agreement with the experimentally measured values reported in the literature [Kim K. C. and Person W. B., Journal of Chemical Physics, Vol. 74 (1), pp. 171-178, (1981)]. The above results demonstrate the consistency of both, the classical and quantum equations used in the calculation of the parameters for the heavy hexafluorides. All the equations given above have been extensively tested with all the available experimental results in the literature and were found to give extremely close values for the heavy hexafluoride molecules.

[0073] Through our derivation of the absorption coefficient we have obtained an expression in terms of the linear frequency, for the absorption cross section of a two-level quantum particle whose vibration is equivalent to that of a perfect harmonic oscillator as

[00058]$\begin{array}{cc}\left(v\right)=\frac{g_o}{g_1}\frac{{}_o}{2{}_{o}c}{.Math.{}_{01}^{\left(z\right)}.Math.}^{2}g\left(v\right)& \left(23\right)\end{array}$

with the value of

[00059]${.Math.{}_{01}^{\left(z\right)}.Math.}^2$

given by eq. (19), or its value can be read off Table 7. Eq. (23) is in agreement with expressions given in the literature [Judd O. P., Journal of Chemical Physics, Vol. 71, No 11, p. 4515, [eq. (17)], [FIG. 1], (1979)]. We see from eq. (23) that the absorption cross section depends only on the lineshape function g(). Substituting the values of

[00060]${.Math.{}_{01}^{\left(z\right)}.Math.}^2$

and .sub.o for the UF.sub.6 .sub.3-mode fundamental vibration from the tables we obtain (.sub.o)=1.16713244210.sup.10 g(.sub.o). The lineshape function g(.sub.o) is given by eq. (16), where (.sub.o).sub..sub.3 is the spread of subbandheads of the Q-branch spectrum listed in Table 10. For the heavy hexafluorides the absorption cross sections can be calculated directly from eq. (23) by substituting the effective spread of the subbandheads (.sub.o).sub..sub.3=Q.sub.A-Q.sub.G representing one single quantum state. For the lighter hexafluorides the absorption cross sections can best be calculated from eq. (23) using the measured values of the induced dipole moments. In Table 11 we summarised the absorption cross sections of some of the basic hexafluorides where the values of (.sub.o).sub..sub.3 are taken from Table 10.

[0074] A similar equation holds for the overtone absorption cross section of the third energy excitation level, where the lineshape function (g(.sub.o)).sub..sub.3 corresponds to ().sub..sub.3 the effective spread of the subbandheads of the third energy excitation level Q-branch spectrum and is listed in Table 10. For the third energy excitation level the degeneracies of the Hexafluorides are g.sub.o=1 and g.sub.3=10, the frequency .sub..sub.3 is given by the m(F.sub.1) line of the 3.sub.3 level Table 4, and the measured dipole moment

[00061]$.Math.{}_{3v_3}^{\left(z\right)}.Math.$

can be found from Table 7. The results are outside the scope of the present account.

TABLE-US-00011 TABLE 11 Absorption Cross Sections of the Fundamental for Some Hexafluorides Subbandhead Absorption cross spread section Absorption cross .sub.3-mode (effective) (calculated, section (from measured frequency (.sub.o).sub.v.sub.3 = Lineshape eq. (25) dipole moments eq. (23)) Hexa- .sub.o (Q.sub.A-Q.sub.G).sub.v.sub.3 function .sub.01(.sub.o) .sub.01(.sub.o) fluoride (10.sup.14 s.sup.1) (10.sup.9 s.sup.1) g(.sub.o) (10.sup.20 m.sup.2) (10.sup.20 m.sup.2) Molecule (cm.sup.1) (cm.sup.1) (10.sup.10 s) (10.sup.16 cm.sup.2) (10.sup.16 cm.sup.2) .sup.238UF.sub.6 1.82371856 5.905911423 1.07793654 1.258095 1.25575 (627.702) (0.197) .sup.100MoF.sub.6 1.397669463 5.99584916 1.0617675 1.2392233 (742) (~0.2) .sup.32SF.sub.6 1.785658362 6.385579355 0.996964781 1.16359 1.765214 (947.977) (0.213)

[0075] We have carried out extensive investigations on the results and the measurements of the absorption cross section available in the literature using powerful probing sources. These bear no consequences to laser isotope separation processes as they pertain to the absorption cross section for the whole vibrational ladder of the molecules. Eq. (23) is not applicable to these cases. There is a clear trend, however, for the absorption cross section of the molecules to increase rapidly as the fluence of the laser beam is lowered.

[0076] In order to obtain reliable values for the absorption cross section of the fundamental of the UF.sub.6 molecule, low fluence probing beams must be employed. Spectrophotometer measurements of the absorption cross section of the UF.sub.6 gas at the fundamental frequency of the .sub.3-vibrational mode were carried out [Maier II W. B., Holland R. F. and Beattie W. H., Journal of Chemical Physics, Vol. 79, No 10, pp. 4794-4804, Table 1, (15, November 1983)]. These resulted in a value for the absorption cross section of the fundamental transition of 1,15510.sup.20 m.sup.2 (1.15510.sup.20 cm.sup.2). This value is extremely close to the value for the cross section of the UF.sub.6 molecule calculated from the harmonic oscillator analysis from eq. (23) and listed in Table 11 ([(.sub.o)].sub.cal.=1.25810.sup.20 m.sup.2). Note that the calculated value is well within the 10% estimated experimental error claimed. We see that in the case of very heavy hexafluorides, such as UF.sub.6, eq. (23) gives values for the absorption cross section which are completely consistent with the observed experimental results. As a further check, in FIG. 2 we have extended the experimental graph of the absorption cross section of the UF.sub.6 molecule at a gas temperature of 105 K and a frequency of 627.6 cm.sup.1 as a function of laser fluence [Alexander et al, Journal de Chimie Physique, Vol. 80, No 4, pp. 331-337, (1983)]. The experimental points are marked by solid rhombus with the absorption cross section increasing steadily as we go towards lower fluences according to the law

[00062]${\left({}_{\mathrm{IR}}\right)}^{-\frac{1}{2}}$

[or .sub.IR(<n>.sub.qu).sup.2 where <n>.sub.qu is the average number of photons actually absorbed by the molecule in the quasicontinuum]. This relation holds for pumping fluence values between 210.sup.1 J/m.sup.2 and 210.sup.2 J/m.sup.2 (see below for a statistical interpretation and also the effects at higher pumping fluences). The graph is on a logarithmic scale. At low fluences of magnitude 410.sup.1 J/m.sup.2 the absorption cross section of UF.sub.6 approaches an experimentally recorded value of 210.sup.21 m.sup.2. When the graph is extended to lower fluences it is seen that at a value of 1.410.sup.2 J/m.sup.2 the absorption cross section is 110.sup.20 m.sup.2 and at even lower fluences of the pumping beam it approaches steadily the calculated value for the absorption cross section of the fundamental of an ideal harmonic oscillator, eq. (23), and the values obtained through the use of an infrared spectrophotometer (triangular point).

[0077] On substituting eq. (19) into eq. (13), we obtain

[00063]$\begin{array}{cc}{S}_o=\frac{N_A}{768}\frac{1}{4{}_o}\left(\frac{e^2}{m{c}^2}\right)=\frac{N_A}{768}{r}_e& \left(24\right)\end{array}$

where N.sub.A is Avogadro's number in mole.sup.1 and r.sub.e=2.8179409210.sup.15 m is the classical radius of the electron. Eq. (24) is a very simple and straightforward result for the vibrational band strength of a molecule whose oscillation is very close to that of a perfect harmonic oscillator. In this case the vibrational band strength depends only on two standard physical constants, Avogadro's number and the classical radius of the electron. On substituting the constants in eq. (24) we obtain the vibrational band strength of the fundamental of a perfect harmonic oscillator to be S.sub.=703.3503114 km mole.sup.1. This is an extremely close value to the experimentally measured value for the vibrational band strength of the .sup.238UF.sub.6 molecule in Table 6. It suggests that the vibration of the lower levels of the 13-mode of the UF.sub.6 molecule is a very close match to the vibration of the energy levels of an ideal harmonic oscillator.

[0078] On expressing the Integrated Absorption coefficient

[00064]${}_{\mathrm{if}}^{\left(l\right)}\left(v\right)\left(m^{-2}\right)$

through the vibrational band strength eqs. (12) and (24) we obtained an expression for the linear absorption coefficient () (m.sup.1) in terms of the lineshape function g(.sub.o) (s), eq. (16), and subsequently for the absorption cross section as

[00065]$\begin{array}{cc}\left(v\right)=\frac{{}_{01}\left(v\right)}{N}=\frac{g_o}{g_1}g\left(v_o\right)\frac{c}{768}{r}_e=\frac{g_o}{g_1}\frac{2}{v}\frac{c}{768}{r}_e& \left(25\right)\end{array}$

where () (m.sup.2) is the absorption cross section, N (m.sup.3) is the number of molecules per unit volume, a.sub.01() (m.sup.1) is the absorption coefficient, (s.sup.1) is the linewidth of the transition, c (m/s) is the velocity of light and r.sub.e (m) is the classical radius of the electron. If we substitute the value of the classical radius of the electron, eq. (25) reduces to eq. (23). Eq. (23) had been derived through the concept of the dipole moment and the harmonic oscillator vibrational parameters, whilst eq. (25) had been arrived at through the concepts of the vibrational band strength and the Integrated Absorption Coefficient. Both approaches lead to the same result for the absorption cross section demonstrating the consistency of the results.

[0079] For the UF.sub.6 molecule if we substitute =0.197 cm.sup.1=5.90591142310.sup.9 s.sup.1 from Table 10 (the effective spread from the A to the G subbandheads at low temperatures) we obtain the absorption cross section of the fundamental to be .sub.238=1.25809470710.sup.20 m.sup.2, the same value as the one obtained from eq. (23) above, and tabulated in Table 11. We have obtained many more equations which give values compatible with the experimental results. All the equations derived above, expressed in SI units and incorporating all the necessary characteristics of the levels such as degeneracies, are fully compatible amongst themselves. The values of the various parameters of the heavy hexafluorides obtained through their application are very close to their experimentally measured values. Their extremely close agreement signifies that the properties of the lower levels of the .sub.3-mode vibrational ladder of the heavy hexafluorides are very close to those of an ideal harmonic oscillator. Thus, all the parameters of the .sub.3-mode vibrational ladder of the UF.sub.6 molecule such as the frequencies, vibrational constants (eqs. (3)-(7)), level and sublevel positions (Tables 3 and 4), the vibrational band strength (eqs. (12, (13) and (24)), the absorption coefficient and the absorption cross sections (eqs. (23) and (25)), the dipole moments (eqs. (14), (15), (17) and (19)), the spontaneous lifetime of the first excited level (eqs. (20) and (21)) and many others, can be obtained through the above equations and they are all fully compatible with the measured experimental results. We have developed further techniques for obtaining the parameters of all the hexafluorides even the lighter ones. It is not possible to give a detailed analysis of all results in the short space of a patent application.

[0080] A detailed analysis of the Rabi theory when a two state quantum particle interacts with a radiation field results in the expression for the power broadening of a spectral line given by (see eq. (1) above)

[00066]$\begin{array}{cc}{v}_P=\frac{{}_{01}^{\left(z\right)}{E}_o}{}& \left(26\right)\end{array}$

where

[00067]${}_{01}^{\left(z\right)}$

is the dipole moment and .sub.P is the Full Width at Half the Maximum of the power broadened transition. E.sub.o is the electric field of the radiation. The probability of a transition for a two-level system under the interaction with a powerful electromagnetic beam, taking into account the power broadening of the levels is

[00068]$\begin{array}{cc}{.Math.c_1\left(t\right).Math.}^2=\frac{{\left(\frac{v_P}{2}\right)}^2}{{\left(v_{01}-v\right)}^2+{\left(\frac{v_P}{2}\right)}^2}=\frac{1}{1+{\left(\frac{v_{01}-v}{\frac{v_P}{2}}\right)}^2}& \left(27\right)\end{array}$

where =2.sub.P. From the field strength E.sub.o and the dipole moment .sub.01 we can calculate .sub.P and we can plot the probability amplitude of the transition. The curve of |c.sub.t(t)|.sup.2 as a function of is a near Bell-shaped Lorentzian curve and has a peak centred at (-.sub.01) with a full width at half the maximum .sub.P given by eq. (26). In the cases of closely spaced isotopes we can plot the curves for the transitions of both isotopes and see the overlapping occurring for various values of the intensity of the pumping radiation.

[0081] The intensity of the beam of a plain sinusoidal electromagnetic wave I as it propagates through a medium can be obtained through the average value of the Poynting vector as

[00069]$\begin{array}{cc}\begin{array}{ccccc}I=\frac{1}{2}& & c& {}_o& {.Math.E_o.Math.}^2\end{array}& \left(28\right)\end{array}$

where it was assumed that the period of oscillation is very rapid, the relative permeability K.sub.m=1, .sub.o=8.854187810.sup.12 F/m is the permittivity of free space and n is the refractive index of the medium. We have elaborated greatly on the propagation and the structure of the laser beams as they travel through the supercooled molecular gas within the Rayleigh range

[00070]$z_o=\frac{w_o^2}{}$

but the details are outside the scope present account.

[0082] We have calculated and tabulated the power broadening of the fundamental transitions of the three molecules UF.sub.6, MoF.sub.6 and SF.sub.6 under the interaction with a powerful electromagnetic beam. With the vibrational ladder of the UF.sub.6 molecules being a very close match to that of an ideal harmonic oscillator, it is imperative that the selectivity of the desired isotope is achieved over the first few vibrational levels. FIG. 3 shows the power broadened absorption probabilities of the fundamental for the two Uranium Hexafluoride isotopes .sup.238UF.sub.6 and .sup.238UF.sub.6, for four different pumping beam intensities ranging from 1010.sup.9 W/m.sup.2 to 6010.sup.9 W/m.sup.2. It demonstrates how the absorption probability of the ground to first energy level of the UF.sub.6 isotopes broadens up as the pumping is increased. The peaks of the curves are located at the centre of the absorption lines. We see that as we increase the pumping intensities the widths of the absorption curves increase rapidly. When the frequency of the pumping beam is set at the absorption frequency of the desired isotope .sup.235UF.sub.6 at 628.306 cm.sup.1 we see that for low intensities the unwanted isotope .sup.238UF.sub.6 is hardly absorbent (first two graphs (a) and (b) in FIG. 3). As the intensity of the pumping beam is increased further we see from the successive graphs that the absorption probability of the unwanted isotope .sup.238UF.sub.6 increases rapidly. But even at high powers (6010.sup.9 W/m.sup.2) it does not severely affect the selectivity of the desired isotope. The vertical lines indicate frequencies between 628.306 cm.sup.1 and 628.7 cm.sup.1 and their relation to the power broadened curves of the fundamental.

[0083] From the standard theory of a harmonic oscillator model of a vibrating molecule [Weissbluth M., Atoms and Molecules, Academic Press, New York, p. 234, (1978)] we have computed the matrix elements between adjacent energy levels and obtained the identity

[00071]$\begin{array}{cc}\sqrt{\frac{m}{}}.Math.{}_{{}^{}}.Math.r.Math.{}_{}.Math.=\sqrt{\frac{+1}{2}}{}_{{}^{},+1}+\sqrt{\frac{}{2}}{}_{{}^{},-1}& \left(29\right)\end{array}$

with ==1. In practice the vibrations of the molecule are anharmonic and the selection rule is replaced by =1, 2, 3, . . . . The most intense absorption band is the fundamental 0.fwdarw.1 transition. Bands corresponding to transitions 0.fwdarw.2, 0.fwdarw.3 etc are called overtones and in general they are very weak. Eq. (29) can be used to calculate the matrix elements between adjacent levels of the vibrational ladder corresponding to a harmonic oscillator, and subsequently the dipole matrix elements between adjacent levels. The results indicate that the value of .sub.|r|.sub. increases according to {square root over (1)}, {square root over (2)}, {square root over (3)}, 4, 5, 6, . . . with increasing vibrational number and the dipole moments between successive higher vibrational levels of a harmonic oscillator increase accordingly with the square root of increasing vibrational number. We will elaborate no more on the subject here.

[0084] To a first approximation the power broadening of a transition line is given by eq. (26). On taking the value of the dipole moment of the UF.sub.6 molecule to be 1.28510.sup.30 C.Math.m (Table 7) we have calculated the power broadening of the first six vibrational levels at various pumping intensities. The electric field was calculated using eq. (28). The results are summarized in Table 12. We have investigated all the deviations from the sublevels of the first, second, third and fourth energy states when pumping at the frequency of the fundamental 628.306 cm.sup.1 and at the frequency corresponding to the exact three-photon resonance with the third energy excitation sublevel [m(A.sub.2):(3.sub.3)] of the desired isotope .sup.235UF.sub.6 at 528.527 cm.sup.1 (Table 4). The conclusion was clear that the three-photon absorption process is determined by the power broadening of the first energy level at this frequency. Comparison with the values of Table 12 shows that the mismatch of four-photon resonance with the fourth energy excitation level is even greater when pumping at 628.527 cm.sup.1 than at 628.306 cm.sup.1 and it is well outside the power broadened curve of the level. Note that the successive photon absorption up the sublevels of the first three vibrational levels of the UF.sub.6 molecule corresponds to perfectly allowed transitions with =1.

TABLE-US-00012 TABLE 12 PUMPING POWERS AND POWER BROADENING OF THE UF.sub.6 ENERGY LEVELS FOR VARIOUS PUMPING INTENSITIES AND FOR SPECIFIC BEAM CHARACTERISTICS Pumping beam parameters: pulse duration = 20 10.sup.9 s beam radius r = 0.004 m = 4 10.sup.3 m Power broadening of the first eight levels 1 2 3 4 5 6 Intensity (I.sub.o) Energy ( ) Electric field [00072]$v_{10}=\frac{\begin{array}{cc}& E_o\end{array}}{}$ [00073]$v_{21}=\frac{\begin{array}{cc}\sqrt{2}& E_o\end{array}}{}$ [00074]$v_{32}=\frac{\begin{array}{cc}\sqrt{3}& E_o\end{array}}{}$ [00075]$v_{43}=\frac{\begin{array}{cc}\sqrt{4}& E_o\end{array}}{}$ [00076]$v_{54}=\frac{\begin{array}{cc}\sqrt{5}& E_o\end{array}}{}$ [00077]$v_{65}=\frac{\begin{array}{cc}\sqrt{6}& E_o\end{array}}{}$ 10.sup.9 10.sup.3 (E.sub.o) 10.sup.6 (s.sup.1) (s.sup.1) (s.sup.1) (s.sup.1) (s.sup.1) (s.sup.1) (W/m.sup.2) (J) (V/m) cm.sup.1 cm.sup.1 cm.sup.1 cm.sup.1 cm.sup.1 cm.sup.1 5 5.03E03 1.940954 0.252 0.356 0.436 0.503 0.562 0.616 10 1.01E02 2.744924 0.356 0.503 0.616 0.711 0.795 0.871 15 1.51E02 3.361831 0.436 0.616 0.755 0.871 0.974 1.067 20 2.01E02 3.881908 0.503 0.711 0.871 1.006 1.125 1.232 30 3.02E02 4.754347 0.616 0.871 1.067 1.232 1.378 1.509 40 4.02E02 5.489847 0.711 1.006 1.232 1.423 1.591 1.742 50 5.03E02 6.137836 0.795 1.125 1.378 1.591 1.778 1.948 60 6.03E02 6.723663 0.871 1,232 1.509 1.742 1.948 2.134 80 8.04E02 7.763817 1.006 1.423 1.742 2.012 2.250 2.464

[0085] It was clear also that when pumping at the frequency of the fundamental 628.306 cm.sup.1 of the desired isotope the selectivity between the two isotopes is inhibited by the mismatch occurring at the second and especially the third energy excitation levels and this mismatch cannot be compensated by the power broadening of the higher level transitions up the vibrational ladder. The reason for the drastic reduction in the selectivity observed experimentally as the pumping intensity was increased is due to the fact that at high pumping powers multiphoton resonances can set up indiscriminately, between the ground level of the undesired isotope or the first excited level of the desired isotope and a higher level in the quasicontinuum of energy states. Subsequently, it is irrelevant whether a molecule exists in the ground or the first excited state and the selectivity between the two isotopes is lost.

[0086] Using eq. (26) with the values of

[00078]${}_{01}^{\left(z\right)},{}_{12}^{\left(z\right)},{}_{23}^{\left(z\right)},{}_{34}^{\left(z\right)}$

higher vibrational transitions of the UF.sub.6 molecule as obtained from the above results and using Table 12, we plotted the power broadening curves according to eq. (27). The results have been plotted in FIG. 4 for different pumping intensities from 1010.sup.9 W/m.sup.2 to 6010.sup.9 W/m.sup.2. The power broadening of the transitions increases progressively with higher vibrational levels as well as with higher pumping powers. The curves define approximately the probability for absorption as a function of the frequency spread. For all practical purposes, they give more than adequate values for determining the absorption characteristics of the transitions up the vibrational ladder. Their practical significance is up to the fourth energy level of the vibrational ladder of the UF.sub.6 molecule i.e. the discrete energy section of its vibrational ladder.

[0087] Experiments with polyatomic molecules have demonstrated the collisionless nature of the excitation process during multiphoton absorption. Normal modes behave initially like harmonic oscillators, but as energy is put into these motions their anharmonic nature becomes more pronounced until dissociation is reached. At higher levels the excited states of the resonant mode mix with other vibrational states of the same energy but of different normal modes. This region of absorption is called the molecular quasicontinuum. The theory assumes (RRKM unimolecular reaction-rate theory) that at high vibrational energies the interaction among the vibrational states is strong enough to continuously maintain a statistical distribution of population among those states giving a huge statistical advantage for absorption. The density of vibrational states is the number of available vibrational states per unit energy interval of the molecule. In this respect there is a profound difference between a diatomic molecule and a larger polyatomic molecule which has considerably larger density of states in the quasicontinuum. There are thus two distinct absorption regions in the excitation process of polyatomic molecules.

[0088] The density of vibrational states as a function of the excitation energy has been investigated and published in detail in the literature for all the hexafluorides [D. Jackson, Statistical Thermodynamic Properties of Hexafluoride Molecules, Los Alamos Scientific Laboratory, Report-LA-6025-MS, (1975)]. From this work the density of vibrational states of a hexafluoride molecule at a particular level of excitation can be obtained. For the UF.sub.6 molecule, even as low a frequency as 1880 cm.sup.1 corresponding to the 3.sub.3 level of the .sub.3-vibrational mode, the density of states is already more than 1000 vibrational states per cm.sup.1. This includes of course the vibrational states of all the vibrational modes as well as their octahedral symmetry sublevels. It is clear that in the case of an isotope separation process, substantial selectivity between the two isotopes must be achieved through absorption by the desired isotope in the first few excitation states. Subsequently, means must be devised to take the selected molecules to the dissociation limit through the quasicontinuum and continuum of energy states.

[0089] Transitions in the lower energy levels of the vibrational ladder are peak power dependent. The electromagnetic radiation interacts with the discrete levels of the vibrational ladder and drives the molecule towards the higher vibrational levels. This is the region of coherent interaction which can best be treated through higher order quantum interaction theory, or else it can be described in terms of the Bloch equations. The first approach, however, can give quantitative results for the first two to three energy levels. The populations of near resonant or resonant laser-coupled energy levels are seen to Rabi oscillate inducing a power broadening in the transitions (eq. 26). Because the Rabi frequency depends on the magnitude of the applied electric field, increasing the intensity decreases the effect of the detuning. The off resonant states behave in a resonant manner with population flow between states. Our task is to find at what energy level the discrete region stops and the quasicontinuum of energy levels begins.

[0090] The probability of a transition from an initial stationary state |.sub.i into a smearing of final states f(=+) clustered around a stationary state |.sub.f can be obtained from Fermi's Golden Rule for transitions as

[00079]$\begin{array}{cc}{.Math.\text{?}.Math.}^2=W\text{?}t=\frac{2}{}{.Math.\text{?}.Math.}^{2}f\left(\text{?}=\text{?}+\right)t& \left(30\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where W.sub.fi is the transition rate and the perturbation Hamiltonian is given by =.sub.fiE, where E is the applied electric field, .sub.fi the dipole matrix element with the ground state and t is the interaction time (pulse duration). The density of final states f() clustered around the energy level |.sub.f is given by

[00080]$f\left(\right)=\left({}_f\right)\frac{\mathrm{dn}\left({}_f\right)}{d}$

where (.sub.f) is some real positive function governing the normalization of the .sub.f states which for practical purposes has been checked to be near unity for the range of intensities necessary for our present applications. Eq. (30) is valid under the condition that

[00081]${.Math.a\text{?}.Math.}^2<<1.$$\text{?}\text{indicates text missing or illegible when filed}$

If this inequality does not hold well, then higher order terms in the time-dependent perturbation expansion must be taken into account involving transitions to higher excitation states. If the inequality is reversed this would correspond to the situation where a quasicontinuum of states sets in and the time dependent perturbation expansion is no longer valid but must be replaced by a statistical thermodynamic approach for the description of the evolution of a quantum system up the quasicontinuum of a vibrational ladder. In the case of a final state in the quasicontinuum, energy can be transferred to other vibrational modes and background states at that particular energy. If is the energy width of the band into which the oscillator strength is smeared and E is the dipole matrix element relative to the ground state then for inequality

[00082]${.Math.a\text{?}.Math.}^2<<1$$\text{?}\text{indicates text missing or illegible when filed}$

to be reversed marking the beginning of the quasicontinuum of energy states, the following condition must be satisfied [Yablonovitch E., in the The Significance of Nonlinearity in the Natural Sciences, edited by A. Perlmutter and L. Scott, Plenum Press, New York and London, pp. 207-226, (1977)]:

[00083]$\begin{array}{cc}\frac{\mathrm{dn}\left({}_f\right)}{d}\frac{}{{.Math.{}_{f}E.Math.}^2}& \left(31\right)\end{array}$

[0091] The left hand side of inequality (31) represents the number of states per unit energy interval whilst on the right hand side , the energy width of the band into which the oscillator strength is smeared, can be considered to be approximately equal to the maximum spread of the sublevels of the octahedral splitting of the final state (Tables 3 and 4).

[0092] We seek to determine the level up the vibrational ladder of the UF.sub.6 molecule from which the quasicontinuum of energy states for the UF.sub.6 molecule starts. From Table 3 the manifold origin of the third vibrational level is at a frequency (3.sub.3)=1877.532 cm.sup.1. The density of vibrational states of the UF.sub.6 molecule at this frequency is 750 states/cm.sup.1=3.7755882910.sup.25 states/J (see D. Jackson above). These states correspond to 15 vibrational modes of which only six are non-degenerate. Therefore the density of vibrational states for the .sub.3-vibrational mode is

[00084]$\frac{\mathrm{dn}\left({}_f\right)}{d}0.251706{10}^{25}$

states/J. The maximum spread of the sublevels of the third energy state (3.sub.3) is equal to (Tables 3 and 4) =m(A.sub.2)m(F.sub.1)=8.17 cm.sup.11.62292610.sup.22 J. For a pumping intensity of 1010.sup.9 W/m.sup.2 corresponding to an electric field E.sub.o=2.74492410.sup.6 V/m (eq. 28) and the dipole moment of the UF.sub.6 molecule being =1.286410.sup.30 C.Math.m (Table 8) we obtain

[00085]$\frac{}{{\left(E_0\right)}^2}=1.30245{10}^{25}{J}^{-1}.$

This value violates inequality (31) and thus at this pumping intensities the third energy excitation level of the UF.sub.6 molecule is a pure discrete level of the .sub.3-vibrational mode, not being at all affected by other vibrational modes. Note that we have used the value for the dipole moment of the fundamental but any further averaging of the dipole moments of the intermediate states would have not made much difference.

[0093] With increasing pumping intensities and at higher vibrational levels the situation changes rapidly. We have repeated the above procedure for higher pumping intensities for the fourth, fifth and sixth energy levels of the UF.sub.6 molecule. The density of states for the various vibrational frequencies are obtained from D. Jackson above. The results for the UF.sub.6 molecule are summarized in Table 13. We observe that for the third energy level the condition for the density of states set by inequality (31) is still smaller than the smearing term up to intensities of 5010.sup.9 W/m.sup.2 (electric fields of up to 6.1410.sup.9 V/m), and in eq. (31) is within the limits of being satisfied (bearing in mind the limit of experimental discrepancies). Up to these levels of pumping intensities the third energy level can be considered to be outside the quasicontinuum of energy states. It is clear that there is a maximum intensity level for the selecting laser for avoiding the quasicontinuum of energy states setting in at the third energy level. Table 13 is explicitly clear.

[0094] At the fourth energy level, however, the density of states of the UF.sub.6 molecule (a very heavy polyatomic molecule) becomes so large that the quasicontinuum of energy states begins to set in at pumping intensities as low as 1010.sup.9 W/m.sup.2 (electric fields of up to 2.7510.sup.6 V/m). At the fifth energy level the density of states of the UF.sub.6 molecule becomes so large that inequality (31) is seen to be satisfied even at very low pumping intensities. At these energy levels the quasicontinuum of energy states is present even at very low pumping intensities.

[0095] When pumping at the frequency of the fundamental of the .sup.235UF.sub.6 isotope (628.306 cm.sup.1) selectivity of the desired isotope .sup.235UF.sub.6 takes place at the fundamental with the subsequent levels serving as intermediaries towards the higher levels. As the power of the applied beam increases in order to selectively elevate more molecules of the desired isotope to the higher levels, the molecules of both isotopes can proceed to the quasicontinuum of energy states through

TABLE-US-00013 TABLE 13 Parameters defining the beginning of the quasicontinuum of energy states for the UF.sub.6 molecule = 1.2864 10.sup.30 C .Math. m, 1 cm.sup.1 1.98644544 10.sup.23 J, 1 state/cm.sup.1 5.034117625 10.sup.22 states/J Third Energy level (3v.sub.3) Fourth Energy level (4v.sub.3) Fifth Energy level (5v.sub.3) = 8.17 cm.sup.1 = = 13.61 cm.sup.1 = = 21.86 cm.sup.1 = 1.623 10.sup.22 J 2.7036 10.sup.22 J 4.3424 10.sup.22 J Density of states = 750/15 = Density of states = 6300/15 = Density of states = 37000/15 = 50 states/cm.sup.1 = 420 states/cm.sup.1 = 2467 states/cm.sup.1 = 0.25171 10.sup.25 states/J 2.1143 10.sup.25 states/J 12.419 10.sup.25 states/J Inten- sity Electric Field (E.sub.o).sup.2 [00086]$\frac{}{{\left(E_0\right)}^2}$ [00087]$\frac{\mathrm{dn}\left({}_f\right)}{d}$ (E.sub.o).sup.2 [00088]$\frac{}{{\left(E_0\right)}^2}$ [00089]$\frac{\mathrm{dn}\left({}_f\right)}{d}$ (E.sub.o).sup.2 [00090]$\frac{}{{\left(E_0\right)}^2}$ [00091]$\frac{\mathrm{dn}\left({}_f\right)}{d}$ 10.sup.9 E.sub.o 10.sup.22 10.sup.47 10.sup.25 10.sup.25 10.sup.22 10.sup.47 10.sup.25 10.sup.25 10.sup.22 10.sup.47 10.sup.25 10.sup.25 W/m.sup.2 10.sup.6 V/m J J.sup.2 J.sup.1 states/J J J.sup.2 J.sup.1 states/J J J.sup.2 J.sup.1 states/J 2.5 1.372462 1623 0.31171 5.20674 0.2517 2.7036 0.31171 8.6734 2.1143 4.3424 0.31171 13.9308 12.419 5 1.940954 0.62207 2.6090 0.62207 4.3461 0.62207 6.9806 10 2.744924 1.24413 1.3045 1.24413 2.1730 1.24413 3.4903 15 3.361831 1.8662 0.86968 1.8662 1.4487 1.8662 2.3269 20 3.881908 2.48827 0.65226 2.48827 1.0865 2.48827 1.7451 30 4.754347 3.7324 0.43484 3.7324 0.72435 3.7324 1.1634 40 5.489847 4.97653 0.32613 4.97653 0.54326 4.97653 0.87257 50 6.137836 6.22067 0.26090 6.22067 0.434614 6.22067 0.69806 60 6.723663 7.4648 0.21742 7.4648 0.36217 7.4648 0.58171 80 7.763817 9.95306 0.16307 9.95306 0.27163 9.95306 0.43628
multiphoton resonances destroying the selectivity. Moreover, the application of a powerful dissociating beam cannot distinguish between the ground levels and the excited levels of the lower states of the two isotopes. Selectivity is lost before substantial numbers of the molecules of the desired isotope are elevated to its higher states. Any resonances with the higher levels, starting from whichever level, destroy selectivity as the molecules are being diffused within the quasicontinuum of energy states.

[0096] Similar calculations have been performed for the SF.sub.6, the MoF.sub.6 and the PuF.sub.6 molecules and the results were tabulated. It was found that in the case of the SF.sub.6 molecule even at intensities as high as 10010.sup.9 W/m.sup.2 (electric fields of up to 8.510.sup.6 V/m) inequality (31) easily holds up to the seventh energy excitation state. Under these conditions the quasicontinuum of energy states starts at the seventh or eighth excited energy state. With the absorbed energy staying within one single vibrational mode up to the seventh energy excitation state it is much easier to preserve selectivity up the .sub.3-mode vibrational ladder of the SF.sub.6 molecules and thereafter drive them to their dissociation limit. The detailed results are outside the scope of the present account.

[0097] Experimental results on the number of photons absorbed per molecule as a function of fluence for widely different pumping pulse lengths but the same energy are available in the literature [Yablonovitch E., in the The Significance of Nonlinearity in the Natural Sciences, edited by A. Perlmutter and L. Scott, Plenum Press, New York and London, pp. 207-226, (1977)]. Simple considerations on the statistical properties of the quasicontinuum, in connection with the available experimental results can reveal the approximate values of the temperature of the molecules and the approximate interaction rate with it. On considering the quasicontinuum of the vibrational ladder to be analogous to a statistical distribution of energy states, the average rate of interaction Rin can be considered to be given by the Arrhenius equation

[00092]$\begin{array}{cc}{R}_{\mathrm{int}}={}_{\mathrm{qu}}\exp \left(-\frac{n_{\mathrm{qu}}{d}_f}{{.Math.n.Math.}_{\mathrm{qu}}}\right){}_{\mathrm{qu}}\exp \left(-\frac{k_{B}T}{}\right)& \left(32\right)\end{array}$

where .sub.qu is the minimum number of photons needed for dissociation from the start of the quasicontinuum onwards with being the minimum number of photons needed for dissociation from the ground state defined by =. The quantity is the average number of photons actually absorbed by the molecule in the quasicontinuum and de is the number of vibrational degrees of freedom which for the hexafluorides is d.sub.f=15. =.sub.qu.sub.qu, is the energy absorbed by the molecule from the start of the quasicontinuum onwards and .sub.qu is an average vibrational frequency. The temperature of the molecule is then given by

[00093]$\begin{array}{cc}{k}_{B}T=\frac{{.Math.n.Math.}_{\mathrm{qu}}{}_{\mathrm{qu}}}{d_f}& \left(33\right)\end{array}$

[0098] This is the temperature attained by an incoherently driven oscillator and eq. (33) is justified if the rate of intramolecular vibrational relaxation is faster than the rate of absorption of photons. Absorption cross section measurements confirm that the rate of photon absorption is indeed slower than the intramolecular vibrational times in polyatomic molecules.

[0099] For transitions in the quasicontinuum only the energy density, not the peak power, is important. A comparison of the dissociation yield for pulses of various durations but fixed energy, indicated that for a 200-fold increase in peak power the fraction of molecules dissociated increased only by 30%. This demonstrated that absorption of radiation in the quasicontinuum is more important to dissociation than the anharmonicity bottleneck in the discrete levels. On applying eqs. (32) and (33) to the results of quantitative experiments on the absorption of photons per pulse as a function of energy fluence we were able to calculate the approximate values of the temperature of the molecules and the approximate interaction rate with the quasicontinuum of energy states. The results for the UF.sub.6 and the SF.sub.6 molecule are summarized in Table 14. All the results pertain to the cases for a near unity probability of the dissociation yield through the quasicontinuum of energy states, which is the minimum flux (J/m.sup.2) through the quasicontinuum required for the dissociation of the molecule i.e. R.sub.int=R.sub.diss, with R.sub.diss(s.sup.1) being the minimum dissociation rate. At higher pumping fluxes the interaction rate with the states of the quasicontinuum R.sub.int>R.sub.diss (s.sup.1) increases because the rate of the number of photons absorbed increases drastically due to heating of the molecule. The values calculated in the last two columns correspond to the two expressions for the R.sub.int given in eq. (32). The two expressions give close values in both cases, for the heavier UF.sub.6 molecule as well as for the much lighter SF.sub.6 molecule.

TABLE-US-00014 TABLE 14 Temperature and dissociation rate through the quasicontinuum of energy states for the SF.sub.6, and UF.sub.6 molecules .sub.qu = 10.sup.14 s.sup.1 .sub.qu 7.sub.qu 7.sub.qu k.sub.BT T R (.sub.qu) R ( ) Molecule 10.sup.19 J (.sub.qu cm.sup.1) 10.sup.20 J 10.sup.20 J 10.sup.19 J .sub.qu .sub.qu d.sub.f 10.sup.20 J K 10.sup.9 s.sup.1 10.sup.9 s.sup.1 SF.sub.6 6.39616 34 48 1.7774136 1.87441 13.12087 5.08406909 27 31 15 3.873781 ~2805.8 ~0.37633 ~0.35448 (943.6) .sub.qu = 10.sup.14 s.sup.1 .sub.qu 4.sub.qu 4.sub.qu k.sub.BT T R (.sub.qu) R ( ) Molecule 10.sup.19 J (.sub.qu cm.sup.1) 10.sup.20 J 10.sup.20 J 10.sup.19 J .sub.qu .sub.qu d.sub.f 10.sup.20 J K 10.sup.9 s.sup.1 10.sup.9 s.sup.1 UF.sub.6 4.72759 38 55 1.174777 1.238886 4.955546 4.23203843 34 44 15 3.634067 ~2632.1 ~1.08665 ~1.02895 (623.67) indicates data missing or illegible when filed

[0100] From Table 14 we observe that: (a) the molecular temperature attained for a near unity probability of the dissociation yield for the SE, molecule is greater than for the UF.sub.6 molecule as a result of the lower density of vibrational states in the quasicontinuum requiring much higher fluences for dissociation to occur. (b) the interaction rate Kit through the quasicontinuum for the UF.sub.6 molecule (1.086610.sup.9 s.sup.1) is much greater than the one for the SF.sub.6 molecule (0.37510.sup.9 s.sup.1). This is the result of the much higher density of states in the quasicontinuum for the UF.sub.6 molecule. It is the transition rate through the entire quasicontinuum resulting in the dissociation of the molecule. (c) The transition rate through the quasicontinuum of energy states of the UF.sub.6 molecule resulting in its dissociation (1.0866510.sup.9 s.sup.1) is lower than the equivalent two-level transition rate at a pumping intensity as low as 0.510.sup.9 W/m.sup.2 (see eq. 18). The intensities used in isotope separation experiments are greater than this value and the driving of the molecules through the quasicontinuum of energy states can readily occur.

[0101] More photons are actually absorbed in the quasicontinuum than the number necessary for dissociation to occur and subsequently many more photons take part in the interaction process with the quasicontinuum region. From eq. (33) we can calculate the corresponding quasicontinuum temperature T for any number of photons higher than the minimum number of photons necessary for dissociation. The temperature T will be proportional to the number of photons taking part in the interaction with the quasicontinuum region. Using the values of the dissociation energy absorbed by the molecule from the start of the quasicontinuum onwards (Table 14) we can plot the interaction rate Ri given by eq. (32) against the number of photons taking part in the interaction within the quasicontinuum region , or against its power

[00094]${\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}.$

At pumping fluxes much higher than those necessary for near unity probability of dissociation, the interaction rate Rim increases, but the rate of the number of photons absorbed by the molecule in the quasicontinuum also increases drastically due to further heating of the molecule. The results for the Uranium Hexafluoride molecule (UF.sub.6) are shown in FIG. 5(a), (b): (a) The number of photons

[00095]${\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}$

interacting with the quasicontinuum of states is proportional to the interaction rate R.sub.int when the number of absorbed photons is 170<<255; (b) The number of photons interacting with the quasicontinuum of states is found not to be proportional to the interaction rate R.sub.int within the same range of absorbed photons. The broken vertical lines on both graphs correspond to an applied number of photons of approximately 170 and 255 interacting with the quasicontinuum

[00096]$R_{\mathrm{int}}{\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}.$

[0102] Since the interaction rate is proportional to the fluence of the pumping beam i.e. R.sub.int.sub.IR, then

[00097]${}_{\mathrm{IR}}{\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}$

resulting in

[00098]${.Math.n.Math.}_{\mathrm{qu}}{\left({}_{\mathrm{IR}}\right)}^{\frac{2}{3}}\left[i.e.{\left({}_{\mathrm{IR}}\right)}^{-\frac{1}{3}}\right].$

This is the result which has been obtained theoretically and experimentally by Okada Y. et al (Journal of Nuclear Science and Tech., Vol. 30, pp. 762-767, August 1993) valid for pumping fluences between 0.110.sup.3 J/m.sup.2 and 310.sup.3 J/m.sup.2. These are the relations which had previously been derived theoretically by Judd O. P. (J, Chem. Phys., Vol. 71, No 11, pp. 4515-4530 December 1979) for a number of polyatomic molecules. We have obtained similar results for the SF.sub.6 molecule with the

[00099]${\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}$

dependency of the fluence starting at a lower value for the interacting photons (120 instead of 170 for the UF.sub.6 molecule) as a result of the fact that, at the intensities considered, the quasicontinuum of energy states starts at a much higher state (7.sup.th or 8th) and the density of energy states is much smaller in the quasicontinuum of the SF.sub.6 molecule. In general the relations are more pronounced for the heavier polyatomic molecules.

[0103] As previously pointed out above Alexander et al have obtained experimentally that at lower pumping fluences, between 0.1 J/m.sup.2 and 0.110.sup.3 J/m.sup.2, the fluence is proportional to the square of the number of interacting photons in the quasicontinuum, i.e. (, .sub.IR (see FIG. 2). We have followed a similar procedure to the one described above using eqs. (32) and (33) and plotted the interaction rate in the quasicontinuum of energy states of the UF.sub.6 molecule against the square of the number of interacting photons. The results indicate again that the interaction rate R.sub.int is proportional to the square of the number of interacting photons in the interval 135<<185. This again is in perfect agreement with the experimental results.

[0104] For the UF.sub.6 molecules the experimental results and observations indicate three distinct intervals for the magnitude of the pumping fluence where the interaction follows specific trends: (a) At low fluences the absorption process is independent of the fluence; (b) For a pumping fluence in the interval between 0.1 J/m.sup.2 and 0.110.sup.3 J/m.sup.2 it is proportional to the square of the number of interacting photons in the quasicontinuum, i.e. .sub.IR (see FIG. 2); (c) For higher pumping fluences in the interval between 0.110.sup.3 J/m.sup.2 and 310.sup.3 J/m.sup.2 it is proportional to

[00100]${}_{\mathrm{IR}}{\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}.$

Thus, on the basis of a simple statistical analysis of the interaction of the electromagnetic beam with the quasicontinuum of energy states during the dissociation process we have demonstrated that these experimental results and observations reported in the literature hold true and are readily explainable. Furthermore, we demonstrated that the number of interacting photons where the law .sub.IR holds is between 135<<185. The number of interacting photons with the quasicontinuum of states where the law

[00101]${}_{\mathrm{IR}}{\left({.Math.n.Math.}_{\mathrm{qu}}\right)}^{\frac{3}{2}}$

holds is between 170<<255. No further analysis on this subject of the quasicontinuum of energy states in the hexafluoride molecules will be presented here. The important point of the above analysis with regard to the isotope separation process is that once high selectivity of the desired isotope is achieved in the distinct energy level section of the .sub.3-vibrational ladder of the UF.sub.6 molecule it is easy to drive the selectively excited molecules to dissociation through the quasicontinuum of energy states by suitably adjusting the intensity, fluence and frequency of the dissociating beam.

[0105] We have carried out complete analyses of all the above aspects of the .sub.3-vibrational mode of the hexafluorides. The vibrational amplitudes of the various nuclei have been drawn to scale, in units of

[00102]$a.Math.m.Math.u^{-1/2}$

[Aldridge et al, Journal of Chemical Physics, Vol. 83(1), pp. 34-48, (1985)] so as to avoid the frequency dependence and have a direct comparison of the amplitudes. A comparison between those of the heavy and the lighter hesafluorides has been made. On all occasions the experimental results for the .sub.3-vibrational mode of the heavy hexafluorides are in perfect agreement with the theoretical expressions given above. By comparing the well established experimental values for the UF.sub.6 molecule (a heavy hexafluoride) with those of the SF.sub.6 molecule (the lightest of the hexafluorides) and using the above analyses it was possible to obtain very good values for the parameters of all the other hexaflorides. These are outside the scope of the present account.

[0106] We very briefly summarize the basic conclusions which were obtained from the above analysis for the .sub.3-vibrational mode of the hexafluoride molecules. These corroborate that on the basis of all the available experimental results and their close agreement to the theoretical analysis developed, the lower levels of the vibrational ladder of the heavy hexafluoride molecules, and in particular of the UF.sub.6 molecule, are a very close match in their behaviour to those of an ideal harmonic oscillator: (1) The vibration of the .sub.3-mode of the heavy hexafluorides, and in particular of the UF.sub.6 molecule, is very close to that of an ideal harmonic oscillator. Its vibrational constants can be determined from an analysis of the Schrdinger equation through the use of the Morse potential and its equivalence to that of a diatomic molecule, to a very high degree of accuracy (Table 1); (ii) The anharmonic vibrational constant X.sub.33 of the heavy hexafluorides is very close to the anharmonicity constant ().sub.ex.sub.e, calculated through the Schrdinger equation and the Morse potential (FIG. 1); (iii) Resonance at the fundamental is vital to the absorption process up the whole vibrational ladder with simultaneous resonances between the ground state and all higher levels (three-photon absorption resonance theory); (iv) By imposing the frequency conditions it is possible to obtain all the constants of the vibrational ladder of the hexafluorides to a very high degree of accuracy (Table 1); (v) The dipole moments of heavy hexafluorides, and in particular that of the UF.sub.6 molecule, obtained using the experimentally measured vibrational band strengths (eq. 12), are extremely close to those calculated from elementary quantum mechanical principles for the dipole moment in terms of the vibrating frequency .sub.k of a harmonic oscillator model and the three fundamental physical constants e, m and (eq. 19); (vi) The experimentally measured value of the spontaneous lifetime of the first excited level of the .sub.3-vibrational mode of the UF.sub.6 molecule is found to be extremely close to that calculated from an elementary expression for the spontaneous lifetime of the fundamental transition of a harmonic oscillator which depends only on the inverse of the square of the frequency and the fundamental physical constants (eq. 20). (vii) The measured absorption cross section at the fundamental of the .sub.3-vibrational mode of the UF.sub.6 molecule using an infrared spectrophotometer with very low probing intensities (1.15510.sup.20 m.sup.2), is very close to the calculated value of the absorption cross section of the fundamental of an ideal harmonic oscillator (1.25810.sup.20 m.sup.2) (eq. (23) and FIG. 2); (vii) The quasicontinuum of energy states in lighter hexafluorides starts at higher vibrational levels than for the heavier hexafluorides for the same pumping intensities and the molecular temperature attained for the lighter hexafluoride molecules is much higher than for the heavier hexafluoride molecules (Table 13); (ix) The interaction rate through the quasicontinuum of the heavier hexafluoride molecules (R.sub.int1.0866510.sup.9 s.sup.1 for the UF.sub.6 molecule) is much greater than the one for the lighter hexafluoride molecules (R.sub.int0.3763310.sup.9 s.sup.1 for the SF.sub.6 molecule) providing an easier elevation of the heavy molecules to dissociation through the quasicontinuum of energy states (Table 14); (x) The vibrational band strength in the case of an ideal harmonic oscillator depends only on two standard physical constants, Avogadro's number and the classical radius of the electron giving an extremely close value to the experimentally measured value for the vibrational band strength of the .sup.238UF.sub.6 molecule (S.sub.o=703.3503114 km mole.sup.1); (xi) The size of the manifold structures of the vibrational levels for the hexafluorides as a function of the square of the vibrational quantum number indicate that the resulting graphs are extremely close to a straight line as required by theory. The widths of the vibrational manifolds for the various hexafluorides varies according to their weight and the vibrational anharmonicity constant X.sub.33, indicating an extreme closeness of the vibration of the lower vibrational levels to that of a harmonic oscillator. Thus, we have established that the vibration of the lower levels of the .sub.3-mode molecular vibrational ladder of the UF.sub.6 molecule is a very close match in its behaviour to that of an ideal harmonic oscillator. Subsequently, we can proceed to use its properties and exploit schemes for the molecular isotope separation of the UF.sub.6 molecule.

[0107] The population of the ground level as a function of temperature is a very important factor in the molecular laser isotope separation process. Following the procedure by Erkens [Erkens J. W., Applied Physics, Vol. 10, pp. 15-31, (1976)] we have calculated the population probabilities of the hot states of the UF.sub.6 gas at various temperatures and for the various energy excitation levels. By using the appropriate frequencies of the fundamentals of the vibrational modes of the UF.sub.6 molecule at various temperatures (gathered from various references) we also calculated the vibrational partition function at various temperatures. Our results, which we have tabulated, are in perfect agreement with those of the above reference but are outside the scope of the present account.

[0108] The population probability of the ground state is characterized by .sub.o=0 giving a statistical weight of w.sub.o=.sub.o+1=1. Subsequently, the population probability of the ground state is

[00103]$\begin{array}{cc}{p}_{\mathrm{ground}}=\frac{1}{\text{?}}\left(\text{?}\right)=\frac{1}{\text{?}}& \left(34\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

from which we have obtained the population probabilities of the ground state of UF.sub.6 at various temperatures using the corresponding values of the vibrational partition function Z.sub.. The results have been tabulated giving the Population probabilities of the ground level p.sub.ground of the UF.sub.6 gas at various temperatures together with the corresponding values of the Partition Function. They are fully compatible with those in the literature. The results are plotted in FIG. 6 where the variation of the population of the UF.sub.6 gas in the ground state as a function of temperature is plotted. It is evident that for temperatures greater than 100 K, less than 40% of the UF.sub.6 molecules are in the ground state. Most of the UF.sub.6 molecular population lies in other higher vibrational states. If the cooling of the UF.sub.6 molecular gas is not below 100 K many of the higher vibrational levels, of the desired and the unwanted isotopes, will hold most of the molecular population at the time of the interaction process with a dissociating laser. The powerful dissociating laser frequency will not be able to distinguish between these molecules and the molecules which had been selectively excited at the fundamental frequency of the .sub.3-vibrational mode. It is thus clear that the temperature of the molecular gas must be less than a 100 K, and preferably much lower, around 60 K to 70% so that the vast majority of molecules are in the ground state during the interaction process. The drastic change in the percentage of the molecules of the ground state population of the UF.sub.6 gas from 85% .sup.235UF.sub.6 molecules at a temperature of 60 K to 30% .sup.238UF.sub.6 molecules at a temperature of 110 K has been overlooked in all the hitherto applications of the MLIS process in prototype experiments. The importance of the expansion supercooling process for application to laser isotope separation systems is evident. This fact has hitherto been overlooked or given very little attention.

[0109] The primary dissociation of the UF.sub.6 molecules occurs via the following schemes:

[00104]$\begin{array}{cc}\begin{array}{cc}{}^{235}{\mathrm{UF}}_6\stackrel{nhv}{.fwdarw.}{}^{235}{\mathrm{UF}}_5+F& {}^{238}{\mathrm{UF}}_6\stackrel{nhv}{.fwdarw.}{}^{238}{\mathrm{UF}}_5+F\end{array}& \left(35\right)\end{array}$

[0110] Dissociation experiments indicated that the reverse reactions i.e. the recombination of UF.sub.5 with the F atoms yielding UF.sub.6 parent molecules (UF.sub.5+F.fwdarw.UF.sub.6) occurs significantly in the reaction system. An upper limit for the rate constant of the reverse reactions has been experimentally found to be k.sub.r<2.010.sup.12 cm.sup.3 molecule.sup.1 s.sup.1 [Lyman J. L. et al, Journal of Chemical Physics, Vol. 82, No 1, pp. 175-182, (January 1985)]. Thus, a scavenger gas should be used which reacts rapidly with the F-atoms and yet does not produce any species which could be reactive towards the parent UF.sub.6 molecules. The intrinsic separation factor S effected by radical reaction has been shown to be [Kato S. et al, Journal of Nuclear Science and Technology, Vol. 26, No 2, pp. 256-260, (February 1989)]

[00105]$\begin{array}{cc}S=1+\frac{}{1+}{S}^{}& \left(36\right)\end{array}$

where S=1+a with 1 is the primary separation factor for eqs. (35), independent of the scavenger gas, and 01 is the fraction of radicals R.sub.ad which react non-selectively with parent UF.sub.6 molecules. Eq. (36) is valid under the condition [.sup.235UF.sub.6]<<[.sup.238UF.sub.6]. Experiments with H.sub.2, C.sub.2H.sub.6 and CH.sub.4 as scavenger gases have resulted in the following values for :H.sub.2:0.56 (S deterioration 30%); C.sub.2H.sub.6: 0.85 (S deterioration 38%); CH.sub.4: 0.07 (S deterioration 5%). The very small value of indicates that CH.sub.3 radicals hardly deteriorate the primary separation factor. Thus, Methane hardly causes any detrimental radical reactions to lower the separation factor in UF.sub.6 laser isotope experiments and it is considered to be the most suitable scavenger gas. It is stable, has no infrared absorption at 16 m, has relatively high vapour pressure at low temperatures, even below 100 K, and has high heat capacity ratio.

[0111] Uranium Hexafluoride has the highest vapour pressure of all known Uranium compounds. Simple calculations indicate that in order to achieve very low temperatures without condensation of the Uranium Hexafluoride gas it is necessary to have extremely low pressures i.e. very few UF.sub.6 gas molecules. Supersonic expansion processes have been devised in order to achieve spectroscopically acceptable temperatures with sufficient UF.sub.6 molecules for interaction. The cooling attainable by an adiabatically expanded working gas through a supersonic jet stream is determined by the ratio of pressures on both sides of the nozzle and also by the ratio of specific heats =c.sub.P/c.sub.. Using available experimental data and expressions for the specific heats of the UF.sub.6 derived from thermodynamic functions we have obtained the values of c.sub.P and as a function of temperature. By using a carrier gas which is a mixture of a monatomic gas with =1.67 (for example Ar) and a diatomic gas with =1.4 (for example N.sub.2) we can obtain an effective =1.58 which is as near as the highest value possible to avoid any practically inherent condensation problems. The gas equation governing the adiabatic expansion is

[00106]$\begin{array}{cc}\frac{T}{T_o}=\frac{P}{P_o}{J}^{\frac{-1}{}}={\left(\frac{}{{}_o}\right)}^{-1}& \left(37\right)\end{array}$

where T.sub.o, P.sub.o, .sub.o and V.sub.o are the initial temperature, pressure, density and volume of the gas respectively, and T, P, , and V are their corresponding values attained after adiabatic expansion through the supersonic nozzle. The introduction of a carrier gas in the expansion flow process facilitates substantial cooling of the gas with only modest initial pressure and nozzle area expansion ratios, and a collisional environment which ensures continuum fluid flow and thermal equilibrium among the vibrational, rotational and translational degrees of freedom of the UF.sub.6 before irradiation. FIG. 7 depicts a gas dynamic expansion through a two dimensional nozzle, having a slit throat. The gas is pulsed through a pipe constriction towards a long thin slit. The slit runs perpendicular to the plane of the paper. The expansion occurs very rapidly within a short distance from the slit throat. High Mach numbers can be achieved in a uniform flow region of the central core, bounded on each side by thin boundary layers. The small circle after the slit represents the region of a uniform high gas density where the laser beams must be applied. This is usually very close to the slit. Hyperbolic, laval or any other nozzle geometry enables the gas to retain all its fluid properties in the collision dominated regime. The mixture of UF.sub.6, the scavenging gas and the inert gases is introduced into a loop of known volume composed of a series of compressors designed for the use of UF.sub.6. The duration of continuous supersonic flow is set by the volume of the damp tank. A valve actuated at an appropriate repetition rate provides a fully supersonic flow of cooled gas lasting for several milliseconds per pulse at the exit nozzle. After enrichment the dissociated UF.sub.5 molecules rapidly form dimers and the scavenger gas makes them relatively immune to the exchange of fluorine atoms with the UF.sub.6. The enriched product is collected by the passage of the gas through a sonic impactor. More details on the design of the expansion nozzle are outside the scope of the present account but more elaborate relations for the supersonic gas expansion can be found in recently published scientific literature, Table 15 lists some experimental and estimated values of the vibrational relaxation times causing de-excitation of the UP, molecules as a consequence of collisions with the inert gas which have been recorded in the literature. These values refer to the average de-excitation time of the molecules in all the excited levels including transfer of energy to the levels of other vibrational modes of the molecule. This last point is one of the reasons why the effectiveness of ultraviolet dissociation of the UF.sub.6 molecules is limited. The vibrational relaxation times decrease roughly in proportion with the temperature. One important result was the measurement of the collisional de-excitation times of the vibrational energy in UF.sub.6: P .sub.c (UF.sub.6-UF.sub.6)=0.5 s.Math.Torr and

TABLE-US-00015 TABLE 15 The vibrational relaxation times of UF.sub.6 gas mixtures with scavenger and inert gases at various temperatures Vibrational relaxation Temperature (P.sub.gas .Math. .sub.c) Gas Mixture K (s .Math. Torr) UF.sub.6:H.sub.2:Ar (measured) 300 2.9 UF.sub.6:H.sub.2:Ar (measured) 90 0.85 UF.sub.6:H.sub.2:Ar (estimated) 60 ~0.57 UF.sub.6:H.sub.2:N.sub.2 (measured) 105 2.5 UF.sub.6:H.sub.2:N.sub.2 (estimated) 60 ~1.43

TABLE-US-00016 TABLE 16 The vibrational relaxation times of UF.sub.6 gas and a mixture with He for various temperatures Vibrational relaxation times Temperature (P.sub.gas .Math. .sub.c) Gas Mixture K s .Math. Torr UF.sub.6 (measured) 300 0.5 UF.sub.6 (estimated) 60 ~0.1 UF.sub.6:He (measured) 300 1.5 UF.sub.6:He (estimated) 60 ~0.3
P .sub.c (UF.sub.6-He)=1.5 s.Math.Torr [Alimpiev S. S. et al., Soviet Journal of Quantum Electronics, Vol. 11, No 3, pp. 375-379, (March 1981)]. All measurements were made at a temperature of 300 K. On assuming a rough proportionality relation with temperature the corresponding values at 60 K. were estimated. Table 16 lists the experimental results obtained and the values estimated at 60 K. Nearly all the experimental results we have searched are consistent with the measurements listed in Tables 15 and 16.

[0112] In most of the experiments described in the literature molecular gas velocities of 450 m/s to 500 m/s have been reported after expansion supercooling through the nozzle. Typical expansion supercooled mixtures consisted of a combination of the inert gases kr, Ar and/or N.sub.2 together with the scavenger gas CH.sub.4 in the following proportions: Basic gas: UF.sub.6 (0.1%-1%); Scavenger gas: CH.sub.4 (0.5%-5%); Inert carrier gas: Kr, Ar and/or N.sub.2 (94%-99%) [see for example, Takeuchi K. et al., Journal of Nuclear Science and Technology, Vol. 26, No 2, pp 301-303, (February 1989)]. Three possible combinations within the above percentages would be: 1) UF.sub.6: 0.1%, CH.sub.4: 0.9%, Inert gas: 99%; 2) UF.sub.6: 0.5%, CH.sub.4: 2.5%, Inert gas: 97%; 3) UF.sub.6: 1%, CH.sub.4: 4%, Inert gas: 95%. The range over which the UF.sub.6 partial pressures in gas mixtures has been reported in expansion supercooling experiments at temperatures below 100% is usually of the order of: (0.3-2.0) Torr.

[0113] By applying eq. (37) to a gas having the proportions of its constituents mentioned above we have evaluated the parameters of the expansion supercooled gas for a number of cases at a temperature of 60 K. The results are summarized in Table 17. Note that the final UF.sub.6 densities

TABLE-US-00017 TABLE 17 Expansion supercooling of a mixture of gases containing UF.sub.6 in various percentages from an initial room temperature of 300 K to a final temperature of 60 K Initial Final Final Initial Initial density Final Final density Density Pressure Temperature .sub.o Pressure Temperature of UF.sub.6 Composition P.sub.o T.sub.o Molecules/m.sup.3 P T Molecules/m.sup.3 Molecules/m.sup.3 of gas mixture Torr K 10.sup.24 Torr K 10.sup.24 10.sup.21 UF.sub.6: 0.1% 50 300 1.6085 0.62356 60 0.1003 0.1003 CH.sub.4: 0.9% 250 300 8.0425 3.11781 60 0.5015 0.5015 Inert gas: 99% 500 300 16.086 6.23562 60 1.003 1.003 1000 300 32.175 12.47123 60 2.006 2.006 2500 300 80.438 311.78076 60 5.015 5.015 UF.sub.6: 0.5% 10 300 0.3217 0.1247 60 0.02006 0.1003 CH.sub.4: 2.5% 50 300 1.6085 0.62356 60 0.1003 0.5015 Inert gas: 97% 100 300 3.217 1.24712 60 0.200 1.003 200 300 6.435 2.49425 60 0.401 2.006 500 300 16.086 6.23561 60 1.003 5.015 UF.sub.6: 1% 5 300 0.16085 0.062356 60 0.01003 0.1003 CH.sub.4: 4% 25 300 0.80425 0.311781 60 0.05015 0.5015 Inert gas: 95% 50 300 1.608 0.62356 60 0.1003 1.003 100 300 3.217 1.24712 60 0.20063 2.006 250 300 8.043 3.11781 60 0.50158 5.015
in the table are in the range where it has been claimed they were achieved experimentally. From Table 15 the vibrational de-excitation times for collisions in the gas mixture of the UF.sub.6 with inert gases at 60 K ranges from P .sub.c (UF.sub.6:H.sub.2:Ar)=0.6 s.Math.Torr to P .sub.c (UF.sub.6:H.sub.2: Na)=1.4 s.Math.Torr. This means that for laser pulses of up to 10010.sup.9 s collisional de-excitation of the vibrational energy during the interaction process is completely avoided. Furthermore, from Table 16 the vibrational de-excitation times for collisions between UF.sub.6 molecules at T=60 K is P .sub.c (UF.sub.6-UF.sub.6)0.1 s.Math.Torr. The

TABLE-US-00018 TABLE 18 Densities of the expansion supercooled UF.sub.6 gas in the ground state at 60 K Final Density Final Density of UF.sub.6 in the Final density of UF.sub.6 ground state .sub.UF6 (.sub.UF6).sub.gr Molecules/ Molecules/ Molecules/ m.sup.3 10.sup.21 m.sup.3 10.sup.21 m.sup.3 10.sup.21 0.01003 0.1003 0.0857 0.05015 0.5015 0.4286 0.1003 1.003 0.857 0.20063 2.006 1.715 0.50158 5.015 4.286
UF.sub.6 partial pressure in the final gas is 1.2470.01=0.01247 Torr. This means that the average time between collisions is 0.1/0.01247=8.02 s which is a very long time compared to the duration of a laser pulse. The vibrational de-excitation due to this kind of collisions is far less important even for very long laser pulses, We have used the five values in Table 17 for the final gas density after expansion supercooling, which have been experimentally achieved without condensation occurring, to calculate the corresponding final densities of UF.sub.6 in the ground state at 60 K. The corresponding values of the partition function have been used in the calculations. The values are tabulated in Table 18. It is not possible to list here more calculations and the conclusions from the extensive analysis which we have carried out.

[0114] Expansion nozzles over 1 m wide with depths (distance between the orifice and the skimmer top) ranging from 12 mm to 20 mm have already been operated successfully, resulting in the expansion supercooled UF.sub.6 gas at T=60 K with the parameters shown in Table 18. The velocities of the expanded gas just after the slit were in the range .sub.exp510.sup.2 m/s, By slightly varying the slit opening and changing the depth of the expansion nozzle (distance between the orifice and the skimmer top) in conjunction with the volume of the dump tank the parameters of the expansion supercooled gas can be kept unaltered, whilst changing the cross

TABLE-US-00019 TABLE 19 Values for the UF.sub.6 gas densities after supersonic expansion at 60 K Density of the UF.sub.6 Natural abundance Tails reprocessing (depleted Uranium) in the expanded gas .sup.238UF.sub.6 = 0.9929 .sup.238UF.sub.6 = 0.0071 .sup.238UF.sub.6 = 0.9975 .sup.238UF.sub.6 = 0.0025 Final Density Final Density Final Density Final Density Final Density Final Density of UF.sub.6 in the of .sup.238UF.sub.6 in the of .sup.235UF.sub.6 in the of .sup.238UF.sub.6 in the of .sup.235UF.sub.6 in the of UF.sub.6 ground state ground state ground state ground state ground state .sub.UF.sub.6 (.sub.UF.sub.6).sub.gr (.sub.238.sub.UF.sub.6).sub.gr (.sub.235.sub.UF.sub.6).sub.gr (.sub.238.sub.UF.sub.6).sub.gr (.sub.235.sub.UF.sub.6).sub.gr molecules/ molecules/ molecules/ molecules/ molecules/ molecules/ m.sup.3 10.sup.21 m.sup.3 10.sup.21 m.sup.3 10.sup.21 m.sup.3 10.sup.19 m.sup.3 10.sup.21 m.sup.3 10.sup.19 0.1003 0.0857 0.0851 0.06085 0.085486 0.02143 0.5015 0.4286 0.42556 0.30431 0.42753 0.10715 1.003 0.857 0.851 0.6085 0.8548 0.2143 2.006 1.715 1.703 1.2177 1.7107 0.4287 5.015 4.286 4.256 3.0431 4.2753 1.0715
sectional area of the volume in which the gas density is uniform, at the values given in Table 18. The same amount of gas will go through this volume at different speeds. Calculations on the parameters of the expansion supercooled UF.sub.6 gas in the irradiation area in a nozzle 1 m wide and cross sectional area 0.001 m.sup.2 flowing with a velocity of 510.sup.2 m/s have been carried out for the densities in Table 18. These are listed in Table 19 where the .sup.238UF.sub.6 and .sup.235UF.sub.6 densities in the expanded gas are shown for two different initial assays. The third and fourth columns list the values for the natural abundance of Uranium whilst the fifth and sixth columns list the values if depleted UF.sub.6 was to be used in the expansion gas (Tails reprocessing). It is not, however possible to give a detailed analysis of the results here.

[0115] For the isotope separation of the UF.sub.6 isotopes we must observe the following steps: (i) We must supercool the gas to temperatures much lower than 105 K, preferably around 60 K so that most molecules are in the ground state (greater than 85%, FIG. 6). Otherwise, absorption by molecules in the higher levels of the unwanted .sup.238UF.sub.6 isotope will take place having a detrimental effect on the selectivity of the process; (ii) We must aim at three-photon resonance with the third energy excitation level [m(A.sub.2):(3.sub.3)] of the desired isotope .sup.235UF.sub.6 i.e. pumping at a frequency of 628.527 cm.sup.1, making use of the power broadening of the first, second and third energy levels for achieving the excitation of the molecules to this level; (iii) The optimum frequencies for attaining maximum overall three-photon absorption selectivity to the third energy excitation level may range between 628.45 cm.sup.1 and 628.56 cm.sup.1 i.e. in the region of establishing three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel; (iv) The intensity of the pumping beam must be limited to such levels that interaction with the quasicontinuum of energy states has no effect at the third energy level or thereafter (Table 13); (v) when pumping within the narrow frequency range capable of establishing three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel the pumping intensity should not exceed 5010.sup.9 W/m.sup.2, otherwise the quasicontinuum of energy states could begin at the third energy level of the UF.sub.6 molecule facilitating the escape of the molecules to other vibrational modes and background states, thereby having a detrimental effect on the selectivity process; (vi) Four-photon resonance with the fourth energy level must be avoided at all costs so that no transfer of energy to other vibrational modes can take place through the quasicontinuum of energy states. On summarizing all the aspects of the properties and interaction parameters described above and which we have fully scrutinized in order to arrive at the above six basic steps, we have: (a) The properties of the .sub.3-mode vibrational levels are very close to those of a harmonic oscillator; (b) The intensity of the selecting pumping beam relevant to the beginning of the quasicontinuum of energy states must be carefully controlled within a certain range; (c) The detrimental effects of pumping the .sup.235UF.sub.6 molecules at the frequency of the fundamental transition at 628.306 cm.sup.1 have been analysed; (d) The collisions amongst the UF.sub.6 molecules and with the carrier gas have been shown to be irrelevant for pumping pulses with duration below 8010.sup.9 s.sup.1 at the appropriate densities of the expansion supercooled gas for the MLIS process; (e) Restriction of the UF.sub.6 molecules from moving into the quasicontinuum of energy levels and other background states by careful controlling of the pumping intensity and the frequency and intensity of the dissociating laser.

[0116] Following the results of the French experiments [Alexander et al, Journal de Chimie Physique, Vol. 80, No 4, pp 331-337, (1983)] all prototype systems have been operated with selecting frequencies at the fundamental of the desired .sup.235UF.sub.6 isotope or on the lower frequency side. It is however a misleading concept. As the pumping intensity is increased in order to excite higher numbers of molecules, selectivity is lost due to the fact that multiphoton resonances cannot distinguish between molecules in the ground state of the undesired isotope and molecules in the first excited state of the desired isotope. Furthermore, at high pumping intensities (>5010.sup.9 W/m.sup.2) the quasicontinuum of energy states for the .sup.235UF.sub.6 molecule can set in at the third energy excitation state, thereby enabling the loss of excited molecules to other vibrational modes and background states (Table 13).

[0117] The method of simultaneously tackling the selectivity and dissociation process is conceptually wrong. Any good chess player knows that in order to launch an effective attack he will have to arrange his pieces in the right position beforehand. In an analogous way we first aim at selectively exciting as many molecules as possible to a particular state, from which they cannot immediately escape to other vibrational modes and background states (in the case of the UF.sub.6 molecule the third energy excitation level [m(A.sub.2):(3.sub.3)]), giving sufficient time to a second beam, carefully chosen for its frequency and intensity, to drive them to dissociation. This is the object of the present process. The important factor in the selectivity process is to locate the intensity level at which direct three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel is readily established whilst at the same time the absorption probability resonance at the first energy excitation level is not inhibited. If the pumping intensity is low, producing insufficient power broadening of the first energy excitation level which results in low absorption probability resonance at this level, then the setting up of direct three-photon resonance with the third energy level will be inhibited. The pumping intensity of the selecting beam must be high enough to produce sufficient power broadening at the fundamental, thus enabling the direct three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the third energy excitation state (3.sub.3) to be readily accessible. Moreover, the pumping pulses must have sufficient intensity for three photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel to be established but at the same time the intensity must be restricted to levels below which the quasicontinuum of energy states will not set in at the third energy excitation level of the .sup.235UF.sub.6 isotope, thereby facilitating the escape of molecules to other vibrational modes and background states.

[0118] With regard to the frequency of the applied radiation we have drawn tables summarizing all the possible frequencies obtainable by stimulated rotational Raman scattering from a CO.sub.2 laser in para-Hydrogen. We have also carried out calculations on the power broadening of the CO.sub.2 laser necessary to obtain Raman frequency shifts to match those required for the present invention. Note that even a 2 arm CO.sub.2 laser can suffice to cover all the required frequencies for the present invention. We also note that one can obtain frequencies near the required one from Raman shifting into the deuterium. Nowadays, however, there exist commercially available isotopic Carbon Dioxide Lasers [.sup.13C.sup.16O.sub.2] and [.sup.12C.sup.18O.sub.2] which could produce the required wavelength directly from a rotational line without the need of pressure broadening or other complicated modifications. For example, the P(38)[.sup.13C.sup.16O.sub.2] at 982.912774 cm.sup.1 when shifted by the parahydrogen line at H.sub.2[S(0)] at 354.275 cm.sup.1 will produce a shifted frequency at 628.5378 cm.sup.1 which is only 0.0108 cm.sup.1 from the three-photon resonance frequency with the [m(A.sub.2):(3.sub.3)] level of .sup.235UF.sub.6 at 628.527 cm.sup.1. The beam diameter of the applied beams can only be as large as they can comfortably cover a cross sectional area of the expanding gas with uniform density, at the maximum concentration numbers of UF.sub.6 available without condensation (FIG. 7). Gaussian spatial profile beams in the interaction region have been used in several experimental works published with spot sizes ranging from 1 mm to 3.2 mm.

[0119] The conversion of CO.sub.2 radiation through Raman scattering in para-Hydrogen using repetitive re-focusing in multipass Raman cells does not provide any flexibility in the control of the parameters of the applied beam. Pulses of duration approximately 75 ns are usually produced. The most suitable pulse duration for application to the present invention is between 1010.sup.9 s and 4010.sup.9 s. The correct way to produce controllable and flexible 16 m pulses with the required intensity, mode profile and frequency control is through the proper construction and the cavity control of Raman oscillators. The design of such oscillators is now being effected.

[0120] A comparison and evaluation of the potentialities of the AVLIS and the MLIS processes has been carried out. It is not possible to present even a summary the extensive analysis here. There are two many engineering problems in the AVLIS process, both in laser technology and in the construction of the interaction systems due to the corrosive nature of atomic Uranium, as well as the fact that the feed material lies outside the fuel cycle rendering an immediate 30% increase in the cost of fuel production due to the conversion process (UF.sub.6.fwdarw.U.sub.metal.fwdarw.UF.sub.6). The problems associated with the MLIS process were specific mainly associated with the interaction process and those are solved with the present invention. It is evident that the AVLIS process was a non starter from the beginning and this is the reason why the MLIS process was deliberately suppressed in the USA as early as 1983. A brief account of the systematic and contradictory statements on the development of the AVLIS process was given as early as 1997 [Andreou D., Nuclear Engineering International, pp 36-39, May 1997]. It is outside the scope of the present account to analyse the deliberate manipulations in the development of the LIS processes here.

[0121] We have also carried out a comparison and evaluation analysis with the centrifuge process. The latter can never be anywhere near as efficient as the MLIS process and more important, it hardly has any prospect of improving its present efficiency. The recent disaster with the construction of the large American centrifuge was to be expected. It was not only the enormous engineering problems which had to be overcome in the construction of such enormous systems such as the strength-to-weight ratio of the rotor material, the tensile strength affecting the maximum peripheral speed, the lifetime of the bearings at either end of the rotor, the characteristic vibrations the long rotor experiences as it spins and so many others arising as the size of the centrifuge is increased. One should have realized from the beginning that the UF.sub.6 gas has a very low self-diffusion coefficient D and any large increase in the size of the centrifuge will have an enormous effect on the circulation, movement and diffusion in the rotor. In all centrifuge equations the self-diffusion coefficient is the proportionality constant in the basic terms concerning both the axial variations and the radial variations in the gas [Cohen K., The Theory of Isotope Separation as Applied to the Large-Scale Production of U.sup.235 Chapter 6: Centrifuges, pp 106-109, McGraw-Hill Book Company, (1951)]. The self-diffusion coefficient of the Uranium Hexafluoride at 288 K is Dur, =0.042810.sup.4 m.sup.2/s. At 320 K it is D.sub.UF.sub.6=0.05310.sup.4 m.sup.2/s. This is a very low self-diffusion coefficient affecting all the processes in the rotor. Compare this with that of Carbon dioxide at 288 K is Dco.sub.2=0.12110.sup.4 m.sup.2/s.

[0122] Having set out the restrictions imposed on the selectivity process we now proceed to define the appropriate pumping frequencies and the applied pumping intensities with which we can attain the maximum selectivity for the .sup.235UF.sub.6 isotope. FIG. 8 (a) is indicative of the detrimental effect of pumping the .sub.3-vibrational ladder of the UF.sub.6 molecule at the frequency of the fundamental. The solid lines correspond to the power broadening of the energy levels of the desired isotope .sup.235UF.sub.6 and the broken line corresponds to the power broadening of the energy levels of the undesired isotope .sup.238UF.sub.6. The thick solid line running through all four graphs corresponds to the pumping frequency at the fundamental of the .sup.235UF.sub.6 isotope at 628.306 cm.sup.1. Its relation to the power broadened resonances of the first four energy levels of the .sub.3-vibrational mode is clearly depicted. The graphs are plotted for a pumping intensity of 3010.sup.9 W/m.sup.2. Even at such high pumping intensities there can only be a limited three-photon resonance with the third energy level and very small discrimination between the three-photon resonances of the two isotopes. We have plotted similar graphs with intensities ranging from 510.sup.9 W/m.sup.2 to 6010.sup.9 W/m.sup.2.

[0123] The conclusion is always that there is very small discrimination between the three-photon resonances of the two isotopes. This means that although one might distinguish and selectively excite the first energy level of the desired isotope it is not easy to drive the molecules to higher levels unless pumping them with very high intensity, but then they will escape into the quasicontinuum of energy states as it can be seen from Table 13. At the same time multiphoton resonance from the ground state of the undesired isotope will elevate the molecules to the quasicontinuum of energy states, and the molecules of the two isotopes will be fully intermixed, making it very difficult to subsequently dissociate them selectively.

[0124] FIG. 8 (b) shows the relations of the first four power broadened levels of the UF.sub.6 molecule to the pumping frequency for three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the third energy state of the .sup.235UF.sub.6 isotope at 628.527 cm.sup.1 at a pumping intensity of 3010.sup.9 W/m.sup.2, as described in the present invention. The solid line curves correspond to the power broadening of the energy levels of the desired isotope .sup.235UF.sub.6 and the broken line curves correspond to the power broadening of the energy levels of the undesired isotope .sup.238UF.sub.6. The thick solid line running through all four graphs corresponds to the pumping frequency for three-photon resonance with the third energy level of the .sup.235UF.sub.6 isotope at 628.527 cm.sup.1, and its relation to the power broadened resonances of the other levels is depicted. We have plotted similar graphs with intensities ranging from 510.sup.9 W/m.sup.2 to 6010.sup.9 W/m.sup.2 and for beams with frequencies ranging from 628.45 cm.sup.1 to 628.56 cm.sup.1. As the pumping intensity is increased the resonance conditions at the first and second energy excitation states improve and the three-photon absorption to the third energy level is readily enhanced. Even at high pumping intensities the levels of the undesired isotope .sup.238UF.sub.6 remain largely unexcited with the population of the third level of the desired isotope .sup.235UF.sub.6 being selectively populated. The high pumping intensities must be limited to levels below which the quasicontinuum of energy states does not acquire importance and detrimental effects to the absorption and dissociation processes cannot set in (Table 13). From Table 5(a) note that the molecules are elevated to the third energy state sublevel [m(A.sub.2):(3.sub.3)] via the pathway

[00107]$\left[m\left(A_1\right):\left(0v_3\right)\right]\left(l=0\right).fwdarw.\left[m\left(F_1\right):\left(1v_3\right)\right]\left(l=1\right).fwdarw.\left[m\left(F_2\right):\left(2v_3\right)\right]\left(l=2\right).fwdarw.\left[m\left(A_2\right):\left(3v_3\right)\right]\left(l=3\right)$

with the quantum transition rule l=1 being perfectly satisfied. A comparison of FIG. 8(a) with FIG. 8(b) demonstrates the enormous difference in selectively elevating the molecules of the desired isotope .sup.235UF.sub.6 to the third energy excitation level when pumping at the three-photon resonance frequency of 628.527 cm.sup.1 as compared with pumping at the frequency of the fundamental at 628.306 cm.sup.1. We have also plotted in more detail the power broadening of the individual levels as the intensity of the pumping beam increases showing the relation of the various pumping frequencies near 628.257 cm.sup.1 to the power broadened level. Furthermore we have tabulated the .sub.3-vibrational transitions of the lower five levels and their deviations from exact multiphoton absorption resonance for the Uranium Hexafluoride .sup.235UF.sub.6 isotope for various frequencies in this region. The sublevels involved have been listed, with all the transitions up the .sub.3-vibrational ladder obeying the selection rule l=1. Similar tables have been drawn for the Uranium Hexafluoride .sup.238UF.sub.6 isotope for the same frequencies indicating that they are very far away from any resonances with the same levels.

[0125] We now summarize the steps of the process which will selectively excite the molecules to a particular excited state from which they will subsequently be able to be dissociated by a variety of methods. The UF.sub.6 mixture with a carrier gas and a scavenger gas is expansion supercooled to a temperature of 60 K in such a way that the final UF.sub.6 (density is between 110.sup.21 and 510.sup.21 molecules/m.sup.3 with the final density of the UF.sub.6 molecules in the ground state being between 0.8610.sup.21 molecules/m.sup.3 and 4.310.sup.21 molecules/m.sup.3 respectively (Tables 18 and 19). The supercooled UF.sub.6 gas mixture is then irradiated with laser beams at the appropriate frequencies and intensities to selectively excite the desired isotope .sup.235UF.sub.6 to the [m(A.sub.2):(3.sub.3)] third energy state sublevel. The following points are of particular importance to the selective excitation of the desired isotope .sup.235UF.sub.6: (i) Good three photon resonance must be achieved with the [m(A.sub.2):(3.sub.3)] sublevel of the third energy excitation state (3.sub.3) of the desired .sup.235UF.sub.6 isotope. The applied selective frequency should therefore be at 628.527 cm.sup.1 or at a nearby frequency; (ii) The applied selective frequency should therefore preferably be between 628.45 cm.sup.1 and 628.6 cm.sup.1 with the most possible preferable range being between 628.49 cm.sup.1 and 628.527 cm.sup.1, values which are dependent on the intensity of the pumping beam; (iii) The intensity of the pumping beam should be between 110.sup.9 W/m.sup.2 and 5010.sup.9 W/m.sup.2 with the most probable range being between 510.sup.9 W/m.sup.2 and 3010.sup.9 W/m.sup.2. This range appears to be preferable so that no effect of the quasicontinuum of energy states sets in at the third energy state of the .sup.235UF.sub.6 molecules enabling them to escape to other vibrational modes and background states; (iv) Restricting the UF.sub.6 molecules from moving into the quasicontinuum of energy levels and other background states enables the effective application of infrared or ultraviolet laser beams to subsequently lead the selectively excited molecules to their dissociation; (v) By applying frequencies in the region of 628.527 cm.sup.1, resonance with the higher energy excitation levels, fourth, fifth etc. of the .sup.235UF.sub.6 isotope is limited, so that the molecules are kept in the third energy excitation level for a considerable amount of time, sufficient for simultaneous or subsequent dissociation processes to be applied; (vi) By applying the selective frequency for the desired .sup.235UF.sub.6 isotope at 628.527 cm.sup.1, or at a nearby frequency, resonances with the lower levels of the unwanted .sup.235UF.sub.6 are practically removed from resonance, further enhancing the selectivity of the desired .sup.235UF.sub.6 isotope to the third energy excitation level; (vii) The selective excitation of the .sup.235UF.sub.6 molecules to the third energy level will occur via three-photon resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the third energy excitation state assisted by the power broadening of the first and second energy excitation states through power broadening of the first, second and third energy level transitions. The positions of the selective frequency is further removed from the resonances with the levels of the unwanted .sup.238UF.sub.6 isotope; (viii) The pressure of the expansion supercooled gas should be such that collisional de-excitation of the molecules is limited to long de-excitation times, more than 150 ns; (ix) For a final pressure of the expansion supercooled gas of 1.25 Torr, the average time between collisions of the UF.sub.6 molecules with the carrier gas is much more than 500 ns which is a safe limit for considering that no collisional de-excitation of the UF.sub.6 molecules takes place during the interaction process. Collisional de-excitation by collisions between the UF.sub.6 molecules is much slower and of no particular consequence; (x) The main object of the pumping process is to selectively excite as many .sup.235UF.sub.6 molecules as possible to the third energy excitation level (nearly all of them can be excited) and keep them there for a sufficiently long time necessary for a subsequent dissociation process to be applied, either by simultaneously irradiating the molecular gas with infrared or ultraviolet lasers, or with a slight delay between them, or by any other dissociation process; (xi) We could also attempt pumping at frequencies slightly higher from the exact three-photon resonance frequency with the third energy excitation level. Resonance at the first and second energy levels may not only occur through power broadening processes. Other effects similar to Raman-type transitions in optically pumped lasers may enhance the transition due to the proximity of the pumping frequency to the resonance frequencies. Classical electrodynamics for near resonance interactions, which encompasses all the quantum effects simultaneously indicates that, during the interaction of electromagnetic radiation with a frequency near the resonance conditions with harmonically bound electrons, the scattering cross section (and similarly absorption) is greatly enhanced. We have taken the power broadening of the levels as a good practical indication of the significance of the level proximity to the absorption process.

[0126] FIG. 9 depicts the selectivity of the desired isotope .sup.235UF.sub.6 to the third energy excitation state through the power broadening of the lower vibrational levels and pumping frequencies near the three photon-resonance with the [m(A.sub.2):(3.sub.3)] sublevel of the state, at 628.527 cm.sup.1. The pumping intensity is set at 20 GW/m.sup.2. The solid line curves correspond to the desired .sup.235UF.sub.6 isotope whilst the broken line curves correspond to the unwanted .sup.238UF.sub.6 isotope. The thick black solid line corresponds to the exact three-photon resonance at 628.527 cm.sup.1. The other frequencies correspond to the lines indicated on the figure. The absorption probability resonances at the first and second energy levels at the three-photon resonance frequency of 628.527 cm.sup.1 and the nearby frequencies now have very substantial values. The probability resonance at the third level remains very near the peak of the curve. At 628.49 cm.sup.1 the resonances at the second and third energy levels are now excellent. The probability resonance at the first energy level is 70% and at the second energy level is more than 90% whilst the three-photon resonance at the third energy level remains more than 95% (97% crossing of the curve, see later). At the frequency of 628.45 cm.sup.1 the probability resonance at the first energy level increases to 80%, at the second energy level is near 100%, whilst the three-photon resonance at the third energy level remains more than 75 (85% crossing of the curve, see later). Four-photon resonances with the fourth energy excitation level still remain extremely low. At the pumping frequency of the fundamental 628.306 cm.sup.1, the first level selectivity remains very good but three-photon resonance with the third energy level is still very poor for both isotopes. We have plotted and investigated many such graphs for pumping intensities between 2.5 GW/m.sup.2 and 50 GW/m.sup.2. The results are always very impressive.

[0127] The next task we have carried out is to check the possibilities of resonances between the sublevels of the first four energy states at the frequencies depicted in FIG. 9. It was found that no such resonances are possible even at high pumping intensities. We have also investigated the possibilities for transitions between the sublevels of the third and fourth energy excitation states, in case there is any fast redistribution of the excited molecules amongst the sublevels of the third energy excitation state within a time comparable to the duration of the pumping pulses. Any escaping of .sup.235UF.sub.6 molecules to the (4.sub.3) state as a result of fast possible redistribution amongst the sublevels of the (3.sub.3) state was found to be negligible. All the results were tabulated but it is not possible to present them in the short space available.

[0128] The next task we considered were the frequencies of the high power infrared beams necessary for the selective dissociation of the .sup.235UF.sub.6 molecules. It is important that the simultaneous application of the dissociating beams avoid any multiphoton resonances with the higher levels of the vibrational ladder as much as possible, when starting from the ground state. At the same time they should match as closely as possible the frequency differences between the sublevels of the 3.sup.rd, 4.sup.th, 5.sup.th, 6.sup.th, 7.sup.th and 8.sup.th energy states of the .sub.3-mode vibrational ladder when starting from the [m(A.sub.2):(3.sub.3)] sublevel of the third energy state. We have drawn many graphs from the 4.sup.th to the 8.sup.th energy states depicting the relevant sublevels in relation to the CO.sub.2 Raman shifted lines in parahydrogen when considered from the ground state for many pumping beam intensities between 60 GW/m.sup.2 and 120 GW/m.sup.2. Most of the frequencies in the range from R(20) to R(12) will miss all multiphoton resonances with the higher vibrational states of the .sub.3-vibrational mode of the undesired isotope .sup.238UF.sub.6 even at high pumping intensities. The most suitable pumping frequencies directly originating from the CO.sub.2 rotational lines shifted in parahydrogen are the R(18)=620.2476393 cm.sup.1 and R(20)=621.5561374 cm.sup.1 lines. We have located two pumping dissociation frequencies at 620.6 cm.sup.1 and 623.3 cm.sup.1 which can match multiphoton resonances between the [m(A.sub.2):(3.sub.3)] sublevel of the third energy excitation state and higher levels of the .sub.3-vibrational mode up to the eighth energy excitation state of the desired isotope .sup.235UF.sub.6, without affecting or resonating with any of the levels of the unwanted isotope .sup.238UF.sub.6 even at high pumping intensities. The details cannot be presented in the short space available.

[0129] We now depict the entire selectivity and dissociation process using selectivity frequencies in the region of 628.527 cm.sup.1 and dissociation frequencies in the region of 620.6 cm.sup.1. The latter is approximately the frequency difference between the [m(A.sub.2):(3.sub.3)] and [m(F.sub.1):(4.sub.3)] of the 3.sup.rd and 4.sup.th energy states respectively of the .sub.3-vibrational mode of the desired .sup.235UF.sub.6 isotope and can match most of the sublevels up to the eighth energy state thus keeping the excitation energy within one single vibrational mode up to very high energies. FIG. 10 depicts the entire selectivity and dissociation process for a selective beam frequency of 628.527 cm.sup.1 (continuous thick black line in the first three energy excitation states) at a pumping intensity of 20 GW/m.sup.2 and a dissociating beam frequency of 620.6 cm.sup.1 (thick broken line from the third to the eighth energy excitation state) at a pumping intensity of 80 GW/m.sup.2. The solid line curves correspond to the desired 333UF.sub.6 isotope whilst the broken line curves correspond to the unwanted .sup.238UF.sub.6 isotope. We see that the dissociating frequency is in direct resonance with three of the five levels keeping the excitation energy within the .sub.3-vibrational mode up to the eighth energy excitation state of the desired isotope .sup.235UF.sub.6 without affecting or resonating with any of the levels of the unwanted isotope .sup.238UF.sub.6. We recall that at these higher levels the selection rules are not very strict. The broken arrow line corresponds to the resonances of the unwanted isotope .sup.238UF.sub.6 when dissociation frequency at 620.6 cm.sup.1 starts from the corresponding third energy sublevel at 1883.77 cm.sup.1. Note that resonances with the higher levels can exist and this is why very good selectivity of the desired isotope must be achieved at the third energy excitation state, hence the importance of not elevating molecules of the unwanted .sup.238UF.sub.6 isotope to the third energy excitation level. Recall that the selection rules for the higher energy excitation states are loosely applicable. By applying this scheme searching for the most suitable frequencies in the neighbourhood of these frequencies and also obtaining the optimum pumping intensities for the two beams the selectivity of the desired isotope will turn out to be outstanding. We have investigated dozens of schemes like the one in FIG. 10 with different selectivity and dissociation frequencies at various pumping intensities. The nearest dissociating frequency directly derivable from the CO.sub.2 Raman shifted line in parahydrogen is the R(18)=620.2476393 cm.sup.1. The power of the dissociating laser must also be restrained to levels which do not originate processes starting from the ground state to the quasicontinuum thereby affecting selectivity.

[0130] We have depicted the position of the absorption frequencies in a number of illustrative ways all of them showing the same trends and values but their presentation is outside the scope of the present patent application.

[0131] We have investigated in more detail the selectivity to the third energy excitation level of the .sub.3-vibrational mode of UF.sub.6 for each of the energy levels up to the fourth energy state, for various selective frequencies ranging from 628.45 cm.sup.1 to 628.56 cm.sup.1 and pumping powers ranging from 1 GW/m.sup.2 to 80 GW/m.sup.2. FIG. 11 shows a summary of the hundreds of graphs drawn for obtaining the selectivity between the two isotopes at the various pumping frequencies and intensities. All four graphs are drawn for a pumping intensity of 20 GW/m.sup.2. The abscissa |c(t)|.sup.2 represents the absorption probability for the corresponding frequency on the ordinate. We have then calculated the relative selectivities to the third energy state for the two isotopes UF.sub.6 and .sup.238UF.sub.6. For the first energy excitation level (fundamental) the absorption cross section follows the power broadened curve of the fundamental transition. The effective selectivity between the two isotopes due to the inherent resonance at the fundamental, would therefore correspond to the ratio of their respective power broadened curves at the position of the pumping frequency. For the second energy excitation level the absorption cross section is proportional to the intensity of the applied beam (two-photon absorption). Since the power broadened curve represents the probability for absorption at a particular frequency, the envelope of the curve is proportional to the intensity of the pumping beam. Again, the effective selectivity between the two isotopes due to inherent two-photon resonance with the second energy excitation level would correspond to the ratio of their respective power broadened curves at the position of the pumping frequency. The situation is different in the case of three-photon resonance with the third energy excitation level. The three-photon absorption cross section with the third energy excitation level is proportional to the square of the pumping intensity (see below eq. (54)). Let I.sub.o be the intensity of the pumping beam at the peak of the absorption power broadened curve. Its value at any other point on the curve will be qI.sub.o where q is the ratio of the abscissa of the curve at the position of the pumping frequency to the peak of the power broadened curve. The three-photon absorption cross section at the peak of the curve is .sub.o=K.sup.2I.sub.o.sup.2. The absorption cross section at the pumping frequency is .sub.1=K.sup.2 (q.sup.2 I.sub.o.sup.2). The ratio of the absorption cross sections at the two frequencies is (.sub.1/.sub.o)=q.sup.2. The effective selectivity between the two isotopes due to three-photon resonance with the third energy excitation level would therefore be proportional to the square of the ratio of their respective power broadened curves at the position of the pumping frequency. Similar relations for the absorption cross sections at the higher energy excitation levels will progress accordingly but at these high excitation states and pumping intensity levels other effects may also become dominant.

[0132] The positions of six pumping frequencies at 628.306 cm.sup.1, 628.45 cm.sup.1, 628.49 cm.sup.1, 628.527 cm.sup.1, 628.56 cm.sup.1 and 628.6 cm.sup.1 are shown on the graphs of FIG. 11. A look at the first graph (a) demonstrates clearly that at 628.306 cm.sup.1 (frequency of the fundamental transition of the .sup.235UF.sub.6 isotope) the selectivity at the ground state absorption remains excellent despite the much increased power broadening of the levels. Although at the second energy excitation state a fair amount of selectivity is indicated [graph (b)], the selectivity at the three-photon resonance remains extremely poor [graph (c)], whilst absorption is also limited, especially since we must consider the square of the absorption selectivity values. Thus, at this frequency overall absorption and selectivity of the desired isotope to the third energy excitation level remains extremely poor. At the selecting frequency of 628.527 cm.sup.1 [three-photon resonance with the third level, graph (c)] excellent absorption probability resonance ensues at the third energy level [graph (c)] whilst maintaining very good two-photon resonance conditions at the second energy excitation level [graph (b)]. The now much improved resonance at the first energy excitation level due to power broadening of the .sup.235UF.sub.6 fundamental, renders

TABLE-US-00020 TABLE 20 Pumping Intensity 20 10.sup.9 W/m.sup.2 Frequency for two- Arbitrary frequency Frequency for three- Slightly higher Fundamental photon resonance slightly on the photon resonance frequency than PUMPING frequency of the with second level at lower side of with third level at the sublevel FREQUENCY .sup.235UF.sub.6 (1/2) m(F.sub.2):(2v.sub.3) (1/3) m(A.sub.2):(3v.sub.3) (1/3) m(A.sub.2)(3v.sub.3) (1/3) m(A.sub.2)(3v.sub.3) 1.sup.st Energy Level Pumping 628.306 (cm.sup.1) 628.45 (cm.sup.1) 628.49 (cm.sup.1) 628.527 (cm.sup.1) 628.6 (cm.sup.1) frequency Selectivity 7 7.31 7.13 6.77 5.91 [00108]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{curve}}{\mathrm{peak}\mathrm{of}{}^{235}{\mathrm{UF}}_6}$ 1 0.74 0.69 0.56 0.42 [00109]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{curve}}{\mathrm{peak}\mathrm{of}{}^{238}{\mathrm{UF}}_6}$ 0.15 0.1 0.097 0.084 0.072 2.sup.nd Energy Level Two photon 1256.612 (cm.sup.1) 1256.9 (cm.sup.1) 1256.98 (cm.sup.1) 1257.054 (cm.sup.1) 1257.2 (cm.sup.1) Pumping frequency Selectivity 5.89 12.62 13.56 12.9 10.79 [00110]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{curve}}{\mathrm{peak}\mathrm{of}{}^{235}{\mathrm{UF}}_6}$ 0.68 1 0.93 0.84 0.52 [00111]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{curve}}{\mathrm{peak}\mathrm{of}{}^{238}{\mathrm{UF}}_6}$ 0.12 0.078 0.07 0.065 0.05 3.sup.rd Energy Level Three photon 1884.918 (cm.sup.1) 1885.35 (cm.sup.1) 1885.47 (cm.sup.1) 1885.58 (cm.sup.1) 1885.8 (cm.sup.1) Pumping frequency Selectivity 2.47 11.27 15 18.33 18.75 [00112]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{curve}}{\mathrm{peak}\mathrm{of}{}^{235}{\mathrm{UF}}_6}\left(\mathrm{squared}\right)$ 0.305 0.093 0.805 0.648 0.94 0.884 1 1 0.786 0.617 [00113]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{curve}}{\mathrm{peak}\mathrm{of}{}^{238}{\mathrm{UF}}_6}\left(\mathrm{squared}\right)$ 0.123 0.015 0.069 4.74 10.sup.3 0.06 3.6 10.sup.3 0.054 2.9 10.sup.3 0.042 1.76 10.sup.3 Absorption probability 0.063 0.48 0.57 0.47 0.134 of the .sup.235UF.sub.6 isotope to the 3.sup.rd energy level Absorption probability 0.27 10.sup.3 0.037 10.sup.3 0.0245 10.sup.3 0.0158 10.sup.3 0.0063 10.sup.3 of the.sup.238UF.sub.6 isotope to the 3.sup.rd energy level. Relative selectivity 0.233 10.sup.3 12.97 10.sup.3 23.26 10.sup.3 29.75 10.sup.3 21.27 10.sup.3 between the two isotopes to the 3.sup.rd energy level Selectivity increase 1 55.7 99.8 127.7 91.29 that of the fundamental .sup.235UF.sub.6 line at 628.306 cm.sup.1 4.sup.th Energy Level Four-photon 2513.22 (cm.sup.1) 2513.8 (cm.sup.1) 2513.96 (cm.sup.1) 2514.11 (cm.sup.1) 2514.4 (cm.sup.1) Pumping frequency [00114]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{curve}}{\begin{array}{c}\mathrm{peak}{\mathrm{of}}^{235}{\mathrm{UF}}_6\\ \left(\mathrm{cubed}\right)\end{array}}$ 0.139 2.68 10.sup.3 0.073 0.385 10.sup.3 0.061 0.227 10.sup.3 0.051 0.13 10.sup.3 0.039 0.06 10.sup.3 [00115]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{curve}}{\begin{array}{c}\mathrm{peak}{\mathrm{of}}^{238}{\mathrm{UF}}_6\\ \left(\mathrm{cubed}\right)\end{array}}$ 0.017 4.81 10.sup.6 0.013 2.19 10.sup.6 0.012 1.6 10.sup.6 0.01 1.12 10.sup.6 0.007 0.27 10.sup.6
the overall selectivity to the third energy excitation, at this pumping frequency and intensity, very good. At the nearby frequencies of 628.45 cm.sup.1 and 628.49 cm.sup.1 the absorption probability resonances are now substantial at all three levels rendering the selectivity process to the third energy excitation level excellent. Especially at the frequency 628.49 cm.sup.1 the resonances become very good at all three levels. The three-photon selectivity process at these

TABLE-US-00021 TABLE 21 Absorption probability resonances and Selectivities to the Third Energy Level of UF.sub.6 Frequency for Frequency for two-photon Arbitrary frequency three photon Slightly higher Fundamental resonance with slightly on the resonance with frequency than Pumping PUMPING frequency of the level at lower side of third level at the Intensity FREQUENCY the .sup.235UF.sub.6 (1/3) m(F.sub.2):(2v.sub.3) (1/3) m(A.sub.2):(3v.sub.3) (1/3) m(A.sub.2):(3v.sub.3) (1/3) m(A.sub.2):(3v.sub.3) (W/m.sup.2) Pumping frequency 628.306 (cm.sup.1) 628.45 (cm.sup.1) 628.49 (cm.sup.1) 628.527 (cm.sup.1) 628.6 (cm.sup.1) 5 10.sup.9 [00116]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{235}{\mathrm{UF}}_6}$ 0.097 0.5 0.81 1 0.45 [00117]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{238}{\mathrm{UF}}_6}$ 0.031 0.014 0.015 0.015 0.008 Selectivity increase to the 3.sup.rd 1 423 604 947 193 level relative to that at 628.306 cm.sup.1 10 10.sup.9 [00118]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{235}{\mathrm{UF}}_6}$ 0.18 0.65 0.857 1 0.67 [00119]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{238}{\mathrm{UF}}_6}$ 0.0675 0.034 0.032 0.028 0.0235 Selectivity increase to the 3.sup.rd 1 187.33 373.1 423.8 154.13 level relative to that at 628.306 cm.sup.1 15 10.sup.9 [00120]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{235}{\mathrm{UF}}_6}$ 0.252 0.75 0.922 1 0.7273 [00121]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{238}{\mathrm{UF}}_6}$ 0.093 0.05 0.04545 0.04 0.0325 Selectivity increase to the 3.sup.rd 1 90 161.2 203 110.17 level relative to that at 628.306 cm.sup.1 20 10.sup.9 [00122]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{235}{\mathrm{UF}}_6}$ 0.305 0.805 0.94 1 0.786 [00123]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{238}{\mathrm{UF}}_6}$ 0.123 0.069 0.06 0.054 0.042 Selectivity increase to the 3.sup.rd 1 55.7 99.8 127.7 91.29 level relative to that at 628.306 cm.sup.1 30 10.sup.9 [00124]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{235}{\mathrm{UF}}_6}$ 0.4 0.85 0.95 1 0.85 [00125]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{238}{\mathrm{UF}}_6}$ 0.171 0.095 0.089 0.08 0.064 Selectivity increase to the 3.sup.rd 1 37.32 54.39 71.9 68.21 level relative to that at 628.306 cm.sup.1 40 10.sup.9 [00126]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{235}{\mathrm{UF}}_6}$ 0.46 0.896 0.96 1 0.89 [00127]$\frac{\mathrm{Crossing}\mathrm{of}\mathrm{the}\mathrm{curve}}{\mathrm{Peak}\mathrm{of}{3}^{rd}{\mathrm{level}}^{238}{\mathrm{UF}}_6}$ 0.227 0.126 0.1143 0.1013 0.084 Selectivity increase to the 3.sup.rd 1 30.21 45.66 61.25 62.35 level relative to that at 628.306 cm.sup.1
frequencies now becomes good and it is only a matter of the level of the pumping intensity to excite the molecules of the .sup.235UF.sub.6 isotope for the laser isotope separation process. Note that the calculations of the selectivity and the graphic representations are relative and approximate but they give a very good indication of the limits for the intensities, pulse lengths and frequencies which must be applied to the molecular gas for obtaining outstanding selectivity.

[0133] All the graphs have been plotted on an enlarged scale and the relative selectivities to the third energy state for the two isotopes .sup.235UF.sub.6 and .sup.238UF.sub.6 have been calculated for each pumping intensity by considering the points at which the various frequencies cross the curves as described above. Table 20 is a typical table at a pumping intensity of 20 GW/m.sup.2. Note the enormous increase in the selectivity to the third energy excitation level when pumping at the three-photon resonance frequency 628.527 cm.sup.1 as compared with pumping at the frequency of the fundamental at 628.306 cm.sup.1 (127 times higher selectivity). Subsequently, we have summarized all the results for all the pumping intensities from 0.5 GW/m.sup.2 to 80 GW/m.sup.2. Table 21 is a typical example for pumping intensities between 510.sup.9 W/m.sup.2 and 4010.sup.9 W/m.sup.2. The third row at each pumping intensity level shows the selectivity increase to the third energy excitation level when pumping at frequencies near the three-photon resonance at 628.527 cm.sup.1 as compared with pumping at the frequency of the fundamental at 628.306 cm.sup.1, as calculated from tables like table 20. Results for pumping intensities below 510.sup.9 W/m.sup.2 may not have any practical significance because the attainment of three-photon resonance may be difficult to establish. The selectivity for pumping intensities higher than 4010.sup.9 W/m.sup.2 may be hampered by absorption in the quasicontinuum of energy states. The trend with which the selectivity decreases with increasing intensity is clear. The reason of the enormous selectivity at lower pumping intensities is due to the fact that all molecules of the desired isotope .sup.235UF.sub.6 can be elevated to the third energy level whilst at lower pumping intensities those of the undesired .sup.238UF.sub.6 isotope remain largely unexcited. Although the results may be considered to be only indicative it is clear that enormous selectivities can be achieved when three photon absorption for the .sup.235UF.sub.6 isotope is attained at lower pumping intensities with shorter duration pulses (see later).

[0134] We have investigated the three-photon absorption process quantitatively by obtaining the transition rate for the three-photon absorption resonance. We have started from the most general form of the Fermi Golden rule for transitions and expanded the terms of the interaction Hamiltonian to higher orders:

[00128]$\begin{array}{cc}{W\text{?}=\frac{2}{}.Math.R\text{?}{.Math.}^2\left(-\right)=\frac{2}{}.Math..Math.\text{?}.Math.R.Math.\text{?}.Math.}^2\left(-\right)& \left(36\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where .sub.F and .sub.I are the final and initial states of the entire quantum system (atom plus radiation) and the -function is given by

[00129]$\begin{array}{cc}\left(-\right)=\left(--{}_k\right)=\frac{1}{}\left(\text{?}-{}_k\right)& \left(37\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where and are the final and initial energy states of the entire quantum system (atom plus radiation), and are the final and initial energies of the atomic states, on is the frequency difference between the final and initial atomic states, .sub.k is the frequency of the interacting radiation and eq. (37) ensures the conservation of energy for the system as a whole (atom plus radiation field), R.sub.FI is the reaction matrix for the whole quantum system as defined mathematically in eq. (36). The reaction matrix R.sub.fi for an atomic system which will reach a final state .sub.f from an initial atomic state .sub.i can be expanded as

[00130]$\begin{array}{cc}{R}_{\mathrm{FI}}.Math.{}_F.Math.R.Math.\text{?}.Math.=R_{\mathrm{FI}}^{\left(1\right)},R_{\mathrm{FI}}^{\left(2\right)},R_{\mathrm{FI}}^{\left(3\right)},R_{\mathrm{FI}}^{\left(4\right)}+.Math.& \left(38\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where each of the terms

[00131]$R_{\mathrm{FI}}^{\left(1\right)},R_{\mathrm{FI}}^{\left(2\right)},R_{\mathrm{FI}}^{\left(3\right)},R_{\mathrm{FI}}^{\left(4\right)}+.Math.$

has been evaluated through the repetitive application of the Lippmann-Schwinger equation describing the evolution of the wavefunction from time t.sub.o=0 to/through the evolution operator U(t,t.sub.o). During the three-photon absorption process three incident photons (k), (k) and (k) with nearly the same energy and specified polarization and propagation direction are absorbed by an atomic or molecular system, the latter making a transition to a final state whose energy is approximately (.sub.k+.sub.k+.sub.k). Thus, in a three-photon absorption process three photons are lost. The evolution of the wavefunction to the third energy state in the case of three-photon absorption occurs through the use of the Dyson chronological operator with the integrations being carried out over all the possible orderings of the times t.sub.1, t.sub.2 and t.sub.3 i.e. the number of ways in which the three photons are being absorbed between levels |.sub.i and |.sub.f. The matrix elements of all six possible pathways have been computed and summarized mathematically. All pathways contribute to the third order term in the interaction Hamiltonian during the three-photon absorption process. They are depicted schematically in FIG. 12(a). Subsequently we have obtained the matrix elements for all the terms that contribute to the third order reaction matrix R.sub.FI.sup.(3) in eq. (38). Our analysis followed similar lines to those suggested by Weissbluth on higher order electromagnetic interactions applied to the three-photon absorption process [Weissbluth M., Atoms and Molecules, Academic Press, New York, pp. 544-547, (1978)]. FIG. 12(b) depicts the situation when the three photons are the same. i.e. .sub.k=.sub.k=.sub.k. Note that the intermediate states |.sub.i and |.sub.m although imaginary they nevertheless constitute solutions to the atomic Schrdinger equation. When the intermediate states |.sub.i and |.sub.m are real atomic or molecular states as in the case of a vibrational ladder the situation is depicted in FIG. 12(c) where their position may differ slightly from exact resonance. This is the case of three-photon resonance with the third energy excitation level [m(A.sub.2):(3.sub.3)] of the .sup.235UF.sub.6 isotope when the vibrational ladder interacts with a one frequency pumping beam i.e with reference to FIGS. 12(a), (b) we set .sub.k=.sub.k=.sub.k, n.sub.k=n.sub.k=n.sub.k, m.sub.1=m.sub.2=m.sub.3=m, l.sub.1=l.sub.2=l.sub.3=l, with the resulting situation being depicted in FIG. 12(c). It is not possible to present the complete derivation here but on noting that in the case of sufficiently powerful pumping beams the electric dipole approximation e.sup.ik.Math.r1 is valid, giving the identity:

[00132]$\begin{array}{cc}.Math.\text{?}.Math.\hat{e}\text{?}.Math.p)e\text{?}.Math.{}_m.Math.=im{}_k\hat{e}\text{?}.Math..Math.{}_f.Math.r.Math.{}_m.Math.& \left(39\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

which when used in the detailed analysis results in the following expression for the three-photon transition rate W.sub.fi as (in MKSA units)

[00133]$\begin{array}{cc}W\text{?}=36{\left(\frac{2{}_{k}n\text{?}}{V\text{?}}\right)}^3\underset{m}{.Math.}\underset{1}{.Math.}\frac{1}{{\left(4{}_o\right)}^3}{\left\{\frac{e^3.Math.{}_f\left|r\right|{}_m.Math..Math.{}_m\left|r\right|{}_l.Math..Math.{}_l\left|r\right|\text{?}.Math.}{\left(+2{}_k\right)\left(+{}_m\right)}\right\}}^{2}g\left(v_f\right)& \left(40\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0135] The units of the terms on the right hand side are

[00134]$Js{\left(\frac{s^{-1}}{m^3}\right)}^3\frac{1}{{\left(F/m\right)}^3}\frac{C^3{m}^3}{J^2})s=s^{-1}$

which is the unit for the transition rate W.sub.fi. Eq. (40) gives the transition rate in s.sup.1 for three-photon absorption from an initial state |.sub.i to a final excited state |.sub.f in terms of the number of photons n.sub.k. The pumping intensity I.sub..sub.k is given by

[00135]$\begin{array}{cc}I\text{?}=\frac{c{n}_{k_{}}{}_k}{V_{\mathrm{ol}}}& \left(41\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

which when substituted into eq. (40) gives the transition rate for the three photon resonance

[00136]$\begin{array}{cc}\begin{array}{c}W\text{?}=\frac{4I_{}^3\text{?}}{c^3{}_o^3{}^2}\underset{m}{.Math.}\underset{l}{.Math.}{\left\{\frac{e^3.Math.{}_f\left|r\right|{}_m.Math..Math.{}_m\left|r\right|{}_l.Math..Math.{}_l\left|r\right|\text{?}.Math.}{\left(+2{}_k\right)\left(+{}_m\right)}\right\}}^{2}g\left(v_f\right)\\ =\frac{4{}_k^3}{c^3{}_o^3}{I}_{}^3\text{?}\left(x,t\right)\underset{m}{.Math.}\underset{l}{.Math.}{\left\{\frac{e^3.Math.{}_f\left|r\right|{}_m.Math..Math.{}_m\left|r\right|{}_l.Math..Math.{}_l\left|r\right|\text{?}.Math.}{\left(+2{}_k\right)\left(+{}_m\right)}\right\}}^{2}g\left(v_f\right)\end{array}& \left(42\right)\end{array}$$\mathrm{with}$$\begin{array}{cc}{I}_{}\text{?}=hv{I}_{}\text{?}\left(x,t\right)={}_k{I}_{}\text{?}\left(x,t\right)& \left(43\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

where W.sub.fi(s.sup.1) is the three-photon transition rate,

[00137]$I_{}\text{?}\left(J/m^{2}s\right)$$\text{?}\text{indicates text missing or illegible when filed}$

is the intensity of the pumping beam, I.sub..sub.k(x,t) (photons/m.sup.2s) is the photon intensity of the pumping beam, .sub.k=2 (s.sup.1) is the frequency of the applied radiation, g(.sub.f) (s) is the lineshape function given by eq. (16) resulting from the smearing out of the position of the final energy state and .sub.f(s.sup.1) is the half width of the absorption spectrum of the level. Eq. (42) gives the transition rate in s.sup.1 for three-photon absorption from an initial state |.sub.i to a final excited state |.sub.f in terms of the intensity I.sub..sub.k and the induced dipole moments. Note the resonances in the denominator. They are resonances of the quantum system between the ground and the 1.sup.st energy level, and the ground and the second energy level. They are not between successive energy levels. Two photons are required to match the energy difference between the ground and the second energy excitation level simultaneously with the photon match at the fundamental. This scientific subtlety has always been overlooked in the rather factitious conception of the multiphoton effect, frequently depicted as the stepwise absorption of photons one by one, up the vibrational ladder during its interaction with a laser beam. The transition rate in eq. (42) becomes infinite when one of the terms in the denominator vanishes. This is a consequence of the assumption of infinitely sharp atomic states. Near resonance it is therefore necessary to include a damping factor which must be incorporated in each of the terms in the denominator. This damping factor will be of the order of

[00138]$\frac{1}{2}.fwdarw.\frac{1}{2}\left(2v\right)$

where is the spectral width of the spontaneous emission line of the transition. Thus, the excited atomic state is broadened to a width as a result of the spontaneous emission process, resulting in a Lorentzian line shape in the emission spectrum whose full width at half maximum is =/. This means that, as a consequence of spontaneous emission, the quantum states cannot be infinitely sharp but must have a finite spread in energy equal to ( ) which corresponds approximately to the natural linewidth of the transition. It is the minimum possible width assuming all other broadening mechanisms have been eliminated. We will not elaborate any further on the theory of three-photon absorption due to extreme shortage of space.

[0136] Following eq. (29) above, the dipole moment between successive higher vibrational levels of a harmonic oscillator increases according to the square root of increasing vibrational numbers. Since the first three energy levels of the .sub.3-vibrational mode of the Hexafluorides are a very close match to a harmonic oscillator, we can write the transition elements of the dipole moments of the levels as e.sub.f|r|.sub.m=e{square root over (3)}.sub.1|r|.sub.i and e.sub.m|r|.sub.1=e{square root over (2)}.sub.i|r|.sub.i with the dipole moment between the ground and the third energy excitation level becoming

[00139]$\begin{array}{cc}{}_{\mathrm{fi}}=.Math.{}_f.Math..Math.{}_i.Math.=e^3.Math.{}_j.Math.r.Math.{}_m.Math..Math.{}_m.Math.r.Math.{}_i.Math..Math.{}_i.Math.r.Math.{}_i.Math.=\sqrt{6}{\left(e.Math.{}_l.Math.r.Math.{}_i.Math.\right)}^3& \left(44\right)\end{array}$