Laser isotope enrichment of 168Er

In the three-step photoionization process under investigation, the atoms present in the 4f¹²6s^{2 3}H₆ (0.0 cm^-1) ground state are excited to the 4f¹¹5d6s² J = 7 (15,846.549 cm^-1) excited level using the first excitation laser tuned to the 631.052 nm transition. Atoms that are excited to the first excitation level are further excited to the 4f¹¹5d²6s J = 8 (32,884.867 cm^-1) level using a second excitation laser tuned to the 586.912 transition. These excited atoms are ionized by further exciting into an auto-ionization level present at 50,552.6 cm^-1 using a third excitation laser tuned to the 566.003 nm transition. For the present work, all the lasers have been considered to be having a pulse width of 30ns, with a pulsed repetition frequency (PRF) of 10 kHz. The lasers have been considered to be co-propagating unless stated otherwise.

The density matrix elements describing the laser-atom interactions in three-step photoionization scheme have been published previously²⁹. The density matrix elements are numerically integrated using the standard integration methods for the entire pulse duration using Gaussian temporal profiles with no temporal delay between pulses.

At the end of the laser-atom interaction, the atomic population of the auto-ionization state undergoes ionization. Therefore, population of the auto-ionization state has been considered as the ionization efficiency. From the ionization efficiency of the constituent isotopes, the degree of enrichment of the target isotope has been calculated using the equation

where, η is the ionization efficiency of an isotope and A is its fractional abundance.

Features of the ionization efficiency contours

First, ionization efficiency of natural erbium has been calculated under Doppler free conditions (i.e., ignoring the angular and velocity distribution of atoms) varying the frequency of the first and second excitation lasers for a bandwidth of 100 MHz and the intensity of 5 W/cm² for all the lasers. For these calculations the third excitation laser is set to the ¹⁶⁸Er resonance. The resultant two-dimensional contour is plotted in Fig. 6. The following observations can be drawn from the contour.

All the resonances of even isotopes have been found at the expected positions (Table 5). Since the third excitation laser is tuned to the resonance of ¹⁶⁸Er isotope, the ionization efficiency of other isotopes at their resonance frequencies positions in two-dimensional reference frame does not correspond to the respective abundances. The horizontal ridge of an isotope corresponds to the line spectra of the first excitation transition while the vertical ridge corresponds to the line spectra of the second excitation transition. The diagonal ridge observed arises due to the coherent two-photon excitation at all frequencies wherein the detuning of the two transitions are equal in magnitude and opposite in sign (i.e., Δ₁ =—Δ₂). The coherent two-photon excitation probability decreases with increase in detuning (\(\left|\Delta \right|\)).

Under the conditions described in Fig. 6, the coherent two-photon excitation is strong and extends to a few GHz. The resonance of the ¹⁶⁸Er target isotope lies on coherent two-photon excitation ridges of the non-target even isotopes. Since the degree of coherent two-photon excitation of the constituent isotopes is determined by the laser (such as bandwidth and intensity) and atom source parameters (such as Doppler broadening) a careful control of all the system parameters is important to obtain adequate degree of enrichment and production rate.

Ionization efficiency contours of even isotopes of Er

In two dimensional reference frame, ¹⁶²Er resonance appears (marked as ¹⁶²Er) when the first and second excitation lasers are tuned to the (-5981 MHz, 5270 MHz) frequencies respectively. Its ionization efficiency is lower as the third excitation laser is detuned by + 398.4 MHz in its reference frame. The net detuning of this transition Δ₁ + Δ₂ + Δ₃ is 0 MHz + 0 MHz + 398.4 MHz =  + 398.4 MHz in three-photon reference frame of ¹⁶²Er isotope. An unexpected additional resonance (marked as #) for ¹⁶²Er has been observed at frequency positions of (-5981 MHz, 4871.6 MHz). This resonance occurs due to the direct two-photon ionization of ¹⁶²Er isotope from the first excitation level. Since the third excitation is laser is detuned by + 398.4 MHz (in ¹⁶²Er reference frame) the resonance occurs at the frequency of 5270 MHz – 398.4 MHz which is equal to + 4871.6 MHz. In this case, the net detuning from the resonance is Δ₁ + Δ₂ + Δ₃ = 0 MHz – 398.4 MHz + 398.4 MHz = 0 MHz. Since the detuning in this case is smaller than the resonance at (-5981 MHz, 5270 MHz), this resonance (marked with #) is stronger. A similar but poorly resolved resonance (due to the small shift of + 187 MHz) can be for observed for ¹⁶⁴Er isotope as well. In fact, this type of resonance can also be observed for all non-target constituent isotopes, although it cannot be resolved due to the saturation broadening and small isotope shifts associated with the third excitation transition. A detailed discussion on the lineshape contours of the three-step photoionization can be found in Ref.³⁰.

Ionization efficiency contour of odd ¹⁶⁷ Er isotope

A weak resonance corresponding to the 15/2–17/2–19/2 hyperfine pathway of ¹⁶⁷Er isotope can be observed at (-1775 MHz, 348.6 MHz) frequency position (marked as * in Fig. 6). Although the natural abundance of ¹⁶⁷Er is relatively high at 22.869%, its ionization efficiency at the frequency position corresponding to the 15/2–17/2–19/2 hyperfine pathway is quite low. This is because of the large spread of the hyperfine spectrum (4378 MHz, 4370 MHz and 5760 MHz for the three excitation transitions respectively) of ¹⁶⁷Er. Furthermore, the most intense resonance of the ¹⁶⁷Er isotope, associated with the 19/2–21/2–23/2 hyperfine pathway, is not observed at the (-1401.6 MHz and -53.1 MHz) frequency position (see Table 5). This cannot be explained using the two-photon reference frame presented in Table 5; instead, a three-photon reference frame must be considered. Using the spectroscopy selection rules, it is possible to formulate 135 hyperfine pathways for the three-step photoionization of ¹⁶⁷Er (see Table 6). To further elucidate, a two-dimensional contour plot of ¹⁶⁷Er has been plotted in Fig. 7A, where the third excitation laser is tuned to the 23/2–25/2 hyperfine transition of ¹⁶⁷Er. The hyperfine pathways in Fig. 7 were marked as per the serial numbers in Table 6. Since the third excitation laser is tuned to the 23/2–25/2 hyperfine transition, the most intense hyperfine pathway 19/2–21/2–23/2–25/2 is distinctly observed (marked as 30 in Fig. 7A) at the expected frequency position of (-1401.6 MHz, -53.1 MHz, 815.9 MHz). It should be noted that when the excitation lasers are tuned to the (-1401.6 MHz, -53.1 MHz, 815.9 MHz) frequencies, ionization also occurs through hyperfine pathways 19/2–21/-23/2–23/2 and 19/2–21/-23/2–21/2 corresponding to the (-1401.6 MHz, -53.1 MHz, 2876.9 MHz) and (-1401.6 MHz, -53.1 MHz, 4481.5 MHz) frequency positions respectively. However, the contribution of ionization of ¹⁶⁷Er through these channels would be rather small as third excitation laser frequency is farther away from these hyperfine pathways of the third excitation transition. Though it is expected to observe (since the third excitation laser is tuned to 23/2–25/2 hyperfine pathway) only one resonance corresponding to 19/2–21/2–23/2–25/2 hyperfine pathway (marked as 30), it is interesting to observe several other resonances (Fig. 7A). It is possible to understand the underlying reasons for the occurrence of all the resonances, nevertheless only two cases are discussed here in detail.

A resonance marked as 22 in Fig. 7A is observed at the frequency position (-1431.9 MHz, 684.3 MHz) which in two dimensional frame corresponds to 13/2–15/2–17/2 hyperfine pathway. At the outset, this is entirely unexpected as the third excitation laser is tuned to 815.9 MHz corresponding to the 23/2–25/2 hyperfine pathway. On the closer look, one can observe that the (-1431.9 MHz, 684.3 MHz) in two-dimensional reference frame corresponds to the three hyperfine pathways 13/2–15/2–17/2–19/2, 13/2–15/2–17/2–17/2 and 13/2–15/2–17/2–15/2 (marked as 21–23 in Table 6). Among them the hyperfine pathway, 13/2–15/2–17/2–17/2 hyperfine pathway (marked as 22) corresponds to the frequency position the (-1431.9 MHz, 684.3 MHz, 690.8 MHz) in three-dimensional reference frame. Since the third excitation laser (at 815.9 MHz) is detuned to as little as 125 MHz, this resonance is observed. The contribution from the hyperfine pathways 21 and 23 is expected to be smaller because the larger detuning associated with the third hyperfine transition reduce the effective coupling.

Similarly, the resonance at (877.8 MHz, 3583.1 MHz) in two-dimensional reference frame (marked as 119 in Fig. 7A) corresponds to the 19/2–19/2–17/2 hyperfine pathway. In three dimensional reference frame this corresponds to 19/2–19/2–17/2–19/2, 19/2–19/2–17/2–17/2 and 19/2–19/2–17/2–15/2 hyperfine pathways (marked as 118–120 in Table 6) at frequency positions (877.8 MHz, 3583.1 MHz, -213.2 MHz), (877.8 MHz, 3583.1 MHz, 690.8 MHz) and (877.8 MHz, 3583.1 MHz, 1337.9 MHz) respectively. As discussed earlier, contribution from the hyperfine pathway 19/2–19/2–17/2–17/2 at (877.8 MHz, 3583.1 MHz, 690.8 MHz) would be much higher than other hyperfine pathways (118 and 120) due to the lower detuning associated with of the third step excitation.

Apart from the relative intensities of the hyperfine transitions, the contribution from the different hyperfine pathways to the ionization of ¹⁶⁷Er varies with variation in all the laser and atom source parameters. Thus, when the third excitation laser is tuned to the ¹⁶⁸Er resonance the relative contribution of all the 135 hyperfine pathways towards the ionization of ¹⁶⁷Er varies (Fig. 7B). In this, case the hyperfine pathway at (-122.9 MHz, 348.6 MHz) in two-dimensional reference frame (marked as 76 in Fig. 7B) corresponds to the hyperfine pathways 17/2–17/2–19/2–21/2, 17/2–17/2–19/2–19/2 and 17/2–17/2–19/2–17/2 (marked as 76–78 in Table 6) at frequency positions (-122.9 MHz, 348.6 MHz, 111.7 MHz), (-122.9 MHz, 348.6 MHz, 1332.6 MHz) and (-122.9 MHz, 348.6 MHz, 2236.6 MHz) respectively. Among them the contribution by the 17/2–17/2–19/2–21/2 at (-122.9 MHz, 348.6 MHz, 111.7 MHz) frequency position would be the highest due to higher intensity and lower detuning corresponding to the hyperfine pathway of the third excitation transition (19/2–21/2).

It is also important to note that when the excitation lasers are tuned to ¹⁶⁸Er, the resonance of ¹⁶⁸Er isotope lies on the vertical ridge (step-wise excitation of the second excitation transition) of hyperfine pathways (marked 76–78 in Table 6) of ¹⁶⁷Er isotope. This is in contrast to the case of even isotopes (Fig. 6), wherein the resonance of ¹⁶⁸Er isotope lies on the diagonal ridges of even isotopes (arising due to coherent two-photon excitation). Therefore, the ionization of ¹⁶⁷Er can be somewhat controlled by controlling the intensity and bandwidth of the second excitation laser. On the other hand, to reduce the ionization of non-target even isotopes, one needs to control the intensity and bandwidth of all excitation lasers.

It is not possible to qualitatively assess the effect of intensity and bandwidth of all the three excitation lasers on the ionization efficiency of target and non-target isotopes. For quantitative assessments, one needs to obtain ionization efficiencies of all constituent isotopes varying the system parameters through numerical integration of coupled differential equations²⁹.

A series of calculations have been conducted by varying the intensities of all three excitation lasers across different bandwidths, all under Doppler-free conditions. The results have been tabulated in Table 7. The table shows that it is possible to enrich the ¹⁶⁸Er isotope to up to 95% for all laser bandwidths up to 750 MHz. Therefore, laser bandwidths need to be controlled to ≤ 750 MHz. Interestingly, the content of ¹⁶⁷Er decreased with an increase in the excitation laser’s bandwidth. This can be attributed to the reduction in the intensity of the second excitation laser (please refer to the earlier discussion in this section).

Effect of Doppler broadening

So far, the Doppler broadening of the atomic ensemble has been ignored for the computations. However, in the laser isotope separation process, the atomic ensemble exhibits Doppler broadening due to the velocity distribution of the atoms.

To include Doppler broadening of the atomic ensemble, atomic flux-velocity distribution is taken as

where, α is the most probable velocity; for the present calculations, the integration is carried out up to 4α, at which the relative flux had dropped to the value of ~ 10^–7 of the maximum.

At 1200 ˚C, the erbium atoms have a most probable velocity of 381.7 m/sec. When these atoms traverse with an angle of “θ” to the laser propagation axis, the Doppler shift of the resonance can be expressed by the equation

where v is the velocity of the atom (m/sec) and λ is the wavelength of the transition (nm) and θ is the divergence angle of the atom relative to the laser propagation axis. When an atom has a divergence angle of 10˚ with reference to the laser propagation axis, the Doppler shift of the transitions is 105.1 MHz, 113.0 MHz, 117.2 MHz. Even when all the co-propagating lasers are tuned to resonance for each excitation step, the net detuning caused by the Doppler shift totals 335.3 MHz, which is the sum of 105.1 MHz, 113.0 MHz, and 117.2 MHz. If the bandwidth of all the excitation lasers is 100 MHz or greater, the atoms will be excited and ionized, though with reduced efficiency. In a practical laser isotope separation process, the atomic ensemble exhibits a distribution of velocities Therefore, to account for Doppler broadening, ionization efficiency calculations have been performed for a range of angular divergences of the atomic ensemble. The results are summarized in Table 8, which includes the degree of enrichment for each isotope and the ionization efficiency of the target isotope. It can be observed that a degree of enrichment greater than 95% can be achieved for all laser bandwidths up to 750 MHz when the atomic ensemble’s full angular divergence is 30° or less. A laser bandwidth of 500 MHz and a full angle divergence of 30° can be considered optimal for the laser isotope separation process. Under these conditions the degree of enrichment of ¹⁶⁸Er is > 96% and the ionization efficiency is 0.2.

Choice of co-propagating vs counter-propagating laser beams

In laser isotope separation, employing co-propagating laser beams ensures simpler and optimal overlap throughout the interaction zone, thereby maximizing the effectiveness of the separation process. While co-propagating beams provide the benefit of effective overlap, laser isotope separation can also be achieved using counter-propagating laser configuration. Since the resonance of the ¹⁶⁸Er target isotope lies on the coherent two-photon excitation lines of non-target even isotopes (Fig. 6), the ionization efficiency of target and non-target isotopes varies with co- and counter-propagating laser beam configurations. Consequently, calculations have been conducted to determine the ionization efficiency of the ¹⁶⁸Er isotope and the enrichment levels of all constituent isotopes across all four possible configurations (see Table 9). Data from Table 9 suggests that the direction of laser-3 propagation (relative to laser-1 or laser-2) does not influence both degree of enrichment and the ionization efficiency of the constituent isotopes. However, Table 9 also shows that when laser-2 propagates in the opposite direction (counter-propagating) relative to laser-1, the degree of enrichment of the target isotope ¹⁶⁸Er decreases by 1%. This decrease is attributed to the increased ionization efficiency of neighbouring non-target even isotopes, particularly ¹⁶⁶Er and ¹⁷⁰Er. When laser-2 counter-propagates (relative to laser-1), it cancels out the velocity-induced detunings in the two-photon reference frame, resulting in higher ionization efficiency of non-target isotopes. Therefore, the co-propagating configuration (↑↑↑) is preferred for the laser isotope separation process.

Charge-exchange collisions

Charge exchange collisions significantly influence the effectiveness of the laser isotope separation process. To achieve a high production rate of the enriched isotope, it is essential to operate the atomic ensemble at the highest feasible atomic number density. However, as the number density increases, the charge-exchange collisions also increase, which negatively impacts the degree of enrichment. The probability of charge-exchange collisions can be calculated using the expression.

Where, σ is the resonant charge exchange cross-section (cm²), d is the distance traversed by photoions prior to collection at the ion collector (cm) and N is the number density of the atoms (atoms / cm³).

Resonant charge exchange cross-section can be calculated using the following formula³¹

where v is the velocity of the ion in cm/sec and IP is the ionization potential of the element in eV.

For the most probable atomic velocity of 381.7 m/sec for erbium at 1200° C, the resonant charge exchange cross-section has been calculated to be 4.2 × 10^–14 cm², which is in reasonable agreement with the value of 2.8 × 10^–14 cm² reported by Smirnov et al.³².

A series of calculations of degree of enrichment have been carried out varying the number density of the atoms in the laser-atom interaction region diameter of 5 cm and the results are plotted in Fig. 8. It can be observed from Fig. 8 that a degree of enrichment of > 90% for ¹⁶⁸Er can be achieved by controlling the number density to ≤ 7 × 10¹¹ atoms/cm³; while a degree of enrichment of 80% for ¹⁶⁸Er can be achieved at the number density of 2 × 10¹² atoms/cm³. From the data in Fig. 8, the appropriate number density value can be chosen based on the desired degree of enrichment for the target isotope. Although ¹⁶⁷Er is adjacent in mass to the target isotope, its degree of enrichment is lower than that of ¹⁶⁶Er. This is attributed to the broader hyperfine spectrum of ¹⁶⁷Er.

Production rate

Production rates can be calculated using the following equation

where b is the laser beam diameter (cm), p is the fractional population of the ground level, l is the length of the laser-atom interaction region (cm), d is the number density of atoms in the interaction region (atoms/ cm³), A is the fractional abundance of the target isotope, f is the fractional flux (flux relative to the flux of unhindered atomic beam), η is the ionization efficiency (derived from the density matrix calculations), i is the irradiation probability, n is the number of passes of the laser beam through the laser-atom interaction region, M is the atomic mass of the target isotope (AMU), N_A is the Avogadro number (6.02214076 × 10²³) and PRF is the pulse repetition frequency of the lasers (Hz).

For the values of b = 5 cm, p = 0.992, l = 100 cm, d = 7 × 10¹¹ atoms/cm³, A = 0.26978, f = 1, η = 0.20, i = 1, n = 1, M = 168, PRF = 10 kHz, the production rate has been calculated to be 740 mg/ hour (or 18 g/day). A summary of the optimum system parameters for the separation of ¹⁶⁸Er isotope is shown in Table 10.

Table 11 compares the degree of enrichment and the production rates of precursor isotopes from previous studies with those from the current work. The production rate of the precursor isotope with the current method is one to two orders of magnitude higher than that of previously reported methods. Therefore, it is envisaged that the enriched isotopes produced by this method will meet the current demand for the enriched isotope precursors.

Irradiation of enriched isotope mixture

In this section, the utility of enriched ¹⁶⁸Er (produced through the AVLIS process) for medical applications has been studied for its suitability to medical application. When the natural erbium is irradiated in a nuclear reactor, after chemical separation erbium, it consists of five erbium radioisotopes namely., ¹⁶³Er, ¹⁶⁵Er, ¹⁶⁹Er, ¹⁷¹Er and ¹⁷²Er. (Fig. 1). On the other hand, when the enriched erbium is irradiated, after chemical separation, it consists of only three ¹⁶⁹Er, ¹⁷¹Er and ¹⁷²Er radioisotopes. The production efficiency erbium radioisotopes during the irradiation in low, medium and high-flux reactors is shown in Fig. 9. The separated radioisotopes of erbium produce decay chain nuclides which are shown in Fig. 10.

Medical fraternity desires the medical isotope to be produced with highest radioisotopic purity (not to be confused with isotopic purity) possible for the optimal control of the dose to the patient. The radioisotopic purity (R_i) of an isotope “i” can be calculated using the expression:

where S_i is the carrier free specific activity of the radioisotope “i” (Bq/g), f_i is the relative fractional abundance of the isotope present in the isotope mixture, n is the number of radioisotopes in the isotope mixture.

Since the radioisotopes of the isotopic mixture have varying half-lives the radioisotopic purity of the isotope mixture varies with time. The radioisotopic purity of the ¹⁶⁹Er produced from the enriched erbium increases from 96% to 99.5% within 24 h which is adequate for the application intended (Fig. 11). Using the enriched ¹⁶⁸Er isotope obtained from the laser isotope separation process, irradiation in low, medium, and high flux reactors can produce 180, 1800, and 18,000 doses per day (each with an activity of 7.4 GBq) respectively.