Numerical study of the laser isotope separation of optically pumped 102Pd

The feasibility of laser isotope separation of 102Pd through pulsed laser optical pumping followed by isotope selective photoionization has been studied through density matrix formalism. The effect of various parameters such as bandwidth of the excitation lasers, intensity of the lasers and Doppler broadening of the atomic ensemble on the efficiency of optical pumping and isotope selective photoionization have been evaluated. The optimum number density in the laser-atom interaction region has been derived from the studies of the effect of charge exchange collisions on the degree of enrichment. It has been shown that it is possible to enrich 102Pd up to ~ 23.7% at a production rate of 1.1 mg /h. The achievable degree of enrichment through this photoionization scheme is higher than the previously reported laser isotope separation process. The radionuclidic purity of the irradiated enriched mixture has been found to be suitable for medical applications.

The feasibility of laser isotope separation of 102 Pd through pulsed laser optical pumping followed by isotope selective photoionization has been studied through density matrix formalism.The effect of various parameters such as bandwidth of the excitation lasers, intensity of the lasers and Doppler broadening of the atomic ensemble on the efficiency of optical pumping and isotope selective photoionization have been evaluated.The optimum number density in the laser-atom interaction region has been derived from the studies of the effect of charge exchange collisions on the degree of enrichment.It has been shown that it is possible to enrich 102 Pd up to ~ 23.7% at a production rate of 1.1 mg /h.The achievable degree of enrichment through this photoionization scheme is higher than the previously reported laser isotope separation process.The radionuclidic purity of the irradiated enriched mixture has been found to be suitable for medical applications.
Palladium has six stable isotopes (Table 1), namely, 102 Pd (natural abundance = 1.02%), 104 Pd (natural abundance = 11.1%), 105Pd (natural abundance = 22.3%), 106 Pd (natural abundance = 27.3%), 108Pd (natural abundance = 26.5%)and 110 Pd (natural abundance = 11.7%).Currently, palladium metal costs about 40,000 USD/kg.Pd is also produced in the nuclear reactor as a fission product.Due to its high value, separation of Pd from the nuclear reactor spent fuel has been studied 1,2 .Apart from the difficulties in the chemical separation of Pd from the spent fuel, the primary limitation lies due to the presence of the long-lived 107 Pd (T 1/2 = 6.5 × 10 6 years) isotope in the nuclear reactor spent fuel.Laser isotope separation method has been studied 3 for the removal 107 Pd from the palladium recovered from nuclear reactor spent fuel.At present, cost of laser cleanup of reactor produced Pd limits practical utility of the method.
The radioisotope 103 Pd (T 1/2 = 16.991days) decays 100% by electron capture to 103m Rh by emitting two characteristic X-rays with energy 20.073 (22.09%) keV and 20.215 (41.83%) keV.The Q-value of the reaction is + 543.1 keV.Due to its favorable decay properties, 103 Pd is used for radiation therapy of the patients suffering from prostate cancer 7 and uveal melanoma 8 . 103Pd is produced by irradiation of its precursor 102 Pd isotope in a nuclear reactor.Owing to the low natural abundance of the parent 102 Pd isotope and low thermal neutron absorption cross-section 9 (σ th = 1.82 b), the amount of 103 Pd isotope produced in the nuclear reactor would be rather small.As a result, the specific activity of the produced 103 Pd isotope would also be small.For example, when one gram of natural Pd is irradiated in a low-flux nuclear reactor (neutron flux = 3 × 10 13 neutrons/cm 2 -sec) for 60 days, at the end of irradiation, the amount of 103 Pd produced is ~ 1 μg.Though the no-carrier added specific activity of 103 Pd (2.76 × 10 15 Bq/gram) is high, the specific activity of the 103 Pd produced from the natural Pd (2.76 × 10 9 Bq/gram) is rather low.Further, during the irradiation, the highly abundant 106 Pd (27.3%) and 108 Pd (26.5%) isotopes produce 107 Pd (T 1/2 = 6.5 × 10 6 years) and 109 Pd (T 1/2 = 13.59 h) radioactive daughter isotopes respectively which degrades the radionuclidic purity of 103 Pd.The radionuclidic purity of an isotopic mixture is defined by the equation where S i = Specific activity of the radioisotope "i" (Bq/gm), f i = Relative fractional abundance of the radioisotope in the isotopic mixture, n = Number of radioisotopes in the isotopic mixture.
No-carrier added specific activity of a radioisotope can be calculated using the following expression
At the end of irradiation, the radionuclidic purity of 103 Pd in the irradiated isotopic mixture is 0.78% while the radionuclidic purity of 109 Pd is 99.21%.Since both isotopes have widely varying half-lives (Table 1), patient dose optimization is a complex task.This necessitates the utilization of enriched 102 Pd for the production of 103 Pd medical isotope.
Laser enrichment of 102 Pd isotope is extremely complex owing to its low natural abundance, high melting (1554.9°C) and boiling (2963 °C) points and large ionization potential (67,241.14 cm −1 or 8.3368 eV).Additionally, all the three known transitions viz., 1 S 0 -1 P°1 (244.8647nm), 1 S 0 -3 D°1 (247.7161nm) and 1 S 0 -3 P°1 (276.3906nm) originating from the 4d 10 1 S 0 (0.0 cm −1 ) ground state have their wavelengths in the UV region and their isotopes shifts have been reported to be extremely small 10 .In general, for the mid-Z elements, the field shift (also known as volume shift) and the mass shift are of comparable magnitudes.When these two components are in opposing direction, the net isotope shift, which is a sum of field shift and the mass shift, would be rather small (Table 2).This makes laser isotope separation of 102 Pd a daunting task.
Researchers at the AM Prokhorov General Physics Institute [12][13][14] , Russia have employed the following photoionization scheme for the laser isotope separation of 102 Pd.
They have reported that the bandwidth of the excitation lasers needs to be controlled to about 60-80 MHz for obtaining 18% enrichment of 102 Pd isotope 12 .Further, the angular divergence of the atomic beam shall be restricted to 0.1 rad (5.7°) for the requisite isotopic selectivity.Additionally, to minimize the overlap of the hyperfine spectrum of the 105 Pd isotope, a magnetic field of 2000G has been employed in the laser-atom interaction region.On the whole, the experimental setup is complex and hence the reported method 12 may not be suitable for laser based separations where the volume of laser-atom interaction region is large.
Upon careful observation of the branching ratios and the decay rates of the transitions (Fig. 1), it can be observed that the 276.3906 nm transition can be used for the efficient optical pumping of atoms from the 4d 10 1 S 0 (0.0 cm −1 ) ground state into the 4d 9 5s 3 D 2 (7755.025cm −1 ) meta-stable state.Due to the large isotope shift, these optically pumped atoms can be selectively photoionized using the 361.0575 nm transition.
The photoionization process is shown below.

Photoionization
In the present work, the photoionization process mentioned above has been studied for its suitability for the laser isotope separation of 102 Pd.A schematic of the proposed experimental setup is shown in Fig. 2. The density matrix formalism accurately describes the laser-atom interactions 16 in the multi-step laser excitation processes; therefore, it has been invoked for the calculation of degree of enrichment and production rates of the laser isotope separation process under various conditions.The optimization of the laser isotope separation process has been done numerically.

Theoretical basis
Palladium has a ground state configuration of 4d 10 1 S 0 (0.0 cm −1 ).At 1500 °C, 96% of the population is available in the ground state.At a temperature (T 0 K), the vapour pressure can be calculated using the following equation 17 , where P = Vapor pressure (bar).For the case of elemental Pd 13 , A = 5.426, B = − 17,889, C = 0 and D = 0.At 1500 °C, the vapor pressure of Pd is calculated to be 22 μbar which corresponds to a number density of 9 × 10 13 atoms/cm 3 .
The population of the ground state can be excited into the 4d 9 5p 3 P°1 state using a nanosecond pulsed broadband laser tuned to the 276.3906 nm transition (Fig. 1).
The atom dynamics of the optical pumping process can be described by the coupled density matrix equations described below.The atoms initially present in the ground fine-structure level |1 � are optical pumped into the level |2 � .Due to the finite life-time, the atoms in level |2 � decay to the high-lying meta-stable level |3 � and to the other trapping levels |T1 � and |T2 � .The atoms in the resonant level |2 � may also decay to the resonant lower level |1 � at a rate denoted as Γ 21 .The decay rate of atoms from the resonant level |2 � to trapped levels |T1 � and |T2 � at a total rate denoted as γ 2T . (3) 4d 101 S 0 0.0 cm −1 276.3906 nm → 4d 9 5p 3 P o 1 36180.677cm −1 radiative decay → 4d 9 5s 3 D 2 7755.025cm −1 (4) where the density matrix element ρ(M, m, N, n) describes the coherence between the states |M, m � and |N, n � when M ≠ N and/or m ≠ n and represents the level population when M = N and m = n.Δ corresponds to the detuning of the laser frequency from the resonance.
The laser is considered to be having a pulse width of 30 ns and a pulse repetition frequency (PRF) of 1 kHz.To include laser bandwidth effects in the calculations the optical pumping laser is considered to have a phase diffusion bandwidth.The laser bandwidth and its line shape have been included in the terms From the above equation, the bandwidth of the lasers at the atomic resonance corresponds to γ L ; and at large detunings, the laser is nearly monochromatic.
Since the optical pumping laser has a temporal intensity profile, the time varying Rabi frequency has been calculated according to the equation where I(t) is the temporal profile of the laser intensity; Ω is the time-independent Rabi frequency.
As discussed earlier, the population of the meta-stable level 4d 9 5s 3 D 2 (7755.025cm −1 ) can be photoionized through the pathway shown below.
The atoms optically pumped into the level |3 � are excited by the excitation laser to the level |4 � .The atoms from this level are incoherently excited into the ionization level |I � by the ionization laser at a rate denoted as γ I .The atoms in level |4 � may decay to the lower level |3 � at a rate denoted as Γ 43 .The atoms in the level |4 � also decay to the trapped level |T3 � at a rate denoted as γ 4T and are lost from the excitation process.The population dynamics corresponding to the photoionization process of an odd isotope can be described by the coupled density matrix equations given below.(10)   ρ 2, j, 2, j1 = −i.
ρ 4, j, 4, j = −i.The ionization rate which is induced by the ionizing laser can be calculated using the equation γ I = σ φ , where σ = Photoionization cross-section and ϕ is the flux of the ionization laser.The photoionization cross-section is considered to be 1 × 10 -16 cm 2 and the corresponding ionization rate has been calculated to be 0.128 × I kHz (where, I = Intensity of the laser in W/cm 2 ).
The coupled differential equations are integrated using standard numerical integration methods.

Doppler broadening and atomic flux-velocity distribution
The most probable velocity (v mp ) of the atoms can be calculated using the expression ( 18) ρ 4, j, 4, j1 = −i.
where k is the Boltzmann Constant (1.380649 × 10 -23 J/K), T is the temperature of the atomic ensemble (°K) and m is the mass of the atom (kg).At 1500 °C, the most probable atomic velocity of 102 Pd isotope is 537.507m/s.The variation in the mean probable velocity between the stable Pd isotopes is ± 8 m/s.At this temperature, a variation of ± 50 °C in the temperature results in ± 8 m/s variation in the most probable velocity.These small variations in the velocities do not induce any significant effect on the laser isotope separation process.Doppler broadening of the atomic transitions arises due to the velocity and angular distributions of atoms effusing from the atom source.The flux-velocity distribution of atomic ensemble can be described by the following expression where, v mp = Most probable velocity.At 4 × v mp , the relative flux drops to the value of ~ 10 -7 of the maximum.
The Doppler broadening of an atomic transition can be calculated using the expression where ν 0 is the resonance frequency of the transition (Hz), T is the temperature of the atomic ensemble (°K) and M is the mass of the isotope (AMU).
When the atom source is heated to a temperature of 1500 °C, the Doppler broadening of the unhindered atoms of the ensemble can be calculated to be 3240 MHz for the 276.3906 nm transition.Since the optical pumping process is intended to pump atoms into the 4d 9 5s 3 D 2 (7755.025cm −1 ) meta-stable state, the Doppler broadening of the transition is not a concern.However, in practice there will be a limit to the Doppler broadening which is governed by the physical dimensions of the self-collimating long canal type atomizer and apertures if any in perpendicular plane to the propagation axis of the effusing atoms.
Doppler broadening of the atomic transition is limited by the angular divergence of the atomic beam.The full angle divergence of the atom source having an aperture diameter "d" and length "l" can be calculated using the expression 19 (27)  In order to account for the Doppler broadening of the atomic transitions, the atom velocities and angular divergence have been segmented into 30 groups each which was sufficient to obtain the convergence of the ionization efficiency values.The segmentation has been done in the following manner.For example, if the full angle divergence is 30°, the angular divergence is varied between the values of − 15° to + 15° with a step size of 1°.Each angular group is further segmented into 30 velocity segments in the range of ± 4 × v mp and the flux of each velocity group is determined by the Eq. ( 27).

Results and discussion
Due to the small isotope shifts of the 276.3906 nm transition (Table 2), it is not possible to obtain any isotopic selectivity in the optical pumping process; therefore, broadband lasers can be utilized.A series of calculations of optical pumping efficiency (i.e.population of the 4d 9 5s 3 D 2 (7755.025cm −1 ) meta-stable state) has been carried out varying the intensity of the optical pumping laser and full angle divergence of the atomic beam and the results are plotted in Fig. 3.The bandwidth of the optical pumping laser was set to 10 GHz for these calculations.From Fig. 3, as expected, optical pumping efficiency increased with an increase in the intensity of the optical pumping laser.It can also be observed that for a given intensity, variation in the full angle divergence of up to a value of 45° did not show any impact on the optical efficiency.This is due to the large bandwidth of the optical pumping laser (10 GHz) which is larger than the Doppler broadening of 2480 MHz at a full angle divergence of the atomic beam of 45°.For a laser intensity of 1800 W/cm 2 , the optical pumping efficiency is found to be 0.60.For a laser with a 30 mm beam diameter and having an intensity of 1800 W/cm 2 the average power corresponds to 0.38 W (calculated based on the constant intensity laser pulse).
The simulated Doppler free frequency spectrum of natural Pd has been computed based on the isotope shifts (Table 2) and the hyperfine structure (Table 3) of 105 Pd isotopes for the 361.0579 nm transition which is shown in Fig. 4. The resonance frequency positions of the even Pd isotopes and hyperfine transitions of odd 105 Pd isotope have been tabulated in Table 4.The resonance frequency positions of even isotopes of Pd isotope lie > 450 MHz away from the resonance of the 102 Pd isotope.Therefore, at low powers, when the bandwidth of the excitation laser is < 200 MHz, the even isotopes are not expected to get ionized significantly.Nevertheless, at high powers, due to the power (saturation) broadening, the even isotopes are also ionized considerably.On the other hand, hyperfine spectrum of the only odd 105 Pd isotope is spread over 2819 MHz, impeding the selective ionization of 102 Pd isotope.Particularly the 7/2-7/2 hyperfine transition of 105 Pd which lies 32 MHz away from the 102 Pd resonance causes significant overlap.As a result, enrichment of 105 Pd along with enrichment of 102 Pd is inevitable.Therefore, the process should be optimized for the depletion of the remaining even isotopes i.e., 104 Pd, 106 Pd, 108 Pd and 110 Pd. lines are Doppler broadened due to finite divergence angle of the atoms and velocity distribution.When the atom source is heated to a temperature of 1500 °C, the Doppler broadening of the unhindered atoms of the ensemble will be 2480 MHz for the 361.0575 nm transition.Since the Doppler broadening is much larger than the frequency difference between the resonances of the target and non-target isotopes, the Doppler broadening along the laser propagation axis must be reduced.
The Doppler broadening along the laser axis can be curtailed by incorporating additional collimators along the atomic beam propagation axis which determines the full angle divergence of the atomic beam.In this case, the flux of the atoms having higher divergence is inhibited to enter the laser-atom interaction region.However, this results in loss of throughput of the system.Alternatively, Doppler broadening of the atomic transition can also be controlled using long canal type atomizers having full angular divergence as per Eq. ( 29).In order to account  for the Doppler broadening, as discussed previously, both full angle divergence and the velocity distribution are segmented into 30 groups each for the calculations.
The overall lineshape of a resonance is a result of complex confluence of the bandwidth of the excitation laser, Doppler broadening and the power broadening which dictate both ionization efficiency and the degree of enrichment.A series of calculations of ionization efficiency and degree of enrichment of 102 Pd have been carried out varying the intensity of the excitation and ionization lasers as well as full angle divergence of atoms.The results are shown in Fig. 7.The dependence of both the ionization efficiency and degree of enrichment of 102 Pd on laser intensities showed similar pattern like in Doppler free conditions (Fig. 7 A and B) except that in the present case, both ionization efficiency and degree of enrichment dropped further which can be attributed to the Doppler broadening.With increase in the full divergence of atoms to ≥ 20°, the degree of enrichment dropped drastically.Therefore, it is required to limit the full angle divergence of the atoms to 10° to obtain adequate enrichment in the laser isotope separation process.

Effect of charge exchange collisions
Charge exchange collisions have an adverse effect on the enrichment process.Photoions generated during laser excitation process undergo charge exchange collisions with neutral atoms prior to the collection at the ion collector.
Resonant charge exchange cross-section can be calculated using the following formula 23 where v is the velocity of the ion in cm/s and IP is the ionization potential of the element in eV.
For the most probable atomic velocity of 537.507 m/s for Pd at 1500 °C, the resonant charge exchange crosssection has been calculated to be 1.6 × 10 -14 cm 2 which is in good agreement with the value reported by Smirnov 24 .
The charge exchange probability can be calculated using the expression Where σ is the resonant charge exchange cross-section (cm 2 ), d is the distance traversed by the photoions prior to the collection at the collector (cm) and N is the number density of the atoms (atoms /cm 3 ).For d = 3 cm, N = 1 × 10 12 atoms/cm 3 (0.24 μbar); the charge exchange probability is 4.7%.
Computations of the degree of enrichment have been carried out varying the number density of atoms in the laser-atom interaction region and the results are plotted in Fig. 8.A gradual reduction in the degree of enrichment with increase in the number density has been observed.At a number density of 5 × 10 12 atoms/cm 3 (1.22 μbar), the degree of enrichment was found to be about 23.7%.Of course it is also possible to choose any other number density based on the requirement of the degree of enrichment.

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/cm 3 ), A is the fractional abundance of the target isotope, f is the fractional flux (flux relative to the flux of unhindered atomic beam), η o is the optical pumping efficiency, η i is the ionization efficiency (both 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 × 10 23 ) and PRF is the pulse repetition frequency of the lasers (Hz).
When the Pd isotope mixture is irradiated in a nuclear reactor, they undergo (n, γ) reactions and produce daughter nuclides.The radioactivity production equations 25  where N i is the amount of Pd isotope, σ i is the thermal neutron absorption cross-section of the isotope (cm 2 ), φ is the thermal neutron flux of the reactor (neutrons/cm 2 /sec) and t is the irradiation time.
When the enriched isotopic mixture is irradiated in a low-flux nuclear reactor (3 × 10 13 neutrons/cm 2 -sec), radioactive 107 Pd and 109 Pd isotopes are also produced along with the desired 103 Pd isotope (Fig. 9).It can be observed from Fig. 9 that, it requires about 60 days of irradiation in order to obtain maximum production yield of 102 Pd isotope.At the end of irradiation (60 days), the amount of radioactive isotopes produced is 25 μg of 102 Pd, 5.3 μg of 107 Pd and 1.4 μg of 109 Pd.Since the radioactive isotopes decay at different rates due to the vast difference in their half-lives, their relative fractions also vary with time; therefore, the radionuclidic purity of the isotopes varies with time.Radionuclidic purity of radioactive isotopes of Pd has been has been calculated varying the cooling time using the Eqs.(1 and 2) and the results are plotted in Fig. 10.
At the end of irradiation time, the radionuclidic purity of 103 Pd, 107 Pd and 109 Pd are 38.6%,5.6 × 10 -6 % and 61.4% respectively.At the outset the radionuclidic purity seems to be low.However, it should be noted that the 109 Pd (T 1/2 = 13.59 h) isotope dies down quickly due to its relatively low half-life as compared to 103 Pd.As a result the radionuclidic purity of 103 Pd increases with cooling time (Fig. 10).After a cooling period of 60 h, the radionuclidic purity of 102 Pd increases to > 92%; while the loss in activity during this period is just 10%.Therefore, enrichment of 102 Pd using atomic vapor laser isotope separation process described above can be used for the production of 103 Pd isotope for cancer therapy.

Conclusions
The feasibility of laser isotope separation of 102 Pd through pulsed laser optical pumping followed by isotope selective photoionization has been studied through density matrix formalism.The effect of various parameters such as bandwidth of the excitation lasers, intensity of the lasers and Doppler broadening of the atomic ensemble on the efficiency of optical pumping and isotope selective photoionization have been evaluated.The optimum number density in the laser-atom interaction has been derived from the studies of the effect of charge exchange collisions on the degree of enrichment.It has been shown that it is possible to enrich the 102 Pd isotope up to ~ 23.7% at a production rate of 1.1 mg /h.The achievable degree of enrichment through this photoionization scheme is higher than previously reported laser isotope separation 12,13 .The radionuclidic purity analysis of irradiated enriched mixture has been found to be suitable for medical applications.
The overall efficiency of the laser isotope separation process has been found to be 5.6 × 10 -2 .Though the optical pumping efficiency (0.6) is adequately high, the low efficiency (9.4 × 10 -2 ) of the isotope selective photoionization process remains a primary impediment at present.This also highlights the need for more experimental work on the suitable second step excitation transitions originating from the 4d 9 5p 3 F°3 (35,451.443cm −1 ) and autoionization transitions from the connected levels which makes the laser isotope separation more efficient.

Figure 1 .
Figure 1.Schematic diagram of the photoionization scheme of palladium (not to scale).

Figure 2 .
Figure 2. Schematic of the experimental geometry for the laser isotope separation of Pd.

Figure 4 .
Figure 4. Simulated Doppler free excitation spectrum of 361.0575 nm transition of natural Pd.Intensity of 102 Pd isotope is multiplied 25 times for better visualization.Please refer to Table 4 for the resonance frequency positions of Pd isotopes.The bandwidth of the excitation laser is taken as 10 MHz and the intensities of both excitation and ionization lasers are 10 W/cm 2 .

Figure 5 .
Figure 5. Lineshape of the 102 Pd isotope for the 361.0575 nm transition.(A) Doppler free, bandwidth of the excitation laser is 0 MHz, intensity of the excitation and ionization lasers are 1mW/cm 2 and 0 W/cm 2 respectively.(B) same as A, except that the intensity of the excitation laser is 10 W/cm 2 .(C) same as B, except that the linewidth of the excitation laser is 10 MHz.(D) same as C, except that the full angle divergence of the atomic beam is 10°.

Figure 6 .
Figure 6.Dependence of ionization efficiency and degree of enrichment of 102 Pd isotope on the intensity of the excitation and ionization lasers under Doppler free conditions.

Figure 7 .
Figure 7. Plot of the dependence of ionization efficiency and degree of enrichment of 102 Pd isotope on the intensity of the excitation, ionization lasers and the full angle divergence of the atoms.Bandwidth of the excitation laser is 100 MHz.

Figure 8 .
Figure 8. Plot of the dependence of the degree of enrichment on the number density of atoms in the laser-atom interaction region.Laser bandwidth is 100 MHz, full angle divergence of atoms is 10° and the intensity of the excitation and ionization lasers are 1.5 W/cm 2 and 1 × 10 5 W/cm 2 respectively.

Table 1 .
Table of atomic and nuclear parameters of Pd isotopes.NA not available.

Table 2 .
Isotope shifts of the transitions of Pd isotopes.

Table 3 .
Hyperfine structure constants of 105 Pd isotope for different energy levels of the photoionization scheme.

Table 4 .
Table of the resonance frequency positions of Pd isotopes for the 361.0575 nm transition.Frequency positions of Pd isotopes are referenced to the resonance frequency of the 102 Pd isotope.

Table 5 .
for Pd isotopes can be written as given below.A brief summary of the optimized parameters for the laser isotope separation of 102 Pd.