A 540-535-618 three-step photoionization scheme has been theoretically investigated for the laser isotope separation of $^{177}{\rm Lu}$ through density matrix formalism. Equations of motion for the three-step photoionization of odd isotopes have been derived. The effects of Doppler broadening, bandwidth and peak power density of excitation lasers, number density of atoms on the degree of enrichment of $^{177}{\rm Lu}$ and production rate have been studied. Optimum conditions for the laser isotope separation of $^{177}{\rm Lu}$ for the cases of initial abundances corresponding to the irradiation of natural Lu in low, medium, and high flux reactors have been derived. Under optimum conditions, the production rates of $^{177}{\rm Lu}$ in ranges of 0.87, 14.5, and 92.5 mg/h can be achieved with degrees of enrichment of 66%, 97%, and 100%, respectively, for the cases of initial abundances corresponding to the irradiation of natural Lu in low, medium, and high flux reactors. It has also been shown that the content of $^{177{\rm m}}{\rm Lu}$ in the enriched isotope mixture will be in the range of ${{10}^{- 2}}\%$ to ${{10}^{- 4}}\%$. This provides a comprehensive solution to the application of $^{177}{\rm Lu}$ for cancer therapy.
Hyperfine structure spacings of 175Lu, 176Lu, and 177mLu for the three step photoionization scheme.
Data availability
Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
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Resonance Frequency Positions of Three-Step Hyperfine Excitation Pathways for 540-535-618 Photoionization Scheme with Reference to Center of Gravity of
Relative Intensity of Hyperfine Transition (%)
Frequency Relative to Center of Gravity (MHz)
S No
F1-F2-F3-F4
540.4068 nm
535.0626 nm
618.0061 nm
540.4068 nm
535.0626 nm
618.0061 nm
Three-Photon Frequency
1
2-1-2-3
23.1
23.1
45.5
−10027
3106
3482
−3439
2
2-2-2-3
23.1
23.1
45.5
−8352
1431
3482
−3439
3
2-2-3-3
23.1
15.4
47.7
−8352
4720
193
−3439
4
2-2-3-4
23.1
15.4
15.9
−8352
4720
7249
3617
5
2-3-2-3
11.5
11.5
45.5
−5726
−1195
3482
−3439
6
2-3-3-3
11.5
33.7
47.7
−5726
2094
193
−3439
7
2-3-3-4
11.5
33.7
15.9
−5726
2094
7249
3617
8
2-3-4-3
11.5
8.7
34.1
−5726
6510
−4224
−3439
9
2-3-4-4
11.5
8.7
47.7
−5726
6510
2832
3617
10
3-2-2-3
15.4
23.1
45.5
−7889
1431
3482
−2976
11
3-2-3-3
15.4
15.4
47.7
−7889
4720
193
−2976
12
3-2-3-4
15.4
15.4
15.9
−7889
4720
7249
4080
13
3-3-2-3
33.7
11.5
45.5
−5263
−1195
3482
−2976
14
3-3-3-3
33.7
33.7
47.7
−5263
2094
193
−2976
15
3-3-3-3
33.7
33.7
47.7
−5263
2094
7249
4080
16
3-3-4-3
33.7
8.7
34.1
−5263
6510
−4224
−2976
17
3-3-4-4
33.7
8.7
47.7
−5263
6510
2832
4080
18
3-4-3-3
31.7
31.7
47.7
–1550
–1620
193
–2976
19
3-4-3-4
31.7
31.7
15.9
–1550
–1620
7249
4080
20
3-4-4-3
31.7
34.3
34.1
–1550
2797
–4224
–2976
21
3-4-4-4
31.7
34.3
47.7
–1550
2797
2832
4080
22
3-4-5-4
31.7
3.2
100
–1550
8369
–2740
4080
23
4-3-2-3
8.7
11.5
45.4
–5623
–1195
3482
–3337
24
4-3-3-3
8.7
33.7
47.7
−5623
2094
193
−3337
25
4-3-3-4
8.7
33.7
15.9
−5623
2094
7249
3719
26
4-3-4-3
8.7
8.7
34.1
−5623
6510
−4224
−3337
27
4-3-4-4
8.7
8.7
47.7
−5623
6510
2832
3719
28
4-4-3-3
34.3
31.7
47.7
−1910
−1620
193
−3337
29
4-4-3-4
34.3
31.7
15.9
−1910
−1620
7249
3719
30
4-4-4-3
34.3
34.3
34.1
−1910
2797
−4224
−3337
31
4-4-4-4
34.3
34.3
47.7
−1910
2797
2832
3719
32
4-4-5-4
34.3
3.2
100
−1910
8369
−2740
3719
33
4-5-4-3
60.9
60.9
34.1
3072
−2185
−4224
−3337
34
4-5-4-4
60.9
60.9
47.7
3072
−2185
2832
3719
35
4-5-5-4
60.9
23.7
100
3072
3388
−2740
3719
36
5-4-3-3
3.2
31.7
47.7
−3932
−1620
193
−5359
37
5-4-3-4
3.2
31.7
15.9
−3932
−1620
7249
1697
38
5-4-4-3
3.2
34.3
34.1
−3932
2797
−4224
−5359
39
5-4-4-4
3.2
34.3
47.7
−3932
2797
2832
1697
40
5-4-5-4
3.2
3.2
100
−3932
8369
−2740
1697
41
5-5-4-3
23.7
60.9
34.1
1049
−2185
−4224
−5359
42
5-5-4-4
23.7
60.9
47.7
1049
−2185
2832
1697
43
5-5-5-4
23.7
23.7
100
1049
3388
−2740
1697
44
5-6-5-4
100
100
100
7527
−3090
−2740
1697
Table 6.
Most Intense Hyperfine Pathways of (Hyperfine Pathways Having Intensity in Each Excitation Step)
Relative Intensity of Hyperfine Transition (%)
Frequency Relative to Center of Gravity (MHz)
S No
F1-F2-F3-F4
540.4068 nm
535.0626 nm
618.0061 nm
540.4068 nm
535.0626 nm
618.0061 nm
Three-Photon Frequency
14
3-3-3-3
33.7
33.7
47.7
−5263
2094
193
−2976
18
3-4-3-3
31.7
31.7
47.7
−1550
−1620
193
−2976
20
3-4-4-3
31.7
34.3
34.1
−1550
2797
−4224
−2976
21
3-4-4-4
31.7
34.3
47.7
−1550
2797
2832
4080
28
4-4-3-3
34.3
31.7
47.7
−1910
−1620
193
−3337
30
4-4-4-3
34.3
34.3
34.1
−1910
2797
−4224
−3337
31
4-4-4-4
34.3
34.3
47.7
−1910
2797
2832
3719
33
4-5-4-3
60.9
60.9
34.1
3072
−2185
−4224
−3337
34
4-5-4-4
60.9
60.9
47.7
3072
−2185
2832
3719
44
5-6-5-4
100
100
100
7527
−3090
−2740
1697
Table 7.
Dependence of Degree of Enrichment on Full Angle Divergence of the Atomic Beama
Bandwidth of Excitation
Bandwidth of Excitation
Bandwidth of Excitation
Degree of Enrichment of
Degree of Enrichment of
Degree of Enrichment of
Full Angle Divergence (degrees)
Ionization Efficiency of
Low Flux Reactors
Medium Flux Reactors
High Flux Reactors
Ionization Efficiency of
Low Flux Reactors
Medium Flux Reactors
High Flux Reactors
Ionization Efficiency of
Low Flux Reactors
Medium Flux Reactors
High Flux Reactors
5°
86.7
99.1
99.9
71.0
97.6
99.7
27.0
86.2
98.0
10°
83.1
98.8
99.8
65.9
97.0
99.6
24.9
84.8
97.8
30°
61.1
96.4
99.5
43.9
93.0
99.0
16.0
76.3
96.2
45°
30.5
88.1
98.3
34.7
90.0
98.6
12.5
70.8
95.0
Lasers are in resonance to the 5-6-5-4 hyperfine pathway of $^{177}{\rm Lu}$. Peak power densities of the lasers are 20, 20, ${1000}\;{{\rm W/cm}^2}$ (10 MHz); 10, 2, ${1000}\;{{\rm W/cm}^2}$ (50 MHz); and 2, 2, ${1000}\;{{\rm W/cm}^2}$ (100 MHz).
Table 8.
Summary of Optimum Conditions for Enrichment of
Description
Low Flux Reactors
Medium Flux Reactors
High Flux Reactors
Hyperfine pathway
5-6-5-4
Pulse width of excitation lasers (ns)
20
Laser propagation
All lasers co-propagating
Delay between pulses (ns)
0
Peak power density of first, second, and third excitation lasers () for laser bandwidth of 50 MHz
10, 2, 1000
Peak power density of first, second, and third excitation lasers () for laser bandwidth of 100 MHz
2, 2, 1000
Laser beam diameter (mm)
30
Laser–atom interaction region length (mm)
270
Number of passes of the laser beam
20
Number density of atoms
Full angle divergence of the atomic beam
10°
Ionization efficiency for lasers with bandwidth 50 MHz
Ionization efficiency for lasers with bandwidth 100 MHz
Degree of enrichment of for lasers with bandwidth 50 MHz (%)
66
97
100
Degree of enrichment of for lasers with bandwidth 100 MHz (%)
25
85
98
Productivity rate (mg/h)
0.87
14.5
92.5
Degree of enrichment of for lasers with bandwidth 50 MHz (%)
Degree of enrichment of for lasers with bandwidth 100 MHz (%)
Amount of injected along with 1.8 µg of (for isotope mixture enriched with lasers having bandwidth of 50 MHz) pico grams
14
20
431
Amount of injected along with 1.8 µg of (for isotope mixture enriched with lasers having bandwidth of 100 MHz) pico grams
56
62
754
Activity of injected (for isotope mixture enriched with lasers having bandwidth of 50 MHz) mBq
2
3
73
Activity of injected (for isotope mixture enriched with lasers having bandwidth of 100 MHz) mBq
10
11
128
Tables (8)
Table 1.
Abundances of Lu Isotopes after Irradiation of Natural Lu for 21 days in Low, Medium, and High Flux Reactors
Resonance Frequency Positions of Three-Step Hyperfine Excitation Pathways for 540-535-618 Photoionization Scheme with Reference to Center of Gravity of
Relative Intensity of Hyperfine Transition (%)
Frequency Relative to Center of Gravity (MHz)
S No
F1-F2-F3-F4
540.4068 nm
535.0626 nm
618.0061 nm
540.4068 nm
535.0626 nm
618.0061 nm
Three-Photon Frequency
1
2-1-2-3
23.1
23.1
45.5
−10027
3106
3482
−3439
2
2-2-2-3
23.1
23.1
45.5
−8352
1431
3482
−3439
3
2-2-3-3
23.1
15.4
47.7
−8352
4720
193
−3439
4
2-2-3-4
23.1
15.4
15.9
−8352
4720
7249
3617
5
2-3-2-3
11.5
11.5
45.5
−5726
−1195
3482
−3439
6
2-3-3-3
11.5
33.7
47.7
−5726
2094
193
−3439
7
2-3-3-4
11.5
33.7
15.9
−5726
2094
7249
3617
8
2-3-4-3
11.5
8.7
34.1
−5726
6510
−4224
−3439
9
2-3-4-4
11.5
8.7
47.7
−5726
6510
2832
3617
10
3-2-2-3
15.4
23.1
45.5
−7889
1431
3482
−2976
11
3-2-3-3
15.4
15.4
47.7
−7889
4720
193
−2976
12
3-2-3-4
15.4
15.4
15.9
−7889
4720
7249
4080
13
3-3-2-3
33.7
11.5
45.5
−5263
−1195
3482
−2976
14
3-3-3-3
33.7
33.7
47.7
−5263
2094
193
−2976
15
3-3-3-3
33.7
33.7
47.7
−5263
2094
7249
4080
16
3-3-4-3
33.7
8.7
34.1
−5263
6510
−4224
−2976
17
3-3-4-4
33.7
8.7
47.7
−5263
6510
2832
4080
18
3-4-3-3
31.7
31.7
47.7
–1550
–1620
193
–2976
19
3-4-3-4
31.7
31.7
15.9
–1550
–1620
7249
4080
20
3-4-4-3
31.7
34.3
34.1
–1550
2797
–4224
–2976
21
3-4-4-4
31.7
34.3
47.7
–1550
2797
2832
4080
22
3-4-5-4
31.7
3.2
100
–1550
8369
–2740
4080
23
4-3-2-3
8.7
11.5
45.4
–5623
–1195
3482
–3337
24
4-3-3-3
8.7
33.7
47.7
−5623
2094
193
−3337
25
4-3-3-4
8.7
33.7
15.9
−5623
2094
7249
3719
26
4-3-4-3
8.7
8.7
34.1
−5623
6510
−4224
−3337
27
4-3-4-4
8.7
8.7
47.7
−5623
6510
2832
3719
28
4-4-3-3
34.3
31.7
47.7
−1910
−1620
193
−3337
29
4-4-3-4
34.3
31.7
15.9
−1910
−1620
7249
3719
30
4-4-4-3
34.3
34.3
34.1
−1910
2797
−4224
−3337
31
4-4-4-4
34.3
34.3
47.7
−1910
2797
2832
3719
32
4-4-5-4
34.3
3.2
100
−1910
8369
−2740
3719
33
4-5-4-3
60.9
60.9
34.1
3072
−2185
−4224
−3337
34
4-5-4-4
60.9
60.9
47.7
3072
−2185
2832
3719
35
4-5-5-4
60.9
23.7
100
3072
3388
−2740
3719
36
5-4-3-3
3.2
31.7
47.7
−3932
−1620
193
−5359
37
5-4-3-4
3.2
31.7
15.9
−3932
−1620
7249
1697
38
5-4-4-3
3.2
34.3
34.1
−3932
2797
−4224
−5359
39
5-4-4-4
3.2
34.3
47.7
−3932
2797
2832
1697
40
5-4-5-4
3.2
3.2
100
−3932
8369
−2740
1697
41
5-5-4-3
23.7
60.9
34.1
1049
−2185
−4224
−5359
42
5-5-4-4
23.7
60.9
47.7
1049
−2185
2832
1697
43
5-5-5-4
23.7
23.7
100
1049
3388
−2740
1697
44
5-6-5-4
100
100
100
7527
−3090
−2740
1697
Table 6.
Most Intense Hyperfine Pathways of (Hyperfine Pathways Having Intensity in Each Excitation Step)
Relative Intensity of Hyperfine Transition (%)
Frequency Relative to Center of Gravity (MHz)
S No
F1-F2-F3-F4
540.4068 nm
535.0626 nm
618.0061 nm
540.4068 nm
535.0626 nm
618.0061 nm
Three-Photon Frequency
14
3-3-3-3
33.7
33.7
47.7
−5263
2094
193
−2976
18
3-4-3-3
31.7
31.7
47.7
−1550
−1620
193
−2976
20
3-4-4-3
31.7
34.3
34.1
−1550
2797
−4224
−2976
21
3-4-4-4
31.7
34.3
47.7
−1550
2797
2832
4080
28
4-4-3-3
34.3
31.7
47.7
−1910
−1620
193
−3337
30
4-4-4-3
34.3
34.3
34.1
−1910
2797
−4224
−3337
31
4-4-4-4
34.3
34.3
47.7
−1910
2797
2832
3719
33
4-5-4-3
60.9
60.9
34.1
3072
−2185
−4224
−3337
34
4-5-4-4
60.9
60.9
47.7
3072
−2185
2832
3719
44
5-6-5-4
100
100
100
7527
−3090
−2740
1697
Table 7.
Dependence of Degree of Enrichment on Full Angle Divergence of the Atomic Beama
Bandwidth of Excitation
Bandwidth of Excitation
Bandwidth of Excitation
Degree of Enrichment of
Degree of Enrichment of
Degree of Enrichment of
Full Angle Divergence (degrees)
Ionization Efficiency of
Low Flux Reactors
Medium Flux Reactors
High Flux Reactors
Ionization Efficiency of
Low Flux Reactors
Medium Flux Reactors
High Flux Reactors
Ionization Efficiency of
Low Flux Reactors
Medium Flux Reactors
High Flux Reactors
5°
86.7
99.1
99.9
71.0
97.6
99.7
27.0
86.2
98.0
10°
83.1
98.8
99.8
65.9
97.0
99.6
24.9
84.8
97.8
30°
61.1
96.4
99.5
43.9
93.0
99.0
16.0
76.3
96.2
45°
30.5
88.1
98.3
34.7
90.0
98.6
12.5
70.8
95.0
Lasers are in resonance to the 5-6-5-4 hyperfine pathway of $^{177}{\rm Lu}$. Peak power densities of the lasers are 20, 20, ${1000}\;{{\rm W/cm}^2}$ (10 MHz); 10, 2, ${1000}\;{{\rm W/cm}^2}$ (50 MHz); and 2, 2, ${1000}\;{{\rm W/cm}^2}$ (100 MHz).
Table 8.
Summary of Optimum Conditions for Enrichment of
Description
Low Flux Reactors
Medium Flux Reactors
High Flux Reactors
Hyperfine pathway
5-6-5-4
Pulse width of excitation lasers (ns)
20
Laser propagation
All lasers co-propagating
Delay between pulses (ns)
0
Peak power density of first, second, and third excitation lasers () for laser bandwidth of 50 MHz
10, 2, 1000
Peak power density of first, second, and third excitation lasers () for laser bandwidth of 100 MHz
2, 2, 1000
Laser beam diameter (mm)
30
Laser–atom interaction region length (mm)
270
Number of passes of the laser beam
20
Number density of atoms
Full angle divergence of the atomic beam
10°
Ionization efficiency for lasers with bandwidth 50 MHz
Ionization efficiency for lasers with bandwidth 100 MHz
Degree of enrichment of for lasers with bandwidth 50 MHz (%)
66
97
100
Degree of enrichment of for lasers with bandwidth 100 MHz (%)
25
85
98
Productivity rate (mg/h)
0.87
14.5
92.5
Degree of enrichment of for lasers with bandwidth 50 MHz (%)
Degree of enrichment of for lasers with bandwidth 100 MHz (%)
Amount of injected along with 1.8 µg of (for isotope mixture enriched with lasers having bandwidth of 50 MHz) pico grams
14
20
431
Amount of injected along with 1.8 µg of (for isotope mixture enriched with lasers having bandwidth of 100 MHz) pico grams
56
62
754
Activity of injected (for isotope mixture enriched with lasers having bandwidth of 50 MHz) mBq
2
3
73
Activity of injected (for isotope mixture enriched with lasers having bandwidth of 100 MHz) mBq
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