Scheme for the excitation of thorium-229 nuclei based on electronic bridge excitation

Scheme for the excitation of thorium-229 nuclei based on electronic bridge excitation 1 2 first-author https://orcid.org/0000-0001-9177-7867 1 2 1 2 1 2 1 corresp huaxia@wipm.ac.cn huaxia@wipm.ac.cn huaxia@wipm.ac.cn 1 3 corresp tongxin@wipm.ac.cn tongxin@wipm.ac.cn tongxin@wipm.ac.cn State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Wuhan Institute of Quantum Technology, Wuhan 430206, China Wuhan Institute of Quantum Technology, Wuhan 430206, China Thorium-229 possesses the lowest first nuclear excited state, with an energy of approximately 8 eV. The extremely narrow linewidth of the first nuclear excited state, with an uncertainty of 53 THz, prevents direct laser excitation and realization of the nuclear clock. We present a proposal using the Coulomb crystal of a linear chain formed by 229 Th 3+ ions, where the nuclei of 229 Th 3+ ions in the ion trap are excited by the electronic bridge (EB) process. The 7 P 1/2 state of the thorium-229 nuclear ground state is chosen for EB excitation. Using the two-level optical Bloch equation under experimental conditions, we calculate that two out of 36 prepared thorium ions in the Coulomb crystal can be excited to the first nuclear excited state, and it takes approximately 2 h to scan over an uncertainty of 0.22 eV. Taking advantage of the transition enhancement of EB and the long stability of the Coulomb crystal, the energy uncertainty of the first excited state can be limited to the order of 1 GHz. Coulomb crystal thorium-229 electronic bridge transition isomeric state Introduction 1 The excitation energy between the nuclear ground state and the first nuclear excited state (isomeric state) of thorium-229 (Th-229) is approximately 8 eV [ 1 1 - 4 4 1-4 ]. The optical transition linewidth of the first nuclear excited state is 10 -4 Hz [ 5 5 , 6 6 ]. The low isomeric energy and narrow linewidth provide the potential for a resonator with a quality of a factor of 10 19 and make Th-229 the most suitable choice for the development of nuclear optical clocks [ 6 6 - 8 8 6-8 ]. Such a Th-229 nuclear clock is expected to be a sensitive probe for the time variation of the fundamental constants of nature and will open opportunities for highly sensitive tests of the fundamental principles of physics, particularly in searches for violations of Einstein’s equivalence principle and new particles [ 9 9 - 11 11 9-11 ]. The isomeric energy is currently measured in two ways: 1) Measuring the kinetic energy of internal conversion electrons from the Th-229 isomeric state. For example, an electron spectrometer has been used to detect the internal conversion electron energy, and the isomeric energy was obtained as 8.28 ± 0.17 eV [ 4 4 ]. 2) Observing the γ radiation from Th-229 excited nuclei. An isomeric energy of 7.8 ± 0.5 eV was obtained using a magnetic microcalorimeter [ 2 2 ]. Subsequently, a 29.2-keV inter-band excitation of the Th-229 nuclear state was observed using synchrotron radiation [ 3 3 ]. Combined with the in-band 29.2-keV transition observed via nuclear rotational spectroscopy, the Th-229 isomeric state energy was determined to be 8.30 ± 0.92 eV [ 12 12 ]. Recently, a more precise magnetic microcalorimeter with improved energy resolution was developed, and an isomeric energy of 8.10 ± 0.17 eV was obtained [ 1 1 ]. Taking different weights for these measurements, the average value of the isomeric state was determined as 8.12 ± 0.11 eV [ 13 13 ]. Its uncertainty corresponds to 53 THz, which is at least 16 orders of magnitude higher than the narrow linewidth of 10 -4 Hz. To improve the precision of the isomeric energy, synchrotron radiation has been used to irradiate Th-229 doped in a vacuum ultraviolet (VUV) transparent crystal, and γ photons were detected from the decay of the Th-229 excited nuclei. This measurement excluded approximately half of the favored transition search area [ 14 14 ]. Another scheme was proposed to achieve Th-229 nuclei excitation via electron capture and detect the isomeric state of ions in heavy-ion storage rings [ 15 15 ]. Recently, an optical frequency comb was proposed to irradiate a Th-229 dioxide film, and the isomeric energy was measured based on the internal conversion process [ 16 16 ]. Furthermore, a new approach has been suggested to excite the isomeric Th-229 nuclear state via laser-driven electron recollision. The advantage of this approach is that it does not require precise knowledge of the isomeric energy [ 17 17 ]. In this paper, we propose the excitation of the nuclei of 229 Th 3+ ions in the ion trap via electronic bridge (EB) excitation and compare it with direct nuclear excitation via a pulsed laser. As shown in the inset of Fig. 1 Fig. 1 , EB excitation starts from an electric dipole ( E 1) transition, which is utilized to excite the electrons from an initial state to a virtual intermediate state. Subsequently, the intermediate state decays via the magnetic dipole ( M 1) transition, and the nuclear ground state 5/2 + [633] is excited to the isomeric state 3/2 + [631] simultaneously. To date, EB schemes for nuclear excitation have been theoretically investigated for 229g Th + [ 18 18 ], 229 Th 2+ [ 19 19 ], 229 Th 3+ [ 20 20 - 22 22 20-22 ], and 229 Th 35+ [ 23 23 ]. Our proposed experiment is based on EB excitation in triply charged 229 Th 3+ ions. Owing to their simple electronic energy levels, the trapped 229 Th 3+ ions can be laser-cooled to form a linear-chain Coulomb crystal using two closed transitions in the nuclear ground state of 229 Th 3+ ( 229g Th 3+ ) [ 24 24 ]. When the 229g Th 3+ ions are prepared in the 7 P 1/2 state, EB excitation is triggered by a 350-nm laser, and isomeric states are expected to be populated. Finally, the isomeric state is detected based on the isomer shift of the cooling lasers [ 7 7 ]. Preparation of 229g Th 3+ ion Coulomb crystals and the 7 P 1/2 state 1 229g Th 3+ ions are typically produced by laser ablation [ 24 24 , 25 25 ]. The linear Paul trap used to trap 229g Th 3+ ions are described in detail in our previous papers [ 26 26 , 27 27 ]. The trapped 229g Th 3+ ions are laser-cooled to form a Coulomb crystal. The energy levels of laser cooling and fluorescence detection of the trapped ions are shown in Fig. 2 Fig. 2 . The 229g Th 3+ ions are Doppler-laser-cooled on a 5 2 F 5/2 ↔ 6 2 D 3/2 transition at 1088 nm. Alternatively, a closed optical transition of the three-level Λ system, 5 2 F 5/2 ↔ 6 2 D 5/2 ↔ 5 2 F 7/2 , can be used to laser-cool the 229g Th 3+ ions with two lasers at 690 nm and 984 nm [ 7 7 ]. The population in the three hyperfine levels of <math id="M1"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>F</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>3,4,5</mn></mrow><mo>〉</mo></mrow></mrow></math> can be transferred to <math id="M2"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>F</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>1</mn></mrow><mo>〉</mo></mrow></mrow></math> and <math id="M3"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>F</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>2</mn></mrow><mo>〉</mo></mrow></mrow></math> via the 6 2 D 3/2 state, and the population in the <math id="M4"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>F</mi><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>1</mn></mrow><mo>〉</mo></mrow></mrow></math> state is cumulated as shown in Fig. 2 Fig. 2 . Therefore, the ions at other hyperfine levels are transferred to the closed system of 5 2 F 5/2 ↔ 6 2 D 5/2 ↔ 5 2 F 7/2 , and the second laser cooling cycle is realized. Once the 229g Th 3+ ions are laser-cooled to 50 μK, the Doppler broadening of the transition 6 2 D 5/2 → 5 2 F 7/2 at 984 nm is approximately 1 MHz [ 24 24 ]. At this temperature, the corresponding Doppler broadenings of direct nuclear excitation at 150 nm and EB excitation at 350 nm are estimated to be 6.5 MHz and 2.8 MHz, respectively. The shapes of Coulomb crystals can be manipulated into a linear chain by adjusting the radiofrequency (RF) and endcap voltages [ 28 28 ]. Two-dimensional and linear-chain Coulomb crystals can be simulated with single-ion resolution [ 29 29 , 30 30 ]. Using the GPU-accelerated LAMMPS [ 31 31 ] wrapped by the LIon package [ 32 32 ], a Coulomb crystal consisting of 36 229g Th 3+ ions are simulated, as shown in Fig. 3 Fig. 3 . When an endcap voltage of 0.2 V and an RF voltage (200 V 0-pk ) of 2 MHz are applied to the ion trap, a linear-chain Coulomb crystal of 36 229g Th 3+ ions ( Fig. 3(c) Fig. 3(c) ) is obtained. When an RF voltage (200 V 0-pk ) of 2 MHz and endcap voltage of 0.025 V are applied to the ion trap, a linear-chain Coulomb crystal of 100 229g Th 3+ ions is obtained. From the simulation results, the motion amplitude of ions in the x- and y-directions (radial direction) is known to be less than 0.6 μM when the temperature is below 5 mK, which is consistent with the motion amplitude of single ions [ 33 33 ]. Therefore, a strongly focused laser spot size of over 1.2 μM is required to increase the EB and direct nuclear excitation rates. To induce EB excitation from the 7 P 1/2 state of the 229g Th 3+ ions, the three-level transition 5 2 F 5/2 → 6 2 D 3/2 → 7 2 P 1/2 can be used to accumulate the population at the 7 P 1/2 state. Once the 229g Th 3+ ions are laser-cooled to 50 μK, the 690-nm and 984-nm lasers are turned off in sequence to maintain the ions at the 5 2 F 5/2 state, and the 1088-nm laser is kept on. To transfer a large population to the 7 2 P 1/2 state with a lifetime of 1 ns, both the 269-nm and 196-nm transitions are excited simultaneously, as shown in Fig. 4 Fig. 4 . The natural linewidths of the 269-nm and 196-nm transitions are approximately 2 π ⋅ 32 MHz and 2 π ⋅ 127 MHz, respectively [ 34 34 ]. Continuous wave (CW) lasers at both wavelengths are commercially available [ 35 35 ]. The spontaneous radiation rate of the hyperfine structure level transitions between two electronic energy levels is [ 36 36 ] <math display="block" id="M10"><mrow><msub><mtext>Γ</mtext><mrow><mtext>f</mtext><mo>→</mo><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><msub><mi>ξ</mi><mtext>f</mtext></msub><msub><mi>J</mi><mtext>f</mtext></msub><msup><mi>I</mi><mtext>g</mtext></msup><msub><mi>F</mi><mtext>f</mtext></msub><mrow><mo>‖</mo><mi>d</mi><mo>‖</mo></mrow><msub><mi>ξ</mi><mtext>i</mtext></msub><msub><mi>J</mi><mtext>i</mtext></msub><msup><mi>I</mi><mtext>g</mtext></msup><msub><mi>F</mi><mtext>i</mtext></msub></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup><mfrac><mrow><mn>4</mn><msup><mi>ω</mi><mn>3</mn></msup></mrow><mrow><mn>3</mn><mi>ℏ</mi><msup><mi>c</mi><mn>3</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where ξ i ( ξ f ) incorporates all other electronic quantum numbers, J i ( J f ) is the total electronic quantum number of the initial (final) state, F i ( F f ) is the total quantum number of the initial (final) state, I g is the nuclear spin of the nuclear ground state, d is the electron dipole, ω is the angular frequency of the electronic transition, <math id="M11"><mi>ℏ</mi></math> is the reduced Planck constant, and c is the speed of light in a vacuum. <math id="M12"><mrow><msup><mrow><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><msub><mi>J</mi><mtext>f</mtext></msub><msup><mi>I</mi><mtext>g</mtext></msup><msub><mi>F</mi><mtext>f</mtext></msub><mrow><mo>‖</mo><mi>d</mi><mo>‖</mo></mrow><msub><mi>J</mi><mtext>i</mtext></msub><msup><mi>I</mi><mtext>g</mtext></msup><msub><mi>F</mi><mtext>i</mtext></msub></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow></math> is the transition strength of the hyperfine structure level, which can be expressed as <math display="block" id="M13"><mtable><mtr><mtd><msup><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><msub><mi>ξ</mi><mtext>f</mtext></msub><msub><mi>J</mi><mtext>f</mtext></msub><msup><mi>I</mi><mtext>g</mtext></msup><msub><mi>F</mi><mtext>f</mtext></msub><mrow><mo>‖</mo><mi>d</mi><mo>‖</mo></mrow><msub><mi>ξ</mi><mtext>i</mtext></msub><msub><mi>J</mi><mtext>i</mtext></msub><msup><mi>I</mi><mtext>g</mtext></msup><msub><mi>F</mi><mtext>i</mtext></msub></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow><mn>2</mn></msup><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>F</mi><mtext>i</mtext></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>F</mi><mtext>f</mtext></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msup><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><msub><mi>J</mi><mtext>f</mtext></msub></mrow></mtd><mtd><mrow><msub><mi>F</mi><mtext>f</mtext></msub></mrow></mtd><mtd><mrow><msup><mi>I</mi><mtext>g</mtext></msup></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>F</mi><mtext>i</mtext></msub></mrow></mtd><mtd><mrow><msub><mi>J</mi><mtext>i</mtext></msub></mrow></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mn>2</mn></msup><msup><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><msub><mi>J</mi><mtext>f</mtext></msub><mrow><mo>‖</mo><mi>d</mi><mo>‖</mo></mrow><msub><mi>J</mi><mtext>i</mtext></msub></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow><mn>2</mn></msup><mn>,</mn></mtd></mtr></mtable></math> where <math id="M14"><mrow><msup><mrow><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><msub><mi>J</mi><mtext>f</mtext></msub><mrow><mo>‖</mo><mi>d</mi><mo>‖</mo></mrow><msub><mi>J</mi><mtext>i</mtext></msub></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup></mrow></math> denotes the transition strength of the electronic energy levels. The spontaneous radiation rate of the electronic energy levels is <math display="block" id="M15"><mrow><msub><mtext>Γ</mtext><mrow><mtext>spont</mtext></mrow></msub><mo>=</mo><msup><mrow><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><msub><mi>ξ</mi><mtext>f</mtext></msub><msub><mi>J</mi><mtext>f</mtext></msub><mrow><mo>‖</mo><mi>d</mi><mo>‖</mo></mrow><msub><mi>ξ</mi><mtext>i</mtext></msub><msub><mi>J</mi><mtext>i</mtext></msub></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow></mrow><mn>2</mn></msup><mfrac><mrow><mn>4</mn><msup><mi>ω</mi><mn>3</mn></msup></mrow><mrow><mn>3</mn><mi>ℏ</mi><msup><mi>c</mi><mn>3</mn></msup></mrow></mfrac><mn>.</mn></mrow></math> The excitation rate of the hyperfine structure levels between the two electronic energy levels is [ 37 37 ] <math display="block" id="M16"><mrow><msub><mtext>Γ</mtext><mrow><mtext>exc</mtext></mrow></msub><mo>=</mo><msub><mtext>Γ</mtext><mrow><mtext>f</mtext><mo>→</mo><mtext>i</mtext></mrow></msub><mfrac><mrow><msup><mi>π</mi><mn>2</mn></msup><msup><mi>c</mi><mn>2</mn></msup></mrow><mrow><mi>ℏ</mi><msup><mi>ω</mi><mn>3</mn></msup></mrow></mfrac><msub><mi>P</mi><mi>ω</mi></msub><mfrac><mrow><mn>2</mn><msub><mi>F</mi><mtext>f</mtext></msub><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msub><mi>F</mi><mtext>i</mtext></msub><mo>+</mo><mn>1</mn></mrow></mfrac><mn>,</mn></mrow></math> where P ω denotes the laser spectral intensity. The stimulated emission rate of the hyperfine structure levels between the two electronic energy levels is <math display="block" id="M17"><mrow><msub><mtext>Γ</mtext><mrow><mtext>sti</mtext></mrow></msub><mo>=</mo><msub><mtext>Γ</mtext><mrow><mtext>exc</mtext></mrow></msub><mfrac><mrow><mn>2</mn><msub><mi>F</mi><mtext>i</mtext></msub><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msub><mi>F</mi><mtext>f</mtext></msub><mo>+</mo><mn>1</mn></mrow></mfrac><mn>.</mn></mrow></math> With a laser power of 1 mW and 100-μM laser spot sizes for 1088 nm, 269 nm, and 196 nm lasers, the steady-state condition can be fulfilled for all 5 2 F 5/2 , 7 2 S 1/2 , 6 2 D 3/2 , and 7 2 P 1/2 states under continuous illumination. For multiple hyperfine structure transitions between electronic transitions 5 2 F 5/2 ↔ 6 2 D 3/2 and 7 2 S 1/2 ↔ 7 2 P 1/2 , the power of the lasers is divided equally. Thus, after state preparation, the population of the <math id="M18"><mrow><mrow><mo>|</mo><mrow><mn>7</mn><msub><mi>P</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>2</mn></mrow><mo>〉</mo></mrow></mrow></math> state accounts for approximately 12% of the total number of trapped ions. Calculation of the population in the nuclear excited state 1 The time-dependent nuclear excitation probability ρ exc ( t ) for a single nucleus under resonant irradiation is given by Torrey’s solution to the optical Bloch equations as follows [ 38 38 - 40 40 38-40 ]: <math display="block" id="M19"><mtable><mtr><mtd><mi>①</mi><msub><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow></msub><mo>></mo><mfrac><mrow><mo>|</mo><mtext>Γ</mtext><mo>−</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo>|</mo></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><msub><mi>ρ</mi><mrow><mtext>exc</mtext></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mtext>Γ</mtext><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo>+</mo><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mtext>Γ</mtext><mo>+</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo stretchy="false">)</mo><mi>t</mi></mrow></msup><mo>×</mo><mrow><mo>(</mo><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mtext>Γ</mtext><mo>+</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover></mrow><mrow><mn>2</mn><mi>λ</mi></mrow></mfrac><mi>sin</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M20"><mtable><mtr><mtd><mi>②</mi><msub><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow></msub><mo><</mo><mfrac><mrow><mo>|</mo><mtext>Γ</mtext><mo>−</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo>|</mo></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><msub><mi>ρ</mi><mrow><mtext>exc</mtext></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mtext>Γ</mtext><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo>+</mo><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mtext>Γ</mtext><mo>+</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo stretchy="false">)</mo><mi>t</mi></mrow></msup><mo>×</mo><mrow><mo>(</mo><mrow><mi>cosh</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mtext>Γ</mtext><mo>+</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover></mrow><mrow><mn>2</mn><mi>λ</mi></mrow></mfrac><mi>sinh</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> Here, Ω eg denotes the Rabi frequency between the isomeric and nuclear ground states. Γ=Γ γ +Γ nr is the total linewidth of the isomeric state, and Γ γ denotes the γ decay rate of the isomeric state. Because the ionization energy of 229g Th 3+ (28.6 eV) is considerably greater than the energy of the isomeric state, the internal conversion process is strongly forbidden [ 41 41 ]. The possible non-radiative transition linewidth Γ nr is dominated by the EB transition for 229g Th 3+ . <math id="M21"><mrow><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo>=</mo><mfrac><mrow><mtext>Γ</mtext><mo>+</mo><msub><mtext>Γ</mtext><mtext>L</mtext></msub></mrow><mn>2</mn></mfrac><mo>+</mo><msub><mtext>Γ</mtext><mrow><mtext>add</mtext></mrow></msub></mrow></math> denotes the total transition linewidth [ 40 40 ], Γ L denotes the laser linewidth used for nuclear excitation, and Γ add is the additional incoherent linewidth, such as phonon coupling of ions in a Coulomb crystal. Because Γ add is significantly smaller than the laser linewidth Γ L , the influence of the incoherent linewidth is negligible. t denotes the interaction time between the laser and 229g Th 3+ ions. Here, <math id="M22"><mrow><mi>λ</mi><mo>=</mo><mo>|</mo><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mtext>Γ</mtext><mo>−</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>/</mo><mn>4</mn><mo>|</mo></mrow></math> . Under the resonant condition, the Rabi frequency between the isomeric state and nuclear ground state is [ 39 39 ] <math display="block" id="M23"><mrow><msub><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow></msub><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>2</mn><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi><msubsup><mi>C</mi><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><mtext>Γ</mtext></mrow><mrow><mi>ℏ</mi><msubsup><mi>ω</mi><mtext>m</mtext><mn>3</mn></msubsup></mrow></mfrac></mrow></msqrt><mn>.</mn></mrow></math> Here, I is the intensity of the excitation laser, and ω m is the angular frequency of the nuclear transition. The Zeeman splitting caused by the geomagnetic field between the two outermost lines is below 1 kHz and is negligible. Therefore, the Clebsch–Gordan coefficient of the sub-state transition between the isomeric state and the nuclear ground state C eg is 1 [ 16 16 ]. Assuming that the intensities of all lasers have a Gaussian distribution, the Rabi frequency can be modified via the following two situations: (i) If the laser linewidth (Γ L ) is greater than the Doppler-broadening linewidth (Γ D ), only part of the laser with a matching frequency range can effectively interact with the triply charged thorium ions. The equivalent Doppler-broadening linewidth is <math display="block" id="M24"><mrow><msubsup><mtext>Γ</mtext><mtext>D</mtext><mo>'</mo></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mrow><mfrac><mi>π</mi><mrow><mi>ln</mi><mn>2</mn></mrow></mfrac></mrow></msqrt><msub><mtext>Γ</mtext><mtext>D</mtext></msub><mn>.</mn></mrow></math> The effective laser intensity is <math display="block" id="M25"><mrow><msub><mi>I</mi><mrow><mtext>eff</mtext></mrow></msub><mo>=</mo><mi>I</mi><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mfrac><mrow><msubsup><mtext>Γ</mtext><mtext>D</mtext><mo>'</mo></msubsup></mrow><mn>2</mn></mfrac></mrow><mrow><mfrac><mrow><msubsup><mtext>Γ</mtext><mtext>D</mtext><mo>'</mo></msubsup></mrow><mn>2</mn></mfrac></mrow></msubsup></mrow></mstyle><msub><mi>g</mi><mtext>L</mtext></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>ω</mi><mn>.</mn></mrow></math> The modified Rabi frequency is <math display="block" id="M26"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>2</mn><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><msub><mi>I</mi><mrow><mtext>eff</mtext></mrow></msub><msubsup><mi>C</mi><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><mtext>Γ</mtext></mrow><mrow><mi>ℏ</mi><msubsup><mi>ω</mi><mtext>m</mtext><mn>3</mn></msubsup></mrow></mfrac></mrow></msqrt><mn>.</mn></mrow></math> (ii) If the laser linewidth (Γ L ) is smaller than the Doppler-broadening linewidth (Γ D ), only part of the ions interact with the laser light at the frequency resonance at a certain time, and the equivalent laser linewidth is <math display="block" id="M27"><mrow><msubsup><mtext>Γ</mtext><mtext>L</mtext><mo>'</mo></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mrow><mfrac><mi>π</mi><mrow><mi>ln</mi><mn>2</mn></mrow></mfrac></mrow></msqrt><msub><mtext>Γ</mtext><mtext>L</mtext></msub><mn>.</mn></mrow></math> The effective Doppler-broadening linewidth is <math display="block" id="M28"><mrow><msub><mtext>Γ</mtext><mrow><mtext>eff</mtext></mrow></msub><mo>=</mo><mtext>Γ</mtext><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mfrac><mrow><msubsup><mtext>Γ</mtext><mtext>L</mtext><mo>'</mo></msubsup></mrow><mn>2</mn></mfrac></mrow><mrow><mfrac><mrow><msubsup><mtext>Γ</mtext><mtext>L</mtext><mo>'</mo></msubsup></mrow><mn>2</mn></mfrac></mrow></msubsup></mrow></mstyle><msub><mi>g</mi><mtext>D</mtext></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>ω</mi><mn>.</mn></mrow></math> The modified Rabi frequency is <math display="block" id="M29"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>2</mn><mi>π</mi><msup><mi>c</mi><mn>2</mn></msup><mi>I</mi><msubsup><mi>C</mi><mrow><mtext>eg</mtext></mrow><mn>2</mn></msubsup><msub><mtext>Γ</mtext><mrow><mtext>eff</mtext></mrow></msub></mrow><mrow><mi>ℏ</mi><msubsup><mi>ω</mi><mtext>m</mtext><mn>3</mn></msubsup></mrow></mfrac></mrow></msqrt><mn>.</mn></mrow></math> Direct nuclear excitation 1 The Th-229 isomeric energy is 8.12 ± 0.11 eV, corresponding to a wavelength of 150.7-154.8 nm. Currently, available light sources such as the VUV pulsed laser [ 42 42 ], synchrotron radiation light source [ 14 14 ], and the 7th harmonic of a Yb-doped fiber laser [ 16 16 , 43 43 - 45 45 43-45 ] are suitable candidates for direct nuclear excitation. Here, based on resonance-enhanced four-wave mixing, the parameters of a tunable and pulsed VUV laser source are a pulse energy of E L = 13.2  μJ around 150 nm, a bandwidth of δν L = 15 GHz, and a repetition rate of R L = 10 Hz, which are used to calculate the direct nuclear excitation rate [ 42 42 ]. Assuming a strongly focused laser with a 3- μ m spot radius to irradiate the linear chain of 100 ions, the corresponding laser intensity <math id="M30"><mrow><mi>I</mi><mo>=</mo><mfrac><mrow><msub><mi>E</mi><mtext>L</mtext></msub><msub><mi>R</mi><mtext>L</mtext></msub></mrow><mrow><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>4.67</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mn>6</mn></msup></mrow></math> W/m 2 can be obtained. The temperature of Th 3+ ions is 50 μ K, and the corresponding Doppler-broadening linewidth Γ D = 6.5 MHz is narrower than the laser linewidth. Therefore, the modified Rabi frequency at resonance <math id="M31"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>δ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> = 107 Hz can be obtained based on Eq. (9)-(11), which is significantly smaller than the total transition linewidth <math id="M32"><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover></math> shown in Table 1 Table 1 . In this case, <math id="M33"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup><mo><</mo><mfrac><mrow><mo>|</mo><mtext>Γ</mtext><mo>−</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo>|</mo></mrow><mn>2</mn></mfrac></mrow></math> is satisfied. The number of excited nuclei is calculated as a function of time using Eq. (7). As shown in Fig. 1 Fig. 1 , the number of excited Th-229 nuclei is only 0.12 out of 100 Th-229 ions and reaches saturation at an irradiation time of 30000 s. The required laser irradiation time is beyond the stable trapping time; therefore, no Th-229 isomeric state can be detected under the current experimental conditions. 1 1 Variable 1 1 Description 1 1 Value 1 1 Comment 1 1 <math id="M34"><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover></math> 1 1 Total transition linewidth 1 1 2 π⋅ 7.5 GHz 1 1 Significantly larger than the nuclear transition linewidth 1 1 Γ γ 1 1 Radiative linewidth 1 1 10 -4 Hz 1 1 Estimated from theory 1 1 Γ D 1 1 Doppler-broadening linewidth 1 1 6.5 MHz 1 1 Doppler broadening of ions from direct nuclear excitation 1 1 N 1 1 Number of ions 1 1 100 1 1 Laser-cooled ions 1 1 Ω eg 1 1 Rabi frequency 1 1 1156 Hz 1 1 Direct excitation between the isomeric and nuclear ground states 1 1 <math id="M35"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>δ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> 1 1 Modified Rabi frequency 1 1 107 Hz 1 1 Modified Rabi frequency at resonance Electronic bridge excitation 1 Direct nuclear excitation is limited by the natural linewidth of the isomeric state and the power of the excitation laser. If EB excitation occurs, the transition linewidth can be increased by approximately 40 times compared with direct nuclear excitation [ 46 46 ]. EB excitation can be utilized by exciting the quantu m state of the electronic state 7 P 1/2 of 229g Th 3+ (7 P 1/2 5/2 + [633]) to the electronic state 7 S 1/2 of 229m Th 3+ (7 S 1/2 3/2 + [631]), as shown in Fig. 1 Fig. 1 . The prepared 229g Th 3+ Coulomb crystal comprising a linear chain of 36 ions is shown in Fig. 3(c) Fig. 3(c) , among which there are four ions (12% of the total number of trapped ions) in the 7 P 1/2 state under steady-state conditions (see Sect. 2). The wavelength of the laser used for EB excitation is 350 nm. A 350-nm laser with a power of 250 mW can be obtained from Toptica DLC TA-SHG pro and focused to a spot size of 10  μmeter. Thus, an intensity of 7.95 × 10 8 Wm -2 can be achieved. Therefore, the Rabi frequency is estimated as 334 kHz. The EB laser linewidth, typically 500 kHz at 350 nm, is narrower than the Doppler broadening of the trapped thorium ions. The modified Rabi frequency <math id="M36"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup></mrow></math> is 140 kHz at resonance and 98 kHz upon the detuning Γ D /2 (shown in Table 2 Table 2 ). Then, the condition <math id="M37"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup><mo><</mo><mfrac><mrow><mo>|</mo><mtext>Γ</mtext><mo>−</mo><mover accent="true"><mtext>Γ</mtext><mo>˜</mo></mover><mo>|</mo></mrow><mn>2</mn></mfrac></mrow></math> is satisfied, and thus the number of excited nuclei as a function of time can be obtained based on Eq. (7), as shown in Fig. 5 Fig. 5 . If the excitation energy from the 7 P 1/2 state of 229g Th 3+ to the 7 S 1/2 state of 229m Th 3+ is at resonance, approximately 2 229g Th 3+ ions can be excited to the isomeric state after 300-μs irradiation by the 350-nm laser. If the excitation laser is detuned by Γ D /2, approximately 1.8 229g Th 3+ ions can be excited to the isomeric state within the same irradiation time. 1 1 Variable 1 1 Value 1 1 Comment 1 1 Γ L 1 1 2π⋅ 500 kHz 1 1 Larger than the nuclear transition linewidth 1 1 Γ γ 1 1 10 -4 Hz 1 1 Estimated from theory 1 1 Γ D 1 1 2.8 MHz 1 1 Doppler broadening of ions from EB excitation 1 1 N 1 1 36 1 1 Laser-cooled ions 1 1 α eb 1 1 40 1 1 EB decay of Th 3+ ions 1 1 Ω eg 1 1 334 kHz 1 1 Rabi frequency 1 1 <math id="M38"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>δ</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> 1 1 140 kHz 1 1 Modified Ω eg at resonance 1 1 <math id="M39"><mrow><msubsup><mtext>Ω</mtext><mrow><mtext>eg</mtext></mrow><mo>'</mo></msubsup><mo stretchy="false">(</mo><mi>δ</mi><mo>=</mo><mfrac><mrow><msub><mtext>Γ</mtext><mtext>D</mtext></msub></mrow><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow></math> 1 1 98 kHz 1 1 Modified Ω eg detune <math id="M40"><mrow><mfrac><mrow><msub><mtext>Γ</mtext><mtext>D</mtext></msub></mrow><mn>2</mn></mfrac></mrow></math> After successful EB excitation, Th-229 ions in the isomeric state can be identified based on the different hyperfine structures of the isomeric and nuclear ground states. Detailed hyperfine structures of the isomeric state are shown in Appendix B. After EB nuclear excitation, the EB laser at 350 nm and the state-preparation lasers at 196 nm and 269 nm are all turned off, and the cooling lasers at 690 nm and 984 nm are immediately switched back on and combined with the 1088-nm laser to cool the ions. The nuclear-excited ions remain in the isomeric states and accumulate in the electronic state of 5 2 F 5/2 , as illustrated in Fig. 6 Fig. 6 . The remaining nuclear-unexcited ions remain in the nuclear ground state of the Doppler cooling cycle of 5 2 F 5/2 , 6 2 D 5/2 , 5 2 F 7/2 and 6 2 D 3/2 . The isomer shifts of the 6 D 5/2 and 6 D 3/2 electronic states are both approximately 400 MHz [ 24 24 , 47 47 ]. The 690-nm and 984-nm lasers are off-resonance for the Doppler cooling of ions in the isomeric state. As a result, the nuclear-excited 229m Th 3+ ions leave the 690-nm and 984-nm lasers’ cooling cycle and appear as dark ions in the Coulomb crystal [ 7 7 ]. In the linear-chain 229 Th 3+ ion Coulomb crystal, single ions are distinguishable, and any dark 229m Th 3+ ions generated by a successful EB excitation event can be detected. The current reported uncertainty on the isomeric state energy is 0.22 eV [ 13 13 ]. To cover this uncertainty range, approximately 1.9 × 10 7 scans are required for each scanning interval of the Doppler-broadening linewidth Γ D . The time required for a single scan step is approximately 300 μs; therefore, the total scan time is approximately 5670 s, as shown in Table 3 Table 3 . The final isomeric energy is determined by the sum of the energy intervals between the electronic state 7 P 1/2 of 229g Th 3+ and the electronic state 7 S 1/2 of 229m Th 3+ and the photon energy of the 350-nm laser. Because the uncertainty on the natural linewidth of the 7 P 1/2 state of 229g Th 3+ is 1 GHz, which is significantly larger than the laser and Doppler-broadening linewidths, the uncertainty on the isomeric energy is of the order of 1 GHz (corresponding to 4 × 10 -6 eV), dominated by the natural linewidth of the 7 P 1/2 state. 1 1 Variable 1 1 Value 1 1 Time per scan step 1 1 ≈ 300 μs 1 1 Number of ions at the 7 P 1/2 state 1 1 4 1 1 Number of laser-cooled ions 1 1 <math id="M41"><mrow><mo>≈</mo><mn>36</mn></mrow></math> 1 1 Number of isomeric states per scan step 1 1 2( δ =0) 1 1 Number of isomeric states per scan step 1 1 1.8( <math id="M42"><mrow><mi>δ</mi><mo>=</mo><mfrac><mrow><msub><mtext>Γ</mtext><mi>D</mi></msub></mrow><mn>2</mn></mfrac></mrow></math> ) 1 1 Time required to scan 0.22 eV 1 1 ≈ 5670 s Conclusion 1 In this paper, considering the Doppler broadening of thorium ions in the ion trap, we show that 2 229g Th 3+ ions out of 36 trapped ions can be excited to the isomeric state under resonant conditions via EB excitation. We propose that a total measurement time of approximately 2 h can be achieved for the current uncertainty on the isomeric state energy of 0.22 eV. If a cryogenic linear Paul trap is used, the ions can be stably trapped for a longer period and the radiative lifetime of the isomeric state can be measured. The utilization of EB excitation can reduce the uncertainty on the Th-229 isomeric energy to approximately 1 GHz. The calculation is based on the theoretical prediction that the transition linewidth of EB excitation is 40 times larger than that of direct nuclear excitation. If EB excitation is not observed, as proposed in this paper, it may indicate that the coupling between the electrons and the nuclear core is not as strong as expected or that the explored nuclear transition energy range is not accurate, and the laser energy used for the EB excitation needs to be increased. 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Lett. 102 , 210801 ( 2009 ). doi: 10.1103/PhysRevLett.102.210801 http://doi.org/10.1103/PhysRevLett.102.210801 Proposed Experimental Method to Determine α Sensitivity of Splitting between Ground and 7.6 eV Isomeric States in 229Th 10.1007/s41365-023-01169-4 This work was supported by the National Natural Science Foundation of China (No. 11804372). 2022-10-16 2022-11-29 2022-12-11 2023-02-24 2023-03-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2023 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-023-01169-4 http://dx.doi.org/10.1007/s41365-023-01169-4 . 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To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4.0/. 2023 Experimental steps 2 As shown in Fig. 7 Fig. 7 , the experiments are divided into three steps. ① Laser cooling: The 1088-nm, 984-nm, and 690 nm lasers are turned on to laser-cool the ions into a Coulomb crystal of a linear chain. Later, the 690-nm and 984-nm lasers are turned off in sequence to keep the entire population at the 5 F 5/2 state. ② State preparation and nuclear excitation: Once the 690-nm laser is turned off, the 269-nm and 196-nm lasers are turned on to prepare the 229g Th 3+ ions at the 7 P 1/2 state, and the 350-nm laser is turned on to excite the Th-229 nuclei from the nuclear ground state to the isomeric state. ③ Detection of the isomeric state: The 350-nm laser is turned off to prohibit the EB transition from 7 S 1/2 of the isomeric state and maintain the population of the isomeric state. Simultaneously, the 196-nm and 269-nm lasers are turned off in sequence to cumulate the ions at the four lowest electronic states. Then, the 690-nm and 984-nm lasers are turned on to detect the isomeric state based on the isomer shift. Hyperfine structures of 229 Th 3+ 2 The hyperfine structures of 229g Th 3+ are measured as described in [ 48 48 ], where μ g =0.360(7) μ N , and Q g =3.11(16)eb. Subsequently, the hyperfine structures of 229g Th 2+ and 229m Th 2+ are measured as described in [ 47 47 ], where μ m =-0.37(6) μ N , and Q m =1.74(6)eb. The magnetic dipole ( A m ) hyperfine coefficients of 229m Th 3+ are determined using <math display="block" id="M43"><mrow><msup><mi>A</mi><mtext>m</mtext></msup><mo>=</mo><mfrac><mrow><msup><mi>A</mi><mtext>g</mtext></msup><msup><mi>I</mi><mtext>g</mtext></msup><msup><mi>μ</mi><mtext>m</mtext></msup></mrow><mrow><msub><mi>μ</mi><mtext>g</mtext></msub><msup><mi>I</mi><mtext>m</mtext></msup></mrow></mfrac><mn>.</mn></mrow></math> The electric quadrupole ( B m ) hyperfine coefficients of 229m Th 3+ are determined using <math display="block" id="M44"><mrow><msup><mi>B</mi><mtext>m</mtext></msup><mo>=</mo><mfrac><mrow><msup><mi>B</mi><mtext>g</mtext></msup><msup><mi>Q</mi><mtext>m</mtext></msup></mrow><mrow><msup><mi>Q</mi><mtext>g</mtext></msup></mrow></mfrac><mn>.</mn></mrow></math> The magnetic dipole and electric quadrupole hyperfine coefficients of the four lowest levels of 229m Th 3+ are calculated, as shown in Table 4 Table 4 . 1 1 Valence orbital 1 1 A m (MHz) 1 1 m (MHz) 1 1 5 F 5/2 1 1 -140(20) 1 1 1270(80) 1 1 5 F 7/2 1 1 -54(9) 1 1 1430(90) 1 1 6 D 3/2 1 1 -270(40) 1 1 1270(80) 1 1 6 D 5/2 1 1 22(4) 1 1 1510(90) Therefore, the hyperfine structures of 229m Th 3+ can be obtained as follows: <math display="block" id="M45"><mrow><msub><mi>E</mi><mrow><mtext>HFS</mtext></mrow></msub><mo stretchy="false">(</mo><mi>J</mi><mn>,</mn><msup><mi>I</mi><mtext>m</mtext></msup><mn>,</mn><mi>F</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>A</mi><mtext>m</mtext></msup><mi>K</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><msup><mi>B</mi><mtext>m</mtext></msup><mo stretchy="false">[</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>K</mi><mo stretchy="false">(</mo><mi>K</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msup><mi>I</mi><mtext>m</mtext></msup><mo stretchy="false">(</mo><msup><mi>I</mi><mtext>m</mtext></msup><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>J</mi><mo stretchy="false">(</mo><mi>J</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mn>2</mn><msup><mi>I</mi><mtext>m</mtext></msup><mo stretchy="false">(</mo><mn>2</mn><msup><mi>I</mi><mtext>m</mtext></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>J</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where K = F ( F +1)- J ( J +1)- I m ( I m +1). Using Eq. 17, the hyperfine structures of the four lowest levels of 229m Th 3+ are calculated, as listed in Table. 5 Table. 5 1 1 Valence orbital 1 1 F m 1 1 229m Th 3+ 1 1 5 F 5/2 1 1 1 1 1 3000(200) 1 1   1 1 2 1 1 140(70) 1 1   1 1 3 1 1 -1700(100) 1 1   1 1 4 1 1 270(90) 1 1 5 F 7/2 1 1 2 1 1 3000(200) 1 1   1 1 3 1 1 -690(70) 1 1   1 1 4 1 1 -2300(100) 1 1   1 1 5 1 1 970(90) 1 1 6 D 3/2 1 1 0 1 1 3400(200) 1 1   1 1 1 1 1 1200(100) 1 1   1 1 2 1 1 -1230(90) 1 1   1 1 3 1 1 -130(90) 1 1 6 D 5/2 1 1 1 1 1 2500(200) 1 1   1 1 2 1 1 -450(30) 1 1   1 1 3 1 1 -2100(100) 1 1   1 1 4 1 1 1030(60) The isomer shift is determined using <math display="block" id="M46"><mrow><mi>Δ</mi><msup><mi>ν</mi><mi>m</mi></msup><mo>=</mo><msup><mi>F</mi><mo>'</mo></msup><mi>δ</mi><mo>〈</mo><msup><mi>r</mi><mn>2</mn></msup><mo>〉</mo><mn>,</mn></mrow></math> where the difference between the mean-square radii of the isomeric and ground states in 229 Th is <math id="M47"><mrow><mi>δ</mi><mo>〈</mo><msup><mi>r</mi><mn>2</mn></msup><mo>〉</mo></mrow></math> = 0.012(2) fm 2 [ 49 49 ], and the field shift constant F ’ can be obtained as shown in Ref. [ 42 42 ]. Therefore, the centers of the hyperfine structures of 229m Th 3+ can be obtained as shown in Table 6 Table 6 . 1 1 Valence orbital 1 1 229g Th 3+ 1 1 229m Th 3+ 1 1 5 F 5/2 1 1 0 1 1 0 1 1 5 F 7/2 1 1 129671430(40) 1 1 129671450(50) 1 1 6 D 3/2 1 1 275596727(31) 1 1 275597100(100) 1 1 6 D 5/2 1 1 434280888(31) 1 1 434281300(100) The hyperfine structures of 229m Th 3+ and 229g Th 3+ are shown in Fig. 7 Fig. 7 , and <math id="M48"><mrow><mi>Δ</mi><msubsup><mi>ν</mi><mtext>i</mtext><mtext>m</mtext></msubsup></mrow></math> indicates the isomer shift. The transition frequency of <math id="M49"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>F</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>1</mn></mrow><mo>〉</mo></mrow></mrow></math> → <math id="M50"><mrow><mrow><mo>|</mo><mrow><mn>6</mn><msub><mi>D</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>0</mn></mrow><mo>〉</mo></mrow></mrow></math> in 229g Th 3+ is 434282480(32) MHz. The hyperfine transitions between the 5 F 5/2 and 6 D 5/2 states of 229m Th 3+ are listed in Table 7 Table 7 . 1 1 Transition 1 1   1 1 Transition 1 1   1 1 1→1 1 1 434280800(300) 1 1 2→1 1 1 434283700(200) 1 1 1→2 1 1 434277900(200) 1 1 2→2 1 1 434280700(100) 1 1 3→2 1 1 434282600(100) 1 1 2→3 1 1 434279100(200) 1 1 3→3 1 1 434280900(200) 1 1 3→4 1 1 434284000(200) 1 1   1 1   The transition frequency of <math id="M51"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>F</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>3</mn></mrow><mo>〉</mo></mrow></mrow></math> → <math id="M52"><mrow><mrow><mo>|</mo><mrow><mn>6</mn><msub><mi>D</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>2</mn></mrow><mo>〉</mo></mrow></mrow></math> in 229m Th 3+ is close to that of <math id="M53"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>F</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>1</mn></mrow><mo>〉</mo></mrow></mrow></math> → <math id="M54"><mrow><mrow><mo>|</mo><mrow><mn>6</mn><msub><mi>D</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>0</mn></mrow><mo>〉</mo></mrow></mrow></math> in 229g Th 3+ . The transition frequency of <math id="M55"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>D</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>0</mn></mrow><mo>〉</mo></mrow></mrow></math> → <math id="M56"><mrow><mrow><mo>|</mo><mrow><mn>5</mn><msub><mi>F</mi><mrow><mn>7</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>1</mn></mrow><mo>〉</mo></mrow></mrow></math> in 229g Th 3+ is 304610430(34) MHz. The hyperfine transitions between the <math id="M57"><mrow><mrow><mo>|</mo><mrow><mn>6</mn><msub><mi>D</mi><mrow><mn>5</mn><mo>/</mo><mn>2</mn></mrow></msub><mn>,</mn><mi>F</mi><mo>=</mo><mn>2</mn></mrow><mo>〉</mo></mrow></mrow></math> and 5 F 7/2 states of 229m Th 3+ are listed in Table 8 Table 8 . As a result, the 690-nm and 984-nm lasers used to cool the 229g Th 3+ ions cannot form the closed optical cycle required for the laser-cooling of 229m Th 3+ ions. 1 1 Transition 1 1   1 1 2→2 1 1 304606900 (200) 1 1 2→3 1 1 304610500 (100) Nuclear Science and Techniques 2 34