The effect of 5f states on the nd → 5f transition energies and spectra of americium ions

Americium (Am) is a radioactive element with atomic number Z = 95. It is the third element past uranium and the fourth element discovered after curium. The main isotopes of Am are known as ²⁴³Am (with a half-lifetime τ_1/2 = 7,400 yr), ²⁴¹Am (τ_1/2 = 433 yr), and ²⁴²Am (τ_1/2 = 152 yr). Am is the product of a series of successive neutron captures plutonium, and its most common use is focused on smoke detectors [1]. Due to the unique behavior of the 5f states of actinides which dominate its electronic structure, it have been of great interest in physics and chemistry [2–6]. Until the present, although a large number of studies have been performed on actinides [7–13], a lot of questions still remain nowadays [14].

Recently, Moore et al [14] studied the N_4,5 a short-hand notation for the $4d\to 5f$ transition spectra of Am metal by using the electron energy-loss spectroscopy (EELS). Butterfield et al [15] examined the O_4,5 ($5d\to 5f$) edge structure of the ground state α-phase of Am metal using the same experimental technique as in [14]. Furthermore, Buck and Fortner [16] have measured the absorption edge energies of Am also with the use of the EELS.

In the present work, the relative intensities of spectra originating from the electric dipole transitions from the nd (n = 3–5) core states excited to the 5f valence states of Am²⁺ to Am⁸⁺ ions have been theoretically studied by using the multiconfiguration Dirac-Hartee-Fock (MCDHF) method [17, 18] and its corresponding computer code GRASP2K [19, 20]. However, GRASP2K is not the only code that can do MCDHF. The Breit interaction, and the quantum-electrodynamic (QED) effect have been taken into account.

As the details of the theoretical procedure of the MCDHF method have been explained in the [21] by Grant, therefore, in the present paper we only present the main points of it. The Dirac-Coulomb Hamiltonian of an atom or ion with N-electron can be given by

$$\begin{eqnarray}&&{H}_{{DC}}=\displaystyle \sum _{i=1}^{N}{h}_{D}({r}_{i})+\displaystyle \sum _{i\lt j}^{N}\left(\displaystyle \frac{1}{{r}_{{ij}}}\right).\end{eqnarray} \tag{ 1 }$$

Here, h_D (r_i ) denotes one-electron Dirac Hamiltonian. while the second term is the electron-electron Coulomb interactions. In the MCDHF method, an atomic state function (ASF) of the system with parity P, total angular momentum J and its component M is approximated by a linear combination of configuration state functions (CSFs) of the same symmetry PJM,

$$\begin{eqnarray}&&| {\psi }_{\alpha }({PJM})\rangle =\displaystyle \sum _{r=1}^{{n}_{c}}{c}_{r}(\alpha )| {\gamma }_{r}({PJM})\rangle ,\end{eqnarray} \tag{ 2 }$$

in which n_c is the number of CSFs and c_r (α) denotes the configuration mixing coefficients corresponding to each individual CSF.

The calculation is started from a single configuration Dirac-Fock solution with the nucleus described as an extended Fermi distribution. A single electron is excited from 3d, 4d, and 5d core hole-states to the 5f states. The trial radial wave functions are estimated by solving the Dirac equation either in the Thomas-Fermi potential or in the screened hydrogenic approximation. Furthermore, the contributions of the Breit interaction and QED effects are considered as a perturbation through relativistic configuration interaction (RCI) calculations.

In the calculations of transition energies △E, transition rates A [22], and weighted oscillator strengths gf [23] of the ${nd}\to 5f$ (n = 3 − 5) transitions of Am²⁺–Am⁸⁺ ions, the configurations [Rn]5f⁷, [Rn]5f⁶, [Rn]5f⁵, [Rn]5f⁴, [Rn]5f³, [Rn]5f², and [Rn]5f¹ are adopted as the ground-state configurations corresponding to Am²⁺, Am³⁺, Am⁴⁺, Am⁵⁺, Am⁶⁺, Am⁷⁺, and Am⁸⁺ ions, respectively. For example, in order to perform calculations for the $3{d}^{10}5{f}^{7}\to 3{d}^{9}5{f}^{8}$, $4{d}^{10}5{f}^{7}\to 4{d}^{9}5{f}^{8}$, and $5{d}^{10}5{f}^{7}\to 5{d}^{9}5{f}^{8}$ transitions of Am²⁺ ions, $1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}3{p}^{6}3{d}^{10}$...$6{s}^{2}6{p}^{6}5{f}^{7}$ is taken as the initial-state configuration, and $1{s}^{2}2{s}^{2}2{p}^{6}3{s}^{2}3{p}^{6}3{d}^{9}$... $6{s}^{2}6{p}^{6}5{f}^{8}$, $1{s}^{2}2{s}^{2}2{p}^{6}$ $3{s}^{2}3{p}^{6}3{d}^{10}4{s}^{2}4{p}^{6}4{d}^{9}$...$6{s}^{2}6{p}^{6}5{f}^{8}$ and $1{s}^{2}2{s}^{2}2{p}^{6}$...$5{d}^{9}6{s}^{2}6{p}^{6}5{f}^{8}$ act as the final-state configurations. In table 1, the numbers of the relativistic CSFs generated for the initial- and final-state configurations of Am${}^{q+}$ (q = 2, 3 ... 8) ions are presented.

Table 1. Numbers of the relativistic CSFs for Am${}^{q+}$ (q = 2, 3 ... 8) ions considered in the calculations.

Ions	Ground config.	CSFs	Excited config.	CSFs
Am2+	${d}^{10}5{f}^{7}$	327	${d}^{9}5{f}^{8}$	2725
Am3+	${d}^{10}5{f}^{6}$	295	${d}^{9}5{f}^{7}$	3106
Am4+	${d}^{10}5{f}^{5}$	198	${d}^{9}5{f}^{6}$	2725
Am5+	${d}^{10}5{f}^{4}$	107	${d}^{9}5{f}^{5}$	1878
Am6+	${d}^{10}5{f}^{3}$	41	${d}^{9}5{f}^{4}$	977
Am7+	${d}^{10}5{f}^{2}$	13	${d}^{9}5{f}^{3}$	386
Am8+	${d}^{10}5{f}^{1}$	2	${d}^{9}5{f}^{2}$	107

In table 2, the presently calculated transition energies △E, transition rates A, and weighted oscillator strengths gf for the ${M}_{4}(3{d}_{3/2}\to 5{f}_{5/2})$, ${M}_{5}(3{d}_{5/2}\to 5{f}_{5/\mathrm{2,7}/2})$, ${N}_{4}(4{d}_{3/2}\to 5{f}_{5/2})$, ${N}_{5}(4{d}_{5/2}\to 5{f}_{5/\mathrm{2,7}/2})$, ${O}_{4}(5{d}_{3/2}\to 5{f}_{5/2})$, and ${O}_{5}(5{d}_{5/2}\to 5{f}_{5/\mathrm{2,7}/2})$ transitions of Am²⁺–Am⁸⁺ ions are shown. To the best of authors knowledge, there are no other available data for Am ions in literature to compare with the present results. In order to justify the present GRASP2K results, similar calculations with the use of the Flexible Atomic Code (FAC) [24] are performed as well, which is also a fully relativistic package based on the Dirac equation for the calculations of atomic transition properties. Regarding the transition energies, the maximum relative discrepancies between the GRASP2K and the FAC results are 0.21 %, 0.41 %, and 3.34 % for the M, N, and O transitions, respectively. As for the A- and gf-values, only relatively slight discrepancies between the GRASP2K and the FAC are found for most of the transitions, while for a few transitions such as the M₅ of Am⁴⁺ and Am⁸⁺ ions such a discrepancy is a little bit large. Since the same configurations have employed in both the GRASP2K and FAC calculations with an inclusion of the Breit interaction and QED effect, these discrepancies are mainly due to the difference of optimization techniques adopted by the two codes.

Table 2. The presently calculated transition energies $\bigtriangleup {\rm{E}}$ (eV), rates A (${s}^{-1}$), and gf values for ${nd}\to 5f(n=3-5)$ transitions of Am²⁺–Am⁸⁺ ions in both cases of the GRASP2K and FAC While the entries in parentheses refer to the power of ten, the symbols M, N, and O are x-ray notations used for inner-shell transitions.

Ions		M4	M5	N4	N5	O4	O5
		GRASP2K
Am2+	△E	4096.51	3892.94	885.26	834.91	129.59	112.92
	A	1.08(12)	1.07(12)	9.28(10)	1.62(11)	2.42(09)	4.13(09)
	gf	2.98(−03)	3.24(−03)	5.46(−03)	1.07(−02)	6.65(−03)	1.49(−02)
Am3+	△E	4098.54	3895.93	887.46	836.79	133.09	113.17
	A	9.45(11)	1.63(12)	1.91(11)	9.52(11)	1.21(09)	1.60(09)
	gf	3.89(−03)	7.44(−03)	1.68(−02)	9.40(−02)	4.73(−03)	8.59(−03)
Am4+	$\bigtriangleup {\rm{E}}$	4100.76	3897.60	887.51	838.71	136.75	114.32
	A	3.57(12)	9.67(09)	2.80(10)	3.74(11)	3.72(11)	6.47(08)
	gf	9.80(−03)	2.94(−05)	1.64(−03)	2.45(−02)	9.16(−01)	2.28(−03)
Am5+	△E	4102.18	3900.24	889.93	841.98	140.17	116.49
	A	5.03(12)	4.87(11)	1.74(10)	1.62(09)	6.65(09)	3.17(08)
	gf	1.07(−02)	2.21(−03)	1.52(−03)	1.58(−04)	3.40(−02)	1.62(−03)
Am6+	$\bigtriangleup {\rm{E}}$	4105.18	3903.20	892.14	843.83	142.20	118.69
	A	9.23(12)	8.02(12)	4.94(10)	9.05(10)	2.80(10)	1.31(−09)
	gf	2.52(−02)	2.43(−02)	2.86(−03)	8.86(−03)	6.37(−02)	4.28(−03)
Am7+	$\bigtriangleup {\rm{E}}$	4108.80	3906.38	895.63	846.48	147.44	122.87
	A	2.28(12)	5.91(12)	2.40(10)	2.74(11)	3.73(11)	4.89(09)
	gf	9.35(−03)	2.68(−02)	2.07(−03)	2.64(−02)	2.82(00)	2.24(−02)
Am8+	$\bigtriangleup {\rm{E}}$	4111.80	3911.93	899.11	848.36	150.57	122.68
	A	8.47(12)	4.86(13)	1.49(12)	1.43(11)	1.75(12)	4.66(09)
	gf	4.62(−02)	2.92(−01)	1.70(−01)	1.83(−02)	1.07(01)	2.85(−02)
		FAC
Am2+	△E	4088.89	3886.03	881.89	832.45	127.13	112.41
	A	4.18(11)	1.07(12)	2.38(10)	1.33(11)	2.00(08)	1.82(09)
	gf	3.99(−03)	4.20(−04)	8.47(−03)	6.18(−02)	3.42(−03)	3.32(−02)
Am3+	△E	4090.06	3887.47	883.30	834.07	130.65	113.21
	A	2.66(11)	1.81(11)	1.36(11)	3.08(11)	7.66(09)	2.55(08)
	gf	1.79(−03)	3.04(−03)	5.43(−03)	1.02(−02)	7.24(−02)	5.96(−03)
Am4+	$\bigtriangleup {\rm{E}}$	4092.26	3890.46	885.45	836.36	134.41	114.22
	A	1.81(12)	2.51(09)	4.62(10)	1.33(11)	2.69(11)	3.54(08)
	gf	3.48(−03)	1.23(−03)	5.43(−03)	1.58(−03)	2.75(00)	4.99(−03)
Am5+	△E	4095.57	3892.49	887.48	838.38	137.80	116.31
	A	2.04(12)	2.20(11)	6.74(10)	1.34(10)	7.92(09)	1.16(09)
	gf	4.20(−02)	3.68(−03)	1.17(−02)	4.84(−03)	6.73(−02)	9.89(−03)
Am6+	$\bigtriangleup {\rm{E}}$	4097.60	3896.80	890.55	841.07	140.90	117.64
	A	2.07(12)	4.29(12)	1.64(10)	2.93(10)	7.22(09)	1.09(09)
	gf	2.84(−02)	6.52(−02)	1.81(−03)	7.64(−03)	5.03(−02)	2.03(−02)
Am7+	$\bigtriangleup {\rm{E}}$	4101.45	3899.72	893.13	844.55	145.30	118.71
	A	3.29(12)	4.57(12)	1.04(11)	2.74(11)	7.67(11)	2.33(09)
	gf	4.95(−02)	2.08(−02)	3.89(−02)	6.19(−02)	2.51(00)	1.90(−02)
Am8+	$\bigtriangleup {\rm{E}}$	4105.14	3908.71	896.47	846.17	147.36	121.37
	A	2.56(11)	1.75(11)	1.72(12)	6.26(11)	3.63(12)	3.23(10)
	gf	2.80(−03)	1.59(−03)	3.94(−01)	2.02(−01)	2.31(01)	4.04(−01)

As can be seen from table 2, the transition energies corresponding to the same inner-shell hole-states are quite close in sequence to each other. For a specific ion, the corresponding transition energies decrease while the inner-shell hole moves towards outer shells. The energy difference between M₄ and M₅, N₄ and N₅ as well as O₄ and O₅ indicates the splitting of the core states, i.e., the fine-structure splitting of $3{d}^{-1}$ ${}^{2}{D}_{3/2}$ and ${}^{2}{D}_{5/2}$.

In figure 1, the presently calculated ${O}_{\mathrm{4,5}}(5d\to 5f)$ spectra of Am²⁺–Am⁸⁺ ions are shown, which are Gaussian line shapes and are obtained by convoluting the corresponding transition rates with the full width at half maximum (FWHM) 5 eV. However, since the separation of the 5f and 5d fine-structure energy levels is too small to be well separated, it appears like a quasi-continuum profile. With the decreasing number of the 5f spectator electrons, such a quasi-continuum profile changes gradually to be a two-peak-like characteristics especially for Am⁷⁺ and Am⁸⁺ ions, and the width of the corresponding transition peaks becomes narrower and the intensity becomes weaker. Moreover, for all of the spectra the peak center shifts towards higher-energy region.

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Moreover, the ${N}_{\mathrm{4,5}}(4d\to 5f)$ spectra of these Am ions as shown in figure 2 are studied. While these spectrum are obtained by the same convolution as in figure 1, the results are ultimately different from the ones of the O_4,5 transition. In the latter case, the core spin-orbit interaction is dominant over the electrostatic interaction of the 4d hole. Based on this fact, the two peaks are well separated in energies corresponding to the N₄ and N₅. The energy difference between these two peaks is approximately 49.10 eV on average. Moreover, the N₄ and N₅ peaks witness a reduction in intensity from Am²⁺ to Am⁸⁺. In figure 3, the ratio N₄:N₅ is plotted versus the number of the 5f electrons. It is found that the ratio N₄:N₅ increases with the decreasing number of the 5f vacancies. Also, it is found that the differences of the ratio N₄:N₅ corresponding to the GRASP2K and FAC results are relatively small for most of the Am ions.

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The ${M}_{\mathrm{4,5}}(3d\to 5f)$ spectra are plotted in figure 4. M₄ and M₅ peaks corresponding respectively to the E1 transitions $3{d}_{3/2}\to 5{f}_{5/2}$ and $3{d}_{5/2}\to 5{f}_{5/\mathrm{2,7}/2}$ are well separated from each other, which arises from the spin-orbit splitting of 3d electrons. The energy separation of these two peaks is about 202.11 eV for all of the ions under study. Furthermore, the separation of energy levels are corresponding to the 3d_3/2 and 3d_5/2 states. Moreover, the peak intensity ratio M₄:M₅ is plotted in figure 5 as a function of the number of the 5f electrons. It is found that the ratio M₄:M₅ decreases as increasing number of the 5f spectator electrons.

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The spectra originating from ${nd}\to 5f(n=3-5)$ transitions of Am²⁺–Am⁸⁺ ions have been calculated by using GRASP2K code. While the obtained ${O}_{\mathrm{4,5}}(5d\to 5f)$ spectra show a broad quasi-continuum profile, both the ${N}_{\mathrm{4,5}}(4d\to 5f)$ and ${M}_{\mathrm{4,5}}(3d\to 5f)$ spectra consist of two peaks that are well separated with respect to energy because of the strong spin-orbit interactions of $3{d}^{-1}$ and $4{d}^{-1}$ hole state.

This work has been supported by the National Key Research and Development Program of China under Grant No. 2017YFA0402300 and the National Natural Science Foundation of China (NSFC) under Grant Nos. 11804280, 11 874 051, U1832126, U1530742, 91 126 007, and the Scientific Research Program of the Higher Education Institutions of Gansu Province of China (Grant No. 2018A-002) and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University under No. NWNU-LKQN-15-3.

All data that support the findings of this study are included within the article (and any supplementary information files).