Wavelengths of the Self-Photopumped Nickel-Like 4f ¹P₁→4d ¹P₁ X-ray Laser Transitions

Abstract

The energies for the lower 3d_3/24d_3/2 [J = 1] and upper 3d_3/24f_5/2 [J = 1] working levels in the self-photopumped X-ray laser are analyzed along the Ni-like sequence. We have found some irregularities in these energy levels in the range Z = 42–49. The causes of the irregularities are studied. The list of elements that lase on the self-photopumped transition can be extended much further than originally known. We calculate the wavelengths of this transition in Ni-like sequence to Z = 79 using the relativistic perturbation theory with a zero approximation model potential. We estimate the wavelength accuracy for Z > 50 as Δλ/λ ≤ 0.005.

1. Introduction

Self-photo pumped (SPP) X-ray lasers (XRL) in Ni-like ions were presented in 1996 [1] as an alternative approach to the standard radiative collisional scheme for inversion creation. We use the term SPP following the name given in literature. This is really a collisionally pumped laser assisted by radiation trapping. Both schemes for Ni-like ions are shown in Figure 1. This new class of SPP in Ni-like XRL was first investigated theoretically in [2], where high gain was predicted for the 4f ¹P₁–4d ¹P₁ transition in Mo¹⁴⁺ at 22.0 nm. It was supposed that preplasma was created by a nanosecond pulse followed by a picosecond pulse to control the temperature and density in plasma, and to achieve high gain. This wavelength was calculated using the multiconfiguration Dirac-Fock atomic physics code by Grant and co-workers in the extended average level mode [3]. In the experiment [4], the Ni-like SPP XRL on the 4f ¹P₁–4d ¹P₁ transition was demonstrated in Zr, Nb, and Mo, and the measured wavelengths for these ions were presented. For Mo¹⁴⁺ a gain of 13 cm⁻¹ was measured at 22.6 nm for a target up to 1 cm long [4]. The wavelengths of this transition for ions from Z = 36 to 54 were predicted in [4] using the experimental data of this work to provide small corrections to their calculations. In the experiment [5], the progress in the optimization and understanding of the collisional pumping of X-ray lasers using an ultrashort subpicosecond heating pulse was reported. Time-integrated and time-resolved lasing signals at the standard 4d ¹S₀–4p ¹P₁ XRL line in Ni-like Ag were studied in detail. Under specific irradiation conditions, strong lasing was obtained on the SPP 4f ¹P₁–4d ¹P₁ transition at 16.1 nm. The strong lasing on the SPP transition in Mo¹⁴⁺ was also observed with very modest (less than 1 J) pump energy at a high repetition rate [6]. Recently, lasing on the SPP 3d ¹P₁–3p ¹P₁ laser line has been observed for Ne-like V, Cr, Fe, and Co, as well as for Ni-like Ru, Pd, and Ag [7]. A strong dependence on the delay between the main and second prepulse was found: the optimum delay shifts towards smaller delays with increasing atomic number Z. Accurate wavelength measurements and calculations were shown to be in excellent agreement. The experiment [7] demonstrated that the list of elements that lase on the SPP transitions can be extended much further than originally known.

Many authors have investigated the spectra of Ni-like ions using vacuum sparks, laser produced plasma and electron beam ion traps as light sources [8,9,10,11,12,13,14]. The 3d⁹4d and 3d⁹4f configurations have been analyzed in the Rb X–Mo XV sequence [10,11]. In [10,11], these configurations were investigated using parameter extrapolations within the Generalized-Least-Squares (GLS) method. This method was used in [12,13] to predict for 3d⁹4d, 3d⁹4f configuration energy levels in Cd XXI and Ag XX. GLS predictions of 3d⁹4d, 3d⁹4f energy levels in the Zr XIII–Pd XIX sequence are tabulated in [14].

Note that lasing wavelength (λ_las) in Mo¹⁴⁺ was determined theoretically [2] and in the experiment [4] using one and the same atomic physics code [3], but results for λ_las were somewhat different (by 4 Å). The 3d_3/24f_5/2 [J = 1] upper working level has the largest oscillator strength and radiative transition probability to the 3d¹⁰ ground level. This fact allows it to achieve high precision in this level energy measurement along the Ni-like sequence up to high Z ~ 84; in some ions, the energy of the transition to the ground state was accurate up to the fourth significant digit. The wavelengths of resonant radiative transitions in heavy Ni-like ions were calculated by us to Z = 83 in [15]. Moreover, in [15], the wavelengths (for Z within 79–82) were predicted with the same accuracy, although they have not yet been measured experimentally.

In the present paper, we analyze the smoothness of the working energy levels of SPP XRL along the Ni-like sequence. We found some irregularities in Ni-like sequence energies in the region Z = 42 (Mo¹⁴⁺) and in the region Z = 49 (In²¹⁺) for the upper 3d_3/24f_5/2 [J = 1] working level. The causes of the irregularities are studied.

The principle purpose of this paper is to predict the wavelengths of SPP XRL lines in Ni-like ions with Z ≤ 79. The calculations are performed by the Relativistic Perturbation Theory with Model Zero Approximation, (RPTMP). The fundamental principles of the RPTMP approach are given in [16]. Energy levels of the 3p⁶3d⁹4l, 3p⁵3d¹⁰4l, (l = 0, 1) configurations and radiative transition rates to the 3p⁶3d¹⁰ ground state in the Kr IX ion are calculated by this method in [16]. The stability of calculations on the approximation used is shown in [16].

2. Features of Lower and Upper Working Levels of SPP XRL along the Ni-Like Sequence

The schematic diagram of three strong XRL transitions is shown in Figure 1: two of them are standard 3d4d [J = 0]–3d_5/24p_3/2 [J = 1] and 3d4d [J = 0]–3d_3/24p_1/2 [J = 1] transitions. The classifications of lower working levels in Figure 1 are valid for Z > 42. The 3d_5/24p_3/2 [J = 1] level is the lower working level of an XRL for the entire nickel isoelectronic sequence, the 3d_3/24p_1/2 [J = 1] level is the lower working level for heavy ions starting with Z = 62. The third 3d_3/24p_3/2 [J = 1] level decays to a ground state significantly weaker than the two mentioned above, and does not provide a significant gain. In our recent work [17], the energies of standard XRL transitions in ions of the Ni-like sequence with Z ≤ 79 are refined by RPTMP calculations. The calculated energies of the two standard 4d–4p, J = 0–1 XRL transitions are corrected by extrapolation of the experimental differentials of XRL transition energies dE_Z^las = E_Z^las–E_Z-1^las, i.e., the differences between transition energies of neighboring ions, which weakly depend on Z (especially in the region Z ≤ 50). It is proven that the accuracy for the final results for large Z is within the experimental error.

The 3d_3/24f_5/2 [J = 1]–3d_3/24d_3/2 [J = 1] transition is optically self-photopumped XRL in all Ni-like ions, the positions of working levels vary with respect to other levels along the sequence. Based on our previous studies of XRL [18,19,20], it can be argued that there are at least four principal differences between standard and self photo-pumped mechanisms:

(1)

In the standard scheme, the upper working level is populated by strong monopole electron collisions: in the SPP scheme it is populated by strong dipole electron collisions, which means high oscillator strength and effective photoabsorption.

(2)

Effective SPP XRL is possible only in optically thick plasma (large electron density n_e and diameter d), while the standard XRL is possible both in optically thick and in optically thin plasma over a wide range of n_e and d.

(3)

In the SPP, the upper working level is quickly emptied due to the large radiative decay rate. Therefore, in this scheme, a laser effect is short-lived; maximum XRL duration may be a few tens of picoseconds. A standard XRL can operate in quasi-continuous mode (under certain conditions).

(4)

In the SPP, the lower and upper working levels do not change their classification along the Ni-like sequence; in the standard scheme the upper working level changes its classification: the 3d_5/24d_5/2 [J = 0] state is dominant in the classification of the upper working level at Z ≤ 51, and the 3d_3/24d_3/2 [J = 0] state is dominant for Z > 51 [17].

Below, we demonstrate the irregularities in the sequence of both the lower and the upper working levels of SPP XRL. Crossing of each working level with another level causes these irregularities. Level crossing is accompanied by a strong interaction at certain Z points. Figure 2a shows the scaled energies along Z of the 3d_3/24d_3/2 [J = 1] lower working level and the 3d_3/24d_5/2 [J = 1] level close to it. In addition to the energy levels calculated here, Figure 2a also shows the corresponding experimental values [14]. Reference [14] does not indicate classification of 3d4d [J = 1] levels, their classification was made earlier in [11]. Note that theoretical and experimental classifications are identical. There are some differences between theoretical and experimental energies, typically a few units in the 4th–5th digits. These differences are conditioned by the shift of the theoretical list of energy levels as a whole, but this shift does not affect the accuracy of λ_las. The energy levels in Figure 2a are scaled by dividing by (Z–23)², so that the behavior of the third and fourth significant digits can be observed. At the beginning of the sequence, the 3d_3/24d_3/2 [J = 1] level is above the 3d_3/24d_5/2 [J = 1] level. The crossing of these levels is in the range 41 < Z < 42 (shown by arrows). The crossing of the corresponding experimental energy levels occurs at exactly the same Z values. At Z = 42, one can observe the “repulsion” of levels caused by their interaction; the “repulsion” is a feature of theoretical and experimental data. Note, that repulsion can be seen due to energy scaling; in fact, the repulsion value is approximately a few thousand cm⁻¹, i.e., a few units in the fourth digit for the 3d_3/24d_5/2 [J = 1] level.

In Figure 2b, we can see hard-to-explain behavior of the 3d_3/24f_5/2 [J = 1] upper working level in the region of Z = 42. The features of this level will be considered below in more detail; however, it is important to note, here, that the energy structure of odd states in the range Z = 40–49 exhibits extremely high instability caused by the interaction of levels with each other, which rapidly changes with Z. In the case at hand, we understand the instability as the ambiguity of the calculation of eigenvectors and eigenenergies. As a result, the calculation in the same approximation leads to different energies at a certain level. The deviation from the smooth curve in Figure 2a is ~10,000 cm⁻¹; however, such a value leads to a sufficiently large deviation from the corresponding experimental values of λ_las shown in Figure 3.

At the point Z = 42, λ_las calculated here is ~222 Å, which is smaller than the experimental and theoretical values of [4] by 4 Å. In a recent experiment [7], the delay time between preliminary and main pump pulses was optimized to achieve the maximum yield of the X-ray laser. In fact, the electron density was optimized in [7]. X-ray lasing occurs in the Ni-like ion ionization mode, so that the lasing times on both transitions were restricted to the ionization time of Ni-like ions to the Co-like state. Time-resolved measurements in [7] allowed high-accuracy wavelength measurements of the SPP and standard X-ray laser lines. Thus, the calculations of the previous work [4] were confirmed: λ_las ≈ 22.61 nm in Ni-like molybdenum (Mo¹⁴⁺, Z = 42). Our calculations are performed for an isolated atom. Based on the studies performed, it can be argued that the interaction of levels at the point Z = 42 is so strong that the energy levels 3d_3/24d_3/2 [J = 1], 3d_3/24d_5/2 [J = 1] in dense hot plasma can differ significantly from the corresponding energy levels in an isolated atom.

The problem is the composition of the 3d_3/24d_3/2 [J = 1] working level, which indicates the strength of level interaction. It is shown in Figure 4 for all 3d4d [J = 1] levels in Ni-like ions with Z = 36–79. Figure 4 shows that contributions of the 3d_3/24d_3/2 [J = 1] and 3d_3/24d_5/2 [J = 1] levels are almost equal at Z = 42, which could lead to levels’ misidentification. Theoretical energies of these levels at Z = 42 are 2,393,554 cm⁻¹ and 2,400,846 cm⁻¹ (51% and 41%, respectively, are the contributions to the 3d_3/24d_3/2 [J = 1] low working level). The contributions of these levels in [11] are 45% and 34%, and the energies are 2,385,902 cm⁻¹ and 2,393,229 cm⁻¹ respectively. (We note that the theoretical list of energies of Ni- like ions in the range of small Z is shifted as a whole by 5000–8000 cm⁻¹). Figure 4 demonstrates the rapid restructuring of lower working level compositions: so that the 3d_5/24d_3/2 [J = 1] level contribution increases by five orders of magnitude in the range Z = 40–42.

Figure 5 shows the scaled energies along Z of the 3d_3/24f_5/2 [J = 1] upper working level and the close 3p_3/24s_1/2 [J = 1] level. Crossing of these levels occurs in the range 48 < Z < 49. At Z = 49 one can see the “repulsion” of levels caused by their interaction; the “repulsion” is a feature of theoretical data. In Figure 5, the corresponding experimental energies for the 3d_3/24f_5/2 [J = 1] level are shown [14]. Unfortunately, we have no available data on the experimental 3p_3/24s_1/2 [J = 1] levels in the Z region under consideration. The value Z = 49 is the point of an abrupt jump (irregularity) in spectroscopic constants of the 3d_3/24f_5/2 [J = 1] upper working level and the 3p_3/24s_1/2 [J = 1] level crossing it, caused by the strong interaction of these levels at this value of Z. This interaction is shown in Figure 6, where we can see the 3d_3/24f_5/2 [J = 1] level composition. The interaction of levels at the point Z = 49 leads to the so-called effect of oscillator strength transfer we considered in [21] for the Ne-like sequence. At this point, the rate of radiative processes abruptly changes: the probabilities of the transition from the 3d_3/24f_5/2 [J = 1] level to the ground state and to the state of the lower working level slightly decrease. At the same time, these probabilities for the 3p_3/24s_1/2 [J = 1] level increase by an order of magnitude and become almost equal in magnitude to the corresponding values of the 3d_3/24f_5/2 [J = 1] level. It can be assumed that there was an incorrect identification at the point Z = 49 when extrapolating the upper working level in [4], and the 3p_3/24s_1/2 [J = 1] level that is close to the 3d_3/24f_5/2 [J = 1] level in energy was used as the upper working level (see Figure 5). If this assumption is correct, λ_las ~ 144.7 Å for Z = 49, which is identical to [4]. When using our value for 3d_3/24f_5/2 [J = 1], λ_las ~ 140.0 Å (here the energy jump shown in Figure 5 is taken into account). Another argument in favor of the incorrect identification in [4], are large jumps of the differential dλ_las (Z) = λ_las (Z)−λ_las (Z−1) in the range Z = 47–50.

3. Wavelengths of the Self-Photopumped Nickel-Like 4f ¹P₁→4d ¹P₁ X-ray Laser Transitions

A comparison of the wavelengths of the self-photopumped nickel-like 4f ¹P₁→4d ¹P₁ X-ray laser transitions, calculated using the RPTMP method with corresponding experimental values and shown in Figure 3, exhibits a deviation of ≤1% in the range Z = 37–46. For Z ≥ 48 Å, our results are identical to experimental data, with an accuracy of several units in the fourth significant digit. Two values of Z are exceptions: (i) the calculation instability point at Z = 42; and (ii) the point Z = 49, where the 3d_3/24f_5/2 [J = 1] and 3p_3/24s_1/2 [J = 1] states are probably incorrectly identified in the calculation by the MCDF method in [4]. We estimated the accuracy of the calculation of the energies of the upper and lower working states for high Z using experimental measurements of various studies. As an example, we compared the experimental energies for Z = 74 (W⁴⁶⁺), obtained using the Super EBIT (electron beam ion trap) [22,23], presented in

Table 1. Energy levels (10³ cm⁻¹) of W XLVII. Comparison of present calculations with experimental data [22,23] and with calculations by GRASP92 [24].

Configuration	Term	J	Experiment	Present Work	GRASP92
3p63d10	1S0	0	0.0	0.0	0.0
3p63d94s	(5/2,1/2)	3	12,601.5	12,600.1
		2	12,616.44	12,615.2	12,591.1
3p63d94s	(3/2,1/2)	1	13,138.66	13,137.8	13,110.8
		2	13,148.2	13,147.4	13,120.7
3p63d94p	(5/2,1/2)	2	13,379.05	13,357.5
		3	13,388.20	13,366.3
3p63d94p	(3/2,1/2)	2	13,916.27	13,894.8
		1	13,940.6	13,922.4	13,930.6
3p63d94p	(5/2,3/2)	1	14,229.0	14,234.9	14,221.0
3p63d94p	(3/2,3/2)	1	14,751.0	14,756.2	14,741.1
3p63d94d	(3/2,3/2)	1		15,935.9	15,924.2
3p63d94d	(5/2,5/2)	1	15,556.1	15,561.3	15,550.2
		2	15,610.2	15,614.9	15,605.0
3p53d104s	(3/2,1/2)	1	16,247.0	16,258.9
3p63d94d	(3/2,3/2)	0	16,256.2	16,284.7	16,282.9
3p63d94f	(5/2,7/2)	1	17,045.9	17,042.2	17,030.6
3p63d94f	(3/2,5/2)	1	17,574.7	17,586.5	17,585.6
			17,580.3 *
3p53d104s	(1/2,1/2)	1	18,727	18,726.4	18,724.4
3p53d104d	(3/2,3/2)	1	19,044.4	19,041.8	19,057.5
3p53d104d	(3/2,5/2)	1	19,244.5	19,234.8	19,244.1
3p53d104f	(3/2,7/2)	2	20,589.0	20,600.1	20,613.8
3p53d104d	(1/2,3/2)	1	21,561.0	21,547.0	21,614.6

* Data from [23].

. There are also listed the theoretical results calculated using the MCDF method called Grasp92 [24]. Here, we do not present earlier calculations of other authors. We also note the impossible comparison to the other calculations [25] in view of the level identification entanglement in this paper.

Good agreement between experimental and theoretical results for the energy levels in Table 1 may be noted: the maximum deviation is two units in the fourth significant digit. For the problem under study, it is important to ascertain the high accuracy of the calculation of the upper and lower working levels. For the experimental energy of the 3d_3/24f_5/2 [J = 1] level, Table 1 gives two values: one obtained in the experiments [22], and the other later [23]. The difference with our calculation is 6 units in the fifth significant digit. We did not find the experimental energy of the 3d_3/24d_3/2 [J = 1] lower working level for high Z in the literature. The energies of two other states of the 3d4d configuration with J = 1, 2, given in Table 1, also agree with high accuracy, which indirectly confirms the calculation reliability. Wavelengths of the 3d_3/24f_5/2 (¹P₁)–3d_3/24d_3/2 (¹P₁) SPP laser transitions in Ni-like sequence calculated by RPTMP are listed in

Table 2. Wavelengths (λ_las, Å) of the 3d_3/24f_5/2 (¹P₁)–3d_3/24d_3/2 (¹P₁) SPP laser transitions in Ni-like sequence calculated by RPTMP.

Z	λlas
50	134.08
51	128.12
52	122.54
53	117.39
54	112.66
55	108.36
56	104.295
57	100.51
58	96.98
59	93.68
60	90.57
61	87.65
62	84.89
63	82.28
64	79.81
65	77.47
66	75.23
67	73.08
68	71.06
69	69.11
70	67.25
71	65.47
72	63.75
73	62.10
74	60.51
75	58.97
76	57.48
77	56.04
78	54.64
79	53.23

.

4. Conclusions

The data on λ_las (see Table 2) were obtained a priori, no fittings were used. The error could be several units in the fourth significant digit. The precision wavelengths of laser transitions are necessary, in particular, to determine ions in which intense laser emission is possible at wavelengths for which multilayer mirrors (MM) with high reflectance are developed. At least three values of λ_las are of interest from the viewpoint of the development of XRL-based sources for nanolithography.

(i)

For Z = 50, λ_las ~ 134.1 Å. For this wavelength region, MMs for nanolithography were developed as early as in 1993 [26]. The maximum normal incidence reflectivity achieved that time was 66% for a Mo/Si MM at λ = 13.4 nm, the reflectivity can be increased to 70%.

(ii)

For Z = 54 λ_las ~ 11.3 nm. A series of normal-incidence reflectance measurements at just longer than the beryllium K-edge (11.1 nm) from Mo/Be MM was reported in [27]. The highest peak reflectance was 68.7 ± 0.2% at =11.3 nm obtained from a MM with 70 bilayers ending in beryllium.

Our model of the high efficient monochromatic radiation sources near λ = 13.5 at 11.3 nm obtained in Xe²⁶⁺, intended for commercial nanolithography, was presented in our recent work [28].

(iii)

In Ni-like Ytterbium (Z = 70), λ_las ≈ 67.25 Å. MM for wavelengths of 6.71–6.89 nm were developed in [29]. Summary of measured and calculated reflectivity of La/B MM for these wavelengths is listed in Table 1 of [29]. The largest reflectivity was observed and calculated for λ = 6.71–6.74 nm.

The crossing region of each working level with another level is characterized by their strong effect on each other, which can cause strong instability of the energy structure in the crossing region. In such regions, jumps in functions of energy levels and probabilities of radiative transition on Z are possible (see Figure 2a). The authors of [30], where the level crossing in the Ni-like sequence and associated irregularities in the functions of energies and probabilities of radiative transitions in the range Z = 74–84 were studied, arrived at the same conclusion. From this, the conclusion regarding the possible incorrect identification of levels in their crossing regions follows.

The SPP XRL can be very sensitive to external fields. It is implied that even an insignificant change in the plasma density can affect the emission spectrum. The remarkable phenomenon (see Figure 4) where a rapid increase in the contribution of the 3d_5/24d_3/2 [J = 1] level to the composition of the lower working level is demonstrated could be an indirect confirmation of this. In the interval Z = 40–42, the contribution of this level increases by five orders of magnitude. A similar pattern is observed in Figure 6, where the contribution of 3p_3/24s_1/2 [J = 1] also rapidly increases to Z = 49, where this level strongly interacts with the upper working level. In this case, the oscillator strength is transferred from the upper working level to the 3p_3/24s_1/2 [J = 1] level.