Electric field ionization (EFI) processes in simpler atoms have been investigated extensively for the last two decades either experimentally^{[1– 3]} or theoretically.^{[1, 4]} The full-scale investigations of the EFI of Rydberg states for both alkali-metal atoms, ^{[4– 6]} such as Li and Na, and alkaline-earth atoms, ^{[7, 8]} e.g., Ba and Sr, have been carried out. The same is not true for rare-earth atoms, however, since the existing studies have almost all been concentrated on Yb atom, on either the EFI threshold or the dynamics of the Stark states.^{[9, 10]}

All rare-earth atoms, except for Yb, have a partially filled 4f subshell, resulting in the complicated spectra of highly excited states due to the very complex structures of the ground state.^{[11– 13]} In recent years, the spectral studies of highly excited states of the Eu atom, a typical complex rare-earth atom with electronic configuration of [Xe]4f⁷6s², in zero field have been reported, ^{[14, 15]} not only on bound states, ^{[16, 17]} but also on some auto-ionizing states.^{[18, 19]}

Due to the high selectivity and efficiency, the EFI method has been used for detection of highly excited states, ^{[20, 21]} and determination of the first ionization limit^[22] and the lifetimes of the Eu atom.^[23] Obviously, the core structure of the Eu atom ultimately dominates the complexity of its properties, leading to the fact that the EFI process of the Eu atom has been hardly touched, especially the behaviors when the intensity of electric field exceeds ionization threshold. Thus, it is more challenging to extend the above studies to its EFI process, which is the goal of the present study. More explicitly, the EFI process of the Eu 4f⁷6snp Rydberg states is systematically investigated, including the exact EFI ionization thresholds, and the scaling law of the EFI threshold with the effective quantum number for the Eu 4f⁷6snp Rydberg states.

In this section, both experimental setup and approach, as well as some related formulas, will be described.

The experimental setup, as shown in Fig. 1, includes three parts: a laser system, an atomic beam system, and a data acquisition system. The laser system includes three tunable dye lasers pumped by an Nd:YAG laser at wavelength of 532 nm. Each of three dye lasers has a pulse width of 5– 8 ns, a line width of 0.5 cm^{− 1}, and a repetition rate of 20 Hz. In order to ensure the given excitation scheme, the pulses of the second and third lasers are delayed by 8 ns and 16 ns relative to the pulse of the first laser, respectively. The polarization of three lasers is perpendicular to the direction of the electric field.

Fig. 1.

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	Fig. 1. Schematic diagram of the experimental setup. Here, P is a linear polarizer.

An atomic beam produced from a heated oven at 750 K ejects into the interacting region between two parallel electric field plates, 1 cm apart, on which an appropriate voltage is applied. The collimated atomic beam intersects with the laser beams to decrease the Doppler broadening effect.

The data acquisition system includes a micro-channel plate (MCP) detector, a digital pulse generator, and a pulsed HV power supply with the rise time of 100 ns and the tunable pulse width from 0.2 μ s to 2.5 μ s. A digital oscilloscope and a high-speed digitizer are connected with a computer. The excited atom is ionized by the pulsed electric field, 0.5 μ s after the laser pulses, from which the ion is extracted by the same electric field to the MCP detector, whose signal is processed by the high-speed digitizer, and downloaded to a computer for further analysis.

The excitation scheme, as shown in Fig. 2, is carried out with the three dye lasers. The first two lasers sequentially excite the Eu atom from the 4f⁷6s² ground state to the 4f⁷6s7s state through the 4f⁷6s6p ¹⁰P_9/2 intermediate state, where their wavelengths are fixed at 686.44 nm and 668.70 nm, respectively. The wavelength of the third laser is scanned within the range of 617– 636 nm so that many Rydberg states can be populated, which can be detected by the EFI technique.

Fig. 2.

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	Fig. 2. The diagram of the experimental scheme.

The EFI method is well known as an effective method for detecting higher members of Rydberg states. If the strength of the electric field exceeds a critical value for a given Rydberg atom, the ionization efficiency can be up to 100%. Namely, the classical EFI threshold F for a hydrogenic atom, ^[10] in the atomic units, is

where n* is the effective quantum number, which can be obtained from the Rydberg formula

where R = 109736.918 cm^{− 1} is the mass-corrected Rydberg constant for the Eu atom; E_n is the level energy obtained from experiment; and I₀ is the ionization limit of the Eu atom.

As mentioned in the previous section, the main concern of the present study is to observe the EFI process of Eu atom. In order to carry out such an investigation, the property of the pulsed power supply has to be controlled to fulfill the EFI method. For instance, the variation of the EFI signal with the electric field strength can be obtained with the computerized power supply.

It is well known that the EFI process can be either diabatic or adiabatic, mainly depending on the rise time of the pulsed power supply used in the experiment. We have studied two different cases when atoms are field ionized: (i) field with almost constant strength, (ii) field strength is rapidly decreased.

Based on the parameters of our pulsed power supply, the rise time of 100 ns and the width varied from 0.2 μ s to 2.5 μ s, it enables us to observe the EFI process within the frame of the adiabatic process, while the diabatic process, requiring a slow varied and linearly rising field ramp, is not the subject of the present study.

It is worth noting that the rise time of pulsed power supply is kept at the same value of 100 ns, which enables us to study the adiabatic EFI process with two different pulse widths, 0.2 μ s and 2.5 μ s. The differences between the two cases mentioned above will be demonstrated and discussed in the next section.

The experiment starts with the spectra of the Eu 4f⁷6snp Rydberg states, acquired by the combination of multi-step resonance excitation and the EFI technique, to obtain the exact value of ionization threshold of Eu, as shown in Fig. 3. An overview of the 4f⁷6snp Rydberg series with n = 26 to 72 and a close-up of the highest members are shown in Figs. 3(a) and 3(b), respectively.

Fig. 3.

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	Fig. 3. The EFI spectra of the Eu 4f76s(9S)np Rydberg series at F = 2 kV· cm− 1. (a) An overview, and (b) the close-up. The dot line in panel (a) denotes the position of the first ionization limit.

A narrow peak, marked as A, is above the first ionization limit, while a wide profile, marked as B, is across the limit, as shown in Fig. 3(a). Obviously, both of them have a much larger width than those of bound states, which is partially due to the profound configuration interaction with the related 4f⁷6s (⁹S)ɛ p continuum states, whose detailed physical mechanism was explained in the related literature.^{[18, 19]}

With the aid of the close-up of highest members shown in Fig. 3(b) and the help of the relevant literature, ^[20] the Eu 4f⁷6s(⁹S)np ⁸P_J Rydberg states with n = 26 to 72 are identified, based on the fact that energy corresponding to each peak is the sum of the three-step laser energy.

Turning to the observation of the EFI spectrum when the strength of electric field is continuously adjusted, we are able to examine the differences between the two cases corresponding to pulse width fixed at 0.2 μ s and 2.5 μ s, respectively. Taking the 4f⁷6s66p state at E = 45705.74 cm^{− 1} as an example, shown in Fig. 4, the EFI signal varies with the strength of electric field. The EFI threshold F₀, corresponding to an abrupt increase in the ion current indicated by an arrow in the figure, is defined by the maximum slope of EFI signal, F₁ is defined as the EFI signal suddenly drops at a specific field strength and indicated by a dotted arrow in the figure.

Namely, for the given state, 4f⁷6s66p, the EFI threshold is F₀ = 32V· cm^{− 1} and 35 V· cm^{− 1} in the two cases corresponding to pulse width of 2.5 μ s and 0.2 μ s, respectively.

Fig. 4.

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	Fig. 4. The EFI process of 4f76s66p 8PJ state. The pulse width is fixed at (a) 2.5 μ s and (b) 0.2 μ s, respectively.

Now, let us make a comparison between the two different EFI processes corresponding to the pulse width of 2.5 μ s and 0.2 μ s. First, the EFI threshold with pulse width of 0.2 μ s is higher than that with pulse width of 2.5 μ s, which is true for all Eu 4f⁷6s(⁹S)np ⁸P_J Rydberg states. Secondly, In the case of pulse width of 2.5 μ s, the EFI signal drops suddenly at a specific field strength, F₁ = 2815 V· cm^{− 1}, while the same phenomenon is not observed with the pulse width of 0.2 μ s within 3 kV· cm^{− 1}, the scale of the pulsed power supply used in experiment. This phenomenon is also true for all Eu 4f⁷6s(⁹S)np ⁸P_J Rydberg states. Thus, it is reasonable to assume that the same phenomenon may appear at the higher field strength, beyond 3 kV· cm^{− 1}. The two different cases when atoms are field ionized: one is field with almost a constant strength, the other field strength is rapidly decreased, this leads to the differences, that are the EFI threshold with pulse width of 0.2 μ s is higher than that with pulse width of 2.5 μ s, and the EFI signal drops suddenly at a specific field strength only with pulse width of 2.5 μ s.

To examine the above assumption, similar observations have also been carried out for other states in the same 4f⁷6s(⁹S)np series, such as n = 26 state, as shown in Fig. 5.

Fig. 5.

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	Fig. 5. The EFI process of 4f76s26p 8PJ state. The pulse width is fixed at (a) 2.5 μ s and (b) 0.2 μ s, respectively.

In terms of the EFI process, there is obviously a difference among the different-n states by comparing their EFI profiles. For instance, there is a flat top in the EFI profile of 26p state, while it is not the case for the 66p state. In other words, the EFI signal of 66p state reaches its maximum before dropping down, while there is no obvious maximum in the case of 26p state. To find out the reason for this, we have examined their positions in the spectrum shown in Fig. 3(a) and found that the 26p state is far away from the wide profile B, while the 66p state is superimposed on it. We have made a further step by checking all the EFI profiles of other np states, and have confirmed the above observation. This fact indicates that the wide profile B in Fig. 3(a) plays an important role on the EFI profile of the 4f⁷6s(⁹S)np states. One may attribute this fact to their interaction with the related 4f⁷6s(⁹S)ɛ p continuum states and the adjacent states.

Another observation has been made by examining the width of the EFI profiles of different np states, defined by the difference between the two thresholds, F₁– F₀, indicated by two arrows in Figs. 4(a) and 5(a). It is found that the more difference in the n values, the more in this width, which may also be seen from Table 1. In the utmost case, taking the 26p and 66p states as a pair, the EFI profile of the 66p state is much broader than that of the 26p state. In other words, the difference, F₁– F₀, is much larger for the higher states, due to the fact that F₀ is much more sensitive to the n-value than the F₁ seems to be.

Table 1.

Table 1.

Table 1. The EFI properties of the Eu 4f76snp Rydberg states, converging to the first ionization limit, 4f76s 9S4. The pulse width is fixed at 2.5 μ s. n E/cm− 1 n* F0/V· cm− 1 F1/V· cm− 1 26 45511.29 22.14 1806 2812 27 45529.97 23.14 1532 2702 27 45534.05 23.37 1527 2707 27 45535.21 23.44 1525 2710 28 45541.83 23.84 1498 2708 28 45546.17 24.11 1308 2698 28 45547.22 24.18 1305 2702 28 45549.23 24.31 1291 2707 29 45559.18 24.99 1158 2714 29 45561.35 25.15 1107 2722 29 45562.92 25.26 1091 2712 30 45570.85 25.86 967 2706 30 45574.18 26.13 939 2708 31 45582.36 26.82 936 2707 31 45585.69 27.12 837 2702 31 45586.82 27.22 827 2714 32 45592.61 27.77 808 2714 32 45596.15 28.18 707 2714 33 45601.82 28.72 696 2697 33 45605.05 29.07 617 2700 34 45614.72 30.22 532 2689 35 45621.31 31.08 470 2695 35 45622.91 31.29 452 2685 36 45629.19 32.22 400 2765 37 45635.17 33.17 361 2702 38 45640.13 34.03 321 2705 39 45645.91 35.12 283 2705 40 45651.05 36.18 251 2815 42 45659.38 38.12 207 2817 44 45666.81 40.15 185 2800 45 45670.04 41.13 158 2819 46 45672.98 42.09 140 2801 50 45683.34 46.13 99 2819 52 45686.61 47.67 84 2818 53 45688.53 48.65 82 2803 54 45690.33 49.62 74 2833 55 45691.97 50.56 69 2800 56 45693.69 51.59 64 2808 57 45695.18 52.56 57.5 2818 58 45696.75 53.64 57.3 2805 59 45697.93 54.48 48 2807 60 45697.34 55.55 44 2810 61 45700.25 56.28 41 2808 62 45701.49 57.31 38 2813 63 45702.62 58.31 36 2813 65 45704.71 60.28 34 2829 66 45705.74 61.34 32 2815 67 45706.71 62.39 28.5 2817

Table 1. The EFI properties of the Eu 4f76snp Rydberg states, converging to the first ionization limit, 4f76s 9S4. The pulse width is fixed at 2.5 μ s.

Now, let us discuss the possible effect from the blackbody radiation (BBR) due to the temperature of the atomic beam.^[24] As shown in Fig. 6, there are some peaks superimposed on the EFI profiles.

Fig. 6.

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	Fig. 6. The EFI process of the 4f76s40p 8PJ state. The pulse width is fixed at 2.5 μ s.

The BBR, originated from the operating temperature at 800 K, may induce some transitions between the nearby Rydberg states. Based on the detailed analysis of the EFI spectrum of the 40p state, shown in Fig. 6, one may observe the correspondence between the peaks in the EFI spectrum and the EFI thresholds of 6snp Rydberg states nearby with the lower n values.

For instance, the peaks at 463 V· cm^{− 1}, 791 V· cm^{− 1}, and 1502 V· cm^{− 1} on the EFI profile of the 40p state are corresponding to the lower 6snp states with n = 35, 32, and 27, respectively. The above peaks^[25] are verified by the results listed in Table 1 within the error bars, which are measured individually with the EFI detection technique. The agreement between the two independent measurements mentioned above in turn interprets the mechanism of the BBR. Namely, the structures above the EFI threshold on the EFI profiles shown in Fig. 4 or Fig. 6 are indeed due to BBR effect, driving the atom from its initial state to some nearby lower states.

It is worth mentioning that the structures above the EFI threshold of 4f⁷6snp Rydberg series lead to the profile being less smooth than what it should be. There is an additional effect in the field ionization process experiment, especially for those states with the higher principal quantum number. In addition, the mutational site of the field ionization signal F₁ should be concerned with self structure of the Eu atom and is affected by the ionization efficiency.

More observations on the EFI threshold of Eu 4f⁷6snp ⁸P_J Rydberg states are systematically made, ^[20] as shown in Table 1. Apart from the level energies for many states with their n and n* values, both the EFI threshold F₀ and the F₁, where the EFI signal drops down, are listed. As seen from the table, the variation of F₁ for different-n states, about 2800 V· cm^{− 1}, is much smaller than those of F₀.

As shown in Table 1, it is demonstrated that for the Eu 4f⁷6snp ⁸P_J Rydberg states, the higher the state is, the easier it ionized, and the EFI threshold increases with the decrease of n* , as expected. The F₀ and F₁ have an uncertainty of 1 V· cm^{− 1}. Ionization efficiency may induce the EFI signal decline about 2800 V· cm^{− 1}. Different J values converging to the same n of the Rydberg states can be distinguished in the lower energy levels of Fig. 3(a), ^[26] so we can see one n member corresponding to multiple EFI threshold F₀ in Table 1, however, the J value of higher n-value cannot be distinguished because of the line width of lasers, so higher n-value of the Rydberg states, one-to-one correspondence in Table 1.

In order to obtain the physical insight into the interior relations between the EFI threshold and the n* , the effective quantum number, it is helpful to plot a diagram that shows the EFI thresholds versus n* , for the Eu 4f⁷6snp (⁸P_J) Rydberg states with n = 26– 67, as shown in Fig. 7.

Fig. 7.

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	Fig. 7. The EFI thresholds versus n* for the Eu 4f76snp (8PJ) Rydberg states.

In Fig. 7, the circles represent the experimental data of the EFI threshold, while the solid line is the curve by fitting the data to scale the n* values. The best fit yields the scaling law for the EFI thresholds of the Eu 4f⁷6snp (⁸P_J) Rydberg states, namely

The obvious difference from Eq. (1) indicates that the Eu atom is by no means a hydrogenic atom. Actually, it is less difficult to ionize the Eu atom than to do so for a counterpart of hydrogenic atom, reflecting the impact of quantum defects.

The electric field ionization process of the Eu 4f⁷6snp Rydberg states has been investigated with the combination of multi-step laser excitation and continuously scanning the strength of the pulsed electric field. Not only the ionization thresholds for the 42 4f⁷6snp Rydberg states are measured systematically with high precision, but also their exact turning point, where the EFI signal drops, is observed for the first time. Many characteristics of the EFI process and possible impact from the black body radiation are discussed either qualitatively or quantitatively. Apart from the observation on field ionization process of the Eu 4f⁷ 6snp Rydberg states, the scaling law of their ionization thresholds of the Eu atom with the n* is also obtained for the first time.