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Abstract
The dynamic polarizabilities of Al atom at 19 wavelengths from 420 nm to 680 nm are measured experimentally for the first time. A 15 µm diameter Al wire is heated to the gas/ microdrop stage using a 3 kA, 25 ns pulsed current, and the energy deposition when the resistive voltage reaches its peak is 4.7 eV·atom−1, which is higher than the atomization enthalpy of Al but much lower than its first ionization potential. Two laser interferometric images of the Al gas are obtained simultaneously using a 532 nm laser and an optical parametric oscillator laser. Then an integrated phase method is used to reconstruct the Al atomic linear density distribution based on the known atomic polarizability at 532 nm, which acts as a bridge between the two interferograms to obtain the relative atomic polarizability value at a specific wavelength with respect to the value at 532 nm. The measured dynamic polarizability of Al atom decreases from 13.5×10−24 cm3 to 9.4×10−24 cm3 as the wavelength increases from 420 nm to 680 nm with a measurement error of approximately ±10%. The experimental result fits well with an uncoupled Hartree-Fock approximation, and the reconstructed static polarizability of 8.13 ± 0.79×10−24 cm3 is well matched with reference measurement results.
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1. Introduction
The dynamic polarizability of an atom, which describes the response of the atom’s electron cloud to an external electromagnetic disturbance that varies over time, is an important spectroscopic characteristic of atoms. The dynamic polarizability is required to estimate induced dipole moments, oscillator strengths, the Stark and Kerr effects, and van der Waals constants. The availability of experimental data for the dynamic and static polarizabilities of the metal atoms is very limited and these values are mainly obtained via calculations.
Theoretically, the atomic polarizability is well defined in the sense that it is a measure of the additional dipole moment induced in an atom or molecule by the external field. Calculations of the polarizabilities have been performed using a variety of techniques, including the perturbation, variation-perturbation, direct ab initio computation, and semiempirical methods. Under relatively weak fields, the response of an atom or molecular system to an external perturbation is primarily linear and is described well by perturbation theory. Calculations of the second-order properties, e.g., the polarizability, require formal knowledge of an infinite number of excited states (including the continuum) and therefore present formidable theoretical difficulties [1]. These problems can be solved using the variation-perturbation technique [2,3] and local density theory [4]. The most straightforward technique for polarizability calculation is direct ab initio computation. However, at present, the area of application of this technique is limited to small molecules and single atoms, among which the hydrogen molecule is the best-known example [5]. For conjugated hydrocarbons, calculations of the excited state polarizabilities have been performed based on semi-empirical methods, but the agreement of these theoretical results with the experimental data is only qualitative [6].
Measurement of the atomic dynamic polarizability is difficult in that it requires generation of an external electromagnetic field around a gas cell. The dielectric constant is measured using low-frequency fields, and the measured value is then used to predict the atomic polarizability for the noble gas atoms. Other bulk diagnostic techniques such as refractive index, Rayleigh scattering, and Kerr effect measurements have also been performed [7]. Beam techniques have notably been applied to the measurement of the alkali metal atoms since 1934 [8]. Use of optical diagnostic techniques in vapor cells or in shock tubes became an effective method following the development of laser technology. The interferometry method in electrical exploding wires is one of the optical approaches that can be used to measure the atomic dynamic polarizability and is based on heating of metal wires using a fast pulse current. Sarkisov proposed an integrated phase method for polarizability measurements [9]. The measured polarizabilities of metal atoms can be obtained at 532 nm and 1064 nm using this method [9,10]. The current rise rate has been shown to affect the energy deposition of the exploding wire significantly [11,12]. Sarkisov et al. also measured the dynamic polarizabilities of Al, Mg, Ag, Au, Cu, and Ti at both 532 nm and 1064 nm using their integrated phase technique for exploding wires in vacuum [9,13,14]. Further works show that the geometry of electrodes can affect sign and value of radial electrical field, increase the current rise rate, and enhance the deposited energy before voltage collapse [15,16]. However, the experimentally measured atomic polarizability data at other wavelengths are absent, because lasers that operate at wavelengths other than 532 nm or 1064 nm are not commonly used in electrical exploding wire experiments.
In this manuscript, an Al wire with a length of 5-10 mm and a diameter of 15 µm is driven by a 3 kA, 25 ns pulsed current. The energy deposition of the exploding wire is estimated to be close to the atomization enthalpy. The atomic dynamic polarizabilities are measured at 19 wavelengths from 420 nm to 680 nm using the laser interferometry method. Two Mach-Zehnder interferometers are used to measure the phase shift distribution in our experiments. An integrated phase method is then used to reconstruct the atomic linear density distribution along the wire axis based on the known atomic polarizability at 532 nm, which acts as a bridge between the two interferograms to obtain the relative atomic polarizability value at a specific wavelength with respect to the atomic polarizability at 532 nm. The measured atomic dynamic polarizabilities are consistent with the values obtained from the uncoupled Hartree-Fock approximation.
2. Experimental description
The load used in this experiment is a 15 µm Al wire with four current return posts around it. The diameter of the Al wire (AL000271, GoodFellow), as measured using a transmission electron microscope (TEM; VE9800S, Keyence) is 15 ± 0.7 µm.
A schematic of the experimental optical setup is shown in Fig. 1(a). The exploded wire is diagnosed using two laser probe beams, which are generated using an 8 ns Nd:YAG laser (SGR) [17,18] and a 7 ns optical parametric oscillator (OPO) laser (VIS, MagicPRISM) [19]. The OPO, which consists of two nonlinear optical crystals placed within an optical cavity, converts a 355 nm pump laser (200 mJ, 7 ns, Gaussian beam) into two beams at longer wavelengths, designated the signal (420–680 nm) and the idler (740–1200 nm). The wavelength adjustment step size is 1 nm, and the specific wavelength value is calibrated using an echelle spectrometer (Aryelle Butterfly) in our experiments. The spectral width at the wavelength of 450 nm is 0.2 nm, and the spectral width at the wavelength of 580 nm is 0.6 nm. The signal laser is used and is referred the OPO laser. Interference of the probe beams is realized using a Mach-Zehnder interferometer [20]. To obtain the interference images, the interference optical path must meet the requirements of the two laser beams for both temporal and spatial coherence. The optical path difference is required to be less than 30 mm for the Nd:YAG laser and to be in the 0.2–2 mm range for the OPO laser, depending on the length of the wave train. Images with shifted fringes are captured using two charge-coupled device (CCD) cameras (600D, Canon). The probing times of the two beams are determined using two photodiodes (DET10A, Thorlabs). When the optical path difference and the cable difference are taken into consideration, the time between the arrivals of the two laser pulses at the exploding wire does not exceed 2 ns.
Fig. 1. (a) Schematic of the experimental optical setup. The blue line and the red line represent the 355 nm pump laser and the OPO laser (beam for polarizability reconstruction), respectively. The green line represents the 532 nm laser (beam for polarizability reconstruction reference). (b) Vacuum chamber and circuit diagram.
The wires are heated in a vacuum chamber with pressure below 0.02 Pa using a pulsed current, with the pulse being sharpened using a self-breakdown switch in air at 1.2 bar, as shown in Fig. 1(b) [11,21]. The load current is measured using a Rogowski coil and a B-dot probe located upstream of the load. A D-dot probe is fixed on the wall of the ground return cylinder, and is separated from the cathode by an insulator. A disk of copper is used as an apron to block the transmission of radiation from the wire load to the D-dot. The waveforms are recorded using an oscilloscope (WaveSurfer 510, Teledyne LeCroy, 1 GHz). The inductance of the region downstream of the D-dot probe with a 1-mm-diameter copper wire acting as the short load is 20 nH, as measured from the current and voltage waveforms.
The joule heating power and the deposited energy can be calculated from the current/voltage waveforms, as shown in Fig. 2. The resistive voltage UR is obtained by subtracting the inductive component (LloaddI/dt) from the load voltage U, which is obtained by integrating the D-dot signal. The deposited energy E is estimated by integrating the Joule heating power P (IUR). The resistive voltage UR is almost zero up to 5 ns because the resistance of the cold Al wire is very low. As the deposition energy increases, the wire temperature increases in tandem, and the wire resistivity then rises. Greater increases in the resistance voltage lead to higher deposition energy under the positive feedback condition, until the voltage collapses [22]. The joule energy deposition that occurs when the resistive voltage rises to its peak is 4.7 eV·atom−1, which is higher than its atomization enthalpy, and the total deposited joule energy reaches 7 eV·atom−1 when the resistive voltage falls to zero. The Al atoms in our experiments are in their ground states because the deposited energy is much lower than its first ionization potential.
Fig. 2. Current I, voltage U, resistive voltage UR of an exploding Al wire with a length of 5 mm and a diameter of 15 µm.
The radii obtained from the 2D laser interferograms from other shots are plotted as the scattered points in Fig. 3. The wire starts to expand since the voltage collapses. The average expansion velocity is ∼6.3 km/s from voltage collapse time. The expansion velocity shows a positive correlation with the vaporization coefficient, which has been studied under a pulse current of 0.12 kA/ns [23]. The wire is supposed to expand in the gas/microdrop stage, according to the reconstructed linear atomic mass obtained in our experiment.
3. Measurement of dynamic polarizability
Because the Al wire is not vaporized completely during our experiment, an integrated phase method is used to reconstruct the linear density distribution along the wire axis, which then acts as a bridge between the two laser interferograms. Then the atomic polarizability value at a specified wavelength related to the known atomic polarizability at the wavelength of 532 nm can then be obtained.
Interferograms of the exploding Al wire measured at 532 nm and 433 nm from shot 150 are shown in Fig. 4. The fringe patterns are both continuous and distinct, and the fringe shift direction corresponds to the neutral atom refraction. In the interpretation of the interferograms, the refraction of the laser beam that occurs when crossing the aluminum gas column is ignored [9,24,25].
Fig. 4. (a) Interferograms for the exploding Al wire at 532 nm and 433 nm from shot 150, acquired at ∼115 ns after the current started. (b) Fringe shifts of the interferograms of the two laser beams. (c) Areal densities of Al reconstructed from the interferograms. (d) Linear density Nl of gas phase Al vs. the height of the z axis, as reconstructed using the 532 nm interferogram. The inferred atomic dynamic polarizability at 433 nm is determined from Nl.
Using the integrated phase method, the areal density distribution of the exploding products is related to the interferometric fringe patterns. The phase shift $ \Delta \varphi $ is given by:
The linear number density Nl of the expanding column is obtained by radial integration of the areal number density as ${N_l}(z )= \smallint {N_s}({y,z} )dy$. Here, y is the direction perpendicular to both the probe laser propagation direction and the wire axis, and z is the direction along the length of the wire.
When the atomic polarizability at the wavelength of 532 nm is α532 = 10.8×10−24±1×10−24 cm3 [9], the atomic linear density Nl(z) along the length of the wire is calculated as shown in Fig. 4(d). Note that the reconstructed Nl changes from 10.4×1016 cm−1 to 11.1×1016 cm−1, where the relative deviation from the standard linear density of the Al wire with the diameter of 15 µm is ±5%. Using the reconstructed atomic linear density curve at 532 nm, the atomic dynamic polarizability α433 at 433 nm can be derived. The average value of the atomic dynamic polarizability curve is regarded as the measured value α433 =13.9×10−24 cm3.
The main measurement error source in a single shot is the manually traced fringe-shift FS. The error in manual tracing of the FS is introduced because the line drawing cannot be located exactly at the center of the dark fringes or bright fringes, and this error is estimated to be 0.1 line for both the 532 nm laser and the OPO laser as shown in Fig. 4(b). The measured deviation in the angle between the laser beams and the wire axis is ∼3 mrad, which leads to an atomic linear density error of 0.001%. The errors from the optical components, such as the lens distortion, are ignored. So the total measurement error in a single shot is estimated to be approximately ±10% as shown in Fig. 4(d).
The atomic polarizabilities at the different wavelengths are measured using the method discussed above. These measurements are performed 70–200 ns after the current starts. To verify the reproducibility of these experiments, the results from three experiments performed with the OPO laser at a wavelength of 430 nm are shown in Fig. 5. In these three shots, and the corresponding times when the laser beam arrives at the wire are 90 ns, 174 ns, and 197 ns, respectively. The inferred atomic polarizabilities of the three shots are 13.4×10−24 cm3, 14.3×10−24 cm3, and 14.0×10−24 cm3, respectively. The uncertainty for the reproducibility tests is thus estimated to be ±5%.
Fig. 5. Linear number density Nl characteristics reconstructed using the 532 nm interferogram for three shots at different instants, and the corresponding atomic dynamic polarizability characteristics at 430 nm calculated from Nl.
4. Theoretical predictions and comparison with experimental results
The experimental data in the work is analyzed using a geometric approximation formula, which is obtained using an uncoupled Hartree-Fock approximation with a first-order perturbation correction [26,27]:
The experimental results at 19 wavelengths are indicated by the scatters shown in Fig. 6, and the error bars are obtained from the error analysis discussed above. The atomic dynamic polarizability decreases from 13.5×10−24 cm3 to 9.4×10−24 cm3 as the wavelength increases from 420 nm to 680 nm. The atomic dynamic polarizability line is fitted from Eq. (4) using the scatter points without error bars and the least squares method. In this way, the values and its confidence interval are α0 = 8.13 ± 0.79×10−24 cm3 and Er = 4.7 ± 0.1 eV. Our atomic static polarizability α0 matches the recommended value from the latest published review with a difference of only 5% [28]. Additionally, α0 matches another reported measurement result with a difference of only 0.6% [9]. When the wavelength of 1064 nm is substituted into this fitting result, the value of α(1064 nm) = 8.47×10−24 cm3 agrees with the experimental results [9]. Therefore, it could be inferred that the fitting curve for the atomic dynamic polarizability can be extended to 1064 nm and to even longer wavelengths.
Fig. 6. Atomic dynamic polarizability vs. wavelength characteristics. The scattered points with the error bars are the results from all shots.
5. Conclusions
The atomic dynamic polarizabilities of the Al atom at 19 wavelengths from 420 nm to 680 nm are measured in this work. When a fast-rising current (3 kA, 25 ns) is used, the energy deposition in a 15µm diameter Al wire when the resistive voltage reaches its peak is 4.7 eV·atom−1, which is higher than its atomization enthalpy but much lower than the first ionization potential. The average expansion velocity of the gas/microdrop column after joule heating is ∼6.3 km/s.
Because the Al wire is not vaporized completely in our experiment, the measurement is performed using two Mach-Zehnder interferometers. A 532 nm probe beam is introduced to reconstruct the linear density characteristics of the expanding gas column based on the fringe patterns on the interferograms. Another OPO probing beam with a specific wavelength within the 420–680 nm range is used to diagnose the column at the same instant. The atomic polarizability at the specified wavelength λ is reconstructed based on the linear density of vaporized part of the exploding wire obtained from the 532 nm laser interferogram using an integrated phase method.
The measured atomic dynamic polarizability of the Al atom decreases from 13.5×10−24 cm3 to 9.4×10−24 cm3 as the wavelength increases from 420 nm to 680 nm, with an error of approximately ±10%. The experimental results are fitted well to an uncoupled Hartree-Fock approximation formula, and the fitting curve for the dynamic polarizability can be extended to 1064 nm. The reconstructed static polarizability of 8.13 ± 0.79×10−24 cm3 matches the reference measurement results well, with a difference of only 0.6%.
Funding
National Natural Science Foundation of China (51790523, 51807155, 51922087); Natural Science Basic Research Program of Shaanxi Province (2021JCW-02).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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