A Critical Compilation of Energy Levels, Spectral Lines, and Transition Probabilities of Singly Ionized Silver, Ag II

All available experimental measurements of the spectrum of the Ag+ ion are critically reviewed. Systematic shifts are removed from the measured wavelengths. The compiled list of critically evaluated wavelengths is used to derive a comprehensive list of energy levels with well-defined uncertainties. Eigenvector compositions and level designations are found in two alternate coupling schemes. Some of the older work is found to be incorrect. A revised value of the ionization energy, 173283(7) cm−1, equivalent to 21.4844(8) eV, is derived from the new energy levels. A set of critically evaluated transition probabilities is given.


Introduction
Ag II, like the isoelectronic neutral palladium, has the ground term 4d 10 1 S 0 . Promotion of one electron from the 4d shell produces a relatively simple system of level series 4d 9 nl. Promotion of two electrons results in complex configurations 4d 8 nln′l′ with three open shells. Two of them, 4d 8 5s 2 and 4d 8 5s5p, appear below the ionization limit and contribute to the complexity of the observed spectrum.
The most complete previous analysis of the Ag II spectrum was made by Kalus et al. [1]. These authors measured the spectrum emitted by a pulsed hollow-cathode discharge in the region 940 Å to 8500 Å. A good review of the previous work can be found in that paper.
Kalus et al. [1] noted that their line list does not include many of the lines reported by earlier observers. In particular, they mentioned that Gilbert [2] was able to observe many more combinations between highly excited levels by using a condensed hollow-cathode discharge. Rasmussen [3,4] reported 40 lines originating from the 4d 9 6p, 4d 8 5s 2 , and 4d 8 5s5p configurations that were not observed by Kalus et al. Benschop et al. [5] listed 65 lines ascribed to Ag II, 43 of which they classified as transitions involving the highly excited 4d 9 8s and 4d 9 9s configurations. They used two different types of spark discharges as light sources, a sliding spark and a three-electrode vacuum spark. Kalus et al. [1] suggested the use of these spark sources as the reason for the appearance of lines from highly excited levels in the spectra observed by Benschop et al. [5]. However, other authors who used spark discharges [6,7] have identified lines from only moderately excited configurations (4d 9 5s, 4d 9 6s, 4d 9 5p, 4d 9 5d). The latter two works contain 32 lines not observed by Kalus et al. 1400 9052.697 (18) 11043.401 4d 9 3/2 6s 2 [ Variations of sensitivity of registration equipment with wavelength have been removed from intensity values (see text).Where several observations were available, they have been averaged. b Observed and Ritz wavelengths below 2000 Å are given in vacuum, above that in standard air. Conversion from vacuum to air was made using the five-parameter formula of Peck and Reeder [12]. Uncertainties at a confidence level of one standard deviation are given in parentheses after the value in units of the last decimal place of the value. c References to observed wavelengths and/or line identifications: B30 -Blair [7]; F95 -Ferrero et al. [32]; K02c -Kalus et al. [1] with a small calibration correction (see text); G35c, R40ac, R40bc, S28c -Gilbert [2], Rasmussen [3,4], and Shenstone [6], respectively, recalibrated using internal wave number standards based on FTS measurements of Kalus et al. [1]. d Accuracy codes for A-values are explained in Table 3. e References to A-values: B97 -Biémont et al. [11]; C05 -Campos et al. [30]; F95c -Ferrero et al. [32], renormalized using experimental lifetimes [11]; F95c1 -Ferrero et al. [32], recalculated from observed intensities using the Boltzmann equation, Eq. (4); TW -present work. f Notes: Las -Lasing was observed on these lines in hollow-cathode discharges [21,22]; P -predicted (not observed); S -this line alone determines one of the levels involved in the transition. http://dx.doi.org/10.6028/jres.118.009

Observations of Kalus et al. [1]
As mentioned in the Introduction, the light source used in Kalus et al. [1] was a pulsed hollow cathode discharge. Argon, neon, and helium mixtures were used as carrier gases. Neon and helium were used as it was found by many authors that collisional transfer of charge and energy from He + and Ne + greatly enhances lines from highly excited levels of Ag + . Two different measurement techniques were used in two wavelength regions. Measurements of 97 lines in the vacuum ultraviolet (VUV) region below 1800 Å were made on photographic recordings of spectra obtained with a 10.7 m vacuum grating spectrograph. Above 1800 Å, the spectrum was recorded with a Fourier transform spectrometer (FTS); 216 lines were measured in this spectral region.
The wavelength scale of the FTS measurements was calibrated by means of Ar II lines measured by Norlén [13]. Uncertainties of the FTS lines, including the calibration uncertainty, ranged from 0.003 cm −1 for strong lines to 0.03 cm −1 for the weakest lines.
For the VUV spectrum, the wavelength scale was initially calibrated using internal standards consisting of Ne II lines from the carrier gas and C I, N II, and O II impurity lines, as well as external Cu II standards provided by an auxiliary Cu-Ne hollow cathode. The final calibration included a large number of Ag II Ritz wavelengths of 5s-6p lines, derived from the energy levels established from the FTS measurements. Uncertainties of the VUV lines were in the range from 0.0005 Å for strong lines to 0.002 Å for weak lines.
Nave and Sansonetti [14] have found that the wave number scale of Norlén [13] has a calibration error, and all wave numbers from his paper have to be increased by 6.7 parts in 10 8 . Since all measurements of Kalus et al. [1] were ultimately based on the scale of Norlén, I applied the above correction factor to all their wave numbers. The correction amounted to 0.0008 cm −1 to 0.0035 cm −1 for the FTS measurements and 0.004 cm −1 to 0.007 cm −1 for the VUV grating measurements. While for the VUV region the correction is well below the stated measurement uncertainties, (7-12)×10 -5 Å for wavelengths, the correction of the FTS measurements is significant, especially for short wavelengths.
To estimate the wave number uncertainties, I assumed that the measurement uncertainties δσ are entirely due to statistical and systematic uncertainties (δσ stat and δσ syst , respectively) in the wave number measurements of symmetric well-resolved features (see, for example, Kramida and Nave [15]): where W is the full width at half-maximum. The signal-to-noise ratios S/N are given in the line list of Kalus et al. [1] in the column of line intensities. The values of W and δσ syst can be estimated as 0.03 cm −1 and 0.003 cm −1 , respectively, from the statement above about the uncertainties of the FTS measurements quoted from [1]. Uncertainties of the VUV grating measurements given in Table 1 were estimated using similar considerations.
To verify the assigned uncertainties, I calculated the energy levels with the least-squares level optimization code LOPT [16] using only the uncorrected wave numbers listed by Kalus et al. [1] with the uncertainties assigned as described above. The resulting energy level values were in close agreement with those given by Kalus et al., which confirms that the assigned uncertainties are close to those used by Kalus et al. in their level optimization.
In Table 1, the observed wavelengths referred to Kalus et al. [1] were obtained from the wave numbers, corrected as described above.

Observations of Rasmussen [3,4]
Rasmussen observed the Ag II spectrum between 2530 Å and 9053 Å with a hollow cathode of pure silver in neon [3,4] and helium [4] discharges. The first paper [3] is a preliminary report of the more extensive studies reported in [4]. Rasmussen specified the uncertainty of his wavelength measurements to be about 0.01 Å. He listed the wavelengths rounded off to two places after the decimal point. However, his listed wave numbers are given with higher precision. Comparison of these wave numbers with the Ritz values based mainly on the FTS measurements of Kalus et al. [1] reveals that they have a relative statistical uncertainty Δσ/σ = 2.0×10 -6 and a systematic shift smoothly varying from +1 part in 10 6 at 18000 cm −1 to +4 parts in 10 6 at 33000 cm −1 (see Fig. 1). Fig. 1. Relative deviations of wave numbers measured by Rasmussen [4] from Ritz wave numbers based on measurements of Kalus et al. [1]. The smooth curve is a cubic polynomial that fits the data points.
I removed this systematic shift by subtracting the values represented by the smooth curve shown in Fig.  1 from the wave numbers of Rasmussen. The statistical uncertainties of the resulting wavelengths vary from ±0.005 Å at 2530 Å to ±0.018 Å at 9000 Å.

Observations of Gilbert [2]
In the work of Gilbert [2], the spectrum was excited in a pulsed hollow cathode discharge with a lowpressure helium carrier gas. Enhancement of lines originating from highly excited levels was achieved by using a spark gap in series with the hollow cathode. Wavelength measurements were made in the region from 728 Å to 2600 Å with a 1.5-meter grating vacuum spectrograph and in the region from 4000 Å to 11000 Å with a 3-prism spectrograph. Impurity lines of oxygen, nitrogen, carbon, helium, and hydrogen, as well as Ag II lines previously measured by Shenstone [6] were used as internal standards in the region below 2600 Å. The measurements were believed to be accurate to 0.05 Å in this region (1.4 cm −1 at 1900 Å to 5 cm −1 at 1000 Å). In the region above 4000 Å, the standards were furnished by an iron comparison spectrum. The measurement uncertainty was estimated as ±1 cm −1 over this region (0.16 Å at 4100 Å and 1.0 Å at 9000 Å).
Three lines listed by Gilbert with wave numbers 94429.6 cm −1 , 89847.3 cm −1 , and 21499.7 cm −1 were classified as transitions to or from the odd-parity level designated by Gilbert as 3 2 , located at 133589.4 cm −1 . Since my energy-level calculations (see Sec. 4) show that this level is not real, and since these three lines cannot be classified as transitions between any known levels, I discarded the spurious level and the lines.

Observations of Blair [7]
Blair [7] photographed the Ag II spectrum in the region 2000 Å to 3400 Å emitted by a hollow cathode. Only 20 newly classified lines were given in this paper. Most of them were remeasured by Kalus et al. [1] with much greater accuracy. However, seven lines were not reported elsewhere. To determine their uncertainties, I compared wavelengths reported by Blair with the Ritz wavelengths derived from the measurements of Kalus et al. [1]. From this comparison, it follows that the measurements of Blair [7] have a statistical uncertainty of 0.04 Å with a small systematic shift of about +0.01 Å. Since this shift is much smaller than the statistical uncertainty, and all the levels involved are precisely determined by the measurements of Kalus et al. [1], I did not remove this shift from the wavelengths reported by Blair.

The Work of Shenstone [6]
All observations reported in the work of Shenstone [6] were made with a spark source. For wavelengths above 2246 Å, Shenstone listed measurements of Exner and Haschek [17], with few exceptions where he gave his own measured values or those from Frings [18]. Most of the lines reported by Shenstone were accurately re-measured by Kalus et al. [1]. However, there are 12 lines not reported by other observers, 10 quoted from [17] and two measured by Shenstone. These lines are between 2111 Å and 3270 Å. As in the previous cases, it was possible to re-calibrate the wavelength scales of Shenstone and of Exner and Haschek by using Ritz wavelengths based on the measurements of Kalus et al. [1] as internal standards. This procedure revealed the presence of large systematic shifts, which are nearly the same in the measurements of Shenstone and of Exner and Haschek and amount to +0.04 Å at 2111 Å and +0.25 Å at 3270 Å. After removing this shift, the resulting wavelengths reported in Table 1 have statistical uncertainties ranging from 0.03 Å at 2111 Å to 0.05 Å at 3270 Å. There is one exception for the weak unresolved line at 2829.0 Å, which deviates from the (much more accurate) Ritz wavelength by 0.4 Å (5 cm −1 ). For this line, I adopted the uncertainty of 0.4 Å.

The Work of Benschop et al. [5]
Benschop et al. [5] observed the VUV spectrum of singly and multiply ionized silver in the region 500 Å to 2200 Å by using two light sources: 1) a sliding spark with a ceramic or quartz spacer, and 2) a triggered vacuum spark. Three high-resolution vacuum grating spectrographs facilitated wavelength measurements with uncertainties about or below 0.005 Å. Cu II lines excited in a copper hollow cathode lamp in a helium atmosphere were used as standards in the region above 700 Å. Between 500 Å and 700 Å, either O, C, and N lines or some silver lines measured in the higher orders were used as internal standards. The total number of observed spectral lines was over 10000. Most of these lines were found to be due to Ag III-V [19,20]. Separation of lines belonging to different ionization stages was achieved by varying the operational conditions of the discharges.
Benschop et al. [5] stated that they re-measured many of the previously known Ag II lines with greater accuracy and confirmed the previous analysis. However, they listed only 64 lines that were newly classified in their work and gave revised values of 74 energy levels. Thirty-two of the new lines were classified as transitions between levels that are now known with high accuracy from Refs. [1,[3][4]. Only one of these lines (1313.809 Å, 4d 9 5p 3 P°1 -4d 9 8s 1 D 2 ) agrees with the Ritz wavelength (1313.804(4) Å) within the combined uncertainties. The root-mean-square (rms) deviation of the measured wavelengths from the Ritz values is 0.09 Å, i.e., 18 times the claimed measurement uncertainty. The other 32 lines ascribed by Benschop et al. to the previously unknown 4d 9 9s configuration and 4d 8 5s 2 1 S 0 level have similarly large deviations of wavelengths from the Ritz values. These deviations appear to be random and have both large positive and large negative values. They do not show any regular trend that could be interpreted as a smoothly varying calibration error. The rms deviation of the energy level values obtained by Benschop et al. from the new, much more accurate values (see Table 2 and Sec. 3) is about 6 cm −1 , while the uncertainty implied by their wavelength measurements should not exceed 0.5 cm −1 . The values of 4d 9 6p 3 P°2 and 4d 9 5d 3 D 2 deviate from those of Kalus et al. by 21 cm −1 and -42 cm −1 , respectively. In addition, the observed relative intensities of the lines have no correlation with calculated radiative rates (see Sec. 6). Therefore, the entire analysis of Benschop et al. [5] must be deemed incorrect, and their results must be disregarded, including the identification of the missing 1 S 0 level of the 4d 8 5s 2 configuration, the 4d 9 9s levels, and their derivation of the ionization energy.
A special note should be made about the line at 1160.887 Å assigned to the 4d 10 1 S 0 − 4d 9 5p 3 P°2 transition [5]. Its wave number does not agree with the energy of the 4d 9 5p 3 P°2 level given by Benschop et al. [5]. The closest match is with the 4d 9 5p 3 P°0 level, which differs from the wave number of this line by 5 cm −1 . Neither classification is feasible, since all 4d 9 5p levels have allowed (electric-dipole) radiative transitions to lower even-parity levels of the 4d 9 5s configuration. Forbidden transitions from these levels must have very small branching ratios.

Other Observations
Several lines of Ag II in the range 2243 Å to 8404 Å were found by Reid et al. [21,22] to exhibit lasing in hollow cathode discharges filled with Ne or He. Eleven lasing lines were identified with known Ag II transitions listed in Refs. [1,2]. Several unidentified lines reported in other papers of the same team [23,24] were found by Reid et al. [21] to be diffraction-grating ghost lines. The identified lasing lines are marked in Table 1 with a corresponding note.
Tables of the Massachusetts Institute of Technology (MIT) [25] include wavelengths and relative intensities of about 340 lines of Ag I and Ag II in the range 2000 Å to 8274 Å. Most of the lines ascribed to Ag II in these tables have been observed by Kalus et al. [1] and Gilbert [2]. The MIT tables should be used with caution, as most Ag II wavelengths referred to MIT measurements, including those given with three places after the decimal point, appear to possess a large systematic shift on the order of 0.2 Å. Because of this shift, I did not make use of these tables.

Optimized Energy Levels
After the systematic shifts have been removed, and wavelength measurement uncertainties have been assessed, the level optimization is a straightforward procedure. I used the least-squares level optimization code LOPT [16] and included only the measurements of Kalus et al. [1], Rasmussen [3,4], Blair [7], and Shenstone [6], which makes a total of 450 lines. The level-optimization problem defined by this line list has 353 degrees of freedom (DF). The resulting ratio of the residual sum of squares (RSS) to the number of degrees of freedom RSS/DF = 0.78, which indicates that the wavelength values and their uncertainties are statistically reasonable. The optimized energy levels and their uncertainties are given in Table 2. http://dx.doi.org/10.6028/jres.118.009 As noted by Kalus et al. [1], the uncertainty of the absolute level values is determined mainly by three observed spectral lines at 1106, 1112, and 1196 Å connecting the 4d 10 1 S 0 ground level with the three J = 1 levels of 4d 9 5p ( 3 D°1, 1 P°1, and 3 P°1, respectively). They estimated this uncertainty as ±0.1 cm −1 and estimated the uncertainties of intervals between the excited levels to be in the range ±0.001 to ±0.01 cm −1 . The newly optimized energy levels given in Table 2 agree very well with those given by Kalus et al. [1]. Excitation energies of all levels are greater than those given by Kalus et al. by 0.04 cm −1 on average, which is well within the uncertainty ±0.1 cm −1 specified by Kalus et al. Separations of all excited levels from 4d 9 5p 3 P°1 are all within the range of uncertainties given by Kalus et al. The level 4d 9 5p 3 P°1 was chosen as the base for determining the relative uncertainties of excited levels, because it has the greatest number of accurately measured connecting lines. In most cases, the number of significant figures given in the energy value is determined by the value of the relative uncertainty. In some cases, an extra digit was required in the energy value in order to exactly match some of the observed wavelengths.
The only questionable level in Table 2 is 4d 9 6p 3 P°0, which is determined by a single line at 9052.70 Å tentatively classified by Rasmussen [4] as a transition from this level to 4d 9 6s 3 D 1 . Another potentially strong transition from this level to 4d 9 5s 3 D 1 is predicted to occur at 1081.9340(3) Å and could have been masked by the much stronger line at 1082.1458 Å in the spectrum observed by Kalus et al. [1].
The four 4d 9 8s levels are determined with uncertainties of about 0.2 cm −1 from lines of transitions to 4d 9 5p and 4d 9 6p measured by Gilbert [2].

Ionization Limit
The previously accepted value of the first ionization limit, 173277.4 cm −1 , was derived by Benschop et al. [5] from the 4d 9 ns (n = 5-8) 1,3 D series. They noted that their newly identified (now discarded) 4d 9 9s 1,3 D levels are "slightly perturbed" and did not include them in the derivation of the limit. Benschop et al. did not specify the uncertainty of their value. Since now the 4d 9 ns series are known much more accurately, it is possible to derive an improved limit value from the same series using similar quantumdefect expansion formulas. For that I used the computer program RITZPL by Sansonetti [27]. Each of the four four-member series was exactly fitted using the three-constant extended Ritz formula where c i are the fitted constants and δ n is the quantum defect describing an empirical correction to the principal quantum number n required for the excitation energy E n to satisfy the hydrogenic formula: where E I is the ionization energy, Z is the charge of the ionic core, and R is the mass-corrected Rydberg constant.
The mean of the four resulting values of the ionization limit is 173283 cm −1 with a standard deviation of 7 cm −1 . In this derivation, I used the known value of the 4d 9 2 D 5/2 -2 D 3/2 separation in Ag III, 4609.2 cm −1 [20].
To verify that higher-order terms omitted in Eq. (1) do not significantly influence the results, I made a similar derivation of the ionization limit for neutral palladium, where an accurate value of the limit, 67241.3(8) cm −1 was derived by Baig et al. [28] from high-member absorption series 4d 10 1 S 0 -4d 9 np (n = 10-30) and nf (n = 9-17). For comparison with Ag II, only the 4d 9 5/2 ns 2 [5/2] 3 series can be used in Pd I, because the n = 8 members are not known in the Pd I 4d 9 3/2 ns 2 [3/2] 1 and 2 [3/2] 2 series, and the 4d 9 5/2 ns 2 [5/2] 2 series is strongly perturbed. From the exact fit of the formulas (1) and (2) with the first four members (n = 5-8) of the 4d 9 5/2 ns 2 [5/2] 3 series precisely measured by Engleman et al. [8], I obtained the limit at 67241.0 cm −1 , differing from the adopted value of Baig et al. by only 0.3 cm −1 . This, as well as the small variation between the limit values obtained from the four different series, confirms the validity of the procedure for Ag II and allows me to adopt the resulting value of 173283(7) cm −1 for the Ag II ionization limit.
I fitted 43 known levels of even parity with 17 free parameters, with a standard deviation of 23 cm −1 , and 55 odd-parity levels with 18 free parameters, with a standard deviation of 63 cm −1 .
In these calculations, I found that the 4d 9 nl configurations are best described in the JK coupling scheme, in which the purity of levels varies from 79 % for 4d 9 5p to about 95 % for 4d 9 5d and all 4d 9 nl with n > 5. Therefore, the configuration and term labels, as well as percentage compositions for these configurations are given in Table 2 in this coupling scheme, providing unambiguous physically meaningful designations. For the 4d 8 5s5p configuration, LS coupling of the type 4d 8 + 5s5p was found to give a slightly better purity of levels, 56 %, compared to the coupling 4d 8 5s + 5p, 54 %. For some of the levels of this configuration, the leading percentage is as small as 25 %. In these cases, the configuration and term labels have little physical meaning and are used for bookkeeping purpose only.

Relative Intensities of Observed Lines
As noted in the Introduction, the relative intensities of lines observed with different light sources and with different registration equipment are vastly different. In order to give a consistent set of relative intensities, they must be converted to the same scale. To account for the different excitation conditions in various light sources, the observed line intensities can be approximated by local thermodynamic equilibrium (LTE) with an effective excitation temperature pertinent to each light source, and then scaled to the same excitation temperature. In reality, the LTE approximation describes the observed intensities only qualitatively, with deviations in both directions up to an order of magnitude. However, this method results in much better qualitative agreement between relative intensities observed by different authors. In addition to the different effective temperatures in the light sources, the observed intensities are strongly affected by different responses of the dispersive elements and detectors to different wavelengths. These variations can also be accounted for and removed. The general method for doing this was described in my recent paper on In II [31]. It relies on radiative transition rates A ki calculated with Cowan's codes (see the previous section), and on the LTE relation between these transition rates and the observed intensities I obs : I obs ∝ (g k A ki /λ)exp(-E k /kT eff ), (4) where g k and E k are the statistical weight and energy of the upper level, λ is the central wavelength of the line in vacuum, k is the Boltzmann constant, and T eff is the effective temperature.

Transition Probabilities
Experimental transition probabilities (A-values) for transitions originating from the 4d 8 5s 2 , 4d 9 6s, and 4d 9 5d configurations have been obtained by Campos et al. [30] using a combination of theoretical lifetimes and measured branching fractions. The calculations were made with Cowan's codes [29] using two models, one with semi-empirical account for core polarization (CP), and the other one without core polarization corrections, but with an extended set of interacting configurations that effectively introduced the corepolarization effects. In both models, the Slater parameters were adjusted by the LSF method to fit experimental energy levels. The CP calculations were deemed to be more accurate and were used to derive the radiative lifetimes. The accuracy of the calculated lifetimes was indirectly confirmed by a close http://dx.doi.org/10.6028/jres.118.009 All adopted A-values from the three published sources and the present work are given in Table 1. For the reasons discussed above, uncertainties of the adopted A-values are specified in Table 1 with a letter code instead of numerical values. The letter code is explained in Table 3. In addition to the papers discussed above, there are several other publications involving calculated transition probabilities and measured radiative lifetimes in Ag II. They are not discussed here because their accuracy was found to be lower than for the used sources. A complete list of relevant papers can be retrieved from the NIST Atomic Transition Probability Bibliographic Database [33]. http://dx.doi.org/10.6028/jres.118.009

Conclusions
In the present paper, critically evaluated energy levels, wavelengths, and transition probabilities have been derived for the Ag II spectrum from available experimental and theoretical data. The level list includes 98 excited energy levels having relative uncertainties ranging from 0.0012 cm −1 to 0.24 cm −1 . The uncertainty of the connection between the excited level system and the ground level is 0.06 cm −1 . The line identifications and level values of the 4d 9 9s configuration from the work of Benschop et al. [5] have been found incorrect. A revised value of the first ionization limit, 173283(7) cm −1 , equivalent to 21.4844(8) eV, has been derived from four 4d 9 ns 1,3 D Rydberg series (n ≤ 8). The complete line list includes 452 observed spectral lines between 728 Å and 9053 Å. Transition probabilities have been critically assessed for 237 transitions; 141 of them have been calculated in the present work. Uncertainties of the assessed transition probabilities range from 2 % to 75 %; for 226 transitions, the uncertainties are smaller than 35%.