IOP PUBLISHING
PHYSICA SCRIPTA
Phys. Scr. 81 (2010) 065301 (8pp)
doi:10.1088/0031-8949/81/06/065301
Measurements of isotope shifts and hyperfine structure in Ti II Z Nouri1 , S D Rosner1 , R Li2 , T J Scholl3,4 and R A Holt1 1
Department of Physics and Astronomy, University of Western Ontario, London, ON N6A 3K7, Canada TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6 T 2A3, Canada 3 Department of Medical Biophysics, University of Western Ontario, London, ON N6A 5C1, Canada 4 Imaging Research Laboratories, Robarts Research Institute, London, ON N6A 5K8, Canada 2
E-mail:
[email protected]
Received 27 January 2010 Accepted for publication 9 April 2010 Published 11 May 2010 Online at stacks.iop.org/PhysScr/81/065301 Abstract We have applied fast-ion-beam laser-fluorescence spectroscopy to measure the isotope shifts of 38 transitions in the wavelength range 429–457 nm and the hyperfine structures (hfs) of 22 levels in Ti II. The isotope shift and hfs measurements are the first for these transitions and levels. These atomic data are essential for astrophysical studies of chemical abundances, allowing correction for saturation and the effects of blended lines. PACS numbers: 32.10.Fn, 32.30.Jc, 95.30.Ky
been attributed to departures from local thermodynamic equilibrium (non-LTE effects) [20, 21]. As a consequence, abundance data from Ti II lines are generally considered more trustworthy than those from Ti I, because typically Ti II is the dominant ionization state in the stellar atmospheres, so that non-LTE effects are expected to be reduced. This view has recently been challenged by Przybilla et al [22], who were testing the hypothesis that hyper-velocity stars originate from the dynamical interaction of binary stars with the supermassive black hole in the galactic centre, giving one of the pairs sufficient velocity to escape the galaxy. They used chemical composition to try to identify the place of origin of HVS 7, which required a state-of-the-art modelling of non-LTE effects. One of their significant findings was large non-LTE effects in Ti II. The discrepancy between the solar and meteoritic abundances of Ti remains a puzzle, with the photospheric abundance being about 0.08 dex above the meteoritic [23, 24]. Although this only represents about one standard deviation, it merits further investigation. Laboratory studies of the Ti II spectrum and energy levels include Russell’s pioneering 1927 work [25], Meggers, Corliss and Scribner’s 1975 monograph [26] and more recent work by Huldt et al [27], Gianfrani et al [28], Zapadlik et al [29], Pickering et al [30] and Aldenius [31]. The highest resolution achieved in the above work was by Gianfrani et al [28], who used laser polarization spectroscopy on a hollow cathode discharge. Unfortunately, linewidths in the
1. Introduction Titanium contributes enormously to stellar spectra; for example, in the Sun it is in third place in terms of number of lines [1]. In B, A and F stars, the first ions of Fe, Cr and Ti dominate the visible spectrum [2]. Usually considered a member of the iron group (20 6 Z 6 30), Ti is perhaps more significantly one of the α elements (O, Mg, Si, S, Ca and Ti), whose principal isotope has an atomic mass that is a multiple of that of 4 He. These elements have enhanced abundances relative to Fe in metal-poor stars [3–5]. The latter are a window to the history of chemical evolution in the early Universe, as a result of the increasing metallicity of succeeding generations of stars, each producing more of the heavier elements in the last stages of their lives, just before spewing them into the interstellar medium (ISM) in supernova explosions [6, 7]. Tinsley [8] suggested that the trend of increasing overabundance of α elements with decreasing metallicity was due to the time delay between Type II supernovae, which produce both α elements and iron-peak elements [9, 10], and Type Ia supernovae, which produce mostly iron-peak elements [11, 12]. Not all old stars show enhanced α elements: the enhancement depends on the environment. Studies of Ti and the other α elements have played an important role in understanding the galactic halo [13, 14], the thin and thick discs [15, 16], the galactic bulge [17, 18] and the ISM [19]. Chemical abundances of Ti derived from Ti I and Ti II spectral lines show a discrepancy in most stars. This has 0031-8949/10/065301+08$30.00
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range 250–460 MHz, which they attributed to homogeneous broadening from collisions between neutral and ionic species, made it impossible to resolve the hyperfine structure (hfs) and, in the case of one transition, to resolve the isotopic structure fully. Isotope shifts have been calculated by Berengut et al [32]. The highest resolution ever achieved and, as far as we know, the only experiment to resolve the hfs, was the 1992 measurement by Berrah et al [33], which used fast-ion-beam laser spectroscopy. They applied our laser-rf double-resonance technique [34] to measure the magnetic dipole (A) and electric quadrupole (B) hfs constants of seven even-parity metastable levels of 49 Ti II with kHz precision, using an isotopically enriched sample. They also employed collinear laser-induced-fluorescence (LIF) optical spectroscopy to measure the hfs constants of five odd-parity levels, achieving MHz precision for the A constants. In the same paper they presented Hartree–Fock and Sandars–Beck effective operator calculations. Radiative lifetimes of Ti II levels with electricdipole-allowed transitions have been measured by Roberts et al [35], Kwiatkowski et al [36], Gosselin et al [37], Bizzarri et al [38] and Langhans et al [39] and have been calculated by Kaijser and Linderberg [40] and Luke [41]. In a landmark series of accelerator-based experiments at the CRYRING ion storage ring of Stockholm University, Mannervik and co-workers [42, 43] measured lifetimes as long as 28 s and also used the Cowan code to compute theoretical values. More recently, Deb et al [44] carried out a relativistic configuration interaction (CI) calculation. Transition probabilities and oscillator strengths have been measured by the classic emission, absorption and hook methods by many groups. A critical evaluation was provided by Savanov et al [45] in 1990. More accurate results were obtained by the combination of lifetime and branching-fraction data by Bizzarri et al [38] and Pickering et al [30]. As mentioned above, the resulting solar abundances are not quite in agreement with meteoritic values. Accurate determination of astrophysical chemical abundances requires atomic data on oscillator strengths, isotope shifts (IS), hfs and absolute wave numbers. Information on IS and hfs is required in order to correct for the desaturation of an absorption line that results from any mechanism that heterogeneously broadens a spectral line or splits it into components, as was first pointed out by Abt [46]. As an example, some of the principal Mn II lines in HgMn stars gave abundances discrepant by two or three orders of magnitude until Jomaron et al [47] showed that an ad hoc model of hfs could resolve the problem; however, actual atomic data were lacking until we measured them [48]. Absolute wave numbers are needed in the case of partial line blending of spectral features from different atomic species. In addition, the transition elements, whose 3d3 , 3d2 4s, and 3d 4s2 configurations are interleaved, resulting in strong CI effects, present a significant but not intractable challenge that is an important test of many-body atomic structure calculations. The IS and hfs parameters are sensitive probes of CI. We have begun a systematic study of the Ti II atomic properties needed for astrophysics and atomic theory.
Table 1. Properties of the Ti isotopes. Isotope 46
Ti Ti 48 Ti 49 Ti 50 Ti
47
Mass (u)a 45.952 6316(9) 46.951 7631(9) 47.947 9463(9) 48.947 8700(9) 49.944 7912(9)
Abundance I c (%)b 8.0(1) 7.3(1) 73.8(1) 5.5(1) 5.4(1)
µ I (nm)c
Q (b)c
0 5/2 −0.788 46(6) + 0.29(1) 0 7/2 −1.10414(9) + 0.24(1) 0
a
Audi et al [58]. De Bièvre and Taylor [59]. c Fuller [60]. b
In this first paper, we present the results of a fast-ion-beam LIF experiment to measure the IS of 38 transitions in the wavelength range 429–457 nm and the hfs of 22 levels.
2. Experimental method In our apparatus, Ti+ ions were produced by a Penning ion source [49] using Ne as the support gas. The cathode and anti-cathode were made of Ti, which was sputtered into the discharge by Ne ions. Under such conditions, Ti+ ions could be produced in metastable states with energies to 25 000 cm−1 or higher. The background pressure in the apparatus just downstream of the ion source is normally ∼1.5 × 10−7 Torr, but increased to ∼5.6 × 10−6 Torr as a result of gas effusing from the ion source. After acceleration to 10 keV, the ions were focused and then mass-filtered by a Wien velocity selector. Ti has five stable isotopes, with natural abundances and other properties listed in table 1. We deliberately reduced the mass resolution of the Wien filter by using lower values of the electric and magnetic fields, to achieve a high transmission for all five isotopes and at the same time effectively separate them from the Ne+ beam. This allowed us to record the signals from all the Ti isotopes on a single spectrum, enabling us to determine the IS and hfs simultaneously. Electrostatic deflection of the ion beam through 5◦ was used to overlap the ion beam with a counter-propagating laser beam for collinear laser spectroscopy; however, the ion-beam energy and laser frequency were set so that the Doppler-shifted laser frequency was not in resonance with the ion’s transition frequency. To bring the ions into resonance with the laser beam over a 3 cm length, a set of electrodes surrounding the beams created a Doppler-tuning region in which the ions were given a small additional acceleration, typically through a potential difference of −478 V. Further downstream, another electrostatic deflection was used to separate the beams, allowing the ion current to be monitored and avoiding damage to the laser beam entrance window of the vacuum system. The typical ion-beam current at a distance of ∼3 m from the ion source was 150 nA. We produced single-frequency CW laser light in the range 429–457 nm with a Stilbene 3 ring dye laser pumped by the all-lines UV output of an argon ion laser. Depending on wavelength, the typical laser power at the exit of the laser was 50–100 mW. Roughly half of this power reached the ion beam; the remainder was used to monitor the wavelength, calibrate the frequency scan or observe the mode structure or was simply lost due to imperfect optics. The laser beam’s 2
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absolute wave number was determined to 1 part in 107 using a travelling-corner-cube-mirror Michelson interferometer of the Hall design [50], with a polarization-stabilized HeNe laser as the reference. Fringe multiplication by a factor of 10 was used to improve the resolution. To calibrate and linearize the laser frequency scan, a portion of the light was sent through a 666.163 MHz free-spectral range (FSR) plane-parallel Fabry–Pérot étalon [51] and detected by an Si diode. LIF from the Doppler tuning region was collected by a ring of 80 multi-mode quartz optical fibres, filtered by a Schott UG-11 UV-pass filter to block scattered laser light and detected by a bialkali photomultiplier (Electron Tubes Ltd model 9235QB) with a gain of roughly 1.8 × 106 . We obtained a spectrum by using a computer with a data acquisition board to increment the laser frequency while recording the photomultiplier anode current, ion current, laser power, transmitted light through the Fabry–Pérot étalon and the error signal from the dye-laser locking circuitry (to monitor the scanning performance of the laser). Depending on the transition being observed, the LIF signals were as large as 1 µA on a background of ∼20 nA arising primarily from collisions between fast Ti ions and the thermal Ne background gas emanating from the ion source. The observed linewidth was in the best cases ∼75 MHz, the Doppler part of which (∼65 MHz) was significantly narrowed as a result of kinematic compression [52]; the other contributions are natural broadening and saturation. This Doppler linewidth implies a nonuniformity of plasma potential within the region of the source from which the ions were drawn of ∼2.8 V (the Doppler-tuning coefficient dν/dV was 22.5 MHz V−1 ). A typical spectrum covered 20–28 GHz in 1024 steps, with a dwell time of 100–300 ms per step, depending on the signal size. The number of steps in a spectral line was ∼15. We took three spectra for each transition, also recording the absolute wave number of the 48 Ti peak in order to determine the ion-beam energy for Doppler-shift corrections to the IS data (see section 3).
Figure 1. LIF spectrum of the transition 3d2 (1 D)4s a2 D5/2 –3d2 (3 F)4p z2 Do5/2 at 429.41 nm in Ti II. The upper curve is the same signal multiplied by 10 and displaced upward for clarity. Note that the hfs is almost completely resolved. The 48 Ti signal was 670 nA. The spectrum is a single scan of 1024 channels at a dwell time of 100 ms per channel.
Figure 2. LIF spectrum of the transition 3d3 a4 P3/2 –3d2 (3 F)4p z4 Do5/2 at 429.02 nm in Ti II. The upper curve is the same signal multiplied by 10 and displaced upward for clarity. The 48 Ti signal was 650 nA. The spectrum is a single scan of 1024 channels at a dwell time of 150 ms per channel.
3. Data and analysis An example of an LIF spectrum is given in figure 1. It is important to understand that most of the displacement between the signals from different isotopes comes from the relative Doppler shift, not from IS. This example shows large hfs, with enough resolution so that the lower- and upper-level magnetic dipole and electric quadrupole parameters A, B and A0 , B 0 , respectively, could be extracted. In other cases, such as that shown in figure 2, the hfs is too small to be resolved; however, that information itself is of use in stellar spectrum analysis. The fitted hyperfine constants are consistent with zero, within two standard deviations. The first step in data analysis was to linearize and calibrate the frequency scale, using the data from the Fabry– Pérot transmission curve. The FSR of the étalon is precisely known from measurements by the method of exact fractions [51]. A computer program was written to identify the transmission peaks automatically and fit a quadratic curve to each to determine its centre accurately. This is simpler than fitting an Airy function and works just as well. It then fitted a
natural cubic spline to the data consisting of integer multiples of the FSR versus peak locations in order to interpolate a frequency versus channel function. Next, we used nonlinear least-squares curve fitting to determine the IS and hfs parameters, plus a set of necessary but uninteresting constants for a background, an overall amplitude and the location of the 48 Ti peak on the scan. The relative amplitudes of the LIF signals from different isotopes were constrained by the isotopic abundances in table 1, and the relative amplitudes a(F, F 0 ) of hyperfine components were constrained by standard angular-momentum recoupling coefficients, 0 2 F F 1 0 −1 0 a F, F ∝ (2I +1) 2F +1 (2F+1) (1) J J0 I 3
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in which I is the nuclear spin and J , F and J 0 , F 0 are the electronic and total angular momenta of the lower and upper levels of the transition, respectively. The spectrum was fitted with a sum of saturationbroadened Lorentzian peaks whose centres were calculated from the hfs and IS, with a correction for the differential Doppler shift of the different isotopes. It should be noted that the correct lineshape function is a saturation-broadened Lorentzian for the natural linewidth convolved with an asymmetric ‘instrumental’ function representing the Doppler contribution from source-potential nonuniformity. This function is wider on the high-frequency side due to the low-speed ‘tail’ from ions produced within the narrow potential-gradient region adjacent to the cathode (the ‘cathode fall’), rather than in the nearly uniform plasma region at approximately anode potential, which produces most of the ions [49]. Within our experimental error, the use of an approximate lineshape should not affect the hfs or IS analysis because the slight asymmetry of the true lineshape is small and affects the ‘centre’ of each peak in the same way; thus, any possible contribution of the asymmetry to the differences between line centres is negligible. For 47 Ti and 49 Ti, the lower-level hfs is given by
parameters led to physically unrealistic values for the B’s. Hence, in fits to such spectra, we set the B’s to zero. When an isotope of mass M is accelerated from rest through a potential difference V , it acquires a speed (in units of c) 2eV 1/2 β= (4) Mc2 and thus different isotopes have different Doppler shifts. To achieve resonance with a transition whose frequency in the ion’s rest frame is ν0 , we must set the laser’s lab-frame frequency to 1 − β 1/2 ν` = ν0 (5) 1+β in anti-parallel geometry. Consequently, the separation in our laser-scanning spectra between lines from isotopes M and M 0 whose rest-frame resonance frequencies are ν0 and ν00 , respectively, can be written as " # 1 − β 0 1/2 1 − β 0 1/2 1 − β 1/2 1ν` = 1ν0 + ν0 − 1 + β0 1 + β0 1+β (6) in which 1ν0 = ν00 − ν0 is the IS relative to reference mass M. In all cases we took 48 Ti as the reference isotope. An accurate knowledge of β and β 0 is thus essential. We cannot simply use the potential difference between the ‘terminal voltage’ of the ion source and the voltage applied to the Doppler-tuning region because of the small difference between the plasma potential and the anode potential. (In the high-pressure mode in which we operated the source, the plasma potential is quite uniform and very close to that of the anode (within ∼1 V); nevertheless, the exact value is unknown.) To overcome this difficulty, we measured the absolute wave number of the laser when it was set to achieve resonance with the 48 Ti peak, using the Michelson interferometer. We then solved for the actual velocity by inverting the Doppler formula
1 3K (K + 1) − 4I (I + 1) J (J + 1) 1 AK + B , 2 2 2I (2I − 1) 2J (2J − 1) (2) K = F (F + 1) − I (I + 1) − J (J + 1)
νhfs (F) = where
with a similar formula for the upper-level hfs, in which A and B are replaced by A0 and B 0 , respectively. For levels with J = 1/2 there is no quadrupole term. In the least-squares fitting we only allowed the hfs constants for 49 Ti to vary. The corresponding constants for 47 Ti were constrained by the relationships A47 Ti /A49 Ti = (µ I /I ) 47 Ti / (µ I /I )49 Ti = 0.999 73(8), B47 Ti /B49 Ti = (Q) 47 Ti / (Q)49 Ti = 1.21(7)
(3)
β=
using the values listed in table 1. In all the cases where accurate values of the hfs constants for either the upper or lower level were available from the work of Berrah et al [33], they were fixed at those values. Fixing A and B, or A0 and B 0 , has the important effect of breaking the high correlations between A and A0 , and between B and B 0 , often allowing a statistically meaningful determination of B or B 0 values, to which the data are less sensitive. The uncertainties in these fixed parameters are not reflected in the statistical uncertainties of the floating constants obtained from a least-squares fit, and so they were propagated to those uncertainties after the fit. This approach was also used when we could obtain a high-precision measurement of hfs constants for a given level from one of our high-resolution spectra: such constants were fixed in subsequent fits to other less resolved spectra in which the same level appeared, and their uncertainties were propagated in the same way. This ‘bootstrap’ approach allowed the determination of a more complete and accurate set of hfs parameters than treating each spectrum entirely independently. For transitions where no prior knowledge of hfs parameters exists and the spectrum was not well resolved, we found that floating both A and B
σ02 − σ`2 σ02 + σ`2
(7)
in which σ0 is the transition wave number in the ion’s rest frame and σ` is the wave number to which the laser must be set to achieve resonance in anti-parallel geometry. Also, β 0 = (M/M 0 )1/2 β. The error budget for the IS measurements includes: (i) the statistical uncertainty from least-squares curve-fitting to data with scatter, (ii) the uncertainty in linearizing and calibrating the laser-frequency scan and (iii) the uncertainty in determining the relative Doppler shift. For even isotopes the curve fitting typically determined the IS to an uncertainty of a few MHz. Odd isotopes have smaller peaks because of hyperfine splitting, and there is a correlation between IS and hfs parameters. As a result, the statistical uncertainty was typically of the order of 10 MHz for these. The FSR of the étalon is known to be better than 1 part in 108 and makes no significant contribution to the uncertainty in the calibration of the laser-frequency scan. The differential nonlinearity of the scan can be defined as the standard deviation of the separations between étalon transmission peaks, which was about 2.7%. The integral 4
Phys. Scr. 81 (2010) 065301
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Table 2. Isotope shifts in Ti II, relative to 48 Ti II. Wavelengths, terms, energies and leading percentages are taken from the NIST Atomic Spectra Database a . Lower level
Upper level
Isotope shifts
λ (nm)
Term
Energy (cm−1 )
Leading percentage
433.792 428.787 444.379 434.429 429.410 445.048 439.503 446.914
a2 D3/2 a2 D3/2 a2 D3/2 a2 D5/2 a2 D5/2 a2 D5/2 a2 D5/2 a2 D5/2
8 710.44 8 710.44 8 710.44 8 744.25 8 744.25 8 744.25 8 744.25 8 744.25
17% 3d3 17% 3d3 17% 3d3 16% 3d3 16% 3d3 16% 3d3 16% 3d3 16% 3d3
3d2 (1 D) 4s → 3d2 (3 F) 4p z2 Do3/2 31 756.51 −379(11) z2 Do5/2 32 025.47 −373(11) z2 Fo5/2 31 207.42 −338(12) z2 Do3/2 31 756.51 −401(12) z2 Do5/2 32 025.47 −426(11) z2 Fo5/2 31 207.42 −430(12) z2 Fo7/2 31 490.82 −444(11) z4 Fo7/2 31 113.65 −445(11)
−177(10) −173(10) −187(10) −205(10) −201(10) −210(12) −212(10) −214(10)
200(10) 207(11) 210(10) 230(10) 231(10) 234(11) 238(10) 239(10)
363(11) 365(11) 380(11) 415(11) 420(11) 421(12) 432(11) 441(11)
433.025 431.680 435.083 437.482 429.035 433.726
b2 P1/2 b2 P1/2 b2 P3/2 b2 P3/2 b2 P3/2 b2 P3/2
16 515.86 16 515.86 16 625.11 16 625.11 16 625.11 16 625.11
28% 3d3 28% 3d3 28% 3d3 28% 3d3 28% 3d3 28% 3d3
3d2 (3 P) 4s → 3d2 (1 D) 4p y2 Do3/2 39 602.75 614(11) z2 Po1/2 39 674.66 601(11) y2 Do3/2 39 602.75 674(11) y2 Do5/2 39 476.80 793(12) y2 Fo5/2 39 926.66 526(12) z2 Po1/2 39 674.66 623(11)
302(10) 280(10) 300(10) 337(10) 246(11) 291(10)
−279(10) −268(10) −281(10) −319(10) −242(14) −267(10)
−554(11) −529(11) −564(11) −625(11) −469(11) −532(11)
3d3 → 3d2 (3 F) 4p z Do5/2 32 025.47 z2 Fo5/2 31 207.42 z2 Fo7/2 31 490.82 z2 Fo7/2 31 490.82 z4 Fo7/2 31 113.65 z2 Do3/2 31 756.51 z4 Do1/2 32 532.21 z4 Do3/2 32 602.55 z2 Do5/2 32 025.47 z4 Do1/2 32 532.21 z4 Do3/2 32 602.55 z4 Do5/2 32 697.99 z4 Do3/2 32 602.55 z4 Do5/2 32 697.99 z4 Do7/2 32 767.07 z2 Do3/2 31 756.51 z4 Do3/2 32 602.55 z2 Do5/2 32 025.47 z4 Do5/2 32 697.99 z2 Go7/2 34 543.26
1782(11) 1764(11) 1785(11) 1745(11) 1734(11) 1837(12) 1858(12) 1799(11) 1803(11) 1781(12) 1778(11) 1780(11) 1874(22) 1886(12) 1857(11) 866(11) 820(11) 918(13) 863(11) 1795(15)
866(10) 850(10) 855(11) 847(11) 849(10) 893(10) 881(11) 871(10) 877(10) 867(10) 857(11) 872(10) 902(10) 900(10) 910(11) 417(10) 405(10) 429(11) 416(10) 857(10)
−844(10) −842(11) −837(11) −841(10) −828(11) −871(10) −859(11) −868(11) −861(10) −847(10) −858(10) −840(10) −879(10) −888(10) −881(10) −407(11) −382(10) −416(11) −399(10) −865(10)
−1648(11) −1632(11) −1627(11) −1624(11) −1614(11) −1692(11) −1670(11) −1677(11) −1664(11) −1646(11) −1647(11) −1646(11) −1712(11) −1717(11) −1712(11) −800(11) −758(11) −813(11) −784(11) −1667(11)
948(11) 884(10)
−902(11) −866(10)
−1775(12) −1689(11)
−1154(10) −1104(10)
1243(11) 1154(11)
2287(12) 2182(11)
2
Term
Energy (cm−1 )
2
46
Ti (MHz)
434.136 450.127 444.456 446.851 454.513 446.445 431.498 430.191 441.772 432.096 430.786 429.022 433.070 431.286 430.005 456.377 439.405 453.397 439.977 457.198
a G7/2 a2 G7/2 a2 G7/2 a2 G9/2 a2 G9/2 a4 P1/2 a4 P1/2 a4 P1/2 a4 P3/2 a4 P3/2 a4 P3/2 a4 P3/2 a4 P5/2 a4 P5/2 a4 P5/2 a2 P1/2 a2 P1/2 a2 P3/2 a2 P3/2 a2 H9/2
8 997.71 8 997.71 8 997.71 9 118.26 9 118.26 9 363.62 9 363.62 9 363.62 9 395.71 9 395.71 9 395.71 9 395.71 9 518.06 9 518.06 9 518.06 9 850.90 9 850.90 9 975.92 9 975.92 12 676.97
438.684 436.766
b2 F5/2 b2 F7/2
20 951.62 20 891.66
3d3 → 3d2 (1 G) 4p y2 Go7/2 43 740.65 1909(12) y2 Go9/2 43 780.79 1822(11)
24 961.03 25 192.79
x2 Fo5/2 x2 Fo7/2
441.107 448.833 a
2
c D3/2 c2 D5/2
24% 3d2 (3 P) 4s 24% 3d2 (3 P) 4s 27% 3d2 (3 P) 4s 27% 3d2 (3 P) 4s
3d 4s2 → 3d2 (1 G) 4p 47 624.88 −2419(11) 47 466.54 −2308(11)
47
Ti (MHz)
49
Ti (MHz)
50 Ti (MHz)
Ralchenko et al [61].
comes from two parameters: σ0 (or ν0 ) and σ` . The uncertainty in ν0 as it appears explicitly in (6) produces an entirely negligible 2 kHz contribution. However, the uncertainties in σ0 and σ` lead by standard error propagation to an uncertainty in β of ±2.4 × 10−7 , which produces an uncertainty of ±1.7 MHz × (M 0 –M) in the relative Doppler shift. The error budget for the hfs measurements includes the statistical uncertainty from least-squares curve fitting to data with scatter and from using fixed parameters in the fit. The contributions to the uncertainties in the hfs parameters from
nonlinearity, defined as the maximum deviation of the actual curve of frequency versus channel number from a straight-line fit, was typically 100 MHz. Of course the spline fit effectively removes most of this nonlinearity. We somewhat arbitrarily chose 10% of this value as an estimate of the amount by which the correction by the spline fitting procedure might be in error. This gives a 10 MHz uncertainty to be added in quadrature to the statistical uncertainty. The contribution to the IS-measurement error budget from the uncertainty in determining the relative Doppler shift 5
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the frequency calibration procedure and the determination of the relative Doppler shift are both negligible. The actual values of the uncertainties in the hfs parameters varied widely, depending on the degree to which the hfs was resolved.
4. Results Table 2 lists the results for the IS measurements. It is clear that many transitions are spread over a spectral region of the order of 3.6 GHz, which can have important effects on spectral synthesis, particularly for slowly rotating cool stars. There are no previous experimental results with which to compare our IS data. In light elements such as Ti, one expects the contribution of the field shift to the IS to be much smaller than that of the specific mass shift (SMS) [53]. Bauche and Crubellier [54] carried out a nonrelativistic Hartree–Fock calculation of SMS in four configurations of the 3d series of transition elements (Sc to Cu). Ti II is isoelectronic to Sc I. They found a number of important regularities, including: (i) the SMS is only large when the transition involves a 3d electron jump, and (ii) the SMS does not vary within a given configuration, for the configurations of interest. Bauche and Crubellier’s calculation yielded nearly identical SMS values, ∼80 mK (2400 MHz) for 1A = 2, for the 3d N +1 4s → 3d N 4s2 transition in all the elements of the series, with much smaller values for the three transitions not involving a 3d jump. Using their results, we can give a surprisingly accurate picture of our data. In all three cases of transitions involving a d-electron jump, we observe IS of 1800–2400 MHz for 1A = 2. Specifically, the IS of the two 3d4s2 → 3d2 (1 G)4p transitions are 2300–2400 MHz, whereas the 3d3 → 3d2 (1 G)4p IS are 1800–1900 MHz. The corresponding 3d3 → 3d2 (3 F)4p IS are almost all concentrated in a narrow range around 1800 MHz. The exceptions are transitions which have an a2 P1/2 or a2 P3/2 lower state which have strong admixtures of configurations that do not lead to a 3d jump and which we can therefore assume to yield a very small IS. Huldt et al [27] carried out a parametric study of the (3d + 4s)3 configuration complex. The 3d3 a2 P1/2 eigenvector composition is 62% 3d3 , 24% 3d2 (3 P)4s b2 P, and 10% 3d3 a4 P, for a total of 72% of the levels that lead to a d-electron jump. (The upper levels are 82 and 94% pure, with unspecified other components.) Thus we predict an IS of about 72% of 1800 MHz, i.e. 1300 MHz. The actual values are in the 800–900 MHz range. The 3d3 a2 P3/2 eigenvector composition is 48% 3d3 , 27% 3d2 (3 P)4s b4 P and 18% 3d2 (3 P)4s b2 P. In this case we predict an IS of only 48% of 1800 MHz, i.e. 860 MHz. The experimental values are in the 860–920 MHz range. The remaining transitions do not involve a 3d jump and thus would be expected to show very small IS; however, the lower states contain admixtures of 16–17 and 28% of the 3d3 configuration, which would lead us to predict 300 and 500 MHz, compared to the experimental shifts of roughly 400 MHz (3d2 (1 D)4s → 3d2 (3 F)4p) and 600 MHz (3d2 (3 P)4s → 3d2 (1 D)4p), respectively. In order to look for evidence of a small field shift, we have plotted in figure 3 the ratio of the relative isotope shift (RIS), 1σαβ /1σγ δ , to its expected value, Aαβ /Aγ δ , where Aαβ = (Aβ − Aα )/Aα Aβ , for the pairs α = 46, β = 48, γ = 50, and δ = 48 from all our transitions. These isotope pairs were
Figure 3. The ratios of relative isotope shifts, 1σαβ /1σγ δ , to their expected values, Aαβ /Aγ δ , where Aαβ = (Aβ − Aα )/Aα Aβ , for the pairs α = 46, β = 48, γ = 50, and δ = 48 from all the transitions in table 2. The legend indicates the configuration of the lower level of the transition. The configuration of the upper level in all cases is 3d2 4p.
chosen to avoid odd isotopes, which have larger uncertainties due to correlation between hfs and IS parameters; in addition, second-order hfs effects can shift the levels. It is well known [53] that, in either of the limits in which the mass shift or field shift dominates the other, the RIS is the same for all transitions. In our case, in which the mass shift is the dominant contribution, the expected value of the RIS is easily calculated as the mass ratio Aαβ /Aγ δ . Since the field shift depends on the probability density of the electronic wavefunction at the nucleus, we expect contributions only from s electrons (and p1/2 in relativistic cases). As can be seen from figure 3, the RIS of the transitions with lower-level configuration 3d3 agree extremely well with the expected value, indicating a negligible field shift. In contrast, all the transitions with one or two 4s electrons in the lower level exhibit significant deviations. We chose a pair of transitions with large deviations and constructed a King plot [55], which is shown in figure 4. 0 The modified residual isotope shift 1σαβ is calculated by subtracting the normal mass shift, σ0 [(1 + m e /Mβ )−1 − (1 + m e /Mα )−1 ], and then scaling the result by the factor A48,50 /Aαβ . As expected, the slope g, which equals the ratio of the field shifts of the two transitions, is well determined: g = 0.89(9). The intercept, which includes both SMS and field-shift information, is also well determined: Sp 0 = −31(2) × 10−3 cm−1 . (We use the notation of [53].) Figure 4 thus shows evidence of the presence of field shifts in Ti II transitions involving an s-electron jump. Table 3 lists the hfs constants we have determined. Here the overall size of the splitting of a spectral line can also be of the order of 4 GHz (see, for example, figure 1), which clearly cannot be ignored in stellar spectroscopy. The theoretical situation for hfs is less favourable. Berrah et al [33] carried out both a nonrelativistic Hartree–Fock calculation and a Sandars–Beck [56] effective-operator analysis. The agreement between their experimental results in 49 Ti II and the Hartree–Fock predictions ranged from good (10%) to poor 6
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large mixing with the 3d2 (3 P)4s configuration. In a later experimental and theoretical study of Zr II, they were able to show that a relativistic CI calculation removed much of the discrepancy for the corresponding 4d3 2 P states [57]. Given this prehistory and the requirements of spectral synthesis in astrophysics, we decided to limit our objectives to providing a convenient and useful description of the hfs splittings in terms of phenomenological constants, in some cases not even including the quadrupole constant.
5. Conclusions The high-resolution spectra taken with our sputter ion source, using the fast-ion-beam laser-fluorescence method, have provided a substantial amount of new atomic data for Ti II. These data provide accurate information for modern spectral synthesis techniques in stellar spectroscopy. In particular, they reveal the underlying line-splitting mechanisms that are unresolved in most stellar spectra but can lead to substantial errors in derived chemical abundances if ignored.
Figure 4. A King plot of the modified residual IS (see text) of two transitions in Ti II. The mass pair 48, 50 has been chosen as the reference. Table 3. The hfs constants in 49 Ti II. A is the magnetic-dipole interaction constant and B is the electric-quadrupole interaction constant. A value of zero for B indicates that the data were inadequate to determine B, which has therefore been fixed at this value. Levels with J = 1/2 have no quadrupole hyperfine interaction, indicated by a blank in the B column. Energy (cm−1 )
A (MHz)
B (MHz)
Even levels 3d3 a4 P3/2 3d3 a4 P5/2 3d3 a2 H9/2 3d2 (3 P)4s b2 P1/2 3d2 (3 P)4s b2 P3/2 3d3 b2 F7/2 3d3 b2 F5/2 3d4s2 c2 D3/2 3d4s2 c2 D5/2
9 395.71 9 518.06 12 676.97 16 515.86 16 625.11 20 891.66 20 951.62 24 961.03 25 192.79
−6.63(31) 11.52(43) −58.4(1.2) −82.8(1.4) 27.2(1.0) −37.9(2.3) −46.23(82) −132.24(94) −43.2(1.4)
−24.1(2.2) 38.4(6.8) 0
Odd levels 3d2 (3 F)4p z4 Fo7/2 3d2 (3 F)4p z4 Do1/2 3d2 (3 F)4p z4 Do5/2 3d2 (3 F)4p z4 Do7/2 3d2 (3 F)4p z2 Go7/2 3d2 (1 D)4p y2 Do5/2 3d2 (1 D)4p y2 Do3/2 3d2 (1 D)4p z2 Po1/2 3d2 (1 D)4p y2 Fo5/2 3d2 (1 G)4p y2 Go7/2 3d2 (1 G)4p y2 Go9/2 3d2 (1 G)4p x2 Fo7/2 3d2 (1 G)4p x2 Fo5/2
31 113.65 32 532.21 32 697.99 32 767.07 34 543.26 39 476.80 39 602.75 39 674.66 39 926.66 43 740.65 43 780.79 47 466.54 47 624.88
−37.99(27) −177.72(52) −23.03(18) −17.56(33) −80.8(1.5) −75.8(1.4) −36.5(1.2) −84.0(1.4) −66.1(1.0) −50.46(57) −58.3(1.9) −59.7(1.0) −83.12(60)
Level
Acknowledgments We thank the Natural Sciences and Engineering Research Council of Canada for financial support. We thank Harry Chen for expert electronics assistance and Brian Dalrymple and Frank Van Sas for expert machining.
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