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Isotope shift in chromium
Spectrochimica Acta Part B 60 (2005) 33 – 40 www.elsevier.com/locate/sab
Isotope shift in chromium B. Furmann*, A. Jarosz, D. Stefan´ska , J. Dembczyn´ski, E. Stachowska Faculty of Technical Physics, Poznan´ University of Technology, Nieszawska 13B, 60-965 Poznan´, Poland Received 17 June 2004; accepted 21 October 2004
Abstract Thirty-three spectral lines of chromium atom in the blue-violet region (425–465 nm) have been investigated with the method of laserinduced resonance fluorescence on an atomic beam. For all the lines, the isotope shifts for every pair of chromium isotopes have been determined. The lines can be divided into six groups, according to the configuration of the upper and lower levels. Electronic factors of the field shift and the specific mass shift ( F ik and M ikSMS, respectively) have been evaluated and the values for each pure configuration involved have been determined. Comparison of the values F ik and M ikSMS to the ab initio calculations results has been performed. The presence of crossed second order (CSO) effects has been observed. D 2004 Elsevier B.V. All rights reserved. Keywords: Isotope shift; Atomic beam
1. Introduction Chromium is an element with four stable isotopes: three even isotopes 50Cr, 52Cr, 54Cr and an odd isotope 53Cr with nuclear spin I=3/2. The natural isotope abundance is as follows [1]: 50Cr—4.35%, 52Cr—83.79%, 53Cr—9.5% and 54 Cr—2.36%. Isotope shifts, observed in chromium spectrum, are interesting from the point of view of both atomic and nuclear physics. The chromium isotope 52Cr possesses a magic neutron number N=28. In this situation, it can be expected (and the results of muonic atoms and electron scattering investigations confirm the expectations [2]) that the mean square charge radius hr 2i should be minimum for this isotope. As the field shift is directly dependent on yhr 2i, an anomalous change of the charge distribution is reflected in the results of measurements of the optical isotope shifts. Chromium is an element with a half-filled 3d electron shell. According to the theoretical analysis of second-order effects in the field shift, performed by Aufmuth [3], the elements
* Corresponding author. Tel.: +48 61 665 3226; fax: +48 61 665 3239. E-mail address: [email protected] (B. Furmann). 0584-8547/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sab.2004.10.007
with half-filled electron shells should exhibit the most pronounced effects of the kind mentioned. Recently, the interest in spectroscopic properties of chromium atom has increased as a result of its application in experiments concerned with cooling of atoms. It appeared that cooling of chromium atoms allows to obtain simultaneously a Bose-Einstein condensate (cooling of 52Cr isotope) and a degenerate Fermi gas (53Cr isotope) [4]. The technique applied in this case (evaporative cooling) takes advantage of elastic collisions of 52 Cr. The fundamental problem is an unfavorable ratio between the elastic and the inelastic (undesirable) collisions cross-sections in the temperature region near 10 mK. Below this temperature, the evaporative cooling technique again becomes effective. The way to overcome this problem is application of laser cooling in the vicinity of 10 mK, which requires the detailed knowledge of isotope shifts [4]. In this context, it seems somewhat surprising that the only papers concerning isotope shifts in chromium, that can be found in the literature, are the ones by Heilig and Wendlandt [5], published in the 1960s of the last century. A likely explanation could be experimental difficulties, as well as problems with satisfactory interpretation of the
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B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
results of the measurements. Experimental difficulties are mainly concerned with relatively small splitting of the chromium spectral lines. A typical width of the spectrum, containing all the four stable isotopes (hyperfine structure of 53Cr isotope included) is of the order of 1–2 GHz. Thus, in order to resolve the spectral components, Doppler-free spectroscopic methods have to be applied. In our experiment, the method of laser-induced fluorescence (LIF) on an atomic beam has been used. This method has, however, an essential disadvantage—it can be used only for those transitions, whose lower level is metastable, while the thermal population of the lower level decreases rapidly with its increasing energy. Unfortunately, the number of low-lying levels in chromium atom is rather small. On the other hand, in order to allow to take into account the configuration mixing effects as well as the second-order interactions in the analysis of the isotope shifts, a possibly large number of experimental data is required. In this work, we present the results of isotope shift measurements for 33 optical transitions in chromium atom. Most of them concern the same terms (the levels differing in their J quantum numbers). After subtraction of normal mass shifts, for all the transitions studied, the residual isotope shifts have been separated into specific mass shifts and field shifts. Using the results of the combined analysis of the optical spectra, muonic atoms and electron scattering measurements [2], the isotope shift electronic factors for individual transitions have been determined. With the use of the sharing rule, the electronic factors for the pairs of pure configurations 3d54p, 3d44s4p, 3d44s2 and 3d54s have been estimated. The results have been compared to the results of nonrelativistic multiconfiguration HartreeFock calculations.
2. Experimental The method of LIF on an atomic beam has been used for measurements of the relative positions of all chromium isotopes. Chromium, placed in a graphite crucible, has been heated and evaporated by the electron beam bombardment. Electron beam emitted from electron gun has been focused and deflected by system of electromagnets. An auxiliary discharge (between additional electrode placed just above the crucible and the melted metallic sample in the crucible) has been applied to allow better population of the high-lying metastable levels. Atomic beam has been formed by a system of diaphragms (each of 1 mm diameter). The divergency of the atomic beam amounted 8d 103 rad. A laser beam generated by a stabilized single-mode ring dye laser (Coherent model CR-699-21), operating on stilbene 3 and pumped by an Ar+ laser, has crossed the atomic beam perpendicularly. The fluorescence light, collected by an optical detection system, has been recorded by a photo-
multiplier positioned perpendicularly to both the laser and the atomic beam directions. Optical system has been carefully shielded against scattered light from the oven. The more detailed description of atomic beam setup can be found in Ref. [6]. For each investigated optical transition on the average, 20 scans have been recorded. An example of a recorded spectrum, containing all the stable chromium isotopes, is presented in Fig. 1. Because of a very high share of 52Cr in the natural abundance, the spectral component belonging to this isotope has always been much stronger than the remaining ones and it has easily become saturated. Therefore, 10 scans with low laser power have been recorded in order to determine exactly the position of this component, and then another 10 scans with higher laser power with the aim of determination of positions of the components belonging to the remaining isotopes. Initially, the recorded spectra have been calibrated with an approximate value of the free spectral range (FSR) of the frequency marker—150 MHz. For each scan, the relative positions of all the isotopes, as well as the hfs parameters for the odd isotope 53Cr, have been fitted simultaneously. A program bFitterQ, received from the University of Bundeswehr in Hamburg, has been used for that purpose. The final values for this stage have been the mean values of those obtained in individual scans, and the errors have been the mean standard deviations. The procedure described has been applied also to the lines involving levels measured very accurately with resonance method by Childs et al. [7] or Ertmer et al. [8]. Intervals, known with high accuracy from Refs. [7,8], have been evaluated from our measurements and compared with the former ones. The FSR value of the marker has been corrected until a least mean difference in the values of intervals has been obtained. The corrected FSR amounts to 149.98 MHz (this value is consistent with the one obtained with a similar method in our earlier works [9,10]). The final values quoted are calibrated with the corrected FSR value of
Fig. 1. Example of a recorded spectrum of chromium atom, containing all the four stable isotopes (k=437.1264 nm). Center of mass of the hyperfine structure of the isotope 53 is marked.
B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
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Table 1 Transitions in chromium, investigated from the point of view of the isotope shifts and results of the measurements k [nm] (in air)
E [cm ]
Designation
E [cm ]
Designation
458.0043 454.5945 434.0122 438.1109 429.7031 434.3158 454.4604 452.6443 454.0488 453.5695 453.0730 453.0679 452.9839 452.7332 428.9733 427.4806 425.4331 461.3357 462.6174 459.1389 465.1282 461.6120 456.5498 465.2152 464.6148 433.9713 435.1054 433.7552 433.9436 437.1264 434.4496 438.4964 435.1755
29420.90 29584.62 44875.19 44666.74 45113.22 44875.19 42515.35 42605.81 42538.81 42564.85 42589.25 42589.25 42589.25 42605.81 23305.01 23386.35 23498.84 29420.90 29420.90 29584.62 29420.90 29584.62 29824.75 29584.62 29824.75 30787.30 30787.30 30858.82 30965.46 30965.46 31106.37 31106.37 31280.35
3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(a3P)4s4p(3Po) v5P2 3d4(a3P)4s4p(3Po) v5P1 3d4(a3P)4s4p(3Po) v5P3 3d4(a3P)4s4p(3Po) v5P2 3d5(4G)4p z5G2 3d5(4G)4p z5G6 3d5(4G)4p z5G3 3d5(4G)4p z5G4 3d5(4G)4p z5G5 3d5(4G)4p z5G5 3d5(4G)4p z5G5 3d5(4G)4p z5G6 3d5(6S)4p z7P2 3d5(6S)4p z7P3 3d5(6S)4p z7P4 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P3 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P3 3d4(5D)4s4p(3Po) z5F1 3d4(5D)4s4p(3Po) z5F1 3d4(5D)4s4p(3Po) z5F2 3d4(5D)4s4p(3Po) z5F3 3d4(5D)4s4p(3Po) z5F3 3d4(5D)4s4p(3Po) z5F4 3d4(5D)4s4p(3Po) z5F4 3d4(5D)4s4p(3Po) z5F5
7593.16 7593.16 21840.84 21847.88 21847.88 21856.94 20517.40 20519.60 20520.92 20523.69 20523.94 20523.69 20519.60 20523.94 0.00 0.00 0.00 7750.78 7810.82 7810.82 7927.47 7927.47 7927.47 8095.21 8307.57 7750.78 7810.82 7810.82 7927.47 8095.21 8095.21 8307.57 8307.57
3d5(6S)4s a5S2 3d5(6S)4s a5S2 3d5(4P)4s a5P3 3d5(4P)4s a5P2 3d5(4P)4s a5P2 3d5(4P)4s a5P1 3d5(4G)4s a5G2 3d5(4G)4s a5G6 3d5(4G)4s a5G3 3d5(4G)4s a5G4 3d5(4G)4s a5G5 3d5(4G)4s a5G4 3d5(4G)4s a5G6 3d5(4G)4s a5G5 3d5(6S)4s a7S3 3d5(6S)4s a7S3 3d5(6S)4s a7S3 3d44s2 a5D0 3d44s2 a5D1 3d44s2 a5D1 3d44s2 a5D2 3d44s2 a5D2 3d44s2 a5D2 3d44s2 a5D3 3d44s2 a5D4 3d44s2 a5D0 3d44s2 a5D1 3d44s2 a5D1 3d44s2 a5D2 3d44s2 a5D3 3d44s2 a5D3 3d44s2 a5D4 3d44s2 a5D4
Upper level 1
Lower level 1
the marker. All measured transitions and obtained results of isotope shift are presented in Table 1.
IS52,50 [MHz]
IS53,50 [MHz]
IS54,50 [MHz]
1898.2(0.7) 1893.6(0.7) 1177.5(0.9) 1153.7(3.8) 1125.8(5.5) 1177.0(0.7) 152.4(1.5) 170.2(0.4) 152.5(0.6) 167.8(1.1) 166.0(4.1) 172.0(1.7) 176.2(3.1) 173.9(2.1) 115.0(2.0) 121.1(2.0) 130.3(2.0) 787.3(0.3) 782.3(0.6) 784.4(0.6) 784.0(1.0) 792.1(2.3) 783.8(0.6) 767.7(3.0) 788.0(0.6) 852.8(1.6) 846.1(1.5) 851.2(0.6) 860.6(1.2) 850.0(0.2) 848.7(1.8) 847.1(3.4) 852.1(0.3)
2722.7(0.9) 2724.1(0.9) 1707.6(0.6) 1686.0(7.7) 1658.6(8.5) 1707.4(0.9) 256.8(1.4) 281.7(0.8) 269.2(2.3) 276.4(0.8) 298.7(5.5) 285.1(1.0) 292.3(3.5) 293.1(2.5) 128.8(2.0) 141.2(2.0) 153.1(2.0) 1071.7(1.4) 1068.1(2.5) 1071.5(1.0) 1072.0(0.8) 1069.6(2.4) 1070.9(1.4) 1065.1(0.8) 1069.7(1.1) 1171.5(1.9) 1163.9(1.6) 1166.8(1.3) 1168.8(1.1) 1167.3(0.5) 1168.0(1.8) 1166.2(1.9) 1170.5(0.3)
3480.4(5.7) 3474.3(0.9) 2202.0(1.3) 2132.3(9.5) 2140.8(7.0) 2202.9(1.2) 372.8(0.8) 403.7(1.7) 386.9(3.6) 399.1(1.5) 408.1(1.0) 412.1(1.8) 415.1(1.2) 415.6(2.0)
1315.9(0.8) 1309.6(2.6) 1311.3(1.2) 1309.8(1.2) 1308.4(1.4) 1310.3(0.6) 1306.2(1.5) 1303.6(0.5) 1436.9(2.6) 1428.8(2.0) 1431.7(0.8) 1442.8(1.8) 1432.7(0.6) 1433.1(0.7) 1430.0(2.1) 1437.8(0.2)
where M ikSMS is an electronic factor for the transition. The field shift can be expressed as: FSAAV Fik yhr2 iAAV ;
3. Fundamental The isotope shift in a transition ik of wavenumber m ik : ISAAV ¼ mAik mAV ik may be split into two terms [11]: ISAAV ¼ MSAAV þ FSAAV ; where: MSAAV —mass shift and FSAAV —field shift. The mass shift consists of two contributions: MSAAV ¼ NMSAAV þ SMSAAV ; NMSAAV ¼
me AAV mik mp A AV
SMSAAV ¼ MikSMS
AAV A AV
normal mass shift ;
specific mass shift ;
where F ik is an electronic factor for the transition and yhr 2iAAV is a change of the mean square charge radius of the nucleus (a nuclear factor). The normal mass shift can easily be calculated and is often subtracted from the total isotope shift to give the residual isotope shift RISAAV RISAAV uISAAV NMSAAV ¼ SMSAAV þ FSAAV AAV þ Fik yhr2 iAAV : ¼ MikSMS A AV Separation of the residual isotope shifts into field shifts and specific mass shifts is conventionally achieved with the use of King diagrams, if the abscissa contains RIS values for a transition, for which F ik (and consequently also M ikSMS) is known from previous measurements, and the ordinate—RIS values of a transition under study. On the basis of the slope and the intersection coefficients of the resulting straight line, F ik and M ikSMS of all other investigated transitions can be
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B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
Fig. 2. Modified King plots (modified RIS vs. modified yhr 2i) for four types of measured transitions: (a) 3d4(5D)4s4p(3Po) y5Po2–3d5(6S)4s a5S2, (b) 3d4(5D)4s4p(3Po) z5Fo2–3d44s2 a5D1, (c) 3d5(4G)4p z5Go5–3d5(4G)4s a5G6, (d) 3d4(a3P)4s4p(3Po) v5Po2–3d5(4G)4s a5G6. The error bars in the abscissa result from the values given in Ref. [13], while those in the ordinate—from the error values given in Table 1 (as already mentioned in text above, they have been calculated as mean standard deviations for 10 values).
estimated, respectively. Since in chromium atom F ik or SMS M ik is not known for any spectral line, we have used an alternative method. Separation of SMSAAV and FSAAV can be also performed with the use of another kind of King plot: modified optical RISAAV vs. modified yhr 2iAAV (for chromium model independent yhr 2iAAV are known from optical measurements, muonic X-ray spectroscopy and electron scattering [12–14]). Modified RISAAV is defined as follows: lRISAAV ¼
A AV 52 50 RISAAV A AV 52 50
(the isotopes 52Cr and 50Cr have been chosen as pair of reference isotopes) and yhr 2iAAV have been modified in the same way. From this kind of King diagram, the F ik value can be determined directly as the slope of the straight line and the M ikSMS value is obtained through multiplication of the intersection coefficient by (5250)/(5250). Some examples of King diagrams are presented in Fig. 2, while the values of M ikSMS and F ik , obtained for all the investigated transitions, are listed in Table 2.
4. Analysis of the results 4.1. Configuration mixing and sharing rule In the first approximation both MSMS and F i for all the i levels belonging to a pure configuration should be equal.
Admittedly, from the results obtained it is not possible to determine the electronic factors SMS and FS for the individual levels participating in the transitions measured (only the factors for transitions are known), but because of the relations between the factors MikSMS ¼ MiSMS MkSMS ; Fik ¼ Fi Fk ; also the electronic factors for transitions between the levels belonging to the same configurations should be equal, provided the levels were pure and the higher order effects were neglected (these effects are discussed in the next section). At the first stage of the analysis of the obtained M ikSMS and F ik factors, the dependence on J quantum number, i.e. variation within the same term, has been investigated. In this case, comparison of the results for transitions with a common upper or lower level has yielded the differences between the factors for the levels with various J. The analysis performed in this way proved, that all the factors for the levels belonging to the same term are indeed equal within the limits of experimental error. Therefore, in the following, average values for each term have been used. SMS The final values of M ik and F ik , obtained in this way, are presented in Table 3. It can be seen that, if the values within one term are equal, the values of both M ikSMS and F ik for various terms belonging to the same configuration differ considerably.
B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
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Table 2 Electronic factors F ik and M ikSMS for the studied transitions k [nm] (in air)
E [cm ]
Designation
E [cm ]
Designation
458.0043 454.5945 434.0122 438.1109 429.7031 434.3158 454.4604 452.6443 454.0488 453.5695 453.0730 453.0679 452.9839 452.7332 428.9733 427.4806 425.4331 461.3357 462.6174 459.1389 465.1282 461.6120 456.5498 465.2152 464.6148 433.9713 435.1054 433.7552 433.9436 437.1264 434.4496 438.4964 435.1755
29420.90 29584.62 44875.19 44666.74 45113.22 44875.19 42515.35 42605.81 42538.81 42564.85 42589.25 42589.25 42589.25 42605.81 23305.01 23386.35 23498.84 29420.90 29420.90 29584.62 29420.90 29584.62 29824.75 29584.62 29824.75 30787.30 30787.30 30858.82 30965.46 30965.46 31106.37 31106.37 31280.35
3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(a3P)4s4p(3Po) v5P2 3d4(a3P)4s4p(3Po) v5P1 3d4(a3P)4s4p(3Po) v5P3 3d4(a3P)4s4p(3Po) v5P2 3d5(4G)4p z5G2 3d5(4G)4p z5G6 3d5(4G)4p z5G3 3d5(4G)4p z5G4 3d5(4G)4p z5G5 3d5(4G)4p z5G5 3d5(4G)4p z5G5 3d5(4G)4p z5G6 3d5(6S)4p z7P2 3d5(6S)4p z7P3 3d5(6S)4p z7P4 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P3 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P3 3d4(5D)4s4p(3Po) z5F1 3d4(5D)4s4p(3Po) z5F1 3d4(5D)4s4p(3Po) z5F2 3d4(5D)4s4p(3Po) z5F3 3d4(5D)4s4p(3Po) z5F3 3d4(5D)4s4p(3Po) z5F4 3d4(5D)4s4p(3Po) z5F4 3d4(5D)4s4p(3Po) z5F5
7593.16 7593.16 21840.84 21847.88 21847.88 21856.94 20517.40 20519.60 20520.92 20523.69 20523.94 20523.69 20519.60 20523.94 0.00 0.00 0.00 7750.78 7810.82 7810.82 7927.47 7927.47 7927.47 8095.21 8307.57 7750.78 7810.82 7810.82 7927.47 8095.21 8095.21 8307.57 8307.57
3d5(6S)4s a5S2 3d5(6S)4s a5S2 3d5(4P)4s a5P3 3d5(4P)4s a5P2 3d5(4P)4s a5P2 3d5(4P)4s a5P1 3d5(4G)4s a5G2 3d5(4G)4s a5G6 3d5(4G)4s a5G3 3d5(4G)4s a5G4 3d5(4G)4s a5G5 3d5(4G)4s a5G4 3d5(4G)4s a5G6 3d5(4G)4s a5G5 3d5(6S)4s a7S3 3d5(6S)4s a7S3 3d5(6S)4s a7S3 3d44s2 a5D0 3d44s2 a5D1 3d44s2 a5D1 3d44s2 a5D2 3d44s2 a5D2 3d44s2 a5D2 3d44s2 a5D3 3d44s2 a5D4 3d44s2 a5D0 3d44s2 a5D1 3d44s2 a5D1 3d44s2 a5D2 3d44s2 a5D3 3d44s2 a5D3 3d44s2 a5D4 3d44s2 a5D4
Upper level 1
Lower level 1
In the table, the results of calculations performed on the basis of the results of the measurements by Heilig and Wendlandt [5] have been included, while in the case of
M ikSMS [GHz]
F ik [MHz/fm2]
2750(4) 2750(4) 1879(7) 1842(7) 1835(7) 1879(7) 590(4) 613(4) 598(4) 609(4) 618(4) 617(4) 621(4) 621(4) 294(10) 279(10) 264(10) 585(2) 581(2) 581(2) 585(2) 584(3) 578(3) 577(3) 583(3) 647(2) 641(2) 644(2) 650(5) 646(3) 644(2) 646(2) 648(3)
782(20) 769(15) 295(15) 390(90) 120(80) 287(20) 348(10) 333(15) 412(30) 332(20) 396(50) 354(20) 333(10) 356(20) 534(10) 557(10) 540(10) 887(13) 871(10) 881(12) 884(20) 962(30) 880(15) 759(20) 946(20) 907(23) 886(25) 917(14) 952(45) 902(20) 889(30) 890(30) 898(20)
the lines no. 4 their spectra have been recorded and the positions of the isotopes 50Cr, 52Cr and 53Cr have been determined (the component belonging to 54Cr isotope
Table 3 Mean values of the electronic factors M SMS and F ik for transitions between ik particular terms No Upper term 1 2 3 4 5 6
3d4(5D)4s4p(3Po) y5Po 3d4(a3P)4s4p(3Po) v5Po 3d5(4G)4p z5Go 3d54p z7Po 3d4(5D)4s4p(3Po) y5Po 3d4(5D)4s4p(3Po) z5Fo
Contents Lower 3d54p term
M ikSMS [GHz]
F ik [MHz/fm2]
0.06
3d5(6S)4s a5S 2750(4)
776(10)
0.28
3d5(4P)4s a5P 1859(18)
273(60)
0.59
3d5(4G)4s a5G
611(9)
0.65 0.06
3d5(6S)4s a7S 3d44s2 a5D
279(12)* 544(12)* 646(3) 884(40)
0.05
3d44s2 a5D
582(3)
358(30)
905(22)
The averaging is possible due to the lack of any J dependence within the limits of experimental error. * Position of 54Cr isotope on the basis of Wendlandt’s results [5], positions of the remaining isotopes from the own measurements.
Fig. 3. Dependence between the electronic factors of the specific mass shift and the field shift in chromium atom for transitions of the type 3d54p–3d54s and 3d44s4p–3d54s. The points are denoted with numbers referring to the numbered groups of rows in Table 3.
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B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
Fig. 4. Dependence of the specific mass shift electronic factor M ikSMS for transitions of the type 3d54p–3d54s and 3d44s4p–3d54s on the contribution of configuration 3d54p.
overlaps with the one of 52Cr isotope and it is not observable with the natural abundance of the sample). According to the well-established sharing rule, if the level under consideration can be described as a mixture of the levels belonging to different configurations X X W¼ an An where an ¼ 1; n
n
then the electronic factors MSMS and F i can be represented i as linear combinations of the factors for pure configurations X X an MnSMS and Fi ¼ an Fn ; MiSMS ¼ n
n
in the case of just two configurations the a n coefficients can be written as a and (1a). Then, if in the abscissa the MSMS i values and in the ordinate the F i values are represented, the points corresponding to the real levels should fall on a straight line. In all the investigated transitions in chromium atom, the lower levels can be treated as approximately pure (according
to the fine structure analysis, the admixture of the second configuration does not exceed 2%). Therefore, the plot of the dependence F ik on M ikSMS for the transitions of the type 3d54p–3d54s and 3d44s4p–3d54s should reflect—like an analogous plot for the transitions of the type 3d54p–3d44s2 and 3d44s4p–3d44s2—configuration mixing of the configurations 3d54p and 3d44s4p for the upper levels of the transitions studied. Because of the number of the results construction of the latter plot is aimless, thus only the former has been analyzed in detail. The plot is presented in Fig. 3. It is clearly visible that the experimental points fall on a straight line to a good accuracy. On the one hand, it indicates the dominant influence of the configuration mixing effects and, on the other hand, it proves that only contribution from the two mentioned configurations is essential. The aim of further analysis has been the estimation of the factors F ik and M ikSMS for transitions between the levels belonging to the pure configurations 3d54p or 3d44s4p and the configuration 3d54s. For this purpose, results of the fine structure analysis have been used, which served to determine coefficients n for particular levels [15]. The resulting dependences of M ikSMS and F ik on a are presented in Figs. 4 and 5. Application of the linear regression method to both dependences has yielded the values of the desired coefficients, which are listed in the first columns of Tables 4 and 5. As the transitions nos. 1 and 5 from Table 3 have a common upper level, it has been possible to determine F ik and M ikSMS for transitions between the levels belonging to the pure configurations 3d54p or 3d44s4p and the configuration 3d44s2. The values obtained are also presented in the second columns of Tables 4 and 5, respectively. 4.2. Crossed second order (CSO) effects Hamiltonian of the interaction responsible for the specific mass shift has a structure very similar to the fine structure
Fig. 5. Dependence of the field shift electronic factor F ik for transitions of the type 3d54p–3d54s and 3d44s4p–3d54s on the contribution of configuration 3d54p.
B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
39
Table 4 M ikSMS for transitions between the pure levels of investigated configurations in chromium atom, determined from experimental results and from ab initio calculations performed by the authors with the use of Cowan code, as well as from ab initio calculations from the work of Bauche and Crubellier [16] Type of transition
M SMS ik ,exp [GHz] without CSO
M SMS ik ,exp corr [GHz] including CSO
M SMS ik,th [GHz] (Cowan code ab initio calc.)
M SMS ik,th [GHz] (ab initio Bauche and Crubellier [16])
3d54pY3d54s 3d44s4pY3d54s 3d54pY3d44s2 3d44s4pY3d44s2
1128(74) 3043(53) 4460(60) 289(74)
552 3071 3884 261
273 2923 3442 246
410 2937 3565 218
hamiltonian. Therefore, it can be parametrized in a similar way. Apart from the part common to all the levels belonging to a certain configuration, some contributions with different coefficients for different levels appear. In the case of configurations in chromium atom three components play an essential role: g(3d,4p), g(4s,4p) and g(3d,4s) with coefficients analogous to the integrals G(3d,4p), G(4s,4p) and G(3d,4s), which appear in the expansion of the fine structure hamiltonian. Of course, the exact determination of all the three components from the measurements would require a much larger number of investigated levels than achieved in the present work. However, it seems interesting to estimate the influence of CSO effects on the results (according to Aufmuth [3], for the elements with a halffilled electron shell this influence can be essential). In order to perform such an estimation the authors have calculated (with the use of Froese-Fischer program [17]), the ratios between the parameters g(3d,4p), g(4s,4p) and g(3d,4s), as well as the respective coefficients for the particular levels. Then, the parameters g(4s,4p) and g(3d,4s) have been represented as multiples of the smallest parameter g(3d,4p). In this way, the number of independent parameters in the equations system increased only by one (the additional parameter being g(3d,4p)). The results of calculations performed with this method are presented in the third column of Table 4. The value obtained for the specific mass shift is g(3d4p)=55 MHz. It appears from Table 4 that taking into account CSO effects improves the agreement between the experimental M ikSMS values with the results of ab initio calculations (Section 5).
where N is the number of nl electrons, NV is the number of nV[l1] electrons and J 2(nl,nV[l1]) is a so-called Vinti integral [16,19]. Fi ¼
pha30 jWð0Þj2 f ðZ Þ Z
where |W(0)|2 is the change of a nonrelativistic electron charge density at the nucleus, while f(Z) is a correction coefficient required when a nonrelativistic electron density at the point nucleus is used (for chromium f(24)=1120 MHz/fm2 [2]). From the electronic factors for the levels, obtained in this way, the electronic factors for the transitions can be obtained by subtraction; these can be compared to the experimental results. Among several verified programs for MCHF calculations, the authors had the possibility to use two following ones: the Cowan code and the ATSP FroeseFischer program [17], both in Windows version available via Internet. The Cowan code admittedly does not determine directly MSMS or F i , but it allows calculation of all the i required quantities, i.e. Vinti integrals with their weighting coefficients and the function |W(0)|2. The Froese-Fischer program contains a module specialized for isotope shift calculations [17]. Results of calculations obtained with the Cowan code are presented in the fourth column of Table 4 and SMS the third one of Table 5. In the case of M ik , the results obtained by Bauche and Crubellier [16] have also been presented for comparison. It can be seen (Table 4) that the ab initio results for M ikSMS are in very good agreement with experimental results, except for transitions of the type 3d54pY3d54s. On the contrary, the calculated F ik (Table 5) are smaller than the experimental ones by a factor of 2. Both
5. Ab initio multiconfiguration Hartree-Fock calculations Both electronic factors considered: for the specific mass shift MSMS and for the field shift F i can be determined ab i initio with the use of multiconfiguration nonrelativistic Hartree-Fock (MCHF) [17] or relativistic (MCDF) [18] approximation for the individual levels. The respective formulae have the following form [17]: MiSMS ¼
XX N
NV
2lN N V J 2 ðnl; nV½l 1 Þ ð4l þ 2Þð4½l 1 þ 2Þ
Table 5 F ik for transitions between the pure levels of investigated configurations in chromium atom, determined from experimental results and from ab initio calculations performed by the authors with the use of Cowan code Type of transition
F ik,exp F ik,th F ik,exp/ F ik ,th [MHz/fm2] F i,th (Cowan code)2 [MHz/fm2] [MHz/fm2] (Cowan code)
3d54pY3d54s 1302(90) 649 3d44s4pY3d54s 863(90) 423 2962(90) 1429 3d54pY3d44s2 3d44s4pY3d44s2 797(90) 357
2.006 2.040 2.070 2.230
1298 845 2858 714
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B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
programs for ab initio calculations have yielded similar results in this respect. The CSO corrections, calculated for the experimental F ik values, are often below the experimental accuracy and do not improve the agreement with ab initio results significantly.
6. Conclusions The results of the performed isotope shift analysis may be used in estimation of the isotope shifts for another spectral lines of chromium atom, on the basis of the SMS determined M ik and F ik values for pure configurations. Because of the scarcity of published results of the measurements by another authors, concerning chromium atom, it is difficult to account for the fact, that the obtained F ik factors are twice as large as the ones derived from ab initio calculations. It might suggest that the yhr 2i values assumed in separation of SMS and FS require some correction, or it might be f(Z) values, that have to be corrected. Since the differences considerably exceed experimental errors, they have to be regarded substantial and might constitute an inspiration for detailed investigations of other spectral lines of chromium atom.
Acknowledgments We would like to thank Prof. W. Ertmer from University of Hannover for presenting us with the apparatus for atomic beam generation and Prof. G. Guthfhrlein from University of Bundeswehr in Hamburg for making the computer program bFitterQ available to us. This work has been supported by Poznan˜ University of Technology under Project DS 63-024/04.
References [1] J.R. Parrington, H.D. Knox, S.L. Breneman, E.M. Baum, F. Feiner, Chart of the Nuclides, 15th edition, General Electric, 1996.
[2] G. Fricke, C. Bernhardt, K. Heilig, L.A. Schaller, L. Schellenberg, E.B. Shera, C.W. de Jager, Nuclear ground state charge radii from electromagnetic interactions, At. Data Nucl. Data Tables 60 (1995) 177 – 285. [3] P. Aufmuth, Field shift crossed-second-order effects in lNs configurations, J. Phys., B At. Mol. Opt. Phys. 15 (1982) 3127 – 3140. [4] J.D. Weinstein, R. de Carvalho, C.I. Hancox, J.M. Doyle, Evaporative cooling of atomic chromium, Phys. Rev., A 65 (2002) 0216041 – 0216044. [5] K. Heilig, D. Wendlandt, Isotope shift in CrI, Phys. Lett. 25A (1967) 277 – 278. [6] J. Dembczynski, W. Ertmer, U. Johann, P. Stinner, High precision measurements of the hyperfine structure of seven metastable atomic states of 57Fe by laser Rf-double resonance, Z. Phys. A 294 (1980) 313 – 317. [7] W.J. Childs, L.S. Goodman, D. von Ehrenstein, Magnetic hyperfine interaction of Cr53, Phys. Rev. 132 (1963) 2128 – 2135. [8] W. Ertmer, U. Johann, R. Mosmann, Measurement of the nuclear quadrupole moment of 53Cr by Laser-Rf double resonance, Z. Phys. A 309 (1982) 1 – 4. [9] A. Krzykowski, B. Furmann, D. Stefanska, A. Jarosz, A. Kajoch, Hyperfine structures in the configuration 4f35d6s of the praseodymium atom, Opt. Commun. 140 (1997) 216 – 219. [10] B. Furmann, D. Stefan´ska, A. Krzykowski, A. Jarosz, A. Kajoch, Isotope shift in titanium atom, Z. Phys., D 37 (1996) 289 – 294. [11] W.H. King, Isotope Shifts in Atomic Spectra, Plenum, New York, 1984. [12] K. Heilig, A. Steudel, Changes in mean-square nuclear charge radii from optical isotope shifts, At. Data Nucl. Data Tables 14 (1974) 613 – 638. [13] P. Aufmuth, K. Heilig, A. Steudel, Changes in mean-square nuclear charge radii from optical isotope shifts, At. Data Nucl. Data Tables 37 (1987) 455 – 490. [14] H.D. Wohlfahrt, E.B. Shera, M.V. Hoehn, Y. Yamazaki, R.M. Steffen, Nuclear charge distributions in 1f7/2 shell nuclei from muonic X-ray measurements, Phys. Rev., C 23 (1981) 533 – 548. [15] J. Sugar, C. Corliss, Atomic energy levels of potassium through nickel, J. Phys. Chem. Ref. Data 14 (Suppl. 2) (1985) 264 – 278. [16] J. Bauche, A. Crubellier, Evaluation theoretique des deplacements isotopiques specifiques dans la serie 3d, J. Phys. 31 (1970) 429 – 434. [17] C. Froese Fischer, L. Smentek-Mielczarek, N. Vaeck, A program to computing isotope shift, Comput. Phys. Commun. 74 (1993) 415 – 431. [18] A.M. Mortensson-Pendrill, L. Pendrill, S. Salomonson, A. Ynnerman, H. Warston, Reanalysis of the isotope shift and nuclear charge radii in radioactive potassium isotopes, J. Phys., B At. Mol. Opt. Phys. 23 (1990) 1749 – 1761. [19] J. Bauche, Hartree-Fock evaluations of specific-mass isotope shifts, J. Phys. 35 (1974) 19 – 26.
Isotope shift in chromium B. Furmann*, A. Jarosz, D. Stefan´ska , J. Dembczyn´ski, E. Stachowska Faculty of Technical Physics, Poznan´ University of Technology, Nieszawska 13B, 60-965 Poznan´, Poland Received 17 June 2004; accepted 21 October 2004
Abstract Thirty-three spectral lines of chromium atom in the blue-violet region (425–465 nm) have been investigated with the method of laserinduced resonance fluorescence on an atomic beam. For all the lines, the isotope shifts for every pair of chromium isotopes have been determined. The lines can be divided into six groups, according to the configuration of the upper and lower levels. Electronic factors of the field shift and the specific mass shift ( F ik and M ikSMS, respectively) have been evaluated and the values for each pure configuration involved have been determined. Comparison of the values F ik and M ikSMS to the ab initio calculations results has been performed. The presence of crossed second order (CSO) effects has been observed. D 2004 Elsevier B.V. All rights reserved. Keywords: Isotope shift; Atomic beam
1. Introduction Chromium is an element with four stable isotopes: three even isotopes 50Cr, 52Cr, 54Cr and an odd isotope 53Cr with nuclear spin I=3/2. The natural isotope abundance is as follows [1]: 50Cr—4.35%, 52Cr—83.79%, 53Cr—9.5% and 54 Cr—2.36%. Isotope shifts, observed in chromium spectrum, are interesting from the point of view of both atomic and nuclear physics. The chromium isotope 52Cr possesses a magic neutron number N=28. In this situation, it can be expected (and the results of muonic atoms and electron scattering investigations confirm the expectations [2]) that the mean square charge radius hr 2i should be minimum for this isotope. As the field shift is directly dependent on yhr 2i, an anomalous change of the charge distribution is reflected in the results of measurements of the optical isotope shifts. Chromium is an element with a half-filled 3d electron shell. According to the theoretical analysis of second-order effects in the field shift, performed by Aufmuth [3], the elements
* Corresponding author. Tel.: +48 61 665 3226; fax: +48 61 665 3239. E-mail address: [email protected] (B. Furmann). 0584-8547/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sab.2004.10.007
with half-filled electron shells should exhibit the most pronounced effects of the kind mentioned. Recently, the interest in spectroscopic properties of chromium atom has increased as a result of its application in experiments concerned with cooling of atoms. It appeared that cooling of chromium atoms allows to obtain simultaneously a Bose-Einstein condensate (cooling of 52Cr isotope) and a degenerate Fermi gas (53Cr isotope) [4]. The technique applied in this case (evaporative cooling) takes advantage of elastic collisions of 52 Cr. The fundamental problem is an unfavorable ratio between the elastic and the inelastic (undesirable) collisions cross-sections in the temperature region near 10 mK. Below this temperature, the evaporative cooling technique again becomes effective. The way to overcome this problem is application of laser cooling in the vicinity of 10 mK, which requires the detailed knowledge of isotope shifts [4]. In this context, it seems somewhat surprising that the only papers concerning isotope shifts in chromium, that can be found in the literature, are the ones by Heilig and Wendlandt [5], published in the 1960s of the last century. A likely explanation could be experimental difficulties, as well as problems with satisfactory interpretation of the
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B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
results of the measurements. Experimental difficulties are mainly concerned with relatively small splitting of the chromium spectral lines. A typical width of the spectrum, containing all the four stable isotopes (hyperfine structure of 53Cr isotope included) is of the order of 1–2 GHz. Thus, in order to resolve the spectral components, Doppler-free spectroscopic methods have to be applied. In our experiment, the method of laser-induced fluorescence (LIF) on an atomic beam has been used. This method has, however, an essential disadvantage—it can be used only for those transitions, whose lower level is metastable, while the thermal population of the lower level decreases rapidly with its increasing energy. Unfortunately, the number of low-lying levels in chromium atom is rather small. On the other hand, in order to allow to take into account the configuration mixing effects as well as the second-order interactions in the analysis of the isotope shifts, a possibly large number of experimental data is required. In this work, we present the results of isotope shift measurements for 33 optical transitions in chromium atom. Most of them concern the same terms (the levels differing in their J quantum numbers). After subtraction of normal mass shifts, for all the transitions studied, the residual isotope shifts have been separated into specific mass shifts and field shifts. Using the results of the combined analysis of the optical spectra, muonic atoms and electron scattering measurements [2], the isotope shift electronic factors for individual transitions have been determined. With the use of the sharing rule, the electronic factors for the pairs of pure configurations 3d54p, 3d44s4p, 3d44s2 and 3d54s have been estimated. The results have been compared to the results of nonrelativistic multiconfiguration HartreeFock calculations.
2. Experimental The method of LIF on an atomic beam has been used for measurements of the relative positions of all chromium isotopes. Chromium, placed in a graphite crucible, has been heated and evaporated by the electron beam bombardment. Electron beam emitted from electron gun has been focused and deflected by system of electromagnets. An auxiliary discharge (between additional electrode placed just above the crucible and the melted metallic sample in the crucible) has been applied to allow better population of the high-lying metastable levels. Atomic beam has been formed by a system of diaphragms (each of 1 mm diameter). The divergency of the atomic beam amounted 8d 103 rad. A laser beam generated by a stabilized single-mode ring dye laser (Coherent model CR-699-21), operating on stilbene 3 and pumped by an Ar+ laser, has crossed the atomic beam perpendicularly. The fluorescence light, collected by an optical detection system, has been recorded by a photo-
multiplier positioned perpendicularly to both the laser and the atomic beam directions. Optical system has been carefully shielded against scattered light from the oven. The more detailed description of atomic beam setup can be found in Ref. [6]. For each investigated optical transition on the average, 20 scans have been recorded. An example of a recorded spectrum, containing all the stable chromium isotopes, is presented in Fig. 1. Because of a very high share of 52Cr in the natural abundance, the spectral component belonging to this isotope has always been much stronger than the remaining ones and it has easily become saturated. Therefore, 10 scans with low laser power have been recorded in order to determine exactly the position of this component, and then another 10 scans with higher laser power with the aim of determination of positions of the components belonging to the remaining isotopes. Initially, the recorded spectra have been calibrated with an approximate value of the free spectral range (FSR) of the frequency marker—150 MHz. For each scan, the relative positions of all the isotopes, as well as the hfs parameters for the odd isotope 53Cr, have been fitted simultaneously. A program bFitterQ, received from the University of Bundeswehr in Hamburg, has been used for that purpose. The final values for this stage have been the mean values of those obtained in individual scans, and the errors have been the mean standard deviations. The procedure described has been applied also to the lines involving levels measured very accurately with resonance method by Childs et al. [7] or Ertmer et al. [8]. Intervals, known with high accuracy from Refs. [7,8], have been evaluated from our measurements and compared with the former ones. The FSR value of the marker has been corrected until a least mean difference in the values of intervals has been obtained. The corrected FSR amounts to 149.98 MHz (this value is consistent with the one obtained with a similar method in our earlier works [9,10]). The final values quoted are calibrated with the corrected FSR value of
Fig. 1. Example of a recorded spectrum of chromium atom, containing all the four stable isotopes (k=437.1264 nm). Center of mass of the hyperfine structure of the isotope 53 is marked.
B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
35
Table 1 Transitions in chromium, investigated from the point of view of the isotope shifts and results of the measurements k [nm] (in air)
E [cm ]
Designation
E [cm ]
Designation
458.0043 454.5945 434.0122 438.1109 429.7031 434.3158 454.4604 452.6443 454.0488 453.5695 453.0730 453.0679 452.9839 452.7332 428.9733 427.4806 425.4331 461.3357 462.6174 459.1389 465.1282 461.6120 456.5498 465.2152 464.6148 433.9713 435.1054 433.7552 433.9436 437.1264 434.4496 438.4964 435.1755
29420.90 29584.62 44875.19 44666.74 45113.22 44875.19 42515.35 42605.81 42538.81 42564.85 42589.25 42589.25 42589.25 42605.81 23305.01 23386.35 23498.84 29420.90 29420.90 29584.62 29420.90 29584.62 29824.75 29584.62 29824.75 30787.30 30787.30 30858.82 30965.46 30965.46 31106.37 31106.37 31280.35
3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(a3P)4s4p(3Po) v5P2 3d4(a3P)4s4p(3Po) v5P1 3d4(a3P)4s4p(3Po) v5P3 3d4(a3P)4s4p(3Po) v5P2 3d5(4G)4p z5G2 3d5(4G)4p z5G6 3d5(4G)4p z5G3 3d5(4G)4p z5G4 3d5(4G)4p z5G5 3d5(4G)4p z5G5 3d5(4G)4p z5G5 3d5(4G)4p z5G6 3d5(6S)4p z7P2 3d5(6S)4p z7P3 3d5(6S)4p z7P4 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P3 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P3 3d4(5D)4s4p(3Po) z5F1 3d4(5D)4s4p(3Po) z5F1 3d4(5D)4s4p(3Po) z5F2 3d4(5D)4s4p(3Po) z5F3 3d4(5D)4s4p(3Po) z5F3 3d4(5D)4s4p(3Po) z5F4 3d4(5D)4s4p(3Po) z5F4 3d4(5D)4s4p(3Po) z5F5
7593.16 7593.16 21840.84 21847.88 21847.88 21856.94 20517.40 20519.60 20520.92 20523.69 20523.94 20523.69 20519.60 20523.94 0.00 0.00 0.00 7750.78 7810.82 7810.82 7927.47 7927.47 7927.47 8095.21 8307.57 7750.78 7810.82 7810.82 7927.47 8095.21 8095.21 8307.57 8307.57
3d5(6S)4s a5S2 3d5(6S)4s a5S2 3d5(4P)4s a5P3 3d5(4P)4s a5P2 3d5(4P)4s a5P2 3d5(4P)4s a5P1 3d5(4G)4s a5G2 3d5(4G)4s a5G6 3d5(4G)4s a5G3 3d5(4G)4s a5G4 3d5(4G)4s a5G5 3d5(4G)4s a5G4 3d5(4G)4s a5G6 3d5(4G)4s a5G5 3d5(6S)4s a7S3 3d5(6S)4s a7S3 3d5(6S)4s a7S3 3d44s2 a5D0 3d44s2 a5D1 3d44s2 a5D1 3d44s2 a5D2 3d44s2 a5D2 3d44s2 a5D2 3d44s2 a5D3 3d44s2 a5D4 3d44s2 a5D0 3d44s2 a5D1 3d44s2 a5D1 3d44s2 a5D2 3d44s2 a5D3 3d44s2 a5D3 3d44s2 a5D4 3d44s2 a5D4
Upper level 1
Lower level 1
the marker. All measured transitions and obtained results of isotope shift are presented in Table 1.
IS52,50 [MHz]
IS53,50 [MHz]
IS54,50 [MHz]
1898.2(0.7) 1893.6(0.7) 1177.5(0.9) 1153.7(3.8) 1125.8(5.5) 1177.0(0.7) 152.4(1.5) 170.2(0.4) 152.5(0.6) 167.8(1.1) 166.0(4.1) 172.0(1.7) 176.2(3.1) 173.9(2.1) 115.0(2.0) 121.1(2.0) 130.3(2.0) 787.3(0.3) 782.3(0.6) 784.4(0.6) 784.0(1.0) 792.1(2.3) 783.8(0.6) 767.7(3.0) 788.0(0.6) 852.8(1.6) 846.1(1.5) 851.2(0.6) 860.6(1.2) 850.0(0.2) 848.7(1.8) 847.1(3.4) 852.1(0.3)
2722.7(0.9) 2724.1(0.9) 1707.6(0.6) 1686.0(7.7) 1658.6(8.5) 1707.4(0.9) 256.8(1.4) 281.7(0.8) 269.2(2.3) 276.4(0.8) 298.7(5.5) 285.1(1.0) 292.3(3.5) 293.1(2.5) 128.8(2.0) 141.2(2.0) 153.1(2.0) 1071.7(1.4) 1068.1(2.5) 1071.5(1.0) 1072.0(0.8) 1069.6(2.4) 1070.9(1.4) 1065.1(0.8) 1069.7(1.1) 1171.5(1.9) 1163.9(1.6) 1166.8(1.3) 1168.8(1.1) 1167.3(0.5) 1168.0(1.8) 1166.2(1.9) 1170.5(0.3)
3480.4(5.7) 3474.3(0.9) 2202.0(1.3) 2132.3(9.5) 2140.8(7.0) 2202.9(1.2) 372.8(0.8) 403.7(1.7) 386.9(3.6) 399.1(1.5) 408.1(1.0) 412.1(1.8) 415.1(1.2) 415.6(2.0)
1315.9(0.8) 1309.6(2.6) 1311.3(1.2) 1309.8(1.2) 1308.4(1.4) 1310.3(0.6) 1306.2(1.5) 1303.6(0.5) 1436.9(2.6) 1428.8(2.0) 1431.7(0.8) 1442.8(1.8) 1432.7(0.6) 1433.1(0.7) 1430.0(2.1) 1437.8(0.2)
where M ikSMS is an electronic factor for the transition. The field shift can be expressed as: FSAAV Fik yhr2 iAAV ;
3. Fundamental The isotope shift in a transition ik of wavenumber m ik : ISAAV ¼ mAik mAV ik may be split into two terms [11]: ISAAV ¼ MSAAV þ FSAAV ; where: MSAAV —mass shift and FSAAV —field shift. The mass shift consists of two contributions: MSAAV ¼ NMSAAV þ SMSAAV ; NMSAAV ¼
me AAV mik mp A AV
SMSAAV ¼ MikSMS
AAV A AV
normal mass shift ;
specific mass shift ;
where F ik is an electronic factor for the transition and yhr 2iAAV is a change of the mean square charge radius of the nucleus (a nuclear factor). The normal mass shift can easily be calculated and is often subtracted from the total isotope shift to give the residual isotope shift RISAAV RISAAV uISAAV NMSAAV ¼ SMSAAV þ FSAAV AAV þ Fik yhr2 iAAV : ¼ MikSMS A AV Separation of the residual isotope shifts into field shifts and specific mass shifts is conventionally achieved with the use of King diagrams, if the abscissa contains RIS values for a transition, for which F ik (and consequently also M ikSMS) is known from previous measurements, and the ordinate—RIS values of a transition under study. On the basis of the slope and the intersection coefficients of the resulting straight line, F ik and M ikSMS of all other investigated transitions can be
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B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
Fig. 2. Modified King plots (modified RIS vs. modified yhr 2i) for four types of measured transitions: (a) 3d4(5D)4s4p(3Po) y5Po2–3d5(6S)4s a5S2, (b) 3d4(5D)4s4p(3Po) z5Fo2–3d44s2 a5D1, (c) 3d5(4G)4p z5Go5–3d5(4G)4s a5G6, (d) 3d4(a3P)4s4p(3Po) v5Po2–3d5(4G)4s a5G6. The error bars in the abscissa result from the values given in Ref. [13], while those in the ordinate—from the error values given in Table 1 (as already mentioned in text above, they have been calculated as mean standard deviations for 10 values).
estimated, respectively. Since in chromium atom F ik or SMS M ik is not known for any spectral line, we have used an alternative method. Separation of SMSAAV and FSAAV can be also performed with the use of another kind of King plot: modified optical RISAAV vs. modified yhr 2iAAV (for chromium model independent yhr 2iAAV are known from optical measurements, muonic X-ray spectroscopy and electron scattering [12–14]). Modified RISAAV is defined as follows: lRISAAV ¼
A AV 52 50 RISAAV A AV 52 50
(the isotopes 52Cr and 50Cr have been chosen as pair of reference isotopes) and yhr 2iAAV have been modified in the same way. From this kind of King diagram, the F ik value can be determined directly as the slope of the straight line and the M ikSMS value is obtained through multiplication of the intersection coefficient by (5250)/(5250). Some examples of King diagrams are presented in Fig. 2, while the values of M ikSMS and F ik , obtained for all the investigated transitions, are listed in Table 2.
4. Analysis of the results 4.1. Configuration mixing and sharing rule In the first approximation both MSMS and F i for all the i levels belonging to a pure configuration should be equal.
Admittedly, from the results obtained it is not possible to determine the electronic factors SMS and FS for the individual levels participating in the transitions measured (only the factors for transitions are known), but because of the relations between the factors MikSMS ¼ MiSMS MkSMS ; Fik ¼ Fi Fk ; also the electronic factors for transitions between the levels belonging to the same configurations should be equal, provided the levels were pure and the higher order effects were neglected (these effects are discussed in the next section). At the first stage of the analysis of the obtained M ikSMS and F ik factors, the dependence on J quantum number, i.e. variation within the same term, has been investigated. In this case, comparison of the results for transitions with a common upper or lower level has yielded the differences between the factors for the levels with various J. The analysis performed in this way proved, that all the factors for the levels belonging to the same term are indeed equal within the limits of experimental error. Therefore, in the following, average values for each term have been used. SMS The final values of M ik and F ik , obtained in this way, are presented in Table 3. It can be seen that, if the values within one term are equal, the values of both M ikSMS and F ik for various terms belonging to the same configuration differ considerably.
B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
37
Table 2 Electronic factors F ik and M ikSMS for the studied transitions k [nm] (in air)
E [cm ]
Designation
E [cm ]
Designation
458.0043 454.5945 434.0122 438.1109 429.7031 434.3158 454.4604 452.6443 454.0488 453.5695 453.0730 453.0679 452.9839 452.7332 428.9733 427.4806 425.4331 461.3357 462.6174 459.1389 465.1282 461.6120 456.5498 465.2152 464.6148 433.9713 435.1054 433.7552 433.9436 437.1264 434.4496 438.4964 435.1755
29420.90 29584.62 44875.19 44666.74 45113.22 44875.19 42515.35 42605.81 42538.81 42564.85 42589.25 42589.25 42589.25 42605.81 23305.01 23386.35 23498.84 29420.90 29420.90 29584.62 29420.90 29584.62 29824.75 29584.62 29824.75 30787.30 30787.30 30858.82 30965.46 30965.46 31106.37 31106.37 31280.35
3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(a3P)4s4p(3Po) v5P2 3d4(a3P)4s4p(3Po) v5P1 3d4(a3P)4s4p(3Po) v5P3 3d4(a3P)4s4p(3Po) v5P2 3d5(4G)4p z5G2 3d5(4G)4p z5G6 3d5(4G)4p z5G3 3d5(4G)4p z5G4 3d5(4G)4p z5G5 3d5(4G)4p z5G5 3d5(4G)4p z5G5 3d5(4G)4p z5G6 3d5(6S)4p z7P2 3d5(6S)4p z7P3 3d5(6S)4p z7P4 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P1 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P3 3d4(5D)4s4p(3Po) y5P2 3d4(5D)4s4p(3Po) y5P3 3d4(5D)4s4p(3Po) z5F1 3d4(5D)4s4p(3Po) z5F1 3d4(5D)4s4p(3Po) z5F2 3d4(5D)4s4p(3Po) z5F3 3d4(5D)4s4p(3Po) z5F3 3d4(5D)4s4p(3Po) z5F4 3d4(5D)4s4p(3Po) z5F4 3d4(5D)4s4p(3Po) z5F5
7593.16 7593.16 21840.84 21847.88 21847.88 21856.94 20517.40 20519.60 20520.92 20523.69 20523.94 20523.69 20519.60 20523.94 0.00 0.00 0.00 7750.78 7810.82 7810.82 7927.47 7927.47 7927.47 8095.21 8307.57 7750.78 7810.82 7810.82 7927.47 8095.21 8095.21 8307.57 8307.57
3d5(6S)4s a5S2 3d5(6S)4s a5S2 3d5(4P)4s a5P3 3d5(4P)4s a5P2 3d5(4P)4s a5P2 3d5(4P)4s a5P1 3d5(4G)4s a5G2 3d5(4G)4s a5G6 3d5(4G)4s a5G3 3d5(4G)4s a5G4 3d5(4G)4s a5G5 3d5(4G)4s a5G4 3d5(4G)4s a5G6 3d5(4G)4s a5G5 3d5(6S)4s a7S3 3d5(6S)4s a7S3 3d5(6S)4s a7S3 3d44s2 a5D0 3d44s2 a5D1 3d44s2 a5D1 3d44s2 a5D2 3d44s2 a5D2 3d44s2 a5D2 3d44s2 a5D3 3d44s2 a5D4 3d44s2 a5D0 3d44s2 a5D1 3d44s2 a5D1 3d44s2 a5D2 3d44s2 a5D3 3d44s2 a5D3 3d44s2 a5D4 3d44s2 a5D4
Upper level 1
Lower level 1
In the table, the results of calculations performed on the basis of the results of the measurements by Heilig and Wendlandt [5] have been included, while in the case of
M ikSMS [GHz]
F ik [MHz/fm2]
2750(4) 2750(4) 1879(7) 1842(7) 1835(7) 1879(7) 590(4) 613(4) 598(4) 609(4) 618(4) 617(4) 621(4) 621(4) 294(10) 279(10) 264(10) 585(2) 581(2) 581(2) 585(2) 584(3) 578(3) 577(3) 583(3) 647(2) 641(2) 644(2) 650(5) 646(3) 644(2) 646(2) 648(3)
782(20) 769(15) 295(15) 390(90) 120(80) 287(20) 348(10) 333(15) 412(30) 332(20) 396(50) 354(20) 333(10) 356(20) 534(10) 557(10) 540(10) 887(13) 871(10) 881(12) 884(20) 962(30) 880(15) 759(20) 946(20) 907(23) 886(25) 917(14) 952(45) 902(20) 889(30) 890(30) 898(20)
the lines no. 4 their spectra have been recorded and the positions of the isotopes 50Cr, 52Cr and 53Cr have been determined (the component belonging to 54Cr isotope
Table 3 Mean values of the electronic factors M SMS and F ik for transitions between ik particular terms No Upper term 1 2 3 4 5 6
3d4(5D)4s4p(3Po) y5Po 3d4(a3P)4s4p(3Po) v5Po 3d5(4G)4p z5Go 3d54p z7Po 3d4(5D)4s4p(3Po) y5Po 3d4(5D)4s4p(3Po) z5Fo
Contents Lower 3d54p term
M ikSMS [GHz]
F ik [MHz/fm2]
0.06
3d5(6S)4s a5S 2750(4)
776(10)
0.28
3d5(4P)4s a5P 1859(18)
273(60)
0.59
3d5(4G)4s a5G
611(9)
0.65 0.06
3d5(6S)4s a7S 3d44s2 a5D
279(12)* 544(12)* 646(3) 884(40)
0.05
3d44s2 a5D
582(3)
358(30)
905(22)
The averaging is possible due to the lack of any J dependence within the limits of experimental error. * Position of 54Cr isotope on the basis of Wendlandt’s results [5], positions of the remaining isotopes from the own measurements.
Fig. 3. Dependence between the electronic factors of the specific mass shift and the field shift in chromium atom for transitions of the type 3d54p–3d54s and 3d44s4p–3d54s. The points are denoted with numbers referring to the numbered groups of rows in Table 3.
38
B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
Fig. 4. Dependence of the specific mass shift electronic factor M ikSMS for transitions of the type 3d54p–3d54s and 3d44s4p–3d54s on the contribution of configuration 3d54p.
overlaps with the one of 52Cr isotope and it is not observable with the natural abundance of the sample). According to the well-established sharing rule, if the level under consideration can be described as a mixture of the levels belonging to different configurations X X W¼ an An where an ¼ 1; n
n
then the electronic factors MSMS and F i can be represented i as linear combinations of the factors for pure configurations X X an MnSMS and Fi ¼ an Fn ; MiSMS ¼ n
n
in the case of just two configurations the a n coefficients can be written as a and (1a). Then, if in the abscissa the MSMS i values and in the ordinate the F i values are represented, the points corresponding to the real levels should fall on a straight line. In all the investigated transitions in chromium atom, the lower levels can be treated as approximately pure (according
to the fine structure analysis, the admixture of the second configuration does not exceed 2%). Therefore, the plot of the dependence F ik on M ikSMS for the transitions of the type 3d54p–3d54s and 3d44s4p–3d54s should reflect—like an analogous plot for the transitions of the type 3d54p–3d44s2 and 3d44s4p–3d44s2—configuration mixing of the configurations 3d54p and 3d44s4p for the upper levels of the transitions studied. Because of the number of the results construction of the latter plot is aimless, thus only the former has been analyzed in detail. The plot is presented in Fig. 3. It is clearly visible that the experimental points fall on a straight line to a good accuracy. On the one hand, it indicates the dominant influence of the configuration mixing effects and, on the other hand, it proves that only contribution from the two mentioned configurations is essential. The aim of further analysis has been the estimation of the factors F ik and M ikSMS for transitions between the levels belonging to the pure configurations 3d54p or 3d44s4p and the configuration 3d54s. For this purpose, results of the fine structure analysis have been used, which served to determine coefficients n for particular levels [15]. The resulting dependences of M ikSMS and F ik on a are presented in Figs. 4 and 5. Application of the linear regression method to both dependences has yielded the values of the desired coefficients, which are listed in the first columns of Tables 4 and 5. As the transitions nos. 1 and 5 from Table 3 have a common upper level, it has been possible to determine F ik and M ikSMS for transitions between the levels belonging to the pure configurations 3d54p or 3d44s4p and the configuration 3d44s2. The values obtained are also presented in the second columns of Tables 4 and 5, respectively. 4.2. Crossed second order (CSO) effects Hamiltonian of the interaction responsible for the specific mass shift has a structure very similar to the fine structure
Fig. 5. Dependence of the field shift electronic factor F ik for transitions of the type 3d54p–3d54s and 3d44s4p–3d54s on the contribution of configuration 3d54p.
B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
39
Table 4 M ikSMS for transitions between the pure levels of investigated configurations in chromium atom, determined from experimental results and from ab initio calculations performed by the authors with the use of Cowan code, as well as from ab initio calculations from the work of Bauche and Crubellier [16] Type of transition
M SMS ik ,exp [GHz] without CSO
M SMS ik ,exp corr [GHz] including CSO
M SMS ik,th [GHz] (Cowan code ab initio calc.)
M SMS ik,th [GHz] (ab initio Bauche and Crubellier [16])
3d54pY3d54s 3d44s4pY3d54s 3d54pY3d44s2 3d44s4pY3d44s2
1128(74) 3043(53) 4460(60) 289(74)
552 3071 3884 261
273 2923 3442 246
410 2937 3565 218
hamiltonian. Therefore, it can be parametrized in a similar way. Apart from the part common to all the levels belonging to a certain configuration, some contributions with different coefficients for different levels appear. In the case of configurations in chromium atom three components play an essential role: g(3d,4p), g(4s,4p) and g(3d,4s) with coefficients analogous to the integrals G(3d,4p), G(4s,4p) and G(3d,4s), which appear in the expansion of the fine structure hamiltonian. Of course, the exact determination of all the three components from the measurements would require a much larger number of investigated levels than achieved in the present work. However, it seems interesting to estimate the influence of CSO effects on the results (according to Aufmuth [3], for the elements with a halffilled electron shell this influence can be essential). In order to perform such an estimation the authors have calculated (with the use of Froese-Fischer program [17]), the ratios between the parameters g(3d,4p), g(4s,4p) and g(3d,4s), as well as the respective coefficients for the particular levels. Then, the parameters g(4s,4p) and g(3d,4s) have been represented as multiples of the smallest parameter g(3d,4p). In this way, the number of independent parameters in the equations system increased only by one (the additional parameter being g(3d,4p)). The results of calculations performed with this method are presented in the third column of Table 4. The value obtained for the specific mass shift is g(3d4p)=55 MHz. It appears from Table 4 that taking into account CSO effects improves the agreement between the experimental M ikSMS values with the results of ab initio calculations (Section 5).
where N is the number of nl electrons, NV is the number of nV[l1] electrons and J 2(nl,nV[l1]) is a so-called Vinti integral [16,19]. Fi ¼
pha30 jWð0Þj2 f ðZ Þ Z
where |W(0)|2 is the change of a nonrelativistic electron charge density at the nucleus, while f(Z) is a correction coefficient required when a nonrelativistic electron density at the point nucleus is used (for chromium f(24)=1120 MHz/fm2 [2]). From the electronic factors for the levels, obtained in this way, the electronic factors for the transitions can be obtained by subtraction; these can be compared to the experimental results. Among several verified programs for MCHF calculations, the authors had the possibility to use two following ones: the Cowan code and the ATSP FroeseFischer program [17], both in Windows version available via Internet. The Cowan code admittedly does not determine directly MSMS or F i , but it allows calculation of all the i required quantities, i.e. Vinti integrals with their weighting coefficients and the function |W(0)|2. The Froese-Fischer program contains a module specialized for isotope shift calculations [17]. Results of calculations obtained with the Cowan code are presented in the fourth column of Table 4 and SMS the third one of Table 5. In the case of M ik , the results obtained by Bauche and Crubellier [16] have also been presented for comparison. It can be seen (Table 4) that the ab initio results for M ikSMS are in very good agreement with experimental results, except for transitions of the type 3d54pY3d54s. On the contrary, the calculated F ik (Table 5) are smaller than the experimental ones by a factor of 2. Both
5. Ab initio multiconfiguration Hartree-Fock calculations Both electronic factors considered: for the specific mass shift MSMS and for the field shift F i can be determined ab i initio with the use of multiconfiguration nonrelativistic Hartree-Fock (MCHF) [17] or relativistic (MCDF) [18] approximation for the individual levels. The respective formulae have the following form [17]: MiSMS ¼
XX N
NV
2lN N V J 2 ðnl; nV½l 1 Þ ð4l þ 2Þð4½l 1 þ 2Þ
Table 5 F ik for transitions between the pure levels of investigated configurations in chromium atom, determined from experimental results and from ab initio calculations performed by the authors with the use of Cowan code Type of transition
F ik,exp F ik,th F ik,exp/ F ik ,th [MHz/fm2] F i,th (Cowan code)2 [MHz/fm2] [MHz/fm2] (Cowan code)
3d54pY3d54s 1302(90) 649 3d44s4pY3d54s 863(90) 423 2962(90) 1429 3d54pY3d44s2 3d44s4pY3d44s2 797(90) 357
2.006 2.040 2.070 2.230
1298 845 2858 714
40
B. Furmann et al. / Spectrochimica Acta Part B 60 (2005) 33–40
programs for ab initio calculations have yielded similar results in this respect. The CSO corrections, calculated for the experimental F ik values, are often below the experimental accuracy and do not improve the agreement with ab initio results significantly.
6. Conclusions The results of the performed isotope shift analysis may be used in estimation of the isotope shifts for another spectral lines of chromium atom, on the basis of the SMS determined M ik and F ik values for pure configurations. Because of the scarcity of published results of the measurements by another authors, concerning chromium atom, it is difficult to account for the fact, that the obtained F ik factors are twice as large as the ones derived from ab initio calculations. It might suggest that the yhr 2i values assumed in separation of SMS and FS require some correction, or it might be f(Z) values, that have to be corrected. Since the differences considerably exceed experimental errors, they have to be regarded substantial and might constitute an inspiration for detailed investigations of other spectral lines of chromium atom.
Acknowledgments We would like to thank Prof. W. Ertmer from University of Hannover for presenting us with the apparatus for atomic beam generation and Prof. G. Guthfhrlein from University of Bundeswehr in Hamburg for making the computer program bFitterQ available to us. This work has been supported by Poznan˜ University of Technology under Project DS 63-024/04.
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