Coulomb displacement energies as a probe for nucleon pairing in the f_{7/2} shell

Coulomb displacement energies as a probe for nucleon pairing in the $f_{7/2}$ shell


Coulomb displacement energies of mirror nuclei have been studied via a series of high-precision -value measurements with the double Penning trap mass spectrometer JYFLTRAP. Most recently, the values of the -shell mirror nuclei V ( keV) and Mn ( keV) have been measured with an unprecedented precision. The data reveal a 16-keV () offset in the adopted Atomic Mass Evaluation 2012 value of Mn suggesting the need for further measurements to verify the breakdown of the quadratic form of the isobaric multiplet mass equation. Precisely measured values confirm that the pairing effect in the Coulomb energies is quenched when entering the shell and reaches a minimum in the midshell.

21.10.Dr, 21.10.Sf, 23.40.-s, 27.40.+z

The strong nuclear force is nearly charge independent, thus nuclei close to the line exhibit exchange symmetry between protons and neutrons. Mirror nuclei have an equal number of nucleons but the proton and neutron numbers are mutually interchanged. They should have similar energy levels assuming that the nuclear force between two protons is identical to the force between two neutrons (charge symmetry). Therefore, the energy difference between the ground states of mirror nuclei ( value) should result from the difference in their Coulomb energies once the neutron-proton mass difference () has been taken into acccount (1); (2). Approximating nuclei as uniformly charged spheres, the Coulomb displacement energies () should increase linearly as a function of , where is the average charge of the mirror pair. The first studies on Coulomb displacement energies between mirror nuclei focused on extracting charge radii from this linear behavior (3); (4); (5). It was soon noted that the CDE values exhibit shell and pairing effects (6); (7); (8); (9); (10). Despite various corrections included in theoretical calculations, about 7 % of the experimental CDE remained unexplained (Nolen-Schiffer anomaly) (11). The effect of charge-symmetry breaking (12) or isospin-nonconserving forces (13) have recently been investigated to explain this anomaly. Accurate values of mirror nuclei are also essential for high-precision beta-decay values (14); (15); (16) to be discussed in a separate paper.

The energy differences between excited states of mirror nuclei in the shell have been studied intensively via in-beam gamma-ray spectroscopy (17). The variation of Coulomb energy differences as a function of excitation energy and spin has been interpreted fairly well with large-scale shell-model calculations albeit an anomalously high two-body Coulomb matrix element for the proton pair has to be implemented to match with the observations (18); (19); (17). To date, the values of most of the shell mirror nuclei have been known with a modest precision of around 10 keV (20); (21). For comparison, mirror energy differences in V (22) and Mn (23) are known up to 7159 keV () and 10726 keV (), respectively, with a precision of around 1 keV or better. In this Rapid Communication, we report on the first high-precision -value measurements of V and Mn. This work continues the successful programme on mirror nuclei at the Ion Guide Isotope Separator On-Line (IGISOL) facility: Mg (24), P (25), S (26); (27), Co, Co, Ni, Cu, and Zn (28).

The ions of interest were produced via TiV and CrMn reactions with a 40-MeV proton beam on enriched Ti and Cr targets at IGISOL (29); (30). They were extracted from the ion-guide gas cell with the help of differential pumping and a sextupole ion guide (SPIG) (31) and subsequently mass-separated with a 55 dipole magnet. The mass-to-charge-separated continuous beam was cooled in the radio-frequency quadrupole cooler (RFQ) (32) and released as short bunches into the JYFLTRAP double Penning trap mass spectrometer (33); (34). The buffer-gas cooling technique (35) was applied in the first trap to select only V or Mn ions for the high-precision mass measurements using the time-of-flight ion cyclotron resonance technique (36); (37) in the second trap. Ion cyclotron excitations were performed via Ramsey’s method of time-separated oscillatory fields (38); (39); (40) with two 25-ms excitation periods separated by a 150-ms waiting time. An example of a time-of-flight spectrum is shown in Fig. 1. The ion cyclotron resonance frequency is , where is the charge of the ion, is the magnetic field in the trap and the mass of the ion of interest. The magnetic field was calibrated with the corresponding beta-decay daughter nuclei, Ti ( u (21)) and Cr ( u (21)), which were simultaneously produced with the decay-parent ions. From the measured parent-daughter frequency ratio, the value was calculated as:


With doublets, this will result in a high-precision value even if the daughter nucleus is known only with a modest precision. Additionally, mass-dependent systematic shifts cancel to a level that is well below the statistical uncertainty quoted in this work (41).

The cyclotron resonance frequencies were fitted with the theoretical lineshape (37); (42); (40) (see Fig. 1). The measured frequencies were corrected for the count-rate-effect (43) whenever possible, similarly as in Ref. (44). In order to take into account fluctuations in the magnetic field, a correction of , where is the time between the two reference measurements, was quadratically added to the statistical uncertainty of each frequency ratio. The cyclotron frequency measurements were carried out interleavedly to reduce the time between the reference measurements. After one frequency scan on the reference ion, 2-5 scans were done for the ion of interest, and the pattern was repeated. Altogether the data were split to 18 frequency pairs of TiV) and 20 pairs of CrMn) to obtain resonance quality as in Fig. 1. The weighted mean of the measured frequency ratios was calculated and used as the final value. The ratio of the outer and inner errors (45) was less than 1 for the data sets, and the larger, inner error was taken as the error of the mean.

Figure 1: A time-of-flight spectrum for V using an excitation pattern  ms (On) -  ms (Off) -  ms (On) at JYFLTRAP. Here, the number of ions has been limited to ions/bunch and the TOF window to  .

Table 1 shows the frequency ratios, and mass-excess values determined in this work. Compared to the previous experiments and mass tables (see Fig. 2), JYFLTRAP provides an improvement in precision of about 20 times for the values. Previously, the masses of V and Mn have been determined via CrHeV and FeHeMn reaction values at the Michigan State University (10). Recently, they have also been measured with the isochronous mass spectrometry method at the experimental cooler storage ring CSRe in Lanzhou (46). The former measurements were the main contributors to the Atomic Mass Evaluation 2003 (AME03) whereas the latter experiment has influenced the new Atomic Mass Evaluation 2012 (AME12) values significantly. As can be seen from Fig. 2, all experiments agree with each other in the case of V. For Mn, the He and the AME03 values agree well with JYFLTRAP. On the contrary, the CSRe result is lower than the JYFLTRAP value. This is reflected in the AME12 value, which deviates by from the JYFLTRAP value. Similar deviations are observed with the mass-excess value of Mn. If the more exotic isobar Fe measured also at CSRe (47) has a similar shift, then the cubic coefficient of the Isobaric Multiplet Mass Equation (IMME) would agree with zero at . Thus, further measurements are essential to confirm the breakdown of the IMME in the shell (47).

Nuclide Reference (keV) (keV) (keV) (keV)
V Ti
Mn Cr
Table 1: Frequency ratios , values and mass-excess values compared to the AME12 values (21).
Figure 2: values of V and Mn from previous measurements compared to the value determined at JYFLTRAP. Values from the CrHeV (Mueller et al. (10)) and CSRe storage ring (Tu et al. (46)) experiments as well as from the Atomic Mass Evaluations 2003 (AME03) (20) and 2012 (AME12) (21) all agree well with JYFLTRAP for V. For Mn, the CSRe value (46), and consequently the AME12 value, deviate from JYFLTRAP significantly.

To study the impact of JYFLTRAP measurements on the CDE values, an error-weighted linear fit on the CDE values based on the AME03 (20) was performed (see Fig. 3). The shell and pairing effects in the Coulomb energies of mirror nuclei are clearly seen in the lower panel of Fig. 3 showing differences from the linear fit. As the nuclear force is much stronger than Coulomb repulsion, protons tend to pair to angular momentum . Thus, mirror nuclei with an even number of protons have spatially closer proton orbitals and therefore higher than average Coulomb energies. At magic proton numbers, this effect is magnified. The magnitude of the odd-even staggering (pairing effect) is substantially reduced when entering the shell. Recently, Kaneko et al. (13) have explained the decrease using isospin-nonconserving isovector and isotensor nuclear interactions in their large-scale shell-model calculations and have compared the results to the AME03 values (20) updated with the storage-ring data from CSRe (46). The JYFLTRAP data show a fairly similar behavior to AME03 with most striking difference being the stronger staggering above . The AME12 value for Mn is 16-19 keV lower than the AME03 and JYFLTRAP values and would therefore imply a stronger odd-even staggering in the shell.

To illustrate the odd-even staggering of the CDE values in the shell, its magnitude has been plotted in Fig. 4. JYFLTRAP data for V, Mn, Co and Ni (using the AME12 values for the remaining data points) suggest a different trend in the pairing effect along the shell than the AME03 and AME12: the minimum is reached at the midshell where mirror nuclei show more collective features and deformation. In the midshell, the ground-state spins are no longer but correspond to deformed Nilsson orbitals , (Cr) and , (Mn and Fe), see e.g. shell-model calculations (48); (49); (50). High quadrupole deformation parameters for V ( (16)), Cr ( (16)), Mn ( (51)), and Fe ( (51)) have also been calculated e.g. with the FRDM (51) and EDF (16) models. The experimental CDE values demonstrate nicely that pairing energies are reduced in midshells for deformed nuclei and show peaks at the closed shells (52).

The change in deformation and ground-state configuration have an influence on pairing. Thus, the pairing effect in the CDE values was also studied for the lowest and states by combining the CDE data with the mirror energy differences (MED) between the excited states: (53); (54); (55); (56); (57); (58); (28) (see Fig. 5). As a result, the pairing effect is around 60 keV for the states throughout the shell. When the protons are coupled to ( states), the even- nuclei have higher CDEs than their odd- neighbors, but an opposite behavior is observed for the maximal angular momentum coupling ( states). Two protons coupled to are likely to be close to each other, whereas for they tend to be further apart as illustrated in Fig. 2 of Ref. (19). Even these subtle changes in the charge distribution are reflected in the CDE values.

Figure 3: Linear fit to the CDE values of the mirror nuclei based on the AME03 (20) (upper panel) and differences from the fit (lower panel). JYFLTRAP measurements are shown as red dots.
Figure 4: Pairing effect in the shell based on the AME03 (20), AME12 (21) and JYFLTRAP values for V, Mn, Co, and Ni (using AME12 values for the other nuclei).
Figure 5: Odd-even staggering of CDE values for () and () states in the shell.

In conclusion, the values of V and Mn have been determined with a superior precision at the JYFLTRAP double Penning trap. The observed difference to the CSRe measurement of Mn suggests that the masses of the isobaric quartets should be remeasured to verify the breakdown of the quadratic form of the IMME in the shell (47). Precise Penning trap measurements have made it possible to explore details in the experimental Coulomb displacement energies at a totally new level. The pairing effect in Coulomb displacement energies corresponding to the ground states in the shell has been shown to be quenched and reaches the minimum at the midshell. Further measurements on Ti, Cr and Fe are anticipated to reach sub-keV precision in the CDE values for all -shell mirror nuclei.

This work has been supported by the EU 6th Framework programme “Integrating Infrastructure Initiative - Transnational Access”, Contract Number: 506065 (EURONS) and by the Academy of Finland under the Finnish Centre of Excellence Programme 2006-2011 (Nuclear and Accelerator Based Physics Programme at JYFL). A.K. acknowledges the support from the Academy of Finland under the project 127301.


  1. preprint: APS/123-QED


  1. W. Heisenberg, Z. Phys. 77, 1 (1932).
  2. E. Wigner, Phys. Rev. 51, 106 (1937).
  3. H. A. Bethe, Phys. Rev. 54, 436 (1938).
  4. D. C. Peaslee, Phys. Rev. 95, 717 (1954).
  5. O. Kofoed-Hansen, Rev. Mod. Phys. 30, 449 (1958).
  6. E. Feenberg and G. Goertzel, Phys. Rev. 70, 597 (1946).
  7. B. C. Carlson and I. Talmi, Phys. Rev. 96, 436 (1954).
  8. J. Jänecke, Phys. Rev. 147, 735 (1966).
  9. J. Jänecke, Z. Phys. 196, 477 (1966).
  10. D. Mueller, E. Kashy, W. Benenson, and H. Nann, Phys. Rev. C 12, 51 (1975).
  11. J. A. Nolen and J. P. Schiffer, Ann. Rev. Nucl. Sci. 19, 471 (1969).
  12. A. P. Zuker, S. M. Lenzi, G. Martínez-Pinedo, and A. Poves, Phys. Rev. Lett. 89, 142502 (2002).
  13. K. Kaneko, Y. Sun, T. Mizusaki, and S. Tazaki, Phys. Rev. Lett. 110, 172505 (2013).
  14. N. Severijns, M. Tandecki, T. Phalet, and I. S. Towner, Phys. Rev. C 78, 055501 (2008).
  15. O. Naviliat-Cuncic and N. Severijns, Phys. Rev. Lett. 102, 142302 (2009).
  16. W. Satuła, J. Dobaczewski, W. Nazarewicz, and T. R. Werner, Phys. Rev. C 86, 054316 (2012).
  17. M. Bentley and S. Lenzi, Progr. in Part. and Nucl. Phys. 59, 497 (2007).
  18. S. J. Williams et al., Phys. Rev. C 68, 011301 (2003).
  19. D. D. Warner, M. A. Bentley, and P. V. Isacker, Nat. Phys. 2, 311 (2006).
  20. G. Audi, A. H. Wapstra, and C. Thibault, Nucl. Phys. A 729, 337 (2003).
  21. M. Wang et al., Chin. Phys. C 36, 1603 (2012).
  22. M. A. Bentley et al., Phys. Rev. C 73, 024304 (2006).
  23. C. D. O’Leary et al., Phys. Rev. Lett. 79, 4349 (1997).
  24. A. Saastamoinen et al., Phys. Rev. C 80, 044330 (2009).
  25. JYFLTRAP, to be published.
  26. A. Kankainen et al., Phys. Rev. C 82, 052501 (2010).
  27. A. Bacquias et al., Eur. Phys. J. A 48, 1 (2012).
  28. A. Kankainen et al., Phys. Rev. C 82, 034311 (2010).
  29. J. Äystö, Nucl. Phys. A 693, 477 (2001).
  30. I. Moore, P. Dendooven, and J. Ärje, Hyperfine Interact. 223, 17 (2014).
  31. P. Karvonen et al., Nucl. Instrum. and Methods Phys. Res. B 266, 4794 (2008).
  32. A. Nieminen et al., Nucl. Instrum. Methods Phys. Res. A 469, 244 (2001).
  33. V. S. Kolhinen et al., Nucl. Instrum. Methods Phys. Res. A 528, 776 (2004).
  34. T. Eronen et al., Eur. Phys. J. A 48, 1 (2012).
  35. G. Savard et al., Phys. Lett. A 158, 247 (1991).
  36. G. Gräff, H. Kalinowsky, and J. Traut, Z. Phys. A 297, 35 (1980).
  37. M. König et al., Int. J. Mass Spectrom. Ion Processes 142, 95 (1995).
  38. N. F. Ramsey, Rev. Mod. Phys. 62, 541 (1990).
  39. G. Bollen et al., Nucl. Instrum. Methods Phys. Res. B 70, 490 (1992).
  40. M. Kretzschmar, Int. J. Mass Spectrom. 264, 122 (2007).
  41. G. Gabrielse, Int. J. Mass Spectrom. 279, 107 (2009).
  42. S. George et al., Int. J. Mass Spectrom. 264, 110 (2007).
  43. A. Kellerbauer et al., Eur. Phys. J. D 22, 53 (2003).
  44. A. Kankainen et al., Phys. Rev. C 87, 024307 (2013).
  45. R. T. Birge, Phys. Rev. 40, 207 (1932).
  46. X. Tu et al., Nucl. Instrum. Methods Phys. Res. A 654, 213 (2011).
  47. Y. H. Zhang et al., Phys. Rev. Lett. 109, 102501 (2012).
  48. G. Martínez-Pinedo, A. P. Zuker, A. Poves, and E. Caurier, Phys. Rev. C 55, 187 (1997).
  49. A. Poves, J. Phys. G: Nucl. and Part. Phys. 25, 589 (1999).
  50. F. Brandolini, Eur. Phys. J. A 41, 115 (2009).
  51. P. Möller, J. Nix, W. Myers, and W. Swiatecki, At. Data and Nucl. Data Tables 59, 185 (1995).
  52. J. Jänecke, T. O’Donnell, and V. Goldanskii, Nucl. Phys. A 728, 23 (2003).
  53. J. A. Cameron and B. Singh, Nucl. Data Sheets 92, 783 (2001).
  54. T. Burrows, Nucl. Data Sheets 109, 171 (2008).
  55. T. Burrows, Nucl. Data Sheets 108, 923 (2007).
  56. T. Burrows, Nucl. Data Sheets 109, 1879 (2008).
  57. H. Xiaolong, Nucl. Data Sheets 107, 2131 (2006).
  58. H. Junde, Nucl. Data Sheets 110, 2689 (2009).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description