Electric Monopole Transitions between 0 States for Nucleithroughout the Periodic Table
T. Kibédi and R.H. Spear
Department of Nuclear Physics, Research School of Physical Sciences and Engineering, The AustralianNational University, Canberra, ACT 0200, Australia
Abstract. Adopted spectroscopic information on 0i 0f pure E0 transitions has been deduced by critical evaluation of the
available experimental data for all even-even nuclei ranging from 42He2 to 250
98 C f 152. Values of q2KE0E2, the ratio of the
K-conversion electron intensity of the 0i 0f , E0 transition to that of the 0i 21 , E2 transition, have been determined.This procedure, together with the most recent theoretical conversion coefficients for internal conversion and electron-positronpair creation, as well electronic factors, produced a large number of new XE0E2 values, defined as the dimensionless ratioof the absolute BE0 and BE2 transition rates. The squared value of the monopole transition strength, ρ2E0, has beendeduced using the best available 0-level half-lives and branching ratios.
INTRODUCTION
It has been known for many years that electric monopole(E0) transitions are possible between states of the samespin and parity in a nucleus enclosed by electrons. Thecharacteristics of E0 transitions provide sensitive tests ofthe various models of nuclear structure. They elucidatesuch matters as volume oscillations (the so-called breath-ing mode, related to nuclear compressibility), shape co-existence, and isotope and isomer shift. Excellent sur-veys of monopole transitions have been given by Al-dushchenkov and Voinova in 1972 [1], Lange, Ku-mar and Hamilton in 1982 [2], Voinova-Elseeva andMitropolsky in 1986 [3], and Wood et al. in 1999 [4].The present paper provides a critical evaluation of allavailable data on E0 transitions between 0 states fornuclei throughout the periodic table. It adopts a consis-tent approach to the calculation of characteristic parame-ters from published experimental data using the most up-to-date information on conversion coefficients and elec-tronic factors. The literature has been covered to March2004. Similar compilations have been published for E2transitions by Raman et al. [5, 6] and for E3 transitionsby Spear [7] and Kibédi and Spear [8]. In Fig. 1 the exci-tation energies of the first excited 0 states are comparedto the energies of the first excited 2 and 3 states as afunction of the neutron number. There is a striking sim-ilarity in the shell structure evident in the three cases,e.g., peaks are clearly evident in each case at N=28, 50,82, and 126.
The electric monopole operator couples the nucleus
Excitation Energy [keV]
Neutron Number N50 100 150
0
5000
10000(c) 3 states - 1
↓N=8
↓N=20↓
N=28
↓N=50 ↓
N=82
↓N=126
0
2000
4000
6000
(b) 2 states + 1
↓N=8
↓N=20
↓N=28
↓N=50
↓N=82
↓N=126
0
4000
8000(a) 0 states + 2definitedoubtful
↓N=8
↓N=20
↓N=28
↓N=50
↓N=82
↓N=126
FIGURE 1. Excitation energy of the first excited 0 (panela), 2 (panel b), and 3 (panel c) states in even-even nuclidesas a function of neutron number N. The lines connect isotopes.
442
to the atomic electrons, giving rise to the internal con-version process. It also couples the nucleus to the Diracbackground to produce electron-positron pairs if the E0transition energy is greater then twice the electron restmass. Simultaneous emission of two photons is a higherorder process (relative probability 103 to 104 [3])and will be neglected in the present work. Single-photonE0 transitions are strictly forbidden by considerations ofangular-momentum conservation.
The E0 transition probability is given by the expres-sion
W E0 1τE0 WicE0WπE0 (1)
where τE0 is the partial mean life of the initial state forE0 decay. The quantities WicE0 and WπE0 are thetransition probabilities for internal-conversion electronand electron-positron pair emission, respectively. Theyare given by the expression
WicE0WπE0 ρ2E0 ΩicE0Ωπ E0(2)
where ΩicE0 and ΩπE0 are electronic factors de-fined by Church and Weneser [9]. They are functions ofatomic number, Z, and transition energy. They can be cal-culated independently of nuclear properties. The quantityρE0 is the dimensionless monopole transition strength.It carries all the information about the nuclear structure,being related to the monopole matrix element accordingto the expression
ρE0 f ME0i
eR2 (3)
where R is the nuclear radius. It will be assumed through-out this paper that R rÆA13, where A is the atomic massnumber and rÆ 120 fm.
The reduced E0 transition probability BE0 is equalto the square of the E0 matrix element, and so
BE0 ρ2E0e2R4 (4)
where e is the electronic charge. Clearly ρE0 is a basiccharacteristic of electric-monopole transitions. Becausethere is often an ambiguity in determining its sign, it iscustomary to use ρ2E0. Since the value of ρ2E0 usu-ally lies in the range 103 to 101, reference is usuallymade to 103ρ2E0. It is evident from Eq. (2) that exper-imental determination of ρ2E0 requires the measure-ment of absolute transition rates and the calculation ofelectronic factors. In some cases the transition rate canbe determined indirectly from that of another transitionde-exciting the same nuclear state, provided that the rel-evant branching ratio is known.
In their discussion of E0 transitions between 2 states,Church, Rose, and Weneser [10] introduced the quantity
q2KE0E2
IKE0IKE2
(5)
where IKE0 and IKE2 represent the intensities of E0and E2 K-conversion electron components of the Ji Jf transition, respectively.
The definition of q2KE0E2 can be extended to the
case of 0i 0f transitions (which can have no E2component) by somewhat arbitrary reference to an E2transition from the 0i state to a 2f state [1, 11, 12]. Inthe present work this will be taken to be the first excited2 state (21 ).
In some cases experimental information other thanIKE0 and IKE2 can be used in conjunction with therelevant conversion coefficients and electronic factors todeduce q2
K . For example,
q2KE0E2
IπE0IπE2
ΩKE0Ωπ E0
απE2αKE2
(6)
where IπE0 and IπE2 are the observed internal pairintensities for the E0 and E2 transitions, respectively,and ΩKπ and αKπ are the relevant electronic factors andconversion coefficients.
A dimensionless ratio of the E0 and E2 reduced-transition probabilities was defined by Rasmussen [13]:
XE0E2BE0BE2
ρ2E0e2R4BE2 (7)
The equivalent experimental value, considering K con-version electrons, can be deduced from the general for-mula:
XE0E2 254109A43q2KE0E2
αKE2ΩKE0
E5γ
(8)where Eγ is the E2 γ-ray energy in MeV.
The experimental monopole strength can be obtaineddirectly if the partial mean life of the E0 transition,τE0, is known
ρ2E0 1
ΩKE0ΩL1E0 ΩπE0 τE0
(9)Alternatively, if the E2 transition rate, WγE2, is knownit can be obtained from the expression
ρ2E0 q2KE0E2
αKE2ΩKE0
WγE2 (10)
Theoretical Conversion Coefficients andElectronic Factors
In order to determine values of the characteristicmonopole transition parameters, conversion coefficientsand electronic factors were required over a broad range
443
Mass Number A
(b)
50 100 150 200 250
-310
-210
-110
010
110
ρ (E0) A
2 2/3
0 0 + + 2 1→
0 0 (i>2) + + i f→
"Single Particle"
(a)
-210
-110
010
110
210X(E0/E2) A
2/3
0 0 + + 2 1→
"Single Particle"
FIGURE 2. Adopted values as a function of mass number Afor (a) XE0E2A23 for 02 01 transitions; (b) ρ2E0
A23 for 02 01 transitions (filled symbols) and for 0i 0f ,i 2 transitions (open symbols). Dashed lines show "single-particle" values [3, 4].
of energies and atomic numbers. A comprehensive con-version electron database and appropriate software havebeen developed primarily for the present compilation(see for details [14]).
Adopted Values of q2KE0E2, XE0E2,
and ρ2E0
Our objective in this study was to present the mostcomplete set of up-to-date values possible for the quan-tities q2
KE0E2, XE0E2, and ρ2E0. Previous re-views have usually adopted the values of these quanti-ties as calculated by the original authors from their ob-served intensity data using a variety of calculated tablesof internal conversion coefficients and electronic factors.We have wherever possible used original intensity data tocalculate q2
KE0E2, XE0E2, and ρ2E0 in a con-sistent fashion using the most up-to-date published cal-culations of conversion coefficients and electronic fac-tors, together with the most recent information on life-times and branching ratios.
Values adopted for 103ρ2E0 range from 019 (172Yb,03 01 , and 194Pt , 02 01 ) to 500 (12C, 02 01 ),except for the remarkably small value for 103ρ2E0:0.66(16) 106 for the ground-state transition from the04 fission-isomeric state of 238U , are two other outstand-ingly small values of for the 02 01 transition in 58Ni,and 0.011(4) for the 02 01 transition in 188Os.
As pointed out in [3] and [4], simple shell-model con-siderations suggest that XE0E2 and ρ2E0 shouldboth be proportional to A23, providing a convenientscaling for the consideration of experimental values:XE0E2 A23 and ρ2E0 A23 should be inde-pendent of mass. Figure 2 shows the A-dependence of(a) XE0E2A23 and (b) ρ2E0A23. It is clearthat there remains an overall decrease with mass in val-ues of ρ2E0 A23. The dashed lines show the so-called single-particle values: XE0E2spA23 17(ref. [3]) and ρ2E0spA23 05 (ref. [4]).
Reliability of Monopole StrengthDeterminations from Electron Scattering
Endt [15, 16] has suggested that there are “disturbing”discrepancies between values of monopole strengths de-termined from electron scattering and those from more“traditional” procedures, such as the measurement ofinternal-pair intensities. In Table 1 we compare valuesof monopole strengths for excitation of 02 states as ob-tained from (e,e) measurements with those from tradi-tional methods. Direct comparison is possible in ninecases (12C, 16O, 18O, 26Mg, 32S, and 40424448Ca). Inseven cases the agreement is very good. For 26Mg thedifference is only about 2 standard deviations. For 42Ca,the difference is about 4 to 5 standard deviations. Thus,there is little or no support for Endt’s suggestion. How-ever, given the model dependence of most, if not all, anal-yses of electron-scattering data, the traditional data are tobe preferred where available.
CONCLUSION
A total of 276 XE0E2 and 141 ρ2E0 values havebeen determined for even-even nuclei ranging from 4
2He2to 250
98 C f 152. A full report has been prepared and submit-ted for publication at the Atomic Data and Nuclear DataTables.
444
TABLE 1. Monopole strengths of transitions between 01 and 02 states for Z20determined from inelastic electron scattering (ee) compared with values from “tradi-tional” procedures (i.e., observation of pairs or conversion electrons). Values for tradi-tional methods are taken from this work. ME0 is the magnitude of the electricmonopole matrix element.
Nuclide 103ρ2E0 ME0 (e fm2) References(e,e) TRAD. (e,e) TRAD. for (e,e) data
4He 19 (8)10 – 1.6 (5) – [17, 18]12C 52 (3)10 50 (8)10 5.46 (16) 5.3 (9) [19, 20]16O 152 (17) 153 (22) 3.56 (20) 3.58 (26) [21, 22]18O [23 - 42] 10 43 (8) 10 4.7 - 6.4 6.5 (12) [23, 24]20Ne 36 (14) 10 – 6.4 (12) – [25, 26]24Mg 294 (19) – 6.50 (20) – [27, 20, 28]26Mg 112 (25) 65 (9) 4.2 (5) 3.1 (2) [29]28Si 26 (3) 10 – 6.8 (4) – [20]32S 22 58
18 19 (5) 2.2 (13) 2.0 (3) [20]40Ca 26 (8) 25.6 (7) 2.7 (4) 2.69 (4) [20, 30]42Ca 90 (14) 140 (12) 5.2 (4) 6.51 (28) [30]44Ca 92 (14) 140 (50) 5.5 (4) 6.7 (12) [30]48Ca 14 (7) 14.5 (9) 2.3 (5) 2.29 (7) [30]
ACKNOWLEDGMENTS
The authors are grateful for useful exchanges with J.L.Wood and W.D. Kulp (School of Physics, Georgia Insti-tute of Technology, Atlanta, Georgia, U.S.A.).
REFERENCES
1. A.V. Aldushchenkov and N.A. Voinova, Nucl. Data Tables11, 299 (1972).
2. J. Lange, K. Kumar and J.H. Hamilton, Rev. Mod. Phys.54, 119 (1982).
3. N.A. Voinova-Elseeva and I.A. Mitropolsky, Izv. Akad.Nauk SSSR, Ser. Fiz. 50, 14 (1986); Bull. Acad. Sci.USSR, Phys. Ser. 50, No.1, 12 (1986).
4. J.L. Wood, E.F. Zganjar, C. De Coster and K. Heyde,Nucl. Phys. A651, 323 (1999).
5. S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor, Jr.and P.H. Stelson, At. Data and Nucl. Data Tables 36, 1(1987).
6. S. Raman, C.W. Nestor, Jr. and P. Tikkanen, At. Data andNucl. Data Tables 78, 1 (2001).
7. R.H. Spear, At. Data and Nucl. Data Tables 42, 55 (1989).8. T. Kibédi and R.H. Spear, At. Data and Nucl. Data Tables
80, 35 (2002).9. E.L. Church and J. Weneser, Phys. Rev. 103, 1035 (1956).10. E.L. Church, M.E. Rose and J. Weneser, Phys. Rev. 109,
1299 (1958).11. J.H. Hamilton, K. Kumar, L. Varnell, A.V. Ramayya,
P.E. Little and N.R. Johnson, Phys. Rev. C10, 2540(1974).
12. T. Kibédi, G.D. Dracoulis, A.P. Byrne, P.M. Davidson andS. Kuyucak, Nucl. Phys. A567, 183 (1994).
13. J.O. Rasmussen, Nucl. Phys. 19, 85 (1960).14. T. Kibédi, T.W. Burrows, M.B. Trzhaskovskaja, C.W.
Nestor, Jr. (In this proceedings).
15. P.M. Endt, At. Data and Nucl. Data Tables 23, 3 (1979).16. P.M. Endt, At. Data and Nucl. Data Tables 55, 171 (1993).17. R.F. Frosch, R.E. Rand, H. Crannell, J.S. McCarthy, L.R.
Suelzle and M.R. Yearian, Nucl. Phys. A110, 657 (1968).18. T. Walcher, Phys. Lett. 31B, 442 (1970).19. H. Crannell, T.A. Griffy, L.R. Suelzle and M.R. Yearian,
Nucl. Phys. A90, 152 (1967).20. P. Strehl, Z. Phys. 234, 416 (1970).21. M. Stroetzel, Z. Phys. 214, 357 (1968).22. H. Miska, H.D. Graf, A. Richter, R. Schneider, D. Schull,
E. Spamer, H. Theissen, O. Titze and T. Walcher, Phys.Lett. 58B, 155 (1975).
23. K.H. Souw, J.C. Adloff, D. Disdier and P. Chevallier,Phys. Rev. C11, 1899 (1975).
24. J.L. Groh, R.P. Singhal, H.S. Caplan and B.S. Dolbilkin,Can. J. Phys. 49, 2743 (1971).
25. S. Mitsunobu and Y. Torizuka, Phys. Rev. Lett. 28, 920(1972).
26. R.P. Singhal, H.S. Caplan, J.R. Moreira and T.E. Drake,Can. J. Phys. 51, 2125 (1973).
27. O. Titze, Z. Physik 220, 66(1969).28. A. Johnston and T.E. Drake, J. Phys. (London) A7, 898
(1974).29. E.W. Lees, A. Johnston, S.W. Brain, C.S. Curran,
W.A. Gillespie and R.P. Singhal, J. of Phys. (London) A7,936 (1974).
30. H.D. Graf, H. Feldmeier, P. Manakos, A. Richter, E.Spamer and D. Strottman, Nucl. Phys. A295, 319 (1978).
445