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.

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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

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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.

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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.).

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