Electric Monopole Transitions between 0 States for Nuclei
throughout the Periodic Table
T. Kibédi and R.H. Spear
£
Department of Nuclear Physics, Research School of Physical Sciences and Engineering, The Australian
National University, Canberra, ACT 0200, Australia
Abstract. Adopted spectroscopic information on 0
i 0 f pure E0 transitions has been deduced by critical evaluation of the
2
available experimental data for all even-even nuclei ranging from 42 He2 to 250
98 C f 152 . Values of qK E0 E2, the ratio of the
K-conversion electron intensity of the 0i 0 f , 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-positron
pair creation, as well electronic factors, produced a large number of new X E0 E2 values, defined as the dimensionless ratio
of the absolute BE0 and BE2 transition rates. The squared value of the monopole transition strength, ρ 2 E0, has been
deduced using the best available 0 -level half-lives and branching ratios.
INTRODUCTION
8000
(a) 0+
2 states
definite
doubtful
N=126
↓
N=8
↓
N=20
↓N=28
↓
4000
N=50
↓
N=82
↓
0
Excitation Energy [keV]
It has been known for many years that electric monopole
(E0) transitions are possible between states of the same
spin and parity in a nucleus enclosed by electrons. The
characteristics of E0 transitions provide sensitive tests of
the various models of nuclear structure. They elucidate
such matters as volume oscillations (the so-called breathing mode, related to nuclear compressibility), shape coexistence, and isotope and isomer shift. Excellent surveys of monopole transitions have been given by Aldushchenkov and Voinova in 1972 [1], Lange, Kumar and Hamilton in 1982 [2], Voinova-Elseeva and
Mitropolsky in 1986 [3], and Wood et al. in 1999 [4].
The present paper provides a critical evaluation of all
available data on E0 transitions between 0 states for
nuclei throughout the periodic table. It adopts a consistent approach to the calculation of characteristic parameters from published experimental data using the most upto-date information on conversion coefficients and electronic factors. The literature has been covered to March
2004. Similar compilations have been published for E2
transitions by Raman et al. [5, 6] and for E3 transitions
by Spear [7] and Kibédi and Spear [8]. In Fig. 1 the excitation energies of the first excited 0 states are compared
to the energies of the first excited 2 and 3 states as a
function of the neutron number. There is a striking similarity 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
(b) 2+
1 states
↓N=8
6000
4000
N=20
↓N=28
↓
N=82
↓
N=50
↓
2000
0
10000
5000
0
(c) 31 states
N=8
↓
N=28
N=20↓
↓
N=50
↓
50
N=82
↓
100
Neutron Number N
N=126
↓
150
FIGURE 1. Excitation energy of the first excited 0 (panel
a), 2 (panel b), and 3 (panel c) states in even-even nuclides
as a function of neutron number N. The lines connect isotopes.
CP769, International Conference on Nuclear Data for Science and Technology,
edited by R. C. Haight, M. B. Chadwick, T. Kawano, and P. Talou
© 2005 American Institute of Physics 0-7354-0254-X/05/$22.50
442
N=126
↓
to the atomic electrons, giving rise to the internal conversion process. It also couples the nucleus to the Dirac
background to produce electron-positron pairs if the E0
transition energy is greater then twice the electron rest
mass. Simultaneous emission of two photons is a higher
order process (relative probability 10 3 to 10 4 [3])
and will be neglected in the present work. Single-photon
E0 transitions are strictly forbidden by considerations of
angular-momentum conservation.
The E0 transition probability is given by the expression
W E0 1 τ E0 Wic E0 Wπ E0
(1)
where τ E0 is the partial mean life of the initial state for
E0 decay. The quantities Wic E0 and Wπ E0 are the
transition probabilities for internal-conversion electron
and electron-positron pair emission, respectively. They
are given by the expression
Wic E0 Wπ E0 ρ 2 E0 Ωic E0 Ωπ E0
(2)
where Ωic E0 and Ωπ E0 are electronic factors defined by Church and Weneser [9]. They are functions of
atomic number, Z, and transition energy. They can be calculated independently of nuclear properties. The quantity
ρ E0 is the dimensionless monopole transition strength.
It carries all the information about the nuclear structure,
being related to the monopole matrix element according
to the expression
f M E0i (3)
eR2
where R is the nuclear radius. It will be assumed throughout this paper that R rÆ A1 3 , where A is the atomic mass
number and rÆ 120 fm.
The reduced E0 transition probability B E0 is equal
to the square of the E0 matrix element, and so
ρ E0 B E0 ρ 2 E0e2 R4 (4)
where e is the electronic charge. Clearly ρ E0 is a basic
characteristic of electric-monopole transitions. Because
there is often an ambiguity in determining its sign, it is
customary to use ρ 2 E0. Since the value of ρ 2 E0 usually lies in the range 10 3 to 10 1 , reference is usually
made to 103 ρ 2 E0. It is evident from Eq. (2) that experimental determination of ρ 2 E0 requires the measurement of absolute transition rates and the calculation of
electronic factors. In some cases the transition rate can
be determined indirectly from that of another transition
de-exciting the same nuclear state, provided that the relevant branching ratio is known.
In their discussion of E0 transitions between 2 states,
Church, Rose, and Weneser [10] introduced the quantity
q2K E0 E2 IK E0
IK E2
(5)
443
where IK E0 and IK E2 represent the intensities of E0
and E2 K-conversion electron components of the Ji J
f transition, respectively.
The definition of q2K E0 E2 can be extended to the
case of 0
i 0 f transitions (which can have no E2
component) by somewhat arbitrary reference to an E2
transition from the 0
i state to a 2 f state [1, 11, 12]. In
the present work this will be taken to be the first excited
2 state (2
1 ).
In some cases experimental information other than
IK E0 and IK E2 can be used in conjunction with the
relevant conversion coefficients and electronic factors to
deduce q2K . For example,
q2K E0 E2 Iπ E0 ΩK E0 απ E2
Iπ E2 Ωπ E0 αK E2
(6)
where Iπ E0 and Iπ E2 are the observed internal pair
intensities for the E0 and E2 transitions, respectively,
and ΩKπ and αKπ are the relevant electronic factors and
conversion coefficients.
A dimensionless ratio of the E0 and E2 reducedtransition probabilities was defined by Rasmussen [13]:
X E0 E2 B E0
B E2
ρ 2 E0e2 R4 B E2
(7)
The equivalent experimental value, considering K conversion electrons, can be deduced from the general formula:
X E0 E2 254 109 A4
3
q2K
E0 E2 αK E2
Eγ5 ΩK E0
(8)
where Eγ is the E2 γ -ray energy in MeV.
The experimental monopole strength can be obtained
directly if the partial mean life of the E0 transition,
τ E0, is known
1
E0 ΩL1 E0 Ωπ E0 τ E0
(9)
Alternatively, if the E2 transition rate, Wγ E2, is known
it can be obtained from the expression
ρ 2 E0 ΩK
ρ 2 E0 q2K E0 E2 αK E2
Wγ E2
ΩK E0
(10)
Theoretical Conversion Coefficients and
Electronic Factors
In order to determine values of the characteristic
monopole transition parameters, conversion coefficients
and electronic factors were required over a broad range
+
0+
2 → 01
102
Values adopted for 103 ρ 2 E0 range from 019 (172Y b,
194 Pt, 0 0 ) to 500 (12C, 0 0 ),
0
3 01 , and
2
1
2
1
except for the remarkably small value for 103 ρ 2 E0:
0.66(16) 10 6 for the ground-state transition from the
238U , are two other outstand0
4 fission-isomeric state of
58 Ni,
ingly small values of for the 0
2 01 transition in
188
Os.
and 0.011(4) for the 02 01 transition in
As pointed out in [3] and [4], simple shell-model considerations suggest that X E0 E2 and ρ 2 E0 should
both be proportional to A 2 3 , providing a convenient
scaling for the consideration of experimental values:
X E0 E2 A2 3 and ρ 2 E0 A2 3 should be independent of mass. Figure 2 shows the A-dependence of
(a) X E0 E2 A2 3 and (b) ρ 2 E0 A2 3 . It is clear
that there remains an overall decrease with mass in values of ρ 2 E0 A2 3 . The dashed lines show the socalled single-particle values: X E0 E2s p A2 3 17
(ref. [3]) and ρ 2 E0s p A2 3 05 (ref. [4]).
(a)
X(E0/E2) A2/3
"Single Particle"
101
100
10-1
10-2
+
(b)
0+
2 → 01
+ (i>2)
0+
→
0
i
f
ρ 2(E0) A2/3
101
100
"Single Particle"
10-1
10-2
Reliability of Monopole Strength
Determinations from Electron Scattering
10-3
50
100
150
Mass Number A
200
250
FIGURE 2. Adopted values as a function of mass number A
2
for (a) X E0 E2 A2 3 for 0
2 01 transitions; (b) ρ E0 2
3
A
for 02 01 transitions (filled symbols) and for 0i 0
f ,
i 2 transitions (open symbols). Dashed lines show "singleparticle" values [3, 4].
of energies and atomic numbers. A comprehensive conversion electron database and appropriate software have
been developed primarily for the present compilation
(see for details [14]).
Adopted Values of q2K E0 E2, X E0 E2,
and ρ 2 E0
Our objective in this study was to present the most
complete set of up-to-date values possible for the quantities q2K E0 E2, X E0 E2, and ρ 2 E0. Previous reviews have usually adopted the values of these quantities as calculated by the original authors from their observed intensity data using a variety of calculated tables
of internal conversion coefficients and electronic factors.
We have wherever possible used original intensity data to
calculate q2K E0 E2, X E0 E2, and ρ 2 E0 in a consistent fashion using the most up-to-date published calculations of conversion coefficients and electronic factors, together with the most recent information on lifetimes and branching ratios.
444
Endt [15, 16] has suggested that there are “disturbing”
discrepancies between values of monopole strengths determined from electron scattering and those from more
“traditional” procedures, such as the measurement of
internal-pair intensities. In Table 1 we compare values
of monopole strengths for excitation of 0
2 states as obtained from (e,e ) measurements with those from traditional methods. Direct comparison is possible in nine
cases (12 C, 16 O, 18 O, 26 Mg, 32 S, and 40424448 Ca). In
seven cases the agreement is very good. For 26 Mg the
difference is only about 2 standard deviations. For 42 Ca,
the difference is about 4 to 5 standard deviations. Thus,
there is little or no support for Endt’s suggestion. However, given the model dependence of most, if not all, analyses of electron-scattering data, the traditional data are to
be preferred where available.
CONCLUSION
A total of 276 X E0 E2 and 141 ρ 2 E0 values have
been determined for even-even nuclei ranging from 42 He2
to 250
98 C f 152 . A full report has been prepared and submitted for publication at the Atomic Data and Nuclear Data
Tables.
TABLE 1. Monopole strengths of transitions between 0
1 and 02 states for Z20
determined from inelastic electron scattering (e e¼) compared with values from “traditional” procedures (i.e., observation of pairs or conversion electrons). Values for traditional methods are taken from this work. ME0 is the magnitude of the electric
monopole matrix element.
Nuclide
4 He
12 C
16 O
18 O
20 Ne
24 Mg
26 Mg
28 Si
32 S
40 Ca
42 Ca
44 Ca
48 Ca
103 ρ 2 E0
(e,e¼ )
TRAD.
19 (8)10
52 (3)10
152 (17)
[23 - 42] 10
36 (14) 10
294 (19)
112 (25)
26 (3) 10
58
22 18
26 (8)
90 (14)
92 (14)
14 (7)
–
50 (8)10
153 (22)
43 (8) 10
–
–
65 (9)
–
19 (5)
25.6 (7)
140 (12)
140 (50)
14.5 (9)
ACKNOWLEDGMENTS
The authors are grateful for useful exchanges with J.L.
Wood and W.D. Kulp (School of Physics, Georgia Institute of Technology, Atlanta, Georgia, U.S.A.).
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