The odd-even staggering of the nuclear charge radii of Pb isotopes

Nuclear Physics @ North-Holland

A451 (1986) 471-480 Publishing Company

THE

ODD-EVEN CHARGE

STAGGERING RADII

M. ANSELMENT, W. FAUBEL*, G. MEISEL, H. REBEL Kemforschungszentrum

OF THE

NUCLEAR

OF Pb ISOTOPES S. GGRING, A. HANSER, and G. SCHATZ

Karisruhe, Institut fiir Kemphysik, POB 3640, D-7500 Karlsruhe, Federal Republic of Germany Received (Revised

12 September 1985 4 November 1985)

Abstract: Previous laser-spectroscopic studies of the hyperfine structure splittings and isotope shifts of the (6~’ 3P,,-6p7s 3P,; 283.3 nm) PbI resonance line have been extended to further radioactive Pb nuclides. .,96,LP7.,97m,2,,,*,4 Pb. Using an improved theoretical result for the specific mass shift, we recalibrate the isotope shifts of the full series (A = 196-214) of measured isotope shifts in terms of the nuclear parameter and determine the odd-even staggering parameters. The variation of the nuclear charge radii of the Pb nuclides exhibits distinct shell effects, and the odd-even staggering shows a conspicuous trend, being more pronounced for neutron-deficient Pb isotopes.

1. Introduction A conspicuous feature, generally observed with the isotope variation of nuclear charge radii, is a distinctive odd-even staggering ‘). The mean-square (ms) charge radii (r*) of the odd-neutron isotopes appear to be slightly smaller (usually of the order of 10e2 fm2) than the averages of their even-neutron neighbours. In contrast to the remarkably long history of experimental observation and the wealth of data, a convincing and consistent theoretical explanation, which quantitatively reproduces the data on the basis of a microscopic structure description, is missing. Several kinds of effects 2-5) h ave been discussed as the origin of the phenomenon, but with limited success when compared with experimental results. In addition, empirical systematic features of normal+ odd-even staggering, which may guide the theoretical investigation, are not very pronounced and not well worked out, since most often other influences on isotopic changes of the ms radii, i.e. fluctuations of the ms deformation and specific shell effects, obscure the trend of the relatively tiny odd-even staggering effects. The nuclei in the neighbourhood of the doubly-magic nucleus “‘Pb, however, may provide a more favourable situation for studies of the polarization of the closed proton shell as there is a distinct and clear microscopic * Institut fur Radiochemie des Kernforschungszentrums Karlsruhe. ’ We exclude by this term exceptional situations such as with the neutron-deficient a prolate-oblate shape staggering leads to unusually large effects 6). 471

Hg isotopes,

where

M. Anselment

472

ei al. / Odd-even

staggering

picture of the nuclear structure and due to the reduced importance effects. With this view, we have extended previous high-resolution measurements

‘) of isotope

shifts and hyperfine

structure

of the deformation laser-spectroscopic

splittings

of the (6~’ 3P0-

6~7s 3P1; 283.3 nm) PbI resonance line to further radioactive Pb nuclides, including some radioactive Pb isotopes with N > 126. In this paper we present a consistent set of results for the isotopic series from 196Pb to ‘14Pb (except ‘13Pb) and look, in particular, for systematic features in the odd-even staggering of the charge the closed N = 126 shell of 2osPb.

radii, when neutrons

are removed

or added

to

2. Experiment The radioactive lead nuclides 196,‘97,197mPb were produced by a-particle irradiation of enriched Hg samples, and subsequently purified by an electromagnetic mass separator. 214Pb( RaB) and 2”Pb(AcB) which are members of the Uran and AcU series, respectively, have been collected as daughter products of freshly prepared, readily emanating 226Ra and 227Ac sources. In these cases, ca. 10 pg Pb samples were available for the spectroscopic measurements. The laser-spectroscopic set-up and the experimental procedures were basically the same as described in refs. 7*8).F rom the samples, an atomic beam was produced. The dye laser beam, exciting the atomic transitions, intersects the atomic beam perpendicularly. The 283 nm (UV) light was generated by frequency-doubling the output of a C.W. dye laser in a temperature-tuned ADA crystal. For accurate measurements a special r.f. technique was applied to tune the dye laser repetitively over the atomic resonance of interest and to provide the long-term stability required for reference to stable isotopes ‘,‘O). 3. Analysis of the measured data Table 1 compiles the hyperfine splittings (h.f.s.), the isotope shifts (IS) relative to the transition in “‘Pb, and the A- and B-factors derived from the observed hyperfine splitting of the resonance line in the odd-Pb isotopes, including the results of ref. ‘). In order to calibrate the measured isotope shifts A-A’ Sv,,, = M+ FA /,/,, AA’ in terms of the nuclear

parameter

h=6(r2)+C2S(r4)i-Cj~(r6)~~~=0.936(r2) [see ref. ‘)I, Thompson et af. ‘) determined the electronic shift factor M by a King plot using independent information

factor F and the mass on h from a combined

M. Anselment et al. / Odd-even staggering TABLE Measured

h.f.s., IS (relative

Isotope

196 191

512 312

1312

1512

-6 083 (30) -19048

1112

-1377

201

-4417

202m

203

512

-14 753.1 (8.0)

512

-9 736.1 (5.0)

712

-2 708.5 (4.0)

10

-7 938.3 (8.0)

9

-6 003.6 (6.0)

8

-4 374.2 (6.0)

312

-12 897.8 (7.0)

512

-7 780.3 (6.0) -651.1

712 204 205

312

-11

-5 807.6 (3.5) 1 570.2 (2.0)

(6)

(39)

-9 10)

-7 727.6 (5.0)

2007.5 (1.3)

1 (5)

-6 193.7 (3.5) -6 230.4 (5.0)

-187.9

(0.5)

-67 (9)

-5 749.0 (5.0)

+2040.3

(1.3)

-11 (6)

-4 212.0 (2.5) -3 712.6 (3.0)

2115.7 (8.0)

3 012.7 (3.0)

-1 390.9 (2.5)

8807.2 (3.0)

0.0

0.0

-26

(4)

-2 226.6 (2.5)

112

112

-10

208 1112

912

198.1 (3.0)

-9 175 (20)

912

4 180 (15)

712

15 165 (15)

210 211

-5322

-54

135.8 (5.0)

712

312 209

(3)

(4.0)

512 206 207

-1263

9 (20)

-8 094.1 (3.5)

-4 212.0 (2.5) 512

-10 827 (17)

-9 748 (9)

(15)

-6 193.7 (3.5) 9

(11)

3 545 (15)

312

202

-5327

-9 848 (3.5)

-8 094.1 (3.5) 512

(19)

-17 733 (15)

112 200

-11402

(30)

-9 848 (4.5) 312

(30)

[MHz1

(30)

-9 530 (30)

512

3/2

-11441

B-factor

1 927 (30)

1312 198 199

[MHz1

line

-19 391(30)

112 197m

A-factor

[MHz1 -11441(30)

312

for the 283.3 nm lead resonance

Position

F

I

(Pb)

1

and A- and B-factors

to *“Pb),

473

1 767 (9)

3 973.7 (3.5) 1112

912

712

31(19)

-2318.3

(1.3)

10 (13)

5 648.2 (5.6)

7 960.1 (11.5) 18403.2(11.5)

212

7 815 (30)

7 815 (30)

214

11 503 (20)

11 503 (20)

analysis of electron values obtained,

(3)

3 973.7 (3.5)

-4 781.8 (6.5)

912

-2433

scattering

and muonic

X-ray data of stable Pb isotopes

F = 22.1 (3.3) GHz . fm-*, A4 = 7.1 (13.1) x NMS

(NMS = normal

mass shift) ,

I’). The

shift

(9) (6)

-0.3068 -0.2832

-0.2075

-0.1097

-0.0685 0.0

-9 848 (4.5)

-9 748 (9)

-8 094.1 (3.5)

-7 727.6 (5.0)

-6 193.7 (3.5)

-6 230.4 (5.0) -5 749.0 (5.0)

-4 212.0 (2.5)

-3 712.6 (3.0)

-2 226.6 (2.5)

-1 390.9 (2.5)

0.0

199

200

201

202

202m 203

204

205

206

207

208

0.4144 (38)

0.3854 (35) 0.5672 (48)

11 503 (20)

0.6099 (52)

0.2995 (25)

0.2785 (23)

5 648.2 (5.6)

212 214

7 815 (30)

211

0.937 (9)

(6)

(10)

(16)

(18)

(27) (25)

(27)

(34)

(35)

(42)

(49)

(47)

(50)

0.2107 (18)

0.0

-0.0737

-0.1179

-0.1967

-0.2231

-0.3299 -0.3045

-0.3280

-0.4093

-0.4286

-0.5163

-0.5214

-0.5739

-0.6038

-0.6057

(52)

s(rY

(283.3 nm)

0.0871 (8)

(15)

2

(p,, Q) as derived

TABLE

WI

moments

0.1959 (16)

1 767.0 (9.0)

3 973.7 (3.5)

209

210

-0.1829

(17)

(25) (23)

(31) (25)

-0.3050

(32)

(39)

(40)

(44)

-0.3806

-0.3986

-0.4801

-0.4849

-0.5331

(46)

198

(48)

-10 827 (17)

-0.5616

-0.5633

197m

(19)

-11402

(30)

-11441

[f&

197

[MHz1

Isotope

A, and of the electromagnetic

196

Pb

Isotope

Values of the nuclear parameter

0.872 (8)

0.889 (5)

0.750 (4)

0.502 (6)

0.447 (8)

0.384 (9)

0.111 (16)

-1.4037

-1.4735

(8)

(16)

0.59258 (1)

0.7117 (4)

-0.2276 (7) 0.6864 (5)

0.6753 (5)

(12)

(27)

-1.0742

(22)

-1.1045

(174)

(43)

(165) 0.087 (62)

-0.269

0.226 (37)

0.581 (86) 0.095 (52)

-0.009

0.078 (86)

0.469 (339)

-0.078

[e.bl

[PII

Y

-1.0753

moment

moment

0.046 (46)

Quadrupole

Magnetic

parameter

of the Pb resonance

Staggering-

from the IS and h.f.s. measurements

line

I

B 2. a?

:

2 3

9 0 T 0 %

1 L 3 3

M. A~el~ent

et al. / Odd-even

staggering

475

and their errors are highly correlated, and the correlation has been properly taken into account when evaluating the A-parameter. However, the staggering parameter

2((r2),4+1 -(r2)~)

‘=

(r2)A+2-_(rz)A

’

A even,

2(5v,+1-6uA-M/A(Afl)) = 8v,4+2 -6v,-2M/A(A+2)

can be deduced without a knowledge of the F-factor, but with a reasonable estimate of M. For that, Thompson et al. ‘) assumed M = 1 (10) x NMS (= 27 (270) MHz) . Recently, Ring and Wilson “) deduced from theoretical calculations of the specific mass shift in the considered atomic transition a value M = 0.19 (0.75) x NMS , with considerably reduced uncertainty. Our present analysis of the data given in table 1 adopts this value. We introduce it as a constraint in determining the F-factor by the Ring plot procedure, resulting in

F = 20.26 (0.18) GHz * fm-’ , M = 0.19 (0.25) x NMS . This value, with reduced uncertainty, is compatible with the previous value. The corresponding Ring plot fits the data quite well. Table 2 compiles the results of the nuclear quantitites extracted. The magnetic moments are calculated assuming a value for p1(207Pb) of +0.5783pN, as given by Fuller 13) [see ref. “I. The interpretation of the B-factors in terms of the quadrupole moments is based on a theoretically derived relation, Q(APb) [e. b] =0.00868 (52)B(APb; 6~7s 3P,), which has been found by an analysis 14) of fine and hyperfine structure splittings and g-factors of the relevant atomic configurations in PbI. Using this relation the extracted values of Q are about 40% larger than with the calibration used in ref. ‘). 4. Discussion The variation of the experimental ms charge radii s(r2) is displayed in fig. 1. The global trend is fairly well described by the droplet model 15), but the distinct discontinuity, observed when crossing the closed shell, is not reproduced. This is a well-known feature seen in various similar cases 16). Fig. 1 also displays a prediction on the basis of a recent semi-empirical approach 2s) for calculating rms charge radii’ ’ E. WesoIowski info~ed us that eqs. (16) and (17) of his paper “) should be corrected by AN = 1 (eq. (16)) and . . .+1/A’” (S,+S,) (eq. (17)).

476

M. Anselmenr

er al. / Odd-even

staggering

I

.6 -

Mean Square Differences

Charge

Radii

of Pb Isotopes

8

AN N L

V I Aa -.2-

N

L

V

/ Fig. 1. Mean-square

,

I

charge radii differences of Pb isotopes compared with the droplet-model and predictions following Wesoiowski’s approach 25).

prediction

Finer details and the limited accuracy of the droplet model (which does not account for any fine structure) are brought out more clearly by a Brix-Kopfermann plot showing the differential variation ( r2)A - ( r2)A+2 (fig. 2). The sudden acceleration of the increase (r2)A+2-(r2)A, with crossing the closed shell, and the smooth, but significant slowing-down after closing the neutron shell, are empirical features which are found to be typical signatures for shell and subshell closures 8,‘7*‘8).The marked discontinuities when passing N = 118 (“‘Pb) and IV = 124 (206Pb) are correlated with a change of the values of the ground-state spin of the odd nuclei [for 199Pb see ref. ‘)] and may indicate the filling up of the 3~,,~ and 2f5,2 neutron orbits. The odd-even staggering, indicated in fig. 1 as a microstructure of the global variation of the charge radii, shows a conspicuous trend, more pronounced when increasing the number of removed neutrons, while the staggering nearly (but not completely) disappears beyond “‘Pb, i.e. when filling up the new shell. This latter feature is common with various other cases [see ref. “)I. The staggering is usually

M. Anseiment

et al. / Odd-even

477

staggering

. Differentlal

-020

Change

Square Charge -019

of the

‘i

Mean

Rod11 in Pb Isotopes

I

I-

-018 -017 - 0.16

i

-015 N i ,^ _ -0.lL :” E P’ -* -013 Y ” -012 -011

Fig. 2. Differential changes s(r’) A,A+2 of the ms charge radii of Pb isotopes. The error bars include statistical uncertainties. The droptet model predicts the value (r2}a -(r’)a+z = -0.108 fm’.

only

represented by the staggering parameter y given in table 2. This parameter can be understood as a measure of the polarization of the proton core of the nucleus by the added (odd) neutron relative to the effect of adding a neutron pair. Fig. 3 displays the isotopic variation of y and shows how the polarization of the closed-shell proton core varies with the number of neutrons holes and with the angular momentum. At a first glance, it resembles the behaviour of the i,3,2 isomers of the neutrondeficient Hg isotopes ‘O). In that case, the observed trend of y has been explained 2’) as a consequence of the transition from strong coupling to Coriolis decoupling of the i,,,,-neutron-core coupling. Looking more closely at the Pb results (fig. 3), small changes in the average slope of y around N = 118 and N = 124 are evident, indicating the onset of filling up of the various orbitals.

478

M. Anselmenr

er al. / Odd-even

staggering

“‘Pb

‘d

N=126

10 Odd - Even

Staggering

of the ms Charge

09

Parameter Radii

08

06-

N=118

o.5_

Talmi’s formula .- --w

1 ~_ t

’ I

0.4 0.30.2 0.1 1

f I I i1 I I 197 199 201

Fig. 3. Isotopic

variation

I

I

203

205

I

I

207 ’ 209

of the staggering

parameter

I

*

211 A y.

A quantitative theoretical description of all these details, which might reveal interesting effects of the interactions within a finite Fermi system, implies a great challenge to nuclear structure theories. Large-scale microscopic structure calculahave been found to be unsuccessful in reproducing the observed features tions 7322*18) correctly. Talmi ‘) has recently studied the polarization of the proton core by valence neutrons as a possible mechanism for producing the odd-even staggering. Simple expressions for j” neutron configurations and various multipole terms of the pn interaction are derived and result in a three-parameter formula of the variation of 6( r’). The three parameters (A, B, C) can be, in principle, microscopically calculated, but due to a lack of knowledge about some details of the microscopic interaction, a phenomenological determination is proposed by fitting the experimental data. In fact, such a procedure has been rather successfully applied to the Ca isotopes 2335) reproducing the experimental variation of the Ca charge radii, better than any other description. While the neutrons in the Ca isotopes occupy a rather pure If,,, orbit and favor Talmi’s theoretical considerations, the situation in Pb is more complex since several orbitals are involved. The theoretical expression for S( r2) as a function of neutron number, applied with fixed parameter values over the full range of data, is certainly less accurate. Nevertheless, and following Talmi’s work, a least-square

h4. Affselme~t et al. / Odd-even

fit of the parameters general tendencies, ing the observed

in Talmi’s expression

479

staggering

to the experimental

data reproduces

some

even of the staggering. However, the limited success in reproducvalues of the staggering parameter correctly is obvious when

comparing

the theoretical

staggering

parameter

values to the experimental

displays

very sensitively

results

(fig. 3). Of course, the

small deviations

between

meas,ured

and calculated values of 6(r*). Though microscopic studies on the basis of HFB and HF+BCS methods failed to account quantitatively for the staggering, there is little doubt 181z4)that the effect is related to pairing correlations which are weakened in odd isotopes due to blocking 2-4). In a recent analysis of the situation, Zawischa *‘f suggests a mechanism which strongly couples the pairing properties of proton- and neutron-like four-body correlations. Breaking up a neutron-pair should consequently reduce the proton pairing, thus reducing the occupation probabilities of the levels above the Fermi energy. At present, the proposed mechanism has been described only in a simplified way 24), but the results are rather promising. The manifestation of an interplay of neutron and proton pairs would be, in fact, an interesting aspect of the observed fine structure in the variation of nuclear charge radii. The authors

would

like to thank

Prof. Dr. I. Talmi

for his encouraging

interest

in these studies and acknowledge the clarifying comments of Prof. Dr. A. Fassler and Dr. D. Zawischa. We are also indebted to Prof. Dr. W. Seelmann-Eggebe~ for his advice in preparing readily emanating 226Ra and *“AC samples. Note added in prooj: I. Talmi has informed us about an improved fit of his theoretical expressions to the present data set, leading to a better reproduction of the overall trend of y (fig. 3).

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staggering

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