Neutron transfer reactions on 204Hg

~

Nuclear Physics A153 (1970)17--31; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

N E U T R O N T R A N S F E R R E A C T I O N S O N '~°'tHg M.-L. ANDERSEN, S. A. ANDERSEN and O. NATHAN

The Niels Bohr Institute, University of Copenhagen, Denmark K. M. BISGARD t

Physics Laboratory, Royal Veterinary and Agricultural University, Copenhagen, Denmark K. GREGERSEN

Institute of Physics, University of Aarhus, Denmark OLE HANSEN

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544 and S. HINDS tt and R. CHAPMAN ttt

A WRE, Aldermaston, Berkshire, England Received 11 May 1970 Abstract: The reactions 2°4Hg(d, p)2°SHg and 2°4Hg(d, t)2°SHg were investigated for Ed = 12-13 MeV, using magnetic spectrographs and photographic detection. Six excited states were observed in 2°SHg below 3.6 MeV and thirteen were found in 2°3Hg below 1.8 MeV. The measured angular distributions were compared with DWBA predictions, and/-values and spectroscopic factors were determined for some of the levels. The data are compared with the corresponding reaction data for the lead nuclei. The nucleus 2°SHg appears to have some features in common with 2°~Pb. In particular, one can identify a 2°SHg state at 1.853 MeV which probably corresponds to the 2.73 MeV g~.-2h-lp state of 2°Tpb. The energy systematics of the ½- and ~+ states near 2°apb is discussed in terms of the pairing vibrational model. It is suggested that the change in excitation energy observed between the g~ states of 2°7pb and 2°SHg is due mainly to the interaction between the proton-hole pair and the neutron-hole pair. I

E

NUCLEAR REACTIONS 2°'*Hg(d, p), (d, t) E = 12-13 MeV; measured tr(Ep, 0), tr(Et, 0), Q. 2°3"2°SHg deduced levels, J, ~, I, spectroscopic factor. Enriched target.

1. Introduction The n u c l e a r shell m o d e l has been applied with r e m a r k a b l e success 1,2) to describe the properties o f nuclei with one or two valence particles (or holes) outside 2°spb. Recently, the calculations of few-nucleon spectra have been extended to three-particle (hole) configurations, such as 2 ° 5 p b [refs. a, 4)] a n d 2°7Bi [ref. 5)1. I n order to extend further the experimental picture o f three-hole configurations, we here present a study of the 204Hg(d ' p)205Hg stripping reaction, i n which the final nucleus has two p r o t o n t Work supported in part by the Carlsberg Foundation, Copenhagen. tt Now at Physics Department, Australian National University, Canberra. ttt Now at Department of Physics, Manchester University. 17

18

M.L. ANDERSENe t al.

holes and one neutron hole outside 2°spb. At the same time, we report some evidence on the nucleus 2°3Hg from the (d, t) pickup reaction on the same 2°4Hg target. No experimental data on excited states of 203, 205Hg have been published so far.

2. Experimental methods and results 2.1. TARGETS AND IRRADIATION PROCEDURES Relatively few nuclear reaction experiments have been reported on isotopes of mercury owing to the difficulties in prodl*.e~ng suitable targets of this element. Such targets should be thin, stable against charged particle bombardment, and the amount of medium weight and heavy impurities should be small. In the present work, two different methods have been used to fabricate 2°4Hg targets. Method one resulted in a target which contained only small amounts of contaminants heavier than oxygen, but with less satisfying stability properties. Method two resulted in a very stable target which, on the other hand, contained a large amount of aluminium. The first method was based on a special form of macromolecular electrospraying of fine grains of mercuro oxide molecules suspended in acetone. The suspension was prepared by treating an acidic solution of mercuro nitrate with sodium hydroxide. The precipitate of mercuro oxide was washed thrice with water and thereafter treated with di-isopropyl ketone to hinder congestion of the grains when suspended in acetone, The resulting clusters of mercuro oxide molecules had a grain diameter of less than about 30 Itm. The electrospraying was done at 4.2 kV with a 5-6 mm distance between the tip of the pipette and the carbon foil used as backing. The foil was preheated and was kept in rotation during the spraying. The target was finally covered with a thin foil of of formvar to reduce the evaporation of the target material during bombardment in the beam. The target had a mercury thickness of about 340/~g/cm 2. The mercuro nitrate solution used for the preparation of the sprayed targets was purchased from the Oak Ridge National Laboratory, USA, and had the following isotopic composition: 68.6 ~o 2°4Hg, 10.7 ~ 2°2Hg, 4 . 4 ~ 2°1rig, 7.6 ~ 2°°/'-Ig, 5.5 ~o 199Hg and 3.2 ~ 198Hg" The main chemical impurities were oxygen and carbon but small amounts of medium weight impurities were also detected. Test runs with a Van de Graaff accelerator indicated only moderate stability against charged particle bombardment, and this type of target therefore was used solely for angular distribution measurements with multiangle detection, not involving absolute cross-section determinations. In the second method, described elsewhere 6), the Aarhus mass separator was used. A preoxydized layer of an aluminium foil was bombarded with 400 keV Hg ions. The targets had a mercury thickness of about 50 #g/cm 2 deposited in a layer of A1203 about 125/~g/cm 2 thick. The isotopic composition was 100 ~ 2°4Hg. The targets were stable towards deuteron bombardment and were convenient for absolute crosssection measurements.

19

2°4Hg NEUTRON TRANSFER REACTIONS

2.2. E X P E R I M E N T A L

PROCEDURES

The deuteron bombardments were made with the Aldermaston tandem accelerator and with the tandem accelerator at the Niels Bohr Institute (NBI), at incident energies between 12 and 13 MeV. In the Aldermaston experiments, the reaction products were momentum analyzed in the multiangle spectrograph 7) and detected in Ex IN MeV 2.5

2.0

1.5

ols

10

o,

{3)

(0) X 1/2

(2) (I/2

/2

300

2°4Hg (d,p)2°SHg Ed = 12.11 MeV Lab.angle =125 ° 10 000 ~C

250

E E X I/2 (~

200

or LLI 12. tO (,-)

150

DISTANCE ALONG PLATE (cm) Fig. 1. T h e p r o t o n s p e c t r u m at Ed : 12.11 M e V a n d 0lab = 125 ° for the reaction 2°4Hg(d, p)2°SHg as m e a s u r e d in t h e N B I experiment. T h e n a r r o w p e a k s labelled (0)-(3) are due to 2°SHg. T h e g r o u p n u m b e r s are defined in table 1. T h e r e m a i n i n g p e a k s are due to impurities, m a i n l y a l u m i n i u m .

photographic Ilford K2 emulsions, in the (d, p) experiment, the emulsions were covered by a layer of polyethene, thick enough to stop all impinging charged particles except the protons. The target was of the electrosprayed type. In the NBI experiments, the mass separated targets were used and the reaction products were analyzed at 90 ° and 125 ° in a single-gap, magnetic spectrograph, with photographic detection. In the latter experiments, absolute reaction cross sections were determined by corn-

0

100

200

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49

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RADIUS OF CURVATURE IN cm

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.

"

i

=

•

&.O

4.5 2.S u

, 51

",.

-

.. i $2

--.

~°4Hg (d,p) Ed'12.96 MeV Lab. angle = 167.5 ° B = 10.334 kG 5950 pC

t...'-,,.~

(s)

3.0 i

-

• -

2.0 !

.

(3)

Fig. 2. T h e p r o t o n s p e c t r u m at Ed = 12.96 M e V and:0~=b = 167.5 ° for the reaction ~°4Hg(d, p)2°SHg as m e a s u r e d in t h e A l d e r m a s t o n experiment. T h e peaks (3)-(6) arc due to 2°SHg. T h e g r o u p n u m b e r s are defined in table 1. T h e cluster o f unresolved peaks in t h e region o f E= = 4-4.5 M e V probably belongs to 2°SHg b u t a detailed interpretation h a s n o t been attempted. T h e isotopic purity o f the target was 68.6 Yo.

I-..-

%.. 0 < rY

o n" L.LI n

t.O

E E

I-Or)

n n,"

300

6.0

Ex IN MeV

m

O

21

20'*Hg NEUTRON TRANSFER REACTIONS

TABLE 1 T h e 2 ° * H g ( d , p)2°SHg results

do'\ Group no.

Ex (MeV±0.05)

0 1 2 3 4 5 6

0 0.376 0.468 1.853 2.917 2.952 3.576

b

T~) ...... o

1

j

1

½-

0.5

4

~+

0.8

)

S

#b/sr 90 °

125 °

278 120 251 1267

133 75 122 738

Qo = 3.4434-0.005 M e V a). T h e b o m b a r d i n g energy was 12.11 M e V in t h e N B I e x p e r i m e n t a n d 12.96 M e V in t h e A l d e r m a s t o n experiment. a) T h i s value o f Qo results f r o m t h e N B I data. T h e s a m e value is o b t a i n e d f r o m t h e A l d e r m a s t o n experiment, T h e t a b u l a t e d value a) is Qo = 3.3184-0.100 MeV, i.e. 0.125 M e V lower t h a n the p r e s e n t result. b) T h e experimental cross sections (NBI data) h a v e a relative u n c e r t a i n t y o f 4-10 ~ . T h e absolute scale h a s a n u n c e r t a i n t y o f 4-30 ~ . l0 s

102 .

(o)

13i

DWL''

DW --1"4 I -'-- 1=2

/

-

_~

-10'

y S

5

O3 I--

t t l l | i t t l I l S l l I l !

, , , , I

I , , , , I , ,

Z 10 s

cr < or

5

(4)

I--

10 s

(s)

tY <

S 10 z

A

t~

3 "O

I l i i l l l l f l | l l

t

0

,

i

I

I

I

50

100

15Q

,

,

10 ~. ~)e,m.

"O 5

DW t'0

.

.

.

.

t'2 ....... L = 4

103 --

i

0

i

i

i

I

50

i

i

i

i

I

100

cm

i

i

i

i

I

150

,

,

Fig. 3. A n g u l a r distributions f r o m t h e re tion z o ¢Hg(d ' p)20SHg as m e a s u r e d with t h e m u l t i g a p spectrograph. T h e curves s h o w n with t h e distributions (0), (3) a n d (6) are D W predictions. In t h e distributions (4) a n d (5), t h e experimental points are connected b y ~trai~ht line~ withont thenretieal ~i~,nificance.

M . L . ANDERSEN et aL

22

parison of the (d, p) and (d, t) intensities with the yield of elastically scattered deuterons. The absolute value of the elastic deuteron cross section at 12.11 MeV is determined previously from measurements with known targets. EXCITATION 2.0 l

ENERGY (MeV)

1.5 ~

n

i

J

[

1.0 n

n

I

n

0.5

i

J

n

[

I

l

0 J

D

=

,

i

(2)

(0)

p

x 1/3

2°~Hg (d,t)Z°SHg Ed = 12.11 MeV Lab.angle = 125 ° 10000jJC

350

x 1/5

(3)

300

(61

i

(13) E E

(

250

(e)

I.O t'xd

b

1:5 n,I./J 12.

1

200

to ,,r t..) rr p-

15(]

(L.)

100

I

(E

E" II 50

80

85

90

95

DISTANCE ALONG PLATE (crn) Fig. 4. T h e t r i t o n s p e c t r u m at Ea : 12.11 MeV a n d 0~a~ = 125 ° for the re a c t i on 2°'~Hg(d, t ) 2 ° a H g as m e a s u r e d in the N B I experiment. The group n u m b e r s are defined in table 2.

2.3. THE (d, p) RESULTS Figs. 1 and 2 show parts of the proton spectra from the (d, p) reaction in the two different setups. In the NBI experiments, the energy resolution was 12 keV FWHM, but the spectra were contaminated by the presence of many strong proton groups from the

t-

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~=~"

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=. ~] ;>

•

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ul

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m

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v

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o

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N

I ("

(do'/dco)c.m. ARBITRARY UNITS I

o

'1 I

m

[3

i

~

v

I

~

m

©

Z

H

M . L . ANDERSEN e t al.

24

27Al(d, p)2aA1 reaction. As a consequence, only four proton groups could be safely identified in this experiment as belonging to 2°SHg, and no conclusions could be drawn on Hg levels above 2 MeV excitation. In the Aldermaston experiment (energy resolution about 25 keV), some additional levels were seen above 2 MeV. Due to the lower isotopic purity of the electrosprayed target, only the stronger of the corresponding proton groups may be ascribed to TABLE 2 The 204Hg(d ' t)20ZHg results. Qo = ( - 1.242 4-0.005) MeV a)

Group no.

Ex (MeV4-0.005)

(dcr) b) ~ ...... p (/zb/sr) 90 °

125 °

1

S

0 1

0 0.048

1097

1279 615

3 1

7.9 ¢) 1.7 j = ½

2

0.221

861

546

1

1.5 j = ½ 1.4 j = ~

3 4 5 6 7 8 9 10 11 12 13

0.544 0.747 0.765 0.925 1.023 1.041 1.109 1.326 1.462 1.640 1.752

175 49 48 74 60 272 6 9 24 15 109

159 48 34 123 51 208 6 7 23 19 129

3

3.9 j = ~ 2.9 j = ]

1.5 j = ~

The bombarding energy was 12.11 MeV in the NBI experiment and 12.00 MeV in the Aldermaston experiment. a) The tabulated value s) is Qo = -1.2414-0.009 MeV. b) The relative cross-section values (NBI data) have an uncertainty of 4-10 %. The absolute scale may have an uncertainty of 4-30 %. c) Using the 1.16 fm triton potential o f Flynn e t al. 21) for Pb, a spectroscopic factor of 7.1 is obtained.

2°5I"Ig with certainty. A cluster of peaks in the spectrum corresponding to the region of excitation between 4 and 4.5 MeV probably belongs to 2°SHg but no detailed interpretation has been attempted. Table 1 lists the excitation energies and observed cross sections for the seven levels of 2o 5Hg which were identified in the (d, p) reaction. The angular distributions of the outgoing protons were measured in the Aldermaston experiment for five of the groups of table 1. These distributions are shown in fig. 3. The groups (I) and (2), corresponding to the first and second excited states of

204Hg NEUTRONTRANSFERREACTIONS

25

2°SHg, were masked by impurity groups at several angles and meaningful angular distribution data could not be obtained in these cases. 2.4. THE (d, t) RESULTS The Q-value for the 27Al(d, t) reaction is less than the Q-value for the 2°4Hg(d, t) reaction by about 5.6 MeV and the triton spectrum for 2°3Hg therefore could be observed in the NBI experiment without interference from 26A1 peaks (fig. 4). A total of 14 levels was observed below 1.8 MeV of excitation; the corresponding excitation energies and cross sections are given in table 2. The same levels were also excited in the Aldermaston experiment but with an energy resolution that was much poorer than in the NBI experiment (50 keV against 10 keV). This bad resolution was caused by a combination of stopping effects in the rather thick target and by a deterioration of the spectrograph performance at the high magnetic field needed to bend the tritons. Angular distribution data, therefore, were obtained only for six triton groups as shown in fig. 5. 3. The D W B A

analysis

3.1. THE (d, p) REACTION In order to determine/-values and to obtain spectroscopic factors from the experimental data, DWBA analyses were performed in the zero-range approximation, without corrections for non-locality, using the Oak Ridge code JULIE. It is our experience lo) from previous experiments on 2°4pb that neutron stripping and pick-up reactions on heavy nuclei in this range of bombarding energies are well described by the DWBA. The proton and deuteron optical-model parameters as well as the bound neutron-well parameters were taken from table III of the work of Muellehner et al.9), in which the (d, p) and (d, t) reactions on 2°spb were analyzed for Ea = 15-20 MeV. The use of these parameters without a lower cutoff in the radial integration was found to give somewhat improved fits to the data as compared to the fits obtained in our previous lo) 13 MeV (d, p) work on 2°4pb. Some calculated curves are shown in fig. 3 for the 2°4Hg(d, p) reaction. The ground state distribution is well fitted by the l = 1 DW curve and no other/-value gives a reasonable fit to this distribution. Similarly, we find a unique DW assignment of l = 4 for group 3, Ex = 1.85 MeV. For the remaining three levels at excitation energies between 2.9 and 3.6 MeV, unique assignments cannot be made since the predicted shapes in the 50°-150 ° interval for l = 2 and l = 4 are quite similar. In table 1 we show the (d, p) spectroscopic factors derived from the 90 ° and 125 ° absolute cross sections, giving equal weight to both, and using the relation do'_ 1.48S(2j+1) dO

~

Dw"

For the calculation of S, we have assumedj = ½ for the ground state, a n d j = ~ for the 1.85 MeV state, by analogy to the reaction 2°6pb(d, p)2°Tpb (see sect. 4).

26

M.L. ANDERSEN et al.

3.2. THE (d, t) REACTION

Also for this reaction the optical model and neutron-well parameters were taken from the work of Muellehner et aL 9), and the calculation was made without a lower cutoffin the radial calculation (see tame v l r I of ref. 9). The value of W o for the triton was put equal to 17 MeV. Guided by the evidence xl)on the neutron states in 2°Spb, we have assumed the low-lying states to have either 1 = 1 or 1 = 3 character. The ground state and the first excited state at 48 keV were resolved in the Aldermaston data and were found to exhibit different angular distribution shapes. The ground state is best fitted by l = 3, whereas the 48 keV state probably has l = 1. (See fig. 5.) Reasonable fits could be obtained also for the 221 keV state (l = 1) and the 1752 keV state (1 = 3). Spectroscopic factors were obtained from equal weighting of the 90 ° and 125 ° data (except for the g.s. where only 125 ° were available) on the basis of the relation dO

Dw"

The results are shown in table 2 for the four states mentioned above. For the l = 3 ground state transition, the spectroscopic factor is calculated on the assumption of j = ~ since j = { can be excluded on the basis of radioactivity data x6,17). The remaining spectroscopic factors are calculated for both of the two possible j-values corresponding to each/-value. 4. Discussion 4.1. THE N U C L E U S 2°SHg

Many features of reaction cross sections and electromagnetic transition probabilities for nuclei close to 2°8pb have been explained successfully by the nuclear shell model. When we move away from 2°spb, the situation rapidly becomes more complicated due to the coupling between the particle and core degrees of freedom. This is well known for the thallium isotopes and also for the light mercury isotopes. For the heavier mercury isotopes, the experimental evidence on the coupling situation is as yet very meagre and few theoretical studies have been made. Two calculations of the 2 ° 3 H g spectrum have been reported 12,13), both using a pairing plus quadrupole residual interaction, but no detailed calculation of 2°5Hg has been made as yet. In the present discussion, we compare the mercury data with the corresponding data for lead, in order to see to which extent the simple shell-model features survive in mercury. Thus we compare the reactions 204Hg(d ' p)205Hg with the lead reaction 2°6pb(d, p)2°Tpb, and the reaction 2°4Hg(d, t)2°aHg with the 2°6pb(d, t)E°Spb reaction. In fig. 6 some relevant (d, p) data for 2°Tpb and 2°SHg are summarized. The (d, p) spectroscopic factors for levels in 2°7pb are taken from the works of Mukherjee and Cohen 14), and of Darcey et al. 15). The states of 207pb excited in the stripping reaction below 2.7 MeV are relatively pure neutron-hole states. The lowest 2h-lp excita-

2°4Hg

NEUTRON

27

TRANSFER REACTIONS

tion in 2°Tpb is the g~ level at 2.73 MeV, which occurs as a very strong state in the (d, p) experiment with about 90 % of the spectroscopic strength allowed by the shell model. The ground state of Z°5Hg has l = 1, which is consistent with the tentative ½assignment reported from radioactivity data 16). Assuming this spin, we find a spectroscopic factor of S = 0.5 which is close to the value found for the 2°Tpb ground state. Presumably the 2o 5Hg ground state therefore is a simple p~ neutron hole excitation of MeV

-

0.76 -

-

Silz

0.70 -

d 5/2

_ _ 0.12 . -

d5/2 111/2 -i

h%

0.g0

".gg/2

__.f-1 0.008

?/z

0.02

. -~

0.06

p_~

0.07

f -1 s/z

0.5/.

-T P 1/2

2

"'-. "'.

0.8

t.Z,(V2÷)

0.5

t - l(Vz-)

i 13/2

3/2

2°7pb

2°SHg

Fig. 6. The levels of 2°SHg seen in this experiment and some relevant levels of 2°TPb, excited in the reaction 2°6pb(d, p)2°TPb. The numbers shown with some of the levels are spectroscopic factors from this work and from refs. 14.25). the 2 ° 6 H g ground state, and the degree of filling of the p~ orbital in 2°4Hg (g.s.) is about the same as in 2°6pb (g.s.). The/-values for the first and second excited states of 2°SHg at 376 and 468 keV unfortunately could not be determined in this experiment. I f we compare to 2°7pb and for a moment assume that these states correspond to the f{ and p~ hole excitations, we find the experimental spectroscopic factors to be of the order 0.1-0.4. These values are, however, considerably higher than those found for 2°7pb (fig. 6). Improved experimental data are needed before the nature of these two states in 2°SHg can be ascertained. The 1.85 MeV state has 1 = 4 and it is likely that this level corresponds to the g~ state is) of 2°Tpb at 2.73 MeV. Assuming a spin of 9, we find S = 0.8 for this state,

28

M. L. ANDERSEN e t

al.

which is close to the value 0.9 observed 15) for 2°Tpb. Thus, it appears that also in 2°SHg a large part of the g~ neutron strength is concentrated in a single state. This is an interesting fact, since another 2 + state is expected about 400 keV above the lowest g~ state, formed by coupling the first excited 2 + state of the 2°4Hg core to the g~ orbital. The coupling between these two states should be strong in view of the large B(E2) value of the 2 + --, 0 + transition in 2°4I-[g. It would be interesting in future experiments to locate the remaining part of the l = 4 strength in 2°SHg. The three strong states found in 2°SHg between 2.9 and 3.6 MeV may correspond to 2h-lp states with the neutron particle in the s or d orbitals t s). The observed cross sections are consistent with this interpretation, but the experimental difficulties in assigning/-values preclude a detailed discussion at the present stage. 4.2. THE NUCLEUS

2°aI-Ig

In the energy range below 1.75 MeV, a total of 13 excited states are seen in the (d, t) reaction leading to 2°3Hg, as compared to 9 states seen 11) in the same energy range in 2°spb from the 2°rPb(d, t)2°Spb experiment. This seems to indicate a larger splitting of the hole strength, but it is hard to make quantitative conclusions since /-values have been determined for only a few states in 2°3Hg. Recently, the ground state of 2°3I-Ig has been assigned i7) ~ - from radioactivity data, which is consistent with the present l = 3 assignment. The experimental spectroscopic factor of 7.9 is 30 ~ larger than the sum-rule limit. In view of the uncertainty of the cross-section scale, and in the determination of the spectroscopic factor itself, this discrepancy may not be serious. The two states of 2°3Hg at 48 and 221 keV both have l = 1, but we have no means of distinguishing between j = ½ and j = ½. In 2°Spb, the first excited state occurs 2 keV above the ~r- ground state and has spin and parity ½-. In the calculations of Kisslinger and Sorensen 12), the same sequence is predicted for 2°aHg, the lowest ½state occurring about 30 keV above the ground state. The second excited state in their model occurs at about 100 keV and has spin and parity ½-. Thus, as concerns level energies and/-values, their calculation shows fair agreement with the data for the lowest states of 20aHg" In the calculations of Lo Iudice et aL 1a), the spin sequence of the fit'st and second excited states is reversed as compared to Kisslinger and Sorensen, and the first excited state occurs somewhat higher, at about 110 keV. A slight preference for the Kisslinger and Sorensen results is thus indicated by the present energy data. 4.3. ENERGY SYSTEMATICS OF ½- AND ~+ STATESNEAR 2°apb We shall discuss the energy systematics of ½- and 2 + states near 2°8pb in the spirit of the pairing vibrational model. The energy of a state of spin and parity J~, E(Z, iV, J'~) in a nucleus with charge Z and neutron number N is defined as

E(Z, N, J'~) = Ex+B(2°apb)-B(Z, N ) + 5 8 0 8 ( N - 1 2 6 ) + 6 0 4 0 ( Z - - 8 2 ) ,

(1)

204HgNEUTRON

TRANSFER REACTIONS

29

where Ex is the excitation energy of the level in question and B(Z, N) is the ground state binding energy of nucleus (Z, N). The zero point of the energy scale is the ground state of 2°spb. The constants in front of the last two terms of eq. (1) are chosen such that E[Z~°Pb(g.s.)] = E[2°6pb(g.s.)] = 2493 keV, (2) E [210po(g.s.) ] = E [2°6Hg(g.s.)] = 3301 keV.

(3)

In the pairing vibrational model is), the four ground states involved in eqs. (2) and (3) constitute the four fundamental one-phonon states and the energies (2) and (3) are the phonon energies for neutron and proton pairing vibrations, respectively. The energies of the 2°7pb and 2°9pb ground states are the Jr- single-hole and 3`+ single-particle energies, respectively. The values are 19) e(Jr-) = E[2°7pb(g.s.)] = 1567 keV,

(4)

8(2~+) - E[2°9pb(g.s.)] = 1865 keV.

(5)

In the pairing vibrational model, the ½- and ~+ states around 2°spb are described as the appropriate number of proton (or proton-hole) phonons plus the appropriate number of neutron (or neutron-hole) phonons coupled to the single particle or single hole. In the limit of vanishing interaction between the phonons internally and between phonons and single particle (hole), we have

Ep.,..(Z, N, J " = 0 +) = JrlZ-8213301 + ½ I N - 12612493,

(6)

Ep.,,.(Z, N, J~ = ½-) = 1567+½1Z-8213301 + ½l( N + 1 ) - 12612493,

(7)

Ep.v.(Z, N, J~ = ~+) 2 = 1865+½1Z-8213301 + Jrl(N- 1 ) - 12612493.

(8)

The energies of eqs. (6)-(8) are in keV. All the relevant data for E and Ep.v. are given in table 3. Also shown are the differences

AE = E(Z, N)-Ep.v.(Z, N). The data for the ~+ states in 21~pb, 2°7pb and 2aapo demonstrate the magnitude of the interaction between the g4 neutron and the various pairing phonons. The interaction between the g~ neutron and the proton-particle pair (~ = + 2 mode or pair addition mode) is particularly large and attractive. The interaction between the proton-hole pair (~ = - 2 mode) and the g~ neutron is not known because the mass of the 2°7I-Ig is not known. The ground state of 2°4Hg contains a proton phonon as well as a neutron phonon. From the 2°4Hg mass it follows that the interaction between the proton ~ = - 2 and the neutron ~ = - 2 phonon is 1708 keV (attractive). Hence, the interaction in the

30

M. L. A N D E R S E N

et al.

TABLE 3 Energy systematics a) Isotope

2°Spb 2°7pb 2°gpb 211pb 2°Tpb 211po 2°THg 2OSHg b) 2°*Hg 213po 212po 2OSHg b) 2°9po 2°SPb 2°gPb 2°3Hg 2°7po 2°apo 2°THg

l~"

0+ ½~+ 3+ ]+ ]+ 3+ {+ 0+ 3+ 0+ ½- 0 ½{½½½0+ ½-

E

Ep.v.

0 1567 1865 4473 4299 4559

dE

~.

0 4359 4359 5166 5166 7659 5794 7659 5794 4868 4868 4060 4060 7361 7361 5794 7361

6078 4086 5778 4354 4225 5145 4770 4020 5824 8933 6307

114 -- 60 -- 607 --1581 --1708 --1881 -- 1440 -- 643 277 710 -- 40 --1537 1572 513

v.

c~= --2

c~=2

0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1

0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 0

co= --2 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0

c~=2 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1

") The second column indicates spin and parity of the states concerned. The state in question is always the lowest of this spin and parity. The quantities E, Ep .... and A E are defined in the text. They are quoted in keV. ~, is the number of proton phonons, whereas vn is the number of neutron phonons, ct = --2 indicates a hole pair, whereas ~ = 2 stands for a particle pair. The ground state masses are from ref. a). The data on excited states are from refs. 16, 20). b) The binding energy is taken to be 1614335 keV on the basis of the present Q-value for the 2°4Hg(d, p)2°SHg reaction (table I) and the mass of 2°4Hg as given in ref. s). c) Ex = 48 keV.

2 ° 5 H g ~+ state s h o u l d be the s u m o f t h e p h o n o n - p h o n o n

interaction (-1708

keV),

t h e i n t e r a c t i o n b e t w e e n the ~ = - 2 n e u t r o n p h o n o n a n d t h e g l p a r t i c l e ( - 6 0 k e V ) a n d the u n k n o w n g~- ~ = 2 p r o t o n - p h o n o n i n t e r a c t i o n . T h e t o t a l i n t e r a c t i o n in t h e 2 o 5 H g (2~+) state is ( - 1581 keV), a n d h e n c e t h e i n t e r a c t i o n ~2+ - ( " = - 2 , 7c), s h o u l d be a b o u t (1708 + 6 0 ) - 1581 = + 187 keV, i.e., q u i t e s m a l l a n d p o s i t i v e . T h i s e s t i m a t e leads to a p r e d i c t e d e n e r g y E[2°7Hg(~+)]

= 5353 k e V

o r B [ 2 ° T H g ( ~ + ) ] = 1624823 keV.

T h e m e t h o d o f a d d i n g i n t e r a c t i o n s is e q u i v a l e n t to the a s s u m p t i o n t h a t t h e effective i n t e r a c t i o n s b e t w e e n t h e different b o s o n s a n d f e r m i o n s are o f t w o - b o d y n a t u r e . T h e p r o c e d u r e c a n be a p p l i e d also t o the A E v a l u e s f o r t h e 213p0(~+), 2 ° 7 p 0 ( ½ - ) a n d 2 ° 3 H g ( ½ - ) states. F o r 213p0, we get A E ' = - 1 9 3 3 k e V b y s u m m i n g t h e a p p r o p r i a t e i n t e r a c t i o n s , as c o m p a r e d to t h e m e a s u r e d v a l u e A E = - 1 8 8 1 keV. F o r

204"Hg NEUTRON TRANSFER REACTIONS

31

2 ° 7 p o ( ½ - ) , we have A E ' = 1500 keV a n d A E = 1572 keV. F o r 2°3Hg(½-), we get A E ' = - 1641 keV a n d A E = -- 1537 keV, a s s u m i n g Ex = 48 keV. F o r Ex = 221 keV, the value o f A E for 2 ° 3 H g ( ½ - ) is - 1 3 6 4 keV. It a p p e a r s t h a t the m e t h o d o f a d d i n g interactions in these three cases is a c c u r a t e to a few h u n d r e d keV a n d this m a y be t a k e n as the a c c u r a c y o f the a b o v e p r e d i c t i o n for the energy o f 2°7Hg(~+). The energy o f the ½- state in 2°7I-Ig c a n n o t be p r e d i c t e d (except in the vanishing interaction limit) since the f u n d a m e n t a l (~ = - 2 , 7 r ) - ( ~ = + 2 , v) p h o n o n intera c t i o n [equivalent to the mass o f 2°SHg(g.s.)] is u n k n o w n . On the basis o f the a b o v e discussion it is suggested t h a t the interaction between the g~ n e u t r o n a n d the p a i r o f p r o t o n holes m a y be quite small, so that the change in excitation energy o f the ~+ state o b s e r v e d between 2 ° 7 p b a n d 2°SHg is due m a i n l y to the i n t e r a c t i o n between the p r o t o n - h o l e p a i r a n d the n e u t r o n - h o l e pair. W e are i n d e b t e d to G u n n a r Sorensen, A a r h u s , for d e v e l o p i n g the m a s s s e p a r a t o r target techniques. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

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