Frank A., Jolie J., Van Isacker P. Symmetries in Atomic Nuclei: From Isospin to Supersymmetry

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5.4 A Case Study: Structure of

196

Au 141

may provide a challenge and motivation for future detailed experiments of

the kind described in the next section.

5.4 A Case Study: Structure of

196

5.4.1 First Transfer Reaction Experiments

A ﬁrst investigation of the structure of

196

Au to test predictions of the

(6/12) ⊗ U

(6/4) extended supersymmetry was performed in the late

1980s [266]. From the known experimental level schemes of the even–even

and odd-mass members of the quartet, the level scheme of

196

Au was

predicted and subsequently the single-particle-transfer amplitudes for the

197

Au(d,t)

196

Au reaction were calculated. All these results were derived in

the SO(6) limit of the U

(6/12) ⊗ U

(6/4) scheme, for which, as always in

the presence of a dynamical symmetry, the wave functions do not depend

on the hamiltonian parameters. Therefore, quantities such as the spectro-

scopic strengths are ﬁngerprints of the underlying symmetries. At the Accel-

erator Laboratory in Munchen the transfer reaction was performed by the

group headed by von Egidy. As the ground state of

197

Au has J

=3/2

transfer with l =1, 3 leads to the negative-parity states described by the

model but also to states based on ν2f

7/2

excitations which are outside the

model. Other negative-parity states, arising from the unique-parity orbitals

ν1i

13/2

× π1h

11/2

occur at higher energies, above the known isomeric J

−

state at 595.66 keV. To distinguish them unambiguously and also the

positive-parity states populated by ν1i

13/2

transfer, the

197

Au(

He,α)

196

reaction can be used which is dominated by l = 6 transfer.

Because the spin and parity of the excited states were unknown, the only

test possible was a comparison of the measured strength function with the one

predicted from the supersymmetry scheme. Theory predicts the excitation of

22 states and a negligible strength distribution above 500 keV. Figure 5.4

shows the results obtained in Ref. [266]. In accordance with the theoretical

predictions, about 20 states share the transfer strength below 500 keV. How-

ever, a clear discrepancy occurs at low energy where theory predicts strength

to two states while the only strength observed at that time was to the ground

state. Also, a huge peak was observed at an excitation energy of 165 keV,

whereas the theoretical prediction showed much more fragmentation. The

running sums of the strength did, however, agree very well as can be seen

from the inset of Fig. 5.4.

Although the transfer experiment sensitively probed states predicted by

the model, one of its drawbacks was that it could not distinguish between

l =1andl = 3 transfer. Therefore, no information could be deduced about

spins. A few years later Vergnes and collaborators performed additional trans-

fer experiments at the Institut de Physique Nucl´eaire in Orsay using the reac-

tion

197

Au(p,d)

196

Au. The resolution in this experiment was 11 keV full width

142 5 Supersymmetries with Neutrons and Protons

Fig. 5.4. Observed transfer strength at 15

◦

for the reaction

197

Au(d,t)

196

(middle), compared to the predictions of the U

(6/12) ⊗ U

(6/4) extended super-

symmetry (top). The inset compares the summed l =1, 3 strength. The character of

the states populated with l = 6 can be deduced from

197

Au(

He,α)

196

Au (bottom).

(Reprinted from J. Jolie et al., Phys. Rev. C43. (1991) R16

1991 by the American

Physical Society, with kind permission.)

at half maximum (FWHM). The measured angular distributions allowed to

distinguish between the l =1andthel = 3 transfer and the corresponding

strengths could now be compared separately to the theoretical results. The

experiment essentially conﬁrmed the discrepancies already mentioned but

also that the summed l =1andl = 3 strengths agree with the predictions.

Because of the considerable observed strength in the ground state and in the

5.4 A Case Study: Structure of

196

Au 143

level at 165 keV, breaking of the dynamical supersymmetry was envisaged to

explain the observed strength distribution [267]. The main problem, shared

by all these experiments, is that the ground-state spin 3/2 of

197

Au did not

allow one to determine the spin values of the excited states in

196

Au nor did

it allow one to distinguish between the transfer of a 3p

1/2

or 3p

3/2

neutron.

5.4.2 New Experiments at the PSI, the Bonn Cyclotron and the

Munchen Q3D Spectrometer

At that moment of despair during the mid-1990s, a new collaboration was

established involving the University of Fribourg, the University of Bonn and

the Ludwig–Maximillian University of Munich. In an ultimate attempt to

determine the structure of

196

Au, in-beam spectroscopy, in which radiation

emitted by

196

Au is measured, was performed at the Philips cyclotron of the

Paul Scherrer Institute (PSI) in Switzerland and at the Bonn cyclotron. In-

beam studies are needed because the unstable

196

Au isotope must be created

by continuously bombarding stable isotopes, such as

196

Pt or

197

Au, with

accelerated protons or deuterons. Under these circumstances the observed

radiation is extremely complex since many states are excited by the diﬀerent

nuclear reactions and are populated in the de-excitation of these states. In

parallel with the in-beam experiments, several sophisticated transfer experi-

ments were performed at the Tandem accelerator in Garching.

The ﬁrst in-beam experiments were performed at the Philips cyclotron of

the PSI with the

196

Pt(d,2n)

196

Au reaction at 12.5 MeV. The very complex

spectra were then analyzed with γ–γ coincidences obtained with a spectrom-

eter consisting of ﬁve BGO shielded Ge detectors [268]. From the coincidence

conditions several γ rays could be interrelated leading to the identiﬁcation

of several structures of connected excited states. Of importance were also

the coincidences between the γ rays associated with these structures and the

x-rays which allowed to identify whether they belonged to Au or Pt isotopes.

The observation of structures associated with

195

Pt showed that beside the

(d,2n) reaction also

196

Pt(d,p)

195

Pt contributed. Therefore, complementary

in-beam γ–γ coincidence experiments were performed at the Cyclotron in

Bonn with use of the

196

Pt(p,n)

196

Au reaction at 9.12 MeV. This very low

energy, well below the Coulomb barrier of 13 MeV, ensured that all other

reaction channels essentially were closed. The cross-section was estimated to

be 30 mb only. To assign spins, an excitation function was measured with

protons at three higher energies.

Internal conversion, in which atomic electrons instead of γ rays carry

away the energy of a nuclear de-excitation, is a strong competitor to γ-ray

emission in heavy odd–odd nuclei. The K-to-L conversion intensity ratio is

strongly dependent on the multipolarity and this mechanism can be used to

determine the latter. Therefore, in addition to the in-beam γ-ray experiments,

conversion electrons were measured with the Orange spectrometers in Bonn

144 5 Supersymmetries with Neutrons and Protons

in both the (d,2n) and the (p,n) reactions. These experiments included the

determination of e–γ and e–e coincidences.

The analysis of the in-beam data quickly revealed that the peak observed

at 165 keV was in fact a closely spaced triplet of levels, decaying to the

−

ground state with an M1 transition [269]. The multipolarity of the tran-

sition was determined from the conversion electron spectra using e–γ co-

incidences to clean up the spectrum. All three states turned out to have

low spin and negative parity and as such belonged to the supersymmetric

model space.

To study the very dense odd–odd nucleus

196

Au, state-of-the-art instru-

mentation provided at the magnetic Q3D spectrometer is needed for transfer

reaction studies. Improvements to the detector and the polarized source made

new transfer experiments worthwhile. The detector measures the energy of

the outgoing particle and as such the excitation energy of the newly formed

excited nucleus which appears as missing energy. As discussed in Sect. 3.5,

the detector developed by Graw and collaborators attained an energy reso-

lution of 3 keV FWHM for the (p,d) reaction, representing an improvement

with a factor of about four compared to the Orsay experiment.

The high-resolution data of the

197

Au(p, d)

196

Au transfer showed for the

ﬁrst time [270] that the strength previously associated with the ground state

196

Au was in fact split into a closely spaced doublet—a crucial ingredient

for the solution of the problems encountered before (see Fig. 5.5). In total 47

states were resolved in the energy interval 0–1,350 keV. With knowledge of

the energies of the excited states the

197

Au(d,t)

196

Au reaction was then used

to determine the nature of the transferred particle. This allows to compare

the spectroscopic strengths with the theoretical predictions, as discussed in

Sect. 3.5. However, as the initial state (ground state of

197

Au) has J =3/2

several orbits can contribute to the strength to an excited state in

196

Au,

and these contributions may add incoherently. This feature was incorporated

in the data analysis which allowed for several contributions to the angular

distributions and analyzing powers. Figure 5.6 illustrates the quality of such

ﬁts. Due to this possibility of transferring nucleons in diﬀerent single-particle

states, the spin of the ﬁnal state cannot be determined unambiguously as was

the case for

195

Pt. Nevertheless, the possible spins can be limited as follows.

Observation of a 3p

3/2

transfer leads to possible spin–parities in the range

from 0

−

to 3

−

;a2f

5/2

transfer gives 1

−

to 4

−

. When both are observed,

only 1

−

and 3

−

are possible. The high-quality data of the (d,t) reaction

allow the observation of extremely weak contributions to the strengths and

this can be used in the spin determination. The starting hypothesis is that

any state that can be reached by a given transfer will show some small com-

ponents of this transfer. Due to the high level density, the non-observation

of a given transfer can be used to restrict the range of possible spins. For

example, if only 3p

3/2

and 2f

5/2

but no 3p

1/2

transfer is observed, the state

is assumed to be 3

−

. The observation of only 3p

3/2

transfer indicates a

possible 0

−

state.

5.4 A Case Study: Structure of

196

Au 145

0 100 200 300 400 500 600

counts

0.0, 2

6.0H

42.1

84.7, 5

162.4H

166.5

197.8

212.9

233.5

252.5

257.9H

287.4

298.3H

307.3

323.4

348.1

355.4H

369.7, 6

375.0H

387.5H

399.2, 6

402.5H

407.4H

413.0

420.1, 8

455.6

465.5

479.8

490.6

502.1

519.8

541.0

550.8

564.1

569.9

586.7

596.9

700 800 900 1000 1100 1200

624.8

635.9

645.0H

650.3

667.4

679.2

688.2H

701.6H

707.9

716.3H

721.3

734.3

746.4

761.1H

769.1

785.7

798.9

806.8

815.3

841.3

850.0

855.2H

881.8

892.1

901.0

907.5

921.9

938.2

950.0H

958.7H

967.9H

973.9H

985.7H

990.7H

1005.4H

1017.9H

1024.9H

1045.0H

1057.9H

1066.1H

1070.6H

1083.7H

1088.6H

1095.8H

1112.0H

1121.5H

1129.7H

1137.1H

1144.0H

1149.2H

1166.3H

1175.4H

1189.6H

1198.5H

1202.4H

1207.2H

1223.2H

1230.9H

1236.6H

1244.2H

1248.2H

(keV)

197

Au(p,d)

196

counts

Fig. 5.5. Spectra measured in the (p,d) reaction from

197

Au to the low-energy

states in

196

Au. Multiplets of levels that were previously unresolved are indicated

by an asterisk. (Reprinted from H.F. Wirth et al., Phys. Rev. C70 (2004) 014610

2004 by the American Physical Society, with kind permission.)

To test this method, a third transfer experiment with 18 MeV polarized

deuterons was performed using the

198

Hg(d,α)

196

Au reaction. This type of

experiment is very diﬃcult due to an increased energy loss of the α particles

in target and detectors. With use of very thin targets (34 μg/cm

of enriched

mercury-sulphid on 7 μg/cm

of carbon) an energy resolution of 7 keV could

be reached which is excellent for this reaction. The low cross-section combined

with the small amount of target atoms led to low statistics which allowed the

observation of 17 states only. The main advantage of the (d,α) reaction is

that it yields ﬁrm spin values because the transfer starts from an even–even

Hg isotope with a 0

ground state and produces as ejectile an α-particle

146 5 Supersymmetries with Neutrons and Protons

10 20 30 40

p1/2

p3/2

f5/2

dσ/dΩ (mb/sr)

–0.4

–0.2

0.2

0.4

10 20 30 40

(θ)

lab

(deg)

Fig. 5.6. Fits of angular distributions and analyzing powers observed in the (d,t)

reaction on

197

Au to the low-energy states in

196

Au. Also indicated are the diﬀerent

contributions of transferred orbits. (Reprinted from H.F. Wirth et al., Phys. Rev.

C70 (2004) 014610

2004 by the American Physical Society, with kind permission.)

with spin J = 0. As the transfer involves a deuteron with spin S =1,the

knowledge of the transferred orbital angular momentum L (obtained from

angular distributions) is insuﬃcient to determine the J

value of the ﬁnal

state. The solution of this problem is to have vector-polarized deuterons which

are in a magnetic substate with either M

= −1orM

= +1. The tensor

analyzing powers then determine the favored relative orientation of L and S

andassuchtheJ

value of the state excited in the reaction. In all 17 cases the

result agreed with the spin assignment obtained from the (d,t) reaction [270].

The results of the transfer reaction analyzed under the weaker spin as-

signment assumption showed a very good agreement with the predictions

of the U

(6/12) ⊗ U

(6/4) extended supersymmetry [270]. The predictions

for the odd–odd nucleus were partially based on the new quantum number

assignments for negative-parity states in

195

Pt which were obtained in par-

allel (see Table 3.2). The most important result of the entire analysis was

the existence of a ground-state doublet with spins 2

−

and 1

−

in agreement

with the supersymmetry prediction, although the order of the two levels was

reversed. Also the resolution of the structures at 165 keV was a success. The

observation of almost all predicted states in the very complex level scheme

of a transitional odd–odd nucleus was particularly spectacular [271].

Following these encouraging results, a collaboration with the group at Yale

University, headed by Casten, was set up to get high-statistics γ data. While

the Bonn and PSI experiments used only ﬁve Compton-shielded detectors,

the third in-beam experiment was performed at the ESTU Tandem acceler-

ator in Yale with the YRAST ball, equipped with four fourfold-segmented

clover detectors, 17 single Ge detectors and two low-energy photon detectors.

5.4 A Case Study: Structure of

196

Au 147

In 3 days of data taking about 6 × 10

coincidences were accumulated. The

combination of all in-beam data led to the identiﬁcation of 53 states below

800 keV of which 10 new in comparison with the transfer reaction work. In

total, 300 transitions between these levels could be placed. The range of pos-

sible spins and parities of these levels could be limited via the multipolarities.

A detailed account of the analysis of all in-beam data is given in Ref. [272].

5.4.3 Recent High-Resolution and Polarized-Transfer

Experiments

In recent years more transfer reactions leading to

196

Au were performed, with

the aim of limiting the possible spins of the observed states. These eﬀorts are

presented in Ref. [273] and rely on higher statistics for

198

Hg(d,α)

196

Au, and

the new reactions

194

Pt(α,d)

196

Au and

195

Pt(

He,d)

196

Au. The latter two re-

actions are theoretically interesting as they involve only nuclei that belong to

one supermultiplet. The proton-transfer reaction

195

Pt(

He,d)

196

Au did pro-

vide important clues. Speciﬁcally, the importance of the π3s

1/2

orbit, which

is neglected in the supersymmetric scheme, could be established. Its inﬂuence

can be deduced from the running sums of the π3d

3/2

and π3s

1/2

strengths,

shown in Fig. 5.7. One observes that π3d

3/2

is dominant at low energies and

π3s

1/2

, which might give rise to additional states, becomes important above

900 keV only. The observation of the weaker π3s

1/2

-transfer contributions

was nevertheless important to limit the possible spin assignments because it

reduced the possible spins of the ﬁnal state in

196

Au to J

−

, 1

−

.The

3/2

1/2

incremented strength (arb. units)

0.2

0.4

0.6

0.8

(keV)

0 200 400 600 800 1000 1200 1400

Fig. 5.7. Incremental plot of the observed π3d

3/2

and π3s

1/2

strengths obtained

from the

195

Pt(

He,d)

196

Au reaction. (Reprinted from H.F. Wirth et al., Phys. Rev.

C70 (2004) 014610

2004 by the American Physical Society, with kind permission.)

148 5 Supersymmetries with Neutrons and Protons

other reaction

194

Pt(α,d)

196

Au had an extremely small cross-section of only

5 μb/sr and needed a very thin target. From these new data many uncertain

spin assignments could be conﬁrmed and this has been an important element

in the clariﬁcation of the comparison with theory to which we now turn.

5.4.4 Comparison with Theory

By combining the information obtained from the diﬀerent experiments de-

scribed in the previous subsections, 27 negative-parity states are now known

below 500 keV, of which 21 have ﬁrm spin values and 6 restricted ones [273].

Figure 5.8 summarizes the present experimental status and displays a com-

parison with theory [273]. One notices that the energies and spins of most

excited states agree well with the prediction, considering that the level den-

sity is very high (compare with Fig. 3.6). Table 5.2 lists two parameter sets

which are obtained either by ﬁtting all four nuclei of the supermultiplet (in-

cluding the 27 levels in

196

Au) or only to the even–even and odd-mass nuclei.

In both ﬁts the six-parameter hamiltonian (5.5) involves 8 levels in

196

Pt, 32

195

Pt and 11 in

195

Au. Signiﬁcant changes are seen to occur only in the

values of κ

and κ



which govern the ﬁne splitting of the levels. Other pa-

rameters remain essentially the same which supports the validity of extended

supersymmetry for this particular quartet of nuclei.

(1-)

(1/2,1/2)

(3/2,1/2)

(3/2,3/2)

(3/2,1/2)

(5/2,1/2)(5/2,1/2)

(3/2,3/2)

(2-)

(3-)

Theory Experiment

200 400 600 800

196

117

<13/2,1/2,1/2>

[6,0]<6,0>

<11/2,1/2,1/2>

[5,1]<5,1>

<11/2,3/2,1/2>

[5,1]<5,1 >

<11/2,3/2,1/2>

[5,1]<5,1>

(0-)

(keV)

253

288

326

356

404

408

259

414

490

521

166

163

480

456

376

324

235

213

198

299

307

349

167

388

(0-)

Fig. 5.8. Observed and calculated negative-parity spectrum of

196

Au. Levels are

labeled by their angular momentum and parity J

(to the right of each level) and by

the other quantum numbers in the classiﬁcation (5.6). (Reprinted from H.F. Wirth

et al., Phys. Rev. C70 (2004) 014610

2004 by the American Physical Society, with

kind permission.)

Table 5.2. Parameters (in keV) for the quartet with and without

196



With

196

Au 52.58.7 −53.948.88.84.5

Without

196

Au 51.27.9 −52.649.06.96.2

5.4 A Case Study: Structure of

196

Au 149

In Table 5.3 the observed and calculated spectroscopic strengths are sum-

marized. The theoretical values for the strengths G

are obtained with the

transfer operator (3.26) with coeﬃcients v

as determined in the study of

195

Pt (see Sect. 3.5). The agreement is reasonable although one might hope

to reach a better fragmentation of strength if more realistic transfer operators

are used. The overall strength for the ν3p

3/2

orbit is observed to be much

weaker than in

195

Pt and a better agreement can be obtained by lowering

the value of v

3/2

Table 5.3. Observed (d,t) transfer strength to negative-parity states in

196

compared with the predictions of U

(6/12) ⊗ U

(6/4)

Experiment Theory

p1/2

p3/2

f5/2

p1/2

p3/2

f5/2

0.0 2

−

24.3(3) 15.0(10) 43. 1.1 6.5

6.48(4) 1

−

1.2(7) 6.3(7) 2.6(15) 26. 3.4 0.9

41.86(6) 0

−

1.68(4) 0 7.6 0

162.56(4) 2

−

, 3

−

5.2(9) 59.3(21) 1.1 6.7 40.2

166.40(4) 1

−

23.8(13)

27.0(9)

16.4(27)

0.66 20. 5.7

167.43(4) 2

−

23.8(13)

27.0(9)

16.4(27)

0 32. 6.9

197.97(4) 1

−

, 2

−

2.84(28) 3.16(24) 4.98(6) 0 4.5 21.

212.80(4) 4

−

56.2(3) 0 0 77.

234.53(4) 3

−

[0.12(12)] 16.6(6) 0 49. 5.2

252.58(4) 1

−

3.1(7) 3.6(5) 2.34(14) 0 1.5 6.9

258.61(5) 1

−

, 2

−

1.21(12) 0 0.28 1.25

288.06(4) 2

−

1.64(12) 0.84(3) 0 10.5 2.3

298.56(5) 1

−

, 2

−

[0.4(3)] 0.72(28) 0 0 0

307.22(4) 2

−

1.7(6) 3.1(6) 11.6(12) 0 0 0

323.83(4) 1

−

1.48(20)

2.0(5)

000

326.09(5) 1

−

, 2

−

, 3

−

1.48(20)

2.0(5)

0 1.53 18.3

349.17(4) 2

−

0.28(7) 0.30(5) 1.19(12) 0 0 0

355.91(11) 0

−

1.40(12) 0 2.5 0

375.61(4) 3

−

19.2(3) 12.7(7) 0 4.5 55.

387.5(7) 0

−

to 3

−

0.32(4) 0 0.45 0

403.79(4) 4

−

7.3(5) 0 0 26.

408.36(5) 2

−

, 3

−

0.91(10) 1.1(3) 0 16. 1.75

413.74(5) 2

−

0.33(9) 3.5(3) 0 2.0 0.42

456.44(5) 2

−

0.34(6) 0.76(8) 0 0 0

465.5(7) 2

−

0.12(4) 0.08(4) — — —

480.29(4) 2

−

0.22(6) 0.08(4) 0 0 0

490.19(4) 3

−

1.12(4) 2.40(12) 0 0.29 3.4

Energy from in-beam studies, in units of keV.

Bold value is the one used to compare with theory.

Spectroscopic strength in units of 10

−2

Unresolved doublet.

150 5 Supersymmetries with Neutrons and Protons

5.4.5 Two-Nucleon-Transfer Reactions

The one-nucleon-transfer reactions discussed so far,

197

Au(d,t)

196

Au and

196

Pt(d,t)

195

Pt, are interpreted, in ﬁrst approximation, with a transfer opera-

tor of the form a

†

. As already argued, these reactions are useful for measuring

energies, spins and parities of states in the residual nucleus. However, they do

not test correlations present in the quartet’s wave functions as is the case for

one-nucleon-transfer reactions inside the supermultiplet. The latter reactions

do provide a direct test of the fermionic sector of the graded Lie algebras

(6/12) and U

(6/4) (operators a

†

b and b

†

a of Sect. 1.2.4).

In contrast to one-nucleon-transfer reactions where the single-particle con-

tent of the states of the ﬁnal nucleus is scrutinized, two-nucleon-transfer re-

actions probe the structure of these states in a more subtle way, through

the exploration of possible two-nucleon correlations [274]. The spectroscopic

strength of the two-nucleon-transfer reaction depends on two factors: the sim-

ilarity between the states in the initial and ﬁnal nucleus which diﬀer by two

nucleons and the correlation of the transferred pair of nucleons. The informa-

tion extracted through these reactions supply a challenging test of the wave

functions of any nuclear structure model.

As an illustration of such a test, we present results for the reaction

198

Hg(d,α)

196

Au [273] and compare them with predictions of U

(6/12) ⊗

(6/4) [275]. The initial nucleus

198

Hg is assumed to have an SO(6) ground

state, while the states of the ﬁnal nucleus

196

Au are characterized by the

labels (6.6) of the U

(6/12)⊗U

(6/4) scheme. In ﬁrst order, the two-nucleon-

transfer operator for the (d,α) reaction is

†

× a

†

)

(λ)

. (5.11)

As in Sect. 5.3, two-nucleon-transfer operators of this type can be constructed

which transform as a given tensor under the various (dynamical) symmetry

algebras of the classiﬁcations. As a result, the transfer operator (5.11) can be

expanded in terms of the following three tensor operators:

1/2,1/2,1/2(1/2,1/2)

1,m

3/2,1/2,1/2(1/2,1/2)

2,m

3/2,1/2,1/2(3/2,1/2)

3,m

. (5.12)

These are the analogs of the one-proton-transfer operators (5.8) except that

now

L and J may be diﬀerent because the transfer involves a neutron as well.

Each of the operators (5.12) has well-deﬁned selection rules for the deuteron

transfer which are given in Table 5.4. Furthermore, closed expressions for

their matrix elements can be derived [275].

The

198

Hg(d,α)

196

Au reaction is characterized by the transfer of a cor-

related neutron–proton pair with spin S = 1. Since the angular momentum