Study of 116sn via neutron pickup and proton stripping reactions

Nuclear Physics A510 (1990) 70-92 North-Holland

STUDY

OF “%n

VIA NEUTRON STRIPPING

J.M.

SCHIPPERS,

J.M.

PICKUP

AND

PROTON

REACTIONS

SCHREUDER’

and

S.Y. VAN

DER

WERF

Kernfysisch Versneller Instituut, Zernikelaan 25, 9747AA Groningen, The Netherlands K. ALLAART Natuurkundig Laboratorium,

Vrije Universiteit, 1007MC Amsterdam,

The Netherlands

N. BLASI Dipartimento di Fisica dell’ Universitd di Milan0 and INFN,

Sezione di Milano, I-20133 Milano, Italy

M. WAROQUIER Instituut ooor Kernfysica, Proeftuinstraat 42, B-9000, Gent, Belgium. Received

11 October

1989

Abstract. The nucleus “%n has been studied via the neutron pickup reactions “‘Sn(d, t)“%n and “7Sn(3He, a)“%n, and via the proton stripping reactions “‘Ir@He, d)“%n and “‘In(cy, t)“%n. Spectroscopic factors have been deduced from the “‘Sn(d, t) and the ‘151n(‘He, d) reactions for final states in “%n below E, = 3.8 MeV. By combining the l-transfers, observed in these reactions, spin assignments for various states can be given or limits on them can be set. Notably an unambiguous assignment of I * = 3+ can be made for the states at 2.997, 3.180,3.416 and 3.709 MeV. Spectroscopic factors and their fragmentation are compared with those derived from the broken pair model and from a large-basis shell-model calculation involving both neutron quasi-particle excitations and proton lplh excitations.

E

“7Sn(3He, a), NUCLEAR REACTIONS “‘Sn(d, t), E = 50 MeV; E = 50 MeV; ~(0). “%n deduced levels, “51n(3He, d), E = 50 MeV; ‘151n(a, t), E = 65 MeV, measured J, r, spectroscopic factors. Comparison with model calculations.

1. Introduction The tin isotopes the other hand the hand the five-neutron at half-filled major shell-gap, all states collective behaviour. ’ Present Netherlands.

address:

0375-9474/90/$03.50 (North-Holland)

are of particular interest in the study of nuclear properties. On protons have the magic 2 = 50 shell-closure and on the other orbitals are close enough in energy to give rise to superfluidity shell. At excitation energies below about 4 MeV, the proton are basically neutron excitations and exhibit various kinds of

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J. M. Schippers et al. / Study of ‘16Sn

71

Especially the nucleus “‘Sn has been studied via many reactions. It is believed that nearly all states below 4 MeV have been located, especially via radioactive decay and in-beam (cu,2ny), (n, y) and (n, n’r) gamma decay studies l-4). However, the available spin assignment apply in vast majority to states of natural parity, because these are favoured in inelastic excitation reaction studies such as (p, p’) [refs. ‘*6)] and (e, e’) [ref. 6)] and b ecause many of them belong to one of the several collective bands identified in gamma decay 2-4). We report here on a high-resolution study of the neutron pickup reactions and the proton stripping reactions ‘l’Sn(d, t)“?Sn and ‘17Sm3He, a)“%n 1151n(3He,d)“%n and ‘IsIn( cy,t)“%n. The purpose of this investigation is threefold: in the first place the ground-state spins of ‘*‘Sn($‘) and ‘*JIn(~f) are non-zero and single-nucleon transfer reactions have no selectivity for either states of natural or unnatural parity. For each (8) transfer the partial sumrule for the spectroscopic strength is proportional to (2&+ 1) (see the appendix). By combining the limits on spin, set by the two reactions, we are able to make additional assignments for states of unnatural parity, mostly with 1; = 3+. Second, proton stripping on “‘In reveals the magnitude of the proton lplh amplitudes in the final wave functions. These are small below the gap and have not been investigated before. Their magnitude relates to the neutron-proton interaction and gives a check on model calculations where these amplitudes have been included ‘*‘). In the third place the fragmentation and spreading of 2qp neutron amplitudes is approximately obtained from the neutron pickup reactions on “‘Sn. We compare the spectroscopic strengths obtained in this work with those derived from the two most extensive model calculations available to-date, those of Waroquier et al. “) and those of Bonsignori et al. ‘). 2. Review of previous work The established level scheme ‘) of “%n consists of a vibrational band whose 4+ member mixes with a rotational band based on 2p2h proton configurations, and a negative-parity band which de-excites mainly through the members of the v(h,,,,g,,,) and Y(hlf,ZdS,Z) multiplets down to the collective 3; state. These states and their decay have been observed via the (LX,2ny) reaction by van Poelgeest et al. 3, and Bron et al. “). More states have been observed via inelastic proton scattering at E, = 28 MeV by Wienke et al. 5), but the states to which a spin could be assigned are all of natural parity, with the exception of a possible 1’ state at 2.587 MeV and a possible 3+ state at 2.997 MeV. Neutron-pickup reactions leading to “%n have been studied by several groups. Yagi et al. lo) presented studies of the ‘l’Sn(p, d)“%n reaction and the ‘18Sn(p, t)‘%n reaction at 55 MeV. They find that these two reactions excite mostly the same states in “%n and that the deuteron spectra show the same intensity distribution as the triton spectra. However, their energy resolution was only about

72

J.M. Schippers et al. / Study of l16Sn

70 keV and they could identify only 12 peaks below E, = 4 MeV. Yagi et al. established the two J” = Ot states at 1.76 and 2.03 MeV, and identified three J” = 2+ states at 3.43(2), 3.72(2) and 3.97(2) MeV, respectively. On the other hand, they did not observe the J” = 3; state contrary to the results of the (p, t) reaction which was performed by Fleming et al. ‘I). A study of the “‘Sn(d, t)*?Sn reaction at Ed = 15 MeV has been reported by Schneid et al. 12) with an energy resolution of 40-60 keV. Most levels observed by these authors show either 1= 0 or I= 2 pickup. No 1= 4 angular distributions were observed in this study. It was concluded that the gT12strength is spread over many levels and therefore not easy to observe. The “‘In(“He, d) reaction has been studied before by several groups 13-15),but their analyses were focused on the region above 3.8 MeV, where the strength of the rd5/2 and rrg,,, orbitals is concentrated. Model calculations of ‘uSn including proton lplh excitations through the 2 = 50 gap have been performed by Clement and Baranger ‘) and Waroquier et al. *), who described the neutron space in terms of two quasi-particles, and by Bonsignori et al. 9), who applied the broken pairs model to this nucleus, but included proton excitations only for the lowest 2+, 3- and 4’ states. According to these calculations, small admixtures of proton lplh excitations are present in the low-lying neutron states and can be populated via proton transfer reactions. 3. Experimental procedure and analysis The experiments were performed with momentum analyzed beams from the AVF cyclotron at the KVI. The pickup reactions were performed using beams of 50 MeV deuterons and 50 MeV 3He particles on a 330 ug/cm2 self-supporting ‘17Sn target enriched to 95.44%. The outgoing particles were detected in the QMG/2 spectrograph. The “‘Sn(d, t)‘?Sn experiment was performed twice: the first time with a 120 cm long resistive wire detector ‘“) (obtained resolution -30 keV) and a second time with a 48 cm MWDC 17) (obtained resolution -18 keV). Also the (3He, a) experiment was performed using the MWDC as focal-plane detector. The energy resolution was about 20 keV in this case. In fig. 1 spectra of the (d, t) and (‘He, tu) reactions are shown on the same excitation energy scale. Angular distributions were measured between 6” and 30” in steps of 2”. Particle discrimination was performed by gating on the pulse height of the scintillation detector placed behind the MWDC and in addition, the time-of-flight signal (T.O.F.) was used to reduce background caused by other particles. For the proton stripping reactions l151m3He, d)‘16Sn at 50 MeVand “SIn(cr, t)‘16Sn at 65 MeV a self-supposing “‘In target of about 300 pg/cm* thickness was used. The outgoing deuterons and tritons were identified and detected with the 120 cm resistive wire detector. The energy resolution was about 35 keV. Spectra on a common excitation energy scale are shown in fig. 2. Measurements at 0” were performed for

J.M. Shippers et al. / Study of ‘16Sn 4x10&

I

I

13

I

'17Sn(d,t)"6Sn? E,=SO MeV 0,,,=6O

Fig. 1. Spectra of “%I obtained with the neutron pickup reactions. The (d, t) and the (3He, a) spectra are displayed on a common excitation energy scale. The ground state and the 2: state are not in the spectra with these spectrograph field settings.

these reactions providing in the case of (3He, d) a strong selectivity for si12 transfer. In this case, the beam was dumped onto a Faraday cup inside the first dipole magnet of the spectrograph. Differential cross sections of elastically scattered deuterons and 3He particles were used to normalize the cross sections of the (d, t) and (3He, d) experiments. The angular distributions of the “51n(3He, d)‘?3n and “‘Sn(d, t)1’6Sn reactions have

been

compared

with

zero-range

DWBA

calculations,

using

the

code

DWUCK4 “). The optical-model parameters for the 3He particles were taken from Gibson et al. 19) and those for the deuterons from Hinterberger et al. 20). They are given in table 1 together with the geometrical parameters of the transferred particle. The optical-model parameters of the tritons were taken equal to those for the 3He particles. In all reactions the target nucleus has a spin (J” = 4’ for i151n; J” = f’ for “‘Sn) and the final states that are produced will be mixtures of various proton particle-hole and/or neutron two and four quasi-particle configurations. If the final state has

J.M. Schippers et al. / Study of “%n

1151n t3He E3,,:50 o,,,=14°

d) ‘%n kleV

“‘In (a,t) ‘%n Em=65 MeV 0,&Z IO0

1 3

2

1

d

0

E, (MeV) Fig. 2. Spectra

of “%n

obtained

via the “%@He, excitation

d)“‘Sn and the ‘151n(cY, t)‘%n energy scale.

reactions

on a common

_I.M. Schippers et al. 1 Study of ‘16Sn TABLET

Optical-model

3He d t P. n

175 99.8 175 varied b,

parameters

“) for the “51n(3He,

17.5 65.5

6.4 17.5 A = 25 ‘)

“) For 3He and t from ref. 19) for d from ref. “). b, Varied to reproduce the binding energy. ‘) The spin-orbit term in the bound-state potential x=(r-r,,A”*)/a,.

d)“%

1.14 1.05 1.14 1.25)

and “‘Sn(d, t)‘%n reactions

1.60 1.32 1.60

1.05

is +A (h/2m,c)*(

0.723 0.913 0.723 0.650

0.86 0.722 0.86

V,/ r)(d/dr)(

0.913

1.4 1.3 1.4 1.2

1 + ex)-‘( I - u), with

negative parity, it has assuming that at these If, moreover, the “‘In transfer cross section

obviously been produced by I= 5, i.e. a h,,,* nucleon transfer, low energies only valence orbits may contribute significantly. ground state is approximated as a g9/2 proton hole, the proton for such a negative-parity state is directly proportional to the magnitude of the (g
Gi “(2j+l)

distributions

(3.1)

CDW9

and No = 4.42 for (3He, d).

4. Experimental Angular

using the relation:

of the most prominent

results peaks in the “‘Sn(d,

t)“%

spectra

and fits from DWBA calculations are shown in fig. 3. In table 2 the deduced spectroscopic strengths are listed. Since the pickup from the d3j2 cannot be distinguished from that of the d5,* orbital and since the results of the neutron pickup reaction on “‘Sn [ref. “)I show that the excitation energies associated with the strengths of these orbitals are similar, usually no decision can be made upon the j-value of the I= 2 transfer. For the s,,, , I= 2 (d 3/2 and c&,2) and the hII occupancies of the neutron

orbitals orbitals

less than the sum rule is found. Adopting the in “%n as found by Van der Werf et al. *‘),

J.M. Schippers et al. / Study of “6Sn

76

Fig. 3. Angular

distributions

ofthe most important

states below E, = 4 MeV, excited in the “‘Sn(d, reaction.

t)“%n

11

J. M. Schippers et al. / Study of “6Sn

TABLE

2

of known Spectroscopic strengths observed with the “‘Sn(d, t) reaction at 50 MeV. Spin assignments strengths that have been obtained in a DWBA fit using states are from refs. ‘-3*22.23). The spectroscopic eq. (3.1), have an uncertainty of 20% due to systematic errors in the normalization and ambiguities of the optical-model parameters

(ME,V) 0.000 ‘) 1.294 “) 1.757 2.027 2.112 2.225 “) 2.266 “) 2.366 2.390 “) 2.529 “) 2.545 2.587 2.650 “) 2.173 2.801 “) 2.843 2.960 2.991 3.046 a) 3.096 “) 3.180a) 3.228 3.315 3.371”) 3.416 3.470 3.513 “) 3.589 3.618 3.709 3.739 “) 3.172 3.950 4.037 4.084 total: “) Also observed

J” 0+

2+ 0+ 0+ 2+ 2+ 35-

%I,,)

G(d,,,

or W

(2+) (l+) 3+ 4+ 4+ 3+

G(h,,,z)

GV,,,

or G/2)

0.32 0.20

(0.16)

0.006 0.24

(0.004) (0.19)

0.06 0.05

0.01

0.01

0.01

0.01

0.40 0.26 0.31

;: (O)+ (l+) (2+) 64+

Gk,,,)

0.04 0.22 0.01

(0.17) (0.006) 0.88 0.70

(0.02)

0.07 0.02

(0.06) 0.06

0.24

(0.19)

0.24

0.07 0.38 0.34 0.08 0.06

(0.19) 0.47 (0.21) (0.07) (0.29) (0.02) 0.75 (0.06) (0.29) (0.28) (0.06) (0.04)

2.95

3.51

0.80 0.50 1.63 1.30 0.07

3+ 0.27 0.09 0.37 0.04 3+ (2,3)+

0.49 in the “sIn(3He,

d) reaction.

0.17

0.69

6.43

1.28

J&f. Schippers et al. / Study of 116~n

4

‘C

10‘

M

Ep3.797 MeV

10-f

cC*

ka-2

10-z 0

Fig. 4. Angular

distributions

from the “51n(3He,

d\“‘%

ceactian

10

20

to the states below

30

4 MeV.

79

J.M. Schippers et al. / Study of ‘16Sn

about 60% of the available strength could be located. In contrast about 90% of the g7/2 strength has been located, assuming that all I=4 strength below E,=4 MeV may be attributed to this orbital. The shapes of the angular distributions obtained with the (3He, a) reaction give little information on I-transfer due to the large momentum mismatch of this reaction. However, since the cross sections for the large I-values are strongly enhanced, this reaction serves to distinguish the hl,,* from the g7,2 transfer. The fits of the DWBA calculations to the 1151n(3He,d)‘16Sn data are shown in fig. 4. In table 3 the spectroscopic strengths of the states observed below E, = 3.8 MeV are given. TABLE

3

in peaks below E,=3.8 MeV, observed with the Spectroscopic strengths “), concentrated “SIn(3He, d)“‘Sn reaction at 50 MeV and cross sections of the peaks in the “‘In(cr, t)“?Sn reaction. The well parameters of the proton form factor are r,, = 1.25 fm and a, = 0.650 fm

(MEe;)

Jn

0.000 1.294 2.225 2.266 2.390 2.529 2.650 2.801 2.843 2.997 3.046 3.096 3.180 3.277 3.311 3.513 3.658 3.139 3.191

0+

total:

2’ 2+ 34+ 4+

WI,,)

0.05 1 0.009

?

G(g9(,),J

‘)

G(h,,/z)

0.011 0.009 0.006 0.019

(0.003) 0.009

0.011 0.012 0.003 0.003 0.007 0.009 (0.005) 0.012 0.070 0.070

0.023

0.307

0.001 0.001

(6+)

G(Pw)

0.46 0.16 0.021

0.006 0.003

(2+) 4+ (2+) 3+ 4+ 4+ 3+

G(ds(s),z)

0.009 0.011 0.044

0.019

(mb/sr, at 5”) 1.64 0.83 0.06 0.14 0.08 0.08 0.14 0.18

0.044 0.007

0.06 0.08 0.05 0.06 0.08 0.07

0.012 0.021

0.14 0.41 0.37 0.148

(only

l37/2)

“) The spectroscopic strengths have an uncertainty of 20% due to systematic errors in the normalization and ambiguities of the optical model parameters. The experimental error is estimated to be smaller than 5%. “) From the angular distributions no distinction can be made between dS12 and ds,* transfer. All spectroscopic strengths have been calculated assuming ds,z transfer. For d,,* all values are about 30% higher. ‘) The transfer to the g.s. is g,,,. This has also been adopted for the transition to the 1.294 MeV 2+ state. For all other [ = 4 transitions g,,, has been adopted.

80

J.M. Schippers et al. / Study

Also in this case, it is not possible Since

the d3,* orbital

strengths

is expected

between

to be located

in our analysis

can be made between that the I= 4 transfer

the d3,2 and d5,* orbitals.

at higher

to states below 4 MeV that are associated

d 5,2. Also, no distinction assumed

to distinguish

of“6Sn

energy,

spectroscopic

with 1= 2 transfer

are given for

the ggi2 and the g7/2 orbitals. to the ground

We

state and to the first

excited 2+ state corresponds to g9,2 transfer (see discussion below), higher excited state to g7,2 transfer. The strong momentum mismatch reaction enhances the cross sections of the hlli2 transfer over that of used for comparing with cross section ratios from the (3He, d) reaction the I= 5 transfers.

and for the of the (a, t) g,/*. This is to establish

Since the strengths observed below 3.8 MeV in the proton stripping reaction are very small, and the 2: and 3; states in “%n are known to be strongly collective, we performed coupled-channel calculations for these two states in order to assess the possible importance of two-step contributions. The coupling potential was assumed to have a deformation expressed in terms of vibrational degrees of freedom, and the optical-model parameters for the different channels were taken equal to those used in the DWBA calculations. A quadrupole deformation parameter p2 of 0.13 was necessary to reproduce the B(E2) value, while for the B(E3) value, /I3 was taken equal to 0.15. The calculations were performed with an extended version of CHUCK 22), which was required because of the large spin value of the target nucleus ground state. The resulting strengths turn out to be the same as those obtained from the DWBA calculations, indicating that two-step processes are, in this case, negligible. The size of the cross sections calculated by assuming two-step processes only is of the order of lo-’ mb/sr. 5. Discussion 5.1. THE

GROUND

STATE

OF “‘In

AND

SUM

RULES

The simplest picture of the “‘In ground state is that of a single proton hole in the g9,2 orbital. By stripping a proton into it, only the ground state of ‘%n should be populated via gg12 transfer with a strength equal to 1. Experimentally this strength is much smaller and in addition a large 1=4 component is present in the angular distribution of the 2: state which, if identified with the g9,2 orbital, points to a wave function. Conversely, a [g,$r2@2f; t’] component in the ‘151n ground-state [g;j2] single hole component appears to be present in the core excited state at E, = 1.449 MeV in “‘In [ref. “)I. Similarly, the 3, state is populated via I = 1 transfer, indicating a [~$~@3;; 5’1 component. by a sum of the components Thus, the ‘%r ground state may be approximated that have a proton-hole state coupled to a physical state in “%n as: (“%r, g.s.) = a/g&;

4’ >+c

i

Pmlg;;2@~+;

+ more complex

terms .

;+>+-A

I

YmlP$z@~;;

4’)

(5.1)

J.M. Schippers et al. / Study of l16Sn

81

This is a correct expansion in an orthogonal basis, i.e. as long as the excited state components. Here we restrict ourselves, in analogy to hole-core coupling models *4-29)to the two core states 1: = 2: and 1; = 3;. Microscopic calculations 8,9) indicate that the 2: has only modest contributions from proton particle-hole components, of the order of 5%. The 3;) on the contrary, is almost equally composed of proton particle-hole and of neutron quasi-particle configurations. Due to its very collective nature, however, none of the proton particle-hole components dominates in the wave functions. The treatment of ~$03; components in (5.1) as orthogonal to the other terms is therefore a not unreasonable first approximation. Thus an indication of the magnitudes of the coefficients /3(2:) and -y(3;) may be gained from spectroscopic strengths of the “51n(3He, d)‘r6Sn reaction. These are listed in table 3, from which the values (Y’= 0.46; p*(2:) = 0.16 and ~~(3;) = 0.019 are deduced. Several calculations 24-29)b ase d on hole-hole coupling, with the core represented by a system of phonons, have been performed for “‘In, evaluating the amplitudes (Y,p(2:) and ~(3;) of the ground-state wave function. In table 4 the results of these calculations are compared with our analysis. It should be noted that in the calculations of Covello et al. 2”) the strength is not located completely in the three components considered. The results of the present work indicate that the contribution of the other components is even larger; approximately 35%. This would imply that there is also a substantial coupling of the proton hole to non-collective excitations 1: and 1; of “‘?Sn. We shall discuss this possibility now in some more detail. When considering the observed 1= 4 strength, in table 3, for the states above 2 MeV one might be tempted to interpret this as associated with the proton g,,, orbit, i.e. as the squares of the coefficients p( 1:) in eq. (5.1). The microscopic calculations of Waroquier et al. “) suggest, however, that these 1= 4 stripping strengths mainly originate from the proton gTj2 orbit. These calculated strengths are listed in table 5. So these strengths should rather be associated with proton (lg$*, lg,,,) admixtures in the wave functions of the various states in “‘Sn. The question therefore remains what the other about 35% of the “‘In ground-state wave function might be. Our suggestion is that a large fraction of this is the g9,* proton IT and I; in “?Sn do not contain gg12or p3/2 proton-hole

TABLE 4 Squared

amplitudes

in the ground

Ref.

a2

present work 24

0.46 0.65 0.66 0.50 0.66 0.75 0.79

zs ; 26 27i 28 29i

state of “%I

PY2:)

r2(3J

0.16 0.29 0.27 0.33 0.27 0.25 0.17

0.02

0.03

0.02

J.M. Schippers et al. / Study of “6Sn

82

TABLE

Spectroscopic

0: 0+ * 1+I 1+ 2: 2: 2:3 3; 3+1 3: 3: 3: 3+ 4: 4; 4: 4+ 5”, 5: 6:

proton

stripping

strengths

0.000 1.920 2.249 3.855 1.542 2.256 2.647

on “%I

5

from the wave functions

0.014 0.007 0.003

0.000 0.001 0.005 0.003 0.001

0.787 0.002 0.000 0.000 0.078 0.014 0.001

0.004 0.001 0.000 0.004 0.354 0.014 0.002 0.006 0.049 0.001 0.640 0.018

0.000 0.001 0.000 0.000 0.000 0.009 0.020 0.043 0.044 0.000 0.003 0.022

0.001 0.000 0.000 0.001 0.000 0.008 0.011 0.007 0.000 0.000 0.001 0.001

of Waroquier

0.006 2.795 2.976 3.132 3.451 3.748 2.287 2.834 3.265 3.479 3.197 3.832 3.313

0.000 0.000 0.000 0.000 0.013 0.003 0.002 0.002 0.001 0.002 0.017 0.005

0.005 0.000 0.001 0.006 0.000 0.006

et al. *)

0.032

hole coupled to a very large set of collective as well as non-collective excitations We mention two arguments in of “‘?Sn, all with rather small coefficients p(1’). favour of this interpretation. The first is that such a strong coupling of the proton hole to other excitations of the nucleus is also observed in recent (e, e’p) experiments 30331).There one also finds only about 50-60% of the expected proton-hole strength, even in an energy region up to 15 MeV excitation energy. Assuming that nevertheless the occupation of the proton orbits is substantially larger, of the order of 80% as borne out by a combined analysis of charge densities and spectroscopic factors, some 20% of the proton-hole strength must be spread over a large energy region, of the order of a hundred MeV. Calculations with realistic meson-exchange potentials support this idea 32). The second argument in favour of substantial other components with a gsi2 proton hole coupled to higher states of “?Sn in the “‘In ground state is provided by the magnetic moment of this state. In the hole-core coupling model of eq. (5.1) one obtains, neglecting the more complicated terms: p (“‘In,

Lj,v +cP(y)

g.s.) = C s2( I:)

Jo(J,+l)-I(I+l)+j(j+l)

2(Jo+ 1)

Jo(Jo+l)+I(I+l)-j(j+l)

i.i.ir

2(Jo+ 1)

1ss.,.(j) 1g(r),

(5.2)

83

J.M. Schippers et al. / Study of “‘Sn

where J,, = 4 is the ‘15In g.s. spin and the symbol 6 denotes the coefficients cr, /3(1’) if j = g,,, and ~(1;) for J = p3/2. The g-factors g,.,.(j) are those for a single-hole state and g(lT) those of the core states in ‘16Sn. The latter are probably not very large, e.g. g(2:) = -0.16*0.10~, has been measured by Hass ef al. 33). Therefore, in our qualitative considerations we shall neglect the second sum in (5.2). The components with the proton gg12hole coupled to the “?Sn ground state and the 2: state, which constitutes 62% in the wave function (5.1), contributes 4.15~~ to the magnetic moment (5.2) if one adopts the Schmidt value p = 6.793~~ for the ggj2 orbit. This is already a slight overestimate because there is a general quenching of single-particle spin magnetic moments. Nevertheless the experimental value p = 5.5408~~ [ref. ‘“)I is about 80% of this Schmidt value. Since the magnetic moments of the other proton-hole orbits are much smaller than those of the ggj2 orbit, this result suggests that other configurations with the gg12proton hole coupled to states in “‘?Sn amount to at least some 20% of the “‘In ground-state wave function. 5.2. THE

GROUND

STATE

OF

“‘Sn

- THE

SPIN-WEIGHTED

SUM

RULE

It is generally accepted in shell model and related theories that pairing is a preponderant feature of semi-magic nuclei and that the lowest-lying states of such nuclei are well approximated by configurations of lowest seniority 35). In projected three-quasi-particle calculations of the Sn isotopes, Bonsignori et al. 36) found that the lowest i’ states in these nuclei are for more than 90% projected one-quasi-particle states; i.e. the odd neutron is in the slj2 orbit and other nucleons are distributed as J” = 0’ pairs over the other shells. A pure configuration of this type would obviously imply an occupation probability u2 = 0.5 for the s,/~ shell and a total 1= 0 pickup strength of exactly 1.0. The observed I= 0 strength, listed in table 2 for the “‘Sn(d, t)‘uSn reaction to the ground state, the excited O+ states at 1.757, 2.027 and 2.545 MeV, and to the assumed l+ state at 2.960 MeV amount to only 0.47. This result suggests that the true ground state deviates substantially from a seniority-one state and/or that s,/~ strength is spread out over higher excited states in fragments too small to be identified experimentally. This situation for the s ,,* neutron quasi-particle seems reminiscent of that of the coupling of the proton g9/2 hole to ‘?Sn excitations discussed in the previous paragraph. This idea again finds support in the experimental value of the magnetic moment of the p ground state of “‘Sn. A pure seniority-one state would yield the single-particle (Schmidt) value of -1.913~~. The spin-weighted sum rule expresses the contribution of the s,/~ neutron to the ground-state magnetic moment in terms of the spectroscopic strengths to O+ and l+ states of “‘Sn as: ~(s,,,)=-1.913~~

[i

C G(O+)-fC

I 1i G(1:)

+$I

G(l+)j~(l;).

(5.3)

Substituting the observed pickup strengths one obtains p (+‘) = -0.89~~) which falls

84

J. M. Schippers et al. / Study

of ‘16Sn

only about 10% short of the experimental value of -1.00104~~ for the magnetic moment of the “‘Sn ground state 34). The remaining 10% of the pickup strength is probably

spread

resulting

from

though

maybe

over many the monopole

states.

Conversely,

sum rule

the estimated

may be considered

occupancy reasonably

v* = 0.25 accurate,

some 10% too low.

In the following we shall stick to the framework of a valence shell model when comparing the data with model calculations. This means that the “‘Sn ground state is then considered as an approximate seniority-one (one-quasi-particle) state. It should be mentioned that in neutron stripping to “‘Sn [ref. 37)] 1= 0 transfer is observed to the ground state only, which supports this approximation.

5.3. EXCITED

STATES

IN “%I

- SPIN ASSIGNMENTS

From neutron pickup reactions 2’) the neutron-shell occupations in “‘?Sn of 0.74(d,,,), 0.88(g7,,), 0.22(h,,,,), 0.40(d3,,) and 0.32(~,,~) can be deduced. These values agree well with those predicted by or adopted in theoretical calculations 7-9). In “?Sn two major y-decay bands have been observed via the ‘14Cd(a, 2ny)‘?Sn reaction at 28 MeV [refs. “,“)]. The levels with positive parity are of rather complex character: they form a quasi-rotational band built upon the 0: state which originates from a non-closed Z = 50 shell “) and are of proton 2p2h nature. Branching ratios observed in the y decay of the J” = 6+ and J” = 4: have been explained by assuming configuration mixing between the band member J” = 4: and the J” = 4: state, which has a predominant two-phonon configuration. This would explain the enhanced transition probability from band members to the 2: state, which is known to have a vibrational character. Jonsson et al. 31) concluded are partly of two-phonon vibrational character.

that also the 2: and 4: states

The negative-parity states ‘) form a band with a two-quasi-particle neutron character (broken pairs), based on (g7,2h11,2) and (d5,2h11,2) neutron configurations. The state with the highest spin is the J” = 9-, which is described as: (g7,2h11,2) plus core-polarization

terms 32).

In the (d, t) reaction we observe I= 4 transfer, contrary to the results of Schneid et al. 12), who observed I= 0 and 1= 2 strength only. This difference is due to our higher beam energy. The g7,2 strength fulfills a large part of the sum rule. It should be noted, however, that part of the 1= 4 strength may originate from the g9,2 orbital which can have some small fragments at the excitation energy range considered here. Although the 1= 4 angular distribution could not be distinguished from the 1= 5 angular distribution in the (d, t) reaction, they are excited with different cross sections in the (3He, a) reaction. The cross-section ratios of the peaks at E, = 2.843 MeV (I = 2), E, = 2.801 MeV (I = 4) and E, = 2.773 MeV (I = 5) are very different in the two spectra in fig. 1. The only 1= 5 states observed are the levels J” = 5 at E, = 2.366 MeV and the J” = 6- at E, = 2.773 MeV.

J&f. Schippers et al. / Study

of ‘*‘Sn

85

New spin assignments in tables 2 and 3 are based on the combined analysis of the proton stripping and the neutron pickup reactions, which are discussed simultaneouly in the following. It should be noted that no members of the two positiveparity bands are observed in the (3He, d) experiment, except for the three J” = 4+ states at 2.390, 2.529 and 2.801 MeV. The y-decay of these states ‘) suggests that these states are coupled to the low-lying collective 2: state. Most of the states observed in the (d, t) reaction could be identified with known states ‘). The 3-, 5- and 6- states at 2.266, 2.366 and 2.773 MeV observed here, are members of the negative-parity band, shown in the level scheme. The 5 and 6states have (hlliz)l. components, which is consistent with the expected Z-neutron quasi-particle character of this band. The I= 5, instead of I = 4, transfer to these states is in agreement with the observations in the (3He, CX)spectrum. With the (3He, d) reaction no states of the negative-parity band are observed, except for the collective J” = 3-, again indicating the rather pure neutron character of the states in this band. The state at 2.587 has been observed via the (p, p’) reaction by Wienke et al. 5), and a tentative l+ spin assignment was proposed for it. This assignment is not in contradiction with a pure I= 2 neutron transfer. In this case, the transfer would be uniquely via a d3,2 orbital. The state at 3.096 MeV, which had a tentative spin assignment (4+) [ref.‘)] is observed in (d, t) with g,/, strength, so that J” = 3+ or 4+. We observe, in addition, a significant s,/, strength for this state in the proton stripping, so that we may give a definite spin assignment of J” = 4+ to it. The partial sum rule for J T = 4+ states observed with the “‘Sn(d, t) reaction is 3~: = 3.96. The observed i = 4 strength for the known 4+ states is already 3.46, so that it is expected that all J” = 4+ states that may be excited are observed. The partial sum rule would be exceeded with almost 20% if the state at 3.180 MeV, which has a large 1= 4 amplitude were assigned as 4+. Therefore the most probable spin assignment for this state is J” = 3’. This yields 2.8 for the total 3+ strength, which is equal to 80% of the partial sum rule for 3+. In the (3He, d) reaction at E, = 3.18 MeV probably the same state is observed. If the states at 3.416 and 3.709 MeV are the same states as the 3.42 (2) and the 3.72 (2) MeV, both identified as J” = 2+ by Yagi et al. lo) with the (p, d) and the (p, t) reaction, our results are different from theirs: the states at 2.997, 3.416 and 3.709 MeV can be assigned as J” = 3+. The presence of gyi2 strength in the neutron pickup restricts the final spin to f * = 3” or 4+. This means that the I= 2 transfer, which is present, has to be ds12, yielding an unambiguous spin assignment of J” = 3+. We therefore confirm the tentative assignment given by Wienke et al. ‘) to the state at 2.997 MeV. The state at 3.739 MeV is also observed in both the proton stripping and the neutron pickup reaction. In the (3He, d) reaction it shows l= 2 strength, so that J” = 2’-7+. The result of the pickup reaction shows that J” value of this state can

86

J.M. Schippers et al. / Study

be l+ or 2+ if the 1= 2 component

is dJj2 and 2+ or 3’ for ds12. If both reactions

really

excite this state, then it possesses

ations

in its wave function.

restricts

the assignment

5.4. COMPARISON

WITH

of‘16Sn

Combination

both strong

neutron

of the stripping

and proton and the pickup

configurresults

of the 3.739 MeV state to J” = 2+ or 3+.

MODEL

CALCULATIONS

We compare our data with two large-scale shell-model calculations that differ in their choice of the basis space. Bonsignori et al ‘) describe the neutron space within the framework of the broken pair model including up to two broken neutron pairs. The active neutron shells are dSIZ, d3,*, g,/*, s,,~ and h11,2. Proton lplh proton excitations have only been included for a few levels. Waroquier et al. ‘*) use an effective Skyrme interaction in a completely selfconsistent microscopic calculation and include in the model space two (neutron) quasi-particles based on the same five shells as above and in addition the g9,* shell. Proton lplh excitations are included with d5,*, d3,2, g7,2, s,,~ and h,,,* as particle the coupled shells and g9/2, P1/2, P3/2, f5,2 and f-7,2 as hole shells. In addition configurations of these neutron and proton excitation modes have been incorporated. This space enlargement provides more collectivity to the lowest 2+ and 3- levels, and larger fragmentation to all states. The ground state of “‘In has been described within the same framework in order to allow a calculation of the strengths for the proton stripping to “%n. In table 5 we show the results for the proton stripping spectroscopic factors. These are to be compared with the experimental results of table 3. The calculated g9/2 strength to the ground state and the first 2+ state indicates that the wave function of the ground state of “% has a too large lg9,J component, while the 12:Og;j2) one is too small. The h,,,2 strength to the 3; state is well reproduced. The spectroscopic strengths for the lowest 4+ states are in fair agreement with experiment and the same is true for the 6: state. For the other states no attempt is made to find a one to one correspondence with the experimental levels, but in general the order of magnitude of the proton admixture in the wave functions of the low-lying states is in good agreement with the data. Note that the states belonging to the rotational band based on 2p2h configurations are not included in the model space and should not be taken into account in this comparison. The neutron pickup strengths, listed in table 2, are compared with two calculations, the results of which are presented in table 6. In the calculation of Waroquier et al. the “‘Sn ground state is predominantly a s,,* one-quasi-particle state with small admixtures of three-quasi-particle components 40). So the s,,* transfer can also reside on excited Ot states, while for a pure one-quasi-particle state in BCS approximation only the ground state would be populated with strength 2v2. For all final states of non-zero spin the three-quasi-particle admixtures turned out to be rather unimportant.

87

J.M. Schippers et al. / Study of ‘%I TABLE Spectroscopic

strengths

Waroquier

6

pickup from “‘Sn, calculated from the wave functions et al. ‘) and of Bonsignori et al. 9,

for neutron

et al.

of Waroquier

et al.

Bonsignori

1; E, WV) 0+

0:

0.00 1.64

2: 1: 2: 2:3 2: 2+ 3,: 3+ 33: 44 3+

2.25 1.54 2.26 2.65 2.89 3.10 2.79 2.98 3.13 2.29 3.45

4: 4: 4:

2.83 3.26 3.48

5; 6;

2.45 2.63

+/2

4/z

4/z

0.334 0.275 0.251 0.037

0.336 0.256 0.640 0.700 0.006 0.002 2.954 0.005 0.034

w12

E, WV)

2.647 0.001 0.006 2.435

2.73 1.36 2.19 2.7 1 2.86 3.16 2.91 3.19 3.26 2.40 3.70

0.742 0.020 0.006

2.98 3.06 3.23

%/2

&,,

4/z

w/2

0.389 0.194 0.259 0.032 0.059 0.037

0.263 0.045 0.007 0.638 0.024 0.182 0.028 2.141

0.075 0.457 1.885 0.622 0.019

0.435 0.419

0.071 1.086 1.233

2.53 2.73

0.493 (vh,,,,) 0.671 (vh,,,,)

0.389 (vh,,,J 0.462 (vh,,,,)

In the calculations of Bonsignori et al. the mixing of one and states (projected two- and four-quasi-particle neutron states) mechanism which causes a further fragmentation of the particle Since no (complicated) computer code was available to compute projected amplitudes

three-

to four-quasi-particle

may be omitted

components,

as a reasonable

we assumed

appoximation.

two broken pair is an important transfer strength. the transfer from here that these

Then only the transfer

from the projected one-quasi-particle to projected two-quasi-particle components (one broken pair) was retained and these matrix elements were calculated in BCS approximation. wave functions way in which

Since these one broken pair components are taken from the complete the essential mechanism of fragmentation is retained. Details of the the spectroscopic factors have been derived can be found in the

appendix. In table 6 the calculated values for the lowest members of the relevant spin families are listed. They are to be compared with the experimental results of the “‘Sn(d, t)“‘Sn reaction given in table 2. The sl/, strength to the ground state is about 30% too large in Waroquiers calculation, probably indicating that the adopted value of 0.55 for u2(s1,J is too large. The calculations agree on the d3,2 strength to the 1: state. This strength and the predicted energies fit the state found at 2.587 MeV and tentatively assigned l+.

J.M. Schippers et al. / Study

88

Both calculations

put a combined

is 2-3 times too large and predict that the fourth,

in agreement

I= 2 (d3,2, d5,J

strength

on the 2: state which

the third 2+ state to be more strongly

with experiment.

is interpreted

9, as a member

of the proton

is expected

to be very weakly

excited;

4+ states the effect of truncation

of'16Sn

populated

Note that the second 2+, at 2.112 MeV

two-particle-two-hole

in agreement

band and therefore

with out data. For the 3+ and

of the basis spaces becomes

evident.

Experimentally

the strength is more fragmented than in the calculations where the spectroscopic strength is exhausted mostly by the first four states of each family. The wave functions of Bonsignori et al. reproduce this fragmentation to some extent, in particular for the 4+ states where the third and the fourth member are predicted to be strongest populated, while most of the gT12 strength resides on the first 4+ state in Waroquiers calculation. It may be concluded that the extension of the basis space with a second broken pair introduces more spreading at these low energies of the strength than the addition of the lplh proton configurations. and in both calculations the 5; and The h11,2 transfer is rather well reproduced, the 6; state exhaust most of the I = 5 strength, in agreement with experiment. In fig. 5 the cumulative strength distributions for 1= 2 (d3,2, d5,2) and I = 4 (gT12) are shown, summed over all final states. This way of presentation demonstrates the spreading individual

of the spectroscopic strength without comparing the strength on each state separately. It is seen that for I= 2 the experimental fragmentation I

7.00

6.00

.

5.00

,i?=2

I

I

I

I

P=4 7.00

-

cup.

-----

Bonsignori

et al.

“““...

,Wtar,lquier

:

c

I -

..__,..:.... i---

-

exp.

-----

Bonsignori

et al.

r--5.00 -

4.00-

3.00 -

E, (MeV) Fig. 5. Cumulative distributions of spectroscopic I = 2 and I= 4 neutron pickup strength, reaction, compared with predictions all final spins, as derived from the “‘Sn(d, t)“% wave functions of Waroquier et al. ‘) and Bosignori et al. 9).

summed over based on the

89

J.M. Schippers et al. / Study of “‘Sn I

I

LO-

I

1;=3+ dc,,*

3.0---_;“,“,;i,“,,i

e+ a,, t’“““““““‘“’

. . . . . ..Waroquier

et al.

__------

;

I

2.0 1.0vl -

u

3

4

I

1

WG I 4.0 1;=3+ I0

IT

.._,,.

. .. . .

97/2

Zkignori

Waroquier

et al.

i”‘”

et al.

2.0 -

I I I

1.0-

Fig. 6. Cumulative

I

I

2

4

E, &kV)

I = 2 and I = 4 distributions

as in fig. 5, but restricted in the present work.

to the final states assigned

3+

is clearly larger than in both calculations. For I = 4 the agreement between experiment and the Bosignori calculation is very good. In fig. 6 a similar cumulative strength function is shown for the 3+ states only. These are the assignments made in the present work. The agreement experiment and the calculation of Bonsignori et al. is very satisfactory. The authors was performed Onderzoek Organisatie

appreciate useful discussions with J. Weil and J. Blachot. This work as part of the research program of the “Stichting voor Fundamenteel

der Materie” (FOM) voor Wetenschappelijk

with financial support Onderzoek” (NWO).

from the “Nederlandse

Appendix SPECTROSCOPIC

between

STRENGTHS

AND

SUM

RULES

IN QUASI-PARTICLE

The relation do -_=N do

a”( DWBA) o

czj+

1)

G”(Ii,

If)

MODELS

J.M. Schippers et al. / Study of “6Sn

90

connects the differential cross section of a single nucleon transfer reaction to the spectroscopic strength G”(I,, If) of the transition rig If. The strength is related to the spectroscopic

factor by:

G”(Ii,

Ir)=](Ii,

Mil[C$OlI,>]Ii,

Mi)]‘=C2S

(pickup)

G”(I,, If) = 21,+ 1

= where

cc is the particle

The strengths

( >

c2s

21,+1

creation

(stripping).

operator.

obey the sum rules: C G”(li, If,f

If) = N”

(pickup)

C G”(I. II If) = (2j+ 1 -iv”) If,f The wave functions of Waroquier neutron sector of the wave functions.

(stripping)

.

et al. “) use quasi-particles to describe the Since in such models particle number is not

strictly conserved it is necessary to verify that the above sumrules are obeyed. consider the neutron pickup reaction on a one quasi-particle state (I’,j’)

We

Iri, Mi)=C s,,,j,, 6,,,rn,a:j,m,IBCS) with a&, = uGc&, + v&,,. The final states in ‘?3n

are described

in terms of two quasi-particle

states and basis states of a coupled scheme the particle annihilation operator I&,,. IA, M,, a) = C &b(&, a.b

that do not connect

(2qp-) basis

to the lqp state by

c.u)A+(a& 4, MAIBCS) . . . ,

with

where LY= (l&m,) and p = (lbjbmb). For pickup the spectroscopic strength G”(J, and the partial If is given by:

Ifa) =

vf

sum rule for pickup

if (Ifa) # g.s.) from shell (Ij) leading

CG”(I,,I,a)=v; n

is then given by:

to final states with spin

J. M. Schippers et al. / Study of “%n

91

The full sum rule is correctly given by: C G”(Ii,I,(Y)=(2j+l)v~=N~. ‘P The corresponding

expressions for the stripping reaction are:

These sum rules are evidently correct if the occupation numbers uf and the emptinesses ui are taken those that apply to the target nucleus. In deriving the sumrules the implicit assumption was made that these occupation numbers and emptinesses were the same as for the neighbouring nuclei. This slight inacurracy is the price that one pays in using the quasi-particle formalism, where the number of nucleons is not strictly a good quantum number. The “‘Sn ground-state wave function actually used in the calculations contains an additional 3qp component. The resulting expressions for the spectroscopic strengths are therefore more complex but the numerical results differ little from those obtained for a pure lqp state. The calculation with the wave functions of Bosignori et al. ‘) could be performed with the same methods, after the approximations mentioned in the text. The calculation of spectroscopic strengths for proton stripping on the “‘In ground state using the wave functions of Waroquier et al. involves the amplitudes lh+ lplh and lhO2qp+ lplh02qp and for the details we refer to ref. “). Considering all proton orbitals above the 2 = 50 closure to be essentially empty the partial sum rules take the simple form C G”(Ii, Ira

Ira)=~I

with Ii=?.

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J. M. Schippers et al. / Study of “%n

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